LMFDB - Embedded newform 850.2.h.c.251.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [850,2,Mod(251,850)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(850, base_ring=CyclotomicField(4))

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("850.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 850 = 2 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	850.h (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:	$6.78728417181$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 34)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			251.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	850.251
Dual form			850.2.h.c.701.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -3.00000i q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -3.00000i q^{9} +(-4.00000 + 4.00000i) q^{11} -4.00000 q^{13} +1.00000 q^{16} +(1.00000 - 4.00000i) q^{17} -3.00000 q^{18} +4.00000i q^{19} +(4.00000 + 4.00000i) q^{22} +(-4.00000 + 4.00000i) q^{23} +4.00000i q^{26} +(-3.00000 - 3.00000i) q^{29} +(-4.00000 - 4.00000i) q^{31} -1.00000i q^{32} +(-4.00000 - 1.00000i) q^{34} +3.00000i q^{36} +(-3.00000 - 3.00000i) q^{37} +4.00000 q^{38} +(1.00000 - 1.00000i) q^{41} +4.00000i q^{43} +(4.00000 - 4.00000i) q^{44} +(4.00000 + 4.00000i) q^{46} -8.00000 q^{47} +7.00000i q^{49} +4.00000 q^{52} +4.00000i q^{53} +(-3.00000 + 3.00000i) q^{58} +4.00000i q^{59} +(-9.00000 + 9.00000i) q^{61} +(-4.00000 + 4.00000i) q^{62} -1.00000 q^{64} +12.0000 q^{67} +(-1.00000 + 4.00000i) q^{68} +(-4.00000 - 4.00000i) q^{71} +3.00000 q^{72} +(-5.00000 - 5.00000i) q^{73} +(-3.00000 + 3.00000i) q^{74} -4.00000i q^{76} +(8.00000 - 8.00000i) q^{79} -9.00000 q^{81} +(-1.00000 - 1.00000i) q^{82} +4.00000i q^{83} +4.00000 q^{86} +(-4.00000 - 4.00000i) q^{88} +(4.00000 - 4.00000i) q^{92} +8.00000i q^{94} +(-3.00000 - 3.00000i) q^{97} +7.00000 q^{98} +(12.0000 + 12.0000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{4}+O(q^{10}) $ 2 * q - 2 * q^4	$ 2 q - 2 q^{4} - 8 q^{11} - 8 q^{13} + 2 q^{16} + 2 q^{17} - 6 q^{18} + 8 q^{22} - 8 q^{23} - 6 q^{29} - 8 q^{31} - 8 q^{34} - 6 q^{37} + 8 q^{38} + 2 q^{41} + 8 q^{44} + 8 q^{46} - 16 q^{47} + 8 q^{52} - 6 q^{58} - 18 q^{61} - 8 q^{62} - 2 q^{64} + 24 q^{67} - 2 q^{68} - 8 q^{71} + 6 q^{72} - 10 q^{73} - 6 q^{74} + 16 q^{79} - 18 q^{81} - 2 q^{82} + 8 q^{86} - 8 q^{88} + 8 q^{92} - 6 q^{97} + 14 q^{98} + 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^4 - 8 * q^11 - 8 * q^13 + 2 * q^16 + 2 * q^17 - 6 * q^18 + 8 * q^22 - 8 * q^23 - 6 * q^29 - 8 * q^31 - 8 * q^34 - 6 * q^37 + 8 * q^38 + 2 * q^41 + 8 * q^44 + 8 * q^46 - 16 * q^47 + 8 * q^52 - 6 * q^58 - 18 * q^61 - 8 * q^62 - 2 * q^64 + 24 * q^67 - 2 * q^68 - 8 * q^71 + 6 * q^72 - 10 * q^73 - 6 * q^74 + 16 * q^79 - 18 * q^81 - 2 * q^82 + 8 * q^86 - 8 * q^88 + 8 * q^92 - 6 * q^97 + 14 * q^98 + 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/850\mathbb{Z}\right)^\times$.

$n$	$477$	$751$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$

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 $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$3$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$4$	−1.00000			−0.500000
$5$		0			0
$5$		0			0
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$			1.00000i			0.353553i
$9$		−	3.00000i		−	1.00000i
$10$		0			0
$11$	−4.00000	+	4.00000i	−1.20605	+	1.20605i	−0.233748	+	0.972297i	$0.575099\pi$
$11$	−4.00000	+	4.00000i	−1.20605	+	1.20605i	−0.972297	+	0.233748i	$0.924901\pi$
$12$		0			0
$13$	−4.00000			−1.10940			−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000			−1.10940			−0.554700	+	0.832050i	$0.687167\pi$
$14$		0			0
$15$		0			0
$16$	1.00000			0.250000
$17$	1.00000	−	4.00000i	0.242536	−	0.970143i
$17$	1.00000	−	4.00000i	0.242536	−	0.970143i
$18$	−3.00000			−0.707107
$19$			4.00000i			0.917663i	0.888523	+	0.458831i	$0.151732\pi$
$19$			4.00000i			0.917663i	−0.888523	+	0.458831i	$0.848268\pi$
$20$		0			0
$21$		0			0
$22$	4.00000	+	4.00000i	0.852803	+	0.852803i
$23$	−4.00000	+	4.00000i	−0.834058	+	0.834058i	−0.988069	−	0.154011i	$-0.950781\pi$
$23$	−4.00000	+	4.00000i	−0.834058	+	0.834058i	0.154011	+	0.988069i	$0.450781\pi$
$24$		0			0
$25$		0			0
$26$			4.00000i			0.784465i
$27$		0			0
$28$		0			0
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	0.371391	−	0.928477i	$-0.378881\pi$
$29$	−3.00000	−	3.00000i	−0.557086	−	0.557086i	−0.928477	+	0.371391i	$0.878881\pi$
$30$		0			0
$31$	−4.00000	−	4.00000i	−0.718421	−	0.718421i	0.249861	−	0.968282i	$-0.419615\pi$
$31$	−4.00000	−	4.00000i	−0.718421	−	0.718421i	−0.968282	+	0.249861i	$0.919615\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$	−4.00000	−	1.00000i	−0.685994	−	0.171499i
$35$		0			0
$36$			3.00000i			0.500000i
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	0.416115	−	0.909312i	$-0.363391\pi$
$37$	−3.00000	−	3.00000i	−0.493197	−	0.493197i	−0.909312	+	0.416115i	$0.863391\pi$
$38$	4.00000			0.648886
$39$		0			0
$40$		0			0
$41$	1.00000	−	1.00000i	0.156174	−	0.156174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	1.00000	−	1.00000i	0.156174	−	0.156174i	0.780869	+	0.624695i	$0.214777\pi$
$42$		0			0
$43$			4.00000i			0.609994i	0.952353	+	0.304997i	$0.0986555\pi$
$43$			4.00000i			0.609994i	−0.952353	+	0.304997i	$0.901344\pi$
$44$	4.00000	−	4.00000i	0.603023	−	0.603023i
$45$		0			0
$46$	4.00000	+	4.00000i	0.589768	+	0.589768i
$47$	−8.00000			−1.16692			−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000			−1.16692			−0.583460	+	0.812142i	$0.698301\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$		0			0
$52$	4.00000			0.554700
$53$			4.00000i			0.549442i	0.961524	+	0.274721i	$0.0885855\pi$
$53$			4.00000i			0.549442i	−0.961524	+	0.274721i	$0.911414\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	−3.00000	+	3.00000i	−0.393919	+	0.393919i
$59$			4.00000i			0.520756i	0.965507	+	0.260378i	$0.0838471\pi$
$59$			4.00000i			0.520756i	−0.965507	+	0.260378i	$0.916153\pi$
$60$		0			0
$61$	−9.00000	+	9.00000i	−1.15233	+	1.15233i	−0.166248	+	0.986084i	$0.553165\pi$
$61$	−9.00000	+	9.00000i	−1.15233	+	1.15233i	−0.986084	+	0.166248i	$0.946835\pi$
$62$	−4.00000	+	4.00000i	−0.508001	+	0.508001i
$63$		0			0
$64$	−1.00000			−0.125000
$65$		0			0
$66$		0			0
$67$	12.0000			1.46603			0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000			1.46603			0.733017	+	0.680211i	$0.238112\pi$
$68$	−1.00000	+	4.00000i	−0.121268	+	0.485071i
$69$		0			0
$70$		0			0
$71$	−4.00000	−	4.00000i	−0.474713	−	0.474713i	0.428723	−	0.903436i	$-0.358964\pi$
$71$	−4.00000	−	4.00000i	−0.474713	−	0.474713i	−0.903436	+	0.428723i	$0.858964\pi$
$72$	3.00000			0.353553
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	0.351123	−	0.936329i	$-0.385800\pi$
$73$	−5.00000	−	5.00000i	−0.585206	−	0.585206i	−0.936329	+	0.351123i	$0.885800\pi$
$74$	−3.00000	+	3.00000i	−0.348743	+	0.348743i
$75$		0			0
$76$		−	4.00000i		−	0.458831i
$77$		0			0
$78$		0			0
$79$	8.00000	−	8.00000i	0.900070	−	0.900070i	−0.0953714	−	0.995442i	$-0.530404\pi$
$79$	8.00000	−	8.00000i	0.900070	−	0.900070i	0.995442	+	0.0953714i	$0.0304039\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$	−1.00000	−	1.00000i	−0.110432	−	0.110432i
$83$			4.00000i			0.439057i	0.975606	+	0.219529i	$0.0704519\pi$
$83$			4.00000i			0.439057i	−0.975606	+	0.219529i	$0.929548\pi$
$84$		0			0
$85$		0			0
$86$	4.00000			0.431331
$87$		0			0
$88$	−4.00000	−	4.00000i	−0.426401	−	0.426401i
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$		0			0
$92$	4.00000	−	4.00000i	0.417029	−	0.417029i
$93$		0			0
$94$			8.00000i			0.825137i
$95$		0			0
$96$		0			0
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	0.538208	−	0.842812i	$-0.319101\pi$
$97$	−3.00000	−	3.00000i	−0.304604	−	0.304604i	−0.842812	+	0.538208i	$0.819101\pi$
$98$	7.00000			0.707107
$99$	12.0000	+	12.0000i	1.20605	+	1.20605i
$100$		0			0
$101$	12.0000			1.19404			0.597022	−	0.802225i	$-0.296350\pi$
$101$	12.0000			1.19404			0.597022	+	0.802225i	$0.296350\pi$
$102$		0			0
$103$	16.0000			1.57653			0.788263	−	0.615338i	$-0.210980\pi$
$103$	16.0000			1.57653			0.788263	+	0.615338i	$0.210980\pi$
$104$		−	4.00000i		−	0.392232i
$105$		0			0
$106$	4.00000			0.388514
$107$	−8.00000	−	8.00000i	−0.773389	−	0.773389i	0.205308	−	0.978697i	$-0.434180\pi$
$107$	−8.00000	−	8.00000i	−0.773389	−	0.773389i	−0.978697	+	0.205308i	$0.934180\pi$
$108$		0			0
$109$	3.00000	−	3.00000i	0.287348	−	0.287348i	−0.548683	−	0.836031i	$-0.684871\pi$
$109$	3.00000	−	3.00000i	0.287348	−	0.287348i	0.836031	+	0.548683i	$0.184871\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.00000	−	1.00000i	0.0940721	−	0.0940721i	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	1.00000	−	1.00000i	0.0940721	−	0.0940721i	0.752577	+	0.658505i	$0.228811\pi$
$114$		0			0
$115$		0			0
$116$	3.00000	+	3.00000i	0.278543	+	0.278543i
$117$			12.0000i			1.10940i
$118$	4.00000			0.368230
$119$		0			0
$120$		0			0
$121$		−	21.0000i		−	1.90909i
$122$	9.00000	+	9.00000i	0.814822	+	0.814822i
$123$		0			0
$124$	4.00000	+	4.00000i	0.359211	+	0.359211i
$125$		0			0
$126$		0			0
$127$		−	8.00000i		−	0.709885i	−0.934888	−	0.354943i	$-0.884500\pi$
$127$		−	8.00000i		−	0.709885i	0.934888	−	0.354943i	$-0.115500\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$		0			0
$131$	−4.00000	−	4.00000i	−0.349482	−	0.349482i	0.510435	−	0.859916i	$-0.329484\pi$
$131$	−4.00000	−	4.00000i	−0.349482	−	0.349482i	−0.859916	+	0.510435i	$0.829484\pi$
$132$		0			0
$133$		0			0
$134$		−	12.0000i		−	1.03664i
$135$		0			0
$136$	4.00000	+	1.00000i	0.342997	+	0.0857493i
$137$	−8.00000			−0.683486			−0.341743	−	0.939793i	$-0.611017\pi$
$137$	−8.00000			−0.683486			−0.341743	+	0.939793i	$0.611017\pi$
$138$		0			0
$139$	−8.00000	−	8.00000i	−0.678551	−	0.678551i	0.281121	−	0.959672i	$-0.409294\pi$
$139$	−8.00000	−	8.00000i	−0.678551	−	0.678551i	−0.959672	+	0.281121i	$0.909294\pi$
$140$		0			0
$141$		0			0
$142$	−4.00000	+	4.00000i	−0.335673	+	0.335673i
$143$	16.0000	−	16.0000i	1.33799	−	1.33799i
$144$		−	3.00000i		−	0.250000i
$145$		0			0
$146$	−5.00000	+	5.00000i	−0.413803	+	0.413803i
$147$		0			0
$148$	3.00000	+	3.00000i	0.246598	+	0.246598i
$149$	−10.0000			−0.819232			−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000			−0.819232			−0.409616	+	0.912258i	$0.634337\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$	−4.00000			−0.324443
$153$	−12.0000	−	3.00000i	−0.970143	−	0.242536i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	2.00000			0.159617			0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000			0.159617			0.0798087	+	0.996810i	$0.474569\pi$
$158$	−8.00000	−	8.00000i	−0.636446	−	0.636446i
$159$		0			0
$160$		0			0
$161$		0			0
$162$			9.00000i			0.707107i
$163$	−4.00000	+	4.00000i	−0.313304	+	0.313304i	−0.846188	−	0.532884i	$-0.821108\pi$
$163$	−4.00000	+	4.00000i	−0.313304	+	0.313304i	0.532884	+	0.846188i	$0.321108\pi$
$164$	−1.00000	+	1.00000i	−0.0780869	+	0.0780869i
$165$		0			0
$166$	4.00000			0.310460
$167$	−8.00000	−	8.00000i	−0.619059	−	0.619059i	0.326231	−	0.945290i	$-0.394221\pi$
$167$	−8.00000	−	8.00000i	−0.619059	−	0.619059i	−0.945290	+	0.326231i	$0.894221\pi$
$168$		0			0
$169$	3.00000			0.230769
$170$		0			0
$171$	12.0000			0.917663
$172$		−	4.00000i		−	0.304997i
$173$	−5.00000	−	5.00000i	−0.380143	−	0.380143i	0.491011	−	0.871154i	$-0.336628\pi$
$173$	−5.00000	−	5.00000i	−0.380143	−	0.380143i	−0.871154	+	0.491011i	$0.836628\pi$
$174$		0			0
$175$		0			0
$176$	−4.00000	+	4.00000i	−0.301511	+	0.301511i
$177$		0			0
$178$		0			0
$179$			4.00000i			0.298974i	0.988764	+	0.149487i	$0.0477622\pi$
$179$			4.00000i			0.298974i	−0.988764	+	0.149487i	$0.952238\pi$
$180$		0			0
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	−0.168140	−	0.985763i	$-0.553776\pi$
$181$	11.0000	−	11.0000i	0.817624	−	0.817624i	0.985763	+	0.168140i	$0.0537759\pi$
$182$		0			0
$183$		0			0
$184$	−4.00000	−	4.00000i	−0.294884	−	0.294884i
$185$		0			0
$186$		0			0
$187$	12.0000	+	20.0000i	0.877527	+	1.46254i
$188$	8.00000			0.583460
$189$		0			0
$190$		0			0
$191$	−8.00000			−0.578860			−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000			−0.578860			−0.289430	+	0.957199i	$0.593466\pi$
$192$		0			0
$193$	11.0000	−	11.0000i	0.791797	−	0.791797i	−0.189989	−	0.981786i	$-0.560845\pi$
$193$	11.0000	−	11.0000i	0.791797	−	0.791797i	0.981786	+	0.189989i	$0.0608452\pi$
$194$	−3.00000	+	3.00000i	−0.215387	+	0.215387i
$195$		0			0
$196$		−	7.00000i		−	0.500000i
$197$	−5.00000	+	5.00000i	−0.356235	+	0.356235i	−0.862423	−	0.506188i	$-0.831054\pi$
$197$	−5.00000	+	5.00000i	−0.356235	+	0.356235i	0.506188	+	0.862423i	$0.331054\pi$
$198$	12.0000	−	12.0000i	0.852803	−	0.852803i
$199$	12.0000	+	12.0000i	0.850657	+	0.850657i	0.990214	−	0.139557i	$-0.0445677\pi$
$199$	12.0000	+	12.0000i	0.850657	+	0.850657i	−0.139557	+	0.990214i	$0.544568\pi$
$200$		0			0
$201$		0			0
$202$		−	12.0000i		−	0.844317i
$203$		0			0
$204$		0			0
$205$		0			0
$206$		−	16.0000i		−	1.11477i
$207$	12.0000	+	12.0000i	0.834058	+	0.834058i
$208$	−4.00000			−0.277350
$209$	−16.0000	−	16.0000i	−1.10674	−	1.10674i
$210$		0			0
$211$	16.0000	−	16.0000i	1.10149	−	1.10149i	0.107254	−	0.994232i	$-0.465794\pi$
$211$	16.0000	−	16.0000i	1.10149	−	1.10149i	0.994232	−	0.107254i	$-0.0342057\pi$
$212$		−	4.00000i		−	0.274721i
$213$		0			0
$214$	−8.00000	+	8.00000i	−0.546869	+	0.546869i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−3.00000	−	3.00000i	−0.203186	−	0.203186i
$219$		0			0
$220$		0			0
$221$	−4.00000	+	16.0000i	−0.269069	+	1.07628i
$222$		0			0
$223$		−	16.0000i		−	1.07144i	−0.844396	−	0.535720i	$-0.820040\pi$
$223$		−	16.0000i		−	1.07144i	0.844396	−	0.535720i	$-0.179960\pi$
$224$		0			0
$225$		0			0
$226$	−1.00000	−	1.00000i	−0.0665190	−	0.0665190i
$227$	−20.0000	+	20.0000i	−1.32745	+	1.32745i	−0.419856	+	0.907591i	$0.637919\pi$
$227$	−20.0000	+	20.0000i	−1.32745	+	1.32745i	−0.907591	+	0.419856i	$0.862081\pi$
$228$		0			0
$229$		−	6.00000i		−	0.396491i	−0.980152	−	0.198246i	$-0.936476\pi$
$229$		−	6.00000i		−	0.396491i	0.980152	−	0.198246i	$-0.0635244\pi$
$230$		0			0
$231$		0			0
$232$	3.00000	−	3.00000i	0.196960	−	0.196960i
$233$	5.00000	+	5.00000i	0.327561	+	0.327561i	0.851658	−	0.524097i	$-0.175597\pi$
$233$	5.00000	+	5.00000i	0.327561	+	0.327561i	−0.524097	+	0.851658i	$0.675597\pi$
$234$	12.0000			0.784465
$235$		0			0
$236$		−	4.00000i		−	0.260378i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	21.0000	+	21.0000i	1.35273	+	1.35273i	0.882595	+	0.470134i	$0.155794\pi$
$241$	21.0000	+	21.0000i	1.35273	+	1.35273i	0.470134	+	0.882595i	$0.344206\pi$
$242$	−21.0000			−1.34993
$243$		0			0
$244$	9.00000	−	9.00000i	0.576166	−	0.576166i
$245$		0			0
$246$		0			0
$247$		−	16.0000i		−	1.01806i
$248$	4.00000	−	4.00000i	0.254000	−	0.254000i
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$		−	32.0000i		−	2.01182i
$254$	−8.00000			−0.501965
$255$		0			0
$256$	1.00000			0.0625000
$257$		−	18.0000i		−	1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$		−	18.0000i		−	1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−9.00000	+	9.00000i	−0.557086	+	0.557086i
$262$	−4.00000	+	4.00000i	−0.247121	+	0.247121i
$263$			24.0000i			1.47990i	0.672660	+	0.739952i	$0.265152\pi$
$263$			24.0000i			1.47990i	−0.672660	+	0.739952i	$0.734848\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−12.0000			−0.733017
$269$	−13.0000	−	13.0000i	−0.792624	−	0.792624i	0.189296	−	0.981920i	$-0.439379\pi$
$269$	−13.0000	−	13.0000i	−0.792624	−	0.792624i	−0.981920	+	0.189296i	$0.939379\pi$
$270$		0			0
$271$	−8.00000			−0.485965			−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000			−0.485965			−0.242983	+	0.970031i	$0.578126\pi$
$272$	1.00000	−	4.00000i	0.0606339	−	0.242536i
$273$		0			0
$274$			8.00000i			0.483298i
$275$		0			0
$276$		0			0
$277$	7.00000	+	7.00000i	0.420589	+	0.420589i	0.885407	−	0.464817i	$-0.153880\pi$
$277$	7.00000	+	7.00000i	0.420589	+	0.420589i	−0.464817	+	0.885407i	$0.653880\pi$
$278$	−8.00000	+	8.00000i	−0.479808	+	0.479808i
$279$	−12.0000	+	12.0000i	−0.718421	+	0.718421i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−4.00000	+	4.00000i	−0.237775	+	0.237775i	−0.815928	−	0.578153i	$-0.803774\pi$
$283$	−4.00000	+	4.00000i	−0.237775	+	0.237775i	0.578153	+	0.815928i	$0.303774\pi$
$284$	4.00000	+	4.00000i	0.237356	+	0.237356i
$285$		0			0
$286$	−16.0000	−	16.0000i	−0.946100	−	0.946100i
$287$		0			0
$288$	−3.00000			−0.176777
$289$	−15.0000	−	8.00000i	−0.882353	−	0.470588i
$290$		0			0
$291$		0			0
$292$	5.00000	+	5.00000i	0.292603	+	0.292603i
$293$	6.00000			0.350524			0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000			0.350524			0.175262	+	0.984522i	$0.443923\pi$
$294$		0			0
$295$		0			0
$296$	3.00000	−	3.00000i	0.174371	−	0.174371i
$297$		0			0
$298$			10.0000i			0.579284i
$299$	16.0000	−	16.0000i	0.925304	−	0.925304i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$			4.00000i			0.229416i
$305$		0			0
$306$	−3.00000	+	12.0000i	−0.171499	+	0.685994i
$307$	12.0000			0.684876			0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000			0.684876			0.342438	+	0.939540i	$0.388747\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−24.0000	−	24.0000i	−1.36092	−	1.36092i	−0.872753	−	0.488162i	$-0.837667\pi$
$311$	−24.0000	−	24.0000i	−1.36092	−	1.36092i	−0.488162	−	0.872753i	$-0.662333\pi$
$312$		0			0
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	−0.914130	−	0.405420i	$-0.867125\pi$
$313$	−9.00000	+	9.00000i	−0.508710	+	0.508710i	0.405420	+	0.914130i	$0.367125\pi$
$314$		−	2.00000i		−	0.112867i
$315$		0			0
$316$	−8.00000	+	8.00000i	−0.450035	+	0.450035i
$317$	15.0000	−	15.0000i	0.842484	−	0.842484i	−0.146697	−	0.989181i	$-0.546864\pi$
$317$	15.0000	−	15.0000i	0.842484	−	0.842484i	0.989181	+	0.146697i	$0.0468644\pi$
$318$		0			0
$319$	24.0000			1.34374
$320$		0			0
$321$		0			0
$322$		0			0
$323$	16.0000	+	4.00000i	0.890264	+	0.222566i
$324$	9.00000			0.500000
$325$		0			0
$326$	4.00000	+	4.00000i	0.221540	+	0.221540i
$327$		0			0
$328$	1.00000	+	1.00000i	0.0552158	+	0.0552158i
$329$		0			0
$330$		0			0
$331$			20.0000i			1.09930i	0.835395	+	0.549650i	$0.185239\pi$
$331$			20.0000i			1.09930i	−0.835395	+	0.549650i	$0.814761\pi$
$332$		−	4.00000i		−	0.219529i
$333$	−9.00000	+	9.00000i	−0.493197	+	0.493197i
$334$	−8.00000	+	8.00000i	−0.437741	+	0.437741i
$335$		0			0
$336$		0			0
$337$	7.00000	+	7.00000i	0.381314	+	0.381314i	0.871576	−	0.490261i	$-0.163099\pi$
$337$	7.00000	+	7.00000i	0.381314	+	0.381314i	−0.490261	+	0.871576i	$0.663099\pi$
$338$		−	3.00000i		−	0.163178i
$339$		0			0
$340$		0			0
$341$	32.0000			1.73290
$342$		−	12.0000i		−	0.648886i
$343$		0			0
$344$	−4.00000			−0.215666
$345$		0			0
$346$	−5.00000	+	5.00000i	−0.268802	+	0.268802i
$347$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$347$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$348$		0			0
$349$			4.00000i			0.214115i	0.994253	+	0.107058i	$0.0341429\pi$
$349$			4.00000i			0.214115i	−0.994253	+	0.107058i	$0.965857\pi$
$350$		0			0
$351$		0			0
$352$	4.00000	+	4.00000i	0.213201	+	0.213201i
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$	4.00000			0.211407
$359$			24.0000i			1.26667i	0.773877	+	0.633336i	$0.218315\pi$
$359$			24.0000i			1.26667i	−0.773877	+	0.633336i	$0.781685\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$	−11.0000	−	11.0000i	−0.578147	−	0.578147i
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$	−4.00000	+	4.00000i	−0.208514	+	0.208514i
$369$	−3.00000	−	3.00000i	−0.156174	−	0.156174i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4.00000			−0.207112			−0.103556	−	0.994624i	$-0.533022\pi$
$373$	−4.00000			−0.207112			−0.103556	+	0.994624i	$0.533022\pi$
$374$	20.0000	−	12.0000i	1.03418	−	0.620505i
$375$		0			0
$376$		−	8.00000i		−	0.412568i
$377$	12.0000	+	12.0000i	0.618031	+	0.618031i
$378$		0			0
$379$	12.0000	+	12.0000i	0.616399	+	0.616399i	0.944606	−	0.328207i	$-0.106444\pi$
$379$	12.0000	+	12.0000i	0.616399	+	0.616399i	−0.328207	+	0.944606i	$0.606444\pi$
$380$		0			0
$381$		0			0
$382$			8.00000i			0.409316i
$383$			24.0000i			1.22634i	0.789950	+	0.613171i	$0.210106\pi$
$383$			24.0000i			1.22634i	−0.789950	+	0.613171i	$0.789894\pi$
$384$		0			0
$385$		0			0
$386$	−11.0000	−	11.0000i	−0.559885	−	0.559885i
$387$	12.0000			0.609994
$388$	3.00000	+	3.00000i	0.152302	+	0.152302i
$389$		−	6.00000i		−	0.304212i	−0.988364	−	0.152106i	$-0.951394\pi$
$389$		−	6.00000i		−	0.304212i	0.988364	−	0.152106i	$-0.0486055\pi$
$390$		0			0
$391$	12.0000	+	20.0000i	0.606866	+	1.01144i
$392$	−7.00000			−0.353553
$393$		0			0
$394$	5.00000	+	5.00000i	0.251896	+	0.251896i
$395$		0			0
$396$	−12.0000	−	12.0000i	−0.603023	−	0.603023i
$397$	−15.0000	+	15.0000i	−0.752828	+	0.752828i	−0.975006	−	0.222178i	$-0.928683\pi$
$397$	−15.0000	+	15.0000i	−0.752828	+	0.752828i	0.222178	+	0.975006i	$0.428683\pi$
$398$	12.0000	−	12.0000i	0.601506	−	0.601506i
$399$		0			0
$400$		0			0
$401$	−19.0000	+	19.0000i	−0.948815	+	0.948815i	−0.998752	−	0.0499376i	$-0.984098\pi$
$401$	−19.0000	+	19.0000i	−0.948815	+	0.948815i	0.0499376	+	0.998752i	$0.484098\pi$
$402$		0			0
$403$	16.0000	+	16.0000i	0.797017	+	0.797017i
$404$	−12.0000			−0.597022
$405$		0			0
$406$		0			0
$407$	24.0000			1.18964
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$		0			0
$412$	−16.0000			−0.788263
$413$		0			0
$414$	12.0000	−	12.0000i	0.589768	−	0.589768i
$415$		0			0
$416$			4.00000i			0.196116i
$417$		0			0
$418$	−16.0000	+	16.0000i	−0.782586	+	0.782586i
$419$	−12.0000	+	12.0000i	−0.586238	+	0.586238i	−0.936611	−	0.350372i	$-0.886055\pi$
$419$	−12.0000	+	12.0000i	−0.586238	+	0.586238i	0.350372	+	0.936611i	$0.386055\pi$
$420$		0			0
$421$	−28.0000			−1.36464			−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000			−1.36464			−0.682318	+	0.731055i	$0.739028\pi$
$422$	−16.0000	−	16.0000i	−0.778868	−	0.778868i
$423$			24.0000i			1.16692i
$424$	−4.00000			−0.194257
$425$		0			0
$426$		0			0
$427$		0			0
$428$	8.00000	+	8.00000i	0.386695	+	0.386695i
$429$		0			0
$430$		0			0
$431$	−24.0000	+	24.0000i	−1.15604	+	1.15604i	−0.170720	+	0.985320i	$0.554609\pi$
$431$	−24.0000	+	24.0000i	−1.15604	+	1.15604i	−0.985320	+	0.170720i	$0.945391\pi$
$432$		0			0
$433$			14.0000i			0.672797i	0.941720	+	0.336399i	$0.109209\pi$
$433$			14.0000i			0.672797i	−0.941720	+	0.336399i	$0.890791\pi$
$434$		0			0
$435$		0			0
$436$	−3.00000	+	3.00000i	−0.143674	+	0.143674i
$437$	−16.0000	−	16.0000i	−0.765384	−	0.765384i
$438$		0			0
$439$	12.0000	+	12.0000i	0.572729	+	0.572729i	0.932890	−	0.360161i	$-0.117278\pi$
$439$	12.0000	+	12.0000i	0.572729	+	0.572729i	−0.360161	+	0.932890i	$0.617278\pi$
$440$		0			0
$441$	21.0000			1.00000
$442$	16.0000	+	4.00000i	0.761042	+	0.190261i
$443$	−4.00000			−0.190046			−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000			−0.190046			−0.0950229	+	0.995475i	$0.530292\pi$
$444$		0			0
$445$		0			0
$446$	−16.0000			−0.757622
$447$		0			0
$448$		0			0
$449$	3.00000	−	3.00000i	0.141579	−	0.141579i	−0.632765	−	0.774344i	$-0.718080\pi$
$449$	3.00000	−	3.00000i	0.141579	−	0.141579i	0.774344	+	0.632765i	$0.218080\pi$
$450$		0			0
$451$			8.00000i			0.376705i
$452$	−1.00000	+	1.00000i	−0.0470360	+	0.0470360i
$453$		0			0
$454$	20.0000	+	20.0000i	0.938647	+	0.938647i
$455$		0			0
$456$		0			0
$457$			22.0000i			1.02912i	0.857455	+	0.514558i	$0.172044\pi$
$457$			22.0000i			1.02912i	−0.857455	+	0.514558i	$0.827956\pi$
$458$	−6.00000			−0.280362
$459$		0			0
$460$		0			0
$461$		−	20.0000i		−	0.931493i	−0.884918	−	0.465746i	$-0.845786\pi$
$461$		−	20.0000i		−	0.931493i	0.884918	−	0.465746i	$-0.154214\pi$
$462$		0			0
$463$	−24.0000			−1.11537			−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000			−1.11537			−0.557687	+	0.830051i	$0.688311\pi$
$464$	−3.00000	−	3.00000i	−0.139272	−	0.139272i
$465$		0			0
$466$	5.00000	−	5.00000i	0.231621	−	0.231621i
$467$		−	28.0000i		−	1.29569i	−0.761774	−	0.647843i	$-0.775671\pi$
$467$		−	28.0000i		−	1.29569i	0.761774	−	0.647843i	$-0.224329\pi$
$468$		−	12.0000i		−	0.554700i
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−4.00000			−0.184115
$473$	−16.0000	−	16.0000i	−0.735681	−	0.735681i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	12.0000			0.549442
$478$		0			0
$479$	−8.00000	−	8.00000i	−0.365529	−	0.365529i	0.500314	−	0.865844i	$-0.333218\pi$
$479$	−8.00000	−	8.00000i	−0.365529	−	0.365529i	−0.865844	+	0.500314i	$0.833218\pi$
$480$		0			0
$481$	12.0000	+	12.0000i	0.547153	+	0.547153i
$482$	21.0000	−	21.0000i	0.956524	−	0.956524i
$483$		0			0
$484$			21.0000i			0.954545i
$485$		0			0
$486$		0			0
$487$	−20.0000	+	20.0000i	−0.906287	+	0.906287i	−0.995970	−	0.0896838i	$-0.971414\pi$
$487$	−20.0000	+	20.0000i	−0.906287	+	0.906287i	0.0896838	+	0.995970i	$0.471414\pi$
$488$	−9.00000	−	9.00000i	−0.407411	−	0.407411i
$489$		0			0
$490$		0			0
$491$			20.0000i			0.902587i	0.892375	+	0.451294i	$0.149037\pi$
$491$			20.0000i			0.902587i	−0.892375	+	0.451294i	$0.850963\pi$
$492$		0			0
$493$	−15.0000	+	9.00000i	−0.675566	+	0.405340i
$494$	−16.0000			−0.719874
$495$		0			0
$496$	−4.00000	−	4.00000i	−0.179605	−	0.179605i
$497$		0			0
$498$		0			0
$499$	8.00000	−	8.00000i	0.358129	−	0.358129i	−0.504994	−	0.863123i	$-0.668505\pi$
$499$	8.00000	−	8.00000i	0.358129	−	0.358129i	0.863123	+	0.504994i	$0.168505\pi$
$500$		0			0
$501$		0			0
$502$		−	12.0000i		−	0.535586i
$503$	−4.00000	+	4.00000i	−0.178351	+	0.178351i	−0.790637	−	0.612286i	$-0.790250\pi$
$503$	−4.00000	+	4.00000i	−0.178351	+	0.178351i	0.612286	+	0.790637i	$0.290250\pi$
$504$		0			0
$505$		0			0
$506$	−32.0000			−1.42257
$507$		0			0
$508$			8.00000i			0.354943i
$509$	−30.0000			−1.32973			−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000			−1.32973			−0.664863	+	0.746965i	$0.731510\pi$
$510$		0			0
$511$		0			0
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$	−18.0000			−0.793946
$515$		0			0
$516$		0			0
$517$	32.0000	−	32.0000i	1.40736	−	1.40736i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.00000	−	1.00000i	0.0438108	−	0.0438108i	−0.684862	−	0.728673i	$-0.740138\pi$
$521$	1.00000	−	1.00000i	0.0438108	−	0.0438108i	0.728673	+	0.684862i	$0.240138\pi$
$522$	9.00000	+	9.00000i	0.393919	+	0.393919i
$523$	36.0000			1.57417			0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000			1.57417			0.787085	+	0.616844i	$0.211589\pi$
$524$	4.00000	+	4.00000i	0.174741	+	0.174741i
$525$		0			0
$526$	24.0000			1.04645
$527$	−20.0000	+	12.0000i	−0.871214	+	0.522728i
$528$		0			0
$529$		−	9.00000i		−	0.391304i
$530$		0			0
$531$	12.0000			0.520756
$532$		0			0
$533$	−4.00000	+	4.00000i	−0.173259	+	0.173259i
$534$		0			0
$535$		0			0
$536$			12.0000i			0.518321i
$537$		0			0
$538$	−13.0000	+	13.0000i	−0.560470	+	0.560470i
$539$	−28.0000	−	28.0000i	−1.20605	−	1.20605i
$540$		0			0
$541$	−9.00000	−	9.00000i	−0.386940	−	0.386940i	0.486654	−	0.873595i	$-0.338217\pi$
$541$	−9.00000	−	9.00000i	−0.386940	−	0.386940i	−0.873595	+	0.486654i	$0.838217\pi$
$542$			8.00000i			0.343629i
$543$		0			0
$544$	−4.00000	−	1.00000i	−0.171499	−	0.0428746i
$545$		0			0
$546$		0			0
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	0.915470	−	0.402387i	$-0.131819\pi$
$547$	12.0000	+	12.0000i	0.513083	+	0.513083i	−0.402387	+	0.915470i	$0.631819\pi$
$548$	8.00000			0.341743
$549$	27.0000	+	27.0000i	1.15233	+	1.15233i
$550$		0			0
$551$	12.0000	−	12.0000i	0.511217	−	0.511217i
$552$		0			0
$553$		0			0
$554$	7.00000	−	7.00000i	0.297402	−	0.297402i
$555$		0			0
$556$	8.00000	+	8.00000i	0.339276	+	0.339276i
$557$	−28.0000			−1.18640			−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000			−1.18640			−0.593199	+	0.805056i	$0.702135\pi$
$558$	12.0000	+	12.0000i	0.508001	+	0.508001i
$559$		−	16.0000i		−	0.676728i
$560$		0			0
$561$		0			0
$562$		0			0
$563$			4.00000i			0.168580i	0.996441	+	0.0842900i	$0.0268622\pi$
$563$			4.00000i			0.168580i	−0.996441	+	0.0842900i	$0.973138\pi$
$564$		0			0
$565$		0			0
$566$	4.00000	+	4.00000i	0.168133	+	0.168133i
$567$		0			0
$568$	4.00000	−	4.00000i	0.167836	−	0.167836i
$569$			24.0000i			1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$			24.0000i			1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$		0			0
$571$	−24.0000	+	24.0000i	−1.00437	+	1.00437i	−0.00437833	+	0.999990i	$0.501394\pi$
$571$	−24.0000	+	24.0000i	−1.00437	+	1.00437i	−0.999990	+	0.00437833i	$0.998606\pi$
$572$	−16.0000	+	16.0000i	−0.668994	+	0.668994i
$573$		0			0
$574$		0			0
$575$		0			0
$576$			3.00000i			0.125000i
$577$	32.0000			1.33218			0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000			1.33218			0.666089	+	0.745873i	$0.267967\pi$
$578$	−8.00000	+	15.0000i	−0.332756	+	0.623918i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−16.0000	−	16.0000i	−0.662652	−	0.662652i
$584$	5.00000	−	5.00000i	0.206901	−	0.206901i
$585$		0			0
$586$		−	6.00000i		−	0.247858i
$587$			12.0000i			0.495293i	0.968850	+	0.247647i	$0.0796572\pi$
$587$			12.0000i			0.495293i	−0.968850	+	0.247647i	$0.920343\pi$
$588$		0			0
$589$	16.0000	−	16.0000i	0.659269	−	0.659269i
$590$		0			0
$591$		0			0
$592$	−3.00000	−	3.00000i	−0.123299	−	0.123299i
$593$		−	16.0000i		−	0.657041i	−0.944497	−	0.328521i	$-0.893450\pi$
$593$		−	16.0000i		−	0.657041i	0.944497	−	0.328521i	$-0.106550\pi$
$594$		0			0
$595$		0			0
$596$	10.0000			0.409616
$597$		0			0
$598$	−16.0000	−	16.0000i	−0.654289	−	0.654289i
$599$	40.0000			1.63436			0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000			1.63436			0.817178	+	0.576386i	$0.195537\pi$
$600$		0			0
$601$	−19.0000	+	19.0000i	−0.775026	+	0.775026i	−0.978980	−	0.203954i	$-0.934621\pi$
$601$	−19.0000	+	19.0000i	−0.775026	+	0.775026i	0.203954	+	0.978980i	$0.434621\pi$
$602$		0			0
$603$		−	36.0000i		−	1.46603i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	−28.0000	−	28.0000i	−1.13648	−	1.13648i	−0.989076	−	0.147409i	$-0.952907\pi$
$607$	−28.0000	−	28.0000i	−1.13648	−	1.13648i	−0.147409	−	0.989076i	$-0.547093\pi$
$608$	4.00000			0.162221
$609$		0			0
$610$		0			0
$611$	32.0000			1.29458
$612$	12.0000	+	3.00000i	0.485071	+	0.121268i
$613$	26.0000			1.05013			0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000			1.05013			0.525065	+	0.851062i	$0.324041\pi$
$614$		−	12.0000i		−	0.484281i
$615$		0			0
$616$		0			0
$617$	−3.00000	−	3.00000i	−0.120775	−	0.120775i	0.644136	−	0.764911i	$-0.277217\pi$
$617$	−3.00000	−	3.00000i	−0.120775	−	0.120775i	−0.764911	+	0.644136i	$0.777217\pi$
$618$		0			0
$619$	8.00000	−	8.00000i	0.321547	−	0.321547i	−0.527813	−	0.849360i	$-0.676988\pi$
$619$	8.00000	−	8.00000i	0.321547	−	0.321547i	0.849360	+	0.527813i	$0.176988\pi$
$620$		0			0
$621$		0			0
$622$	−24.0000	+	24.0000i	−0.962312	+	0.962312i
$623$		0			0
$624$		0			0
$625$		0			0
$626$	9.00000	+	9.00000i	0.359712	+	0.359712i
$627$		0			0
$628$	−2.00000			−0.0798087
$629$	−15.0000	+	9.00000i	−0.598089	+	0.358854i
$630$		0			0
$631$		−	40.0000i		−	1.59237i	−0.605050	−	0.796187i	$-0.706847\pi$
$631$		−	40.0000i		−	1.59237i	0.605050	−	0.796187i	$-0.293153\pi$
$632$	8.00000	+	8.00000i	0.318223	+	0.318223i
$633$		0			0
$634$	−15.0000	−	15.0000i	−0.595726	−	0.595726i
$635$		0			0
$636$		0			0
$637$		−	28.0000i		−	1.10940i
$638$		−	24.0000i		−	0.950169i
$639$	−12.0000	+	12.0000i	−0.474713	+	0.474713i
$640$		0			0
$641$	11.0000	+	11.0000i	0.434474	+	0.434474i	0.890147	−	0.455673i	$-0.150601\pi$
$641$	11.0000	+	11.0000i	0.434474	+	0.434474i	−0.455673	+	0.890147i	$0.650601\pi$
$642$		0			0
$643$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$643$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$644$		0			0
$645$		0			0
$646$	4.00000	−	16.0000i	0.157378	−	0.629512i
$647$	32.0000			1.25805			0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000			1.25805			0.629025	+	0.777385i	$0.283454\pi$
$648$		−	9.00000i		−	0.353553i
$649$	−16.0000	−	16.0000i	−0.628055	−	0.628055i
$650$		0			0
$651$		0			0
$652$	4.00000	−	4.00000i	0.156652	−	0.156652i
$653$	−29.0000	+	29.0000i	−1.13486	+	1.13486i	−0.145499	+	0.989358i	$0.546479\pi$
$653$	−29.0000	+	29.0000i	−1.13486	+	1.13486i	−0.989358	+	0.145499i	$0.953521\pi$
$654$		0			0
$655$		0			0
$656$	1.00000	−	1.00000i	0.0390434	−	0.0390434i
$657$	−15.0000	+	15.0000i	−0.585206	+	0.585206i
$658$		0			0
$659$	−20.0000			−0.779089			−0.389545	−	0.921008i	$-0.627368\pi$
$659$	−20.0000			−0.779089			−0.389545	+	0.921008i	$0.627368\pi$
$660$		0			0
$661$		−	20.0000i		−	0.777910i	−0.921257	−	0.388955i	$-0.872836\pi$
$661$		−	20.0000i		−	0.777910i	0.921257	−	0.388955i	$-0.127164\pi$
$662$	20.0000			0.777322
$663$		0			0
$664$	−4.00000			−0.155230
$665$		0			0
$666$	9.00000	+	9.00000i	0.348743	+	0.348743i
$667$	24.0000			0.929284
$668$	8.00000	+	8.00000i	0.309529	+	0.309529i
$669$		0			0
$670$		0			0
$671$		−	72.0000i		−	2.77953i
$672$		0			0
$673$	−19.0000	+	19.0000i	−0.732396	+	0.732396i	−0.971094	−	0.238698i	$-0.923279\pi$
$673$	−19.0000	+	19.0000i	−0.732396	+	0.732396i	0.238698	+	0.971094i	$0.423279\pi$
$674$	7.00000	−	7.00000i	0.269630	−	0.269630i
$675$		0			0
$676$	−3.00000			−0.115385
$677$	−13.0000	−	13.0000i	−0.499631	−	0.499631i	0.411692	−	0.911323i	$-0.364938\pi$
$677$	−13.0000	−	13.0000i	−0.499631	−	0.499631i	−0.911323	+	0.411692i	$0.864938\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		−	32.0000i		−	1.22534i
$683$	−20.0000	−	20.0000i	−0.765279	−	0.765279i	0.211993	−	0.977271i	$-0.432005\pi$
$683$	−20.0000	−	20.0000i	−0.765279	−	0.765279i	−0.977271	+	0.211993i	$0.932005\pi$
$684$	−12.0000			−0.458831
$685$		0			0
$686$		0			0
$687$		0			0
$688$			4.00000i			0.152499i
$689$		−	16.0000i		−	0.609551i
$690$		0			0
$691$	−24.0000	+	24.0000i	−0.913003	+	0.913003i	−0.996507	−	0.0835044i	$-0.973389\pi$
$691$	−24.0000	+	24.0000i	−0.913003	+	0.913003i	0.0835044	+	0.996507i	$0.473389\pi$
$692$	5.00000	+	5.00000i	0.190071	+	0.190071i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−3.00000	−	5.00000i	−0.113633	−	0.189389i
$698$	4.00000			0.151402
$699$		0			0
$700$		0			0
$701$	12.0000			0.453234			0.226617	−	0.973984i	$-0.427233\pi$
$701$	12.0000			0.453234			0.226617	+	0.973984i	$0.427233\pi$
$702$		0			0
$703$	12.0000	−	12.0000i	0.452589	−	0.452589i
$704$	4.00000	−	4.00000i	0.150756	−	0.150756i
$705$		0			0
$706$			14.0000i			0.526897i
$707$		0			0
$708$		0			0
$709$	−23.0000	−	23.0000i	−0.863783	−	0.863783i	0.127992	−	0.991775i	$-0.459147\pi$
$709$	−23.0000	−	23.0000i	−0.863783	−	0.863783i	−0.991775	+	0.127992i	$0.959147\pi$
$710$		0			0
$711$	−24.0000	−	24.0000i	−0.900070	−	0.900070i
$712$		0			0
$713$	32.0000			1.19841
$714$		0			0
$715$		0			0
$716$		−	4.00000i		−	0.149487i
$717$		0			0
$718$	24.0000			0.895672
$719$	−8.00000	−	8.00000i	−0.298350	−	0.298350i	0.542018	−	0.840367i	$-0.317661\pi$
$719$	−8.00000	−	8.00000i	−0.298350	−	0.298350i	−0.840367	+	0.542018i	$0.817661\pi$
$720$		0			0
$721$		0			0
$722$		−	3.00000i		−	0.111648i
$723$		0			0
$724$	−11.0000	+	11.0000i	−0.408812	+	0.408812i
$725$		0			0
$726$		0			0
$727$	−48.0000			−1.78022			−0.890111	−	0.455744i	$-0.849373\pi$
$727$	−48.0000			−1.78022			−0.890111	+	0.455744i	$0.849373\pi$
$728$		0			0
$729$			27.0000i			1.00000i
$730$		0			0
$731$	16.0000	+	4.00000i	0.591781	+	0.147945i
$732$		0			0
$733$		−	36.0000i		−	1.32969i	−0.746981	−	0.664845i	$-0.768498\pi$
$733$		−	36.0000i		−	1.32969i	0.746981	−	0.664845i	$-0.231502\pi$
$734$		0			0
$735$		0			0
$736$	4.00000	+	4.00000i	0.147442	+	0.147442i
$737$	−48.0000	+	48.0000i	−1.76810	+	1.76810i
$738$	−3.00000	+	3.00000i	−0.110432	+	0.110432i
$739$			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
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$			4.00000i			0.146450i
$747$	12.0000			0.439057
$748$	−12.0000	−	20.0000i	−0.438763	−	0.731272i
$749$		0			0
$750$		0			0
$751$	16.0000	+	16.0000i	0.583848	+	0.583848i	0.935959	−	0.352110i	$-0.114536\pi$
$751$	16.0000	+	16.0000i	0.583848	+	0.583848i	−0.352110	+	0.935959i	$0.614536\pi$
$752$	−8.00000			−0.291730
$753$		0			0
$754$	12.0000	−	12.0000i	0.437014	−	0.437014i
$755$		0			0
$756$		0			0
$757$			22.0000i			0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$757$			22.0000i			0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$758$	12.0000	−	12.0000i	0.435860	−	0.435860i
$759$		0			0
$760$		0			0
$761$	−8.00000			−0.290000			−0.145000	−	0.989432i	$-0.546318\pi$
$761$	−8.00000			−0.290000			−0.145000	+	0.989432i	$0.546318\pi$
$762$		0			0
$763$		0			0
$764$	8.00000			0.289430
$765$		0			0
$766$	24.0000			0.867155
$767$		−	16.0000i		−	0.577727i
$768$		0			0
$769$	40.0000			1.44244			0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000			1.44244			0.721218	+	0.692708i	$0.243582\pi$
$770$		0			0
$771$		0			0
$772$	−11.0000	+	11.0000i	−0.395899	+	0.395899i
$773$		−	6.00000i		−	0.215805i	−0.994161	−	0.107903i	$-0.965587\pi$
$773$		−	6.00000i		−	0.215805i	0.994161	−	0.107903i	$-0.0344134\pi$
$774$		−	12.0000i		−	0.431331i
$775$		0			0
$776$	3.00000	−	3.00000i	0.107694	−	0.107694i
$777$		0			0
$778$	−6.00000			−0.215110
$779$	4.00000	+	4.00000i	0.143315	+	0.143315i
$780$		0			0
$781$	32.0000			1.14505
$782$	20.0000	−	12.0000i	0.715199	−	0.429119i
$783$		0			0
$784$			7.00000i			0.250000i
$785$		0			0
$786$		0			0
$787$	12.0000	+	12.0000i	0.427754	+	0.427754i	0.887863	−	0.460109i	$-0.152190\pi$
$787$	12.0000	+	12.0000i	0.427754	+	0.427754i	−0.460109	+	0.887863i	$0.652190\pi$
$788$	5.00000	−	5.00000i	0.178118	−	0.178118i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−12.0000	+	12.0000i	−0.426401	+	0.426401i
$793$	36.0000	−	36.0000i	1.27840	−	1.27840i
$794$	15.0000	+	15.0000i	0.532330	+	0.532330i
$795$		0			0
$796$	−12.0000	−	12.0000i	−0.425329	−	0.425329i
$797$			12.0000i			0.425062i	0.977154	+	0.212531i	$0.0681706\pi$
$797$			12.0000i			0.425062i	−0.977154	+	0.212531i	$0.931829\pi$
$798$		0			0
$799$	−8.00000	+	32.0000i	−0.283020	+	1.13208i
$800$		0			0
$801$		0			0
$802$	19.0000	+	19.0000i	0.670913	+	0.670913i
$803$	40.0000			1.41157
$804$		0			0
$805$		0			0
$806$	16.0000	−	16.0000i	0.563576	−	0.563576i
$807$		0			0
$808$			12.0000i			0.422159i
$809$	23.0000	−	23.0000i	0.808637	−	0.808637i	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	23.0000	−	23.0000i	0.808637	−	0.808637i	0.984428	+	0.175791i	$0.0562482\pi$
$810$		0			0
$811$	−4.00000	−	4.00000i	−0.140459	−	0.140459i	0.633381	−	0.773840i	$-0.281667\pi$
$811$	−4.00000	−	4.00000i	−0.140459	−	0.140459i	−0.773840	+	0.633381i	$0.781667\pi$
$812$		0			0
$813$		0			0
$814$		−	24.0000i		−	0.841200i
$815$		0			0
$816$		0			0
$817$	−16.0000			−0.559769
$818$			10.0000i			0.349642i
$819$		0			0
$820$		0			0
$821$	−19.0000	−	19.0000i	−0.663105	−	0.663105i	0.293006	−	0.956111i	$-0.405344\pi$
$821$	−19.0000	−	19.0000i	−0.663105	−	0.663105i	−0.956111	+	0.293006i	$0.905344\pi$
$822$		0			0
$823$	36.0000	−	36.0000i	1.25488	−	1.25488i	0.301376	−	0.953506i	$-0.402554\pi$
$823$	36.0000	−	36.0000i	1.25488	−	1.25488i	0.953506	−	0.301376i	$-0.0974458\pi$
$824$			16.0000i			0.557386i
$825$		0			0
$826$		0			0
$827$	20.0000	−	20.0000i	0.695468	−	0.695468i	−0.267961	−	0.963430i	$-0.586350\pi$
$827$	20.0000	−	20.0000i	0.695468	−	0.695468i	0.963430	+	0.267961i	$0.0863500\pi$
$828$	−12.0000	−	12.0000i	−0.417029	−	0.417029i
$829$	30.0000			1.04194			0.520972	−	0.853574i	$-0.325570\pi$
$829$	30.0000			1.04194			0.520972	+	0.853574i	$0.325570\pi$
$830$		0			0
$831$		0			0
$832$	4.00000			0.138675
$833$	28.0000	+	7.00000i	0.970143	+	0.242536i
$834$		0			0
$835$		0			0
$836$	16.0000	+	16.0000i	0.553372	+	0.553372i
$837$		0			0
$838$	12.0000	+	12.0000i	0.414533	+	0.414533i
$839$	−32.0000	+	32.0000i	−1.10476	+	1.10476i	−0.110935	+	0.993828i	$0.535385\pi$
$839$	−32.0000	+	32.0000i	−1.10476	+	1.10476i	−0.993828	+	0.110935i	$0.964615\pi$
$840$		0			0
$841$		−	11.0000i		−	0.379310i
$842$			28.0000i			0.964944i
$843$		0			0
$844$	−16.0000	+	16.0000i	−0.550743	+	0.550743i
$845$		0			0
$846$	24.0000			0.825137
$847$		0			0
$848$			4.00000i			0.137361i
$849$		0			0
$850$		0			0
$851$	24.0000			0.822709
$852$		0			0
$853$	25.0000	+	25.0000i	0.855984	+	0.855984i	0.990862	−	0.134878i	$-0.0430644\pi$
$853$	25.0000	+	25.0000i	0.855984	+	0.855984i	−0.134878	+	0.990862i	$0.543064\pi$
$854$		0			0
$855$		0			0
$856$	8.00000	−	8.00000i	0.273434	−	0.273434i
$857$	15.0000	−	15.0000i	0.512390	−	0.512390i	−0.402868	−	0.915258i	$-0.631987\pi$
$857$	15.0000	−	15.0000i	0.512390	−	0.512390i	0.915258	+	0.402868i	$0.131987\pi$
$858$		0			0
$859$		−	36.0000i		−	1.22830i	−0.789188	−	0.614152i	$-0.789498\pi$
$859$		−	36.0000i		−	1.22830i	0.789188	−	0.614152i	$-0.210502\pi$
$860$		0			0
$861$		0			0
$862$	24.0000	+	24.0000i	0.817443	+	0.817443i
$863$	−24.0000			−0.816970			−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000			−0.816970			−0.408485	+	0.912765i	$0.633943\pi$
$864$		0			0
$865$		0			0
$866$	14.0000			0.475739
$867$		0			0
$868$		0			0
$869$			64.0000i			2.17105i
$870$		0			0
$871$	−48.0000			−1.62642
$872$	3.00000	+	3.00000i	0.101593	+	0.101593i
$873$	−9.00000	+	9.00000i	−0.304604	+	0.304604i
$874$	−16.0000	+	16.0000i	−0.541208	+	0.541208i
$875$		0			0
$876$		0			0
$877$	−35.0000	+	35.0000i	−1.18187	+	1.18187i	−0.202606	+	0.979260i	$0.564941\pi$
$877$	−35.0000	+	35.0000i	−1.18187	+	1.18187i	−0.979260	+	0.202606i	$0.935059\pi$
$878$	12.0000	−	12.0000i	0.404980	−	0.404980i
$879$		0			0
$880$		0			0
$881$	−39.0000	−	39.0000i	−1.31394	−	1.31394i	−0.918483	−	0.395460i	$-0.870585\pi$
$881$	−39.0000	−	39.0000i	−1.31394	−	1.31394i	−0.395460	−	0.918483i	$-0.629415\pi$
$882$		−	21.0000i		−	0.707107i
$883$	−44.0000			−1.48072			−0.740359	−	0.672212i	$-0.765344\pi$
$883$	−44.0000			−1.48072			−0.740359	+	0.672212i	$0.765344\pi$
$884$	4.00000	−	16.0000i	0.134535	−	0.538138i
$885$		0			0
$886$			4.00000i			0.134383i
$887$	−28.0000	−	28.0000i	−0.940148	−	0.940148i	0.0581593	−	0.998307i	$-0.481477\pi$
$887$	−28.0000	−	28.0000i	−0.940148	−	0.940148i	−0.998307	+	0.0581593i	$0.981477\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	36.0000	−	36.0000i	1.20605	−	1.20605i
$892$			16.0000i			0.535720i
$893$		−	32.0000i		−	1.07084i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	−3.00000	−	3.00000i	−0.100111	−	0.100111i
$899$			24.0000i			0.800445i
$900$		0			0
$901$	16.0000	+	4.00000i	0.533037	+	0.133259i
$902$	8.00000			0.266371
$903$		0			0
$904$	1.00000	+	1.00000i	0.0332595	+	0.0332595i
$905$		0			0
$906$		0			0
$907$	−20.0000	+	20.0000i	−0.664089	+	0.664089i	−0.956341	−	0.292252i	$-0.905595\pi$
$907$	−20.0000	+	20.0000i	−0.664089	+	0.664089i	0.292252	+	0.956341i	$0.405595\pi$
$908$	20.0000	−	20.0000i	0.663723	−	0.663723i
$909$		−	36.0000i		−	1.19404i
$910$		0			0
$911$	−4.00000	+	4.00000i	−0.132526	+	0.132526i	−0.770258	−	0.637732i	$-0.779873\pi$
$911$	−4.00000	+	4.00000i	−0.132526	+	0.132526i	0.637732	+	0.770258i	$0.279873\pi$
$912$		0			0
$913$	−16.0000	−	16.0000i	−0.529523	−	0.529523i
$914$	22.0000			0.727695
$915$		0			0
$916$			6.00000i			0.198246i
$917$		0			0
$918$		0			0
$919$	40.0000			1.31948			0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000			1.31948			0.659739	+	0.751495i	$0.270667\pi$
$920$		0			0
$921$		0			0
$922$	−20.0000			−0.658665
$923$	16.0000	+	16.0000i	0.526646	+	0.526646i
$924$		0			0
$925$		0			0
$926$			24.0000i			0.788689i
$927$		−	48.0000i		−	1.57653i
$928$	−3.00000	+	3.00000i	−0.0984798	+	0.0984798i
$929$	13.0000	−	13.0000i	0.426516	−	0.426516i	−0.460924	−	0.887440i	$-0.652482\pi$
$929$	13.0000	−	13.0000i	0.426516	−	0.426516i	0.887440	+	0.460924i	$0.152482\pi$
$930$		0			0
$931$	−28.0000			−0.917663
$932$	−5.00000	−	5.00000i	−0.163780	−	0.163780i
$933$		0			0
$934$	−28.0000			−0.916188
$935$		0			0
$936$	−12.0000			−0.392232
$937$		−	48.0000i		−	1.56809i	−0.620703	−	0.784046i	$-0.713153\pi$
$937$		−	48.0000i		−	1.56809i	0.620703	−	0.784046i	$-0.286847\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.00000	−	1.00000i	0.0325991	−	0.0325991i	−0.690619	−	0.723218i	$-0.742662\pi$
$941$	1.00000	−	1.00000i	0.0325991	−	0.0325991i	0.723218	+	0.690619i	$0.242662\pi$
$942$		0			0
$943$			8.00000i			0.260516i
$944$			4.00000i			0.130189i
$945$		0			0
$946$	−16.0000	+	16.0000i	−0.520205	+	0.520205i
$947$	12.0000	+	12.0000i	0.389948	+	0.389948i	0.874669	−	0.484721i	$-0.161079\pi$
$947$	12.0000	+	12.0000i	0.389948	+	0.389948i	−0.484721	+	0.874669i	$0.661079\pi$
$948$		0			0
$949$	20.0000	+	20.0000i	0.649227	+	0.649227i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−24.0000			−0.777436			−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000			−0.777436			−0.388718	+	0.921357i	$0.627082\pi$
$954$		−	12.0000i		−	0.388514i
$955$		0			0
$956$		0			0
$957$		0			0
$958$	−8.00000	+	8.00000i	−0.258468	+	0.258468i
$959$		0			0
$960$		0			0
$961$			1.00000i			0.0322581i
$962$	12.0000	−	12.0000i	0.386896	−	0.386896i
$963$	−24.0000	+	24.0000i	−0.773389	+	0.773389i
$964$	−21.0000	−	21.0000i	−0.676364	−	0.676364i
$965$		0			0
$966$		0			0
$967$			32.0000i			1.02905i	0.857475	+	0.514525i	$0.172032\pi$
$967$			32.0000i			1.02905i	−0.857475	+	0.514525i	$0.827968\pi$
$968$	21.0000			0.674966
$969$		0			0
$970$		0			0
$971$		−	20.0000i		−	0.641831i	−0.947108	−	0.320915i	$-0.896010\pi$
$971$		−	20.0000i		−	0.641831i	0.947108	−	0.320915i	$-0.103990\pi$
$972$		0			0
$973$		0			0
$974$	20.0000	+	20.0000i	0.640841	+	0.640841i
$975$		0			0
$976$	−9.00000	+	9.00000i	−0.288083	+	0.288083i
$977$		−	8.00000i		−	0.255943i	−0.991778	−	0.127971i	$-0.959153\pi$
$977$		−	8.00000i		−	0.255943i	0.991778	−	0.127971i	$-0.0408466\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−9.00000	−	9.00000i	−0.287348	−	0.287348i
$982$	20.0000			0.638226
$983$	20.0000	+	20.0000i	0.637901	+	0.637901i	0.950037	−	0.312136i	$-0.101045\pi$
$983$	20.0000	+	20.0000i	0.637901	+	0.637901i	−0.312136	+	0.950037i	$0.601045\pi$
$984$		0			0
$985$		0			0
$986$	9.00000	+	15.0000i	0.286618	+	0.477697i
$987$		0			0
$988$			16.0000i			0.509028i
$989$	−16.0000	−	16.0000i	−0.508770	−	0.508770i
$990$		0			0
$991$	−24.0000	−	24.0000i	−0.762385	−	0.762385i	0.214368	−	0.976753i	$-0.431231\pi$
$991$	−24.0000	−	24.0000i	−0.762385	−	0.762385i	−0.976753	+	0.214368i	$0.931231\pi$
$992$	−4.00000	+	4.00000i	−0.127000	+	0.127000i
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$	35.0000	−	35.0000i	1.10846	−	1.10846i	0.115108	−	0.993353i	$-0.463279\pi$
$997$	35.0000	−	35.0000i	1.10846	−	1.10846i	0.993353	−	0.115108i	$-0.0367215\pi$
$998$	−8.00000	−	8.00000i	−0.253236	−	0.253236i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	850.2.h.c.251.1		2
5.2	odd	4		850.2.g.d.149.1		2
5.3	odd	4		850.2.g.a.149.1		2
5.4	even	2		34.2.c.b.13.1	✓	2
15.14	odd	2		306.2.g.d.217.1		2
17.4	even	4	inner	850.2.h.c.701.1		2
20.19	odd	2		272.2.o.c.81.1		2
40.19	odd	2		1088.2.o.i.897.1		2
40.29	even	2		1088.2.o.k.897.1		2
60.59	even	2		2448.2.be.j.1441.1		2
85.4	even	4		34.2.c.b.21.1	yes	2
85.9	even	8		578.2.b.c.577.2		2
85.14	odd	16		578.2.d.f.423.1		8
85.19	even	8		578.2.a.b.1.2		2
85.24	odd	16		578.2.d.f.399.2		8
85.29	odd	16		578.2.d.f.155.1		8
85.38	odd	4		850.2.g.d.599.1		2
85.39	odd	16		578.2.d.f.155.2		8
85.44	odd	16		578.2.d.f.399.1		8
85.49	even	8		578.2.a.b.1.1		2
85.54	odd	16		578.2.d.f.423.2		8
85.59	even	8		578.2.b.c.577.1		2
85.64	even	4		578.2.c.b.327.1		2
85.72	odd	4		850.2.g.a.599.1		2
85.74	odd	16		578.2.d.f.179.1		8
85.79	odd	16		578.2.d.f.179.2		8
85.84	even	2		578.2.c.b.251.1		2
255.89	odd	4		306.2.g.d.55.1		2
255.104	odd	8		5202.2.a.bb.1.1		2
255.134	odd	8		5202.2.a.bb.1.2		2
340.19	odd	8		4624.2.a.l.1.2		2
340.219	odd	8		4624.2.a.l.1.1		2
340.259	odd	4		272.2.o.c.225.1		2
680.259	odd	4		1088.2.o.i.769.1		2
680.429	even	4		1088.2.o.k.769.1		2
1020.599	even	4		2448.2.be.j.1585.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
34.2.c.b.13.1	✓	2	5.4	even	2
34.2.c.b.21.1	yes	2	85.4	even	4
272.2.o.c.81.1		2	20.19	odd	2
272.2.o.c.225.1		2	340.259	odd	4
306.2.g.d.55.1		2	255.89	odd	4
306.2.g.d.217.1		2	15.14	odd	2
578.2.a.b.1.1		2	85.49	even	8
578.2.a.b.1.2		2	85.19	even	8
578.2.b.c.577.1		2	85.59	even	8
578.2.b.c.577.2		2	85.9	even	8
578.2.c.b.251.1		2	85.84	even	2
578.2.c.b.327.1		2	85.64	even	4
578.2.d.f.155.1		8	85.29	odd	16
578.2.d.f.155.2		8	85.39	odd	16
578.2.d.f.179.1		8	85.74	odd	16
578.2.d.f.179.2		8	85.79	odd	16
578.2.d.f.399.1		8	85.44	odd	16
578.2.d.f.399.2		8	85.24	odd	16
578.2.d.f.423.1		8	85.14	odd	16
578.2.d.f.423.2		8	85.54	odd	16
850.2.g.a.149.1		2	5.3	odd	4
850.2.g.a.599.1		2	85.72	odd	4
850.2.g.d.149.1		2	5.2	odd	4
850.2.g.d.599.1		2	85.38	odd	4
850.2.h.c.251.1		2	1.1	even	1	trivial
850.2.h.c.701.1		2	17.4	even	4	inner
1088.2.o.i.769.1		2	680.259	odd	4
1088.2.o.i.897.1		2	40.19	odd	2
1088.2.o.k.769.1		2	680.429	even	4
1088.2.o.k.897.1		2	40.29	even	2
2448.2.be.j.1441.1		2	60.59	even	2
2448.2.be.j.1585.1		2	1020.599	even	4
4624.2.a.l.1.1		2	340.219	odd	8
4624.2.a.l.1.2		2	340.19	odd	8
5202.2.a.bb.1.1		2	255.104	odd	8
5202.2.a.bb.1.2		2	255.134	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:	\( N \)	\(=\)	\( 850 = 2 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	850.h (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -3.00000i q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{4} +1.00000i q^{8} -3.00000i q^{9} +(-4.00000 + 4.00000i) q^{11} -4.00000 q^{13} +1.00000 q^{16} +(1.00000 - 4.00000i) q^{17} -3.00000 q^{18} +4.00000i q^{19} +(4.00000 + 4.00000i) q^{22} +(-4.00000 + 4.00000i) q^{23} +4.00000i q^{26} +(-3.00000 - 3.00000i) q^{29} +(-4.00000 - 4.00000i) q^{31} -1.00000i q^{32} +(-4.00000 - 1.00000i) q^{34} +3.00000i q^{36} +(-3.00000 - 3.00000i) q^{37} +4.00000 q^{38} +(1.00000 - 1.00000i) q^{41} +4.00000i q^{43} +(4.00000 - 4.00000i) q^{44} +(4.00000 + 4.00000i) q^{46} -8.00000 q^{47} +7.00000i q^{49} +4.00000 q^{52} +4.00000i q^{53} +(-3.00000 + 3.00000i) q^{58} +4.00000i q^{59} +(-9.00000 + 9.00000i) q^{61} +(-4.00000 + 4.00000i) q^{62} -1.00000 q^{64} +12.0000 q^{67} +(-1.00000 + 4.00000i) q^{68} +(-4.00000 - 4.00000i) q^{71} +3.00000 q^{72} +(-5.00000 - 5.00000i) q^{73} +(-3.00000 + 3.00000i) q^{74} -4.00000i q^{76} +(8.00000 - 8.00000i) q^{79} -9.00000 q^{81} +(-1.00000 - 1.00000i) q^{82} +4.00000i q^{83} +4.00000 q^{86} +(-4.00000 - 4.00000i) q^{88} +(4.00000 - 4.00000i) q^{92} +8.00000i q^{94} +(-3.00000 - 3.00000i) q^{97} +7.00000 q^{98} +(12.0000 + 12.0000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{4}+O(q^{10}) \) 2 * q - 2 * q^4	\( 2 q - 2 q^{4} - 8 q^{11} - 8 q^{13} + 2 q^{16} + 2 q^{17} - 6 q^{18} + 8 q^{22} - 8 q^{23} - 6 q^{29} - 8 q^{31} - 8 q^{34} - 6 q^{37} + 8 q^{38} + 2 q^{41} + 8 q^{44} + 8 q^{46} - 16 q^{47} + 8 q^{52} - 6 q^{58} - 18 q^{61} - 8 q^{62} - 2 q^{64} + 24 q^{67} - 2 q^{68} - 8 q^{71} + 6 q^{72} - 10 q^{73} - 6 q^{74} + 16 q^{79} - 18 q^{81} - 2 q^{82} + 8 q^{86} - 8 q^{88} + 8 q^{92} - 6 q^{97} + 14 q^{98} + 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^4 - 8 * q^11 - 8 * q^13 + 2 * q^16 + 2 * q^17 - 6 * q^18 + 8 * q^22 - 8 * q^23 - 6 * q^29 - 8 * q^31 - 8 * q^34 - 6 * q^37 + 8 * q^38 + 2 * q^41 + 8 * q^44 + 8 * q^46 - 16 * q^47 + 8 * q^52 - 6 * q^58 - 18 * q^61 - 8 * q^62 - 2 * q^64 + 24 * q^67 - 2 * q^68 - 8 * q^71 + 6 * q^72 - 10 * q^73 - 6 * q^74 + 16 * q^79 - 18 * q^81 - 2 * q^82 + 8 * q^86 - 8 * q^88 + 8 * q^92 - 6 * q^97 + 14 * q^98 + 24 * q^99

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