LMFDB - Embedded newform 294.4.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,4,Mod(1,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("294.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 294 = 2 \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	294.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:	$17.3465615417$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{1345}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 336 $ x^2 - x - 336
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.8371$ of defining polynomial
Character	$\chi$	$=$	294.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +20.8371 q^{5} +6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} +41.6742 q^{10} +15.1629 q^{11} +12.0000 q^{12} +2.16288 q^{13} +62.5114 q^{15} +16.0000 q^{16} -119.348 q^{17} +18.0000 q^{18} -33.5114 q^{19} +83.3485 q^{20} +30.3258 q^{22} +0.651517 q^{23} +24.0000 q^{24} +309.186 q^{25} +4.32576 q^{26} +27.0000 q^{27} -163.208 q^{29} +125.023 q^{30} -223.326 q^{31} +32.0000 q^{32} +45.4886 q^{33} -238.697 q^{34} +36.0000 q^{36} +168.534 q^{37} -67.0227 q^{38} +6.48864 q^{39} +166.697 q^{40} -323.023 q^{41} +221.557 q^{43} +60.6515 q^{44} +187.534 q^{45} +1.30303 q^{46} +508.045 q^{47} +48.0000 q^{48} +618.371 q^{50} -358.045 q^{51} +8.65152 q^{52} -176.511 q^{53} +54.0000 q^{54} +315.951 q^{55} -100.534 q^{57} -326.417 q^{58} +454.928 q^{59} +250.045 q^{60} +38.6515 q^{61} -446.652 q^{62} +64.0000 q^{64} +45.0682 q^{65} +90.9773 q^{66} +141.792 q^{67} -477.394 q^{68} +1.95455 q^{69} +602.742 q^{71} +72.0000 q^{72} -1102.30 q^{73} +337.068 q^{74} +927.557 q^{75} -134.045 q^{76} +12.9773 q^{78} -116.303 q^{79} +333.394 q^{80} +81.0000 q^{81} -646.045 q^{82} -568.928 q^{83} -2486.88 q^{85} +443.114 q^{86} -489.625 q^{87} +121.303 q^{88} -383.159 q^{89} +375.068 q^{90} +2.60607 q^{92} -669.977 q^{93} +1016.09 q^{94} -698.280 q^{95} +96.0000 q^{96} +334.701 q^{97} +136.466 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 5 q^{5} + 12 q^{6} + 16 q^{8} + 18 q^{9} + 10 q^{10} + 67 q^{11} + 24 q^{12} + 41 q^{13} + 15 q^{15} + 32 q^{16} - 92 q^{17} + 36 q^{18} + 43 q^{19} + 20 q^{20} + 134 q^{22}+ \cdots + 603 q^{99}+O(q^{100}) $ 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 5 * q^5 + 12 * q^6 + 16 * q^8 + 18 * q^9 + 10 * q^10 + 67 * q^11 + 24 * q^12 + 41 * q^13 + 15 * q^15 + 32 * q^16 - 92 * q^17 + 36 * q^18 + 43 * q^19 + 20 * q^20 + 134 * q^22 + 148 * q^23 + 48 * q^24 + 435 * q^25 + 82 * q^26 + 54 * q^27 + 77 * q^29 + 30 * q^30 - 520 * q^31 + 64 * q^32 + 201 * q^33 - 184 * q^34 + 72 * q^36 + 7 * q^37 + 86 * q^38 + 123 * q^39 + 40 * q^40 - 426 * q^41 - 107 * q^43 + 268 * q^44 + 45 * q^45 + 296 * q^46 + 576 * q^47 + 96 * q^48 + 870 * q^50 - 276 * q^51 + 164 * q^52 - 243 * q^53 + 108 * q^54 - 505 * q^55 + 129 * q^57 + 154 * q^58 - 7 * q^59 + 60 * q^60 + 224 * q^61 - 1040 * q^62 + 128 * q^64 - 570 * q^65 + 402 * q^66 + 687 * q^67 - 368 * q^68 + 444 * q^69 + 472 * q^71 + 144 * q^72 - 921 * q^73 + 14 * q^74 + 1305 * q^75 + 172 * q^76 + 246 * q^78 - 526 * q^79 + 80 * q^80 + 162 * q^81 - 852 * q^82 - 221 * q^83 - 2920 * q^85 - 214 * q^86 + 231 * q^87 + 536 * q^88 + 774 * q^89 + 90 * q^90 + 592 * q^92 - 1560 * q^93 + 1152 * q^94 - 1910 * q^95 + 192 * q^96 + 1953 * q^97 + 603 * q^99

Coefficient data

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

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

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

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	4.00000	0.500000
$5$	20.8371	1.86373	0.931864	−	0.362807i	$-0.118182\pi$
$5$	20.8371	1.86373	0.931864	+	0.362807i	$0.118182\pi$
$6$	6.00000	0.408248
$7$	0	0
$7$	0	0
$8$	8.00000	0.353553
$9$	9.00000	0.333333
$10$	41.6742	1.31786
$11$	15.1629	0.415616	0.207808	−	0.978170i	$-0.433367\pi$
$11$	15.1629	0.415616	0.207808	+	0.978170i	$0.433367\pi$
$12$	12.0000	0.288675
$13$	2.16288	0.0461442	0.0230721	−	0.999734i	$-0.492655\pi$
$13$	2.16288	0.0461442	0.0230721	+	0.999734i	$0.492655\pi$
$14$	0	0
$15$	62.5114	1.07602
$16$	16.0000	0.250000
$17$	−119.348	−1.70272	−0.851361	−	0.524581i	$-0.824222\pi$
$17$	−119.348	−1.70272	−0.851361	+	0.524581i	$0.824222\pi$
$18$	18.0000	0.235702
$19$	−33.5114	−0.404633	−0.202317	−	0.979320i	$-0.564847\pi$
$19$	−33.5114	−0.404633	−0.202317	+	0.979320i	$0.564847\pi$
$20$	83.3485	0.931864
$21$	0	0
$22$	30.3258	0.293885
$23$	0.651517	0.00590655	0.00295327	−	0.999996i	$-0.499060\pi$
$23$	0.651517	0.00590655	0.00295327	+	0.999996i	$0.499060\pi$
$24$	24.0000	0.204124
$25$	309.186	2.47348
$26$	4.32576	0.0326289
$27$	27.0000	0.192450
$28$	0	0
$29$	−163.208	−1.04507	−0.522535	−	0.852618i	$-0.675014\pi$
$29$	−163.208	−1.04507	−0.522535	+	0.852618i	$0.675014\pi$
$30$	125.023	0.760864
$31$	−223.326	−1.29389	−0.646943	−	0.762538i	$-0.723953\pi$
$31$	−223.326	−1.29389	−0.646943	+	0.762538i	$0.723953\pi$
$32$	32.0000	0.176777
$33$	45.4886	0.239956
$34$	−238.697	−1.20401
$35$	0	0
$36$	36.0000	0.166667
$37$	168.534	0.748833	0.374417	−	0.927261i	$-0.377843\pi$
$37$	168.534	0.748833	0.374417	+	0.927261i	$0.377843\pi$
$38$	−67.0227	−0.286119
$39$	6.48864	0.0266414
$40$	166.697	0.658928
$41$	−323.023	−1.23043	−0.615216	−	0.788359i	$-0.710931\pi$
$41$	−323.023	−1.23043	−0.615216	+	0.788359i	$0.710931\pi$
$42$	0	0
$43$	221.557	0.785746	0.392873	−	0.919593i	$-0.371481\pi$
$43$	221.557	0.785746	0.392873	+	0.919593i	$0.371481\pi$
$44$	60.6515	0.207808
$45$	187.534	0.621243
$46$	1.30303	0.00417656
$47$	508.045	1.57672	0.788362	−	0.615211i	$-0.210929\pi$
$47$	508.045	1.57672	0.788362	+	0.615211i	$0.210929\pi$
$48$	48.0000	0.144338
$49$	0	0
$50$	618.371	1.74902
$51$	−358.045	−0.983066
$52$	8.65152	0.0230721
$53$	−176.511	−0.457466	−0.228733	−	0.973489i	$-0.573458\pi$
$53$	−176.511	−0.457466	−0.228733	+	0.973489i	$0.573458\pi$
$54$	54.0000	0.136083
$55$	315.951	0.774596
$56$	0	0
$57$	−100.534	−0.233615
$58$	−326.417	−0.738976
$59$	454.928	1.00384	0.501920	−	0.864914i	$-0.332627\pi$
$59$	454.928	1.00384	0.501920	+	0.864914i	$0.332627\pi$
$60$	250.045	0.538012
$61$	38.6515	0.0811282	0.0405641	−	0.999177i	$-0.487085\pi$
$61$	38.6515	0.0811282	0.0405641	+	0.999177i	$0.487085\pi$
$62$	−446.652	−0.914916
$63$	0	0
$64$	64.0000	0.125000
$65$	45.0682	0.0860003
$66$	90.9773	0.169675
$67$	141.792	0.258546	0.129273	−	0.991609i	$-0.458736\pi$
$67$	141.792	0.258546	0.129273	+	0.991609i	$0.458736\pi$
$68$	−477.394	−0.851361
$69$	1.95455	0.00341015
$70$	0	0
$71$	602.742	1.00750	0.503749	−	0.863850i	$-0.331954\pi$
$71$	602.742	1.00750	0.503749	+	0.863850i	$0.331954\pi$
$72$	72.0000	0.117851
$73$	−1102.30	−1.76732	−0.883660	−	0.468129i	$-0.844928\pi$
$73$	−1102.30	−1.76732	−0.883660	+	0.468129i	$0.844928\pi$
$74$	337.068	0.529505
$75$	927.557	1.42807
$76$	−134.045	−0.202317
$77$	0	0
$78$	12.9773	0.0188383
$79$	−116.303	−0.165634	−0.0828172	−	0.996565i	$-0.526392\pi$
$79$	−116.303	−0.165634	−0.0828172	+	0.996565i	$0.526392\pi$
$80$	333.394	0.465932
$81$	81.0000	0.111111
$82$	−646.045	−0.870046
$83$	−568.928	−0.752385	−0.376193	−	0.926542i	$-0.622767\pi$
$83$	−568.928	−0.752385	−0.376193	+	0.926542i	$0.622767\pi$
$84$	0	0
$85$	−2486.88	−3.17341
$86$	443.114	0.555607
$87$	−489.625	−0.603371
$88$	121.303	0.146943
$89$	−383.159	−0.456346	−0.228173	−	0.973621i	$-0.573275\pi$
$89$	−383.159	−0.456346	−0.228173	+	0.973621i	$0.573275\pi$
$90$	375.068	0.439285
$91$	0	0
$92$	2.60607	0.00295327
$93$	−669.977	−0.747026
$94$	1016.09	1.11491
$95$	−698.280	−0.754127
$96$	96.0000	0.102062
$97$	334.701	0.350348	0.175174	−	0.984538i	$-0.443951\pi$
$97$	334.701	0.350348	0.175174	+	0.984538i	$0.443951\pi$
$98$	0	0
$99$	136.466	0.138539
$100$	1236.74	1.23674
$101$	−14.7424	−0.0145240	−0.00726201	−	0.999974i	$-0.502312\pi$
$101$	−14.7424	−0.0145240	−0.00726201	+	0.999974i	$0.502312\pi$
$102$	−716.091	−0.695133
$103$	−841.420	−0.804928	−0.402464	−	0.915436i	$-0.631846\pi$
$103$	−841.420	−0.804928	−0.402464	+	0.915436i	$0.631846\pi$
$104$	17.3030	0.0163144
$105$	0	0
$106$	−353.023	−0.323477
$107$	−715.670	−0.646603	−0.323301	−	0.946296i	$-0.604793\pi$
$107$	−715.670	−0.646603	−0.323301	+	0.946296i	$0.604793\pi$
$108$	108.000	0.0962250
$109$	600.019	0.527260	0.263630	−	0.964624i	$-0.415080\pi$
$109$	600.019	0.527260	0.263630	+	0.964624i	$0.415080\pi$
$110$	631.901	0.547722
$111$	505.602	0.432339
$112$	0	0
$113$	622.644	0.518349	0.259174	−	0.965831i	$-0.416550\pi$
$113$	622.644	0.518349	0.259174	+	0.965831i	$0.416550\pi$
$114$	−201.068	−0.165191
$115$	13.5757	0.0110082
$116$	−652.833	−0.522535
$117$	19.4659	0.0153814
$118$	909.856	0.709822
$119$	0	0
$120$	500.091	0.380432
$121$	−1101.09	−0.827263
$122$	77.3030	0.0573663
$123$	−969.068	−0.710390
$124$	−893.303	−0.646943
$125$	3837.90	2.74618
$126$	0	0
$127$	−180.076	−0.125820	−0.0629100	−	0.998019i	$-0.520038\pi$
$127$	−180.076	−0.125820	−0.0629100	+	0.998019i	$0.520038\pi$
$128$	128.000	0.0883883
$129$	664.670	0.453651
$130$	90.1363	0.0608114
$131$	217.860	0.145302	0.0726508	−	0.997357i	$-0.476854\pi$
$131$	217.860	0.145302	0.0726508	+	0.997357i	$0.476854\pi$
$132$	181.955	0.119978
$133$	0	0
$134$	283.583	0.182820
$135$	562.602	0.358675
$136$	−954.788	−0.602003
$137$	−2601.86	−1.62257	−0.811283	−	0.584654i	$-0.801230\pi$
$137$	−2601.86	−1.62257	−0.811283	+	0.584654i	$0.801230\pi$
$138$	3.90910	0.00241134
$139$	−2651.55	−1.61800	−0.808998	−	0.587811i	$-0.799990\pi$
$139$	−2651.55	−1.61800	−0.808998	+	0.587811i	$0.799990\pi$
$140$	0	0
$141$	1524.14	0.910322
$142$	1205.48	0.712409
$143$	32.7955	0.0191783
$144$	144.000	0.0833333
$145$	−3400.79	−1.94773
$146$	−2204.60	−1.24968
$147$	0	0
$148$	674.136	0.374417
$149$	581.023	0.319458	0.159729	−	0.987161i	$-0.448938\pi$
$149$	581.023	0.319458	0.159729	+	0.987161i	$0.448938\pi$
$150$	1855.11	1.00980
$151$	−615.390	−0.331654	−0.165827	−	0.986155i	$-0.553029\pi$
$151$	−615.390	−0.331654	−0.165827	+	0.986155i	$0.553029\pi$
$152$	−268.091	−0.143059
$153$	−1074.14	−0.567574
$154$	0	0
$155$	−4653.47	−2.41145
$156$	25.9546	0.0133207
$157$	−306.932	−0.156024	−0.0780122	−	0.996952i	$-0.524857\pi$
$157$	−306.932	−0.156024	−0.0780122	+	0.996952i	$0.524857\pi$
$158$	−232.606	−0.117121
$159$	−529.534	−0.264118
$160$	666.788	0.329464
$161$	0	0
$162$	162.000	0.0785674
$163$	3514.50	1.68882	0.844408	−	0.535701i	$-0.179953\pi$
$163$	3514.50	1.68882	0.844408	+	0.535701i	$0.179953\pi$
$164$	−1292.09	−0.615216
$165$	947.852	0.447213
$166$	−1137.86	−0.532017
$167$	1123.30	0.520502	0.260251	−	0.965541i	$-0.416195\pi$
$167$	1123.30	0.520502	0.260251	+	0.965541i	$0.416195\pi$
$168$	0	0
$169$	−2192.32	−0.997871
$170$	−4973.76	−2.24394
$171$	−301.602	−0.134878
$172$	886.227	0.392873
$173$	1530.60	0.672655	0.336327	−	0.941745i	$-0.390815\pi$
$173$	1530.60	0.672655	0.336327	+	0.941745i	$0.390815\pi$
$174$	−979.250	−0.426648
$175$	0	0
$176$	242.606	0.103904
$177$	1364.78	0.579568
$178$	−766.318	−0.322685
$179$	3413.43	1.42532	0.712659	−	0.701511i	$-0.247491\pi$
$179$	3413.43	1.42532	0.712659	+	0.701511i	$0.247491\pi$
$180$	750.136	0.310621
$181$	1286.71	0.528399	0.264200	−	0.964468i	$-0.414892\pi$
$181$	1286.71	0.528399	0.264200	+	0.964468i	$0.414892\pi$
$182$	0	0
$183$	115.955	0.0468394
$184$	5.21213	0.00208828
$185$	3511.77	1.39562
$186$	−1339.95	−0.528227
$187$	−1809.67	−0.707679
$188$	2032.18	0.788362
$189$	0	0
$190$	−1396.56	−0.533248
$191$	1055.30	0.399783	0.199891	−	0.979818i	$-0.435941\pi$
$191$	1055.30	0.399783	0.199891	+	0.979818i	$0.435941\pi$
$192$	192.000	0.0721688
$193$	−4770.84	−1.77934	−0.889670	−	0.456604i	$-0.849066\pi$
$193$	−4770.84	−1.77934	−0.889670	+	0.456604i	$0.849066\pi$
$194$	669.402	0.247733
$195$	135.205	0.0496523
$196$	0	0
$197$	1622.31	0.586725	0.293363	−	0.956001i	$-0.405226\pi$
$197$	1622.31	0.586725	0.293363	+	0.956001i	$0.405226\pi$
$198$	272.932	0.0979617
$199$	−3550.14	−1.26464	−0.632318	−	0.774709i	$-0.717896\pi$
$199$	−3550.14	−1.26464	−0.632318	+	0.774709i	$0.717896\pi$
$200$	2473.48	0.874509
$201$	425.375	0.149272
$202$	−29.4848	−0.0102700
$203$	0	0
$204$	−1432.18	−0.491533
$205$	−6730.86	−2.29319
$206$	−1682.84	−0.569170
$207$	5.86365	0.00196885
$208$	34.6061	0.0115361
$209$	−508.129	−0.168172
$210$	0	0
$211$	4653.39	1.51826	0.759129	−	0.650941i	$-0.225625\pi$
$211$	4653.39	1.51826	0.759129	+	0.650941i	$0.225625\pi$
$212$	−706.045	−0.228733
$213$	1808.23	0.581679
$214$	−1431.34	−0.457217
$215$	4616.61	1.46442
$216$	216.000	0.0680414
$217$	0	0
$218$	1200.04	0.372829
$219$	−3306.90	−1.02036
$220$	1263.80	0.387298
$221$	−258.136	−0.0785707
$222$	1011.20	0.305710
$223$	−4649.53	−1.39621	−0.698107	−	0.715993i	$-0.745974\pi$
$223$	−4649.53	−1.39621	−0.698107	+	0.715993i	$0.745974\pi$
$224$	0	0
$225$	2782.67	0.824495
$226$	1245.29	0.366528
$227$	4151.72	1.21392	0.606958	−	0.794734i	$-0.292389\pi$
$227$	4151.72	1.21392	0.606958	+	0.794734i	$0.292389\pi$
$228$	−402.136	−0.116808
$229$	4263.63	1.23034	0.615172	−	0.788393i	$-0.289087\pi$
$229$	4263.63	1.23034	0.615172	+	0.788393i	$0.289087\pi$
$230$	27.1515	0.00778398
$231$	0	0
$232$	−1305.67	−0.369488
$233$	3049.90	0.857535	0.428768	−	0.903415i	$-0.358948\pi$
$233$	3049.90	0.857535	0.428768	+	0.903415i	$0.358948\pi$
$234$	38.9318	0.0108763
$235$	10586.2	2.93859
$236$	1819.71	0.501920
$237$	−348.909	−0.0956290
$238$	0	0
$239$	3987.20	1.07912	0.539562	−	0.841946i	$-0.318590\pi$
$239$	3987.20	1.07912	0.539562	+	0.841946i	$0.318590\pi$
$240$	1000.18	0.269006
$241$	−624.648	−0.166959	−0.0834795	−	0.996509i	$-0.526603\pi$
$241$	−624.648	−0.166959	−0.0834795	+	0.996509i	$0.526603\pi$
$242$	−2202.17	−0.584963
$243$	243.000	0.0641500
$244$	154.606	0.0405641
$245$	0	0
$246$	−1938.14	−0.502321
$247$	−72.4810	−0.0186715
$248$	−1786.61	−0.457458
$249$	−1706.78	−0.434390
$250$	7675.80	1.94184
$251$	−1328.78	−0.334152	−0.167076	−	0.985944i	$-0.553432\pi$
$251$	−1328.78	−0.334152	−0.167076	+	0.985944i	$0.553432\pi$
$252$	0	0
$253$	9.87887	0.00245486
$254$	−360.152	−0.0889682
$255$	−7460.64	−1.83217
$256$	256.000	0.0625000
$257$	−3226.18	−0.783049	−0.391525	−	0.920168i	$-0.628052\pi$
$257$	−3226.18	−0.783049	−0.391525	+	0.920168i	$0.628052\pi$
$258$	1329.34	0.320780
$259$	0	0
$260$	180.273	0.0430001
$261$	−1468.87	−0.348357
$262$	435.720	0.102744
$263$	3250.61	0.762135	0.381067	−	0.924547i	$-0.375557\pi$
$263$	3250.61	0.762135	0.381067	+	0.924547i	$0.375557\pi$
$264$	363.909	0.0848373
$265$	−3677.99	−0.852593
$266$	0	0
$267$	−1149.48	−0.263471
$268$	567.167	0.129273
$269$	2826.04	0.640546	0.320273	−	0.947325i	$-0.396225\pi$
$269$	2826.04	0.640546	0.320273	+	0.947325i	$0.396225\pi$
$270$	1125.20	0.253621
$271$	−2396.77	−0.537245	−0.268622	−	0.963246i	$-0.586568\pi$
$271$	−2396.77	−0.537245	−0.268622	+	0.963246i	$0.586568\pi$
$272$	−1909.58	−0.425680
$273$	0	0
$274$	−5203.71	−1.14733
$275$	4688.14	1.02802
$276$	7.81820	0.00170507
$277$	1820.47	0.394878	0.197439	−	0.980315i	$-0.436738\pi$
$277$	1820.47	0.394878	0.197439	+	0.980315i	$0.436738\pi$
$278$	−5303.10	−1.14410
$279$	−2009.93	−0.431296
$280$	0	0
$281$	3083.81	0.654679	0.327339	−	0.944907i	$-0.393848\pi$
$281$	3083.81	0.654679	0.327339	+	0.944907i	$0.393848\pi$
$282$	3048.27	0.643695
$283$	2554.77	0.536626	0.268313	−	0.963332i	$-0.413534\pi$
$283$	2554.77	0.536626	0.268313	+	0.963332i	$0.413534\pi$
$284$	2410.97	0.503749
$285$	−2094.84	−0.435395
$286$	65.5910	0.0135611
$287$	0	0
$288$	288.000	0.0589256
$289$	9331.06	1.89926
$290$	−6801.58	−1.37725
$291$	1004.10	0.202273
$292$	−4409.20	−0.883660
$293$	−1846.47	−0.368163	−0.184081	−	0.982911i	$-0.558931\pi$
$293$	−1846.47	−0.368163	−0.184081	+	0.982911i	$0.558931\pi$
$294$	0	0
$295$	9479.39	1.87089
$296$	1348.27	0.264753
$297$	409.398	0.0799854
$298$	1162.05	0.225891
$299$	1.40915	0.000272553 0
$300$	3710.23	0.714034
$301$	0	0
$302$	−1230.78	−0.234515
$303$	−44.2272	−0.00838544
$304$	−536.182	−0.101158
$305$	805.386	0.151201
$306$	−2148.27	−0.401335
$307$	7041.50	1.30905	0.654527	−	0.756039i	$-0.272868\pi$
$307$	7041.50	1.30905	0.654527	+	0.756039i	$0.272868\pi$
$308$	0	0
$309$	−2524.26	−0.464726
$310$	−9306.93	−1.70516
$311$	−2685.99	−0.489738	−0.244869	−	0.969556i	$-0.578745\pi$
$311$	−2685.99	−0.489738	−0.244869	+	0.969556i	$0.578745\pi$
$312$	51.9091	0.00941915
$313$	2219.19	0.400754	0.200377	−	0.979719i	$-0.435783\pi$
$313$	2219.19	0.400754	0.200377	+	0.979719i	$0.435783\pi$
$314$	−613.864	−0.110326
$315$	0	0
$316$	−465.212	−0.0828172
$317$	2221.26	0.393560	0.196780	−	0.980448i	$-0.436952\pi$
$317$	2221.26	0.393560	0.196780	+	0.980448i	$0.436952\pi$
$318$	−1059.07	−0.186760
$319$	−2474.71	−0.434348
$320$	1333.58	0.232966
$321$	−2147.01	−0.373316
$322$	0	0
$323$	3999.53	0.688978
$324$	324.000	0.0555556
$325$	668.731	0.114137
$326$	7029.00	1.19417
$327$	1800.06	0.304414
$328$	−2584.18	−0.435023
$329$	0	0
$330$	1895.70	0.316228
$331$	4154.06	0.689812	0.344906	−	0.938637i	$-0.387911\pi$
$331$	4154.06	0.689812	0.344906	+	0.938637i	$0.387911\pi$
$332$	−2275.71	−0.376193
$333$	1516.81	0.249611
$334$	2246.61	0.368050
$335$	2954.53	0.481860
$336$	0	0
$337$	−254.167	−0.0410841	−0.0205420	−	0.999789i	$-0.506539\pi$
$337$	−254.167	−0.0410841	−0.0205420	+	0.999789i	$0.506539\pi$
$338$	−4384.64	−0.705601
$339$	1867.93	0.299269
$340$	−9947.52	−1.58671
$341$	−3386.26	−0.537761
$342$	−603.205	−0.0953730
$343$	0	0
$344$	1772.45	0.277803
$345$	40.7272	0.00635559
$346$	3061.20	0.475639
$347$	−6224.64	−0.962986	−0.481493	−	0.876450i	$-0.659905\pi$
$347$	−6224.64	−0.962986	−0.481493	+	0.876450i	$0.659905\pi$
$348$	−1958.50	−0.301686
$349$	9732.21	1.49270	0.746352	−	0.665552i	$-0.231804\pi$
$349$	9732.21	1.49270	0.746352	+	0.665552i	$0.231804\pi$
$350$	0	0
$351$	58.3977	0.00888046
$352$	485.212	0.0734713
$353$	1425.61	0.214951	0.107476	−	0.994208i	$-0.465723\pi$
$353$	1425.61	0.214951	0.107476	+	0.994208i	$0.465723\pi$
$354$	2729.57	0.409816
$355$	12559.4	1.87770
$356$	−1532.64	−0.228173
$357$	0	0
$358$	6826.86	1.00785
$359$	−5766.49	−0.847754	−0.423877	−	0.905720i	$-0.639331\pi$
$359$	−5766.49	−0.847754	−0.423877	+	0.905720i	$0.639331\pi$
$360$	1500.27	0.219643
$361$	−5735.99	−0.836272
$362$	2573.42	0.373635
$363$	−3303.26	−0.477621
$364$	0	0
$365$	−22968.7	−3.29381
$366$	231.909	0.0331204
$367$	11545.3	1.64213	0.821065	−	0.570834i	$-0.193380\pi$
$367$	11545.3	1.64213	0.821065	+	0.570834i	$0.193380\pi$
$368$	10.4243	0.00147664
$369$	−2907.20	−0.410144
$370$	7023.53	0.986854
$371$	0	0
$372$	−2679.91	−0.373513
$373$	−6479.57	−0.899463	−0.449731	−	0.893164i	$-0.648480\pi$
$373$	−6479.57	−0.899463	−0.449731	+	0.893164i	$0.648480\pi$
$374$	−3619.33	−0.500404
$375$	11513.7	1.58551
$376$	4064.36	0.557456
$377$	−353.000	−0.0482239
$378$	0	0
$379$	611.996	0.0829449	0.0414725	−	0.999140i	$-0.486795\pi$
$379$	611.996	0.0829449	0.0414725	+	0.999140i	$0.486795\pi$
$380$	−2793.12	−0.377063
$381$	−540.227	−0.0726422
$382$	2110.59	0.282689
$383$	4360.81	0.581794	0.290897	−	0.956754i	$-0.406046\pi$
$383$	4360.81	0.581794	0.290897	+	0.956754i	$0.406046\pi$
$384$	384.000	0.0510310
$385$	0	0
$386$	−9541.68	−1.25818
$387$	1994.01	0.261915
$388$	1338.80	0.175174
$389$	−13146.9	−1.71356	−0.856781	−	0.515681i	$-0.827539\pi$
$389$	−13146.9	−1.71356	−0.856781	+	0.515681i	$0.827539\pi$
$390$	270.409	0.0351095
$391$	−77.7575	−0.0100572
$392$	0	0
$393$	653.580	0.0838899
$394$	3244.62	0.414877
$395$	−2423.42	−0.308697
$396$	545.864	0.0692694
$397$	−8478.04	−1.07179	−0.535895	−	0.844285i	$-0.680026\pi$
$397$	−8478.04	−1.07179	−0.535895	+	0.844285i	$0.680026\pi$
$398$	−7100.27	−0.894232
$399$	0	0
$400$	4946.97	0.618371
$401$	2803.00	0.349065	0.174533	−	0.984651i	$-0.444159\pi$
$401$	2803.00	0.349065	0.174533	+	0.984651i	$0.444159\pi$
$402$	850.750	0.105551
$403$	−483.027	−0.0597054
$404$	−58.9697	−0.00726201
$405$	1687.81	0.207081
$406$	0	0
$407$	2555.46	0.311227
$408$	−2864.36	−0.347566
$409$	−6385.39	−0.771973	−0.385987	−	0.922504i	$-0.626139\pi$
$409$	−6385.39	−0.771973	−0.385987	+	0.922504i	$0.626139\pi$
$410$	−13461.7	−1.62153
$411$	−7805.57	−0.936789
$412$	−3365.68	−0.402464
$413$	0	0
$414$	11.7273	0.00139219
$415$	−11854.8	−1.40224
$416$	69.2121	0.00815722
$417$	−7954.65	−0.934151
$418$	−1016.26	−0.118916
$419$	4831.66	0.563346	0.281673	−	0.959510i	$-0.409111\pi$
$419$	4831.66	0.563346	0.281673	+	0.959510i	$0.409111\pi$
$420$	0	0
$421$	7475.37	0.865385	0.432693	−	0.901542i	$-0.357564\pi$
$421$	7475.37	0.865385	0.432693	+	0.901542i	$0.357564\pi$
$422$	9306.77	1.07357
$423$	4572.41	0.525575
$424$	−1412.09	−0.161739
$425$	−36900.8	−4.21165
$426$	3616.45	0.411309
$427$	0	0
$428$	−2862.68	−0.323301
$429$	98.3864	0.0110726
$430$	9233.21	1.03550
$431$	6991.93	0.781414	0.390707	−	0.920515i	$-0.372231\pi$
$431$	6991.93	0.781414	0.390707	+	0.920515i	$0.372231\pi$
$432$	432.000	0.0481125
$433$	−7699.26	−0.854510	−0.427255	−	0.904131i	$-0.640519\pi$
$433$	−7699.26	−0.854510	−0.427255	+	0.904131i	$0.640519\pi$
$434$	0	0
$435$	−10202.4	−1.12452
$436$	2400.08	0.263630
$437$	−21.8332	−0.00238999
$438$	−6613.80	−0.721505
$439$	9412.32	1.02329	0.511646	−	0.859196i	$-0.329036\pi$
$439$	9412.32	1.02329	0.511646	+	0.859196i	$0.329036\pi$
$440$	2527.61	0.273861
$441$	0	0
$442$	−516.273	−0.0555579
$443$	−6258.18	−0.671185	−0.335593	−	0.942007i	$-0.608937\pi$
$443$	−6258.18	−0.671185	−0.335593	+	0.942007i	$0.608937\pi$
$444$	2022.41	0.216170
$445$	−7983.93	−0.850505
$446$	−9299.07	−0.987273
$447$	1743.07	0.184439
$448$	0	0
$449$	−11633.8	−1.22279	−0.611396	−	0.791325i	$-0.709392\pi$
$449$	−11633.8	−1.22279	−0.611396	+	0.791325i	$0.709392\pi$
$450$	5565.34	0.583006
$451$	−4897.95	−0.511387
$452$	2490.58	0.259174
$453$	−1846.17	−0.191480
$454$	8303.43	0.858369
$455$	0	0
$456$	−804.273	−0.0825954
$457$	−13104.6	−1.34138	−0.670688	−	0.741740i	$-0.734001\pi$
$457$	−13104.6	−1.34138	−0.670688	+	0.741740i	$0.734001\pi$
$458$	8527.26	0.869985
$459$	−3222.41	−0.327689
$460$	54.3029	0.00550410
$461$	−2594.63	−0.262134	−0.131067	−	0.991373i	$-0.541840\pi$
$461$	−2594.63	−0.262134	−0.131067	+	0.991373i	$0.541840\pi$
$462$	0	0
$463$	−14136.2	−1.41893	−0.709465	−	0.704741i	$-0.751063\pi$
$463$	−14136.2	−1.41893	−0.709465	+	0.704741i	$0.751063\pi$
$464$	−2611.33	−0.261267
$465$	−13960.4	−1.39225
$466$	6099.80	0.606369
$467$	15590.2	1.54482	0.772409	−	0.635125i	$-0.219052\pi$
$467$	15590.2	1.54482	0.772409	+	0.635125i	$0.219052\pi$
$468$	77.8637	0.00769070
$469$	0	0
$470$	21172.4	2.07789
$471$	−920.795	−0.0900807
$472$	3639.42	0.354911
$473$	3359.44	0.326569
$474$	−697.818	−0.0676199
$475$	−10361.2	−1.00085
$476$	0	0
$477$	−1588.60	−0.152489
$478$	7974.41	0.763056
$479$	−8453.51	−0.806369	−0.403184	−	0.915119i	$-0.632097\pi$
$479$	−8453.51	−0.806369	−0.403184	+	0.915119i	$0.632097\pi$
$480$	2000.36	0.190216
$481$	364.519	0.0345543
$482$	−1249.30	−0.118058
$483$	0	0
$484$	−4404.35	−0.413632
$485$	6974.20	0.652953
$486$	486.000	0.0453609
$487$	−4011.07	−0.373221	−0.186611	−	0.982434i	$-0.559750\pi$
$487$	−4011.07	−0.373221	−0.186611	+	0.982434i	$0.559750\pi$
$488$	309.212	0.0286831
$489$	10543.5	0.975038
$490$	0	0
$491$	13927.9	1.28016	0.640079	−	0.768309i	$-0.278902\pi$
$491$	13927.9	1.28016	0.640079	+	0.768309i	$0.278902\pi$
$492$	−3876.27	−0.355195
$493$	19478.7	1.77946
$494$	−144.962	−0.0132027
$495$	2843.56	0.258199
$496$	−3573.21	−0.323472
$497$	0	0
$498$	−3413.57	−0.307160
$499$	−3947.55	−0.354141	−0.177071	−	0.984198i	$-0.556662\pi$
$499$	−3947.55	−0.354141	−0.177071	+	0.984198i	$0.556662\pi$
$500$	15351.6	1.37309
$501$	3369.91	0.300512
$502$	−2657.57	−0.236281
$503$	−13725.3	−1.21666	−0.608331	−	0.793684i	$-0.708161\pi$
$503$	−13725.3	−1.21666	−0.608331	+	0.793684i	$0.708161\pi$
$504$	0	0
$505$	−307.190	−0.0270688
$506$	19.7577	0.00173585
$507$	−6576.97	−0.576121
$508$	−720.303	−0.0629100
$509$	7830.10	0.681853	0.340926	−	0.940090i	$-0.389259\pi$
$509$	7830.10	0.681853	0.340926	+	0.940090i	$0.389259\pi$
$510$	−14921.3	−1.29554
$511$	0	0
$512$	512.000	0.0441942
$513$	−904.807	−0.0778717
$514$	−6452.36	−0.553700
$515$	−17532.8	−1.50017
$516$	2658.68	0.226825
$517$	7703.43	0.655312
$518$	0	0
$519$	4591.80	0.388357
$520$	360.545	0.0304057
$521$	−5907.39	−0.496751	−0.248376	−	0.968664i	$-0.579897\pi$
$521$	−5907.39	−0.496751	−0.248376	+	0.968664i	$0.579897\pi$
$522$	−2937.75	−0.246325
$523$	7908.06	0.661176	0.330588	−	0.943775i	$-0.392753\pi$
$523$	7908.06	0.661176	0.330588	+	0.943775i	$0.392753\pi$
$524$	871.439	0.0726508
$525$	0	0
$526$	6501.23	0.538911
$527$	26653.6	2.20313
$528$	727.818	0.0599891
$529$	−12166.6	−0.999965
$530$	−7355.98	−0.602874
$531$	4094.35	0.334613
$532$	0	0
$533$	−698.659	−0.0567773
$534$	−2298.95	−0.186302
$535$	−14912.5	−1.20509
$536$	1134.33	0.0914100
$537$	10240.3	0.822908
$538$	5652.08	0.452934
$539$	0	0
$540$	2250.41	0.179337
$541$	−3941.04	−0.313195	−0.156598	−	0.987662i	$-0.550053\pi$
$541$	−3941.04	−0.313195	−0.156598	+	0.987662i	$0.550053\pi$
$542$	−4793.54	−0.379889
$543$	3860.12	0.305071
$544$	−3819.15	−0.301001
$545$	12502.7	0.982670
$546$	0	0
$547$	−1828.71	−0.142943	−0.0714717	−	0.997443i	$-0.522770\pi$
$547$	−1828.71	−0.142943	−0.0714717	+	0.997443i	$0.522770\pi$
$548$	−10407.4	−0.811283
$549$	347.864	0.0270427
$550$	9376.29	0.726920
$551$	5469.33	0.422870
$552$	15.6364	0.00120567
$553$	0	0
$554$	3640.93	0.279221
$555$	10535.3	0.805763
$556$	−10606.2	−0.808998
$557$	22532.0	1.71402	0.857011	−	0.515298i	$-0.172319\pi$
$557$	22532.0	1.71402	0.857011	+	0.515298i	$0.172319\pi$
$558$	−4019.86	−0.304972
$559$	479.201	0.0362577
$560$	0	0
$561$	−5429.00	−0.408579
$562$	6167.62	0.462928
$563$	23355.7	1.74836	0.874179	−	0.485604i	$-0.161400\pi$
$563$	23355.7	1.74836	0.874179	+	0.485604i	$0.161400\pi$
$564$	6096.55	0.455161
$565$	12974.1	0.966062
$566$	5109.54	0.379452
$567$	0	0
$568$	4821.94	0.356204
$569$	−20887.6	−1.53894	−0.769468	−	0.638686i	$-0.779478\pi$
$569$	−20887.6	−1.53894	−0.769468	+	0.638686i	$0.779478\pi$
$570$	−4189.68	−0.307871
$571$	23745.3	1.74029	0.870147	−	0.492792i	$-0.164024\pi$
$571$	23745.3	1.74029	0.870147	+	0.492792i	$0.164024\pi$
$572$	131.182	0.00958914
$573$	3165.89	0.230815
$574$	0	0
$575$	201.440	0.0146098
$576$	576.000	0.0416667
$577$	2454.39	0.177084	0.0885422	−	0.996072i	$-0.471779\pi$
$577$	2454.39	0.177084	0.0885422	+	0.996072i	$0.471779\pi$
$578$	18662.1	1.34298
$579$	−14312.5	−1.02730
$580$	−13603.2	−0.973863
$581$	0	0
$582$	2008.20	0.143029
$583$	−2676.42	−0.190130
$584$	−8818.39	−0.624842
$585$	405.614	0.0286668
$586$	−3692.93	−0.260330
$587$	18567.5	1.30556	0.652780	−	0.757547i	$-0.273603\pi$
$587$	18567.5	1.30556	0.652780	+	0.757547i	$0.273603\pi$
$588$	0	0
$589$	7483.95	0.523550
$590$	18958.8	1.32292
$591$	4866.93	0.338746
$592$	2696.55	0.187208
$593$	17112.9	1.18507	0.592533	−	0.805546i	$-0.298128\pi$
$593$	17112.9	1.18507	0.592533	+	0.805546i	$0.298128\pi$
$594$	818.795	0.0565582
$595$	0	0
$596$	2324.09	0.159729
$597$	−10650.4	−0.730138
$598$	2.81830	0.000192724 0
$599$	23264.8	1.58694	0.793469	−	0.608611i	$-0.208273\pi$
$599$	23264.8	1.58694	0.793469	+	0.608611i	$0.208273\pi$
$600$	7420.45	0.504898
$601$	25322.3	1.71867	0.859334	−	0.511416i	$-0.170879\pi$
$601$	25322.3	1.71867	0.859334	+	0.511416i	$0.170879\pi$
$602$	0	0
$603$	1276.13	0.0861821
$604$	−2461.56	−0.165827
$605$	−22943.5	−1.54179
$606$	−88.4545	−0.00592940
$607$	−21734.4	−1.45333	−0.726665	−	0.686992i	$-0.758931\pi$
$607$	−21734.4	−1.45333	−0.726665	+	0.686992i	$0.758931\pi$
$608$	−1072.36	−0.0715297
$609$	0	0
$610$	1610.77	0.106915
$611$	1098.84	0.0727567
$612$	−4296.55	−0.283787
$613$	−13572.4	−0.894262	−0.447131	−	0.894468i	$-0.647554\pi$
$613$	−13572.4	−0.894262	−0.447131	+	0.894468i	$0.647554\pi$
$614$	14083.0	0.925641
$615$	−20192.6	−1.32397
$616$	0	0
$617$	−8497.12	−0.554427	−0.277213	−	0.960808i	$-0.589411\pi$
$617$	−8497.12	−0.554427	−0.277213	+	0.960808i	$0.589411\pi$
$618$	−5048.52	−0.328611
$619$	−22982.9	−1.49235	−0.746173	−	0.665752i	$-0.768111\pi$
$619$	−22982.9	−1.49235	−0.746173	+	0.665752i	$0.768111\pi$
$620$	−18613.9	−1.20573
$621$	17.5910	0.00113672
$622$	−5371.98	−0.346297
$623$	0	0
$624$	103.818	0.00666034
$625$	41322.5	2.64464
$626$	4438.38	0.283376
$627$	−1524.39	−0.0970943
$628$	−1227.73	−0.0780122
$629$	−20114.3	−1.27505
$630$	0	0
$631$	−15717.9	−0.991635	−0.495817	−	0.868427i	$-0.665131\pi$
$631$	−15717.9	−0.991635	−0.495817	+	0.868427i	$0.665131\pi$
$632$	−930.424	−0.0585606
$633$	13960.2	0.876566
$634$	4442.52	0.278289
$635$	−3752.26	−0.234494
$636$	−2118.14	−0.132059
$637$	0	0
$638$	−4949.42	−0.307131
$639$	5424.68	0.335833
$640$	2667.15	0.164732
$641$	29107.4	1.79356	0.896780	−	0.442478i	$-0.145900\pi$
$641$	29107.4	1.79356	0.896780	+	0.442478i	$0.145900\pi$
$642$	−4294.02	−0.263974
$643$	−3112.26	−0.190880	−0.0954398	−	0.995435i	$-0.530426\pi$
$643$	−3112.26	−0.190880	−0.0954398	+	0.995435i	$0.530426\pi$
$644$	0	0
$645$	13849.8	0.845482
$646$	7999.06	0.487181
$647$	−7857.59	−0.477456	−0.238728	−	0.971087i	$-0.576730\pi$
$647$	−7857.59	−0.477456	−0.238728	+	0.971087i	$0.576730\pi$
$648$	648.000	0.0392837
$649$	6898.02	0.417213
$650$	1337.46	0.0807071
$651$	0	0
$652$	14058.0	0.844408
$653$	−19522.0	−1.16992	−0.584958	−	0.811063i	$-0.698889\pi$
$653$	−19522.0	−1.16992	−0.584958	+	0.811063i	$0.698889\pi$
$654$	3600.11	0.215253
$655$	4539.57	0.270803
$656$	−5168.36	−0.307608
$657$	−9920.69	−0.589107
$658$	0	0
$659$	664.061	0.0392536	0.0196268	−	0.999807i	$-0.493752\pi$
$659$	664.061	0.0392536	0.0196268	+	0.999807i	$0.493752\pi$
$660$	3791.41	0.223607
$661$	15921.6	0.936883	0.468442	−	0.883494i	$-0.344816\pi$
$661$	15921.6	0.936883	0.468442	+	0.883494i	$0.344816\pi$
$662$	8308.11	0.487770
$663$	−774.409	−0.0453628
$664$	−4551.42	−0.266008
$665$	0	0
$666$	3033.61	0.176502
$667$	−106.333	−0.00617276
$668$	4493.21	0.260251
$669$	−13948.6	−0.806105
$670$	5909.06	0.340727
$671$	586.068	0.0337182
$672$	0	0
$673$	24631.0	1.41078	0.705391	−	0.708819i	$-0.250771\pi$
$673$	24631.0	1.41078	0.705391	+	0.708819i	$0.250771\pi$
$674$	−508.333	−0.0290508
$675$	8348.01	0.476022
$676$	−8769.29	−0.498935
$677$	17092.8	0.970353	0.485177	−	0.874416i	$-0.338755\pi$
$677$	17092.8	0.970353	0.485177	+	0.874416i	$0.338755\pi$
$678$	3735.86	0.211615
$679$	0	0
$680$	−19895.0	−1.12197
$681$	12455.1	0.700855
$682$	−6772.52	−0.380254
$683$	−19163.6	−1.07361	−0.536804	−	0.843707i	$-0.680368\pi$
$683$	−19163.6	−1.07361	−0.536804	+	0.843707i	$0.680368\pi$
$684$	−1206.41	−0.0674389
$685$	−54215.2	−3.02402
$686$	0	0
$687$	12790.9	0.710339
$688$	3544.91	0.196437
$689$	−381.773	−0.0211094
$690$	81.4544	0.00449408
$691$	8095.87	0.445704	0.222852	−	0.974852i	$-0.428463\pi$
$691$	8095.87	0.445704	0.222852	+	0.974852i	$0.428463\pi$
$692$	6122.39	0.336327
$693$	0	0
$694$	−12449.3	−0.680934
$695$	−55250.7	−3.01551
$696$	−3917.00	−0.213324
$697$	38552.3	2.09508
$698$	19464.4	1.05550
$699$	9149.70	0.495098
$700$	0	0
$701$	12354.7	0.665664	0.332832	−	0.942986i	$-0.391996\pi$
$701$	12354.7	0.665664	0.332832	+	0.942986i	$0.391996\pi$
$702$	116.795	0.00627943
$703$	−5647.81	−0.303003
$704$	970.424	0.0519520
$705$	31758.6	1.69659
$706$	2851.23	0.151993
$707$	0	0
$708$	5459.14	0.289784
$709$	3828.82	0.202813	0.101406	−	0.994845i	$-0.467666\pi$
$709$	3828.82	0.202813	0.101406	+	0.994845i	$0.467666\pi$
$710$	25118.8	1.32774
$711$	−1046.73	−0.0552114
$712$	−3065.27	−0.161343
$713$	−145.500	−0.00764241
$714$	0	0
$715$	683.363	0.0357431
$716$	13653.7	0.712659
$717$	11961.6	0.623033
$718$	−11533.0	−0.599453
$719$	1223.00	0.0634356	0.0317178	−	0.999497i	$-0.489902\pi$
$719$	1223.00	0.0634356	0.0317178	+	0.999497i	$0.489902\pi$
$720$	3000.55	0.155311
$721$	0	0
$722$	−11472.0	−0.591333
$723$	−1873.94	−0.0963938
$724$	5146.83	0.264200
$725$	−50461.7	−2.58496
$726$	−6606.52	−0.337729
$727$	6368.21	0.324875	0.162437	−	0.986719i	$-0.448064\pi$
$727$	6368.21	0.324875	0.162437	+	0.986719i	$0.448064\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	−45937.5	−2.32907
$731$	−26442.5	−1.33791
$732$	463.818	0.0234197
$733$	−25154.0	−1.26751	−0.633753	−	0.773535i	$-0.718487\pi$
$733$	−25154.0	−1.26751	−0.633753	+	0.773535i	$0.718487\pi$
$734$	23090.7	1.16116
$735$	0	0
$736$	20.8485	0.00104414
$737$	2149.97	0.107456
$738$	−5814.41	−0.290015
$739$	−10739.1	−0.534566	−0.267283	−	0.963618i	$-0.586126\pi$
$739$	−10739.1	−0.534566	−0.267283	+	0.963618i	$0.586126\pi$
$740$	14047.1	0.697811
$741$	−217.443	−0.0107800
$742$	0	0
$743$	28166.3	1.39074	0.695370	−	0.718652i	$-0.255240\pi$
$743$	28166.3	1.39074	0.695370	+	0.718652i	$0.255240\pi$
$744$	−5359.82	−0.264114
$745$	12106.8	0.595383
$746$	−12959.1	−0.636016
$747$	−5120.35	−0.250795
$748$	−7238.67	−0.353839
$749$	0	0
$750$	23027.4	1.12112
$751$	28657.0	1.39242	0.696211	−	0.717837i	$-0.254868\pi$
$751$	28657.0	1.39242	0.696211	+	0.717837i	$0.254868\pi$
$752$	8128.73	0.394181
$753$	−3986.35	−0.192923
$754$	−706.000	−0.0340995
$755$	−12823.0	−0.618113
$756$	0	0
$757$	−23604.1	−1.13330	−0.566648	−	0.823960i	$-0.691760\pi$
$757$	−23604.1	−1.13330	−0.566648	+	0.823960i	$0.691760\pi$
$758$	1223.99	0.0586509
$759$	29.6366	0.00141731
$760$	−5586.24	−0.266624
$761$	4630.97	0.220595	0.110297	−	0.993899i	$-0.464820\pi$
$761$	4630.97	0.220595	0.110297	+	0.993899i	$0.464820\pi$
$762$	−1080.45	−0.0513658
$763$	0	0
$764$	4221.18	0.199891
$765$	−22381.9	−1.05780
$766$	8721.62	0.411390
$767$	983.954	0.0463214
$768$	768.000	0.0360844
$769$	33276.8	1.56046	0.780228	−	0.625495i	$-0.215103\pi$
$769$	33276.8	1.56046	0.780228	+	0.625495i	$0.215103\pi$
$770$	0	0
$771$	−9678.55	−0.452094
$772$	−19083.4	−0.889670
$773$	−22938.8	−1.06734	−0.533668	−	0.845694i	$-0.679187\pi$
$773$	−22938.8	−1.06734	−0.533668	+	0.845694i	$0.679187\pi$
$774$	3988.02	0.185202
$775$	−69049.1	−3.20041
$776$	2677.61	0.123867
$777$	0	0
$778$	−26293.8	−1.21167
$779$	10824.9	0.497873
$780$	540.818	0.0248261
$781$	9139.31	0.418733
$782$	−155.515	−0.00711152
$783$	−4406.62	−0.201124
$784$	0	0
$785$	−6395.58	−0.290787
$786$	1307.16	0.0593191
$787$	−13514.5	−0.612120	−0.306060	−	0.952012i	$-0.599011\pi$
$787$	−13514.5	−0.612120	−0.306060	+	0.952012i	$0.599011\pi$
$788$	6489.24	0.293363
$789$	9751.84	0.440019
$790$	−4846.84	−0.218282
$791$	0	0
$792$	1091.73	0.0489809
$793$	83.5986	0.00374360
$794$	−16956.1	−0.757870
$795$	−11034.0	−0.492245
$796$	−14200.5	−0.632318
$797$	10473.4	0.465480	0.232740	−	0.972539i	$-0.425231\pi$
$797$	10473.4	0.465480	0.232740	+	0.972539i	$0.425231\pi$
$798$	0	0
$799$	−60634.5	−2.68472
$800$	9893.94	0.437254
$801$	−3448.43	−0.152115
$802$	5606.00	0.246826
$803$	−16714.0	−0.734527
$804$	1701.50	0.0746359
$805$	0	0
$806$	−966.053	−0.0422181
$807$	8478.12	0.369819
$808$	−117.939	−0.00513501
$809$	23568.0	1.02423	0.512117	−	0.858916i	$-0.328861\pi$
$809$	23568.0	1.02423	0.512117	+	0.858916i	$0.328861\pi$
$810$	3375.61	0.146428
$811$	−6704.22	−0.290280	−0.145140	−	0.989411i	$-0.546363\pi$
$811$	−6704.22	−0.290280	−0.145140	+	0.989411i	$0.546363\pi$
$812$	0	0
$813$	−7190.31	−0.310178
$814$	5110.92	0.220071
$815$	73232.1	3.14749
$816$	−5728.73	−0.245767
$817$	−7424.67	−0.317939
$818$	−12770.8	−0.545867
$819$	0	0
$820$	−26923.5	−1.14659
$821$	24539.4	1.04316	0.521579	−	0.853203i	$-0.325343\pi$
$821$	24539.4	1.04316	0.521579	+	0.853203i	$0.325343\pi$
$822$	−15611.1	−0.662410
$823$	−31117.0	−1.31795	−0.658973	−	0.752167i	$-0.729009\pi$
$823$	−31117.0	−1.31795	−0.658973	+	0.752167i	$0.729009\pi$
$824$	−6731.36	−0.284585
$825$	14064.4	0.593528
$826$	0	0
$827$	31244.9	1.31377	0.656887	−	0.753989i	$-0.271873\pi$
$827$	31244.9	1.31377	0.656887	+	0.753989i	$0.271873\pi$
$828$	23.4546	0.000984425 0
$829$	4231.50	0.177281	0.0886405	−	0.996064i	$-0.471748\pi$
$829$	4231.50	0.177281	0.0886405	+	0.996064i	$0.471748\pi$
$830$	−23709.6	−0.991535
$831$	5461.40	0.227983
$832$	138.424	0.00576803
$833$	0	0
$834$	−15909.3	−0.660544
$835$	23406.4	0.970074
$836$	−2032.51	−0.0840861
$837$	−6029.80	−0.249009
$838$	9663.32	0.398346
$839$	−38670.4	−1.59124	−0.795621	−	0.605795i	$-0.792855\pi$
$839$	−38670.4	−1.59124	−0.795621	+	0.605795i	$0.792855\pi$
$840$	0	0
$841$	2247.96	0.0921710
$842$	14950.7	0.611920
$843$	9251.43	0.377979
$844$	18613.5	0.759129
$845$	−45681.7	−1.85976
$846$	9144.82	0.371637
$847$	0	0
$848$	−2824.18	−0.114367
$849$	7664.31	0.309821
$850$	−73801.7	−2.97809
$851$	109.803	0.00442302
$852$	7232.91	0.290840
$853$	−19944.4	−0.800565	−0.400282	−	0.916392i	$-0.631088\pi$
$853$	−19944.4	−0.800565	−0.400282	+	0.916392i	$0.631088\pi$
$854$	0	0
$855$	−6284.52	−0.251376
$856$	−5725.36	−0.228609
$857$	13882.2	0.553334	0.276667	−	0.960966i	$-0.410770\pi$
$857$	13882.2	0.553334	0.276667	+	0.960966i	$0.410770\pi$
$858$	196.773	0.00782950
$859$	−4157.16	−0.165123	−0.0825614	−	0.996586i	$-0.526310\pi$
$859$	−4157.16	−0.165123	−0.0825614	+	0.996586i	$0.526310\pi$
$860$	18466.4	0.732209
$861$	0	0
$862$	13983.9	0.552543
$863$	−16237.9	−0.640493	−0.320246	−	0.947334i	$-0.603766\pi$
$863$	−16237.9	−0.640493	−0.320246	+	0.947334i	$0.603766\pi$
$864$	864.000	0.0340207
$865$	31893.3	1.25365
$866$	−15398.5	−0.604230
$867$	27993.2	1.09654
$868$	0	0
$869$	−1763.49	−0.0688403
$870$	−20404.8	−0.795156
$871$	306.678	0.0119304
$872$	4800.15	0.186415
$873$	3012.31	0.116783
$874$	−43.6664	−0.00168998
$875$	0	0
$876$	−13227.6	−0.510181
$877$	−16489.4	−0.634900	−0.317450	−	0.948275i	$-0.602827\pi$
$877$	−16489.4	−0.634900	−0.317450	+	0.948275i	$0.602827\pi$
$878$	18824.6	0.723577
$879$	−5539.40	−0.212559
$880$	5055.21	0.193649
$881$	45411.7	1.73662	0.868309	−	0.496023i	$-0.165207\pi$
$881$	45411.7	1.73662	0.868309	+	0.496023i	$0.165207\pi$
$882$	0	0
$883$	−2206.85	−0.0841070	−0.0420535	−	0.999115i	$-0.513390\pi$
$883$	−2206.85	−0.0841070	−0.0420535	+	0.999115i	$0.513390\pi$
$884$	−1032.55	−0.0392854
$885$	28438.2	1.08016
$886$	−12516.4	−0.474600
$887$	28146.2	1.06545	0.532727	−	0.846287i	$-0.321167\pi$
$887$	28146.2	1.06545	0.532727	+	0.846287i	$0.321167\pi$
$888$	4044.82	0.152855
$889$	0	0
$890$	−15967.9	−0.601398
$891$	1228.19	0.0461796
$892$	−18598.1	−0.698107
$893$	−17025.3	−0.637995
$894$	3486.14	0.130418
$895$	71126.1	2.65641
$896$	0	0
$897$	4.22746	0.000157359 0
$898$	−23267.6	−0.864644
$899$	36448.6	1.35220
$900$	11130.7	0.412247
$901$	21066.4	0.778937
$902$	−9795.91	−0.361605
$903$	0	0
$904$	4981.15	0.183264
$905$	26811.3	0.984793
$906$	−3692.34	−0.135397
$907$	−5042.25	−0.184592	−0.0922960	−	0.995732i	$-0.529421\pi$
$907$	−5042.25	−0.184592	−0.0922960	+	0.995732i	$0.529421\pi$
$908$	16606.9	0.606958
$909$	−132.682	−0.00484134
$910$	0	0
$911$	29647.3	1.07822	0.539110	−	0.842235i	$-0.318761\pi$
$911$	29647.3	1.07822	0.539110	+	0.842235i	$0.318761\pi$
$912$	−1608.55	−0.0584038
$913$	−8626.59	−0.312704
$914$	−26209.3	−0.948496
$915$	2416.16	0.0872959
$916$	17054.5	0.615172
$917$	0	0
$918$	−6444.82	−0.231711
$919$	−11891.3	−0.426830	−0.213415	−	0.976962i	$-0.568459\pi$
$919$	−11891.3	−0.426830	−0.213415	+	0.976962i	$0.568459\pi$
$920$	108.606	0.00389199
$921$	21124.5	0.755782
$922$	−5189.26	−0.185357
$923$	1303.66	0.0464902
$924$	0	0
$925$	52108.3	1.85223
$926$	−28272.4	−1.00333
$927$	−7572.78	−0.268309
$928$	−5222.67	−0.184744
$929$	−39188.5	−1.38400	−0.691999	−	0.721898i	$-0.743270\pi$
$929$	−39188.5	−1.38400	−0.691999	+	0.721898i	$0.743270\pi$
$930$	−27920.8	−0.984472
$931$	0	0
$932$	12199.6	0.428768
$933$	−8057.98	−0.282751
$934$	31180.5	1.09235
$935$	−37708.2	−1.31892
$936$	155.727	0.00543815
$937$	9716.23	0.338757	0.169379	−	0.985551i	$-0.445824\pi$
$937$	9716.23	0.338757	0.169379	+	0.985551i	$0.445824\pi$
$938$	0	0
$939$	6657.57	0.231375
$940$	42344.8	1.46929
$941$	−6995.87	−0.242358	−0.121179	−	0.992631i	$-0.538667\pi$
$941$	−6995.87	−0.242358	−0.121179	+	0.992631i	$0.538667\pi$
$942$	−1841.59	−0.0636967
$943$	−210.455	−0.00726760
$944$	7278.85	0.250960
$945$	0	0
$946$	6718.88	0.230919
$947$	14979.2	0.514002	0.257001	−	0.966411i	$-0.417266\pi$
$947$	14979.2	0.514002	0.257001	+	0.966411i	$0.417266\pi$
$948$	−1395.64	−0.0478145
$949$	−2384.14	−0.0815516
$950$	−20722.5	−0.707711
$951$	6663.78	0.227222
$952$	0	0
$953$	29393.3	0.999100	0.499550	−	0.866285i	$-0.333499\pi$
$953$	29393.3	0.999100	0.499550	+	0.866285i	$0.333499\pi$
$954$	−3177.20	−0.107826
$955$	21989.3	0.745087
$956$	15948.8	0.539562
$957$	−7424.12	−0.250771
$958$	−16907.0	−0.570189
$959$	0	0
$960$	4000.73	0.134503
$961$	20083.4	0.674143
$962$	729.038	0.0244336
$963$	−6441.03	−0.215534
$964$	−2498.59	−0.0834795
$965$	−99410.6	−3.31621
$966$	0	0
$967$	−7133.95	−0.237241	−0.118621	−	0.992940i	$-0.537847\pi$
$967$	−7133.95	−0.237241	−0.118621	+	0.992940i	$0.537847\pi$
$968$	−8808.70	−0.292482
$969$	11998.6	0.397781
$970$	13948.4	0.461707
$971$	−9688.13	−0.320192	−0.160096	−	0.987101i	$-0.551180\pi$
$971$	−9688.13	−0.320192	−0.160096	+	0.987101i	$0.551180\pi$
$972$	972.000	0.0320750
$973$	0	0
$974$	−8022.14	−0.263907
$975$	2006.19	0.0658970
$976$	618.424	0.0202820
$977$	21305.7	0.697676	0.348838	−	0.937183i	$-0.386576\pi$
$977$	21305.7	0.697676	0.348838	+	0.937183i	$0.386576\pi$
$978$	21087.0	0.689456
$979$	−5809.79	−0.189665
$980$	0	0
$981$	5400.17	0.175753
$982$	27855.8	0.905208
$983$	−37280.8	−1.20964	−0.604818	−	0.796364i	$-0.706754\pi$
$983$	−37280.8	−1.20964	−0.604818	+	0.796364i	$0.706754\pi$
$984$	−7752.55	−0.251161
$985$	33804.3	1.09350
$986$	38957.3	1.25827
$987$	0	0
$988$	−289.924	−0.00933574
$989$	144.348	0.00464105
$990$	5687.11	0.182574
$991$	−51397.1	−1.64751	−0.823755	−	0.566946i	$-0.808125\pi$
$991$	−51397.1	−1.64751	−0.823755	+	0.566946i	$0.808125\pi$
$992$	−7146.42	−0.228729
$993$	12462.2	0.398263
$994$	0	0
$995$	−73974.6	−2.35694
$996$	−6827.14	−0.217195
$997$	−34373.8	−1.09191	−0.545953	−	0.837816i	$-0.683832\pi$
$997$	−34373.8	−1.09191	−0.545953	+	0.837816i	$0.683832\pi$
$998$	−7895.10	−0.250416
$999$	4550.42	0.144113

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	294.4.a.n.1.2		2
3.2	odd	2		882.4.a.v.1.1		2
4.3	odd	2		2352.4.a.bq.1.2		2
7.2	even	3		42.4.e.c.25.1	✓	4
7.3	odd	6		294.4.e.l.79.2		4
7.4	even	3		42.4.e.c.37.1	yes	4
7.5	odd	6		294.4.e.l.67.2		4
7.6	odd	2		294.4.a.m.1.1		2
21.2	odd	6		126.4.g.g.109.2		4
21.5	even	6		882.4.g.bf.361.1		4
21.11	odd	6		126.4.g.g.37.2		4
21.17	even	6		882.4.g.bf.667.1		4
21.20	even	2		882.4.a.z.1.2		2
28.11	odd	6		336.4.q.j.289.1		4
28.23	odd	6		336.4.q.j.193.1		4
28.27	even	2		2352.4.a.ca.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.e.c.25.1	✓	4	7.2	even	3
42.4.e.c.37.1	yes	4	7.4	even	3
126.4.g.g.37.2		4	21.11	odd	6
126.4.g.g.109.2		4	21.2	odd	6
294.4.a.m.1.1		2	7.6	odd	2
294.4.a.n.1.2		2	1.1	even	1	trivial
294.4.e.l.67.2		4	7.5	odd	6
294.4.e.l.79.2		4	7.3	odd	6
336.4.q.j.193.1		4	28.23	odd	6
336.4.q.j.289.1		4	28.11	odd	6
882.4.a.v.1.1		2	3.2	odd	2
882.4.a.z.1.2		2	21.20	even	2
882.4.g.bf.361.1		4	21.5	even	6
882.4.g.bf.667.1		4	21.17	even	6
2352.4.a.bq.1.2		2	4.3	odd	2
2352.4.a.ca.1.1		2	28.27	even	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:	\( N \)	\(=\)	\( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	294.a (trivial)

\(f(q)\)	\(=\)	\(q+2.00000 q^{2} +3.00000 q^{3} +4.00000 q^{4} +20.8371 q^{5} +6.00000 q^{6} +8.00000 q^{8} +9.00000 q^{9} +41.6742 q^{10} +15.1629 q^{11} +12.0000 q^{12} +2.16288 q^{13} +62.5114 q^{15} +16.0000 q^{16} -119.348 q^{17} +18.0000 q^{18} -33.5114 q^{19} +83.3485 q^{20} +30.3258 q^{22} +0.651517 q^{23} +24.0000 q^{24} +309.186 q^{25} +4.32576 q^{26} +27.0000 q^{27} -163.208 q^{29} +125.023 q^{30} -223.326 q^{31} +32.0000 q^{32} +45.4886 q^{33} -238.697 q^{34} +36.0000 q^{36} +168.534 q^{37} -67.0227 q^{38} +6.48864 q^{39} +166.697 q^{40} -323.023 q^{41} +221.557 q^{43} +60.6515 q^{44} +187.534 q^{45} +1.30303 q^{46} +508.045 q^{47} +48.0000 q^{48} +618.371 q^{50} -358.045 q^{51} +8.65152 q^{52} -176.511 q^{53} +54.0000 q^{54} +315.951 q^{55} -100.534 q^{57} -326.417 q^{58} +454.928 q^{59} +250.045 q^{60} +38.6515 q^{61} -446.652 q^{62} +64.0000 q^{64} +45.0682 q^{65} +90.9773 q^{66} +141.792 q^{67} -477.394 q^{68} +1.95455 q^{69} +602.742 q^{71} +72.0000 q^{72} -1102.30 q^{73} +337.068 q^{74} +927.557 q^{75} -134.045 q^{76} +12.9773 q^{78} -116.303 q^{79} +333.394 q^{80} +81.0000 q^{81} -646.045 q^{82} -568.928 q^{83} -2486.88 q^{85} +443.114 q^{86} -489.625 q^{87} +121.303 q^{88} -383.159 q^{89} +375.068 q^{90} +2.60607 q^{92} -669.977 q^{93} +1016.09 q^{94} -698.280 q^{95} +96.0000 q^{96} +334.701 q^{97} +136.466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{2} + 6 q^{3} + 8 q^{4} + 5 q^{5} + 12 q^{6} + 16 q^{8} + 18 q^{9} + 10 q^{10} + 67 q^{11} + 24 q^{12} + 41 q^{13} + 15 q^{15} + 32 q^{16} - 92 q^{17} + 36 q^{18} + 43 q^{19} + 20 q^{20} + 134 q^{22}+ \cdots + 603 q^{99}+O(q^{100}) \) 2 * q + 4 * q^2 + 6 * q^3 + 8 * q^4 + 5 * q^5 + 12 * q^6 + 16 * q^8 + 18 * q^9 + 10 * q^10 + 67 * q^11 + 24 * q^12 + 41 * q^13 + 15 * q^15 + 32 * q^16 - 92 * q^17 + 36 * q^18 + 43 * q^19 + 20 * q^20 + 134 * q^22 + 148 * q^23 + 48 * q^24 + 435 * q^25 + 82 * q^26 + 54 * q^27 + 77 * q^29 + 30 * q^30 - 520 * q^31 + 64 * q^32 + 201 * q^33 - 184 * q^34 + 72 * q^36 + 7 * q^37 + 86 * q^38 + 123 * q^39 + 40 * q^40 - 426 * q^41 - 107 * q^43 + 268 * q^44 + 45 * q^45 + 296 * q^46 + 576 * q^47 + 96 * q^48 + 870 * q^50 - 276 * q^51 + 164 * q^52 - 243 * q^53 + 108 * q^54 - 505 * q^55 + 129 * q^57 + 154 * q^58 - 7 * q^59 + 60 * q^60 + 224 * q^61 - 1040 * q^62 + 128 * q^64 - 570 * q^65 + 402 * q^66 + 687 * q^67 - 368 * q^68 + 444 * q^69 + 472 * q^71 + 144 * q^72 - 921 * q^73 + 14 * q^74 + 1305 * q^75 + 172 * q^76 + 246 * q^78 - 526 * q^79 + 80 * q^80 + 162 * q^81 - 852 * q^82 - 221 * q^83 - 2920 * q^85 - 214 * q^86 + 231 * q^87 + 536 * q^88 + 774 * q^89 + 90 * q^90 + 592 * q^92 - 1560 * q^93 + 1152 * q^94 - 1910 * q^95 + 192 * q^96 + 1953 * q^97 + 603 * q^99

Introduction

L-functions

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