LMFDB - Embedded newform 256.12.a.j.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [256,12,Mod(1,256)] 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("256.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(256, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	256.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,-3256,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$196.695854223$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 1640x^{4} + 512875x^{2} - 42187500 $ x^6 - 1640x^4 + 512875x^2 - 42187500
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{41}\cdot 3\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 128)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$35.4873$ of defining polynomial
Character	$\chi$	$=$	256.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+580.453 q^{3} +6948.91 q^{5} +45098.0 q^{7} +159779. q^{9} +289556. q^{11} +1.75734e6 q^{13} +4.03352e6 q^{15} +1.03003e7 q^{17} +1.82205e7 q^{19} +2.61773e7 q^{21} +7.69390e6 q^{23} -540832. q^{25} -1.00813e7 q^{27} -5.73199e7 q^{29} -1.12000e8 q^{31} +1.68073e8 q^{33} +3.13382e8 q^{35} +3.18512e8 q^{37} +1.02005e9 q^{39} +4.00743e7 q^{41} -2.22418e8 q^{43} +1.11029e9 q^{45} -1.84115e9 q^{47} +5.65042e7 q^{49} +5.97887e9 q^{51} -2.89667e9 q^{53} +2.01209e9 q^{55} +1.05762e10 q^{57} -5.07972e9 q^{59} -8.50164e9 q^{61} +7.20572e9 q^{63} +1.22116e10 q^{65} -1.69897e10 q^{67} +4.46595e9 q^{69} +2.80844e10 q^{71} -1.17976e10 q^{73} -3.13928e8 q^{75} +1.30584e10 q^{77} +5.01575e10 q^{79} -3.41561e10 q^{81} -3.23224e10 q^{83} +7.15761e10 q^{85} -3.32715e10 q^{87} +7.05746e9 q^{89} +7.92523e10 q^{91} -6.50105e10 q^{93} +1.26613e11 q^{95} -6.57214e10 q^{97} +4.62649e10 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3256 q^{5} + 158366 q^{9} + 1974152 q^{13} + 5650868 q^{17} + 5622272 q^{21} + 50201138 q^{25} + 78510600 q^{29} - 185111168 q^{33} + 187521320 q^{37} - 729393436 q^{41} - 94716312 q^{45} - 1212515210 q^{49}+ \cdots - 198387795308 q^{97}+O(q^{100}) $ 6 * q - 3256 * q^5 + 158366 * q^9 + 1974152 * q^13 + 5650868 * q^17 + 5622272 * q^21 + 50201138 * q^25 + 78510600 * q^29 - 185111168 * q^33 + 187521320 * q^37 - 729393436 * q^41 - 94716312 * q^45 - 1212515210 * q^49 + 1345806024 * q^53 + 4211057792 * q^57 - 13682701080 * q^61 - 9028891040 * q^65 + 12746864128 * q^69 + 20889754044 * q^73 - 6919916032 * q^77 - 106544843242 * q^81 + 149487054448 * q^85 + 165819876316 * q^89 - 263298174976 * q^93 - 198387795308 * q^97

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
$3$	580.453	1.37911	0.689557	−	0.724231i	$-0.257805\pi$
$3$	580.453	1.37911	0.689557	+	0.724231i	$0.257805\pi$
$4$	0	0
$5$	6948.91	0.994446	0.497223	−	0.867623i	$-0.334353\pi$
$5$	6948.91	0.994446	0.497223	+	0.867623i	$0.334353\pi$
$6$	0	0
$7$	45098.0	1.01419	0.507094	−	0.861891i	$-0.330720\pi$
$7$	45098.0	1.01419	0.507094	+	0.861891i	$0.330720\pi$
$8$	0	0
$9$	159779.	0.901957
$10$	0	0
$11$	289556.	0.542091	0.271046	−	0.962567i	$-0.412631\pi$
$11$	289556.	0.542091	0.271046	+	0.962567i	$0.412631\pi$
$12$	0	0
$13$	1.75734e6	1.31270	0.656351	−	0.754456i	$-0.272099\pi$
$13$	1.75734e6	1.31270	0.656351	+	0.754456i	$0.272099\pi$
$14$	0	0
$15$	4.03352e6	1.37146
$16$	0	0
$17$	1.03003e7	1.75947	0.879737	−	0.475461i	$-0.157719\pi$
$17$	1.03003e7	1.75947	0.879737	+	0.475461i	$0.157719\pi$
$18$	0	0
$19$	1.82205e7	1.68817	0.844084	−	0.536211i	$-0.180145\pi$
$19$	1.82205e7	1.68817	0.844084	+	0.536211i	$0.180145\pi$
$20$	0	0
$21$	2.61773e7	1.39868
$22$	0	0
$23$	7.69390e6	0.249255	0.124627	−	0.992204i	$-0.460226\pi$
$23$	7.69390e6	0.249255	0.124627	+	0.992204i	$0.460226\pi$
$24$	0	0
$25$	−540832.	−0.0110762
$26$	0	0
$27$	−1.00813e7	−0.135212
$28$	0	0
$29$	−5.73199e7	−0.518939	−0.259469	−	0.965751i	$-0.583548\pi$
$29$	−5.73199e7	−0.518939	−0.259469	+	0.965751i	$0.583548\pi$
$30$	0	0
$31$	−1.12000e8	−0.702630	−0.351315	−	0.936257i	$-0.614265\pi$
$31$	−1.12000e8	−0.702630	−0.351315	+	0.936257i	$0.614265\pi$
$32$	0	0
$33$	1.68073e8	0.747606
$34$	0	0
$35$	3.13382e8	1.00856
$36$	0	0
$37$	3.18512e8	0.755121	0.377560	−	0.925985i	$-0.376763\pi$
$37$	3.18512e8	0.755121	0.377560	+	0.925985i	$0.376763\pi$
$38$	0	0
$39$	1.02005e9	1.81037
$40$	0	0
$41$	4.00743e7	0.0540200	0.0270100	−	0.999635i	$-0.491401\pi$
$41$	4.00743e7	0.0540200	0.0270100	+	0.999635i	$0.491401\pi$
$42$	0	0
$43$	−2.22418e8	−0.230724	−0.115362	−	0.993323i	$-0.536803\pi$
$43$	−2.22418e8	−0.230724	−0.115362	+	0.993323i	$0.536803\pi$
$44$	0	0
$45$	1.11029e9	0.896948
$46$	0	0
$47$	−1.84115e9	−1.17098	−0.585491	−	0.810679i	$-0.699098\pi$
$47$	−1.84115e9	−1.17098	−0.585491	+	0.810679i	$0.699098\pi$
$48$	0	0
$49$	5.65042e7	0.0285761
$50$	0	0
$51$	5.97887e9	2.42652
$52$	0	0
$53$	−2.89667e9	−0.951440	−0.475720	−	0.879597i	$-0.657813\pi$
$53$	−2.89667e9	−0.951440	−0.475720	+	0.879597i	$0.657813\pi$
$54$	0	0
$55$	2.01209e9	0.539080
$56$	0	0
$57$	1.05762e10	2.32818
$58$	0	0
$59$	−5.07972e9	−0.925026	−0.462513	−	0.886612i	$-0.653052\pi$
$59$	−5.07972e9	−0.925026	−0.462513	+	0.886612i	$0.653052\pi$
$60$	0	0
$61$	−8.50164e9	−1.28881	−0.644404	−	0.764685i	$-0.722895\pi$
$61$	−8.50164e9	−1.28881	−0.644404	+	0.764685i	$0.722895\pi$
$62$	0	0
$63$	7.20572e9	0.914754
$64$	0	0
$65$	1.22116e10	1.30541
$66$	0	0
$67$	−1.69897e10	−1.53736	−0.768680	−	0.639634i	$-0.779086\pi$
$67$	−1.69897e10	−1.53736	−0.768680	+	0.639634i	$0.779086\pi$
$68$	0	0
$69$	4.46595e9	0.343751
$70$	0	0
$71$	2.80844e10	1.84733	0.923665	−	0.383201i	$-0.125178\pi$
$71$	2.80844e10	1.84733	0.923665	+	0.383201i	$0.125178\pi$
$72$	0	0
$73$	−1.17976e10	−0.666068	−0.333034	−	0.942915i	$-0.608072\pi$
$73$	−1.17976e10	−0.666068	−0.333034	+	0.942915i	$0.608072\pi$
$74$	0	0
$75$	−3.13928e8	−0.0152754
$76$	0	0
$77$	1.30584e10	0.549782
$78$	0	0
$79$	5.01575e10	1.83395	0.916973	−	0.398948i	$-0.130625\pi$
$79$	5.01575e10	1.83395	0.916973	+	0.398948i	$0.130625\pi$
$80$	0	0
$81$	−3.41561e10	−1.08843
$82$	0	0
$83$	−3.23224e10	−0.900688	−0.450344	−	0.892855i	$-0.648699\pi$
$83$	−3.23224e10	−0.900688	−0.450344	+	0.892855i	$0.648699\pi$
$84$	0	0
$85$	7.15761e10	1.74970
$86$	0	0
$87$	−3.32715e10	−0.715676
$88$	0	0
$89$	7.05746e9	0.133969	0.0669843	−	0.997754i	$-0.478662\pi$
$89$	7.05746e9	0.133969	0.0669843	+	0.997754i	$0.478662\pi$
$90$	0	0
$91$	7.92523e10	1.33133
$92$	0	0
$93$	−6.50105e10	−0.969008
$94$	0	0
$95$	1.26613e11	1.67879
$96$	0	0
$97$	−6.57214e10	−0.777074	−0.388537	−	0.921433i	$-0.627019\pi$
$97$	−6.57214e10	−0.777074	−0.388537	+	0.921433i	$0.627019\pi$
$98$	0	0
$99$	4.62649e10	0.488943
$100$	0	0
$101$	−7.41643e10	−0.702146	−0.351073	−	0.936348i	$-0.614183\pi$
$101$	−7.41643e10	−0.702146	−0.351073	+	0.936348i	$0.614183\pi$
$102$	0	0
$103$	1.57349e11	1.33739	0.668696	−	0.743536i	$-0.266853\pi$
$103$	1.57349e11	1.33739	0.668696	+	0.743536i	$0.266853\pi$
$104$	0	0
$105$	1.81904e11	1.39091
$106$	0	0
$107$	−6.47704e10	−0.446443	−0.223221	−	0.974768i	$-0.571657\pi$
$107$	−6.47704e10	−0.446443	−0.223221	+	0.974768i	$0.571657\pi$
$108$	0	0
$109$	−3.21051e10	−0.199861	−0.0999304	−	0.994994i	$-0.531862\pi$
$109$	−3.21051e10	−0.199861	−0.0999304	+	0.994994i	$0.531862\pi$
$110$	0	0
$111$	1.84881e11	1.04140
$112$	0	0
$113$	−3.59862e11	−1.83740	−0.918701	−	0.394954i	$-0.870761\pi$
$113$	−3.59862e11	−1.83740	−0.918701	+	0.394954i	$0.870761\pi$
$114$	0	0
$115$	5.34642e10	0.247870
$116$	0	0
$117$	2.80785e11	1.18400
$118$	0	0
$119$	4.64525e11	1.78444
$120$	0	0
$121$	−2.01469e11	−0.706137
$122$	0	0
$123$	2.32613e10	0.0744998
$124$	0	0
$125$	−3.43060e11	−1.00546
$126$	0	0
$127$	−6.32223e11	−1.69805	−0.849024	−	0.528355i	$-0.822809\pi$
$127$	−6.32223e11	−1.69805	−0.849024	+	0.528355i	$0.822809\pi$
$128$	0	0
$129$	−1.29103e11	−0.318195
$130$	0	0
$131$	3.83538e11	0.868593	0.434297	−	0.900770i	$-0.356997\pi$
$131$	3.83538e11	0.868593	0.434297	+	0.900770i	$0.356997\pi$
$132$	0	0
$133$	8.21709e11	1.71212
$134$	0	0
$135$	−7.00541e10	−0.134461
$136$	0	0
$137$	9.59331e11	1.69826	0.849132	−	0.528180i	$-0.177126\pi$
$137$	9.59331e11	1.69826	0.849132	+	0.528180i	$0.177126\pi$
$138$	0	0
$139$	−5.48647e11	−0.896832	−0.448416	−	0.893825i	$-0.648012\pi$
$139$	−5.48647e11	−0.896832	−0.448416	+	0.893825i	$0.648012\pi$
$140$	0	0
$141$	−1.06870e12	−1.61492
$142$	0	0
$143$	5.08846e11	0.711604
$144$	0	0
$145$	−3.98310e11	−0.516057
$146$	0	0
$147$	3.27981e10	0.0394097
$148$	0	0
$149$	9.63639e11	1.07495	0.537477	−	0.843278i	$-0.319377\pi$
$149$	9.63639e11	1.07495	0.537477	+	0.843278i	$0.319377\pi$
$150$	0	0
$151$	−9.05063e11	−0.938222	−0.469111	−	0.883139i	$-0.655426\pi$
$151$	−9.05063e11	−0.938222	−0.469111	+	0.883139i	$0.655426\pi$
$152$	0	0
$153$	1.64578e12	1.58697
$154$	0	0
$155$	−7.78274e11	−0.698728
$156$	0	0
$157$	1.53256e12	1.28224	0.641121	−	0.767440i	$-0.278470\pi$
$157$	1.53256e12	1.28224	0.641121	+	0.767440i	$0.278470\pi$
$158$	0	0
$159$	−1.68138e12	−1.31215
$160$	0	0
$161$	3.46980e11	0.252791
$162$	0	0
$163$	−2.46200e12	−1.67593	−0.837966	−	0.545722i	$-0.816255\pi$
$163$	−2.46200e12	−1.67593	−0.837966	+	0.545722i	$0.816255\pi$
$164$	0	0
$165$	1.16793e12	0.743454
$166$	0	0
$167$	−2.23163e12	−1.32948	−0.664741	−	0.747074i	$-0.731458\pi$
$167$	−2.23163e12	−1.32948	−0.664741	+	0.747074i	$0.731458\pi$
$168$	0	0
$169$	1.29607e12	0.723187
$170$	0	0
$171$	2.91126e12	1.52266
$172$	0	0
$173$	−8.86199e11	−0.434788	−0.217394	−	0.976084i	$-0.569756\pi$
$173$	−8.86199e11	−0.434788	−0.217394	+	0.976084i	$0.569756\pi$
$174$	0	0
$175$	−2.43904e10	−0.0112334
$176$	0	0
$177$	−2.94854e12	−1.27572
$178$	0	0
$179$	−8.75197e11	−0.355971	−0.177985	−	0.984033i	$-0.556958\pi$
$179$	−8.75197e11	−0.355971	−0.177985	+	0.984033i	$0.556958\pi$
$180$	0	0
$181$	3.55818e12	1.36143	0.680716	−	0.732547i	$-0.261669\pi$
$181$	3.55818e12	1.36143	0.680716	+	0.732547i	$0.261669\pi$
$182$	0	0
$183$	−4.93480e12	−1.77741
$184$	0	0
$185$	2.21331e12	0.750927
$186$	0	0
$187$	2.98252e12	0.953795
$188$	0	0
$189$	−4.54647e11	−0.137131
$190$	0	0
$191$	−1.67284e12	−0.476179	−0.238090	−	0.971243i	$-0.576521\pi$
$191$	−1.67284e12	−0.476179	−0.238090	+	0.971243i	$0.576521\pi$
$192$	0	0
$193$	−2.82751e12	−0.760045	−0.380023	−	0.924977i	$-0.624084\pi$
$193$	−2.82751e12	−0.760045	−0.380023	+	0.924977i	$0.624084\pi$
$194$	0	0
$195$	7.08824e12	1.80031
$196$	0	0
$197$	−4.26195e12	−1.02340	−0.511699	−	0.859165i	$-0.670984\pi$
$197$	−4.26195e12	−1.02340	−0.511699	+	0.859165i	$0.670984\pi$
$198$	0	0
$199$	4.84379e12	1.10026	0.550128	−	0.835080i	$-0.314579\pi$
$199$	4.84379e12	1.10026	0.550128	+	0.835080i	$0.314579\pi$
$200$	0	0
$201$	−9.86175e12	−2.12019
$202$	0	0
$203$	−2.58501e12	−0.526301
$204$	0	0
$205$	2.78473e11	0.0537200
$206$	0	0
$207$	1.22932e12	0.224817
$208$	0	0
$209$	5.27585e12	0.915141
$210$	0	0
$211$	−3.10465e12	−0.511045	−0.255522	−	0.966803i	$-0.582247\pi$
$211$	−3.10465e12	−0.511045	−0.255522	+	0.966803i	$0.582247\pi$
$212$	0	0
$213$	1.63017e13	2.54768
$214$	0	0
$215$	−1.54556e12	−0.229443
$216$	0	0
$217$	−5.05096e12	−0.712599
$218$	0	0
$219$	−6.84796e12	−0.918584
$220$	0	0
$221$	1.81012e13	2.30966
$222$	0	0
$223$	−6.22755e12	−0.756207	−0.378104	−	0.925763i	$-0.623424\pi$
$223$	−6.22755e12	−0.756207	−0.378104	+	0.925763i	$0.623424\pi$
$224$	0	0
$225$	−8.64136e10	−0.00999029
$226$	0	0
$227$	6.95550e12	0.765924	0.382962	−	0.923764i	$-0.374904\pi$
$227$	6.95550e12	0.765924	0.382962	+	0.923764i	$0.374904\pi$
$228$	0	0
$229$	−2.14549e12	−0.225129	−0.112565	−	0.993644i	$-0.535907\pi$
$229$	−2.14549e12	−0.225129	−0.112565	+	0.993644i	$0.535907\pi$
$230$	0	0
$231$	7.57978e12	0.758212
$232$	0	0
$233$	1.30485e13	1.24481	0.622407	−	0.782694i	$-0.286155\pi$
$233$	1.30485e13	1.24481	0.622407	+	0.782694i	$0.286155\pi$
$234$	0	0
$235$	−1.27940e13	−1.16448
$236$	0	0
$237$	2.91141e13	2.52922
$238$	0	0
$239$	−6.34372e12	−0.526206	−0.263103	−	0.964768i	$-0.584746\pi$
$239$	−6.34372e12	−0.526206	−0.263103	+	0.964768i	$0.584746\pi$
$240$	0	0
$241$	1.18515e13	0.939027	0.469514	−	0.882925i	$-0.344429\pi$
$241$	1.18515e13	0.939027	0.469514	+	0.882925i	$0.344429\pi$
$242$	0	0
$243$	−1.80401e13	−1.36586
$244$	0	0
$245$	3.92643e11	0.0284174
$246$	0	0
$247$	3.20195e13	2.21606
$248$	0	0
$249$	−1.87617e13	−1.24215
$250$	0	0
$251$	1.60684e13	1.01804	0.509021	−	0.860754i	$-0.330007\pi$
$251$	1.60684e13	1.01804	0.509021	+	0.860754i	$0.330007\pi$
$252$	0	0
$253$	2.22781e12	0.135119
$254$	0	0
$255$	4.15466e13	2.41304
$256$	0	0
$257$	−7.37440e12	−0.410293	−0.205147	−	0.978731i	$-0.565767\pi$
$257$	−7.37440e12	−0.410293	−0.205147	+	0.978731i	$0.565767\pi$
$258$	0	0
$259$	1.43643e13	0.765834
$260$	0	0
$261$	−9.15851e12	−0.468061
$262$	0	0
$263$	−6.23304e12	−0.305452	−0.152726	−	0.988269i	$-0.548805\pi$
$263$	−6.23304e12	−0.305452	−0.152726	+	0.988269i	$0.548805\pi$
$264$	0	0
$265$	−2.01287e13	−0.946157
$266$	0	0
$267$	4.09652e12	0.184758
$268$	0	0
$269$	3.04654e13	1.31877	0.659384	−	0.751806i	$-0.270817\pi$
$269$	3.04654e13	1.31877	0.659384	+	0.751806i	$0.270817\pi$
$270$	0	0
$271$	3.51737e13	1.46180	0.730899	−	0.682486i	$-0.239101\pi$
$271$	3.51737e13	1.46180	0.730899	+	0.682486i	$0.239101\pi$
$272$	0	0
$273$	4.60023e13	1.83605
$274$	0	0
$275$	−1.56601e11	−0.00600433
$276$	0	0
$277$	1.82082e13	0.670854	0.335427	−	0.942066i	$-0.391119\pi$
$277$	1.82082e13	0.670854	0.335427	+	0.942066i	$0.391119\pi$
$278$	0	0
$279$	−1.78952e13	−0.633743
$280$	0	0
$281$	1.12053e13	0.381539	0.190770	−	0.981635i	$-0.438902\pi$
$281$	1.12053e13	0.381539	0.190770	+	0.981635i	$0.438902\pi$
$282$	0	0
$283$	1.55481e13	0.509157	0.254579	−	0.967052i	$-0.418063\pi$
$283$	1.55481e13	0.509157	0.254579	+	0.967052i	$0.418063\pi$
$284$	0	0
$285$	7.34927e13	2.31525
$286$	0	0
$287$	1.80727e12	0.0547864
$288$	0	0
$289$	7.18252e13	2.09575
$290$	0	0
$291$	−3.81482e13	−1.07167
$292$	0	0
$293$	3.12722e13	0.846030	0.423015	−	0.906123i	$-0.360972\pi$
$293$	3.12722e13	0.846030	0.423015	+	0.906123i	$0.360972\pi$
$294$	0	0
$295$	−3.52985e13	−0.919889
$296$	0	0
$297$	−2.91910e12	−0.0732974
$298$	0	0
$299$	1.35208e13	0.327197
$300$	0	0
$301$	−1.00306e13	−0.233998
$302$	0	0
$303$	−4.30489e13	−0.968340
$304$	0	0
$305$	−5.90771e13	−1.28165
$306$	0	0
$307$	−1.92015e13	−0.401859	−0.200930	−	0.979606i	$-0.564396\pi$
$307$	−1.92015e13	−0.401859	−0.200930	+	0.979606i	$0.564396\pi$
$308$	0	0
$309$	9.13336e13	1.84442
$310$	0	0
$311$	5.30647e13	1.03425	0.517123	−	0.855911i	$-0.327003\pi$
$311$	5.30647e13	1.03425	0.517123	+	0.855911i	$0.327003\pi$
$312$	0	0
$313$	1.49736e11	0.00281730	0.00140865	−	0.999999i	$-0.499552\pi$
$313$	1.49736e11	0.00281730	0.00140865	+	0.999999i	$0.499552\pi$
$314$	0	0
$315$	5.00718e13	0.909673
$316$	0	0
$317$	7.50072e13	1.31606	0.658032	−	0.752990i	$-0.271389\pi$
$317$	7.50072e13	1.31606	0.658032	+	0.752990i	$0.271389\pi$
$318$	0	0
$319$	−1.65973e13	−0.281312
$320$	0	0
$321$	−3.75962e13	−0.615696
$322$	0	0
$323$	1.87678e14	2.97029
$324$	0	0
$325$	−9.50423e11	−0.0145398
$326$	0	0
$327$	−1.86355e13	−0.275631
$328$	0	0
$329$	−8.30321e13	−1.18760
$330$	0	0
$331$	−1.27975e14	−1.77040	−0.885202	−	0.465207i	$-0.845980\pi$
$331$	−1.27975e14	−1.77040	−0.885202	+	0.465207i	$0.845980\pi$
$332$	0	0
$333$	5.08915e13	0.681086
$334$	0	0
$335$	−1.18060e14	−1.52882
$336$	0	0
$337$	−1.12134e14	−1.40531	−0.702654	−	0.711531i	$-0.748002\pi$
$337$	−1.12134e14	−1.40531	−0.702654	+	0.711531i	$0.748002\pi$
$338$	0	0
$339$	−2.08883e14	−2.53399
$340$	0	0
$341$	−3.24301e13	−0.380890
$342$	0	0
$343$	−8.66253e13	−0.985206
$344$	0	0
$345$	3.10335e13	0.341842
$346$	0	0
$347$	−3.56620e13	−0.380534	−0.190267	−	0.981732i	$-0.560935\pi$
$347$	−3.56620e13	−0.380534	−0.190267	+	0.981732i	$0.560935\pi$
$348$	0	0
$349$	5.99525e13	0.619823	0.309911	−	0.950765i	$-0.399701\pi$
$349$	5.99525e13	0.619823	0.309911	+	0.950765i	$0.399701\pi$
$350$	0	0
$351$	−1.77162e13	−0.177493
$352$	0	0
$353$	−6.42007e13	−0.623417	−0.311709	−	0.950178i	$-0.600901\pi$
$353$	−6.42007e13	−0.623417	−0.311709	+	0.950178i	$0.600901\pi$
$354$	0	0
$355$	1.95156e14	1.83707
$356$	0	0
$357$	2.69635e14	2.46094
$358$	0	0
$359$	4.25028e13	0.376182	0.188091	−	0.982152i	$-0.439770\pi$
$359$	4.25028e13	0.376182	0.188091	+	0.982152i	$0.439770\pi$
$360$	0	0
$361$	2.15497e14	1.84991
$362$	0	0
$363$	−1.16943e14	−0.973844
$364$	0	0
$365$	−8.19805e13	−0.662369
$366$	0	0
$367$	−1.70912e14	−1.34001	−0.670007	−	0.742354i	$-0.733709\pi$
$367$	−1.70912e14	−1.34001	−0.670007	+	0.742354i	$0.733709\pi$
$368$	0	0
$369$	6.40303e12	0.0487238
$370$	0	0
$371$	−1.30634e14	−0.964939
$372$	0	0
$373$	1.09528e14	0.785464	0.392732	−	0.919653i	$-0.371530\pi$
$373$	1.09528e14	0.785464	0.392732	+	0.919653i	$0.371530\pi$
$374$	0	0
$375$	−1.99130e14	−1.38665
$376$	0	0
$377$	−1.00730e14	−0.681212
$378$	0	0
$379$	−1.99830e13	−0.131264	−0.0656318	−	0.997844i	$-0.520906\pi$
$379$	−1.99830e13	−0.131264	−0.0656318	+	0.997844i	$0.520906\pi$
$380$	0	0
$381$	−3.66976e14	−2.34180
$382$	0	0
$383$	−1.17178e13	−0.0726528	−0.0363264	−	0.999340i	$-0.511566\pi$
$383$	−1.17178e13	−0.0726528	−0.0363264	+	0.999340i	$0.511566\pi$
$384$	0	0
$385$	9.07415e13	0.546729
$386$	0	0
$387$	−3.55377e13	−0.208103
$388$	0	0
$389$	1.21767e14	0.693119	0.346559	−	0.938028i	$-0.387350\pi$
$389$	1.21767e14	0.693119	0.346559	+	0.938028i	$0.387350\pi$
$390$	0	0
$391$	7.92498e13	0.438557
$392$	0	0
$393$	2.22626e14	1.19789
$394$	0	0
$395$	3.48540e14	1.82376
$396$	0	0
$397$	1.21271e14	0.617178	0.308589	−	0.951195i	$-0.400143\pi$
$397$	1.21271e14	0.617178	0.308589	+	0.951195i	$0.400143\pi$
$398$	0	0
$399$	4.76964e14	2.36121
$400$	0	0
$401$	−7.44203e13	−0.358424	−0.179212	−	0.983810i	$-0.557355\pi$
$401$	−7.44203e13	−0.358424	−0.179212	+	0.983810i	$0.557355\pi$
$402$	0	0
$403$	−1.96821e14	−0.922344
$404$	0	0
$405$	−2.37348e14	−1.08239
$406$	0	0
$407$	9.22269e13	0.409344
$408$	0	0
$409$	2.43794e14	1.05328	0.526640	−	0.850088i	$-0.323451\pi$
$409$	2.43794e14	1.05328	0.526640	+	0.850088i	$0.323451\pi$
$410$	0	0
$411$	5.56847e14	2.34210
$412$	0	0
$413$	−2.29085e14	−0.938150
$414$	0	0
$415$	−2.24605e14	−0.895686
$416$	0	0
$417$	−3.18464e14	−1.23683
$418$	0	0
$419$	4.28661e14	1.62158	0.810788	−	0.585340i	$-0.199039\pi$
$419$	4.28661e14	1.62158	0.810788	+	0.585340i	$0.199039\pi$
$420$	0	0
$421$	2.58105e14	0.951142	0.475571	−	0.879677i	$-0.342241\pi$
$421$	2.58105e14	0.951142	0.475571	+	0.879677i	$0.342241\pi$
$422$	0	0
$423$	−2.94177e14	−1.05618
$424$	0	0
$425$	−5.57076e12	−0.0194883
$426$	0	0
$427$	−3.83407e14	−1.30709
$428$	0	0
$429$	2.95361e14	0.981383
$430$	0	0
$431$	3.60983e14	1.16913	0.584564	−	0.811348i	$-0.301266\pi$
$431$	3.60983e14	1.16913	0.584564	+	0.811348i	$0.301266\pi$
$432$	0	0
$433$	3.86281e14	1.21961	0.609804	−	0.792552i	$-0.291248\pi$
$433$	3.86281e14	1.21961	0.609804	+	0.792552i	$0.291248\pi$
$434$	0	0
$435$	−2.31201e14	−0.711702
$436$	0	0
$437$	1.40187e14	0.420784
$438$	0	0
$439$	2.77212e14	0.811442	0.405721	−	0.913997i	$-0.367021\pi$
$439$	2.77212e14	0.811442	0.405721	+	0.913997i	$0.367021\pi$
$440$	0	0
$441$	9.02819e12	0.0257744
$442$	0	0
$443$	−4.15207e14	−1.15623	−0.578115	−	0.815955i	$-0.696211\pi$
$443$	−4.15207e14	−1.15623	−0.578115	+	0.815955i	$0.696211\pi$
$444$	0	0
$445$	4.90416e13	0.133225
$446$	0	0
$447$	5.59348e14	1.48249
$448$	0	0
$449$	−2.13366e14	−0.551786	−0.275893	−	0.961188i	$-0.588974\pi$
$449$	−2.13366e14	−0.551786	−0.275893	+	0.961188i	$0.588974\pi$
$450$	0	0
$451$	1.16037e13	0.0292838
$452$	0	0
$453$	−5.25347e14	−1.29392
$454$	0	0
$455$	5.50717e14	1.32393
$456$	0	0
$457$	6.15476e13	0.144435	0.0722174	−	0.997389i	$-0.476992\pi$
$457$	6.15476e13	0.144435	0.0722174	+	0.997389i	$0.476992\pi$
$458$	0	0
$459$	−1.03841e14	−0.237902
$460$	0	0
$461$	−5.98791e14	−1.33943	−0.669715	−	0.742618i	$-0.733584\pi$
$461$	−5.98791e14	−1.33943	−0.669715	+	0.742618i	$0.733584\pi$
$462$	0	0
$463$	−2.94203e11	−0.000642617 0	−0.000321308	−	1.00000i	$-0.500102\pi$
$463$	−2.94203e11	−0.000642617 0	−0.000321308		1.00000i	$0.500102\pi$
$464$	0	0
$465$	−4.51752e14	−0.963627
$466$	0	0
$467$	−1.06508e14	−0.221891	−0.110945	−	0.993827i	$-0.535388\pi$
$467$	−1.06508e14	−0.221891	−0.110945	+	0.993827i	$0.535388\pi$
$468$	0	0
$469$	−7.66204e14	−1.55917
$470$	0	0
$471$	8.89580e14	1.76836
$472$	0	0
$473$	−6.44024e13	−0.125074
$474$	0	0
$475$	−9.85423e12	−0.0186986
$476$	0	0
$477$	−4.62827e14	−0.858159
$478$	0	0
$479$	−8.44379e14	−1.53000	−0.765001	−	0.644029i	$-0.777262\pi$
$479$	−8.44379e14	−1.53000	−0.765001	+	0.644029i	$0.777262\pi$
$480$	0	0
$481$	5.59732e14	0.991248
$482$	0	0
$483$	2.01405e14	0.348628
$484$	0	0
$485$	−4.56692e14	−0.772759
$486$	0	0
$487$	4.82693e14	0.798475	0.399238	−	0.916848i	$-0.369275\pi$
$487$	4.82693e14	0.798475	0.399238	+	0.916848i	$0.369275\pi$
$488$	0	0
$489$	−1.42908e15	−2.31130
$490$	0	0
$491$	−7.65357e14	−1.21036	−0.605182	−	0.796087i	$-0.706900\pi$
$491$	−7.65357e14	−1.21036	−0.605182	+	0.796087i	$0.706900\pi$
$492$	0	0
$493$	−5.90414e14	−0.913059
$494$	0	0
$495$	3.21490e14	0.486227
$496$	0	0
$497$	1.26655e15	1.87354
$498$	0	0
$499$	−1.12419e14	−0.162663	−0.0813314	−	0.996687i	$-0.525917\pi$
$499$	−1.12419e14	−0.162663	−0.0813314	+	0.996687i	$0.525917\pi$
$500$	0	0
$501$	−1.29536e15	−1.83351
$502$	0	0
$503$	−1.22185e15	−1.69197	−0.845985	−	0.533207i	$-0.820987\pi$
$503$	−1.22185e15	−1.69197	−0.845985	+	0.533207i	$0.820987\pi$
$504$	0	0
$505$	−5.15361e14	−0.698247
$506$	0	0
$507$	7.52306e14	0.997358
$508$	0	0
$509$	4.96425e14	0.644030	0.322015	−	0.946735i	$-0.395640\pi$
$509$	4.96425e14	0.644030	0.322015	+	0.946735i	$0.395640\pi$
$510$	0	0
$511$	−5.32049e14	−0.675517
$512$	0	0
$513$	−1.83687e14	−0.228261
$514$	0	0
$515$	1.09340e15	1.32996
$516$	0	0
$517$	−5.33114e14	−0.634779
$518$	0	0
$519$	−5.14397e14	−0.599623
$520$	0	0
$521$	−5.24045e14	−0.598082	−0.299041	−	0.954240i	$-0.596667\pi$
$521$	−5.24045e14	−0.598082	−0.299041	+	0.954240i	$0.596667\pi$
$522$	0	0
$523$	−5.29678e13	−0.0591906	−0.0295953	−	0.999562i	$-0.509422\pi$
$523$	−5.29678e13	−0.0591906	−0.0295953	+	0.999562i	$0.509422\pi$
$524$	0	0
$525$	−1.41575e13	−0.0154921
$526$	0	0
$527$	−1.15363e15	−1.23626
$528$	0	0
$529$	−8.93614e14	−0.937872
$530$	0	0
$531$	−8.11633e14	−0.834334
$532$	0	0
$533$	7.04240e13	0.0709122
$534$	0	0
$535$	−4.50084e14	−0.443964
$536$	0	0
$537$	−5.08011e14	−0.490924
$538$	0	0
$539$	1.63611e13	0.0154908
$540$	0	0
$541$	5.26351e14	0.488305	0.244152	−	0.969737i	$-0.421490\pi$
$541$	5.26351e14	0.488305	0.244152	+	0.969737i	$0.421490\pi$
$542$	0	0
$543$	2.06536e15	1.87757
$544$	0	0
$545$	−2.23095e14	−0.198751
$546$	0	0
$547$	−5.02844e14	−0.439038	−0.219519	−	0.975608i	$-0.570449\pi$
$547$	−5.02844e14	−0.439038	−0.219519	+	0.975608i	$0.570449\pi$
$548$	0	0
$549$	−1.35838e15	−1.16245
$550$	0	0
$551$	−1.04440e15	−0.876056
$552$	0	0
$553$	2.26200e15	1.85997
$554$	0	0
$555$	1.28472e15	1.03561
$556$	0	0
$557$	2.18116e14	0.172379	0.0861896	−	0.996279i	$-0.472531\pi$
$557$	2.18116e14	0.172379	0.0861896	+	0.996279i	$0.472531\pi$
$558$	0	0
$559$	−3.90863e14	−0.302872
$560$	0	0
$561$	1.73122e15	1.31539
$562$	0	0
$563$	−1.31875e14	−0.0982576	−0.0491288	−	0.998792i	$-0.515644\pi$
$563$	−1.31875e14	−0.0982576	−0.0491288	+	0.998792i	$0.515644\pi$
$564$	0	0
$565$	−2.50065e15	−1.82720
$566$	0	0
$567$	−1.54037e15	−1.10387
$568$	0	0
$569$	−1.26560e15	−0.889568	−0.444784	−	0.895638i	$-0.646720\pi$
$569$	−1.26560e15	−0.889568	−0.444784	+	0.895638i	$0.646720\pi$
$570$	0	0
$571$	8.30574e14	0.572638	0.286319	−	0.958134i	$-0.407568\pi$
$571$	8.30574e14	0.572638	0.286319	+	0.958134i	$0.407568\pi$
$572$	0	0
$573$	−9.71005e14	−0.656706
$574$	0	0
$575$	−4.16111e12	−0.00276080
$576$	0	0
$577$	3.00156e15	1.95380	0.976900	−	0.213698i	$-0.0685509\pi$
$577$	3.00156e15	1.95380	0.976900	+	0.213698i	$0.0685509\pi$
$578$	0	0
$579$	−1.64124e15	−1.04819
$580$	0	0
$581$	−1.45768e15	−0.913466
$582$	0	0
$583$	−8.38747e14	−0.515767
$584$	0	0
$585$	1.95115e15	1.17743
$586$	0	0
$587$	1.25889e15	0.745555	0.372778	−	0.927921i	$-0.378405\pi$
$587$	1.25889e15	0.745555	0.372778	+	0.927921i	$0.378405\pi$
$588$	0	0
$589$	−2.04069e15	−1.18616
$590$	0	0
$591$	−2.47386e15	−1.41138
$592$	0	0
$593$	1.15002e14	0.0644026	0.0322013	−	0.999481i	$-0.489748\pi$
$593$	1.15002e14	0.0644026	0.0322013	+	0.999481i	$0.489748\pi$
$594$	0	0
$595$	3.22794e15	1.77453
$596$	0	0
$597$	2.81160e15	1.51738
$598$	0	0
$599$	1.32935e15	0.704353	0.352176	−	0.935934i	$-0.385442\pi$
$599$	1.32935e15	0.704353	0.352176	+	0.935934i	$0.385442\pi$
$600$	0	0
$601$	−8.19476e14	−0.426311	−0.213155	−	0.977018i	$-0.568374\pi$
$601$	−8.19476e14	−0.426311	−0.213155	+	0.977018i	$0.568374\pi$
$602$	0	0
$603$	−2.71460e15	−1.38663
$604$	0	0
$605$	−1.39999e15	−0.702216
$606$	0	0
$607$	−2.77138e15	−1.36508	−0.682541	−	0.730848i	$-0.739125\pi$
$607$	−2.77138e15	−1.36508	−0.682541	+	0.730848i	$0.739125\pi$
$608$	0	0
$609$	−1.50048e15	−0.725830
$610$	0	0
$611$	−3.23551e15	−1.53715
$612$	0	0
$613$	−9.59996e14	−0.447957	−0.223979	−	0.974594i	$-0.571905\pi$
$613$	−9.59996e14	−0.447957	−0.223979	+	0.974594i	$0.571905\pi$
$614$	0	0
$615$	1.61640e14	0.0740861
$616$	0	0
$617$	7.29160e14	0.328288	0.164144	−	0.986436i	$-0.447514\pi$
$617$	7.29160e14	0.328288	0.164144	+	0.986436i	$0.447514\pi$
$618$	0	0
$619$	3.06883e14	0.135729	0.0678646	−	0.997695i	$-0.478381\pi$
$619$	3.06883e14	0.135729	0.0678646	+	0.997695i	$0.478381\pi$
$620$	0	0
$621$	−7.75646e13	−0.0337023
$622$	0	0
$623$	3.18277e14	0.135869
$624$	0	0
$625$	−2.35749e15	−0.988801
$626$	0	0
$627$	3.06238e15	1.26208
$628$	0	0
$629$	3.28078e15	1.32861
$630$	0	0
$631$	−2.59220e14	−0.103159	−0.0515795	−	0.998669i	$-0.516426\pi$
$631$	−2.59220e14	−0.103159	−0.0515795	+	0.998669i	$0.516426\pi$
$632$	0	0
$633$	−1.80210e15	−0.704789
$634$	0	0
$635$	−4.39326e15	−1.68862
$636$	0	0
$637$	9.92969e13	0.0375119
$638$	0	0
$639$	4.48730e15	1.66621
$640$	0	0
$641$	−4.45616e15	−1.62645	−0.813227	−	0.581946i	$-0.802291\pi$
$641$	−4.45616e15	−1.62645	−0.813227	+	0.581946i	$0.802291\pi$
$642$	0	0
$643$	−4.16268e14	−0.149353	−0.0746763	−	0.997208i	$-0.523792\pi$
$643$	−4.16268e14	−0.149353	−0.0746763	+	0.997208i	$0.523792\pi$
$644$	0	0
$645$	−8.97126e14	−0.316428
$646$	0	0
$647$	1.88622e15	0.654061	0.327030	−	0.945014i	$-0.393952\pi$
$647$	1.88622e15	0.654061	0.327030	+	0.945014i	$0.393952\pi$
$648$	0	0
$649$	−1.47086e15	−0.501448
$650$	0	0
$651$	−2.93184e15	−0.982756
$652$	0	0
$653$	−2.47844e15	−0.816876	−0.408438	−	0.912786i	$-0.633926\pi$
$653$	−2.47844e15	−0.816876	−0.408438	+	0.912786i	$0.633926\pi$
$654$	0	0
$655$	2.66517e15	0.863769
$656$	0	0
$657$	−1.88501e15	−0.600765
$658$	0	0
$659$	5.70088e14	0.178678	0.0893392	−	0.996001i	$-0.471524\pi$
$659$	5.70088e14	0.178678	0.0893392	+	0.996001i	$0.471524\pi$
$660$	0	0
$661$	5.29360e15	1.63171	0.815855	−	0.578256i	$-0.196267\pi$
$661$	5.29360e15	1.63171	0.815855	+	0.578256i	$0.196267\pi$
$662$	0	0
$663$	1.05069e16	3.18529
$664$	0	0
$665$	5.70998e15	1.70261
$666$	0	0
$667$	−4.41013e14	−0.129348
$668$	0	0
$669$	−3.61480e15	−1.04290
$670$	0	0
$671$	−2.46170e15	−0.698652
$672$	0	0
$673$	3.40210e15	0.949871	0.474935	−	0.880021i	$-0.342471\pi$
$673$	3.40210e15	0.949871	0.474935	+	0.880021i	$0.342471\pi$
$674$	0	0
$675$	5.45229e12	0.00149764
$676$	0	0
$677$	−9.87821e14	−0.266957	−0.133478	−	0.991052i	$-0.542615\pi$
$677$	−9.87821e14	−0.266957	−0.133478	+	0.991052i	$0.542615\pi$
$678$	0	0
$679$	−2.96391e15	−0.788099
$680$	0	0
$681$	4.03734e15	1.05630
$682$	0	0
$683$	5.81162e15	1.49618	0.748090	−	0.663598i	$-0.230971\pi$
$683$	5.81162e15	1.49618	0.748090	+	0.663598i	$0.230971\pi$
$684$	0	0
$685$	6.66630e15	1.68883
$686$	0	0
$687$	−1.24536e15	−0.310479
$688$	0	0
$689$	−5.09042e15	−1.24896
$690$	0	0
$691$	1.37529e15	0.332097	0.166049	−	0.986118i	$-0.446899\pi$
$691$	1.37529e15	0.332097	0.166049	+	0.986118i	$0.446899\pi$
$692$	0	0
$693$	2.08646e15	0.495880
$694$	0	0
$695$	−3.81249e15	−0.891852
$696$	0	0
$697$	4.12779e14	0.0950468
$698$	0	0
$699$	7.57407e15	1.71674
$700$	0	0
$701$	8.41846e15	1.87838	0.939191	−	0.343396i	$-0.111577\pi$
$701$	8.41846e15	1.87838	0.939191	+	0.343396i	$0.111577\pi$
$702$	0	0
$703$	5.80345e15	1.27477
$704$	0	0
$705$	−7.42629e15	−1.60595
$706$	0	0
$707$	−3.34466e15	−0.712108
$708$	0	0
$709$	8.56874e15	1.79623	0.898116	−	0.439758i	$-0.144936\pi$
$709$	8.56874e15	1.79623	0.898116	+	0.439758i	$0.144936\pi$
$710$	0	0
$711$	8.01411e15	1.65414
$712$	0	0
$713$	−8.61713e14	−0.175134
$714$	0	0
$715$	3.53592e15	0.707652
$716$	0	0
$717$	−3.68224e15	−0.725698
$718$	0	0
$719$	8.53831e15	1.65715	0.828577	−	0.559874i	$-0.189151\pi$
$719$	8.53831e15	1.65715	0.828577	+	0.559874i	$0.189151\pi$
$720$	0	0
$721$	7.09612e15	1.35637
$722$	0	0
$723$	6.87922e15	1.29503
$724$	0	0
$725$	3.10004e13	0.00574789
$726$	0	0
$727$	−7.28132e15	−1.32975	−0.664876	−	0.746953i	$-0.731516\pi$
$727$	−7.28132e15	−1.32975	−0.664876	+	0.746953i	$0.731516\pi$
$728$	0	0
$729$	−4.42081e15	−0.795244
$730$	0	0
$731$	−2.29098e15	−0.405953
$732$	0	0
$733$	−3.98737e15	−0.696009	−0.348004	−	0.937493i	$-0.613141\pi$
$733$	−3.98737e15	−0.696009	−0.348004	+	0.937493i	$0.613141\pi$
$734$	0	0
$735$	2.27911e14	0.0391908
$736$	0	0
$737$	−4.91947e15	−0.833388
$738$	0	0
$739$	−8.32659e15	−1.38971	−0.694853	−	0.719152i	$-0.744531\pi$
$739$	−8.32659e15	−1.38971	−0.694853	+	0.719152i	$0.744531\pi$
$740$	0	0
$741$	1.85858e16	3.05620
$742$	0	0
$743$	−5.71778e15	−0.926380	−0.463190	−	0.886259i	$-0.653295\pi$
$743$	−5.71778e15	−0.926380	−0.463190	+	0.886259i	$0.653295\pi$
$744$	0	0
$745$	6.69624e15	1.06898
$746$	0	0
$747$	−5.16444e15	−0.812382
$748$	0	0
$749$	−2.92102e15	−0.452777
$750$	0	0
$751$	1.40138e15	0.214060	0.107030	−	0.994256i	$-0.465866\pi$
$751$	1.40138e15	0.214060	0.107030	+	0.994256i	$0.465866\pi$
$752$	0	0
$753$	9.32693e15	1.40400
$754$	0	0
$755$	−6.28920e15	−0.933011
$756$	0	0
$757$	−6.08851e15	−0.890192	−0.445096	−	0.895483i	$-0.646830\pi$
$757$	−6.08851e15	−0.890192	−0.445096	+	0.895483i	$0.646830\pi$
$758$	0	0
$759$	1.29314e15	0.186344
$760$	0	0
$761$	−5.37439e15	−0.763332	−0.381666	−	0.924300i	$-0.624650\pi$
$761$	−5.37439e15	−0.763332	−0.381666	+	0.924300i	$0.624650\pi$
$762$	0	0
$763$	−1.44787e15	−0.202696
$764$	0	0
$765$	1.14364e16	1.57816
$766$	0	0
$767$	−8.92678e15	−1.21428
$768$	0	0
$769$	2.80951e15	0.376734	0.188367	−	0.982099i	$-0.439681\pi$
$769$	2.80951e15	0.376734	0.188367	+	0.982099i	$0.439681\pi$
$770$	0	0
$771$	−4.28049e15	−0.565841
$772$	0	0
$773$	−9.58543e15	−1.24918	−0.624589	−	0.780954i	$-0.714733\pi$
$773$	−9.58543e15	−1.24918	−0.624589	+	0.780954i	$0.714733\pi$
$774$	0	0
$775$	6.05729e13	0.00778250
$776$	0	0
$777$	8.33778e15	1.05617
$778$	0	0
$779$	7.30174e14	0.0911949
$780$	0	0
$781$	8.13200e15	1.00142
$782$	0	0
$783$	5.77859e14	0.0701669
$784$	0	0
$785$	1.06496e16	1.27512
$786$	0	0
$787$	1.40930e16	1.66395	0.831977	−	0.554810i	$-0.187209\pi$
$787$	1.40930e16	1.66395	0.831977	+	0.554810i	$0.187209\pi$
$788$	0	0
$789$	−3.61799e15	−0.421254
$790$	0	0
$791$	−1.62290e16	−1.86347
$792$	0	0
$793$	−1.49402e16	−1.69182
$794$	0	0
$795$	−1.16838e16	−1.30486
$796$	0	0
$797$	5.39857e15	0.594645	0.297322	−	0.954777i	$-0.403906\pi$
$797$	5.39857e15	0.594645	0.297322	+	0.954777i	$0.403906\pi$
$798$	0	0
$799$	−1.89645e16	−2.06031
$800$	0	0
$801$	1.12763e15	0.120834
$802$	0	0
$803$	−3.41606e15	−0.361069
$804$	0	0
$805$	2.41113e15	0.251387
$806$	0	0
$807$	1.76837e16	1.81873
$808$	0	0
$809$	1.05031e16	1.06561	0.532806	−	0.846237i	$-0.321137\pi$
$809$	1.05031e16	1.06561	0.532806	+	0.846237i	$0.321137\pi$
$810$	0	0
$811$	2.25918e15	0.226119	0.113059	−	0.993588i	$-0.463935\pi$
$811$	2.25918e15	0.226119	0.113059	+	0.993588i	$0.463935\pi$
$812$	0	0
$813$	2.04167e16	2.01599
$814$	0	0
$815$	−1.71082e16	−1.66662
$816$	0	0
$817$	−4.05257e15	−0.389502
$818$	0	0
$819$	1.26629e16	1.20080
$820$	0	0
$821$	2.03003e16	1.89939	0.949697	−	0.313171i	$-0.101391\pi$
$821$	2.03003e16	1.89939	0.949697	+	0.313171i	$0.101391\pi$
$822$	0	0
$823$	−1.87497e14	−0.0173099	−0.00865497	−	0.999963i	$-0.502755\pi$
$823$	−1.87497e14	−0.0173099	−0.00865497	+	0.999963i	$0.502755\pi$
$824$	0	0
$825$	−9.08995e13	−0.00828066
$826$	0	0
$827$	−2.19523e15	−0.197333	−0.0986666	−	0.995121i	$-0.531458\pi$
$827$	−2.19523e15	−0.197333	−0.0986666	+	0.995121i	$0.531458\pi$
$828$	0	0
$829$	1.25652e16	1.11460	0.557302	−	0.830310i	$-0.311837\pi$
$829$	1.25652e16	1.11460	0.557302	+	0.830310i	$0.311837\pi$
$830$	0	0
$831$	1.05690e16	0.925185
$832$	0	0
$833$	5.82013e14	0.0502789
$834$	0	0
$835$	−1.55074e16	−1.32210
$836$	0	0
$837$	1.12910e15	0.0950043
$838$	0	0
$839$	−1.16843e16	−0.970309	−0.485155	−	0.874428i	$-0.661237\pi$
$839$	−1.16843e16	−0.970309	−0.485155	+	0.874428i	$0.661237\pi$
$840$	0	0
$841$	−8.91494e15	−0.730702
$842$	0	0
$843$	6.50416e15	0.526186
$844$	0	0
$845$	9.00625e15	0.719171
$846$	0	0
$847$	−9.08586e15	−0.716156
$848$	0	0
$849$	9.02495e15	0.702186
$850$	0	0
$851$	2.45060e15	0.188217
$852$	0	0
$853$	−2.18789e16	−1.65884	−0.829421	−	0.558624i	$-0.811330\pi$
$853$	−2.18789e16	−1.65884	−0.829421	+	0.558624i	$0.811330\pi$
$854$	0	0
$855$	2.02300e16	1.51420
$856$	0	0
$857$	−2.10716e16	−1.55705	−0.778524	−	0.627614i	$-0.784032\pi$
$857$	−2.10716e16	−1.55705	−0.778524	+	0.627614i	$0.784032\pi$
$858$	0	0
$859$	8.11417e15	0.591946	0.295973	−	0.955196i	$-0.404356\pi$
$859$	8.11417e15	0.591946	0.295973	+	0.955196i	$0.404356\pi$
$860$	0	0
$861$	1.04904e15	0.0755568
$862$	0	0
$863$	−3.04030e15	−0.216201	−0.108100	−	0.994140i	$-0.534477\pi$
$863$	−3.04030e15	−0.216201	−0.108100	+	0.994140i	$0.534477\pi$
$864$	0	0
$865$	−6.15811e15	−0.432373
$866$	0	0
$867$	4.16912e16	2.89028
$868$	0	0
$869$	1.45234e16	0.994166
$870$	0	0
$871$	−2.98567e16	−2.01809
$872$	0	0
$873$	−1.05009e16	−0.700888
$874$	0	0
$875$	−1.54713e16	−1.01973
$876$	0	0
$877$	1.29894e16	0.845455	0.422727	−	0.906257i	$-0.361073\pi$
$877$	1.29894e16	0.845455	0.422727	+	0.906257i	$0.361073\pi$
$878$	0	0
$879$	1.81520e16	1.16677
$880$	0	0
$881$	−5.15273e14	−0.0327092	−0.0163546	−	0.999866i	$-0.505206\pi$
$881$	−5.15273e14	−0.0327092	−0.0163546	+	0.999866i	$0.505206\pi$
$882$	0	0
$883$	−3.14075e16	−1.96901	−0.984507	−	0.175344i	$-0.943896\pi$
$883$	−3.14075e16	−1.96901	−0.984507	+	0.175344i	$0.943896\pi$
$884$	0	0
$885$	−2.04891e16	−1.26863
$886$	0	0
$887$	−1.12736e15	−0.0689420	−0.0344710	−	0.999406i	$-0.510975\pi$
$887$	−1.12736e15	−0.0689420	−0.0344710	+	0.999406i	$0.510975\pi$
$888$	0	0
$889$	−2.85120e16	−1.72214
$890$	0	0
$891$	−9.89009e15	−0.590028
$892$	0	0
$893$	−3.35466e16	−1.97681
$894$	0	0
$895$	−6.08166e15	−0.353994
$896$	0	0
$897$	7.84817e15	0.451242
$898$	0	0
$899$	6.41980e15	0.364622
$900$	0	0
$901$	−2.98367e16	−1.67403
$902$	0	0
$903$	−5.82230e15	−0.322710
$904$	0	0
$905$	2.47255e16	1.35387
$906$	0	0
$907$	1.89493e16	1.02507	0.512536	−	0.858666i	$-0.328706\pi$
$907$	1.89493e16	1.02507	0.512536	+	0.858666i	$0.328706\pi$
$908$	0	0
$909$	−1.18499e16	−0.633306
$910$	0	0
$911$	1.45138e16	0.766353	0.383177	−	0.923675i	$-0.374830\pi$
$911$	1.45138e16	0.766353	0.383177	+	0.923675i	$0.374830\pi$
$912$	0	0
$913$	−9.35914e15	−0.488255
$914$	0	0
$915$	−3.42915e16	−1.76754
$916$	0	0
$917$	1.72968e16	0.880916
$918$	0	0
$919$	−1.00596e16	−0.506226	−0.253113	−	0.967437i	$-0.581454\pi$
$919$	−1.00596e16	−0.506226	−0.253113	+	0.967437i	$0.581454\pi$
$920$	0	0
$921$	−1.11456e16	−0.554210
$922$	0	0
$923$	4.93537e16	2.42499
$924$	0	0
$925$	−1.72261e14	−0.00836390
$926$	0	0
$927$	2.51410e16	1.20627
$928$	0	0
$929$	2.09205e16	0.991940	0.495970	−	0.868340i	$-0.334813\pi$
$929$	2.09205e16	0.991940	0.495970	+	0.868340i	$0.334813\pi$
$930$	0	0
$931$	1.02954e15	0.0482412
$932$	0	0
$933$	3.08016e16	1.42634
$934$	0	0
$935$	2.07253e16	0.948498
$936$	0	0
$937$	−9.87940e15	−0.446851	−0.223426	−	0.974721i	$-0.571724\pi$
$937$	−9.87940e15	−0.446851	−0.223426	+	0.974721i	$0.571724\pi$
$938$	0	0
$939$	8.69150e13	0.00388538
$940$	0	0
$941$	8.95836e15	0.395809	0.197904	−	0.980221i	$-0.436586\pi$
$941$	8.95836e15	0.395809	0.197904	+	0.980221i	$0.436586\pi$
$942$	0	0
$943$	3.08328e14	0.0134647
$944$	0	0
$945$	−3.15930e15	−0.136369
$946$	0	0
$947$	2.63888e15	0.112589	0.0562944	−	0.998414i	$-0.482071\pi$
$947$	2.63888e15	0.112589	0.0562944	+	0.998414i	$0.482071\pi$
$948$	0	0
$949$	−2.07324e16	−0.874348
$950$	0	0
$951$	4.35381e16	1.81500
$952$	0	0
$953$	3.22131e16	1.32746	0.663729	−	0.747973i	$-0.268973\pi$
$953$	3.22131e16	1.32746	0.663729	+	0.747973i	$0.268973\pi$
$954$	0	0
$955$	−1.16244e16	−0.473535
$956$	0	0
$957$	−9.63395e15	−0.387962
$958$	0	0
$959$	4.32639e16	1.72236
$960$	0	0
$961$	−1.28646e16	−0.506310
$962$	0	0
$963$	−1.03490e16	−0.402672
$964$	0	0
$965$	−1.96481e16	−0.755824
$966$	0	0
$967$	3.32975e16	1.26638	0.633192	−	0.773995i	$-0.281744\pi$
$967$	3.32975e16	1.26638	0.633192	+	0.773995i	$0.281744\pi$
$968$	0	0
$969$	1.08938e17	4.09637
$970$	0	0
$971$	−2.03551e16	−0.756777	−0.378388	−	0.925647i	$-0.623522\pi$
$971$	−2.03551e16	−0.756777	−0.378388	+	0.925647i	$0.623522\pi$
$972$	0	0
$973$	−2.47429e16	−0.909556
$974$	0	0
$975$	−5.51676e14	−0.0200521
$976$	0	0
$977$	1.14008e16	0.409746	0.204873	−	0.978789i	$-0.434322\pi$
$977$	1.14008e16	0.409746	0.204873	+	0.978789i	$0.434322\pi$
$978$	0	0
$979$	2.04353e15	0.0726232
$980$	0	0
$981$	−5.12971e15	−0.180266
$982$	0	0
$983$	−8.12549e15	−0.282361	−0.141181	−	0.989984i	$-0.545090\pi$
$983$	−8.12549e15	−0.282361	−0.141181	+	0.989984i	$0.545090\pi$
$984$	0	0
$985$	−2.96159e16	−1.01771
$986$	0	0
$987$	−4.81962e16	−1.63783
$988$	0	0
$989$	−1.71126e15	−0.0575091
$990$	0	0
$991$	1.87482e16	0.623095	0.311547	−	0.950231i	$-0.399153\pi$
$991$	1.87482e16	0.623095	0.311547	+	0.950231i	$0.399153\pi$
$992$	0	0
$993$	−7.42837e16	−2.44159
$994$	0	0
$995$	3.36591e16	1.09415
$996$	0	0
$997$	2.16114e15	0.0694800	0.0347400	−	0.999396i	$-0.488940\pi$
$997$	2.16114e15	0.0694800	0.0347400	+	0.999396i	$0.488940\pi$
$998$	0	0
$999$	−3.21102e15	−0.102102

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.12.a.j.1.6		6
4.3	odd	2	inner	256.12.a.j.1.1		6
8.3	odd	2		256.12.a.k.1.6		6
8.5	even	2		256.12.a.k.1.1		6
16.3	odd	4		128.12.b.f.65.1	✓	12
16.5	even	4		128.12.b.f.65.2	yes	12
16.11	odd	4		128.12.b.f.65.12	yes	12
16.13	even	4		128.12.b.f.65.11	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.12.b.f.65.1	✓	12	16.3	odd	4
128.12.b.f.65.2	yes	12	16.5	even	4
128.12.b.f.65.11	yes	12	16.13	even	4
128.12.b.f.65.12	yes	12	16.11	odd	4
256.12.a.j.1.1		6	4.3	odd	2	inner
256.12.a.j.1.6		6	1.1	even	1	trivial
256.12.a.k.1.1		6	8.5	even	2
256.12.a.k.1.6		6	8.3	odd	2

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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	256.a (trivial)

\(f(q)\)	\(=\)	\(q+580.453 q^{3} +6948.91 q^{5} +45098.0 q^{7} +159779. q^{9} +289556. q^{11} +1.75734e6 q^{13} +4.03352e6 q^{15} +1.03003e7 q^{17} +1.82205e7 q^{19} +2.61773e7 q^{21} +7.69390e6 q^{23} -540832. q^{25} -1.00813e7 q^{27} -5.73199e7 q^{29} -1.12000e8 q^{31} +1.68073e8 q^{33} +3.13382e8 q^{35} +3.18512e8 q^{37} +1.02005e9 q^{39} +4.00743e7 q^{41} -2.22418e8 q^{43} +1.11029e9 q^{45} -1.84115e9 q^{47} +5.65042e7 q^{49} +5.97887e9 q^{51} -2.89667e9 q^{53} +2.01209e9 q^{55} +1.05762e10 q^{57} -5.07972e9 q^{59} -8.50164e9 q^{61} +7.20572e9 q^{63} +1.22116e10 q^{65} -1.69897e10 q^{67} +4.46595e9 q^{69} +2.80844e10 q^{71} -1.17976e10 q^{73} -3.13928e8 q^{75} +1.30584e10 q^{77} +5.01575e10 q^{79} -3.41561e10 q^{81} -3.23224e10 q^{83} +7.15761e10 q^{85} -3.32715e10 q^{87} +7.05746e9 q^{89} +7.92523e10 q^{91} -6.50105e10 q^{93} +1.26613e11 q^{95} -6.57214e10 q^{97} +4.62649e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3256 q^{5} + 158366 q^{9} + 1974152 q^{13} + 5650868 q^{17} + 5622272 q^{21} + 50201138 q^{25} + 78510600 q^{29} - 185111168 q^{33} + 187521320 q^{37} - 729393436 q^{41} - 94716312 q^{45} - 1212515210 q^{49}+ \cdots - 198387795308 q^{97}+O(q^{100}) \) 6 * q - 3256 * q^5 + 158366 * q^9 + 1974152 * q^13 + 5650868 * q^17 + 5622272 * q^21 + 50201138 * q^25 + 78510600 * q^29 - 185111168 * q^33 + 187521320 * q^37 - 729393436 * q^41 - 94716312 * q^45 - 1212515210 * q^49 + 1345806024 * q^53 + 4211057792 * q^57 - 13682701080 * q^61 - 9028891040 * q^65 + 12746864128 * q^69 + 20889754044 * q^73 - 6919916032 * q^77 - 106544843242 * q^81 + 149487054448 * q^85 + 165819876316 * q^89 - 263298174976 * q^93 - 198387795308 * q^97

Introduction

L-functions

Modular forms

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Database

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