LMFDB - Embedded newform 8281.2.a.cs.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 42x^{14} + 641x^{12} - 4448x^{10} + 14076x^{8} - 17900x^{6} + 6960x^{4} - 416x^{2} + 4 $ x^16 - 42x^14 + 641x^12 - 4448x^10 + 14076x^8 - 17900x^6 + 6960x^4 - 416*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-2.09441$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.680196 q^{2} -2.33413 q^{3} -1.53733 q^{4} +3.24613 q^{5} +1.58766 q^{6} +2.40608 q^{8} +2.44815 q^{9} +O(q^{10})$	\(q-0.680196 q^{2} -2.33413 q^{3} -1.53733 q^{4} +3.24613 q^{5} +1.58766 q^{6} +2.40608 q^{8} +2.44815 q^{9} -2.20800 q^{10} -5.33157 q^{11} +3.58833 q^{12} -7.57687 q^{15} +1.43806 q^{16} +2.66842 q^{17} -1.66522 q^{18} +0.696519 q^{19} -4.99038 q^{20} +3.62652 q^{22} +8.96087 q^{23} -5.61610 q^{24} +5.53733 q^{25} +1.28809 q^{27} -5.28644 q^{29} +5.15376 q^{30} +5.45951 q^{31} -5.79032 q^{32} +12.4446 q^{33} -1.81505 q^{34} -3.76362 q^{36} -6.69197 q^{37} -0.473769 q^{38} +7.81044 q^{40} +2.21339 q^{41} -2.39014 q^{43} +8.19640 q^{44} +7.94700 q^{45} -6.09515 q^{46} -5.79573 q^{47} -3.35661 q^{48} -3.76647 q^{50} -6.22843 q^{51} -4.71570 q^{53} -0.876153 q^{54} -17.3070 q^{55} -1.62576 q^{57} +3.59581 q^{58} +2.98070 q^{59} +11.6482 q^{60} +2.19465 q^{61} -3.71354 q^{62} +1.06244 q^{64} -8.46475 q^{66} +8.70594 q^{67} -4.10225 q^{68} -20.9158 q^{69} +8.56190 q^{71} +5.89045 q^{72} +5.88152 q^{73} +4.55185 q^{74} -12.9248 q^{75} -1.07078 q^{76} +0.910817 q^{79} +4.66812 q^{80} -10.3510 q^{81} -1.50554 q^{82} +11.7481 q^{83} +8.66202 q^{85} +1.62576 q^{86} +12.3392 q^{87} -12.8282 q^{88} +0.986019 q^{89} -5.40552 q^{90} -13.7758 q^{92} -12.7432 q^{93} +3.94224 q^{94} +2.26099 q^{95} +13.5154 q^{96} -15.3454 q^{97} -13.0525 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 16 q^{9}+O(q^{10}) $ 16 * q + 20 * q^4 + 16 * q^9	$ 16 q + 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} + 36 q^{23} + 44 q^{25} + 36 q^{29} - 52 q^{36} + 36 q^{43} + 72 q^{51} + 12 q^{53} + 164 q^{64} + 96 q^{74} + 36 q^{79} + 16 q^{81} - 136 q^{88} + 24 q^{92} + 84 q^{95}+O(q^{100}) $ 16 * q + 20 * q^4 + 16 * q^9 + 28 * q^16 - 8 * q^22 + 36 * q^23 + 44 * q^25 + 36 * q^29 - 52 * q^36 + 36 * q^43 + 72 * q^51 + 12 * q^53 + 164 * q^64 + 96 * q^74 + 36 * q^79 + 16 * q^81 - 136 * q^88 + 24 * q^92 + 84 * q^95

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 $

$2$	−0.680196	−0.480971	−0.240486	−	0.970653i	$-0.577307\pi$
$2$	−0.680196	−0.480971	−0.240486	+	0.970653i	$0.577307\pi$
$3$	−2.33413	−1.34761	−0.673804	−	0.738910i	$-0.735341\pi$
$3$	−2.33413	−1.34761	−0.673804	+	0.738910i	$0.735341\pi$
$4$	−1.53733	−0.768667
$5$	3.24613	1.45171	0.725856	−	0.687847i	$-0.241444\pi$
$5$	3.24613	1.45171	0.725856	+	0.687847i	$0.241444\pi$
$6$	1.58766	0.648161
$7$	0	0
$7$	0	0
$8$	2.40608	0.850678
$9$	2.44815	0.816050
$10$	−2.20800	−0.698232
$11$	−5.33157	−1.60753	−0.803765	−	0.594947i	$-0.797173\pi$
$11$	−5.33157	−1.60753	−0.803765	+	0.594947i	$0.797173\pi$
$12$	3.58833	1.03586
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−7.57687	−1.95634
$16$	1.43806	0.359515
$17$	2.66842	0.647186	0.323593	−	0.946196i	$-0.395109\pi$
$17$	2.66842	0.647186	0.323593	+	0.946196i	$0.395109\pi$
$18$	−1.66522	−0.392497
$19$	0.696519	0.159792	0.0798962	−	0.996803i	$-0.474541\pi$
$19$	0.696519	0.159792	0.0798962	+	0.996803i	$0.474541\pi$
$20$	−4.99038	−1.11588
$21$	0	0
$22$	3.62652	0.773176
$23$	8.96087	1.86847	0.934236	−	0.356656i	$-0.116083\pi$
$23$	8.96087	1.86847	0.934236	+	0.356656i	$0.116083\pi$
$24$	−5.61610	−1.14638
$25$	5.53733	1.10747
$26$	0	0
$27$	1.28809	0.247893
$28$	0	0
$29$	−5.28644	−0.981667	−0.490833	−	0.871253i	$-0.663308\pi$
$29$	−5.28644	−0.981667	−0.490833	+	0.871253i	$0.663308\pi$
$30$	5.15376	0.940943
$31$	5.45951	0.980557	0.490279	−	0.871566i	$-0.336895\pi$
$31$	5.45951	0.980557	0.490279	+	0.871566i	$0.336895\pi$
$32$	−5.79032	−1.02359
$33$	12.4446	2.16632
$34$	−1.81505	−0.311278
$35$	0	0
$36$	−3.76362	−0.627270
$37$	−6.69197	−1.10015	−0.550076	−	0.835114i	$-0.685401\pi$
$37$	−6.69197	−1.10015	−0.550076	+	0.835114i	$0.685401\pi$
$38$	−0.473769	−0.0768555
$39$	0	0
$40$	7.81044	1.23494
$41$	2.21339	0.345673	0.172836	−	0.984951i	$-0.444707\pi$
$41$	2.21339	0.345673	0.172836	+	0.984951i	$0.444707\pi$
$42$	0	0
$43$	−2.39014	−0.364493	−0.182246	−	0.983253i	$-0.558337\pi$
$43$	−2.39014	−0.364493	−0.182246	+	0.983253i	$0.558337\pi$
$44$	8.19640	1.23565
$45$	7.94700	1.18467
$46$	−6.09515	−0.898681
$47$	−5.79573	−0.845395	−0.422697	−	0.906271i	$-0.638917\pi$
$47$	−5.79573	−0.845395	−0.422697	+	0.906271i	$0.638917\pi$
$48$	−3.35661	−0.484485
$49$	0	0
$50$	−3.76647	−0.532660
$51$	−6.22843	−0.872154
$52$	0	0
$53$	−4.71570	−0.647751	−0.323876	−	0.946100i	$-0.604986\pi$
$53$	−4.71570	−0.647751	−0.323876	+	0.946100i	$0.604986\pi$
$54$	−0.876153	−0.119229
$55$	−17.3070	−2.33367
$56$	0	0
$57$	−1.62576	−0.215338
$58$	3.59581	0.472154
$59$	2.98070	0.388054	0.194027	−	0.980996i	$-0.437845\pi$
$59$	2.98070	0.388054	0.194027	+	0.980996i	$0.437845\pi$
$60$	11.6482	1.50377
$61$	2.19465	0.280996	0.140498	−	0.990081i	$-0.455130\pi$
$61$	2.19465	0.280996	0.140498	+	0.990081i	$0.455130\pi$
$62$	−3.71354	−0.471620
$63$	0	0
$64$	1.06244	0.132805
$65$	0	0
$66$	−8.46475	−1.04194
$67$	8.70594	1.06360	0.531800	−	0.846870i	$-0.321516\pi$
$67$	8.70594	1.06360	0.531800	+	0.846870i	$0.321516\pi$
$68$	−4.10225	−0.497470
$69$	−20.9158	−2.51797
$70$	0	0
$71$	8.56190	1.01611	0.508055	−	0.861325i	$-0.330365\pi$
$71$	8.56190	1.01611	0.508055	+	0.861325i	$0.330365\pi$
$72$	5.89045	0.694196
$73$	5.88152	0.688380	0.344190	−	0.938900i	$-0.388154\pi$
$73$	5.88152	0.688380	0.344190	+	0.938900i	$0.388154\pi$
$74$	4.55185	0.529142
$75$	−12.9248	−1.49243
$76$	−1.07078	−0.122827
$77$	0	0
$78$	0	0
$79$	0.910817	0.102475	0.0512374	−	0.998687i	$-0.483683\pi$
$79$	0.910817	0.102475	0.0512374	+	0.998687i	$0.483683\pi$
$80$	4.66812	0.521912
$81$	−10.3510	−1.15011
$82$	−1.50554	−0.166259
$83$	11.7481	1.28952	0.644758	−	0.764387i	$-0.276958\pi$
$83$	11.7481	1.28952	0.644758	+	0.764387i	$0.276958\pi$
$84$	0	0
$85$	8.66202	0.939528
$86$	1.62576	0.175311
$87$	12.3392	1.32290
$88$	−12.8282	−1.36749
$89$	0.986019	0.104518	0.0522589	−	0.998634i	$-0.483358\pi$
$89$	0.986019	0.104518	0.0522589	+	0.998634i	$0.483358\pi$
$90$	−5.40552	−0.569792
$91$	0	0
$92$	−13.7758	−1.43623
$93$	−12.7432	−1.32141
$94$	3.94224	0.406611
$95$	2.26099	0.231972
$96$	13.5154	1.37940
$97$	−15.3454	−1.55809	−0.779044	−	0.626969i	$-0.784295\pi$
$97$	−15.3454	−1.55809	−0.779044	+	0.626969i	$0.784295\pi$
$98$	0	0
$99$	−13.0525	−1.31182
$100$	−8.51273	−0.851273
$101$	−3.90927	−0.388987	−0.194493	−	0.980904i	$-0.562306\pi$
$101$	−3.90927	−0.388987	−0.194493	+	0.980904i	$0.562306\pi$
$102$	4.23655	0.419481
$103$	18.5612	1.82889	0.914443	−	0.404715i	$-0.132629\pi$
$103$	18.5612	1.82889	0.914443	+	0.404715i	$0.132629\pi$
$104$	0	0
$105$	0	0
$106$	3.20760	0.311550
$107$	13.1058	1.26699	0.633495	−	0.773747i	$-0.281620\pi$
$107$	13.1058	1.26699	0.633495	+	0.773747i	$0.281620\pi$
$108$	−1.98022	−0.190547
$109$	−15.3979	−1.47485	−0.737426	−	0.675428i	$-0.763959\pi$
$109$	−15.3979	−1.47485	−0.737426	+	0.675428i	$0.763959\pi$
$110$	11.7721	1.12243
$111$	15.6199	1.48258
$112$	0	0
$113$	12.7346	1.19797	0.598985	−	0.800761i	$-0.295571\pi$
$113$	12.7346	1.19797	0.598985	+	0.800761i	$0.295571\pi$
$114$	1.10584	0.103571
$115$	29.0881	2.71248
$116$	8.12701	0.754574
$117$	0	0
$118$	−2.02746	−0.186643
$119$	0	0
$120$	−18.2306	−1.66422
$121$	17.4257	1.58415
$122$	−1.49279	−0.135151
$123$	−5.16633	−0.465832
$124$	−8.39309	−0.753722
$125$	1.74425	0.156011
$126$	0	0
$127$	−10.2763	−0.911878	−0.455939	−	0.890011i	$-0.650697\pi$
$127$	−10.2763	−0.911878	−0.455939	+	0.890011i	$0.650697\pi$
$128$	10.8580	0.959719
$129$	5.57889	0.491193
$130$	0	0
$131$	−12.5813	−1.09923	−0.549615	−	0.835418i	$-0.685226\pi$
$131$	−12.5813	−1.09923	−0.549615	+	0.835418i	$0.685226\pi$
$132$	−19.1315	−1.66518
$133$	0	0
$134$	−5.92175	−0.511561
$135$	4.18130	0.359869
$136$	6.42043	0.550547
$137$	−1.72588	−0.147452	−0.0737261	−	0.997279i	$-0.523489\pi$
$137$	−1.72588	−0.147452	−0.0737261	+	0.997279i	$0.523489\pi$
$138$	14.2269	1.21107
$139$	2.93311	0.248783	0.124391	−	0.992233i	$-0.460302\pi$
$139$	2.93311	0.248783	0.124391	+	0.992233i	$0.460302\pi$
$140$	0	0
$141$	13.5280	1.13926
$142$	−5.82377	−0.488720
$143$	0	0
$144$	3.52058	0.293382
$145$	−17.1604	−1.42510
$146$	−4.00059	−0.331091
$147$	0	0
$148$	10.2878	0.845650
$149$	−13.2399	−1.08465	−0.542327	−	0.840168i	$-0.682457\pi$
$149$	−13.2399	−1.08465	−0.542327	+	0.840168i	$0.682457\pi$
$150$	8.79143	0.717817
$151$	−9.25883	−0.753473	−0.376736	−	0.926320i	$-0.622954\pi$
$151$	−9.25883	−0.753473	−0.376736	+	0.926320i	$0.622954\pi$
$152$	1.67588	0.135932
$153$	6.53268	0.528136
$154$	0	0
$155$	17.7223	1.42349
$156$	0	0
$157$	1.65626	0.132184	0.0660922	−	0.997814i	$-0.478947\pi$
$157$	1.65626	0.132184	0.0660922	+	0.997814i	$0.478947\pi$
$158$	−0.619534	−0.0492875
$159$	11.0070	0.872915
$160$	−18.7961	−1.48596
$161$	0	0
$162$	7.04072	0.553171
$163$	9.89957	0.775394	0.387697	−	0.921787i	$-0.373271\pi$
$163$	9.89957	0.775394	0.387697	+	0.921787i	$0.373271\pi$
$164$	−3.40271	−0.265707
$165$	40.3966	3.14487
$166$	−7.99098	−0.620220
$167$	−6.10891	−0.472721	−0.236361	−	0.971665i	$-0.575955\pi$
$167$	−6.10891	−0.472721	−0.236361	+	0.971665i	$0.575955\pi$
$168$	0	0
$169$	0	0
$170$	−5.89187	−0.451886
$171$	1.70518	0.130398
$172$	3.67444	0.280173
$173$	−18.7212	−1.42334	−0.711672	−	0.702512i	$-0.752062\pi$
$173$	−18.7212	−1.42334	−0.711672	+	0.702512i	$0.752062\pi$
$174$	−8.39309	−0.636278
$175$	0	0
$176$	−7.66712	−0.577931
$177$	−6.95734	−0.522945
$178$	−0.670687	−0.0502701
$179$	−9.93756	−0.742768	−0.371384	−	0.928479i	$-0.621117\pi$
$179$	−9.93756	−0.742768	−0.371384	+	0.928479i	$0.621117\pi$
$180$	−12.2172	−0.910615
$181$	−22.6991	−1.68721	−0.843606	−	0.536962i	$-0.819572\pi$
$181$	−22.6991	−1.68721	−0.843606	+	0.536962i	$0.819572\pi$
$182$	0	0
$183$	−5.12259	−0.378673
$184$	21.5606	1.58947
$185$	−21.7230	−1.59710
$186$	8.66787	0.635559
$187$	−14.2269	−1.04037
$188$	8.90997	0.649827
$189$	0	0
$190$	−1.53791	−0.111572
$191$	5.42926	0.392848	0.196424	−	0.980519i	$-0.437067\pi$
$191$	5.42926	0.392848	0.196424	+	0.980519i	$0.437067\pi$
$192$	−2.47987	−0.178969
$193$	1.97005	0.141807	0.0709037	−	0.997483i	$-0.477412\pi$
$193$	1.97005	0.141807	0.0709037	+	0.997483i	$0.477412\pi$
$194$	10.4379	0.749396
$195$	0	0
$196$	0	0
$197$	19.1476	1.36421	0.682107	−	0.731253i	$-0.261064\pi$
$197$	19.1476	1.36421	0.682107	+	0.731253i	$0.261064\pi$
$198$	8.87825	0.630950
$199$	0.194811	0.0138098	0.00690489	−	0.999976i	$-0.497802\pi$
$199$	0.194811	0.0138098	0.00690489	+	0.999976i	$0.497802\pi$
$200$	13.3233	0.942097
$201$	−20.3208	−1.43332
$202$	2.65907	0.187091
$203$	0	0
$204$	9.57516	0.670396
$205$	7.18493	0.501817
$206$	−12.6252	−0.879642
$207$	21.9376	1.52477
$208$	0	0
$209$	−3.71354	−0.256871
$210$	0	0
$211$	12.2618	0.844139	0.422070	−	0.906563i	$-0.361304\pi$
$211$	12.2618	0.844139	0.422070	+	0.906563i	$0.361304\pi$
$212$	7.24960	0.497905
$213$	−19.9846	−1.36932
$214$	−8.91454	−0.609386
$215$	−7.75869	−0.529138
$216$	3.09925	0.210877
$217$	0	0
$218$	10.4736	0.709362
$219$	−13.7282	−0.927667
$220$	26.6066	1.79381
$221$	0	0
$222$	−10.6246	−0.713076
$223$	27.1733	1.81966	0.909830	−	0.414981i	$-0.136212\pi$
$223$	27.1733	1.81966	0.909830	+	0.414981i	$0.136212\pi$
$224$	0	0
$225$	13.5562	0.903748
$226$	−8.66202	−0.576189
$227$	−7.21285	−0.478733	−0.239367	−	0.970929i	$-0.576940\pi$
$227$	−7.21285	−0.478733	−0.239367	+	0.970929i	$0.576940\pi$
$228$	2.49934	0.165523
$229$	−17.1127	−1.13084	−0.565419	−	0.824804i	$-0.691286\pi$
$229$	−17.1127	−1.13084	−0.565419	+	0.824804i	$0.691286\pi$
$230$	−19.7856	−1.30463
$231$	0	0
$232$	−12.7196	−0.835082
$233$	−12.6353	−0.827767	−0.413883	−	0.910330i	$-0.635828\pi$
$233$	−12.6353	−0.827767	−0.413883	+	0.910330i	$0.635828\pi$
$234$	0	0
$235$	−18.8137	−1.22727
$236$	−4.58233	−0.298284
$237$	−2.12596	−0.138096
$238$	0	0
$239$	7.70807	0.498594	0.249297	−	0.968427i	$-0.419801\pi$
$239$	7.70807	0.498594	0.249297	+	0.968427i	$0.419801\pi$
$240$	−10.8960	−0.703333
$241$	−19.2022	−1.23693	−0.618463	−	0.785814i	$-0.712244\pi$
$241$	−19.2022	−1.23693	−0.618463	+	0.785814i	$0.712244\pi$
$242$	−11.8529	−0.761932
$243$	20.2963	1.30201
$244$	−3.37390	−0.215992
$245$	0	0
$246$	3.51412	0.224052
$247$	0	0
$248$	13.1360	0.834139
$249$	−27.4214	−1.73776
$250$	−1.18643	−0.0750366
$251$	−17.6761	−1.11570	−0.557851	−	0.829941i	$-0.688374\pi$
$251$	−17.6761	−1.11570	−0.557851	+	0.829941i	$0.688374\pi$
$252$	0	0
$253$	−47.7756	−3.00362
$254$	6.98993	0.438587
$255$	−20.2183	−1.26612
$256$	−9.51044	−0.594402
$257$	2.61935	0.163390	0.0816952	−	0.996657i	$-0.473967\pi$
$257$	2.61935	0.163390	0.0816952	+	0.996657i	$0.473967\pi$
$258$	−3.79474	−0.236250
$259$	0	0
$260$	0	0
$261$	−12.9420	−0.801089
$262$	8.55773	0.528698
$263$	5.62875	0.347084	0.173542	−	0.984827i	$-0.444479\pi$
$263$	5.62875	0.347084	0.173542	+	0.984827i	$0.444479\pi$
$264$	29.9426	1.84284
$265$	−15.3078	−0.940348
$266$	0	0
$267$	−2.30149	−0.140849
$268$	−13.3839	−0.817554
$269$	13.8581	0.844944	0.422472	−	0.906376i	$-0.361163\pi$
$269$	13.8581	0.844944	0.422472	+	0.906376i	$0.361163\pi$
$270$	−2.84410	−0.173087
$271$	−8.00871	−0.486495	−0.243247	−	0.969964i	$-0.578213\pi$
$271$	−8.00871	−0.486495	−0.243247	+	0.969964i	$0.578213\pi$
$272$	3.83734	0.232673
$273$	0	0
$274$	1.17394	0.0709203
$275$	−29.5227	−1.78029
$276$	32.1546	1.93548
$277$	18.2930	1.09912	0.549560	−	0.835454i	$-0.314795\pi$
$277$	18.2930	1.09912	0.549560	+	0.835454i	$0.314795\pi$
$278$	−1.99509	−0.119657
$279$	13.3657	0.800184
$280$	0	0
$281$	19.3790	1.15605	0.578026	−	0.816018i	$-0.303823\pi$
$281$	19.3790	1.15605	0.578026	+	0.816018i	$0.303823\pi$
$282$	−9.20168	−0.547952
$283$	−6.43729	−0.382657	−0.191329	−	0.981526i	$-0.561280\pi$
$283$	−6.43729	−0.382657	−0.191329	+	0.981526i	$0.561280\pi$
$284$	−13.1625	−0.781050
$285$	−5.27743	−0.312608
$286$	0	0
$287$	0	0
$288$	−14.1756	−0.835304
$289$	−9.87955	−0.581150
$290$	11.6725	0.685431
$291$	35.8181	2.09969
$292$	−9.04186	−0.529135
$293$	−7.66381	−0.447725	−0.223862	−	0.974621i	$-0.571867\pi$
$293$	−7.66381	−0.447725	−0.223862	+	0.974621i	$0.571867\pi$
$294$	0	0
$295$	9.67573	0.563343
$296$	−16.1014	−0.935876
$297$	−6.86754	−0.398495
$298$	9.00572	0.521687
$299$	0	0
$300$	19.8698	1.14718
$301$	0	0
$302$	6.29782	0.362399
$303$	9.12472	0.524202
$304$	1.00163	0.0574477
$305$	7.12410	0.407925
$306$	−4.44351	−0.254018
$307$	0.312144	0.0178150	0.00890751	−	0.999960i	$-0.497165\pi$
$307$	0.312144	0.0178150	0.00890751	+	0.999960i	$0.497165\pi$
$308$	0	0
$309$	−43.3241	−2.46462
$310$	−12.0546	−0.684656
$311$	2.81507	0.159628	0.0798141	−	0.996810i	$-0.474567\pi$
$311$	2.81507	0.159628	0.0798141	+	0.996810i	$0.474567\pi$
$312$	0	0
$313$	13.2930	0.751366	0.375683	−	0.926748i	$-0.377408\pi$
$313$	13.2930	0.751366	0.375683	+	0.926748i	$0.377408\pi$
$314$	−1.12658	−0.0635769
$315$	0	0
$316$	−1.40023	−0.0787690
$317$	0.972179	0.0546030	0.0273015	−	0.999627i	$-0.491309\pi$
$317$	0.972179	0.0546030	0.0273015	+	0.999627i	$0.491309\pi$
$318$	−7.48695	−0.419847
$319$	28.1850	1.57806
$320$	3.44881	0.192794
$321$	−30.5907	−1.70741
$322$	0	0
$323$	1.85860	0.103415
$324$	15.9130	0.884053
$325$	0	0
$326$	−6.73365	−0.372942
$327$	35.9407	1.98752
$328$	5.32559	0.294056
$329$	0	0
$330$	−27.4776	−1.51259
$331$	7.72092	0.424380	0.212190	−	0.977228i	$-0.431940\pi$
$331$	7.72092	0.424380	0.212190	+	0.977228i	$0.431940\pi$
$332$	−18.0607	−0.991208
$333$	−16.3829	−0.897779
$334$	4.15525	0.227365
$335$	28.2606	1.54404
$336$	0	0
$337$	27.4858	1.49725	0.748624	−	0.662995i	$-0.230715\pi$
$337$	27.4858	1.49725	0.748624	+	0.662995i	$0.230715\pi$
$338$	0	0
$339$	−29.7241	−1.61439
$340$	−13.3164	−0.722184
$341$	−29.1078	−1.57628
$342$	−1.15986	−0.0627179
$343$	0	0
$344$	−5.75086	−0.310066
$345$	−67.8954	−3.65536
$346$	12.7341	0.684588
$347$	−3.69546	−0.198383	−0.0991914	−	0.995068i	$-0.531626\pi$
$347$	−3.69546	−0.198383	−0.0991914	+	0.995068i	$0.531626\pi$
$348$	−18.9695	−1.01687
$349$	26.7513	1.43197	0.715983	−	0.698118i	$-0.245979\pi$
$349$	26.7513	1.43197	0.715983	+	0.698118i	$0.245979\pi$
$350$	0	0
$351$	0	0
$352$	30.8715	1.64546
$353$	−2.75332	−0.146544	−0.0732722	−	0.997312i	$-0.523344\pi$
$353$	−2.75332	−0.146544	−0.0732722	+	0.997312i	$0.523344\pi$
$354$	4.73235	0.251522
$355$	27.7930	1.47510
$356$	−1.51584	−0.0803394
$357$	0	0
$358$	6.75949	0.357250
$359$	35.0545	1.85011	0.925053	−	0.379837i	$-0.124020\pi$
$359$	35.0545	1.85011	0.925053	+	0.379837i	$0.124020\pi$
$360$	19.1211	1.00777
$361$	−18.5149	−0.974466
$362$	15.4399	0.811501
$363$	−40.6737	−2.13482
$364$	0	0
$365$	19.0922	0.999329
$366$	3.48436	0.182131
$367$	−15.0005	−0.783022	−0.391511	−	0.920173i	$-0.628047\pi$
$367$	−15.0005	−0.783022	−0.391511	+	0.920173i	$0.628047\pi$
$368$	12.8863	0.671743
$369$	5.41870	0.282086
$370$	14.7759	0.768161
$371$	0	0
$372$	19.5905	1.01572
$373$	20.9943	1.08704	0.543521	−	0.839395i	$-0.317091\pi$
$373$	20.9943	1.08704	0.543521	+	0.839395i	$0.317091\pi$
$374$	9.67706	0.500389
$375$	−4.07130	−0.210241
$376$	−13.9450	−0.719159
$377$	0	0
$378$	0	0
$379$	−15.2425	−0.782956	−0.391478	−	0.920187i	$-0.628036\pi$
$379$	−15.2425	−0.782956	−0.391478	+	0.920187i	$0.628036\pi$
$380$	−3.47589	−0.178309
$381$	23.9863	1.22886
$382$	−3.69296	−0.188948
$383$	−24.5760	−1.25577	−0.627887	−	0.778305i	$-0.716080\pi$
$383$	−24.5760	−1.25577	−0.627887	+	0.778305i	$0.716080\pi$
$384$	−25.3439	−1.29333
$385$	0	0
$386$	−1.34002	−0.0682053
$387$	−5.85141	−0.297444
$388$	23.5910	1.19765
$389$	19.2407	0.975545	0.487772	−	0.872971i	$-0.337810\pi$
$389$	19.2407	0.975545	0.487772	+	0.872971i	$0.337810\pi$
$390$	0	0
$391$	23.9114	1.20925
$392$	0	0
$393$	29.3663	1.48133
$394$	−13.0242	−0.656147
$395$	2.95663	0.148764
$396$	20.0660	1.00836
$397$	−0.596764	−0.0299507	−0.0149754	−	0.999888i	$-0.504767\pi$
$397$	−0.596764	−0.0299507	−0.0149754	+	0.999888i	$0.504767\pi$
$398$	−0.132510	−0.00664211
$399$	0	0
$400$	7.96301	0.398151
$401$	−1.59469	−0.0796348	−0.0398174	−	0.999207i	$-0.512678\pi$
$401$	−1.59469	−0.0796348	−0.0398174	+	0.999207i	$0.512678\pi$
$402$	13.8221	0.689384
$403$	0	0
$404$	6.00984	0.299001
$405$	−33.6007	−1.66963
$406$	0	0
$407$	35.6787	1.76853
$408$	−14.9861	−0.741922
$409$	28.9884	1.43338	0.716691	−	0.697390i	$-0.245656\pi$
$409$	28.9884	1.43338	0.716691	+	0.697390i	$0.245656\pi$
$410$	−4.88716	−0.241360
$411$	4.02843	0.198708
$412$	−28.5347	−1.40580
$413$	0	0
$414$	−14.9218	−0.733369
$415$	38.1357	1.87201
$416$	0	0
$417$	−6.84624	−0.335262
$418$	2.52594	0.123548
$419$	5.18611	0.253358	0.126679	−	0.991944i	$-0.459568\pi$
$419$	5.18611	0.253358	0.126679	+	0.991944i	$0.459568\pi$
$420$	0	0
$421$	2.84363	0.138590	0.0692950	−	0.997596i	$-0.477925\pi$
$421$	2.84363	0.138590	0.0692950	+	0.997596i	$0.477925\pi$
$422$	−8.34045	−0.406007
$423$	−14.1888	−0.689884
$424$	−11.3464	−0.551028
$425$	14.7759	0.716737
$426$	13.5934	0.658603
$427$	0	0
$428$	−20.1480	−0.973892
$429$	0	0
$430$	5.27743	0.254500
$431$	3.98106	0.191761	0.0958804	−	0.995393i	$-0.469433\pi$
$431$	3.98106	0.191761	0.0958804	+	0.995393i	$0.469433\pi$
$432$	1.85235	0.0891211
$433$	7.42082	0.356622	0.178311	−	0.983974i	$-0.442937\pi$
$433$	7.42082	0.356622	0.178311	+	0.983974i	$0.442937\pi$
$434$	0	0
$435$	40.0546	1.92047
$436$	23.6717	1.13367
$437$	6.24142	0.298567
$438$	9.33789	0.446181
$439$	30.5670	1.45888	0.729442	−	0.684043i	$-0.239780\pi$
$439$	30.5670	1.45888	0.729442	+	0.684043i	$0.239780\pi$
$440$	−41.6419	−1.98520
$441$	0	0
$442$	0	0
$443$	−4.30975	−0.204762	−0.102381	−	0.994745i	$-0.532646\pi$
$443$	−4.30975	−0.204762	−0.102381	+	0.994745i	$0.532646\pi$
$444$	−24.0130	−1.13961
$445$	3.20074	0.151730
$446$	−18.4832	−0.875205
$447$	30.9036	1.46169
$448$	0	0
$449$	−30.7830	−1.45274	−0.726369	−	0.687305i	$-0.758794\pi$
$449$	−30.7830	−1.45274	−0.726369	+	0.687305i	$0.758794\pi$
$450$	−9.22089	−0.434677
$451$	−11.8008	−0.555680
$452$	−19.5773	−0.920839
$453$	21.6113	1.01539
$454$	4.90615	0.230257
$455$	0	0
$456$	−3.91172	−0.183183
$457$	13.4954	0.631287	0.315644	−	0.948878i	$-0.397780\pi$
$457$	13.4954	0.631287	0.315644	+	0.948878i	$0.397780\pi$
$458$	11.6400	0.543901
$459$	3.43716	0.160433
$460$	−44.7181	−2.08499
$461$	−27.4778	−1.27977	−0.639885	−	0.768471i	$-0.721018\pi$
$461$	−27.4778	−1.27977	−0.639885	+	0.768471i	$0.721018\pi$
$462$	0	0
$463$	−8.06521	−0.374822	−0.187411	−	0.982282i	$-0.560010\pi$
$463$	−8.06521	−0.374822	−0.187411	+	0.982282i	$0.560010\pi$
$464$	−7.60221	−0.352924
$465$	−41.3660	−1.91830
$466$	8.59449	0.398132
$467$	12.6325	0.584564	0.292282	−	0.956332i	$-0.405585\pi$
$467$	12.6325	0.584564	0.292282	+	0.956332i	$0.405585\pi$
$468$	0	0
$469$	0	0
$470$	12.7970	0.590281
$471$	−3.86593	−0.178133
$472$	7.17181	0.330109
$473$	12.7432	0.585933
$474$	1.44607	0.0664202
$475$	3.85685	0.176965
$476$	0	0
$477$	−11.5447	−0.528597
$478$	−5.24300	−0.239809
$479$	−31.5380	−1.44101	−0.720504	−	0.693451i	$-0.756089\pi$
$479$	−31.5380	−1.44101	−0.720504	+	0.693451i	$0.756089\pi$
$480$	43.8725	2.00250
$481$	0	0
$482$	13.0613	0.594926
$483$	0	0
$484$	−26.7891	−1.21769
$485$	−49.8131	−2.26190
$486$	−13.8055	−0.626229
$487$	42.1218	1.90872	0.954362	−	0.298654i	$-0.0965375\pi$
$487$	42.1218	1.90872	0.954362	+	0.298654i	$0.0965375\pi$
$488$	5.28050	0.239037
$489$	−23.1068	−1.04493
$490$	0	0
$491$	30.2284	1.36419	0.682095	−	0.731264i	$-0.261069\pi$
$491$	30.2284	1.36419	0.682095	+	0.731264i	$0.261069\pi$
$492$	7.94236	0.358069
$493$	−14.1064	−0.635321
$494$	0	0
$495$	−42.3700	−1.90439
$496$	7.85110	0.352525
$497$	0	0
$498$	18.6520	0.835815
$499$	−3.50797	−0.157038	−0.0785191	−	0.996913i	$-0.525019\pi$
$499$	−3.50797	−0.157038	−0.0785191	+	0.996913i	$0.525019\pi$
$500$	−2.68149	−0.119920
$501$	14.2590	0.637043
$502$	12.0232	0.536621
$503$	−22.8622	−1.01937	−0.509687	−	0.860360i	$-0.670239\pi$
$503$	−22.8622	−1.01937	−0.509687	+	0.860360i	$0.670239\pi$
$504$	0	0
$505$	−12.6900	−0.564696
$506$	32.4968	1.44466
$507$	0	0
$508$	15.7982	0.700930
$509$	21.7755	0.965183	0.482592	−	0.875846i	$-0.339696\pi$
$509$	21.7755	0.965183	0.482592	+	0.875846i	$0.339696\pi$
$510$	13.7524	0.608966
$511$	0	0
$512$	−15.2470	−0.673829
$513$	0.897178	0.0396114
$514$	−1.78167	−0.0785861
$515$	60.2519	2.65501
$516$	−8.57660	−0.377564
$517$	30.9004	1.35900
$518$	0	0
$519$	43.6976	1.91811
$520$	0	0
$521$	14.5235	0.636287	0.318144	−	0.948042i	$-0.396941\pi$
$521$	14.5235	0.636287	0.318144	+	0.948042i	$0.396941\pi$
$522$	8.80309	0.385301
$523$	16.4188	0.717945	0.358973	−	0.933348i	$-0.383127\pi$
$523$	16.4188	0.717945	0.358973	+	0.933348i	$0.383127\pi$
$524$	19.3416	0.844942
$525$	0	0
$526$	−3.82865	−0.166937
$527$	14.5683	0.634603
$528$	17.8960	0.778825
$529$	57.2973	2.49119
$530$	10.4123	0.452280
$531$	7.29720	0.316672
$532$	0	0
$533$	0	0
$534$	1.56547	0.0677444
$535$	42.5432	1.83930
$536$	20.9472	0.904781
$537$	23.1955	1.00096
$538$	−9.42623	−0.406394
$539$	0	0
$540$	−6.42805	−0.276619
$541$	−13.4954	−0.580212	−0.290106	−	0.956995i	$-0.593691\pi$
$541$	−13.4954	−0.580212	−0.290106	+	0.956995i	$0.593691\pi$
$542$	5.44750	0.233990
$543$	52.9826	2.27370
$544$	−15.4510	−0.662456
$545$	−49.9835	−2.14106
$546$	0	0
$547$	13.5615	0.579847	0.289924	−	0.957050i	$-0.406370\pi$
$547$	13.5615	0.579847	0.289924	+	0.957050i	$0.406370\pi$
$548$	2.65326	0.113342
$549$	5.37283	0.229307
$550$	20.0812	0.856267
$551$	−3.68210	−0.156863
$552$	−50.3252	−2.14198
$553$	0	0
$554$	−12.4428	−0.528645
$555$	50.7042	2.15227
$556$	−4.50916	−0.191231
$557$	35.2736	1.49459	0.747294	−	0.664493i	$-0.231352\pi$
$557$	35.2736	1.49459	0.747294	+	0.664493i	$0.231352\pi$
$558$	−9.09130	−0.384865
$559$	0	0
$560$	0	0
$561$	33.2073	1.40201
$562$	−13.1815	−0.556028
$563$	36.1102	1.52186	0.760931	−	0.648833i	$-0.224742\pi$
$563$	36.1102	1.52186	0.760931	+	0.648833i	$0.224742\pi$
$564$	−20.7970	−0.875712
$565$	41.3381	1.73911
$566$	4.37862	0.184047
$567$	0	0
$568$	20.6006	0.864383
$569$	11.8362	0.496197	0.248099	−	0.968735i	$-0.420194\pi$
$569$	11.8362	0.496197	0.248099	+	0.968735i	$0.420194\pi$
$570$	3.58969	0.150356
$571$	−45.9414	−1.92259	−0.961294	−	0.275526i	$-0.911148\pi$
$571$	−45.9414	−1.92259	−0.961294	+	0.275526i	$0.911148\pi$
$572$	0	0
$573$	−12.6726	−0.529405
$574$	0	0
$575$	49.6193	2.06927
$576$	2.60101	0.108375
$577$	−29.7470	−1.23838	−0.619192	−	0.785240i	$-0.712540\pi$
$577$	−29.7470	−1.23838	−0.619192	+	0.785240i	$0.712540\pi$
$578$	6.72003	0.279517
$579$	−4.59835	−0.191101
$580$	26.3813	1.09542
$581$	0	0
$582$	−24.3633	−1.00989
$583$	25.1421	1.04128
$584$	14.1514	0.585590
$585$	0	0
$586$	5.21290	0.215343
$587$	−11.3406	−0.468078	−0.234039	−	0.972227i	$-0.575194\pi$
$587$	−11.3406	−0.468078	−0.234039	+	0.972227i	$0.575194\pi$
$588$	0	0
$589$	3.80265	0.156686
$590$	−6.58140	−0.270952
$591$	−44.6930	−1.83843
$592$	−9.62344	−0.395521
$593$	−2.87084	−0.117891	−0.0589455	−	0.998261i	$-0.518774\pi$
$593$	−2.87084	−0.117891	−0.0589455	+	0.998261i	$0.518774\pi$
$594$	4.67128	0.191665
$595$	0	0
$596$	20.3541	0.833737
$597$	−0.454713	−0.0186102
$598$	0	0
$599$	−5.41559	−0.221275	−0.110637	−	0.993861i	$-0.535289\pi$
$599$	−5.41559	−0.221275	−0.110637	+	0.993861i	$0.535289\pi$
$600$	−31.0982	−1.26958
$601$	34.1648	1.39361	0.696805	−	0.717261i	$-0.254604\pi$
$601$	34.1648	1.39361	0.696805	+	0.717261i	$0.254604\pi$
$602$	0	0
$603$	21.3134	0.867951
$604$	14.2339	0.579169
$605$	56.5659	2.29973
$606$	−6.20660	−0.252126
$607$	22.0430	0.894697	0.447349	−	0.894360i	$-0.352368\pi$
$607$	22.0430	0.894697	0.447349	+	0.894360i	$0.352368\pi$
$608$	−4.03307	−0.163562
$609$	0	0
$610$	−4.84579	−0.196200
$611$	0	0
$612$	−10.0429	−0.405961
$613$	24.7446	0.999424	0.499712	−	0.866191i	$-0.333439\pi$
$613$	24.7446	0.999424	0.499712	+	0.866191i	$0.333439\pi$
$614$	−0.212320	−0.00856852
$615$	−16.7705	−0.676254
$616$	0	0
$617$	19.9659	0.803797	0.401898	−	0.915684i	$-0.368350\pi$
$617$	19.9659	0.803797	0.401898	+	0.915684i	$0.368350\pi$
$618$	29.4689	1.18541
$619$	10.0750	0.404949	0.202474	−	0.979288i	$-0.435102\pi$
$619$	10.0750	0.404949	0.202474	+	0.979288i	$0.435102\pi$
$620$	−27.2450	−1.09419
$621$	11.5424	0.463181
$622$	−1.91480	−0.0767766
$623$	0	0
$624$	0	0
$625$	−22.0246	−0.880984
$626$	−9.04186	−0.361385
$627$	8.66787	0.346162
$628$	−2.54623	−0.101606
$629$	−17.8570	−0.712004
$630$	0	0
$631$	26.2386	1.04454	0.522271	−	0.852780i	$-0.325085\pi$
$631$	26.2386	1.04454	0.522271	+	0.852780i	$0.325085\pi$
$632$	2.19150	0.0871731
$633$	−28.6207	−1.13757
$634$	−0.661273	−0.0262625
$635$	−33.3583	−1.32378
$636$	−16.9215	−0.670981
$637$	0	0
$638$	−19.1714	−0.759001
$639$	20.9608	0.829197
$640$	35.2464	1.39324
$641$	37.2213	1.47015	0.735077	−	0.677984i	$-0.237146\pi$
$641$	37.2213	1.47015	0.735077	+	0.677984i	$0.237146\pi$
$642$	20.8077	0.821213
$643$	10.8569	0.428155	0.214077	−	0.976817i	$-0.431326\pi$
$643$	10.8569	0.428155	0.214077	+	0.976817i	$0.431326\pi$
$644$	0	0
$645$	18.1098	0.713071
$646$	−1.26421	−0.0497398
$647$	11.2388	0.441841	0.220921	−	0.975292i	$-0.429094\pi$
$647$	11.2388	0.441841	0.220921	+	0.975292i	$0.429094\pi$
$648$	−24.9054	−0.978375
$649$	−15.8918	−0.623809
$650$	0	0
$651$	0	0
$652$	−15.2189	−0.596019
$653$	−15.8246	−0.619265	−0.309632	−	0.950856i	$-0.600206\pi$
$653$	−15.8246	−0.619265	−0.309632	+	0.950856i	$0.600206\pi$
$654$	−24.4467	−0.955942
$655$	−40.8404	−1.59577
$656$	3.18298	0.124274
$657$	14.3988	0.561753
$658$	0	0
$659$	−27.8237	−1.08386	−0.541928	−	0.840425i	$-0.682306\pi$
$659$	−27.8237	−1.08386	−0.541928	+	0.840425i	$0.682306\pi$
$660$	−62.1031	−2.41736
$661$	17.4521	0.678808	0.339404	−	0.940641i	$-0.389775\pi$
$661$	17.4521	0.678808	0.339404	+	0.940641i	$0.389775\pi$
$662$	−5.25174	−0.204115
$663$	0	0
$664$	28.2668	1.09696
$665$	0	0
$666$	11.1436	0.431806
$667$	−47.3711	−1.83422
$668$	9.39142	0.363365
$669$	−63.4260	−2.45219
$670$	−19.2227	−0.742639
$671$	−11.7009	−0.451709
$672$	0	0
$673$	30.6883	1.18295	0.591474	−	0.806324i	$-0.298546\pi$
$673$	30.6883	1.18295	0.591474	+	0.806324i	$0.298546\pi$
$674$	−18.6958	−0.720134
$675$	7.13258	0.274533
$676$	0	0
$677$	−3.47556	−0.133577	−0.0667883	−	0.997767i	$-0.521275\pi$
$677$	−3.47556	−0.133577	−0.0667883	+	0.997767i	$0.521275\pi$
$678$	20.2183	0.776477
$679$	0	0
$680$	20.8415	0.799236
$681$	16.8357	0.645145
$682$	19.7990	0.758143
$683$	−2.98767	−0.114320	−0.0571601	−	0.998365i	$-0.518205\pi$
$683$	−2.98767	−0.114320	−0.0571601	+	0.998365i	$0.518205\pi$
$684$	−2.62143	−0.100233
$685$	−5.60244	−0.214058
$686$	0	0
$687$	39.9432	1.52393
$688$	−3.43716	−0.131040
$689$	0	0
$690$	46.1822	1.75813
$691$	−11.4913	−0.437150	−0.218575	−	0.975820i	$-0.570141\pi$
$691$	−11.4913	−0.437150	−0.218575	+	0.975820i	$0.570141\pi$
$692$	28.7807	1.09408
$693$	0	0
$694$	2.51364	0.0954164
$695$	9.52123	0.361161
$696$	29.6891	1.12536
$697$	5.90624	0.223715
$698$	−18.1961	−0.688734
$699$	29.4924	1.11551
$700$	0	0
$701$	0.977624	0.0369244	0.0184622	−	0.999830i	$-0.494123\pi$
$701$	0.977624	0.0369244	0.0184622	+	0.999830i	$0.494123\pi$
$702$	0	0
$703$	−4.66108	−0.175796
$704$	−5.66447	−0.213488
$705$	43.9135	1.65388
$706$	1.87280	0.0704836
$707$	0	0
$708$	10.6957	0.401971
$709$	−9.85564	−0.370136	−0.185068	−	0.982726i	$-0.559251\pi$
$709$	−9.85564	−0.370136	−0.185068	+	0.982726i	$0.559251\pi$
$710$	−18.9047	−0.709480
$711$	2.22982	0.0836246
$712$	2.37244	0.0889110
$713$	48.9220	1.83214
$714$	0	0
$715$	0	0
$716$	15.2773	0.570941
$717$	−17.9916	−0.671909
$718$	−23.8440	−0.889848
$719$	19.5346	0.728517	0.364258	−	0.931298i	$-0.381322\pi$
$719$	19.5346	0.728517	0.364258	+	0.931298i	$0.381322\pi$
$720$	11.4283	0.425906
$721$	0	0
$722$	12.5937	0.468690
$723$	44.8205	1.66689
$724$	34.8961	1.29690
$725$	−29.2728	−1.08716
$726$	27.6661	1.02679
$727$	24.4958	0.908498	0.454249	−	0.890875i	$-0.349908\pi$
$727$	24.4958	0.908498	0.454249	+	0.890875i	$0.349908\pi$
$728$	0	0
$729$	−16.3211	−0.604486
$730$	−12.9864	−0.480649
$731$	−6.37788	−0.235895
$732$	7.87512	0.291073
$733$	30.8279	1.13865	0.569327	−	0.822111i	$-0.307204\pi$
$733$	30.8279	1.13865	0.569327	+	0.822111i	$0.307204\pi$
$734$	10.2033	0.376611
$735$	0	0
$736$	−51.8864	−1.91256
$737$	−46.4164	−1.70977
$738$	−3.68578	−0.135675
$739$	−2.69864	−0.0992710	−0.0496355	−	0.998767i	$-0.515806\pi$
$739$	−2.69864	−0.0992710	−0.0496355	+	0.998767i	$0.515806\pi$
$740$	33.3954	1.22764
$741$	0	0
$742$	0	0
$743$	−15.2204	−0.558382	−0.279191	−	0.960236i	$-0.590066\pi$
$743$	−15.2204	−0.558382	−0.279191	+	0.960236i	$0.590066\pi$
$744$	−30.6612	−1.12409
$745$	−42.9783	−1.57460
$746$	−14.2802	−0.522836
$747$	28.7610	1.05231
$748$	21.8714	0.799699
$749$	0	0
$750$	2.76929	0.101120
$751$	−24.5482	−0.895777	−0.447888	−	0.894089i	$-0.647824\pi$
$751$	−24.5482	−0.895777	−0.447888	+	0.894089i	$0.647824\pi$
$752$	−8.33461	−0.303932
$753$	41.2581	1.50353
$754$	0	0
$755$	−30.0553	−1.09383
$756$	0	0
$757$	11.0892	0.403043	0.201522	−	0.979484i	$-0.435411\pi$
$757$	11.0892	0.403043	0.201522	+	0.979484i	$0.435411\pi$
$758$	10.3679	0.376580
$759$	111.514	4.04771
$760$	5.44012	0.197334
$761$	16.9512	0.614481	0.307241	−	0.951632i	$-0.400594\pi$
$761$	16.9512	0.614481	0.307241	+	0.951632i	$0.400594\pi$
$762$	−16.3154	−0.591044
$763$	0	0
$764$	−8.34659	−0.301969
$765$	21.2059	0.766702
$766$	16.7165	0.603991
$767$	0	0
$768$	22.1986	0.801022
$769$	50.7371	1.82962	0.914812	−	0.403879i	$-0.132338\pi$
$769$	50.7371	1.82962	0.914812	+	0.403879i	$0.132338\pi$
$770$	0	0
$771$	−6.11389	−0.220186
$772$	−3.02863	−0.109003
$773$	−30.8560	−1.10981	−0.554907	−	0.831912i	$-0.687246\pi$
$773$	−30.8560	−1.10981	−0.554907	+	0.831912i	$0.687246\pi$
$774$	3.98011	0.143062
$775$	30.2311	1.08593
$776$	−36.9223	−1.32543
$777$	0	0
$778$	−13.0875	−0.469209
$779$	1.54166	0.0552359
$780$	0	0
$781$	−45.6484	−1.63343
$782$	−16.2644	−0.581614
$783$	−6.80940	−0.243348
$784$	0	0
$785$	5.37644	0.191893
$786$	−19.9748	−0.712479
$787$	−4.00436	−0.142740	−0.0713700	−	0.997450i	$-0.522737\pi$
$787$	−4.00436	−0.142740	−0.0713700	+	0.997450i	$0.522737\pi$
$788$	−29.4363	−1.04862
$789$	−13.1382	−0.467733
$790$	−2.01109	−0.0715512
$791$	0	0
$792$	−31.4053	−1.11594
$793$	0	0
$794$	0.405917	0.0144054
$795$	35.7302	1.26722
$796$	−0.299489	−0.0106151
$797$	17.3993	0.616316	0.308158	−	0.951335i	$-0.400287\pi$
$797$	17.3993	0.616316	0.308158	+	0.951335i	$0.400287\pi$
$798$	0	0
$799$	−15.4654	−0.547128
$800$	−32.0630	−1.13360
$801$	2.41392	0.0852918
$802$	1.08470	0.0383021
$803$	−31.3578	−1.10659
$804$	31.2398	1.10174
$805$	0	0
$806$	0	0
$807$	−32.3466	−1.13865
$808$	−9.40601	−0.330902
$809$	45.3185	1.59331	0.796656	−	0.604433i	$-0.206600\pi$
$809$	45.3185	1.59331	0.796656	+	0.604433i	$0.206600\pi$
$810$	22.8551	0.803045
$811$	3.47248	0.121935	0.0609676	−	0.998140i	$-0.480581\pi$
$811$	3.47248	0.121935	0.0609676	+	0.998140i	$0.480581\pi$
$812$	0	0
$813$	18.6934	0.655605
$814$	−24.2685	−0.850611
$815$	32.1352	1.12565
$816$	−8.95684	−0.313552
$817$	−1.66478	−0.0582431
$818$	−19.7178	−0.689416
$819$	0	0
$820$	−11.0456	−0.385730
$821$	12.1108	0.422670	0.211335	−	0.977414i	$-0.432219\pi$
$821$	12.1108	0.422670	0.211335	+	0.977414i	$0.432219\pi$
$822$	−2.74013	−0.0955729
$823$	12.3444	0.430300	0.215150	−	0.976581i	$-0.430976\pi$
$823$	12.3444	0.430300	0.215150	+	0.976581i	$0.430976\pi$
$824$	44.6597	1.55579
$825$	68.9097	2.39913
$826$	0	0
$827$	−33.0214	−1.14827	−0.574133	−	0.818762i	$-0.694661\pi$
$827$	−33.0214	−1.14827	−0.574133	+	0.818762i	$0.694661\pi$
$828$	−33.7253	−1.17204
$829$	25.8518	0.897870	0.448935	−	0.893565i	$-0.351804\pi$
$829$	25.8518	0.897870	0.448935	+	0.893565i	$0.351804\pi$
$830$	−25.9397	−0.900381
$831$	−42.6982	−1.48118
$832$	0	0
$833$	0	0
$834$	4.65679	0.161251
$835$	−19.8303	−0.686255
$836$	5.70895	0.197448
$837$	7.03234	0.243073
$838$	−3.52757	−0.121858
$839$	−24.3723	−0.841424	−0.420712	−	0.907194i	$-0.638220\pi$
$839$	−24.3723	−0.841424	−0.420712	+	0.907194i	$0.638220\pi$
$840$	0	0
$841$	−1.05359	−0.0363305
$842$	−1.93423	−0.0666578
$843$	−45.2330	−1.55791
$844$	−18.8505	−0.648861
$845$	0	0
$846$	9.65118	0.331815
$847$	0	0
$848$	−6.78145	−0.232876
$849$	15.0255	0.515672
$850$	−10.0505	−0.344730
$851$	−59.9659	−2.05560
$852$	30.7229	1.05255
$853$	16.2919	0.557825	0.278913	−	0.960316i	$-0.410026\pi$
$853$	16.2919	0.557825	0.278913	+	0.960316i	$0.410026\pi$
$854$	0	0
$855$	5.53523	0.189301
$856$	31.5337	1.07780
$857$	−28.2966	−0.966593	−0.483296	−	0.875457i	$-0.660561\pi$
$857$	−28.2966	−0.966593	−0.483296	+	0.875457i	$0.660561\pi$
$858$	0	0
$859$	−5.30710	−0.181076	−0.0905380	−	0.995893i	$-0.528859\pi$
$859$	−5.30710	−0.181076	−0.0905380	+	0.995893i	$0.528859\pi$
$860$	11.9277	0.406731
$861$	0	0
$862$	−2.70790	−0.0922314
$863$	35.4867	1.20798	0.603990	−	0.796992i	$-0.293577\pi$
$863$	35.4867	1.20798	0.603990	+	0.796992i	$0.293577\pi$
$864$	−7.45845	−0.253742
$865$	−60.7713	−2.06629
$866$	−5.04761	−0.171525
$867$	23.0601	0.783163
$868$	0	0
$869$	−4.85609	−0.164731
$870$	−27.2450	−0.923693
$871$	0	0
$872$	−37.0486	−1.25462
$873$	−37.5678	−1.27148
$874$	−4.24539	−0.143602
$875$	0	0
$876$	21.1048	0.713067
$877$	42.3986	1.43170	0.715850	−	0.698254i	$-0.246040\pi$
$877$	42.3986	1.43170	0.715850	+	0.698254i	$0.246040\pi$
$878$	−20.7916	−0.701681
$879$	17.8883	0.603358
$880$	−24.8884	−0.838989
$881$	33.1997	1.11853	0.559263	−	0.828990i	$-0.311084\pi$
$881$	33.1997	1.11853	0.559263	+	0.828990i	$0.311084\pi$
$882$	0	0
$883$	11.9618	0.402546	0.201273	−	0.979535i	$-0.435492\pi$
$883$	11.9618	0.402546	0.201273	+	0.979535i	$0.435492\pi$
$884$	0	0
$885$	−22.5844	−0.759166
$886$	2.93148	0.0984849
$887$	6.32663	0.212427	0.106214	−	0.994343i	$-0.466127\pi$
$887$	6.32663	0.212427	0.106214	+	0.994343i	$0.466127\pi$
$888$	37.5827	1.26119
$889$	0	0
$890$	−2.17713	−0.0729777
$891$	55.1872	1.84884
$892$	−41.7745	−1.39871
$893$	−4.03684	−0.135088
$894$	−21.0205	−0.703031
$895$	−32.2586	−1.07829
$896$	0	0
$897$	0	0
$898$	20.9385	0.698726
$899$	−28.8614	−0.962580
$900$	−20.8404	−0.694681
$901$	−12.5835	−0.419216
$902$	8.02688	0.267266
$903$	0	0
$904$	30.6404	1.01909
$905$	−73.6842	−2.44935
$906$	−14.6999	−0.488372
$907$	9.31333	0.309244	0.154622	−	0.987974i	$-0.450584\pi$
$907$	9.31333	0.309244	0.154622	+	0.987974i	$0.450584\pi$
$908$	11.0885	0.367986
$909$	−9.57047	−0.317432
$910$	0	0
$911$	28.4873	0.943826	0.471913	−	0.881645i	$-0.343564\pi$
$911$	28.4873	0.943826	0.471913	+	0.881645i	$0.343564\pi$
$912$	−2.33794	−0.0774170
$913$	−62.6356	−2.07294
$914$	−9.17951	−0.303631
$915$	−16.6286	−0.549723
$916$	26.3079	0.869238
$917$	0	0
$918$	−2.33794	−0.0771636
$919$	54.7833	1.80713	0.903566	−	0.428448i	$-0.140940\pi$
$919$	54.7833	1.80713	0.903566	+	0.428448i	$0.140940\pi$
$920$	69.9884	2.30745
$921$	−0.728585	−0.0240077
$922$	18.6903	0.615533
$923$	0	0
$924$	0	0
$925$	−37.0556	−1.21838
$926$	5.48592	0.180279
$927$	45.4405	1.49246
$928$	30.6102	1.00483
$929$	−13.3363	−0.437551	−0.218776	−	0.975775i	$-0.570206\pi$
$929$	−13.3363	−0.437551	−0.218776	+	0.975775i	$0.570206\pi$
$930$	28.1370	0.922649
$931$	0	0
$932$	19.4247	0.636277
$933$	−6.57074	−0.215116
$934$	−8.59260	−0.281158
$935$	−46.1822	−1.51032
$936$	0	0
$937$	−15.4989	−0.506327	−0.253164	−	0.967423i	$-0.581471\pi$
$937$	−15.4989	−0.506327	−0.253164	+	0.967423i	$0.581471\pi$
$938$	0	0
$939$	−31.0276	−1.01255
$940$	28.9229	0.943361
$941$	21.7664	0.709566	0.354783	−	0.934949i	$-0.384555\pi$
$941$	21.7664	0.709566	0.354783	+	0.934949i	$0.384555\pi$
$942$	2.62959	0.0856768
$943$	19.8339	0.645880
$944$	4.28642	0.139511
$945$	0	0
$946$	−8.66787	−0.281817
$947$	−16.2266	−0.527294	−0.263647	−	0.964619i	$-0.584925\pi$
$947$	−16.2266	−0.527294	−0.263647	+	0.964619i	$0.584925\pi$
$948$	3.26831	0.106150
$949$	0	0
$950$	−2.62342	−0.0851149
$951$	−2.26919	−0.0735835
$952$	0	0
$953$	−25.2125	−0.816712	−0.408356	−	0.912823i	$-0.633898\pi$
$953$	−25.2125	−0.816712	−0.408356	+	0.912823i	$0.633898\pi$
$954$	7.85269	0.254240
$955$	17.6241	0.570302
$956$	−11.8499	−0.383252
$957$	−65.7874	−2.12661
$958$	21.4520	0.693083
$959$	0	0
$960$	−8.04996	−0.259811
$961$	−1.19373	−0.0385073
$962$	0	0
$963$	32.0851	1.03393
$964$	29.5202	0.950783
$965$	6.39504	0.205863
$966$	0	0
$967$	−32.6302	−1.04932	−0.524658	−	0.851313i	$-0.675807\pi$
$967$	−32.6302	−1.04932	−0.524658	+	0.851313i	$0.675807\pi$
$968$	41.9276	1.34760
$969$	−4.33821	−0.139363
$970$	33.8827	1.08791
$971$	34.0324	1.09215	0.546077	−	0.837735i	$-0.316121\pi$
$971$	34.0324	1.09215	0.546077	+	0.837735i	$0.316121\pi$
$972$	−31.2022	−1.00081
$973$	0	0
$974$	−28.6511	−0.918041
$975$	0	0
$976$	3.15603	0.101022
$977$	4.85551	0.155342	0.0776708	−	0.996979i	$-0.475252\pi$
$977$	4.85551	0.155342	0.0776708	+	0.996979i	$0.475252\pi$
$978$	15.7172	0.502580
$979$	−5.25703	−0.168016
$980$	0	0
$981$	−37.6964	−1.20355
$982$	−20.5613	−0.656136
$983$	27.5856	0.879845	0.439922	−	0.898036i	$-0.355006\pi$
$983$	27.5856	0.879845	0.439922	+	0.898036i	$0.355006\pi$
$984$	−12.4306	−0.396273
$985$	62.1557	1.98044
$986$	9.59513	0.305571
$987$	0	0
$988$	0	0
$989$	−21.4177	−0.681044
$990$	28.8199	0.915958
$991$	−1.99210	−0.0632813	−0.0316406	−	0.999499i	$-0.510073\pi$
$991$	−1.99210	−0.0632813	−0.0316406	+	0.999499i	$0.510073\pi$
$992$	−31.6123	−1.00369
$993$	−18.0216	−0.571898
$994$	0	0
$995$	0.632380	0.0200478
$996$	42.1559	1.33576
$997$	−31.5600	−0.999516	−0.499758	−	0.866165i	$-0.666578\pi$
$997$	−31.5600	−0.999516	−0.499758	+	0.866165i	$0.666578\pi$
$998$	2.38611	0.0755309
$999$	−8.61985	−0.272720

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cs.1.7		16
7.6	odd	2	inner	8281.2.a.cs.1.8		16
13.5	odd	4		637.2.c.g.246.9	yes	16
13.8	odd	4		637.2.c.g.246.7	✓	16
13.12	even	2	inner	8281.2.a.cs.1.9		16
91.5	even	12		637.2.r.g.116.9		32
91.18	odd	12		637.2.r.g.324.8		32
91.31	even	12		637.2.r.g.324.7		32
91.34	even	4		637.2.c.g.246.8	yes	16
91.44	odd	12		637.2.r.g.116.10		32
91.47	even	12		637.2.r.g.116.7		32
91.60	odd	12		637.2.r.g.324.10		32
91.73	even	12		637.2.r.g.324.9		32
91.83	even	4		637.2.c.g.246.10	yes	16
91.86	odd	12		637.2.r.g.116.8		32
91.90	odd	2	inner	8281.2.a.cs.1.10		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.c.g.246.7	✓	16	13.8	odd	4
637.2.c.g.246.8	yes	16	91.34	even	4
637.2.c.g.246.9	yes	16	13.5	odd	4
637.2.c.g.246.10	yes	16	91.83	even	4
637.2.r.g.116.7		32	91.47	even	12
637.2.r.g.116.8		32	91.86	odd	12
637.2.r.g.116.9		32	91.5	even	12
637.2.r.g.116.10		32	91.44	odd	12
637.2.r.g.324.7		32	91.31	even	12
637.2.r.g.324.8		32	91.18	odd	12
637.2.r.g.324.9		32	91.73	even	12
637.2.r.g.324.10		32	91.60	odd	12
8281.2.a.cs.1.7		16	1.1	even	1	trivial
8281.2.a.cs.1.8		16	7.6	odd	2	inner
8281.2.a.cs.1.9		16	13.12	even	2	inner
8281.2.a.cs.1.10		16	91.90	odd	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q-0.680196 q^{2} -2.33413 q^{3} -1.53733 q^{4} +3.24613 q^{5} +1.58766 q^{6} +2.40608 q^{8} +2.44815 q^{9} +O(q^{10})\)	\(q-0.680196 q^{2} -2.33413 q^{3} -1.53733 q^{4} +3.24613 q^{5} +1.58766 q^{6} +2.40608 q^{8} +2.44815 q^{9} -2.20800 q^{10} -5.33157 q^{11} +3.58833 q^{12} -7.57687 q^{15} +1.43806 q^{16} +2.66842 q^{17} -1.66522 q^{18} +0.696519 q^{19} -4.99038 q^{20} +3.62652 q^{22} +8.96087 q^{23} -5.61610 q^{24} +5.53733 q^{25} +1.28809 q^{27} -5.28644 q^{29} +5.15376 q^{30} +5.45951 q^{31} -5.79032 q^{32} +12.4446 q^{33} -1.81505 q^{34} -3.76362 q^{36} -6.69197 q^{37} -0.473769 q^{38} +7.81044 q^{40} +2.21339 q^{41} -2.39014 q^{43} +8.19640 q^{44} +7.94700 q^{45} -6.09515 q^{46} -5.79573 q^{47} -3.35661 q^{48} -3.76647 q^{50} -6.22843 q^{51} -4.71570 q^{53} -0.876153 q^{54} -17.3070 q^{55} -1.62576 q^{57} +3.59581 q^{58} +2.98070 q^{59} +11.6482 q^{60} +2.19465 q^{61} -3.71354 q^{62} +1.06244 q^{64} -8.46475 q^{66} +8.70594 q^{67} -4.10225 q^{68} -20.9158 q^{69} +8.56190 q^{71} +5.89045 q^{72} +5.88152 q^{73} +4.55185 q^{74} -12.9248 q^{75} -1.07078 q^{76} +0.910817 q^{79} +4.66812 q^{80} -10.3510 q^{81} -1.50554 q^{82} +11.7481 q^{83} +8.66202 q^{85} +1.62576 q^{86} +12.3392 q^{87} -12.8282 q^{88} +0.986019 q^{89} -5.40552 q^{90} -13.7758 q^{92} -12.7432 q^{93} +3.94224 q^{94} +2.26099 q^{95} +13.5154 q^{96} -15.3454 q^{97} -13.0525 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{9}+O(q^{10}) \) 16 * q + 20 * q^4 + 16 * q^9	\( 16 q + 20 q^{4} + 16 q^{9} + 28 q^{16} - 8 q^{22} + 36 q^{23} + 44 q^{25} + 36 q^{29} - 52 q^{36} + 36 q^{43} + 72 q^{51} + 12 q^{53} + 164 q^{64} + 96 q^{74} + 36 q^{79} + 16 q^{81} - 136 q^{88} + 24 q^{92} + 84 q^{95}+O(q^{100}) \) 16 * q + 20 * q^4 + 16 * q^9 + 28 * q^16 - 8 * q^22 + 36 * q^23 + 44 * q^25 + 36 * q^29 - 52 * q^36 + 36 * q^43 + 72 * q^51 + 12 * q^53 + 164 * q^64 + 96 * q^74 + 36 * q^79 + 16 * q^81 - 136 * q^88 + 24 * q^92 + 84 * q^95

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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