LMFDB - Embedded newform 16.11.c.b.15.4

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			15.4
Root			$-5.36805 - 9.29774i$ of defining polynomial
Character	$\chi$	$=$	16.15
Dual form			16.11.c.b.15.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+408.379i q^{3} +5364.66 q^{5} +9544.75i q^{7} -107724. q^{9} +O(q^{10})$	$q+408.379i q^{3} +5364.66 q^{5} +9544.75i q^{7} -107724. q^{9} +101347. i q^{11} +37465.8 q^{13} +2.19082e6i q^{15} -1.39953e6 q^{17} -2.17175e6i q^{19} -3.89788e6 q^{21} +4.54199e6i q^{23} +1.90140e7 q^{25} -1.98780e7i q^{27} +7.75400e6 q^{29} -4.37669e7i q^{31} -4.13880e7 q^{33} +5.12044e7i q^{35} +2.45115e7 q^{37} +1.53002e7i q^{39} +1.44623e8 q^{41} +2.85632e8i q^{43} -5.77905e8 q^{45} -5.16819e6i q^{47} +1.91373e8 q^{49} -5.71538e8i q^{51} +2.90947e8 q^{53} +5.43692e8i q^{55} +8.86895e8 q^{57} -7.53488e8i q^{59} -5.34830e8 q^{61} -1.02820e9i q^{63} +2.00991e8 q^{65} -8.34206e8i q^{67} -1.85485e9 q^{69} +1.40737e9i q^{71} +1.64246e9 q^{73} +7.76491e9i q^{75} -9.67332e8 q^{77} -2.71488e9i q^{79} +1.75673e9 q^{81} -4.09183e9i q^{83} -7.50800e9 q^{85} +3.16657e9i q^{87} -4.62029e9 q^{89} +3.57602e8i q^{91} +1.78735e10 q^{93} -1.16507e10i q^{95} +2.28945e9 q^{97} -1.09175e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	408.379i	1.68057i	0.542144	+	0.840286i	$0.317613\pi$
$3$	408.379i	1.68057i	−0.542144	+	0.840286i	$0.682387\pi$
$4$	0	0
$5$	5364.66	1.71669	0.858346	−	0.513071i	$-0.171492\pi$
$5$	5364.66	1.71669	0.858346	+	0.513071i	$0.171492\pi$
$6$	0	0
$7$	9544.75i	0.567903i	0.958839	+	0.283952i	$0.0916455\pi$
$7$	9544.75i	0.567903i	−0.958839	+	0.283952i	$0.908354\pi$
$8$	0	0
$9$	−107724.	−1.82432
$10$	0	0
$11$	101347.i	0.629285i	0.949210	+	0.314642i	$0.101885\pi$
$11$	101347.i	0.629285i	−0.949210	+	0.314642i	$0.898115\pi$
$12$	0	0
$13$	37465.8	0.100906	0.0504531	−	0.998726i	$-0.483933\pi$
$13$	37465.8	0.100906	0.0504531	+	0.998726i	$0.483933\pi$
$14$	0	0
$15$	2.19082e6i	2.88502i
$16$	0	0
$17$	−1.39953e6	−0.985683	−0.492841	−	0.870119i	$-0.664042\pi$
$17$	−1.39953e6	−0.985683	−0.492841	+	0.870119i	$0.664042\pi$
$18$	0	0
$19$	− 2.17175e6i	− 0.877084i	−0.898711	−	0.438542i	$-0.855495\pi$
$19$	− 2.17175e6i	− 0.877084i	0.898711	−	0.438542i	$-0.144505\pi$
$20$	0	0
$21$	−3.89788e6	−0.954402
$22$	0	0
$23$	4.54199e6i	0.705679i	0.935684	+	0.352840i	$0.114784\pi$
$23$	4.54199e6i	0.705679i	−0.935684	+	0.352840i	$0.885216\pi$
$24$	0	0
$25$	1.90140e7	1.94703
$26$	0	0
$27$	− 1.98780e7i	− 1.38533i
$28$	0	0
$29$	7.75400e6	0.378038	0.189019	−	0.981973i	$-0.439469\pi$
$29$	7.75400e6	0.378038	0.189019	+	0.981973i	$0.439469\pi$
$30$	0	0
$31$	− 4.37669e7i	− 1.52875i	−0.644769	−	0.764377i	$-0.723047\pi$
$31$	− 4.37669e7i	− 1.52875i	0.644769	−	0.764377i	$-0.276953\pi$
$32$	0	0
$33$	−4.13880e7	−1.05756
$34$	0	0
$35$	5.12044e7i	0.974915i
$36$	0	0
$37$	2.45115e7	0.353477	0.176739	−	0.984258i	$-0.443445\pi$
$37$	2.45115e7	0.353477	0.176739	+	0.984258i	$0.443445\pi$
$38$	0	0
$39$	1.53002e7i	0.169580i
$40$	0	0
$41$	1.44623e8	1.24830	0.624148	−	0.781306i	$-0.285446\pi$
$41$	1.44623e8	1.24830	0.624148	+	0.781306i	$0.285446\pi$
$42$	0	0
$43$	2.85632e8i	1.94296i	0.237119	+	0.971481i	$0.423797\pi$
$43$	2.85632e8i	1.94296i	−0.237119	+	0.971481i	$0.576203\pi$
$44$	0	0
$45$	−5.77905e8	−3.13180
$46$	0	0
$47$	− 5.16819e6i	− 0.0225346i	−0.999937	−	0.0112673i	$-0.996413\pi$
$47$	− 5.16819e6i	− 0.0225346i	0.999937	−	0.0112673i	$-0.00358657\pi$
$48$	0	0
$49$	1.91373e8	0.677486
$50$	0	0
$51$	− 5.71538e8i	− 1.65651i
$52$	0	0
$53$	2.90947e8	0.695720	0.347860	−	0.937546i	$-0.386908\pi$
$53$	2.90947e8	0.695720	0.347860	+	0.937546i	$0.386908\pi$
$54$	0	0
$55$	5.43692e8i	1.08029i
$56$	0	0
$57$	8.86895e8	1.47400
$58$	0	0
$59$	− 7.53488e8i	− 1.05394i	−0.849884	−	0.526970i	$-0.823328\pi$
$59$	− 7.53488e8i	− 1.05394i	0.849884	−	0.526970i	$-0.176672\pi$
$60$	0	0
$61$	−5.34830e8	−0.633238	−0.316619	−	0.948553i	$-0.602548\pi$
$61$	−5.34830e8	−0.633238	−0.316619	+	0.948553i	$0.602548\pi$
$62$	0	0
$63$	− 1.02820e9i	− 1.03604i
$64$	0	0
$65$	2.00991e8	0.173225
$66$	0	0
$67$	− 8.34206e8i	− 0.617873i	−0.951083	−	0.308937i	$-0.900027\pi$
$67$	− 8.34206e8i	− 0.617873i	0.951083	−	0.308937i	$-0.0999731\pi$
$68$	0	0
$69$	−1.85485e9	−1.18594
$70$	0	0
$71$	1.40737e9i	0.780042i	0.920806	+	0.390021i	$0.127532\pi$
$71$	1.40737e9i	0.780042i	−0.920806	+	0.390021i	$0.872468\pi$
$72$	0	0
$73$	1.64246e9	0.792284	0.396142	−	0.918189i	$-0.370349\pi$
$73$	1.64246e9	0.792284	0.396142	+	0.918189i	$0.370349\pi$
$74$	0	0
$75$	7.76491e9i	3.27213i
$76$	0	0
$77$	−9.67332e8	−0.357373
$78$	0	0
$79$	− 2.71488e9i	− 0.882296i	−0.897434	−	0.441148i	$-0.854571\pi$
$79$	− 2.71488e9i	− 0.882296i	0.897434	−	0.441148i	$-0.145429\pi$
$80$	0	0
$81$	1.75673e9	0.503824
$82$	0	0
$83$	− 4.09183e9i	− 1.03879i	−0.854535	−	0.519394i	$-0.826158\pi$
$83$	− 4.09183e9i	− 1.03879i	0.854535	−	0.519394i	$-0.173842\pi$
$84$	0	0
$85$	−7.50800e9	−1.69211
$86$	0	0
$87$	3.16657e9i	0.635320i
$88$	0	0
$89$	−4.62029e9	−0.827406	−0.413703	−	0.910412i	$-0.635765\pi$
$89$	−4.62029e9	−0.827406	−0.413703	+	0.910412i	$0.635765\pi$
$90$	0	0
$91$	3.57602e8i	0.0573050i
$92$	0	0
$93$	1.78735e10	2.56918
$94$	0	0
$95$	− 1.16507e10i	− 1.50568i
$96$	0	0
$97$	2.28945e9	0.266608	0.133304	−	0.991075i	$-0.457441\pi$
$97$	2.28945e9	0.266608	0.133304	+	0.991075i	$0.457441\pi$
$98$	0	0
$99$	− 1.09175e10i	− 1.14802i
$100$	0	0
$101$	1.67692e10	1.59554	0.797768	−	0.602965i	$-0.206014\pi$
$101$	1.67692e10	1.59554	0.797768	+	0.602965i	$0.206014\pi$
$102$	0	0
$103$	1.83520e10i	1.58306i	0.611133	+	0.791528i	$0.290714\pi$
$103$	1.83520e10i	1.58306i	−0.611133	+	0.791528i	$0.709286\pi$
$104$	0	0
$105$	−2.09108e10	−1.63842
$106$	0	0
$107$	− 1.88451e10i	− 1.34363i	−0.740719	−	0.671815i	$-0.765515\pi$
$107$	− 1.88451e10i	− 1.34363i	0.740719	−	0.671815i	$-0.234485\pi$
$108$	0	0
$109$	−2.10995e10	−1.37132	−0.685661	−	0.727921i	$-0.740487\pi$
$109$	−2.10995e10	−1.37132	−0.685661	+	0.727921i	$0.740487\pi$
$110$	0	0
$111$	1.00100e10i	0.594044i
$112$	0	0
$113$	2.61654e9	0.142015	0.0710076	−	0.997476i	$-0.477379\pi$
$113$	2.61654e9	0.142015	0.0710076	+	0.997476i	$0.477379\pi$
$114$	0	0
$115$	2.43663e10i	1.21143i
$116$	0	0
$117$	−4.03597e9	−0.184085
$118$	0	0
$119$	− 1.33582e10i	− 0.559773i
$120$	0	0
$121$	1.56662e10	0.604000
$122$	0	0
$123$	5.90609e10i	2.09785i
$124$	0	0
$125$	4.96144e10	1.62576
$126$	0	0
$127$	1.95517e10i	0.591788i	0.955221	+	0.295894i	$0.0956176\pi$
$127$	1.95517e10i	0.591788i	−0.955221	+	0.295894i	$0.904382\pi$
$128$	0	0
$129$	−1.16646e11	−3.26528
$130$	0	0
$131$	− 1.81134e10i	− 0.469507i	−0.972055	−	0.234754i	$-0.924572\pi$
$131$	− 1.81134e10i	− 0.469507i	0.972055	−	0.234754i	$-0.0754283\pi$
$132$	0	0
$133$	2.07288e10	0.498099
$134$	0	0
$135$	− 1.06639e11i	− 2.37818i
$136$	0	0
$137$	5.58290e10	1.15680	0.578398	−	0.815754i	$-0.303678\pi$
$137$	5.58290e10	1.15680	0.578398	+	0.815754i	$0.303678\pi$
$138$	0	0
$139$	− 6.47341e10i	− 1.24755i	−0.781602	−	0.623777i	$-0.785597\pi$
$139$	− 6.47341e10i	− 1.24755i	0.781602	−	0.623777i	$-0.214403\pi$
$140$	0	0
$141$	2.11058e9	0.0378710
$142$	0	0
$143$	3.79704e9i	0.0634988i
$144$	0	0
$145$	4.15976e10	0.648975
$146$	0	0
$147$	7.81527e10i	1.13856i
$148$	0	0
$149$	−9.48338e9	−0.129131	−0.0645657	−	0.997913i	$-0.520566\pi$
$149$	−9.48338e9	−0.129131	−0.0645657	+	0.997913i	$0.520566\pi$
$150$	0	0
$151$	8.42597e10i	1.07333i	0.843794	+	0.536667i	$0.180317\pi$
$151$	8.42597e10i	1.07333i	−0.843794	+	0.536667i	$0.819683\pi$
$152$	0	0
$153$	1.50763e11	1.79820
$154$	0	0
$155$	− 2.34795e11i	− 2.62440i
$156$	0	0
$157$	−1.13978e11	−1.19488	−0.597440	−	0.801914i	$-0.703815\pi$
$157$	−1.13978e11	−1.19488	−0.597440	+	0.801914i	$0.703815\pi$
$158$	0	0
$159$	1.18817e11i	1.16921i
$160$	0	0
$161$	−4.33522e10	−0.400758
$162$	0	0
$163$	7.57172e10i	0.658047i	0.944322	+	0.329023i	$0.106719\pi$
$163$	7.57172e10i	0.658047i	−0.944322	+	0.329023i	$0.893281\pi$
$164$	0	0
$165$	−2.22032e11	−1.81550
$166$	0	0
$167$	− 4.56344e10i	− 0.351326i	−0.984450	−	0.175663i	$-0.943793\pi$
$167$	− 4.56344e10i	− 0.351326i	0.984450	−	0.175663i	$-0.0562069\pi$
$168$	0	0
$169$	−1.36455e11	−0.989818
$170$	0	0
$171$	2.33950e11i	1.60008i
$172$	0	0
$173$	9.76852e10	0.630374	0.315187	−	0.949030i	$-0.397933\pi$
$173$	9.76852e10	0.630374	0.315187	+	0.949030i	$0.397933\pi$
$174$	0	0
$175$	1.81484e11i	1.10573i
$176$	0	0
$177$	3.07708e11	1.77122
$178$	0	0
$179$	1.40698e10i	0.0765636i	0.999267	+	0.0382818i	$0.0121885\pi$
$179$	1.40698e10i	0.0765636i	−0.999267	+	0.0382818i	$0.987812\pi$
$180$	0	0
$181$	−8.91364e10	−0.458841	−0.229420	−	0.973327i	$-0.573683\pi$
$181$	−8.91364e10	−0.458841	−0.229420	+	0.973327i	$0.573683\pi$
$182$	0	0
$183$	− 2.18413e11i	− 1.06420i
$184$	0	0
$185$	1.31496e11	0.606812
$186$	0	0
$187$	− 1.41838e11i	− 0.620275i
$188$	0	0
$189$	1.89730e11	0.786733
$190$	0	0
$191$	− 3.62252e11i	− 1.42510i	−0.701624	−	0.712548i	$-0.747541\pi$
$191$	− 3.62252e11i	− 1.42510i	0.701624	−	0.712548i	$-0.252459\pi$
$192$	0	0
$193$	−3.73165e11	−1.39352	−0.696761	−	0.717303i	$-0.745376\pi$
$193$	−3.73165e11	−1.39352	−0.696761	+	0.717303i	$0.745376\pi$
$194$	0	0
$195$	8.20806e10i	0.291117i
$196$	0	0
$197$	−4.59783e11	−1.54961	−0.774803	−	0.632202i	$-0.782151\pi$
$197$	−4.59783e11	−1.54961	−0.774803	+	0.632202i	$0.782151\pi$
$198$	0	0
$199$	− 1.46666e11i	− 0.469965i	−0.972000	−	0.234983i	$-0.924497\pi$
$199$	− 1.46666e11i	− 0.469965i	0.972000	−	0.234983i	$-0.0755033\pi$
$200$	0	0
$201$	3.40672e11	1.03838
$202$	0	0
$203$	7.40100e10i	0.214689i
$204$	0	0
$205$	7.75853e11	2.14294
$206$	0	0
$207$	− 4.89283e11i	− 1.28738i
$208$	0	0
$209$	2.20100e11	0.551936
$210$	0	0
$211$	− 4.85432e11i	− 1.16069i	−0.814371	−	0.580345i	$-0.802918\pi$
$211$	− 4.85432e11i	− 1.16069i	0.814371	−	0.580345i	$-0.197082\pi$
$212$	0	0
$213$	−5.74742e11	−1.31092
$214$	0	0
$215$	1.53232e12i	3.33547i
$216$	0	0
$217$	4.17745e11	0.868185
$218$	0	0
$219$	6.70747e11i	1.33149i
$220$	0	0
$221$	−5.24344e10	−0.0994615
$222$	0	0
$223$	2.12853e11i	0.385973i	0.981201	+	0.192986i	$0.0618173\pi$
$223$	2.12853e11i	0.385973i	−0.981201	+	0.192986i	$0.938183\pi$
$224$	0	0
$225$	−2.04827e12	−3.55201
$226$	0	0
$227$	− 6.38820e11i	− 1.05986i	−0.848041	−	0.529931i	$-0.822218\pi$
$227$	− 6.38820e11i	− 1.05986i	0.848041	−	0.529931i	$-0.177782\pi$
$228$	0	0
$229$	5.55410e11	0.881934	0.440967	−	0.897523i	$-0.354636\pi$
$229$	5.55410e11	0.881934	0.440967	+	0.897523i	$0.354636\pi$
$230$	0	0
$231$	− 3.95038e11i	− 0.600591i
$232$	0	0
$233$	−6.59015e11	−0.959656	−0.479828	−	0.877363i	$-0.659301\pi$
$233$	−6.59015e11	−0.959656	−0.479828	+	0.877363i	$0.659301\pi$
$234$	0	0
$235$	− 2.77256e10i	− 0.0386849i
$236$	0	0
$237$	1.10870e12	1.48276
$238$	0	0
$239$	1.14216e12i	1.46466i	0.680948	+	0.732332i	$0.261568\pi$
$239$	1.14216e12i	1.46466i	−0.680948	+	0.732332i	$0.738432\pi$
$240$	0	0
$241$	−6.28604e11	−0.773200	−0.386600	−	0.922247i	$-0.626351\pi$
$241$	−6.28604e11	−0.773200	−0.386600	+	0.922247i	$0.626351\pi$
$242$	0	0
$243$	− 4.56364e11i	− 0.538617i
$244$	0	0
$245$	1.02665e12	1.16303
$246$	0	0
$247$	− 8.13661e10i	− 0.0885032i
$248$	0	0
$249$	1.67102e12	1.74576
$250$	0	0
$251$	1.39499e12i	1.40024i	0.714025	+	0.700120i	$0.246870\pi$
$251$	1.39499e12i	1.40024i	−0.714025	+	0.700120i	$0.753130\pi$
$252$	0	0
$253$	−4.60317e11	−0.444073
$254$	0	0
$255$	− 3.06611e12i	− 2.84372i
$256$	0	0
$257$	−4.75742e11	−0.424332	−0.212166	−	0.977234i	$-0.568052\pi$
$257$	−4.75742e11	−0.424332	−0.212166	+	0.977234i	$0.568052\pi$
$258$	0	0
$259$	2.33956e11i	0.200741i
$260$	0	0
$261$	−8.35294e11	−0.689663
$262$	0	0
$263$	− 1.71591e12i	− 1.36369i	−0.731495	−	0.681847i	$-0.761177\pi$
$263$	− 1.71591e12i	− 1.36369i	0.731495	−	0.681847i	$-0.238823\pi$
$264$	0	0
$265$	1.56083e12	1.19434
$266$	0	0
$267$	− 1.88683e12i	− 1.39052i
$268$	0	0
$269$	−8.08461e11	−0.573982	−0.286991	−	0.957933i	$-0.592655\pi$
$269$	−8.08461e11	−0.573982	−0.286991	+	0.957933i	$0.592655\pi$
$270$	0	0
$271$	− 3.37521e11i	− 0.230916i	−0.993312	−	0.115458i	$-0.963166\pi$
$271$	− 3.37521e11i	− 0.230916i	0.993312	−	0.115458i	$-0.0368336\pi$
$272$	0	0
$273$	−1.46037e11	−0.0963051
$274$	0	0
$275$	1.92701e12i	1.22524i
$276$	0	0
$277$	−1.13402e12	−0.695379	−0.347689	−	0.937610i	$-0.613034\pi$
$277$	−1.13402e12	−0.695379	−0.347689	+	0.937610i	$0.613034\pi$
$278$	0	0
$279$	4.71476e12i	2.78894i
$280$	0	0
$281$	−1.56535e12	−0.893473	−0.446736	−	0.894666i	$-0.647414\pi$
$281$	−1.56535e12	−0.893473	−0.446736	+	0.894666i	$0.647414\pi$
$282$	0	0
$283$	− 2.45625e11i	− 0.135313i	−0.997709	−	0.0676567i	$-0.978448\pi$
$283$	− 2.45625e11i	− 0.135313i	0.997709	−	0.0676567i	$-0.0215523\pi$
$284$	0	0
$285$	4.75789e12	2.53041
$286$	0	0
$287$	1.38039e12i	0.708911i
$288$	0	0
$289$	−5.73141e10	−0.0284297
$290$	0	0
$291$	9.34963e11i	0.448053i
$292$	0	0
$293$	−3.14707e12	−1.45737	−0.728683	−	0.684851i	$-0.759867\pi$
$293$	−3.14707e12	−1.45737	−0.728683	+	0.684851i	$0.759867\pi$
$294$	0	0
$295$	− 4.04221e12i	− 1.80929i
$296$	0	0
$297$	2.01457e12	0.871767
$298$	0	0
$299$	1.70169e11i	0.0712074i
$300$	0	0
$301$	−2.72628e12	−1.10341
$302$	0	0
$303$	6.84820e12i	2.68141i
$304$	0	0
$305$	−2.86918e12	−1.08707
$306$	0	0
$307$	− 2.49023e12i	− 0.913161i	−0.889682	−	0.456580i	$-0.849074\pi$
$307$	− 2.49023e12i	− 0.913161i	0.889682	−	0.456580i	$-0.150926\pi$
$308$	0	0
$309$	−7.49455e12	−2.66044
$310$	0	0
$311$	− 8.16113e11i	− 0.280510i	−0.990115	−	0.140255i	$-0.955208\pi$
$311$	− 8.16113e11i	− 0.280510i	0.990115	−	0.140255i	$-0.0447923\pi$
$312$	0	0
$313$	4.78703e12	1.59347	0.796735	−	0.604328i	$-0.206558\pi$
$313$	4.78703e12	1.59347	0.796735	+	0.604328i	$0.206558\pi$
$314$	0	0
$315$	− 5.51596e12i	− 1.77856i
$316$	0	0
$317$	5.90995e12	1.84624	0.923119	−	0.384515i	$-0.125631\pi$
$317$	5.90995e12	1.84624	0.923119	+	0.384515i	$0.125631\pi$
$318$	0	0
$319$	7.85844e11i	0.237894i
$320$	0	0
$321$	7.69594e12	2.25806
$322$	0	0
$323$	3.03942e12i	0.864526i
$324$	0	0
$325$	7.12374e11	0.196468
$326$	0	0
$327$	− 8.61659e12i	− 2.30460i
$328$	0	0
$329$	4.93291e10	0.0127975
$330$	0	0
$331$	3.30442e12i	0.831679i	0.909438	+	0.415840i	$0.136512\pi$
$331$	3.30442e12i	0.831679i	−0.909438	+	0.415840i	$0.863488\pi$
$332$	0	0
$333$	−2.64049e12	−0.644856
$334$	0	0
$335$	− 4.47524e12i	− 1.06070i
$336$	0	0
$337$	6.57151e11	0.151187	0.0755937	−	0.997139i	$-0.475915\pi$
$337$	6.57151e11	0.151187	0.0755937	+	0.997139i	$0.475915\pi$
$338$	0	0
$339$	1.06854e12i	0.238667i
$340$	0	0
$341$	4.43565e12	0.962022
$342$	0	0
$343$	4.52276e12i	0.952650i
$344$	0	0
$345$	−9.95067e12	−2.03590
$346$	0	0
$347$	− 5.18721e12i	− 1.03107i	−0.856870	−	0.515533i	$-0.827594\pi$
$347$	− 5.18721e12i	− 1.03107i	0.856870	−	0.515533i	$-0.172406\pi$
$348$	0	0
$349$	−9.56219e12	−1.84684	−0.923422	−	0.383786i	$-0.874620\pi$
$349$	−9.56219e12	−1.84684	−0.923422	+	0.383786i	$0.874620\pi$
$350$	0	0
$351$	− 7.44743e11i	− 0.139788i
$352$	0	0
$353$	1.87625e12	0.342308	0.171154	−	0.985244i	$-0.445250\pi$
$353$	1.87625e12	0.342308	0.171154	+	0.985244i	$0.445250\pi$
$354$	0	0
$355$	7.55009e12i	1.33909i
$356$	0	0
$357$	5.45519e12	0.940738
$358$	0	0
$359$	− 3.84621e12i	− 0.645001i	−0.946569	−	0.322501i	$-0.895477\pi$
$359$	− 3.84621e12i	− 0.645001i	0.946569	−	0.322501i	$-0.104523\pi$
$360$	0	0
$361$	1.41459e12	0.230724
$362$	0	0
$363$	6.39775e12i	1.01507i
$364$	0	0
$365$	8.81125e12	1.36011
$366$	0	0
$367$	3.03515e12i	0.455880i	0.973675	+	0.227940i	$0.0731990\pi$
$367$	3.03515e12i	0.455880i	−0.973675	+	0.227940i	$0.926801\pi$
$368$	0	0
$369$	−1.55794e13	−2.27729
$370$	0	0
$371$	2.77702e12i	0.395102i
$372$	0	0
$373$	9.37252e12	1.29811	0.649056	−	0.760741i	$-0.275164\pi$
$373$	9.37252e12	1.29811	0.649056	+	0.760741i	$0.275164\pi$
$374$	0	0
$375$	2.02615e13i	2.73221i
$376$	0	0
$377$	2.90510e11	0.0381464
$378$	0	0
$379$	7.33971e12i	0.938605i	0.883037	+	0.469303i	$0.155495\pi$
$379$	7.33971e12i	0.938605i	−0.883037	+	0.469303i	$0.844505\pi$
$380$	0	0
$381$	−7.98451e12	−0.994543
$382$	0	0
$383$	− 6.80691e12i	− 0.825955i	−0.910741	−	0.412977i	$-0.864489\pi$
$383$	− 6.80691e12i	− 0.825955i	0.910741	−	0.412977i	$-0.135511\pi$
$384$	0	0
$385$	−5.18941e12	−0.613500
$386$	0	0
$387$	− 3.07695e13i	− 3.54458i
$388$	0	0
$389$	−1.12422e12	−0.126212	−0.0631061	−	0.998007i	$-0.520101\pi$
$389$	−1.12422e12	−0.126212	−0.0631061	+	0.998007i	$0.520101\pi$
$390$	0	0
$391$	− 6.35665e12i	− 0.695576i
$392$	0	0
$393$	7.39711e12	0.789041
$394$	0	0
$395$	− 1.45644e13i	− 1.51463i
$396$	0	0
$397$	−1.06436e12	−0.107928	−0.0539641	−	0.998543i	$-0.517186\pi$
$397$	−1.06436e12	−0.107928	−0.0539641	+	0.998543i	$0.517186\pi$
$398$	0	0
$399$	8.46519e12i	0.837091i
$400$	0	0
$401$	1.54122e12	0.148643	0.0743214	−	0.997234i	$-0.476321\pi$
$401$	1.54122e12	0.148643	0.0743214	+	0.997234i	$0.476321\pi$
$402$	0	0
$403$	− 1.63976e12i	− 0.154261i
$404$	0	0
$405$	9.42424e12	0.864911
$406$	0	0
$407$	2.48417e12i	0.222438i
$408$	0	0
$409$	−3.99185e12	−0.348785	−0.174392	−	0.984676i	$-0.555796\pi$
$409$	−3.99185e12	−0.348785	−0.174392	+	0.984676i	$0.555796\pi$
$410$	0	0
$411$	2.27994e13i	1.94408i
$412$	0	0
$413$	7.19185e12	0.598536
$414$	0	0
$415$	− 2.19513e13i	− 1.78328i
$416$	0	0
$417$	2.64361e13	2.09660
$418$	0	0
$419$	3.25017e11i	0.0251673i	0.999921	+	0.0125836i	$0.00400561\pi$
$419$	3.25017e11i	0.0251673i	−0.999921	+	0.0125836i	$0.995994\pi$
$420$	0	0
$421$	9.20728e12	0.696179	0.348089	−	0.937461i	$-0.386831\pi$
$421$	9.20728e12	0.696179	0.348089	+	0.937461i	$0.386831\pi$
$422$	0	0
$423$	5.56740e11i	0.0411103i
$424$	0	0
$425$	−2.66106e13	−1.91916
$426$	0	0
$427$	− 5.10482e12i	− 0.359618i
$428$	0	0
$429$	−1.55063e12	−0.106714
$430$	0	0
$431$	− 1.63975e13i	− 1.10253i	−0.834330	−	0.551265i	$-0.814145\pi$
$431$	− 1.63975e13i	− 1.10253i	0.834330	−	0.551265i	$-0.185855\pi$
$432$	0	0
$433$	−2.09044e13	−1.37341	−0.686703	−	0.726938i	$-0.740943\pi$
$433$	−2.09044e13	−1.37341	−0.686703	+	0.726938i	$0.740943\pi$
$434$	0	0
$435$	1.69876e13i	1.09065i
$436$	0	0
$437$	9.86406e12	0.618940
$438$	0	0
$439$	1.92926e13i	1.18323i	0.806221	+	0.591615i	$0.201509\pi$
$439$	1.92926e13i	1.18323i	−0.806221	+	0.591615i	$0.798491\pi$
$440$	0	0
$441$	−2.06155e13	−1.23595
$442$	0	0
$443$	9.36078e12i	0.548648i	0.961637	+	0.274324i	$0.0884540\pi$
$443$	9.36078e12i	0.548648i	−0.961637	+	0.274324i	$0.911546\pi$
$444$	0	0
$445$	−2.47863e13	−1.42040
$446$	0	0
$447$	− 3.87281e12i	− 0.217015i
$448$	0	0
$449$	2.15502e13	1.18092	0.590458	−	0.807068i	$-0.298947\pi$
$449$	2.15502e13	1.18092	0.590458	+	0.807068i	$0.298947\pi$
$450$	0	0
$451$	1.46571e13i	0.785534i
$452$	0	0
$453$	−3.44099e13	−1.80382
$454$	0	0
$455$	1.91841e12i	0.0983750i
$456$	0	0
$457$	1.60141e13	0.803383	0.401692	−	0.915775i	$-0.368422\pi$
$457$	1.60141e13	0.803383	0.401692	+	0.915775i	$0.368422\pi$
$458$	0	0
$459$	2.78198e13i	1.36549i
$460$	0	0
$461$	−1.16723e13	−0.560600	−0.280300	−	0.959912i	$-0.590434\pi$
$461$	−1.16723e13	−0.560600	−0.280300	+	0.959912i	$0.590434\pi$
$462$	0	0
$463$	3.57648e13i	1.68094i	0.541861	+	0.840468i	$0.317720\pi$
$463$	3.57648e13i	1.68094i	−0.541861	+	0.840468i	$0.682280\pi$
$464$	0	0
$465$	9.58853e13	4.41049
$466$	0	0
$467$	− 3.33050e13i	− 1.49942i	−0.661764	−	0.749712i	$-0.730192\pi$
$467$	− 3.33050e13i	− 1.49942i	0.661764	−	0.749712i	$-0.269808\pi$
$468$	0	0
$469$	7.96229e12	0.350892
$470$	0	0
$471$	− 4.65464e13i	− 2.00808i
$472$	0	0
$473$	−2.89479e13	−1.22268
$474$	0	0
$475$	− 4.12935e13i	− 1.70771i
$476$	0	0
$477$	−3.13421e13	−1.26922
$478$	0	0
$479$	4.62581e13i	1.83447i	0.398346	+	0.917235i	$0.369584\pi$
$479$	4.62581e13i	1.83447i	−0.398346	+	0.917235i	$0.630416\pi$
$480$	0	0
$481$	9.18343e11	0.0356681
$482$	0	0
$483$	− 1.77041e13i	− 0.673502i
$484$	0	0
$485$	1.22821e13	0.457683
$486$	0	0
$487$	2.86260e12i	0.104500i	0.998634	+	0.0522500i	$0.0166393\pi$
$487$	2.86260e12i	0.104500i	−0.998634	+	0.0522500i	$0.983361\pi$
$488$	0	0
$489$	−3.09213e13	−1.10589
$490$	0	0
$491$	− 3.25328e11i	− 0.0114002i	−0.999984	−	0.00570012i	$-0.998186\pi$
$491$	− 3.25328e11i	− 0.0114002i	0.999984	−	0.00570012i	$-0.00181441\pi$
$492$	0	0
$493$	−1.08519e13	−0.372626
$494$	0	0
$495$	− 5.85689e13i	− 1.97079i
$496$	0	0
$497$	−1.34330e13	−0.442989
$498$	0	0
$499$	− 1.42774e13i	− 0.461472i	−0.973016	−	0.230736i	$-0.925887\pi$
$499$	− 1.42774e13i	− 0.461472i	0.973016	−	0.230736i	$-0.0741133\pi$
$500$	0	0
$501$	1.86361e13	0.590428
$502$	0	0
$503$	3.77395e12i	0.117208i	0.998281	+	0.0586039i	$0.0186649\pi$
$503$	3.77395e12i	0.117208i	−0.998281	+	0.0586039i	$0.981335\pi$
$504$	0	0
$505$	8.99613e13	2.73904
$506$	0	0
$507$	− 5.57253e13i	− 1.66346i
$508$	0	0
$509$	6.27306e13	1.83608	0.918038	−	0.396492i	$-0.129773\pi$
$509$	6.27306e13	1.83608	0.918038	+	0.396492i	$0.129773\pi$
$510$	0	0
$511$	1.56769e13i	0.449941i
$512$	0	0
$513$	−4.31699e13	−1.21505
$514$	0	0
$515$	9.84521e13i	2.71762i
$516$	0	0
$517$	5.23781e11	0.0141807
$518$	0	0
$519$	3.98926e13i	1.05939i
$520$	0	0
$521$	−3.50191e13	−0.912255	−0.456128	−	0.889914i	$-0.650764\pi$
$521$	−3.50191e13	−0.912255	−0.456128	+	0.889914i	$0.650764\pi$
$522$	0	0
$523$	− 3.75567e13i	− 0.959797i	−0.877324	−	0.479898i	$-0.840674\pi$
$523$	− 3.75567e13i	− 0.959797i	0.877324	−	0.479898i	$-0.159326\pi$
$524$	0	0
$525$	−7.41142e13	−1.85825
$526$	0	0
$527$	6.12531e13i	1.50687i
$528$	0	0
$529$	2.07968e13	0.502017
$530$	0	0
$531$	8.11689e13i	1.92273i
$532$	0	0
$533$	5.41840e12	0.125961
$534$	0	0
$535$	− 1.01098e14i	− 2.30660i
$536$	0	0
$537$	−5.74581e12	−0.128671
$538$	0	0
$539$	1.93951e13i	0.426332i
$540$	0	0
$541$	−5.03894e13	−1.08731	−0.543655	−	0.839309i	$-0.682960\pi$
$541$	−5.03894e13	−1.08731	−0.543655	+	0.839309i	$0.682960\pi$
$542$	0	0
$543$	− 3.64014e13i	− 0.771115i
$544$	0	0
$545$	−1.13192e14	−2.35414
$546$	0	0
$547$	2.26584e13i	0.462692i	0.972872	+	0.231346i	$0.0743130\pi$
$547$	2.26584e13i	0.462692i	−0.972872	+	0.231346i	$0.925687\pi$
$548$	0	0
$549$	5.76142e13	1.15523
$550$	0	0
$551$	− 1.68397e13i	− 0.331571i
$552$	0	0
$553$	2.59128e13	0.501059
$554$	0	0
$555$	5.37002e13i	1.01979i
$556$	0	0
$557$	1.29640e13	0.241804	0.120902	−	0.992664i	$-0.461421\pi$
$557$	1.29640e13	0.241804	0.120902	+	0.992664i	$0.461421\pi$
$558$	0	0
$559$	1.07014e13i	0.196057i
$560$	0	0
$561$	5.79236e13	1.04242
$562$	0	0
$563$	− 5.44815e13i	− 0.963179i	−0.876397	−	0.481590i	$-0.840060\pi$
$563$	− 5.44815e13i	− 0.963179i	0.876397	−	0.481590i	$-0.159940\pi$
$564$	0	0
$565$	1.40368e13	0.243796
$566$	0	0
$567$	1.67675e13i	0.286123i
$568$	0	0
$569$	−5.01943e13	−0.841576	−0.420788	−	0.907159i	$-0.638246\pi$
$569$	−5.01943e13	−0.841576	−0.420788	+	0.907159i	$0.638246\pi$
$570$	0	0
$571$	3.64208e13i	0.600024i	0.953935	+	0.300012i	$0.0969906\pi$
$571$	3.64208e13i	0.600024i	−0.953935	+	0.300012i	$0.903009\pi$
$572$	0	0
$573$	1.47936e14	2.39498
$574$	0	0
$575$	8.63614e13i	1.37398i
$576$	0	0
$577$	5.92697e12	0.0926730	0.0463365	−	0.998926i	$-0.485245\pi$
$577$	5.92697e12	0.0926730	0.0463365	+	0.998926i	$0.485245\pi$
$578$	0	0
$579$	− 1.52393e14i	− 2.34191i
$580$	0	0
$581$	3.90555e13	0.589932
$582$	0	0
$583$	2.94866e13i	0.437806i
$584$	0	0
$585$	−2.16516e13	−0.316018
$586$	0	0
$587$	− 8.77913e13i	− 1.25968i	−0.776724	−	0.629841i	$-0.783120\pi$
$587$	− 8.77913e13i	− 1.25968i	0.776724	−	0.629841i	$-0.216880\pi$
$588$	0	0
$589$	−9.50507e13	−1.34085
$590$	0	0
$591$	− 1.87766e14i	− 2.60423i
$592$	0	0
$593$	−3.75046e13	−0.511460	−0.255730	−	0.966748i	$-0.582316\pi$
$593$	−3.75046e13	−0.511460	−0.255730	+	0.966748i	$0.582316\pi$
$594$	0	0
$595$	− 7.16620e13i	− 0.960957i
$596$	0	0
$597$	5.98955e13	0.789810
$598$	0	0
$599$	− 6.79928e13i	− 0.881716i	−0.897577	−	0.440858i	$-0.854674\pi$
$599$	− 6.79928e13i	− 0.881716i	0.897577	−	0.440858i	$-0.145326\pi$
$600$	0	0
$601$	−5.11703e13	−0.652598	−0.326299	−	0.945267i	$-0.605802\pi$
$601$	−5.11703e13	−0.652598	−0.326299	+	0.945267i	$0.605802\pi$
$602$	0	0
$603$	8.98643e13i	1.12720i
$604$	0	0
$605$	8.40440e13	1.03688
$606$	0	0
$607$	1.20018e14i	1.45647i	0.685328	+	0.728234i	$0.259659\pi$
$607$	1.20018e14i	1.45647i	−0.685328	+	0.728234i	$0.740341\pi$
$608$	0	0
$609$	−3.02241e13	−0.360801
$610$	0	0
$611$	− 1.93630e11i	− 0.00227388i
$612$	0	0
$613$	−9.46038e13	−1.09297	−0.546483	−	0.837470i	$-0.684034\pi$
$613$	−9.46038e13	−1.09297	−0.546483	+	0.837470i	$0.684034\pi$
$614$	0	0
$615$	3.16842e14i	3.60136i
$616$	0	0
$617$	−7.11726e13	−0.795952	−0.397976	−	0.917396i	$-0.630287\pi$
$617$	−7.11726e13	−0.795952	−0.397976	+	0.917396i	$0.630287\pi$
$618$	0	0
$619$	− 6.60896e13i	− 0.727243i	−0.931547	−	0.363622i	$-0.881540\pi$
$619$	− 6.60896e13i	− 0.727243i	0.931547	−	0.363622i	$-0.118460\pi$
$620$	0	0
$621$	9.02856e13	0.977598
$622$	0	0
$623$	− 4.40995e13i	− 0.469887i
$624$	0	0
$625$	8.04808e13	0.843903
$626$	0	0
$627$	8.98841e13i	0.927567i
$628$	0	0
$629$	−3.43046e13	−0.348416
$630$	0	0
$631$	− 7.71751e13i	− 0.771491i	−0.922605	−	0.385745i	$-0.873944\pi$
$631$	− 7.71751e13i	− 0.771491i	0.922605	−	0.385745i	$-0.126056\pi$
$632$	0	0
$633$	1.98240e14	1.95062
$634$	0	0
$635$	1.04888e14i	1.01592i
$636$	0	0
$637$	7.16993e12	0.0683625
$638$	0	0
$639$	− 1.51608e14i	− 1.42305i
$640$	0	0
$641$	2.38322e13	0.220229	0.110114	−	0.993919i	$-0.464878\pi$
$641$	2.38322e13	0.220229	0.110114	+	0.993919i	$0.464878\pi$
$642$	0	0
$643$	1.44200e14i	1.31193i	0.754791	+	0.655965i	$0.227738\pi$
$643$	1.44200e14i	1.31193i	−0.754791	+	0.655965i	$0.772262\pi$
$644$	0	0
$645$	−6.25766e14	−5.60549
$646$	0	0
$647$	2.15203e13i	0.189813i	0.995486	+	0.0949066i	$0.0302553\pi$
$647$	2.15203e13i	0.189813i	−0.995486	+	0.0949066i	$0.969745\pi$
$648$	0	0
$649$	7.63637e13	0.663229
$650$	0	0
$651$	1.70598e14i	1.45905i
$652$	0	0
$653$	1.35263e14	1.13923	0.569617	−	0.821910i	$-0.307091\pi$
$653$	1.35263e14	1.13923	0.569617	+	0.821910i	$0.307091\pi$
$654$	0	0
$655$	− 9.71720e13i	− 0.806000i
$656$	0	0
$657$	−1.76933e14	−1.44538
$658$	0	0
$659$	− 9.66511e13i	− 0.777642i	−0.921313	−	0.388821i	$-0.872882\pi$
$659$	− 9.66511e13i	− 0.777642i	0.921313	−	0.388821i	$-0.127118\pi$
$660$	0	0
$661$	6.57555e13	0.521104	0.260552	−	0.965460i	$-0.416095\pi$
$661$	6.57555e13	0.521104	0.260552	+	0.965460i	$0.416095\pi$
$662$	0	0
$663$	− 2.14131e13i	− 0.167152i
$664$	0	0
$665$	1.11203e14	0.855082
$666$	0	0
$667$	3.52186e13i	0.266774i
$668$	0	0
$669$	−8.69248e13	−0.648654
$670$	0	0
$671$	− 5.42034e13i	− 0.398487i
$672$	0	0
$673$	−1.77433e11	−0.00128516	−0.000642581	−	1.00000i	$-0.500205\pi$
$673$	−1.77433e11	−0.00128516	−0.000642581		1.00000i	$0.500205\pi$
$674$	0	0
$675$	− 3.77959e14i	− 2.69728i
$676$	0	0
$677$	1.73344e14	1.21889	0.609447	−	0.792827i	$-0.291391\pi$
$677$	1.73344e14	1.21889	0.609447	+	0.792827i	$0.291391\pi$
$678$	0	0
$679$	2.18522e13i	0.151407i
$680$	0	0
$681$	2.60880e14	1.78117
$682$	0	0
$683$	− 6.71421e12i	− 0.0451743i	−0.999745	−	0.0225872i	$-0.992810\pi$
$683$	− 6.71421e12i	− 0.0451743i	0.999745	−	0.0225872i	$-0.00719033\pi$
$684$	0	0
$685$	2.99504e14	1.98586
$686$	0	0
$687$	2.26818e14i	1.48215i
$688$	0	0
$689$	1.09006e13	0.0702025
$690$	0	0
$691$	1.77666e14i	1.12775i	0.825860	+	0.563876i	$0.190690\pi$
$691$	1.77666e14i	1.12775i	−0.825860	+	0.563876i	$0.809310\pi$
$692$	0	0
$693$	1.04205e14	0.651963
$694$	0	0
$695$	− 3.47277e14i	− 2.14167i
$696$	0	0
$697$	−2.02404e14	−1.23042
$698$	0	0
$699$	− 2.69128e14i	− 1.61277i
$700$	0	0
$701$	−2.19484e14	−1.29662	−0.648311	−	0.761375i	$-0.724524\pi$
$701$	−2.19484e14	−1.29662	−0.648311	+	0.761375i	$0.724524\pi$
$702$	0	0
$703$	− 5.32328e13i	− 0.310029i
$704$	0	0
$705$	1.13226e13	0.0650128
$706$	0	0
$707$	1.60058e14i	0.906110i
$708$	0	0
$709$	−2.03719e14	−1.13710	−0.568551	−	0.822648i	$-0.692496\pi$
$709$	−2.03719e14	−1.13710	−0.568551	+	0.822648i	$0.692496\pi$
$710$	0	0
$711$	2.92458e14i	1.60959i
$712$	0	0
$713$	1.98789e14	1.07881
$714$	0	0
$715$	2.03699e13i	0.109008i
$716$	0	0
$717$	−4.66435e14	−2.46147
$718$	0	0
$719$	− 9.57688e13i	− 0.498402i	−0.968452	−	0.249201i	$-0.919832\pi$
$719$	− 9.57688e13i	− 0.498402i	0.968452	−	0.249201i	$-0.0801679\pi$
$720$	0	0
$721$	−1.75165e14	−0.899023
$722$	0	0
$723$	− 2.56709e14i	− 1.29942i
$724$	0	0
$725$	1.47434e14	0.736053
$726$	0	0
$727$	2.55764e14i	1.25941i	0.776835	+	0.629704i	$0.216824\pi$
$727$	2.55764e14i	1.25941i	−0.776835	+	0.629704i	$0.783176\pi$
$728$	0	0
$729$	2.90102e14	1.40901
$730$	0	0
$731$	− 3.99750e14i	− 1.91514i
$732$	0	0
$733$	7.64347e12	0.0361219	0.0180610	−	0.999837i	$-0.494251\pi$
$733$	7.64347e12	0.0361219	0.0180610	+	0.999837i	$0.494251\pi$
$734$	0	0
$735$	4.19263e14i	1.95456i
$736$	0	0
$737$	8.45443e13	0.388818
$738$	0	0
$739$	3.70566e14i	1.68129i	0.541584	+	0.840646i	$0.317825\pi$
$739$	3.70566e14i	1.68129i	−0.541584	+	0.840646i	$0.682175\pi$
$740$	0	0
$741$	3.32282e13	0.148736
$742$	0	0
$743$	− 1.30916e14i	− 0.578163i	−0.957305	−	0.289081i	$-0.906650\pi$
$743$	− 1.30916e14i	− 0.578163i	0.957305	−	0.289081i	$-0.0933498\pi$
$744$	0	0
$745$	−5.08752e13	−0.221679
$746$	0	0
$747$	4.40790e14i	1.89508i
$748$	0	0
$749$	1.79872e14	0.763052
$750$	0	0
$751$	1.89841e14i	0.794676i	0.917672	+	0.397338i	$0.130066\pi$
$751$	1.89841e14i	0.794676i	−0.917672	+	0.397338i	$0.869934\pi$
$752$	0	0
$753$	−5.69684e14	−2.35320
$754$	0	0
$755$	4.52025e14i	1.84259i
$756$	0	0
$757$	−9.34488e13	−0.375919	−0.187960	−	0.982177i	$-0.560187\pi$
$757$	−9.34488e13	−0.375919	−0.187960	+	0.982177i	$0.560187\pi$
$758$	0	0
$759$	− 1.87984e14i	− 0.746297i
$760$	0	0
$761$	−2.26330e14	−0.886787	−0.443393	−	0.896327i	$-0.646226\pi$
$761$	−2.26330e14	−0.886787	−0.443393	+	0.896327i	$0.646226\pi$
$762$	0	0
$763$	− 2.01389e14i	− 0.778779i
$764$	0	0
$765$	8.08794e14	3.08696
$766$	0	0
$767$	− 2.82300e13i	− 0.106349i
$768$	0	0
$769$	7.69613e13	0.286181	0.143090	−	0.989710i	$-0.454296\pi$
$769$	7.69613e13	0.286181	0.143090	+	0.989710i	$0.454296\pi$
$770$	0	0
$771$	− 1.94283e14i	− 0.713120i
$772$	0	0
$773$	−2.51431e14	−0.911006	−0.455503	−	0.890234i	$-0.650541\pi$
$773$	−2.51431e14	−0.911006	−0.455503	+	0.890234i	$0.650541\pi$
$774$	0	0
$775$	− 8.32184e14i	− 2.97653i
$776$	0	0
$777$	−9.55428e13	−0.337360
$778$	0	0
$779$	− 3.14084e14i	− 1.09486i
$780$	0	0
$781$	−1.42633e14	−0.490869
$782$	0	0
$783$	− 1.54134e14i	− 0.523707i
$784$	0	0
$785$	−6.11456e14	−2.05124
$786$	0	0
$787$	− 9.19393e13i	− 0.304528i	−0.988340	−	0.152264i	$-0.951344\pi$
$787$	− 9.19393e13i	− 0.304528i	0.988340	−	0.152264i	$-0.0486564\pi$
$788$	0	0
$789$	7.00743e14	2.29178
$790$	0	0
$791$	2.49742e13i	0.0806509i
$792$	0	0
$793$	−2.00378e13	−0.0638976
$794$	0	0
$795$	6.37411e14i	2.00717i
$796$	0	0
$797$	1.15010e14	0.357639	0.178820	−	0.983882i	$-0.442772\pi$
$797$	1.15010e14	0.357639	0.178820	+	0.983882i	$0.442772\pi$
$798$	0	0
$799$	7.23303e12i	0.0222119i
$800$	0	0
$801$	4.97717e14	1.50945
$802$	0	0
$803$	1.66459e14i	0.498573i
$804$	0	0
$805$	−2.32570e14	−0.687978
$806$	0	0
$807$	− 3.30159e14i	− 0.964617i
$808$	0	0
$809$	2.90895e14	0.839449	0.419724	−	0.907652i	$-0.362127\pi$
$809$	2.90895e14	0.839449	0.419724	+	0.907652i	$0.362127\pi$
$810$	0	0
$811$	− 6.21469e14i	− 1.77140i	−0.464263	−	0.885698i	$-0.653681\pi$
$811$	− 6.21469e14i	− 1.77140i	0.464263	−	0.885698i	$-0.346319\pi$
$812$	0	0
$813$	1.37836e14	0.388071
$814$	0	0
$815$	4.06197e14i	1.12966i
$816$	0	0
$817$	6.20319e14	1.70414
$818$	0	0
$819$	− 3.85224e13i	− 0.104543i
$820$	0	0
$821$	1.11703e12	0.00299466	0.00149733	−	0.999999i	$-0.499523\pi$
$821$	1.11703e12	0.00299466	0.00149733	+	0.999999i	$0.499523\pi$
$822$	0	0
$823$	3.80626e14i	1.00809i	0.863677	+	0.504045i	$0.168155\pi$
$823$	3.80626e14i	1.00809i	−0.863677	+	0.504045i	$0.831845\pi$
$824$	0	0
$825$	−7.86950e14	−2.05910
$826$	0	0
$827$	1.39585e14i	0.360837i	0.983590	+	0.180419i	$0.0577453\pi$
$827$	1.39585e14i	0.360837i	−0.983590	+	0.180419i	$0.942255\pi$
$828$	0	0
$829$	6.88035e14	1.75727	0.878633	−	0.477497i	$-0.158456\pi$
$829$	6.88035e14	1.75727	0.878633	+	0.477497i	$0.158456\pi$
$830$	0	0
$831$	− 4.63109e14i	− 1.16863i
$832$	0	0
$833$	−2.67832e14	−0.667786
$834$	0	0
$835$	− 2.44813e14i	− 0.603118i
$836$	0	0
$837$	−8.69997e14	−2.11783
$838$	0	0
$839$	6.40774e14i	1.54133i	0.637241	+	0.770664i	$0.280075\pi$
$839$	6.40774e14i	1.54133i	−0.637241	+	0.770664i	$0.719925\pi$
$840$	0	0
$841$	−3.60583e14	−0.857087
$842$	0	0
$843$	− 6.39258e14i	− 1.50154i
$844$	0	0
$845$	−7.32034e14	−1.69921
$846$	0	0
$847$	1.49530e14i	0.343014i
$848$	0	0
$849$	1.00308e14	0.227404
$850$	0	0
$851$	1.11331e14i	0.249442i
$852$	0	0
$853$	−7.90408e12	−0.0175027	−0.00875137	−	0.999962i	$-0.502786\pi$
$853$	−7.90408e12	−0.0175027	−0.00875137	+	0.999962i	$0.502786\pi$
$854$	0	0
$855$	1.25506e15i	2.74685i
$856$	0	0
$857$	2.88063e14	0.623136	0.311568	−	0.950224i	$-0.399146\pi$
$857$	2.88063e14	0.623136	0.311568	+	0.950224i	$0.399146\pi$
$858$	0	0
$859$	6.19296e14i	1.32414i	0.749444	+	0.662068i	$0.230321\pi$
$859$	6.19296e14i	1.32414i	−0.749444	+	0.662068i	$0.769679\pi$
$860$	0	0
$861$	−5.63722e14	−1.19138
$862$	0	0
$863$	5.51098e13i	0.115126i	0.998342	+	0.0575632i	$0.0183331\pi$
$863$	5.51098e13i	0.115126i	−0.998342	+	0.0575632i	$0.981667\pi$
$864$	0	0
$865$	5.24048e14	1.08216
$866$	0	0
$867$	− 2.34058e13i	− 0.0477781i
$868$	0	0
$869$	2.75144e14	0.555216
$870$	0	0
$871$	− 3.12542e13i	− 0.0623473i
$872$	0	0
$873$	−2.46629e14	−0.486378
$874$	0	0
$875$	4.73557e14i	0.923277i
$876$	0	0
$877$	−3.90462e14	−0.752630	−0.376315	−	0.926492i	$-0.622809\pi$
$877$	−3.90462e14	−0.752630	−0.376315	+	0.926492i	$0.622809\pi$
$878$	0	0
$879$	− 1.28520e15i	− 2.44921i
$880$	0	0
$881$	7.63305e14	1.43820	0.719099	−	0.694908i	$-0.244555\pi$
$881$	7.63305e14	1.43820	0.719099	+	0.694908i	$0.244555\pi$
$882$	0	0
$883$	− 2.23813e14i	− 0.416948i	−0.978028	−	0.208474i	$-0.933150\pi$
$883$	− 2.23813e14i	− 0.416948i	0.978028	−	0.208474i	$-0.0668497\pi$
$884$	0	0
$885$	1.65075e15	3.04064
$886$	0	0
$887$	2.92112e13i	0.0532024i	0.999646	+	0.0266012i	$0.00846842\pi$
$887$	2.92112e13i	0.0532024i	−0.999646	+	0.0266012i	$0.991532\pi$
$888$	0	0
$889$	−1.86616e14	−0.336079
$890$	0	0
$891$	1.78039e14i	0.317049i
$892$	0	0
$893$	−1.12240e13	−0.0197647
$894$	0	0
$895$	7.54797e13i	0.131436i
$896$	0	0
$897$	−6.94935e13	−0.119669
$898$	0	0
$899$	− 3.39369e14i	− 0.577928i
$900$	0	0
$901$	−4.07189e14	−0.685759
$902$	0	0
$903$	− 1.11336e15i	− 1.85437i
$904$	0	0
$905$	−4.78187e14	−0.787688
$906$	0	0
$907$	6.04217e14i	0.984365i	0.870492	+	0.492183i	$0.163801\pi$
$907$	6.04217e14i	0.984365i	−0.870492	+	0.492183i	$0.836199\pi$
$908$	0	0
$909$	−1.80645e15	−2.91077
$910$	0	0
$911$	1.07958e15i	1.72053i	0.509849	+	0.860264i	$0.329701\pi$
$911$	1.07958e15i	1.72053i	−0.509849	+	0.860264i	$0.670299\pi$
$912$	0	0
$913$	4.14695e14	0.653694
$914$	0	0
$915$	− 1.17171e15i	− 1.82691i
$916$	0	0
$917$	1.72887e14	0.266635
$918$	0	0
$919$	− 6.38247e14i	− 0.973669i	−0.873494	−	0.486835i	$-0.838151\pi$
$919$	− 6.38247e14i	− 0.973669i	0.873494	−	0.486835i	$-0.161849\pi$
$920$	0	0
$921$	1.01696e15	1.53463
$922$	0	0
$923$	5.27284e13i	0.0787111i
$924$	0	0
$925$	4.66062e14	0.688232
$926$	0	0
$927$	− 1.97695e15i	− 2.88800i
$928$	0	0
$929$	−3.37039e14	−0.487082	−0.243541	−	0.969891i	$-0.578309\pi$
$929$	−3.37039e14	−0.487082	−0.243541	+	0.969891i	$0.578309\pi$
$930$	0	0
$931$	− 4.15613e14i	− 0.594212i
$932$	0	0
$933$	3.33283e14	0.471417
$934$	0	0
$935$	− 7.60913e14i	− 1.06482i
$936$	0	0
$937$	9.12403e14	1.26325	0.631624	−	0.775275i	$-0.282389\pi$
$937$	9.12403e14	1.26325	0.631624	+	0.775275i	$0.282389\pi$
$938$	0	0
$939$	1.95492e15i	2.67794i
$940$	0	0
$941$	−7.10467e14	−0.962932	−0.481466	−	0.876465i	$-0.659895\pi$
$941$	−7.10467e14	−0.962932	−0.481466	+	0.876465i	$0.659895\pi$
$942$	0	0
$943$	6.56876e14i	0.880896i
$944$	0	0
$945$	1.01784e15	1.35058
$946$	0	0
$947$	5.47329e14i	0.718618i	0.933219	+	0.359309i	$0.116988\pi$
$947$	5.47329e14i	0.718618i	−0.933219	+	0.359309i	$0.883012\pi$
$948$	0	0
$949$	6.15361e13	0.0799464
$950$	0	0
$951$	2.41350e15i	3.10273i
$952$	0	0
$953$	1.03538e15	1.31715	0.658577	−	0.752513i	$-0.271159\pi$
$953$	1.03538e15	1.31715	0.658577	+	0.752513i	$0.271159\pi$
$954$	0	0
$955$	− 1.94336e15i	− 2.44645i
$956$	0	0
$957$	−3.20922e14	−0.399798
$958$	0	0
$959$	5.32874e14i	0.656949i
$960$	0	0
$961$	−1.09592e15	−1.33709
$962$	0	0
$963$	2.03007e15i	2.45121i
$964$	0	0
$965$	−2.00190e15	−2.39225
$966$	0	0
$967$	6.12691e13i	0.0724618i	0.999343	+	0.0362309i	$0.0115352\pi$
$967$	6.12691e13i	0.0724618i	−0.999343	+	0.0362309i	$0.988465\pi$
$968$	0	0
$969$	−1.24123e15	−1.45290
$970$	0	0
$971$	− 1.48384e15i	− 1.71906i	−0.511089	−	0.859528i	$-0.670758\pi$
$971$	− 1.48384e15i	− 1.71906i	0.511089	−	0.859528i	$-0.329242\pi$
$972$	0	0
$973$	6.17871e14	0.708490
$974$	0	0
$975$	2.90918e14i	0.330178i
$976$	0	0
$977$	1.32452e14	0.148794	0.0743969	−	0.997229i	$-0.476297\pi$
$977$	1.32452e14	0.148794	0.0743969	+	0.997229i	$0.476297\pi$
$978$	0	0
$979$	− 4.68252e14i	− 0.520674i
$980$	0	0
$981$	2.27293e15	2.50173
$982$	0	0
$983$	9.11244e14i	0.992812i	0.868090	+	0.496406i	$0.165347\pi$
$983$	9.11244e14i	0.992812i	−0.868090	+	0.496406i	$0.834653\pi$
$984$	0	0
$985$	−2.46658e15	−2.66020
$986$	0	0
$987$	2.01450e13i	0.0215071i
$988$	0	0
$989$	−1.29734e15	−1.37111
$990$	0	0
$991$	− 6.42680e13i	− 0.0672398i	−0.999435	−	0.0336199i	$-0.989296\pi$
$991$	− 6.42680e13i	− 0.0672398i	0.999435	−	0.0336199i	$-0.0107036\pi$
$992$	0	0
$993$	−1.34946e15	−1.39770
$994$	0	0
$995$	− 7.86816e14i	− 0.806785i
$996$	0	0
$997$	1.33145e15	1.35160	0.675801	−	0.737085i	$-0.263798\pi$
$997$	1.33145e15	1.35160	0.675801	+	0.737085i	$0.263798\pi$
$998$	0	0
$999$	− 4.87239e14i	− 0.489682i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	16.11.c.b.15.4	yes	4
3.2	odd	2		144.11.g.d.127.2		4
4.3	odd	2	inner	16.11.c.b.15.1	✓	4
8.3	odd	2		64.11.c.b.63.4		4
8.5	even	2		64.11.c.b.63.1		4
12.11	even	2		144.11.g.d.127.1		4
16.3	odd	4		256.11.d.g.127.7		8
16.5	even	4		256.11.d.g.127.8		8
16.11	odd	4		256.11.d.g.127.2		8
16.13	even	4		256.11.d.g.127.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
16.11.c.b.15.1	✓	4	4.3	odd	2	inner
16.11.c.b.15.4	yes	4	1.1	even	1	trivial
64.11.c.b.63.1		4	8.5	even	2
64.11.c.b.63.4		4	8.3	odd	2
144.11.g.d.127.1		4	12.11	even	2
144.11.g.d.127.2		4	3.2	odd	2
256.11.d.g.127.1		8	16.13	even	4
256.11.d.g.127.2		8	16.11	odd	4
256.11.d.g.127.7		8	16.3	odd	4
256.11.d.g.127.8		8	16.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 16 = 2^{4} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	16.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+408.379i q^{3} +5364.66 q^{5} +9544.75i q^{7} -107724. q^{9} +O(q^{10})\)	\(q+408.379i q^{3} +5364.66 q^{5} +9544.75i q^{7} -107724. q^{9} +101347. i q^{11} +37465.8 q^{13} +2.19082e6i q^{15} -1.39953e6 q^{17} -2.17175e6i q^{19} -3.89788e6 q^{21} +4.54199e6i q^{23} +1.90140e7 q^{25} -1.98780e7i q^{27} +7.75400e6 q^{29} -4.37669e7i q^{31} -4.13880e7 q^{33} +5.12044e7i q^{35} +2.45115e7 q^{37} +1.53002e7i q^{39} +1.44623e8 q^{41} +2.85632e8i q^{43} -5.77905e8 q^{45} -5.16819e6i q^{47} +1.91373e8 q^{49} -5.71538e8i q^{51} +2.90947e8 q^{53} +5.43692e8i q^{55} +8.86895e8 q^{57} -7.53488e8i q^{59} -5.34830e8 q^{61} -1.02820e9i q^{63} +2.00991e8 q^{65} -8.34206e8i q^{67} -1.85485e9 q^{69} +1.40737e9i q^{71} +1.64246e9 q^{73} +7.76491e9i q^{75} -9.67332e8 q^{77} -2.71488e9i q^{79} +1.75673e9 q^{81} -4.09183e9i q^{83} -7.50800e9 q^{85} +3.16657e9i q^{87} -4.62029e9 q^{89} +3.57602e8i q^{91} +1.78735e10 q^{93} -1.16507e10i q^{95} +2.28945e9 q^{97} -1.09175e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4200 q^{5} - 189276 q^{9}+O(q^{10}) \) 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

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