LMFDB - Embedded newform 9065.2.a.s.1.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9065,2,Mod(1,9065)] 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("9065.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9065, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9065 = 5 \cdot 7^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9065.a (trivial)

Newform invariants

comment:select newform

sage:traces = [19,-3,1,11,-19,4,0,-9,12,3,-13] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.3843894323$
Analytic rank:	$1$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 3 x^{18} - 20 x^{17} + 62 x^{16} + 161 x^{15} - 522 x^{14} - 670 x^{13} + 2318 x^{12} + \cdots + 9 $ x^19 - 3x^18 - 20x^17 + 62x^16 + 161x^15 - 522x^14 - 670x^13 + 2318x^12 + 1529x^11 - 5872x^10 - 1822x^9 + 8582x^8 + 804x^7 - 6991x^6 + 383x^5 + 2893x^4 - 460x^3 - 477x^2 + 102x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1295)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-0.879722$ of defining polynomial
Character	$\chi$	$=$	9065.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.879722 q^{2} +1.40421 q^{3} -1.22609 q^{4} -1.00000 q^{5} +1.23531 q^{6} -2.83806 q^{8} -1.02820 q^{9} -0.879722 q^{10} +2.61877 q^{11} -1.72168 q^{12} +1.85392 q^{13} -1.40421 q^{15} -0.0445300 q^{16} -0.662500 q^{17} -0.904534 q^{18} +1.45539 q^{19} +1.22609 q^{20} +2.30379 q^{22} +4.51770 q^{23} -3.98523 q^{24} +1.00000 q^{25} +1.63094 q^{26} -5.65643 q^{27} -10.4288 q^{29} -1.23531 q^{30} -2.65051 q^{31} +5.63695 q^{32} +3.67729 q^{33} -0.582816 q^{34} +1.26067 q^{36} +1.00000 q^{37} +1.28034 q^{38} +2.60329 q^{39} +2.83806 q^{40} +5.51742 q^{41} +4.15401 q^{43} -3.21084 q^{44} +1.02820 q^{45} +3.97432 q^{46} -2.70302 q^{47} -0.0625294 q^{48} +0.879722 q^{50} -0.930286 q^{51} -2.27307 q^{52} -11.8740 q^{53} -4.97609 q^{54} -2.61877 q^{55} +2.04367 q^{57} -9.17448 q^{58} -8.94848 q^{59} +1.72168 q^{60} -6.51506 q^{61} -2.33171 q^{62} +5.04801 q^{64} -1.85392 q^{65} +3.23499 q^{66} -1.22004 q^{67} +0.812283 q^{68} +6.34378 q^{69} +11.7319 q^{71} +2.91811 q^{72} +2.31731 q^{73} +0.879722 q^{74} +1.40421 q^{75} -1.78444 q^{76} +2.29017 q^{78} +5.23574 q^{79} +0.0445300 q^{80} -4.85819 q^{81} +4.85380 q^{82} +11.5498 q^{83} +0.662500 q^{85} +3.65437 q^{86} -14.6442 q^{87} -7.43222 q^{88} -7.60131 q^{89} +0.904534 q^{90} -5.53910 q^{92} -3.72186 q^{93} -2.37791 q^{94} -1.45539 q^{95} +7.91544 q^{96} +0.968186 q^{97} -2.69262 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 3 q^{2} + q^{3} + 11 q^{4} - 19 q^{5} + 4 q^{6} - 9 q^{8} + 12 q^{9} + 3 q^{10} - 13 q^{11} + 4 q^{12} + 3 q^{13} - q^{15} - 5 q^{16} + 2 q^{17} - 11 q^{18} + 12 q^{19} - 11 q^{20} - 10 q^{22}+ \cdots - 71 q^{99}+O(q^{100}) $ 19 * q - 3 * q^2 + q^3 + 11 * q^4 - 19 * q^5 + 4 * q^6 - 9 * q^8 + 12 * q^9 + 3 * q^10 - 13 * q^11 + 4 * q^12 + 3 * q^13 - q^15 - 5 * q^16 + 2 * q^17 - 11 * q^18 + 12 * q^19 - 11 * q^20 - 10 * q^22 - 16 * q^23 - 15 * q^24 + 19 * q^25 + 3 * q^26 + 7 * q^27 - q^29 - 4 * q^30 + 8 * q^31 - 16 * q^32 + 2 * q^33 + 14 * q^34 + 5 * q^36 + 19 * q^37 - 2 * q^38 - 45 * q^39 + 9 * q^40 + 18 * q^41 - 37 * q^43 + q^44 - 12 * q^45 - 29 * q^46 - 2 * q^47 + 35 * q^48 - 3 * q^50 - 43 * q^51 - 40 * q^52 - 10 * q^53 + 38 * q^54 + 13 * q^55 + 12 * q^57 - 13 * q^58 + 28 * q^59 - 4 * q^60 - 22 * q^61 + 10 * q^62 - 7 * q^64 - 3 * q^65 - 34 * q^66 - 61 * q^67 + 17 * q^68 + 20 * q^69 - 17 * q^71 + 17 * q^72 - 16 * q^73 - 3 * q^74 + q^75 + 45 * q^76 + 53 * q^78 - 63 * q^79 + 5 * q^80 - 5 * q^81 - 37 * q^82 + 24 * q^83 - 2 * q^85 - 19 * q^86 - 32 * q^87 - 58 * q^88 + 16 * q^89 + 11 * q^90 + 7 * q^92 - 31 * q^93 - 12 * q^95 - 40 * q^96 + 24 * q^97 - 71 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0.879722	0.622058	0.311029	−	0.950400i	$-0.399326\pi$
$2$	0.879722	0.622058	0.311029	+	0.950400i	$0.399326\pi$
$3$	1.40421	0.810719	0.405360	−	0.914157i	$-0.367146\pi$
$3$	1.40421	0.810719	0.405360	+	0.914157i	$0.367146\pi$
$4$	−1.22609	−0.613044
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.23531	0.504314
$7$	0	0
$7$	0	0
$8$	−2.83806	−1.00341
$9$	−1.02820	−0.342734
$10$	−0.879722	−0.278193
$11$	2.61877	0.789588	0.394794	−	0.918770i	$-0.370816\pi$
$11$	2.61877	0.789588	0.394794	+	0.918770i	$0.370816\pi$
$12$	−1.72168	−0.497007
$13$	1.85392	0.514186	0.257093	−	0.966387i	$-0.417235\pi$
$13$	1.85392	0.514186	0.257093	+	0.966387i	$0.417235\pi$
$14$	0	0
$15$	−1.40421	−0.362565
$16$	−0.0445300	−0.0111325
$17$	−0.662500	−0.160680	−0.0803399	−	0.996768i	$-0.525601\pi$
$17$	−0.662500	−0.160680	−0.0803399	+	0.996768i	$0.525601\pi$
$18$	−0.904534	−0.213201
$19$	1.45539	0.333890	0.166945	−	0.985966i	$-0.446610\pi$
$19$	1.45539	0.333890	0.166945	+	0.985966i	$0.446610\pi$
$20$	1.22609	0.274162
$21$	0	0
$22$	2.30379	0.491169
$23$	4.51770	0.942005	0.471003	−	0.882132i	$-0.343892\pi$
$23$	4.51770	0.942005	0.471003	+	0.882132i	$0.343892\pi$
$24$	−3.98523	−0.813481
$25$	1.00000	0.200000
$26$	1.63094	0.319853
$27$	−5.65643	−1.08858
$28$	0	0
$29$	−10.4288	−1.93659	−0.968293	−	0.249818i	$-0.919629\pi$
$29$	−10.4288	−1.93659	−0.968293	+	0.249818i	$0.919629\pi$
$30$	−1.23531	−0.225536
$31$	−2.65051	−0.476045	−0.238023	−	0.971260i	$-0.576499\pi$
$31$	−2.65051	−0.476045	−0.238023	+	0.971260i	$0.576499\pi$
$32$	5.63695	0.996481
$33$	3.67729	0.640134
$34$	−0.582816	−0.0999521
$35$	0	0
$36$	1.26067	0.210111
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	1.28034	0.207699
$39$	2.60329	0.416860
$40$	2.83806	0.448737
$41$	5.51742	0.861677	0.430838	−	0.902429i	$-0.358218\pi$
$41$	5.51742	0.861677	0.430838	+	0.902429i	$0.358218\pi$
$42$	0	0
$43$	4.15401	0.633480	0.316740	−	0.948512i	$-0.397412\pi$
$43$	4.15401	0.633480	0.316740	+	0.948512i	$0.397412\pi$
$44$	−3.21084	−0.484052
$45$	1.02820	0.153276
$46$	3.97432	0.585982
$47$	−2.70302	−0.394276	−0.197138	−	0.980376i	$-0.563165\pi$
$47$	−2.70302	−0.394276	−0.197138	+	0.980376i	$0.563165\pi$
$48$	−0.0625294	−0.00902534
$49$	0	0
$50$	0.879722	0.124412
$51$	−0.930286	−0.130266
$52$	−2.27307	−0.315219
$53$	−11.8740	−1.63101	−0.815507	−	0.578748i	$-0.803542\pi$
$53$	−11.8740	−1.63101	−0.815507	+	0.578748i	$0.803542\pi$
$54$	−4.97609	−0.677160
$55$	−2.61877	−0.353114
$56$	0	0
$57$	2.04367	0.270691
$58$	−9.17448	−1.20467
$59$	−8.94848	−1.16499	−0.582496	−	0.812833i	$-0.697924\pi$
$59$	−8.94848	−1.16499	−0.582496	+	0.812833i	$0.697924\pi$
$60$	1.72168	0.222268
$61$	−6.51506	−0.834168	−0.417084	−	0.908868i	$-0.636948\pi$
$61$	−6.51506	−0.834168	−0.417084	+	0.908868i	$0.636948\pi$
$62$	−2.33171	−0.296128
$63$	0	0
$64$	5.04801	0.631001
$65$	−1.85392	−0.229951
$66$	3.23499	0.398200
$67$	−1.22004	−0.149052	−0.0745259	−	0.997219i	$-0.523744\pi$
$67$	−1.22004	−0.149052	−0.0745259	+	0.997219i	$0.523744\pi$
$68$	0.812283	0.0985038
$69$	6.34378	0.763702
$70$	0	0
$71$	11.7319	1.39231	0.696157	−	0.717889i	$-0.254892\pi$
$71$	11.7319	1.39231	0.696157	+	0.717889i	$0.254892\pi$
$72$	2.91811	0.343902
$73$	2.31731	0.271220	0.135610	−	0.990762i	$-0.456701\pi$
$73$	2.31731	0.271220	0.135610	+	0.990762i	$0.456701\pi$
$74$	0.879722	0.102266
$75$	1.40421	0.162144
$76$	−1.78444	−0.204689
$77$	0	0
$78$	2.29017	0.259311
$79$	5.23574	0.589066	0.294533	−	0.955641i	$-0.404836\pi$
$79$	5.23574	0.589066	0.294533	+	0.955641i	$0.404836\pi$
$80$	0.0445300	0.00497861
$81$	−4.85819	−0.539799
$82$	4.85380	0.536013
$83$	11.5498	1.26776	0.633880	−	0.773431i	$-0.281461\pi$
$83$	11.5498	1.26776	0.633880	+	0.773431i	$0.281461\pi$
$84$	0	0
$85$	0.662500	0.0718582
$86$	3.65437	0.394061
$87$	−14.6442	−1.57003
$88$	−7.43222	−0.792278
$89$	−7.60131	−0.805738	−0.402869	−	0.915258i	$-0.631987\pi$
$89$	−7.60131	−0.805738	−0.402869	+	0.915258i	$0.631987\pi$
$90$	0.904534	0.0953462
$91$	0	0
$92$	−5.53910	−0.577491
$93$	−3.72186	−0.385939
$94$	−2.37791	−0.245262
$95$	−1.45539	−0.149320
$96$	7.91544	0.807867
$97$	0.968186	0.0983044	0.0491522	−	0.998791i	$-0.484348\pi$
$97$	0.968186	0.0983044	0.0491522	+	0.998791i	$0.484348\pi$
$98$	0	0
$99$	−2.69262	−0.270619
$100$	−1.22609	−0.122609
$101$	−0.787221	−0.0783314	−0.0391657	−	0.999233i	$-0.512470\pi$
$101$	−0.787221	−0.0783314	−0.0391657	+	0.999233i	$0.512470\pi$
$102$	−0.818394	−0.0810331
$103$	0.715193	0.0704700	0.0352350	−	0.999379i	$-0.488782\pi$
$103$	0.715193	0.0704700	0.0352350	+	0.999379i	$0.488782\pi$
$104$	−5.26155	−0.515938
$105$	0	0
$106$	−10.4458	−1.01458
$107$	2.91415	0.281721	0.140861	−	0.990029i	$-0.455013\pi$
$107$	2.91415	0.281721	0.140861	+	0.990029i	$0.455013\pi$
$108$	6.93528	0.667348
$109$	−16.6276	−1.59264	−0.796319	−	0.604877i	$-0.793222\pi$
$109$	−16.6276	−1.59264	−0.796319	+	0.604877i	$0.793222\pi$
$110$	−2.30379	−0.219658
$111$	1.40421	0.133281
$112$	0	0
$113$	8.86284	0.833746	0.416873	−	0.908965i	$-0.363126\pi$
$113$	8.86284	0.833746	0.416873	+	0.908965i	$0.363126\pi$
$114$	1.79786	0.168385
$115$	−4.51770	−0.421278
$116$	12.7867	1.18721
$117$	−1.90621	−0.176229
$118$	−7.87218	−0.724692
$119$	0	0
$120$	3.98523	0.363800
$121$	−4.14206	−0.376551
$122$	−5.73145	−0.518901
$123$	7.74760	0.698578
$124$	3.24976	0.291837
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−17.2232	−1.52831	−0.764157	−	0.645030i	$-0.776845\pi$
$127$	−17.2232	−1.52831	−0.764157	+	0.645030i	$0.776845\pi$
$128$	−6.83305	−0.603962
$129$	5.83309	0.513575
$130$	−1.63094	−0.143043
$131$	−18.6561	−1.62999	−0.814994	−	0.579470i	$-0.803260\pi$
$131$	−18.6561	−1.62999	−0.814994	+	0.579470i	$0.803260\pi$
$132$	−4.50868	−0.392430
$133$	0	0
$134$	−1.07330	−0.0927188
$135$	5.65643	0.486828
$136$	1.88022	0.161227
$137$	8.75628	0.748100	0.374050	−	0.927409i	$-0.377969\pi$
$137$	8.75628	0.748100	0.374050	+	0.927409i	$0.377969\pi$
$138$	5.58077	0.475067
$139$	−8.49623	−0.720640	−0.360320	−	0.932829i	$-0.617333\pi$
$139$	−8.49623	−0.720640	−0.360320	+	0.932829i	$0.617333\pi$
$140$	0	0
$141$	−3.79560	−0.319647
$142$	10.3208	0.866100
$143$	4.85499	0.405995
$144$	0.0457859	0.00381549
$145$	10.4288	0.866067
$146$	2.03859	0.168715
$147$	0	0
$148$	−1.22609	−0.100784
$149$	−16.4471	−1.34740	−0.673700	−	0.739005i	$-0.735296\pi$
$149$	−16.4471	−1.34740	−0.673700	+	0.739005i	$0.735296\pi$
$150$	1.23531	0.100863
$151$	−3.82972	−0.311659	−0.155829	−	0.987784i	$-0.549805\pi$
$151$	−3.82972	−0.311659	−0.155829	+	0.987784i	$0.549805\pi$
$152$	−4.13050	−0.335028
$153$	0.681184	0.0550705
$154$	0	0
$155$	2.65051	0.212894
$156$	−3.19187	−0.255554
$157$	1.36906	0.109263	0.0546315	−	0.998507i	$-0.482602\pi$
$157$	1.36906	0.109263	0.0546315	+	0.998507i	$0.482602\pi$
$158$	4.60599	0.366433
$159$	−16.6735	−1.32229
$160$	−5.63695	−0.445640
$161$	0	0
$162$	−4.27386	−0.335786
$163$	−3.75902	−0.294429	−0.147215	−	0.989105i	$-0.547031\pi$
$163$	−3.75902	−0.294429	−0.147215	+	0.989105i	$0.547031\pi$
$164$	−6.76485	−0.528246
$165$	−3.67729	−0.286277
$166$	10.1607	0.788620
$167$	22.0658	1.70751	0.853753	−	0.520678i	$-0.174321\pi$
$167$	22.0658	1.70751	0.853753	+	0.520678i	$0.174321\pi$
$168$	0	0
$169$	−9.56297	−0.735613
$170$	0.582816	0.0446999
$171$	−1.49644	−0.114436
$172$	−5.09318	−0.388352
$173$	−13.5155	−1.02756	−0.513782	−	0.857921i	$-0.671756\pi$
$173$	−13.5155	−1.02756	−0.513782	+	0.857921i	$0.671756\pi$
$174$	−12.8829	−0.976647
$175$	0	0
$176$	−0.116614	−0.00879009
$177$	−12.5655	−0.944482
$178$	−6.68704	−0.501215
$179$	23.0584	1.72347	0.861734	−	0.507361i	$-0.169379\pi$
$179$	23.0584	1.72347	0.861734	+	0.507361i	$0.169379\pi$
$180$	−1.26067	−0.0939647
$181$	0.407733	0.0303066	0.0151533	−	0.999885i	$-0.495176\pi$
$181$	0.407733	0.0303066	0.0151533	+	0.999885i	$0.495176\pi$
$182$	0	0
$183$	−9.14849	−0.676276
$184$	−12.8215	−0.945214
$185$	−1.00000	−0.0735215
$186$	−3.27420	−0.240076
$187$	−1.73493	−0.126871
$188$	3.31414	0.241709
$189$	0	0
$190$	−1.28034	−0.0928858
$191$	−16.6147	−1.20220	−0.601098	−	0.799175i	$-0.705270\pi$
$191$	−16.6147	−1.20220	−0.601098	+	0.799175i	$0.705270\pi$
$192$	7.08845	0.511565
$193$	−21.1323	−1.52113	−0.760567	−	0.649259i	$-0.775079\pi$
$193$	−21.1323	−1.52113	−0.760567	+	0.649259i	$0.775079\pi$
$194$	0.851735	0.0611510
$195$	−2.60329	−0.186426
$196$	0	0
$197$	−15.7226	−1.12019	−0.560095	−	0.828428i	$-0.689235\pi$
$197$	−15.7226	−1.12019	−0.560095	+	0.828428i	$0.689235\pi$
$198$	−2.36876	−0.168341
$199$	3.20391	0.227119	0.113560	−	0.993531i	$-0.463775\pi$
$199$	3.20391	0.227119	0.113560	+	0.993531i	$0.463775\pi$
$200$	−2.83806	−0.200681
$201$	−1.71319	−0.120839
$202$	−0.692536	−0.0487266
$203$	0	0
$204$	1.14061	0.0798589
$205$	−5.51742	−0.385354
$206$	0.629171	0.0438364
$207$	−4.64511	−0.322858
$208$	−0.0825553	−0.00572418
$209$	3.81133	0.263636
$210$	0	0
$211$	−7.07225	−0.486874	−0.243437	−	0.969917i	$-0.578275\pi$
$211$	−7.07225	−0.486874	−0.243437	+	0.969917i	$0.578275\pi$
$212$	14.5585	0.999883
$213$	16.4739	1.12878
$214$	2.56364	0.175247
$215$	−4.15401	−0.283301
$216$	16.0533	1.09229
$217$	0	0
$218$	−14.6277	−0.990713
$219$	3.25398	0.219883
$220$	3.21084	0.216475
$221$	−1.22822	−0.0826193
$222$	1.23531	0.0829087
$223$	−2.44022	−0.163409	−0.0817045	−	0.996657i	$-0.526036\pi$
$223$	−2.44022	−0.163409	−0.0817045	+	0.996657i	$0.526036\pi$
$224$	0	0
$225$	−1.02820	−0.0685469
$226$	7.79684	0.518638
$227$	−13.6836	−0.908212	−0.454106	−	0.890948i	$-0.650041\pi$
$227$	−13.6836	−0.908212	−0.454106	+	0.890948i	$0.650041\pi$
$228$	−2.50572	−0.165946
$229$	−7.71274	−0.509672	−0.254836	−	0.966984i	$-0.582021\pi$
$229$	−7.71274	−0.509672	−0.254836	+	0.966984i	$0.582021\pi$
$230$	−3.97432	−0.262059
$231$	0	0
$232$	29.5977	1.94318
$233$	−9.32576	−0.610951	−0.305475	−	0.952200i	$-0.598815\pi$
$233$	−9.32576	−0.610951	−0.305475	+	0.952200i	$0.598815\pi$
$234$	−1.67694	−0.109625
$235$	2.70302	0.176326
$236$	10.9716	0.714192
$237$	7.35206	0.477567
$238$	0	0
$239$	−21.9382	−1.41907	−0.709533	−	0.704672i	$-0.751094\pi$
$239$	−21.9382	−1.41907	−0.709533	+	0.704672i	$0.751094\pi$
$240$	0.0625294	0.00403625
$241$	1.51458	0.0975630	0.0487815	−	0.998809i	$-0.484466\pi$
$241$	1.51458	0.0975630	0.0487815	+	0.998809i	$0.484466\pi$
$242$	−3.64387	−0.234237
$243$	10.1474	0.650955
$244$	7.98804	0.511382
$245$	0	0
$246$	6.81574	0.434556
$247$	2.69819	0.171682
$248$	7.52231	0.477667
$249$	16.2184	1.02780
$250$	−0.879722	−0.0556385
$251$	28.9786	1.82911	0.914556	−	0.404459i	$-0.132540\pi$
$251$	28.9786	1.82911	0.914556	+	0.404459i	$0.132540\pi$
$252$	0	0
$253$	11.8308	0.743796
$254$	−15.1517	−0.950700
$255$	0.930286	0.0582568
$256$	−16.1072	−1.00670
$257$	−25.5519	−1.59388	−0.796941	−	0.604057i	$-0.793550\pi$
$257$	−25.5519	−1.59388	−0.796941	+	0.604057i	$0.793550\pi$
$258$	5.13150	0.319473
$259$	0	0
$260$	2.27307	0.140970
$261$	10.7230	0.663735
$262$	−16.4122	−1.01395
$263$	−18.9298	−1.16726	−0.583630	−	0.812020i	$-0.698368\pi$
$263$	−18.9298	−1.16726	−0.583630	+	0.812020i	$0.698368\pi$
$264$	−10.4364	−0.642315
$265$	11.8740	0.729411
$266$	0	0
$267$	−10.6738	−0.653227
$268$	1.49588	0.0913753
$269$	21.4422	1.30735	0.653676	−	0.756775i	$-0.273226\pi$
$269$	21.4422	1.30735	0.653676	+	0.756775i	$0.273226\pi$
$270$	4.97609	0.302835
$271$	−30.1732	−1.83289	−0.916444	−	0.400162i	$-0.868954\pi$
$271$	−30.1732	−1.83289	−0.916444	+	0.400162i	$0.868954\pi$
$272$	0.0295011	0.00178877
$273$	0	0
$274$	7.70310	0.465361
$275$	2.61877	0.157918
$276$	−7.77804	−0.468183
$277$	−11.8692	−0.713149	−0.356574	−	0.934267i	$-0.616055\pi$
$277$	−11.8692	−0.713149	−0.356574	+	0.934267i	$0.616055\pi$
$278$	−7.47432	−0.448280
$279$	2.72526	0.163157
$280$	0	0
$281$	−1.92744	−0.114981	−0.0574907	−	0.998346i	$-0.518310\pi$
$281$	−1.92744	−0.114981	−0.0574907	+	0.998346i	$0.518310\pi$
$282$	−3.33907	−0.198839
$283$	18.1887	1.08121	0.540603	−	0.841278i	$-0.318196\pi$
$283$	18.1887	1.08121	0.540603	+	0.841278i	$0.318196\pi$
$284$	−14.3843	−0.853551
$285$	−2.04367	−0.121057
$286$	4.27105	0.252552
$287$	0	0
$288$	−5.79593	−0.341529
$289$	−16.5611	−0.974182
$290$	9.17448	0.538744
$291$	1.35953	0.0796972
$292$	−2.84122	−0.166270
$293$	−23.8905	−1.39570	−0.697850	−	0.716244i	$-0.745860\pi$
$293$	−23.8905	−1.39570	−0.697850	+	0.716244i	$0.745860\pi$
$294$	0	0
$295$	8.94848	0.521000
$296$	−2.83806	−0.164959
$297$	−14.8129	−0.859530
$298$	−14.4689	−0.838160
$299$	8.37547	0.484366
$300$	−1.72168	−0.0994013
$301$	0	0
$302$	−3.36909	−0.193870
$303$	−1.10542	−0.0635047
$304$	−0.0648087	−0.00371703
$305$	6.51506	0.373051
$306$	0.599253	0.0342570
$307$	−5.64062	−0.321927	−0.160964	−	0.986960i	$-0.551460\pi$
$307$	−5.64062	−0.321927	−0.160964	+	0.986960i	$0.551460\pi$
$308$	0	0
$309$	1.00428	0.0571314
$310$	2.33171	0.132432
$311$	15.1742	0.860450	0.430225	−	0.902722i	$-0.358434\pi$
$311$	15.1742	0.860450	0.430225	+	0.902722i	$0.358434\pi$
$312$	−7.38831	−0.418281
$313$	−32.4050	−1.83164	−0.915819	−	0.401592i	$-0.868457\pi$
$313$	−32.4050	−1.83164	−0.915819	+	0.401592i	$0.868457\pi$
$314$	1.20439	0.0679679
$315$	0	0
$316$	−6.41947	−0.361124
$317$	−2.31606	−0.130083	−0.0650416	−	0.997883i	$-0.520718\pi$
$317$	−2.31606	−0.130083	−0.0650416	+	0.997883i	$0.520718\pi$
$318$	−14.6680	−0.822543
$319$	−27.3107	−1.52910
$320$	−5.04801	−0.282192
$321$	4.09206	0.228397
$322$	0	0
$323$	−0.964197	−0.0536494
$324$	5.95657	0.330920
$325$	1.85392	0.102837
$326$	−3.30690	−0.183152
$327$	−23.3486	−1.29118
$328$	−15.6588	−0.864612
$329$	0	0
$330$	−3.23499	−0.178081
$331$	1.76521	0.0970248	0.0485124	−	0.998823i	$-0.484552\pi$
$331$	1.76521	0.0970248	0.0485124	+	0.998823i	$0.484552\pi$
$332$	−14.1611	−0.777193
$333$	−1.02820	−0.0563452
$334$	19.4118	1.06217
$335$	1.22004	0.0666580
$336$	0	0
$337$	−13.8878	−0.756517	−0.378258	−	0.925700i	$-0.623477\pi$
$337$	−13.8878	−0.756517	−0.378258	+	0.925700i	$0.623477\pi$
$338$	−8.41275	−0.457594
$339$	12.4453	0.675934
$340$	−0.812283	−0.0440522
$341$	−6.94106	−0.375879
$342$	−1.31645	−0.0711856
$343$	0	0
$344$	−11.7893	−0.635638
$345$	−6.34378	−0.341538
$346$	−11.8899	−0.639204
$347$	−8.02642	−0.430881	−0.215440	−	0.976517i	$-0.569119\pi$
$347$	−8.02642	−0.430881	−0.215440	+	0.976517i	$0.569119\pi$
$348$	17.9551	0.962496
$349$	16.2924	0.872115	0.436057	−	0.899919i	$-0.356374\pi$
$349$	16.2924	0.872115	0.436057	+	0.899919i	$0.356374\pi$
$350$	0	0
$351$	−10.4866	−0.559733
$352$	14.7619	0.786810
$353$	13.4808	0.717511	0.358756	−	0.933432i	$-0.383201\pi$
$353$	13.4808	0.717511	0.358756	+	0.933432i	$0.383201\pi$
$354$	−11.0542	−0.587522
$355$	−11.7319	−0.622662
$356$	9.31988	0.493953
$357$	0	0
$358$	20.2850	1.07210
$359$	−31.6816	−1.67209	−0.836045	−	0.548661i	$-0.815138\pi$
$359$	−31.6816	−1.67209	−0.836045	+	0.548661i	$0.815138\pi$
$360$	−2.91811	−0.153798
$361$	−16.8818	−0.888517
$362$	0.358692	0.0188524
$363$	−5.81631	−0.305277
$364$	0	0
$365$	−2.31731	−0.121293
$366$	−8.04813	−0.420683
$367$	−27.7286	−1.44742	−0.723711	−	0.690103i	$-0.757565\pi$
$367$	−27.7286	−1.44742	−0.723711	+	0.690103i	$0.757565\pi$
$368$	−0.201173	−0.0104869
$369$	−5.67303	−0.295326
$370$	−0.879722	−0.0457346
$371$	0	0
$372$	4.56333	0.236598
$373$	17.5203	0.907166	0.453583	−	0.891214i	$-0.350146\pi$
$373$	17.5203	0.907166	0.453583	+	0.891214i	$0.350146\pi$
$374$	−1.52626	−0.0789209
$375$	−1.40421	−0.0725129
$376$	7.67134	0.395619
$377$	−19.3343	−0.995765
$378$	0	0
$379$	22.4681	1.15411	0.577054	−	0.816706i	$-0.304202\pi$
$379$	22.4681	1.15411	0.577054	+	0.816706i	$0.304202\pi$
$380$	1.78444	0.0915399
$381$	−24.1850	−1.23903
$382$	−14.6163	−0.747835
$383$	28.8633	1.47485	0.737423	−	0.675431i	$-0.236043\pi$
$383$	28.8633	1.47485	0.737423	+	0.675431i	$0.236043\pi$
$384$	−9.59502	−0.489644
$385$	0	0
$386$	−18.5905	−0.946233
$387$	−4.27117	−0.217116
$388$	−1.18708	−0.0602649
$389$	−6.13581	−0.311098	−0.155549	−	0.987828i	$-0.549715\pi$
$389$	−6.13581	−0.311098	−0.155549	+	0.987828i	$0.549715\pi$
$390$	−2.29017	−0.115968
$391$	−2.99297	−0.151361
$392$	0	0
$393$	−26.1970	−1.32146
$394$	−13.8315	−0.696823
$395$	−5.23574	−0.263438
$396$	3.30140	0.165901
$397$	18.8253	0.944812	0.472406	−	0.881381i	$-0.343386\pi$
$397$	18.8253	0.944812	0.472406	+	0.881381i	$0.343386\pi$
$398$	2.81855	0.141281
$399$	0	0
$400$	−0.0445300	−0.00222650
$401$	−35.5937	−1.77746	−0.888732	−	0.458427i	$-0.848413\pi$
$401$	−35.5937	−1.77746	−0.888732	+	0.458427i	$0.848413\pi$
$402$	−1.50713	−0.0751689
$403$	−4.91384	−0.244776
$404$	0.965202	0.0480206
$405$	4.85819	0.241405
$406$	0	0
$407$	2.61877	0.129807
$408$	2.64021	0.130710
$409$	19.9848	0.988185	0.494093	−	0.869409i	$-0.335500\pi$
$409$	19.9848	0.988185	0.494093	+	0.869409i	$0.335500\pi$
$410$	−4.85380	−0.239712
$411$	12.2956	0.606499
$412$	−0.876889	−0.0432012
$413$	0	0
$414$	−4.08641	−0.200836
$415$	−11.5498	−0.566959
$416$	10.4505	0.512377
$417$	−11.9305	−0.584237
$418$	3.35292	0.163997
$419$	−5.57361	−0.272288	−0.136144	−	0.990689i	$-0.543471\pi$
$419$	−5.57361	−0.272288	−0.136144	+	0.990689i	$0.543471\pi$
$420$	0	0
$421$	15.7722	0.768691	0.384346	−	0.923189i	$-0.374427\pi$
$421$	15.7722	0.768691	0.384346	+	0.923189i	$0.374427\pi$
$422$	−6.22161	−0.302863
$423$	2.77925	0.135132
$424$	33.6990	1.63657
$425$	−0.662500	−0.0321360
$426$	14.4925	0.702164
$427$	0	0
$428$	−3.57300	−0.172707
$429$	6.81742	0.329148
$430$	−3.65437	−0.176230
$431$	5.72242	0.275639	0.137820	−	0.990457i	$-0.455991\pi$
$431$	5.72242	0.275639	0.137820	+	0.990457i	$0.455991\pi$
$432$	0.251881	0.0121186
$433$	3.41219	0.163979	0.0819896	−	0.996633i	$-0.473873\pi$
$433$	3.41219	0.163979	0.0819896	+	0.996633i	$0.473873\pi$
$434$	0	0
$435$	14.6442	0.702137
$436$	20.3869	0.976358
$437$	6.57503	0.314526
$438$	2.86260	0.136780
$439$	21.4458	1.02355	0.511776	−	0.859119i	$-0.328988\pi$
$439$	21.4458	1.02355	0.511776	+	0.859119i	$0.328988\pi$
$440$	7.43222	0.354317
$441$	0	0
$442$	−1.08050	−0.0513940
$443$	31.8502	1.51325	0.756624	−	0.653851i	$-0.226848\pi$
$443$	31.8502	1.51325	0.756624	+	0.653851i	$0.226848\pi$
$444$	−1.72168	−0.0817074
$445$	7.60131	0.360337
$446$	−2.14671	−0.101650
$447$	−23.0951	−1.09236
$448$	0	0
$449$	−14.2049	−0.670370	−0.335185	−	0.942152i	$-0.608799\pi$
$449$	−14.2049	−0.670370	−0.335185	+	0.942152i	$0.608799\pi$
$450$	−0.904534	−0.0426401
$451$	14.4488	0.680369
$452$	−10.8666	−0.511123
$453$	−5.37772	−0.252668
$454$	−12.0378	−0.564960
$455$	0	0
$456$	−5.80007	−0.271613
$457$	15.9495	0.746084	0.373042	−	0.927814i	$-0.378315\pi$
$457$	15.9495	0.746084	0.373042	+	0.927814i	$0.378315\pi$
$458$	−6.78507	−0.317045
$459$	3.74738	0.174913
$460$	5.53910	0.258262
$461$	29.2431	1.36199	0.680993	−	0.732290i	$-0.261548\pi$
$461$	29.2431	1.36199	0.680993	+	0.732290i	$0.261548\pi$
$462$	0	0
$463$	−11.6477	−0.541313	−0.270657	−	0.962676i	$-0.587241\pi$
$463$	−11.6477	−0.541313	−0.270657	+	0.962676i	$0.587241\pi$
$464$	0.464396	0.0215591
$465$	3.72186	0.172597
$466$	−8.20408	−0.380047
$467$	−6.21356	−0.287529	−0.143765	−	0.989612i	$-0.545921\pi$
$467$	−6.21356	−0.287529	−0.143765	+	0.989612i	$0.545921\pi$
$468$	2.33718	0.108036
$469$	0	0
$470$	2.37791	0.109685
$471$	1.92245	0.0885816
$472$	25.3963	1.16896
$473$	10.8784	0.500188
$474$	6.46777	0.297074
$475$	1.45539	0.0667780
$476$	0	0
$477$	12.2088	0.559004
$478$	−19.2995	−0.882741
$479$	0.355614	0.0162484	0.00812422	−	0.999967i	$-0.497414\pi$
$479$	0.355614	0.0162484	0.00812422	+	0.999967i	$0.497414\pi$
$480$	−7.91544	−0.361289
$481$	1.85392	0.0845317
$482$	1.33241	0.0606898
$483$	0	0
$484$	5.07854	0.230843
$485$	−0.968186	−0.0439630
$486$	8.92689	0.404932
$487$	−0.485476	−0.0219990	−0.0109995	−	0.999940i	$-0.503501\pi$
$487$	−0.485476	−0.0219990	−0.0109995	+	0.999940i	$0.503501\pi$
$488$	18.4902	0.837010
$489$	−5.27845	−0.238700
$490$	0	0
$491$	20.6425	0.931584	0.465792	−	0.884894i	$-0.345769\pi$
$491$	20.6425	0.931584	0.465792	+	0.884894i	$0.345769\pi$
$492$	−9.49925	−0.428259
$493$	6.90910	0.311170
$494$	2.37366	0.106796
$495$	2.69262	0.121024
$496$	0.118027	0.00529958
$497$	0	0
$498$	14.2677	0.639349
$499$	12.3036	0.550783	0.275391	−	0.961332i	$-0.411193\pi$
$499$	12.3036	0.550783	0.275391	+	0.961332i	$0.411193\pi$
$500$	1.22609	0.0548323
$501$	30.9850	1.38431
$502$	25.4931	1.13781
$503$	16.4357	0.732830	0.366415	−	0.930451i	$-0.380585\pi$
$503$	16.4357	0.732830	0.366415	+	0.930451i	$0.380585\pi$
$504$	0	0
$505$	0.787221	0.0350309
$506$	10.4078	0.462684
$507$	−13.4284	−0.596375
$508$	21.1172	0.936925
$509$	−13.0898	−0.580197	−0.290099	−	0.956997i	$-0.593688\pi$
$509$	−13.0898	−0.580197	−0.290099	+	0.956997i	$0.593688\pi$
$510$	0.818394	0.0362391
$511$	0	0
$512$	−0.503772	−0.0222638
$513$	−8.23233	−0.363466
$514$	−22.4786	−0.991487
$515$	−0.715193	−0.0315152
$516$	−7.15188	−0.314844
$517$	−7.07858	−0.311315
$518$	0	0
$519$	−18.9786	−0.833066
$520$	5.26155	0.230734
$521$	36.5549	1.60150	0.800750	−	0.598999i	$-0.204434\pi$
$521$	36.5549	1.60150	0.800750	+	0.598999i	$0.204434\pi$
$522$	9.43323	0.412881
$523$	34.6300	1.51426	0.757132	−	0.653262i	$-0.226600\pi$
$523$	34.6300	1.51426	0.757132	+	0.653262i	$0.226600\pi$
$524$	22.8740	0.999254
$525$	0	0
$526$	−16.6529	−0.726103
$527$	1.75596	0.0764908
$528$	−0.163750	−0.00712630
$529$	−2.59040	−0.112626
$530$	10.4458	0.453736
$531$	9.20085	0.399283
$532$	0	0
$533$	10.2289	0.443062
$534$	−9.38999	−0.406345
$535$	−2.91415	−0.125989
$536$	3.46255	0.149560
$537$	32.3788	1.39725
$538$	18.8631	0.813248
$539$	0	0
$540$	−6.93528	−0.298447
$541$	−43.2188	−1.85812	−0.929061	−	0.369927i	$-0.879383\pi$
$541$	−43.2188	−1.85812	−0.929061	+	0.369927i	$0.879383\pi$
$542$	−26.5440	−1.14016
$543$	0.572542	0.0245701
$544$	−3.73448	−0.160114
$545$	16.6276	0.712249
$546$	0	0
$547$	11.9097	0.509220	0.254610	−	0.967044i	$-0.418053\pi$
$547$	11.9097	0.509220	0.254610	+	0.967044i	$0.418053\pi$
$548$	−10.7360	−0.458618
$549$	6.69881	0.285898
$550$	2.30379	0.0982338
$551$	−15.1781	−0.646607
$552$	−18.0041	−0.766303
$553$	0	0
$554$	−10.4416	−0.443620
$555$	−1.40421	−0.0596053
$556$	10.4171	0.441784
$557$	7.25288	0.307314	0.153657	−	0.988124i	$-0.450895\pi$
$557$	7.25288	0.307314	0.153657	+	0.988124i	$0.450895\pi$
$558$	2.39747	0.101493
$559$	7.70122	0.325727
$560$	0	0
$561$	−2.43620	−0.102857
$562$	−1.69561	−0.0715250
$563$	29.4309	1.24036	0.620181	−	0.784459i	$-0.287059\pi$
$563$	29.4309	1.24036	0.620181	+	0.784459i	$0.287059\pi$
$564$	4.65374	0.195958
$565$	−8.86284	−0.372863
$566$	16.0010	0.672572
$567$	0	0
$568$	−33.2957	−1.39706
$569$	−22.6006	−0.947466	−0.473733	−	0.880669i	$-0.657094\pi$
$569$	−22.6006	−0.947466	−0.473733	+	0.880669i	$0.657094\pi$
$570$	−1.79786	−0.0753043
$571$	−22.7982	−0.954077	−0.477038	−	0.878882i	$-0.658290\pi$
$571$	−22.7982	−0.954077	−0.477038	+	0.878882i	$0.658290\pi$
$572$	−5.95265	−0.248893
$573$	−23.3304	−0.974643
$574$	0	0
$575$	4.51770	0.188401
$576$	−5.19038	−0.216266
$577$	−27.8583	−1.15976	−0.579879	−	0.814703i	$-0.696900\pi$
$577$	−27.8583	−1.15976	−0.579879	+	0.814703i	$0.696900\pi$
$578$	−14.5692	−0.605997
$579$	−29.6741	−1.23321
$580$	−12.7867	−0.530938
$581$	0	0
$582$	1.19601	0.0495763
$583$	−31.0951	−1.28783
$584$	−6.57666	−0.272144
$585$	1.90621	0.0788121
$586$	−21.0170	−0.868206
$587$	30.0899	1.24194	0.620972	−	0.783833i	$-0.286738\pi$
$587$	30.0899	1.24194	0.620972	+	0.783833i	$0.286738\pi$
$588$	0	0
$589$	−3.85753	−0.158947
$590$	7.87218	0.324092
$591$	−22.0778	−0.908160
$592$	−0.0445300	−0.00183017
$593$	11.5979	0.476269	0.238134	−	0.971232i	$-0.423464\pi$
$593$	11.5979	0.476269	0.238134	+	0.971232i	$0.423464\pi$
$594$	−13.0312	−0.534677
$595$	0	0
$596$	20.1656	0.826016
$597$	4.49895	0.184130
$598$	7.36809	0.301304
$599$	−10.5940	−0.432859	−0.216429	−	0.976298i	$-0.569441\pi$
$599$	−10.5940	−0.432859	−0.216429	+	0.976298i	$0.569441\pi$
$600$	−3.98523	−0.162696
$601$	−19.9386	−0.813312	−0.406656	−	0.913581i	$-0.633305\pi$
$601$	−19.9386	−0.813312	−0.406656	+	0.913581i	$0.633305\pi$
$602$	0	0
$603$	1.25445	0.0510852
$604$	4.69558	0.191060
$605$	4.14206	0.168399
$606$	−0.972463	−0.0395036
$607$	19.6674	0.798275	0.399138	−	0.916891i	$-0.369310\pi$
$607$	19.6674	0.798275	0.399138	+	0.916891i	$0.369310\pi$
$608$	8.20398	0.332715
$609$	0	0
$610$	5.73145	0.232059
$611$	−5.01119	−0.202731
$612$	−0.835192	−0.0337606
$613$	−17.0577	−0.688954	−0.344477	−	0.938795i	$-0.611944\pi$
$613$	−17.0577	−0.688954	−0.344477	+	0.938795i	$0.611944\pi$
$614$	−4.96218	−0.200257
$615$	−7.74760	−0.312414
$616$	0	0
$617$	15.0250	0.604886	0.302443	−	0.953168i	$-0.402198\pi$
$617$	15.0250	0.604886	0.302443	+	0.953168i	$0.402198\pi$
$618$	0.883486	0.0355390
$619$	−8.86844	−0.356453	−0.178226	−	0.983990i	$-0.557036\pi$
$619$	−8.86844	−0.356453	−0.178226	+	0.983990i	$0.557036\pi$
$620$	−3.24976	−0.130513
$621$	−25.5541	−1.02545
$622$	13.3491	0.535249
$623$	0	0
$624$	−0.115925	−0.00464070
$625$	1.00000	0.0400000
$626$	−28.5074	−1.13938
$627$	5.35190	0.213734
$628$	−1.67859	−0.0669831
$629$	−0.662500	−0.0264156
$630$	0	0
$631$	4.20432	0.167371	0.0836856	−	0.996492i	$-0.473331\pi$
$631$	4.20432	0.167371	0.0836856	+	0.996492i	$0.473331\pi$
$632$	−14.8593	−0.591073
$633$	−9.93090	−0.394718
$634$	−2.03749	−0.0809192
$635$	17.2232	0.683483
$636$	20.4432	0.810624
$637$	0	0
$638$	−24.0258	−0.951191
$639$	−12.0627	−0.477194
$640$	6.83305	0.270100
$641$	27.8569	1.10028	0.550141	−	0.835072i	$-0.314574\pi$
$641$	27.8569	1.10028	0.550141	+	0.835072i	$0.314574\pi$
$642$	3.59988	0.142076
$643$	18.4951	0.729374	0.364687	−	0.931130i	$-0.381176\pi$
$643$	18.4951	0.729374	0.364687	+	0.931130i	$0.381176\pi$
$644$	0	0
$645$	−5.83309	−0.229678
$646$	−0.848226	−0.0333730
$647$	−15.1084	−0.593972	−0.296986	−	0.954882i	$-0.595981\pi$
$647$	−15.1084	−0.593972	−0.296986	+	0.954882i	$0.595981\pi$
$648$	13.7878	0.541637
$649$	−23.4340	−0.919864
$650$	1.63094	0.0639707
$651$	0	0
$652$	4.60890	0.180498
$653$	−19.8998	−0.778739	−0.389369	−	0.921082i	$-0.627307\pi$
$653$	−19.8998	−0.778739	−0.389369	+	0.921082i	$0.627307\pi$
$654$	−20.5403	−0.803190
$655$	18.6561	0.728952
$656$	−0.245691	−0.00959262
$657$	−2.38266	−0.0929565
$658$	0	0
$659$	−22.9674	−0.894683	−0.447342	−	0.894363i	$-0.647629\pi$
$659$	−22.9674	−0.894683	−0.447342	+	0.894363i	$0.647629\pi$
$660$	4.50868	0.175500
$661$	−9.31371	−0.362261	−0.181131	−	0.983459i	$-0.557976\pi$
$661$	−9.31371	−0.362261	−0.181131	+	0.983459i	$0.557976\pi$
$662$	1.55290	0.0603550
$663$	−1.72468	−0.0669810
$664$	−32.7792	−1.27208
$665$	0	0
$666$	−0.904534	−0.0350500
$667$	−47.1143	−1.82427
$668$	−27.0547	−1.04678
$669$	−3.42657	−0.132479
$670$	1.07330	0.0414651
$671$	−17.0614	−0.658649
$672$	0	0
$673$	29.2926	1.12915	0.564573	−	0.825383i	$-0.309041\pi$
$673$	29.2926	1.12915	0.564573	+	0.825383i	$0.309041\pi$
$674$	−12.2174	−0.470597
$675$	−5.65643	−0.217716
$676$	11.7250	0.450963
$677$	−14.0090	−0.538411	−0.269206	−	0.963083i	$-0.586761\pi$
$677$	−14.0090	−0.538411	−0.269206	+	0.963083i	$0.586761\pi$
$678$	10.9484	0.420470
$679$	0	0
$680$	−1.88022	−0.0721030
$681$	−19.2146	−0.736305
$682$	−6.10621	−0.233819
$683$	8.32325	0.318480	0.159240	−	0.987240i	$-0.449096\pi$
$683$	8.32325	0.318480	0.159240	+	0.987240i	$0.449096\pi$
$684$	1.83477	0.0701541
$685$	−8.75628	−0.334560
$686$	0	0
$687$	−10.8303	−0.413201
$688$	−0.184978	−0.00705223
$689$	−22.0134	−0.838644
$690$	−5.58077	−0.212456
$691$	26.6380	1.01336	0.506678	−	0.862135i	$-0.330873\pi$
$691$	26.6380	1.01336	0.506678	+	0.862135i	$0.330873\pi$
$692$	16.5712	0.629943
$693$	0	0
$694$	−7.06102	−0.268033
$695$	8.49623	0.322280
$696$	41.5613	1.57538
$697$	−3.65529	−0.138454
$698$	14.3328	0.542506
$699$	−13.0953	−0.495310
$700$	0	0
$701$	24.1312	0.911423	0.455711	−	0.890128i	$-0.349385\pi$
$701$	24.1312	0.911423	0.455711	+	0.890128i	$0.349385\pi$
$702$	−9.22529	−0.348186
$703$	1.45539	0.0548912
$704$	13.2196	0.498231
$705$	3.79560	0.142951
$706$	11.8594	0.446333
$707$	0	0
$708$	15.4064	0.579009
$709$	17.7141	0.665268	0.332634	−	0.943056i	$-0.392063\pi$
$709$	17.7141	0.665268	0.332634	+	0.943056i	$0.392063\pi$
$710$	−10.3208	−0.387332
$711$	−5.38340	−0.201893
$712$	21.5730	0.808482
$713$	−11.9742	−0.448437
$714$	0	0
$715$	−4.85499	−0.181566
$716$	−28.2717	−1.05656
$717$	−30.8058	−1.15046
$718$	−27.8710	−1.04014
$719$	37.5186	1.39921	0.699603	−	0.714531i	$-0.253360\pi$
$719$	37.5186	1.39921	0.699603	+	0.714531i	$0.253360\pi$
$720$	−0.0457859	−0.00170634
$721$	0	0
$722$	−14.8513	−0.552709
$723$	2.12679	0.0790962
$724$	−0.499917	−0.0185793
$725$	−10.4288	−0.387317
$726$	−5.11674	−0.189900
$727$	37.0709	1.37488	0.687442	−	0.726239i	$-0.258733\pi$
$727$	37.0709	1.37488	0.687442	+	0.726239i	$0.258733\pi$
$728$	0	0
$729$	28.8236	1.06754
$730$	−2.03859	−0.0754515
$731$	−2.75203	−0.101787
$732$	11.2169	0.414587
$733$	−1.97390	−0.0729078	−0.0364539	−	0.999335i	$-0.511606\pi$
$733$	−1.97390	−0.0729078	−0.0364539	+	0.999335i	$0.511606\pi$
$734$	−24.3935	−0.900381
$735$	0	0
$736$	25.4660	0.938691
$737$	−3.19500	−0.117689
$738$	−4.99069	−0.183710
$739$	−22.4746	−0.826742	−0.413371	−	0.910563i	$-0.635649\pi$
$739$	−22.4746	−0.826742	−0.413371	+	0.910563i	$0.635649\pi$
$740$	1.22609	0.0450719
$741$	3.78881	0.139186
$742$	0	0
$743$	29.4497	1.08040	0.540202	−	0.841535i	$-0.318348\pi$
$743$	29.4497	1.08040	0.540202	+	0.841535i	$0.318348\pi$
$744$	10.5629	0.387254
$745$	16.4471	0.602575
$746$	15.4130	0.564309
$747$	−11.8756	−0.434505
$748$	2.12718	0.0777774
$749$	0	0
$750$	−1.23531	−0.0451072
$751$	2.24684	0.0819884	0.0409942	−	0.999159i	$-0.486947\pi$
$751$	2.24684	0.0819884	0.0409942	+	0.999159i	$0.486947\pi$
$752$	0.120366	0.00438928
$753$	40.6919	1.48290
$754$	−17.0088	−0.619423
$755$	3.82972	0.139378
$756$	0	0
$757$	30.1017	1.09406	0.547032	−	0.837112i	$-0.315757\pi$
$757$	30.1017	1.09406	0.547032	+	0.837112i	$0.315757\pi$
$758$	19.7657	0.717921
$759$	16.6129	0.603010
$760$	4.13050	0.149829
$761$	1.00475	0.0364223	0.0182111	−	0.999834i	$-0.494203\pi$
$761$	1.00475	0.0364223	0.0182111	+	0.999834i	$0.494203\pi$
$762$	−21.2761	−0.770751
$763$	0	0
$764$	20.3711	0.736999
$765$	−0.681184	−0.0246283
$766$	25.3917	0.917439
$767$	−16.5898	−0.599023
$768$	−22.6179	−0.816152
$769$	34.7250	1.25222	0.626108	−	0.779737i	$-0.284647\pi$
$769$	34.7250	1.25222	0.626108	+	0.779737i	$0.284647\pi$
$770$	0	0
$771$	−35.8801	−1.29219
$772$	25.9100	0.932523
$773$	−6.07902	−0.218647	−0.109324	−	0.994006i	$-0.534868\pi$
$773$	−6.07902	−0.218647	−0.109324	+	0.994006i	$0.534868\pi$
$774$	−3.75744	−0.135058
$775$	−2.65051	−0.0952090
$776$	−2.74777	−0.0986392
$777$	0	0
$778$	−5.39781	−0.193521
$779$	8.03002	0.287705
$780$	3.19187	0.114287
$781$	30.7230	1.09935
$782$	−2.63299	−0.0941554
$783$	58.9900	2.10813
$784$	0	0
$785$	−1.36906	−0.0488639
$786$	−23.0461	−0.822026
$787$	−2.37626	−0.0847044	−0.0423522	−	0.999103i	$-0.513485\pi$
$787$	−2.37626	−0.0847044	−0.0423522	+	0.999103i	$0.513485\pi$
$788$	19.2773	0.686726
$789$	−26.5813	−0.946320
$790$	−4.60599	−0.163874
$791$	0	0
$792$	7.64184	0.271541
$793$	−12.0784	−0.428918
$794$	16.5610	0.587728
$795$	16.6735	0.591348
$796$	−3.92828	−0.139234
$797$	42.8422	1.51755	0.758774	−	0.651354i	$-0.225799\pi$
$797$	42.8422	1.51755	0.758774	+	0.651354i	$0.225799\pi$
$798$	0	0
$799$	1.79075	0.0633522
$800$	5.63695	0.199296
$801$	7.81570	0.276154
$802$	−31.3126	−1.10569
$803$	6.06848	0.214152
$804$	2.10052	0.0740797
$805$	0	0
$806$	−4.32281	−0.152265
$807$	30.1092	1.05989
$808$	2.23418	0.0785982
$809$	−45.3307	−1.59374	−0.796871	−	0.604149i	$-0.793513\pi$
$809$	−45.3307	−1.59374	−0.796871	+	0.604149i	$0.793513\pi$
$810$	4.27386	0.150168
$811$	−13.4410	−0.471977	−0.235988	−	0.971756i	$-0.575833\pi$
$811$	−13.4410	−0.471977	−0.235988	+	0.971756i	$0.575833\pi$
$812$	0	0
$813$	−42.3693	−1.48596
$814$	2.30379	0.0807477
$815$	3.75902	0.131673
$816$	0.0414257	0.00145019
$817$	6.04572	0.211513
$818$	17.5811	0.614708
$819$	0	0
$820$	6.76485	0.236239
$821$	−15.3028	−0.534072	−0.267036	−	0.963686i	$-0.586044\pi$
$821$	−15.3028	−0.534072	−0.267036	+	0.963686i	$0.586044\pi$
$822$	10.8167	0.377277
$823$	41.0609	1.43129	0.715646	−	0.698463i	$-0.246132\pi$
$823$	41.0609	1.43129	0.715646	+	0.698463i	$0.246132\pi$
$824$	−2.02976	−0.0707101
$825$	3.67729	0.128027
$826$	0	0
$827$	18.9159	0.657769	0.328884	−	0.944370i	$-0.393327\pi$
$827$	18.9159	0.657769	0.328884	+	0.944370i	$0.393327\pi$
$828$	5.69532	0.197926
$829$	−34.4593	−1.19682	−0.598411	−	0.801189i	$-0.704201\pi$
$829$	−34.4593	−1.19682	−0.598411	+	0.801189i	$0.704201\pi$
$830$	−10.1607	−0.352681
$831$	−16.6668	−0.578164
$832$	9.35863	0.324452
$833$	0	0
$834$	−10.4955	−0.363429
$835$	−22.0658	−0.763620
$836$	−4.67303	−0.161620
$837$	14.9924	0.518214
$838$	−4.90323	−0.169379
$839$	−1.24446	−0.0429634	−0.0214817	−	0.999769i	$-0.506838\pi$
$839$	−1.24446	−0.0429634	−0.0214817	+	0.999769i	$0.506838\pi$
$840$	0	0
$841$	79.7606	2.75036
$842$	13.8752	0.478170
$843$	−2.70652	−0.0932176
$844$	8.67120	0.298475
$845$	9.56297	0.328976
$846$	2.44497	0.0840599
$847$	0	0
$848$	0.528748	0.0181573
$849$	25.5407	0.876554
$850$	−0.582816	−0.0199904
$851$	4.51770	0.154865
$852$	−20.1985	−0.691990
$853$	−38.3071	−1.31161	−0.655804	−	0.754931i	$-0.727670\pi$
$853$	−38.3071	−1.31161	−0.655804	+	0.754931i	$0.727670\pi$
$854$	0	0
$855$	1.49644	0.0511772
$856$	−8.27053	−0.282681
$857$	−11.9468	−0.408094	−0.204047	−	0.978961i	$-0.565410\pi$
$857$	−11.9468	−0.408094	−0.204047	+	0.978961i	$0.565410\pi$
$858$	5.99743	0.204749
$859$	−12.5285	−0.427468	−0.213734	−	0.976892i	$-0.568563\pi$
$859$	−12.5285	−0.427468	−0.213734	+	0.976892i	$0.568563\pi$
$860$	5.09318	0.173676
$861$	0	0
$862$	5.03414	0.171464
$863$	34.8951	1.18784	0.593921	−	0.804523i	$-0.297579\pi$
$863$	34.8951	1.18784	0.593921	+	0.804523i	$0.297579\pi$
$864$	−31.8850	−1.08475
$865$	13.5155	0.459541
$866$	3.00178	0.102005
$867$	−23.2552	−0.789788
$868$	0	0
$869$	13.7112	0.465119
$870$	12.8829	0.436770
$871$	−2.26186	−0.0766403
$872$	47.1902	1.59806
$873$	−0.995492	−0.0336923
$874$	5.78420	0.195653
$875$	0	0
$876$	−3.98966	−0.134798
$877$	25.8780	0.873839	0.436919	−	0.899501i	$-0.356069\pi$
$877$	25.8780	0.873839	0.436919	+	0.899501i	$0.356069\pi$
$878$	18.8664	0.636709
$879$	−33.5473	−1.13152
$880$	0.116614	0.00393105
$881$	−43.0360	−1.44992	−0.724960	−	0.688790i	$-0.758142\pi$
$881$	−43.0360	−1.44992	−0.724960	+	0.688790i	$0.758142\pi$
$882$	0	0
$883$	−26.8171	−0.902467	−0.451233	−	0.892406i	$-0.649016\pi$
$883$	−26.8171	−0.902467	−0.451233	+	0.892406i	$0.649016\pi$
$884$	1.50591	0.0506493
$885$	12.5655	0.422385
$886$	28.0193	0.941327
$887$	43.9701	1.47637	0.738186	−	0.674597i	$-0.235682\pi$
$887$	43.9701	1.47637	0.738186	+	0.674597i	$0.235682\pi$
$888$	−3.98523	−0.133735
$889$	0	0
$890$	6.68704	0.224150
$891$	−12.7225	−0.426218
$892$	2.99192	0.100177
$893$	−3.93396	−0.131645
$894$	−20.3173	−0.679513
$895$	−23.0584	−0.770758
$896$	0	0
$897$	11.7609	0.392685
$898$	−12.4964	−0.417009
$899$	27.6417	0.921902
$900$	1.26067	0.0420223
$901$	7.86649	0.262071
$902$	12.7110	0.423229
$903$	0	0
$904$	−25.1533	−0.836586
$905$	−0.407733	−0.0135535
$906$	−4.73090	−0.157174
$907$	−19.5143	−0.647961	−0.323980	−	0.946064i	$-0.605021\pi$
$907$	−19.5143	−0.647961	−0.323980	+	0.946064i	$0.605021\pi$
$908$	16.7773	0.556774
$909$	0.809423	0.0268469
$910$	0	0
$911$	−16.7608	−0.555311	−0.277655	−	0.960681i	$-0.589557\pi$
$911$	−16.7608	−0.555311	−0.277655	+	0.960681i	$0.589557\pi$
$912$	−0.0910048	−0.00301347
$913$	30.2463	1.00101
$914$	14.0311	0.464107
$915$	9.14849	0.302440
$916$	9.45650	0.312451
$917$	0	0
$918$	3.29666	0.108806
$919$	12.4241	0.409835	0.204917	−	0.978779i	$-0.434307\pi$
$919$	12.4241	0.409835	0.204917	+	0.978779i	$0.434307\pi$
$920$	12.8215	0.422713
$921$	−7.92059	−0.260992
$922$	25.7258	0.847234
$923$	21.7500	0.715909
$924$	0	0
$925$	1.00000	0.0328798
$926$	−10.2467	−0.336728
$927$	−0.735363	−0.0241525
$928$	−58.7868	−1.92977
$929$	−39.4594	−1.29462	−0.647310	−	0.762227i	$-0.724106\pi$
$929$	−39.4594	−1.29462	−0.647310	+	0.762227i	$0.724106\pi$
$930$	3.27420	0.107365
$931$	0	0
$932$	11.4342	0.374540
$933$	21.3077	0.697583
$934$	−5.46621	−0.178860
$935$	1.73493	0.0567383
$936$	5.40995	0.176830
$937$	7.43644	0.242938	0.121469	−	0.992595i	$-0.461239\pi$
$937$	7.43644	0.242938	0.121469	+	0.992595i	$0.461239\pi$
$938$	0	0
$939$	−45.5033	−1.48494
$940$	−3.31414	−0.108095
$941$	10.1584	0.331154	0.165577	−	0.986197i	$-0.447051\pi$
$941$	10.1584	0.331154	0.165577	+	0.986197i	$0.447051\pi$
$942$	1.69122	0.0551029
$943$	24.9261	0.811704
$944$	0.398476	0.0129693
$945$	0	0
$946$	9.56995	0.311146
$947$	52.3157	1.70003	0.850015	−	0.526758i	$-0.176593\pi$
$947$	52.3157	1.70003	0.850015	+	0.526758i	$0.176593\pi$
$948$	−9.01427	−0.292770
$949$	4.29611	0.139458
$950$	1.28034	0.0415398
$951$	−3.25223	−0.105461
$952$	0	0
$953$	16.5930	0.537499	0.268750	−	0.963210i	$-0.413390\pi$
$953$	16.5930	0.537499	0.268750	+	0.963210i	$0.413390\pi$
$954$	10.7404	0.347733
$955$	16.6147	0.537638
$956$	26.8982	0.869950
$957$	−38.3498	−1.23967
$958$	0.312842	0.0101075
$959$	0	0
$960$	−7.08845	−0.228779
$961$	−23.9748	−0.773381
$962$	1.63094	0.0525836
$963$	−2.99633	−0.0965555
$964$	−1.85701	−0.0598104
$965$	21.1323	0.680272
$966$	0	0
$967$	13.1962	0.424360	0.212180	−	0.977231i	$-0.431944\pi$
$967$	13.1962	0.424360	0.212180	+	0.977231i	$0.431944\pi$
$968$	11.7554	0.377834
$969$	−1.35393	−0.0434946
$970$	−0.851735	−0.0273475
$971$	−4.51276	−0.144821	−0.0724107	−	0.997375i	$-0.523069\pi$
$971$	−4.51276	−0.144821	−0.0724107	+	0.997375i	$0.523069\pi$
$972$	−12.4416	−0.399065
$973$	0	0
$974$	−0.427084	−0.0136847
$975$	2.60329	0.0833721
$976$	0.290116	0.00928638
$977$	−44.9853	−1.43921	−0.719604	−	0.694385i	$-0.755677\pi$
$977$	−44.9853	−1.43921	−0.719604	+	0.694385i	$0.755677\pi$
$978$	−4.64357	−0.148485
$979$	−19.9061	−0.636200
$980$	0	0
$981$	17.0966	0.545852
$982$	18.1597	0.579499
$983$	−24.0442	−0.766892	−0.383446	−	0.923563i	$-0.625263\pi$
$983$	−24.0442	−0.766892	−0.383446	+	0.923563i	$0.625263\pi$
$984$	−21.9882	−0.700958
$985$	15.7226	0.500964
$986$	6.07809	0.193566
$987$	0	0
$988$	−3.30822	−0.105248
$989$	18.7666	0.596742
$990$	2.36876	0.0752842
$991$	−42.3789	−1.34621	−0.673105	−	0.739547i	$-0.735040\pi$
$991$	−42.3789	−1.34621	−0.673105	+	0.739547i	$0.735040\pi$
$992$	−14.9408	−0.474370
$993$	2.47872	0.0786599
$994$	0	0
$995$	−3.20391	−0.101571
$996$	−19.8852	−0.630085
$997$	−20.4053	−0.646244	−0.323122	−	0.946357i	$-0.604732\pi$
$997$	−20.4053	−0.646244	−0.323122	+	0.946357i	$0.604732\pi$
$998$	10.8237	0.342619
$999$	−5.65643	−0.178962

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.s.1.13		19
7.3	odd	6		1295.2.j.a.926.7	yes	38
7.5	odd	6		1295.2.j.a.186.7	✓	38
7.6	odd	2		9065.2.a.r.1.13		19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1295.2.j.a.186.7	✓	38	7.5	odd	6
1295.2.j.a.926.7	yes	38	7.3	odd	6
9065.2.a.r.1.13		19	7.6	odd	2
9065.2.a.s.1.13		19	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:	\( N \)	\(=\)	\( 9065 = 5 \cdot 7^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9065.a (trivial)

\(f(q)\)	\(=\)	\(q+0.879722 q^{2} +1.40421 q^{3} -1.22609 q^{4} -1.00000 q^{5} +1.23531 q^{6} -2.83806 q^{8} -1.02820 q^{9} -0.879722 q^{10} +2.61877 q^{11} -1.72168 q^{12} +1.85392 q^{13} -1.40421 q^{15} -0.0445300 q^{16} -0.662500 q^{17} -0.904534 q^{18} +1.45539 q^{19} +1.22609 q^{20} +2.30379 q^{22} +4.51770 q^{23} -3.98523 q^{24} +1.00000 q^{25} +1.63094 q^{26} -5.65643 q^{27} -10.4288 q^{29} -1.23531 q^{30} -2.65051 q^{31} +5.63695 q^{32} +3.67729 q^{33} -0.582816 q^{34} +1.26067 q^{36} +1.00000 q^{37} +1.28034 q^{38} +2.60329 q^{39} +2.83806 q^{40} +5.51742 q^{41} +4.15401 q^{43} -3.21084 q^{44} +1.02820 q^{45} +3.97432 q^{46} -2.70302 q^{47} -0.0625294 q^{48} +0.879722 q^{50} -0.930286 q^{51} -2.27307 q^{52} -11.8740 q^{53} -4.97609 q^{54} -2.61877 q^{55} +2.04367 q^{57} -9.17448 q^{58} -8.94848 q^{59} +1.72168 q^{60} -6.51506 q^{61} -2.33171 q^{62} +5.04801 q^{64} -1.85392 q^{65} +3.23499 q^{66} -1.22004 q^{67} +0.812283 q^{68} +6.34378 q^{69} +11.7319 q^{71} +2.91811 q^{72} +2.31731 q^{73} +0.879722 q^{74} +1.40421 q^{75} -1.78444 q^{76} +2.29017 q^{78} +5.23574 q^{79} +0.0445300 q^{80} -4.85819 q^{81} +4.85380 q^{82} +11.5498 q^{83} +0.662500 q^{85} +3.65437 q^{86} -14.6442 q^{87} -7.43222 q^{88} -7.60131 q^{89} +0.904534 q^{90} -5.53910 q^{92} -3.72186 q^{93} -2.37791 q^{94} -1.45539 q^{95} +7.91544 q^{96} +0.968186 q^{97} -2.69262 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 3 q^{2} + q^{3} + 11 q^{4} - 19 q^{5} + 4 q^{6} - 9 q^{8} + 12 q^{9} + 3 q^{10} - 13 q^{11} + 4 q^{12} + 3 q^{13} - q^{15} - 5 q^{16} + 2 q^{17} - 11 q^{18} + 12 q^{19} - 11 q^{20} - 10 q^{22}+ \cdots - 71 q^{99}+O(q^{100}) \) 19 * q - 3 * q^2 + q^3 + 11 * q^4 - 19 * q^5 + 4 * q^6 - 9 * q^8 + 12 * q^9 + 3 * q^10 - 13 * q^11 + 4 * q^12 + 3 * q^13 - q^15 - 5 * q^16 + 2 * q^17 - 11 * q^18 + 12 * q^19 - 11 * q^20 - 10 * q^22 - 16 * q^23 - 15 * q^24 + 19 * q^25 + 3 * q^26 + 7 * q^27 - q^29 - 4 * q^30 + 8 * q^31 - 16 * q^32 + 2 * q^33 + 14 * q^34 + 5 * q^36 + 19 * q^37 - 2 * q^38 - 45 * q^39 + 9 * q^40 + 18 * q^41 - 37 * q^43 + q^44 - 12 * q^45 - 29 * q^46 - 2 * q^47 + 35 * q^48 - 3 * q^50 - 43 * q^51 - 40 * q^52 - 10 * q^53 + 38 * q^54 + 13 * q^55 + 12 * q^57 - 13 * q^58 + 28 * q^59 - 4 * q^60 - 22 * q^61 + 10 * q^62 - 7 * q^64 - 3 * q^65 - 34 * q^66 - 61 * q^67 + 17 * q^68 + 20 * q^69 - 17 * q^71 + 17 * q^72 - 16 * q^73 - 3 * q^74 + q^75 + 45 * q^76 + 53 * q^78 - 63 * q^79 + 5 * q^80 - 5 * q^81 - 37 * q^82 + 24 * q^83 - 2 * q^85 - 19 * q^86 - 32 * q^87 - 58 * q^88 + 16 * q^89 + 11 * q^90 + 7 * q^92 - 31 * q^93 - 12 * q^95 - 40 * q^96 + 24 * q^97 - 71 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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