LMFDB - Embedded newform 315.8.a.q.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [315,8,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, 8); 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, 8, names="a")

Level:	$ N $	$=$	$ 315 = 3^{2} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	315.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,5] 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:	$98.4012830275$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 3x^{7} - 832x^{6} + 1124x^{5} + 214693x^{4} - 78223x^{3} - 18019982x^{2} - 2181234x + 146417544 $ x^8 - 3x^7 - 832x^6 + 1124x^5 + 214693x^4 - 78223x^3 - 18019982x^2 - 2181234*x + 146417544
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{7}\cdot 3^{4} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$12.3623$ 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)

$f(q)$	$=$	\(q-11.3623 q^{2} +1.10101 q^{4} -125.000 q^{5} +343.000 q^{7} +1441.86 q^{8} +1420.28 q^{10} +7083.95 q^{11} -9009.21 q^{13} -3897.26 q^{14} -16523.7 q^{16} -34857.1 q^{17} +9352.33 q^{19} -137.626 q^{20} -80489.7 q^{22} -13824.7 q^{23} +15625.0 q^{25} +102365. q^{26} +377.645 q^{28} -187074. q^{29} +211239. q^{31} +3188.77 q^{32} +396056. q^{34} -42875.0 q^{35} +541911. q^{37} -106264. q^{38} -180232. q^{40} +776141. q^{41} -400320. q^{43} +7799.47 q^{44} +157080. q^{46} +490421. q^{47} +117649. q^{49} -177535. q^{50} -9919.20 q^{52} -1.83533e6 q^{53} -885494. q^{55} +494558. q^{56} +2.12559e6 q^{58} +28206.5 q^{59} -1.31536e6 q^{61} -2.40015e6 q^{62} +2.07880e6 q^{64} +1.12615e6 q^{65} +259266. q^{67} -38377.9 q^{68} +487157. q^{70} +1.63724e6 q^{71} +2.83182e6 q^{73} -6.15733e6 q^{74} +10297.0 q^{76} +2.42980e6 q^{77} +5.67086e6 q^{79} +2.06546e6 q^{80} -8.81871e6 q^{82} -2.02481e6 q^{83} +4.35714e6 q^{85} +4.54855e6 q^{86} +1.02141e7 q^{88} -4.34277e6 q^{89} -3.09016e6 q^{91} -15221.0 q^{92} -5.57229e6 q^{94} -1.16904e6 q^{95} +1.45480e7 q^{97} -1.33676e6 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 5 q^{2} + 651 q^{4} - 1000 q^{5} + 2744 q^{7} - 405 q^{8} - 625 q^{10} - 1040 q^{11} - 10412 q^{13} + 1715 q^{14} + 23555 q^{16} - 33600 q^{17} + 25508 q^{19} - 81375 q^{20} + 27440 q^{22} - 110100 q^{23}+ \cdots + 588245 q^{98}+O(q^{100}) $ 8 * q + 5 * q^2 + 651 * q^4 - 1000 * q^5 + 2744 * q^7 - 405 * q^8 - 625 * q^10 - 1040 * q^11 - 10412 * q^13 + 1715 * q^14 + 23555 * q^16 - 33600 * q^17 + 25508 * q^19 - 81375 * q^20 + 27440 * q^22 - 110100 * q^23 + 125000 * q^25 - 255890 * q^26 + 223293 * q^28 - 348980 * q^29 + 243684 * q^31 - 456065 * q^32 + 200968 * q^34 - 343000 * q^35 + 185216 * q^37 - 993110 * q^38 + 50625 * q^40 - 700000 * q^41 + 264684 * q^43 - 338760 * q^44 + 92668 * q^46 - 47220 * q^47 + 941192 * q^49 + 78125 * q^50 - 2621774 * q^52 - 1582520 * q^53 + 130000 * q^55 - 138915 * q^56 - 429406 * q^58 - 1618120 * q^59 - 1554236 * q^61 - 5213900 * q^62 - 55953 * q^64 + 1301500 * q^65 - 5215444 * q^67 - 3199720 * q^68 - 214375 * q^70 - 5827000 * q^71 + 18532 * q^73 - 52740 * q^74 - 11979466 * q^76 - 356720 * q^77 - 236880 * q^79 - 2944375 * q^80 - 12362418 * q^82 - 7424160 * q^83 + 4200000 * q^85 + 5147530 * q^86 + 5080880 * q^88 - 10567360 * q^89 - 3571316 * q^91 - 2262100 * q^92 + 1581266 * q^94 - 3188500 * q^95 + 11406932 * q^97 + 588245 * 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 $

$2$	−11.3623	−1.00429	−0.502146	−	0.864783i	$-0.667456\pi$
$2$	−11.3623	−1.00429	−0.502146	+	0.864783i	$0.667456\pi$
$3$	0	0
$3$	0	0
$4$	1.10101	0.00860161
$5$	−125.000	−0.447214
$5$	−125.000	−0.447214
$6$	0	0
$7$	343.000	0.377964
$7$	343.000	0.377964
$8$	1441.86	0.995653
$9$	0	0
$10$	1420.28	0.449133
$11$	7083.95	1.60473	0.802364	−	0.596835i	$-0.203576\pi$
$11$	7083.95	1.60473	0.802364	+	0.596835i	$0.203576\pi$
$12$	0	0
$13$	−9009.21	−1.13733	−0.568663	−	0.822570i	$-0.692539\pi$
$13$	−9009.21	−1.13733	−0.568663	+	0.822570i	$0.692539\pi$
$14$	−3897.26	−0.379587
$15$	0	0
$16$	−16523.7	−1.00853
$17$	−34857.1	−1.72076	−0.860380	−	0.509654i	$-0.829773\pi$
$17$	−34857.1	−1.72076	−0.860380	+	0.509654i	$0.829773\pi$
$18$	0	0
$19$	9352.33	0.312811	0.156406	−	0.987693i	$-0.450009\pi$
$19$	9352.33	0.312811	0.156406	+	0.987693i	$0.450009\pi$
$20$	−137.626	−0.00384676
$21$	0	0
$22$	−80489.7	−1.61161
$23$	−13824.7	−0.236923	−0.118462	−	0.992959i	$-0.537796\pi$
$23$	−13824.7	−0.236923	−0.118462	+	0.992959i	$0.537796\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	102365.	1.14221
$27$	0	0
$28$	377.645	0.00325110
$29$	−187074.	−1.42436	−0.712182	−	0.701995i	$-0.752293\pi$
$29$	−187074.	−1.42436	−0.712182	+	0.701995i	$0.752293\pi$
$30$	0	0
$31$	211239.	1.27353	0.636764	−	0.771059i	$-0.280273\pi$
$31$	211239.	1.27353	0.636764	+	0.771059i	$0.280273\pi$
$32$	3188.77	0.0172027
$33$	0	0
$34$	396056.	1.72814
$35$	−42875.0	−0.169031
$36$	0	0
$37$	541911.	1.75882	0.879410	−	0.476065i	$-0.157937\pi$
$37$	541911.	1.75882	0.879410	+	0.476065i	$0.157937\pi$
$38$	−106264.	−0.314154
$39$	0	0
$40$	−180232.	−0.445270
$41$	776141.	1.75872	0.879360	−	0.476157i	$-0.157970\pi$
$41$	776141.	1.75872	0.879360	+	0.476157i	$0.157970\pi$
$42$	0	0
$43$	−400320.	−0.767836	−0.383918	−	0.923367i	$-0.625425\pi$
$43$	−400320.	−0.767836	−0.383918	+	0.923367i	$0.625425\pi$
$44$	7799.47	0.0138032
$45$	0	0
$46$	157080.	0.237940
$47$	490421.	0.689012	0.344506	−	0.938784i	$-0.388046\pi$
$47$	490421.	0.689012	0.344506	+	0.938784i	$0.388046\pi$
$48$	0	0
$49$	117649.	0.142857
$50$	−177535.	−0.200858
$51$	0	0
$52$	−9919.20	−0.00978284
$53$	−1.83533e6	−1.69336	−0.846678	−	0.532106i	$-0.821401\pi$
$53$	−1.83533e6	−1.69336	−0.846678	+	0.532106i	$0.821401\pi$
$54$	0	0
$55$	−885494.	−0.717656
$56$	494558.	0.376321
$57$	0	0
$58$	2.12559e6	1.43048
$59$	28206.5	0.0178800	0.00893998	−	0.999960i	$-0.497154\pi$
$59$	28206.5	0.0178800	0.00893998	+	0.999960i	$0.497154\pi$
$60$	0	0
$61$	−1.31536e6	−0.741974	−0.370987	−	0.928638i	$-0.620981\pi$
$61$	−1.31536e6	−0.741974	−0.370987	+	0.928638i	$0.620981\pi$
$62$	−2.40015e6	−1.27899
$63$	0	0
$64$	2.07880e6	0.991251
$65$	1.12615e6	0.508628
$66$	0	0
$67$	259266.	0.105314	0.0526568	−	0.998613i	$-0.483231\pi$
$67$	259266.	0.105314	0.0526568	+	0.998613i	$0.483231\pi$
$68$	−38377.9	−0.0148013
$69$	0	0
$70$	487157.	0.169756
$71$	1.63724e6	0.542885	0.271443	−	0.962455i	$-0.412499\pi$
$71$	1.63724e6	0.542885	0.271443	+	0.962455i	$0.412499\pi$
$72$	0	0
$73$	2.83182e6	0.851992	0.425996	−	0.904725i	$-0.359924\pi$
$73$	2.83182e6	0.851992	0.425996	+	0.904725i	$0.359924\pi$
$74$	−6.15733e6	−1.76637
$75$	0	0
$76$	10297.0	0.00269068
$77$	2.42980e6	0.606530
$78$	0	0
$79$	5.67086e6	1.29406	0.647029	−	0.762465i	$-0.276011\pi$
$79$	5.67086e6	1.29406	0.647029	+	0.762465i	$0.276011\pi$
$80$	2.06546e6	0.451027
$81$	0	0
$82$	−8.81871e6	−1.76627
$83$	−2.02481e6	−0.388696	−0.194348	−	0.980933i	$-0.562259\pi$
$83$	−2.02481e6	−0.388696	−0.194348	+	0.980933i	$0.562259\pi$
$84$	0	0
$85$	4.35714e6	0.769547
$86$	4.54855e6	0.771131
$87$	0	0
$88$	1.02141e7	1.59775
$89$	−4.34277e6	−0.652982	−0.326491	−	0.945200i	$-0.605866\pi$
$89$	−4.34277e6	−0.652982	−0.326491	+	0.945200i	$0.605866\pi$
$90$	0	0
$91$	−3.09016e6	−0.429869
$92$	−15221.0	−0.00203792
$93$	0	0
$94$	−5.57229e6	−0.691969
$95$	−1.16904e6	−0.139893
$96$	0	0
$97$	1.45480e7	1.61846	0.809228	−	0.587495i	$-0.199886\pi$
$97$	1.45480e7	1.61846	0.809228	+	0.587495i	$0.199886\pi$
$98$	−1.33676e6	−0.143470
$99$	0	0
$100$	17203.2	0.00172032
$101$	−1.47112e7	−1.42077	−0.710384	−	0.703815i	$-0.751479\pi$
$101$	−1.47112e7	−1.42077	−0.710384	+	0.703815i	$0.751479\pi$
$102$	0	0
$103$	−7.31348e6	−0.659468	−0.329734	−	0.944074i	$-0.606959\pi$
$103$	−7.31348e6	−0.659468	−0.329734	+	0.944074i	$0.606959\pi$
$104$	−1.29900e7	−1.13238
$105$	0	0
$106$	2.08535e7	1.70062
$107$	−1.95454e7	−1.54242	−0.771208	−	0.636583i	$-0.780347\pi$
$107$	−1.95454e7	−1.54242	−0.771208	+	0.636583i	$0.780347\pi$
$108$	0	0
$109$	−1.69281e6	−0.125203	−0.0626016	−	0.998039i	$-0.519940\pi$
$109$	−1.69281e6	−0.125203	−0.0626016	+	0.998039i	$0.519940\pi$
$110$	1.00612e7	0.720736
$111$	0	0
$112$	−5.66763e6	−0.381188
$113$	1.24951e7	0.814640	0.407320	−	0.913286i	$-0.366463\pi$
$113$	1.24951e7	0.814640	0.407320	+	0.913286i	$0.366463\pi$
$114$	0	0
$115$	1.72808e6	0.105955
$116$	−205970.	−0.0122518
$117$	0	0
$118$	−320489.	−0.0179567
$119$	−1.19560e7	−0.650386
$120$	0	0
$121$	3.06952e7	1.57515
$122$	1.49454e7	0.745158
$123$	0	0
$124$	232575.	0.0109544
$125$	−1.95312e6	−0.0894427
$126$	0	0
$127$	2.01112e7	0.871215	0.435608	−	0.900137i	$-0.356534\pi$
$127$	2.01112e7	0.871215	0.435608	+	0.900137i	$0.356534\pi$
$128$	−2.40281e7	−1.01271
$129$	0	0
$130$	−1.27956e7	−0.510811
$131$	−2.16090e7	−0.839816	−0.419908	−	0.907567i	$-0.637938\pi$
$131$	−2.16090e7	−0.839816	−0.419908	+	0.907567i	$0.637938\pi$
$132$	0	0
$133$	3.20785e6	0.118232
$134$	−2.94585e6	−0.105766
$135$	0	0
$136$	−5.02591e7	−1.71328
$137$	5.04631e6	0.167669	0.0838343	−	0.996480i	$-0.473283\pi$
$137$	5.04631e6	0.167669	0.0838343	+	0.996480i	$0.473283\pi$
$138$	0	0
$139$	−1.70685e7	−0.539067	−0.269534	−	0.962991i	$-0.586870\pi$
$139$	−1.70685e7	−0.539067	−0.269534	+	0.962991i	$0.586870\pi$
$140$	−47205.6	−0.00145394
$141$	0	0
$142$	−1.86027e7	−0.545215
$143$	−6.38208e7	−1.82510
$144$	0	0
$145$	2.33843e7	0.636995
$146$	−3.21759e7	−0.855648
$147$	0	0
$148$	596647.	0.0151287
$149$	2.17186e7	0.537873	0.268937	−	0.963158i	$-0.413328\pi$
$149$	2.17186e7	0.537873	0.268937	+	0.963158i	$0.413328\pi$
$150$	0	0
$151$	−6.56456e7	−1.55162	−0.775812	−	0.630965i	$-0.782659\pi$
$151$	−6.56456e7	−1.55162	−0.775812	+	0.630965i	$0.782659\pi$
$152$	1.34847e7	0.311451
$153$	0	0
$154$	−2.76080e7	−0.609133
$155$	−2.64049e7	−0.569539
$156$	0	0
$157$	3.13893e7	0.647341	0.323671	−	0.946170i	$-0.395083\pi$
$157$	3.13893e7	0.647341	0.323671	+	0.946170i	$0.395083\pi$
$158$	−6.44338e7	−1.29961
$159$	0	0
$160$	−398596.	−0.00769330
$161$	−4.74186e6	−0.0895485
$162$	0	0
$163$	−1.08639e8	−1.96486	−0.982428	−	0.186644i	$-0.940239\pi$
$163$	−1.08639e8	−1.96486	−0.982428	+	0.186644i	$0.940239\pi$
$164$	854535.	0.0151278
$165$	0	0
$166$	2.30064e7	0.390364
$167$	1.60597e7	0.266827	0.133413	−	0.991060i	$-0.457406\pi$
$167$	1.60597e7	0.266827	0.133413	+	0.991060i	$0.457406\pi$
$168$	0	0
$169$	1.84174e7	0.293512
$170$	−4.95069e7	−0.772849
$171$	0	0
$172$	−440755.	−0.00660462
$173$	4.03262e6	0.0592142	0.0296071	−	0.999562i	$-0.490574\pi$
$173$	4.03262e6	0.0592142	0.0296071	+	0.999562i	$0.490574\pi$
$174$	0	0
$175$	5.35938e6	0.0755929
$176$	−1.17053e8	−1.61841
$177$	0	0
$178$	4.93436e7	0.655785
$179$	−5.98324e6	−0.0779742	−0.0389871	−	0.999240i	$-0.512413\pi$
$179$	−5.98324e6	−0.0779742	−0.0389871	+	0.999240i	$0.512413\pi$
$180$	0	0
$181$	−9.29872e7	−1.16560	−0.582798	−	0.812617i	$-0.698042\pi$
$181$	−9.29872e7	−1.16560	−0.582798	+	0.812617i	$0.698042\pi$
$182$	3.51112e7	0.431714
$183$	0	0
$184$	−1.99332e7	−0.235893
$185$	−6.77388e7	−0.786569
$186$	0	0
$187$	−2.46926e8	−2.76135
$188$	539956.	0.00592661
$189$	0	0
$190$	1.32830e7	0.140494
$191$	4.73858e6	0.0492075	0.0246038	−	0.999697i	$-0.492168\pi$
$191$	4.73858e6	0.0492075	0.0246038	+	0.999697i	$0.492168\pi$
$192$	0	0
$193$	6.37490e7	0.638297	0.319149	−	0.947705i	$-0.396603\pi$
$193$	6.37490e7	0.638297	0.319149	+	0.947705i	$0.396603\pi$
$194$	−1.65298e8	−1.62540
$195$	0	0
$196$	129532.	0.00122880
$197$	−1.50904e8	−1.40627	−0.703134	−	0.711058i	$-0.748216\pi$
$197$	−1.50904e8	−1.40627	−0.703134	+	0.711058i	$0.748216\pi$
$198$	0	0
$199$	−6.02814e7	−0.542247	−0.271124	−	0.962545i	$-0.587395\pi$
$199$	−6.02814e7	−0.542247	−0.271124	+	0.962545i	$0.587395\pi$
$200$	2.25291e7	0.199131
$201$	0	0
$202$	1.67152e8	1.42686
$203$	−6.41664e7	−0.538359
$204$	0	0
$205$	−9.70176e7	−0.786524
$206$	8.30977e7	0.662299
$207$	0	0
$208$	1.48866e8	1.14703
$209$	6.62515e7	0.501977
$210$	0	0
$211$	2.21018e8	1.61972	0.809860	−	0.586624i	$-0.199543\pi$
$211$	2.21018e8	1.61972	0.809860	+	0.586624i	$0.199543\pi$
$212$	−2.02071e6	−0.0145656
$213$	0	0
$214$	2.22080e8	1.54904
$215$	5.00401e7	0.343386
$216$	0	0
$217$	7.24550e7	0.481348
$218$	1.92342e7	0.125741
$219$	0	0
$220$	−974934.	−0.00617299
$221$	3.14035e8	1.95706
$222$	0	0
$223$	−2.88131e8	−1.73989	−0.869947	−	0.493145i	$-0.835847\pi$
$223$	−2.88131e8	−1.73989	−0.869947	+	0.493145i	$0.835847\pi$
$224$	1.09375e6	0.00650202
$225$	0	0
$226$	−1.41973e8	−0.818136
$227$	−1.66127e8	−0.942647	−0.471323	−	0.881960i	$-0.656224\pi$
$227$	−1.66127e8	−0.942647	−0.471323	+	0.881960i	$0.656224\pi$
$228$	0	0
$229$	−1.97789e8	−1.08837	−0.544186	−	0.838965i	$-0.683161\pi$
$229$	−1.97789e8	−1.08837	−0.544186	+	0.838965i	$0.683161\pi$
$230$	−1.96349e7	−0.106410
$231$	0	0
$232$	−2.69735e8	−1.41817
$233$	−1.49491e8	−0.774231	−0.387115	−	0.922031i	$-0.626528\pi$
$233$	−1.49491e8	−0.774231	−0.387115	+	0.922031i	$0.626528\pi$
$234$	0	0
$235$	−6.13026e7	−0.308135
$236$	31055.5	0.000153796 0
$237$	0	0
$238$	1.35847e8	0.653177
$239$	1.72111e8	0.815486	0.407743	−	0.913097i	$-0.366316\pi$
$239$	1.72111e8	0.815486	0.407743	+	0.913097i	$0.366316\pi$
$240$	0	0
$241$	5.78274e7	0.266118	0.133059	−	0.991108i	$-0.457520\pi$
$241$	5.78274e7	0.266118	0.133059	+	0.991108i	$0.457520\pi$
$242$	−3.48767e8	−1.58191
$243$	0	0
$244$	−1.44821e6	−0.00638217
$245$	−1.47061e7	−0.0638877
$246$	0	0
$247$	−8.42572e7	−0.355768
$248$	3.04577e8	1.26799
$249$	0	0
$250$	2.21919e7	0.0898266
$251$	−2.29474e8	−0.915957	−0.457978	−	0.888963i	$-0.651426\pi$
$251$	−2.29474e8	−0.915957	−0.457978	+	0.888963i	$0.651426\pi$
$252$	0	0
$253$	−9.79333e7	−0.380197
$254$	−2.28509e8	−0.874954
$255$	0	0
$256$	6.92642e6	0.0258029
$257$	−4.11193e8	−1.51105	−0.755527	−	0.655117i	$-0.772619\pi$
$257$	−4.11193e8	−1.51105	−0.755527	+	0.655117i	$0.772619\pi$
$258$	0	0
$259$	1.85875e8	0.664772
$260$	1.23990e6	0.00437502
$261$	0	0
$262$	2.45527e8	0.843420
$263$	−4.26697e8	−1.44635	−0.723177	−	0.690662i	$-0.757319\pi$
$263$	−4.26697e8	−1.44635	−0.723177	+	0.690662i	$0.757319\pi$
$264$	0	0
$265$	2.29416e8	0.757292
$266$	−3.64484e7	−0.118739
$267$	0	0
$268$	285454.	0.000905866 0
$269$	4.26051e8	1.33453	0.667266	−	0.744820i	$-0.267465\pi$
$269$	4.26051e8	1.33453	0.667266	+	0.744820i	$0.267465\pi$
$270$	0	0
$271$	5.72893e8	1.74856	0.874282	−	0.485419i	$-0.161333\pi$
$271$	5.72893e8	1.74856	0.874282	+	0.485419i	$0.161333\pi$
$272$	5.75969e8	1.73543
$273$	0	0
$274$	−5.73375e7	−0.168388
$275$	1.10687e8	0.320945
$276$	0	0
$277$	2.05790e8	0.581760	0.290880	−	0.956759i	$-0.406052\pi$
$277$	2.05790e8	0.581760	0.290880	+	0.956759i	$0.406052\pi$
$278$	1.93937e8	0.541381
$279$	0	0
$280$	−6.18197e7	−0.168296
$281$	4.79814e8	1.29003	0.645016	−	0.764169i	$-0.276851\pi$
$281$	4.79814e8	1.29003	0.645016	+	0.764169i	$0.276851\pi$
$282$	0	0
$283$	−1.37563e8	−0.360785	−0.180393	−	0.983595i	$-0.557737\pi$
$283$	−1.37563e8	−0.360785	−0.180393	+	0.983595i	$0.557737\pi$
$284$	1.80261e6	0.00466969
$285$	0	0
$286$	7.25149e8	1.83293
$287$	2.66216e8	0.664734
$288$	0	0
$289$	8.04679e8	1.96101
$290$	−2.65698e8	−0.639729
$291$	0	0
$292$	3.11785e6	0.00732850
$293$	−6.49956e8	−1.50955	−0.754775	−	0.655984i	$-0.772254\pi$
$293$	−6.49956e8	−1.50955	−0.754775	+	0.655984i	$0.772254\pi$
$294$	0	0
$295$	−3.52581e6	−0.00799616
$296$	7.81359e8	1.75118
$297$	0	0
$298$	−2.46772e8	−0.540182
$299$	1.24549e8	0.269459
$300$	0	0
$301$	−1.37310e8	−0.290215
$302$	7.45882e8	1.55828
$303$	0	0
$304$	−1.54535e8	−0.315479
$305$	1.64419e8	0.331821
$306$	0	0
$307$	−4.19831e7	−0.0828114	−0.0414057	−	0.999142i	$-0.513184\pi$
$307$	−4.19831e7	−0.0828114	−0.0414057	+	0.999142i	$0.513184\pi$
$308$	2.67522e6	0.00521713
$309$	0	0
$310$	3.00019e8	0.571983
$311$	−2.89966e8	−0.546620	−0.273310	−	0.961926i	$-0.588119\pi$
$311$	−2.89966e8	−0.546620	−0.273310	+	0.961926i	$0.588119\pi$
$312$	0	0
$313$	−4.93896e7	−0.0910395	−0.0455198	−	0.998963i	$-0.514494\pi$
$313$	−4.93896e7	−0.0910395	−0.0455198	+	0.998963i	$0.514494\pi$
$314$	−3.56654e8	−0.650119
$315$	0	0
$316$	6.24365e6	0.0111310
$317$	4.30076e8	0.758294	0.379147	−	0.925337i	$-0.376218\pi$
$317$	4.30076e8	0.758294	0.379147	+	0.925337i	$0.376218\pi$
$318$	0	0
$319$	−1.32522e9	−2.28572
$320$	−2.59851e8	−0.443301
$321$	0	0
$322$	5.38783e7	0.0899328
$323$	−3.25995e8	−0.538273
$324$	0	0
$325$	−1.40769e8	−0.227465
$326$	1.23439e9	1.97329
$327$	0	0
$328$	1.11909e9	1.75108
$329$	1.68214e8	0.260422
$330$	0	0
$331$	−1.18542e9	−1.79669	−0.898344	−	0.439293i	$-0.855229\pi$
$331$	−1.18542e9	−1.79669	−0.898344	+	0.439293i	$0.855229\pi$
$332$	−2.22932e6	−0.00334341
$333$	0	0
$334$	−1.82474e8	−0.267972
$335$	−3.24083e7	−0.0470977
$336$	0	0
$337$	3.87597e8	0.551666	0.275833	−	0.961206i	$-0.411046\pi$
$337$	3.87597e8	0.551666	0.275833	+	0.961206i	$0.411046\pi$
$338$	−2.09264e8	−0.294771
$339$	0	0
$340$	4.79723e6	0.00661934
$341$	1.49641e9	2.04366
$342$	0	0
$343$	4.03536e7	0.0539949
$344$	−5.77206e8	−0.764498
$345$	0	0
$346$	−4.58197e7	−0.0594683
$347$	5.12544e8	0.658534	0.329267	−	0.944237i	$-0.393198\pi$
$347$	5.12544e8	0.658534	0.329267	+	0.944237i	$0.393198\pi$
$348$	0	0
$349$	−4.81520e8	−0.606352	−0.303176	−	0.952935i	$-0.598047\pi$
$349$	−4.81520e8	−0.606352	−0.303176	+	0.952935i	$0.598047\pi$
$350$	−6.08946e7	−0.0759173
$351$	0	0
$352$	2.25891e7	0.0276057
$353$	−1.08368e9	−1.31126	−0.655632	−	0.755080i	$-0.727598\pi$
$353$	−1.08368e9	−1.31126	−0.655632	+	0.755080i	$0.727598\pi$
$354$	0	0
$355$	−2.04655e8	−0.242786
$356$	−4.78141e6	−0.00561670
$357$	0	0
$358$	6.79831e7	0.0783088
$359$	7.94916e8	0.906757	0.453378	−	0.891318i	$-0.350219\pi$
$359$	7.94916e8	0.906757	0.453378	+	0.891318i	$0.350219\pi$
$360$	0	0
$361$	−8.06406e8	−0.902149
$362$	1.05654e9	1.17060
$363$	0	0
$364$	−3.40228e6	−0.00369756
$365$	−3.53977e8	−0.381022
$366$	0	0
$367$	−1.84819e8	−0.195171	−0.0975857	−	0.995227i	$-0.531112\pi$
$367$	−1.84819e8	−0.195171	−0.0975857	+	0.995227i	$0.531112\pi$
$368$	2.28435e8	0.238944
$369$	0	0
$370$	7.69666e8	0.789944
$371$	−6.29517e8	−0.640028
$372$	0	0
$373$	3.00820e8	0.300141	0.150071	−	0.988675i	$-0.452050\pi$
$373$	3.00820e8	0.300141	0.150071	+	0.988675i	$0.452050\pi$
$374$	2.80564e9	2.77320
$375$	0	0
$376$	7.07118e8	0.686017
$377$	1.68539e9	1.61997
$378$	0	0
$379$	1.64486e9	1.55200	0.775999	−	0.630734i	$-0.217246\pi$
$379$	1.64486e9	1.55200	0.775999	+	0.630734i	$0.217246\pi$
$380$	−1.28712e6	−0.00120331
$381$	0	0
$382$	−5.38410e7	−0.0494187
$383$	−9.93750e8	−0.903819	−0.451909	−	0.892064i	$-0.649257\pi$
$383$	−9.93750e8	−0.903819	−0.451909	+	0.892064i	$0.649257\pi$
$384$	0	0
$385$	−3.03724e8	−0.271248
$386$	−7.24333e8	−0.641037
$387$	0	0
$388$	1.60174e7	0.0139213
$389$	−1.61369e9	−1.38994	−0.694971	−	0.719038i	$-0.744583\pi$
$389$	−1.61369e9	−1.38994	−0.694971	+	0.719038i	$0.744583\pi$
$390$	0	0
$391$	4.81888e8	0.407688
$392$	1.69633e8	0.142236
$393$	0	0
$394$	1.71461e9	1.41230
$395$	−7.08857e8	−0.578721
$396$	0	0
$397$	−1.87653e9	−1.50518	−0.752591	−	0.658489i	$-0.771196\pi$
$397$	−1.87653e9	−1.50518	−0.752591	+	0.658489i	$0.771196\pi$
$398$	6.84933e8	0.544575
$399$	0	0
$400$	−2.58183e8	−0.201706
$401$	−4.69102e8	−0.363297	−0.181649	−	0.983364i	$-0.558143\pi$
$401$	−4.69102e8	−0.363297	−0.181649	+	0.983364i	$0.558143\pi$
$402$	0	0
$403$	−1.90310e9	−1.44842
$404$	−1.61971e7	−0.0122209
$405$	0	0
$406$	7.29076e8	0.540669
$407$	3.83887e9	2.82243
$408$	0	0
$409$	−1.40800e9	−1.01758	−0.508791	−	0.860890i	$-0.669908\pi$
$409$	−1.40800e9	−1.01758	−0.508791	+	0.860890i	$0.669908\pi$
$410$	1.10234e9	0.789899
$411$	0	0
$412$	−8.05219e6	−0.00567249
$413$	9.67482e6	0.00675799
$414$	0	0
$415$	2.53101e8	0.173830
$416$	−2.87283e7	−0.0195651
$417$	0	0
$418$	−7.52767e8	−0.504131
$419$	4.88680e8	0.324545	0.162273	−	0.986746i	$-0.448118\pi$
$419$	4.88680e8	0.324545	0.162273	+	0.986746i	$0.448118\pi$
$420$	0	0
$421$	9.74301e6	0.00636365	0.00318182	−	0.999995i	$-0.498987\pi$
$421$	9.74301e6	0.00636365	0.00318182	+	0.999995i	$0.498987\pi$
$422$	−2.51127e9	−1.62667
$423$	0	0
$424$	−2.64629e9	−1.68599
$425$	−5.44642e8	−0.344152
$426$	0	0
$427$	−4.51167e8	−0.280440
$428$	−2.15196e7	−0.0132673
$429$	0	0
$430$	−5.68568e8	−0.344860
$431$	6.14059e8	0.369436	0.184718	−	0.982792i	$-0.440863\pi$
$431$	6.14059e8	0.369436	0.184718	+	0.982792i	$0.440863\pi$
$432$	0	0
$433$	−1.26904e9	−0.751223	−0.375612	−	0.926777i	$-0.622567\pi$
$433$	−1.26904e9	−0.751223	−0.375612	+	0.926777i	$0.622567\pi$
$434$	−8.23253e8	−0.483414
$435$	0	0
$436$	−1.86379e6	−0.00107695
$437$	−1.29293e8	−0.0741122
$438$	0	0
$439$	−2.43834e9	−1.37553	−0.687764	−	0.725935i	$-0.741407\pi$
$439$	−2.43834e9	−1.37553	−0.687764	+	0.725935i	$0.741407\pi$
$440$	−1.27676e9	−0.714536
$441$	0	0
$442$	−3.56815e9	−1.96546
$443$	−7.12131e8	−0.389177	−0.194588	−	0.980885i	$-0.562337\pi$
$443$	−7.12131e8	−0.389177	−0.194588	+	0.980885i	$0.562337\pi$
$444$	0	0
$445$	5.42846e8	0.292023
$446$	3.27382e9	1.74736
$447$	0	0
$448$	7.13030e8	0.374658
$449$	−1.62831e9	−0.848934	−0.424467	−	0.905443i	$-0.639538\pi$
$449$	−1.62831e9	−0.848934	−0.424467	+	0.905443i	$0.639538\pi$
$450$	0	0
$451$	5.49814e9	2.82227
$452$	1.37572e7	0.00700721
$453$	0	0
$454$	1.88758e9	0.946692
$455$	3.86270e8	0.192243
$456$	0	0
$457$	−3.35306e9	−1.64337	−0.821683	−	0.569945i	$-0.806965\pi$
$457$	−3.35306e9	−1.64337	−0.821683	+	0.569945i	$0.806965\pi$
$458$	2.24733e9	1.09304
$459$	0	0
$460$	1.90263e6	0.000911386 0
$461$	7.99292e8	0.379973	0.189986	−	0.981787i	$-0.439156\pi$
$461$	7.99292e8	0.379973	0.189986	+	0.981787i	$0.439156\pi$
$462$	0	0
$463$	6.18703e8	0.289700	0.144850	−	0.989454i	$-0.453730\pi$
$463$	6.18703e8	0.289700	0.144850	+	0.989454i	$0.453730\pi$
$464$	3.09116e9	1.43651
$465$	0	0
$466$	1.69856e9	0.777553
$467$	−3.96700e9	−1.80241	−0.901204	−	0.433395i	$-0.857315\pi$
$467$	−3.96700e9	−1.80241	−0.901204	+	0.433395i	$0.857315\pi$
$468$	0	0
$469$	8.89284e7	0.0398048
$470$	6.96537e8	0.309458
$471$	0	0
$472$	4.06698e7	0.0178022
$473$	−2.83585e9	−1.23217
$474$	0	0
$475$	1.46130e8	0.0625622
$476$	−1.31636e7	−0.00559436
$477$	0	0
$478$	−1.95557e9	−0.818986
$479$	−1.75023e8	−0.0727647	−0.0363823	−	0.999338i	$-0.511583\pi$
$479$	−1.75023e8	−0.0727647	−0.0363823	+	0.999338i	$0.511583\pi$
$480$	0	0
$481$	−4.88219e9	−2.00035
$482$	−6.57050e8	−0.267260
$483$	0	0
$484$	3.37956e7	0.0135488
$485$	−1.81849e9	−0.723796
$486$	0	0
$487$	−2.08394e9	−0.817585	−0.408793	−	0.912627i	$-0.634050\pi$
$487$	−2.08394e9	−0.817585	−0.408793	+	0.912627i	$0.634050\pi$
$488$	−1.89656e9	−0.738749
$489$	0	0
$490$	1.67095e8	0.0641618
$491$	3.60299e9	1.37366	0.686828	−	0.726820i	$-0.259003\pi$
$491$	3.60299e9	1.37366	0.686828	+	0.726820i	$0.259003\pi$
$492$	0	0
$493$	6.52086e9	2.45099
$494$	9.57352e8	0.357295
$495$	0	0
$496$	−3.49045e9	−1.28439
$497$	5.61573e8	0.205191
$498$	0	0
$499$	−2.49479e9	−0.898840	−0.449420	−	0.893321i	$-0.648369\pi$
$499$	−2.49479e9	−0.898840	−0.449420	+	0.893321i	$0.648369\pi$
$500$	−2.15040e6	−0.000769351 0
$501$	0	0
$502$	2.60734e9	0.919887
$503$	−3.61891e9	−1.26791	−0.633957	−	0.773368i	$-0.718571\pi$
$503$	−3.61891e9	−1.26791	−0.633957	+	0.773368i	$0.718571\pi$
$504$	0	0
$505$	1.83890e9	0.635387
$506$	1.11274e9	0.381829
$507$	0	0
$508$	2.21426e7	0.00749385
$509$	−3.92673e9	−1.31983	−0.659916	−	0.751340i	$-0.729408\pi$
$509$	−3.92673e9	−1.31983	−0.659916	+	0.751340i	$0.729408\pi$
$510$	0	0
$511$	9.71314e8	0.322023
$512$	2.99689e9	0.986794
$513$	0	0
$514$	4.67209e9	1.51754
$515$	9.14185e8	0.294923
$516$	0	0
$517$	3.47412e9	1.10568
$518$	−2.11196e9	−0.667625
$519$	0	0
$520$	1.62375e9	0.506417
$521$	2.42452e9	0.751094	0.375547	−	0.926803i	$-0.377455\pi$
$521$	2.42452e9	0.751094	0.375547	+	0.926803i	$0.377455\pi$
$522$	0	0
$523$	2.34068e9	0.715462	0.357731	−	0.933825i	$-0.383550\pi$
$523$	2.34068e9	0.715462	0.357731	+	0.933825i	$0.383550\pi$
$524$	−2.37916e7	−0.00722377
$525$	0	0
$526$	4.84825e9	1.45256
$527$	−7.36318e9	−2.19143
$528$	0	0
$529$	−3.21370e9	−0.943867
$530$	−2.60668e9	−0.760541
$531$	0	0
$532$	3.53186e6	0.00101698
$533$	−6.99242e9	−2.00024
$534$	0	0
$535$	2.44318e9	0.689789
$536$	3.73826e8	0.104856
$537$	0	0
$538$	−4.84090e9	−1.34026
$539$	8.33420e8	0.229247
$540$	0	0
$541$	6.95764e9	1.88917	0.944587	−	0.328262i	$-0.106463\pi$
$541$	6.95764e9	1.88917	0.944587	+	0.328262i	$0.106463\pi$
$542$	−6.50937e9	−1.75607
$543$	0	0
$544$	−1.11151e8	−0.0296018
$545$	2.11601e8	0.0559926
$546$	0	0
$547$	5.57680e9	1.45690	0.728449	−	0.685100i	$-0.240241\pi$
$547$	5.57680e9	1.45690	0.728449	+	0.685100i	$0.240241\pi$
$548$	5.55602e6	0.00144222
$549$	0	0
$550$	−1.25765e9	−0.322323
$551$	−1.74958e9	−0.445557
$552$	0	0
$553$	1.94510e9	0.489108
$554$	−2.33823e9	−0.584257
$555$	0	0
$556$	−1.87925e7	−0.00463685
$557$	4.18187e9	1.02536	0.512681	−	0.858579i	$-0.328653\pi$
$557$	4.18187e9	1.02536	0.512681	+	0.858579i	$0.328653\pi$
$558$	0	0
$559$	3.60657e9	0.873280
$560$	7.08454e8	0.170472
$561$	0	0
$562$	−5.45177e9	−1.29557
$563$	−1.03544e9	−0.244538	−0.122269	−	0.992497i	$-0.539017\pi$
$563$	−1.03544e9	−0.244538	−0.122269	+	0.992497i	$0.539017\pi$
$564$	0	0
$565$	−1.56189e9	−0.364318
$566$	1.56303e9	0.362334
$567$	0	0
$568$	2.36067e9	0.540525
$569$	4.67891e9	1.06476	0.532380	−	0.846505i	$-0.321298\pi$
$569$	4.67891e9	1.06476	0.532380	+	0.846505i	$0.321298\pi$
$570$	0	0
$571$	−2.09498e9	−0.470927	−0.235463	−	0.971883i	$-0.575661\pi$
$571$	−2.09498e9	−0.470927	−0.235463	+	0.971883i	$0.575661\pi$
$572$	−7.02671e7	−0.0156988
$573$	0	0
$574$	−3.02482e9	−0.667587
$575$	−2.16011e8	−0.0473846
$576$	0	0
$577$	5.97079e9	1.29395	0.646974	−	0.762512i	$-0.276034\pi$
$577$	5.97079e9	1.29395	0.646974	+	0.762512i	$0.276034\pi$
$578$	−9.14297e9	−1.96943
$579$	0	0
$580$	2.57462e7	0.00547918
$581$	−6.94508e8	−0.146913
$582$	0	0
$583$	−1.30014e10	−2.71737
$584$	4.08309e9	0.848288
$585$	0	0
$586$	7.38497e9	1.51603
$587$	−3.25018e9	−0.663244	−0.331622	−	0.943412i	$-0.607596\pi$
$587$	−3.25018e9	−0.663244	−0.331622	+	0.943412i	$0.607596\pi$
$588$	0	0
$589$	1.97558e9	0.398374
$590$	4.00612e7	0.00803048
$591$	0	0
$592$	−8.95438e9	−1.77382
$593$	8.02960e9	1.58126	0.790629	−	0.612296i	$-0.209754\pi$
$593$	8.02960e9	1.58126	0.790629	+	0.612296i	$0.209754\pi$
$594$	0	0
$595$	1.49450e9	0.290861
$596$	2.39123e7	0.00462657
$597$	0	0
$598$	−1.41516e9	−0.270615
$599$	−9.80486e8	−0.186401	−0.0932003	−	0.995647i	$-0.529710\pi$
$599$	−9.80486e8	−0.186401	−0.0932003	+	0.995647i	$0.529710\pi$
$600$	0	0
$601$	9.69399e9	1.82155	0.910777	−	0.412899i	$-0.135484\pi$
$601$	9.69399e9	1.82155	0.910777	+	0.412899i	$0.135484\pi$
$602$	1.56015e9	0.291460
$603$	0	0
$604$	−7.22762e7	−0.0133465
$605$	−3.83690e9	−0.704428
$606$	0	0
$607$	−5.43368e9	−0.986128	−0.493064	−	0.869993i	$-0.664123\pi$
$607$	−5.43368e9	−0.986128	−0.493064	+	0.869993i	$0.664123\pi$
$608$	2.98224e7	0.00538121
$609$	0	0
$610$	−1.86818e9	−0.333245
$611$	−4.41831e9	−0.783631
$612$	0	0
$613$	−1.46693e8	−0.0257215	−0.0128608	−	0.999917i	$-0.504094\pi$
$613$	−1.46693e8	−0.0257215	−0.0128608	+	0.999917i	$0.504094\pi$
$614$	4.77023e8	0.0831668
$615$	0	0
$616$	3.50342e9	0.603893
$617$	6.19027e8	0.106099	0.0530495	−	0.998592i	$-0.483106\pi$
$617$	6.19027e8	0.106099	0.0530495	+	0.998592i	$0.483106\pi$
$618$	0	0
$619$	−9.45693e9	−1.60263	−0.801314	−	0.598244i	$-0.795865\pi$
$619$	−9.45693e9	−1.60263	−0.801314	+	0.598244i	$0.795865\pi$
$620$	−2.90719e7	−0.00489895
$621$	0	0
$622$	3.29467e9	0.548966
$623$	−1.48957e9	−0.246804
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	5.61177e8	0.0914303
$627$	0	0
$628$	3.45598e7	0.00556817
$629$	−1.88894e10	−3.02651
$630$	0	0
$631$	−2.88883e8	−0.0457741	−0.0228870	−	0.999738i	$-0.507286\pi$
$631$	−2.88883e8	−0.0457741	−0.0228870	+	0.999738i	$0.507286\pi$
$632$	8.17658e9	1.28843
$633$	0	0
$634$	−4.88663e9	−0.761548
$635$	−2.51390e9	−0.389619
$636$	0	0
$637$	−1.05993e9	−0.162475
$638$	1.50575e10	2.29552
$639$	0	0
$640$	3.00351e9	0.452897
$641$	−1.15705e10	−1.73520	−0.867600	−	0.497263i	$-0.834339\pi$
$641$	−1.15705e10	−1.73520	−0.867600	+	0.497263i	$0.834339\pi$
$642$	0	0
$643$	1.23393e10	1.83043	0.915215	−	0.402966i	$-0.132021\pi$
$643$	1.23393e10	1.83043	0.915215	+	0.402966i	$0.132021\pi$
$644$	−5.22082e6	−0.000770261 0
$645$	0	0
$646$	3.70404e9	0.540583
$647$	−7.08241e9	−1.02805	−0.514027	−	0.857774i	$-0.671847\pi$
$647$	−7.08241e9	−1.02805	−0.514027	+	0.857774i	$0.671847\pi$
$648$	0	0
$649$	1.99813e8	0.0286925
$650$	1.59945e9	0.228441
$651$	0	0
$652$	−1.19613e8	−0.0169009
$653$	−6.93556e9	−0.974732	−0.487366	−	0.873198i	$-0.662042\pi$
$653$	−6.93556e9	−0.974732	−0.487366	+	0.873198i	$0.662042\pi$
$654$	0	0
$655$	2.70112e9	0.375577
$656$	−1.28247e10	−1.77372
$657$	0	0
$658$	−1.91130e9	−0.261540
$659$	−5.62562e9	−0.765723	−0.382861	−	0.923806i	$-0.625061\pi$
$659$	−5.62562e9	−0.765723	−0.382861	+	0.923806i	$0.625061\pi$
$660$	0	0
$661$	5.51506e9	0.742754	0.371377	−	0.928482i	$-0.378886\pi$
$661$	5.51506e9	0.742754	0.371377	+	0.928482i	$0.378886\pi$
$662$	1.34690e10	1.80440
$663$	0	0
$664$	−2.91949e9	−0.387006
$665$	−4.00981e8	−0.0528747
$666$	0	0
$667$	2.58624e9	0.337465
$668$	1.76818e7	0.00229514
$669$	0	0
$670$	3.68232e8	0.0472998
$671$	−9.31791e9	−1.19067
$672$	0	0
$673$	−1.04597e10	−1.32271	−0.661355	−	0.750073i	$-0.730018\pi$
$673$	−1.04597e10	−1.32271	−0.661355	+	0.750073i	$0.730018\pi$
$674$	−4.40398e9	−0.554033
$675$	0	0
$676$	2.02777e7	0.00252467
$677$	2.38983e9	0.296010	0.148005	−	0.988987i	$-0.452715\pi$
$677$	2.38983e9	0.296010	0.148005	+	0.988987i	$0.452715\pi$
$678$	0	0
$679$	4.98995e9	0.611719
$680$	6.28238e9	0.766202
$681$	0	0
$682$	−1.70026e10	−2.05243
$683$	8.97218e9	1.07752	0.538760	−	0.842459i	$-0.318893\pi$
$683$	8.97218e9	1.07752	0.538760	+	0.842459i	$0.318893\pi$
$684$	0	0
$685$	−6.30789e8	−0.0749837
$686$	−4.58508e8	−0.0542266
$687$	0	0
$688$	6.61478e9	0.774383
$689$	1.65349e10	1.92590
$690$	0	0
$691$	−6.69457e9	−0.771879	−0.385940	−	0.922524i	$-0.626123\pi$
$691$	−6.69457e9	−0.771879	−0.385940	+	0.922524i	$0.626123\pi$
$692$	4.43994e6	0.000509337 0
$693$	0	0
$694$	−5.82366e9	−0.661360
$695$	2.13356e9	0.241078
$696$	0	0
$697$	−2.70540e10	−3.02633
$698$	5.47115e9	0.608955
$699$	0	0
$700$	5.90070e6	0.000650220 0
$701$	4.81924e9	0.528404	0.264202	−	0.964467i	$-0.414891\pi$
$701$	4.81924e9	0.528404	0.264202	+	0.964467i	$0.414891\pi$
$702$	0	0
$703$	5.06813e9	0.550179
$704$	1.47261e10	1.59069
$705$	0	0
$706$	1.23131e10	1.31689
$707$	−5.04594e9	−0.537000
$708$	0	0
$709$	1.00519e10	1.05922	0.529612	−	0.848240i	$-0.322338\pi$
$709$	1.00519e10	1.05922	0.529612	+	0.848240i	$0.322338\pi$
$710$	2.32534e9	0.243828
$711$	0	0
$712$	−6.26166e9	−0.650144
$713$	−2.92031e9	−0.301728
$714$	0	0
$715$	7.97761e9	0.816209
$716$	−6.58758e6	−0.000670703 0
$717$	0	0
$718$	−9.03205e9	−0.910648
$719$	−1.86383e10	−1.87005	−0.935027	−	0.354576i	$-0.884625\pi$
$719$	−1.86383e10	−1.87005	−0.935027	+	0.354576i	$0.884625\pi$
$720$	0	0
$721$	−2.50852e9	−0.249256
$722$	9.16259e9	0.906021
$723$	0	0
$724$	−1.02379e8	−0.0100260
$725$	−2.92303e9	−0.284873
$726$	0	0
$727$	−1.72431e10	−1.66435	−0.832175	−	0.554512i	$-0.812905\pi$
$727$	−1.72431e10	−1.66435	−0.832175	+	0.554512i	$0.812905\pi$
$728$	−4.45558e9	−0.428000
$729$	0	0
$730$	4.02198e9	0.382658
$731$	1.39540e10	1.32126
$732$	0	0
$733$	−1.16999e10	−1.09728	−0.548639	−	0.836059i	$-0.684854\pi$
$733$	−1.16999e10	−1.09728	−0.548639	+	0.836059i	$0.684854\pi$
$734$	2.09997e9	0.196009
$735$	0	0
$736$	−4.40836e7	−0.00407573
$737$	1.83663e9	0.169000
$738$	0	0
$739$	3.61810e9	0.329781	0.164890	−	0.986312i	$-0.447273\pi$
$739$	3.61810e9	0.329781	0.164890	+	0.986312i	$0.447273\pi$
$740$	−7.45808e7	−0.00676575
$741$	0	0
$742$	7.15274e9	0.642775
$743$	1.33275e10	1.19203	0.596017	−	0.802971i	$-0.296749\pi$
$743$	1.33275e10	1.19203	0.596017	+	0.802971i	$0.296749\pi$
$744$	0	0
$745$	−2.71483e9	−0.240544
$746$	−3.41800e9	−0.301430
$747$	0	0
$748$	−2.71867e8	−0.0237520
$749$	−6.70407e9	−0.582978
$750$	0	0
$751$	6.15962e9	0.530657	0.265328	−	0.964158i	$-0.414520\pi$
$751$	6.15962e9	0.530657	0.265328	+	0.964158i	$0.414520\pi$
$752$	−8.10358e9	−0.694887
$753$	0	0
$754$	−1.91499e10	−1.62692
$755$	8.20570e9	0.693907
$756$	0	0
$757$	−6.89179e9	−0.577426	−0.288713	−	0.957416i	$-0.593227\pi$
$757$	−6.89179e9	−0.577426	−0.288713	+	0.957416i	$0.593227\pi$
$758$	−1.86893e10	−1.55866
$759$	0	0
$760$	−1.68559e9	−0.139285
$761$	−5.78765e9	−0.476054	−0.238027	−	0.971259i	$-0.576501\pi$
$761$	−5.78765e9	−0.476054	−0.238027	+	0.971259i	$0.576501\pi$
$762$	0	0
$763$	−5.80634e8	−0.0473224
$764$	5.21721e6	0.000423264 0
$765$	0	0
$766$	1.12912e10	0.907698
$767$	−2.54118e8	−0.0203354
$768$	0	0
$769$	6.81983e9	0.540794	0.270397	−	0.962749i	$-0.412845\pi$
$769$	6.81983e9	0.540794	0.270397	+	0.962749i	$0.412845\pi$
$770$	3.45100e9	0.272413
$771$	0	0
$772$	7.01881e7	0.00549038
$773$	3.94610e8	0.0307284	0.0153642	−	0.999882i	$-0.495109\pi$
$773$	3.94610e8	0.0307284	0.0153642	+	0.999882i	$0.495109\pi$
$774$	0	0
$775$	3.30061e9	0.254705
$776$	2.09761e10	1.61142
$777$	0	0
$778$	1.83352e10	1.39591
$779$	7.25872e9	0.550148
$780$	0	0
$781$	1.15981e10	0.871183
$782$	−5.47534e9	−0.409437
$783$	0	0
$784$	−1.94400e9	−0.144075
$785$	−3.92367e9	−0.289500
$786$	0	0
$787$	−3.03861e9	−0.222210	−0.111105	−	0.993809i	$-0.535439\pi$
$787$	−3.03861e9	−0.222210	−0.111105	+	0.993809i	$0.535439\pi$
$788$	−1.66146e8	−0.0120962
$789$	0	0
$790$	8.05422e9	0.581204
$791$	4.28582e9	0.307905
$792$	0	0
$793$	1.18503e10	0.843867
$794$	2.13216e10	1.51164
$795$	0	0
$796$	−6.63702e7	−0.00466420
$797$	1.02857e10	0.719663	0.359831	−	0.933017i	$-0.382834\pi$
$797$	1.02857e10	0.719663	0.359831	+	0.933017i	$0.382834\pi$
$798$	0	0
$799$	−1.70947e10	−1.18562
$800$	4.98245e7	0.00344055
$801$	0	0
$802$	5.33006e9	0.364856
$803$	2.00605e10	1.36721
$804$	0	0
$805$	5.92733e8	0.0400473
$806$	2.16235e10	1.45463
$807$	0	0
$808$	−2.12115e10	−1.41459
$809$	2.39679e9	0.159151	0.0795755	−	0.996829i	$-0.474644\pi$
$809$	2.39679e9	0.159151	0.0795755	+	0.996829i	$0.474644\pi$
$810$	0	0
$811$	2.41910e10	1.59250	0.796251	−	0.604966i	$-0.206813\pi$
$811$	2.41910e10	1.59250	0.796251	+	0.604966i	$0.206813\pi$
$812$	−7.06476e7	−0.00463075
$813$	0	0
$814$	−4.36182e10	−2.83454
$815$	1.35799e10	0.878710
$816$	0	0
$817$	−3.74393e9	−0.240188
$818$	1.59980e10	1.02195
$819$	0	0
$820$	−1.06817e8	−0.00676537
$821$	−2.87851e10	−1.81537	−0.907687	−	0.419648i	$-0.862154\pi$
$821$	−2.87851e10	−1.81537	−0.907687	+	0.419648i	$0.862154\pi$
$822$	0	0
$823$	−8.45606e9	−0.528772	−0.264386	−	0.964417i	$-0.585169\pi$
$823$	−8.45606e9	−0.528772	−0.264386	+	0.964417i	$0.585169\pi$
$824$	−1.05450e10	−0.656602
$825$	0	0
$826$	−1.09928e8	−0.00678699
$827$	−3.68371e9	−0.226473	−0.113236	−	0.993568i	$-0.536122\pi$
$827$	−3.68371e9	−0.226473	−0.113236	+	0.993568i	$0.536122\pi$
$828$	0	0
$829$	6.49057e9	0.395678	0.197839	−	0.980235i	$-0.436608\pi$
$829$	6.49057e9	0.395678	0.197839	+	0.980235i	$0.436608\pi$
$830$	−2.87580e9	−0.174576
$831$	0	0
$832$	−1.87284e10	−1.12738
$833$	−4.10090e9	−0.245823
$834$	0	0
$835$	−2.00746e9	−0.119329
$836$	7.29432e7	0.00431781
$837$	0	0
$838$	−5.55251e9	−0.325938
$839$	−1.12657e10	−0.658554	−0.329277	−	0.944233i	$-0.606805\pi$
$839$	−1.12657e10	−0.658554	−0.329277	+	0.944233i	$0.606805\pi$
$840$	0	0
$841$	1.77469e10	1.02881
$842$	−1.10703e8	−0.00639096
$843$	0	0
$844$	2.43342e8	0.0139322
$845$	−2.30218e9	−0.131262
$846$	0	0
$847$	1.05285e10	0.595351
$848$	3.03264e10	1.70780
$849$	0	0
$850$	6.18837e9	0.345629
$851$	−7.49174e9	−0.416705
$852$	0	0
$853$	1.11602e10	0.615675	0.307838	−	0.951439i	$-0.400395\pi$
$853$	1.11602e10	0.615675	0.307838	+	0.951439i	$0.400395\pi$
$854$	5.12628e9	0.281643
$855$	0	0
$856$	−2.81817e10	−1.53571
$857$	−3.34670e9	−0.181629	−0.0908144	−	0.995868i	$-0.528947\pi$
$857$	−3.34670e9	−0.181629	−0.0908144	+	0.995868i	$0.528947\pi$
$858$	0	0
$859$	1.47916e10	0.796233	0.398117	−	0.917335i	$-0.369664\pi$
$859$	1.47916e10	0.796233	0.398117	+	0.917335i	$0.369664\pi$
$860$	5.50944e7	0.00295368
$861$	0	0
$862$	−6.97710e9	−0.371022
$863$	2.61067e10	1.38266	0.691329	−	0.722540i	$-0.257026\pi$
$863$	2.61067e10	1.38266	0.691329	+	0.722540i	$0.257026\pi$
$864$	0	0
$865$	−5.04078e8	−0.0264814
$866$	1.44192e10	0.754447
$867$	0	0
$868$	7.97734e7	0.00414037
$869$	4.01721e10	2.07661
$870$	0	0
$871$	−2.33579e9	−0.119776
$872$	−2.44079e9	−0.124659
$873$	0	0
$874$	1.46906e9	0.0744303
$875$	−6.69922e8	−0.0338062
$876$	0	0
$877$	9.77952e9	0.489575	0.244787	−	0.969577i	$-0.421282\pi$
$877$	9.77952e9	0.489575	0.244787	+	0.969577i	$0.421282\pi$
$878$	2.77051e10	1.38143
$879$	0	0
$880$	1.46317e10	0.723776
$881$	2.43721e9	0.120082	0.0600409	−	0.998196i	$-0.480877\pi$
$881$	2.43721e9	0.120082	0.0600409	+	0.998196i	$0.480877\pi$
$882$	0	0
$883$	1.46066e9	0.0713982	0.0356991	−	0.999363i	$-0.488634\pi$
$883$	1.46066e9	0.0713982	0.0356991	+	0.999363i	$0.488634\pi$
$884$	3.45754e8	0.0168339
$885$	0	0
$886$	8.09142e9	0.390847
$887$	3.37167e9	0.162223	0.0811115	−	0.996705i	$-0.474153\pi$
$887$	3.37167e9	0.162223	0.0811115	+	0.996705i	$0.474153\pi$
$888$	0	0
$889$	6.89815e9	0.329288
$890$	−6.16796e9	−0.293276
$891$	0	0
$892$	−3.17234e8	−0.0149659
$893$	4.58658e9	0.215531
$894$	0	0
$895$	7.47905e8	0.0348711
$896$	−8.24163e9	−0.382768
$897$	0	0
$898$	1.85012e10	0.852577
$899$	−3.95174e10	−1.81397
$900$	0	0
$901$	6.39742e10	2.91386
$902$	−6.24713e10	−2.83438
$903$	0	0
$904$	1.80162e10	0.811098
$905$	1.16234e10	0.521270
$906$	0	0
$907$	3.26298e10	1.45207	0.726037	−	0.687656i	$-0.241360\pi$
$907$	3.26298e10	1.45207	0.726037	+	0.687656i	$0.241360\pi$
$908$	−1.82906e8	−0.00810828
$909$	0	0
$910$	−4.38890e9	−0.193068
$911$	−1.18670e9	−0.0520027	−0.0260013	−	0.999662i	$-0.508277\pi$
$911$	−1.18670e9	−0.0520027	−0.0260013	+	0.999662i	$0.508277\pi$
$912$	0	0
$913$	−1.43436e10	−0.623751
$914$	3.80983e10	1.65042
$915$	0	0
$916$	−2.17767e8	−0.00936175
$917$	−7.41187e9	−0.317421
$918$	0	0
$919$	−2.05295e10	−0.872517	−0.436258	−	0.899821i	$-0.643697\pi$
$919$	−2.05295e10	−0.872517	−0.436258	+	0.899821i	$0.643697\pi$
$920$	2.49166e9	0.105495
$921$	0	0
$922$	−9.08177e9	−0.381603
$923$	−1.47502e10	−0.617438
$924$	0	0
$925$	8.46735e9	0.351764
$926$	−7.02987e9	−0.290944
$927$	0	0
$928$	−5.96536e8	−0.0245030
$929$	−2.59782e10	−1.06305	−0.531524	−	0.847043i	$-0.678381\pi$
$929$	−2.59782e10	−1.06305	−0.531524	+	0.847043i	$0.678381\pi$
$930$	0	0
$931$	1.10029e9	0.0446873
$932$	−1.64591e8	−0.00665963
$933$	0	0
$934$	4.50741e10	1.81014
$935$	3.08658e10	1.23491
$936$	0	0
$937$	−2.87793e10	−1.14286	−0.571428	−	0.820652i	$-0.693610\pi$
$937$	−2.87793e10	−1.14286	−0.571428	+	0.820652i	$0.693610\pi$
$938$	−1.01043e9	−0.0399756
$939$	0	0
$940$	−6.74945e7	−0.00265046
$941$	−8.32316e9	−0.325630	−0.162815	−	0.986657i	$-0.552057\pi$
$941$	−8.32316e9	−0.325630	−0.162815	+	0.986657i	$0.552057\pi$
$942$	0	0
$943$	−1.07299e10	−0.416682
$944$	−4.66076e8	−0.0180324
$945$	0	0
$946$	3.22217e10	1.23745
$947$	−3.47394e10	−1.32922	−0.664611	−	0.747190i	$-0.731403\pi$
$947$	−3.47394e10	−1.32922	−0.664611	+	0.747190i	$0.731403\pi$
$948$	0	0
$949$	−2.55125e10	−0.968993
$950$	−1.66037e9	−0.0628307
$951$	0	0
$952$	−1.72389e10	−0.647559
$953$	2.15908e10	0.808059	0.404030	−	0.914746i	$-0.367609\pi$
$953$	2.15908e10	0.808059	0.404030	+	0.914746i	$0.367609\pi$
$954$	0	0
$955$	−5.92323e8	−0.0220063
$956$	1.89495e8	0.00701449
$957$	0	0
$958$	1.98866e9	0.0730770
$959$	1.73088e9	0.0633728
$960$	0	0
$961$	1.71093e10	0.621872
$962$	5.54727e10	2.00894
$963$	0	0
$964$	6.36683e7	0.00228904
$965$	−7.96863e9	−0.285455
$966$	0	0
$967$	−1.09511e10	−0.389463	−0.194731	−	0.980857i	$-0.562383\pi$
$967$	−1.09511e10	−0.389463	−0.194731	+	0.980857i	$0.562383\pi$
$968$	4.42582e10	1.56830
$969$	0	0
$970$	2.06622e10	0.726902
$971$	−3.82109e10	−1.33943	−0.669714	−	0.742619i	$-0.733583\pi$
$971$	−3.82109e10	−1.33943	−0.669714	+	0.742619i	$0.733583\pi$
$972$	0	0
$973$	−5.85449e9	−0.203748
$974$	2.36782e10	0.821094
$975$	0	0
$976$	2.17346e10	0.748301
$977$	5.99153e8	0.0205545	0.0102772	−	0.999947i	$-0.496729\pi$
$977$	5.99153e8	0.0205545	0.0102772	+	0.999947i	$0.496729\pi$
$978$	0	0
$979$	−3.07639e10	−1.04786
$980$	−1.61915e7	−0.000549537 0
$981$	0	0
$982$	−4.09381e10	−1.37955
$983$	2.33267e10	0.783278	0.391639	−	0.920119i	$-0.371908\pi$
$983$	2.33267e10	0.783278	0.391639	+	0.920119i	$0.371908\pi$
$984$	0	0
$985$	1.88630e10	0.628902
$986$	−7.40918e10	−2.46151
$987$	0	0
$988$	−9.27676e7	−0.00306018
$989$	5.53430e9	0.181918
$990$	0	0
$991$	−7.22304e9	−0.235756	−0.117878	−	0.993028i	$-0.537609\pi$
$991$	−7.22304e9	−0.235756	−0.117878	+	0.993028i	$0.537609\pi$
$992$	6.73592e8	0.0219082
$993$	0	0
$994$	−6.38074e9	−0.206072
$995$	7.53517e9	0.242500
$996$	0	0
$997$	−2.26921e10	−0.725174	−0.362587	−	0.931950i	$-0.618106\pi$
$997$	−2.26921e10	−0.725174	−0.362587	+	0.931950i	$0.618106\pi$
$998$	2.83465e10	0.902697
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	315.8.a.q.1.3	yes	8
3.2	odd	2		315.8.a.p.1.6	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
315.8.a.p.1.6	✓	8	3.2	odd	2
315.8.a.q.1.3	yes	8	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:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	315.a (trivial)

\(f(q)\)	\(=\)	\(q-11.3623 q^{2} +1.10101 q^{4} -125.000 q^{5} +343.000 q^{7} +1441.86 q^{8} +1420.28 q^{10} +7083.95 q^{11} -9009.21 q^{13} -3897.26 q^{14} -16523.7 q^{16} -34857.1 q^{17} +9352.33 q^{19} -137.626 q^{20} -80489.7 q^{22} -13824.7 q^{23} +15625.0 q^{25} +102365. q^{26} +377.645 q^{28} -187074. q^{29} +211239. q^{31} +3188.77 q^{32} +396056. q^{34} -42875.0 q^{35} +541911. q^{37} -106264. q^{38} -180232. q^{40} +776141. q^{41} -400320. q^{43} +7799.47 q^{44} +157080. q^{46} +490421. q^{47} +117649. q^{49} -177535. q^{50} -9919.20 q^{52} -1.83533e6 q^{53} -885494. q^{55} +494558. q^{56} +2.12559e6 q^{58} +28206.5 q^{59} -1.31536e6 q^{61} -2.40015e6 q^{62} +2.07880e6 q^{64} +1.12615e6 q^{65} +259266. q^{67} -38377.9 q^{68} +487157. q^{70} +1.63724e6 q^{71} +2.83182e6 q^{73} -6.15733e6 q^{74} +10297.0 q^{76} +2.42980e6 q^{77} +5.67086e6 q^{79} +2.06546e6 q^{80} -8.81871e6 q^{82} -2.02481e6 q^{83} +4.35714e6 q^{85} +4.54855e6 q^{86} +1.02141e7 q^{88} -4.34277e6 q^{89} -3.09016e6 q^{91} -15221.0 q^{92} -5.57229e6 q^{94} -1.16904e6 q^{95} +1.45480e7 q^{97} -1.33676e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 5 q^{2} + 651 q^{4} - 1000 q^{5} + 2744 q^{7} - 405 q^{8} - 625 q^{10} - 1040 q^{11} - 10412 q^{13} + 1715 q^{14} + 23555 q^{16} - 33600 q^{17} + 25508 q^{19} - 81375 q^{20} + 27440 q^{22} - 110100 q^{23}+ \cdots + 588245 q^{98}+O(q^{100}) \) 8 * q + 5 * q^2 + 651 * q^4 - 1000 * q^5 + 2744 * q^7 - 405 * q^8 - 625 * q^10 - 1040 * q^11 - 10412 * q^13 + 1715 * q^14 + 23555 * q^16 - 33600 * q^17 + 25508 * q^19 - 81375 * q^20 + 27440 * q^22 - 110100 * q^23 + 125000 * q^25 - 255890 * q^26 + 223293 * q^28 - 348980 * q^29 + 243684 * q^31 - 456065 * q^32 + 200968 * q^34 - 343000 * q^35 + 185216 * q^37 - 993110 * q^38 + 50625 * q^40 - 700000 * q^41 + 264684 * q^43 - 338760 * q^44 + 92668 * q^46 - 47220 * q^47 + 941192 * q^49 + 78125 * q^50 - 2621774 * q^52 - 1582520 * q^53 + 130000 * q^55 - 138915 * q^56 - 429406 * q^58 - 1618120 * q^59 - 1554236 * q^61 - 5213900 * q^62 - 55953 * q^64 + 1301500 * q^65 - 5215444 * q^67 - 3199720 * q^68 - 214375 * q^70 - 5827000 * q^71 + 18532 * q^73 - 52740 * q^74 - 11979466 * q^76 - 356720 * q^77 - 236880 * q^79 - 2944375 * q^80 - 12362418 * q^82 - 7424160 * q^83 + 4200000 * q^85 + 5147530 * q^86 + 5080880 * q^88 - 10567360 * q^89 - 3571316 * q^91 - 2262100 * q^92 + 1581266 * q^94 - 3188500 * q^95 + 11406932 * q^97 + 588245 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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