LMFDB - Embedded newform 16.36.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,36,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, 36, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 36 $
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:	$124.152209014$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\mathbb{Q}[x]/(x^{3} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 12422194x - 2645665785 $ x^3 - 12422194*x - 2645665785
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{24}\cdot 3^{3}\cdot 5\cdot 7 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3412.77$ 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)

$f(q)$	$=$	$q-3.95729e8 q^{3} +8.21401e11 q^{5} -6.06942e14 q^{7} +1.06570e17 q^{9} +O(q^{10})$	$q-3.95729e8 q^{3} +8.21401e11 q^{5} -6.06942e14 q^{7} +1.06570e17 q^{9} -1.23316e18 q^{11} -8.46079e17 q^{13} -3.25052e20 q^{15} -3.92226e21 q^{17} +3.52757e20 q^{19} +2.40184e23 q^{21} +8.58137e23 q^{23} -2.23568e24 q^{25} -2.23738e25 q^{27} -1.38664e25 q^{29} +3.33324e25 q^{31} +4.87995e26 q^{33} -4.98543e26 q^{35} +1.99674e27 q^{37} +3.34818e26 q^{39} +2.10955e28 q^{41} +4.57475e28 q^{43} +8.75365e28 q^{45} +1.72608e29 q^{47} -1.04400e28 q^{49} +1.55215e30 q^{51} +1.81145e30 q^{53} -1.01292e30 q^{55} -1.39596e29 q^{57} +5.02395e30 q^{59} -3.25742e29 q^{61} -6.46817e31 q^{63} -6.94970e29 q^{65} +4.01253e31 q^{67} -3.39589e32 q^{69} -3.85444e32 q^{71} +9.93637e31 q^{73} +8.84724e32 q^{75} +7.48454e32 q^{77} +2.62804e33 q^{79} +3.52211e33 q^{81} -6.17326e33 q^{83} -3.22175e33 q^{85} +5.48733e33 q^{87} +6.75031e33 q^{89} +5.13521e32 q^{91} -1.31906e34 q^{93} +2.89755e32 q^{95} -8.10218e34 q^{97} -1.31417e35 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+O(q^{10}) $ 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9	$ 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) $ 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9 + 1157945428549987044 * q^11 - 62139610550998650558 * q^13 - 706062479352366674520 * q^15 - 3932636930193139724394 * q^17 + 32303396242934620550940 * q^19 + 222122458753667398807776 * q^21 + 514822069421375217224808 * q^23 + 646885156836409574920725 * q^25 - 27373531144823515220747400 * q^27 - 38460399143984437598248110 * q^29 - 103731594563724689730029856 * q^31 + 1184565114402567148957069584 * q^33 - 1596564453000564139676248560 * q^35 + 2464613172948888017021063706 * q^37 - 18611981954062877742706387512 * q^39 + 23522986584532768129037087406 * q^41 + 47516297135443858184325274308 * q^43 - 110576300682246572226345828030 * q^45 - 164346768841318947159729902256 * q^47 - 595089038084925731622037450821 * q^49 + 1803044074850794727352259927704 * q^51 - 1670480551111723484744458809558 * q^53 - 3067178595313878889174894808520 * q^55 + 4048712721984947152977307522800 * q^57 - 4370528583092646699845243631180 * q^59 + 23537078276516028947219080488306 * q^61 - 45614247892704690123948892506792 * q^63 + 75762337609865790449417835274620 * q^65 + 188845033384156918977315211921644 * q^67 - 325983212396274250308653315934048 * q^69 - 348774602295358210538621162006856 * q^71 - 285174964854822786615792218987058 * q^73 + 1577819189216049218913461709237300 * q^75 + 691501421964810422051915307575712 * q^77 + 426212862591061661901736630874160 * q^79 + 1867221150796978455595408934723163 * q^81 - 14867187331664065317008883105766692 * q^83 - 8752040931978292430566755304420140 * q^85 + 7833119385476156993771606256673800 * q^87 + 30583590846193751335205992418034270 * q^89 - 1003391702519487362566122156997296 * q^91 - 40914650682166972019906140161446016 * q^93 + 84521392849966457182141665888160200 * q^95 - 106165667044630951063328618701091994 * q^97 - 30288686341597731928441860752034252 * 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
$3$	−3.95729e8	−1.76920	−0.884598	−	0.466355i	$-0.845567\pi$
$3$	−3.95729e8	−1.76920	−0.884598	+	0.466355i	$0.845567\pi$
$4$	0	0
$5$	8.21401e11	0.481482	0.240741	−	0.970589i	$-0.422610\pi$
$5$	8.21401e11	0.481482	0.240741	+	0.970589i	$0.422610\pi$
$6$	0	0
$7$	−6.06942e14	−0.986124	−0.493062	−	0.869994i	$-0.664122\pi$
$7$	−6.06942e14	−0.986124	−0.493062	+	0.869994i	$0.664122\pi$
$8$	0	0
$9$	1.06570e17	2.13005
$10$	0	0
$11$	−1.23316e18	−0.735607	−0.367804	−	0.929904i	$-0.619890\pi$
$11$	−1.23316e18	−0.735607	−0.367804	+	0.929904i	$0.619890\pi$
$12$	0	0
$13$	−8.46079e17	−0.0271270	−0.0135635	−	0.999908i	$-0.504318\pi$
$13$	−8.46079e17	−0.0271270	−0.0135635	+	0.999908i	$0.504318\pi$
$14$	0	0
$15$	−3.25052e20	−0.851836
$16$	0	0
$17$	−3.92226e21	−1.14995	−0.574976	−	0.818170i	$-0.694989\pi$
$17$	−3.92226e21	−1.14995	−0.574976	+	0.818170i	$0.694989\pi$
$18$	0	0
$19$	3.52757e20	0.0147668	0.00738342	−	0.999973i	$-0.497650\pi$
$19$	3.52757e20	0.0147668	0.00738342	+	0.999973i	$0.497650\pi$
$20$	0	0
$21$	2.40184e23	1.74465
$22$	0	0
$23$	8.58137e23	1.26858	0.634292	−	0.773093i	$-0.281292\pi$
$23$	8.58137e23	1.26858	0.634292	+	0.773093i	$0.281292\pi$
$24$	0	0
$25$	−2.23568e24	−0.768175
$26$	0	0
$27$	−2.23738e25	−1.99928
$28$	0	0
$29$	−1.38664e25	−0.354812	−0.177406	−	0.984138i	$-0.556771\pi$
$29$	−1.38664e25	−0.354812	−0.177406	+	0.984138i	$0.556771\pi$
$30$	0	0
$31$	3.33324e25	0.265483	0.132741	−	0.991151i	$-0.457622\pi$
$31$	3.33324e25	0.265483	0.132741	+	0.991151i	$0.457622\pi$
$32$	0	0
$33$	4.87995e26	1.30143
$34$	0	0
$35$	−4.98543e26	−0.474801
$36$	0	0
$37$	1.99674e27	0.719102	0.359551	−	0.933125i	$-0.382930\pi$
$37$	1.99674e27	0.719102	0.359551	+	0.933125i	$0.382930\pi$
$38$	0	0
$39$	3.34818e26	0.0479930
$40$	0	0
$41$	2.10955e28	1.26029	0.630146	−	0.776476i	$-0.282995\pi$
$41$	2.10955e28	1.26029	0.630146	+	0.776476i	$0.282995\pi$
$42$	0	0
$43$	4.57475e28	1.18760	0.593799	−	0.804614i	$-0.297628\pi$
$43$	4.57475e28	1.18760	0.593799	+	0.804614i	$0.297628\pi$
$44$	0	0
$45$	8.75365e28	1.02558
$46$	0	0
$47$	1.72608e29	0.944816	0.472408	−	0.881380i	$-0.343385\pi$
$47$	1.72608e29	0.944816	0.472408	+	0.881380i	$0.343385\pi$
$48$	0	0
$49$	−1.04400e28	−0.0275593
$50$	0	0
$51$	1.55215e30	2.03449
$52$	0	0
$53$	1.81145e30	1.21115	0.605577	−	0.795786i	$-0.292942\pi$
$53$	1.81145e30	1.21115	0.605577	+	0.795786i	$0.292942\pi$
$54$	0	0
$55$	−1.01292e30	−0.354182
$56$	0	0
$57$	−1.39596e29	−0.0261254
$58$	0	0
$59$	5.02395e30	0.514204	0.257102	−	0.966384i	$-0.417232\pi$
$59$	5.02395e30	0.514204	0.257102	+	0.966384i	$0.417232\pi$
$60$	0	0
$61$	−3.25742e29	−0.0186038	−0.00930189	−	0.999957i	$-0.502961\pi$
$61$	−3.25742e29	−0.0186038	−0.00930189	+	0.999957i	$0.502961\pi$
$62$	0	0
$63$	−6.46817e31	−2.10050
$64$	0	0
$65$	−6.94970e29	−0.0130612
$66$	0	0
$67$	4.01253e31	0.443720	0.221860	−	0.975079i	$-0.428787\pi$
$67$	4.01253e31	0.443720	0.221860	+	0.975079i	$0.428787\pi$
$68$	0	0
$69$	−3.39589e32	−2.24437
$70$	0	0
$71$	−3.85444e32	−1.54504	−0.772520	−	0.634990i	$-0.781004\pi$
$71$	−3.85444e32	−1.54504	−0.772520	+	0.634990i	$0.781004\pi$
$72$	0	0
$73$	9.93637e31	0.244950	0.122475	−	0.992472i	$-0.460917\pi$
$73$	9.93637e31	0.244950	0.122475	+	0.992472i	$0.460917\pi$
$74$	0	0
$75$	8.84724e32	1.35905
$76$	0	0
$77$	7.48454e32	0.725400
$78$	0	0
$79$	2.62804e33	1.62614	0.813069	−	0.582167i	$-0.197795\pi$
$79$	2.62804e33	1.62614	0.813069	+	0.582167i	$0.197795\pi$
$80$	0	0
$81$	3.52211e33	1.40707
$82$	0	0
$83$	−6.17326e33	−1.60934	−0.804671	−	0.593722i	$-0.797658\pi$
$83$	−6.17326e33	−1.60934	−0.804671	+	0.593722i	$0.797658\pi$
$84$	0	0
$85$	−3.22175e33	−0.553682
$86$	0	0
$87$	5.48733e33	0.627732
$88$	0	0
$89$	6.75031e33	0.518799	0.259399	−	0.965770i	$-0.416475\pi$
$89$	6.75031e33	0.518799	0.259399	+	0.965770i	$0.416475\pi$
$90$	0	0
$91$	5.13521e32	0.0267506
$92$	0	0
$93$	−1.31906e34	−0.469691
$94$	0	0
$95$	2.89755e32	0.00710997
$96$	0	0
$97$	−8.10218e34	−1.38069	−0.690346	−	0.723480i	$-0.742542\pi$
$97$	−8.10218e34	−1.38069	−0.690346	+	0.723480i	$0.742542\pi$
$98$	0	0
$99$	−1.31417e35	−1.56688
$100$	0	0
$101$	−1.08776e35	−0.913923	−0.456962	−	0.889486i	$-0.651062\pi$
$101$	−1.08776e35	−0.913923	−0.456962	+	0.889486i	$0.651062\pi$
$102$	0	0
$103$	2.99910e33	0.0178788	0.00893941	−	0.999960i	$-0.497154\pi$
$103$	2.99910e33	0.0178788	0.00893941	+	0.999960i	$0.497154\pi$
$104$	0	0
$105$	1.97288e35	0.840016
$106$	0	0
$107$	3.51492e35	1.07572	0.537860	−	0.843034i	$-0.319233\pi$
$107$	3.51492e35	1.07572	0.537860	+	0.843034i	$0.319233\pi$
$108$	0	0
$109$	6.20112e35	1.37248	0.686240	−	0.727375i	$-0.259260\pi$
$109$	6.20112e35	1.37248	0.686240	+	0.727375i	$0.259260\pi$
$110$	0	0
$111$	−7.90166e35	−1.27223
$112$	0	0
$113$	−7.30280e35	−0.860234	−0.430117	−	0.902773i	$-0.641528\pi$
$113$	−7.30280e35	−0.860234	−0.430117	+	0.902773i	$0.641528\pi$
$114$	0	0
$115$	7.04874e35	0.610801
$116$	0	0
$117$	−9.01664e34	−0.0577819
$118$	0	0
$119$	2.38058e36	1.13400
$120$	0	0
$121$	−1.28957e36	−0.458882
$122$	0	0
$123$	−8.34808e36	−2.22970
$124$	0	0
$125$	−4.22698e36	−0.851345
$126$	0	0
$127$	−5.89251e35	−0.0898949	−0.0449475	−	0.998989i	$-0.514312\pi$
$127$	−5.89251e35	−0.0898949	−0.0449475	+	0.998989i	$0.514312\pi$
$128$	0	0
$129$	−1.81036e37	−2.10109
$130$	0	0
$131$	−2.10664e37	−1.86785	−0.933925	−	0.357468i	$-0.883640\pi$
$131$	−2.10664e37	−1.86785	−0.933925	+	0.357468i	$0.883640\pi$
$132$	0	0
$133$	−2.14103e35	−0.0145619
$134$	0	0
$135$	−1.83779e37	−0.962618
$136$	0	0
$137$	2.66291e37	1.07832	0.539158	−	0.842204i	$-0.318742\pi$
$137$	2.66291e37	1.07832	0.539158	+	0.842204i	$0.318742\pi$
$138$	0	0
$139$	1.89155e37	0.594374	0.297187	−	0.954819i	$-0.403952\pi$
$139$	1.89155e37	0.594374	0.297187	+	0.954819i	$0.403952\pi$
$140$	0	0
$141$	−6.83058e37	−1.67156
$142$	0	0
$143$	1.04335e36	0.0199548
$144$	0	0
$145$	−1.13899e37	−0.170836
$146$	0	0
$147$	4.13140e36	0.0487578
$148$	0	0
$149$	6.86254e37	0.639330	0.319665	−	0.947531i	$-0.396430\pi$
$149$	6.86254e37	0.639330	0.319665	+	0.947531i	$0.396430\pi$
$150$	0	0
$151$	−1.75463e38	−1.29446	−0.647229	−	0.762295i	$-0.724072\pi$
$151$	−1.75463e38	−1.29446	−0.647229	+	0.762295i	$0.724072\pi$
$152$	0	0
$153$	−4.17994e38	−2.44946
$154$	0	0
$155$	2.73792e37	0.127825
$156$	0	0
$157$	−1.13686e36	−0.00424096	−0.00212048	−	0.999998i	$-0.500675\pi$
$157$	−1.13686e36	−0.00424096	−0.00212048	+	0.999998i	$0.500675\pi$
$158$	0	0
$159$	−7.16845e38	−2.14277
$160$	0	0
$161$	−5.20839e38	−1.25098
$162$	0	0
$163$	−7.96117e38	−1.54061	−0.770307	−	0.637673i	$-0.779897\pi$
$163$	−7.96117e38	−1.54061	−0.770307	+	0.637673i	$0.779897\pi$
$164$	0	0
$165$	4.00840e38	0.626616
$166$	0	0
$167$	9.18990e38	1.16352	0.581758	−	0.813362i	$-0.302365\pi$
$167$	9.18990e38	1.16352	0.581758	+	0.813362i	$0.302365\pi$
$168$	0	0
$169$	−9.72070e38	−0.999264
$170$	0	0
$171$	3.75932e37	0.0314541
$172$	0	0
$173$	1.70459e39	1.16363	0.581813	−	0.813323i	$-0.302344\pi$
$173$	1.70459e39	1.16363	0.581813	+	0.813323i	$0.302344\pi$
$174$	0	0
$175$	1.35693e39	0.757516
$176$	0	0
$177$	−1.98812e39	−0.909727
$178$	0	0
$179$	9.48476e38	0.356532	0.178266	−	0.983982i	$-0.442951\pi$
$179$	9.48476e38	0.356532	0.178266	+	0.983982i	$0.442951\pi$
$180$	0	0
$181$	3.12075e39	0.965791	0.482896	−	0.875678i	$-0.339585\pi$
$181$	3.12075e39	0.965791	0.482896	+	0.875678i	$0.339585\pi$
$182$	0	0
$183$	1.28905e38	0.0329137
$184$	0	0
$185$	1.64012e39	0.346235
$186$	0	0
$187$	4.83675e39	0.845913
$188$	0	0
$189$	1.35796e40	1.97154
$190$	0	0
$191$	3.57216e38	0.0431367	0.0215683	−	0.999767i	$-0.493134\pi$
$191$	3.57216e38	0.0431367	0.0215683	+	0.999767i	$0.493134\pi$
$192$	0	0
$193$	−1.46241e40	−1.47169	−0.735847	−	0.677148i	$-0.763216\pi$
$193$	−1.46241e40	−1.47169	−0.735847	+	0.677148i	$0.763216\pi$
$194$	0	0
$195$	2.75020e38	0.0231078
$196$	0	0
$197$	1.11047e40	0.780453	0.390226	−	0.920719i	$-0.372397\pi$
$197$	1.11047e40	0.780453	0.390226	+	0.920719i	$0.372397\pi$
$198$	0	0
$199$	−1.81867e40	−1.07109	−0.535545	−	0.844507i	$-0.679894\pi$
$199$	−1.81867e40	−1.07109	−0.535545	+	0.844507i	$0.679894\pi$
$200$	0	0
$201$	−1.58787e40	−0.785027
$202$	0	0
$203$	8.41610e39	0.349889
$204$	0	0
$205$	1.73278e40	0.606808
$206$	0	0
$207$	9.14514e40	2.70215
$208$	0	0
$209$	−4.35004e38	−0.0108626
$210$	0	0
$211$	6.95408e39	0.146993	0.0734965	−	0.997295i	$-0.476584\pi$
$211$	6.95408e39	0.146993	0.0734965	+	0.997295i	$0.476584\pi$
$212$	0	0
$213$	1.52531e41	2.73348
$214$	0	0
$215$	3.75770e40	0.571807
$216$	0	0
$217$	−2.02308e40	−0.261799
$218$	0	0
$219$	−3.93211e40	−0.433365
$220$	0	0
$221$	3.31854e39	0.0311948
$222$	0	0
$223$	−7.64864e40	−0.614115	−0.307057	−	0.951691i	$-0.599344\pi$
$223$	−7.64864e40	−0.614115	−0.307057	+	0.951691i	$0.599344\pi$
$224$	0	0
$225$	−2.38256e41	−1.63625
$226$	0	0
$227$	1.63339e40	0.0960808	0.0480404	−	0.998845i	$-0.484702\pi$
$227$	1.63339e40	0.0960808	0.0480404	+	0.998845i	$0.484702\pi$
$228$	0	0
$229$	1.58770e41	0.801026	0.400513	−	0.916291i	$-0.368832\pi$
$229$	1.58770e41	0.801026	0.400513	+	0.916291i	$0.368832\pi$
$230$	0	0
$231$	−2.96185e41	−1.28337
$232$	0	0
$233$	1.16673e41	0.434751	0.217375	−	0.976088i	$-0.430250\pi$
$233$	1.16673e41	0.434751	0.217375	+	0.976088i	$0.430250\pi$
$234$	0	0
$235$	1.41780e41	0.454912
$236$	0	0
$237$	−1.03999e42	−2.87696
$238$	0	0
$239$	−5.09777e41	−1.21735	−0.608677	−	0.793418i	$-0.708299\pi$
$239$	−5.09777e41	−1.21735	−0.608677	+	0.793418i	$0.708299\pi$
$240$	0	0
$241$	−7.27282e41	−1.50108	−0.750541	−	0.660824i	$-0.770207\pi$
$241$	−7.27282e41	−1.50108	−0.750541	+	0.660824i	$0.770207\pi$
$242$	0	0
$243$	−2.74404e41	−0.490096
$244$	0	0
$245$	−8.57541e39	−0.0132693
$246$	0	0
$247$	−2.98460e38	−0.000400580 0
$248$	0	0
$249$	2.44294e42	2.84724
$250$	0	0
$251$	−4.56359e40	−0.0462399	−0.0231199	−	0.999733i	$-0.507360\pi$
$251$	−4.56359e40	−0.0462399	−0.0231199	+	0.999733i	$0.507360\pi$
$252$	0	0
$253$	−1.05822e42	−0.933180
$254$	0	0
$255$	1.27494e42	0.979571
$256$	0	0
$257$	2.36058e41	0.158194	0.0790968	−	0.996867i	$-0.474796\pi$
$257$	2.36058e41	0.158194	0.0790968	+	0.996867i	$0.474796\pi$
$258$	0	0
$259$	−1.21190e42	−0.709124
$260$	0	0
$261$	−1.47774e42	−0.755768
$262$	0	0
$263$	−7.96602e41	−0.356464	−0.178232	−	0.983988i	$-0.557038\pi$
$263$	−7.96602e41	−0.356464	−0.178232	+	0.983988i	$0.557038\pi$
$264$	0	0
$265$	1.48793e42	0.583149
$266$	0	0
$267$	−2.67129e42	−0.917856
$268$	0	0
$269$	−4.82372e42	−1.45451	−0.727256	−	0.686366i	$-0.759205\pi$
$269$	−4.82372e42	−1.45451	−0.727256	+	0.686366i	$0.759205\pi$
$270$	0	0
$271$	2.22025e42	0.588083	0.294042	−	0.955793i	$-0.405000\pi$
$271$	2.22025e42	0.588083	0.294042	+	0.955793i	$0.405000\pi$
$272$	0	0
$273$	−2.03215e41	−0.0473270
$274$	0	0
$275$	2.75695e42	0.565075
$276$	0	0
$277$	1.62622e42	0.293619	0.146809	−	0.989165i	$-0.453100\pi$
$277$	1.62622e42	0.293619	0.146809	+	0.989165i	$0.453100\pi$
$278$	0	0
$279$	3.55222e42	0.565492
$280$	0	0
$281$	6.29076e42	0.883777	0.441888	−	0.897070i	$-0.354309\pi$
$281$	6.29076e42	0.883777	0.441888	+	0.897070i	$0.354309\pi$
$282$	0	0
$283$	−4.07259e41	−0.0505368	−0.0252684	−	0.999681i	$-0.508044\pi$
$283$	−4.07259e41	−0.0505368	−0.0252684	+	0.999681i	$0.508044\pi$
$284$	0	0
$285$	−1.14664e41	−0.0125789
$286$	0	0
$287$	−1.28037e43	−1.24281
$288$	0	0
$289$	3.75055e42	0.322391
$290$	0	0
$291$	3.20627e43	2.44271
$292$	0	0
$293$	2.65171e43	1.79202	0.896010	−	0.444033i	$-0.146453\pi$
$293$	2.65171e43	1.79202	0.896010	+	0.444033i	$0.146453\pi$
$294$	0	0
$295$	4.12668e42	0.247580
$296$	0	0
$297$	2.75904e43	1.47069
$298$	0	0
$299$	−7.26051e41	−0.0344129
$300$	0	0
$301$	−2.77661e43	−1.17112
$302$	0	0
$303$	4.30459e43	1.61691
$304$	0	0
$305$	−2.67565e41	−0.00895739
$306$	0	0
$307$	−2.04163e43	−0.609615	−0.304808	−	0.952414i	$-0.598592\pi$
$307$	−2.04163e43	−0.609615	−0.304808	+	0.952414i	$0.598592\pi$
$308$	0	0
$309$	−1.18683e42	−0.0316311
$310$	0	0
$311$	−1.35407e43	−0.322355	−0.161178	−	0.986925i	$-0.551529\pi$
$311$	−1.35407e43	−0.322355	−0.161178	+	0.986925i	$0.551529\pi$
$312$	0	0
$313$	−6.11170e43	−1.30058	−0.650289	−	0.759687i	$-0.725352\pi$
$313$	−6.11170e43	−1.30058	−0.650289	+	0.759687i	$0.725352\pi$
$314$	0	0
$315$	−5.31296e43	−1.01135
$316$	0	0
$317$	7.44500e43	1.26861	0.634305	−	0.773083i	$-0.281286\pi$
$317$	7.44500e43	1.26861	0.634305	+	0.773083i	$0.281286\pi$
$318$	0	0
$319$	1.70994e43	0.261002
$320$	0	0
$321$	−1.39095e44	−1.90316
$322$	0	0
$323$	−1.38360e42	−0.0169812
$324$	0	0
$325$	1.89156e42	0.0208383
$326$	0	0
$327$	−2.45396e44	−2.42819
$328$	0	0
$329$	−1.04763e44	−0.931706
$330$	0	0
$331$	−2.60186e43	−0.208110	−0.104055	−	0.994572i	$-0.533182\pi$
$331$	−2.60186e43	−0.208110	−0.104055	+	0.994572i	$0.533182\pi$
$332$	0	0
$333$	2.12792e44	1.53172
$334$	0	0
$335$	3.29590e43	0.213643
$336$	0	0
$337$	−3.62826e43	−0.211921	−0.105961	−	0.994370i	$-0.533792\pi$
$337$	−3.62826e43	−0.211921	−0.105961	+	0.994370i	$0.533792\pi$
$338$	0	0
$339$	2.88993e44	1.52192
$340$	0	0
$341$	−4.11040e43	−0.195291
$342$	0	0
$343$	2.36257e44	1.01330
$344$	0	0
$345$	−2.78939e44	−1.08063
$346$	0	0
$347$	−3.83287e44	−1.34202	−0.671009	−	0.741449i	$-0.734139\pi$
$347$	−3.83287e44	−1.34202	−0.671009	+	0.741449i	$0.734139\pi$
$348$	0	0
$349$	1.99513e44	0.631723	0.315862	−	0.948805i	$-0.397706\pi$
$349$	1.99513e44	0.631723	0.315862	+	0.948805i	$0.397706\pi$
$350$	0	0
$351$	1.89300e43	0.0542345
$352$	0	0
$353$	2.67595e44	0.694098	0.347049	−	0.937847i	$-0.387184\pi$
$353$	2.67595e44	0.694098	0.347049	+	0.937847i	$0.387184\pi$
$354$	0	0
$355$	−3.16604e44	−0.743909
$356$	0	0
$357$	−9.42065e44	−2.00626
$358$	0	0
$359$	−7.93674e44	−1.53281	−0.766407	−	0.642355i	$-0.777957\pi$
$359$	−7.93674e44	−1.53281	−0.766407	+	0.642355i	$0.777957\pi$
$360$	0	0
$361$	−5.70534e44	−0.999782
$362$	0	0
$363$	5.10320e44	0.811852
$364$	0	0
$365$	8.16175e43	0.117939
$366$	0	0
$367$	4.95841e44	0.651158	0.325579	−	0.945515i	$-0.394441\pi$
$367$	4.95841e44	0.651158	0.325579	+	0.945515i	$0.394441\pi$
$368$	0	0
$369$	2.24814e45	2.68449
$370$	0	0
$371$	−1.09945e45	−1.19435
$372$	0	0
$373$	1.46949e44	0.145299	0.0726496	−	0.997358i	$-0.476855\pi$
$373$	1.46949e44	0.145299	0.0726496	+	0.997358i	$0.476855\pi$
$374$	0	0
$375$	1.67274e45	1.50619
$376$	0	0
$377$	1.17321e43	0.00962500
$378$	0	0
$379$	7.55442e44	0.564956	0.282478	−	0.959274i	$-0.408844\pi$
$379$	7.55442e44	0.564956	0.282478	+	0.959274i	$0.408844\pi$
$380$	0	0
$381$	2.33184e44	0.159042
$382$	0	0
$383$	5.97753e44	0.372000	0.186000	−	0.982550i	$-0.440448\pi$
$383$	5.97753e44	0.372000	0.186000	+	0.982550i	$0.440448\pi$
$384$	0	0
$385$	6.14781e44	0.349267
$386$	0	0
$387$	4.87530e45	2.52964
$388$	0	0
$389$	−2.41609e45	−1.14550	−0.572750	−	0.819730i	$-0.694123\pi$
$389$	−2.41609e45	−1.14550	−0.572750	+	0.819730i	$0.694123\pi$
$390$	0	0
$391$	−3.36583e45	−1.45881
$392$	0	0
$393$	8.33657e45	3.30459
$394$	0	0
$395$	2.15867e45	0.782957
$396$	0	0
$397$	−1.25639e45	−0.417148	−0.208574	−	0.978007i	$-0.566882\pi$
$397$	−1.25639e45	−0.417148	−0.208574	+	0.978007i	$0.566882\pi$
$398$	0	0
$399$	8.47267e43	0.0257629
$400$	0	0
$401$	−3.77150e45	−1.05072	−0.525361	−	0.850880i	$-0.676070\pi$
$401$	−3.77150e45	−1.05072	−0.525361	+	0.850880i	$0.676070\pi$
$402$	0	0
$403$	−2.82018e43	−0.00720176
$404$	0	0
$405$	2.89306e45	0.677478
$406$	0	0
$407$	−2.46229e45	−0.528977
$408$	0	0
$409$	−5.28767e45	−1.04257	−0.521286	−	0.853382i	$-0.674548\pi$
$409$	−5.28767e45	−1.04257	−0.521286	+	0.853382i	$0.674548\pi$
$410$	0	0
$411$	−1.05379e46	−1.90775
$412$	0	0
$413$	−3.04925e45	−0.507069
$414$	0	0
$415$	−5.07072e45	−0.774869
$416$	0	0
$417$	−7.48542e45	−1.05156
$418$	0	0
$419$	−3.73280e45	−0.482270	−0.241135	−	0.970492i	$-0.577520\pi$
$419$	−3.73280e45	−0.482270	−0.241135	+	0.970492i	$0.577520\pi$
$420$	0	0
$421$	−1.18084e45	−0.140364	−0.0701819	−	0.997534i	$-0.522358\pi$
$421$	−1.18084e45	−0.140364	−0.0701819	+	0.997534i	$0.522358\pi$
$422$	0	0
$423$	1.83947e46	2.01251
$424$	0	0
$425$	8.76893e45	0.883365
$426$	0	0
$427$	1.97706e44	0.0183456
$428$	0	0
$429$	−4.12882e44	−0.0353040
$430$	0	0
$431$	−5.00269e45	−0.394322	−0.197161	−	0.980371i	$-0.563172\pi$
$431$	−5.00269e45	−0.394322	−0.197161	+	0.980371i	$0.563172\pi$
$432$	0	0
$433$	−1.22691e46	−0.891817	−0.445909	−	0.895079i	$-0.647119\pi$
$433$	−1.22691e46	−0.891817	−0.445909	+	0.895079i	$0.647119\pi$
$434$	0	0
$435$	4.50730e45	0.302242
$436$	0	0
$437$	3.02714e44	0.0187330
$438$	0	0
$439$	−1.91016e46	−1.09129	−0.545645	−	0.838016i	$-0.683715\pi$
$439$	−1.91016e46	−1.09129	−0.545645	+	0.838016i	$0.683715\pi$
$440$	0	0
$441$	−1.11259e45	−0.0587027
$442$	0	0
$443$	1.54265e46	0.751973	0.375987	−	0.926625i	$-0.377304\pi$
$443$	1.54265e46	0.751973	0.375987	+	0.926625i	$0.377304\pi$
$444$	0	0
$445$	5.54471e45	0.249792
$446$	0	0
$447$	−2.71570e46	−1.13110
$448$	0	0
$449$	−3.71387e46	−1.43059	−0.715296	−	0.698822i	$-0.753708\pi$
$449$	−3.71387e46	−1.43059	−0.715296	+	0.698822i	$0.753708\pi$
$450$	0	0
$451$	−2.60140e46	−0.927080
$452$	0	0
$453$	6.94356e46	2.29015
$454$	0	0
$455$	4.21806e44	0.0128799
$456$	0	0
$457$	−1.10776e46	−0.313266	−0.156633	−	0.987657i	$-0.550064\pi$
$457$	−1.10776e46	−0.313266	−0.156633	+	0.987657i	$0.550064\pi$
$458$	0	0
$459$	8.77558e46	2.29908
$460$	0	0
$461$	−3.91769e46	−0.951183	−0.475592	−	0.879666i	$-0.657766\pi$
$461$	−3.91769e46	−0.951183	−0.475592	+	0.879666i	$0.657766\pi$
$462$	0	0
$463$	−5.50481e46	−1.23901	−0.619505	−	0.784992i	$-0.712667\pi$
$463$	−5.50481e46	−1.23901	−0.619505	+	0.784992i	$0.712667\pi$
$464$	0	0
$465$	−1.08348e46	−0.226148
$466$	0	0
$467$	−2.96872e46	−0.574808	−0.287404	−	0.957809i	$-0.592792\pi$
$467$	−2.96872e46	−0.574808	−0.287404	+	0.957809i	$0.592792\pi$
$468$	0	0
$469$	−2.43537e46	−0.437563
$470$	0	0
$471$	4.49889e44	0.00750308
$472$	0	0
$473$	−5.64137e46	−0.873605
$474$	0	0
$475$	−7.88652e44	−0.0113435
$476$	0	0
$477$	1.93046e47	2.57982
$478$	0	0
$479$	7.25716e46	0.901354	0.450677	−	0.892687i	$-0.351183\pi$
$479$	7.25716e46	0.901354	0.450677	+	0.892687i	$0.351183\pi$
$480$	0	0
$481$	−1.68940e45	−0.0195071
$482$	0	0
$483$	2.06111e47	2.21323
$484$	0	0
$485$	−6.65514e46	−0.664778
$486$	0	0
$487$	1.32479e47	1.23137	0.615686	−	0.787992i	$-0.288879\pi$
$487$	1.32479e47	1.23137	0.615686	+	0.787992i	$0.288879\pi$
$488$	0	0
$489$	3.15046e47	2.72565
$490$	0	0
$491$	−5.04170e46	−0.406117	−0.203058	−	0.979167i	$-0.565088\pi$
$491$	−5.04170e46	−0.406117	−0.203058	+	0.979167i	$0.565088\pi$
$492$	0	0
$493$	5.43876e46	0.408017
$494$	0	0
$495$	−1.07946e47	−0.754425
$496$	0	0
$497$	2.33942e47	1.52360
$498$	0	0
$499$	−3.13288e46	−0.190189	−0.0950944	−	0.995468i	$-0.530315\pi$
$499$	−3.13288e46	−0.190189	−0.0950944	+	0.995468i	$0.530315\pi$
$500$	0	0
$501$	−3.63671e47	−2.05849
$502$	0	0
$503$	−2.79560e47	−1.47582	−0.737912	−	0.674897i	$-0.764188\pi$
$503$	−2.79560e47	−1.47582	−0.737912	+	0.674897i	$0.764188\pi$
$504$	0	0
$505$	−8.93488e46	−0.440038
$506$	0	0
$507$	3.84676e47	1.76789
$508$	0	0
$509$	−2.02483e47	−0.868614	−0.434307	−	0.900765i	$-0.643007\pi$
$509$	−2.02483e47	−0.868614	−0.434307	+	0.900765i	$0.643007\pi$
$510$	0	0
$511$	−6.03080e46	−0.241551
$512$	0	0
$513$	−7.89251e45	−0.0295231
$514$	0	0
$515$	2.46346e45	0.00860833
$516$	0	0
$517$	−2.12852e47	−0.695014
$518$	0	0
$519$	−6.74556e47	−2.05868
$520$	0	0
$521$	−7.62521e46	−0.217566	−0.108783	−	0.994066i	$-0.534695\pi$
$521$	−7.62521e46	−0.217566	−0.108783	+	0.994066i	$0.534695\pi$
$522$	0	0
$523$	−2.14532e47	−0.572417	−0.286208	−	0.958167i	$-0.592395\pi$
$523$	−2.14532e47	−0.572417	−0.286208	+	0.958167i	$0.592395\pi$
$524$	0	0
$525$	−5.36976e47	−1.34019
$526$	0	0
$527$	−1.30738e47	−0.305293
$528$	0	0
$529$	2.78811e47	0.609307
$530$	0	0
$531$	5.35401e47	1.09528
$532$	0	0
$533$	−1.78484e46	−0.0341880
$534$	0	0
$535$	2.88716e47	0.517940
$536$	0	0
$537$	−3.75339e47	−0.630774
$538$	0	0
$539$	1.28741e46	0.0202728
$540$	0	0
$541$	1.23166e48	1.81777	0.908884	−	0.417049i	$-0.136936\pi$
$541$	1.23166e48	1.81777	0.908884	+	0.417049i	$0.136936\pi$
$542$	0	0
$543$	−1.23497e48	−1.70867
$544$	0	0
$545$	5.09360e47	0.660825
$546$	0	0
$547$	7.20961e47	0.877271	0.438635	−	0.898665i	$-0.355462\pi$
$547$	7.20961e47	0.877271	0.438635	+	0.898665i	$0.355462\pi$
$548$	0	0
$549$	−3.47142e46	−0.0396270
$550$	0	0
$551$	−4.89146e45	−0.00523945
$552$	0	0
$553$	−1.59507e48	−1.60357
$554$	0	0
$555$	−6.49043e47	−0.612557
$556$	0	0
$557$	5.88528e47	0.521556	0.260778	−	0.965399i	$-0.416021\pi$
$557$	5.88528e47	0.521556	0.260778	+	0.965399i	$0.416021\pi$
$558$	0	0
$559$	−3.87060e46	−0.0322160
$560$	0	0
$561$	−1.91404e48	−1.49659
$562$	0	0
$563$	1.44205e48	1.05946	0.529730	−	0.848166i	$-0.322293\pi$
$563$	1.44205e48	1.05946	0.529730	+	0.848166i	$0.322293\pi$
$564$	0	0
$565$	−5.99852e47	−0.414187
$566$	0	0
$567$	−2.13772e48	−1.38754
$568$	0	0
$569$	1.87904e48	1.14676	0.573379	−	0.819290i	$-0.305632\pi$
$569$	1.87904e48	1.14676	0.573379	+	0.819290i	$0.305632\pi$
$570$	0	0
$571$	1.57690e48	0.905050	0.452525	−	0.891752i	$-0.350523\pi$
$571$	1.57690e48	0.905050	0.452525	+	0.891752i	$0.350523\pi$
$572$	0	0
$573$	−1.41361e47	−0.0763172
$574$	0	0
$575$	−1.91852e48	−0.974495
$576$	0	0
$577$	−7.02869e47	−0.335968	−0.167984	−	0.985790i	$-0.553726\pi$
$577$	−7.02869e47	−0.335968	−0.167984	+	0.985790i	$0.553726\pi$
$578$	0	0
$579$	5.78720e48	2.60371
$580$	0	0
$581$	3.74681e48	1.58701
$582$	0	0
$583$	−2.23381e48	−0.890934
$584$	0	0
$585$	−7.40628e46	−0.0278210
$586$	0	0
$587$	1.10752e48	0.391908	0.195954	−	0.980613i	$-0.437220\pi$
$587$	1.10752e48	0.391908	0.195954	+	0.980613i	$0.437220\pi$
$588$	0	0
$589$	1.17582e46	0.00392034
$590$	0	0
$591$	−4.39444e48	−1.38077
$592$	0	0
$593$	−2.15841e48	−0.639258	−0.319629	−	0.947543i	$-0.603558\pi$
$593$	−2.15841e48	−0.639258	−0.319629	+	0.947543i	$0.603558\pi$
$594$	0	0
$595$	1.95541e48	0.545999
$596$	0	0
$597$	7.19701e48	1.89497
$598$	0	0
$599$	5.67895e48	1.41026	0.705129	−	0.709079i	$-0.250889\pi$
$599$	5.67895e48	1.41026	0.705129	+	0.709079i	$0.250889\pi$
$600$	0	0
$601$	6.99029e48	1.63754	0.818770	−	0.574122i	$-0.194656\pi$
$601$	6.99029e48	1.63754	0.818770	+	0.574122i	$0.194656\pi$
$602$	0	0
$603$	4.27614e48	0.945147
$604$	0	0
$605$	−1.05925e48	−0.220944
$606$	0	0
$607$	4.92572e48	0.969768	0.484884	−	0.874578i	$-0.338862\pi$
$607$	4.92572e48	0.969768	0.484884	+	0.874578i	$0.338862\pi$
$608$	0	0
$609$	−3.33049e48	−0.619022
$610$	0	0
$611$	−1.46040e47	−0.0256300
$612$	0	0
$613$	3.59765e48	0.596295	0.298148	−	0.954520i	$-0.403631\pi$
$613$	3.59765e48	0.596295	0.298148	+	0.954520i	$0.403631\pi$
$614$	0	0
$615$	−6.85712e48	−1.07356
$616$	0	0
$617$	−3.73363e48	−0.552257	−0.276128	−	0.961121i	$-0.589051\pi$
$617$	−3.73363e48	−0.552257	−0.276128	+	0.961121i	$0.589051\pi$
$618$	0	0
$619$	−2.84346e48	−0.397430	−0.198715	−	0.980057i	$-0.563677\pi$
$619$	−2.84346e48	−0.397430	−0.198715	+	0.980057i	$0.563677\pi$
$620$	0	0
$621$	−1.91998e49	−2.53626
$622$	0	0
$623$	−4.09705e48	−0.511600
$624$	0	0
$625$	3.03465e48	0.358268
$626$	0	0
$627$	1.72144e47	0.0192180
$628$	0	0
$629$	−7.83171e48	−0.826933
$630$	0	0
$631$	−8.72640e48	−0.871608	−0.435804	−	0.900041i	$-0.643536\pi$
$631$	−8.72640e48	−0.871608	−0.435804	+	0.900041i	$0.643536\pi$
$632$	0	0
$633$	−2.75193e48	−0.260059
$634$	0	0
$635$	−4.84011e47	−0.0432828
$636$	0	0
$637$	8.83305e45	0.000747602 0
$638$	0	0
$639$	−4.10766e49	−3.29101
$640$	0	0
$641$	1.94871e49	1.47820	0.739098	−	0.673598i	$-0.235252\pi$
$641$	1.94871e49	1.47820	0.739098	+	0.673598i	$0.235252\pi$
$642$	0	0
$643$	2.20357e49	1.58283	0.791417	−	0.611277i	$-0.209344\pi$
$643$	2.20357e49	1.58283	0.791417	+	0.611277i	$0.209344\pi$
$644$	0	0
$645$	−1.48703e49	−1.01164
$646$	0	0
$647$	1.15002e49	0.741106	0.370553	−	0.928811i	$-0.379168\pi$
$647$	1.15002e49	0.741106	0.370553	+	0.928811i	$0.379168\pi$
$648$	0	0
$649$	−6.19531e48	−0.378252
$650$	0	0
$651$	8.00592e48	0.463174
$652$	0	0
$653$	−3.36604e48	−0.184560	−0.0922801	−	0.995733i	$-0.529416\pi$
$653$	−3.36604e48	−0.184560	−0.0922801	+	0.995733i	$0.529416\pi$
$654$	0	0
$655$	−1.73039e49	−0.899337
$656$	0	0
$657$	1.05892e49	0.521757
$658$	0	0
$659$	1.07733e49	0.503334	0.251667	−	0.967814i	$-0.419021\pi$
$659$	1.07733e49	0.503334	0.251667	+	0.967814i	$0.419021\pi$
$660$	0	0
$661$	−1.67936e49	−0.744079	−0.372040	−	0.928217i	$-0.621341\pi$
$661$	−1.67936e49	−0.744079	−0.372040	+	0.928217i	$0.621341\pi$
$662$	0	0
$663$	−1.31324e48	−0.0551897
$664$	0	0
$665$	−1.75864e47	−0.00701131
$666$	0	0
$667$	−1.18993e49	−0.450109
$668$	0	0
$669$	3.02679e49	1.08649
$670$	0	0
$671$	4.01690e47	0.0136851
$672$	0	0
$673$	2.60356e49	0.841984	0.420992	−	0.907064i	$-0.361682\pi$
$673$	2.60356e49	0.841984	0.420992	+	0.907064i	$0.361682\pi$
$674$	0	0
$675$	5.00207e49	1.53580
$676$	0	0
$677$	−5.14137e49	−1.49891	−0.749457	−	0.662053i	$-0.769685\pi$
$677$	−5.14137e49	−1.49891	−0.749457	+	0.662053i	$0.769685\pi$
$678$	0	0
$679$	4.91756e49	1.36153
$680$	0	0
$681$	−6.46378e48	−0.169986
$682$	0	0
$683$	−3.66408e47	−0.00915382	−0.00457691	−	0.999990i	$-0.501457\pi$
$683$	−3.66408e47	−0.00915382	−0.00457691	+	0.999990i	$0.501457\pi$
$684$	0	0
$685$	2.18732e49	0.519190
$686$	0	0
$687$	−6.28298e49	−1.41717
$688$	0	0
$689$	−1.53263e48	−0.0328550
$690$	0	0
$691$	−8.07844e49	−1.64612	−0.823060	−	0.567954i	$-0.807735\pi$
$691$	−8.07844e49	−1.64612	−0.823060	+	0.567954i	$0.807735\pi$
$692$	0	0
$693$	7.97626e49	1.54514
$694$	0	0
$695$	1.55372e49	0.286180
$696$	0	0
$697$	−8.27418e49	−1.44928
$698$	0	0
$699$	−4.61707e49	−0.769159
$700$	0	0
$701$	1.36918e49	0.216968	0.108484	−	0.994098i	$-0.465400\pi$
$701$	1.36918e49	0.216968	0.108484	+	0.994098i	$0.465400\pi$
$702$	0	0
$703$	7.04362e47	0.0106189
$704$	0	0
$705$	−5.61065e49	−0.804828
$706$	0	0
$707$	6.60208e49	0.901242
$708$	0	0
$709$	9.81931e49	1.27577	0.637884	−	0.770132i	$-0.279810\pi$
$709$	9.81931e49	1.27577	0.637884	+	0.770132i	$0.279810\pi$
$710$	0	0
$711$	2.80069e50	3.46376
$712$	0	0
$713$	2.86037e49	0.336787
$714$	0	0
$715$	8.57006e47	0.00960789
$716$	0	0
$717$	2.01733e50	2.15374
$718$	0	0
$719$	−1.63037e50	−1.65780	−0.828899	−	0.559398i	$-0.811032\pi$
$719$	−1.63037e50	−1.65780	−0.828899	+	0.559398i	$0.811032\pi$
$720$	0	0
$721$	−1.82028e48	−0.0176307
$722$	0	0
$723$	2.87806e50	2.65571
$724$	0	0
$725$	3.10009e49	0.272558
$726$	0	0
$727$	−3.34890e49	−0.280576	−0.140288	−	0.990111i	$-0.544803\pi$
$727$	−3.34890e49	−0.280576	−0.140288	+	0.990111i	$0.544803\pi$
$728$	0	0
$729$	−6.76269e49	−0.539993
$730$	0	0
$731$	−1.79433e50	−1.36568
$732$	0	0
$733$	−2.00965e50	−1.45814	−0.729071	−	0.684438i	$-0.760048\pi$
$733$	−2.00965e50	−1.45814	−0.729071	+	0.684438i	$0.760048\pi$
$734$	0	0
$735$	3.39354e48	0.0234760
$736$	0	0
$737$	−4.94807e49	−0.326404
$738$	0	0
$739$	1.06372e50	0.669190	0.334595	−	0.942362i	$-0.391401\pi$
$739$	1.06372e50	0.669190	0.334595	+	0.942362i	$0.391401\pi$
$740$	0	0
$741$	1.18109e47	0.000708704 0
$742$	0	0
$743$	−7.25015e49	−0.414995	−0.207498	−	0.978236i	$-0.566532\pi$
$743$	−7.25015e49	−0.414995	−0.207498	+	0.978236i	$0.566532\pi$
$744$	0	0
$745$	5.63690e49	0.307826
$746$	0	0
$747$	−6.57883e50	−3.42798
$748$	0	0
$749$	−2.13335e50	−1.06079
$750$	0	0
$751$	4.51226e49	0.214139	0.107069	−	0.994252i	$-0.465853\pi$
$751$	4.51226e49	0.214139	0.107069	+	0.994252i	$0.465853\pi$
$752$	0	0
$753$	1.80595e49	0.0818074
$754$	0	0
$755$	−1.44125e50	−0.623259
$756$	0	0
$757$	−1.21415e50	−0.501296	−0.250648	−	0.968078i	$-0.580644\pi$
$757$	−1.21415e50	−0.501296	−0.250648	+	0.968078i	$0.580644\pi$
$758$	0	0
$759$	4.18767e50	1.65098
$760$	0	0
$761$	3.14799e50	1.18523	0.592614	−	0.805486i	$-0.298096\pi$
$761$	3.14799e50	1.18523	0.592614	+	0.805486i	$0.298096\pi$
$762$	0	0
$763$	−3.76372e50	−1.35344
$764$	0	0
$765$	−3.43341e50	−1.17937
$766$	0	0
$767$	−4.25066e48	−0.0139488
$768$	0	0
$769$	−5.16540e50	−1.61954	−0.809772	−	0.586744i	$-0.800409\pi$
$769$	−5.16540e50	−1.61954	−0.809772	+	0.586744i	$0.800409\pi$
$770$	0	0
$771$	−9.34149e49	−0.279875
$772$	0	0
$773$	−4.51959e50	−1.29407	−0.647035	−	0.762460i	$-0.723991\pi$
$773$	−4.51959e50	−1.29407	−0.647035	+	0.762460i	$0.723991\pi$
$774$	0	0
$775$	−7.45206e49	−0.203937
$776$	0	0
$777$	4.79585e50	1.25458
$778$	0	0
$779$	7.44157e48	0.0186105
$780$	0	0
$781$	4.75312e50	1.13654
$782$	0	0
$783$	3.10244e50	0.709369
$784$	0	0
$785$	−9.33820e47	−0.00204195
$786$	0	0
$787$	7.76649e50	1.62430	0.812151	−	0.583447i	$-0.198296\pi$
$787$	7.76649e50	1.62430	0.812151	+	0.583447i	$0.198296\pi$
$788$	0	0
$789$	3.15238e50	0.630655
$790$	0	0
$791$	4.43237e50	0.848297
$792$	0	0
$793$	2.75603e47	0.000504665 0
$794$	0	0
$795$	−5.88817e50	−1.03171
$796$	0	0
$797$	−7.96535e49	−0.133562	−0.0667812	−	0.997768i	$-0.521273\pi$
$797$	−7.96535e49	−0.133562	−0.0667812	+	0.997768i	$0.521273\pi$
$798$	0	0
$799$	−6.77011e50	−1.08649
$800$	0	0
$801$	7.19379e50	1.10507
$802$	0	0
$803$	−1.22531e50	−0.180187
$804$	0	0
$805$	−4.27818e50	−0.602325
$806$	0	0
$807$	1.90889e51	2.57332
$808$	0	0
$809$	8.80158e50	1.13622	0.568109	−	0.822954i	$-0.307675\pi$
$809$	8.80158e50	1.13622	0.568109	+	0.822954i	$0.307675\pi$
$810$	0	0
$811$	−1.29798e51	−1.60473	−0.802364	−	0.596835i	$-0.796425\pi$
$811$	−1.29798e51	−1.60473	−0.802364	+	0.596835i	$0.796425\pi$
$812$	0	0
$813$	−8.78616e50	−1.04043
$814$	0	0
$815$	−6.53931e50	−0.741778
$816$	0	0
$817$	1.61377e49	0.0175370
$818$	0	0
$819$	5.47258e49	0.0569802
$820$	0	0
$821$	8.37740e50	0.835804	0.417902	−	0.908492i	$-0.362766\pi$
$821$	8.37740e50	0.835804	0.417902	+	0.908492i	$0.362766\pi$
$822$	0	0
$823$	1.79490e51	1.71610	0.858051	−	0.513564i	$-0.171675\pi$
$823$	1.79490e51	1.71610	0.858051	+	0.513564i	$0.171675\pi$
$824$	0	0
$825$	−1.09100e51	−0.999728
$826$	0	0
$827$	−6.78828e50	−0.596230	−0.298115	−	0.954530i	$-0.596358\pi$
$827$	−6.78828e50	−0.596230	−0.298115	+	0.954530i	$0.596358\pi$
$828$	0	0
$829$	1.62563e51	1.36873	0.684363	−	0.729141i	$-0.260080\pi$
$829$	1.62563e51	1.36873	0.684363	+	0.729141i	$0.260080\pi$
$830$	0	0
$831$	−6.43543e50	−0.519469
$832$	0	0
$833$	4.09483e49	0.0316919
$834$	0	0
$835$	7.54860e50	0.560213
$836$	0	0
$837$	−7.45772e50	−0.530775
$838$	0	0
$839$	2.62276e51	1.79029	0.895147	−	0.445771i	$-0.147070\pi$
$839$	2.62276e51	1.79029	0.895147	+	0.445771i	$0.147070\pi$
$840$	0	0
$841$	−1.33504e51	−0.874108
$842$	0	0
$843$	−2.48944e51	−1.56357
$844$	0	0
$845$	−7.98459e50	−0.481128
$846$	0	0
$847$	7.82695e50	0.452515
$848$	0	0
$849$	1.61164e50	0.0894094
$850$	0	0
$851$	1.71347e51	0.912242
$852$	0	0
$853$	2.63497e51	1.34638	0.673191	−	0.739469i	$-0.264923\pi$
$853$	2.63497e51	1.34638	0.673191	+	0.739469i	$0.264923\pi$
$854$	0	0
$855$	3.08791e49	0.0151446
$856$	0	0
$857$	1.21045e51	0.569880	0.284940	−	0.958545i	$-0.408026\pi$
$857$	1.21045e51	0.569880	0.284940	+	0.958545i	$0.408026\pi$
$858$	0	0
$859$	−1.22181e51	−0.552234	−0.276117	−	0.961124i	$-0.589048\pi$
$859$	−1.22181e51	−0.552234	−0.276117	+	0.961124i	$0.589048\pi$
$860$	0	0
$861$	5.06680e51	2.19876
$862$	0	0
$863$	1.06383e51	0.443284	0.221642	−	0.975128i	$-0.428858\pi$
$863$	1.06383e51	0.443284	0.221642	+	0.975128i	$0.428858\pi$
$864$	0	0
$865$	1.40015e51	0.560265
$866$	0	0
$867$	−1.48420e51	−0.570373
$868$	0	0
$869$	−3.24078e51	−1.19620
$870$	0	0
$871$	−3.39491e49	−0.0120368
$872$	0	0
$873$	−8.63448e51	−2.94094
$874$	0	0
$875$	2.56553e51	0.839531
$876$	0	0
$877$	2.07831e51	0.653459	0.326729	−	0.945118i	$-0.394053\pi$
$877$	2.07831e51	0.653459	0.326729	+	0.945118i	$0.394053\pi$
$878$	0	0
$879$	−1.04936e52	−3.17043
$880$	0	0
$881$	−5.23747e51	−1.52070	−0.760350	−	0.649514i	$-0.774973\pi$
$881$	−5.23747e51	−1.52070	−0.760350	+	0.649514i	$0.774973\pi$
$882$	0	0
$883$	2.39426e51	0.668126	0.334063	−	0.942551i	$-0.391580\pi$
$883$	2.39426e51	0.668126	0.334063	+	0.942551i	$0.391580\pi$
$884$	0	0
$885$	−1.63305e51	−0.438017
$886$	0	0
$887$	−4.11591e51	−1.06121	−0.530606	−	0.847618i	$-0.678036\pi$
$887$	−4.11591e51	−1.06121	−0.530606	+	0.847618i	$0.678036\pi$
$888$	0	0
$889$	3.57641e50	0.0886476
$890$	0	0
$891$	−4.34331e51	−1.03505
$892$	0	0
$893$	6.08885e49	0.0139519
$894$	0	0
$895$	7.79079e50	0.171664
$896$	0	0
$897$	2.87319e50	0.0608832
$898$	0	0
$899$	−4.62200e50	−0.0941966
$900$	0	0
$901$	−7.10499e51	−1.39277
$902$	0	0
$903$	1.09878e52	2.07194
$904$	0	0
$905$	2.56338e51	0.465011
$906$	0	0
$907$	−1.27732e51	−0.222932	−0.111466	−	0.993768i	$-0.535555\pi$
$907$	−1.27732e51	−0.222932	−0.111466	+	0.993768i	$0.535555\pi$
$908$	0	0
$909$	−1.15922e52	−1.94670
$910$	0	0
$911$	−4.62833e51	−0.747915	−0.373958	−	0.927446i	$-0.621999\pi$
$911$	−4.62833e51	−0.747915	−0.373958	+	0.927446i	$0.621999\pi$
$912$	0	0
$913$	7.61259e51	1.18384
$914$	0	0
$915$	1.05883e50	0.0158474
$916$	0	0
$917$	1.27861e52	1.84193
$918$	0	0
$919$	−3.73013e50	−0.0517253	−0.0258626	−	0.999666i	$-0.508233\pi$
$919$	−3.73013e50	−0.0517253	−0.0258626	+	0.999666i	$0.508233\pi$
$920$	0	0
$921$	8.07933e51	1.07853
$922$	0	0
$923$	3.26116e50	0.0419123
$924$	0	0
$925$	−4.46407e51	−0.552396
$926$	0	0
$927$	3.19613e50	0.0380828
$928$	0	0
$929$	3.82307e51	0.438669	0.219334	−	0.975650i	$-0.429611\pi$
$929$	3.82307e51	0.438669	0.219334	+	0.975650i	$0.429611\pi$
$930$	0	0
$931$	−3.68277e48	−0.000406964 0
$932$	0	0
$933$	5.35845e51	0.570309
$934$	0	0
$935$	3.97291e51	0.407292
$936$	0	0
$937$	1.21639e52	1.20124	0.600619	−	0.799536i	$-0.294921\pi$
$937$	1.21639e52	1.20124	0.600619	+	0.799536i	$0.294921\pi$
$938$	0	0
$939$	2.41858e52	2.30098
$940$	0	0
$941$	−1.11091e52	−1.01827	−0.509133	−	0.860688i	$-0.670034\pi$
$941$	−1.11091e52	−1.01827	−0.509133	+	0.860688i	$0.670034\pi$
$942$	0	0
$943$	1.81028e52	1.59879
$944$	0	0
$945$	1.11543e52	0.949261
$946$	0	0
$947$	−3.07714e51	−0.252362	−0.126181	−	0.992007i	$-0.540272\pi$
$947$	−3.07714e51	−0.252362	−0.126181	+	0.992007i	$0.540272\pi$
$948$	0	0
$949$	−8.40695e49	−0.00664477
$950$	0	0
$951$	−2.94620e52	−2.24442
$952$	0	0
$953$	4.41309e50	0.0324053	0.0162027	−	0.999869i	$-0.494842\pi$
$953$	4.41309e50	0.0324053	0.0162027	+	0.999869i	$0.494842\pi$
$954$	0	0
$955$	2.93417e50	0.0207695
$956$	0	0
$957$	−6.76673e51	−0.461764
$958$	0	0
$959$	−1.61623e52	−1.06335
$960$	0	0
$961$	−1.46527e52	−0.929519
$962$	0	0
$963$	3.74584e52	2.29134
$964$	0	0
$965$	−1.20123e52	−0.708594
$966$	0	0
$967$	−2.95259e52	−1.67973	−0.839866	−	0.542795i	$-0.817366\pi$
$967$	−2.95259e52	−1.67973	−0.839866	+	0.542795i	$0.817366\pi$
$968$	0	0
$969$	5.47532e50	0.0300430
$970$	0	0
$971$	8.31094e51	0.439859	0.219930	−	0.975516i	$-0.429417\pi$
$971$	8.31094e51	0.439859	0.219930	+	0.975516i	$0.429417\pi$
$972$	0	0
$973$	−1.14806e52	−0.586126
$974$	0	0
$975$	−7.48546e50	−0.0368670
$976$	0	0
$977$	−2.16398e52	−1.02825	−0.514125	−	0.857715i	$-0.671883\pi$
$977$	−2.16398e52	−1.02825	−0.514125	+	0.857715i	$0.671883\pi$
$978$	0	0
$979$	−8.32419e51	−0.381632
$980$	0	0
$981$	6.60851e52	2.92345
$982$	0	0
$983$	−1.17980e51	−0.0503643	−0.0251822	−	0.999683i	$-0.508017\pi$
$983$	−1.17980e51	−0.0503643	−0.0251822	+	0.999683i	$0.508017\pi$
$984$	0	0
$985$	9.12139e51	0.375774
$986$	0	0
$987$	4.14577e52	1.64837
$988$	0	0
$989$	3.92576e52	1.50657
$990$	0	0
$991$	1.92041e52	0.711387	0.355694	−	0.934603i	$-0.384245\pi$
$991$	1.92041e52	0.711387	0.355694	+	0.934603i	$0.384245\pi$
$992$	0	0
$993$	1.02963e52	0.368188
$994$	0	0
$995$	−1.49386e52	−0.515710
$996$	0	0
$997$	−4.51709e52	−1.50554	−0.752772	−	0.658281i	$-0.771284\pi$
$997$	−4.51709e52	−1.50554	−0.752772	+	0.658281i	$0.771284\pi$
$998$	0	0
$999$	−4.46746e52	−1.43769

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.36.a.d.1.1		3
4.3	odd	2		1.36.a.a.1.2	✓	3
12.11	even	2		9.36.a.b.1.2		3
20.3	even	4		25.36.b.a.24.4		6
20.7	even	4		25.36.b.a.24.3		6
20.19	odd	2		25.36.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.36.a.a.1.2	✓	3	4.3	odd	2
9.36.a.b.1.2		3	12.11	even	2
16.36.a.d.1.1		3	1.1	even	1	trivial
25.36.a.a.1.2		3	20.19	odd	2
25.36.b.a.24.3		6	20.7	even	4
25.36.b.a.24.4		6	20.3	even	4

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:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-3.95729e8 q^{3} +8.21401e11 q^{5} -6.06942e14 q^{7} +1.06570e17 q^{9} +O(q^{10})\)	\(q-3.95729e8 q^{3} +8.21401e11 q^{5} -6.06942e14 q^{7} +1.06570e17 q^{9} -1.23316e18 q^{11} -8.46079e17 q^{13} -3.25052e20 q^{15} -3.92226e21 q^{17} +3.52757e20 q^{19} +2.40184e23 q^{21} +8.58137e23 q^{23} -2.23568e24 q^{25} -2.23738e25 q^{27} -1.38664e25 q^{29} +3.33324e25 q^{31} +4.87995e26 q^{33} -4.98543e26 q^{35} +1.99674e27 q^{37} +3.34818e26 q^{39} +2.10955e28 q^{41} +4.57475e28 q^{43} +8.75365e28 q^{45} +1.72608e29 q^{47} -1.04400e28 q^{49} +1.55215e30 q^{51} +1.81145e30 q^{53} -1.01292e30 q^{55} -1.39596e29 q^{57} +5.02395e30 q^{59} -3.25742e29 q^{61} -6.46817e31 q^{63} -6.94970e29 q^{65} +4.01253e31 q^{67} -3.39589e32 q^{69} -3.85444e32 q^{71} +9.93637e31 q^{73} +8.84724e32 q^{75} +7.48454e32 q^{77} +2.62804e33 q^{79} +3.52211e33 q^{81} -6.17326e33 q^{83} -3.22175e33 q^{85} +5.48733e33 q^{87} +6.75031e33 q^{89} +5.13521e32 q^{91} -1.31906e34 q^{93} +2.89755e32 q^{95} -8.10218e34 q^{97} -1.31417e35 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+O(q^{10}) \) 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9	\( 3 q + 104875308 q^{3} + 892652054010 q^{5} - 878422149346056 q^{7} + 15\!\cdots\!51 q^{9}+ \cdots - 30\!\cdots\!52 q^{99}+O(q^{100}) \) 3 * q + 104875308 * q^3 + 892652054010 * q^5 - 878422149346056 * q^7 + 150091978876243551 * q^9 + 1157945428549987044 * q^11 - 62139610550998650558 * q^13 - 706062479352366674520 * q^15 - 3932636930193139724394 * q^17 + 32303396242934620550940 * q^19 + 222122458753667398807776 * q^21 + 514822069421375217224808 * q^23 + 646885156836409574920725 * q^25 - 27373531144823515220747400 * q^27 - 38460399143984437598248110 * q^29 - 103731594563724689730029856 * q^31 + 1184565114402567148957069584 * q^33 - 1596564453000564139676248560 * q^35 + 2464613172948888017021063706 * q^37 - 18611981954062877742706387512 * q^39 + 23522986584532768129037087406 * q^41 + 47516297135443858184325274308 * q^43 - 110576300682246572226345828030 * q^45 - 164346768841318947159729902256 * q^47 - 595089038084925731622037450821 * q^49 + 1803044074850794727352259927704 * q^51 - 1670480551111723484744458809558 * q^53 - 3067178595313878889174894808520 * q^55 + 4048712721984947152977307522800 * q^57 - 4370528583092646699845243631180 * q^59 + 23537078276516028947219080488306 * q^61 - 45614247892704690123948892506792 * q^63 + 75762337609865790449417835274620 * q^65 + 188845033384156918977315211921644 * q^67 - 325983212396274250308653315934048 * q^69 - 348774602295358210538621162006856 * q^71 - 285174964854822786615792218987058 * q^73 + 1577819189216049218913461709237300 * q^75 + 691501421964810422051915307575712 * q^77 + 426212862591061661901736630874160 * q^79 + 1867221150796978455595408934723163 * q^81 - 14867187331664065317008883105766692 * q^83 - 8752040931978292430566755304420140 * q^85 + 7833119385476156993771606256673800 * q^87 + 30583590846193751335205992418034270 * q^89 - 1003391702519487362566122156997296 * q^91 - 40914650682166972019906140161446016 * q^93 + 84521392849966457182141665888160200 * q^95 - 106165667044630951063328618701091994 * q^97 - 30288686341597731928441860752034252 * q^99

Introduction

L-functions

Modular forms

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