LMFDB - Embedded newform 7569.2.a.bt.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:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 2 x^{10} + 38 x^{9} - 30 x^{8} - 90 x^{7} + 55 x^{6} + 90 x^{5} - 30 x^{4} + \cdots + 1 $ x^12 - 6x^11 + 2x^10 + 38x^9 - 30x^8 - 90x^7 + 55x^6 + 90x^5 - 30x^4 - 38x^3 + 2x^2 + 6*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
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			1.2
Root			$2.90759$ 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)

$f(q)$	$=$	$q-1.90759 q^{2} +1.63890 q^{4} +1.71673 q^{5} -1.24174 q^{7} +0.688829 q^{8} -3.27481 q^{10} -2.54849 q^{11} -6.09491 q^{13} +2.36873 q^{14} -4.59181 q^{16} +1.70189 q^{17} +0.426606 q^{19} +2.81354 q^{20} +4.86148 q^{22} +5.66024 q^{23} -2.05286 q^{25} +11.6266 q^{26} -2.03509 q^{28} +1.85090 q^{31} +7.38163 q^{32} -3.24652 q^{34} -2.13172 q^{35} -7.56191 q^{37} -0.813789 q^{38} +1.18253 q^{40} -6.80500 q^{41} +10.4809 q^{43} -4.17672 q^{44} -10.7974 q^{46} +12.0970 q^{47} -5.45809 q^{49} +3.91601 q^{50} -9.98896 q^{52} +3.20805 q^{53} -4.37506 q^{55} -0.855345 q^{56} +3.90939 q^{59} +14.5607 q^{61} -3.53077 q^{62} -4.89751 q^{64} -10.4633 q^{65} -2.00896 q^{67} +2.78924 q^{68} +4.06646 q^{70} +11.3543 q^{71} +6.86914 q^{73} +14.4250 q^{74} +0.699164 q^{76} +3.16456 q^{77} +5.87993 q^{79} -7.88287 q^{80} +12.9812 q^{82} -6.66375 q^{83} +2.92168 q^{85} -19.9932 q^{86} -1.75547 q^{88} +3.27600 q^{89} +7.56828 q^{91} +9.27658 q^{92} -23.0761 q^{94} +0.732364 q^{95} +2.49744 q^{97} +10.4118 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{2} + 8 q^{4} - 2 q^{5} - 10 q^{7} - 20 q^{10} + 14 q^{11} - 16 q^{13} - 4 q^{16} + 22 q^{17} - 16 q^{19} - 4 q^{20} + 12 q^{22} - 2 q^{23} - 2 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{31} + 16 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) $ 12 * q + 6 * q^2 + 8 * q^4 - 2 * q^5 - 10 * q^7 - 20 * q^10 + 14 * q^11 - 16 * q^13 - 4 * q^16 + 22 * q^17 - 16 * q^19 - 4 * q^20 + 12 * q^22 - 2 * q^23 - 2 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^31 + 16 * q^32 - 10 * q^34 + 14 * q^35 - 28 * q^37 - 16 * q^38 - 32 * q^40 - 14 * q^41 - 36 * q^43 + 26 * q^44 - 24 * q^46 + 44 * q^47 - 18 * q^49 + 24 * q^50 - 20 * q^52 + 18 * q^53 - 4 * q^55 - 10 * q^56 - 14 * q^59 - 24 * q^61 - 4 * q^62 - 18 * q^64 - 2 * q^65 - 18 * q^67 + 52 * q^68 + 36 * q^70 - 14 * q^71 - 44 * q^73 - 12 * q^74 - 56 * q^76 + 32 * q^77 - 24 * q^79 - 52 * q^80 + 52 * q^82 - 26 * q^83 - 48 * q^85 - 64 * q^86 - 10 * q^88 + 10 * q^89 - 10 * q^91 - 64 * q^92 - 8 * q^94 - 4 * q^95 - 28 * q^97 + 6 * 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 $

$2$	−1.90759	−1.34887	−0.674435	−	0.738334i	$-0.735613\pi$
$2$	−1.90759	−1.34887	−0.674435	+	0.738334i	$0.735613\pi$
$3$	0	0
$3$	0	0
$4$	1.63890	0.819451
$5$	1.71673	0.767743	0.383871	−	0.923387i	$-0.374591\pi$
$5$	1.71673	0.767743	0.383871	+	0.923387i	$0.374591\pi$
$6$	0	0
$7$	−1.24174	−0.469333	−0.234666	−	0.972076i	$-0.575400\pi$
$7$	−1.24174	−0.469333	−0.234666	+	0.972076i	$0.575400\pi$
$8$	0.688829	0.243538
$9$	0	0
$10$	−3.27481	−1.03559
$11$	−2.54849	−0.768399	−0.384199	−	0.923250i	$-0.625522\pi$
$11$	−2.54849	−0.768399	−0.384199	+	0.923250i	$0.625522\pi$
$12$	0	0
$13$	−6.09491	−1.69042	−0.845212	−	0.534431i	$-0.820526\pi$
$13$	−6.09491	−1.69042	−0.845212	+	0.534431i	$0.820526\pi$
$14$	2.36873	0.633069
$15$	0	0
$16$	−4.59181	−1.14795
$17$	1.70189	0.412770	0.206385	−	0.978471i	$-0.433830\pi$
$17$	1.70189	0.412770	0.206385	+	0.978471i	$0.433830\pi$
$18$	0	0
$19$	0.426606	0.0978700	0.0489350	−	0.998802i	$-0.484417\pi$
$19$	0.426606	0.0978700	0.0489350	+	0.998802i	$0.484417\pi$
$20$	2.81354	0.629127
$21$	0	0
$22$	4.86148	1.03647
$23$	5.66024	1.18024	0.590121	−	0.807315i	$-0.299080\pi$
$23$	5.66024	1.18024	0.590121	+	0.807315i	$0.299080\pi$
$24$	0	0
$25$	−2.05286	−0.410571
$26$	11.6266	2.28016
$27$	0	0
$28$	−2.03509	−0.384595
$29$	0	0
$29$	0	0
$30$	0	0
$31$	1.85090	0.332432	0.166216	−	0.986089i	$-0.446845\pi$
$31$	1.85090	0.332432	0.166216	+	0.986089i	$0.446845\pi$
$32$	7.38163	1.30490
$33$	0	0
$34$	−3.24652	−0.556773
$35$	−2.13172	−0.360327
$36$	0	0
$37$	−7.56191	−1.24317	−0.621585	−	0.783347i	$-0.713511\pi$
$37$	−7.56191	−1.24317	−0.621585	+	0.783347i	$0.713511\pi$
$38$	−0.813789	−0.132014
$39$	0	0
$40$	1.18253	0.186974
$41$	−6.80500	−1.06276	−0.531382	−	0.847133i	$-0.678327\pi$
$41$	−6.80500	−1.06276	−0.531382	+	0.847133i	$0.678327\pi$
$42$	0	0
$43$	10.4809	1.59832	0.799158	−	0.601120i	$-0.205279\pi$
$43$	10.4809	1.59832	0.799158	+	0.601120i	$0.205279\pi$
$44$	−4.17672	−0.629665
$45$	0	0
$46$	−10.7974	−1.59199
$47$	12.0970	1.76452	0.882262	−	0.470759i	$-0.156020\pi$
$47$	12.0970	1.76452	0.882262	+	0.470759i	$0.156020\pi$
$48$	0	0
$49$	−5.45809	−0.779727
$50$	3.91601	0.553807
$51$	0	0
$52$	−9.98896	−1.38522
$53$	3.20805	0.440660	0.220330	−	0.975425i	$-0.429287\pi$
$53$	3.20805	0.440660	0.220330	+	0.975425i	$0.429287\pi$
$54$	0	0
$55$	−4.37506	−0.589933
$56$	−0.855345	−0.114300
$57$	0	0
$58$	0	0
$59$	3.90939	0.508959	0.254479	−	0.967078i	$-0.418096\pi$
$59$	3.90939	0.508959	0.254479	+	0.967078i	$0.418096\pi$
$60$	0	0
$61$	14.5607	1.86430	0.932151	−	0.362069i	$-0.117930\pi$
$61$	14.5607	1.86430	0.932151	+	0.362069i	$0.117930\pi$
$62$	−3.53077	−0.448408
$63$	0	0
$64$	−4.89751	−0.612188
$65$	−10.4633	−1.29781
$66$	0	0
$67$	−2.00896	−0.245433	−0.122717	−	0.992442i	$-0.539161\pi$
$67$	−2.00896	−0.245433	−0.122717	+	0.992442i	$0.539161\pi$
$68$	2.78924	0.338244
$69$	0	0
$70$	4.06646	0.486034
$71$	11.3543	1.34751	0.673755	−	0.738955i	$-0.264680\pi$
$71$	11.3543	1.34751	0.673755	+	0.738955i	$0.264680\pi$
$72$	0	0
$73$	6.86914	0.803972	0.401986	−	0.915646i	$-0.368320\pi$
$73$	6.86914	0.803972	0.401986	+	0.915646i	$0.368320\pi$
$74$	14.4250	1.67687
$75$	0	0
$76$	0.699164	0.0801996
$77$	3.16456	0.360635
$78$	0	0
$79$	5.87993	0.661544	0.330772	−	0.943711i	$-0.392691\pi$
$79$	5.87993	0.661544	0.330772	+	0.943711i	$0.392691\pi$
$80$	−7.88287	−0.881331
$81$	0	0
$82$	12.9812	1.43353
$83$	−6.66375	−0.731441	−0.365721	−	0.930725i	$-0.619177\pi$
$83$	−6.66375	−0.731441	−0.365721	+	0.930725i	$0.619177\pi$
$84$	0	0
$85$	2.92168	0.316901
$86$	−19.9932	−2.15592
$87$	0	0
$88$	−1.75547	−0.187134
$89$	3.27600	0.347255	0.173628	−	0.984811i	$-0.444451\pi$
$89$	3.27600	0.347255	0.173628	+	0.984811i	$0.444451\pi$
$90$	0	0
$91$	7.56828	0.793372
$92$	9.27658	0.967150
$93$	0	0
$94$	−23.0761	−2.38011
$95$	0.732364	0.0751390
$96$	0	0
$97$	2.49744	0.253576	0.126788	−	0.991930i	$-0.459533\pi$
$97$	2.49744	0.253576	0.126788	+	0.991930i	$0.459533\pi$
$98$	10.4118	1.05175
$99$	0	0
$100$	−3.36443	−0.336443
$101$	17.2626	1.71770	0.858848	−	0.512230i	$-0.171181\pi$
$101$	17.2626	1.71770	0.858848	+	0.512230i	$0.171181\pi$
$102$	0	0
$103$	−17.6813	−1.74219	−0.871097	−	0.491112i	$-0.836591\pi$
$103$	−17.6813	−1.74219	−0.871097	+	0.491112i	$0.836591\pi$
$104$	−4.19835	−0.411682
$105$	0	0
$106$	−6.11965	−0.594393
$107$	−4.64732	−0.449273	−0.224637	−	0.974443i	$-0.572119\pi$
$107$	−4.64732	−0.449273	−0.224637	+	0.974443i	$0.572119\pi$
$108$	0	0
$109$	−0.941184	−0.0901491	−0.0450745	−	0.998984i	$-0.514353\pi$
$109$	−0.941184	−0.0901491	−0.0450745	+	0.998984i	$0.514353\pi$
$110$	8.34582	0.795742
$111$	0	0
$112$	5.70182	0.538771
$113$	−7.30986	−0.687654	−0.343827	−	0.939033i	$-0.611723\pi$
$113$	−7.30986	−0.687654	−0.343827	+	0.939033i	$0.611723\pi$
$114$	0	0
$115$	9.71708	0.906123
$116$	0	0
$117$	0	0
$118$	−7.45751	−0.686519
$119$	−2.11331	−0.193727
$120$	0	0
$121$	−4.50520	−0.409563
$122$	−27.7758	−2.51470
$123$	0	0
$124$	3.03345	0.272412
$125$	−12.1078	−1.08296
$126$	0	0
$127$	−7.45690	−0.661693	−0.330846	−	0.943685i	$-0.607334\pi$
$127$	−7.45690	−0.661693	−0.330846	+	0.943685i	$0.607334\pi$
$128$	−5.42081	−0.479137
$129$	0	0
$130$	19.9597	1.75058
$131$	−12.2316	−1.06868	−0.534342	−	0.845268i	$-0.679441\pi$
$131$	−12.2316	−1.06868	−0.534342	+	0.845268i	$0.679441\pi$
$132$	0	0
$133$	−0.529732	−0.0459336
$134$	3.83227	0.331057
$135$	0	0
$136$	1.17231	0.100525
$137$	−6.61576	−0.565223	−0.282611	−	0.959234i	$-0.591201\pi$
$137$	−6.61576	−0.565223	−0.282611	+	0.959234i	$0.591201\pi$
$138$	0	0
$139$	8.30734	0.704620	0.352310	−	0.935883i	$-0.385396\pi$
$139$	8.30734	0.704620	0.352310	+	0.935883i	$0.385396\pi$
$140$	−3.49368	−0.295270
$141$	0	0
$142$	−21.6594	−1.81762
$143$	15.5328	1.29892
$144$	0	0
$145$	0	0
$146$	−13.1035	−1.08445
$147$	0	0
$148$	−12.3932	−1.01872
$149$	−20.9098	−1.71300	−0.856500	−	0.516147i	$-0.827366\pi$
$149$	−20.9098	−1.71300	−0.856500	+	0.516147i	$0.827366\pi$
$150$	0	0
$151$	−23.4397	−1.90750	−0.953748	−	0.300608i	$-0.902810\pi$
$151$	−23.4397	−1.90750	−0.953748	+	0.300608i	$0.902810\pi$
$152$	0.293858	0.0238351
$153$	0	0
$154$	−6.03668	−0.486450
$155$	3.17749	0.255222
$156$	0	0
$157$	−15.7821	−1.25955	−0.629775	−	0.776778i	$-0.716853\pi$
$157$	−15.7821	−1.25955	−0.629775	+	0.776778i	$0.716853\pi$
$158$	−11.2165	−0.892337
$159$	0	0
$160$	12.6722	1.00183
$161$	−7.02854	−0.553927
$162$	0	0
$163$	−10.0692	−0.788679	−0.394340	−	0.918965i	$-0.629027\pi$
$163$	−10.0692	−0.788679	−0.394340	+	0.918965i	$0.629027\pi$
$164$	−11.1527	−0.870882
$165$	0	0
$166$	12.7117	0.986619
$167$	−0.303139	−0.0234576	−0.0117288	−	0.999931i	$-0.503733\pi$
$167$	−0.303139	−0.0234576	−0.0117288	+	0.999931i	$0.503733\pi$
$168$	0	0
$169$	24.1479	1.85753
$170$	−5.57338	−0.427458
$171$	0	0
$172$	17.1771	1.30974
$173$	−9.95056	−0.756527	−0.378264	−	0.925698i	$-0.623479\pi$
$173$	−9.95056	−0.756527	−0.378264	+	0.925698i	$0.623479\pi$
$174$	0	0
$175$	2.54911	0.192695
$176$	11.7022	0.882084
$177$	0	0
$178$	−6.24927	−0.468402
$179$	4.67839	0.349679	0.174839	−	0.984597i	$-0.444059\pi$
$179$	4.67839	0.349679	0.174839	+	0.984597i	$0.444059\pi$
$180$	0	0
$181$	−3.54773	−0.263700	−0.131850	−	0.991270i	$-0.542092\pi$
$181$	−3.54773	−0.263700	−0.131850	+	0.991270i	$0.542092\pi$
$182$	−14.4372	−1.07016
$183$	0	0
$184$	3.89894	0.287434
$185$	−12.9817	−0.954435
$186$	0	0
$187$	−4.33726	−0.317172
$188$	19.8257	1.44594
$189$	0	0
$190$	−1.39705	−0.101353
$191$	−4.17518	−0.302106	−0.151053	−	0.988526i	$-0.548266\pi$
$191$	−4.17518	−0.302106	−0.151053	+	0.988526i	$0.548266\pi$
$192$	0	0
$193$	−13.3631	−0.961894	−0.480947	−	0.876750i	$-0.659707\pi$
$193$	−13.3631	−0.961894	−0.480947	+	0.876750i	$0.659707\pi$
$194$	−4.76408	−0.342041
$195$	0	0
$196$	−8.94526	−0.638947
$197$	2.59538	0.184913	0.0924564	−	0.995717i	$-0.470528\pi$
$197$	2.59538	0.184913	0.0924564	+	0.995717i	$0.470528\pi$
$198$	0	0
$199$	−6.19886	−0.439425	−0.219713	−	0.975565i	$-0.570512\pi$
$199$	−6.19886	−0.439425	−0.219713	+	0.975565i	$0.570512\pi$
$200$	−1.41407	−0.0999896
$201$	0	0
$202$	−32.9300	−2.31695
$203$	0	0
$204$	0	0
$205$	−11.6823	−0.815929
$206$	33.7287	2.34999
$207$	0	0
$208$	27.9866	1.94052
$209$	−1.08720	−0.0752032
$210$	0	0
$211$	−20.9045	−1.43912	−0.719562	−	0.694428i	$-0.755658\pi$
$211$	−20.9045	−1.43912	−0.719562	+	0.694428i	$0.755658\pi$
$212$	5.25768	0.361099
$213$	0	0
$214$	8.86518	0.606011
$215$	17.9928	1.22710
$216$	0	0
$217$	−2.29834	−0.156021
$218$	1.79539	0.121599
$219$	0	0
$220$	−7.17028	−0.483421
$221$	−10.3729	−0.697756
$222$	0	0
$223$	22.2000	1.48662	0.743312	−	0.668945i	$-0.233254\pi$
$223$	22.2000	1.48662	0.743312	+	0.668945i	$0.233254\pi$
$224$	−9.16605	−0.612432
$225$	0	0
$226$	13.9442	0.927556
$227$	−4.91192	−0.326015	−0.163008	−	0.986625i	$-0.552120\pi$
$227$	−4.91192	−0.326015	−0.163008	+	0.986625i	$0.552120\pi$
$228$	0	0
$229$	21.6047	1.42768	0.713840	−	0.700309i	$-0.246955\pi$
$229$	21.6047	1.42768	0.713840	+	0.700309i	$0.246955\pi$
$230$	−18.5362	−1.22224
$231$	0	0
$232$	0	0
$233$	−21.5539	−1.41204	−0.706021	−	0.708191i	$-0.749512\pi$
$233$	−21.5539	−1.41204	−0.706021	+	0.708191i	$0.749512\pi$
$234$	0	0
$235$	20.7672	1.35470
$236$	6.40710	0.417066
$237$	0	0
$238$	4.03132	0.261312
$239$	−4.73822	−0.306490	−0.153245	−	0.988188i	$-0.548972\pi$
$239$	−4.73822	−0.306490	−0.153245	+	0.988188i	$0.548972\pi$
$240$	0	0
$241$	5.81935	0.374857	0.187429	−	0.982278i	$-0.439985\pi$
$241$	5.81935	0.374857	0.187429	+	0.982278i	$0.439985\pi$
$242$	8.59407	0.552448
$243$	0	0
$244$	23.8635	1.52770
$245$	−9.37003	−0.598629
$246$	0	0
$247$	−2.60012	−0.165442
$248$	1.27496	0.0809598
$249$	0	0
$250$	23.0967	1.46077
$251$	14.0940	0.889606	0.444803	−	0.895629i	$-0.353274\pi$
$251$	14.0940	0.889606	0.444803	+	0.895629i	$0.353274\pi$
$252$	0	0
$253$	−14.4251	−0.906897
$254$	14.2247	0.892537
$255$	0	0
$256$	20.1357	1.25848
$257$	8.61361	0.537302	0.268651	−	0.963238i	$-0.413422\pi$
$257$	8.61361	0.537302	0.268651	+	0.963238i	$0.413422\pi$
$258$	0	0
$259$	9.38991	0.583461
$260$	−17.1483	−1.06349
$261$	0	0
$262$	23.3330	1.44152
$263$	−7.52746	−0.464163	−0.232082	−	0.972696i	$-0.574554\pi$
$263$	−7.52746	−0.464163	−0.232082	+	0.972696i	$0.574554\pi$
$264$	0	0
$265$	5.50734	0.338313
$266$	1.01051	0.0619585
$267$	0	0
$268$	−3.29248	−0.201120
$269$	−13.3188	−0.812060	−0.406030	−	0.913860i	$-0.633087\pi$
$269$	−13.3188	−0.812060	−0.406030	+	0.913860i	$0.633087\pi$
$270$	0	0
$271$	16.1608	0.981696	0.490848	−	0.871245i	$-0.336687\pi$
$271$	16.1608	0.981696	0.490848	+	0.871245i	$0.336687\pi$
$272$	−7.81476	−0.473840
$273$	0	0
$274$	12.6202	0.762412
$275$	5.23168	0.315482
$276$	0	0
$277$	21.5548	1.29510	0.647552	−	0.762021i	$-0.275793\pi$
$277$	21.5548	1.29510	0.647552	+	0.762021i	$0.275793\pi$
$278$	−15.8470	−0.950440
$279$	0	0
$280$	−1.46839	−0.0877533
$281$	1.08043	0.0644528	0.0322264	−	0.999481i	$-0.489740\pi$
$281$	1.08043	0.0644528	0.0322264	+	0.999481i	$0.489740\pi$
$282$	0	0
$283$	0.280757	0.0166893	0.00834464	−	0.999965i	$-0.497344\pi$
$283$	0.280757	0.0166893	0.00834464	+	0.999965i	$0.497344\pi$
$284$	18.6086	1.10422
$285$	0	0
$286$	−29.6303	−1.75207
$287$	8.45004	0.498790
$288$	0	0
$289$	−14.1036	−0.829621
$290$	0	0
$291$	0	0
$292$	11.2578	0.658816
$293$	−1.28096	−0.0748342	−0.0374171	−	0.999300i	$-0.511913\pi$
$293$	−1.28096	−0.0748342	−0.0374171	+	0.999300i	$0.511913\pi$
$294$	0	0
$295$	6.71134	0.390749
$296$	−5.20886	−0.302759
$297$	0	0
$298$	39.8874	2.31061
$299$	−34.4987	−1.99511
$300$	0	0
$301$	−13.0145	−0.750143
$302$	44.7133	2.57296
$303$	0	0
$304$	−1.95889	−0.112350
$305$	24.9967	1.43130
$306$	0	0
$307$	−9.48636	−0.541415	−0.270708	−	0.962662i	$-0.587258\pi$
$307$	−9.48636	−0.541415	−0.270708	+	0.962662i	$0.587258\pi$
$308$	5.18640	0.295522
$309$	0	0
$310$	−6.06135	−0.344262
$311$	−10.7333	−0.608632	−0.304316	−	0.952571i	$-0.598428\pi$
$311$	−10.7333	−0.608632	−0.304316	+	0.952571i	$0.598428\pi$
$312$	0	0
$313$	6.54110	0.369725	0.184862	−	0.982764i	$-0.440816\pi$
$313$	6.54110	0.369725	0.184862	+	0.982764i	$0.440816\pi$
$314$	30.1058	1.69897
$315$	0	0
$316$	9.63663	0.542103
$317$	10.5645	0.593359	0.296680	−	0.954977i	$-0.404121\pi$
$317$	10.5645	0.593359	0.296680	+	0.954977i	$0.404121\pi$
$318$	0	0
$319$	0	0
$320$	−8.40767	−0.470003
$321$	0	0
$322$	13.4076	0.747175
$323$	0.726037	0.0403978
$324$	0	0
$325$	12.5120	0.694039
$326$	19.2079	1.06383
$327$	0	0
$328$	−4.68748	−0.258823
$329$	−15.0213	−0.828149
$330$	0	0
$331$	−23.0562	−1.26728	−0.633641	−	0.773627i	$-0.718440\pi$
$331$	−23.0562	−1.26728	−0.633641	+	0.773627i	$0.718440\pi$
$332$	−10.9212	−0.599380
$333$	0	0
$334$	0.578264	0.0316412
$335$	−3.44883	−0.188429
$336$	0	0
$337$	−22.3538	−1.21769	−0.608844	−	0.793290i	$-0.708366\pi$
$337$	−22.3538	−1.21769	−0.608844	+	0.793290i	$0.708366\pi$
$338$	−46.0644	−2.50557
$339$	0	0
$340$	4.78835	0.259685
$341$	−4.71701	−0.255440
$342$	0	0
$343$	15.4697	0.835284
$344$	7.21952	0.389251
$345$	0	0
$346$	18.9816	1.02046
$347$	17.5923	0.944404	0.472202	−	0.881490i	$-0.343459\pi$
$347$	17.5923	0.944404	0.472202	+	0.881490i	$0.343459\pi$
$348$	0	0
$349$	−8.06121	−0.431507	−0.215753	−	0.976448i	$-0.569221\pi$
$349$	−8.06121	−0.431507	−0.215753	+	0.976448i	$0.569221\pi$
$350$	−4.86266	−0.259920
$351$	0	0
$352$	−18.8120	−1.00268
$353$	16.6752	0.887532	0.443766	−	0.896143i	$-0.353642\pi$
$353$	16.6752	0.887532	0.443766	+	0.896143i	$0.353642\pi$
$354$	0	0
$355$	19.4922	1.03454
$356$	5.36904	0.284559
$357$	0	0
$358$	−8.92444	−0.471671
$359$	−13.1953	−0.696419	−0.348210	−	0.937417i	$-0.613210\pi$
$359$	−13.1953	−0.696419	−0.348210	+	0.937417i	$0.613210\pi$
$360$	0	0
$361$	−18.8180	−0.990421
$362$	6.76761	0.355698
$363$	0	0
$364$	12.4037	0.650129
$365$	11.7924	0.617244
$366$	0	0
$367$	−1.47179	−0.0768269	−0.0384135	−	0.999262i	$-0.512230\pi$
$367$	−1.47179	−0.0768269	−0.0384135	+	0.999262i	$0.512230\pi$
$368$	−25.9907	−1.35486
$369$	0	0
$370$	24.7638	1.28741
$371$	−3.98356	−0.206816
$372$	0	0
$373$	17.2512	0.893235	0.446617	−	0.894725i	$-0.352629\pi$
$373$	17.2512	0.893235	0.446617	+	0.894725i	$0.352629\pi$
$374$	8.27371	0.427824
$375$	0	0
$376$	8.33274	0.429728
$377$	0	0
$378$	0	0
$379$	−7.50752	−0.385635	−0.192818	−	0.981235i	$-0.561763\pi$
$379$	−7.50752	−0.385635	−0.192818	+	0.981235i	$0.561763\pi$
$380$	1.20027	0.0615727
$381$	0	0
$382$	7.96454	0.407501
$383$	−2.39899	−0.122583	−0.0612915	−	0.998120i	$-0.519522\pi$
$383$	−2.39899	−0.122583	−0.0612915	+	0.998120i	$0.519522\pi$
$384$	0	0
$385$	5.43268	0.276875
$386$	25.4912	1.29747
$387$	0	0
$388$	4.09305	0.207793
$389$	26.6147	1.34942	0.674709	−	0.738084i	$-0.264269\pi$
$389$	26.6147	1.34942	0.674709	+	0.738084i	$0.264269\pi$
$390$	0	0
$391$	9.63313	0.487168
$392$	−3.75969	−0.189893
$393$	0	0
$394$	−4.95091	−0.249423
$395$	10.0942	0.507896
$396$	0	0
$397$	−2.71330	−0.136176	−0.0680882	−	0.997679i	$-0.521690\pi$
$397$	−2.71330	−0.136176	−0.0680882	+	0.997679i	$0.521690\pi$
$398$	11.8249	0.592728
$399$	0	0
$400$	9.42631	0.471316
$401$	7.43927	0.371499	0.185750	−	0.982597i	$-0.440529\pi$
$401$	7.43927	0.371499	0.185750	+	0.982597i	$0.440529\pi$
$402$	0	0
$403$	−11.2811	−0.561951
$404$	28.2917	1.40757
$405$	0	0
$406$	0	0
$407$	19.2714	0.955250
$408$	0	0
$409$	−2.97967	−0.147335	−0.0736675	−	0.997283i	$-0.523470\pi$
$409$	−2.97967	−0.147335	−0.0736675	+	0.997283i	$0.523470\pi$
$410$	22.2851	1.10058
$411$	0	0
$412$	−28.9780	−1.42764
$413$	−4.85443	−0.238871
$414$	0	0
$415$	−11.4398	−0.561559
$416$	−44.9904	−2.20583
$417$	0	0
$418$	2.07393	0.101439
$419$	−35.1174	−1.71560	−0.857798	−	0.513987i	$-0.828168\pi$
$419$	−35.1174	−1.71560	−0.857798	+	0.513987i	$0.828168\pi$
$420$	0	0
$421$	−29.0918	−1.41785	−0.708923	−	0.705286i	$-0.750819\pi$
$421$	−29.0918	−1.41785	−0.708923	+	0.705286i	$0.750819\pi$
$422$	39.8772	1.94119
$423$	0	0
$424$	2.20980	0.107317
$425$	−3.49374	−0.169471
$426$	0	0
$427$	−18.0805	−0.874979
$428$	−7.61649	−0.368157
$429$	0	0
$430$	−34.3228	−1.65519
$431$	−20.9780	−1.01047	−0.505236	−	0.862981i	$-0.668595\pi$
$431$	−20.9780	−1.01047	−0.505236	+	0.862981i	$0.668595\pi$
$432$	0	0
$433$	−28.3258	−1.36125	−0.680626	−	0.732631i	$-0.738292\pi$
$433$	−28.3258	−1.36125	−0.680626	+	0.732631i	$0.738292\pi$
$434$	4.38429	0.210452
$435$	0	0
$436$	−1.54251	−0.0738727
$437$	2.41469	0.115510
$438$	0	0
$439$	−16.2271	−0.774475	−0.387238	−	0.921980i	$-0.626571\pi$
$439$	−16.2271	−0.774475	−0.387238	+	0.921980i	$0.626571\pi$
$440$	−3.01367	−0.143671
$441$	0	0
$442$	19.7872	0.941182
$443$	14.8967	0.707765	0.353882	−	0.935290i	$-0.384861\pi$
$443$	14.8967	0.707765	0.353882	+	0.935290i	$0.384861\pi$
$444$	0	0
$445$	5.62399	0.266603
$446$	−42.3486	−2.00526
$447$	0	0
$448$	6.08142	0.287320
$449$	6.90118	0.325687	0.162843	−	0.986652i	$-0.447933\pi$
$449$	6.90118	0.325687	0.162843	+	0.986652i	$0.447933\pi$
$450$	0	0
$451$	17.3425	0.816626
$452$	−11.9801	−0.563499
$453$	0	0
$454$	9.36993	0.439752
$455$	12.9927	0.609105
$456$	0	0
$457$	−36.5848	−1.71136	−0.855682	−	0.517503i	$-0.826862\pi$
$457$	−36.5848	−1.71136	−0.855682	+	0.517503i	$0.826862\pi$
$458$	−41.2129	−1.92575
$459$	0	0
$460$	15.9253	0.742523
$461$	−6.43285	−0.299608	−0.149804	−	0.988716i	$-0.547864\pi$
$461$	−6.43285	−0.299608	−0.149804	+	0.988716i	$0.547864\pi$
$462$	0	0
$463$	−5.95015	−0.276527	−0.138263	−	0.990395i	$-0.544152\pi$
$463$	−5.95015	−0.276527	−0.138263	+	0.990395i	$0.544152\pi$
$464$	0	0
$465$	0	0
$466$	41.1160	1.90466
$467$	−24.1307	−1.11663	−0.558317	−	0.829628i	$-0.688553\pi$
$467$	−24.1307	−1.11663	−0.558317	+	0.829628i	$0.688553\pi$
$468$	0	0
$469$	2.49460	0.115190
$470$	−39.6152	−1.82731
$471$	0	0
$472$	2.69290	0.123951
$473$	−26.7104	−1.22814
$474$	0	0
$475$	−0.875759	−0.0401826
$476$	−3.46350	−0.158749
$477$	0	0
$478$	9.03859	0.413416
$479$	−14.2737	−0.652182	−0.326091	−	0.945338i	$-0.605732\pi$
$479$	−14.2737	−0.652182	−0.326091	+	0.945338i	$0.605732\pi$
$480$	0	0
$481$	46.0891	2.10148
$482$	−11.1009	−0.505634
$483$	0	0
$484$	−7.38357	−0.335617
$485$	4.28741	0.194681
$486$	0	0
$487$	1.92761	0.0873482	0.0436741	−	0.999046i	$-0.486094\pi$
$487$	1.92761	0.0873482	0.0436741	+	0.999046i	$0.486094\pi$
$488$	10.0298	0.454028
$489$	0	0
$490$	17.8742	0.807473
$491$	10.5430	0.475798	0.237899	−	0.971290i	$-0.423541\pi$
$491$	10.5430	0.475798	0.237899	+	0.971290i	$0.423541\pi$
$492$	0	0
$493$	0	0
$494$	4.95997	0.223160
$495$	0	0
$496$	−8.49899	−0.381616
$497$	−14.0991	−0.632431
$498$	0	0
$499$	−15.5558	−0.696373	−0.348186	−	0.937425i	$-0.613202\pi$
$499$	−15.5558	−0.696373	−0.348186	+	0.937425i	$0.613202\pi$
$500$	−19.8435	−0.887429
$501$	0	0
$502$	−26.8856	−1.19996
$503$	37.7207	1.68188	0.840941	−	0.541127i	$-0.182002\pi$
$503$	37.7207	1.68188	0.840941	+	0.541127i	$0.182002\pi$
$504$	0	0
$505$	29.6352	1.31875
$506$	27.5171	1.22329
$507$	0	0
$508$	−12.2211	−0.542224
$509$	−2.04844	−0.0907953	−0.0453977	−	0.998969i	$-0.514455\pi$
$509$	−2.04844	−0.0907953	−0.0453977	+	0.998969i	$0.514455\pi$
$510$	0	0
$511$	−8.52968	−0.377331
$512$	−27.5691	−1.21839
$513$	0	0
$514$	−16.4312	−0.724751
$515$	−30.3540	−1.33756
$516$	0	0
$517$	−30.8290	−1.35586
$518$	−17.9121	−0.787013
$519$	0	0
$520$	−7.20741	−0.316066
$521$	−14.7934	−0.648109	−0.324055	−	0.946038i	$-0.605046\pi$
$521$	−14.7934	−0.648109	−0.324055	+	0.946038i	$0.605046\pi$
$522$	0	0
$523$	−39.6013	−1.73164	−0.865821	−	0.500354i	$-0.833203\pi$
$523$	−39.6013	−1.73164	−0.865821	+	0.500354i	$0.833203\pi$
$524$	−20.0465	−0.875734
$525$	0	0
$526$	14.3593	0.626096
$527$	3.15004	0.137218
$528$	0	0
$529$	9.03835	0.392972
$530$	−10.5058	−0.456341
$531$	0	0
$532$	−0.868179	−0.0376403
$533$	41.4759	1.79652
$534$	0	0
$535$	−7.97817	−0.344926
$536$	−1.38383	−0.0597722
$537$	0	0
$538$	25.4068	1.09536
$539$	13.9099	0.599141
$540$	0	0
$541$	16.1188	0.693003	0.346501	−	0.938049i	$-0.387370\pi$
$541$	16.1188	0.693003	0.346501	+	0.938049i	$0.387370\pi$
$542$	−30.8281	−1.32418
$543$	0	0
$544$	12.5627	0.538623
$545$	−1.61575	−0.0692113
$546$	0	0
$547$	0.445342	0.0190414	0.00952072	−	0.999955i	$-0.496969\pi$
$547$	0.445342	0.0190414	0.00952072	+	0.999955i	$0.496969\pi$
$548$	−10.8426	−0.463172
$549$	0	0
$550$	−9.97990	−0.425545
$551$	0	0
$552$	0	0
$553$	−7.30134	−0.310484
$554$	−41.1178	−1.74693
$555$	0	0
$556$	13.6149	0.577401
$557$	42.0616	1.78221	0.891103	−	0.453802i	$-0.149933\pi$
$557$	42.0616	1.78221	0.891103	+	0.453802i	$0.149933\pi$
$558$	0	0
$559$	−63.8799	−2.70183
$560$	9.78846	0.413638
$561$	0	0
$562$	−2.06101	−0.0869384
$563$	−29.9081	−1.26048	−0.630239	−	0.776401i	$-0.717043\pi$
$563$	−29.9081	−1.26048	−0.630239	+	0.776401i	$0.717043\pi$
$564$	0	0
$565$	−12.5490	−0.527942
$566$	−0.535570	−0.0225117
$567$	0	0
$568$	7.82119	0.328170
$569$	−36.4098	−1.52638	−0.763189	−	0.646175i	$-0.776368\pi$
$569$	−36.4098	−1.52638	−0.763189	+	0.646175i	$0.776368\pi$
$570$	0	0
$571$	−42.0605	−1.76018	−0.880088	−	0.474810i	$-0.842517\pi$
$571$	−42.0605	−1.76018	−0.880088	+	0.474810i	$0.842517\pi$
$572$	25.4568	1.06440
$573$	0	0
$574$	−16.1192	−0.672803
$575$	−11.6197	−0.484573
$576$	0	0
$577$	20.2054	0.841162	0.420581	−	0.907255i	$-0.361826\pi$
$577$	20.2054	0.841162	0.420581	+	0.907255i	$0.361826\pi$
$578$	26.9038	1.11905
$579$	0	0
$580$	0	0
$581$	8.27463	0.343289
$582$	0	0
$583$	−8.17569	−0.338602
$584$	4.73167	0.195798
$585$	0	0
$586$	2.44354	0.100942
$587$	38.4897	1.58864	0.794319	−	0.607501i	$-0.207828\pi$
$587$	38.4897	1.58864	0.794319	+	0.607501i	$0.207828\pi$
$588$	0	0
$589$	0.789606	0.0325351
$590$	−12.8025	−0.527070
$591$	0	0
$592$	34.7228	1.42710
$593$	37.0241	1.52040	0.760198	−	0.649691i	$-0.225102\pi$
$593$	37.0241	1.52040	0.760198	+	0.649691i	$0.225102\pi$
$594$	0	0
$595$	−3.62797	−0.148732
$596$	−34.2691	−1.40372
$597$	0	0
$598$	65.8093	2.69114
$599$	−43.8397	−1.79124	−0.895621	−	0.444818i	$-0.853268\pi$
$599$	−43.8397	−1.79124	−0.895621	+	0.444818i	$0.853268\pi$
$600$	0	0
$601$	31.2193	1.27346	0.636731	−	0.771086i	$-0.280286\pi$
$601$	31.2193	1.27346	0.636731	+	0.771086i	$0.280286\pi$
$602$	24.8263	1.01185
$603$	0	0
$604$	−38.4153	−1.56310
$605$	−7.73419	−0.314439
$606$	0	0
$607$	0.493466	0.0200291	0.0100146	−	0.999950i	$-0.496812\pi$
$607$	0.493466	0.0200291	0.0100146	+	0.999950i	$0.496812\pi$
$608$	3.14904	0.127711
$609$	0	0
$610$	−47.6834	−1.93064
$611$	−73.7299	−2.98279
$612$	0	0
$613$	−16.2384	−0.655864	−0.327932	−	0.944701i	$-0.606352\pi$
$613$	−16.2384	−0.655864	−0.327932	+	0.944701i	$0.606352\pi$
$614$	18.0961	0.730299
$615$	0	0
$616$	2.17984	0.0878282
$617$	25.7102	1.03506	0.517528	−	0.855666i	$-0.326852\pi$
$617$	25.7102	1.03506	0.517528	+	0.855666i	$0.326852\pi$
$618$	0	0
$619$	17.6792	0.710585	0.355293	−	0.934755i	$-0.384381\pi$
$619$	17.6792	0.710585	0.355293	+	0.934755i	$0.384381\pi$
$620$	5.20759	0.209142
$621$	0	0
$622$	20.4748	0.820965
$623$	−4.06794	−0.162978
$624$	0	0
$625$	−10.5215	−0.420860
$626$	−12.4777	−0.498711
$627$	0	0
$628$	−25.8653	−1.03214
$629$	−12.8696	−0.513143
$630$	0	0
$631$	−24.9109	−0.991688	−0.495844	−	0.868412i	$-0.665141\pi$
$631$	−24.9109	−0.991688	−0.495844	+	0.868412i	$0.665141\pi$
$632$	4.05027	0.161111
$633$	0	0
$634$	−20.1527	−0.800365
$635$	−12.8014	−0.508010
$636$	0	0
$637$	33.2665	1.31807
$638$	0	0
$639$	0	0
$640$	−9.30605	−0.367854
$641$	22.3113	0.881242	0.440621	−	0.897693i	$-0.354758\pi$
$641$	22.3113	0.881242	0.440621	+	0.897693i	$0.354758\pi$
$642$	0	0
$643$	−31.8680	−1.25675	−0.628375	−	0.777911i	$-0.716280\pi$
$643$	−31.8680	−1.25675	−0.628375	+	0.777911i	$0.716280\pi$
$644$	−11.5191	−0.453915
$645$	0	0
$646$	−1.38498	−0.0544914
$647$	39.4579	1.55125	0.775625	−	0.631194i	$-0.217435\pi$
$647$	39.4579	1.55125	0.775625	+	0.631194i	$0.217435\pi$
$648$	0	0
$649$	−9.96303	−0.391083
$650$	−23.8677	−0.936169
$651$	0	0
$652$	−16.5024	−0.646284
$653$	−3.48476	−0.136369	−0.0681845	−	0.997673i	$-0.521721\pi$
$653$	−3.48476	−0.136369	−0.0681845	+	0.997673i	$0.521721\pi$
$654$	0	0
$655$	−20.9984	−0.820475
$656$	31.2473	1.22000
$657$	0	0
$658$	28.6544	1.11707
$659$	22.3709	0.871446	0.435723	−	0.900081i	$-0.356493\pi$
$659$	22.3709	0.871446	0.435723	+	0.900081i	$0.356493\pi$
$660$	0	0
$661$	2.43331	0.0946449	0.0473225	−	0.998880i	$-0.484931\pi$
$661$	2.43331	0.0946449	0.0473225	+	0.998880i	$0.484931\pi$
$662$	43.9817	1.70940
$663$	0	0
$664$	−4.59018	−0.178134
$665$	−0.909405	−0.0352652
$666$	0	0
$667$	0	0
$668$	−0.496814	−0.0192223
$669$	0	0
$670$	6.57895	0.254167
$671$	−37.1077	−1.43253
$672$	0	0
$673$	29.5001	1.13715	0.568573	−	0.822633i	$-0.307496\pi$
$673$	29.5001	1.13715	0.568573	+	0.822633i	$0.307496\pi$
$674$	42.6418	1.64250
$675$	0	0
$676$	39.5761	1.52216
$677$	−2.01910	−0.0776003	−0.0388002	−	0.999247i	$-0.512354\pi$
$677$	−2.01910	−0.0776003	−0.0388002	+	0.999247i	$0.512354\pi$
$678$	0	0
$679$	−3.10116	−0.119012
$680$	2.01254	0.0771774
$681$	0	0
$682$	8.99812	0.344556
$683$	24.2990	0.929776	0.464888	−	0.885370i	$-0.346095\pi$
$683$	24.2990	0.929776	0.464888	+	0.885370i	$0.346095\pi$
$684$	0	0
$685$	−11.3574	−0.433946
$686$	−29.5098	−1.12669
$687$	0	0
$688$	−48.1261	−1.83479
$689$	−19.5528	−0.744902
$690$	0	0
$691$	−18.4208	−0.700758	−0.350379	−	0.936608i	$-0.613947\pi$
$691$	−18.4208	−0.700758	−0.350379	+	0.936608i	$0.613947\pi$
$692$	−16.3080	−0.619937
$693$	0	0
$694$	−33.5589	−1.27388
$695$	14.2614	0.540967
$696$	0	0
$697$	−11.5814	−0.438677
$698$	15.3775	0.582047
$699$	0	0
$700$	4.17774	0.157904
$701$	−22.7157	−0.857961	−0.428981	−	0.903314i	$-0.641127\pi$
$701$	−22.7157	−0.857961	−0.428981	+	0.903314i	$0.641127\pi$
$702$	0	0
$703$	−3.22595	−0.121669
$704$	12.4812	0.470405
$705$	0	0
$706$	−31.8095	−1.19716
$707$	−21.4357	−0.806171
$708$	0	0
$709$	41.0465	1.54153	0.770767	−	0.637117i	$-0.219873\pi$
$709$	41.0465	1.54153	0.770767	+	0.637117i	$0.219873\pi$
$710$	−37.1832	−1.39546
$711$	0	0
$712$	2.25660	0.0845698
$713$	10.4766	0.392350
$714$	0	0
$715$	26.6656	0.997236
$716$	7.66741	0.286545
$717$	0	0
$718$	25.1712	0.939379
$719$	9.71121	0.362167	0.181084	−	0.983468i	$-0.442040\pi$
$719$	9.71121	0.362167	0.181084	+	0.983468i	$0.442040\pi$
$720$	0	0
$721$	21.9556	0.817669
$722$	35.8971	1.33595
$723$	0	0
$724$	−5.81437	−0.216089
$725$	0	0
$726$	0	0
$727$	11.2890	0.418687	0.209344	−	0.977842i	$-0.432867\pi$
$727$	11.2890	0.418687	0.209344	+	0.977842i	$0.432867\pi$
$728$	5.21325	0.193216
$729$	0	0
$730$	−22.4951	−0.832582
$731$	17.8373	0.659737
$732$	0	0
$733$	8.47930	0.313190	0.156595	−	0.987663i	$-0.449948\pi$
$733$	8.47930	0.313190	0.156595	+	0.987663i	$0.449948\pi$
$734$	2.80758	0.103630
$735$	0	0
$736$	41.7818	1.54010
$737$	5.11981	0.188590
$738$	0	0
$739$	−11.4841	−0.422449	−0.211225	−	0.977438i	$-0.567745\pi$
$739$	−11.4841	−0.422449	−0.211225	+	0.977438i	$0.567745\pi$
$740$	−21.2757	−0.782112
$741$	0	0
$742$	7.59900	0.278968
$743$	−19.9907	−0.733387	−0.366694	−	0.930342i	$-0.619510\pi$
$743$	−19.9907	−0.733387	−0.366694	+	0.930342i	$0.619510\pi$
$744$	0	0
$745$	−35.8964	−1.31514
$746$	−32.9083	−1.20486
$747$	0	0
$748$	−7.10834	−0.259907
$749$	5.77075	0.210859
$750$	0	0
$751$	−24.5548	−0.896016	−0.448008	−	0.894030i	$-0.647866\pi$
$751$	−24.5548	−0.896016	−0.448008	+	0.894030i	$0.647866\pi$
$752$	−55.5469	−2.02559
$753$	0	0
$754$	0	0
$755$	−40.2395	−1.46447
$756$	0	0
$757$	−28.1411	−1.02281	−0.511403	−	0.859341i	$-0.670874\pi$
$757$	−28.1411	−1.02281	−0.511403	+	0.859341i	$0.670874\pi$
$758$	14.3213	0.520172
$759$	0	0
$760$	0.504474	0.0182992
$761$	−20.9629	−0.759904	−0.379952	−	0.925006i	$-0.624059\pi$
$761$	−20.9629	−0.759904	−0.379952	+	0.925006i	$0.624059\pi$
$762$	0	0
$763$	1.16870	0.0423099
$764$	−6.84271	−0.247561
$765$	0	0
$766$	4.57630	0.165348
$767$	−23.8274	−0.860356
$768$	0	0
$769$	−14.7809	−0.533014	−0.266507	−	0.963833i	$-0.585870\pi$
$769$	−14.7809	−0.533014	−0.266507	+	0.963833i	$0.585870\pi$
$770$	−10.3633	−0.373468
$771$	0	0
$772$	−21.9007	−0.788224
$773$	26.3677	0.948381	0.474190	−	0.880422i	$-0.342741\pi$
$773$	26.3677	0.948381	0.474190	+	0.880422i	$0.342741\pi$
$774$	0	0
$775$	−3.79964	−0.136487
$776$	1.72031	0.0617554
$777$	0	0
$778$	−50.7699	−1.82019
$779$	−2.90305	−0.104013
$780$	0	0
$781$	−28.9364	−1.03542
$782$	−18.3761	−0.657127
$783$	0	0
$784$	25.0625	0.895088
$785$	−27.0936	−0.967010
$786$	0	0
$787$	−2.71416	−0.0967493	−0.0483747	−	0.998829i	$-0.515404\pi$
$787$	−2.71416	−0.0967493	−0.0483747	+	0.998829i	$0.515404\pi$
$788$	4.25356	0.151527
$789$	0	0
$790$	−19.2557	−0.685085
$791$	9.07694	0.322739
$792$	0	0
$793$	−88.7460	−3.15146
$794$	5.17586	0.183684
$795$	0	0
$796$	−10.1593	−0.360087
$797$	43.3542	1.53568	0.767841	−	0.640640i	$-0.221331\pi$
$797$	43.3542	1.53568	0.767841	+	0.640640i	$0.221331\pi$
$798$	0	0
$799$	20.5877	0.728342
$800$	−15.1534	−0.535754
$801$	0	0
$802$	−14.1911	−0.501104
$803$	−17.5059	−0.617771
$804$	0	0
$805$	−12.0661	−0.425273
$806$	21.5197	0.757999
$807$	0	0
$808$	11.8910	0.418324
$809$	10.8368	0.381000	0.190500	−	0.981687i	$-0.438989\pi$
$809$	10.8368	0.381000	0.190500	+	0.981687i	$0.438989\pi$
$810$	0	0
$811$	25.5684	0.897828	0.448914	−	0.893575i	$-0.351811\pi$
$811$	25.5684	0.897828	0.448914	+	0.893575i	$0.351811\pi$
$812$	0	0
$813$	0	0
$814$	−36.7620	−1.28851
$815$	−17.2860	−0.605503
$816$	0	0
$817$	4.47119	0.156427
$818$	5.68399	0.198736
$819$	0	0
$820$	−19.1462	−0.668613
$821$	32.2144	1.12429	0.562145	−	0.827039i	$-0.309976\pi$
$821$	32.2144	1.12429	0.562145	+	0.827039i	$0.309976\pi$
$822$	0	0
$823$	−28.8996	−1.00738	−0.503689	−	0.863885i	$-0.668024\pi$
$823$	−28.8996	−1.00738	−0.503689	+	0.863885i	$0.668024\pi$
$824$	−12.1794	−0.424290
$825$	0	0
$826$	9.26027	0.322206
$827$	−43.1409	−1.50016	−0.750078	−	0.661350i	$-0.769984\pi$
$827$	−43.1409	−1.50016	−0.750078	+	0.661350i	$0.769984\pi$
$828$	0	0
$829$	−4.73634	−0.164500	−0.0822500	−	0.996612i	$-0.526211\pi$
$829$	−4.73634	−0.164500	−0.0822500	+	0.996612i	$0.526211\pi$
$830$	21.8225	0.757470
$831$	0	0
$832$	29.8499	1.03486
$833$	−9.28908	−0.321848
$834$	0	0
$835$	−0.520405	−0.0180094
$836$	−1.78181	−0.0616253
$837$	0	0
$838$	66.9896	2.31412
$839$	−50.5499	−1.74517	−0.872587	−	0.488458i	$-0.837560\pi$
$839$	−50.5499	−1.74517	−0.872587	+	0.488458i	$0.837560\pi$
$840$	0	0
$841$	0	0
$842$	55.4952	1.91249
$843$	0	0
$844$	−34.2604	−1.17929
$845$	41.4554	1.42611
$846$	0	0
$847$	5.59428	0.192222
$848$	−14.7308	−0.505856
$849$	0	0
$850$	6.66463	0.228595
$851$	−42.8022	−1.46724
$852$	0	0
$853$	−16.6034	−0.568489	−0.284244	−	0.958752i	$-0.591743\pi$
$853$	−16.6034	−0.568489	−0.284244	+	0.958752i	$0.591743\pi$
$854$	34.4903	1.18023
$855$	0	0
$856$	−3.20121	−0.109415
$857$	−34.4467	−1.17668	−0.588338	−	0.808615i	$-0.700217\pi$
$857$	−34.4467	−1.17668	−0.588338	+	0.808615i	$0.700217\pi$
$858$	0	0
$859$	−17.1112	−0.583826	−0.291913	−	0.956445i	$-0.594292\pi$
$859$	−17.1112	−0.583826	−0.291913	+	0.956445i	$0.594292\pi$
$860$	29.4884	1.00554
$861$	0	0
$862$	40.0174	1.36300
$863$	−29.3576	−0.999344	−0.499672	−	0.866215i	$-0.666546\pi$
$863$	−29.3576	−0.999344	−0.499672	+	0.866215i	$0.666546\pi$
$864$	0	0
$865$	−17.0824	−0.580818
$866$	54.0340	1.83615
$867$	0	0
$868$	−3.76675	−0.127852
$869$	−14.9850	−0.508330
$870$	0	0
$871$	12.2444	0.414886
$872$	−0.648315	−0.0219547
$873$	0	0
$874$	−4.60624	−0.155808
$875$	15.0347	0.508267
$876$	0	0
$877$	51.3508	1.73399	0.866997	−	0.498314i	$-0.166047\pi$
$877$	51.3508	1.73399	0.866997	+	0.498314i	$0.166047\pi$
$878$	30.9546	1.04467
$879$	0	0
$880$	20.0894	0.677214
$881$	−16.0968	−0.542314	−0.271157	−	0.962535i	$-0.587406\pi$
$881$	−16.0968	−0.542314	−0.271157	+	0.962535i	$0.587406\pi$
$882$	0	0
$883$	−17.4335	−0.586683	−0.293341	−	0.956008i	$-0.594767\pi$
$883$	−17.4335	−0.586683	−0.293341	+	0.956008i	$0.594767\pi$
$884$	−17.0001	−0.571777
$885$	0	0
$886$	−28.4169	−0.954683
$887$	40.4547	1.35834	0.679168	−	0.733983i	$-0.262341\pi$
$887$	40.4547	1.35834	0.679168	+	0.733983i	$0.262341\pi$
$888$	0	0
$889$	9.25952	0.310554
$890$	−10.7283	−0.359613
$891$	0	0
$892$	36.3836	1.21821
$893$	5.16063	0.172694
$894$	0	0
$895$	8.03150	0.268463
$896$	6.73123	0.224875
$897$	0	0
$898$	−13.1646	−0.439309
$899$	0	0
$900$	0	0
$901$	5.45976	0.181891
$902$	−33.0824	−1.10152
$903$	0	0
$904$	−5.03525	−0.167470
$905$	−6.09047	−0.202454
$906$	0	0
$907$	38.0589	1.26373	0.631863	−	0.775080i	$-0.282291\pi$
$907$	38.0589	1.26373	0.631863	+	0.775080i	$0.282291\pi$
$908$	−8.05015	−0.267153
$909$	0	0
$910$	−24.7847	−0.821604
$911$	−1.63752	−0.0542534	−0.0271267	−	0.999632i	$-0.508636\pi$
$911$	−1.63752	−0.0542534	−0.0271267	+	0.999632i	$0.508636\pi$
$912$	0	0
$913$	16.9825	0.562038
$914$	69.7887	2.30841
$915$	0	0
$916$	35.4080	1.16991
$917$	15.1885	0.501569
$918$	0	0
$919$	−7.82279	−0.258050	−0.129025	−	0.991641i	$-0.541185\pi$
$919$	−7.82279	−0.258050	−0.129025	+	0.991641i	$0.541185\pi$
$920$	6.69341	0.220675
$921$	0	0
$922$	12.2712	0.404132
$923$	−69.2036	−2.27786
$924$	0	0
$925$	15.5235	0.510409
$926$	11.3504	0.372999
$927$	0	0
$928$	0	0
$929$	−18.3159	−0.600926	−0.300463	−	0.953794i	$-0.597141\pi$
$929$	−18.3159	−0.600926	−0.300463	+	0.953794i	$0.597141\pi$
$930$	0	0
$931$	−2.32845	−0.0763119
$932$	−35.3247	−1.15710
$933$	0	0
$934$	46.0314	1.50619
$935$	−7.44588	−0.243506
$936$	0	0
$937$	0.405863	0.0132590	0.00662948	−	0.999978i	$-0.497890\pi$
$937$	0.405863	0.0132590	0.00662948	+	0.999978i	$0.497890\pi$
$938$	−4.75867	−0.155376
$939$	0	0
$940$	34.0353	1.11011
$941$	−21.8792	−0.713242	−0.356621	−	0.934249i	$-0.616071\pi$
$941$	−21.8792	−0.713242	−0.356621	+	0.934249i	$0.616071\pi$
$942$	0	0
$943$	−38.5180	−1.25432
$944$	−17.9511	−0.584260
$945$	0	0
$946$	50.9525	1.65661
$947$	36.2764	1.17882	0.589412	−	0.807833i	$-0.299359\pi$
$947$	36.2764	1.17882	0.589412	+	0.807833i	$0.299359\pi$
$948$	0	0
$949$	−41.8668	−1.35905
$950$	1.67059	0.0542011
$951$	0	0
$952$	−1.45571	−0.0471797
$953$	−39.6858	−1.28555	−0.642775	−	0.766055i	$-0.722217\pi$
$953$	−39.6858	−1.28555	−0.642775	+	0.766055i	$0.722217\pi$
$954$	0	0
$955$	−7.16764	−0.231939
$956$	−7.76548	−0.251154
$957$	0	0
$958$	27.2284	0.879708
$959$	8.21505	0.265278
$960$	0	0
$961$	−27.5742	−0.889489
$962$	−87.9192	−2.83463
$963$	0	0
$964$	9.53734	0.307177
$965$	−22.9407	−0.738487
$966$	0	0
$967$	17.3969	0.559447	0.279723	−	0.960081i	$-0.409757\pi$
$967$	17.3969	0.559447	0.279723	+	0.960081i	$0.409757\pi$
$968$	−3.10331	−0.0997442
$969$	0	0
$970$	−8.17862	−0.262600
$971$	−48.7992	−1.56604	−0.783020	−	0.621996i	$-0.786322\pi$
$971$	−48.7992	−1.56604	−0.783020	+	0.621996i	$0.786322\pi$
$972$	0	0
$973$	−10.3155	−0.330701
$974$	−3.67708	−0.117821
$975$	0	0
$976$	−66.8598	−2.14013
$977$	51.9072	1.66066	0.830329	−	0.557274i	$-0.188153\pi$
$977$	51.9072	1.66066	0.830329	+	0.557274i	$0.188153\pi$
$978$	0	0
$979$	−8.34885	−0.266831
$980$	−15.3566	−0.490547
$981$	0	0
$982$	−20.1117	−0.641790
$983$	−16.0820	−0.512936	−0.256468	−	0.966553i	$-0.582559\pi$
$983$	−16.0820	−0.512936	−0.256468	+	0.966553i	$0.582559\pi$
$984$	0	0
$985$	4.45555	0.141966
$986$	0	0
$987$	0	0
$988$	−4.26134	−0.135571
$989$	59.3242	1.88640
$990$	0	0
$991$	−24.4350	−0.776202	−0.388101	−	0.921617i	$-0.626869\pi$
$991$	−24.4350	−0.776202	−0.388101	+	0.921617i	$0.626869\pi$
$992$	13.6627	0.433790
$993$	0	0
$994$	26.8953	0.853067
$995$	−10.6417	−0.337366
$996$	0	0
$997$	−55.5897	−1.76054	−0.880272	−	0.474469i	$-0.842640\pi$
$997$	−55.5897	−1.76054	−0.880272	+	0.474469i	$0.842640\pi$
$998$	29.6741	0.939316
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bt.1.2		12
3.2	odd	2		2523.2.a.s.1.11		12
29.3	odd	28		261.2.o.b.154.2		24
29.10	odd	28		261.2.o.b.100.2		24
29.28	even	2		7569.2.a.bn.1.11		12
87.32	even	28		87.2.i.a.67.3	yes	24
87.68	even	28		87.2.i.a.13.3	✓	24
87.86	odd	2		2523.2.a.v.1.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.i.a.13.3	✓	24	87.68	even	28
87.2.i.a.67.3	yes	24	87.32	even	28
261.2.o.b.100.2		24	29.10	odd	28
261.2.o.b.154.2		24	29.3	odd	28
2523.2.a.s.1.11		12	3.2	odd	2
2523.2.a.v.1.2		12	87.86	odd	2
7569.2.a.bn.1.11		12	29.28	even	2
7569.2.a.bt.1.2		12	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-1.90759 q^{2} +1.63890 q^{4} +1.71673 q^{5} -1.24174 q^{7} +0.688829 q^{8} -3.27481 q^{10} -2.54849 q^{11} -6.09491 q^{13} +2.36873 q^{14} -4.59181 q^{16} +1.70189 q^{17} +0.426606 q^{19} +2.81354 q^{20} +4.86148 q^{22} +5.66024 q^{23} -2.05286 q^{25} +11.6266 q^{26} -2.03509 q^{28} +1.85090 q^{31} +7.38163 q^{32} -3.24652 q^{34} -2.13172 q^{35} -7.56191 q^{37} -0.813789 q^{38} +1.18253 q^{40} -6.80500 q^{41} +10.4809 q^{43} -4.17672 q^{44} -10.7974 q^{46} +12.0970 q^{47} -5.45809 q^{49} +3.91601 q^{50} -9.98896 q^{52} +3.20805 q^{53} -4.37506 q^{55} -0.855345 q^{56} +3.90939 q^{59} +14.5607 q^{61} -3.53077 q^{62} -4.89751 q^{64} -10.4633 q^{65} -2.00896 q^{67} +2.78924 q^{68} +4.06646 q^{70} +11.3543 q^{71} +6.86914 q^{73} +14.4250 q^{74} +0.699164 q^{76} +3.16456 q^{77} +5.87993 q^{79} -7.88287 q^{80} +12.9812 q^{82} -6.66375 q^{83} +2.92168 q^{85} -19.9932 q^{86} -1.75547 q^{88} +3.27600 q^{89} +7.56828 q^{91} +9.27658 q^{92} -23.0761 q^{94} +0.732364 q^{95} +2.49744 q^{97} +10.4118 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 8 q^{4} - 2 q^{5} - 10 q^{7} - 20 q^{10} + 14 q^{11} - 16 q^{13} - 4 q^{16} + 22 q^{17} - 16 q^{19} - 4 q^{20} + 12 q^{22} - 2 q^{23} - 2 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{31} + 16 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) 12 * q + 6 * q^2 + 8 * q^4 - 2 * q^5 - 10 * q^7 - 20 * q^10 + 14 * q^11 - 16 * q^13 - 4 * q^16 + 22 * q^17 - 16 * q^19 - 4 * q^20 + 12 * q^22 - 2 * q^23 - 2 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^31 + 16 * q^32 - 10 * q^34 + 14 * q^35 - 28 * q^37 - 16 * q^38 - 32 * q^40 - 14 * q^41 - 36 * q^43 + 26 * q^44 - 24 * q^46 + 44 * q^47 - 18 * q^49 + 24 * q^50 - 20 * q^52 + 18 * q^53 - 4 * q^55 - 10 * q^56 - 14 * q^59 - 24 * q^61 - 4 * q^62 - 18 * q^64 - 2 * q^65 - 18 * q^67 + 52 * q^68 + 36 * q^70 - 14 * q^71 - 44 * q^73 - 12 * q^74 - 56 * q^76 + 32 * q^77 - 24 * q^79 - 52 * q^80 + 52 * q^82 - 26 * q^83 - 48 * q^85 - 64 * q^86 - 10 * q^88 + 10 * q^89 - 10 * q^91 - 64 * q^92 - 8 * q^94 - 4 * q^95 - 28 * q^97 + 6 * q^98

Introduction

L-functions

Modular forms

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