LMFDB - Embedded newform 7569.2.a.bg.1.2

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:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.08529$ 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)

$f(q)$	$=$	\(q-2.08529 q^{2} +2.34841 q^{4} +0.992398 q^{5} +2.45140 q^{7} -0.726543 q^{8} -2.06943 q^{10} -0.745407 q^{11} +0.730380 q^{13} -5.11187 q^{14} -3.18178 q^{16} +4.76333 q^{17} -1.38197 q^{19} +2.33056 q^{20} +1.55439 q^{22} +5.31562 q^{23} -4.01515 q^{25} -1.52305 q^{26} +5.75690 q^{28} +0.563746 q^{31} +8.08800 q^{32} -9.93290 q^{34} +2.43277 q^{35} -9.49664 q^{37} +2.88179 q^{38} -0.721020 q^{40} -5.63637 q^{41} -11.0117 q^{43} -1.75052 q^{44} -11.0846 q^{46} -11.3183 q^{47} -0.990640 q^{49} +8.37272 q^{50} +1.71523 q^{52} +4.53691 q^{53} -0.739740 q^{55} -1.78105 q^{56} +8.54697 q^{59} -8.56375 q^{61} -1.17557 q^{62} -10.5022 q^{64} +0.724828 q^{65} +11.0939 q^{67} +11.1863 q^{68} -5.07301 q^{70} -6.10935 q^{71} -5.41226 q^{73} +19.8032 q^{74} -3.24543 q^{76} -1.82729 q^{77} +7.67465 q^{79} -3.15759 q^{80} +11.7534 q^{82} -15.9212 q^{83} +4.72712 q^{85} +22.9625 q^{86} +0.541570 q^{88} +9.90572 q^{89} +1.79045 q^{91} +12.4833 q^{92} +23.6020 q^{94} -1.37146 q^{95} -13.5775 q^{97} +2.06577 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 12 q^{10} + 2 q^{13} - 2 q^{16} - 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} - 10 q^{31} - 36 q^{34} - 18 q^{37} + 10 q^{40} - 28 q^{43} - 26 q^{46} + 4 q^{49} + 44 q^{52} - 14 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) $ 8 * q + 6 * q^4 + 12 * q^10 + 2 * q^13 - 2 * q^16 - 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 - 10 * q^31 - 36 * q^34 - 18 * q^37 + 10 * q^40 - 28 * q^43 - 26 * q^46 + 4 * q^49 + 44 * q^52 - 14 * q^55 - 54 * q^61 - 44 * q^64 + 38 * q^67 - 70 * q^70 + 16 * q^73 - 20 * q^76 - 14 * q^79 + 2 * q^82 - 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 + 6 * 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 $

$2$	−2.08529	−1.47452	−0.737260	−	0.675610i	$-0.763881\pi$
$2$	−2.08529	−1.47452	−0.737260	+	0.675610i	$0.763881\pi$
$3$	0	0
$3$	0	0
$4$	2.34841	1.17421
$5$	0.992398	0.443814	0.221907	−	0.975068i	$-0.428772\pi$
$5$	0.992398	0.443814	0.221907	+	0.975068i	$0.428772\pi$
$6$	0	0
$7$	2.45140	0.926542	0.463271	−	0.886217i	$-0.346676\pi$
$7$	2.45140	0.926542	0.463271	+	0.886217i	$0.346676\pi$
$8$	−0.726543	−0.256872
$9$	0	0
$10$	−2.06943	−0.654412
$11$	−0.745407	−0.224749	−0.112374	−	0.993666i	$-0.535846\pi$
$11$	−0.745407	−0.224749	−0.112374	+	0.993666i	$0.535846\pi$
$12$	0	0
$13$	0.730380	0.202571	0.101285	−	0.994857i	$-0.467704\pi$
$13$	0.730380	0.202571	0.101285	+	0.994857i	$0.467704\pi$
$14$	−5.11187	−1.36620
$15$	0	0
$16$	−3.18178	−0.795445
$17$	4.76333	1.15528	0.577638	−	0.816293i	$-0.303974\pi$
$17$	4.76333	1.15528	0.577638	+	0.816293i	$0.303974\pi$
$18$	0	0
$19$	−1.38197	−0.317045	−0.158522	−	0.987355i	$-0.550673\pi$
$19$	−1.38197	−0.317045	−0.158522	+	0.987355i	$0.550673\pi$
$20$	2.33056	0.521130
$21$	0	0
$22$	1.55439	0.331396
$23$	5.31562	1.10838	0.554191	−	0.832389i	$-0.313028\pi$
$23$	5.31562	1.10838	0.554191	+	0.832389i	$0.313028\pi$
$24$	0	0
$25$	−4.01515	−0.803029
$26$	−1.52305	−0.298695
$27$	0	0
$28$	5.75690	1.08795
$29$	0	0
$29$	0	0
$30$	0	0
$31$	0.563746	0.101252	0.0506259	−	0.998718i	$-0.483878\pi$
$31$	0.563746	0.101252	0.0506259	+	0.998718i	$0.483878\pi$
$32$	8.08800	1.42977
$33$	0	0
$34$	−9.93290	−1.70348
$35$	2.43277	0.411212
$36$	0	0
$37$	−9.49664	−1.56124	−0.780619	−	0.625007i	$-0.785096\pi$
$37$	−9.49664	−1.56124	−0.780619	+	0.625007i	$0.785096\pi$
$38$	2.88179	0.467489
$39$	0	0
$40$	−0.721020	−0.114003
$41$	−5.63637	−0.880254	−0.440127	−	0.897936i	$-0.645067\pi$
$41$	−5.63637	−0.880254	−0.440127	+	0.897936i	$0.645067\pi$
$42$	0	0
$43$	−11.0117	−1.67927	−0.839633	−	0.543153i	$-0.817230\pi$
$43$	−11.0117	−1.67927	−0.839633	+	0.543153i	$0.817230\pi$
$44$	−1.75052	−0.263901
$45$	0	0
$46$	−11.0846	−1.63433
$47$	−11.3183	−1.65095	−0.825474	−	0.564439i	$-0.809092\pi$
$47$	−11.3183	−1.65095	−0.825474	+	0.564439i	$0.809092\pi$
$48$	0	0
$49$	−0.990640	−0.141520
$50$	8.37272	1.18408
$51$	0	0
$52$	1.71523	0.237860
$53$	4.53691	0.623193	0.311597	−	0.950214i	$-0.399136\pi$
$53$	4.53691	0.623193	0.311597	+	0.950214i	$0.399136\pi$
$54$	0	0
$55$	−0.739740	−0.0997466
$56$	−1.78105	−0.238002
$57$	0	0
$58$	0	0
$59$	8.54697	1.11272	0.556361	−	0.830941i	$-0.312197\pi$
$59$	8.54697	1.11272	0.556361	+	0.830941i	$0.312197\pi$
$60$	0	0
$61$	−8.56375	−1.09648	−0.548238	−	0.836323i	$-0.684701\pi$
$61$	−8.56375	−1.09648	−0.548238	+	0.836323i	$0.684701\pi$
$62$	−1.17557	−0.149298
$63$	0	0
$64$	−10.5022	−1.31278
$65$	0.724828	0.0899038
$66$	0	0
$67$	11.0939	1.35534	0.677670	−	0.735366i	$-0.262990\pi$
$67$	11.0939	1.35534	0.677670	+	0.735366i	$0.262990\pi$
$68$	11.1863	1.35653
$69$	0	0
$70$	−5.07301	−0.606341
$71$	−6.10935	−0.725047	−0.362523	−	0.931975i	$-0.618085\pi$
$71$	−6.10935	−0.725047	−0.362523	+	0.931975i	$0.618085\pi$
$72$	0	0
$73$	−5.41226	−0.633457	−0.316728	−	0.948516i	$-0.602584\pi$
$73$	−5.41226	−0.633457	−0.316728	+	0.948516i	$0.602584\pi$
$74$	19.8032	2.30208
$75$	0	0
$76$	−3.24543	−0.372276
$77$	−1.82729	−0.208239
$78$	0	0
$79$	7.67465	0.863466	0.431733	−	0.902001i	$-0.357902\pi$
$79$	7.67465	0.863466	0.431733	+	0.902001i	$0.357902\pi$
$80$	−3.15759	−0.353030
$81$	0	0
$82$	11.7534	1.29795
$83$	−15.9212	−1.74757	−0.873787	−	0.486309i	$-0.838343\pi$
$83$	−15.9212	−1.74757	−0.873787	+	0.486309i	$0.838343\pi$
$84$	0	0
$85$	4.72712	0.512728
$86$	22.9625	2.47611
$87$	0	0
$88$	0.541570	0.0577315
$89$	9.90572	1.05000	0.525002	−	0.851101i	$-0.324065\pi$
$89$	9.90572	1.05000	0.525002	+	0.851101i	$0.324065\pi$
$90$	0	0
$91$	1.79045	0.187691
$92$	12.4833	1.30147
$93$	0	0
$94$	23.6020	2.43436
$95$	−1.37146	−0.140709
$96$	0	0
$97$	−13.5775	−1.37858	−0.689291	−	0.724485i	$-0.742078\pi$
$97$	−13.5775	−1.37858	−0.689291	+	0.724485i	$0.742078\pi$
$98$	2.06577	0.208674
$99$	0	0
$100$	−9.42922	−0.942922
$101$	−13.0106	−1.29460	−0.647299	−	0.762236i	$-0.724102\pi$
$101$	−13.0106	−1.29460	−0.647299	+	0.762236i	$0.724102\pi$
$102$	0	0
$103$	−6.23048	−0.613907	−0.306954	−	0.951724i	$-0.599310\pi$
$103$	−6.23048	−0.613907	−0.306954	+	0.951724i	$0.599310\pi$
$104$	−0.530652	−0.0520347
$105$	0	0
$106$	−9.46076	−0.918910
$107$	−4.83435	−0.467355	−0.233677	−	0.972314i	$-0.575076\pi$
$107$	−4.83435	−0.467355	−0.233677	+	0.972314i	$0.575076\pi$
$108$	0	0
$109$	1.10299	0.105647	0.0528234	−	0.998604i	$-0.483178\pi$
$109$	1.10299	0.105647	0.0528234	+	0.998604i	$0.483178\pi$
$110$	1.54257	0.147078
$111$	0	0
$112$	−7.79981	−0.737013
$113$	−17.8067	−1.67512	−0.837559	−	0.546347i	$-0.816018\pi$
$113$	−17.8067	−1.67512	−0.837559	+	0.546347i	$0.816018\pi$
$114$	0	0
$115$	5.27521	0.491916
$116$	0	0
$117$	0	0
$118$	−17.8229	−1.64073
$119$	11.6768	1.07041
$120$	0	0
$121$	−10.4444	−0.949488
$122$	17.8579	1.61677
$123$	0	0
$124$	1.32391	0.118890
$125$	−8.94662	−0.800210
$126$	0	0
$127$	−4.24166	−0.376386	−0.188193	−	0.982132i	$-0.560263\pi$
$127$	−4.24166	−0.376386	−0.188193	+	0.982132i	$0.560263\pi$
$128$	5.72414	0.505948
$129$	0	0
$130$	−1.51147	−0.132565
$131$	8.64644	0.755443	0.377721	−	0.925919i	$-0.376708\pi$
$131$	8.64644	0.755443	0.377721	+	0.925919i	$0.376708\pi$
$132$	0	0
$133$	−3.38775	−0.293755
$134$	−23.1340	−1.99848
$135$	0	0
$136$	−3.46076	−0.296758
$137$	14.2650	1.21874	0.609369	−	0.792886i	$-0.291423\pi$
$137$	14.2650	1.21874	0.609369	+	0.792886i	$0.291423\pi$
$138$	0	0
$139$	−14.3624	−1.21821	−0.609103	−	0.793091i	$-0.708470\pi$
$139$	−14.3624	−1.21821	−0.609103	+	0.793091i	$0.708470\pi$
$140$	5.71314	0.482848
$141$	0	0
$142$	12.7397	1.06910
$143$	−0.544430	−0.0455275
$144$	0	0
$145$	0	0
$146$	11.2861	0.934044
$147$	0	0
$148$	−22.3020	−1.83322
$149$	−20.8910	−1.71146	−0.855729	−	0.517424i	$-0.826891\pi$
$149$	−20.8910	−1.71146	−0.855729	+	0.517424i	$0.826891\pi$
$150$	0	0
$151$	−2.14213	−0.174324	−0.0871620	−	0.996194i	$-0.527780\pi$
$151$	−2.14213	−0.174324	−0.0871620	+	0.996194i	$0.527780\pi$
$152$	1.00406	0.0814398
$153$	0	0
$154$	3.81042	0.307052
$155$	0.559460	0.0449369
$156$	0	0
$157$	−14.3784	−1.14752	−0.573760	−	0.819023i	$-0.694516\pi$
$157$	−14.3784	−1.14752	−0.573760	+	0.819023i	$0.694516\pi$
$158$	−16.0038	−1.27320
$159$	0	0
$160$	8.02652	0.634552
$161$	13.0307	1.02696
$162$	0	0
$163$	−7.01860	−0.549739	−0.274870	−	0.961482i	$-0.588635\pi$
$163$	−7.01860	−0.549739	−0.274870	+	0.961482i	$0.588635\pi$
$164$	−13.2365	−1.03360
$165$	0	0
$166$	33.2001	2.57683
$167$	−23.3159	−1.80424	−0.902120	−	0.431486i	$-0.857989\pi$
$167$	−23.3159	−1.80424	−0.902120	+	0.431486i	$0.857989\pi$
$168$	0	0
$169$	−12.4665	−0.958965
$170$	−9.85739	−0.756027
$171$	0	0
$172$	−25.8600	−1.97181
$173$	11.7018	0.889674	0.444837	−	0.895612i	$-0.353262\pi$
$173$	11.7018	0.889674	0.444837	+	0.895612i	$0.353262\pi$
$174$	0	0
$175$	−9.84273	−0.744040
$176$	2.37172	0.178775
$177$	0	0
$178$	−20.6562	−1.54825
$179$	1.11218	0.0831281	0.0415640	−	0.999136i	$-0.486766\pi$
$179$	1.11218	0.0831281	0.0415640	+	0.999136i	$0.486766\pi$
$180$	0	0
$181$	2.68281	0.199412	0.0997058	−	0.995017i	$-0.468210\pi$
$181$	2.68281	0.199412	0.0997058	+	0.995017i	$0.468210\pi$
$182$	−3.73361	−0.276753
$183$	0	0
$184$	−3.86202	−0.284712
$185$	−9.42445	−0.692900
$186$	0	0
$187$	−3.55062	−0.259647
$188$	−26.5801	−1.93856
$189$	0	0
$190$	2.85989	0.207478
$191$	22.2037	1.60661	0.803303	−	0.595571i	$-0.203074\pi$
$191$	22.2037	1.60661	0.803303	+	0.595571i	$0.203074\pi$
$192$	0	0
$193$	5.48383	0.394734	0.197367	−	0.980330i	$-0.436761\pi$
$193$	5.48383	0.394734	0.197367	+	0.980330i	$0.436761\pi$
$194$	28.3129	2.03274
$195$	0	0
$196$	−2.32643	−0.166174
$197$	17.3694	1.23752	0.618759	−	0.785581i	$-0.287636\pi$
$197$	17.3694	1.23752	0.618759	+	0.785581i	$0.287636\pi$
$198$	0	0
$199$	11.6747	0.827594	0.413797	−	0.910369i	$-0.364202\pi$
$199$	11.6747	0.827594	0.413797	+	0.910369i	$0.364202\pi$
$200$	2.91717	0.206275
$201$	0	0
$202$	27.1307	1.90891
$203$	0	0
$204$	0	0
$205$	−5.59353	−0.390669
$206$	12.9923	0.905218
$207$	0	0
$208$	−2.32391	−0.161134
$209$	1.03013	0.0712553
$210$	0	0
$211$	7.66906	0.527960	0.263980	−	0.964528i	$-0.414965\pi$
$211$	7.66906	0.527960	0.263980	+	0.964528i	$0.414965\pi$
$212$	10.6546	0.731758
$213$	0	0
$214$	10.0810	0.689123
$215$	−10.9280	−0.745282
$216$	0	0
$217$	1.38197	0.0938140
$218$	−2.30004	−0.155778
$219$	0	0
$220$	−1.73722	−0.117123
$221$	3.47904	0.234026
$222$	0	0
$223$	−13.7010	−0.917485	−0.458743	−	0.888569i	$-0.651700\pi$
$223$	−13.7010	−0.917485	−0.458743	+	0.888569i	$0.651700\pi$
$224$	19.8269	1.32474
$225$	0	0
$226$	37.1322	2.46999
$227$	−4.83709	−0.321049	−0.160525	−	0.987032i	$-0.551319\pi$
$227$	−4.83709	−0.321049	−0.160525	+	0.987032i	$0.551319\pi$
$228$	0	0
$229$	11.9291	0.788299	0.394149	−	0.919046i	$-0.371039\pi$
$229$	11.9291	0.788299	0.394149	+	0.919046i	$0.371039\pi$
$230$	−11.0003	−0.725340
$231$	0	0
$232$	0	0
$233$	25.9346	1.69903	0.849517	−	0.527562i	$-0.176894\pi$
$233$	25.9346	1.69903	0.849517	+	0.527562i	$0.176894\pi$
$234$	0	0
$235$	−11.2323	−0.732714
$236$	20.0718	1.30657
$237$	0	0
$238$	−24.3495	−1.57834
$239$	−3.50336	−0.226613	−0.113307	−	0.993560i	$-0.536144\pi$
$239$	−3.50336	−0.226613	−0.113307	+	0.993560i	$0.536144\pi$
$240$	0	0
$241$	−1.32077	−0.0850781	−0.0425390	−	0.999095i	$-0.513545\pi$
$241$	−1.32077	−0.0850781	−0.0425390	+	0.999095i	$0.513545\pi$
$242$	21.7795	1.40004
$243$	0	0
$244$	−20.1112	−1.28749
$245$	−0.983109	−0.0628085
$246$	0	0
$247$	−1.00936	−0.0642241
$248$	−0.409585	−0.0260087
$249$	0	0
$250$	18.6562	1.17992
$251$	2.64346	0.166854	0.0834268	−	0.996514i	$-0.473414\pi$
$251$	2.64346	0.166854	0.0834268	+	0.996514i	$0.473414\pi$
$252$	0	0
$253$	−3.96230	−0.249107
$254$	8.84507	0.554989
$255$	0	0
$256$	9.06799	0.566750
$257$	−4.76819	−0.297431	−0.148716	−	0.988880i	$-0.547514\pi$
$257$	−4.76819	−0.297431	−0.148716	+	0.988880i	$0.547514\pi$
$258$	0	0
$259$	−23.2801	−1.44655
$260$	1.70220	0.105566
$261$	0	0
$262$	−18.0303	−1.11392
$263$	−14.5370	−0.896388	−0.448194	−	0.893936i	$-0.647933\pi$
$263$	−14.5370	−0.896388	−0.448194	+	0.893936i	$0.647933\pi$
$264$	0	0
$265$	4.50243	0.276582
$266$	7.06443	0.433148
$267$	0	0
$268$	26.0532	1.59145
$269$	−10.8477	−0.661394	−0.330697	−	0.943737i	$-0.607284\pi$
$269$	−10.8477	−0.661394	−0.330697	+	0.943737i	$0.607284\pi$
$270$	0	0
$271$	−0.956894	−0.0581271	−0.0290636	−	0.999578i	$-0.509253\pi$
$271$	−0.956894	−0.0581271	−0.0290636	+	0.999578i	$0.509253\pi$
$272$	−15.1559	−0.918959
$273$	0	0
$274$	−29.7465	−1.79705
$275$	2.99292	0.180480
$276$	0	0
$277$	18.5533	1.11476	0.557379	−	0.830258i	$-0.311807\pi$
$277$	18.5533	1.11476	0.557379	+	0.830258i	$0.311807\pi$
$278$	29.9498	1.79627
$279$	0	0
$280$	−1.76751	−0.105629
$281$	7.34600	0.438226	0.219113	−	0.975700i	$-0.429684\pi$
$281$	7.34600	0.438226	0.219113	+	0.975700i	$0.429684\pi$
$282$	0	0
$283$	9.15413	0.544157	0.272078	−	0.962275i	$-0.412289\pi$
$283$	9.15413	0.544157	0.272078	+	0.962275i	$0.412289\pi$
$284$	−14.3473	−0.851355
$285$	0	0
$286$	1.13529	0.0671312
$287$	−13.8170	−0.815592
$288$	0	0
$289$	5.68929	0.334664
$290$	0	0
$291$	0	0
$292$	−12.7102	−0.743809
$293$	19.0582	1.11339	0.556695	−	0.830717i	$-0.312069\pi$
$293$	19.0582	1.11339	0.556695	+	0.830717i	$0.312069\pi$
$294$	0	0
$295$	8.48200	0.493841
$296$	6.89971	0.401038
$297$	0	0
$298$	43.5637	2.52358
$299$	3.88242	0.224526
$300$	0	0
$301$	−26.9941	−1.55591
$302$	4.46695	0.257044
$303$	0	0
$304$	4.39711	0.252192
$305$	−8.49865	−0.486631
$306$	0	0
$307$	26.3761	1.50537	0.752683	−	0.658383i	$-0.228759\pi$
$307$	26.3761	1.50537	0.752683	+	0.658383i	$0.228759\pi$
$308$	−4.29123	−0.244516
$309$	0	0
$310$	−1.16663	−0.0662604
$311$	8.19847	0.464893	0.232446	−	0.972609i	$-0.425327\pi$
$311$	8.19847	0.464893	0.232446	+	0.972609i	$0.425327\pi$
$312$	0	0
$313$	11.4853	0.649186	0.324593	−	0.945854i	$-0.394773\pi$
$313$	11.4853	0.649186	0.324593	+	0.945854i	$0.394773\pi$
$314$	29.9830	1.69204
$315$	0	0
$316$	18.0233	1.01389
$317$	26.7178	1.50062	0.750309	−	0.661087i	$-0.229905\pi$
$317$	26.7178	1.50062	0.750309	+	0.661087i	$0.229905\pi$
$318$	0	0
$319$	0	0
$320$	−10.4224	−0.582630
$321$	0	0
$322$	−27.1727	−1.51428
$323$	−6.58276	−0.366274
$324$	0	0
$325$	−2.93258	−0.162670
$326$	14.6358	0.810601
$327$	0	0
$328$	4.09507	0.226112
$329$	−27.7458	−1.52967
$330$	0	0
$331$	10.1135	0.555891	0.277945	−	0.960597i	$-0.410347\pi$
$331$	10.1135	0.555891	0.277945	+	0.960597i	$0.410347\pi$
$332$	−37.3895	−2.05201
$333$	0	0
$334$	48.6203	2.66039
$335$	11.0096	0.601519
$336$	0	0
$337$	0.987184	0.0537754	0.0268877	−	0.999638i	$-0.491440\pi$
$337$	0.987184	0.0537754	0.0268877	+	0.999638i	$0.491440\pi$
$338$	25.9963	1.41401
$339$	0	0
$340$	11.1012	0.602049
$341$	−0.420220	−0.0227562
$342$	0	0
$343$	−19.5883	−1.05767
$344$	8.00046	0.431356
$345$	0	0
$346$	−24.4017	−1.31184
$347$	11.7941	0.633138	0.316569	−	0.948569i	$-0.397469\pi$
$347$	11.7941	0.633138	0.316569	+	0.948569i	$0.397469\pi$
$348$	0	0
$349$	24.5011	1.31151	0.655757	−	0.754972i	$-0.272350\pi$
$349$	24.5011	1.31151	0.655757	+	0.754972i	$0.272350\pi$
$350$	20.5249	1.09710
$351$	0	0
$352$	−6.02885	−0.321339
$353$	2.52368	0.134322	0.0671609	−	0.997742i	$-0.478606\pi$
$353$	2.52368	0.134322	0.0671609	+	0.997742i	$0.478606\pi$
$354$	0	0
$355$	−6.06291	−0.321786
$356$	23.2627	1.23292
$357$	0	0
$358$	−2.31921	−0.122574
$359$	1.67121	0.0882031	0.0441016	−	0.999027i	$-0.485957\pi$
$359$	1.67121	0.0882031	0.0441016	+	0.999027i	$0.485957\pi$
$360$	0	0
$361$	−17.0902	−0.899483
$362$	−5.59442	−0.294036
$363$	0	0
$364$	4.20473	0.220388
$365$	−5.37112	−0.281137
$366$	0	0
$367$	−27.7998	−1.45114	−0.725569	−	0.688149i	$-0.758423\pi$
$367$	−27.7998	−1.45114	−0.725569	+	0.688149i	$0.758423\pi$
$368$	−16.9131	−0.881658
$369$	0	0
$370$	19.6527	1.02169
$371$	11.1218	0.577415
$372$	0	0
$373$	16.3437	0.846245	0.423123	−	0.906072i	$-0.360934\pi$
$373$	16.3437	0.846245	0.423123	+	0.906072i	$0.360934\pi$
$374$	7.40405	0.382854
$375$	0	0
$376$	8.22325	0.424082
$377$	0	0
$378$	0	0
$379$	−11.0324	−0.566698	−0.283349	−	0.959017i	$-0.591445\pi$
$379$	−11.0324	−0.566698	−0.283349	+	0.959017i	$0.591445\pi$
$380$	−3.22076	−0.165221
$381$	0	0
$382$	−46.3011	−2.36897
$383$	−17.0153	−0.869443	−0.434721	−	0.900565i	$-0.643153\pi$
$383$	−17.0153	−0.869443	−0.434721	+	0.900565i	$0.643153\pi$
$384$	0	0
$385$	−1.81340	−0.0924194
$386$	−11.4353	−0.582044
$387$	0	0
$388$	−31.8855	−1.61874
$389$	−1.93366	−0.0980404	−0.0490202	−	0.998798i	$-0.515610\pi$
$389$	−1.93366	−0.0980404	−0.0490202	+	0.998798i	$0.515610\pi$
$390$	0	0
$391$	25.3200	1.28049
$392$	0.719742	0.0363525
$393$	0	0
$394$	−36.2201	−1.82474
$395$	7.61631	0.383218
$396$	0	0
$397$	−20.5396	−1.03085	−0.515425	−	0.856934i	$-0.672366\pi$
$397$	−20.5396	−1.03085	−0.515425	+	0.856934i	$0.672366\pi$
$398$	−24.3450	−1.22030
$399$	0	0
$400$	12.7753	0.638765
$401$	−21.3490	−1.06612	−0.533058	−	0.846079i	$-0.678957\pi$
$401$	−21.3490	−1.06612	−0.533058	+	0.846079i	$0.678957\pi$
$402$	0	0
$403$	0.411749	0.0205107
$404$	−30.5542	−1.52013
$405$	0	0
$406$	0	0
$407$	7.07886	0.350886
$408$	0	0
$409$	22.0616	1.09088	0.545438	−	0.838151i	$-0.316363\pi$
$409$	22.0616	1.09088	0.545438	+	0.838151i	$0.316363\pi$
$410$	11.6641	0.576049
$411$	0	0
$412$	−14.6317	−0.720854
$413$	20.9521	1.03098
$414$	0	0
$415$	−15.8001	−0.775598
$416$	5.90732	0.289630
$417$	0	0
$418$	−2.14811	−0.105067
$419$	37.0004	1.80759	0.903794	−	0.427967i	$-0.140770\pi$
$419$	37.0004	1.80759	0.903794	+	0.427967i	$0.140770\pi$
$420$	0	0
$421$	20.0872	0.978991	0.489496	−	0.872006i	$-0.337181\pi$
$421$	20.0872	0.978991	0.489496	+	0.872006i	$0.337181\pi$
$422$	−15.9922	−0.778487
$423$	0	0
$424$	−3.29626	−0.160081
$425$	−19.1255	−0.927721
$426$	0	0
$427$	−20.9932	−1.01593
$428$	−11.3531	−0.548771
$429$	0	0
$430$	22.7880	1.09893
$431$	−28.4380	−1.36981	−0.684905	−	0.728633i	$-0.740156\pi$
$431$	−28.4380	−1.36981	−0.684905	+	0.728633i	$0.740156\pi$
$432$	0	0
$433$	−38.0368	−1.82793	−0.913965	−	0.405792i	$-0.866996\pi$
$433$	−38.0368	−1.82793	−0.913965	+	0.405792i	$0.866996\pi$
$434$	−2.88179	−0.138330
$435$	0	0
$436$	2.59027	0.124051
$437$	−7.34600	−0.351407
$438$	0	0
$439$	−16.6576	−0.795022	−0.397511	−	0.917597i	$-0.630126\pi$
$439$	−16.6576	−0.795022	−0.397511	+	0.917597i	$0.630126\pi$
$440$	0.537453	0.0256221
$441$	0	0
$442$	−7.25479	−0.345075
$443$	−1.37489	−0.0653230	−0.0326615	−	0.999466i	$-0.510398\pi$
$443$	−1.37489	−0.0653230	−0.0326615	+	0.999466i	$0.510398\pi$
$444$	0	0
$445$	9.83042	0.466007
$446$	28.5704	1.35285
$447$	0	0
$448$	−25.7452	−1.21634
$449$	39.1073	1.84559	0.922794	−	0.385294i	$-0.125900\pi$
$449$	39.1073	1.84559	0.922794	+	0.385294i	$0.125900\pi$
$450$	0	0
$451$	4.20139	0.197836
$452$	−41.8176	−1.96694
$453$	0	0
$454$	10.0867	0.473393
$455$	1.77684	0.0832997
$456$	0	0
$457$	−27.4493	−1.28403	−0.642013	−	0.766694i	$-0.721901\pi$
$457$	−27.4493	−1.28403	−0.642013	+	0.766694i	$0.721901\pi$
$458$	−24.8756	−1.16236
$459$	0	0
$460$	12.3884	0.577611
$461$	−4.54620	−0.211738	−0.105869	−	0.994380i	$-0.533762\pi$
$461$	−4.54620	−0.211738	−0.105869	+	0.994380i	$0.533762\pi$
$462$	0	0
$463$	35.1190	1.63212	0.816060	−	0.577968i	$-0.196154\pi$
$463$	35.1190	1.63212	0.816060	+	0.577968i	$0.196154\pi$
$464$	0	0
$465$	0	0
$466$	−54.0811	−2.50526
$467$	−34.4640	−1.59480	−0.797402	−	0.603448i	$-0.793793\pi$
$467$	−34.4640	−1.59480	−0.797402	+	0.603448i	$0.793793\pi$
$468$	0	0
$469$	27.1957	1.25578
$470$	23.4225	1.08040
$471$	0	0
$472$	−6.20974	−0.285826
$473$	8.20819	0.377413
$474$	0	0
$475$	5.54879	0.254596
$476$	27.4220	1.25689
$477$	0	0
$478$	7.30550	0.334146
$479$	30.4715	1.39228	0.696140	−	0.717906i	$-0.254899\pi$
$479$	30.4715	1.39228	0.696140	+	0.717906i	$0.254899\pi$
$480$	0	0
$481$	−6.93616	−0.316262
$482$	2.75418	0.125449
$483$	0	0
$484$	−24.5277	−1.11490
$485$	−13.4742	−0.611834
$486$	0	0
$487$	25.0119	1.13340	0.566698	−	0.823925i	$-0.308221\pi$
$487$	25.0119	1.13340	0.566698	+	0.823925i	$0.308221\pi$
$488$	6.22193	0.281653
$489$	0	0
$490$	2.05006	0.0926124
$491$	26.3863	1.19080	0.595399	−	0.803430i	$-0.296994\pi$
$491$	26.3863	1.19080	0.595399	+	0.803430i	$0.296994\pi$
$492$	0	0
$493$	0	0
$494$	2.10480	0.0946996
$495$	0	0
$496$	−1.79371	−0.0805402
$497$	−14.9765	−0.671786
$498$	0	0
$499$	−17.2139	−0.770600	−0.385300	−	0.922791i	$-0.625902\pi$
$499$	−17.2139	−0.770600	−0.385300	+	0.922791i	$0.625902\pi$
$500$	−21.0104	−0.939612
$501$	0	0
$502$	−5.51236	−0.246029
$503$	−2.51651	−0.112206	−0.0561028	−	0.998425i	$-0.517867\pi$
$503$	−2.51651	−0.112206	−0.0561028	+	0.998425i	$0.517867\pi$
$504$	0	0
$505$	−12.9117	−0.574561
$506$	8.26252	0.367314
$507$	0	0
$508$	−9.96117	−0.441956
$509$	24.6901	1.09437	0.547185	−	0.837012i	$-0.315699\pi$
$509$	24.6901	1.09437	0.547185	+	0.837012i	$0.315699\pi$
$510$	0	0
$511$	−13.2676	−0.586924
$512$	−30.3576	−1.34163
$513$	0	0
$514$	9.94303	0.438568
$515$	−6.18312	−0.272461
$516$	0	0
$517$	8.43676	0.371048
$518$	48.5456	2.13297
$519$	0	0
$520$	−0.526618	−0.0230937
$521$	−40.0205	−1.75333	−0.876664	−	0.481103i	$-0.840236\pi$
$521$	−40.0205	−1.75333	−0.876664	+	0.481103i	$0.840236\pi$
$522$	0	0
$523$	6.67862	0.292035	0.146018	−	0.989282i	$-0.453354\pi$
$523$	6.67862	0.292035	0.146018	+	0.989282i	$0.453354\pi$
$524$	20.3054	0.887046
$525$	0	0
$526$	30.3137	1.32174
$527$	2.68531	0.116974
$528$	0	0
$529$	5.25579	0.228513
$530$	−9.38884	−0.407825
$531$	0	0
$532$	−7.95584	−0.344929
$533$	−4.11669	−0.178314
$534$	0	0
$535$	−4.79760	−0.207419
$536$	−8.06022	−0.348148
$537$	0	0
$538$	22.6205	0.975238
$539$	0.738429	0.0318064
$540$	0	0
$541$	2.72906	0.117331	0.0586657	−	0.998278i	$-0.481315\pi$
$541$	2.72906	0.117331	0.0586657	+	0.998278i	$0.481315\pi$
$542$	1.99540	0.0857096
$543$	0	0
$544$	38.5258	1.65178
$545$	1.09460	0.0468876
$546$	0	0
$547$	41.6350	1.78018	0.890091	−	0.455783i	$-0.150641\pi$
$547$	41.6350	1.78018	0.890091	+	0.455783i	$0.150641\pi$
$548$	33.5001	1.43105
$549$	0	0
$550$	−6.24108	−0.266121
$551$	0	0
$552$	0	0
$553$	18.8136	0.800037
$554$	−38.6888	−1.64373
$555$	0	0
$556$	−33.7289	−1.43043
$557$	−38.8044	−1.64419	−0.822097	−	0.569347i	$-0.807196\pi$
$557$	−38.8044	−1.64419	−0.822097	+	0.569347i	$0.807196\pi$
$558$	0	0
$559$	−8.04272	−0.340171
$560$	−7.74052	−0.327097
$561$	0	0
$562$	−15.3185	−0.646172
$563$	−17.6548	−0.744061	−0.372030	−	0.928221i	$-0.621338\pi$
$563$	−17.6548	−0.744061	−0.372030	+	0.928221i	$0.621338\pi$
$564$	0	0
$565$	−17.6714	−0.743441
$566$	−19.0890	−0.802370
$567$	0	0
$568$	4.43870	0.186244
$569$	12.0256	0.504139	0.252069	−	0.967709i	$-0.418889\pi$
$569$	12.0256	0.504139	0.252069	+	0.967709i	$0.418889\pi$
$570$	0	0
$571$	−3.70261	−0.154950	−0.0774748	−	0.996994i	$-0.524686\pi$
$571$	−3.70261	−0.154950	−0.0774748	+	0.996994i	$0.524686\pi$
$572$	−1.27855	−0.0534587
$573$	0	0
$574$	28.8124	1.20261
$575$	−21.3430	−0.890064
$576$	0	0
$577$	44.4757	1.85155	0.925773	−	0.378079i	$-0.123415\pi$
$577$	44.4757	1.85155	0.925773	+	0.378079i	$0.123415\pi$
$578$	−11.8638	−0.493469
$579$	0	0
$580$	0	0
$581$	−39.0291	−1.61920
$582$	0	0
$583$	−3.38185	−0.140062
$584$	3.93223	0.162717
$585$	0	0
$586$	−39.7417	−1.64172
$587$	−23.1650	−0.956120	−0.478060	−	0.878327i	$-0.658660\pi$
$587$	−23.1650	−0.956120	−0.478060	+	0.878327i	$0.658660\pi$
$588$	0	0
$589$	−0.779077	−0.0321013
$590$	−17.6874	−0.728179
$591$	0	0
$592$	30.2162	1.24188
$593$	31.9010	1.31002	0.655009	−	0.755621i	$-0.272665\pi$
$593$	31.9010	1.31002	0.655009	+	0.755621i	$0.272665\pi$
$594$	0	0
$595$	11.5881	0.475064
$596$	−49.0607	−2.00961
$597$	0	0
$598$	−8.09595	−0.331068
$599$	4.65253	0.190097	0.0950485	−	0.995473i	$-0.469699\pi$
$599$	4.65253	0.190097	0.0950485	+	0.995473i	$0.469699\pi$
$600$	0	0
$601$	−23.4167	−0.955188	−0.477594	−	0.878581i	$-0.658491\pi$
$601$	−23.4167	−0.955188	−0.477594	+	0.878581i	$0.658491\pi$
$602$	56.2903	2.29422
$603$	0	0
$604$	−5.03060	−0.204692
$605$	−10.3650	−0.421396
$606$	0	0
$607$	37.3093	1.51434	0.757170	−	0.653218i	$-0.226581\pi$
$607$	37.3093	1.51434	0.757170	+	0.653218i	$0.226581\pi$
$608$	−11.1773	−0.453301
$609$	0	0
$610$	17.7221	0.717547
$611$	−8.26669	−0.334434
$612$	0	0
$613$	6.83915	0.276231	0.138115	−	0.990416i	$-0.455896\pi$
$613$	6.83915	0.276231	0.138115	+	0.990416i	$0.455896\pi$
$614$	−55.0018	−2.21969
$615$	0	0
$616$	1.32760	0.0534907
$617$	13.4967	0.543356	0.271678	−	0.962388i	$-0.412421\pi$
$617$	13.4967	0.543356	0.271678	+	0.962388i	$0.412421\pi$
$618$	0	0
$619$	−34.3963	−1.38250	−0.691252	−	0.722614i	$-0.742941\pi$
$619$	−34.3963	−1.38250	−0.691252	+	0.722614i	$0.742941\pi$
$620$	1.31384	0.0527653
$621$	0	0
$622$	−17.0961	−0.685493
$623$	24.2829	0.972873
$624$	0	0
$625$	11.1971	0.447885
$626$	−23.9501	−0.957237
$627$	0	0
$628$	−33.7664	−1.34743
$629$	−45.2356	−1.80366
$630$	0	0
$631$	−14.6283	−0.582343	−0.291171	−	0.956671i	$-0.594045\pi$
$631$	−14.6283	−0.582343	−0.291171	+	0.956671i	$0.594045\pi$
$632$	−5.57596	−0.221800
$633$	0	0
$634$	−55.7141	−2.21269
$635$	−4.20942	−0.167046
$636$	0	0
$637$	−0.723543	−0.0286678
$638$	0	0
$639$	0	0
$640$	5.68063	0.224547
$641$	2.49007	0.0983518	0.0491759	−	0.998790i	$-0.484341\pi$
$641$	2.49007	0.0983518	0.0491759	+	0.998790i	$0.484341\pi$
$642$	0	0
$643$	−17.5372	−0.691599	−0.345799	−	0.938308i	$-0.612392\pi$
$643$	−17.5372	−0.691599	−0.345799	+	0.938308i	$0.612392\pi$
$644$	30.6015	1.20587
$645$	0	0
$646$	13.7269	0.540079
$647$	−34.0582	−1.33897	−0.669483	−	0.742827i	$-0.733484\pi$
$647$	−34.0582	−1.33897	−0.669483	+	0.742827i	$0.733484\pi$
$648$	0	0
$649$	−6.37097	−0.250083
$650$	6.11527	0.239861
$651$	0	0
$652$	−16.4826	−0.645508
$653$	4.91571	0.192367	0.0961834	−	0.995364i	$-0.469336\pi$
$653$	4.91571	0.192367	0.0961834	+	0.995364i	$0.469336\pi$
$654$	0	0
$655$	8.58071	0.335276
$656$	17.9337	0.700193
$657$	0	0
$658$	57.8578	2.25553
$659$	−29.2987	−1.14132	−0.570658	−	0.821188i	$-0.693312\pi$
$659$	−29.2987	−1.14132	−0.570658	+	0.821188i	$0.693312\pi$
$660$	0	0
$661$	−45.1181	−1.75489	−0.877445	−	0.479677i	$-0.840754\pi$
$661$	−45.1181	−1.75489	−0.877445	+	0.479677i	$0.840754\pi$
$662$	−21.0896	−0.819672
$663$	0	0
$664$	11.5674	0.448902
$665$	−3.36200	−0.130373
$666$	0	0
$667$	0	0
$668$	−54.7554	−2.11855
$669$	0	0
$670$	−22.9582	−0.886952
$671$	6.38347	0.246431
$672$	0	0
$673$	17.3306	0.668045	0.334022	−	0.942565i	$-0.391594\pi$
$673$	17.3306	0.668045	0.334022	+	0.942565i	$0.391594\pi$
$674$	−2.05856	−0.0792928
$675$	0	0
$676$	−29.2766	−1.12602
$677$	−15.9644	−0.613560	−0.306780	−	0.951780i	$-0.599252\pi$
$677$	−15.9644	−0.613560	−0.306780	+	0.951780i	$0.599252\pi$
$678$	0	0
$679$	−33.2838	−1.27731
$680$	−3.43445	−0.131705
$681$	0	0
$682$	0.876278	0.0335544
$683$	−51.9292	−1.98701	−0.993507	−	0.113768i	$-0.963708\pi$
$683$	−51.9292	−1.98701	−0.993507	+	0.113768i	$0.963708\pi$
$684$	0	0
$685$	14.1565	0.540893
$686$	40.8471	1.55955
$687$	0	0
$688$	35.0368	1.33576
$689$	3.31367	0.126241
$690$	0	0
$691$	−5.92354	−0.225342	−0.112671	−	0.993632i	$-0.535941\pi$
$691$	−5.92354	−0.225342	−0.112671	+	0.993632i	$0.535941\pi$
$692$	27.4807	1.04466
$693$	0	0
$694$	−24.5940	−0.933574
$695$	−14.2533	−0.540657
$696$	0	0
$697$	−26.8479	−1.01694
$698$	−51.0918	−1.93385
$699$	0	0
$700$	−23.1148	−0.873657
$701$	−23.4563	−0.885931	−0.442966	−	0.896539i	$-0.646074\pi$
$701$	−23.4563	−0.885931	−0.442966	+	0.896539i	$0.646074\pi$
$702$	0	0
$703$	13.1240	0.494982
$704$	7.82843	0.295045
$705$	0	0
$706$	−5.26259	−0.198060
$707$	−31.8941	−1.19950
$708$	0	0
$709$	1.14986	0.0431840	0.0215920	−	0.999767i	$-0.493127\pi$
$709$	1.14986	0.0431840	0.0215920	+	0.999767i	$0.493127\pi$
$710$	12.6429	0.474480
$711$	0	0
$712$	−7.19693	−0.269716
$713$	2.99666	0.112226
$714$	0	0
$715$	−0.540292	−0.0202058
$716$	2.61185	0.0976095
$717$	0	0
$718$	−3.48495	−0.130057
$719$	−8.56227	−0.319319	−0.159659	−	0.987172i	$-0.551040\pi$
$719$	−8.56227	−0.319319	−0.159659	+	0.987172i	$0.551040\pi$
$720$	0	0
$721$	−15.2734	−0.568811
$722$	35.6379	1.32630
$723$	0	0
$724$	6.30034	0.234150
$725$	0	0
$726$	0	0
$727$	8.30010	0.307834	0.153917	−	0.988084i	$-0.450811\pi$
$727$	8.30010	0.307834	0.153917	+	0.988084i	$0.450811\pi$
$728$	−1.30084	−0.0482124
$729$	0	0
$730$	11.2003	0.414542
$731$	−52.4523	−1.94002
$732$	0	0
$733$	−3.63299	−0.134187	−0.0670937	−	0.997747i	$-0.521373\pi$
$733$	−3.63299	−0.134187	−0.0670937	+	0.997747i	$0.521373\pi$
$734$	57.9705	2.13973
$735$	0	0
$736$	42.9927	1.58473
$737$	−8.26950	−0.304611
$738$	0	0
$739$	14.5044	0.533555	0.266777	−	0.963758i	$-0.414041\pi$
$739$	14.5044	0.533555	0.266777	+	0.963758i	$0.414041\pi$
$740$	−22.1325	−0.813607
$741$	0	0
$742$	−23.1921	−0.851409
$743$	33.9157	1.24425	0.622124	−	0.782919i	$-0.286270\pi$
$743$	33.9157	1.24425	0.622124	+	0.782919i	$0.286270\pi$
$744$	0	0
$745$	−20.7322	−0.759569
$746$	−34.0813	−1.24781
$747$	0	0
$748$	−8.33832	−0.304879
$749$	−11.8509	−0.433024
$750$	0	0
$751$	−11.7583	−0.429068	−0.214534	−	0.976717i	$-0.568823\pi$
$751$	−11.7583	−0.429068	−0.214534	+	0.976717i	$0.568823\pi$
$752$	36.0125	1.31324
$753$	0	0
$754$	0	0
$755$	−2.12585	−0.0773674
$756$	0	0
$757$	0.654727	0.0237965	0.0118982	−	0.999929i	$-0.496213\pi$
$757$	0.654727	0.0237965	0.0118982	+	0.999929i	$0.496213\pi$
$758$	23.0058	0.835607
$759$	0	0
$760$	0.996425	0.0361441
$761$	−0.209264	−0.00758580	−0.00379290	−	0.999993i	$-0.501207\pi$
$761$	−0.209264	−0.00758580	−0.00379290	+	0.999993i	$0.501207\pi$
$762$	0	0
$763$	2.70386	0.0978863
$764$	52.1436	1.88649
$765$	0	0
$766$	35.4818	1.28201
$767$	6.24254	0.225405
$768$	0	0
$769$	−53.1893	−1.91805	−0.959027	−	0.283314i	$-0.908566\pi$
$769$	−53.1893	−1.91805	−0.959027	+	0.283314i	$0.908566\pi$
$770$	3.78145	0.136274
$771$	0	0
$772$	12.8783	0.463500
$773$	40.0741	1.44136	0.720682	−	0.693266i	$-0.243829\pi$
$773$	40.0741	1.44136	0.720682	+	0.693266i	$0.243829\pi$
$774$	0	0
$775$	−2.26352	−0.0813081
$776$	9.86460	0.354118
$777$	0	0
$778$	4.03223	0.144563
$779$	7.78928	0.279080
$780$	0	0
$781$	4.55395	0.162953
$782$	−52.7995	−1.88811
$783$	0	0
$784$	3.15200	0.112571
$785$	−14.2691	−0.509286
$786$	0	0
$787$	11.3024	0.402886	0.201443	−	0.979500i	$-0.435437\pi$
$787$	11.3024	0.402886	0.201443	+	0.979500i	$0.435437\pi$
$788$	40.7905	1.45310
$789$	0	0
$790$	−15.8822	−0.565063
$791$	−43.6515	−1.55207
$792$	0	0
$793$	−6.25479	−0.222114
$794$	42.8308	1.52001
$795$	0	0
$796$	27.4169	0.971767
$797$	39.8500	1.41156	0.705779	−	0.708432i	$-0.250597\pi$
$797$	39.8500	1.41156	0.705779	+	0.708432i	$0.250597\pi$
$798$	0	0
$799$	−53.9129	−1.90730
$800$	−32.4745	−1.14815
$801$	0	0
$802$	44.5187	1.57201
$803$	4.03433	0.142368
$804$	0	0
$805$	12.9317	0.455781
$806$	−0.858613	−0.0302434
$807$	0	0
$808$	9.45272	0.332546
$809$	6.93364	0.243774	0.121887	−	0.992544i	$-0.461105\pi$
$809$	6.93364	0.243774	0.121887	+	0.992544i	$0.461105\pi$
$810$	0	0
$811$	34.2757	1.20358	0.601792	−	0.798653i	$-0.294454\pi$
$811$	34.2757	1.20358	0.601792	+	0.798653i	$0.294454\pi$
$812$	0	0
$813$	0	0
$814$	−14.7614	−0.517388
$815$	−6.96525	−0.243982
$816$	0	0
$817$	15.2178	0.532403
$818$	−46.0047	−1.60852
$819$	0	0
$820$	−13.1359	−0.458726
$821$	−7.47322	−0.260817	−0.130409	−	0.991460i	$-0.541629\pi$
$821$	−7.47322	−0.260817	−0.130409	+	0.991460i	$0.541629\pi$
$822$	0	0
$823$	7.85554	0.273827	0.136913	−	0.990583i	$-0.456282\pi$
$823$	7.85554	0.273827	0.136913	+	0.990583i	$0.456282\pi$
$824$	4.52671	0.157695
$825$	0	0
$826$	−43.6910	−1.52020
$827$	−51.8242	−1.80211	−0.901053	−	0.433710i	$-0.857204\pi$
$827$	−51.8242	−1.80211	−0.901053	+	0.433710i	$0.857204\pi$
$828$	0	0
$829$	−28.1574	−0.977945	−0.488973	−	0.872299i	$-0.662628\pi$
$829$	−28.1574	−0.977945	−0.488973	+	0.872299i	$0.662628\pi$
$830$	32.9478	1.14363
$831$	0	0
$832$	−7.67062	−0.265931
$833$	−4.71874	−0.163495
$834$	0	0
$835$	−23.1387	−0.800747
$836$	2.41916	0.0836685
$837$	0	0
$838$	−77.1564	−2.66532
$839$	−0.155652	−0.00537369	−0.00268685	−	0.999996i	$-0.500855\pi$
$839$	−0.155652	−0.00537369	−0.00268685	+	0.999996i	$0.500855\pi$
$840$	0	0
$841$	0	0
$842$	−41.8876	−1.44354
$843$	0	0
$844$	18.0101	0.619934
$845$	−12.3718	−0.425602
$846$	0	0
$847$	−25.6033	−0.879741
$848$	−14.4355	−0.495716
$849$	0	0
$850$	39.8820	1.36794
$851$	−50.4805	−1.73045
$852$	0	0
$853$	−45.4412	−1.55588	−0.777939	−	0.628340i	$-0.783735\pi$
$853$	−45.4412	−1.55588	−0.777939	+	0.628340i	$0.783735\pi$
$854$	43.7767	1.49801
$855$	0	0
$856$	3.51236	0.120050
$857$	−43.7310	−1.49382	−0.746911	−	0.664924i	$-0.768464\pi$
$857$	−43.7310	−1.49382	−0.746911	+	0.664924i	$0.768464\pi$
$858$	0	0
$859$	−34.9206	−1.19148	−0.595738	−	0.803179i	$-0.703140\pi$
$859$	−34.9206	−1.19148	−0.595738	+	0.803179i	$0.703140\pi$
$860$	−25.6634	−0.875116
$861$	0	0
$862$	59.3013	2.01981
$863$	34.4638	1.17316	0.586580	−	0.809891i	$-0.300474\pi$
$863$	34.4638	1.17316	0.586580	+	0.809891i	$0.300474\pi$
$864$	0	0
$865$	11.6129	0.394850
$866$	79.3175	2.69532
$867$	0	0
$868$	3.24543	0.110157
$869$	−5.72074	−0.194063
$870$	0	0
$871$	8.10279	0.274553
$872$	−0.801366	−0.0271377
$873$	0	0
$874$	15.3185	0.518156
$875$	−21.9317	−0.741428
$876$	0	0
$877$	36.0203	1.21632	0.608160	−	0.793815i	$-0.291908\pi$
$877$	36.0203	1.21632	0.608160	+	0.793815i	$0.291908\pi$
$878$	34.7358	1.17228
$879$	0	0
$880$	2.35369	0.0793429
$881$	20.4125	0.687714	0.343857	−	0.939022i	$-0.388266\pi$
$881$	20.4125	0.687714	0.343857	+	0.939022i	$0.388266\pi$
$882$	0	0
$883$	42.2458	1.42168	0.710842	−	0.703352i	$-0.248314\pi$
$883$	42.2458	1.42168	0.710842	+	0.703352i	$0.248314\pi$
$884$	8.17022	0.274794
$885$	0	0
$886$	2.86704	0.0963200
$887$	−29.9288	−1.00491	−0.502455	−	0.864603i	$-0.667570\pi$
$887$	−29.9288	−1.00491	−0.502455	+	0.864603i	$0.667570\pi$
$888$	0	0
$889$	−10.3980	−0.348738
$890$	−20.4992	−0.687136
$891$	0	0
$892$	−32.1756	−1.07732
$893$	15.6416	0.523425
$894$	0	0
$895$	1.10372	0.0368934
$896$	14.0322	0.468782
$897$	0	0
$898$	−81.5499	−2.72135
$899$	0	0
$900$	0	0
$901$	21.6108	0.719960
$902$	−8.76110	−0.291713
$903$	0	0
$904$	12.9374	0.430290
$905$	2.66241	0.0885017
$906$	0	0
$907$	−42.5943	−1.41432	−0.707160	−	0.707053i	$-0.750024\pi$
$907$	−42.5943	−1.41432	−0.707160	+	0.707053i	$0.750024\pi$
$908$	−11.3595	−0.376978
$909$	0	0
$910$	−3.70522	−0.122827
$911$	−37.3420	−1.23720	−0.618598	−	0.785708i	$-0.712299\pi$
$911$	−37.3420	−1.23720	−0.618598	+	0.785708i	$0.712299\pi$
$912$	0	0
$913$	11.8677	0.392765
$914$	57.2397	1.89332
$915$	0	0
$916$	28.0145	0.925626
$917$	21.1959	0.699950
$918$	0	0
$919$	9.86001	0.325252	0.162626	−	0.986688i	$-0.448004\pi$
$919$	9.86001	0.325252	0.162626	+	0.986688i	$0.448004\pi$
$920$	−3.83267	−0.126359
$921$	0	0
$922$	9.48013	0.312211
$923$	−4.46215	−0.146873
$924$	0	0
$925$	38.1304	1.25372
$926$	−73.2332	−2.40659
$927$	0	0
$928$	0	0
$929$	28.6956	0.941471	0.470736	−	0.882274i	$-0.343989\pi$
$929$	28.6956	0.941471	0.470736	+	0.882274i	$0.343989\pi$
$930$	0	0
$931$	1.36903	0.0448682
$932$	60.9052	1.99502
$933$	0	0
$934$	71.8673	2.35157
$935$	−3.52363	−0.115235
$936$	0	0
$937$	36.9075	1.20572	0.602858	−	0.797848i	$-0.294028\pi$
$937$	36.9075	1.20572	0.602858	+	0.797848i	$0.294028\pi$
$938$	−56.7107	−1.85167
$939$	0	0
$940$	−26.3781	−0.860358
$941$	31.1116	1.01421	0.507104	−	0.861885i	$-0.330716\pi$
$941$	31.1116	1.01421	0.507104	+	0.861885i	$0.330716\pi$
$942$	0	0
$943$	−29.9608	−0.975658
$944$	−27.1946	−0.885109
$945$	0	0
$946$	−17.1164	−0.556502
$947$	28.3337	0.920720	0.460360	−	0.887732i	$-0.347720\pi$
$947$	28.3337	0.920720	0.460360	+	0.887732i	$0.347720\pi$
$948$	0	0
$949$	−3.95300	−0.128320
$950$	−11.5708	−0.375407
$951$	0	0
$952$	−8.48371	−0.274958
$953$	8.84041	0.286369	0.143184	−	0.989696i	$-0.454266\pi$
$953$	8.84041	0.286369	0.143184	+	0.989696i	$0.454266\pi$
$954$	0	0
$955$	22.0350	0.713034
$956$	−8.22734	−0.266091
$957$	0	0
$958$	−63.5418	−2.05294
$959$	34.9691	1.12921
$960$	0	0
$961$	−30.6822	−0.989748
$962$	14.4639	0.466334
$963$	0	0
$964$	−3.10171	−0.0998993
$965$	5.44214	0.175189
$966$	0	0
$967$	49.4538	1.59033	0.795164	−	0.606395i	$-0.207385\pi$
$967$	49.4538	1.59033	0.795164	+	0.606395i	$0.207385\pi$
$968$	7.58828	0.243896
$969$	0	0
$970$	28.0976	0.902161
$971$	41.4194	1.32921	0.664606	−	0.747194i	$-0.268599\pi$
$971$	41.4194	1.32921	0.664606	+	0.747194i	$0.268599\pi$
$972$	0	0
$973$	−35.2081	−1.12872
$974$	−52.1569	−1.67122
$975$	0	0
$976$	27.2480	0.872186
$977$	−24.5426	−0.785187	−0.392593	−	0.919712i	$-0.628422\pi$
$977$	−24.5426	−0.785187	−0.392593	+	0.919712i	$0.628422\pi$
$978$	0	0
$979$	−7.38379	−0.235987
$980$	−2.30875	−0.0737502
$981$	0	0
$982$	−55.0230	−1.75586
$983$	−29.5289	−0.941825	−0.470913	−	0.882180i	$-0.656075\pi$
$983$	−29.5289	−0.941825	−0.470913	+	0.882180i	$0.656075\pi$
$984$	0	0
$985$	17.2373	0.549228
$986$	0	0
$987$	0	0
$988$	−2.37040	−0.0754123
$989$	−58.5339	−1.86127
$990$	0	0
$991$	2.23362	0.0709532	0.0354766	−	0.999371i	$-0.488705\pi$
$991$	2.23362	0.0709532	0.0354766	+	0.999371i	$0.488705\pi$
$992$	4.55958	0.144767
$993$	0	0
$994$	31.2302	0.990562
$995$	11.5859	0.367298
$996$	0	0
$997$	25.3556	0.803021	0.401511	−	0.915854i	$-0.368485\pi$
$997$	25.3556	0.803021	0.401511	+	0.915854i	$0.368485\pi$
$998$	35.8959	1.13626
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bg.1.2	yes	8
3.2	odd	2	inner	7569.2.a.bg.1.7	yes	8
29.28	even	2		7569.2.a.bf.1.7	yes	8
87.86	odd	2		7569.2.a.bf.1.2	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7569.2.a.bf.1.2	✓	8	87.86	odd	2
7569.2.a.bf.1.7	yes	8	29.28	even	2
7569.2.a.bg.1.2	yes	8	1.1	even	1	trivial
7569.2.a.bg.1.7	yes	8	3.2	odd	2	inner

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)

\(f(q)\)	\(=\)	\(q-2.08529 q^{2} +2.34841 q^{4} +0.992398 q^{5} +2.45140 q^{7} -0.726543 q^{8} -2.06943 q^{10} -0.745407 q^{11} +0.730380 q^{13} -5.11187 q^{14} -3.18178 q^{16} +4.76333 q^{17} -1.38197 q^{19} +2.33056 q^{20} +1.55439 q^{22} +5.31562 q^{23} -4.01515 q^{25} -1.52305 q^{26} +5.75690 q^{28} +0.563746 q^{31} +8.08800 q^{32} -9.93290 q^{34} +2.43277 q^{35} -9.49664 q^{37} +2.88179 q^{38} -0.721020 q^{40} -5.63637 q^{41} -11.0117 q^{43} -1.75052 q^{44} -11.0846 q^{46} -11.3183 q^{47} -0.990640 q^{49} +8.37272 q^{50} +1.71523 q^{52} +4.53691 q^{53} -0.739740 q^{55} -1.78105 q^{56} +8.54697 q^{59} -8.56375 q^{61} -1.17557 q^{62} -10.5022 q^{64} +0.724828 q^{65} +11.0939 q^{67} +11.1863 q^{68} -5.07301 q^{70} -6.10935 q^{71} -5.41226 q^{73} +19.8032 q^{74} -3.24543 q^{76} -1.82729 q^{77} +7.67465 q^{79} -3.15759 q^{80} +11.7534 q^{82} -15.9212 q^{83} +4.72712 q^{85} +22.9625 q^{86} +0.541570 q^{88} +9.90572 q^{89} +1.79045 q^{91} +12.4833 q^{92} +23.6020 q^{94} -1.37146 q^{95} -13.5775 q^{97} +2.06577 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 12 q^{10} + 2 q^{13} - 2 q^{16} - 20 q^{19} - 14 q^{22} + 2 q^{25} - 20 q^{28} - 10 q^{31} - 36 q^{34} - 18 q^{37} + 10 q^{40} - 28 q^{43} - 26 q^{46} + 4 q^{49} + 44 q^{52} - 14 q^{55}+ \cdots + 6 q^{97}+O(q^{100}) \) 8 * q + 6 * q^4 + 12 * q^10 + 2 * q^13 - 2 * q^16 - 20 * q^19 - 14 * q^22 + 2 * q^25 - 20 * q^28 - 10 * q^31 - 36 * q^34 - 18 * q^37 + 10 * q^40 - 28 * q^43 - 26 * q^46 + 4 * q^49 + 44 * q^52 - 14 * q^55 - 54 * q^61 - 44 * q^64 + 38 * q^67 - 70 * q^70 + 16 * q^73 - 20 * q^76 - 14 * q^79 + 2 * q^82 - 56 * q^85 - 60 * q^88 - 30 * q^91 + 52 * q^94 + 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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