LMFDB - Embedded newform 7497.2.a.ce.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7497,2,Mod(1,7497)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7497, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("7497.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7497 = 3^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7497.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$59.8638463954$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.17314349056.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 12x^{6} + 40x^{4} - 32x^{2} + 4 $ x^8 - 12x^6 + 40x^4 - 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.966796$ of defining polynomial
Character	$\chi$	$=$	7497.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.966796 q^{2} -1.06531 q^{4} +4.16083 q^{5} +2.96352 q^{8} +O(q^{10})$	$q-0.966796 q^{2} -1.06531 q^{4} +4.16083 q^{5} +2.96352 q^{8} -4.02268 q^{10} +4.23619 q^{11} +6.09782 q^{13} -0.734511 q^{16} +1.00000 q^{17} -2.14710 q^{19} -4.43256 q^{20} -4.09553 q^{22} -1.31540 q^{23} +12.3125 q^{25} -5.89535 q^{26} -6.40369 q^{29} +4.29287 q^{31} -5.21693 q^{32} -0.966796 q^{34} +3.49163 q^{37} +2.07581 q^{38} +12.3307 q^{40} -5.84976 q^{41} -3.49301 q^{43} -4.51284 q^{44} +1.27173 q^{46} +13.0120 q^{47} -11.9037 q^{50} -6.49604 q^{52} +10.0421 q^{53} +17.6261 q^{55} +6.19106 q^{58} +8.11525 q^{59} +2.38101 q^{61} -4.15033 q^{62} +6.51273 q^{64} +25.3720 q^{65} +14.7254 q^{67} -1.06531 q^{68} -10.3117 q^{71} -13.8373 q^{73} -3.37569 q^{74} +2.28732 q^{76} +17.3428 q^{79} -3.05618 q^{80} +5.65552 q^{82} -3.05618 q^{83} +4.16083 q^{85} +3.37702 q^{86} +12.5540 q^{88} -7.27608 q^{89} +1.40131 q^{92} -12.5799 q^{94} -8.93374 q^{95} -14.3994 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} + 12 q^{5}+O(q^{10}) $ 8 * q + 8 * q^4 + 12 * q^5	$ 8 q + 8 q^{4} + 12 q^{5} + 16 q^{16} + 8 q^{17} - 8 q^{20} - 28 q^{22} + 8 q^{25} - 4 q^{26} + 12 q^{37} + 60 q^{38} + 28 q^{41} - 24 q^{43} + 4 q^{46} + 20 q^{47} + 40 q^{58} + 12 q^{59} + 48 q^{62} + 48 q^{67} + 8 q^{68} + 60 q^{79} + 40 q^{80} + 40 q^{83} + 12 q^{85} - 8 q^{88} + 16 q^{89}+O(q^{100}) $ 8 * q + 8 * q^4 + 12 * q^5 + 16 * q^16 + 8 * q^17 - 8 * q^20 - 28 * q^22 + 8 * q^25 - 4 * q^26 + 12 * q^37 + 60 * q^38 + 28 * q^41 - 24 * q^43 + 4 * q^46 + 20 * q^47 + 40 * q^58 + 12 * q^59 + 48 * q^62 + 48 * q^67 + 8 * q^68 + 60 * q^79 + 40 * q^80 + 40 * q^83 + 12 * q^85 - 8 * q^88 + 16 * q^89

Display 100 coefficients 10 coefficients

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 $

$2$	−0.966796	−0.683628	−0.341814	−	0.939768i	$-0.611041\pi$
$2$	−0.966796	−0.683628	−0.341814	+	0.939768i	$0.611041\pi$
$3$	0	0
$3$	0	0
$4$	−1.06531	−0.532653
$5$	4.16083	1.86078	0.930391	−	0.366569i	$-0.119468\pi$
$5$	4.16083	1.86078	0.930391	+	0.366569i	$0.119468\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.96352	1.04776
$9$	0	0
$10$	−4.02268	−1.27208
$11$	4.23619	1.27726	0.638629	−	0.769514i	$-0.279502\pi$
$11$	4.23619	1.27726	0.638629	+	0.769514i	$0.279502\pi$
$12$	0	0
$13$	6.09782	1.69123	0.845615	−	0.533793i	$-0.179234\pi$
$13$	6.09782	1.69123	0.845615	+	0.533793i	$0.179234\pi$
$14$	0	0
$15$	0	0
$16$	−0.734511	−0.183628
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−2.14710	−0.492579	−0.246290	−	0.969196i	$-0.579211\pi$
$19$	−2.14710	−0.492579	−0.246290	+	0.969196i	$0.579211\pi$
$20$	−4.43256	−0.991151
$21$	0	0
$22$	−4.09553	−0.873170
$23$	−1.31540	−0.274281	−0.137140	−	0.990552i	$-0.543791\pi$
$23$	−1.31540	−0.274281	−0.137140	+	0.990552i	$0.543791\pi$
$24$	0	0
$25$	12.3125	2.46251
$26$	−5.89535	−1.15617
$27$	0	0
$28$	0	0
$29$	−6.40369	−1.18913	−0.594567	−	0.804046i	$-0.702677\pi$
$29$	−6.40369	−1.18913	−0.594567	+	0.804046i	$0.702677\pi$
$30$	0	0
$31$	4.29287	0.771023	0.385511	−	0.922703i	$-0.374025\pi$
$31$	4.29287	0.771023	0.385511	+	0.922703i	$0.374025\pi$
$32$	−5.21693	−0.922231
$33$	0	0
$34$	−0.966796	−0.165804
$35$	0	0
$36$	0	0
$37$	3.49163	0.574020	0.287010	−	0.957928i	$-0.407339\pi$
$37$	3.49163	0.574020	0.287010	+	0.957928i	$0.407339\pi$
$38$	2.07581	0.336741
$39$	0	0
$40$	12.3307	1.94966
$41$	−5.84976	−0.913579	−0.456789	−	0.889575i	$-0.651001\pi$
$41$	−5.84976	−0.913579	−0.456789	+	0.889575i	$0.651001\pi$
$42$	0	0
$43$	−3.49301	−0.532679	−0.266339	−	0.963879i	$-0.585814\pi$
$43$	−3.49301	−0.532679	−0.266339	+	0.963879i	$0.585814\pi$
$44$	−4.51284	−0.680336
$45$	0	0
$46$	1.27173	0.187506
$47$	13.0120	1.89799	0.948996	−	0.315290i	$-0.102102\pi$
$47$	13.0120	1.89799	0.948996	+	0.315290i	$0.102102\pi$
$48$	0	0
$49$	0	0
$50$	−11.9037	−1.68344
$51$	0	0
$52$	−6.49604	−0.900839
$53$	10.0421	1.37939	0.689693	−	0.724102i	$-0.257745\pi$
$53$	10.0421	1.37939	0.689693	+	0.724102i	$0.257745\pi$
$54$	0	0
$55$	17.6261	2.37670
$56$	0	0
$57$	0	0
$58$	6.19106	0.812926
$59$	8.11525	1.05652	0.528258	−	0.849084i	$-0.322846\pi$
$59$	8.11525	1.05652	0.528258	+	0.849084i	$0.322846\pi$
$60$	0	0
$61$	2.38101	0.304857	0.152429	−	0.988315i	$-0.451291\pi$
$61$	2.38101	0.304857	0.152429	+	0.988315i	$0.451291\pi$
$62$	−4.15033	−0.527093
$63$	0	0
$64$	6.51273	0.814091
$65$	25.3720	3.14701
$66$	0	0
$67$	14.7254	1.79899	0.899496	−	0.436929i	$-0.143934\pi$
$67$	14.7254	1.79899	0.899496	+	0.436929i	$0.143934\pi$
$68$	−1.06531	−0.129187
$69$	0	0
$70$	0	0
$71$	−10.3117	−1.22377	−0.611885	−	0.790947i	$-0.709588\pi$
$71$	−10.3117	−1.22377	−0.611885	+	0.790947i	$0.709588\pi$
$72$	0	0
$73$	−13.8373	−1.61953	−0.809767	−	0.586751i	$-0.800407\pi$
$73$	−13.8373	−1.61953	−0.809767	+	0.586751i	$0.800407\pi$
$74$	−3.37569	−0.392416
$75$	0	0
$76$	2.28732	0.262374
$77$	0	0
$78$	0	0
$79$	17.3428	1.95121	0.975607	−	0.219525i	$-0.0704509\pi$
$79$	17.3428	1.95121	0.975607	+	0.219525i	$0.0704509\pi$
$80$	−3.05618	−0.341691
$81$	0	0
$82$	5.65552	0.624548
$83$	−3.05618	−0.335459	−0.167730	−	0.985833i	$-0.553644\pi$
$83$	−3.05618	−0.335459	−0.167730	+	0.985833i	$0.553644\pi$
$84$	0	0
$85$	4.16083	0.451306
$86$	3.37702	0.364154
$87$	0	0
$88$	12.5540	1.33827
$89$	−7.27608	−0.771263	−0.385632	−	0.922653i	$-0.626016\pi$
$89$	−7.27608	−0.771263	−0.385632	+	0.922653i	$0.626016\pi$
$90$	0	0
$91$	0	0
$92$	1.40131	0.146096
$93$	0	0
$94$	−12.5799	−1.29752
$95$	−8.93374	−0.916582
$96$	0	0
$97$	−14.3994	−1.46204	−0.731020	−	0.682356i	$-0.760955\pi$
$97$	−14.3994	−1.46204	−0.731020	+	0.682356i	$0.760955\pi$
$98$	0	0
$99$	0	0
$100$	−13.1166	−1.31166
$101$	−6.91020	−0.687591	−0.343796	−	0.939045i	$-0.611713\pi$
$101$	−6.91020	−0.687591	−0.343796	+	0.939045i	$0.611713\pi$
$102$	0	0
$103$	−12.5442	−1.23602	−0.618011	−	0.786170i	$-0.712061\pi$
$103$	−12.5442	−1.23602	−0.618011	+	0.786170i	$0.712061\pi$
$104$	18.0710	1.77201
$105$	0	0
$106$	−9.70864	−0.942987
$107$	6.07547	0.587338	0.293669	−	0.955907i	$-0.405124\pi$
$107$	6.07547	0.587338	0.293669	+	0.955907i	$0.405124\pi$
$108$	0	0
$109$	10.1973	0.976724	0.488362	−	0.872641i	$-0.337595\pi$
$109$	10.1973	0.976724	0.488362	+	0.872641i	$0.337595\pi$
$110$	−17.0408	−1.62478
$111$	0	0
$112$	0	0
$113$	−4.40592	−0.414474	−0.207237	−	0.978291i	$-0.566447\pi$
$113$	−4.40592	−0.414474	−0.207237	+	0.978291i	$0.566447\pi$
$114$	0	0
$115$	−5.47318	−0.510377
$116$	6.82189	0.633396
$117$	0	0
$118$	−7.84579	−0.722263
$119$	0	0
$120$	0	0
$121$	6.94529	0.631390
$122$	−2.30195	−0.208409
$123$	0	0
$124$	−4.57322	−0.410688
$125$	30.4263	2.72141
$126$	0	0
$127$	−15.8982	−1.41074	−0.705368	−	0.708841i	$-0.749218\pi$
$127$	−15.8982	−1.41074	−0.705368	+	0.708841i	$0.749218\pi$
$128$	4.13738	0.365696
$129$	0	0
$130$	−24.5296	−2.15138
$131$	9.96978	0.871063	0.435532	−	0.900173i	$-0.356560\pi$
$131$	9.96978	0.871063	0.435532	+	0.900173i	$0.356560\pi$
$132$	0	0
$133$	0	0
$134$	−14.2364	−1.22984
$135$	0	0
$136$	2.96352	0.254120
$137$	2.92827	0.250179	0.125090	−	0.992145i	$-0.460078\pi$
$137$	2.92827	0.250179	0.125090	+	0.992145i	$0.460078\pi$
$138$	0	0
$139$	−10.1140	−0.857862	−0.428931	−	0.903337i	$-0.641110\pi$
$139$	−10.1140	−0.857862	−0.428931	+	0.903337i	$0.641110\pi$
$140$	0	0
$141$	0	0
$142$	9.96927	0.836603
$143$	25.8315	2.16014
$144$	0	0
$145$	−26.6447	−2.21272
$146$	13.3778	1.10716
$147$	0	0
$148$	−3.71965	−0.305754
$149$	−13.7965	−1.13025	−0.565127	−	0.825004i	$-0.691173\pi$
$149$	−13.7965	−1.13025	−0.565127	+	0.825004i	$0.691173\pi$
$150$	0	0
$151$	−4.84352	−0.394160	−0.197080	−	0.980387i	$-0.563146\pi$
$151$	−4.84352	−0.394160	−0.197080	+	0.980387i	$0.563146\pi$
$152$	−6.36299	−0.516107
$153$	0	0
$154$	0	0
$155$	17.8619	1.43470
$156$	0	0
$157$	−5.44334	−0.434426	−0.217213	−	0.976124i	$-0.569697\pi$
$157$	−5.44334	−0.434426	−0.217213	+	0.976124i	$0.569697\pi$
$158$	−16.7669	−1.33390
$159$	0	0
$160$	−21.7068	−1.71607
$161$	0	0
$162$	0	0
$163$	−16.1440	−1.26449	−0.632246	−	0.774767i	$-0.717867\pi$
$163$	−16.1440	−1.26449	−0.632246	+	0.774767i	$0.717867\pi$
$164$	6.23178	0.486620
$165$	0	0
$166$	2.95470	0.229329
$167$	6.99376	0.541194	0.270597	−	0.962693i	$-0.412779\pi$
$167$	6.99376	0.541194	0.270597	+	0.962693i	$0.412779\pi$
$168$	0	0
$169$	24.1834	1.86026
$170$	−4.02268	−0.308525
$171$	0	0
$172$	3.72112	0.283733
$173$	−8.83490	−0.671705	−0.335853	−	0.941915i	$-0.609024\pi$
$173$	−8.83490	−0.671705	−0.335853	+	0.941915i	$0.609024\pi$
$174$	0	0
$175$	0	0
$176$	−3.11153	−0.234540
$177$	0	0
$178$	7.03449	0.527257
$179$	7.18515	0.537044	0.268522	−	0.963274i	$-0.413465\pi$
$179$	7.18515	0.537044	0.268522	+	0.963274i	$0.413465\pi$
$180$	0	0
$181$	19.4366	1.44471	0.722354	−	0.691523i	$-0.243060\pi$
$181$	19.4366	1.44471	0.722354	+	0.691523i	$0.243060\pi$
$182$	0	0
$183$	0	0
$184$	−3.89823	−0.287382
$185$	14.5281	1.06813
$186$	0	0
$187$	4.23619	0.309781
$188$	−13.8617	−1.01097
$189$	0	0
$190$	8.63710	0.626601
$191$	11.6552	0.843344	0.421672	−	0.906748i	$-0.361443\pi$
$191$	11.6552	0.843344	0.421672	+	0.906748i	$0.361443\pi$
$192$	0	0
$193$	−19.3112	−1.39005	−0.695024	−	0.718986i	$-0.744606\pi$
$193$	−19.3112	−1.39005	−0.695024	+	0.718986i	$0.744606\pi$
$194$	13.9213	0.999491
$195$	0	0
$196$	0	0
$197$	−7.71909	−0.549962	−0.274981	−	0.961450i	$-0.588672\pi$
$197$	−7.71909	−0.549962	−0.274981	+	0.961450i	$0.588672\pi$
$198$	0	0
$199$	1.96689	0.139429	0.0697147	−	0.997567i	$-0.477791\pi$
$199$	1.96689	0.139429	0.0697147	+	0.997567i	$0.477791\pi$
$200$	36.4885	2.58013
$201$	0	0
$202$	6.68076	0.470056
$203$	0	0
$204$	0	0
$205$	−24.3399	−1.69997
$206$	12.1277	0.844979
$207$	0	0
$208$	−4.47892	−0.310557
$209$	−9.09553	−0.629151
$210$	0	0
$211$	−2.74075	−0.188681	−0.0943405	−	0.995540i	$-0.530074\pi$
$211$	−2.74075	−0.188681	−0.0943405	+	0.995540i	$0.530074\pi$
$212$	−10.6979	−0.734734
$213$	0	0
$214$	−5.87374	−0.401521
$215$	−14.5338	−0.991198
$216$	0	0
$217$	0	0
$218$	−9.85870	−0.667716
$219$	0	0
$220$	−18.7772	−1.26596
$221$	6.09782	0.410184
$222$	0	0
$223$	−18.8021	−1.25908	−0.629540	−	0.776968i	$-0.716756\pi$
$223$	−18.8021	−1.25908	−0.629540	+	0.776968i	$0.716756\pi$
$224$	0	0
$225$	0	0
$226$	4.25962	0.283346
$227$	−0.0408160	−0.00270905	−0.00135453	−	0.999999i	$-0.500431\pi$
$227$	−0.0408160	−0.00270905	−0.00135453	+	0.999999i	$0.500431\pi$
$228$	0	0
$229$	−12.3436	−0.815691	−0.407845	−	0.913051i	$-0.633720\pi$
$229$	−12.3436	−0.815691	−0.407845	+	0.913051i	$0.633720\pi$
$230$	5.29145	0.348908
$231$	0	0
$232$	−18.9775	−1.24593
$233$	−7.80950	−0.511617	−0.255809	−	0.966727i	$-0.582342\pi$
$233$	−7.80950	−0.511617	−0.255809	+	0.966727i	$0.582342\pi$
$234$	0	0
$235$	54.1407	3.53175
$236$	−8.64522	−0.562756
$237$	0	0
$238$	0	0
$239$	0.683695	0.0442245	0.0221123	−	0.999755i	$-0.492961\pi$
$239$	0.683695	0.0442245	0.0221123	+	0.999755i	$0.492961\pi$
$240$	0	0
$241$	14.6467	0.943474	0.471737	−	0.881739i	$-0.343627\pi$
$241$	14.6467	0.943474	0.471737	+	0.881739i	$0.343627\pi$
$242$	−6.71467	−0.431636
$243$	0	0
$244$	−2.53650	−0.162383
$245$	0	0
$246$	0	0
$247$	−13.0926	−0.833065
$248$	12.7220	0.807850
$249$	0	0
$250$	−29.4160	−1.86043
$251$	−1.20303	−0.0759345	−0.0379672	−	0.999279i	$-0.512088\pi$
$251$	−1.20303	−0.0759345	−0.0379672	+	0.999279i	$0.512088\pi$
$252$	0	0
$253$	−5.57230	−0.350327
$254$	15.3703	0.964419
$255$	0	0
$256$	−17.0255	−1.06409
$257$	27.2003	1.69671	0.848353	−	0.529431i	$-0.177594\pi$
$257$	27.2003	1.69671	0.848353	+	0.529431i	$0.177594\pi$
$258$	0	0
$259$	0	0
$260$	−27.0290	−1.67626
$261$	0	0
$262$	−9.63874	−0.595483
$263$	−2.01907	−0.124501	−0.0622507	−	0.998061i	$-0.519828\pi$
$263$	−2.01907	−0.124501	−0.0622507	+	0.998061i	$0.519828\pi$
$264$	0	0
$265$	41.7834	2.56674
$266$	0	0
$267$	0	0
$268$	−15.6870	−0.958239
$269$	−24.4651	−1.49166	−0.745832	−	0.666134i	$-0.767948\pi$
$269$	−24.4651	−1.49166	−0.745832	+	0.666134i	$0.767948\pi$
$270$	0	0
$271$	1.53537	0.0932669	0.0466335	−	0.998912i	$-0.485151\pi$
$271$	1.53537	0.0932669	0.0466335	+	0.998912i	$0.485151\pi$
$272$	−0.734511	−0.0445363
$273$	0	0
$274$	−2.83104	−0.171030
$275$	52.1582	3.14526
$276$	0	0
$277$	−11.4987	−0.690892	−0.345446	−	0.938439i	$-0.612272\pi$
$277$	−11.4987	−0.690892	−0.345446	+	0.938439i	$0.612272\pi$
$278$	9.77821	0.586458
$279$	0	0
$280$	0	0
$281$	−12.2582	−0.731260	−0.365630	−	0.930760i	$-0.619147\pi$
$281$	−12.2582	−0.731260	−0.365630	+	0.930760i	$0.619147\pi$
$282$	0	0
$283$	−8.22258	−0.488781	−0.244391	−	0.969677i	$-0.578588\pi$
$283$	−8.22258	−0.488781	−0.244391	+	0.969677i	$0.578588\pi$
$284$	10.9851	0.651844
$285$	0	0
$286$	−24.9738	−1.47673
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	25.7600	1.51268
$291$	0	0
$292$	14.7410	0.862650
$293$	−30.4349	−1.77803	−0.889013	−	0.457882i	$-0.848608\pi$
$293$	−30.4349	−1.77803	−0.889013	+	0.457882i	$0.848608\pi$
$294$	0	0
$295$	33.7662	1.96594
$296$	10.3475	0.601438
$297$	0	0
$298$	13.3384	0.772673
$299$	−8.02110	−0.463872
$300$	0	0
$301$	0	0
$302$	4.68270	0.269459
$303$	0	0
$304$	1.57707	0.0904512
$305$	9.90699	0.567272
$306$	0	0
$307$	−12.0457	−0.687484	−0.343742	−	0.939064i	$-0.611695\pi$
$307$	−12.0457	−0.687484	−0.343742	+	0.939064i	$0.611695\pi$
$308$	0	0
$309$	0	0
$310$	−17.2688	−0.980804
$311$	15.5267	0.880439	0.440220	−	0.897890i	$-0.354901\pi$
$311$	15.5267	0.880439	0.440220	+	0.897890i	$0.354901\pi$
$312$	0	0
$313$	12.6390	0.714397	0.357199	−	0.934028i	$-0.383732\pi$
$313$	12.6390	0.714397	0.357199	+	0.934028i	$0.383732\pi$
$314$	5.26260	0.296986
$315$	0	0
$316$	−18.4754	−1.03932
$317$	8.25783	0.463806	0.231903	−	0.972739i	$-0.425505\pi$
$317$	8.25783	0.463806	0.231903	+	0.972739i	$0.425505\pi$
$318$	0	0
$319$	−27.1272	−1.51883
$320$	27.0984	1.51485
$321$	0	0
$322$	0	0
$323$	−2.14710	−0.119468
$324$	0	0
$325$	75.0797	4.16467
$326$	15.6079	0.864442
$327$	0	0
$328$	−17.3359	−0.957215
$329$	0	0
$330$	0	0
$331$	6.22563	0.342192	0.171096	−	0.985254i	$-0.445269\pi$
$331$	6.22563	0.342192	0.171096	+	0.985254i	$0.445269\pi$
$332$	3.25577	0.178683
$333$	0	0
$334$	−6.76154	−0.369975
$335$	61.2699	3.34753
$336$	0	0
$337$	−10.2243	−0.556951	−0.278475	−	0.960443i	$-0.589829\pi$
$337$	−10.2243	−0.556951	−0.278475	+	0.960443i	$0.589829\pi$
$338$	−23.3804	−1.27173
$339$	0	0
$340$	−4.43256	−0.240389
$341$	18.1854	0.984795
$342$	0	0
$343$	0	0
$344$	−10.3516	−0.558122
$345$	0	0
$346$	8.54154	0.459196
$347$	20.4208	1.09624	0.548122	−	0.836398i	$-0.315343\pi$
$347$	20.4208	1.09624	0.548122	+	0.836398i	$0.315343\pi$
$348$	0	0
$349$	6.37684	0.341344	0.170672	−	0.985328i	$-0.445406\pi$
$349$	6.37684	0.341344	0.170672	+	0.985328i	$0.445406\pi$
$350$	0	0
$351$	0	0
$352$	−22.0999	−1.17793
$353$	8.64622	0.460192	0.230096	−	0.973168i	$-0.426096\pi$
$353$	8.64622	0.460192	0.230096	+	0.973168i	$0.426096\pi$
$354$	0	0
$355$	−42.9051	−2.27717
$356$	7.75125	0.410816
$357$	0	0
$358$	−6.94658	−0.367138
$359$	−32.5047	−1.71553	−0.857766	−	0.514040i	$-0.828148\pi$
$359$	−32.5047	−1.71553	−0.857766	+	0.514040i	$0.828148\pi$
$360$	0	0
$361$	−14.3900	−0.757366
$362$	−18.7912	−0.987643
$363$	0	0
$364$	0	0
$365$	−57.5747	−3.01360
$366$	0	0
$367$	32.1726	1.67940	0.839698	−	0.543054i	$-0.182732\pi$
$367$	32.1726	1.67940	0.839698	+	0.543054i	$0.182732\pi$
$368$	0.966179	0.0503656
$369$	0	0
$370$	−14.0457	−0.730201
$371$	0	0
$372$	0	0
$373$	−22.8982	−1.18562	−0.592812	−	0.805341i	$-0.701982\pi$
$373$	−22.8982	−1.18562	−0.592812	+	0.805341i	$0.701982\pi$
$374$	−4.09553	−0.211775
$375$	0	0
$376$	38.5613	1.98865
$377$	−39.0485	−2.01110
$378$	0	0
$379$	36.1335	1.85606	0.928028	−	0.372511i	$-0.121503\pi$
$379$	36.1335	1.85606	0.928028	+	0.372511i	$0.121503\pi$
$380$	9.51716	0.488220
$381$	0	0
$382$	−11.2682	−0.576534
$383$	8.81816	0.450587	0.225293	−	0.974291i	$-0.427666\pi$
$383$	8.81816	0.450587	0.225293	+	0.974291i	$0.427666\pi$
$384$	0	0
$385$	0	0
$386$	18.6700	0.950276
$387$	0	0
$388$	15.3398	0.778760
$389$	6.06707	0.307613	0.153806	−	0.988101i	$-0.450847\pi$
$389$	6.06707	0.307613	0.153806	+	0.988101i	$0.450847\pi$
$390$	0	0
$391$	−1.31540	−0.0665228
$392$	0	0
$393$	0	0
$394$	7.46278	0.375970
$395$	72.1604	3.63078
$396$	0	0
$397$	−1.97305	−0.0990247	−0.0495123	−	0.998774i	$-0.515767\pi$
$397$	−1.97305	−0.0990247	−0.0495123	+	0.998774i	$0.515767\pi$
$398$	−1.90158	−0.0953178
$399$	0	0
$400$	−9.04370	−0.452185
$401$	−13.3027	−0.664303	−0.332151	−	0.943226i	$-0.607774\pi$
$401$	−13.3027	−0.664303	−0.332151	+	0.943226i	$0.607774\pi$
$402$	0	0
$403$	26.1772	1.30398
$404$	7.36148	0.366247
$405$	0	0
$406$	0	0
$407$	14.7912	0.733172
$408$	0	0
$409$	−23.5490	−1.16443	−0.582213	−	0.813037i	$-0.697813\pi$
$409$	−23.5490	−1.16443	−0.582213	+	0.813037i	$0.697813\pi$
$410$	23.5317	1.16215
$411$	0	0
$412$	13.3635	0.658370
$413$	0	0
$414$	0	0
$415$	−12.7163	−0.624217
$416$	−31.8119	−1.55971
$417$	0	0
$418$	8.79352	0.430105
$419$	−6.77524	−0.330992	−0.165496	−	0.986210i	$-0.552922\pi$
$419$	−6.77524	−0.330992	−0.165496	+	0.986210i	$0.552922\pi$
$420$	0	0
$421$	11.5738	0.564071	0.282035	−	0.959404i	$-0.408990\pi$
$421$	11.5738	0.564071	0.282035	+	0.959404i	$0.408990\pi$
$422$	2.64975	0.128988
$423$	0	0
$424$	29.7600	1.44527
$425$	12.3125	0.597246
$426$	0	0
$427$	0	0
$428$	−6.47224	−0.312847
$429$	0	0
$430$	14.0512	0.677611
$431$	−15.1699	−0.730710	−0.365355	−	0.930868i	$-0.619052\pi$
$431$	−15.1699	−0.730710	−0.365355	+	0.930868i	$0.619052\pi$
$432$	0	0
$433$	10.0009	0.480611	0.240306	−	0.970697i	$-0.422752\pi$
$433$	10.0009	0.480611	0.240306	+	0.970697i	$0.422752\pi$
$434$	0	0
$435$	0	0
$436$	−10.8632	−0.520255
$437$	2.82431	0.135105
$438$	0	0
$439$	−27.4974	−1.31238	−0.656189	−	0.754596i	$-0.727833\pi$
$439$	−27.4974	−1.31238	−0.656189	+	0.754596i	$0.727833\pi$
$440$	52.2353	2.49022
$441$	0	0
$442$	−5.89535	−0.280413
$443$	−5.74788	−0.273090	−0.136545	−	0.990634i	$-0.543600\pi$
$443$	−5.74788	−0.273090	−0.136545	+	0.990634i	$0.543600\pi$
$444$	0	0
$445$	−30.2746	−1.43515
$446$	18.1778	0.860742
$447$	0	0
$448$	0	0
$449$	14.7192	0.694644	0.347322	−	0.937746i	$-0.387091\pi$
$449$	14.7192	0.694644	0.347322	+	0.937746i	$0.387091\pi$
$450$	0	0
$451$	−24.7807	−1.16688
$452$	4.69365	0.220771
$453$	0	0
$454$	0.0394607	0.00185198
$455$	0	0
$456$	0	0
$457$	−22.5680	−1.05569	−0.527844	−	0.849341i	$-0.676999\pi$
$457$	−22.5680	−1.05569	−0.527844	+	0.849341i	$0.676999\pi$
$458$	11.9338	0.557629
$459$	0	0
$460$	5.83061	0.271854
$461$	19.0688	0.888122	0.444061	−	0.895997i	$-0.353537\pi$
$461$	19.0688	0.888122	0.444061	+	0.895997i	$0.353537\pi$
$462$	0	0
$463$	−26.2214	−1.21861	−0.609305	−	0.792936i	$-0.708552\pi$
$463$	−26.2214	−1.21861	−0.609305	+	0.792936i	$0.708552\pi$
$464$	4.70358	0.218358
$465$	0	0
$466$	7.55019	0.349756
$467$	38.6567	1.78882	0.894409	−	0.447250i	$-0.147597\pi$
$467$	38.6567	1.78882	0.894409	+	0.447250i	$0.147597\pi$
$468$	0	0
$469$	0	0
$470$	−52.3430	−2.41440
$471$	0	0
$472$	24.0497	1.10698
$473$	−14.7970	−0.680368
$474$	0	0
$475$	−26.4363	−1.21298
$476$	0	0
$477$	0	0
$478$	−0.660993	−0.0302331
$479$	−18.8023	−0.859098	−0.429549	−	0.903043i	$-0.641328\pi$
$479$	−18.8023	−0.859098	−0.429549	+	0.903043i	$0.641328\pi$
$480$	0	0
$481$	21.2913	0.970801
$482$	−14.1603	−0.644985
$483$	0	0
$484$	−7.39886	−0.336312
$485$	−59.9136	−2.72054
$486$	0	0
$487$	16.0449	0.727066	0.363533	−	0.931581i	$-0.381570\pi$
$487$	16.0449	0.727066	0.363533	+	0.931581i	$0.381570\pi$
$488$	7.05618	0.319418
$489$	0	0
$490$	0	0
$491$	19.4663	0.878501	0.439250	−	0.898365i	$-0.355244\pi$
$491$	19.4663	0.878501	0.439250	+	0.898365i	$0.355244\pi$
$492$	0	0
$493$	−6.40369	−0.288408
$494$	12.6579	0.569506
$495$	0	0
$496$	−3.15316	−0.141581
$497$	0	0
$498$	0	0
$499$	−24.7024	−1.10583	−0.552915	−	0.833238i	$-0.686485\pi$
$499$	−24.7024	−1.10583	−0.552915	+	0.833238i	$0.686485\pi$
$500$	−32.4133	−1.44957
$501$	0	0
$502$	1.16308	0.0519109
$503$	33.7000	1.50261	0.751305	−	0.659955i	$-0.229425\pi$
$503$	33.7000	1.50261	0.751305	+	0.659955i	$0.229425\pi$
$504$	0	0
$505$	−28.7522	−1.27946
$506$	5.38728	0.239494
$507$	0	0
$508$	16.9364	0.751433
$509$	−22.2035	−0.984154	−0.492077	−	0.870552i	$-0.663762\pi$
$509$	−22.2035	−0.984154	−0.492077	+	0.870552i	$0.663762\pi$
$510$	0	0
$511$	0	0
$512$	8.18538	0.361746
$513$	0	0
$514$	−26.2971	−1.15992
$515$	−52.1945	−2.29997
$516$	0	0
$517$	55.1212	2.42423
$518$	0	0
$519$	0	0
$520$	75.1906	3.29733
$521$	1.62462	0.0711760	0.0355880	−	0.999367i	$-0.488670\pi$
$521$	1.62462	0.0711760	0.0355880	+	0.999367i	$0.488670\pi$
$522$	0	0
$523$	6.13154	0.268114	0.134057	−	0.990974i	$-0.457200\pi$
$523$	6.13154	0.268114	0.134057	+	0.990974i	$0.457200\pi$
$524$	−10.6209	−0.463975
$525$	0	0
$526$	1.95203	0.0851126
$527$	4.29287	0.187000
$528$	0	0
$529$	−21.2697	−0.924770
$530$	−40.3961	−1.75469
$531$	0	0
$532$	0	0
$533$	−35.6708	−1.54507
$534$	0	0
$535$	25.2790	1.09291
$536$	43.6390	1.88492
$537$	0	0
$538$	23.6528	1.01974
$539$	0	0
$540$	0	0
$541$	24.8055	1.06647	0.533237	−	0.845966i	$-0.320975\pi$
$541$	24.8055	1.06647	0.533237	+	0.845966i	$0.320975\pi$
$542$	−1.48439	−0.0637599
$543$	0	0
$544$	−5.21693	−0.223674
$545$	42.4293	1.81747
$546$	0	0
$547$	16.3404	0.698664	0.349332	−	0.936999i	$-0.386408\pi$
$547$	16.3404	0.698664	0.349332	+	0.936999i	$0.386408\pi$
$548$	−3.11951	−0.133259
$549$	0	0
$550$	−50.4264	−2.15019
$551$	13.7494	0.585743
$552$	0	0
$553$	0	0
$554$	11.1169	0.472313
$555$	0	0
$556$	10.7746	0.456943
$557$	−37.3840	−1.58401	−0.792005	−	0.610515i	$-0.790963\pi$
$557$	−37.3840	−1.58401	−0.792005	+	0.610515i	$0.790963\pi$
$558$	0	0
$559$	−21.2997	−0.900882
$560$	0	0
$561$	0	0
$562$	11.8511	0.499910
$563$	19.4374	0.819190	0.409595	−	0.912268i	$-0.365670\pi$
$563$	19.4374	0.819190	0.409595	+	0.912268i	$0.365670\pi$
$564$	0	0
$565$	−18.3323	−0.771246
$566$	7.94955	0.334145
$567$	0	0
$568$	−30.5589	−1.28222
$569$	8.49819	0.356263	0.178131	−	0.984007i	$-0.442995\pi$
$569$	8.49819	0.356263	0.178131	+	0.984007i	$0.442995\pi$
$570$	0	0
$571$	15.0640	0.630410	0.315205	−	0.949024i	$-0.397927\pi$
$571$	15.0640	0.630410	0.315205	+	0.949024i	$0.397927\pi$
$572$	−27.5185	−1.15060
$573$	0	0
$574$	0	0
$575$	−16.1960	−0.675419
$576$	0	0
$577$	−9.44776	−0.393315	−0.196658	−	0.980472i	$-0.563009\pi$
$577$	−9.44776	−0.393315	−0.196658	+	0.980472i	$0.563009\pi$
$578$	−0.966796	−0.0402134
$579$	0	0
$580$	28.3847	1.17861
$581$	0	0
$582$	0	0
$583$	42.5402	1.76183
$584$	−41.0072	−1.69689
$585$	0	0
$586$	29.4243	1.21551
$587$	13.8953	0.573522	0.286761	−	0.958002i	$-0.407421\pi$
$587$	13.8953	0.573522	0.286761	+	0.958002i	$0.407421\pi$
$588$	0	0
$589$	−9.21724	−0.379790
$590$	−32.6450	−1.34397
$591$	0	0
$592$	−2.56464	−0.105406
$593$	3.81371	0.156610	0.0783052	−	0.996929i	$-0.475049\pi$
$593$	3.81371	0.156610	0.0783052	+	0.996929i	$0.475049\pi$
$594$	0	0
$595$	0	0
$596$	14.6975	0.602033
$597$	0	0
$598$	7.75476	0.317116
$599$	14.8503	0.606765	0.303382	−	0.952869i	$-0.401884\pi$
$599$	14.8503	0.606765	0.303382	+	0.952869i	$0.401884\pi$
$600$	0	0
$601$	−1.40192	−0.0571857	−0.0285928	−	0.999591i	$-0.509103\pi$
$601$	−1.40192	−0.0571857	−0.0285928	+	0.999591i	$0.509103\pi$
$602$	0	0
$603$	0	0
$604$	5.15983	0.209951
$605$	28.8982	1.17488
$606$	0	0
$607$	12.9797	0.526829	0.263414	−	0.964683i	$-0.415151\pi$
$607$	12.9797	0.526829	0.263414	+	0.964683i	$0.415151\pi$
$608$	11.2013	0.454272
$609$	0	0
$610$	−9.57803	−0.387803
$611$	79.3446	3.20994
$612$	0	0
$613$	−17.7892	−0.718498	−0.359249	−	0.933242i	$-0.616967\pi$
$613$	−17.7892	−0.718498	−0.359249	+	0.933242i	$0.616967\pi$
$614$	11.6457	0.469983
$615$	0	0
$616$	0	0
$617$	−43.5337	−1.75260	−0.876300	−	0.481767i	$-0.839995\pi$
$617$	−43.5337	−1.75260	−0.876300	+	0.481767i	$0.839995\pi$
$618$	0	0
$619$	−3.25288	−0.130744	−0.0653722	−	0.997861i	$-0.520823\pi$
$619$	−3.25288	−0.130744	−0.0653722	+	0.997861i	$0.520823\pi$
$620$	−19.0284	−0.764200
$621$	0	0
$622$	−15.0112	−0.601893
$623$	0	0
$624$	0	0
$625$	65.0360	2.60144
$626$	−12.2193	−0.488382
$627$	0	0
$628$	5.79883	0.231398
$629$	3.49163	0.139220
$630$	0	0
$631$	17.9650	0.715176	0.357588	−	0.933880i	$-0.383599\pi$
$631$	17.9650	0.715176	0.357588	+	0.933880i	$0.383599\pi$
$632$	51.3957	2.04441
$633$	0	0
$634$	−7.98363	−0.317071
$635$	−66.1497	−2.62507
$636$	0	0
$637$	0	0
$638$	26.2265	1.03832
$639$	0	0
$640$	17.2149	0.680481
$641$	15.6016	0.616226	0.308113	−	0.951350i	$-0.400303\pi$
$641$	15.6016	0.616226	0.308113	+	0.951350i	$0.400303\pi$
$642$	0	0
$643$	27.9497	1.10223	0.551115	−	0.834429i	$-0.314203\pi$
$643$	27.9497	1.10223	0.551115	+	0.834429i	$0.314203\pi$
$644$	0	0
$645$	0	0
$646$	2.07581	0.0816716
$647$	27.6188	1.08581	0.542903	−	0.839796i	$-0.317325\pi$
$647$	27.6188	1.08581	0.542903	+	0.839796i	$0.317325\pi$
$648$	0	0
$649$	34.3777	1.34944
$650$	−72.5867	−2.84708
$651$	0	0
$652$	17.1983	0.673536
$653$	34.5010	1.35013	0.675064	−	0.737759i	$-0.264116\pi$
$653$	34.5010	1.35013	0.675064	+	0.737759i	$0.264116\pi$
$654$	0	0
$655$	41.4826	1.62086
$656$	4.29671	0.167759
$657$	0	0
$658$	0	0
$659$	−16.5985	−0.646586	−0.323293	−	0.946299i	$-0.604790\pi$
$659$	−16.5985	−0.646586	−0.323293	+	0.946299i	$0.604790\pi$
$660$	0	0
$661$	−34.8608	−1.35593	−0.677964	−	0.735095i	$-0.737138\pi$
$661$	−34.8608	−1.35593	−0.677964	+	0.735095i	$0.737138\pi$
$662$	−6.01892	−0.233932
$663$	0	0
$664$	−9.05707	−0.351482
$665$	0	0
$666$	0	0
$667$	8.42344	0.326157
$668$	−7.45050	−0.288268
$669$	0	0
$670$	−59.2355	−2.28847
$671$	10.0864	0.389381
$672$	0	0
$673$	−33.5910	−1.29484	−0.647418	−	0.762135i	$-0.724151\pi$
$673$	−33.5910	−1.29484	−0.647418	+	0.762135i	$0.724151\pi$
$674$	9.88477	0.380747
$675$	0	0
$676$	−25.7627	−0.990874
$677$	8.12227	0.312164	0.156082	−	0.987744i	$-0.450114\pi$
$677$	8.12227	0.312164	0.156082	+	0.987744i	$0.450114\pi$
$678$	0	0
$679$	0	0
$680$	12.3307	0.472862
$681$	0	0
$682$	−17.5816	−0.673234
$683$	6.45690	0.247067	0.123533	−	0.992340i	$-0.460577\pi$
$683$	6.45690	0.247067	0.123533	+	0.992340i	$0.460577\pi$
$684$	0	0
$685$	12.1841	0.465529
$686$	0	0
$687$	0	0
$688$	2.56565	0.0978146
$689$	61.2348	2.33286
$690$	0	0
$691$	39.2191	1.49196	0.745982	−	0.665967i	$-0.231981\pi$
$691$	39.2191	1.49196	0.745982	+	0.665967i	$0.231981\pi$
$692$	9.41187	0.357786
$693$	0	0
$694$	−19.7427	−0.749424
$695$	−42.0829	−1.59629
$696$	0	0
$697$	−5.84976	−0.221575
$698$	−6.16510	−0.233352
$699$	0	0
$700$	0	0
$701$	10.8369	0.409304	0.204652	−	0.978835i	$-0.434394\pi$
$701$	10.8369	0.409304	0.204652	+	0.978835i	$0.434394\pi$
$702$	0	0
$703$	−7.49688	−0.282750
$704$	27.5891	1.03980
$705$	0	0
$706$	−8.35913	−0.314600
$707$	0	0
$708$	0	0
$709$	−24.7057	−0.927841	−0.463920	−	0.885877i	$-0.653558\pi$
$709$	−24.7057	−0.927841	−0.463920	+	0.885877i	$0.653558\pi$
$710$	41.4805	1.55673
$711$	0	0
$712$	−21.5628	−0.808102
$713$	−5.64686	−0.211477
$714$	0	0
$715$	107.481	4.01955
$716$	−7.65439	−0.286058
$717$	0	0
$718$	31.4254	1.17279
$719$	1.39936	0.0521874	0.0260937	−	0.999660i	$-0.491693\pi$
$719$	1.39936	0.0521874	0.0260937	+	0.999660i	$0.491693\pi$
$720$	0	0
$721$	0	0
$722$	13.9121	0.517756
$723$	0	0
$724$	−20.7059	−0.769528
$725$	−78.8457	−2.92825
$726$	0	0
$727$	−28.5587	−1.05918	−0.529591	−	0.848253i	$-0.677655\pi$
$727$	−28.5587	−1.05918	−0.529591	+	0.848253i	$0.677655\pi$
$728$	0	0
$729$	0	0
$730$	55.6630	2.06018
$731$	−3.49301	−0.129194
$732$	0	0
$733$	13.3546	0.493265	0.246632	−	0.969109i	$-0.420676\pi$
$733$	13.3546	0.493265	0.246632	+	0.969109i	$0.420676\pi$
$734$	−31.1043	−1.14808
$735$	0	0
$736$	6.86237	0.252950
$737$	62.3795	2.29778
$738$	0	0
$739$	26.4650	0.973530	0.486765	−	0.873533i	$-0.338177\pi$
$739$	26.4650	0.973530	0.486765	+	0.873533i	$0.338177\pi$
$740$	−15.4769	−0.568941
$741$	0	0
$742$	0	0
$743$	27.7474	1.01795	0.508976	−	0.860781i	$-0.330024\pi$
$743$	27.7474	1.01795	0.508976	+	0.860781i	$0.330024\pi$
$744$	0	0
$745$	−57.4050	−2.10316
$746$	22.1379	0.810525
$747$	0	0
$748$	−4.51284	−0.165006
$749$	0	0
$750$	0	0
$751$	−9.00963	−0.328766	−0.164383	−	0.986397i	$-0.552563\pi$
$751$	−9.00963	−0.328766	−0.164383	+	0.986397i	$0.552563\pi$
$752$	−9.55744	−0.348524
$753$	0	0
$754$	37.7519	1.37484
$755$	−20.1531	−0.733446
$756$	0	0
$757$	39.9798	1.45309	0.726546	−	0.687118i	$-0.241125\pi$
$757$	39.9798	1.45309	0.726546	+	0.687118i	$0.241125\pi$
$758$	−34.9338	−1.26885
$759$	0	0
$760$	−26.4754	−0.960362
$761$	−9.51993	−0.345097	−0.172549	−	0.985001i	$-0.555200\pi$
$761$	−9.51993	−0.345097	−0.172549	+	0.985001i	$0.555200\pi$
$762$	0	0
$763$	0	0
$764$	−12.4164	−0.449210
$765$	0	0
$766$	−8.52536	−0.308034
$767$	49.4853	1.78681
$768$	0	0
$769$	−33.4219	−1.20522	−0.602612	−	0.798034i	$-0.705873\pi$
$769$	−33.4219	−1.20522	−0.602612	+	0.798034i	$0.705873\pi$
$770$	0	0
$771$	0	0
$772$	20.5723	0.740413
$773$	8.47617	0.304867	0.152433	−	0.988314i	$-0.451289\pi$
$773$	8.47617	0.304867	0.152433	+	0.988314i	$0.451289\pi$
$774$	0	0
$775$	52.8562	1.89865
$776$	−42.6731	−1.53187
$777$	0	0
$778$	−5.86562	−0.210293
$779$	12.5600	0.450010
$780$	0	0
$781$	−43.6821	−1.56307
$782$	1.27173	0.0454769
$783$	0	0
$784$	0	0
$785$	−22.6489	−0.808372
$786$	0	0
$787$	−55.6425	−1.98344	−0.991720	−	0.128418i	$-0.959010\pi$
$787$	−55.6425	−1.98344	−0.991720	+	0.128418i	$0.959010\pi$
$788$	8.22319	0.292939
$789$	0	0
$790$	−69.7643	−2.48210
$791$	0	0
$792$	0	0
$793$	14.5190	0.515584
$794$	1.90754	0.0676960
$795$	0	0
$796$	−2.09534	−0.0742675
$797$	41.6217	1.47432	0.737158	−	0.675720i	$-0.236167\pi$
$797$	41.6217	1.47432	0.737158	+	0.675720i	$0.236167\pi$
$798$	0	0
$799$	13.0120	0.460330
$800$	−64.2336	−2.27100
$801$	0	0
$802$	12.8609	0.454136
$803$	−58.6174	−2.06856
$804$	0	0
$805$	0	0
$806$	−25.3080	−0.891435
$807$	0	0
$808$	−20.4786	−0.720433
$809$	22.1639	0.779242	0.389621	−	0.920975i	$-0.372606\pi$
$809$	22.1639	0.779242	0.389621	+	0.920975i	$0.372606\pi$
$810$	0	0
$811$	23.4795	0.824475	0.412238	−	0.911076i	$-0.364747\pi$
$811$	23.4795	0.824475	0.412238	+	0.911076i	$0.364747\pi$
$812$	0	0
$813$	0	0
$814$	−14.3001	−0.501217
$815$	−67.1723	−2.35294
$816$	0	0
$817$	7.49984	0.262386
$818$	22.7671	0.796034
$819$	0	0
$820$	25.9294	0.905495
$821$	28.4586	0.993213	0.496607	−	0.867976i	$-0.334579\pi$
$821$	28.4586	0.993213	0.496607	+	0.867976i	$0.334579\pi$
$822$	0	0
$823$	19.5251	0.680603	0.340301	−	0.940316i	$-0.389471\pi$
$823$	19.5251	0.680603	0.340301	+	0.940316i	$0.389471\pi$
$824$	−37.1752	−1.29506
$825$	0	0
$826$	0	0
$827$	8.28601	0.288133	0.144066	−	0.989568i	$-0.453982\pi$
$827$	8.28601	0.288133	0.144066	+	0.989568i	$0.453982\pi$
$828$	0	0
$829$	−4.29368	−0.149125	−0.0745627	−	0.997216i	$-0.523756\pi$
$829$	−4.29368	−0.149125	−0.0745627	+	0.997216i	$0.523756\pi$
$830$	12.2940	0.426732
$831$	0	0
$832$	39.7134	1.37682
$833$	0	0
$834$	0	0
$835$	29.0999	1.00704
$836$	9.68952	0.335119
$837$	0	0
$838$	6.55027	0.226275
$839$	6.12227	0.211364	0.105682	−	0.994400i	$-0.466297\pi$
$839$	6.12227	0.211364	0.105682	+	0.994400i	$0.466297\pi$
$840$	0	0
$841$	12.0072	0.414041
$842$	−11.1895	−0.385615
$843$	0	0
$844$	2.91974	0.100501
$845$	100.623	3.46154
$846$	0	0
$847$	0	0
$848$	−7.37602	−0.253294
$849$	0	0
$850$	−11.9037	−0.408294
$851$	−4.59290	−0.157443
$852$	0	0
$853$	19.0351	0.651748	0.325874	−	0.945413i	$-0.394341\pi$
$853$	19.0351	0.651748	0.325874	+	0.945413i	$0.394341\pi$
$854$	0	0
$855$	0	0
$856$	18.0048	0.615392
$857$	−22.5829	−0.771417	−0.385708	−	0.922621i	$-0.626043\pi$
$857$	−22.5829	−0.771417	−0.385708	+	0.922621i	$0.626043\pi$
$858$	0	0
$859$	45.8884	1.56569	0.782846	−	0.622215i	$-0.213767\pi$
$859$	45.8884	1.56569	0.782846	+	0.622215i	$0.213767\pi$
$860$	15.4830	0.527965
$861$	0	0
$862$	14.6662	0.499534
$863$	−48.4683	−1.64988	−0.824940	−	0.565221i	$-0.808791\pi$
$863$	−48.4683	−1.64988	−0.824940	+	0.565221i	$0.808791\pi$
$864$	0	0
$865$	−36.7606	−1.24990
$866$	−9.66880	−0.328559
$867$	0	0
$868$	0	0
$869$	73.4672	2.49220
$870$	0	0
$871$	89.7927	3.04251
$872$	30.2199	1.02338
$873$	0	0
$874$	−2.73053	−0.0923615
$875$	0	0
$876$	0	0
$877$	23.9333	0.808171	0.404085	−	0.914721i	$-0.367590\pi$
$877$	23.9333	0.808171	0.404085	+	0.914721i	$0.367590\pi$
$878$	26.5844	0.897179
$879$	0	0
$880$	−12.9466	−0.436428
$881$	36.6338	1.23422	0.617112	−	0.786875i	$-0.288303\pi$
$881$	36.6338	1.23422	0.617112	+	0.786875i	$0.288303\pi$
$882$	0	0
$883$	−52.2818	−1.75942	−0.879711	−	0.475509i	$-0.842264\pi$
$883$	−52.2818	−1.75942	−0.879711	+	0.475509i	$0.842264\pi$
$884$	−6.49604	−0.218486
$885$	0	0
$886$	5.55702	0.186692
$887$	40.1074	1.34667	0.673337	−	0.739336i	$-0.264860\pi$
$887$	40.1074	1.34667	0.673337	+	0.739336i	$0.264860\pi$
$888$	0	0
$889$	0	0
$890$	29.2693	0.981110
$891$	0	0
$892$	20.0300	0.670652
$893$	−27.9380	−0.934911
$894$	0	0
$895$	29.8962	0.999321
$896$	0	0
$897$	0	0
$898$	−14.2305	−0.474878
$899$	−27.4902	−0.916850
$900$	0	0
$901$	10.0421	0.334550
$902$	23.9579	0.797709
$903$	0	0
$904$	−13.0571	−0.434271
$905$	80.8723	2.68829
$906$	0	0
$907$	−28.6953	−0.952810	−0.476405	−	0.879226i	$-0.658060\pi$
$907$	−28.6953	−0.952810	−0.476405	+	0.879226i	$0.658060\pi$
$908$	0.0434815	0.00144298
$909$	0	0
$910$	0	0
$911$	27.5785	0.913715	0.456858	−	0.889540i	$-0.348975\pi$
$911$	27.5785	0.913715	0.456858	+	0.889540i	$0.348975\pi$
$912$	0	0
$913$	−12.9466	−0.428468
$914$	21.8187	0.721698
$915$	0	0
$916$	13.1498	0.434480
$917$	0	0
$918$	0	0
$919$	−33.7979	−1.11489	−0.557445	−	0.830214i	$-0.688218\pi$
$919$	−33.7979	−1.11489	−0.557445	+	0.830214i	$0.688218\pi$
$920$	−16.2199	−0.534754
$921$	0	0
$922$	−18.4356	−0.607145
$923$	−62.8787	−2.06968
$924$	0	0
$925$	42.9908	1.41353
$926$	25.3507	0.833076
$927$	0	0
$928$	33.4076	1.09666
$929$	28.2106	0.925560	0.462780	−	0.886473i	$-0.346852\pi$
$929$	28.2106	0.925560	0.462780	+	0.886473i	$0.346852\pi$
$930$	0	0
$931$	0	0
$932$	8.31951	0.272514
$933$	0	0
$934$	−37.3731	−1.22289
$935$	17.6261	0.576434
$936$	0	0
$937$	43.8764	1.43338	0.716690	−	0.697392i	$-0.245656\pi$
$937$	43.8764	1.43338	0.716690	+	0.697392i	$0.245656\pi$
$938$	0	0
$939$	0	0
$940$	−57.6764	−1.88120
$941$	54.3765	1.77262	0.886312	−	0.463089i	$-0.153259\pi$
$941$	54.3765	1.77262	0.886312	+	0.463089i	$0.153259\pi$
$942$	0	0
$943$	7.69480	0.250577
$944$	−5.96074	−0.194006
$945$	0	0
$946$	14.3057	0.465119
$947$	−57.7853	−1.87777	−0.938884	−	0.344233i	$-0.888139\pi$
$947$	−57.7853	−1.87777	−0.938884	+	0.344233i	$0.888139\pi$
$948$	0	0
$949$	−84.3774	−2.73901
$950$	25.5585	0.829227
$951$	0	0
$952$	0	0
$953$	44.8014	1.45126	0.725630	−	0.688086i	$-0.241549\pi$
$953$	44.8014	1.45126	0.725630	+	0.688086i	$0.241549\pi$
$954$	0	0
$955$	48.4956	1.56928
$956$	−0.728344	−0.0235563
$957$	0	0
$958$	18.1780	0.587304
$959$	0	0
$960$	0	0
$961$	−12.5712	−0.405524
$962$	−20.5844	−0.663666
$963$	0	0
$964$	−15.6032	−0.502544
$965$	−80.3506	−2.58658
$966$	0	0
$967$	−8.31130	−0.267273	−0.133637	−	0.991030i	$-0.542666\pi$
$967$	−8.31130	−0.267273	−0.133637	+	0.991030i	$0.542666\pi$
$968$	20.5825	0.661548
$969$	0	0
$970$	57.9242	1.85984
$971$	58.0327	1.86236	0.931178	−	0.364564i	$-0.118782\pi$
$971$	58.0327	1.86236	0.931178	+	0.364564i	$0.118782\pi$
$972$	0	0
$973$	0	0
$974$	−15.5122	−0.497043
$975$	0	0
$976$	−1.74888	−0.0559802
$977$	−40.5215	−1.29640	−0.648200	−	0.761471i	$-0.724478\pi$
$977$	−40.5215	−1.29640	−0.648200	+	0.761471i	$0.724478\pi$
$978$	0	0
$979$	−30.8229	−0.985103
$980$	0	0
$981$	0	0
$982$	−18.8199	−0.600567
$983$	−0.523229	−0.0166884	−0.00834421	−	0.999965i	$-0.502656\pi$
$983$	−0.523229	−0.0166884	−0.00834421	+	0.999965i	$0.502656\pi$
$984$	0	0
$985$	−32.1179	−1.02336
$986$	6.19106	0.197163
$987$	0	0
$988$	13.9477	0.443734
$989$	4.59472	0.146103
$990$	0	0
$991$	43.6287	1.38591	0.692955	−	0.720981i	$-0.256308\pi$
$991$	43.6287	1.38591	0.692955	+	0.720981i	$0.256308\pi$
$992$	−22.3956	−0.711061
$993$	0	0
$994$	0	0
$995$	8.18392	0.259448
$996$	0	0
$997$	14.7896	0.468390	0.234195	−	0.972190i	$-0.424755\pi$
$997$	14.7896	0.468390	0.234195	+	0.972190i	$0.424755\pi$
$998$	23.8821	0.755976
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7497.2.a.ce.1.3	yes	8
3.2	odd	2		7497.2.a.cd.1.6	yes	8
7.6	odd	2		7497.2.a.cd.1.3	✓	8
21.20	even	2	inner	7497.2.a.ce.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7497.2.a.cd.1.3	✓	8	7.6	odd	2
7497.2.a.cd.1.6	yes	8	3.2	odd	2
7497.2.a.ce.1.3	yes	8	1.1	even	1	trivial
7497.2.a.ce.1.6	yes	8	21.20	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7497 = 3^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7497.a (trivial)

\(f(q)\)	\(=\)	\(q-0.966796 q^{2} -1.06531 q^{4} +4.16083 q^{5} +2.96352 q^{8} +O(q^{10})\)	\(q-0.966796 q^{2} -1.06531 q^{4} +4.16083 q^{5} +2.96352 q^{8} -4.02268 q^{10} +4.23619 q^{11} +6.09782 q^{13} -0.734511 q^{16} +1.00000 q^{17} -2.14710 q^{19} -4.43256 q^{20} -4.09553 q^{22} -1.31540 q^{23} +12.3125 q^{25} -5.89535 q^{26} -6.40369 q^{29} +4.29287 q^{31} -5.21693 q^{32} -0.966796 q^{34} +3.49163 q^{37} +2.07581 q^{38} +12.3307 q^{40} -5.84976 q^{41} -3.49301 q^{43} -4.51284 q^{44} +1.27173 q^{46} +13.0120 q^{47} -11.9037 q^{50} -6.49604 q^{52} +10.0421 q^{53} +17.6261 q^{55} +6.19106 q^{58} +8.11525 q^{59} +2.38101 q^{61} -4.15033 q^{62} +6.51273 q^{64} +25.3720 q^{65} +14.7254 q^{67} -1.06531 q^{68} -10.3117 q^{71} -13.8373 q^{73} -3.37569 q^{74} +2.28732 q^{76} +17.3428 q^{79} -3.05618 q^{80} +5.65552 q^{82} -3.05618 q^{83} +4.16083 q^{85} +3.37702 q^{86} +12.5540 q^{88} -7.27608 q^{89} +1.40131 q^{92} -12.5799 q^{94} -8.93374 q^{95} -14.3994 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 12 q^{5}+O(q^{10}) \) 8 * q + 8 * q^4 + 12 * q^5	\( 8 q + 8 q^{4} + 12 q^{5} + 16 q^{16} + 8 q^{17} - 8 q^{20} - 28 q^{22} + 8 q^{25} - 4 q^{26} + 12 q^{37} + 60 q^{38} + 28 q^{41} - 24 q^{43} + 4 q^{46} + 20 q^{47} + 40 q^{58} + 12 q^{59} + 48 q^{62} + 48 q^{67} + 8 q^{68} + 60 q^{79} + 40 q^{80} + 40 q^{83} + 12 q^{85} - 8 q^{88} + 16 q^{89}+O(q^{100}) \) 8 * q + 8 * q^4 + 12 * q^5 + 16 * q^16 + 8 * q^17 - 8 * q^20 - 28 * q^22 + 8 * q^25 - 4 * q^26 + 12 * q^37 + 60 * q^38 + 28 * q^41 - 24 * q^43 + 4 * q^46 + 20 * q^47 + 40 * q^58 + 12 * q^59 + 48 * q^62 + 48 * q^67 + 8 * q^68 + 60 * q^79 + 40 * q^80 + 40 * q^83 + 12 * q^85 - 8 * q^88 + 16 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more