LMFDB - Embedded newform 256.11.c.f.255.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [256,11,Mod(255,256)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(256, 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("256.255");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 256 = 2^{8} $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	256.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:	$162.651456684$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 2954x^{2} + 753424 $ x^4 + 2954*x^2 + 753424
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{16}\cdot 3 $
Twist minimal:	no (minimal twist has level 128)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			255.3
Root			$-51.6917i$ of defining polynomial
Character	$\chi$	$=$	256.255
Dual form			256.11.c.f.255.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+418.798i q^{3} -5463.01 q^{5} -26297.7i q^{7} -116343. q^{9} +O(q^{10})$	$q+418.798i q^{3} -5463.01 q^{5} -26297.7i q^{7} -116343. q^{9} -110423. i q^{11} -333244. q^{13} -2.28790e6i q^{15} +1.74912e6 q^{17} +579477. i q^{19} +1.10134e7 q^{21} -4.39171e6i q^{23} +2.00789e7 q^{25} -2.39946e7i q^{27} -1.17400e7 q^{29} +1.47267e6i q^{31} +4.62450e7 q^{33} +1.43665e8i q^{35} +4.17975e7 q^{37} -1.39562e8i q^{39} -1.23793e8 q^{41} -1.56278e8i q^{43} +6.35583e8 q^{45} -4.45115e8i q^{47} -4.09093e8 q^{49} +7.32528e8i q^{51} +4.27115e8 q^{53} +6.03243e8i q^{55} -2.42684e8 q^{57} +9.43855e8i q^{59} -3.87311e8 q^{61} +3.05955e9i q^{63} +1.82051e9 q^{65} -4.47781e8i q^{67} +1.83924e9 q^{69} +6.97704e8i q^{71} -1.66084e9 q^{73} +8.40899e9i q^{75} -2.90387e9 q^{77} -5.43347e9i q^{79} +3.17897e9 q^{81} +4.16025e9i q^{83} -9.55545e9 q^{85} -4.91670e9i q^{87} +2.50799e9 q^{89} +8.76354e9i q^{91} -6.16752e8 q^{93} -3.16569e9i q^{95} +7.97785e9 q^{97} +1.28470e10i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 465372 q^{9}+O(q^{10}) $ 4 * q - 465372 * q^9	$ 4 q - 465372 q^{9} + 6996472 q^{17} + 80315420 q^{25} + 184980096 q^{33} - 495171656 q^{41} - 1636373564 q^{49} - 970736256 q^{57} + 7282053120 q^{65} - 6643358008 q^{73} + 12715885764 q^{81} + 10031942984 q^{89} + 31911414136 q^{97}+O(q^{100}) $ 4 * q - 465372 * q^9 + 6996472 * q^17 + 80315420 * q^25 + 184980096 * q^33 - 495171656 * q^41 - 1636373564 * q^49 - 970736256 * q^57 + 7282053120 * q^65 - 6643358008 * q^73 + 12715885764 * q^81 + 10031942984 * q^89 + 31911414136 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/256\mathbb{Z}\right)^\times$.

$n$	$5$	$255$
$\chi(n)$	$1$	$-1$

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
$3$	418.798i	1.72345i	0.507376	+	0.861725i	$0.330616\pi$
$3$	418.798i	1.72345i	−0.507376	+	0.861725i	$0.669384\pi$
$4$	0	0
$5$	−5463.01	−1.74816	−0.874082	−	0.485779i	$-0.838536\pi$
$5$	−5463.01	−1.74816	−0.874082	+	0.485779i	$0.838536\pi$
$6$	0	0
$7$	− 26297.7i	− 1.56469i	−0.622847	−	0.782343i	$-0.714024\pi$
$7$	− 26297.7i	− 1.56469i	0.622847	−	0.782343i	$-0.285976\pi$
$8$	0	0
$9$	−116343.	−1.97028
$10$	0	0
$11$	− 110423.i	− 0.685641i	−0.939401	−	0.342820i	$-0.888618\pi$
$11$	− 110423.i	− 0.685641i	0.939401	−	0.342820i	$-0.111382\pi$
$12$	0	0
$13$	−333244.	−0.897522	−0.448761	−	0.893652i	$-0.648135\pi$
$13$	−333244.	−0.897522	−0.448761	+	0.893652i	$0.648135\pi$
$14$	0	0
$15$	− 2.28790e6i	− 3.01287i
$16$	0	0
$17$	1.74912e6	1.23190	0.615949	−	0.787786i	$-0.288773\pi$
$17$	1.74912e6	1.23190	0.615949	+	0.787786i	$0.288773\pi$
$18$	0	0
$19$	579477.i	0.234028i	0.993130	+	0.117014i	$0.0373323\pi$
$19$	579477.i	0.234028i	−0.993130	+	0.117014i	$0.962668\pi$
$20$	0	0
$21$	1.10134e7	2.69666
$22$	0	0
$23$	− 4.39171e6i	− 0.682331i	−0.940003	−	0.341165i	$-0.889178\pi$
$23$	− 4.39171e6i	− 0.682331i	0.940003	−	0.341165i	$-0.110822\pi$
$24$	0	0
$25$	2.00789e7	2.05607
$26$	0	0
$27$	− 2.39946e7i	− 1.67223i
$28$	0	0
$29$	−1.17400e7	−0.572372	−0.286186	−	0.958174i	$-0.592388\pi$
$29$	−1.17400e7	−0.572372	−0.286186	+	0.958174i	$0.592388\pi$
$30$	0	0
$31$	1.47267e6i	0.0514396i	0.999669	+	0.0257198i	$0.00818776\pi$
$31$	1.47267e6i	0.0514396i	−0.999669	+	0.0257198i	$0.991812\pi$
$32$	0	0
$33$	4.62450e7	1.18167
$34$	0	0
$35$	1.43665e8i	2.73533i
$36$	0	0
$37$	4.17975e7	0.602756	0.301378	−	0.953505i	$-0.402553\pi$
$37$	4.17975e7	0.602756	0.301378	+	0.953505i	$0.402553\pi$
$38$	0	0
$39$	− 1.39562e8i	− 1.54683i
$40$	0	0
$41$	−1.23793e8	−1.06850	−0.534252	−	0.845325i	$-0.679407\pi$
$41$	−1.23793e8	−1.06850	−0.534252	+	0.845325i	$0.679407\pi$
$42$	0	0
$43$	− 1.56278e8i	− 1.06306i	−0.847041	−	0.531528i	$-0.821618\pi$
$43$	− 1.56278e8i	− 1.06306i	0.847041	−	0.531528i	$-0.178382\pi$
$44$	0	0
$45$	6.35583e8	3.44437
$46$	0	0
$47$	− 4.45115e8i	− 1.94081i	−0.241486	−	0.970404i	$-0.577635\pi$
$47$	− 4.45115e8i	− 1.94081i	0.241486	−	0.970404i	$-0.422365\pi$
$48$	0	0
$49$	−4.09093e8	−1.44825
$50$	0	0
$51$	7.32528e8i	2.12311i
$52$	0	0
$53$	4.27115e8	1.02133	0.510664	−	0.859781i	$-0.329400\pi$
$53$	4.27115e8	1.02133	0.510664	+	0.859781i	$0.329400\pi$
$54$	0	0
$55$	6.03243e8i	1.19861i
$56$	0	0
$57$	−2.42684e8	−0.403336
$58$	0	0
$59$	9.43855e8i	1.32022i	0.751171	+	0.660108i	$0.229490\pi$
$59$	9.43855e8i	1.32022i	−0.751171	+	0.660108i	$0.770510\pi$
$60$	0	0
$61$	−3.87311e8	−0.458575	−0.229288	−	0.973359i	$-0.573640\pi$
$61$	−3.87311e8	−0.458575	−0.229288	+	0.973359i	$0.573640\pi$
$62$	0	0
$63$	3.05955e9i	3.08287i
$64$	0	0
$65$	1.82051e9	1.56901
$66$	0	0
$67$	− 4.47781e8i	− 0.331659i	−0.986154	−	0.165830i	$-0.946970\pi$
$67$	− 4.47781e8i	− 0.331659i	0.986154	−	0.165830i	$-0.0530302\pi$
$68$	0	0
$69$	1.83924e9	1.17596
$70$	0	0
$71$	6.97704e8i	0.386705i	0.981129	+	0.193352i	$0.0619361\pi$
$71$	6.97704e8i	0.386705i	−0.981129	+	0.193352i	$0.938064\pi$
$72$	0	0
$73$	−1.66084e9	−0.801149	−0.400575	−	0.916264i	$-0.631189\pi$
$73$	−1.66084e9	−0.801149	−0.400575	+	0.916264i	$0.631189\pi$
$74$	0	0
$75$	8.40899e9i	3.54354i
$76$	0	0
$77$	−2.90387e9	−1.07281
$78$	0	0
$79$	− 5.43347e9i	− 1.76580i	−0.469559	−	0.882901i	$-0.655587\pi$
$79$	− 5.43347e9i	− 1.76580i	0.469559	−	0.882901i	$-0.344413\pi$
$80$	0	0
$81$	3.17897e9	0.911720
$82$	0	0
$83$	4.16025e9i	1.05616i	0.849196	+	0.528079i	$0.177087\pi$
$83$	4.16025e9i	1.05616i	−0.849196	+	0.528079i	$0.822913\pi$
$84$	0	0
$85$	−9.55545e9	−2.15356
$86$	0	0
$87$	− 4.91670e9i	− 0.986454i
$88$	0	0
$89$	2.50799e9	0.449133	0.224567	−	0.974459i	$-0.427903\pi$
$89$	2.50799e9	0.449133	0.224567	+	0.974459i	$0.427903\pi$
$90$	0	0
$91$	8.76354e9i	1.40434i
$92$	0	0
$93$	−6.16752e8	−0.0886535
$94$	0	0
$95$	− 3.16569e9i	− 0.409120i
$96$	0	0
$97$	7.97785e9	0.929025	0.464512	−	0.885567i	$-0.346230\pi$
$97$	7.97785e9	0.929025	0.464512	+	0.885567i	$0.346230\pi$
$98$	0	0
$99$	1.28470e10i	1.35090i
$100$	0	0
$101$	−1.21818e10	−1.15905	−0.579526	−	0.814954i	$-0.696762\pi$
$101$	−1.21818e10	−1.15905	−0.579526	+	0.814954i	$0.696762\pi$
$102$	0	0
$103$	8.61384e9i	0.743037i	0.928426	+	0.371519i	$0.121163\pi$
$103$	8.61384e9i	0.743037i	−0.928426	+	0.371519i	$0.878837\pi$
$104$	0	0
$105$	−6.01665e10	−4.71420
$106$	0	0
$107$	− 1.41140e10i	− 1.00631i	−0.864197	−	0.503154i	$-0.832173\pi$
$107$	− 1.41140e10i	− 1.00631i	0.864197	−	0.503154i	$-0.167827\pi$
$108$	0	0
$109$	5.47096e9	0.355575	0.177787	−	0.984069i	$-0.443106\pi$
$109$	5.47096e9	0.355575	0.177787	+	0.984069i	$0.443106\pi$
$110$	0	0
$111$	1.75047e10i	1.03882i
$112$	0	0
$113$	1.84408e9	0.100089	0.0500446	−	0.998747i	$-0.484064\pi$
$113$	1.84408e9	0.100089	0.0500446	+	0.998747i	$0.484064\pi$
$114$	0	0
$115$	2.39920e10i	1.19283i
$116$	0	0
$117$	3.87706e10	1.76837
$118$	0	0
$119$	− 4.59978e10i	− 1.92753i
$120$	0	0
$121$	1.37442e10	0.529897
$122$	0	0
$123$	− 5.18443e10i	− 1.84151i
$124$	0	0
$125$	−5.63413e10	−1.84619
$126$	0	0
$127$	− 3.14784e10i	− 0.952784i	−0.879233	−	0.476392i	$-0.841944\pi$
$127$	− 3.14784e10i	− 0.952784i	0.879233	−	0.476392i	$-0.158056\pi$
$128$	0	0
$129$	6.54490e10	1.83212
$130$	0	0
$131$	− 2.56736e10i	− 0.665472i	−0.943020	−	0.332736i	$-0.892028\pi$
$131$	− 2.56736e10i	− 0.665472i	0.943020	−	0.332736i	$-0.107972\pi$
$132$	0	0
$133$	1.52389e10	0.366181
$134$	0	0
$135$	1.31083e11i	2.92333i
$136$	0	0
$137$	−4.81372e10	−0.997420	−0.498710	−	0.866769i	$-0.666193\pi$
$137$	−4.81372e10	−0.997420	−0.498710	+	0.866769i	$0.666193\pi$
$138$	0	0
$139$	1.78108e10i	0.343249i	0.985162	+	0.171625i	$0.0549016\pi$
$139$	1.78108e10i	0.343249i	−0.985162	+	0.171625i	$0.945098\pi$
$140$	0	0
$141$	1.86413e11	3.34489
$142$	0	0
$143$	3.67978e10i	0.615378i
$144$	0	0
$145$	6.41358e10	1.00060
$146$	0	0
$147$	− 1.71328e11i	− 2.49598i
$148$	0	0
$149$	8.31198e10	1.13181	0.565904	−	0.824471i	$-0.308527\pi$
$149$	8.31198e10	1.13181	0.565904	+	0.824471i	$0.308527\pi$
$150$	0	0
$151$	− 5.45559e10i	− 0.694956i	−0.937688	−	0.347478i	$-0.887038\pi$
$151$	− 5.45559e10i	− 0.694956i	0.937688	−	0.347478i	$-0.112962\pi$
$152$	0	0
$153$	−2.03498e11	−2.42718
$154$	0	0
$155$	− 8.04522e9i	− 0.0899247i
$156$	0	0
$157$	6.04811e9	0.0634047	0.0317024	−	0.999497i	$-0.489907\pi$
$157$	6.04811e9	0.0634047	0.0317024	+	0.999497i	$0.489907\pi$
$158$	0	0
$159$	1.78875e11i	1.76021i
$160$	0	0
$161$	−1.15492e11	−1.06763
$162$	0	0
$163$	− 1.70110e11i	− 1.47840i	−0.673487	−	0.739199i	$-0.735204\pi$
$163$	− 1.70110e11i	− 1.47840i	0.673487	−	0.739199i	$-0.264796\pi$
$164$	0	0
$165$	−2.52637e11	−2.06575
$166$	0	0
$167$	− 1.34438e11i	− 1.03500i	−0.855684	−	0.517498i	$-0.826863\pi$
$167$	− 1.34438e11i	− 1.03500i	0.855684	−	0.517498i	$-0.173137\pi$
$168$	0	0
$169$	−2.68072e10	−0.194454
$170$	0	0
$171$	− 6.74181e10i	− 0.461101i
$172$	0	0
$173$	−1.71335e11	−1.10564	−0.552822	−	0.833300i	$-0.686449\pi$
$173$	−1.71335e11	−1.10564	−0.552822	+	0.833300i	$0.686449\pi$
$174$	0	0
$175$	− 5.28028e11i	− 3.21711i
$176$	0	0
$177$	−3.95285e11	−2.27533
$178$	0	0
$179$	− 1.91319e11i	− 1.04110i	−0.853831	−	0.520550i	$-0.825727\pi$
$179$	− 1.91319e11i	− 1.04110i	0.853831	−	0.520550i	$-0.174273\pi$
$180$	0	0
$181$	−3.93580e10	−0.202601	−0.101300	−	0.994856i	$-0.532300\pi$
$181$	−3.93580e10	−0.202601	−0.101300	+	0.994856i	$0.532300\pi$
$182$	0	0
$183$	− 1.62205e11i	− 0.790332i
$184$	0	0
$185$	−2.28340e11	−1.05372
$186$	0	0
$187$	− 1.93143e11i	− 0.844639i
$188$	0	0
$189$	−6.31003e11	−2.61651
$190$	0	0
$191$	7.19954e10i	0.283229i	0.989922	+	0.141615i	$0.0452294\pi$
$191$	7.19954e10i	0.283229i	−0.989922	+	0.141615i	$0.954771\pi$
$192$	0	0
$193$	−2.03210e11	−0.758856	−0.379428	−	0.925221i	$-0.623879\pi$
$193$	−2.03210e11	−0.758856	−0.379428	+	0.925221i	$0.623879\pi$
$194$	0	0
$195$	7.62428e11i	2.70412i
$196$	0	0
$197$	3.24561e11	1.09387	0.546934	−	0.837176i	$-0.315795\pi$
$197$	3.24561e11	1.09387	0.546934	+	0.837176i	$0.315795\pi$
$198$	0	0
$199$	2.70113e11i	0.865526i	0.901508	+	0.432763i	$0.142461\pi$
$199$	2.70113e11i	0.865526i	−0.901508	+	0.432763i	$0.857539\pi$
$200$	0	0
$201$	1.87530e11	0.571598
$202$	0	0
$203$	3.08735e11i	0.895583i
$204$	0	0
$205$	6.76282e11	1.86792
$206$	0	0
$207$	5.10945e11i	1.34438i
$208$	0	0
$209$	6.39877e10	0.160459
$210$	0	0
$211$	7.32007e11i	1.75026i	0.483887	+	0.875130i	$0.339225\pi$
$211$	7.32007e11i	1.75026i	−0.483887	+	0.875130i	$0.660775\pi$
$212$	0	0
$213$	−2.92197e11	−0.666466
$214$	0	0
$215$	8.53749e11i	1.85839i
$216$	0	0
$217$	3.87278e10	0.0804868
$218$	0	0
$219$	− 6.95557e11i	− 1.38074i
$220$	0	0
$221$	−5.82882e11	−1.10565
$222$	0	0
$223$	3.09427e11i	0.561092i	0.959841	+	0.280546i	$0.0905156\pi$
$223$	3.09427e11i	0.561092i	−0.959841	+	0.280546i	$0.909484\pi$
$224$	0	0
$225$	−2.33603e12	−4.05104
$226$	0	0
$227$	3.02491e11i	0.501860i	0.968005	+	0.250930i	$0.0807364\pi$
$227$	3.02491e11i	0.501860i	−0.968005	+	0.250930i	$0.919264\pi$
$228$	0	0
$229$	−1.06890e12	−1.69731	−0.848654	−	0.528949i	$-0.822586\pi$
$229$	−1.06890e12	−1.69731	−0.848654	+	0.528949i	$0.822586\pi$
$230$	0	0
$231$	− 1.21614e12i	− 1.84894i
$232$	0	0
$233$	7.35352e11	1.07082	0.535409	−	0.844593i	$-0.320157\pi$
$233$	7.35352e11	1.07082	0.535409	+	0.844593i	$0.320157\pi$
$234$	0	0
$235$	2.43167e12i	3.39285i
$236$	0	0
$237$	2.27553e12	3.04327
$238$	0	0
$239$	7.29086e11i	0.934951i	0.884006	+	0.467476i	$0.154836\pi$
$239$	7.29086e11i	0.934951i	−0.884006	+	0.467476i	$0.845164\pi$
$240$	0	0
$241$	−1.23180e12	−1.51514	−0.757572	−	0.652751i	$-0.773615\pi$
$241$	−1.23180e12	−1.51514	−0.757572	+	0.652751i	$0.773615\pi$
$242$	0	0
$243$	− 8.55111e10i	− 0.100923i
$244$	0	0
$245$	2.23488e12	2.53177
$246$	0	0
$247$	− 1.93107e11i	− 0.210046i
$248$	0	0
$249$	−1.74230e12	−1.82023
$250$	0	0
$251$	2.95292e11i	0.296404i	0.988957	+	0.148202i	$0.0473485\pi$
$251$	2.95292e11i	0.296404i	−0.988957	+	0.148202i	$0.952651\pi$
$252$	0	0
$253$	−4.84947e11	−0.467834
$254$	0	0
$255$	− 4.00181e12i	− 3.71155i
$256$	0	0
$257$	−5.00862e11	−0.446738	−0.223369	−	0.974734i	$-0.571705\pi$
$257$	−5.00862e11	−0.446738	−0.223369	+	0.974734i	$0.571705\pi$
$258$	0	0
$259$	− 1.09918e12i	− 0.943125i
$260$	0	0
$261$	1.36587e12	1.12773
$262$	0	0
$263$	1.41474e12i	1.12434i	0.827022	+	0.562170i	$0.190033\pi$
$263$	1.41474e12i	1.12434i	−0.827022	+	0.562170i	$0.809967\pi$
$264$	0	0
$265$	−2.33333e12	−1.78545
$266$	0	0
$267$	1.05034e12i	0.774058i
$268$	0	0
$269$	−1.93262e12	−1.37210	−0.686048	−	0.727557i	$-0.740656\pi$
$269$	−1.93262e12	−1.37210	−0.686048	+	0.727557i	$0.740656\pi$
$270$	0	0
$271$	2.14923e12i	1.47040i	0.677848	+	0.735202i	$0.262913\pi$
$271$	2.14923e12i	1.47040i	−0.677848	+	0.735202i	$0.737087\pi$
$272$	0	0
$273$	−3.67015e12	−2.42031
$274$	0	0
$275$	− 2.21717e12i	− 1.40973i
$276$	0	0
$277$	−1.21431e12	−0.744616	−0.372308	−	0.928109i	$-0.621433\pi$
$277$	−1.21431e12	−0.744616	−0.372308	+	0.928109i	$0.621433\pi$
$278$	0	0
$279$	− 1.71335e11i	− 0.101350i
$280$	0	0
$281$	1.78582e12	1.01931	0.509656	−	0.860378i	$-0.329773\pi$
$281$	1.78582e12	1.01931	0.509656	+	0.860378i	$0.329773\pi$
$282$	0	0
$283$	− 2.04553e12i	− 1.12687i	−0.826161	−	0.563434i	$-0.809480\pi$
$283$	− 2.04553e12i	− 1.12687i	0.826161	−	0.563434i	$-0.190520\pi$
$284$	0	0
$285$	1.32579e12	0.705097
$286$	0	0
$287$	3.25547e12i	1.67188i
$288$	0	0
$289$	1.04342e12	0.517571
$290$	0	0
$291$	3.34111e12i	1.60113i
$292$	0	0
$293$	−4.06244e11	−0.188126	−0.0940629	−	0.995566i	$-0.529985\pi$
$293$	−4.06244e11	−0.188126	−0.0940629	+	0.995566i	$0.529985\pi$
$294$	0	0
$295$	− 5.15629e12i	− 2.30795i
$296$	0	0
$297$	−2.64956e12	−1.14655
$298$	0	0
$299$	1.46351e12i	0.612407i
$300$	0	0
$301$	−4.10975e12	−1.66335
$302$	0	0
$303$	− 5.10170e12i	− 1.99757i
$304$	0	0
$305$	2.11588e12	0.801665
$306$	0	0
$307$	− 4.06751e12i	− 1.49155i	−0.666199	−	0.745774i	$-0.732080\pi$
$307$	− 4.06751e12i	− 1.49155i	0.666199	−	0.745774i	$-0.267920\pi$
$308$	0	0
$309$	−3.60746e12	−1.28059
$310$	0	0
$311$	4.24403e12i	1.45873i	0.684123	+	0.729367i	$0.260185\pi$
$311$	4.24403e12i	1.45873i	−0.684123	+	0.729367i	$0.739815\pi$
$312$	0	0
$313$	2.08562e12	0.694246	0.347123	−	0.937820i	$-0.387159\pi$
$313$	2.08562e12	0.694246	0.347123	+	0.937820i	$0.387159\pi$
$314$	0	0
$315$	− 1.67144e13i	− 5.38936i
$316$	0	0
$317$	5.76448e12	1.80079	0.900396	−	0.435072i	$-0.143277\pi$
$317$	5.76448e12	1.80079	0.900396	+	0.435072i	$0.143277\pi$
$318$	0	0
$319$	1.29637e12i	0.392442i
$320$	0	0
$321$	5.91092e12	1.73432
$322$	0	0
$323$	1.01357e12i	0.288299i
$324$	0	0
$325$	−6.69115e12	−1.84537
$326$	0	0
$327$	2.29123e12i	0.612815i
$328$	0	0
$329$	−1.17055e13	−3.03676
$330$	0	0
$331$	2.77682e12i	0.698889i	0.936957	+	0.349445i	$0.113630\pi$
$331$	2.77682e12i	0.698889i	−0.936957	+	0.349445i	$0.886370\pi$
$332$	0	0
$333$	−4.86285e12	−1.18760
$334$	0	0
$335$	2.44623e12i	0.579795i
$336$	0	0
$337$	−4.18090e11	−0.0961880	−0.0480940	−	0.998843i	$-0.515315\pi$
$337$	−4.18090e11	−0.0961880	−0.0480940	+	0.998843i	$0.515315\pi$
$338$	0	0
$339$	7.72297e11i	0.172499i
$340$	0	0
$341$	1.62617e11	0.0352691
$342$	0	0
$343$	3.32976e12i	0.701363i
$344$	0	0
$345$	−1.00478e13	−2.05577
$346$	0	0
$347$	1.12418e12i	0.223454i	0.993739	+	0.111727i	$0.0356383\pi$
$347$	1.12418e12i	0.223454i	−0.993739	+	0.111727i	$0.964362\pi$
$348$	0	0
$349$	−6.05546e12	−1.16955	−0.584777	−	0.811194i	$-0.698818\pi$
$349$	−6.05546e12	−1.16955	−0.584777	+	0.811194i	$0.698818\pi$
$350$	0	0
$351$	7.99606e12i	1.50086i
$352$	0	0
$353$	−1.01400e13	−1.84997	−0.924987	−	0.380000i	$-0.875924\pi$
$353$	−1.01400e13	−1.84997	−0.924987	+	0.380000i	$0.875924\pi$
$354$	0	0
$355$	− 3.81156e12i	− 0.676023i
$356$	0	0
$357$	1.92638e13	3.32201
$358$	0	0
$359$	7.27823e12i	1.22054i	0.792192	+	0.610272i	$0.208940\pi$
$359$	7.27823e12i	1.22054i	−0.792192	+	0.610272i	$0.791060\pi$
$360$	0	0
$361$	5.79527e12	0.945231
$362$	0	0
$363$	5.75603e12i	0.913250i
$364$	0	0
$365$	9.07318e12	1.40054
$366$	0	0
$367$	− 9.72622e12i	− 1.46088i	−0.682979	−	0.730438i	$-0.739316\pi$
$367$	− 9.72622e12i	− 1.46088i	0.682979	−	0.730438i	$-0.260684\pi$
$368$	0	0
$369$	1.44024e13	2.10525
$370$	0	0
$371$	− 1.12321e13i	− 1.59806i
$372$	0	0
$373$	4.97590e12	0.689172	0.344586	−	0.938755i	$-0.388019\pi$
$373$	4.97590e12	0.689172	0.344586	+	0.938755i	$0.388019\pi$
$374$	0	0
$375$	− 2.35956e13i	− 3.18182i
$376$	0	0
$377$	3.91228e12	0.513717
$378$	0	0
$379$	1.03988e13i	1.32980i	0.746933	+	0.664899i	$0.231526\pi$
$379$	1.03988e13i	1.32980i	−0.746933	+	0.664899i	$0.768474\pi$
$380$	0	0
$381$	1.31831e13	1.64208
$382$	0	0
$383$	− 7.33165e12i	− 0.889626i	−0.895623	−	0.444813i	$-0.853270\pi$
$383$	− 7.33165e12i	− 0.889626i	0.895623	−	0.444813i	$-0.146730\pi$
$384$	0	0
$385$	1.58639e13	1.87545
$386$	0	0
$387$	1.81819e13i	2.09451i
$388$	0	0
$389$	9.44295e11	0.106013	0.0530066	−	0.998594i	$-0.483120\pi$
$389$	9.44295e11	0.106013	0.0530066	+	0.998594i	$0.483120\pi$
$390$	0	0
$391$	− 7.68163e12i	− 0.840561i
$392$	0	0
$393$	1.07521e13	1.14691
$394$	0	0
$395$	2.96831e13i	3.08691i
$396$	0	0
$397$	−1.50155e13	−1.52261	−0.761304	−	0.648396i	$-0.775440\pi$
$397$	−1.50155e13	−1.52261	−0.761304	+	0.648396i	$0.775440\pi$
$398$	0	0
$399$	6.38203e12i	0.631095i
$400$	0	0
$401$	1.49213e13	1.43908	0.719540	−	0.694451i	$-0.244353\pi$
$401$	1.49213e13	1.43908	0.719540	+	0.694451i	$0.244353\pi$
$402$	0	0
$403$	− 4.90758e11i	− 0.0461681i
$404$	0	0
$405$	−1.73668e13	−1.59384
$406$	0	0
$407$	− 4.61541e12i	− 0.413274i
$408$	0	0
$409$	−1.08558e13	−0.948517	−0.474259	−	0.880386i	$-0.657284\pi$
$409$	−1.08558e13	−0.948517	−0.474259	+	0.880386i	$0.657284\pi$
$410$	0	0
$411$	− 2.01598e13i	− 1.71900i
$412$	0	0
$413$	2.48212e13	2.06573
$414$	0	0
$415$	− 2.27275e13i	− 1.84633i
$416$	0	0
$417$	−7.45913e12	−0.591572
$418$	0	0
$419$	3.67981e12i	0.284941i	0.989799	+	0.142470i	$0.0455046\pi$
$419$	3.67981e12i	0.284941i	−0.989799	+	0.142470i	$0.954495\pi$
$420$	0	0
$421$	−1.30822e13	−0.989171	−0.494585	−	0.869129i	$-0.664680\pi$
$421$	−1.30822e13	−0.989171	−0.494585	+	0.869129i	$0.664680\pi$
$422$	0	0
$423$	5.17860e13i	3.82393i
$424$	0	0
$425$	3.51203e13	2.53287
$426$	0	0
$427$	1.01854e13i	0.717527i
$428$	0	0
$429$	−1.54109e13	−1.06057
$430$	0	0
$431$	2.18557e13i	1.46953i	0.678321	+	0.734766i	$0.262708\pi$
$431$	2.18557e13i	1.46953i	−0.678321	+	0.734766i	$0.737292\pi$
$432$	0	0
$433$	−1.28808e13	−0.846258	−0.423129	−	0.906070i	$-0.639068\pi$
$433$	−1.28808e13	−0.846258	−0.423129	+	0.906070i	$0.639068\pi$
$434$	0	0
$435$	2.68600e13i	1.72448i
$436$	0	0
$437$	2.54490e12	0.159685
$438$	0	0
$439$	7.00117e12i	0.429386i	0.976682	+	0.214693i	$0.0688751\pi$
$439$	7.00117e12i	0.429386i	−0.976682	+	0.214693i	$0.931125\pi$
$440$	0	0
$441$	4.75952e13	2.85345
$442$	0	0
$443$	− 6.04112e12i	− 0.354078i	−0.984204	−	0.177039i	$-0.943348\pi$
$443$	− 6.04112e12i	− 0.354078i	0.984204	−	0.177039i	$-0.0566519\pi$
$444$	0	0
$445$	−1.37012e13	−0.785158
$446$	0	0
$447$	3.48104e13i	1.95061i
$448$	0	0
$449$	−2.98910e12	−0.163798	−0.0818991	−	0.996641i	$-0.526099\pi$
$449$	−2.98910e12	−0.163798	−0.0818991	+	0.996641i	$0.526099\pi$
$450$	0	0
$451$	1.36696e13i	0.732611i
$452$	0	0
$453$	2.28479e13	1.19772
$454$	0	0
$455$	− 4.78753e13i	− 2.45502i
$456$	0	0
$457$	−9.55205e12	−0.479198	−0.239599	−	0.970872i	$-0.577016\pi$
$457$	−9.55205e12	−0.479198	−0.239599	+	0.970872i	$0.577016\pi$
$458$	0	0
$459$	− 4.19694e13i	− 2.06001i
$460$	0	0
$461$	−1.18162e12	−0.0567510	−0.0283755	−	0.999597i	$-0.509033\pi$
$461$	−1.18162e12	−0.0567510	−0.0283755	+	0.999597i	$0.509033\pi$
$462$	0	0
$463$	1.54135e12i	0.0724429i	0.999344	+	0.0362214i	$0.0115322\pi$
$463$	1.54135e12i	0.0724429i	−0.999344	+	0.0362214i	$0.988468\pi$
$464$	0	0
$465$	3.36932e12	0.154981
$466$	0	0
$467$	2.73781e13i	1.23259i	0.787515	+	0.616296i	$0.211368\pi$
$467$	2.73781e13i	1.23259i	−0.787515	+	0.616296i	$0.788632\pi$
$468$	0	0
$469$	−1.17756e13	−0.518943
$470$	0	0
$471$	2.53294e12i	0.109275i
$472$	0	0
$473$	−1.72567e13	−0.728874
$474$	0	0
$475$	1.16352e13i	0.481180i
$476$	0	0
$477$	−4.96918e13	−2.01230
$478$	0	0
$479$	− 1.91529e13i	− 0.759551i	−0.925079	−	0.379776i	$-0.876001\pi$
$479$	− 1.91529e13i	− 0.759551i	0.925079	−	0.379776i	$-0.123999\pi$
$480$	0	0
$481$	−1.39287e13	−0.540987
$482$	0	0
$483$	− 4.83678e13i	− 1.84001i
$484$	0	0
$485$	−4.35831e13	−1.62409
$486$	0	0
$487$	4.50643e12i	0.164508i	0.996611	+	0.0822542i	$0.0262120\pi$
$487$	4.50643e12i	0.164508i	−0.996611	+	0.0822542i	$0.973788\pi$
$488$	0	0
$489$	7.12417e13	2.54794
$490$	0	0
$491$	1.13324e12i	0.0397115i	0.999803	+	0.0198557i	$0.00632069\pi$
$491$	1.13324e12i	0.0397115i	−0.999803	+	0.0198557i	$0.993679\pi$
$492$	0	0
$493$	−2.05347e13	−0.705104
$494$	0	0
$495$	− 7.01831e13i	− 2.36160i
$496$	0	0
$497$	1.83480e13	0.605072
$498$	0	0
$499$	− 4.68466e13i	− 1.51417i	−0.653316	−	0.757085i	$-0.726623\pi$
$499$	− 4.68466e13i	− 1.51417i	0.653316	−	0.757085i	$-0.273377\pi$
$500$	0	0
$501$	5.63023e13	1.78376
$502$	0	0
$503$	2.81899e13i	0.875494i	0.899098	+	0.437747i	$0.144223\pi$
$503$	2.81899e13i	0.875494i	−0.899098	+	0.437747i	$0.855777\pi$
$504$	0	0
$505$	6.65490e13	2.02621
$506$	0	0
$507$	− 1.12268e13i	− 0.335132i
$508$	0	0
$509$	1.81784e13	0.532067	0.266034	−	0.963964i	$-0.414287\pi$
$509$	1.81784e13	0.532067	0.266034	+	0.963964i	$0.414287\pi$
$510$	0	0
$511$	4.36762e13i	1.25355i
$512$	0	0
$513$	1.39043e13	0.391348
$514$	0	0
$515$	− 4.70575e13i	− 1.29895i
$516$	0	0
$517$	−4.91510e13	−1.33070
$518$	0	0
$519$	− 7.17547e13i	− 1.90552i
$520$	0	0
$521$	2.74447e12	0.0714941	0.0357471	−	0.999361i	$-0.488619\pi$
$521$	2.74447e12	0.0714941	0.0357471	+	0.999361i	$0.488619\pi$
$522$	0	0
$523$	− 1.78045e13i	− 0.455011i	−0.973777	−	0.227506i	$-0.926943\pi$
$523$	− 1.78045e13i	− 0.455011i	0.973777	−	0.227506i	$-0.0730570\pi$
$524$	0	0
$525$	2.21137e14	5.54453
$526$	0	0
$527$	2.57587e12i	0.0633682i
$528$	0	0
$529$	2.21394e13	0.534425
$530$	0	0
$531$	− 1.09811e14i	− 2.60119i
$532$	0	0
$533$	4.12532e13	0.959007
$534$	0	0
$535$	7.71049e13i	1.75919i
$536$	0	0
$537$	8.01240e13	1.79428
$538$	0	0
$539$	4.51734e13i	0.992976i
$540$	0	0
$541$	−6.16455e13	−1.33020	−0.665098	−	0.746756i	$-0.731610\pi$
$541$	−6.16455e13	−1.33020	−0.665098	+	0.746756i	$0.731610\pi$
$542$	0	0
$543$	− 1.64831e13i	− 0.349172i
$544$	0	0
$545$	−2.98879e13	−0.621603
$546$	0	0
$547$	3.59569e13i	0.734253i	0.930171	+	0.367126i	$0.119658\pi$
$547$	3.59569e13i	0.734253i	−0.930171	+	0.367126i	$0.880342\pi$
$548$	0	0
$549$	4.50609e13	0.903521
$550$	0	0
$551$	− 6.80307e12i	− 0.133951i
$552$	0	0
$553$	−1.42888e14	−2.76293
$554$	0	0
$555$	− 9.56284e13i	− 1.81603i
$556$	0	0
$557$	−5.76776e13	−1.07580	−0.537900	−	0.843009i	$-0.680782\pi$
$557$	−5.76776e13	−1.07580	−0.537900	+	0.843009i	$0.680782\pi$
$558$	0	0
$559$	5.20787e13i	0.954115i
$560$	0	0
$561$	8.08880e13	1.45569
$562$	0	0
$563$	6.07106e13i	1.07330i	0.843804	+	0.536652i	$0.180311\pi$
$563$	6.07106e13i	1.07330i	−0.843804	+	0.536652i	$0.819689\pi$
$564$	0	0
$565$	−1.00742e13	−0.174972
$566$	0	0
$567$	− 8.35996e13i	− 1.42656i
$568$	0	0
$569$	−7.65838e12	−0.128403	−0.0642016	−	0.997937i	$-0.520450\pi$
$569$	−7.65838e12	−0.128403	−0.0642016	+	0.997937i	$0.520450\pi$
$570$	0	0
$571$	1.12203e14i	1.84852i	0.381762	+	0.924261i	$0.375317\pi$
$571$	1.12203e14i	1.84852i	−0.381762	+	0.924261i	$0.624683\pi$
$572$	0	0
$573$	−3.01516e13	−0.488131
$574$	0	0
$575$	− 8.81806e13i	− 1.40292i
$576$	0	0
$577$	5.13832e13	0.803418	0.401709	−	0.915767i	$-0.368416\pi$
$577$	5.13832e13	0.803418	0.401709	+	0.915767i	$0.368416\pi$
$578$	0	0
$579$	− 8.51041e13i	− 1.30785i
$580$	0	0
$581$	1.09405e14	1.65256
$582$	0	0
$583$	− 4.71633e13i	− 0.700264i
$584$	0	0
$585$	−2.11804e14	−3.09140
$586$	0	0
$587$	− 1.29068e14i	− 1.85194i	−0.377595	−	0.925971i	$-0.623249\pi$
$587$	− 1.29068e14i	− 1.85194i	0.377595	−	0.925971i	$-0.376751\pi$
$588$	0	0
$589$	−8.53379e11	−0.0120383
$590$	0	0
$591$	1.35926e14i	1.88523i
$592$	0	0
$593$	−9.04040e13	−1.23286	−0.616430	−	0.787409i	$-0.711422\pi$
$593$	−9.04040e13	−1.23286	−0.616430	+	0.787409i	$0.711422\pi$
$594$	0	0
$595$	2.51286e14i	3.36964i
$596$	0	0
$597$	−1.13123e14	−1.49169
$598$	0	0
$599$	8.31908e13i	1.07880i	0.842050	+	0.539400i	$0.181349\pi$
$599$	8.31908e13i	1.07880i	−0.842050	+	0.539400i	$0.818651\pi$
$600$	0	0
$601$	−1.06845e13	−0.136265	−0.0681324	−	0.997676i	$-0.521704\pi$
$601$	−1.06845e13	−0.136265	−0.0681324	+	0.997676i	$0.521704\pi$
$602$	0	0
$603$	5.20962e13i	0.653461i
$604$	0	0
$605$	−7.50844e13	−0.926346
$606$	0	0
$607$	1.31564e13i	0.159660i	0.996809	+	0.0798298i	$0.0254377\pi$
$607$	1.31564e13i	0.159660i	−0.996809	+	0.0798298i	$0.974562\pi$
$608$	0	0
$609$	−1.29298e14	−1.54349
$610$	0	0
$611$	1.48332e14i	1.74192i
$612$	0	0
$613$	−4.88203e13	−0.564025	−0.282013	−	0.959411i	$-0.591002\pi$
$613$	−4.88203e13	−0.564025	−0.282013	+	0.959411i	$0.591002\pi$
$614$	0	0
$615$	2.83226e14i	3.21927i
$616$	0	0
$617$	5.86214e13	0.655587	0.327794	−	0.944749i	$-0.393695\pi$
$617$	5.86214e13	0.655587	0.327794	+	0.944749i	$0.393695\pi$
$618$	0	0
$619$	− 2.91307e13i	− 0.320552i	−0.987072	−	0.160276i	$-0.948762\pi$
$619$	− 2.91307e13i	− 0.320552i	0.987072	−	0.160276i	$-0.0512384\pi$
$620$	0	0
$621$	−1.05378e14	−1.14101
$622$	0	0
$623$	− 6.59542e13i	− 0.702753i
$624$	0	0
$625$	1.11710e14	1.17137
$626$	0	0
$627$	2.67979e13i	0.276544i
$628$	0	0
$629$	7.31087e13	0.742534
$630$	0	0
$631$	− 1.26025e13i	− 0.125983i	−0.998014	−	0.0629913i	$-0.979936\pi$
$631$	− 1.26025e13i	− 0.125983i	0.998014	−	0.0629913i	$-0.0200640\pi$
$632$	0	0
$633$	−3.06563e14	−3.01649
$634$	0	0
$635$	1.71967e14i	1.66562i
$636$	0	0
$637$	1.36328e14	1.29983
$638$	0	0
$639$	− 8.11730e13i	− 0.761916i
$640$	0	0
$641$	1.27909e14	1.18198	0.590990	−	0.806679i	$-0.298737\pi$
$641$	1.27909e14	1.18198	0.590990	+	0.806679i	$0.298737\pi$
$642$	0	0
$643$	6.94385e13i	0.631751i	0.948801	+	0.315875i	$0.102298\pi$
$643$	6.94385e13i	0.631751i	−0.948801	+	0.315875i	$0.897702\pi$
$644$	0	0
$645$	−3.57548e14	−3.20285
$646$	0	0
$647$	− 1.86746e14i	− 1.64714i	−0.567216	−	0.823569i	$-0.691979\pi$
$647$	− 1.86746e14i	− 1.64714i	0.567216	−	0.823569i	$-0.308021\pi$
$648$	0	0
$649$	1.04223e14	0.905194
$650$	0	0
$651$	1.62192e13i	0.138715i
$652$	0	0
$653$	−5.62934e12	−0.0474124	−0.0237062	−	0.999719i	$-0.507547\pi$
$653$	−5.62934e12	−0.0474124	−0.0237062	+	0.999719i	$0.507547\pi$
$654$	0	0
$655$	1.40255e14i	1.16335i
$656$	0	0
$657$	1.93227e14	1.57849
$658$	0	0
$659$	− 8.19186e12i	− 0.0659106i	−0.999457	−	0.0329553i	$-0.989508\pi$
$659$	− 8.19186e12i	− 0.0659106i	0.999457	−	0.0329553i	$-0.0104919\pi$
$660$	0	0
$661$	−1.39893e14	−1.10864	−0.554319	−	0.832304i	$-0.687021\pi$
$661$	−1.39893e14	−1.10864	−0.554319	+	0.832304i	$0.687021\pi$
$662$	0	0
$663$	− 2.44110e14i	− 1.90554i
$664$	0	0
$665$	−8.32503e13	−0.640144
$666$	0	0
$667$	5.15588e13i	0.390547i
$668$	0	0
$669$	−1.29588e14	−0.967014
$670$	0	0
$671$	4.27681e13i	0.314418i
$672$	0	0
$673$	−8.92018e13	−0.646098	−0.323049	−	0.946382i	$-0.604708\pi$
$673$	−8.92018e13	−0.646098	−0.323049	+	0.946382i	$0.604708\pi$
$674$	0	0
$675$	− 4.81785e14i	− 3.43822i
$676$	0	0
$677$	2.33670e14	1.64308	0.821541	−	0.570150i	$-0.193115\pi$
$677$	2.33670e14	1.64308	0.821541	+	0.570150i	$0.193115\pi$
$678$	0	0
$679$	− 2.09799e14i	− 1.45363i
$680$	0	0
$681$	−1.26683e14	−0.864931
$682$	0	0
$683$	1.68249e14i	1.13201i	0.824403	+	0.566003i	$0.191511\pi$
$683$	1.68249e14i	1.13201i	−0.824403	+	0.566003i	$0.808489\pi$
$684$	0	0
$685$	2.62974e14	1.74365
$686$	0	0
$687$	− 4.47654e14i	− 2.92522i
$688$	0	0
$689$	−1.42333e14	−0.916664
$690$	0	0
$691$	1.30725e14i	0.829788i	0.909870	+	0.414894i	$0.136181\pi$
$691$	1.30725e14i	0.829788i	−0.909870	+	0.414894i	$0.863819\pi$
$692$	0	0
$693$	3.37845e14	2.11374
$694$	0	0
$695$	− 9.73006e13i	− 0.600055i
$696$	0	0
$697$	−2.16528e14	−1.31629
$698$	0	0
$699$	3.07964e14i	1.84550i
$700$	0	0
$701$	1.70089e14	1.00482	0.502408	−	0.864631i	$-0.332448\pi$
$701$	1.70089e14	1.00482	0.502408	+	0.864631i	$0.332448\pi$
$702$	0	0
$703$	2.42207e13i	0.141062i
$704$	0	0
$705$	−1.01838e15	−5.84741
$706$	0	0
$707$	3.20352e14i	1.81355i
$708$	0	0
$709$	2.77876e14	1.55103	0.775514	−	0.631330i	$-0.217491\pi$
$709$	2.77876e14	1.55103	0.775514	+	0.631330i	$0.217491\pi$
$710$	0	0
$711$	6.32146e14i	3.47912i
$712$	0	0
$713$	6.46755e12	0.0350988
$714$	0	0
$715$	− 2.01027e14i	− 1.07578i
$716$	0	0
$717$	−3.05340e14	−1.61134
$718$	0	0
$719$	− 2.87814e13i	− 0.149785i	−0.997192	−	0.0748923i	$-0.976139\pi$
$719$	− 2.87814e13i	− 0.149785i	0.997192	−	0.0748923i	$-0.0238613\pi$
$720$	0	0
$721$	2.26524e14	1.16262
$722$	0	0
$723$	− 5.15875e14i	− 2.61128i
$724$	0	0
$725$	−2.35726e14	−1.17684
$726$	0	0
$727$	2.74931e14i	1.35379i	0.736079	+	0.676896i	$0.236675\pi$
$727$	2.74931e14i	1.35379i	−0.736079	+	0.676896i	$0.763325\pi$
$728$	0	0
$729$	2.23527e14	1.08566
$730$	0	0
$731$	− 2.73349e14i	− 1.30957i
$732$	0	0
$733$	−7.09317e13	−0.335213	−0.167606	−	0.985854i	$-0.553604\pi$
$733$	−7.09317e13	−0.335213	−0.167606	+	0.985854i	$0.553604\pi$
$734$	0	0
$735$	9.35964e14i	4.36338i
$736$	0	0
$737$	−4.94454e13	−0.227399
$738$	0	0
$739$	− 2.82339e14i	− 1.28100i	−0.767959	−	0.640499i	$-0.778728\pi$
$739$	− 2.82339e14i	− 1.28100i	0.767959	−	0.640499i	$-0.221272\pi$
$740$	0	0
$741$	8.08729e13	0.362003
$742$	0	0
$743$	− 5.16921e13i	− 0.228286i	−0.993464	−	0.114143i	$-0.963588\pi$
$743$	− 5.16921e13i	− 0.228286i	0.993464	−	0.114143i	$-0.0364123\pi$
$744$	0	0
$745$	−4.54084e14	−1.97859
$746$	0	0
$747$	− 4.84015e14i	− 2.08092i
$748$	0	0
$749$	−3.71166e14	−1.57456
$750$	0	0
$751$	− 2.72827e14i	− 1.14206i	−0.820930	−	0.571029i	$-0.806544\pi$
$751$	− 2.72827e14i	− 1.14206i	0.820930	−	0.571029i	$-0.193456\pi$
$752$	0	0
$753$	−1.23668e14	−0.510837
$754$	0	0
$755$	2.98039e14i	1.21490i
$756$	0	0
$757$	3.07665e14	1.23765	0.618826	−	0.785528i	$-0.287609\pi$
$757$	3.07665e14	1.23765	0.618826	+	0.785528i	$0.287609\pi$
$758$	0	0
$759$	− 2.03095e14i	− 0.806288i
$760$	0	0
$761$	4.11540e14	1.61246	0.806229	−	0.591603i	$-0.201505\pi$
$761$	4.11540e14	1.61246	0.806229	+	0.591603i	$0.201505\pi$
$762$	0	0
$763$	− 1.43874e14i	− 0.556363i
$764$	0	0
$765$	1.11171e15	4.24311
$766$	0	0
$767$	− 3.14534e14i	− 1.18492i
$768$	0	0
$769$	−1.10107e14	−0.409434	−0.204717	−	0.978821i	$-0.565627\pi$
$769$	−1.10107e14	−0.409434	−0.204717	+	0.978821i	$0.565627\pi$
$770$	0	0
$771$	− 2.09760e14i	− 0.769930i
$772$	0	0
$773$	−4.24016e14	−1.53633	−0.768165	−	0.640252i	$-0.778830\pi$
$773$	−4.24016e14	−1.53633	−0.768165	+	0.640252i	$0.778830\pi$
$774$	0	0
$775$	2.95695e13i	0.105764i
$776$	0	0
$777$	4.60334e14	1.62543
$778$	0	0
$779$	− 7.17352e13i	− 0.250060i
$780$	0	0
$781$	7.70427e13	0.265141
$782$	0	0
$783$	2.81697e14i	0.957136i
$784$	0	0
$785$	−3.30409e13	−0.110842
$786$	0	0
$787$	4.59104e14i	1.52068i	0.649527	+	0.760339i	$0.274967\pi$
$787$	4.59104e14i	1.52068i	−0.649527	+	0.760339i	$0.725033\pi$
$788$	0	0
$789$	−5.92490e14	−1.93774
$790$	0	0
$791$	− 4.84950e13i	− 0.156608i
$792$	0	0
$793$	1.29069e14	0.411581
$794$	0	0
$795$	− 9.77195e14i	− 3.07713i
$796$	0	0
$797$	2.46075e13	0.0765201	0.0382601	−	0.999268i	$-0.487818\pi$
$797$	2.46075e13	0.0765201	0.0382601	+	0.999268i	$0.487818\pi$
$798$	0	0
$799$	− 7.78558e14i	− 2.39088i
$800$	0	0
$801$	−2.91787e14	−0.884917
$802$	0	0
$803$	1.83395e14i	0.549301i
$804$	0	0
$805$	6.30934e14	1.86640
$806$	0	0
$807$	− 8.09376e14i	− 2.36474i
$808$	0	0
$809$	−4.69463e13	−0.135475	−0.0677375	−	0.997703i	$-0.521578\pi$
$809$	−4.69463e13	−0.135475	−0.0677375	+	0.997703i	$0.521578\pi$
$810$	0	0
$811$	3.49009e14i	0.994792i	0.867524	+	0.497396i	$0.165710\pi$
$811$	3.49009e14i	0.994792i	−0.867524	+	0.497396i	$0.834290\pi$
$812$	0	0
$813$	−9.00094e14	−2.53417
$814$	0	0
$815$	9.29312e14i	2.58448i
$816$	0	0
$817$	9.05596e13	0.248785
$818$	0	0
$819$	− 1.01958e15i	− 2.76694i
$820$	0	0
$821$	−6.89603e14	−1.84877	−0.924387	−	0.381457i	$-0.875422\pi$
$821$	−6.89603e14	−1.84877	−0.924387	+	0.381457i	$0.875422\pi$
$822$	0	0
$823$	5.64368e13i	0.149473i	0.997203	+	0.0747366i	$0.0238116\pi$
$823$	5.64368e13i	0.149473i	−0.997203	+	0.0747366i	$0.976188\pi$
$824$	0	0
$825$	9.28547e14	2.42960
$826$	0	0
$827$	− 1.88514e13i	− 0.0487323i	−0.999703	−	0.0243662i	$-0.992243\pi$
$827$	− 1.88514e13i	− 0.0487323i	0.999703	−	0.0243662i	$-0.00775676\pi$
$828$	0	0
$829$	−3.49152e14	−0.891748	−0.445874	−	0.895096i	$-0.647107\pi$
$829$	−3.49152e14	−0.891748	−0.445874	+	0.895096i	$0.647107\pi$
$830$	0	0
$831$	− 5.08553e14i	− 1.28331i
$832$	0	0
$833$	−7.15553e14	−1.78409
$834$	0	0
$835$	7.34435e14i	1.80934i
$836$	0	0
$837$	3.53362e13	0.0860186
$838$	0	0
$839$	6.06902e14i	1.45985i	0.683526	+	0.729926i	$0.260446\pi$
$839$	6.06902e14i	1.45985i	−0.683526	+	0.729926i	$0.739554\pi$
$840$	0	0
$841$	−2.82879e14	−0.672390
$842$	0	0
$843$	7.47900e14i	1.75673i
$844$	0	0
$845$	1.46448e14	0.339938
$846$	0	0
$847$	− 3.61440e14i	− 0.829122i
$848$	0	0
$849$	8.56664e14	1.94210
$850$	0	0
$851$	− 1.83563e14i	− 0.411279i
$852$	0	0
$853$	−2.94475e14	−0.652084	−0.326042	−	0.945355i	$-0.605715\pi$
$853$	−2.94475e14	−0.652084	−0.326042	+	0.945355i	$0.605715\pi$
$854$	0	0
$855$	3.68306e14i	0.806080i
$856$	0	0
$857$	2.94947e14	0.638028	0.319014	−	0.947750i	$-0.396648\pi$
$857$	2.94947e14	0.638028	0.319014	+	0.947750i	$0.396648\pi$
$858$	0	0
$859$	8.55967e14i	1.83017i	0.403261	+	0.915085i	$0.367877\pi$
$859$	8.55967e14i	1.83017i	−0.403261	+	0.915085i	$0.632123\pi$
$860$	0	0
$861$	−1.36338e15	−2.88139
$862$	0	0
$863$	− 2.16575e14i	− 0.452432i	−0.974077	−	0.226216i	$-0.927364\pi$
$863$	− 2.16575e14i	− 0.452432i	0.974077	−	0.226216i	$-0.0726355\pi$
$864$	0	0
$865$	9.36004e14	1.93284
$866$	0	0
$867$	4.36982e14i	0.892008i
$868$	0	0
$869$	−5.99981e14	−1.21071
$870$	0	0
$871$	1.49220e14i	0.297671i
$872$	0	0
$873$	−9.28167e14	−1.83044
$874$	0	0
$875$	1.48165e15i	2.88871i
$876$	0	0
$877$	6.44007e14	1.24134	0.620672	−	0.784070i	$-0.286860\pi$
$877$	6.44007e14	1.24134	0.620672	+	0.784070i	$0.286860\pi$
$878$	0	0
$879$	− 1.70134e14i	− 0.324225i
$880$	0	0
$881$	2.61867e14	0.493402	0.246701	−	0.969092i	$-0.420653\pi$
$881$	2.61867e14	0.493402	0.246701	+	0.969092i	$0.420653\pi$
$882$	0	0
$883$	− 5.55178e14i	− 1.03426i	−0.855908	−	0.517128i	$-0.827001\pi$
$883$	− 5.55178e14i	− 1.03426i	0.855908	−	0.517128i	$-0.172999\pi$
$884$	0	0
$885$	2.15944e15	3.97764
$886$	0	0
$887$	1.50444e14i	0.274004i	0.990571	+	0.137002i	$0.0437466\pi$
$887$	1.50444e14i	0.274004i	−0.990571	+	0.137002i	$0.956253\pi$
$888$	0	0
$889$	−8.27810e14	−1.49081
$890$	0	0
$891$	− 3.51032e14i	− 0.625113i
$892$	0	0
$893$	2.57934e14	0.454204
$894$	0	0
$895$	1.04518e15i	1.82001i
$896$	0	0
$897$	−6.12916e14	−1.05545
$898$	0	0
$899$	− 1.72892e13i	− 0.0294426i
$900$	0	0
$901$	7.47074e14	1.25817
$902$	0	0
$903$	− 1.72116e15i	− 2.86670i
$904$	0	0
$905$	2.15013e14	0.354179
$906$	0	0
$907$	− 1.74391e14i	− 0.284110i	−0.989859	−	0.142055i	$-0.954629\pi$
$907$	− 1.74391e14i	− 0.284110i	0.989859	−	0.142055i	$-0.0453710\pi$
$908$	0	0
$909$	1.41726e15	2.28366
$910$	0	0
$911$	6.63100e14i	1.05679i	0.849000	+	0.528393i	$0.177205\pi$
$911$	6.63100e14i	1.05679i	−0.849000	+	0.528393i	$0.822795\pi$
$912$	0	0
$913$	4.59387e14	0.724144
$914$	0	0
$915$	8.86129e14i	1.38163i
$916$	0	0
$917$	−6.75156e14	−1.04126
$918$	0	0
$919$	− 6.20205e14i	− 0.946145i	−0.881024	−	0.473072i	$-0.843145\pi$
$919$	− 6.20205e14i	− 0.946145i	0.881024	−	0.473072i	$-0.156855\pi$
$920$	0	0
$921$	1.70347e15	2.57061
$922$	0	0
$923$	− 2.32505e14i	− 0.347076i
$924$	0	0
$925$	8.39246e14	1.23931
$926$	0	0
$927$	− 1.00216e15i	− 1.46399i
$928$	0	0
$929$	2.38065e14	0.344046	0.172023	−	0.985093i	$-0.444970\pi$
$929$	2.38065e14	0.344046	0.172023	+	0.985093i	$0.444970\pi$
$930$	0	0
$931$	− 2.37060e14i	− 0.338930i
$932$	0	0
$933$	−1.77739e15	−2.51405
$934$	0	0
$935$	1.05514e15i	1.47657i
$936$	0	0
$937$	−1.82267e14	−0.252354	−0.126177	−	0.992008i	$-0.540271\pi$
$937$	−1.82267e14	−0.252354	−0.126177	+	0.992008i	$0.540271\pi$
$938$	0	0
$939$	8.73454e14i	1.19650i
$940$	0	0
$941$	2.90715e14	0.394021	0.197010	−	0.980401i	$-0.436877\pi$
$941$	2.90715e14	0.394021	0.197010	+	0.980401i	$0.436877\pi$
$942$	0	0
$943$	5.43663e14i	0.729074i
$944$	0	0
$945$	3.44718e15	4.57409
$946$	0	0
$947$	8.87325e14i	1.16502i	0.812824	+	0.582509i	$0.197929\pi$
$947$	8.87325e14i	1.16502i	−0.812824	+	0.582509i	$0.802071\pi$
$948$	0	0
$949$	5.53464e14	0.719049
$950$	0	0
$951$	2.41415e15i	3.10357i
$952$	0	0
$953$	8.84641e14	1.12539	0.562694	−	0.826665i	$-0.309765\pi$
$953$	8.84641e14	1.12539	0.562694	+	0.826665i	$0.309765\pi$
$954$	0	0
$955$	− 3.93312e14i	− 0.495131i
$956$	0	0
$957$	−5.42917e14	−0.676354
$958$	0	0
$959$	1.26590e15i	1.56065i
$960$	0	0
$961$	8.17460e14	0.997354
$962$	0	0
$963$	1.64207e15i	1.98271i
$964$	0	0
$965$	1.11014e15	1.32660
$966$	0	0
$967$	− 1.84782e14i	− 0.218538i	−0.994012	−	0.109269i	$-0.965149\pi$
$967$	− 1.84782e14i	− 0.218538i	0.994012	−	0.109269i	$-0.0348510\pi$
$968$	0	0
$969$	−4.24483e14	−0.496869
$970$	0	0
$971$	1.91821e14i	0.222229i	0.993808	+	0.111115i	$0.0354421\pi$
$971$	1.91821e14i	0.222229i	−0.993808	+	0.111115i	$0.964558\pi$
$972$	0	0
$973$	4.68383e14	0.537077
$974$	0	0
$975$	− 2.80224e15i	− 3.18041i
$976$	0	0
$977$	1.56955e15	1.76320	0.881601	−	0.471995i	$-0.156466\pi$
$977$	1.56955e15	1.76320	0.881601	+	0.471995i	$0.156466\pi$
$978$	0	0
$979$	− 2.76940e14i	− 0.307944i
$980$	0	0
$981$	−6.36508e14	−0.700582
$982$	0	0
$983$	2.01567e14i	0.219610i	0.993953	+	0.109805i	$0.0350227\pi$
$983$	2.01567e14i	0.219610i	−0.993953	+	0.109805i	$0.964977\pi$
$984$	0	0
$985$	−1.77308e15	−1.91226
$986$	0	0
$987$	− 4.90224e15i	− 5.23370i
$988$	0	0
$989$	−6.86329e14	−0.725355
$990$	0	0
$991$	1.45574e15i	1.52306i	0.648132	+	0.761528i	$0.275550\pi$
$991$	1.45574e15i	1.52306i	−0.648132	+	0.761528i	$0.724450\pi$
$992$	0	0
$993$	−1.16293e15	−1.20450
$994$	0	0
$995$	− 1.47563e15i	− 1.51308i
$996$	0	0
$997$	2.19831e14	0.223158	0.111579	−	0.993756i	$-0.464409\pi$
$997$	2.19831e14	0.223158	0.111579	+	0.993756i	$0.464409\pi$
$998$	0	0
$999$	− 1.00292e15i	− 1.00794i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	256.11.c.f.255.3		4
4.3	odd	2	inner	256.11.c.f.255.1		4
8.3	odd	2	inner	256.11.c.f.255.4		4
8.5	even	2	inner	256.11.c.f.255.2		4
16.3	odd	4		128.11.d.d.63.4	yes	4
16.5	even	4		128.11.d.d.63.3	yes	4
16.11	odd	4		128.11.d.d.63.1	✓	4
16.13	even	4		128.11.d.d.63.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
128.11.d.d.63.1	✓	4	16.11	odd	4
128.11.d.d.63.2	yes	4	16.13	even	4
128.11.d.d.63.3	yes	4	16.5	even	4
128.11.d.d.63.4	yes	4	16.3	odd	4
256.11.c.f.255.1		4	4.3	odd	2	inner
256.11.c.f.255.2		4	8.5	even	2	inner
256.11.c.f.255.3		4	1.1	even	1	trivial
256.11.c.f.255.4		4	8.3	odd	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:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+418.798i q^{3} -5463.01 q^{5} -26297.7i q^{7} -116343. q^{9} +O(q^{10})\)	\(q+418.798i q^{3} -5463.01 q^{5} -26297.7i q^{7} -116343. q^{9} -110423. i q^{11} -333244. q^{13} -2.28790e6i q^{15} +1.74912e6 q^{17} +579477. i q^{19} +1.10134e7 q^{21} -4.39171e6i q^{23} +2.00789e7 q^{25} -2.39946e7i q^{27} -1.17400e7 q^{29} +1.47267e6i q^{31} +4.62450e7 q^{33} +1.43665e8i q^{35} +4.17975e7 q^{37} -1.39562e8i q^{39} -1.23793e8 q^{41} -1.56278e8i q^{43} +6.35583e8 q^{45} -4.45115e8i q^{47} -4.09093e8 q^{49} +7.32528e8i q^{51} +4.27115e8 q^{53} +6.03243e8i q^{55} -2.42684e8 q^{57} +9.43855e8i q^{59} -3.87311e8 q^{61} +3.05955e9i q^{63} +1.82051e9 q^{65} -4.47781e8i q^{67} +1.83924e9 q^{69} +6.97704e8i q^{71} -1.66084e9 q^{73} +8.40899e9i q^{75} -2.90387e9 q^{77} -5.43347e9i q^{79} +3.17897e9 q^{81} +4.16025e9i q^{83} -9.55545e9 q^{85} -4.91670e9i q^{87} +2.50799e9 q^{89} +8.76354e9i q^{91} -6.16752e8 q^{93} -3.16569e9i q^{95} +7.97785e9 q^{97} +1.28470e10i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 465372 q^{9}+O(q^{10}) \) 4 * q - 465372 * q^9	\( 4 q - 465372 q^{9} + 6996472 q^{17} + 80315420 q^{25} + 184980096 q^{33} - 495171656 q^{41} - 1636373564 q^{49} - 970736256 q^{57} + 7282053120 q^{65} - 6643358008 q^{73} + 12715885764 q^{81} + 10031942984 q^{89} + 31911414136 q^{97}+O(q^{100}) \) 4 * q - 465372 * q^9 + 6996472 * q^17 + 80315420 * q^25 + 184980096 * q^33 - 495171656 * q^41 - 1636373564 * q^49 - 970736256 * q^57 + 7282053120 * q^65 - 6643358008 * q^73 + 12715885764 * q^81 + 10031942984 * q^89 + 31911414136 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more