LMFDB - Embedded newform 315.10.a.c.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,10,Mod(1,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$162.236288392$
Analytic rank:	$1$
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} - 307x^{2} - 270x + 8836 $ x^4 - 307x^2 - 270x + 8836
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{2}\cdot 5 $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$5.15226$ of defining polynomial
Character	$\chi$	$=$	315.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.30453 q^{2} -443.035 q^{4} +625.000 q^{5} -2401.00 q^{7} -7931.11 q^{8} +5190.33 q^{10} -14902.2 q^{11} -85007.1 q^{13} -19939.2 q^{14} +160970. q^{16} +582410. q^{17} -264118. q^{19} -276897. q^{20} -123756. q^{22} +768571. q^{23} +390625. q^{25} -705944. q^{26} +1.06373e6 q^{28} -188493. q^{29} +6.59032e6 q^{31} +5.39751e6 q^{32} +4.83664e6 q^{34} -1.50062e6 q^{35} -1.94757e6 q^{37} -2.19338e6 q^{38} -4.95695e6 q^{40} -1.41517e7 q^{41} -9.39531e6 q^{43} +6.60218e6 q^{44} +6.38262e6 q^{46} -2.23972e7 q^{47} +5.76480e6 q^{49} +3.24396e6 q^{50} +3.76611e7 q^{52} -3.32886e7 q^{53} -9.31386e6 q^{55} +1.90426e7 q^{56} -1.56534e6 q^{58} +9.10729e7 q^{59} +9.84479e7 q^{61} +5.47295e7 q^{62} -3.75927e7 q^{64} -5.31294e7 q^{65} +1.15591e8 q^{67} -2.58028e8 q^{68} -1.24620e7 q^{70} -1.86826e8 q^{71} +1.59090e8 q^{73} -1.61737e7 q^{74} +1.17013e8 q^{76} +3.57801e7 q^{77} +2.05688e8 q^{79} +1.00606e8 q^{80} -1.17524e8 q^{82} -5.94838e8 q^{83} +3.64006e8 q^{85} -7.80236e7 q^{86} +1.18191e8 q^{88} -1.02347e9 q^{89} +2.04102e8 q^{91} -3.40504e8 q^{92} -1.85998e8 q^{94} -1.65074e8 q^{95} +1.35649e9 q^{97} +4.78740e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{2} + 424 q^{4} + 2500 q^{5} - 9604 q^{7} - 96 q^{8} - 5000 q^{10} - 9832 q^{11} - 68264 q^{13} + 19208 q^{14} - 290784 q^{16} - 86272 q^{17} + 807672 q^{19} + 265000 q^{20} + 4293352 q^{22} - 683032 q^{23}+ \cdots - 46118408 q^{98}+O(q^{100}) $ 4 * q - 8 * q^2 + 424 * q^4 + 2500 * q^5 - 9604 * q^7 - 96 * q^8 - 5000 * q^10 - 9832 * q^11 - 68264 * q^13 + 19208 * q^14 - 290784 * q^16 - 86272 * q^17 + 807672 * q^19 + 265000 * q^20 + 4293352 * q^22 - 683032 * q^23 + 1562500 * q^25 - 4583544 * q^26 - 1018024 * q^28 - 5029312 * q^29 + 9802992 * q^31 - 497280 * q^32 + 429792 * q^34 - 6002500 * q^35 - 4536432 * q^37 - 21534872 * q^38 - 60000 * q^40 - 11081000 * q^41 - 34442248 * q^43 - 35679024 * q^44 + 67652080 * q^46 + 39230168 * q^47 + 23059204 * q^49 - 3125000 * q^50 - 14778976 * q^52 - 76012528 * q^53 - 6145000 * q^55 + 230496 * q^56 + 57813344 * q^58 - 74595424 * q^59 + 101246592 * q^61 - 392291880 * q^62 - 276909952 * q^64 - 42665000 * q^65 - 45694264 * q^67 - 345896144 * q^68 + 12005000 * q^70 - 447187600 * q^71 + 141527624 * q^73 - 480388368 * q^74 + 1081835504 * q^76 + 23606632 * q^77 + 789659136 * q^79 - 181740000 * q^80 + 937199856 * q^82 - 586282352 * q^83 - 53920000 * q^85 - 132711648 * q^86 + 1042925024 * q^88 - 1050230088 * q^89 + 163901864 * q^91 - 44316096 * q^92 - 1211140960 * q^94 + 504795000 * q^95 + 788061288 * q^97 - 46118408 * q^98

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 $

$2$	8.30453	0.367012	0.183506	−	0.983019i	$-0.441255\pi$
$2$	8.30453	0.367012	0.183506	+	0.983019i	$0.441255\pi$
$3$	0	0
$3$	0	0
$4$	−443.035	−0.865302
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−7931.11	−0.684588
$9$	0	0
$10$	5190.33	0.164133
$11$	−14902.2	−0.306890	−0.153445	−	0.988157i	$-0.549037\pi$
$11$	−14902.2	−0.306890	−0.153445	+	0.988157i	$0.549037\pi$
$12$	0	0
$13$	−85007.1	−0.825487	−0.412743	−	0.910847i	$-0.635429\pi$
$13$	−85007.1	−0.825487	−0.412743	+	0.910847i	$0.635429\pi$
$14$	−19939.2	−0.138717
$15$	0	0
$16$	160970.	0.614050
$17$	582410.	1.69125	0.845626	−	0.533775i	$-0.179227\pi$
$17$	582410.	1.69125	0.845626	+	0.533775i	$0.179227\pi$
$18$	0	0
$19$	−264118.	−0.464951	−0.232475	−	0.972602i	$-0.574682\pi$
$19$	−264118.	−0.464951	−0.232475	+	0.972602i	$0.574682\pi$
$20$	−276897.	−0.386975
$21$	0	0
$22$	−123756.	−0.112632
$23$	768571.	0.572675	0.286338	−	0.958129i	$-0.407562\pi$
$23$	768571.	0.572675	0.286338	+	0.958129i	$0.407562\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	−705944.	−0.302963
$27$	0	0
$28$	1.06373e6	0.327054
$29$	−188493.	−0.0494884	−0.0247442	−	0.999694i	$-0.507877\pi$
$29$	−188493.	−0.0494884	−0.0247442	+	0.999694i	$0.507877\pi$
$30$	0	0
$31$	6.59032e6	1.28168	0.640839	−	0.767675i	$-0.278587\pi$
$31$	6.59032e6	1.28168	0.640839	+	0.767675i	$0.278587\pi$
$32$	5.39751e6	0.909952
$33$	0	0
$34$	4.83664e6	0.620710
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	−1.94757e6	−0.170839	−0.0854193	−	0.996345i	$-0.527223\pi$
$37$	−1.94757e6	−0.170839	−0.0854193	+	0.996345i	$0.527223\pi$
$38$	−2.19338e6	−0.170642
$39$	0	0
$40$	−4.95695e6	−0.306157
$41$	−1.41517e7	−0.782137	−0.391068	−	0.920362i	$-0.627894\pi$
$41$	−1.41517e7	−0.782137	−0.391068	+	0.920362i	$0.627894\pi$
$42$	0	0
$43$	−9.39531e6	−0.419086	−0.209543	−	0.977799i	$-0.567198\pi$
$43$	−9.39531e6	−0.419086	−0.209543	+	0.977799i	$0.567198\pi$
$44$	6.60218e6	0.265552
$45$	0	0
$46$	6.38262e6	0.210179
$47$	−2.23972e7	−0.669505	−0.334753	−	0.942306i	$-0.608653\pi$
$47$	−2.23972e7	−0.669505	−0.334753	+	0.942306i	$0.608653\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	3.24396e6	0.0734024
$51$	0	0
$52$	3.76611e7	0.714295
$53$	−3.32886e7	−0.579501	−0.289750	−	0.957102i	$-0.593572\pi$
$53$	−3.32886e7	−0.579501	−0.289750	+	0.957102i	$0.593572\pi$
$54$	0	0
$55$	−9.31386e6	−0.137245
$56$	1.90426e7	0.258750
$57$	0	0
$58$	−1.56534e6	−0.0181628
$59$	9.10729e7	0.978487	0.489244	−	0.872147i	$-0.337273\pi$
$59$	9.10729e7	0.978487	0.489244	+	0.872147i	$0.337273\pi$
$60$	0	0
$61$	9.84479e7	0.910379	0.455190	−	0.890395i	$-0.349571\pi$
$61$	9.84479e7	0.910379	0.455190	+	0.890395i	$0.349571\pi$
$62$	5.47295e7	0.470391
$63$	0	0
$64$	−3.75927e7	−0.280087
$65$	−5.31294e7	−0.369169
$66$	0	0
$67$	1.15591e8	0.700788	0.350394	−	0.936602i	$-0.386048\pi$
$67$	1.15591e8	0.700788	0.350394	+	0.936602i	$0.386048\pi$
$68$	−2.58028e8	−1.46344
$69$	0	0
$70$	−1.24620e7	−0.0620363
$71$	−1.86826e8	−0.872519	−0.436260	−	0.899821i	$-0.643697\pi$
$71$	−1.86826e8	−0.872519	−0.436260	+	0.899821i	$0.643697\pi$
$72$	0	0
$73$	1.59090e8	0.655677	0.327838	−	0.944734i	$-0.393680\pi$
$73$	1.59090e8	0.655677	0.327838	+	0.944734i	$0.393680\pi$
$74$	−1.61737e7	−0.0626998
$75$	0	0
$76$	1.17013e8	0.402323
$77$	3.57801e7	0.115993
$78$	0	0
$79$	2.05688e8	0.594138	0.297069	−	0.954856i	$-0.403991\pi$
$79$	2.05688e8	0.594138	0.297069	+	0.954856i	$0.403991\pi$
$80$	1.00606e8	0.274612
$81$	0	0
$82$	−1.17524e8	−0.287053
$83$	−5.94838e8	−1.37577	−0.687887	−	0.725818i	$-0.741462\pi$
$83$	−5.94838e8	−1.37577	−0.687887	+	0.725818i	$0.741462\pi$
$84$	0	0
$85$	3.64006e8	0.756351
$86$	−7.80236e7	−0.153809
$87$	0	0
$88$	1.18191e8	0.210093
$89$	−1.02347e9	−1.72910	−0.864552	−	0.502543i	$-0.832398\pi$
$89$	−1.02347e9	−1.72910	−0.864552	+	0.502543i	$0.832398\pi$
$90$	0	0
$91$	2.04102e8	0.312005
$92$	−3.40504e8	−0.495537
$93$	0	0
$94$	−1.85998e8	−0.245716
$95$	−1.65074e8	−0.207932
$96$	0	0
$97$	1.35649e9	1.55577	0.777883	−	0.628409i	$-0.216294\pi$
$97$	1.35649e9	1.55577	0.777883	+	0.628409i	$0.216294\pi$
$98$	4.78740e7	0.0524303
$99$	0	0
$100$	−1.73060e8	−0.173060
$101$	6.05703e8	0.579180	0.289590	−	0.957151i	$-0.406481\pi$
$101$	6.05703e8	0.579180	0.289590	+	0.957151i	$0.406481\pi$
$102$	0	0
$103$	−3.74165e8	−0.327564	−0.163782	−	0.986497i	$-0.552369\pi$
$103$	−3.74165e8	−0.327564	−0.163782	+	0.986497i	$0.552369\pi$
$104$	6.74201e8	0.565118
$105$	0	0
$106$	−2.76446e8	−0.212684
$107$	1.76404e9	1.30101	0.650507	−	0.759500i	$-0.274556\pi$
$107$	1.76404e9	1.30101	0.650507	+	0.759500i	$0.274556\pi$
$108$	0	0
$109$	−2.43042e8	−0.164916	−0.0824580	−	0.996595i	$-0.526277\pi$
$109$	−2.43042e8	−0.164916	−0.0824580	+	0.996595i	$0.526277\pi$
$110$	−7.73472e7	−0.0503706
$111$	0	0
$112$	−3.86488e8	−0.232089
$113$	−1.06491e9	−0.614411	−0.307206	−	0.951643i	$-0.599394\pi$
$113$	−1.06491e9	−0.614411	−0.307206	+	0.951643i	$0.599394\pi$
$114$	0	0
$115$	4.80357e8	0.256108
$116$	8.35088e7	0.0428224
$117$	0	0
$118$	7.56317e8	0.359116
$119$	−1.39837e9	−0.639233
$120$	0	0
$121$	−2.13587e9	−0.905819
$122$	8.17564e8	0.334120
$123$	0	0
$124$	−2.91974e9	−1.10904
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	−3.31480e9	−1.13068	−0.565341	−	0.824857i	$-0.691256\pi$
$127$	−3.31480e9	−1.13068	−0.565341	+	0.824857i	$0.691256\pi$
$128$	−3.07571e9	−1.01275
$129$	0	0
$130$	−4.41215e8	−0.135489
$131$	−5.40328e9	−1.60301	−0.801506	−	0.597986i	$-0.795968\pi$
$131$	−5.40328e9	−1.60301	−0.801506	+	0.597986i	$0.795968\pi$
$132$	0	0
$133$	6.34147e8	0.175735
$134$	9.59928e8	0.257198
$135$	0	0
$136$	−4.61916e9	−1.15781
$137$	−4.08237e9	−0.990078	−0.495039	−	0.868871i	$-0.664846\pi$
$137$	−4.08237e9	−0.990078	−0.495039	+	0.868871i	$0.664846\pi$
$138$	0	0
$139$	−7.54578e9	−1.71450	−0.857250	−	0.514900i	$-0.827829\pi$
$139$	−7.54578e9	−1.71450	−0.857250	+	0.514900i	$0.827829\pi$
$140$	6.64829e8	0.146263
$141$	0	0
$142$	−1.55150e9	−0.320225
$143$	1.26679e9	0.253333
$144$	0	0
$145$	−1.17808e8	−0.0221319
$146$	1.32117e9	0.240641
$147$	0	0
$148$	8.62843e8	0.147827
$149$	−4.86426e8	−0.0808498	−0.0404249	−	0.999183i	$-0.512871\pi$
$149$	−4.86426e8	−0.0808498	−0.0404249	+	0.999183i	$0.512871\pi$
$150$	0	0
$151$	6.20733e9	0.971647	0.485824	−	0.874057i	$-0.338520\pi$
$151$	6.20733e9	0.971647	0.485824	+	0.874057i	$0.338520\pi$
$152$	2.09475e9	0.318300
$153$	0	0
$154$	2.97137e8	0.0425710
$155$	4.11895e9	0.573184
$156$	0	0
$157$	−9.71573e9	−1.27622	−0.638112	−	0.769943i	$-0.720285\pi$
$157$	−9.71573e9	−1.27622	−0.638112	+	0.769943i	$0.720285\pi$
$158$	1.70814e9	0.218056
$159$	0	0
$160$	3.37344e9	0.406943
$161$	−1.84534e9	−0.216451
$162$	0	0
$163$	3.98507e9	0.442173	0.221086	−	0.975254i	$-0.429040\pi$
$163$	3.98507e9	0.442173	0.221086	+	0.975254i	$0.429040\pi$
$164$	6.26971e9	0.676785
$165$	0	0
$166$	−4.93985e9	−0.504925
$167$	−7.63594e9	−0.759694	−0.379847	−	0.925049i	$-0.624023\pi$
$167$	−7.63594e9	−0.759694	−0.379847	+	0.925049i	$0.624023\pi$
$168$	0	0
$169$	−3.37830e9	−0.318572
$170$	3.02290e9	0.277590
$171$	0	0
$172$	4.16245e9	0.362636
$173$	−1.39108e9	−0.118071	−0.0590355	−	0.998256i	$-0.518803\pi$
$173$	−1.39108e9	−0.118071	−0.0590355	+	0.998256i	$0.518803\pi$
$174$	0	0
$175$	−9.37891e8	−0.0755929
$176$	−2.39880e9	−0.188446
$177$	0	0
$178$	−8.49946e9	−0.634602
$179$	1.87616e9	0.136594	0.0682969	−	0.997665i	$-0.478243\pi$
$179$	1.87616e9	0.136594	0.0682969	+	0.997665i	$0.478243\pi$
$180$	0	0
$181$	−2.42614e9	−0.168021	−0.0840104	−	0.996465i	$-0.526773\pi$
$181$	−2.42614e9	−0.168021	−0.0840104	+	0.996465i	$0.526773\pi$
$182$	1.69497e9	0.114509
$183$	0	0
$184$	−6.09562e9	−0.392047
$185$	−1.21723e9	−0.0764014
$186$	0	0
$187$	−8.67917e9	−0.519028
$188$	9.92275e9	0.579324
$189$	0	0
$190$	−1.37086e9	−0.0763136
$191$	−9.41804e9	−0.512048	−0.256024	−	0.966670i	$-0.582413\pi$
$191$	−9.41804e9	−0.512048	−0.256024	+	0.966670i	$0.582413\pi$
$192$	0	0
$193$	−1.49145e10	−0.773749	−0.386874	−	0.922132i	$-0.626445\pi$
$193$	−1.49145e10	−0.773749	−0.386874	+	0.922132i	$0.626445\pi$
$194$	1.12650e10	0.570984
$195$	0	0
$196$	−2.55401e9	−0.123615
$197$	1.49680e10	0.708054	0.354027	−	0.935235i	$-0.384812\pi$
$197$	1.49680e10	0.708054	0.354027	+	0.935235i	$0.384812\pi$
$198$	0	0
$199$	−8.64658e9	−0.390846	−0.195423	−	0.980719i	$-0.562608\pi$
$199$	−8.64658e9	−0.390846	−0.195423	+	0.980719i	$0.562608\pi$
$200$	−3.09809e9	−0.136918
$201$	0	0
$202$	5.03008e9	0.212566
$203$	4.52571e8	0.0187048
$204$	0	0
$205$	−8.84484e9	−0.349782
$206$	−3.10727e9	−0.120220
$207$	0	0
$208$	−1.36836e10	−0.506890
$209$	3.93593e9	0.142689
$210$	0	0
$211$	4.04960e10	1.40651	0.703253	−	0.710940i	$-0.251730\pi$
$211$	4.04960e10	1.40651	0.703253	+	0.710940i	$0.251730\pi$
$212$	1.47480e10	0.501443
$213$	0	0
$214$	1.46495e10	0.477487
$215$	−5.87207e9	−0.187421
$216$	0	0
$217$	−1.58234e10	−0.484429
$218$	−2.01835e9	−0.0605261
$219$	0	0
$220$	4.12636e9	0.118759
$221$	−4.95090e10	−1.39611
$222$	0	0
$223$	−4.98356e10	−1.34948	−0.674742	−	0.738053i	$-0.735745\pi$
$223$	−4.98356e10	−1.34948	−0.674742	+	0.738053i	$0.735745\pi$
$224$	−1.29594e10	−0.343929
$225$	0	0
$226$	−8.84356e9	−0.225496
$227$	1.87758e10	0.469334	0.234667	−	0.972076i	$-0.424600\pi$
$227$	1.87758e10	0.469334	0.234667	+	0.972076i	$0.424600\pi$
$228$	0	0
$229$	7.34599e10	1.76519	0.882594	−	0.470136i	$-0.155795\pi$
$229$	7.34599e10	1.76519	0.882594	+	0.470136i	$0.155795\pi$
$230$	3.98914e9	0.0939947
$231$	0	0
$232$	1.49496e9	0.0338791
$233$	−5.23351e10	−1.16330	−0.581650	−	0.813439i	$-0.697592\pi$
$233$	−5.23351e10	−1.16330	−0.581650	+	0.813439i	$0.697592\pi$
$234$	0	0
$235$	−1.39983e10	−0.299412
$236$	−4.03485e10	−0.846687
$237$	0	0
$238$	−1.16128e10	−0.234606
$239$	−6.86346e9	−0.136067	−0.0680334	−	0.997683i	$-0.521672\pi$
$239$	−6.86346e9	−0.136067	−0.0680334	+	0.997683i	$0.521672\pi$
$240$	0	0
$241$	−5.02151e10	−0.958865	−0.479433	−	0.877579i	$-0.659157\pi$
$241$	−5.02151e10	−0.958865	−0.479433	+	0.877579i	$0.659157\pi$
$242$	−1.77374e10	−0.332446
$243$	0	0
$244$	−4.36159e10	−0.787753
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	2.24519e10	0.383811
$248$	−5.22686e10	−0.877421
$249$	0	0
$250$	2.02747e9	0.0328265
$251$	7.34378e9	0.116785	0.0583926	−	0.998294i	$-0.481402\pi$
$251$	7.34378e9	0.116785	0.0583926	+	0.998294i	$0.481402\pi$
$252$	0	0
$253$	−1.14534e10	−0.175748
$254$	−2.75279e10	−0.414974
$255$	0	0
$256$	−6.29489e9	−0.0916027
$257$	−9.49808e10	−1.35812	−0.679058	−	0.734085i	$-0.737611\pi$
$257$	−9.49808e10	−1.35812	−0.679058	+	0.734085i	$0.737611\pi$
$258$	0	0
$259$	4.67612e9	0.0645709
$260$	2.35382e10	0.319443
$261$	0	0
$262$	−4.48717e10	−0.588325
$263$	6.93155e10	0.893366	0.446683	−	0.894692i	$-0.352605\pi$
$263$	6.93155e10	0.893366	0.446683	+	0.894692i	$0.352605\pi$
$264$	0	0
$265$	−2.08054e10	−0.259161
$266$	5.26630e9	0.0644968
$267$	0	0
$268$	−5.12108e10	−0.606394
$269$	−9.53527e10	−1.11032	−0.555159	−	0.831744i	$-0.687343\pi$
$269$	−9.53527e10	−1.11032	−0.555159	+	0.831744i	$0.687343\pi$
$270$	0	0
$271$	−6.38320e10	−0.718914	−0.359457	−	0.933162i	$-0.617038\pi$
$271$	−6.38320e10	−0.718914	−0.359457	+	0.933162i	$0.617038\pi$
$272$	9.37503e10	1.03851
$273$	0	0
$274$	−3.39021e10	−0.363370
$275$	−5.82116e9	−0.0613780
$276$	0	0
$277$	−1.77106e11	−1.80748	−0.903742	−	0.428078i	$-0.859191\pi$
$277$	−1.77106e11	−1.80748	−0.903742	+	0.428078i	$0.859191\pi$
$278$	−6.26642e10	−0.629242
$279$	0	0
$280$	1.19016e10	0.115716
$281$	−1.61938e10	−0.154942	−0.0774710	−	0.996995i	$-0.524685\pi$
$281$	−1.61938e10	−0.154942	−0.0774710	+	0.996995i	$0.524685\pi$
$282$	0	0
$283$	1.05574e11	0.978404	0.489202	−	0.872170i	$-0.337288\pi$
$283$	1.05574e11	0.978404	0.489202	+	0.872170i	$0.337288\pi$
$284$	8.27705e10	0.754993
$285$	0	0
$286$	1.05201e10	0.0929764
$287$	3.39783e10	0.295620
$288$	0	0
$289$	2.20613e11	1.86034
$290$	−9.78339e8	−0.00812266
$291$	0	0
$292$	−7.04824e10	−0.567359
$293$	−1.71735e11	−1.36131	−0.680653	−	0.732606i	$-0.738304\pi$
$293$	−1.71735e11	−1.36131	−0.680653	+	0.732606i	$0.738304\pi$
$294$	0	0
$295$	5.69206e10	0.437593
$296$	1.54464e10	0.116954
$297$	0	0
$298$	−4.03954e9	−0.0296728
$299$	−6.53339e10	−0.472736
$300$	0	0
$301$	2.25581e10	0.158400
$302$	5.15489e10	0.356606
$303$	0	0
$304$	−4.25150e10	−0.285503
$305$	6.15300e10	0.407134
$306$	0	0
$307$	−2.84858e11	−1.83023	−0.915114	−	0.403194i	$-0.867900\pi$
$307$	−2.84858e11	−1.83023	−0.915114	+	0.403194i	$0.867900\pi$
$308$	−1.58518e10	−0.100369
$309$	0	0
$310$	3.42059e10	0.210365
$311$	−7.98673e10	−0.484113	−0.242057	−	0.970262i	$-0.577822\pi$
$311$	−7.98673e10	−0.484113	−0.242057	+	0.970262i	$0.577822\pi$
$312$	0	0
$313$	−1.43657e10	−0.0846016	−0.0423008	−	0.999105i	$-0.513469\pi$
$313$	−1.43657e10	−0.0846016	−0.0423008	+	0.999105i	$0.513469\pi$
$314$	−8.06846e10	−0.468390
$315$	0	0
$316$	−9.11271e10	−0.514109
$317$	−4.97249e9	−0.0276572	−0.0138286	−	0.999904i	$-0.504402\pi$
$317$	−4.97249e9	−0.0276572	−0.0138286	+	0.999904i	$0.504402\pi$
$318$	0	0
$319$	2.80895e9	0.0151875
$320$	−2.34954e10	−0.125259
$321$	0	0
$322$	−1.53247e10	−0.0794401
$323$	−1.53825e11	−0.786349
$324$	0	0
$325$	−3.32059e10	−0.165097
$326$	3.30942e10	0.162283
$327$	0	0
$328$	1.12239e11	0.535441
$329$	5.37757e10	0.253049
$330$	0	0
$331$	−1.22488e11	−0.560879	−0.280439	−	0.959872i	$-0.590480\pi$
$331$	−1.22488e11	−0.560879	−0.280439	+	0.959872i	$0.590480\pi$
$332$	2.63534e11	1.19046
$333$	0	0
$334$	−6.34129e10	−0.278817
$335$	7.22443e10	0.313402
$336$	0	0
$337$	−3.27544e10	−0.138336	−0.0691680	−	0.997605i	$-0.522034\pi$
$337$	−3.27544e10	−0.138336	−0.0691680	+	0.997605i	$0.522034\pi$
$338$	−2.80552e10	−0.116920
$339$	0	0
$340$	−1.61267e11	−0.654472
$341$	−9.82101e10	−0.393334
$342$	0	0
$343$	−1.38413e10	−0.0539949
$344$	7.45153e10	0.286901
$345$	0	0
$346$	−1.15522e10	−0.0433335
$347$	1.96183e11	0.726405	0.363203	−	0.931710i	$-0.381683\pi$
$347$	1.96183e11	0.726405	0.363203	+	0.931710i	$0.381683\pi$
$348$	0	0
$349$	4.28405e11	1.54575	0.772876	−	0.634557i	$-0.218818\pi$
$349$	4.28405e11	1.54575	0.772876	+	0.634557i	$0.218818\pi$
$350$	−7.78874e9	−0.0277435
$351$	0	0
$352$	−8.04346e10	−0.279255
$353$	−2.10417e11	−0.721264	−0.360632	−	0.932708i	$-0.617439\pi$
$353$	−2.10417e11	−0.721264	−0.360632	+	0.932708i	$0.617439\pi$
$354$	0	0
$355$	−1.16766e11	−0.390203
$356$	4.53434e11	1.49620
$357$	0	0
$358$	1.55806e10	0.0501315
$359$	2.64740e11	0.841190	0.420595	−	0.907248i	$-0.361821\pi$
$359$	2.64740e11	0.841190	0.420595	+	0.907248i	$0.361821\pi$
$360$	0	0
$361$	−2.52929e11	−0.783821
$362$	−2.01480e10	−0.0616656
$363$	0	0
$364$	−9.04243e10	−0.269978
$365$	9.94312e10	0.293228
$366$	0	0
$367$	−3.85439e11	−1.10907	−0.554534	−	0.832161i	$-0.687103\pi$
$367$	−3.85439e11	−1.10907	−0.554534	+	0.832161i	$0.687103\pi$
$368$	1.23717e11	0.351652
$369$	0	0
$370$	−1.01086e10	−0.0280402
$371$	7.99259e10	0.219031
$372$	0	0
$373$	−6.14857e10	−0.164469	−0.0822346	−	0.996613i	$-0.526206\pi$
$373$	−6.14857e10	−0.164469	−0.0822346	+	0.996613i	$0.526206\pi$
$374$	−7.20764e10	−0.190489
$375$	0	0
$376$	1.77635e11	0.458335
$377$	1.60232e10	0.0408520
$378$	0	0
$379$	3.85377e11	0.959421	0.479710	−	0.877427i	$-0.340742\pi$
$379$	3.85377e11	0.959421	0.479710	+	0.877427i	$0.340742\pi$
$380$	7.31334e10	0.179924
$381$	0	0
$382$	−7.82124e10	−0.187928
$383$	−2.30751e11	−0.547961	−0.273981	−	0.961735i	$-0.588340\pi$
$383$	−2.30751e11	−0.547961	−0.273981	+	0.961735i	$0.588340\pi$
$384$	0	0
$385$	2.23626e10	0.0518739
$386$	−1.23858e11	−0.283975
$387$	0	0
$388$	−6.00973e11	−1.34621
$389$	1.36395e11	0.302014	0.151007	−	0.988533i	$-0.451748\pi$
$389$	1.36395e11	0.302014	0.151007	+	0.988533i	$0.451748\pi$
$390$	0	0
$391$	4.47623e11	0.968539
$392$	−4.57213e10	−0.0977983
$393$	0	0
$394$	1.24302e11	0.259864
$395$	1.28555e11	0.265707
$396$	0	0
$397$	5.45803e11	1.10275	0.551377	−	0.834256i	$-0.314102\pi$
$397$	5.45803e11	1.10275	0.551377	+	0.834256i	$0.314102\pi$
$398$	−7.18058e10	−0.143445
$399$	0	0
$400$	6.28788e10	0.122810
$401$	2.89087e11	0.558315	0.279157	−	0.960245i	$-0.409945\pi$
$401$	2.89087e11	0.558315	0.279157	+	0.960245i	$0.409945\pi$
$402$	0	0
$403$	−5.60224e11	−1.05801
$404$	−2.68347e11	−0.501166
$405$	0	0
$406$	3.75839e9	0.00686490
$407$	2.90231e10	0.0524287
$408$	0	0
$409$	−1.92007e11	−0.339282	−0.169641	−	0.985506i	$-0.554261\pi$
$409$	−1.92007e11	−0.339282	−0.169641	+	0.985506i	$0.554261\pi$
$410$	−7.34522e10	−0.128374
$411$	0	0
$412$	1.65768e11	0.283442
$413$	−2.18666e11	−0.369833
$414$	0	0
$415$	−3.71774e11	−0.615265
$416$	−4.58826e11	−0.751153
$417$	0	0
$418$	3.26861e10	0.0523684
$419$	−1.41343e11	−0.224033	−0.112017	−	0.993706i	$-0.535731\pi$
$419$	−1.41343e11	−0.224033	−0.112017	+	0.993706i	$0.535731\pi$
$420$	0	0
$421$	8.37353e11	1.29909	0.649545	−	0.760323i	$-0.274959\pi$
$421$	8.37353e11	1.29909	0.649545	+	0.760323i	$0.274959\pi$
$422$	3.36301e11	0.516204
$423$	0	0
$424$	2.64016e11	0.396719
$425$	2.27504e11	0.338251
$426$	0	0
$427$	−2.36373e11	−0.344091
$428$	−7.81532e11	−1.12577
$429$	0	0
$430$	−4.87648e10	−0.0687857
$431$	−6.27369e11	−0.875740	−0.437870	−	0.899038i	$-0.644267\pi$
$431$	−6.27369e11	−0.875740	−0.437870	+	0.899038i	$0.644267\pi$
$432$	0	0
$433$	1.32269e12	1.80827	0.904133	−	0.427251i	$-0.140518\pi$
$433$	1.32269e12	1.80827	0.904133	+	0.427251i	$0.140518\pi$
$434$	−1.31406e11	−0.177791
$435$	0	0
$436$	1.07676e11	0.142702
$437$	−2.02993e11	−0.266266
$438$	0	0
$439$	9.03428e11	1.16092	0.580461	−	0.814288i	$-0.302872\pi$
$439$	9.03428e11	1.16092	0.580461	+	0.814288i	$0.302872\pi$
$440$	7.38693e10	0.0939565
$441$	0	0
$442$	−4.11149e11	−0.512388
$443$	−2.06273e11	−0.254464	−0.127232	−	0.991873i	$-0.540609\pi$
$443$	−2.06273e11	−0.254464	−0.127232	+	0.991873i	$0.540609\pi$
$444$	0	0
$445$	−6.39670e11	−0.773279
$446$	−4.13861e11	−0.495277
$447$	0	0
$448$	9.02601e10	0.105863
$449$	−4.66846e11	−0.542082	−0.271041	−	0.962568i	$-0.587368\pi$
$449$	−4.66846e11	−0.542082	−0.271041	+	0.962568i	$0.587368\pi$
$450$	0	0
$451$	2.10892e11	0.240030
$452$	4.71791e11	0.531651
$453$	0	0
$454$	1.55924e11	0.172251
$455$	1.27564e11	0.139533
$456$	0	0
$457$	8.24857e11	0.884618	0.442309	−	0.896863i	$-0.354159\pi$
$457$	8.24857e11	0.884618	0.442309	+	0.896863i	$0.354159\pi$
$458$	6.10050e11	0.647845
$459$	0	0
$460$	−2.12815e11	−0.221611
$461$	−8.87190e10	−0.0914876	−0.0457438	−	0.998953i	$-0.514566\pi$
$461$	−8.87190e10	−0.0914876	−0.0457438	+	0.998953i	$0.514566\pi$
$462$	0	0
$463$	1.10664e12	1.11916	0.559581	−	0.828776i	$-0.310962\pi$
$463$	1.10664e12	1.11916	0.559581	+	0.828776i	$0.310962\pi$
$464$	−3.03416e10	−0.0303884
$465$	0	0
$466$	−4.34618e11	−0.426944
$467$	1.11998e12	1.08964	0.544821	−	0.838552i	$-0.316597\pi$
$467$	1.11998e12	1.08964	0.544821	+	0.838552i	$0.316597\pi$
$468$	0	0
$469$	−2.77534e11	−0.264873
$470$	−1.16249e11	−0.109888
$471$	0	0
$472$	−7.22310e11	−0.669860
$473$	1.40010e11	0.128613
$474$	0	0
$475$	−1.03171e11	−0.0929901
$476$	6.19525e11	0.553130
$477$	0	0
$478$	−5.69978e10	−0.0499381
$479$	−6.71179e11	−0.582544	−0.291272	−	0.956640i	$-0.594078\pi$
$479$	−6.71179e11	−0.582544	−0.291272	+	0.956640i	$0.594078\pi$
$480$	0	0
$481$	1.65558e11	0.141025
$482$	−4.17013e11	−0.351915
$483$	0	0
$484$	9.46266e11	0.783807
$485$	8.47807e11	0.695760
$486$	0	0
$487$	2.01428e12	1.62270	0.811352	−	0.584558i	$-0.198732\pi$
$487$	2.01428e12	1.62270	0.811352	+	0.584558i	$0.198732\pi$
$488$	−7.80802e11	−0.623235
$489$	0	0
$490$	2.99212e10	0.0234475
$491$	−1.13665e12	−0.882594	−0.441297	−	0.897361i	$-0.645481\pi$
$491$	−1.13665e12	−0.882594	−0.441297	+	0.897361i	$0.645481\pi$
$492$	0	0
$493$	−1.09780e11	−0.0836973
$494$	1.86452e11	0.140863
$495$	0	0
$496$	1.06084e12	0.787015
$497$	4.48570e11	0.329781
$498$	0	0
$499$	1.82340e12	1.31653	0.658263	−	0.752788i	$-0.271292\pi$
$499$	1.82340e12	1.31653	0.658263	+	0.752788i	$0.271292\pi$
$500$	−1.08163e11	−0.0773950
$501$	0	0
$502$	6.09867e10	0.0428616
$503$	5.72073e11	0.398470	0.199235	−	0.979952i	$-0.436154\pi$
$503$	5.72073e11	0.398470	0.199235	+	0.979952i	$0.436154\pi$
$504$	0	0
$505$	3.78564e11	0.259017
$506$	−9.51149e10	−0.0645017
$507$	0	0
$508$	1.46857e12	0.978382
$509$	−2.95989e12	−1.95455	−0.977273	−	0.211986i	$-0.932007\pi$
$509$	−2.95989e12	−1.95455	−0.977273	+	0.211986i	$0.932007\pi$
$510$	0	0
$511$	−3.81975e11	−0.247822
$512$	1.52249e12	0.979128
$513$	0	0
$514$	−7.88771e11	−0.498445
$515$	−2.33853e11	−0.146491
$516$	0	0
$517$	3.33767e11	0.205464
$518$	3.88330e10	0.0236983
$519$	0	0
$520$	4.21376e11	0.252729
$521$	2.85602e12	1.69821	0.849107	−	0.528222i	$-0.177141\pi$
$521$	2.85602e12	1.69821	0.849107	+	0.528222i	$0.177141\pi$
$522$	0	0
$523$	3.83909e11	0.224373	0.112187	−	0.993687i	$-0.464215\pi$
$523$	3.83909e11	0.224373	0.112187	+	0.993687i	$0.464215\pi$
$524$	2.39384e12	1.38709
$525$	0	0
$526$	5.75632e11	0.327876
$527$	3.83827e12	2.16764
$528$	0	0
$529$	−1.21045e12	−0.672043
$530$	−1.72779e11	−0.0951150
$531$	0	0
$532$	−2.80949e11	−0.152064
$533$	1.20300e12	0.645643
$534$	0	0
$535$	1.10253e12	0.581831
$536$	−9.16764e11	−0.479751
$537$	0	0
$538$	−7.91859e11	−0.407500
$539$	−8.59081e10	−0.0438414
$540$	0	0
$541$	−8.49593e11	−0.426406	−0.213203	−	0.977008i	$-0.568390\pi$
$541$	−8.49593e11	−0.426406	−0.213203	+	0.977008i	$0.568390\pi$
$542$	−5.30095e11	−0.263850
$543$	0	0
$544$	3.14356e12	1.53896
$545$	−1.51901e11	−0.0737526
$546$	0	0
$547$	−1.94856e12	−0.930618	−0.465309	−	0.885148i	$-0.654057\pi$
$547$	−1.94856e12	−0.930618	−0.465309	+	0.885148i	$0.654057\pi$
$548$	1.80863e12	0.856717
$549$	0	0
$550$	−4.83420e10	−0.0225264
$551$	4.97843e10	0.0230097
$552$	0	0
$553$	−4.93858e11	−0.224563
$554$	−1.47078e12	−0.663368
$555$	0	0
$556$	3.34304e12	1.48356
$557$	3.99623e12	1.75915	0.879574	−	0.475762i	$-0.157828\pi$
$557$	3.99623e12	1.75915	0.879574	+	0.475762i	$0.157828\pi$
$558$	0	0
$559$	7.98668e11	0.345950
$560$	−2.41555e11	−0.103793
$561$	0	0
$562$	−1.34482e11	−0.0568656
$563$	2.64531e12	1.10966	0.554829	−	0.831964i	$-0.312784\pi$
$563$	2.64531e12	1.10966	0.554829	+	0.831964i	$0.312784\pi$
$564$	0	0
$565$	−6.65568e11	−0.274773
$566$	8.76743e11	0.359086
$567$	0	0
$568$	1.48174e12	0.597316
$569$	2.09982e12	0.839801	0.419900	−	0.907570i	$-0.362065\pi$
$569$	2.09982e12	0.839801	0.419900	+	0.907570i	$0.362065\pi$
$570$	0	0
$571$	−4.39896e12	−1.73176	−0.865881	−	0.500251i	$-0.833241\pi$
$571$	−4.39896e12	−1.73176	−0.865881	+	0.500251i	$0.833241\pi$
$572$	−5.61232e11	−0.219210
$573$	0	0
$574$	2.82174e11	0.108496
$575$	3.00223e11	0.114535
$576$	0	0
$577$	−3.17807e12	−1.19364	−0.596818	−	0.802377i	$-0.703569\pi$
$577$	−3.17807e12	−1.19364	−0.596818	+	0.802377i	$0.703569\pi$
$578$	1.83209e12	0.682765
$579$	0	0
$580$	5.21930e10	0.0191508
$581$	1.42821e12	0.519994
$582$	0	0
$583$	4.96072e11	0.177843
$584$	−1.26176e12	−0.448868
$585$	0	0
$586$	−1.42618e12	−0.499615
$587$	−4.91687e12	−1.70930	−0.854648	−	0.519207i	$-0.826227\pi$
$587$	−4.91687e12	−1.70930	−0.854648	+	0.519207i	$0.826227\pi$
$588$	0	0
$589$	−1.74062e12	−0.595917
$590$	4.72698e11	0.160602
$591$	0	0
$592$	−3.13500e11	−0.104904
$593$	−4.74114e12	−1.57448	−0.787239	−	0.616648i	$-0.788490\pi$
$593$	−4.74114e12	−1.57448	−0.787239	+	0.616648i	$0.788490\pi$
$594$	0	0
$595$	−8.73979e11	−0.285874
$596$	2.15504e11	0.0699595
$597$	0	0
$598$	−5.42568e11	−0.173500
$599$	−2.19748e12	−0.697436	−0.348718	−	0.937228i	$-0.613383\pi$
$599$	−2.19748e12	−0.697436	−0.348718	+	0.937228i	$0.613383\pi$
$600$	0	0
$601$	3.12379e12	0.976667	0.488334	−	0.872657i	$-0.337605\pi$
$601$	3.12379e12	0.976667	0.488334	+	0.872657i	$0.337605\pi$
$602$	1.87335e11	0.0581345
$603$	0	0
$604$	−2.75006e12	−0.840768
$605$	−1.33492e12	−0.405094
$606$	0	0
$607$	5.39395e12	1.61272	0.806358	−	0.591428i	$-0.201436\pi$
$607$	5.39395e12	1.61272	0.806358	+	0.591428i	$0.201436\pi$
$608$	−1.42558e12	−0.423083
$609$	0	0
$610$	5.10977e11	0.149423
$611$	1.90392e12	0.552667
$612$	0	0
$613$	−3.40302e12	−0.973402	−0.486701	−	0.873569i	$-0.661800\pi$
$613$	−3.40302e12	−0.973402	−0.486701	+	0.873569i	$0.661800\pi$
$614$	−2.36561e12	−0.671716
$615$	0	0
$616$	−2.83776e11	−0.0794077
$617$	1.74671e12	0.485220	0.242610	−	0.970124i	$-0.421997\pi$
$617$	1.74671e12	0.485220	0.242610	+	0.970124i	$0.421997\pi$
$618$	0	0
$619$	−1.47701e12	−0.404368	−0.202184	−	0.979348i	$-0.564804\pi$
$619$	−1.47701e12	−0.404368	−0.202184	+	0.979348i	$0.564804\pi$
$620$	−1.82484e12	−0.495977
$621$	0	0
$622$	−6.63260e11	−0.177675
$623$	2.45736e12	0.653540
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	−1.19301e11	−0.0310498
$627$	0	0
$628$	4.30441e12	1.10432
$629$	−1.13429e12	−0.288931
$630$	0	0
$631$	5.31333e12	1.33424	0.667121	−	0.744949i	$-0.267526\pi$
$631$	5.31333e12	1.33424	0.667121	+	0.744949i	$0.267526\pi$
$632$	−1.63134e12	−0.406740
$633$	0	0
$634$	−4.12942e10	−0.0101505
$635$	−2.07175e12	−0.505657
$636$	0	0
$637$	−4.90049e11	−0.117927
$638$	2.33270e10	0.00557398
$639$	0	0
$640$	−1.92232e12	−0.452914
$641$	−1.02026e12	−0.238699	−0.119349	−	0.992852i	$-0.538081\pi$
$641$	−1.02026e12	−0.238699	−0.119349	+	0.992852i	$0.538081\pi$
$642$	0	0
$643$	−6.29061e12	−1.45125	−0.725627	−	0.688088i	$-0.758450\pi$
$643$	−6.29061e12	−1.45125	−0.725627	+	0.688088i	$0.758450\pi$
$644$	8.17549e11	0.187296
$645$	0	0
$646$	−1.27744e12	−0.288599
$647$	6.48657e12	1.45528	0.727639	−	0.685960i	$-0.240618\pi$
$647$	6.48657e12	1.45528	0.727639	+	0.685960i	$0.240618\pi$
$648$	0	0
$649$	−1.35718e12	−0.300288
$650$	−2.75759e11	−0.0605927
$651$	0	0
$652$	−1.76553e12	−0.382613
$653$	−6.85042e12	−1.47437	−0.737187	−	0.675689i	$-0.763846\pi$
$653$	−6.85042e12	−1.47437	−0.737187	+	0.675689i	$0.763846\pi$
$654$	0	0
$655$	−3.37705e12	−0.716889
$656$	−2.27800e12	−0.480271
$657$	0	0
$658$	4.46582e11	0.0928720
$659$	6.33720e12	1.30892	0.654460	−	0.756097i	$-0.272896\pi$
$659$	6.33720e12	1.30892	0.654460	+	0.756097i	$0.272896\pi$
$660$	0	0
$661$	1.13408e12	0.231066	0.115533	−	0.993304i	$-0.463142\pi$
$661$	1.13408e12	0.231066	0.115533	+	0.993304i	$0.463142\pi$
$662$	−1.01721e12	−0.205849
$663$	0	0
$664$	4.71773e12	0.941839
$665$	3.96342e11	0.0785910
$666$	0	0
$667$	−1.44870e11	−0.0283408
$668$	3.38299e12	0.657365
$669$	0	0
$670$	5.99955e11	0.115022
$671$	−1.46709e12	−0.279386
$672$	0	0
$673$	−8.81447e12	−1.65626	−0.828129	−	0.560537i	$-0.810595\pi$
$673$	−8.81447e12	−1.65626	−0.828129	+	0.560537i	$0.810595\pi$
$674$	−2.72010e11	−0.0507709
$675$	0	0
$676$	1.49670e12	0.275661
$677$	−7.77532e12	−1.42256	−0.711278	−	0.702911i	$-0.751883\pi$
$677$	−7.77532e12	−1.42256	−0.711278	+	0.702911i	$0.751883\pi$
$678$	0	0
$679$	−3.25694e12	−0.588024
$680$	−2.88697e12	−0.517789
$681$	0	0
$682$	−8.15588e11	−0.144358
$683$	1.63173e12	0.286916	0.143458	−	0.989656i	$-0.454178\pi$
$683$	1.63173e12	0.286916	0.143458	+	0.989656i	$0.454178\pi$
$684$	0	0
$685$	−2.55148e12	−0.442776
$686$	−1.14945e11	−0.0198168
$687$	0	0
$688$	−1.51236e12	−0.257340
$689$	2.82977e12	0.478370
$690$	0	0
$691$	−4.11023e12	−0.685828	−0.342914	−	0.939367i	$-0.611414\pi$
$691$	−4.11023e12	−0.685828	−0.342914	+	0.939367i	$0.611414\pi$
$692$	6.16295e11	0.102167
$693$	0	0
$694$	1.62921e12	0.266599
$695$	−4.71611e12	−0.766748
$696$	0	0
$697$	−8.24211e12	−1.32279
$698$	3.55770e12	0.567309
$699$	0	0
$700$	4.15518e11	0.0654107
$701$	−1.05286e13	−1.64680	−0.823398	−	0.567464i	$-0.807924\pi$
$701$	−1.05286e13	−1.64680	−0.823398	+	0.567464i	$0.807924\pi$
$702$	0	0
$703$	5.14389e11	0.0794315
$704$	5.60213e11	0.0859560
$705$	0	0
$706$	−1.74741e12	−0.264712
$707$	−1.45429e12	−0.218909
$708$	0	0
$709$	−1.03943e13	−1.54485	−0.772425	−	0.635106i	$-0.780956\pi$
$709$	−1.03943e13	−1.54485	−0.772425	+	0.635106i	$0.780956\pi$
$710$	−9.69689e11	−0.143209
$711$	0	0
$712$	8.11728e12	1.18372
$713$	5.06513e12	0.733985
$714$	0	0
$715$	7.91744e11	0.113294
$716$	−8.31203e11	−0.118195
$717$	0	0
$718$	2.19854e12	0.308727
$719$	−2.48589e11	−0.0346898	−0.0173449	−	0.999850i	$-0.505521\pi$
$719$	−2.48589e11	−0.0346898	−0.0173449	+	0.999850i	$0.505521\pi$
$720$	0	0
$721$	8.98371e11	0.123807
$722$	−2.10046e12	−0.287672
$723$	0	0
$724$	1.07487e12	0.145389
$725$	−7.36299e10	−0.00989767
$726$	0	0
$727$	6.33150e12	0.840624	0.420312	−	0.907380i	$-0.361921\pi$
$727$	6.33150e12	0.840624	0.420312	+	0.907380i	$0.361921\pi$
$728$	−1.61876e12	−0.213595
$729$	0	0
$730$	8.25729e11	0.107618
$731$	−5.47192e12	−0.708780
$732$	0	0
$733$	2.64023e10	0.00337812	0.00168906	−	0.999999i	$-0.499462\pi$
$733$	2.64023e10	0.00337812	0.00168906	+	0.999999i	$0.499462\pi$
$734$	−3.20089e12	−0.407041
$735$	0	0
$736$	4.14837e12	0.521107
$737$	−1.72256e12	−0.215065
$738$	0	0
$739$	−2.86810e11	−0.0353748	−0.0176874	−	0.999844i	$-0.505630\pi$
$739$	−2.86810e11	−0.0353748	−0.0176874	+	0.999844i	$0.505630\pi$
$740$	5.39277e11	0.0661103
$741$	0	0
$742$	6.63747e11	0.0803868
$743$	−6.53255e12	−0.786381	−0.393190	−	0.919457i	$-0.628629\pi$
$743$	−6.53255e12	−0.786381	−0.393190	+	0.919457i	$0.628629\pi$
$744$	0	0
$745$	−3.04016e11	−0.0361571
$746$	−5.10610e11	−0.0603621
$747$	0	0
$748$	3.84517e12	0.449116
$749$	−4.23547e12	−0.491737
$750$	0	0
$751$	−1.25151e13	−1.43567	−0.717833	−	0.696216i	$-0.754866\pi$
$751$	−1.25151e13	−1.43567	−0.717833	+	0.696216i	$0.754866\pi$
$752$	−3.60527e12	−0.411110
$753$	0	0
$754$	1.33065e11	0.0149932
$755$	3.87958e12	0.434534
$756$	0	0
$757$	4.70847e12	0.521132	0.260566	−	0.965456i	$-0.416091\pi$
$757$	4.70847e12	0.521132	0.260566	+	0.965456i	$0.416091\pi$
$758$	3.20037e12	0.352119
$759$	0	0
$760$	1.30922e12	0.142348
$761$	3.14056e12	0.339450	0.169725	−	0.985491i	$-0.445712\pi$
$761$	3.14056e12	0.339450	0.169725	+	0.985491i	$0.445712\pi$
$762$	0	0
$763$	5.83545e11	0.0623324
$764$	4.17252e12	0.443076
$765$	0	0
$766$	−1.91628e12	−0.201108
$767$	−7.74184e12	−0.807728
$768$	0	0
$769$	1.54826e12	0.159652	0.0798260	−	0.996809i	$-0.474564\pi$
$769$	1.54826e12	0.159652	0.0798260	+	0.996809i	$0.474564\pi$
$770$	1.85711e11	0.0190383
$771$	0	0
$772$	6.60763e12	0.669527
$773$	3.03796e12	0.306038	0.153019	−	0.988223i	$-0.451100\pi$
$773$	3.03796e12	0.306038	0.153019	+	0.988223i	$0.451100\pi$
$774$	0	0
$775$	2.57434e12	0.256336
$776$	−1.07585e13	−1.06506
$777$	0	0
$778$	1.13270e12	0.110843
$779$	3.73773e12	0.363655
$780$	0	0
$781$	2.78412e12	0.267767
$782$	3.71730e12	0.355465
$783$	0	0
$784$	9.27958e11	0.0877215
$785$	−6.07233e12	−0.570745
$786$	0	0
$787$	1.23112e13	1.14397	0.571983	−	0.820266i	$-0.306174\pi$
$787$	1.23112e13	1.14397	0.571983	+	0.820266i	$0.306174\pi$
$788$	−6.63136e12	−0.612681
$789$	0	0
$790$	1.06759e12	0.0975175
$791$	2.55684e12	0.232226
$792$	0	0
$793$	−8.36877e12	−0.751506
$794$	4.53264e12	0.404724
$795$	0	0
$796$	3.83074e12	0.338200
$797$	−3.93564e12	−0.345504	−0.172752	−	0.984965i	$-0.555266\pi$
$797$	−3.93564e12	−0.345504	−0.172752	+	0.984965i	$0.555266\pi$
$798$	0	0
$799$	−1.30444e13	−1.13230
$800$	2.10840e12	0.181990
$801$	0	0
$802$	2.40073e12	0.204908
$803$	−2.37079e12	−0.201221
$804$	0	0
$805$	−1.15334e12	−0.0967998
$806$	−4.65239e12	−0.388301
$807$	0	0
$808$	−4.80390e12	−0.396500
$809$	1.20730e13	0.990936	0.495468	−	0.868626i	$-0.334997\pi$
$809$	1.20730e13	0.990936	0.495468	+	0.868626i	$0.334997\pi$
$810$	0	0
$811$	3.10097e12	0.251712	0.125856	−	0.992049i	$-0.459832\pi$
$811$	3.10097e12	0.251712	0.125856	+	0.992049i	$0.459832\pi$
$812$	−2.00505e11	−0.0161853
$813$	0	0
$814$	2.41023e11	0.0192419
$815$	2.49067e12	0.197746
$816$	0	0
$817$	2.48147e12	0.194854
$818$	−1.59452e12	−0.124521
$819$	0	0
$820$	3.91857e12	0.302667
$821$	2.69315e12	0.206879	0.103439	−	0.994636i	$-0.467015\pi$
$821$	2.69315e12	0.206879	0.103439	+	0.994636i	$0.467015\pi$
$822$	0	0
$823$	−7.54132e12	−0.572992	−0.286496	−	0.958081i	$-0.592490\pi$
$823$	−7.54132e12	−0.572992	−0.286496	+	0.958081i	$0.592490\pi$
$824$	2.96755e12	0.224246
$825$	0	0
$826$	−1.81592e12	−0.135733
$827$	−3.97671e12	−0.295630	−0.147815	−	0.989015i	$-0.547224\pi$
$827$	−3.97671e12	−0.295630	−0.147815	+	0.989015i	$0.547224\pi$
$828$	0	0
$829$	−2.22856e13	−1.63881	−0.819405	−	0.573215i	$-0.805696\pi$
$829$	−2.22856e13	−1.63881	−0.819405	+	0.573215i	$0.805696\pi$
$830$	−3.08741e12	−0.225810
$831$	0	0
$832$	3.19565e12	0.231208
$833$	3.35748e12	0.241608
$834$	0	0
$835$	−4.77246e12	−0.339745
$836$	−1.74376e12	−0.123469
$837$	0	0
$838$	−1.17379e12	−0.0822229
$839$	1.40634e13	0.979857	0.489929	−	0.871762i	$-0.337023\pi$
$839$	1.40634e13	0.979857	0.489929	+	0.871762i	$0.337023\pi$
$840$	0	0
$841$	−1.44716e13	−0.997551
$842$	6.95383e12	0.476782
$843$	0	0
$844$	−1.79412e13	−1.21705
$845$	−2.11143e12	−0.142470
$846$	0	0
$847$	5.12823e12	0.342367
$848$	−5.35845e12	−0.355843
$849$	0	0
$850$	1.88931e12	0.124142
$851$	−1.49685e12	−0.0978351
$852$	0	0
$853$	−3.03343e13	−1.96184	−0.980920	−	0.194411i	$-0.937720\pi$
$853$	−3.03343e13	−1.96184	−0.980920	+	0.194411i	$0.937720\pi$
$854$	−1.96297e12	−0.126285
$855$	0	0
$856$	−1.39908e13	−0.890659
$857$	2.53281e13	1.60394	0.801971	−	0.597363i	$-0.203785\pi$
$857$	2.53281e13	1.60394	0.801971	+	0.597363i	$0.203785\pi$
$858$	0	0
$859$	6.84041e12	0.428660	0.214330	−	0.976761i	$-0.431243\pi$
$859$	6.84041e12	0.428660	0.214330	+	0.976761i	$0.431243\pi$
$860$	2.60153e12	0.162176
$861$	0	0
$862$	−5.21000e12	−0.321407
$863$	−1.75856e13	−1.07922	−0.539608	−	0.841916i	$-0.681428\pi$
$863$	−1.75856e13	−1.07922	−0.539608	+	0.841916i	$0.681428\pi$
$864$	0	0
$865$	−8.69422e11	−0.0528030
$866$	1.09843e13	0.663655
$867$	0	0
$868$	7.01030e12	0.419177
$869$	−3.06520e12	−0.182335
$870$	0	0
$871$	−9.82604e12	−0.578491
$872$	1.92760e12	0.112899
$873$	0	0
$874$	−1.68576e12	−0.0977227
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	8.22398e12	0.469444	0.234722	−	0.972063i	$-0.424582\pi$
$877$	8.22398e12	0.469444	0.234722	+	0.972063i	$0.424582\pi$
$878$	7.50255e12	0.426072
$879$	0	0
$880$	−1.49925e12	−0.0842755
$881$	6.30199e12	0.352441	0.176220	−	0.984351i	$-0.443613\pi$
$881$	6.30199e12	0.352441	0.176220	+	0.984351i	$0.443613\pi$
$882$	0	0
$883$	4.44356e12	0.245985	0.122992	−	0.992408i	$-0.460751\pi$
$883$	4.44356e12	0.245985	0.122992	+	0.992408i	$0.460751\pi$
$884$	2.19342e13	1.20805
$885$	0	0
$886$	−1.71300e12	−0.0933912
$887$	−1.23069e13	−0.667562	−0.333781	−	0.942651i	$-0.608325\pi$
$887$	−1.23069e13	−0.667562	−0.333781	+	0.942651i	$0.608325\pi$
$888$	0	0
$889$	7.95884e12	0.427358
$890$	−5.31216e12	−0.283802
$891$	0	0
$892$	2.20789e13	1.16771
$893$	5.91551e12	0.311287
$894$	0	0
$895$	1.17260e12	0.0610866
$896$	7.38479e12	0.382782
$897$	0	0
$898$	−3.87694e12	−0.198951
$899$	−1.24223e12	−0.0634281
$900$	0	0
$901$	−1.93876e13	−0.980082
$902$	1.75136e12	0.0880938
$903$	0	0
$904$	8.44591e12	0.420619
$905$	−1.51634e12	−0.0751412
$906$	0	0
$907$	−3.62565e12	−0.177891	−0.0889453	−	0.996037i	$-0.528350\pi$
$907$	−3.62565e12	−0.177891	−0.0889453	+	0.996037i	$0.528350\pi$
$908$	−8.31834e12	−0.406116
$909$	0	0
$910$	1.05936e12	0.0512101
$911$	−9.19296e12	−0.442204	−0.221102	−	0.975251i	$-0.570965\pi$
$911$	−9.19296e12	−0.442204	−0.221102	+	0.975251i	$0.570965\pi$
$912$	0	0
$913$	8.86438e12	0.422211
$914$	6.85005e12	0.324665
$915$	0	0
$916$	−3.25453e13	−1.52742
$917$	1.29733e13	0.605882
$918$	0	0
$919$	5.32373e12	0.246205	0.123102	−	0.992394i	$-0.460716\pi$
$919$	5.32373e12	0.246205	0.123102	+	0.992394i	$0.460716\pi$
$920$	−3.80976e12	−0.175329
$921$	0	0
$922$	−7.36769e11	−0.0335770
$923$	1.58815e13	0.720253
$924$	0	0
$925$	−7.60771e11	−0.0341677
$926$	9.19014e12	0.410745
$927$	0	0
$928$	−1.01739e12	−0.0450320
$929$	2.28515e13	1.00657	0.503285	−	0.864121i	$-0.332125\pi$
$929$	2.28515e13	1.00657	0.503285	+	0.864121i	$0.332125\pi$
$930$	0	0
$931$	−1.52259e12	−0.0664215
$932$	2.31863e13	1.00661
$933$	0	0
$934$	9.30090e12	0.399912
$935$	−5.42448e12	−0.232116
$936$	0	0
$937$	4.49728e13	1.90600	0.952998	−	0.302976i	$-0.0979801\pi$
$937$	4.49728e13	1.90600	0.952998	+	0.302976i	$0.0979801\pi$
$938$	−2.30479e12	−0.0972116
$939$	0	0
$940$	6.20172e12	0.259082
$941$	−4.05411e13	−1.68555	−0.842775	−	0.538265i	$-0.819080\pi$
$941$	−4.05411e13	−1.68555	−0.842775	+	0.538265i	$0.819080\pi$
$942$	0	0
$943$	−1.08766e13	−0.447910
$944$	1.46600e13	0.600840
$945$	0	0
$946$	1.16272e12	0.0472026
$947$	−3.97581e13	−1.60639	−0.803194	−	0.595718i	$-0.796868\pi$
$947$	−3.97581e13	−1.60639	−0.803194	+	0.595718i	$0.796868\pi$
$948$	0	0
$949$	−1.35238e13	−0.541252
$950$	−8.56787e11	−0.0341285
$951$	0	0
$952$	1.10906e13	0.437612
$953$	−2.02769e13	−0.796311	−0.398155	−	0.917318i	$-0.630350\pi$
$953$	−2.02769e13	−0.796311	−0.398155	+	0.917318i	$0.630350\pi$
$954$	0	0
$955$	−5.88628e12	−0.228995
$956$	3.04075e12	0.117739
$957$	0	0
$958$	−5.57383e12	−0.213801
$959$	9.80176e12	0.374214
$960$	0	0
$961$	1.69927e13	0.642698
$962$	1.37488e12	0.0517578
$963$	0	0
$964$	2.22470e13	0.829708
$965$	−9.32155e12	−0.346031
$966$	0	0
$967$	−2.76625e13	−1.01736	−0.508678	−	0.860957i	$-0.669866\pi$
$967$	−2.76625e13	−1.01736	−0.508678	+	0.860957i	$0.669866\pi$
$968$	1.69399e13	0.620113
$969$	0	0
$970$	7.04064e12	0.255352
$971$	−4.14078e13	−1.49484	−0.747421	−	0.664351i	$-0.768708\pi$
$971$	−4.14078e13	−1.49484	−0.747421	+	0.664351i	$0.768708\pi$
$972$	0	0
$973$	1.81174e13	0.648020
$974$	1.67276e13	0.595552
$975$	0	0
$976$	1.58471e13	0.559019
$977$	−1.16278e13	−0.408292	−0.204146	−	0.978940i	$-0.565442\pi$
$977$	−1.16278e13	−0.408292	−0.204146	+	0.978940i	$0.565442\pi$
$978$	0	0
$979$	1.52520e13	0.530644
$980$	−1.59625e12	−0.0552821
$981$	0	0
$982$	−9.43936e12	−0.323922
$983$	1.24150e12	0.0424089	0.0212044	−	0.999775i	$-0.493250\pi$
$983$	1.24150e12	0.0424089	0.0212044	+	0.999775i	$0.493250\pi$
$984$	0	0
$985$	9.35502e12	0.316652
$986$	−9.11670e11	−0.0307179
$987$	0	0
$988$	−9.94697e12	−0.332112
$989$	−7.22096e12	−0.240000
$990$	0	0
$991$	4.90415e13	1.61522	0.807611	−	0.589715i	$-0.200760\pi$
$991$	4.90415e13	1.61522	0.807611	+	0.589715i	$0.200760\pi$
$992$	3.55713e13	1.16626
$993$	0	0
$994$	3.72516e12	0.121034
$995$	−5.40412e12	−0.174792
$996$	0	0
$997$	9.18652e12	0.294458	0.147229	−	0.989102i	$-0.452965\pi$
$997$	9.18652e12	0.294458	0.147229	+	0.989102i	$0.452965\pi$
$998$	1.51425e13	0.483180
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.c.1.3		4
3.2	odd	2		105.10.a.f.1.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.f.1.2	✓	4	3.2	odd	2
315.10.a.c.1.3		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q+8.30453 q^{2} -443.035 q^{4} +625.000 q^{5} -2401.00 q^{7} -7931.11 q^{8} +5190.33 q^{10} -14902.2 q^{11} -85007.1 q^{13} -19939.2 q^{14} +160970. q^{16} +582410. q^{17} -264118. q^{19} -276897. q^{20} -123756. q^{22} +768571. q^{23} +390625. q^{25} -705944. q^{26} +1.06373e6 q^{28} -188493. q^{29} +6.59032e6 q^{31} +5.39751e6 q^{32} +4.83664e6 q^{34} -1.50062e6 q^{35} -1.94757e6 q^{37} -2.19338e6 q^{38} -4.95695e6 q^{40} -1.41517e7 q^{41} -9.39531e6 q^{43} +6.60218e6 q^{44} +6.38262e6 q^{46} -2.23972e7 q^{47} +5.76480e6 q^{49} +3.24396e6 q^{50} +3.76611e7 q^{52} -3.32886e7 q^{53} -9.31386e6 q^{55} +1.90426e7 q^{56} -1.56534e6 q^{58} +9.10729e7 q^{59} +9.84479e7 q^{61} +5.47295e7 q^{62} -3.75927e7 q^{64} -5.31294e7 q^{65} +1.15591e8 q^{67} -2.58028e8 q^{68} -1.24620e7 q^{70} -1.86826e8 q^{71} +1.59090e8 q^{73} -1.61737e7 q^{74} +1.17013e8 q^{76} +3.57801e7 q^{77} +2.05688e8 q^{79} +1.00606e8 q^{80} -1.17524e8 q^{82} -5.94838e8 q^{83} +3.64006e8 q^{85} -7.80236e7 q^{86} +1.18191e8 q^{88} -1.02347e9 q^{89} +2.04102e8 q^{91} -3.40504e8 q^{92} -1.85998e8 q^{94} -1.65074e8 q^{95} +1.35649e9 q^{97} +4.78740e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{2} + 424 q^{4} + 2500 q^{5} - 9604 q^{7} - 96 q^{8} - 5000 q^{10} - 9832 q^{11} - 68264 q^{13} + 19208 q^{14} - 290784 q^{16} - 86272 q^{17} + 807672 q^{19} + 265000 q^{20} + 4293352 q^{22} - 683032 q^{23}+ \cdots - 46118408 q^{98}+O(q^{100}) \) 4 * q - 8 * q^2 + 424 * q^4 + 2500 * q^5 - 9604 * q^7 - 96 * q^8 - 5000 * q^10 - 9832 * q^11 - 68264 * q^13 + 19208 * q^14 - 290784 * q^16 - 86272 * q^17 + 807672 * q^19 + 265000 * q^20 + 4293352 * q^22 - 683032 * q^23 + 1562500 * q^25 - 4583544 * q^26 - 1018024 * q^28 - 5029312 * q^29 + 9802992 * q^31 - 497280 * q^32 + 429792 * q^34 - 6002500 * q^35 - 4536432 * q^37 - 21534872 * q^38 - 60000 * q^40 - 11081000 * q^41 - 34442248 * q^43 - 35679024 * q^44 + 67652080 * q^46 + 39230168 * q^47 + 23059204 * q^49 - 3125000 * q^50 - 14778976 * q^52 - 76012528 * q^53 - 6145000 * q^55 + 230496 * q^56 + 57813344 * q^58 - 74595424 * q^59 + 101246592 * q^61 - 392291880 * q^62 - 276909952 * q^64 - 42665000 * q^65 - 45694264 * q^67 - 345896144 * q^68 + 12005000 * q^70 - 447187600 * q^71 + 141527624 * q^73 - 480388368 * q^74 + 1081835504 * q^76 + 23606632 * q^77 + 789659136 * q^79 - 181740000 * q^80 + 937199856 * q^82 - 586282352 * q^83 - 53920000 * q^85 - 132711648 * q^86 + 1042925024 * q^88 - 1050230088 * q^89 + 163901864 * q^91 - 44316096 * q^92 - 1211140960 * q^94 + 504795000 * q^95 + 788061288 * q^97 - 46118408 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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