LMFDB - Embedded newform 9065.2.a.o.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 = [15,-1,1,23,15,-6,0,-3,30,-1,17] 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:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 26 x^{13} + 24 x^{12} + 266 x^{11} - 222 x^{10} - 1368 x^{9} + 998 x^{8} + 3770 x^{7} + \cdots - 158 $ x^15 - x^14 - 26x^13 + 24x^12 + 266x^11 - 222x^10 - 1368x^9 + 998x^8 + 3770x^7 - 2257x^6 - 5537x^5 + 2394x^4 + 3982x^3 - 824x^2 - 1058*x - 158
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 1295)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$-2.37325$ 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)

$f(q)$	$=$	\(q+2.37325 q^{2} -3.18420 q^{3} +3.63232 q^{4} +1.00000 q^{5} -7.55690 q^{6} +3.87391 q^{8} +7.13912 q^{9} +2.37325 q^{10} +3.49997 q^{11} -11.5660 q^{12} +6.91394 q^{13} -3.18420 q^{15} +1.92912 q^{16} +1.52255 q^{17} +16.9429 q^{18} +6.63261 q^{19} +3.63232 q^{20} +8.30631 q^{22} +8.57518 q^{23} -12.3353 q^{24} +1.00000 q^{25} +16.4085 q^{26} -13.1798 q^{27} +0.503093 q^{29} -7.55690 q^{30} +4.42021 q^{31} -3.16953 q^{32} -11.1446 q^{33} +3.61340 q^{34} +25.9316 q^{36} +1.00000 q^{37} +15.7409 q^{38} -22.0154 q^{39} +3.87391 q^{40} -5.21974 q^{41} -6.10336 q^{43} +12.7130 q^{44} +7.13912 q^{45} +20.3511 q^{46} -0.975847 q^{47} -6.14270 q^{48} +2.37325 q^{50} -4.84811 q^{51} +25.1137 q^{52} -10.9329 q^{53} -31.2789 q^{54} +3.49997 q^{55} -21.1196 q^{57} +1.19397 q^{58} -6.64903 q^{59} -11.5660 q^{60} +5.63846 q^{61} +10.4903 q^{62} -11.3803 q^{64} +6.91394 q^{65} -26.4489 q^{66} +12.7317 q^{67} +5.53040 q^{68} -27.3051 q^{69} +4.14221 q^{71} +27.6563 q^{72} +12.1214 q^{73} +2.37325 q^{74} -3.18420 q^{75} +24.0918 q^{76} -52.2480 q^{78} -1.02367 q^{79} +1.92912 q^{80} +20.5497 q^{81} -12.3877 q^{82} -14.4374 q^{83} +1.52255 q^{85} -14.4848 q^{86} -1.60195 q^{87} +13.5586 q^{88} -7.52820 q^{89} +16.9429 q^{90} +31.1478 q^{92} -14.0748 q^{93} -2.31593 q^{94} +6.63261 q^{95} +10.0924 q^{96} -12.9860 q^{97} +24.9867 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{2} + q^{3} + 23 q^{4} + 15 q^{5} - 6 q^{6} - 3 q^{8} + 30 q^{9} - q^{10} + 17 q^{11} - 8 q^{12} + 5 q^{13} + q^{15} + 39 q^{16} + 7 q^{17} + 12 q^{18} - 6 q^{19} + 23 q^{20} + 18 q^{22} - 2 q^{23}+ \cdots + 90 q^{99}+O(q^{100}) $ 15 * q - q^2 + q^3 + 23 * q^4 + 15 * q^5 - 6 * q^6 - 3 * q^8 + 30 * q^9 - q^10 + 17 * q^11 - 8 * q^12 + 5 * q^13 + q^15 + 39 * q^16 + 7 * q^17 + 12 * q^18 - 6 * q^19 + 23 * q^20 + 18 * q^22 - 2 * q^23 - 4 * q^24 + 15 * q^25 - 4 * q^26 - 5 * q^27 + 13 * q^29 - 6 * q^30 + 4 * q^31 - 7 * q^32 - 7 * q^33 - 38 * q^34 + 49 * q^36 + 15 * q^37 + 26 * q^38 + 33 * q^39 - 3 * q^40 - 22 * q^41 + 10 * q^43 + 58 * q^44 + 30 * q^45 + 32 * q^46 + 9 * q^47 - 43 * q^48 - q^50 + 39 * q^51 - 6 * q^52 - 16 * q^53 - 30 * q^54 + 17 * q^55 + 16 * q^57 + 44 * q^58 - 6 * q^59 - 8 * q^60 + 11 * q^62 + 119 * q^64 + 5 * q^65 - 3 * q^66 + 4 * q^67 + 70 * q^68 - 14 * q^69 + 48 * q^71 - 25 * q^72 - 38 * q^73 - q^74 + q^75 + 3 * q^76 - 40 * q^78 + 7 * q^79 + 39 * q^80 + 111 * q^81 - 6 * q^82 + 6 * q^83 + 7 * q^85 - 40 * q^86 + 47 * q^87 + 23 * q^88 + 2 * q^89 + 12 * q^90 + 7 * q^92 + 26 * q^94 - 6 * q^95 + 29 * q^96 - 79 * q^97 + 90 * 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 $

$2$	2.37325	1.67814	0.839071	−	0.544022i	$-0.183099\pi$
$2$	2.37325	1.67814	0.839071	+	0.544022i	$0.183099\pi$
$3$	−3.18420	−1.83840	−0.919199	−	0.393794i	$-0.871162\pi$
$3$	−3.18420	−1.83840	−0.919199	+	0.393794i	$0.871162\pi$
$4$	3.63232	1.81616
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−7.55690	−3.08509
$7$	0	0
$7$	0	0
$8$	3.87391	1.36963
$9$	7.13912	2.37971
$10$	2.37325	0.750488
$11$	3.49997	1.05528	0.527640	−	0.849468i	$-0.323077\pi$
$11$	3.49997	1.05528	0.527640	+	0.849468i	$0.323077\pi$
$12$	−11.5660	−3.33883
$13$	6.91394	1.91758	0.958791	−	0.284111i	$-0.0916984\pi$
$13$	6.91394	1.91758	0.958791	+	0.284111i	$0.0916984\pi$
$14$	0	0
$15$	−3.18420	−0.822156
$16$	1.92912	0.482280
$17$	1.52255	0.369273	0.184636	−	0.982807i	$-0.440889\pi$
$17$	1.52255	0.369273	0.184636	+	0.982807i	$0.440889\pi$
$18$	16.9429	3.99349
$19$	6.63261	1.52163	0.760813	−	0.648971i	$-0.224801\pi$
$19$	6.63261	1.52163	0.760813	+	0.648971i	$0.224801\pi$
$20$	3.63232	0.812212
$21$	0	0
$22$	8.30631	1.77091
$23$	8.57518	1.78805	0.894025	−	0.448018i	$-0.147870\pi$
$23$	8.57518	1.78805	0.894025	+	0.448018i	$0.147870\pi$
$24$	−12.3353	−2.51793
$25$	1.00000	0.200000
$26$	16.4085	3.21798
$27$	−13.1798	−2.53645
$28$	0	0
$29$	0.503093	0.0934220	0.0467110	−	0.998908i	$-0.485126\pi$
$29$	0.503093	0.0934220	0.0467110	+	0.998908i	$0.485126\pi$
$30$	−7.55690	−1.37970
$31$	4.42021	0.793892	0.396946	−	0.917842i	$-0.370070\pi$
$31$	4.42021	0.793892	0.396946	+	0.917842i	$0.370070\pi$
$32$	−3.16953	−0.560300
$33$	−11.1446	−1.94002
$34$	3.61340	0.619692
$35$	0	0
$36$	25.9316	4.32193
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	15.7409	2.55350
$39$	−22.0154	−3.52528
$40$	3.87391	0.612519
$41$	−5.21974	−0.815186	−0.407593	−	0.913164i	$-0.633632\pi$
$41$	−5.21974	−0.815186	−0.407593	+	0.913164i	$0.633632\pi$
$42$	0	0
$43$	−6.10336	−0.930753	−0.465377	−	0.885113i	$-0.654081\pi$
$43$	−6.10336	−0.930753	−0.465377	+	0.885113i	$0.654081\pi$
$44$	12.7130	1.91656
$45$	7.13912	1.06424
$46$	20.3511	3.00060
$47$	−0.975847	−0.142342	−0.0711709	−	0.997464i	$-0.522674\pi$
$47$	−0.975847	−0.142342	−0.0711709	+	0.997464i	$0.522674\pi$
$48$	−6.14270	−0.886622
$49$	0	0
$50$	2.37325	0.335628
$51$	−4.84811	−0.678871
$52$	25.1137	3.48264
$53$	−10.9329	−1.50175	−0.750877	−	0.660442i	$-0.770369\pi$
$53$	−10.9329	−1.50175	−0.750877	+	0.660442i	$0.770369\pi$
$54$	−31.2789	−4.25652
$55$	3.49997	0.471936
$56$	0	0
$57$	−21.1196	−2.79735
$58$	1.19397	0.156775
$59$	−6.64903	−0.865630	−0.432815	−	0.901483i	$-0.642480\pi$
$59$	−6.64903	−0.865630	−0.432815	+	0.901483i	$0.642480\pi$
$60$	−11.5660	−1.49317
$61$	5.63846	0.721931	0.360966	−	0.932579i	$-0.382447\pi$
$61$	5.63846	0.721931	0.360966	+	0.932579i	$0.382447\pi$
$62$	10.4903	1.33226
$63$	0	0
$64$	−11.3803	−1.42254
$65$	6.91394	0.857569
$66$	−26.4489	−3.25564
$67$	12.7317	1.55542	0.777710	−	0.628623i	$-0.216381\pi$
$67$	12.7317	1.55542	0.777710	+	0.628623i	$0.216381\pi$
$68$	5.53040	0.670659
$69$	−27.3051	−3.28715
$70$	0	0
$71$	4.14221	0.491589	0.245795	−	0.969322i	$-0.420951\pi$
$71$	4.14221	0.491589	0.245795	+	0.969322i	$0.420951\pi$
$72$	27.6563	3.25933
$73$	12.1214	1.41870	0.709349	−	0.704858i	$-0.248989\pi$
$73$	12.1214	1.41870	0.709349	+	0.704858i	$0.248989\pi$
$74$	2.37325	0.275885
$75$	−3.18420	−0.367680
$76$	24.0918	2.76352
$77$	0	0
$78$	−52.2480	−5.91592
$79$	−1.02367	−0.115172	−0.0575861	−	0.998341i	$-0.518340\pi$
$79$	−1.02367	−0.115172	−0.0575861	+	0.998341i	$0.518340\pi$
$80$	1.92912	0.215682
$81$	20.5497	2.28330
$82$	−12.3877	−1.36800
$83$	−14.4374	−1.58471	−0.792354	−	0.610062i	$-0.791145\pi$
$83$	−14.4374	−1.58471	−0.792354	+	0.610062i	$0.791145\pi$
$84$	0	0
$85$	1.52255	0.165144
$86$	−14.4848	−1.56194
$87$	−1.60195	−0.171747
$88$	13.5586	1.44535
$89$	−7.52820	−0.797988	−0.398994	−	0.916954i	$-0.630641\pi$
$89$	−7.52820	−0.797988	−0.398994	+	0.916954i	$0.630641\pi$
$90$	16.9429	1.78594
$91$	0	0
$92$	31.1478	3.24739
$93$	−14.0748	−1.45949
$94$	−2.31593	−0.238870
$95$	6.63261	0.680492
$96$	10.0924	1.03005
$97$	−12.9860	−1.31853	−0.659265	−	0.751911i	$-0.729132\pi$
$97$	−12.9860	−1.31853	−0.659265	+	0.751911i	$0.729132\pi$
$98$	0	0
$99$	24.9867	2.51126
$100$	3.63232	0.363232
$101$	−11.7610	−1.17027	−0.585134	−	0.810937i	$-0.698958\pi$
$101$	−11.7610	−1.17027	−0.585134	+	0.810937i	$0.698958\pi$
$102$	−11.5058	−1.13924
$103$	7.05670	0.695317	0.347659	−	0.937621i	$-0.386977\pi$
$103$	7.05670	0.695317	0.347659	+	0.937621i	$0.386977\pi$
$104$	26.7840	2.62639
$105$	0	0
$106$	−25.9466	−2.52016
$107$	−6.38327	−0.617094	−0.308547	−	0.951209i	$-0.599843\pi$
$107$	−6.38327	−0.617094	−0.308547	+	0.951209i	$0.599843\pi$
$108$	−47.8732	−4.60660
$109$	−15.2987	−1.46535	−0.732676	−	0.680578i	$-0.761729\pi$
$109$	−15.2987	−1.46535	−0.732676	+	0.680578i	$0.761729\pi$
$110$	8.30631	0.791975
$111$	−3.18420	−0.302231
$112$	0	0
$113$	1.86234	0.175194	0.0875972	−	0.996156i	$-0.472081\pi$
$113$	1.86234	0.175194	0.0875972	+	0.996156i	$0.472081\pi$
$114$	−50.1220	−4.69436
$115$	8.57518	0.799640
$116$	1.82739	0.169669
$117$	49.3595	4.56328
$118$	−15.7798	−1.45265
$119$	0	0
$120$	−12.3353	−1.12605
$121$	1.24978	0.113616
$122$	13.3815	1.21150
$123$	16.6207	1.49864
$124$	16.0556	1.44184
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.03777	0.0920873	0.0460436	−	0.998939i	$-0.485339\pi$
$127$	1.03777	0.0920873	0.0460436	+	0.998939i	$0.485339\pi$
$128$	−20.6693	−1.82693
$129$	19.4343	1.71109
$130$	16.4085	1.43912
$131$	−9.18844	−0.802798	−0.401399	−	0.915903i	$-0.631476\pi$
$131$	−9.18844	−0.802798	−0.401399	+	0.915903i	$0.631476\pi$
$132$	−40.4808	−3.52340
$133$	0	0
$134$	30.2155	2.61022
$135$	−13.1798	−1.13433
$136$	5.89823	0.505769
$137$	−11.1697	−0.954288	−0.477144	−	0.878825i	$-0.658328\pi$
$137$	−11.1697	−0.954288	−0.477144	+	0.878825i	$0.658328\pi$
$138$	−64.8018	−5.51630
$139$	14.1571	1.20079	0.600395	−	0.799704i	$-0.295010\pi$
$139$	14.1571	1.20079	0.600395	+	0.799704i	$0.295010\pi$
$140$	0	0
$141$	3.10729	0.261681
$142$	9.83050	0.824957
$143$	24.1986	2.02359
$144$	13.7722	1.14768
$145$	0.503093	0.0417796
$146$	28.7670	2.38078
$147$	0	0
$148$	3.63232	0.298575
$149$	1.17907	0.0965931	0.0482966	−	0.998833i	$-0.484621\pi$
$149$	1.17907	0.0965931	0.0482966	+	0.998833i	$0.484621\pi$
$150$	−7.55690	−0.617019
$151$	21.4683	1.74707	0.873533	−	0.486765i	$-0.161823\pi$
$151$	21.4683	1.74707	0.873533	+	0.486765i	$0.161823\pi$
$152$	25.6942	2.08407
$153$	10.8697	0.878761
$154$	0	0
$155$	4.42021	0.355039
$156$	−79.9669	−6.40248
$157$	11.4933	0.917264	0.458632	−	0.888626i	$-0.348340\pi$
$157$	11.4933	0.917264	0.458632	+	0.888626i	$0.348340\pi$
$158$	−2.42943	−0.193275
$159$	34.8126	2.76082
$160$	−3.16953	−0.250574
$161$	0	0
$162$	48.7695	3.83169
$163$	−14.4986	−1.13562	−0.567808	−	0.823161i	$-0.692208\pi$
$163$	−14.4986	−1.13562	−0.567808	+	0.823161i	$0.692208\pi$
$164$	−18.9598	−1.48051
$165$	−11.1446	−0.867605
$166$	−34.2635	−2.65936
$167$	−17.4815	−1.35276	−0.676378	−	0.736554i	$-0.736452\pi$
$167$	−17.4815	−1.35276	−0.676378	+	0.736554i	$0.736452\pi$
$168$	0	0
$169$	34.8026	2.67712
$170$	3.61340	0.277135
$171$	47.3510	3.62102
$172$	−22.1694	−1.69040
$173$	−18.6424	−1.41736	−0.708679	−	0.705531i	$-0.750708\pi$
$173$	−18.6424	−1.41736	−0.708679	+	0.705531i	$0.750708\pi$
$174$	−3.80182	−0.288215
$175$	0	0
$176$	6.75186	0.508941
$177$	21.1718	1.59137
$178$	−17.8663	−1.33914
$179$	11.8371	0.884744	0.442372	−	0.896832i	$-0.354137\pi$
$179$	11.8371	0.884744	0.442372	+	0.896832i	$0.354137\pi$
$180$	25.9316	1.93283
$181$	1.23510	0.0918042	0.0459021	−	0.998946i	$-0.485384\pi$
$181$	1.23510	0.0918042	0.0459021	+	0.998946i	$0.485384\pi$
$182$	0	0
$183$	−17.9540	−1.32720
$184$	33.2195	2.44897
$185$	1.00000	0.0735215
$186$	−33.4031	−2.44923
$187$	5.32888	0.389686
$188$	−3.54459	−0.258516
$189$	0	0
$190$	15.7409	1.14196
$191$	9.42006	0.681611	0.340806	−	0.940134i	$-0.389300\pi$
$191$	9.42006	0.681611	0.340806	+	0.940134i	$0.389300\pi$
$192$	36.2373	2.61520
$193$	6.27210	0.451476	0.225738	−	0.974188i	$-0.427521\pi$
$193$	6.27210	0.451476	0.225738	+	0.974188i	$0.427521\pi$
$194$	−30.8191	−2.21268
$195$	−22.0154	−1.57655
$196$	0	0
$197$	−9.11416	−0.649357	−0.324679	−	0.945824i	$-0.605256\pi$
$197$	−9.11416	−0.649357	−0.324679	+	0.945824i	$0.605256\pi$
$198$	59.2997	4.21425
$199$	9.94184	0.704758	0.352379	−	0.935857i	$-0.385373\pi$
$199$	9.94184	0.704758	0.352379	+	0.935857i	$0.385373\pi$
$200$	3.87391	0.273927
$201$	−40.5402	−2.85948
$202$	−27.9119	−1.96388
$203$	0	0
$204$	−17.6099	−1.23294
$205$	−5.21974	−0.364562
$206$	16.7473	1.16684
$207$	61.2193	4.25503
$208$	13.3378	0.924812
$209$	23.2139	1.60574
$210$	0	0
$211$	7.93915	0.546554	0.273277	−	0.961935i	$-0.411892\pi$
$211$	7.93915	0.546554	0.273277	+	0.961935i	$0.411892\pi$
$212$	−39.7120	−2.72743
$213$	−13.1896	−0.903737
$214$	−15.1491	−1.03557
$215$	−6.10336	−0.416245
$216$	−51.0573	−3.47401
$217$	0	0
$218$	−36.3077	−2.45907
$219$	−38.5968	−2.60813
$220$	12.7130	0.857111
$221$	10.5268	0.708111
$222$	−7.55690	−0.507186
$223$	−5.70280	−0.381888	−0.190944	−	0.981601i	$-0.561155\pi$
$223$	−5.70280	−0.381888	−0.190944	+	0.981601i	$0.561155\pi$
$224$	0	0
$225$	7.13912	0.475941
$226$	4.41981	0.294001
$227$	1.47436	0.0978565	0.0489283	−	0.998802i	$-0.484419\pi$
$227$	1.47436	0.0978565	0.0489283	+	0.998802i	$0.484419\pi$
$228$	−76.7130	−5.08044
$229$	5.80227	0.383425	0.191713	−	0.981451i	$-0.438596\pi$
$229$	5.80227	0.383425	0.191713	+	0.981451i	$0.438596\pi$
$230$	20.3511	1.34191
$231$	0	0
$232$	1.94894	0.127954
$233$	−19.3814	−1.26972	−0.634860	−	0.772627i	$-0.718942\pi$
$233$	−19.3814	−1.26972	−0.634860	+	0.772627i	$0.718942\pi$
$234$	117.142	7.65784
$235$	−0.975847	−0.0636572
$236$	−24.1514	−1.57212
$237$	3.25958	0.211732
$238$	0	0
$239$	−4.83338	−0.312645	−0.156323	−	0.987706i	$-0.549964\pi$
$239$	−4.83338	−0.312645	−0.156323	+	0.987706i	$0.549964\pi$
$240$	−6.14270	−0.396510
$241$	−10.6196	−0.684071	−0.342036	−	0.939687i	$-0.611116\pi$
$241$	−10.6196	−0.684071	−0.342036	+	0.939687i	$0.611116\pi$
$242$	2.96604	0.190664
$243$	−25.8949	−1.66116
$244$	20.4807	1.31114
$245$	0	0
$246$	39.4450	2.51492
$247$	45.8575	2.91784
$248$	17.1235	1.08734
$249$	45.9715	2.91332
$250$	2.37325	0.150098
$251$	−0.370128	−0.0233623	−0.0116812	−	0.999932i	$-0.503718\pi$
$251$	−0.370128	−0.0233623	−0.0116812	+	0.999932i	$0.503718\pi$
$252$	0	0
$253$	30.0129	1.88689
$254$	2.46289	0.154536
$255$	−4.84811	−0.303600
$256$	−26.2929	−1.64330
$257$	9.98570	0.622891	0.311446	−	0.950264i	$-0.399187\pi$
$257$	9.98570	0.622891	0.311446	+	0.950264i	$0.399187\pi$
$258$	46.1225	2.87146
$259$	0	0
$260$	25.1137	1.55748
$261$	3.59164	0.222317
$262$	−21.8065	−1.34721
$263$	−12.7295	−0.784934	−0.392467	−	0.919766i	$-0.628378\pi$
$263$	−12.7295	−0.784934	−0.392467	+	0.919766i	$0.628378\pi$
$264$	−43.1732	−2.65712
$265$	−10.9329	−0.671605
$266$	0	0
$267$	23.9713	1.46702
$268$	46.2455	2.82489
$269$	27.2708	1.66273	0.831364	−	0.555729i	$-0.187561\pi$
$269$	27.2708	1.66273	0.831364	+	0.555729i	$0.187561\pi$
$270$	−31.2789	−1.90357
$271$	−13.9946	−0.850113	−0.425056	−	0.905167i	$-0.639746\pi$
$271$	−13.9946	−0.850113	−0.425056	+	0.905167i	$0.639746\pi$
$272$	2.93718	0.178093
$273$	0	0
$274$	−26.5084	−1.60143
$275$	3.49997	0.211056
$276$	−99.1809	−5.96999
$277$	−18.5423	−1.11410	−0.557050	−	0.830479i	$-0.688067\pi$
$277$	−18.5423	−1.11410	−0.557050	+	0.830479i	$0.688067\pi$
$278$	33.5984	2.01510
$279$	31.5564	1.88923
$280$	0	0
$281$	6.01847	0.359032	0.179516	−	0.983755i	$-0.442547\pi$
$281$	6.01847	0.359032	0.179516	+	0.983755i	$0.442547\pi$
$282$	7.37438	0.439138
$283$	−4.14590	−0.246448	−0.123224	−	0.992379i	$-0.539323\pi$
$283$	−4.14590	−0.246448	−0.123224	+	0.992379i	$0.539323\pi$
$284$	15.0458	0.892806
$285$	−21.1196	−1.25101
$286$	57.4293	3.39587
$287$	0	0
$288$	−22.6277	−1.33335
$289$	−14.6818	−0.863637
$290$	1.19397	0.0701121
$291$	41.3500	2.42398
$292$	44.0287	2.57658
$293$	17.1686	1.00300	0.501499	−	0.865158i	$-0.332782\pi$
$293$	17.1686	1.00300	0.501499	+	0.865158i	$0.332782\pi$
$294$	0	0
$295$	−6.64903	−0.387122
$296$	3.87391	0.225166
$297$	−46.1288	−2.67666
$298$	2.79823	0.162097
$299$	59.2883	3.42873
$300$	−11.5660	−0.667765
$301$	0	0
$302$	50.9497	2.93182
$303$	37.4495	2.15142
$304$	12.7951	0.733850
$305$	5.63846	0.322857
$306$	25.7965	1.47469
$307$	−4.55808	−0.260143	−0.130072	−	0.991505i	$-0.541521\pi$
$307$	−4.55808	−0.260143	−0.130072	+	0.991505i	$0.541521\pi$
$308$	0	0
$309$	−22.4699	−1.27827
$310$	10.4903	0.595807
$311$	−30.9809	−1.75676	−0.878382	−	0.477959i	$-0.841377\pi$
$311$	−30.9809	−1.75676	−0.878382	+	0.477959i	$0.841377\pi$
$312$	−85.2856	−4.82834
$313$	−25.3720	−1.43411	−0.717055	−	0.697017i	$-0.754510\pi$
$313$	−25.3720	−1.43411	−0.717055	+	0.697017i	$0.754510\pi$
$314$	27.2765	1.53930
$315$	0	0
$316$	−3.71831	−0.209171
$317$	17.9251	1.00677	0.503386	−	0.864062i	$-0.332087\pi$
$317$	17.9251	1.00677	0.503386	+	0.864062i	$0.332087\pi$
$318$	82.6192	4.63305
$319$	1.76081	0.0985864
$320$	−11.3803	−0.636181
$321$	20.3256	1.13446
$322$	0	0
$323$	10.0985	0.561895
$324$	74.6430	4.14683
$325$	6.91394	0.383517
$326$	−34.4088	−1.90573
$327$	48.7141	2.69390
$328$	−20.2208	−1.11651
$329$	0	0
$330$	−26.4489	−1.45597
$331$	15.3376	0.843032	0.421516	−	0.906821i	$-0.361498\pi$
$331$	15.3376	0.843032	0.421516	+	0.906821i	$0.361498\pi$
$332$	−52.4412	−2.87808
$333$	7.13912	0.391221
$334$	−41.4879	−2.27012
$335$	12.7317	0.695605
$336$	0	0
$337$	−13.9485	−0.759821	−0.379910	−	0.925023i	$-0.624045\pi$
$337$	−13.9485	−0.759821	−0.379910	+	0.925023i	$0.624045\pi$
$338$	82.5953	4.49259
$339$	−5.93007	−0.322077
$340$	5.53040	0.299928
$341$	15.4706	0.837779
$342$	112.376	6.07659
$343$	0	0
$344$	−23.6439	−1.27479
$345$	−27.3051	−1.47006
$346$	−44.2432	−2.37853
$347$	12.6597	0.679609	0.339805	−	0.940496i	$-0.389639\pi$
$347$	12.6597	0.679609	0.339805	+	0.940496i	$0.389639\pi$
$348$	−5.81879	−0.311920
$349$	−5.86107	−0.313736	−0.156868	−	0.987620i	$-0.550140\pi$
$349$	−5.86107	−0.313736	−0.156868	+	0.987620i	$0.550140\pi$
$350$	0	0
$351$	−91.1242	−4.86385
$352$	−11.0933	−0.591273
$353$	−16.1780	−0.861067	−0.430533	−	0.902575i	$-0.641674\pi$
$353$	−16.1780	−0.861067	−0.430533	+	0.902575i	$0.641674\pi$
$354$	50.2461	2.67055
$355$	4.14221	0.219845
$356$	−27.3448	−1.44927
$357$	0	0
$358$	28.0923	1.48473
$359$	22.7126	1.19873	0.599363	−	0.800478i	$-0.295421\pi$
$359$	22.7126	1.19873	0.599363	+	0.800478i	$0.295421\pi$
$360$	27.6563	1.45762
$361$	24.9916	1.31534
$362$	2.93120	0.154061
$363$	−3.97955	−0.208872
$364$	0	0
$365$	12.1214	0.634461
$366$	−42.6093	−2.22722
$367$	9.08738	0.474357	0.237179	−	0.971466i	$-0.423777\pi$
$367$	9.08738	0.474357	0.237179	+	0.971466i	$0.423777\pi$
$368$	16.5426	0.862340
$369$	−37.2643	−1.93990
$370$	2.37325	0.123379
$371$	0	0
$372$	−51.1243	−2.65067
$373$	−8.98414	−0.465181	−0.232591	−	0.972575i	$-0.574720\pi$
$373$	−8.98414	−0.465181	−0.232591	+	0.972575i	$0.574720\pi$
$374$	12.6468	0.653949
$375$	−3.18420	−0.164431
$376$	−3.78034	−0.194956
$377$	3.47835	0.179144
$378$	0	0
$379$	1.50404	0.0772576	0.0386288	−	0.999254i	$-0.487701\pi$
$379$	1.50404	0.0772576	0.0386288	+	0.999254i	$0.487701\pi$
$380$	24.0918	1.23588
$381$	−3.30447	−0.169293
$382$	22.3562	1.14384
$383$	6.47324	0.330767	0.165384	−	0.986229i	$-0.447114\pi$
$383$	6.47324	0.330767	0.165384	+	0.986229i	$0.447114\pi$
$384$	65.8153	3.35862
$385$	0	0
$386$	14.8853	0.757640
$387$	−43.5726	−2.21492
$388$	−47.1694	−2.39466
$389$	−33.5422	−1.70066	−0.850328	−	0.526253i	$-0.823597\pi$
$389$	−33.5422	−1.70066	−0.850328	+	0.526253i	$0.823597\pi$
$390$	−52.2480	−2.64568
$391$	13.0562	0.660278
$392$	0	0
$393$	29.2578	1.47586
$394$	−21.6302	−1.08971
$395$	−1.02367	−0.0515066
$396$	90.7597	4.56085
$397$	−7.68199	−0.385548	−0.192774	−	0.981243i	$-0.561748\pi$
$397$	−7.68199	−0.385548	−0.192774	+	0.981243i	$0.561748\pi$
$398$	23.5945	1.18268
$399$	0	0
$400$	1.92912	0.0964560
$401$	−2.91491	−0.145564	−0.0727818	−	0.997348i	$-0.523188\pi$
$401$	−2.91491	−0.145564	−0.0727818	+	0.997348i	$0.523188\pi$
$402$	−96.2120	−4.79862
$403$	30.5611	1.52235
$404$	−42.7199	−2.12540
$405$	20.5497	1.02112
$406$	0	0
$407$	3.49997	0.173487
$408$	−18.7811	−0.929804
$409$	24.4302	1.20800	0.603999	−	0.796985i	$-0.293573\pi$
$409$	24.4302	1.20800	0.603999	+	0.796985i	$0.293573\pi$
$410$	−12.3877	−0.611787
$411$	35.5664	1.75436
$412$	25.6322	1.26281
$413$	0	0
$414$	145.289	7.14055
$415$	−14.4374	−0.708703
$416$	−21.9140	−1.07442
$417$	−45.0790	−2.20753
$418$	55.0925	2.69466
$419$	−13.9295	−0.680501	−0.340250	−	0.940335i	$-0.610512\pi$
$419$	−13.9295	−0.680501	−0.340250	+	0.940335i	$0.610512\pi$
$420$	0	0
$421$	−11.3830	−0.554772	−0.277386	−	0.960759i	$-0.589468\pi$
$421$	−11.3830	−0.554772	−0.277386	+	0.960759i	$0.589468\pi$
$422$	18.8416	0.917195
$423$	−6.96669	−0.338732
$424$	−42.3532	−2.05685
$425$	1.52255	0.0738546
$426$	−31.3022	−1.51660
$427$	0	0
$428$	−23.1861	−1.12074
$429$	−77.0531	−3.72016
$430$	−14.4848	−0.698519
$431$	−3.97735	−0.191582	−0.0957911	−	0.995401i	$-0.530538\pi$
$431$	−3.97735	−0.191582	−0.0957911	+	0.995401i	$0.530538\pi$
$432$	−25.4254	−1.22328
$433$	−9.41302	−0.452361	−0.226181	−	0.974085i	$-0.572624\pi$
$433$	−9.41302	−0.452361	−0.226181	+	0.974085i	$0.572624\pi$
$434$	0	0
$435$	−1.60195	−0.0768075
$436$	−55.5699	−2.66131
$437$	56.8759	2.72074
$438$	−91.5999	−4.37681
$439$	4.22300	0.201553	0.100776	−	0.994909i	$-0.467867\pi$
$439$	4.22300	0.201553	0.100776	+	0.994909i	$0.467867\pi$
$440$	13.5586	0.646379
$441$	0	0
$442$	24.9828	1.18831
$443$	−18.0837	−0.859185	−0.429592	−	0.903023i	$-0.641343\pi$
$443$	−18.0837	−0.859185	−0.429592	+	0.903023i	$0.641343\pi$
$444$	−11.5660	−0.548900
$445$	−7.52820	−0.356871
$446$	−13.5342	−0.640862
$447$	−3.75439	−0.177577
$448$	0	0
$449$	−32.6153	−1.53921	−0.769607	−	0.638518i	$-0.779548\pi$
$449$	−32.6153	−1.53921	−0.769607	+	0.638518i	$0.779548\pi$
$450$	16.9429	0.798697
$451$	−18.2689	−0.860250
$452$	6.76463	0.318181
$453$	−68.3593	−3.21180
$454$	3.49902	0.164217
$455$	0	0
$456$	−81.8153	−3.83135
$457$	9.18611	0.429708	0.214854	−	0.976646i	$-0.431072\pi$
$457$	9.18611	0.429708	0.214854	+	0.976646i	$0.431072\pi$
$458$	13.7703	0.643442
$459$	−20.0669	−0.936642
$460$	31.1478	1.45228
$461$	−6.85424	−0.319234	−0.159617	−	0.987179i	$-0.551026\pi$
$461$	−6.85424	−0.319234	−0.159617	+	0.987179i	$0.551026\pi$
$462$	0	0
$463$	−21.7528	−1.01094	−0.505470	−	0.862844i	$-0.668681\pi$
$463$	−21.7528	−1.01094	−0.505470	+	0.862844i	$0.668681\pi$
$464$	0.970526	0.0450555
$465$	−14.0748	−0.652704
$466$	−45.9970	−2.13077
$467$	−1.77003	−0.0819072	−0.0409536	−	0.999161i	$-0.513040\pi$
$467$	−1.77003	−0.0819072	−0.0409536	+	0.999161i	$0.513040\pi$
$468$	179.289	8.28766
$469$	0	0
$470$	−2.31593	−0.106826
$471$	−36.5969	−1.68630
$472$	−25.7578	−1.18560
$473$	−21.3616	−0.982205
$474$	7.73580	0.355317
$475$	6.63261	0.304325
$476$	0	0
$477$	−78.0516	−3.57374
$478$	−11.4708	−0.524663
$479$	−13.6174	−0.622196	−0.311098	−	0.950378i	$-0.600697\pi$
$479$	−13.6174	−0.622196	−0.311098	+	0.950378i	$0.600697\pi$
$480$	10.0924	0.460654
$481$	6.91394	0.315249
$482$	−25.2031	−1.14797
$483$	0	0
$484$	4.53960	0.206346
$485$	−12.9860	−0.589664
$486$	−61.4550	−2.78766
$487$	37.5416	1.70117	0.850585	−	0.525837i	$-0.176248\pi$
$487$	37.5416	1.70117	0.850585	+	0.525837i	$0.176248\pi$
$488$	21.8429	0.988782
$489$	46.1663	2.08771
$490$	0	0
$491$	35.1288	1.58534	0.792669	−	0.609652i	$-0.208691\pi$
$491$	35.1288	1.58534	0.792669	+	0.609652i	$0.208691\pi$
$492$	60.3717	2.72176
$493$	0.765984	0.0344982
$494$	108.831	4.89656
$495$	24.9867	1.12307
$496$	8.52711	0.382878
$497$	0	0
$498$	109.102	4.88897
$499$	18.6646	0.835541	0.417770	−	0.908553i	$-0.362812\pi$
$499$	18.6646	0.835541	0.417770	+	0.908553i	$0.362812\pi$
$500$	3.63232	0.162442
$501$	55.6645	2.48691
$502$	−0.878408	−0.0392053
$503$	−12.9220	−0.576163	−0.288081	−	0.957606i	$-0.593017\pi$
$503$	−12.9220	−0.576163	−0.288081	+	0.957606i	$0.593017\pi$
$504$	0	0
$505$	−11.7610	−0.523360
$506$	71.2281	3.16647
$507$	−110.818	−4.92162
$508$	3.76952	0.167245
$509$	−1.26772	−0.0561908	−0.0280954	−	0.999605i	$-0.508944\pi$
$509$	−1.26772	−0.0561908	−0.0280954	+	0.999605i	$0.508944\pi$
$510$	−11.5058	−0.509484
$511$	0	0
$512$	−21.0609	−0.930769
$513$	−87.4163	−3.85953
$514$	23.6986	1.04530
$515$	7.05670	0.310955
$516$	70.5916	3.10762
$517$	−3.41543	−0.150211
$518$	0	0
$519$	59.3612	2.60567
$520$	26.7840	1.17456
$521$	−2.89326	−0.126756	−0.0633779	−	0.997990i	$-0.520187\pi$
$521$	−2.89326	−0.126756	−0.0633779	+	0.997990i	$0.520187\pi$
$522$	8.52386	0.373079
$523$	17.7639	0.776762	0.388381	−	0.921499i	$-0.373035\pi$
$523$	17.7639	0.776762	0.388381	+	0.921499i	$0.373035\pi$
$524$	−33.3754	−1.45801
$525$	0	0
$526$	−30.2103	−1.31723
$527$	6.72999	0.293163
$528$	−21.4993	−0.935635
$529$	50.5338	2.19712
$530$	−25.9466	−1.12705
$531$	−47.4682	−2.05995
$532$	0	0
$533$	−36.0890	−1.56319
$534$	56.8899	2.46187
$535$	−6.38327	−0.275973
$536$	49.3214	2.13036
$537$	−37.6916	−1.62651
$538$	64.7204	2.79029
$539$	0	0
$540$	−47.8732	−2.06013
$541$	3.59778	0.154681	0.0773404	−	0.997005i	$-0.475357\pi$
$541$	3.59778	0.154681	0.0773404	+	0.997005i	$0.475357\pi$
$542$	−33.2128	−1.42661
$543$	−3.93280	−0.168773
$544$	−4.82578	−0.206904
$545$	−15.2987	−0.655325
$546$	0	0
$547$	−35.5831	−1.52142	−0.760712	−	0.649090i	$-0.775150\pi$
$547$	−35.5831	−1.52142	−0.760712	+	0.649090i	$0.775150\pi$
$548$	−40.5718	−1.73314
$549$	40.2537	1.71798
$550$	8.30631	0.354182
$551$	3.33682	0.142153
$552$	−105.777	−4.50219
$553$	0	0
$554$	−44.0056	−1.86962
$555$	−3.18420	−0.135162
$556$	51.4232	2.18083
$557$	−33.5378	−1.42104	−0.710521	−	0.703676i	$-0.751540\pi$
$557$	−33.5378	−1.42104	−0.710521	+	0.703676i	$0.751540\pi$
$558$	74.8912	3.17040
$559$	−42.1983	−1.78480
$560$	0	0
$561$	−16.9682	−0.716399
$562$	14.2833	0.602507
$563$	18.3460	0.773190	0.386595	−	0.922250i	$-0.373651\pi$
$563$	18.3460	0.773190	0.386595	+	0.922250i	$0.373651\pi$
$564$	11.2867	0.475255
$565$	1.86234	0.0783493
$566$	−9.83926	−0.413575
$567$	0	0
$568$	16.0465	0.673298
$569$	−0.941599	−0.0394739	−0.0197369	−	0.999805i	$-0.506283\pi$
$569$	−0.941599	−0.0394739	−0.0197369	+	0.999805i	$0.506283\pi$
$570$	−50.1220	−2.09938
$571$	−25.1787	−1.05370	−0.526848	−	0.849960i	$-0.676626\pi$
$571$	−25.1787	−1.05370	−0.526848	+	0.849960i	$0.676626\pi$
$572$	87.8971	3.67516
$573$	−29.9953	−1.25307
$574$	0	0
$575$	8.57518	0.357610
$576$	−81.2456	−3.38523
$577$	−25.0899	−1.04451	−0.522253	−	0.852791i	$-0.674908\pi$
$577$	−25.0899	−1.04451	−0.522253	+	0.852791i	$0.674908\pi$
$578$	−34.8437	−1.44931
$579$	−19.9716	−0.829992
$580$	1.82739	0.0758784
$581$	0	0
$582$	98.1340	4.06779
$583$	−38.2649	−1.58477
$584$	46.9571	1.94310
$585$	49.3595	2.04076
$586$	40.7453	1.68317
$587$	−21.2093	−0.875401	−0.437700	−	0.899121i	$-0.644207\pi$
$587$	−21.2093	−0.875401	−0.437700	+	0.899121i	$0.644207\pi$
$588$	0	0
$589$	29.3175	1.20801
$590$	−15.7798	−0.649645
$591$	29.0213	1.19378
$592$	1.92912	0.0792863
$593$	−15.9503	−0.655001	−0.327501	−	0.944851i	$-0.606206\pi$
$593$	−15.9503	−0.655001	−0.327501	+	0.944851i	$0.606206\pi$
$594$	−109.475	−4.49182
$595$	0	0
$596$	4.28276	0.175429
$597$	−31.6568	−1.29563
$598$	140.706	5.75390
$599$	36.7056	1.49975	0.749876	−	0.661578i	$-0.230113\pi$
$599$	36.7056	1.49975	0.749876	+	0.661578i	$0.230113\pi$
$600$	−12.3353	−0.503587
$601$	32.9738	1.34503	0.672514	−	0.740084i	$-0.265214\pi$
$601$	32.9738	1.34503	0.672514	+	0.740084i	$0.265214\pi$
$602$	0	0
$603$	90.8929	3.70144
$604$	77.9798	3.17295
$605$	1.24978	0.0508108
$606$	88.8771	3.61039
$607$	−44.2623	−1.79655	−0.898276	−	0.439432i	$-0.855180\pi$
$607$	−44.2623	−1.79655	−0.898276	+	0.439432i	$0.855180\pi$
$608$	−21.0223	−0.852567
$609$	0	0
$610$	13.3815	0.541801
$611$	−6.74695	−0.272952
$612$	39.4822	1.59597
$613$	25.1406	1.01542	0.507710	−	0.861528i	$-0.330492\pi$
$613$	25.1406	1.01542	0.507710	+	0.861528i	$0.330492\pi$
$614$	−10.8175	−0.436557
$615$	16.6207	0.670210
$616$	0	0
$617$	11.5779	0.466107	0.233053	−	0.972464i	$-0.425128\pi$
$617$	11.5779	0.466107	0.233053	+	0.972464i	$0.425128\pi$
$618$	−53.3268	−2.14512
$619$	33.3134	1.33898	0.669490	−	0.742821i	$-0.266513\pi$
$619$	33.3134	1.33898	0.669490	+	0.742821i	$0.266513\pi$
$620$	16.0556	0.644809
$621$	−113.019	−4.53530
$622$	−73.5254	−2.94810
$623$	0	0
$624$	−42.4703	−1.70017
$625$	1.00000	0.0400000
$626$	−60.2141	−2.40664
$627$	−73.9178	−2.95199
$628$	41.7473	1.66590
$629$	1.52255	0.0607081
$630$	0	0
$631$	−9.59380	−0.381923	−0.190962	−	0.981597i	$-0.561161\pi$
$631$	−9.59380	−0.381923	−0.190962	+	0.981597i	$0.561161\pi$
$632$	−3.96562	−0.157744
$633$	−25.2798	−1.00478
$634$	42.5407	1.68951
$635$	1.03777	0.0411827
$636$	126.451	5.01410
$637$	0	0
$638$	4.17884	0.165442
$639$	29.5717	1.16984
$640$	−20.6693	−0.817028
$641$	−0.0694761	−0.00274414	−0.00137207	−	0.999999i	$-0.500437\pi$
$641$	−0.0694761	−0.00274414	−0.00137207	+	0.999999i	$0.500437\pi$
$642$	48.2377	1.90379
$643$	41.3636	1.63122	0.815610	−	0.578602i	$-0.196401\pi$
$643$	41.3636	1.63122	0.815610	+	0.578602i	$0.196401\pi$
$644$	0	0
$645$	19.4343	0.765225
$646$	23.9663	0.942940
$647$	32.3134	1.27037	0.635185	−	0.772360i	$-0.280924\pi$
$647$	32.3134	1.27037	0.635185	+	0.772360i	$0.280924\pi$
$648$	79.6076	3.12728
$649$	−23.2714	−0.913483
$650$	16.4085	0.643595
$651$	0	0
$652$	−52.6635	−2.06246
$653$	−10.0609	−0.393713	−0.196857	−	0.980432i	$-0.563073\pi$
$653$	−10.0609	−0.393713	−0.196857	+	0.980432i	$0.563073\pi$
$654$	115.611	4.52074
$655$	−9.18844	−0.359022
$656$	−10.0695	−0.393148
$657$	86.5358	3.37608
$658$	0	0
$659$	18.1987	0.708921	0.354461	−	0.935071i	$-0.384665\pi$
$659$	18.1987	0.708921	0.354461	+	0.935071i	$0.384665\pi$
$660$	−40.4808	−1.57571
$661$	2.55287	0.0992950	0.0496475	−	0.998767i	$-0.484190\pi$
$661$	2.55287	0.0992950	0.0496475	+	0.998767i	$0.484190\pi$
$662$	36.4000	1.41473
$663$	−33.5195	−1.30179
$664$	−55.9291	−2.17047
$665$	0	0
$666$	16.9429	0.656525
$667$	4.31411	0.167043
$668$	−63.4983	−2.45682
$669$	18.1589	0.702062
$670$	30.2155	1.16732
$671$	19.7344	0.761840
$672$	0	0
$673$	−34.7725	−1.34038	−0.670191	−	0.742188i	$-0.733788\pi$
$673$	−34.7725	−1.34038	−0.670191	+	0.742188i	$0.733788\pi$
$674$	−33.1032	−1.27509
$675$	−13.1798	−0.507290
$676$	126.414	4.86209
$677$	43.3732	1.66697	0.833483	−	0.552545i	$-0.186343\pi$
$677$	43.3732	1.66697	0.833483	+	0.552545i	$0.186343\pi$
$678$	−14.0735	−0.540491
$679$	0	0
$680$	5.89823	0.226187
$681$	−4.69465	−0.179899
$682$	36.7156	1.40591
$683$	21.3140	0.815556	0.407778	−	0.913081i	$-0.366304\pi$
$683$	21.3140	0.815556	0.407778	+	0.913081i	$0.366304\pi$
$684$	171.994	6.57636
$685$	−11.1697	−0.426770
$686$	0	0
$687$	−18.4756	−0.704888
$688$	−11.7741	−0.448884
$689$	−75.5897	−2.87974
$690$	−64.8018	−2.46696
$691$	30.4053	1.15667	0.578336	−	0.815799i	$-0.303702\pi$
$691$	30.4053	1.15667	0.578336	+	0.815799i	$0.303702\pi$
$692$	−67.7153	−2.57415
$693$	0	0
$694$	30.0447	1.14048
$695$	14.1571	0.537010
$696$	−6.20580	−0.235230
$697$	−7.94732	−0.301026
$698$	−13.9098	−0.526494
$699$	61.7144	2.33425
$700$	0	0
$701$	30.2055	1.14085	0.570423	−	0.821351i	$-0.306779\pi$
$701$	30.2055	1.14085	0.570423	+	0.821351i	$0.306779\pi$
$702$	−216.261	−8.16223
$703$	6.63261	0.250154
$704$	−39.8308	−1.50118
$705$	3.10729	0.117027
$706$	−38.3944	−1.44499
$707$	0	0
$708$	76.9029	2.89019
$709$	−20.2828	−0.761735	−0.380868	−	0.924630i	$-0.624375\pi$
$709$	−20.2828	−0.761735	−0.380868	+	0.924630i	$0.624375\pi$
$710$	9.83050	0.368932
$711$	−7.30813	−0.274076
$712$	−29.1636	−1.09295
$713$	37.9041	1.41952
$714$	0	0
$715$	24.1986	0.904976
$716$	42.9960	1.60684
$717$	15.3904	0.574767
$718$	53.9027	2.01163
$719$	28.6503	1.06847	0.534237	−	0.845334i	$-0.320599\pi$
$719$	28.6503	1.06847	0.534237	+	0.845334i	$0.320599\pi$
$720$	13.7722	0.513260
$721$	0	0
$722$	59.3112	2.20734
$723$	33.8150	1.25759
$724$	4.48628	0.166731
$725$	0.503093	0.0186844
$726$	−9.44447	−0.350517
$727$	35.5428	1.31821	0.659105	−	0.752051i	$-0.270935\pi$
$727$	35.5428	1.31821	0.659105	+	0.752051i	$0.270935\pi$
$728$	0	0
$729$	20.8054	0.770571
$730$	28.7670	1.06472
$731$	−9.29267	−0.343702
$732$	−65.2146	−2.41040
$733$	39.3766	1.45441	0.727204	−	0.686422i	$-0.240819\pi$
$733$	39.3766	1.45441	0.727204	+	0.686422i	$0.240819\pi$
$734$	21.5666	0.796039
$735$	0	0
$736$	−27.1793	−1.00184
$737$	44.5604	1.64140
$738$	−88.4376	−3.25543
$739$	25.2704	0.929587	0.464793	−	0.885419i	$-0.346129\pi$
$739$	25.2704	0.929587	0.464793	+	0.885419i	$0.346129\pi$
$740$	3.63232	0.133527
$741$	−146.019	−5.36416
$742$	0	0
$743$	−28.0425	−1.02878	−0.514390	−	0.857556i	$-0.671982\pi$
$743$	−28.0425	−1.02878	−0.514390	+	0.857556i	$0.671982\pi$
$744$	−54.5246	−1.99897
$745$	1.17907	0.0431978
$746$	−21.3216	−0.780640
$747$	−103.070	−3.77114
$748$	19.3562	0.707733
$749$	0	0
$750$	−7.55690	−0.275939
$751$	−35.0194	−1.27788	−0.638938	−	0.769258i	$-0.720626\pi$
$751$	−35.0194	−1.27788	−0.638938	+	0.769258i	$0.720626\pi$
$752$	−1.88253	−0.0686486
$753$	1.17856	0.0429492
$754$	8.25501	0.300630
$755$	21.4683	0.781312
$756$	0	0
$757$	−1.69731	−0.0616897	−0.0308448	−	0.999524i	$-0.509820\pi$
$757$	−1.69731	−0.0616897	−0.0308448	+	0.999524i	$0.509820\pi$
$758$	3.56948	0.129649
$759$	−95.5669	−3.46886
$760$	25.6942	0.932025
$761$	43.4145	1.57378	0.786888	−	0.617096i	$-0.211691\pi$
$761$	43.4145	1.57378	0.786888	+	0.617096i	$0.211691\pi$
$762$	−7.84234	−0.284098
$763$	0	0
$764$	34.2167	1.23792
$765$	10.8697	0.392994
$766$	15.3626	0.555075
$767$	−45.9710	−1.65992
$768$	83.7217	3.02105
$769$	−32.8931	−1.18615	−0.593077	−	0.805146i	$-0.702087\pi$
$769$	−32.8931	−1.18615	−0.593077	+	0.805146i	$0.702087\pi$
$770$	0	0
$771$	−31.7965	−1.14512
$772$	22.7823	0.819953
$773$	−44.4645	−1.59928	−0.799638	−	0.600482i	$-0.794975\pi$
$773$	−44.4645	−1.59928	−0.799638	+	0.600482i	$0.794975\pi$
$774$	−103.409	−3.71695
$775$	4.42021	0.158778
$776$	−50.3067	−1.80590
$777$	0	0
$778$	−79.6040	−2.85394
$779$	−34.6205	−1.24041
$780$	−79.9669	−2.86327
$781$	14.4976	0.518765
$782$	30.9855	1.10804
$783$	−6.63065	−0.236960
$784$	0	0
$785$	11.4933	0.410213
$786$	69.4362	2.47671
$787$	9.22725	0.328916	0.164458	−	0.986384i	$-0.447413\pi$
$787$	9.22725	0.328916	0.164458	+	0.986384i	$0.447413\pi$
$788$	−33.1056	−1.17934
$789$	40.5332	1.44302
$790$	−2.42943	−0.0864354
$791$	0	0
$792$	96.7962	3.43950
$793$	38.9840	1.38436
$794$	−18.2313	−0.647004
$795$	34.8126	1.23468
$796$	36.1120	1.27995
$797$	−25.4924	−0.902986	−0.451493	−	0.892275i	$-0.649108\pi$
$797$	−25.4924	−0.902986	−0.451493	+	0.892275i	$0.649108\pi$
$798$	0	0
$799$	−1.48578	−0.0525630
$800$	−3.16953	−0.112060
$801$	−53.7447	−1.89898
$802$	−6.91781	−0.244276
$803$	42.4244	1.49712
$804$	−147.255	−5.19328
$805$	0	0
$806$	72.5291	2.55473
$807$	−86.8355	−3.05675
$808$	−45.5613	−1.60284
$809$	31.6435	1.11253	0.556263	−	0.831006i	$-0.312235\pi$
$809$	31.6435	1.11253	0.556263	+	0.831006i	$0.312235\pi$
$810$	48.7695	1.71359
$811$	−52.3379	−1.83783	−0.918916	−	0.394453i	$-0.870934\pi$
$811$	−52.3379	−1.83783	−0.918916	+	0.394453i	$0.870934\pi$
$812$	0	0
$813$	44.5617	1.56285
$814$	8.30631	0.291136
$815$	−14.4986	−0.507863
$816$	−9.35258	−0.327406
$817$	−40.4812	−1.41626
$818$	57.9791	2.02719
$819$	0	0
$820$	−18.9598	−0.662104
$821$	45.4358	1.58572	0.792860	−	0.609404i	$-0.208591\pi$
$821$	45.4358	1.58572	0.792860	+	0.609404i	$0.208591\pi$
$822$	84.4080	2.94407
$823$	26.1766	0.912457	0.456229	−	0.889863i	$-0.349200\pi$
$823$	26.1766	0.912457	0.456229	+	0.889863i	$0.349200\pi$
$824$	27.3370	0.952330
$825$	−11.1446	−0.388005
$826$	0	0
$827$	−41.1176	−1.42980	−0.714899	−	0.699228i	$-0.753527\pi$
$827$	−41.1176	−1.42980	−0.714899	+	0.699228i	$0.753527\pi$
$828$	222.368	7.72782
$829$	−37.5994	−1.30588	−0.652941	−	0.757409i	$-0.726465\pi$
$829$	−37.5994	−1.30588	−0.652941	+	0.757409i	$0.726465\pi$
$830$	−34.2635	−1.18930
$831$	59.0424	2.04816
$832$	−78.6830	−2.72784
$833$	0	0
$834$	−106.984	−3.70455
$835$	−17.4815	−0.604971
$836$	84.3205	2.91629
$837$	−58.2573	−2.01367
$838$	−33.0582	−1.14198
$839$	−0.179212	−0.00618708	−0.00309354	−	0.999995i	$-0.500985\pi$
$839$	−0.179212	−0.00618708	−0.00309354	+	0.999995i	$0.500985\pi$
$840$	0	0
$841$	−28.7469	−0.991272
$842$	−27.0146	−0.930986
$843$	−19.1640	−0.660044
$844$	28.8376	0.992630
$845$	34.8026	1.19725
$846$	−16.5337	−0.568440
$847$	0	0
$848$	−21.0910	−0.724266
$849$	13.2014	0.453069
$850$	3.61340	0.123938
$851$	8.57518	0.293954
$852$	−47.9089	−1.64133
$853$	14.5685	0.498816	0.249408	−	0.968398i	$-0.419764\pi$
$853$	14.5685	0.498816	0.249408	+	0.968398i	$0.419764\pi$
$854$	0	0
$855$	47.3510	1.61937
$856$	−24.7282	−0.845193
$857$	−11.4981	−0.392767	−0.196384	−	0.980527i	$-0.562920\pi$
$857$	−11.4981	−0.392767	−0.196384	+	0.980527i	$0.562920\pi$
$858$	−182.866	−6.24295
$859$	20.9960	0.716376	0.358188	−	0.933649i	$-0.383395\pi$
$859$	20.9960	0.716376	0.358188	+	0.933649i	$0.383395\pi$
$860$	−22.1694	−0.755969
$861$	0	0
$862$	−9.43925	−0.321502
$863$	−22.4888	−0.765529	−0.382764	−	0.923846i	$-0.625028\pi$
$863$	−22.4888	−0.765529	−0.382764	+	0.923846i	$0.625028\pi$
$864$	41.7738	1.42117
$865$	−18.6424	−0.633862
$866$	−22.3395	−0.759126
$867$	46.7499	1.58771
$868$	0	0
$869$	−3.58282	−0.121539
$870$	−3.80182	−0.128894
$871$	88.0260	2.98265
$872$	−59.2659	−2.00700
$873$	−92.7087	−3.13771
$874$	134.981	4.56579
$875$	0	0
$876$	−140.196	−4.73678
$877$	23.6933	0.800067	0.400033	−	0.916501i	$-0.368999\pi$
$877$	23.6933	0.800067	0.400033	+	0.916501i	$0.368999\pi$
$878$	10.0222	0.338234
$879$	−54.6682	−1.84391
$880$	6.75186	0.227605
$881$	−4.69148	−0.158060	−0.0790299	−	0.996872i	$-0.525182\pi$
$881$	−4.69148	−0.158060	−0.0790299	+	0.996872i	$0.525182\pi$
$882$	0	0
$883$	6.65954	0.224111	0.112056	−	0.993702i	$-0.464256\pi$
$883$	6.65954	0.224111	0.112056	+	0.993702i	$0.464256\pi$
$884$	38.2368	1.28604
$885$	21.1718	0.711684
$886$	−42.9173	−1.44183
$887$	−10.0059	−0.335965	−0.167982	−	0.985790i	$-0.553725\pi$
$887$	−10.0059	−0.335965	−0.167982	+	0.985790i	$0.553725\pi$
$888$	−12.3353	−0.413946
$889$	0	0
$890$	−17.8663	−0.598880
$891$	71.9232	2.40952
$892$	−20.7144	−0.693570
$893$	−6.47241	−0.216591
$894$	−8.91011	−0.297999
$895$	11.8371	0.395670
$896$	0	0
$897$	−188.786	−6.30337
$898$	−77.4044	−2.58302
$899$	2.22377	0.0741670
$900$	25.9316	0.864386
$901$	−16.6460	−0.554557
$902$	−43.3567	−1.44362
$903$	0	0
$904$	7.21455	0.239952
$905$	1.23510	0.0410561
$906$	−162.234	−5.38986
$907$	21.7136	0.720988	0.360494	−	0.932761i	$-0.382608\pi$
$907$	21.7136	0.720988	0.360494	+	0.932761i	$0.382608\pi$
$908$	5.35534	0.177723
$909$	−83.9635	−2.78489
$910$	0	0
$911$	5.52661	0.183105	0.0915524	−	0.995800i	$-0.470817\pi$
$911$	5.52661	0.183105	0.0915524	+	0.995800i	$0.470817\pi$
$912$	−40.7422	−1.34911
$913$	−50.5304	−1.67231
$914$	21.8009	0.721111
$915$	−17.9540	−0.593540
$916$	21.0757	0.696362
$917$	0	0
$918$	−47.6238	−1.57182
$919$	16.0649	0.529932	0.264966	−	0.964258i	$-0.414639\pi$
$919$	16.0649	0.529932	0.264966	+	0.964258i	$0.414639\pi$
$920$	33.2195	1.09521
$921$	14.5138	0.478247
$922$	−16.2668	−0.535719
$923$	28.6390	0.942663
$924$	0	0
$925$	1.00000	0.0328798
$926$	−51.6249	−1.69650
$927$	50.3786	1.65465
$928$	−1.59457	−0.0523443
$929$	18.5488	0.608567	0.304284	−	0.952582i	$-0.401583\pi$
$929$	18.5488	0.608567	0.304284	+	0.952582i	$0.401583\pi$
$930$	−33.4031	−1.09533
$931$	0	0
$932$	−70.3996	−2.30602
$933$	98.6492	3.22963
$934$	−4.20073	−0.137452
$935$	5.32888	0.174273
$936$	191.214	6.25003
$937$	35.8245	1.17034	0.585168	−	0.810912i	$-0.301029\pi$
$937$	35.8245	1.17034	0.585168	+	0.810912i	$0.301029\pi$
$938$	0	0
$939$	80.7895	2.63646
$940$	−3.54459	−0.115612
$941$	−35.9464	−1.17182	−0.585910	−	0.810376i	$-0.699263\pi$
$941$	−35.9464	−1.17182	−0.585910	+	0.810376i	$0.699263\pi$
$942$	−86.8537	−2.82984
$943$	−44.7602	−1.45759
$944$	−12.8268	−0.417476
$945$	0	0
$946$	−50.6963	−1.64828
$947$	−38.2245	−1.24213	−0.621065	−	0.783759i	$-0.713300\pi$
$947$	−38.2245	−1.24213	−0.621065	+	0.783759i	$0.713300\pi$
$948$	11.8398	0.384540
$949$	83.8064	2.72047
$950$	15.7409	0.510701
$951$	−57.0769	−1.85085
$952$	0	0
$953$	31.8773	1.03261	0.516304	−	0.856405i	$-0.327307\pi$
$953$	31.8773	1.03261	0.516304	+	0.856405i	$0.327307\pi$
$954$	−185.236	−5.99724
$955$	9.42006	0.304826
$956$	−17.5564	−0.567814
$957$	−5.60676	−0.181241
$958$	−32.3176	−1.04413
$959$	0	0
$960$	36.2373	1.16955
$961$	−11.4618	−0.369735
$962$	16.4085	0.529032
$963$	−45.5709	−1.46850
$964$	−38.5740	−1.24238
$965$	6.27210	0.201906
$966$	0	0
$967$	−2.52974	−0.0813510	−0.0406755	−	0.999172i	$-0.512951\pi$
$967$	−2.52974	−0.0813510	−0.0406755	+	0.999172i	$0.512951\pi$
$968$	4.84154	0.155613
$969$	−32.1556	−1.03299
$970$	−30.8191	−0.989541
$971$	58.8714	1.88927	0.944637	−	0.328117i	$-0.106414\pi$
$971$	58.8714	1.88927	0.944637	+	0.328117i	$0.106414\pi$
$972$	−94.0585	−3.01693
$973$	0	0
$974$	89.0956	2.85481
$975$	−22.0154	−0.705056
$976$	10.8773	0.348173
$977$	56.5992	1.81077	0.905385	−	0.424592i	$-0.139582\pi$
$977$	56.5992	1.81077	0.905385	+	0.424592i	$0.139582\pi$
$978$	109.564	3.50348
$979$	−26.3485	−0.842101
$980$	0	0
$981$	−109.219	−3.48711
$982$	83.3694	2.66042
$983$	−3.61521	−0.115307	−0.0576537	−	0.998337i	$-0.518362\pi$
$983$	−3.61521	−0.115307	−0.0576537	+	0.998337i	$0.518362\pi$
$984$	64.3870	2.05258
$985$	−9.11416	−0.290401
$986$	1.81787	0.0578929
$987$	0	0
$988$	166.569	5.29927
$989$	−52.3374	−1.66423
$990$	59.2997	1.88467
$991$	−58.3476	−1.85347	−0.926737	−	0.375711i	$-0.877398\pi$
$991$	−58.3476	−1.85347	−0.926737	+	0.375711i	$0.877398\pi$
$992$	−14.0100	−0.444818
$993$	−48.8380	−1.54983
$994$	0	0
$995$	9.94184	0.315178
$996$	166.983	5.29106
$997$	−40.0226	−1.26753	−0.633764	−	0.773527i	$-0.718491\pi$
$997$	−40.0226	−1.26753	−0.633764	+	0.773527i	$0.718491\pi$
$998$	44.2957	1.40216
$999$	−13.1798	−0.416990

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9065.2.a.o.1.13		15
7.6	odd	2		1295.2.a.j.1.13	✓	15
35.34	odd	2		6475.2.a.u.1.3		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1295.2.a.j.1.13	✓	15	7.6	odd	2
6475.2.a.u.1.3		15	35.34	odd	2
9065.2.a.o.1.13		15	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)

\(f(q)\)	\(=\)	\(q+2.37325 q^{2} -3.18420 q^{3} +3.63232 q^{4} +1.00000 q^{5} -7.55690 q^{6} +3.87391 q^{8} +7.13912 q^{9} +2.37325 q^{10} +3.49997 q^{11} -11.5660 q^{12} +6.91394 q^{13} -3.18420 q^{15} +1.92912 q^{16} +1.52255 q^{17} +16.9429 q^{18} +6.63261 q^{19} +3.63232 q^{20} +8.30631 q^{22} +8.57518 q^{23} -12.3353 q^{24} +1.00000 q^{25} +16.4085 q^{26} -13.1798 q^{27} +0.503093 q^{29} -7.55690 q^{30} +4.42021 q^{31} -3.16953 q^{32} -11.1446 q^{33} +3.61340 q^{34} +25.9316 q^{36} +1.00000 q^{37} +15.7409 q^{38} -22.0154 q^{39} +3.87391 q^{40} -5.21974 q^{41} -6.10336 q^{43} +12.7130 q^{44} +7.13912 q^{45} +20.3511 q^{46} -0.975847 q^{47} -6.14270 q^{48} +2.37325 q^{50} -4.84811 q^{51} +25.1137 q^{52} -10.9329 q^{53} -31.2789 q^{54} +3.49997 q^{55} -21.1196 q^{57} +1.19397 q^{58} -6.64903 q^{59} -11.5660 q^{60} +5.63846 q^{61} +10.4903 q^{62} -11.3803 q^{64} +6.91394 q^{65} -26.4489 q^{66} +12.7317 q^{67} +5.53040 q^{68} -27.3051 q^{69} +4.14221 q^{71} +27.6563 q^{72} +12.1214 q^{73} +2.37325 q^{74} -3.18420 q^{75} +24.0918 q^{76} -52.2480 q^{78} -1.02367 q^{79} +1.92912 q^{80} +20.5497 q^{81} -12.3877 q^{82} -14.4374 q^{83} +1.52255 q^{85} -14.4848 q^{86} -1.60195 q^{87} +13.5586 q^{88} -7.52820 q^{89} +16.9429 q^{90} +31.1478 q^{92} -14.0748 q^{93} -2.31593 q^{94} +6.63261 q^{95} +10.0924 q^{96} -12.9860 q^{97} +24.9867 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{2} + q^{3} + 23 q^{4} + 15 q^{5} - 6 q^{6} - 3 q^{8} + 30 q^{9} - q^{10} + 17 q^{11} - 8 q^{12} + 5 q^{13} + q^{15} + 39 q^{16} + 7 q^{17} + 12 q^{18} - 6 q^{19} + 23 q^{20} + 18 q^{22} - 2 q^{23}+ \cdots + 90 q^{99}+O(q^{100}) \) 15 * q - q^2 + q^3 + 23 * q^4 + 15 * q^5 - 6 * q^6 - 3 * q^8 + 30 * q^9 - q^10 + 17 * q^11 - 8 * q^12 + 5 * q^13 + q^15 + 39 * q^16 + 7 * q^17 + 12 * q^18 - 6 * q^19 + 23 * q^20 + 18 * q^22 - 2 * q^23 - 4 * q^24 + 15 * q^25 - 4 * q^26 - 5 * q^27 + 13 * q^29 - 6 * q^30 + 4 * q^31 - 7 * q^32 - 7 * q^33 - 38 * q^34 + 49 * q^36 + 15 * q^37 + 26 * q^38 + 33 * q^39 - 3 * q^40 - 22 * q^41 + 10 * q^43 + 58 * q^44 + 30 * q^45 + 32 * q^46 + 9 * q^47 - 43 * q^48 - q^50 + 39 * q^51 - 6 * q^52 - 16 * q^53 - 30 * q^54 + 17 * q^55 + 16 * q^57 + 44 * q^58 - 6 * q^59 - 8 * q^60 + 11 * q^62 + 119 * q^64 + 5 * q^65 - 3 * q^66 + 4 * q^67 + 70 * q^68 - 14 * q^69 + 48 * q^71 - 25 * q^72 - 38 * q^73 - q^74 + q^75 + 3 * q^76 - 40 * q^78 + 7 * q^79 + 39 * q^80 + 111 * q^81 - 6 * q^82 + 6 * q^83 + 7 * q^85 - 40 * q^86 + 47 * q^87 + 23 * q^88 + 2 * q^89 + 12 * q^90 + 7 * q^92 + 26 * q^94 - 6 * q^95 + 29 * q^96 - 79 * q^97 + 90 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more