LMFDB - Embedded newform 16.48.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,48,Mod(1,16)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(16, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 48, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 48);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 48 $
Character orbit:	$[\chi]$	$=$	16.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:	$223.852260248$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 $ x^4 - x^3 - 832803191366x^2 + 3710135215485780x + 13175318942671469337000
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{46}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$901372.$ of defining polynomial
Character	$\chi$	$=$	16.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.28510e11 q^{3} -1.15086e16 q^{5} -1.54211e19 q^{7} -1.00741e22 q^{9} +O(q^{10})$	$q-1.28510e11 q^{3} -1.15086e16 q^{5} -1.54211e19 q^{7} -1.00741e22 q^{9} +3.19474e24 q^{11} +1.57743e25 q^{13} +1.47897e27 q^{15} -3.42618e28 q^{17} -1.00019e30 q^{19} +1.98176e30 q^{21} -5.17765e31 q^{23} -5.78095e32 q^{25} +4.71154e33 q^{27} -6.90432e33 q^{29} +1.79630e35 q^{31} -4.10556e35 q^{33} +1.77475e35 q^{35} -2.68894e36 q^{37} -2.02715e36 q^{39} +1.17249e38 q^{41} -1.58085e38 q^{43} +1.15938e38 q^{45} +1.34555e39 q^{47} -5.00553e39 q^{49} +4.40298e39 q^{51} +4.68526e40 q^{53} -3.67670e40 q^{55} +1.28534e41 q^{57} -3.84916e41 q^{59} +4.37844e41 q^{61} +1.55353e41 q^{63} -1.81540e41 q^{65} -4.64892e42 q^{67} +6.65378e42 q^{69} -1.91856e43 q^{71} +8.25143e43 q^{73} +7.42908e43 q^{75} -4.92663e43 q^{77} +5.45765e44 q^{79} -3.37621e44 q^{81} +1.02727e45 q^{83} +3.94306e44 q^{85} +8.87272e44 q^{87} +1.06667e46 q^{89} -2.43256e44 q^{91} -2.30842e46 q^{93} +1.15107e46 q^{95} +3.72976e46 q^{97} -3.21841e46 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 17\!\cdots\!72 q^{9}+O(q^{10}) $ 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9	$ 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 53\!\cdots\!16 q^{99}+O(q^{100}) $ 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9 + 1902165515717740404742512 * q^11 + 126959149195199610466858520 * q^13 - 12437812896841241272880733600 * q^15 + 210426285758836687099688769480 * q^17 + 1059119811802437301721159140240 * q^19 - 26054906182888686128940663165312 * q^21 - 137531418344493639994288859179680 * q^23 + 1009909900344396506188154274977500 * q^25 + 7345952425324536073931587214783520 * q^27 - 22743132490172744184500850941726760 * q^29 - 75786540814706434901630981303934848 * q^31 + 263381387470144952341649359926294720 * q^33 + 1313148822925052387484546800067979200 * q^35 - 11291457045120157869429951769632191560 * q^37 - 39294375338503805226442587829551714336 * q^39 + 130658126291089294119301677048020391528 * q^41 + 445357287738427943170332944488133090800 * q^43 + 86022747271219179966948746072020632600 * q^45 - 2031912465958423919194866366997063590720 * q^47 + 9153745744156628626499502447643951451172 * q^49 + 18534366050002170683962116343746386369952 * q^51 + 29171719896619533806321793123448457437560 * q^53 - 192172580604633896452606078657416872901600 * q^55 + 558743644452458098451223285784560964733760 * q^57 - 478242858216320686410790860443879577596880 * q^59 + 626683213833907852009669690320782900857688 * q^61 - 5811636950821577036406237472810763671724640 * q^63 + 12996735432890074297558542478104126872331600 * q^65 - 18701411419127220668869505853739529194216880 * q^67 - 2096956664577882726873494616728974107851904 * q^69 - 22196594617762006008028198712153461686540768 * q^71 + 104911201720506650033573412943840886878191080 * q^73 - 32388703324174028323482311940739905891330000 * q^75 - 263490744376507692433815778071300370399209600 * q^77 + 1327995310109394234162785254414226288730352960 * q^79 - 1061448205755511803861508510986654404170081116 * q^81 + 1439130627532364606968733068855111784504073360 * q^83 - 1001895286410663502447652611446390227458237200 * q^85 + 4326251093943206485496474339443178030108648160 * q^87 - 7960590522674945421108320896849737731421281880 * q^89 - 5325194479227227497182497798631911205603381568 * q^91 + 10728611999875738337226031496188630272675064320 * q^93 - 75421152415106939771624998522249518123547668000 * q^95 + 95066248961258904822565865801364599382905757320 * q^97 - 530900415919166351015781326273177630089784016 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	0	0
$2$	0	0
$3$	−1.28510e11	−0.788109	−0.394055	−	0.919087i	$-0.628928\pi$
$3$	−1.28510e11	−0.788109	−0.394055	+	0.919087i	$0.628928\pi$
$4$	0	0
$5$	−1.15086e16	−0.431745	−0.215873	−	0.976422i	$-0.569260\pi$
$5$	−1.15086e16	−0.431745	−0.215873	+	0.976422i	$0.569260\pi$
$6$	0	0
$7$	−1.54211e19	−0.212966	−0.106483	−	0.994315i	$-0.533959\pi$
$7$	−1.54211e19	−0.212966	−0.106483	+	0.994315i	$0.533959\pi$
$8$	0	0
$9$	−1.00741e22	−0.378884
$10$	0	0
$11$	3.19474e24	1.07574	0.537871	−	0.843027i	$-0.319229\pi$
$11$	3.19474e24	1.07574	0.537871	+	0.843027i	$0.319229\pi$
$12$	0	0
$13$	1.57743e25	0.104781	0.0523903	−	0.998627i	$-0.483316\pi$
$13$	1.57743e25	0.104781	0.0523903	+	0.998627i	$0.483316\pi$
$14$	0	0
$15$	1.47897e27	0.340262
$16$	0	0
$17$	−3.42618e28	−0.416160	−0.208080	−	0.978112i	$-0.566722\pi$
$17$	−3.42618e28	−0.416160	−0.208080	+	0.978112i	$0.566722\pi$
$18$	0	0
$19$	−1.00019e30	−0.889961	−0.444980	−	0.895540i	$-0.646789\pi$
$19$	−1.00019e30	−0.889961	−0.444980	+	0.895540i	$0.646789\pi$
$20$	0	0
$21$	1.98176e30	0.167841
$22$	0	0
$23$	−5.17765e31	−0.517045	−0.258522	−	0.966005i	$-0.583236\pi$
$23$	−5.17765e31	−0.517045	−0.258522	+	0.966005i	$0.583236\pi$
$24$	0	0
$25$	−5.78095e32	−0.813596
$26$	0	0
$27$	4.71154e33	1.08671
$28$	0	0
$29$	−6.90432e33	−0.297008	−0.148504	−	0.988912i	$-0.547446\pi$
$29$	−6.90432e33	−0.297008	−0.148504	+	0.988912i	$0.547446\pi$
$30$	0	0
$31$	1.79630e35	1.61205	0.806025	−	0.591882i	$-0.201615\pi$
$31$	1.79630e35	1.61205	0.806025	+	0.591882i	$0.201615\pi$
$32$	0	0
$33$	−4.10556e35	−0.847802
$34$	0	0
$35$	1.77475e35	0.0919471
$36$	0	0
$37$	−2.68894e36	−0.377434	−0.188717	−	0.982032i	$-0.560433\pi$
$37$	−2.68894e36	−0.377434	−0.188717	+	0.982032i	$0.560433\pi$
$38$	0	0
$39$	−2.02715e36	−0.0825786
$40$	0	0
$41$	1.17249e38	1.47465	0.737325	−	0.675538i	$-0.236088\pi$
$41$	1.17249e38	1.47465	0.737325	+	0.675538i	$0.236088\pi$
$42$	0	0
$43$	−1.58085e38	−0.649204	−0.324602	−	0.945851i	$-0.605230\pi$
$43$	−1.58085e38	−0.649204	−0.324602	+	0.945851i	$0.605230\pi$
$44$	0	0
$45$	1.15938e38	0.163581
$46$	0	0
$47$	1.34555e39	0.683282	0.341641	−	0.939831i	$-0.389017\pi$
$47$	1.34555e39	0.683282	0.341641	+	0.939831i	$0.389017\pi$
$48$	0	0
$49$	−5.00553e39	−0.954645
$50$	0	0
$51$	4.40298e39	0.327980
$52$	0	0
$53$	4.68526e40	1.41335	0.706677	−	0.707536i	$-0.250193\pi$
$53$	4.68526e40	1.41335	0.706677	+	0.707536i	$0.250193\pi$
$54$	0	0
$55$	−3.67670e40	−0.464446
$56$	0	0
$57$	1.28534e41	0.701387
$58$	0	0
$59$	−3.84916e41	−0.933993	−0.466997	−	0.884259i	$-0.654664\pi$
$59$	−3.84916e41	−0.933993	−0.466997	+	0.884259i	$0.654664\pi$
$60$	0	0
$61$	4.37844e41	0.485365	0.242683	−	0.970106i	$-0.421973\pi$
$61$	4.37844e41	0.485365	0.242683	+	0.970106i	$0.421973\pi$
$62$	0	0
$63$	1.55353e41	0.0806893
$64$	0	0
$65$	−1.81540e41	−0.0452385
$66$	0	0
$67$	−4.64892e42	−0.568321	−0.284161	−	0.958777i	$-0.591715\pi$
$67$	−4.64892e42	−0.568321	−0.284161	+	0.958777i	$0.591715\pi$
$68$	0	0
$69$	6.65378e42	0.407488
$70$	0	0
$71$	−1.91856e43	−0.600349	−0.300175	−	0.953884i	$-0.597045\pi$
$71$	−1.91856e43	−0.600349	−0.300175	+	0.953884i	$0.597045\pi$
$72$	0	0
$73$	8.25143e43	1.34413	0.672066	−	0.740491i	$-0.265407\pi$
$73$	8.25143e43	1.34413	0.672066	+	0.740491i	$0.265407\pi$
$74$	0	0
$75$	7.42908e43	0.641203
$76$	0	0
$77$	−4.92663e43	−0.229096
$78$	0	0
$79$	5.45765e44	1.38921	0.694607	−	0.719389i	$-0.255578\pi$
$79$	5.45765e44	1.38921	0.694607	+	0.719389i	$0.255578\pi$
$80$	0	0
$81$	−3.37621e44	−0.477564
$82$	0	0
$83$	1.02727e45	0.819125	0.409563	−	0.912282i	$-0.365681\pi$
$83$	1.02727e45	0.819125	0.409563	+	0.912282i	$0.365681\pi$
$84$	0	0
$85$	3.94306e44	0.179675
$86$	0	0
$87$	8.87272e44	0.234075
$88$	0	0
$89$	1.06667e46	1.64954	0.824771	−	0.565467i	$-0.191304\pi$
$89$	1.06667e46	1.64954	0.824771	+	0.565467i	$0.191304\pi$
$90$	0	0
$91$	−2.43256e44	−0.0223147
$92$	0	0
$93$	−2.30842e46	−1.27047
$94$	0	0
$95$	1.15107e46	0.384236
$96$	0	0
$97$	3.72976e46	0.763037	0.381518	−	0.924361i	$-0.375401\pi$
$97$	3.72976e46	0.763037	0.381518	+	0.924361i	$0.375401\pi$
$98$	0	0
$99$	−3.21841e46	−0.407581
$100$	0	0
$101$	9.27720e46	0.734285	0.367142	−	0.930165i	$-0.380336\pi$
$101$	9.27720e46	0.734285	0.367142	+	0.930165i	$0.380336\pi$
$102$	0	0
$103$	−1.00960e47	−0.504051	−0.252026	−	0.967721i	$-0.581097\pi$
$103$	−1.00960e47	−0.504051	−0.252026	+	0.967721i	$0.581097\pi$
$104$	0	0
$105$	−2.28072e46	−0.0724643
$106$	0	0
$107$	−1.99211e47	−0.406252	−0.203126	−	0.979153i	$-0.565110\pi$
$107$	−1.99211e47	−0.406252	−0.203126	+	0.979153i	$0.565110\pi$
$108$	0	0
$109$	−6.24516e47	−0.824177	−0.412089	−	0.911144i	$-0.635201\pi$
$109$	−6.24516e47	−0.824177	−0.412089	+	0.911144i	$0.635201\pi$
$110$	0	0
$111$	3.45555e47	0.297459
$112$	0	0
$113$	−2.17631e48	−1.23134	−0.615669	−	0.788005i	$-0.711114\pi$
$113$	−2.17631e48	−1.23134	−0.615669	+	0.788005i	$0.711114\pi$
$114$	0	0
$115$	5.95875e47	0.223232
$116$	0	0
$117$	−1.58911e47	−0.0396996
$118$	0	0
$119$	5.28354e47	0.0886280
$120$	0	0
$121$	1.38664e48	0.157220
$122$	0	0
$123$	−1.50677e49	−1.16219
$124$	0	0
$125$	1.48304e49	0.783011
$126$	0	0
$127$	1.42624e49	0.518568	0.259284	−	0.965801i	$-0.416513\pi$
$127$	1.42624e49	0.518568	0.259284	+	0.965801i	$0.416513\pi$
$128$	0	0
$129$	2.03154e49	0.511644
$130$	0	0
$131$	7.28871e49	1.27872	0.639359	−	0.768908i	$-0.279200\pi$
$131$	7.28871e49	1.27872	0.639359	+	0.768908i	$0.279200\pi$
$132$	0	0
$133$	1.54239e49	0.189531
$134$	0	0
$135$	−5.42232e49	−0.469182
$136$	0	0
$137$	2.99893e49	0.183668	0.0918338	−	0.995774i	$-0.470727\pi$
$137$	2.99893e49	0.183668	0.0918338	+	0.995774i	$0.470727\pi$
$138$	0	0
$139$	−1.25821e50	−0.548159	−0.274079	−	0.961707i	$-0.588373\pi$
$139$	−1.25821e50	−0.548159	−0.274079	+	0.961707i	$0.588373\pi$
$140$	0	0
$141$	−1.72916e50	−0.538501
$142$	0	0
$143$	5.03948e49	0.112717
$144$	0	0
$145$	7.94590e49	0.128232
$146$	0	0
$147$	6.43259e50	0.752365
$148$	0	0
$149$	−1.60599e51	−1.36730	−0.683652	−	0.729809i	$-0.739609\pi$
$149$	−1.60599e51	−1.36730	−0.683652	+	0.729809i	$0.739609\pi$
$150$	0	0
$151$	−2.76267e49	−0.0171938	−0.00859689	−	0.999963i	$-0.502737\pi$
$151$	−2.76267e49	−0.0171938	−0.00859689	+	0.999963i	$0.502737\pi$
$152$	0	0
$153$	3.45156e50	0.157676
$154$	0	0
$155$	−2.06729e51	−0.695995
$156$	0	0
$157$	3.26071e51	0.812214	0.406107	−	0.913825i	$-0.366886\pi$
$157$	3.26071e51	0.812214	0.406107	+	0.913825i	$0.366886\pi$
$158$	0	0
$159$	−6.02102e51	−1.11388
$160$	0	0
$161$	7.98448e50	0.110113
$162$	0	0
$163$	1.71469e51	0.176920	0.0884602	−	0.996080i	$-0.471805\pi$
$163$	1.71469e51	0.176920	0.0884602	+	0.996080i	$0.471805\pi$
$164$	0	0
$165$	4.72492e51	0.366034
$166$	0	0
$167$	1.35054e52	0.788264	0.394132	−	0.919054i	$-0.371045\pi$
$167$	1.35054e52	0.788264	0.394132	+	0.919054i	$0.371045\pi$
$168$	0	0
$169$	−2.24152e52	−0.989021
$170$	0	0
$171$	1.00759e52	0.337192
$172$	0	0
$173$	9.07124e51	0.230985	0.115492	−	0.993308i	$-0.463155\pi$
$173$	9.07124e51	0.230985	0.115492	+	0.993308i	$0.463155\pi$
$174$	0	0
$175$	8.91484e51	0.173268
$176$	0	0
$177$	4.94655e52	0.736089
$178$	0	0
$179$	2.83103e52	0.323518	0.161759	−	0.986830i	$-0.448283\pi$
$179$	2.83103e52	0.323518	0.161759	+	0.986830i	$0.448283\pi$
$180$	0	0
$181$	−7.28434e52	−0.641127	−0.320563	−	0.947227i	$-0.603872\pi$
$181$	−7.28434e52	−0.641127	−0.320563	+	0.947227i	$0.603872\pi$
$182$	0	0
$183$	−5.62672e52	−0.382521
$184$	0	0
$185$	3.09460e52	0.162955
$186$	0	0
$187$	−1.09458e53	−0.447681
$188$	0	0
$189$	−7.26569e52	−0.231433
$190$	0	0
$191$	−2.15414e53	−0.535785	−0.267893	−	0.963449i	$-0.586327\pi$
$191$	−2.15414e53	−0.535785	−0.267893	+	0.963449i	$0.586327\pi$
$192$	0	0
$193$	−4.58281e53	−0.892351	−0.446175	−	0.894946i	$-0.647214\pi$
$193$	−4.58281e53	−0.892351	−0.446175	+	0.894946i	$0.647214\pi$
$194$	0	0
$195$	2.33296e52	0.0356529
$196$	0	0
$197$	2.29005e53	0.275353	0.137676	−	0.990477i	$-0.456037\pi$
$197$	2.29005e53	0.275353	0.137676	+	0.990477i	$0.456037\pi$
$198$	0	0
$199$	−1.26295e54	−1.19767	−0.598835	−	0.800872i	$-0.704370\pi$
$199$	−1.26295e54	−1.19767	−0.598835	+	0.800872i	$0.704370\pi$
$200$	0	0
$201$	5.97431e53	0.447899
$202$	0	0
$203$	1.06472e53	0.0632525
$204$	0	0
$205$	−1.34937e54	−0.636673
$206$	0	0
$207$	5.21599e53	0.195900
$208$	0	0
$209$	−3.19534e54	−0.957368
$210$	0	0
$211$	4.97549e54	1.19178	0.595892	−	0.803064i	$-0.296799\pi$
$211$	4.97549e54	1.19178	0.595892	+	0.803064i	$0.296799\pi$
$212$	0	0
$213$	2.46554e54	0.473141
$214$	0	0
$215$	1.81934e54	0.280291
$216$	0	0
$217$	−2.77008e54	−0.343312
$218$	0	0
$219$	−1.06039e55	−1.05932
$220$	0	0
$221$	−5.40455e53	−0.0436055
$222$	0	0
$223$	−2.79892e55	−1.82737	−0.913686	−	0.406421i	$-0.866777\pi$
$223$	−2.79892e55	−1.82737	−0.913686	+	0.406421i	$0.866777\pi$
$224$	0	0
$225$	5.82376e54	0.308258
$226$	0	0
$227$	−1.40478e55	−0.603949	−0.301974	−	0.953316i	$-0.597646\pi$
$227$	−1.40478e55	−0.603949	−0.301974	+	0.953316i	$0.597646\pi$
$228$	0	0
$229$	−4.09641e55	−1.43308	−0.716539	−	0.697547i	$-0.754275\pi$
$229$	−4.09641e55	−1.43308	−0.716539	+	0.697547i	$0.754275\pi$
$230$	0	0
$231$	6.33120e54	0.180553
$232$	0	0
$233$	−1.21897e55	−0.283876	−0.141938	−	0.989876i	$-0.545333\pi$
$233$	−1.21897e55	−0.283876	−0.141938	+	0.989876i	$0.545333\pi$
$234$	0	0
$235$	−1.54854e55	−0.295004
$236$	0	0
$237$	−7.01362e55	−1.09485
$238$	0	0
$239$	−6.34948e55	−0.813556	−0.406778	−	0.913527i	$-0.633348\pi$
$239$	−6.34948e55	−0.813556	−0.406778	+	0.913527i	$0.633348\pi$
$240$	0	0
$241$	−1.11357e56	−1.17305	−0.586525	−	0.809931i	$-0.699504\pi$
$241$	−1.11357e56	−1.17305	−0.586525	+	0.809931i	$0.699504\pi$
$242$	0	0
$243$	−8.18866e55	−0.710339
$244$	0	0
$245$	5.76066e55	0.412164
$246$	0	0
$247$	−1.57772e55	−0.0932506
$248$	0	0
$249$	−1.32014e56	−0.645560
$250$	0	0
$251$	−1.81960e56	−0.737299	−0.368649	−	0.929569i	$-0.620180\pi$
$251$	−1.81960e56	−0.737299	−0.368649	+	0.929569i	$0.620180\pi$
$252$	0	0
$253$	−1.65413e56	−0.556207
$254$	0	0
$255$	−5.06721e55	−0.141604
$256$	0	0
$257$	−3.46416e56	−0.805694	−0.402847	−	0.915267i	$-0.631979\pi$
$257$	−3.46416e56	−0.805694	−0.402847	+	0.915267i	$0.631979\pi$
$258$	0	0
$259$	4.14663e55	0.0803806
$260$	0	0
$261$	6.95545e55	0.112531
$262$	0	0
$263$	−1.25535e57	−1.69748	−0.848738	−	0.528814i	$-0.822637\pi$
$263$	−1.25535e57	−1.69748	−0.848738	+	0.528814i	$0.822637\pi$
$264$	0	0
$265$	−5.39208e56	−0.610209
$266$	0	0
$267$	−1.37077e57	−1.30002
$268$	0	0
$269$	−2.60597e55	−0.0207391	−0.0103696	−	0.999946i	$-0.503301\pi$
$269$	−2.60597e55	−0.0207391	−0.0103696	+	0.999946i	$0.503301\pi$
$270$	0	0
$271$	1.42492e57	0.952822	0.476411	−	0.879223i	$-0.341937\pi$
$271$	1.42492e57	0.952822	0.476411	+	0.879223i	$0.341937\pi$
$272$	0	0
$273$	3.12608e55	0.0175864
$274$	0	0
$275$	−1.84686e57	−0.875219
$276$	0	0
$277$	−3.96329e57	−1.58409	−0.792047	−	0.610461i	$-0.790984\pi$
$277$	−3.96329e57	−1.58409	−0.792047	+	0.610461i	$0.790984\pi$
$278$	0	0
$279$	−1.80960e57	−0.610779
$280$	0	0
$281$	−4.27684e57	−1.22046	−0.610231	−	0.792223i	$-0.708924\pi$
$281$	−4.27684e57	−1.22046	−0.610231	+	0.792223i	$0.708924\pi$
$282$	0	0
$283$	9.57574e56	0.231308	0.115654	−	0.993290i	$-0.463104\pi$
$283$	9.57574e56	0.231308	0.115654	+	0.993290i	$0.463104\pi$
$284$	0	0
$285$	−1.47924e57	−0.302820
$286$	0	0
$287$	−1.80811e57	−0.314050
$288$	0	0
$289$	−5.60409e57	−0.826811
$290$	0	0
$291$	−4.79311e57	−0.601357
$292$	0	0
$293$	1.40107e58	1.49648	0.748241	−	0.663427i	$-0.230899\pi$
$293$	1.40107e58	1.49648	0.748241	+	0.663427i	$0.230899\pi$
$294$	0	0
$295$	4.42985e57	0.403247
$296$	0	0
$297$	1.50522e58	1.16902
$298$	0	0
$299$	−8.16736e56	−0.0541762
$300$	0	0
$301$	2.43784e57	0.138258
$302$	0	0
$303$	−1.19221e58	−0.578697
$304$	0	0
$305$	−5.03897e57	−0.209554
$306$	0	0
$307$	4.70339e58	1.67749	0.838743	−	0.544527i	$-0.183291\pi$
$307$	4.70339e58	1.67749	0.838743	+	0.544527i	$0.183291\pi$
$308$	0	0
$309$	1.29743e58	0.397248
$310$	0	0
$311$	2.66280e58	0.700600	0.350300	−	0.936638i	$-0.386080\pi$
$311$	2.66280e58	0.700600	0.350300	+	0.936638i	$0.386080\pi$
$312$	0	0
$313$	−5.63252e58	−1.27471	−0.637354	−	0.770571i	$-0.719971\pi$
$313$	−5.63252e58	−1.27471	−0.637354	+	0.770571i	$0.719971\pi$
$314$	0	0
$315$	−1.78789e57	−0.0348372
$316$	0	0
$317$	9.99715e58	1.67874	0.839371	−	0.543560i	$-0.182924\pi$
$317$	9.99715e58	1.67874	0.839371	+	0.543560i	$0.182924\pi$
$318$	0	0
$319$	−2.20575e58	−0.319503
$320$	0	0
$321$	2.56006e58	0.320171
$322$	0	0
$323$	3.42682e58	0.370366
$324$	0	0
$325$	−9.11903e57	−0.0852491
$326$	0	0
$327$	8.02563e58	0.649542
$328$	0	0
$329$	−2.07498e58	−0.145516
$330$	0	0
$331$	1.86524e58	0.113443	0.0567215	−	0.998390i	$-0.481935\pi$
$331$	1.86524e58	0.113443	0.0567215	+	0.998390i	$0.481935\pi$
$332$	0	0
$333$	2.70886e58	0.143004
$334$	0	0
$335$	5.35025e58	0.245370
$336$	0	0
$337$	2.11425e59	0.843050	0.421525	−	0.906817i	$-0.361495\pi$
$337$	2.11425e59	0.843050	0.421525	+	0.906817i	$0.361495\pi$
$338$	0	0
$339$	2.79677e59	0.970429
$340$	0	0
$341$	5.73871e59	1.73415
$342$	0	0
$343$	1.58048e59	0.416273
$344$	0	0
$345$	−7.65757e58	−0.175931
$346$	0	0
$347$	−1.60147e59	−0.321200	−0.160600	−	0.987020i	$-0.551343\pi$
$347$	−1.60147e59	−0.321200	−0.160600	+	0.987020i	$0.551343\pi$
$348$	0	0
$349$	8.79943e59	1.54190	0.770951	−	0.636894i	$-0.219781\pi$
$349$	8.79943e59	1.54190	0.770951	+	0.636894i	$0.219781\pi$
$350$	0	0
$351$	7.43211e58	0.113866
$352$	0	0
$353$	−9.28528e59	−1.24477	−0.622386	−	0.782711i	$-0.713836\pi$
$353$	−9.28528e59	−1.24477	−0.622386	+	0.782711i	$0.713836\pi$
$354$	0	0
$355$	2.20799e59	0.259198
$356$	0	0
$357$	−6.78986e58	−0.0698486
$358$	0	0
$359$	1.74133e60	1.57095	0.785475	−	0.618893i	$-0.212419\pi$
$359$	1.74133e60	1.57095	0.785475	+	0.618893i	$0.212419\pi$
$360$	0	0
$361$	−2.62675e59	−0.207970
$362$	0	0
$363$	−1.78197e59	−0.123906
$364$	0	0
$365$	−9.49624e59	−0.580323
$366$	0	0
$367$	−2.18975e60	−1.17691	−0.588453	−	0.808531i	$-0.700263\pi$
$367$	−2.18975e60	−1.17691	−0.588453	+	0.808531i	$0.700263\pi$
$368$	0	0
$369$	−1.18118e60	−0.558721
$370$	0	0
$371$	−7.22517e59	−0.300996
$372$	0	0
$373$	1.69293e60	0.621556	0.310778	−	0.950482i	$-0.399410\pi$
$373$	1.69293e60	0.621556	0.310778	+	0.950482i	$0.399410\pi$
$374$	0	0
$375$	−1.90585e60	−0.617099
$376$	0	0
$377$	−1.08911e59	−0.0311206
$378$	0	0
$379$	−4.02001e60	−1.01439	−0.507196	−	0.861831i	$-0.669318\pi$
$379$	−4.02001e60	−1.01439	−0.507196	+	0.861831i	$0.669318\pi$
$380$	0	0
$381$	−1.83286e60	−0.408688
$382$	0	0
$383$	−2.58725e60	−0.510113	−0.255057	−	0.966926i	$-0.582094\pi$
$383$	−2.58725e60	−0.510113	−0.255057	+	0.966926i	$0.582094\pi$
$384$	0	0
$385$	5.66987e59	0.0989113
$386$	0	0
$387$	1.59256e60	0.245973
$388$	0	0
$389$	−7.66990e60	−1.04948	−0.524740	−	0.851263i	$-0.675837\pi$
$389$	−7.66990e60	−1.04948	−0.524740	+	0.851263i	$0.675837\pi$
$390$	0	0
$391$	1.77396e60	0.215174
$392$	0	0
$393$	−9.36670e60	−1.00777
$394$	0	0
$395$	−6.28100e60	−0.599787
$396$	0	0
$397$	2.69403e60	0.228468	0.114234	−	0.993454i	$-0.463559\pi$
$397$	2.69403e60	0.228468	0.114234	+	0.993454i	$0.463559\pi$
$398$	0	0
$399$	−1.98212e60	−0.149372
$400$	0	0
$401$	−8.61448e60	−0.577213	−0.288607	−	0.957448i	$-0.593192\pi$
$401$	−8.61448e60	−0.577213	−0.288607	+	0.957448i	$0.593192\pi$
$402$	0	0
$403$	2.83353e60	0.168912
$404$	0	0
$405$	3.88554e60	0.206186
$406$	0	0
$407$	−8.59048e60	−0.406021
$408$	0	0
$409$	−4.89023e60	−0.205983	−0.102992	−	0.994682i	$-0.532841\pi$
$409$	−4.89023e60	−0.205983	−0.102992	+	0.994682i	$0.532841\pi$
$410$	0	0
$411$	−3.85391e60	−0.144750
$412$	0	0
$413$	5.93581e60	0.198909
$414$	0	0
$415$	−1.18225e61	−0.353653
$416$	0	0
$417$	1.61693e61	0.432009
$418$	0	0
$419$	4.08353e60	0.0975003	0.0487501	−	0.998811i	$-0.484476\pi$
$419$	4.08353e60	0.0975003	0.0487501	+	0.998811i	$0.484476\pi$
$420$	0	0
$421$	−8.09919e61	−1.72907	−0.864534	−	0.502574i	$-0.832386\pi$
$421$	−8.09919e61	−1.72907	−0.864534	+	0.502574i	$0.832386\pi$
$422$	0	0
$423$	−1.35551e61	−0.258884
$424$	0	0
$425$	1.98066e61	0.338586
$426$	0	0
$427$	−6.75202e60	−0.103366
$428$	0	0
$429$	−6.47622e60	−0.0888332
$430$	0	0
$431$	−5.41910e61	−0.666364	−0.333182	−	0.942863i	$-0.608122\pi$
$431$	−5.41910e61	−0.666364	−0.333182	+	0.942863i	$0.608122\pi$
$432$	0	0
$433$	1.10085e62	1.21412	0.607060	−	0.794656i	$-0.292349\pi$
$433$	1.10085e62	1.21412	0.607060	+	0.794656i	$0.292349\pi$
$434$	0	0
$435$	−1.02113e61	−0.101061
$436$	0	0
$437$	5.17861e61	0.460150
$438$	0	0
$439$	1.32013e62	1.05366	0.526832	−	0.849969i	$-0.323380\pi$
$439$	1.32013e62	1.05366	0.526832	+	0.849969i	$0.323380\pi$
$440$	0	0
$441$	5.04260e61	0.361699
$442$	0	0
$443$	8.89175e61	0.573455	0.286728	−	0.958012i	$-0.407433\pi$
$443$	8.89175e61	0.573455	0.286728	+	0.958012i	$0.407433\pi$
$444$	0	0
$445$	−1.22758e62	−0.712181
$446$	0	0
$447$	2.06385e62	1.07758
$448$	0	0
$449$	9.45855e61	0.444668	0.222334	−	0.974971i	$-0.428633\pi$
$449$	9.45855e61	0.444668	0.222334	+	0.974971i	$0.428633\pi$
$450$	0	0
$451$	3.74581e62	1.58634
$452$	0	0
$453$	3.55030e60	0.0135506
$454$	0	0
$455$	2.79954e60	0.00963426
$456$	0	0
$457$	3.37293e62	1.04707	0.523536	−	0.852003i	$-0.324612\pi$
$457$	3.37293e62	1.04707	0.523536	+	0.852003i	$0.324612\pi$
$458$	0	0
$459$	−1.61426e62	−0.452246
$460$	0	0
$461$	4.97092e62	1.25738	0.628688	−	0.777658i	$-0.283592\pi$
$461$	4.97092e62	1.25738	0.628688	+	0.777658i	$0.283592\pi$
$462$	0	0
$463$	4.42973e62	1.01210	0.506051	−	0.862504i	$-0.331105\pi$
$463$	4.42973e62	1.01210	0.506051	+	0.862504i	$0.331105\pi$
$464$	0	0
$465$	2.65667e62	0.548520
$466$	0	0
$467$	−7.17000e62	−1.33836	−0.669179	−	0.743101i	$-0.733354\pi$
$467$	−7.17000e62	−1.33836	−0.669179	+	0.743101i	$0.733354\pi$
$468$	0	0
$469$	7.16912e61	0.121033
$470$	0	0
$471$	−4.19033e62	−0.640114
$472$	0	0
$473$	−5.05040e62	−0.698376
$474$	0	0
$475$	5.78202e62	0.724069
$476$	0	0
$477$	−4.71996e62	−0.535497
$478$	0	0
$479$	1.03149e62	0.106067	0.0530337	−	0.998593i	$-0.483111\pi$
$479$	1.03149e62	0.106067	0.0530337	+	0.998593i	$0.483111\pi$
$480$	0	0
$481$	−4.24161e61	−0.0395477
$482$	0	0
$483$	−1.02608e62	−0.0867811
$484$	0	0
$485$	−4.29244e62	−0.329438
$486$	0	0
$487$	2.41720e61	0.0168415	0.00842076	−	0.999965i	$-0.497320\pi$
$487$	2.41720e61	0.0168415	0.00842076	+	0.999965i	$0.497320\pi$
$488$	0	0
$489$	−2.20355e62	−0.139433
$490$	0	0
$491$	1.48334e63	0.852760	0.426380	−	0.904544i	$-0.359789\pi$
$491$	1.48334e63	0.852760	0.426380	+	0.904544i	$0.359789\pi$
$492$	0	0
$493$	2.36554e62	0.123603
$494$	0	0
$495$	3.70394e62	0.175971
$496$	0	0
$497$	2.95862e62	0.127854
$498$	0	0
$499$	−3.58476e63	−1.40960	−0.704802	−	0.709404i	$-0.748964\pi$
$499$	−3.58476e63	−1.40960	−0.704802	+	0.709404i	$0.748964\pi$
$500$	0	0
$501$	−1.73558e63	−0.621238
$502$	0	0
$503$	−3.59373e63	−1.17138	−0.585691	−	0.810535i	$-0.699177\pi$
$503$	−3.59373e63	−1.17138	−0.585691	+	0.810535i	$0.699177\pi$
$504$	0	0
$505$	−1.06768e63	−0.317024
$506$	0	0
$507$	2.88057e63	0.779457
$508$	0	0
$509$	1.51124e63	0.372790	0.186395	−	0.982475i	$-0.440320\pi$
$509$	1.51124e63	0.372790	0.186395	+	0.982475i	$0.440320\pi$
$510$	0	0
$511$	−1.27246e63	−0.286255
$512$	0	0
$513$	−4.71241e63	−0.967130
$514$	0	0
$515$	1.16191e63	0.217622
$516$	0	0
$517$	4.29868e63	0.735034
$518$	0	0
$519$	−1.16574e63	−0.182041
$520$	0	0
$521$	−5.89563e63	−0.841093	−0.420546	−	0.907271i	$-0.638162\pi$
$521$	−5.89563e63	−0.841093	−0.420546	+	0.907271i	$0.638162\pi$
$522$	0	0
$523$	−1.19071e64	−1.55245	−0.776224	−	0.630457i	$-0.782868\pi$
$523$	−1.19071e64	−1.55245	−0.776224	+	0.630457i	$0.782868\pi$
$524$	0	0
$525$	−1.14564e63	−0.136554
$526$	0	0
$527$	−6.15444e63	−0.670871
$528$	0	0
$529$	−7.34706e63	−0.732665
$530$	0	0
$531$	3.87767e63	0.353875
$532$	0	0
$533$	1.84952e63	0.154515
$534$	0	0
$535$	2.29264e63	0.175397
$536$	0	0
$537$	−3.63815e63	−0.254968
$538$	0	0
$539$	−1.59914e64	−1.02695
$540$	0	0
$541$	−1.24707e64	−0.734101	−0.367051	−	0.930201i	$-0.619632\pi$
$541$	−1.24707e64	−0.734101	−0.367051	+	0.930201i	$0.619632\pi$
$542$	0	0
$543$	9.36109e63	0.505278
$544$	0	0
$545$	7.18730e63	0.355835
$546$	0	0
$547$	3.38744e64	1.53876	0.769378	−	0.638794i	$-0.220566\pi$
$547$	3.38744e64	1.53876	0.769378	+	0.638794i	$0.220566\pi$
$548$	0	0
$549$	−4.41087e63	−0.183897
$550$	0	0
$551$	6.90560e63	0.264325
$552$	0	0
$553$	−8.41628e63	−0.295856
$554$	0	0
$555$	−3.97686e63	−0.128427
$556$	0	0
$557$	4.80284e64	1.42528	0.712641	−	0.701528i	$-0.247499\pi$
$557$	4.80284e64	1.42528	0.712641	+	0.701528i	$0.247499\pi$
$558$	0	0
$559$	−2.49367e63	−0.0680240
$560$	0	0
$561$	1.40664e64	0.352822
$562$	0	0
$563$	−3.05103e64	−0.703882	−0.351941	−	0.936022i	$-0.614478\pi$
$563$	−3.05103e64	−0.703882	−0.351941	+	0.936022i	$0.614478\pi$
$564$	0	0
$565$	2.50463e64	0.531624
$566$	0	0
$567$	5.20647e63	0.101705
$568$	0	0
$569$	7.49437e64	1.34771	0.673856	−	0.738863i	$-0.264637\pi$
$569$	7.49437e64	1.34771	0.673856	+	0.738863i	$0.264637\pi$
$570$	0	0
$571$	5.47320e64	0.906344	0.453172	−	0.891423i	$-0.350292\pi$
$571$	5.47320e64	0.906344	0.453172	+	0.891423i	$0.350292\pi$
$572$	0	0
$573$	2.76828e64	0.422257
$574$	0	0
$575$	2.99317e64	0.420666
$576$	0	0
$577$	−4.19523e63	−0.0543406	−0.0271703	−	0.999631i	$-0.508650\pi$
$577$	−4.19523e63	−0.0543406	−0.0271703	+	0.999631i	$0.508650\pi$
$578$	0	0
$579$	5.88936e64	0.703270
$580$	0	0
$581$	−1.58416e64	−0.174446
$582$	0	0
$583$	1.49682e65	1.52040
$584$	0	0
$585$	1.82884e63	0.0171401
$586$	0	0
$587$	−3.17978e64	−0.275043	−0.137522	−	0.990499i	$-0.543914\pi$
$587$	−3.17978e64	−0.275043	−0.137522	+	0.990499i	$0.543914\pi$
$588$	0	0
$589$	−1.79663e65	−1.43466
$590$	0	0
$591$	−2.94294e64	−0.217008
$592$	0	0
$593$	4.66511e64	0.317744	0.158872	−	0.987299i	$-0.449214\pi$
$593$	4.66511e64	0.317744	0.158872	+	0.987299i	$0.449214\pi$
$594$	0	0
$595$	−6.08061e63	−0.0382647
$596$	0	0
$597$	1.62301e65	0.943896
$598$	0	0
$599$	−2.50137e65	−1.34477	−0.672384	−	0.740202i	$-0.734730\pi$
$599$	−2.50137e65	−1.34477	−0.672384	+	0.740202i	$0.734730\pi$
$600$	0	0
$601$	4.33262e64	0.215377	0.107689	−	0.994185i	$-0.465655\pi$
$601$	4.33262e64	0.215377	0.107689	+	0.994185i	$0.465655\pi$
$602$	0	0
$603$	4.68335e64	0.215328
$604$	0	0
$605$	−1.59583e64	−0.0678789
$606$	0	0
$607$	2.09462e64	0.0824464	0.0412232	−	0.999150i	$-0.486875\pi$
$607$	2.09462e64	0.0824464	0.0412232	+	0.999150i	$0.486875\pi$
$608$	0	0
$609$	−1.36827e64	−0.0498499
$610$	0	0
$611$	2.12251e64	0.0715946
$612$	0	0
$613$	−4.02606e65	−1.25765	−0.628824	−	0.777548i	$-0.716463\pi$
$613$	−4.02606e65	−1.25765	−0.628824	+	0.777548i	$0.716463\pi$
$614$	0	0
$615$	1.73408e65	0.501768
$616$	0	0
$617$	4.25894e65	1.14183	0.570914	−	0.821010i	$-0.306589\pi$
$617$	4.25894e65	1.14183	0.570914	+	0.821010i	$0.306589\pi$
$618$	0	0
$619$	−3.05847e65	−0.759931	−0.379965	−	0.925001i	$-0.624064\pi$
$619$	−3.05847e65	−0.759931	−0.379965	+	0.925001i	$0.624064\pi$
$620$	0	0
$621$	−2.43947e65	−0.561878
$622$	0	0
$623$	−1.64491e65	−0.351296
$624$	0	0
$625$	2.40084e65	0.475535
$626$	0	0
$627$	4.10632e65	0.754511
$628$	0	0
$629$	9.21280e64	0.157073
$630$	0	0
$631$	−6.21302e65	−0.983134	−0.491567	−	0.870840i	$-0.663576\pi$
$631$	−6.21302e65	−0.983134	−0.491567	+	0.870840i	$0.663576\pi$
$632$	0	0
$633$	−6.39398e65	−0.939257
$634$	0	0
$635$	−1.64141e65	−0.223889
$636$	0	0
$637$	−7.89586e64	−0.100028
$638$	0	0
$639$	1.93277e65	0.227463
$640$	0	0
$641$	−6.05212e65	−0.661826	−0.330913	−	0.943661i	$-0.607357\pi$
$641$	−6.05212e65	−0.661826	−0.330913	+	0.943661i	$0.607357\pi$
$642$	0	0
$643$	−8.65041e65	−0.879183	−0.439591	−	0.898198i	$-0.644877\pi$
$643$	−8.65041e65	−0.879183	−0.439591	+	0.898198i	$0.644877\pi$
$644$	0	0
$645$	−2.33802e65	−0.220900
$646$	0	0
$647$	−2.75625e65	−0.242141	−0.121070	−	0.992644i	$-0.538633\pi$
$647$	−2.75625e65	−0.242141	−0.121070	+	0.992644i	$0.538633\pi$
$648$	0	0
$649$	−1.22971e66	−1.00474
$650$	0	0
$651$	3.55982e65	0.270567
$652$	0	0
$653$	−1.27020e66	−0.898282	−0.449141	−	0.893461i	$-0.648270\pi$
$653$	−1.27020e66	−0.898282	−0.449141	+	0.893461i	$0.648270\pi$
$654$	0	0
$655$	−8.38828e65	−0.552081
$656$	0	0
$657$	−8.31254e65	−0.509270
$658$	0	0
$659$	−1.08832e66	−0.620794	−0.310397	−	0.950607i	$-0.600462\pi$
$659$	−1.08832e66	−0.620794	−0.310397	+	0.950607i	$0.600462\pi$
$660$	0	0
$661$	1.19166e66	0.633020	0.316510	−	0.948589i	$-0.397489\pi$
$661$	1.19166e66	0.633020	0.316510	+	0.948589i	$0.397489\pi$
$662$	0	0
$663$	6.94538e64	0.0343659
$664$	0	0
$665$	−1.77508e65	−0.0818293
$666$	0	0
$667$	3.57481e65	0.153566
$668$	0	0
$669$	3.59689e66	1.44017
$670$	0	0
$671$	1.39880e66	0.522127
$672$	0	0
$673$	2.35019e66	0.817993	0.408997	−	0.912536i	$-0.365879\pi$
$673$	2.35019e66	0.817993	0.408997	+	0.912536i	$0.365879\pi$
$674$	0	0
$675$	−2.72371e66	−0.884144
$676$	0	0
$677$	−1.35609e66	−0.410633	−0.205316	−	0.978696i	$-0.565822\pi$
$677$	−1.35609e66	−0.410633	−0.205316	+	0.978696i	$0.565822\pi$
$678$	0	0
$679$	−5.75169e65	−0.162501
$680$	0	0
$681$	1.80528e66	0.475978
$682$	0	0
$683$	2.30165e66	0.566438	0.283219	−	0.959055i	$-0.408598\pi$
$683$	2.30165e66	0.566438	0.283219	+	0.959055i	$0.408598\pi$
$684$	0	0
$685$	−3.45135e65	−0.0792976
$686$	0	0
$687$	5.26429e66	1.12942
$688$	0	0
$689$	7.39066e65	0.148092
$690$	0	0
$691$	8.06352e66	1.50935	0.754676	−	0.656098i	$-0.227794\pi$
$691$	8.06352e66	1.50935	0.754676	+	0.656098i	$0.227794\pi$
$692$	0	0
$693$	4.96312e65	0.0868009
$694$	0	0
$695$	1.44803e66	0.236665
$696$	0	0
$697$	−4.01717e66	−0.613691
$698$	0	0
$699$	1.56649e66	0.223725
$700$	0	0
$701$	5.85991e66	0.782560	0.391280	−	0.920272i	$-0.372032\pi$
$701$	5.85991e66	0.782560	0.391280	+	0.920272i	$0.372032\pi$
$702$	0	0
$703$	2.68944e66	0.335901
$704$	0	0
$705$	1.99002e66	0.232495
$706$	0	0
$707$	−1.43064e66	−0.156378
$708$	0	0
$709$	1.17722e67	1.20413	0.602064	−	0.798448i	$-0.294345\pi$
$709$	1.17722e67	1.20413	0.602064	+	0.798448i	$0.294345\pi$
$710$	0	0
$711$	−5.49808e66	−0.526351
$712$	0	0
$713$	−9.30059e66	−0.833502
$714$	0	0
$715$	−5.79974e65	−0.0486649
$716$	0	0
$717$	8.15970e66	0.641171
$718$	0	0
$719$	−1.34473e67	−0.989709	−0.494854	−	0.868976i	$-0.664779\pi$
$719$	−1.34473e67	−0.989709	−0.494854	+	0.868976i	$0.664779\pi$
$720$	0	0
$721$	1.55691e66	0.107346
$722$	0	0
$723$	1.43104e67	0.924491
$724$	0	0
$725$	3.99135e66	0.241644
$726$	0	0
$727$	6.52974e66	0.370542	0.185271	−	0.982687i	$-0.440684\pi$
$727$	6.52974e66	0.370542	0.185271	+	0.982687i	$0.440684\pi$
$728$	0	0
$729$	1.95002e67	1.03739
$730$	0	0
$731$	5.41627e66	0.270173
$732$	0	0
$733$	−4.16163e67	−1.94679	−0.973397	−	0.229127i	$-0.926413\pi$
$733$	−4.16163e67	−1.94679	−0.973397	+	0.229127i	$0.926413\pi$
$734$	0	0
$735$	−7.40301e66	−0.324830
$736$	0	0
$737$	−1.48521e67	−0.611367
$738$	0	0
$739$	−1.83231e67	−0.707709	−0.353854	−	0.935301i	$-0.615129\pi$
$739$	−1.83231e67	−0.707709	−0.353854	+	0.935301i	$0.615129\pi$
$740$	0	0
$741$	2.02752e66	0.0734917
$742$	0	0
$743$	2.26817e67	0.771681	0.385840	−	0.922566i	$-0.373912\pi$
$743$	2.26817e67	0.771681	0.385840	+	0.922566i	$0.373912\pi$
$744$	0	0
$745$	1.84827e67	0.590327
$746$	0	0
$747$	−1.03488e67	−0.310353
$748$	0	0
$749$	3.07205e66	0.0865178
$750$	0	0
$751$	−3.95970e67	−1.04743	−0.523713	−	0.851895i	$-0.675454\pi$
$751$	−3.95970e67	−1.04743	−0.523713	+	0.851895i	$0.675454\pi$
$752$	0	0
$753$	2.33836e67	0.581072
$754$	0	0
$755$	3.17945e65	0.00742333
$756$	0	0
$757$	−3.90670e67	−0.857149	−0.428575	−	0.903506i	$-0.640984\pi$
$757$	−3.90670e67	−0.857149	−0.428575	+	0.903506i	$0.640984\pi$
$758$	0	0
$759$	2.12571e67	0.438352
$760$	0	0
$761$	−8.64597e67	−1.67600	−0.838001	−	0.545669i	$-0.816276\pi$
$761$	−8.64597e67	−1.67600	−0.838001	+	0.545669i	$0.816276\pi$
$762$	0	0
$763$	9.63069e66	0.175522
$764$	0	0
$765$	−3.97226e66	−0.0680760
$766$	0	0
$767$	−6.07177e66	−0.0978643
$768$	0	0
$769$	8.45845e67	1.28240	0.641198	−	0.767376i	$-0.278438\pi$
$769$	8.45845e67	1.28240	0.641198	+	0.767376i	$0.278438\pi$
$770$	0	0
$771$	4.45179e67	0.634975
$772$	0	0
$773$	−9.32204e67	−1.25110	−0.625551	−	0.780183i	$-0.715126\pi$
$773$	−9.32204e67	−1.25110	−0.625551	+	0.780183i	$0.715126\pi$
$774$	0	0
$775$	−1.03843e68	−1.31156
$776$	0	0
$777$	−5.32883e66	−0.0633487
$778$	0	0
$779$	−1.17271e68	−1.31238
$780$	0	0
$781$	−6.12931e67	−0.645821
$782$	0	0
$783$	−3.25299e67	−0.322762
$784$	0	0
$785$	−3.75262e67	−0.350670
$786$	0	0
$787$	−1.40543e68	−1.23709	−0.618547	−	0.785748i	$-0.712278\pi$
$787$	−1.40543e68	−1.23709	−0.618547	+	0.785748i	$0.712278\pi$
$788$	0	0
$789$	1.61324e68	1.33780
$790$	0	0
$791$	3.35610e67	0.262233
$792$	0	0
$793$	6.90668e66	0.0508568
$794$	0	0
$795$	6.92935e67	0.480911
$796$	0	0
$797$	−6.41377e67	−0.419607	−0.209804	−	0.977744i	$-0.567282\pi$
$797$	−6.41377e67	−0.419607	−0.209804	+	0.977744i	$0.567282\pi$
$798$	0	0
$799$	−4.61009e67	−0.284355
$800$	0	0
$801$	−1.07457e68	−0.624984
$802$	0	0
$803$	2.63612e68	1.44594
$804$	0	0
$805$	−9.18902e66	−0.0475407
$806$	0	0
$807$	3.34892e66	0.0163447
$808$	0	0
$809$	7.54417e67	0.347393	0.173696	−	0.984799i	$-0.444429\pi$
$809$	7.54417e67	0.347393	0.173696	+	0.984799i	$0.444429\pi$
$810$	0	0
$811$	2.25843e68	0.981332	0.490666	−	0.871348i	$-0.336753\pi$
$811$	2.25843e68	0.981332	0.490666	+	0.871348i	$0.336753\pi$
$812$	0	0
$813$	−1.83116e68	−0.750928
$814$	0	0
$815$	−1.97337e67	−0.0763845
$816$	0	0
$817$	1.58114e68	0.577766
$818$	0	0
$819$	2.45058e66	0.00845467
$820$	0	0
$821$	2.57889e68	0.840174	0.420087	−	0.907484i	$-0.362000\pi$
$821$	2.57889e68	0.840174	0.420087	+	0.907484i	$0.362000\pi$
$822$	0	0
$823$	8.83986e67	0.271988	0.135994	−	0.990710i	$-0.456577\pi$
$823$	8.83986e67	0.271988	0.135994	+	0.990710i	$0.456577\pi$
$824$	0	0
$825$	2.37340e68	0.689768
$826$	0	0
$827$	2.82593e68	0.775856	0.387928	−	0.921690i	$-0.373191\pi$
$827$	2.82593e68	0.775856	0.387928	+	0.921690i	$0.373191\pi$
$828$	0	0
$829$	2.91606e68	0.756423	0.378211	−	0.925719i	$-0.376539\pi$
$829$	2.91606e68	0.756423	0.378211	+	0.925719i	$0.376539\pi$
$830$	0	0
$831$	5.09321e68	1.24844
$832$	0	0
$833$	1.71499e68	0.397286
$834$	0	0
$835$	−1.55429e68	−0.340329
$836$	0	0
$837$	8.46332e68	1.75183
$838$	0	0
$839$	−9.42111e68	−1.84372	−0.921862	−	0.387518i	$-0.873333\pi$
$839$	−9.42111e68	−1.84372	−0.921862	+	0.387518i	$0.873333\pi$
$840$	0	0
$841$	−4.92719e68	−0.911786
$842$	0	0
$843$	5.49616e68	0.961858
$844$	0	0
$845$	2.57968e68	0.427005
$846$	0	0
$847$	−2.13835e67	−0.0334825
$848$	0	0
$849$	−1.23058e68	−0.182296
$850$	0	0
$851$	1.39224e68	0.195150
$852$	0	0
$853$	−1.22430e69	−1.62400	−0.811999	−	0.583659i	$-0.801620\pi$
$853$	−1.22430e69	−1.62400	−0.811999	+	0.583659i	$0.801620\pi$
$854$	0	0
$855$	−1.15960e68	−0.145581
$856$	0	0
$857$	−4.45209e68	−0.529071	−0.264536	−	0.964376i	$-0.585219\pi$
$857$	−4.45209e68	−0.529071	−0.264536	+	0.964376i	$0.585219\pi$
$858$	0	0
$859$	2.77149e68	0.311799	0.155899	−	0.987773i	$-0.450172\pi$
$859$	2.77149e68	0.311799	0.155899	+	0.987773i	$0.450172\pi$
$860$	0	0
$861$	2.32359e68	0.247506
$862$	0	0
$863$	−2.86475e68	−0.288957	−0.144479	−	0.989508i	$-0.546150\pi$
$863$	−2.86475e68	−0.288957	−0.144479	+	0.989508i	$0.546150\pi$
$864$	0	0
$865$	−1.04397e68	−0.0997265
$866$	0	0
$867$	7.20180e68	0.651617
$868$	0	0
$869$	1.74358e69	1.49444
$870$	0	0
$871$	−7.33333e67	−0.0595490
$872$	0	0
$873$	−3.75739e68	−0.289102
$874$	0	0
$875$	−2.28701e68	−0.166755
$876$	0	0
$877$	−6.75109e68	−0.466534	−0.233267	−	0.972413i	$-0.574942\pi$
$877$	−6.75109e68	−0.466534	−0.233267	+	0.972413i	$0.574942\pi$
$878$	0	0
$879$	−1.80052e69	−1.17939
$880$	0	0
$881$	−2.56812e67	−0.0159471	−0.00797354	−	0.999968i	$-0.502538\pi$
$881$	−2.56812e67	−0.0159471	−0.00797354	+	0.999968i	$0.502538\pi$
$882$	0	0
$883$	1.81395e69	1.06794	0.533972	−	0.845502i	$-0.320699\pi$
$883$	1.81395e69	1.06794	0.533972	+	0.845502i	$0.320699\pi$
$884$	0	0
$885$	−5.69278e68	−0.317803
$886$	0	0
$887$	−1.74355e68	−0.0923059	−0.0461529	−	0.998934i	$-0.514696\pi$
$887$	−1.74355e68	−0.0923059	−0.0461529	+	0.998934i	$0.514696\pi$
$888$	0	0
$889$	−2.19942e68	−0.110437
$890$	0	0
$891$	−1.07861e69	−0.513735
$892$	0	0
$893$	−1.34580e69	−0.608094
$894$	0	0
$895$	−3.25812e68	−0.139677
$896$	0	0
$897$	1.04959e68	0.0426968
$898$	0	0
$899$	−1.24022e69	−0.478791
$900$	0	0
$901$	−1.60526e69	−0.588182
$902$	0	0
$903$	−3.13286e68	−0.108963
$904$	0	0
$905$	8.38326e68	0.276803
$906$	0	0
$907$	7.97555e68	0.250028	0.125014	−	0.992155i	$-0.460102\pi$
$907$	7.97555e68	0.250028	0.125014	+	0.992155i	$0.460102\pi$
$908$	0	0
$909$	−9.34591e68	−0.278208
$910$	0	0
$911$	−6.39008e69	−1.80644	−0.903222	−	0.429173i	$-0.858805\pi$
$911$	−6.39008e69	−1.80644	−0.903222	+	0.429173i	$0.858805\pi$
$912$	0	0
$913$	3.28187e69	0.881167
$914$	0	0
$915$	6.47557e68	0.165151
$916$	0	0
$917$	−1.12400e69	−0.272324
$918$	0	0
$919$	−3.86963e69	−0.890747	−0.445373	−	0.895345i	$-0.646929\pi$
$919$	−3.86963e69	−0.890747	−0.445373	+	0.895345i	$0.646929\pi$
$920$	0	0
$921$	−6.04431e69	−1.32204
$922$	0	0
$923$	−3.02639e68	−0.0629050
$924$	0	0
$925$	1.55446e69	0.307079
$926$	0	0
$927$	1.01708e69	0.190977
$928$	0	0
$929$	−4.53114e69	−0.808795	−0.404398	−	0.914583i	$-0.632519\pi$
$929$	−4.53114e69	−0.808795	−0.404398	+	0.914583i	$0.632519\pi$
$930$	0	0
$931$	5.00646e69	0.849597
$932$	0	0
$933$	−3.42196e69	−0.552149
$934$	0	0
$935$	1.25971e69	0.193284
$936$	0	0
$937$	2.69082e69	0.392648	0.196324	−	0.980539i	$-0.437100\pi$
$937$	2.69082e69	0.392648	0.196324	+	0.980539i	$0.437100\pi$
$938$	0	0
$939$	7.23833e69	1.00461
$940$	0	0
$941$	−3.38983e69	−0.447530	−0.223765	−	0.974643i	$-0.571835\pi$
$941$	−3.38983e69	−0.447530	−0.223765	+	0.974643i	$0.571835\pi$
$942$	0	0
$943$	−6.07075e69	−0.762460
$944$	0	0
$945$	8.36179e68	0.0999199
$946$	0	0
$947$	1.24658e70	1.41742	0.708708	−	0.705502i	$-0.249279\pi$
$947$	1.24658e70	1.41742	0.708708	+	0.705502i	$0.249279\pi$
$948$	0	0
$949$	1.30160e69	0.140839
$950$	0	0
$951$	−1.28473e70	−1.32303
$952$	0	0
$953$	1.47472e70	1.44553	0.722766	−	0.691093i	$-0.242870\pi$
$953$	1.47472e70	1.44553	0.722766	+	0.691093i	$0.242870\pi$
$954$	0	0
$955$	2.47912e69	0.231323
$956$	0	0
$957$	2.83461e69	0.251804
$958$	0	0
$959$	−4.62466e68	−0.0391150
$960$	0	0
$961$	1.98503e70	1.59871
$962$	0	0
$963$	2.00687e69	0.153922
$964$	0	0
$965$	5.27418e69	0.385268
$966$	0	0
$967$	−9.72062e69	−0.676351	−0.338175	−	0.941083i	$-0.609810\pi$
$967$	−9.72062e69	−0.676351	−0.338175	+	0.941083i	$0.609810\pi$
$968$	0	0
$969$	−4.40379e69	−0.291889
$970$	0	0
$971$	2.15826e70	1.36286	0.681431	−	0.731883i	$-0.261358\pi$
$971$	2.15826e70	1.36286	0.681431	+	0.731883i	$0.261358\pi$
$972$	0	0
$973$	1.94030e69	0.116739
$974$	0	0
$975$	1.17188e69	0.0671856
$976$	0	0
$977$	−1.11352e69	−0.0608384	−0.0304192	−	0.999537i	$-0.509684\pi$
$977$	−1.11352e69	−0.0608384	−0.0304192	+	0.999537i	$0.509684\pi$
$978$	0	0
$979$	3.40772e70	1.77448
$980$	0	0
$981$	6.29141e69	0.312267
$982$	0	0
$983$	3.46672e69	0.164025	0.0820127	−	0.996631i	$-0.473865\pi$
$983$	3.46672e69	0.164025	0.0820127	+	0.996631i	$0.473865\pi$
$984$	0	0
$985$	−2.63553e69	−0.118882
$986$	0	0
$987$	2.66655e69	0.114682
$988$	0	0
$989$	8.18507e69	0.335668
$990$	0	0
$991$	−1.50226e70	−0.587510	−0.293755	−	0.955881i	$-0.594905\pi$
$991$	−1.50226e70	−0.587510	−0.293755	+	0.955881i	$0.594905\pi$
$992$	0	0
$993$	−2.39702e69	−0.0894055
$994$	0	0
$995$	1.45347e70	0.517089
$996$	0	0
$997$	−9.94591e69	−0.337527	−0.168763	−	0.985657i	$-0.553977\pi$
$997$	−9.94591e69	−0.337527	−0.168763	+	0.985657i	$0.553977\pi$
$998$	0	0
$999$	−1.26690e70	−0.410162

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.48.a.d.1.2		4
4.3	odd	2		1.48.a.a.1.4	✓	4
12.11	even	2		9.48.a.c.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.48.a.a.1.4	✓	4	4.3	odd	2
9.48.a.c.1.1		4	12.11	even	2
16.48.a.d.1.2		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:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 48 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-1.28510e11 q^{3} -1.15086e16 q^{5} -1.54211e19 q^{7} -1.00741e22 q^{9} +O(q^{10})\)	\(q-1.28510e11 q^{3} -1.15086e16 q^{5} -1.54211e19 q^{7} -1.00741e22 q^{9} +3.19474e24 q^{11} +1.57743e25 q^{13} +1.47897e27 q^{15} -3.42618e28 q^{17} -1.00019e30 q^{19} +1.98176e30 q^{21} -5.17765e31 q^{23} -5.78095e32 q^{25} +4.71154e33 q^{27} -6.90432e33 q^{29} +1.79630e35 q^{31} -4.10556e35 q^{33} +1.77475e35 q^{35} -2.68894e36 q^{37} -2.02715e36 q^{39} +1.17249e38 q^{41} -1.58085e38 q^{43} +1.15938e38 q^{45} +1.34555e39 q^{47} -5.00553e39 q^{49} +4.40298e39 q^{51} +4.68526e40 q^{53} -3.67670e40 q^{55} +1.28534e41 q^{57} -3.84916e41 q^{59} +4.37844e41 q^{61} +1.55353e41 q^{63} -1.81540e41 q^{65} -4.64892e42 q^{67} +6.65378e42 q^{69} -1.91856e43 q^{71} +8.25143e43 q^{73} +7.42908e43 q^{75} -4.92663e43 q^{77} +5.45765e44 q^{79} -3.37621e44 q^{81} +1.02727e45 q^{83} +3.94306e44 q^{85} +8.87272e44 q^{87} +1.06667e46 q^{89} -2.43256e44 q^{91} -2.30842e46 q^{93} +1.15107e46 q^{95} +3.72976e46 q^{97} -3.21841e46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 17\!\cdots\!72 q^{9}+O(q^{10}) \) 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9	\( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 53\!\cdots\!16 q^{99}+O(q^{100}) \) 4 * q - 38461494960 * q^3 - 31114680242272200 * q^5 + 39169218725888423200 * q^7 - 17071972417358142200172 * q^9 + 1902165515717740404742512 * q^11 + 126959149195199610466858520 * q^13 - 12437812896841241272880733600 * q^15 + 210426285758836687099688769480 * q^17 + 1059119811802437301721159140240 * q^19 - 26054906182888686128940663165312 * q^21 - 137531418344493639994288859179680 * q^23 + 1009909900344396506188154274977500 * q^25 + 7345952425324536073931587214783520 * q^27 - 22743132490172744184500850941726760 * q^29 - 75786540814706434901630981303934848 * q^31 + 263381387470144952341649359926294720 * q^33 + 1313148822925052387484546800067979200 * q^35 - 11291457045120157869429951769632191560 * q^37 - 39294375338503805226442587829551714336 * q^39 + 130658126291089294119301677048020391528 * q^41 + 445357287738427943170332944488133090800 * q^43 + 86022747271219179966948746072020632600 * q^45 - 2031912465958423919194866366997063590720 * q^47 + 9153745744156628626499502447643951451172 * q^49 + 18534366050002170683962116343746386369952 * q^51 + 29171719896619533806321793123448457437560 * q^53 - 192172580604633896452606078657416872901600 * q^55 + 558743644452458098451223285784560964733760 * q^57 - 478242858216320686410790860443879577596880 * q^59 + 626683213833907852009669690320782900857688 * q^61 - 5811636950821577036406237472810763671724640 * q^63 + 12996735432890074297558542478104126872331600 * q^65 - 18701411419127220668869505853739529194216880 * q^67 - 2096956664577882726873494616728974107851904 * q^69 - 22196594617762006008028198712153461686540768 * q^71 + 104911201720506650033573412943840886878191080 * q^73 - 32388703324174028323482311940739905891330000 * q^75 - 263490744376507692433815778071300370399209600 * q^77 + 1327995310109394234162785254414226288730352960 * q^79 - 1061448205755511803861508510986654404170081116 * q^81 + 1439130627532364606968733068855111784504073360 * q^83 - 1001895286410663502447652611446390227458237200 * q^85 + 4326251093943206485496474339443178030108648160 * q^87 - 7960590522674945421108320896849737731421281880 * q^89 - 5325194479227227497182497798631911205603381568 * q^91 + 10728611999875738337226031496188630272675064320 * q^93 - 75421152415106939771624998522249518123547668000 * q^95 + 95066248961258904822565865801364599382905757320 * q^97 - 530900415919166351015781326273177630089784016 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

Properties

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