LMFDB - Embedded newform 308.4.a.d.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [308,4,Mod(1,308)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(308, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("308.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 308 = 2^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	308.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:	$18.1725882818$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 43x^{2} - 11x + 222 $ x^4 - x^3 - 43x^2 - 11x + 222
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.21425$ of defining polynomial
Character	$\chi$	$=$	308.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.21425 q^{3} -20.5256 q^{5} -7.00000 q^{7} -25.5256 q^{9} +11.0000 q^{11} +66.7086 q^{13} -24.9233 q^{15} -59.7289 q^{17} +131.071 q^{19} -8.49978 q^{21} +163.117 q^{23} +296.300 q^{25} -63.7794 q^{27} -77.2961 q^{29} +206.358 q^{31} +13.3568 q^{33} +143.679 q^{35} -215.238 q^{37} +81.0011 q^{39} -403.218 q^{41} +43.5690 q^{43} +523.928 q^{45} +69.1706 q^{47} +49.0000 q^{49} -72.5261 q^{51} +334.969 q^{53} -225.781 q^{55} +159.154 q^{57} +681.283 q^{59} -24.4899 q^{61} +178.679 q^{63} -1369.23 q^{65} -285.338 q^{67} +198.065 q^{69} -522.204 q^{71} +592.444 q^{73} +359.783 q^{75} -77.0000 q^{77} -92.7989 q^{79} +611.747 q^{81} +480.928 q^{83} +1225.97 q^{85} -93.8570 q^{87} +1197.02 q^{89} -466.960 q^{91} +250.571 q^{93} -2690.31 q^{95} +308.934 q^{97} -280.781 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.21425	0.233683	0.116842	−	0.993151i	$-0.462723\pi$
$3$	1.21425	0.233683	0.116842	+	0.993151i	$0.462723\pi$
$4$	0	0
$5$	−20.5256	−1.83586	−0.917932	−	0.396737i	$-0.870142\pi$
$5$	−20.5256	−1.83586	−0.917932	+	0.396737i	$0.870142\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	−25.5256	−0.945392
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	66.7086	1.42320	0.711601	−	0.702584i	$-0.247970\pi$
$13$	66.7086	1.42320	0.711601	+	0.702584i	$0.247970\pi$
$14$	0	0
$15$	−24.9233	−0.429011
$16$	0	0
$17$	−59.7289	−0.852141	−0.426070	−	0.904690i	$-0.640102\pi$
$17$	−59.7289	−0.852141	−0.426070	+	0.904690i	$0.640102\pi$
$18$	0	0
$19$	131.071	1.58262	0.791310	−	0.611416i	$-0.209400\pi$
$19$	131.071	1.58262	0.791310	+	0.611416i	$0.209400\pi$
$20$	0	0
$21$	−8.49978	−0.0883240
$22$	0	0
$23$	163.117	1.47879	0.739395	−	0.673272i	$-0.235112\pi$
$23$	163.117	1.47879	0.739395	+	0.673272i	$0.235112\pi$
$24$	0	0
$25$	296.300	2.37040
$26$	0	0
$27$	−63.7794	−0.454605
$28$	0	0
$29$	−77.2961	−0.494949	−0.247474	−	0.968894i	$-0.579601\pi$
$29$	−77.2961	−0.494949	−0.247474	+	0.968894i	$0.579601\pi$
$30$	0	0
$31$	206.358	1.19558	0.597791	−	0.801652i	$-0.296045\pi$
$31$	206.358	1.19558	0.597791	+	0.801652i	$0.296045\pi$
$32$	0	0
$33$	13.3568	0.0704581
$34$	0	0
$35$	143.679	0.693892
$36$	0	0
$37$	−215.238	−0.956348	−0.478174	−	0.878265i	$-0.658701\pi$
$37$	−215.238	−0.956348	−0.478174	+	0.878265i	$0.658701\pi$
$38$	0	0
$39$	81.0011	0.332578
$40$	0	0
$41$	−403.218	−1.53590	−0.767952	−	0.640508i	$-0.778724\pi$
$41$	−403.218	−1.53590	−0.767952	+	0.640508i	$0.778724\pi$
$42$	0	0
$43$	43.5690	0.154517	0.0772583	−	0.997011i	$-0.475383\pi$
$43$	43.5690	0.154517	0.0772583	+	0.997011i	$0.475383\pi$
$44$	0	0
$45$	523.928	1.73561
$46$	0	0
$47$	69.1706	0.214672	0.107336	−	0.994223i	$-0.465768\pi$
$47$	69.1706	0.214672	0.107336	+	0.994223i	$0.465768\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	−72.5261	−0.199131
$52$	0	0
$53$	334.969	0.868143	0.434071	−	0.900878i	$-0.357077\pi$
$53$	334.969	0.868143	0.434071	+	0.900878i	$0.357077\pi$
$54$	0	0
$55$	−225.781	−0.553534
$56$	0	0
$57$	159.154	0.369832
$58$	0	0
$59$	681.283	1.50331	0.751656	−	0.659555i	$-0.229255\pi$
$59$	681.283	1.50331	0.751656	+	0.659555i	$0.229255\pi$
$60$	0	0
$61$	−24.4899	−0.0514034	−0.0257017	−	0.999670i	$-0.508182\pi$
$61$	−24.4899	−0.0514034	−0.0257017	+	0.999670i	$0.508182\pi$
$62$	0	0
$63$	178.679	0.357325
$64$	0	0
$65$	−1369.23	−2.61281
$66$	0	0
$67$	−285.338	−0.520292	−0.260146	−	0.965569i	$-0.583771\pi$
$67$	−285.338	−0.520292	−0.260146	+	0.965569i	$0.583771\pi$
$68$	0	0
$69$	198.065	0.345568
$70$	0	0
$71$	−522.204	−0.872876	−0.436438	−	0.899734i	$-0.643760\pi$
$71$	−522.204	−0.872876	−0.436438	+	0.899734i	$0.643760\pi$
$72$	0	0
$73$	592.444	0.949867	0.474934	−	0.880022i	$-0.342472\pi$
$73$	592.444	0.949867	0.474934	+	0.880022i	$0.342472\pi$
$74$	0	0
$75$	359.783	0.553922
$76$	0	0
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−92.7989	−0.132161	−0.0660803	−	0.997814i	$-0.521049\pi$
$79$	−92.7989	−0.132161	−0.0660803	+	0.997814i	$0.521049\pi$
$80$	0	0
$81$	611.747	0.839158
$82$	0	0
$83$	480.928	0.636008	0.318004	−	0.948089i	$-0.396987\pi$
$83$	480.928	0.636008	0.318004	+	0.948089i	$0.396987\pi$
$84$	0	0
$85$	1225.97	1.56441
$86$	0	0
$87$	−93.8570	−0.115661
$88$	0	0
$89$	1197.02	1.42566	0.712831	−	0.701336i	$-0.247413\pi$
$89$	1197.02	1.42566	0.712831	+	0.701336i	$0.247413\pi$
$90$	0	0
$91$	−466.960	−0.537920
$92$	0	0
$93$	250.571	0.279388
$94$	0	0
$95$	−2690.31	−2.90547
$96$	0	0
$97$	308.934	0.323377	0.161688	−	0.986842i	$-0.448306\pi$
$97$	308.934	0.323377	0.161688	+	0.986842i	$0.448306\pi$
$98$	0	0
$99$	−280.781	−0.285046
$100$	0	0
$101$	1742.73	1.71691	0.858454	−	0.512891i	$-0.171425\pi$
$101$	1742.73	1.71691	0.858454	+	0.512891i	$0.171425\pi$
$102$	0	0
$103$	1043.91	0.998635	0.499318	−	0.866419i	$-0.333584\pi$
$103$	1043.91	0.998635	0.499318	+	0.866419i	$0.333584\pi$
$104$	0	0
$105$	174.463	0.162151
$106$	0	0
$107$	204.369	0.184646	0.0923230	−	0.995729i	$-0.470571\pi$
$107$	204.369	0.184646	0.0923230	+	0.995729i	$0.470571\pi$
$108$	0	0
$109$	944.275	0.829772	0.414886	−	0.909873i	$-0.363821\pi$
$109$	944.275	0.829772	0.414886	+	0.909873i	$0.363821\pi$
$110$	0	0
$111$	−261.353	−0.223483
$112$	0	0
$113$	−1905.29	−1.58615	−0.793073	−	0.609127i	$-0.791520\pi$
$113$	−1905.29	−1.58615	−0.793073	+	0.609127i	$0.791520\pi$
$114$	0	0
$115$	−3348.06	−2.71486
$116$	0	0
$117$	−1702.78	−1.34548
$118$	0	0
$119$	418.102	0.322079
$120$	0	0
$121$	121.000	0.0909091
$122$	0	0
$123$	−489.609	−0.358915
$124$	0	0
$125$	−3516.03	−2.51587
$126$	0	0
$127$	547.690	0.382674	0.191337	−	0.981524i	$-0.438718\pi$
$127$	547.690	0.382674	0.191337	+	0.981524i	$0.438718\pi$
$128$	0	0
$129$	52.9038	0.0361079
$130$	0	0
$131$	−1926.59	−1.28494	−0.642468	−	0.766312i	$-0.722090\pi$
$131$	−1926.59	−1.28494	−0.642468	+	0.766312i	$0.722090\pi$
$132$	0	0
$133$	−917.498	−0.598174
$134$	0	0
$135$	1309.11	0.834594
$136$	0	0
$137$	−1440.77	−0.898491	−0.449245	−	0.893408i	$-0.648307\pi$
$137$	−1440.77	−0.898491	−0.449245	+	0.893408i	$0.648307\pi$
$138$	0	0
$139$	−2430.80	−1.48330	−0.741648	−	0.670790i	$-0.765955\pi$
$139$	−2430.80	−1.48330	−0.741648	+	0.670790i	$0.765955\pi$
$140$	0	0
$141$	83.9906	0.0501652
$142$	0	0
$143$	733.794	0.429112
$144$	0	0
$145$	1586.55	0.908659
$146$	0	0
$147$	59.4984	0.0333833
$148$	0	0
$149$	686.765	0.377597	0.188798	−	0.982016i	$-0.439541\pi$
$149$	686.765	0.377597	0.188798	+	0.982016i	$0.439541\pi$
$150$	0	0
$151$	1759.45	0.948225	0.474113	−	0.880464i	$-0.342769\pi$
$151$	1759.45	0.948225	0.474113	+	0.880464i	$0.342769\pi$
$152$	0	0
$153$	1524.62	0.805607
$154$	0	0
$155$	−4235.63	−2.19493
$156$	0	0
$157$	1833.85	0.932211	0.466105	−	0.884729i	$-0.345657\pi$
$157$	1833.85	0.932211	0.466105	+	0.884729i	$0.345657\pi$
$158$	0	0
$159$	406.738	0.202870
$160$	0	0
$161$	−1141.82	−0.558930
$162$	0	0
$163$	−2306.20	−1.10820	−0.554098	−	0.832452i	$-0.686937\pi$
$163$	−2306.20	−1.10820	−0.554098	+	0.832452i	$0.686937\pi$
$164$	0	0
$165$	−274.156	−0.129352
$166$	0	0
$167$	2243.47	1.03955	0.519774	−	0.854304i	$-0.326016\pi$
$167$	2243.47	1.03955	0.519774	+	0.854304i	$0.326016\pi$
$168$	0	0
$169$	2253.03	1.02550
$170$	0	0
$171$	−3345.67	−1.49620
$172$	0	0
$173$	−2499.07	−1.09827	−0.549136	−	0.835733i	$-0.685043\pi$
$173$	−2499.07	−1.09827	−0.549136	+	0.835733i	$0.685043\pi$
$174$	0	0
$175$	−2074.10	−0.895926
$176$	0	0
$177$	827.250	0.351299
$178$	0	0
$179$	270.765	0.113061	0.0565304	−	0.998401i	$-0.481996\pi$
$179$	270.765	0.113061	0.0565304	+	0.998401i	$0.481996\pi$
$180$	0	0
$181$	1480.59	0.608018	0.304009	−	0.952669i	$-0.401675\pi$
$181$	1480.59	0.608018	0.304009	+	0.952669i	$0.401675\pi$
$182$	0	0
$183$	−29.7369	−0.0120121
$184$	0	0
$185$	4417.88	1.75573
$186$	0	0
$187$	−657.018	−0.256930
$188$	0	0
$189$	446.456	0.171825
$190$	0	0
$191$	2461.01	0.932317	0.466158	−	0.884701i	$-0.345638\pi$
$191$	2461.01	0.932317	0.466158	+	0.884701i	$0.345638\pi$
$192$	0	0
$193$	−1419.76	−0.529518	−0.264759	−	0.964315i	$-0.585292\pi$
$193$	−1419.76	−0.529518	−0.264759	+	0.964315i	$0.585292\pi$
$194$	0	0
$195$	−1662.60	−0.610569
$196$	0	0
$197$	2918.28	1.05542	0.527712	−	0.849423i	$-0.323050\pi$
$197$	2918.28	1.05542	0.527712	+	0.849423i	$0.323050\pi$
$198$	0	0
$199$	3767.60	1.34210	0.671050	−	0.741412i	$-0.265844\pi$
$199$	3767.60	1.34210	0.671050	+	0.741412i	$0.265844\pi$
$200$	0	0
$201$	−346.472	−0.121583
$202$	0	0
$203$	541.072	0.187073
$204$	0	0
$205$	8276.28	2.81971
$206$	0	0
$207$	−4163.65	−1.39804
$208$	0	0
$209$	1441.78	0.477178
$210$	0	0
$211$	−3565.15	−1.16320	−0.581600	−	0.813475i	$-0.697573\pi$
$211$	−3565.15	−1.16320	−0.581600	+	0.813475i	$0.697573\pi$
$212$	0	0
$213$	−634.088	−0.203976
$214$	0	0
$215$	−894.280	−0.283672
$216$	0	0
$217$	−1444.51	−0.451888
$218$	0	0
$219$	719.377	0.221968
$220$	0	0
$221$	−3984.43	−1.21277
$222$	0	0
$223$	2531.28	0.760122	0.380061	−	0.924962i	$-0.375903\pi$
$223$	2531.28	0.760122	0.380061	+	0.924962i	$0.375903\pi$
$224$	0	0
$225$	−7563.23	−2.24096
$226$	0	0
$227$	1207.47	0.353050	0.176525	−	0.984296i	$-0.443514\pi$
$227$	1207.47	0.353050	0.176525	+	0.984296i	$0.443514\pi$
$228$	0	0
$229$	−3998.70	−1.15389	−0.576946	−	0.816782i	$-0.695756\pi$
$229$	−3998.70	−1.15389	−0.576946	+	0.816782i	$0.695756\pi$
$230$	0	0
$231$	−93.4975	−0.0266307
$232$	0	0
$233$	1429.63	0.401966	0.200983	−	0.979595i	$-0.435586\pi$
$233$	1429.63	0.401966	0.200983	+	0.979595i	$0.435586\pi$
$234$	0	0
$235$	−1419.77	−0.394108
$236$	0	0
$237$	−112.681	−0.0308837
$238$	0	0
$239$	3223.01	0.872297	0.436148	−	0.899875i	$-0.356342\pi$
$239$	3223.01	0.872297	0.436148	+	0.899875i	$0.356342\pi$
$240$	0	0
$241$	580.569	0.155177	0.0775887	−	0.996985i	$-0.475278\pi$
$241$	580.569	0.155177	0.0775887	+	0.996985i	$0.475278\pi$
$242$	0	0
$243$	2464.86	0.650703
$244$	0	0
$245$	−1005.75	−0.262266
$246$	0	0
$247$	8743.56	2.25239
$248$	0	0
$249$	583.968	0.148624
$250$	0	0
$251$	1277.24	0.321190	0.160595	−	0.987020i	$-0.448659\pi$
$251$	1277.24	0.321190	0.160595	+	0.987020i	$0.448659\pi$
$252$	0	0
$253$	1794.28	0.445872
$254$	0	0
$255$	1488.64	0.365577
$256$	0	0
$257$	2517.10	0.610943	0.305471	−	0.952201i	$-0.401186\pi$
$257$	2517.10	0.610943	0.305471	+	0.952201i	$0.401186\pi$
$258$	0	0
$259$	1506.67	0.361466
$260$	0	0
$261$	1973.03	0.467921
$262$	0	0
$263$	−5684.42	−1.33276	−0.666381	−	0.745611i	$-0.732158\pi$
$263$	−5684.42	−1.33276	−0.666381	+	0.745611i	$0.732158\pi$
$264$	0	0
$265$	−6875.44	−1.59379
$266$	0	0
$267$	1453.49	0.333153
$268$	0	0
$269$	870.076	0.197210	0.0986049	−	0.995127i	$-0.468562\pi$
$269$	870.076	0.197210	0.0986049	+	0.995127i	$0.468562\pi$
$270$	0	0
$271$	190.969	0.0428065	0.0214033	−	0.999771i	$-0.493187\pi$
$271$	190.969	0.0428065	0.0214033	+	0.999771i	$0.493187\pi$
$272$	0	0
$273$	−567.008	−0.125703
$274$	0	0
$275$	3259.30	0.714702
$276$	0	0
$277$	5601.27	1.21497	0.607487	−	0.794330i	$-0.292178\pi$
$277$	5601.27	1.21497	0.607487	+	0.794330i	$0.292178\pi$
$278$	0	0
$279$	−5267.42	−1.13029
$280$	0	0
$281$	6878.31	1.46023	0.730117	−	0.683322i	$-0.239466\pi$
$281$	6878.31	1.46023	0.730117	+	0.683322i	$0.239466\pi$
$282$	0	0
$283$	5722.15	1.20193	0.600965	−	0.799275i	$-0.294783\pi$
$283$	5722.15	1.20193	0.600965	+	0.799275i	$0.294783\pi$
$284$	0	0
$285$	−3266.72	−0.678961
$286$	0	0
$287$	2822.53	0.580517
$288$	0	0
$289$	−1345.46	−0.273856
$290$	0	0
$291$	375.125	0.0755677
$292$	0	0
$293$	8278.49	1.65063	0.825315	−	0.564673i	$-0.190998\pi$
$293$	8278.49	1.65063	0.825315	+	0.564673i	$0.190998\pi$
$294$	0	0
$295$	−13983.7	−2.75988
$296$	0	0
$297$	−701.573	−0.137069
$298$	0	0
$299$	10881.3	2.10462
$300$	0	0
$301$	−304.983	−0.0584018
$302$	0	0
$303$	2116.11	0.401213
$304$	0	0
$305$	502.669	0.0943696
$306$	0	0
$307$	−645.184	−0.119943	−0.0599717	−	0.998200i	$-0.519101\pi$
$307$	−645.184	−0.119943	−0.0599717	+	0.998200i	$0.519101\pi$
$308$	0	0
$309$	1267.57	0.233364
$310$	0	0
$311$	7930.96	1.44606	0.723028	−	0.690819i	$-0.242750\pi$
$311$	7930.96	1.44606	0.723028	+	0.690819i	$0.242750\pi$
$312$	0	0
$313$	6600.26	1.19191	0.595956	−	0.803017i	$-0.296773\pi$
$313$	6600.26	1.19191	0.595956	+	0.803017i	$0.296773\pi$
$314$	0	0
$315$	−3667.49	−0.656000
$316$	0	0
$317$	−10260.0	−1.81785	−0.908923	−	0.416963i	$-0.863094\pi$
$317$	−10260.0	−1.81785	−0.908923	+	0.416963i	$0.863094\pi$
$318$	0	0
$319$	−850.257	−0.149233
$320$	0	0
$321$	248.156	0.0431487
$322$	0	0
$323$	−7828.73	−1.34861
$324$	0	0
$325$	19765.7	3.37356
$326$	0	0
$327$	1146.59	0.193904
$328$	0	0
$329$	−484.194	−0.0811383
$330$	0	0
$331$	−9625.35	−1.59836	−0.799180	−	0.601092i	$-0.794732\pi$
$331$	−9625.35	−1.59836	−0.799180	+	0.601092i	$0.794732\pi$
$332$	0	0
$333$	5494.07	0.904124
$334$	0	0
$335$	5856.72	0.955185
$336$	0	0
$337$	5877.46	0.950046	0.475023	−	0.879973i	$-0.342440\pi$
$337$	5877.46	0.950046	0.475023	+	0.879973i	$0.342440\pi$
$338$	0	0
$339$	−2313.50	−0.370656
$340$	0	0
$341$	2269.94	0.360482
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	−4065.40	−0.634416
$346$	0	0
$347$	−10051.4	−1.55501	−0.777506	−	0.628875i	$-0.783516\pi$
$347$	−10051.4	−1.55501	−0.777506	+	0.628875i	$0.783516\pi$
$348$	0	0
$349$	−12242.2	−1.87768	−0.938839	−	0.344356i	$-0.888097\pi$
$349$	−12242.2	−1.87768	−0.938839	+	0.344356i	$0.888097\pi$
$350$	0	0
$351$	−4254.63	−0.646996
$352$	0	0
$353$	−11468.0	−1.72912	−0.864559	−	0.502532i	$-0.832402\pi$
$353$	−11468.0	−1.72912	−0.864559	+	0.502532i	$0.832402\pi$
$354$	0	0
$355$	10718.5	1.60248
$356$	0	0
$357$	507.682	0.0752644
$358$	0	0
$359$	−8566.44	−1.25939	−0.629693	−	0.776844i	$-0.716819\pi$
$359$	−8566.44	−1.25939	−0.629693	+	0.776844i	$0.716819\pi$
$360$	0	0
$361$	10320.6	1.50468
$362$	0	0
$363$	146.925	0.0212439
$364$	0	0
$365$	−12160.3	−1.74383
$366$	0	0
$367$	6925.30	0.985008	0.492504	−	0.870310i	$-0.336082\pi$
$367$	6925.30	0.985008	0.492504	+	0.870310i	$0.336082\pi$
$368$	0	0
$369$	10292.4	1.45203
$370$	0	0
$371$	−2344.78	−0.328127
$372$	0	0
$373$	5752.39	0.798518	0.399259	−	0.916838i	$-0.369267\pi$
$373$	5752.39	0.798518	0.399259	+	0.916838i	$0.369267\pi$
$374$	0	0
$375$	−4269.35	−0.587915
$376$	0	0
$377$	−5156.31	−0.704412
$378$	0	0
$379$	−10010.2	−1.35670	−0.678348	−	0.734740i	$-0.737304\pi$
$379$	−10010.2	−1.35670	−0.678348	+	0.734740i	$0.737304\pi$
$380$	0	0
$381$	665.034	0.0894245
$382$	0	0
$383$	4686.10	0.625192	0.312596	−	0.949886i	$-0.398801\pi$
$383$	4686.10	0.625192	0.312596	+	0.949886i	$0.398801\pi$
$384$	0	0
$385$	1580.47	0.209216
$386$	0	0
$387$	−1112.13	−0.146079
$388$	0	0
$389$	−11018.9	−1.43619	−0.718095	−	0.695945i	$-0.754986\pi$
$389$	−11018.9	−1.43619	−0.718095	+	0.695945i	$0.754986\pi$
$390$	0	0
$391$	−9742.77	−1.26014
$392$	0	0
$393$	−2339.37	−0.300268
$394$	0	0
$395$	1904.75	0.242629
$396$	0	0
$397$	−9780.07	−1.23639	−0.618196	−	0.786024i	$-0.712136\pi$
$397$	−9780.07	−1.23639	−0.618196	+	0.786024i	$0.712136\pi$
$398$	0	0
$399$	−1114.07	−0.139783
$400$	0	0
$401$	5288.90	0.658641	0.329320	−	0.944218i	$-0.393180\pi$
$401$	5288.90	0.658641	0.329320	+	0.944218i	$0.393180\pi$
$402$	0	0
$403$	13765.9	1.70156
$404$	0	0
$405$	−12556.5	−1.54058
$406$	0	0
$407$	−2367.62	−0.288350
$408$	0	0
$409$	7276.22	0.879672	0.439836	−	0.898078i	$-0.355037\pi$
$409$	7276.22	0.879672	0.439836	+	0.898078i	$0.355037\pi$
$410$	0	0
$411$	−1749.46	−0.209962
$412$	0	0
$413$	−4768.98	−0.568199
$414$	0	0
$415$	−9871.32	−1.16762
$416$	0	0
$417$	−2951.61	−0.346621
$418$	0	0
$419$	−11032.7	−1.28636	−0.643178	−	0.765717i	$-0.722384\pi$
$419$	−11032.7	−1.28636	−0.643178	+	0.765717i	$0.722384\pi$
$420$	0	0
$421$	16046.8	1.85765	0.928826	−	0.370517i	$-0.120820\pi$
$421$	16046.8	1.85765	0.928826	+	0.370517i	$0.120820\pi$
$422$	0	0
$423$	−1765.62	−0.202949
$424$	0	0
$425$	−17697.7	−2.01991
$426$	0	0
$427$	171.429	0.0194286
$428$	0	0
$429$	891.012	0.100276
$430$	0	0
$431$	−6726.13	−0.751709	−0.375854	−	0.926679i	$-0.622651\pi$
$431$	−6726.13	−0.751709	−0.375854	+	0.926679i	$0.622651\pi$
$432$	0	0
$433$	4622.57	0.513041	0.256520	−	0.966539i	$-0.417424\pi$
$433$	4622.57	0.513041	0.256520	+	0.966539i	$0.417424\pi$
$434$	0	0
$435$	1926.47	0.212338
$436$	0	0
$437$	21379.9	2.34036
$438$	0	0
$439$	11917.6	1.29566	0.647830	−	0.761785i	$-0.275677\pi$
$439$	11917.6	1.29566	0.647830	+	0.761785i	$0.275677\pi$
$440$	0	0
$441$	−1250.75	−0.135056
$442$	0	0
$443$	5606.28	0.601270	0.300635	−	0.953739i	$-0.402801\pi$
$443$	5606.28	0.601270	0.300635	+	0.953739i	$0.402801\pi$
$444$	0	0
$445$	−24569.6	−2.61732
$446$	0	0
$447$	833.906	0.0882381
$448$	0	0
$449$	−2144.37	−0.225388	−0.112694	−	0.993630i	$-0.535948\pi$
$449$	−2144.37	−0.225388	−0.112694	+	0.993630i	$0.535948\pi$
$450$	0	0
$451$	−4435.40	−0.463092
$452$	0	0
$453$	2136.42	0.221584
$454$	0	0
$455$	9584.63	0.987548
$456$	0	0
$457$	8524.38	0.872546	0.436273	−	0.899814i	$-0.356298\pi$
$457$	8524.38	0.872546	0.436273	+	0.899814i	$0.356298\pi$
$458$	0	0
$459$	3809.47	0.387388
$460$	0	0
$461$	−11768.7	−1.18899	−0.594494	−	0.804100i	$-0.702647\pi$
$461$	−11768.7	−1.18899	−0.594494	+	0.804100i	$0.702647\pi$
$462$	0	0
$463$	393.751	0.0395230	0.0197615	−	0.999805i	$-0.493709\pi$
$463$	393.751	0.0395230	0.0197615	+	0.999805i	$0.493709\pi$
$464$	0	0
$465$	−5143.13	−0.512918
$466$	0	0
$467$	−3258.65	−0.322896	−0.161448	−	0.986881i	$-0.551616\pi$
$467$	−3258.65	−0.322896	−0.161448	+	0.986881i	$0.551616\pi$
$468$	0	0
$469$	1997.36	0.196652
$470$	0	0
$471$	2226.76	0.217842
$472$	0	0
$473$	479.259	0.0465885
$474$	0	0
$475$	38836.3	3.75144
$476$	0	0
$477$	−8550.29	−0.820735
$478$	0	0
$479$	7913.65	0.754873	0.377436	−	0.926036i	$-0.376806\pi$
$479$	7913.65	0.754873	0.377436	+	0.926036i	$0.376806\pi$
$480$	0	0
$481$	−14358.2	−1.36108
$482$	0	0
$483$	−1386.45	−0.130612
$484$	0	0
$485$	−6341.06	−0.593676
$486$	0	0
$487$	5667.52	0.527351	0.263676	−	0.964611i	$-0.415065\pi$
$487$	5667.52	0.527351	0.263676	+	0.964611i	$0.415065\pi$
$488$	0	0
$489$	−2800.32	−0.258967
$490$	0	0
$491$	−389.830	−0.0358305	−0.0179152	−	0.999840i	$-0.505703\pi$
$491$	−389.830	−0.0358305	−0.0179152	+	0.999840i	$0.505703\pi$
$492$	0	0
$493$	4616.81	0.421766
$494$	0	0
$495$	5763.20	0.523307
$496$	0	0
$497$	3655.43	0.329916
$498$	0	0
$499$	13968.3	1.25312	0.626558	−	0.779375i	$-0.284463\pi$
$499$	13968.3	1.25312	0.626558	+	0.779375i	$0.284463\pi$
$500$	0	0
$501$	2724.14	0.242925
$502$	0	0
$503$	15152.6	1.34318	0.671591	−	0.740922i	$-0.265611\pi$
$503$	15152.6	1.34318	0.671591	+	0.740922i	$0.265611\pi$
$504$	0	0
$505$	−35770.5	−3.15201
$506$	0	0
$507$	2735.75	0.239643
$508$	0	0
$509$	−11774.4	−1.02532	−0.512662	−	0.858591i	$-0.671341\pi$
$509$	−11774.4	−1.02532	−0.512662	+	0.858591i	$0.671341\pi$
$510$	0	0
$511$	−4147.11	−0.359016
$512$	0	0
$513$	−8359.63	−0.719467
$514$	0	0
$515$	−21426.9	−1.83336
$516$	0	0
$517$	760.877	0.0647259
$518$	0	0
$519$	−3034.51	−0.256648
$520$	0	0
$521$	−7972.13	−0.670375	−0.335188	−	0.942151i	$-0.608800\pi$
$521$	−7972.13	−0.670375	−0.335188	+	0.942151i	$0.608800\pi$
$522$	0	0
$523$	−20833.0	−1.74181	−0.870904	−	0.491454i	$-0.836466\pi$
$523$	−20833.0	−1.74181	−0.870904	+	0.491454i	$0.836466\pi$
$524$	0	0
$525$	−2518.48	−0.209363
$526$	0	0
$527$	−12325.6	−1.01880
$528$	0	0
$529$	14440.0	1.18682
$530$	0	0
$531$	−17390.1	−1.42122
$532$	0	0
$533$	−26898.1	−2.18590
$534$	0	0
$535$	−4194.80	−0.338985
$536$	0	0
$537$	328.777	0.0264204
$538$	0	0
$539$	539.000	0.0430730
$540$	0	0
$541$	9288.86	0.738187	0.369093	−	0.929392i	$-0.379668\pi$
$541$	9288.86	0.738187	0.369093	+	0.929392i	$0.379668\pi$
$542$	0	0
$543$	1797.81	0.142084
$544$	0	0
$545$	−19381.8	−1.52335
$546$	0	0
$547$	22942.9	1.79336	0.896681	−	0.442678i	$-0.145972\pi$
$547$	22942.9	1.79336	0.896681	+	0.442678i	$0.145972\pi$
$548$	0	0
$549$	625.118	0.0485963
$550$	0	0
$551$	−10131.3	−0.783316
$552$	0	0
$553$	649.592	0.0499520
$554$	0	0
$555$	5364.43	0.410284
$556$	0	0
$557$	−4841.26	−0.368278	−0.184139	−	0.982900i	$-0.558950\pi$
$557$	−4841.26	−0.368278	−0.184139	+	0.982900i	$0.558950\pi$
$558$	0	0
$559$	2906.43	0.219908
$560$	0	0
$561$	−797.787	−0.0600402
$562$	0	0
$563$	−2784.32	−0.208428	−0.104214	−	0.994555i	$-0.533233\pi$
$563$	−2784.32	−0.208428	−0.104214	+	0.994555i	$0.533233\pi$
$564$	0	0
$565$	39107.2	2.91195
$566$	0	0
$567$	−4282.23	−0.317172
$568$	0	0
$569$	−2229.27	−0.164246	−0.0821228	−	0.996622i	$-0.526170\pi$
$569$	−2229.27	−0.164246	−0.0821228	+	0.996622i	$0.526170\pi$
$570$	0	0
$571$	−6752.10	−0.494863	−0.247431	−	0.968905i	$-0.579586\pi$
$571$	−6752.10	−0.494863	−0.247431	+	0.968905i	$0.579586\pi$
$572$	0	0
$573$	2988.29	0.217867
$574$	0	0
$575$	48331.4	3.50532
$576$	0	0
$577$	−18454.6	−1.33150	−0.665749	−	0.746175i	$-0.731888\pi$
$577$	−18454.6	−1.33150	−0.665749	+	0.746175i	$0.731888\pi$
$578$	0	0
$579$	−1723.95	−0.123739
$580$	0	0
$581$	−3366.49	−0.240388
$582$	0	0
$583$	3684.66	0.261755
$584$	0	0
$585$	34950.5	2.47013
$586$	0	0
$587$	11807.7	0.830250	0.415125	−	0.909764i	$-0.363738\pi$
$587$	11807.7	0.830250	0.415125	+	0.909764i	$0.363738\pi$
$588$	0	0
$589$	27047.6	1.89215
$590$	0	0
$591$	3543.53	0.246635
$592$	0	0
$593$	−2300.54	−0.159311	−0.0796557	−	0.996822i	$-0.525382\pi$
$593$	−2300.54	−0.159311	−0.0796557	+	0.996822i	$0.525382\pi$
$594$	0	0
$595$	−8581.80	−0.591293
$596$	0	0
$597$	4574.82	0.313626
$598$	0	0
$599$	−6654.41	−0.453909	−0.226955	−	0.973905i	$-0.572877\pi$
$599$	−6654.41	−0.453909	−0.226955	+	0.973905i	$0.572877\pi$
$600$	0	0
$601$	−10631.7	−0.721588	−0.360794	−	0.932646i	$-0.617494\pi$
$601$	−10631.7	−0.721588	−0.360794	+	0.932646i	$0.617494\pi$
$602$	0	0
$603$	7283.41	0.491880
$604$	0	0
$605$	−2483.60	−0.166897
$606$	0	0
$607$	14834.8	0.991971	0.495985	−	0.868331i	$-0.334807\pi$
$607$	14834.8	0.991971	0.495985	+	0.868331i	$0.334807\pi$
$608$	0	0
$609$	656.999	0.0437158
$610$	0	0
$611$	4614.27	0.305521
$612$	0	0
$613$	−21769.1	−1.43433	−0.717167	−	0.696902i	$-0.754561\pi$
$613$	−21769.1	−1.43433	−0.717167	+	0.696902i	$0.754561\pi$
$614$	0	0
$615$	10049.5	0.658919
$616$	0	0
$617$	11615.5	0.757899	0.378949	−	0.925417i	$-0.376285\pi$
$617$	11615.5	0.757899	0.378949	+	0.925417i	$0.376285\pi$
$618$	0	0
$619$	25928.4	1.68360	0.841800	−	0.539789i	$-0.181496\pi$
$619$	25928.4	1.68360	0.841800	+	0.539789i	$0.181496\pi$
$620$	0	0
$621$	−10403.5	−0.672266
$622$	0	0
$623$	−8379.15	−0.538850
$624$	0	0
$625$	35131.1	2.24839
$626$	0	0
$627$	1750.69	0.111508
$628$	0	0
$629$	12855.9	0.814943
$630$	0	0
$631$	−1291.82	−0.0815003	−0.0407502	−	0.999169i	$-0.512975\pi$
$631$	−1291.82	−0.0815003	−0.0407502	+	0.999169i	$0.512975\pi$
$632$	0	0
$633$	−4329.00	−0.271820
$634$	0	0
$635$	−11241.7	−0.702538
$636$	0	0
$637$	3268.72	0.203315
$638$	0	0
$639$	13329.6	0.825210
$640$	0	0
$641$	−5836.16	−0.359617	−0.179808	−	0.983702i	$-0.557548\pi$
$641$	−5836.16	−0.359617	−0.179808	+	0.983702i	$0.557548\pi$
$642$	0	0
$643$	−1068.38	−0.0655254	−0.0327627	−	0.999463i	$-0.510431\pi$
$643$	−1068.38	−0.0655254	−0.0327627	+	0.999463i	$0.510431\pi$
$644$	0	0
$645$	−1085.88	−0.0662893
$646$	0	0
$647$	−14064.4	−0.854607	−0.427303	−	0.904108i	$-0.640536\pi$
$647$	−14064.4	−0.854607	−0.427303	+	0.904108i	$0.640536\pi$
$648$	0	0
$649$	7494.11	0.453266
$650$	0	0
$651$	−1754.00	−0.105599
$652$	0	0
$653$	23428.7	1.40404	0.702018	−	0.712160i	$-0.252283\pi$
$653$	23428.7	1.40404	0.702018	+	0.712160i	$0.252283\pi$
$654$	0	0
$655$	39544.3	2.35897
$656$	0	0
$657$	−15122.5	−0.897997
$658$	0	0
$659$	23122.9	1.36683	0.683414	−	0.730031i	$-0.260494\pi$
$659$	23122.9	1.36683	0.683414	+	0.730031i	$0.260494\pi$
$660$	0	0
$661$	−30074.8	−1.76970	−0.884851	−	0.465873i	$-0.845740\pi$
$661$	−30074.8	−1.76970	−0.884851	+	0.465873i	$0.845740\pi$
$662$	0	0
$663$	−4838.11	−0.283404
$664$	0	0
$665$	18832.2	1.09817
$666$	0	0
$667$	−12608.3	−0.731925
$668$	0	0
$669$	3073.62	0.177628
$670$	0	0
$671$	−269.388	−0.0154987
$672$	0	0
$673$	−27970.6	−1.60206	−0.801032	−	0.598621i	$-0.795715\pi$
$673$	−27970.6	−1.60206	−0.801032	+	0.598621i	$0.795715\pi$
$674$	0	0
$675$	−18897.8	−1.07760
$676$	0	0
$677$	−3911.49	−0.222054	−0.111027	−	0.993817i	$-0.535414\pi$
$677$	−3911.49	−0.222054	−0.111027	+	0.993817i	$0.535414\pi$
$678$	0	0
$679$	−2162.54	−0.122225
$680$	0	0
$681$	1466.17	0.0825019
$682$	0	0
$683$	−14125.7	−0.791367	−0.395683	−	0.918387i	$-0.629492\pi$
$683$	−14125.7	−0.791367	−0.395683	+	0.918387i	$0.629492\pi$
$684$	0	0
$685$	29572.6	1.64951
$686$	0	0
$687$	−4855.43	−0.269645
$688$	0	0
$689$	22345.3	1.23554
$690$	0	0
$691$	20549.5	1.13132	0.565658	−	0.824640i	$-0.308622\pi$
$691$	20549.5	1.13132	0.565658	+	0.824640i	$0.308622\pi$
$692$	0	0
$693$	1965.47	0.107737
$694$	0	0
$695$	49893.7	2.72313
$696$	0	0
$697$	24083.8	1.30881
$698$	0	0
$699$	1735.93	0.0939328
$700$	0	0
$701$	24071.4	1.29695	0.648475	−	0.761236i	$-0.275407\pi$
$701$	24071.4	1.29695	0.648475	+	0.761236i	$0.275407\pi$
$702$	0	0
$703$	−28211.5	−1.51354
$704$	0	0
$705$	−1723.96	−0.0920964
$706$	0	0
$707$	−12199.1	−0.648930
$708$	0	0
$709$	−29125.6	−1.54278	−0.771392	−	0.636360i	$-0.780439\pi$
$709$	−29125.6	−1.54278	−0.771392	+	0.636360i	$0.780439\pi$
$710$	0	0
$711$	2368.75	0.124944
$712$	0	0
$713$	33660.5	1.76801
$714$	0	0
$715$	−15061.6	−0.787791
$716$	0	0
$717$	3913.55	0.203841
$718$	0	0
$719$	−34567.0	−1.79295	−0.896475	−	0.443094i	$-0.853881\pi$
$719$	−34567.0	−1.79295	−0.896475	+	0.443094i	$0.853881\pi$
$720$	0	0
$721$	−7307.36	−0.377449
$722$	0	0
$723$	704.958	0.0362623
$724$	0	0
$725$	−22902.8	−1.17323
$726$	0	0
$727$	−16450.2	−0.839206	−0.419603	−	0.907708i	$-0.637831\pi$
$727$	−16450.2	−0.839206	−0.419603	+	0.907708i	$0.637831\pi$
$728$	0	0
$729$	−13524.2	−0.687100
$730$	0	0
$731$	−2602.33	−0.131670
$732$	0	0
$733$	−21032.1	−1.05981	−0.529904	−	0.848058i	$-0.677772\pi$
$733$	−21032.1	−1.05981	−0.529904	+	0.848058i	$0.677772\pi$
$734$	0	0
$735$	−1221.24	−0.0612872
$736$	0	0
$737$	−3138.71	−0.156874
$738$	0	0
$739$	−33830.2	−1.68398	−0.841991	−	0.539491i	$-0.818617\pi$
$739$	−33830.2	−1.68398	−0.841991	+	0.539491i	$0.818617\pi$
$740$	0	0
$741$	10616.9	0.526345
$742$	0	0
$743$	30148.6	1.48862	0.744311	−	0.667833i	$-0.232778\pi$
$743$	30148.6	1.48862	0.744311	+	0.667833i	$0.232778\pi$
$744$	0	0
$745$	−14096.2	−0.693217
$746$	0	0
$747$	−12276.0	−0.601277
$748$	0	0
$749$	−1430.58	−0.0697896
$750$	0	0
$751$	−25821.0	−1.25462	−0.627311	−	0.778769i	$-0.715844\pi$
$751$	−25821.0	−1.25462	−0.627311	+	0.778769i	$0.715844\pi$
$752$	0	0
$753$	1550.90	0.0750568
$754$	0	0
$755$	−36113.8	−1.74081
$756$	0	0
$757$	8504.19	0.408309	0.204155	−	0.978939i	$-0.434555\pi$
$757$	8504.19	0.408309	0.204155	+	0.978939i	$0.434555\pi$
$758$	0	0
$759$	2178.71	0.104193
$760$	0	0
$761$	4597.99	0.219024	0.109512	−	0.993985i	$-0.465071\pi$
$761$	4597.99	0.219024	0.109512	+	0.993985i	$0.465071\pi$
$762$	0	0
$763$	−6609.93	−0.313624
$764$	0	0
$765$	−31293.6	−1.47899
$766$	0	0
$767$	45447.4	2.13952
$768$	0	0
$769$	38871.6	1.82282	0.911408	−	0.411505i	$-0.134997\pi$
$769$	38871.6	1.82282	0.911408	+	0.411505i	$0.134997\pi$
$770$	0	0
$771$	3056.39	0.142767
$772$	0	0
$773$	21974.1	1.02245	0.511224	−	0.859447i	$-0.329192\pi$
$773$	21974.1	1.02245	0.511224	+	0.859447i	$0.329192\pi$
$774$	0	0
$775$	61144.0	2.83401
$776$	0	0
$777$	1829.47	0.0844685
$778$	0	0
$779$	−52850.2	−2.43075
$780$	0	0
$781$	−5744.24	−0.263182
$782$	0	0
$783$	4929.90	0.225007
$784$	0	0
$785$	−37640.8	−1.71141
$786$	0	0
$787$	−14241.7	−0.645061	−0.322531	−	0.946559i	$-0.604534\pi$
$787$	−14241.7	−0.645061	−0.322531	+	0.946559i	$0.604534\pi$
$788$	0	0
$789$	−6902.33	−0.311444
$790$	0	0
$791$	13337.0	0.599507
$792$	0	0
$793$	−1633.68	−0.0731574
$794$	0	0
$795$	−8348.53	−0.372443
$796$	0	0
$797$	39470.7	1.75424	0.877118	−	0.480276i	$-0.159463\pi$
$797$	39470.7	1.75424	0.877118	+	0.480276i	$0.159463\pi$
$798$	0	0
$799$	−4131.48	−0.182930
$800$	0	0
$801$	−30554.7	−1.34781
$802$	0	0
$803$	6516.88	0.286396
$804$	0	0
$805$	23436.4	1.02612
$806$	0	0
$807$	1056.49	0.0460846
$808$	0	0
$809$	11073.5	0.481238	0.240619	−	0.970620i	$-0.422650\pi$
$809$	11073.5	0.481238	0.240619	+	0.970620i	$0.422650\pi$
$810$	0	0
$811$	−38855.7	−1.68238	−0.841189	−	0.540741i	$-0.818144\pi$
$811$	−38855.7	−1.68238	−0.841189	+	0.540741i	$0.818144\pi$
$812$	0	0
$813$	231.885	0.0100032
$814$	0	0
$815$	47336.2	2.03450
$816$	0	0
$817$	5710.64	0.244541
$818$	0	0
$819$	11919.4	0.508545
$820$	0	0
$821$	−9523.67	−0.404846	−0.202423	−	0.979298i	$-0.564882\pi$
$821$	−9523.67	−0.404846	−0.202423	+	0.979298i	$0.564882\pi$
$822$	0	0
$823$	−2542.13	−0.107671	−0.0538355	−	0.998550i	$-0.517145\pi$
$823$	−2542.13	−0.107671	−0.0538355	+	0.998550i	$0.517145\pi$
$824$	0	0
$825$	3957.61	0.167014
$826$	0	0
$827$	8794.30	0.369779	0.184890	−	0.982759i	$-0.440807\pi$
$827$	8794.30	0.369779	0.184890	+	0.982759i	$0.440807\pi$
$828$	0	0
$829$	−18268.3	−0.765362	−0.382681	−	0.923881i	$-0.624999\pi$
$829$	−18268.3	−0.765362	−0.382681	+	0.923881i	$0.624999\pi$
$830$	0	0
$831$	6801.36	0.283919
$832$	0	0
$833$	−2926.72	−0.121734
$834$	0	0
$835$	−46048.5	−1.90847
$836$	0	0
$837$	−13161.4	−0.543519
$838$	0	0
$839$	6216.04	0.255783	0.127891	−	0.991788i	$-0.459179\pi$
$839$	6216.04	0.255783	0.127891	+	0.991788i	$0.459179\pi$
$840$	0	0
$841$	−18414.3	−0.755026
$842$	0	0
$843$	8352.02	0.341232
$844$	0	0
$845$	−46244.8	−1.88269
$846$	0	0
$847$	−847.000	−0.0343604
$848$	0	0
$849$	6948.14	0.280871
$850$	0	0
$851$	−35108.9	−1.41424
$852$	0	0
$853$	3081.20	0.123679	0.0618396	−	0.998086i	$-0.480303\pi$
$853$	3081.20	0.123679	0.0618396	+	0.998086i	$0.480303\pi$
$854$	0	0
$855$	68671.8	2.74681
$856$	0	0
$857$	2992.90	0.119295	0.0596473	−	0.998220i	$-0.481002\pi$
$857$	2992.90	0.119295	0.0596473	+	0.998220i	$0.481002\pi$
$858$	0	0
$859$	1740.43	0.0691300	0.0345650	−	0.999402i	$-0.488995\pi$
$859$	1740.43	0.0691300	0.0345650	+	0.999402i	$0.488995\pi$
$860$	0	0
$861$	3427.26	0.135657
$862$	0	0
$863$	−35296.5	−1.39225	−0.696123	−	0.717923i	$-0.745093\pi$
$863$	−35296.5	−1.39225	−0.696123	+	0.717923i	$0.745093\pi$
$864$	0	0
$865$	51294.9	2.01628
$866$	0	0
$867$	−1633.73	−0.0639956
$868$	0	0
$869$	−1020.79	−0.0398479
$870$	0	0
$871$	−19034.5	−0.740480
$872$	0	0
$873$	−7885.73	−0.305718
$874$	0	0
$875$	24612.2	0.950908
$876$	0	0
$877$	1103.09	0.0424727	0.0212364	−	0.999774i	$-0.493240\pi$
$877$	1103.09	0.0424727	0.0212364	+	0.999774i	$0.493240\pi$
$878$	0	0
$879$	10052.2	0.385725
$880$	0	0
$881$	−37691.0	−1.44136	−0.720682	−	0.693266i	$-0.756171\pi$
$881$	−37691.0	−1.44136	−0.720682	+	0.693266i	$0.756171\pi$
$882$	0	0
$883$	23031.4	0.877769	0.438884	−	0.898544i	$-0.355374\pi$
$883$	23031.4	0.877769	0.438884	+	0.898544i	$0.355374\pi$
$884$	0	0
$885$	−16979.8	−0.644937
$886$	0	0
$887$	47839.7	1.81094	0.905468	−	0.424414i	$-0.139520\pi$
$887$	47839.7	1.81094	0.905468	+	0.424414i	$0.139520\pi$
$888$	0	0
$889$	−3833.83	−0.144637
$890$	0	0
$891$	6729.21	0.253016
$892$	0	0
$893$	9066.26	0.339744
$894$	0	0
$895$	−5557.60	−0.207564
$896$	0	0
$897$	13212.6	0.491813
$898$	0	0
$899$	−15950.7	−0.591752
$900$	0	0
$901$	−20007.4	−0.739780
$902$	0	0
$903$	−370.327	−0.0136475
$904$	0	0
$905$	−30389.9	−1.11624
$906$	0	0
$907$	35960.6	1.31649	0.658243	−	0.752805i	$-0.271300\pi$
$907$	35960.6	1.31649	0.658243	+	0.752805i	$0.271300\pi$
$908$	0	0
$909$	−44484.1	−1.62315
$910$	0	0
$911$	−25190.4	−0.916132	−0.458066	−	0.888918i	$-0.651458\pi$
$911$	−25190.4	−0.916132	−0.458066	+	0.888918i	$0.651458\pi$
$912$	0	0
$913$	5290.20	0.191764
$914$	0	0
$915$	610.367	0.0220526
$916$	0	0
$917$	13486.1	0.485660
$918$	0	0
$919$	−42686.9	−1.53222	−0.766111	−	0.642709i	$-0.777811\pi$
$919$	−42686.9	−1.53222	−0.766111	+	0.642709i	$0.777811\pi$
$920$	0	0
$921$	−783.417	−0.0280287
$922$	0	0
$923$	−34835.5	−1.24228
$924$	0	0
$925$	−63774.9	−2.26693
$926$	0	0
$927$	−26646.4	−0.944102
$928$	0	0
$929$	29206.6	1.03147	0.515736	−	0.856748i	$-0.327519\pi$
$929$	29206.6	1.03147	0.515736	+	0.856748i	$0.327519\pi$
$930$	0	0
$931$	6422.48	0.226088
$932$	0	0
$933$	9630.19	0.337919
$934$	0	0
$935$	13485.7	0.471689
$936$	0	0
$937$	−27326.2	−0.952730	−0.476365	−	0.879248i	$-0.658046\pi$
$937$	−27326.2	−0.952730	−0.476365	+	0.879248i	$0.658046\pi$
$938$	0	0
$939$	8014.39	0.278530
$940$	0	0
$941$	6802.35	0.235654	0.117827	−	0.993034i	$-0.462407\pi$
$941$	6802.35	0.235654	0.117827	+	0.993034i	$0.462407\pi$
$942$	0	0
$943$	−65771.5	−2.27128
$944$	0	0
$945$	−9163.77	−0.315447
$946$	0	0
$947$	1090.60	0.0374231	0.0187116	−	0.999825i	$-0.494044\pi$
$947$	1090.60	0.0374231	0.0187116	+	0.999825i	$0.494044\pi$
$948$	0	0
$949$	39521.1	1.35185
$950$	0	0
$951$	−12458.2	−0.424800
$952$	0	0
$953$	48392.3	1.64489	0.822446	−	0.568843i	$-0.192609\pi$
$953$	48392.3	1.64489	0.822446	+	0.568843i	$0.192609\pi$
$954$	0	0
$955$	−50513.7	−1.71161
$956$	0	0
$957$	−1032.43	−0.0348732
$958$	0	0
$959$	10085.4	0.339598
$960$	0	0
$961$	12792.8	0.429418
$962$	0	0
$963$	−5216.65	−0.174563
$964$	0	0
$965$	29141.5	0.972123
$966$	0	0
$967$	27678.0	0.920438	0.460219	−	0.887805i	$-0.347771\pi$
$967$	27678.0	0.920438	0.460219	+	0.887805i	$0.347771\pi$
$968$	0	0
$969$	−9506.07	−0.315149
$970$	0	0
$971$	−17642.8	−0.583096	−0.291548	−	0.956556i	$-0.594170\pi$
$971$	−17642.8	−0.583096	−0.291548	+	0.956556i	$0.594170\pi$
$972$	0	0
$973$	17015.6	0.560633
$974$	0	0
$975$	24000.6	0.788343
$976$	0	0
$977$	−4539.48	−0.148650	−0.0743249	−	0.997234i	$-0.523680\pi$
$977$	−4539.48	−0.148650	−0.0743249	+	0.997234i	$0.523680\pi$
$978$	0	0
$979$	13167.2	0.429853
$980$	0	0
$981$	−24103.2	−0.784460
$982$	0	0
$983$	7250.49	0.235254	0.117627	−	0.993058i	$-0.462471\pi$
$983$	7250.49	0.235254	0.117627	+	0.993058i	$0.462471\pi$
$984$	0	0
$985$	−59899.4	−1.93762
$986$	0	0
$987$	−587.935	−0.0189607
$988$	0	0
$989$	7106.83	0.228497
$990$	0	0
$991$	851.834	0.0273052	0.0136526	−	0.999907i	$-0.495654\pi$
$991$	851.834	0.0273052	0.0136526	+	0.999907i	$0.495654\pi$
$992$	0	0
$993$	−11687.6	−0.373510
$994$	0	0
$995$	−77332.1	−2.46391
$996$	0	0
$997$	52462.3	1.66650	0.833249	−	0.552897i	$-0.186478\pi$
$997$	52462.3	1.66650	0.833249	+	0.552897i	$0.186478\pi$
$998$	0	0
$999$	13727.7	0.434761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	308.4.a.d.1.3	✓	4
4.3	odd	2		1232.4.a.v.1.2		4
7.6	odd	2		2156.4.a.f.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
308.4.a.d.1.3	✓	4	1.1	even	1	trivial
1232.4.a.v.1.2		4	4.3	odd	2
2156.4.a.f.1.2		4	7.6	odd	2

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:	\( N \)	\(=\)	\( 308 = 2^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	308.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21425 q^{3} -20.5256 q^{5} -7.00000 q^{7} -25.5256 q^{9} +11.0000 q^{11} +66.7086 q^{13} -24.9233 q^{15} -59.7289 q^{17} +131.071 q^{19} -8.49978 q^{21} +163.117 q^{23} +296.300 q^{25} -63.7794 q^{27} -77.2961 q^{29} +206.358 q^{31} +13.3568 q^{33} +143.679 q^{35} -215.238 q^{37} +81.0011 q^{39} -403.218 q^{41} +43.5690 q^{43} +523.928 q^{45} +69.1706 q^{47} +49.0000 q^{49} -72.5261 q^{51} +334.969 q^{53} -225.781 q^{55} +159.154 q^{57} +681.283 q^{59} -24.4899 q^{61} +178.679 q^{63} -1369.23 q^{65} -285.338 q^{67} +198.065 q^{69} -522.204 q^{71} +592.444 q^{73} +359.783 q^{75} -77.0000 q^{77} -92.7989 q^{79} +611.747 q^{81} +480.928 q^{83} +1225.97 q^{85} -93.8570 q^{87} +1197.02 q^{89} -466.960 q^{91} +250.571 q^{93} -2690.31 q^{95} +308.934 q^{97} -280.781 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + q^{5} - 28 q^{7} - 19 q^{9} + 44 q^{11} + 98 q^{13} - 33 q^{15} + 64 q^{17} + 114 q^{19} + 21 q^{21} + 231 q^{23} + 417 q^{25} + 63 q^{27} + 268 q^{29} + 33 q^{31} - 33 q^{33} - 7 q^{35}+ \cdots - 209 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + q^5 - 28 * q^7 - 19 * q^9 + 44 * q^11 + 98 * q^13 - 33 * q^15 + 64 * q^17 + 114 * q^19 + 21 * q^21 + 231 * q^23 + 417 * q^25 + 63 * q^27 + 268 * q^29 + 33 * q^31 - 33 * q^33 - 7 * q^35 + 357 * q^37 + 140 * q^39 + 364 * q^41 + 44 * q^43 + 912 * q^45 + 720 * q^47 + 196 * q^49 + 794 * q^51 + 740 * q^53 + 11 * q^55 + 138 * q^57 + 787 * q^59 + 1020 * q^61 + 133 * q^63 + 296 * q^65 - 995 * q^67 + 1059 * q^69 - 1011 * q^71 + 1592 * q^73 - 324 * q^75 - 308 * q^77 - 178 * q^79 - 1396 * q^81 + 324 * q^83 - 94 * q^85 - 1262 * q^87 + 19 * q^89 - 686 * q^91 + 637 * q^93 - 2418 * q^95 - 555 * q^97 - 209 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

Properties

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