LMFDB - Embedded newform 7616.2.a.bz.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7616,2,Mod(1,7616)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7616, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7616.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7616 = 2^{6} \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7616.a (trivial)

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:	yes
Analytic conductor:	$60.8140661794$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.93059344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 41x^{2} - 8x - 20 $ x^6 - 13x^4 + 41x^2 - 8*x - 20
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3808)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.71138$ of defining polynomial
Character	$\chi$	$=$	7616.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.71138 q^{3} +2.50542 q^{5} +1.00000 q^{7} -0.0711684 q^{9} +O(q^{10})$	$q+1.71138 q^{3} +2.50542 q^{5} +1.00000 q^{7} -0.0711684 q^{9} +1.68166 q^{11} -0.720803 q^{13} +4.28773 q^{15} -1.00000 q^{17} +5.17536 q^{19} +1.71138 q^{21} -2.40246 q^{23} +1.27713 q^{25} -5.25595 q^{27} -0.318339 q^{29} -4.96733 q^{31} +2.87797 q^{33} +2.50542 q^{35} +10.2456 q^{37} -1.23357 q^{39} +9.60754 q^{41} +1.30892 q^{43} -0.178307 q^{45} +2.49684 q^{47} +1.00000 q^{49} -1.71138 q^{51} +1.99912 q^{53} +4.21327 q^{55} +8.85702 q^{57} +2.29739 q^{59} +0.620150 q^{61} -0.0711684 q^{63} -1.80591 q^{65} +12.4845 q^{67} -4.11154 q^{69} -8.00769 q^{71} +1.93411 q^{73} +2.18566 q^{75} +1.68166 q^{77} +1.23357 q^{79} -8.78143 q^{81} +17.6470 q^{83} -2.50542 q^{85} -0.544800 q^{87} -1.87684 q^{89} -0.720803 q^{91} -8.50100 q^{93} +12.9664 q^{95} +14.8561 q^{97} -0.119681 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{5} + 6 q^{7} + 8 q^{9}+O(q^{10}) $ 6 * q - 4 * q^5 + 6 * q^7 + 8 * q^9	$ 6 q - 4 q^{5} + 6 q^{7} + 8 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} - 6 q^{17} + 6 q^{19} + 8 q^{25} - 8 q^{29} + 12 q^{31} + 28 q^{33} - 4 q^{35} - 10 q^{37} + 14 q^{39} + 14 q^{41} + 12 q^{43} - 22 q^{45} + 2 q^{47} + 6 q^{49} - 26 q^{53} + 10 q^{55} + 22 q^{57} - 22 q^{59} - 2 q^{61} + 8 q^{63} + 8 q^{65} + 14 q^{67} - 14 q^{69} - 4 q^{71} + 18 q^{73} + 28 q^{75} + 4 q^{77} - 14 q^{79} - 6 q^{81} - 8 q^{83} + 4 q^{85} + 28 q^{87} + 6 q^{89} + 4 q^{91} - 8 q^{93} - 2 q^{95} + 20 q^{97} + 26 q^{99}+O(q^{100}) $ 6 * q - 4 * q^5 + 6 * q^7 + 8 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^15 - 6 * q^17 + 6 * q^19 + 8 * q^25 - 8 * q^29 + 12 * q^31 + 28 * q^33 - 4 * q^35 - 10 * q^37 + 14 * q^39 + 14 * q^41 + 12 * q^43 - 22 * q^45 + 2 * q^47 + 6 * q^49 - 26 * q^53 + 10 * q^55 + 22 * q^57 - 22 * q^59 - 2 * q^61 + 8 * q^63 + 8 * q^65 + 14 * q^67 - 14 * q^69 - 4 * q^71 + 18 * q^73 + 28 * q^75 + 4 * q^77 - 14 * q^79 - 6 * q^81 - 8 * q^83 + 4 * q^85 + 28 * q^87 + 6 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 20 * q^97 + 26 * q^99

Display 100 coefficients 10 coefficients

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
$3$	1.71138	0.988067	0.494034	−	0.869443i	$-0.335522\pi$
$3$	1.71138	0.988067	0.494034	+	0.869443i	$0.335522\pi$
$4$	0	0
$5$	2.50542	1.12046	0.560229	−	0.828338i	$-0.310713\pi$
$5$	2.50542	1.12046	0.560229	+	0.828338i	$0.310713\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.0711684	−0.0237228
$10$	0	0
$11$	1.68166	0.507040	0.253520	−	0.967330i	$-0.418412\pi$
$11$	1.68166	0.507040	0.253520	+	0.967330i	$0.418412\pi$
$12$	0	0
$13$	−0.720803	−0.199915	−0.0999574	−	0.994992i	$-0.531871\pi$
$13$	−0.720803	−0.199915	−0.0999574	+	0.994992i	$0.531871\pi$
$14$	0	0
$15$	4.28773	1.10709
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.17536	1.18731	0.593654	−	0.804720i	$-0.297685\pi$
$19$	5.17536	1.18731	0.593654	+	0.804720i	$0.297685\pi$
$20$	0	0
$21$	1.71138	0.373454
$22$	0	0
$23$	−2.40246	−0.500948	−0.250474	−	0.968123i	$-0.580587\pi$
$23$	−2.40246	−0.500948	−0.250474	+	0.968123i	$0.580587\pi$
$24$	0	0
$25$	1.27713	0.255426
$26$	0	0
$27$	−5.25595	−1.01151
$28$	0	0
$29$	−0.318339	−0.0591141	−0.0295571	−	0.999563i	$-0.509410\pi$
$29$	−0.318339	−0.0591141	−0.0295571	+	0.999563i	$0.509410\pi$
$30$	0	0
$31$	−4.96733	−0.892158	−0.446079	−	0.894993i	$-0.647180\pi$
$31$	−4.96733	−0.892158	−0.446079	+	0.894993i	$0.647180\pi$
$32$	0	0
$33$	2.87797	0.500990
$34$	0	0
$35$	2.50542	0.423493
$36$	0	0
$37$	10.2456	1.68437	0.842187	−	0.539186i	$-0.181268\pi$
$37$	10.2456	1.68437	0.842187	+	0.539186i	$0.181268\pi$
$38$	0	0
$39$	−1.23357	−0.197529
$40$	0	0
$41$	9.60754	1.50045	0.750223	−	0.661185i	$-0.229946\pi$
$41$	9.60754	1.50045	0.750223	+	0.661185i	$0.229946\pi$
$42$	0	0
$43$	1.30892	0.199608	0.0998042	−	0.995007i	$-0.468178\pi$
$43$	1.30892	0.199608	0.0998042	+	0.995007i	$0.468178\pi$
$44$	0	0
$45$	−0.178307	−0.0265804
$46$	0	0
$47$	2.49684	0.364202	0.182101	−	0.983280i	$-0.441710\pi$
$47$	2.49684	0.364202	0.182101	+	0.983280i	$0.441710\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.71138	−0.239642
$52$	0	0
$53$	1.99912	0.274600	0.137300	−	0.990530i	$-0.456158\pi$
$53$	1.99912	0.274600	0.137300	+	0.990530i	$0.456158\pi$
$54$	0	0
$55$	4.21327	0.568117
$56$	0	0
$57$	8.85702	1.17314
$58$	0	0
$59$	2.29739	0.299095	0.149547	−	0.988755i	$-0.452218\pi$
$59$	2.29739	0.299095	0.149547	+	0.988755i	$0.452218\pi$
$60$	0	0
$61$	0.620150	0.0794021	0.0397010	−	0.999212i	$-0.487359\pi$
$61$	0.620150	0.0794021	0.0397010	+	0.999212i	$0.487359\pi$
$62$	0	0
$63$	−0.0711684	−0.00896637
$64$	0	0
$65$	−1.80591	−0.223996
$66$	0	0
$67$	12.4845	1.52522	0.762611	−	0.646858i	$-0.223917\pi$
$67$	12.4845	1.52522	0.762611	+	0.646858i	$0.223917\pi$
$68$	0	0
$69$	−4.11154	−0.494971
$70$	0	0
$71$	−8.00769	−0.950339	−0.475169	−	0.879894i	$-0.657613\pi$
$71$	−8.00769	−0.950339	−0.475169	+	0.879894i	$0.657613\pi$
$72$	0	0
$73$	1.93411	0.226371	0.113185	−	0.993574i	$-0.463895\pi$
$73$	1.93411	0.226371	0.113185	+	0.993574i	$0.463895\pi$
$74$	0	0
$75$	2.18566	0.252378
$76$	0	0
$77$	1.68166	0.191643
$78$	0	0
$79$	1.23357	0.138787	0.0693937	−	0.997589i	$-0.477894\pi$
$79$	1.23357	0.138787	0.0693937	+	0.997589i	$0.477894\pi$
$80$	0	0
$81$	−8.78143	−0.975714
$82$	0	0
$83$	17.6470	1.93701	0.968504	−	0.248998i	$-0.0801012\pi$
$83$	17.6470	1.93701	0.968504	+	0.248998i	$0.0801012\pi$
$84$	0	0
$85$	−2.50542	−0.271751
$86$	0	0
$87$	−0.544800	−0.0584087
$88$	0	0
$89$	−1.87684	−0.198945	−0.0994725	−	0.995040i	$-0.531716\pi$
$89$	−1.87684	−0.198945	−0.0994725	+	0.995040i	$0.531716\pi$
$90$	0	0
$91$	−0.720803	−0.0755607
$92$	0	0
$93$	−8.50100	−0.881513
$94$	0	0
$95$	12.9664	1.33033
$96$	0	0
$97$	14.8561	1.50841	0.754206	−	0.656638i	$-0.228022\pi$
$97$	14.8561	1.50841	0.754206	+	0.656638i	$0.228022\pi$
$98$	0	0
$99$	−0.119681	−0.0120284
$100$	0	0
$101$	−5.83548	−0.580652	−0.290326	−	0.956928i	$-0.593764\pi$
$101$	−5.83548	−0.580652	−0.290326	+	0.956928i	$0.593764\pi$
$102$	0	0
$103$	9.90799	0.976263	0.488131	−	0.872770i	$-0.337679\pi$
$103$	9.90799	0.976263	0.488131	+	0.872770i	$0.337679\pi$
$104$	0	0
$105$	4.28773	0.418440
$106$	0	0
$107$	13.1945	1.27556	0.637782	−	0.770217i	$-0.279852\pi$
$107$	13.1945	1.27556	0.637782	+	0.770217i	$0.279852\pi$
$108$	0	0
$109$	−7.57782	−0.725824	−0.362912	−	0.931823i	$-0.618217\pi$
$109$	−7.57782	−0.725824	−0.362912	+	0.931823i	$0.618217\pi$
$110$	0	0
$111$	17.5342	1.66427
$112$	0	0
$113$	12.9706	1.22017	0.610085	−	0.792336i	$-0.291135\pi$
$113$	12.9706	1.22017	0.610085	+	0.792336i	$0.291135\pi$
$114$	0	0
$115$	−6.01918	−0.561292
$116$	0	0
$117$	0.0512984	0.00474254
$118$	0	0
$119$	−1.00000	−0.0916698
$120$	0	0
$121$	−8.17202	−0.742911
$122$	0	0
$123$	16.4422	1.48254
$124$	0	0
$125$	−9.32735	−0.834264
$126$	0	0
$127$	3.96845	0.352143	0.176071	−	0.984377i	$-0.443661\pi$
$127$	3.96845	0.352143	0.176071	+	0.984377i	$0.443661\pi$
$128$	0	0
$129$	2.24006	0.197226
$130$	0	0
$131$	−14.6420	−1.27928	−0.639638	−	0.768676i	$-0.720916\pi$
$131$	−14.6420	−1.27928	−0.639638	+	0.768676i	$0.720916\pi$
$132$	0	0
$133$	5.17536	0.448760
$134$	0	0
$135$	−13.1684	−1.13335
$136$	0	0
$137$	7.06322	0.603451	0.301726	−	0.953395i	$-0.402437\pi$
$137$	7.06322	0.603451	0.301726	+	0.953395i	$0.402437\pi$
$138$	0	0
$139$	4.61661	0.391576	0.195788	−	0.980646i	$-0.437274\pi$
$139$	4.61661	0.391576	0.195788	+	0.980646i	$0.437274\pi$
$140$	0	0
$141$	4.27305	0.359856
$142$	0	0
$143$	−1.21215	−0.101365
$144$	0	0
$145$	−0.797574	−0.0662349
$146$	0	0
$147$	1.71138	0.141152
$148$	0	0
$149$	−24.0907	−1.97358	−0.986792	−	0.161994i	$-0.948207\pi$
$149$	−24.0907	−1.97358	−0.986792	+	0.161994i	$0.948207\pi$
$150$	0	0
$151$	−14.8952	−1.21216	−0.606078	−	0.795405i	$-0.707258\pi$
$151$	−14.8952	−1.21216	−0.606078	+	0.795405i	$0.707258\pi$
$152$	0	0
$153$	0.0711684	0.00575362
$154$	0	0
$155$	−12.4452	−0.999626
$156$	0	0
$157$	−13.9453	−1.11295	−0.556476	−	0.830864i	$-0.687847\pi$
$157$	−13.9453	−1.11295	−0.556476	+	0.830864i	$0.687847\pi$
$158$	0	0
$159$	3.42125	0.271323
$160$	0	0
$161$	−2.40246	−0.189341
$162$	0	0
$163$	12.2900	0.962630	0.481315	−	0.876548i	$-0.340159\pi$
$163$	12.2900	0.962630	0.481315	+	0.876548i	$0.340159\pi$
$164$	0	0
$165$	7.21051	0.561338
$166$	0	0
$167$	12.1851	0.942913	0.471456	−	0.881889i	$-0.343728\pi$
$167$	12.1851	0.942913	0.471456	+	0.881889i	$0.343728\pi$
$168$	0	0
$169$	−12.4804	−0.960034
$170$	0	0
$171$	−0.368322	−0.0281663
$172$	0	0
$173$	−19.3211	−1.46896	−0.734478	−	0.678632i	$-0.762573\pi$
$173$	−19.3211	−1.46896	−0.734478	+	0.678632i	$0.762573\pi$
$174$	0	0
$175$	1.27713	0.0965420
$176$	0	0
$177$	3.93172	0.295526
$178$	0	0
$179$	1.99496	0.149110	0.0745551	−	0.997217i	$-0.476246\pi$
$179$	1.99496	0.149110	0.0745551	+	0.997217i	$0.476246\pi$
$180$	0	0
$181$	4.42900	0.329205	0.164602	−	0.986360i	$-0.447366\pi$
$181$	4.42900	0.329205	0.164602	+	0.986360i	$0.447366\pi$
$182$	0	0
$183$	1.06131	0.0784546
$184$	0	0
$185$	25.6696	1.88727
$186$	0	0
$187$	−1.68166	−0.122975
$188$	0	0
$189$	−5.25595	−0.382314
$190$	0	0
$191$	12.2363	0.885386	0.442693	−	0.896673i	$-0.354023\pi$
$191$	12.2363	0.885386	0.442693	+	0.896673i	$0.354023\pi$
$192$	0	0
$193$	19.4518	1.40017	0.700086	−	0.714059i	$-0.253145\pi$
$193$	19.4518	1.40017	0.700086	+	0.714059i	$0.253145\pi$
$194$	0	0
$195$	−3.09061	−0.221323
$196$	0	0
$197$	−5.95816	−0.424502	−0.212251	−	0.977215i	$-0.568079\pi$
$197$	−5.95816	−0.424502	−0.212251	+	0.977215i	$0.568079\pi$
$198$	0	0
$199$	18.7181	1.32689	0.663445	−	0.748225i	$-0.269094\pi$
$199$	18.7181	1.32689	0.663445	+	0.748225i	$0.269094\pi$
$200$	0	0
$201$	21.3657	1.50702
$202$	0	0
$203$	−0.318339	−0.0223430
$204$	0	0
$205$	24.0709	1.68119
$206$	0	0
$207$	0.170979	0.0118839
$208$	0	0
$209$	8.70320	0.602013
$210$	0	0
$211$	−13.7801	−0.948660	−0.474330	−	0.880347i	$-0.657310\pi$
$211$	−13.7801	−0.948660	−0.474330	+	0.880347i	$0.657310\pi$
$212$	0	0
$213$	−13.7042	−0.938999
$214$	0	0
$215$	3.27939	0.223653
$216$	0	0
$217$	−4.96733	−0.337204
$218$	0	0
$219$	3.31001	0.223669
$220$	0	0
$221$	0.720803	0.0484864
$222$	0	0
$223$	−15.0846	−1.01014	−0.505069	−	0.863079i	$-0.668533\pi$
$223$	−15.0846	−1.01014	−0.505069	+	0.863079i	$0.668533\pi$
$224$	0	0
$225$	−0.0908914	−0.00605942
$226$	0	0
$227$	−7.79118	−0.517119	−0.258559	−	0.965995i	$-0.583248\pi$
$227$	−7.79118	−0.517119	−0.258559	+	0.965995i	$0.583248\pi$
$228$	0	0
$229$	20.6673	1.36573	0.682866	−	0.730543i	$-0.260733\pi$
$229$	20.6673	1.36573	0.682866	+	0.730543i	$0.260733\pi$
$230$	0	0
$231$	2.87797	0.189356
$232$	0	0
$233$	6.98383	0.457526	0.228763	−	0.973482i	$-0.426532\pi$
$233$	6.98383	0.457526	0.228763	+	0.973482i	$0.426532\pi$
$234$	0	0
$235$	6.25564	0.408073
$236$	0	0
$237$	2.11111	0.137131
$238$	0	0
$239$	11.9866	0.775348	0.387674	−	0.921796i	$-0.373279\pi$
$239$	11.9866	0.775348	0.387674	+	0.921796i	$0.373279\pi$
$240$	0	0
$241$	−19.3467	−1.24623	−0.623116	−	0.782129i	$-0.714134\pi$
$241$	−19.3467	−1.24623	−0.623116	+	0.782129i	$0.714134\pi$
$242$	0	0
$243$	0.739446	0.0474355
$244$	0	0
$245$	2.50542	0.160065
$246$	0	0
$247$	−3.73041	−0.237360
$248$	0	0
$249$	30.2007	1.91389
$250$	0	0
$251$	−13.6513	−0.861659	−0.430830	−	0.902433i	$-0.641779\pi$
$251$	−13.6513	−0.861659	−0.430830	+	0.902433i	$0.641779\pi$
$252$	0	0
$253$	−4.04013	−0.254001
$254$	0	0
$255$	−4.28773	−0.268508
$256$	0	0
$257$	−4.52842	−0.282475	−0.141238	−	0.989976i	$-0.545108\pi$
$257$	−4.52842	−0.282475	−0.141238	+	0.989976i	$0.545108\pi$
$258$	0	0
$259$	10.2456	0.636633
$260$	0	0
$261$	0.0226557	0.00140235
$262$	0	0
$263$	−5.61105	−0.345992	−0.172996	−	0.984923i	$-0.555345\pi$
$263$	−5.61105	−0.345992	−0.172996	+	0.984923i	$0.555345\pi$
$264$	0	0
$265$	5.00863	0.307678
$266$	0	0
$267$	−3.21200	−0.196571
$268$	0	0
$269$	−21.7291	−1.32485	−0.662424	−	0.749129i	$-0.730472\pi$
$269$	−21.7291	−1.32485	−0.662424	+	0.749129i	$0.730472\pi$
$270$	0	0
$271$	16.8402	1.02297	0.511485	−	0.859292i	$-0.329096\pi$
$271$	16.8402	1.02297	0.511485	+	0.859292i	$0.329096\pi$
$272$	0	0
$273$	−1.23357	−0.0746590
$274$	0	0
$275$	2.14770	0.129511
$276$	0	0
$277$	−15.1519	−0.910390	−0.455195	−	0.890392i	$-0.650430\pi$
$277$	−15.1519	−0.910390	−0.455195	+	0.890392i	$0.650430\pi$
$278$	0	0
$279$	0.353517	0.0211645
$280$	0	0
$281$	0.718687	0.0428733	0.0214366	−	0.999770i	$-0.493176\pi$
$281$	0.718687	0.0428733	0.0214366	+	0.999770i	$0.493176\pi$
$282$	0	0
$283$	5.49568	0.326684	0.163342	−	0.986569i	$-0.447773\pi$
$283$	5.49568	0.326684	0.163342	+	0.986569i	$0.447773\pi$
$284$	0	0
$285$	22.1906	1.31445
$286$	0	0
$287$	9.60754	0.567115
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	25.4245	1.49041
$292$	0	0
$293$	3.81204	0.222701	0.111351	−	0.993781i	$-0.464482\pi$
$293$	3.81204	0.222701	0.111351	+	0.993781i	$0.464482\pi$
$294$	0	0
$295$	5.75593	0.335123
$296$	0	0
$297$	−8.83872	−0.512874
$298$	0	0
$299$	1.73170	0.100147
$300$	0	0
$301$	1.30892	0.0754449
$302$	0	0
$303$	−9.98675	−0.573724
$304$	0	0
$305$	1.55374	0.0889667
$306$	0	0
$307$	−24.3206	−1.38805	−0.694025	−	0.719951i	$-0.744164\pi$
$307$	−24.3206	−1.38805	−0.694025	+	0.719951i	$0.744164\pi$
$308$	0	0
$309$	16.9564	0.964614
$310$	0	0
$311$	1.50404	0.0852864	0.0426432	−	0.999090i	$-0.486422\pi$
$311$	1.50404	0.0852864	0.0426432	+	0.999090i	$0.486422\pi$
$312$	0	0
$313$	−24.1918	−1.36740	−0.683700	−	0.729763i	$-0.739630\pi$
$313$	−24.1918	−1.36740	−0.683700	+	0.729763i	$0.739630\pi$
$314$	0	0
$315$	−0.178307	−0.0100464
$316$	0	0
$317$	10.7953	0.606324	0.303162	−	0.952939i	$-0.401958\pi$
$317$	10.7953	0.606324	0.303162	+	0.952939i	$0.401958\pi$
$318$	0	0
$319$	−0.535339	−0.0299732
$320$	0	0
$321$	22.5809	1.26034
$322$	0	0
$323$	−5.17536	−0.287965
$324$	0	0
$325$	−0.920560	−0.0510635
$326$	0	0
$327$	−12.9686	−0.717163
$328$	0	0
$329$	2.49684	0.137655
$330$	0	0
$331$	−11.4765	−0.630804	−0.315402	−	0.948958i	$-0.602139\pi$
$331$	−11.4765	−0.630804	−0.315402	+	0.948958i	$0.602139\pi$
$332$	0	0
$333$	−0.729166	−0.0399580
$334$	0	0
$335$	31.2789	1.70895
$336$	0	0
$337$	−17.6082	−0.959178	−0.479589	−	0.877493i	$-0.659214\pi$
$337$	−17.6082	−0.959178	−0.479589	+	0.877493i	$0.659214\pi$
$338$	0	0
$339$	22.1976	1.20561
$340$	0	0
$341$	−8.35336	−0.452360
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−10.3011	−0.554594
$346$	0	0
$347$	5.59372	0.300287	0.150143	−	0.988664i	$-0.452026\pi$
$347$	5.59372	0.300287	0.150143	+	0.988664i	$0.452026\pi$
$348$	0	0
$349$	6.74791	0.361207	0.180604	−	0.983556i	$-0.442195\pi$
$349$	6.74791	0.361207	0.180604	+	0.983556i	$0.442195\pi$
$350$	0	0
$351$	3.78850	0.202215
$352$	0	0
$353$	−21.5847	−1.14884	−0.574418	−	0.818562i	$-0.694772\pi$
$353$	−21.5847	−1.14884	−0.574418	+	0.818562i	$0.694772\pi$
$354$	0	0
$355$	−20.0626	−1.06481
$356$	0	0
$357$	−1.71138	−0.0905760
$358$	0	0
$359$	7.92560	0.418297	0.209149	−	0.977884i	$-0.432931\pi$
$359$	7.92560	0.418297	0.209149	+	0.977884i	$0.432931\pi$
$360$	0	0
$361$	7.78432	0.409701
$362$	0	0
$363$	−13.9855	−0.734046
$364$	0	0
$365$	4.84576	0.253639
$366$	0	0
$367$	17.3807	0.907263	0.453632	−	0.891189i	$-0.350128\pi$
$367$	17.3807	0.907263	0.453632	+	0.891189i	$0.350128\pi$
$368$	0	0
$369$	−0.683753	−0.0355948
$370$	0	0
$371$	1.99912	0.103789
$372$	0	0
$373$	−23.2882	−1.20582	−0.602908	−	0.797810i	$-0.705992\pi$
$373$	−23.2882	−1.20582	−0.602908	+	0.797810i	$0.705992\pi$
$374$	0	0
$375$	−15.9627	−0.824309
$376$	0	0
$377$	0.229460	0.0118178
$378$	0	0
$379$	−18.3617	−0.943177	−0.471589	−	0.881819i	$-0.656319\pi$
$379$	−18.3617	−0.943177	−0.471589	+	0.881819i	$0.656319\pi$
$380$	0	0
$381$	6.79154	0.347941
$382$	0	0
$383$	−14.0603	−0.718447	−0.359223	−	0.933252i	$-0.616958\pi$
$383$	−14.0603	−0.718447	−0.359223	+	0.933252i	$0.616958\pi$
$384$	0	0
$385$	4.21327	0.214728
$386$	0	0
$387$	−0.0931537	−0.00473527
$388$	0	0
$389$	−9.99912	−0.506975	−0.253488	−	0.967339i	$-0.581578\pi$
$389$	−9.99912	−0.506975	−0.253488	+	0.967339i	$0.581578\pi$
$390$	0	0
$391$	2.40246	0.121498
$392$	0	0
$393$	−25.0580	−1.26401
$394$	0	0
$395$	3.09061	0.155505
$396$	0	0
$397$	−2.69013	−0.135014	−0.0675068	−	0.997719i	$-0.521504\pi$
$397$	−2.69013	−0.135014	−0.0675068	+	0.997719i	$0.521504\pi$
$398$	0	0
$399$	8.85702	0.443405
$400$	0	0
$401$	−12.6560	−0.632009	−0.316004	−	0.948758i	$-0.602341\pi$
$401$	−12.6560	−0.632009	−0.316004	+	0.948758i	$0.602341\pi$
$402$	0	0
$403$	3.58046	0.178356
$404$	0	0
$405$	−22.0012	−1.09325
$406$	0	0
$407$	17.2297	0.854044
$408$	0	0
$409$	−11.5780	−0.572495	−0.286247	−	0.958156i	$-0.592408\pi$
$409$	−11.5780	−0.572495	−0.286247	+	0.958156i	$0.592408\pi$
$410$	0	0
$411$	12.0879	0.596251
$412$	0	0
$413$	2.29739	0.113047
$414$	0	0
$415$	44.2131	2.17034
$416$	0	0
$417$	7.90079	0.386904
$418$	0	0
$419$	−17.1204	−0.836386	−0.418193	−	0.908358i	$-0.637336\pi$
$419$	−17.1204	−0.836386	−0.418193	+	0.908358i	$0.637336\pi$
$420$	0	0
$421$	10.7936	0.526049	0.263025	−	0.964789i	$-0.415280\pi$
$421$	10.7936	0.526049	0.263025	+	0.964789i	$0.415280\pi$
$422$	0	0
$423$	−0.177696	−0.00863989
$424$	0	0
$425$	−1.27713	−0.0619500
$426$	0	0
$427$	0.620150	0.0300112
$428$	0	0
$429$	−2.07445	−0.100155
$430$	0	0
$431$	17.4396	0.840034	0.420017	−	0.907516i	$-0.362024\pi$
$431$	17.4396	0.840034	0.420017	+	0.907516i	$0.362024\pi$
$432$	0	0
$433$	−1.59256	−0.0765337	−0.0382668	−	0.999268i	$-0.512184\pi$
$433$	−1.59256	−0.0765337	−0.0382668	+	0.999268i	$0.512184\pi$
$434$	0	0
$435$	−1.36495	−0.0654445
$436$	0	0
$437$	−12.4336	−0.594780
$438$	0	0
$439$	−27.4643	−1.31080	−0.655399	−	0.755283i	$-0.727499\pi$
$439$	−27.4643	−1.31080	−0.655399	+	0.755283i	$0.727499\pi$
$440$	0	0
$441$	−0.0711684	−0.00338897
$442$	0	0
$443$	9.08714	0.431743	0.215871	−	0.976422i	$-0.430741\pi$
$443$	9.08714	0.431743	0.215871	+	0.976422i	$0.430741\pi$
$444$	0	0
$445$	−4.70228	−0.222910
$446$	0	0
$447$	−41.2283	−1.95003
$448$	0	0
$449$	−31.2208	−1.47340	−0.736699	−	0.676220i	$-0.763617\pi$
$449$	−31.2208	−1.47340	−0.736699	+	0.676220i	$0.763617\pi$
$450$	0	0
$451$	16.1566	0.760786
$452$	0	0
$453$	−25.4914	−1.19769
$454$	0	0
$455$	−1.80591	−0.0846625
$456$	0	0
$457$	6.99457	0.327192	0.163596	−	0.986527i	$-0.447691\pi$
$457$	6.99457	0.327192	0.163596	+	0.986527i	$0.447691\pi$
$458$	0	0
$459$	5.25595	0.245327
$460$	0	0
$461$	−17.5770	−0.818641	−0.409321	−	0.912391i	$-0.634234\pi$
$461$	−17.5770	−0.818641	−0.409321	+	0.912391i	$0.634234\pi$
$462$	0	0
$463$	9.53781	0.443259	0.221630	−	0.975131i	$-0.428862\pi$
$463$	9.53781	0.443259	0.221630	+	0.975131i	$0.428862\pi$
$464$	0	0
$465$	−21.2986	−0.987698
$466$	0	0
$467$	−41.6493	−1.92730	−0.963649	−	0.267172i	$-0.913911\pi$
$467$	−41.6493	−1.92730	−0.963649	+	0.267172i	$0.913911\pi$
$468$	0	0
$469$	12.4845	0.576479
$470$	0	0
$471$	−23.8657	−1.09967
$472$	0	0
$473$	2.20116	0.101209
$474$	0	0
$475$	6.60961	0.303270
$476$	0	0
$477$	−0.142274	−0.00651427
$478$	0	0
$479$	22.4506	1.02579	0.512896	−	0.858450i	$-0.328572\pi$
$479$	22.4506	1.02579	0.512896	+	0.858450i	$0.328572\pi$
$480$	0	0
$481$	−7.38509	−0.336731
$482$	0	0
$483$	−4.11154	−0.187081
$484$	0	0
$485$	37.2209	1.69011
$486$	0	0
$487$	−24.0332	−1.08905	−0.544523	−	0.838746i	$-0.683289\pi$
$487$	−24.0332	−1.08905	−0.544523	+	0.838746i	$0.683289\pi$
$488$	0	0
$489$	21.0330	0.951143
$490$	0	0
$491$	26.1151	1.17856	0.589280	−	0.807929i	$-0.299412\pi$
$491$	26.1151	1.17856	0.589280	+	0.807929i	$0.299412\pi$
$492$	0	0
$493$	0.318339	0.0143373
$494$	0	0
$495$	−0.299851	−0.0134773
$496$	0	0
$497$	−8.00769	−0.359194
$498$	0	0
$499$	32.1995	1.44145	0.720723	−	0.693223i	$-0.243810\pi$
$499$	32.1995	1.44145	0.720723	+	0.693223i	$0.243810\pi$
$500$	0	0
$501$	20.8534	0.931661
$502$	0	0
$503$	−5.52771	−0.246468	−0.123234	−	0.992378i	$-0.539327\pi$
$503$	−5.52771	−0.246468	−0.123234	+	0.992378i	$0.539327\pi$
$504$	0	0
$505$	−14.6203	−0.650597
$506$	0	0
$507$	−21.3588	−0.948578
$508$	0	0
$509$	10.3472	0.458633	0.229317	−	0.973352i	$-0.426351\pi$
$509$	10.3472	0.458633	0.229317	+	0.973352i	$0.426351\pi$
$510$	0	0
$511$	1.93411	0.0855601
$512$	0	0
$513$	−27.2014	−1.20097
$514$	0	0
$515$	24.8237	1.09386
$516$	0	0
$517$	4.19884	0.184665
$518$	0	0
$519$	−33.0658	−1.45143
$520$	0	0
$521$	20.7908	0.910860	0.455430	−	0.890272i	$-0.349486\pi$
$521$	20.7908	0.910860	0.455430	+	0.890272i	$0.349486\pi$
$522$	0	0
$523$	16.0410	0.701425	0.350712	−	0.936483i	$-0.385939\pi$
$523$	16.0410	0.701425	0.350712	+	0.936483i	$0.385939\pi$
$524$	0	0
$525$	2.18566	0.0953900
$526$	0	0
$527$	4.96733	0.216380
$528$	0	0
$529$	−17.2282	−0.749051
$530$	0	0
$531$	−0.163502	−0.00709537
$532$	0	0
$533$	−6.92514	−0.299961
$534$	0	0
$535$	33.0579	1.42922
$536$	0	0
$537$	3.41414	0.147331
$538$	0	0
$539$	1.68166	0.0724343
$540$	0	0
$541$	−27.1601	−1.16770	−0.583851	−	0.811861i	$-0.698455\pi$
$541$	−27.1601	−1.16770	−0.583851	+	0.811861i	$0.698455\pi$
$542$	0	0
$543$	7.57971	0.325277
$544$	0	0
$545$	−18.9856	−0.813255
$546$	0	0
$547$	−4.95580	−0.211895	−0.105947	−	0.994372i	$-0.533787\pi$
$547$	−4.95580	−0.211895	−0.105947	+	0.994372i	$0.533787\pi$
$548$	0	0
$549$	−0.0441351	−0.00188364
$550$	0	0
$551$	−1.64752	−0.0701867
$552$	0	0
$553$	1.23357	0.0524567
$554$	0	0
$555$	43.9306	1.86475
$556$	0	0
$557$	−35.7252	−1.51373	−0.756863	−	0.653574i	$-0.773269\pi$
$557$	−35.7252	−1.51373	−0.756863	+	0.653574i	$0.773269\pi$
$558$	0	0
$559$	−0.943473	−0.0399046
$560$	0	0
$561$	−2.87797	−0.121508
$562$	0	0
$563$	27.2269	1.14748	0.573738	−	0.819039i	$-0.305493\pi$
$563$	27.2269	1.14748	0.573738	+	0.819039i	$0.305493\pi$
$564$	0	0
$565$	32.4967	1.36715
$566$	0	0
$567$	−8.78143	−0.368785
$568$	0	0
$569$	15.1009	0.633063	0.316531	−	0.948582i	$-0.397482\pi$
$569$	15.1009	0.633063	0.316531	+	0.948582i	$0.397482\pi$
$570$	0	0
$571$	36.6452	1.53356	0.766778	−	0.641913i	$-0.221859\pi$
$571$	36.6452	1.53356	0.766778	+	0.641913i	$0.221859\pi$
$572$	0	0
$573$	20.9409	0.874821
$574$	0	0
$575$	−3.06826	−0.127955
$576$	0	0
$577$	−32.9557	−1.37196	−0.685982	−	0.727619i	$-0.740627\pi$
$577$	−32.9557	−1.37196	−0.685982	+	0.727619i	$0.740627\pi$
$578$	0	0
$579$	33.2895	1.38346
$580$	0	0
$581$	17.6470	0.732120
$582$	0	0
$583$	3.36184	0.139233
$584$	0	0
$585$	0.128524	0.00531381
$586$	0	0
$587$	21.3492	0.881175	0.440588	−	0.897710i	$-0.354770\pi$
$587$	21.3492	0.881175	0.440588	+	0.897710i	$0.354770\pi$
$588$	0	0
$589$	−25.7077	−1.05927
$590$	0	0
$591$	−10.1967	−0.419436
$592$	0	0
$593$	17.6869	0.726315	0.363158	−	0.931728i	$-0.381699\pi$
$593$	17.6869	0.726315	0.363158	+	0.931728i	$0.381699\pi$
$594$	0	0
$595$	−2.50542	−0.102712
$596$	0	0
$597$	32.0338	1.31106
$598$	0	0
$599$	−16.0922	−0.657508	−0.328754	−	0.944416i	$-0.606629\pi$
$599$	−16.0922	−0.657508	−0.328754	+	0.944416i	$0.606629\pi$
$600$	0	0
$601$	41.0378	1.67397	0.836984	−	0.547227i	$-0.184317\pi$
$601$	41.0378	1.67397	0.836984	+	0.547227i	$0.184317\pi$
$602$	0	0
$603$	−0.888500	−0.0361825
$604$	0	0
$605$	−20.4743	−0.832400
$606$	0	0
$607$	−16.1224	−0.654388	−0.327194	−	0.944957i	$-0.606103\pi$
$607$	−16.1224	−0.654388	−0.327194	+	0.944957i	$0.606103\pi$
$608$	0	0
$609$	−0.544800	−0.0220764
$610$	0	0
$611$	−1.79973	−0.0728093
$612$	0	0
$613$	38.3222	1.54782	0.773909	−	0.633297i	$-0.218299\pi$
$613$	38.3222	1.54782	0.773909	+	0.633297i	$0.218299\pi$
$614$	0	0
$615$	41.1946	1.66113
$616$	0	0
$617$	3.67506	0.147952	0.0739762	−	0.997260i	$-0.476431\pi$
$617$	3.67506	0.147952	0.0739762	+	0.997260i	$0.476431\pi$
$618$	0	0
$619$	−11.5909	−0.465878	−0.232939	−	0.972491i	$-0.574834\pi$
$619$	−11.5909	−0.465878	−0.232939	+	0.972491i	$0.574834\pi$
$620$	0	0
$621$	12.6272	0.506713
$622$	0	0
$623$	−1.87684	−0.0751942
$624$	0	0
$625$	−29.7546	−1.19018
$626$	0	0
$627$	14.8945	0.594829
$628$	0	0
$629$	−10.2456	−0.408521
$630$	0	0
$631$	−17.8784	−0.711727	−0.355863	−	0.934538i	$-0.615813\pi$
$631$	−17.8784	−0.711727	−0.355863	+	0.934538i	$0.615813\pi$
$632$	0	0
$633$	−23.5830	−0.937340
$634$	0	0
$635$	9.94263	0.394561
$636$	0	0
$637$	−0.720803	−0.0285592
$638$	0	0
$639$	0.569895	0.0225447
$640$	0	0
$641$	1.99265	0.0787048	0.0393524	−	0.999225i	$-0.487471\pi$
$641$	1.99265	0.0787048	0.0393524	+	0.999225i	$0.487471\pi$
$642$	0	0
$643$	−22.6582	−0.893551	−0.446776	−	0.894646i	$-0.647428\pi$
$643$	−22.6582	−0.893551	−0.446776	+	0.894646i	$0.647428\pi$
$644$	0	0
$645$	5.61230	0.220984
$646$	0	0
$647$	11.3812	0.447442	0.223721	−	0.974653i	$-0.428180\pi$
$647$	11.3812	0.447442	0.223721	+	0.974653i	$0.428180\pi$
$648$	0	0
$649$	3.86343	0.151653
$650$	0	0
$651$	−8.50100	−0.333180
$652$	0	0
$653$	−7.78618	−0.304697	−0.152348	−	0.988327i	$-0.548684\pi$
$653$	−7.78618	−0.304697	−0.152348	+	0.988327i	$0.548684\pi$
$654$	0	0
$655$	−36.6843	−1.43338
$656$	0	0
$657$	−0.137648	−0.00537014
$658$	0	0
$659$	13.9880	0.544894	0.272447	−	0.962171i	$-0.412167\pi$
$659$	13.9880	0.544894	0.272447	+	0.962171i	$0.412167\pi$
$660$	0	0
$661$	−2.26909	−0.0882574	−0.0441287	−	0.999026i	$-0.514051\pi$
$661$	−2.26909	−0.0882574	−0.0441287	+	0.999026i	$0.514051\pi$
$662$	0	0
$663$	1.23357	0.0479079
$664$	0	0
$665$	12.9664	0.502817
$666$	0	0
$667$	0.764798	0.0296131
$668$	0	0
$669$	−25.8155	−0.998084
$670$	0	0
$671$	1.04288	0.0402600
$672$	0	0
$673$	31.0901	1.19843	0.599217	−	0.800586i	$-0.295479\pi$
$673$	31.0901	1.19843	0.599217	+	0.800586i	$0.295479\pi$
$674$	0	0
$675$	−6.71253	−0.258365
$676$	0	0
$677$	−22.9483	−0.881976	−0.440988	−	0.897513i	$-0.645372\pi$
$677$	−22.9483	−0.881976	−0.440988	+	0.897513i	$0.645372\pi$
$678$	0	0
$679$	14.8561	0.570126
$680$	0	0
$681$	−13.3337	−0.510948
$682$	0	0
$683$	21.5257	0.823659	0.411829	−	0.911261i	$-0.364890\pi$
$683$	21.5257	0.823659	0.411829	+	0.911261i	$0.364890\pi$
$684$	0	0
$685$	17.6963	0.676142
$686$	0	0
$687$	35.3696	1.34944
$688$	0	0
$689$	−1.44097	−0.0548965
$690$	0	0
$691$	−19.0875	−0.726124	−0.363062	−	0.931765i	$-0.618269\pi$
$691$	−19.0875	−0.726124	−0.363062	+	0.931765i	$0.618269\pi$
$692$	0	0
$693$	−0.119681	−0.00454631
$694$	0	0
$695$	11.5666	0.438745
$696$	0	0
$697$	−9.60754	−0.363912
$698$	0	0
$699$	11.9520	0.452066
$700$	0	0
$701$	−3.21461	−0.121414	−0.0607071	−	0.998156i	$-0.519336\pi$
$701$	−3.21461	−0.121414	−0.0607071	+	0.998156i	$0.519336\pi$
$702$	0	0
$703$	53.0249	1.99987
$704$	0	0
$705$	10.7058	0.403204
$706$	0	0
$707$	−5.83548	−0.219466
$708$	0	0
$709$	5.63120	0.211484	0.105742	−	0.994394i	$-0.466278\pi$
$709$	5.63120	0.211484	0.105742	+	0.994394i	$0.466278\pi$
$710$	0	0
$711$	−0.0877912	−0.00329243
$712$	0	0
$713$	11.9338	0.446925
$714$	0	0
$715$	−3.03693	−0.113575
$716$	0	0
$717$	20.5137	0.766096
$718$	0	0
$719$	−27.2458	−1.01610	−0.508048	−	0.861329i	$-0.669633\pi$
$719$	−27.2458	−1.01610	−0.508048	+	0.861329i	$0.669633\pi$
$720$	0	0
$721$	9.90799	0.368993
$722$	0	0
$723$	−33.1096	−1.23136
$724$	0	0
$725$	−0.406561	−0.0150993
$726$	0	0
$727$	13.0626	0.484463	0.242232	−	0.970218i	$-0.422121\pi$
$727$	13.0626	0.484463	0.242232	+	0.970218i	$0.422121\pi$
$728$	0	0
$729$	27.6098	1.02258
$730$	0	0
$731$	−1.30892	−0.0484121
$732$	0	0
$733$	22.5648	0.833449	0.416724	−	0.909033i	$-0.363178\pi$
$733$	22.5648	0.833449	0.416724	+	0.909033i	$0.363178\pi$
$734$	0	0
$735$	4.28773	0.158155
$736$	0	0
$737$	20.9946	0.773348
$738$	0	0
$739$	−10.2180	−0.375877	−0.187938	−	0.982181i	$-0.560181\pi$
$739$	−10.2180	−0.375877	−0.187938	+	0.982181i	$0.560181\pi$
$740$	0	0
$741$	−6.38416	−0.234528
$742$	0	0
$743$	−9.52705	−0.349513	−0.174757	−	0.984612i	$-0.555914\pi$
$743$	−9.52705	−0.349513	−0.174757	+	0.984612i	$0.555914\pi$
$744$	0	0
$745$	−60.3572	−2.21132
$746$	0	0
$747$	−1.25591	−0.0459512
$748$	0	0
$749$	13.1945	0.482118
$750$	0	0
$751$	18.6623	0.680999	0.340499	−	0.940245i	$-0.389404\pi$
$751$	18.6623	0.680999	0.340499	+	0.940245i	$0.389404\pi$
$752$	0	0
$753$	−23.3625	−0.851378
$754$	0	0
$755$	−37.3188	−1.35817
$756$	0	0
$757$	2.79325	0.101522	0.0507612	−	0.998711i	$-0.483835\pi$
$757$	2.79325	0.101522	0.0507612	+	0.998711i	$0.483835\pi$
$758$	0	0
$759$	−6.91421	−0.250970
$760$	0	0
$761$	−38.4932	−1.39538	−0.697688	−	0.716402i	$-0.745788\pi$
$761$	−38.4932	−1.39538	−0.697688	+	0.716402i	$0.745788\pi$
$762$	0	0
$763$	−7.57782	−0.274336
$764$	0	0
$765$	0.178307	0.00644669
$766$	0	0
$767$	−1.65597	−0.0597935
$768$	0	0
$769$	−6.36882	−0.229666	−0.114833	−	0.993385i	$-0.536633\pi$
$769$	−6.36882	−0.229666	−0.114833	+	0.993385i	$0.536633\pi$
$770$	0	0
$771$	−7.74987	−0.279105
$772$	0	0
$773$	−21.1663	−0.761300	−0.380650	−	0.924719i	$-0.624300\pi$
$773$	−21.1663	−0.761300	−0.380650	+	0.924719i	$0.624300\pi$
$774$	0	0
$775$	−6.34393	−0.227881
$776$	0	0
$777$	17.5342	0.629037
$778$	0	0
$779$	49.7225	1.78149
$780$	0	0
$781$	−13.4662	−0.481859
$782$	0	0
$783$	1.67317	0.0597943
$784$	0	0
$785$	−34.9387	−1.24702
$786$	0	0
$787$	−25.1790	−0.897534	−0.448767	−	0.893649i	$-0.648137\pi$
$787$	−25.1790	−0.897534	−0.448767	+	0.893649i	$0.648137\pi$
$788$	0	0
$789$	−9.60265	−0.341863
$790$	0	0
$791$	12.9706	0.461181
$792$	0	0
$793$	−0.447006	−0.0158736
$794$	0	0
$795$	8.57168	0.304006
$796$	0	0
$797$	−40.6542	−1.44004	−0.720022	−	0.693951i	$-0.755868\pi$
$797$	−40.6542	−1.44004	−0.720022	+	0.693951i	$0.755868\pi$
$798$	0	0
$799$	−2.49684	−0.0883319
$800$	0	0
$801$	0.133572	0.00471953
$802$	0	0
$803$	3.25252	0.114779
$804$	0	0
$805$	−6.01918	−0.212148
$806$	0	0
$807$	−37.1868	−1.30904
$808$	0	0
$809$	−37.5327	−1.31958	−0.659789	−	0.751451i	$-0.729354\pi$
$809$	−37.5327	−1.31958	−0.659789	+	0.751451i	$0.729354\pi$
$810$	0	0
$811$	10.3230	0.362490	0.181245	−	0.983438i	$-0.441987\pi$
$811$	10.3230	0.362490	0.181245	+	0.983438i	$0.441987\pi$
$812$	0	0
$813$	28.8200	1.01076
$814$	0	0
$815$	30.7917	1.07859
$816$	0	0
$817$	6.77413	0.236997
$818$	0	0
$819$	0.0512984	0.00179251
$820$	0	0
$821$	49.2789	1.71984	0.859922	−	0.510425i	$-0.170512\pi$
$821$	49.2789	1.71984	0.859922	+	0.510425i	$0.170512\pi$
$822$	0	0
$823$	7.23440	0.252175	0.126088	−	0.992019i	$-0.459758\pi$
$823$	7.23440	0.252175	0.126088	+	0.992019i	$0.459758\pi$
$824$	0	0
$825$	3.67554	0.127966
$826$	0	0
$827$	−16.6133	−0.577699	−0.288850	−	0.957374i	$-0.593273\pi$
$827$	−16.6133	−0.577699	−0.288850	+	0.957374i	$0.593273\pi$
$828$	0	0
$829$	7.27515	0.252676	0.126338	−	0.991987i	$-0.459678\pi$
$829$	7.27515	0.252676	0.126338	+	0.991987i	$0.459678\pi$
$830$	0	0
$831$	−25.9307	−0.899527
$832$	0	0
$833$	−1.00000	−0.0346479
$834$	0	0
$835$	30.5288	1.05649
$836$	0	0
$837$	26.1080	0.902425
$838$	0	0
$839$	−7.47949	−0.258221	−0.129110	−	0.991630i	$-0.541212\pi$
$839$	−7.47949	−0.258221	−0.129110	+	0.991630i	$0.541212\pi$
$840$	0	0
$841$	−28.8987	−0.996506
$842$	0	0
$843$	1.22995	0.0423617
$844$	0	0
$845$	−31.2688	−1.07568
$846$	0	0
$847$	−8.17202	−0.280794
$848$	0	0
$849$	9.40521	0.322786
$850$	0	0
$851$	−24.6148	−0.843784
$852$	0	0
$853$	5.42256	0.185665	0.0928324	−	0.995682i	$-0.470408\pi$
$853$	5.42256	0.185665	0.0928324	+	0.995682i	$0.470408\pi$
$854$	0	0
$855$	−0.922801	−0.0315591
$856$	0	0
$857$	43.8806	1.49893	0.749466	−	0.662042i	$-0.230310\pi$
$857$	43.8806	1.49893	0.749466	+	0.662042i	$0.230310\pi$
$858$	0	0
$859$	−14.5861	−0.497670	−0.248835	−	0.968546i	$-0.580048\pi$
$859$	−14.5861	−0.497670	−0.248835	+	0.968546i	$0.580048\pi$
$860$	0	0
$861$	16.4422	0.560348
$862$	0	0
$863$	−10.6121	−0.361241	−0.180620	−	0.983553i	$-0.557811\pi$
$863$	−10.6121	−0.361241	−0.180620	+	0.983553i	$0.557811\pi$
$864$	0	0
$865$	−48.4075	−1.64590
$866$	0	0
$867$	1.71138	0.0581216
$868$	0	0
$869$	2.07445	0.0703707
$870$	0	0
$871$	−8.99884	−0.304914
$872$	0	0
$873$	−1.05729	−0.0357837
$874$	0	0
$875$	−9.32735	−0.315322
$876$	0	0
$877$	−33.5478	−1.13283	−0.566414	−	0.824121i	$-0.691670\pi$
$877$	−33.5478	−1.13283	−0.566414	+	0.824121i	$0.691670\pi$
$878$	0	0
$879$	6.52385	0.220044
$880$	0	0
$881$	3.90585	0.131591	0.0657957	−	0.997833i	$-0.479041\pi$
$881$	3.90585	0.131591	0.0657957	+	0.997833i	$0.479041\pi$
$882$	0	0
$883$	29.3052	0.986197	0.493099	−	0.869973i	$-0.335864\pi$
$883$	29.3052	0.986197	0.493099	+	0.869973i	$0.335864\pi$
$884$	0	0
$885$	9.85060	0.331124
$886$	0	0
$887$	−12.5083	−0.419987	−0.209994	−	0.977703i	$-0.567344\pi$
$887$	−12.5083	−0.419987	−0.209994	+	0.977703i	$0.567344\pi$
$888$	0	0
$889$	3.96845	0.133098
$890$	0	0
$891$	−14.7674	−0.494726
$892$	0	0
$893$	12.9220	0.432420
$894$	0	0
$895$	4.99821	0.167072
$896$	0	0
$897$	2.96361	0.0989519
$898$	0	0
$899$	1.58130	0.0527392
$900$	0	0
$901$	−1.99912	−0.0666002
$902$	0	0
$903$	2.24006	0.0745446
$904$	0	0
$905$	11.0965	0.368860
$906$	0	0
$907$	18.2568	0.606207	0.303103	−	0.952958i	$-0.401977\pi$
$907$	18.2568	0.606207	0.303103	+	0.952958i	$0.401977\pi$
$908$	0	0
$909$	0.415302	0.0137747
$910$	0	0
$911$	33.9347	1.12431	0.562154	−	0.827033i	$-0.309973\pi$
$911$	33.9347	1.12431	0.562154	+	0.827033i	$0.309973\pi$
$912$	0	0
$913$	29.6762	0.982140
$914$	0	0
$915$	2.65904	0.0879051
$916$	0	0
$917$	−14.6420	−0.483521
$918$	0	0
$919$	−4.06647	−0.134141	−0.0670703	−	0.997748i	$-0.521365\pi$
$919$	−4.06647	−0.134141	−0.0670703	+	0.997748i	$0.521365\pi$
$920$	0	0
$921$	−41.6219	−1.37149
$922$	0	0
$923$	5.77197	0.189987
$924$	0	0
$925$	13.0850	0.430233
$926$	0	0
$927$	−0.705135	−0.0231597
$928$	0	0
$929$	24.3400	0.798570	0.399285	−	0.916827i	$-0.369258\pi$
$929$	24.3400	0.798570	0.399285	+	0.916827i	$0.369258\pi$
$930$	0	0
$931$	5.17536	0.169615
$932$	0	0
$933$	2.57399	0.0842687
$934$	0	0
$935$	−4.21327	−0.137789
$936$	0	0
$937$	−30.0158	−0.980573	−0.490286	−	0.871561i	$-0.663108\pi$
$937$	−30.0158	−0.980573	−0.490286	+	0.871561i	$0.663108\pi$
$938$	0	0
$939$	−41.4014	−1.35108
$940$	0	0
$941$	−57.1921	−1.86441	−0.932204	−	0.361932i	$-0.882117\pi$
$941$	−57.1921	−1.86441	−0.932204	+	0.361932i	$0.882117\pi$
$942$	0	0
$943$	−23.0818	−0.751646
$944$	0	0
$945$	−13.1684	−0.428367
$946$	0	0
$947$	−5.20830	−0.169247	−0.0846236	−	0.996413i	$-0.526969\pi$
$947$	−5.20830	−0.169247	−0.0846236	+	0.996413i	$0.526969\pi$
$948$	0	0
$949$	−1.39411	−0.0452548
$950$	0	0
$951$	18.4749	0.599089
$952$	0	0
$953$	31.0876	1.00703	0.503514	−	0.863987i	$-0.332040\pi$
$953$	31.0876	1.00703	0.503514	+	0.863987i	$0.332040\pi$
$954$	0	0
$955$	30.6570	0.992037
$956$	0	0
$957$	−0.916169	−0.0296155
$958$	0	0
$959$	7.06322	0.228083
$960$	0	0
$961$	−6.32565	−0.204053
$962$	0	0
$963$	−0.939034	−0.0302599
$964$	0	0
$965$	48.7349	1.56883
$966$	0	0
$967$	42.6747	1.37232	0.686162	−	0.727449i	$-0.259294\pi$
$967$	42.6747	1.37232	0.686162	+	0.727449i	$0.259294\pi$
$968$	0	0
$969$	−8.85702	−0.284528
$970$	0	0
$971$	50.8041	1.63038	0.815191	−	0.579193i	$-0.196632\pi$
$971$	50.8041	1.63038	0.815191	+	0.579193i	$0.196632\pi$
$972$	0	0
$973$	4.61661	0.148002
$974$	0	0
$975$	−1.57543	−0.0504541
$976$	0	0
$977$	47.8633	1.53128	0.765642	−	0.643267i	$-0.222421\pi$
$977$	47.8633	1.53128	0.765642	+	0.643267i	$0.222421\pi$
$978$	0	0
$979$	−3.15622	−0.100873
$980$	0	0
$981$	0.539301	0.0172186
$982$	0	0
$983$	−38.0735	−1.21436	−0.607178	−	0.794566i	$-0.707699\pi$
$983$	−38.0735	−1.21436	−0.607178	+	0.794566i	$0.707699\pi$
$984$	0	0
$985$	−14.9277	−0.475636
$986$	0	0
$987$	4.27305	0.136013
$988$	0	0
$989$	−3.14463	−0.0999935
$990$	0	0
$991$	−1.40607	−0.0446653	−0.0223327	−	0.999751i	$-0.507109\pi$
$991$	−1.40607	−0.0446653	−0.0223327	+	0.999751i	$0.507109\pi$
$992$	0	0
$993$	−19.6406	−0.623277
$994$	0	0
$995$	46.8967	1.48672
$996$	0	0
$997$	−34.6376	−1.09698	−0.548492	−	0.836156i	$-0.684798\pi$
$997$	−34.6376	−1.09698	−0.548492	+	0.836156i	$0.684798\pi$
$998$	0	0
$999$	−53.8505	−1.70376

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7616.2.a.bz.1.5		6
4.3	odd	2		7616.2.a.by.1.2		6
8.3	odd	2		3808.2.a.k.1.5	✓	6
8.5	even	2		3808.2.a.l.1.2	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3808.2.a.k.1.5	✓	6	8.3	odd	2
3808.2.a.l.1.2	yes	6	8.5	even	2
7616.2.a.by.1.2		6	4.3	odd	2
7616.2.a.bz.1.5		6	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7616 = 2^{6} \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7616.a (trivial)

\(f(q)\)	\(=\)	\(q+1.71138 q^{3} +2.50542 q^{5} +1.00000 q^{7} -0.0711684 q^{9} +O(q^{10})\)	\(q+1.71138 q^{3} +2.50542 q^{5} +1.00000 q^{7} -0.0711684 q^{9} +1.68166 q^{11} -0.720803 q^{13} +4.28773 q^{15} -1.00000 q^{17} +5.17536 q^{19} +1.71138 q^{21} -2.40246 q^{23} +1.27713 q^{25} -5.25595 q^{27} -0.318339 q^{29} -4.96733 q^{31} +2.87797 q^{33} +2.50542 q^{35} +10.2456 q^{37} -1.23357 q^{39} +9.60754 q^{41} +1.30892 q^{43} -0.178307 q^{45} +2.49684 q^{47} +1.00000 q^{49} -1.71138 q^{51} +1.99912 q^{53} +4.21327 q^{55} +8.85702 q^{57} +2.29739 q^{59} +0.620150 q^{61} -0.0711684 q^{63} -1.80591 q^{65} +12.4845 q^{67} -4.11154 q^{69} -8.00769 q^{71} +1.93411 q^{73} +2.18566 q^{75} +1.68166 q^{77} +1.23357 q^{79} -8.78143 q^{81} +17.6470 q^{83} -2.50542 q^{85} -0.544800 q^{87} -1.87684 q^{89} -0.720803 q^{91} -8.50100 q^{93} +12.9664 q^{95} +14.8561 q^{97} -0.119681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{5} + 6 q^{7} + 8 q^{9}+O(q^{10}) \) 6 * q - 4 * q^5 + 6 * q^7 + 8 * q^9	\( 6 q - 4 q^{5} + 6 q^{7} + 8 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} - 6 q^{17} + 6 q^{19} + 8 q^{25} - 8 q^{29} + 12 q^{31} + 28 q^{33} - 4 q^{35} - 10 q^{37} + 14 q^{39} + 14 q^{41} + 12 q^{43} - 22 q^{45} + 2 q^{47} + 6 q^{49} - 26 q^{53} + 10 q^{55} + 22 q^{57} - 22 q^{59} - 2 q^{61} + 8 q^{63} + 8 q^{65} + 14 q^{67} - 14 q^{69} - 4 q^{71} + 18 q^{73} + 28 q^{75} + 4 q^{77} - 14 q^{79} - 6 q^{81} - 8 q^{83} + 4 q^{85} + 28 q^{87} + 6 q^{89} + 4 q^{91} - 8 q^{93} - 2 q^{95} + 20 q^{97} + 26 q^{99}+O(q^{100}) \) 6 * q - 4 * q^5 + 6 * q^7 + 8 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^15 - 6 * q^17 + 6 * q^19 + 8 * q^25 - 8 * q^29 + 12 * q^31 + 28 * q^33 - 4 * q^35 - 10 * q^37 + 14 * q^39 + 14 * q^41 + 12 * q^43 - 22 * q^45 + 2 * q^47 + 6 * q^49 - 26 * q^53 + 10 * q^55 + 22 * q^57 - 22 * q^59 - 2 * q^61 + 8 * q^63 + 8 * q^65 + 14 * q^67 - 14 * q^69 - 4 * q^71 + 18 * q^73 + 28 * q^75 + 4 * q^77 - 14 * q^79 - 6 * q^81 - 8 * q^83 + 4 * q^85 + 28 * q^87 + 6 * q^89 + 4 * q^91 - 8 * q^93 - 2 * q^95 + 20 * q^97 + 26 * q^99

Introduction

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