LMFDB - Embedded newform 7569.2.a.bt.1.11

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			1.11
Root			$-1.32079$ of defining polynomial
Character	$\chi$	$=$	7569.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.32079 q^{2} +3.38605 q^{4} -1.90061 q^{5} -3.42040 q^{7} +3.21673 q^{8} -4.41092 q^{10} -0.827442 q^{11} +5.42009 q^{13} -7.93802 q^{14} +0.693250 q^{16} +6.71892 q^{17} -4.41458 q^{19} -6.43558 q^{20} -1.92032 q^{22} -0.154530 q^{23} -1.38766 q^{25} +12.5789 q^{26} -11.5817 q^{28} +5.73382 q^{31} -4.82458 q^{32} +15.5932 q^{34} +6.50087 q^{35} -9.73479 q^{37} -10.2453 q^{38} -6.11377 q^{40} +5.43514 q^{41} -4.46599 q^{43} -2.80176 q^{44} -0.358631 q^{46} -1.21724 q^{47} +4.69915 q^{49} -3.22047 q^{50} +18.3527 q^{52} -0.466064 q^{53} +1.57265 q^{55} -11.0025 q^{56} -11.5925 q^{59} -11.2197 q^{61} +13.3070 q^{62} -12.5833 q^{64} -10.3015 q^{65} -12.9498 q^{67} +22.7506 q^{68} +15.0871 q^{70} -3.37310 q^{71} +0.980407 q^{73} -22.5924 q^{74} -14.9480 q^{76} +2.83018 q^{77} -4.86752 q^{79} -1.31760 q^{80} +12.6138 q^{82} +7.72823 q^{83} -12.7701 q^{85} -10.3646 q^{86} -2.66166 q^{88} +4.65736 q^{89} -18.5389 q^{91} -0.523246 q^{92} -2.82496 q^{94} +8.39042 q^{95} -10.0143 q^{97} +10.9057 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.32079	1.64104	0.820522	−	0.571615i	$-0.193683\pi$
$2$	2.32079	1.64104	0.820522	+	0.571615i	$0.193683\pi$
$3$	0	0
$3$	0	0
$4$	3.38605	1.69303
$5$	−1.90061	−0.849981	−0.424990	−	0.905198i	$-0.639723\pi$
$5$	−1.90061	−0.849981	−0.424990	+	0.905198i	$0.639723\pi$
$6$	0	0
$7$	−3.42040	−1.29279	−0.646395	−	0.763003i	$-0.723724\pi$
$7$	−3.42040	−1.29279	−0.646395	+	0.763003i	$0.723724\pi$
$8$	3.21673	1.13729
$9$	0	0
$10$	−4.41092	−1.39486
$11$	−0.827442	−0.249483	−0.124742	−	0.992189i	$-0.539810\pi$
$11$	−0.827442	−0.249483	−0.124742	+	0.992189i	$0.539810\pi$
$12$	0	0
$13$	5.42009	1.50326	0.751631	−	0.659584i	$-0.229268\pi$
$13$	5.42009	1.50326	0.751631	+	0.659584i	$0.229268\pi$
$14$	−7.93802	−2.12153
$15$	0	0
$16$	0.693250	0.173313
$17$	6.71892	1.62958	0.814788	−	0.579758i	$-0.196853\pi$
$17$	6.71892	1.62958	0.814788	+	0.579758i	$0.196853\pi$
$18$	0	0
$19$	−4.41458	−1.01278	−0.506388	−	0.862306i	$-0.669019\pi$
$19$	−4.41458	−1.01278	−0.506388	+	0.862306i	$0.669019\pi$
$20$	−6.43558	−1.43904
$21$	0	0
$22$	−1.92032	−0.409413
$23$	−0.154530	−0.0322217	−0.0161109	−	0.999870i	$-0.505128\pi$
$23$	−0.154530	−0.0322217	−0.0161109	+	0.999870i	$0.505128\pi$
$24$	0	0
$25$	−1.38766	−0.277533
$26$	12.5789	2.46692
$27$	0	0
$28$	−11.5817	−2.18873
$29$	0	0
$29$	0	0
$30$	0	0
$31$	5.73382	1.02982	0.514912	−	0.857243i	$-0.327825\pi$
$31$	5.73382	1.02982	0.514912	+	0.857243i	$0.327825\pi$
$32$	−4.82458	−0.852874
$33$	0	0
$34$	15.5932	2.67421
$35$	6.50087	1.09885
$36$	0	0
$37$	−9.73479	−1.60039	−0.800194	−	0.599741i	$-0.795270\pi$
$37$	−9.73479	−1.60039	−0.800194	+	0.599741i	$0.795270\pi$
$38$	−10.2453	−1.66201
$39$	0	0
$40$	−6.11377	−0.966673
$41$	5.43514	0.848827	0.424413	−	0.905469i	$-0.360480\pi$
$41$	5.43514	0.848827	0.424413	+	0.905469i	$0.360480\pi$
$42$	0	0
$43$	−4.46599	−0.681058	−0.340529	−	0.940234i	$-0.610606\pi$
$43$	−4.46599	−0.681058	−0.340529	+	0.940234i	$0.610606\pi$
$44$	−2.80176	−0.422381
$45$	0	0
$46$	−0.358631	−0.0528772
$47$	−1.21724	−0.177553	−0.0887764	−	0.996052i	$-0.528296\pi$
$47$	−1.21724	−0.177553	−0.0887764	+	0.996052i	$0.528296\pi$
$48$	0	0
$49$	4.69915	0.671307
$50$	−3.22047	−0.455443
$51$	0	0
$52$	18.3527	2.54506
$53$	−0.466064	−0.0640189	−0.0320094	−	0.999488i	$-0.510191\pi$
$53$	−0.466064	−0.0640189	−0.0320094	+	0.999488i	$0.510191\pi$
$54$	0	0
$55$	1.57265	0.212056
$56$	−11.0025	−1.47027
$57$	0	0
$58$	0	0
$59$	−11.5925	−1.50921	−0.754605	−	0.656179i	$-0.772172\pi$
$59$	−11.5925	−1.50921	−0.754605	+	0.656179i	$0.772172\pi$
$60$	0	0
$61$	−11.2197	−1.43653	−0.718265	−	0.695769i	$-0.755064\pi$
$61$	−11.2197	−1.43653	−0.718265	+	0.695769i	$0.755064\pi$
$62$	13.3070	1.68999
$63$	0	0
$64$	−12.5833	−1.57292
$65$	−10.3015	−1.27774
$66$	0	0
$67$	−12.9498	−1.58206	−0.791032	−	0.611774i	$-0.790456\pi$
$67$	−12.9498	−1.58206	−0.791032	+	0.611774i	$0.790456\pi$
$68$	22.7506	2.75892
$69$	0	0
$70$	15.0871	1.80326
$71$	−3.37310	−0.400313	−0.200156	−	0.979764i	$-0.564145\pi$
$71$	−3.37310	−0.400313	−0.200156	+	0.979764i	$0.564145\pi$
$72$	0	0
$73$	0.980407	0.114748	0.0573740	−	0.998353i	$-0.481727\pi$
$73$	0.980407	0.114748	0.0573740	+	0.998353i	$0.481727\pi$
$74$	−22.5924	−2.62631
$75$	0	0
$76$	−14.9480	−1.71466
$77$	2.83018	0.322529
$78$	0	0
$79$	−4.86752	−0.547639	−0.273820	−	0.961781i	$-0.588287\pi$
$79$	−4.86752	−0.547639	−0.273820	+	0.961781i	$0.588287\pi$
$80$	−1.31760	−0.147312
$81$	0	0
$82$	12.6138	1.39296
$83$	7.72823	0.848284	0.424142	−	0.905596i	$-0.360576\pi$
$83$	7.72823	0.848284	0.424142	+	0.905596i	$0.360576\pi$
$84$	0	0
$85$	−12.7701	−1.38511
$86$	−10.3646	−1.11765
$87$	0	0
$88$	−2.66166	−0.283734
$89$	4.65736	0.493679	0.246839	−	0.969056i	$-0.420608\pi$
$89$	4.65736	0.493679	0.246839	+	0.969056i	$0.420608\pi$
$90$	0	0
$91$	−18.5389	−1.94340
$92$	−0.523246	−0.0545522
$93$	0	0
$94$	−2.82496	−0.291372
$95$	8.39042	0.860839
$96$	0	0
$97$	−10.0143	−1.01679	−0.508397	−	0.861123i	$-0.669762\pi$
$97$	−10.0143	−1.01679	−0.508397	+	0.861123i	$0.669762\pi$
$98$	10.9057	1.10164
$99$	0	0
$100$	−4.69870	−0.469870
$101$	−15.1822	−1.51069	−0.755344	−	0.655328i	$-0.772530\pi$
$101$	−15.1822	−1.51069	−0.755344	+	0.655328i	$0.772530\pi$
$102$	0	0
$103$	0.208861	0.0205797	0.0102898	−	0.999947i	$-0.496725\pi$
$103$	0.208861	0.0205797	0.0102898	+	0.999947i	$0.496725\pi$
$104$	17.4350	1.70964
$105$	0	0
$106$	−1.08164	−0.105058
$107$	2.57414	0.248852	0.124426	−	0.992229i	$-0.460291\pi$
$107$	2.57414	0.248852	0.124426	+	0.992229i	$0.460291\pi$
$108$	0	0
$109$	−1.15217	−0.110358	−0.0551789	−	0.998476i	$-0.517573\pi$
$109$	−1.15217	−0.110358	−0.0551789	+	0.998476i	$0.517573\pi$
$110$	3.64978	0.347993
$111$	0	0
$112$	−2.37119	−0.224057
$113$	−4.34913	−0.409132	−0.204566	−	0.978853i	$-0.565578\pi$
$113$	−4.34913	−0.409132	−0.204566	+	0.978853i	$0.565578\pi$
$114$	0	0
$115$	0.293702	0.0273878
$116$	0	0
$117$	0	0
$118$	−26.9036	−2.47668
$119$	−22.9814	−2.10670
$120$	0	0
$121$	−10.3153	−0.937758
$122$	−26.0385	−2.35741
$123$	0	0
$124$	19.4150	1.74352
$125$	12.1405	1.08588
$126$	0	0
$127$	0.389534	0.0345655	0.0172828	−	0.999851i	$-0.494498\pi$
$127$	0.389534	0.0345655	0.0172828	+	0.999851i	$0.494498\pi$
$128$	−19.5541	−1.72835
$129$	0	0
$130$	−23.9076	−2.09683
$131$	−0.760404	−0.0664368	−0.0332184	−	0.999448i	$-0.510576\pi$
$131$	−0.760404	−0.0664368	−0.0332184	+	0.999448i	$0.510576\pi$
$132$	0	0
$133$	15.0997	1.30931
$134$	−30.0536	−2.59624
$135$	0	0
$136$	21.6130	1.85330
$137$	−14.9332	−1.27583	−0.637914	−	0.770107i	$-0.720203\pi$
$137$	−14.9332	−1.27583	−0.637914	+	0.770107i	$0.720203\pi$
$138$	0	0
$139$	3.65961	0.310404	0.155202	−	0.987883i	$-0.450397\pi$
$139$	3.65961	0.310404	0.155202	+	0.987883i	$0.450397\pi$
$140$	22.0123	1.86038
$141$	0	0
$142$	−7.82824	−0.656931
$143$	−4.48481	−0.375038
$144$	0	0
$145$	0	0
$146$	2.27532	0.188307
$147$	0	0
$148$	−32.9625	−2.70950
$149$	15.5231	1.27171	0.635853	−	0.771810i	$-0.280648\pi$
$149$	15.5231	1.27171	0.635853	+	0.771810i	$0.280648\pi$
$150$	0	0
$151$	6.58551	0.535921	0.267960	−	0.963430i	$-0.413650\pi$
$151$	6.58551	0.535921	0.267960	+	0.963430i	$0.413650\pi$
$152$	−14.2005	−1.15182
$153$	0	0
$154$	6.56825	0.529285
$155$	−10.8978	−0.875331
$156$	0	0
$157$	10.0667	0.803407	0.401704	−	0.915770i	$-0.368418\pi$
$157$	10.0667	0.803407	0.401704	+	0.915770i	$0.368418\pi$
$158$	−11.2965	−0.898700
$159$	0	0
$160$	9.16968	0.724926
$161$	0.528554	0.0416559
$162$	0	0
$163$	−1.47844	−0.115801	−0.0579004	−	0.998322i	$-0.518441\pi$
$163$	−1.47844	−0.115801	−0.0579004	+	0.998322i	$0.518441\pi$
$164$	18.4037	1.43709
$165$	0	0
$166$	17.9356	1.39207
$167$	−12.4760	−0.965423	−0.482711	−	0.875779i	$-0.660348\pi$
$167$	−12.4760	−0.965423	−0.482711	+	0.875779i	$0.660348\pi$
$168$	0	0
$169$	16.3773	1.25980
$170$	−29.6366	−2.27303
$171$	0	0
$172$	−15.1221	−1.15305
$173$	−2.64519	−0.201110	−0.100555	−	0.994931i	$-0.532062\pi$
$173$	−2.64519	−0.201110	−0.100555	+	0.994931i	$0.532062\pi$
$174$	0	0
$175$	4.74636	0.358791
$176$	−0.573624	−0.0432385
$177$	0	0
$178$	10.8087	0.810149
$179$	−6.35825	−0.475238	−0.237619	−	0.971358i	$-0.576367\pi$
$179$	−6.35825	−0.475238	−0.237619	+	0.971358i	$0.576367\pi$
$180$	0	0
$181$	−17.2344	−1.28102	−0.640512	−	0.767948i	$-0.721278\pi$
$181$	−17.2344	−1.28102	−0.640512	+	0.767948i	$0.721278\pi$
$182$	−43.0248	−3.18921
$183$	0	0
$184$	−0.497082	−0.0366453
$185$	18.5021	1.36030
$186$	0	0
$187$	−5.55951	−0.406552
$188$	−4.12164	−0.300602
$189$	0	0
$190$	19.4724	1.41268
$191$	−16.4482	−1.19015	−0.595074	−	0.803671i	$-0.702877\pi$
$191$	−16.4482	−1.19015	−0.595074	+	0.803671i	$0.702877\pi$
$192$	0	0
$193$	−12.5041	−0.900067	−0.450033	−	0.893012i	$-0.648588\pi$
$193$	−12.5041	−0.900067	−0.450033	+	0.893012i	$0.648588\pi$
$194$	−23.2410	−1.66860
$195$	0	0
$196$	15.9116	1.13654
$197$	14.1460	1.00786	0.503930	−	0.863745i	$-0.331887\pi$
$197$	14.1460	1.00786	0.503930	+	0.863745i	$0.331887\pi$
$198$	0	0
$199$	12.9194	0.915830	0.457915	−	0.888996i	$-0.348596\pi$
$199$	12.9194	0.915830	0.457915	+	0.888996i	$0.348596\pi$
$200$	−4.46374	−0.315634
$201$	0	0
$202$	−35.2347	−2.47911
$203$	0	0
$204$	0	0
$205$	−10.3301	−0.721487
$206$	0.484721	0.0337721
$207$	0	0
$208$	3.75748	0.260534
$209$	3.65281	0.252670
$210$	0	0
$211$	13.7490	0.946518	0.473259	−	0.880923i	$-0.343078\pi$
$211$	13.7490	0.946518	0.473259	+	0.880923i	$0.343078\pi$
$212$	−1.57812	−0.108386
$213$	0	0
$214$	5.97404	0.408377
$215$	8.48813	0.578886
$216$	0	0
$217$	−19.6120	−1.33135
$218$	−2.67394	−0.181102
$219$	0	0
$220$	5.32507	0.359016
$221$	36.4171	2.44968
$222$	0	0
$223$	−13.4989	−0.903955	−0.451978	−	0.892029i	$-0.649281\pi$
$223$	−13.4989	−0.903955	−0.451978	+	0.892029i	$0.649281\pi$
$224$	16.5020	1.10259
$225$	0	0
$226$	−10.0934	−0.671403
$227$	26.9707	1.79011	0.895053	−	0.445959i	$-0.147137\pi$
$227$	26.9707	1.79011	0.895053	+	0.445959i	$0.147137\pi$
$228$	0	0
$229$	10.9171	0.721424	0.360712	−	0.932677i	$-0.382534\pi$
$229$	10.9171	0.721424	0.360712	+	0.932677i	$0.382534\pi$
$230$	0.681619	0.0449446
$231$	0	0
$232$	0	0
$233$	5.48306	0.359207	0.179604	−	0.983739i	$-0.442518\pi$
$233$	5.48306	0.359207	0.179604	+	0.983739i	$0.442518\pi$
$234$	0	0
$235$	2.31351	0.150917
$236$	−39.2527	−2.55513
$237$	0	0
$238$	−53.3349	−3.45719
$239$	0.760826	0.0492138	0.0246069	−	0.999697i	$-0.492167\pi$
$239$	0.760826	0.0492138	0.0246069	+	0.999697i	$0.492167\pi$
$240$	0	0
$241$	−25.0342	−1.61259	−0.806297	−	0.591511i	$-0.798532\pi$
$241$	−25.0342	−1.61259	−0.806297	+	0.591511i	$0.798532\pi$
$242$	−23.9397	−1.53890
$243$	0	0
$244$	−37.9904	−2.43208
$245$	−8.93127	−0.570598
$246$	0	0
$247$	−23.9274	−1.52247
$248$	18.4442	1.17121
$249$	0	0
$250$	28.1755	1.78197
$251$	10.4978	0.662617	0.331308	−	0.943523i	$-0.392510\pi$
$251$	10.4978	0.662617	0.331308	+	0.943523i	$0.392510\pi$
$252$	0	0
$253$	0.127864	0.00803877
$254$	0.904025	0.0567236
$255$	0	0
$256$	−20.2142	−1.26339
$257$	0.271801	0.0169545	0.00847725	−	0.999964i	$-0.497302\pi$
$257$	0.271801	0.0169545	0.00847725	+	0.999964i	$0.497302\pi$
$258$	0	0
$259$	33.2969	2.06897
$260$	−34.8814	−2.16325
$261$	0	0
$262$	−1.76474	−0.109026
$263$	7.34264	0.452767	0.226383	−	0.974038i	$-0.427310\pi$
$263$	7.34264	0.452767	0.226383	+	0.974038i	$0.427310\pi$
$264$	0	0
$265$	0.885809	0.0544148
$266$	35.0431	2.14863
$267$	0	0
$268$	−43.8486	−2.67848
$269$	30.0231	1.83054	0.915269	−	0.402843i	$-0.131978\pi$
$269$	30.0231	1.83054	0.915269	+	0.402843i	$0.131978\pi$
$270$	0	0
$271$	0.175497	0.0106607	0.00533036	−	0.999986i	$-0.498303\pi$
$271$	0.175497	0.0106607	0.00533036	+	0.999986i	$0.498303\pi$
$272$	4.65789	0.282426
$273$	0	0
$274$	−34.6568	−2.09369
$275$	1.14821	0.0692397
$276$	0	0
$277$	2.93089	0.176100	0.0880502	−	0.996116i	$-0.471936\pi$
$277$	2.93089	0.176100	0.0880502	+	0.996116i	$0.471936\pi$
$278$	8.49318	0.509387
$279$	0	0
$280$	20.9116	1.24970
$281$	10.1937	0.608105	0.304052	−	0.952655i	$-0.401660\pi$
$281$	10.1937	0.608105	0.304052	+	0.952655i	$0.401660\pi$
$282$	0	0
$283$	−19.2316	−1.14320	−0.571599	−	0.820533i	$-0.693677\pi$
$283$	−19.2316	−1.14320	−0.571599	+	0.820533i	$0.693677\pi$
$284$	−11.4215	−0.677740
$285$	0	0
$286$	−10.4083	−0.615454
$287$	−18.5904	−1.09736
$288$	0	0
$289$	28.1439	1.65552
$290$	0	0
$291$	0	0
$292$	3.31971	0.194271
$293$	11.1238	0.649860	0.324930	−	0.945738i	$-0.394659\pi$
$293$	11.1238	0.649860	0.324930	+	0.945738i	$0.394659\pi$
$294$	0	0
$295$	22.0328	1.28280
$296$	−31.3142	−1.82010
$297$	0	0
$298$	36.0259	2.08693
$299$	−0.837565	−0.0484377
$300$	0	0
$301$	15.2755	0.880465
$302$	15.2836	0.879470
$303$	0	0
$304$	−3.06041	−0.175527
$305$	21.3243	1.22102
$306$	0	0
$307$	−3.09566	−0.176678	−0.0883392	−	0.996090i	$-0.528156\pi$
$307$	−3.09566	−0.176678	−0.0883392	+	0.996090i	$0.528156\pi$
$308$	9.58315	0.546051
$309$	0	0
$310$	−25.2914	−1.43646
$311$	−14.6432	−0.830342	−0.415171	−	0.909743i	$-0.636278\pi$
$311$	−14.6432	−0.830342	−0.415171	+	0.909743i	$0.636278\pi$
$312$	0	0
$313$	−6.68806	−0.378031	−0.189016	−	0.981974i	$-0.560530\pi$
$313$	−6.68806	−0.378031	−0.189016	+	0.981974i	$0.560530\pi$
$314$	23.3626	1.31843
$315$	0	0
$316$	−16.4817	−0.927168
$317$	5.56851	0.312759	0.156379	−	0.987697i	$-0.450018\pi$
$317$	5.56851	0.312759	0.156379	+	0.987697i	$0.450018\pi$
$318$	0	0
$319$	0	0
$320$	23.9161	1.33695
$321$	0	0
$322$	1.22666	0.0683592
$323$	−29.6612	−1.65039
$324$	0	0
$325$	−7.52125	−0.417204
$326$	−3.43116	−0.190034
$327$	0	0
$328$	17.4834	0.965360
$329$	4.16345	0.229539
$330$	0	0
$331$	−34.6315	−1.90352	−0.951759	−	0.306847i	$-0.900726\pi$
$331$	−34.6315	−1.90352	−0.951759	+	0.306847i	$0.900726\pi$
$332$	26.1682	1.43617
$333$	0	0
$334$	−28.9542	−1.58430
$335$	24.6125	1.34472
$336$	0	0
$337$	−8.82837	−0.480912	−0.240456	−	0.970660i	$-0.577297\pi$
$337$	−8.82837	−0.480912	−0.240456	+	0.970660i	$0.577297\pi$
$338$	38.0083	2.06738
$339$	0	0
$340$	−43.2402	−2.34503
$341$	−4.74440	−0.256924
$342$	0	0
$343$	7.86984	0.424931
$344$	−14.3659	−0.774558
$345$	0	0
$346$	−6.13892	−0.330031
$347$	16.3136	0.875759	0.437880	−	0.899034i	$-0.355730\pi$
$347$	16.3136	0.875759	0.437880	+	0.899034i	$0.355730\pi$
$348$	0	0
$349$	−9.44632	−0.505650	−0.252825	−	0.967512i	$-0.581360\pi$
$349$	−9.44632	−0.505650	−0.252825	+	0.967512i	$0.581360\pi$
$350$	11.0153	0.588793
$351$	0	0
$352$	3.99206	0.212778
$353$	14.9647	0.796490	0.398245	−	0.917279i	$-0.369619\pi$
$353$	14.9647	0.796490	0.398245	+	0.917279i	$0.369619\pi$
$354$	0	0
$355$	6.41096	0.340258
$356$	15.7701	0.835811
$357$	0	0
$358$	−14.7562	−0.779887
$359$	−15.4219	−0.813934	−0.406967	−	0.913443i	$-0.633414\pi$
$359$	−15.4219	−0.813934	−0.406967	+	0.913443i	$0.633414\pi$
$360$	0	0
$361$	0.488553	0.0257133
$362$	−39.9974	−2.10222
$363$	0	0
$364$	−62.7736	−3.29023
$365$	−1.86338	−0.0975336
$366$	0	0
$367$	−33.3456	−1.74063	−0.870313	−	0.492499i	$-0.836083\pi$
$367$	−33.3456	−1.74063	−0.870313	+	0.492499i	$0.836083\pi$
$368$	−0.107128	−0.00558442
$369$	0	0
$370$	42.9394	2.23231
$371$	1.59413	0.0827630
$372$	0	0
$373$	−3.25578	−0.168578	−0.0842888	−	0.996441i	$-0.526862\pi$
$373$	−3.25578	−0.168578	−0.0842888	+	0.996441i	$0.526862\pi$
$374$	−12.9024	−0.667170
$375$	0	0
$376$	−3.91554	−0.201929
$377$	0	0
$378$	0	0
$379$	25.9336	1.33212	0.666059	−	0.745899i	$-0.267980\pi$
$379$	25.9336	1.33212	0.666059	+	0.745899i	$0.267980\pi$
$380$	28.4104	1.45742
$381$	0	0
$382$	−38.1727	−1.95309
$383$	−29.5945	−1.51221	−0.756105	−	0.654450i	$-0.772900\pi$
$383$	−29.5945	−1.51221	−0.756105	+	0.654450i	$0.772900\pi$
$384$	0	0
$385$	−5.37909	−0.274144
$386$	−29.0194	−1.47705
$387$	0	0
$388$	−33.9088	−1.72146
$389$	−11.6584	−0.591103	−0.295551	−	0.955327i	$-0.595503\pi$
$389$	−11.6584	−0.591103	−0.295551	+	0.955327i	$0.595503\pi$
$390$	0	0
$391$	−1.03827	−0.0525077
$392$	15.1159	0.763469
$393$	0	0
$394$	32.8298	1.65394
$395$	9.25129	0.465483
$396$	0	0
$397$	21.5610	1.08211	0.541057	−	0.840986i	$-0.318024\pi$
$397$	21.5610	1.08211	0.541057	+	0.840986i	$0.318024\pi$
$398$	29.9831	1.50292
$399$	0	0
$400$	−0.961997	−0.0480999
$401$	−20.0052	−0.999010	−0.499505	−	0.866311i	$-0.666485\pi$
$401$	−20.0052	−0.999010	−0.499505	+	0.866311i	$0.666485\pi$
$402$	0	0
$403$	31.0778	1.54810
$404$	−51.4079	−2.55764
$405$	0	0
$406$	0	0
$407$	8.05497	0.399270
$408$	0	0
$409$	28.4623	1.40737	0.703686	−	0.710511i	$-0.251536\pi$
$409$	28.4623	1.40737	0.703686	+	0.710511i	$0.251536\pi$
$410$	−23.9740	−1.18399
$411$	0	0
$412$	0.707214	0.0348419
$413$	39.6509	1.95109
$414$	0	0
$415$	−14.6884	−0.721025
$416$	−26.1497	−1.28209
$417$	0	0
$418$	8.47740	0.414643
$419$	−11.0690	−0.540754	−0.270377	−	0.962755i	$-0.587148\pi$
$419$	−11.0690	−0.540754	−0.270377	+	0.962755i	$0.587148\pi$
$420$	0	0
$421$	26.7707	1.30472	0.652362	−	0.757907i	$-0.273778\pi$
$421$	26.7707	1.30472	0.652362	+	0.757907i	$0.273778\pi$
$422$	31.9084	1.55328
$423$	0	0
$424$	−1.49921	−0.0728078
$425$	−9.32359	−0.452261
$426$	0	0
$427$	38.3758	1.85713
$428$	8.71619	0.421313
$429$	0	0
$430$	19.6992	0.949978
$431$	−33.1348	−1.59605	−0.798025	−	0.602625i	$-0.794122\pi$
$431$	−33.1348	−1.59605	−0.798025	+	0.602625i	$0.794122\pi$
$432$	0	0
$433$	−5.32076	−0.255699	−0.127850	−	0.991794i	$-0.540808\pi$
$433$	−5.32076	−0.255699	−0.127850	+	0.991794i	$0.540808\pi$
$434$	−45.5152	−2.18480
$435$	0	0
$436$	−3.90131	−0.186839
$437$	0.682185	0.0326333
$438$	0	0
$439$	−31.1536	−1.48688	−0.743439	−	0.668804i	$-0.766807\pi$
$439$	−31.1536	−1.48688	−0.743439	+	0.668804i	$0.766807\pi$
$440$	5.05879	0.241168
$441$	0	0
$442$	84.5164	4.02003
$443$	12.9023	0.613005	0.306503	−	0.951870i	$-0.400841\pi$
$443$	12.9023	0.613005	0.306503	+	0.951870i	$0.400841\pi$
$444$	0	0
$445$	−8.85184	−0.419618
$446$	−31.3281	−1.48343
$447$	0	0
$448$	43.0400	2.03345
$449$	33.8506	1.59751	0.798754	−	0.601658i	$-0.205493\pi$
$449$	33.8506	1.59751	0.798754	+	0.601658i	$0.205493\pi$
$450$	0	0
$451$	−4.49726	−0.211768
$452$	−14.7264	−0.692671
$453$	0	0
$454$	62.5932	2.93764
$455$	35.2353	1.65185
$456$	0	0
$457$	−7.63489	−0.357145	−0.178572	−	0.983927i	$-0.557148\pi$
$457$	−7.63489	−0.357145	−0.178572	+	0.983927i	$0.557148\pi$
$458$	25.3363	1.18389
$459$	0	0
$460$	0.994490	0.0463683
$461$	16.9643	0.790108	0.395054	−	0.918658i	$-0.370726\pi$
$461$	16.9643	0.790108	0.395054	+	0.918658i	$0.370726\pi$
$462$	0	0
$463$	9.31660	0.432979	0.216490	−	0.976285i	$-0.430539\pi$
$463$	9.31660	0.432979	0.216490	+	0.976285i	$0.430539\pi$
$464$	0	0
$465$	0	0
$466$	12.7250	0.589475
$467$	−25.3522	−1.17316	−0.586579	−	0.809892i	$-0.699526\pi$
$467$	−25.3522	−1.17316	−0.586579	+	0.809892i	$0.699526\pi$
$468$	0	0
$469$	44.2934	2.04528
$470$	5.36916	0.247661
$471$	0	0
$472$	−37.2899	−1.71641
$473$	3.69535	0.169912
$474$	0	0
$475$	6.12595	0.281078
$476$	−77.8162	−3.56670
$477$	0	0
$478$	1.76572	0.0807620
$479$	−23.0247	−1.05203	−0.526013	−	0.850476i	$-0.676314\pi$
$479$	−23.0247	−1.05203	−0.526013	+	0.850476i	$0.676314\pi$
$480$	0	0
$481$	−52.7634	−2.40580
$482$	−58.0990	−2.64634
$483$	0	0
$484$	−34.9283	−1.58765
$485$	19.0332	0.864255
$486$	0	0
$487$	1.53425	0.0695234	0.0347617	−	0.999396i	$-0.488933\pi$
$487$	1.53425	0.0695234	0.0347617	+	0.999396i	$0.488933\pi$
$488$	−36.0907	−1.63375
$489$	0	0
$490$	−20.7276	−0.936376
$491$	20.1498	0.909346	0.454673	−	0.890658i	$-0.349756\pi$
$491$	20.1498	0.909346	0.454673	+	0.890658i	$0.349756\pi$
$492$	0	0
$493$	0	0
$494$	−55.5305	−2.49843
$495$	0	0
$496$	3.97497	0.178481
$497$	11.5373	0.517521
$498$	0	0
$499$	34.7057	1.55364	0.776821	−	0.629721i	$-0.216831\pi$
$499$	34.7057	1.55364	0.776821	+	0.629721i	$0.216831\pi$
$500$	41.1083	1.83842
$501$	0	0
$502$	24.3632	1.08738
$503$	28.8361	1.28574	0.642870	−	0.765975i	$-0.277743\pi$
$503$	28.8361	1.28574	0.642870	+	0.765975i	$0.277743\pi$
$504$	0	0
$505$	28.8556	1.28406
$506$	0.296746	0.0131920
$507$	0	0
$508$	1.31898	0.0585204
$509$	3.10101	0.137450	0.0687249	−	0.997636i	$-0.478107\pi$
$509$	3.10101	0.137450	0.0687249	+	0.997636i	$0.478107\pi$
$510$	0	0
$511$	−3.35339	−0.148345
$512$	−7.80465	−0.344920
$513$	0	0
$514$	0.630793	0.0278231
$515$	−0.396964	−0.0174923
$516$	0	0
$517$	1.00720	0.0442964
$518$	77.2750	3.39527
$519$	0	0
$520$	−33.1372	−1.45316
$521$	33.4475	1.46536	0.732680	−	0.680573i	$-0.238269\pi$
$521$	33.4475	1.46536	0.732680	+	0.680573i	$0.238269\pi$
$522$	0	0
$523$	−23.5081	−1.02794	−0.513969	−	0.857809i	$-0.671825\pi$
$523$	−23.5081	−1.02794	−0.513969	+	0.857809i	$0.671825\pi$
$524$	−2.57477	−0.112479
$525$	0	0
$526$	17.0407	0.743010
$527$	38.5251	1.67818
$528$	0	0
$529$	−22.9761	−0.998962
$530$	2.05577	0.0892971
$531$	0	0
$532$	51.1282	2.21669
$533$	29.4590	1.27601
$534$	0	0
$535$	−4.89245	−0.211519
$536$	−41.6559	−1.79926
$537$	0	0
$538$	69.6771	3.00399
$539$	−3.88827	−0.167480
$540$	0	0
$541$	32.7727	1.40901	0.704503	−	0.709701i	$-0.251170\pi$
$541$	32.7727	1.40901	0.704503	+	0.709701i	$0.251170\pi$
$542$	0.407292	0.0174947
$543$	0	0
$544$	−32.4160	−1.38982
$545$	2.18983	0.0938020
$546$	0	0
$547$	−28.1424	−1.20328	−0.601642	−	0.798766i	$-0.705486\pi$
$547$	−28.1424	−1.20328	−0.601642	+	0.798766i	$0.705486\pi$
$548$	−50.5646	−2.16001
$549$	0	0
$550$	2.66475	0.113625
$551$	0	0
$552$	0	0
$553$	16.6489	0.707983
$554$	6.80198	0.288989
$555$	0	0
$556$	12.3916	0.525523
$557$	13.2791	0.562652	0.281326	−	0.959612i	$-0.409226\pi$
$557$	13.2791	0.562652	0.281326	+	0.959612i	$0.409226\pi$
$558$	0	0
$559$	−24.2061	−1.02381
$560$	4.50673	0.190444
$561$	0	0
$562$	23.6574	0.997927
$563$	−12.5201	−0.527658	−0.263829	−	0.964569i	$-0.584986\pi$
$563$	−12.5201	−0.527658	−0.263829	+	0.964569i	$0.584986\pi$
$564$	0	0
$565$	8.26602	0.347754
$566$	−44.6324	−1.87604
$567$	0	0
$568$	−10.8504	−0.455271
$569$	12.9397	0.542461	0.271230	−	0.962514i	$-0.412569\pi$
$569$	12.9397	0.542461	0.271230	+	0.962514i	$0.412569\pi$
$570$	0	0
$571$	20.4961	0.857734	0.428867	−	0.903368i	$-0.358913\pi$
$571$	20.4961	0.857734	0.428867	+	0.903368i	$0.358913\pi$
$572$	−15.1858	−0.634950
$573$	0	0
$574$	−43.1443	−1.80081
$575$	0.214435	0.00894257
$576$	0	0
$577$	2.35603	0.0980827	0.0490414	−	0.998797i	$-0.484383\pi$
$577$	2.35603	0.0980827	0.0490414	+	0.998797i	$0.484383\pi$
$578$	65.3159	2.71678
$579$	0	0
$580$	0	0
$581$	−26.4337	−1.09665
$582$	0	0
$583$	0.385641	0.0159716
$584$	3.15371	0.130501
$585$	0	0
$586$	25.8160	1.06645
$587$	−8.14265	−0.336083	−0.168041	−	0.985780i	$-0.553744\pi$
$587$	−8.14265	−0.336083	−0.168041	+	0.985780i	$0.553744\pi$
$588$	0	0
$589$	−25.3124	−1.04298
$590$	51.1335	2.10513
$591$	0	0
$592$	−6.74864	−0.277367
$593$	35.0659	1.43998	0.719991	−	0.693983i	$-0.244146\pi$
$593$	35.0659	1.43998	0.719991	+	0.693983i	$0.244146\pi$
$594$	0	0
$595$	43.6788	1.79066
$596$	52.5622	2.15303
$597$	0	0
$598$	−1.94381	−0.0794883
$599$	−25.7373	−1.05160	−0.525798	−	0.850609i	$-0.676233\pi$
$599$	−25.7373	−1.05160	−0.525798	+	0.850609i	$0.676233\pi$
$600$	0	0
$601$	−0.894006	−0.0364673	−0.0182336	−	0.999834i	$-0.505804\pi$
$601$	−0.894006	−0.0364673	−0.0182336	+	0.999834i	$0.505804\pi$
$602$	35.4512	1.44488
$603$	0	0
$604$	22.2989	0.907328
$605$	19.6055	0.797077
$606$	0	0
$607$	−4.25639	−0.172762	−0.0863808	−	0.996262i	$-0.527530\pi$
$607$	−4.25639	−0.172762	−0.0863808	+	0.996262i	$0.527530\pi$
$608$	21.2985	0.863769
$609$	0	0
$610$	49.4891	2.00375
$611$	−6.59755	−0.266908
$612$	0	0
$613$	−13.8078	−0.557691	−0.278845	−	0.960336i	$-0.589952\pi$
$613$	−13.8078	−0.557691	−0.278845	+	0.960336i	$0.589952\pi$
$614$	−7.18436	−0.289937
$615$	0	0
$616$	9.10395	0.366808
$617$	−5.11993	−0.206121	−0.103060	−	0.994675i	$-0.532863\pi$
$617$	−5.11993	−0.206121	−0.103060	+	0.994675i	$0.532863\pi$
$618$	0	0
$619$	40.3193	1.62057	0.810284	−	0.586037i	$-0.199313\pi$
$619$	40.3193	1.62057	0.810284	+	0.586037i	$0.199313\pi$
$620$	−36.9005	−1.48196
$621$	0	0
$622$	−33.9838	−1.36263
$623$	−15.9300	−0.638223
$624$	0	0
$625$	−16.1361	−0.645443
$626$	−15.5216	−0.620366
$627$	0	0
$628$	34.0863	1.36019
$629$	−65.4072	−2.60796
$630$	0	0
$631$	−2.46097	−0.0979698	−0.0489849	−	0.998800i	$-0.515599\pi$
$631$	−2.46097	−0.0979698	−0.0489849	+	0.998800i	$0.515599\pi$
$632$	−15.6575	−0.622823
$633$	0	0
$634$	12.9233	0.513251
$635$	−0.740354	−0.0293800
$636$	0	0
$637$	25.4698	1.00915
$638$	0	0
$639$	0	0
$640$	37.1648	1.46907
$641$	17.1138	0.675953	0.337977	−	0.941154i	$-0.390258\pi$
$641$	17.1138	0.675953	0.337977	+	0.941154i	$0.390258\pi$
$642$	0	0
$643$	−39.9949	−1.57724	−0.788622	−	0.614879i	$-0.789205\pi$
$643$	−39.9949	−1.57724	−0.788622	+	0.614879i	$0.789205\pi$
$644$	1.78971	0.0705246
$645$	0	0
$646$	−68.8374	−2.70837
$647$	−7.53650	−0.296290	−0.148145	−	0.988966i	$-0.547330\pi$
$647$	−7.53650	−0.296290	−0.148145	+	0.988966i	$0.547330\pi$
$648$	0	0
$649$	9.59209	0.376522
$650$	−17.4552	−0.684650
$651$	0	0
$652$	−5.00609	−0.196054
$653$	39.8823	1.56072	0.780358	−	0.625332i	$-0.215037\pi$
$653$	39.8823	1.56072	0.780358	+	0.625332i	$0.215037\pi$
$654$	0	0
$655$	1.44524	0.0564700
$656$	3.76791	0.147112
$657$	0	0
$658$	9.66249	0.376683
$659$	8.97996	0.349810	0.174905	−	0.984585i	$-0.444038\pi$
$659$	8.97996	0.349810	0.174905	+	0.984585i	$0.444038\pi$
$660$	0	0
$661$	24.5234	0.953851	0.476926	−	0.878944i	$-0.341751\pi$
$661$	24.5234	0.953851	0.476926	+	0.878944i	$0.341751\pi$
$662$	−80.3723	−3.12376
$663$	0	0
$664$	24.8597	0.964742
$665$	−28.6986	−1.11288
$666$	0	0
$667$	0	0
$668$	−42.2444	−1.63449
$669$	0	0
$670$	57.1204	2.20675
$671$	9.28362	0.358390
$672$	0	0
$673$	21.3524	0.823074	0.411537	−	0.911393i	$-0.364992\pi$
$673$	21.3524	0.823074	0.411537	+	0.911393i	$0.364992\pi$
$674$	−20.4888	−0.789198
$675$	0	0
$676$	55.4545	2.13287
$677$	−8.83784	−0.339666	−0.169833	−	0.985473i	$-0.554323\pi$
$677$	−8.83784	−0.339666	−0.169833	+	0.985473i	$0.554323\pi$
$678$	0	0
$679$	34.2528	1.31450
$680$	−41.0779	−1.57527
$681$	0	0
$682$	−11.0107	−0.421623
$683$	−31.8703	−1.21948	−0.609741	−	0.792601i	$-0.708726\pi$
$683$	−31.8703	−1.21948	−0.609741	+	0.792601i	$0.708726\pi$
$684$	0	0
$685$	28.3822	1.08443
$686$	18.2642	0.697331
$687$	0	0
$688$	−3.09605	−0.118036
$689$	−2.52611	−0.0962371
$690$	0	0
$691$	−6.92284	−0.263357	−0.131679	−	0.991292i	$-0.542037\pi$
$691$	−6.92284	−0.263357	−0.131679	+	0.991292i	$0.542037\pi$
$692$	−8.95675	−0.340485
$693$	0	0
$694$	37.8604	1.43716
$695$	−6.95551	−0.263838
$696$	0	0
$697$	36.5183	1.38323
$698$	−21.9229	−0.829794
$699$	0	0
$700$	16.0714	0.607443
$701$	−44.7487	−1.69013	−0.845067	−	0.534661i	$-0.820439\pi$
$701$	−44.7487	−1.69013	−0.845067	+	0.534661i	$0.820439\pi$
$702$	0	0
$703$	42.9750	1.62083
$704$	10.4120	0.392416
$705$	0	0
$706$	34.7299	1.30708
$707$	51.9293	1.95300
$708$	0	0
$709$	6.32860	0.237676	0.118838	−	0.992914i	$-0.462083\pi$
$709$	6.32860	0.237676	0.118838	+	0.992914i	$0.462083\pi$
$710$	14.8785	0.558379
$711$	0	0
$712$	14.9815	0.561455
$713$	−0.886046	−0.0331827
$714$	0	0
$715$	8.52389	0.318775
$716$	−21.5294	−0.804591
$717$	0	0
$718$	−35.7908	−1.33570
$719$	−46.2583	−1.72514	−0.862571	−	0.505936i	$-0.831147\pi$
$719$	−46.2583	−1.72514	−0.862571	+	0.505936i	$0.831147\pi$
$720$	0	0
$721$	−0.714388	−0.0266052
$722$	1.13383	0.0421967
$723$	0	0
$724$	−58.3566	−2.16881
$725$	0	0
$726$	0	0
$727$	21.0732	0.781561	0.390780	−	0.920484i	$-0.372205\pi$
$727$	21.0732	0.781561	0.390780	+	0.920484i	$0.372205\pi$
$728$	−59.6346	−2.21021
$729$	0	0
$730$	−4.32450	−0.160057
$731$	−30.0066	−1.10984
$732$	0	0
$733$	31.3184	1.15677	0.578386	−	0.815763i	$-0.303683\pi$
$733$	31.3184	1.15677	0.578386	+	0.815763i	$0.303683\pi$
$734$	−77.3880	−2.85644
$735$	0	0
$736$	0.745542	0.0274811
$737$	10.7152	0.394698
$738$	0	0
$739$	46.7013	1.71793	0.858967	−	0.512032i	$-0.171107\pi$
$739$	46.7013	1.71793	0.858967	+	0.512032i	$0.171107\pi$
$740$	62.6490	2.30302
$741$	0	0
$742$	3.69963	0.135818
$743$	−7.62970	−0.279906	−0.139953	−	0.990158i	$-0.544695\pi$
$743$	−7.62970	−0.279906	−0.139953	+	0.990158i	$0.544695\pi$
$744$	0	0
$745$	−29.5035	−1.08093
$746$	−7.55596	−0.276643
$747$	0	0
$748$	−18.8248	−0.688303
$749$	−8.80460	−0.321713
$750$	0	0
$751$	−52.6341	−1.92064	−0.960322	−	0.278893i	$-0.910032\pi$
$751$	−52.6341	−1.92064	−0.960322	+	0.278893i	$0.910032\pi$
$752$	−0.843853	−0.0307721
$753$	0	0
$754$	0	0
$755$	−12.5165	−0.455522
$756$	0	0
$757$	−17.6192	−0.640382	−0.320191	−	0.947353i	$-0.603747\pi$
$757$	−17.6192	−0.640382	−0.320191	+	0.947353i	$0.603747\pi$
$758$	60.1863	2.18606
$759$	0	0
$760$	26.9898	0.979022
$761$	1.36919	0.0496332	0.0248166	−	0.999692i	$-0.492100\pi$
$761$	1.36919	0.0496332	0.0248166	+	0.999692i	$0.492100\pi$
$762$	0	0
$763$	3.94088	0.142669
$764$	−55.6944	−2.01495
$765$	0	0
$766$	−68.6826	−2.48160
$767$	−62.8322	−2.26874
$768$	0	0
$769$	16.0129	0.577442	0.288721	−	0.957413i	$-0.406770\pi$
$769$	16.0129	0.577442	0.288721	+	0.957413i	$0.406770\pi$
$770$	−12.4837	−0.449882
$771$	0	0
$772$	−42.3397	−1.52384
$773$	−34.0107	−1.22328	−0.611640	−	0.791136i	$-0.709490\pi$
$773$	−34.0107	−1.22328	−0.611640	+	0.791136i	$0.709490\pi$
$774$	0	0
$775$	−7.95661	−0.285810
$776$	−32.2132	−1.15639
$777$	0	0
$778$	−27.0566	−0.970025
$779$	−23.9939	−0.859671
$780$	0	0
$781$	2.79104	0.0998713
$782$	−2.40961	−0.0861675
$783$	0	0
$784$	3.25768	0.116346
$785$	−19.1328	−0.682881
$786$	0	0
$787$	50.0028	1.78241	0.891203	−	0.453604i	$-0.149862\pi$
$787$	50.0028	1.78241	0.891203	+	0.453604i	$0.149862\pi$
$788$	47.8991	1.70633
$789$	0	0
$790$	21.4703	0.763878
$791$	14.8758	0.528921
$792$	0	0
$793$	−60.8115	−2.15948
$794$	50.0384	1.77580
$795$	0	0
$796$	43.7457	1.55052
$797$	−21.8515	−0.774019	−0.387010	−	0.922076i	$-0.626492\pi$
$797$	−21.8515	−0.774019	−0.387010	+	0.922076i	$0.626492\pi$
$798$	0	0
$799$	−8.17854	−0.289336
$800$	6.69490	0.236700
$801$	0	0
$802$	−46.4277	−1.63942
$803$	−0.811230	−0.0286277
$804$	0	0
$805$	−1.00458	−0.0354067
$806$	72.1250	2.54049
$807$	0	0
$808$	−48.8372	−1.71809
$809$	−4.61865	−0.162383	−0.0811915	−	0.996699i	$-0.525873\pi$
$809$	−4.61865	−0.162383	−0.0811915	+	0.996699i	$0.525873\pi$
$810$	0	0
$811$	5.86604	0.205985	0.102992	−	0.994682i	$-0.467158\pi$
$811$	5.86604	0.205985	0.102992	+	0.994682i	$0.467158\pi$
$812$	0	0
$813$	0	0
$814$	18.6939	0.655220
$815$	2.80995	0.0984284
$816$	0	0
$817$	19.7155	0.689758
$818$	66.0550	2.30956
$819$	0	0
$820$	−34.9783	−1.22150
$821$	11.9154	0.415850	0.207925	−	0.978145i	$-0.433329\pi$
$821$	11.9154	0.415850	0.207925	+	0.978145i	$0.433329\pi$
$822$	0	0
$823$	9.37146	0.326669	0.163334	−	0.986571i	$-0.447775\pi$
$823$	9.37146	0.326669	0.163334	+	0.986571i	$0.447775\pi$
$824$	0.671850	0.0234050
$825$	0	0
$826$	92.0213	3.20183
$827$	39.0974	1.35955	0.679776	−	0.733420i	$-0.262077\pi$
$827$	39.0974	1.35955	0.679776	+	0.733420i	$0.262077\pi$
$828$	0	0
$829$	6.57154	0.228239	0.114119	−	0.993467i	$-0.463595\pi$
$829$	6.57154	0.228239	0.114119	+	0.993467i	$0.463595\pi$
$830$	−34.0886	−1.18323
$831$	0	0
$832$	−68.2027	−2.36450
$833$	31.5732	1.09395
$834$	0	0
$835$	23.7121	0.820591
$836$	12.3686	0.427777
$837$	0	0
$838$	−25.6887	−0.887401
$839$	−18.5940	−0.641935	−0.320967	−	0.947090i	$-0.604008\pi$
$839$	−18.5940	−0.641935	−0.320967	+	0.947090i	$0.604008\pi$
$840$	0	0
$841$	0	0
$842$	62.1291	2.14111
$843$	0	0
$844$	46.5547	1.60248
$845$	−31.1270	−1.07080
$846$	0	0
$847$	35.2826	1.21232
$848$	−0.323099	−0.0110953
$849$	0	0
$850$	−21.6381	−0.742180
$851$	1.50432	0.0515673
$852$	0	0
$853$	15.0581	0.515578	0.257789	−	0.966201i	$-0.417006\pi$
$853$	15.0581	0.515578	0.257789	+	0.966201i	$0.417006\pi$
$854$	89.0620	3.04764
$855$	0	0
$856$	8.28033	0.283016
$857$	−6.59884	−0.225412	−0.112706	−	0.993628i	$-0.535952\pi$
$857$	−6.59884	−0.225412	−0.112706	+	0.993628i	$0.535952\pi$
$858$	0	0
$859$	10.6264	0.362569	0.181284	−	0.983431i	$-0.441975\pi$
$859$	10.6264	0.362569	0.181284	+	0.983431i	$0.441975\pi$
$860$	28.7413	0.980069
$861$	0	0
$862$	−76.8989	−2.61919
$863$	−41.2826	−1.40528	−0.702638	−	0.711548i	$-0.747994\pi$
$863$	−41.2826	−1.40528	−0.702638	+	0.711548i	$0.747994\pi$
$864$	0	0
$865$	5.02749	0.170940
$866$	−12.3483	−0.419614
$867$	0	0
$868$	−66.4072	−2.25401
$869$	4.02759	0.136627
$870$	0	0
$871$	−70.1888	−2.37826
$872$	−3.70622	−0.125509
$873$	0	0
$874$	1.58321	0.0535528
$875$	−41.5253	−1.40381
$876$	0	0
$877$	24.4873	0.826878	0.413439	−	0.910532i	$-0.364327\pi$
$877$	24.4873	0.826878	0.413439	+	0.910532i	$0.364327\pi$
$878$	−72.3008	−2.44003
$879$	0	0
$880$	1.09024	0.0367519
$881$	−12.3751	−0.416929	−0.208465	−	0.978030i	$-0.566847\pi$
$881$	−12.3751	−0.416929	−0.208465	+	0.978030i	$0.566847\pi$
$882$	0	0
$883$	28.8587	0.971171	0.485586	−	0.874189i	$-0.338606\pi$
$883$	28.8587	0.971171	0.485586	+	0.874189i	$0.338606\pi$
$884$	123.310	4.14737
$885$	0	0
$886$	29.9434	1.00597
$887$	46.9153	1.57526	0.787631	−	0.616147i	$-0.211307\pi$
$887$	46.9153	1.57526	0.787631	+	0.616147i	$0.211307\pi$
$888$	0	0
$889$	−1.33236	−0.0446860
$890$	−20.5432	−0.688611
$891$	0	0
$892$	−45.7081	−1.53042
$893$	5.37361	0.179821
$894$	0	0
$895$	12.0846	0.403943
$896$	66.8828	2.23440
$897$	0	0
$898$	78.5600	2.62158
$899$	0	0
$900$	0	0
$901$	−3.13145	−0.104324
$902$	−10.4372	−0.347521
$903$	0	0
$904$	−13.9900	−0.465300
$905$	32.7560	1.08885
$906$	0	0
$907$	21.2233	0.704707	0.352354	−	0.935867i	$-0.385381\pi$
$907$	21.2233	0.704707	0.352354	+	0.935867i	$0.385381\pi$
$908$	91.3242	3.03070
$909$	0	0
$910$	81.7735	2.71077
$911$	46.1944	1.53049	0.765245	−	0.643739i	$-0.222618\pi$
$911$	46.1944	1.53049	0.765245	+	0.643739i	$0.222618\pi$
$912$	0	0
$913$	−6.39466	−0.211632
$914$	−17.7189	−0.586091
$915$	0	0
$916$	36.9660	1.22139
$917$	2.60089	0.0858888
$918$	0	0
$919$	−49.8555	−1.64458	−0.822290	−	0.569068i	$-0.807304\pi$
$919$	−49.8555	−1.64458	−0.822290	+	0.569068i	$0.807304\pi$
$920$	0.944761	0.0311478
$921$	0	0
$922$	39.3706	1.29660
$923$	−18.2825	−0.601775
$924$	0	0
$925$	13.5086	0.444160
$926$	21.6219	0.710538
$927$	0	0
$928$	0	0
$929$	−16.6870	−0.547484	−0.273742	−	0.961803i	$-0.588261\pi$
$929$	−16.6870	−0.547484	−0.273742	+	0.961803i	$0.588261\pi$
$930$	0	0
$931$	−20.7448	−0.679883
$932$	18.5659	0.608147
$933$	0	0
$934$	−58.8370	−1.92521
$935$	10.5665	0.345561
$936$	0	0
$937$	22.6452	0.739787	0.369893	−	0.929074i	$-0.379394\pi$
$937$	22.6452	0.739787	0.369893	+	0.929074i	$0.379394\pi$
$938$	102.795	3.35639
$939$	0	0
$940$	7.83366	0.255506
$941$	−4.03174	−0.131431	−0.0657155	−	0.997838i	$-0.520933\pi$
$941$	−4.03174	−0.131431	−0.0657155	+	0.997838i	$0.520933\pi$
$942$	0	0
$943$	−0.839892	−0.0273506
$944$	−8.03648	−0.261565
$945$	0	0
$946$	8.57612	0.278834
$947$	−28.1604	−0.915090	−0.457545	−	0.889186i	$-0.651271\pi$
$947$	−28.1604	−0.915090	−0.457545	+	0.889186i	$0.651271\pi$
$948$	0	0
$949$	5.31389	0.172496
$950$	14.2170	0.461262
$951$	0	0
$952$	−73.9251	−2.39592
$953$	−32.5042	−1.05291	−0.526457	−	0.850201i	$-0.676480\pi$
$953$	−32.5042	−1.05291	−0.526457	+	0.850201i	$0.676480\pi$
$954$	0	0
$955$	31.2616	1.01160
$956$	2.57620	0.0833202
$957$	0	0
$958$	−53.4355	−1.72642
$959$	51.0775	1.64938
$960$	0	0
$961$	1.87669	0.0605384
$962$	−122.453	−3.94803
$963$	0	0
$964$	−84.7671	−2.73016
$965$	23.7655	0.765040
$966$	0	0
$967$	−1.32129	−0.0424899	−0.0212450	−	0.999774i	$-0.506763\pi$
$967$	−1.32129	−0.0424899	−0.0212450	+	0.999774i	$0.506763\pi$
$968$	−33.1817	−1.06650
$969$	0	0
$970$	44.1721	1.41828
$971$	39.4627	1.26642	0.633209	−	0.773981i	$-0.281737\pi$
$971$	39.4627	1.26642	0.633209	+	0.773981i	$0.281737\pi$
$972$	0	0
$973$	−12.5173	−0.401288
$974$	3.56066	0.114091
$975$	0	0
$976$	−7.77803	−0.248969
$977$	−18.7051	−0.598428	−0.299214	−	0.954186i	$-0.596724\pi$
$977$	−18.7051	−0.598428	−0.299214	+	0.954186i	$0.596724\pi$
$978$	0	0
$979$	−3.85369	−0.123164
$980$	−30.2418	−0.966037
$981$	0	0
$982$	46.7633	1.49228
$983$	8.05331	0.256861	0.128430	−	0.991719i	$-0.459006\pi$
$983$	8.05331	0.256861	0.128430	+	0.991719i	$0.459006\pi$
$984$	0	0
$985$	−26.8861	−0.856662
$986$	0	0
$987$	0	0
$988$	−81.0195	−2.57758
$989$	0.690129	0.0219448
$990$	0	0
$991$	−60.4922	−1.92160	−0.960800	−	0.277244i	$-0.910579\pi$
$991$	−60.4922	−1.92160	−0.960800	+	0.277244i	$0.910579\pi$
$992$	−27.6633	−0.878310
$993$	0	0
$994$	26.7757	0.849274
$995$	−24.5547	−0.778438
$996$	0	0
$997$	2.96516	0.0939076	0.0469538	−	0.998897i	$-0.485049\pi$
$997$	2.96516	0.0939076	0.0469538	+	0.998897i	$0.485049\pi$
$998$	80.5446	2.54960
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bt.1.11		12
3.2	odd	2		2523.2.a.s.1.2		12
29.19	odd	28		261.2.o.b.100.1		24
29.26	odd	28		261.2.o.b.154.1		24
29.28	even	2		7569.2.a.bn.1.2		12
87.26	even	28		87.2.i.a.67.4	yes	24
87.77	even	28		87.2.i.a.13.4	✓	24
87.86	odd	2		2523.2.a.v.1.11		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.i.a.13.4	✓	24	87.77	even	28
87.2.i.a.67.4	yes	24	87.26	even	28
261.2.o.b.100.1		24	29.19	odd	28
261.2.o.b.154.1		24	29.26	odd	28
2523.2.a.s.1.2		12	3.2	odd	2
2523.2.a.v.1.11		12	87.86	odd	2
7569.2.a.bn.1.2		12	29.28	even	2
7569.2.a.bt.1.11		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)

\(f(q)\)	\(=\)	\(q+2.32079 q^{2} +3.38605 q^{4} -1.90061 q^{5} -3.42040 q^{7} +3.21673 q^{8} -4.41092 q^{10} -0.827442 q^{11} +5.42009 q^{13} -7.93802 q^{14} +0.693250 q^{16} +6.71892 q^{17} -4.41458 q^{19} -6.43558 q^{20} -1.92032 q^{22} -0.154530 q^{23} -1.38766 q^{25} +12.5789 q^{26} -11.5817 q^{28} +5.73382 q^{31} -4.82458 q^{32} +15.5932 q^{34} +6.50087 q^{35} -9.73479 q^{37} -10.2453 q^{38} -6.11377 q^{40} +5.43514 q^{41} -4.46599 q^{43} -2.80176 q^{44} -0.358631 q^{46} -1.21724 q^{47} +4.69915 q^{49} -3.22047 q^{50} +18.3527 q^{52} -0.466064 q^{53} +1.57265 q^{55} -11.0025 q^{56} -11.5925 q^{59} -11.2197 q^{61} +13.3070 q^{62} -12.5833 q^{64} -10.3015 q^{65} -12.9498 q^{67} +22.7506 q^{68} +15.0871 q^{70} -3.37310 q^{71} +0.980407 q^{73} -22.5924 q^{74} -14.9480 q^{76} +2.83018 q^{77} -4.86752 q^{79} -1.31760 q^{80} +12.6138 q^{82} +7.72823 q^{83} -12.7701 q^{85} -10.3646 q^{86} -2.66166 q^{88} +4.65736 q^{89} -18.5389 q^{91} -0.523246 q^{92} -2.82496 q^{94} +8.39042 q^{95} -10.0143 q^{97} +10.9057 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{2} + 8 q^{4} - 2 q^{5} - 10 q^{7} - 20 q^{10} + 14 q^{11} - 16 q^{13} - 4 q^{16} + 22 q^{17} - 16 q^{19} - 4 q^{20} + 12 q^{22} - 2 q^{23} - 2 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{31} + 16 q^{32}+ \cdots + 6 q^{98}+O(q^{100}) \) 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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