LMFDB - Embedded newform 2925.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2925,2,Mod(1,2925)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2925, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2925.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2925 = 3^{2} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2925.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:	$23.3562425912$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1509051136.3
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 43x^{2} - 19 $ x^6 - 13x^4 + 43x^2 - 19
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.79260$ of defining polynomial
Character	$\chi$	$=$	2925.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.79260 q^{2} +5.79859 q^{4} -3.71465 q^{7} -10.6079 q^{8} +O(q^{10})$	$q-2.79260 q^{2} +5.79859 q^{4} -3.71465 q^{7} -10.6079 q^{8} -5.02274 q^{11} -1.00000 q^{13} +10.3735 q^{14} +18.0265 q^{16} +2.55817 q^{17} +6.79859 q^{19} +14.0265 q^{22} +2.23014 q^{23} +2.79260 q^{26} -21.5397 q^{28} -8.14336 q^{29} -4.51324 q^{31} -29.1248 q^{32} -7.14394 q^{34} -0.630700 q^{37} -18.9857 q^{38} -2.23014 q^{41} -2.79859 q^{43} -29.1248 q^{44} -6.22788 q^{46} +2.79260 q^{47} +6.79859 q^{49} -5.79859 q^{52} -5.91322 q^{53} +39.4047 q^{56} +22.7411 q^{58} -8.37779 q^{59} -8.02647 q^{61} +12.6036 q^{62} +45.2808 q^{64} +2.51324 q^{67} +14.8338 q^{68} +8.28418 q^{71} -14.3958 q^{73} +1.76129 q^{74} +39.4222 q^{76} +18.6577 q^{77} -8.39577 q^{79} +6.22788 q^{82} +7.90894 q^{83} +7.81533 q^{86} +53.2808 q^{88} -4.92913 q^{89} +3.71465 q^{91} +12.9317 q^{92} -7.79859 q^{94} +3.42929 q^{97} -18.9857 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} - 6 q^{7}+O(q^{10}) $ 6 * q + 14 * q^4 - 6 * q^7	$ 6 q + 14 q^{4} - 6 q^{7} - 6 q^{13} + 34 q^{16} + 20 q^{19} + 10 q^{22} - 18 q^{28} + 10 q^{31} + 6 q^{34} + 8 q^{37} + 4 q^{43} + 16 q^{46} + 20 q^{49} - 14 q^{52} + 46 q^{58} + 26 q^{61} + 70 q^{64} - 22 q^{67} - 24 q^{73} + 100 q^{76} + 12 q^{79} - 16 q^{82} + 118 q^{88} + 6 q^{91} - 26 q^{94} - 12 q^{97}+O(q^{100}) $ 6 * q + 14 * q^4 - 6 * q^7 - 6 * q^13 + 34 * q^16 + 20 * q^19 + 10 * q^22 - 18 * q^28 + 10 * q^31 + 6 * q^34 + 8 * q^37 + 4 * q^43 + 16 * q^46 + 20 * q^49 - 14 * q^52 + 46 * q^58 + 26 * q^61 + 70 * q^64 - 22 * q^67 - 24 * q^73 + 100 * q^76 + 12 * q^79 - 16 * q^82 + 118 * q^88 + 6 * q^91 - 26 * q^94 - 12 * q^97

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 $

$2$	−2.79260	−1.97466	−0.987332	−	0.158670i	$-0.949279\pi$
$2$	−2.79260	−1.97466	−0.987332	+	0.158670i	$0.949279\pi$
$3$	0	0
$3$	0	0
$4$	5.79859	2.89930
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.71465	−1.40400	−0.702002	−	0.712175i	$-0.747710\pi$
$7$	−3.71465	−1.40400	−0.702002	+	0.712175i	$0.747710\pi$
$8$	−10.6079	−3.75047
$9$	0	0
$10$	0	0
$11$	−5.02274	−1.51441	−0.757206	−	0.653176i	$-0.773436\pi$
$11$	−5.02274	−1.51441	−0.757206	+	0.653176i	$0.773436\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	10.3735	2.77244
$15$	0	0
$16$	18.0265	4.50662
$17$	2.55817	0.620448	0.310224	−	0.950664i	$-0.399596\pi$
$17$	2.55817	0.620448	0.310224	+	0.950664i	$0.399596\pi$
$18$	0	0
$19$	6.79859	1.55970	0.779852	−	0.625964i	$-0.215294\pi$
$19$	6.79859	1.55970	0.779852	+	0.625964i	$0.215294\pi$
$20$	0	0
$21$	0	0
$22$	14.0265	2.99045
$23$	2.23014	0.465016	0.232508	−	0.972594i	$-0.425307\pi$
$23$	2.23014	0.465016	0.232508	+	0.972594i	$0.425307\pi$
$24$	0	0
$25$	0	0
$26$	2.79260	0.547673
$27$	0	0
$28$	−21.5397	−4.07062
$29$	−8.14336	−1.51218	−0.756092	−	0.654465i	$-0.772894\pi$
$29$	−8.14336	−1.51218	−0.756092	+	0.654465i	$0.772894\pi$
$30$	0	0
$31$	−4.51324	−0.810601	−0.405301	−	0.914183i	$-0.632833\pi$
$31$	−4.51324	−0.810601	−0.405301	+	0.914183i	$0.632833\pi$
$32$	−29.1248	−5.14858
$33$	0	0
$34$	−7.14394	−1.22517
$35$	0	0
$36$	0	0
$37$	−0.630700	−0.103686	−0.0518432	−	0.998655i	$-0.516510\pi$
$37$	−0.630700	−0.103686	−0.0518432	+	0.998655i	$0.516510\pi$
$38$	−18.9857	−3.07989
$39$	0	0
$40$	0	0
$41$	−2.23014	−0.348289	−0.174145	−	0.984720i	$-0.555716\pi$
$41$	−2.23014	−0.348289	−0.174145	+	0.984720i	$0.555716\pi$
$42$	0	0
$43$	−2.79859	−0.426781	−0.213391	−	0.976967i	$-0.568451\pi$
$43$	−2.79859	−0.426781	−0.213391	+	0.976967i	$0.568451\pi$
$44$	−29.1248	−4.39073
$45$	0	0
$46$	−6.22788	−0.918251
$47$	2.79260	0.407342	0.203671	−	0.979039i	$-0.434713\pi$
$47$	2.79260	0.407342	0.203671	+	0.979039i	$0.434713\pi$
$48$	0	0
$49$	6.79859	0.971227
$50$	0	0
$51$	0	0
$52$	−5.79859	−0.804120
$53$	−5.91322	−0.812243	−0.406122	−	0.913819i	$-0.633119\pi$
$53$	−5.91322	−0.812243	−0.406122	+	0.913819i	$0.633119\pi$
$54$	0	0
$55$	0	0
$56$	39.4047	5.26567
$57$	0	0
$58$	22.7411	2.98606
$59$	−8.37779	−1.09069	−0.545347	−	0.838210i	$-0.683602\pi$
$59$	−8.37779	−1.09069	−0.545347	+	0.838210i	$0.683602\pi$
$60$	0	0
$61$	−8.02647	−1.02768	−0.513842	−	0.857885i	$-0.671778\pi$
$61$	−8.02647	−1.02768	−0.513842	+	0.857885i	$0.671778\pi$
$62$	12.6036	1.60066
$63$	0	0
$64$	45.2808	5.66010
$65$	0	0
$66$	0	0
$67$	2.51324	0.307041	0.153520	−	0.988145i	$-0.450939\pi$
$67$	2.51324	0.307041	0.153520	+	0.988145i	$0.450939\pi$
$68$	14.8338	1.79886
$69$	0	0
$70$	0	0
$71$	8.28418	0.983152	0.491576	−	0.870835i	$-0.336421\pi$
$71$	8.28418	0.983152	0.491576	+	0.870835i	$0.336421\pi$
$72$	0	0
$73$	−14.3958	−1.68490	−0.842449	−	0.538776i	$-0.818887\pi$
$73$	−14.3958	−1.68490	−0.842449	+	0.538776i	$0.818887\pi$
$74$	1.76129	0.204746
$75$	0	0
$76$	39.4222	4.52204
$77$	18.6577	2.12624
$78$	0	0
$79$	−8.39577	−0.944598	−0.472299	−	0.881438i	$-0.656576\pi$
$79$	−8.39577	−0.944598	−0.472299	+	0.881438i	$0.656576\pi$
$80$	0	0
$81$	0	0
$82$	6.22788	0.687754
$83$	7.90894	0.868119	0.434059	−	0.900884i	$-0.357081\pi$
$83$	7.90894	0.868119	0.434059	+	0.900884i	$0.357081\pi$
$84$	0	0
$85$	0	0
$86$	7.81533	0.842749
$87$	0	0
$88$	53.2808	5.67976
$89$	−4.92913	−0.522487	−0.261243	−	0.965273i	$-0.584133\pi$
$89$	−4.92913	−0.522487	−0.261243	+	0.965273i	$0.584133\pi$
$90$	0	0
$91$	3.71465	0.389401
$92$	12.9317	1.34822
$93$	0	0
$94$	−7.79859	−0.804363
$95$	0	0
$96$	0	0
$97$	3.42929	0.348192	0.174096	−	0.984729i	$-0.444300\pi$
$97$	3.42929	0.348192	0.174096	+	0.984729i	$0.444300\pi$
$98$	−18.9857	−1.91785
$99$	0	0
$100$	0	0
$101$	11.4984	1.14413	0.572067	−	0.820207i	$-0.306142\pi$
$101$	11.4984	1.14413	0.572067	+	0.820207i	$0.306142\pi$
$102$	0	0
$103$	8.79859	0.866951	0.433475	−	0.901165i	$-0.357287\pi$
$103$	8.79859	0.866951	0.433475	+	0.901165i	$0.357287\pi$
$104$	10.6079	1.04019
$105$	0	0
$106$	16.5132	1.60391
$107$	−10.0455	−0.971133	−0.485566	−	0.874200i	$-0.661387\pi$
$107$	−10.0455	−0.971133	−0.485566	+	0.874200i	$0.661387\pi$
$108$	0	0
$109$	−4.39577	−0.421039	−0.210519	−	0.977590i	$-0.567515\pi$
$109$	−4.39577	−0.421039	−0.210519	+	0.977590i	$0.567515\pi$
$110$	0	0
$111$	0	0
$112$	−66.9620	−6.32731
$113$	0.468850	0.0441057	0.0220529	−	0.999757i	$-0.492980\pi$
$113$	0.468850	0.0441057	0.0220529	+	0.999757i	$0.492980\pi$
$114$	0	0
$115$	0	0
$116$	−47.2200	−4.38427
$117$	0	0
$118$	23.3958	2.15376
$119$	−9.50270	−0.871111
$120$	0	0
$121$	14.2279	1.29344
$122$	22.4147	2.02933
$123$	0	0
$124$	−26.1704	−2.35017
$125$	0	0
$126$	0	0
$127$	6.96648	0.618175	0.309088	−	0.951034i	$-0.399976\pi$
$127$	6.96648	0.618175	0.309088	+	0.951034i	$0.399976\pi$
$128$	−68.2015	−6.02821
$129$	0	0
$130$	0	0
$131$	18.5169	1.61783	0.808913	−	0.587928i	$-0.200056\pi$
$131$	18.5169	1.61783	0.808913	+	0.587928i	$0.200056\pi$
$132$	0	0
$133$	−25.2544	−2.18983
$134$	−7.01845	−0.606302
$135$	0	0
$136$	−27.1369	−2.32697
$137$	0.468850	0.0400566	0.0200283	−	0.999799i	$-0.493624\pi$
$137$	0.468850	0.0400566	0.0200283	+	0.999799i	$0.493624\pi$
$138$	0	0
$139$	3.26140	0.276628	0.138314	−	0.990388i	$-0.455832\pi$
$139$	3.26140	0.276628	0.138314	+	0.990388i	$0.455832\pi$
$140$	0	0
$141$	0	0
$142$	−23.1344	−1.94139
$143$	5.02274	0.420022
$144$	0	0
$145$	0	0
$146$	40.2016	3.32711
$147$	0	0
$148$	−3.65717	−0.300618
$149$	21.2159	1.73807	0.869035	−	0.494750i	$-0.164740\pi$
$149$	21.2159	1.73807	0.869035	+	0.494750i	$0.164740\pi$
$150$	0	0
$151$	−16.9090	−1.37604	−0.688018	−	0.725694i	$-0.741519\pi$
$151$	−16.9090	−1.37604	−0.688018	+	0.725694i	$0.741519\pi$
$152$	−72.1190	−5.84962
$153$	0	0
$154$	−52.1034	−4.19861
$155$	0	0
$156$	0	0
$157$	17.3958	1.38833	0.694167	−	0.719814i	$-0.255773\pi$
$157$	17.3958	1.38833	0.694167	+	0.719814i	$0.255773\pi$
$158$	23.4460	1.86526
$159$	0	0
$160$	0	0
$161$	−8.28418	−0.652885
$162$	0	0
$163$	17.6572	1.38302	0.691508	−	0.722369i	$-0.256947\pi$
$163$	17.6572	1.38302	0.691508	+	0.722369i	$0.256947\pi$
$164$	−12.9317	−1.00979
$165$	0	0
$166$	−22.0865	−1.71424
$167$	14.5254	1.12401	0.562006	−	0.827133i	$-0.310030\pi$
$167$	14.5254	1.12401	0.562006	+	0.827133i	$0.310030\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0	0
$172$	−16.2279	−1.23736
$173$	11.4984	0.874208	0.437104	−	0.899411i	$-0.356004\pi$
$173$	11.4984	0.874208	0.437104	+	0.899411i	$0.356004\pi$
$174$	0	0
$175$	0	0
$176$	−90.5422	−6.82488
$177$	0	0
$178$	13.7651	1.03174
$179$	18.9857	1.41906	0.709529	−	0.704676i	$-0.248908\pi$
$179$	18.9857	1.41906	0.709529	+	0.704676i	$0.248908\pi$
$180$	0	0
$181$	−25.1608	−1.87019	−0.935095	−	0.354396i	$-0.884686\pi$
$181$	−25.1608	−1.87019	−0.935095	+	0.354396i	$0.884686\pi$
$182$	−10.3735	−0.768935
$183$	0	0
$184$	−23.6572	−1.74403
$185$	0	0
$186$	0	0
$187$	−12.8490	−0.939613
$188$	16.1931	1.18100
$189$	0	0
$190$	0	0
$191$	10.5143	0.760790	0.380395	−	0.924824i	$-0.375788\pi$
$191$	10.5143	0.760790	0.380395	+	0.924824i	$0.375788\pi$
$192$	0	0
$193$	1.53719	0.110649	0.0553247	−	0.998468i	$-0.482381\pi$
$193$	1.53719	0.110649	0.0553247	+	0.998468i	$0.482381\pi$
$194$	−9.57662	−0.687561
$195$	0	0
$196$	39.4222	2.81587
$197$	6.69042	0.476673	0.238336	−	0.971183i	$-0.423398\pi$
$197$	6.69042	0.476673	0.238336	+	0.971183i	$0.423398\pi$
$198$	0	0
$199$	11.2614	0.798300	0.399150	−	0.916886i	$-0.369305\pi$
$199$	11.2614	0.798300	0.399150	+	0.916886i	$0.369305\pi$
$200$	0	0
$201$	0	0
$202$	−32.1104	−2.25928
$203$	30.2497	2.12311
$204$	0	0
$205$	0	0
$206$	−24.5709	−1.71194
$207$	0	0
$208$	−18.0265	−1.24991
$209$	−34.1475	−2.36203
$210$	0	0
$211$	−0.227881	−0.0156880	−0.00784400	−	0.999969i	$-0.502497\pi$
$211$	−0.227881	−0.0156880	−0.00784400	+	0.999969i	$0.502497\pi$
$212$	−34.2883	−2.35493
$213$	0	0
$214$	28.0529	1.91766
$215$	0	0
$216$	0	0
$217$	16.7651	1.13809
$218$	12.2756	0.831409
$219$	0	0
$220$	0	0
$221$	−2.55817	−0.172081
$222$	0	0
$223$	9.65717	0.646692	0.323346	−	0.946281i	$-0.395192\pi$
$223$	9.65717	0.646692	0.323346	+	0.946281i	$0.395192\pi$
$224$	108.188	7.22863
$225$	0	0
$226$	−1.30931	−0.0870939
$227$	6.80371	0.451578	0.225789	−	0.974176i	$-0.427504\pi$
$227$	6.80371	0.451578	0.225789	+	0.974176i	$0.427504\pi$
$228$	0	0
$229$	6.33578	0.418680	0.209340	−	0.977843i	$-0.432868\pi$
$229$	6.33578	0.418680	0.209340	+	0.977843i	$0.432868\pi$
$230$	0	0
$231$	0	0
$232$	86.3842	5.67140
$233$	−16.0995	−1.05471	−0.527357	−	0.849644i	$-0.676817\pi$
$233$	−16.0995	−1.05471	−0.527357	+	0.849644i	$0.676817\pi$
$234$	0	0
$235$	0	0
$236$	−48.5794	−3.16225
$237$	0	0
$238$	26.5372	1.72015
$239$	6.78403	0.438822	0.219411	−	0.975632i	$-0.429586\pi$
$239$	6.78403	0.438822	0.219411	+	0.975632i	$0.429586\pi$
$240$	0	0
$241$	21.6572	1.39506	0.697531	−	0.716555i	$-0.254282\pi$
$241$	21.6572	1.39506	0.697531	+	0.716555i	$0.254282\pi$
$242$	−39.7327	−2.55412
$243$	0	0
$244$	−46.5422	−2.97956
$245$	0	0
$246$	0	0
$247$	−6.79859	−0.432584
$248$	47.8761	3.04013
$249$	0	0
$250$	0	0
$251$	27.9063	1.76143	0.880714	−	0.473648i	$-0.157063\pi$
$251$	27.9063	1.76143	0.880714	+	0.473648i	$0.157063\pi$
$252$	0	0
$253$	−11.2014	−0.704227
$254$	−19.4546	−1.22069
$255$	0	0
$256$	99.8974	6.24359
$257$	17.0836	1.06565	0.532823	−	0.846227i	$-0.321131\pi$
$257$	17.0836	1.06565	0.532823	+	0.846227i	$0.321131\pi$
$258$	0	0
$259$	2.34283	0.145576
$260$	0	0
$261$	0	0
$262$	−51.7101	−3.19466
$263$	−18.0480	−1.11289	−0.556444	−	0.830885i	$-0.687835\pi$
$263$	−18.0480	−1.11289	−0.556444	+	0.830885i	$0.687835\pi$
$264$	0	0
$265$	0	0
$266$	70.5252	4.32418
$267$	0	0
$268$	14.5732	0.890202
$269$	−4.80799	−0.293148	−0.146574	−	0.989200i	$-0.546825\pi$
$269$	−4.80799	−0.293148	−0.146574	+	0.989200i	$0.546825\pi$
$270$	0	0
$271$	2.85606	0.173494	0.0867468	−	0.996230i	$-0.472353\pi$
$271$	2.85606	0.173494	0.0867468	+	0.996230i	$0.472353\pi$
$272$	46.1148	2.79612
$273$	0	0
$274$	−1.30931	−0.0790982
$275$	0	0
$276$	0	0
$277$	−21.4822	−1.29074	−0.645371	−	0.763869i	$-0.723297\pi$
$277$	−21.4822	−1.29074	−0.645371	+	0.763869i	$0.723297\pi$
$278$	−9.10777	−0.546248
$279$	0	0
$280$	0	0
$281$	10.5143	0.627232	0.313616	−	0.949550i	$-0.398460\pi$
$281$	10.5143	0.627232	0.313616	+	0.949550i	$0.398460\pi$
$282$	0	0
$283$	−14.7915	−0.879266	−0.439633	−	0.898177i	$-0.644891\pi$
$283$	−14.7915	−0.879266	−0.439633	+	0.898177i	$0.644891\pi$
$284$	48.0366	2.85045
$285$	0	0
$286$	−14.0265	−0.829403
$287$	8.28418	0.489000
$288$	0	0
$289$	−10.4558	−0.615045
$290$	0	0
$291$	0	0
$292$	−83.4752	−4.88502
$293$	−29.6676	−1.73320	−0.866599	−	0.499005i	$-0.833699\pi$
$293$	−29.6676	−1.73320	−0.866599	+	0.499005i	$0.833699\pi$
$294$	0	0
$295$	0	0
$296$	6.69042	0.388873
$297$	0	0
$298$	−59.2473	−3.43210
$299$	−2.23014	−0.128972
$300$	0	0
$301$	10.3958	0.599202
$302$	47.2200	2.71721
$303$	0	0
$304$	122.555	7.02899
$305$	0	0
$306$	0	0
$307$	19.0865	1.08932	0.544661	−	0.838656i	$-0.316658\pi$
$307$	19.0865	1.08932	0.544661	+	0.838656i	$0.316658\pi$
$308$	108.188	6.16460
$309$	0	0
$310$	0	0
$311$	−21.6847	−1.22963	−0.614813	−	0.788673i	$-0.710769\pi$
$311$	−21.6847	−1.22963	−0.614813	+	0.788673i	$0.710769\pi$
$312$	0	0
$313$	−11.0600	−0.625148	−0.312574	−	0.949893i	$-0.601191\pi$
$313$	−11.0600	−0.625148	−0.312574	+	0.949893i	$0.601191\pi$
$314$	−48.5794	−2.74149
$315$	0	0
$316$	−48.6836	−2.73867
$317$	14.5254	0.815830	0.407915	−	0.913020i	$-0.366256\pi$
$317$	14.5254	0.815830	0.407915	+	0.913020i	$0.366256\pi$
$318$	0	0
$319$	40.9020	2.29007
$320$	0	0
$321$	0	0
$322$	23.1344	1.28923
$323$	17.3920	0.967714
$324$	0	0
$325$	0	0
$326$	−49.3093	−2.73099
$327$	0	0
$328$	23.6572	1.30625
$329$	−10.3735	−0.571910
$330$	0	0
$331$	5.20141	0.285895	0.142948	−	0.989730i	$-0.454342\pi$
$331$	5.20141	0.285895	0.142948	+	0.989730i	$0.454342\pi$
$332$	45.8607	2.51693
$333$	0	0
$334$	−40.5637	−2.21954
$335$	0	0
$336$	0	0
$337$	−0.705081	−0.0384082	−0.0192041	−	0.999816i	$-0.506113\pi$
$337$	−0.705081	−0.0384082	−0.0192041	+	0.999816i	$0.506113\pi$
$338$	−2.79260	−0.151897
$339$	0	0
$340$	0	0
$341$	22.6688	1.22758
$342$	0	0
$343$	0.748164	0.0403971
$344$	29.6872	1.60063
$345$	0	0
$346$	−32.1104	−1.72627
$347$	25.2073	1.35320	0.676599	−	0.736352i	$-0.263453\pi$
$347$	25.2073	1.35320	0.676599	+	0.736352i	$0.263453\pi$
$348$	0	0
$349$	4.63070	0.247876	0.123938	−	0.992290i	$-0.460448\pi$
$349$	4.63070	0.247876	0.123938	+	0.992290i	$0.460448\pi$
$350$	0	0
$351$	0	0
$352$	146.286	7.79708
$353$	31.2810	1.66492	0.832460	−	0.554085i	$-0.186932\pi$
$353$	31.2810	1.66492	0.832460	+	0.554085i	$0.186932\pi$
$354$	0	0
$355$	0	0
$356$	−28.5820	−1.51484
$357$	0	0
$358$	−53.0194	−2.80216
$359$	−17.7672	−0.937717	−0.468858	−	0.883273i	$-0.655335\pi$
$359$	−17.7672	−0.937717	−0.468858	+	0.883273i	$0.655335\pi$
$360$	0	0
$361$	27.2208	1.43268
$362$	70.2641	3.69300
$363$	0	0
$364$	21.5397	1.12899
$365$	0	0
$366$	0	0
$367$	28.4558	1.48538	0.742689	−	0.669636i	$-0.233550\pi$
$367$	28.4558	1.48538	0.742689	+	0.669636i	$0.233550\pi$
$368$	40.2016	2.09565
$369$	0	0
$370$	0	0
$371$	21.9655	1.14039
$372$	0	0
$373$	−3.17494	−0.164392	−0.0821960	−	0.996616i	$-0.526193\pi$
$373$	−3.17494	−0.164392	−0.0821960	+	0.996616i	$0.526193\pi$
$374$	35.8821	1.85542
$375$	0	0
$376$	−29.6237	−1.52772
$377$	8.14336	0.419404
$378$	0	0
$379$	24.6811	1.26778	0.633892	−	0.773421i	$-0.281456\pi$
$379$	24.6811	1.26778	0.633892	+	0.773421i	$0.281456\pi$
$380$	0	0
$381$	0	0
$382$	−29.3623	−1.50230
$383$	3.82390	0.195392	0.0976961	−	0.995216i	$-0.468853\pi$
$383$	3.82390	0.195392	0.0976961	+	0.995216i	$0.468853\pi$
$384$	0	0
$385$	0	0
$386$	−4.29275	−0.218495
$387$	0	0
$388$	19.8851	1.00951
$389$	−9.38941	−0.476062	−0.238031	−	0.971258i	$-0.576502\pi$
$389$	−9.38941	−0.476062	−0.238031	+	0.971258i	$0.576502\pi$
$390$	0	0
$391$	5.70508	0.288518
$392$	−72.1190	−3.64256
$393$	0	0
$394$	−18.6836	−0.941268
$395$	0	0
$396$	0	0
$397$	14.1679	0.711066	0.355533	−	0.934664i	$-0.384299\pi$
$397$	14.1679	0.711066	0.355533	+	0.934664i	$0.384299\pi$
$398$	−31.4485	−1.57637
$399$	0	0
$400$	0	0
$401$	−22.9771	−1.14742	−0.573712	−	0.819057i	$-0.694497\pi$
$401$	−22.9771	−1.14742	−0.573712	+	0.819057i	$0.694497\pi$
$402$	0	0
$403$	4.51324	0.224820
$404$	66.6746	3.31718
$405$	0	0
$406$	−84.4752	−4.19243
$407$	3.16784	0.157024
$408$	0	0
$409$	−6.29492	−0.311264	−0.155632	−	0.987815i	$-0.549741\pi$
$409$	−6.29492	−0.311264	−0.155632	+	0.987815i	$0.549741\pi$
$410$	0	0
$411$	0	0
$412$	51.0194	2.51355
$413$	31.1205	1.53134
$414$	0	0
$415$	0	0
$416$	29.1248	1.42796
$417$	0	0
$418$	95.3602	4.66422
$419$	−1.10523	−0.0539941	−0.0269970	−	0.999636i	$-0.508594\pi$
$419$	−1.10523	−0.0539941	−0.0269970	+	0.999636i	$0.508594\pi$
$420$	0	0
$421$	6.05294	0.295002	0.147501	−	0.989062i	$-0.452877\pi$
$421$	6.05294	0.295002	0.147501	+	0.989062i	$0.452877\pi$
$422$	0.636381	0.0309785
$423$	0	0
$424$	62.7270	3.04629
$425$	0	0
$426$	0	0
$427$	29.8155	1.44287
$428$	−58.2496	−2.81560
$429$	0	0
$430$	0	0
$431$	6.03436	0.290665	0.145333	−	0.989383i	$-0.453575\pi$
$431$	6.03436	0.290665	0.145333	+	0.989383i	$0.453575\pi$
$432$	0	0
$433$	−0.906490	−0.0435631	−0.0217816	−	0.999763i	$-0.506934\pi$
$433$	−0.906490	−0.0435631	−0.0217816	+	0.999763i	$0.506934\pi$
$434$	−46.8181	−2.24734
$435$	0	0
$436$	−25.4893	−1.22072
$437$	15.1618	0.725288
$438$	0	0
$439$	−15.4222	−0.736064	−0.368032	−	0.929813i	$-0.619968\pi$
$439$	−15.4222	−0.736064	−0.368032	+	0.929813i	$0.619968\pi$
$440$	0	0
$441$	0	0
$442$	7.14394	0.339802
$443$	−28.5623	−1.35704	−0.678519	−	0.734583i	$-0.737378\pi$
$443$	−28.5623	−1.35704	−0.678519	+	0.734583i	$0.737378\pi$
$444$	0	0
$445$	0	0
$446$	−26.9686	−1.27700
$447$	0	0
$448$	−168.202	−7.94681
$449$	−30.3236	−1.43106	−0.715530	−	0.698582i	$-0.753815\pi$
$449$	−30.3236	−1.43106	−0.715530	+	0.698582i	$0.753815\pi$
$450$	0	0
$451$	11.2014	0.527454
$452$	2.71867	0.127875
$453$	0	0
$454$	−19.0000	−0.891714
$455$	0	0
$456$	0	0
$457$	18.6307	0.871507	0.435754	−	0.900066i	$-0.356482\pi$
$457$	18.6307	0.871507	0.435754	+	0.900066i	$0.356482\pi$
$458$	−17.6933	−0.826752
$459$	0	0
$460$	0	0
$461$	−25.2073	−1.17402	−0.587010	−	0.809580i	$-0.699695\pi$
$461$	−25.2073	−1.17402	−0.587010	+	0.809580i	$0.699695\pi$
$462$	0	0
$463$	−27.7076	−1.28768	−0.643841	−	0.765160i	$-0.722660\pi$
$463$	−27.7076	−1.28768	−0.643841	+	0.765160i	$0.722660\pi$
$464$	−146.796	−6.81484
$465$	0	0
$466$	44.9594	2.08271
$467$	33.0423	1.52902	0.764508	−	0.644615i	$-0.222982\pi$
$467$	33.0423	1.52902	0.764508	+	0.644615i	$0.222982\pi$
$468$	0	0
$469$	−9.33578	−0.431086
$470$	0	0
$471$	0	0
$472$	88.8710	4.09062
$473$	14.0566	0.646322
$474$	0	0
$475$	0	0
$476$	−55.1022	−2.52561
$477$	0	0
$478$	−18.9450	−0.866526
$479$	16.4748	0.752751	0.376375	−	0.926467i	$-0.377170\pi$
$479$	16.4748	0.752751	0.376375	+	0.926467i	$0.377170\pi$
$480$	0	0
$481$	0.630700	0.0287575
$482$	−60.4797	−2.75478
$483$	0	0
$484$	82.5017	3.75008
$485$	0	0
$486$	0	0
$487$	−17.0169	−0.771110	−0.385555	−	0.922685i	$-0.625990\pi$
$487$	−17.0169	−0.771110	−0.385555	+	0.922685i	$0.625990\pi$
$488$	85.1442	3.85430
$489$	0	0
$490$	0	0
$491$	1.59376	0.0719254	0.0359627	−	0.999353i	$-0.488550\pi$
$491$	1.59376	0.0719254	0.0359627	+	0.999353i	$0.488550\pi$
$492$	0	0
$493$	−20.8321	−0.938231
$494$	18.9857	0.854208
$495$	0	0
$496$	−81.3577	−3.65307
$497$	−30.7728	−1.38035
$498$	0	0
$499$	35.3118	1.58077	0.790387	−	0.612608i	$-0.209879\pi$
$499$	35.3118	1.58077	0.790387	+	0.612608i	$0.209879\pi$
$500$	0	0
$501$	0	0
$502$	−77.9310	−3.47823
$503$	−17.2047	−0.767122	−0.383561	−	0.923516i	$-0.625302\pi$
$503$	−17.2047	−0.767122	−0.383561	+	0.923516i	$0.625302\pi$
$504$	0	0
$505$	0	0
$506$	31.2810	1.39061
$507$	0	0
$508$	40.3958	1.79227
$509$	26.3125	1.16628	0.583141	−	0.812371i	$-0.301824\pi$
$509$	26.3125	1.16628	0.583141	+	0.812371i	$0.301824\pi$
$510$	0	0
$511$	53.4752	2.36560
$512$	−142.570	−6.30077
$513$	0	0
$514$	−47.7076	−2.10429
$515$	0	0
$516$	0	0
$517$	−14.0265	−0.616884
$518$	−6.54257	−0.287464
$519$	0	0
$520$	0	0
$521$	4.46028	0.195408	0.0977042	−	0.995215i	$-0.468850\pi$
$521$	4.46028	0.195408	0.0977042	+	0.995215i	$0.468850\pi$
$522$	0	0
$523$	39.0794	1.70882	0.854412	−	0.519596i	$-0.173918\pi$
$523$	39.0794	1.70882	0.854412	+	0.519596i	$0.173918\pi$
$524$	107.372	4.69056
$525$	0	0
$526$	50.4008	2.19758
$527$	−11.5456	−0.502935
$528$	0	0
$529$	−18.0265	−0.783760
$530$	0	0
$531$	0	0
$532$	−146.440	−6.34896
$533$	2.23014	0.0965981
$534$	0	0
$535$	0	0
$536$	−26.6602	−1.15155
$537$	0	0
$538$	13.4268	0.578869
$539$	−34.1475	−1.47084
$540$	0	0
$541$	−41.2544	−1.77366	−0.886832	−	0.462093i	$-0.847099\pi$
$541$	−41.2544	−1.77366	−0.886832	+	0.462093i	$0.847099\pi$
$542$	−7.97583	−0.342591
$543$	0	0
$544$	−74.5062	−3.19443
$545$	0	0
$546$	0	0
$547$	12.3428	0.527741	0.263871	−	0.964558i	$-0.415001\pi$
$547$	12.3428	0.527741	0.263871	+	0.964558i	$0.415001\pi$
$548$	2.71867	0.116136
$549$	0	0
$550$	0	0
$551$	−55.3634	−2.35856
$552$	0	0
$553$	31.1873	1.32622
$554$	59.9912	2.54878
$555$	0	0
$556$	18.9115	0.802027
$557$	−18.3297	−0.776652	−0.388326	−	0.921522i	$-0.626947\pi$
$557$	−18.3297	−0.776652	−0.388326	+	0.921522i	$0.626947\pi$
$558$	0	0
$559$	2.79859	0.118368
$560$	0	0
$561$	0	0
$562$	−29.3623	−1.23857
$563$	−30.3433	−1.27882	−0.639409	−	0.768867i	$-0.720821\pi$
$563$	−30.3433	−1.27882	−0.639409	+	0.768867i	$0.720821\pi$
$564$	0	0
$565$	0	0
$566$	41.3068	1.73625
$567$	0	0
$568$	−87.8780	−3.68728
$569$	6.10043	0.255743	0.127872	−	0.991791i	$-0.459185\pi$
$569$	6.10043	0.255743	0.127872	+	0.991791i	$0.459185\pi$
$570$	0	0
$571$	27.4222	1.14759	0.573793	−	0.819001i	$-0.305472\pi$
$571$	27.4222	1.14759	0.573793	+	0.819001i	$0.305472\pi$
$572$	29.1248	1.21777
$573$	0	0
$574$	−23.1344	−0.965610
$575$	0	0
$576$	0	0
$577$	17.8321	0.742360	0.371180	−	0.928561i	$-0.378953\pi$
$577$	17.8321	0.742360	0.371180	+	0.928561i	$0.378953\pi$
$578$	29.1987	1.21451
$579$	0	0
$580$	0	0
$581$	−29.3789	−1.21884
$582$	0	0
$583$	29.7006	1.23007
$584$	152.709	6.31916
$585$	0	0
$586$	82.8495	3.42248
$587$	−1.03130	−0.0425665	−0.0212833	−	0.999773i	$-0.506775\pi$
$587$	−1.03130	−0.0425665	−0.0212833	+	0.999773i	$0.506775\pi$
$588$	0	0
$589$	−30.6836	−1.26430
$590$	0	0
$591$	0	0
$592$	−11.3693	−0.467275
$593$	8.26450	0.339382	0.169691	−	0.985497i	$-0.445723\pi$
$593$	8.26450	0.339382	0.169691	+	0.985497i	$0.445723\pi$
$594$	0	0
$595$	0	0
$596$	123.022	5.03918
$597$	0	0
$598$	6.22788	0.254677
$599$	5.13602	0.209852	0.104926	−	0.994480i	$-0.466539\pi$
$599$	5.13602	0.209852	0.104926	+	0.994480i	$0.466539\pi$
$600$	0	0
$601$	−6.94001	−0.283089	−0.141544	−	0.989932i	$-0.545207\pi$
$601$	−6.94001	−0.283089	−0.141544	+	0.989932i	$0.545207\pi$
$602$	−29.0312	−1.18322
$603$	0	0
$604$	−98.0484	−3.98953
$605$	0	0
$606$	0	0
$607$	−4.23493	−0.171890	−0.0859452	−	0.996300i	$-0.527391\pi$
$607$	−4.23493	−0.171890	−0.0859452	+	0.996300i	$0.527391\pi$
$608$	−198.008	−8.03027
$609$	0	0
$610$	0	0
$611$	−2.79260	−0.112976
$612$	0	0
$613$	−26.1200	−1.05498	−0.527488	−	0.849562i	$-0.676866\pi$
$613$	−26.1200	−1.05498	−0.527488	+	0.849562i	$0.676866\pi$
$614$	−53.3008	−2.15104
$615$	0	0
$616$	−197.919	−7.97440
$617$	−16.7556	−0.674554	−0.337277	−	0.941405i	$-0.609506\pi$
$617$	−16.7556	−0.674554	−0.337277	+	0.941405i	$0.609506\pi$
$618$	0	0
$619$	17.3693	0.698131	0.349066	−	0.937098i	$-0.386499\pi$
$619$	17.3693	0.698131	0.349066	+	0.937098i	$0.386499\pi$
$620$	0	0
$621$	0	0
$622$	60.5566	2.42810
$623$	18.3100	0.733574
$624$	0	0
$625$	0	0
$626$	30.8861	1.23446
$627$	0	0
$628$	100.871	4.02519
$629$	−1.61344	−0.0643320
$630$	0	0
$631$	23.8921	0.951130	0.475565	−	0.879681i	$-0.342244\pi$
$631$	23.8921	0.951130	0.475565	+	0.879681i	$0.342244\pi$
$632$	89.0617	3.54269
$633$	0	0
$634$	−40.5637	−1.61099
$635$	0	0
$636$	0	0
$637$	−6.79859	−0.269370
$638$	−114.223	−4.52212
$639$	0	0
$640$	0	0
$641$	15.0407	0.594071	0.297035	−	0.954866i	$-0.404002\pi$
$641$	15.0407	0.594071	0.297035	+	0.954866i	$0.404002\pi$
$642$	0	0
$643$	−16.8656	−0.665115	−0.332558	−	0.943083i	$-0.607912\pi$
$643$	−16.8656	−0.665115	−0.332558	+	0.943083i	$0.607912\pi$
$644$	−48.0366	−1.89291
$645$	0	0
$646$	−48.5687	−1.91091
$647$	−38.8147	−1.52596	−0.762982	−	0.646420i	$-0.776265\pi$
$647$	−38.8147	−1.52596	−0.762982	+	0.646420i	$0.776265\pi$
$648$	0	0
$649$	42.0794	1.65176
$650$	0	0
$651$	0	0
$652$	102.387	4.00977
$653$	−25.8366	−1.01107	−0.505533	−	0.862807i	$-0.668704\pi$
$653$	−25.8366	−1.01107	−0.505533	+	0.862807i	$0.668704\pi$
$654$	0	0
$655$	0	0
$656$	−40.2016	−1.56961
$657$	0	0
$658$	28.9690	1.12933
$659$	−33.0620	−1.28791	−0.643956	−	0.765063i	$-0.722708\pi$
$659$	−33.0620	−1.28791	−0.643956	+	0.765063i	$0.722708\pi$
$660$	0	0
$661$	−13.0335	−0.506945	−0.253473	−	0.967343i	$-0.581573\pi$
$661$	−13.0335	−0.506945	−0.253473	+	0.967343i	$0.581573\pi$
$662$	−14.5254	−0.564547
$663$	0	0
$664$	−83.8974	−3.25585
$665$	0	0
$666$	0	0
$667$	−18.1608	−0.703191
$668$	84.2270	3.25884
$669$	0	0
$670$	0	0
$671$	40.3149	1.55634
$672$	0	0
$673$	−16.3693	−0.630990	−0.315495	−	0.948927i	$-0.602171\pi$
$673$	−16.3693	−0.630990	−0.315495	+	0.948927i	$0.602171\pi$
$674$	1.96901	0.0758432
$675$	0	0
$676$	5.79859	0.223023
$677$	37.9714	1.45936	0.729680	−	0.683788i	$-0.239669\pi$
$677$	37.9714	1.45936	0.729680	+	0.683788i	$0.239669\pi$
$678$	0	0
$679$	−12.7386	−0.488863
$680$	0	0
$681$	0	0
$682$	−63.3048	−2.42407
$683$	−5.86601	−0.224456	−0.112228	−	0.993682i	$-0.535799\pi$
$683$	−5.86601	−0.224456	−0.112228	+	0.993682i	$0.535799\pi$
$684$	0	0
$685$	0	0
$686$	−2.08932	−0.0797706
$687$	0	0
$688$	−50.4487	−1.92334
$689$	5.91322	0.225276
$690$	0	0
$691$	−44.6262	−1.69766	−0.848830	−	0.528666i	$-0.822692\pi$
$691$	−44.6262	−1.69766	−0.848830	+	0.528666i	$0.822692\pi$
$692$	66.6746	2.53459
$693$	0	0
$694$	−70.3938	−2.67211
$695$	0	0
$696$	0	0
$697$	−5.70508	−0.216095
$698$	−12.9317	−0.489471
$699$	0	0
$700$	0	0
$701$	27.5782	1.04162	0.520808	−	0.853674i	$-0.325631\pi$
$701$	27.5782	1.04162	0.520808	+	0.853674i	$0.325631\pi$
$702$	0	0
$703$	−4.28787	−0.161720
$704$	−227.434	−8.57173
$705$	0	0
$706$	−87.3552	−3.28766
$707$	−42.7125	−1.60637
$708$	0	0
$709$	13.4893	0.506601	0.253300	−	0.967388i	$-0.418484\pi$
$709$	13.4893	0.506601	0.253300	+	0.967388i	$0.418484\pi$
$710$	0	0
$711$	0	0
$712$	52.2879	1.95957
$713$	−10.0652	−0.376943
$714$	0	0
$715$	0	0
$716$	110.090	4.11427
$717$	0	0
$718$	49.6166	1.85168
$719$	−19.9234	−0.743018	−0.371509	−	0.928429i	$-0.621160\pi$
$719$	−19.9234	−0.743018	−0.371509	+	0.928429i	$0.621160\pi$
$720$	0	0
$721$	−32.6836	−1.21720
$722$	−76.0168	−2.82905
$723$	0	0
$724$	−145.897	−5.42224
$725$	0	0
$726$	0	0
$727$	12.7986	0.474673	0.237337	−	0.971427i	$-0.423726\pi$
$727$	12.7986	0.474673	0.237337	+	0.971427i	$0.423726\pi$
$728$	−39.4047	−1.46043
$729$	0	0
$730$	0	0
$731$	−7.15927	−0.264795
$732$	0	0
$733$	−40.4417	−1.49375	−0.746874	−	0.664966i	$-0.768446\pi$
$733$	−40.4417	−1.49375	−0.746874	+	0.664966i	$0.768446\pi$
$734$	−79.4654	−2.93312
$735$	0	0
$736$	−64.9524	−2.39418
$737$	−12.6233	−0.464986
$738$	0	0
$739$	47.8276	1.75937	0.879683	−	0.475561i	$-0.157755\pi$
$739$	47.8276	1.75937	0.879683	+	0.475561i	$0.157755\pi$
$740$	0	0
$741$	0	0
$742$	−61.3408	−2.25189
$743$	17.0364	0.625004	0.312502	−	0.949917i	$-0.398833\pi$
$743$	17.0364	0.625004	0.312502	+	0.949917i	$0.398833\pi$
$744$	0	0
$745$	0	0
$746$	8.86632	0.324619
$747$	0	0
$748$	−74.5062	−2.72422
$749$	37.3154	1.36347
$750$	0	0
$751$	23.4222	0.854690	0.427345	−	0.904089i	$-0.359449\pi$
$751$	23.4222	0.854690	0.427345	+	0.904089i	$0.359449\pi$
$752$	50.3406	1.83573
$753$	0	0
$754$	−22.7411	−0.828183
$755$	0	0
$756$	0	0
$757$	−22.6501	−0.823233	−0.411616	−	0.911357i	$-0.635036\pi$
$757$	−22.6501	−0.823233	−0.411616	+	0.911357i	$0.635036\pi$
$758$	−68.9244	−2.50345
$759$	0	0
$760$	0	0
$761$	44.4746	1.61220	0.806102	−	0.591776i	$-0.201573\pi$
$761$	44.4746	1.61220	0.806102	+	0.591776i	$0.201573\pi$
$762$	0	0
$763$	16.3287	0.591140
$764$	60.9683	2.20575
$765$	0	0
$766$	−10.6786	−0.385834
$767$	8.37779	0.302504
$768$	0	0
$769$	3.02647	0.109137	0.0545687	−	0.998510i	$-0.482622\pi$
$769$	3.02647	0.109137	0.0545687	+	0.998510i	$0.482622\pi$
$770$	0	0
$771$	0	0
$772$	8.91354	0.320805
$773$	3.33537	0.119965	0.0599825	−	0.998199i	$-0.480896\pi$
$773$	3.33537	0.119965	0.0599825	+	0.998199i	$0.480896\pi$
$774$	0	0
$775$	0	0
$776$	−36.3777	−1.30588
$777$	0	0
$778$	26.2208	0.940062
$779$	−15.1618	−0.543228
$780$	0	0
$781$	−41.6093	−1.48890
$782$	−15.9320	−0.569727
$783$	0	0
$784$	122.555	4.37695
$785$	0	0
$786$	0	0
$787$	−17.5397	−0.625223	−0.312612	−	0.949881i	$-0.601204\pi$
$787$	−17.5397	−0.625223	−0.312612	+	0.949881i	$0.601204\pi$
$788$	38.7950	1.38202
$789$	0	0
$790$	0	0
$791$	−1.74161	−0.0619246
$792$	0	0
$793$	8.02647	0.285028
$794$	−39.5652	−1.40412
$795$	0	0
$796$	65.3003	2.31451
$797$	22.4619	0.795642	0.397821	−	0.917463i	$-0.369767\pi$
$797$	22.4619	0.795642	0.397821	+	0.917463i	$0.369767\pi$
$798$	0	0
$799$	7.14394	0.252734
$800$	0	0
$801$	0	0
$802$	64.1659	2.26578
$803$	72.3062	2.55163
$804$	0	0
$805$	0	0
$806$	−12.6036	−0.443944
$807$	0	0
$808$	−121.974	−4.29104
$809$	29.5197	1.03786	0.518929	−	0.854817i	$-0.326331\pi$
$809$	29.5197	1.03786	0.518929	+	0.854817i	$0.326331\pi$
$810$	0	0
$811$	20.3383	0.714174	0.357087	−	0.934071i	$-0.383770\pi$
$811$	20.3383	0.714174	0.357087	+	0.934071i	$0.383770\pi$
$812$	175.406	6.15553
$813$	0	0
$814$	−8.84650	−0.310070
$815$	0	0
$816$	0	0
$817$	−19.0265	−0.665652
$818$	17.5792	0.614641
$819$	0	0
$820$	0	0
$821$	14.0566	0.490578	0.245289	−	0.969450i	$-0.421117\pi$
$821$	14.0566	0.490578	0.245289	+	0.969450i	$0.421117\pi$
$822$	0	0
$823$	−14.6766	−0.511594	−0.255797	−	0.966731i	$-0.582338\pi$
$823$	−14.6766	−0.511594	−0.255797	+	0.966731i	$0.582338\pi$
$824$	−93.3348	−3.25147
$825$	0	0
$826$	−86.9070	−3.02388
$827$	−4.59324	−0.159723	−0.0798614	−	0.996806i	$-0.525448\pi$
$827$	−4.59324	−0.159723	−0.0798614	+	0.996806i	$0.525448\pi$
$828$	0	0
$829$	44.5493	1.54726	0.773630	−	0.633637i	$-0.218439\pi$
$829$	44.5493	1.54726	0.773630	+	0.633637i	$0.218439\pi$
$830$	0	0
$831$	0	0
$832$	−45.2808	−1.56983
$833$	17.3920	0.602596
$834$	0	0
$835$	0	0
$836$	−198.008	−6.84823
$837$	0	0
$838$	3.08646	0.106620
$839$	31.8977	1.10123	0.550616	−	0.834759i	$-0.314393\pi$
$839$	31.8977	1.10123	0.550616	+	0.834759i	$0.314393\pi$
$840$	0	0
$841$	37.3143	1.28670
$842$	−16.9034	−0.582531
$843$	0	0
$844$	−1.32139	−0.0454842
$845$	0	0
$846$	0	0
$847$	−52.8515	−1.81600
$848$	−106.595	−3.66047
$849$	0	0
$850$	0	0
$851$	−1.40655	−0.0482159
$852$	0	0
$853$	33.5831	1.14986	0.574932	−	0.818202i	$-0.305029\pi$
$853$	33.5831	1.14986	0.574932	+	0.818202i	$0.305029\pi$
$854$	−83.2626	−2.84919
$855$	0	0
$856$	106.562	3.64220
$857$	−51.3523	−1.75416	−0.877080	−	0.480344i	$-0.840512\pi$
$857$	−51.3523	−1.75416	−0.877080	+	0.480344i	$0.840512\pi$
$858$	0	0
$859$	−9.32139	−0.318042	−0.159021	−	0.987275i	$-0.550834\pi$
$859$	−9.32139	−0.318042	−0.159021	+	0.987275i	$0.550834\pi$
$860$	0	0
$861$	0	0
$862$	−16.8515	−0.573966
$863$	−31.8238	−1.08329	−0.541647	−	0.840606i	$-0.682199\pi$
$863$	−31.8238	−1.08329	−0.541647	+	0.840606i	$0.682199\pi$
$864$	0	0
$865$	0	0
$866$	2.53146	0.0860225
$867$	0	0
$868$	97.2138	3.29965
$869$	42.1697	1.43051
$870$	0	0
$871$	−2.51324	−0.0851578
$872$	46.6300	1.57909
$873$	0	0
$874$	−42.3408	−1.43220
$875$	0	0
$876$	0	0
$877$	10.9065	0.368286	0.184143	−	0.982899i	$-0.441049\pi$
$877$	10.9065	0.368286	0.184143	+	0.982899i	$0.441049\pi$
$878$	43.0681	1.45348
$879$	0	0
$880$	0	0
$881$	27.5979	0.929798	0.464899	−	0.885364i	$-0.346091\pi$
$881$	27.5979	0.929798	0.464899	+	0.885364i	$0.346091\pi$
$882$	0	0
$883$	7.49633	0.252272	0.126136	−	0.992013i	$-0.459743\pi$
$883$	7.49633	0.252272	0.126136	+	0.992013i	$0.459743\pi$
$884$	−14.8338	−0.498914
$885$	0	0
$886$	79.7631	2.67969
$887$	10.7212	0.359983	0.179992	−	0.983668i	$-0.442393\pi$
$887$	10.7212	0.359983	0.179992	+	0.983668i	$0.442393\pi$
$888$	0	0
$889$	−25.8780	−0.867920
$890$	0	0
$891$	0	0
$892$	55.9980	1.87495
$893$	18.9857	0.635333
$894$	0	0
$895$	0	0
$896$	253.344	8.46364
$897$	0	0
$898$	84.6816	2.82586
$899$	36.7529	1.22578
$900$	0	0
$901$	−15.1270	−0.503954
$902$	−31.2810	−1.04154
$903$	0	0
$904$	−4.97353	−0.165417
$905$	0	0
$906$	0	0
$907$	−30.0529	−0.997892	−0.498946	−	0.866633i	$-0.666279\pi$
$907$	−30.0529	−0.997892	−0.498946	+	0.866633i	$0.666279\pi$
$908$	39.4519	1.30926
$909$	0	0
$910$	0	0
$911$	−26.8207	−0.888610	−0.444305	−	0.895876i	$-0.646549\pi$
$911$	−26.8207	−0.888610	−0.444305	+	0.895876i	$0.646549\pi$
$912$	0	0
$913$	−39.7245	−1.31469
$914$	−52.0280	−1.72093
$915$	0	0
$916$	36.7386	1.21388
$917$	−68.7836	−2.27143
$918$	0	0
$919$	40.5566	1.33784	0.668920	−	0.743335i	$-0.266757\pi$
$919$	40.5566	1.33784	0.668920	+	0.743335i	$0.266757\pi$
$920$	0	0
$921$	0	0
$922$	70.3938	2.31830
$923$	−8.28418	−0.272677
$924$	0	0
$925$	0	0
$926$	77.3761	2.54274
$927$	0	0
$928$	237.174	7.78561
$929$	14.0369	0.460536	0.230268	−	0.973127i	$-0.426040\pi$
$929$	14.0369	0.460536	0.230268	+	0.973127i	$0.426040\pi$
$930$	0	0
$931$	46.2208	1.51483
$932$	−93.3545	−3.05793
$933$	0	0
$934$	−92.2738	−3.01929
$935$	0	0
$936$	0	0
$937$	49.2879	1.61016	0.805082	−	0.593163i	$-0.202121\pi$
$937$	49.2879	1.61016	0.805082	+	0.593163i	$0.202121\pi$
$938$	26.0711	0.851250
$939$	0	0
$940$	0	0
$941$	−5.60487	−0.182714	−0.0913568	−	0.995818i	$-0.529120\pi$
$941$	−5.60487	−0.182714	−0.0913568	+	0.995818i	$0.529120\pi$
$942$	0	0
$943$	−4.97353	−0.161960
$944$	−151.022	−4.91535
$945$	0	0
$946$	−39.2544	−1.27627
$947$	38.0650	1.23695	0.618474	−	0.785805i	$-0.287751\pi$
$947$	38.0650	1.23695	0.618474	+	0.785805i	$0.287751\pi$
$948$	0	0
$949$	14.3958	0.467307
$950$	0	0
$951$	0	0
$952$	100.804	3.26707
$953$	−8.63189	−0.279614	−0.139807	−	0.990179i	$-0.544648\pi$
$953$	−8.63189	−0.279614	−0.139807	+	0.990179i	$0.544648\pi$
$954$	0	0
$955$	0	0
$956$	39.3378	1.27228
$957$	0	0
$958$	−46.0073	−1.48643
$959$	−1.74161	−0.0562396
$960$	0	0
$961$	−10.6307	−0.342926
$962$	−1.76129	−0.0567863
$963$	0	0
$964$	125.581	4.04469
$965$	0	0
$966$	0	0
$967$	−48.8011	−1.56934	−0.784669	−	0.619915i	$-0.787167\pi$
$967$	−48.8011	−1.56934	−0.784669	+	0.619915i	$0.787167\pi$
$968$	−150.928	−4.85102
$969$	0	0
$970$	0	0
$971$	−53.4149	−1.71417	−0.857083	−	0.515179i	$-0.827725\pi$
$971$	−53.4149	−1.71417	−0.857083	+	0.515179i	$0.827725\pi$
$972$	0	0
$973$	−12.1149	−0.388387
$974$	47.5213	1.52268
$975$	0	0
$976$	−144.689	−4.63138
$977$	13.3808	0.428091	0.214046	−	0.976824i	$-0.431336\pi$
$977$	13.3808	0.428091	0.214046	+	0.976824i	$0.431336\pi$
$978$	0	0
$979$	24.7577	0.791260
$980$	0	0
$981$	0	0
$982$	−4.45073	−0.142028
$983$	−35.3464	−1.12737	−0.563687	−	0.825988i	$-0.690618\pi$
$983$	−35.3464	−1.12737	−0.563687	+	0.825988i	$0.690618\pi$
$984$	0	0
$985$	0	0
$986$	58.1757	1.85269
$987$	0	0
$988$	−39.4222	−1.25419
$989$	−6.24125	−0.198460
$990$	0	0
$991$	54.1608	1.72048	0.860238	−	0.509893i	$-0.170315\pi$
$991$	54.1608	1.72048	0.860238	+	0.509893i	$0.170315\pi$
$992$	131.447	4.17345
$993$	0	0
$994$	85.9360	2.72572
$995$	0	0
$996$	0	0
$997$	6.64509	0.210452	0.105226	−	0.994448i	$-0.466443\pi$
$997$	6.64509	0.210452	0.105226	+	0.994448i	$0.466443\pi$
$998$	−98.6117	−3.12150
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2925.2.a.bp.1.1	✓	6
3.2	odd	2	inner	2925.2.a.bp.1.6	yes	6
5.2	odd	4		2925.2.c.z.2224.1		12
5.3	odd	4		2925.2.c.z.2224.12		12
5.4	even	2		2925.2.a.bq.1.6	yes	6
15.2	even	4		2925.2.c.z.2224.11		12
15.8	even	4		2925.2.c.z.2224.2		12
15.14	odd	2		2925.2.a.bq.1.1	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2925.2.a.bp.1.1	✓	6	1.1	even	1	trivial
2925.2.a.bp.1.6	yes	6	3.2	odd	2	inner
2925.2.a.bq.1.1	yes	6	15.14	odd	2
2925.2.a.bq.1.6	yes	6	5.4	even	2
2925.2.c.z.2224.1		12	5.2	odd	4
2925.2.c.z.2224.2		12	15.8	even	4
2925.2.c.z.2224.11		12	15.2	even	4
2925.2.c.z.2224.12		12	5.3	odd	4

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:	\( N \)	\(=\)	\( 2925 = 3^{2} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2925.a (trivial)

\(f(q)\)	\(=\)	\(q-2.79260 q^{2} +5.79859 q^{4} -3.71465 q^{7} -10.6079 q^{8} +O(q^{10})\)	\(q-2.79260 q^{2} +5.79859 q^{4} -3.71465 q^{7} -10.6079 q^{8} -5.02274 q^{11} -1.00000 q^{13} +10.3735 q^{14} +18.0265 q^{16} +2.55817 q^{17} +6.79859 q^{19} +14.0265 q^{22} +2.23014 q^{23} +2.79260 q^{26} -21.5397 q^{28} -8.14336 q^{29} -4.51324 q^{31} -29.1248 q^{32} -7.14394 q^{34} -0.630700 q^{37} -18.9857 q^{38} -2.23014 q^{41} -2.79859 q^{43} -29.1248 q^{44} -6.22788 q^{46} +2.79260 q^{47} +6.79859 q^{49} -5.79859 q^{52} -5.91322 q^{53} +39.4047 q^{56} +22.7411 q^{58} -8.37779 q^{59} -8.02647 q^{61} +12.6036 q^{62} +45.2808 q^{64} +2.51324 q^{67} +14.8338 q^{68} +8.28418 q^{71} -14.3958 q^{73} +1.76129 q^{74} +39.4222 q^{76} +18.6577 q^{77} -8.39577 q^{79} +6.22788 q^{82} +7.90894 q^{83} +7.81533 q^{86} +53.2808 q^{88} -4.92913 q^{89} +3.71465 q^{91} +12.9317 q^{92} -7.79859 q^{94} +3.42929 q^{97} -18.9857 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} - 6 q^{7}+O(q^{10}) \) 6 * q + 14 * q^4 - 6 * q^7	\( 6 q + 14 q^{4} - 6 q^{7} - 6 q^{13} + 34 q^{16} + 20 q^{19} + 10 q^{22} - 18 q^{28} + 10 q^{31} + 6 q^{34} + 8 q^{37} + 4 q^{43} + 16 q^{46} + 20 q^{49} - 14 q^{52} + 46 q^{58} + 26 q^{61} + 70 q^{64} - 22 q^{67} - 24 q^{73} + 100 q^{76} + 12 q^{79} - 16 q^{82} + 118 q^{88} + 6 q^{91} - 26 q^{94} - 12 q^{97}+O(q^{100}) \) 6 * q + 14 * q^4 - 6 * q^7 - 6 * q^13 + 34 * q^16 + 20 * q^19 + 10 * q^22 - 18 * q^28 + 10 * q^31 + 6 * q^34 + 8 * q^37 + 4 * q^43 + 16 * q^46 + 20 * q^49 - 14 * q^52 + 46 * q^58 + 26 * q^61 + 70 * q^64 - 22 * q^67 - 24 * q^73 + 100 * q^76 + 12 * q^79 - 16 * q^82 + 118 * q^88 + 6 * q^91 - 26 * q^94 - 12 * q^97

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