LMFDB - Embedded newform 9025.2.a.bi.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{20})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 5 $ x^4 - 5*x^2 + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.90211$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.726543 q^{3} -2.00000 q^{4} -1.61803 q^{7} -2.47214 q^{9} +O(q^{10})$	$q+0.726543 q^{3} -2.00000 q^{4} -1.61803 q^{7} -2.47214 q^{9} +0.854102 q^{11} -1.45309 q^{12} -0.726543 q^{13} +4.00000 q^{16} -3.85410 q^{17} -1.17557 q^{21} +3.47214 q^{23} -3.97574 q^{27} +3.23607 q^{28} -2.35114 q^{29} +6.88191 q^{31} +0.620541 q^{33} +4.94427 q^{36} +7.77997 q^{37} -0.527864 q^{39} +5.98385 q^{41} +9.61803 q^{43} -1.70820 q^{44} +8.70820 q^{47} +2.90617 q^{48} -4.38197 q^{49} -2.80017 q^{51} +1.45309 q^{52} -12.5882 q^{53} -2.62866 q^{59} -2.76393 q^{61} +4.00000 q^{63} -8.00000 q^{64} +3.35520 q^{67} +7.70820 q^{68} +2.52265 q^{69} -13.0373 q^{71} +11.0000 q^{73} -1.38197 q^{77} +16.2865 q^{79} +4.52786 q^{81} +2.70820 q^{83} +2.35114 q^{84} -1.70820 q^{87} -13.3148 q^{89} +1.17557 q^{91} -6.94427 q^{92} +5.00000 q^{93} +7.77997 q^{97} -2.11146 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9	$ 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} - 2 q^{17} - 4 q^{23} + 4 q^{28} - 16 q^{36} - 20 q^{39} + 34 q^{43} + 20 q^{44} + 8 q^{47} - 22 q^{49} - 20 q^{61} + 16 q^{63} - 32 q^{64} + 4 q^{68} + 44 q^{73} - 10 q^{77} + 36 q^{81} - 16 q^{83} + 20 q^{87} + 8 q^{92} + 20 q^{93} - 80 q^{99}+O(q^{100}) $ 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9 - 10 * q^11 + 16 * q^16 - 2 * q^17 - 4 * q^23 + 4 * q^28 - 16 * q^36 - 20 * q^39 + 34 * q^43 + 20 * q^44 + 8 * q^47 - 22 * q^49 - 20 * q^61 + 16 * q^63 - 32 * q^64 + 4 * q^68 + 44 * q^73 - 10 * q^77 + 36 * q^81 - 16 * q^83 + 20 * q^87 + 8 * q^92 + 20 * q^93 - 80 * 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		−	1.00000i	$-0.5\pi$
$2$	0	0			1.00000i	$0.5\pi$
$3$	0.726543	0.419470	0.209735	−	0.977758i	$-0.432740\pi$
$3$	0.726543	0.419470	0.209735	+	0.977758i	$0.432740\pi$
$4$	−2.00000	−1.00000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.61803	−0.611559	−0.305780	−	0.952102i	$-0.598917\pi$
$7$	−1.61803	−0.611559	−0.305780	+	0.952102i	$0.598917\pi$
$8$	0	0
$9$	−2.47214	−0.824045
$10$	0	0
$11$	0.854102	0.257521	0.128761	−	0.991676i	$-0.458900\pi$
$11$	0.854102	0.257521	0.128761	+	0.991676i	$0.458900\pi$
$12$	−1.45309	−0.419470
$13$	−0.726543	−0.201507	−0.100753	−	0.994911i	$-0.532125\pi$
$13$	−0.726543	−0.201507	−0.100753	+	0.994911i	$0.532125\pi$
$14$	0	0
$15$	0	0
$16$	4.00000	1.00000
$17$	−3.85410	−0.934757	−0.467379	−	0.884057i	$-0.654801\pi$
$17$	−3.85410	−0.934757	−0.467379	+	0.884057i	$0.654801\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.17557	−0.256531
$22$	0	0
$23$	3.47214	0.723990	0.361995	−	0.932180i	$-0.382096\pi$
$23$	3.47214	0.723990	0.361995	+	0.932180i	$0.382096\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.97574	−0.765131
$28$	3.23607	0.611559
$29$	−2.35114	−0.436596	−0.218298	−	0.975882i	$-0.570050\pi$
$29$	−2.35114	−0.436596	−0.218298	+	0.975882i	$0.570050\pi$
$30$	0	0
$31$	6.88191	1.23603	0.618014	−	0.786167i	$-0.287938\pi$
$31$	6.88191	1.23603	0.618014	+	0.786167i	$0.287938\pi$
$32$	0	0
$33$	0.620541	0.108022
$34$	0	0
$35$	0	0
$36$	4.94427	0.824045
$37$	7.77997	1.27902	0.639509	−	0.768783i	$-0.279138\pi$
$37$	7.77997	1.27902	0.639509	+	0.768783i	$0.279138\pi$
$38$	0	0
$39$	−0.527864	−0.0845259
$40$	0	0
$41$	5.98385	0.934521	0.467260	−	0.884120i	$-0.345241\pi$
$41$	5.98385	0.934521	0.467260	+	0.884120i	$0.345241\pi$
$42$	0	0
$43$	9.61803	1.46674	0.733368	−	0.679832i	$-0.237947\pi$
$43$	9.61803	1.46674	0.733368	+	0.679832i	$0.237947\pi$
$44$	−1.70820	−0.257521
$45$	0	0
$46$	0	0
$47$	8.70820	1.27022	0.635111	−	0.772421i	$-0.280954\pi$
$47$	8.70820	1.27022	0.635111	+	0.772421i	$0.280954\pi$
$48$	2.90617	0.419470
$49$	−4.38197	−0.625995
$50$	0	0
$51$	−2.80017	−0.392102
$52$	1.45309	0.201507
$53$	−12.5882	−1.72913	−0.864564	−	0.502522i	$-0.832406\pi$
$53$	−12.5882	−1.72913	−0.864564	+	0.502522i	$0.832406\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.62866	−0.342222	−0.171111	−	0.985252i	$-0.554736\pi$
$59$	−2.62866	−0.342222	−0.171111	+	0.985252i	$0.554736\pi$
$60$	0	0
$61$	−2.76393	−0.353885	−0.176943	−	0.984221i	$-0.556621\pi$
$61$	−2.76393	−0.353885	−0.176943	+	0.984221i	$0.556621\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	−8.00000	−1.00000
$65$	0	0
$66$	0	0
$67$	3.35520	0.409903	0.204951	−	0.978772i	$-0.434296\pi$
$67$	3.35520	0.409903	0.204951	+	0.978772i	$0.434296\pi$
$68$	7.70820	0.934757
$69$	2.52265	0.303692
$70$	0	0
$71$	−13.0373	−1.54724	−0.773620	−	0.633650i	$-0.781556\pi$
$71$	−13.0373	−1.54724	−0.773620	+	0.633650i	$0.781556\pi$
$72$	0	0
$73$	11.0000	1.28745	0.643726	−	0.765256i	$-0.277388\pi$
$73$	11.0000	1.28745	0.643726	+	0.765256i	$0.277388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.38197	−0.157490
$78$	0	0
$79$	16.2865	1.83237	0.916186	−	0.400754i	$-0.131252\pi$
$79$	16.2865	1.83237	0.916186	+	0.400754i	$0.131252\pi$
$80$	0	0
$81$	4.52786	0.503096
$82$	0	0
$83$	2.70820	0.297264	0.148632	−	0.988893i	$-0.452513\pi$
$83$	2.70820	0.297264	0.148632	+	0.988893i	$0.452513\pi$
$84$	2.35114	0.256531
$85$	0	0
$86$	0	0
$87$	−1.70820	−0.183139
$88$	0	0
$89$	−13.3148	−1.41137	−0.705683	−	0.708528i	$-0.749360\pi$
$89$	−13.3148	−1.41137	−0.705683	+	0.708528i	$0.749360\pi$
$90$	0	0
$91$	1.17557	0.123233
$92$	−6.94427	−0.723990
$93$	5.00000	0.518476
$94$	0	0
$95$	0	0
$96$	0	0
$97$	7.77997	0.789936	0.394968	−	0.918695i	$-0.370756\pi$
$97$	7.77997	0.789936	0.394968	+	0.918695i	$0.370756\pi$
$98$	0	0
$99$	−2.11146	−0.212209
$100$	0	0
$101$	−13.0902	−1.30252	−0.651260	−	0.758854i	$-0.725759\pi$
$101$	−13.0902	−1.30252	−0.651260	+	0.758854i	$0.725759\pi$
$102$	0	0
$103$	10.8576	1.06984	0.534918	−	0.844904i	$-0.320343\pi$
$103$	10.8576	1.06984	0.534918	+	0.844904i	$0.320343\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.2905	−1.67154	−0.835769	−	0.549081i	$-0.814978\pi$
$107$	−17.2905	−1.67154	−0.835769	+	0.549081i	$0.814978\pi$
$108$	7.95148	0.765131
$109$	−13.5923	−1.30191	−0.650953	−	0.759118i	$-0.725631\pi$
$109$	−13.5923	−1.30191	−0.650953	+	0.759118i	$0.725631\pi$
$110$	0	0
$111$	5.65248	0.536509
$112$	−6.47214	−0.611559
$113$	−8.78402	−0.826331	−0.413166	−	0.910656i	$-0.635577\pi$
$113$	−8.78402	−0.826331	−0.413166	+	0.910656i	$0.635577\pi$
$114$	0	0
$115$	0	0
$116$	4.70228	0.436596
$117$	1.79611	0.166051
$118$	0	0
$119$	6.23607	0.571659
$120$	0	0
$121$	−10.2705	−0.933683
$122$	0	0
$123$	4.34752	0.392003
$124$	−13.7638	−1.23603
$125$	0	0
$126$	0	0
$127$	−18.0171	−1.59876	−0.799378	−	0.600828i	$-0.794838\pi$
$127$	−18.0171	−1.59876	−0.799378	+	0.600828i	$0.794838\pi$
$128$	0	0
$129$	6.98791	0.615251
$130$	0	0
$131$	0.763932	0.0667451	0.0333725	−	0.999443i	$-0.489375\pi$
$131$	0.763932	0.0667451	0.0333725	+	0.999443i	$0.489375\pi$
$132$	−1.24108	−0.108022
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.56231	0.475220	0.237610	−	0.971361i	$-0.423636\pi$
$137$	5.56231	0.475220	0.237610	+	0.971361i	$0.423636\pi$
$138$	0	0
$139$	−2.61803	−0.222059	−0.111029	−	0.993817i	$-0.535415\pi$
$139$	−2.61803	−0.222059	−0.111029	+	0.993817i	$0.535415\pi$
$140$	0	0
$141$	6.32688	0.532819
$142$	0	0
$143$	−0.620541	−0.0518923
$144$	−9.88854	−0.824045
$145$	0	0
$146$	0	0
$147$	−3.18368	−0.262586
$148$	−15.5599	−1.27902
$149$	−8.23607	−0.674725	−0.337362	−	0.941375i	$-0.609535\pi$
$149$	−8.23607	−0.674725	−0.337362	+	0.941375i	$0.609535\pi$
$150$	0	0
$151$	−7.33094	−0.596583	−0.298292	−	0.954475i	$-0.596417\pi$
$151$	−7.33094	−0.596583	−0.298292	+	0.954475i	$0.596417\pi$
$152$	0	0
$153$	9.52786	0.770282
$154$	0	0
$155$	0	0
$156$	1.05573	0.0845259
$157$	14.0344	1.12007	0.560035	−	0.828469i	$-0.310788\pi$
$157$	14.0344	1.12007	0.560035	+	0.828469i	$0.310788\pi$
$158$	0	0
$159$	−9.14590	−0.725317
$160$	0	0
$161$	−5.61803	−0.442763
$162$	0	0
$163$	0.909830	0.0712634	0.0356317	−	0.999365i	$-0.488656\pi$
$163$	0.909830	0.0712634	0.0356317	+	0.999365i	$0.488656\pi$
$164$	−11.9677	−0.934521
$165$	0	0
$166$	0	0
$167$	17.2905	1.33798	0.668991	−	0.743271i	$-0.266727\pi$
$167$	17.2905	1.33798	0.668991	+	0.743271i	$0.266727\pi$
$168$	0	0
$169$	−12.4721	−0.959395
$170$	0	0
$171$	0	0
$172$	−19.2361	−1.46674
$173$	6.60440	0.502123	0.251061	−	0.967971i	$-0.419220\pi$
$173$	6.60440	0.502123	0.251061	+	0.967971i	$0.419220\pi$
$174$	0	0
$175$	0	0
$176$	3.41641	0.257521
$177$	−1.90983	−0.143552
$178$	0	0
$179$	−4.53077	−0.338646	−0.169323	−	0.985561i	$-0.554158\pi$
$179$	−4.53077	−0.338646	−0.169323	+	0.985561i	$0.554158\pi$
$180$	0	0
$181$	−7.43694	−0.552783	−0.276392	−	0.961045i	$-0.589139\pi$
$181$	−7.43694	−0.552783	−0.276392	+	0.961045i	$0.589139\pi$
$182$	0	0
$183$	−2.00811	−0.148444
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−3.29180	−0.240720
$188$	−17.4164	−1.27022
$189$	6.43288	0.467923
$190$	0	0
$191$	−5.09017	−0.368312	−0.184156	−	0.982897i	$-0.558955\pi$
$191$	−5.09017	−0.368312	−0.184156	+	0.982897i	$0.558955\pi$
$192$	−5.81234	−0.419470
$193$	−3.91023	−0.281464	−0.140732	−	0.990048i	$-0.544946\pi$
$193$	−3.91023	−0.281464	−0.140732	+	0.990048i	$0.544946\pi$
$194$	0	0
$195$	0	0
$196$	8.76393	0.625995
$197$	−22.6525	−1.61392	−0.806961	−	0.590605i	$-0.798889\pi$
$197$	−22.6525	−1.61392	−0.806961	+	0.590605i	$0.798889\pi$
$198$	0	0
$199$	12.5623	0.890518	0.445259	−	0.895402i	$-0.353112\pi$
$199$	12.5623	0.890518	0.445259	+	0.895402i	$0.353112\pi$
$200$	0	0
$201$	2.43769	0.171942
$202$	0	0
$203$	3.80423	0.267004
$204$	5.60034	0.392102
$205$	0	0
$206$	0	0
$207$	−8.58359	−0.596601
$208$	−2.90617	−0.201507
$209$	0	0
$210$	0	0
$211$	2.80017	0.192772	0.0963858	−	0.995344i	$-0.469272\pi$
$211$	2.80017	0.192772	0.0963858	+	0.995344i	$0.469272\pi$
$212$	25.1765	1.72913
$213$	−9.47214	−0.649020
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−11.1352	−0.755904
$218$	0	0
$219$	7.99197	0.540047
$220$	0	0
$221$	2.80017	0.188360
$222$	0	0
$223$	−5.15131	−0.344957	−0.172479	−	0.985013i	$-0.555178\pi$
$223$	−5.15131	−0.344957	−0.172479	+	0.985013i	$0.555178\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	26.9726	1.79023	0.895117	−	0.445830i	$-0.147092\pi$
$227$	26.9726	1.79023	0.895117	+	0.445830i	$0.147092\pi$
$228$	0	0
$229$	−4.85410	−0.320768	−0.160384	−	0.987055i	$-0.551273\pi$
$229$	−4.85410	−0.320768	−0.160384	+	0.987055i	$0.551273\pi$
$230$	0	0
$231$	−1.00406	−0.0660621
$232$	0	0
$233$	−12.9443	−0.848007	−0.424004	−	0.905660i	$-0.639376\pi$
$233$	−12.9443	−0.848007	−0.424004	+	0.905660i	$0.639376\pi$
$234$	0	0
$235$	0	0
$236$	5.25731	0.342222
$237$	11.8328	0.768624
$238$	0	0
$239$	3.94427	0.255134	0.127567	−	0.991830i	$-0.459283\pi$
$239$	3.94427	0.255134	0.127567	+	0.991830i	$0.459283\pi$
$240$	0	0
$241$	20.3682	1.31203	0.656016	−	0.754747i	$-0.272240\pi$
$241$	20.3682	1.31203	0.656016	+	0.754747i	$0.272240\pi$
$242$	0	0
$243$	15.2169	0.976165
$244$	5.52786	0.353885
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.96763	0.124693
$250$	0	0
$251$	6.14590	0.387926	0.193963	−	0.981009i	$-0.437866\pi$
$251$	6.14590	0.387926	0.193963	+	0.981009i	$0.437866\pi$
$252$	−8.00000	−0.503953
$253$	2.96556	0.186443
$254$	0	0
$255$	0	0
$256$	16.0000	1.00000
$257$	−29.6013	−1.84648	−0.923238	−	0.384228i	$-0.874468\pi$
$257$	−29.6013	−1.84648	−0.923238	+	0.384228i	$0.874468\pi$
$258$	0	0
$259$	−12.5882	−0.782196
$260$	0	0
$261$	5.81234	0.359775
$262$	0	0
$263$	−10.9098	−0.672729	−0.336364	−	0.941732i	$-0.609197\pi$
$263$	−10.9098	−0.672729	−0.336364	+	0.941732i	$0.609197\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−9.67376	−0.592025
$268$	−6.71040	−0.409903
$269$	19.0866	1.16373	0.581867	−	0.813284i	$-0.302323\pi$
$269$	19.0866	1.16373	0.581867	+	0.813284i	$0.302323\pi$
$270$	0	0
$271$	−18.4164	−1.11872	−0.559359	−	0.828926i	$-0.688952\pi$
$271$	−18.4164	−1.11872	−0.559359	+	0.828926i	$0.688952\pi$
$272$	−15.4164	−0.934757
$273$	0.854102	0.0516926
$274$	0	0
$275$	0	0
$276$	−5.04531	−0.303692
$277$	−25.3607	−1.52378	−0.761888	−	0.647709i	$-0.775727\pi$
$277$	−25.3607	−1.52378	−0.761888	+	0.647709i	$0.775727\pi$
$278$	0	0
$279$	−17.0130	−1.01854
$280$	0	0
$281$	−4.42477	−0.263959	−0.131980	−	0.991252i	$-0.542133\pi$
$281$	−4.42477	−0.263959	−0.131980	+	0.991252i	$0.542133\pi$
$282$	0	0
$283$	24.0000	1.42665	0.713326	−	0.700832i	$-0.247188\pi$
$283$	24.0000	1.42665	0.713326	+	0.700832i	$0.247188\pi$
$284$	26.0746	1.54724
$285$	0	0
$286$	0	0
$287$	−9.68208	−0.571515
$288$	0	0
$289$	−2.14590	−0.126229
$290$	0	0
$291$	5.65248	0.331354
$292$	−22.0000	−1.28745
$293$	6.04937	0.353408	0.176704	−	0.984264i	$-0.443457\pi$
$293$	6.04937	0.353408	0.176704	+	0.984264i	$0.443457\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−3.39569	−0.197038
$298$	0	0
$299$	−2.52265	−0.145889
$300$	0	0
$301$	−15.5623	−0.896996
$302$	0	0
$303$	−9.51057	−0.546368
$304$	0	0
$305$	0	0
$306$	0	0
$307$	24.0009	1.36981	0.684903	−	0.728635i	$-0.259845\pi$
$307$	24.0009	1.36981	0.684903	+	0.728635i	$0.259845\pi$
$308$	2.76393	0.157490
$309$	7.88854	0.448764
$310$	0	0
$311$	2.03444	0.115363	0.0576813	−	0.998335i	$-0.481629\pi$
$311$	2.03444	0.115363	0.0576813	+	0.998335i	$0.481629\pi$
$312$	0	0
$313$	−29.3262	−1.65762	−0.828808	−	0.559532i	$-0.810981\pi$
$313$	−29.3262	−1.65762	−0.828808	+	0.559532i	$0.810981\pi$
$314$	0	0
$315$	0	0
$316$	−32.5729	−1.83237
$317$	−8.12299	−0.456233	−0.228116	−	0.973634i	$-0.573257\pi$
$317$	−8.12299	−0.456233	−0.228116	+	0.973634i	$0.573257\pi$
$318$	0	0
$319$	−2.00811	−0.112433
$320$	0	0
$321$	−12.5623	−0.701160
$322$	0	0
$323$	0	0
$324$	−9.05573	−0.503096
$325$	0	0
$326$	0	0
$327$	−9.87539	−0.546110
$328$	0	0
$329$	−14.0902	−0.776816
$330$	0	0
$331$	−17.1845	−0.944547	−0.472274	−	0.881452i	$-0.656567\pi$
$331$	−17.1845	−0.944547	−0.472274	+	0.881452i	$0.656567\pi$
$332$	−5.41641	−0.297264
$333$	−19.2331	−1.05397
$334$	0	0
$335$	0	0
$336$	−4.70228	−0.256531
$337$	−13.0373	−0.710186	−0.355093	−	0.934831i	$-0.615551\pi$
$337$	−13.0373	−0.710186	−0.355093	+	0.934831i	$0.615551\pi$
$338$	0	0
$339$	−6.38197	−0.346621
$340$	0	0
$341$	5.87785	0.318304
$342$	0	0
$343$	18.4164	0.994393
$344$	0	0
$345$	0	0
$346$	0	0
$347$	11.2918	0.606175	0.303088	−	0.952963i	$-0.401982\pi$
$347$	11.2918	0.606175	0.303088	+	0.952963i	$0.401982\pi$
$348$	3.41641	0.183139
$349$	−30.1246	−1.61253	−0.806267	−	0.591552i	$-0.798515\pi$
$349$	−30.1246	−1.61253	−0.806267	+	0.591552i	$0.798515\pi$
$350$	0	0
$351$	2.88854	0.154179
$352$	0	0
$353$	−17.6180	−0.937713	−0.468857	−	0.883274i	$-0.655334\pi$
$353$	−17.6180	−0.937713	−0.468857	+	0.883274i	$0.655334\pi$
$354$	0	0
$355$	0	0
$356$	26.6296	1.41137
$357$	4.53077	0.239794
$358$	0	0
$359$	−26.5623	−1.40190	−0.700952	−	0.713208i	$-0.747241\pi$
$359$	−26.5623	−1.40190	−0.700952	+	0.713208i	$0.747241\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−7.46196	−0.391651
$364$	−2.35114	−0.123233
$365$	0	0
$366$	0	0
$367$	−17.6525	−0.921452	−0.460726	−	0.887542i	$-0.652411\pi$
$367$	−17.6525	−0.921452	−0.460726	+	0.887542i	$0.652411\pi$
$368$	13.8885	0.723990
$369$	−14.7929	−0.770088
$370$	0	0
$371$	20.3682	1.05746
$372$	−10.0000	−0.518476
$373$	30.9888	1.60454	0.802271	−	0.596961i	$-0.203625\pi$
$373$	30.9888	1.60454	0.802271	+	0.596961i	$0.203625\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.70820	0.0879770
$378$	0	0
$379$	7.26543	0.373200	0.186600	−	0.982436i	$-0.440253\pi$
$379$	7.26543	0.373200	0.186600	+	0.982436i	$0.440253\pi$
$380$	0	0
$381$	−13.0902	−0.670630
$382$	0	0
$383$	−20.9888	−1.07248	−0.536238	−	0.844067i	$-0.680155\pi$
$383$	−20.9888	−1.07248	−0.536238	+	0.844067i	$0.680155\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−23.7771	−1.20866
$388$	−15.5599	−0.789936
$389$	27.5623	1.39746	0.698732	−	0.715383i	$-0.253748\pi$
$389$	27.5623	1.39746	0.698732	+	0.715383i	$0.253748\pi$
$390$	0	0
$391$	−13.3820	−0.676755
$392$	0	0
$393$	0.555029	0.0279975
$394$	0	0
$395$	0	0
$396$	4.22291	0.212209
$397$	7.52786	0.377813	0.188906	−	0.981995i	$-0.439506\pi$
$397$	7.52786	0.377813	0.188906	+	0.981995i	$0.439506\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.8172	−1.03956	−0.519782	−	0.854299i	$-0.673987\pi$
$401$	−20.8172	−1.03956	−0.519782	+	0.854299i	$0.673987\pi$
$402$	0	0
$403$	−5.00000	−0.249068
$404$	26.1803	1.30252
$405$	0	0
$406$	0	0
$407$	6.64488	0.329375
$408$	0	0
$409$	−2.62866	−0.129979	−0.0649893	−	0.997886i	$-0.520701\pi$
$409$	−2.62866	−0.129979	−0.0649893	+	0.997886i	$0.520701\pi$
$410$	0	0
$411$	4.04125	0.199340
$412$	−21.7153	−1.06984
$413$	4.25325	0.209289
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−1.90211	−0.0931469
$418$	0	0
$419$	−36.7082	−1.79331	−0.896657	−	0.442727i	$-0.854011\pi$
$419$	−36.7082	−1.79331	−0.896657	+	0.442727i	$0.854011\pi$
$420$	0	0
$421$	−2.52265	−0.122947	−0.0614733	−	0.998109i	$-0.519580\pi$
$421$	−2.52265	−0.122947	−0.0614733	+	0.998109i	$0.519580\pi$
$422$	0	0
$423$	−21.5279	−1.04672
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.47214	0.216422
$428$	34.5811	1.67154
$429$	−0.450850	−0.0217672
$430$	0	0
$431$	−30.6053	−1.47421	−0.737103	−	0.675780i	$-0.763807\pi$
$431$	−30.6053	−1.47421	−0.737103	+	0.675780i	$0.763807\pi$
$432$	−15.9030	−0.765131
$433$	22.5478	1.08358	0.541790	−	0.840514i	$-0.317747\pi$
$433$	22.5478	1.08358	0.541790	+	0.840514i	$0.317747\pi$
$434$	0	0
$435$	0	0
$436$	27.1846	1.30191
$437$	0	0
$438$	0	0
$439$	28.5972	1.36487	0.682435	−	0.730946i	$-0.260921\pi$
$439$	28.5972	1.36487	0.682435	+	0.730946i	$0.260921\pi$
$440$	0	0
$441$	10.8328	0.515848
$442$	0	0
$443$	−0.0557281	−0.00264772	−0.00132386	−	0.999999i	$-0.500421\pi$
$443$	−0.0557281	−0.00264772	−0.00132386	+	0.999999i	$0.500421\pi$
$444$	−11.3050	−0.536509
$445$	0	0
$446$	0	0
$447$	−5.98385	−0.283027
$448$	12.9443	0.611559
$449$	−39.1118	−1.84580	−0.922901	−	0.385038i	$-0.874188\pi$
$449$	−39.1118	−1.84580	−0.922901	+	0.385038i	$0.874188\pi$
$450$	0	0
$451$	5.11082	0.240659
$452$	17.5680	0.826331
$453$	−5.32624	−0.250248
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.6525	−0.872526	−0.436263	−	0.899819i	$-0.643698\pi$
$457$	−18.6525	−0.872526	−0.436263	+	0.899819i	$0.643698\pi$
$458$	0	0
$459$	15.3229	0.715212
$460$	0	0
$461$	20.4164	0.950887	0.475443	−	0.879746i	$-0.342288\pi$
$461$	20.4164	0.950887	0.475443	+	0.879746i	$0.342288\pi$
$462$	0	0
$463$	30.7082	1.42713	0.713566	−	0.700588i	$-0.247079\pi$
$463$	30.7082	1.42713	0.713566	+	0.700588i	$0.247079\pi$
$464$	−9.40456	−0.436596
$465$	0	0
$466$	0	0
$467$	6.94427	0.321343	0.160671	−	0.987008i	$-0.448634\pi$
$467$	6.94427	0.321343	0.160671	+	0.987008i	$0.448634\pi$
$468$	−3.59222	−0.166051
$469$	−5.42882	−0.250680
$470$	0	0
$471$	10.1966	0.469835
$472$	0	0
$473$	8.21478	0.377716
$474$	0	0
$475$	0	0
$476$	−12.4721	−0.571659
$477$	31.1199	1.42488
$478$	0	0
$479$	31.8541	1.45545	0.727726	−	0.685868i	$-0.240577\pi$
$479$	31.8541	1.45545	0.727726	+	0.685868i	$0.240577\pi$
$480$	0	0
$481$	−5.65248	−0.257731
$482$	0	0
$483$	−4.08174	−0.185726
$484$	20.5410	0.933683
$485$	0	0
$486$	0	0
$487$	1.79611	0.0813896	0.0406948	−	0.999172i	$-0.487043\pi$
$487$	1.79611	0.0813896	0.0406948	+	0.999172i	$0.487043\pi$
$488$	0	0
$489$	0.661030	0.0298928
$490$	0	0
$491$	−28.3820	−1.28086	−0.640430	−	0.768016i	$-0.721244\pi$
$491$	−28.3820	−1.28086	−0.640430	+	0.768016i	$0.721244\pi$
$492$	−8.69505	−0.392003
$493$	9.06154	0.408111
$494$	0	0
$495$	0	0
$496$	27.5276	1.23603
$497$	21.0948	0.946229
$498$	0	0
$499$	−24.1246	−1.07997	−0.539983	−	0.841676i	$-0.681569\pi$
$499$	−24.1246	−1.07997	−0.539983	+	0.841676i	$0.681569\pi$
$500$	0	0
$501$	12.5623	0.561242
$502$	0	0
$503$	−31.8541	−1.42030	−0.710152	−	0.704048i	$-0.751374\pi$
$503$	−31.8541	−1.42030	−0.710152	+	0.704048i	$0.751374\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−9.06154	−0.402437
$508$	36.0341	1.59876
$509$	32.1239	1.42387	0.711934	−	0.702247i	$-0.247820\pi$
$509$	32.1239	1.42387	0.711934	+	0.702247i	$0.247820\pi$
$510$	0	0
$511$	−17.7984	−0.787354
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	−13.9758	−0.615251
$517$	7.43769	0.327109
$518$	0	0
$519$	4.79837	0.210625
$520$	0	0
$521$	−4.76779	−0.208881	−0.104440	−	0.994531i	$-0.533305\pi$
$521$	−4.76779	−0.208881	−0.104440	+	0.994531i	$0.533305\pi$
$522$	0	0
$523$	−11.9677	−0.523311	−0.261656	−	0.965161i	$-0.584268\pi$
$523$	−11.9677	−0.523311	−0.261656	+	0.965161i	$0.584268\pi$
$524$	−1.52786	−0.0667451
$525$	0	0
$526$	0	0
$527$	−26.5236	−1.15539
$528$	2.48217	0.108022
$529$	−10.9443	−0.475838
$530$	0	0
$531$	6.49839	0.282006
$532$	0	0
$533$	−4.34752	−0.188312
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−3.29180	−0.142051
$538$	0	0
$539$	−3.74265	−0.161207
$540$	0	0
$541$	−11.0557	−0.475323	−0.237661	−	0.971348i	$-0.576381\pi$
$541$	−11.0557	−0.475323	−0.237661	+	0.971348i	$0.576381\pi$
$542$	0	0
$543$	−5.40325	−0.231876
$544$	0	0
$545$	0	0
$546$	0	0
$547$	9.61657	0.411175	0.205587	−	0.978639i	$-0.434090\pi$
$547$	9.61657	0.411175	0.205587	+	0.978639i	$0.434090\pi$
$548$	−11.1246	−0.475220
$549$	6.83282	0.291617
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−26.3521	−1.12060
$554$	0	0
$555$	0	0
$556$	5.23607	0.222059
$557$	−10.3607	−0.438996	−0.219498	−	0.975613i	$-0.570442\pi$
$557$	−10.3607	−0.438996	−0.219498	+	0.975613i	$0.570442\pi$
$558$	0	0
$559$	−6.98791	−0.295557
$560$	0	0
$561$	−2.39163	−0.100975
$562$	0	0
$563$	5.98385	0.252189	0.126095	−	0.992018i	$-0.459756\pi$
$563$	5.98385	0.252189	0.126095	+	0.992018i	$0.459756\pi$
$564$	−12.6538	−0.532819
$565$	0	0
$566$	0	0
$567$	−7.32624	−0.307673
$568$	0	0
$569$	17.3560	0.727603	0.363802	−	0.931476i	$-0.381479\pi$
$569$	17.3560	0.727603	0.363802	+	0.931476i	$0.381479\pi$
$570$	0	0
$571$	28.5410	1.19440	0.597202	−	0.802091i	$-0.296279\pi$
$571$	28.5410	1.19440	0.597202	+	0.802091i	$0.296279\pi$
$572$	1.24108	0.0518923
$573$	−3.69822	−0.154496
$574$	0	0
$575$	0	0
$576$	19.7771	0.824045
$577$	−28.1246	−1.17084	−0.585421	−	0.810729i	$-0.699071\pi$
$577$	−28.1246	−1.17084	−0.585421	+	0.810729i	$0.699071\pi$
$578$	0	0
$579$	−2.84095	−0.118066
$580$	0	0
$581$	−4.38197	−0.181795
$582$	0	0
$583$	−10.7516	−0.445288
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.8885	−0.655790	−0.327895	−	0.944714i	$-0.606339\pi$
$587$	−15.8885	−0.655790	−0.327895	+	0.944714i	$0.606339\pi$
$588$	6.36737	0.262586
$589$	0	0
$590$	0	0
$591$	−16.4580	−0.676991
$592$	31.1199	1.27902
$593$	−27.2148	−1.11758	−0.558789	−	0.829310i	$-0.688734\pi$
$593$	−27.2148	−1.11758	−0.558789	+	0.829310i	$0.688734\pi$
$594$	0	0
$595$	0	0
$596$	16.4721	0.674725
$597$	9.12705	0.373545
$598$	0	0
$599$	−1.55909	−0.0637025	−0.0318513	−	0.999493i	$-0.510140\pi$
$599$	−1.55909	−0.0637025	−0.0318513	+	0.999493i	$0.510140\pi$
$600$	0	0
$601$	−1.24108	−0.0506248	−0.0253124	−	0.999680i	$-0.508058\pi$
$601$	−1.24108	−0.0506248	−0.0253124	+	0.999680i	$0.508058\pi$
$602$	0	0
$603$	−8.29451	−0.337778
$604$	14.6619	0.596583
$605$	0	0
$606$	0	0
$607$	26.2866	1.06694	0.533469	−	0.845819i	$-0.320888\pi$
$607$	26.2866	1.06694	0.533469	+	0.845819i	$0.320888\pi$
$608$	0	0
$609$	2.76393	0.112000
$610$	0	0
$611$	−6.32688	−0.255958
$612$	−19.0557	−0.770282
$613$	−10.2705	−0.414822	−0.207411	−	0.978254i	$-0.566504\pi$
$613$	−10.2705	−0.414822	−0.207411	+	0.978254i	$0.566504\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	34.8885	1.40456	0.702280	−	0.711901i	$-0.252165\pi$
$617$	34.8885	1.40456	0.702280	+	0.711901i	$0.252165\pi$
$618$	0	0
$619$	−32.0902	−1.28981	−0.644906	−	0.764262i	$-0.723104\pi$
$619$	−32.0902	−1.28981	−0.644906	+	0.764262i	$0.723104\pi$
$620$	0	0
$621$	−13.8043	−0.553948
$622$	0	0
$623$	21.5438	0.863134
$624$	−2.11146	−0.0845259
$625$	0	0
$626$	0	0
$627$	0	0
$628$	−28.0689	−1.12007
$629$	−29.9848	−1.19557
$630$	0	0
$631$	−15.5279	−0.618155	−0.309077	−	0.951037i	$-0.600020\pi$
$631$	−15.5279	−0.618155	−0.309077	+	0.951037i	$0.600020\pi$
$632$	0	0
$633$	2.03444	0.0808618
$634$	0	0
$635$	0	0
$636$	18.2918	0.725317
$637$	3.18368	0.126142
$638$	0	0
$639$	32.2299	1.27500
$640$	0	0
$641$	−14.3188	−0.565561	−0.282780	−	0.959185i	$-0.591257\pi$
$641$	−14.3188	−0.565561	−0.282780	+	0.959185i	$0.591257\pi$
$642$	0	0
$643$	2.94427	0.116111	0.0580554	−	0.998313i	$-0.481510\pi$
$643$	2.94427	0.116111	0.0580554	+	0.998313i	$0.481510\pi$
$644$	11.2361	0.442763
$645$	0	0
$646$	0	0
$647$	−0.639320	−0.0251343	−0.0125671	−	0.999921i	$-0.504000\pi$
$647$	−0.639320	−0.0251343	−0.0125671	+	0.999921i	$0.504000\pi$
$648$	0	0
$649$	−2.24514	−0.0881294
$650$	0	0
$651$	−8.09017	−0.317079
$652$	−1.81966	−0.0712634
$653$	−6.32624	−0.247565	−0.123782	−	0.992309i	$-0.539502\pi$
$653$	−6.32624	−0.247565	−0.123782	+	0.992309i	$0.539502\pi$
$654$	0	0
$655$	0	0
$656$	23.9354	0.934521
$657$	−27.1935	−1.06092
$658$	0	0
$659$	5.36331	0.208925	0.104462	−	0.994529i	$-0.466688\pi$
$659$	5.36331	0.208925	0.104462	+	0.994529i	$0.466688\pi$
$660$	0	0
$661$	11.2412	0.437231	0.218615	−	0.975811i	$-0.429846\pi$
$661$	11.2412	0.437231	0.218615	+	0.975811i	$0.429846\pi$
$662$	0	0
$663$	2.03444	0.0790112
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−8.16348	−0.316091
$668$	−34.5811	−1.33798
$669$	−3.74265	−0.144699
$670$	0	0
$671$	−2.36068	−0.0911330
$672$	0	0
$673$	−37.2097	−1.43433	−0.717165	−	0.696904i	$-0.754560\pi$
$673$	−37.2097	−1.43433	−0.717165	+	0.696904i	$0.754560\pi$
$674$	0	0
$675$	0	0
$676$	24.9443	0.959395
$677$	−44.8182	−1.72250	−0.861251	−	0.508180i	$-0.830319\pi$
$677$	−44.8182	−1.72250	−0.861251	+	0.508180i	$0.830319\pi$
$678$	0	0
$679$	−12.5882	−0.483093
$680$	0	0
$681$	19.5967	0.750949
$682$	0	0
$683$	3.42071	0.130890	0.0654449	−	0.997856i	$-0.479153\pi$
$683$	3.42071	0.130890	0.0654449	+	0.997856i	$0.479153\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−3.52671	−0.134552
$688$	38.4721	1.46674
$689$	9.14590	0.348431
$690$	0	0
$691$	−11.0557	−0.420580	−0.210290	−	0.977639i	$-0.567441\pi$
$691$	−11.0557	−0.420580	−0.210290	+	0.977639i	$0.567441\pi$
$692$	−13.2088	−0.502123
$693$	3.41641	0.129779
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−23.0624	−0.873550
$698$	0	0
$699$	−9.40456	−0.355713
$700$	0	0
$701$	25.0000	0.944237	0.472118	−	0.881535i	$-0.343489\pi$
$701$	25.0000	0.944237	0.472118	+	0.881535i	$0.343489\pi$
$702$	0	0
$703$	0	0
$704$	−6.83282	−0.257521
$705$	0	0
$706$	0	0
$707$	21.1803	0.796569
$708$	3.81966	0.143552
$709$	−34.4721	−1.29463	−0.647314	−	0.762223i	$-0.724108\pi$
$709$	−34.4721	−1.29463	−0.647314	+	0.762223i	$0.724108\pi$
$710$	0	0
$711$	−40.2624	−1.50996
$712$	0	0
$713$	23.8949	0.894872
$714$	0	0
$715$	0	0
$716$	9.06154	0.338646
$717$	2.86568	0.107021
$718$	0	0
$719$	3.76393	0.140371	0.0701855	−	0.997534i	$-0.477641\pi$
$719$	3.76393	0.140371	0.0701855	+	0.997534i	$0.477641\pi$
$720$	0	0
$721$	−17.5680	−0.654268
$722$	0	0
$723$	14.7984	0.550357
$724$	14.8739	0.552783
$725$	0	0
$726$	0	0
$727$	37.0689	1.37481	0.687404	−	0.726275i	$-0.258750\pi$
$727$	37.0689	1.37481	0.687404	+	0.726275i	$0.258750\pi$
$728$	0	0
$729$	−2.52786	−0.0936246
$730$	0	0
$731$	−37.0689	−1.37104
$732$	4.01623	0.148444
$733$	1.20163	0.0443831	0.0221915	−	0.999754i	$-0.492936\pi$
$733$	1.20163	0.0443831	0.0221915	+	0.999754i	$0.492936\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.86568	0.105559
$738$	0	0
$739$	−11.5836	−0.426109	−0.213055	−	0.977040i	$-0.568341\pi$
$739$	−11.5836	−0.426109	−0.213055	+	0.977040i	$0.568341\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	36.6952	1.34622	0.673108	−	0.739544i	$-0.264959\pi$
$743$	36.6952	1.34622	0.673108	+	0.739544i	$0.264959\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6.69505	−0.244959
$748$	6.58359	0.240720
$749$	27.9767	1.02225
$750$	0	0
$751$	−27.6992	−1.01076	−0.505378	−	0.862898i	$-0.668647\pi$
$751$	−27.6992	−1.01076	−0.505378	+	0.862898i	$0.668647\pi$
$752$	34.8328	1.27022
$753$	4.46526	0.162723
$754$	0	0
$755$	0	0
$756$	−12.8658	−0.467923
$757$	18.7771	0.682465	0.341232	−	0.939979i	$-0.389156\pi$
$757$	18.7771	0.682465	0.341232	+	0.939979i	$0.389156\pi$
$758$	0	0
$759$	2.15460	0.0782072
$760$	0	0
$761$	19.0689	0.691246	0.345623	−	0.938373i	$-0.387668\pi$
$761$	19.0689	0.691246	0.345623	+	0.938373i	$0.387668\pi$
$762$	0	0
$763$	21.9928	0.796193
$764$	10.1803	0.368312
$765$	0	0
$766$	0	0
$767$	1.90983	0.0689600
$768$	11.6247	0.419470
$769$	−26.3607	−0.950590	−0.475295	−	0.879826i	$-0.657659\pi$
$769$	−26.3607	−0.950590	−0.475295	+	0.879826i	$0.657659\pi$
$770$	0	0
$771$	−21.5066	−0.774540
$772$	7.82045	0.281464
$773$	7.95148	0.285995	0.142997	−	0.989723i	$-0.454326\pi$
$773$	7.95148	0.285995	0.142997	+	0.989723i	$0.454326\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−9.14590	−0.328107
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−11.1352	−0.398447
$782$	0	0
$783$	9.34752	0.334053
$784$	−17.5279	−0.625995
$785$	0	0
$786$	0	0
$787$	−44.1976	−1.57548	−0.787738	−	0.616011i	$-0.788748\pi$
$787$	−44.1976	−1.57548	−0.787738	+	0.616011i	$0.788748\pi$
$788$	45.3050	1.61392
$789$	−7.92646	−0.282189
$790$	0	0
$791$	14.2128	0.505351
$792$	0	0
$793$	2.00811	0.0713102
$794$	0	0
$795$	0	0
$796$	−25.1246	−0.890518
$797$	15.9434	0.564746	0.282373	−	0.959305i	$-0.408878\pi$
$797$	15.9434	0.564746	0.282373	+	0.959305i	$0.408878\pi$
$798$	0	0
$799$	−33.5623	−1.18735
$800$	0	0
$801$	32.9160	1.16303
$802$	0	0
$803$	9.39512	0.331547
$804$	−4.87539	−0.171942
$805$	0	0
$806$	0	0
$807$	13.8673	0.488151
$808$	0	0
$809$	37.7639	1.32771	0.663855	−	0.747862i	$-0.268919\pi$
$809$	37.7639	1.32771	0.663855	+	0.747862i	$0.268919\pi$
$810$	0	0
$811$	9.16754	0.321916	0.160958	−	0.986961i	$-0.448542\pi$
$811$	9.16754	0.321916	0.160958	+	0.986961i	$0.448542\pi$
$812$	−7.60845	−0.267004
$813$	−13.3803	−0.469268
$814$	0	0
$815$	0	0
$816$	−11.2007	−0.392102
$817$	0	0
$818$	0	0
$819$	−2.90617	−0.101550
$820$	0	0
$821$	−31.5967	−1.10273	−0.551367	−	0.834263i	$-0.685894\pi$
$821$	−31.5967	−1.10273	−0.551367	+	0.834263i	$0.685894\pi$
$822$	0	0
$823$	16.8885	0.588698	0.294349	−	0.955698i	$-0.404897\pi$
$823$	16.8885	0.588698	0.294349	+	0.955698i	$0.404897\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.7921	−0.375279	−0.187640	−	0.982238i	$-0.560084\pi$
$827$	−10.7921	−0.375279	−0.187640	+	0.982238i	$0.560084\pi$
$828$	17.1672	0.596601
$829$	−11.9272	−0.414249	−0.207125	−	0.978315i	$-0.566411\pi$
$829$	−11.9272	−0.414249	−0.207125	+	0.978315i	$0.566411\pi$
$830$	0	0
$831$	−18.4256	−0.639177
$832$	5.81234	0.201507
$833$	16.8885	0.585153
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−27.3607	−0.945723
$838$	0	0
$839$	18.8496	0.650761	0.325381	−	0.945583i	$-0.394508\pi$
$839$	18.8496	0.650761	0.325381	+	0.945583i	$0.394508\pi$
$840$	0	0
$841$	−23.4721	−0.809384
$842$	0	0
$843$	−3.21478	−0.110723
$844$	−5.60034	−0.192772
$845$	0	0
$846$	0	0
$847$	16.6180	0.571002
$848$	−50.3530	−1.72913
$849$	17.4370	0.598437
$850$	0	0
$851$	27.0131	0.925997
$852$	18.9443	0.649020
$853$	−3.27051	−0.111980	−0.0559901	−	0.998431i	$-0.517832\pi$
$853$	−3.27051	−0.111980	−0.0559901	+	0.998431i	$0.517832\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−0.237026	−0.00809664	−0.00404832	−	0.999992i	$-0.501289\pi$
$857$	−0.237026	−0.00809664	−0.00404832	+	0.999992i	$0.501289\pi$
$858$	0	0
$859$	−46.8328	−1.59792	−0.798958	−	0.601387i	$-0.794615\pi$
$859$	−46.8328	−1.59792	−0.798958	+	0.601387i	$0.794615\pi$
$860$	0	0
$861$	−7.03444	−0.239733
$862$	0	0
$863$	44.3691	1.51034	0.755172	−	0.655527i	$-0.227554\pi$
$863$	44.3691	1.51034	0.755172	+	0.655527i	$0.227554\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.55909	−0.0529493
$868$	22.2703	0.755904
$869$	13.9103	0.471875
$870$	0	0
$871$	−2.43769	−0.0825981
$872$	0	0
$873$	−19.2331	−0.650943
$874$	0	0
$875$	0	0
$876$	−15.9839	−0.540047
$877$	−29.4953	−0.995984	−0.497992	−	0.867182i	$-0.665929\pi$
$877$	−29.4953	−0.995984	−0.497992	+	0.867182i	$0.665929\pi$
$878$	0	0
$879$	4.39512	0.148244
$880$	0	0
$881$	−19.5967	−0.660231	−0.330116	−	0.943941i	$-0.607088\pi$
$881$	−19.5967	−0.660231	−0.330116	+	0.943941i	$0.607088\pi$
$882$	0	0
$883$	20.9098	0.703672	0.351836	−	0.936062i	$-0.385558\pi$
$883$	20.9098	0.703672	0.351836	+	0.936062i	$0.385558\pi$
$884$	−5.60034	−0.188360
$885$	0	0
$886$	0	0
$887$	49.4800	1.66137	0.830687	−	0.556739i	$-0.187948\pi$
$887$	49.4800	1.66137	0.830687	+	0.556739i	$0.187948\pi$
$888$	0	0
$889$	29.1522	0.977735
$890$	0	0
$891$	3.86726	0.129558
$892$	10.3026	0.344957
$893$	0	0
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.83282	−0.0611959
$898$	0	0
$899$	−16.1803	−0.539645
$900$	0	0
$901$	48.5164	1.61632
$902$	0	0
$903$	−11.3067	−0.376263
$904$	0	0
$905$	0	0
$906$	0	0
$907$	51.4881	1.70963	0.854817	−	0.518930i	$-0.173669\pi$
$907$	51.4881	1.70963	0.854817	+	0.518930i	$0.173669\pi$
$908$	−53.9452	−1.79023
$909$	32.3607	1.07334
$910$	0	0
$911$	1.69011	0.0559959	0.0279979	−	0.999608i	$-0.491087\pi$
$911$	1.69011	0.0559959	0.0279979	+	0.999608i	$0.491087\pi$
$912$	0	0
$913$	2.31308	0.0765519
$914$	0	0
$915$	0	0
$916$	9.70820	0.320768
$917$	−1.23607	−0.0408186
$918$	0	0
$919$	−39.5967	−1.30618	−0.653088	−	0.757282i	$-0.726527\pi$
$919$	−39.5967	−1.30618	−0.653088	+	0.757282i	$0.726527\pi$
$920$	0	0
$921$	17.4377	0.574592
$922$	0	0
$923$	9.47214	0.311779
$924$	2.00811	0.0660621
$925$	0	0
$926$	0	0
$927$	−26.8416	−0.881593
$928$	0	0
$929$	43.5410	1.42853	0.714267	−	0.699873i	$-0.246760\pi$
$929$	43.5410	1.42853	0.714267	+	0.699873i	$0.246760\pi$
$930$	0	0
$931$	0	0
$932$	25.8885	0.848007
$933$	1.47811	0.0483911
$934$	0	0
$935$	0	0
$936$	0	0
$937$	39.5623	1.29244	0.646222	−	0.763149i	$-0.276348\pi$
$937$	39.5623	1.29244	0.646222	+	0.763149i	$0.276348\pi$
$938$	0	0
$939$	−21.3068	−0.695320
$940$	0	0
$941$	−49.7980	−1.62337	−0.811684	−	0.584097i	$-0.801449\pi$
$941$	−49.7980	−1.62337	−0.811684	+	0.584097i	$0.801449\pi$
$942$	0	0
$943$	20.7768	0.676584
$944$	−10.5146	−0.342222
$945$	0	0
$946$	0	0
$947$	−49.1033	−1.59564	−0.797822	−	0.602893i	$-0.794014\pi$
$947$	−49.1033	−1.59564	−0.797822	+	0.602893i	$0.794014\pi$
$948$	−23.6656	−0.768624
$949$	−7.99197	−0.259430
$950$	0	0
$951$	−5.90170	−0.191376
$952$	0	0
$953$	−30.2218	−0.978980	−0.489490	−	0.872009i	$-0.662817\pi$
$953$	−30.2218	−0.978980	−0.489490	+	0.872009i	$0.662817\pi$
$954$	0	0
$955$	0	0
$956$	−7.88854	−0.255134
$957$	−1.45898	−0.0471621
$958$	0	0
$959$	−9.00000	−0.290625
$960$	0	0
$961$	16.3607	0.527764
$962$	0	0
$963$	42.7445	1.37742
$964$	−40.7364	−1.31203
$965$	0	0
$966$	0	0
$967$	−16.0689	−0.516740	−0.258370	−	0.966046i	$-0.583185\pi$
$967$	−16.0689	−0.516740	−0.258370	+	0.966046i	$0.583185\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	58.1985	1.86768	0.933839	−	0.357694i	$-0.116437\pi$
$971$	58.1985	1.86768	0.933839	+	0.357694i	$0.116437\pi$
$972$	−30.4338	−0.976165
$973$	4.23607	0.135802
$974$	0	0
$975$	0	0
$976$	−11.0557	−0.353885
$977$	19.1271	0.611931	0.305966	−	0.952043i	$-0.401021\pi$
$977$	19.1271	0.611931	0.305966	+	0.952043i	$0.401021\pi$
$978$	0	0
$979$	−11.3722	−0.363457
$980$	0	0
$981$	33.6020	1.07283
$982$	0	0
$983$	−24.6215	−0.785303	−0.392651	−	0.919687i	$-0.628442\pi$
$983$	−24.6215	−0.785303	−0.392651	+	0.919687i	$0.628442\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−10.2371	−0.325851
$988$	0	0
$989$	33.3951	1.06190
$990$	0	0
$991$	28.3602	0.900891	0.450445	−	0.892804i	$-0.351265\pi$
$991$	28.3602	0.900891	0.450445	+	0.892804i	$0.351265\pi$
$992$	0	0
$993$	−12.4853	−0.396209
$994$	0	0
$995$	0	0
$996$	−3.93525	−0.124693
$997$	−4.76393	−0.150875	−0.0754376	−	0.997151i	$-0.524035\pi$
$997$	−4.76393	−0.150875	−0.0754376	+	0.997151i	$0.524035\pi$
$998$	0	0
$999$	−30.9311	−0.978617

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bi.1.3		4
5.4	even	2		1805.2.a.k.1.2	✓	4
19.18	odd	2	inner	9025.2.a.bi.1.2		4
95.94	odd	2		1805.2.a.k.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.k.1.2	✓	4	5.4	even	2
1805.2.a.k.1.3	yes	4	95.94	odd	2
9025.2.a.bi.1.2		4	19.18	odd	2	inner
9025.2.a.bi.1.3		4	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+0.726543 q^{3} -2.00000 q^{4} -1.61803 q^{7} -2.47214 q^{9} +O(q^{10})\)	\(q+0.726543 q^{3} -2.00000 q^{4} -1.61803 q^{7} -2.47214 q^{9} +0.854102 q^{11} -1.45309 q^{12} -0.726543 q^{13} +4.00000 q^{16} -3.85410 q^{17} -1.17557 q^{21} +3.47214 q^{23} -3.97574 q^{27} +3.23607 q^{28} -2.35114 q^{29} +6.88191 q^{31} +0.620541 q^{33} +4.94427 q^{36} +7.77997 q^{37} -0.527864 q^{39} +5.98385 q^{41} +9.61803 q^{43} -1.70820 q^{44} +8.70820 q^{47} +2.90617 q^{48} -4.38197 q^{49} -2.80017 q^{51} +1.45309 q^{52} -12.5882 q^{53} -2.62866 q^{59} -2.76393 q^{61} +4.00000 q^{63} -8.00000 q^{64} +3.35520 q^{67} +7.70820 q^{68} +2.52265 q^{69} -13.0373 q^{71} +11.0000 q^{73} -1.38197 q^{77} +16.2865 q^{79} +4.52786 q^{81} +2.70820 q^{83} +2.35114 q^{84} -1.70820 q^{87} -13.3148 q^{89} +1.17557 q^{91} -6.94427 q^{92} +5.00000 q^{93} +7.77997 q^{97} -2.11146 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9	\( 4 q - 8 q^{4} - 2 q^{7} + 8 q^{9} - 10 q^{11} + 16 q^{16} - 2 q^{17} - 4 q^{23} + 4 q^{28} - 16 q^{36} - 20 q^{39} + 34 q^{43} + 20 q^{44} + 8 q^{47} - 22 q^{49} - 20 q^{61} + 16 q^{63} - 32 q^{64} + 4 q^{68} + 44 q^{73} - 10 q^{77} + 36 q^{81} - 16 q^{83} + 20 q^{87} + 8 q^{92} + 20 q^{93} - 80 q^{99}+O(q^{100}) \) 4 * q - 8 * q^4 - 2 * q^7 + 8 * q^9 - 10 * q^11 + 16 * q^16 - 2 * q^17 - 4 * q^23 + 4 * q^28 - 16 * q^36 - 20 * q^39 + 34 * q^43 + 20 * q^44 + 8 * q^47 - 22 * q^49 - 20 * q^61 + 16 * q^63 - 32 * q^64 + 4 * q^68 + 44 * q^73 - 10 * q^77 + 36 * q^81 - 16 * q^83 + 20 * q^87 + 8 * q^92 + 20 * q^93 - 80 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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