LMFDB - Embedded newform 64.22.b.c.33.8

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			33.8
Character	$\chi$	$=$	64.33
Dual form			64.22.b.c.33.21

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-113044. i q^{3} +2.14177e7i q^{5} +5.47923e8 q^{7} -2.31853e9 q^{9} +O(q^{10})$	$q-113044. i q^{3} +2.14177e7i q^{5} +5.47923e8 q^{7} -2.31853e9 q^{9} -1.01141e11i q^{11} +4.95652e11i q^{13} +2.42114e12 q^{15} -1.25452e13 q^{17} -2.73037e12i q^{19} -6.19392e13i q^{21} +1.32428e14 q^{23} +1.81193e13 q^{25} -9.20382e14i q^{27} +2.72124e15i q^{29} -4.42476e15 q^{31} -1.14333e16 q^{33} +1.17353e16i q^{35} +8.33163e15i q^{37} +5.60304e16 q^{39} -7.49071e16 q^{41} +2.77506e16i q^{43} -4.96575e16i q^{45} +1.56846e17 q^{47} -2.58326e17 q^{49} +1.41816e18i q^{51} +6.67135e17i q^{53} +2.16620e18 q^{55} -3.08651e17 q^{57} -5.80845e18i q^{59} -9.87978e18i q^{61} -1.27037e18 q^{63} -1.06157e19 q^{65} -2.22140e19i q^{67} -1.49701e19i q^{69} +2.93837e19 q^{71} +3.26872e19 q^{73} -2.04827e18i q^{75} -5.54173e19i q^{77} +5.86091e19 q^{79} -1.28296e20 q^{81} -9.06380e19i q^{83} -2.68690e20i q^{85} +3.07619e20 q^{87} -1.01252e20 q^{89} +2.71579e20i q^{91} +5.00191e20i q^{93} +5.84782e19 q^{95} -6.59651e20 q^{97} +2.34497e20i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 113044.i	− 1.10528i	−0.833419	−	0.552641i	$-0.813620\pi$
$3$	− 113044.i	− 1.10528i	0.833419	−	0.552641i	$-0.186380\pi$
$4$	0	0
$5$	2.14177e7i	0.980817i	0.871493	+	0.490408i	$0.163152\pi$
$5$	2.14177e7i	0.980817i	−0.871493	+	0.490408i	$0.836848\pi$
$6$	0	0
$7$	5.47923e8	0.733145	0.366573	−	0.930389i	$-0.380531\pi$
$7$	5.47923e8	0.733145	0.366573	+	0.930389i	$0.380531\pi$
$8$	0	0
$9$	−2.31853e9	−0.221649
$10$	0	0
$11$	− 1.01141e11i	− 1.17572i	−0.808964	−	0.587858i	$-0.799971\pi$
$11$	− 1.01141e11i	− 1.17572i	0.808964	−	0.587858i	$-0.200029\pi$
$12$	0	0
$13$	4.95652e11i	0.997176i	0.866839	+	0.498588i	$0.166148\pi$
$13$	4.95652e11i	0.997176i	−0.866839	+	0.498588i	$0.833852\pi$
$14$	0	0
$15$	2.42114e12	1.08408
$16$	0	0
$17$	−1.25452e13	−1.50926	−0.754632	−	0.656149i	$-0.772184\pi$
$17$	−1.25452e13	−1.50926	−0.754632	+	0.656149i	$0.772184\pi$
$18$	0	0
$19$	− 2.73037e12i	− 0.102166i	−0.998694	−	0.0510832i	$-0.983733\pi$
$19$	− 2.73037e12i	− 0.102166i	0.998694	−	0.0510832i	$-0.0162674\pi$
$20$	0	0
$21$	− 6.19392e13i	− 0.810333i
$22$	0	0
$23$	1.32428e14	0.666557	0.333279	−	0.942828i	$-0.391845\pi$
$23$	1.32428e14	0.666557	0.333279	+	0.942828i	$0.391845\pi$
$24$	0	0
$25$	1.81193e13	0.0379989
$26$	0	0
$27$	− 9.20382e14i	− 0.860298i
$28$	0	0
$29$	2.72124e15i	1.20112i	0.799578	+	0.600562i	$0.205056\pi$
$29$	2.72124e15i	1.20112i	−0.799578	+	0.600562i	$0.794944\pi$
$30$	0	0
$31$	−4.42476e15	−0.969598	−0.484799	−	0.874626i	$-0.661107\pi$
$31$	−4.42476e15	−0.969598	−0.484799	+	0.874626i	$0.661107\pi$
$32$	0	0
$33$	−1.14333e16	−1.29950
$34$	0	0
$35$	1.17353e16i	0.719081i
$36$	0	0
$37$	8.33163e15i	0.284847i	0.989806	+	0.142424i	$0.0454895\pi$
$37$	8.33163e15i	0.284847i	−0.989806	+	0.142424i	$0.954510\pi$
$38$	0	0
$39$	5.60304e16	1.10216
$40$	0	0
$41$	−7.49071e16	−0.871550	−0.435775	−	0.900056i	$-0.643526\pi$
$41$	−7.49071e16	−0.871550	−0.435775	+	0.900056i	$0.643526\pi$
$42$	0	0
$43$	2.77506e16i	0.195818i	0.995195	+	0.0979091i	$0.0312154\pi$
$43$	2.77506e16i	0.195818i	−0.995195	+	0.0979091i	$0.968785\pi$
$44$	0	0
$45$	− 4.96575e16i	− 0.217397i
$46$	0	0
$47$	1.56846e17	0.434958	0.217479	−	0.976065i	$-0.430217\pi$
$47$	1.56846e17	0.434958	0.217479	+	0.976065i	$0.430217\pi$
$48$	0	0
$49$	−2.58326e17	−0.462498
$50$	0	0
$51$	1.41816e18i	1.66816i
$52$	0	0
$53$	6.67135e17i	0.523983i	0.965070	+	0.261991i	$0.0843792\pi$
$53$	6.67135e17i	0.523983i	−0.965070	+	0.261991i	$0.915621\pi$
$54$	0	0
$55$	2.16620e18	1.15316
$56$	0	0
$57$	−3.08651e17	−0.112923
$58$	0	0
$59$	− 5.80845e18i	− 1.47950i	−0.672883	−	0.739749i	$-0.734944\pi$
$59$	− 5.80845e18i	− 1.47950i	0.672883	−	0.739749i	$-0.265056\pi$
$60$	0	0
$61$	− 9.87978e18i	− 1.77331i	−0.462433	−	0.886654i	$-0.653024\pi$
$61$	− 9.87978e18i	− 1.77331i	0.462433	−	0.886654i	$-0.346976\pi$
$62$	0	0
$63$	−1.27037e18	−0.162501
$64$	0	0
$65$	−1.06157e19	−0.978047
$66$	0	0
$67$	− 2.22140e19i	− 1.48882i	−0.667725	−	0.744408i	$-0.732732\pi$
$67$	− 2.22140e19i	− 1.48882i	0.667725	−	0.744408i	$-0.267268\pi$
$68$	0	0
$69$	− 1.49701e19i	− 0.736734i
$70$	0	0
$71$	2.93837e19	1.07126	0.535628	−	0.844454i	$-0.320075\pi$
$71$	2.93837e19	1.07126	0.535628	+	0.844454i	$0.320075\pi$
$72$	0	0
$73$	3.26872e19	0.890201	0.445101	−	0.895481i	$-0.353168\pi$
$73$	3.26872e19	0.890201	0.445101	+	0.895481i	$0.353168\pi$
$74$	0	0
$75$	− 2.04827e18i	− 0.0419995i
$76$	0	0
$77$	− 5.54173e19i	− 0.861971i
$78$	0	0
$79$	5.86091e19	0.696435	0.348217	−	0.937414i	$-0.386787\pi$
$79$	5.86091e19	0.696435	0.348217	+	0.937414i	$0.386787\pi$
$80$	0	0
$81$	−1.28296e20	−1.17252
$82$	0	0
$83$	− 9.06380e19i	− 0.641195i	−0.947215	−	0.320598i	$-0.896116\pi$
$83$	− 9.06380e19i	− 0.641195i	0.947215	−	0.320598i	$-0.103884\pi$
$84$	0	0
$85$	− 2.68690e20i	− 1.48031i
$86$	0	0
$87$	3.07619e20	1.32758
$88$	0	0
$89$	−1.01252e20	−0.344197	−0.172099	−	0.985080i	$-0.555055\pi$
$89$	−1.01252e20	−0.344197	−0.172099	+	0.985080i	$0.555055\pi$
$90$	0	0
$91$	2.71579e20i	0.731075i
$92$	0	0
$93$	5.00191e20i	1.07168i
$94$	0	0
$95$	5.84782e19	0.100207
$96$	0	0
$97$	−6.59651e20	−0.908262	−0.454131	−	0.890935i	$-0.650050\pi$
$97$	−6.59651e20	−0.908262	−0.454131	+	0.890935i	$0.650050\pi$
$98$	0	0
$99$	2.34497e20i	0.260596i
$100$	0	0
$101$	3.88306e20i	0.349784i	0.984588	+	0.174892i	$0.0559576\pi$
$101$	3.88306e20i	0.349784i	−0.984588	+	0.174892i	$0.944042\pi$
$102$	0	0
$103$	−2.34779e21	−1.72135	−0.860675	−	0.509155i	$-0.829958\pi$
$103$	−2.34779e21	−1.72135	−0.860675	+	0.509155i	$0.829958\pi$
$104$	0	0
$105$	1.32660e21	0.794788
$106$	0	0
$107$	− 1.48184e21i	− 0.728237i	−0.931353	−	0.364119i	$-0.881370\pi$
$107$	− 1.48184e21i	− 0.728237i	0.931353	−	0.364119i	$-0.118630\pi$
$108$	0	0
$109$	6.49101e20i	0.262624i	0.991341	+	0.131312i	$0.0419189\pi$
$109$	6.49101e20i	0.262624i	−0.991341	+	0.131312i	$0.958081\pi$
$110$	0	0
$111$	9.41839e20	0.314837
$112$	0	0
$113$	6.00001e21	1.66276	0.831378	−	0.555708i	$-0.187553\pi$
$113$	6.00001e21	1.66276	0.831378	+	0.555708i	$0.187553\pi$
$114$	0	0
$115$	2.83630e21i	0.653770i
$116$	0	0
$117$	− 1.14918e21i	− 0.221023i
$118$	0	0
$119$	−6.87382e21	−1.10651
$120$	0	0
$121$	−2.82918e21	−0.382309
$122$	0	0
$123$	8.46777e21i	0.963309i
$124$	0	0
$125$	1.06008e22i	1.01809i
$126$	0	0
$127$	−1.32176e22	−1.07452	−0.537260	−	0.843417i	$-0.680541\pi$
$127$	−1.32176e22	−1.07452	−0.537260	+	0.843417i	$0.680541\pi$
$128$	0	0
$129$	3.13703e21	0.216434
$130$	0	0
$131$	− 1.18192e22i	− 0.693810i	−0.937900	−	0.346905i	$-0.887233\pi$
$131$	− 1.18192e22i	− 0.693810i	0.937900	−	0.346905i	$-0.112767\pi$
$132$	0	0
$133$	− 1.49603e21i	− 0.0749029i
$134$	0	0
$135$	1.97125e22	0.843794
$136$	0	0
$137$	−2.98802e22	−1.09602	−0.548008	−	0.836473i	$-0.684614\pi$
$137$	−2.98802e22	−1.09602	−0.548008	+	0.836473i	$0.684614\pi$
$138$	0	0
$139$	9.83642e21i	0.309872i	0.987925	+	0.154936i	$0.0495171\pi$
$139$	9.83642e21i	0.309872i	−0.987925	+	0.154936i	$0.950483\pi$
$140$	0	0
$141$	− 1.77305e22i	− 0.480751i
$142$	0	0
$143$	5.01306e22	1.17240
$144$	0	0
$145$	−5.82827e22	−1.17808
$146$	0	0
$147$	2.92022e22i	0.511191i
$148$	0	0
$149$	− 1.17376e23i	− 1.78288i	−0.453136	−	0.891442i	$-0.649695\pi$
$149$	− 1.17376e23i	− 1.78288i	0.453136	−	0.891442i	$-0.350305\pi$
$150$	0	0
$151$	−6.42882e22	−0.848933	−0.424467	−	0.905444i	$-0.639538\pi$
$151$	−6.42882e22	−0.848933	−0.424467	+	0.905444i	$0.639538\pi$
$152$	0	0
$153$	2.90865e22	0.334527
$154$	0	0
$155$	− 9.47681e22i	− 0.950997i
$156$	0	0
$157$	− 4.13381e21i	− 0.0362580i	−0.999836	−	0.0181290i	$-0.994229\pi$
$157$	− 4.13381e21i	− 0.0362580i	0.999836	−	0.0181290i	$-0.00577096\pi$
$158$	0	0
$159$	7.54154e22	0.579149
$160$	0	0
$161$	7.25603e22	0.488683
$162$	0	0
$163$	3.60202e22i	0.213097i	0.994308	+	0.106548i	$0.0339799\pi$
$163$	3.60202e22i	0.213097i	−0.994308	+	0.106548i	$0.966020\pi$
$164$	0	0
$165$	− 2.44875e23i	− 1.27457i
$166$	0	0
$167$	−2.98093e23	−1.36719	−0.683596	−	0.729861i	$-0.739585\pi$
$167$	−2.98093e23	−1.36719	−0.683596	+	0.729861i	$0.739585\pi$
$168$	0	0
$169$	1.39333e21	0.00563954
$170$	0	0
$171$	6.33044e21i	0.0226451i
$172$	0	0
$173$	− 3.60739e23i	− 1.14211i	−0.820910	−	0.571057i	$-0.806534\pi$
$173$	− 3.60739e23i	− 1.14211i	0.820910	−	0.571057i	$-0.193466\pi$
$174$	0	0
$175$	9.92797e21	0.0278587
$176$	0	0
$177$	−6.56609e23	−1.63526
$178$	0	0
$179$	1.18642e23i	0.262593i	0.991343	+	0.131296i	$0.0419139\pi$
$179$	1.18642e23i	0.262593i	−0.991343	+	0.131296i	$0.958086\pi$
$180$	0	0
$181$	9.59757e23i	1.89032i	0.326601	+	0.945162i	$0.394097\pi$
$181$	9.59757e23i	1.89032i	−0.326601	+	0.945162i	$0.605903\pi$
$182$	0	0
$183$	−1.11685e24	−1.96001
$184$	0	0
$185$	−1.78444e23	−0.279383
$186$	0	0
$187$	1.26883e24i	1.77447i
$188$	0	0
$189$	− 5.04299e23i	− 0.630723i
$190$	0	0
$191$	−5.44317e23	−0.609540	−0.304770	−	0.952426i	$-0.598580\pi$
$191$	−5.44317e23	−0.609540	−0.304770	+	0.952426i	$0.598580\pi$
$192$	0	0
$193$	1.28420e24	1.28909	0.644543	−	0.764568i	$-0.277047\pi$
$193$	1.28420e24	1.28909	0.644543	+	0.764568i	$0.277047\pi$
$194$	0	0
$195$	1.20004e24i	1.08102i
$196$	0	0
$197$	− 1.10906e24i	− 0.897549i	−0.893645	−	0.448775i	$-0.851861\pi$
$197$	− 1.10906e24i	− 0.897549i	0.893645	−	0.448775i	$-0.148139\pi$
$198$	0	0
$199$	−2.35928e24	−1.71721	−0.858604	−	0.512640i	$-0.828668\pi$
$199$	−2.35928e24	−1.71721	−0.858604	+	0.512640i	$0.828668\pi$
$200$	0	0
$201$	−2.51115e24	−1.64556
$202$	0	0
$203$	1.49103e24i	0.880599i
$204$	0	0
$205$	− 1.60434e24i	− 0.854831i
$206$	0	0
$207$	−3.07038e23	−0.147742
$208$	0	0
$209$	−2.76151e23	−0.120119
$210$	0	0
$211$	− 1.16947e24i	− 0.460282i	−0.973157	−	0.230141i	$-0.926081\pi$
$211$	− 1.16947e24i	− 0.460282i	0.973157	−	0.230141i	$-0.0739188\pi$
$212$	0	0
$213$	− 3.32164e24i	− 1.18404i
$214$	0	0
$215$	−5.94353e23	−0.192062
$216$	0	0
$217$	−2.42443e24	−0.710856
$218$	0	0
$219$	− 3.69509e24i	− 0.983924i
$220$	0	0
$221$	− 6.21807e24i	− 1.50500i
$222$	0	0
$223$	−7.94569e24	−1.74957	−0.874784	−	0.484513i	$-0.838997\pi$
$223$	−7.94569e24	−1.74957	−0.874784	+	0.484513i	$0.838997\pi$
$224$	0	0
$225$	−4.20101e22	−0.00842242
$226$	0	0
$227$	5.62705e24i	1.02804i	0.857779	+	0.514019i	$0.171844\pi$
$227$	5.62705e24i	1.02804i	−0.857779	+	0.514019i	$0.828156\pi$
$228$	0	0
$229$	− 1.06522e25i	− 1.77487i	−0.460938	−	0.887433i	$-0.652487\pi$
$229$	− 1.06522e25i	− 1.77487i	0.460938	−	0.887433i	$-0.347513\pi$
$230$	0	0
$231$	−6.26458e24	−0.952721
$232$	0	0
$233$	−9.29267e24	−1.29093	−0.645466	−	0.763789i	$-0.723337\pi$
$233$	−9.29267e24	−1.29093	−0.645466	+	0.763789i	$0.723337\pi$
$234$	0	0
$235$	3.35929e24i	0.426614i
$236$	0	0
$237$	− 6.62539e24i	− 0.769757i
$238$	0	0
$239$	7.78342e24	0.827928	0.413964	−	0.910293i	$-0.364144\pi$
$239$	7.78342e24	0.827928	0.413964	+	0.910293i	$0.364144\pi$
$240$	0	0
$241$	−1.28258e25	−1.24999	−0.624995	−	0.780629i	$-0.714899\pi$
$241$	−1.28258e25	−1.24999	−0.624995	+	0.780629i	$0.714899\pi$
$242$	0	0
$243$	4.87554e24i	0.435669i
$244$	0	0
$245$	− 5.53275e24i	− 0.453626i
$246$	0	0
$247$	1.35331e24	0.101878
$248$	0	0
$249$	−1.02461e25	−0.708702
$250$	0	0
$251$	2.31193e25i	1.47028i	0.677915	+	0.735141i	$0.262884\pi$
$251$	2.31193e25i	1.47028i	−0.677915	+	0.735141i	$0.737116\pi$
$252$	0	0
$253$	− 1.33938e25i	− 0.783682i
$254$	0	0
$255$	−3.03737e25	−1.63616
$256$	0	0
$257$	1.91470e25	0.950172	0.475086	−	0.879939i	$-0.342417\pi$
$257$	1.91470e25	0.950172	0.475086	+	0.879939i	$0.342417\pi$
$258$	0	0
$259$	4.56509e24i	0.208834i
$260$	0	0
$261$	− 6.30927e24i	− 0.266228i
$262$	0	0
$263$	−3.04029e25	−1.18408	−0.592039	−	0.805910i	$-0.701677\pi$
$263$	−3.04029e25	−1.18408	−0.592039	+	0.805910i	$0.701677\pi$
$264$	0	0
$265$	−1.42885e25	−0.513931
$266$	0	0
$267$	1.14459e25i	0.380435i
$268$	0	0
$269$	3.63656e25i	1.11761i	0.829298	+	0.558807i	$0.188741\pi$
$269$	3.63656e25i	1.11761i	−0.829298	+	0.558807i	$0.811259\pi$
$270$	0	0
$271$	−5.62095e25	−1.59820	−0.799101	−	0.601196i	$-0.794691\pi$
$271$	−5.62095e25	−1.59820	−0.799101	+	0.601196i	$0.794691\pi$
$272$	0	0
$273$	3.07003e25	0.808044
$274$	0	0
$275$	− 1.83260e24i	− 0.0446759i
$276$	0	0
$277$	− 2.98642e24i	− 0.0674703i	−0.999431	−	0.0337352i	$-0.989260\pi$
$277$	− 2.98642e24i	− 0.0674703i	0.999431	−	0.0337352i	$-0.0107403\pi$
$278$	0	0
$279$	1.02589e25	0.214910
$280$	0	0
$281$	−4.37979e25	−0.851210	−0.425605	−	0.904909i	$-0.639939\pi$
$281$	−4.37979e25	−0.851210	−0.425605	+	0.904909i	$0.639939\pi$
$282$	0	0
$283$	− 7.10913e25i	− 1.28250i	−0.767331	−	0.641252i	$-0.778415\pi$
$283$	− 7.10913e25i	− 1.28250i	0.767331	−	0.641252i	$-0.221585\pi$
$284$	0	0
$285$	− 6.61059e24i	− 0.110757i
$286$	0	0
$287$	−4.10433e25	−0.638973
$288$	0	0
$289$	8.82909e25	1.27788
$290$	0	0
$291$	7.45695e25i	1.00389i
$292$	0	0
$293$	− 8.62661e25i	− 1.08076i	−0.841421	−	0.540381i	$-0.818280\pi$
$293$	− 8.62661e25i	− 1.08076i	0.841421	−	0.540381i	$-0.181720\pi$
$294$	0	0
$295$	1.24404e26	1.45112
$296$	0	0
$297$	−9.30880e25	−1.01147
$298$	0	0
$299$	6.56382e25i	0.664675i
$300$	0	0
$301$	1.52052e25i	0.143563i
$302$	0	0
$303$	4.38956e25	0.386610
$304$	0	0
$305$	2.11602e26	1.73929
$306$	0	0
$307$	− 1.85195e26i	− 1.42127i	−0.703563	−	0.710633i	$-0.748409\pi$
$307$	− 1.85195e26i	− 1.42127i	0.703563	−	0.710633i	$-0.251591\pi$
$308$	0	0
$309$	2.65403e26i	1.90258i
$310$	0	0
$311$	1.69425e26	1.13500	0.567498	−	0.823375i	$-0.307912\pi$
$311$	1.69425e26	1.13500	0.567498	+	0.823375i	$0.307912\pi$
$312$	0	0
$313$	−2.63392e26	−1.64963	−0.824817	−	0.565400i	$-0.808722\pi$
$313$	−2.63392e26	−1.64963	−0.824817	+	0.565400i	$0.808722\pi$
$314$	0	0
$315$	− 2.72085e25i	− 0.159384i
$316$	0	0
$317$	− 7.44745e25i	− 0.408212i	−0.978949	−	0.204106i	$-0.934571\pi$
$317$	− 7.44745e25i	− 0.408212i	0.978949	−	0.204106i	$-0.0654287\pi$
$318$	0	0
$319$	2.75228e26	1.41218
$320$	0	0
$321$	−1.67513e26	−0.804908
$322$	0	0
$323$	3.42531e25i	0.154196i
$324$	0	0
$325$	8.98086e24i	0.0378916i
$326$	0	0
$327$	7.33768e25	0.290273
$328$	0	0
$329$	8.59397e25	0.318887
$330$	0	0
$331$	− 2.19401e26i	− 0.763914i	−0.924180	−	0.381957i	$-0.875250\pi$
$331$	− 2.19401e26i	− 0.763914i	0.924180	−	0.381957i	$-0.124750\pi$
$332$	0	0
$333$	− 1.93171e25i	− 0.0631362i
$334$	0	0
$335$	4.75772e26	1.46025
$336$	0	0
$337$	6.44130e26	1.85720	0.928602	−	0.371076i	$-0.121011\pi$
$337$	6.44130e26	1.85720	0.928602	+	0.371076i	$0.121011\pi$
$338$	0	0
$339$	− 6.78264e26i	− 1.83781i
$340$	0	0
$341$	4.47523e26i	1.13997i
$342$	0	0
$343$	−4.47583e26	−1.07222
$344$	0	0
$345$	3.20626e26	0.722601
$346$	0	0
$347$	− 2.20424e26i	− 0.467519i	−0.972294	−	0.233759i	$-0.924897\pi$
$347$	− 2.20424e26i	− 0.467519i	0.972294	−	0.233759i	$-0.0751028\pi$
$348$	0	0
$349$	6.75335e26i	1.34850i	0.738501	+	0.674252i	$0.235534\pi$
$349$	6.75335e26i	1.34850i	−0.738501	+	0.674252i	$0.764466\pi$
$350$	0	0
$351$	4.56190e26	0.857868
$352$	0	0
$353$	3.73260e26	0.661268	0.330634	−	0.943759i	$-0.392737\pi$
$353$	3.73260e26	0.661268	0.330634	+	0.943759i	$0.392737\pi$
$354$	0	0
$355$	6.29331e26i	1.05071i
$356$	0	0
$357$	7.77042e26i	1.22301i
$358$	0	0
$359$	1.12264e26	0.166628	0.0833141	−	0.996523i	$-0.473450\pi$
$359$	1.12264e26	0.166628	0.0833141	+	0.996523i	$0.473450\pi$
$360$	0	0
$361$	7.06755e26	0.989562
$362$	0	0
$363$	3.19821e26i	0.422559i
$364$	0	0
$365$	7.00085e26i	0.873124i
$366$	0	0
$367$	−6.14616e26	−0.723787	−0.361893	−	0.932220i	$-0.617870\pi$
$367$	−6.14616e26	−0.723787	−0.361893	+	0.932220i	$0.617870\pi$
$368$	0	0
$369$	1.73674e26	0.193178
$370$	0	0
$371$	3.65538e26i	0.384155i
$372$	0	0
$373$	− 6.32059e26i	− 0.627791i	−0.949458	−	0.313896i	$-0.898366\pi$
$373$	− 6.32059e26i	− 0.627791i	0.949458	−	0.313896i	$-0.101634\pi$
$374$	0	0
$375$	1.19836e27	1.12527
$376$	0	0
$377$	−1.34879e27	−1.19773
$378$	0	0
$379$	− 1.51223e27i	− 1.27030i	−0.772388	−	0.635151i	$-0.780938\pi$
$379$	− 1.51223e27i	− 1.27030i	0.772388	−	0.635151i	$-0.219062\pi$
$380$	0	0
$381$	1.49417e27i	1.18765i
$382$	0	0
$383$	1.54176e27	1.15993	0.579963	−	0.814643i	$-0.303067\pi$
$383$	1.54176e27	1.15993	0.579963	+	0.814643i	$0.303067\pi$
$384$	0	0
$385$	1.18691e27	0.845435
$386$	0	0
$387$	− 6.43405e25i	− 0.0434029i
$388$	0	0
$389$	7.81828e26i	0.499621i	0.968295	+	0.249810i	$0.0803683\pi$
$389$	7.81828e26i	0.499621i	−0.968295	+	0.249810i	$0.919632\pi$
$390$	0	0
$391$	−1.66134e27	−1.00601
$392$	0	0
$393$	−1.33609e27	−0.766856
$394$	0	0
$395$	1.25527e27i	0.683075i
$396$	0	0
$397$	− 6.54287e26i	− 0.337651i	−0.985646	−	0.168826i	$-0.946003\pi$
$397$	− 6.54287e26i	− 0.337651i	0.985646	−	0.168826i	$-0.0539975\pi$
$398$	0	0
$399$	−1.69117e26	−0.0827888
$400$	0	0
$401$	−2.99175e27	−1.38966	−0.694832	−	0.719172i	$-0.744521\pi$
$401$	−2.99175e27	−1.38966	−0.694832	+	0.719172i	$0.744521\pi$
$402$	0	0
$403$	− 2.19314e27i	− 0.966860i
$404$	0	0
$405$	− 2.74781e27i	− 1.15003i
$406$	0	0
$407$	8.42667e26	0.334900
$408$	0	0
$409$	3.26081e27	1.23092	0.615461	−	0.788167i	$-0.288970\pi$
$409$	3.26081e27	1.23092	0.615461	+	0.788167i	$0.288970\pi$
$410$	0	0
$411$	3.37776e27i	1.21141i
$412$	0	0
$413$	− 3.18258e27i	− 1.08469i
$414$	0	0
$415$	1.94126e27	0.628895
$416$	0	0
$417$	1.11195e27	0.342496
$418$	0	0
$419$	− 2.55201e27i	− 0.747541i	−0.927521	−	0.373770i	$-0.878065\pi$
$419$	− 2.55201e27i	− 0.747541i	0.927521	−	0.373770i	$-0.121935\pi$
$420$	0	0
$421$	− 3.58655e27i	− 0.999345i	−0.866214	−	0.499673i	$-0.833454\pi$
$421$	− 3.58655e27i	− 0.999345i	0.866214	−	0.499673i	$-0.166546\pi$
$422$	0	0
$423$	−3.63653e26	−0.0964080
$424$	0	0
$425$	−2.27311e26	−0.0573503
$426$	0	0
$427$	− 5.41336e27i	− 1.30009i
$428$	0	0
$429$	− 5.66695e27i	− 1.29583i
$430$	0	0
$431$	−6.07118e27	−1.32209	−0.661046	−	0.750346i	$-0.729887\pi$
$431$	−6.07118e27	−1.32209	−0.661046	+	0.750346i	$0.729887\pi$
$432$	0	0
$433$	−6.45425e27	−1.33882	−0.669411	−	0.742892i	$-0.733454\pi$
$433$	−6.45425e27	−1.33882	−0.669411	+	0.742892i	$0.733454\pi$
$434$	0	0
$435$	6.58849e27i	1.30211i
$436$	0	0
$437$	− 3.61577e26i	− 0.0680998i
$438$	0	0
$439$	−1.29991e27	−0.233365	−0.116682	−	0.993169i	$-0.537226\pi$
$439$	−1.29991e27	−0.233365	−0.116682	+	0.993169i	$0.537226\pi$
$440$	0	0
$441$	5.98937e26	0.102512
$442$	0	0
$443$	3.19692e27i	0.521786i	0.965368	+	0.260893i	$0.0840169\pi$
$443$	3.19692e27i	0.521786i	−0.965368	+	0.260893i	$0.915983\pi$
$444$	0	0
$445$	− 2.16858e27i	− 0.337594i
$446$	0	0
$447$	−1.32686e28	−1.97059
$448$	0	0
$449$	−1.18224e28	−1.67540	−0.837702	−	0.546128i	$-0.816101\pi$
$449$	−1.18224e28	−1.67540	−0.837702	+	0.546128i	$0.816101\pi$
$450$	0	0
$451$	7.57615e27i	1.02470i
$452$	0	0
$453$	7.26738e27i	0.938311i
$454$	0	0
$455$	−5.81660e27	−0.717051
$456$	0	0
$457$	1.06899e28	1.25850	0.629248	−	0.777205i	$-0.283363\pi$
$457$	1.06899e28	1.25850	0.629248	+	0.777205i	$0.283363\pi$
$458$	0	0
$459$	1.15464e28i	1.29842i
$460$	0	0
$461$	6.69871e27i	0.719667i	0.933017	+	0.359833i	$0.117166\pi$
$461$	6.69871e27i	0.719667i	−0.933017	+	0.359833i	$0.882834\pi$
$462$	0	0
$463$	−1.53210e27	−0.157285	−0.0786424	−	0.996903i	$-0.525059\pi$
$463$	−1.53210e27	−0.157285	−0.0786424	+	0.996903i	$0.525059\pi$
$464$	0	0
$465$	−1.07129e28	−1.05112
$466$	0	0
$467$	3.79701e27i	0.356135i	0.984018	+	0.178068i	$0.0569846\pi$
$467$	3.79701e27i	0.356135i	−0.984018	+	0.178068i	$0.943015\pi$
$468$	0	0
$469$	− 1.21715e28i	− 1.09152i
$470$	0	0
$471$	−4.67301e26	−0.0400753
$472$	0	0
$473$	2.80671e27	0.230227
$474$	0	0
$475$	− 4.94723e25i	− 0.00388221i
$476$	0	0
$477$	− 1.54677e27i	− 0.116140i
$478$	0	0
$479$	−1.80986e28	−1.30053	−0.650266	−	0.759707i	$-0.725343\pi$
$479$	−1.80986e28	−1.30053	−0.650266	+	0.759707i	$0.725343\pi$
$480$	0	0
$481$	−4.12959e27	−0.284043
$482$	0	0
$483$	− 8.20249e27i	− 0.540133i
$484$	0	0
$485$	− 1.41282e28i	− 0.890839i
$486$	0	0
$487$	−1.00750e28	−0.608402	−0.304201	−	0.952608i	$-0.598389\pi$
$487$	−1.00750e28	−0.608402	−0.304201	+	0.952608i	$0.598389\pi$
$488$	0	0
$489$	4.07186e27	0.235532
$490$	0	0
$491$	7.32090e27i	0.405703i	0.979209	+	0.202852i	$0.0650209\pi$
$491$	7.32090e27i	0.405703i	−0.979209	+	0.202852i	$0.934979\pi$
$492$	0	0
$493$	− 3.41386e28i	− 1.81281i
$494$	0	0
$495$	−5.02240e27	−0.255597
$496$	0	0
$497$	1.61000e28	0.785387
$498$	0	0
$499$	7.65061e27i	0.357800i	0.983867	+	0.178900i	$0.0572538\pi$
$499$	7.65061e27i	0.357800i	−0.983867	+	0.178900i	$0.942746\pi$
$500$	0	0
$501$	3.36976e28i	1.51113i
$502$	0	0
$503$	3.57060e28	1.53560	0.767799	−	0.640691i	$-0.221352\pi$
$503$	3.57060e28	1.53560	0.767799	+	0.640691i	$0.221352\pi$
$504$	0	0
$505$	−8.31662e27	−0.343074
$506$	0	0
$507$	− 1.57507e26i	− 0.00623329i
$508$	0	0
$509$	2.64613e27i	0.100479i	0.998737	+	0.0502394i	$0.0159984\pi$
$509$	2.64613e27i	0.100479i	−0.998737	+	0.0502394i	$0.984002\pi$
$510$	0	0
$511$	1.79101e28	0.652647
$512$	0	0
$513$	−2.51298e27	−0.0878936
$514$	0	0
$515$	− 5.02844e28i	− 1.68833i
$516$	0	0
$517$	− 1.58635e28i	− 0.511387i
$518$	0	0
$519$	−4.07793e28	−1.26236
$520$	0	0
$521$	−1.65583e28	−0.492288	−0.246144	−	0.969233i	$-0.579164\pi$
$521$	−1.65583e28	−0.492288	−0.246144	+	0.969233i	$0.579164\pi$
$522$	0	0
$523$	− 4.65365e28i	− 1.32900i	−0.747287	−	0.664502i	$-0.768644\pi$
$523$	− 4.65365e28i	− 1.32900i	0.747287	−	0.664502i	$-0.231356\pi$
$524$	0	0
$525$	− 1.12229e27i	− 0.0307917i
$526$	0	0
$527$	5.55096e28	1.46338
$528$	0	0
$529$	−2.19344e28	−0.555702
$530$	0	0
$531$	1.34671e28i	0.327929i
$532$	0	0
$533$	− 3.71279e28i	− 0.869089i
$534$	0	0
$535$	3.17377e28	0.714267
$536$	0	0
$537$	1.34118e28	0.290239
$538$	0	0
$539$	2.61273e28i	0.543766i
$540$	0	0
$541$	9.63648e28i	1.92907i	0.263963	+	0.964533i	$0.414970\pi$
$541$	9.63648e28i	1.92907i	−0.263963	+	0.964533i	$0.585030\pi$
$542$	0	0
$543$	1.08495e29	2.08934
$544$	0	0
$545$	−1.39022e28	−0.257586
$546$	0	0
$547$	2.31842e28i	0.413357i	0.978409	+	0.206678i	$0.0662654\pi$
$547$	2.31842e28i	0.413357i	−0.978409	+	0.206678i	$0.933735\pi$
$548$	0	0
$549$	2.29066e28i	0.393052i
$550$	0	0
$551$	7.42999e27	0.122715
$552$	0	0
$553$	3.21133e28	0.510588
$554$	0	0
$555$	2.01720e28i	0.308797i
$556$	0	0
$557$	6.35606e28i	0.936932i	0.883481	+	0.468466i	$0.155193\pi$
$557$	6.35606e28i	0.936932i	−0.883481	+	0.468466i	$0.844807\pi$
$558$	0	0
$559$	−1.37546e28	−0.195265
$560$	0	0
$561$	1.43434e29	1.96129
$562$	0	0
$563$	− 1.72019e28i	− 0.226589i	−0.993561	−	0.113295i	$-0.963860\pi$
$563$	− 1.72019e28i	− 0.226589i	0.993561	−	0.113295i	$-0.0361404\pi$
$564$	0	0
$565$	1.28506e29i	1.63086i
$566$	0	0
$567$	−7.02963e28	−0.859628
$568$	0	0
$569$	2.38764e28	0.281377	0.140689	−	0.990054i	$-0.455068\pi$
$569$	2.38764e28	0.281377	0.140689	+	0.990054i	$0.455068\pi$
$570$	0	0
$571$	− 8.74933e27i	− 0.0993792i	−0.998765	−	0.0496896i	$-0.984177\pi$
$571$	− 8.74933e27i	− 0.0993792i	0.998765	−	0.0496896i	$-0.0158232\pi$
$572$	0	0
$573$	6.15317e28i	0.673713i
$574$	0	0
$575$	2.39950e27	0.0253284
$576$	0	0
$577$	3.06705e28	0.312158	0.156079	−	0.987745i	$-0.450114\pi$
$577$	3.06705e28	0.312158	0.156079	+	0.987745i	$0.450114\pi$
$578$	0	0
$579$	− 1.45171e29i	− 1.42481i
$580$	0	0
$581$	− 4.96626e28i	− 0.470089i
$582$	0	0
$583$	6.74744e28	0.616055
$584$	0	0
$585$	2.46129e28	0.216783
$586$	0	0
$587$	− 1.96975e28i	− 0.167383i	−0.996492	−	0.0836914i	$-0.973329\pi$
$587$	− 1.96975e28i	− 0.167383i	0.996492	−	0.0836914i	$-0.0266710\pi$
$588$	0	0
$589$	1.20812e28i	0.0990604i
$590$	0	0
$591$	−1.25372e29	−0.992045
$592$	0	0
$593$	1.31524e29	1.00446	0.502228	−	0.864735i	$-0.332514\pi$
$593$	1.31524e29	1.00446	0.502228	+	0.864735i	$0.332514\pi$
$594$	0	0
$595$	− 1.47221e29i	− 1.08528i
$596$	0	0
$597$	2.66702e29i	1.89800i
$598$	0	0
$599$	2.14257e29	1.47216	0.736078	−	0.676897i	$-0.236676\pi$
$599$	2.14257e29	1.47216	0.736078	+	0.676897i	$0.236676\pi$
$600$	0	0
$601$	−1.12340e29	−0.745335	−0.372667	−	0.927965i	$-0.621557\pi$
$601$	−1.12340e29	−0.745335	−0.372667	+	0.927965i	$0.621557\pi$
$602$	0	0
$603$	5.15037e28i	0.329995i
$604$	0	0
$605$	− 6.05945e28i	− 0.374975i
$606$	0	0
$607$	1.87733e29	1.12217	0.561084	−	0.827759i	$-0.310384\pi$
$607$	1.87733e29	1.12217	0.561084	+	0.827759i	$0.310384\pi$
$608$	0	0
$609$	1.68552e29	0.973310
$610$	0	0
$611$	7.77413e28i	0.433729i
$612$	0	0
$613$	− 7.00320e28i	− 0.377539i	−0.982021	−	0.188769i	$-0.939550\pi$
$613$	− 7.00320e28i	− 0.377539i	0.982021	−	0.188769i	$-0.0604499\pi$
$614$	0	0
$615$	−1.81360e29	−0.944829
$616$	0	0
$617$	−2.24600e29	−1.13088	−0.565439	−	0.824790i	$-0.691293\pi$
$617$	−2.24600e29	−1.13088	−0.565439	+	0.824790i	$0.691293\pi$
$618$	0	0
$619$	1.70814e29i	0.831328i	0.909518	+	0.415664i	$0.136451\pi$
$619$	1.70814e29i	0.831328i	−0.909518	+	0.415664i	$0.863549\pi$
$620$	0	0
$621$	− 1.21884e29i	− 0.573437i
$622$	0	0
$623$	−5.54782e28	−0.252347
$624$	0	0
$625$	−2.18405e29	−0.960557
$626$	0	0
$627$	3.12172e28i	0.132765i
$628$	0	0
$629$	− 1.04522e29i	− 0.429910i
$630$	0	0
$631$	2.09697e29	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$631$	2.09697e29	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$632$	0	0
$633$	−1.32202e29	−0.508742
$634$	0	0
$635$	− 2.83091e29i	− 1.05391i
$636$	0	0
$637$	− 1.28040e29i	− 0.461192i
$638$	0	0
$639$	−6.81269e28	−0.237443
$640$	0	0
$641$	4.29875e29	1.44988	0.724941	−	0.688811i	$-0.241867\pi$
$641$	4.29875e29	1.44988	0.724941	+	0.688811i	$0.241867\pi$
$642$	0	0
$643$	2.53411e29i	0.827200i	0.910459	+	0.413600i	$0.135729\pi$
$643$	2.53411e29i	0.827200i	−0.910459	+	0.413600i	$0.864271\pi$
$644$	0	0
$645$	6.71879e28i	0.212282i
$646$	0	0
$647$	3.69344e29	1.12963	0.564814	−	0.825218i	$-0.308948\pi$
$647$	3.69344e29	1.12963	0.564814	+	0.825218i	$0.308948\pi$
$648$	0	0
$649$	−5.87471e29	−1.73947
$650$	0	0
$651$	2.74066e29i	0.785697i
$652$	0	0
$653$	− 6.30520e29i	− 1.75029i	−0.483858	−	0.875146i	$-0.660765\pi$
$653$	− 6.30520e29i	− 1.75029i	0.483858	−	0.875146i	$-0.339235\pi$
$654$	0	0
$655$	2.53141e29	0.680500
$656$	0	0
$657$	−7.57863e28	−0.197312
$658$	0	0
$659$	− 5.82887e29i	− 1.46990i	−0.678121	−	0.734950i	$-0.737206\pi$
$659$	− 5.82887e29i	− 1.46990i	0.678121	−	0.734950i	$-0.262794\pi$
$660$	0	0
$661$	− 7.38100e29i	− 1.80302i	−0.432761	−	0.901509i	$-0.642460\pi$
$661$	− 7.38100e29i	− 1.80302i	0.432761	−	0.901509i	$-0.357540\pi$
$662$	0	0
$663$	−7.02914e29	−1.66345
$664$	0	0
$665$	3.20416e28	0.0734660
$666$	0	0
$667$	3.60368e29i	0.800618i
$668$	0	0
$669$	8.98211e29i	1.93377i
$670$	0	0
$671$	−9.99248e29	−2.08491
$672$	0	0
$673$	−6.09143e29	−1.23186	−0.615930	−	0.787801i	$-0.711219\pi$
$673$	−6.09143e29	−1.23186	−0.615930	+	0.787801i	$0.711219\pi$
$674$	0	0
$675$	− 1.66767e28i	− 0.0326903i
$676$	0	0
$677$	− 2.44707e29i	− 0.465013i	−0.972595	−	0.232506i	$-0.925307\pi$
$677$	− 2.44707e29i	− 0.465013i	0.972595	−	0.232506i	$-0.0746926\pi$
$678$	0	0
$679$	−3.61438e29	−0.665888
$680$	0	0
$681$	6.36103e29	1.13627
$682$	0	0
$683$	5.95435e29i	1.03138i	0.856777	+	0.515688i	$0.172464\pi$
$683$	5.95435e29i	1.03138i	−0.856777	+	0.515688i	$0.827536\pi$
$684$	0	0
$685$	− 6.39964e29i	− 1.07499i
$686$	0	0
$687$	−1.20416e30	−1.96173
$688$	0	0
$689$	−3.30667e29	−0.522503
$690$	0	0
$691$	− 7.09780e29i	− 1.08794i	−0.839105	−	0.543969i	$-0.816921\pi$
$691$	− 7.09780e29i	− 1.08794i	0.839105	−	0.543969i	$-0.183079\pi$
$692$	0	0
$693$	1.28487e29i	0.191055i
$694$	0	0
$695$	−2.10674e29	−0.303927
$696$	0	0
$697$	9.39727e29	1.31540
$698$	0	0
$699$	1.05048e30i	1.42684i
$700$	0	0
$701$	2.52807e29i	0.333235i	0.986022	+	0.166618i	$0.0532845\pi$
$701$	2.52807e29i	0.333235i	−0.986022	+	0.166618i	$0.946715\pi$
$702$	0	0
$703$	2.27484e28	0.0291018
$704$	0	0
$705$	3.79746e29	0.471528
$706$	0	0
$707$	2.12762e29i	0.256442i
$708$	0	0
$709$	− 3.06144e29i	− 0.358212i	−0.983830	−	0.179106i	$-0.942680\pi$
$709$	− 3.06144e29i	− 0.358212i	0.983830	−	0.179106i	$-0.0573204\pi$
$710$	0	0
$711$	−1.35887e29	−0.154364
$712$	0	0
$713$	−5.85961e29	−0.646292
$714$	0	0
$715$	1.07368e30i	1.14991i
$716$	0	0
$717$	− 8.79867e29i	− 0.915095i
$718$	0	0
$719$	5.62728e29	0.568389	0.284194	−	0.958767i	$-0.408274\pi$
$719$	5.62728e29	0.568389	0.284194	+	0.958767i	$0.408274\pi$
$720$	0	0
$721$	−1.28641e30	−1.26200
$722$	0	0
$723$	1.44988e30i	1.38159i
$724$	0	0
$725$	4.93069e28i	0.0456414i
$726$	0	0
$727$	−1.58100e29	−0.142174	−0.0710870	−	0.997470i	$-0.522647\pi$
$727$	−1.58100e29	−0.142174	−0.0710870	+	0.997470i	$0.522647\pi$
$728$	0	0
$729$	−7.90873e29	−0.690983
$730$	0	0
$731$	− 3.48137e29i	− 0.295541i
$732$	0	0
$733$	1.12300e30i	0.926380i	0.886259	+	0.463190i	$0.153295\pi$
$733$	1.12300e30i	0.926380i	−0.886259	+	0.463190i	$0.846705\pi$
$734$	0	0
$735$	−6.25443e29	−0.501384
$736$	0	0
$737$	−2.24674e30	−1.75042
$738$	0	0
$739$	− 2.20129e30i	− 1.66691i	−0.552590	−	0.833453i	$-0.686360\pi$
$739$	− 2.20129e30i	− 1.66691i	0.552590	−	0.833453i	$-0.313640\pi$
$740$	0	0
$741$	− 1.52984e29i	− 0.112604i
$742$	0	0
$743$	−2.12300e29	−0.151904	−0.0759518	−	0.997111i	$-0.524200\pi$
$743$	−2.12300e29	−0.151904	−0.0759518	+	0.997111i	$0.524200\pi$
$744$	0	0
$745$	2.51392e30	1.74868
$746$	0	0
$747$	2.10147e29i	0.142120i
$748$	0	0
$749$	− 8.11937e29i	− 0.533904i
$750$	0	0
$751$	2.48060e30	1.58613	0.793063	−	0.609140i	$-0.208485\pi$
$751$	2.48060e30	1.58613	0.793063	+	0.609140i	$0.208485\pi$
$752$	0	0
$753$	2.61349e30	1.62508
$754$	0	0
$755$	− 1.37691e30i	− 0.832648i
$756$	0	0
$757$	2.16389e29i	0.127271i	0.997973	+	0.0636354i	$0.0202695\pi$
$757$	2.16389e29i	0.127271i	−0.997973	+	0.0636354i	$0.979731\pi$
$758$	0	0
$759$	−1.51409e30	−0.866190
$760$	0	0
$761$	−1.71830e30	−0.956223	−0.478111	−	0.878299i	$-0.658678\pi$
$761$	−1.71830e30	−0.956223	−0.478111	+	0.878299i	$0.658678\pi$
$762$	0	0
$763$	3.55657e29i	0.192541i
$764$	0	0
$765$	6.22965e29i	0.328110i
$766$	0	0
$767$	2.87897e30	1.47532
$768$	0	0
$769$	−1.45774e30	−0.726863	−0.363431	−	0.931621i	$-0.618395\pi$
$769$	−1.45774e30	−0.726863	−0.363431	+	0.931621i	$0.618395\pi$
$770$	0	0
$771$	− 2.16444e30i	− 1.05021i
$772$	0	0
$773$	− 1.73699e30i	− 0.820185i	−0.912044	−	0.410093i	$-0.865496\pi$
$773$	− 1.73699e30i	− 0.820185i	0.912044	−	0.410093i	$-0.134504\pi$
$774$	0	0
$775$	−8.01734e28	−0.0368436
$776$	0	0
$777$	5.16055e29	0.230821
$778$	0	0
$779$	2.04524e29i	0.0890432i
$780$	0	0
$781$	− 2.97188e30i	− 1.25949i
$782$	0	0
$783$	2.50458e30	1.03332
$784$	0	0
$785$	8.85366e28	0.0355624
$786$	0	0
$787$	− 3.48708e30i	− 1.36373i	−0.731480	−	0.681863i	$-0.761170\pi$
$787$	− 3.48708e30i	− 1.36373i	0.731480	−	0.681863i	$-0.238830\pi$
$788$	0	0
$789$	3.43686e30i	1.30874i
$790$	0	0
$791$	3.28754e30	1.21904
$792$	0	0
$793$	4.89694e30	1.76830
$794$	0	0
$795$	1.61522e30i	0.568039i
$796$	0	0
$797$	2.86176e30i	0.980214i	0.871662	+	0.490107i	$0.163042\pi$
$797$	2.86176e30i	0.980214i	−0.871662	+	0.490107i	$0.836958\pi$
$798$	0	0
$799$	−1.96767e30	−0.656466
$800$	0	0
$801$	2.34755e29	0.0762910
$802$	0	0
$803$	− 3.30601e30i	− 1.04662i
$804$	0	0
$805$	1.55407e30i	0.479309i
$806$	0	0
$807$	4.11090e30	1.23528
$808$	0	0
$809$	2.71150e30	0.793871	0.396935	−	0.917847i	$-0.370074\pi$
$809$	2.71150e30	0.793871	0.396935	+	0.917847i	$0.370074\pi$
$810$	0	0
$811$	4.27827e30i	1.22053i	0.792196	+	0.610267i	$0.208938\pi$
$811$	4.27827e30i	1.22053i	−0.792196	+	0.610267i	$0.791062\pi$
$812$	0	0
$813$	6.35413e30i	1.76647i
$814$	0	0
$815$	−7.71470e29	−0.209009
$816$	0	0
$817$	7.57693e28	0.0200061
$818$	0	0
$819$	− 6.29664e29i	− 0.162042i
$820$	0	0
$821$	4.93734e30i	1.23848i	0.785201	+	0.619241i	$0.212560\pi$
$821$	4.93734e30i	1.23848i	−0.785201	+	0.619241i	$0.787440\pi$
$822$	0	0
$823$	−2.40348e30	−0.587682	−0.293841	−	0.955854i	$-0.594934\pi$
$823$	−2.40348e30	−0.587682	−0.293841	+	0.955854i	$0.594934\pi$
$824$	0	0
$825$	−2.07163e29	−0.0493795
$826$	0	0
$827$	4.76245e30i	1.10668i	0.832955	+	0.553341i	$0.186647\pi$
$827$	4.76245e30i	1.10668i	−0.832955	+	0.553341i	$0.813353\pi$
$828$	0	0
$829$	− 1.94883e29i	− 0.0441520i	−0.999756	−	0.0220760i	$-0.992972\pi$
$829$	− 1.94883e29i	− 0.0441520i	0.999756	−	0.0220760i	$-0.00702759\pi$
$830$	0	0
$831$	−3.37596e29	−0.0745737
$832$	0	0
$833$	3.24076e30	0.698031
$834$	0	0
$835$	− 6.38447e30i	− 1.34096i
$836$	0	0
$837$	4.07247e30i	0.834142i
$838$	0	0
$839$	2.88849e30	0.576992	0.288496	−	0.957481i	$-0.406845\pi$
$839$	2.88849e30	0.576992	0.288496	+	0.957481i	$0.406845\pi$
$840$	0	0
$841$	−2.27230e30	−0.442699
$842$	0	0
$843$	4.95107e30i	0.940827i
$844$	0	0
$845$	2.98419e28i	0.00553136i
$846$	0	0
$847$	−1.55017e30	−0.280288
$848$	0	0
$849$	−8.03642e30	−1.41753
$850$	0	0
$851$	1.10334e30i	0.189867i
$852$	0	0
$853$	4.45549e30i	0.748050i	0.927419	+	0.374025i	$0.122023\pi$
$853$	4.45549e30i	0.748050i	−0.927419	+	0.374025i	$0.877977\pi$
$854$	0	0
$855$	−1.35583e29	−0.0222107
$856$	0	0
$857$	1.85423e30	0.296390	0.148195	−	0.988958i	$-0.452654\pi$
$857$	1.85423e30	0.296390	0.148195	+	0.988958i	$0.452654\pi$
$858$	0	0
$859$	− 6.19354e30i	− 0.966075i	−0.875600	−	0.483038i	$-0.839533\pi$
$859$	− 6.19354e30i	− 0.966075i	0.875600	−	0.483038i	$-0.160467\pi$
$860$	0	0
$861$	4.63969e30i	0.706245i
$862$	0	0
$863$	3.78094e30	0.561677	0.280838	−	0.959755i	$-0.409387\pi$
$863$	3.78094e30	0.561677	0.280838	+	0.959755i	$0.409387\pi$
$864$	0	0
$865$	7.72621e30	1.12020
$866$	0	0
$867$	− 9.98074e30i	− 1.41241i
$868$	0	0
$869$	− 5.92776e30i	− 0.818810i
$870$	0	0
$871$	1.10104e31	1.48461
$872$	0	0
$873$	1.52942e30	0.201316
$874$	0	0
$875$	5.80844e30i	0.746405i
$876$	0	0
$877$	− 8.12478e30i	− 1.01933i	−0.860372	−	0.509666i	$-0.829769\pi$
$877$	− 8.12478e30i	− 1.01933i	0.860372	−	0.509666i	$-0.170231\pi$
$878$	0	0
$879$	−9.75183e30	−1.19455
$880$	0	0
$881$	1.18922e31	1.42238	0.711189	−	0.703001i	$-0.248157\pi$
$881$	1.18922e31	1.42238	0.711189	+	0.703001i	$0.248157\pi$
$882$	0	0
$883$	− 1.45797e30i	− 0.170279i	−0.996369	−	0.0851394i	$-0.972866\pi$
$883$	− 1.45797e30i	− 0.170279i	0.996369	−	0.0851394i	$-0.0271336\pi$
$884$	0	0
$885$	− 1.40631e31i	− 1.60389i
$886$	0	0
$887$	−8.15596e30	−0.908399	−0.454200	−	0.890900i	$-0.650075\pi$
$887$	−8.15596e30	−0.908399	−0.454200	+	0.890900i	$0.650075\pi$
$888$	0	0
$889$	−7.24224e30	−0.787779
$890$	0	0
$891$	1.29759e31i	1.37855i
$892$	0	0
$893$	− 4.28248e29i	− 0.0444381i
$894$	0	0
$895$	−2.54104e30	−0.257555
$896$	0	0
$897$	7.41999e30	0.734653
$898$	0	0
$899$	− 1.20408e31i	− 1.16461i
$900$	0	0
$901$	− 8.36936e30i	− 0.790828i
$902$	0	0
$903$	1.71885e30	0.158678
$904$	0	0
$905$	−2.05558e31	−1.85406
$906$	0	0
$907$	1.13763e31i	1.00260i	0.865275	+	0.501298i	$0.167144\pi$
$907$	1.13763e31i	1.00260i	−0.865275	+	0.501298i	$0.832856\pi$
$908$	0	0
$909$	− 9.00298e29i	− 0.0775293i
$910$	0	0
$911$	3.13153e30	0.263520	0.131760	−	0.991282i	$-0.457937\pi$
$911$	3.13153e30	0.263520	0.131760	+	0.991282i	$0.457937\pi$
$912$	0	0
$913$	−9.16718e30	−0.753864
$914$	0	0
$915$	− 2.39203e31i	− 1.92241i
$916$	0	0
$917$	− 6.47603e30i	− 0.508664i
$918$	0	0
$919$	7.58539e30	0.582324	0.291162	−	0.956674i	$-0.405958\pi$
$919$	7.58539e30	0.582324	0.291162	+	0.956674i	$0.405958\pi$
$920$	0	0
$921$	−2.09351e31	−1.57090
$922$	0	0
$923$	1.45641e31i	1.06823i
$924$	0	0
$925$	1.50963e29i	0.0108239i
$926$	0	0
$927$	5.44343e30	0.381536
$928$	0	0
$929$	−4.63632e30	−0.317693	−0.158847	−	0.987303i	$-0.550778\pi$
$929$	−4.63632e30	−0.317693	−0.158847	+	0.987303i	$0.550778\pi$
$930$	0	0
$931$	7.05326e29i	0.0472518i
$932$	0	0
$933$	− 1.91525e31i	− 1.25449i
$934$	0	0
$935$	−2.71755e31	−1.74043
$936$	0	0
$937$	−8.77613e30	−0.549588	−0.274794	−	0.961503i	$-0.588610\pi$
$937$	−8.77613e30	−0.549588	−0.274794	+	0.961503i	$0.588610\pi$
$938$	0	0
$939$	2.97749e31i	1.82331i
$940$	0	0
$941$	− 5.53870e30i	− 0.331678i	−0.986153	−	0.165839i	$-0.946967\pi$
$941$	− 5.53870e30i	− 0.331678i	0.986153	−	0.165839i	$-0.0530332\pi$
$942$	0	0
$943$	−9.91979e30	−0.580938
$944$	0	0
$945$	1.08009e31	0.618624
$946$	0	0
$947$	9.78123e30i	0.547922i	0.961741	+	0.273961i	$0.0883339\pi$
$947$	9.78123e30i	0.547922i	−0.961741	+	0.273961i	$0.911666\pi$
$948$	0	0
$949$	1.62015e31i	0.887687i
$950$	0	0
$951$	−8.41887e30	−0.451189
$952$	0	0
$953$	4.26179e30	0.223417	0.111709	−	0.993741i	$-0.464368\pi$
$953$	4.26179e30	0.223417	0.111709	+	0.993741i	$0.464368\pi$
$954$	0	0
$955$	− 1.16580e31i	− 0.597846i
$956$	0	0
$957$	− 3.11128e31i	− 1.56086i
$958$	0	0
$959$	−1.63720e31	−0.803539
$960$	0	0
$961$	−1.24704e30	−0.0598804
$962$	0	0
$963$	3.43570e30i	0.161413i
$964$	0	0
$965$	2.75047e31i	1.26436i
$966$	0	0
$967$	−4.20137e31	−1.88979	−0.944893	−	0.327379i	$-0.893835\pi$
$967$	−4.20137e31	−1.88979	−0.944893	+	0.327379i	$0.893835\pi$
$968$	0	0
$969$	3.87210e30	0.170430
$970$	0	0
$971$	3.88373e31i	1.67281i	0.548110	+	0.836406i	$0.315347\pi$
$971$	3.88373e31i	1.67281i	−0.548110	+	0.836406i	$0.684653\pi$
$972$	0	0
$973$	5.38960e30i	0.227181i
$974$	0	0
$975$	1.01523e30	0.0418809
$976$	0	0
$977$	4.18924e29	0.0169138	0.00845691	−	0.999964i	$-0.497308\pi$
$977$	4.18924e29	0.0169138	0.00845691	+	0.999964i	$0.497308\pi$
$978$	0	0
$979$	1.02407e31i	0.404678i
$980$	0	0
$981$	− 1.50496e30i	− 0.0582103i
$982$	0	0
$983$	1.91079e31	0.723437	0.361718	−	0.932287i	$-0.382190\pi$
$983$	1.91079e31	0.723437	0.361718	+	0.932287i	$0.382190\pi$
$984$	0	0
$985$	2.37534e31	0.880331
$986$	0	0
$987$	− 9.71494e30i	− 0.352460i
$988$	0	0
$989$	3.67495e30i	0.130524i
$990$	0	0
$991$	−3.21119e31	−1.11659	−0.558294	−	0.829643i	$-0.688544\pi$
$991$	−3.21119e31	−1.11659	−0.558294	+	0.829643i	$0.688544\pi$
$992$	0	0
$993$	−2.48019e31	−0.844341
$994$	0	0
$995$	− 5.05304e31i	− 1.68427i
$996$	0	0
$997$	8.49310e30i	0.277183i	0.990350	+	0.138592i	$0.0442575\pi$
$997$	8.49310e30i	0.277183i	−0.990350	+	0.138592i	$0.955742\pi$
$998$	0	0
$999$	7.66829e30	0.245053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.22.b.c.33.8	yes	28
4.3	odd	2	inner	64.22.b.c.33.22	yes	28
8.3	odd	2	inner	64.22.b.c.33.7	✓	28
8.5	even	2	inner	64.22.b.c.33.21	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
64.22.b.c.33.7	✓	28	8.3	odd	2	inner
64.22.b.c.33.8	yes	28	1.1	even	1	trivial
64.22.b.c.33.21	yes	28	8.5	even	2	inner
64.22.b.c.33.22	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)

\(f(q)\)	\(=\)	\(q-113044. i q^{3} +2.14177e7i q^{5} +5.47923e8 q^{7} -2.31853e9 q^{9} +O(q^{10})\)	\(q-113044. i q^{3} +2.14177e7i q^{5} +5.47923e8 q^{7} -2.31853e9 q^{9} -1.01141e11i q^{11} +4.95652e11i q^{13} +2.42114e12 q^{15} -1.25452e13 q^{17} -2.73037e12i q^{19} -6.19392e13i q^{21} +1.32428e14 q^{23} +1.81193e13 q^{25} -9.20382e14i q^{27} +2.72124e15i q^{29} -4.42476e15 q^{31} -1.14333e16 q^{33} +1.17353e16i q^{35} +8.33163e15i q^{37} +5.60304e16 q^{39} -7.49071e16 q^{41} +2.77506e16i q^{43} -4.96575e16i q^{45} +1.56846e17 q^{47} -2.58326e17 q^{49} +1.41816e18i q^{51} +6.67135e17i q^{53} +2.16620e18 q^{55} -3.08651e17 q^{57} -5.80845e18i q^{59} -9.87978e18i q^{61} -1.27037e18 q^{63} -1.06157e19 q^{65} -2.22140e19i q^{67} -1.49701e19i q^{69} +2.93837e19 q^{71} +3.26872e19 q^{73} -2.04827e18i q^{75} -5.54173e19i q^{77} +5.86091e19 q^{79} -1.28296e20 q^{81} -9.06380e19i q^{83} -2.68690e20i q^{85} +3.07619e20 q^{87} -1.01252e20 q^{89} +2.71579e20i q^{91} +5.00191e20i q^{93} +5.84782e19 q^{95} -6.59651e20 q^{97} +2.34497e20i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q - 77960422492 q^{9}+O(q^{10}) \) 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

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