LMFDB - Embedded newform 200.14.c.a.49.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,14,Mod(49,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 14, names="a")

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

chi := DirichletCharacter("200.49");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 14 $
Character orbit:	$[\chi]$	$=$	200.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$214.461857904$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	200.49
Dual form			200.14.c.a.49.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.0000i q^{3} +139992. i q^{7} +1.59418e6 q^{9} +O(q^{10})$	$q-12.0000i q^{3} +139992. i q^{7} +1.59418e6 q^{9} -6.48432e6 q^{11} -2.25880e7i q^{13} +2.37323e7i q^{17} -3.25345e8 q^{19} +1.67990e6 q^{21} +9.21601e8i q^{23} -3.82620e7i q^{27} +3.86588e9 q^{29} -2.25340e9 q^{31} +7.78119e7i q^{33} -1.82504e10i q^{37} -2.71056e8 q^{39} +3.44228e10 q^{41} -1.71925e10i q^{43} +6.73717e10i q^{47} +7.72913e10 q^{49} +2.84787e8 q^{51} -8.72812e10i q^{53} +3.90414e9i q^{57} -5.40215e11 q^{59} -5.12766e10 q^{61} +2.23172e11i q^{63} -2.55199e10i q^{67} +1.10592e10 q^{69} -1.38750e12 q^{71} -8.19049e11i q^{73} -9.07753e11i q^{77} +4.03094e12 q^{79} +2.54118e12 q^{81} +4.18082e12i q^{83} -4.63906e10i q^{87} -2.67703e12 q^{89} +3.16214e12 q^{91} +2.70408e10i q^{93} +1.40395e13i q^{97} -1.03372e13 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3188358 q^{9}+O(q^{10}) $ 2 * q + 3188358 * q^9	$ 2 q + 3188358 q^{9} - 12968648 q^{11} - 650689672 q^{19} + 3359808 q^{21} + 7731758436 q^{29} - 4506802880 q^{31} - 542112816 q^{39} + 68845690644 q^{41} + 154582500686 q^{49} + 569574480 q^{51} - 1080429037336 q^{59} - 102553137700 q^{61} + 22118415168 q^{69} - 2775001398064 q^{71} + 8061871230688 q^{79} + 5082354203058 q^{81} - 5354055596532 q^{89} + 6324288111456 q^{91} - 20674346299992 q^{99}+O(q^{100}) $ 2 * q + 3188358 * q^9 - 12968648 * q^11 - 650689672 * q^19 + 3359808 * q^21 + 7731758436 * q^29 - 4506802880 * q^31 - 542112816 * q^39 + 68845690644 * q^41 + 154582500686 * q^49 + 569574480 * q^51 - 1080429037336 * q^59 - 102553137700 * q^61 + 22118415168 * q^69 - 2775001398064 * q^71 + 8061871230688 * q^79 + 5082354203058 * q^81 - 5354055596532 * q^89 + 6324288111456 * q^91 - 20674346299992 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/200\mathbb{Z}\right)^\times$.

$n$	$101$	$151$	$177$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 12.0000i	− 0.00950371i	−0.999989	−	0.00475185i	$-0.998487\pi$
$3$	− 12.0000i	− 0.00950371i	0.999989	−	0.00475185i	$-0.00151257\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	139992.i	0.449745i	0.974388	+	0.224872i	$0.0721965\pi$
$7$	139992.i	0.449745i	−0.974388	+	0.224872i	$0.927804\pi$
$8$	0	0
$9$	1.59418e6	0.999910
$10$	0	0
$11$	−6.48432e6	−1.10360	−0.551801	−	0.833976i	$-0.686059\pi$
$11$	−6.48432e6	−1.10360	−0.551801	+	0.833976i	$0.686059\pi$
$12$	0	0
$13$	− 2.25880e7i	− 1.29792i	−0.760824	−	0.648958i	$-0.775205\pi$
$13$	− 2.25880e7i	− 1.29792i	0.760824	−	0.648958i	$-0.224795\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.37323e7i	0.238463i	0.992866	+	0.119232i	$0.0380431\pi$
$17$	2.37323e7i	0.238463i	−0.992866	+	0.119232i	$0.961957\pi$
$18$	0	0
$19$	−3.25345e8	−1.58652	−0.793260	−	0.608883i	$-0.791618\pi$
$19$	−3.25345e8	−1.58652	−0.793260	+	0.608883i	$0.791618\pi$
$20$	0	0
$21$	1.67990e6	0.00427424
$22$	0	0
$23$	9.21601e8i	1.29811i	0.760741	+	0.649055i	$0.224836\pi$
$23$	9.21601e8i	1.29811i	−0.760741	+	0.649055i	$0.775164\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 3.82620e7i	− 0.0190066i
$28$	0	0
$29$	3.86588e9	1.20687	0.603436	−	0.797411i	$-0.293798\pi$
$29$	3.86588e9	1.20687	0.603436	+	0.797411i	$0.293798\pi$
$30$	0	0
$31$	−2.25340e9	−0.456024	−0.228012	−	0.973658i	$-0.573223\pi$
$31$	−2.25340e9	−0.456024	−0.228012	+	0.973658i	$0.573223\pi$
$32$	0	0
$33$	7.78119e7i	0.0104883i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 1.82504e10i	− 1.16939i	−0.811252	−	0.584697i	$-0.801213\pi$
$37$	− 1.82504e10i	− 1.16939i	0.811252	−	0.584697i	$-0.198787\pi$
$38$	0	0
$39$	−2.71056e8	−0.0123350
$40$	0	0
$41$	3.44228e10	1.13175	0.565877	−	0.824490i	$-0.308538\pi$
$41$	3.44228e10	1.13175	0.565877	+	0.824490i	$0.308538\pi$
$42$	0	0
$43$	− 1.71925e10i	− 0.414757i	−0.978261	−	0.207379i	$-0.933507\pi$
$43$	− 1.71925e10i	− 0.414757i	0.978261	−	0.207379i	$-0.0664932\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	6.73717e10i	0.911679i	0.890062	+	0.455839i	$0.150661\pi$
$47$	6.73717e10i	0.911679i	−0.890062	+	0.455839i	$0.849339\pi$
$48$	0	0
$49$	7.72913e10	0.797730
$50$	0	0
$51$	2.84787e8	0.00226628
$52$	0	0
$53$	− 8.72812e10i	− 0.540913i	−0.962732	−	0.270457i	$-0.912825\pi$
$53$	− 8.72812e10i	− 0.540913i	0.962732	−	0.270457i	$-0.0871747\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	3.90414e9i	0.0150778i
$58$	0	0
$59$	−5.40215e11	−1.66736	−0.833678	−	0.552251i	$-0.813769\pi$
$59$	−5.40215e11	−1.66736	−0.833678	+	0.552251i	$0.813769\pi$
$60$	0	0
$61$	−5.12766e10	−0.127431	−0.0637155	−	0.997968i	$-0.520295\pi$
$61$	−5.12766e10	−0.127431	−0.0637155	+	0.997968i	$0.520295\pi$
$62$	0	0
$63$	2.23172e11i	0.449704i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2.55199e10i	− 0.0344662i	−0.999851	−	0.0172331i	$-0.994514\pi$
$67$	− 2.55199e10i	− 0.0344662i	0.999851	−	0.0172331i	$-0.00548574\pi$
$68$	0	0
$69$	1.10592e10	0.0123369
$70$	0	0
$71$	−1.38750e12	−1.28545	−0.642723	−	0.766099i	$-0.722195\pi$
$71$	−1.38750e12	−1.28545	−0.642723	+	0.766099i	$0.722195\pi$
$72$	0	0
$73$	− 8.19049e11i	− 0.633449i	−0.948518	−	0.316724i	$-0.897417\pi$
$73$	− 8.19049e11i	− 0.633449i	0.948518	−	0.316724i	$-0.102583\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9.07753e11i	− 0.496339i
$78$	0	0
$79$	4.03094e12	1.86565	0.932824	−	0.360332i	$-0.117337\pi$
$79$	4.03094e12	1.86565	0.932824	+	0.360332i	$0.117337\pi$
$80$	0	0
$81$	2.54118e12	0.999729
$82$	0	0
$83$	4.18082e12i	1.40364i	0.712357	+	0.701818i	$0.247628\pi$
$83$	4.18082e12i	1.40364i	−0.712357	+	0.701818i	$0.752372\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 4.63906e10i	− 0.0114698i
$88$	0	0
$89$	−2.67703e12	−0.570976	−0.285488	−	0.958382i	$-0.592156\pi$
$89$	−2.67703e12	−0.570976	−0.285488	+	0.958382i	$0.592156\pi$
$90$	0	0
$91$	3.16214e12	0.583731
$92$	0	0
$93$	2.70408e10i	0.00433392i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.40395e13i	1.71133i	0.517528	+	0.855666i	$0.326852\pi$
$97$	1.40395e13i	1.71133i	−0.517528	+	0.855666i	$0.673148\pi$
$98$	0	0
$99$	−1.03372e13	−1.10350
$100$	0	0
$101$	−6.22392e12	−0.583411	−0.291706	−	0.956508i	$-0.594223\pi$
$101$	−6.22392e12	−0.583411	−0.291706	+	0.956508i	$0.594223\pi$
$102$	0	0
$103$	− 2.11756e13i	− 1.74740i	−0.486461	−	0.873702i	$-0.661712\pi$
$103$	− 2.11756e13i	− 1.74740i	0.486461	−	0.873702i	$-0.338288\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.87895e13i	− 1.21038i	−0.796083	−	0.605188i	$-0.793098\pi$
$107$	− 1.87895e13i	− 1.21038i	0.796083	−	0.605188i	$-0.206902\pi$
$108$	0	0
$109$	9.95159e12	0.568356	0.284178	−	0.958772i	$-0.408279\pi$
$109$	9.95159e12	0.568356	0.284178	+	0.958772i	$0.408279\pi$
$110$	0	0
$111$	−2.19005e11	−0.0111136
$112$	0	0
$113$	2.08879e13i	0.943812i	0.881649	+	0.471906i	$0.156434\pi$
$113$	2.08879e13i	0.943812i	−0.881649	+	0.471906i	$0.843566\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.60094e13i	− 1.29780i
$118$	0	0
$119$	−3.32233e12	−0.107248
$120$	0	0
$121$	7.52375e12	0.217936
$122$	0	0
$123$	− 4.13074e11i	− 0.0107559i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	6.26814e13i	1.32561i	0.748794	+	0.662803i	$0.230633\pi$
$127$	6.26814e13i	1.32561i	−0.748794	+	0.662803i	$0.769367\pi$
$128$	0	0
$129$	−2.06310e11	−0.00394173
$130$	0	0
$131$	−6.97773e12	−0.120629	−0.0603144	−	0.998179i	$-0.519210\pi$
$131$	−6.97773e12	−0.120629	−0.0603144	+	0.998179i	$0.519210\pi$
$132$	0	0
$133$	− 4.55457e13i	− 0.713529i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.78591e13i	0.747632i	0.927503	+	0.373816i	$0.121951\pi$
$137$	5.78591e13i	0.747632i	−0.927503	+	0.373816i	$0.878049\pi$
$138$	0	0
$139$	3.93498e13	0.462750	0.231375	−	0.972865i	$-0.425678\pi$
$139$	3.93498e13	0.462750	0.231375	+	0.972865i	$0.425678\pi$
$140$	0	0
$141$	8.08461e11	0.00866433
$142$	0	0
$143$	1.46468e14i	1.43238i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 9.27495e11i	− 0.00758139i
$148$	0	0
$149$	4.31370e13	0.322953	0.161476	−	0.986877i	$-0.448374\pi$
$149$	4.31370e13	0.322953	0.161476	+	0.986877i	$0.448374\pi$
$150$	0	0
$151$	2.19599e14	1.50758	0.753788	−	0.657117i	$-0.228224\pi$
$151$	2.19599e14	1.50758	0.753788	+	0.657117i	$0.228224\pi$
$152$	0	0
$153$	3.78335e13i	0.238442i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 6.61866e13i	− 0.352714i	−0.984326	−	0.176357i	$-0.943569\pi$
$157$	− 6.61866e13i	− 0.352714i	0.984326	−	0.176357i	$-0.0564313\pi$
$158$	0	0
$159$	−1.04737e12	−0.00514068
$160$	0	0
$161$	−1.29017e14	−0.583818
$162$	0	0
$163$	− 2.47622e14i	− 1.03412i	−0.855950	−	0.517058i	$-0.827027\pi$
$163$	− 2.47622e14i	− 1.03412i	0.855950	−	0.517058i	$-0.172973\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.05226e13i	0.215904i	0.994156	+	0.107952i	$0.0344292\pi$
$167$	6.05226e13i	0.215904i	−0.994156	+	0.107952i	$0.965571\pi$
$168$	0	0
$169$	−2.07344e14	−0.684586
$170$	0	0
$171$	−5.18658e14	−1.58638
$172$	0	0
$173$	− 2.82357e14i	− 0.800752i	−0.916351	−	0.400376i	$-0.868879\pi$
$173$	− 2.82357e14i	− 0.800752i	0.916351	−	0.400376i	$-0.131121\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	6.48257e12i	0.0158461i
$178$	0	0
$179$	1.65618e14	0.376324	0.188162	−	0.982138i	$-0.439747\pi$
$179$	1.65618e14	0.376324	0.188162	+	0.982138i	$0.439747\pi$
$180$	0	0
$181$	8.52134e14	1.80135	0.900673	−	0.434497i	$-0.143074\pi$
$181$	8.52134e14	1.80135	0.900673	+	0.434497i	$0.143074\pi$
$182$	0	0
$183$	6.15319e11i	0.00121107i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 1.53888e14i	− 0.263168i
$188$	0	0
$189$	5.35638e12	0.00854810
$190$	0	0
$191$	9.30990e14	1.38748	0.693742	−	0.720223i	$-0.255961\pi$
$191$	9.30990e14	1.38748	0.693742	+	0.720223i	$0.255961\pi$
$192$	0	0
$193$	4.17870e14i	0.581994i	0.956724	+	0.290997i	$0.0939870\pi$
$193$	4.17870e14i	0.581994i	−0.956724	+	0.290997i	$0.906013\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.17420e15i	1.43124i	0.698489	+	0.715621i	$0.253856\pi$
$197$	1.17420e15i	1.43124i	−0.698489	+	0.715621i	$0.746144\pi$
$198$	0	0
$199$	−9.52478e13	−0.108720	−0.0543601	−	0.998521i	$-0.517312\pi$
$199$	−9.52478e13	−0.108720	−0.0543601	+	0.998521i	$0.517312\pi$
$200$	0	0
$201$	−3.06239e11	−0.000327557 0
$202$	0	0
$203$	5.41192e14i	0.542784i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.46920e15i	1.29799i
$208$	0	0
$209$	2.10964e15	1.75089
$210$	0	0
$211$	7.61637e14	0.594172	0.297086	−	0.954851i	$-0.403985\pi$
$211$	7.61637e14	0.594172	0.297086	+	0.954851i	$0.403985\pi$
$212$	0	0
$213$	1.66500e13i	0.0122165i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 3.15458e14i	− 0.205094i
$218$	0	0
$219$	−9.82859e12	−0.00602011
$220$	0	0
$221$	5.36065e14	0.309505
$222$	0	0
$223$	− 1.62673e15i	− 0.885796i	−0.896572	−	0.442898i	$-0.853950\pi$
$223$	− 1.62673e15i	− 0.885796i	0.896572	−	0.442898i	$-0.146050\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.50895e15i	− 0.731991i	−0.930617	−	0.365996i	$-0.880729\pi$
$227$	− 1.50895e15i	− 0.731991i	0.930617	−	0.365996i	$-0.119271\pi$
$228$	0	0
$229$	−1.69644e15	−0.777334	−0.388667	−	0.921378i	$-0.627064\pi$
$229$	−1.69644e15	−0.777334	−0.388667	+	0.921378i	$0.627064\pi$
$230$	0	0
$231$	−1.08930e13	−0.00471706
$232$	0	0
$233$	1.27819e15i	0.523339i	0.965158	+	0.261670i	$0.0842730\pi$
$233$	1.27819e15i	0.523339i	−0.965158	+	0.261670i	$0.915727\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 4.83712e13i	− 0.0177306i
$238$	0	0
$239$	1.10138e15	0.382252	0.191126	−	0.981566i	$-0.438786\pi$
$239$	1.10138e15	0.382252	0.191126	+	0.981566i	$0.438786\pi$
$240$	0	0
$241$	9.15325e14	0.300929	0.150465	−	0.988615i	$-0.451923\pi$
$241$	9.15325e14	0.300929	0.150465	+	0.988615i	$0.451923\pi$
$242$	0	0
$243$	− 9.14962e13i	− 0.0285077i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	7.34890e15i	2.05917i
$248$	0	0
$249$	5.01699e13	0.0133397
$250$	0	0
$251$	5.68301e15	1.43450	0.717248	−	0.696818i	$-0.245402\pi$
$251$	5.68301e15	1.43450	0.717248	+	0.696818i	$0.245402\pi$
$252$	0	0
$253$	− 5.97596e15i	− 1.43260i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7.40938e15i	1.60404i	0.597294	+	0.802022i	$0.296242\pi$
$257$	7.40938e15i	1.60404i	−0.597294	+	0.802022i	$0.703758\pi$
$258$	0	0
$259$	2.55491e15	0.525929
$260$	0	0
$261$	6.16290e15	1.20676
$262$	0	0
$263$	4.44830e15i	0.828861i	0.910081	+	0.414431i	$0.136019\pi$
$263$	4.44830e15i	0.828861i	−0.910081	+	0.414431i	$0.863981\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	3.21243e13i	0.00542639i
$268$	0	0
$269$	3.99525e15	0.642916	0.321458	−	0.946924i	$-0.395827\pi$
$269$	3.99525e15	0.642916	0.321458	+	0.946924i	$0.395827\pi$
$270$	0	0
$271$	5.58778e15	0.856917	0.428458	−	0.903562i	$-0.359057\pi$
$271$	5.58778e15	0.856917	0.428458	+	0.903562i	$0.359057\pi$
$272$	0	0
$273$	− 3.79457e13i	− 0.00554761i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	9.01343e14i	0.119887i	0.998202	+	0.0599435i	$0.0190920\pi$
$277$	9.01343e14i	0.119887i	−0.998202	+	0.0599435i	$0.980908\pi$
$278$	0	0
$279$	−3.59233e15	−0.455983
$280$	0	0
$281$	1.24836e16	1.51268	0.756342	−	0.654176i	$-0.226985\pi$
$281$	1.24836e16	1.51268	0.756342	+	0.654176i	$0.226985\pi$
$282$	0	0
$283$	− 5.47980e15i	− 0.634093i	−0.948410	−	0.317046i	$-0.897309\pi$
$283$	− 5.47980e15i	− 0.634093i	0.948410	−	0.317046i	$-0.102691\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.81892e15i	0.509000i
$288$	0	0
$289$	9.34136e15	0.943135
$290$	0	0
$291$	1.68474e14	0.0162640
$292$	0	0
$293$	4.61630e15i	0.426240i	0.977026	+	0.213120i	$0.0683626\pi$
$293$	4.61630e15i	0.426240i	−0.977026	+	0.213120i	$0.931637\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	2.48103e14i	0.0209757i
$298$	0	0
$299$	2.08171e16	1.68484
$300$	0	0
$301$	2.40681e15	0.186535
$302$	0	0
$303$	7.46870e13i	0.00554457i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.22392e16i	1.51607i	0.652211	+	0.758037i	$0.273842\pi$
$307$	2.22392e16i	1.51607i	−0.652211	+	0.758037i	$0.726158\pi$
$308$	0	0
$309$	−2.54107e14	−0.0166068
$310$	0	0
$311$	1.51173e16	0.947393	0.473697	−	0.880688i	$-0.342919\pi$
$311$	1.51173e16	0.947393	0.473697	+	0.880688i	$0.342919\pi$
$312$	0	0
$313$	8.36531e15i	0.502856i	0.967876	+	0.251428i	$0.0809002\pi$
$313$	8.36531e15i	0.502856i	−0.967876	+	0.251428i	$0.919100\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	9.34425e15i	0.517201i	0.965984	+	0.258600i	$0.0832613\pi$
$317$	9.34425e15i	0.517201i	−0.965984	+	0.258600i	$0.916739\pi$
$318$	0	0
$319$	−2.50676e16	−1.33191
$320$	0	0
$321$	−2.25474e14	−0.0115031
$322$	0	0
$323$	− 7.72117e15i	− 0.378327i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 1.19419e14i	− 0.00540149i
$328$	0	0
$329$	−9.43151e15	−0.410023
$330$	0	0
$331$	3.92466e16	1.64029	0.820144	−	0.572157i	$-0.193893\pi$
$331$	3.92466e16	1.64029	0.820144	+	0.572157i	$0.193893\pi$
$332$	0	0
$333$	− 2.90944e16i	− 1.16929i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.01727e16i	1.12207i	0.827792	+	0.561036i	$0.189597\pi$
$337$	3.01727e16i	1.12207i	−0.827792	+	0.561036i	$0.810403\pi$
$338$	0	0
$339$	2.50655e14	0.00896972
$340$	0	0
$341$	1.46118e16	0.503269
$342$	0	0
$343$	2.43838e16i	0.808519i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4.02701e16i	1.23834i	0.785255	+	0.619172i	$0.212532\pi$
$347$	4.02701e16i	1.23834i	−0.785255	+	0.619172i	$0.787468\pi$
$348$	0	0
$349$	4.36418e16	1.29282	0.646409	−	0.762991i	$-0.276270\pi$
$349$	4.36418e16	1.29282	0.646409	+	0.762991i	$0.276270\pi$
$350$	0	0
$351$	−8.64264e14	−0.0246689
$352$	0	0
$353$	− 3.08122e16i	− 0.847593i	−0.905758	−	0.423796i	$-0.860697\pi$
$353$	− 3.08122e16i	− 0.847593i	0.905758	−	0.423796i	$-0.139303\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.98679e13i	0.00101925i
$358$	0	0
$359$	4.06926e16	1.00323	0.501616	−	0.865090i	$-0.332739\pi$
$359$	4.06926e16	1.00323	0.501616	+	0.865090i	$0.332739\pi$
$360$	0	0
$361$	6.37963e16	1.51705
$362$	0	0
$363$	− 9.02849e13i	− 0.00207120i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.06190e16i	0.226857i	0.993546	+	0.113429i	$0.0361833\pi$
$367$	1.06190e16i	0.226857i	−0.993546	+	0.113429i	$0.963817\pi$
$368$	0	0
$369$	5.48762e16	1.13165
$370$	0	0
$371$	1.22187e16	0.243273
$372$	0	0
$373$	7.41221e16i	1.42508i	0.701630	+	0.712542i	$0.252456\pi$
$373$	7.41221e16i	1.42508i	−0.701630	+	0.712542i	$0.747544\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.73226e16i	− 1.56642i
$378$	0	0
$379$	−4.60131e16	−0.797493	−0.398746	−	0.917061i	$-0.630555\pi$
$379$	−4.60131e16	−0.797493	−0.398746	+	0.917061i	$0.630555\pi$
$380$	0	0
$381$	7.52177e14	0.0125982
$382$	0	0
$383$	− 7.36970e16i	− 1.19305i	−0.802595	−	0.596524i	$-0.796548\pi$
$383$	− 7.36970e16i	− 1.19305i	0.802595	−	0.596524i	$-0.203452\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 2.74079e16i	− 0.414720i
$388$	0	0
$389$	−7.03726e16	−1.02975	−0.514874	−	0.857266i	$-0.672161\pi$
$389$	−7.03726e16	−1.02975	−0.514874	+	0.857266i	$0.672161\pi$
$390$	0	0
$391$	−2.18717e16	−0.309552
$392$	0	0
$393$	8.37328e13i	0.00114642i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	1.07601e17i	1.37936i	0.724115	+	0.689680i	$0.242249\pi$
$397$	1.07601e17i	1.37936i	−0.724115	+	0.689680i	$0.757751\pi$
$398$	0	0
$399$	−5.46548e14	−0.00678117
$400$	0	0
$401$	−3.85722e16	−0.463272	−0.231636	−	0.972802i	$-0.574408\pi$
$401$	−3.85722e16	−0.463272	−0.231636	+	0.972802i	$0.574408\pi$
$402$	0	0
$403$	5.08999e16i	0.591881i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	1.18341e17i	1.29054i
$408$	0	0
$409$	5.34639e16	0.564753	0.282377	−	0.959304i	$-0.408877\pi$
$409$	5.34639e16	0.564753	0.282377	+	0.959304i	$0.408877\pi$
$410$	0	0
$411$	6.94309e14	0.00710527
$412$	0	0
$413$	− 7.56257e16i	− 0.749884i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 4.72197e14i	− 0.00439784i
$418$	0	0
$419$	−1.04340e17	−0.942016	−0.471008	−	0.882129i	$-0.656110\pi$
$419$	−1.04340e17	−0.942016	−0.471008	+	0.882129i	$0.656110\pi$
$420$	0	0
$421$	−7.03173e16	−0.615500	−0.307750	−	0.951467i	$-0.599576\pi$
$421$	−7.03173e16	−0.615500	−0.307750	+	0.951467i	$0.599576\pi$
$422$	0	0
$423$	1.07403e17i	0.911596i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 7.17831e15i	− 0.0573114i
$428$	0	0
$429$	1.75762e15	0.0136129
$430$	0	0
$431$	6.40437e16	0.481254	0.240627	−	0.970618i	$-0.422647\pi$
$431$	6.40437e16	0.481254	0.240627	+	0.970618i	$0.422647\pi$
$432$	0	0
$433$	− 1.12484e17i	− 0.820201i	−0.912040	−	0.410101i	$-0.865494\pi$
$433$	− 1.12484e17i	− 0.820201i	0.912040	−	0.410101i	$-0.134506\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 2.99838e17i	− 2.05948i
$438$	0	0
$439$	3.14978e16	0.210020	0.105010	−	0.994471i	$-0.466513\pi$
$439$	3.14978e16	0.210020	0.105010	+	0.994471i	$0.466513\pi$
$440$	0	0
$441$	1.23216e17	0.797658
$442$	0	0
$443$	1.14104e17i	0.717261i	0.933480	+	0.358631i	$0.116756\pi$
$443$	1.14104e17i	0.717261i	−0.933480	+	0.358631i	$0.883244\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 5.17644e14i	− 0.00306925i
$448$	0	0
$449$	7.86758e16	0.453148	0.226574	−	0.973994i	$-0.427247\pi$
$449$	7.86758e16	0.453148	0.226574	+	0.973994i	$0.427247\pi$
$450$	0	0
$451$	−2.23209e17	−1.24900
$452$	0	0
$453$	− 2.63518e15i	− 0.0143276i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 2.35679e17i	− 1.21022i	−0.796140	−	0.605112i	$-0.793128\pi$
$457$	− 2.35679e17i	− 1.21022i	0.796140	−	0.605112i	$-0.206872\pi$
$458$	0	0
$459$	9.08045e14	0.00453236
$460$	0	0
$461$	−2.33465e17	−1.13283	−0.566415	−	0.824120i	$-0.691670\pi$
$461$	−2.33465e17	−1.13283	−0.566415	+	0.824120i	$0.691670\pi$
$462$	0	0
$463$	− 1.43911e17i	− 0.678920i	−0.940620	−	0.339460i	$-0.889756\pi$
$463$	− 1.43911e17i	− 0.678920i	0.940620	−	0.339460i	$-0.110244\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	2.72838e17i	1.21715i	0.793496	+	0.608576i	$0.208259\pi$
$467$	2.72838e17i	1.21715i	−0.793496	+	0.608576i	$0.791741\pi$
$468$	0	0
$469$	3.57259e15	0.0155010
$470$	0	0
$471$	−7.94239e14	−0.00335209
$472$	0	0
$473$	1.11482e17i	0.457727i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 1.39142e17i	− 0.540864i
$478$	0	0
$479$	−3.61844e17	−1.36880	−0.684401	−	0.729106i	$-0.739936\pi$
$479$	−3.61844e17	−1.36880	−0.684401	+	0.729106i	$0.739936\pi$
$480$	0	0
$481$	−4.12240e17	−1.51778
$482$	0	0
$483$	1.54820e15i	0.00554844i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.33934e17i	− 1.13429i	−0.823619	−	0.567143i	$-0.808049\pi$
$487$	− 3.33934e17i	− 1.13429i	0.823619	−	0.567143i	$-0.191951\pi$
$488$	0	0
$489$	−2.97146e15	−0.00982793
$490$	0	0
$491$	−1.79534e16	−0.0578251	−0.0289125	−	0.999582i	$-0.509204\pi$
$491$	−1.79534e16	−0.0578251	−0.0289125	+	0.999582i	$0.509204\pi$
$492$	0	0
$493$	9.17461e16i	0.287795i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.94239e17i	− 0.578123i
$498$	0	0
$499$	−4.21623e17	−1.22256	−0.611281	−	0.791414i	$-0.709345\pi$
$499$	−4.21623e17	−1.22256	−0.611281	+	0.791414i	$0.709345\pi$
$500$	0	0
$501$	7.26271e14	0.00205189
$502$	0	0
$503$	3.53513e17i	0.973225i	0.873618	+	0.486612i	$0.161768\pi$
$503$	3.53513e17i	0.973225i	−0.873618	+	0.486612i	$0.838232\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	2.48813e15i	0.00650611i
$508$	0	0
$509$	5.95547e17	1.51792	0.758962	−	0.651135i	$-0.225707\pi$
$509$	5.95547e17	1.51792	0.758962	+	0.651135i	$0.225707\pi$
$510$	0	0
$511$	1.14660e17	0.284890
$512$	0	0
$513$	1.24484e16i	0.0301543i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.36860e17i	− 1.00613i
$518$	0	0
$519$	−3.38828e15	−0.00761012
$520$	0	0
$521$	−6.18807e17	−1.35553	−0.677767	−	0.735277i	$-0.737052\pi$
$521$	−6.18807e17	−1.35553	−0.677767	+	0.735277i	$0.737052\pi$
$522$	0	0
$523$	− 3.97661e17i	− 0.849674i	−0.905270	−	0.424837i	$-0.860331\pi$
$523$	− 3.97661e17i	− 0.849674i	0.905270	−	0.424837i	$-0.139669\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 5.34783e16i	− 0.108745i
$528$	0	0
$529$	−3.45311e17	−0.685092
$530$	0	0
$531$	−8.61199e17	−1.66720
$532$	0	0
$533$	− 7.77544e17i	− 1.46892i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.98741e15i	− 0.00357647i
$538$	0	0
$539$	−5.01182e17	−0.880376
$540$	0	0
$541$	4.06842e17	0.697660	0.348830	−	0.937186i	$-0.386579\pi$
$541$	4.06842e17	0.697660	0.348830	+	0.937186i	$0.386579\pi$
$542$	0	0
$543$	− 1.02256e16i	− 0.0171195i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.81721e17i	0.449678i	0.974396	+	0.224839i	$0.0721856\pi$
$547$	2.81721e17i	0.449678i	−0.974396	+	0.224839i	$0.927814\pi$
$548$	0	0
$549$	−8.17440e16	−0.127419
$550$	0	0
$551$	−1.25774e18	−1.91473
$552$	0	0
$553$	5.64299e17i	0.839065i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.23400e18i	− 1.75088i	−0.483324	−	0.875442i	$-0.660571\pi$
$557$	− 1.23400e18i	− 1.75088i	0.483324	−	0.875442i	$-0.339429\pi$
$558$	0	0
$559$	−3.88345e17	−0.538320
$560$	0	0
$561$	−1.84665e15	−0.00250107
$562$	0	0
$563$	1.42261e18i	1.88270i	0.337437	+	0.941348i	$0.390440\pi$
$563$	1.42261e18i	1.88270i	−0.337437	+	0.941348i	$0.609560\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	3.55744e17i	0.449623i
$568$	0	0
$569$	4.50797e17	0.556866	0.278433	−	0.960456i	$-0.410185\pi$
$569$	4.50797e17	0.556866	0.278433	+	0.960456i	$0.410185\pi$
$570$	0	0
$571$	7.13748e17	0.861807	0.430904	−	0.902398i	$-0.358195\pi$
$571$	7.13748e17	0.861807	0.430904	+	0.902398i	$0.358195\pi$
$572$	0	0
$573$	− 1.11719e16i	− 0.0131863i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 1.28447e18i	− 1.44905i	−0.689250	−	0.724524i	$-0.742060\pi$
$577$	− 1.28447e18i	− 1.44905i	0.689250	−	0.724524i	$-0.257940\pi$
$578$	0	0
$579$	5.01444e15	0.00553110
$580$	0	0
$581$	−5.85282e17	−0.631277
$582$	0	0
$583$	5.65960e17i	0.596953i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.80690e16i	0.0585864i	0.999571	+	0.0292932i	$0.00932565\pi$
$587$	5.80690e16i	0.0585864i	−0.999571	+	0.0292932i	$0.990674\pi$
$588$	0	0
$589$	7.33133e17	0.723491
$590$	0	0
$591$	1.40904e16	0.0136021
$592$	0	0
$593$	− 1.39716e18i	− 1.31944i	−0.751509	−	0.659722i	$-0.770674\pi$
$593$	− 1.39716e18i	− 1.31944i	0.751509	−	0.659722i	$-0.229326\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	1.14297e15i	0.00103324i
$598$	0	0
$599$	−5.23231e17	−0.462827	−0.231414	−	0.972855i	$-0.574335\pi$
$599$	−5.23231e17	−0.462827	−0.231414	+	0.972855i	$0.574335\pi$
$600$	0	0
$601$	−1.51221e18	−1.30897	−0.654484	−	0.756076i	$-0.727114\pi$
$601$	−1.51221e18	−1.30897	−0.654484	+	0.756076i	$0.727114\pi$
$602$	0	0
$603$	− 4.06833e16i	− 0.0344631i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.70314e18i	1.38205i	0.722832	+	0.691024i	$0.242840\pi$
$607$	1.70314e18i	1.38205i	−0.722832	+	0.691024i	$0.757160\pi$
$608$	0	0
$609$	6.49431e15	0.00515846
$610$	0	0
$611$	1.52180e18	1.18328
$612$	0	0
$613$	− 6.08885e17i	− 0.463492i	−0.972776	−	0.231746i	$-0.925556\pi$
$613$	− 6.08885e17i	− 0.463492i	0.972776	−	0.231746i	$-0.0744438\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 5.93917e17i	− 0.433383i	−0.976240	−	0.216692i	$-0.930473\pi$
$617$	− 5.93917e17i	− 0.433383i	0.976240	−	0.216692i	$-0.0695266\pi$
$618$	0	0
$619$	−1.00496e18	−0.718055	−0.359028	−	0.933327i	$-0.616892\pi$
$619$	−1.00496e18	−0.718055	−0.359028	+	0.933327i	$0.616892\pi$
$620$	0	0
$621$	3.52623e16	0.0246726
$622$	0	0
$623$	− 3.74762e17i	− 0.256793i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 2.53157e16i	− 0.0166399i
$628$	0	0
$629$	4.33123e17	0.278857
$630$	0	0
$631$	−7.35349e16	−0.0463770	−0.0231885	−	0.999731i	$-0.507382\pi$
$631$	−7.35349e16	−0.0463770	−0.0231885	+	0.999731i	$0.507382\pi$
$632$	0	0
$633$	− 9.13964e15i	− 0.00564683i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 1.74586e18i	− 1.03539i
$638$	0	0
$639$	−2.21192e18	−1.28533
$640$	0	0
$641$	2.82655e18	1.60946	0.804728	−	0.593644i	$-0.202311\pi$
$641$	2.82655e18	1.60946	0.804728	+	0.593644i	$0.202311\pi$
$642$	0	0
$643$	− 1.57731e18i	− 0.880126i	−0.897967	−	0.440063i	$-0.854956\pi$
$643$	− 1.57731e18i	− 0.880126i	0.897967	−	0.440063i	$-0.145044\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.78710e18i	0.957789i	0.877872	+	0.478895i	$0.158962\pi$
$647$	1.78710e18i	0.957789i	−0.877872	+	0.478895i	$0.841038\pi$
$648$	0	0
$649$	3.50293e18	1.84010
$650$	0	0
$651$	−3.78550e15	−0.00194916
$652$	0	0
$653$	3.34506e18i	1.68837i	0.536051	+	0.844185i	$0.319915\pi$
$653$	3.34506e18i	1.68837i	−0.536051	+	0.844185i	$0.680085\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 1.30571e18i	− 0.633392i
$658$	0	0
$659$	1.69802e18	0.807584	0.403792	−	0.914851i	$-0.367692\pi$
$659$	1.69802e18	0.807584	0.403792	+	0.914851i	$0.367692\pi$
$660$	0	0
$661$	−1.72815e18	−0.805883	−0.402942	−	0.915226i	$-0.632012\pi$
$661$	−1.72815e18	−0.805883	−0.402942	+	0.915226i	$0.632012\pi$
$662$	0	0
$663$	− 6.43278e15i	− 0.00294145i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	3.56280e18i	1.56665i
$668$	0	0
$669$	−1.95208e16	−0.00841835
$670$	0	0
$671$	3.32494e17	0.140633
$672$	0	0
$673$	− 1.82377e18i	− 0.756608i	−0.925681	−	0.378304i	$-0.876507\pi$
$673$	− 1.82377e18i	− 0.756608i	0.925681	−	0.378304i	$-0.123493\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	3.09138e18i	1.23403i	0.786952	+	0.617014i	$0.211658\pi$
$677$	3.09138e18i	1.23403i	−0.786952	+	0.617014i	$0.788342\pi$
$678$	0	0
$679$	−1.96541e18	−0.769662
$680$	0	0
$681$	−1.81074e16	−0.00695663
$682$	0	0
$683$	3.14496e17i	0.118544i	0.998242	+	0.0592722i	$0.0188780\pi$
$683$	3.14496e17i	0.118544i	−0.998242	+	0.0592722i	$0.981122\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	2.03573e16i	0.00738756i
$688$	0	0
$689$	−1.97151e18	−0.702060
$690$	0	0
$691$	−4.48311e17	−0.156665	−0.0783325	−	0.996927i	$-0.524960\pi$
$691$	−4.48311e17	−0.156665	−0.0783325	+	0.996927i	$0.524960\pi$
$692$	0	0
$693$	− 1.44712e18i	− 0.496294i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	8.16932e17i	0.269881i
$698$	0	0
$699$	1.53383e16	0.00497366
$700$	0	0
$701$	3.21704e18	1.02398	0.511988	−	0.858992i	$-0.328909\pi$
$701$	3.21704e18	1.02398	0.511988	+	0.858992i	$0.328909\pi$
$702$	0	0
$703$	5.93767e18i	1.85527i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 8.71299e17i	− 0.262386i
$708$	0	0
$709$	3.21328e18	0.950053	0.475026	−	0.879971i	$-0.342438\pi$
$709$	3.21328e18	0.950053	0.475026	+	0.879971i	$0.342438\pi$
$710$	0	0
$711$	6.42603e18	1.86548
$712$	0	0
$713$	− 2.07674e18i	− 0.591970i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 1.32165e16i	− 0.00363281i
$718$	0	0
$719$	2.51628e18	0.679237	0.339619	−	0.940563i	$-0.389702\pi$
$719$	2.51628e18	0.679237	0.339619	+	0.940563i	$0.389702\pi$
$720$	0	0
$721$	2.96441e18	0.785886
$722$	0	0
$723$	− 1.09839e16i	− 0.00285994i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.20284e18i	0.302157i	0.988522	+	0.151078i	$0.0482746\pi$
$727$	1.20284e18i	0.302157i	−0.988522	+	0.151078i	$0.951725\pi$
$728$	0	0
$729$	4.05036e18	0.999458
$730$	0	0
$731$	4.08017e17	0.0989044
$732$	0	0
$733$	− 1.79874e18i	− 0.428345i	−0.976796	−	0.214173i	$-0.931295\pi$
$733$	− 1.79874e18i	− 0.428345i	0.976796	−	0.214173i	$-0.0687055\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.65479e17i	0.0380369i
$738$	0	0
$739$	2.14261e18	0.483898	0.241949	−	0.970289i	$-0.422213\pi$
$739$	2.14261e18	0.483898	0.241949	+	0.970289i	$0.422213\pi$
$740$	0	0
$741$	8.81868e16	0.0195697
$742$	0	0
$743$	− 5.34550e18i	− 1.16563i	−0.812604	−	0.582816i	$-0.801951\pi$
$743$	− 5.34550e18i	− 1.16563i	0.812604	−	0.582816i	$-0.198049\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6.66498e18i	1.40351i
$748$	0	0
$749$	2.63038e18	0.544360
$750$	0	0
$751$	3.42693e18	0.697021	0.348511	−	0.937305i	$-0.386688\pi$
$751$	3.42693e18	0.697021	0.348511	+	0.937305i	$0.386688\pi$
$752$	0	0
$753$	− 6.81961e16i	− 0.0136330i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 1.77769e18i	− 0.343347i	−0.985154	−	0.171674i	$-0.945083\pi$
$757$	− 1.77769e18i	− 0.343347i	0.985154	−	0.171674i	$-0.0549175\pi$
$758$	0	0
$759$	−7.17115e16	−0.0136150
$760$	0	0
$761$	4.61017e18	0.860432	0.430216	−	0.902726i	$-0.358437\pi$
$761$	4.61017e18	0.860432	0.430216	+	0.902726i	$0.358437\pi$
$762$	0	0
$763$	1.39314e18i	0.255615i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.22024e19i	2.16409i
$768$	0	0
$769$	−4.06261e18	−0.708409	−0.354204	−	0.935168i	$-0.615248\pi$
$769$	−4.06261e18	−0.708409	−0.354204	+	0.935168i	$0.615248\pi$
$770$	0	0
$771$	8.89125e16	0.0152444
$772$	0	0
$773$	− 2.77091e18i	− 0.467149i	−0.972339	−	0.233574i	$-0.924958\pi$
$773$	− 2.77091e18i	− 0.467149i	0.972339	−	0.233574i	$-0.0750422\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.06589e16i	− 0.00499827i
$778$	0	0
$779$	−1.11993e19	−1.79555
$780$	0	0
$781$	8.99700e18	1.41862
$782$	0	0
$783$	− 1.47916e17i	− 0.0229385i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 4.81906e18i	− 0.722980i	−0.932376	−	0.361490i	$-0.882268\pi$
$787$	− 4.81906e18i	− 0.722980i	0.932376	−	0.361490i	$-0.117732\pi$
$788$	0	0
$789$	5.33796e16	0.00787726
$790$	0	0
$791$	−2.92414e18	−0.424475
$792$	0	0
$793$	1.15824e18i	0.165395i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	8.13421e17i	0.112418i	0.998419	+	0.0562090i	$0.0179013\pi$
$797$	8.13421e17i	0.112418i	−0.998419	+	0.0562090i	$0.982099\pi$
$798$	0	0
$799$	−1.59888e18	−0.217402
$800$	0	0
$801$	−4.26766e18	−0.570924
$802$	0	0
$803$	5.31098e18i	0.699075i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 4.79430e16i	− 0.00611009i
$808$	0	0
$809$	6.03069e18	0.756313	0.378156	−	0.925742i	$-0.376558\pi$
$809$	6.03069e18	0.756313	0.378156	+	0.925742i	$0.376558\pi$
$810$	0	0
$811$	3.20788e18	0.395897	0.197948	−	0.980212i	$-0.436572\pi$
$811$	3.20788e18	0.395897	0.197948	+	0.980212i	$0.436572\pi$
$812$	0	0
$813$	− 6.70533e16i	− 0.00814389i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	5.59349e18i	0.658021i
$818$	0	0
$819$	5.04102e18	0.583678
$820$	0	0
$821$	−1.08418e19	−1.23558	−0.617789	−	0.786344i	$-0.711971\pi$
$821$	−1.08418e19	−1.23558	−0.617789	+	0.786344i	$0.711971\pi$
$822$	0	0
$823$	− 3.62368e18i	− 0.406491i	−0.979128	−	0.203246i	$-0.934851\pi$
$823$	− 3.62368e18i	− 0.406491i	0.979128	−	0.203246i	$-0.0651490\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 5.78670e18i	− 0.628992i	−0.949259	−	0.314496i	$-0.898164\pi$
$827$	− 5.78670e18i	− 0.628992i	0.949259	−	0.314496i	$-0.101836\pi$
$828$	0	0
$829$	8.10871e18	0.867655	0.433828	−	0.900996i	$-0.357163\pi$
$829$	8.10871e18	0.867655	0.433828	+	0.900996i	$0.357163\pi$
$830$	0	0
$831$	1.08161e16	0.00113937
$832$	0	0
$833$	1.83430e18i	0.190229i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	8.62197e16i	0.00866744i
$838$	0	0
$839$	1.53236e19	1.51673	0.758365	−	0.651830i	$-0.225998\pi$
$839$	1.53236e19	1.51673	0.758365	+	0.651830i	$0.225998\pi$
$840$	0	0
$841$	4.68439e18	0.456541
$842$	0	0
$843$	− 1.49803e17i	− 0.0143761i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.05326e18i	0.0980156i
$848$	0	0
$849$	−6.57576e16	−0.00602623
$850$	0	0
$851$	1.68196e19	1.51800
$852$	0	0
$853$	9.91139e18i	0.880979i	0.897758	+	0.440490i	$0.145195\pi$
$853$	9.91139e18i	0.880979i	−0.897758	+	0.440490i	$0.854805\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	3.72370e18i	0.321069i	0.987030	+	0.160535i	$0.0513219\pi$
$857$	3.72370e18i	0.321069i	−0.987030	+	0.160535i	$0.948678\pi$
$858$	0	0
$859$	3.07811e18	0.261414	0.130707	−	0.991421i	$-0.458275\pi$
$859$	3.07811e18	0.261414	0.130707	+	0.991421i	$0.458275\pi$
$860$	0	0
$861$	5.78271e16	0.00483739
$862$	0	0
$863$	− 4.68138e18i	− 0.385748i	−0.981224	−	0.192874i	$-0.938219\pi$
$863$	− 4.68138e18i	− 0.385748i	0.981224	−	0.192874i	$-0.0617809\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.12096e17i	− 0.00896328i
$868$	0	0
$869$	−2.61379e19	−2.05893
$870$	0	0
$871$	−5.76445e17	−0.0447342
$872$	0	0
$873$	2.23814e19i	1.71118i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	3.27869e18i	0.243334i	0.992571	+	0.121667i	$0.0388240\pi$
$877$	3.27869e18i	0.243334i	−0.992571	+	0.121667i	$0.961176\pi$
$878$	0	0
$879$	5.53956e16	0.00405086
$880$	0	0
$881$	−8.02437e18	−0.578186	−0.289093	−	0.957301i	$-0.593354\pi$
$881$	−8.02437e18	−0.578186	−0.289093	+	0.957301i	$0.593354\pi$
$882$	0	0
$883$	− 2.07925e19i	− 1.47626i	−0.674661	−	0.738128i	$-0.735710\pi$
$883$	− 2.07925e19i	− 1.47626i	0.674661	−	0.738128i	$-0.264290\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 2.61191e19i	− 1.80075i	−0.435112	−	0.900376i	$-0.643291\pi$
$887$	− 2.61191e19i	− 1.80075i	0.435112	−	0.900376i	$-0.356709\pi$
$888$	0	0
$889$	−8.77489e18	−0.596184
$890$	0	0
$891$	−1.64778e19	−1.10330
$892$	0	0
$893$	− 2.19191e19i	− 1.44640i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 2.49806e17i	− 0.0160122i
$898$	0	0
$899$	−8.71138e18	−0.550363
$900$	0	0
$901$	2.07138e18	0.128988
$902$	0	0
$903$	− 2.88818e16i	− 0.00177277i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.55190e19i	1.52201i	0.648747	+	0.761004i	$0.275293\pi$
$907$	2.55190e19i	1.52201i	−0.648747	+	0.761004i	$0.724707\pi$
$908$	0	0
$909$	−9.92204e18	−0.583359
$910$	0	0
$911$	−1.62951e19	−0.944466	−0.472233	−	0.881474i	$-0.656552\pi$
$911$	−1.62951e19	−0.944466	−0.472233	+	0.881474i	$0.656552\pi$
$912$	0	0
$913$	− 2.71098e19i	− 1.54905i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.76826e17i	− 0.0542521i
$918$	0	0
$919$	2.25963e18	0.123733	0.0618667	−	0.998084i	$-0.480295\pi$
$919$	2.25963e18	0.123733	0.0618667	+	0.998084i	$0.480295\pi$
$920$	0	0
$921$	2.66871e17	0.0144083
$922$	0	0
$923$	3.13409e19i	1.66840i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.37577e19i	− 1.74725i
$928$	0	0
$929$	1.38768e19	0.708250	0.354125	−	0.935198i	$-0.384779\pi$
$929$	1.38768e19	0.708250	0.354125	+	0.935198i	$0.384779\pi$
$930$	0	0
$931$	−2.51463e19	−1.26561
$932$	0	0
$933$	− 1.81407e17i	− 0.00900375i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 1.27938e19i	− 0.617578i	−0.951131	−	0.308789i	$-0.900076\pi$
$937$	− 1.27938e19i	− 0.617578i	0.951131	−	0.308789i	$-0.0999237\pi$
$938$	0	0
$939$	1.00384e17	0.00477900
$940$	0	0
$941$	−1.00758e19	−0.473093	−0.236547	−	0.971620i	$-0.576016\pi$
$941$	−1.00758e19	−0.473093	−0.236547	+	0.971620i	$0.576016\pi$
$942$	0	0
$943$	3.17241e19i	1.46914i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.66672e19i	− 0.750912i	−0.926840	−	0.375456i	$-0.877486\pi$
$947$	− 1.66672e19i	− 0.750912i	0.926840	−	0.375456i	$-0.122514\pi$
$948$	0	0
$949$	−1.85007e19	−0.822163
$950$	0	0
$951$	1.12131e17	0.00491532
$952$	0	0
$953$	2.79060e19i	1.20668i	0.797483	+	0.603341i	$0.206164\pi$
$953$	2.79060e19i	1.20668i	−0.797483	+	0.603341i	$0.793836\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	3.00811e17i	0.0126580i
$958$	0	0
$959$	−8.09981e18	−0.336243
$960$	0	0
$961$	−1.93397e19	−0.792042
$962$	0	0
$963$	− 2.99538e19i	− 1.21027i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.06377e19i	− 1.59829i	−0.601136	−	0.799147i	$-0.705285\pi$
$967$	− 4.06377e19i	− 1.59829i	0.601136	−	0.799147i	$-0.294715\pi$
$968$	0	0
$969$	−9.26541e16	−0.00359550
$970$	0	0
$971$	3.91648e18	0.149958	0.0749792	−	0.997185i	$-0.476111\pi$
$971$	3.91648e18	0.149958	0.0749792	+	0.997185i	$0.476111\pi$
$972$	0	0
$973$	5.50865e18i	0.208119i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 8.91758e18i	− 0.328044i	−0.986457	−	0.164022i	$-0.947553\pi$
$977$	− 8.91758e18i	− 0.328044i	0.986457	−	0.164022i	$-0.0524468\pi$
$978$	0	0
$979$	1.73587e19	0.630130
$980$	0	0
$981$	1.58646e19	0.568304
$982$	0	0
$983$	− 3.03354e19i	− 1.07239i	−0.844095	−	0.536194i	$-0.819862\pi$
$983$	− 3.03354e19i	− 1.07239i	0.844095	−	0.536194i	$-0.180138\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.13178e17i	0.00389673i
$988$	0	0
$989$	1.58446e19	0.538401
$990$	0	0
$991$	−4.90085e19	−1.64359	−0.821794	−	0.569785i	$-0.807027\pi$
$991$	−4.90085e19	−1.64359	−0.821794	+	0.569785i	$0.807027\pi$
$992$	0	0
$993$	− 4.70959e17i	− 0.0155888i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.57448e19i	0.507715i	0.967242	+	0.253857i	$0.0816994\pi$
$997$	1.57448e19i	0.507715i	−0.967242	+	0.253857i	$0.918301\pi$
$998$	0	0
$999$	−6.98297e17	−0.0222262

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.14.c.a.49.1		2
5.2	odd	4		8.14.a.a.1.1	✓	1
5.3	odd	4		200.14.a.a.1.1		1
5.4	even	2	inner	200.14.c.a.49.2		2
15.2	even	4		72.14.a.a.1.1		1
20.7	even	4		16.14.a.c.1.1		1
40.27	even	4		64.14.a.d.1.1		1
40.37	odd	4		64.14.a.f.1.1		1
60.47	odd	4		144.14.a.f.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.14.a.a.1.1	✓	1	5.2	odd	4
16.14.a.c.1.1		1	20.7	even	4
64.14.a.d.1.1		1	40.27	even	4
64.14.a.f.1.1		1	40.37	odd	4
72.14.a.a.1.1		1	15.2	even	4
144.14.a.f.1.1		1	60.47	odd	4
200.14.a.a.1.1		1	5.3	odd	4
200.14.c.a.49.1		2	1.1	even	1	trivial
200.14.c.a.49.2		2	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	200.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.0000i q^{3} +139992. i q^{7} +1.59418e6 q^{9} +O(q^{10})\)	\(q-12.0000i q^{3} +139992. i q^{7} +1.59418e6 q^{9} -6.48432e6 q^{11} -2.25880e7i q^{13} +2.37323e7i q^{17} -3.25345e8 q^{19} +1.67990e6 q^{21} +9.21601e8i q^{23} -3.82620e7i q^{27} +3.86588e9 q^{29} -2.25340e9 q^{31} +7.78119e7i q^{33} -1.82504e10i q^{37} -2.71056e8 q^{39} +3.44228e10 q^{41} -1.71925e10i q^{43} +6.73717e10i q^{47} +7.72913e10 q^{49} +2.84787e8 q^{51} -8.72812e10i q^{53} +3.90414e9i q^{57} -5.40215e11 q^{59} -5.12766e10 q^{61} +2.23172e11i q^{63} -2.55199e10i q^{67} +1.10592e10 q^{69} -1.38750e12 q^{71} -8.19049e11i q^{73} -9.07753e11i q^{77} +4.03094e12 q^{79} +2.54118e12 q^{81} +4.18082e12i q^{83} -4.63906e10i q^{87} -2.67703e12 q^{89} +3.16214e12 q^{91} +2.70408e10i q^{93} +1.40395e13i q^{97} -1.03372e13 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3188358 q^{9}+O(q^{10}) \) 2 * q + 3188358 * q^9	\( 2 q + 3188358 q^{9} - 12968648 q^{11} - 650689672 q^{19} + 3359808 q^{21} + 7731758436 q^{29} - 4506802880 q^{31} - 542112816 q^{39} + 68845690644 q^{41} + 154582500686 q^{49} + 569574480 q^{51} - 1080429037336 q^{59} - 102553137700 q^{61} + 22118415168 q^{69} - 2775001398064 q^{71} + 8061871230688 q^{79} + 5082354203058 q^{81} - 5354055596532 q^{89} + 6324288111456 q^{91} - 20674346299992 q^{99}+O(q^{100}) \) 2 * q + 3188358 * q^9 - 12968648 * q^11 - 650689672 * q^19 + 3359808 * q^21 + 7731758436 * q^29 - 4506802880 * q^31 - 542112816 * q^39 + 68845690644 * q^41 + 154582500686 * q^49 + 569574480 * q^51 - 1080429037336 * q^59 - 102553137700 * q^61 + 22118415168 * q^69 - 2775001398064 * q^71 + 8061871230688 * q^79 + 5082354203058 * q^81 - 5354055596532 * q^89 + 6324288111456 * q^91 - 20674346299992 * q^99

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