LMFDB - Embedded newform 16.30.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,30,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, 30, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 30);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 30 $
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:	$85.2448678129$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{51349}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 12837 $ x^2 - x - 12837
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{9}\cdot 3 $
Twist minimal:	no (minimal twist has level 1)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$113.802$ 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)

$f(q)$	$=$	$q-9.52434e6 q^{3} -1.12623e10 q^{5} +2.98628e12 q^{7} +2.20826e13 q^{9} +O(q^{10})$	$q-9.52434e6 q^{3} -1.12623e10 q^{5} +2.98628e12 q^{7} +2.20826e13 q^{9} +1.25811e15 q^{11} +6.09036e15 q^{13} +1.07266e17 q^{15} +8.28952e16 q^{17} -3.72257e18 q^{19} -2.84423e19 q^{21} +1.69513e19 q^{23} -5.94240e19 q^{25} +4.43337e20 q^{27} -8.81402e20 q^{29} -3.71093e21 q^{31} -1.19827e22 q^{33} -3.36325e22 q^{35} +1.97746e22 q^{37} -5.80066e22 q^{39} -1.00635e23 q^{41} +1.61886e23 q^{43} -2.48702e23 q^{45} +2.61745e24 q^{47} +5.69796e24 q^{49} -7.89521e23 q^{51} -1.44856e25 q^{53} -1.41693e25 q^{55} +3.54550e25 q^{57} -5.87125e24 q^{59} -6.18692e25 q^{61} +6.59448e25 q^{63} -6.85917e25 q^{65} -1.29243e26 q^{67} -1.61450e26 q^{69} +3.63672e26 q^{71} +1.45255e27 q^{73} +5.65975e26 q^{75} +3.75707e27 q^{77} +3.93709e27 q^{79} -5.73802e27 q^{81} -6.61171e27 q^{83} -9.33594e26 q^{85} +8.39477e27 q^{87} +6.47986e27 q^{89} +1.81875e28 q^{91} +3.53442e28 q^{93} +4.19249e28 q^{95} +9.52029e28 q^{97} +2.77823e28 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9}+O(q^{10}) $ 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9	$ 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9} + 20\!\cdots\!16 q^{11}+ \cdots + 14\!\cdots\!48 q^{99}+O(q^{100}) $ 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9 + 2055380519588616 * q^11 + 17139371179332220 * q^13 + 17192351392506000 * q^15 - 664795927907749980 * q^17 - 1232169445452155080 * q^19 - 27949113476248821696 * q^21 + 18588430504374379920 * q^23 - 207056854502617281250 * q^25 + 1497723629507205824880 * q^27 + 996096417765761595420 * q^29 + 1087258664181924030656 * q^31 - 428627571739162757280 * q^33 - 33844046536902649068000 * q^35 + 98878127759336138311660 * q^37 + 102115380027539490485328 * q^39 - 106699837259610025757676 * q^41 - 510640433114974031366200 * q^43 - 1127484267514124881450500 * q^45 + 4521107671109825795039520 * q^47 + 2479210387392854586312114 * q^49 - 11625042659032475755122384 * q^51 - 16141544629666790250663540 * q^53 - 19124657798421120971058000 * q^55 + 71545878452212360967003040 * q^57 + 83497291515660150276062760 * q^59 - 15876762943230179624113316 * q^61 + 70756669640265175617870960 * q^63 - 137266226551726798673511000 * q^65 - 24468754434128844890782120 * q^67 - 137724750477330905859409728 * q^69 + 188428428253360067209577136 * q^71 + 995872721297568533052080980 * q^73 - 1573516410323456985690375000 * q^75 + 3784200478484579386131729600 * q^77 + 7218460852587735797823642080 * q^79 - 161317068789001769884865358 * q^81 - 2210846225248155523694669640 * q^83 + 3713635892367734904614943000 * q^85 + 35603430680717182696869516240 * q^87 + 5872964746600969363962031860 * q^89 + 18563550073842319069958420896 * q^91 + 104879397265778648689061310720 * q^93 + 26445947767121183571013410000 * q^95 + 109649982219471997093227005380 * q^97 + 140506044809020243496724809448 * 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
$3$	−9.52434e6	−1.14968	−0.574839	−	0.818266i	$-0.694935\pi$
$3$	−9.52434e6	−1.14968	−0.574839	+	0.818266i	$0.694935\pi$
$4$	0	0
$5$	−1.12623e10	−0.825209	−0.412604	−	0.910910i	$-0.635381\pi$
$5$	−1.12623e10	−0.825209	−0.412604	+	0.910910i	$0.635381\pi$
$6$	0	0
$7$	2.98628e12	1.66421	0.832106	−	0.554616i	$-0.187135\pi$
$7$	2.98628e12	1.66421	0.832106	+	0.554616i	$0.187135\pi$
$8$	0	0
$9$	2.20826e13	0.321761
$10$	0	0
$11$	1.25811e15	0.998906	0.499453	−	0.866341i	$-0.333534\pi$
$11$	1.25811e15	0.998906	0.499453	+	0.866341i	$0.333534\pi$
$12$	0	0
$13$	6.09036e15	0.429007	0.214503	−	0.976723i	$-0.431187\pi$
$13$	6.09036e15	0.429007	0.214503	+	0.976723i	$0.431187\pi$
$14$	0	0
$15$	1.07266e17	0.948725
$16$	0	0
$17$	8.28952e16	0.119404	0.0597021	−	0.998216i	$-0.480985\pi$
$17$	8.28952e16	0.119404	0.0597021	+	0.998216i	$0.480985\pi$
$18$	0	0
$19$	−3.72257e18	−1.06885	−0.534424	−	0.845217i	$-0.679471\pi$
$19$	−3.72257e18	−1.06885	−0.534424	+	0.845217i	$0.679471\pi$
$20$	0	0
$21$	−2.84423e19	−1.91331
$22$	0	0
$23$	1.69513e19	0.304894	0.152447	−	0.988312i	$-0.451285\pi$
$23$	1.69513e19	0.304894	0.152447	+	0.988312i	$0.451285\pi$
$24$	0	0
$25$	−5.94240e19	−0.319030
$26$	0	0
$27$	4.43337e20	0.779757
$28$	0	0
$29$	−8.81402e20	−0.550051	−0.275026	−	0.961437i	$-0.588686\pi$
$29$	−8.81402e20	−0.550051	−0.275026	+	0.961437i	$0.588686\pi$
$30$	0	0
$31$	−3.71093e21	−0.880518	−0.440259	−	0.897871i	$-0.645113\pi$
$31$	−3.71093e21	−0.880518	−0.440259	+	0.897871i	$0.645113\pi$
$32$	0	0
$33$	−1.19827e22	−1.14842
$34$	0	0
$35$	−3.36325e22	−1.37332
$36$	0	0
$37$	1.97746e22	0.360730	0.180365	−	0.983600i	$-0.442272\pi$
$37$	1.97746e22	0.360730	0.180365	+	0.983600i	$0.442272\pi$
$38$	0	0
$39$	−5.80066e22	−0.493220
$40$	0	0
$41$	−1.00635e23	−0.414367	−0.207183	−	0.978302i	$-0.566430\pi$
$41$	−1.00635e23	−0.414367	−0.207183	+	0.978302i	$0.566430\pi$
$42$	0	0
$43$	1.61886e23	0.334131	0.167066	−	0.985946i	$-0.446571\pi$
$43$	1.61886e23	0.334131	0.167066	+	0.985946i	$0.446571\pi$
$44$	0	0
$45$	−2.48702e23	−0.265520
$46$	0	0
$47$	2.61745e24	1.48751	0.743755	−	0.668453i	$-0.233043\pi$
$47$	2.61745e24	1.48751	0.743755	+	0.668453i	$0.233043\pi$
$48$	0	0
$49$	5.69796e24	1.76960
$50$	0	0
$51$	−7.89521e23	−0.137277
$52$	0	0
$53$	−1.44856e25	−1.44190	−0.720951	−	0.692986i	$-0.756295\pi$
$53$	−1.44856e25	−1.44190	−0.720951	+	0.692986i	$0.756295\pi$
$54$	0	0
$55$	−1.41693e25	−0.824306
$56$	0	0
$57$	3.54550e25	1.22883
$58$	0	0
$59$	−5.87125e24	−0.123417	−0.0617087	−	0.998094i	$-0.519655\pi$
$59$	−5.87125e24	−0.123417	−0.0617087	+	0.998094i	$0.519655\pi$
$60$	0	0
$61$	−6.18692e25	−0.802032	−0.401016	−	0.916071i	$-0.631343\pi$
$61$	−6.18692e25	−0.802032	−0.401016	+	0.916071i	$0.631343\pi$
$62$	0	0
$63$	6.59448e25	0.535479
$64$	0	0
$65$	−6.85917e25	−0.354020
$66$	0	0
$67$	−1.29243e26	−0.429854	−0.214927	−	0.976630i	$-0.568951\pi$
$67$	−1.29243e26	−0.429854	−0.214927	+	0.976630i	$0.568951\pi$
$68$	0	0
$69$	−1.61450e26	−0.350531
$70$	0	0
$71$	3.63672e26	0.521751	0.260876	−	0.965372i	$-0.415989\pi$
$71$	3.63672e26	0.521751	0.260876	+	0.965372i	$0.415989\pi$
$72$	0	0
$73$	1.45255e27	1.39299	0.696496	−	0.717561i	$-0.254741\pi$
$73$	1.45255e27	1.39299	0.696496	+	0.717561i	$0.254741\pi$
$74$	0	0
$75$	5.65975e26	0.366783
$76$	0	0
$77$	3.75707e27	1.66239
$78$	0	0
$79$	3.93709e27	1.20111	0.600554	−	0.799584i	$-0.294947\pi$
$79$	3.93709e27	1.20111	0.600554	+	0.799584i	$0.294947\pi$
$80$	0	0
$81$	−5.73802e27	−1.21823
$82$	0	0
$83$	−6.61171e27	−0.985556	−0.492778	−	0.870155i	$-0.664018\pi$
$83$	−6.61171e27	−0.985556	−0.492778	+	0.870155i	$0.664018\pi$
$84$	0	0
$85$	−9.33594e26	−0.0985335
$86$	0	0
$87$	8.39477e27	0.632382
$88$	0	0
$89$	6.47986e27	0.351084	0.175542	−	0.984472i	$-0.443832\pi$
$89$	6.47986e27	0.351084	0.175542	+	0.984472i	$0.443832\pi$
$90$	0	0
$91$	1.81875e28	0.713959
$92$	0	0
$93$	3.53442e28	1.01231
$94$	0	0
$95$	4.19249e28	0.882023
$96$	0	0
$97$	9.52029e28	1.48068	0.740338	−	0.672235i	$-0.234666\pi$
$97$	9.52029e28	1.48068	0.740338	+	0.672235i	$0.234666\pi$
$98$	0	0
$99$	2.77823e28	0.321409
$100$	0	0
$101$	4.65862e28	0.403271	0.201635	−	0.979461i	$-0.435374\pi$
$101$	4.65862e28	0.403271	0.201635	+	0.979461i	$0.435374\pi$
$102$	0	0
$103$	1.34288e29	0.874775	0.437388	−	0.899273i	$-0.355904\pi$
$103$	1.34288e29	0.874775	0.437388	+	0.899273i	$0.355904\pi$
$104$	0	0
$105$	3.20327e29	1.57888
$106$	0	0
$107$	4.69036e29	1.75850	0.879248	−	0.476364i	$-0.158045\pi$
$107$	4.69036e29	1.75850	0.879248	+	0.476364i	$0.158045\pi$
$108$	0	0
$109$	−5.33865e29	−1.53020	−0.765098	−	0.643913i	$-0.777310\pi$
$109$	−5.33865e29	−1.53020	−0.765098	+	0.643913i	$0.777310\pi$
$110$	0	0
$111$	−1.88340e29	−0.414723
$112$	0	0
$113$	7.95436e29	1.35197	0.675987	−	0.736914i	$-0.263718\pi$
$113$	7.95436e29	1.35197	0.675987	+	0.736914i	$0.263718\pi$
$114$	0	0
$115$	−1.90911e29	−0.251602
$116$	0	0
$117$	1.34491e29	0.138038
$118$	0	0
$119$	2.47548e29	0.198714
$120$	0	0
$121$	−3.46951e27	−0.00218716
$122$	0	0
$123$	9.58486e29	0.476389
$124$	0	0
$125$	2.76703e30	1.08848
$126$	0	0
$127$	−1.93923e29	−0.0606003	−0.0303002	−	0.999541i	$-0.509646\pi$
$127$	−1.93923e29	−0.0606003	−0.0303002	+	0.999541i	$0.509646\pi$
$128$	0	0
$129$	−1.54186e30	−0.384143
$130$	0	0
$131$	2.32229e30	0.462894	0.231447	−	0.972847i	$-0.425654\pi$
$131$	2.32229e30	0.462894	0.231447	+	0.972847i	$0.425654\pi$
$132$	0	0
$133$	−1.11166e31	−1.77879
$134$	0	0
$135$	−4.99301e30	−0.643462
$136$	0	0
$137$	1.26990e31	1.32228	0.661139	−	0.750263i	$-0.270073\pi$
$137$	1.26990e31	1.32228	0.661139	+	0.750263i	$0.270073\pi$
$138$	0	0
$139$	−1.14547e31	−0.966648	−0.483324	−	0.875442i	$-0.660571\pi$
$139$	−1.14547e31	−0.966648	−0.483324	+	0.875442i	$0.660571\pi$
$140$	0	0
$141$	−2.49295e31	−1.71016
$142$	0	0
$143$	7.66234e30	0.428538
$144$	0	0
$145$	9.92666e30	0.453907
$146$	0	0
$147$	−5.42693e31	−2.03448
$148$	0	0
$149$	−5.74280e31	−1.76979	−0.884897	−	0.465787i	$-0.845771\pi$
$149$	−5.74280e31	−1.76979	−0.884897	+	0.465787i	$0.845771\pi$
$150$	0	0
$151$	−5.07974e31	−1.29026	−0.645128	−	0.764075i	$-0.723196\pi$
$151$	−5.07974e31	−1.29026	−0.645128	+	0.764075i	$0.723196\pi$
$152$	0	0
$153$	1.83054e30	0.0384197
$154$	0	0
$155$	4.17938e31	0.726611
$156$	0	0
$157$	−9.31373e30	−0.134455	−0.0672277	−	0.997738i	$-0.521415\pi$
$157$	−9.31373e30	−0.134455	−0.0672277	+	0.997738i	$0.521415\pi$
$158$	0	0
$159$	1.37965e32	1.65772
$160$	0	0
$161$	5.06213e31	0.507409
$162$	0	0
$163$	4.45351e31	0.373235	0.186617	−	0.982433i	$-0.440248\pi$
$163$	4.45351e31	0.373235	0.186617	+	0.982433i	$0.440248\pi$
$164$	0	0
$165$	1.34953e32	0.947687
$166$	0	0
$167$	−1.74608e32	−1.02962	−0.514808	−	0.857305i	$-0.672137\pi$
$167$	−1.74608e32	−1.02962	−0.514808	+	0.857305i	$0.672137\pi$
$168$	0	0
$169$	−1.64446e32	−0.815953
$170$	0	0
$171$	−8.22040e31	−0.343914
$172$	0	0
$173$	4.75360e31	0.168017	0.0840086	−	0.996465i	$-0.473228\pi$
$173$	4.75360e31	0.168017	0.0840086	+	0.996465i	$0.473228\pi$
$174$	0	0
$175$	−1.77457e32	−0.530935
$176$	0	0
$177$	5.59197e31	0.141890
$178$	0	0
$179$	6.60198e32	1.42333	0.711664	−	0.702520i	$-0.247942\pi$
$179$	6.60198e32	1.42333	0.711664	+	0.702520i	$0.247942\pi$
$180$	0	0
$181$	5.34310e32	0.980515	0.490257	−	0.871578i	$-0.336903\pi$
$181$	5.34310e32	0.980515	0.490257	+	0.871578i	$0.336903\pi$
$182$	0	0
$183$	5.89263e32	0.922079
$184$	0	0
$185$	−2.22708e32	−0.297677
$186$	0	0
$187$	1.04291e32	0.119274
$188$	0	0
$189$	1.32393e33	1.29768
$190$	0	0
$191$	9.58754e32	0.806721	0.403360	−	0.915041i	$-0.367842\pi$
$191$	9.58754e32	0.806721	0.403360	+	0.915041i	$0.367842\pi$
$192$	0	0
$193$	7.81683e31	0.0565522	0.0282761	−	0.999600i	$-0.490998\pi$
$193$	7.81683e31	0.0565522	0.0282761	+	0.999600i	$0.490998\pi$
$194$	0	0
$195$	6.53291e32	0.407010
$196$	0	0
$197$	−5.60393e32	−0.301115	−0.150557	−	0.988601i	$-0.548107\pi$
$197$	−5.60393e32	−0.301115	−0.150557	+	0.988601i	$0.548107\pi$
$198$	0	0
$199$	−8.91542e32	−0.413783	−0.206891	−	0.978364i	$-0.566335\pi$
$199$	−8.91542e32	−0.413783	−0.206891	+	0.978364i	$0.566335\pi$
$200$	0	0
$201$	1.23095e33	0.494194
$202$	0	0
$203$	−2.63211e33	−0.915402
$204$	0	0
$205$	1.13339e33	0.341939
$206$	0	0
$207$	3.74329e32	0.0981032
$208$	0	0
$209$	−4.68340e33	−1.06768
$210$	0	0
$211$	−1.07645e33	−0.213746	−0.106873	−	0.994273i	$-0.534084\pi$
$211$	−1.07645e33	−0.213746	−0.106873	+	0.994273i	$0.534084\pi$
$212$	0	0
$213$	−3.46373e33	−0.599846
$214$	0	0
$215$	−1.82322e33	−0.275728
$216$	0	0
$217$	−1.10819e34	−1.46537
$218$	0	0
$219$	−1.38345e34	−1.60149
$220$	0	0
$221$	5.04861e32	0.0512253
$222$	0	0
$223$	3.94073e33	0.350878	0.175439	−	0.984490i	$-0.443866\pi$
$223$	3.94073e33	0.350878	0.175439	+	0.984490i	$0.443866\pi$
$224$	0	0
$225$	−1.31224e33	−0.102652
$226$	0	0
$227$	2.60531e34	1.79260	0.896302	−	0.443445i	$-0.146244\pi$
$227$	2.60531e34	1.79260	0.896302	+	0.443445i	$0.146244\pi$
$228$	0	0
$229$	1.02479e34	0.620898	0.310449	−	0.950590i	$-0.399521\pi$
$229$	1.02479e34	0.620898	0.310449	+	0.950590i	$0.399521\pi$
$230$	0	0
$231$	−3.57836e34	−1.91122
$232$	0	0
$233$	1.94574e34	0.917113	0.458557	−	0.888665i	$-0.348367\pi$
$233$	1.94574e34	0.917113	0.458557	+	0.888665i	$0.348367\pi$
$234$	0	0
$235$	−2.94786e34	−1.22751
$236$	0	0
$237$	−3.74981e34	−1.38089
$238$	0	0
$239$	4.56285e34	1.48753	0.743766	−	0.668440i	$-0.233037\pi$
$239$	4.56285e34	1.48753	0.743766	+	0.668440i	$0.233037\pi$
$240$	0	0
$241$	2.90791e34	0.840107	0.420053	−	0.907499i	$-0.362011\pi$
$241$	2.90791e34	0.840107	0.420053	+	0.907499i	$0.362011\pi$
$242$	0	0
$243$	2.42245e34	0.620817
$244$	0	0
$245$	−6.41724e34	−1.46029
$246$	0	0
$247$	−2.26718e34	−0.458543
$248$	0	0
$249$	6.29721e34	1.13307
$250$	0	0
$251$	1.72418e34	0.276256	0.138128	−	0.990414i	$-0.455891\pi$
$251$	1.72418e34	0.276256	0.138128	+	0.990414i	$0.455891\pi$
$252$	0	0
$253$	2.13266e34	0.304561
$254$	0	0
$255$	8.89186e33	0.113282
$256$	0	0
$257$	4.22826e33	0.0480985	0.0240493	−	0.999711i	$-0.492344\pi$
$257$	4.22826e33	0.0480985	0.0240493	+	0.999711i	$0.492344\pi$
$258$	0	0
$259$	5.90524e34	0.600331
$260$	0	0
$261$	−1.94636e34	−0.176985
$262$	0	0
$263$	3.28330e34	0.267271	0.133635	−	0.991031i	$-0.457335\pi$
$263$	3.28330e34	0.267271	0.133635	+	0.991031i	$0.457335\pi$
$264$	0	0
$265$	1.63142e35	1.18987
$266$	0	0
$267$	−6.17164e34	−0.403633
$268$	0	0
$269$	2.79966e35	1.64322	0.821612	−	0.570048i	$-0.193075\pi$
$269$	2.79966e35	1.64322	0.821612	+	0.570048i	$0.193075\pi$
$270$	0	0
$271$	3.70768e35	1.95455	0.977277	−	0.211965i	$-0.0679862\pi$
$271$	3.70768e35	1.95455	0.977277	+	0.211965i	$0.0679862\pi$
$272$	0	0
$273$	−1.73224e35	−0.820823
$274$	0	0
$275$	−7.47620e34	−0.318681
$276$	0	0
$277$	3.33125e35	1.27835	0.639176	−	0.769060i	$-0.279275\pi$
$277$	3.33125e35	1.27835	0.639176	+	0.769060i	$0.279275\pi$
$278$	0	0
$279$	−8.19470e34	−0.283317
$280$	0	0
$281$	−5.19345e35	−1.61888	−0.809440	−	0.587203i	$-0.800229\pi$
$281$	−5.19345e35	−1.61888	−0.809440	+	0.587203i	$0.800229\pi$
$282$	0	0
$283$	1.04587e34	0.0294154	0.0147077	−	0.999892i	$-0.495318\pi$
$283$	1.04587e34	0.0294154	0.0147077	+	0.999892i	$0.495318\pi$
$284$	0	0
$285$	−3.99307e35	−1.01404
$286$	0	0
$287$	−3.00526e35	−0.689595
$288$	0	0
$289$	−4.75097e35	−0.985743
$290$	0	0
$291$	−9.06744e35	−1.70230
$292$	0	0
$293$	−3.74269e35	−0.636214	−0.318107	−	0.948055i	$-0.603047\pi$
$293$	−3.74269e35	−0.636214	−0.318107	+	0.948055i	$0.603047\pi$
$294$	0	0
$295$	6.61240e34	0.101845
$296$	0	0
$297$	5.57766e35	0.778904
$298$	0	0
$299$	1.03240e35	0.130802
$300$	0	0
$301$	4.83437e35	0.556065
$302$	0	0
$303$	−4.43702e35	−0.463632
$304$	0	0
$305$	6.96792e35	0.661844
$306$	0	0
$307$	5.37563e35	0.464433	0.232217	−	0.972664i	$-0.425402\pi$
$307$	5.37563e35	0.464433	0.232217	+	0.972664i	$0.425402\pi$
$308$	0	0
$309$	−1.27900e36	−1.00571
$310$	0	0
$311$	4.11459e35	0.294646	0.147323	−	0.989088i	$-0.452934\pi$
$311$	4.11459e35	0.294646	0.147323	+	0.989088i	$0.452934\pi$
$312$	0	0
$313$	−1.66582e36	−1.08702	−0.543508	−	0.839404i	$-0.682904\pi$
$313$	−1.66582e36	−1.08702	−0.543508	+	0.839404i	$0.682904\pi$
$314$	0	0
$315$	−7.42693e35	−0.441882
$316$	0	0
$317$	2.49069e36	1.35195	0.675975	−	0.736925i	$-0.263723\pi$
$317$	2.49069e36	1.35195	0.675975	+	0.736925i	$0.263723\pi$
$318$	0	0
$319$	−1.10890e36	−0.549449
$320$	0	0
$321$	−4.46726e36	−2.02171
$322$	0	0
$323$	−3.08583e35	−0.127625
$324$	0	0
$325$	−3.61914e35	−0.136866
$326$	0	0
$327$	5.08471e36	1.75923
$328$	0	0
$329$	7.81644e36	2.47553
$330$	0	0
$331$	−1.70323e36	−0.494046	−0.247023	−	0.969010i	$-0.579452\pi$
$331$	−1.70323e36	−0.494046	−0.247023	+	0.969010i	$0.579452\pi$
$332$	0	0
$333$	4.36674e35	0.116069
$334$	0	0
$335$	1.45557e36	0.354719
$336$	0	0
$337$	1.36026e36	0.304080	0.152040	−	0.988374i	$-0.451416\pi$
$337$	1.36026e36	0.304080	0.152040	+	0.988374i	$0.451416\pi$
$338$	0	0
$339$	−7.57600e36	−1.55433
$340$	0	0
$341$	−4.66876e36	−0.879555
$342$	0	0
$343$	7.40016e36	1.28078
$344$	0	0
$345$	1.81831e36	0.289261
$346$	0	0
$347$	6.83587e36	1.00004	0.500019	−	0.866015i	$-0.333326\pi$
$347$	6.83587e36	1.00004	0.500019	+	0.866015i	$0.333326\pi$
$348$	0	0
$349$	4.60521e36	0.619842	0.309921	−	0.950762i	$-0.399697\pi$
$349$	4.60521e36	0.619842	0.309921	+	0.950762i	$0.399697\pi$
$350$	0	0
$351$	2.70008e36	0.334521
$352$	0	0
$353$	1.14489e37	1.30627	0.653134	−	0.757243i	$-0.273454\pi$
$353$	1.14489e37	1.30627	0.653134	+	0.757243i	$0.273454\pi$
$354$	0	0
$355$	−4.09580e36	−0.430554
$356$	0	0
$357$	−2.35773e36	−0.228457
$358$	0	0
$359$	1.33540e37	1.19328	0.596638	−	0.802510i	$-0.296503\pi$
$359$	1.33540e37	1.19328	0.596638	+	0.802510i	$0.296503\pi$
$360$	0	0
$361$	1.72772e36	0.142436
$362$	0	0
$363$	3.30447e34	0.00251453
$364$	0	0
$365$	−1.63591e37	−1.14951
$366$	0	0
$367$	1.51022e37	0.980353	0.490177	−	0.871623i	$-0.336932\pi$
$367$	1.51022e37	0.980353	0.490177	+	0.871623i	$0.336932\pi$
$368$	0	0
$369$	−2.22229e36	−0.133327
$370$	0	0
$371$	−4.32580e37	−2.39963
$372$	0	0
$373$	−1.09297e36	−0.0560828	−0.0280414	−	0.999607i	$-0.508927\pi$
$373$	−1.09297e36	−0.0560828	−0.0280414	+	0.999607i	$0.508927\pi$
$374$	0	0
$375$	−2.63541e37	−1.25140
$376$	0	0
$377$	−5.36806e36	−0.235976
$378$	0	0
$379$	−1.45750e37	−0.593392	−0.296696	−	0.954972i	$-0.595885\pi$
$379$	−1.45750e37	−0.593392	−0.296696	+	0.954972i	$0.595885\pi$
$380$	0	0
$381$	1.84699e36	0.0696709
$382$	0	0
$383$	5.82118e36	0.203530	0.101765	−	0.994808i	$-0.467551\pi$
$383$	5.82118e36	0.203530	0.101765	+	0.994808i	$0.467551\pi$
$384$	0	0
$385$	−4.23134e37	−1.37182
$386$	0	0
$387$	3.57486e36	0.107510
$388$	0	0
$389$	−6.42121e36	−0.179204	−0.0896020	−	0.995978i	$-0.528560\pi$
$389$	−6.42121e36	−0.179204	−0.0896020	+	0.995978i	$0.528560\pi$
$390$	0	0
$391$	1.40518e36	0.0364057
$392$	0	0
$393$	−2.21182e37	−0.532180
$394$	0	0
$395$	−4.43408e37	−0.991165
$396$	0	0
$397$	6.76912e37	1.40627	0.703137	−	0.711054i	$-0.251782\pi$
$397$	6.76912e37	1.40627	0.703137	+	0.711054i	$0.251782\pi$
$398$	0	0
$399$	1.05879e38	2.04504
$400$	0	0
$401$	−8.28993e37	−1.48922	−0.744608	−	0.667502i	$-0.767364\pi$
$401$	−8.28993e37	−1.48922	−0.744608	+	0.667502i	$0.767364\pi$
$402$	0	0
$403$	−2.26009e37	−0.377748
$404$	0	0
$405$	6.46236e37	1.00529
$406$	0	0
$407$	2.48786e37	0.360335
$408$	0	0
$409$	2.19435e37	0.296018	0.148009	−	0.988986i	$-0.452714\pi$
$409$	2.19435e37	0.296018	0.148009	+	0.988986i	$0.452714\pi$
$410$	0	0
$411$	−1.20950e38	−1.52019
$412$	0	0
$413$	−1.75332e37	−0.205393
$414$	0	0
$415$	7.44633e37	0.813289
$416$	0	0
$417$	1.09098e38	1.11133
$418$	0	0
$419$	1.94941e37	0.185268	0.0926338	−	0.995700i	$-0.470471\pi$
$419$	1.94941e37	0.185268	0.0926338	+	0.995700i	$0.470471\pi$
$420$	0	0
$421$	−3.14435e37	−0.278895	−0.139447	−	0.990229i	$-0.544533\pi$
$421$	−3.14435e37	−0.278895	−0.139447	+	0.990229i	$0.544533\pi$
$422$	0	0
$423$	5.78001e37	0.478623
$424$	0	0
$425$	−4.92597e36	−0.0380936
$426$	0	0
$427$	−1.84759e38	−1.33475
$428$	0	0
$429$	−7.29787e37	−0.492680
$430$	0	0
$431$	2.71956e38	1.71624	0.858120	−	0.513449i	$-0.171633\pi$
$431$	2.71956e38	1.71624	0.858120	+	0.513449i	$0.171633\pi$
$432$	0	0
$433$	1.20292e38	0.709842	0.354921	−	0.934896i	$-0.384508\pi$
$433$	1.20292e38	0.709842	0.354921	+	0.934896i	$0.384508\pi$
$434$	0	0
$435$	−9.45448e37	−0.521847
$436$	0	0
$437$	−6.31025e37	−0.325886
$438$	0	0
$439$	6.48441e37	0.313426	0.156713	−	0.987644i	$-0.449910\pi$
$439$	6.48441e37	0.313426	0.156713	+	0.987644i	$0.449910\pi$
$440$	0	0
$441$	1.25826e38	0.569390
$442$	0	0
$443$	1.25901e38	0.533550	0.266775	−	0.963759i	$-0.414042\pi$
$443$	1.25901e38	0.533550	0.266775	+	0.963759i	$0.414042\pi$
$444$	0	0
$445$	−7.29785e37	−0.289717
$446$	0	0
$447$	5.46963e38	2.03469
$448$	0	0
$449$	4.32701e38	1.50874	0.754372	−	0.656447i	$-0.227941\pi$
$449$	4.32701e38	1.50874	0.754372	+	0.656447i	$0.227941\pi$
$450$	0	0
$451$	−1.26610e38	−0.413914
$452$	0	0
$453$	4.83812e38	1.48338
$454$	0	0
$455$	−2.04834e38	−0.589165
$456$	0	0
$457$	−5.34446e38	−1.44251	−0.721255	−	0.692670i	$-0.756434\pi$
$457$	−5.34446e38	−1.44251	−0.721255	+	0.692670i	$0.756434\pi$
$458$	0	0
$459$	3.67505e37	0.0931063
$460$	0	0
$461$	−2.70831e38	−0.644220	−0.322110	−	0.946702i	$-0.604392\pi$
$461$	−2.70831e38	−0.644220	−0.322110	+	0.946702i	$0.604392\pi$
$462$	0	0
$463$	−5.36275e38	−1.19802	−0.599009	−	0.800742i	$-0.704439\pi$
$463$	−5.36275e38	−1.19802	−0.599009	+	0.800742i	$0.704439\pi$
$464$	0	0
$465$	−3.98058e38	−0.835370
$466$	0	0
$467$	−9.88390e38	−1.94910	−0.974549	−	0.224177i	$-0.928031\pi$
$467$	−9.88390e38	−1.94910	−0.974549	+	0.224177i	$0.928031\pi$
$468$	0	0
$469$	−3.85954e38	−0.715368
$470$	0	0
$471$	8.87071e37	0.154581
$472$	0	0
$473$	2.03670e38	0.333765
$474$	0	0
$475$	2.21210e38	0.340995
$476$	0	0
$477$	−3.19879e38	−0.463948
$478$	0	0
$479$	1.66145e38	0.226789	0.113395	−	0.993550i	$-0.463828\pi$
$479$	1.66145e38	0.226789	0.113395	+	0.993550i	$0.463828\pi$
$480$	0	0
$481$	1.20434e38	0.154756
$482$	0	0
$483$	−4.82135e38	−0.583358
$484$	0	0
$485$	−1.07221e39	−1.22187
$486$	0	0
$487$	5.21209e38	0.559554	0.279777	−	0.960065i	$-0.409740\pi$
$487$	5.21209e38	0.559554	0.279777	+	0.960065i	$0.409740\pi$
$488$	0	0
$489$	−4.24167e38	−0.429100
$490$	0	0
$491$	6.08070e38	0.579791	0.289896	−	0.957058i	$-0.406379\pi$
$491$	6.08070e38	0.579791	0.289896	+	0.957058i	$0.406379\pi$
$492$	0	0
$493$	−7.30640e37	−0.0656785
$494$	0	0
$495$	−3.12894e38	−0.265230
$496$	0	0
$497$	1.08602e39	0.868305
$498$	0	0
$499$	−1.92541e39	−1.45233	−0.726165	−	0.687521i	$-0.758699\pi$
$499$	−1.92541e39	−1.45233	−0.726165	+	0.687521i	$0.758699\pi$
$500$	0	0
$501$	1.66303e39	1.18373
$502$	0	0
$503$	−1.47205e39	−0.988981	−0.494490	−	0.869183i	$-0.664645\pi$
$503$	−1.47205e39	−0.988981	−0.494490	+	0.869183i	$0.664645\pi$
$504$	0	0
$505$	−5.24669e38	−0.332783
$506$	0	0
$507$	1.56624e39	0.938084
$508$	0	0
$509$	9.86933e37	0.0558316	0.0279158	−	0.999610i	$-0.491113\pi$
$509$	9.86933e37	0.0558316	0.0279158	+	0.999610i	$0.491113\pi$
$510$	0	0
$511$	4.33771e39	2.31823
$512$	0	0
$513$	−1.65035e39	−0.833441
$514$	0	0
$515$	−1.51239e39	−0.721872
$516$	0	0
$517$	3.29304e39	1.48588
$518$	0	0
$519$	−4.52749e38	−0.193166
$520$	0	0
$521$	1.36208e39	0.549613	0.274806	−	0.961500i	$-0.411386\pi$
$521$	1.36208e39	0.549613	0.274806	+	0.961500i	$0.411386\pi$
$522$	0	0
$523$	3.92619e39	1.49864	0.749320	−	0.662208i	$-0.230381\pi$
$523$	3.92619e39	1.49864	0.749320	+	0.662208i	$0.230381\pi$
$524$	0	0
$525$	1.69016e39	0.610404
$526$	0	0
$527$	−3.07618e38	−0.105138
$528$	0	0
$529$	−2.80371e39	−0.907039
$530$	0	0
$531$	−1.29652e38	−0.0397109
$532$	0	0
$533$	−6.12906e38	−0.177766
$534$	0	0
$535$	−5.28245e39	−1.45113
$536$	0	0
$537$	−6.28794e39	−1.63637
$538$	0	0
$539$	7.16866e39	1.76767
$540$	0	0
$541$	3.64331e39	0.851405	0.425702	−	0.904863i	$-0.360027\pi$
$541$	3.64331e39	0.851405	0.425702	+	0.904863i	$0.360027\pi$
$542$	0	0
$543$	−5.08895e39	−1.12728
$544$	0	0
$545$	6.01257e39	1.26273
$546$	0	0
$547$	9.64716e38	0.192125	0.0960624	−	0.995375i	$-0.469375\pi$
$547$	9.64716e38	0.192125	0.0960624	+	0.995375i	$0.469375\pi$
$548$	0	0
$549$	−1.36623e39	−0.258063
$550$	0	0
$551$	3.28108e39	0.587921
$552$	0	0
$553$	1.17572e40	1.99890
$554$	0	0
$555$	2.12115e39	0.342233
$556$	0	0
$557$	−4.34857e39	−0.665956	−0.332978	−	0.942935i	$-0.608054\pi$
$557$	−4.34857e39	−0.665956	−0.332978	+	0.942935i	$0.608054\pi$
$558$	0	0
$559$	9.85944e38	0.143345
$560$	0	0
$561$	−9.93305e38	−0.137126
$562$	0	0
$563$	5.22292e39	0.684765	0.342382	−	0.939561i	$-0.388766\pi$
$563$	5.22292e39	0.684765	0.342382	+	0.939561i	$0.388766\pi$
$564$	0	0
$565$	−8.95848e39	−1.11566
$566$	0	0
$567$	−1.71353e40	−2.02740
$568$	0	0
$569$	8.52031e39	0.957916	0.478958	−	0.877838i	$-0.341015\pi$
$569$	8.52031e39	0.957916	0.478958	+	0.877838i	$0.341015\pi$
$570$	0	0
$571$	−8.06636e38	−0.0861894	−0.0430947	−	0.999071i	$-0.513722\pi$
$571$	−8.06636e38	−0.0861894	−0.0430947	+	0.999071i	$0.513722\pi$
$572$	0	0
$573$	−9.13149e39	−0.927470
$574$	0	0
$575$	−1.00732e39	−0.0972706
$576$	0	0
$577$	−1.60079e40	−1.46989	−0.734947	−	0.678125i	$-0.762793\pi$
$577$	−1.60079e40	−1.46989	−0.734947	+	0.678125i	$0.762793\pi$
$578$	0	0
$579$	−7.44501e38	−0.0650169
$580$	0	0
$581$	−1.97444e40	−1.64017
$582$	0	0
$583$	−1.82244e40	−1.44032
$584$	0	0
$585$	−1.51468e39	−0.113910
$586$	0	0
$587$	7.25470e39	0.519239	0.259619	−	0.965711i	$-0.416403\pi$
$587$	7.25470e39	0.519239	0.259619	+	0.965711i	$0.416403\pi$
$588$	0	0
$589$	1.38142e40	0.941140
$590$	0	0
$591$	5.33737e39	0.346185
$592$	0	0
$593$	4.67095e39	0.288478	0.144239	−	0.989543i	$-0.453927\pi$
$593$	4.67095e39	0.288478	0.144239	+	0.989543i	$0.453927\pi$
$594$	0	0
$595$	−2.78797e39	−0.163981
$596$	0	0
$597$	8.49135e39	0.475717
$598$	0	0
$599$	−3.02320e40	−1.61353	−0.806766	−	0.590871i	$-0.798784\pi$
$599$	−3.02320e40	−1.61353	−0.806766	+	0.590871i	$0.798784\pi$
$600$	0	0
$601$	−1.75696e40	−0.893476	−0.446738	−	0.894665i	$-0.647414\pi$
$601$	−1.75696e40	−0.893476	−0.446738	+	0.894665i	$0.647414\pi$
$602$	0	0
$603$	−2.85401e39	−0.138310
$604$	0	0
$605$	3.90748e37	0.00180486
$606$	0	0
$607$	2.78325e40	1.22551	0.612755	−	0.790273i	$-0.290061\pi$
$607$	2.78325e40	1.22551	0.612755	+	0.790273i	$0.290061\pi$
$608$	0	0
$609$	2.50691e40	1.05242
$610$	0	0
$611$	1.59412e40	0.638152
$612$	0	0
$613$	2.63728e40	1.00688	0.503441	−	0.864030i	$-0.332067\pi$
$613$	2.63728e40	1.00688	0.503441	+	0.864030i	$0.332067\pi$
$614$	0	0
$615$	−1.07948e40	−0.393120
$616$	0	0
$617$	9.21532e38	0.0320166	0.0160083	−	0.999872i	$-0.494904\pi$
$617$	9.21532e38	0.0320166	0.0160083	+	0.999872i	$0.494904\pi$
$618$	0	0
$619$	4.42763e40	1.46777	0.733883	−	0.679276i	$-0.237706\pi$
$619$	4.42763e40	1.46777	0.733883	+	0.679276i	$0.237706\pi$
$620$	0	0
$621$	7.51514e39	0.237744
$622$	0	0
$623$	1.93507e40	0.584278
$624$	0	0
$625$	−2.00947e40	−0.579189
$626$	0	0
$627$	4.46063e40	1.22749
$628$	0	0
$629$	1.63922e39	0.0430727
$630$	0	0
$631$	−4.23643e40	−1.06310	−0.531551	−	0.847026i	$-0.678391\pi$
$631$	−4.23643e40	−1.06310	−0.531551	+	0.847026i	$0.678391\pi$
$632$	0	0
$633$	1.02525e40	0.245739
$634$	0	0
$635$	2.18403e39	0.0500079
$636$	0	0
$637$	3.47026e40	0.759172
$638$	0	0
$639$	8.03081e39	0.167879
$640$	0	0
$641$	−4.13775e40	−0.826652	−0.413326	−	0.910583i	$-0.635633\pi$
$641$	−4.13775e40	−0.826652	−0.413326	+	0.910583i	$0.635633\pi$
$642$	0	0
$643$	−9.84722e40	−1.88042	−0.940209	−	0.340598i	$-0.889370\pi$
$643$	−9.84722e40	−1.88042	−0.940209	+	0.340598i	$0.889370\pi$
$644$	0	0
$645$	1.73649e40	0.316998
$646$	0	0
$647$	6.78001e39	0.118336	0.0591682	−	0.998248i	$-0.481155\pi$
$647$	6.78001e39	0.118336	0.0591682	+	0.998248i	$0.481155\pi$
$648$	0	0
$649$	−7.38667e39	−0.123282
$650$	0	0
$651$	1.05548e41	1.68470
$652$	0	0
$653$	3.23913e40	0.494524	0.247262	−	0.968949i	$-0.420469\pi$
$653$	3.23913e40	0.494524	0.247262	+	0.968949i	$0.420469\pi$
$654$	0	0
$655$	−2.61544e40	−0.381984
$656$	0	0
$657$	3.20760e40	0.448211
$658$	0	0
$659$	1.40926e40	0.188431	0.0942156	−	0.995552i	$-0.469966\pi$
$659$	1.40926e40	0.188431	0.0942156	+	0.995552i	$0.469966\pi$
$660$	0	0
$661$	−9.56737e40	−1.22425	−0.612127	−	0.790759i	$-0.709686\pi$
$661$	−9.56737e40	−1.22425	−0.612127	+	0.790759i	$0.709686\pi$
$662$	0	0
$663$	−4.80847e39	−0.0588926
$664$	0	0
$665$	1.25199e41	1.46787
$666$	0	0
$667$	−1.49409e40	−0.167708
$668$	0	0
$669$	−3.75328e40	−0.403397
$670$	0	0
$671$	−7.78382e40	−0.801154
$672$	0	0
$673$	−1.43434e40	−0.141394	−0.0706972	−	0.997498i	$-0.522522\pi$
$673$	−1.43434e40	−0.141394	−0.0706972	+	0.997498i	$0.522522\pi$
$674$	0	0
$675$	−2.63449e40	−0.248766
$676$	0	0
$677$	8.89678e40	0.804817	0.402408	−	0.915460i	$-0.368173\pi$
$677$	8.89678e40	0.804817	0.402408	+	0.915460i	$0.368173\pi$
$678$	0	0
$679$	2.84302e41	2.46416
$680$	0	0
$681$	−2.48138e41	−2.06092
$682$	0	0
$683$	−1.49228e41	−1.18782	−0.593910	−	0.804531i	$-0.702417\pi$
$683$	−1.49228e41	−1.18782	−0.593910	+	0.804531i	$0.702417\pi$
$684$	0	0
$685$	−1.43021e41	−1.09116
$686$	0	0
$687$	−9.76045e40	−0.713834
$688$	0	0
$689$	−8.82224e40	−0.618586
$690$	0	0
$691$	−2.54015e41	−1.70776	−0.853880	−	0.520469i	$-0.825757\pi$
$691$	−2.54015e41	−1.70776	−0.853880	+	0.520469i	$0.825757\pi$
$692$	0	0
$693$	8.29658e40	0.534893
$694$	0	0
$695$	1.29007e41	0.797686
$696$	0	0
$697$	−8.34219e39	−0.0494772
$698$	0	0
$699$	−1.85318e41	−1.05439
$700$	0	0
$701$	−1.56415e40	−0.0853824	−0.0426912	−	0.999088i	$-0.513593\pi$
$701$	−1.56415e40	−0.0853824	−0.0426912	+	0.999088i	$0.513593\pi$
$702$	0	0
$703$	−7.36123e40	−0.385565
$704$	0	0
$705$	2.80765e41	1.41124
$706$	0	0
$707$	1.39119e41	0.671129
$708$	0	0
$709$	−7.96882e40	−0.368998	−0.184499	−	0.982833i	$-0.559066\pi$
$709$	−7.96882e40	−0.368998	−0.184499	+	0.982833i	$0.559066\pi$
$710$	0	0
$711$	8.69411e40	0.386470
$712$	0	0
$713$	−6.29051e40	−0.268465
$714$	0	0
$715$	−8.62959e40	−0.353633
$716$	0	0
$717$	−4.34581e41	−1.71018
$718$	0	0
$719$	−6.70217e39	−0.0253306	−0.0126653	−	0.999920i	$-0.504032\pi$
$719$	−6.70217e39	−0.0253306	−0.0126653	+	0.999920i	$0.504032\pi$
$720$	0	0
$721$	4.01021e41	1.45581
$722$	0	0
$723$	−2.76960e41	−0.965853
$724$	0	0
$725$	5.23765e40	0.175483
$726$	0	0
$727$	3.08325e41	0.992565	0.496283	−	0.868161i	$-0.334698\pi$
$727$	3.08325e41	0.992565	0.496283	+	0.868161i	$0.334698\pi$
$728$	0	0
$729$	1.63080e41	0.504490
$730$	0	0
$731$	1.34196e40	0.0398967
$732$	0	0
$733$	−2.79480e41	−0.798626	−0.399313	−	0.916815i	$-0.630751\pi$
$733$	−2.79480e41	−0.798626	−0.399313	+	0.916815i	$0.630751\pi$
$734$	0	0
$735$	6.11199e41	1.67887
$736$	0	0
$737$	−1.62601e41	−0.429384
$738$	0	0
$739$	4.36349e41	1.10787	0.553936	−	0.832560i	$-0.313125\pi$
$739$	4.36349e41	1.10787	0.553936	+	0.832560i	$0.313125\pi$
$740$	0	0
$741$	2.15934e41	0.527177
$742$	0	0
$743$	1.06242e41	0.249437	0.124718	−	0.992192i	$-0.460197\pi$
$743$	1.06242e41	0.249437	0.124718	+	0.992192i	$0.460197\pi$
$744$	0	0
$745$	6.46774e41	1.46045
$746$	0	0
$747$	−1.46004e41	−0.317114
$748$	0	0
$749$	1.40067e42	2.92651
$750$	0	0
$751$	1.53315e41	0.308181	0.154091	−	0.988057i	$-0.450755\pi$
$751$	1.53315e41	0.308181	0.154091	+	0.988057i	$0.450755\pi$
$752$	0	0
$753$	−1.64216e41	−0.317606
$754$	0	0
$755$	5.72098e41	1.06473
$756$	0	0
$757$	−1.29063e41	−0.231160	−0.115580	−	0.993298i	$-0.536873\pi$
$757$	−1.29063e41	−0.231160	−0.115580	+	0.993298i	$0.536873\pi$
$758$	0	0
$759$	−2.03122e41	−0.350147
$760$	0	0
$761$	−1.01002e42	−1.67592	−0.837962	−	0.545728i	$-0.816253\pi$
$761$	−1.01002e42	−1.67592	−0.837962	+	0.545728i	$0.816253\pi$
$762$	0	0
$763$	−1.59427e42	−2.54657
$764$	0	0
$765$	−2.06162e40	−0.0317042
$766$	0	0
$767$	−3.57580e40	−0.0529469
$768$	0	0
$769$	−8.10252e41	−1.15528	−0.577641	−	0.816291i	$-0.696027\pi$
$769$	−8.10252e41	−1.15528	−0.577641	+	0.816291i	$0.696027\pi$
$770$	0	0
$771$	−4.02714e40	−0.0552979
$772$	0	0
$773$	3.13830e41	0.415042	0.207521	−	0.978231i	$-0.433461\pi$
$773$	3.13830e41	0.415042	0.207521	+	0.978231i	$0.433461\pi$
$774$	0	0
$775$	2.20519e41	0.280912
$776$	0	0
$777$	−5.62435e41	−0.690188
$778$	0	0
$779$	3.74623e41	0.442895
$780$	0	0
$781$	4.57539e41	0.521180
$782$	0	0
$783$	−3.90758e41	−0.428906
$784$	0	0
$785$	1.04894e41	0.110954
$786$	0	0
$787$	−8.07245e41	−0.822947	−0.411473	−	0.911422i	$-0.634986\pi$
$787$	−8.07245e41	−0.822947	−0.411473	+	0.911422i	$0.634986\pi$
$788$	0	0
$789$	−3.12712e41	−0.307276
$790$	0	0
$791$	2.37540e42	2.24997
$792$	0	0
$793$	−3.76805e41	−0.344077
$794$	0	0
$795$	−1.55382e42	−1.36797
$796$	0	0
$797$	2.35976e41	0.200319	0.100160	−	0.994971i	$-0.468065\pi$
$797$	2.35976e41	0.200319	0.100160	+	0.994971i	$0.468065\pi$
$798$	0	0
$799$	2.16974e41	0.177615
$800$	0	0
$801$	1.43092e41	0.112965
$802$	0	0
$803$	1.82746e42	1.39147
$804$	0	0
$805$	−5.70115e41	−0.418719
$806$	0	0
$807$	−2.66649e42	−1.88918
$808$	0	0
$809$	2.25512e42	1.54140	0.770700	−	0.637198i	$-0.219907\pi$
$809$	2.25512e42	1.54140	0.770700	+	0.637198i	$0.219907\pi$
$810$	0	0
$811$	−8.17269e41	−0.538967	−0.269484	−	0.963005i	$-0.586853\pi$
$811$	−8.17269e41	−0.538967	−0.269484	+	0.963005i	$0.586853\pi$
$812$	0	0
$813$	−3.53132e42	−2.24711
$814$	0	0
$815$	−5.01570e41	−0.307997
$816$	0	0
$817$	−6.02632e41	−0.357135
$818$	0	0
$819$	4.01627e41	0.229724
$820$	0	0
$821$	8.86919e39	0.00489675	0.00244838	−	0.999997i	$-0.499221\pi$
$821$	8.86919e39	0.00489675	0.00244838	+	0.999997i	$0.499221\pi$
$822$	0	0
$823$	−4.50695e41	−0.240207	−0.120103	−	0.992761i	$-0.538323\pi$
$823$	−4.50695e41	−0.240207	−0.120103	+	0.992761i	$0.538323\pi$
$824$	0	0
$825$	7.12058e41	0.366381
$826$	0	0
$827$	−1.64298e42	−0.816211	−0.408106	−	0.912935i	$-0.633810\pi$
$827$	−1.64298e42	−0.816211	−0.408106	+	0.912935i	$0.633810\pi$
$828$	0	0
$829$	9.28693e41	0.445483	0.222742	−	0.974878i	$-0.428499\pi$
$829$	9.28693e41	0.445483	0.222742	+	0.974878i	$0.428499\pi$
$830$	0	0
$831$	−3.17280e42	−1.46970
$832$	0	0
$833$	4.72333e41	0.211298
$834$	0	0
$835$	1.96650e42	0.849648
$836$	0	0
$837$	−1.64519e42	−0.686590
$838$	0	0
$839$	1.59653e42	0.643620	0.321810	−	0.946804i	$-0.395709\pi$
$839$	1.59653e42	0.643620	0.321810	+	0.946804i	$0.395709\pi$
$840$	0	0
$841$	−1.79082e42	−0.697444
$842$	0	0
$843$	4.94642e42	1.86119
$844$	0	0
$845$	1.85204e42	0.673332
$846$	0	0
$847$	−1.03609e40	−0.00363989
$848$	0	0
$849$	−9.96121e40	−0.0338182
$850$	0	0
$851$	3.35205e41	0.109985
$852$	0	0
$853$	−2.50802e42	−0.795373	−0.397686	−	0.917521i	$-0.630187\pi$
$853$	−2.50802e42	−0.795373	−0.397686	+	0.917521i	$0.630187\pi$
$854$	0	0
$855$	9.25810e41	0.283801
$856$	0	0
$857$	4.74569e42	1.40630	0.703149	−	0.711043i	$-0.251777\pi$
$857$	4.74569e42	1.40630	0.703149	+	0.711043i	$0.251777\pi$
$858$	0	0
$859$	1.19413e42	0.342098	0.171049	−	0.985263i	$-0.445284\pi$
$859$	1.19413e42	0.342098	0.171049	+	0.985263i	$0.445284\pi$
$860$	0	0
$861$	2.86231e42	0.792812
$862$	0	0
$863$	−5.51388e42	−1.47673	−0.738364	−	0.674402i	$-0.764401\pi$
$863$	−5.51388e42	−1.47673	−0.738364	+	0.674402i	$0.764401\pi$
$864$	0	0
$865$	−5.35367e41	−0.138649
$866$	0	0
$867$	4.52498e42	1.13329
$868$	0	0
$869$	4.95329e42	1.19979
$870$	0	0
$871$	−7.87134e41	−0.184410
$872$	0	0
$873$	2.10233e42	0.476424
$874$	0	0
$875$	8.26312e42	1.81145
$876$	0	0
$877$	3.25016e42	0.689304	0.344652	−	0.938731i	$-0.387997\pi$
$877$	3.25016e42	0.689304	0.344652	+	0.938731i	$0.387997\pi$
$878$	0	0
$879$	3.56466e42	0.731441
$880$	0	0
$881$	1.07364e42	0.213162	0.106581	−	0.994304i	$-0.466010\pi$
$881$	1.07364e42	0.213162	0.106581	+	0.994304i	$0.466010\pi$
$882$	0	0
$883$	−4.03526e42	−0.775250	−0.387625	−	0.921817i	$-0.626704\pi$
$883$	−4.03526e42	−0.775250	−0.387625	+	0.921817i	$0.626704\pi$
$884$	0	0
$885$	−6.29787e41	−0.117089
$886$	0	0
$887$	1.14783e42	0.206532	0.103266	−	0.994654i	$-0.467071\pi$
$887$	1.14783e42	0.206532	0.103266	+	0.994654i	$0.467071\pi$
$888$	0	0
$889$	−5.79108e41	−0.100852
$890$	0	0
$891$	−7.21906e42	−1.21690
$892$	0	0
$893$	−9.74365e42	−1.58992
$894$	0	0
$895$	−7.43537e42	−1.17454
$896$	0	0
$897$	−9.83288e41	−0.150380
$898$	0	0
$899$	3.27082e42	0.484330
$900$	0	0
$901$	−1.20078e42	−0.172169
$902$	0	0
$903$	−4.60442e42	−0.639296
$904$	0	0
$905$	−6.01759e42	−0.809129
$906$	0	0
$907$	1.34894e43	1.75666	0.878328	−	0.478058i	$-0.158659\pi$
$907$	1.34894e43	1.75666	0.878328	+	0.478058i	$0.158659\pi$
$908$	0	0
$909$	1.02874e42	0.129757
$910$	0	0
$911$	−3.01049e42	−0.367807	−0.183904	−	0.982944i	$-0.558873\pi$
$911$	−3.01049e42	−0.367807	−0.183904	+	0.982944i	$0.558873\pi$
$912$	0	0
$913$	−8.31825e42	−0.984477
$914$	0	0
$915$	−6.63648e42	−0.760908
$916$	0	0
$917$	6.93499e42	0.770355
$918$	0	0
$919$	−6.28761e42	−0.676723	−0.338361	−	0.941016i	$-0.609873\pi$
$919$	−6.28761e42	−0.676723	−0.338361	+	0.941016i	$0.609873\pi$
$920$	0	0
$921$	−5.11993e42	−0.533949
$922$	0	0
$923$	2.21489e42	0.223835
$924$	0	0
$925$	−1.17509e42	−0.115084
$926$	0	0
$927$	2.96542e42	0.281469
$928$	0	0
$929$	1.66684e43	1.53344	0.766720	−	0.641982i	$-0.221888\pi$
$929$	1.66684e43	1.53344	0.766720	+	0.641982i	$0.221888\pi$
$930$	0	0
$931$	−2.12111e43	−1.89144
$932$	0	0
$933$	−3.91887e42	−0.338748
$934$	0	0
$935$	−1.17456e42	−0.0984256
$936$	0	0
$937$	−1.87621e43	−1.52426	−0.762129	−	0.647425i	$-0.775846\pi$
$937$	−1.87621e43	−1.52426	−0.762129	+	0.647425i	$0.775846\pi$
$938$	0	0
$939$	1.58659e43	1.24972
$940$	0	0
$941$	5.14783e42	0.393165	0.196582	−	0.980487i	$-0.437016\pi$
$941$	5.14783e42	0.393165	0.196582	+	0.980487i	$0.437016\pi$
$942$	0	0
$943$	−1.70590e42	−0.126338
$944$	0	0
$945$	−1.49105e43	−1.07086
$946$	0	0
$947$	2.96240e42	0.206333	0.103167	−	0.994664i	$-0.467103\pi$
$947$	2.96240e42	0.206333	0.103167	+	0.994664i	$0.467103\pi$
$948$	0	0
$949$	8.84653e42	0.597603
$950$	0	0
$951$	−2.37222e43	−1.55431
$952$	0	0
$953$	1.30942e43	0.832208	0.416104	−	0.909317i	$-0.363395\pi$
$953$	1.30942e43	0.832208	0.416104	+	0.909317i	$0.363395\pi$
$954$	0	0
$955$	−1.07978e43	−0.665713
$956$	0	0
$957$	1.05615e43	0.631690
$958$	0	0
$959$	3.79229e43	2.20055
$960$	0	0
$961$	−3.99088e42	−0.224688
$962$	0	0
$963$	1.03575e43	0.565816
$964$	0	0
$965$	−8.80359e41	−0.0466674
$966$	0	0
$967$	−2.64169e43	−1.35894	−0.679468	−	0.733706i	$-0.737789\pi$
$967$	−2.64169e43	−1.35894	−0.679468	+	0.733706i	$0.737789\pi$
$968$	0	0
$969$	2.93905e42	0.146728
$970$	0	0
$971$	3.41602e43	1.65517	0.827583	−	0.561343i	$-0.189715\pi$
$971$	3.41602e43	1.65517	0.827583	+	0.561343i	$0.189715\pi$
$972$	0	0
$973$	−3.42069e43	−1.60871
$974$	0	0
$975$	3.44699e42	0.157352
$976$	0	0
$977$	2.06179e43	0.913636	0.456818	−	0.889560i	$-0.348989\pi$
$977$	2.06179e43	0.913636	0.456818	+	0.889560i	$0.348989\pi$
$978$	0	0
$979$	8.15238e42	0.350700
$980$	0	0
$981$	−1.17891e43	−0.492358
$982$	0	0
$983$	1.21841e43	0.494047	0.247023	−	0.969009i	$-0.420548\pi$
$983$	1.21841e43	0.494047	0.247023	+	0.969009i	$0.420548\pi$
$984$	0	0
$985$	6.31134e42	0.248483
$986$	0	0
$987$	−7.44464e43	−2.84607
$988$	0	0
$989$	2.74418e42	0.101875
$990$	0	0
$991$	1.94686e43	0.701888	0.350944	−	0.936397i	$-0.385861\pi$
$991$	1.94686e43	0.701888	0.350944	+	0.936397i	$0.385861\pi$
$992$	0	0
$993$	1.62221e43	0.567994
$994$	0	0
$995$	1.00409e43	0.341457
$996$	0	0
$997$	2.12798e43	0.702891	0.351446	−	0.936208i	$-0.385690\pi$
$997$	2.12798e43	0.702891	0.351446	+	0.936208i	$0.385690\pi$
$998$	0	0
$999$	8.76679e42	0.281282

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.30.a.c.1.1		2
4.3	odd	2		1.30.a.a.1.1	✓	2
12.11	even	2		9.30.a.a.1.2		2
20.3	even	4		25.30.b.a.24.3		4
20.7	even	4		25.30.b.a.24.2		4
20.19	odd	2		25.30.a.a.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1.30.a.a.1.1	✓	2	4.3	odd	2
9.30.a.a.1.2		2	12.11	even	2
16.30.a.c.1.1		2	1.1	even	1	trivial
25.30.a.a.1.2		2	20.19	odd	2
25.30.b.a.24.2		4	20.7	even	4
25.30.b.a.24.3		4	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:	\( k \)	\(=\)	\( 30 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

\(f(q)\)	\(=\)	\(q-9.52434e6 q^{3} -1.12623e10 q^{5} +2.98628e12 q^{7} +2.20826e13 q^{9} +O(q^{10})\)	\(q-9.52434e6 q^{3} -1.12623e10 q^{5} +2.98628e12 q^{7} +2.20826e13 q^{9} +1.25811e15 q^{11} +6.09036e15 q^{13} +1.07266e17 q^{15} +8.28952e16 q^{17} -3.72257e18 q^{19} -2.84423e19 q^{21} +1.69513e19 q^{23} -5.94240e19 q^{25} +4.43337e20 q^{27} -8.81402e20 q^{29} -3.71093e21 q^{31} -1.19827e22 q^{33} -3.36325e22 q^{35} +1.97746e22 q^{37} -5.80066e22 q^{39} -1.00635e23 q^{41} +1.61886e23 q^{43} -2.48702e23 q^{45} +2.61745e24 q^{47} +5.69796e24 q^{49} -7.89521e23 q^{51} -1.44856e25 q^{53} -1.41693e25 q^{55} +3.54550e25 q^{57} -5.87125e24 q^{59} -6.18692e25 q^{61} +6.59448e25 q^{63} -6.85917e25 q^{65} -1.29243e26 q^{67} -1.61450e26 q^{69} +3.63672e26 q^{71} +1.45255e27 q^{73} +5.65975e26 q^{75} +3.75707e27 q^{77} +3.93709e27 q^{79} -5.73802e27 q^{81} -6.61171e27 q^{83} -9.33594e26 q^{85} +8.39477e27 q^{87} +6.47986e27 q^{89} +1.81875e28 q^{91} +3.53442e28 q^{93} +4.19249e28 q^{95} +9.52029e28 q^{97} +2.77823e28 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9}+O(q^{10}) \) 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9	\( 2 q + 4967640 q^{3} - 17477788500 q^{5} + 3020312682800 q^{7} + 163469569523706 q^{9} + 20\!\cdots\!16 q^{11}+ \cdots + 14\!\cdots\!48 q^{99}+O(q^{100}) \) 2 * q + 4967640 * q^3 - 17477788500 * q^5 + 3020312682800 * q^7 + 163469569523706 * q^9 + 2055380519588616 * q^11 + 17139371179332220 * q^13 + 17192351392506000 * q^15 - 664795927907749980 * q^17 - 1232169445452155080 * q^19 - 27949113476248821696 * q^21 + 18588430504374379920 * q^23 - 207056854502617281250 * q^25 + 1497723629507205824880 * q^27 + 996096417765761595420 * q^29 + 1087258664181924030656 * q^31 - 428627571739162757280 * q^33 - 33844046536902649068000 * q^35 + 98878127759336138311660 * q^37 + 102115380027539490485328 * q^39 - 106699837259610025757676 * q^41 - 510640433114974031366200 * q^43 - 1127484267514124881450500 * q^45 + 4521107671109825795039520 * q^47 + 2479210387392854586312114 * q^49 - 11625042659032475755122384 * q^51 - 16141544629666790250663540 * q^53 - 19124657798421120971058000 * q^55 + 71545878452212360967003040 * q^57 + 83497291515660150276062760 * q^59 - 15876762943230179624113316 * q^61 + 70756669640265175617870960 * q^63 - 137266226551726798673511000 * q^65 - 24468754434128844890782120 * q^67 - 137724750477330905859409728 * q^69 + 188428428253360067209577136 * q^71 + 995872721297568533052080980 * q^73 - 1573516410323456985690375000 * q^75 + 3784200478484579386131729600 * q^77 + 7218460852587735797823642080 * q^79 - 161317068789001769884865358 * q^81 - 2210846225248155523694669640 * q^83 + 3713635892367734904614943000 * q^85 + 35603430680717182696869516240 * q^87 + 5872964746600969363962031860 * q^89 + 18563550073842319069958420896 * q^91 + 104879397265778648689061310720 * q^93 + 26445947767121183571013410000 * q^95 + 109649982219471997093227005380 * q^97 + 140506044809020243496724809448 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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