LMFDB - Embedded newform 16.11.c.b.15.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [16,11,Mod(15,16)]

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([1, 0]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("16.15");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 16 = 2^{4} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	16.c (of order $2$, degree $1$, 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:	$10.1657160428$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{505})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 127x^{2} + 126x + 15876 $ x^4 - x^3 + 127x^2 + 126x + 15876
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{25}\cdot 3 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			15.3
Root			$5.86805 - 10.1638i$ of defining polynomial
Character	$\chi$	$=$	16.15
Dual form			16.11.c.b.15.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+214.389i q^{3} -3264.66 q^{5} -5808.15i q^{7} +13086.3 q^{9} +O(q^{10})$	$q+214.389i q^{3} -3264.66 q^{5} -5808.15i q^{7} +13086.3 q^{9} -271363. i q^{11} -557958. q^{13} -699908. i q^{15} -795476. q^{17} -4.01795e6i q^{19} +1.24520e6 q^{21} +7.93454e6i q^{23} +892402. q^{25} +1.54650e7i q^{27} +7.81440e6 q^{29} +1.15574e7i q^{31} +5.81772e7 q^{33} +1.89616e7i q^{35} -1.07595e8 q^{37} -1.19620e8i q^{39} -5.82354e7 q^{41} -7.58193e7i q^{43} -4.27223e7 q^{45} -1.19493e8i q^{47} +2.48741e8 q^{49} -1.70541e8i q^{51} -4.68891e8 q^{53} +8.85908e8i q^{55} +8.61404e8 q^{57} +8.13126e8i q^{59} -4.39744e8 q^{61} -7.60071e7i q^{63} +1.82154e9 q^{65} -1.14136e9i q^{67} -1.70108e9 q^{69} -9.19220e6i q^{71} -8.89952e8 q^{73} +1.91321e8i q^{75} -1.57611e9 q^{77} -5.96194e9i q^{79} -2.54280e9 q^{81} +2.00196e8i q^{83} +2.59696e9 q^{85} +1.67532e9i q^{87} +7.64100e9 q^{89} +3.24070e9i q^{91} -2.47778e9 q^{93} +1.31172e10i q^{95} -6.43557e9 q^{97} -3.55113e9i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4200 q^{5} - 189276 q^{9}+O(q^{10}) $ 4 * q + 4200 * q^5 - 189276 * q^9	$ 4 q + 4200 q^{5} - 189276 q^{9} - 1040984 q^{13} - 4390008 q^{17} - 5305344 q^{21} + 39812780 q^{25} + 31136808 q^{29} + 33578496 q^{33} - 166166680 q^{37} + 172774728 q^{41} - 1241253720 q^{45} + 880227268 q^{49} - 355888152 q^{53} + 3496598016 q^{57} - 1949148376 q^{61} + 4045071120 q^{65} - 7111867392 q^{69} + 1505019208 q^{73} - 5086891008 q^{77} - 1572150204 q^{81} - 9822078000 q^{85} + 6041434056 q^{89} + 30791430144 q^{93} - 8292247544 q^{97}+O(q^{100}) $ 4 * q + 4200 * q^5 - 189276 * q^9 - 1040984 * q^13 - 4390008 * q^17 - 5305344 * q^21 + 39812780 * q^25 + 31136808 * q^29 + 33578496 * q^33 - 166166680 * q^37 + 172774728 * q^41 - 1241253720 * q^45 + 880227268 * q^49 - 355888152 * q^53 + 3496598016 * q^57 - 1949148376 * q^61 + 4045071120 * q^65 - 7111867392 * q^69 + 1505019208 * q^73 - 5086891008 * q^77 - 1572150204 * q^81 - 9822078000 * q^85 + 6041434056 * q^89 + 30791430144 * q^93 - 8292247544 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/16\mathbb{Z}\right)^\times$.

$n$	$5$	$15$
$\chi(n)$	$1$	$-1$

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
$3$	214.389i	0.882260i	0.897443	+	0.441130i	$0.145422\pi$
$3$	214.389i	0.882260i	−0.897443	+	0.441130i	$0.854578\pi$
$4$	0	0
$5$	−3264.66	−1.04469	−0.522346	−	0.852734i	$-0.674943\pi$
$5$	−3264.66	−1.04469	−0.522346	+	0.852734i	$0.674943\pi$
$6$	0	0
$7$	− 5808.15i	− 0.345579i	−0.984959	−	0.172789i	$-0.944722\pi$
$7$	− 5808.15i	− 0.345579i	0.984959	−	0.172789i	$-0.0552780\pi$
$8$	0	0
$9$	13086.3	0.221617
$10$	0	0
$11$	− 271363.i	− 1.68495i	−0.538737	−	0.842474i	$-0.681098\pi$
$11$	− 271363.i	− 1.68495i	0.538737	−	0.842474i	$-0.318902\pi$
$12$	0	0
$13$	−557958.	−1.50274	−0.751371	−	0.659880i	$-0.770607\pi$
$13$	−557958.	−1.50274	−0.751371	+	0.659880i	$0.770607\pi$
$14$	0	0
$15$	− 699908.i	− 0.921690i
$16$	0	0
$17$	−795476.	−0.560250	−0.280125	−	0.959963i	$-0.590376\pi$
$17$	−795476.	−0.560250	−0.280125	+	0.959963i	$0.590376\pi$
$18$	0	0
$19$	− 4.01795e6i	− 1.62269i	−0.584566	−	0.811346i	$-0.698735\pi$
$19$	− 4.01795e6i	− 1.62269i	0.584566	−	0.811346i	$-0.301265\pi$
$20$	0	0
$21$	1.24520e6	0.304890
$22$	0	0
$23$	7.93454e6i	1.23277i	0.787444	+	0.616386i	$0.211404\pi$
$23$	7.93454e6i	1.23277i	−0.787444	+	0.616386i	$0.788596\pi$
$24$	0	0
$25$	892402.	0.0913820
$26$	0	0
$27$	1.54650e7i	1.07778i
$28$	0	0
$29$	7.81440e6	0.380983	0.190492	−	0.981689i	$-0.438992\pi$
$29$	7.81440e6	0.380983	0.190492	+	0.981689i	$0.438992\pi$
$30$	0	0
$31$	1.15574e7i	0.403692i	0.979417	+	0.201846i	$0.0646941\pi$
$31$	1.15574e7i	0.403692i	−0.979417	+	0.201846i	$0.935306\pi$
$32$	0	0
$33$	5.81772e7	1.48656
$34$	0	0
$35$	1.89616e7i	0.361024i
$36$	0	0
$37$	−1.07595e8	−1.55161	−0.775806	−	0.630972i	$-0.782656\pi$
$37$	−1.07595e8	−1.55161	−0.775806	+	0.630972i	$0.782656\pi$
$38$	0	0
$39$	− 1.19620e8i	− 1.32581i
$40$	0	0
$41$	−5.82354e7	−0.502653	−0.251326	−	0.967902i	$-0.580867\pi$
$41$	−5.82354e7	−0.502653	−0.251326	+	0.967902i	$0.580867\pi$
$42$	0	0
$43$	− 7.58193e7i	− 0.515748i	−0.966179	−	0.257874i	$-0.916978\pi$
$43$	− 7.58193e7i	− 0.515748i	0.966179	−	0.257874i	$-0.0830219\pi$
$44$	0	0
$45$	−4.27223e7	−0.231522
$46$	0	0
$47$	− 1.19493e8i	− 0.521016i	−0.965472	−	0.260508i	$-0.916110\pi$
$47$	− 1.19493e8i	− 0.521016i	0.965472	−	0.260508i	$-0.0838901\pi$
$48$	0	0
$49$	2.48741e8	0.880575
$50$	0	0
$51$	− 1.70541e8i	− 0.494287i
$52$	0	0
$53$	−4.68891e8	−1.12122	−0.560612	−	0.828078i	$-0.689434\pi$
$53$	−4.68891e8	−1.12122	−0.560612	+	0.828078i	$0.689434\pi$
$54$	0	0
$55$	8.85908e8i	1.76025i
$56$	0	0
$57$	8.61404e8	1.43164
$58$	0	0
$59$	8.13126e8i	1.13736i	0.822559	+	0.568680i	$0.192546\pi$
$59$	8.13126e8i	1.13736i	−0.822559	+	0.568680i	$0.807454\pi$
$60$	0	0
$61$	−4.39744e8	−0.520656	−0.260328	−	0.965520i	$-0.583831\pi$
$61$	−4.39744e8	−0.520656	−0.260328	+	0.965520i	$0.583831\pi$
$62$	0	0
$63$	− 7.60071e7i	− 0.0765863i
$64$	0	0
$65$	1.82154e9	1.56990
$66$	0	0
$67$	− 1.14136e9i	− 0.845375i	−0.906275	−	0.422688i	$-0.861087\pi$
$67$	− 1.14136e9i	− 0.845375i	0.906275	−	0.422688i	$-0.138913\pi$
$68$	0	0
$69$	−1.70108e9	−1.08762
$70$	0	0
$71$	− 9.19220e6i	− 0.00509481i	−0.999997	−	0.00254740i	$-0.999189\pi$
$71$	− 9.19220e6i	− 0.00509481i	0.999997	−	0.00254740i	$-0.000810864\pi$
$72$	0	0
$73$	−8.89952e8	−0.429292	−0.214646	−	0.976692i	$-0.568860\pi$
$73$	−8.89952e8	−0.429292	−0.214646	+	0.976692i	$0.568860\pi$
$74$	0	0
$75$	1.91321e8i	0.0806226i
$76$	0	0
$77$	−1.57611e9	−0.582283
$78$	0	0
$79$	− 5.96194e9i	− 1.93755i	−0.247948	−	0.968773i	$-0.579756\pi$
$79$	− 5.96194e9i	− 1.93755i	0.247948	−	0.968773i	$-0.420244\pi$
$80$	0	0
$81$	−2.54280e9	−0.729268
$82$	0	0
$83$	2.00196e8i	0.0508236i	0.999677	+	0.0254118i	$0.00808970\pi$
$83$	2.00196e8i	0.0508236i	−0.999677	+	0.0254118i	$0.991910\pi$
$84$	0	0
$85$	2.59696e9	0.585289
$86$	0	0
$87$	1.67532e9i	0.336126i
$88$	0	0
$89$	7.64100e9	1.36836	0.684180	−	0.729313i	$-0.260160\pi$
$89$	7.64100e9	1.36836	0.684180	+	0.729313i	$0.260160\pi$
$90$	0	0
$91$	3.24070e9i	0.519316i
$92$	0	0
$93$	−2.47778e9	−0.356162
$94$	0	0
$95$	1.31172e10i	1.69521i
$96$	0	0
$97$	−6.43557e9	−0.749426	−0.374713	−	0.927141i	$-0.622259\pi$
$97$	−6.43557e9	−0.749426	−0.374713	+	0.927141i	$0.622259\pi$
$98$	0	0
$99$	− 3.55113e9i	− 0.373414i
$100$	0	0
$101$	1.13632e10	1.08117	0.540586	−	0.841289i	$-0.318203\pi$
$101$	1.13632e10	1.08117	0.540586	+	0.841289i	$0.318203\pi$
$102$	0	0
$103$	5.64423e9i	0.486876i	0.969916	+	0.243438i	$0.0782753\pi$
$103$	5.64423e9i	0.486876i	−0.969916	+	0.243438i	$0.921725\pi$
$104$	0	0
$105$	−4.06517e9	−0.318517
$106$	0	0
$107$	7.55013e9i	0.538314i	0.963096	+	0.269157i	$0.0867450\pi$
$107$	7.55013e9i	0.538314i	−0.963096	+	0.269157i	$0.913255\pi$
$108$	0	0
$109$	1.83583e10	1.19316	0.596581	−	0.802553i	$-0.296526\pi$
$109$	1.83583e10	1.19316	0.596581	+	0.802553i	$0.296526\pi$
$110$	0	0
$111$	− 2.30672e10i	− 1.36892i
$112$	0	0
$113$	−2.06245e10	−1.11942	−0.559708	−	0.828690i	$-0.689087\pi$
$113$	−2.06245e10	−1.11942	−0.559708	+	0.828690i	$0.689087\pi$
$114$	0	0
$115$	− 2.59036e10i	− 1.28787i
$116$	0	0
$117$	−7.30160e9	−0.333034
$118$	0	0
$119$	4.62024e9i	0.193611i
$120$	0	0
$121$	−4.77003e10	−1.83905
$122$	0	0
$123$	− 1.24850e10i	− 0.443470i
$124$	0	0
$125$	2.89681e10	0.949226
$126$	0	0
$127$	− 1.64084e10i	− 0.496646i	−0.968677	−	0.248323i	$-0.920121\pi$
$127$	− 1.64084e10i	− 0.496646i	0.968677	−	0.248323i	$-0.0798794\pi$
$128$	0	0
$129$	1.62548e10	0.455023
$130$	0	0
$131$	7.30966e10i	1.89470i	0.320197	+	0.947351i	$0.396251\pi$
$131$	7.30966e10i	1.89470i	−0.320197	+	0.947351i	$0.603749\pi$
$132$	0	0
$133$	−2.33368e10	−0.560768
$134$	0	0
$135$	− 5.04881e10i	− 1.12595i
$136$	0	0
$137$	6.96229e10	1.44261	0.721306	−	0.692617i	$-0.243542\pi$
$137$	6.96229e10	1.44261	0.721306	+	0.692617i	$0.243542\pi$
$138$	0	0
$139$	− 2.72919e10i	− 0.525968i	−0.964800	−	0.262984i	$-0.915293\pi$
$139$	− 2.72919e10i	− 0.525968i	0.964800	−	0.262984i	$-0.0847066\pi$
$140$	0	0
$141$	2.56179e10	0.459672
$142$	0	0
$143$	1.51409e11i	2.53204i
$144$	0	0
$145$	−2.55114e10	−0.398010
$146$	0	0
$147$	5.33273e10i	0.776896i
$148$	0	0
$149$	−5.59806e9	−0.0762265	−0.0381132	−	0.999273i	$-0.512135\pi$
$149$	−5.59806e9	−0.0762265	−0.0381132	+	0.999273i	$0.512135\pi$
$150$	0	0
$151$	9.57307e8i	0.0121946i	0.999981	+	0.00609728i	$0.00194084\pi$
$151$	9.57307e8i	0.0121946i	−0.999981	+	0.00609728i	$0.998059\pi$
$152$	0	0
$153$	−1.04098e10	−0.124161
$154$	0	0
$155$	− 3.77309e10i	− 0.421734i
$156$	0	0
$157$	−4.46354e9	−0.0467930	−0.0233965	−	0.999726i	$-0.507448\pi$
$157$	−4.46354e9	−0.0467930	−0.0233965	+	0.999726i	$0.507448\pi$
$158$	0	0
$159$	− 1.00525e11i	− 0.989212i
$160$	0	0
$161$	4.60850e10	0.426020
$162$	0	0
$163$	− 1.51244e11i	− 1.31444i	−0.753701	−	0.657218i	$-0.771733\pi$
$163$	− 1.51244e11i	− 1.31444i	0.753701	−	0.657218i	$-0.228267\pi$
$164$	0	0
$165$	−1.89929e11	−1.55300
$166$	0	0
$167$	3.55365e9i	0.0273585i	0.999906	+	0.0136792i	$0.00435437\pi$
$167$	3.55365e9i	0.0273585i	−0.999906	+	0.0136792i	$0.995646\pi$
$168$	0	0
$169$	1.73458e11	1.25824
$170$	0	0
$171$	− 5.25800e10i	− 0.359617i
$172$	0	0
$173$	−6.02607e10	−0.388869	−0.194435	−	0.980915i	$-0.562287\pi$
$173$	−6.02607e10	−0.388869	−0.194435	+	0.980915i	$0.562287\pi$
$174$	0	0
$175$	− 5.18320e9i	− 0.0315797i
$176$	0	0
$177$	−1.74325e11	−1.00345
$178$	0	0
$179$	2.71175e10i	0.147566i	0.997274	+	0.0737828i	$0.0235072\pi$
$179$	2.71175e10i	0.147566i	−0.997274	+	0.0737828i	$0.976493\pi$
$180$	0	0
$181$	−2.51187e11	−1.29302	−0.646509	−	0.762906i	$-0.723772\pi$
$181$	−2.51187e11	−1.29302	−0.646509	+	0.762906i	$0.723772\pi$
$182$	0	0
$183$	− 9.42763e10i	− 0.459354i
$184$	0	0
$185$	3.51261e11	1.62096
$186$	0	0
$187$	2.15862e11i	0.943993i
$188$	0	0
$189$	8.98231e10	0.372459
$190$	0	0
$191$	5.07107e10i	0.199496i	0.995013	+	0.0997478i	$0.0318036\pi$
$191$	5.07107e10i	0.199496i	−0.995013	+	0.0997478i	$0.968196\pi$
$192$	0	0
$193$	1.34867e11	0.503640	0.251820	−	0.967774i	$-0.418971\pi$
$193$	1.34867e11	0.503640	0.251820	+	0.967774i	$0.418971\pi$
$194$	0	0
$195$	3.90519e11i	1.38506i
$196$	0	0
$197$	2.06474e11	0.695880	0.347940	−	0.937517i	$-0.386881\pi$
$197$	2.06474e11	0.695880	0.347940	+	0.937517i	$0.386881\pi$
$198$	0	0
$199$	7.68901e10i	0.246380i	0.992383	+	0.123190i	$0.0393124\pi$
$199$	7.68901e10i	0.246380i	−0.992383	+	0.123190i	$0.960688\pi$
$200$	0	0
$201$	2.44696e11	0.745841
$202$	0	0
$203$	− 4.53872e10i	− 0.131660i
$204$	0	0
$205$	1.90119e11	0.525117
$206$	0	0
$207$	1.03834e11i	0.273204i
$208$	0	0
$209$	−1.09032e12	−2.73415
$210$	0	0
$211$	− 3.36839e11i	− 0.805398i	−0.915332	−	0.402699i	$-0.868072\pi$
$211$	− 3.36839e11i	− 0.805398i	0.915332	−	0.402699i	$-0.131928\pi$
$212$	0	0
$213$	1.97071e9	0.00449494
$214$	0	0
$215$	2.47524e11i	0.538798i
$216$	0	0
$217$	6.71269e10	0.139508
$218$	0	0
$219$	− 1.90796e11i	− 0.378747i
$220$	0	0
$221$	4.43842e11	0.841912
$222$	0	0
$223$	− 3.53395e11i	− 0.640820i	−0.947279	−	0.320410i	$-0.896179\pi$
$223$	− 3.53395e11i	− 0.640820i	0.947279	−	0.320410i	$-0.103821\pi$
$224$	0	0
$225$	1.16782e10	0.0202518
$226$	0	0
$227$	− 9.19956e11i	− 1.52629i	−0.646226	−	0.763146i	$-0.723654\pi$
$227$	− 9.19956e11i	− 1.52629i	0.646226	−	0.763146i	$-0.276346\pi$
$228$	0	0
$229$	−8.41416e11	−1.33608	−0.668041	−	0.744124i	$-0.732867\pi$
$229$	−8.41416e11	−1.33608	−0.668041	+	0.744124i	$0.732867\pi$
$230$	0	0
$231$	− 3.37902e11i	− 0.513725i
$232$	0	0
$233$	−3.89085e11	−0.566585	−0.283293	−	0.959033i	$-0.591427\pi$
$233$	−3.89085e11	−0.566585	−0.283293	+	0.959033i	$0.591427\pi$
$234$	0	0
$235$	3.90103e11i	0.544302i
$236$	0	0
$237$	1.27818e12	1.70942
$238$	0	0
$239$	− 9.15245e11i	− 1.17367i	−0.809705	−	0.586837i	$-0.800373\pi$
$239$	− 9.15245e11i	− 1.17367i	0.809705	−	0.586837i	$-0.199627\pi$
$240$	0	0
$241$	−5.33079e11	−0.655702	−0.327851	−	0.944729i	$-0.606324\pi$
$241$	−5.33079e11	−0.655702	−0.327851	+	0.944729i	$0.606324\pi$
$242$	0	0
$243$	3.68045e11i	0.434380i
$244$	0	0
$245$	−8.12055e11	−0.919930
$246$	0	0
$247$	2.24184e12i	2.43849i
$248$	0	0
$249$	−4.29199e10	−0.0448397
$250$	0	0
$251$	2.72571e11i	0.273597i	0.990599	+	0.136799i	$0.0436813\pi$
$251$	2.72571e11i	0.273597i	−0.990599	+	0.136799i	$0.956319\pi$
$252$	0	0
$253$	2.15314e12	2.07716
$254$	0	0
$255$	5.56760e11i	0.516377i
$256$	0	0
$257$	−7.57292e11	−0.675457	−0.337729	−	0.941244i	$-0.609659\pi$
$257$	−7.57292e11	−0.675457	−0.337729	+	0.941244i	$0.609659\pi$
$258$	0	0
$259$	6.24927e11i	0.536204i
$260$	0	0
$261$	1.02262e11	0.0844325
$262$	0	0
$263$	− 2.26776e12i	− 1.80226i	−0.433548	−	0.901130i	$-0.642739\pi$
$263$	− 2.26776e12i	− 1.80226i	0.433548	−	0.901130i	$-0.357261\pi$
$264$	0	0
$265$	1.53077e12	1.17133
$266$	0	0
$267$	1.63815e12i	1.20725i
$268$	0	0
$269$	3.43059e11	0.243561	0.121780	−	0.992557i	$-0.461140\pi$
$269$	3.43059e11	0.243561	0.121780	+	0.992557i	$0.461140\pi$
$270$	0	0
$271$	3.43493e11i	0.235002i	0.993073	+	0.117501i	$0.0374883\pi$
$271$	3.43493e11i	0.235002i	−0.993073	+	0.117501i	$0.962512\pi$
$272$	0	0
$273$	−6.94771e11	−0.458172
$274$	0	0
$275$	− 2.42165e11i	− 0.153974i
$276$	0	0
$277$	−4.83205e11	−0.296300	−0.148150	−	0.988965i	$-0.547332\pi$
$277$	−4.83205e11	−0.296300	−0.148150	+	0.988965i	$0.547332\pi$
$278$	0	0
$279$	1.51243e11i	0.0894653i
$280$	0	0
$281$	6.01290e11	0.343204	0.171602	−	0.985166i	$-0.445106\pi$
$281$	6.01290e11	0.343204	0.171602	+	0.985166i	$0.445106\pi$
$282$	0	0
$283$	− 1.62113e12i	− 0.893068i	−0.894767	−	0.446534i	$-0.852658\pi$
$283$	− 1.62113e12i	− 0.893068i	0.894767	−	0.446534i	$-0.147342\pi$
$284$	0	0
$285$	−2.81219e12	−1.49562
$286$	0	0
$287$	3.38240e11i	0.173706i
$288$	0	0
$289$	−1.38321e12	−0.686119
$290$	0	0
$291$	− 1.37972e12i	− 0.661188i
$292$	0	0
$293$	−2.88785e12	−1.33732	−0.668662	−	0.743566i	$-0.733133\pi$
$293$	−2.88785e12	−1.33732	−0.668662	+	0.743566i	$0.733133\pi$
$294$	0	0
$295$	− 2.65458e12i	− 1.18819i
$296$	0	0
$297$	4.19663e12	1.81601
$298$	0	0
$299$	− 4.42714e12i	− 1.85254i
$300$	0	0
$301$	−4.40369e11	−0.178232
$302$	0	0
$303$	2.43615e12i	0.953875i
$304$	0	0
$305$	1.43562e12	0.543925
$306$	0	0
$307$	3.23799e12i	1.18736i	0.804700	+	0.593682i	$0.202326\pi$
$307$	3.23799e12i	1.18736i	−0.804700	+	0.593682i	$0.797674\pi$
$308$	0	0
$309$	−1.21006e12	−0.429551
$310$	0	0
$311$	1.36616e12i	0.469569i	0.972047	+	0.234784i	$0.0754384\pi$
$311$	1.36616e12i	0.469569i	−0.972047	+	0.234784i	$0.924562\pi$
$312$	0	0
$313$	1.12701e12	0.375153	0.187576	−	0.982250i	$-0.439937\pi$
$313$	1.12701e12	0.375153	0.187576	+	0.982250i	$0.439937\pi$
$314$	0	0
$315$	2.48137e11i	0.0800091i
$316$	0	0
$317$	2.60457e12	0.813654	0.406827	−	0.913505i	$-0.366635\pi$
$317$	2.60457e12	0.813654	0.406827	+	0.913505i	$0.366635\pi$
$318$	0	0
$319$	− 2.12054e12i	− 0.641937i
$320$	0	0
$321$	−1.61867e12	−0.474933
$322$	0	0
$323$	3.19618e12i	0.909114i
$324$	0	0
$325$	−4.97923e11	−0.137324
$326$	0	0
$327$	3.93581e12i	1.05268i
$328$	0	0
$329$	−6.94030e11	−0.180052
$330$	0	0
$331$	3.29607e12i	0.829576i	0.909918	+	0.414788i	$0.136144\pi$
$331$	3.29607e12i	0.829576i	−0.909918	+	0.414788i	$0.863856\pi$
$332$	0	0
$333$	−1.40802e12	−0.343864
$334$	0	0
$335$	3.72616e12i	0.883157i
$336$	0	0
$337$	4.20420e12	0.967240	0.483620	−	0.875278i	$-0.339322\pi$
$337$	4.20420e12	0.967240	0.483620	+	0.875278i	$0.339322\pi$
$338$	0	0
$339$	− 4.42167e12i	− 0.987616i
$340$	0	0
$341$	3.13624e12	0.680201
$342$	0	0
$343$	− 3.08538e12i	− 0.649887i
$344$	0	0
$345$	5.55345e12	1.13623
$346$	0	0
$347$	6.66462e12i	1.32473i	0.749180	+	0.662366i	$0.230448\pi$
$347$	6.66462e12i	1.32473i	−0.749180	+	0.662366i	$0.769552\pi$
$348$	0	0
$349$	−2.54127e12	−0.490822	−0.245411	−	0.969419i	$-0.578923\pi$
$349$	−2.54127e12	−0.490822	−0.245411	+	0.969419i	$0.578923\pi$
$350$	0	0
$351$	− 8.62883e12i	− 1.61963i
$352$	0	0
$353$	−5.66543e12	−1.03362	−0.516808	−	0.856101i	$-0.672880\pi$
$353$	−5.66543e12	−1.03362	−0.516808	+	0.856101i	$0.672880\pi$
$354$	0	0
$355$	3.00094e10i	0.00532250i
$356$	0	0
$357$	−9.90529e11	−0.170815
$358$	0	0
$359$	− 7.85553e12i	− 1.31736i	−0.752425	−	0.658678i	$-0.771116\pi$
$359$	− 7.85553e12i	− 1.31736i	0.752425	−	0.658678i	$-0.228884\pi$
$360$	0	0
$361$	−1.00128e13	−1.63313
$362$	0	0
$363$	− 1.02264e13i	− 1.62252i
$364$	0	0
$365$	2.90539e12	0.448478
$366$	0	0
$367$	7.32521e12i	1.10025i	0.835084	+	0.550123i	$0.185419\pi$
$367$	7.32521e12i	1.10025i	−0.835084	+	0.550123i	$0.814581\pi$
$368$	0	0
$369$	−7.62085e11	−0.111397
$370$	0	0
$371$	2.72339e12i	0.387472i
$372$	0	0
$373$	−3.96652e12	−0.549371	−0.274685	−	0.961534i	$-0.588574\pi$
$373$	−3.96652e12	−0.549371	−0.274685	+	0.961534i	$0.588574\pi$
$374$	0	0
$375$	6.21044e12i	0.837464i
$376$	0	0
$377$	−4.36011e12	−0.572520
$378$	0	0
$379$	− 1.25996e13i	− 1.61124i	−0.592432	−	0.805621i	$-0.701832\pi$
$379$	− 1.25996e13i	− 1.61124i	0.592432	−	0.805621i	$-0.298168\pi$
$380$	0	0
$381$	3.51778e12	0.438171
$382$	0	0
$383$	2.66109e12i	0.322898i	0.986881	+	0.161449i	$0.0516168\pi$
$383$	2.66109e12i	0.322898i	−0.986881	+	0.161449i	$0.948383\pi$
$384$	0	0
$385$	5.14548e12	0.608306
$386$	0	0
$387$	− 9.92193e11i	− 0.114299i
$388$	0	0
$389$	1.59316e13	1.78859	0.894293	−	0.447481i	$-0.147679\pi$
$389$	1.59316e13	1.78859	0.894293	+	0.447481i	$0.147679\pi$
$390$	0	0
$391$	− 6.31173e12i	− 0.690661i
$392$	0	0
$393$	−1.56711e13	−1.67162
$394$	0	0
$395$	1.94637e13i	2.02414i
$396$	0	0
$397$	7.33345e12	0.743628	0.371814	−	0.928307i	$-0.378736\pi$
$397$	7.33345e12	0.743628	0.371814	+	0.928307i	$0.378736\pi$
$398$	0	0
$399$	− 5.00316e12i	− 0.494743i
$400$	0	0
$401$	3.75733e12	0.362374	0.181187	−	0.983449i	$-0.442006\pi$
$401$	3.75733e12	0.362374	0.181187	+	0.983449i	$0.442006\pi$
$402$	0	0
$403$	− 6.44853e12i	− 0.606646i
$404$	0	0
$405$	8.30139e12	0.761861
$406$	0	0
$407$	2.91972e13i	2.61438i
$408$	0	0
$409$	−5.93389e12	−0.518469	−0.259234	−	0.965814i	$-0.583470\pi$
$409$	−5.93389e12	−0.518469	−0.259234	+	0.965814i	$0.583470\pi$
$410$	0	0
$411$	1.49264e13i	1.27276i
$412$	0	0
$413$	4.72275e12	0.393047
$414$	0	0
$415$	− 6.53574e11i	− 0.0530951i
$416$	0	0
$417$	5.85108e12	0.464040
$418$	0	0
$419$	− 9.29143e12i	− 0.719470i	−0.933055	−	0.359735i	$-0.882867\pi$
$419$	− 9.29143e12i	− 0.719470i	0.933055	−	0.359735i	$-0.117133\pi$
$420$	0	0
$421$	−1.23618e13	−0.934700	−0.467350	−	0.884072i	$-0.654791\pi$
$421$	−1.23618e13	−0.934700	−0.467350	+	0.884072i	$0.654791\pi$
$422$	0	0
$423$	− 1.56371e12i	− 0.115466i
$424$	0	0
$425$	−7.09884e11	−0.0511968
$426$	0	0
$427$	2.55410e12i	0.179928i
$428$	0	0
$429$	−3.24604e13	−2.23392
$430$	0	0
$431$	− 2.18593e13i	− 1.46977i	−0.678190	−	0.734887i	$-0.737235\pi$
$431$	− 2.18593e13i	− 1.46977i	0.678190	−	0.734887i	$-0.262765\pi$
$432$	0	0
$433$	2.12719e13	1.39755	0.698774	−	0.715343i	$-0.253729\pi$
$433$	2.12719e13	1.39755	0.698774	+	0.715343i	$0.253729\pi$
$434$	0	0
$435$	− 5.46937e12i	− 0.351149i
$436$	0	0
$437$	3.18806e13	2.00041
$438$	0	0
$439$	− 1.56063e13i	− 0.957142i	−0.878049	−	0.478571i	$-0.841155\pi$
$439$	− 1.56063e13i	− 0.957142i	0.878049	−	0.478571i	$-0.158845\pi$
$440$	0	0
$441$	3.25509e12	0.195151
$442$	0	0
$443$	− 9.61292e12i	− 0.563426i	−0.959499	−	0.281713i	$-0.909097\pi$
$443$	− 9.61292e12i	− 0.563426i	0.959499	−	0.281713i	$-0.0909026\pi$
$444$	0	0
$445$	−2.49453e13	−1.42952
$446$	0	0
$447$	− 1.20016e12i	− 0.0672516i
$448$	0	0
$449$	−2.80049e13	−1.53463	−0.767313	−	0.641273i	$-0.778407\pi$
$449$	−2.80049e13	−1.53463	−0.767313	+	0.641273i	$0.778407\pi$
$450$	0	0
$451$	1.58029e13i	0.846944i
$452$	0	0
$453$	−2.05236e11	−0.0107588
$454$	0	0
$455$	− 1.05798e13i	− 0.542526i
$456$	0	0
$457$	1.32562e13	0.665023	0.332512	−	0.943099i	$-0.392104\pi$
$457$	1.32562e13	0.665023	0.332512	+	0.943099i	$0.392104\pi$
$458$	0	0
$459$	− 1.23020e13i	− 0.603829i
$460$	0	0
$461$	−1.78513e13	−0.857364	−0.428682	−	0.903456i	$-0.641022\pi$
$461$	−1.78513e13	−0.857364	−0.428682	+	0.903456i	$0.641022\pi$
$462$	0	0
$463$	1.32158e12i	0.0621141i	0.999518	+	0.0310570i	$0.00988735\pi$
$463$	1.32158e12i	0.0621141i	−0.999518	+	0.0310570i	$0.990113\pi$
$464$	0	0
$465$	8.08910e12	0.372079
$466$	0	0
$467$	7.54561e10i	0.00339711i	0.999999	+	0.00169856i	$0.000540668\pi$
$467$	7.54561e10i	0.00339711i	−0.999999	+	0.00169856i	$0.999459\pi$
$468$	0	0
$469$	−6.62920e12	−0.292144
$470$	0	0
$471$	− 9.56934e11i	− 0.0412836i
$472$	0	0
$473$	−2.05745e13	−0.869008
$474$	0	0
$475$	− 3.58562e12i	− 0.148285i
$476$	0	0
$477$	−6.13604e12	−0.248483
$478$	0	0
$479$	3.89488e13i	1.54460i	0.635256	+	0.772301i	$0.280894\pi$
$479$	3.89488e13i	1.54460i	−0.635256	+	0.772301i	$0.719106\pi$
$480$	0	0
$481$	6.00334e13	2.33167
$482$	0	0
$483$	9.88012e12i	0.375860i
$484$	0	0
$485$	2.10100e13	0.782919
$486$	0	0
$487$	4.16055e13i	1.51882i	0.650613	+	0.759410i	$0.274512\pi$
$487$	4.16055e13i	1.51882i	−0.650613	+	0.759410i	$0.725488\pi$
$488$	0	0
$489$	3.24250e13	1.15967
$490$	0	0
$491$	− 2.34592e13i	− 0.822065i	−0.911621	−	0.411033i	$-0.865168\pi$
$491$	− 2.34592e13i	− 0.822065i	0.911621	−	0.411033i	$-0.134832\pi$
$492$	0	0
$493$	−6.21617e12	−0.213446
$494$	0	0
$495$	1.15932e13i	0.390103i
$496$	0	0
$497$	−5.33896e10	−0.00176066
$498$	0	0
$499$	1.87544e13i	0.606180i	0.952962	+	0.303090i	$0.0980182\pi$
$499$	1.87544e13i	0.606180i	−0.952962	+	0.303090i	$0.901982\pi$
$500$	0	0
$501$	−7.61863e11	−0.0241373
$502$	0	0
$503$	− 3.93000e13i	− 1.22054i	−0.792193	−	0.610271i	$-0.791061\pi$
$503$	− 3.93000e13i	− 1.22054i	0.792193	−	0.610271i	$-0.208939\pi$
$504$	0	0
$505$	−3.70971e13	−1.12949
$506$	0	0
$507$	3.71876e13i	1.11009i
$508$	0	0
$509$	−2.74761e13	−0.804203	−0.402102	−	0.915595i	$-0.631720\pi$
$509$	−2.74761e13	−0.804203	−0.402102	+	0.915595i	$0.631720\pi$
$510$	0	0
$511$	5.16897e12i	0.148354i
$512$	0	0
$513$	6.21376e13	1.74891
$514$	0	0
$515$	− 1.84265e13i	− 0.508636i
$516$	0	0
$517$	−3.24258e13	−0.877886
$518$	0	0
$519$	− 1.29192e13i	− 0.343084i
$520$	0	0
$521$	−4.29715e13	−1.11942	−0.559708	−	0.828690i	$-0.689087\pi$
$521$	−4.29715e13	−1.11942	−0.559708	+	0.828690i	$0.689087\pi$
$522$	0	0
$523$	− 5.57231e12i	− 0.142406i	−0.997462	−	0.0712028i	$-0.977316\pi$
$523$	− 5.57231e12i	− 0.142406i	0.997462	−	0.0712028i	$-0.0226837\pi$
$524$	0	0
$525$	1.11122e12	0.0278615
$526$	0	0
$527$	− 9.19361e12i	− 0.226169i
$528$	0	0
$529$	−2.15304e13	−0.519726
$530$	0	0
$531$	1.06408e13i	0.252059i
$532$	0	0
$533$	3.24929e13	0.755357
$534$	0	0
$535$	− 2.46486e13i	− 0.562372i
$536$	0	0
$537$	−5.81371e12	−0.130191
$538$	0	0
$539$	− 6.74989e13i	− 1.48372i
$540$	0	0
$541$	5.50948e13	1.18884	0.594421	−	0.804154i	$-0.297381\pi$
$541$	5.50948e13	1.18884	0.594421	+	0.804154i	$0.297381\pi$
$542$	0	0
$543$	− 5.38518e13i	− 1.14078i
$544$	0	0
$545$	−5.99336e13	−1.24649
$546$	0	0
$547$	− 4.22238e13i	− 0.862225i	−0.902298	−	0.431113i	$-0.858121\pi$
$547$	− 4.22238e13i	− 0.862225i	0.902298	−	0.431113i	$-0.141879\pi$
$548$	0	0
$549$	−5.75461e12	−0.115386
$550$	0	0
$551$	− 3.13979e13i	− 0.618218i
$552$	0	0
$553$	−3.46278e13	−0.669575
$554$	0	0
$555$	7.53065e13i	1.43010i
$556$	0	0
$557$	1.79435e13	0.334681	0.167340	−	0.985899i	$-0.446482\pi$
$557$	1.79435e13	0.334681	0.167340	+	0.985899i	$0.446482\pi$
$558$	0	0
$559$	4.23039e13i	0.775036i
$560$	0	0
$561$	−4.62785e13	−0.832847
$562$	0	0
$563$	5.45215e13i	0.963887i	0.876202	+	0.481943i	$0.160069\pi$
$563$	5.45215e13i	0.963887i	−0.876202	+	0.481943i	$0.839931\pi$
$564$	0	0
$565$	6.73321e13	1.16945
$566$	0	0
$567$	1.47690e13i	0.252020i
$568$	0	0
$569$	−9.00977e12	−0.151061	−0.0755305	−	0.997143i	$-0.524065\pi$
$569$	−9.00977e12	−0.151061	−0.0755305	+	0.997143i	$0.524065\pi$
$570$	0	0
$571$	1.62036e13i	0.266951i	0.991052	+	0.133476i	$0.0426138\pi$
$571$	1.62036e13i	0.266951i	−0.991052	+	0.133476i	$0.957386\pi$
$572$	0	0
$573$	−1.08718e13	−0.176007
$574$	0	0
$575$	7.08080e12i	0.112653i
$576$	0	0
$577$	6.01800e13	0.940964	0.470482	−	0.882410i	$-0.344080\pi$
$577$	6.01800e13	0.940964	0.470482	+	0.882410i	$0.344080\pi$
$578$	0	0
$579$	2.89141e13i	0.444341i
$580$	0	0
$581$	1.16277e12	0.0175636
$582$	0	0
$583$	1.27240e14i	1.88921i
$584$	0	0
$585$	2.38373e13	0.347918
$586$	0	0
$587$	− 9.25085e13i	− 1.32737i	−0.748013	−	0.663684i	$-0.768992\pi$
$587$	− 9.25085e13i	− 1.32737i	0.748013	−	0.663684i	$-0.231008\pi$
$588$	0	0
$589$	4.64369e13	0.655068
$590$	0	0
$591$	4.42658e13i	0.613947i
$592$	0	0
$593$	−5.49299e13	−0.749092	−0.374546	−	0.927208i	$-0.622201\pi$
$593$	−5.49299e13	−0.749092	−0.374546	+	0.927208i	$0.622201\pi$
$594$	0	0
$595$	− 1.50835e13i	− 0.202264i
$596$	0	0
$597$	−1.64844e13	−0.217371
$598$	0	0
$599$	− 3.23562e13i	− 0.419589i	−0.977746	−	0.209794i	$-0.932721\pi$
$599$	− 3.23562e13i	− 0.419589i	0.977746	−	0.209794i	$-0.0672795\pi$
$600$	0	0
$601$	6.36515e13	0.811776	0.405888	−	0.913923i	$-0.366962\pi$
$601$	6.36515e13	0.811776	0.405888	+	0.913923i	$0.366962\pi$
$602$	0	0
$603$	− 1.49362e13i	− 0.187350i
$604$	0	0
$605$	1.55725e14	1.92124
$606$	0	0
$607$	− 3.20644e13i	− 0.389117i	−0.980891	−	0.194559i	$-0.937673\pi$
$607$	− 3.20644e13i	− 0.389117i	0.980891	−	0.194559i	$-0.0623274\pi$
$608$	0	0
$609$	9.73052e12	0.116158
$610$	0	0
$611$	6.66718e13i	0.782954i
$612$	0	0
$613$	−1.22368e14	−1.41372	−0.706862	−	0.707352i	$-0.749890\pi$
$613$	−1.22368e14	−1.41372	−0.706862	+	0.707352i	$0.749890\pi$
$614$	0	0
$615$	4.07595e13i	0.463290i
$616$	0	0
$617$	−6.30683e13	−0.705318	−0.352659	−	0.935752i	$-0.614722\pi$
$617$	−6.30683e13	−0.705318	−0.352659	+	0.935752i	$0.614722\pi$
$618$	0	0
$619$	8.58755e13i	0.944965i	0.881340	+	0.472483i	$0.156642\pi$
$619$	8.58755e13i	0.944965i	−0.881340	+	0.472483i	$0.843358\pi$
$620$	0	0
$621$	−1.22708e14	−1.32866
$622$	0	0
$623$	− 4.43801e13i	− 0.472876i
$624$	0	0
$625$	−1.03286e14	−1.08303
$626$	0	0
$627$	− 2.33753e14i	− 2.41223i
$628$	0	0
$629$	8.55891e13	0.869291
$630$	0	0
$631$	− 1.73536e14i	− 1.73477i	−0.497636	−	0.867386i	$-0.665799\pi$
$631$	− 1.73536e14i	− 1.73477i	0.497636	−	0.867386i	$-0.334201\pi$
$632$	0	0
$633$	7.22147e13	0.710571
$634$	0	0
$635$	5.35678e13i	0.518842i
$636$	0	0
$637$	−1.38787e14	−1.32328
$638$	0	0
$639$	− 1.20292e11i	− 0.00112910i
$640$	0	0
$641$	−1.14463e14	−1.05773	−0.528865	−	0.848706i	$-0.677382\pi$
$641$	−1.14463e14	−1.05773	−0.528865	+	0.848706i	$0.677382\pi$
$642$	0	0
$643$	− 8.51666e13i	− 0.774844i	−0.921902	−	0.387422i	$-0.873366\pi$
$643$	− 8.51666e13i	− 0.774844i	0.921902	−	0.387422i	$-0.126634\pi$
$644$	0	0
$645$	−5.30665e13	−0.475360
$646$	0	0
$647$	1.12675e14i	0.993817i	0.867803	+	0.496909i	$0.165532\pi$
$647$	1.12675e14i	0.993817i	−0.867803	+	0.496909i	$0.834468\pi$
$648$	0	0
$649$	2.20652e14	1.91639
$650$	0	0
$651$	1.43913e13i	0.123082i
$652$	0	0
$653$	1.18452e14	0.997650	0.498825	−	0.866703i	$-0.333765\pi$
$653$	1.18452e14	0.997650	0.498825	+	0.866703i	$0.333765\pi$
$654$	0	0
$655$	− 2.38636e14i	− 1.97938i
$656$	0	0
$657$	−1.16462e13	−0.0951385
$658$	0	0
$659$	9.87434e13i	0.794477i	0.917715	+	0.397238i	$0.130031\pi$
$659$	9.87434e13i	0.794477i	−0.917715	+	0.397238i	$0.869969\pi$
$660$	0	0
$661$	−3.64683e13	−0.289007	−0.144503	−	0.989504i	$-0.546158\pi$
$661$	−3.64683e13	−0.289007	−0.144503	+	0.989504i	$0.546158\pi$
$662$	0	0
$663$	9.51549e13i	0.742785i
$664$	0	0
$665$	7.61868e13	0.585830
$666$	0	0
$667$	6.20037e13i	0.469665i
$668$	0	0
$669$	7.57641e13	0.565370
$670$	0	0
$671$	1.19330e14i	0.877278i
$672$	0	0
$673$	−2.32949e13	−0.168727	−0.0843635	−	0.996435i	$-0.526886\pi$
$673$	−2.32949e13	−0.168727	−0.0843635	+	0.996435i	$0.526886\pi$
$674$	0	0
$675$	1.38010e13i	0.0984900i
$676$	0	0
$677$	−2.80321e14	−1.97112	−0.985558	−	0.169341i	$-0.945836\pi$
$677$	−2.80321e14	−1.97112	−0.985558	+	0.169341i	$0.945836\pi$
$678$	0	0
$679$	3.73788e13i	0.258986i
$680$	0	0
$681$	1.97229e14	1.34659
$682$	0	0
$683$	1.78641e14i	1.20192i	0.799277	+	0.600962i	$0.205216\pi$
$683$	1.78641e14i	1.20192i	−0.799277	+	0.600962i	$0.794784\pi$
$684$	0	0
$685$	−2.27295e14	−1.50708
$686$	0	0
$687$	− 1.80390e14i	− 1.17877i
$688$	0	0
$689$	2.61621e14	1.68491
$690$	0	0
$691$	1.57529e14i	0.999934i	0.866044	+	0.499967i	$0.166655\pi$
$691$	1.57529e14i	0.999934i	−0.866044	+	0.499967i	$0.833345\pi$
$692$	0	0
$693$	−2.06255e13	−0.129044
$694$	0	0
$695$	8.90988e13i	0.549475i
$696$	0	0
$697$	4.63249e13	0.281611
$698$	0	0
$699$	− 8.34157e13i	− 0.499876i
$700$	0	0
$701$	2.85468e14	1.68643	0.843213	−	0.537579i	$-0.180661\pi$
$701$	2.85468e14	1.68643	0.843213	+	0.537579i	$0.180661\pi$
$702$	0	0
$703$	4.32310e14i	2.51779i
$704$	0	0
$705$	−8.36338e13	−0.480216
$706$	0	0
$707$	− 6.59993e13i	− 0.373630i
$708$	0	0
$709$	−1.37963e14	−0.770074	−0.385037	−	0.922901i	$-0.625811\pi$
$709$	−1.37963e14	−0.770074	−0.385037	+	0.922901i	$0.625811\pi$
$710$	0	0
$711$	− 7.80197e13i	− 0.429394i
$712$	0	0
$713$	−9.17024e13	−0.497661
$714$	0	0
$715$	− 4.94299e14i	− 2.64521i
$716$	0	0
$717$	1.96219e14	1.03549
$718$	0	0
$719$	2.17642e14i	1.13266i	0.824179	+	0.566329i	$0.191637\pi$
$719$	2.17642e14i	1.13266i	−0.824179	+	0.566329i	$0.808363\pi$
$720$	0	0
$721$	3.27825e13	0.168254
$722$	0	0
$723$	− 1.14286e14i	− 0.578500i
$724$	0	0
$725$	6.97359e12	0.0348150
$726$	0	0
$727$	− 4.03817e14i	− 1.98844i	−0.107361	−	0.994220i	$-0.534240\pi$
$727$	− 4.03817e14i	− 1.98844i	0.107361	−	0.994220i	$-0.465760\pi$
$728$	0	0
$729$	−2.29055e14	−1.11250
$730$	0	0
$731$	6.03124e13i	0.288948i
$732$	0	0
$733$	1.88036e14	0.888631	0.444315	−	0.895870i	$-0.353447\pi$
$733$	1.88036e14	0.888631	0.444315	+	0.895870i	$0.353447\pi$
$734$	0	0
$735$	− 1.74096e14i	− 0.811617i
$736$	0	0
$737$	−3.09723e14	−1.42441
$738$	0	0
$739$	2.43068e14i	1.10282i	0.834233	+	0.551412i	$0.185911\pi$
$739$	2.43068e14i	1.10282i	−0.834233	+	0.551412i	$0.814089\pi$
$740$	0	0
$741$	−4.80627e14	−2.15138
$742$	0	0
$743$	− 2.43594e14i	− 1.07578i	−0.843015	−	0.537890i	$-0.819222\pi$
$743$	− 2.43594e14i	− 1.07578i	0.843015	−	0.537890i	$-0.180778\pi$
$744$	0	0
$745$	1.82758e13	0.0796332
$746$	0	0
$747$	2.61983e12i	0.0112634i
$748$	0	0
$749$	4.38522e13	0.186030
$750$	0	0
$751$	2.56493e13i	0.107368i	0.998558	+	0.0536841i	$0.0170964\pi$
$751$	2.56493e13i	0.107368i	−0.998558	+	0.0536841i	$0.982904\pi$
$752$	0	0
$753$	−5.84363e13	−0.241384
$754$	0	0
$755$	− 3.12528e12i	− 0.0127396i
$756$	0	0
$757$	2.04584e14	0.822986	0.411493	−	0.911413i	$-0.365008\pi$
$757$	2.04584e14	0.822986	0.411493	+	0.911413i	$0.365008\pi$
$758$	0	0
$759$	4.61609e14i	1.83259i
$760$	0	0
$761$	1.91807e14	0.751523	0.375761	−	0.926716i	$-0.377381\pi$
$761$	1.91807e14	0.751523	0.375761	+	0.926716i	$0.377381\pi$
$762$	0	0
$763$	− 1.06627e14i	− 0.412331i
$764$	0	0
$765$	3.39846e13	0.129710
$766$	0	0
$767$	− 4.53690e14i	− 1.70916i
$768$	0	0
$769$	−2.24807e14	−0.835948	−0.417974	−	0.908459i	$-0.637260\pi$
$769$	−2.24807e14	−0.835948	−0.417974	+	0.908459i	$0.637260\pi$
$770$	0	0
$771$	− 1.62355e14i	− 0.595929i
$772$	0	0
$773$	3.11685e14	1.12932	0.564662	−	0.825323i	$-0.309007\pi$
$773$	3.11685e14	1.12932	0.564662	+	0.825323i	$0.309007\pi$
$774$	0	0
$775$	1.03138e13i	0.0368902i
$776$	0	0
$777$	−1.33977e14	−0.473071
$778$	0	0
$779$	2.33987e14i	0.815650i
$780$	0	0
$781$	−2.49442e12	−0.00858448
$782$	0	0
$783$	1.20850e14i	0.410618i
$784$	0	0
$785$	1.45719e13	0.0488843
$786$	0	0
$787$	2.22225e14i	0.736070i	0.929812	+	0.368035i	$0.119969\pi$
$787$	2.22225e14i	0.736070i	−0.929812	+	0.368035i	$0.880031\pi$
$788$	0	0
$789$	4.86182e14	1.59006
$790$	0	0
$791$	1.19790e14i	0.386847i
$792$	0	0
$793$	2.45358e14	0.782411
$794$	0	0
$795$	3.28181e14i	1.03342i
$796$	0	0
$797$	−5.53959e13	−0.172261	−0.0861303	−	0.996284i	$-0.527450\pi$
$797$	−5.53959e13	−0.172261	−0.0861303	+	0.996284i	$0.527450\pi$
$798$	0	0
$799$	9.50534e13i	0.291900i
$800$	0	0
$801$	9.99924e13	0.303252
$802$	0	0
$803$	2.41500e14i	0.723334i
$804$	0	0
$805$	−1.50452e14	−0.445060
$806$	0	0
$807$	7.35481e13i	0.214884i
$808$	0	0
$809$	−2.98842e14	−0.862381	−0.431191	−	0.902261i	$-0.641906\pi$
$809$	−2.98842e14	−0.862381	−0.431191	+	0.902261i	$0.641906\pi$
$810$	0	0
$811$	5.43786e11i	0.00154997i	1.00000		0.000774986i	$0.000246686\pi$
$811$	5.43786e11i	0.00154997i	−1.00000		0.000774986i	$0.999753\pi$
$812$	0	0
$813$	−7.36411e13	−0.207333
$814$	0	0
$815$	4.93760e14i	1.37318i
$816$	0	0
$817$	−3.04638e14	−0.836899
$818$	0	0
$819$	4.24087e13i	0.115090i
$820$	0	0
$821$	3.15262e14	0.845192	0.422596	−	0.906318i	$-0.361119\pi$
$821$	3.15262e14	0.845192	0.422596	+	0.906318i	$0.361119\pi$
$822$	0	0
$823$	− 4.15807e14i	− 1.10127i	−0.834747	−	0.550634i	$-0.814386\pi$
$823$	− 4.15807e14i	− 1.10127i	0.834747	−	0.550634i	$-0.185614\pi$
$824$	0	0
$825$	5.19175e13	0.135845
$826$	0	0
$827$	− 7.26600e14i	− 1.87831i	−0.343490	−	0.939156i	$-0.611609\pi$
$827$	− 7.26600e14i	− 1.87831i	0.343490	−	0.939156i	$-0.388391\pi$
$828$	0	0
$829$	3.63960e14	0.929567	0.464784	−	0.885424i	$-0.346132\pi$
$829$	3.63960e14	0.929567	0.464784	+	0.885424i	$0.346132\pi$
$830$	0	0
$831$	− 1.03594e14i	− 0.261414i
$832$	0	0
$833$	−1.97867e14	−0.493343
$834$	0	0
$835$	− 1.16015e13i	− 0.0285812i
$836$	0	0
$837$	−1.78735e14	−0.435093
$838$	0	0
$839$	− 4.39813e14i	− 1.05793i	−0.848642	−	0.528967i	$-0.822579\pi$
$839$	− 4.39813e14i	− 1.05793i	0.848642	−	0.528967i	$-0.177421\pi$
$840$	0	0
$841$	−3.59642e14	−0.854852
$842$	0	0
$843$	1.28910e14i	0.302795i
$844$	0	0
$845$	−5.66283e14	−1.31447
$846$	0	0
$847$	2.77050e14i	0.635537i
$848$	0	0
$849$	3.47552e14	0.787918
$850$	0	0
$851$	− 8.53716e14i	− 1.91278i
$852$	0	0
$853$	7.98926e14	1.76914	0.884568	−	0.466411i	$-0.154453\pi$
$853$	7.98926e14	1.76914	0.884568	+	0.466411i	$0.154453\pi$
$854$	0	0
$855$	1.71656e14i	0.375689i
$856$	0	0
$857$	1.58695e14	0.343287	0.171644	−	0.985159i	$-0.445092\pi$
$857$	1.58695e14	0.343287	0.171644	+	0.985159i	$0.445092\pi$
$858$	0	0
$859$	2.82497e13i	0.0604016i	0.999544	+	0.0302008i	$0.00961468\pi$
$859$	2.82497e13i	0.0604016i	−0.999544	+	0.0302008i	$0.990385\pi$
$860$	0	0
$861$	−7.25149e13	−0.153254
$862$	0	0
$863$	− 3.71130e14i	− 0.775305i	−0.921806	−	0.387652i	$-0.873286\pi$
$863$	− 3.71130e14i	− 0.775305i	0.921806	−	0.387652i	$-0.126714\pi$
$864$	0	0
$865$	1.96731e14	0.406249
$866$	0	0
$867$	− 2.96546e14i	− 0.605336i
$868$	0	0
$869$	−1.61785e15	−3.26467
$870$	0	0
$871$	6.36832e14i	1.27038i
$872$	0	0
$873$	−8.42178e13	−0.166086
$874$	0	0
$875$	− 1.68251e14i	− 0.328033i
$876$	0	0
$877$	−6.90301e14	−1.33058	−0.665289	−	0.746586i	$-0.731692\pi$
$877$	−6.90301e14	−1.33058	−0.665289	+	0.746586i	$0.731692\pi$
$878$	0	0
$879$	− 6.19124e14i	− 1.17987i
$880$	0	0
$881$	4.19676e14	0.790741	0.395371	−	0.918522i	$-0.370616\pi$
$881$	4.19676e14	0.790741	0.395371	+	0.918522i	$0.370616\pi$
$882$	0	0
$883$	4.96457e14i	0.924864i	0.886655	+	0.462432i	$0.153023\pi$
$883$	4.96457e14i	0.924864i	−0.886655	+	0.462432i	$0.846977\pi$
$884$	0	0
$885$	5.69114e14	1.04829
$886$	0	0
$887$	8.83447e13i	0.160902i	0.996759	+	0.0804512i	$0.0256361\pi$
$887$	8.83447e13i	0.160902i	−0.996759	+	0.0804512i	$0.974364\pi$
$888$	0	0
$889$	−9.53022e13	−0.171630
$890$	0	0
$891$	6.90021e14i	1.22878i
$892$	0	0
$893$	−4.80114e14	−0.845449
$894$	0	0
$895$	− 8.85296e13i	− 0.154161i
$896$	0	0
$897$	9.49131e14	1.63442
$898$	0	0
$899$	9.03140e13i	0.153800i
$900$	0	0
$901$	3.72991e14	0.628167
$902$	0	0
$903$	− 9.44104e13i	− 0.157247i
$904$	0	0
$905$	8.20041e14	1.35081
$906$	0	0
$907$	6.74003e14i	1.09806i	0.835803	+	0.549030i	$0.185003\pi$
$907$	6.74003e14i	1.09806i	−0.835803	+	0.549030i	$0.814997\pi$
$908$	0	0
$909$	1.48702e14	0.239607
$910$	0	0
$911$	7.09659e13i	0.113099i	0.998400	+	0.0565493i	$0.0180098\pi$
$911$	7.09659e13i	0.113099i	−0.998400	+	0.0565493i	$0.981990\pi$
$912$	0	0
$913$	5.43258e13	0.0856352
$914$	0	0
$915$	3.07780e14i	0.479883i
$916$	0	0
$917$	4.24556e14	0.654769
$918$	0	0
$919$	1.07969e15i	1.64710i	0.567244	+	0.823550i	$0.308010\pi$
$919$	1.07969e15i	1.64710i	−0.567244	+	0.823550i	$0.691990\pi$
$920$	0	0
$921$	−6.94190e14	−1.04756
$922$	0	0
$923$	5.12886e12i	0.00765618i
$924$	0	0
$925$	−9.60179e13	−0.141789
$926$	0	0
$927$	7.38620e13i	0.107900i
$928$	0	0
$929$	9.74339e14	1.40809	0.704047	−	0.710154i	$-0.251375\pi$
$929$	9.74339e14	1.40809	0.704047	+	0.710154i	$0.251375\pi$
$930$	0	0
$931$	− 9.99427e14i	− 1.42890i
$932$	0	0
$933$	−2.92890e14	−0.414282
$934$	0	0
$935$	− 7.04718e14i	− 0.986182i
$936$	0	0
$937$	−9.84048e13	−0.136244	−0.0681222	−	0.997677i	$-0.521701\pi$
$937$	−9.84048e13	−0.136244	−0.0681222	+	0.997677i	$0.521701\pi$
$938$	0	0
$939$	2.41620e14i	0.330982i
$940$	0	0
$941$	−1.00184e15	−1.35785	−0.678924	−	0.734209i	$-0.737553\pi$
$941$	−1.00184e15	−1.35785	−0.678924	+	0.734209i	$0.737553\pi$
$942$	0	0
$943$	− 4.62071e14i	− 0.619656i
$944$	0	0
$945$	−2.93242e14	−0.389106
$946$	0	0
$947$	1.08733e15i	1.42761i	0.700344	+	0.713805i	$0.253030\pi$
$947$	1.08733e15i	1.42761i	−0.700344	+	0.713805i	$0.746970\pi$
$948$	0	0
$949$	4.96556e14	0.645115
$950$	0	0
$951$	5.58392e14i	0.717854i
$952$	0	0
$953$	−1.33790e15	−1.70200	−0.851002	−	0.525162i	$-0.824005\pi$
$953$	−1.33790e15	−1.70200	−0.851002	+	0.525162i	$0.824005\pi$
$954$	0	0
$955$	− 1.65554e14i	− 0.208411i
$956$	0	0
$957$	4.54620e14	0.566355
$958$	0	0
$959$	− 4.04380e14i	− 0.498536i
$960$	0	0
$961$	6.86055e14	0.837032
$962$	0	0
$963$	9.88032e13i	0.119300i
$964$	0	0
$965$	−4.40296e14	−0.526149
$966$	0	0
$967$	− 7.58551e14i	− 0.897124i	−0.893752	−	0.448562i	$-0.851936\pi$
$967$	− 7.58551e14i	− 0.897124i	0.893752	−	0.448562i	$-0.148064\pi$
$968$	0	0
$969$	−6.85226e14	−0.802075
$970$	0	0
$971$	− 1.11065e15i	− 1.28671i	−0.765568	−	0.643355i	$-0.777542\pi$
$971$	− 1.11065e15i	− 1.28671i	0.765568	−	0.643355i	$-0.222458\pi$
$972$	0	0
$973$	−1.58515e14	−0.181763
$974$	0	0
$975$	− 1.06749e14i	− 0.121155i
$976$	0	0
$977$	7.64097e14	0.858372	0.429186	−	0.903216i	$-0.358800\pi$
$977$	7.64097e14	0.858372	0.429186	+	0.903216i	$0.358800\pi$
$978$	0	0
$979$	− 2.07348e15i	− 2.30562i
$980$	0	0
$981$	2.40242e14	0.264425
$982$	0	0
$983$	5.51019e14i	0.600342i	0.953885	+	0.300171i	$0.0970438\pi$
$983$	5.51019e14i	0.600342i	−0.953885	+	0.300171i	$0.902956\pi$
$984$	0	0
$985$	−6.74068e14	−0.726981
$986$	0	0
$987$	− 1.48792e14i	− 0.158853i
$988$	0	0
$989$	6.01591e14	0.635799
$990$	0	0
$991$	7.54791e14i	0.789693i	0.918747	+	0.394847i	$0.129202\pi$
$991$	7.54791e14i	0.789693i	−0.918747	+	0.394847i	$0.870798\pi$
$992$	0	0
$993$	−7.06642e14	−0.731902
$994$	0	0
$995$	− 2.51020e14i	− 0.257391i
$996$	0	0
$997$	−6.04154e14	−0.613298	−0.306649	−	0.951823i	$-0.599208\pi$
$997$	−6.04154e14	−0.613298	−0.306649	+	0.951823i	$0.599208\pi$
$998$	0	0
$999$	− 1.66396e15i	− 1.67230i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.c.b.15.3	yes	4
3.2	odd	2		144.11.g.d.127.3		4
4.3	odd	2	inner	16.11.c.b.15.2	✓	4
8.3	odd	2		64.11.c.b.63.3		4
8.5	even	2		64.11.c.b.63.2		4
12.11	even	2		144.11.g.d.127.4		4
16.3	odd	4		256.11.d.g.127.6		8
16.5	even	4		256.11.d.g.127.5		8
16.11	odd	4		256.11.d.g.127.3		8
16.13	even	4		256.11.d.g.127.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.c.b.15.2	✓	4	4.3	odd	2	inner
16.11.c.b.15.3	yes	4	1.1	even	1	trivial
64.11.c.b.63.2		4	8.5	even	2
64.11.c.b.63.3		4	8.3	odd	2
144.11.g.d.127.3		4	3.2	odd	2
144.11.g.d.127.4		4	12.11	even	2
256.11.d.g.127.3		8	16.11	odd	4
256.11.d.g.127.4		8	16.13	even	4
256.11.d.g.127.5		8	16.5	even	4
256.11.d.g.127.6		8	16.3	odd	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:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+214.389i q^{3} -3264.66 q^{5} -5808.15i q^{7} +13086.3 q^{9} +O(q^{10})\)	\(q+214.389i q^{3} -3264.66 q^{5} -5808.15i q^{7} +13086.3 q^{9} -271363. i q^{11} -557958. q^{13} -699908. i q^{15} -795476. q^{17} -4.01795e6i q^{19} +1.24520e6 q^{21} +7.93454e6i q^{23} +892402. q^{25} +1.54650e7i q^{27} +7.81440e6 q^{29} +1.15574e7i q^{31} +5.81772e7 q^{33} +1.89616e7i q^{35} -1.07595e8 q^{37} -1.19620e8i q^{39} -5.82354e7 q^{41} -7.58193e7i q^{43} -4.27223e7 q^{45} -1.19493e8i q^{47} +2.48741e8 q^{49} -1.70541e8i q^{51} -4.68891e8 q^{53} +8.85908e8i q^{55} +8.61404e8 q^{57} +8.13126e8i q^{59} -4.39744e8 q^{61} -7.60071e7i q^{63} +1.82154e9 q^{65} -1.14136e9i q^{67} -1.70108e9 q^{69} -9.19220e6i q^{71} -8.89952e8 q^{73} +1.91321e8i q^{75} -1.57611e9 q^{77} -5.96194e9i q^{79} -2.54280e9 q^{81} +2.00196e8i q^{83} +2.59696e9 q^{85} +1.67532e9i q^{87} +7.64100e9 q^{89} +3.24070e9i q^{91} -2.47778e9 q^{93} +1.31172e10i q^{95} -6.43557e9 q^{97} -3.55113e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4200 q^{5} - 189276 q^{9}+O(q^{10}) \) 4 * q + 4200 * q^5 - 189276 * q^9	\( 4 q + 4200 q^{5} - 189276 q^{9} - 1040984 q^{13} - 4390008 q^{17} - 5305344 q^{21} + 39812780 q^{25} + 31136808 q^{29} + 33578496 q^{33} - 166166680 q^{37} + 172774728 q^{41} - 1241253720 q^{45} + 880227268 q^{49} - 355888152 q^{53} + 3496598016 q^{57} - 1949148376 q^{61} + 4045071120 q^{65} - 7111867392 q^{69} + 1505019208 q^{73} - 5086891008 q^{77} - 1572150204 q^{81} - 9822078000 q^{85} + 6041434056 q^{89} + 30791430144 q^{93} - 8292247544 q^{97}+O(q^{100}) \) 4 * q + 4200 * q^5 - 189276 * q^9 - 1040984 * q^13 - 4390008 * q^17 - 5305344 * q^21 + 39812780 * q^25 + 31136808 * q^29 + 33578496 * q^33 - 166166680 * q^37 + 172774728 * q^41 - 1241253720 * q^45 + 880227268 * q^49 - 355888152 * q^53 + 3496598016 * q^57 - 1949148376 * q^61 + 4045071120 * q^65 - 7111867392 * q^69 + 1505019208 * q^73 - 5086891008 * q^77 - 1572150204 * q^81 - 9822078000 * q^85 + 6041434056 * q^89 + 30791430144 * q^93 - 8292247544 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more