LMFDB - Embedded newform 7569.2.a.br.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7569,2,Mod(1,7569)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7569, 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("7569.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7569 = 3^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7569.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$60.4387692899$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 16x^{10} + 93x^{8} - 241x^{6} + 282x^{4} - 149x^{2} + 29 $ x^12 - 16x^10 + 93x^8 - 241x^6 + 282x^4 - 149*x^2 + 29
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 261)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.854850$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.854850 q^{2} -1.26923 q^{4} +3.95934 q^{5} -4.38464 q^{7} -2.79470 q^{8} +3.38464 q^{10} +2.89336 q^{11} -2.95135 q^{13} -3.74821 q^{14} +0.149412 q^{16} +3.86068 q^{17} -2.14757 q^{19} -5.02532 q^{20} +2.47339 q^{22} -0.0986600 q^{23} +10.6764 q^{25} -2.52296 q^{26} +5.56513 q^{28} -0.0777611 q^{31} +5.71713 q^{32} +3.30030 q^{34} -17.3603 q^{35} -10.4189 q^{37} -1.83585 q^{38} -11.0652 q^{40} -6.48468 q^{41} +1.79584 q^{43} -3.67235 q^{44} -0.0843395 q^{46} -9.87873 q^{47} +12.2251 q^{49} +9.12671 q^{50} +3.74595 q^{52} +7.41795 q^{53} +11.4558 q^{55} +12.2538 q^{56} +2.09902 q^{59} +3.55691 q^{61} -0.0664741 q^{62} +4.58846 q^{64} -11.6854 q^{65} +4.65082 q^{67} -4.90010 q^{68} -14.8405 q^{70} -10.2521 q^{71} -6.78225 q^{73} -8.90661 q^{74} +2.72576 q^{76} -12.6864 q^{77} -6.58508 q^{79} +0.591574 q^{80} -5.54343 q^{82} -15.3808 q^{83} +15.2858 q^{85} +1.53517 q^{86} -8.08609 q^{88} +12.9155 q^{89} +12.9406 q^{91} +0.125222 q^{92} -8.44483 q^{94} -8.50295 q^{95} +12.6506 q^{97} +10.4506 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 8 q^{4} - 14 q^{7} + 2 q^{10} - 12 q^{13} - 4 q^{16} + 14 q^{19} + 8 q^{22} + 2 q^{25} - 12 q^{28} - 28 q^{31} - 18 q^{34} - 36 q^{37} + 20 q^{43} - 20 q^{46} - 10 q^{49} - 12 q^{52} + 58 q^{55}+ \cdots - 26 q^{97}+O(q^{100}) $ 12 * q + 8 * q^4 - 14 * q^7 + 2 * q^10 - 12 * q^13 - 4 * q^16 + 14 * q^19 + 8 * q^22 + 2 * q^25 - 12 * q^28 - 28 * q^31 - 18 * q^34 - 36 * q^37 + 20 * q^43 - 20 * q^46 - 10 * q^49 - 12 * q^52 + 58 * q^55 - 4 * q^61 - 18 * q^64 - 54 * q^67 - 60 * q^70 - 32 * q^73 + 56 * q^76 + 2 * q^79 - 96 * q^82 + 4 * q^85 - 10 * q^88 + 6 * q^91 - 32 * q^94 - 26 * q^97

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 $

$2$	0.854850	0.604470	0.302235	−	0.953233i	$-0.402267\pi$
$2$	0.854850	0.604470	0.302235	+	0.953233i	$0.402267\pi$
$3$	0	0
$3$	0	0
$4$	−1.26923	−0.634616
$5$	3.95934	1.77067	0.885336	−	0.464952i	$-0.153928\pi$
$5$	3.95934	1.77067	0.885336	+	0.464952i	$0.153928\pi$
$6$	0	0
$7$	−4.38464	−1.65724	−0.828620	−	0.559812i	$-0.810873\pi$
$7$	−4.38464	−1.65724	−0.828620	+	0.559812i	$0.810873\pi$
$8$	−2.79470	−0.988076
$9$	0	0
$10$	3.38464	1.07032
$11$	2.89336	0.872382	0.436191	−	0.899854i	$-0.356327\pi$
$11$	2.89336	0.872382	0.436191	+	0.899854i	$0.356327\pi$
$12$	0	0
$13$	−2.95135	−0.818557	−0.409279	−	0.912409i	$-0.634220\pi$
$13$	−2.95135	−0.818557	−0.409279	+	0.912409i	$0.634220\pi$
$14$	−3.74821	−1.00175
$15$	0	0
$16$	0.149412	0.0373531
$17$	3.86068	0.936353	0.468177	−	0.883635i	$-0.344911\pi$
$17$	3.86068	0.936353	0.468177	+	0.883635i	$0.344911\pi$
$18$	0	0
$19$	−2.14757	−0.492686	−0.246343	−	0.969183i	$-0.579229\pi$
$19$	−2.14757	−0.492686	−0.246343	+	0.969183i	$0.579229\pi$
$20$	−5.02532	−1.12370
$21$	0	0
$22$	2.47339	0.527329
$23$	−0.0986600	−0.0205720	−0.0102860	−	0.999947i	$-0.503274\pi$
$23$	−0.0986600	−0.0205720	−0.0102860	+	0.999947i	$0.503274\pi$
$24$	0	0
$25$	10.6764	2.13528
$26$	−2.52296	−0.494793
$27$	0	0
$28$	5.56513	1.05171
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−0.0777611	−0.0139663	−0.00698315	−	0.999976i	$-0.502223\pi$
$31$	−0.0777611	−0.0139663	−0.00698315	+	0.999976i	$0.502223\pi$
$32$	5.71713	1.01066
$33$	0	0
$34$	3.30030	0.565998
$35$	−17.3603	−2.93443
$36$	0	0
$37$	−10.4189	−1.71286	−0.856429	−	0.516264i	$-0.827322\pi$
$37$	−10.4189	−1.71286	−0.856429	+	0.516264i	$0.827322\pi$
$38$	−1.83585	−0.297814
$39$	0	0
$40$	−11.0652	−1.74956
$41$	−6.48468	−1.01274	−0.506369	−	0.862317i	$-0.669012\pi$
$41$	−6.48468	−1.01274	−0.506369	+	0.862317i	$0.669012\pi$
$42$	0	0
$43$	1.79584	0.273863	0.136932	−	0.990581i	$-0.456276\pi$
$43$	1.79584	0.273863	0.136932	+	0.990581i	$0.456276\pi$
$44$	−3.67235	−0.553627
$45$	0	0
$46$	−0.0843395	−0.0124352
$47$	−9.87873	−1.44096	−0.720480	−	0.693476i	$-0.756079\pi$
$47$	−9.87873	−1.44096	−0.720480	+	0.693476i	$0.756079\pi$
$48$	0	0
$49$	12.2251	1.74644
$50$	9.12671	1.29071
$51$	0	0
$52$	3.74595	0.519469
$53$	7.41795	1.01893	0.509467	−	0.860490i	$-0.329843\pi$
$53$	7.41795	1.01893	0.509467	+	0.860490i	$0.329843\pi$
$54$	0	0
$55$	11.4558	1.54470
$56$	12.2538	1.63748
$57$	0	0
$58$	0	0
$59$	2.09902	0.273269	0.136635	−	0.990622i	$-0.456371\pi$
$59$	2.09902	0.273269	0.136635	+	0.990622i	$0.456371\pi$
$60$	0	0
$61$	3.55691	0.455416	0.227708	−	0.973730i	$-0.426877\pi$
$61$	3.55691	0.455416	0.227708	+	0.973730i	$0.426877\pi$
$62$	−0.0664741	−0.00844222
$63$	0	0
$64$	4.58846	0.573558
$65$	−11.6854	−1.44940
$66$	0	0
$67$	4.65082	0.568188	0.284094	−	0.958796i	$-0.408307\pi$
$67$	4.65082	0.568188	0.284094	+	0.958796i	$0.408307\pi$
$68$	−4.90010	−0.594224
$69$	0	0
$70$	−14.8405	−1.77377
$71$	−10.2521	−1.21671	−0.608353	−	0.793667i	$-0.708169\pi$
$71$	−10.2521	−1.21671	−0.608353	+	0.793667i	$0.708169\pi$
$72$	0	0
$73$	−6.78225	−0.793802	−0.396901	−	0.917861i	$-0.629914\pi$
$73$	−6.78225	−0.793802	−0.396901	+	0.917861i	$0.629914\pi$
$74$	−8.90661	−1.03537
$75$	0	0
$76$	2.72576	0.312666
$77$	−12.6864	−1.44575
$78$	0	0
$79$	−6.58508	−0.740879	−0.370440	−	0.928857i	$-0.620793\pi$
$79$	−6.58508	−0.740879	−0.370440	+	0.928857i	$0.620793\pi$
$80$	0.591574	0.0661400
$81$	0	0
$82$	−5.54343	−0.612169
$83$	−15.3808	−1.68827	−0.844133	−	0.536134i	$-0.819884\pi$
$83$	−15.3808	−1.68827	−0.844133	+	0.536134i	$0.819884\pi$
$84$	0	0
$85$	15.2858	1.65797
$86$	1.53517	0.165542
$87$	0	0
$88$	−8.08609	−0.861980
$89$	12.9155	1.36904	0.684520	−	0.728995i	$-0.260012\pi$
$89$	12.9155	1.36904	0.684520	+	0.728995i	$0.260012\pi$
$90$	0	0
$91$	12.9406	1.35655
$92$	0.125222	0.0130553
$93$	0	0
$94$	−8.44483	−0.871017
$95$	−8.50295	−0.872385
$96$	0	0
$97$	12.6506	1.28447	0.642237	−	0.766506i	$-0.278006\pi$
$97$	12.6506	1.28447	0.642237	+	0.766506i	$0.278006\pi$
$98$	10.4506	1.05567
$99$	0	0
$100$	−13.5508	−1.35508
$101$	−9.85634	−0.980743	−0.490371	−	0.871514i	$-0.663139\pi$
$101$	−9.85634	−0.980743	−0.490371	+	0.871514i	$0.663139\pi$
$102$	0	0
$103$	3.07274	0.302766	0.151383	−	0.988475i	$-0.451627\pi$
$103$	3.07274	0.302766	0.151383	+	0.988475i	$0.451627\pi$
$104$	8.24814	0.808797
$105$	0	0
$106$	6.34124	0.615915
$107$	−11.5675	−1.11828	−0.559138	−	0.829074i	$-0.688868\pi$
$107$	−11.5675	−1.11828	−0.559138	+	0.829074i	$0.688868\pi$
$108$	0	0
$109$	−2.82529	−0.270614	−0.135307	−	0.990804i	$-0.543202\pi$
$109$	−2.82529	−0.270614	−0.135307	+	0.990804i	$0.543202\pi$
$110$	9.79300	0.933726
$111$	0	0
$112$	−0.655120	−0.0619030
$113$	9.07625	0.853822	0.426911	−	0.904294i	$-0.359602\pi$
$113$	9.07625	0.853822	0.426911	+	0.904294i	$0.359602\pi$
$114$	0	0
$115$	−0.390629	−0.0364263
$116$	0	0
$117$	0	0
$118$	1.79435	0.165183
$119$	−16.9277	−1.55176
$120$	0	0
$121$	−2.62845	−0.238950
$122$	3.04062	0.275285
$123$	0	0
$124$	0.0986969	0.00886324
$125$	22.4748	2.01021
$126$	0	0
$127$	3.28599	0.291584	0.145792	−	0.989315i	$-0.453427\pi$
$127$	3.28599	0.291584	0.145792	+	0.989315i	$0.453427\pi$
$128$	−7.51181	−0.663957
$129$	0	0
$130$	−9.98927	−0.876117
$131$	−9.04174	−0.789981	−0.394990	−	0.918685i	$-0.629252\pi$
$131$	−9.04174	−0.789981	−0.394990	+	0.918685i	$0.629252\pi$
$132$	0	0
$133$	9.41632	0.816498
$134$	3.97575	0.343453
$135$	0	0
$136$	−10.7895	−0.925188
$137$	−1.83041	−0.156382	−0.0781911	−	0.996938i	$-0.524914\pi$
$137$	−1.83041	−0.156382	−0.0781911	+	0.996938i	$0.524914\pi$
$138$	0	0
$139$	−22.1499	−1.87873	−0.939366	−	0.342915i	$-0.888586\pi$
$139$	−22.1499	−1.87873	−0.939366	+	0.342915i	$0.888586\pi$
$140$	22.0343	1.86223
$141$	0	0
$142$	−8.76405	−0.735462
$143$	−8.53933	−0.714094
$144$	0	0
$145$	0	0
$146$	−5.79780	−0.479830
$147$	0	0
$148$	13.2240	1.08701
$149$	−8.46364	−0.693369	−0.346684	−	0.937982i	$-0.612692\pi$
$149$	−8.46364	−0.693369	−0.346684	+	0.937982i	$0.612692\pi$
$150$	0	0
$151$	−10.4844	−0.853212	−0.426606	−	0.904438i	$-0.640291\pi$
$151$	−10.4844	−0.853212	−0.426606	+	0.904438i	$0.640291\pi$
$152$	6.00181	0.486811
$153$	0	0
$154$	−10.8449	−0.873910
$155$	−0.307883	−0.0247298
$156$	0	0
$157$	−15.6706	−1.25065	−0.625324	−	0.780366i	$-0.715033\pi$
$157$	−15.6706	−1.25065	−0.625324	+	0.780366i	$0.715033\pi$
$158$	−5.62925	−0.447839
$159$	0	0
$160$	22.6361	1.78954
$161$	0.432589	0.0340928
$162$	0	0
$163$	24.8052	1.94290	0.971448	−	0.237252i	$-0.0762466\pi$
$163$	24.8052	1.94290	0.971448	+	0.237252i	$0.0762466\pi$
$164$	8.23056	0.642699
$165$	0	0
$166$	−13.1483	−1.02051
$167$	−2.65683	−0.205592	−0.102796	−	0.994702i	$-0.532779\pi$
$167$	−2.65683	−0.205592	−0.102796	+	0.994702i	$0.532779\pi$
$168$	0	0
$169$	−4.28953	−0.329964
$170$	13.0670	1.00220
$171$	0	0
$172$	−2.27934	−0.173798
$173$	−4.85004	−0.368742	−0.184371	−	0.982857i	$-0.559025\pi$
$173$	−4.85004	−0.368742	−0.184371	+	0.982857i	$0.559025\pi$
$174$	0	0
$175$	−46.8122	−3.53867
$176$	0.432304	0.0325861
$177$	0	0
$178$	11.0408	0.827543
$179$	−8.72679	−0.652271	−0.326135	−	0.945323i	$-0.605746\pi$
$179$	−8.72679	−0.652271	−0.326135	+	0.945323i	$0.605746\pi$
$180$	0	0
$181$	−8.70449	−0.647000	−0.323500	−	0.946228i	$-0.604860\pi$
$181$	−8.70449	−0.647000	−0.323500	+	0.946228i	$0.604860\pi$
$182$	11.0623	0.819991
$183$	0	0
$184$	0.275725	0.0203267
$185$	−41.2520	−3.03291
$186$	0	0
$187$	11.1704	0.816857
$188$	12.5384	0.914456
$189$	0	0
$190$	−7.26875	−0.527331
$191$	1.77576	0.128489	0.0642446	−	0.997934i	$-0.479536\pi$
$191$	1.77576	0.128489	0.0642446	+	0.997934i	$0.479536\pi$
$192$	0	0
$193$	−0.800725	−0.0576375	−0.0288187	−	0.999585i	$-0.509175\pi$
$193$	−0.800725	−0.0576375	−0.0288187	+	0.999585i	$0.509175\pi$
$194$	10.8144	0.776426
$195$	0	0
$196$	−15.5165	−1.10832
$197$	−7.89840	−0.562738	−0.281369	−	0.959600i	$-0.590788\pi$
$197$	−7.89840	−0.562738	−0.281369	+	0.959600i	$0.590788\pi$
$198$	0	0
$199$	−21.7842	−1.54424	−0.772120	−	0.635477i	$-0.780804\pi$
$199$	−21.7842	−1.54424	−0.772120	+	0.635477i	$0.780804\pi$
$200$	−29.8373	−2.10982
$201$	0	0
$202$	−8.42569	−0.592830
$203$	0	0
$204$	0	0
$205$	−25.6751	−1.79323
$206$	2.62673	0.183013
$207$	0	0
$208$	−0.440968	−0.0305756
$209$	−6.21369	−0.429810
$210$	0	0
$211$	4.64303	0.319639	0.159820	−	0.987146i	$-0.448909\pi$
$211$	4.64303	0.319639	0.159820	+	0.987146i	$0.448909\pi$
$212$	−9.41510	−0.646632
$213$	0	0
$214$	−9.88851	−0.675965
$215$	7.11035	0.484922
$216$	0	0
$217$	0.340955	0.0231455
$218$	−2.41520	−0.163578
$219$	0	0
$220$	−14.5401	−0.980292
$221$	−11.3942	−0.766459
$222$	0	0
$223$	−2.78463	−0.186473	−0.0932364	−	0.995644i	$-0.529721\pi$
$223$	−2.78463	−0.186473	−0.0932364	+	0.995644i	$0.529721\pi$
$224$	−25.0676	−1.67490
$225$	0	0
$226$	7.75883	0.516110
$227$	−22.2087	−1.47404	−0.737022	−	0.675869i	$-0.763769\pi$
$227$	−22.2087	−1.47404	−0.737022	+	0.675869i	$0.763769\pi$
$228$	0	0
$229$	9.52046	0.629129	0.314565	−	0.949236i	$-0.398141\pi$
$229$	9.52046	0.629129	0.314565	+	0.949236i	$0.398141\pi$
$230$	−0.333929	−0.0220186
$231$	0	0
$232$	0	0
$233$	−17.1025	−1.12042	−0.560210	−	0.828350i	$-0.689280\pi$
$233$	−17.1025	−1.12042	−0.560210	+	0.828350i	$0.689280\pi$
$234$	0	0
$235$	−39.1133	−2.55147
$236$	−2.66414	−0.173421
$237$	0	0
$238$	−14.4707	−0.937993
$239$	13.9576	0.902839	0.451419	−	0.892312i	$-0.350918\pi$
$239$	13.9576	0.902839	0.451419	+	0.892312i	$0.350918\pi$
$240$	0	0
$241$	4.05067	0.260927	0.130463	−	0.991453i	$-0.458354\pi$
$241$	4.05067	0.260927	0.130463	+	0.991453i	$0.458354\pi$
$242$	−2.24693	−0.144438
$243$	0	0
$244$	−4.51454	−0.289014
$245$	48.4034	3.09238
$246$	0	0
$247$	6.33822	0.403291
$248$	0.217319	0.0137998
$249$	0	0
$250$	19.2126	1.21511
$251$	0.931184	0.0587758	0.0293879	−	0.999568i	$-0.490644\pi$
$251$	0.931184	0.0587758	0.0293879	+	0.999568i	$0.490644\pi$
$252$	0	0
$253$	−0.285459	−0.0179467
$254$	2.80903	0.176254
$255$	0	0
$256$	−15.5984	−0.974900
$257$	7.08616	0.442023	0.221011	−	0.975271i	$-0.429064\pi$
$257$	7.08616	0.442023	0.221011	+	0.975271i	$0.429064\pi$
$258$	0	0
$259$	45.6832	2.83862
$260$	14.8315	0.919810
$261$	0	0
$262$	−7.72933	−0.477520
$263$	12.6970	0.782932	0.391466	−	0.920192i	$-0.371968\pi$
$263$	12.6970	0.782932	0.391466	+	0.920192i	$0.371968\pi$
$264$	0	0
$265$	29.3702	1.80420
$266$	8.04954	0.493549
$267$	0	0
$268$	−5.90297	−0.360581
$269$	10.3950	0.633796	0.316898	−	0.948460i	$-0.397359\pi$
$269$	10.3950	0.633796	0.316898	+	0.948460i	$0.397359\pi$
$270$	0	0
$271$	−24.6121	−1.49508	−0.747538	−	0.664219i	$-0.768764\pi$
$271$	−24.6121	−1.49508	−0.747538	+	0.664219i	$0.768764\pi$
$272$	0.576833	0.0349757
$273$	0	0
$274$	−1.56472	−0.0945284
$275$	30.8907	1.86278
$276$	0	0
$277$	13.3979	0.805000	0.402500	−	0.915420i	$-0.368141\pi$
$277$	13.3979	0.805000	0.402500	+	0.915420i	$0.368141\pi$
$278$	−18.9349	−1.13564
$279$	0	0
$280$	48.5169	2.89944
$281$	−4.44947	−0.265433	−0.132716	−	0.991154i	$-0.542370\pi$
$281$	−4.44947	−0.265433	−0.132716	+	0.991154i	$0.542370\pi$
$282$	0	0
$283$	−11.4395	−0.680009	−0.340005	−	0.940424i	$-0.610429\pi$
$283$	−11.4395	−0.680009	−0.340005	+	0.940424i	$0.610429\pi$
$284$	13.0123	0.772141
$285$	0	0
$286$	−7.29984	−0.431649
$287$	28.4330	1.67835
$288$	0	0
$289$	−2.09513	−0.123243
$290$	0	0
$291$	0	0
$292$	8.60824	0.503759
$293$	−13.3520	−0.780030	−0.390015	−	0.920808i	$-0.627530\pi$
$293$	−13.3520	−0.780030	−0.390015	+	0.920808i	$0.627530\pi$
$294$	0	0
$295$	8.31074	0.483870
$296$	29.1178	1.69244
$297$	0	0
$298$	−7.23514	−0.419121
$299$	0.291180	0.0168394
$300$	0	0
$301$	−7.87412	−0.453857
$302$	−8.96262	−0.515741
$303$	0	0
$304$	−0.320873	−0.0184033
$305$	14.0830	0.806391
$306$	0	0
$307$	14.9596	0.853787	0.426894	−	0.904302i	$-0.359608\pi$
$307$	14.9596	0.853787	0.426894	+	0.904302i	$0.359608\pi$
$308$	16.1019	0.917493
$309$	0	0
$310$	−0.263194	−0.0149484
$311$	−19.7839	−1.12184	−0.560921	−	0.827869i	$-0.689553\pi$
$311$	−19.7839	−1.12184	−0.560921	+	0.827869i	$0.689553\pi$
$312$	0	0
$313$	4.99856	0.282535	0.141268	−	0.989971i	$-0.454882\pi$
$313$	4.99856	0.282535	0.141268	+	0.989971i	$0.454882\pi$
$314$	−13.3960	−0.755979
$315$	0	0
$316$	8.35799	0.470174
$317$	−25.4435	−1.42905	−0.714524	−	0.699611i	$-0.753357\pi$
$317$	−25.4435	−1.42905	−0.714524	+	0.699611i	$0.753357\pi$
$318$	0	0
$319$	0	0
$320$	18.1673	1.01558
$321$	0	0
$322$	0.369799	0.0206081
$323$	−8.29108	−0.461328
$324$	0	0
$325$	−31.5098	−1.74785
$326$	21.2048	1.17442
$327$	0	0
$328$	18.1228	1.00066
$329$	43.3147	2.38802
$330$	0	0
$331$	6.50771	0.357696	0.178848	−	0.983877i	$-0.442763\pi$
$331$	6.50771	0.357696	0.178848	+	0.983877i	$0.442763\pi$
$332$	19.5218	1.07140
$333$	0	0
$334$	−2.27119	−0.124274
$335$	18.4142	1.00607
$336$	0	0
$337$	17.5303	0.954939	0.477470	−	0.878648i	$-0.341554\pi$
$337$	17.5303	0.954939	0.477470	+	0.878648i	$0.341554\pi$
$338$	−3.66691	−0.199453
$339$	0	0
$340$	−19.4012	−1.05218
$341$	−0.224991	−0.0121840
$342$	0	0
$343$	−22.9102	−1.23703
$344$	−5.01884	−0.270598
$345$	0	0
$346$	−4.14606	−0.222893
$347$	30.8369	1.65541	0.827706	−	0.561163i	$-0.189646\pi$
$347$	30.8369	1.65541	0.827706	+	0.561163i	$0.189646\pi$
$348$	0	0
$349$	−5.77523	−0.309141	−0.154570	−	0.987982i	$-0.549399\pi$
$349$	−5.77523	−0.309141	−0.154570	+	0.987982i	$0.549399\pi$
$350$	−40.0174	−2.13902
$351$	0	0
$352$	16.5417	0.881677
$353$	17.8845	0.951894	0.475947	−	0.879474i	$-0.342105\pi$
$353$	17.8845	0.951894	0.475947	+	0.879474i	$0.342105\pi$
$354$	0	0
$355$	−40.5918	−2.15439
$356$	−16.3927	−0.868814
$357$	0	0
$358$	−7.46009	−0.394278
$359$	8.77761	0.463265	0.231632	−	0.972803i	$-0.425593\pi$
$359$	8.77761	0.463265	0.231632	+	0.972803i	$0.425593\pi$
$360$	0	0
$361$	−14.3880	−0.757261
$362$	−7.44103	−0.391092
$363$	0	0
$364$	−16.4246	−0.860885
$365$	−26.8532	−1.40556
$366$	0	0
$367$	−1.80426	−0.0941814	−0.0470907	−	0.998891i	$-0.514995\pi$
$367$	−1.80426	−0.0941814	−0.0470907	+	0.998891i	$0.514995\pi$
$368$	−0.0147410	−0.000768429 0
$369$	0	0
$370$	−35.2643	−1.83330
$371$	−32.5251	−1.68862
$372$	0	0
$373$	27.5219	1.42503	0.712515	−	0.701657i	$-0.247556\pi$
$373$	27.5219	1.42503	0.712515	+	0.701657i	$0.247556\pi$
$374$	9.54898	0.493766
$375$	0	0
$376$	27.6081	1.42378
$377$	0	0
$378$	0	0
$379$	−32.5757	−1.67330	−0.836651	−	0.547737i	$-0.815490\pi$
$379$	−32.5757	−1.67330	−0.836651	+	0.547737i	$0.815490\pi$
$380$	10.7922	0.553629
$381$	0	0
$382$	1.51800	0.0776679
$383$	25.3043	1.29299	0.646496	−	0.762917i	$-0.276234\pi$
$383$	25.3043	1.29299	0.646496	+	0.762917i	$0.276234\pi$
$384$	0	0
$385$	−50.2297	−2.55994
$386$	−0.684500	−0.0348401
$387$	0	0
$388$	−16.0565	−0.815148
$389$	25.9152	1.31395	0.656976	−	0.753912i	$-0.271835\pi$
$389$	25.9152	1.31395	0.656976	+	0.753912i	$0.271835\pi$
$390$	0	0
$391$	−0.380895	−0.0192627
$392$	−34.1655	−1.72562
$393$	0	0
$394$	−6.75195	−0.340158
$395$	−26.0726	−1.31185
$396$	0	0
$397$	4.17910	0.209743	0.104872	−	0.994486i	$-0.466557\pi$
$397$	4.17910	0.209743	0.104872	+	0.994486i	$0.466557\pi$
$398$	−18.6222	−0.933447
$399$	0	0
$400$	1.59518	0.0797592
$401$	−12.6370	−0.631062	−0.315531	−	0.948915i	$-0.602183\pi$
$401$	−12.6370	−0.631062	−0.315531	+	0.948915i	$0.602183\pi$
$402$	0	0
$403$	0.229500	0.0114322
$404$	12.5100	0.622395
$405$	0	0
$406$	0	0
$407$	−30.1457	−1.49427
$408$	0	0
$409$	−20.3988	−1.00866	−0.504328	−	0.863512i	$-0.668260\pi$
$409$	−20.3988	−1.00866	−0.504328	+	0.863512i	$0.668260\pi$
$410$	−21.9483	−1.08395
$411$	0	0
$412$	−3.90002	−0.192140
$413$	−9.20346	−0.452873
$414$	0	0
$415$	−60.8980	−2.98936
$416$	−16.8733	−0.827279
$417$	0	0
$418$	−5.31177	−0.259807
$419$	−22.2066	−1.08486	−0.542432	−	0.840099i	$-0.682497\pi$
$419$	−22.2066	−1.08486	−0.542432	+	0.840099i	$0.682497\pi$
$420$	0	0
$421$	1.61392	0.0786576	0.0393288	−	0.999226i	$-0.487478\pi$
$421$	1.61392	0.0786576	0.0393288	+	0.999226i	$0.487478\pi$
$422$	3.96909	0.193212
$423$	0	0
$424$	−20.7310	−1.00678
$425$	41.2182	1.99938
$426$	0	0
$427$	−15.5958	−0.754733
$428$	14.6819	0.709676
$429$	0	0
$430$	6.07828	0.293121
$431$	−31.7335	−1.52855	−0.764275	−	0.644891i	$-0.776903\pi$
$431$	−31.7335	−1.52855	−0.764275	+	0.644891i	$0.776903\pi$
$432$	0	0
$433$	−28.5005	−1.36965	−0.684823	−	0.728709i	$-0.740121\pi$
$433$	−28.5005	−1.36965	−0.684823	+	0.728709i	$0.740121\pi$
$434$	0.291465	0.0139908
$435$	0	0
$436$	3.58595	0.171736
$437$	0.211879	0.0101355
$438$	0	0
$439$	−22.4252	−1.07030	−0.535148	−	0.844758i	$-0.679744\pi$
$439$	−22.4252	−1.07030	−0.535148	+	0.844758i	$0.679744\pi$
$440$	−32.0156	−1.52628
$441$	0	0
$442$	−9.74035	−0.463301
$443$	26.9746	1.28160	0.640801	−	0.767707i	$-0.278602\pi$
$443$	26.9746	1.28160	0.640801	+	0.767707i	$0.278602\pi$
$444$	0	0
$445$	51.1368	2.42412
$446$	−2.38044	−0.112717
$447$	0	0
$448$	−20.1188	−0.950523
$449$	23.3909	1.10388	0.551942	−	0.833883i	$-0.313887\pi$
$449$	23.3909	1.10388	0.551942	+	0.833883i	$0.313887\pi$
$450$	0	0
$451$	−18.7625	−0.883493
$452$	−11.5199	−0.541849
$453$	0	0
$454$	−18.9851	−0.891016
$455$	51.2363	2.40200
$456$	0	0
$457$	−3.99072	−0.186678	−0.0933389	−	0.995634i	$-0.529754\pi$
$457$	−3.99072	−0.186678	−0.0933389	+	0.995634i	$0.529754\pi$
$458$	8.13856	0.380290
$459$	0	0
$460$	0.495798	0.0231167
$461$	−25.6053	−1.19256	−0.596279	−	0.802777i	$-0.703355\pi$
$461$	−25.6053	−1.19256	−0.596279	+	0.802777i	$0.703355\pi$
$462$	0	0
$463$	12.5384	0.582708	0.291354	−	0.956615i	$-0.405894\pi$
$463$	12.5384	0.582708	0.291354	+	0.956615i	$0.405894\pi$
$464$	0	0
$465$	0	0
$466$	−14.6201	−0.677261
$467$	18.1338	0.839134	0.419567	−	0.907724i	$-0.362182\pi$
$467$	18.1338	0.839134	0.419567	+	0.907724i	$0.362182\pi$
$468$	0	0
$469$	−20.3922	−0.941623
$470$	−33.4360	−1.54229
$471$	0	0
$472$	−5.86614	−0.270011
$473$	5.19602	0.238913
$474$	0	0
$475$	−22.9283	−1.05202
$476$	21.4852	0.984772
$477$	0	0
$478$	11.9316	0.545739
$479$	12.0320	0.549758	0.274879	−	0.961479i	$-0.411362\pi$
$479$	12.0320	0.549758	0.274879	+	0.961479i	$0.411362\pi$
$480$	0	0
$481$	30.7499	1.40207
$482$	3.46272	0.157722
$483$	0	0
$484$	3.33612	0.151642
$485$	50.0881	2.27438
$486$	0	0
$487$	−23.6834	−1.07320	−0.536599	−	0.843838i	$-0.680291\pi$
$487$	−23.6834	−1.07320	−0.536599	+	0.843838i	$0.680291\pi$
$488$	−9.94050	−0.449985
$489$	0	0
$490$	41.3776	1.86925
$491$	−8.55934	−0.386277	−0.193139	−	0.981171i	$-0.561867\pi$
$491$	−8.55934	−0.386277	−0.193139	+	0.981171i	$0.561867\pi$
$492$	0	0
$493$	0	0
$494$	5.41823	0.243778
$495$	0	0
$496$	−0.0116185	−0.000521685 0
$497$	44.9520	2.01637
$498$	0	0
$499$	25.2356	1.12970	0.564850	−	0.825194i	$-0.308934\pi$
$499$	25.2356	1.12970	0.564850	+	0.825194i	$0.308934\pi$
$500$	−28.5257	−1.27571
$501$	0	0
$502$	0.796022	0.0355282
$503$	−6.30927	−0.281317	−0.140658	−	0.990058i	$-0.544922\pi$
$503$	−6.30927	−0.281317	−0.140658	+	0.990058i	$0.544922\pi$
$504$	0	0
$505$	−39.0246	−1.73657
$506$	−0.244025	−0.0108482
$507$	0	0
$508$	−4.17068	−0.185044
$509$	6.06578	0.268861	0.134431	−	0.990923i	$-0.457079\pi$
$509$	6.06578	0.268861	0.134431	+	0.990923i	$0.457079\pi$
$510$	0	0
$511$	29.7377	1.31552
$512$	1.68934	0.0746588
$513$	0	0
$514$	6.05760	0.267189
$515$	12.1660	0.536099
$516$	0	0
$517$	−28.5827	−1.25707
$518$	39.0523	1.71586
$519$	0	0
$520$	32.6572	1.43211
$521$	−1.78915	−0.0783842	−0.0391921	−	0.999232i	$-0.512478\pi$
$521$	−1.78915	−0.0783842	−0.0391921	+	0.999232i	$0.512478\pi$
$522$	0	0
$523$	1.03803	0.0453899	0.0226950	−	0.999742i	$-0.492775\pi$
$523$	1.03803	0.0453899	0.0226950	+	0.999742i	$0.492775\pi$
$524$	11.4761	0.501334
$525$	0	0
$526$	10.8541	0.473259
$527$	−0.300211	−0.0130774
$528$	0	0
$529$	−22.9903	−0.999577
$530$	25.1071	1.09058
$531$	0	0
$532$	−11.9515	−0.518163
$533$	19.1386	0.828983
$534$	0	0
$535$	−45.7999	−1.98010
$536$	−12.9977	−0.561413
$537$	0	0
$538$	8.88619	0.383111
$539$	35.3716	1.52356
$540$	0	0
$541$	36.9316	1.58781	0.793907	−	0.608039i	$-0.208044\pi$
$541$	36.9316	1.58781	0.793907	+	0.608039i	$0.208044\pi$
$542$	−21.0396	−0.903729
$543$	0	0
$544$	22.0720	0.946330
$545$	−11.1863	−0.479168
$546$	0	0
$547$	39.4434	1.68648	0.843239	−	0.537539i	$-0.180646\pi$
$547$	39.4434	1.68648	0.843239	+	0.537539i	$0.180646\pi$
$548$	2.32321	0.0992427
$549$	0	0
$550$	26.4069	1.12599
$551$	0	0
$552$	0	0
$553$	28.8732	1.22781
$554$	11.4532	0.486599
$555$	0	0
$556$	28.1134	1.19227
$557$	3.06343	0.129802	0.0649009	−	0.997892i	$-0.479327\pi$
$557$	3.06343	0.129802	0.0649009	+	0.997892i	$0.479327\pi$
$558$	0	0
$559$	−5.30016	−0.224173
$560$	−2.59384	−0.109610
$561$	0	0
$562$	−3.80363	−0.160446
$563$	−20.6277	−0.869355	−0.434678	−	0.900586i	$-0.643138\pi$
$563$	−20.6277	−0.869355	−0.434678	+	0.900586i	$0.643138\pi$
$564$	0	0
$565$	35.9360	1.51184
$566$	−9.77908	−0.411045
$567$	0	0
$568$	28.6517	1.20220
$569$	43.6297	1.82905	0.914525	−	0.404529i	$-0.132565\pi$
$569$	43.6297	1.82905	0.914525	+	0.404529i	$0.132565\pi$
$570$	0	0
$571$	12.9521	0.542028	0.271014	−	0.962575i	$-0.412641\pi$
$571$	12.9521	0.542028	0.271014	+	0.962575i	$0.412641\pi$
$572$	10.8384	0.453176
$573$	0	0
$574$	24.3060	1.01451
$575$	−1.05333	−0.0439270
$576$	0	0
$577$	0.395788	0.0164769	0.00823843	−	0.999966i	$-0.497378\pi$
$577$	0.395788	0.0164769	0.00823843	+	0.999966i	$0.497378\pi$
$578$	−1.79102	−0.0744966
$579$	0	0
$580$	0	0
$581$	67.4395	2.79786
$582$	0	0
$583$	21.4628	0.888899
$584$	18.9544	0.784337
$585$	0	0
$586$	−11.4139	−0.471505
$587$	−5.35996	−0.221229	−0.110615	−	0.993863i	$-0.535282\pi$
$587$	−5.35996	−0.221229	−0.110615	+	0.993863i	$0.535282\pi$
$588$	0	0
$589$	0.166997	0.00688100
$590$	7.10444	0.292485
$591$	0	0
$592$	−1.55671	−0.0639805
$593$	−25.3529	−1.04112	−0.520558	−	0.853826i	$-0.674276\pi$
$593$	−25.3529	−1.04112	−0.520558	+	0.853826i	$0.674276\pi$
$594$	0	0
$595$	−67.0226	−2.74766
$596$	10.7423	0.440023
$597$	0	0
$598$	0.248915	0.0101789
$599$	−18.8602	−0.770606	−0.385303	−	0.922790i	$-0.625903\pi$
$599$	−18.8602	−0.770606	−0.385303	+	0.922790i	$0.625903\pi$
$600$	0	0
$601$	32.5256	1.32675	0.663374	−	0.748288i	$-0.269124\pi$
$601$	32.5256	1.32675	0.663374	+	0.748288i	$0.269124\pi$
$602$	−6.73119	−0.274343
$603$	0	0
$604$	13.3072	0.541462
$605$	−10.4070	−0.423103
$606$	0	0
$607$	1.44552	0.0586719	0.0293360	−	0.999570i	$-0.490661\pi$
$607$	1.44552	0.0586719	0.0293360	+	0.999570i	$0.490661\pi$
$608$	−12.2779	−0.497935
$609$	0	0
$610$	12.0389	0.487440
$611$	29.1556	1.17951
$612$	0	0
$613$	−7.05074	−0.284777	−0.142388	−	0.989811i	$-0.545478\pi$
$613$	−7.05074	−0.284777	−0.142388	+	0.989811i	$0.545478\pi$
$614$	12.7882	0.516089
$615$	0	0
$616$	35.4546	1.42851
$617$	−44.5362	−1.79296	−0.896480	−	0.443084i	$-0.853884\pi$
$617$	−44.5362	−1.79296	−0.896480	+	0.443084i	$0.853884\pi$
$618$	0	0
$619$	18.1811	0.730759	0.365379	−	0.930859i	$-0.380939\pi$
$619$	18.1811	0.730759	0.365379	+	0.930859i	$0.380939\pi$
$620$	0.390775	0.0156939
$621$	0	0
$622$	−16.9123	−0.678120
$623$	−56.6298	−2.26883
$624$	0	0
$625$	35.6034	1.42414
$626$	4.27302	0.170784
$627$	0	0
$628$	19.8896	0.793681
$629$	−40.2241	−1.60384
$630$	0	0
$631$	−22.2589	−0.886111	−0.443056	−	0.896494i	$-0.646106\pi$
$631$	−22.2589	−0.886111	−0.443056	+	0.896494i	$0.646106\pi$
$632$	18.4033	0.732045
$633$	0	0
$634$	−21.7504	−0.863817
$635$	13.0104	0.516300
$636$	0	0
$637$	−36.0806	−1.42956
$638$	0	0
$639$	0	0
$640$	−29.7418	−1.17565
$641$	−27.8977	−1.10189	−0.550946	−	0.834541i	$-0.685733\pi$
$641$	−27.8977	−1.10189	−0.550946	+	0.834541i	$0.685733\pi$
$642$	0	0
$643$	39.5665	1.56035	0.780176	−	0.625560i	$-0.215129\pi$
$643$	39.5665	1.56035	0.780176	+	0.625560i	$0.215129\pi$
$644$	−0.549056	−0.0216358
$645$	0	0
$646$	−7.08763	−0.278859
$647$	−19.4957	−0.766454	−0.383227	−	0.923654i	$-0.625187\pi$
$647$	−19.4957	−0.766454	−0.383227	+	0.923654i	$0.625187\pi$
$648$	0	0
$649$	6.07323	0.238395
$650$	−26.9361	−1.05652
$651$	0	0
$652$	−31.4836	−1.23299
$653$	7.08315	0.277185	0.138593	−	0.990349i	$-0.455742\pi$
$653$	7.08315	0.277185	0.138593	+	0.990349i	$0.455742\pi$
$654$	0	0
$655$	−35.7994	−1.39880
$656$	−0.968891	−0.0378289
$657$	0	0
$658$	37.0276	1.44348
$659$	−8.05571	−0.313806	−0.156903	−	0.987614i	$-0.550151\pi$
$659$	−8.05571	−0.313806	−0.156903	+	0.987614i	$0.550151\pi$
$660$	0	0
$661$	−8.38772	−0.326244	−0.163122	−	0.986606i	$-0.552156\pi$
$661$	−8.38772	−0.326244	−0.163122	+	0.986606i	$0.552156\pi$
$662$	5.56311	0.216216
$663$	0	0
$664$	42.9849	1.66814
$665$	37.2824	1.44575
$666$	0	0
$667$	0	0
$668$	3.37213	0.130472
$669$	0	0
$670$	15.7414	0.608142
$671$	10.2914	0.397296
$672$	0	0
$673$	−4.63335	−0.178602	−0.0893012	−	0.996005i	$-0.528463\pi$
$673$	−4.63335	−0.178602	−0.0893012	+	0.996005i	$0.528463\pi$
$674$	14.9858	0.577232
$675$	0	0
$676$	5.44441	0.209400
$677$	−32.8006	−1.26063	−0.630315	−	0.776339i	$-0.717074\pi$
$677$	−32.8006	−1.26063	−0.630315	+	0.776339i	$0.717074\pi$
$678$	0	0
$679$	−55.4684	−2.12868
$680$	−42.7192	−1.63821
$681$	0	0
$682$	−0.192334	−0.00736483
$683$	−5.26590	−0.201494	−0.100747	−	0.994912i	$-0.532123\pi$
$683$	−5.26590	−0.201494	−0.100747	+	0.994912i	$0.532123\pi$
$684$	0	0
$685$	−7.24721	−0.276902
$686$	−19.5848	−0.747751
$687$	0	0
$688$	0.268321	0.0102296
$689$	−21.8930	−0.834056
$690$	0	0
$691$	12.9033	0.490864	0.245432	−	0.969414i	$-0.421070\pi$
$691$	12.9033	0.490864	0.245432	+	0.969414i	$0.421070\pi$
$692$	6.15583	0.234009
$693$	0	0
$694$	26.3609	1.00065
$695$	−87.6992	−3.32662
$696$	0	0
$697$	−25.0353	−0.948280
$698$	−4.93695	−0.186866
$699$	0	0
$700$	59.4155	2.24570
$701$	23.0739	0.871487	0.435744	−	0.900071i	$-0.356485\pi$
$701$	23.0739	0.871487	0.435744	+	0.900071i	$0.356485\pi$
$702$	0	0
$703$	22.3753	0.843901
$704$	13.2761	0.500361
$705$	0	0
$706$	15.2885	0.575392
$707$	43.2166	1.62533
$708$	0	0
$709$	33.6636	1.26426	0.632132	−	0.774861i	$-0.282180\pi$
$709$	33.6636	1.26426	0.632132	+	0.774861i	$0.282180\pi$
$710$	−34.6999	−1.30226
$711$	0	0
$712$	−36.0949	−1.35272
$713$	0.00767191	0.000287315 0
$714$	0	0
$715$	−33.8101	−1.26443
$716$	11.0763	0.413941
$717$	0	0
$718$	7.50354	0.280030
$719$	17.6394	0.657839	0.328920	−	0.944358i	$-0.393315\pi$
$719$	17.6394	0.657839	0.328920	+	0.944358i	$0.393315\pi$
$720$	0	0
$721$	−13.4729	−0.501756
$722$	−12.2995	−0.457742
$723$	0	0
$724$	11.0480	0.410596
$725$	0	0
$726$	0	0
$727$	30.3820	1.12681	0.563403	−	0.826182i	$-0.309492\pi$
$727$	30.3820	1.12681	0.563403	+	0.826182i	$0.309492\pi$
$728$	−36.1652	−1.34037
$729$	0	0
$730$	−22.9555	−0.849621
$731$	6.93317	0.256433
$732$	0	0
$733$	−21.7102	−0.801884	−0.400942	−	0.916103i	$-0.631317\pi$
$733$	−21.7102	−0.801884	−0.400942	+	0.916103i	$0.631317\pi$
$734$	−1.54237	−0.0569298
$735$	0	0
$736$	−0.564052	−0.0207912
$737$	13.4565	0.495677
$738$	0	0
$739$	36.3858	1.33847	0.669237	−	0.743049i	$-0.266621\pi$
$739$	36.3858	1.33847	0.669237	+	0.743049i	$0.266621\pi$
$740$	52.3584	1.92473
$741$	0	0
$742$	−27.8041	−1.02072
$743$	15.3305	0.562421	0.281210	−	0.959646i	$-0.409264\pi$
$743$	15.3305	0.562421	0.281210	+	0.959646i	$0.409264\pi$
$744$	0	0
$745$	−33.5105	−1.22773
$746$	23.5271	0.861388
$747$	0	0
$748$	−14.1778	−0.518390
$749$	50.7195	1.85325
$750$	0	0
$751$	−24.8068	−0.905214	−0.452607	−	0.891710i	$-0.649506\pi$
$751$	−24.8068	−0.905214	−0.452607	+	0.891710i	$0.649506\pi$
$752$	−1.47600	−0.0538243
$753$	0	0
$754$	0	0
$755$	−41.5115	−1.51076
$756$	0	0
$757$	29.1867	1.06081	0.530404	−	0.847745i	$-0.322040\pi$
$757$	29.1867	1.06081	0.530404	+	0.847745i	$0.322040\pi$
$758$	−27.8473	−1.01146
$759$	0	0
$760$	23.7632	0.861983
$761$	−42.0242	−1.52338	−0.761689	−	0.647943i	$-0.775629\pi$
$761$	−42.0242	−1.52338	−0.761689	+	0.647943i	$0.775629\pi$
$762$	0	0
$763$	12.3879	0.448472
$764$	−2.25385	−0.0815413
$765$	0	0
$766$	21.6314	0.781575
$767$	−6.19495	−0.223687
$768$	0	0
$769$	55.1537	1.98889	0.994447	−	0.105235i	$-0.0335596\pi$
$769$	55.1537	1.98889	0.994447	+	0.105235i	$0.0335596\pi$
$770$	−42.9388	−1.54741
$771$	0	0
$772$	1.01631	0.0365777
$773$	1.29858	0.0467068	0.0233534	−	0.999727i	$-0.492566\pi$
$773$	1.29858	0.0467068	0.0233534	+	0.999727i	$0.492566\pi$
$774$	0	0
$775$	−0.830208	−0.0298220
$776$	−35.3547	−1.26916
$777$	0	0
$778$	22.1536	0.794245
$779$	13.9263	0.498961
$780$	0	0
$781$	−29.6632	−1.06143
$782$	−0.325608	−0.0116437
$783$	0	0
$784$	1.82658	0.0652350
$785$	−62.0451	−2.21449
$786$	0	0
$787$	−28.1412	−1.00313	−0.501563	−	0.865121i	$-0.667241\pi$
$787$	−28.1412	−1.00313	−0.501563	+	0.865121i	$0.667241\pi$
$788$	10.0249	0.357122
$789$	0	0
$790$	−22.2881	−0.792977
$791$	−39.7961	−1.41499
$792$	0	0
$793$	−10.4977	−0.372784
$794$	3.57251	0.126784
$795$	0	0
$796$	27.6492	0.979999
$797$	35.8082	1.26839	0.634196	−	0.773173i	$-0.281331\pi$
$797$	35.8082	1.26839	0.634196	+	0.773173i	$0.281331\pi$
$798$	0	0
$799$	−38.1386	−1.34925
$800$	61.0383	2.15803
$801$	0	0
$802$	−10.8027	−0.381458
$803$	−19.6235	−0.692498
$804$	0	0
$805$	1.71277	0.0603671
$806$	0.196188	0.00691044
$807$	0	0
$808$	27.5455	0.969049
$809$	−11.9323	−0.419517	−0.209759	−	0.977753i	$-0.567268\pi$
$809$	−11.9323	−0.419517	−0.209759	+	0.977753i	$0.567268\pi$
$810$	0	0
$811$	−27.0319	−0.949218	−0.474609	−	0.880197i	$-0.657410\pi$
$811$	−27.0319	−0.949218	−0.474609	+	0.880197i	$0.657410\pi$
$812$	0	0
$813$	0	0
$814$	−25.7700	−0.903239
$815$	98.2125	3.44023
$816$	0	0
$817$	−3.85669	−0.134928
$818$	−17.4379	−0.609702
$819$	0	0
$820$	32.5876	1.13801
$821$	−27.0561	−0.944264	−0.472132	−	0.881528i	$-0.656516\pi$
$821$	−27.0561	−0.944264	−0.472132	+	0.881528i	$0.656516\pi$
$822$	0	0
$823$	−27.3770	−0.954301	−0.477150	−	0.878822i	$-0.658330\pi$
$823$	−27.3770	−0.954301	−0.477150	+	0.878822i	$0.658330\pi$
$824$	−8.58739	−0.299156
$825$	0	0
$826$	−7.86758	−0.273748
$827$	47.2162	1.64187	0.820934	−	0.571024i	$-0.193454\pi$
$827$	47.2162	1.64187	0.820934	+	0.571024i	$0.193454\pi$
$828$	0	0
$829$	16.6695	0.578957	0.289479	−	0.957185i	$-0.406518\pi$
$829$	16.6695	0.578957	0.289479	+	0.957185i	$0.406518\pi$
$830$	−52.0586	−1.80698
$831$	0	0
$832$	−13.5422	−0.469490
$833$	47.1972	1.63529
$834$	0	0
$835$	−10.5193	−0.364035
$836$	7.88661	0.272764
$837$	0	0
$838$	−18.9833	−0.655768
$839$	31.6460	1.09254	0.546271	−	0.837609i	$-0.316047\pi$
$839$	31.6460	1.09254	0.546271	+	0.837609i	$0.316047\pi$
$840$	0	0
$841$	0	0
$842$	1.37966	0.0475462
$843$	0	0
$844$	−5.89308	−0.202848
$845$	−16.9837	−0.584258
$846$	0	0
$847$	11.5248	0.395998
$848$	1.10833	0.0380603
$849$	0	0
$850$	35.2354	1.20856
$851$	1.02793	0.0352370
$852$	0	0
$853$	21.5091	0.736458	0.368229	−	0.929735i	$-0.379964\pi$
$853$	21.5091	0.736458	0.368229	+	0.929735i	$0.379964\pi$
$854$	−13.3320	−0.456213
$855$	0	0
$856$	32.3278	1.10494
$857$	−46.9045	−1.60223	−0.801114	−	0.598512i	$-0.795759\pi$
$857$	−46.9045	−1.60223	−0.801114	+	0.598512i	$0.795759\pi$
$858$	0	0
$859$	2.47726	0.0845232	0.0422616	−	0.999107i	$-0.486544\pi$
$859$	2.47726	0.0845232	0.0422616	+	0.999107i	$0.486544\pi$
$860$	−9.02468	−0.307739
$861$	0	0
$862$	−27.1274	−0.923963
$863$	9.69575	0.330047	0.165024	−	0.986290i	$-0.447230\pi$
$863$	9.69575	0.330047	0.165024	+	0.986290i	$0.447230\pi$
$864$	0	0
$865$	−19.2030	−0.652921
$866$	−24.3637	−0.827911
$867$	0	0
$868$	−0.432751	−0.0146885
$869$	−19.0530	−0.646330
$870$	0	0
$871$	−13.7262	−0.465094
$872$	7.89584	0.267387
$873$	0	0
$874$	0.181125	0.00612663
$875$	−98.5440	−3.33139
$876$	0	0
$877$	−9.19027	−0.310334	−0.155167	−	0.987888i	$-0.549591\pi$
$877$	−9.19027	−0.310334	−0.155167	+	0.987888i	$0.549591\pi$
$878$	−19.1702	−0.646962
$879$	0	0
$880$	1.71164	0.0576994
$881$	−27.8621	−0.938698	−0.469349	−	0.883013i	$-0.655511\pi$
$881$	−27.8621	−0.938698	−0.469349	+	0.883013i	$0.655511\pi$
$882$	0	0
$883$	−17.0461	−0.573646	−0.286823	−	0.957984i	$-0.592599\pi$
$883$	−17.0461	−0.573646	−0.286823	+	0.957984i	$0.592599\pi$
$884$	14.4619	0.486407
$885$	0	0
$886$	23.0592	0.774690
$887$	19.6228	0.658871	0.329435	−	0.944178i	$-0.393142\pi$
$887$	19.6228	0.658871	0.329435	+	0.944178i	$0.393142\pi$
$888$	0	0
$889$	−14.4079	−0.483225
$890$	43.7143	1.46531
$891$	0	0
$892$	3.53434	0.118339
$893$	21.2152	0.709940
$894$	0	0
$895$	−34.5523	−1.15496
$896$	32.9366	1.10034
$897$	0	0
$898$	19.9957	0.667265
$899$	0	0
$900$	0	0
$901$	28.6384	0.954082
$902$	−16.0392	−0.534045
$903$	0	0
$904$	−25.3654	−0.843641
$905$	−34.4641	−1.14562
$906$	0	0
$907$	45.3685	1.50644	0.753218	−	0.657771i	$-0.228501\pi$
$907$	45.3685	1.50644	0.753218	+	0.657771i	$0.228501\pi$
$908$	28.1880	0.935452
$909$	0	0
$910$	43.7994	1.45194
$911$	−2.87604	−0.0952875	−0.0476437	−	0.998864i	$-0.515171\pi$
$911$	−2.87604	−0.0952875	−0.0476437	+	0.998864i	$0.515171\pi$
$912$	0	0
$913$	−44.5023	−1.47281
$914$	−3.41146	−0.112841
$915$	0	0
$916$	−12.0837	−0.399256
$917$	39.6448	1.30919
$918$	0	0
$919$	−27.5647	−0.909277	−0.454638	−	0.890676i	$-0.650232\pi$
$919$	−27.5647	−0.909277	−0.454638	+	0.890676i	$0.650232\pi$
$920$	1.09169	0.0359920
$921$	0	0
$922$	−21.8887	−0.720865
$923$	30.2577	0.995943
$924$	0	0
$925$	−111.236	−3.65743
$926$	10.7184	0.352230
$927$	0	0
$928$	0	0
$929$	21.8994	0.718497	0.359248	−	0.933242i	$-0.383033\pi$
$929$	21.8994	0.718497	0.359248	+	0.933242i	$0.383033\pi$
$930$	0	0
$931$	−26.2542	−0.860447
$932$	21.7070	0.711037
$933$	0	0
$934$	15.5017	0.507231
$935$	44.2273	1.44639
$936$	0	0
$937$	2.96293	0.0967946	0.0483973	−	0.998828i	$-0.484589\pi$
$937$	2.96293	0.0967946	0.0483973	+	0.998828i	$0.484589\pi$
$938$	−17.4323	−0.569183
$939$	0	0
$940$	49.6438	1.61920
$941$	0.422587	0.0137759	0.00688797	−	0.999976i	$-0.497807\pi$
$941$	0.422587	0.0137759	0.00688797	+	0.999976i	$0.497807\pi$
$942$	0	0
$943$	0.639779	0.0208341
$944$	0.313620	0.0102074
$945$	0	0
$946$	4.44182	0.144416
$947$	−55.3487	−1.79859	−0.899296	−	0.437341i	$-0.855920\pi$
$947$	−55.3487	−1.79859	−0.899296	+	0.437341i	$0.855920\pi$
$948$	0	0
$949$	20.0168	0.649772
$950$	−19.6002	−0.635916
$951$	0	0
$952$	47.3079	1.53326
$953$	40.4654	1.31080	0.655402	−	0.755280i	$-0.272499\pi$
$953$	40.4654	1.31080	0.655402	+	0.755280i	$0.272499\pi$
$954$	0	0
$955$	7.03083	0.227512
$956$	−17.7154	−0.572956
$957$	0	0
$958$	10.2856	0.332312
$959$	8.02568	0.259163
$960$	0	0
$961$	−30.9940	−0.999805
$962$	26.2865	0.847511
$963$	0	0
$964$	−5.14124	−0.165588
$965$	−3.17035	−0.102057
$966$	0	0
$967$	−11.5025	−0.369896	−0.184948	−	0.982748i	$-0.559212\pi$
$967$	−11.5025	−0.369896	−0.184948	+	0.982748i	$0.559212\pi$
$968$	7.34575	0.236101
$969$	0	0
$970$	42.8178	1.37480
$971$	−52.7268	−1.69208	−0.846042	−	0.533116i	$-0.821021\pi$
$971$	−52.7268	−1.69208	−0.846042	+	0.533116i	$0.821021\pi$
$972$	0	0
$973$	97.1196	3.11351
$974$	−20.2457	−0.648716
$975$	0	0
$976$	0.531446	0.0170112
$977$	−5.72799	−0.183255	−0.0916273	−	0.995793i	$-0.529207\pi$
$977$	−5.72799	−0.183255	−0.0916273	+	0.995793i	$0.529207\pi$
$978$	0	0
$979$	37.3692	1.19432
$980$	−61.4351	−1.96247
$981$	0	0
$982$	−7.31695	−0.233493
$983$	52.2866	1.66768	0.833841	−	0.552005i	$-0.186137\pi$
$983$	52.2866	1.66768	0.833841	+	0.552005i	$0.186137\pi$
$984$	0	0
$985$	−31.2725	−0.996424
$986$	0	0
$987$	0	0
$988$	−8.04467	−0.255935
$989$	−0.177178	−0.00563392
$990$	0	0
$991$	−14.5007	−0.460629	−0.230315	−	0.973116i	$-0.573976\pi$
$991$	−14.5007	−0.460629	−0.230315	+	0.973116i	$0.573976\pi$
$992$	−0.444570	−0.0141151
$993$	0	0
$994$	38.4272	1.21884
$995$	−86.2511	−2.73434
$996$	0	0
$997$	8.53368	0.270264	0.135132	−	0.990828i	$-0.456854\pi$
$997$	8.53368	0.270264	0.135132	+	0.990828i	$0.456854\pi$
$998$	21.5726	0.682870
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.br.1.8		12
3.2	odd	2	inner	7569.2.a.br.1.5		12
29.4	even	14		261.2.k.d.190.3	yes	24
29.22	even	14		261.2.k.d.136.3	yes	24
29.28	even	2		7569.2.a.bq.1.5		12
87.62	odd	14		261.2.k.d.190.2	yes	24
87.80	odd	14		261.2.k.d.136.2	✓	24
87.86	odd	2		7569.2.a.bq.1.8		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
261.2.k.d.136.2	✓	24	87.80	odd	14
261.2.k.d.136.3	yes	24	29.22	even	14
261.2.k.d.190.2	yes	24	87.62	odd	14
261.2.k.d.190.3	yes	24	29.4	even	14
7569.2.a.bq.1.5		12	29.28	even	2
7569.2.a.bq.1.8		12	87.86	odd	2
7569.2.a.br.1.5		12	3.2	odd	2	inner
7569.2.a.br.1.8		12	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q+0.854850 q^{2} -1.26923 q^{4} +3.95934 q^{5} -4.38464 q^{7} -2.79470 q^{8} +3.38464 q^{10} +2.89336 q^{11} -2.95135 q^{13} -3.74821 q^{14} +0.149412 q^{16} +3.86068 q^{17} -2.14757 q^{19} -5.02532 q^{20} +2.47339 q^{22} -0.0986600 q^{23} +10.6764 q^{25} -2.52296 q^{26} +5.56513 q^{28} -0.0777611 q^{31} +5.71713 q^{32} +3.30030 q^{34} -17.3603 q^{35} -10.4189 q^{37} -1.83585 q^{38} -11.0652 q^{40} -6.48468 q^{41} +1.79584 q^{43} -3.67235 q^{44} -0.0843395 q^{46} -9.87873 q^{47} +12.2251 q^{49} +9.12671 q^{50} +3.74595 q^{52} +7.41795 q^{53} +11.4558 q^{55} +12.2538 q^{56} +2.09902 q^{59} +3.55691 q^{61} -0.0664741 q^{62} +4.58846 q^{64} -11.6854 q^{65} +4.65082 q^{67} -4.90010 q^{68} -14.8405 q^{70} -10.2521 q^{71} -6.78225 q^{73} -8.90661 q^{74} +2.72576 q^{76} -12.6864 q^{77} -6.58508 q^{79} +0.591574 q^{80} -5.54343 q^{82} -15.3808 q^{83} +15.2858 q^{85} +1.53517 q^{86} -8.08609 q^{88} +12.9155 q^{89} +12.9406 q^{91} +0.125222 q^{92} -8.44483 q^{94} -8.50295 q^{95} +12.6506 q^{97} +10.4506 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 8 q^{4} - 14 q^{7} + 2 q^{10} - 12 q^{13} - 4 q^{16} + 14 q^{19} + 8 q^{22} + 2 q^{25} - 12 q^{28} - 28 q^{31} - 18 q^{34} - 36 q^{37} + 20 q^{43} - 20 q^{46} - 10 q^{49} - 12 q^{52} + 58 q^{55}+ \cdots - 26 q^{97}+O(q^{100}) \) 12 * q + 8 * q^4 - 14 * q^7 + 2 * q^10 - 12 * q^13 - 4 * q^16 + 14 * q^19 + 8 * q^22 + 2 * q^25 - 12 * q^28 - 28 * q^31 - 18 * q^34 - 36 * q^37 + 20 * q^43 - 20 * q^46 - 10 * q^49 - 12 * q^52 + 58 * q^55 - 4 * q^61 - 18 * q^64 - 54 * q^67 - 60 * q^70 - 32 * q^73 + 56 * q^76 + 2 * q^79 - 96 * q^82 + 4 * q^85 - 10 * q^88 + 6 * q^91 - 32 * q^94 - 26 * q^97

Introduction

L-functions

Modular forms

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Database

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