LMFDB - Embedded newform 315.10.a.e.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [315,10,Mod(1,315)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	315.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:	$162.236288392$
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} - 1253x^{2} - 1039x + 42996 $ x^4 - x^3 - 1253x^2 - 1039x + 42996
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 105)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$35.8379$ 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)

$f(q)$	$=$	\(q+38.8379 q^{2} +996.380 q^{4} +625.000 q^{5} +2401.00 q^{7} +18812.3 q^{8} +24273.7 q^{10} -6971.86 q^{11} +45601.2 q^{13} +93249.7 q^{14} +220483. q^{16} +154849. q^{17} -380792. q^{19} +622737. q^{20} -270772. q^{22} +1.66968e6 q^{23} +390625. q^{25} +1.77105e6 q^{26} +2.39231e6 q^{28} +2.23565e6 q^{29} +5.92992e6 q^{31} -1.06882e6 q^{32} +6.01401e6 q^{34} +1.50062e6 q^{35} +4.08925e6 q^{37} -1.47892e7 q^{38} +1.17577e7 q^{40} -1.62091e6 q^{41} +2.61736e7 q^{43} -6.94663e6 q^{44} +6.48468e7 q^{46} +3.12418e7 q^{47} +5.76480e6 q^{49} +1.51710e7 q^{50} +4.54361e7 q^{52} -8.31434e7 q^{53} -4.35742e6 q^{55} +4.51683e7 q^{56} +8.68279e7 q^{58} +3.03472e7 q^{59} -1.10999e8 q^{61} +2.30305e8 q^{62} -1.54398e8 q^{64} +2.85007e7 q^{65} -1.01836e8 q^{67} +1.54289e8 q^{68} +5.82811e7 q^{70} +3.91346e8 q^{71} +2.16261e8 q^{73} +1.58818e8 q^{74} -3.79414e8 q^{76} -1.67394e7 q^{77} +1.00652e8 q^{79} +1.37802e8 q^{80} -6.29525e7 q^{82} +3.07287e8 q^{83} +9.67807e7 q^{85} +1.01653e9 q^{86} -1.31157e8 q^{88} -2.71171e8 q^{89} +1.09488e8 q^{91} +1.66364e9 q^{92} +1.21337e9 q^{94} -2.37995e8 q^{95} -2.51010e8 q^{97} +2.23893e8 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) $ 4 * q + 13 * q^2 + 501 * q^4 + 2500 * q^5 + 9604 * q^7 + 16263 * q^8 + 8125 * q^10 + 87062 * q^11 + 39494 * q^13 + 31213 * q^14 + 328849 * q^16 + 291756 * q^17 + 50482 * q^19 + 313125 * q^20 - 1003016 * q^22 + 1048656 * q^23 + 1562500 * q^25 + 7622594 * q^26 + 1202901 * q^28 + 8018996 * q^29 - 1148130 * q^31 + 9765119 * q^32 + 1255358 * q^34 + 6002500 * q^35 - 23416328 * q^37 - 11334892 * q^38 + 10164375 * q^40 + 8000860 * q^41 + 27520044 * q^43 - 42192 * q^44 + 90166880 * q^46 + 78277092 * q^47 + 23059204 * q^49 + 5078125 * q^50 - 62776870 * q^52 - 36168514 * q^53 + 54413750 * q^55 + 39047463 * q^56 - 32819954 * q^58 + 132924868 * q^59 - 119014876 * q^61 + 76291172 * q^62 - 400777983 * q^64 + 24683750 * q^65 - 170018492 * q^67 + 345488158 * q^68 + 19508125 * q^70 + 743169022 * q^71 - 360015502 * q^73 + 547147386 * q^74 - 472117316 * q^76 + 209035862 * q^77 - 613400256 * q^79 + 205530625 * q^80 - 714522510 * q^82 + 1222551024 * q^83 + 182347500 * q^85 + 593721752 * q^86 - 855674336 * q^88 - 835315808 * q^89 + 94825094 * q^91 + 1276931776 * q^92 + 950602540 * q^94 + 31551250 * q^95 - 704297762 * q^97 + 74942413 * 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 $

$2$	38.8379	1.71641	0.858204	−	0.513309i	$-0.171581\pi$
$2$	38.8379	1.71641	0.858204	+	0.513309i	$0.171581\pi$
$3$	0	0
$3$	0	0
$4$	996.380	1.94605
$5$	625.000	0.447214
$5$	625.000	0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	18812.3	1.62382
$9$	0	0
$10$	24273.7	0.767601
$11$	−6971.86	−0.143576	−0.0717880	−	0.997420i	$-0.522871\pi$
$11$	−6971.86	−0.143576	−0.0717880	+	0.997420i	$0.522871\pi$
$12$	0	0
$13$	45601.2	0.442824	0.221412	−	0.975180i	$-0.428933\pi$
$13$	45601.2	0.442824	0.221412	+	0.975180i	$0.428933\pi$
$14$	93249.7	0.648741
$15$	0	0
$16$	220483.	0.841074
$17$	154849.	0.449664	0.224832	−	0.974398i	$-0.427817\pi$
$17$	154849.	0.449664	0.224832	+	0.974398i	$0.427817\pi$
$18$	0	0
$19$	−380792.	−0.670343	−0.335171	−	0.942157i	$-0.608794\pi$
$19$	−380792.	−0.670343	−0.335171	+	0.942157i	$0.608794\pi$
$20$	622737.	0.870302
$21$	0	0
$22$	−270772.	−0.246435
$23$	1.66968e6	1.24411	0.622054	−	0.782974i	$-0.286298\pi$
$23$	1.66968e6	1.24411	0.622054	+	0.782974i	$0.286298\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	1.77105e6	0.760066
$27$	0	0
$28$	2.39231e6	0.735540
$29$	2.23565e6	0.586966	0.293483	−	0.955964i	$-0.405186\pi$
$29$	2.23565e6	0.586966	0.293483	+	0.955964i	$0.405186\pi$
$30$	0	0
$31$	5.92992e6	1.15324	0.576622	−	0.817011i	$-0.304370\pi$
$31$	5.92992e6	1.15324	0.576622	+	0.817011i	$0.304370\pi$
$32$	−1.06882e6	−0.180190
$33$	0	0
$34$	6.01401e6	0.771807
$35$	1.50062e6	0.169031
$36$	0	0
$37$	4.08925e6	0.358703	0.179352	−	0.983785i	$-0.442600\pi$
$37$	4.08925e6	0.358703	0.179352	+	0.983785i	$0.442600\pi$
$38$	−1.47892e7	−1.15058
$39$	0	0
$40$	1.17577e7	0.726192
$41$	−1.62091e6	−0.0895840	−0.0447920	−	0.998996i	$-0.514263\pi$
$41$	−1.62091e6	−0.0895840	−0.0447920	+	0.998996i	$0.514263\pi$
$42$	0	0
$43$	2.61736e7	1.16750	0.583749	−	0.811934i	$-0.301585\pi$
$43$	2.61736e7	1.16750	0.583749	+	0.811934i	$0.301585\pi$
$44$	−6.94663e6	−0.279407
$45$	0	0
$46$	6.48468e7	2.13540
$47$	3.12418e7	0.933891	0.466945	−	0.884286i	$-0.345355\pi$
$47$	3.12418e7	0.933891	0.466945	+	0.884286i	$0.345355\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	1.51710e7	0.343281
$51$	0	0
$52$	4.54361e7	0.861759
$53$	−8.31434e7	−1.44739	−0.723696	−	0.690119i	$-0.757558\pi$
$53$	−8.31434e7	−1.44739	−0.723696	+	0.690119i	$0.757558\pi$
$54$	0	0
$55$	−4.35742e6	−0.0642091
$56$	4.51683e7	0.613744
$57$	0	0
$58$	8.68279e7	1.00747
$59$	3.03472e7	0.326051	0.163025	−	0.986622i	$-0.447875\pi$
$59$	3.03472e7	0.326051	0.163025	+	0.986622i	$0.447875\pi$
$60$	0	0
$61$	−1.10999e8	−1.02644	−0.513221	−	0.858256i	$-0.671548\pi$
$61$	−1.10999e8	−1.02644	−0.513221	+	0.858256i	$0.671548\pi$
$62$	2.30305e8	1.97944
$63$	0	0
$64$	−1.54398e8	−1.15035
$65$	2.85007e7	0.198037
$66$	0	0
$67$	−1.01836e8	−0.617400	−0.308700	−	0.951160i	$-0.599894\pi$
$67$	−1.01836e8	−0.617400	−0.308700	+	0.951160i	$0.599894\pi$
$68$	1.54289e8	0.875071
$69$	0	0
$70$	5.82811e7	0.290126
$71$	3.91346e8	1.82767	0.913837	−	0.406082i	$-0.133105\pi$
$71$	3.91346e8	1.82767	0.913837	+	0.406082i	$0.133105\pi$
$72$	0	0
$73$	2.16261e8	0.891304	0.445652	−	0.895206i	$-0.352972\pi$
$73$	2.16261e8	0.891304	0.445652	+	0.895206i	$0.352972\pi$
$74$	1.58818e8	0.615681
$75$	0	0
$76$	−3.79414e8	−1.30452
$77$	−1.67394e7	−0.0542666
$78$	0	0
$79$	1.00652e8	0.290738	0.145369	−	0.989377i	$-0.453563\pi$
$79$	1.00652e8	0.290738	0.145369	+	0.989377i	$0.453563\pi$
$80$	1.37802e8	0.376140
$81$	0	0
$82$	−6.29525e7	−0.153763
$83$	3.07287e8	0.710710	0.355355	−	0.934731i	$-0.384360\pi$
$83$	3.07287e8	0.710710	0.355355	+	0.934731i	$0.384360\pi$
$84$	0	0
$85$	9.67807e7	0.201096
$86$	1.01653e9	2.00390
$87$	0	0
$88$	−1.31157e8	−0.233141
$89$	−2.71171e8	−0.458130	−0.229065	−	0.973411i	$-0.573567\pi$
$89$	−2.71171e8	−0.458130	−0.229065	+	0.973411i	$0.573567\pi$
$90$	0	0
$91$	1.09488e8	0.167372
$92$	1.66364e9	2.42110
$93$	0	0
$94$	1.21337e9	1.60294
$95$	−2.37995e8	−0.299786
$96$	0	0
$97$	−2.51010e8	−0.287885	−0.143942	−	0.989586i	$-0.545978\pi$
$97$	−2.51010e8	−0.287885	−0.143942	+	0.989586i	$0.545978\pi$
$98$	2.23893e8	0.245201
$99$	0	0
$100$	3.89211e8	0.389211
$101$	7.39084e8	0.706721	0.353360	−	0.935487i	$-0.385039\pi$
$101$	7.39084e8	0.706721	0.353360	+	0.935487i	$0.385039\pi$
$102$	0	0
$103$	−2.34196e8	−0.205028	−0.102514	−	0.994732i	$-0.532689\pi$
$103$	−2.34196e8	−0.205028	−0.102514	+	0.994732i	$0.532689\pi$
$104$	8.57862e8	0.719064
$105$	0	0
$106$	−3.22911e9	−2.48432
$107$	9.62581e8	0.709922	0.354961	−	0.934881i	$-0.384494\pi$
$107$	9.62581e8	0.709922	0.354961	+	0.934881i	$0.384494\pi$
$108$	0	0
$109$	−1.80446e9	−1.22441	−0.612205	−	0.790699i	$-0.709717\pi$
$109$	−1.80446e9	−1.22441	−0.612205	+	0.790699i	$0.709717\pi$
$110$	−1.69233e8	−0.110209
$111$	0	0
$112$	5.29378e8	0.317896
$113$	2.49288e8	0.143830	0.0719148	−	0.997411i	$-0.477089\pi$
$113$	2.49288e8	0.143830	0.0719148	+	0.997411i	$0.477089\pi$
$114$	0	0
$115$	1.04355e9	0.556382
$116$	2.22756e9	1.14227
$117$	0	0
$118$	1.17862e9	0.559636
$119$	3.71793e8	0.169957
$120$	0	0
$121$	−2.30934e9	−0.979386
$122$	−4.31096e9	−1.76179
$123$	0	0
$124$	5.90845e9	2.24428
$125$	2.44141e8	0.0894427
$126$	0	0
$127$	2.06763e9	0.705270	0.352635	−	0.935761i	$-0.385286\pi$
$127$	2.06763e9	0.705270	0.352635	+	0.935761i	$0.385286\pi$
$128$	−5.44924e9	−1.79428
$129$	0	0
$130$	1.10691e9	0.339912
$131$	−2.31344e8	−0.0686338	−0.0343169	−	0.999411i	$-0.510926\pi$
$131$	−2.31344e8	−0.0686338	−0.0343169	+	0.999411i	$0.510926\pi$
$132$	0	0
$133$	−9.14282e8	−0.253366
$134$	−3.95511e9	−1.05971
$135$	0	0
$136$	2.91306e9	0.730172
$137$	4.04100e9	0.980046	0.490023	−	0.871709i	$-0.336988\pi$
$137$	4.04100e9	0.980046	0.490023	+	0.871709i	$0.336988\pi$
$138$	0	0
$139$	−5.97793e9	−1.35827	−0.679133	−	0.734016i	$-0.737644\pi$
$139$	−5.97793e9	−1.35827	−0.679133	+	0.734016i	$0.737644\pi$
$140$	1.49519e9	0.328943
$141$	0	0
$142$	1.51991e10	3.13703
$143$	−3.17925e8	−0.0635789
$144$	0	0
$145$	1.39728e9	0.262499
$146$	8.39913e9	1.52984
$147$	0	0
$148$	4.07444e9	0.698056
$149$	7.04279e9	1.17059	0.585297	−	0.810819i	$-0.300978\pi$
$149$	7.04279e9	1.17059	0.585297	+	0.810819i	$0.300978\pi$
$150$	0	0
$151$	−1.00816e10	−1.57809	−0.789045	−	0.614336i	$-0.789424\pi$
$151$	−1.00816e10	−1.57809	−0.789045	+	0.614336i	$0.789424\pi$
$152$	−7.16357e9	−1.08851
$153$	0	0
$154$	−6.50124e8	−0.0931436
$155$	3.70620e9	0.515747
$156$	0	0
$157$	9.31829e9	1.22402	0.612009	−	0.790851i	$-0.290362\pi$
$157$	9.31829e9	1.22402	0.612009	+	0.790851i	$0.290362\pi$
$158$	3.90912e9	0.499025
$159$	0	0
$160$	−6.68012e8	−0.0805832
$161$	4.00890e9	0.470228
$162$	0	0
$163$	4.38753e9	0.486828	0.243414	−	0.969922i	$-0.421733\pi$
$163$	4.38753e9	0.486828	0.243414	+	0.969922i	$0.421733\pi$
$164$	−1.61504e9	−0.174335
$165$	0	0
$166$	1.19344e10	1.21987
$167$	1.22903e10	1.22275	0.611377	−	0.791339i	$-0.290616\pi$
$167$	1.22903e10	1.22275	0.611377	+	0.791339i	$0.290616\pi$
$168$	0	0
$169$	−8.52503e9	−0.803907
$170$	3.75875e9	0.345163
$171$	0	0
$172$	2.60789e10	2.27201
$173$	1.03424e10	0.877838	0.438919	−	0.898527i	$-0.355362\pi$
$173$	1.03424e10	0.877838	0.438919	+	0.898527i	$0.355362\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	−1.53717e9	−0.120758
$177$	0	0
$178$	−1.05317e10	−0.786337
$179$	8.58820e9	0.625264	0.312632	−	0.949874i	$-0.398789\pi$
$179$	8.58820e9	0.625264	0.312632	+	0.949874i	$0.398789\pi$
$180$	0	0
$181$	−8.83994e9	−0.612203	−0.306102	−	0.951999i	$-0.599025\pi$
$181$	−8.83994e9	−0.612203	−0.306102	+	0.951999i	$0.599025\pi$
$182$	4.25230e9	0.287278
$183$	0	0
$184$	3.14105e10	2.02020
$185$	2.55578e9	0.160417
$186$	0	0
$187$	−1.07959e9	−0.0645610
$188$	3.11287e10	1.81740
$189$	0	0
$190$	−9.24323e9	−0.514556
$191$	−7.26678e9	−0.395086	−0.197543	−	0.980294i	$-0.563296\pi$
$191$	−7.26678e9	−0.395086	−0.197543	+	0.980294i	$0.563296\pi$
$192$	0	0
$193$	2.35437e9	0.122143	0.0610713	−	0.998133i	$-0.480548\pi$
$193$	2.35437e9	0.122143	0.0610713	+	0.998133i	$0.480548\pi$
$194$	−9.74870e9	−0.494127
$195$	0	0
$196$	5.74393e9	0.278008
$197$	−6.99832e9	−0.331052	−0.165526	−	0.986205i	$-0.552932\pi$
$197$	−6.99832e9	−0.331052	−0.165526	+	0.986205i	$0.552932\pi$
$198$	0	0
$199$	1.92323e10	0.869347	0.434674	−	0.900588i	$-0.356864\pi$
$199$	1.92323e10	0.869347	0.434674	+	0.900588i	$0.356864\pi$
$200$	7.34855e9	0.324763
$201$	0	0
$202$	2.87045e10	1.21302
$203$	5.36780e9	0.221852
$204$	0	0
$205$	−1.01307e9	−0.0400632
$206$	−9.09569e9	−0.351911
$207$	0	0
$208$	1.00543e10	0.372448
$209$	2.65483e9	0.0962452
$210$	0	0
$211$	−4.79906e9	−0.166680	−0.0833402	−	0.996521i	$-0.526559\pi$
$211$	−4.79906e9	−0.166680	−0.0833402	+	0.996521i	$0.526559\pi$
$212$	−8.28424e10	−2.81670
$213$	0	0
$214$	3.73846e10	1.21851
$215$	1.63585e10	0.522121
$216$	0	0
$217$	1.42377e10	0.435885
$218$	−7.00812e10	−2.10159
$219$	0	0
$220$	−4.34164e9	−0.124954
$221$	7.06130e9	0.199122
$222$	0	0
$223$	−4.10110e10	−1.11053	−0.555263	−	0.831675i	$-0.687382\pi$
$223$	−4.10110e10	−1.11053	−0.555263	+	0.831675i	$0.687382\pi$
$224$	−2.56624e9	−0.0681052
$225$	0	0
$226$	9.68182e9	0.246870
$227$	−7.44961e10	−1.86216	−0.931081	−	0.364813i	$-0.881133\pi$
$227$	−7.44961e10	−1.86216	−0.931081	+	0.364813i	$0.881133\pi$
$228$	0	0
$229$	4.45316e10	1.07006	0.535031	−	0.844833i	$-0.320300\pi$
$229$	4.45316e10	1.07006	0.535031	+	0.844833i	$0.320300\pi$
$230$	4.05293e10	0.954978
$231$	0	0
$232$	4.20577e10	0.953124
$233$	−3.18512e10	−0.707986	−0.353993	−	0.935248i	$-0.615176\pi$
$233$	−3.18512e10	−0.707986	−0.353993	+	0.935248i	$0.615176\pi$
$234$	0	0
$235$	1.95261e10	0.417649
$236$	3.02374e10	0.634512
$237$	0	0
$238$	1.44396e10	0.291716
$239$	−6.19905e10	−1.22895	−0.614476	−	0.788936i	$-0.710632\pi$
$239$	−6.19905e10	−1.22895	−0.614476	+	0.788936i	$0.710632\pi$
$240$	0	0
$241$	−8.58415e10	−1.63916	−0.819579	−	0.572967i	$-0.805792\pi$
$241$	−8.58415e10	−1.63916	−0.819579	+	0.572967i	$0.805792\pi$
$242$	−8.96899e10	−1.68103
$243$	0	0
$244$	−1.10597e11	−1.99751
$245$	3.60300e9	0.0638877
$246$	0	0
$247$	−1.73646e10	−0.296844
$248$	1.11555e11	1.87266
$249$	0	0
$250$	9.48190e9	0.153520
$251$	7.93380e10	1.26168	0.630840	−	0.775913i	$-0.282710\pi$
$251$	7.93380e10	1.26168	0.630840	+	0.775913i	$0.282710\pi$
$252$	0	0
$253$	−1.16408e10	−0.178624
$254$	8.03022e10	1.21053
$255$	0	0
$256$	−1.32585e11	−1.92937
$257$	6.33351e10	0.905619	0.452809	−	0.891607i	$-0.350422\pi$
$257$	6.33351e10	0.905619	0.452809	+	0.891607i	$0.350422\pi$
$258$	0	0
$259$	9.81828e9	0.135577
$260$	2.83976e10	0.385390
$261$	0	0
$262$	−8.98492e9	−0.117804
$263$	−1.19647e11	−1.54206	−0.771028	−	0.636801i	$-0.780257\pi$
$263$	−1.19647e11	−1.54206	−0.771028	+	0.636801i	$0.780257\pi$
$264$	0	0
$265$	−5.19646e10	−0.647294
$266$	−3.55088e10	−0.434879
$267$	0	0
$268$	−1.01468e11	−1.20149
$269$	1.41728e11	1.65033	0.825166	−	0.564891i	$-0.191082\pi$
$269$	1.41728e11	1.65033	0.825166	+	0.564891i	$0.191082\pi$
$270$	0	0
$271$	−3.59020e10	−0.404349	−0.202174	−	0.979350i	$-0.564801\pi$
$271$	−3.59020e10	−0.404349	−0.202174	+	0.979350i	$0.564801\pi$
$272$	3.41415e10	0.378201
$273$	0	0
$274$	1.56944e11	1.68216
$275$	−2.72338e9	−0.0287152
$276$	0	0
$277$	−1.20184e11	−1.22656	−0.613278	−	0.789867i	$-0.710150\pi$
$277$	−1.20184e11	−1.22656	−0.613278	+	0.789867i	$0.710150\pi$
$278$	−2.32170e11	−2.33134
$279$	0	0
$280$	2.82302e10	0.274475
$281$	−1.34904e11	−1.29077	−0.645383	−	0.763859i	$-0.723302\pi$
$281$	−1.34904e11	−1.29077	−0.645383	+	0.763859i	$0.723302\pi$
$282$	0	0
$283$	−6.33817e9	−0.0587388	−0.0293694	−	0.999569i	$-0.509350\pi$
$283$	−6.33817e9	−0.0587388	−0.0293694	+	0.999569i	$0.509350\pi$
$284$	3.89930e11	3.55675
$285$	0	0
$286$	−1.23475e10	−0.109127
$287$	−3.89180e9	−0.0338596
$288$	0	0
$289$	−9.46096e10	−0.797802
$290$	5.42674e10	0.450556
$291$	0	0
$292$	2.15478e11	1.73453
$293$	−2.64542e10	−0.209696	−0.104848	−	0.994488i	$-0.533436\pi$
$293$	−2.64542e10	−0.209696	−0.104848	+	0.994488i	$0.533436\pi$
$294$	0	0
$295$	1.89670e10	0.145814
$296$	7.69280e10	0.582468
$297$	0	0
$298$	2.73527e11	2.00922
$299$	7.61394e10	0.550920
$300$	0	0
$301$	6.28429e10	0.441273
$302$	−3.91546e11	−2.70864
$303$	0	0
$304$	−8.39581e10	−0.563808
$305$	−6.93743e10	−0.459039
$306$	0	0
$307$	−1.05442e11	−0.677473	−0.338737	−	0.940881i	$-0.610000\pi$
$307$	−1.05442e11	−0.677473	−0.338737	+	0.940881i	$0.610000\pi$
$308$	−1.66788e10	−0.105606
$309$	0	0
$310$	1.43941e11	0.885231
$311$	−4.94542e10	−0.299765	−0.149883	−	0.988704i	$-0.547890\pi$
$311$	−4.94542e10	−0.299765	−0.149883	+	0.988704i	$0.547890\pi$
$312$	0	0
$313$	2.25492e11	1.32795	0.663975	−	0.747755i	$-0.268868\pi$
$313$	2.25492e11	1.32795	0.663975	+	0.747755i	$0.268868\pi$
$314$	3.61902e11	2.10091
$315$	0	0
$316$	1.00288e11	0.565792
$317$	2.28187e10	0.126919	0.0634593	−	0.997984i	$-0.479787\pi$
$317$	2.28187e10	0.126919	0.0634593	+	0.997984i	$0.479787\pi$
$318$	0	0
$319$	−1.55867e10	−0.0842742
$320$	−9.64986e10	−0.514453
$321$	0	0
$322$	1.55697e11	0.807104
$323$	−5.89653e10	−0.301429
$324$	0	0
$325$	1.78130e10	0.0885648
$326$	1.70402e11	0.835595
$327$	0	0
$328$	−3.04929e10	−0.145468
$329$	7.50116e10	0.352977
$330$	0	0
$331$	−3.79965e11	−1.73987	−0.869937	−	0.493163i	$-0.835841\pi$
$331$	−3.79965e11	−1.73987	−0.869937	+	0.493163i	$0.835841\pi$
$332$	3.06174e11	1.38308
$333$	0	0
$334$	4.77330e11	2.09874
$335$	−6.36477e10	−0.276110
$336$	0	0
$337$	9.46687e10	0.399827	0.199913	−	0.979814i	$-0.435934\pi$
$337$	9.46687e10	0.399827	0.199913	+	0.979814i	$0.435934\pi$
$338$	−3.31094e11	−1.37983
$339$	0	0
$340$	9.64303e10	0.391344
$341$	−4.13426e10	−0.165578
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	4.92386e11	1.89580
$345$	0	0
$346$	4.01677e11	1.50673
$347$	1.00957e11	0.373812	0.186906	−	0.982378i	$-0.440154\pi$
$347$	1.00957e11	0.373812	0.186906	+	0.982378i	$0.440154\pi$
$348$	0	0
$349$	6.18343e10	0.223108	0.111554	−	0.993758i	$-0.464417\pi$
$349$	6.18343e10	0.223108	0.111554	+	0.993758i	$0.464417\pi$
$350$	3.64257e10	0.129748
$351$	0	0
$352$	7.45167e9	0.0258709
$353$	1.91858e11	0.657647	0.328823	−	0.944391i	$-0.393348\pi$
$353$	1.91858e11	0.657647	0.328823	+	0.944391i	$0.393348\pi$
$354$	0	0
$355$	2.44591e11	0.817360
$356$	−2.70189e11	−0.891545
$357$	0	0
$358$	3.33547e11	1.07321
$359$	−4.76805e11	−1.51501	−0.757505	−	0.652830i	$-0.773582\pi$
$359$	−4.76805e11	−1.51501	−0.757505	+	0.652830i	$0.773582\pi$
$360$	0	0
$361$	−1.77685e11	−0.550640
$362$	−3.43324e11	−1.05079
$363$	0	0
$364$	1.09092e11	0.325714
$365$	1.35163e11	0.398603
$366$	0	0
$367$	2.22747e11	0.640935	0.320467	−	0.947260i	$-0.396160\pi$
$367$	2.22747e11	0.640935	0.320467	+	0.947260i	$0.396160\pi$
$368$	3.68135e11	1.04639
$369$	0	0
$370$	9.92610e10	0.275341
$371$	−1.99627e11	−0.547063
$372$	0	0
$373$	−1.57925e11	−0.422437	−0.211219	−	0.977439i	$-0.567743\pi$
$373$	−1.57925e11	−0.422437	−0.211219	+	0.977439i	$0.567743\pi$
$374$	−4.19288e10	−0.110813
$375$	0	0
$376$	5.87730e11	1.51647
$377$	1.01948e11	0.259923
$378$	0	0
$379$	3.54764e11	0.883208	0.441604	−	0.897210i	$-0.354410\pi$
$379$	3.54764e11	0.883208	0.441604	+	0.897210i	$0.354410\pi$
$380$	−2.37134e11	−0.583401
$381$	0	0
$382$	−2.82226e11	−0.678129
$383$	−9.78887e9	−0.0232455	−0.0116227	−	0.999932i	$-0.503700\pi$
$383$	−9.78887e9	−0.0232455	−0.0116227	+	0.999932i	$0.503700\pi$
$384$	0	0
$385$	−1.04622e10	−0.0242688
$386$	9.14388e10	0.209646
$387$	0	0
$388$	−2.50102e11	−0.560239
$389$	5.83029e11	1.29097	0.645487	−	0.763772i	$-0.276655\pi$
$389$	5.83029e11	1.29097	0.645487	+	0.763772i	$0.276655\pi$
$390$	0	0
$391$	2.58548e11	0.559431
$392$	1.08449e11	0.231974
$393$	0	0
$394$	−2.71800e11	−0.568219
$395$	6.29077e10	0.130022
$396$	0	0
$397$	−3.67206e11	−0.741911	−0.370956	−	0.928651i	$-0.620970\pi$
$397$	−3.67206e11	−0.741911	−0.370956	+	0.928651i	$0.620970\pi$
$398$	7.46943e11	1.49215
$399$	0	0
$400$	8.61260e10	0.168215
$401$	−9.04311e11	−1.74650	−0.873249	−	0.487274i	$-0.837991\pi$
$401$	−9.04311e11	−1.74650	−0.873249	+	0.487274i	$0.837991\pi$
$402$	0	0
$403$	2.70411e11	0.510684
$404$	7.36409e11	1.37532
$405$	0	0
$406$	2.08474e11	0.380789
$407$	−2.85097e10	−0.0515012
$408$	0	0
$409$	5.60857e11	0.991053	0.495527	−	0.868593i	$-0.334975\pi$
$409$	5.60857e11	0.991053	0.495527	+	0.868593i	$0.334975\pi$
$410$	−3.93453e10	−0.0687648
$411$	0	0
$412$	−2.33349e11	−0.398995
$413$	7.28637e10	0.123236
$414$	0	0
$415$	1.92054e11	0.317839
$416$	−4.87394e10	−0.0797922
$417$	0	0
$418$	1.03108e11	0.165196
$419$	1.01774e11	0.161314	0.0806571	−	0.996742i	$-0.474298\pi$
$419$	1.01774e11	0.161314	0.0806571	+	0.996742i	$0.474298\pi$
$420$	0	0
$421$	−1.29266e11	−0.200546	−0.100273	−	0.994960i	$-0.531972\pi$
$421$	−1.29266e11	−0.200546	−0.100273	+	0.994960i	$0.531972\pi$
$422$	−1.86385e11	−0.286092
$423$	0	0
$424$	−1.56412e12	−2.35030
$425$	6.04879e10	0.0899329
$426$	0	0
$427$	−2.66508e11	−0.387959
$428$	9.59097e11	1.38155
$429$	0	0
$430$	6.35330e11	0.896172
$431$	5.12625e11	0.715570	0.357785	−	0.933804i	$-0.383532\pi$
$431$	5.12625e11	0.715570	0.357785	+	0.933804i	$0.383532\pi$
$432$	0	0
$433$	−1.29131e12	−1.76537	−0.882685	−	0.469964i	$-0.844267\pi$
$433$	−1.29131e12	−1.76537	−0.882685	+	0.469964i	$0.844267\pi$
$434$	5.52964e11	0.748157
$435$	0	0
$436$	−1.79792e12	−2.38277
$437$	−6.35801e11	−0.833979
$438$	0	0
$439$	−1.36592e12	−1.75523	−0.877615	−	0.479367i	$-0.840866\pi$
$439$	−1.36592e12	−1.75523	−0.877615	+	0.479367i	$0.840866\pi$
$440$	−8.19729e10	−0.104264
$441$	0	0
$442$	2.74246e11	0.341775
$443$	−2.42875e11	−0.299617	−0.149808	−	0.988715i	$-0.547866\pi$
$443$	−2.42875e11	−0.299617	−0.149808	+	0.988715i	$0.547866\pi$
$444$	0	0
$445$	−1.69482e11	−0.204882
$446$	−1.59278e12	−1.90611
$447$	0	0
$448$	−3.70709e11	−0.434792
$449$	8.03565e11	0.933067	0.466533	−	0.884504i	$-0.345503\pi$
$449$	8.03565e11	0.933067	0.466533	+	0.884504i	$0.345503\pi$
$450$	0	0
$451$	1.13007e10	0.0128621
$452$	2.48386e11	0.279900
$453$	0	0
$454$	−2.89327e12	−3.19623
$455$	6.84303e10	0.0748509
$456$	0	0
$457$	−1.30176e12	−1.39607	−0.698037	−	0.716062i	$-0.745943\pi$
$457$	−1.30176e12	−1.39607	−0.698037	+	0.716062i	$0.745943\pi$
$458$	1.72951e12	1.83666
$459$	0	0
$460$	1.03977e12	1.08275
$461$	−1.57807e12	−1.62732	−0.813661	−	0.581340i	$-0.802529\pi$
$461$	−1.57807e12	−1.62732	−0.813661	+	0.581340i	$0.802529\pi$
$462$	0	0
$463$	−9.58693e11	−0.969539	−0.484769	−	0.874642i	$-0.661096\pi$
$463$	−9.58693e11	−0.969539	−0.484769	+	0.874642i	$0.661096\pi$
$464$	4.92922e11	0.493682
$465$	0	0
$466$	−1.23703e12	−1.21519
$467$	3.07820e11	0.299482	0.149741	−	0.988725i	$-0.452156\pi$
$467$	3.07820e11	0.299482	0.149741	+	0.988725i	$0.452156\pi$
$468$	0	0
$469$	−2.44509e11	−0.233355
$470$	7.58354e11	0.716855
$471$	0	0
$472$	5.70901e11	0.529446
$473$	−1.82479e11	−0.167625
$474$	0	0
$475$	−1.48747e11	−0.134069
$476$	3.70447e11	0.330746
$477$	0	0
$478$	−2.40758e12	−2.10938
$479$	−1.13441e12	−0.984598	−0.492299	−	0.870426i	$-0.663843\pi$
$479$	−1.13441e12	−0.984598	−0.492299	+	0.870426i	$0.663843\pi$
$480$	0	0
$481$	1.86474e11	0.158842
$482$	−3.33390e12	−2.81346
$483$	0	0
$484$	−2.30098e12	−1.90594
$485$	−1.56881e11	−0.128746
$486$	0	0
$487$	4.63079e11	0.373057	0.186528	−	0.982450i	$-0.440276\pi$
$487$	4.63079e11	0.373057	0.186528	+	0.982450i	$0.440276\pi$
$488$	−2.08814e12	−1.66675
$489$	0	0
$490$	1.39933e11	0.109657
$491$	−7.03123e11	−0.545965	−0.272982	−	0.962019i	$-0.588010\pi$
$491$	−7.03123e11	−0.545965	−0.272982	+	0.962019i	$0.588010\pi$
$492$	0	0
$493$	3.46188e11	0.263938
$494$	−6.74403e11	−0.509505
$495$	0	0
$496$	1.30744e12	0.969964
$497$	9.39622e11	0.690796
$498$	0	0
$499$	−1.64409e12	−1.18706	−0.593532	−	0.804810i	$-0.702267\pi$
$499$	−1.64409e12	−1.18706	−0.593532	+	0.804810i	$0.702267\pi$
$500$	2.43257e11	0.174060
$501$	0	0
$502$	3.08132e12	2.16556
$503$	6.66858e11	0.464491	0.232246	−	0.972657i	$-0.425393\pi$
$503$	6.66858e11	0.464491	0.232246	+	0.972657i	$0.425393\pi$
$504$	0	0
$505$	4.61928e11	0.316055
$506$	−4.52103e11	−0.306592
$507$	0	0
$508$	2.06014e12	1.37249
$509$	−2.71000e12	−1.78953	−0.894766	−	0.446536i	$-0.852657\pi$
$509$	−2.71000e12	−1.78953	−0.894766	+	0.446536i	$0.852657\pi$
$510$	0	0
$511$	5.19243e11	0.336881
$512$	−2.35932e12	−1.51730
$513$	0	0
$514$	2.45980e12	1.55441
$515$	−1.46373e11	−0.0916912
$516$	0	0
$517$	−2.17814e11	−0.134084
$518$	3.81321e11	0.232706
$519$	0	0
$520$	5.36164e11	0.321575
$521$	2.09535e12	1.24591	0.622956	−	0.782257i	$-0.285932\pi$
$521$	2.09535e12	1.24591	0.622956	+	0.782257i	$0.285932\pi$
$522$	0	0
$523$	1.14725e12	0.670505	0.335253	−	0.942128i	$-0.391178\pi$
$523$	1.14725e12	0.670505	0.335253	+	0.942128i	$0.391178\pi$
$524$	−2.30507e11	−0.133565
$525$	0	0
$526$	−4.64683e12	−2.64680
$527$	9.18243e11	0.518573
$528$	0	0
$529$	9.86678e11	0.547804
$530$	−2.01820e12	−1.11102
$531$	0	0
$532$	−9.10973e11	−0.493064
$533$	−7.39152e10	−0.0396699
$534$	0	0
$535$	6.01613e11	0.317487
$536$	−1.91578e12	−1.00254
$537$	0	0
$538$	5.50442e12	2.83264
$539$	−4.01914e10	−0.0205109
$540$	0	0
$541$	−1.34131e12	−0.673196	−0.336598	−	0.941648i	$-0.609276\pi$
$541$	−1.34131e12	−0.673196	−0.336598	+	0.941648i	$0.609276\pi$
$542$	−1.39436e12	−0.694028
$543$	0	0
$544$	−1.65506e11	−0.0810248
$545$	−1.12778e12	−0.547573
$546$	0	0
$547$	2.40564e12	1.14891	0.574457	−	0.818535i	$-0.305213\pi$
$547$	2.40564e12	1.14891	0.574457	+	0.818535i	$0.305213\pi$
$548$	4.02637e12	1.90722
$549$	0	0
$550$	−1.05770e11	−0.0492870
$551$	−8.51319e11	−0.393469
$552$	0	0
$553$	2.41666e11	0.109889
$554$	−4.66769e12	−2.10527
$555$	0	0
$556$	−5.95629e12	−2.64326
$557$	−1.37511e12	−0.605328	−0.302664	−	0.953097i	$-0.597876\pi$
$557$	−1.37511e12	−0.605328	−0.302664	+	0.953097i	$0.597876\pi$
$558$	0	0
$559$	1.19355e12	0.516996
$560$	3.30862e11	0.142167
$561$	0	0
$562$	−5.23940e12	−2.21548
$563$	6.09186e11	0.255542	0.127771	−	0.991804i	$-0.459218\pi$
$563$	6.09186e11	0.255542	0.127771	+	0.991804i	$0.459218\pi$
$564$	0	0
$565$	1.55805e11	0.0643226
$566$	−2.46161e11	−0.100820
$567$	0	0
$568$	7.36212e12	2.96780
$569$	−2.05650e12	−0.822477	−0.411238	−	0.911528i	$-0.634904\pi$
$569$	−2.05650e12	−0.822477	−0.411238	+	0.911528i	$0.634904\pi$
$570$	0	0
$571$	−2.06365e12	−0.812408	−0.406204	−	0.913782i	$-0.633148\pi$
$571$	−2.06365e12	−0.812408	−0.406204	+	0.913782i	$0.633148\pi$
$572$	−3.16774e11	−0.123728
$573$	0	0
$574$	−1.51149e11	−0.0581168
$575$	6.52219e11	0.248822
$576$	0	0
$577$	−5.64677e11	−0.212084	−0.106042	−	0.994362i	$-0.533818\pi$
$577$	−5.64677e11	−0.212084	−0.106042	+	0.994362i	$0.533818\pi$
$578$	−3.67444e12	−1.36935
$579$	0	0
$580$	1.39222e12	0.510838
$581$	7.37795e11	0.268623
$582$	0	0
$583$	5.79664e11	0.207811
$584$	4.06837e12	1.44731
$585$	0	0
$586$	−1.02742e12	−0.359924
$587$	−2.31562e12	−0.805001	−0.402501	−	0.915420i	$-0.631859\pi$
$587$	−2.31562e12	−0.805001	−0.402501	+	0.915420i	$0.631859\pi$
$588$	0	0
$589$	−2.25807e12	−0.773069
$590$	7.36638e11	0.250277
$591$	0	0
$592$	9.01607e11	0.301696
$593$	4.63777e12	1.54015	0.770075	−	0.637953i	$-0.220219\pi$
$593$	4.63777e12	1.54015	0.770075	+	0.637953i	$0.220219\pi$
$594$	0	0
$595$	2.32370e11	0.0760071
$596$	7.01729e12	2.27804
$597$	0	0
$598$	2.95709e12	0.945604
$599$	−5.43529e12	−1.72505	−0.862525	−	0.506014i	$-0.831118\pi$
$599$	−5.43529e12	−1.72505	−0.862525	+	0.506014i	$0.831118\pi$
$600$	0	0
$601$	3.29328e12	1.02966	0.514830	−	0.857292i	$-0.327855\pi$
$601$	3.29328e12	1.02966	0.514830	+	0.857292i	$0.327855\pi$
$602$	2.44068e12	0.757403
$603$	0	0
$604$	−1.00451e13	−3.07105
$605$	−1.44334e12	−0.437995
$606$	0	0
$607$	−6.38098e12	−1.90782	−0.953911	−	0.300089i	$-0.902984\pi$
$607$	−6.38098e12	−1.90782	−0.953911	+	0.300089i	$0.902984\pi$
$608$	4.06998e11	0.120789
$609$	0	0
$610$	−2.69435e12	−0.787898
$611$	1.42466e12	0.413549
$612$	0	0
$613$	−2.75511e11	−0.0788075	−0.0394037	−	0.999223i	$-0.512546\pi$
$613$	−2.75511e11	−0.0788075	−0.0394037	+	0.999223i	$0.512546\pi$
$614$	−4.09515e12	−1.16282
$615$	0	0
$616$	−3.14907e11	−0.0881190
$617$	−4.64493e12	−1.29031	−0.645157	−	0.764050i	$-0.723208\pi$
$617$	−4.64493e12	−1.29031	−0.645157	+	0.764050i	$0.723208\pi$
$618$	0	0
$619$	3.81630e12	1.04480	0.522401	−	0.852700i	$-0.325036\pi$
$619$	3.81630e12	1.04480	0.522401	+	0.852700i	$0.325036\pi$
$620$	3.69278e12	1.00367
$621$	0	0
$622$	−1.92070e12	−0.514519
$623$	−6.51082e11	−0.173157
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	8.75763e12	2.27930
$627$	0	0
$628$	9.28455e12	2.38200
$629$	6.33216e11	0.161296
$630$	0	0
$631$	−7.70753e12	−1.93546	−0.967728	−	0.251997i	$-0.918913\pi$
$631$	−7.70753e12	−1.93546	−0.967728	+	0.251997i	$0.918913\pi$
$632$	1.89350e12	0.472105
$633$	0	0
$634$	8.86231e11	0.217844
$635$	1.29227e12	0.315406
$636$	0	0
$637$	2.62882e11	0.0632605
$638$	−6.05352e11	−0.144649
$639$	0	0
$640$	−3.40578e12	−0.802428
$641$	−5.61631e12	−1.31398	−0.656992	−	0.753898i	$-0.728171\pi$
$641$	−5.61631e12	−1.31398	−0.656992	+	0.753898i	$0.728171\pi$
$642$	0	0
$643$	8.17255e12	1.88542	0.942710	−	0.333613i	$-0.108268\pi$
$643$	8.17255e12	1.88542	0.942710	+	0.333613i	$0.108268\pi$
$644$	3.99439e12	0.915090
$645$	0	0
$646$	−2.29009e12	−0.517376
$647$	−1.11023e12	−0.249083	−0.124541	−	0.992214i	$-0.539746\pi$
$647$	−1.11023e12	−0.249083	−0.124541	+	0.992214i	$0.539746\pi$
$648$	0	0
$649$	−2.11577e11	−0.0468130
$650$	6.91817e11	0.152013
$651$	0	0
$652$	4.37164e12	0.947394
$653$	7.67144e11	0.165108	0.0825539	−	0.996587i	$-0.473692\pi$
$653$	7.67144e11	0.165108	0.0825539	+	0.996587i	$0.473692\pi$
$654$	0	0
$655$	−1.44590e11	−0.0306940
$656$	−3.57381e11	−0.0753468
$657$	0	0
$658$	2.91329e12	0.605853
$659$	−6.50728e12	−1.34405	−0.672024	−	0.740529i	$-0.734575\pi$
$659$	−6.50728e12	−1.34405	−0.672024	+	0.740529i	$0.734575\pi$
$660$	0	0
$661$	−2.96745e12	−0.604613	−0.302306	−	0.953211i	$-0.597757\pi$
$661$	−2.96745e12	−0.604613	−0.302306	+	0.953211i	$0.597757\pi$
$662$	−1.47570e13	−2.98633
$663$	0	0
$664$	5.78076e12	1.15406
$665$	−5.71427e11	−0.113309
$666$	0	0
$667$	3.73282e12	0.730249
$668$	1.22458e13	2.37955
$669$	0	0
$670$	−2.47194e12	−0.473916
$671$	7.73869e11	0.147372
$672$	0	0
$673$	5.01760e12	0.942818	0.471409	−	0.881915i	$-0.343746\pi$
$673$	5.01760e12	0.942818	0.471409	+	0.881915i	$0.343746\pi$
$674$	3.67673e12	0.686265
$675$	0	0
$676$	−8.49417e12	−1.56445
$677$	−6.45425e11	−0.118086	−0.0590428	−	0.998255i	$-0.518805\pi$
$677$	−6.45425e11	−0.118086	−0.0590428	+	0.998255i	$0.518805\pi$
$678$	0	0
$679$	−6.02676e11	−0.108810
$680$	1.82067e12	0.326543
$681$	0	0
$682$	−1.60566e12	−0.284200
$683$	8.16738e12	1.43612	0.718058	−	0.695983i	$-0.245031\pi$
$683$	8.16738e12	1.43612	0.718058	+	0.695983i	$0.245031\pi$
$684$	0	0
$685$	2.52563e12	0.438290
$686$	5.37566e11	0.0926773
$687$	0	0
$688$	5.77083e12	0.981952
$689$	−3.79144e12	−0.640940
$690$	0	0
$691$	9.92408e12	1.65592	0.827959	−	0.560788i	$-0.189502\pi$
$691$	9.92408e12	1.65592	0.827959	+	0.560788i	$0.189502\pi$
$692$	1.03050e13	1.70832
$693$	0	0
$694$	3.92095e12	0.641613
$695$	−3.73621e12	−0.607435
$696$	0	0
$697$	−2.50996e11	−0.0402827
$698$	2.40151e12	0.382944
$699$	0	0
$700$	9.34495e11	0.147108
$701$	1.13064e13	1.76845	0.884223	−	0.467066i	$-0.154689\pi$
$701$	1.13064e13	1.76845	0.884223	+	0.467066i	$0.154689\pi$
$702$	0	0
$703$	−1.55715e12	−0.240454
$704$	1.07644e12	0.165163
$705$	0	0
$706$	7.45134e12	1.12879
$707$	1.77454e12	0.267115
$708$	0	0
$709$	1.24336e13	1.84795	0.923974	−	0.382456i	$-0.124922\pi$
$709$	1.24336e13	1.84795	0.923974	+	0.382456i	$0.124922\pi$
$710$	9.49941e12	1.40292
$711$	0	0
$712$	−5.10135e12	−0.743918
$713$	9.90107e12	1.43476
$714$	0	0
$715$	−1.98703e11	−0.0284333
$716$	8.55711e12	1.21680
$717$	0	0
$718$	−1.85181e13	−2.60037
$719$	−9.25121e12	−1.29098	−0.645489	−	0.763770i	$-0.723346\pi$
$719$	−9.25121e12	−1.29098	−0.645489	+	0.763770i	$0.723346\pi$
$720$	0	0
$721$	−5.62306e11	−0.0774932
$722$	−6.90090e12	−0.945123
$723$	0	0
$724$	−8.80794e12	−1.19138
$725$	8.73301e11	0.117393
$726$	0	0
$727$	−3.05496e12	−0.405602	−0.202801	−	0.979220i	$-0.565004\pi$
$727$	−3.05496e12	−0.405602	−0.202801	+	0.979220i	$0.565004\pi$
$728$	2.05973e12	0.271781
$729$	0	0
$730$	5.24946e12	0.684166
$731$	4.05296e12	0.524982
$732$	0	0
$733$	1.02202e13	1.30765	0.653823	−	0.756647i	$-0.273164\pi$
$733$	1.02202e13	1.30765	0.653823	+	0.756647i	$0.273164\pi$
$734$	8.65100e12	1.10010
$735$	0	0
$736$	−1.78459e12	−0.224175
$737$	7.09989e11	0.0886438
$738$	0	0
$739$	5.72258e12	0.705817	0.352909	−	0.935658i	$-0.385193\pi$
$739$	5.72258e12	0.705817	0.352909	+	0.935658i	$0.385193\pi$
$740$	2.54653e12	0.312180
$741$	0	0
$742$	−7.75310e12	−0.938983
$743$	6.47714e12	0.779711	0.389856	−	0.920876i	$-0.372525\pi$
$743$	6.47714e12	0.779711	0.389856	+	0.920876i	$0.372525\pi$
$744$	0	0
$745$	4.40174e12	0.523506
$746$	−6.13348e12	−0.725074
$747$	0	0
$748$	−1.07568e12	−0.125639
$749$	2.31116e12	0.268325
$750$	0	0
$751$	9.43753e12	1.08263	0.541313	−	0.840821i	$-0.317927\pi$
$751$	9.43753e12	1.08263	0.541313	+	0.840821i	$0.317927\pi$
$752$	6.88827e12	0.785471
$753$	0	0
$754$	3.95945e12	0.446133
$755$	−6.30098e12	−0.705743
$756$	0	0
$757$	2.22866e12	0.246667	0.123334	−	0.992365i	$-0.460641\pi$
$757$	2.22866e12	0.246667	0.123334	+	0.992365i	$0.460641\pi$
$758$	1.37783e13	1.51595
$759$	0	0
$760$	−4.47723e12	−0.486798
$761$	−1.71728e13	−1.85614	−0.928070	−	0.372407i	$-0.878533\pi$
$761$	−1.71728e13	−1.85614	−0.928070	+	0.372407i	$0.878533\pi$
$762$	0	0
$763$	−4.33250e12	−0.462784
$764$	−7.24047e12	−0.768859
$765$	0	0
$766$	−3.80179e11	−0.0398987
$767$	1.38387e12	0.144383
$768$	0	0
$769$	1.02149e13	1.05333	0.526667	−	0.850072i	$-0.323442\pi$
$769$	1.02149e13	1.05333	0.526667	+	0.850072i	$0.323442\pi$
$770$	−4.06328e11	−0.0416551
$771$	0	0
$772$	2.34585e12	0.237696
$773$	−1.64381e13	−1.65594	−0.827970	−	0.560772i	$-0.810505\pi$
$773$	−1.64381e13	−1.65594	−0.827970	+	0.560772i	$0.810505\pi$
$774$	0	0
$775$	2.31638e12	0.230649
$776$	−4.72208e12	−0.467472
$777$	0	0
$778$	2.26436e13	2.21584
$779$	6.17229e11	0.0600520
$780$	0	0
$781$	−2.72841e12	−0.262410
$782$	1.00415e13	0.960211
$783$	0	0
$784$	1.27104e12	0.120153
$785$	5.82393e12	0.547397
$786$	0	0
$787$	1.15989e13	1.07778	0.538889	−	0.842377i	$-0.318844\pi$
$787$	1.15989e13	1.07778	0.538889	+	0.842377i	$0.318844\pi$
$788$	−6.97298e12	−0.644245
$789$	0	0
$790$	2.44320e12	0.223171
$791$	5.98541e11	0.0543625
$792$	0	0
$793$	−5.06168e12	−0.454533
$794$	−1.42615e13	−1.27342
$795$	0	0
$796$	1.91627e13	1.69180
$797$	1.74592e13	1.53271	0.766357	−	0.642415i	$-0.222067\pi$
$797$	1.74592e13	1.53271	0.766357	+	0.642415i	$0.222067\pi$
$798$	0	0
$799$	4.83777e12	0.419937
$800$	−4.17508e11	−0.0360379
$801$	0	0
$802$	−3.51215e13	−2.99770
$803$	−1.50774e12	−0.127970
$804$	0	0
$805$	2.50556e12	0.210293
$806$	1.05022e13	0.876542
$807$	0	0
$808$	1.39039e13	1.14758
$809$	1.56333e12	0.128317	0.0641583	−	0.997940i	$-0.479564\pi$
$809$	1.56333e12	0.128317	0.0641583	+	0.997940i	$0.479564\pi$
$810$	0	0
$811$	−8.96295e12	−0.727541	−0.363770	−	0.931489i	$-0.618511\pi$
$811$	−8.96295e12	−0.727541	−0.363770	+	0.931489i	$0.618511\pi$
$812$	5.34837e12	0.431737
$813$	0	0
$814$	−1.10725e12	−0.0883970
$815$	2.74220e12	0.217716
$816$	0	0
$817$	−9.96672e12	−0.782624
$818$	2.17825e13	1.70105
$819$	0	0
$820$	−1.00940e12	−0.0779652
$821$	−1.08280e13	−0.831774	−0.415887	−	0.909416i	$-0.636529\pi$
$821$	−1.08280e13	−0.831774	−0.415887	+	0.909416i	$0.636529\pi$
$822$	0	0
$823$	−1.86017e13	−1.41336	−0.706680	−	0.707533i	$-0.749808\pi$
$823$	−1.86017e13	−1.41336	−0.706680	+	0.707533i	$0.749808\pi$
$824$	−4.40577e12	−0.332927
$825$	0	0
$826$	2.82987e12	0.211522
$827$	1.27422e13	0.947261	0.473631	−	0.880724i	$-0.342943\pi$
$827$	1.27422e13	0.947261	0.473631	+	0.880724i	$0.342943\pi$
$828$	0	0
$829$	1.66886e13	1.22722	0.613612	−	0.789608i	$-0.289716\pi$
$829$	1.66886e13	1.22722	0.613612	+	0.789608i	$0.289716\pi$
$830$	7.45897e12	0.545541
$831$	0	0
$832$	−7.04072e12	−0.509404
$833$	8.92674e11	0.0642378
$834$	0	0
$835$	7.68145e12	0.546832
$836$	2.64522e12	0.187298
$837$	0	0
$838$	3.95268e12	0.276881
$839$	1.35503e13	0.944103	0.472051	−	0.881571i	$-0.343514\pi$
$839$	1.35503e13	0.944103	0.472051	+	0.881571i	$0.343514\pi$
$840$	0	0
$841$	−9.50901e12	−0.655471
$842$	−5.02040e12	−0.344218
$843$	0	0
$844$	−4.78168e12	−0.324369
$845$	−5.32815e12	−0.359518
$846$	0	0
$847$	−5.54473e12	−0.370173
$848$	−1.83317e13	−1.21736
$849$	0	0
$850$	2.34922e12	0.154361
$851$	6.82773e12	0.446266
$852$	0	0
$853$	1.77943e12	0.115083	0.0575413	−	0.998343i	$-0.481674\pi$
$853$	1.77943e12	0.115083	0.0575413	+	0.998343i	$0.481674\pi$
$854$	−1.03506e13	−0.665895
$855$	0	0
$856$	1.81084e13	1.15278
$857$	1.81805e13	1.15131	0.575654	−	0.817693i	$-0.304747\pi$
$857$	1.81805e13	1.15131	0.575654	+	0.817693i	$0.304747\pi$
$858$	0	0
$859$	−2.50942e13	−1.57255	−0.786274	−	0.617878i	$-0.787992\pi$
$859$	−2.50942e13	−1.57255	−0.786274	+	0.617878i	$0.787992\pi$
$860$	1.62993e13	1.01608
$861$	0	0
$862$	1.99093e13	1.22821
$863$	−2.65824e13	−1.63134	−0.815671	−	0.578516i	$-0.803632\pi$
$863$	−2.65824e13	−1.63134	−0.815671	+	0.578516i	$0.803632\pi$
$864$	0	0
$865$	6.46400e12	0.392581
$866$	−5.01518e13	−3.03010
$867$	0	0
$868$	1.41862e13	0.848257
$869$	−7.01735e11	−0.0417430
$870$	0	0
$871$	−4.64386e12	−0.273399
$872$	−3.39459e13	−1.98822
$873$	0	0
$874$	−2.46932e13	−1.43145
$875$	5.86182e11	0.0338062
$876$	0	0
$877$	2.22625e13	1.27080	0.635398	−	0.772185i	$-0.280836\pi$
$877$	2.22625e13	1.27080	0.635398	+	0.772185i	$0.280836\pi$
$878$	−5.30493e13	−3.01269
$879$	0	0
$880$	−9.60734e11	−0.0540046
$881$	9.84025e12	0.550319	0.275160	−	0.961399i	$-0.411269\pi$
$881$	9.84025e12	0.550319	0.275160	+	0.961399i	$0.411269\pi$
$882$	0	0
$883$	−7.23541e11	−0.0400535	−0.0200267	−	0.999799i	$-0.506375\pi$
$883$	−7.23541e11	−0.0400535	−0.0200267	+	0.999799i	$0.506375\pi$
$884$	7.03574e12	0.387502
$885$	0	0
$886$	−9.43275e12	−0.514264
$887$	4.19479e12	0.227538	0.113769	−	0.993507i	$-0.463708\pi$
$887$	4.19479e12	0.227538	0.113769	+	0.993507i	$0.463708\pi$
$888$	0	0
$889$	4.96437e12	0.266567
$890$	−6.58232e12	−0.351661
$891$	0	0
$892$	−4.08626e13	−2.16114
$893$	−1.18966e13	−0.626027
$894$	0	0
$895$	5.36763e12	0.279627
$896$	−1.30836e13	−0.678176
$897$	0	0
$898$	3.12088e13	1.60152
$899$	1.32572e13	0.676915
$900$	0	0
$901$	−1.28747e13	−0.650841
$902$	4.38897e11	0.0220766
$903$	0	0
$904$	4.68968e12	0.233553
$905$	−5.52496e12	−0.273786
$906$	0	0
$907$	−2.27774e13	−1.11756	−0.558781	−	0.829315i	$-0.688731\pi$
$907$	−2.27774e13	−1.11756	−0.558781	+	0.829315i	$0.688731\pi$
$908$	−7.42265e13	−3.62387
$909$	0	0
$910$	2.65769e12	0.128475
$911$	3.61101e13	1.73699	0.868493	−	0.495702i	$-0.165089\pi$
$911$	3.61101e13	1.73699	0.868493	+	0.495702i	$0.165089\pi$
$912$	0	0
$913$	−2.14236e12	−0.102041
$914$	−5.05576e13	−2.39623
$915$	0	0
$916$	4.43704e13	2.08240
$917$	−5.55458e11	−0.0259411
$918$	0	0
$919$	2.36288e12	0.109275	0.0546376	−	0.998506i	$-0.482600\pi$
$919$	2.36288e12	0.109275	0.0546376	+	0.998506i	$0.482600\pi$
$920$	1.96316e13	0.903461
$921$	0	0
$922$	−6.12890e13	−2.79315
$923$	1.78458e13	0.809337
$924$	0	0
$925$	1.59736e12	0.0717407
$926$	−3.72336e13	−1.66412
$927$	0	0
$928$	−2.38951e12	−0.105765
$929$	−2.11552e13	−0.931852	−0.465926	−	0.884824i	$-0.654279\pi$
$929$	−2.11552e13	−0.931852	−0.465926	+	0.884824i	$0.654279\pi$
$930$	0	0
$931$	−2.19519e12	−0.0957633
$932$	−3.17359e13	−1.37778
$933$	0	0
$934$	1.19551e13	0.514033
$935$	−6.74742e11	−0.0288726
$936$	0	0
$937$	1.08882e13	0.461455	0.230728	−	0.973018i	$-0.425889\pi$
$937$	1.08882e13	0.461455	0.230728	+	0.973018i	$0.425889\pi$
$938$	−9.49621e12	−0.400532
$939$	0	0
$940$	1.94555e13	0.812767
$941$	1.12541e13	0.467905	0.233952	−	0.972248i	$-0.424834\pi$
$941$	1.12541e13	0.467905	0.233952	+	0.972248i	$0.424834\pi$
$942$	0	0
$943$	−2.70639e12	−0.111452
$944$	6.69103e12	0.274233
$945$	0	0
$946$	−7.08710e12	−0.287712
$947$	−1.93129e13	−0.780320	−0.390160	−	0.920747i	$-0.627580\pi$
$947$	−1.93129e13	−0.780320	−0.390160	+	0.920747i	$0.627580\pi$
$948$	0	0
$949$	9.86177e12	0.394691
$950$	−5.77702e12	−0.230116
$951$	0	0
$952$	6.99427e12	0.275979
$953$	1.08910e13	0.427709	0.213855	−	0.976866i	$-0.431398\pi$
$953$	1.08910e13	0.427709	0.213855	+	0.976866i	$0.431398\pi$
$954$	0	0
$955$	−4.54174e12	−0.176688
$956$	−6.17661e13	−2.39161
$957$	0	0
$958$	−4.40580e13	−1.68997
$959$	9.70245e12	0.370423
$960$	0	0
$961$	8.72434e12	0.329972
$962$	7.24227e12	0.272638
$963$	0	0
$964$	−8.55307e13	−3.18989
$965$	1.47148e12	0.0546238
$966$	0	0
$967$	−4.04400e12	−0.148728	−0.0743639	−	0.997231i	$-0.523693\pi$
$967$	−4.04400e12	−0.148728	−0.0743639	+	0.997231i	$0.523693\pi$
$968$	−4.34440e13	−1.59034
$969$	0	0
$970$	−6.09294e12	−0.220981
$971$	3.21445e13	1.16043	0.580217	−	0.814462i	$-0.302968\pi$
$971$	3.21445e13	1.16043	0.580217	+	0.814462i	$0.302968\pi$
$972$	0	0
$973$	−1.43530e13	−0.513376
$974$	1.79850e13	0.640318
$975$	0	0
$976$	−2.44733e13	−0.863314
$977$	2.04345e13	0.717526	0.358763	−	0.933429i	$-0.383199\pi$
$977$	2.04345e13	0.717526	0.358763	+	0.933429i	$0.383199\pi$
$978$	0	0
$979$	1.89057e12	0.0657764
$980$	3.58996e12	0.124329
$981$	0	0
$982$	−2.73078e13	−0.937098
$983$	3.90327e13	1.33333	0.666665	−	0.745357i	$-0.267721\pi$
$983$	3.90327e13	1.33333	0.666665	+	0.745357i	$0.267721\pi$
$984$	0	0
$985$	−4.37395e12	−0.148051
$986$	1.34452e13	0.453025
$987$	0	0
$988$	−1.73017e13	−0.577674
$989$	4.37016e13	1.45249
$990$	0	0
$991$	−4.57560e13	−1.50701	−0.753506	−	0.657441i	$-0.771639\pi$
$991$	−4.57560e13	−1.50701	−0.753506	+	0.657441i	$0.771639\pi$
$992$	−6.33802e12	−0.207803
$993$	0	0
$994$	3.64929e13	1.18569
$995$	1.20202e13	0.388784
$996$	0	0
$997$	−3.09400e12	−0.0991726	−0.0495863	−	0.998770i	$-0.515790\pi$
$997$	−3.09400e12	−0.0991726	−0.0495863	+	0.998770i	$0.515790\pi$
$998$	−6.38531e13	−2.03749
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.e.1.4		4
3.2	odd	2		105.10.a.d.1.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
105.10.a.d.1.1	✓	4	3.2	odd	2
315.10.a.e.1.4		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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+38.8379 q^{2} +996.380 q^{4} +625.000 q^{5} +2401.00 q^{7} +18812.3 q^{8} +24273.7 q^{10} -6971.86 q^{11} +45601.2 q^{13} +93249.7 q^{14} +220483. q^{16} +154849. q^{17} -380792. q^{19} +622737. q^{20} -270772. q^{22} +1.66968e6 q^{23} +390625. q^{25} +1.77105e6 q^{26} +2.39231e6 q^{28} +2.23565e6 q^{29} +5.92992e6 q^{31} -1.06882e6 q^{32} +6.01401e6 q^{34} +1.50062e6 q^{35} +4.08925e6 q^{37} -1.47892e7 q^{38} +1.17577e7 q^{40} -1.62091e6 q^{41} +2.61736e7 q^{43} -6.94663e6 q^{44} +6.48468e7 q^{46} +3.12418e7 q^{47} +5.76480e6 q^{49} +1.51710e7 q^{50} +4.54361e7 q^{52} -8.31434e7 q^{53} -4.35742e6 q^{55} +4.51683e7 q^{56} +8.68279e7 q^{58} +3.03472e7 q^{59} -1.10999e8 q^{61} +2.30305e8 q^{62} -1.54398e8 q^{64} +2.85007e7 q^{65} -1.01836e8 q^{67} +1.54289e8 q^{68} +5.82811e7 q^{70} +3.91346e8 q^{71} +2.16261e8 q^{73} +1.58818e8 q^{74} -3.79414e8 q^{76} -1.67394e7 q^{77} +1.00652e8 q^{79} +1.37802e8 q^{80} -6.29525e7 q^{82} +3.07287e8 q^{83} +9.67807e7 q^{85} +1.01653e9 q^{86} -1.31157e8 q^{88} -2.71171e8 q^{89} +1.09488e8 q^{91} +1.66364e9 q^{92} +1.21337e9 q^{94} -2.37995e8 q^{95} -2.51010e8 q^{97} +2.23893e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 13 q^{2} + 501 q^{4} + 2500 q^{5} + 9604 q^{7} + 16263 q^{8} + 8125 q^{10} + 87062 q^{11} + 39494 q^{13} + 31213 q^{14} + 328849 q^{16} + 291756 q^{17} + 50482 q^{19} + 313125 q^{20} - 1003016 q^{22}+ \cdots + 74942413 q^{98}+O(q^{100}) \) 4 * q + 13 * q^2 + 501 * q^4 + 2500 * q^5 + 9604 * q^7 + 16263 * q^8 + 8125 * q^10 + 87062 * q^11 + 39494 * q^13 + 31213 * q^14 + 328849 * q^16 + 291756 * q^17 + 50482 * q^19 + 313125 * q^20 - 1003016 * q^22 + 1048656 * q^23 + 1562500 * q^25 + 7622594 * q^26 + 1202901 * q^28 + 8018996 * q^29 - 1148130 * q^31 + 9765119 * q^32 + 1255358 * q^34 + 6002500 * q^35 - 23416328 * q^37 - 11334892 * q^38 + 10164375 * q^40 + 8000860 * q^41 + 27520044 * q^43 - 42192 * q^44 + 90166880 * q^46 + 78277092 * q^47 + 23059204 * q^49 + 5078125 * q^50 - 62776870 * q^52 - 36168514 * q^53 + 54413750 * q^55 + 39047463 * q^56 - 32819954 * q^58 + 132924868 * q^59 - 119014876 * q^61 + 76291172 * q^62 - 400777983 * q^64 + 24683750 * q^65 - 170018492 * q^67 + 345488158 * q^68 + 19508125 * q^70 + 743169022 * q^71 - 360015502 * q^73 + 547147386 * q^74 - 472117316 * q^76 + 209035862 * q^77 - 613400256 * q^79 + 205530625 * q^80 - 714522510 * q^82 + 1222551024 * q^83 + 182347500 * q^85 + 593721752 * q^86 - 855674336 * q^88 - 835315808 * q^89 + 94825094 * q^91 + 1276931776 * q^92 + 950602540 * q^94 + 31551250 * q^95 - 704297762 * q^97 + 74942413 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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