LMFDB - Embedded newform 7700.2.a.bb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7700,2,Mod(1,7700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7700, 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("7700.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7700 = 2^{2} \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7700.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:	$61.4848095564$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.111028.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 10x^{2} - 2x + 4 $ x^4 - 10x^2 - 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1540)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.766757$ of defining polynomial
Character	$\chi$	$=$	7700.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.766757 q^{3} +1.00000 q^{7} -2.41208 q^{9} +O(q^{10})$	$q-0.766757 q^{3} +1.00000 q^{7} -2.41208 q^{9} +1.00000 q^{11} +4.45002 q^{13} -2.76676 q^{17} +7.21678 q^{19} -0.766757 q^{21} -7.09535 q^{23} +4.14975 q^{27} +2.00000 q^{29} +4.64533 q^{31} -0.766757 q^{33} -11.0953 q^{37} -3.41208 q^{39} +8.17884 q^{41} -10.6289 q^{43} +4.91651 q^{47} +1.00000 q^{49} +2.12143 q^{51} +10.1624 q^{53} -5.53351 q^{57} +3.11181 q^{59} +8.57145 q^{61} -2.41208 q^{63} +12.6289 q^{67} +5.44041 q^{69} -7.36653 q^{71} -5.23324 q^{73} +1.00000 q^{77} -15.3792 q^{79} +4.05440 q^{81} +5.14090 q^{83} -1.53351 q^{87} -14.5074 q^{89} +4.45002 q^{91} -3.56184 q^{93} -3.68327 q^{97} -2.41208 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q + 4 * q^7 + 8 * q^9	$ 4 q + 4 q^{7} + 8 q^{9} + 4 q^{11} - 2 q^{13} - 8 q^{17} + 6 q^{19} + 6 q^{23} + 6 q^{27} + 8 q^{29} + 4 q^{31} - 10 q^{37} + 4 q^{39} + 12 q^{41} - 2 q^{43} + 6 q^{47} + 4 q^{49} + 20 q^{51} - 6 q^{53} - 16 q^{57} + 4 q^{59} + 26 q^{61} + 8 q^{63} + 10 q^{67} - 18 q^{69} + 4 q^{71} - 24 q^{73} + 4 q^{77} + 8 q^{79} + 40 q^{81} + 2 q^{83} - 6 q^{89} - 2 q^{91} + 14 q^{93} + 2 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q + 4 * q^7 + 8 * q^9 + 4 * q^11 - 2 * q^13 - 8 * q^17 + 6 * q^19 + 6 * q^23 + 6 * q^27 + 8 * q^29 + 4 * q^31 - 10 * q^37 + 4 * q^39 + 12 * q^41 - 2 * q^43 + 6 * q^47 + 4 * q^49 + 20 * q^51 - 6 * q^53 - 16 * q^57 + 4 * q^59 + 26 * q^61 + 8 * q^63 + 10 * q^67 - 18 * q^69 + 4 * q^71 - 24 * q^73 + 4 * q^77 + 8 * q^79 + 40 * q^81 + 2 * q^83 - 6 * q^89 - 2 * q^91 + 14 * q^93 + 2 * q^97 + 8 * 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
$3$	−0.766757	−0.442687	−0.221344	−	0.975196i	$-0.571044\pi$
$3$	−0.766757	−0.442687	−0.221344	+	0.975196i	$0.571044\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.41208	−0.804028
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.45002	1.23421	0.617107	−	0.786879i	$-0.288305\pi$
$13$	4.45002	1.23421	0.617107	+	0.786879i	$0.288305\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.76676	−0.671037	−0.335519	−	0.942034i	$-0.608912\pi$
$17$	−2.76676	−0.671037	−0.335519	+	0.942034i	$0.608912\pi$
$18$	0	0
$19$	7.21678	1.65564	0.827821	−	0.560992i	$-0.189580\pi$
$19$	7.21678	1.65564	0.827821	+	0.560992i	$0.189580\pi$
$20$	0	0
$21$	−0.766757	−0.167320
$22$	0	0
$23$	−7.09535	−1.47948	−0.739741	−	0.672891i	$-0.765052\pi$
$23$	−7.09535	−1.47948	−0.739741	+	0.672891i	$0.765052\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.14975	0.798620
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.64533	0.834325	0.417163	−	0.908832i	$-0.363025\pi$
$31$	4.64533	0.834325	0.417163	+	0.908832i	$0.363025\pi$
$32$	0	0
$33$	−0.766757	−0.133475
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.0953	−1.82406	−0.912032	−	0.410119i	$-0.865487\pi$
$37$	−11.0953	−1.82406	−0.912032	+	0.410119i	$0.865487\pi$
$38$	0	0
$39$	−3.41208	−0.546371
$40$	0	0
$41$	8.17884	1.27732	0.638660	−	0.769489i	$-0.279489\pi$
$41$	8.17884	1.27732	0.638660	+	0.769489i	$0.279489\pi$
$42$	0	0
$43$	−10.6289	−1.62089	−0.810443	−	0.585817i	$-0.800774\pi$
$43$	−10.6289	−1.62089	−0.810443	+	0.585817i	$0.800774\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.91651	0.717146	0.358573	−	0.933502i	$-0.383263\pi$
$47$	4.91651	0.717146	0.358573	+	0.933502i	$0.383263\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	2.12143	0.297060
$52$	0	0
$53$	10.1624	1.39591	0.697955	−	0.716142i	$-0.254094\pi$
$53$	10.1624	1.39591	0.697955	+	0.716142i	$0.254094\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.53351	−0.732932
$58$	0	0
$59$	3.11181	0.405124	0.202562	−	0.979269i	$-0.435073\pi$
$59$	3.11181	0.405124	0.202562	+	0.979269i	$0.435073\pi$
$60$	0	0
$61$	8.57145	1.09746	0.548731	−	0.835999i	$-0.315111\pi$
$61$	8.57145	1.09746	0.548731	+	0.835999i	$0.315111\pi$
$62$	0	0
$63$	−2.41208	−0.303894
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.6289	1.54286	0.771431	−	0.636313i	$-0.219542\pi$
$67$	12.6289	1.54286	0.771431	+	0.636313i	$0.219542\pi$
$68$	0	0
$69$	5.44041	0.654948
$70$	0	0
$71$	−7.36653	−0.874246	−0.437123	−	0.899402i	$-0.644003\pi$
$71$	−7.36653	−0.874246	−0.437123	+	0.899402i	$0.644003\pi$
$72$	0	0
$73$	−5.23324	−0.612505	−0.306252	−	0.951950i	$-0.599075\pi$
$73$	−5.23324	−0.612505	−0.306252	+	0.951950i	$0.599075\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−15.3792	−1.73029	−0.865145	−	0.501522i	$-0.832774\pi$
$79$	−15.3792	−1.73029	−0.865145	+	0.501522i	$0.832774\pi$
$80$	0	0
$81$	4.05440	0.450489
$82$	0	0
$83$	5.14090	0.564287	0.282144	−	0.959372i	$-0.408955\pi$
$83$	5.14090	0.564287	0.282144	+	0.959372i	$0.408955\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.53351	−0.164410
$88$	0	0
$89$	−14.5074	−1.53778	−0.768892	−	0.639378i	$-0.779192\pi$
$89$	−14.5074	−1.53778	−0.768892	+	0.639378i	$0.779192\pi$
$90$	0	0
$91$	4.45002	0.466489
$92$	0	0
$93$	−3.56184	−0.369345
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.68327	−0.373979	−0.186989	−	0.982362i	$-0.559873\pi$
$97$	−3.68327	−0.373979	−0.186989	+	0.982362i	$0.559873\pi$
$98$	0	0
$99$	−2.41208	−0.242424
$100$	0	0
$101$	7.57830	0.754069	0.377035	−	0.926199i	$-0.376944\pi$
$101$	7.57830	0.754069	0.377035	+	0.926199i	$0.376944\pi$
$102$	0	0
$103$	13.6668	1.34663	0.673315	−	0.739356i	$-0.264870\pi$
$103$	13.6668	1.34663	0.673315	+	0.739356i	$0.264870\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.82417	0.659717	0.329859	−	0.944030i	$-0.392999\pi$
$107$	6.82417	0.659717	0.329859	+	0.944030i	$0.392999\pi$
$108$	0	0
$109$	0.466487	0.0446813	0.0223407	−	0.999750i	$-0.492888\pi$
$109$	0.466487	0.0446813	0.0223407	+	0.999750i	$0.492888\pi$
$110$	0	0
$111$	8.50743	0.807490
$112$	0	0
$113$	−18.6572	−1.75512	−0.877560	−	0.479467i	$-0.840830\pi$
$113$	−18.6572	−1.75512	−0.877560	+	0.479467i	$0.840830\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−10.7338	−0.992343
$118$	0	0
$119$	−2.76676	−0.253628
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−6.27118	−0.565453
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−4.90004	−0.434809	−0.217404	−	0.976082i	$-0.569759\pi$
$127$	−4.90004	−0.434809	−0.217404	+	0.976082i	$0.569759\pi$
$128$	0	0
$129$	8.14975	0.717546
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	7.21678	0.625774
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−10.1624	−0.868230	−0.434115	−	0.900857i	$-0.642939\pi$
$137$	−10.1624	−0.868230	−0.434115	+	0.900857i	$0.642939\pi$
$138$	0	0
$139$	−12.9000	−1.09417	−0.547084	−	0.837078i	$-0.684262\pi$
$139$	−12.9000	−1.09417	−0.547084	+	0.837078i	$0.684262\pi$
$140$	0	0
$141$	−3.76977	−0.317472
$142$	0	0
$143$	4.45002	0.372130
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−0.766757	−0.0632410
$148$	0	0
$149$	−7.14290	−0.585169	−0.292585	−	0.956240i	$-0.594515\pi$
$149$	−7.14290	−0.585169	−0.292585	+	0.956240i	$0.594515\pi$
$150$	0	0
$151$	15.1692	1.23445	0.617227	−	0.786785i	$-0.288256\pi$
$151$	15.1692	1.23445	0.617227	+	0.786785i	$0.288256\pi$
$152$	0	0
$153$	6.67365	0.539533
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.5833	1.32349	0.661746	−	0.749728i	$-0.269816\pi$
$157$	16.5833	1.32349	0.661746	+	0.749728i	$0.269816\pi$
$158$	0	0
$159$	−7.79207	−0.617951
$160$	0	0
$161$	−7.09535	−0.559192
$162$	0	0
$163$	19.4530	1.52368	0.761839	−	0.647766i	$-0.224296\pi$
$163$	19.4530	1.52368	0.761839	+	0.647766i	$0.224296\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.6503	−1.05629	−0.528147	−	0.849153i	$-0.677113\pi$
$167$	−13.6503	−1.05629	−0.528147	+	0.849153i	$0.677113\pi$
$168$	0	0
$169$	6.80270	0.523284
$170$	0	0
$171$	−17.4075	−1.33118
$172$	0	0
$173$	10.3761	0.788884	0.394442	−	0.918921i	$-0.370938\pi$
$173$	10.3761	0.788884	0.394442	+	0.918921i	$0.370938\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.38600	−0.179343
$178$	0	0
$179$	24.6243	1.84050	0.920252	−	0.391327i	$-0.127984\pi$
$179$	24.6243	1.84050	0.920252	+	0.391327i	$0.127984\pi$
$180$	0	0
$181$	0.283805	0.0210951	0.0105475	−	0.999944i	$-0.496643\pi$
$181$	0.283805	0.0210951	0.0105475	+	0.999944i	$0.496643\pi$
$182$	0	0
$183$	−6.57222	−0.485832
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.76676	−0.202325
$188$	0	0
$189$	4.14975	0.301850
$190$	0	0
$191$	13.8330	1.00092	0.500461	−	0.865759i	$-0.333164\pi$
$191$	13.8330	1.00092	0.500461	+	0.865759i	$0.333164\pi$
$192$	0	0
$193$	13.6959	0.985852	0.492926	−	0.870071i	$-0.335927\pi$
$193$	13.6959	0.985852	0.492926	+	0.870071i	$0.335927\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	5.99539	0.427154	0.213577	−	0.976926i	$-0.431489\pi$
$197$	5.99539	0.427154	0.213577	+	0.976926i	$0.431489\pi$
$198$	0	0
$199$	−4.49558	−0.318683	−0.159341	−	0.987224i	$-0.550937\pi$
$199$	−4.49558	−0.318683	−0.159341	+	0.987224i	$0.550937\pi$
$200$	0	0
$201$	−9.68327	−0.683005
$202$	0	0
$203$	2.00000	0.140372
$204$	0	0
$205$	0	0
$206$	0	0
$207$	17.1146	1.18955
$208$	0	0
$209$	7.21678	0.499195
$210$	0	0
$211$	9.83302	0.676933	0.338466	−	0.940978i	$-0.390092\pi$
$211$	9.83302	0.676933	0.338466	+	0.940978i	$0.390092\pi$
$212$	0	0
$213$	5.64834	0.387018
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4.64533	0.315345
$218$	0	0
$219$	4.01262	0.271148
$220$	0	0
$221$	−12.3121	−0.828203
$222$	0	0
$223$	16.8816	1.13047	0.565237	−	0.824928i	$-0.308785\pi$
$223$	16.8816	1.13047	0.565237	+	0.824928i	$0.308785\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−7.36653	−0.488934	−0.244467	−	0.969658i	$-0.578613\pi$
$227$	−7.36653	−0.488934	−0.244467	+	0.969658i	$0.578613\pi$
$228$	0	0
$229$	20.1578	1.33206	0.666031	−	0.745924i	$-0.267992\pi$
$229$	20.1578	1.33206	0.666031	+	0.745924i	$0.267992\pi$
$230$	0	0
$231$	−0.766757	−0.0504489
$232$	0	0
$233$	−12.8525	−0.841995	−0.420997	−	0.907062i	$-0.638320\pi$
$233$	−12.8525	−0.841995	−0.420997	+	0.907062i	$0.638320\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	11.7921	0.765977
$238$	0	0
$239$	−22.5484	−1.45853	−0.729267	−	0.684230i	$-0.760139\pi$
$239$	−22.5484	−1.45853	−0.729267	+	0.684230i	$0.760139\pi$
$240$	0	0
$241$	30.2218	1.94676	0.973378	−	0.229205i	$-0.0736126\pi$
$241$	30.2218	1.94676	0.973378	+	0.229205i	$0.0736126\pi$
$242$	0	0
$243$	−15.5580	−0.998046
$244$	0	0
$245$	0	0
$246$	0	0
$247$	32.1148	2.04342
$248$	0	0
$249$	−3.94182	−0.249803
$250$	0	0
$251$	−16.6124	−1.04857	−0.524283	−	0.851544i	$-0.675666\pi$
$251$	−16.6124	−1.04857	−0.524283	+	0.851544i	$0.675666\pi$
$252$	0	0
$253$	−7.09535	−0.446081
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.97392	0.310265	0.155132	−	0.987894i	$-0.450420\pi$
$257$	4.97392	0.310265	0.155132	+	0.987894i	$0.450420\pi$
$258$	0	0
$259$	−11.0953	−0.689431
$260$	0	0
$261$	−4.82417	−0.298609
$262$	0	0
$263$	−20.9000	−1.28875	−0.644376	−	0.764709i	$-0.722883\pi$
$263$	−20.9000	−1.28875	−0.644376	+	0.764709i	$0.722883\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	11.1237	0.680758
$268$	0	0
$269$	12.8262	0.782025	0.391013	−	0.920385i	$-0.372125\pi$
$269$	12.8262	0.782025	0.391013	+	0.920385i	$0.372125\pi$
$270$	0	0
$271$	0.600540	0.0364802	0.0182401	−	0.999834i	$-0.494194\pi$
$271$	0.600540	0.0364802	0.0182401	+	0.999834i	$0.494194\pi$
$272$	0	0
$273$	−3.41208	−0.206509
$274$	0	0
$275$	0	0
$276$	0	0
$277$	3.83762	0.230580	0.115290	−	0.993332i	$-0.463220\pi$
$277$	3.83762	0.230580	0.115290	+	0.993332i	$0.463220\pi$
$278$	0	0
$279$	−11.2049	−0.670821
$280$	0	0
$281$	26.9582	1.60819	0.804096	−	0.594499i	$-0.202650\pi$
$281$	26.9582	1.60819	0.804096	+	0.594499i	$0.202650\pi$
$282$	0	0
$283$	3.60939	0.214556	0.107278	−	0.994229i	$-0.465787\pi$
$283$	3.60939	0.214556	0.107278	+	0.994229i	$0.465787\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	8.17884	0.482782
$288$	0	0
$289$	−9.34506	−0.549709
$290$	0	0
$291$	2.82417	0.165556
$292$	0	0
$293$	−6.37615	−0.372498	−0.186249	−	0.982503i	$-0.559633\pi$
$293$	−6.37615	−0.372498	−0.186249	+	0.982503i	$0.559633\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	4.14975	0.240793
$298$	0	0
$299$	−31.5745	−1.82600
$300$	0	0
$301$	−10.6289	−0.612637
$302$	0	0
$303$	−5.81071	−0.333817
$304$	0	0
$305$	0	0
$306$	0	0
$307$	7.45964	0.425744	0.212872	−	0.977080i	$-0.431718\pi$
$307$	7.45964	0.425744	0.212872	+	0.977080i	$0.431718\pi$
$308$	0	0
$309$	−10.4791	−0.596136
$310$	0	0
$311$	−23.2286	−1.31718	−0.658588	−	0.752504i	$-0.728846\pi$
$311$	−23.2286	−1.31718	−0.658588	+	0.752504i	$0.728846\pi$
$312$	0	0
$313$	28.8242	1.62924	0.814619	−	0.579996i	$-0.196946\pi$
$313$	28.8242	1.62924	0.814619	+	0.579996i	$0.196946\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	30.0819	1.68957	0.844784	−	0.535108i	$-0.179729\pi$
$317$	30.0819	1.68957	0.844784	+	0.535108i	$0.179729\pi$
$318$	0	0
$319$	2.00000	0.111979
$320$	0	0
$321$	−5.23248	−0.292048
$322$	0	0
$323$	−19.9671	−1.11100
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−0.357682	−0.0197798
$328$	0	0
$329$	4.91651	0.271056
$330$	0	0
$331$	−5.77637	−0.317498	−0.158749	−	0.987319i	$-0.550746\pi$
$331$	−5.77637	−0.317498	−0.158749	+	0.987319i	$0.550746\pi$
$332$	0	0
$333$	26.7629	1.46660
$334$	0	0
$335$	0	0
$336$	0	0
$337$	8.73767	0.475971	0.237986	−	0.971269i	$-0.423513\pi$
$337$	8.73767	0.475971	0.237986	+	0.971269i	$0.423513\pi$
$338$	0	0
$339$	14.3055	0.776969
$340$	0	0
$341$	4.64533	0.251559
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.8717	0.583625	0.291812	−	0.956476i	$-0.405742\pi$
$347$	10.8717	0.583625	0.291812	+	0.956476i	$0.405742\pi$
$348$	0	0
$349$	−15.7533	−0.843255	−0.421627	−	0.906769i	$-0.638541\pi$
$349$	−15.7533	−0.843255	−0.421627	+	0.906769i	$0.638541\pi$
$350$	0	0
$351$	18.4665	0.985668
$352$	0	0
$353$	−1.60939	−0.0856591	−0.0428296	−	0.999082i	$-0.513637\pi$
$353$	−1.60939	−0.0856591	−0.0428296	+	0.999082i	$0.513637\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	2.12143	0.112278
$358$	0	0
$359$	0.945598	0.0499067	0.0249534	−	0.999689i	$-0.492056\pi$
$359$	0.945598	0.0499067	0.0249534	+	0.999689i	$0.492056\pi$
$360$	0	0
$361$	33.0819	1.74115
$362$	0	0
$363$	−0.766757	−0.0402443
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11.4412	0.597224	0.298612	−	0.954375i	$-0.403476\pi$
$367$	11.4412	0.597224	0.298612	+	0.954375i	$0.403476\pi$
$368$	0	0
$369$	−19.7281	−1.02700
$370$	0	0
$371$	10.1624	0.527604
$372$	0	0
$373$	−21.2102	−1.09822	−0.549111	−	0.835750i	$-0.685033\pi$
$373$	−21.2102	−1.09822	−0.549111	+	0.835750i	$0.685033\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.90004	0.458376
$378$	0	0
$379$	13.8001	0.708863	0.354431	−	0.935082i	$-0.384674\pi$
$379$	13.8001	0.708863	0.354431	+	0.935082i	$0.384674\pi$
$380$	0	0
$381$	3.75714	0.192484
$382$	0	0
$383$	−2.05741	−0.105129	−0.0525644	−	0.998618i	$-0.516739\pi$
$383$	−2.05741	−0.105129	−0.0525644	+	0.998618i	$0.516739\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	25.6377	1.30324
$388$	0	0
$389$	33.4247	1.69470	0.847350	−	0.531035i	$-0.178197\pi$
$389$	33.4247	1.69470	0.847350	+	0.531035i	$0.178197\pi$
$390$	0	0
$391$	19.6311	0.992788
$392$	0	0
$393$	−3.06703	−0.154711
$394$	0	0
$395$	0	0
$396$	0	0
$397$	34.1750	1.71519	0.857597	−	0.514322i	$-0.171956\pi$
$397$	34.1750	1.71519	0.857597	+	0.514322i	$0.171956\pi$
$398$	0	0
$399$	−5.53351	−0.277022
$400$	0	0
$401$	6.78522	0.338838	0.169419	−	0.985544i	$-0.445811\pi$
$401$	6.78522	0.338838	0.169419	+	0.985544i	$0.445811\pi$
$402$	0	0
$403$	20.6718	1.02974
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11.0953	−0.549976
$408$	0	0
$409$	−28.0199	−1.38549	−0.692747	−	0.721181i	$-0.743600\pi$
$409$	−28.0199	−1.38549	−0.692747	+	0.721181i	$0.743600\pi$
$410$	0	0
$411$	7.79207	0.384354
$412$	0	0
$413$	3.11181	0.153122
$414$	0	0
$415$	0	0
$416$	0	0
$417$	9.89120	0.484374
$418$	0	0
$419$	−37.1198	−1.81342	−0.906711	−	0.421752i	$-0.861415\pi$
$419$	−37.1198	−1.81342	−0.906711	+	0.421752i	$0.861415\pi$
$420$	0	0
$421$	15.7308	0.766673	0.383337	−	0.923609i	$-0.374775\pi$
$421$	15.7308	0.766673	0.383337	+	0.923609i	$0.374775\pi$
$422$	0	0
$423$	−11.8590	−0.576606
$424$	0	0
$425$	0	0
$426$	0	0
$427$	8.57145	0.414802
$428$	0	0
$429$	−3.41208	−0.164737
$430$	0	0
$431$	−12.0126	−0.578628	−0.289314	−	0.957234i	$-0.593427\pi$
$431$	−12.0126	−0.578628	−0.289314	+	0.957234i	$0.593427\pi$
$432$	0	0
$433$	1.09996	0.0528605	0.0264303	−	0.999651i	$-0.491586\pi$
$433$	1.09996	0.0528605	0.0264303	+	0.999651i	$0.491586\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−51.2056	−2.44949
$438$	0	0
$439$	19.5765	0.934333	0.467167	−	0.884169i	$-0.345275\pi$
$439$	19.5765	0.934333	0.467167	+	0.884169i	$0.345275\pi$
$440$	0	0
$441$	−2.41208	−0.114861
$442$	0	0
$443$	−0.514040	−0.0244228	−0.0122114	−	0.999925i	$-0.503887\pi$
$443$	−0.514040	−0.0244228	−0.0122114	+	0.999925i	$0.503887\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	5.47687	0.259047
$448$	0	0
$449$	18.6461	0.879964	0.439982	−	0.898007i	$-0.354985\pi$
$449$	18.6461	0.879964	0.439982	+	0.898007i	$0.354985\pi$
$450$	0	0
$451$	8.17884	0.385127
$452$	0	0
$453$	−11.6311	−0.546477
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.1283	0.614115	0.307057	−	0.951691i	$-0.400656\pi$
$457$	13.1283	0.614115	0.307057	+	0.951691i	$0.400656\pi$
$458$	0	0
$459$	−11.4814	−0.535904
$460$	0	0
$461$	40.8942	1.90463	0.952316	−	0.305112i	$-0.0986939\pi$
$461$	40.8942	1.90463	0.952316	+	0.305112i	$0.0986939\pi$
$462$	0	0
$463$	28.7806	1.33755	0.668774	−	0.743466i	$-0.266819\pi$
$463$	28.7806	1.33755	0.668774	+	0.743466i	$0.266819\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−5.09919	−0.235962	−0.117981	−	0.993016i	$-0.537642\pi$
$467$	−5.09919	−0.235962	−0.117981	+	0.993016i	$0.537642\pi$
$468$	0	0
$469$	12.6289	0.583147
$470$	0	0
$471$	−12.7154	−0.585893
$472$	0	0
$473$	−10.6289	−0.488716
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−24.5125	−1.12235
$478$	0	0
$479$	−40.1750	−1.83564	−0.917821	−	0.396994i	$-0.870053\pi$
$479$	−40.1750	−1.83564	−0.917821	+	0.396994i	$0.870053\pi$
$480$	0	0
$481$	−49.3745	−2.25129
$482$	0	0
$483$	5.44041	0.247547
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−6.94760	−0.314826	−0.157413	−	0.987533i	$-0.550315\pi$
$487$	−6.94760	−0.314826	−0.157413	+	0.987533i	$0.550315\pi$
$488$	0	0
$489$	−14.9157	−0.674513
$490$	0	0
$491$	37.2451	1.68085	0.840424	−	0.541930i	$-0.182306\pi$
$491$	37.2451	1.68085	0.840424	+	0.541930i	$0.182306\pi$
$492$	0	0
$493$	−5.53351	−0.249217
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−7.36653	−0.330434
$498$	0	0
$499$	13.8912	0.621855	0.310928	−	0.950434i	$-0.399360\pi$
$499$	13.8912	0.621855	0.310928	+	0.950434i	$0.399360\pi$
$500$	0	0
$501$	10.4665	0.467608
$502$	0	0
$503$	4.99315	0.222634	0.111317	−	0.993785i	$-0.464493\pi$
$503$	4.99315	0.222634	0.111317	+	0.993785i	$0.464493\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−5.21601	−0.231651
$508$	0	0
$509$	−25.0341	−1.10962	−0.554808	−	0.831978i	$-0.687208\pi$
$509$	−25.0341	−1.10962	−0.554808	+	0.831978i	$0.687208\pi$
$510$	0	0
$511$	−5.23324	−0.231505
$512$	0	0
$513$	29.9478	1.32223
$514$	0	0
$515$	0	0
$516$	0	0
$517$	4.91651	0.216228
$518$	0	0
$519$	−7.95598	−0.349229
$520$	0	0
$521$	30.5656	1.33910	0.669552	−	0.742765i	$-0.266486\pi$
$521$	30.5656	1.33910	0.669552	+	0.742765i	$0.266486\pi$
$522$	0	0
$523$	17.7414	0.775779	0.387890	−	0.921706i	$-0.373204\pi$
$523$	17.7414	0.775779	0.387890	+	0.921706i	$0.373204\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−12.8525	−0.559863
$528$	0	0
$529$	27.3440	1.18887
$530$	0	0
$531$	−7.50596	−0.325731
$532$	0	0
$533$	36.3960	1.57649
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−18.8808	−0.814767
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−12.0759	−0.519182	−0.259591	−	0.965719i	$-0.583588\pi$
$541$	−12.0759	−0.519182	−0.259591	+	0.965719i	$0.583588\pi$
$542$	0	0
$543$	−0.217610	−0.00933853
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−41.8583	−1.78973	−0.894865	−	0.446337i	$-0.852728\pi$
$547$	−41.8583	−1.78973	−0.894865	+	0.446337i	$0.852728\pi$
$548$	0	0
$549$	−20.6751	−0.882390
$550$	0	0
$551$	14.4336	0.614890
$552$	0	0
$553$	−15.3792	−0.653988
$554$	0	0
$555$	0	0
$556$	0	0
$557$	17.2431	0.730614	0.365307	−	0.930887i	$-0.380964\pi$
$557$	17.2431	0.730614	0.365307	+	0.930887i	$0.380964\pi$
$558$	0	0
$559$	−47.2987	−2.00052
$560$	0	0
$561$	2.12143	0.0895668
$562$	0	0
$563$	22.1578	0.933839	0.466919	−	0.884300i	$-0.345364\pi$
$563$	22.1578	0.933839	0.466919	+	0.884300i	$0.345364\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.05440	0.170269
$568$	0	0
$569$	13.2517	0.555540	0.277770	−	0.960648i	$-0.410405\pi$
$569$	13.2517	0.555540	0.277770	+	0.960648i	$0.410405\pi$
$570$	0	0
$571$	−33.2814	−1.39278	−0.696392	−	0.717661i	$-0.745213\pi$
$571$	−33.2814	−1.39278	−0.696392	+	0.717661i	$0.745213\pi$
$572$	0	0
$573$	−10.6066	−0.443095
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.3597	1.43041	0.715206	−	0.698914i	$-0.246333\pi$
$577$	34.3597	1.43041	0.715206	+	0.698914i	$0.246333\pi$
$578$	0	0
$579$	−10.5014	−0.436424
$580$	0	0
$581$	5.14090	0.213281
$582$	0	0
$583$	10.1624	0.420883
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.60862	0.0663950	0.0331975	−	0.999449i	$-0.489431\pi$
$587$	1.60862	0.0663950	0.0331975	+	0.999449i	$0.489431\pi$
$588$	0	0
$589$	33.5243	1.38134
$590$	0	0
$591$	−4.59701	−0.189096
$592$	0	0
$593$	20.5319	0.843145	0.421572	−	0.906795i	$-0.361478\pi$
$593$	20.5319	0.843145	0.421572	+	0.906795i	$0.361478\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	3.44701	0.141077
$598$	0	0
$599$	−19.4232	−0.793609	−0.396805	−	0.917903i	$-0.629881\pi$
$599$	−19.4232	−0.793609	−0.396805	+	0.917903i	$0.629881\pi$
$600$	0	0
$601$	−10.3114	−0.420609	−0.210305	−	0.977636i	$-0.567446\pi$
$601$	−10.3114	−0.420609	−0.210305	+	0.977636i	$0.567446\pi$
$602$	0	0
$603$	−30.4619	−1.24050
$604$	0	0
$605$	0	0
$606$	0	0
$607$	31.9321	1.29609	0.648043	−	0.761604i	$-0.275588\pi$
$607$	31.9321	1.29609	0.648043	+	0.761604i	$0.275588\pi$
$608$	0	0
$609$	−1.53351	−0.0621411
$610$	0	0
$611$	21.8786	0.885112
$612$	0	0
$613$	−46.0191	−1.85869	−0.929347	−	0.369207i	$-0.879629\pi$
$613$	−46.0191	−1.85869	−0.929347	+	0.369207i	$0.879629\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−10.5277	−0.423831	−0.211915	−	0.977288i	$-0.567970\pi$
$617$	−10.5277	−0.423831	−0.211915	+	0.977288i	$0.567970\pi$
$618$	0	0
$619$	31.3764	1.26112	0.630562	−	0.776139i	$-0.282825\pi$
$619$	31.3764	1.26112	0.630562	+	0.776139i	$0.282825\pi$
$620$	0	0
$621$	−29.4439	−1.18154
$622$	0	0
$623$	−14.5074	−0.581228
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.53351	−0.220987
$628$	0	0
$629$	30.6981	1.22401
$630$	0	0
$631$	−43.7479	−1.74158	−0.870789	−	0.491657i	$-0.836391\pi$
$631$	−43.7479	−1.74158	−0.870789	+	0.491657i	$0.836391\pi$
$632$	0	0
$633$	−7.53953	−0.299669
$634$	0	0
$635$	0	0
$636$	0	0
$637$	4.45002	0.176316
$638$	0	0
$639$	17.7687	0.702919
$640$	0	0
$641$	−8.51204	−0.336205	−0.168103	−	0.985769i	$-0.553764\pi$
$641$	−8.51204	−0.336205	−0.168103	+	0.985769i	$0.553764\pi$
$642$	0	0
$643$	−1.72698	−0.0681054	−0.0340527	−	0.999420i	$-0.510841\pi$
$643$	−1.72698	−0.0681054	−0.0340527	+	0.999420i	$0.510841\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.2742	1.15089	0.575444	−	0.817841i	$-0.304829\pi$
$647$	29.2742	1.15089	0.575444	+	0.817841i	$0.304829\pi$
$648$	0	0
$649$	3.11181	0.122149
$650$	0	0
$651$	−3.56184	−0.139599
$652$	0	0
$653$	7.22480	0.282728	0.141364	−	0.989958i	$-0.454851\pi$
$653$	7.22480	0.282728	0.141364	+	0.989958i	$0.454851\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	12.6230	0.492471
$658$	0	0
$659$	28.7609	1.12037	0.560183	−	0.828369i	$-0.310731\pi$
$659$	28.7609	1.12037	0.560183	+	0.828369i	$0.310731\pi$
$660$	0	0
$661$	27.7652	1.07994	0.539970	−	0.841684i	$-0.318436\pi$
$661$	27.7652	1.07994	0.539970	+	0.841684i	$0.318436\pi$
$662$	0	0
$663$	9.44041	0.366635
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−14.1907	−0.549466
$668$	0	0
$669$	−12.9441	−0.500446
$670$	0	0
$671$	8.57145	0.330897
$672$	0	0
$673$	−4.19531	−0.161717	−0.0808586	−	0.996726i	$-0.525766\pi$
$673$	−4.19531	−0.161717	−0.0808586	+	0.996726i	$0.525766\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	2.07311	0.0796761	0.0398380	−	0.999206i	$-0.487316\pi$
$677$	2.07311	0.0796761	0.0398380	+	0.999206i	$0.487316\pi$
$678$	0	0
$679$	−3.68327	−0.141351
$680$	0	0
$681$	5.64834	0.216445
$682$	0	0
$683$	−32.3860	−1.23922	−0.619608	−	0.784911i	$-0.712708\pi$
$683$	−32.3860	−1.23922	−0.619608	+	0.784911i	$0.712708\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−15.4561	−0.589687
$688$	0	0
$689$	45.2228	1.72285
$690$	0	0
$691$	3.20492	0.121921	0.0609605	−	0.998140i	$-0.480584\pi$
$691$	3.20492	0.121921	0.0609605	+	0.998140i	$0.480584\pi$
$692$	0	0
$693$	−2.41208	−0.0916275
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−22.6289	−0.857129
$698$	0	0
$699$	9.85473	0.372740
$700$	0	0
$701$	42.7391	1.61423	0.807116	−	0.590392i	$-0.201027\pi$
$701$	42.7391	1.61423	0.807116	+	0.590392i	$0.201027\pi$
$702$	0	0
$703$	−80.0727	−3.02000
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.57830	0.285011
$708$	0	0
$709$	−18.5676	−0.697321	−0.348661	−	0.937249i	$-0.613363\pi$
$709$	−18.5676	−0.697321	−0.348661	+	0.937249i	$0.613363\pi$
$710$	0	0
$711$	37.0958	1.39120
$712$	0	0
$713$	−32.9602	−1.23437
$714$	0	0
$715$	0	0
$716$	0	0
$717$	17.2891	0.645674
$718$	0	0
$719$	−19.2616	−0.718335	−0.359168	−	0.933273i	$-0.616939\pi$
$719$	−19.2616	−0.718335	−0.359168	+	0.933273i	$0.616939\pi$
$720$	0	0
$721$	13.6668	0.508978
$722$	0	0
$723$	−23.1728	−0.861804
$724$	0	0
$725$	0	0
$726$	0	0
$727$	18.6929	0.693281	0.346640	−	0.937998i	$-0.387322\pi$
$727$	18.6929	0.693281	0.346640	+	0.937998i	$0.387322\pi$
$728$	0	0
$729$	−0.234010	−0.00866703
$730$	0	0
$731$	29.4075	1.08767
$732$	0	0
$733$	37.6491	1.39060	0.695301	−	0.718719i	$-0.255271\pi$
$733$	37.6491	1.39060	0.695301	+	0.718719i	$0.255271\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	12.6289	0.465190
$738$	0	0
$739$	39.0012	1.43468	0.717341	−	0.696723i	$-0.245359\pi$
$739$	39.0012	1.43468	0.717341	+	0.696723i	$0.245359\pi$
$740$	0	0
$741$	−24.6243	−0.904595
$742$	0	0
$743$	47.2056	1.73180	0.865902	−	0.500213i	$-0.166745\pi$
$743$	47.2056	1.73180	0.865902	+	0.500213i	$0.166745\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.4003	−0.453703
$748$	0	0
$749$	6.82417	0.249350
$750$	0	0
$751$	−28.9000	−1.05458	−0.527289	−	0.849686i	$-0.676791\pi$
$751$	−28.9000	−1.05458	−0.527289	+	0.849686i	$0.676791\pi$
$752$	0	0
$753$	12.7377	0.464186
$754$	0	0
$755$	0	0
$756$	0	0
$757$	27.9149	1.01458	0.507292	−	0.861774i	$-0.330646\pi$
$757$	27.9149	1.01458	0.507292	+	0.861774i	$0.330646\pi$
$758$	0	0
$759$	5.44041	0.197474
$760$	0	0
$761$	−19.3956	−0.703091	−0.351545	−	0.936171i	$-0.614344\pi$
$761$	−19.3956	−0.703091	−0.351545	+	0.936171i	$0.614344\pi$
$762$	0	0
$763$	0.466487	0.0168879
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13.8476	0.500009
$768$	0	0
$769$	50.0938	1.80643	0.903213	−	0.429192i	$-0.141202\pi$
$769$	50.0938	1.80643	0.903213	+	0.429192i	$0.141202\pi$
$770$	0	0
$771$	−3.81379	−0.137350
$772$	0	0
$773$	9.33160	0.335634	0.167817	−	0.985818i	$-0.446328\pi$
$773$	9.33160	0.335634	0.167817	+	0.985818i	$0.446328\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.50743	0.305202
$778$	0	0
$779$	59.0249	2.11479
$780$	0	0
$781$	−7.36653	−0.263595
$782$	0	0
$783$	8.29950	0.296600
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.0060	0.927014	0.463507	−	0.886093i	$-0.346591\pi$
$787$	26.0060	0.927014	0.463507	+	0.886093i	$0.346591\pi$
$788$	0	0
$789$	16.0252	0.570514
$790$	0	0
$791$	−18.6572	−0.663373
$792$	0	0
$793$	38.1431	1.35450
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.0839	−1.06563	−0.532813	−	0.846233i	$-0.678865\pi$
$797$	−30.0839	−1.06563	−0.532813	+	0.846233i	$0.678865\pi$
$798$	0	0
$799$	−13.6028	−0.481232
$800$	0	0
$801$	34.9932	1.23642
$802$	0	0
$803$	−5.23324	−0.184677
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−9.83455	−0.346193
$808$	0	0
$809$	30.6243	1.07669	0.538346	−	0.842724i	$-0.319049\pi$
$809$	30.6243	1.07669	0.538346	+	0.842724i	$0.319049\pi$
$810$	0	0
$811$	3.15860	0.110913	0.0554567	−	0.998461i	$-0.482339\pi$
$811$	3.15860	0.110913	0.0554567	+	0.998461i	$0.482339\pi$
$812$	0	0
$813$	−0.460468	−0.0161493
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−76.7061	−2.68361
$818$	0	0
$819$	−10.7338	−0.375070
$820$	0	0
$821$	−1.06549	−0.0371860	−0.0185930	−	0.999827i	$-0.505919\pi$
$821$	−1.06549	−0.0371860	−0.0185930	+	0.999827i	$0.505919\pi$
$822$	0	0
$823$	−15.0624	−0.525043	−0.262521	−	0.964926i	$-0.584554\pi$
$823$	−15.0624	−0.525043	−0.262521	+	0.964926i	$0.584554\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−38.6541	−1.34414	−0.672068	−	0.740490i	$-0.734594\pi$
$827$	−38.6541	−1.34414	−0.672068	+	0.740490i	$0.734594\pi$
$828$	0	0
$829$	7.26694	0.252391	0.126196	−	0.992005i	$-0.459723\pi$
$829$	7.26694	0.252391	0.126196	+	0.992005i	$0.459723\pi$
$830$	0	0
$831$	−2.94252	−0.102075
$832$	0	0
$833$	−2.76676	−0.0958624
$834$	0	0
$835$	0	0
$836$	0	0
$837$	19.2770	0.666309
$838$	0	0
$839$	−46.4105	−1.60227	−0.801134	−	0.598485i	$-0.795770\pi$
$839$	−46.4105	−1.60227	−0.801134	+	0.598485i	$0.795770\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	−20.6704	−0.711926
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−2.76752	−0.0949812
$850$	0	0
$851$	78.7254	2.69867
$852$	0	0
$853$	27.8166	0.952421	0.476210	−	0.879331i	$-0.342010\pi$
$853$	27.8166	0.952421	0.476210	+	0.879331i	$0.342010\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17.7756	−0.607203	−0.303602	−	0.952799i	$-0.598189\pi$
$857$	−17.7756	−0.607203	−0.303602	+	0.952799i	$0.598189\pi$
$858$	0	0
$859$	−6.72120	−0.229324	−0.114662	−	0.993405i	$-0.536579\pi$
$859$	−6.72120	−0.229324	−0.114662	+	0.993405i	$0.536579\pi$
$860$	0	0
$861$	−6.27118	−0.213721
$862$	0	0
$863$	−12.6618	−0.431012	−0.215506	−	0.976502i	$-0.569140\pi$
$863$	−12.6618	−0.431012	−0.215506	+	0.976502i	$0.569140\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	7.16539	0.243349
$868$	0	0
$869$	−15.3792	−0.521702
$870$	0	0
$871$	56.1987	1.90422
$872$	0	0
$873$	8.88435	0.300690
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.33821	−0.0451881	−0.0225940	−	0.999745i	$-0.507193\pi$
$877$	−1.33821	−0.0451881	−0.0225940	+	0.999745i	$0.507193\pi$
$878$	0	0
$879$	4.88895	0.164900
$880$	0	0
$881$	6.93497	0.233645	0.116823	−	0.993153i	$-0.462729\pi$
$881$	6.93497	0.233645	0.116823	+	0.993153i	$0.462729\pi$
$882$	0	0
$883$	14.4037	0.484723	0.242362	−	0.970186i	$-0.422078\pi$
$883$	14.4037	0.484723	0.242362	+	0.970186i	$0.422078\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.33360	0.111931	0.0559657	−	0.998433i	$-0.482176\pi$
$887$	3.33360	0.111931	0.0559657	+	0.998433i	$0.482176\pi$
$888$	0	0
$889$	−4.90004	−0.164342
$890$	0	0
$891$	4.05440	0.135828
$892$	0	0
$893$	35.4814	1.18734
$894$	0	0
$895$	0	0
$896$	0	0
$897$	24.2099	0.808346
$898$	0	0
$899$	9.29065	0.309861
$900$	0	0
$901$	−28.1168	−0.936707
$902$	0	0
$903$	8.14975	0.271207
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−24.0283	−0.797847	−0.398924	−	0.916984i	$-0.630616\pi$
$907$	−24.0283	−0.797847	−0.398924	+	0.916984i	$0.630616\pi$
$908$	0	0
$909$	−18.2795	−0.606293
$910$	0	0
$911$	34.8808	1.15565	0.577826	−	0.816160i	$-0.303901\pi$
$911$	34.8808	1.15565	0.577826	+	0.816160i	$0.303901\pi$
$912$	0	0
$913$	5.14090	0.170139
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.00000	0.132092
$918$	0	0
$919$	−5.83302	−0.192413	−0.0962067	−	0.995361i	$-0.530671\pi$
$919$	−5.83302	−0.192413	−0.0962067	+	0.995361i	$0.530671\pi$
$920$	0	0
$921$	−5.71973	−0.188471
$922$	0	0
$923$	−32.7812	−1.07901
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−32.9655	−1.08273
$928$	0	0
$929$	−13.2168	−0.433628	−0.216814	−	0.976213i	$-0.569567\pi$
$929$	−13.2168	−0.433628	−0.216814	+	0.976213i	$0.569567\pi$
$930$	0	0
$931$	7.21678	0.236520
$932$	0	0
$933$	17.8107	0.583097
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.1176	−1.04924	−0.524618	−	0.851338i	$-0.675792\pi$
$937$	−32.1176	−1.04924	−0.524618	+	0.851338i	$0.675792\pi$
$938$	0	0
$939$	−22.1011	−0.721243
$940$	0	0
$941$	−27.6582	−0.901631	−0.450816	−	0.892617i	$-0.648867\pi$
$941$	−27.6582	−0.901631	−0.450816	+	0.892617i	$0.648867\pi$
$942$	0	0
$943$	−58.0317	−1.88977
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−1.98538	−0.0645161	−0.0322580	−	0.999480i	$-0.510270\pi$
$947$	−1.98538	−0.0645161	−0.0322580	+	0.999480i	$0.510270\pi$
$948$	0	0
$949$	−23.2880	−0.755962
$950$	0	0
$951$	−23.0655	−0.747950
$952$	0	0
$953$	−37.4767	−1.21399	−0.606995	−	0.794705i	$-0.707625\pi$
$953$	−37.4767	−1.21399	−0.606995	+	0.794705i	$0.707625\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−1.53351	−0.0495714
$958$	0	0
$959$	−10.1624	−0.328160
$960$	0	0
$961$	−9.42093	−0.303901
$962$	0	0
$963$	−16.4605	−0.530431
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−46.4113	−1.49249	−0.746243	−	0.665674i	$-0.768144\pi$
$967$	−46.4113	−1.49249	−0.746243	+	0.665674i	$0.768144\pi$
$968$	0	0
$969$	15.3099	0.491824
$970$	0	0
$971$	12.1869	0.391095	0.195547	−	0.980694i	$-0.437352\pi$
$971$	12.1869	0.391095	0.195547	+	0.980694i	$0.437352\pi$
$972$	0	0
$973$	−12.9000	−0.413556
$974$	0	0
$975$	0	0
$976$	0	0
$977$	17.7901	0.569155	0.284577	−	0.958653i	$-0.408147\pi$
$977$	17.7901	0.569155	0.284577	+	0.958653i	$0.408147\pi$
$978$	0	0
$979$	−14.5074	−0.463660
$980$	0	0
$981$	−1.12520	−0.0359250
$982$	0	0
$983$	−36.0325	−1.14926	−0.574629	−	0.818414i	$-0.694854\pi$
$983$	−36.0325	−1.14926	−0.574629	+	0.818414i	$0.694854\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−3.76977	−0.119993
$988$	0	0
$989$	75.4155	2.39807
$990$	0	0
$991$	−36.5913	−1.16236	−0.581181	−	0.813774i	$-0.697409\pi$
$991$	−36.5913	−1.16236	−0.581181	+	0.813774i	$0.697409\pi$
$992$	0	0
$993$	4.42907	0.140552
$994$	0	0
$995$	0	0
$996$	0	0
$997$	25.7993	0.817073	0.408536	−	0.912742i	$-0.366039\pi$
$997$	25.7993	0.817073	0.408536	+	0.912742i	$0.366039\pi$
$998$	0	0
$999$	−46.0429	−1.45673

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7700.2.a.bb.1.2		4
5.2	odd	4		7700.2.e.t.1849.6		8
5.3	odd	4		7700.2.e.t.1849.3		8
5.4	even	2		1540.2.a.i.1.3	✓	4
20.19	odd	2		6160.2.a.bs.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1540.2.a.i.1.3	✓	4	5.4	even	2
6160.2.a.bs.1.2		4	20.19	odd	2
7700.2.a.bb.1.2		4	1.1	even	1	trivial
7700.2.e.t.1849.3		8	5.3	odd	4
7700.2.e.t.1849.6		8	5.2	odd	4

Overview	Random
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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:	\( N \)	\(=\)	\( 7700 = 2^{2} \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7700.a (trivial)

\(f(q)\)	\(=\)	\(q-0.766757 q^{3} +1.00000 q^{7} -2.41208 q^{9} +O(q^{10})\)	\(q-0.766757 q^{3} +1.00000 q^{7} -2.41208 q^{9} +1.00000 q^{11} +4.45002 q^{13} -2.76676 q^{17} +7.21678 q^{19} -0.766757 q^{21} -7.09535 q^{23} +4.14975 q^{27} +2.00000 q^{29} +4.64533 q^{31} -0.766757 q^{33} -11.0953 q^{37} -3.41208 q^{39} +8.17884 q^{41} -10.6289 q^{43} +4.91651 q^{47} +1.00000 q^{49} +2.12143 q^{51} +10.1624 q^{53} -5.53351 q^{57} +3.11181 q^{59} +8.57145 q^{61} -2.41208 q^{63} +12.6289 q^{67} +5.44041 q^{69} -7.36653 q^{71} -5.23324 q^{73} +1.00000 q^{77} -15.3792 q^{79} +4.05440 q^{81} +5.14090 q^{83} -1.53351 q^{87} -14.5074 q^{89} +4.45002 q^{91} -3.56184 q^{93} -3.68327 q^{97} -2.41208 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q + 4 * q^7 + 8 * q^9	\( 4 q + 4 q^{7} + 8 q^{9} + 4 q^{11} - 2 q^{13} - 8 q^{17} + 6 q^{19} + 6 q^{23} + 6 q^{27} + 8 q^{29} + 4 q^{31} - 10 q^{37} + 4 q^{39} + 12 q^{41} - 2 q^{43} + 6 q^{47} + 4 q^{49} + 20 q^{51} - 6 q^{53} - 16 q^{57} + 4 q^{59} + 26 q^{61} + 8 q^{63} + 10 q^{67} - 18 q^{69} + 4 q^{71} - 24 q^{73} + 4 q^{77} + 8 q^{79} + 40 q^{81} + 2 q^{83} - 6 q^{89} - 2 q^{91} + 14 q^{93} + 2 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q + 4 * q^7 + 8 * q^9 + 4 * q^11 - 2 * q^13 - 8 * q^17 + 6 * q^19 + 6 * q^23 + 6 * q^27 + 8 * q^29 + 4 * q^31 - 10 * q^37 + 4 * q^39 + 12 * q^41 - 2 * q^43 + 6 * q^47 + 4 * q^49 + 20 * q^51 - 6 * q^53 - 16 * q^57 + 4 * q^59 + 26 * q^61 + 8 * q^63 + 10 * q^67 - 18 * q^69 + 4 * q^71 - 24 * q^73 + 4 * q^77 + 8 * q^79 + 40 * q^81 + 2 * q^83 - 6 * q^89 - 2 * q^91 + 14 * q^93 + 2 * q^97 + 8 * q^99

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