LMFDB - Embedded newform 16.14.a.d.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,14,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, 14, names="a")

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

chi := DirichletCharacter("16.1");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 14 $
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:	$17.1569486323$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$

$=$

$q+1836.00 q^{3} +3990.00 q^{5} +433432. q^{7} +1.77657e6 q^{9} +O(q^{10})$

$q+1836.00 q^{3} +3990.00 q^{5} +433432. q^{7} +1.77657e6 q^{9} -1.61977e6 q^{11} -1.08785e7 q^{13} +7.32564e6 q^{15} +6.05693e7 q^{17} +2.43132e8 q^{19} +7.95781e8 q^{21} +6.06096e8 q^{23} -1.20478e9 q^{25} +3.34611e8 q^{27} +5.25864e9 q^{29} +1.82431e9 q^{31} -2.97390e9 q^{33} +1.72939e9 q^{35} -3.00588e9 q^{37} -1.99729e10 q^{39} -4.97049e10 q^{41} -5.87667e10 q^{43} +7.08853e9 q^{45} +4.20959e10 q^{47} +9.09743e10 q^{49} +1.11205e11 q^{51} -1.81141e11 q^{53} -6.46289e9 q^{55} +4.46390e11 q^{57} -2.06731e11 q^{59} -1.24479e11 q^{61} +7.70024e11 q^{63} -4.34051e10 q^{65} -9.56651e10 q^{67} +1.11279e12 q^{69} +3.71436e11 q^{71} -1.80058e12 q^{73} -2.21198e12 q^{75} -7.02061e11 q^{77} -1.55793e12 q^{79} -2.21809e12 q^{81} -2.49279e12 q^{83} +2.41671e11 q^{85} +9.65486e12 q^{87} +2.99424e12 q^{89} -4.71508e12 q^{91} +3.34944e12 q^{93} +9.70096e11 q^{95} +4.38249e12 q^{97} -2.87764e12 q^{99} +O(q^{100})$

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
$3$	1836.00	1.45407	0.727034	−	0.686602i	$-0.240898\pi$
$3$	1836.00	1.45407	0.727034	+	0.686602i	$0.240898\pi$
$4$	0	0
$5$	3990.00	0.114200	0.0571002	−	0.998368i	$-0.481815\pi$
$5$	3990.00	0.114200	0.0571002	+	0.998368i	$0.481815\pi$
$6$	0	0
$7$	433432.	1.39246	0.696232	−	0.717817i	$-0.254859\pi$
$7$	433432.	1.39246	0.696232	+	0.717817i	$0.254859\pi$
$8$	0	0
$9$	1.77657e6	1.11431
$10$	0	0
$11$	−1.61977e6	−0.275678	−0.137839	−	0.990455i	$-0.544016\pi$
$11$	−1.61977e6	−0.275678	−0.137839	+	0.990455i	$0.544016\pi$
$12$	0	0
$13$	−1.08785e7	−0.625080	−0.312540	−	0.949905i	$-0.601180\pi$
$13$	−1.08785e7	−0.625080	−0.312540	+	0.949905i	$0.601180\pi$
$14$	0	0
$15$	7.32564e6	0.166055
$16$	0	0
$17$	6.05693e7	0.608604	0.304302	−	0.952576i	$-0.401577\pi$
$17$	6.05693e7	0.608604	0.304302	+	0.952576i	$0.401577\pi$
$18$	0	0
$19$	2.43132e8	1.18561	0.592807	−	0.805345i	$-0.298020\pi$
$19$	2.43132e8	1.18561	0.592807	+	0.805345i	$0.298020\pi$
$20$	0	0
$21$	7.95781e8	2.02474
$22$	0	0
$23$	6.06096e8	0.853711	0.426855	−	0.904320i	$-0.359621\pi$
$23$	6.06096e8	0.853711	0.426855	+	0.904320i	$0.359621\pi$
$24$	0	0
$25$	−1.20478e9	−0.986958
$26$	0	0
$27$	3.34611e8	0.166217
$28$	0	0
$29$	5.25864e9	1.64167	0.820836	−	0.571164i	$-0.193508\pi$
$29$	5.25864e9	1.64167	0.820836	+	0.571164i	$0.193508\pi$
$30$	0	0
$31$	1.82431e9	0.369189	0.184594	−	0.982815i	$-0.440903\pi$
$31$	1.82431e9	0.369189	0.184594	+	0.982815i	$0.440903\pi$
$32$	0	0
$33$	−2.97390e9	−0.400854
$34$	0	0
$35$	1.72939e9	0.159020
$36$	0	0
$37$	−3.00588e9	−0.192602	−0.0963008	−	0.995352i	$-0.530701\pi$
$37$	−3.00588e9	−0.192602	−0.0963008	+	0.995352i	$0.530701\pi$
$38$	0	0
$39$	−1.99729e10	−0.908909
$40$	0	0
$41$	−4.97049e10	−1.63420	−0.817098	−	0.576499i	$-0.804418\pi$
$41$	−4.97049e10	−1.63420	−0.817098	+	0.576499i	$0.804418\pi$
$42$	0	0
$43$	−5.87667e10	−1.41771	−0.708853	−	0.705356i	$-0.750787\pi$
$43$	−5.87667e10	−1.41771	−0.708853	+	0.705356i	$0.750787\pi$
$44$	0	0
$45$	7.08853e9	0.127255
$46$	0	0
$47$	4.20959e10	0.569644	0.284822	−	0.958580i	$-0.408066\pi$
$47$	4.20959e10	0.569644	0.284822	+	0.958580i	$0.408066\pi$
$48$	0	0
$49$	9.09743e10	0.938954
$50$	0	0
$51$	1.11205e11	0.884951
$52$	0	0
$53$	−1.81141e11	−1.12259	−0.561297	−	0.827614i	$-0.689698\pi$
$53$	−1.81141e11	−1.12259	−0.561297	+	0.827614i	$0.689698\pi$
$54$	0	0
$55$	−6.46289e9	−0.0314825
$56$	0	0
$57$	4.46390e11	1.72396
$58$	0	0
$59$	−2.06731e11	−0.638068	−0.319034	−	0.947743i	$-0.603358\pi$
$59$	−2.06731e11	−0.638068	−0.319034	+	0.947743i	$0.603358\pi$
$60$	0	0
$61$	−1.24479e11	−0.309351	−0.154676	−	0.987965i	$-0.549433\pi$
$61$	−1.24479e11	−0.309351	−0.154676	+	0.987965i	$0.549433\pi$
$62$	0	0
$63$	7.70024e11	1.55164
$64$	0	0
$65$	−4.34051e10	−0.0713845
$66$	0	0
$67$	−9.56651e10	−0.129201	−0.0646007	−	0.997911i	$-0.520577\pi$
$67$	−9.56651e10	−0.129201	−0.0646007	+	0.997911i	$0.520577\pi$
$68$	0	0
$69$	1.11279e12	1.24135
$70$	0	0
$71$	3.71436e11	0.344116	0.172058	−	0.985087i	$-0.444958\pi$
$71$	3.71436e11	0.344116	0.172058	+	0.985087i	$0.444958\pi$
$72$	0	0
$73$	−1.80058e12	−1.39256	−0.696278	−	0.717772i	$-0.745162\pi$
$73$	−1.80058e12	−1.39256	−0.696278	+	0.717772i	$0.745162\pi$
$74$	0	0
$75$	−2.21198e12	−1.43510
$76$	0	0
$77$	−7.02061e11	−0.383871
$78$	0	0
$79$	−1.55793e12	−0.721062	−0.360531	−	0.932747i	$-0.617404\pi$
$79$	−1.55793e12	−0.721062	−0.360531	+	0.932747i	$0.617404\pi$
$80$	0	0
$81$	−2.21809e12	−0.872621
$82$	0	0
$83$	−2.49279e12	−0.836909	−0.418455	−	0.908238i	$-0.637428\pi$
$83$	−2.49279e12	−0.836909	−0.418455	+	0.908238i	$0.637428\pi$
$84$	0	0
$85$	2.41671e11	0.0695028
$86$	0	0
$87$	9.65486e12	2.38710
$88$	0	0
$89$	2.99424e12	0.638632	0.319316	−	0.947648i	$-0.396547\pi$
$89$	2.99424e12	0.638632	0.319316	+	0.947648i	$0.396547\pi$
$90$	0	0
$91$	−4.71508e12	−0.870401
$92$	0	0
$93$	3.34944e12	0.536825
$94$	0	0
$95$	9.70096e11	0.135398
$96$	0	0
$97$	4.38249e12	0.534201	0.267101	−	0.963669i	$-0.413934\pi$
$97$	4.38249e12	0.534201	0.267101	+	0.963669i	$0.413934\pi$
$98$	0	0
$99$	−2.87764e12	−0.307191
$100$	0	0
$101$	−5.66110e12	−0.530655	−0.265327	−	0.964158i	$-0.585480\pi$
$101$	−5.66110e12	−0.530655	−0.265327	+	0.964158i	$0.585480\pi$
$102$	0	0
$103$	2.16104e13	1.78329	0.891644	−	0.452737i	$-0.149552\pi$
$103$	2.16104e13	1.78329	0.891644	+	0.452737i	$0.149552\pi$
$104$	0	0
$105$	3.17517e12	0.231226
$106$	0	0
$107$	−1.09348e13	−0.704397	−0.352198	−	0.935925i	$-0.614566\pi$
$107$	−1.09348e13	−0.704397	−0.352198	+	0.935925i	$0.614566\pi$
$108$	0	0
$109$	−4.20682e12	−0.240260	−0.120130	−	0.992758i	$-0.538331\pi$
$109$	−4.20682e12	−0.240260	−0.120130	+	0.992758i	$0.538331\pi$
$110$	0	0
$111$	−5.51879e12	−0.280056
$112$	0	0
$113$	2.87650e13	1.29974	0.649868	−	0.760047i	$-0.274824\pi$
$113$	2.87650e13	1.29974	0.649868	+	0.760047i	$0.274824\pi$
$114$	0	0
$115$	2.41832e12	0.0974942
$116$	0	0
$117$	−1.93264e13	−0.696534
$118$	0	0
$119$	2.62527e13	0.847458
$120$	0	0
$121$	−3.18991e13	−0.924002
$122$	0	0
$123$	−9.12582e13	−2.37623
$124$	0	0
$125$	−9.67769e12	−0.226912
$126$	0	0
$127$	−4.78013e13	−1.01092	−0.505458	−	0.862851i	$-0.668677\pi$
$127$	−4.78013e13	−1.01092	−0.505458	+	0.862851i	$0.668677\pi$
$128$	0	0
$129$	−1.07896e14	−2.06144
$130$	0	0
$131$	1.03694e14	1.79262	0.896312	−	0.443425i	$-0.146237\pi$
$131$	1.03694e14	1.79262	0.896312	+	0.443425i	$0.146237\pi$
$132$	0	0
$133$	1.05381e14	1.65092
$134$	0	0
$135$	1.33510e12	0.0189821
$136$	0	0
$137$	5.05383e13	0.653035	0.326518	−	0.945191i	$-0.394125\pi$
$137$	5.05383e13	0.653035	0.326518	+	0.945191i	$0.394125\pi$
$138$	0	0
$139$	−2.14878e13	−0.252694	−0.126347	−	0.991986i	$-0.540325\pi$
$139$	−2.14878e13	−0.252694	−0.126347	+	0.991986i	$0.540325\pi$
$140$	0	0
$141$	7.72880e13	0.828301
$142$	0	0
$143$	1.76206e13	0.172321
$144$	0	0
$145$	2.09820e13	0.187480
$146$	0	0
$147$	1.67029e14	1.36530
$148$	0	0
$149$	−8.79741e13	−0.658634	−0.329317	−	0.944219i	$-0.606818\pi$
$149$	−8.79741e13	−0.658634	−0.329317	+	0.944219i	$0.606818\pi$
$150$	0	0
$151$	2.07435e13	0.142407	0.0712037	−	0.997462i	$-0.477316\pi$
$151$	2.07435e13	0.142407	0.0712037	+	0.997462i	$0.477316\pi$
$152$	0	0
$153$	1.07606e14	0.678174
$154$	0	0
$155$	7.27901e12	0.0421615
$156$	0	0
$157$	−2.27523e14	−1.21249	−0.606243	−	0.795279i	$-0.707324\pi$
$157$	−2.27523e14	−1.21249	−0.606243	+	0.795279i	$0.707324\pi$
$158$	0	0
$159$	−3.32574e14	−1.63233
$160$	0	0
$161$	2.62702e14	1.18876
$162$	0	0
$163$	1.41533e14	0.591070	0.295535	−	0.955332i	$-0.404502\pi$
$163$	1.41533e14	0.591070	0.295535	+	0.955332i	$0.404502\pi$
$164$	0	0
$165$	−1.18659e13	−0.0457777
$166$	0	0
$167$	−5.58000e13	−0.199057	−0.0995284	−	0.995035i	$-0.531733\pi$
$167$	−5.58000e13	−0.199057	−0.0995284	+	0.995035i	$0.531733\pi$
$168$	0	0
$169$	−1.84534e14	−0.609275
$170$	0	0
$171$	4.31941e14	1.32114
$172$	0	0
$173$	1.26777e14	0.359535	0.179767	−	0.983709i	$-0.442465\pi$
$173$	1.26777e14	0.359535	0.179767	+	0.983709i	$0.442465\pi$
$174$	0	0
$175$	−5.22192e14	−1.37430
$176$	0	0
$177$	−3.79557e14	−0.927793
$178$	0	0
$179$	−3.08976e14	−0.702070	−0.351035	−	0.936362i	$-0.614170\pi$
$179$	−3.08976e14	−0.702070	−0.351035	+	0.936362i	$0.614170\pi$
$180$	0	0
$181$	−4.74336e14	−1.00271	−0.501355	−	0.865242i	$-0.667165\pi$
$181$	−4.74336e14	−1.00271	−0.501355	+	0.865242i	$0.667165\pi$
$182$	0	0
$183$	−2.28543e14	−0.449818
$184$	0	0
$185$	−1.19934e13	−0.0219952
$186$	0	0
$187$	−9.81085e13	−0.167778
$188$	0	0
$189$	1.45031e14	0.231451
$190$	0	0
$191$	6.58450e14	0.981310	0.490655	−	0.871354i	$-0.336758\pi$
$191$	6.58450e14	0.981310	0.490655	+	0.871354i	$0.336758\pi$
$192$	0	0
$193$	2.10092e14	0.292609	0.146305	−	0.989240i	$-0.453262\pi$
$193$	2.10092e14	0.292609	0.146305	+	0.989240i	$0.453262\pi$
$194$	0	0
$195$	−7.96917e13	−0.103798
$196$	0	0
$197$	1.56559e15	1.90831	0.954153	−	0.299320i	$-0.0967599\pi$
$197$	1.56559e15	1.90831	0.954153	+	0.299320i	$0.0967599\pi$
$198$	0	0
$199$	−3.66804e14	−0.418686	−0.209343	−	0.977842i	$-0.567133\pi$
$199$	−3.66804e14	−0.418686	−0.209343	+	0.977842i	$0.567133\pi$
$200$	0	0
$201$	−1.75641e14	−0.187868
$202$	0	0
$203$	2.27926e15	2.28597
$204$	0	0
$205$	−1.98322e14	−0.186626
$206$	0	0
$207$	1.07677e15	0.951300
$208$	0	0
$209$	−3.93818e14	−0.326847
$210$	0	0
$211$	−1.34485e15	−1.04915	−0.524574	−	0.851365i	$-0.675776\pi$
$211$	−1.34485e15	−1.04915	−0.524574	+	0.851365i	$0.675776\pi$
$212$	0	0
$213$	6.81957e14	0.500368
$214$	0	0
$215$	−2.34479e14	−0.161903
$216$	0	0
$217$	7.90716e14	0.514082
$218$	0	0
$219$	−3.30586e15	−2.02487
$220$	0	0
$221$	−6.58901e14	−0.380426
$222$	0	0
$223$	2.04083e15	1.11128	0.555642	−	0.831421i	$-0.312472\pi$
$223$	2.04083e15	1.11128	0.555642	+	0.831421i	$0.312472\pi$
$224$	0	0
$225$	−2.14038e15	−1.09978
$226$	0	0
$227$	−2.91994e14	−0.141647	−0.0708233	−	0.997489i	$-0.522563\pi$
$227$	−2.91994e14	−0.141647	−0.0708233	+	0.997489i	$0.522563\pi$
$228$	0	0
$229$	−2.81485e15	−1.28981	−0.644903	−	0.764264i	$-0.723102\pi$
$229$	−2.81485e15	−1.28981	−0.644903	+	0.764264i	$0.723102\pi$
$230$	0	0
$231$	−1.28898e15	−0.558174
$232$	0	0
$233$	4.39377e15	1.79897	0.899486	−	0.436950i	$-0.143941\pi$
$233$	4.39377e15	1.79897	0.899486	+	0.436950i	$0.143941\pi$
$234$	0	0
$235$	1.67963e14	0.0650536
$236$	0	0
$237$	−2.86036e15	−1.04847
$238$	0	0
$239$	4.40953e15	1.53040	0.765202	−	0.643791i	$-0.222639\pi$
$239$	4.40953e15	1.53040	0.765202	+	0.643791i	$0.222639\pi$
$240$	0	0
$241$	1.92940e15	0.634324	0.317162	−	0.948371i	$-0.397270\pi$
$241$	1.92940e15	0.634324	0.317162	+	0.948371i	$0.397270\pi$
$242$	0	0
$243$	−4.60588e15	−1.43507
$244$	0	0
$245$	3.62987e14	0.107229
$246$	0	0
$247$	−2.64490e15	−0.741104
$248$	0	0
$249$	−4.57676e15	−1.21692
$250$	0	0
$251$	−3.46331e14	−0.0874202	−0.0437101	−	0.999044i	$-0.513918\pi$
$251$	−3.46331e14	−0.0874202	−0.0437101	+	0.999044i	$0.513918\pi$
$252$	0	0
$253$	−9.81738e14	−0.235349
$254$	0	0
$255$	4.43709e14	0.101062
$256$	0	0
$257$	4.59506e14	0.0994778	0.0497389	−	0.998762i	$-0.484161\pi$
$257$	4.59506e14	0.0994778	0.0497389	+	0.998762i	$0.484161\pi$
$258$	0	0
$259$	−1.30284e15	−0.268191
$260$	0	0
$261$	9.34236e15	1.82933
$262$	0	0
$263$	8.08966e15	1.50736	0.753681	−	0.657240i	$-0.228276\pi$
$263$	8.08966e15	1.50736	0.753681	+	0.657240i	$0.228276\pi$
$264$	0	0
$265$	−7.22752e14	−0.128201
$266$	0	0
$267$	5.49742e15	0.928615
$268$	0	0
$269$	5.06059e15	0.814350	0.407175	−	0.913350i	$-0.366514\pi$
$269$	5.06059e15	0.814350	0.407175	+	0.913350i	$0.366514\pi$
$270$	0	0
$271$	4.91453e14	0.0753671	0.0376835	−	0.999290i	$-0.488002\pi$
$271$	4.91453e14	0.0753671	0.0376835	+	0.999290i	$0.488002\pi$
$272$	0	0
$273$	−8.65688e15	−1.26562
$274$	0	0
$275$	1.95147e15	0.272082
$276$	0	0
$277$	3.92917e15	0.522615	0.261308	−	0.965256i	$-0.415846\pi$
$277$	3.92917e15	0.522615	0.261308	+	0.965256i	$0.415846\pi$
$278$	0	0
$279$	3.24103e15	0.411391
$280$	0	0
$281$	1.14417e16	1.38644	0.693219	−	0.720727i	$-0.256192\pi$
$281$	1.14417e16	1.38644	0.693219	+	0.720727i	$0.256192\pi$
$282$	0	0
$283$	−2.12246e15	−0.245599	−0.122800	−	0.992431i	$-0.539187\pi$
$283$	−2.12246e15	−0.245599	−0.122800	+	0.992431i	$0.539187\pi$
$284$	0	0
$285$	1.78110e15	0.196877
$286$	0	0
$287$	−2.15437e16	−2.27556
$288$	0	0
$289$	−6.23594e15	−0.629602
$290$	0	0
$291$	8.04626e15	0.776765
$292$	0	0
$293$	−1.99050e16	−1.83790	−0.918951	−	0.394372i	$-0.870962\pi$
$293$	−1.99050e16	−1.83790	−0.918951	+	0.394372i	$0.870962\pi$
$294$	0	0
$295$	−8.24855e14	−0.0728676
$296$	0	0
$297$	−5.41994e14	−0.0458223
$298$	0	0
$299$	−6.59340e15	−0.533638
$300$	0	0
$301$	−2.54714e16	−1.97410
$302$	0	0
$303$	−1.03938e16	−0.771608
$304$	0	0
$305$	−4.96671e14	−0.0353281
$306$	0	0
$307$	1.26611e16	0.863118	0.431559	−	0.902085i	$-0.357964\pi$
$307$	1.26611e16	0.863118	0.431559	+	0.902085i	$0.357964\pi$
$308$	0	0
$309$	3.96768e16	2.59302
$310$	0	0
$311$	−1.08678e15	−0.0681081	−0.0340541	−	0.999420i	$-0.510842\pi$
$311$	−1.08678e15	−0.0681081	−0.0340541	+	0.999420i	$0.510842\pi$
$312$	0	0
$313$	2.13383e16	1.28269	0.641346	−	0.767252i	$-0.278376\pi$
$313$	2.13383e16	1.28269	0.641346	+	0.767252i	$0.278376\pi$
$314$	0	0
$315$	3.07239e15	0.177198
$316$	0	0
$317$	−3.00971e16	−1.66586	−0.832932	−	0.553375i	$-0.813340\pi$
$317$	−3.00971e16	−1.66586	−0.832932	+	0.553375i	$0.813340\pi$
$318$	0	0
$319$	−8.51780e15	−0.452572
$320$	0	0
$321$	−2.00763e16	−1.02424
$322$	0	0
$323$	1.47263e16	0.721569
$324$	0	0
$325$	1.31062e16	0.616928
$326$	0	0
$327$	−7.72373e15	−0.349355
$328$	0	0
$329$	1.82457e16	0.793208
$330$	0	0
$331$	−1.00468e16	−0.419900	−0.209950	−	0.977712i	$-0.567330\pi$
$331$	−1.00468e16	−0.419900	−0.209950	+	0.977712i	$0.567330\pi$
$332$	0	0
$333$	−5.34016e15	−0.214618
$334$	0	0
$335$	−3.81704e14	−0.0147549
$336$	0	0
$337$	4.00487e15	0.148934	0.0744671	−	0.997223i	$-0.476274\pi$
$337$	4.00487e15	0.148934	0.0744671	+	0.997223i	$0.476274\pi$
$338$	0	0
$339$	5.28126e16	1.88990
$340$	0	0
$341$	−2.95497e15	−0.101777
$342$	0	0
$343$	−2.56363e15	−0.0850048
$344$	0	0
$345$	4.44004e15	0.141763
$346$	0	0
$347$	−6.06273e13	−0.00186435	−0.000932173	−	1.00000i	$-0.500297\pi$
$347$	−6.06273e13	−0.00186435	−0.000932173		1.00000i	$0.500297\pi$
$348$	0	0
$349$	3.07578e16	0.911151	0.455576	−	0.890197i	$-0.349433\pi$
$349$	3.07578e16	0.911151	0.455576	+	0.890197i	$0.349433\pi$
$350$	0	0
$351$	−3.64005e15	−0.103899
$352$	0	0
$353$	−2.08897e16	−0.574640	−0.287320	−	0.957835i	$-0.592764\pi$
$353$	−2.08897e16	−0.574640	−0.287320	+	0.957835i	$0.592764\pi$
$354$	0	0
$355$	1.48203e15	0.0392982
$356$	0	0
$357$	4.81999e16	1.23226
$358$	0	0
$359$	2.46366e16	0.607390	0.303695	−	0.952769i	$-0.401780\pi$
$359$	2.46366e16	0.607390	0.303695	+	0.952769i	$0.401780\pi$
$360$	0	0
$361$	1.70601e16	0.405680
$362$	0	0
$363$	−5.85667e16	−1.34356
$364$	0	0
$365$	−7.18430e15	−0.159031
$366$	0	0
$367$	5.28651e16	1.12938	0.564689	−	0.825304i	$-0.308996\pi$
$367$	5.28651e16	1.12938	0.564689	+	0.825304i	$0.308996\pi$
$368$	0	0
$369$	−8.83043e16	−1.82100
$370$	0	0
$371$	−7.85122e16	−1.56317
$372$	0	0
$373$	−6.44569e15	−0.123926	−0.0619630	−	0.998078i	$-0.519736\pi$
$373$	−6.44569e15	−0.123926	−0.0619630	+	0.998078i	$0.519736\pi$
$374$	0	0
$375$	−1.77682e16	−0.329945
$376$	0	0
$377$	−5.72059e16	−1.02618
$378$	0	0
$379$	6.58787e16	1.14180	0.570900	−	0.821019i	$-0.306594\pi$
$379$	6.58787e16	1.14180	0.570900	+	0.821019i	$0.306594\pi$
$380$	0	0
$381$	−8.77632e16	−1.46994
$382$	0	0
$383$	−8.87457e16	−1.43666	−0.718332	−	0.695700i	$-0.755094\pi$
$383$	−8.87457e16	−1.43666	−0.718332	+	0.695700i	$0.755094\pi$
$384$	0	0
$385$	−2.80122e15	−0.0438382
$386$	0	0
$387$	−1.04403e17	−1.57977
$388$	0	0
$389$	6.27464e16	0.918156	0.459078	−	0.888396i	$-0.348180\pi$
$389$	6.27464e16	0.918156	0.459078	+	0.888396i	$0.348180\pi$
$390$	0	0
$391$	3.67108e16	0.519572
$392$	0	0
$393$	1.90382e17	2.60659
$394$	0	0
$395$	−6.21615e15	−0.0823456
$396$	0	0
$397$	−9.86261e16	−1.26431	−0.632155	−	0.774842i	$-0.717829\pi$
$397$	−9.86261e16	−1.26431	−0.632155	+	0.774842i	$0.717829\pi$
$398$	0	0
$399$	1.93480e17	2.40055
$400$	0	0
$401$	−3.66088e15	−0.0439691	−0.0219845	−	0.999758i	$-0.506998\pi$
$401$	−3.66088e15	−0.0439691	−0.0219845	+	0.999758i	$0.506998\pi$
$402$	0	0
$403$	−1.98457e16	−0.230773
$404$	0	0
$405$	−8.85016e15	−0.0996537
$406$	0	0
$407$	4.86883e15	0.0530959
$408$	0	0
$409$	1.41675e17	1.49655	0.748273	−	0.663390i	$-0.230883\pi$
$409$	1.41675e17	1.49655	0.748273	+	0.663390i	$0.230883\pi$
$410$	0	0
$411$	9.27882e16	0.949557
$412$	0	0
$413$	−8.96037e16	−0.888486
$414$	0	0
$415$	−9.94624e15	−0.0955754
$416$	0	0
$417$	−3.94515e16	−0.367434
$418$	0	0
$419$	−1.44390e17	−1.30360	−0.651801	−	0.758390i	$-0.725986\pi$
$419$	−1.44390e17	−1.30360	−0.651801	+	0.758390i	$0.725986\pi$
$420$	0	0
$421$	1.32169e17	1.15690	0.578451	−	0.815717i	$-0.303657\pi$
$421$	1.32169e17	1.15690	0.578451	+	0.815717i	$0.303657\pi$
$422$	0	0
$423$	7.47864e16	0.634761
$424$	0	0
$425$	−7.29729e16	−0.600666
$426$	0	0
$427$	−5.39532e16	−0.430761
$428$	0	0
$429$	3.23515e16	0.250566
$430$	0	0
$431$	1.80535e17	1.35662	0.678312	−	0.734774i	$-0.262712\pi$
$431$	1.80535e17	1.35662	0.678312	+	0.734774i	$0.262712\pi$
$432$	0	0
$433$	−1.33563e17	−0.973900	−0.486950	−	0.873430i	$-0.661891\pi$
$433$	−1.33563e17	−0.973900	−0.486950	+	0.873430i	$0.661891\pi$
$434$	0	0
$435$	3.85229e16	0.272608
$436$	0	0
$437$	1.47361e17	1.01217
$438$	0	0
$439$	−2.34588e17	−1.56418	−0.782090	−	0.623165i	$-0.785846\pi$
$439$	−2.34588e17	−1.56418	−0.782090	+	0.623165i	$0.785846\pi$
$440$	0	0
$441$	1.61622e17	1.04629
$442$	0	0
$443$	1.26522e17	0.795318	0.397659	−	0.917533i	$-0.369823\pi$
$443$	1.26522e17	0.795318	0.397659	+	0.917533i	$0.369823\pi$
$444$	0	0
$445$	1.19470e16	0.0729321
$446$	0	0
$447$	−1.61520e17	−0.957698
$448$	0	0
$449$	−1.91607e17	−1.10359	−0.551797	−	0.833978i	$-0.686058\pi$
$449$	−1.91607e17	−1.10359	−0.551797	+	0.833978i	$0.686058\pi$
$450$	0	0
$451$	8.05106e16	0.450511
$452$	0	0
$453$	3.80851e16	0.207070
$454$	0	0
$455$	−1.88132e16	−0.0994002
$456$	0	0
$457$	−2.62243e17	−1.34663	−0.673316	−	0.739355i	$-0.735131\pi$
$457$	−2.62243e17	−1.34663	−0.673316	+	0.739355i	$0.735131\pi$
$458$	0	0
$459$	2.02672e16	0.101160
$460$	0	0
$461$	−6.94961e16	−0.337213	−0.168607	−	0.985683i	$-0.553927\pi$
$461$	−6.94961e16	−0.337213	−0.168607	+	0.985683i	$0.553927\pi$
$462$	0	0
$463$	2.31198e17	1.09071	0.545353	−	0.838207i	$-0.316396\pi$
$463$	2.31198e17	1.09071	0.545353	+	0.838207i	$0.316396\pi$
$464$	0	0
$465$	1.33643e16	0.0613057
$466$	0	0
$467$	−1.70771e17	−0.761825	−0.380913	−	0.924611i	$-0.624390\pi$
$467$	−1.70771e17	−0.761825	−0.380913	+	0.924611i	$0.624390\pi$
$468$	0	0
$469$	−4.14643e16	−0.179908
$470$	0	0
$471$	−4.17732e17	−1.76304
$472$	0	0
$473$	9.51886e16	0.390830
$474$	0	0
$475$	−2.92921e17	−1.17015
$476$	0	0
$477$	−3.21810e17	−1.25092
$478$	0	0
$479$	−3.22880e17	−1.22141	−0.610703	−	0.791859i	$-0.709113\pi$
$479$	−3.22880e17	−1.22141	−0.610703	+	0.791859i	$0.709113\pi$
$480$	0	0
$481$	3.26993e16	0.120391
$482$	0	0
$483$	4.82320e17	1.72854
$484$	0	0
$485$	1.74861e16	0.0610060
$486$	0	0
$487$	2.41562e16	0.0820521	0.0410261	−	0.999158i	$-0.486937\pi$
$487$	2.41562e16	0.0820521	0.0410261	+	0.999158i	$0.486937\pi$
$488$	0	0
$489$	2.59855e17	0.859456
$490$	0	0
$491$	4.61064e17	1.48502	0.742508	−	0.669837i	$-0.233636\pi$
$491$	4.61064e17	1.48502	0.742508	+	0.669837i	$0.233636\pi$
$492$	0	0
$493$	3.18512e17	0.999128
$494$	0	0
$495$	−1.14818e16	−0.0350813
$496$	0	0
$497$	1.60992e17	0.479169
$498$	0	0
$499$	5.85158e17	1.69676	0.848379	−	0.529390i	$-0.177579\pi$
$499$	5.85158e17	1.69676	0.848379	+	0.529390i	$0.177579\pi$
$500$	0	0
$501$	−1.02449e17	−0.289442
$502$	0	0
$503$	5.03073e17	1.38497	0.692483	−	0.721434i	$-0.256517\pi$
$503$	5.03073e17	1.38497	0.692483	+	0.721434i	$0.256517\pi$
$504$	0	0
$505$	−2.25878e16	−0.0606010
$506$	0	0
$507$	−3.38805e17	−0.885926
$508$	0	0
$509$	2.39181e17	0.609622	0.304811	−	0.952413i	$-0.401407\pi$
$509$	2.39181e17	0.609622	0.304811	+	0.952413i	$0.401407\pi$
$510$	0	0
$511$	−7.80427e17	−1.93908
$512$	0	0
$513$	8.13546e16	0.197069
$514$	0	0
$515$	8.62256e16	0.203652
$516$	0	0
$517$	−6.81857e16	−0.157038
$518$	0	0
$519$	2.32763e17	0.522788
$520$	0	0
$521$	5.84939e17	1.28134	0.640671	−	0.767815i	$-0.278656\pi$
$521$	5.84939e17	1.28134	0.640671	+	0.767815i	$0.278656\pi$
$522$	0	0
$523$	1.38091e17	0.295055	0.147528	−	0.989058i	$-0.452868\pi$
$523$	1.38091e17	0.295055	0.147528	+	0.989058i	$0.452868\pi$
$524$	0	0
$525$	−9.58744e17	−1.99833
$526$	0	0
$527$	1.10497e17	0.224690
$528$	0	0
$529$	−1.36683e17	−0.271178
$530$	0	0
$531$	−3.67272e17	−0.711006
$532$	0	0
$533$	5.40713e17	1.02150
$534$	0	0
$535$	−4.36299e16	−0.0804424
$536$	0	0
$537$	−5.67281e17	−1.02086
$538$	0	0
$539$	−1.47358e17	−0.258848
$540$	0	0
$541$	2.87308e16	0.0492680	0.0246340	−	0.999697i	$-0.492158\pi$
$541$	2.87308e16	0.0492680	0.0246340	+	0.999697i	$0.492158\pi$
$542$	0	0
$543$	−8.70880e17	−1.45801
$544$	0	0
$545$	−1.67852e16	−0.0274378
$546$	0	0
$547$	−1.77510e17	−0.283338	−0.141669	−	0.989914i	$-0.545247\pi$
$547$	−1.77510e17	−0.283338	−0.141669	+	0.989914i	$0.545247\pi$
$548$	0	0
$549$	−2.21146e17	−0.344714
$550$	0	0
$551$	1.27854e18	1.94639
$552$	0	0
$553$	−6.75258e17	−1.00405
$554$	0	0
$555$	−2.20200e16	−0.0319825
$556$	0	0
$557$	6.70558e17	0.951432	0.475716	−	0.879599i	$-0.342189\pi$
$557$	6.70558e17	0.951432	0.475716	+	0.879599i	$0.342189\pi$
$558$	0	0
$559$	6.39291e17	0.886180
$560$	0	0
$561$	−1.80127e17	−0.243961
$562$	0	0
$563$	−3.10511e17	−0.410934	−0.205467	−	0.978664i	$-0.565871\pi$
$563$	−3.10511e17	−0.410934	−0.205467	+	0.978664i	$0.565871\pi$
$564$	0	0
$565$	1.14772e17	0.148430
$566$	0	0
$567$	−9.61389e17	−1.21509
$568$	0	0
$569$	−2.58365e17	−0.319157	−0.159578	−	0.987185i	$-0.551013\pi$
$569$	−2.58365e17	−0.319157	−0.159578	+	0.987185i	$0.551013\pi$
$570$	0	0
$571$	−9.91259e17	−1.19688	−0.598442	−	0.801166i	$-0.704213\pi$
$571$	−9.91259e17	−1.19688	−0.598442	+	0.801166i	$0.704213\pi$
$572$	0	0
$573$	1.20891e18	1.42689
$574$	0	0
$575$	−7.30215e17	−0.842577
$576$	0	0
$577$	−9.99359e17	−1.12740	−0.563701	−	0.825979i	$-0.690623\pi$
$577$	−9.99359e17	−1.12740	−0.563701	+	0.825979i	$0.690623\pi$
$578$	0	0
$579$	3.85730e17	0.425473
$580$	0	0
$581$	−1.08046e18	−1.16536
$582$	0	0
$583$	2.93407e17	0.309474
$584$	0	0
$585$	−7.71123e16	−0.0795446
$586$	0	0
$587$	−7.79668e17	−0.786614	−0.393307	−	0.919407i	$-0.628669\pi$
$587$	−7.79668e17	−0.786614	−0.393307	+	0.919407i	$0.628669\pi$
$588$	0	0
$589$	4.43548e17	0.437715
$590$	0	0
$591$	2.87443e18	2.77480
$592$	0	0
$593$	8.86852e17	0.837521	0.418761	−	0.908097i	$-0.362465\pi$
$593$	8.86852e17	0.837521	0.418761	+	0.908097i	$0.362465\pi$
$594$	0	0
$595$	1.04748e17	0.0967801
$596$	0	0
$597$	−6.73452e17	−0.608798
$598$	0	0
$599$	1.56613e18	1.38533	0.692665	−	0.721260i	$-0.256437\pi$
$599$	1.56613e18	1.38533	0.692665	+	0.721260i	$0.256437\pi$
$600$	0	0
$601$	1.44186e18	1.24807	0.624036	−	0.781396i	$-0.285492\pi$
$601$	1.44186e18	1.24807	0.624036	+	0.781396i	$0.285492\pi$
$602$	0	0
$603$	−1.69956e17	−0.143971
$604$	0	0
$605$	−1.27277e17	−0.105521
$606$	0	0
$607$	−4.08280e17	−0.331307	−0.165654	−	0.986184i	$-0.552973\pi$
$607$	−4.08280e17	−0.331307	−0.165654	+	0.986184i	$0.552973\pi$
$608$	0	0
$609$	4.18473e18	3.32395
$610$	0	0
$611$	−4.57939e17	−0.356073
$612$	0	0
$613$	−1.01386e18	−0.771768	−0.385884	−	0.922547i	$-0.626104\pi$
$613$	−1.01386e18	−0.771768	−0.385884	+	0.922547i	$0.626104\pi$
$614$	0	0
$615$	−3.64120e17	−0.271367
$616$	0	0
$617$	−2.38764e18	−1.74227	−0.871133	−	0.491047i	$-0.836614\pi$
$617$	−2.38764e18	−1.74227	−0.871133	+	0.491047i	$0.836614\pi$
$618$	0	0
$619$	8.05525e17	0.575559	0.287780	−	0.957697i	$-0.407083\pi$
$619$	8.05525e17	0.575559	0.287780	+	0.957697i	$0.407083\pi$
$620$	0	0
$621$	2.02807e17	0.141901
$622$	0	0
$623$	1.29780e18	0.889272
$624$	0	0
$625$	1.43207e18	0.961045
$626$	0	0
$627$	−7.23050e17	−0.475258
$628$	0	0
$629$	−1.82064e17	−0.117218
$630$	0	0
$631$	−2.64035e18	−1.66521	−0.832607	−	0.553865i	$-0.813153\pi$
$631$	−2.64035e18	−1.66521	−0.832607	+	0.553865i	$0.813153\pi$
$632$	0	0
$633$	−2.46914e18	−1.52553
$634$	0	0
$635$	−1.90727e17	−0.115447
$636$	0	0
$637$	−9.89661e17	−0.586921
$638$	0	0
$639$	6.59884e17	0.383453
$640$	0	0
$641$	−1.06429e18	−0.606015	−0.303007	−	0.952988i	$-0.597991\pi$
$641$	−1.06429e18	−0.606015	−0.303007	+	0.952988i	$0.597991\pi$
$642$	0	0
$643$	−6.81097e17	−0.380047	−0.190024	−	0.981780i	$-0.560856\pi$
$643$	−6.81097e17	−0.380047	−0.190024	+	0.981780i	$0.560856\pi$
$644$	0	0
$645$	−4.30504e17	−0.235417
$646$	0	0
$647$	−2.04457e18	−1.09578	−0.547891	−	0.836550i	$-0.684569\pi$
$647$	−2.04457e18	−1.09578	−0.547891	+	0.836550i	$0.684569\pi$
$648$	0	0
$649$	3.34856e17	0.175901
$650$	0	0
$651$	1.45175e18	0.747509
$652$	0	0
$653$	2.08196e18	1.05084	0.525419	−	0.850844i	$-0.323909\pi$
$653$	2.08196e18	1.05084	0.525419	+	0.850844i	$0.323909\pi$
$654$	0	0
$655$	4.13738e17	0.204718
$656$	0	0
$657$	−3.19885e18	−1.55174
$658$	0	0
$659$	−2.11908e18	−1.00784	−0.503920	−	0.863750i	$-0.668109\pi$
$659$	−2.11908e18	−1.00784	−0.503920	+	0.863750i	$0.668109\pi$
$660$	0	0
$661$	−1.46459e18	−0.682980	−0.341490	−	0.939885i	$-0.610931\pi$
$661$	−1.46459e18	−0.682980	−0.341490	+	0.939885i	$0.610931\pi$
$662$	0	0
$663$	−1.20974e18	−0.553165
$664$	0	0
$665$	4.20470e17	0.188536
$666$	0	0
$667$	3.18724e18	1.40151
$668$	0	0
$669$	3.74696e18	1.61588
$670$	0	0
$671$	2.01628e17	0.0852813
$672$	0	0
$673$	7.25358e17	0.300922	0.150461	−	0.988616i	$-0.451924\pi$
$673$	7.25358e17	0.300922	0.150461	+	0.988616i	$0.451924\pi$
$674$	0	0
$675$	−4.03134e17	−0.164049
$676$	0	0
$677$	4.22902e17	0.168816	0.0844079	−	0.996431i	$-0.473100\pi$
$677$	4.22902e17	0.168816	0.0844079	+	0.996431i	$0.473100\pi$
$678$	0	0
$679$	1.89951e18	0.743856
$680$	0	0
$681$	−5.36102e17	−0.205964
$682$	0	0
$683$	−1.37469e18	−0.518168	−0.259084	−	0.965855i	$-0.583421\pi$
$683$	−1.37469e18	−0.518168	−0.259084	+	0.965855i	$0.583421\pi$
$684$	0	0
$685$	2.01648e17	0.0745769
$686$	0	0
$687$	−5.16806e18	−1.87546
$688$	0	0
$689$	1.97053e18	0.701712
$690$	0	0
$691$	3.23918e18	1.13195	0.565976	−	0.824422i	$-0.308500\pi$
$691$	3.23918e18	1.13195	0.565976	+	0.824422i	$0.308500\pi$
$692$	0	0
$693$	−1.24726e18	−0.427752
$694$	0	0
$695$	−8.57361e16	−0.0288578
$696$	0	0
$697$	−3.01059e18	−0.994577
$698$	0	0
$699$	8.06697e18	2.61583
$700$	0	0
$701$	−4.32730e18	−1.37737	−0.688683	−	0.725062i	$-0.741811\pi$
$701$	−4.32730e18	−1.37737	−0.688683	+	0.725062i	$0.741811\pi$
$702$	0	0
$703$	−7.30824e17	−0.228351
$704$	0	0
$705$	3.08379e17	0.0945923
$706$	0	0
$707$	−2.45370e18	−0.738917
$708$	0	0
$709$	3.72211e18	1.10050	0.550248	−	0.835001i	$-0.314533\pi$
$709$	3.72211e18	1.10050	0.550248	+	0.835001i	$0.314533\pi$
$710$	0	0
$711$	−2.76778e18	−0.803488
$712$	0	0
$713$	1.10571e18	0.315180
$714$	0	0
$715$	7.03063e16	0.0196791
$716$	0	0
$717$	8.09590e18	2.22531
$718$	0	0
$719$	2.69743e18	0.728135	0.364068	−	0.931373i	$-0.381388\pi$
$719$	2.69743e18	0.728135	0.364068	+	0.931373i	$0.381388\pi$
$720$	0	0
$721$	9.36665e18	2.48316
$722$	0	0
$723$	3.54237e18	0.922349
$724$	0	0
$725$	−6.33552e18	−1.62026
$726$	0	0
$727$	1.30338e18	0.327414	0.163707	−	0.986509i	$-0.447655\pi$
$727$	1.30338e18	0.327414	0.163707	+	0.986509i	$0.447655\pi$
$728$	0	0
$729$	−4.92006e18	−1.21406
$730$	0	0
$731$	−3.55946e18	−0.862821
$732$	0	0
$733$	1.23214e18	0.293417	0.146708	−	0.989180i	$-0.453132\pi$
$733$	1.23214e18	0.293417	0.146708	+	0.989180i	$0.453132\pi$
$734$	0	0
$735$	6.66445e17	0.155918
$736$	0	0
$737$	1.54956e17	0.0356179
$738$	0	0
$739$	2.37998e18	0.537508	0.268754	−	0.963209i	$-0.413388\pi$
$739$	2.37998e18	0.537508	0.268754	+	0.963209i	$0.413388\pi$
$740$	0	0
$741$	−4.85604e18	−1.07762
$742$	0	0
$743$	1.71342e18	0.373626	0.186813	−	0.982396i	$-0.440184\pi$
$743$	1.71342e18	0.373626	0.186813	+	0.982396i	$0.440184\pi$
$744$	0	0
$745$	−3.51017e17	−0.0752163
$746$	0	0
$747$	−4.42863e18	−0.932578
$748$	0	0
$749$	−4.73950e18	−0.980846
$750$	0	0
$751$	−8.15576e18	−1.65884	−0.829421	−	0.558624i	$-0.811329\pi$
$751$	−8.15576e18	−1.65884	−0.829421	+	0.558624i	$0.811329\pi$
$752$	0	0
$753$	−6.35863e17	−0.127115
$754$	0	0
$755$	8.27667e16	0.0162630
$756$	0	0
$757$	−4.08819e18	−0.789601	−0.394800	−	0.918767i	$-0.629186\pi$
$757$	−4.08819e18	−0.789601	−0.394800	+	0.918767i	$0.629186\pi$
$758$	0	0
$759$	−1.80247e18	−0.342213
$760$	0	0
$761$	4.83340e18	0.902094	0.451047	−	0.892500i	$-0.351051\pi$
$761$	4.83340e18	0.902094	0.451047	+	0.892500i	$0.351051\pi$
$762$	0	0
$763$	−1.82337e18	−0.334554
$764$	0	0
$765$	4.29347e17	0.0774478
$766$	0	0
$767$	2.24891e18	0.398844
$768$	0	0
$769$	−5.13768e18	−0.895873	−0.447936	−	0.894065i	$-0.647841\pi$
$769$	−5.13768e18	−0.895873	−0.447936	+	0.894065i	$0.647841\pi$
$770$	0	0
$771$	8.43654e17	0.144647
$772$	0	0
$773$	−2.81948e18	−0.475337	−0.237669	−	0.971346i	$-0.576383\pi$
$773$	−2.81948e18	−0.475337	−0.237669	+	0.971346i	$0.576383\pi$
$774$	0	0
$775$	−2.19790e18	−0.364374
$776$	0	0
$777$	−2.39202e18	−0.389967
$778$	0	0
$779$	−1.20848e19	−1.93752
$780$	0	0
$781$	−6.01642e17	−0.0948652
$782$	0	0
$783$	1.75960e18	0.272874
$784$	0	0
$785$	−9.07816e17	−0.138467
$786$	0	0
$787$	5.90600e18	0.886050	0.443025	−	0.896509i	$-0.353905\pi$
$787$	5.90600e18	0.886050	0.443025	+	0.896509i	$0.353905\pi$
$788$	0	0
$789$	1.48526e19	2.19181
$790$	0	0
$791$	1.24677e19	1.80983
$792$	0	0
$793$	1.35414e18	0.193370
$794$	0	0
$795$	−1.32697e18	−0.186413
$796$	0	0
$797$	6.00465e18	0.829867	0.414933	−	0.909852i	$-0.363805\pi$
$797$	6.00465e18	0.829867	0.414933	+	0.909852i	$0.363805\pi$
$798$	0	0
$799$	2.54972e18	0.346687
$800$	0	0
$801$	5.31948e18	0.711636
$802$	0	0
$803$	2.91652e18	0.383897
$804$	0	0
$805$	1.04818e18	0.135757
$806$	0	0
$807$	9.29124e18	1.18412
$808$	0	0
$809$	−1.03281e19	−1.29525	−0.647624	−	0.761960i	$-0.724237\pi$
$809$	−1.03281e19	−1.29525	−0.647624	+	0.761960i	$0.724237\pi$
$810$	0	0
$811$	−2.37489e18	−0.293094	−0.146547	−	0.989204i	$-0.546816\pi$
$811$	−2.37489e18	−0.293094	−0.146547	+	0.989204i	$0.546816\pi$
$812$	0	0
$813$	9.02308e17	0.109589
$814$	0	0
$815$	5.64718e17	0.0675005
$816$	0	0
$817$	−1.42880e19	−1.68085
$818$	0	0
$819$	−8.37668e18	−0.969899
$820$	0	0
$821$	6.17741e18	0.704006	0.352003	−	0.935999i	$-0.385501\pi$
$821$	6.17741e18	0.704006	0.352003	+	0.935999i	$0.385501\pi$
$822$	0	0
$823$	3.84246e18	0.431033	0.215516	−	0.976500i	$-0.430857\pi$
$823$	3.84246e18	0.431033	0.215516	+	0.976500i	$0.430857\pi$
$824$	0	0
$825$	3.58291e18	0.395626
$826$	0	0
$827$	−1.35926e19	−1.47746	−0.738731	−	0.674001i	$-0.764574\pi$
$827$	−1.35926e19	−1.47746	−0.738731	+	0.674001i	$0.764574\pi$
$828$	0	0
$829$	4.07966e17	0.0436536	0.0218268	−	0.999762i	$-0.493052\pi$
$829$	4.07966e17	0.0436536	0.0218268	+	0.999762i	$0.493052\pi$
$830$	0	0
$831$	7.21395e18	0.759917
$832$	0	0
$833$	5.51025e18	0.571451
$834$	0	0
$835$	−2.22642e17	−0.0227324
$836$	0	0
$837$	6.10435e17	0.0613655
$838$	0	0
$839$	−4.60930e18	−0.456229	−0.228114	−	0.973634i	$-0.573256\pi$
$839$	−4.60930e18	−0.456229	−0.228114	+	0.973634i	$0.573256\pi$
$840$	0	0
$841$	1.73927e19	1.69509
$842$	0	0
$843$	2.10070e19	2.01598
$844$	0	0
$845$	−7.36291e17	−0.0695794
$846$	0	0
$847$	−1.38261e19	−1.28664
$848$	0	0
$849$	−3.89683e18	−0.357118
$850$	0	0
$851$	−1.82185e18	−0.164426
$852$	0	0
$853$	−5.13496e18	−0.456424	−0.228212	−	0.973612i	$-0.573288\pi$
$853$	−5.13496e18	−0.456424	−0.228212	+	0.973612i	$0.573288\pi$
$854$	0	0
$855$	1.72345e18	0.150875
$856$	0	0
$857$	8.29128e18	0.714901	0.357451	−	0.933932i	$-0.383646\pi$
$857$	8.29128e18	0.714901	0.357451	+	0.933932i	$0.383646\pi$
$858$	0	0
$859$	6.79619e17	0.0577179	0.0288589	−	0.999583i	$-0.490813\pi$
$859$	6.79619e17	0.0577179	0.0288589	+	0.999583i	$0.490813\pi$
$860$	0	0
$861$	−3.95542e19	−3.30881
$862$	0	0
$863$	2.11875e19	1.74586	0.872929	−	0.487846i	$-0.162217\pi$
$863$	2.11875e19	1.74586	0.872929	+	0.487846i	$0.162217\pi$
$864$	0	0
$865$	5.05840e17	0.0410590
$866$	0	0
$867$	−1.14492e19	−0.915483
$868$	0	0
$869$	2.52349e18	0.198781
$870$	0	0
$871$	1.04069e18	0.0807613
$872$	0	0
$873$	7.78582e18	0.595267
$874$	0	0
$875$	−4.19462e18	−0.315966
$876$	0	0
$877$	5.29243e18	0.392788	0.196394	−	0.980525i	$-0.437077\pi$
$877$	5.29243e18	0.392788	0.196394	+	0.980525i	$0.437077\pi$
$878$	0	0
$879$	−3.65455e19	−2.67243
$880$	0	0
$881$	2.31832e18	0.167043	0.0835217	−	0.996506i	$-0.473383\pi$
$881$	2.31832e18	0.167043	0.0835217	+	0.996506i	$0.473383\pi$
$882$	0	0
$883$	−1.38961e19	−0.986615	−0.493307	−	0.869855i	$-0.664212\pi$
$883$	−1.38961e19	−0.986615	−0.493307	+	0.869855i	$0.664212\pi$
$884$	0	0
$885$	−1.51443e18	−0.105954
$886$	0	0
$887$	1.68484e19	1.16159	0.580797	−	0.814048i	$-0.302741\pi$
$887$	1.68484e19	1.16159	0.580797	+	0.814048i	$0.302741\pi$
$888$	0	0
$889$	−2.07186e19	−1.40766
$890$	0	0
$891$	3.59279e18	0.240562
$892$	0	0
$893$	1.02348e19	0.675378
$894$	0	0
$895$	−1.23282e18	−0.0801768
$896$	0	0
$897$	−1.21055e19	−0.775945
$898$	0	0
$899$	9.59340e18	0.606087
$900$	0	0
$901$	−1.09716e19	−0.683215
$902$	0	0
$903$	−4.67654e19	−2.87048
$904$	0	0
$905$	−1.89260e18	−0.114510
$906$	0	0
$907$	2.54854e19	1.52000	0.760000	−	0.649923i	$-0.225199\pi$
$907$	2.54854e19	1.52000	0.760000	+	0.649923i	$0.225199\pi$
$908$	0	0
$909$	−1.00574e19	−0.591315
$910$	0	0
$911$	1.75641e19	1.01802	0.509010	−	0.860761i	$-0.330012\pi$
$911$	1.75641e19	1.01802	0.509010	+	0.860761i	$0.330012\pi$
$912$	0	0
$913$	4.03775e18	0.230717
$914$	0	0
$915$	−9.11888e17	−0.0513694
$916$	0	0
$917$	4.49442e19	2.49616
$918$	0	0
$919$	−2.76291e18	−0.151292	−0.0756460	−	0.997135i	$-0.524102\pi$
$919$	−2.76291e18	−0.151292	−0.0756460	+	0.997135i	$0.524102\pi$
$920$	0	0
$921$	2.32457e19	1.25503
$922$	0	0
$923$	−4.04066e18	−0.215100
$924$	0	0
$925$	3.62143e18	0.190090
$926$	0	0
$927$	3.83925e19	1.98714
$928$	0	0
$929$	−2.02252e19	−1.03226	−0.516131	−	0.856510i	$-0.672628\pi$
$929$	−2.02252e19	−1.03226	−0.516131	+	0.856510i	$0.672628\pi$
$930$	0	0
$931$	2.21187e19	1.11324
$932$	0	0
$933$	−1.99533e18	−0.0990338
$934$	0	0
$935$	−3.91453e17	−0.0191604
$936$	0	0
$937$	9.12409e18	0.440436	0.220218	−	0.975451i	$-0.429323\pi$
$937$	9.12409e18	0.440436	0.220218	+	0.975451i	$0.429323\pi$
$938$	0	0
$939$	3.91772e19	1.86512
$940$	0	0
$941$	9.47005e18	0.444651	0.222326	−	0.974972i	$-0.428635\pi$
$941$	9.47005e18	0.444651	0.222326	+	0.974972i	$0.428635\pi$
$942$	0	0
$943$	−3.01260e19	−1.39513
$944$	0	0
$945$	5.78674e17	0.0264318
$946$	0	0
$947$	1.07060e19	0.482340	0.241170	−	0.970483i	$-0.422469\pi$
$947$	1.07060e19	0.482340	0.241170	+	0.970483i	$0.422469\pi$
$948$	0	0
$949$	1.95875e19	0.870460
$950$	0	0
$951$	−5.52583e19	−2.42228
$952$	0	0
$953$	−1.17296e17	−0.00507199	−0.00253599	−	0.999997i	$-0.500807\pi$
$953$	−1.17296e17	−0.00507199	−0.00253599	+	0.999997i	$0.500807\pi$
$954$	0	0
$955$	2.62722e18	0.112066
$956$	0	0
$957$	−1.56387e19	−0.658070
$958$	0	0
$959$	2.19049e19	0.909328
$960$	0	0
$961$	−2.10894e19	−0.863700
$962$	0	0
$963$	−1.94265e19	−0.784917
$964$	0	0
$965$	8.38269e17	0.0334161
$966$	0	0
$967$	−1.48196e19	−0.582860	−0.291430	−	0.956592i	$-0.594131\pi$
$967$	−1.48196e19	−0.582860	−0.291430	+	0.956592i	$0.594131\pi$
$968$	0	0
$969$	2.70375e19	1.04921
$970$	0	0
$971$	−3.03687e19	−1.16279	−0.581395	−	0.813622i	$-0.697493\pi$
$971$	−3.03687e19	−1.16279	−0.581395	+	0.813622i	$0.697493\pi$
$972$	0	0
$973$	−9.31348e18	−0.351867
$974$	0	0
$975$	2.40630e19	0.897055
$976$	0	0
$977$	−3.42204e19	−1.25884	−0.629419	−	0.777066i	$-0.716707\pi$
$977$	−3.42204e19	−1.25884	−0.629419	+	0.777066i	$0.716707\pi$
$978$	0	0
$979$	−4.84998e18	−0.176057
$980$	0	0
$981$	−7.47373e18	−0.267725
$982$	0	0
$983$	2.19778e19	0.776939	0.388469	−	0.921462i	$-0.373004\pi$
$983$	2.19778e19	0.776939	0.388469	+	0.921462i	$0.373004\pi$
$984$	0	0
$985$	6.24671e18	0.217929
$986$	0	0
$987$	3.34991e19	1.15338
$988$	0	0
$989$	−3.56183e19	−1.21031
$990$	0	0
$991$	−1.72348e19	−0.578001	−0.289000	−	0.957329i	$-0.593323\pi$
$991$	−1.72348e19	−0.578001	−0.289000	+	0.957329i	$0.593323\pi$
$992$	0	0
$993$	−1.84459e19	−0.610564
$994$	0	0
$995$	−1.46355e18	−0.0478142
$996$	0	0
$997$	5.30366e19	1.71024	0.855119	−	0.518431i	$-0.173484\pi$
$997$	5.30366e19	1.71024	0.855119	+	0.518431i	$0.173484\pi$
$998$	0	0
$999$	−1.00580e18	−0.0320137

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.14.a.d.1.1		1
3.2	odd	2		144.14.a.d.1.1		1
4.3	odd	2		2.14.a.a.1.1	✓	1
8.3	odd	2		64.14.a.i.1.1		1
8.5	even	2		64.14.a.a.1.1		1
12.11	even	2		18.14.a.d.1.1		1
20.3	even	4		50.14.b.e.49.2		2
20.7	even	4		50.14.b.e.49.1		2
20.19	odd	2		50.14.a.e.1.1		1
28.3	even	6		98.14.c.e.79.1		2
28.11	odd	6		98.14.c.h.79.1		2
28.19	even	6		98.14.c.e.67.1		2
28.23	odd	6		98.14.c.h.67.1		2
28.27	even	2		98.14.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2.14.a.a.1.1	✓	1	4.3	odd	2
16.14.a.d.1.1		1	1.1	even	1	trivial
18.14.a.d.1.1		1	12.11	even	2
50.14.a.e.1.1		1	20.19	odd	2
50.14.b.e.49.1		2	20.7	even	4
50.14.b.e.49.2		2	20.3	even	4
64.14.a.a.1.1		1	8.5	even	2
64.14.a.i.1.1		1	8.3	odd	2
98.14.a.b.1.1		1	28.27	even	2
98.14.c.e.67.1		2	28.19	even	6
98.14.c.e.79.1		2	28.3	even	6
98.14.c.h.67.1		2	28.23	odd	6
98.14.c.h.79.1		2	28.11	odd	6
144.14.a.d.1.1		1	3.2	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	16.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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