LMFDB - Embedded newform 64.22.b.c.33.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,22,Mod(33,64)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(64, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 22, names="a")

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

chi := DirichletCharacter("64.33");

S:= CuspForms(chi, 22);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 22 $
Character orbit:	$[\chi]$	$=$	64.b (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:	$178.865500344$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			33.4
Character	$\chi$	$=$	64.33
Dual form			64.22.b.c.33.25

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-155316. i q^{3} +8.02569e6i q^{5} +7.41453e8 q^{7} -1.36627e10 q^{9} +O(q^{10})$	$q-155316. i q^{3} +8.02569e6i q^{5} +7.41453e8 q^{7} -1.36627e10 q^{9} +5.97707e10i q^{11} +4.24293e11i q^{13} +1.24652e12 q^{15} +1.25673e13 q^{17} -1.26199e13i q^{19} -1.15159e14i q^{21} -2.08109e14 q^{23} +4.12425e14 q^{25} +4.97370e14i q^{27} +5.67956e14i q^{29} -5.21413e15 q^{31} +9.28334e15 q^{33} +5.95068e15i q^{35} +4.07415e16i q^{37} +6.58995e16 q^{39} -2.32703e16 q^{41} +1.35853e15i q^{43} -1.09652e17i q^{45} -3.31769e17 q^{47} -8.79320e15 q^{49} -1.95190e18i q^{51} +7.09069e17i q^{53} -4.79701e17 q^{55} -1.96007e18 q^{57} +3.66701e18i q^{59} +7.38001e18i q^{61} -1.01302e19 q^{63} -3.40525e18 q^{65} -4.59630e18i q^{67} +3.23226e19i q^{69} -4.10103e19 q^{71} -4.51378e19 q^{73} -6.40562e19i q^{75} +4.43172e19i q^{77} -1.57430e20 q^{79} -6.56668e19 q^{81} -1.74399e20i q^{83} +1.00861e20i q^{85} +8.82125e19 q^{87} -2.32052e20 q^{89} +3.14594e20i q^{91} +8.09837e20i q^{93} +1.01283e20 q^{95} +8.12195e20 q^{97} -8.16627e20i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q - 77960422492 q^{9}+O(q^{10}) $ 28 * q - 77960422492 * q^9	$ 28 q - 77960422492 q^{9} + 16832040195288 q^{17} + 202504130118092 q^{25} - 55\!\cdots\!92 q^{33}+ \cdots - 19\!\cdots\!16 q^{97}+O(q^{100}) $ 28 * q - 77960422492 * q^9 + 16832040195288 * q^17 + 202504130118092 * q^25 - 55633095046450992 * q^33 + 265230847130533848 * q^41 + 5944576136704810940 * q^49 - 4544453360963256304 * q^57 - 63340883112821103360 * q^65 + 2760069698648092472 * q^73 + 934709246686586470348 * q^81 - 375882191672218968072 * q^89 - 1953791530192678912616 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/64\mathbb{Z}\right)^\times$.

$n$	$5$	$63$
$\chi(n)$	$-1$	$1$

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
$3$	− 155316.i	− 1.51860i	−0.650742	−	0.759299i	$-0.725542\pi$
$3$	− 155316.i	− 1.51860i	0.650742	−	0.759299i	$-0.274458\pi$
$4$	0	0
$5$	8.02569e6i	0.367534i	0.982970	+	0.183767i	$0.0588292\pi$
$5$	8.02569e6i	0.367534i	−0.982970	+	0.183767i	$0.941171\pi$
$6$	0	0
$7$	7.41453e8	0.992097	0.496049	−	0.868295i	$-0.334784\pi$
$7$	7.41453e8	0.992097	0.496049	+	0.868295i	$0.334784\pi$
$8$	0	0
$9$	−1.36627e10	−1.30614
$10$	0	0
$11$	5.97707e10i	0.694809i	0.937715	+	0.347404i	$0.112937\pi$
$11$	5.97707e10i	0.694809i	−0.937715	+	0.347404i	$0.887063\pi$
$12$	0	0
$13$	4.24293e11i	0.853613i	0.904343	+	0.426807i	$0.140361\pi$
$13$	4.24293e11i	0.853613i	−0.904343	+	0.426807i	$0.859639\pi$
$14$	0	0
$15$	1.24652e12	0.558136
$16$	0	0
$17$	1.25673e13	1.51191	0.755957	−	0.654622i	$-0.227172\pi$
$17$	1.25673e13	1.51191	0.755957	+	0.654622i	$0.227172\pi$
$18$	0	0
$19$	− 1.26199e13i	− 0.472218i	−0.971727	−	0.236109i	$-0.924128\pi$
$19$	− 1.26199e13i	− 0.472218i	0.971727	−	0.236109i	$-0.0758723\pi$
$20$	0	0
$21$	− 1.15159e14i	− 1.50660i
$22$	0	0
$23$	−2.08109e14	−1.04749	−0.523743	−	0.851876i	$-0.675465\pi$
$23$	−2.08109e14	−1.04749	−0.523743	+	0.851876i	$0.675465\pi$
$24$	0	0
$25$	4.12425e14	0.864919
$26$	0	0
$27$	4.97370e14i	0.464901i
$28$	0	0
$29$	5.67956e14i	0.250689i	0.992113	+	0.125345i	$0.0400036\pi$
$29$	5.67956e14i	0.250689i	−0.992113	+	0.125345i	$0.959996\pi$
$30$	0	0
$31$	−5.21413e15	−1.14257	−0.571286	−	0.820751i	$-0.693555\pi$
$31$	−5.21413e15	−1.14257	−0.571286	+	0.820751i	$0.693555\pi$
$32$	0	0
$33$	9.28334e15	1.05513
$34$	0	0
$35$	5.95068e15i	0.364630i
$36$	0	0
$37$	4.07415e16i	1.39290i	0.717607	+	0.696448i	$0.245237\pi$
$37$	4.07415e16i	1.39290i	−0.717607	+	0.696448i	$0.754763\pi$
$38$	0	0
$39$	6.58995e16	1.29629
$40$	0	0
$41$	−2.32703e16	−0.270752	−0.135376	−	0.990794i	$-0.543224\pi$
$41$	−2.32703e16	−0.270752	−0.135376	+	0.990794i	$0.543224\pi$
$42$	0	0
$43$	1.35853e15i	0.00958631i	0.999989	+	0.00479315i	$0.00152571\pi$
$43$	1.35853e15i	0.00958631i	−0.999989	+	0.00479315i	$0.998474\pi$
$44$	0	0
$45$	− 1.09652e17i	− 0.480050i
$46$	0	0
$47$	−3.31769e17	−0.920042	−0.460021	−	0.887908i	$-0.652158\pi$
$47$	−3.31769e17	−0.920042	−0.460021	+	0.887908i	$0.652158\pi$
$48$	0	0
$49$	−8.79320e15	−0.0157430
$50$	0	0
$51$	− 1.95190e18i	− 2.29599i
$52$	0	0
$53$	7.09069e17i	0.556918i	0.960448	+	0.278459i	$0.0898237\pi$
$53$	7.09069e17i	0.556918i	−0.960448	+	0.278459i	$0.910176\pi$
$54$	0	0
$55$	−4.79701e17	−0.255366
$56$	0	0
$57$	−1.96007e18	−0.717109
$58$	0	0
$59$	3.66701e18i	0.934040i	0.884247	+	0.467020i	$0.154672\pi$
$59$	3.66701e18i	0.934040i	−0.884247	+	0.467020i	$0.845328\pi$
$60$	0	0
$61$	7.38001e18i	1.32463i	0.749227	+	0.662313i	$0.230425\pi$
$61$	7.38001e18i	1.32463i	−0.749227	+	0.662313i	$0.769575\pi$
$62$	0	0
$63$	−1.01302e19	−1.29582
$64$	0	0
$65$	−3.40525e18	−0.313732
$66$	0	0
$67$	− 4.59630e18i	− 0.308051i	−0.988067	−	0.154026i	$-0.950776\pi$
$67$	− 4.59630e18i	− 0.308051i	0.988067	−	0.154026i	$-0.0492239\pi$
$68$	0	0
$69$	3.23226e19i	1.59071i
$70$	0	0
$71$	−4.10103e19	−1.49513	−0.747567	−	0.664186i	$-0.768778\pi$
$71$	−4.10103e19	−1.49513	−0.747567	+	0.664186i	$0.768778\pi$
$72$	0	0
$73$	−4.51378e19	−1.22928	−0.614640	−	0.788808i	$-0.710699\pi$
$73$	−4.51378e19	−1.22928	−0.614640	+	0.788808i	$0.710699\pi$
$74$	0	0
$75$	− 6.40562e19i	− 1.31346i
$76$	0	0
$77$	4.43172e19i	0.689318i
$78$	0	0
$79$	−1.57430e20	−1.87069	−0.935347	−	0.353732i	$-0.884912\pi$
$79$	−1.57430e20	−1.87069	−0.935347	+	0.353732i	$0.884912\pi$
$80$	0	0
$81$	−6.56668e19	−0.600141
$82$	0	0
$83$	− 1.74399e20i	− 1.23374i	−0.787064	−	0.616871i	$-0.788400\pi$
$83$	− 1.74399e20i	− 1.23374i	0.787064	−	0.616871i	$-0.211600\pi$
$84$	0	0
$85$	1.00861e20i	0.555680i
$86$	0	0
$87$	8.82125e19	0.380696
$88$	0	0
$89$	−2.32052e20	−0.788841	−0.394421	−	0.918930i	$-0.629055\pi$
$89$	−2.32052e20	−0.788841	−0.394421	+	0.918930i	$0.629055\pi$
$90$	0	0
$91$	3.14594e20i	0.846867i
$92$	0	0
$93$	8.09837e20i	1.73511i
$94$	0	0
$95$	1.01283e20	0.173556
$96$	0	0
$97$	8.12195e20	1.11830	0.559149	−	0.829068i	$-0.311128\pi$
$97$	8.12195e20	1.11830	0.559149	+	0.829068i	$0.311128\pi$
$98$	0	0
$99$	− 8.16627e20i	− 0.907516i
$100$	0	0
$101$	− 1.50868e21i	− 1.35901i	−0.733670	−	0.679506i	$-0.762194\pi$
$101$	− 1.50868e21i	− 1.35901i	0.733670	−	0.679506i	$-0.237806\pi$
$102$	0	0
$103$	−1.19264e21	−0.874414	−0.437207	−	0.899361i	$-0.644032\pi$
$103$	−1.19264e21	−0.874414	−0.437207	+	0.899361i	$0.644032\pi$
$104$	0	0
$105$	9.24234e20	0.553725
$106$	0	0
$107$	3.88023e21i	1.90690i	0.301549	+	0.953451i	$0.402496\pi$
$107$	3.88023e21i	1.90690i	−0.301549	+	0.953451i	$0.597504\pi$
$108$	0	0
$109$	− 3.11424e21i	− 1.26001i	−0.776592	−	0.630004i	$-0.783053\pi$
$109$	− 3.11424e21i	− 1.26001i	0.776592	−	0.630004i	$-0.216947\pi$
$110$	0	0
$111$	6.32780e21	2.11525
$112$	0	0
$113$	−1.81808e21	−0.503835	−0.251918	−	0.967749i	$-0.581061\pi$
$113$	−1.81808e21	−0.503835	−0.251918	+	0.967749i	$0.581061\pi$
$114$	0	0
$115$	− 1.67022e21i	− 0.384987i
$116$	0	0
$117$	− 5.79698e21i	− 1.11494i
$118$	0	0
$119$	9.31803e21	1.49997
$120$	0	0
$121$	3.82771e21	0.517241
$122$	0	0
$123$	3.61425e21i	0.411164i
$124$	0	0
$125$	7.13695e21i	0.685421i
$126$	0	0
$127$	−4.60080e21	−0.374020	−0.187010	−	0.982358i	$-0.559880\pi$
$127$	−4.60080e21	−0.374020	−0.187010	+	0.982358i	$0.559880\pi$
$128$	0	0
$129$	2.11002e20	0.0145577
$130$	0	0
$131$	1.82824e22i	1.07321i	0.843835	+	0.536603i	$0.180293\pi$
$131$	1.82824e22i	1.07321i	−0.843835	+	0.536603i	$0.819707\pi$
$132$	0	0
$133$	− 9.35706e21i	− 0.468486i
$134$	0	0
$135$	−3.99174e21	−0.170867
$136$	0	0
$137$	9.59926e21	0.352105	0.176052	−	0.984381i	$-0.443667\pi$
$137$	9.59926e21	0.352105	0.176052	+	0.984381i	$0.443667\pi$
$138$	0	0
$139$	− 4.27129e22i	− 1.34556i	−0.739842	−	0.672781i	$-0.765100\pi$
$139$	− 4.27129e22i	− 1.34556i	0.739842	−	0.672781i	$-0.234900\pi$
$140$	0	0
$141$	5.15289e22i	1.39717i
$142$	0	0
$143$	−2.53603e22	−0.593098
$144$	0	0
$145$	−4.55824e21	−0.0921368
$146$	0	0
$147$	1.36572e21i	0.0239073i
$148$	0	0
$149$	4.37222e22i	0.664119i	0.943258	+	0.332060i	$0.107743\pi$
$149$	4.37222e22i	0.664119i	−0.943258	+	0.332060i	$0.892257\pi$
$150$	0	0
$151$	−3.94929e22	−0.521508	−0.260754	−	0.965405i	$-0.583971\pi$
$151$	−3.94929e22	−0.521508	−0.260754	+	0.965405i	$0.583971\pi$
$152$	0	0
$153$	−1.71702e23	−1.97477
$154$	0	0
$155$	− 4.18470e22i	− 0.419934i
$156$	0	0
$157$	2.14364e23i	1.88021i	0.340891	+	0.940103i	$0.389271\pi$
$157$	2.14364e23i	1.88021i	−0.340891	+	0.940103i	$0.610729\pi$
$158$	0	0
$159$	1.10130e23	0.845735
$160$	0	0
$161$	−1.54303e23	−1.03921
$162$	0	0
$163$	3.86474e22i	0.228639i	0.993444	+	0.114320i	$0.0364688\pi$
$163$	3.86474e22i	0.228639i	−0.993444	+	0.114320i	$0.963531\pi$
$164$	0	0
$165$	7.45052e22i	0.387798i
$166$	0	0
$167$	−1.66648e23	−0.764325	−0.382162	−	0.924095i	$-0.624821\pi$
$167$	−1.66648e23	−0.764325	−0.382162	+	0.924095i	$0.624821\pi$
$168$	0	0
$169$	6.70396e22	0.271345
$170$	0	0
$171$	1.72421e23i	0.616782i
$172$	0	0
$173$	− 4.20966e23i	− 1.33279i	−0.745597	−	0.666397i	$-0.767835\pi$
$173$	− 4.20966e23i	− 1.33279i	0.745597	−	0.666397i	$-0.232165\pi$
$174$	0	0
$175$	3.05794e23	0.858084
$176$	0	0
$177$	5.69544e23	1.41843
$178$	0	0
$179$	3.54027e23i	0.783574i	0.920056	+	0.391787i	$0.128143\pi$
$179$	3.54027e23i	0.783574i	−0.920056	+	0.391787i	$0.871857\pi$
$180$	0	0
$181$	6.43404e23i	1.26724i	0.773645	+	0.633619i	$0.218431\pi$
$181$	6.43404e23i	1.26724i	−0.773645	+	0.633619i	$0.781569\pi$
$182$	0	0
$183$	1.14623e24	2.01157
$184$	0	0
$185$	−3.26979e23	−0.511937
$186$	0	0
$187$	7.51154e23i	1.05049i
$188$	0	0
$189$	3.68777e23i	0.461227i
$190$	0	0
$191$	1.77844e24	1.99154	0.995770	−	0.0918828i	$-0.0292885\pi$
$191$	1.77844e24	1.99154	0.995770	+	0.0918828i	$0.0292885\pi$
$192$	0	0
$193$	−8.76012e23	−0.879343	−0.439672	−	0.898159i	$-0.644905\pi$
$193$	−8.76012e23	−0.879343	−0.439672	+	0.898159i	$0.644905\pi$
$194$	0	0
$195$	5.28889e23i	0.476432i
$196$	0	0
$197$	− 2.09257e23i	− 0.169350i	−0.996409	−	0.0846748i	$-0.973015\pi$
$197$	− 2.09257e23i	− 0.169350i	0.996409	−	0.0846748i	$-0.0269851\pi$
$198$	0	0
$199$	1.06690e24	0.776544	0.388272	−	0.921545i	$-0.373072\pi$
$199$	1.06690e24	0.776544	0.388272	+	0.921545i	$0.373072\pi$
$200$	0	0
$201$	−7.13879e23	−0.467806
$202$	0	0
$203$	4.21113e23i	0.248708i
$204$	0	0
$205$	− 1.86761e23i	− 0.0995107i
$206$	0	0
$207$	2.84332e24	1.36816
$208$	0	0
$209$	7.54300e23	0.328101
$210$	0	0
$211$	− 4.57109e23i	− 0.179909i	−0.995946	−	0.0899546i	$-0.971328\pi$
$211$	− 4.57109e23i	− 0.179909i	0.995946	−	0.0899546i	$-0.0286722\pi$
$212$	0	0
$213$	6.36955e24i	2.27051i
$214$	0	0
$215$	−1.09032e22	−0.00352329
$216$	0	0
$217$	−3.86603e24	−1.13354
$218$	0	0
$219$	7.01062e24i	1.86678i
$220$	0	0
$221$	5.33221e24i	1.29059i
$222$	0	0
$223$	−5.47918e24	−1.20646	−0.603232	−	0.797566i	$-0.706121\pi$
$223$	−5.47918e24	−1.20646	−0.603232	+	0.797566i	$0.706121\pi$
$224$	0	0
$225$	−5.63483e24	−1.12970
$226$	0	0
$227$	− 2.16943e24i	− 0.396346i	−0.980167	−	0.198173i	$-0.936499\pi$
$227$	− 2.16943e24i	− 0.396346i	0.980167	−	0.198173i	$-0.0635008\pi$
$228$	0	0
$229$	8.35573e24i	1.39223i	0.717929	+	0.696116i	$0.245090\pi$
$229$	8.35573e24i	1.39223i	−0.717929	+	0.696116i	$0.754910\pi$
$230$	0	0
$231$	6.88316e24	1.04680
$232$	0	0
$233$	−1.20380e25	−1.67231	−0.836155	−	0.548493i	$-0.815202\pi$
$233$	−1.20380e25	−1.67231	−0.836155	+	0.548493i	$0.815202\pi$
$234$	0	0
$235$	− 2.66267e24i	− 0.338147i
$236$	0	0
$237$	2.44514e25i	2.84083i
$238$	0	0
$239$	1.42697e25	1.51787	0.758937	−	0.651164i	$-0.225719\pi$
$239$	1.42697e25	1.51787	0.758937	+	0.651164i	$0.225719\pi$
$240$	0	0
$241$	1.08220e25	1.05470	0.527348	−	0.849649i	$-0.323186\pi$
$241$	1.08220e25	1.05470	0.527348	+	0.849649i	$0.323186\pi$
$242$	0	0
$243$	1.54018e25i	1.37627i
$244$	0	0
$245$	− 7.05715e22i	− 0.00578610i
$246$	0	0
$247$	5.35454e24	0.403092
$248$	0	0
$249$	−2.70869e25	−1.87356
$250$	0	0
$251$	− 2.48195e25i	− 1.57841i	−0.614132	−	0.789203i	$-0.710494\pi$
$251$	− 2.48195e25i	− 1.57841i	0.614132	−	0.789203i	$-0.289506\pi$
$252$	0	0
$253$	− 1.24388e25i	− 0.727802i
$254$	0	0
$255$	1.56653e25	0.843854
$256$	0	0
$257$	1.80623e25	0.896347	0.448174	−	0.893947i	$-0.352075\pi$
$257$	1.80623e25	0.896347	0.448174	+	0.893947i	$0.352075\pi$
$258$	0	0
$259$	3.02079e25i	1.38189i
$260$	0	0
$261$	− 7.75979e24i	− 0.327435i
$262$	0	0
$263$	1.85763e25	0.723476	0.361738	−	0.932280i	$-0.382184\pi$
$263$	1.85763e25	0.723476	0.361738	+	0.932280i	$0.382184\pi$
$264$	0	0
$265$	−5.69077e24	−0.204686
$266$	0	0
$267$	3.60413e25i	1.19793i
$268$	0	0
$269$	2.24247e25i	0.689172i	0.938755	+	0.344586i	$0.111981\pi$
$269$	2.24247e25i	0.689172i	−0.938755	+	0.344586i	$0.888019\pi$
$270$	0	0
$271$	2.95313e25	0.839662	0.419831	−	0.907602i	$-0.362089\pi$
$271$	2.95313e25	0.839662	0.419831	+	0.907602i	$0.362089\pi$
$272$	0	0
$273$	4.88614e25	1.28605
$274$	0	0
$275$	2.46510e25i	0.600953i
$276$	0	0
$277$	7.89797e24i	0.178434i	0.996012	+	0.0892170i	$0.0284365\pi$
$277$	7.89797e24i	0.178434i	−0.996012	+	0.0892170i	$0.971564\pi$
$278$	0	0
$279$	7.12389e25	1.49236
$280$	0	0
$281$	−8.32103e25	−1.61719	−0.808595	−	0.588366i	$-0.799771\pi$
$281$	−8.32103e25	−1.61719	−0.808595	+	0.588366i	$0.799771\pi$
$282$	0	0
$283$	8.25057e25i	1.48842i	0.667944	+	0.744212i	$0.267175\pi$
$283$	8.25057e25i	1.48842i	−0.667944	+	0.744212i	$0.732825\pi$
$284$	0	0
$285$	− 1.57309e25i	− 0.263562i
$286$	0	0
$287$	−1.72539e25	−0.268613
$288$	0	0
$289$	8.88441e25	1.28588
$290$	0	0
$291$	− 1.26147e26i	− 1.69824i
$292$	0	0
$293$	− 1.13375e26i	− 1.42038i	−0.704008	−	0.710192i	$-0.748608\pi$
$293$	− 1.13375e26i	− 1.42038i	0.704008	−	0.710192i	$-0.251392\pi$
$294$	0	0
$295$	−2.94303e25	−0.343291
$296$	0	0
$297$	−2.97282e25	−0.323017
$298$	0	0
$299$	− 8.82992e25i	− 0.894148i
$300$	0	0
$301$	1.00729e24i	0.00951055i
$302$	0	0
$303$	−2.34322e26	−2.06379
$304$	0	0
$305$	−5.92297e25	−0.486845
$306$	0	0
$307$	− 7.56790e25i	− 0.580794i	−0.956906	−	0.290397i	$-0.906213\pi$
$307$	− 7.56790e25i	− 0.580794i	0.956906	−	0.290397i	$-0.0937874\pi$
$308$	0	0
$309$	1.85235e26i	1.32788i
$310$	0	0
$311$	−1.06651e26	−0.714463	−0.357232	−	0.934016i	$-0.616279\pi$
$311$	−1.06651e26	−0.714463	−0.357232	+	0.934016i	$0.616279\pi$
$312$	0	0
$313$	−1.96851e26	−1.23288	−0.616441	−	0.787401i	$-0.711426\pi$
$313$	−1.96851e26	−1.23288	−0.616441	+	0.787401i	$0.711426\pi$
$314$	0	0
$315$	− 8.13021e25i	− 0.476257i
$316$	0	0
$317$	− 1.44455e25i	− 0.0791788i	−0.999216	−	0.0395894i	$-0.987395\pi$
$317$	− 1.44455e25i	− 0.0791788i	0.999216	−	0.0395894i	$-0.0126050\pi$
$318$	0	0
$319$	−3.39471e25	−0.174181
$320$	0	0
$321$	6.02662e26	2.89582
$322$	0	0
$323$	− 1.58597e26i	− 0.713953i
$324$	0	0
$325$	1.74989e26i	0.738306i
$326$	0	0
$327$	−4.83690e26	−1.91344
$328$	0	0
$329$	−2.45991e26	−0.912771
$330$	0	0
$331$	− 2.80375e25i	− 0.0976213i	−0.998808	−	0.0488107i	$-0.984457\pi$
$331$	− 2.80375e25i	− 0.0976213i	0.998808	−	0.0488107i	$-0.0155431\pi$
$332$	0	0
$333$	− 5.56637e26i	− 1.81931i
$334$	0	0
$335$	3.68885e25	0.113219
$336$	0	0
$337$	1.87792e26	0.541456	0.270728	−	0.962656i	$-0.412736\pi$
$337$	1.87792e26	0.541456	0.270728	+	0.962656i	$0.412736\pi$
$338$	0	0
$339$	2.82376e26i	0.765123i
$340$	0	0
$341$	− 3.11652e26i	− 0.793869i
$342$	0	0
$343$	−4.20655e26	−1.00772
$344$	0	0
$345$	−2.59411e26	−0.584640
$346$	0	0
$347$	− 3.63992e26i	− 0.772028i	−0.922493	−	0.386014i	$-0.873852\pi$
$347$	− 3.63992e26i	− 0.772028i	0.922493	−	0.386014i	$-0.126148\pi$
$348$	0	0
$349$	− 5.33453e26i	− 1.06519i	−0.846369	−	0.532597i	$-0.821216\pi$
$349$	− 5.33453e26i	− 1.06519i	0.846369	−	0.532597i	$-0.178784\pi$
$350$	0	0
$351$	−2.11031e26	−0.396845
$352$	0	0
$353$	9.21746e26	1.63296	0.816482	−	0.577370i	$-0.195921\pi$
$353$	9.21746e26	1.63296	0.816482	+	0.577370i	$0.195921\pi$
$354$	0	0
$355$	− 3.29136e26i	− 0.549513i
$356$	0	0
$357$	− 1.44724e27i	− 2.27784i
$358$	0	0
$359$	3.98584e26	0.591600	0.295800	−	0.955250i	$-0.404414\pi$
$359$	3.98584e26	0.591600	0.295800	+	0.955250i	$0.404414\pi$
$360$	0	0
$361$	5.54948e26	0.777010
$362$	0	0
$363$	− 5.94505e26i	− 0.785481i
$364$	0	0
$365$	− 3.62262e26i	− 0.451802i
$366$	0	0
$367$	−2.45579e26	−0.289199	−0.144600	−	0.989490i	$-0.546189\pi$
$367$	−2.45579e26	−0.289199	−0.144600	+	0.989490i	$0.546189\pi$
$368$	0	0
$369$	3.17935e26	0.353640
$370$	0	0
$371$	5.25741e26i	0.552517i
$372$	0	0
$373$	− 9.37340e26i	− 0.931010i	−0.885045	−	0.465505i	$-0.845873\pi$
$373$	− 9.37340e26i	− 0.931010i	0.885045	−	0.465505i	$-0.154127\pi$
$374$	0	0
$375$	1.10848e27	1.04088
$376$	0	0
$377$	−2.40980e26	−0.213992
$378$	0	0
$379$	1.28680e27i	1.08093i	0.841366	+	0.540465i	$0.181752\pi$
$379$	1.28680e27i	1.08093i	−0.841366	+	0.540465i	$0.818248\pi$
$380$	0	0
$381$	7.14577e26i	0.567985i
$382$	0	0
$383$	3.90228e26	0.293583	0.146792	−	0.989167i	$-0.453105\pi$
$383$	3.90228e26	0.293583	0.146792	+	0.989167i	$0.453105\pi$
$384$	0	0
$385$	−3.55676e26	−0.253348
$386$	0	0
$387$	− 1.85612e25i	− 0.0125210i
$388$	0	0
$389$	1.30099e27i	0.831386i	0.909505	+	0.415693i	$0.136461\pi$
$389$	1.30099e27i	0.831386i	−0.909505	+	0.415693i	$0.863539\pi$
$390$	0	0
$391$	−2.61536e27	−1.58371
$392$	0	0
$393$	2.83954e27	1.62977
$394$	0	0
$395$	− 1.26348e27i	− 0.687544i
$396$	0	0
$397$	6.48401e26i	0.334613i	0.985905	+	0.167307i	$0.0535070\pi$
$397$	6.48401e26i	0.334613i	−0.985905	+	0.167307i	$0.946493\pi$
$398$	0	0
$399$	−1.45330e27	−0.711442
$400$	0	0
$401$	−4.71600e26	−0.219057	−0.109529	−	0.993984i	$-0.534934\pi$
$401$	−4.71600e26	−0.219057	−0.109529	+	0.993984i	$0.534934\pi$
$402$	0	0
$403$	− 2.21232e27i	− 0.975315i
$404$	0	0
$405$	− 5.27022e26i	− 0.220572i
$406$	0	0
$407$	−2.43515e27	−0.967796
$408$	0	0
$409$	−4.84138e27	−1.82757	−0.913786	−	0.406197i	$-0.866855\pi$
$409$	−4.84138e27	−1.82757	−0.913786	+	0.406197i	$0.866855\pi$
$410$	0	0
$411$	− 1.49092e27i	− 0.534705i
$412$	0	0
$413$	2.71891e27i	0.926658i
$414$	0	0
$415$	1.39967e27	0.453442
$416$	0	0
$417$	−6.63399e27	−2.04337
$418$	0	0
$419$	5.88048e26i	0.172252i	0.996284	+	0.0861262i	$0.0274488\pi$
$419$	5.88048e26i	0.172252i	−0.996284	+	0.0861262i	$0.972551\pi$
$420$	0	0
$421$	4.19942e27i	1.17011i	0.810993	+	0.585056i	$0.198927\pi$
$421$	4.19942e27i	1.17011i	−0.810993	+	0.585056i	$0.801073\pi$
$422$	0	0
$423$	4.53284e27	1.20170
$424$	0	0
$425$	5.18306e27	1.30768
$426$	0	0
$427$	5.47193e27i	1.31416i
$428$	0	0
$429$	3.93886e27i	0.900677i
$430$	0	0
$431$	−3.21497e26	−0.0700108	−0.0350054	−	0.999387i	$-0.511145\pi$
$431$	−3.21497e26	−0.0700108	−0.0350054	+	0.999387i	$0.511145\pi$
$432$	0	0
$433$	−6.65247e27	−1.37994	−0.689970	−	0.723838i	$-0.742376\pi$
$433$	−6.65247e27	−1.37994	−0.689970	+	0.723838i	$0.742376\pi$
$434$	0	0
$435$	7.07967e26i	0.139919i
$436$	0	0
$437$	2.62631e27i	0.494642i
$438$	0	0
$439$	7.43312e27	1.33442	0.667212	−	0.744868i	$-0.267487\pi$
$439$	7.43312e27	1.33442	0.667212	+	0.744868i	$0.267487\pi$
$440$	0	0
$441$	1.20139e26	0.0205626
$442$	0	0
$443$	5.73845e27i	0.936602i	0.883569	+	0.468301i	$0.155134\pi$
$443$	5.73845e27i	0.936602i	−0.883569	+	0.468301i	$0.844866\pi$
$444$	0	0
$445$	− 1.86238e27i	− 0.289926i
$446$	0	0
$447$	6.79075e27	1.00853
$448$	0	0
$449$	1.97502e27	0.279889	0.139944	−	0.990159i	$-0.455308\pi$
$449$	1.97502e27	0.279889	0.139944	+	0.990159i	$0.455308\pi$
$450$	0	0
$451$	− 1.39088e27i	− 0.188121i
$452$	0	0
$453$	6.13387e27i	0.791960i
$454$	0	0
$455$	−2.52483e27	−0.311253
$456$	0	0
$457$	1.32251e28	1.55696	0.778480	−	0.627670i	$-0.215991\pi$
$457$	1.32251e28	1.55696	0.778480	+	0.627670i	$0.215991\pi$
$458$	0	0
$459$	6.25058e27i	0.702890i
$460$	0	0
$461$	− 3.35147e26i	− 0.0360061i	−0.999838	−	0.0180030i	$-0.994269\pi$
$461$	− 3.35147e26i	− 0.0360061i	0.999838	−	0.0180030i	$-0.00573085\pi$
$462$	0	0
$463$	9.67503e27	0.993235	0.496618	−	0.867969i	$-0.334575\pi$
$463$	9.67503e27	0.993235	0.496618	+	0.867969i	$0.334575\pi$
$464$	0	0
$465$	−6.49950e27	−0.637711
$466$	0	0
$467$	9.32923e27i	0.875021i	0.899213	+	0.437511i	$0.144140\pi$
$467$	9.32923e27i	0.875021i	−0.899213	+	0.437511i	$0.855860\pi$
$468$	0	0
$469$	− 3.40794e27i	− 0.305617i
$470$	0	0
$471$	3.32941e28	2.85528
$472$	0	0
$473$	−8.12005e25	−0.00666065
$474$	0	0
$475$	− 5.20476e27i	− 0.408430i
$476$	0	0
$477$	− 9.68777e27i	− 0.727412i
$478$	0	0
$479$	3.58049e27	0.257288	0.128644	−	0.991691i	$-0.458938\pi$
$479$	3.58049e27	0.257288	0.128644	+	0.991691i	$0.458938\pi$
$480$	0	0
$481$	−1.72863e28	−1.18899
$482$	0	0
$483$	2.39657e28i	1.57814i
$484$	0	0
$485$	6.51843e27i	0.411012i
$486$	0	0
$487$	−1.47812e28	−0.892595	−0.446297	−	0.894885i	$-0.647258\pi$
$487$	−1.47812e28	−0.892595	−0.446297	+	0.894885i	$0.647258\pi$
$488$	0	0
$489$	6.00255e27	0.347211
$490$	0	0
$491$	− 1.38118e28i	− 0.765408i	−0.923871	−	0.382704i	$-0.874993\pi$
$491$	− 1.38118e28i	− 0.765408i	0.923871	−	0.382704i	$-0.125007\pi$
$492$	0	0
$493$	7.13765e27i	0.379020i
$494$	0	0
$495$	6.55400e27	0.333543
$496$	0	0
$497$	−3.04072e28	−1.48332
$498$	0	0
$499$	− 8.80069e27i	− 0.411586i	−0.978596	−	0.205793i	$-0.934023\pi$
$499$	− 8.80069e27i	− 0.411586i	0.978596	−	0.205793i	$-0.0659774\pi$
$500$	0	0
$501$	2.58831e28i	1.16070i
$502$	0	0
$503$	1.03870e28	0.446709	0.223355	−	0.974737i	$-0.428299\pi$
$503$	1.03870e28	0.446709	0.223355	+	0.974737i	$0.428299\pi$
$504$	0	0
$505$	1.21082e28	0.499483
$506$	0	0
$507$	− 1.04123e28i	− 0.412063i
$508$	0	0
$509$	3.99571e28i	1.51725i	0.651527	+	0.758625i	$0.274129\pi$
$509$	3.99571e28i	1.51725i	−0.651527	+	0.758625i	$0.725871\pi$
$510$	0	0
$511$	−3.34676e28	−1.21956
$512$	0	0
$513$	6.27676e27	0.219535
$514$	0	0
$515$	− 9.57173e27i	− 0.321377i
$516$	0	0
$517$	− 1.98300e28i	− 0.639253i
$518$	0	0
$519$	−6.53828e28	−2.02398
$520$	0	0
$521$	−5.98202e27	−0.177849	−0.0889245	−	0.996038i	$-0.528343\pi$
$521$	−5.98202e27	−0.177849	−0.0889245	+	0.996038i	$0.528343\pi$
$522$	0	0
$523$	2.46038e28i	0.702643i	0.936255	+	0.351321i	$0.114268\pi$
$523$	2.46038e28i	0.702643i	−0.936255	+	0.351321i	$0.885732\pi$
$524$	0	0
$525$	− 4.74947e28i	− 1.30308i
$526$	0	0
$527$	−6.55273e28	−1.72747
$528$	0	0
$529$	3.83770e27	0.0972270
$530$	0	0
$531$	− 5.01011e28i	− 1.21999i
$532$	0	0
$533$	− 9.87345e27i	− 0.231118i
$534$	0	0
$535$	−3.11416e28	−0.700851
$536$	0	0
$537$	5.49861e28	1.18993
$538$	0	0
$539$	− 5.25576e26i	− 0.0109384i
$540$	0	0
$541$	7.44966e28i	1.49130i	0.666337	+	0.745650i	$0.267861\pi$
$541$	7.44966e28i	1.49130i	−0.666337	+	0.745650i	$0.732139\pi$
$542$	0	0
$543$	9.99308e28	1.92443
$544$	0	0
$545$	2.49939e28	0.463096
$546$	0	0
$547$	− 8.92286e28i	− 1.59088i	−0.606033	−	0.795440i	$-0.707240\pi$
$547$	− 8.92286e28i	− 1.59088i	0.606033	−	0.795440i	$-0.292760\pi$
$548$	0	0
$549$	− 1.00831e29i	− 1.73015i
$550$	0	0
$551$	7.16754e27	0.118380
$552$	0	0
$553$	−1.16727e29	−1.85591
$554$	0	0
$555$	5.07850e28i	0.777426i
$556$	0	0
$557$	− 4.03914e28i	− 0.595400i	−0.954659	−	0.297700i	$-0.903780\pi$
$557$	− 4.03914e28i	− 0.595400i	0.954659	−	0.297700i	$-0.0962196\pi$
$558$	0	0
$559$	−5.76416e26	−0.00818300
$560$	0	0
$561$	1.16666e29	1.59527
$562$	0	0
$563$	− 6.10245e28i	− 0.803834i	−0.915676	−	0.401917i	$-0.868344\pi$
$563$	− 6.10245e28i	− 0.803834i	0.915676	−	0.401917i	$-0.131656\pi$
$564$	0	0
$565$	− 1.45913e28i	− 0.185177i
$566$	0	0
$567$	−4.86889e28	−0.595398
$568$	0	0
$569$	−7.58725e27	−0.0894140	−0.0447070	−	0.999000i	$-0.514235\pi$
$569$	−7.58725e27	−0.0894140	−0.0447070	+	0.999000i	$0.514235\pi$
$570$	0	0
$571$	2.74131e28i	0.311372i	0.987807	+	0.155686i	$0.0497588\pi$
$571$	2.74131e28i	0.311372i	−0.987807	+	0.155686i	$0.950241\pi$
$572$	0	0
$573$	− 2.76220e29i	− 3.02435i
$574$	0	0
$575$	−8.58294e28	−0.905990
$576$	0	0
$577$	1.77354e29	1.80508	0.902538	−	0.430611i	$-0.141702\pi$
$577$	1.77354e29	1.80508	0.902538	+	0.430611i	$0.141702\pi$
$578$	0	0
$579$	1.36059e29i	1.33537i
$580$	0	0
$581$	− 1.29309e29i	− 1.22399i
$582$	0	0
$583$	−4.23815e28	−0.386952
$584$	0	0
$585$	4.65248e28	0.409777
$586$	0	0
$587$	− 1.31950e29i	− 1.12127i	−0.828064	−	0.560633i	$-0.810558\pi$
$587$	− 1.31950e29i	− 1.12127i	0.828064	−	0.560633i	$-0.189442\pi$
$588$	0	0
$589$	6.58017e28i	0.539544i
$590$	0	0
$591$	−3.25009e28	−0.257174
$592$	0	0
$593$	7.89285e28	0.602781	0.301391	−	0.953501i	$-0.402549\pi$
$593$	7.89285e28	0.602781	0.301391	+	0.953501i	$0.402549\pi$
$594$	0	0
$595$	7.47837e28i	0.551288i
$596$	0	0
$597$	− 1.65706e29i	− 1.17926i
$598$	0	0
$599$	1.70040e29	1.16834	0.584170	−	0.811631i	$-0.301420\pi$
$599$	1.70040e29	1.16834	0.584170	+	0.811631i	$0.301420\pi$
$600$	0	0
$601$	−4.56043e28	−0.302568	−0.151284	−	0.988490i	$-0.548341\pi$
$601$	−4.56043e28	−0.302568	−0.151284	+	0.988490i	$0.548341\pi$
$602$	0	0
$603$	6.27978e28i	0.402358i
$604$	0	0
$605$	3.07201e28i	0.190104i
$606$	0	0
$607$	−1.34867e29	−0.806166	−0.403083	−	0.915163i	$-0.632061\pi$
$607$	−1.34867e29	−0.806166	−0.403083	+	0.915163i	$0.632061\pi$
$608$	0	0
$609$	6.54055e28	0.377687
$610$	0	0
$611$	− 1.40767e29i	− 0.785360i
$612$	0	0
$613$	3.23026e29i	1.74142i	0.491799	+	0.870709i	$0.336339\pi$
$613$	3.23026e29i	1.74142i	−0.491799	+	0.870709i	$0.663661\pi$
$614$	0	0
$615$	−2.90069e28	−0.151117
$616$	0	0
$617$	−3.44557e29	−1.73487	−0.867435	−	0.497551i	$-0.834233\pi$
$617$	−3.44557e29	−1.73487	−0.867435	+	0.497551i	$0.834233\pi$
$618$	0	0
$619$	− 8.81433e28i	− 0.428980i	−0.976726	−	0.214490i	$-0.931191\pi$
$619$	− 8.81433e28i	− 0.428980i	0.976726	−	0.214490i	$-0.0688090\pi$
$620$	0	0
$621$	− 1.03507e29i	− 0.486977i
$622$	0	0
$623$	−1.72055e29	−0.782607
$624$	0	0
$625$	1.39381e29	0.613003
$626$	0	0
$627$	− 1.17155e29i	− 0.498254i
$628$	0	0
$629$	5.12009e29i	2.10594i
$630$	0	0
$631$	−2.15900e29	−0.858904	−0.429452	−	0.903090i	$-0.641293\pi$
$631$	−2.15900e29	−0.858904	−0.429452	+	0.903090i	$0.641293\pi$
$632$	0	0
$633$	−7.09962e28	−0.273210
$634$	0	0
$635$	− 3.69246e28i	− 0.137465i
$636$	0	0
$637$	− 3.73090e27i	− 0.0134384i
$638$	0	0
$639$	5.60310e29	1.95285
$640$	0	0
$641$	−5.39720e29	−1.82037	−0.910185	−	0.414203i	$-0.864061\pi$
$641$	−5.39720e29	−1.82037	−0.910185	+	0.414203i	$0.864061\pi$
$642$	0	0
$643$	− 5.02459e29i	− 1.64016i	−0.572252	−	0.820078i	$-0.693930\pi$
$643$	− 5.02459e29i	− 1.64016i	0.572252	−	0.820078i	$-0.306070\pi$
$644$	0	0
$645$	1.69343e27i	0.00535046i
$646$	0	0
$647$	−4.34074e29	−1.32761	−0.663803	−	0.747907i	$-0.731059\pi$
$647$	−4.34074e29	−1.32761	−0.663803	+	0.747907i	$0.731059\pi$
$648$	0	0
$649$	−2.19180e29	−0.648979
$650$	0	0
$651$	6.00456e29i	1.72140i
$652$	0	0
$653$	2.00564e29i	0.556754i	0.960472	+	0.278377i	$0.0897965\pi$
$653$	2.00564e29i	0.556754i	−0.960472	+	0.278377i	$0.910203\pi$
$654$	0	0
$655$	−1.46729e29	−0.394440
$656$	0	0
$657$	6.16703e29	1.60561
$658$	0	0
$659$	3.24797e29i	0.819059i	0.912297	+	0.409529i	$0.134307\pi$
$659$	3.24797e29i	0.819059i	−0.912297	+	0.409529i	$0.865693\pi$
$660$	0	0
$661$	− 1.03582e29i	− 0.253029i	−0.991965	−	0.126514i	$-0.959621\pi$
$661$	− 1.03582e29i	− 0.253029i	0.991965	−	0.126514i	$-0.0403790\pi$
$662$	0	0
$663$	8.28176e29	1.95989
$664$	0	0
$665$	7.50969e28	0.172185
$666$	0	0
$667$	− 1.18197e29i	− 0.262593i
$668$	0	0
$669$	8.51003e29i	1.83213i
$670$	0	0
$671$	−4.41108e29	−0.920362
$672$	0	0
$673$	−1.37867e29	−0.278806	−0.139403	−	0.990236i	$-0.544518\pi$
$673$	−1.37867e29	−0.278806	−0.139403	+	0.990236i	$0.544518\pi$
$674$	0	0
$675$	2.05128e29i	0.402101i
$676$	0	0
$677$	5.59803e28i	0.106379i	0.998584	+	0.0531893i	$0.0169387\pi$
$677$	5.59803e28i	0.106379i	−0.998584	+	0.0531893i	$0.983061\pi$
$678$	0	0
$679$	6.02205e29	1.10946
$680$	0	0
$681$	−3.36947e29	−0.601890
$682$	0	0
$683$	7.24482e29i	1.25490i	0.778656	+	0.627451i	$0.215902\pi$
$683$	7.24482e29i	1.25490i	−0.778656	+	0.627451i	$0.784098\pi$
$684$	0	0
$685$	7.70407e28i	0.129410i
$686$	0	0
$687$	1.29778e30	2.11424
$688$	0	0
$689$	−3.00853e29	−0.475393
$690$	0	0
$691$	2.63071e29i	0.403230i	0.979465	+	0.201615i	$0.0646190\pi$
$691$	2.63071e29i	0.403230i	−0.979465	+	0.201615i	$0.935381\pi$
$692$	0	0
$693$	− 6.05491e29i	− 0.900344i
$694$	0	0
$695$	3.42801e29	0.494540
$696$	0	0
$697$	−2.92444e29	−0.409354
$698$	0	0
$699$	1.86969e30i	2.53957i
$700$	0	0
$701$	5.12839e29i	0.675993i	0.941147	+	0.337996i	$0.109749\pi$
$701$	5.12839e29i	0.675993i	−0.941147	+	0.337996i	$0.890251\pi$
$702$	0	0
$703$	5.14153e29	0.657751
$704$	0	0
$705$	−4.13555e29	−0.513509
$706$	0	0
$707$	− 1.11862e30i	− 1.34827i
$708$	0	0
$709$	5.44783e29i	0.637439i	0.947849	+	0.318719i	$0.103253\pi$
$709$	5.44783e29i	0.637439i	−0.947849	+	0.318719i	$0.896747\pi$
$710$	0	0
$711$	2.15091e30	2.44338
$712$	0	0
$713$	1.08511e30	1.19683
$714$	0	0
$715$	− 2.03534e29i	− 0.217984i
$716$	0	0
$717$	− 2.21631e30i	− 2.30504i
$718$	0	0
$719$	−1.11680e29	−0.112803	−0.0564015	−	0.998408i	$-0.517963\pi$
$719$	−1.11680e29	−0.112803	−0.0564015	+	0.998408i	$0.517963\pi$
$720$	0	0
$721$	−8.84284e29	−0.867504
$722$	0	0
$723$	− 1.68082e30i	− 1.60166i
$724$	0	0
$725$	2.34239e29i	0.216826i
$726$	0	0
$727$	−1.84896e30	−1.66271	−0.831356	−	0.555741i	$-0.812435\pi$
$727$	−1.84896e30	−1.66271	−0.831356	+	0.555741i	$0.812435\pi$
$728$	0	0
$729$	1.70524e30	1.48986
$730$	0	0
$731$	1.70730e28i	0.0144937i
$732$	0	0
$733$	1.69547e30i	1.39861i	0.714822	+	0.699306i	$0.246508\pi$
$733$	1.69547e30i	1.39861i	−0.714822	+	0.699306i	$0.753492\pi$
$734$	0	0
$735$	−1.09609e28	−0.00878675
$736$	0	0
$737$	2.74724e29	0.214037
$738$	0	0
$739$	7.77837e29i	0.589009i	0.955650	+	0.294505i	$0.0951546\pi$
$739$	7.77837e29i	0.589009i	−0.955650	+	0.294505i	$0.904845\pi$
$740$	0	0
$741$	− 8.31644e29i	− 0.612134i
$742$	0	0
$743$	−9.12146e29	−0.652652	−0.326326	−	0.945257i	$-0.605811\pi$
$743$	−9.12146e29	−0.652652	−0.326326	+	0.945257i	$0.605811\pi$
$744$	0	0
$745$	−3.50901e29	−0.244086
$746$	0	0
$747$	2.38276e30i	1.61144i
$748$	0	0
$749$	2.87701e30i	1.89183i
$750$	0	0
$751$	3.47217e29	0.222014	0.111007	−	0.993820i	$-0.464592\pi$
$751$	3.47217e29	0.222014	0.111007	+	0.993820i	$0.464592\pi$
$752$	0	0
$753$	−3.85486e30	−2.39696
$754$	0	0
$755$	− 3.16958e29i	− 0.191672i
$756$	0	0
$757$	2.38797e30i	1.40450i	0.711930	+	0.702250i	$0.247821\pi$
$757$	2.38797e30i	1.40450i	−0.711930	+	0.702250i	$0.752179\pi$
$758$	0	0
$759$	−1.93194e30	−1.10524
$760$	0	0
$761$	4.61747e29	0.256960	0.128480	−	0.991712i	$-0.458990\pi$
$761$	4.61747e29	0.256960	0.128480	+	0.991712i	$0.458990\pi$
$762$	0	0
$763$	− 2.30906e30i	− 1.25005i
$764$	0	0
$765$	− 1.37803e30i	− 0.725795i
$766$	0	0
$767$	−1.55589e30	−0.797309
$768$	0	0
$769$	−1.62563e29	−0.0810579	−0.0405289	−	0.999178i	$-0.512904\pi$
$769$	−1.62563e29	−0.0810579	−0.0405289	+	0.999178i	$0.512904\pi$
$770$	0	0
$771$	− 2.80537e30i	− 1.36119i
$772$	0	0
$773$	4.84051e26i	0 0.000228563i	1.00000		0.000114282i	$3.63770e-5\pi$
$773$	4.84051e26i	0 0.000228563i	−1.00000		0.000114282i	$0.999964\pi$
$774$	0	0
$775$	−2.15044e30	−0.988232
$776$	0	0
$777$	4.69176e30	2.09853
$778$	0	0
$779$	2.93669e29i	0.127854i
$780$	0	0
$781$	− 2.45121e30i	− 1.03883i
$782$	0	0
$783$	−2.82484e29	−0.116546
$784$	0	0
$785$	−1.72042e30	−0.691040
$786$	0	0
$787$	− 1.41039e30i	− 0.551576i	−0.961218	−	0.275788i	$-0.911061\pi$
$787$	− 1.41039e30i	− 0.551576i	0.961218	−	0.275788i	$-0.0889388\pi$
$788$	0	0
$789$	− 2.88520e30i	− 1.09867i
$790$	0	0
$791$	−1.34802e30	−0.499854
$792$	0	0
$793$	−3.13129e30	−1.13072
$794$	0	0
$795$	8.83867e29i	0.310836i
$796$	0	0
$797$	− 2.46502e30i	− 0.844321i	−0.906521	−	0.422161i	$-0.861272\pi$
$797$	− 2.46502e30i	− 0.844321i	0.906521	−	0.422161i	$-0.138728\pi$
$798$	0	0
$799$	−4.16942e30	−1.39102
$800$	0	0
$801$	3.17044e30	1.03034
$802$	0	0
$803$	− 2.69792e30i	− 0.854114i
$804$	0	0
$805$	− 1.23839e30i	− 0.381944i
$806$	0	0
$807$	3.48291e30	1.04657
$808$	0	0
$809$	3.32880e30	0.974605	0.487303	−	0.873233i	$-0.337981\pi$
$809$	3.32880e30	0.974605	0.487303	+	0.873233i	$0.337981\pi$
$810$	0	0
$811$	1.29792e30i	0.370278i	0.982712	+	0.185139i	$0.0592735\pi$
$811$	1.29792e30i	0.370278i	−0.982712	+	0.185139i	$0.940727\pi$
$812$	0	0
$813$	− 4.58667e30i	− 1.27511i
$814$	0	0
$815$	−3.10172e29	−0.0840326
$816$	0	0
$817$	1.71445e28	0.00452683
$818$	0	0
$819$	− 4.29819e30i	− 1.10613i
$820$	0	0
$821$	4.21291e28i	0.0105677i	0.999986	+	0.00528383i	$0.00168190\pi$
$821$	4.21291e28i	0.0105677i	−0.999986	+	0.00528383i	$0.998318\pi$
$822$	0	0
$823$	−1.14078e30	−0.278936	−0.139468	−	0.990227i	$-0.544539\pi$
$823$	−1.14078e30	−0.278936	−0.139468	+	0.990227i	$0.544539\pi$
$824$	0	0
$825$	3.82868e30	0.912606
$826$	0	0
$827$	3.20061e30i	0.743747i	0.928283	+	0.371874i	$0.121285\pi$
$827$	3.20061e30i	0.743747i	−0.928283	+	0.371874i	$0.878715\pi$
$828$	0	0
$829$	− 4.65198e30i	− 1.05394i	−0.849884	−	0.526969i	$-0.823328\pi$
$829$	− 4.65198e30i	− 1.05394i	0.849884	−	0.526969i	$-0.176672\pi$
$830$	0	0
$831$	1.22668e30	0.270969
$832$	0	0
$833$	−1.10506e29	−0.0238021
$834$	0	0
$835$	− 1.33747e30i	− 0.280915i
$836$	0	0
$837$	− 2.59335e30i	− 0.531183i
$838$	0	0
$839$	7.23132e29	0.144450	0.0722250	−	0.997388i	$-0.476990\pi$
$839$	7.23132e29	0.144450	0.0722250	+	0.997388i	$0.476990\pi$
$840$	0	0
$841$	4.81027e30	0.937155
$842$	0	0
$843$	1.29239e31i	2.45586i
$844$	0	0
$845$	5.38039e29i	0.0997284i
$846$	0	0
$847$	2.83807e30	0.513153
$848$	0	0
$849$	1.28144e31	2.26032
$850$	0	0
$851$	− 8.47866e30i	− 1.45904i
$852$	0	0
$853$	− 2.05760e30i	− 0.345459i	−0.984969	−	0.172729i	$-0.944741\pi$
$853$	− 2.05760e30i	− 0.345459i	0.984969	−	0.172729i	$-0.0552586\pi$
$854$	0	0
$855$	−1.38380e30	−0.226688
$856$	0	0
$857$	−1.04014e31	−1.66262	−0.831310	−	0.555809i	$-0.812409\pi$
$857$	−1.04014e31	−1.66262	−0.831310	+	0.555809i	$0.812409\pi$
$858$	0	0
$859$	6.88208e30i	1.07347i	0.843749	+	0.536737i	$0.180343\pi$
$859$	6.88208e30i	1.07347i	−0.843749	+	0.536737i	$0.819657\pi$
$860$	0	0
$861$	2.67980e30i	0.407915i
$862$	0	0
$863$	2.39428e30	0.355681	0.177841	−	0.984059i	$-0.443089\pi$
$863$	2.39428e30	0.355681	0.177841	+	0.984059i	$0.443089\pi$
$864$	0	0
$865$	3.37855e30	0.489847
$866$	0	0
$867$	− 1.37989e31i	− 1.95274i
$868$	0	0
$869$	− 9.40969e30i	− 1.29977i
$870$	0	0
$871$	1.95018e30	0.262957
$872$	0	0
$873$	−1.10968e31	−1.46065
$874$	0	0
$875$	5.29171e30i	0.680004i
$876$	0	0
$877$	4.33796e30i	0.544240i	0.962263	+	0.272120i	$0.0877247\pi$
$877$	4.33796e30i	0.544240i	−0.962263	+	0.272120i	$0.912275\pi$
$878$	0	0
$879$	−1.76089e31	−2.15699
$880$	0	0
$881$	−2.77986e30	−0.332488	−0.166244	−	0.986085i	$-0.553164\pi$
$881$	−2.77986e30	−0.332488	−0.166244	+	0.986085i	$0.553164\pi$
$882$	0	0
$883$	− 8.03207e30i	− 0.938080i	−0.883177	−	0.469040i	$-0.844600\pi$
$883$	− 8.03207e30i	− 0.938080i	0.883177	−	0.469040i	$-0.155400\pi$
$884$	0	0
$885$	4.57099e30i	0.521322i
$886$	0	0
$887$	−3.53837e30	−0.394099	−0.197049	−	0.980394i	$-0.563136\pi$
$887$	−3.53837e30	−0.394099	−0.197049	+	0.980394i	$0.563136\pi$
$888$	0	0
$889$	−3.41128e30	−0.371064
$890$	0	0
$891$	− 3.92495e30i	− 0.416983i
$892$	0	0
$893$	4.18688e30i	0.434461i
$894$	0	0
$895$	−2.84132e30	−0.287990
$896$	0	0
$897$	−1.37143e31	−1.35785
$898$	0	0
$899$	− 2.96139e30i	− 0.286431i
$900$	0	0
$901$	8.91105e30i	0.842013i
$902$	0	0
$903$	1.56448e29	0.0144427
$904$	0	0
$905$	−5.16376e30	−0.465753
$906$	0	0
$907$	1.64232e31i	1.44738i	0.690126	+	0.723690i	$0.257555\pi$
$907$	1.64232e31i	1.44738i	−0.690126	+	0.723690i	$0.742445\pi$
$908$	0	0
$909$	2.06126e31i	1.77506i
$910$	0	0
$911$	1.29241e31	1.08757	0.543787	−	0.839223i	$-0.316990\pi$
$911$	1.29241e31	1.08757	0.543787	+	0.839223i	$0.316990\pi$
$912$	0	0
$913$	1.04240e31	0.857214
$914$	0	0
$915$	9.19931e30i	0.739322i
$916$	0	0
$917$	1.35555e31i	1.06473i
$918$	0	0
$919$	1.12583e31	0.864291	0.432145	−	0.901804i	$-0.357757\pi$
$919$	1.12583e31	0.864291	0.432145	+	0.901804i	$0.357757\pi$
$920$	0	0
$921$	−1.17541e31	−0.881992
$922$	0	0
$923$	− 1.74004e31i	− 1.27627i
$924$	0	0
$925$	1.68028e31i	1.20474i
$926$	0	0
$927$	1.62946e31	1.14211
$928$	0	0
$929$	1.55904e31	1.06830	0.534148	−	0.845391i	$-0.320633\pi$
$929$	1.55904e31	1.06830	0.534148	+	0.845391i	$0.320633\pi$
$930$	0	0
$931$	1.10969e29i	0.00743414i
$932$	0	0
$933$	1.65646e31i	1.08498i
$934$	0	0
$935$	−6.02853e30	−0.386091
$936$	0	0
$937$	−1.48709e31	−0.931262	−0.465631	−	0.884979i	$-0.654173\pi$
$937$	−1.48709e31	−0.931262	−0.465631	+	0.884979i	$0.654173\pi$
$938$	0	0
$939$	3.05740e31i	1.87225i
$940$	0	0
$941$	− 1.57059e31i	− 0.940531i	−0.882525	−	0.470265i	$-0.844158\pi$
$941$	− 1.57059e31i	− 0.940531i	0.882525	−	0.470265i	$-0.155842\pi$
$942$	0	0
$943$	4.84276e30	0.283609
$944$	0	0
$945$	−2.95969e30	−0.169517
$946$	0	0
$947$	1.17080e31i	0.655857i	0.944703	+	0.327929i	$0.106351\pi$
$947$	1.17080e31i	0.655857i	−0.944703	+	0.327929i	$0.893649\pi$
$948$	0	0
$949$	− 1.91517e31i	− 1.04933i
$950$	0	0
$951$	−2.24361e30	−0.120241
$952$	0	0
$953$	8.63744e30	0.452803	0.226402	−	0.974034i	$-0.427304\pi$
$953$	8.63744e30	0.452803	0.226402	+	0.974034i	$0.427304\pi$
$954$	0	0
$955$	1.42732e31i	0.731959i
$956$	0	0
$957$	5.27253e30i	0.264511i
$958$	0	0
$959$	7.11740e30	0.349322
$960$	0	0
$961$	6.36161e30	0.305472
$962$	0	0
$963$	− 5.30143e31i	− 2.49068i
$964$	0	0
$965$	− 7.03061e30i	− 0.323189i
$966$	0	0
$967$	3.06873e31	1.38032	0.690162	−	0.723654i	$-0.257539\pi$
$967$	3.06873e31	1.38032	0.690162	+	0.723654i	$0.257539\pi$
$968$	0	0
$969$	−2.46327e31	−1.08421
$970$	0	0
$971$	3.59284e31i	1.54752i	0.633480	+	0.773759i	$0.281626\pi$
$971$	3.59284e31i	1.54752i	−0.633480	+	0.773759i	$0.718374\pi$
$972$	0	0
$973$	− 3.16696e31i	− 1.33493i
$974$	0	0
$975$	2.71786e31	1.12119
$976$	0	0
$977$	1.48196e31	0.598335	0.299167	−	0.954201i	$-0.403291\pi$
$977$	1.48196e31	0.598335	0.299167	+	0.954201i	$0.403291\pi$
$978$	0	0
$979$	− 1.38699e31i	− 0.548094i
$980$	0	0
$981$	4.25488e31i	1.64574i
$982$	0	0
$983$	5.23919e31	1.98359	0.991796	−	0.127832i	$-0.0408017\pi$
$983$	5.23919e31	1.98359	0.991796	+	0.127832i	$0.0408017\pi$
$984$	0	0
$985$	1.67943e30	0.0622417
$986$	0	0
$987$	3.82063e31i	1.38613i
$988$	0	0
$989$	− 2.82723e29i	− 0.0100415i
$990$	0	0
$991$	2.75667e30	0.0958541	0.0479271	−	0.998851i	$-0.484738\pi$
$991$	2.75667e30	0.0958541	0.0479271	+	0.998851i	$0.484738\pi$
$992$	0	0
$993$	−4.35466e30	−0.148247
$994$	0	0
$995$	8.56260e30i	0.285406i
$996$	0	0
$997$	8.03985e30i	0.262391i	0.991357	+	0.131195i	$0.0418815\pi$
$997$	8.03985e30i	0.262391i	−0.991357	+	0.131195i	$0.958118\pi$
$998$	0	0
$999$	−2.02636e31	−0.647558

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.b.c.33.4	yes	28
4.3	odd	2	inner	64.22.b.c.33.26	yes	28
8.3	odd	2	inner	64.22.b.c.33.3	✓	28
8.5	even	2	inner	64.22.b.c.33.25	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.22.b.c.33.3	✓	28	8.3	odd	2	inner
64.22.b.c.33.4	yes	28	1.1	even	1	trivial
64.22.b.c.33.25	yes	28	8.5	even	2	inner
64.22.b.c.33.26	yes	28	4.3	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:	\( N \)	\(=\)	\( 64 = 2^{6} \)
Weight:	\( k \)	\(=\)	\( 22 \)
Character orbit:	\([\chi]\)	\(=\)	64.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-155316. i q^{3} +8.02569e6i q^{5} +7.41453e8 q^{7} -1.36627e10 q^{9} +O(q^{10})\)	\(q-155316. i q^{3} +8.02569e6i q^{5} +7.41453e8 q^{7} -1.36627e10 q^{9} +5.97707e10i q^{11} +4.24293e11i q^{13} +1.24652e12 q^{15} +1.25673e13 q^{17} -1.26199e13i q^{19} -1.15159e14i q^{21} -2.08109e14 q^{23} +4.12425e14 q^{25} +4.97370e14i q^{27} +5.67956e14i q^{29} -5.21413e15 q^{31} +9.28334e15 q^{33} +5.95068e15i q^{35} +4.07415e16i q^{37} +6.58995e16 q^{39} -2.32703e16 q^{41} +1.35853e15i q^{43} -1.09652e17i q^{45} -3.31769e17 q^{47} -8.79320e15 q^{49} -1.95190e18i q^{51} +7.09069e17i q^{53} -4.79701e17 q^{55} -1.96007e18 q^{57} +3.66701e18i q^{59} +7.38001e18i q^{61} -1.01302e19 q^{63} -3.40525e18 q^{65} -4.59630e18i q^{67} +3.23226e19i q^{69} -4.10103e19 q^{71} -4.51378e19 q^{73} -6.40562e19i q^{75} +4.43172e19i q^{77} -1.57430e20 q^{79} -6.56668e19 q^{81} -1.74399e20i q^{83} +1.00861e20i q^{85} +8.82125e19 q^{87} -2.32052e20 q^{89} +3.14594e20i q^{91} +8.09837e20i q^{93} +1.01283e20 q^{95} +8.12195e20 q^{97} -8.16627e20i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 77960422492 q^{9}+O(q^{10}) \) 28 * q - 77960422492 * q^9	\( 28 q - 77960422492 q^{9} + 16832040195288 q^{17} + 202504130118092 q^{25} - 55\!\cdots\!92 q^{33}+ \cdots - 19\!\cdots\!16 q^{97}+O(q^{100}) \) 28 * q - 77960422492 * q^9 + 16832040195288 * q^17 + 202504130118092 * q^25 - 55633095046450992 * q^33 + 265230847130533848 * q^41 + 5944576136704810940 * q^49 - 4544453360963256304 * q^57 - 63340883112821103360 * q^65 + 2760069698648092472 * q^73 + 934709246686586470348 * q^81 - 375882191672218968072 * q^89 - 1953791530192678912616 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more