LMFDB - Embedded newform 315.10.a.b.1.2

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 = [2,24] 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:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 35)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ 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)

$f(q)$	$=$	\(q+14.8284 q^{2} -292.118 q^{4} -625.000 q^{5} +2401.00 q^{7} -11923.8 q^{8} -9267.77 q^{10} +16523.6 q^{11} +26311.4 q^{13} +35603.1 q^{14} -27246.9 q^{16} +144003. q^{17} -159710. q^{19} +182574. q^{20} +245019. q^{22} -2.07393e6 q^{23} +390625. q^{25} +390156. q^{26} -701375. q^{28} +4.94938e6 q^{29} +4.22040e6 q^{31} +5.70096e6 q^{32} +2.13533e6 q^{34} -1.50062e6 q^{35} -1.29081e7 q^{37} -2.36824e6 q^{38} +7.45238e6 q^{40} +2.87518e7 q^{41} +3.54825e7 q^{43} -4.82683e6 q^{44} -3.07532e7 q^{46} -5.95633e7 q^{47} +5.76480e6 q^{49} +5.79235e6 q^{50} -7.68602e6 q^{52} -2.31161e6 q^{53} -1.03272e7 q^{55} -2.86290e7 q^{56} +7.33915e7 q^{58} +1.68651e8 q^{59} -6.70167e7 q^{61} +6.25819e7 q^{62} +9.84867e7 q^{64} -1.64446e7 q^{65} -1.56259e8 q^{67} -4.20657e7 q^{68} -2.22519e7 q^{70} -6.95067e7 q^{71} -7.83438e7 q^{73} -1.91407e8 q^{74} +4.66540e7 q^{76} +3.96731e7 q^{77} -4.26957e8 q^{79} +1.70293e7 q^{80} +4.26344e8 q^{82} +5.31242e8 q^{83} -9.00016e7 q^{85} +5.26149e8 q^{86} -1.97024e8 q^{88} +1.14168e8 q^{89} +6.31736e7 q^{91} +6.05833e8 q^{92} -8.83231e8 q^{94} +9.98185e7 q^{95} -1.46573e9 q^{97} +8.54829e7 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22}+ \cdots + 138355224 q^{98}+O(q^{100}) $ 2 * q + 24 * q^2 - 720 * q^4 - 1250 * q^5 + 4802 * q^7 - 20544 * q^8 - 15000 * q^10 - 18566 * q^11 - 51090 * q^13 + 57624 * q^14 + 112768 * q^16 + 373910 * q^17 - 143276 * q^19 + 450000 * q^20 - 76808 * q^22 + 498908 * q^23 + 781250 * q^25 - 319736 * q^26 - 1728720 * q^28 + 11577554 * q^29 - 3953760 * q^31 + 11398656 * q^32 + 4243944 * q^34 - 3001250 * q^35 - 3205412 * q^37 - 2217520 * q^38 + 12840000 * q^40 - 1058992 * q^41 + 15948180 * q^43 + 10187376 * q^44 - 7156176 * q^46 - 65501290 * q^47 + 11529602 * q^49 + 9375000 * q^50 + 25432656 * q^52 + 25114688 * q^53 + 11603750 * q^55 - 49326144 * q^56 + 134182296 * q^58 + 116159208 * q^59 - 44688544 * q^61 - 12388000 * q^62 + 79055872 * q^64 + 31931250 * q^65 + 118092496 * q^67 - 140439024 * q^68 - 36015000 * q^70 + 294165824 * q^71 - 57419332 * q^73 - 102418064 * q^74 + 39622368 * q^76 - 44576966 * q^77 - 692852854 * q^79 - 70480000 * q^80 + 152932128 * q^82 + 540679928 * q^83 - 233693750 * q^85 + 346989008 * q^86 + 105455296 * q^88 + 779043704 * q^89 - 122667090 * q^91 - 495040608 * q^92 - 937691032 * q^94 + 89547500 * q^95 - 2673039406 * q^97 + 138355224 * 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 $

$2$	14.8284	0.655330	0.327665	−	0.944794i	$-0.393738\pi$
$2$	14.8284	0.655330	0.327665	+	0.944794i	$0.393738\pi$
$3$	0	0
$3$	0	0
$4$	−292.118	−0.570542
$5$	−625.000	−0.447214
$5$	−625.000	−0.447214
$6$	0	0
$7$	2401.00	0.377964
$7$	2401.00	0.377964
$8$	−11923.8	−1.02922
$9$	0	0
$10$	−9267.77	−0.293073
$11$	16523.6	0.340280	0.170140	−	0.985420i	$-0.445578\pi$
$11$	16523.6	0.340280	0.170140	+	0.985420i	$0.445578\pi$
$12$	0	0
$13$	26311.4	0.255505	0.127752	−	0.991806i	$-0.459224\pi$
$13$	26311.4	0.255505	0.127752	+	0.991806i	$0.459224\pi$
$14$	35603.1	0.247691
$15$	0	0
$16$	−27246.9	−0.103939
$17$	144003.	0.418167	0.209084	−	0.977898i	$-0.432952\pi$
$17$	144003.	0.418167	0.209084	+	0.977898i	$0.432952\pi$
$18$	0	0
$19$	−159710.	−0.281151	−0.140576	−	0.990070i	$-0.544895\pi$
$19$	−159710.	−0.281151	−0.140576	+	0.990070i	$0.544895\pi$
$20$	182574.	0.255154
$21$	0	0
$22$	245019.	0.222996
$23$	−2.07393e6	−1.54533	−0.772663	−	0.634817i	$-0.781075\pi$
$23$	−2.07393e6	−1.54533	−0.772663	+	0.634817i	$0.781075\pi$
$24$	0	0
$25$	390625.	0.200000
$26$	390156.	0.167440
$27$	0	0
$28$	−701375.	−0.215645
$29$	4.94938e6	1.29945	0.649725	−	0.760169i	$-0.274884\pi$
$29$	4.94938e6	1.29945	0.649725	+	0.760169i	$0.274884\pi$
$30$	0	0
$31$	4.22040e6	0.820779	0.410389	−	0.911910i	$-0.365393\pi$
$31$	4.22040e6	0.820779	0.410389	+	0.911910i	$0.365393\pi$
$32$	5.70096e6	0.961110
$33$	0	0
$34$	2.13533e6	0.274038
$35$	−1.50062e6	−0.169031
$36$	0	0
$37$	−1.29081e7	−1.13228	−0.566142	−	0.824308i	$-0.691565\pi$
$37$	−1.29081e7	−1.13228	−0.566142	+	0.824308i	$0.691565\pi$
$38$	−2.36824e6	−0.184247
$39$	0	0
$40$	7.45238e6	0.460283
$41$	2.87518e7	1.58905	0.794525	−	0.607231i	$-0.207720\pi$
$41$	2.87518e7	1.58905	0.794525	+	0.607231i	$0.207720\pi$
$42$	0	0
$43$	3.54825e7	1.58273	0.791363	−	0.611347i	$-0.209372\pi$
$43$	3.54825e7	1.58273	0.791363	+	0.611347i	$0.209372\pi$
$44$	−4.82683e6	−0.194144
$45$	0	0
$46$	−3.07532e7	−1.01270
$47$	−5.95633e7	−1.78049	−0.890243	−	0.455485i	$-0.849466\pi$
$47$	−5.95633e7	−1.78049	−0.890243	+	0.455485i	$0.849466\pi$
$48$	0	0
$49$	5.76480e6	0.142857
$50$	5.79235e6	0.131066
$51$	0	0
$52$	−7.68602e6	−0.145776
$53$	−2.31161e6	−0.0402414	−0.0201207	−	0.999798i	$-0.506405\pi$
$53$	−2.31161e6	−0.0402414	−0.0201207	+	0.999798i	$0.506405\pi$
$54$	0	0
$55$	−1.03272e7	−0.152178
$56$	−2.86290e7	−0.389010
$57$	0	0
$58$	7.33915e7	0.851569
$59$	1.68651e8	1.81198	0.905992	−	0.423296i	$-0.139127\pi$
$59$	1.68651e8	1.81198	0.905992	+	0.423296i	$0.139127\pi$
$60$	0	0
$61$	−6.70167e7	−0.619725	−0.309863	−	0.950781i	$-0.600283\pi$
$61$	−6.70167e7	−0.619725	−0.309863	+	0.950781i	$0.600283\pi$
$62$	6.25819e7	0.537881
$63$	0	0
$64$	9.84867e7	0.733783
$65$	−1.64446e7	−0.114265
$66$	0	0
$67$	−1.56259e8	−0.947343	−0.473671	−	0.880702i	$-0.657072\pi$
$67$	−1.56259e8	−0.947343	−0.473671	+	0.880702i	$0.657072\pi$
$68$	−4.20657e7	−0.238582
$69$	0	0
$70$	−2.22519e7	−0.110771
$71$	−6.95067e7	−0.324612	−0.162306	−	0.986740i	$-0.551893\pi$
$71$	−6.95067e7	−0.324612	−0.162306	+	0.986740i	$0.551893\pi$
$72$	0	0
$73$	−7.83438e7	−0.322888	−0.161444	−	0.986882i	$-0.551615\pi$
$73$	−7.83438e7	−0.322888	−0.161444	+	0.986882i	$0.551615\pi$
$74$	−1.91407e8	−0.742020
$75$	0	0
$76$	4.66540e7	0.160409
$77$	3.96731e7	0.128614
$78$	0	0
$79$	−4.26957e8	−1.23328	−0.616641	−	0.787245i	$-0.711507\pi$
$79$	−4.26957e8	−1.23328	−0.616641	+	0.787245i	$0.711507\pi$
$80$	1.70293e7	0.0464828
$81$	0	0
$82$	4.26344e8	1.04135
$83$	5.31242e8	1.22869	0.614343	−	0.789039i	$-0.289421\pi$
$83$	5.31242e8	1.22869	0.614343	+	0.789039i	$0.289421\pi$
$84$	0	0
$85$	−9.00016e7	−0.187010
$86$	5.26149e8	1.03721
$87$	0	0
$88$	−1.97024e8	−0.350225
$89$	1.14168e8	0.192881	0.0964404	−	0.995339i	$-0.469254\pi$
$89$	1.14168e8	0.192881	0.0964404	+	0.995339i	$0.469254\pi$
$90$	0	0
$91$	6.31736e7	0.0965716
$92$	6.05833e8	0.881674
$93$	0	0
$94$	−8.83231e8	−1.16681
$95$	9.98185e7	0.125735
$96$	0	0
$97$	−1.46573e9	−1.68105	−0.840524	−	0.541774i	$-0.817753\pi$
$97$	−1.46573e9	−1.68105	−0.840524	+	0.541774i	$0.817753\pi$
$98$	8.54829e7	0.0936186
$99$	0	0
$100$	−1.14108e8	−0.114108
$101$	−7.53733e8	−0.720728	−0.360364	−	0.932812i	$-0.617348\pi$
$101$	−7.53733e8	−0.720728	−0.360364	+	0.932812i	$0.617348\pi$
$102$	0	0
$103$	−1.32143e9	−1.15685	−0.578425	−	0.815736i	$-0.696332\pi$
$103$	−1.32143e9	−1.15685	−0.578425	+	0.815736i	$0.696332\pi$
$104$	−3.13732e8	−0.262971
$105$	0	0
$106$	−3.42775e7	−0.0263714
$107$	−1.07364e9	−0.791831	−0.395915	−	0.918287i	$-0.629573\pi$
$107$	−1.07364e9	−0.791831	−0.395915	+	0.918287i	$0.629573\pi$
$108$	0	0
$109$	1.08650e9	0.737245	0.368623	−	0.929579i	$-0.379830\pi$
$109$	1.08650e9	0.737245	0.368623	+	0.929579i	$0.379830\pi$
$110$	−1.53137e8	−0.0997268
$111$	0	0
$112$	−6.54199e7	−0.0392852
$113$	−2.85559e9	−1.64757	−0.823783	−	0.566906i	$-0.808140\pi$
$113$	−2.85559e9	−1.64757	−0.823783	+	0.566906i	$0.808140\pi$
$114$	0	0
$115$	1.29621e9	0.691090
$116$	−1.44580e9	−0.741392
$117$	0	0
$118$	2.50083e9	1.18745
$119$	3.45750e8	0.158052
$120$	0	0
$121$	−2.08492e9	−0.884209
$122$	−9.93753e8	−0.406125
$123$	0	0
$124$	−1.23285e9	−0.468289
$125$	−2.44141e8	−0.0894427
$126$	0	0
$127$	−3.86082e9	−1.31693	−0.658465	−	0.752611i	$-0.728794\pi$
$127$	−3.86082e9	−1.31693	−0.658465	+	0.752611i	$0.728794\pi$
$128$	−1.45849e9	−0.480240
$129$	0	0
$130$	−2.43848e8	−0.0748814
$131$	−4.01900e8	−0.119233	−0.0596166	−	0.998221i	$-0.518988\pi$
$131$	−4.01900e8	−0.119233	−0.0596166	+	0.998221i	$0.518988\pi$
$132$	0	0
$133$	−3.83463e8	−0.106265
$134$	−2.31707e9	−0.620822
$135$	0	0
$136$	−1.71706e9	−0.430388
$137$	−3.04172e9	−0.737695	−0.368847	−	0.929490i	$-0.620247\pi$
$137$	−3.04172e9	−0.737695	−0.368847	+	0.929490i	$0.620247\pi$
$138$	0	0
$139$	8.36421e8	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$139$	8.36421e8	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$140$	4.38359e8	0.0964393
$141$	0	0
$142$	−1.03068e9	−0.212728
$143$	4.34758e8	0.0869431
$144$	0	0
$145$	−3.09336e9	−0.581132
$146$	−1.16172e9	−0.211598
$147$	0	0
$148$	3.77069e9	0.646016
$149$	7.76217e9	1.29016	0.645082	−	0.764114i	$-0.276823\pi$
$149$	7.76217e9	1.29016	0.645082	+	0.764114i	$0.276823\pi$
$150$	0	0
$151$	8.38042e9	1.31181	0.655903	−	0.754846i	$-0.272288\pi$
$151$	8.38042e9	1.31181	0.655903	+	0.754846i	$0.272288\pi$
$152$	1.90435e9	0.289367
$153$	0	0
$154$	5.88290e8	0.0842845
$155$	−2.63775e9	−0.367063
$156$	0	0
$157$	−8.49699e9	−1.11613	−0.558067	−	0.829796i	$-0.688457\pi$
$157$	−8.49699e9	−1.11613	−0.558067	+	0.829796i	$0.688457\pi$
$158$	−6.33110e9	−0.808206
$159$	0	0
$160$	−3.56310e9	−0.429821
$161$	−4.97952e9	−0.584078
$162$	0	0
$163$	4.25863e9	0.472526	0.236263	−	0.971689i	$-0.424077\pi$
$163$	4.25863e9	0.472526	0.236263	+	0.971689i	$0.424077\pi$
$164$	−8.39891e9	−0.906621
$165$	0	0
$166$	7.87749e9	0.805195
$167$	−9.23536e8	−0.0918818	−0.0459409	−	0.998944i	$-0.514629\pi$
$167$	−9.23536e8	−0.0918818	−0.0459409	+	0.998944i	$0.514629\pi$
$168$	0	0
$169$	−9.91221e9	−0.934717
$170$	−1.33458e9	−0.122553
$171$	0	0
$172$	−1.03651e10	−0.903012
$173$	4.76006e9	0.404022	0.202011	−	0.979383i	$-0.435252\pi$
$173$	4.76006e9	0.404022	0.202011	+	0.979383i	$0.435252\pi$
$174$	0	0
$175$	9.37891e8	0.0755929
$176$	−4.50217e8	−0.0353683
$177$	0	0
$178$	1.69293e9	0.126401
$179$	9.47187e9	0.689599	0.344800	−	0.938676i	$-0.387947\pi$
$179$	9.47187e9	0.689599	0.344800	+	0.938676i	$0.387947\pi$
$180$	0	0
$181$	−7.85395e9	−0.543919	−0.271960	−	0.962309i	$-0.587672\pi$
$181$	−7.85395e9	−0.543919	−0.271960	+	0.962309i	$0.587672\pi$
$182$	9.36766e8	0.0632863
$183$	0	0
$184$	2.47292e10	1.59049
$185$	8.06758e9	0.506373
$186$	0	0
$187$	2.37944e9	0.142294
$188$	1.73995e10	1.01584
$189$	0	0
$190$	1.48015e9	0.0823976
$191$	−1.78210e10	−0.968909	−0.484454	−	0.874817i	$-0.660982\pi$
$191$	−1.78210e10	−0.968909	−0.484454	+	0.874817i	$0.660982\pi$
$192$	0	0
$193$	−1.93101e9	−0.100179	−0.0500894	−	0.998745i	$-0.515951\pi$
$193$	−1.93101e9	−0.100179	−0.0500894	+	0.998745i	$0.515951\pi$
$194$	−2.17344e10	−1.10164
$195$	0	0
$196$	−1.68400e9	−0.0815061
$197$	−3.24598e10	−1.53549	−0.767746	−	0.640755i	$-0.778622\pi$
$197$	−3.24598e10	−1.53549	−0.767746	+	0.640755i	$0.778622\pi$
$198$	0	0
$199$	−2.75463e10	−1.24516	−0.622578	−	0.782558i	$-0.713915\pi$
$199$	−2.75463e10	−1.24516	−0.622578	+	0.782558i	$0.713915\pi$
$200$	−4.65773e9	−0.205845
$201$	0	0
$202$	−1.11767e10	−0.472315
$203$	1.18835e10	0.491146
$204$	0	0
$205$	−1.79699e10	−0.710645
$206$	−1.95947e10	−0.758118
$207$	0	0
$208$	−7.16904e8	−0.0265568
$209$	−2.63897e9	−0.0956702
$210$	0	0
$211$	1.94450e10	0.675362	0.337681	−	0.941261i	$-0.390357\pi$
$211$	1.94450e10	0.675362	0.337681	+	0.941261i	$0.390357\pi$
$212$	6.75261e8	0.0229594
$213$	0	0
$214$	−1.59204e10	−0.518911
$215$	−2.21765e10	−0.707817
$216$	0	0
$217$	1.01332e10	0.310225
$218$	1.61111e10	0.483139
$219$	0	0
$220$	3.01677e9	0.0868240
$221$	3.78891e9	0.106844
$222$	0	0
$223$	−4.90459e10	−1.32810	−0.664050	−	0.747688i	$-0.731164\pi$
$223$	−4.90459e10	−1.32810	−0.664050	+	0.747688i	$0.731164\pi$
$224$	1.36880e10	0.363265
$225$	0	0
$226$	−4.23439e10	−1.07970
$227$	6.87990e10	1.71975	0.859876	−	0.510503i	$-0.170541\pi$
$227$	6.87990e10	1.71975	0.859876	+	0.510503i	$0.170541\pi$
$228$	0	0
$229$	−3.08456e9	−0.0741198	−0.0370599	−	0.999313i	$-0.511799\pi$
$229$	−3.08456e9	−0.0741198	−0.0370599	+	0.999313i	$0.511799\pi$
$230$	1.92207e10	0.452892
$231$	0	0
$232$	−5.90154e10	−1.33743
$233$	1.48643e10	0.330402	0.165201	−	0.986260i	$-0.447173\pi$
$233$	1.48643e10	0.330402	0.165201	+	0.986260i	$0.447173\pi$
$234$	0	0
$235$	3.72271e10	0.796258
$236$	−4.92659e10	−1.03381
$237$	0	0
$238$	5.12693e9	0.103576
$239$	−4.06416e10	−0.805713	−0.402857	−	0.915263i	$-0.631983\pi$
$239$	−4.06416e10	−0.805713	−0.402857	+	0.915263i	$0.631983\pi$
$240$	0	0
$241$	−2.09799e10	−0.400615	−0.200308	−	0.979733i	$-0.564194\pi$
$241$	−2.09799e10	−0.400615	−0.200308	+	0.979733i	$0.564194\pi$
$242$	−3.09161e10	−0.579449
$243$	0	0
$244$	1.95768e10	0.353580
$245$	−3.60300e9	−0.0638877
$246$	0	0
$247$	−4.20218e9	−0.0718354
$248$	−5.03232e10	−0.844765
$249$	0	0
$250$	−3.62022e9	−0.0586145
$251$	9.95566e10	1.58321	0.791605	−	0.611034i	$-0.209246\pi$
$251$	9.95566e10	1.58321	0.791605	+	0.611034i	$0.209246\pi$
$252$	0	0
$253$	−3.42688e10	−0.525844
$254$	−5.72499e10	−0.863024
$255$	0	0
$256$	−7.20522e10	−1.04850
$257$	1.30781e11	1.87002	0.935009	−	0.354624i	$-0.115391\pi$
$257$	1.30781e11	1.87002	0.935009	+	0.354624i	$0.115391\pi$
$258$	0	0
$259$	−3.09924e10	−0.427963
$260$	4.80376e9	0.0651931
$261$	0	0
$262$	−5.95954e9	−0.0781371
$263$	3.27938e10	0.422660	0.211330	−	0.977415i	$-0.432221\pi$
$263$	3.27938e10	0.422660	0.211330	+	0.977415i	$0.432221\pi$
$264$	0	0
$265$	1.44475e9	0.0179965
$266$	−5.68615e9	−0.0696387
$267$	0	0
$268$	4.56459e10	0.540499
$269$	−7.47690e9	−0.0870636	−0.0435318	−	0.999052i	$-0.513861\pi$
$269$	−7.47690e9	−0.0870636	−0.0435318	+	0.999052i	$0.513861\pi$
$270$	0	0
$271$	6.72735e9	0.0757674	0.0378837	−	0.999282i	$-0.487938\pi$
$271$	6.72735e9	0.0757674	0.0378837	+	0.999282i	$0.487938\pi$
$272$	−3.92363e9	−0.0434638
$273$	0	0
$274$	−4.51039e10	−0.483434
$275$	6.45452e9	0.0680561
$276$	0	0
$277$	−1.52676e11	−1.55815	−0.779077	−	0.626928i	$-0.784312\pi$
$277$	−1.52676e11	−1.55815	−0.779077	+	0.626928i	$0.784312\pi$
$278$	1.24028e10	0.124543
$279$	0	0
$280$	1.78932e10	0.173971
$281$	−3.09796e10	−0.296413	−0.148206	−	0.988956i	$-0.547350\pi$
$281$	−3.09796e10	−0.296413	−0.148206	+	0.988956i	$0.547350\pi$
$282$	0	0
$283$	−8.92078e10	−0.826731	−0.413365	−	0.910565i	$-0.635647\pi$
$283$	−8.92078e10	−0.826731	−0.413365	+	0.910565i	$0.635647\pi$
$284$	2.03041e10	0.185205
$285$	0	0
$286$	6.44678e9	0.0569765
$287$	6.90331e10	0.600605
$288$	0	0
$289$	−9.78512e10	−0.825136
$290$	−4.58697e10	−0.380833
$291$	0	0
$292$	2.28856e10	0.184221
$293$	−6.30090e10	−0.499457	−0.249728	−	0.968316i	$-0.580341\pi$
$293$	−6.30090e10	−0.499457	−0.249728	+	0.968316i	$0.580341\pi$
$294$	0	0
$295$	−1.05407e11	−0.810344
$296$	1.53914e11	1.16537
$297$	0	0
$298$	1.15101e11	0.845483
$299$	−5.45681e10	−0.394838
$300$	0	0
$301$	8.51934e10	0.598214
$302$	1.24268e11	0.859665
$303$	0	0
$304$	4.35160e9	0.0292225
$305$	4.18855e10	0.277150
$306$	0	0
$307$	−1.22461e11	−0.786821	−0.393411	−	0.919363i	$-0.628705\pi$
$307$	−1.22461e11	−0.786821	−0.393411	+	0.919363i	$0.628705\pi$
$308$	−1.15892e10	−0.0733797
$309$	0	0
$310$	−3.91137e10	−0.240548
$311$	−2.77926e11	−1.68464	−0.842321	−	0.538975i	$-0.818812\pi$
$311$	−2.77926e11	−1.68464	−0.842321	+	0.538975i	$0.818812\pi$
$312$	0	0
$313$	1.22493e11	0.721374	0.360687	−	0.932687i	$-0.382542\pi$
$313$	1.22493e11	0.721374	0.360687	+	0.932687i	$0.382542\pi$
$314$	−1.25997e11	−0.731436
$315$	0	0
$316$	1.24722e11	0.703639
$317$	−2.32000e11	−1.29039	−0.645197	−	0.764016i	$-0.723225\pi$
$317$	−2.32000e11	−1.29039	−0.645197	+	0.764016i	$0.723225\pi$
$318$	0	0
$319$	8.17814e10	0.442177
$320$	−6.15542e10	−0.328158
$321$	0	0
$322$	−7.38384e10	−0.382764
$323$	−2.29986e10	−0.117568
$324$	0	0
$325$	1.02779e10	0.0511009
$326$	6.31487e10	0.309660
$327$	0	0
$328$	−3.42831e11	−1.63549
$329$	−1.43012e11	−0.672961
$330$	0	0
$331$	−3.74253e11	−1.71372	−0.856859	−	0.515551i	$-0.827587\pi$
$331$	−3.74253e11	−1.71372	−0.856859	+	0.515551i	$0.827587\pi$
$332$	−1.55185e11	−0.701018
$333$	0	0
$334$	−1.36946e10	−0.0602129
$335$	9.76616e10	0.423665
$336$	0	0
$337$	3.35878e11	1.41856	0.709280	−	0.704927i	$-0.249020\pi$
$337$	3.35878e11	1.41856	0.709280	+	0.704927i	$0.249020\pi$
$338$	−1.46982e11	−0.612548
$339$	0	0
$340$	2.62911e10	0.106697
$341$	6.97361e10	0.279295
$342$	0	0
$343$	1.38413e10	0.0539949
$344$	−4.23086e11	−1.62898
$345$	0	0
$346$	7.05842e10	0.264768
$347$	3.63698e11	1.34666	0.673331	−	0.739341i	$-0.264863\pi$
$347$	3.63698e11	1.34666	0.673331	+	0.739341i	$0.264863\pi$
$348$	0	0
$349$	1.08740e11	0.392350	0.196175	−	0.980569i	$-0.437148\pi$
$349$	1.08740e11	0.392350	0.196175	+	0.980569i	$0.437148\pi$
$350$	1.39074e10	0.0495383
$351$	0	0
$352$	9.42002e10	0.327047
$353$	3.36249e11	1.15259	0.576296	−	0.817241i	$-0.304498\pi$
$353$	3.36249e11	1.15259	0.576296	+	0.817241i	$0.304498\pi$
$354$	0	0
$355$	4.34417e10	0.145171
$356$	−3.33505e10	−0.110047
$357$	0	0
$358$	1.40453e11	0.451915
$359$	−5.49944e11	−1.74741	−0.873703	−	0.486460i	$-0.838288\pi$
$359$	−5.49944e11	−1.74741	−0.873703	+	0.486460i	$0.838288\pi$
$360$	0	0
$361$	−2.97181e11	−0.920954
$362$	−1.16462e11	−0.356447
$363$	0	0
$364$	−1.84541e10	−0.0550982
$365$	4.89649e10	0.144400
$366$	0	0
$367$	−1.32204e11	−0.380405	−0.190203	−	0.981745i	$-0.560914\pi$
$367$	−1.32204e11	−0.380405	−0.190203	+	0.981745i	$0.560914\pi$
$368$	5.65084e10	0.160619
$369$	0	0
$370$	1.19630e11	0.331841
$371$	−5.55017e9	−0.0152098
$372$	0	0
$373$	4.88064e11	1.30553	0.652765	−	0.757560i	$-0.273609\pi$
$373$	4.88064e11	1.30553	0.652765	+	0.757560i	$0.273609\pi$
$374$	3.52833e10	0.0932496
$375$	0	0
$376$	7.10222e11	1.83252
$377$	1.30225e11	0.332015
$378$	0	0
$379$	3.00162e11	0.747274	0.373637	−	0.927575i	$-0.378111\pi$
$379$	3.00162e11	0.747274	0.373637	+	0.927575i	$0.378111\pi$
$380$	−2.91587e10	−0.0717369
$381$	0	0
$382$	−2.64258e11	−0.634955
$383$	−7.74202e11	−1.83848	−0.919241	−	0.393694i	$-0.871197\pi$
$383$	−7.74202e11	−1.83848	−0.919241	+	0.393694i	$0.871197\pi$
$384$	0	0
$385$	−2.47957e10	−0.0575179
$386$	−2.86338e10	−0.0656501
$387$	0	0
$388$	4.28165e11	0.959109
$389$	1.70127e11	0.376703	0.188352	−	0.982102i	$-0.439686\pi$
$389$	1.70127e11	0.376703	0.188352	+	0.982102i	$0.439686\pi$
$390$	0	0
$391$	−2.98652e11	−0.646204
$392$	−6.87383e10	−0.147032
$393$	0	0
$394$	−4.81327e11	−1.00625
$395$	2.66848e11	0.551540
$396$	0	0
$397$	−5.47242e11	−1.10566	−0.552831	−	0.833293i	$-0.686452\pi$
$397$	−5.47242e11	−1.10566	−0.552831	+	0.833293i	$0.686452\pi$
$398$	−4.08468e11	−0.815988
$399$	0	0
$400$	−1.06433e10	−0.0207878
$401$	4.57174e11	0.882941	0.441471	−	0.897276i	$-0.354457\pi$
$401$	4.57174e11	0.882941	0.441471	+	0.897276i	$0.354457\pi$
$402$	0	0
$403$	1.11045e11	0.209713
$404$	2.20179e11	0.411206
$405$	0	0
$406$	1.76213e11	0.321863
$407$	−2.13288e11	−0.385294
$408$	0	0
$409$	9.42996e10	0.166631	0.0833153	−	0.996523i	$-0.473449\pi$
$409$	9.42996e10	0.166631	0.0833153	+	0.996523i	$0.473449\pi$
$410$	−2.66465e11	−0.465707
$411$	0	0
$412$	3.86013e11	0.660032
$413$	4.04930e11	0.684865
$414$	0	0
$415$	−3.32026e11	−0.549485
$416$	1.50000e11	0.245568
$417$	0	0
$418$	−3.91318e10	−0.0626955
$419$	−1.66874e11	−0.264500	−0.132250	−	0.991216i	$-0.542220\pi$
$419$	−1.66874e11	−0.264500	−0.132250	+	0.991216i	$0.542220\pi$
$420$	0	0
$421$	−5.52779e11	−0.857594	−0.428797	−	0.903401i	$-0.641062\pi$
$421$	−5.52779e11	−0.857594	−0.428797	+	0.903401i	$0.641062\pi$
$422$	2.88339e11	0.442585
$423$	0	0
$424$	2.75631e10	0.0414174
$425$	5.62510e10	0.0836334
$426$	0	0
$427$	−1.60907e11	−0.234234
$428$	3.13630e11	0.451773
$429$	0	0
$430$	−3.28843e11	−0.463853
$431$	4.37408e11	0.610575	0.305288	−	0.952260i	$-0.401247\pi$
$431$	4.37408e11	0.610575	0.305288	+	0.952260i	$0.401247\pi$
$432$	0	0
$433$	2.79837e11	0.382569	0.191285	−	0.981535i	$-0.438735\pi$
$433$	2.79837e11	0.382569	0.191285	+	0.981535i	$0.438735\pi$
$434$	1.50259e11	0.203300
$435$	0	0
$436$	−3.17387e11	−0.420630
$437$	3.31227e11	0.434470
$438$	0	0
$439$	8.87150e11	1.14000	0.570002	−	0.821643i	$-0.306942\pi$
$439$	8.87150e11	1.14000	0.570002	+	0.821643i	$0.306942\pi$
$440$	1.23140e11	0.156625
$441$	0	0
$442$	5.61835e10	0.0700178
$443$	5.87852e11	0.725189	0.362594	−	0.931947i	$-0.381891\pi$
$443$	5.87852e11	0.725189	0.362594	+	0.931947i	$0.381891\pi$
$444$	0	0
$445$	−7.13549e10	−0.0862589
$446$	−7.27274e11	−0.870344
$447$	0	0
$448$	2.36466e11	0.277344
$449$	5.19918e11	0.603707	0.301854	−	0.953354i	$-0.402395\pi$
$449$	5.19918e11	0.603707	0.301854	+	0.953354i	$0.402395\pi$
$450$	0	0
$451$	4.75082e11	0.540723
$452$	8.34168e11	0.940006
$453$	0	0
$454$	1.02018e12	1.12701
$455$	−3.94835e10	−0.0431881
$456$	0	0
$457$	−1.03055e12	−1.10521	−0.552607	−	0.833442i	$-0.686367\pi$
$457$	−1.03055e12	−1.10521	−0.552607	+	0.833442i	$0.686367\pi$
$458$	−4.57392e10	−0.0485729
$459$	0	0
$460$	−3.78646e11	−0.394296
$461$	−9.37018e11	−0.966259	−0.483130	−	0.875549i	$-0.660500\pi$
$461$	−9.37018e11	−0.966259	−0.483130	+	0.875549i	$0.660500\pi$
$462$	0	0
$463$	1.08281e12	1.09506	0.547529	−	0.836787i	$-0.315569\pi$
$463$	1.08281e12	1.09506	0.547529	+	0.836787i	$0.315569\pi$
$464$	−1.34855e11	−0.135063
$465$	0	0
$466$	2.20414e11	0.216522
$467$	−1.52419e11	−0.148290	−0.0741451	−	0.997247i	$-0.523623\pi$
$467$	−1.52419e11	−0.148290	−0.0741451	+	0.997247i	$0.523623\pi$
$468$	0	0
$469$	−3.75177e11	−0.358062
$470$	5.52019e11	0.521812
$471$	0	0
$472$	−2.01096e12	−1.86494
$473$	5.86297e11	0.538570
$474$	0	0
$475$	−6.23865e10	−0.0562302
$476$	−1.01000e11	−0.0901756
$477$	0	0
$478$	−6.02652e11	−0.528008
$479$	2.02244e11	0.175535	0.0877677	−	0.996141i	$-0.472027\pi$
$479$	2.02244e11	0.175535	0.0877677	+	0.996141i	$0.472027\pi$
$480$	0	0
$481$	−3.39631e11	−0.289304
$482$	−3.11099e11	−0.262535
$483$	0	0
$484$	6.09042e11	0.504479
$485$	9.16079e11	0.751788
$486$	0	0
$487$	8.35572e11	0.673137	0.336568	−	0.941659i	$-0.390734\pi$
$487$	8.35572e11	0.673137	0.336568	+	0.941659i	$0.390734\pi$
$488$	7.99094e11	0.637836
$489$	0	0
$490$	−5.34268e10	−0.0418675
$491$	1.06242e12	0.824953	0.412476	−	0.910968i	$-0.364664\pi$
$491$	1.06242e12	0.824953	0.412476	+	0.910968i	$0.364664\pi$
$492$	0	0
$493$	7.12723e11	0.543387
$494$	−6.23117e10	−0.0470759
$495$	0	0
$496$	−1.14993e11	−0.0853108
$497$	−1.66886e11	−0.122692
$498$	0	0
$499$	−3.33938e11	−0.241109	−0.120554	−	0.992707i	$-0.538467\pi$
$499$	−3.33938e11	−0.241109	−0.120554	+	0.992707i	$0.538467\pi$
$500$	7.13178e10	0.0510309
$501$	0	0
$502$	1.47627e12	1.03752
$503$	3.60934e11	0.251404	0.125702	−	0.992068i	$-0.459882\pi$
$503$	3.60934e11	0.251404	0.125702	+	0.992068i	$0.459882\pi$
$504$	0	0
$505$	4.71083e11	0.322319
$506$	−5.08152e11	−0.344601
$507$	0	0
$508$	1.12781e12	0.751365
$509$	−4.09926e11	−0.270692	−0.135346	−	0.990798i	$-0.543215\pi$
$509$	−4.09926e11	−0.270692	−0.135346	+	0.990798i	$0.543215\pi$
$510$	0	0
$511$	−1.88104e11	−0.122040
$512$	−3.21676e11	−0.206873
$513$	0	0
$514$	1.93928e12	1.22548
$515$	8.25894e11	0.517359
$516$	0	0
$517$	−9.84199e11	−0.605865
$518$	−4.59569e11	−0.280457
$519$	0	0
$520$	1.96082e11	0.117604
$521$	1.09884e12	0.653377	0.326688	−	0.945132i	$-0.394067\pi$
$521$	1.09884e12	0.653377	0.326688	+	0.945132i	$0.394067\pi$
$522$	0	0
$523$	2.97552e12	1.73902	0.869511	−	0.493914i	$-0.164434\pi$
$523$	2.97552e12	1.73902	0.869511	+	0.493914i	$0.164434\pi$
$524$	1.17402e11	0.0680276
$525$	0	0
$526$	4.86280e11	0.276982
$527$	6.07748e11	0.343223
$528$	0	0
$529$	2.50005e12	1.38803
$530$	2.14234e10	0.0117936
$531$	0	0
$532$	1.12016e11	0.0606288
$533$	7.56500e11	0.406010
$534$	0	0
$535$	6.71026e11	0.354118
$536$	1.86320e12	0.975028
$537$	0	0
$538$	−1.10871e11	−0.0570554
$539$	9.52551e10	0.0486115
$540$	0	0
$541$	2.97095e12	1.49110	0.745551	−	0.666449i	$-0.232187\pi$
$541$	2.97095e12	1.49110	0.745551	+	0.666449i	$0.232187\pi$
$542$	9.97561e10	0.0496527
$543$	0	0
$544$	8.20952e11	0.401904
$545$	−6.79065e11	−0.329706
$546$	0	0
$547$	−1.50514e12	−0.718842	−0.359421	−	0.933176i	$-0.617026\pi$
$547$	−1.50514e12	−0.718842	−0.359421	+	0.933176i	$0.617026\pi$
$548$	8.88540e11	0.420886
$549$	0	0
$550$	9.57104e10	0.0445992
$551$	−7.90463e11	−0.365342
$552$	0	0
$553$	−1.02512e12	−0.466136
$554$	−2.26394e12	−1.02111
$555$	0	0
$556$	−2.44333e11	−0.108429
$557$	2.17354e12	0.956797	0.478399	−	0.878143i	$-0.341217\pi$
$557$	2.17354e12	0.956797	0.478399	+	0.878143i	$0.341217\pi$
$558$	0	0
$559$	9.33593e11	0.404394
$560$	4.08874e10	0.0175689
$561$	0	0
$562$	−4.59379e11	−0.194248
$563$	−1.40104e12	−0.587710	−0.293855	−	0.955850i	$-0.594938\pi$
$563$	−1.40104e12	−0.587710	−0.293855	+	0.955850i	$0.594938\pi$
$564$	0	0
$565$	1.78474e12	0.736814
$566$	−1.32281e12	−0.541781
$567$	0	0
$568$	8.28784e11	0.334098
$569$	1.81226e12	0.724795	0.362397	−	0.932024i	$-0.381958\pi$
$569$	1.81226e12	0.724795	0.362397	+	0.932024i	$0.381958\pi$
$570$	0	0
$571$	−3.38885e12	−1.33410	−0.667052	−	0.745012i	$-0.732444\pi$
$571$	−3.38885e12	−1.33410	−0.667052	+	0.745012i	$0.732444\pi$
$572$	−1.27001e11	−0.0496048
$573$	0	0
$574$	1.02365e12	0.393594
$575$	−8.10131e11	−0.309065
$576$	0	0
$577$	2.86129e12	1.07466	0.537329	−	0.843373i	$-0.319433\pi$
$577$	2.86129e12	1.07466	0.537329	+	0.843373i	$0.319433\pi$
$578$	−1.45098e12	−0.540737
$579$	0	0
$580$	9.03626e11	0.331560
$581$	1.27551e12	0.464400
$582$	0	0
$583$	−3.81960e10	−0.0136933
$584$	9.34156e11	0.332324
$585$	0	0
$586$	−9.34324e11	−0.327309
$587$	7.85729e11	0.273150	0.136575	−	0.990630i	$-0.456391\pi$
$587$	7.85729e11	0.273150	0.136575	+	0.990630i	$0.456391\pi$
$588$	0	0
$589$	−6.74039e11	−0.230763
$590$	−1.56302e12	−0.531043
$591$	0	0
$592$	3.51707e11	0.117688
$593$	−8.24271e11	−0.273731	−0.136865	−	0.990590i	$-0.543703\pi$
$593$	−8.24271e11	−0.273731	−0.136865	+	0.990590i	$0.543703\pi$
$594$	0	0
$595$	−2.16094e11	−0.0706831
$596$	−2.26747e12	−0.736093
$597$	0	0
$598$	−8.09159e11	−0.258749
$599$	2.60594e10	0.00827073	0.00413537	−	0.999991i	$-0.498684\pi$
$599$	2.60594e10	0.00827073	0.00413537	+	0.999991i	$0.498684\pi$
$600$	0	0
$601$	1.90372e11	0.0595207	0.0297603	−	0.999557i	$-0.490526\pi$
$601$	1.90372e11	0.0595207	0.0297603	+	0.999557i	$0.490526\pi$
$602$	1.26328e12	0.392028
$603$	0	0
$604$	−2.44807e12	−0.748441
$605$	1.30307e12	0.395430
$606$	0	0
$607$	1.77675e12	0.531222	0.265611	−	0.964080i	$-0.414426\pi$
$607$	1.77675e12	0.531222	0.265611	+	0.964080i	$0.414426\pi$
$608$	−9.10497e11	−0.270217
$609$	0	0
$610$	6.21096e11	0.181624
$611$	−1.56719e12	−0.454922
$612$	0	0
$613$	−2.48518e11	−0.0710863	−0.0355431	−	0.999368i	$-0.511316\pi$
$613$	−2.48518e11	−0.0710863	−0.0355431	+	0.999368i	$0.511316\pi$
$614$	−1.81591e12	−0.515628
$615$	0	0
$616$	−4.73054e11	−0.132372
$617$	−3.52347e12	−0.978786	−0.489393	−	0.872063i	$-0.662782\pi$
$617$	−3.52347e12	−0.978786	−0.489393	+	0.872063i	$0.662782\pi$
$618$	0	0
$619$	−6.69712e12	−1.83350	−0.916748	−	0.399466i	$-0.869196\pi$
$619$	−6.69712e12	−1.83350	−0.916748	+	0.399466i	$0.869196\pi$
$620$	7.70534e11	0.209425
$621$	0	0
$622$	−4.12121e12	−1.10400
$623$	2.74117e11	0.0729021
$624$	0	0
$625$	1.52588e11	0.0400000
$626$	1.81637e12	0.472738
$627$	0	0
$628$	2.48212e12	0.636802
$629$	−1.85880e12	−0.473484
$630$	0	0
$631$	6.29917e11	0.158180	0.0790900	−	0.996867i	$-0.474799\pi$
$631$	6.29917e11	0.158180	0.0790900	+	0.996867i	$0.474799\pi$
$632$	5.09095e12	1.26932
$633$	0	0
$634$	−3.44020e12	−0.845634
$635$	2.41301e12	0.588949
$636$	0	0
$637$	1.51680e11	0.0365006
$638$	1.21269e12	0.289772
$639$	0	0
$640$	9.11555e11	0.214770
$641$	−2.82049e12	−0.659877	−0.329938	−	0.944002i	$-0.607028\pi$
$641$	−2.82049e12	−0.659877	−0.329938	+	0.944002i	$0.607028\pi$
$642$	0	0
$643$	7.19804e12	1.66060	0.830300	−	0.557317i	$-0.188169\pi$
$643$	7.19804e12	1.66060	0.830300	+	0.557317i	$0.188169\pi$
$644$	1.45461e12	0.333241
$645$	0	0
$646$	−3.41033e11	−0.0770459
$647$	−7.17803e12	−1.61041	−0.805204	−	0.592998i	$-0.797944\pi$
$647$	−7.17803e12	−1.61041	−0.805204	+	0.592998i	$0.797944\pi$
$648$	0	0
$649$	2.78671e12	0.616582
$650$	1.52405e11	0.0334880
$651$	0	0
$652$	−1.24402e12	−0.269596
$653$	1.54599e12	0.332734	0.166367	−	0.986064i	$-0.446796\pi$
$653$	1.54599e12	0.332734	0.166367	+	0.986064i	$0.446796\pi$
$654$	0	0
$655$	2.51187e11	0.0533227
$656$	−7.83398e11	−0.165164
$657$	0	0
$658$	−2.12064e12	−0.441011
$659$	−2.38624e12	−0.492868	−0.246434	−	0.969160i	$-0.579259\pi$
$659$	−2.38624e12	−0.492868	−0.246434	+	0.969160i	$0.579259\pi$
$660$	0	0
$661$	−2.35488e12	−0.479802	−0.239901	−	0.970797i	$-0.577115\pi$
$661$	−2.35488e12	−0.479802	−0.239901	+	0.970797i	$0.577115\pi$
$662$	−5.54958e12	−1.12305
$663$	0	0
$664$	−6.33443e12	−1.26459
$665$	2.39664e11	0.0475232
$666$	0	0
$667$	−1.02647e13	−2.00807
$668$	2.69781e11	0.0524225
$669$	0	0
$670$	1.44817e12	0.277640
$671$	−1.10736e12	−0.210880
$672$	0	0
$673$	−6.32709e12	−1.18887	−0.594437	−	0.804142i	$-0.702625\pi$
$673$	−6.32709e12	−1.18887	−0.594437	+	0.804142i	$0.702625\pi$
$674$	4.98055e12	0.929625
$675$	0	0
$676$	2.89553e12	0.533296
$677$	−3.16670e12	−0.579372	−0.289686	−	0.957122i	$-0.593551\pi$
$677$	−3.16670e12	−0.579372	−0.289686	+	0.957122i	$0.593551\pi$
$678$	0	0
$679$	−3.51921e12	−0.635376
$680$	1.07316e12	0.192475
$681$	0	0
$682$	1.03408e12	0.183030
$683$	6.06916e12	1.06717	0.533587	−	0.845745i	$-0.320844\pi$
$683$	6.06916e12	1.06717	0.533587	+	0.845745i	$0.320844\pi$
$684$	0	0
$685$	1.90107e12	0.329907
$686$	2.05245e11	0.0353845
$687$	0	0
$688$	−9.66788e11	−0.164507
$689$	−6.08216e10	−0.0102818
$690$	0	0
$691$	−1.36410e12	−0.227611	−0.113806	−	0.993503i	$-0.536304\pi$
$691$	−1.36410e12	−0.227611	−0.113806	+	0.993503i	$0.536304\pi$
$692$	−1.39050e12	−0.230512
$693$	0	0
$694$	5.39307e12	0.882508
$695$	−5.22763e11	−0.0849911
$696$	0	0
$697$	4.14033e12	0.664489
$698$	1.61244e12	0.257119
$699$	0	0
$700$	−2.73974e11	−0.0431290
$701$	−4.27032e12	−0.667927	−0.333964	−	0.942586i	$-0.608386\pi$
$701$	−4.27032e12	−0.667927	−0.333964	+	0.942586i	$0.608386\pi$
$702$	0	0
$703$	2.06155e12	0.318343
$704$	1.62735e12	0.249692
$705$	0	0
$706$	4.98605e12	0.755328
$707$	−1.80971e12	−0.272410
$708$	0	0
$709$	4.76449e12	0.708122	0.354061	−	0.935222i	$-0.384800\pi$
$709$	4.76449e12	0.708122	0.354061	+	0.935222i	$0.384800\pi$
$710$	6.44172e11	0.0951348
$711$	0	0
$712$	−1.36132e12	−0.198517
$713$	−8.75284e12	−1.26837
$714$	0	0
$715$	−2.71724e11	−0.0388822
$716$	−2.76690e12	−0.393446
$717$	0	0
$718$	−8.15481e12	−1.14513
$719$	2.64192e12	0.368671	0.184336	−	0.982863i	$-0.440987\pi$
$719$	2.64192e12	0.368671	0.184336	+	0.982863i	$0.440987\pi$
$720$	0	0
$721$	−3.17276e12	−0.437248
$722$	−4.40672e12	−0.603529
$723$	0	0
$724$	2.29428e12	0.310329
$725$	1.93335e12	0.259890
$726$	0	0
$727$	−3.54861e12	−0.471144	−0.235572	−	0.971857i	$-0.575696\pi$
$727$	−3.54861e12	−0.471144	−0.235572	+	0.971857i	$0.575696\pi$
$728$	−7.53270e11	−0.0993938
$729$	0	0
$730$	7.26072e11	0.0946296
$731$	5.10956e12	0.661844
$732$	0	0
$733$	−1.34909e13	−1.72613	−0.863063	−	0.505095i	$-0.831457\pi$
$733$	−1.34909e13	−1.72613	−0.863063	+	0.505095i	$0.831457\pi$
$734$	−1.96037e12	−0.249291
$735$	0	0
$736$	−1.18234e13	−1.48523
$737$	−2.58195e12	−0.322362
$738$	0	0
$739$	−1.33151e13	−1.64227	−0.821133	−	0.570736i	$-0.806658\pi$
$739$	−1.33151e13	−1.64227	−0.821133	+	0.570736i	$0.806658\pi$
$740$	−2.35668e12	−0.288907
$741$	0	0
$742$	−8.23003e10	−0.00996744
$743$	−1.37811e13	−1.65896	−0.829479	−	0.558538i	$-0.811363\pi$
$743$	−1.37811e13	−1.65896	−0.829479	+	0.558538i	$0.811363\pi$
$744$	0	0
$745$	−4.85135e12	−0.576979
$746$	7.23722e12	0.855553
$747$	0	0
$748$	−6.95075e11	−0.0811848
$749$	−2.57781e12	−0.299284
$750$	0	0
$751$	−1.31166e12	−0.150467	−0.0752334	−	0.997166i	$-0.523970\pi$
$751$	−1.31166e12	−0.150467	−0.0752334	+	0.997166i	$0.523970\pi$
$752$	1.62292e12	0.185062
$753$	0	0
$754$	1.93103e12	0.217580
$755$	−5.23776e12	−0.586657
$756$	0	0
$757$	1.31405e13	1.45439	0.727196	−	0.686430i	$-0.240823\pi$
$757$	1.31405e13	1.45439	0.727196	+	0.686430i	$0.240823\pi$
$758$	4.45093e12	0.489711
$759$	0	0
$760$	−1.19022e12	−0.129409
$761$	−4.85816e12	−0.525099	−0.262549	−	0.964919i	$-0.584563\pi$
$761$	−4.85816e12	−0.525099	−0.262549	+	0.964919i	$0.584563\pi$
$762$	0	0
$763$	2.60869e12	0.278652
$764$	5.20584e12	0.552804
$765$	0	0
$766$	−1.14802e13	−1.20481
$767$	4.43743e12	0.462970
$768$	0	0
$769$	−1.57384e13	−1.62290	−0.811452	−	0.584420i	$-0.801322\pi$
$769$	−1.57384e13	−1.62290	−0.811452	+	0.584420i	$0.801322\pi$
$770$	−3.67681e11	−0.0376932
$771$	0	0
$772$	5.64081e11	0.0571562
$773$	−1.97088e12	−0.198542	−0.0992710	−	0.995060i	$-0.531651\pi$
$773$	−1.97088e12	−0.198542	−0.0992710	+	0.995060i	$0.531651\pi$
$774$	0	0
$775$	1.64859e12	0.164156
$776$	1.74770e13	1.73017
$777$	0	0
$778$	2.52271e12	0.246865
$779$	−4.59194e12	−0.446763
$780$	0	0
$781$	−1.14850e12	−0.110459
$782$	−4.42854e12	−0.423477
$783$	0	0
$784$	−1.57073e11	−0.0148484
$785$	5.31062e12	0.499150
$786$	0	0
$787$	−1.29627e13	−1.20451	−0.602253	−	0.798306i	$-0.705730\pi$
$787$	−1.29627e13	−1.20451	−0.602253	+	0.798306i	$0.705730\pi$
$788$	9.48207e12	0.876063
$789$	0	0
$790$	3.95694e12	0.361441
$791$	−6.85627e12	−0.622721
$792$	0	0
$793$	−1.76330e12	−0.158343
$794$	−8.11474e12	−0.724573
$795$	0	0
$796$	8.04675e12	0.710414
$797$	−1.04429e13	−0.916762	−0.458381	−	0.888756i	$-0.651571\pi$
$797$	−1.04429e13	−0.916762	−0.458381	+	0.888756i	$0.651571\pi$
$798$	0	0
$799$	−8.57727e12	−0.744541
$800$	2.22694e12	0.192222
$801$	0	0
$802$	6.77917e12	0.578618
$803$	−1.29452e12	−0.109872
$804$	0	0
$805$	3.11220e12	0.261208
$806$	1.64662e12	0.137431
$807$	0	0
$808$	8.98737e12	0.741791
$809$	6.99142e12	0.573848	0.286924	−	0.957953i	$-0.407367\pi$
$809$	6.99142e12	0.573848	0.286924	+	0.957953i	$0.407367\pi$
$810$	0	0
$811$	−1.14380e13	−0.928442	−0.464221	−	0.885719i	$-0.653666\pi$
$811$	−1.14380e13	−0.928442	−0.464221	+	0.885719i	$0.653666\pi$
$812$	−3.47137e12	−0.280220
$813$	0	0
$814$	−3.16273e12	−0.252495
$815$	−2.66164e12	−0.211320
$816$	0	0
$817$	−5.66689e12	−0.444985
$818$	1.39831e12	0.109198
$819$	0	0
$820$	5.24932e12	0.405453
$821$	4.42435e12	0.339864	0.169932	−	0.985456i	$-0.445645\pi$
$821$	4.42435e12	0.339864	0.169932	+	0.985456i	$0.445645\pi$
$822$	0	0
$823$	1.71948e13	1.30647	0.653234	−	0.757156i	$-0.273412\pi$
$823$	1.71948e13	1.30647	0.653234	+	0.757156i	$0.273412\pi$
$824$	1.57565e13	1.19066
$825$	0	0
$826$	6.00448e12	0.448813
$827$	−3.04498e12	−0.226365	−0.113182	−	0.993574i	$-0.536104\pi$
$827$	−3.04498e12	−0.226365	−0.113182	+	0.993574i	$0.536104\pi$
$828$	0	0
$829$	−8.03416e12	−0.590806	−0.295403	−	0.955373i	$-0.595454\pi$
$829$	−8.03416e12	−0.590806	−0.295403	+	0.955373i	$0.595454\pi$
$830$	−4.92343e12	−0.360094
$831$	0	0
$832$	2.59132e12	0.187485
$833$	8.30146e11	0.0597382
$834$	0	0
$835$	5.77210e11	0.0410908
$836$	7.70891e11	0.0545839
$837$	0	0
$838$	−2.47448e12	−0.173335
$839$	−1.36209e13	−0.949020	−0.474510	−	0.880250i	$-0.657375\pi$
$839$	−1.36209e13	−0.949020	−0.474510	+	0.880250i	$0.657375\pi$
$840$	0	0
$841$	9.98921e12	0.688571
$842$	−8.19684e12	−0.562007
$843$	0	0
$844$	−5.68023e12	−0.385323
$845$	6.19513e12	0.418018
$846$	0	0
$847$	−5.00589e12	−0.334200
$848$	6.29842e10	0.00418264
$849$	0	0
$850$	8.34114e11	0.0548075
$851$	2.67706e13	1.74975
$852$	0	0
$853$	3.64211e12	0.235549	0.117775	−	0.993040i	$-0.462424\pi$
$853$	3.64211e12	0.235549	0.117775	+	0.993040i	$0.462424\pi$
$854$	−2.38600e12	−0.153501
$855$	0	0
$856$	1.28019e13	0.814971
$857$	−1.62518e13	−1.02917	−0.514587	−	0.857438i	$-0.672055\pi$
$857$	−1.62518e13	−1.02917	−0.514587	+	0.857438i	$0.672055\pi$
$858$	0	0
$859$	1.69063e13	1.05944	0.529722	−	0.848171i	$-0.322296\pi$
$859$	1.69063e13	1.05944	0.529722	+	0.848171i	$0.322296\pi$
$860$	6.47816e12	0.403839
$861$	0	0
$862$	6.48608e12	0.400128
$863$	1.56857e13	0.962619	0.481310	−	0.876551i	$-0.340161\pi$
$863$	1.56857e13	0.962619	0.481310	+	0.876551i	$0.340161\pi$
$864$	0	0
$865$	−2.97504e12	−0.180684
$866$	4.14955e12	0.250709
$867$	0	0
$868$	−2.96008e12	−0.176997
$869$	−7.05485e12	−0.419661
$870$	0	0
$871$	−4.11138e12	−0.242050
$872$	−1.29553e13	−0.758790
$873$	0	0
$874$	4.91158e12	0.284721
$875$	−5.86182e11	−0.0338062
$876$	0	0
$877$	2.37521e13	1.35583	0.677914	−	0.735141i	$-0.262884\pi$
$877$	2.37521e13	1.35583	0.677914	+	0.735141i	$0.262884\pi$
$878$	1.31550e13	0.747079
$879$	0	0
$880$	2.81385e11	0.0158172
$881$	−1.13698e13	−0.635858	−0.317929	−	0.948115i	$-0.602987\pi$
$881$	−1.13698e13	−0.635858	−0.317929	+	0.948115i	$0.602987\pi$
$882$	0	0
$883$	−1.21711e12	−0.0673762	−0.0336881	−	0.999432i	$-0.510725\pi$
$883$	−1.21711e12	−0.0673762	−0.0336881	+	0.999432i	$0.510725\pi$
$884$	−1.10681e12	−0.0609588
$885$	0	0
$886$	8.71692e12	0.475238
$887$	−3.43743e13	−1.86457	−0.932283	−	0.361730i	$-0.882186\pi$
$887$	−3.43743e13	−1.86457	−0.932283	+	0.361730i	$0.882186\pi$
$888$	0	0
$889$	−9.26983e12	−0.497753
$890$	−1.05808e12	−0.0565281
$891$	0	0
$892$	1.43272e13	0.757738
$893$	9.51284e12	0.500586
$894$	0	0
$895$	−5.91992e12	−0.308398
$896$	−3.50183e12	−0.181513
$897$	0	0
$898$	7.70957e12	0.395627
$899$	2.08884e13	1.06656
$900$	0	0
$901$	−3.32877e11	−0.0168276
$902$	7.04472e12	0.354352
$903$	0	0
$904$	3.40495e13	1.69571
$905$	4.90872e12	0.243248
$906$	0	0
$907$	−3.08586e13	−1.51406	−0.757030	−	0.653381i	$-0.773350\pi$
$907$	−3.08586e13	−1.51406	−0.757030	+	0.653381i	$0.773350\pi$
$908$	−2.00974e13	−0.981192
$909$	0	0
$910$	−5.85478e11	−0.0283025
$911$	3.36006e13	1.61627	0.808136	−	0.588996i	$-0.200477\pi$
$911$	3.36006e13	1.61627	0.808136	+	0.588996i	$0.200477\pi$
$912$	0	0
$913$	8.77802e12	0.418098
$914$	−1.52814e13	−0.724280
$915$	0	0
$916$	9.01056e11	0.0422885
$917$	−9.64961e11	−0.0450659
$918$	0	0
$919$	−1.46200e13	−0.676127	−0.338063	−	0.941123i	$-0.609772\pi$
$919$	−1.46200e13	−0.676127	−0.338063	+	0.941123i	$0.609772\pi$
$920$	−1.54557e13	−0.711287
$921$	0	0
$922$	−1.38945e13	−0.633219
$923$	−1.82882e12	−0.0829398
$924$	0	0
$925$	−5.04224e12	−0.226457
$926$	1.60563e13	0.717625
$927$	0	0
$928$	2.82162e13	1.24891
$929$	1.36244e13	0.600130	0.300065	−	0.953919i	$-0.402992\pi$
$929$	1.36244e13	0.600130	0.300065	+	0.953919i	$0.402992\pi$
$930$	0	0
$931$	−9.20694e11	−0.0401644
$932$	−4.34213e12	−0.188508
$933$	0	0
$934$	−2.26013e12	−0.0971790
$935$	−1.48715e12	−0.0636358
$936$	0	0
$937$	−1.06580e13	−0.451697	−0.225848	−	0.974162i	$-0.572515\pi$
$937$	−1.06580e13	−0.451697	−0.225848	+	0.974162i	$0.572515\pi$
$938$	−5.56328e12	−0.234649
$939$	0	0
$940$	−1.08747e13	−0.454299
$941$	−1.54253e13	−0.641328	−0.320664	−	0.947193i	$-0.603906\pi$
$941$	−1.54253e13	−0.641328	−0.320664	+	0.947193i	$0.603906\pi$
$942$	0	0
$943$	−5.96294e13	−2.45560
$944$	−4.59522e12	−0.188335
$945$	0	0
$946$	8.69386e12	0.352941
$947$	−1.97267e13	−0.797039	−0.398519	−	0.917160i	$-0.630476\pi$
$947$	−1.97267e13	−0.797039	−0.398519	+	0.917160i	$0.630476\pi$
$948$	0	0
$949$	−2.06133e12	−0.0824993
$950$	−9.25094e11	−0.0368493
$951$	0	0
$952$	−4.12265e12	−0.162671
$953$	1.39214e12	0.0546721	0.0273361	−	0.999626i	$-0.491298\pi$
$953$	1.39214e12	0.0546721	0.0273361	+	0.999626i	$0.491298\pi$
$954$	0	0
$955$	1.11381e13	0.433309
$956$	1.18721e13	0.459694
$957$	0	0
$958$	2.99895e12	0.115034
$959$	−7.30317e12	−0.278822
$960$	0	0
$961$	−8.62783e12	−0.326322
$962$	−5.03619e12	−0.189589
$963$	0	0
$964$	6.12861e12	0.228568
$965$	1.20688e12	0.0448013
$966$	0	0
$967$	5.29760e12	0.194832	0.0974160	−	0.995244i	$-0.468942\pi$
$967$	5.29760e12	0.194832	0.0974160	+	0.995244i	$0.468942\pi$
$968$	2.48602e13	0.910049
$969$	0	0
$970$	1.35840e13	0.492669
$971$	7.97890e12	0.288042	0.144021	−	0.989575i	$-0.453997\pi$
$971$	7.97890e12	0.288042	0.144021	+	0.989575i	$0.453997\pi$
$972$	0	0
$973$	2.00825e12	0.0718306
$974$	1.23902e13	0.441127
$975$	0	0
$976$	1.82600e12	0.0644135
$977$	6.27831e12	0.220454	0.110227	−	0.993906i	$-0.464842\pi$
$977$	6.27831e12	0.220454	0.110227	+	0.993906i	$0.464842\pi$
$978$	0	0
$979$	1.88646e12	0.0656335
$980$	1.05250e12	0.0364506
$981$	0	0
$982$	1.57540e13	0.540616
$983$	−1.25053e12	−0.0427173	−0.0213586	−	0.999772i	$-0.506799\pi$
$983$	−1.25053e12	−0.0427173	−0.0213586	+	0.999772i	$0.506799\pi$
$984$	0	0
$985$	2.02874e13	0.686692
$986$	1.05686e13	0.356098
$987$	0	0
$988$	1.22753e12	0.0409851
$989$	−7.35883e13	−2.44583
$990$	0	0
$991$	1.89095e13	0.622801	0.311400	−	0.950279i	$-0.399202\pi$
$991$	1.89095e13	0.622801	0.311400	+	0.950279i	$0.399202\pi$
$992$	2.40603e13	0.788858
$993$	0	0
$994$	−2.47465e12	−0.0804036
$995$	1.72164e13	0.556851
$996$	0	0
$997$	−7.17482e12	−0.229976	−0.114988	−	0.993367i	$-0.536683\pi$
$997$	−7.17482e12	−0.229976	−0.114988	+	0.993367i	$0.536683\pi$
$998$	−4.95178e12	−0.158006
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.10.a.b.1.2		2
3.2	odd	2		35.10.a.b.1.1	✓	2
15.2	even	4		175.10.b.c.99.1		4
15.8	even	4		175.10.b.c.99.4		4
15.14	odd	2		175.10.a.c.1.2		2
21.20	even	2		245.10.a.c.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.10.a.b.1.1	✓	2	3.2	odd	2
175.10.a.c.1.2		2	15.14	odd	2
175.10.b.c.99.1		4	15.2	even	4
175.10.b.c.99.4		4	15.8	even	4
245.10.a.c.1.1		2	21.20	even	2
315.10.a.b.1.2		2	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)

\(f(q)\)	\(=\)	\(q+14.8284 q^{2} -292.118 q^{4} -625.000 q^{5} +2401.00 q^{7} -11923.8 q^{8} -9267.77 q^{10} +16523.6 q^{11} +26311.4 q^{13} +35603.1 q^{14} -27246.9 q^{16} +144003. q^{17} -159710. q^{19} +182574. q^{20} +245019. q^{22} -2.07393e6 q^{23} +390625. q^{25} +390156. q^{26} -701375. q^{28} +4.94938e6 q^{29} +4.22040e6 q^{31} +5.70096e6 q^{32} +2.13533e6 q^{34} -1.50062e6 q^{35} -1.29081e7 q^{37} -2.36824e6 q^{38} +7.45238e6 q^{40} +2.87518e7 q^{41} +3.54825e7 q^{43} -4.82683e6 q^{44} -3.07532e7 q^{46} -5.95633e7 q^{47} +5.76480e6 q^{49} +5.79235e6 q^{50} -7.68602e6 q^{52} -2.31161e6 q^{53} -1.03272e7 q^{55} -2.86290e7 q^{56} +7.33915e7 q^{58} +1.68651e8 q^{59} -6.70167e7 q^{61} +6.25819e7 q^{62} +9.84867e7 q^{64} -1.64446e7 q^{65} -1.56259e8 q^{67} -4.20657e7 q^{68} -2.22519e7 q^{70} -6.95067e7 q^{71} -7.83438e7 q^{73} -1.91407e8 q^{74} +4.66540e7 q^{76} +3.96731e7 q^{77} -4.26957e8 q^{79} +1.70293e7 q^{80} +4.26344e8 q^{82} +5.31242e8 q^{83} -9.00016e7 q^{85} +5.26149e8 q^{86} -1.97024e8 q^{88} +1.14168e8 q^{89} +6.31736e7 q^{91} +6.05833e8 q^{92} -8.83231e8 q^{94} +9.98185e7 q^{95} -1.46573e9 q^{97} +8.54829e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 24 q^{2} - 720 q^{4} - 1250 q^{5} + 4802 q^{7} - 20544 q^{8} - 15000 q^{10} - 18566 q^{11} - 51090 q^{13} + 57624 q^{14} + 112768 q^{16} + 373910 q^{17} - 143276 q^{19} + 450000 q^{20} - 76808 q^{22}+ \cdots + 138355224 q^{98}+O(q^{100}) \) 2 * q + 24 * q^2 - 720 * q^4 - 1250 * q^5 + 4802 * q^7 - 20544 * q^8 - 15000 * q^10 - 18566 * q^11 - 51090 * q^13 + 57624 * q^14 + 112768 * q^16 + 373910 * q^17 - 143276 * q^19 + 450000 * q^20 - 76808 * q^22 + 498908 * q^23 + 781250 * q^25 - 319736 * q^26 - 1728720 * q^28 + 11577554 * q^29 - 3953760 * q^31 + 11398656 * q^32 + 4243944 * q^34 - 3001250 * q^35 - 3205412 * q^37 - 2217520 * q^38 + 12840000 * q^40 - 1058992 * q^41 + 15948180 * q^43 + 10187376 * q^44 - 7156176 * q^46 - 65501290 * q^47 + 11529602 * q^49 + 9375000 * q^50 + 25432656 * q^52 + 25114688 * q^53 + 11603750 * q^55 - 49326144 * q^56 + 134182296 * q^58 + 116159208 * q^59 - 44688544 * q^61 - 12388000 * q^62 + 79055872 * q^64 + 31931250 * q^65 + 118092496 * q^67 - 140439024 * q^68 - 36015000 * q^70 + 294165824 * q^71 - 57419332 * q^73 - 102418064 * q^74 + 39622368 * q^76 - 44576966 * q^77 - 692852854 * q^79 - 70480000 * q^80 + 152932128 * q^82 + 540679928 * q^83 - 233693750 * q^85 + 346989008 * q^86 + 105455296 * q^88 + 779043704 * q^89 - 122667090 * q^91 - 495040608 * q^92 - 937691032 * q^94 + 89547500 * q^95 - 2673039406 * q^97 + 138355224 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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