LMFDB - Embedded newform 7569.2.a.bm.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7569,2,Mod(1,7569)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [9,5,0,11,4,0,5,24,0,0,-1,0,1,9,0,35,2,0,-9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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} - 4x^{8} - 6x^{7} + 33x^{6} + 6x^{5} - 90x^{4} + 21x^{3} + 84x^{2} - 36x - 8 $ x^9 - 4x^8 - 6x^7 + 33x^6 + 6x^5 - 90x^4 + 21x^3 + 84x^2 - 36x - 8
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			1.2
Root			$2.21072$ 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)

$f(q)$	$=$	$q-1.21072 q^{2} -0.534161 q^{4} -1.79844 q^{5} +4.27693 q^{7} +3.06816 q^{8} +2.17740 q^{10} +1.05626 q^{11} -5.28193 q^{13} -5.17816 q^{14} -2.64635 q^{16} +5.61769 q^{17} -6.41175 q^{19} +0.960655 q^{20} -1.27883 q^{22} -2.04633 q^{23} -1.76563 q^{25} +6.39492 q^{26} -2.28457 q^{28} -3.82471 q^{31} -2.93233 q^{32} -6.80144 q^{34} -7.69178 q^{35} +0.350297 q^{37} +7.76282 q^{38} -5.51788 q^{40} -3.62240 q^{41} -1.74900 q^{43} -0.564212 q^{44} +2.47753 q^{46} +6.98005 q^{47} +11.2921 q^{49} +2.13768 q^{50} +2.82140 q^{52} +7.81869 q^{53} -1.89961 q^{55} +13.1223 q^{56} -0.382668 q^{59} +5.46644 q^{61} +4.63064 q^{62} +8.84292 q^{64} +9.49920 q^{65} +7.81166 q^{67} -3.00075 q^{68} +9.31258 q^{70} +15.6115 q^{71} +2.54289 q^{73} -0.424110 q^{74} +3.42491 q^{76} +4.51754 q^{77} -10.2680 q^{79} +4.75929 q^{80} +4.38571 q^{82} -1.52139 q^{83} -10.1030 q^{85} +2.11755 q^{86} +3.24076 q^{88} -17.9444 q^{89} -22.5904 q^{91} +1.09307 q^{92} -8.45087 q^{94} +11.5311 q^{95} +6.80968 q^{97} -13.6716 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) $ 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * 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 $

$2$	−1.21072	−0.856107	−0.428054	−	0.903753i	$-0.640801\pi$
$2$	−1.21072	−0.856107	−0.428054	+	0.903753i	$0.640801\pi$
$3$	0	0
$3$	0	0
$4$	−0.534161	−0.267081
$5$	−1.79844	−0.804285	−0.402142	−	0.915577i	$-0.631734\pi$
$5$	−1.79844	−0.804285	−0.402142	+	0.915577i	$0.631734\pi$
$6$	0	0
$7$	4.27693	1.61653	0.808264	−	0.588821i	$-0.200408\pi$
$7$	4.27693	1.61653	0.808264	+	0.588821i	$0.200408\pi$
$8$	3.06816	1.08476
$9$	0	0
$10$	2.17740	0.688554
$11$	1.05626	0.318474	0.159237	−	0.987240i	$-0.449097\pi$
$11$	1.05626	0.318474	0.159237	+	0.987240i	$0.449097\pi$
$12$	0	0
$13$	−5.28193	−1.46494	−0.732471	−	0.680798i	$-0.761633\pi$
$13$	−5.28193	−1.46494	−0.732471	+	0.680798i	$0.761633\pi$
$14$	−5.17816	−1.38392
$15$	0	0
$16$	−2.64635	−0.661587
$17$	5.61769	1.36249	0.681245	−	0.732056i	$-0.261439\pi$
$17$	5.61769	1.36249	0.681245	+	0.732056i	$0.261439\pi$
$18$	0	0
$19$	−6.41175	−1.47096	−0.735478	−	0.677548i	$-0.763042\pi$
$19$	−6.41175	−1.47096	−0.735478	+	0.677548i	$0.763042\pi$
$20$	0.960655	0.214809
$21$	0	0
$22$	−1.27883	−0.272648
$23$	−2.04633	−0.426689	−0.213345	−	0.976977i	$-0.568436\pi$
$23$	−2.04633	−0.426689	−0.213345	+	0.976977i	$0.568436\pi$
$24$	0	0
$25$	−1.76563	−0.353126
$26$	6.39492	1.25415
$27$	0	0
$28$	−2.28457	−0.431743
$29$	0	0
$29$	0	0
$30$	0	0
$31$	−3.82471	−0.686937	−0.343469	−	0.939164i	$-0.611602\pi$
$31$	−3.82471	−0.686937	−0.343469	+	0.939164i	$0.611602\pi$
$32$	−2.93233	−0.518367
$33$	0	0
$34$	−6.80144	−1.16644
$35$	−7.69178	−1.30015
$36$	0	0
$37$	0.350297	0.0575884	0.0287942	−	0.999585i	$-0.490833\pi$
$37$	0.350297	0.0575884	0.0287942	+	0.999585i	$0.490833\pi$
$38$	7.76282	1.25930
$39$	0	0
$40$	−5.51788	−0.872453
$41$	−3.62240	−0.565725	−0.282862	−	0.959161i	$-0.591284\pi$
$41$	−3.62240	−0.565725	−0.282862	+	0.959161i	$0.591284\pi$
$42$	0	0
$43$	−1.74900	−0.266721	−0.133360	−	0.991068i	$-0.542577\pi$
$43$	−1.74900	−0.266721	−0.133360	+	0.991068i	$0.542577\pi$
$44$	−0.564212	−0.0850582
$45$	0	0
$46$	2.47753	0.365292
$47$	6.98005	1.01814	0.509072	−	0.860724i	$-0.329989\pi$
$47$	6.98005	1.01814	0.509072	+	0.860724i	$0.329989\pi$
$48$	0	0
$49$	11.2921	1.61316
$50$	2.13768	0.302314
$51$	0	0
$52$	2.82140	0.391258
$53$	7.81869	1.07398	0.536990	−	0.843589i	$-0.319561\pi$
$53$	7.81869	1.07398	0.536990	+	0.843589i	$0.319561\pi$
$54$	0	0
$55$	−1.89961	−0.256144
$56$	13.1223	1.75354
$57$	0	0
$58$	0	0
$59$	−0.382668	−0.0498191	−0.0249096	−	0.999690i	$-0.507930\pi$
$59$	−0.382668	−0.0498191	−0.0249096	+	0.999690i	$0.507930\pi$
$60$	0	0
$61$	5.46644	0.699906	0.349953	−	0.936767i	$-0.386197\pi$
$61$	5.46644	0.699906	0.349953	+	0.936767i	$0.386197\pi$
$62$	4.63064	0.588092
$63$	0	0
$64$	8.84292	1.10537
$65$	9.49920	1.17823
$66$	0	0
$67$	7.81166	0.954346	0.477173	−	0.878809i	$-0.341662\pi$
$67$	7.81166	0.954346	0.477173	+	0.878809i	$0.341662\pi$
$68$	−3.00075	−0.363895
$69$	0	0
$70$	9.31258	1.11307
$71$	15.6115	1.85274	0.926370	−	0.376615i	$-0.122912\pi$
$71$	15.6115	1.85274	0.926370	+	0.376615i	$0.122912\pi$
$72$	0	0
$73$	2.54289	0.297623	0.148811	−	0.988866i	$-0.452455\pi$
$73$	2.54289	0.297623	0.148811	+	0.988866i	$0.452455\pi$
$74$	−0.424110	−0.0493018
$75$	0	0
$76$	3.42491	0.392864
$77$	4.51754	0.514822
$78$	0	0
$79$	−10.2680	−1.15524	−0.577619	−	0.816306i	$-0.696018\pi$
$79$	−10.2680	−1.15524	−0.577619	+	0.816306i	$0.696018\pi$
$80$	4.75929	0.532104
$81$	0	0
$82$	4.38571	0.484321
$83$	−1.52139	−0.166994	−0.0834971	−	0.996508i	$-0.526609\pi$
$83$	−1.52139	−0.166994	−0.0834971	+	0.996508i	$0.526609\pi$
$84$	0	0
$85$	−10.1030	−1.09583
$86$	2.11755	0.228341
$87$	0	0
$88$	3.24076	0.345467
$89$	−17.9444	−1.90210	−0.951049	−	0.309041i	$-0.899992\pi$
$89$	−17.9444	−1.90210	−0.951049	+	0.309041i	$0.899992\pi$
$90$	0	0
$91$	−22.5904	−2.36812
$92$	1.09307	0.113960
$93$	0	0
$94$	−8.45087	−0.871641
$95$	11.5311	1.18307
$96$	0	0
$97$	6.80968	0.691418	0.345709	−	0.938342i	$-0.387638\pi$
$97$	6.80968	0.691418	0.345709	+	0.938342i	$0.387638\pi$
$98$	−13.6716	−1.38104
$99$	0	0
$100$	0.943132	0.0943132
$101$	4.72573	0.470228	0.235114	−	0.971968i	$-0.424454\pi$
$101$	4.72573	0.470228	0.235114	+	0.971968i	$0.424454\pi$
$102$	0	0
$103$	3.49491	0.344364	0.172182	−	0.985065i	$-0.444918\pi$
$103$	3.49491	0.344364	0.172182	+	0.985065i	$0.444918\pi$
$104$	−16.2058	−1.58911
$105$	0	0
$106$	−9.46624	−0.919442
$107$	4.92192	0.475820	0.237910	−	0.971287i	$-0.423538\pi$
$107$	4.92192	0.475820	0.237910	+	0.971287i	$0.423538\pi$
$108$	0	0
$109$	−14.8847	−1.42569	−0.712846	−	0.701321i	$-0.752594\pi$
$109$	−14.8847	−1.42569	−0.712846	+	0.701321i	$0.752594\pi$
$110$	2.29989	0.219286
$111$	0	0
$112$	−11.3182	−1.06947
$113$	−6.34278	−0.596678	−0.298339	−	0.954460i	$-0.596433\pi$
$113$	−6.34278	−0.596678	−0.298339	+	0.954460i	$0.596433\pi$
$114$	0	0
$115$	3.68019	0.343180
$116$	0	0
$117$	0	0
$118$	0.463303	0.0426505
$119$	24.0265	2.20250
$120$	0	0
$121$	−9.88432	−0.898574
$122$	−6.61832	−0.599195
$123$	0	0
$124$	2.04301	0.183468
$125$	12.1675	1.08830
$126$	0	0
$127$	4.46569	0.396266	0.198133	−	0.980175i	$-0.436512\pi$
$127$	4.46569	0.396266	0.198133	+	0.980175i	$0.436512\pi$
$128$	−4.84163	−0.427944
$129$	0	0
$130$	−11.5009	−1.00869
$131$	0.00995943	0.000870160 0	0.000435080	−	1.00000i	$-0.499862\pi$
$131$	0.00995943	0.000870160 0	0.000435080		1.00000i	$0.499862\pi$
$132$	0	0
$133$	−27.4226	−2.37784
$134$	−9.45772	−0.817022
$135$	0	0
$136$	17.2359	1.47797
$137$	12.2134	1.04346	0.521732	−	0.853109i	$-0.325286\pi$
$137$	12.2134	1.04346	0.521732	+	0.853109i	$0.325286\pi$
$138$	0	0
$139$	4.94561	0.419481	0.209741	−	0.977757i	$-0.432738\pi$
$139$	4.94561	0.419481	0.209741	+	0.977757i	$0.432738\pi$
$140$	4.10865	0.347245
$141$	0	0
$142$	−18.9011	−1.58614
$143$	−5.57908	−0.466546
$144$	0	0
$145$	0	0
$146$	−3.07873	−0.254797
$147$	0	0
$148$	−0.187115	−0.0153808
$149$	−14.0140	−1.14807	−0.574035	−	0.818831i	$-0.694623\pi$
$149$	−14.0140	−1.14807	−0.574035	+	0.818831i	$0.694623\pi$
$150$	0	0
$151$	2.93392	0.238759	0.119379	−	0.992849i	$-0.461910\pi$
$151$	2.93392	0.238759	0.119379	+	0.992849i	$0.461910\pi$
$152$	−19.6722	−1.59563
$153$	0	0
$154$	−5.46947	−0.440742
$155$	6.87849	0.552493
$156$	0	0
$157$	−16.1919	−1.29226	−0.646128	−	0.763229i	$-0.723613\pi$
$157$	−16.1919	−1.29226	−0.646128	+	0.763229i	$0.723613\pi$
$158$	12.4316	0.989008
$159$	0	0
$160$	5.27360	0.416915
$161$	−8.75201	−0.689755
$162$	0	0
$163$	−13.5734	−1.06315	−0.531575	−	0.847011i	$-0.678400\pi$
$163$	−13.5734	−1.06315	−0.531575	+	0.847011i	$0.678400\pi$
$164$	1.93495	0.151094
$165$	0	0
$166$	1.84197	0.142965
$167$	16.3616	1.26610	0.633050	−	0.774111i	$-0.281803\pi$
$167$	16.3616	1.26610	0.633050	+	0.774111i	$0.281803\pi$
$168$	0	0
$169$	14.8987	1.14606
$170$	12.2319	0.938147
$171$	0	0
$172$	0.934250	0.0712359
$173$	4.46374	0.339372	0.169686	−	0.985498i	$-0.445725\pi$
$173$	4.46374	0.339372	0.169686	+	0.985498i	$0.445725\pi$
$174$	0	0
$175$	−7.55148	−0.570838
$176$	−2.79523	−0.210698
$177$	0	0
$178$	21.7256	1.62840
$179$	−3.40063	−0.254175	−0.127088	−	0.991891i	$-0.540563\pi$
$179$	−3.40063	−0.254175	−0.127088	+	0.991891i	$0.540563\pi$
$180$	0	0
$181$	6.62134	0.492160	0.246080	−	0.969250i	$-0.420857\pi$
$181$	6.62134	0.492160	0.246080	+	0.969250i	$0.420857\pi$
$182$	27.3506	2.02736
$183$	0	0
$184$	−6.27846	−0.462854
$185$	−0.629986	−0.0463175
$186$	0	0
$187$	5.93373	0.433917
$188$	−3.72847	−0.271927
$189$	0	0
$190$	−13.9609	−1.01283
$191$	17.6196	1.27491	0.637454	−	0.770489i	$-0.279988\pi$
$191$	17.6196	1.27491	0.637454	+	0.770489i	$0.279988\pi$
$192$	0	0
$193$	13.3986	0.964454	0.482227	−	0.876046i	$-0.339828\pi$
$193$	13.3986	0.964454	0.482227	+	0.876046i	$0.339828\pi$
$194$	−8.24460	−0.591928
$195$	0	0
$196$	−6.03182	−0.430844
$197$	8.20476	0.584565	0.292282	−	0.956332i	$-0.405585\pi$
$197$	8.20476	0.584565	0.292282	+	0.956332i	$0.405585\pi$
$198$	0	0
$199$	24.4954	1.73643	0.868216	−	0.496187i	$-0.165267\pi$
$199$	24.4954	1.73643	0.868216	+	0.496187i	$0.165267\pi$
$200$	−5.41723	−0.383056
$201$	0	0
$202$	−5.72153	−0.402565
$203$	0	0
$204$	0	0
$205$	6.51466	0.455004
$206$	−4.23135	−0.294812
$207$	0	0
$208$	13.9778	0.969187
$209$	−6.77246	−0.468461
$210$	0	0
$211$	2.21143	0.152241	0.0761205	−	0.997099i	$-0.475747\pi$
$211$	2.21143	0.152241	0.0761205	+	0.997099i	$0.475747\pi$
$212$	−4.17644	−0.286839
$213$	0	0
$214$	−5.95905	−0.407353
$215$	3.14547	0.214519
$216$	0	0
$217$	−16.3580	−1.11045
$218$	18.0211	1.22054
$219$	0	0
$220$	1.01470	0.0684110
$221$	−29.6722	−1.99597
$222$	0	0
$223$	2.70437	0.181098	0.0905491	−	0.995892i	$-0.471138\pi$
$223$	2.70437	0.181098	0.0905491	+	0.995892i	$0.471138\pi$
$224$	−12.5414	−0.837955
$225$	0	0
$226$	7.67932	0.510820
$227$	−14.4225	−0.957258	−0.478629	−	0.878017i	$-0.658866\pi$
$227$	−14.4225	−0.957258	−0.478629	+	0.878017i	$0.658866\pi$
$228$	0	0
$229$	−25.7467	−1.70139	−0.850694	−	0.525662i	$-0.823818\pi$
$229$	−25.7467	−1.70139	−0.850694	+	0.525662i	$0.823818\pi$
$230$	−4.45567	−0.293798
$231$	0	0
$232$	0	0
$233$	22.6039	1.48083	0.740417	−	0.672148i	$-0.234628\pi$
$233$	22.6039	1.48083	0.740417	+	0.672148i	$0.234628\pi$
$234$	0	0
$235$	−12.5532	−0.818878
$236$	0.204406	0.0133057
$237$	0	0
$238$	−29.0893	−1.88558
$239$	−23.4233	−1.51513	−0.757565	−	0.652760i	$-0.773611\pi$
$239$	−23.4233	−1.51513	−0.757565	+	0.652760i	$0.773611\pi$
$240$	0	0
$241$	22.3826	1.44179	0.720896	−	0.693043i	$-0.243730\pi$
$241$	22.3826	1.44179	0.720896	+	0.693043i	$0.243730\pi$
$242$	11.9671	0.769276
$243$	0	0
$244$	−2.91996	−0.186932
$245$	−20.3082	−1.29744
$246$	0	0
$247$	33.8664	2.15487
$248$	−11.7348	−0.745160
$249$	0	0
$250$	−14.7315	−0.931700
$251$	−5.95981	−0.376180	−0.188090	−	0.982152i	$-0.560230\pi$
$251$	−5.95981	−0.376180	−0.188090	+	0.982152i	$0.560230\pi$
$252$	0	0
$253$	−2.16145	−0.135889
$254$	−5.40669	−0.339246
$255$	0	0
$256$	−11.8240	−0.739000
$257$	15.0723	0.940185	0.470092	−	0.882617i	$-0.344221\pi$
$257$	15.0723	0.940185	0.470092	+	0.882617i	$0.344221\pi$
$258$	0	0
$259$	1.49819	0.0930932
$260$	−5.07411	−0.314683
$261$	0	0
$262$	−0.0120581	−0.000744950 0
$263$	−18.7252	−1.15465	−0.577324	−	0.816515i	$-0.695903\pi$
$263$	−18.7252	−1.15465	−0.577324	+	0.816515i	$0.695903\pi$
$264$	0	0
$265$	−14.0614	−0.863786
$266$	33.2010	2.03569
$267$	0	0
$268$	−4.17269	−0.254887
$269$	−23.1259	−1.41001	−0.705006	−	0.709202i	$-0.749056\pi$
$269$	−23.1259	−1.41001	−0.705006	+	0.709202i	$0.749056\pi$
$270$	0	0
$271$	−21.3469	−1.29673	−0.648365	−	0.761330i	$-0.724547\pi$
$271$	−21.3469	−1.29673	−0.648365	+	0.761330i	$0.724547\pi$
$272$	−14.8664	−0.901406
$273$	0	0
$274$	−14.7870	−0.893317
$275$	−1.86496	−0.112461
$276$	0	0
$277$	−8.32181	−0.500009	−0.250004	−	0.968245i	$-0.580432\pi$
$277$	−8.32181	−0.500009	−0.250004	+	0.968245i	$0.580432\pi$
$278$	−5.98774	−0.359121
$279$	0	0
$280$	−23.5996	−1.41034
$281$	28.9202	1.72524	0.862618	−	0.505856i	$-0.168823\pi$
$281$	28.9202	1.72524	0.862618	+	0.505856i	$0.168823\pi$
$282$	0	0
$283$	29.9799	1.78212	0.891060	−	0.453885i	$-0.149962\pi$
$283$	29.9799	1.78212	0.891060	+	0.453885i	$0.149962\pi$
$284$	−8.33904	−0.494831
$285$	0	0
$286$	6.75469	0.399413
$287$	−15.4928	−0.914509
$288$	0	0
$289$	14.5584	0.856378
$290$	0	0
$291$	0	0
$292$	−1.35831	−0.0794893
$293$	8.75406	0.511418	0.255709	−	0.966754i	$-0.417691\pi$
$293$	8.75406	0.511418	0.255709	+	0.966754i	$0.417691\pi$
$294$	0	0
$295$	0.688204	0.0400688
$296$	1.07476	0.0624694
$297$	0	0
$298$	16.9670	0.982871
$299$	10.8086	0.625075
$300$	0	0
$301$	−7.48036	−0.431161
$302$	−3.55215	−0.204403
$303$	0	0
$304$	16.9677	0.973166
$305$	−9.83105	−0.562924
$306$	0	0
$307$	33.9161	1.93570	0.967848	−	0.251536i	$-0.0809358\pi$
$307$	33.9161	1.93570	0.967848	+	0.251536i	$0.0809358\pi$
$308$	−2.41310	−0.137499
$309$	0	0
$310$	−8.32791	−0.472993
$311$	8.12574	0.460769	0.230384	−	0.973100i	$-0.426002\pi$
$311$	8.12574	0.460769	0.230384	+	0.973100i	$0.426002\pi$
$312$	0	0
$313$	2.52738	0.142856	0.0714280	−	0.997446i	$-0.477244\pi$
$313$	2.52738	0.142856	0.0714280	+	0.997446i	$0.477244\pi$
$314$	19.6039	1.10631
$315$	0	0
$316$	5.48476	0.308542
$317$	16.6383	0.934499	0.467250	−	0.884125i	$-0.345245\pi$
$317$	16.6383	0.934499	0.467250	+	0.884125i	$0.345245\pi$
$318$	0	0
$319$	0	0
$320$	−15.9034	−0.889028
$321$	0	0
$322$	10.5962	0.590504
$323$	−36.0192	−2.00416
$324$	0	0
$325$	9.32593	0.517309
$326$	16.4336	0.910170
$327$	0	0
$328$	−11.1141	−0.613673
$329$	29.8532	1.64586
$330$	0	0
$331$	−12.9286	−0.710619	−0.355309	−	0.934749i	$-0.615625\pi$
$331$	−12.9286	−0.710619	−0.355309	+	0.934749i	$0.615625\pi$
$332$	0.812667	0.0446009
$333$	0	0
$334$	−19.8093	−1.08392
$335$	−14.0488	−0.767566
$336$	0	0
$337$	5.56273	0.303021	0.151511	−	0.988456i	$-0.451586\pi$
$337$	5.56273	0.303021	0.151511	+	0.988456i	$0.451586\pi$
$338$	−18.0382	−0.981148
$339$	0	0
$340$	5.39666	0.292675
$341$	−4.03988	−0.218772
$342$	0	0
$343$	18.3571	0.991192
$344$	−5.36621	−0.289327
$345$	0	0
$346$	−5.40433	−0.290539
$347$	8.78686	0.471703	0.235852	−	0.971789i	$-0.424212\pi$
$347$	8.78686	0.471703	0.235852	+	0.971789i	$0.424212\pi$
$348$	0	0
$349$	15.1943	0.813332	0.406666	−	0.913577i	$-0.366691\pi$
$349$	15.1943	0.813332	0.406666	+	0.913577i	$0.366691\pi$
$350$	9.14271	0.488698
$351$	0	0
$352$	−3.09729	−0.165086
$353$	12.3667	0.658214	0.329107	−	0.944293i	$-0.393252\pi$
$353$	12.3667	0.658214	0.329107	+	0.944293i	$0.393252\pi$
$354$	0	0
$355$	−28.0762	−1.49013
$356$	9.58518	0.508014
$357$	0	0
$358$	4.11721	0.217601
$359$	−16.0164	−0.845312	−0.422656	−	0.906290i	$-0.638902\pi$
$359$	−16.0164	−0.845312	−0.422656	+	0.906290i	$0.638902\pi$
$360$	0	0
$361$	22.1105	1.16371
$362$	−8.01657	−0.421342
$363$	0	0
$364$	12.0669	0.632479
$365$	−4.57323	−0.239374
$366$	0	0
$367$	22.2447	1.16116	0.580581	−	0.814203i	$-0.302826\pi$
$367$	22.2447	1.16116	0.580581	+	0.814203i	$0.302826\pi$
$368$	5.41530	0.282292
$369$	0	0
$370$	0.762735	0.0396527
$371$	33.4400	1.73612
$372$	0	0
$373$	6.93029	0.358837	0.179418	−	0.983773i	$-0.442578\pi$
$373$	6.93029	0.358837	0.179418	+	0.983773i	$0.442578\pi$
$374$	−7.18407	−0.371480
$375$	0	0
$376$	21.4159	1.10444
$377$	0	0
$378$	0	0
$379$	7.78848	0.400067	0.200034	−	0.979789i	$-0.435895\pi$
$379$	7.78848	0.400067	0.200034	+	0.979789i	$0.435895\pi$
$380$	−6.15948	−0.315975
$381$	0	0
$382$	−21.3323	−1.09146
$383$	−25.4625	−1.30107	−0.650537	−	0.759475i	$-0.725456\pi$
$383$	−25.4625	−1.30107	−0.650537	+	0.759475i	$0.725456\pi$
$384$	0	0
$385$	−8.12451	−0.414063
$386$	−16.2220	−0.825676
$387$	0	0
$388$	−3.63747	−0.184664
$389$	10.9838	0.556902	0.278451	−	0.960450i	$-0.410179\pi$
$389$	10.9838	0.556902	0.278451	+	0.960450i	$0.410179\pi$
$390$	0	0
$391$	−11.4956	−0.581359
$392$	34.6460	1.74989
$393$	0	0
$394$	−9.93365	−0.500450
$395$	18.4663	0.929140
$396$	0	0
$397$	5.10047	0.255985	0.127993	−	0.991775i	$-0.459147\pi$
$397$	5.10047	0.255985	0.127993	+	0.991775i	$0.459147\pi$
$398$	−29.6570	−1.48657
$399$	0	0
$400$	4.67247	0.233624
$401$	19.5421	0.975886	0.487943	−	0.872875i	$-0.337747\pi$
$401$	19.5421	0.975886	0.487943	+	0.872875i	$0.337747\pi$
$402$	0	0
$403$	20.2018	1.00632
$404$	−2.52430	−0.125589
$405$	0	0
$406$	0	0
$407$	0.370004	0.0183404
$408$	0	0
$409$	17.9244	0.886305	0.443152	−	0.896446i	$-0.353860\pi$
$409$	17.9244	0.886305	0.443152	+	0.896446i	$0.353860\pi$
$410$	−7.88742	−0.389532
$411$	0	0
$412$	−1.86685	−0.0919730
$413$	−1.63664	−0.0805340
$414$	0	0
$415$	2.73612	0.134311
$416$	15.4883	0.759378
$417$	0	0
$418$	8.19954	0.401053
$419$	−3.73821	−0.182623	−0.0913116	−	0.995822i	$-0.529106\pi$
$419$	−3.73821	−0.182623	−0.0913116	+	0.995822i	$0.529106\pi$
$420$	0	0
$421$	−8.28779	−0.403922	−0.201961	−	0.979394i	$-0.564731\pi$
$421$	−8.28779	−0.403922	−0.201961	+	0.979394i	$0.564731\pi$
$422$	−2.67742	−0.130335
$423$	0	0
$424$	23.9890	1.16501
$425$	−9.91876	−0.481131
$426$	0	0
$427$	23.3796	1.13142
$428$	−2.62910	−0.127082
$429$	0	0
$430$	−3.80828	−0.183651
$431$	−19.1294	−0.921432	−0.460716	−	0.887548i	$-0.652407\pi$
$431$	−19.1294	−0.921432	−0.460716	+	0.887548i	$0.652407\pi$
$432$	0	0
$433$	16.1509	0.776163	0.388081	−	0.921625i	$-0.373138\pi$
$433$	16.1509	0.776163	0.388081	+	0.921625i	$0.373138\pi$
$434$	19.8049	0.950667
$435$	0	0
$436$	7.95081	0.380775
$437$	13.1206	0.627641
$438$	0	0
$439$	2.96836	0.141672	0.0708360	−	0.997488i	$-0.477433\pi$
$439$	2.96836	0.141672	0.0708360	+	0.997488i	$0.477433\pi$
$440$	−5.82830	−0.277853
$441$	0	0
$442$	35.9247	1.70876
$443$	10.4916	0.498471	0.249235	−	0.968443i	$-0.419821\pi$
$443$	10.4916	0.498471	0.249235	+	0.968443i	$0.419821\pi$
$444$	0	0
$445$	32.2718	1.52983
$446$	−3.27423	−0.155039
$447$	0	0
$448$	37.8206	1.78685
$449$	−4.40325	−0.207802	−0.103901	−	0.994588i	$-0.533133\pi$
$449$	−4.40325	−0.207802	−0.103901	+	0.994588i	$0.533133\pi$
$450$	0	0
$451$	−3.82619	−0.180168
$452$	3.38807	0.159361
$453$	0	0
$454$	17.4616	0.819515
$455$	40.6274	1.90464
$456$	0	0
$457$	11.3485	0.530860	0.265430	−	0.964130i	$-0.414486\pi$
$457$	11.3485	0.530860	0.265430	+	0.964130i	$0.414486\pi$
$458$	31.1720	1.45657
$459$	0	0
$460$	−1.96582	−0.0916566
$461$	19.7092	0.917947	0.458973	−	0.888450i	$-0.348217\pi$
$461$	19.7092	0.917947	0.458973	+	0.888450i	$0.348217\pi$
$462$	0	0
$463$	0.753315	0.0350095	0.0175048	−	0.999847i	$-0.494428\pi$
$463$	0.753315	0.0350095	0.0175048	+	0.999847i	$0.494428\pi$
$464$	0	0
$465$	0	0
$466$	−27.3670	−1.26775
$467$	10.4450	0.483339	0.241670	−	0.970359i	$-0.422305\pi$
$467$	10.4450	0.483339	0.241670	+	0.970359i	$0.422305\pi$
$468$	0	0
$469$	33.4099	1.54273
$470$	15.1983	0.701047
$471$	0	0
$472$	−1.17408	−0.0540416
$473$	−1.84740	−0.0849435
$474$	0	0
$475$	11.3208	0.519433
$476$	−12.8340	−0.588246
$477$	0	0
$478$	28.3591	1.29711
$479$	−13.9781	−0.638673	−0.319337	−	0.947641i	$-0.603460\pi$
$479$	−13.9781	−0.638673	−0.319337	+	0.947641i	$0.603460\pi$
$480$	0	0
$481$	−1.85024	−0.0843637
$482$	−27.0991	−1.23433
$483$	0	0
$484$	5.27982	0.239992
$485$	−12.2468	−0.556097
$486$	0	0
$487$	−1.46028	−0.0661718	−0.0330859	−	0.999453i	$-0.510533\pi$
$487$	−1.46028	−0.0661718	−0.0330859	+	0.999453i	$0.510533\pi$
$488$	16.7719	0.759228
$489$	0	0
$490$	24.5875	1.11075
$491$	−12.1522	−0.548421	−0.274210	−	0.961670i	$-0.588416\pi$
$491$	−12.1522	−0.548421	−0.274210	+	0.961670i	$0.588416\pi$
$492$	0	0
$493$	0	0
$494$	−41.0027	−1.84480
$495$	0	0
$496$	10.1215	0.454469
$497$	66.7691	2.99501
$498$	0	0
$499$	−35.9679	−1.61014	−0.805071	−	0.593178i	$-0.797873\pi$
$499$	−35.9679	−1.61014	−0.805071	+	0.593178i	$0.797873\pi$
$500$	−6.49943	−0.290664
$501$	0	0
$502$	7.21565	0.322050
$503$	−14.3546	−0.640041	−0.320020	−	0.947411i	$-0.603690\pi$
$503$	−14.3546	−0.640041	−0.320020	+	0.947411i	$0.603690\pi$
$504$	0	0
$505$	−8.49892	−0.378197
$506$	2.61691	0.116336
$507$	0	0
$508$	−2.38540	−0.105835
$509$	31.0049	1.37427	0.687135	−	0.726530i	$-0.258868\pi$
$509$	31.0049	1.37427	0.687135	+	0.726530i	$0.258868\pi$
$510$	0	0
$511$	10.8758	0.481116
$512$	23.9988	1.06061
$513$	0	0
$514$	−18.2483	−0.804899
$515$	−6.28537	−0.276967
$516$	0	0
$517$	7.37273	0.324252
$518$	−1.81389	−0.0796978
$519$	0	0
$520$	29.1450	1.27809
$521$	14.9084	0.653150	0.326575	−	0.945171i	$-0.394105\pi$
$521$	14.9084	0.653150	0.326575	+	0.945171i	$0.394105\pi$
$522$	0	0
$523$	−12.9505	−0.566284	−0.283142	−	0.959078i	$-0.591377\pi$
$523$	−12.9505	−0.566284	−0.283142	+	0.959078i	$0.591377\pi$
$524$	−0.00531995	−0.000232403 0
$525$	0	0
$526$	22.6710	0.988502
$527$	−21.4860	−0.935945
$528$	0	0
$529$	−18.8125	−0.817936
$530$	17.0244	0.739493
$531$	0	0
$532$	14.6481	0.635076
$533$	19.1333	0.828754
$534$	0	0
$535$	−8.85175	−0.382694
$536$	23.9674	1.03523
$537$	0	0
$538$	27.9990	1.20712
$539$	11.9274	0.513750
$540$	0	0
$541$	−10.8732	−0.467475	−0.233737	−	0.972300i	$-0.575096\pi$
$541$	−10.8732	−0.467475	−0.233737	+	0.972300i	$0.575096\pi$
$542$	25.8450	1.11014
$543$	0	0
$544$	−16.4729	−0.706270
$545$	26.7691	1.14666
$546$	0	0
$547$	−23.1301	−0.988970	−0.494485	−	0.869186i	$-0.664643\pi$
$547$	−23.1301	−0.988970	−0.494485	+	0.869186i	$0.664643\pi$
$548$	−6.52394	−0.278689
$549$	0	0
$550$	2.25794	0.0962790
$551$	0	0
$552$	0	0
$553$	−43.9154	−1.86747
$554$	10.0754	0.428061
$555$	0	0
$556$	−2.64175	−0.112035
$557$	43.1297	1.82746	0.913731	−	0.406319i	$-0.133188\pi$
$557$	43.1297	1.82746	0.913731	+	0.406319i	$0.133188\pi$
$558$	0	0
$559$	9.23811	0.390730
$560$	20.3551	0.860162
$561$	0	0
$562$	−35.0142	−1.47699
$563$	34.1036	1.43729	0.718647	−	0.695375i	$-0.244762\pi$
$563$	34.1036	1.43729	0.718647	+	0.695375i	$0.244762\pi$
$564$	0	0
$565$	11.4071	0.479899
$566$	−36.2972	−1.52569
$567$	0	0
$568$	47.8984	2.00977
$569$	−19.0810	−0.799917	−0.399958	−	0.916533i	$-0.630975\pi$
$569$	−19.0810	−0.799917	−0.399958	+	0.916533i	$0.630975\pi$
$570$	0	0
$571$	−6.56641	−0.274796	−0.137398	−	0.990516i	$-0.543874\pi$
$571$	−6.56641	−0.274796	−0.137398	+	0.990516i	$0.543874\pi$
$572$	2.98013	0.124605
$573$	0	0
$574$	18.7574	0.782918
$575$	3.61306	0.150675
$576$	0	0
$577$	20.5602	0.855934	0.427967	−	0.903794i	$-0.359230\pi$
$577$	20.5602	0.855934	0.427967	+	0.903794i	$0.359230\pi$
$578$	−17.6261	−0.733151
$579$	0	0
$580$	0	0
$581$	−6.50687	−0.269951
$582$	0	0
$583$	8.25856	0.342035
$584$	7.80199	0.322849
$585$	0	0
$586$	−10.5987	−0.437828
$587$	9.69142	0.400008	0.200004	−	0.979795i	$-0.435905\pi$
$587$	9.69142	0.400008	0.200004	+	0.979795i	$0.435905\pi$
$588$	0	0
$589$	24.5231	1.01046
$590$	−0.833221	−0.0343032
$591$	0	0
$592$	−0.927007	−0.0380997
$593$	13.1810	0.541279	0.270639	−	0.962681i	$-0.412765\pi$
$593$	13.1810	0.541279	0.270639	+	0.962681i	$0.412765\pi$
$594$	0	0
$595$	−43.2100	−1.77144
$596$	7.48573	0.306627
$597$	0	0
$598$	−13.0861	−0.535131
$599$	−14.4397	−0.589991	−0.294995	−	0.955499i	$-0.595318\pi$
$599$	−14.4397	−0.589991	−0.294995	+	0.955499i	$0.595318\pi$
$600$	0	0
$601$	−4.88823	−0.199395	−0.0996976	−	0.995018i	$-0.531788\pi$
$601$	−4.88823	−0.199395	−0.0996976	+	0.995018i	$0.531788\pi$
$602$	9.05661	0.369120
$603$	0	0
$604$	−1.56718	−0.0637678
$605$	17.7763	0.722710
$606$	0	0
$607$	12.5548	0.509583	0.254792	−	0.966996i	$-0.417993\pi$
$607$	12.5548	0.509583	0.254792	+	0.966996i	$0.417993\pi$
$608$	18.8014	0.762496
$609$	0	0
$610$	11.9026	0.481923
$611$	−36.8681	−1.49152
$612$	0	0
$613$	3.49761	0.141267	0.0706336	−	0.997502i	$-0.477498\pi$
$613$	3.49761	0.141267	0.0706336	+	0.997502i	$0.477498\pi$
$614$	−41.0629	−1.65716
$615$	0	0
$616$	13.8605	0.558456
$617$	−27.2896	−1.09864	−0.549318	−	0.835613i	$-0.685113\pi$
$617$	−27.2896	−1.09864	−0.549318	+	0.835613i	$0.685113\pi$
$618$	0	0
$619$	25.5410	1.02658	0.513289	−	0.858216i	$-0.328427\pi$
$619$	25.5410	1.02658	0.513289	+	0.858216i	$0.328427\pi$
$620$	−3.67422	−0.147560
$621$	0	0
$622$	−9.83798	−0.394467
$623$	−76.7467	−3.07479
$624$	0	0
$625$	−13.0544	−0.522176
$626$	−3.05995	−0.122300
$627$	0	0
$628$	8.64911	0.345137
$629$	1.96786	0.0784636
$630$	0	0
$631$	20.7956	0.827858	0.413929	−	0.910309i	$-0.364156\pi$
$631$	20.7956	0.827858	0.413929	+	0.910309i	$0.364156\pi$
$632$	−31.5038	−1.25315
$633$	0	0
$634$	−20.1443	−0.800031
$635$	−8.03126	−0.318711
$636$	0	0
$637$	−59.6442	−2.36319
$638$	0	0
$639$	0	0
$640$	8.70736	0.344188
$641$	−14.1393	−0.558468	−0.279234	−	0.960223i	$-0.590080\pi$
$641$	−14.1393	−0.558468	−0.279234	+	0.960223i	$0.590080\pi$
$642$	0	0
$643$	35.4943	1.39976	0.699878	−	0.714262i	$-0.253238\pi$
$643$	35.4943	1.39976	0.699878	+	0.714262i	$0.253238\pi$
$644$	4.67498	0.184220
$645$	0	0
$646$	43.6091	1.71578
$647$	20.0626	0.788743	0.394371	−	0.918951i	$-0.370962\pi$
$647$	20.0626	0.788743	0.394371	+	0.918951i	$0.370962\pi$
$648$	0	0
$649$	−0.404196	−0.0158661
$650$	−11.2911	−0.442872
$651$	0	0
$652$	7.25038	0.283947
$653$	13.3537	0.522569	0.261285	−	0.965262i	$-0.415854\pi$
$653$	13.3537	0.522569	0.261285	+	0.965262i	$0.415854\pi$
$654$	0	0
$655$	−0.0179114	−0.000699856 0
$656$	9.58615	0.374276
$657$	0	0
$658$	−36.1438	−1.40903
$659$	11.3791	0.443267	0.221633	−	0.975130i	$-0.428861\pi$
$659$	11.3791	0.443267	0.221633	+	0.975130i	$0.428861\pi$
$660$	0	0
$661$	−37.0298	−1.44029	−0.720145	−	0.693823i	$-0.755925\pi$
$661$	−37.0298	−1.44029	−0.720145	+	0.693823i	$0.755925\pi$
$662$	15.6529	0.608366
$663$	0	0
$664$	−4.66786	−0.181148
$665$	49.3178	1.91246
$666$	0	0
$667$	0	0
$668$	−8.73974	−0.338151
$669$	0	0
$670$	17.0091	0.657119
$671$	5.77398	0.222902
$672$	0	0
$673$	37.2436	1.43563	0.717817	−	0.696232i	$-0.245141\pi$
$673$	37.2436	1.43563	0.717817	+	0.696232i	$0.245141\pi$
$674$	−6.73490	−0.259419
$675$	0	0
$676$	−7.95833	−0.306090
$677$	44.9350	1.72699	0.863496	−	0.504356i	$-0.168270\pi$
$677$	44.9350	1.72699	0.863496	+	0.504356i	$0.168270\pi$
$678$	0	0
$679$	29.1245	1.11770
$680$	−30.9977	−1.18871
$681$	0	0
$682$	4.89115	0.187292
$683$	−22.6022	−0.864848	−0.432424	−	0.901670i	$-0.642342\pi$
$683$	−22.6022	−0.864848	−0.432424	+	0.901670i	$0.642342\pi$
$684$	0	0
$685$	−21.9651	−0.839242
$686$	−22.2253	−0.848567
$687$	0	0
$688$	4.62847	0.176459
$689$	−41.2978	−1.57332
$690$	0	0
$691$	−38.8727	−1.47879	−0.739394	−	0.673273i	$-0.764888\pi$
$691$	−38.8727	−1.47879	−0.739394	+	0.673273i	$0.764888\pi$
$692$	−2.38436	−0.0906397
$693$	0	0
$694$	−10.6384	−0.403828
$695$	−8.89436	−0.337382
$696$	0	0
$697$	−20.3495	−0.770794
$698$	−18.3960	−0.696299
$699$	0	0
$700$	4.03371	0.152460
$701$	39.4427	1.48973	0.744865	−	0.667215i	$-0.232514\pi$
$701$	39.4427	1.48973	0.744865	+	0.667215i	$0.232514\pi$
$702$	0	0
$703$	−2.24601	−0.0847100
$704$	9.34041	0.352030
$705$	0	0
$706$	−14.9726	−0.563501
$707$	20.2116	0.760136
$708$	0	0
$709$	−4.07382	−0.152996	−0.0764978	−	0.997070i	$-0.524374\pi$
$709$	−4.07382	−0.152996	−0.0764978	+	0.997070i	$0.524374\pi$
$710$	33.9924	1.27571
$711$	0	0
$712$	−55.0561	−2.06331
$713$	7.82661	0.293109
$714$	0	0
$715$	10.0336	0.375236
$716$	1.81649	0.0678853
$717$	0	0
$718$	19.3913	0.723677
$719$	18.4303	0.687335	0.343667	−	0.939091i	$-0.388331\pi$
$719$	18.4303	0.687335	0.343667	+	0.939091i	$0.388331\pi$
$720$	0	0
$721$	14.9475	0.556674
$722$	−26.7696	−0.996263
$723$	0	0
$724$	−3.53686	−0.131446
$725$	0	0
$726$	0	0
$727$	−17.6150	−0.653306	−0.326653	−	0.945144i	$-0.605921\pi$
$727$	−17.6150	−0.653306	−0.326653	+	0.945144i	$0.605921\pi$
$728$	−69.3109	−2.56883
$729$	0	0
$730$	5.53689	0.204929
$731$	−9.82535	−0.363404
$732$	0	0
$733$	8.22724	0.303880	0.151940	−	0.988390i	$-0.451448\pi$
$733$	8.22724	0.303880	0.151940	+	0.988390i	$0.451448\pi$
$734$	−26.9320	−0.994079
$735$	0	0
$736$	6.00051	0.221182
$737$	8.25113	0.303934
$738$	0	0
$739$	−35.8226	−1.31776	−0.658878	−	0.752250i	$-0.728969\pi$
$739$	−35.8226	−1.31776	−0.658878	+	0.752250i	$0.728969\pi$
$740$	0.336514	0.0123705
$741$	0	0
$742$	−40.4864	−1.48630
$743$	9.29461	0.340986	0.170493	−	0.985359i	$-0.445464\pi$
$743$	9.29461	0.340986	0.170493	+	0.985359i	$0.445464\pi$
$744$	0	0
$745$	25.2032	0.923375
$746$	−8.39063	−0.307203
$747$	0	0
$748$	−3.16957	−0.115891
$749$	21.0507	0.769175
$750$	0	0
$751$	44.8712	1.63737	0.818687	−	0.574240i	$-0.194702\pi$
$751$	44.8712	1.63737	0.818687	+	0.574240i	$0.194702\pi$
$752$	−18.4716	−0.673592
$753$	0	0
$754$	0	0
$755$	−5.27646	−0.192030
$756$	0	0
$757$	−21.6178	−0.785712	−0.392856	−	0.919600i	$-0.628513\pi$
$757$	−21.6178	−0.785712	−0.392856	+	0.919600i	$0.628513\pi$
$758$	−9.42966	−0.342501
$759$	0	0
$760$	35.3793	1.28334
$761$	39.3758	1.42737	0.713685	−	0.700466i	$-0.247025\pi$
$761$	39.3758	1.42737	0.713685	+	0.700466i	$0.247025\pi$
$762$	0	0
$763$	−63.6606	−2.30467
$764$	−9.41169	−0.340503
$765$	0	0
$766$	30.8279	1.11386
$767$	2.02122	0.0729822
$768$	0	0
$769$	−4.29550	−0.154900	−0.0774499	−	0.996996i	$-0.524678\pi$
$769$	−4.29550	−0.154900	−0.0774499	+	0.996996i	$0.524678\pi$
$770$	9.83649	0.354482
$771$	0	0
$772$	−7.15703	−0.257587
$773$	27.9002	1.00350	0.501751	−	0.865012i	$-0.332690\pi$
$773$	27.9002	1.00350	0.501751	+	0.865012i	$0.332690\pi$
$774$	0	0
$775$	6.75302	0.242576
$776$	20.8932	0.750021
$777$	0	0
$778$	−13.2983	−0.476768
$779$	23.2260	0.832156
$780$	0	0
$781$	16.4897	0.590049
$782$	13.9180	0.497706
$783$	0	0
$784$	−29.8829	−1.06725
$785$	29.1201	1.03934
$786$	0	0
$787$	−5.90689	−0.210558	−0.105279	−	0.994443i	$-0.533574\pi$
$787$	−5.90689	−0.210558	−0.105279	+	0.994443i	$0.533574\pi$
$788$	−4.38266	−0.156126
$789$	0	0
$790$	−22.3575	−0.795444
$791$	−27.1276	−0.964547
$792$	0	0
$793$	−28.8734	−1.02532
$794$	−6.17523	−0.219151
$795$	0	0
$796$	−13.0845	−0.463767
$797$	−7.59899	−0.269170	−0.134585	−	0.990902i	$-0.542970\pi$
$797$	−7.59899	−0.269170	−0.134585	+	0.990902i	$0.542970\pi$
$798$	0	0
$799$	39.2117	1.38721
$800$	5.17741	0.183049
$801$	0	0
$802$	−23.6600	−0.835463
$803$	2.68595	0.0947851
$804$	0	0
$805$	15.7399	0.554759
$806$	−24.4587	−0.861521
$807$	0	0
$808$	14.4993	0.510083
$809$	−9.85027	−0.346317	−0.173159	−	0.984894i	$-0.555397\pi$
$809$	−9.85027	−0.346317	−0.173159	+	0.984894i	$0.555397\pi$
$810$	0	0
$811$	46.3250	1.62669	0.813346	−	0.581780i	$-0.197644\pi$
$811$	46.3250	1.62669	0.813346	+	0.581780i	$0.197644\pi$
$812$	0	0
$813$	0	0
$814$	−0.447970	−0.0157013
$815$	24.4109	0.855075
$816$	0	0
$817$	11.2142	0.392334
$818$	−21.7014	−0.758772
$819$	0	0
$820$	−3.47988	−0.121523
$821$	−18.0554	−0.630136	−0.315068	−	0.949069i	$-0.602027\pi$
$821$	−18.0554	−0.630136	−0.315068	+	0.949069i	$0.602027\pi$
$822$	0	0
$823$	33.6347	1.17243	0.586216	−	0.810155i	$-0.300617\pi$
$823$	33.6347	1.17243	0.586216	+	0.810155i	$0.300617\pi$
$824$	10.7229	0.373551
$825$	0	0
$826$	1.98151	0.0689457
$827$	41.1338	1.43036	0.715181	−	0.698939i	$-0.246344\pi$
$827$	41.1338	1.43036	0.715181	+	0.698939i	$0.246344\pi$
$828$	0	0
$829$	−7.95542	−0.276303	−0.138152	−	0.990411i	$-0.544116\pi$
$829$	−7.95542	−0.276303	−0.138152	+	0.990411i	$0.544116\pi$
$830$	−3.31267	−0.114984
$831$	0	0
$832$	−46.7077	−1.61930
$833$	63.4357	2.19792
$834$	0	0
$835$	−29.4253	−1.01830
$836$	3.61759	0.125117
$837$	0	0
$838$	4.52591	0.156345
$839$	−11.3228	−0.390908	−0.195454	−	0.980713i	$-0.562618\pi$
$839$	−11.3228	−0.390908	−0.195454	+	0.980713i	$0.562618\pi$
$840$	0	0
$841$	0	0
$842$	10.0342	0.345801
$843$	0	0
$844$	−1.18126	−0.0406606
$845$	−26.7944	−0.921756
$846$	0	0
$847$	−42.2745	−1.45257
$848$	−20.6910	−0.710532
$849$	0	0
$850$	12.0088	0.411899
$851$	−0.716822	−0.0245723
$852$	0	0
$853$	−14.5146	−0.496971	−0.248486	−	0.968636i	$-0.579933\pi$
$853$	−14.5146	−0.496971	−0.248486	+	0.968636i	$0.579933\pi$
$854$	−28.3061	−0.968615
$855$	0	0
$856$	15.1012	0.516148
$857$	10.7169	0.366083	0.183042	−	0.983105i	$-0.441406\pi$
$857$	10.7169	0.366083	0.183042	+	0.983105i	$0.441406\pi$
$858$	0	0
$859$	−15.7139	−0.536150	−0.268075	−	0.963398i	$-0.586388\pi$
$859$	−15.7139	−0.536150	−0.268075	+	0.963398i	$0.586388\pi$
$860$	−1.68019	−0.0572939
$861$	0	0
$862$	23.1603	0.788844
$863$	14.4146	0.490679	0.245340	−	0.969437i	$-0.421101\pi$
$863$	14.4146	0.490679	0.245340	+	0.969437i	$0.421101\pi$
$864$	0	0
$865$	−8.02775	−0.272952
$866$	−19.5542	−0.664478
$867$	0	0
$868$	8.73781	0.296581
$869$	−10.8456	−0.367913
$870$	0	0
$871$	−41.2606	−1.39806
$872$	−45.6684	−1.54653
$873$	0	0
$874$	−15.8853	−0.537328
$875$	52.0398	1.75926
$876$	0	0
$877$	13.5829	0.458662	0.229331	−	0.973348i	$-0.426346\pi$
$877$	13.5829	0.458662	0.229331	+	0.973348i	$0.426346\pi$
$878$	−3.59384	−0.121286
$879$	0	0
$880$	5.02703	0.169461
$881$	21.3996	0.720972	0.360486	−	0.932765i	$-0.382611\pi$
$881$	21.3996	0.720972	0.360486	+	0.932765i	$0.382611\pi$
$882$	0	0
$883$	−11.1221	−0.374288	−0.187144	−	0.982332i	$-0.559923\pi$
$883$	−11.1221	−0.374288	−0.187144	+	0.982332i	$0.559923\pi$
$884$	15.8498	0.533085
$885$	0	0
$886$	−12.7024	−0.426744
$887$	−1.53319	−0.0514795	−0.0257397	−	0.999669i	$-0.508194\pi$
$887$	−1.53319	−0.0514795	−0.0257397	+	0.999669i	$0.508194\pi$
$888$	0	0
$889$	19.0995	0.640575
$890$	−39.0720	−1.30970
$891$	0	0
$892$	−1.44457	−0.0483678
$893$	−44.7543	−1.49765
$894$	0	0
$895$	6.11582	0.204429
$896$	−20.7073	−0.691783
$897$	0	0
$898$	5.33110	0.177901
$899$	0	0
$900$	0	0
$901$	43.9230	1.46329
$902$	4.63244	0.154243
$903$	0	0
$904$	−19.4606	−0.647251
$905$	−11.9080	−0.395837
$906$	0	0
$907$	−7.68948	−0.255325	−0.127663	−	0.991818i	$-0.540747\pi$
$907$	−7.68948	−0.255325	−0.127663	+	0.991818i	$0.540747\pi$
$908$	7.70396	0.255665
$909$	0	0
$910$	−49.1884	−1.63058
$911$	−29.5727	−0.979788	−0.489894	−	0.871782i	$-0.662964\pi$
$911$	−29.5727	−0.979788	−0.489894	+	0.871782i	$0.662964\pi$
$912$	0	0
$913$	−1.60698	−0.0531832
$914$	−13.7398	−0.454473
$915$	0	0
$916$	13.7529	0.454408
$917$	0.0425958	0.00140664
$918$	0	0
$919$	−34.2295	−1.12913	−0.564564	−	0.825389i	$-0.690956\pi$
$919$	−34.2295	−1.12913	−0.564564	+	0.825389i	$0.690956\pi$
$920$	11.2914	0.372266
$921$	0	0
$922$	−23.8622	−0.785861
$923$	−82.4586	−2.71416
$924$	0	0
$925$	−0.618494	−0.0203360
$926$	−0.912053	−0.0299719
$927$	0	0
$928$	0	0
$929$	−48.1477	−1.57967	−0.789837	−	0.613317i	$-0.789835\pi$
$929$	−48.1477	−1.57967	−0.789837	+	0.613317i	$0.789835\pi$
$930$	0	0
$931$	−72.4023	−2.37289
$932$	−12.0742	−0.395502
$933$	0	0
$934$	−12.6460	−0.413790
$935$	−10.6714	−0.348993
$936$	0	0
$937$	−13.7375	−0.448784	−0.224392	−	0.974499i	$-0.572040\pi$
$937$	−13.7375	−0.448784	−0.224392	+	0.974499i	$0.572040\pi$
$938$	−40.4500	−1.32074
$939$	0	0
$940$	6.70542	0.218707
$941$	37.2514	1.21436	0.607181	−	0.794563i	$-0.292300\pi$
$941$	37.2514	1.21436	0.607181	+	0.794563i	$0.292300\pi$
$942$	0	0
$943$	7.41263	0.241389
$944$	1.01267	0.0329597
$945$	0	0
$946$	2.23668	0.0727207
$947$	16.4787	0.535486	0.267743	−	0.963490i	$-0.413722\pi$
$947$	16.4787	0.535486	0.267743	+	0.963490i	$0.413722\pi$
$948$	0	0
$949$	−13.4314	−0.436001
$950$	−13.7063	−0.444690
$951$	0	0
$952$	73.7169	2.38918
$953$	27.7913	0.900247	0.450124	−	0.892966i	$-0.351380\pi$
$953$	27.7913	0.900247	0.450124	+	0.892966i	$0.351380\pi$
$954$	0	0
$955$	−31.6877	−1.02539
$956$	12.5118	0.404662
$957$	0	0
$958$	16.9235	0.546773
$959$	52.2360	1.68679
$960$	0	0
$961$	−16.3716	−0.528117
$962$	2.24012	0.0722244
$963$	0	0
$964$	−11.9559	−0.385075
$965$	−24.0966	−0.775696
$966$	0	0
$967$	51.9876	1.67181	0.835904	−	0.548876i	$-0.184944\pi$
$967$	51.9876	1.67181	0.835904	+	0.548876i	$0.184944\pi$
$968$	−30.3266	−0.974735
$969$	0	0
$970$	14.8274	0.476079
$971$	26.6145	0.854099	0.427049	−	0.904228i	$-0.359553\pi$
$971$	26.6145	0.854099	0.427049	+	0.904228i	$0.359553\pi$
$972$	0	0
$973$	21.1520	0.678103
$974$	1.76799	0.0566502
$975$	0	0
$976$	−14.4661	−0.463049
$977$	−60.8959	−1.94823	−0.974116	−	0.226050i	$-0.927419\pi$
$977$	−60.8959	−1.94823	−0.974116	+	0.226050i	$0.927419\pi$
$978$	0	0
$979$	−18.9539	−0.605768
$980$	10.8478	0.346521
$981$	0	0
$982$	14.7129	0.469507
$983$	46.9308	1.49686	0.748430	−	0.663214i	$-0.230808\pi$
$983$	46.9308	1.49686	0.748430	+	0.663214i	$0.230808\pi$
$984$	0	0
$985$	−14.7557	−0.470156
$986$	0	0
$987$	0	0
$988$	−18.0901	−0.575523
$989$	3.57904	0.113807
$990$	0	0
$991$	50.7905	1.61341	0.806707	−	0.590952i	$-0.201248\pi$
$991$	50.7905	1.61341	0.806707	+	0.590952i	$0.201248\pi$
$992$	11.2153	0.356086
$993$	0	0
$994$	−80.8386	−2.56405
$995$	−44.0534	−1.39659
$996$	0	0
$997$	45.8087	1.45078	0.725389	−	0.688339i	$-0.241660\pi$
$997$	45.8087	1.45078	0.725389	+	0.688339i	$0.241660\pi$
$998$	43.5469	1.37845
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7569.2.a.bm.1.2		9
3.2	odd	2		2523.2.a.o.1.8		9
29.9	even	14		261.2.k.c.226.3		18
29.13	even	14		261.2.k.c.82.3		18
29.28	even	2		7569.2.a.bj.1.8		9
87.38	odd	14		87.2.g.a.52.1	✓	18
87.71	odd	14		87.2.g.a.82.1	yes	18
87.86	odd	2		2523.2.a.r.1.2		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
87.2.g.a.52.1	✓	18	87.38	odd	14
87.2.g.a.82.1	yes	18	87.71	odd	14
261.2.k.c.82.3		18	29.13	even	14
261.2.k.c.226.3		18	29.9	even	14
2523.2.a.o.1.8		9	3.2	odd	2
2523.2.a.r.1.2		9	87.86	odd	2
7569.2.a.bj.1.8		9	29.28	even	2
7569.2.a.bm.1.2		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-1.21072 q^{2} -0.534161 q^{4} -1.79844 q^{5} +4.27693 q^{7} +3.06816 q^{8} +2.17740 q^{10} +1.05626 q^{11} -5.28193 q^{13} -5.17816 q^{14} -2.64635 q^{16} +5.61769 q^{17} -6.41175 q^{19} +0.960655 q^{20} -1.27883 q^{22} -2.04633 q^{23} -1.76563 q^{25} +6.39492 q^{26} -2.28457 q^{28} -3.82471 q^{31} -2.93233 q^{32} -6.80144 q^{34} -7.69178 q^{35} +0.350297 q^{37} +7.76282 q^{38} -5.51788 q^{40} -3.62240 q^{41} -1.74900 q^{43} -0.564212 q^{44} +2.47753 q^{46} +6.98005 q^{47} +11.2921 q^{49} +2.13768 q^{50} +2.82140 q^{52} +7.81869 q^{53} -1.89961 q^{55} +13.1223 q^{56} -0.382668 q^{59} +5.46644 q^{61} +4.63064 q^{62} +8.84292 q^{64} +9.49920 q^{65} +7.81166 q^{67} -3.00075 q^{68} +9.31258 q^{70} +15.6115 q^{71} +2.54289 q^{73} -0.424110 q^{74} +3.42491 q^{76} +4.51754 q^{77} -10.2680 q^{79} +4.75929 q^{80} +4.38571 q^{82} -1.52139 q^{83} -10.1030 q^{85} +2.11755 q^{86} +3.24076 q^{88} -17.9444 q^{89} -22.5904 q^{91} +1.09307 q^{92} -8.45087 q^{94} +11.5311 q^{95} +6.80968 q^{97} -13.6716 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 5 q^{2} + 11 q^{4} + 4 q^{5} + 5 q^{7} + 24 q^{8} - q^{11} + q^{13} + 9 q^{14} + 35 q^{16} + 2 q^{17} - 9 q^{19} + 18 q^{20} - 4 q^{22} + 4 q^{23} + q^{25} - 8 q^{26} + 40 q^{28} - 8 q^{31} + 43 q^{32}+ \cdots - 30 q^{98}+O(q^{100}) \) 9 * q + 5 * q^2 + 11 * q^4 + 4 * q^5 + 5 * q^7 + 24 * q^8 - q^11 + q^13 + 9 * q^14 + 35 * q^16 + 2 * q^17 - 9 * q^19 + 18 * q^20 - 4 * q^22 + 4 * q^23 + q^25 - 8 * q^26 + 40 * q^28 - 8 * q^31 + 43 * q^32 - 4 * q^34 - 22 * q^35 - 27 * q^37 + 30 * q^38 + 29 * q^40 + 12 * q^41 - 16 * q^43 + 37 * q^44 + 22 * q^46 - 8 * q^47 - 6 * q^49 - 7 * q^50 + 33 * q^52 + 8 * q^53 - 9 * q^55 + 40 * q^56 + 16 * q^59 - 21 * q^61 + 32 * q^62 + 36 * q^64 + 31 * q^65 + 3 * q^67 + 33 * q^68 + 6 * q^70 + 33 * q^71 - 3 * q^73 - 28 * q^74 + 26 * q^76 + 24 * q^77 - 3 * q^79 + 64 * q^80 + 13 * q^82 - 13 * q^83 - 6 * q^85 - 58 * q^86 + 27 * q^88 + 6 * q^89 + q^91 + 29 * q^92 - 18 * q^94 + 48 * q^95 - 4 * q^97 - 30 * q^98

Introduction

L-functions

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