LMFDB - Embedded newform 976.2.a.f.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [976,2,Mod(1,976)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(976, 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("976.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 976 = 2^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	976.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:	$7.79339923728$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 61)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.17009$ of defining polynomial
Character	$\chi$	$=$	976.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.70928 q^{3} -1.63090 q^{5} +0.460811 q^{7} -0.0783777 q^{9} +O(q^{10})$	$q+1.70928 q^{3} -1.63090 q^{5} +0.460811 q^{7} -0.0783777 q^{9} -6.17009 q^{11} -4.07838 q^{13} -2.78765 q^{15} +0.630898 q^{17} -7.12783 q^{19} +0.787653 q^{21} +0.170086 q^{23} -2.34017 q^{25} -5.26180 q^{27} +2.63090 q^{29} +10.3896 q^{31} -10.5464 q^{33} -0.751536 q^{35} +5.12783 q^{37} -6.97107 q^{39} +3.15676 q^{41} +3.36910 q^{43} +0.127826 q^{45} -0.183417 q^{47} -6.78765 q^{49} +1.07838 q^{51} -4.34017 q^{53} +10.0628 q^{55} -12.1834 q^{57} -1.78047 q^{59} +1.00000 q^{61} -0.0361173 q^{63} +6.65142 q^{65} +8.55971 q^{67} +0.290725 q^{69} -14.3896 q^{71} -0.552520 q^{73} -4.00000 q^{75} -2.84324 q^{77} +7.00719 q^{79} -8.75872 q^{81} -6.83710 q^{83} -1.02893 q^{85} +4.49693 q^{87} +4.49693 q^{89} -1.87936 q^{91} +17.7587 q^{93} +11.6248 q^{95} +8.52359 q^{97} +0.483597 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 - q^5 + 3 * q^7 + 3 * q^9	$ 3 q - 2 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 13 q^{11} - 9 q^{13} + 2 q^{15} - 2 q^{17} - 8 q^{21} - 5 q^{23} + 4 q^{25} - 8 q^{27} + 4 q^{29} + 2 q^{31} + 4 q^{33} - 11 q^{35} - 6 q^{37} - 6 q^{39} + 3 q^{41} + 14 q^{43} - 21 q^{45} + 4 q^{47} - 10 q^{49} - 2 q^{53} + 13 q^{55} - 32 q^{57} - 29 q^{59} + 3 q^{61} + 19 q^{63} - 17 q^{65} - 9 q^{67} + 8 q^{69} - 14 q^{71} - q^{73} - 12 q^{75} - 15 q^{77} - 13 q^{79} - q^{81} + 8 q^{83} - 18 q^{85} - 4 q^{87} - 4 q^{89} + 7 q^{91} + 28 q^{93} - 4 q^{95} + 10 q^{97} - 17 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 - q^5 + 3 * q^7 + 3 * q^9 - 13 * q^11 - 9 * q^13 + 2 * q^15 - 2 * q^17 - 8 * q^21 - 5 * q^23 + 4 * q^25 - 8 * q^27 + 4 * q^29 + 2 * q^31 + 4 * q^33 - 11 * q^35 - 6 * q^37 - 6 * q^39 + 3 * q^41 + 14 * q^43 - 21 * q^45 + 4 * q^47 - 10 * q^49 - 2 * q^53 + 13 * q^55 - 32 * q^57 - 29 * q^59 + 3 * q^61 + 19 * q^63 - 17 * q^65 - 9 * q^67 + 8 * q^69 - 14 * q^71 - q^73 - 12 * q^75 - 15 * q^77 - 13 * q^79 - q^81 + 8 * q^83 - 18 * q^85 - 4 * q^87 - 4 * q^89 + 7 * q^91 + 28 * q^93 - 4 * q^95 + 10 * q^97 - 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.70928	0.986851	0.493425	−	0.869788i	$-0.335745\pi$
$3$	1.70928	0.986851	0.493425	+	0.869788i	$0.335745\pi$
$4$	0	0
$5$	−1.63090	−0.729360	−0.364680	−	0.931133i	$-0.618822\pi$
$5$	−1.63090	−0.729360	−0.364680	+	0.931133i	$0.618822\pi$
$6$	0	0
$7$	0.460811	0.174170	0.0870851	−	0.996201i	$-0.472245\pi$
$7$	0.460811	0.174170	0.0870851	+	0.996201i	$0.472245\pi$
$8$	0	0
$9$	−0.0783777	−0.0261259
$10$	0	0
$11$	−6.17009	−1.86035	−0.930176	−	0.367115i	$-0.880346\pi$
$11$	−6.17009	−1.86035	−0.930176	+	0.367115i	$0.880346\pi$
$12$	0	0
$13$	−4.07838	−1.13114	−0.565569	−	0.824701i	$-0.691343\pi$
$13$	−4.07838	−1.13114	−0.565569	+	0.824701i	$0.691343\pi$
$14$	0	0
$15$	−2.78765	−0.719769
$16$	0	0
$17$	0.630898	0.153015	0.0765076	−	0.997069i	$-0.475623\pi$
$17$	0.630898	0.153015	0.0765076	+	0.997069i	$0.475623\pi$
$18$	0	0
$19$	−7.12783	−1.63524	−0.817618	−	0.575761i	$-0.804706\pi$
$19$	−7.12783	−1.63524	−0.817618	+	0.575761i	$0.804706\pi$
$20$	0	0
$21$	0.787653	0.171880
$22$	0	0
$23$	0.170086	0.0354655	0.0177327	−	0.999843i	$-0.494355\pi$
$23$	0.170086	0.0354655	0.0177327	+	0.999843i	$0.494355\pi$
$24$	0	0
$25$	−2.34017	−0.468035
$26$	0	0
$27$	−5.26180	−1.01263
$28$	0	0
$29$	2.63090	0.488545	0.244273	−	0.969707i	$-0.421451\pi$
$29$	2.63090	0.488545	0.244273	+	0.969707i	$0.421451\pi$
$30$	0	0
$31$	10.3896	1.86603	0.933016	−	0.359836i	$-0.117167\pi$
$31$	10.3896	1.86603	0.933016	+	0.359836i	$0.117167\pi$
$32$	0	0
$33$	−10.5464	−1.83589
$34$	0	0
$35$	−0.751536	−0.127033
$36$	0	0
$37$	5.12783	0.843009	0.421505	−	0.906826i	$-0.361502\pi$
$37$	5.12783	0.843009	0.421505	+	0.906826i	$0.361502\pi$
$38$	0	0
$39$	−6.97107	−1.11626
$40$	0	0
$41$	3.15676	0.493002	0.246501	−	0.969142i	$-0.420719\pi$
$41$	3.15676	0.493002	0.246501	+	0.969142i	$0.420719\pi$
$42$	0	0
$43$	3.36910	0.513783	0.256892	−	0.966440i	$-0.417302\pi$
$43$	3.36910	0.513783	0.256892	+	0.966440i	$0.417302\pi$
$44$	0	0
$45$	0.127826	0.0190552
$46$	0	0
$47$	−0.183417	−0.0267542	−0.0133771	−	0.999911i	$-0.504258\pi$
$47$	−0.183417	−0.0267542	−0.0133771	+	0.999911i	$0.504258\pi$
$48$	0	0
$49$	−6.78765	−0.969665
$50$	0	0
$51$	1.07838	0.151003
$52$	0	0
$53$	−4.34017	−0.596169	−0.298084	−	0.954540i	$-0.596348\pi$
$53$	−4.34017	−0.596169	−0.298084	+	0.954540i	$0.596348\pi$
$54$	0	0
$55$	10.0628	1.35686
$56$	0	0
$57$	−12.1834	−1.61373
$58$	0	0
$59$	−1.78047	−0.231797	−0.115898	−	0.993261i	$-0.536975\pi$
$59$	−1.78047	−0.231797	−0.115898	+	0.993261i	$0.536975\pi$
$60$	0	0
$61$	1.00000	0.128037
$61$	1.00000	0.128037
$62$	0	0
$63$	−0.0361173	−0.00455036
$64$	0	0
$65$	6.65142	0.825007
$66$	0	0
$67$	8.55971	1.04573	0.522867	−	0.852414i	$-0.324862\pi$
$67$	8.55971	1.04573	0.522867	+	0.852414i	$0.324862\pi$
$68$	0	0
$69$	0.290725	0.0349991
$70$	0	0
$71$	−14.3896	−1.70773	−0.853867	−	0.520491i	$-0.825749\pi$
$71$	−14.3896	−1.70773	−0.853867	+	0.520491i	$0.825749\pi$
$72$	0	0
$73$	−0.552520	−0.0646676	−0.0323338	−	0.999477i	$-0.510294\pi$
$73$	−0.552520	−0.0646676	−0.0323338	+	0.999477i	$0.510294\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	−2.84324	−0.324018
$78$	0	0
$79$	7.00719	0.788370	0.394185	−	0.919031i	$-0.371027\pi$
$79$	7.00719	0.788370	0.394185	+	0.919031i	$0.371027\pi$
$80$	0	0
$81$	−8.75872	−0.973192
$82$	0	0
$83$	−6.83710	−0.750469	−0.375235	−	0.926930i	$-0.622438\pi$
$83$	−6.83710	−0.750469	−0.375235	+	0.926930i	$0.622438\pi$
$84$	0	0
$85$	−1.02893	−0.111603
$86$	0	0
$87$	4.49693	0.482121
$88$	0	0
$89$	4.49693	0.476673	0.238337	−	0.971183i	$-0.423398\pi$
$89$	4.49693	0.476673	0.238337	+	0.971183i	$0.423398\pi$
$90$	0	0
$91$	−1.87936	−0.197011
$92$	0	0
$93$	17.7587	1.84149
$94$	0	0
$95$	11.6248	1.19267
$96$	0	0
$97$	8.52359	0.865439	0.432720	−	0.901528i	$-0.357554\pi$
$97$	8.52359	0.865439	0.432720	+	0.901528i	$0.357554\pi$
$98$	0	0
$99$	0.483597	0.0486034
$100$	0	0
$101$	−6.78765	−0.675397	−0.337698	−	0.941254i	$-0.609648\pi$
$101$	−6.78765	−0.675397	−0.337698	+	0.941254i	$0.609648\pi$
$102$	0	0
$103$	−11.1278	−1.09646	−0.548229	−	0.836328i	$-0.684698\pi$
$103$	−11.1278	−1.09646	−0.548229	+	0.836328i	$0.684698\pi$
$104$	0	0
$105$	−1.28458	−0.125362
$106$	0	0
$107$	−15.5753	−1.50572	−0.752861	−	0.658180i	$-0.771327\pi$
$107$	−15.5753	−1.50572	−0.752861	+	0.658180i	$0.771327\pi$
$108$	0	0
$109$	11.6803	1.11877	0.559387	−	0.828907i	$-0.311037\pi$
$109$	11.6803	1.11877	0.559387	+	0.828907i	$0.311037\pi$
$110$	0	0
$111$	8.76487	0.831924
$112$	0	0
$113$	−12.9421	−1.21749	−0.608747	−	0.793364i	$-0.708328\pi$
$113$	−12.9421	−1.21749	−0.608747	+	0.793364i	$0.708328\pi$
$114$	0	0
$115$	−0.277394	−0.0258671
$116$	0	0
$117$	0.319654	0.0295520
$118$	0	0
$119$	0.290725	0.0266507
$120$	0	0
$121$	27.0700	2.46091
$122$	0	0
$123$	5.39576	0.486520
$124$	0	0
$125$	11.9711	1.07073
$126$	0	0
$127$	1.02893	0.0913027	0.0456514	−	0.998957i	$-0.485464\pi$
$127$	1.02893	0.0913027	0.0456514	+	0.998957i	$0.485464\pi$
$128$	0	0
$129$	5.75872	0.507027
$130$	0	0
$131$	−2.44748	−0.213837	−0.106919	−	0.994268i	$-0.534098\pi$
$131$	−2.44748	−0.213837	−0.106919	+	0.994268i	$0.534098\pi$
$132$	0	0
$133$	−3.28458	−0.284809
$134$	0	0
$135$	8.58145	0.738574
$136$	0	0
$137$	−16.0700	−1.37295	−0.686475	−	0.727153i	$-0.740843\pi$
$137$	−16.0700	−1.37295	−0.686475	+	0.727153i	$0.740843\pi$
$138$	0	0
$139$	1.53919	0.130552	0.0652761	−	0.997867i	$-0.479207\pi$
$139$	1.53919	0.130552	0.0652761	+	0.997867i	$0.479207\pi$
$140$	0	0
$141$	−0.313511	−0.0264024
$142$	0	0
$143$	25.1639	2.10431
$144$	0	0
$145$	−4.29072	−0.356325
$146$	0	0
$147$	−11.6020	−0.956914
$148$	0	0
$149$	1.92162	0.157425	0.0787127	−	0.996897i	$-0.474919\pi$
$149$	1.92162	0.157425	0.0787127	+	0.996897i	$0.474919\pi$
$150$	0	0
$151$	−9.32457	−0.758823	−0.379412	−	0.925228i	$-0.623874\pi$
$151$	−9.32457	−0.758823	−0.379412	+	0.925228i	$0.623874\pi$
$152$	0	0
$153$	−0.0494483	−0.00399766
$154$	0	0
$155$	−16.9444	−1.36101
$156$	0	0
$157$	−13.4947	−1.07699	−0.538496	−	0.842628i	$-0.681007\pi$
$157$	−13.4947	−1.07699	−0.538496	+	0.842628i	$0.681007\pi$
$158$	0	0
$159$	−7.41855	−0.588329
$160$	0	0
$161$	0.0783777	0.00617703
$162$	0	0
$163$	10.7031	0.838334	0.419167	−	0.907909i	$-0.362322\pi$
$163$	10.7031	0.838334	0.419167	+	0.907909i	$0.362322\pi$
$164$	0	0
$165$	17.2001	1.33902
$166$	0	0
$167$	−4.88655	−0.378133	−0.189066	−	0.981964i	$-0.560546\pi$
$167$	−4.88655	−0.378133	−0.189066	+	0.981964i	$0.560546\pi$
$168$	0	0
$169$	3.63317	0.279474
$170$	0	0
$171$	0.558663	0.0427220
$172$	0	0
$173$	−0.738205	−0.0561247	−0.0280623	−	0.999606i	$-0.508934\pi$
$173$	−0.738205	−0.0561247	−0.0280623	+	0.999606i	$0.508934\pi$
$174$	0	0
$175$	−1.07838	−0.0815177
$176$	0	0
$177$	−3.04331	−0.228749
$178$	0	0
$179$	−1.90110	−0.142095	−0.0710476	−	0.997473i	$-0.522634\pi$
$179$	−1.90110	−0.142095	−0.0710476	+	0.997473i	$0.522634\pi$
$180$	0	0
$181$	−21.3607	−1.58773	−0.793864	−	0.608096i	$-0.791934\pi$
$181$	−21.3607	−1.58773	−0.793864	+	0.608096i	$0.791934\pi$
$182$	0	0
$183$	1.70928	0.126353
$184$	0	0
$185$	−8.36296	−0.614857
$186$	0	0
$187$	−3.89269	−0.284662
$188$	0	0
$189$	−2.42469	−0.176371
$190$	0	0
$191$	−2.88550	−0.208788	−0.104394	−	0.994536i	$-0.533290\pi$
$191$	−2.88550	−0.208788	−0.104394	+	0.994536i	$0.533290\pi$
$192$	0	0
$193$	0.680346	0.0489724	0.0244862	−	0.999700i	$-0.492205\pi$
$193$	0.680346	0.0489724	0.0244862	+	0.999700i	$0.492205\pi$
$194$	0	0
$195$	11.3691	0.814158
$196$	0	0
$197$	−19.3896	−1.38145	−0.690727	−	0.723116i	$-0.742709\pi$
$197$	−19.3896	−1.38145	−0.690727	+	0.723116i	$0.742709\pi$
$198$	0	0
$199$	21.4186	1.51832	0.759160	−	0.650904i	$-0.225610\pi$
$199$	21.4186	1.51832	0.759160	+	0.650904i	$0.225610\pi$
$200$	0	0
$201$	14.6309	1.03198
$202$	0	0
$203$	1.21235	0.0850901
$204$	0	0
$205$	−5.14834	−0.359576
$206$	0	0
$207$	−0.0133310	−0.000926568 0
$208$	0	0
$209$	43.9793	3.04211
$210$	0	0
$211$	−5.39576	−0.371460	−0.185730	−	0.982601i	$-0.559465\pi$
$211$	−5.39576	−0.371460	−0.185730	+	0.982601i	$0.559465\pi$
$212$	0	0
$213$	−24.5958	−1.68528
$214$	0	0
$215$	−5.49466	−0.374733
$216$	0	0
$217$	4.78765	0.325007
$218$	0	0
$219$	−0.944409	−0.0638172
$220$	0	0
$221$	−2.57304	−0.173081
$222$	0	0
$223$	−12.4257	−0.832089	−0.416045	−	0.909344i	$-0.636584\pi$
$223$	−12.4257	−0.832089	−0.416045	+	0.909344i	$0.636584\pi$
$224$	0	0
$225$	0.183417	0.0122278
$226$	0	0
$227$	22.3679	1.48461	0.742304	−	0.670063i	$-0.233733\pi$
$227$	22.3679	1.48461	0.742304	+	0.670063i	$0.233733\pi$
$228$	0	0
$229$	−0.185685	−0.0122704	−0.00613520	−	0.999981i	$-0.501953\pi$
$229$	−0.185685	−0.0122704	−0.00613520	+	0.999981i	$0.501953\pi$
$230$	0	0
$231$	−4.85989	−0.319757
$232$	0	0
$233$	−17.8660	−1.17044	−0.585221	−	0.810874i	$-0.698992\pi$
$233$	−17.8660	−1.17044	−0.585221	+	0.810874i	$0.698992\pi$
$234$	0	0
$235$	0.299135	0.0195134
$236$	0	0
$237$	11.9772	0.778004
$238$	0	0
$239$	22.3896	1.44826	0.724132	−	0.689661i	$-0.242241\pi$
$239$	22.3896	1.44826	0.724132	+	0.689661i	$0.242241\pi$
$240$	0	0
$241$	20.0433	1.29110	0.645551	−	0.763717i	$-0.276628\pi$
$241$	20.0433	1.29110	0.645551	+	0.763717i	$0.276628\pi$
$242$	0	0
$243$	0.814315	0.0522383
$244$	0	0
$245$	11.0700	0.707234
$246$	0	0
$247$	29.0700	1.84968
$248$	0	0
$249$	−11.6865	−0.740601
$250$	0	0
$251$	−7.05559	−0.445345	−0.222672	−	0.974893i	$-0.571478\pi$
$251$	−7.05559	−0.445345	−0.222672	+	0.974893i	$0.571478\pi$
$252$	0	0
$253$	−1.04945	−0.0659783
$254$	0	0
$255$	−1.75872	−0.110136
$256$	0	0
$257$	3.26180	0.203465	0.101733	−	0.994812i	$-0.467561\pi$
$257$	3.26180	0.203465	0.101733	+	0.994812i	$0.467561\pi$
$258$	0	0
$259$	2.36296	0.146827
$260$	0	0
$261$	−0.206204	−0.0127637
$262$	0	0
$263$	20.5730	1.26859	0.634294	−	0.773092i	$-0.281291\pi$
$263$	20.5730	1.26859	0.634294	+	0.773092i	$0.281291\pi$
$264$	0	0
$265$	7.07838	0.434821
$266$	0	0
$267$	7.68649	0.470405
$268$	0	0
$269$	−3.97334	−0.242259	−0.121129	−	0.992637i	$-0.538652\pi$
$269$	−3.97334	−0.242259	−0.121129	+	0.992637i	$0.538652\pi$
$270$	0	0
$271$	4.99773	0.303591	0.151795	−	0.988412i	$-0.451495\pi$
$271$	4.99773	0.303591	0.151795	+	0.988412i	$0.451495\pi$
$272$	0	0
$273$	−3.21235	−0.194420
$274$	0	0
$275$	14.4391	0.870709
$276$	0	0
$277$	17.9649	1.07941	0.539704	−	0.841855i	$-0.318536\pi$
$277$	17.9649	1.07941	0.539704	+	0.841855i	$0.318536\pi$
$278$	0	0
$279$	−0.814315	−0.0487518
$280$	0	0
$281$	15.6020	0.930735	0.465368	−	0.885117i	$-0.345922\pi$
$281$	15.6020	0.930735	0.465368	+	0.885117i	$0.345922\pi$
$282$	0	0
$283$	−24.7298	−1.47003	−0.735017	−	0.678049i	$-0.762826\pi$
$283$	−24.7298	−1.47003	−0.735017	+	0.678049i	$0.762826\pi$
$284$	0	0
$285$	19.8699	1.17699
$286$	0	0
$287$	1.45467	0.0858663
$288$	0	0
$289$	−16.6020	−0.976586
$290$	0	0
$291$	14.5692	0.854059
$292$	0	0
$293$	24.1711	1.41209	0.706046	−	0.708166i	$-0.250477\pi$
$293$	24.1711	1.41209	0.706046	+	0.708166i	$0.250477\pi$
$294$	0	0
$295$	2.90376	0.169063
$296$	0	0
$297$	32.4657	1.88385
$298$	0	0
$299$	−0.693677	−0.0401164
$300$	0	0
$301$	1.55252	0.0894858
$302$	0	0
$303$	−11.6020	−0.666516
$304$	0	0
$305$	−1.63090	−0.0933849
$306$	0	0
$307$	−19.5041	−1.11316	−0.556579	−	0.830794i	$-0.687886\pi$
$307$	−19.5041	−1.11316	−0.556579	+	0.830794i	$0.687886\pi$
$308$	0	0
$309$	−19.0205	−1.08204
$310$	0	0
$311$	−8.35350	−0.473684	−0.236842	−	0.971548i	$-0.576112\pi$
$311$	−8.35350	−0.473684	−0.236842	+	0.971548i	$0.576112\pi$
$312$	0	0
$313$	2.09890	0.118637	0.0593183	−	0.998239i	$-0.481107\pi$
$313$	2.09890	0.118637	0.0593183	+	0.998239i	$0.481107\pi$
$314$	0	0
$315$	0.0589037	0.00331885
$316$	0	0
$317$	27.7587	1.55909	0.779543	−	0.626349i	$-0.215452\pi$
$317$	27.7587	1.55909	0.779543	+	0.626349i	$0.215452\pi$
$318$	0	0
$319$	−16.2329	−0.908866
$320$	0	0
$321$	−26.6225	−1.48592
$322$	0	0
$323$	−4.49693	−0.250216
$324$	0	0
$325$	9.54411	0.529412
$326$	0	0
$327$	19.9649	1.10406
$328$	0	0
$329$	−0.0845208	−0.00465978
$330$	0	0
$331$	1.19902	0.0659039	0.0329519	−	0.999457i	$-0.489509\pi$
$331$	1.19902	0.0659039	0.0329519	+	0.999457i	$0.489509\pi$
$332$	0	0
$333$	−0.401907	−0.0220244
$334$	0	0
$335$	−13.9600	−0.762717
$336$	0	0
$337$	20.9672	1.14216	0.571078	−	0.820896i	$-0.306525\pi$
$337$	20.9672	1.14216	0.571078	+	0.820896i	$0.306525\pi$
$338$	0	0
$339$	−22.1217	−1.20148
$340$	0	0
$341$	−64.1049	−3.47147
$342$	0	0
$343$	−6.35350	−0.343057
$344$	0	0
$345$	−0.474142	−0.0255270
$346$	0	0
$347$	−20.3896	−1.09457	−0.547286	−	0.836946i	$-0.684339\pi$
$347$	−20.3896	−1.09457	−0.547286	+	0.836946i	$0.684339\pi$
$348$	0	0
$349$	−7.65142	−0.409571	−0.204785	−	0.978807i	$-0.565650\pi$
$349$	−7.65142	−0.409571	−0.204785	+	0.978807i	$0.565650\pi$
$350$	0	0
$351$	21.4596	1.14543
$352$	0	0
$353$	−19.6765	−1.04727	−0.523636	−	0.851942i	$-0.675425\pi$
$353$	−19.6765	−1.04727	−0.523636	+	0.851942i	$0.675425\pi$
$354$	0	0
$355$	23.4680	1.24555
$356$	0	0
$357$	0.496928	0.0263002
$358$	0	0
$359$	−20.8599	−1.10094	−0.550471	−	0.834854i	$-0.685552\pi$
$359$	−20.8599	−1.10094	−0.550471	+	0.834854i	$0.685552\pi$
$360$	0	0
$361$	31.8059	1.67399
$362$	0	0
$363$	46.2700	2.42855
$364$	0	0
$365$	0.901103	0.0471659
$366$	0	0
$367$	5.31965	0.277684	0.138842	−	0.990315i	$-0.455662\pi$
$367$	5.31965	0.277684	0.138842	+	0.990315i	$0.455662\pi$
$368$	0	0
$369$	−0.247419	−0.0128801
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−20.0905	−1.04025	−0.520123	−	0.854091i	$-0.674114\pi$
$373$	−20.0905	−1.04025	−0.520123	+	0.854091i	$0.674114\pi$
$374$	0	0
$375$	20.4619	1.05665
$376$	0	0
$377$	−10.7298	−0.552613
$378$	0	0
$379$	26.4040	1.35628	0.678141	−	0.734932i	$-0.262786\pi$
$379$	26.4040	1.35628	0.678141	+	0.734932i	$0.262786\pi$
$380$	0	0
$381$	1.75872	0.0901021
$382$	0	0
$383$	10.9350	0.558750	0.279375	−	0.960182i	$-0.409873\pi$
$383$	10.9350	0.558750	0.279375	+	0.960182i	$0.409873\pi$
$384$	0	0
$385$	4.63704	0.236325
$386$	0	0
$387$	−0.264063	−0.0134231
$388$	0	0
$389$	−5.50307	−0.279017	−0.139508	−	0.990221i	$-0.544552\pi$
$389$	−5.50307	−0.279017	−0.139508	+	0.990221i	$0.544552\pi$
$390$	0	0
$391$	0.107307	0.00542676
$392$	0	0
$393$	−4.18342	−0.211025
$394$	0	0
$395$	−11.4280	−0.575005
$396$	0	0
$397$	14.7565	0.740605	0.370303	−	0.928911i	$-0.379254\pi$
$397$	14.7565	0.740605	0.370303	+	0.928911i	$0.379254\pi$
$398$	0	0
$399$	−5.61425	−0.281064
$400$	0	0
$401$	−17.7587	−0.886828	−0.443414	−	0.896317i	$-0.646233\pi$
$401$	−17.7587	−0.886828	−0.443414	+	0.896317i	$0.646233\pi$
$402$	0	0
$403$	−42.3728	−2.11074
$404$	0	0
$405$	14.2846	0.709807
$406$	0	0
$407$	−31.6391	−1.56829
$408$	0	0
$409$	−16.4391	−0.812860	−0.406430	−	0.913682i	$-0.633226\pi$
$409$	−16.4391	−0.812860	−0.406430	+	0.913682i	$0.633226\pi$
$410$	0	0
$411$	−27.4680	−1.35490
$412$	0	0
$413$	−0.820458	−0.0403721
$414$	0	0
$415$	11.1506	0.547362
$416$	0	0
$417$	2.63090	0.128836
$418$	0	0
$419$	−32.0183	−1.56419	−0.782097	−	0.623157i	$-0.785850\pi$
$419$	−32.0183	−1.56419	−0.782097	+	0.623157i	$0.785850\pi$
$420$	0	0
$421$	−0.630898	−0.0307481	−0.0153740	−	0.999882i	$-0.504894\pi$
$421$	−0.630898	−0.0307481	−0.0153740	+	0.999882i	$0.504894\pi$
$422$	0	0
$423$	0.0143758	0.000698978 0
$424$	0	0
$425$	−1.47641	−0.0716164
$426$	0	0
$427$	0.460811	0.0223002
$428$	0	0
$429$	43.0121	2.07664
$430$	0	0
$431$	16.6225	0.800677	0.400339	−	0.916367i	$-0.368893\pi$
$431$	16.6225	0.800677	0.400339	+	0.916367i	$0.368893\pi$
$432$	0	0
$433$	−20.4969	−0.985020	−0.492510	−	0.870307i	$-0.663920\pi$
$433$	−20.4969	−0.985020	−0.492510	+	0.870307i	$0.663920\pi$
$434$	0	0
$435$	−7.33403	−0.351640
$436$	0	0
$437$	−1.21235	−0.0579944
$438$	0	0
$439$	2.34858	0.112092	0.0560459	−	0.998428i	$-0.482151\pi$
$439$	2.34858	0.112092	0.0560459	+	0.998428i	$0.482151\pi$
$440$	0	0
$441$	0.532001	0.0253334
$442$	0	0
$443$	−16.5958	−0.788491	−0.394246	−	0.919005i	$-0.628994\pi$
$443$	−16.5958	−0.788491	−0.394246	+	0.919005i	$0.628994\pi$
$444$	0	0
$445$	−7.33403	−0.347666
$446$	0	0
$447$	3.28458	0.155355
$448$	0	0
$449$	−20.4680	−0.965945	−0.482972	−	0.875636i	$-0.660443\pi$
$449$	−20.4680	−0.965945	−0.482972	+	0.875636i	$0.660443\pi$
$450$	0	0
$451$	−19.4775	−0.917158
$452$	0	0
$453$	−15.9383	−0.748845
$454$	0	0
$455$	3.06505	0.143692
$456$	0	0
$457$	23.9506	1.12036	0.560180	−	0.828371i	$-0.310732\pi$
$457$	23.9506	1.12036	0.560180	+	0.828371i	$0.310732\pi$
$458$	0	0
$459$	−3.31965	−0.154948
$460$	0	0
$461$	−26.5113	−1.23475	−0.617377	−	0.786667i	$-0.711805\pi$
$461$	−26.5113	−1.23475	−0.617377	+	0.786667i	$0.711805\pi$
$462$	0	0
$463$	−34.0410	−1.58202	−0.791011	−	0.611802i	$-0.790445\pi$
$463$	−34.0410	−1.58202	−0.791011	+	0.611802i	$0.790445\pi$
$464$	0	0
$465$	−28.9627	−1.34311
$466$	0	0
$467$	−4.55971	−0.210998	−0.105499	−	0.994419i	$-0.533644\pi$
$467$	−4.55971	−0.210998	−0.105499	+	0.994419i	$0.533644\pi$
$468$	0	0
$469$	3.94441	0.182136
$470$	0	0
$471$	−23.0661	−1.06283
$472$	0	0
$473$	−20.7877	−0.955817
$474$	0	0
$475$	16.6803	0.765347
$476$	0	0
$477$	0.340173	0.0155755
$478$	0	0
$479$	0.653684	0.0298676	0.0149338	−	0.999888i	$-0.495246\pi$
$479$	0.653684	0.0298676	0.0149338	+	0.999888i	$0.495246\pi$
$480$	0	0
$481$	−20.9132	−0.953560
$482$	0	0
$483$	0.133969	0.00609581
$484$	0	0
$485$	−13.9011	−0.631217
$486$	0	0
$487$	0.0227863	0.00103255	0.000516274	−	1.00000i	$-0.499836\pi$
$487$	0.0227863	0.00103255	0.000516274		1.00000i	$0.499836\pi$
$488$	0	0
$489$	18.2946	0.827310
$490$	0	0
$491$	2.94441	0.132879	0.0664396	−	0.997790i	$-0.478836\pi$
$491$	2.94441	0.132879	0.0664396	+	0.997790i	$0.478836\pi$
$492$	0	0
$493$	1.65983	0.0747548
$494$	0	0
$495$	−0.788698	−0.0354493
$496$	0	0
$497$	−6.63090	−0.297436
$498$	0	0
$499$	−23.0878	−1.03355	−0.516777	−	0.856120i	$-0.672868\pi$
$499$	−23.0878	−1.03355	−0.516777	+	0.856120i	$0.672868\pi$
$500$	0	0
$501$	−8.35246	−0.373160
$502$	0	0
$503$	−5.62475	−0.250795	−0.125398	−	0.992107i	$-0.540021\pi$
$503$	−5.62475	−0.250795	−0.125398	+	0.992107i	$0.540021\pi$
$504$	0	0
$505$	11.0700	0.492607
$506$	0	0
$507$	6.21008	0.275799
$508$	0	0
$509$	−19.5525	−0.866650	−0.433325	−	0.901238i	$-0.642660\pi$
$509$	−19.5525	−0.866650	−0.433325	+	0.901238i	$0.642660\pi$
$510$	0	0
$511$	−0.254607	−0.0112632
$512$	0	0
$513$	37.5052	1.65589
$514$	0	0
$515$	18.1483	0.799712
$516$	0	0
$517$	1.13170	0.0497722
$518$	0	0
$519$	−1.26180	−0.0553867
$520$	0	0
$521$	16.7442	0.733575	0.366788	−	0.930305i	$-0.380458\pi$
$521$	16.7442	0.733575	0.366788	+	0.930305i	$0.380458\pi$
$522$	0	0
$523$	1.69982	0.0743279	0.0371640	−	0.999309i	$-0.488168\pi$
$523$	1.69982	0.0743279	0.0371640	+	0.999309i	$0.488168\pi$
$524$	0	0
$525$	−1.84324	−0.0804458
$526$	0	0
$527$	6.55479	0.285531
$528$	0	0
$529$	−22.9711	−0.998742
$530$	0	0
$531$	0.139549	0.00605590
$532$	0	0
$533$	−12.8744	−0.557654
$534$	0	0
$535$	25.4017	1.09821
$536$	0	0
$537$	−3.24951	−0.140227
$538$	0	0
$539$	41.8804	1.80392
$540$	0	0
$541$	13.3074	0.572128	0.286064	−	0.958210i	$-0.407653\pi$
$541$	13.3074	0.572128	0.286064	+	0.958210i	$0.407653\pi$
$542$	0	0
$543$	−36.5113	−1.56685
$544$	0	0
$545$	−19.0494	−0.815989
$546$	0	0
$547$	24.2690	1.03767	0.518833	−	0.854875i	$-0.326366\pi$
$547$	24.2690	1.03767	0.518833	+	0.854875i	$0.326366\pi$
$548$	0	0
$549$	−0.0783777	−0.00334508
$550$	0	0
$551$	−18.7526	−0.798887
$552$	0	0
$553$	3.22899	0.137311
$554$	0	0
$555$	−14.2946	−0.606772
$556$	0	0
$557$	1.36069	0.0576544	0.0288272	−	0.999584i	$-0.490823\pi$
$557$	1.36069	0.0576544	0.0288272	+	0.999584i	$0.490823\pi$
$558$	0	0
$559$	−13.7405	−0.581160
$560$	0	0
$561$	−6.65368	−0.280919
$562$	0	0
$563$	−14.4885	−0.610618	−0.305309	−	0.952253i	$-0.598760\pi$
$563$	−14.4885	−0.610618	−0.305309	+	0.952253i	$0.598760\pi$
$564$	0	0
$565$	21.1073	0.887991
$566$	0	0
$567$	−4.03612	−0.169501
$568$	0	0
$569$	−3.78765	−0.158787	−0.0793933	−	0.996843i	$-0.525298\pi$
$569$	−3.78765	−0.158787	−0.0793933	+	0.996843i	$0.525298\pi$
$570$	0	0
$571$	3.87444	0.162140	0.0810702	−	0.996708i	$-0.474166\pi$
$571$	3.87444	0.162140	0.0810702	+	0.996708i	$0.474166\pi$
$572$	0	0
$573$	−4.93212	−0.206042
$574$	0	0
$575$	−0.398032	−0.0165991
$576$	0	0
$577$	−23.9071	−0.995264	−0.497632	−	0.867388i	$-0.665797\pi$
$577$	−23.9071	−0.995264	−0.497632	+	0.867388i	$0.665797\pi$
$578$	0	0
$579$	1.16290	0.0483284
$580$	0	0
$581$	−3.15061	−0.130709
$582$	0	0
$583$	26.7792	1.10908
$584$	0	0
$585$	−0.521323	−0.0215541
$586$	0	0
$587$	−12.3318	−0.508986	−0.254493	−	0.967075i	$-0.581909\pi$
$587$	−12.3318	−0.508986	−0.254493	+	0.967075i	$0.581909\pi$
$588$	0	0
$589$	−74.0554	−3.05140
$590$	0	0
$591$	−33.1422	−1.36329
$592$	0	0
$593$	11.1278	0.456965	0.228483	−	0.973548i	$-0.426624\pi$
$593$	11.1278	0.456965	0.228483	+	0.973548i	$0.426624\pi$
$594$	0	0
$595$	−0.474142	−0.0194379
$596$	0	0
$597$	36.6102	1.49836
$598$	0	0
$599$	−23.0878	−0.943343	−0.471672	−	0.881774i	$-0.656349\pi$
$599$	−23.0878	−0.943343	−0.471672	+	0.881774i	$0.656349\pi$
$600$	0	0
$601$	−10.3340	−0.421534	−0.210767	−	0.977536i	$-0.567596\pi$
$601$	−10.3340	−0.421534	−0.210767	+	0.977536i	$0.567596\pi$
$602$	0	0
$603$	−0.670891	−0.0273208
$604$	0	0
$605$	−44.1483	−1.79489
$606$	0	0
$607$	24.8371	1.00811	0.504053	−	0.863672i	$-0.331841\pi$
$607$	24.8371	1.00811	0.504053	+	0.863672i	$0.331841\pi$
$608$	0	0
$609$	2.07223	0.0839712
$610$	0	0
$611$	0.748046	0.0302627
$612$	0	0
$613$	−23.9877	−0.968855	−0.484427	−	0.874832i	$-0.660972\pi$
$613$	−23.9877	−0.968855	−0.484427	+	0.874832i	$0.660972\pi$
$614$	0	0
$615$	−8.79994	−0.354848
$616$	0	0
$617$	21.6020	0.869662	0.434831	−	0.900512i	$-0.356808\pi$
$617$	21.6020	0.869662	0.434831	+	0.900512i	$0.356808\pi$
$618$	0	0
$619$	46.2388	1.85850	0.929248	−	0.369457i	$-0.120456\pi$
$619$	46.2388	1.85850	0.929248	+	0.369457i	$0.120456\pi$
$620$	0	0
$621$	−0.894960	−0.0359135
$622$	0	0
$623$	2.07223	0.0830223
$624$	0	0
$625$	−7.82273	−0.312909
$626$	0	0
$627$	75.1727	3.00211
$628$	0	0
$629$	3.23513	0.128993
$630$	0	0
$631$	−33.2628	−1.32417	−0.662086	−	0.749428i	$-0.730329\pi$
$631$	−33.2628	−1.32417	−0.662086	+	0.749428i	$0.730329\pi$
$632$	0	0
$633$	−9.22285	−0.366575
$634$	0	0
$635$	−1.67808	−0.0665925
$636$	0	0
$637$	27.6826	1.09683
$638$	0	0
$639$	1.12783	0.0446161
$640$	0	0
$641$	33.7009	1.33110	0.665552	−	0.746351i	$-0.268196\pi$
$641$	33.7009	1.33110	0.665552	+	0.746351i	$0.268196\pi$
$642$	0	0
$643$	6.97107	0.274912	0.137456	−	0.990508i	$-0.456107\pi$
$643$	6.97107	0.274912	0.137456	+	0.990508i	$0.456107\pi$
$644$	0	0
$645$	−9.39189	−0.369805
$646$	0	0
$647$	16.0049	0.629218	0.314609	−	0.949221i	$-0.398127\pi$
$647$	16.0049	0.629218	0.314609	+	0.949221i	$0.398127\pi$
$648$	0	0
$649$	10.9856	0.431223
$650$	0	0
$651$	8.18342	0.320733
$652$	0	0
$653$	43.2762	1.69353	0.846764	−	0.531969i	$-0.178548\pi$
$653$	43.2762	1.69353	0.846764	+	0.531969i	$0.178548\pi$
$654$	0	0
$655$	3.99159	0.155964
$656$	0	0
$657$	0.0433053	0.00168950
$658$	0	0
$659$	16.9276	0.659405	0.329703	−	0.944085i	$-0.393052\pi$
$659$	16.9276	0.659405	0.329703	+	0.944085i	$0.393052\pi$
$660$	0	0
$661$	15.6781	0.609807	0.304903	−	0.952383i	$-0.401376\pi$
$661$	15.6781	0.609807	0.304903	+	0.952383i	$0.401376\pi$
$662$	0	0
$663$	−4.39803	−0.170805
$664$	0	0
$665$	5.35682	0.207728
$666$	0	0
$667$	0.447480	0.0173265
$668$	0	0
$669$	−21.2390	−0.821148
$670$	0	0
$671$	−6.17009	−0.238194
$672$	0	0
$673$	−50.6330	−1.95176	−0.975879	−	0.218312i	$-0.929945\pi$
$673$	−50.6330	−1.95176	−0.975879	+	0.218312i	$0.929945\pi$
$674$	0	0
$675$	12.3135	0.473947
$676$	0	0
$677$	38.1711	1.46704	0.733518	−	0.679670i	$-0.237877\pi$
$677$	38.1711	1.46704	0.733518	+	0.679670i	$0.237877\pi$
$678$	0	0
$679$	3.92777	0.150734
$680$	0	0
$681$	38.2329	1.46509
$682$	0	0
$683$	−36.6309	−1.40164	−0.700821	−	0.713337i	$-0.747183\pi$
$683$	−36.6309	−1.40164	−0.700821	+	0.713337i	$0.747183\pi$
$684$	0	0
$685$	26.2085	1.00137
$686$	0	0
$687$	−0.317387	−0.0121091
$688$	0	0
$689$	17.7009	0.674349
$690$	0	0
$691$	17.4764	0.664834	0.332417	−	0.943133i	$-0.392136\pi$
$691$	17.4764	0.664834	0.332417	+	0.943133i	$0.392136\pi$
$692$	0	0
$693$	0.222847	0.00846526
$694$	0	0
$695$	−2.51026	−0.0952196
$696$	0	0
$697$	1.99159	0.0754368
$698$	0	0
$699$	−30.5380	−1.15505
$700$	0	0
$701$	−9.83096	−0.371310	−0.185655	−	0.982615i	$-0.559441\pi$
$701$	−9.83096	−0.371310	−0.185655	+	0.982615i	$0.559441\pi$
$702$	0	0
$703$	−36.5503	−1.37852
$704$	0	0
$705$	0.511304	0.0192568
$706$	0	0
$707$	−3.12783	−0.117634
$708$	0	0
$709$	35.9214	1.34906	0.674529	−	0.738248i	$-0.264347\pi$
$709$	35.9214	1.34906	0.674529	+	0.738248i	$0.264347\pi$
$710$	0	0
$711$	−0.549208	−0.0205969
$712$	0	0
$713$	1.76713	0.0661797
$714$	0	0
$715$	−41.0398	−1.53480
$716$	0	0
$717$	38.2700	1.42922
$718$	0	0
$719$	20.3773	0.759946	0.379973	−	0.924998i	$-0.375933\pi$
$719$	20.3773	0.759946	0.379973	+	0.924998i	$0.375933\pi$
$720$	0	0
$721$	−5.12783	−0.190970
$722$	0	0
$723$	34.2595	1.27413
$724$	0	0
$725$	−6.15676	−0.228656
$726$	0	0
$727$	−30.0722	−1.11532	−0.557659	−	0.830070i	$-0.688300\pi$
$727$	−30.0722	−1.11532	−0.557659	+	0.830070i	$0.688300\pi$
$728$	0	0
$729$	27.6681	1.02474
$730$	0	0
$731$	2.12556	0.0786166
$732$	0	0
$733$	1.44360	0.0533207	0.0266604	−	0.999645i	$-0.491513\pi$
$733$	1.44360	0.0533207	0.0266604	+	0.999645i	$0.491513\pi$
$734$	0	0
$735$	18.9216	0.697935
$736$	0	0
$737$	−52.8141	−1.94543
$738$	0	0
$739$	16.1122	0.592698	0.296349	−	0.955080i	$-0.404231\pi$
$739$	16.1122	0.592698	0.296349	+	0.955080i	$0.404231\pi$
$740$	0	0
$741$	49.6886	1.82536
$742$	0	0
$743$	−37.1100	−1.36143	−0.680716	−	0.732547i	$-0.738331\pi$
$743$	−37.1100	−1.36143	−0.680716	+	0.732547i	$0.738331\pi$
$744$	0	0
$745$	−3.13397	−0.114820
$746$	0	0
$747$	0.535877	0.0196067
$748$	0	0
$749$	−7.17727	−0.262252
$750$	0	0
$751$	−15.6430	−0.570821	−0.285411	−	0.958405i	$-0.592130\pi$
$751$	−15.6430	−0.570821	−0.285411	+	0.958405i	$0.592130\pi$
$752$	0	0
$753$	−12.0599	−0.439489
$754$	0	0
$755$	15.2074	0.553455
$756$	0	0
$757$	−41.3728	−1.50372	−0.751860	−	0.659323i	$-0.770843\pi$
$757$	−41.3728	−1.50372	−0.751860	+	0.659323i	$0.770843\pi$
$758$	0	0
$759$	−1.79380	−0.0651107
$760$	0	0
$761$	−39.4947	−1.43168	−0.715840	−	0.698264i	$-0.753956\pi$
$761$	−39.4947	−1.43168	−0.715840	+	0.698264i	$0.753956\pi$
$762$	0	0
$763$	5.38243	0.194857
$764$	0	0
$765$	0.0806452	0.00291573
$766$	0	0
$767$	7.26141	0.262194
$768$	0	0
$769$	−34.7031	−1.25143	−0.625713	−	0.780053i	$-0.715192\pi$
$769$	−34.7031	−1.25143	−0.625713	+	0.780053i	$0.715192\pi$
$770$	0	0
$771$	5.57531	0.200790
$772$	0	0
$773$	3.34632	0.120359	0.0601793	−	0.998188i	$-0.480833\pi$
$773$	3.34632	0.120359	0.0601793	+	0.998188i	$0.480833\pi$
$774$	0	0
$775$	−24.3135	−0.873367
$776$	0	0
$777$	4.03895	0.144896
$778$	0	0
$779$	−22.5008	−0.806175
$780$	0	0
$781$	88.7852	3.17698
$782$	0	0
$783$	−13.8432	−0.494717
$784$	0	0
$785$	22.0084	0.785514
$786$	0	0
$787$	32.0950	1.14406	0.572032	−	0.820231i	$-0.306155\pi$
$787$	32.0950	1.14406	0.572032	+	0.820231i	$0.306155\pi$
$788$	0	0
$789$	35.1650	1.25191
$790$	0	0
$791$	−5.96388	−0.212051
$792$	0	0
$793$	−4.07838	−0.144827
$794$	0	0
$795$	12.0989	0.429104
$796$	0	0
$797$	38.3979	1.36012	0.680061	−	0.733156i	$-0.261953\pi$
$797$	38.3979	1.36012	0.680061	+	0.733156i	$0.261953\pi$
$798$	0	0
$799$	−0.115718	−0.00409380
$800$	0	0
$801$	−0.352459	−0.0124535
$802$	0	0
$803$	3.40910	0.120304
$804$	0	0
$805$	−0.127826	−0.00450528
$806$	0	0
$807$	−6.79153	−0.239073
$808$	0	0
$809$	−22.0577	−0.775507	−0.387753	−	0.921763i	$-0.626749\pi$
$809$	−22.0577	−0.775507	−0.387753	+	0.921763i	$0.626749\pi$
$810$	0	0
$811$	31.1100	1.09242	0.546209	−	0.837649i	$-0.316070\pi$
$811$	31.1100	1.09242	0.546209	+	0.837649i	$0.316070\pi$
$812$	0	0
$813$	8.54250	0.299599
$814$	0	0
$815$	−17.4557	−0.611447
$816$	0	0
$817$	−24.0144	−0.840157
$818$	0	0
$819$	0.147300	0.00514708
$820$	0	0
$821$	−35.4641	−1.23771	−0.618853	−	0.785507i	$-0.712402\pi$
$821$	−35.4641	−1.23771	−0.618853	+	0.785507i	$0.712402\pi$
$822$	0	0
$823$	4.89884	0.170763	0.0853813	−	0.996348i	$-0.472789\pi$
$823$	4.89884	0.170763	0.0853813	+	0.996348i	$0.472789\pi$
$824$	0	0
$825$	24.6803	0.859259
$826$	0	0
$827$	−13.9506	−0.485108	−0.242554	−	0.970138i	$-0.577985\pi$
$827$	−13.9506	−0.485108	−0.242554	+	0.970138i	$0.577985\pi$
$828$	0	0
$829$	42.4863	1.47561	0.737804	−	0.675015i	$-0.235863\pi$
$829$	42.4863	1.47561	0.737804	+	0.675015i	$0.235863\pi$
$830$	0	0
$831$	30.7070	1.06521
$832$	0	0
$833$	−4.28231	−0.148373
$834$	0	0
$835$	7.96946	0.275795
$836$	0	0
$837$	−54.6681	−1.88960
$838$	0	0
$839$	11.9421	0.412288	0.206144	−	0.978522i	$-0.433908\pi$
$839$	11.9421	0.412288	0.206144	+	0.978522i	$0.433908\pi$
$840$	0	0
$841$	−22.0784	−0.761323
$842$	0	0
$843$	26.6681	0.918497
$844$	0	0
$845$	−5.92532	−0.203837
$846$	0	0
$847$	12.4741	0.428617
$848$	0	0
$849$	−42.2700	−1.45070
$850$	0	0
$851$	0.872174	0.0298977
$852$	0	0
$853$	16.2800	0.557418	0.278709	−	0.960376i	$-0.410093\pi$
$853$	16.2800	0.557418	0.278709	+	0.960376i	$0.410093\pi$
$854$	0	0
$855$	−0.911122	−0.0311597
$856$	0	0
$857$	26.2085	0.895264	0.447632	−	0.894218i	$-0.352267\pi$
$857$	26.2085	0.895264	0.447632	+	0.894218i	$0.352267\pi$
$858$	0	0
$859$	0.0350725	0.00119666	0.000598329	−	1.00000i	$-0.499810\pi$
$859$	0.0350725	0.00119666	0.000598329		1.00000i	$0.499810\pi$
$860$	0	0
$861$	2.48643	0.0847372
$862$	0	0
$863$	−10.8515	−0.369389	−0.184694	−	0.982796i	$-0.559129\pi$
$863$	−10.8515	−0.369389	−0.184694	+	0.982796i	$0.559129\pi$
$864$	0	0
$865$	1.20394	0.0409351
$866$	0	0
$867$	−28.3773	−0.963745
$868$	0	0
$869$	−43.2350	−1.46665
$870$	0	0
$871$	−34.9097	−1.18287
$872$	0	0
$873$	−0.668060	−0.0226104
$874$	0	0
$875$	5.51640	0.186488
$876$	0	0
$877$	0.665970	0.0224882	0.0112441	−	0.999937i	$-0.496421\pi$
$877$	0.665970	0.0224882	0.0112441	+	0.999937i	$0.496421\pi$
$878$	0	0
$879$	41.3151	1.39352
$880$	0	0
$881$	14.4101	0.485490	0.242745	−	0.970090i	$-0.421952\pi$
$881$	14.4101	0.485490	0.242745	+	0.970090i	$0.421952\pi$
$882$	0	0
$883$	−57.8937	−1.94828	−0.974140	−	0.225947i	$-0.927453\pi$
$883$	−57.8937	−1.94828	−0.974140	+	0.225947i	$0.927453\pi$
$884$	0	0
$885$	4.96332	0.166840
$886$	0	0
$887$	38.4040	1.28948	0.644740	−	0.764402i	$-0.276966\pi$
$887$	38.4040	1.28948	0.644740	+	0.764402i	$0.276966\pi$
$888$	0	0
$889$	0.474142	0.0159022
$890$	0	0
$891$	54.0421	1.81048
$892$	0	0
$893$	1.30737	0.0437494
$894$	0	0
$895$	3.10050	0.103638
$896$	0	0
$897$	−1.18568	−0.0395889
$898$	0	0
$899$	27.3340	0.911641
$900$	0	0
$901$	−2.73820	−0.0912228
$902$	0	0
$903$	2.65368	0.0883091
$904$	0	0
$905$	34.8371	1.15802
$906$	0	0
$907$	42.6407	1.41586	0.707931	−	0.706281i	$-0.249629\pi$
$907$	42.6407	1.41586	0.707931	+	0.706281i	$0.249629\pi$
$908$	0	0
$909$	0.532001	0.0176454
$910$	0	0
$911$	−18.6042	−0.616386	−0.308193	−	0.951324i	$-0.599724\pi$
$911$	−18.6042	−0.616386	−0.308193	+	0.951324i	$0.599724\pi$
$912$	0	0
$913$	42.1855	1.39614
$914$	0	0
$915$	−2.78765	−0.0921570
$916$	0	0
$917$	−1.12783	−0.0372441
$918$	0	0
$919$	−33.2306	−1.09618	−0.548088	−	0.836421i	$-0.684644\pi$
$919$	−33.2306	−1.09618	−0.548088	+	0.836421i	$0.684644\pi$
$920$	0	0
$921$	−33.3379	−1.09852
$922$	0	0
$923$	58.6863	1.93168
$924$	0	0
$925$	−12.0000	−0.394558
$926$	0	0
$927$	0.872174	0.0286460
$928$	0	0
$929$	13.0989	0.429761	0.214880	−	0.976640i	$-0.431064\pi$
$929$	13.0989	0.429761	0.214880	+	0.976640i	$0.431064\pi$
$930$	0	0
$931$	48.3812	1.58563
$932$	0	0
$933$	−14.2784	−0.467455
$934$	0	0
$935$	6.34858	0.207621
$936$	0	0
$937$	−21.0144	−0.686510	−0.343255	−	0.939242i	$-0.611529\pi$
$937$	−21.0144	−0.686510	−0.343255	+	0.939242i	$0.611529\pi$
$938$	0	0
$939$	3.58759	0.117077
$940$	0	0
$941$	−37.3523	−1.21765	−0.608825	−	0.793305i	$-0.708359\pi$
$941$	−37.3523	−1.21765	−0.608825	+	0.793305i	$0.708359\pi$
$942$	0	0
$943$	0.536921	0.0174846
$944$	0	0
$945$	3.95443	0.128638
$946$	0	0
$947$	32.2729	1.04873	0.524363	−	0.851495i	$-0.324303\pi$
$947$	32.2729	1.04873	0.524363	+	0.851495i	$0.324303\pi$
$948$	0	0
$949$	2.25338	0.0731480
$950$	0	0
$951$	47.4473	1.53858
$952$	0	0
$953$	51.3400	1.66307	0.831533	−	0.555476i	$-0.187464\pi$
$953$	51.3400	1.66307	0.831533	+	0.555476i	$0.187464\pi$
$954$	0	0
$955$	4.70596	0.152281
$956$	0	0
$957$	−27.7464	−0.896915
$958$	0	0
$959$	−7.40522	−0.239127
$960$	0	0
$961$	76.9442	2.48207
$962$	0	0
$963$	1.22076	0.0393384
$964$	0	0
$965$	−1.10957	−0.0357185
$966$	0	0
$967$	0.746615	0.0240095	0.0120048	−	0.999928i	$-0.496179\pi$
$967$	0.746615	0.0240095	0.0120048	+	0.999928i	$0.496179\pi$
$968$	0	0
$969$	−7.68649	−0.246926
$970$	0	0
$971$	−58.1276	−1.86541	−0.932703	−	0.360647i	$-0.882556\pi$
$971$	−58.1276	−1.86541	−0.932703	+	0.360647i	$0.882556\pi$
$972$	0	0
$973$	0.709275	0.0227383
$974$	0	0
$975$	16.3135	0.522450
$976$	0	0
$977$	6.86376	0.219591	0.109796	−	0.993954i	$-0.464980\pi$
$977$	6.86376	0.219591	0.109796	+	0.993954i	$0.464980\pi$
$978$	0	0
$979$	−27.7464	−0.886780
$980$	0	0
$981$	−0.915479	−0.0292290
$982$	0	0
$983$	27.8660	0.888788	0.444394	−	0.895831i	$-0.353419\pi$
$983$	27.8660	0.888788	0.444394	+	0.895831i	$0.353419\pi$
$984$	0	0
$985$	31.6225	1.00758
$986$	0	0
$987$	−0.144469	−0.00459851
$988$	0	0
$989$	0.573039	0.0182216
$990$	0	0
$991$	20.0312	0.636312	0.318156	−	0.948038i	$-0.396936\pi$
$991$	20.0312	0.636312	0.318156	+	0.948038i	$0.396936\pi$
$992$	0	0
$993$	2.04945	0.0650373
$994$	0	0
$995$	−34.9315	−1.10740
$996$	0	0
$997$	0.500804	0.0158606	0.00793031	−	0.999969i	$-0.497476\pi$
$997$	0.500804	0.0158606	0.00793031	+	0.999969i	$0.497476\pi$
$998$	0	0
$999$	−26.9816	−0.853659

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	976.2.a.f.1.3		3
3.2	odd	2		8784.2.a.bn.1.2		3
4.3	odd	2		61.2.a.b.1.3	✓	3
8.3	odd	2		3904.2.a.r.1.3		3
8.5	even	2		3904.2.a.w.1.1		3
12.11	even	2		549.2.a.g.1.1		3
20.3	even	4		1525.2.b.b.1099.1		6
20.7	even	4		1525.2.b.b.1099.6		6
20.19	odd	2		1525.2.a.d.1.1		3
28.27	even	2		2989.2.a.i.1.3		3
44.43	even	2		7381.2.a.f.1.1		3
244.243	odd	2		3721.2.a.c.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
61.2.a.b.1.3	✓	3	4.3	odd	2
549.2.a.g.1.1		3	12.11	even	2
976.2.a.f.1.3		3	1.1	even	1	trivial
1525.2.a.d.1.1		3	20.19	odd	2
1525.2.b.b.1099.1		6	20.3	even	4
1525.2.b.b.1099.6		6	20.7	even	4
2989.2.a.i.1.3		3	28.27	even	2
3721.2.a.c.1.1		3	244.243	odd	2
3904.2.a.r.1.3		3	8.3	odd	2
3904.2.a.w.1.1		3	8.5	even	2
7381.2.a.f.1.1		3	44.43	even	2
8784.2.a.bn.1.2		3	3.2	odd	2

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:	\( N \)	\(=\)	\( 976 = 2^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	976.a (trivial)

\(f(q)\)	\(=\)	\(q+1.70928 q^{3} -1.63090 q^{5} +0.460811 q^{7} -0.0783777 q^{9} +O(q^{10})\)	\(q+1.70928 q^{3} -1.63090 q^{5} +0.460811 q^{7} -0.0783777 q^{9} -6.17009 q^{11} -4.07838 q^{13} -2.78765 q^{15} +0.630898 q^{17} -7.12783 q^{19} +0.787653 q^{21} +0.170086 q^{23} -2.34017 q^{25} -5.26180 q^{27} +2.63090 q^{29} +10.3896 q^{31} -10.5464 q^{33} -0.751536 q^{35} +5.12783 q^{37} -6.97107 q^{39} +3.15676 q^{41} +3.36910 q^{43} +0.127826 q^{45} -0.183417 q^{47} -6.78765 q^{49} +1.07838 q^{51} -4.34017 q^{53} +10.0628 q^{55} -12.1834 q^{57} -1.78047 q^{59} +1.00000 q^{61} -0.0361173 q^{63} +6.65142 q^{65} +8.55971 q^{67} +0.290725 q^{69} -14.3896 q^{71} -0.552520 q^{73} -4.00000 q^{75} -2.84324 q^{77} +7.00719 q^{79} -8.75872 q^{81} -6.83710 q^{83} -1.02893 q^{85} +4.49693 q^{87} +4.49693 q^{89} -1.87936 q^{91} +17.7587 q^{93} +11.6248 q^{95} +8.52359 q^{97} +0.483597 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} - q^{5} + 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 - q^5 + 3 * q^7 + 3 * q^9	\( 3 q - 2 q^{3} - q^{5} + 3 q^{7} + 3 q^{9} - 13 q^{11} - 9 q^{13} + 2 q^{15} - 2 q^{17} - 8 q^{21} - 5 q^{23} + 4 q^{25} - 8 q^{27} + 4 q^{29} + 2 q^{31} + 4 q^{33} - 11 q^{35} - 6 q^{37} - 6 q^{39} + 3 q^{41} + 14 q^{43} - 21 q^{45} + 4 q^{47} - 10 q^{49} - 2 q^{53} + 13 q^{55} - 32 q^{57} - 29 q^{59} + 3 q^{61} + 19 q^{63} - 17 q^{65} - 9 q^{67} + 8 q^{69} - 14 q^{71} - q^{73} - 12 q^{75} - 15 q^{77} - 13 q^{79} - q^{81} + 8 q^{83} - 18 q^{85} - 4 q^{87} - 4 q^{89} + 7 q^{91} + 28 q^{93} - 4 q^{95} + 10 q^{97} - 17 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 - q^5 + 3 * q^7 + 3 * q^9 - 13 * q^11 - 9 * q^13 + 2 * q^15 - 2 * q^17 - 8 * q^21 - 5 * q^23 + 4 * q^25 - 8 * q^27 + 4 * q^29 + 2 * q^31 + 4 * q^33 - 11 * q^35 - 6 * q^37 - 6 * q^39 + 3 * q^41 + 14 * q^43 - 21 * q^45 + 4 * q^47 - 10 * q^49 - 2 * q^53 + 13 * q^55 - 32 * q^57 - 29 * q^59 + 3 * q^61 + 19 * q^63 - 17 * q^65 - 9 * q^67 + 8 * q^69 - 14 * q^71 - q^73 - 12 * q^75 - 15 * q^77 - 13 * q^79 - q^81 + 8 * q^83 - 18 * q^85 - 4 * q^87 - 4 * q^89 + 7 * q^91 + 28 * q^93 - 4 * q^95 + 10 * q^97 - 17 * q^99

Introduction

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