LMFDB - Embedded newform 136.2.k.b.89.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,2,Mod(81,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 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("136.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	136.k (of order $4$, degree $2$, 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:	$1.08596546749$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			89.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	136.89
Dual form			136.2.k.b.81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} +1.00000i q^{9} +O(q^{10})$	\(q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} +1.00000i q^{9} +(1.00000 + 1.00000i) q^{11} +2.00000 q^{13} +4.00000i q^{15} +(-4.00000 - 1.00000i) q^{17} -4.00000i q^{19} -4.00000 q^{21} +(-4.00000 - 4.00000i) q^{23} -3.00000i q^{25} +(-4.00000 - 4.00000i) q^{27} +(6.00000 - 6.00000i) q^{29} +(-6.00000 + 6.00000i) q^{31} -2.00000 q^{33} +8.00000 q^{35} +(-8.00000 + 8.00000i) q^{37} +(-2.00000 + 2.00000i) q^{39} +(1.00000 + 1.00000i) q^{41} -2.00000i q^{43} +(2.00000 + 2.00000i) q^{45} +1.00000i q^{49} +(5.00000 - 3.00000i) q^{51} +6.00000i q^{53} +4.00000 q^{55} +(4.00000 + 4.00000i) q^{57} -14.0000i q^{59} +(-4.00000 - 4.00000i) q^{61} +(-2.00000 + 2.00000i) q^{63} +(4.00000 - 4.00000i) q^{65} +6.00000 q^{67} +8.00000 q^{69} +(-9.00000 + 9.00000i) q^{73} +(3.00000 + 3.00000i) q^{75} +4.00000i q^{77} +(4.00000 + 4.00000i) q^{79} +5.00000 q^{81} -6.00000i q^{83} +(-10.0000 + 6.00000i) q^{85} +12.0000i q^{87} +16.0000 q^{89} +(4.00000 + 4.00000i) q^{91} -12.0000i q^{93} +(-8.00000 - 8.00000i) q^{95} +(3.00000 - 3.00000i) q^{97} +(-1.00000 + 1.00000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10}) $ 2 * q - 2 * q^3 + 4 * q^5 + 4 * q^7	$ 2 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{11} + 4 q^{13} - 8 q^{17} - 8 q^{21} - 8 q^{23} - 8 q^{27} + 12 q^{29} - 12 q^{31} - 4 q^{33} + 16 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} + 4 q^{45} + 10 q^{51} + 8 q^{55} + 8 q^{57} - 8 q^{61} - 4 q^{63} + 8 q^{65} + 12 q^{67} + 16 q^{69} - 18 q^{73} + 6 q^{75} + 8 q^{79} + 10 q^{81} - 20 q^{85} + 32 q^{89} + 8 q^{91} - 16 q^{95} + 6 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^11 + 4 * q^13 - 8 * q^17 - 8 * q^21 - 8 * q^23 - 8 * q^27 + 12 * q^29 - 12 * q^31 - 4 * q^33 + 16 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 + 4 * q^45 + 10 * q^51 + 8 * q^55 + 8 * q^57 - 8 * q^61 - 4 * q^63 + 8 * q^65 + 12 * q^67 + 16 * q^69 - 18 * q^73 + 6 * q^75 + 8 * q^79 + 10 * q^81 - 20 * q^85 + 32 * q^89 + 8 * q^91 - 16 * q^95 + 6 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	−0.934172	−	0.356822i	$-0.883860\pi$
$3$	−1.00000	+	1.00000i	−0.577350	+	0.577350i	0.356822	+	0.934172i	$0.383860\pi$
$4$		0			0
$5$	2.00000	−	2.00000i	0.894427	−	0.894427i	−0.100509	−	0.994936i	$-0.532047\pi$
$5$	2.00000	−	2.00000i	0.894427	−	0.894427i	0.994936	+	0.100509i	$0.0320471\pi$
$6$		0			0
$7$	2.00000	+	2.00000i	0.755929	+	0.755929i	0.975579	−	0.219650i	$-0.0704915\pi$
$7$	2.00000	+	2.00000i	0.755929	+	0.755929i	−0.219650	+	0.975579i	$0.570491\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	1.00000	+	1.00000i	0.301511	+	0.301511i	0.841605	−	0.540094i	$-0.181611\pi$
$11$	1.00000	+	1.00000i	0.301511	+	0.301511i	−0.540094	+	0.841605i	$0.681611\pi$
$12$		0			0
$13$	2.00000			0.554700			0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000			0.554700			0.277350	+	0.960769i	$0.410544\pi$
$14$		0			0
$15$			4.00000i			1.03280i
$16$		0			0
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$17$	−4.00000	−	1.00000i	−0.970143	−	0.242536i
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$	−4.00000			−0.872872
$22$		0			0
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	0.154011	−	0.988069i	$-0.450781\pi$
$23$	−4.00000	−	4.00000i	−0.834058	−	0.834058i	−0.988069	+	0.154011i	$0.950781\pi$
$24$		0			0
$25$		−	3.00000i		−	0.600000i
$26$		0			0
$27$	−4.00000	−	4.00000i	−0.769800	−	0.769800i
$28$		0			0
$29$	6.00000	−	6.00000i	1.11417	−	1.11417i	0.121592	−	0.992580i	$-0.461200\pi$
$29$	6.00000	−	6.00000i	1.11417	−	1.11417i	0.992580	−	0.121592i	$-0.0387999\pi$
$30$		0			0
$31$	−6.00000	+	6.00000i	−1.07763	+	1.07763i	−0.0809104	+	0.996721i	$0.525783\pi$
$31$	−6.00000	+	6.00000i	−1.07763	+	1.07763i	−0.996721	+	0.0809104i	$0.974217\pi$
$32$		0			0
$33$	−2.00000			−0.348155
$34$		0			0
$35$	8.00000			1.35225
$36$		0			0
$37$	−8.00000	+	8.00000i	−1.31519	+	1.31519i	−0.397658	+	0.917534i	$0.630177\pi$
$37$	−8.00000	+	8.00000i	−1.31519	+	1.31519i	−0.917534	+	0.397658i	$0.869823\pi$
$38$		0			0
$39$	−2.00000	+	2.00000i	−0.320256	+	0.320256i
$40$		0			0
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	0.780869	−	0.624695i	$-0.214777\pi$
$41$	1.00000	+	1.00000i	0.156174	+	0.156174i	−0.624695	+	0.780869i	$0.714777\pi$
$42$		0			0
$43$		−	2.00000i		−	0.304997i	−0.988304	−	0.152499i	$-0.951268\pi$
$43$		−	2.00000i		−	0.304997i	0.988304	−	0.152499i	$-0.0487319\pi$
$44$		0			0
$45$	2.00000	+	2.00000i	0.298142	+	0.298142i
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$			1.00000i			0.142857i
$50$		0			0
$51$	5.00000	−	3.00000i	0.700140	−	0.420084i
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$	4.00000			0.539360
$56$		0			0
$57$	4.00000	+	4.00000i	0.529813	+	0.529813i
$58$		0			0
$59$		−	14.0000i		−	1.82264i	−0.411693	−	0.911322i	$-0.635063\pi$
$59$		−	14.0000i		−	1.82264i	0.411693	−	0.911322i	$-0.364937\pi$
$60$		0			0
$61$	−4.00000	−	4.00000i	−0.512148	−	0.512148i	0.403036	−	0.915184i	$-0.367955\pi$
$61$	−4.00000	−	4.00000i	−0.512148	−	0.512148i	−0.915184	+	0.403036i	$0.867955\pi$
$62$		0			0
$63$	−2.00000	+	2.00000i	−0.251976	+	0.251976i
$64$		0			0
$65$	4.00000	−	4.00000i	0.496139	−	0.496139i
$66$		0			0
$67$	6.00000			0.733017			0.366508	−	0.930415i	$-0.380553\pi$
$67$	6.00000			0.733017			0.366508	+	0.930415i	$0.380553\pi$
$68$		0			0
$69$	8.00000			0.963087
$70$		0			0
$71$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$71$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$72$		0			0
$73$	−9.00000	+	9.00000i	−1.05337	+	1.05337i	−0.0548772	+	0.998493i	$0.517477\pi$
$73$	−9.00000	+	9.00000i	−1.05337	+	1.05337i	−0.998493	+	0.0548772i	$0.982523\pi$
$74$		0			0
$75$	3.00000	+	3.00000i	0.346410	+	0.346410i
$76$		0			0
$77$			4.00000i			0.455842i
$78$		0			0
$79$	4.00000	+	4.00000i	0.450035	+	0.450035i	0.895366	−	0.445331i	$-0.146914\pi$
$79$	4.00000	+	4.00000i	0.450035	+	0.450035i	−0.445331	+	0.895366i	$0.646914\pi$
$80$		0			0
$81$	5.00000			0.555556
$82$		0			0
$83$		−	6.00000i		−	0.658586i	−0.944228	−	0.329293i	$-0.893190\pi$
$83$		−	6.00000i		−	0.658586i	0.944228	−	0.329293i	$-0.106810\pi$
$84$		0			0
$85$	−10.0000	+	6.00000i	−1.08465	+	0.650791i
$86$		0			0
$87$			12.0000i			1.28654i
$88$		0			0
$89$	16.0000			1.69600			0.847998	−	0.529999i	$-0.177808\pi$
$89$	16.0000			1.69600			0.847998	+	0.529999i	$0.177808\pi$
$90$		0			0
$91$	4.00000	+	4.00000i	0.419314	+	0.419314i
$92$		0			0
$93$		−	12.0000i		−	1.24434i
$94$		0			0
$95$	−8.00000	−	8.00000i	−0.820783	−	0.820783i
$96$		0			0
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	−0.538208	−	0.842812i	$-0.680899\pi$
$97$	3.00000	−	3.00000i	0.304604	−	0.304604i	0.842812	+	0.538208i	$0.180899\pi$
$98$		0			0
$99$	−1.00000	+	1.00000i	−0.100504	+	0.100504i
$100$		0			0
$101$	−6.00000			−0.597022			−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000			−0.597022			−0.298511	+	0.954406i	$0.596490\pi$
$102$		0			0
$103$	20.0000			1.97066			0.985329	−	0.170664i	$-0.0545913\pi$
$103$	20.0000			1.97066			0.985329	+	0.170664i	$0.0545913\pi$
$104$		0			0
$105$	−8.00000	+	8.00000i	−0.780720	+	0.780720i
$106$		0			0
$107$	−13.0000	+	13.0000i	−1.25676	+	1.25676i	−0.304125	+	0.952632i	$0.598364\pi$
$107$	−13.0000	+	13.0000i	−1.25676	+	1.25676i	−0.952632	+	0.304125i	$0.901636\pi$
$108$		0			0
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	0.358741	−	0.933437i	$-0.383206\pi$
$109$	−6.00000	−	6.00000i	−0.574696	−	0.574696i	−0.933437	+	0.358741i	$0.883206\pi$
$110$		0			0
$111$		−	16.0000i		−	1.51865i
$112$		0			0
$113$	−5.00000	−	5.00000i	−0.470360	−	0.470360i	0.431671	−	0.902031i	$-0.357924\pi$
$113$	−5.00000	−	5.00000i	−0.470360	−	0.470360i	−0.902031	+	0.431671i	$0.857924\pi$
$114$		0			0
$115$	−16.0000			−1.49201
$116$		0			0
$117$			2.00000i			0.184900i
$118$		0			0
$119$	−6.00000	−	10.0000i	−0.550019	−	0.916698i
$120$		0			0
$121$		−	9.00000i		−	0.818182i
$122$		0			0
$123$	−2.00000			−0.180334
$124$		0			0
$125$	4.00000	+	4.00000i	0.357771	+	0.357771i
$126$		0			0
$127$			16.0000i			1.41977i	0.704317	+	0.709885i	$0.251253\pi$
$127$			16.0000i			1.41977i	−0.704317	+	0.709885i	$0.748747\pi$
$128$		0			0
$129$	2.00000	+	2.00000i	0.176090	+	0.176090i
$130$		0			0
$131$	1.00000	−	1.00000i	0.0873704	−	0.0873704i	−0.662071	−	0.749441i	$-0.730322\pi$
$131$	1.00000	−	1.00000i	0.0873704	−	0.0873704i	0.749441	+	0.662071i	$0.230322\pi$
$132$		0			0
$133$	8.00000	−	8.00000i	0.693688	−	0.693688i
$134$		0			0
$135$	−16.0000			−1.37706
$136$		0			0
$137$	−14.0000			−1.19610			−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000			−1.19610			−0.598050	+	0.801459i	$0.704058\pi$
$138$		0			0
$139$	1.00000	−	1.00000i	0.0848189	−	0.0848189i	−0.663424	−	0.748243i	$-0.730898\pi$
$139$	1.00000	−	1.00000i	0.0848189	−	0.0848189i	0.748243	+	0.663424i	$0.230898\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.00000	+	2.00000i	0.167248	+	0.167248i
$144$		0			0
$145$		−	24.0000i		−	1.99309i
$146$		0			0
$147$	−1.00000	−	1.00000i	−0.0824786	−	0.0824786i
$148$		0			0
$149$	10.0000			0.819232			0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000			0.819232			0.409616	+	0.912258i	$0.365663\pi$
$150$		0			0
$151$			12.0000i			0.976546i	0.872691	+	0.488273i	$0.162373\pi$
$151$			12.0000i			0.976546i	−0.872691	+	0.488273i	$0.837627\pi$
$152$		0			0
$153$	1.00000	−	4.00000i	0.0808452	−	0.323381i
$154$		0			0
$155$			24.0000i			1.92773i
$156$		0			0
$157$	10.0000			0.798087			0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000			0.798087			0.399043	+	0.916932i	$0.369342\pi$
$158$		0			0
$159$	−6.00000	−	6.00000i	−0.475831	−	0.475831i
$160$		0			0
$161$		−	16.0000i		−	1.26098i
$162$		0			0
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	0.260531	−	0.965465i	$-0.416102\pi$
$163$	−9.00000	−	9.00000i	−0.704934	−	0.704934i	−0.965465	+	0.260531i	$0.916102\pi$
$164$		0			0
$165$	−4.00000	+	4.00000i	−0.311400	+	0.311400i
$166$		0			0
$167$	2.00000	−	2.00000i	0.154765	−	0.154765i	−0.625478	−	0.780242i	$-0.715096\pi$
$167$	2.00000	−	2.00000i	0.154765	−	0.154765i	0.780242	+	0.625478i	$0.215096\pi$
$168$		0			0
$169$	−9.00000			−0.692308
$170$		0			0
$171$	4.00000			0.305888
$172$		0			0
$173$	−2.00000	+	2.00000i	−0.152057	+	0.152057i	−0.779036	−	0.626979i	$-0.784291\pi$
$173$	−2.00000	+	2.00000i	−0.152057	+	0.152057i	0.626979	+	0.779036i	$0.284291\pi$
$174$		0			0
$175$	6.00000	−	6.00000i	0.453557	−	0.453557i
$176$		0			0
$177$	14.0000	+	14.0000i	1.05230	+	1.05230i
$178$		0			0
$179$			12.0000i			0.896922i	0.893802	+	0.448461i	$0.148028\pi$
$179$			12.0000i			0.896922i	−0.893802	+	0.448461i	$0.851972\pi$
$180$		0			0
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	0.839962	−	0.542645i	$-0.182577\pi$
$181$	4.00000	+	4.00000i	0.297318	+	0.297318i	−0.542645	+	0.839962i	$0.682577\pi$
$182$		0			0
$183$	8.00000			0.591377
$184$		0			0
$185$			32.0000i			2.35269i
$186$		0			0
$187$	−3.00000	−	5.00000i	−0.219382	−	0.365636i
$188$		0			0
$189$		−	16.0000i		−	1.16383i
$190$		0			0
$191$	4.00000			0.289430			0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000			0.289430			0.144715	+	0.989473i	$0.453773\pi$
$192$		0			0
$193$	9.00000	+	9.00000i	0.647834	+	0.647834i	0.952469	−	0.304635i	$-0.0985345\pi$
$193$	9.00000	+	9.00000i	0.647834	+	0.647834i	−0.304635	+	0.952469i	$0.598534\pi$
$194$		0			0
$195$			8.00000i			0.572892i
$196$		0			0
$197$	4.00000	+	4.00000i	0.284988	+	0.284988i	0.835095	−	0.550106i	$-0.185413\pi$
$197$	4.00000	+	4.00000i	0.284988	+	0.284988i	−0.550106	+	0.835095i	$0.685413\pi$
$198$		0			0
$199$	12.0000	−	12.0000i	0.850657	−	0.850657i	−0.139557	−	0.990214i	$-0.544568\pi$
$199$	12.0000	−	12.0000i	0.850657	−	0.850657i	0.990214	+	0.139557i	$0.0445677\pi$
$200$		0			0
$201$	−6.00000	+	6.00000i	−0.423207	+	0.423207i
$202$		0			0
$203$	24.0000			1.68447
$204$		0			0
$205$	4.00000			0.279372
$206$		0			0
$207$	4.00000	−	4.00000i	0.278019	−	0.278019i
$208$		0			0
$209$	4.00000	−	4.00000i	0.276686	−	0.276686i
$210$		0			0
$211$	15.0000	+	15.0000i	1.03264	+	1.03264i	0.999449	+	0.0331936i	$0.0105678\pi$
$211$	15.0000	+	15.0000i	1.03264	+	1.03264i	0.0331936	+	0.999449i	$0.489432\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000	−	4.00000i	−0.272798	−	0.272798i
$216$		0			0
$217$	−24.0000			−1.62923
$218$		0			0
$219$		−	18.0000i		−	1.21633i
$220$		0			0
$221$	−8.00000	−	2.00000i	−0.538138	−	0.134535i
$222$		0			0
$223$		−	4.00000i		−	0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$		−	4.00000i		−	0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$		0			0
$225$	3.00000			0.200000
$226$		0			0
$227$	5.00000	+	5.00000i	0.331862	+	0.331862i	0.853293	−	0.521431i	$-0.174602\pi$
$227$	5.00000	+	5.00000i	0.331862	+	0.331862i	−0.521431	+	0.853293i	$0.674602\pi$
$228$		0			0
$229$			6.00000i			0.396491i	0.980152	+	0.198246i	$0.0635244\pi$
$229$			6.00000i			0.396491i	−0.980152	+	0.198246i	$0.936476\pi$
$230$		0			0
$231$	−4.00000	−	4.00000i	−0.263181	−	0.263181i
$232$		0			0
$233$	1.00000	−	1.00000i	0.0655122	−	0.0655122i	−0.673592	−	0.739104i	$-0.735249\pi$
$233$	1.00000	−	1.00000i	0.0655122	−	0.0655122i	0.739104	+	0.673592i	$0.235249\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−8.00000			−0.519656
$238$		0			0
$239$	−12.0000			−0.776215			−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000			−0.776215			−0.388108	+	0.921614i	$0.626871\pi$
$240$		0			0
$241$	−1.00000	+	1.00000i	−0.0644157	+	0.0644157i	−0.738581	−	0.674165i	$-0.764504\pi$
$241$	−1.00000	+	1.00000i	−0.0644157	+	0.0644157i	0.674165	+	0.738581i	$0.264504\pi$
$242$		0			0
$243$	7.00000	−	7.00000i	0.449050	−	0.449050i
$244$		0			0
$245$	2.00000	+	2.00000i	0.127775	+	0.127775i
$246$		0			0
$247$		−	8.00000i		−	0.509028i
$248$		0			0
$249$	6.00000	+	6.00000i	0.380235	+	0.380235i
$250$		0			0
$251$	−10.0000			−0.631194			−0.315597	−	0.948893i	$-0.602205\pi$
$251$	−10.0000			−0.631194			−0.315597	+	0.948893i	$0.602205\pi$
$252$		0			0
$253$		−	8.00000i		−	0.502956i
$254$		0			0
$255$	4.00000	−	16.0000i	0.250490	−	1.00196i
$256$		0			0
$257$		−	20.0000i		−	1.24757i	−0.781598	−	0.623783i	$-0.785595\pi$
$257$		−	20.0000i		−	1.24757i	0.781598	−	0.623783i	$-0.214405\pi$
$258$		0			0
$259$	−32.0000			−1.98838
$260$		0			0
$261$	6.00000	+	6.00000i	0.371391	+	0.371391i
$262$		0			0
$263$			8.00000i			0.493301i	0.969104	+	0.246651i	$0.0793300\pi$
$263$			8.00000i			0.493301i	−0.969104	+	0.246651i	$0.920670\pi$
$264$		0			0
$265$	12.0000	+	12.0000i	0.737154	+	0.737154i
$266$		0			0
$267$	−16.0000	+	16.0000i	−0.979184	+	0.979184i
$268$		0			0
$269$	−2.00000	+	2.00000i	−0.121942	+	0.121942i	−0.765444	−	0.643502i	$-0.777481\pi$
$269$	−2.00000	+	2.00000i	−0.121942	+	0.121942i	0.643502	+	0.765444i	$0.277481\pi$
$270$		0			0
$271$	28.0000			1.70088			0.850439	−	0.526073i	$-0.176336\pi$
$271$	28.0000			1.70088			0.850439	+	0.526073i	$0.176336\pi$
$272$		0			0
$273$	−8.00000			−0.484182
$274$		0			0
$275$	3.00000	−	3.00000i	0.180907	−	0.180907i
$276$		0			0
$277$	−6.00000	+	6.00000i	−0.360505	+	0.360505i	−0.863999	−	0.503494i	$-0.832048\pi$
$277$	−6.00000	+	6.00000i	−0.360505	+	0.360505i	0.503494	+	0.863999i	$0.332048\pi$
$278$		0			0
$279$	−6.00000	−	6.00000i	−0.359211	−	0.359211i
$280$		0			0
$281$			22.0000i			1.31241i	0.754583	+	0.656205i	$0.227839\pi$
$281$			22.0000i			1.31241i	−0.754583	+	0.656205i	$0.772161\pi$
$282$		0			0
$283$	5.00000	+	5.00000i	0.297219	+	0.297219i	0.839924	−	0.542705i	$-0.182600\pi$
$283$	5.00000	+	5.00000i	0.297219	+	0.297219i	−0.542705	+	0.839924i	$0.682600\pi$
$284$		0			0
$285$	16.0000			0.947758
$286$		0			0
$287$			4.00000i			0.236113i
$288$		0			0
$289$	15.0000	+	8.00000i	0.882353	+	0.470588i
$290$		0			0
$291$			6.00000i			0.351726i
$292$		0			0
$293$	−22.0000			−1.28525			−0.642627	−	0.766179i	$-0.722155\pi$
$293$	−22.0000			−1.28525			−0.642627	+	0.766179i	$0.722155\pi$
$294$		0			0
$295$	−28.0000	−	28.0000i	−1.63022	−	1.63022i
$296$		0			0
$297$		−	8.00000i		−	0.464207i
$298$		0			0
$299$	−8.00000	−	8.00000i	−0.462652	−	0.462652i
$300$		0			0
$301$	4.00000	−	4.00000i	0.230556	−	0.230556i
$302$		0			0
$303$	6.00000	−	6.00000i	0.344691	−	0.344691i
$304$		0			0
$305$	−16.0000			−0.916157
$306$		0			0
$307$	−12.0000			−0.684876			−0.342438	−	0.939540i	$-0.611253\pi$
$307$	−12.0000			−0.684876			−0.342438	+	0.939540i	$0.611253\pi$
$308$		0			0
$309$	−20.0000	+	20.0000i	−1.13776	+	1.13776i
$310$		0			0
$311$	−6.00000	+	6.00000i	−0.340229	+	0.340229i	−0.856453	−	0.516225i	$-0.827337\pi$
$311$	−6.00000	+	6.00000i	−0.340229	+	0.340229i	0.516225	+	0.856453i	$0.327337\pi$
$312$		0			0
$313$	−21.0000	−	21.0000i	−1.18699	−	1.18699i	−0.977895	−	0.209095i	$-0.932948\pi$
$313$	−21.0000	−	21.0000i	−1.18699	−	1.18699i	−0.209095	−	0.977895i	$-0.567052\pi$
$314$		0			0
$315$			8.00000i			0.450749i
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$	12.0000			0.671871
$320$		0			0
$321$		−	26.0000i		−	1.45118i
$322$		0			0
$323$	−4.00000	+	16.0000i	−0.222566	+	0.890264i
$324$		0			0
$325$		−	6.00000i		−	0.332820i
$326$		0			0
$327$	12.0000			0.663602
$328$		0			0
$329$		0			0
$330$		0			0
$331$		−	22.0000i		−	1.20923i	−0.796518	−	0.604615i	$-0.793327\pi$
$331$		−	22.0000i		−	1.20923i	0.796518	−	0.604615i	$-0.206673\pi$
$332$		0			0
$333$	−8.00000	−	8.00000i	−0.438397	−	0.438397i
$334$		0			0
$335$	12.0000	−	12.0000i	0.655630	−	0.655630i
$336$		0			0
$337$	−5.00000	+	5.00000i	−0.272367	+	0.272367i	−0.830053	−	0.557685i	$-0.811690\pi$
$337$	−5.00000	+	5.00000i	−0.272367	+	0.272367i	0.557685	+	0.830053i	$0.311690\pi$
$338$		0			0
$339$	10.0000			0.543125
$340$		0			0
$341$	−12.0000			−0.649836
$342$		0			0
$343$	12.0000	−	12.0000i	0.647939	−	0.647939i
$344$		0			0
$345$	16.0000	−	16.0000i	0.861411	−	0.861411i
$346$		0			0
$347$	−3.00000	−	3.00000i	−0.161048	−	0.161048i	0.621983	−	0.783031i	$-0.286327\pi$
$347$	−3.00000	−	3.00000i	−0.161048	−	0.161048i	−0.783031	+	0.621983i	$0.786327\pi$
$348$		0			0
$349$		−	2.00000i		−	0.107058i	−0.998566	−	0.0535288i	$-0.982953\pi$
$349$		−	2.00000i		−	0.107058i	0.998566	−	0.0535288i	$-0.0170469\pi$
$350$		0			0
$351$	−8.00000	−	8.00000i	−0.427008	−	0.427008i
$352$		0			0
$353$	4.00000			0.212899			0.106449	−	0.994318i	$-0.466052\pi$
$353$	4.00000			0.212899			0.106449	+	0.994318i	$0.466052\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	16.0000	+	4.00000i	0.846810	+	0.211702i
$358$		0			0
$359$			16.0000i			0.844448i	0.906492	+	0.422224i	$0.138750\pi$
$359$			16.0000i			0.844448i	−0.906492	+	0.422224i	$0.861250\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$	9.00000	+	9.00000i	0.472377	+	0.472377i
$364$		0			0
$365$			36.0000i			1.88433i
$366$		0			0
$367$	16.0000	+	16.0000i	0.835193	+	0.835193i	0.988222	−	0.153029i	$-0.0489027\pi$
$367$	16.0000	+	16.0000i	0.835193	+	0.835193i	−0.153029	+	0.988222i	$0.548903\pi$
$368$		0			0
$369$	−1.00000	+	1.00000i	−0.0520579	+	0.0520579i
$370$		0			0
$371$	−12.0000	+	12.0000i	−0.623009	+	0.623009i
$372$		0			0
$373$	6.00000			0.310668			0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000			0.310668			0.155334	+	0.987862i	$0.450355\pi$
$374$		0			0
$375$	−8.00000			−0.413118
$376$		0			0
$377$	12.0000	−	12.0000i	0.618031	−	0.618031i
$378$		0			0
$379$	−5.00000	+	5.00000i	−0.256833	+	0.256833i	−0.823765	−	0.566932i	$-0.808130\pi$
$379$	−5.00000	+	5.00000i	−0.256833	+	0.256833i	0.566932	+	0.823765i	$0.308130\pi$
$380$		0			0
$381$	−16.0000	−	16.0000i	−0.819705	−	0.819705i
$382$		0			0
$383$			36.0000i			1.83951i	0.392488	+	0.919757i	$0.371614\pi$
$383$			36.0000i			1.83951i	−0.392488	+	0.919757i	$0.628386\pi$
$384$		0			0
$385$	8.00000	+	8.00000i	0.407718	+	0.407718i
$386$		0			0
$387$	2.00000			0.101666
$388$		0			0
$389$		−	14.0000i		−	0.709828i	−0.934899	−	0.354914i	$-0.884510\pi$
$389$		−	14.0000i		−	0.709828i	0.934899	−	0.354914i	$-0.115490\pi$
$390$		0			0
$391$	12.0000	+	20.0000i	0.606866	+	1.01144i
$392$		0			0
$393$			2.00000i			0.100887i
$394$		0			0
$395$	16.0000			0.805047
$396$		0			0
$397$	12.0000	+	12.0000i	0.602263	+	0.602263i	0.940913	−	0.338650i	$-0.109970\pi$
$397$	12.0000	+	12.0000i	0.602263	+	0.602263i	−0.338650	+	0.940913i	$0.609970\pi$
$398$		0			0
$399$			16.0000i			0.801002i
$400$		0			0
$401$	19.0000	+	19.0000i	0.948815	+	0.948815i	0.998752	−	0.0499376i	$-0.0159023\pi$
$401$	19.0000	+	19.0000i	0.948815	+	0.948815i	−0.0499376	+	0.998752i	$0.515902\pi$
$402$		0			0
$403$	−12.0000	+	12.0000i	−0.597763	+	0.597763i
$404$		0			0
$405$	10.0000	−	10.0000i	0.496904	−	0.496904i
$406$		0			0
$407$	−16.0000			−0.793091
$408$		0			0
$409$	32.0000			1.58230			0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000			1.58230			0.791149	+	0.611623i	$0.209483\pi$
$410$		0			0
$411$	14.0000	−	14.0000i	0.690569	−	0.690569i
$412$		0			0
$413$	28.0000	−	28.0000i	1.37779	−	1.37779i
$414$		0			0
$415$	−12.0000	−	12.0000i	−0.589057	−	0.589057i
$416$		0			0
$417$			2.00000i			0.0979404i
$418$		0			0
$419$	−15.0000	−	15.0000i	−0.732798	−	0.732798i	0.238375	−	0.971173i	$-0.423385\pi$
$419$	−15.0000	−	15.0000i	−0.732798	−	0.732798i	−0.971173	+	0.238375i	$0.923385\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3.00000	+	12.0000i	−0.145521	+	0.582086i
$426$		0			0
$427$		−	16.0000i		−	0.774294i
$428$		0			0
$429$	−4.00000			−0.193122
$430$		0			0
$431$	−12.0000	−	12.0000i	−0.578020	−	0.578020i	0.356338	−	0.934357i	$-0.384025\pi$
$431$	−12.0000	−	12.0000i	−0.578020	−	0.578020i	−0.934357	+	0.356338i	$0.884025\pi$
$432$		0			0
$433$			2.00000i			0.0961139i	0.998845	+	0.0480569i	$0.0153029\pi$
$433$			2.00000i			0.0961139i	−0.998845	+	0.0480569i	$0.984697\pi$
$434$		0			0
$435$	24.0000	+	24.0000i	1.15071	+	1.15071i
$436$		0			0
$437$	−16.0000	+	16.0000i	−0.765384	+	0.765384i
$438$		0			0
$439$	−12.0000	+	12.0000i	−0.572729	+	0.572729i	−0.932890	−	0.360161i	$-0.882722\pi$
$439$	−12.0000	+	12.0000i	−0.572729	+	0.572729i	0.360161	+	0.932890i	$0.382722\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	18.0000			0.855206			0.427603	−	0.903967i	$-0.359358\pi$
$443$	18.0000			0.855206			0.427603	+	0.903967i	$0.359358\pi$
$444$		0			0
$445$	32.0000	−	32.0000i	1.51695	−	1.51695i
$446$		0			0
$447$	−10.0000	+	10.0000i	−0.472984	+	0.472984i
$448$		0			0
$449$	−13.0000	−	13.0000i	−0.613508	−	0.613508i	0.330350	−	0.943858i	$-0.392833\pi$
$449$	−13.0000	−	13.0000i	−0.613508	−	0.613508i	−0.943858	+	0.330350i	$0.892833\pi$
$450$		0			0
$451$			2.00000i			0.0941763i
$452$		0			0
$453$	−12.0000	−	12.0000i	−0.563809	−	0.563809i
$454$		0			0
$455$	16.0000			0.750092
$456$		0			0
$457$		−	10.0000i		−	0.467780i	−0.972263	−	0.233890i	$-0.924854\pi$
$457$		−	10.0000i		−	0.467780i	0.972263	−	0.233890i	$-0.0751456\pi$
$458$		0			0
$459$	12.0000	+	20.0000i	0.560112	+	0.933520i
$460$		0			0
$461$			2.00000i			0.0931493i	0.998915	+	0.0465746i	$0.0148305\pi$
$461$			2.00000i			0.0931493i	−0.998915	+	0.0465746i	$0.985169\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$		0			0
$465$	−24.0000	−	24.0000i	−1.11297	−	1.11297i
$466$		0			0
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		0			0
$469$	12.0000	+	12.0000i	0.554109	+	0.554109i
$470$		0			0
$471$	−10.0000	+	10.0000i	−0.460776	+	0.460776i
$472$		0			0
$473$	2.00000	−	2.00000i	0.0919601	−	0.0919601i
$474$		0			0
$475$	−12.0000			−0.550598
$476$		0			0
$477$	−6.00000			−0.274721
$478$		0			0
$479$	−4.00000	+	4.00000i	−0.182765	+	0.182765i	−0.792559	−	0.609795i	$-0.791252\pi$
$479$	−4.00000	+	4.00000i	−0.182765	+	0.182765i	0.609795	+	0.792559i	$0.291252\pi$
$480$		0			0
$481$	−16.0000	+	16.0000i	−0.729537	+	0.729537i
$482$		0			0
$483$	16.0000	+	16.0000i	0.728025	+	0.728025i
$484$		0			0
$485$		−	12.0000i		−	0.544892i
$486$		0			0
$487$	8.00000	+	8.00000i	0.362515	+	0.362515i	0.864738	−	0.502223i	$-0.167484\pi$
$487$	8.00000	+	8.00000i	0.362515	+	0.362515i	−0.502223	+	0.864738i	$0.667484\pi$
$488$		0			0
$489$	18.0000			0.813988
$490$		0			0
$491$		−	12.0000i		−	0.541552i	−0.962642	−	0.270776i	$-0.912720\pi$
$491$		−	12.0000i		−	0.541552i	0.962642	−	0.270776i	$-0.0872803\pi$
$492$		0			0
$493$	−30.0000	+	18.0000i	−1.35113	+	0.810679i
$494$		0			0
$495$			4.00000i			0.179787i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−27.0000	−	27.0000i	−1.20869	−	1.20869i	−0.971453	−	0.237233i	$-0.923759\pi$
$499$	−27.0000	−	27.0000i	−1.20869	−	1.20869i	−0.237233	−	0.971453i	$-0.576241\pi$
$500$		0			0
$501$			4.00000i			0.178707i
$502$		0			0
$503$	−4.00000	−	4.00000i	−0.178351	−	0.178351i	0.612286	−	0.790637i	$-0.290250\pi$
$503$	−4.00000	−	4.00000i	−0.178351	−	0.178351i	−0.790637	+	0.612286i	$0.790250\pi$
$504$		0			0
$505$	−12.0000	+	12.0000i	−0.533993	+	0.533993i
$506$		0			0
$507$	9.00000	−	9.00000i	0.399704	−	0.399704i
$508$		0			0
$509$	6.00000			0.265945			0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000			0.265945			0.132973	+	0.991120i	$0.457548\pi$
$510$		0			0
$511$	−36.0000			−1.59255
$512$		0			0
$513$	−16.0000	+	16.0000i	−0.706417	+	0.706417i
$514$		0			0
$515$	40.0000	−	40.0000i	1.76261	−	1.76261i
$516$		0			0
$517$		0			0
$518$		0			0
$519$		−	4.00000i		−	0.175581i
$520$		0			0
$521$	−13.0000	−	13.0000i	−0.569540	−	0.569540i	0.362459	−	0.932000i	$-0.381937\pi$
$521$	−13.0000	−	13.0000i	−0.569540	−	0.569540i	−0.932000	+	0.362459i	$0.881937\pi$
$522$		0			0
$523$	−30.0000			−1.31181			−0.655904	−	0.754844i	$-0.727712\pi$
$523$	−30.0000			−1.31181			−0.655904	+	0.754844i	$0.727712\pi$
$524$		0			0
$525$			12.0000i			0.523723i
$526$		0			0
$527$	30.0000	−	18.0000i	1.30682	−	0.784092i
$528$		0			0
$529$			9.00000i			0.391304i
$530$		0			0
$531$	14.0000			0.607548
$532$		0			0
$533$	2.00000	+	2.00000i	0.0866296	+	0.0866296i
$534$		0			0
$535$			52.0000i			2.24816i
$536$		0			0
$537$	−12.0000	−	12.0000i	−0.517838	−	0.517838i
$538$		0			0
$539$	−1.00000	+	1.00000i	−0.0430730	+	0.0430730i
$540$		0			0
$541$	16.0000	−	16.0000i	0.687894	−	0.687894i	−0.273872	−	0.961766i	$-0.588305\pi$
$541$	16.0000	−	16.0000i	0.687894	−	0.687894i	0.961766	+	0.273872i	$0.0883046\pi$
$542$		0			0
$543$	−8.00000			−0.343313
$544$		0			0
$545$	−24.0000			−1.02805
$546$		0			0
$547$	25.0000	−	25.0000i	1.06892	−	1.06892i	0.0714808	−	0.997442i	$-0.477228\pi$
$547$	25.0000	−	25.0000i	1.06892	−	1.06892i	0.997442	−	0.0714808i	$-0.0227725\pi$
$548$		0			0
$549$	4.00000	−	4.00000i	0.170716	−	0.170716i
$550$		0			0
$551$	−24.0000	−	24.0000i	−1.02243	−	1.02243i
$552$		0			0
$553$			16.0000i			0.680389i
$554$		0			0
$555$	−32.0000	−	32.0000i	−1.35832	−	1.35832i
$556$		0			0
$557$	−22.0000			−0.932170			−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000			−0.932170			−0.466085	+	0.884740i	$0.654336\pi$
$558$		0			0
$559$		−	4.00000i		−	0.169182i
$560$		0			0
$561$	8.00000	+	2.00000i	0.337760	+	0.0844401i
$562$		0			0
$563$		−	18.0000i		−	0.758610i	−0.925272	−	0.379305i	$-0.876163\pi$
$563$		−	18.0000i		−	0.758610i	0.925272	−	0.379305i	$-0.123837\pi$
$564$		0			0
$565$	−20.0000			−0.841406
$566$		0			0
$567$	10.0000	+	10.0000i	0.419961	+	0.419961i
$568$		0			0
$569$			36.0000i			1.50920i	0.656186	+	0.754599i	$0.272169\pi$
$569$			36.0000i			1.50920i	−0.656186	+	0.754599i	$0.727831\pi$
$570$		0			0
$571$	33.0000	+	33.0000i	1.38101	+	1.38101i	0.842835	+	0.538172i	$0.180885\pi$
$571$	33.0000	+	33.0000i	1.38101	+	1.38101i	0.538172	+	0.842835i	$0.319115\pi$
$572$		0			0
$573$	−4.00000	+	4.00000i	−0.167102	+	0.167102i
$574$		0			0
$575$	−12.0000	+	12.0000i	−0.500435	+	0.500435i
$576$		0			0
$577$	14.0000			0.582828			0.291414	−	0.956597i	$-0.405874\pi$
$577$	14.0000			0.582828			0.291414	+	0.956597i	$0.405874\pi$
$578$		0			0
$579$	−18.0000			−0.748054
$580$		0			0
$581$	12.0000	−	12.0000i	0.497844	−	0.497844i
$582$		0			0
$583$	−6.00000	+	6.00000i	−0.248495	+	0.248495i
$584$		0			0
$585$	4.00000	+	4.00000i	0.165380	+	0.165380i
$586$		0			0
$587$		−	12.0000i		−	0.495293i	−0.968850	−	0.247647i	$-0.920343\pi$
$587$		−	12.0000i		−	0.495293i	0.968850	−	0.247647i	$-0.0796572\pi$
$588$		0			0
$589$	24.0000	+	24.0000i	0.988903	+	0.988903i
$590$		0			0
$591$	−8.00000			−0.329076
$592$		0			0
$593$		−	24.0000i		−	0.985562i	−0.870153	−	0.492781i	$-0.835980\pi$
$593$		−	24.0000i		−	0.985562i	0.870153	−	0.492781i	$-0.164020\pi$
$594$		0			0
$595$	−32.0000	−	8.00000i	−1.31187	−	0.327968i
$596$		0			0
$597$			24.0000i			0.982255i
$598$		0			0
$599$	32.0000			1.30748			0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000			1.30748			0.653742	+	0.756717i	$0.273198\pi$
$600$		0			0
$601$	−9.00000	−	9.00000i	−0.367118	−	0.367118i	0.499307	−	0.866425i	$-0.333588\pi$
$601$	−9.00000	−	9.00000i	−0.367118	−	0.367118i	−0.866425	+	0.499307i	$0.833588\pi$
$602$		0			0
$603$			6.00000i			0.244339i
$604$		0			0
$605$	−18.0000	−	18.0000i	−0.731804	−	0.731804i
$606$		0			0
$607$	26.0000	−	26.0000i	1.05531	−	1.05531i	0.0569292	−	0.998378i	$-0.481869\pi$
$607$	26.0000	−	26.0000i	1.05531	−	1.05531i	0.998378	−	0.0569292i	$-0.0181309\pi$
$608$		0			0
$609$	−24.0000	+	24.0000i	−0.972529	+	0.972529i
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−22.0000			−0.888572			−0.444286	−	0.895885i	$-0.646543\pi$
$613$	−22.0000			−0.888572			−0.444286	+	0.895885i	$0.646543\pi$
$614$		0			0
$615$	−4.00000	+	4.00000i	−0.161296	+	0.161296i
$616$		0			0
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	−0.337462	−	0.941339i	$-0.609568\pi$
$617$	15.0000	−	15.0000i	0.603877	−	0.603877i	0.941339	+	0.337462i	$0.109568\pi$
$618$		0			0
$619$	−5.00000	−	5.00000i	−0.200967	−	0.200967i	0.599447	−	0.800414i	$-0.295387\pi$
$619$	−5.00000	−	5.00000i	−0.200967	−	0.200967i	−0.800414	+	0.599447i	$0.795387\pi$
$620$		0			0
$621$			32.0000i			1.28412i
$622$		0			0
$623$	32.0000	+	32.0000i	1.28205	+	1.28205i
$624$		0			0
$625$	31.0000			1.24000
$626$		0			0
$627$			8.00000i			0.319489i
$628$		0			0
$629$	40.0000	−	24.0000i	1.59490	−	0.956943i
$630$		0			0
$631$		−	32.0000i		−	1.27390i	−0.770905	−	0.636950i	$-0.780196\pi$
$631$		−	32.0000i		−	1.27390i	0.770905	−	0.636950i	$-0.219804\pi$
$632$		0			0
$633$	−30.0000			−1.19239
$634$		0			0
$635$	32.0000	+	32.0000i	1.26988	+	1.26988i
$636$		0			0
$637$			2.00000i			0.0792429i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−11.0000	+	11.0000i	−0.434474	+	0.434474i	−0.890147	−	0.455673i	$-0.849399\pi$
$641$	−11.0000	+	11.0000i	−0.434474	+	0.434474i	0.455673	+	0.890147i	$0.349399\pi$
$642$		0			0
$643$	21.0000	−	21.0000i	0.828159	−	0.828159i	−0.159103	−	0.987262i	$-0.550860\pi$
$643$	21.0000	−	21.0000i	0.828159	−	0.828159i	0.987262	+	0.159103i	$0.0508601\pi$
$644$		0			0
$645$	8.00000			0.315000
$646$		0			0
$647$	−4.00000			−0.157256			−0.0786281	−	0.996904i	$-0.525054\pi$
$647$	−4.00000			−0.157256			−0.0786281	+	0.996904i	$0.525054\pi$
$648$		0			0
$649$	14.0000	−	14.0000i	0.549548	−	0.549548i
$650$		0			0
$651$	24.0000	−	24.0000i	0.940634	−	0.940634i
$652$		0			0
$653$	14.0000	+	14.0000i	0.547862	+	0.547862i	0.925822	−	0.377960i	$-0.123374\pi$
$653$	14.0000	+	14.0000i	0.547862	+	0.547862i	−0.377960	+	0.925822i	$0.623374\pi$
$654$		0			0
$655$		−	4.00000i		−	0.156293i
$656$		0			0
$657$	−9.00000	−	9.00000i	−0.351123	−	0.351123i
$658$		0			0
$659$	4.00000			0.155818			0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000			0.155818			0.0779089	+	0.996960i	$0.475176\pi$
$660$		0			0
$661$			26.0000i			1.01128i	0.862744	+	0.505641i	$0.168744\pi$
$661$			26.0000i			1.01128i	−0.862744	+	0.505641i	$0.831256\pi$
$662$		0			0
$663$	10.0000	−	6.00000i	0.388368	−	0.233021i
$664$		0			0
$665$		−	32.0000i		−	1.24091i
$666$		0			0
$667$	−48.0000			−1.85857
$668$		0			0
$669$	4.00000	+	4.00000i	0.154649	+	0.154649i
$670$		0			0
$671$		−	8.00000i		−	0.308837i
$672$		0			0
$673$	9.00000	+	9.00000i	0.346925	+	0.346925i	0.858963	−	0.512038i	$-0.171109\pi$
$673$	9.00000	+	9.00000i	0.346925	+	0.346925i	−0.512038	+	0.858963i	$0.671109\pi$
$674$		0			0
$675$	−12.0000	+	12.0000i	−0.461880	+	0.461880i
$676$		0			0
$677$	16.0000	−	16.0000i	0.614930	−	0.614930i	−0.329297	−	0.944227i	$-0.606812\pi$
$677$	16.0000	−	16.0000i	0.614930	−	0.614930i	0.944227	+	0.329297i	$0.106812\pi$
$678$		0			0
$679$	12.0000			0.460518
$680$		0			0
$681$	−10.0000			−0.383201
$682$		0			0
$683$	−11.0000	+	11.0000i	−0.420903	+	0.420903i	−0.885515	−	0.464611i	$-0.846194\pi$
$683$	−11.0000	+	11.0000i	−0.420903	+	0.420903i	0.464611	+	0.885515i	$0.346194\pi$
$684$		0			0
$685$	−28.0000	+	28.0000i	−1.06983	+	1.06983i
$686$		0			0
$687$	−6.00000	−	6.00000i	−0.228914	−	0.228914i
$688$		0			0
$689$			12.0000i			0.457164i
$690$		0			0
$691$	9.00000	+	9.00000i	0.342376	+	0.342376i	0.857260	−	0.514884i	$-0.172165\pi$
$691$	9.00000	+	9.00000i	0.342376	+	0.342376i	−0.514884	+	0.857260i	$0.672165\pi$
$692$		0			0
$693$	−4.00000			−0.151947
$694$		0			0
$695$		−	4.00000i		−	0.151729i
$696$		0			0
$697$	−3.00000	−	5.00000i	−0.113633	−	0.189389i
$698$		0			0
$699$			2.00000i			0.0756469i
$700$		0			0
$701$	18.0000			0.679851			0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000			0.679851			0.339925	+	0.940452i	$0.389598\pi$
$702$		0			0
$703$	32.0000	+	32.0000i	1.20690	+	1.20690i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−12.0000	−	12.0000i	−0.451306	−	0.451306i
$708$		0			0
$709$	18.0000	−	18.0000i	0.676004	−	0.676004i	−0.283089	−	0.959094i	$-0.591359\pi$
$709$	18.0000	−	18.0000i	0.676004	−	0.676004i	0.959094	+	0.283089i	$0.0913593\pi$
$710$		0			0
$711$	−4.00000	+	4.00000i	−0.150012	+	0.150012i
$712$		0			0
$713$	48.0000			1.79761
$714$		0			0
$715$	8.00000			0.299183
$716$		0			0
$717$	12.0000	−	12.0000i	0.448148	−	0.448148i
$718$		0			0
$719$	−28.0000	+	28.0000i	−1.04422	+	1.04422i	−0.0452480	+	0.998976i	$0.514408\pi$
$719$	−28.0000	+	28.0000i	−1.04422	+	1.04422i	−0.998976	+	0.0452480i	$0.985592\pi$
$720$		0			0
$721$	40.0000	+	40.0000i	1.48968	+	1.48968i
$722$		0			0
$723$		−	2.00000i		−	0.0743808i
$724$		0			0
$725$	−18.0000	−	18.0000i	−0.668503	−	0.668503i
$726$		0			0
$727$	8.00000			0.296704			0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000			0.296704			0.148352	+	0.988935i	$0.452603\pi$
$728$		0			0
$729$			29.0000i			1.07407i
$730$		0			0
$731$	−2.00000	+	8.00000i	−0.0739727	+	0.295891i
$732$		0			0
$733$			18.0000i			0.664845i	0.943131	+	0.332423i	$0.107866\pi$
$733$			18.0000i			0.664845i	−0.943131	+	0.332423i	$0.892134\pi$
$734$		0			0
$735$	−4.00000			−0.147542
$736$		0			0
$737$	6.00000	+	6.00000i	0.221013	+	0.221013i
$738$		0			0
$739$			4.00000i			0.147142i	0.997290	+	0.0735712i	$0.0234396\pi$
$739$			4.00000i			0.147142i	−0.997290	+	0.0735712i	$0.976560\pi$
$740$		0			0
$741$	8.00000	+	8.00000i	0.293887	+	0.293887i
$742$		0			0
$743$	−10.0000	+	10.0000i	−0.366864	+	0.366864i	−0.866332	−	0.499468i	$-0.833529\pi$
$743$	−10.0000	+	10.0000i	−0.366864	+	0.366864i	0.499468	+	0.866332i	$0.333529\pi$
$744$		0			0
$745$	20.0000	−	20.0000i	0.732743	−	0.732743i
$746$		0			0
$747$	6.00000			0.219529
$748$		0			0
$749$	−52.0000			−1.90004
$750$		0			0
$751$	−36.0000	+	36.0000i	−1.31366	+	1.31366i	−0.394961	+	0.918698i	$0.629242\pi$
$751$	−36.0000	+	36.0000i	−1.31366	+	1.31366i	−0.918698	+	0.394961i	$0.870758\pi$
$752$		0			0
$753$	10.0000	−	10.0000i	0.364420	−	0.364420i
$754$		0			0
$755$	24.0000	+	24.0000i	0.873449	+	0.873449i
$756$		0			0
$757$		−	46.0000i		−	1.67190i	−0.548807	−	0.835949i	$-0.684918\pi$
$757$		−	46.0000i		−	1.67190i	0.548807	−	0.835949i	$-0.315082\pi$
$758$		0			0
$759$	8.00000	+	8.00000i	0.290382	+	0.290382i
$760$		0			0
$761$	−42.0000			−1.52250			−0.761249	−	0.648459i	$-0.775414\pi$
$761$	−42.0000			−1.52250			−0.761249	+	0.648459i	$0.775414\pi$
$762$		0			0
$763$		−	24.0000i		−	0.868858i
$764$		0			0
$765$	−6.00000	−	10.0000i	−0.216930	−	0.361551i
$766$		0			0
$767$		−	28.0000i		−	1.01102i
$768$		0			0
$769$	−10.0000			−0.360609			−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000			−0.360609			−0.180305	+	0.983611i	$0.557708\pi$
$770$		0			0
$771$	20.0000	+	20.0000i	0.720282	+	0.720282i
$772$		0			0
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		0			0
$775$	18.0000	+	18.0000i	0.646579	+	0.646579i
$776$		0			0
$777$	32.0000	−	32.0000i	1.14799	−	1.14799i
$778$		0			0
$779$	4.00000	−	4.00000i	0.143315	−	0.143315i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	−48.0000			−1.71538
$784$		0			0
$785$	20.0000	−	20.0000i	0.713831	−	0.713831i
$786$		0			0
$787$	15.0000	−	15.0000i	0.534692	−	0.534692i	−0.387273	−	0.921965i	$-0.626583\pi$
$787$	15.0000	−	15.0000i	0.534692	−	0.534692i	0.921965	+	0.387273i	$0.126583\pi$
$788$		0			0
$789$	−8.00000	−	8.00000i	−0.284808	−	0.284808i
$790$		0			0
$791$		−	20.0000i		−	0.711118i
$792$		0			0
$793$	−8.00000	−	8.00000i	−0.284088	−	0.284088i
$794$		0			0
$795$	−24.0000			−0.851192
$796$		0			0
$797$		−	30.0000i		−	1.06265i	−0.847167	−	0.531327i	$-0.821693\pi$
$797$		−	30.0000i		−	1.06265i	0.847167	−	0.531327i	$-0.178307\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$			16.0000i			0.565332i
$802$		0			0
$803$	−18.0000			−0.635206
$804$		0			0
$805$	−32.0000	−	32.0000i	−1.12785	−	1.12785i
$806$		0			0
$807$		−	4.00000i		−	0.140807i
$808$		0			0
$809$	−13.0000	−	13.0000i	−0.457056	−	0.457056i	0.440632	−	0.897688i	$-0.354754\pi$
$809$	−13.0000	−	13.0000i	−0.457056	−	0.457056i	−0.897688	+	0.440632i	$0.854754\pi$
$810$		0			0
$811$	−27.0000	+	27.0000i	−0.948098	+	0.948098i	−0.998718	−	0.0506198i	$-0.983880\pi$
$811$	−27.0000	+	27.0000i	−0.948098	+	0.948098i	0.0506198	+	0.998718i	$0.483880\pi$
$812$		0			0
$813$	−28.0000	+	28.0000i	−0.982003	+	0.982003i
$814$		0			0
$815$	−36.0000			−1.26102
$816$		0			0
$817$	−8.00000			−0.279885
$818$		0			0
$819$	−4.00000	+	4.00000i	−0.139771	+	0.139771i
$820$		0			0
$821$	−4.00000	+	4.00000i	−0.139601	+	0.139601i	−0.773454	−	0.633853i	$-0.781473\pi$
$821$	−4.00000	+	4.00000i	−0.139601	+	0.139601i	0.633853	+	0.773454i	$0.281473\pi$
$822$		0			0
$823$	−20.0000	−	20.0000i	−0.697156	−	0.697156i	0.266640	−	0.963796i	$-0.414087\pi$
$823$	−20.0000	−	20.0000i	−0.697156	−	0.697156i	−0.963796	+	0.266640i	$0.914087\pi$
$824$		0			0
$825$			6.00000i			0.208893i
$826$		0			0
$827$	−5.00000	−	5.00000i	−0.173867	−	0.173867i	0.614809	−	0.788676i	$-0.289233\pi$
$827$	−5.00000	−	5.00000i	−0.173867	−	0.173867i	−0.788676	+	0.614809i	$0.789233\pi$
$828$		0			0
$829$	−42.0000			−1.45872			−0.729360	−	0.684130i	$-0.760182\pi$
$829$	−42.0000			−1.45872			−0.729360	+	0.684130i	$0.760182\pi$
$830$		0			0
$831$		−	12.0000i		−	0.416275i
$832$		0			0
$833$	1.00000	−	4.00000i	0.0346479	−	0.138592i
$834$		0			0
$835$		−	8.00000i		−	0.276851i
$836$		0			0
$837$	48.0000			1.65912
$838$		0			0
$839$	−2.00000	−	2.00000i	−0.0690477	−	0.0690477i	0.671740	−	0.740787i	$-0.265547\pi$
$839$	−2.00000	−	2.00000i	−0.0690477	−	0.0690477i	−0.740787	+	0.671740i	$0.765547\pi$
$840$		0			0
$841$		−	43.0000i		−	1.48276i
$842$		0			0
$843$	−22.0000	−	22.0000i	−0.757720	−	0.757720i
$844$		0			0
$845$	−18.0000	+	18.0000i	−0.619219	+	0.619219i
$846$		0			0
$847$	18.0000	−	18.0000i	0.618487	−	0.618487i
$848$		0			0
$849$	−10.0000			−0.343199
$850$		0			0
$851$	64.0000			2.19389
$852$		0			0
$853$	−10.0000	+	10.0000i	−0.342393	+	0.342393i	−0.857266	−	0.514873i	$-0.827839\pi$
$853$	−10.0000	+	10.0000i	−0.342393	+	0.342393i	0.514873	+	0.857266i	$0.327839\pi$
$854$		0			0
$855$	8.00000	−	8.00000i	0.273594	−	0.273594i
$856$		0			0
$857$	−27.0000	−	27.0000i	−0.922302	−	0.922302i	0.0748894	−	0.997192i	$-0.476140\pi$
$857$	−27.0000	−	27.0000i	−0.922302	−	0.922302i	−0.997192	+	0.0748894i	$0.976140\pi$
$858$		0			0
$859$			18.0000i			0.614152i	0.951685	+	0.307076i	$0.0993506\pi$
$859$			18.0000i			0.614152i	−0.951685	+	0.307076i	$0.900649\pi$
$860$		0			0
$861$	−4.00000	−	4.00000i	−0.136320	−	0.136320i
$862$		0			0
$863$	32.0000			1.08929			0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000			1.08929			0.544646	+	0.838666i	$0.316664\pi$
$864$		0			0
$865$			8.00000i			0.272008i
$866$		0			0
$867$	−23.0000	+	7.00000i	−0.781121	+	0.237732i
$868$		0			0
$869$			8.00000i			0.271381i
$870$		0			0
$871$	12.0000			0.406604
$872$		0			0
$873$	3.00000	+	3.00000i	0.101535	+	0.101535i
$874$		0			0
$875$			16.0000i			0.540899i
$876$		0			0
$877$	22.0000	+	22.0000i	0.742887	+	0.742887i	0.973133	−	0.230245i	$-0.0739529\pi$
$877$	22.0000	+	22.0000i	0.742887	+	0.742887i	−0.230245	+	0.973133i	$0.573953\pi$
$878$		0			0
$879$	22.0000	−	22.0000i	0.742042	−	0.742042i
$880$		0			0
$881$	−23.0000	+	23.0000i	−0.774890	+	0.774890i	−0.978957	−	0.204067i	$-0.934584\pi$
$881$	−23.0000	+	23.0000i	−0.774890	+	0.774890i	0.204067	+	0.978957i	$0.434584\pi$
$882$		0			0
$883$	−46.0000			−1.54802			−0.774012	−	0.633171i	$-0.781753\pi$
$883$	−46.0000			−1.54802			−0.774012	+	0.633171i	$0.781753\pi$
$884$		0			0
$885$	56.0000			1.88242
$886$		0			0
$887$	−16.0000	+	16.0000i	−0.537227	+	0.537227i	−0.922714	−	0.385486i	$-0.874034\pi$
$887$	−16.0000	+	16.0000i	−0.537227	+	0.537227i	0.385486	+	0.922714i	$0.374034\pi$
$888$		0			0
$889$	−32.0000	+	32.0000i	−1.07325	+	1.07325i
$890$		0			0
$891$	5.00000	+	5.00000i	0.167506	+	0.167506i
$892$		0			0
$893$		0			0
$894$		0			0
$895$	24.0000	+	24.0000i	0.802232	+	0.802232i
$896$		0			0
$897$	16.0000			0.534224
$898$		0			0
$899$			72.0000i			2.40133i
$900$		0			0
$901$	6.00000	−	24.0000i	0.199889	−	0.799556i
$902$		0			0
$903$			8.00000i			0.266223i
$904$		0			0
$905$	16.0000			0.531858
$906$		0			0
$907$	−7.00000	−	7.00000i	−0.232431	−	0.232431i	0.581276	−	0.813707i	$-0.302554\pi$
$907$	−7.00000	−	7.00000i	−0.232431	−	0.232431i	−0.813707	+	0.581276i	$0.802554\pi$
$908$		0			0
$909$		−	6.00000i		−	0.199007i
$910$		0			0
$911$	−34.0000	−	34.0000i	−1.12647	−	1.12647i	−0.990747	−	0.135724i	$-0.956664\pi$
$911$	−34.0000	−	34.0000i	−1.12647	−	1.12647i	−0.135724	−	0.990747i	$-0.543336\pi$
$912$		0			0
$913$	6.00000	−	6.00000i	0.198571	−	0.198571i
$914$		0			0
$915$	16.0000	−	16.0000i	0.528944	−	0.528944i
$916$		0			0
$917$	4.00000			0.132092
$918$		0			0
$919$	−16.0000			−0.527791			−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000			−0.527791			−0.263896	+	0.964551i	$0.585007\pi$
$920$		0			0
$921$	12.0000	−	12.0000i	0.395413	−	0.395413i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	24.0000	+	24.0000i	0.789115	+	0.789115i
$926$		0			0
$927$			20.0000i			0.656886i
$928$		0			0
$929$	13.0000	+	13.0000i	0.426516	+	0.426516i	0.887440	−	0.460924i	$-0.152482\pi$
$929$	13.0000	+	13.0000i	0.426516	+	0.426516i	−0.460924	+	0.887440i	$0.652482\pi$
$930$		0			0
$931$	4.00000			0.131095
$932$		0			0
$933$		−	12.0000i		−	0.392862i
$934$		0			0
$935$	−16.0000	−	4.00000i	−0.523256	−	0.130814i
$936$		0			0
$937$			38.0000i			1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$			38.0000i			1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$		0			0
$939$	42.0000			1.37062
$940$		0			0
$941$	−12.0000	−	12.0000i	−0.391189	−	0.391189i	0.483922	−	0.875111i	$-0.339212\pi$
$941$	−12.0000	−	12.0000i	−0.391189	−	0.391189i	−0.875111	+	0.483922i	$0.839212\pi$
$942$		0			0
$943$		−	8.00000i		−	0.260516i
$944$		0			0
$945$	−32.0000	−	32.0000i	−1.04096	−	1.04096i
$946$		0			0
$947$	−33.0000	+	33.0000i	−1.07236	+	1.07236i	−0.0751864	+	0.997169i	$0.523955\pi$
$947$	−33.0000	+	33.0000i	−1.07236	+	1.07236i	−0.997169	+	0.0751864i	$0.976045\pi$
$948$		0			0
$949$	−18.0000	+	18.0000i	−0.584305	+	0.584305i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	60.0000			1.94359			0.971795	−	0.235826i	$-0.0757795\pi$
$953$	60.0000			1.94359			0.971795	+	0.235826i	$0.0757795\pi$
$954$		0			0
$955$	8.00000	−	8.00000i	0.258874	−	0.258874i
$956$		0			0
$957$	−12.0000	+	12.0000i	−0.387905	+	0.387905i
$958$		0			0
$959$	−28.0000	−	28.0000i	−0.904167	−	0.904167i
$960$		0			0
$961$		−	41.0000i		−	1.32258i
$962$		0			0
$963$	−13.0000	−	13.0000i	−0.418919	−	0.418919i
$964$		0			0
$965$	36.0000			1.15888
$966$		0			0
$967$			8.00000i			0.257263i	0.991692	+	0.128631i	$0.0410584\pi$
$967$			8.00000i			0.257263i	−0.991692	+	0.128631i	$0.958942\pi$
$968$		0			0
$969$	−12.0000	−	20.0000i	−0.385496	−	0.642493i
$970$		0			0
$971$		−	6.00000i		−	0.192549i	−0.995355	−	0.0962746i	$-0.969307\pi$
$971$		−	6.00000i		−	0.192549i	0.995355	−	0.0962746i	$-0.0306927\pi$
$972$		0			0
$973$	4.00000			0.128234
$974$		0			0
$975$	6.00000	+	6.00000i	0.192154	+	0.192154i
$976$		0			0
$977$			28.0000i			0.895799i	0.894084	+	0.447900i	$0.147828\pi$
$977$			28.0000i			0.895799i	−0.894084	+	0.447900i	$0.852172\pi$
$978$		0			0
$979$	16.0000	+	16.0000i	0.511362	+	0.511362i
$980$		0			0
$981$	6.00000	−	6.00000i	0.191565	−	0.191565i
$982$		0			0
$983$	28.0000	−	28.0000i	0.893061	−	0.893061i	−0.101749	−	0.994810i	$-0.532444\pi$
$983$	28.0000	−	28.0000i	0.893061	−	0.893061i	0.994810	+	0.101749i	$0.0324438\pi$
$984$		0			0
$985$	16.0000			0.509802
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−8.00000	+	8.00000i	−0.254385	+	0.254385i
$990$		0			0
$991$	−24.0000	+	24.0000i	−0.762385	+	0.762385i	−0.976753	−	0.214368i	$-0.931231\pi$
$991$	−24.0000	+	24.0000i	−0.762385	+	0.762385i	0.214368	+	0.976753i	$0.431231\pi$
$992$		0			0
$993$	22.0000	+	22.0000i	0.698149	+	0.698149i
$994$		0			0
$995$		−	48.0000i		−	1.52170i
$996$		0			0
$997$	−28.0000	−	28.0000i	−0.886769	−	0.886769i	0.107442	−	0.994211i	$-0.465734\pi$
$997$	−28.0000	−	28.0000i	−0.886769	−	0.886769i	−0.994211	+	0.107442i	$0.965734\pi$
$998$		0			0
$999$	64.0000			2.02487

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	136.2.k.b.89.1	yes	2
3.2	odd	2		1224.2.w.a.361.1		2
4.3	odd	2		272.2.o.e.225.1		2
8.3	odd	2		1088.2.o.c.769.1		2
8.5	even	2		1088.2.o.m.769.1		2
12.11	even	2		2448.2.be.a.1585.1		2
17.2	even	8		2312.2.b.d.577.2		2
17.8	even	8		2312.2.a.j.1.2		2
17.9	even	8		2312.2.a.j.1.1		2
17.13	even	4	inner	136.2.k.b.81.1	✓	2
17.15	even	8		2312.2.b.d.577.1		2
51.47	odd	4		1224.2.w.a.217.1		2
68.43	odd	8		4624.2.a.o.1.2		2
68.47	odd	4		272.2.o.e.81.1		2
68.59	odd	8		4624.2.a.o.1.1		2
136.13	even	4		1088.2.o.m.897.1		2
136.115	odd	4		1088.2.o.c.897.1		2
204.47	even	4		2448.2.be.a.1441.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.k.b.81.1	✓	2	17.13	even	4	inner
136.2.k.b.89.1	yes	2	1.1	even	1	trivial
272.2.o.e.81.1		2	68.47	odd	4
272.2.o.e.225.1		2	4.3	odd	2
1088.2.o.c.769.1		2	8.3	odd	2
1088.2.o.c.897.1		2	136.115	odd	4
1088.2.o.m.769.1		2	8.5	even	2
1088.2.o.m.897.1		2	136.13	even	4
1224.2.w.a.217.1		2	51.47	odd	4
1224.2.w.a.361.1		2	3.2	odd	2
2312.2.a.j.1.1		2	17.9	even	8
2312.2.a.j.1.2		2	17.8	even	8
2312.2.b.d.577.1		2	17.15	even	8
2312.2.b.d.577.2		2	17.2	even	8
2448.2.be.a.1441.1		2	204.47	even	4
2448.2.be.a.1585.1		2	12.11	even	2
4624.2.a.o.1.1		2	68.59	odd	8
4624.2.a.o.1.2		2	68.43	odd	8

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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	136.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} +1.00000i q^{9} +O(q^{10})\)	\(q+(-1.00000 + 1.00000i) q^{3} +(2.00000 - 2.00000i) q^{5} +(2.00000 + 2.00000i) q^{7} +1.00000i q^{9} +(1.00000 + 1.00000i) q^{11} +2.00000 q^{13} +4.00000i q^{15} +(-4.00000 - 1.00000i) q^{17} -4.00000i q^{19} -4.00000 q^{21} +(-4.00000 - 4.00000i) q^{23} -3.00000i q^{25} +(-4.00000 - 4.00000i) q^{27} +(6.00000 - 6.00000i) q^{29} +(-6.00000 + 6.00000i) q^{31} -2.00000 q^{33} +8.00000 q^{35} +(-8.00000 + 8.00000i) q^{37} +(-2.00000 + 2.00000i) q^{39} +(1.00000 + 1.00000i) q^{41} -2.00000i q^{43} +(2.00000 + 2.00000i) q^{45} +1.00000i q^{49} +(5.00000 - 3.00000i) q^{51} +6.00000i q^{53} +4.00000 q^{55} +(4.00000 + 4.00000i) q^{57} -14.0000i q^{59} +(-4.00000 - 4.00000i) q^{61} +(-2.00000 + 2.00000i) q^{63} +(4.00000 - 4.00000i) q^{65} +6.00000 q^{67} +8.00000 q^{69} +(-9.00000 + 9.00000i) q^{73} +(3.00000 + 3.00000i) q^{75} +4.00000i q^{77} +(4.00000 + 4.00000i) q^{79} +5.00000 q^{81} -6.00000i q^{83} +(-10.0000 + 6.00000i) q^{85} +12.0000i q^{87} +16.0000 q^{89} +(4.00000 + 4.00000i) q^{91} -12.0000i q^{93} +(-8.00000 - 8.00000i) q^{95} +(3.00000 - 3.00000i) q^{97} +(-1.00000 + 1.00000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 4 q^{5} + 4 q^{7}+O(q^{10}) \) 2 * q - 2 * q^3 + 4 * q^5 + 4 * q^7	\( 2 q - 2 q^{3} + 4 q^{5} + 4 q^{7} + 2 q^{11} + 4 q^{13} - 8 q^{17} - 8 q^{21} - 8 q^{23} - 8 q^{27} + 12 q^{29} - 12 q^{31} - 4 q^{33} + 16 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} + 4 q^{45} + 10 q^{51} + 8 q^{55} + 8 q^{57} - 8 q^{61} - 4 q^{63} + 8 q^{65} + 12 q^{67} + 16 q^{69} - 18 q^{73} + 6 q^{75} + 8 q^{79} + 10 q^{81} - 20 q^{85} + 32 q^{89} + 8 q^{91} - 16 q^{95} + 6 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 4 * q^5 + 4 * q^7 + 2 * q^11 + 4 * q^13 - 8 * q^17 - 8 * q^21 - 8 * q^23 - 8 * q^27 + 12 * q^29 - 12 * q^31 - 4 * q^33 + 16 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 + 4 * q^45 + 10 * q^51 + 8 * q^55 + 8 * q^57 - 8 * q^61 - 4 * q^63 + 8 * q^65 + 12 * q^67 + 16 * q^69 - 18 * q^73 + 6 * q^75 + 8 * q^79 + 10 * q^81 - 20 * q^85 + 32 * q^89 + 8 * q^91 - 16 * q^95 + 6 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more