LMFDB - Embedded newform 68.2.h.a.53.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,2,Mod(9,68)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(8))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("68.9");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	68.h (of order $8$, degree $4$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$0.542982733745$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label			53.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	68.53
Dual form			68.2.h.a.9.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.292893i) q^{3} +(0.292893 + 0.707107i) q^{5} +(0.707107 - 1.70711i) q^{7} +(-1.70711 + 1.70711i) q^{9} +O(q^{10})$	\(q+(0.707107 - 0.292893i) q^{3} +(0.292893 + 0.707107i) q^{5} +(0.707107 - 1.70711i) q^{7} +(-1.70711 + 1.70711i) q^{9} +(-2.70711 - 1.12132i) q^{11} +1.17157i q^{13} +(0.414214 + 0.414214i) q^{15} +(-3.00000 + 2.82843i) q^{17} +(-3.82843 - 3.82843i) q^{19} -1.41421i q^{21} +(1.29289 + 0.535534i) q^{23} +(3.12132 - 3.12132i) q^{25} +(-1.58579 + 3.82843i) q^{27} +(2.29289 + 5.53553i) q^{29} +(9.53553 - 3.94975i) q^{31} -2.24264 q^{33} +1.41421 q^{35} +(4.53553 - 1.87868i) q^{37} +(0.343146 + 0.828427i) q^{39} +(3.12132 - 7.53553i) q^{41} +(-3.00000 + 3.00000i) q^{43} +(-1.70711 - 0.707107i) q^{45} +7.65685i q^{47} +(2.53553 + 2.53553i) q^{49} +(-1.29289 + 2.87868i) q^{51} +(-9.82843 - 9.82843i) q^{53} -2.24264i q^{55} +(-3.82843 - 1.58579i) q^{57} +(-4.17157 + 4.17157i) q^{59} +(0.292893 - 0.707107i) q^{61} +(1.70711 + 4.12132i) q^{63} +(-0.828427 + 0.343146i) q^{65} -11.3137 q^{67} +1.07107 q^{69} +(7.53553 - 3.12132i) q^{71} +(1.94975 + 4.70711i) q^{73} +(1.29289 - 3.12132i) q^{75} +(-3.82843 + 3.82843i) q^{77} +(-8.36396 - 3.46447i) q^{79} -4.07107i q^{81} +(5.82843 + 5.82843i) q^{83} +(-2.87868 - 1.29289i) q^{85} +(3.24264 + 3.24264i) q^{87} -5.17157i q^{89} +(2.00000 + 0.828427i) q^{91} +(5.58579 - 5.58579i) q^{93} +(1.58579 - 3.82843i) q^{95} +(5.12132 + 12.3640i) q^{97} +(6.53553 - 2.70711i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^5 - 4 * q^9	$ 4 q + 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 12 q^{27} + 12 q^{29} + 24 q^{31} + 8 q^{33} + 4 q^{37} + 24 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} - 4 q^{49} - 8 q^{51} - 28 q^{53} - 4 q^{57} - 28 q^{59} + 4 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69} + 16 q^{71} - 12 q^{73} + 8 q^{75} - 4 q^{77} - 8 q^{79} + 12 q^{83} - 20 q^{85} - 4 q^{87} + 8 q^{91} + 28 q^{93} + 12 q^{95} + 12 q^{97} + 12 q^{99}+O(q^{100}) $ 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^11 - 4 * q^15 - 12 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 12 * q^27 + 12 * q^29 + 24 * q^31 + 8 * q^33 + 4 * q^37 + 24 * q^39 + 4 * q^41 - 12 * q^43 - 4 * q^45 - 4 * q^49 - 8 * q^51 - 28 * q^53 - 4 * q^57 - 28 * q^59 + 4 * q^61 + 4 * q^63 + 8 * q^65 - 24 * q^69 + 16 * q^71 - 12 * q^73 + 8 * q^75 - 4 * q^77 - 8 * q^79 + 12 * q^83 - 20 * q^85 - 4 * q^87 + 8 * q^91 + 28 * q^93 + 12 * q^95 + 12 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/68\mathbb{Z}\right)^\times$.

$n$	$35$	$37$
$\chi(n)$	$1$	$e\left(\frac{7}{8}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	0.707107	−	0.292893i	0.408248	−	0.169102i	−0.169102	−	0.985599i	$-0.554087\pi$
$3$	0.707107	−	0.292893i	0.408248	−	0.169102i	0.577350	+	0.816497i	$0.304087\pi$
$4$		0			0
$5$	0.292893	+	0.707107i	0.130986	+	0.316228i	0.975742	−	0.218924i	$-0.0702546\pi$
$5$	0.292893	+	0.707107i	0.130986	+	0.316228i	−0.844756	+	0.535151i	$0.820255\pi$
$6$		0			0
$7$	0.707107	−	1.70711i	0.267261	−	0.645226i	−0.732091	−	0.681207i	$-0.761456\pi$
$7$	0.707107	−	1.70711i	0.267261	−	0.645226i	0.999352	+	0.0359809i	$0.0114555\pi$
$8$		0			0
$9$	−1.70711	+	1.70711i	−0.569036	+	0.569036i
$10$		0			0
$11$	−2.70711	−	1.12132i	−0.816223	−	0.338091i	−0.0647893	−	0.997899i	$-0.520638\pi$
$11$	−2.70711	−	1.12132i	−0.816223	−	0.338091i	−0.751434	+	0.659808i	$0.770638\pi$
$12$		0			0
$13$			1.17157i			0.324936i	0.986714	+	0.162468i	$0.0519454\pi$
$13$			1.17157i			0.324936i	−0.986714	+	0.162468i	$0.948055\pi$
$14$		0			0
$15$	0.414214	+	0.414214i	0.106949	+	0.106949i
$16$		0			0
$17$	−3.00000	+	2.82843i	−0.727607	+	0.685994i
$17$	−3.00000	+	2.82843i	−0.727607	+	0.685994i
$18$		0			0
$19$	−3.82843	−	3.82843i	−0.878301	−	0.878301i	0.115057	−	0.993359i	$-0.463295\pi$
$19$	−3.82843	−	3.82843i	−0.878301	−	0.878301i	−0.993359	+	0.115057i	$0.963295\pi$
$20$		0			0
$21$		−	1.41421i		−	0.308607i
$22$		0			0
$23$	1.29289	+	0.535534i	0.269587	+	0.111667i	0.513382	−	0.858160i	$-0.328392\pi$
$23$	1.29289	+	0.535534i	0.269587	+	0.111667i	−0.243795	+	0.969827i	$0.578392\pi$
$24$		0			0
$25$	3.12132	−	3.12132i	0.624264	−	0.624264i
$26$		0			0
$27$	−1.58579	+	3.82843i	−0.305185	+	0.736781i
$28$		0			0
$29$	2.29289	+	5.53553i	0.425780	+	1.02792i	0.980612	+	0.195961i	$0.0627825\pi$
$29$	2.29289	+	5.53553i	0.425780	+	1.02792i	−0.554832	+	0.831962i	$0.687218\pi$
$30$		0			0
$31$	9.53553	−	3.94975i	1.71263	−	0.709396i	0.712664	−	0.701506i	$-0.247489\pi$
$31$	9.53553	−	3.94975i	1.71263	−	0.709396i	0.999969	−	0.00788961i	$-0.00251137\pi$
$32$		0			0
$33$	−2.24264			−0.390394
$34$		0			0
$35$	1.41421			0.239046
$36$		0			0
$37$	4.53553	−	1.87868i	0.745637	−	0.308853i	0.0226771	−	0.999743i	$-0.492781\pi$
$37$	4.53553	−	1.87868i	0.745637	−	0.308853i	0.722960	+	0.690890i	$0.242781\pi$
$38$		0			0
$39$	0.343146	+	0.828427i	0.0549473	+	0.132655i
$40$		0			0
$41$	3.12132	−	7.53553i	0.487468	−	1.17685i	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	3.12132	−	7.53553i	0.487468	−	1.17685i	0.955990	−	0.293400i	$-0.0947869\pi$
$42$		0			0
$43$	−3.00000	+	3.00000i	−0.457496	+	0.457496i	−0.897833	−	0.440337i	$-0.854859\pi$
$43$	−3.00000	+	3.00000i	−0.457496	+	0.457496i	0.440337	+	0.897833i	$0.354859\pi$
$44$		0			0
$45$	−1.70711	−	0.707107i	−0.254480	−	0.105409i
$46$		0			0
$47$			7.65685i			1.11687i	0.829549	+	0.558433i	$0.188597\pi$
$47$			7.65685i			1.11687i	−0.829549	+	0.558433i	$0.811403\pi$
$48$		0			0
$49$	2.53553	+	2.53553i	0.362219	+	0.362219i
$50$		0			0
$51$	−1.29289	+	2.87868i	−0.181041	+	0.403096i
$52$		0			0
$53$	−9.82843	−	9.82843i	−1.35004	−	1.35004i	−0.885612	−	0.464427i	$-0.846260\pi$
$53$	−9.82843	−	9.82843i	−1.35004	−	1.35004i	−0.464427	−	0.885612i	$-0.653740\pi$
$54$		0			0
$55$		−	2.24264i		−	0.302398i
$56$		0			0
$57$	−3.82843	−	1.58579i	−0.507088	−	0.210043i
$58$		0			0
$59$	−4.17157	+	4.17157i	−0.543093	+	0.543093i	−0.924434	−	0.381342i	$-0.875462\pi$
$59$	−4.17157	+	4.17157i	−0.543093	+	0.543093i	0.381342	+	0.924434i	$0.375462\pi$
$60$		0			0
$61$	0.292893	−	0.707107i	0.0375011	−	0.0905357i	−0.904019	−	0.427492i	$-0.859397\pi$
$61$	0.292893	−	0.707107i	0.0375011	−	0.0905357i	0.941520	+	0.336956i	$0.109397\pi$
$62$		0			0
$63$	1.70711	+	4.12132i	0.215075	+	0.519238i
$64$		0			0
$65$	−0.828427	+	0.343146i	−0.102754	+	0.0425620i
$66$		0			0
$67$	−11.3137			−1.38219			−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137			−1.38219			−0.691095	+	0.722764i	$0.742871\pi$
$68$		0			0
$69$	1.07107			0.128941
$70$		0			0
$71$	7.53553	−	3.12132i	0.894303	−	0.370433i	0.112276	−	0.993677i	$-0.464186\pi$
$71$	7.53553	−	3.12132i	0.894303	−	0.370433i	0.782027	+	0.623244i	$0.214186\pi$
$72$		0			0
$73$	1.94975	+	4.70711i	0.228201	+	0.550925i	0.995958	−	0.0898150i	$-0.0286276\pi$
$73$	1.94975	+	4.70711i	0.228201	+	0.550925i	−0.767758	+	0.640740i	$0.778628\pi$
$74$		0			0
$75$	1.29289	−	3.12132i	0.149290	−	0.360419i
$76$		0			0
$77$	−3.82843	+	3.82843i	−0.436290	+	0.436290i
$78$		0			0
$79$	−8.36396	−	3.46447i	−0.941019	−	0.389783i	−0.141171	−	0.989985i	$-0.545087\pi$
$79$	−8.36396	−	3.46447i	−0.941019	−	0.389783i	−0.799848	+	0.600202i	$0.795087\pi$
$80$		0			0
$81$		−	4.07107i		−	0.452341i
$82$		0			0
$83$	5.82843	+	5.82843i	0.639753	+	0.639753i	0.950494	−	0.310741i	$-0.100577\pi$
$83$	5.82843	+	5.82843i	0.639753	+	0.639753i	−0.310741	+	0.950494i	$0.600577\pi$
$84$		0			0
$85$	−2.87868	−	1.29289i	−0.312237	−	0.140234i
$86$		0			0
$87$	3.24264	+	3.24264i	0.347648	+	0.347648i
$88$		0			0
$89$		−	5.17157i		−	0.548186i	−0.961703	−	0.274093i	$-0.911622\pi$
$89$		−	5.17157i		−	0.548186i	0.961703	−	0.274093i	$-0.0883776\pi$
$90$		0			0
$91$	2.00000	+	0.828427i	0.209657	+	0.0868428i
$92$		0			0
$93$	5.58579	−	5.58579i	0.579219	−	0.579219i
$94$		0			0
$95$	1.58579	−	3.82843i	0.162698	−	0.392788i
$96$		0			0
$97$	5.12132	+	12.3640i	0.519991	+	1.25537i	0.937908	+	0.346885i	$0.112760\pi$
$97$	5.12132	+	12.3640i	0.519991	+	1.25537i	−0.417917	+	0.908485i	$0.637240\pi$
$98$		0			0
$99$	6.53553	−	2.70711i	0.656846	−	0.272074i
$100$		0			0
$101$	7.17157			0.713598			0.356799	−	0.934181i	$-0.383868\pi$
$101$	7.17157			0.713598			0.356799	+	0.934181i	$0.383868\pi$
$102$		0			0
$103$	4.00000			0.394132			0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000			0.394132			0.197066	+	0.980390i	$0.436859\pi$
$104$		0			0
$105$	1.00000	−	0.414214i	0.0975900	−	0.0404231i
$106$		0			0
$107$	−3.53553	−	8.53553i	−0.341793	−	0.825161i	−0.997535	−	0.0701759i	$-0.977644\pi$
$107$	−3.53553	−	8.53553i	−0.341793	−	0.825161i	0.655742	−	0.754985i	$-0.272356\pi$
$108$		0			0
$109$	−2.53553	+	6.12132i	−0.242860	+	0.586316i	−0.997565	−	0.0697487i	$-0.977780\pi$
$109$	−2.53553	+	6.12132i	−0.242860	+	0.586316i	0.754704	+	0.656065i	$0.227780\pi$
$110$		0			0
$111$	2.65685	−	2.65685i	0.252177	−	0.252177i
$112$		0			0
$113$	6.53553	+	2.70711i	0.614811	+	0.254663i	0.668284	−	0.743906i	$-0.267029\pi$
$113$	6.53553	+	2.70711i	0.614811	+	0.254663i	−0.0534728	+	0.998569i	$0.517029\pi$
$114$		0			0
$115$			1.07107i			0.0998776i
$116$		0			0
$117$	−2.00000	−	2.00000i	−0.184900	−	0.184900i
$118$		0			0
$119$	2.70711	+	7.12132i	0.248160	+	0.652810i
$120$		0			0
$121$	−1.70711	−	1.70711i	−0.155192	−	0.155192i
$122$		0			0
$123$		−	6.24264i		−	0.562880i
$124$		0			0
$125$	6.65685	+	2.75736i	0.595407	+	0.246626i
$126$		0			0
$127$	−5.82843	+	5.82843i	−0.517189	+	0.517189i	−0.916720	−	0.399531i	$-0.869173\pi$
$127$	−5.82843	+	5.82843i	−0.517189	+	0.517189i	0.399531	+	0.916720i	$0.369173\pi$
$128$		0			0
$129$	−1.24264	+	3.00000i	−0.109408	+	0.264135i
$130$		0			0
$131$	−5.87868	−	14.1924i	−0.513623	−	1.23999i	−0.941761	−	0.336282i	$-0.890831\pi$
$131$	−5.87868	−	14.1924i	−0.513623	−	1.23999i	0.428139	−	0.903713i	$-0.359169\pi$
$132$		0			0
$133$	−9.24264	+	3.82843i	−0.801439	+	0.331967i
$134$		0			0
$135$	−3.17157			−0.272966
$136$		0			0
$137$	−16.1421			−1.37912			−0.689558	−	0.724231i	$-0.742195\pi$
$137$	−16.1421			−1.37912			−0.689558	+	0.724231i	$0.742195\pi$
$138$		0			0
$139$	−5.29289	+	2.19239i	−0.448937	+	0.185956i	−0.595685	−	0.803218i	$-0.703119\pi$
$139$	−5.29289	+	2.19239i	−0.448937	+	0.185956i	0.146748	+	0.989174i	$0.453119\pi$
$140$		0			0
$141$	2.24264	+	5.41421i	0.188864	+	0.455959i
$142$		0			0
$143$	1.31371	−	3.17157i	0.109858	−	0.265220i
$144$		0			0
$145$	−3.24264	+	3.24264i	−0.269287	+	0.269287i
$146$		0			0
$147$	2.53553	+	1.05025i	0.209127	+	0.0866234i
$148$		0			0
$149$		−	1.65685i		−	0.135735i	−0.997694	−	0.0678674i	$-0.978381\pi$
$149$		−	1.65685i		−	0.135735i	0.997694	−	0.0678674i	$-0.0216195\pi$
$150$		0			0
$151$	−12.3137	−	12.3137i	−1.00208	−	1.00208i	−0.999998	−	0.00207754i	$-0.999339\pi$
$151$	−12.3137	−	12.3137i	−1.00208	−	1.00208i	−0.00207754	−	0.999998i	$-0.500661\pi$
$152$		0			0
$153$	0.292893	−	9.94975i	0.0236790	−	0.804389i
$154$		0			0
$155$	5.58579	+	5.58579i	0.448661	+	0.448661i
$156$		0			0
$157$			13.6569i			1.08994i	0.838457	+	0.544968i	$0.183458\pi$
$157$			13.6569i			1.08994i	−0.838457	+	0.544968i	$0.816542\pi$
$158$		0			0
$159$	−9.82843	−	4.07107i	−0.779445	−	0.322857i
$160$		0			0
$161$	1.82843	−	1.82843i	0.144100	−	0.144100i
$162$		0			0
$163$	−5.29289	+	12.7782i	−0.414571	+	1.00086i	0.569323	+	0.822114i	$0.307205\pi$
$163$	−5.29289	+	12.7782i	−0.414571	+	1.00086i	−0.983895	+	0.178750i	$0.942795\pi$
$164$		0			0
$165$	−0.656854	−	1.58579i	−0.0511360	−	0.123453i
$166$		0			0
$167$	−3.29289	+	1.36396i	−0.254812	+	0.105546i	−0.506433	−	0.862279i	$-0.669036\pi$
$167$	−3.29289	+	1.36396i	−0.254812	+	0.105546i	0.251621	+	0.967826i	$0.419036\pi$
$168$		0			0
$169$	11.6274			0.894417
$170$		0			0
$171$	13.0711			0.999570
$172$		0			0
$173$	−11.6066	+	4.80761i	−0.882434	+	0.365516i	−0.777440	−	0.628957i	$-0.783482\pi$
$173$	−11.6066	+	4.80761i	−0.882434	+	0.365516i	−0.104993	+	0.994473i	$0.533482\pi$
$174$		0			0
$175$	−3.12132	−	7.53553i	−0.235950	−	0.569633i
$176$		0			0
$177$	−1.72792	+	4.17157i	−0.129879	+	0.313555i
$178$		0			0
$179$	13.0000	−	13.0000i	0.971666	−	0.971666i	−0.0279439	−	0.999609i	$-0.508896\pi$
$179$	13.0000	−	13.0000i	0.971666	−	0.971666i	0.999609	+	0.0279439i	$0.00889597\pi$
$180$		0			0
$181$	8.53553	+	3.53553i	0.634441	+	0.262794i	0.676639	−	0.736315i	$-0.263436\pi$
$181$	8.53553	+	3.53553i	0.634441	+	0.262794i	−0.0421975	+	0.999109i	$0.513436\pi$
$182$		0			0
$183$		−	0.585786i		−	0.0433026i
$184$		0			0
$185$	2.65685	+	2.65685i	0.195336	+	0.195336i
$186$		0			0
$187$	11.2929	−	4.29289i	0.825818	−	0.313927i
$188$		0			0
$189$	5.41421	+	5.41421i	0.393826	+	0.393826i
$190$		0			0
$191$			11.6569i			0.843460i	0.906721	+	0.421730i	$0.138577\pi$
$191$			11.6569i			0.843460i	−0.906721	+	0.421730i	$0.861423\pi$
$192$		0			0
$193$	−3.94975	−	1.63604i	−0.284309	−	0.117765i	0.235972	−	0.971760i	$-0.424173\pi$
$193$	−3.94975	−	1.63604i	−0.284309	−	0.117765i	−0.520281	+	0.853995i	$0.674173\pi$
$194$		0			0
$195$	−0.485281	+	0.485281i	−0.0347517	+	0.0347517i
$196$		0			0
$197$	9.60660	−	23.1924i	0.684442	−	1.65239i	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	9.60660	−	23.1924i	0.684442	−	1.65239i	0.755689	−	0.654931i	$-0.227302\pi$
$198$		0			0
$199$	−5.53553	−	13.3640i	−0.392404	−	0.947346i	−0.989415	−	0.145113i	$-0.953645\pi$
$199$	−5.53553	−	13.3640i	−0.392404	−	0.947346i	0.597011	−	0.802233i	$-0.296355\pi$
$200$		0			0
$201$	−8.00000	+	3.31371i	−0.564276	+	0.233731i
$202$		0			0
$203$	11.0711			0.777037
$204$		0			0
$205$	6.24264			0.436005
$206$		0			0
$207$	−3.12132	+	1.29289i	−0.216947	+	0.0898623i
$208$		0			0
$209$	6.07107	+	14.6569i	0.419945	+	1.01384i
$210$		0			0
$211$	−6.12132	+	14.7782i	−0.421409	+	1.01737i	0.560523	+	0.828139i	$0.310600\pi$
$211$	−6.12132	+	14.7782i	−0.421409	+	1.01737i	−0.981932	+	0.189233i	$0.939400\pi$
$212$		0			0
$213$	4.41421	−	4.41421i	0.302457	−	0.302457i
$214$		0			0
$215$	−3.00000	−	1.24264i	−0.204598	−	0.0847474i
$216$		0			0
$217$		−	19.0711i		−	1.29463i
$218$		0			0
$219$	2.75736	+	2.75736i	0.186325	+	0.186325i
$220$		0			0
$221$	−3.31371	−	3.51472i	−0.222904	−	0.236426i
$222$		0			0
$223$	17.8284	+	17.8284i	1.19388	+	1.19388i	0.975969	+	0.217911i	$0.0699243\pi$
$223$	17.8284	+	17.8284i	1.19388	+	1.19388i	0.217911	+	0.975969i	$0.430076\pi$
$224$		0			0
$225$			10.6569i			0.710457i
$226$		0			0
$227$	19.4350	+	8.05025i	1.28995	+	0.534314i	0.918970	−	0.394328i	$-0.129022\pi$
$227$	19.4350	+	8.05025i	1.28995	+	0.534314i	0.370978	+	0.928642i	$0.379022\pi$
$228$		0			0
$229$	8.65685	−	8.65685i	0.572061	−	0.572061i	−0.360643	−	0.932704i	$-0.617443\pi$
$229$	8.65685	−	8.65685i	0.572061	−	0.572061i	0.932704	+	0.360643i	$0.117443\pi$
$230$		0			0
$231$	−1.58579	+	3.82843i	−0.104337	+	0.251892i
$232$		0			0
$233$	−9.70711	−	23.4350i	−0.635934	−	1.53528i	−0.832051	−	0.554699i	$-0.812833\pi$
$233$	−9.70711	−	23.4350i	−0.635934	−	1.53528i	0.196117	−	0.980580i	$-0.437167\pi$
$234$		0			0
$235$	−5.41421	+	2.24264i	−0.353184	+	0.146294i
$236$		0			0
$237$	−6.92893			−0.450083
$238$		0			0
$239$	−22.6274			−1.46365			−0.731823	−	0.681495i	$-0.761330\pi$
$239$	−22.6274			−1.46365			−0.731823	+	0.681495i	$0.761330\pi$
$240$		0			0
$241$	−13.6066	+	5.63604i	−0.876478	+	0.363049i	−0.775130	−	0.631802i	$-0.782316\pi$
$241$	−13.6066	+	5.63604i	−0.876478	+	0.363049i	−0.101348	+	0.994851i	$0.532316\pi$
$242$		0			0
$243$	−5.94975	−	14.3640i	−0.381676	−	0.921449i
$244$		0			0
$245$	−1.05025	+	2.53553i	−0.0670982	+	0.161989i
$246$		0			0
$247$	4.48528	−	4.48528i	0.285392	−	0.285392i
$248$		0			0
$249$	5.82843	+	2.41421i	0.369362	+	0.152995i
$250$		0			0
$251$			10.9706i			0.692456i	0.938150	+	0.346228i	$0.112538\pi$
$251$			10.9706i			0.692456i	−0.938150	+	0.346228i	$0.887462\pi$
$252$		0			0
$253$	−2.89949	−	2.89949i	−0.182290	−	0.182290i
$254$		0			0
$255$	−2.41421	−	0.0710678i	−0.151184	−	0.00445044i
$256$		0			0
$257$	8.65685	+	8.65685i	0.540000	+	0.540000i	0.923529	−	0.383529i	$-0.125291\pi$
$257$	8.65685	+	8.65685i	0.540000	+	0.540000i	−0.383529	+	0.923529i	$0.625291\pi$
$258$		0			0
$259$		−	9.07107i		−	0.563649i
$260$		0			0
$261$	−13.3640	−	5.53553i	−0.827208	−	0.342641i
$262$		0			0
$263$	−20.6569	+	20.6569i	−1.27376	+	1.27376i	−0.329655	+	0.944102i	$0.606932\pi$
$263$	−20.6569	+	20.6569i	−1.27376	+	1.27376i	−0.944102	+	0.329655i	$0.893068\pi$
$264$		0			0
$265$	4.07107	−	9.82843i	0.250084	−	0.603755i
$266$		0			0
$267$	−1.51472	−	3.65685i	−0.0926993	−	0.223796i
$268$		0			0
$269$	−18.7782	+	7.77817i	−1.14493	+	0.474244i	−0.872829	−	0.488026i	$-0.837717\pi$
$269$	−18.7782	+	7.77817i	−1.14493	+	0.474244i	−0.272097	+	0.962270i	$0.587717\pi$
$270$		0			0
$271$	11.3137			0.687259			0.343629	−	0.939105i	$-0.388344\pi$
$271$	11.3137			0.687259			0.343629	+	0.939105i	$0.388344\pi$
$272$		0			0
$273$	1.65685			0.100277
$274$		0			0
$275$	−11.9497	+	4.94975i	−0.720597	+	0.298481i
$276$		0			0
$277$	−4.05025	−	9.77817i	−0.243356	−	0.587514i	0.754256	−	0.656581i	$-0.227998\pi$
$277$	−4.05025	−	9.77817i	−0.243356	−	0.587514i	−0.997612	+	0.0690669i	$0.977998\pi$
$278$		0			0
$279$	−9.53553	+	23.0208i	−0.570877	+	1.37822i
$280$		0			0
$281$	−0.656854	+	0.656854i	−0.0391846	+	0.0391846i	−0.726428	−	0.687243i	$-0.758821\pi$
$281$	−0.656854	+	0.656854i	−0.0391846	+	0.0391846i	0.687243	+	0.726428i	$0.258821\pi$
$282$		0			0
$283$	1.29289	+	0.535534i	0.0768545	+	0.0318342i	0.420780	−	0.907163i	$-0.361757\pi$
$283$	1.29289	+	0.535534i	0.0768545	+	0.0318342i	−0.343925	+	0.938997i	$0.611757\pi$
$284$		0			0
$285$		−	3.17157i		−	0.187868i
$286$		0			0
$287$	−10.6569	−	10.6569i	−0.629054	−	0.629054i
$288$		0			0
$289$	1.00000	−	16.9706i	0.0588235	−	0.998268i
$290$		0			0
$291$	7.24264	+	7.24264i	0.424571	+	0.424571i
$292$		0			0
$293$		−	8.00000i		−	0.467365i	−0.972313	−	0.233682i	$-0.924922\pi$
$293$		−	8.00000i		−	0.467365i	0.972313	−	0.233682i	$-0.0750776\pi$
$294$		0			0
$295$	−4.17157	−	1.72792i	−0.242878	−	0.100604i
$296$		0			0
$297$	8.58579	−	8.58579i	0.498198	−	0.498198i
$298$		0			0
$299$	−0.627417	+	1.51472i	−0.0362845	+	0.0875984i
$300$		0			0
$301$	3.00000	+	7.24264i	0.172917	+	0.417459i
$302$		0			0
$303$	5.07107	−	2.10051i	0.291325	−	0.120671i
$304$		0			0
$305$	0.585786			0.0335420
$306$		0			0
$307$	17.6569			1.00773			0.503865	−	0.863782i	$-0.331911\pi$
$307$	17.6569			1.00773			0.503865	+	0.863782i	$0.331911\pi$
$308$		0			0
$309$	2.82843	−	1.17157i	0.160904	−	0.0666485i
$310$		0			0
$311$	3.77817	+	9.12132i	0.214241	+	0.517223i	0.994067	−	0.108774i	$-0.0346924\pi$
$311$	3.77817	+	9.12132i	0.214241	+	0.517223i	−0.779826	+	0.625996i	$0.784692\pi$
$312$		0			0
$313$	−3.02082	+	7.29289i	−0.170747	+	0.412219i	−0.985969	−	0.166930i	$-0.946615\pi$
$313$	−3.02082	+	7.29289i	−0.170747	+	0.412219i	0.815222	+	0.579148i	$0.196615\pi$
$314$		0			0
$315$	−2.41421	+	2.41421i	−0.136026	+	0.136026i
$316$		0			0
$317$	29.8492	+	12.3640i	1.67650	+	0.694429i	0.999150	−	0.0412309i	$-0.0131279\pi$
$317$	29.8492	+	12.3640i	1.67650	+	0.694429i	0.677351	+	0.735660i	$0.263128\pi$
$318$		0			0
$319$		−	17.5563i		−	0.982967i
$320$		0			0
$321$	−5.00000	−	5.00000i	−0.279073	−	0.279073i
$322$		0			0
$323$	22.3137	+	0.656854i	1.24157	+	0.0365483i
$324$		0			0
$325$	3.65685	+	3.65685i	0.202846	+	0.202846i
$326$		0			0
$327$			5.07107i			0.280431i
$328$		0			0
$329$	13.0711	+	5.41421i	0.720631	+	0.298495i
$330$		0			0
$331$	−2.31371	+	2.31371i	−0.127173	+	0.127173i	−0.767828	−	0.640656i	$-0.778663\pi$
$331$	−2.31371	+	2.31371i	−0.127173	+	0.127173i	0.640656	+	0.767828i	$0.278663\pi$
$332$		0			0
$333$	−4.53553	+	10.9497i	−0.248546	+	0.600042i
$334$		0			0
$335$	−3.31371	−	8.00000i	−0.181047	−	0.437087i
$336$		0			0
$337$	0.0502525	−	0.0208153i	0.00273743	−	0.00113388i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	0.0502525	−	0.0208153i	0.00273743	−	0.00113388i	0.384052	+	0.923312i	$0.374528\pi$
$338$		0			0
$339$	5.41421			0.294060
$340$		0			0
$341$	−30.2426			−1.63773
$342$		0			0
$343$	18.0711	−	7.48528i	0.975746	−	0.404167i
$344$		0			0
$345$	0.313708	+	0.757359i	0.0168895	+	0.0407749i
$346$		0			0
$347$	11.1924	−	27.0208i	0.600839	−	1.45055i	−0.271880	−	0.962331i	$-0.587646\pi$
$347$	11.1924	−	27.0208i	0.600839	−	1.45055i	0.872719	−	0.488222i	$-0.162354\pi$
$348$		0			0
$349$	14.1716	−	14.1716i	0.758587	−	0.758587i	−0.217478	−	0.976065i	$-0.569783\pi$
$349$	14.1716	−	14.1716i	0.758587	−	0.758587i	0.976065	+	0.217478i	$0.0697831\pi$
$350$		0			0
$351$	−4.48528	−	1.85786i	−0.239407	−	0.0991655i
$352$		0			0
$353$			7.31371i			0.389269i	0.980876	+	0.194635i	$0.0623521\pi$
$353$			7.31371i			0.389269i	−0.980876	+	0.194635i	$0.937648\pi$
$354$		0			0
$355$	4.41421	+	4.41421i	0.234282	+	0.234282i
$356$		0			0
$357$	4.00000	+	4.24264i	0.211702	+	0.224544i
$358$		0			0
$359$	−4.51472	−	4.51472i	−0.238278	−	0.238278i	0.577859	−	0.816137i	$-0.303888\pi$
$359$	−4.51472	−	4.51472i	−0.238278	−	0.238278i	−0.816137	+	0.577859i	$0.803888\pi$
$360$		0			0
$361$			10.3137i			0.542827i
$362$		0			0
$363$	−1.70711	−	0.707107i	−0.0895999	−	0.0371135i
$364$		0			0
$365$	−2.75736	+	2.75736i	−0.144327	+	0.144327i
$366$		0			0
$367$	3.53553	−	8.53553i	0.184553	−	0.445551i	−0.804342	−	0.594167i	$-0.797482\pi$
$367$	3.53553	−	8.53553i	0.184553	−	0.445551i	0.988895	+	0.148616i	$0.0474818\pi$
$368$		0			0
$369$	7.53553	+	18.1924i	0.392284	+	0.947058i
$370$		0			0
$371$	−23.7279	+	9.82843i	−1.23189	+	0.510267i
$372$		0			0
$373$	1.51472			0.0784292			0.0392146	−	0.999231i	$-0.487514\pi$
$373$	1.51472			0.0784292			0.0392146	+	0.999231i	$0.487514\pi$
$374$		0			0
$375$	5.51472			0.284779
$376$		0			0
$377$	−6.48528	+	2.68629i	−0.334009	+	0.138351i
$378$		0			0
$379$	−5.87868	−	14.1924i	−0.301967	−	0.729014i	−0.999917	−	0.0128672i	$-0.995904\pi$
$379$	−5.87868	−	14.1924i	−0.301967	−	0.729014i	0.697950	−	0.716147i	$-0.254096\pi$
$380$		0			0
$381$	−2.41421	+	5.82843i	−0.123684	+	0.298599i
$382$		0			0
$383$	4.31371	−	4.31371i	0.220420	−	0.220420i	−0.588255	−	0.808675i	$-0.700185\pi$
$383$	4.31371	−	4.31371i	0.220420	−	0.220420i	0.808675	+	0.588255i	$0.200185\pi$
$384$		0			0
$385$	−3.82843	−	1.58579i	−0.195115	−	0.0808192i
$386$		0			0
$387$		−	10.2426i		−	0.520663i
$388$		0			0
$389$	15.4853	+	15.4853i	0.785135	+	0.785135i	0.980692	−	0.195557i	$-0.0626516\pi$
$389$	15.4853	+	15.4853i	0.785135	+	0.785135i	−0.195557	+	0.980692i	$0.562652\pi$
$390$		0			0
$391$	−5.39340	+	2.05025i	−0.272756	+	0.103686i
$392$		0			0
$393$	−8.31371	−	8.31371i	−0.419371	−	0.419371i
$394$		0			0
$395$		−	6.92893i		−	0.348632i
$396$		0			0
$397$	−22.7782	−	9.43503i	−1.14320	−	0.473531i	−0.270955	−	0.962592i	$-0.587339\pi$
$397$	−22.7782	−	9.43503i	−1.14320	−	0.473531i	−0.872249	+	0.489062i	$0.837339\pi$
$398$		0			0
$399$	−5.41421	+	5.41421i	−0.271050	+	0.271050i
$400$		0			0
$401$	1.46447	−	3.53553i	0.0731319	−	0.176556i	−0.883086	−	0.469211i	$-0.844539\pi$
$401$	1.46447	−	3.53553i	0.0731319	−	0.176556i	0.956218	+	0.292654i	$0.0945385\pi$
$402$		0			0
$403$	4.62742	+	11.1716i	0.230508	+	0.556496i
$404$		0			0
$405$	2.87868	−	1.19239i	0.143043	−	0.0592502i
$406$		0			0
$407$	−14.3848			−0.713027
$408$		0			0
$409$	−0.343146			−0.0169675			−0.00848373	−	0.999964i	$-0.502700\pi$
$409$	−0.343146			−0.0169675			−0.00848373	+	0.999964i	$0.502700\pi$
$410$		0			0
$411$	−11.4142	+	4.72792i	−0.563022	+	0.233211i
$412$		0			0
$413$	4.17157	+	10.0711i	0.205270	+	0.495565i
$414$		0			0
$415$	−2.41421	+	5.82843i	−0.118509	+	0.286106i
$416$		0			0
$417$	−3.10051	+	3.10051i	−0.151832	+	0.151832i
$418$		0			0
$419$	−26.3640	−	10.9203i	−1.28796	−	0.533492i	−0.369586	−	0.929197i	$-0.620500\pi$
$419$	−26.3640	−	10.9203i	−1.28796	−	0.533492i	−0.918378	+	0.395705i	$0.870500\pi$
$420$		0			0
$421$			6.82843i			0.332797i	0.986059	+	0.166399i	$0.0532138\pi$
$421$			6.82843i			0.332797i	−0.986059	+	0.166399i	$0.946786\pi$
$422$		0			0
$423$	−13.0711	−	13.0711i	−0.635537	−	0.635537i
$424$		0			0
$425$	−0.535534	+	18.1924i	−0.0259772	+	0.882460i
$426$		0			0
$427$	−1.00000	−	1.00000i	−0.0483934	−	0.0483934i
$428$		0			0
$429$		−	2.62742i		−	0.126853i
$430$		0			0
$431$	12.1213	+	5.02082i	0.583863	+	0.241844i	0.655008	−	0.755622i	$-0.272665\pi$
$431$	12.1213	+	5.02082i	0.583863	+	0.241844i	−0.0711447	+	0.997466i	$0.522665\pi$
$432$		0			0
$433$	0.171573	−	0.171573i	0.00824527	−	0.00824527i	−0.702972	−	0.711217i	$-0.748144\pi$
$433$	0.171573	−	0.171573i	0.00824527	−	0.00824527i	0.711217	+	0.702972i	$0.248144\pi$
$434$		0			0
$435$	−1.34315	+	3.24264i	−0.0643989	+	0.155473i
$436$		0			0
$437$	−2.89949	−	7.00000i	−0.138702	−	0.334855i
$438$		0			0
$439$	29.6777	−	12.2929i	1.41644	−	0.586708i	0.462475	−	0.886632i	$-0.346962\pi$
$439$	29.6777	−	12.2929i	1.41644	−	0.586708i	0.953963	+	0.299925i	$0.0969615\pi$
$440$		0			0
$441$	−8.65685			−0.412231
$442$		0			0
$443$	−10.3431			−0.491418			−0.245709	−	0.969344i	$-0.579021\pi$
$443$	−10.3431			−0.491418			−0.245709	+	0.969344i	$0.579021\pi$
$444$		0			0
$445$	3.65685	−	1.51472i	0.173352	−	0.0718045i
$446$		0			0
$447$	−0.485281	−	1.17157i	−0.0229530	−	0.0554135i
$448$		0			0
$449$	1.60660	−	3.87868i	0.0758202	−	0.183046i	−0.881425	−	0.472324i	$-0.843415\pi$
$449$	1.60660	−	3.87868i	0.0758202	−	0.183046i	0.957245	+	0.289278i	$0.0934152\pi$
$450$		0			0
$451$	−16.8995	+	16.8995i	−0.795766	+	0.795766i
$452$		0			0
$453$	−12.3137	−	5.10051i	−0.578548	−	0.239643i
$454$		0			0
$455$			1.65685i			0.0776745i
$456$		0			0
$457$	3.00000	+	3.00000i	0.140334	+	0.140334i	0.773784	−	0.633450i	$-0.218362\pi$
$457$	3.00000	+	3.00000i	0.140334	+	0.140334i	−0.633450	+	0.773784i	$0.718362\pi$
$458$		0			0
$459$	−6.07107	−	15.9706i	−0.283373	−	0.745442i
$460$		0			0
$461$	−11.4853	−	11.4853i	−0.534923	−	0.534923i	0.387110	−	0.922033i	$-0.373473\pi$
$461$	−11.4853	−	11.4853i	−0.534923	−	0.534923i	−0.922033	+	0.387110i	$0.873473\pi$
$462$		0			0
$463$		−	3.65685i		−	0.169948i	−0.996383	−	0.0849742i	$-0.972919\pi$
$463$		−	3.65685i		−	0.169948i	0.996383	−	0.0849742i	$-0.0270808\pi$
$464$		0			0
$465$	5.58579	+	2.31371i	0.259035	+	0.107296i
$466$		0			0
$467$	13.9706	−	13.9706i	0.646481	−	0.646481i	−0.305660	−	0.952141i	$-0.598877\pi$
$467$	13.9706	−	13.9706i	0.646481	−	0.646481i	0.952141	+	0.305660i	$0.0988771\pi$
$468$		0			0
$469$	−8.00000	+	19.3137i	−0.369406	+	0.891824i
$470$		0			0
$471$	4.00000	+	9.65685i	0.184310	+	0.444964i
$472$		0			0
$473$	11.4853	−	4.75736i	0.528094	−	0.218744i
$474$		0			0
$475$	−23.8995			−1.09658
$476$		0			0
$477$	33.5563			1.53644
$478$		0			0
$479$	32.8492	−	13.6066i	1.50092	−	0.621702i	0.527260	−	0.849704i	$-0.323220\pi$
$479$	32.8492	−	13.6066i	1.50092	−	0.621702i	0.973661	+	0.228003i	$0.0732195\pi$
$480$		0			0
$481$	2.20101	+	5.31371i	0.100357	+	0.242284i
$482$		0			0
$483$	0.757359	−	1.82843i	0.0344610	−	0.0831963i
$484$		0			0
$485$	−7.24264	+	7.24264i	−0.328871	+	0.328871i
$486$		0			0
$487$	22.6066	+	9.36396i	1.02440	+	0.424322i	0.830689	−	0.556737i	$-0.187947\pi$
$487$	22.6066	+	9.36396i	1.02440	+	0.424322i	0.193714	+	0.981058i	$0.437947\pi$
$488$		0			0
$489$			10.5858i			0.478706i
$490$		0			0
$491$	22.7990	+	22.7990i	1.02890	+	1.02890i	0.999570	+	0.0293344i	$0.00933875\pi$
$491$	22.7990	+	22.7990i	1.02890	+	1.02890i	0.0293344	+	0.999570i	$0.490661\pi$
$492$		0			0
$493$	−22.5355	−	10.1213i	−1.01495	−	0.455841i
$494$		0			0
$495$	3.82843	+	3.82843i	0.172075	+	0.172075i
$496$		0			0
$497$		−	15.0711i		−	0.676030i
$498$		0			0
$499$	1.63604	+	0.677670i	0.0732392	+	0.0303367i	0.419002	−	0.907985i	$-0.362380\pi$
$499$	1.63604	+	0.677670i	0.0732392	+	0.0303367i	−0.345763	+	0.938322i	$0.612380\pi$
$500$		0			0
$501$	−1.92893	+	1.92893i	−0.0861783	+	0.0861783i
$502$		0			0
$503$	8.70711	−	21.0208i	0.388231	−	0.937272i	−0.602084	−	0.798433i	$-0.705663\pi$
$503$	8.70711	−	21.0208i	0.388231	−	0.937272i	0.990315	−	0.138839i	$-0.0443371\pi$
$504$		0			0
$505$	2.10051	+	5.07107i	0.0934712	+	0.225660i
$506$		0			0
$507$	8.22183	−	3.40559i	0.365144	−	0.151248i
$508$		0			0
$509$	29.3137			1.29931			0.649654	−	0.760230i	$-0.274914\pi$
$509$	29.3137			1.29931			0.649654	+	0.760230i	$0.274914\pi$
$510$		0			0
$511$	9.41421			0.416460
$512$		0			0
$513$	20.7279	−	8.58579i	0.915160	−	0.379072i
$514$		0			0
$515$	1.17157	+	2.82843i	0.0516257	+	0.124635i
$516$		0			0
$517$	8.58579	−	20.7279i	0.377602	−	0.911613i
$518$		0			0
$519$	−6.79899	+	6.79899i	−0.298443	+	0.298443i
$520$		0			0
$521$	−3.46447	−	1.43503i	−0.151781	−	0.0628698i	0.305500	−	0.952192i	$-0.401177\pi$
$521$	−3.46447	−	1.43503i	−0.151781	−	0.0628698i	−0.457281	+	0.889322i	$0.651177\pi$
$522$		0			0
$523$			34.0000i			1.48672i	0.668894	+	0.743358i	$0.266768\pi$
$523$			34.0000i			1.48672i	−0.668894	+	0.743358i	$0.733232\pi$
$524$		0			0
$525$	−4.41421	−	4.41421i	−0.192652	−	0.192652i
$526$		0			0
$527$	−17.4350	+	38.8198i	−0.759482	+	1.69102i
$528$		0			0
$529$	−14.8787	−	14.8787i	−0.646899	−	0.646899i
$530$		0			0
$531$		−	14.2426i		−	0.618078i
$532$		0			0
$533$	8.82843	+	3.65685i	0.382402	+	0.158396i
$534$		0			0
$535$	5.00000	−	5.00000i	0.216169	−	0.216169i
$536$		0			0
$537$	5.38478	−	13.0000i	0.232370	−	0.560991i
$538$		0			0
$539$	−4.02082	−	9.70711i	−0.173189	−	0.418115i
$540$		0			0
$541$	21.0208	−	8.70711i	0.903755	−	0.374348i	0.118093	−	0.993003i	$-0.462322\pi$
$541$	21.0208	−	8.70711i	0.903755	−	0.374348i	0.785663	+	0.618655i	$0.212322\pi$
$542$		0			0
$543$	7.07107			0.303449
$544$		0			0
$545$	−5.07107			−0.217221
$546$		0			0
$547$	−30.6066	+	12.6777i	−1.30864	+	0.542058i	−0.924488	−	0.381210i	$-0.875507\pi$
$547$	−30.6066	+	12.6777i	−1.30864	+	0.542058i	−0.384155	+	0.923268i	$0.625507\pi$
$548$		0			0
$549$	0.707107	+	1.70711i	0.0301786	+	0.0728575i
$550$		0			0
$551$	12.4142	−	29.9706i	0.528863	−	1.27679i
$552$		0			0
$553$	−11.8284	+	11.8284i	−0.502996	+	0.502996i
$554$		0			0
$555$	2.65685	+	1.10051i	0.112777	+	0.0467138i
$556$		0			0
$557$			17.1716i			0.727583i	0.931480	+	0.363791i	$0.118518\pi$
$557$			17.1716i			0.727583i	−0.931480	+	0.363791i	$0.881482\pi$
$558$		0			0
$559$	−3.51472	−	3.51472i	−0.148657	−	0.148657i
$560$		0			0
$561$	6.72792	−	6.34315i	0.284053	−	0.267808i
$562$		0			0
$563$	−29.2843	−	29.2843i	−1.23418	−	1.23418i	−0.962342	−	0.271843i	$-0.912367\pi$
$563$	−29.2843	−	29.2843i	−1.23418	−	1.23418i	−0.271843	−	0.962342i	$-0.587633\pi$
$564$		0			0
$565$			5.41421i			0.227778i
$566$		0			0
$567$	−6.94975	−	2.87868i	−0.291862	−	0.120893i
$568$		0			0
$569$	−2.51472	+	2.51472i	−0.105422	+	0.105422i	−0.757851	−	0.652428i	$-0.773750\pi$
$569$	−2.51472	+	2.51472i	−0.105422	+	0.105422i	0.652428	+	0.757851i	$0.273750\pi$
$570$		0			0
$571$	−16.4645	+	39.7487i	−0.689016	+	1.66343i	0.0577369	+	0.998332i	$0.481612\pi$
$571$	−16.4645	+	39.7487i	−0.689016	+	1.66343i	−0.746753	+	0.665101i	$0.768388\pi$
$572$		0			0
$573$	3.41421	+	8.24264i	0.142631	+	0.344341i
$574$		0			0
$575$	5.70711	−	2.36396i	0.238003	−	0.0985840i
$576$		0			0
$577$	−37.7990			−1.57359			−0.786796	−	0.617213i	$-0.788262\pi$
$577$	−37.7990			−1.57359			−0.786796	+	0.617213i	$0.788262\pi$
$578$		0			0
$579$	−3.27208			−0.135983
$580$		0			0
$581$	14.0711	−	5.82843i	0.583766	−	0.241804i
$582$		0			0
$583$	15.5858	+	37.6274i	0.645497	+	1.55837i
$584$		0			0
$585$	0.828427	−	2.00000i	0.0342512	−	0.0826898i
$586$		0			0
$587$	−9.34315	+	9.34315i	−0.385633	+	0.385633i	−0.873127	−	0.487494i	$-0.837911\pi$
$587$	−9.34315	+	9.34315i	−0.385633	+	0.385633i	0.487494	+	0.873127i	$0.337911\pi$
$588$		0			0
$589$	−51.6274	−	21.3848i	−2.12727	−	0.881144i
$590$		0			0
$591$		−	19.2132i		−	0.790326i
$592$		0			0
$593$	−6.51472	−	6.51472i	−0.267527	−	0.267527i	0.560576	−	0.828103i	$-0.310580\pi$
$593$	−6.51472	−	6.51472i	−0.267527	−	0.267527i	−0.828103	+	0.560576i	$0.810580\pi$
$594$		0			0
$595$	−4.24264	+	4.00000i	−0.173931	+	0.163984i
$596$		0			0
$597$	−7.82843	−	7.82843i	−0.320396	−	0.320396i
$598$		0			0
$599$			14.6863i			0.600066i	0.953929	+	0.300033i	$0.0969976\pi$
$599$			14.6863i			0.600066i	−0.953929	+	0.300033i	$0.903002\pi$
$600$		0			0
$601$	−7.94975	−	3.29289i	−0.324277	−	0.134320i	0.214606	−	0.976701i	$-0.431153\pi$
$601$	−7.94975	−	3.29289i	−0.324277	−	0.134320i	−0.538883	+	0.842381i	$0.681153\pi$
$602$		0			0
$603$	19.3137	−	19.3137i	0.786515	−	0.786515i
$604$		0			0
$605$	0.707107	−	1.70711i	0.0287480	−	0.0694038i
$606$		0			0
$607$	13.2929	+	32.0919i	0.539542	+	1.30257i	0.925043	+	0.379862i	$0.124029\pi$
$607$	13.2929	+	32.0919i	0.539542	+	1.30257i	−0.385501	+	0.922707i	$0.625971\pi$
$608$		0			0
$609$	7.82843	−	3.24264i	0.317224	−	0.131398i
$610$		0			0
$611$	−8.97056			−0.362910
$612$		0			0
$613$	11.6569			0.470816			0.235408	−	0.971897i	$-0.424357\pi$
$613$	11.6569			0.470816			0.235408	+	0.971897i	$0.424357\pi$
$614$		0			0
$615$	4.41421	−	1.82843i	0.177998	−	0.0737293i
$616$		0			0
$617$	−9.50610	−	22.9497i	−0.382701	−	0.923922i	−0.991441	−	0.130553i	$-0.958325\pi$
$617$	−9.50610	−	22.9497i	−0.382701	−	0.923922i	0.608740	−	0.793370i	$-0.291675\pi$
$618$		0			0
$619$	0.363961	−	0.878680i	0.0146288	−	0.0353171i	−0.916396	−	0.400272i	$-0.868916\pi$
$619$	0.363961	−	0.878680i	0.0146288	−	0.0353171i	0.931025	+	0.364955i	$0.118916\pi$
$620$		0			0
$621$	−4.10051	+	4.10051i	−0.164548	+	0.164548i
$622$		0			0
$623$	−8.82843	−	3.65685i	−0.353703	−	0.146509i
$624$		0			0
$625$		−	16.5563i		−	0.662254i
$626$		0			0
$627$	8.58579	+	8.58579i	0.342883	+	0.342883i
$628$		0			0
$629$	−8.29289	+	18.4645i	−0.330659	+	0.736226i
$630$		0			0
$631$	−12.3137	−	12.3137i	−0.490201	−	0.490201i	0.418168	−	0.908369i	$-0.362672\pi$
$631$	−12.3137	−	12.3137i	−0.490201	−	0.490201i	−0.908369	+	0.418168i	$0.862672\pi$
$632$		0			0
$633$			12.2426i			0.486601i
$634$		0			0
$635$	−5.82843	−	2.41421i	−0.231294	−	0.0958051i
$636$		0			0
$637$	−2.97056	+	2.97056i	−0.117698	+	0.117698i
$638$		0			0
$639$	−7.53553	+	18.1924i	−0.298101	+	0.719680i
$640$		0			0
$641$	0.920310	+	2.22183i	0.0363501	+	0.0877568i	0.941012	−	0.338373i	$-0.109877\pi$
$641$	0.920310	+	2.22183i	0.0363501	+	0.0877568i	−0.904662	+	0.426130i	$0.859877\pi$
$642$		0			0
$643$	−2.12132	+	0.878680i	−0.0836567	+	0.0346517i	−0.424119	−	0.905606i	$-0.639416\pi$
$643$	−2.12132	+	0.878680i	−0.0836567	+	0.0346517i	0.340463	+	0.940258i	$0.389416\pi$
$644$		0			0
$645$	−2.48528			−0.0978579
$646$		0			0
$647$	−0.970563			−0.0381568			−0.0190784	−	0.999818i	$-0.506073\pi$
$647$	−0.970563			−0.0381568			−0.0190784	+	0.999818i	$0.506073\pi$
$648$		0			0
$649$	15.9706	−	6.61522i	0.626899	−	0.259670i
$650$		0			0
$651$	−5.58579	−	13.4853i	−0.218924	−	0.528530i
$652$		0			0
$653$	7.74874	−	18.7071i	0.303232	−	0.732066i	−0.696661	−	0.717401i	$-0.745332\pi$
$653$	7.74874	−	18.7071i	0.303232	−	0.732066i	0.999892	−	0.0146651i	$-0.00466822\pi$
$654$		0			0
$655$	8.31371	−	8.31371i	0.324843	−	0.324843i
$656$		0			0
$657$	−11.3640	−	4.70711i	−0.443350	−	0.183642i
$658$		0			0
$659$			25.3137i			0.986082i	0.870006	+	0.493041i	$0.164115\pi$
$659$			25.3137i			0.986082i	−0.870006	+	0.493041i	$0.835885\pi$
$660$		0			0
$661$	9.00000	+	9.00000i	0.350059	+	0.350059i	0.860132	−	0.510072i	$-0.170381\pi$
$661$	9.00000	+	9.00000i	0.350059	+	0.350059i	−0.510072	+	0.860132i	$0.670381\pi$
$662$		0			0
$663$	−3.37258	−	1.51472i	−0.130980	−	0.0588268i
$664$		0			0
$665$	−5.41421	−	5.41421i	−0.209954	−	0.209954i
$666$		0			0
$667$			8.38478i			0.324660i
$668$		0			0
$669$	17.8284	+	7.38478i	0.689287	+	0.285512i
$670$		0			0
$671$	−1.58579	+	1.58579i	−0.0612186	+	0.0612186i
$672$		0			0
$673$	−6.67767	+	16.1213i	−0.257405	+	0.621431i	−0.998765	−	0.0496762i	$-0.984181\pi$
$673$	−6.67767	+	16.1213i	−0.257405	+	0.621431i	0.741360	+	0.671107i	$0.234181\pi$
$674$		0			0
$675$	7.00000	+	16.8995i	0.269430	+	0.650462i
$676$		0			0
$677$	13.3640	−	5.53553i	0.513619	−	0.212748i	−0.110793	−	0.993844i	$-0.535339\pi$
$677$	13.3640	−	5.53553i	0.513619	−	0.212748i	0.624411	+	0.781096i	$0.285339\pi$
$678$		0			0
$679$	24.7279			0.948971
$680$		0			0
$681$	16.1005			0.616973
$682$		0			0
$683$	14.3640	−	5.94975i	0.549622	−	0.227661i	−0.0905510	−	0.995892i	$-0.528863\pi$
$683$	14.3640	−	5.94975i	0.549622	−	0.227661i	0.640173	+	0.768231i	$0.278863\pi$
$684$		0			0
$685$	−4.72792	−	11.4142i	−0.180645	−	0.436115i
$686$		0			0
$687$	3.58579	−	8.65685i	0.136806	−	0.330280i
$688$		0			0
$689$	11.5147	−	11.5147i	0.438676	−	0.438676i
$690$		0			0
$691$	1.29289	+	0.535534i	0.0491840	+	0.0203727i	0.407140	−	0.913366i	$-0.366526\pi$
$691$	1.29289	+	0.535534i	0.0491840	+	0.0203727i	−0.357956	+	0.933739i	$0.616526\pi$
$692$		0			0
$693$		−	13.0711i		−	0.496529i
$694$		0			0
$695$	−3.10051	−	3.10051i	−0.117609	−	0.117609i
$696$		0			0
$697$	11.9497	+	31.4350i	0.452629	+	1.19069i
$698$		0			0
$699$	−13.7279	−	13.7279i	−0.519238	−	0.519238i
$700$		0			0
$701$		−	36.4853i		−	1.37803i	−0.724747	−	0.689015i	$-0.758043\pi$
$701$		−	36.4853i		−	1.37803i	0.724747	−	0.689015i	$-0.241957\pi$
$702$		0			0
$703$	−24.5563	−	10.1716i	−0.926160	−	0.383628i
$704$		0			0
$705$	−3.17157	+	3.17157i	−0.119448	+	0.119448i
$706$		0			0
$707$	5.07107	−	12.2426i	0.190717	−	0.460432i
$708$		0			0
$709$	1.12132	+	2.70711i	0.0421121	+	0.101668i	0.943536	−	0.331270i	$-0.107477\pi$
$709$	1.12132	+	2.70711i	0.0421121	+	0.101668i	−0.901424	+	0.432937i	$0.857477\pi$
$710$		0			0
$711$	20.1924	−	8.36396i	0.757274	−	0.313673i
$712$		0			0
$713$	14.4437			0.540919
$714$		0			0
$715$	2.62742			0.0982598
$716$		0			0
$717$	−16.0000	+	6.62742i	−0.597531	+	0.247505i
$718$		0			0
$719$	18.1213	+	43.7487i	0.675811	+	1.63155i	0.771567	+	0.636148i	$0.219473\pi$
$719$	18.1213	+	43.7487i	0.675811	+	1.63155i	−0.0957560	+	0.995405i	$0.530527\pi$
$720$		0			0
$721$	2.82843	−	6.82843i	0.105336	−	0.254304i
$722$		0			0
$723$	−7.97056	+	7.97056i	−0.296428	+	0.296428i
$724$		0			0
$725$	24.4350	+	10.1213i	0.907494	+	0.375896i
$726$		0			0
$727$		−	8.34315i		−	0.309430i	−0.987959	−	0.154715i	$-0.950554\pi$
$727$		−	8.34315i		−	0.309430i	0.987959	−	0.154715i	$-0.0494460\pi$
$728$		0			0
$729$	0.221825	+	0.221825i	0.00821576	+	0.00821576i
$730$		0			0
$731$	0.514719	−	17.4853i	0.0190376	−	0.646716i
$732$		0			0
$733$	14.6569	+	14.6569i	0.541363	+	0.541363i	0.923928	−	0.382565i	$-0.124959\pi$
$733$	14.6569	+	14.6569i	0.541363	+	0.541363i	−0.382565	+	0.923928i	$0.624959\pi$
$734$		0			0
$735$			2.10051i			0.0774783i
$736$		0			0
$737$	30.6274	+	12.6863i	1.12818	+	0.467306i
$738$		0			0
$739$	21.9706	−	21.9706i	0.808200	−	0.808200i	−0.176161	−	0.984361i	$-0.556368\pi$
$739$	21.9706	−	21.9706i	0.808200	−	0.808200i	0.984361	+	0.176161i	$0.0563680\pi$
$740$		0			0
$741$	1.85786	−	4.48528i	0.0682504	−	0.164771i
$742$		0			0
$743$	1.97918	+	4.77817i	0.0726092	+	0.175294i	0.956017	−	0.293310i	$-0.0947569\pi$
$743$	1.97918	+	4.77817i	0.0726092	+	0.175294i	−0.883408	+	0.468605i	$0.844757\pi$
$744$		0			0
$745$	1.17157	−	0.485281i	0.0429231	−	0.0177793i
$746$		0			0
$747$	−19.8995			−0.728084
$748$		0			0
$749$	−17.0711			−0.623763
$750$		0			0
$751$	30.3640	−	12.5772i	1.10800	−	0.458947i	0.247749	−	0.968824i	$-0.420309\pi$
$751$	30.3640	−	12.5772i	1.10800	−	0.458947i	0.860247	+	0.509877i	$0.170309\pi$
$752$		0			0
$753$	3.21320	+	7.75736i	0.117096	+	0.282694i
$754$		0			0
$755$	5.10051	−	12.3137i	0.185626	−	0.448142i
$756$		0			0
$757$	−33.0000	+	33.0000i	−1.19941	+	1.19941i	−0.225061	+	0.974345i	$0.572258\pi$
$757$	−33.0000	+	33.0000i	−1.19941	+	1.19941i	−0.974345	+	0.225061i	$0.927742\pi$
$758$		0			0
$759$	−2.89949	−	1.20101i	−0.105245	−	0.0435939i
$760$		0			0
$761$			46.1421i			1.67265i	0.548233	+	0.836326i	$0.315301\pi$
$761$			46.1421i			1.67265i	−0.548233	+	0.836326i	$0.684699\pi$
$762$		0			0
$763$	8.65685	+	8.65685i	0.313399	+	0.313399i
$764$		0			0
$765$	7.12132	−	2.70711i	0.257472	−	0.0978757i
$766$		0			0
$767$	−4.88730	−	4.88730i	−0.176470	−	0.176470i
$768$		0			0
$769$		−	35.7990i		−	1.29094i	−0.763784	−	0.645472i	$-0.776661\pi$
$769$		−	35.7990i		−	1.29094i	0.763784	−	0.645472i	$-0.223339\pi$
$770$		0			0
$771$	8.65685	+	3.58579i	0.311769	+	0.129139i
$772$		0			0
$773$	−24.7990	+	24.7990i	−0.891958	+	0.891958i	−0.994707	−	0.102750i	$-0.967236\pi$
$773$	−24.7990	+	24.7990i	−0.891958	+	0.891958i	0.102750	+	0.994707i	$0.467236\pi$
$774$		0			0
$775$	17.4350	−	42.0919i	0.626285	−	1.51199i
$776$		0			0
$777$	−2.65685	−	6.41421i	−0.0953141	−	0.230109i
$778$		0			0
$779$	−40.7990	+	16.8995i	−1.46178	+	0.605487i
$780$		0			0
$781$	−23.8995			−0.855191
$782$		0			0
$783$	−24.8284			−0.887296
$784$		0			0
$785$	−9.65685	+	4.00000i	−0.344668	+	0.142766i
$786$		0			0
$787$	−11.3934	−	27.5061i	−0.406131	−	0.980486i	−0.986146	−	0.165881i	$-0.946953\pi$
$787$	−11.3934	−	27.5061i	−0.406131	−	0.980486i	0.580015	−	0.814606i	$-0.303047\pi$
$788$		0			0
$789$	−8.55635	+	20.6569i	−0.304614	+	0.735403i
$790$		0			0
$791$	9.24264	−	9.24264i	0.328630	−	0.328630i
$792$		0			0
$793$	0.828427	+	0.343146i	0.0294183	+	0.0121855i
$794$		0			0
$795$		−	8.14214i		−	0.288772i
$796$		0			0
$797$	−11.6863	−	11.6863i	−0.413950	−	0.413950i	0.469162	−	0.883112i	$-0.344556\pi$
$797$	−11.6863	−	11.6863i	−0.413950	−	0.413950i	−0.883112	+	0.469162i	$0.844556\pi$
$798$		0			0
$799$	−21.6569	−	22.9706i	−0.766164	−	0.812640i
$800$		0			0
$801$	8.82843	+	8.82843i	0.311937	+	0.311937i
$802$		0			0
$803$		−	14.9289i		−	0.526831i
$804$		0			0
$805$	1.82843	+	0.757359i	0.0644436	+	0.0266934i
$806$		0			0
$807$	−11.0000	+	11.0000i	−0.387218	+	0.387218i
$808$		0			0
$809$	−13.5061	+	32.6066i	−0.474849	+	1.14639i	0.487146	+	0.873321i	$0.338038\pi$
$809$	−13.5061	+	32.6066i	−0.474849	+	1.14639i	−0.961995	+	0.273067i	$0.911962\pi$
$810$		0			0
$811$	−2.02082	−	4.87868i	−0.0709604	−	0.171314i	0.884420	−	0.466692i	$-0.154554\pi$
$811$	−2.02082	−	4.87868i	−0.0709604	−	0.171314i	−0.955380	+	0.295378i	$0.904554\pi$
$812$		0			0
$813$	8.00000	−	3.31371i	0.280572	−	0.116217i
$814$		0			0
$815$	−10.5858			−0.370804
$816$		0			0
$817$	22.9706			0.803638
$818$		0			0
$819$	−4.82843	+	2.00000i	−0.168719	+	0.0698857i
$820$		0			0
$821$	−12.5355	−	30.2635i	−0.437493	−	1.05620i	−0.976812	−	0.214100i	$-0.931318\pi$
$821$	−12.5355	−	30.2635i	−0.437493	−	1.05620i	0.539319	−	0.842102i	$-0.318682\pi$
$822$		0			0
$823$	−15.9203	+	38.4350i	−0.554947	+	1.33976i	0.358776	+	0.933424i	$0.383194\pi$
$823$	−15.9203	+	38.4350i	−0.554947	+	1.33976i	−0.913723	+	0.406337i	$0.866806\pi$
$824$		0			0
$825$	−7.00000	+	7.00000i	−0.243709	+	0.243709i
$826$		0			0
$827$	22.6066	+	9.36396i	0.786109	+	0.325617i	0.739378	−	0.673291i	$-0.235120\pi$
$827$	22.6066	+	9.36396i	0.786109	+	0.325617i	0.0467307	+	0.998908i	$0.485120\pi$
$828$		0			0
$829$		−	18.3431i		−	0.637084i	−0.947909	−	0.318542i	$-0.896807\pi$
$829$		−	18.3431i		−	0.637084i	0.947909	−	0.318542i	$-0.103193\pi$
$830$		0			0
$831$	−5.72792	−	5.72792i	−0.198699	−	0.198699i
$832$		0			0
$833$	−14.7782	−	0.435029i	−0.512033	−	0.0150729i
$834$		0			0
$835$	−1.92893	−	1.92893i	−0.0667535	−	0.0667535i
$836$		0			0
$837$			42.7696i			1.47833i
$838$		0			0
$839$	−37.6777	−	15.6066i	−1.30078	−	0.538800i	−0.378598	−	0.925561i	$-0.623594\pi$
$839$	−37.6777	−	15.6066i	−1.30078	−	0.538800i	−0.922180	+	0.386762i	$0.873594\pi$
$840$		0			0
$841$	−4.87868	+	4.87868i	−0.168230	+	0.168230i
$842$		0			0
$843$	−0.272078	+	0.656854i	−0.00937086	+	0.0226233i
$844$		0			0
$845$	3.40559	+	8.22183i	0.117156	+	0.282839i
$846$		0			0
$847$	−4.12132	+	1.70711i	−0.141610	+	0.0586569i
$848$		0			0
$849$	1.07107			0.0367590
$850$		0			0
$851$	6.87006			0.235503
$852$		0			0
$853$	9.70711	−	4.02082i	0.332365	−	0.137670i	−0.210259	−	0.977646i	$-0.567431\pi$
$853$	9.70711	−	4.02082i	0.332365	−	0.137670i	0.542624	+	0.839976i	$0.317431\pi$
$854$		0			0
$855$	3.82843	+	9.24264i	0.130929	+	0.316092i
$856$		0			0
$857$	−2.05025	+	4.94975i	−0.0700353	+	0.169080i	−0.955021	−	0.296538i	$-0.904168\pi$
$857$	−2.05025	+	4.94975i	−0.0700353	+	0.169080i	0.884986	+	0.465618i	$0.154168\pi$
$858$		0			0
$859$	6.17157	−	6.17157i	0.210571	−	0.210571i	−0.593939	−	0.804510i	$-0.702428\pi$
$859$	6.17157	−	6.17157i	0.210571	−	0.210571i	0.804510	+	0.593939i	$0.202428\pi$
$860$		0			0
$861$	−10.6569	−	4.41421i	−0.363185	−	0.150436i
$862$		0			0
$863$		−	14.0000i		−	0.476566i	−0.971196	−	0.238283i	$-0.923415\pi$
$863$		−	14.0000i		−	0.476566i	0.971196	−	0.238283i	$-0.0765845\pi$
$864$		0			0
$865$	−6.79899	−	6.79899i	−0.231173	−	0.231173i
$866$		0			0
$867$	−4.26346	−	12.2929i	−0.144795	−	0.417489i
$868$		0			0
$869$	18.7574	+	18.7574i	0.636300	+	0.636300i
$870$		0			0
$871$		−	13.2548i		−	0.449123i
$872$		0			0
$873$	−29.8492	−	12.3640i	−1.01024	−	0.418457i
$874$		0			0
$875$	9.41421	−	9.41421i	0.318259	−	0.318259i
$876$		0			0
$877$	1.94975	−	4.70711i	0.0658383	−	0.158948i	−0.887536	−	0.460739i	$-0.847585\pi$
$877$	1.94975	−	4.70711i	0.0658383	−	0.158948i	0.953374	+	0.301791i	$0.0975845\pi$
$878$		0			0
$879$	−2.34315	−	5.65685i	−0.0790323	−	0.190801i
$880$		0			0
$881$	−2.29289	+	0.949747i	−0.0772495	+	0.0319978i	−0.420974	−	0.907073i	$-0.638312\pi$
$881$	−2.29289	+	0.949747i	−0.0772495	+	0.0319978i	0.343724	+	0.939071i	$0.388312\pi$
$882$		0			0
$883$	25.6569			0.863422			0.431711	−	0.902012i	$-0.357910\pi$
$883$	25.6569			0.863422			0.431711	+	0.902012i	$0.357910\pi$
$884$		0			0
$885$	−3.45584			−0.116167
$886$		0			0
$887$	−41.5772	+	17.2218i	−1.39602	+	0.578252i	−0.948717	−	0.316128i	$-0.897617\pi$
$887$	−41.5772	+	17.2218i	−1.39602	+	0.578252i	−0.447308	+	0.894380i	$0.647617\pi$
$888$		0			0
$889$	5.82843	+	14.0711i	0.195479	+	0.471928i
$890$		0			0
$891$	−4.56497	+	11.0208i	−0.152932	+	0.369211i
$892$		0			0
$893$	29.3137	−	29.3137i	0.980946	−	0.980946i
$894$		0			0
$895$	13.0000	+	5.38478i	0.434542	+	0.179993i
$896$		0			0
$897$			1.25483i			0.0418977i
$898$		0			0
$899$	43.7279	+	43.7279i	1.45841	+	1.45841i
$900$		0			0
$901$	57.2843	+	1.68629i	1.90842	+	0.0561785i
$902$		0			0
$903$	4.24264	+	4.24264i	0.141186	+	0.141186i
$904$		0			0
$905$			7.07107i			0.235050i
$906$		0			0
$907$	41.0919	+	17.0208i	1.36443	+	0.565167i	0.940273	−	0.340420i	$-0.110569\pi$
$907$	41.0919	+	17.0208i	1.36443	+	0.565167i	0.424160	+	0.905587i	$0.360569\pi$
$908$		0			0
$909$	−12.2426	+	12.2426i	−0.406063	+	0.406063i
$910$		0			0
$911$	21.1924	−	51.1630i	0.702135	−	1.69510i	−0.0166400	−	0.999862i	$-0.505297\pi$
$911$	21.1924	−	51.1630i	0.702135	−	1.69510i	0.718775	−	0.695243i	$-0.244703\pi$
$912$		0			0
$913$	−9.24264	−	22.3137i	−0.305887	−	0.738476i
$914$		0			0
$915$	0.414214	−	0.171573i	0.0136935	−	0.00567202i
$916$		0			0
$917$	−28.3848			−0.937348
$918$		0			0
$919$	−25.6569			−0.846342			−0.423171	−	0.906050i	$-0.639083\pi$
$919$	−25.6569			−0.846342			−0.423171	+	0.906050i	$0.639083\pi$
$920$		0			0
$921$	12.4853	−	5.17157i	0.411404	−	0.170409i
$922$		0			0
$923$	3.65685	+	8.82843i	0.120367	+	0.290591i
$924$		0			0
$925$	8.29289	−	20.0208i	0.272669	−	0.658280i
$926$		0			0
$927$	−6.82843	+	6.82843i	−0.224275	+	0.224275i
$928$		0			0
$929$	−24.9203	−	10.3223i	−0.817609	−	0.338665i	−0.0656235	−	0.997844i	$-0.520904\pi$
$929$	−24.9203	−	10.3223i	−0.817609	−	0.338665i	−0.751985	+	0.659180i	$0.770904\pi$
$930$		0			0
$931$		−	19.4142i		−	0.636275i
$932$		0			0
$933$	5.34315	+	5.34315i	0.174927	+	0.174927i
$934$		0			0
$935$	6.34315	+	6.72792i	0.207443	+	0.220027i
$936$		0			0
$937$	33.4853	+	33.4853i	1.09392	+	1.09392i	0.995106	+	0.0988102i	$0.0315037\pi$
$937$	33.4853	+	33.4853i	1.09392	+	1.09392i	0.0988102	+	0.995106i	$0.468496\pi$
$938$		0			0
$939$			6.04163i			0.197161i
$940$		0			0
$941$	13.3640	+	5.53553i	0.435653	+	0.180453i	0.589721	−	0.807607i	$-0.299238\pi$
$941$	13.3640	+	5.53553i	0.435653	+	0.180453i	−0.154068	+	0.988060i	$0.549238\pi$
$942$		0			0
$943$	8.07107	−	8.07107i	0.262830	−	0.262830i
$944$		0			0
$945$	−2.24264	+	5.41421i	−0.0729531	+	0.176124i
$946$		0			0
$947$	−4.36396	−	10.5355i	−0.141810	−	0.342359i	0.836978	−	0.547237i	$-0.184320\pi$
$947$	−4.36396	−	10.5355i	−0.141810	−	0.342359i	−0.978788	+	0.204878i	$0.934320\pi$
$948$		0			0
$949$	−5.51472	+	2.28427i	−0.179015	+	0.0741506i
$950$		0			0
$951$	24.7279			0.801858
$952$		0			0
$953$	−16.1421			−0.522895			−0.261448	−	0.965218i	$-0.584200\pi$
$953$	−16.1421			−0.522895			−0.261448	+	0.965218i	$0.584200\pi$
$954$		0			0
$955$	−8.24264	+	3.41421i	−0.266726	+	0.110481i
$956$		0			0
$957$	−5.14214	−	12.4142i	−0.166222	−	0.401295i
$958$		0			0
$959$	−11.4142	+	27.5563i	−0.368584	+	0.889841i
$960$		0			0
$961$	53.4056	−	53.4056i	1.72276	−	1.72276i
$962$		0			0
$963$	20.6066	+	8.53553i	0.664038	+	0.275054i
$964$		0			0
$965$		−	3.27208i		−	0.105332i
$966$		0			0
$967$	−34.6569	−	34.6569i	−1.11449	−	1.11449i	−0.992536	−	0.121953i	$-0.961084\pi$
$967$	−34.6569	−	34.6569i	−1.11449	−	1.11449i	−0.121953	−	0.992536i	$-0.538916\pi$
$968$		0			0
$969$	15.9706	−	6.07107i	0.513048	−	0.195031i
$970$		0			0
$971$	3.68629	+	3.68629i	0.118299	+	0.118299i	0.763778	−	0.645479i	$-0.223342\pi$
$971$	3.68629	+	3.68629i	0.118299	+	0.118299i	−0.645479	+	0.763778i	$0.723342\pi$
$972$		0			0
$973$			10.5858i			0.339365i
$974$		0			0
$975$	3.65685	+	1.51472i	0.117113	+	0.0485098i
$976$		0			0
$977$	−1.14214	+	1.14214i	−0.0365402	+	0.0365402i	−0.725141	−	0.688601i	$-0.758225\pi$
$977$	−1.14214	+	1.14214i	−0.0365402	+	0.0365402i	0.688601	+	0.725141i	$0.258225\pi$
$978$		0			0
$979$	−5.79899	+	14.0000i	−0.185337	+	0.447442i
$980$		0			0
$981$	−6.12132	−	14.7782i	−0.195439	−	0.471831i
$982$		0			0
$983$	−24.9497	+	10.3345i	−0.795773	+	0.329620i	−0.743262	−	0.669000i	$-0.766723\pi$
$983$	−24.9497	+	10.3345i	−0.795773	+	0.329620i	−0.0525112	+	0.998620i	$0.516723\pi$
$984$		0			0
$985$	19.2132			0.612184
$986$		0			0
$987$	10.8284			0.344673
$988$		0			0
$989$	−5.48528	+	2.27208i	−0.174422	+	0.0722479i
$990$		0			0
$991$	1.92031	+	4.63604i	0.0610007	+	0.147269i	0.951441	−	0.307832i	$-0.0996034\pi$
$991$	1.92031	+	4.63604i	0.0610007	+	0.147269i	−0.890440	+	0.455100i	$0.849603\pi$
$992$		0			0
$993$	−0.958369	+	2.31371i	−0.0304129	+	0.0734233i
$994$		0			0
$995$	7.82843	−	7.82843i	0.248178	−	0.248178i
$996$		0			0
$997$	17.8492	+	7.39340i	0.565291	+	0.234151i	0.646980	−	0.762507i	$-0.276032\pi$
$997$	17.8492	+	7.39340i	0.565291	+	0.234151i	−0.0816893	+	0.996658i	$0.526032\pi$
$998$		0			0
$999$			20.3431i			0.643629i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.2.h.a.53.1	yes	4
3.2	odd	2		612.2.w.a.325.1		4
4.3	odd	2		272.2.v.c.257.1		4
17.2	even	8		1156.2.h.c.977.1		4
17.3	odd	16		1156.2.a.g.1.3		4
17.4	even	4		1156.2.h.a.1001.1		4
17.5	odd	16		1156.2.b.d.577.3		4
17.6	odd	16		1156.2.e.f.905.3		8
17.7	odd	16		1156.2.e.f.829.2		8
17.8	even	8		1156.2.h.b.757.1		4
17.9	even	8	inner	68.2.h.a.9.1	✓	4
17.10	odd	16		1156.2.e.f.829.3		8
17.11	odd	16		1156.2.e.f.905.2		8
17.12	odd	16		1156.2.b.d.577.2		4
17.13	even	4		1156.2.h.c.1001.1		4
17.14	odd	16		1156.2.a.g.1.2		4
17.15	even	8		1156.2.h.a.977.1		4
17.16	even	2		1156.2.h.b.733.1		4
51.26	odd	8		612.2.w.a.145.1		4
68.3	even	16		4624.2.a.bl.1.2		4
68.31	even	16		4624.2.a.bl.1.3		4
68.43	odd	8		272.2.v.c.145.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.h.a.9.1	✓	4	17.9	even	8	inner
68.2.h.a.53.1	yes	4	1.1	even	1	trivial
272.2.v.c.145.1		4	68.43	odd	8
272.2.v.c.257.1		4	4.3	odd	2
612.2.w.a.145.1		4	51.26	odd	8
612.2.w.a.325.1		4	3.2	odd	2
1156.2.a.g.1.2		4	17.14	odd	16
1156.2.a.g.1.3		4	17.3	odd	16
1156.2.b.d.577.2		4	17.12	odd	16
1156.2.b.d.577.3		4	17.5	odd	16
1156.2.e.f.829.2		8	17.7	odd	16
1156.2.e.f.829.3		8	17.10	odd	16
1156.2.e.f.905.2		8	17.11	odd	16
1156.2.e.f.905.3		8	17.6	odd	16
1156.2.h.a.977.1		4	17.15	even	8
1156.2.h.a.1001.1		4	17.4	even	4
1156.2.h.b.733.1		4	17.16	even	2
1156.2.h.b.757.1		4	17.8	even	8
1156.2.h.c.977.1		4	17.2	even	8
1156.2.h.c.1001.1		4	17.13	even	4
4624.2.a.bl.1.2		4	68.3	even	16
4624.2.a.bl.1.3		4	68.31	even	16

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	68.h (of order \(8\), degree \(4\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.292893i) q^{3} +(0.292893 + 0.707107i) q^{5} +(0.707107 - 1.70711i) q^{7} +(-1.70711 + 1.70711i) q^{9} +O(q^{10})\)	\(q+(0.707107 - 0.292893i) q^{3} +(0.292893 + 0.707107i) q^{5} +(0.707107 - 1.70711i) q^{7} +(-1.70711 + 1.70711i) q^{9} +(-2.70711 - 1.12132i) q^{11} +1.17157i q^{13} +(0.414214 + 0.414214i) q^{15} +(-3.00000 + 2.82843i) q^{17} +(-3.82843 - 3.82843i) q^{19} -1.41421i q^{21} +(1.29289 + 0.535534i) q^{23} +(3.12132 - 3.12132i) q^{25} +(-1.58579 + 3.82843i) q^{27} +(2.29289 + 5.53553i) q^{29} +(9.53553 - 3.94975i) q^{31} -2.24264 q^{33} +1.41421 q^{35} +(4.53553 - 1.87868i) q^{37} +(0.343146 + 0.828427i) q^{39} +(3.12132 - 7.53553i) q^{41} +(-3.00000 + 3.00000i) q^{43} +(-1.70711 - 0.707107i) q^{45} +7.65685i q^{47} +(2.53553 + 2.53553i) q^{49} +(-1.29289 + 2.87868i) q^{51} +(-9.82843 - 9.82843i) q^{53} -2.24264i q^{55} +(-3.82843 - 1.58579i) q^{57} +(-4.17157 + 4.17157i) q^{59} +(0.292893 - 0.707107i) q^{61} +(1.70711 + 4.12132i) q^{63} +(-0.828427 + 0.343146i) q^{65} -11.3137 q^{67} +1.07107 q^{69} +(7.53553 - 3.12132i) q^{71} +(1.94975 + 4.70711i) q^{73} +(1.29289 - 3.12132i) q^{75} +(-3.82843 + 3.82843i) q^{77} +(-8.36396 - 3.46447i) q^{79} -4.07107i q^{81} +(5.82843 + 5.82843i) q^{83} +(-2.87868 - 1.29289i) q^{85} +(3.24264 + 3.24264i) q^{87} -5.17157i q^{89} +(2.00000 + 0.828427i) q^{91} +(5.58579 - 5.58579i) q^{93} +(1.58579 - 3.82843i) q^{95} +(5.12132 + 12.3640i) q^{97} +(6.53553 - 2.70711i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{5} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^5 - 4 * q^9	\( 4 q + 4 q^{5} - 4 q^{9} - 8 q^{11} - 4 q^{15} - 12 q^{17} - 4 q^{19} + 8 q^{23} + 4 q^{25} - 12 q^{27} + 12 q^{29} + 24 q^{31} + 8 q^{33} + 4 q^{37} + 24 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} - 4 q^{49} - 8 q^{51} - 28 q^{53} - 4 q^{57} - 28 q^{59} + 4 q^{61} + 4 q^{63} + 8 q^{65} - 24 q^{69} + 16 q^{71} - 12 q^{73} + 8 q^{75} - 4 q^{77} - 8 q^{79} + 12 q^{83} - 20 q^{85} - 4 q^{87} + 8 q^{91} + 28 q^{93} + 12 q^{95} + 12 q^{97} + 12 q^{99}+O(q^{100}) \) 4 * q + 4 * q^5 - 4 * q^9 - 8 * q^11 - 4 * q^15 - 12 * q^17 - 4 * q^19 + 8 * q^23 + 4 * q^25 - 12 * q^27 + 12 * q^29 + 24 * q^31 + 8 * q^33 + 4 * q^37 + 24 * q^39 + 4 * q^41 - 12 * q^43 - 4 * q^45 - 4 * q^49 - 8 * q^51 - 28 * q^53 - 4 * q^57 - 28 * q^59 + 4 * q^61 + 4 * q^63 + 8 * q^65 - 24 * q^69 + 16 * q^71 - 12 * q^73 + 8 * q^75 - 4 * q^77 - 8 * q^79 + 12 * q^83 - 20 * q^85 - 4 * q^87 + 8 * q^91 + 28 * q^93 + 12 * q^95 + 12 * q^97 + 12 * q^99

Introduction

L-functions

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Database

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