LMFDB - Embedded newform 64.24.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [64,24,Mod(1,64)]

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, 0]))

N = Newforms(chi, 24, names="a")

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

chi := DirichletCharacter("64.1");

S:= CuspForms(chi, 24);

N := Newforms(S);

Level:	$ N $	$=$	$ 64 = 2^{6} $
Weight:	$ k $	$=$	$ 24 $
Character orbit:	$[\chi]$	$=$	64.a (trivial)

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:	yes
Analytic conductor:	$214.530583901$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 389656213x^{4} + 47522643058672215x^{2} - 1756479932541937262634975 $ x^6 - 389656213x^4 + 47522643058672215x^2 - 1756479932541937262634975
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{71}\cdot 3^{6}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-8337.36$ of defining polynomial
Character	$\chi$	$=$	64.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-484042. q^{3} +2.23919e7 q^{5} -2.34623e9 q^{7} +1.40154e11 q^{9} +O(q^{10})$	$q-484042. q^{3} +2.23919e7 q^{5} -2.34623e9 q^{7} +1.40154e11 q^{9} +7.12641e11 q^{11} -7.52179e12 q^{13} -1.08386e13 q^{15} +8.04922e13 q^{17} +1.95786e13 q^{19} +1.13567e15 q^{21} -2.11936e15 q^{23} -1.14195e16 q^{25} -2.22710e16 q^{27} -1.03962e17 q^{29} +1.45070e17 q^{31} -3.44948e17 q^{33} -5.25364e16 q^{35} +3.23591e17 q^{37} +3.64086e18 q^{39} +4.65737e18 q^{41} -1.07004e19 q^{43} +3.13830e18 q^{45} +2.32701e19 q^{47} -2.18640e19 q^{49} -3.89616e19 q^{51} -2.42257e19 q^{53} +1.59574e19 q^{55} -9.47688e18 q^{57} +2.61958e20 q^{59} -2.11128e20 q^{61} -3.28833e20 q^{63} -1.68427e20 q^{65} +1.38684e21 q^{67} +1.02586e21 q^{69} +1.11917e20 q^{71} +2.49327e21 q^{73} +5.52754e21 q^{75} -1.67202e21 q^{77} +1.17261e22 q^{79} -2.41439e21 q^{81} -2.34706e21 q^{83} +1.80237e21 q^{85} +5.03218e22 q^{87} -2.06580e21 q^{89} +1.76478e22 q^{91} -7.02201e22 q^{93} +4.38402e20 q^{95} +8.47772e22 q^{97} +9.98793e22 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 163121700 q^{5} + 14120212542 q^{9}+O(q^{10}) $ 6 * q - 163121700 * q^5 + 14120212542 * q^9	$ 6 q - 163121700 q^{5} + 14120212542 q^{9} - 1101295489524 q^{13} + 56435517243468 q^{17} - 14\!\cdots\!84 q^{21}+ \cdots + 25\!\cdots\!16 q^{97}+O(q^{100}) $ 6 * q - 163121700 * q^5 + 14120212542 * q^9 - 1101295489524 * q^13 + 56435517243468 * q^17 - 1447661324841984 * q^21 + 17512304669291850 * q^25 - 87161858953770708 * q^29 + 48462097564970496 * q^33 + 1542766250573484636 * q^37 - 865941451897399716 * q^41 + 9286254082406210700 * q^45 + 52551978665245784886 * q^49 + 10735070383373044668 * q^53 + 155104978099636154880 * q^57 + 432169207960503656364 * q^61 - 316526040451899107400 * q^65 - 695880182828346153984 * q^69 + 1758226166115151758204 * q^73 - 11926085657398731979776 * q^77 + 4010608843245537630678 * q^81 - 20251531460297137542600 * q^85 + 20087131519604301542940 * q^89 - 67310470168488243892224 * q^93 + 254665147475934678043116 * q^97

Display 100 coefficients 10 coefficients

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
$3$	−484042.	−1.57757	−0.788785	−	0.614669i	$-0.789290\pi$
$3$	−484042.	−1.57757	−0.788785	+	0.614669i	$0.789290\pi$
$4$	0	0
$5$	2.23919e7	0.205086	0.102543	−	0.994729i	$-0.467302\pi$
$5$	2.23919e7	0.205086	0.102543	+	0.994729i	$0.467302\pi$
$6$	0	0
$7$	−2.34623e9	−0.448480	−0.224240	−	0.974534i	$-0.571990\pi$
$7$	−2.34623e9	−0.448480	−0.224240	+	0.974534i	$0.571990\pi$
$8$	0	0
$9$	1.40154e11	1.48873
$10$	0	0
$11$	7.12641e11	0.753104	0.376552	−	0.926395i	$-0.377110\pi$
$11$	7.12641e11	0.753104	0.376552	+	0.926395i	$0.377110\pi$
$12$	0	0
$13$	−7.52179e12	−1.16405	−0.582027	−	0.813170i	$-0.697740\pi$
$13$	−7.52179e12	−1.16405	−0.582027	+	0.813170i	$0.697740\pi$
$14$	0	0
$15$	−1.08386e13	−0.323537
$16$	0	0
$17$	8.04922e13	0.569628	0.284814	−	0.958583i	$-0.408068\pi$
$17$	8.04922e13	0.569628	0.284814	+	0.958583i	$0.408068\pi$
$18$	0	0
$19$	1.95786e13	0.0385581	0.0192791	−	0.999814i	$-0.493863\pi$
$19$	1.95786e13	0.0385581	0.0192791	+	0.999814i	$0.493863\pi$
$20$	0	0
$21$	1.13567e15	0.707508
$22$	0	0
$23$	−2.11936e15	−0.463803	−0.231902	−	0.972739i	$-0.574495\pi$
$23$	−2.11936e15	−0.463803	−0.231902	+	0.972739i	$0.574495\pi$
$24$	0	0
$25$	−1.14195e16	−0.957940
$26$	0	0
$27$	−2.22710e16	−0.771005
$28$	0	0
$29$	−1.03962e17	−1.58233	−0.791163	−	0.611605i	$-0.790524\pi$
$29$	−1.03962e17	−1.58233	−0.791163	+	0.611605i	$0.790524\pi$
$30$	0	0
$31$	1.45070e17	1.02546	0.512730	−	0.858550i	$-0.328634\pi$
$31$	1.45070e17	1.02546	0.512730	+	0.858550i	$0.328634\pi$
$32$	0	0
$33$	−3.44948e17	−1.18807
$34$	0	0
$35$	−5.25364e16	−0.0919767
$36$	0	0
$37$	3.23591e17	0.299003	0.149502	−	0.988761i	$-0.452233\pi$
$37$	3.23591e17	0.299003	0.149502	+	0.988761i	$0.452233\pi$
$38$	0	0
$39$	3.64086e18	1.83638
$40$	0	0
$41$	4.65737e18	1.32168	0.660840	−	0.750526i	$-0.270200\pi$
$41$	4.65737e18	1.32168	0.660840	+	0.750526i	$0.270200\pi$
$42$	0	0
$43$	−1.07004e19	−1.75595	−0.877974	−	0.478708i	$-0.841105\pi$
$43$	−1.07004e19	−1.75595	−0.877974	+	0.478708i	$0.841105\pi$
$44$	0	0
$45$	3.13830e18	0.305317
$46$	0	0
$47$	2.32701e19	1.37301	0.686505	−	0.727125i	$-0.259144\pi$
$47$	2.32701e19	1.37301	0.686505	+	0.727125i	$0.259144\pi$
$48$	0	0
$49$	−2.18640e19	−0.798866
$50$	0	0
$51$	−3.89616e19	−0.898628
$52$	0	0
$53$	−2.42257e19	−0.359007	−0.179504	−	0.983757i	$-0.557449\pi$
$53$	−2.42257e19	−0.359007	−0.179504	+	0.983757i	$0.557449\pi$
$54$	0	0
$55$	1.59574e19	0.154451
$56$	0	0
$57$	−9.47688e18	−0.0608281
$58$	0	0
$59$	2.61958e20	1.13092	0.565461	−	0.824775i	$-0.308698\pi$
$59$	2.61958e20	1.13092	0.565461	+	0.824775i	$0.308698\pi$
$60$	0	0
$61$	−2.11128e20	−0.621230	−0.310615	−	0.950536i	$-0.600535\pi$
$61$	−2.11128e20	−0.621230	−0.310615	+	0.950536i	$0.600535\pi$
$62$	0	0
$63$	−3.28833e20	−0.667665
$64$	0	0
$65$	−1.68427e20	−0.238730
$66$	0	0
$67$	1.38684e21	1.38729	0.693643	−	0.720319i	$-0.256005\pi$
$67$	1.38684e21	1.38729	0.693643	+	0.720319i	$0.256005\pi$
$68$	0	0
$69$	1.02586e21	0.731683
$70$	0	0
$71$	1.11917e20	0.0574681	0.0287340	−	0.999587i	$-0.490852\pi$
$71$	1.11917e20	0.0574681	0.0287340	+	0.999587i	$0.490852\pi$
$72$	0	0
$73$	2.49327e21	0.930159	0.465080	−	0.885269i	$-0.346026\pi$
$73$	2.49327e21	0.930159	0.465080	+	0.885269i	$0.346026\pi$
$74$	0	0
$75$	5.52754e21	1.51122
$76$	0	0
$77$	−1.67202e21	−0.337752
$78$	0	0
$79$	1.17261e22	1.76378	0.881889	−	0.471457i	$-0.156272\pi$
$79$	1.17261e22	1.76378	0.881889	+	0.471457i	$0.156272\pi$
$80$	0	0
$81$	−2.41439e21	−0.272415
$82$	0	0
$83$	−2.34706e21	−0.200044	−0.100022	−	0.994985i	$-0.531891\pi$
$83$	−2.34706e21	−0.200044	−0.100022	+	0.994985i	$0.531891\pi$
$84$	0	0
$85$	1.80237e21	0.116822
$86$	0	0
$87$	5.03218e22	2.49623
$88$	0	0
$89$	−2.06580e21	−0.0789046	−0.0394523	−	0.999221i	$-0.512561\pi$
$89$	−2.06580e21	−0.0789046	−0.0394523	+	0.999221i	$0.512561\pi$
$90$	0	0
$91$	1.76478e22	0.522054
$92$	0	0
$93$	−7.02201e22	−1.61773
$94$	0	0
$95$	4.38402e20	0.00790771
$96$	0	0
$97$	8.47772e22	1.20338	0.601692	−	0.798728i	$-0.294494\pi$
$97$	8.47772e22	1.20338	0.601692	+	0.798728i	$0.294494\pi$
$98$	0	0
$99$	9.98793e22	1.12117
$100$	0	0
$101$	1.31267e23	1.17074	0.585370	−	0.810766i	$-0.300949\pi$
$101$	1.31267e23	1.17074	0.585370	+	0.810766i	$0.300949\pi$
$102$	0	0
$103$	4.59660e22	0.327197	0.163598	−	0.986527i	$-0.447690\pi$
$103$	4.59660e22	0.327197	0.163598	+	0.986527i	$0.447690\pi$
$104$	0	0
$105$	2.54298e22	0.145100
$106$	0	0
$107$	3.62290e23	1.66396	0.831981	−	0.554804i	$-0.187207\pi$
$107$	3.62290e23	1.66396	0.831981	+	0.554804i	$0.187207\pi$
$108$	0	0
$109$	−4.09834e23	−1.52126	−0.760630	−	0.649186i	$-0.775110\pi$
$109$	−4.09834e23	−1.52126	−0.760630	+	0.649186i	$0.775110\pi$
$110$	0	0
$111$	−1.56632e23	−0.471699
$112$	0	0
$113$	5.12388e23	1.25660	0.628299	−	0.777972i	$-0.283751\pi$
$113$	5.12388e23	1.25660	0.628299	+	0.777972i	$0.283751\pi$
$114$	0	0
$115$	−4.74563e22	−0.0951194
$116$	0	0
$117$	−1.05421e24	−1.73296
$118$	0	0
$119$	−1.88853e23	−0.255466
$120$	0	0
$121$	−3.87573e23	−0.432834
$122$	0	0
$123$	−2.25437e24	−2.08505
$124$	0	0
$125$	−5.22636e23	−0.401545
$126$	0	0
$127$	−5.33347e23	−0.341402	−0.170701	−	0.985323i	$-0.554603\pi$
$127$	−5.33347e23	−0.341402	−0.170701	+	0.985323i	$0.554603\pi$
$128$	0	0
$129$	5.17943e24	2.77013
$130$	0	0
$131$	−3.10169e24	−1.38988	−0.694942	−	0.719066i	$-0.744570\pi$
$131$	−3.10169e24	−1.38988	−0.694942	+	0.719066i	$0.744570\pi$
$132$	0	0
$133$	−4.59359e22	−0.0172925
$134$	0	0
$135$	−4.98690e23	−0.158122
$136$	0	0
$137$	3.04983e24	0.816561	0.408281	−	0.912857i	$-0.366129\pi$
$137$	3.04983e24	0.816561	0.408281	+	0.912857i	$0.366129\pi$
$138$	0	0
$139$	−2.35238e24	−0.533135	−0.266567	−	0.963816i	$-0.585889\pi$
$139$	−2.35238e24	−0.533135	−0.266567	+	0.963816i	$0.585889\pi$
$140$	0	0
$141$	−1.12637e25	−2.16602
$142$	0	0
$143$	−5.36034e24	−0.876653
$144$	0	0
$145$	−2.32790e24	−0.324512
$146$	0	0
$147$	1.05831e25	1.26027
$148$	0	0
$149$	−6.01863e24	−0.613558	−0.306779	−	0.951781i	$-0.599251\pi$
$149$	−6.01863e24	−0.613558	−0.306779	+	0.951781i	$0.599251\pi$
$150$	0	0
$151$	−1.38513e25	−1.21131	−0.605654	−	0.795728i	$-0.707088\pi$
$151$	−1.38513e25	−1.21131	−0.605654	+	0.795728i	$0.707088\pi$
$152$	0	0
$153$	1.12813e25	0.848022
$154$	0	0
$155$	3.24839e24	0.210307
$156$	0	0
$157$	−1.14941e25	−0.642137	−0.321069	−	0.947056i	$-0.604042\pi$
$157$	−1.14941e25	−0.642137	−0.321069	+	0.947056i	$0.604042\pi$
$158$	0	0
$159$	1.17262e25	0.566359
$160$	0	0
$161$	4.97249e24	0.208006
$162$	0	0
$163$	1.39746e25	0.507203	0.253602	−	0.967309i	$-0.418385\pi$
$163$	1.39746e25	0.507203	0.253602	+	0.967309i	$0.418385\pi$
$164$	0	0
$165$	−7.72403e24	−0.243657
$166$	0	0
$167$	7.17392e25	1.97023	0.985115	−	0.171896i	$-0.0549892\pi$
$167$	7.17392e25	1.97023	0.985115	+	0.171896i	$0.0549892\pi$
$168$	0	0
$169$	1.48234e25	0.355019
$170$	0	0
$171$	2.74402e24	0.0574026
$172$	0	0
$173$	7.10628e25	1.30051	0.650253	−	0.759718i	$-0.274663\pi$
$173$	7.10628e25	1.30051	0.650253	+	0.759718i	$0.274663\pi$
$174$	0	0
$175$	2.67928e25	0.429617
$176$	0	0
$177$	−1.26799e26	−1.78411
$178$	0	0
$179$	−9.30919e24	−0.115107	−0.0575536	−	0.998342i	$-0.518330\pi$
$179$	−9.30919e24	−0.115107	−0.0575536	+	0.998342i	$0.518330\pi$
$180$	0	0
$181$	−7.43574e25	−0.809134	−0.404567	−	0.914508i	$-0.632578\pi$
$181$	−7.43574e25	−0.809134	−0.404567	+	0.914508i	$0.632578\pi$
$182$	0	0
$183$	1.02195e26	0.980034
$184$	0	0
$185$	7.24580e24	0.0613213
$186$	0	0
$187$	5.73620e25	0.428989
$188$	0	0
$189$	5.22529e25	0.345780
$190$	0	0
$191$	−9.29936e25	−0.545217	−0.272608	−	0.962125i	$-0.587886\pi$
$191$	−9.29936e25	−0.545217	−0.272608	+	0.962125i	$0.587886\pi$
$192$	0	0
$193$	−2.30225e26	−1.19741	−0.598705	−	0.800970i	$-0.704318\pi$
$193$	−2.30225e26	−1.19741	−0.598705	+	0.800970i	$0.704318\pi$
$194$	0	0
$195$	8.15257e25	0.376614
$196$	0	0
$197$	2.19548e26	0.901921	0.450960	−	0.892544i	$-0.351082\pi$
$197$	2.19548e26	0.901921	0.450960	+	0.892544i	$0.351082\pi$
$198$	0	0
$199$	−3.82426e26	−1.39874	−0.699370	−	0.714760i	$-0.746536\pi$
$199$	−3.82426e26	−1.39874	−0.699370	+	0.714760i	$0.746536\pi$
$200$	0	0
$201$	−6.71289e26	−2.18854
$202$	0	0
$203$	2.43918e26	0.709641
$204$	0	0
$205$	1.04287e26	0.271058
$206$	0	0
$207$	−2.97036e26	−0.690478
$208$	0	0
$209$	1.39525e25	0.0290383
$210$	0	0
$211$	−7.74159e26	−1.44405	−0.722024	−	0.691868i	$-0.756788\pi$
$211$	−7.74159e26	−1.44405	−0.722024	+	0.691868i	$0.756788\pi$
$212$	0	0
$213$	−5.41727e25	−0.0906599
$214$	0	0
$215$	−2.39601e26	−0.360120
$216$	0	0
$217$	−3.40368e26	−0.459898
$218$	0	0
$219$	−1.20685e27	−1.46739
$220$	0	0
$221$	−6.05446e26	−0.663077
$222$	0	0
$223$	1.61067e26	0.159037	0.0795187	−	0.996833i	$-0.474662\pi$
$223$	1.61067e26	0.159037	0.0795187	+	0.996833i	$0.474662\pi$
$224$	0	0
$225$	−1.60049e27	−1.42611
$226$	0	0
$227$	−9.81040e26	−0.789568	−0.394784	−	0.918774i	$-0.629181\pi$
$227$	−9.81040e26	−0.789568	−0.394784	+	0.918774i	$0.629181\pi$
$228$	0	0
$229$	1.47010e25	0.0106965	0.00534823	−	0.999986i	$-0.498298\pi$
$229$	1.47010e25	0.0106965	0.00534823	+	0.999986i	$0.498298\pi$
$230$	0	0
$231$	8.09327e26	0.532827
$232$	0	0
$233$	−1.36571e27	−0.814266	−0.407133	−	0.913369i	$-0.633471\pi$
$233$	−1.36571e27	−0.814266	−0.407133	+	0.913369i	$0.633471\pi$
$234$	0	0
$235$	5.21062e26	0.281584
$236$	0	0
$237$	−5.67595e27	−2.78248
$238$	0	0
$239$	8.75240e26	0.389539	0.194770	−	0.980849i	$-0.437604\pi$
$239$	8.75240e26	0.389539	0.194770	+	0.980849i	$0.437604\pi$
$240$	0	0
$241$	1.54538e27	0.624941	0.312471	−	0.949927i	$-0.398843\pi$
$241$	1.54538e27	0.624941	0.312471	+	0.949927i	$0.398843\pi$
$242$	0	0
$243$	3.26533e27	1.20076
$244$	0	0
$245$	−4.89575e26	−0.163836
$246$	0	0
$247$	−1.47266e26	−0.0448837
$248$	0	0
$249$	1.13608e27	0.315584
$250$	0	0
$251$	7.86019e27	1.99152	0.995762	−	0.0919684i	$-0.0293159\pi$
$251$	7.86019e27	1.99152	0.995762	+	0.0919684i	$0.0293159\pi$
$252$	0	0
$253$	−1.51034e27	−0.349292
$254$	0	0
$255$	−8.72423e26	−0.184296
$256$	0	0
$257$	−1.94684e27	−0.375923	−0.187961	−	0.982176i	$-0.560188\pi$
$257$	−1.94684e27	−0.375923	−0.187961	+	0.982176i	$0.560188\pi$
$258$	0	0
$259$	−7.59217e26	−0.134097
$260$	0	0
$261$	−1.45706e28	−2.35566
$262$	0	0
$263$	5.15195e27	0.762922	0.381461	−	0.924385i	$-0.375421\pi$
$263$	5.15195e27	0.762922	0.381461	+	0.924385i	$0.375421\pi$
$264$	0	0
$265$	−5.42458e26	−0.0736272
$266$	0	0
$267$	9.99933e26	0.124478
$268$	0	0
$269$	−1.07636e28	−1.22972	−0.614859	−	0.788637i	$-0.710787\pi$
$269$	−1.07636e28	−1.22972	−0.614859	+	0.788637i	$0.710787\pi$
$270$	0	0
$271$	−4.59879e27	−0.482499	−0.241250	−	0.970463i	$-0.577557\pi$
$271$	−4.59879e27	−0.482499	−0.241250	+	0.970463i	$0.577557\pi$
$272$	0	0
$273$	−8.54230e27	−0.823577
$274$	0	0
$275$	−8.13803e27	−0.721428
$276$	0	0
$277$	−2.37346e28	−1.93582	−0.967909	−	0.251300i	$-0.919142\pi$
$277$	−2.37346e28	−1.93582	−0.967909	+	0.251300i	$0.919142\pi$
$278$	0	0
$279$	2.03321e28	1.52663
$280$	0	0
$281$	−9.71493e27	−0.671919	−0.335960	−	0.941876i	$-0.609061\pi$
$281$	−9.71493e27	−0.671919	−0.335960	+	0.941876i	$0.609061\pi$
$282$	0	0
$283$	4.36477e26	0.0278239	0.0139119	−	0.999903i	$-0.495572\pi$
$283$	4.36477e26	0.0278239	0.0139119	+	0.999903i	$0.495572\pi$
$284$	0	0
$285$	−2.12205e26	−0.0124750
$286$	0	0
$287$	−1.09273e28	−0.592747
$288$	0	0
$289$	−1.34886e28	−0.675524
$290$	0	0
$291$	−4.10357e28	−1.89842
$292$	0	0
$293$	7.53351e27	0.322122	0.161061	−	0.986944i	$-0.448508\pi$
$293$	7.53351e27	0.322122	0.161061	+	0.986944i	$0.448508\pi$
$294$	0	0
$295$	5.86572e27	0.231936
$296$	0	0
$297$	−1.58713e28	−0.580647
$298$	0	0
$299$	1.59414e28	0.539892
$300$	0	0
$301$	2.51055e28	0.787507
$302$	0	0
$303$	−6.35389e28	−1.84693
$304$	0	0
$305$	−4.72755e27	−0.127405
$306$	0	0
$307$	7.65129e27	0.191269	0.0956343	−	0.995417i	$-0.469512\pi$
$307$	7.65129e27	0.191269	0.0956343	+	0.995417i	$0.469512\pi$
$308$	0	0
$309$	−2.22495e28	−0.516176
$310$	0	0
$311$	−5.29552e28	−1.14068	−0.570340	−	0.821408i	$-0.693189\pi$
$311$	−5.29552e28	−1.14068	−0.570340	+	0.821408i	$0.693189\pi$
$312$	0	0
$313$	−4.43677e28	−0.887784	−0.443892	−	0.896080i	$-0.646403\pi$
$313$	−4.43677e28	−0.887784	−0.443892	+	0.896080i	$0.646403\pi$
$314$	0	0
$315$	−7.36317e27	−0.136928
$316$	0	0
$317$	−4.15115e28	−0.717772	−0.358886	−	0.933381i	$-0.616843\pi$
$317$	−4.15115e28	−0.717772	−0.358886	+	0.933381i	$0.616843\pi$
$318$	0	0
$319$	−7.40873e28	−1.19166
$320$	0	0
$321$	−1.75364e29	−2.62502
$322$	0	0
$323$	1.57593e27	0.0219638
$324$	0	0
$325$	8.58954e28	1.11509
$326$	0	0
$327$	1.98377e29	2.39989
$328$	0	0
$329$	−5.45970e28	−0.615767
$330$	0	0
$331$	1.14105e29	1.20028	0.600139	−	0.799896i	$-0.295112\pi$
$331$	1.14105e29	1.20028	0.600139	+	0.799896i	$0.295112\pi$
$332$	0	0
$333$	4.53524e28	0.445135
$334$	0	0
$335$	3.10539e28	0.284512
$336$	0	0
$337$	1.63110e29	1.39552	0.697760	−	0.716332i	$-0.254180\pi$
$337$	1.63110e29	1.39552	0.697760	+	0.716332i	$0.254180\pi$
$338$	0	0
$339$	−2.48017e29	−1.98237
$340$	0	0
$341$	1.03383e29	0.772278
$342$	0	0
$343$	1.15511e29	0.806755
$344$	0	0
$345$	2.29709e28	0.150058
$346$	0	0
$347$	2.46678e29	1.50779	0.753895	−	0.656995i	$-0.228173\pi$
$347$	2.46678e29	1.50779	0.753895	+	0.656995i	$0.228173\pi$
$348$	0	0
$349$	−6.60521e28	−0.377915	−0.188957	−	0.981985i	$-0.560511\pi$
$349$	−6.60521e28	−0.377915	−0.188957	+	0.981985i	$0.560511\pi$
$350$	0	0
$351$	1.67518e29	0.897490
$352$	0	0
$353$	1.33282e29	0.668900	0.334450	−	0.942414i	$-0.391449\pi$
$353$	1.33282e29	0.668900	0.334450	+	0.942414i	$0.391449\pi$
$354$	0	0
$355$	2.50604e27	0.0117859
$356$	0	0
$357$	9.14128e28	0.403016
$358$	0	0
$359$	2.22382e29	0.919419	0.459710	−	0.888069i	$-0.347954\pi$
$359$	2.22382e29	0.919419	0.459710	+	0.888069i	$0.347954\pi$
$360$	0	0
$361$	−2.57446e29	−0.998513
$362$	0	0
$363$	1.87602e29	0.682827
$364$	0	0
$365$	5.58291e28	0.190762
$366$	0	0
$367$	−5.78887e29	−1.85752	−0.928761	−	0.370679i	$-0.879125\pi$
$367$	−5.78887e29	−1.85752	−0.928761	+	0.370679i	$0.879125\pi$
$368$	0	0
$369$	6.52748e29	1.96763
$370$	0	0
$371$	5.68389e28	0.161007
$372$	0	0
$373$	−5.31843e29	−1.41622	−0.708112	−	0.706100i	$-0.750453\pi$
$373$	−5.31843e29	−1.41622	−0.708112	+	0.706100i	$0.750453\pi$
$374$	0	0
$375$	2.52978e29	0.633466
$376$	0	0
$377$	7.81978e29	1.84191
$378$	0	0
$379$	6.70608e29	1.48634	0.743169	−	0.669104i	$-0.233322\pi$
$379$	6.70608e29	1.48634	0.743169	+	0.669104i	$0.233322\pi$
$380$	0	0
$381$	2.58162e29	0.538587
$382$	0	0
$383$	−9.69216e29	−1.90386	−0.951929	−	0.306319i	$-0.900903\pi$
$383$	−9.69216e29	−1.90386	−0.951929	+	0.306319i	$0.900903\pi$
$384$	0	0
$385$	−3.74396e28	−0.0692680
$386$	0	0
$387$	−1.49970e30	−2.61413
$388$	0	0
$389$	3.63076e29	0.596455	0.298227	−	0.954495i	$-0.403605\pi$
$389$	3.63076e29	0.596455	0.298227	+	0.954495i	$0.403605\pi$
$390$	0	0
$391$	−1.70592e29	−0.264195
$392$	0	0
$393$	1.50135e30	2.19264
$394$	0	0
$395$	2.62570e29	0.361725
$396$	0	0
$397$	4.97332e29	0.646481	0.323241	−	0.946317i	$-0.395228\pi$
$397$	4.97332e29	0.646481	0.323241	+	0.946317i	$0.395228\pi$
$398$	0	0
$399$	2.22349e28	0.0272802
$400$	0	0
$401$	−3.92102e29	−0.454191	−0.227095	−	0.973873i	$-0.572923\pi$
$401$	−3.92102e29	−0.454191	−0.227095	+	0.973873i	$0.572923\pi$
$402$	0	0
$403$	−1.09119e30	−1.19369
$404$	0	0
$405$	−5.40628e28	−0.0558683
$406$	0	0
$407$	2.30604e29	0.225181
$408$	0	0
$409$	1.29914e30	1.19905	0.599525	−	0.800356i	$-0.295356\pi$
$409$	1.29914e30	1.19905	0.599525	+	0.800356i	$0.295356\pi$
$410$	0	0
$411$	−1.47625e30	−1.28818
$412$	0	0
$413$	−6.14612e29	−0.507196
$414$	0	0
$415$	−5.25550e28	−0.0410262
$416$	0	0
$417$	1.13865e30	0.841057
$418$	0	0
$419$	1.02435e30	0.716121	0.358060	−	0.933698i	$-0.383438\pi$
$419$	1.02435e30	0.716121	0.358060	+	0.933698i	$0.383438\pi$
$420$	0	0
$421$	1.63752e30	1.08379	0.541893	−	0.840447i	$-0.317708\pi$
$421$	1.63752e30	1.08379	0.541893	+	0.840447i	$0.317708\pi$
$422$	0	0
$423$	3.26139e30	2.04404
$424$	0	0
$425$	−9.19183e29	−0.545669
$426$	0	0
$427$	4.95354e29	0.278609
$428$	0	0
$429$	2.59463e30	1.38298
$430$	0	0
$431$	−2.00643e30	−1.01376	−0.506879	−	0.862017i	$-0.669201\pi$
$431$	−2.00643e30	−1.01376	−0.506879	+	0.862017i	$0.669201\pi$
$432$	0	0
$433$	−1.55758e30	−0.746172	−0.373086	−	0.927797i	$-0.621700\pi$
$433$	−1.55758e30	−0.746172	−0.373086	+	0.927797i	$0.621700\pi$
$434$	0	0
$435$	1.12680e30	0.511941
$436$	0	0
$437$	−4.14941e28	−0.0178834
$438$	0	0
$439$	−3.73044e30	−1.52552	−0.762761	−	0.646681i	$-0.776157\pi$
$439$	−3.73044e30	−1.52552	−0.762761	+	0.646681i	$0.776157\pi$
$440$	0	0
$441$	−3.06432e30	−1.18930
$442$	0	0
$443$	4.01531e30	1.47937	0.739683	−	0.672955i	$-0.234975\pi$
$443$	4.01531e30	1.47937	0.739683	+	0.672955i	$0.234975\pi$
$444$	0	0
$445$	−4.62570e28	−0.0161822
$446$	0	0
$447$	2.91327e30	0.967931
$448$	0	0
$449$	1.67632e30	0.529084	0.264542	−	0.964374i	$-0.414779\pi$
$449$	1.67632e30	0.529084	0.264542	+	0.964374i	$0.414779\pi$
$450$	0	0
$451$	3.31903e30	0.995363
$452$	0	0
$453$	6.70459e30	1.91092
$454$	0	0
$455$	3.95168e29	0.107066
$456$	0	0
$457$	9.87055e29	0.254276	0.127138	−	0.991885i	$-0.459421\pi$
$457$	9.87055e29	0.254276	0.127138	+	0.991885i	$0.459421\pi$
$458$	0	0
$459$	−1.79264e30	−0.439186
$460$	0	0
$461$	1.70262e30	0.396786	0.198393	−	0.980123i	$-0.436428\pi$
$461$	1.70262e30	0.396786	0.198393	+	0.980123i	$0.436428\pi$
$462$	0	0
$463$	2.62359e30	0.581720	0.290860	−	0.956766i	$-0.406059\pi$
$463$	2.62359e30	0.581720	0.290860	+	0.956766i	$0.406059\pi$
$464$	0	0
$465$	−1.57236e30	−0.331774
$466$	0	0
$467$	−6.49369e29	−0.130421	−0.0652105	−	0.997872i	$-0.520772\pi$
$467$	−6.49369e29	−0.130421	−0.0652105	+	0.997872i	$0.520772\pi$
$468$	0	0
$469$	−3.25384e30	−0.622169
$470$	0	0
$471$	5.56362e30	1.01302
$472$	0	0
$473$	−7.62553e30	−1.32241
$474$	0	0
$475$	−2.23579e29	−0.0369364
$476$	0	0
$477$	−3.39532e30	−0.534465
$478$	0	0
$479$	1.63915e30	0.245901	0.122951	−	0.992413i	$-0.460764\pi$
$479$	1.63915e30	0.245901	0.122951	+	0.992413i	$0.460764\pi$
$480$	0	0
$481$	−2.43398e30	−0.348056
$482$	0	0
$483$	−2.40690e30	−0.328145
$484$	0	0
$485$	1.89832e30	0.246797
$486$	0	0
$487$	−3.67400e30	−0.455572	−0.227786	−	0.973711i	$-0.573149\pi$
$487$	−3.67400e30	−0.455572	−0.227786	+	0.973711i	$0.573149\pi$
$488$	0	0
$489$	−6.76430e30	−0.800149
$490$	0	0
$491$	−1.49168e31	−1.68360	−0.841799	−	0.539791i	$-0.818503\pi$
$491$	−1.49168e31	−1.68360	−0.841799	+	0.539791i	$0.818503\pi$
$492$	0	0
$493$	−8.36810e30	−0.901338
$494$	0	0
$495$	2.23648e30	0.229935
$496$	0	0
$497$	−2.62584e29	−0.0257733
$498$	0	0
$499$	2.41095e30	0.225960	0.112980	−	0.993597i	$-0.463960\pi$
$499$	2.41095e30	0.225960	0.112980	+	0.993597i	$0.463960\pi$
$500$	0	0
$501$	−3.47248e31	−3.10818
$502$	0	0
$503$	−1.81343e31	−1.55049	−0.775243	−	0.631663i	$-0.782373\pi$
$503$	−1.81343e31	−1.55049	−0.775243	+	0.631663i	$0.782373\pi$
$504$	0	0
$505$	2.93932e30	0.240102
$506$	0	0
$507$	−7.17517e30	−0.560068
$508$	0	0
$509$	−1.39439e30	−0.104023	−0.0520117	−	0.998646i	$-0.516563\pi$
$509$	−1.39439e30	−0.104023	−0.0520117	+	0.998646i	$0.516563\pi$
$510$	0	0
$511$	−5.84979e30	−0.417157
$512$	0	0
$513$	−4.36036e29	−0.0297285
$514$	0	0
$515$	1.02926e30	0.0671033
$516$	0	0
$517$	1.65832e31	1.03402
$518$	0	0
$519$	−3.43974e31	−2.05164
$520$	0	0
$521$	2.40815e31	1.37420	0.687099	−	0.726564i	$-0.258884\pi$
$521$	2.40815e31	1.37420	0.687099	+	0.726564i	$0.258884\pi$
$522$	0	0
$523$	1.27191e30	0.0694522	0.0347261	−	0.999397i	$-0.488944\pi$
$523$	1.27191e30	0.0694522	0.0347261	+	0.999397i	$0.488944\pi$
$524$	0	0
$525$	−1.29689e31	−0.677750
$526$	0	0
$527$	1.16770e31	0.584130
$528$	0	0
$529$	−1.63888e31	−0.784886
$530$	0	0
$531$	3.67143e31	1.68364
$532$	0	0
$533$	−3.50318e31	−1.53851
$534$	0	0
$535$	8.11236e30	0.341255
$536$	0	0
$537$	4.50604e30	0.181590
$538$	0	0
$539$	−1.55812e31	−0.601629
$540$	0	0
$541$	−4.69046e31	−1.73559	−0.867795	−	0.496922i	$-0.834463\pi$
$541$	−4.69046e31	−1.73559	−0.867795	+	0.496922i	$0.834463\pi$
$542$	0	0
$543$	3.59921e31	1.27647
$544$	0	0
$545$	−9.17695e30	−0.311988
$546$	0	0
$547$	−3.34092e30	−0.108896	−0.0544479	−	0.998517i	$-0.517340\pi$
$547$	−3.34092e30	−0.108896	−0.0544479	+	0.998517i	$0.517340\pi$
$548$	0	0
$549$	−2.95904e31	−0.924843
$550$	0	0
$551$	−2.03543e30	−0.0610115
$552$	0	0
$553$	−2.75122e31	−0.791018
$554$	0	0
$555$	−3.50727e30	−0.0967387
$556$	0	0
$557$	5.21142e31	1.37918	0.689591	−	0.724199i	$-0.257790\pi$
$557$	5.21142e31	1.37918	0.689591	+	0.724199i	$0.257790\pi$
$558$	0	0
$559$	8.04860e31	2.04402
$560$	0	0
$561$	−2.77657e31	−0.676760
$562$	0	0
$563$	1.76773e31	0.413590	0.206795	−	0.978384i	$-0.433697\pi$
$563$	1.76773e31	0.413590	0.206795	+	0.978384i	$0.433697\pi$
$564$	0	0
$565$	1.14733e31	0.257710
$566$	0	0
$567$	5.66472e30	0.122172
$568$	0	0
$569$	−1.79586e31	−0.371947	−0.185974	−	0.982555i	$-0.559544\pi$
$569$	−1.79586e31	−0.371947	−0.185974	+	0.982555i	$0.559544\pi$
$570$	0	0
$571$	−5.05096e31	−1.00475	−0.502376	−	0.864649i	$-0.667541\pi$
$571$	−5.05096e31	−1.00475	−0.502376	+	0.864649i	$0.667541\pi$
$572$	0	0
$573$	4.50128e31	0.860118
$574$	0	0
$575$	2.42021e31	0.444296
$576$	0	0
$577$	5.72937e31	1.01061	0.505307	−	0.862940i	$-0.331379\pi$
$577$	5.72937e31	1.01061	0.505307	+	0.862940i	$0.331379\pi$
$578$	0	0
$579$	1.11438e32	1.88900
$580$	0	0
$581$	5.50673e30	0.0897157
$582$	0	0
$583$	−1.72642e31	−0.270370
$584$	0	0
$585$	−2.36057e31	−0.355405
$586$	0	0
$587$	−3.05611e31	−0.442417	−0.221208	−	0.975227i	$-0.571000\pi$
$587$	−3.05611e31	−0.442417	−0.221208	+	0.975227i	$0.571000\pi$
$588$	0	0
$589$	2.84027e30	0.0395398
$590$	0	0
$591$	−1.06271e32	−1.42284
$592$	0	0
$593$	1.80853e31	0.232915	0.116458	−	0.993196i	$-0.462846\pi$
$593$	1.80853e31	0.232915	0.116458	+	0.993196i	$0.462846\pi$
$594$	0	0
$595$	−4.22877e30	−0.0523925
$596$	0	0
$597$	1.85110e32	2.20661
$598$	0	0
$599$	1.95180e30	0.0223886	0.0111943	−	0.999937i	$-0.496437\pi$
$599$	1.95180e30	0.0223886	0.0111943	+	0.999937i	$0.496437\pi$
$600$	0	0
$601$	−6.99778e31	−0.772510	−0.386255	−	0.922392i	$-0.626231\pi$
$601$	−6.99778e31	−0.772510	−0.386255	+	0.922392i	$0.626231\pi$
$602$	0	0
$603$	1.94371e32	2.06529
$604$	0	0
$605$	−8.67848e30	−0.0887681
$606$	0	0
$607$	−1.22226e32	−1.20363	−0.601814	−	0.798636i	$-0.705555\pi$
$607$	−1.22226e32	−1.20363	−0.601814	+	0.798636i	$0.705555\pi$
$608$	0	0
$609$	−1.18066e32	−1.11951
$610$	0	0
$611$	−1.75033e32	−1.59826
$612$	0	0
$613$	6.93405e31	0.609806	0.304903	−	0.952383i	$-0.401376\pi$
$613$	6.93405e31	0.609806	0.304903	+	0.952383i	$0.401376\pi$
$614$	0	0
$615$	−5.04794e31	−0.427613
$616$	0	0
$617$	−1.76301e31	−0.143872	−0.0719359	−	0.997409i	$-0.522918\pi$
$617$	−1.76301e31	−0.143872	−0.0719359	+	0.997409i	$0.522918\pi$
$618$	0	0
$619$	−2.34366e32	−1.84269	−0.921346	−	0.388743i	$-0.872909\pi$
$619$	−2.34366e32	−1.84269	−0.921346	+	0.388743i	$0.872909\pi$
$620$	0	0
$621$	4.72003e31	0.357595
$622$	0	0
$623$	4.84683e30	0.0353871
$624$	0	0
$625$	1.24429e32	0.875589
$626$	0	0
$627$	−6.75361e30	−0.0458099
$628$	0	0
$629$	2.60465e31	0.170321
$630$	0	0
$631$	3.97324e31	0.250499	0.125250	−	0.992125i	$-0.460027\pi$
$631$	3.97324e31	0.250499	0.125250	+	0.992125i	$0.460027\pi$
$632$	0	0
$633$	3.74725e32	2.27809
$634$	0	0
$635$	−1.19426e31	−0.0700167
$636$	0	0
$637$	1.64456e32	0.929922
$638$	0	0
$639$	1.56856e31	0.0855544
$640$	0	0
$641$	−2.87505e31	−0.151279	−0.0756393	−	0.997135i	$-0.524100\pi$
$641$	−2.87505e31	−0.151279	−0.0756393	+	0.997135i	$0.524100\pi$
$642$	0	0
$643$	−1.94333e32	−0.986555	−0.493277	−	0.869872i	$-0.664201\pi$
$643$	−1.94333e32	−0.986555	−0.493277	+	0.869872i	$0.664201\pi$
$644$	0	0
$645$	1.15977e32	0.568114
$646$	0	0
$647$	1.60264e32	0.757594	0.378797	−	0.925480i	$-0.376338\pi$
$647$	1.60264e32	0.757594	0.378797	+	0.925480i	$0.376338\pi$
$648$	0	0
$649$	1.86682e32	0.851702
$650$	0	0
$651$	1.64752e32	0.725521
$652$	0	0
$653$	−1.51268e31	−0.0643052	−0.0321526	−	0.999483i	$-0.510236\pi$
$653$	−1.51268e31	−0.0643052	−0.0321526	+	0.999483i	$0.510236\pi$
$654$	0	0
$655$	−6.94526e31	−0.285045
$656$	0	0
$657$	3.49442e32	1.38476
$658$	0	0
$659$	−2.02346e32	−0.774304	−0.387152	−	0.922016i	$-0.626541\pi$
$659$	−2.02346e32	−0.774304	−0.387152	+	0.922016i	$0.626541\pi$
$660$	0	0
$661$	−7.89573e31	−0.291793	−0.145897	−	0.989300i	$-0.546607\pi$
$661$	−7.89573e31	−0.291793	−0.145897	+	0.989300i	$0.546607\pi$
$662$	0	0
$663$	2.93061e32	1.04605
$664$	0	0
$665$	−1.02859e30	−0.00354645
$666$	0	0
$667$	2.20332e32	0.733889
$668$	0	0
$669$	−7.79630e31	−0.250893
$670$	0	0
$671$	−1.50458e32	−0.467851
$672$	0	0
$673$	−1.70303e32	−0.511740	−0.255870	−	0.966711i	$-0.582362\pi$
$673$	−1.70303e32	−0.511740	−0.255870	+	0.966711i	$0.582362\pi$
$674$	0	0
$675$	2.54325e32	0.738576
$676$	0	0
$677$	−6.13486e32	−1.72201	−0.861004	−	0.508598i	$-0.830164\pi$
$677$	−6.13486e32	−1.72201	−0.861004	+	0.508598i	$0.830164\pi$
$678$	0	0
$679$	−1.98907e32	−0.539693
$680$	0	0
$681$	4.74865e32	1.24560
$682$	0	0
$683$	6.04814e32	1.53385	0.766926	−	0.641735i	$-0.221785\pi$
$683$	6.04814e32	1.53385	0.766926	+	0.641735i	$0.221785\pi$
$684$	0	0
$685$	6.82913e31	0.167465
$686$	0	0
$687$	−7.11592e30	−0.0168744
$688$	0	0
$689$	1.82220e32	0.417903
$690$	0	0
$691$	9.02250e31	0.200138	0.100069	−	0.994981i	$-0.468094\pi$
$691$	9.02250e31	0.200138	0.100069	+	0.994981i	$0.468094\pi$
$692$	0	0
$693$	−2.34340e32	−0.502821
$694$	0	0
$695$	−5.26741e31	−0.109338
$696$	0	0
$697$	3.74882e32	0.752866
$698$	0	0
$699$	6.61063e32	1.28456
$700$	0	0
$701$	9.49911e32	1.78618	0.893091	−	0.449877i	$-0.148532\pi$
$701$	9.49911e32	1.78618	0.893091	+	0.449877i	$0.148532\pi$
$702$	0	0
$703$	6.33546e30	0.0115290
$704$	0	0
$705$	−2.52216e32	−0.444219
$706$	0	0
$707$	−3.07983e32	−0.525053
$708$	0	0
$709$	−8.56123e32	−1.41288	−0.706438	−	0.707775i	$-0.749699\pi$
$709$	−8.56123e32	−1.41288	−0.706438	+	0.707775i	$0.749699\pi$
$710$	0	0
$711$	1.64346e33	2.62579
$712$	0	0
$713$	−3.07455e32	−0.475612
$714$	0	0
$715$	−1.20028e32	−0.179789
$716$	0	0
$717$	−4.23653e32	−0.614525
$718$	0	0
$719$	−5.82101e32	−0.817741	−0.408871	−	0.912592i	$-0.634077\pi$
$719$	−5.82101e32	−0.817741	−0.408871	+	0.912592i	$0.634077\pi$
$720$	0	0
$721$	−1.07847e32	−0.146741
$722$	0	0
$723$	−7.48029e32	−0.985889
$724$	0	0
$725$	1.18719e33	1.51577
$726$	0	0
$727$	−2.03403e32	−0.251600	−0.125800	−	0.992056i	$-0.540150\pi$
$727$	−2.03403e32	−0.251600	−0.125800	+	0.992056i	$0.540150\pi$
$728$	0	0
$729$	−1.35326e33	−1.62187
$730$	0	0
$731$	−8.61297e32	−1.00024
$732$	0	0
$733$	−6.38041e32	−0.718047	−0.359024	−	0.933328i	$-0.616890\pi$
$733$	−6.38041e32	−0.718047	−0.359024	+	0.933328i	$0.616890\pi$
$734$	0	0
$735$	2.36975e32	0.258463
$736$	0	0
$737$	9.88318e32	1.04477
$738$	0	0
$739$	−1.21576e33	−1.24577	−0.622884	−	0.782314i	$-0.714039\pi$
$739$	−1.21576e33	−1.24577	−0.622884	+	0.782314i	$0.714039\pi$
$740$	0	0
$741$	7.12831e31	0.0708072
$742$	0	0
$743$	−8.76371e32	−0.843950	−0.421975	−	0.906607i	$-0.638663\pi$
$743$	−8.76371e32	−0.843950	−0.421975	+	0.906607i	$0.638663\pi$
$744$	0	0
$745$	−1.34768e32	−0.125832
$746$	0	0
$747$	−3.28949e32	−0.297812
$748$	0	0
$749$	−8.50016e32	−0.746253
$750$	0	0
$751$	−1.55769e33	−1.32624	−0.663121	−	0.748512i	$-0.730769\pi$
$751$	−1.55769e33	−1.32624	−0.663121	+	0.748512i	$0.730769\pi$
$752$	0	0
$753$	−3.80467e33	−3.14177
$754$	0	0
$755$	−3.10155e32	−0.248422
$756$	0	0
$757$	1.62779e33	1.26472	0.632362	−	0.774673i	$-0.282086\pi$
$757$	1.62779e33	1.26472	0.632362	+	0.774673i	$0.282086\pi$
$758$	0	0
$759$	7.31069e32	0.551033
$760$	0	0
$761$	−1.89057e33	−1.38251	−0.691257	−	0.722609i	$-0.742943\pi$
$761$	−1.89057e33	−1.38251	−0.691257	+	0.722609i	$0.742943\pi$
$762$	0	0
$763$	9.61564e32	0.682254
$764$	0	0
$765$	2.52609e32	0.173917
$766$	0	0
$767$	−1.97039e33	−1.31645
$768$	0	0
$769$	−1.42586e33	−0.924537	−0.462268	−	0.886740i	$-0.652964\pi$
$769$	−1.42586e33	−0.924537	−0.462268	+	0.886740i	$0.652964\pi$
$770$	0	0
$771$	9.42351e32	0.593045
$772$	0	0
$773$	−2.49392e33	−1.52341	−0.761707	−	0.647921i	$-0.775639\pi$
$773$	−2.49392e33	−1.52341	−0.761707	+	0.647921i	$0.775639\pi$
$774$	0	0
$775$	−1.65663e33	−0.982329
$776$	0	0
$777$	3.67493e32	0.211547
$778$	0	0
$779$	9.11849e31	0.0509615
$780$	0	0
$781$	7.97569e31	0.0432794
$782$	0	0
$783$	2.31533e33	1.21998
$784$	0	0
$785$	−2.57374e32	−0.131693
$786$	0	0
$787$	1.68998e33	0.839793	0.419897	−	0.907572i	$-0.362066\pi$
$787$	1.68998e33	0.839793	0.419897	+	0.907572i	$0.362066\pi$
$788$	0	0
$789$	−2.49376e33	−1.20356
$790$	0	0
$791$	−1.20218e33	−0.563559
$792$	0	0
$793$	1.58806e33	0.723145
$794$	0	0
$795$	2.62572e32	0.116152
$796$	0	0
$797$	2.67384e33	1.14912	0.574560	−	0.818462i	$-0.305173\pi$
$797$	2.67384e33	1.14912	0.574560	+	0.818462i	$0.305173\pi$
$798$	0	0
$799$	1.87306e33	0.782104
$800$	0	0
$801$	−2.89529e32	−0.117468
$802$	0	0
$803$	1.77681e33	0.700506
$804$	0	0
$805$	1.11343e32	0.0426591
$806$	0	0
$807$	5.21003e33	1.93997
$808$	0	0
$809$	−1.39778e33	−0.505862	−0.252931	−	0.967484i	$-0.581395\pi$
$809$	−1.39778e33	−0.505862	−0.252931	+	0.967484i	$0.581395\pi$
$810$	0	0
$811$	−3.45610e33	−1.21576	−0.607879	−	0.794030i	$-0.707979\pi$
$811$	−3.45610e33	−1.21576	−0.607879	+	0.794030i	$0.707979\pi$
$812$	0	0
$813$	2.22601e33	0.761177
$814$	0	0
$815$	3.12917e32	0.104020
$816$	0	0
$817$	−2.09499e32	−0.0677061
$818$	0	0
$819$	2.47341e33	0.777197
$820$	0	0
$821$	−4.67399e33	−1.42804	−0.714022	−	0.700124i	$-0.753128\pi$
$821$	−4.67399e33	−1.42804	−0.714022	+	0.700124i	$0.753128\pi$
$822$	0	0
$823$	−4.44526e33	−1.32068	−0.660342	−	0.750965i	$-0.729589\pi$
$823$	−4.44526e33	−1.32068	−0.660342	+	0.750965i	$0.729589\pi$
$824$	0	0
$825$	3.93915e33	1.13810
$826$	0	0
$827$	−4.08749e33	−1.14853	−0.574266	−	0.818669i	$-0.694713\pi$
$827$	−4.08749e33	−1.14853	−0.574266	+	0.818669i	$0.694713\pi$
$828$	0	0
$829$	−4.84125e33	−1.32306	−0.661532	−	0.749917i	$-0.730093\pi$
$829$	−4.84125e33	−1.32306	−0.661532	+	0.749917i	$0.730093\pi$
$830$	0	0
$831$	1.14886e34	3.05389
$832$	0	0
$833$	−1.75988e33	−0.455056
$834$	0	0
$835$	1.60637e33	0.404066
$836$	0	0
$837$	−3.23086e33	−0.790634
$838$	0	0
$839$	−3.43729e33	−0.818377	−0.409189	−	0.912450i	$-0.634188\pi$
$839$	−3.43729e33	−0.818377	−0.409189	+	0.912450i	$0.634188\pi$
$840$	0	0
$841$	6.49131e33	1.50376
$842$	0	0
$843$	4.70244e33	1.06000
$844$	0	0
$845$	3.31925e32	0.0728094
$846$	0	0
$847$	9.09335e32	0.194117
$848$	0	0
$849$	−2.11273e32	−0.0438941
$850$	0	0
$851$	−6.85804e32	−0.138679
$852$	0	0
$853$	−4.38561e33	−0.863209	−0.431604	−	0.902063i	$-0.642052\pi$
$853$	−4.38561e33	−0.863209	−0.431604	+	0.902063i	$0.642052\pi$
$854$	0	0
$855$	6.14436e31	0.0117724
$856$	0	0
$857$	3.11224e33	0.580489	0.290244	−	0.956953i	$-0.406263\pi$
$857$	3.11224e33	0.580489	0.290244	+	0.956953i	$0.406263\pi$
$858$	0	0
$859$	−5.56272e33	−1.01010	−0.505052	−	0.863089i	$-0.668527\pi$
$859$	−5.56272e33	−1.01010	−0.505052	+	0.863089i	$0.668527\pi$
$860$	0	0
$861$	5.28925e33	0.935100
$862$	0	0
$863$	−7.66403e33	−1.31927	−0.659634	−	0.751587i	$-0.729289\pi$
$863$	−7.66403e33	−1.31927	−0.659634	+	0.751587i	$0.729289\pi$
$864$	0	0
$865$	1.59123e33	0.266715
$866$	0	0
$867$	6.52904e33	1.06569
$868$	0	0
$869$	8.35653e33	1.32831
$870$	0	0
$871$	−1.04315e34	−1.61487
$872$	0	0
$873$	1.18818e34	1.79151
$874$	0	0
$875$	1.22622e33	0.180085
$876$	0	0
$877$	6.31038e33	0.902735	0.451368	−	0.892338i	$-0.350936\pi$
$877$	6.31038e33	0.902735	0.451368	+	0.892338i	$0.350936\pi$
$878$	0	0
$879$	−3.64654e33	−0.508170
$880$	0	0
$881$	4.30450e33	0.584386	0.292193	−	0.956359i	$-0.405615\pi$
$881$	4.30450e33	0.584386	0.292193	+	0.956359i	$0.405615\pi$
$882$	0	0
$883$	2.20567e33	0.291738	0.145869	−	0.989304i	$-0.453402\pi$
$883$	2.20567e33	0.291738	0.145869	+	0.989304i	$0.453402\pi$
$884$	0	0
$885$	−2.83926e33	−0.365895
$886$	0	0
$887$	1.09533e33	0.137538	0.0687689	−	0.997633i	$-0.478093\pi$
$887$	1.09533e33	0.137538	0.0687689	+	0.997633i	$0.478093\pi$
$888$	0	0
$889$	1.25135e33	0.153112
$890$	0	0
$891$	−1.72060e33	−0.205156
$892$	0	0
$893$	4.55597e32	0.0529406
$894$	0	0
$895$	−2.08450e32	−0.0236068
$896$	0	0
$897$	−7.71629e33	−0.851718
$898$	0	0
$899$	−1.50817e34	−1.62261
$900$	0	0
$901$	−1.94998e33	−0.204501
$902$	0	0
$903$	−1.21521e34	−1.24235
$904$	0	0
$905$	−1.66500e33	−0.165942
$906$	0	0
$907$	−1.63673e34	−1.59035	−0.795176	−	0.606378i	$-0.792622\pi$
$907$	−1.63673e34	−1.59035	−0.795176	+	0.606378i	$0.792622\pi$
$908$	0	0
$909$	1.83976e34	1.74292
$910$	0	0
$911$	1.59706e34	1.47523	0.737616	−	0.675220i	$-0.235952\pi$
$911$	1.59706e34	1.47523	0.737616	+	0.675220i	$0.235952\pi$
$912$	0	0
$913$	−1.67261e33	−0.150654
$914$	0	0
$915$	2.28833e33	0.200991
$916$	0	0
$917$	7.27727e33	0.623335
$918$	0	0
$919$	8.92501e33	0.745556	0.372778	−	0.927921i	$-0.378405\pi$
$919$	8.92501e33	0.745556	0.372778	+	0.927921i	$0.378405\pi$
$920$	0	0
$921$	−3.70355e33	−0.301740
$922$	0	0
$923$	−8.41819e32	−0.0668959
$924$	0	0
$925$	−3.69525e33	−0.286427
$926$	0	0
$927$	6.44231e33	0.487107
$928$	0	0
$929$	−7.87435e33	−0.580810	−0.290405	−	0.956904i	$-0.593790\pi$
$929$	−7.87435e33	−0.580810	−0.290405	+	0.956904i	$0.593790\pi$
$930$	0	0
$931$	−4.28066e32	−0.0308028
$932$	0	0
$933$	2.56326e34	1.79950
$934$	0	0
$935$	1.28444e33	0.0879795
$936$	0	0
$937$	8.14387e33	0.544284	0.272142	−	0.962257i	$-0.412268\pi$
$937$	8.14387e33	0.544284	0.272142	+	0.962257i	$0.412268\pi$
$938$	0	0
$939$	2.14759e34	1.40054
$940$	0	0
$941$	1.77249e34	1.12799	0.563993	−	0.825780i	$-0.309265\pi$
$941$	1.77249e34	1.12799	0.563993	+	0.825780i	$0.309265\pi$
$942$	0	0
$943$	−9.87064e33	−0.613000
$944$	0	0
$945$	1.17004e33	0.0709145
$946$	0	0
$947$	−6.86532e33	−0.406103	−0.203052	−	0.979168i	$-0.565086\pi$
$947$	−6.86532e33	−0.406103	−0.203052	+	0.979168i	$0.565086\pi$
$948$	0	0
$949$	−1.87539e34	−1.08275
$950$	0	0
$951$	2.00933e34	1.13234
$952$	0	0
$953$	1.17491e34	0.646305	0.323153	−	0.946347i	$-0.395257\pi$
$953$	1.17491e34	0.646305	0.323153	+	0.946347i	$0.395257\pi$
$954$	0	0
$955$	−2.08230e33	−0.111816
$956$	0	0
$957$	3.58614e34	1.87992
$958$	0	0
$959$	−7.15559e33	−0.366211
$960$	0	0
$961$	1.03203e33	0.0515671
$962$	0	0
$963$	5.07763e34	2.47719
$964$	0	0
$965$	−5.15516e33	−0.245571
$966$	0	0
$967$	2.08642e34	0.970503	0.485251	−	0.874375i	$-0.338728\pi$
$967$	2.08642e34	0.970503	0.485251	+	0.874375i	$0.338728\pi$
$968$	0	0
$969$	−7.62815e32	−0.0346494
$970$	0	0
$971$	5.25352e33	0.233039	0.116520	−	0.993188i	$-0.462826\pi$
$971$	5.25352e33	0.233039	0.116520	+	0.993188i	$0.462826\pi$
$972$	0	0
$973$	5.51921e33	0.239100
$974$	0	0
$975$	−4.15770e34	−1.75914
$976$	0	0
$977$	−3.27459e34	−1.35322	−0.676611	−	0.736341i	$-0.736552\pi$
$977$	−3.27459e34	−1.35322	−0.676611	+	0.736341i	$0.736552\pi$
$978$	0	0
$979$	−1.47217e33	−0.0594234
$980$	0	0
$981$	−5.74398e34	−2.26474
$982$	0	0
$983$	−4.04688e33	−0.155867	−0.0779336	−	0.996959i	$-0.524832\pi$
$983$	−4.04688e33	−0.155867	−0.0779336	+	0.996959i	$0.524832\pi$
$984$	0	0
$985$	4.91609e33	0.184971
$986$	0	0
$987$	2.64273e34	0.971415
$988$	0	0
$989$	2.26779e34	0.814415
$990$	0	0
$991$	4.51095e34	1.58278	0.791390	−	0.611311i	$-0.209358\pi$
$991$	4.51095e34	1.58278	0.791390	+	0.611311i	$0.209358\pi$
$992$	0	0
$993$	−5.52315e34	−1.89352
$994$	0	0
$995$	−8.56323e33	−0.286861
$996$	0	0
$997$	−5.61344e34	−1.83753	−0.918766	−	0.394802i	$-0.870813\pi$
$997$	−5.61344e34	−1.83753	−0.918766	+	0.394802i	$0.870813\pi$
$998$	0	0
$999$	−7.20670e33	−0.230533

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.n.1.1		6
4.3	odd	2	inner	64.24.a.n.1.6		6
8.3	odd	2		32.24.a.d.1.1	✓	6
8.5	even	2		32.24.a.d.1.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.d.1.1	✓	6	8.3	odd	2
32.24.a.d.1.6	yes	6	8.5	even	2
64.24.a.n.1.1		6	1.1	even	1	trivial
64.24.a.n.1.6		6	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:	\( k \)	\(=\)	\( 24 \)
Character orbit:	\([\chi]\)	\(=\)	64.a (trivial)

\(f(q)\)	\(=\)	\(q-484042. q^{3} +2.23919e7 q^{5} -2.34623e9 q^{7} +1.40154e11 q^{9} +O(q^{10})\)	\(q-484042. q^{3} +2.23919e7 q^{5} -2.34623e9 q^{7} +1.40154e11 q^{9} +7.12641e11 q^{11} -7.52179e12 q^{13} -1.08386e13 q^{15} +8.04922e13 q^{17} +1.95786e13 q^{19} +1.13567e15 q^{21} -2.11936e15 q^{23} -1.14195e16 q^{25} -2.22710e16 q^{27} -1.03962e17 q^{29} +1.45070e17 q^{31} -3.44948e17 q^{33} -5.25364e16 q^{35} +3.23591e17 q^{37} +3.64086e18 q^{39} +4.65737e18 q^{41} -1.07004e19 q^{43} +3.13830e18 q^{45} +2.32701e19 q^{47} -2.18640e19 q^{49} -3.89616e19 q^{51} -2.42257e19 q^{53} +1.59574e19 q^{55} -9.47688e18 q^{57} +2.61958e20 q^{59} -2.11128e20 q^{61} -3.28833e20 q^{63} -1.68427e20 q^{65} +1.38684e21 q^{67} +1.02586e21 q^{69} +1.11917e20 q^{71} +2.49327e21 q^{73} +5.52754e21 q^{75} -1.67202e21 q^{77} +1.17261e22 q^{79} -2.41439e21 q^{81} -2.34706e21 q^{83} +1.80237e21 q^{85} +5.03218e22 q^{87} -2.06580e21 q^{89} +1.76478e22 q^{91} -7.02201e22 q^{93} +4.38402e20 q^{95} +8.47772e22 q^{97} +9.98793e22 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 163121700 q^{5} + 14120212542 q^{9}+O(q^{10}) \) 6 * q - 163121700 * q^5 + 14120212542 * q^9	\( 6 q - 163121700 q^{5} + 14120212542 q^{9} - 1101295489524 q^{13} + 56435517243468 q^{17} - 14\!\cdots\!84 q^{21}+ \cdots + 25\!\cdots\!16 q^{97}+O(q^{100}) \) 6 * q - 163121700 * q^5 + 14120212542 * q^9 - 1101295489524 * q^13 + 56435517243468 * q^17 - 1447661324841984 * q^21 + 17512304669291850 * q^25 - 87161858953770708 * q^29 + 48462097564970496 * q^33 + 1542766250573484636 * q^37 - 865941451897399716 * q^41 + 9286254082406210700 * q^45 + 52551978665245784886 * q^49 + 10735070383373044668 * q^53 + 155104978099636154880 * q^57 + 432169207960503656364 * q^61 - 316526040451899107400 * q^65 - 695880182828346153984 * q^69 + 1758226166115151758204 * q^73 - 11926085657398731979776 * q^77 + 4010608843245537630678 * q^81 - 20251531460297137542600 * q^85 + 20087131519604301542940 * q^89 - 67310470168488243892224 * q^93 + 254665147475934678043116 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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