LMFDB - Embedded newform 7569.2.a.bk.1.5

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:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 $ x^9 - x^8 - 14x^7 + 9x^6 + 70x^5 - 23x^4 - 141x^3 + 14x^2 + 84*x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 87)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.0831558$ 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)

$f(q)$	$=$	\(q-0.0831558 q^{2} -1.99309 q^{4} -1.93038 q^{5} +0.343925 q^{7} +0.332048 q^{8} +0.160522 q^{10} -2.42414 q^{11} +3.63120 q^{13} -0.0285993 q^{14} +3.95856 q^{16} -6.15354 q^{17} +1.35067 q^{19} +3.84741 q^{20} +0.201582 q^{22} +3.46619 q^{23} -1.27364 q^{25} -0.301955 q^{26} -0.685471 q^{28} -10.9901 q^{31} -0.993273 q^{32} +0.511703 q^{34} -0.663905 q^{35} +3.08722 q^{37} -0.112316 q^{38} -0.640979 q^{40} +5.28313 q^{41} +0.00844158 q^{43} +4.83153 q^{44} -0.288234 q^{46} -5.79676 q^{47} -6.88172 q^{49} +0.105911 q^{50} -7.23729 q^{52} +4.14234 q^{53} +4.67952 q^{55} +0.114200 q^{56} -5.76819 q^{59} +6.23784 q^{61} +0.913892 q^{62} -7.83452 q^{64} -7.00959 q^{65} +11.0142 q^{67} +12.2645 q^{68} +0.0552076 q^{70} -12.8548 q^{71} -0.459207 q^{73} -0.256721 q^{74} -2.69201 q^{76} -0.833724 q^{77} +12.4792 q^{79} -7.64152 q^{80} -0.439323 q^{82} -2.87846 q^{83} +11.8787 q^{85} -0.000701966 q^{86} -0.804933 q^{88} -12.2239 q^{89} +1.24886 q^{91} -6.90842 q^{92} +0.482034 q^{94} -2.60731 q^{95} +2.72420 q^{97} +0.572255 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{2} + 11 q^{4} + 5 q^{7} - 12 q^{8} + 4 q^{10} - 3 q^{11} + 5 q^{13} - 15 q^{14} - 5 q^{16} - 16 q^{17} - q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} - 12 q^{26} - 24 q^{28} - 4 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) $ 9 * q - q^2 + 11 * q^4 + 5 * q^7 - 12 * q^8 + 4 * q^10 - 3 * q^11 + 5 * q^13 - 15 * q^14 - 5 * q^16 - 16 * q^17 - q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 - 12 * q^26 - 24 * q^28 - 4 * q^31 - 25 * q^32 + 24 * q^34 + 44 * q^35 + 25 * q^37 + 10 * q^38 + 5 * q^40 - 34 * q^41 + 12 * q^43 + 23 * q^44 + 6 * q^46 - 8 * q^47 + 26 * q^49 - 27 * q^50 - 23 * q^52 + 32 * q^53 - 5 * q^55 + 14 * q^56 - 10 * q^59 + 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 - 11 * q^68 - 14 * q^70 - 7 * q^71 + 17 * q^73 + 62 * q^74 - 6 * q^76 - 64 * q^77 + 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 + 42 * q^85 + 70 * q^86 - 29 * q^88 + 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 + 20 * q^95 + 16 * q^97 + 12 * q^98

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 $

$2$	−0.0831558	−0.0588000	−0.0294000	−	0.999568i	$-0.509360\pi$
$2$	−0.0831558	−0.0588000	−0.0294000	+	0.999568i	$0.509360\pi$
$3$	0	0
$3$	0	0
$4$	−1.99309	−0.996543
$5$	−1.93038	−0.863291	−0.431646	−	0.902043i	$-0.642067\pi$
$5$	−1.93038	−0.863291	−0.431646	+	0.902043i	$0.642067\pi$
$6$	0	0
$7$	0.343925	0.129991	0.0649957	−	0.997886i	$-0.479297\pi$
$7$	0.343925	0.129991	0.0649957	+	0.997886i	$0.479297\pi$
$8$	0.332048	0.117397
$9$	0	0
$10$	0.160522	0.0507616
$11$	−2.42414	−0.730907	−0.365454	−	0.930830i	$-0.619086\pi$
$11$	−2.42414	−0.730907	−0.365454	+	0.930830i	$0.619086\pi$
$12$	0	0
$13$	3.63120	1.00711	0.503557	−	0.863962i	$-0.332024\pi$
$13$	3.63120	1.00711	0.503557	+	0.863962i	$0.332024\pi$
$14$	−0.0285993	−0.00764350
$15$	0	0
$16$	3.95856	0.989640
$17$	−6.15354	−1.49245	−0.746227	−	0.665692i	$-0.768137\pi$
$17$	−6.15354	−1.49245	−0.746227	+	0.665692i	$0.768137\pi$
$18$	0	0
$19$	1.35067	0.309866	0.154933	−	0.987925i	$-0.450484\pi$
$19$	1.35067	0.309866	0.154933	+	0.987925i	$0.450484\pi$
$20$	3.84741	0.860307
$21$	0	0
$22$	0.201582	0.0429774
$23$	3.46619	0.722751	0.361376	−	0.932420i	$-0.382307\pi$
$23$	3.46619	0.722751	0.361376	+	0.932420i	$0.382307\pi$
$24$	0	0
$25$	−1.27364	−0.254728
$26$	−0.301955	−0.0592183
$27$	0	0
$28$	−0.685471	−0.129542
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−10.9901	−1.97388	−0.986941	−	0.161080i	$-0.948502\pi$
$31$	−10.9901	−1.97388	−0.986941	+	0.161080i	$0.948502\pi$
$32$	−0.993273	−0.175588
$33$	0	0
$34$	0.511703	0.0877563
$35$	−0.663905	−0.112220
$36$	0	0
$37$	3.08722	0.507536	0.253768	−	0.967265i	$-0.418330\pi$
$37$	3.08722	0.507536	0.253768	+	0.967265i	$0.418330\pi$
$38$	−0.112316	−0.0182201
$39$	0	0
$40$	−0.640979	−0.101348
$41$	5.28313	0.825086	0.412543	−	0.910938i	$-0.364641\pi$
$41$	5.28313	0.825086	0.412543	+	0.910938i	$0.364641\pi$
$42$	0	0
$43$	0.00844158	0.00128733	0.000643664	−	1.00000i	$-0.499795\pi$
$43$	0.00844158	0.00128733	0.000643664		1.00000i	$0.499795\pi$
$44$	4.83153	0.728380
$45$	0	0
$46$	−0.288234	−0.0424978
$47$	−5.79676	−0.845544	−0.422772	−	0.906236i	$-0.638943\pi$
$47$	−5.79676	−0.845544	−0.422772	+	0.906236i	$0.638943\pi$
$48$	0	0
$49$	−6.88172	−0.983102
$50$	0.105911	0.0149780
$51$	0	0
$52$	−7.23729	−1.00363
$53$	4.14234	0.568994	0.284497	−	0.958677i	$-0.408173\pi$
$53$	4.14234	0.568994	0.284497	+	0.958677i	$0.408173\pi$
$54$	0	0
$55$	4.67952	0.630986
$56$	0.114200	0.0152606
$57$	0	0
$58$	0	0
$59$	−5.76819	−0.750955	−0.375477	−	0.926832i	$-0.622521\pi$
$59$	−5.76819	−0.750955	−0.375477	+	0.926832i	$0.622521\pi$
$60$	0	0
$61$	6.23784	0.798673	0.399337	−	0.916804i	$-0.369240\pi$
$61$	6.23784	0.798673	0.399337	+	0.916804i	$0.369240\pi$
$62$	0.913892	0.116064
$63$	0	0
$64$	−7.83452	−0.979315
$65$	−7.00959	−0.869432
$66$	0	0
$67$	11.0142	1.34560	0.672800	−	0.739824i	$-0.265091\pi$
$67$	11.0142	1.34560	0.672800	+	0.739824i	$0.265091\pi$
$68$	12.2645	1.48729
$69$	0	0
$70$	0.0552076	0.00659856
$71$	−12.8548	−1.52558	−0.762791	−	0.646645i	$-0.776171\pi$
$71$	−12.8548	−1.52558	−0.762791	+	0.646645i	$0.776171\pi$
$72$	0	0
$73$	−0.459207	−0.0537461	−0.0268731	−	0.999639i	$-0.508555\pi$
$73$	−0.459207	−0.0537461	−0.0268731	+	0.999639i	$0.508555\pi$
$74$	−0.256721	−0.0298432
$75$	0	0
$76$	−2.69201	−0.308794
$77$	−0.833724	−0.0950116
$78$	0	0
$79$	12.4792	1.40402	0.702010	−	0.712167i	$-0.252286\pi$
$79$	12.4792	1.40402	0.702010	+	0.712167i	$0.252286\pi$
$80$	−7.64152	−0.854347
$81$	0	0
$82$	−0.439323	−0.0485151
$83$	−2.87846	−0.315952	−0.157976	−	0.987443i	$-0.550497\pi$
$83$	−2.87846	−0.315952	−0.157976	+	0.987443i	$0.550497\pi$
$84$	0	0
$85$	11.8787	1.28842
$86$	−0.000701966 0	−7.56950e−5 0
$87$	0	0
$88$	−0.804933	−0.0858061
$89$	−12.2239	−1.29574	−0.647868	−	0.761753i	$-0.724339\pi$
$89$	−12.2239	−1.29574	−0.647868	+	0.761753i	$0.724339\pi$
$90$	0	0
$91$	1.24886	0.130916
$92$	−6.90842	−0.720253
$93$	0	0
$94$	0.482034	0.0497180
$95$	−2.60731	−0.267504
$96$	0	0
$97$	2.72420	0.276601	0.138300	−	0.990390i	$-0.455836\pi$
$97$	2.72420	0.276601	0.138300	+	0.990390i	$0.455836\pi$
$98$	0.572255	0.0578064
$99$	0	0
$100$	2.53847	0.253847
$101$	−18.2696	−1.81789	−0.908947	−	0.416911i	$-0.863113\pi$
$101$	−18.2696	−1.81789	−0.908947	+	0.416911i	$0.863113\pi$
$102$	0	0
$103$	−16.0696	−1.58339	−0.791694	−	0.610918i	$-0.790800\pi$
$103$	−16.0696	−1.58339	−0.791694	+	0.610918i	$0.790800\pi$
$104$	1.20573	0.118232
$105$	0	0
$106$	−0.344459	−0.0334569
$107$	16.1760	1.56379	0.781897	−	0.623408i	$-0.214252\pi$
$107$	16.1760	1.56379	0.781897	+	0.623408i	$0.214252\pi$
$108$	0	0
$109$	9.19953	0.881155	0.440577	−	0.897715i	$-0.354774\pi$
$109$	9.19953	0.881155	0.440577	+	0.897715i	$0.354774\pi$
$110$	−0.389129	−0.0371020
$111$	0	0
$112$	1.36145	0.128645
$113$	−20.2563	−1.90555	−0.952774	−	0.303679i	$-0.901785\pi$
$113$	−20.2563	−1.90555	−0.952774	+	0.303679i	$0.901785\pi$
$114$	0	0
$115$	−6.69107	−0.623945
$116$	0	0
$117$	0	0
$118$	0.479659	0.0441562
$119$	−2.11636	−0.194006
$120$	0	0
$121$	−5.12352	−0.465775
$122$	−0.518713	−0.0469620
$123$	0	0
$124$	21.9042	1.96706
$125$	12.1105	1.08320
$126$	0	0
$127$	−9.56458	−0.848719	−0.424360	−	0.905494i	$-0.639501\pi$
$127$	−9.56458	−0.848719	−0.424360	+	0.905494i	$0.639501\pi$
$128$	2.63803	0.233171
$129$	0	0
$130$	0.582888	0.0511227
$131$	16.8025	1.46804	0.734019	−	0.679129i	$-0.237642\pi$
$131$	16.8025	1.46804	0.734019	+	0.679129i	$0.237642\pi$
$132$	0	0
$133$	0.464530	0.0402799
$134$	−0.915896	−0.0791213
$135$	0	0
$136$	−2.04327	−0.175209
$137$	−5.75126	−0.491363	−0.245682	−	0.969351i	$-0.579012\pi$
$137$	−5.75126	−0.491363	−0.245682	+	0.969351i	$0.579012\pi$
$138$	0	0
$139$	1.56684	0.132898	0.0664488	−	0.997790i	$-0.478833\pi$
$139$	1.56684	0.132898	0.0664488	+	0.997790i	$0.478833\pi$
$140$	1.32322	0.111832
$141$	0	0
$142$	1.06895	0.0897043
$143$	−8.80255	−0.736106
$144$	0	0
$145$	0	0
$146$	0.0381857	0.00316027
$147$	0	0
$148$	−6.15310	−0.505782
$149$	−12.6315	−1.03481	−0.517406	−	0.855740i	$-0.673102\pi$
$149$	−12.6315	−1.03481	−0.517406	+	0.855740i	$0.673102\pi$
$150$	0	0
$151$	−10.2677	−0.835572	−0.417786	−	0.908545i	$-0.637194\pi$
$151$	−10.2677	−0.835572	−0.417786	+	0.908545i	$0.637194\pi$
$152$	0.448489	0.0363772
$153$	0	0
$154$	0.0693290	0.00558669
$155$	21.2151	1.70404
$156$	0	0
$157$	3.78665	0.302207	0.151104	−	0.988518i	$-0.451717\pi$
$157$	3.78665	0.302207	0.151104	+	0.988518i	$0.451717\pi$
$158$	−1.03772	−0.0825564
$159$	0	0
$160$	1.91739	0.151583
$161$	1.19211	0.0939514
$162$	0	0
$163$	6.08181	0.476364	0.238182	−	0.971221i	$-0.423448\pi$
$163$	6.08181	0.476364	0.238182	+	0.971221i	$0.423448\pi$
$164$	−10.5297	−0.822233
$165$	0	0
$166$	0.239361	0.0185780
$167$	7.67550	0.593948	0.296974	−	0.954886i	$-0.404023\pi$
$167$	7.67550	0.593948	0.296974	+	0.954886i	$0.404023\pi$
$168$	0	0
$169$	0.185607	0.0142774
$170$	−0.987780	−0.0757593
$171$	0	0
$172$	−0.0168248	−0.00128288
$173$	14.9932	1.13991	0.569956	−	0.821675i	$-0.306960\pi$
$173$	14.9932	1.13991	0.569956	+	0.821675i	$0.306960\pi$
$174$	0	0
$175$	−0.438036	−0.0331124
$176$	−9.59612	−0.723335
$177$	0	0
$178$	1.01649	0.0761893
$179$	16.0802	1.20189	0.600947	−	0.799289i	$-0.294790\pi$
$179$	16.0802	1.20189	0.600947	+	0.799289i	$0.294790\pi$
$180$	0	0
$181$	−5.67342	−0.421702	−0.210851	−	0.977518i	$-0.567623\pi$
$181$	−5.67342	−0.421702	−0.210851	+	0.977518i	$0.567623\pi$
$182$	−0.103850	−0.00769787
$183$	0	0
$184$	1.15094	0.0848487
$185$	−5.95951	−0.438152
$186$	0	0
$187$	14.9171	1.09084
$188$	11.5534	0.842620
$189$	0	0
$190$	0.216813	0.0157293
$191$	0.636893	0.0460839	0.0230420	−	0.999734i	$-0.492665\pi$
$191$	0.636893	0.0460839	0.0230420	+	0.999734i	$0.492665\pi$
$192$	0	0
$193$	11.3433	0.816510	0.408255	−	0.912868i	$-0.366137\pi$
$193$	11.3433	0.816510	0.408255	+	0.912868i	$0.366137\pi$
$194$	−0.226533	−0.0162641
$195$	0	0
$196$	13.7158	0.979703
$197$	16.9281	1.20608	0.603038	−	0.797712i	$-0.293957\pi$
$197$	16.9281	1.20608	0.603038	+	0.797712i	$0.293957\pi$
$198$	0	0
$199$	8.67794	0.615163	0.307581	−	0.951522i	$-0.400480\pi$
$199$	8.67794	0.615163	0.307581	+	0.951522i	$0.400480\pi$
$200$	−0.422910	−0.0299042
$201$	0	0
$202$	1.51922	0.106892
$203$	0	0
$204$	0	0
$205$	−10.1984	−0.712290
$206$	1.33628	0.0931032
$207$	0	0
$208$	14.3743	0.996679
$209$	−3.27423	−0.226483
$210$	0	0
$211$	−12.7515	−0.877850	−0.438925	−	0.898524i	$-0.644641\pi$
$211$	−12.7515	−0.877850	−0.438925	+	0.898524i	$0.644641\pi$
$212$	−8.25603	−0.567027
$213$	0	0
$214$	−1.34513	−0.0919511
$215$	−0.0162954	−0.00111134
$216$	0	0
$217$	−3.77977	−0.256588
$218$	−0.764994	−0.0518119
$219$	0	0
$220$	−9.32667	−0.628804
$221$	−22.3447	−1.50307
$222$	0	0
$223$	19.1389	1.28164	0.640819	−	0.767692i	$-0.278595\pi$
$223$	19.1389	1.28164	0.640819	+	0.767692i	$0.278595\pi$
$224$	−0.341611	−0.0228249
$225$	0	0
$226$	1.68443	0.112046
$227$	24.7509	1.64278	0.821389	−	0.570368i	$-0.193200\pi$
$227$	24.7509	1.64278	0.821389	+	0.570368i	$0.193200\pi$
$228$	0	0
$229$	19.4336	1.28421	0.642106	−	0.766616i	$-0.278061\pi$
$229$	19.4336	1.28421	0.642106	+	0.766616i	$0.278061\pi$
$230$	0.556401	0.0366880
$231$	0	0
$232$	0	0
$233$	−18.2842	−1.19784	−0.598918	−	0.800810i	$-0.704402\pi$
$233$	−18.2842	−1.19784	−0.598918	+	0.800810i	$0.704402\pi$
$234$	0	0
$235$	11.1899	0.729951
$236$	11.4965	0.748359
$237$	0	0
$238$	0.175987	0.0114076
$239$	−8.19384	−0.530015	−0.265008	−	0.964246i	$-0.585374\pi$
$239$	−8.19384	−0.530015	−0.265008	+	0.964246i	$0.585374\pi$
$240$	0	0
$241$	−7.14233	−0.460078	−0.230039	−	0.973181i	$-0.573885\pi$
$241$	−7.14233	−0.460078	−0.230039	+	0.973181i	$0.573885\pi$
$242$	0.426051	0.0273876
$243$	0	0
$244$	−12.4325	−0.795912
$245$	13.2843	0.848704
$246$	0	0
$247$	4.90456	0.312070
$248$	−3.64925	−0.231727
$249$	0	0
$250$	−1.00706	−0.0636919
$251$	2.32177	0.146549	0.0732744	−	0.997312i	$-0.476655\pi$
$251$	2.32177	0.146549	0.0732744	+	0.997312i	$0.476655\pi$
$252$	0	0
$253$	−8.40256	−0.528264
$254$	0.795350	0.0499047
$255$	0	0
$256$	15.4497	0.965605
$257$	22.7822	1.42111	0.710557	−	0.703640i	$-0.248443\pi$
$257$	22.7822	1.42111	0.710557	+	0.703640i	$0.248443\pi$
$258$	0	0
$259$	1.06177	0.0659754
$260$	13.9707	0.866426
$261$	0	0
$262$	−1.39722	−0.0863207
$263$	−0.955452	−0.0589157	−0.0294578	−	0.999566i	$-0.509378\pi$
$263$	−0.955452	−0.0589157	−0.0294578	+	0.999566i	$0.509378\pi$
$264$	0	0
$265$	−7.99628	−0.491208
$266$	−0.0386284	−0.00236846
$267$	0	0
$268$	−21.9523	−1.34095
$269$	15.1464	0.923493	0.461746	−	0.887012i	$-0.347223\pi$
$269$	15.1464	0.923493	0.461746	+	0.887012i	$0.347223\pi$
$270$	0	0
$271$	8.25078	0.501199	0.250600	−	0.968091i	$-0.419372\pi$
$271$	8.25078	0.501199	0.250600	+	0.968091i	$0.419372\pi$
$272$	−24.3592	−1.47699
$273$	0	0
$274$	0.478250	0.0288922
$275$	3.08749	0.186182
$276$	0	0
$277$	16.7332	1.00540	0.502700	−	0.864461i	$-0.332340\pi$
$277$	16.7332	1.00540	0.502700	+	0.864461i	$0.332340\pi$
$278$	−0.130292	−0.00781439
$279$	0	0
$280$	−0.220448	−0.0131743
$281$	22.1439	1.32099	0.660496	−	0.750830i	$-0.270346\pi$
$281$	22.1439	1.32099	0.660496	+	0.750830i	$0.270346\pi$
$282$	0	0
$283$	−17.6596	−1.04976	−0.524878	−	0.851177i	$-0.675889\pi$
$283$	−17.6596	−1.04976	−0.524878	+	0.851177i	$0.675889\pi$
$284$	25.6207	1.52031
$285$	0	0
$286$	0.731983	0.0432831
$287$	1.81700	0.107254
$288$	0	0
$289$	20.8661	1.22742
$290$	0	0
$291$	0	0
$292$	0.915239	0.0535603
$293$	23.1447	1.35213	0.676063	−	0.736844i	$-0.263685\pi$
$293$	23.1447	1.35213	0.676063	+	0.736844i	$0.263685\pi$
$294$	0	0
$295$	11.1348	0.648293
$296$	1.02511	0.0595831
$297$	0	0
$298$	1.05038	0.0608470
$299$	12.5864	0.727893
$300$	0	0
$301$	0.00290327	0.000167342 0
$302$	0.853817	0.0491317
$303$	0	0
$304$	5.34672	0.306655
$305$	−12.0414	−0.689488
$306$	0	0
$307$	22.9984	1.31259	0.656293	−	0.754506i	$-0.272124\pi$
$307$	22.9984	1.31259	0.656293	+	0.754506i	$0.272124\pi$
$308$	1.66168	0.0946831
$309$	0	0
$310$	−1.76416	−0.100197
$311$	4.08071	0.231396	0.115698	−	0.993284i	$-0.463090\pi$
$311$	4.08071	0.231396	0.115698	+	0.993284i	$0.463090\pi$
$312$	0	0
$313$	28.7526	1.62519	0.812597	−	0.582826i	$-0.198053\pi$
$313$	28.7526	1.62519	0.812597	+	0.582826i	$0.198053\pi$
$314$	−0.314882	−0.0177698
$315$	0	0
$316$	−24.8721	−1.39917
$317$	11.5299	0.647583	0.323792	−	0.946128i	$-0.395042\pi$
$317$	11.5299	0.647583	0.323792	+	0.946128i	$0.395042\pi$
$318$	0	0
$319$	0	0
$320$	15.1236	0.845434
$321$	0	0
$322$	−0.0991309	−0.00552435
$323$	−8.31143	−0.462460
$324$	0	0
$325$	−4.62484	−0.256540
$326$	−0.505738	−0.0280102
$327$	0	0
$328$	1.75425	0.0968624
$329$	−1.99365	−0.109913
$330$	0	0
$331$	−12.3038	−0.676277	−0.338139	−	0.941096i	$-0.609797\pi$
$331$	−12.3038	−0.676277	−0.338139	+	0.941096i	$0.609797\pi$
$332$	5.73702	0.314860
$333$	0	0
$334$	−0.638262	−0.0349241
$335$	−21.2616	−1.16165
$336$	0	0
$337$	−30.5882	−1.66624	−0.833122	−	0.553089i	$-0.813449\pi$
$337$	−30.5882	−1.66624	−0.833122	+	0.553089i	$0.813449\pi$
$338$	−0.0154343	−0.000839513 0
$339$	0	0
$340$	−23.6752	−1.28397
$341$	26.6416	1.44273
$342$	0	0
$343$	−4.77427	−0.257786
$344$	0.00280301	0.000151128 0
$345$	0	0
$346$	−1.24677	−0.0670268
$347$	2.30431	0.123702	0.0618508	−	0.998085i	$-0.480300\pi$
$347$	2.30431	0.123702	0.0618508	+	0.998085i	$0.480300\pi$
$348$	0	0
$349$	9.12305	0.488345	0.244173	−	0.969732i	$-0.421484\pi$
$349$	9.12305	0.488345	0.244173	+	0.969732i	$0.421484\pi$
$350$	0.0364253	0.00194701
$351$	0	0
$352$	2.40784	0.128338
$353$	14.5658	0.775262	0.387631	−	0.921815i	$-0.373294\pi$
$353$	14.5658	0.775262	0.387631	+	0.921815i	$0.373294\pi$
$354$	0	0
$355$	24.8146	1.31702
$356$	24.3634	1.29126
$357$	0	0
$358$	−1.33717	−0.0706714
$359$	3.75107	0.197974	0.0989869	−	0.995089i	$-0.468440\pi$
$359$	3.75107	0.197974	0.0989869	+	0.995089i	$0.468440\pi$
$360$	0	0
$361$	−17.1757	−0.903983
$362$	0.471778	0.0247961
$363$	0	0
$364$	−2.48908	−0.130463
$365$	0.886443	0.0463986
$366$	0	0
$367$	−13.9488	−0.728119	−0.364060	−	0.931376i	$-0.618610\pi$
$367$	−13.9488	−0.728119	−0.364060	+	0.931376i	$0.618610\pi$
$368$	13.7211	0.715263
$369$	0	0
$370$	0.495568	0.0257633
$371$	1.42465	0.0739643
$372$	0	0
$373$	−7.66007	−0.396624	−0.198312	−	0.980139i	$-0.563546\pi$
$373$	−7.66007	−0.396624	−0.198312	+	0.980139i	$0.563546\pi$
$374$	−1.24044	−0.0641417
$375$	0	0
$376$	−1.92480	−0.0992641
$377$	0	0
$378$	0	0
$379$	12.5700	0.645676	0.322838	−	0.946454i	$-0.395363\pi$
$379$	12.5700	0.645676	0.322838	+	0.946454i	$0.395363\pi$
$380$	5.19659	0.266580
$381$	0	0
$382$	−0.0529613	−0.00270974
$383$	−23.5509	−1.20340	−0.601698	−	0.798724i	$-0.705509\pi$
$383$	−23.5509	−1.20340	−0.601698	+	0.798724i	$0.705509\pi$
$384$	0	0
$385$	1.60940	0.0820227
$386$	−0.943263	−0.0480108
$387$	0	0
$388$	−5.42956	−0.275644
$389$	−21.2485	−1.07734	−0.538671	−	0.842516i	$-0.681073\pi$
$389$	−21.2485	−1.07734	−0.538671	+	0.842516i	$0.681073\pi$
$390$	0	0
$391$	−21.3294	−1.07867
$392$	−2.28506	−0.115413
$393$	0	0
$394$	−1.40767	−0.0709173
$395$	−24.0896	−1.21208
$396$	0	0
$397$	14.7291	0.739233	0.369617	−	0.929184i	$-0.379489\pi$
$397$	14.7291	0.739233	0.369617	+	0.929184i	$0.379489\pi$
$398$	−0.721621	−0.0361716
$399$	0	0
$400$	−5.04178	−0.252089
$401$	12.6297	0.630696	0.315348	−	0.948976i	$-0.397879\pi$
$401$	12.6297	0.630696	0.315348	+	0.948976i	$0.397879\pi$
$402$	0	0
$403$	−39.9073	−1.98792
$404$	36.4129	1.81161
$405$	0	0
$406$	0	0
$407$	−7.48388	−0.370962
$408$	0	0
$409$	33.7573	1.66919	0.834597	−	0.550862i	$-0.185701\pi$
$409$	33.7573	1.66919	0.834597	+	0.550862i	$0.185701\pi$
$410$	0.848059	0.0418827
$411$	0	0
$412$	32.0281	1.57791
$413$	−1.98383	−0.0976177
$414$	0	0
$415$	5.55652	0.272759
$416$	−3.60677	−0.176837
$417$	0	0
$418$	0.272271	0.0133172
$419$	24.6416	1.20382	0.601911	−	0.798563i	$-0.294406\pi$
$419$	24.6416	1.20382	0.601911	+	0.798563i	$0.294406\pi$
$420$	0	0
$421$	−12.2557	−0.597309	−0.298654	−	0.954361i	$-0.596538\pi$
$421$	−12.2557	−0.597309	−0.298654	+	0.954361i	$0.596538\pi$
$422$	1.06036	0.0516176
$423$	0	0
$424$	1.37546	0.0667981
$425$	7.83740	0.380170
$426$	0	0
$427$	2.14535	0.103821
$428$	−32.2402	−1.55839
$429$	0	0
$430$	0.00135506	6.53468e−5 0
$431$	8.39549	0.404397	0.202198	−	0.979345i	$-0.435191\pi$
$431$	8.39549	0.404397	0.202198	+	0.979345i	$0.435191\pi$
$432$	0	0
$433$	−9.24157	−0.444122	−0.222061	−	0.975033i	$-0.571278\pi$
$433$	−9.24157	−0.444122	−0.222061	+	0.975033i	$0.571278\pi$
$434$	0.314310	0.0150874
$435$	0	0
$436$	−18.3354	−0.878108
$437$	4.68170	0.223956
$438$	0	0
$439$	17.4230	0.831552	0.415776	−	0.909467i	$-0.363510\pi$
$439$	17.4230	0.831552	0.415776	+	0.909467i	$0.363510\pi$
$440$	1.55382	0.0740757
$441$	0	0
$442$	1.85809	0.0883806
$443$	28.9972	1.37770	0.688850	−	0.724904i	$-0.258116\pi$
$443$	28.9972	1.37770	0.688850	+	0.724904i	$0.258116\pi$
$444$	0	0
$445$	23.5968	1.11860
$446$	−1.59151	−0.0753603
$447$	0	0
$448$	−2.69449	−0.127303
$449$	−13.9198	−0.656914	−0.328457	−	0.944519i	$-0.606529\pi$
$449$	−13.9198	−0.656914	−0.328457	+	0.944519i	$0.606529\pi$
$450$	0	0
$451$	−12.8071	−0.603061
$452$	40.3724	1.89896
$453$	0	0
$454$	−2.05818	−0.0965954
$455$	−2.41077	−0.113019
$456$	0	0
$457$	−0.999387	−0.0467494	−0.0233747	−	0.999727i	$-0.507441\pi$
$457$	−0.999387	−0.0467494	−0.0233747	+	0.999727i	$0.507441\pi$
$458$	−1.61602	−0.0755117
$459$	0	0
$460$	13.3359	0.621788
$461$	15.5235	0.723003	0.361502	−	0.932371i	$-0.382264\pi$
$461$	15.5235	0.723003	0.361502	+	0.932371i	$0.382264\pi$
$462$	0	0
$463$	35.8295	1.66514	0.832568	−	0.553923i	$-0.186870\pi$
$463$	35.8295	1.66514	0.832568	+	0.553923i	$0.186870\pi$
$464$	0	0
$465$	0	0
$466$	1.52043	0.0704328
$467$	−4.17634	−0.193258	−0.0966290	−	0.995320i	$-0.530806\pi$
$467$	−4.17634	−0.193258	−0.0966290	+	0.995320i	$0.530806\pi$
$468$	0	0
$469$	3.78806	0.174916
$470$	−0.930508	−0.0429211
$471$	0	0
$472$	−1.91532	−0.0881597
$473$	−0.0204636	−0.000940918 0
$474$	0	0
$475$	−1.72027	−0.0789314
$476$	4.21808	0.193335
$477$	0	0
$478$	0.681365	0.0311649
$479$	−18.5452	−0.847351	−0.423675	−	0.905814i	$-0.639260\pi$
$479$	−18.5452	−0.847351	−0.423675	+	0.905814i	$0.639260\pi$
$480$	0	0
$481$	11.2103	0.511147
$482$	0.593927	0.0270526
$483$	0	0
$484$	10.2116	0.464164
$485$	−5.25874	−0.238787
$486$	0	0
$487$	19.5628	0.886474	0.443237	−	0.896404i	$-0.353830\pi$
$487$	19.5628	0.886474	0.443237	+	0.896404i	$0.353830\pi$
$488$	2.07126	0.0937617
$489$	0	0
$490$	−1.10467	−0.0499038
$491$	2.39192	0.107946	0.0539728	−	0.998542i	$-0.482812\pi$
$491$	2.39192	0.107946	0.0539728	+	0.998542i	$0.482812\pi$
$492$	0	0
$493$	0	0
$494$	−0.407843	−0.0183497
$495$	0	0
$496$	−43.5050	−1.95343
$497$	−4.42108	−0.198312
$498$	0	0
$499$	−17.3897	−0.778468	−0.389234	−	0.921139i	$-0.627260\pi$
$499$	−17.3897	−0.778468	−0.389234	+	0.921139i	$0.627260\pi$
$500$	−24.1373	−1.07945
$501$	0	0
$502$	−0.193069	−0.00861707
$503$	20.1486	0.898381	0.449191	−	0.893436i	$-0.351712\pi$
$503$	20.1486	0.898381	0.449191	+	0.893436i	$0.351712\pi$
$504$	0	0
$505$	35.2673	1.56937
$506$	0.698721	0.0310619
$507$	0	0
$508$	19.0630	0.845785
$509$	−28.9399	−1.28274	−0.641369	−	0.767232i	$-0.721633\pi$
$509$	−28.9399	−1.28274	−0.641369	+	0.767232i	$0.721633\pi$
$510$	0	0
$511$	−0.157933	−0.00698653
$512$	−6.56080	−0.289949
$513$	0	0
$514$	−1.89447	−0.0835615
$515$	31.0205	1.36692
$516$	0	0
$517$	14.0522	0.618014
$518$	−0.0882926	−0.00387935
$519$	0	0
$520$	−2.32752	−0.102069
$521$	3.28970	0.144125	0.0720623	−	0.997400i	$-0.477042\pi$
$521$	3.28970	0.144125	0.0720623	+	0.997400i	$0.477042\pi$
$522$	0	0
$523$	−0.291341	−0.0127395	−0.00636974	−	0.999980i	$-0.502028\pi$
$523$	−0.291341	−0.0127395	−0.00636974	+	0.999980i	$0.502028\pi$
$524$	−33.4887	−1.46296
$525$	0	0
$526$	0.0794513	0.00346424
$527$	67.6281	2.94593
$528$	0	0
$529$	−10.9855	−0.477630
$530$	0.664937	0.0288830
$531$	0	0
$532$	−0.925848	−0.0401406
$533$	19.1841	0.830955
$534$	0	0
$535$	−31.2258	−1.35001
$536$	3.65725	0.157969
$537$	0	0
$538$	−1.25951	−0.0543014
$539$	16.6823	0.718556
$540$	0	0
$541$	−27.6565	−1.18905	−0.594523	−	0.804078i	$-0.702659\pi$
$541$	−27.6565	−1.18905	−0.594523	+	0.804078i	$0.702659\pi$
$542$	−0.686101	−0.0294705
$543$	0	0
$544$	6.11215	0.262056
$545$	−17.7586	−0.760693
$546$	0	0
$547$	5.06702	0.216650	0.108325	−	0.994116i	$-0.465451\pi$
$547$	5.06702	0.216650	0.108325	+	0.994116i	$0.465451\pi$
$548$	11.4627	0.489664
$549$	0	0
$550$	−0.256742	−0.0109475
$551$	0	0
$552$	0	0
$553$	4.29191	0.182510
$554$	−1.39146	−0.0591175
$555$	0	0
$556$	−3.12285	−0.132438
$557$	−16.1639	−0.684885	−0.342443	−	0.939539i	$-0.611254\pi$
$557$	−16.1639	−0.684885	−0.342443	+	0.939539i	$0.611254\pi$
$558$	0	0
$559$	0.0306531	0.00129649
$560$	−2.62811	−0.111058
$561$	0	0
$562$	−1.84139	−0.0776744
$563$	7.45712	0.314280	0.157140	−	0.987576i	$-0.449773\pi$
$563$	7.45712	0.314280	0.157140	+	0.987576i	$0.449773\pi$
$564$	0	0
$565$	39.1022	1.64504
$566$	1.46850	0.0617257
$567$	0	0
$568$	−4.26841	−0.179098
$569$	17.6440	0.739675	0.369837	−	0.929097i	$-0.379413\pi$
$569$	17.6440	0.739675	0.369837	+	0.929097i	$0.379413\pi$
$570$	0	0
$571$	29.2430	1.22378	0.611890	−	0.790943i	$-0.290410\pi$
$571$	29.2430	1.22378	0.611890	+	0.790943i	$0.290410\pi$
$572$	17.5442	0.733561
$573$	0	0
$574$	−0.151094	−0.00630654
$575$	−4.41468	−0.184105
$576$	0	0
$577$	17.0793	0.711023	0.355511	−	0.934672i	$-0.384307\pi$
$577$	17.0793	0.711023	0.355511	+	0.934672i	$0.384307\pi$
$578$	−1.73514	−0.0721722
$579$	0	0
$580$	0	0
$581$	−0.989975	−0.0410711
$582$	0	0
$583$	−10.0416	−0.415882
$584$	−0.152479	−0.00630962
$585$	0	0
$586$	−1.92461	−0.0795051
$587$	34.0645	1.40599	0.702996	−	0.711193i	$-0.251845\pi$
$587$	34.0645	1.40599	0.702996	+	0.711193i	$0.251845\pi$
$588$	0	0
$589$	−14.8441	−0.611639
$590$	−0.925923	−0.0381196
$591$	0	0
$592$	12.2210	0.502278
$593$	42.8511	1.75969	0.879843	−	0.475265i	$-0.157648\pi$
$593$	42.8511	1.75969	0.879843	+	0.475265i	$0.157648\pi$
$594$	0	0
$595$	4.08537	0.167484
$596$	25.1756	1.03123
$597$	0	0
$598$	−1.04664	−0.0428001
$599$	5.69345	0.232628	0.116314	−	0.993212i	$-0.462892\pi$
$599$	5.69345	0.232628	0.116314	+	0.993212i	$0.462892\pi$
$600$	0	0
$601$	8.96083	0.365520	0.182760	−	0.983158i	$-0.441497\pi$
$601$	8.96083	0.365520	0.182760	+	0.983158i	$0.441497\pi$
$602$	−0.000241424 0	−9.83969e−6 0
$603$	0	0
$604$	20.4644	0.832683
$605$	9.89034	0.402099
$606$	0	0
$607$	−28.3175	−1.14937	−0.574685	−	0.818374i	$-0.694876\pi$
$607$	−28.3175	−1.14937	−0.574685	+	0.818374i	$0.694876\pi$
$608$	−1.34159	−0.0544086
$609$	0	0
$610$	1.00131	0.0405419
$611$	−21.0492	−0.851558
$612$	0	0
$613$	−32.0123	−1.29296	−0.646482	−	0.762929i	$-0.723760\pi$
$613$	−32.0123	−1.29296	−0.646482	+	0.762929i	$0.723760\pi$
$614$	−1.91245	−0.0771801
$615$	0	0
$616$	−0.276836	−0.0111541
$617$	−6.10052	−0.245598	−0.122799	−	0.992432i	$-0.539187\pi$
$617$	−6.10052	−0.245598	−0.122799	+	0.992432i	$0.539187\pi$
$618$	0	0
$619$	−0.710979	−0.0285766	−0.0142883	−	0.999898i	$-0.504548\pi$
$619$	−0.710979	−0.0285766	−0.0142883	+	0.999898i	$0.504548\pi$
$620$	−42.2835	−1.69814
$621$	0	0
$622$	−0.339335	−0.0136061
$623$	−4.20412	−0.168434
$624$	0	0
$625$	−17.0096	−0.680386
$626$	−2.39095	−0.0955615
$627$	0	0
$628$	−7.54711	−0.301162
$629$	−18.9974	−0.757475
$630$	0	0
$631$	−27.0861	−1.07828	−0.539140	−	0.842216i	$-0.681251\pi$
$631$	−27.0861	−1.07828	−0.539140	+	0.842216i	$0.681251\pi$
$632$	4.14370	0.164827
$633$	0	0
$634$	−0.958777	−0.0380779
$635$	18.4633	0.732692
$636$	0	0
$637$	−24.9889	−0.990095
$638$	0	0
$639$	0	0
$640$	−5.09240	−0.201295
$641$	27.4704	1.08501	0.542507	−	0.840051i	$-0.317475\pi$
$641$	27.4704	1.08501	0.542507	+	0.840051i	$0.317475\pi$
$642$	0	0
$643$	−26.5921	−1.04869	−0.524345	−	0.851506i	$-0.675690\pi$
$643$	−26.5921	−1.04869	−0.524345	+	0.851506i	$0.675690\pi$
$644$	−2.37598	−0.0936266
$645$	0	0
$646$	0.691143	0.0271927
$647$	24.3504	0.957314	0.478657	−	0.878002i	$-0.341124\pi$
$647$	24.3504	0.957314	0.478657	+	0.878002i	$0.341124\pi$
$648$	0	0
$649$	13.9829	0.548878
$650$	0.384582	0.0150846
$651$	0	0
$652$	−12.1216	−0.474717
$653$	20.7490	0.811973	0.405986	−	0.913879i	$-0.366928\pi$
$653$	20.7490	0.811973	0.405986	+	0.913879i	$0.366928\pi$
$654$	0	0
$655$	−32.4351	−1.26734
$656$	20.9136	0.816538
$657$	0	0
$658$	0.165783	0.00646291
$659$	−13.3618	−0.520504	−0.260252	−	0.965541i	$-0.583806\pi$
$659$	−13.3618	−0.520504	−0.260252	+	0.965541i	$0.583806\pi$
$660$	0	0
$661$	34.6065	1.34604	0.673018	−	0.739626i	$-0.264998\pi$
$661$	34.6065	1.34604	0.673018	+	0.739626i	$0.264998\pi$
$662$	1.02313	0.0397651
$663$	0	0
$664$	−0.955788	−0.0370918
$665$	−0.896719	−0.0347733
$666$	0	0
$667$	0	0
$668$	−15.2979	−0.591894
$669$	0	0
$670$	1.76803	0.0683048
$671$	−15.1214	−0.583756
$672$	0	0
$673$	36.9298	1.42354	0.711770	−	0.702413i	$-0.247894\pi$
$673$	36.9298	1.42354	0.711770	+	0.702413i	$0.247894\pi$
$674$	2.54358	0.0979752
$675$	0	0
$676$	−0.369930	−0.0142281
$677$	−3.93509	−0.151238	−0.0756189	−	0.997137i	$-0.524093\pi$
$677$	−3.93509	−0.151238	−0.0756189	+	0.997137i	$0.524093\pi$
$678$	0	0
$679$	0.936920	0.0359557
$680$	3.94429	0.151257
$681$	0	0
$682$	−2.21541	−0.0848323
$683$	−3.88519	−0.148663	−0.0743314	−	0.997234i	$-0.523682\pi$
$683$	−3.88519	−0.148663	−0.0743314	+	0.997234i	$0.523682\pi$
$684$	0	0
$685$	11.1021	0.424190
$686$	0.397008	0.0151578
$687$	0	0
$688$	0.0334165	0.00127399
$689$	15.0417	0.573041
$690$	0	0
$691$	7.66976	0.291771	0.145886	−	0.989301i	$-0.453397\pi$
$691$	7.66976	0.291771	0.145886	+	0.989301i	$0.453397\pi$
$692$	−29.8827	−1.13597
$693$	0	0
$694$	−0.191616	−0.00727366
$695$	−3.02459	−0.114729
$696$	0	0
$697$	−32.5100	−1.23140
$698$	−0.758634	−0.0287147
$699$	0	0
$700$	0.873043	0.0329979
$701$	−39.3445	−1.48602	−0.743011	−	0.669279i	$-0.766603\pi$
$701$	−39.3445	−1.48602	−0.743011	+	0.669279i	$0.766603\pi$
$702$	0	0
$703$	4.16983	0.157268
$704$	18.9920	0.715788
$705$	0	0
$706$	−1.21123	−0.0455854
$707$	−6.28337	−0.236311
$708$	0	0
$709$	37.5849	1.41153	0.705765	−	0.708446i	$-0.250603\pi$
$709$	37.5849	1.41153	0.705765	+	0.708446i	$0.250603\pi$
$710$	−2.06348	−0.0774409
$711$	0	0
$712$	−4.05894	−0.152115
$713$	−38.0939	−1.42663
$714$	0	0
$715$	16.9923	0.635474
$716$	−32.0493	−1.19774
$717$	0	0
$718$	−0.311923	−0.0116409
$719$	−4.63300	−0.172782	−0.0863910	−	0.996261i	$-0.527533\pi$
$719$	−4.63300	−0.172782	−0.0863910	+	0.996261i	$0.527533\pi$
$720$	0	0
$721$	−5.52674	−0.205827
$722$	1.42826	0.0531542
$723$	0	0
$724$	11.3076	0.420244
$725$	0	0
$726$	0	0
$727$	−5.51391	−0.204500	−0.102250	−	0.994759i	$-0.532604\pi$
$727$	−5.51391	−0.204500	−0.102250	+	0.994759i	$0.532604\pi$
$728$	0.414682	0.0153691
$729$	0	0
$730$	−0.0737129	−0.00272824
$731$	−0.0519456	−0.00192128
$732$	0	0
$733$	−43.4421	−1.60457	−0.802285	−	0.596941i	$-0.796383\pi$
$733$	−43.4421	−1.60457	−0.802285	+	0.596941i	$0.796383\pi$
$734$	1.15992	0.0428134
$735$	0	0
$736$	−3.44288	−0.126906
$737$	−26.7000	−0.983509
$738$	0	0
$739$	−44.2848	−1.62904	−0.814522	−	0.580133i	$-0.803001\pi$
$739$	−44.2848	−1.62904	−0.814522	+	0.580133i	$0.803001\pi$
$740$	11.8778	0.436637
$741$	0	0
$742$	−0.118468	−0.00434910
$743$	−35.4033	−1.29882	−0.649411	−	0.760438i	$-0.724984\pi$
$743$	−35.4033	−1.29882	−0.649411	+	0.760438i	$0.724984\pi$
$744$	0	0
$745$	24.3836	0.893345
$746$	0.636980	0.0233215
$747$	0	0
$748$	−29.7310	−1.08707
$749$	5.56333	0.203280
$750$	0	0
$751$	3.55544	0.129740	0.0648698	−	0.997894i	$-0.479337\pi$
$751$	3.55544	0.129740	0.0648698	+	0.997894i	$0.479337\pi$
$752$	−22.9468	−0.836784
$753$	0	0
$754$	0	0
$755$	19.8205	0.721342
$756$	0	0
$757$	−21.6056	−0.785268	−0.392634	−	0.919695i	$-0.628436\pi$
$757$	−21.6056	−0.785268	−0.392634	+	0.919695i	$0.628436\pi$
$758$	−1.04527	−0.0379658
$759$	0	0
$760$	−0.865753	−0.0314041
$761$	−16.5313	−0.599260	−0.299630	−	0.954055i	$-0.596863\pi$
$761$	−16.5313	−0.599260	−0.299630	+	0.954055i	$0.596863\pi$
$762$	0	0
$763$	3.16395	0.114543
$764$	−1.26938	−0.0459246
$765$	0	0
$766$	1.95839	0.0707597
$767$	−20.9455	−0.756297
$768$	0	0
$769$	0.202502	0.00730241	0.00365120	−	0.999993i	$-0.498838\pi$
$769$	0.202502	0.00730241	0.00365120	+	0.999993i	$0.498838\pi$
$770$	−0.133831	−0.00482294
$771$	0	0
$772$	−22.6082	−0.813687
$773$	−14.5159	−0.522100	−0.261050	−	0.965325i	$-0.584069\pi$
$773$	−14.5159	−0.522100	−0.261050	+	0.965325i	$0.584069\pi$
$774$	0	0
$775$	13.9974	0.502803
$776$	0.904565	0.0324720
$777$	0	0
$778$	1.76694	0.0633477
$779$	7.13578	0.255666
$780$	0	0
$781$	31.1618	1.11506
$782$	1.77366	0.0634260
$783$	0	0
$784$	−27.2417	−0.972917
$785$	−7.30966	−0.260893
$786$	0	0
$787$	−34.0857	−1.21503	−0.607513	−	0.794310i	$-0.707833\pi$
$787$	−34.0857	−1.21503	−0.607513	+	0.794310i	$0.707833\pi$
$788$	−33.7391	−1.20191
$789$	0	0
$790$	2.00319	0.0712702
$791$	−6.96663	−0.247705
$792$	0	0
$793$	22.6508	0.804355
$794$	−1.22481	−0.0434669
$795$	0	0
$796$	−17.2959	−0.613036
$797$	14.3592	0.508628	0.254314	−	0.967122i	$-0.418150\pi$
$797$	14.3592	0.508628	0.254314	+	0.967122i	$0.418150\pi$
$798$	0	0
$799$	35.6706	1.26193
$800$	1.26507	0.0447271
$801$	0	0
$802$	−1.05023	−0.0370849
$803$	1.11318	0.0392834
$804$	0	0
$805$	−2.30122	−0.0811075
$806$	3.31852	0.116890
$807$	0	0
$808$	−6.06639	−0.213415
$809$	−28.3982	−0.998429	−0.499214	−	0.866479i	$-0.666378\pi$
$809$	−28.3982	−0.998429	−0.499214	+	0.866479i	$0.666378\pi$
$810$	0	0
$811$	−41.4768	−1.45645	−0.728224	−	0.685339i	$-0.759654\pi$
$811$	−41.4768	−1.45645	−0.728224	+	0.685339i	$0.759654\pi$
$812$	0	0
$813$	0	0
$814$	0.622328	0.0218126
$815$	−11.7402	−0.411241
$816$	0	0
$817$	0.0114018	0.000398899 0
$818$	−2.80712	−0.0981486
$819$	0	0
$820$	20.3264	0.709827
$821$	45.2453	1.57907	0.789535	−	0.613705i	$-0.210322\pi$
$821$	45.2453	1.57907	0.789535	+	0.613705i	$0.210322\pi$
$822$	0	0
$823$	1.80834	0.0630349	0.0315174	−	0.999503i	$-0.489966\pi$
$823$	1.80834	0.0630349	0.0315174	+	0.999503i	$0.489966\pi$
$824$	−5.33589	−0.185885
$825$	0	0
$826$	0.164967	0.00573992
$827$	−22.9341	−0.797497	−0.398748	−	0.917060i	$-0.630555\pi$
$827$	−22.9341	−0.797497	−0.398748	+	0.917060i	$0.630555\pi$
$828$	0	0
$829$	9.99341	0.347085	0.173543	−	0.984826i	$-0.444479\pi$
$829$	9.99341	0.347085	0.173543	+	0.984826i	$0.444479\pi$
$830$	−0.462057	−0.0160382
$831$	0	0
$832$	−28.4487	−0.986281
$833$	42.3469	1.46723
$834$	0	0
$835$	−14.8166	−0.512750
$836$	6.52581	0.225700
$837$	0	0
$838$	−2.04909	−0.0707848
$839$	24.1584	0.834042	0.417021	−	0.908897i	$-0.363074\pi$
$839$	24.1584	0.834042	0.417021	+	0.908897i	$0.363074\pi$
$840$	0	0
$841$	0	0
$842$	1.01914	0.0351218
$843$	0	0
$844$	25.4149	0.874815
$845$	−0.358291	−0.0123256
$846$	0	0
$847$	−1.76211	−0.0605467
$848$	16.3977	0.563099
$849$	0	0
$850$	−0.651725	−0.0223540
$851$	10.7009	0.366823
$852$	0	0
$853$	−12.9629	−0.443842	−0.221921	−	0.975065i	$-0.571233\pi$
$853$	−12.9629	−0.443842	−0.221921	+	0.975065i	$0.571233\pi$
$854$	−0.178398	−0.00610466
$855$	0	0
$856$	5.37121	0.183584
$857$	21.5162	0.734980	0.367490	−	0.930027i	$-0.380217\pi$
$857$	21.5162	0.734980	0.367490	+	0.930027i	$0.380217\pi$
$858$	0	0
$859$	4.60632	0.157165	0.0785827	−	0.996908i	$-0.474961\pi$
$859$	4.60632	0.157165	0.0785827	+	0.996908i	$0.474961\pi$
$860$	0.0324782	0.00110750
$861$	0	0
$862$	−0.698134	−0.0237785
$863$	19.2332	0.654707	0.327354	−	0.944902i	$-0.393843\pi$
$863$	19.2332	0.654707	0.327354	+	0.944902i	$0.393843\pi$
$864$	0	0
$865$	−28.9425	−0.984076
$866$	0.768490	0.0261144
$867$	0	0
$868$	7.53341	0.255701
$869$	−30.2514	−1.02621
$870$	0	0
$871$	39.9948	1.35517
$872$	3.05469	0.103445
$873$	0	0
$874$	−0.389310	−0.0131686
$875$	4.16510	0.140806
$876$	0	0
$877$	−27.8752	−0.941279	−0.470640	−	0.882326i	$-0.655977\pi$
$877$	−27.8752	−0.941279	−0.470640	+	0.882326i	$0.655977\pi$
$878$	−1.44882	−0.0488953
$879$	0	0
$880$	18.5241	0.624449
$881$	27.7676	0.935514	0.467757	−	0.883857i	$-0.345062\pi$
$881$	27.7676	0.935514	0.467757	+	0.883857i	$0.345062\pi$
$882$	0	0
$883$	−0.320213	−0.0107760	−0.00538802	−	0.999985i	$-0.501715\pi$
$883$	−0.320213	−0.0107760	−0.00538802	+	0.999985i	$0.501715\pi$
$884$	44.5350	1.49787
$885$	0	0
$886$	−2.41129	−0.0810088
$887$	−24.7909	−0.832397	−0.416198	−	0.909274i	$-0.636638\pi$
$887$	−24.7909	−0.832397	−0.416198	+	0.909274i	$0.636638\pi$
$888$	0	0
$889$	−3.28950	−0.110326
$890$	−1.96221	−0.0657735
$891$	0	0
$892$	−38.1455	−1.27721
$893$	−7.82952	−0.262005
$894$	0	0
$895$	−31.0410	−1.03758
$896$	0.907285	0.0303103
$897$	0	0
$898$	1.15751	0.0386266
$899$	0	0
$900$	0	0
$901$	−25.4901	−0.849197
$902$	1.06498	0.0354600
$903$	0	0
$904$	−6.72605	−0.223705
$905$	10.9518	0.364052
$906$	0	0
$907$	44.0205	1.46168	0.730839	−	0.682550i	$-0.239129\pi$
$907$	44.0205	1.46168	0.730839	+	0.682550i	$0.239129\pi$
$908$	−49.3307	−1.63710
$909$	0	0
$910$	0.200470	0.00664550
$911$	58.6374	1.94275	0.971373	−	0.237561i	$-0.0763480\pi$
$911$	58.6374	1.94275	0.971373	+	0.237561i	$0.0763480\pi$
$912$	0	0
$913$	6.97781	0.230932
$914$	0.0831048	0.00274886
$915$	0	0
$916$	−38.7329	−1.27977
$917$	5.77879	0.190832
$918$	0	0
$919$	−36.3681	−1.19967	−0.599837	−	0.800122i	$-0.704768\pi$
$919$	−36.3681	−1.19967	−0.599837	+	0.800122i	$0.704768\pi$
$920$	−2.22176	−0.0732491
$921$	0	0
$922$	−1.29087	−0.0425126
$923$	−46.6783	−1.53643
$924$	0	0
$925$	−3.93201	−0.129284
$926$	−2.97943	−0.0979101
$927$	0	0
$928$	0	0
$929$	−0.895129	−0.0293682	−0.0146841	−	0.999892i	$-0.504674\pi$
$929$	−0.895129	−0.0293682	−0.0146841	+	0.999892i	$0.504674\pi$
$930$	0	0
$931$	−9.29495	−0.304630
$932$	36.4419	1.19369
$933$	0	0
$934$	0.347287	0.0113636
$935$	−28.7956	−0.941717
$936$	0	0
$937$	23.2168	0.758460	0.379230	−	0.925302i	$-0.376189\pi$
$937$	23.2168	0.758460	0.379230	+	0.925302i	$0.376189\pi$
$938$	−0.314999	−0.0102851
$939$	0	0
$940$	−22.3025	−0.727427
$941$	−26.3658	−0.859501	−0.429750	−	0.902948i	$-0.641398\pi$
$941$	−26.3658	−0.859501	−0.429750	+	0.902948i	$0.641398\pi$
$942$	0	0
$943$	18.3123	0.596332
$944$	−22.8337	−0.743175
$945$	0	0
$946$	0.00170167	5.53260e−5 0
$947$	50.9638	1.65610	0.828051	−	0.560653i	$-0.189450\pi$
$947$	50.9638	1.65610	0.828051	+	0.560653i	$0.189450\pi$
$948$	0	0
$949$	−1.66747	−0.0541284
$950$	0.143050	0.00464117
$951$	0	0
$952$	−0.702732	−0.0227757
$953$	25.1414	0.814410	0.407205	−	0.913337i	$-0.366503\pi$
$953$	25.1414	0.814410	0.407205	+	0.913337i	$0.366503\pi$
$954$	0	0
$955$	−1.22944	−0.0397839
$956$	16.3310	0.528183
$957$	0	0
$958$	1.54214	0.0498243
$959$	−1.97800	−0.0638730
$960$	0	0
$961$	89.7826	2.89621
$962$	−0.932203	−0.0300554
$963$	0	0
$964$	14.2353	0.458488
$965$	−21.8969	−0.704886
$966$	0	0
$967$	−50.2105	−1.61466	−0.807330	−	0.590100i	$-0.799088\pi$
$967$	−50.2105	−1.61466	−0.807330	+	0.590100i	$0.799088\pi$
$968$	−1.70126	−0.0546804
$969$	0	0
$970$	0.437294	0.0140407
$971$	−32.5354	−1.04411	−0.522055	−	0.852912i	$-0.674834\pi$
$971$	−32.5354	−1.04411	−0.522055	+	0.852912i	$0.674834\pi$
$972$	0	0
$973$	0.538875	0.0172755
$974$	−1.62676	−0.0521247
$975$	0	0
$976$	24.6929	0.790399
$977$	32.6634	1.04500	0.522498	−	0.852641i	$-0.325000\pi$
$977$	32.6634	1.04500	0.522498	+	0.852641i	$0.325000\pi$
$978$	0	0
$979$	29.6326	0.947062
$980$	−26.4768	−0.845769
$981$	0	0
$982$	−0.198902	−0.00634721
$983$	1.79101	0.0571245	0.0285622	−	0.999592i	$-0.490907\pi$
$983$	1.79101	0.0571245	0.0285622	+	0.999592i	$0.490907\pi$
$984$	0	0
$985$	−32.6776	−1.04120
$986$	0	0
$987$	0	0
$988$	−9.77521	−0.310991
$989$	0.0292602	0.000930419 0
$990$	0	0
$991$	28.6772	0.910960	0.455480	−	0.890246i	$-0.349468\pi$
$991$	28.6772	0.910960	0.455480	+	0.890246i	$0.349468\pi$
$992$	10.9162	0.346589
$993$	0	0
$994$	0.367638	0.0116608
$995$	−16.7517	−0.531065
$996$	0	0
$997$	34.6674	1.09793	0.548964	−	0.835846i	$-0.315023\pi$
$997$	34.6674	1.09793	0.548964	+	0.835846i	$0.315023\pi$
$998$	1.44605	0.0457739
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bk.1.5		9
3.2	odd	2		2523.2.a.q.1.5		9
29.9	even	14		261.2.k.b.226.2		18
29.13	even	14		261.2.k.b.82.2		18
29.28	even	2		7569.2.a.bl.1.5		9
87.38	odd	14		87.2.g.b.52.2	✓	18
87.71	odd	14		87.2.g.b.82.2	yes	18
87.86	odd	2		2523.2.a.p.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.52.2	✓	18	87.38	odd	14
87.2.g.b.82.2	yes	18	87.71	odd	14
261.2.k.b.82.2		18	29.13	even	14
261.2.k.b.226.2		18	29.9	even	14
2523.2.a.p.1.5		9	87.86	odd	2
2523.2.a.q.1.5		9	3.2	odd	2
7569.2.a.bk.1.5		9	1.1	even	1	trivial
7569.2.a.bl.1.5		9	29.28	even	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:	\( N \)	\(=\)	\( 7569 = 3^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7569.a (trivial)

\(f(q)\)	\(=\)	\(q-0.0831558 q^{2} -1.99309 q^{4} -1.93038 q^{5} +0.343925 q^{7} +0.332048 q^{8} +0.160522 q^{10} -2.42414 q^{11} +3.63120 q^{13} -0.0285993 q^{14} +3.95856 q^{16} -6.15354 q^{17} +1.35067 q^{19} +3.84741 q^{20} +0.201582 q^{22} +3.46619 q^{23} -1.27364 q^{25} -0.301955 q^{26} -0.685471 q^{28} -10.9901 q^{31} -0.993273 q^{32} +0.511703 q^{34} -0.663905 q^{35} +3.08722 q^{37} -0.112316 q^{38} -0.640979 q^{40} +5.28313 q^{41} +0.00844158 q^{43} +4.83153 q^{44} -0.288234 q^{46} -5.79676 q^{47} -6.88172 q^{49} +0.105911 q^{50} -7.23729 q^{52} +4.14234 q^{53} +4.67952 q^{55} +0.114200 q^{56} -5.76819 q^{59} +6.23784 q^{61} +0.913892 q^{62} -7.83452 q^{64} -7.00959 q^{65} +11.0142 q^{67} +12.2645 q^{68} +0.0552076 q^{70} -12.8548 q^{71} -0.459207 q^{73} -0.256721 q^{74} -2.69201 q^{76} -0.833724 q^{77} +12.4792 q^{79} -7.64152 q^{80} -0.439323 q^{82} -2.87846 q^{83} +11.8787 q^{85} -0.000701966 q^{86} -0.804933 q^{88} -12.2239 q^{89} +1.24886 q^{91} -6.90842 q^{92} +0.482034 q^{94} -2.60731 q^{95} +2.72420 q^{97} +0.572255 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{2} + 11 q^{4} + 5 q^{7} - 12 q^{8} + 4 q^{10} - 3 q^{11} + 5 q^{13} - 15 q^{14} - 5 q^{16} - 16 q^{17} - q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} - 12 q^{26} - 24 q^{28} - 4 q^{31}+ \cdots + 12 q^{98}+O(q^{100}) \) 9 * q - q^2 + 11 * q^4 + 5 * q^7 - 12 * q^8 + 4 * q^10 - 3 * q^11 + 5 * q^13 - 15 * q^14 - 5 * q^16 - 16 * q^17 - q^19 - 8 * q^20 + 24 * q^22 + 10 * q^23 + 9 * q^25 - 12 * q^26 - 24 * q^28 - 4 * q^31 - 25 * q^32 + 24 * q^34 + 44 * q^35 + 25 * q^37 + 10 * q^38 + 5 * q^40 - 34 * q^41 + 12 * q^43 + 23 * q^44 + 6 * q^46 - 8 * q^47 + 26 * q^49 - 27 * q^50 - 23 * q^52 + 32 * q^53 - 5 * q^55 + 14 * q^56 - 10 * q^59 + 51 * q^61 - 8 * q^62 - 8 * q^64 + 11 * q^65 + 7 * q^67 - 11 * q^68 - 14 * q^70 - 7 * q^71 + 17 * q^73 + 62 * q^74 - 6 * q^76 - 64 * q^77 + 13 * q^79 - 54 * q^80 + 37 * q^82 + 31 * q^83 + 42 * q^85 + 70 * q^86 - 29 * q^88 + 32 * q^89 + 45 * q^91 - 9 * q^92 + 38 * q^94 + 20 * q^95 + 16 * q^97 + 12 * q^98

Introduction

L-functions

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