LMFDB - Embedded newform 7569.2.a.bl.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	$-0.490640\pi$
$2$	0.0831558	0.0588000	0.0294000	+	0.999568i	$0.490640\pi$
$3$	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	$-0.380914\pi$
$11$	2.42414	0.730907	0.365454	+	0.930830i	$0.380914\pi$
$12$	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	$-0.231863\pi$
$17$	6.15354	1.49245	0.746227	+	0.665692i	$0.231863\pi$
$18$	0	0
$19$	−1.35067	−0.309866	−0.154933	−	0.987925i	$-0.549516\pi$
$19$	−1.35067	−0.309866	−0.154933	+	0.987925i	$0.549516\pi$
$20$	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	$-0.0514976\pi$
$31$	10.9901	1.97388	0.986941	+	0.161080i	$0.0514976\pi$
$32$	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	$-0.581670\pi$
$37$	−3.08722	−0.507536	−0.253768	+	0.967265i	$0.581670\pi$
$38$	−0.112316	−0.0182201
$39$	0	0
$40$	0.640979	0.101348
$41$	−5.28313	−0.825086	−0.412543	−	0.910938i	$-0.635359\pi$
$41$	−5.28313	−0.825086	−0.412543	+	0.910938i	$0.635359\pi$
$42$	0	0
$43$	−0.00844158	−0.00128733	−0.000643664	−	1.00000i	$-0.500205\pi$
$43$	−0.00844158	−0.00128733	−0.000643664		1.00000i	$0.500205\pi$
$44$	−4.83153	−0.728380
$45$	0	0
$46$	0.288234	0.0424978
$47$	5.79676	0.845544	0.422772	−	0.906236i	$-0.361057\pi$
$47$	5.79676	0.845544	0.422772	+	0.906236i	$0.361057\pi$
$48$	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	$-0.630760\pi$
$61$	−6.23784	−0.798673	−0.399337	+	0.916804i	$0.630760\pi$
$62$	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	$-0.491445\pi$
$73$	0.459207	0.0537461	0.0268731	+	0.999639i	$0.491445\pi$
$74$	−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	$-0.747714\pi$
$79$	−12.4792	−1.40402	−0.702010	+	0.712167i	$0.747714\pi$
$80$	−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	$-0.275661\pi$
$89$	12.2239	1.29574	0.647868	+	0.761753i	$0.275661\pi$
$90$	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	$-0.544164\pi$
$97$	−2.72420	−0.276601	−0.138300	+	0.990390i	$0.544164\pi$
$98$	−0.572255	−0.0578064
$99$	0	0
$100$	2.53847	0.253847
$101$	18.2696	1.81789	0.908947	−	0.416911i	$-0.136887\pi$
$101$	18.2696	1.81789	0.908947	+	0.416911i	$0.136887\pi$
$102$	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	$-0.0982152\pi$
$113$	20.2563	1.90555	0.952774	+	0.303679i	$0.0982152\pi$
$114$	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	$-0.360499\pi$
$127$	9.56458	0.848719	0.424360	+	0.905494i	$0.360499\pi$
$128$	−2.63803	−0.233171
$129$	0	0
$130$	−0.582888	−0.0511227
$131$	−16.8025	−1.46804	−0.734019	−	0.679129i	$-0.762358\pi$
$131$	−16.8025	−1.46804	−0.734019	+	0.679129i	$0.762358\pi$
$132$	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	$-0.420988\pi$
$137$	5.75126	0.491363	0.245682	+	0.969351i	$0.420988\pi$
$138$	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	$-0.548283\pi$
$157$	−3.78665	−0.302207	−0.151104	+	0.988518i	$0.548283\pi$
$158$	−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	$-0.576552\pi$
$163$	−6.08181	−0.476364	−0.238182	+	0.971221i	$0.576552\pi$
$164$	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	$-0.507335\pi$
$191$	−0.636893	−0.0460839	−0.0230420	+	0.999734i	$0.507335\pi$
$192$	0	0
$193$	−11.3433	−0.816510	−0.408255	−	0.912868i	$-0.633863\pi$
$193$	−11.3433	−0.816510	−0.408255	+	0.912868i	$0.633863\pi$
$194$	−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	$-0.355359\pi$
$211$	12.7515	0.877850	0.438925	+	0.898524i	$0.355359\pi$
$212$	−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	$-0.721939\pi$
$229$	−19.4336	−1.28421	−0.642106	+	0.766616i	$0.721939\pi$
$230$	−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	$-0.523345\pi$
$251$	−2.32177	−0.146549	−0.0732744	+	0.997312i	$0.523345\pi$
$252$	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	$-0.490622\pi$
$263$	0.955452	0.0589157	0.0294578	+	0.999566i	$0.490622\pi$
$264$	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	$-0.652777\pi$
$269$	−15.1464	−0.923493	−0.461746	+	0.887012i	$0.652777\pi$
$270$	0	0
$271$	−8.25078	−0.501199	−0.250600	−	0.968091i	$-0.580628\pi$
$271$	−8.25078	−0.501199	−0.250600	+	0.968091i	$0.580628\pi$
$272$	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	$-0.736315\pi$
$293$	−23.1447	−1.35213	−0.676063	+	0.736844i	$0.736315\pi$
$294$	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	$-0.727876\pi$
$307$	−22.9984	−1.31259	−0.656293	+	0.754506i	$0.727876\pi$
$308$	−1.66168	−0.0946831
$309$	0	0
$310$	−1.76416	−0.100197
$311$	−4.08071	−0.231396	−0.115698	−	0.993284i	$-0.536910\pi$
$311$	−4.08071	−0.231396	−0.115698	+	0.993284i	$0.536910\pi$
$312$	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	$-0.604958\pi$
$317$	−11.5299	−0.647583	−0.323792	+	0.946128i	$0.604958\pi$
$318$	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	$-0.390203\pi$
$331$	12.3038	0.676277	0.338139	+	0.941096i	$0.390203\pi$
$332$	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	$-0.186551\pi$
$337$	30.5882	1.66624	0.833122	+	0.553089i	$0.186551\pi$
$338$	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	$-0.531560\pi$
$359$	−3.75107	−0.197974	−0.0989869	+	0.995089i	$0.531560\pi$
$360$	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	$-0.381390\pi$
$367$	13.9488	0.728119	0.364060	+	0.931376i	$0.381390\pi$
$368$	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	$-0.604637\pi$
$379$	−12.5700	−0.645676	−0.322838	+	0.946454i	$0.604637\pi$
$380$	−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	$-0.318927\pi$
$389$	21.2485	1.07734	0.538671	+	0.842516i	$0.318927\pi$
$390$	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	$-0.814299\pi$
$409$	−33.7573	−1.66919	−0.834597	+	0.550862i	$0.814299\pi$
$410$	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	$-0.403462\pi$
$421$	12.2557	0.597309	0.298654	+	0.954361i	$0.403462\pi$
$422$	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	$-0.428722\pi$
$433$	9.24157	0.444122	0.222061	+	0.975033i	$0.428722\pi$
$434$	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	$-0.741884\pi$
$443$	−28.9972	−1.37770	−0.688850	+	0.724904i	$0.741884\pi$
$444$	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	$-0.393471\pi$
$449$	13.9198	0.656914	0.328457	+	0.944519i	$0.393471\pi$
$450$	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	$-0.617736\pi$
$461$	−15.5235	−0.723003	−0.361502	+	0.932371i	$0.617736\pi$
$462$	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	$-0.469194\pi$
$467$	4.17634	0.193258	0.0966290	+	0.995320i	$0.469194\pi$
$468$	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	$-0.360740\pi$
$479$	18.5452	0.847351	0.423675	+	0.905814i	$0.360740\pi$
$480$	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	$-0.517188\pi$
$491$	−2.39192	−0.107946	−0.0539728	+	0.998542i	$0.517188\pi$
$492$	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	$-0.648288\pi$
$503$	−20.1486	−0.898381	−0.449191	+	0.893436i	$0.648288\pi$
$504$	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	$-0.297341\pi$
$541$	27.6565	1.18905	0.594523	+	0.804078i	$0.297341\pi$
$542$	−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	$-0.550227\pi$
$563$	−7.45712	−0.314280	−0.157140	+	0.987576i	$0.550227\pi$
$564$	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	$-0.620587\pi$
$569$	−17.6440	−0.739675	−0.369837	+	0.929097i	$0.620587\pi$
$570$	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	$-0.615693\pi$
$577$	−17.0793	−0.711023	−0.355511	+	0.934672i	$0.615693\pi$
$578$	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	$-0.537108\pi$
$599$	−5.69345	−0.232628	−0.116314	+	0.993212i	$0.537108\pi$
$600$	0	0
$601$	−8.96083	−0.365520	−0.182760	−	0.983158i	$-0.558503\pi$
$601$	−8.96083	−0.365520	−0.182760	+	0.983158i	$0.558503\pi$
$602$	−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	$-0.305124\pi$
$607$	28.3175	1.14937	0.574685	+	0.818374i	$0.305124\pi$
$608$	−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	$-0.460813\pi$
$617$	6.10052	0.245598	0.122799	+	0.992432i	$0.460813\pi$
$618$	0	0
$619$	0.710979	0.0285766	0.0142883	−	0.999898i	$-0.495452\pi$
$619$	0.710979	0.0285766	0.0142883	+	0.999898i	$0.495452\pi$
$620$	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	$-0.682525\pi$
$641$	−27.4704	−1.08501	−0.542507	+	0.840051i	$0.682525\pi$
$642$	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	$-0.633072\pi$
$653$	−20.7490	−0.811973	−0.405986	+	0.913879i	$0.633072\pi$
$654$	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	$-0.416194\pi$
$659$	13.3618	0.520504	0.260252	+	0.965541i	$0.416194\pi$
$660$	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	$-0.475907\pi$
$677$	3.93509	0.151238	0.0756189	+	0.997137i	$0.475907\pi$
$678$	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	$-0.467396\pi$
$727$	5.51391	0.204500	0.102250	+	0.994759i	$0.467396\pi$
$728$	−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	$-0.203617\pi$
$733$	43.4421	1.60457	0.802285	+	0.596941i	$0.203617\pi$
$734$	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	$-0.196999\pi$
$739$	44.2848	1.62904	0.814522	+	0.580133i	$0.196999\pi$
$740$	−11.8778	−0.436637
$741$	0	0
$742$	0.118468	0.00434910
$743$	35.4033	1.29882	0.649411	−	0.760438i	$-0.275016\pi$
$743$	35.4033	1.29882	0.649411	+	0.760438i	$0.275016\pi$
$744$	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	$-0.520663\pi$
$751$	−3.55544	−0.129740	−0.0648698	+	0.997894i	$0.520663\pi$
$752$	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	$-0.371564\pi$
$757$	21.6056	0.785268	0.392634	+	0.919695i	$0.371564\pi$
$758$	−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	$-0.501162\pi$
$769$	−0.202502	−0.00730241	−0.00365120	+	0.999993i	$0.501162\pi$
$770$	−0.133831	−0.00482294
$771$	0	0
$772$	22.6082	0.813687
$773$	14.5159	0.522100	0.261050	−	0.965325i	$-0.415931\pi$
$773$	14.5159	0.522100	0.261050	+	0.965325i	$0.415931\pi$
$774$	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	$-0.581850\pi$
$797$	−14.3592	−0.508628	−0.254314	+	0.967122i	$0.581850\pi$
$798$	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	$-0.333622\pi$
$809$	28.3982	0.998429	0.499214	+	0.866479i	$0.333622\pi$
$810$	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	$-0.510034\pi$
$823$	−1.80834	−0.0630349	−0.0315174	+	0.999503i	$0.510034\pi$
$824$	5.33589	0.185885
$825$	0	0
$826$	−0.164967	−0.00573992
$827$	22.9341	0.797497	0.398748	−	0.917060i	$-0.369445\pi$
$827$	22.9341	0.797497	0.398748	+	0.917060i	$0.369445\pi$
$828$	0	0
$829$	−9.99341	−0.347085	−0.173543	−	0.984826i	$-0.555521\pi$
$829$	−9.99341	−0.347085	−0.173543	+	0.984826i	$0.555521\pi$
$830$	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	$-0.636926\pi$
$839$	−24.1584	−0.834042	−0.417021	+	0.908897i	$0.636926\pi$
$840$	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	$-0.428767\pi$
$853$	12.9629	0.443842	0.221921	+	0.975065i	$0.428767\pi$
$854$	−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	$-0.525039\pi$
$859$	−4.60632	−0.157165	−0.0785827	+	0.996908i	$0.525039\pi$
$860$	−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	$-0.654938\pi$
$881$	−27.7676	−0.935514	−0.467757	+	0.883857i	$0.654938\pi$
$882$	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	$-0.363362\pi$
$887$	24.7909	0.832397	0.416198	+	0.909274i	$0.363362\pi$
$888$	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	$-0.760871\pi$
$907$	−44.0205	−1.46168	−0.730839	+	0.682550i	$0.760871\pi$
$908$	−49.3307	−1.63710
$909$	0	0
$910$	−0.200470	−0.00664550
$911$	−58.6374	−1.94275	−0.971373	−	0.237561i	$-0.923652\pi$
$911$	−58.6374	−1.94275	−0.971373	+	0.237561i	$0.923652\pi$
$912$	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	$-0.810550\pi$
$947$	−50.9638	−1.65610	−0.828051	+	0.560653i	$0.810550\pi$
$948$	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	$-0.200912\pi$
$967$	50.2105	1.61466	0.807330	+	0.590100i	$0.200912\pi$
$968$	1.70126	0.0546804
$969$	0	0
$970$	0.437294	0.0140407
$971$	32.5354	1.04411	0.522055	−	0.852912i	$-0.325166\pi$
$971$	32.5354	1.04411	0.522055	+	0.852912i	$0.325166\pi$
$972$	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	$-0.509093\pi$
$983$	−1.79101	−0.0571245	−0.0285622	+	0.999592i	$0.509093\pi$
$984$	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	$-0.684977\pi$
$997$	−34.6674	−1.09793	−0.548964	+	0.835846i	$0.684977\pi$
$998$	−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.bl.1.5		9
3.2	odd	2		2523.2.a.p.1.5		9
29.16	even	7		261.2.k.b.82.2		18
29.20	even	7		261.2.k.b.226.2		18
29.28	even	2		7569.2.a.bk.1.5		9
87.20	odd	14		87.2.g.b.52.2	✓	18
87.74	odd	14		87.2.g.b.82.2	yes	18
87.86	odd	2		2523.2.a.q.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.b.52.2	✓	18	87.20	odd	14
87.2.g.b.82.2	yes	18	87.74	odd	14
261.2.k.b.82.2		18	29.16	even	7
261.2.k.b.226.2		18	29.20	even	7
2523.2.a.p.1.5		9	3.2	odd	2
2523.2.a.q.1.5		9	87.86	odd	2
7569.2.a.bk.1.5		9	29.28	even	2
7569.2.a.bl.1.5		9	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)

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

Modular forms

Varieties

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Representations

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Database

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