LMFDB - Embedded newform 64.24.a.o.1.6

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:	$0$
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} - 2 x^{5} - 376388081 x^{4} + 1624987949956 x^{3} + \cdots + 26\!\cdots\!25 $ x^6 - 2x^5 - 376388081x^4 + 1624987949956x^3 + 32369842380258997x^2 - 236529072927913245962*x + 267804601125256722828825
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{71}\cdot 3^{4}\cdot 5^{4} $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1422.41$ 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)

$f(q)$	$=$	$q+599828. q^{3} -1.94144e8 q^{5} +4.67609e9 q^{7} +2.65650e11 q^{9} +O(q^{10})$	$q+599828. q^{3} -1.94144e8 q^{5} +4.67609e9 q^{7} +2.65650e11 q^{9} +9.07574e11 q^{11} +6.04180e12 q^{13} -1.16453e14 q^{15} -2.80218e13 q^{17} -9.06389e13 q^{19} +2.80485e15 q^{21} +2.89339e14 q^{23} +2.57710e16 q^{25} +1.02875e17 q^{27} +4.09254e16 q^{29} -1.20741e17 q^{31} +5.44388e17 q^{33} -9.07836e17 q^{35} -1.40502e18 q^{37} +3.62404e18 q^{39} +5.50074e17 q^{41} +8.04411e18 q^{43} -5.15744e19 q^{45} +2.67290e18 q^{47} -5.50294e18 q^{49} -1.68082e19 q^{51} -4.15012e19 q^{53} -1.76200e20 q^{55} -5.43677e19 q^{57} +1.95904e20 q^{59} +3.54867e20 q^{61} +1.24220e21 q^{63} -1.17298e21 q^{65} -1.08440e21 q^{67} +1.73554e20 q^{69} -2.49651e20 q^{71} -4.40680e21 q^{73} +1.54582e22 q^{75} +4.24390e21 q^{77} +9.78251e21 q^{79} +3.66979e22 q^{81} +2.67780e20 q^{83} +5.44027e21 q^{85} +2.45482e22 q^{87} +1.67871e22 q^{89} +2.82520e22 q^{91} -7.24236e22 q^{93} +1.75970e22 q^{95} -9.64012e22 q^{97} +2.41097e23 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9}+O(q^{10}) $ 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9	$ 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9} + 926871857520 q^{11} + 3684897167820 q^{13} - 105662245358560 q^{15} + 71064722424780 q^{17} - 453921923982960 q^{19} + 17\!\cdots\!80 q^{21}+ \cdots + 39\!\cdots\!40 q^{99}+O(q^{100}) $ 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9 + 926871857520 * q^11 + 3684897167820 * q^13 - 105662245358560 * q^15 + 71064722424780 * q^17 - 453921923982960 * q^19 + 1712900116737280 * q^21 + 7126581767144736 * q^23 + 21617163289317450 * q^25 + 41851914072529184 * q^27 + 90718382478351852 * q^29 - 53660517982277760 * q^31 + 607667723496123520 * q^33 - 843485446441350720 * q^35 - 1178800099078290660 * q^37 - 387144388087503200 * q^39 + 1672300630976121180 * q^41 + 9707130536958833904 * q^43 - 31072675924013630900 * q^45 + 12637324971625378368 * q^47 + 20589514456941037878 * q^49 - 56125446355277416800 * q^51 + 25955354185131109500 * q^53 - 79415390320029033120 * q^55 + 169187096172727794560 * q^57 + 407713345239006043440 * q^59 + 615884547554394402156 * q^61 + 578320926136902243296 * q^63 - 1095923787012414183240 * q^65 - 523768762996317716400 * q^67 - 596596128814415845632 * q^69 - 948379808687003287200 * q^71 - 4925435605783876580100 * q^73 + 11105120520135150430000 * q^75 - 1628144191735745368320 * q^77 + 6474134783625939495360 * q^79 + 23375719407022646959702 * q^81 + 12413209217367249862032 * q^83 + 5978612007137723165880 * q^85 - 17230630950338425832032 * q^87 + 7892765755425674057628 * q^89 + 117701223538009065704640 * q^91 - 65760567831273830333440 * q^93 - 112886120912929344409440 * q^95 - 12935113600510578486420 * q^97 + 397484583772859161352240 * q^99

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
$3$	599828.	1.95493	0.977467	−	0.211088i	$-0.0677005\pi$
$3$	599828.	1.95493	0.977467	+	0.211088i	$0.0677005\pi$
$4$	0	0
$5$	−1.94144e8	−1.77815	−0.889077	−	0.457758i	$-0.848653\pi$
$5$	−1.94144e8	−1.77815	−0.889077	+	0.457758i	$0.848653\pi$
$6$	0	0
$7$	4.67609e9	0.893831	0.446915	−	0.894576i	$-0.352522\pi$
$7$	4.67609e9	0.893831	0.446915	+	0.894576i	$0.352522\pi$
$8$	0	0
$9$	2.65650e11	2.82177
$10$	0	0
$11$	9.07574e11	0.959105	0.479553	−	0.877513i	$-0.340799\pi$
$11$	9.07574e11	0.959105	0.479553	+	0.877513i	$0.340799\pi$
$12$	0	0
$13$	6.04180e12	0.935013	0.467507	−	0.883990i	$-0.345152\pi$
$13$	6.04180e12	0.935013	0.467507	+	0.883990i	$0.345152\pi$
$14$	0	0
$15$	−1.16453e14	−3.47617
$16$	0	0
$17$	−2.80218e13	−0.198305	−0.0991524	−	0.995072i	$-0.531613\pi$
$17$	−2.80218e13	−0.198305	−0.0991524	+	0.995072i	$0.531613\pi$
$18$	0	0
$19$	−9.06389e13	−0.178504	−0.0892520	−	0.996009i	$-0.528448\pi$
$19$	−9.06389e13	−0.178504	−0.0892520	+	0.996009i	$0.528448\pi$
$20$	0	0
$21$	2.80485e15	1.74738
$22$	0	0
$23$	2.89339e14	0.0633195	0.0316597	−	0.999499i	$-0.489921\pi$
$23$	2.89339e14	0.0633195	0.0316597	+	0.999499i	$0.489921\pi$
$24$	0	0
$25$	2.57710e16	2.16183
$26$	0	0
$27$	1.02875e17	3.56144
$28$	0	0
$29$	4.09254e16	0.622896	0.311448	−	0.950263i	$-0.399186\pi$
$29$	4.09254e16	0.622896	0.311448	+	0.950263i	$0.399186\pi$
$30$	0	0
$31$	−1.20741e17	−0.853481	−0.426741	−	0.904374i	$-0.640338\pi$
$31$	−1.20741e17	−0.853481	−0.426741	+	0.904374i	$0.640338\pi$
$32$	0	0
$33$	5.44388e17	1.87499
$34$	0	0
$35$	−9.07836e17	−1.58937
$36$	0	0
$37$	−1.40502e18	−1.29826	−0.649132	−	0.760676i	$-0.724868\pi$
$37$	−1.40502e18	−1.29826	−0.649132	+	0.760676i	$0.724868\pi$
$38$	0	0
$39$	3.62404e18	1.82789
$40$	0	0
$41$	5.50074e17	0.156101	0.0780507	−	0.996949i	$-0.475130\pi$
$41$	5.50074e17	0.156101	0.0780507	+	0.996949i	$0.475130\pi$
$42$	0	0
$43$	8.04411e18	1.32005	0.660025	−	0.751243i	$-0.270545\pi$
$43$	8.04411e18	1.32005	0.660025	+	0.751243i	$0.270545\pi$
$44$	0	0
$45$	−5.15744e19	−5.01754
$46$	0	0
$47$	2.67290e18	0.157709	0.0788546	−	0.996886i	$-0.474874\pi$
$47$	2.67290e18	0.157709	0.0788546	+	0.996886i	$0.474874\pi$
$48$	0	0
$49$	−5.50294e18	−0.201066
$50$	0	0
$51$	−1.68082e19	−0.387673
$52$	0	0
$53$	−4.15012e19	−0.615018	−0.307509	−	0.951545i	$-0.599495\pi$
$53$	−4.15012e19	−0.615018	−0.307509	+	0.951545i	$0.599495\pi$
$54$	0	0
$55$	−1.76200e20	−1.70544
$56$	0	0
$57$	−5.43677e19	−0.348964
$58$	0	0
$59$	1.95904e20	0.845755	0.422877	−	0.906187i	$-0.361020\pi$
$59$	1.95904e20	0.845755	0.422877	+	0.906187i	$0.361020\pi$
$60$	0	0
$61$	3.54867e20	1.04417	0.522087	−	0.852892i	$-0.325154\pi$
$61$	3.54867e20	1.04417	0.522087	+	0.852892i	$0.325154\pi$
$62$	0	0
$63$	1.24220e21	2.52218
$64$	0	0
$65$	−1.17298e21	−1.66260
$66$	0	0
$67$	−1.08440e21	−1.08475	−0.542373	−	0.840138i	$-0.682474\pi$
$67$	−1.08440e21	−1.08475	−0.542373	+	0.840138i	$0.682474\pi$
$68$	0	0
$69$	1.73554e20	0.123785
$70$	0	0
$71$	−2.49651e20	−0.128193	−0.0640963	−	0.997944i	$-0.520416\pi$
$71$	−2.49651e20	−0.128193	−0.0640963	+	0.997944i	$0.520416\pi$
$72$	0	0
$73$	−4.40680e21	−1.64403	−0.822016	−	0.569464i	$-0.807151\pi$
$73$	−4.40680e21	−1.64403	−0.822016	+	0.569464i	$0.807151\pi$
$74$	0	0
$75$	1.54582e22	4.22624
$76$	0	0
$77$	4.24390e21	0.857278
$78$	0	0
$79$	9.78251e21	1.47143	0.735714	−	0.677293i	$-0.236847\pi$
$79$	9.78251e21	1.47143	0.735714	+	0.677293i	$0.236847\pi$
$80$	0	0
$81$	3.66979e22	4.14061
$82$	0	0
$83$	2.67780e20	0.0228234	0.0114117	−	0.999935i	$-0.496367\pi$
$83$	2.67780e20	0.0228234	0.0114117	+	0.999935i	$0.496367\pi$
$84$	0	0
$85$	5.44027e21	0.352617
$86$	0	0
$87$	2.45482e22	1.21772
$88$	0	0
$89$	1.67871e22	0.641196	0.320598	−	0.947215i	$-0.396116\pi$
$89$	1.67871e22	0.641196	0.320598	+	0.947215i	$0.396116\pi$
$90$	0	0
$91$	2.82520e22	0.835744
$92$	0	0
$93$	−7.24236e22	−1.66850
$94$	0	0
$95$	1.75970e22	0.317408
$96$	0	0
$97$	−9.64012e22	−1.36838	−0.684192	−	0.729302i	$-0.739845\pi$
$97$	−9.64012e22	−1.36838	−0.684192	+	0.729302i	$0.739845\pi$
$98$	0	0
$99$	2.41097e23	2.70637
$100$	0	0
$101$	5.29395e22	0.472154	0.236077	−	0.971734i	$-0.424138\pi$
$101$	5.29395e22	0.472154	0.236077	+	0.971734i	$0.424138\pi$
$102$	0	0
$103$	1.61441e23	1.14918	0.574588	−	0.818443i	$-0.305162\pi$
$103$	1.61441e23	1.14918	0.574588	+	0.818443i	$0.305162\pi$
$104$	0	0
$105$	−5.44545e23	−3.10711
$106$	0	0
$107$	2.61981e23	1.20325	0.601626	−	0.798778i	$-0.294520\pi$
$107$	2.61981e23	1.20325	0.601626	+	0.798778i	$0.294520\pi$
$108$	0	0
$109$	−1.90343e23	−0.706532	−0.353266	−	0.935523i	$-0.614929\pi$
$109$	−1.90343e23	−0.706532	−0.353266	+	0.935523i	$0.614929\pi$
$110$	0	0
$111$	−8.42771e23	−2.53802
$112$	0	0
$113$	7.41158e23	1.81764	0.908821	−	0.417185i	$-0.136983\pi$
$113$	7.41158e23	1.81764	0.908821	+	0.417185i	$0.136983\pi$
$114$	0	0
$115$	−5.61735e22	−0.112592
$116$	0	0
$117$	1.60501e24	2.63839
$118$	0	0
$119$	−1.31032e23	−0.177251
$120$	0	0
$121$	−7.17390e22	−0.0801168
$122$	0	0
$123$	3.29950e23	0.305168
$124$	0	0
$125$	−2.68892e24	−2.06592
$126$	0	0
$127$	−9.29329e23	−0.594876	−0.297438	−	0.954741i	$-0.596132\pi$
$127$	−9.29329e23	−0.594876	−0.297438	+	0.954741i	$0.596132\pi$
$128$	0	0
$129$	4.82508e24	2.58061
$130$	0	0
$131$	−1.03815e24	−0.465199	−0.232600	−	0.972573i	$-0.574723\pi$
$131$	−1.03815e24	−0.465199	−0.232600	+	0.972573i	$0.574723\pi$
$132$	0	0
$133$	−4.23835e23	−0.159552
$134$	0	0
$135$	−1.99725e25	−6.33278
$136$	0	0
$137$	−3.15678e24	−0.845198	−0.422599	−	0.906317i	$-0.638882\pi$
$137$	−3.15678e24	−0.845198	−0.422599	+	0.906317i	$0.638882\pi$
$138$	0	0
$139$	4.58709e24	1.03960	0.519801	−	0.854287i	$-0.326006\pi$
$139$	4.58709e24	1.03960	0.519801	+	0.854287i	$0.326006\pi$
$140$	0	0
$141$	1.60328e24	0.308311
$142$	0	0
$143$	5.48338e24	0.896776
$144$	0	0
$145$	−7.94542e24	−1.10761
$146$	0	0
$147$	−3.30081e24	−0.393072
$148$	0	0
$149$	9.18593e24	0.936442	0.468221	−	0.883611i	$-0.344895\pi$
$149$	9.18593e24	0.936442	0.468221	+	0.883611i	$0.344895\pi$
$150$	0	0
$151$	1.49523e25	1.30759	0.653796	−	0.756671i	$-0.273176\pi$
$151$	1.49523e25	1.30759	0.653796	+	0.756671i	$0.273176\pi$
$152$	0	0
$153$	−7.44400e24	−0.559570
$154$	0	0
$155$	2.34411e25	1.51762
$156$	0	0
$157$	2.47998e25	1.38549	0.692743	−	0.721185i	$-0.256402\pi$
$157$	2.47998e25	1.38549	0.692743	+	0.721185i	$0.256402\pi$
$158$	0	0
$159$	−2.48936e25	−1.20232
$160$	0	0
$161$	1.35298e24	0.0565969
$162$	0	0
$163$	3.23378e25	1.17369	0.586845	−	0.809699i	$-0.300370\pi$
$163$	3.23378e25	1.17369	0.586845	+	0.809699i	$0.300370\pi$
$164$	0	0
$165$	−1.05690e26	−3.33402
$166$	0	0
$167$	−1.21986e25	−0.335019	−0.167510	−	0.985870i	$-0.553573\pi$
$167$	−1.21986e25	−0.335019	−0.167510	+	0.985870i	$0.553573\pi$
$168$	0	0
$169$	−5.25056e24	−0.125750
$170$	0	0
$171$	−2.40782e25	−0.503697
$172$	0	0
$173$	6.07896e25	1.11250	0.556249	−	0.831016i	$-0.312240\pi$
$173$	6.07896e25	1.11250	0.556249	+	0.831016i	$0.312240\pi$
$174$	0	0
$175$	1.20508e26	1.93231
$176$	0	0
$177$	1.17508e26	1.65339
$178$	0	0
$179$	1.04414e26	1.29107	0.645534	−	0.763731i	$-0.276635\pi$
$179$	1.04414e26	1.29107	0.645534	+	0.763731i	$0.276635\pi$
$180$	0	0
$181$	−2.04405e25	−0.222427	−0.111213	−	0.993797i	$-0.535474\pi$
$181$	−2.04405e25	−0.222427	−0.111213	+	0.993797i	$0.535474\pi$
$182$	0	0
$183$	2.12859e26	2.04129
$184$	0	0
$185$	2.72777e26	2.30851
$186$	0	0
$187$	−2.54319e25	−0.190195
$188$	0	0
$189$	4.81051e26	3.18332
$190$	0	0
$191$	2.97979e25	0.174703	0.0873517	−	0.996178i	$-0.472160\pi$
$191$	2.97979e25	0.174703	0.0873517	+	0.996178i	$0.472160\pi$
$192$	0	0
$193$	−1.16456e26	−0.605693	−0.302846	−	0.953039i	$-0.597937\pi$
$193$	−1.16456e26	−0.605693	−0.302846	+	0.953039i	$0.597937\pi$
$194$	0	0
$195$	−7.03586e26	−3.25027
$196$	0	0
$197$	−9.94818e25	−0.408679	−0.204339	−	0.978900i	$-0.565505\pi$
$197$	−9.94818e25	−0.408679	−0.204339	+	0.978900i	$0.565505\pi$
$198$	0	0
$199$	6.40995e25	0.234447	0.117223	−	0.993106i	$-0.462601\pi$
$199$	6.40995e25	0.234447	0.117223	+	0.993106i	$0.462601\pi$
$200$	0	0
$201$	−6.50452e26	−2.12061
$202$	0	0
$203$	1.91371e26	0.556764
$204$	0	0
$205$	−1.06794e26	−0.277572
$206$	0	0
$207$	7.68630e25	0.178673
$208$	0	0
$209$	−8.22615e25	−0.171204
$210$	0	0
$211$	−2.41261e26	−0.450027	−0.225014	−	0.974356i	$-0.572243\pi$
$211$	−2.41261e26	−0.450027	−0.225014	+	0.974356i	$0.572243\pi$
$212$	0	0
$213$	−1.49748e26	−0.250608
$214$	0	0
$215$	−1.56172e27	−2.34725
$216$	0	0
$217$	−5.64594e26	−0.762868
$218$	0	0
$219$	−2.64332e27	−3.21397
$220$	0	0
$221$	−1.69302e26	−0.185418
$222$	0	0
$223$	6.69576e26	0.661141	0.330570	−	0.943781i	$-0.392759\pi$
$223$	6.69576e26	0.661141	0.330570	+	0.943781i	$0.392759\pi$
$224$	0	0
$225$	6.84608e27	6.10019
$226$	0	0
$227$	2.01029e26	0.161793	0.0808967	−	0.996722i	$-0.474222\pi$
$227$	2.01029e26	0.161793	0.0808967	+	0.996722i	$0.474222\pi$
$228$	0	0
$229$	7.20954e26	0.524565	0.262282	−	0.964991i	$-0.415525\pi$
$229$	7.20954e26	0.524565	0.262282	+	0.964991i	$0.415525\pi$
$230$	0	0
$231$	2.54561e27	1.67592
$232$	0	0
$233$	−1.47024e27	−0.876588	−0.438294	−	0.898832i	$-0.644417\pi$
$233$	−1.47024e27	−0.876588	−0.438294	+	0.898832i	$0.644417\pi$
$234$	0	0
$235$	−5.18928e26	−0.280431
$236$	0	0
$237$	5.86782e27	2.87654
$238$	0	0
$239$	−4.12811e27	−1.83728	−0.918639	−	0.395098i	$-0.870711\pi$
$239$	−4.12811e27	−1.83728	−0.918639	+	0.395098i	$0.870711\pi$
$240$	0	0
$241$	1.52090e27	0.615040	0.307520	−	0.951542i	$-0.400501\pi$
$241$	1.52090e27	0.615040	0.307520	+	0.951542i	$0.400501\pi$
$242$	0	0
$243$	1.23275e28	4.53318
$244$	0	0
$245$	1.06836e27	0.357527
$246$	0	0
$247$	−5.47622e26	−0.166904
$248$	0	0
$249$	1.60622e26	0.0446182
$250$	0	0
$251$	3.05800e27	0.774800	0.387400	−	0.921912i	$-0.373373\pi$
$251$	3.05800e27	0.774800	0.387400	+	0.921912i	$0.373373\pi$
$252$	0	0
$253$	2.62597e26	0.0607300
$254$	0	0
$255$	3.26322e27	0.689342
$256$	0	0
$257$	−5.17216e27	−0.998714	−0.499357	−	0.866396i	$-0.666430\pi$
$257$	−5.17216e27	−0.998714	−0.499357	+	0.866396i	$0.666430\pi$
$258$	0	0
$259$	−6.57000e27	−1.16043
$260$	0	0
$261$	1.08718e28	1.75767
$262$	0	0
$263$	−3.45175e27	−0.511150	−0.255575	−	0.966789i	$-0.582265\pi$
$263$	−3.45175e27	−0.511150	−0.255575	+	0.966789i	$0.582265\pi$
$264$	0	0
$265$	8.05721e27	1.09360
$266$	0	0
$267$	1.00694e28	1.25350
$268$	0	0
$269$	6.42405e27	0.733935	0.366968	−	0.930234i	$-0.380396\pi$
$269$	6.42405e27	0.733935	0.366968	+	0.930234i	$0.380396\pi$
$270$	0	0
$271$	1.47170e28	1.54409	0.772043	−	0.635571i	$-0.219235\pi$
$271$	1.47170e28	1.54409	0.772043	+	0.635571i	$0.219235\pi$
$272$	0	0
$273$	1.69463e28	1.63382
$274$	0	0
$275$	2.33891e28	2.07342
$276$	0	0
$277$	−1.65582e28	−1.35050	−0.675251	−	0.737588i	$-0.735965\pi$
$277$	−1.65582e28	−1.35050	−0.675251	+	0.737588i	$0.735965\pi$
$278$	0	0
$279$	−3.20748e28	−2.40833
$280$	0	0
$281$	2.20721e28	1.52659	0.763293	−	0.646052i	$-0.223581\pi$
$281$	2.20721e28	1.52659	0.763293	+	0.646052i	$0.223581\pi$
$282$	0	0
$283$	−1.17978e28	−0.752067	−0.376033	−	0.926606i	$-0.622712\pi$
$283$	−1.17978e28	−0.752067	−0.376033	+	0.926606i	$0.622712\pi$
$284$	0	0
$285$	1.05552e28	0.620511
$286$	0	0
$287$	2.57219e27	0.139528
$288$	0	0
$289$	−1.91823e28	−0.960675
$290$	0	0
$291$	−5.78241e28	−2.67510
$292$	0	0
$293$	3.63485e27	0.155421	0.0777103	−	0.996976i	$-0.475239\pi$
$293$	3.63485e27	0.155421	0.0777103	+	0.996976i	$0.475239\pi$
$294$	0	0
$295$	−3.80336e28	−1.50388
$296$	0	0
$297$	9.33664e28	3.41579
$298$	0	0
$299$	1.74813e27	0.0592045
$300$	0	0
$301$	3.76150e28	1.17990
$302$	0	0
$303$	3.17546e28	0.923030
$304$	0	0
$305$	−6.88954e28	−1.85670
$306$	0	0
$307$	5.61941e28	1.40475	0.702375	−	0.711807i	$-0.252123\pi$
$307$	5.61941e28	1.40475	0.702375	+	0.711807i	$0.252123\pi$
$308$	0	0
$309$	9.68370e28	2.24656
$310$	0	0
$311$	5.56812e28	1.19940	0.599700	−	0.800225i	$-0.295287\pi$
$311$	5.56812e28	1.19940	0.599700	+	0.800225i	$0.295287\pi$
$312$	0	0
$313$	−5.59551e28	−1.11964	−0.559821	−	0.828613i	$-0.689130\pi$
$313$	−5.59551e28	−1.11964	−0.559821	+	0.828613i	$0.689130\pi$
$314$	0	0
$315$	−2.41167e29	−4.48483
$316$	0	0
$317$	2.26956e28	0.392428	0.196214	−	0.980561i	$-0.437135\pi$
$317$	2.26956e28	0.392428	0.196214	+	0.980561i	$0.437135\pi$
$318$	0	0
$319$	3.71428e28	0.597423
$320$	0	0
$321$	1.57144e29	2.35228
$322$	0	0
$323$	2.53986e27	0.0353982
$324$	0	0
$325$	1.55703e29	2.02134
$326$	0	0
$327$	−1.14173e29	−1.38122
$328$	0	0
$329$	1.24987e28	0.140965
$330$	0	0
$331$	−1.03207e29	−1.08564	−0.542822	−	0.839848i	$-0.682644\pi$
$331$	−1.03207e29	−1.08564	−0.542822	+	0.839848i	$0.682644\pi$
$332$	0	0
$333$	−3.73244e29	−3.66340
$334$	0	0
$335$	2.10530e29	1.92885
$336$	0	0
$337$	−1.24757e29	−1.06739	−0.533693	−	0.845678i	$-0.679196\pi$
$337$	−1.24757e29	−1.06739	−0.533693	+	0.845678i	$0.679196\pi$
$338$	0	0
$339$	4.44567e29	3.55337
$340$	0	0
$341$	−1.09581e29	−0.818579
$342$	0	0
$343$	−1.53711e29	−1.07355
$344$	0	0
$345$	−3.36945e28	−0.220109
$346$	0	0
$347$	8.62677e28	0.527302	0.263651	−	0.964618i	$-0.415073\pi$
$347$	8.62677e28	0.527302	0.263651	+	0.964618i	$0.415073\pi$
$348$	0	0
$349$	−3.03023e29	−1.73373	−0.866867	−	0.498539i	$-0.833870\pi$
$349$	−3.03023e29	−1.73373	−0.866867	+	0.498539i	$0.833870\pi$
$350$	0	0
$351$	6.21548e29	3.32999
$352$	0	0
$353$	−1.04105e28	−0.0522471	−0.0261236	−	0.999659i	$-0.508316\pi$
$353$	−1.04105e28	−0.0522471	−0.0261236	+	0.999659i	$0.508316\pi$
$354$	0	0
$355$	4.84683e28	0.227946
$356$	0	0
$357$	−7.85969e28	−0.346514
$358$	0	0
$359$	3.25448e29	1.34554	0.672768	−	0.739853i	$-0.265105\pi$
$359$	3.25448e29	1.34554	0.672768	+	0.739853i	$0.265105\pi$
$360$	0	0
$361$	−2.49614e29	−0.968136
$362$	0	0
$363$	−4.30311e28	−0.156623
$364$	0	0
$365$	8.55554e29	2.92334
$366$	0	0
$367$	−4.17075e29	−1.33830	−0.669151	−	0.743126i	$-0.733342\pi$
$367$	−4.17075e29	−1.33830	−0.669151	+	0.743126i	$0.733342\pi$
$368$	0	0
$369$	1.46127e29	0.440482
$370$	0	0
$371$	−1.94063e29	−0.549722
$372$	0	0
$373$	1.31946e29	0.351353	0.175677	−	0.984448i	$-0.443789\pi$
$373$	1.31946e29	0.351353	0.175677	+	0.984448i	$0.443789\pi$
$374$	0	0
$375$	−1.61289e30	−4.03873
$376$	0	0
$377$	2.47263e29	0.582416
$378$	0	0
$379$	3.04625e28	0.0675173	0.0337587	−	0.999430i	$-0.489252\pi$
$379$	3.04625e28	0.0675173	0.0337587	+	0.999430i	$0.489252\pi$
$380$	0	0
$381$	−5.57437e29	−1.16294
$382$	0	0
$383$	−1.53729e29	−0.301975	−0.150988	−	0.988536i	$-0.548245\pi$
$383$	−1.53729e29	−0.301975	−0.150988	+	0.988536i	$0.548245\pi$
$384$	0	0
$385$	−8.23928e29	−1.52437
$386$	0	0
$387$	2.13692e30	3.72488
$388$	0	0
$389$	7.59912e29	1.24837	0.624184	−	0.781277i	$-0.285432\pi$
$389$	7.59912e29	1.24837	0.624184	+	0.781277i	$0.285432\pi$
$390$	0	0
$391$	−8.10780e27	−0.0125566
$392$	0	0
$393$	−6.22710e29	−0.909434
$394$	0	0
$395$	−1.89922e30	−2.61642
$396$	0	0
$397$	−2.02577e29	−0.263329	−0.131664	−	0.991294i	$-0.542032\pi$
$397$	−2.02577e29	−0.263329	−0.131664	+	0.991294i	$0.542032\pi$
$398$	0	0
$399$	−2.54228e29	−0.311914
$400$	0	0
$401$	−1.23216e30	−1.42727	−0.713634	−	0.700519i	$-0.752952\pi$
$401$	−1.23216e30	−1.42727	−0.713634	+	0.700519i	$0.752952\pi$
$402$	0	0
$403$	−7.29491e29	−0.798016
$404$	0	0
$405$	−7.12469e30	−7.36264
$406$	0	0
$407$	−1.27516e30	−1.24517
$408$	0	0
$409$	−1.05468e30	−0.973427	−0.486714	−	0.873562i	$-0.661805\pi$
$409$	−1.05468e30	−0.973427	−0.486714	+	0.873562i	$0.661805\pi$
$410$	0	0
$411$	−1.89353e30	−1.65231
$412$	0	0
$413$	9.16063e29	0.755962
$414$	0	0
$415$	−5.19879e28	−0.0405834
$416$	0	0
$417$	2.75146e30	2.03235
$418$	0	0
$419$	2.47003e30	1.72680	0.863399	−	0.504522i	$-0.168331\pi$
$419$	2.47003e30	1.72680	0.863399	+	0.504522i	$0.168331\pi$
$420$	0	0
$421$	1.13091e30	0.748489	0.374244	−	0.927330i	$-0.377902\pi$
$421$	1.13091e30	0.748489	0.374244	+	0.927330i	$0.377902\pi$
$422$	0	0
$423$	7.10056e29	0.445019
$424$	0	0
$425$	−7.22151e29	−0.428702
$426$	0	0
$427$	1.65939e30	0.933315
$428$	0	0
$429$	3.28909e30	1.75314
$430$	0	0
$431$	3.30818e30	1.67148	0.835738	−	0.549128i	$-0.185040\pi$
$431$	3.30818e30	1.67148	0.835738	+	0.549128i	$0.185040\pi$
$432$	0	0
$433$	−1.35303e30	−0.648182	−0.324091	−	0.946026i	$-0.605058\pi$
$433$	−1.35303e30	−0.648182	−0.324091	+	0.946026i	$0.605058\pi$
$434$	0	0
$435$	−4.76589e30	−2.16530
$436$	0	0
$437$	−2.62254e28	−0.0113028
$438$	0	0
$439$	6.01224e29	0.245864	0.122932	−	0.992415i	$-0.460770\pi$
$439$	6.01224e29	0.245864	0.122932	+	0.992415i	$0.460770\pi$
$440$	0	0
$441$	−1.46186e30	−0.567363
$442$	0	0
$443$	4.30546e30	1.58627	0.793133	−	0.609048i	$-0.208448\pi$
$443$	4.30546e30	1.58627	0.793133	+	0.609048i	$0.208448\pi$
$444$	0	0
$445$	−3.25912e30	−1.14015
$446$	0	0
$447$	5.50997e30	1.83068
$448$	0	0
$449$	−1.92894e30	−0.608815	−0.304407	−	0.952542i	$-0.598458\pi$
$449$	−1.92894e30	−0.608815	−0.304407	+	0.952542i	$0.598458\pi$
$450$	0	0
$451$	4.99233e29	0.149718
$452$	0	0
$453$	8.96878e30	2.55625
$454$	0	0
$455$	−5.48496e30	−1.48608
$456$	0	0
$457$	−6.87387e30	−1.77078	−0.885391	−	0.464847i	$-0.846109\pi$
$457$	−6.87387e30	−1.77078	−0.885391	+	0.464847i	$0.846109\pi$
$458$	0	0
$459$	−2.88273e30	−0.706250
$460$	0	0
$461$	−1.64871e30	−0.384224	−0.192112	−	0.981373i	$-0.561534\pi$
$461$	−1.64871e30	−0.384224	−0.192112	+	0.981373i	$0.561534\pi$
$462$	0	0
$463$	8.97092e30	1.98910	0.994548	−	0.104281i	$-0.0332540\pi$
$463$	8.97092e30	1.98910	0.994548	+	0.104281i	$0.0332540\pi$
$464$	0	0
$465$	1.40606e31	2.96685
$466$	0	0
$467$	−3.45656e30	−0.694224	−0.347112	−	0.937824i	$-0.612838\pi$
$467$	−3.45656e30	−0.694224	−0.347112	+	0.937824i	$0.612838\pi$
$468$	0	0
$469$	−5.07074e30	−0.969580
$470$	0	0
$471$	1.48756e31	2.70853
$472$	0	0
$473$	7.30063e30	1.26607
$474$	0	0
$475$	−2.33586e30	−0.385896
$476$	0	0
$477$	−1.10248e31	−1.73544
$478$	0	0
$479$	−8.85165e30	−1.32790	−0.663951	−	0.747776i	$-0.731121\pi$
$479$	−8.85165e30	−1.32790	−0.663951	+	0.747776i	$0.731121\pi$
$480$	0	0
$481$	−8.48886e30	−1.21389
$482$	0	0
$483$	8.11553e29	0.110643
$484$	0	0
$485$	1.87157e31	2.43320
$486$	0	0
$487$	6.19608e30	0.768306	0.384153	−	0.923270i	$-0.374494\pi$
$487$	6.19608e30	0.768306	0.384153	+	0.923270i	$0.374494\pi$
$488$	0	0
$489$	1.93971e31	2.29449
$490$	0	0
$491$	1.35301e31	1.52709	0.763546	−	0.645753i	$-0.223456\pi$
$491$	1.35301e31	1.52709	0.763546	+	0.645753i	$0.223456\pi$
$492$	0	0
$493$	−1.14680e30	−0.123523
$494$	0	0
$495$	−4.68076e31	−4.81235
$496$	0	0
$497$	−1.16739e30	−0.114582
$498$	0	0
$499$	−7.41381e30	−0.694841	−0.347420	−	0.937710i	$-0.612942\pi$
$499$	−7.41381e30	−0.694841	−0.347420	+	0.937710i	$0.612942\pi$
$500$	0	0
$501$	−7.31705e30	−0.654941
$502$	0	0
$503$	−5.44447e30	−0.465505	−0.232752	−	0.972536i	$-0.574773\pi$
$503$	−5.44447e30	−0.465505	−0.232752	+	0.972536i	$0.574773\pi$
$504$	0	0
$505$	−1.02779e31	−0.839563
$506$	0	0
$507$	−3.14943e30	−0.245833
$508$	0	0
$509$	6.04960e30	0.451307	0.225654	−	0.974208i	$-0.427548\pi$
$509$	6.04960e30	0.451307	0.225654	+	0.974208i	$0.427548\pi$
$510$	0	0
$511$	−2.06066e31	−1.46949
$512$	0	0
$513$	−9.32444e30	−0.635731
$514$	0	0
$515$	−3.13429e31	−2.04341
$516$	0	0
$517$	2.42585e30	0.151260
$518$	0	0
$519$	3.64633e31	2.17486
$520$	0	0
$521$	1.44858e31	0.826627	0.413313	−	0.910589i	$-0.364371\pi$
$521$	1.44858e31	0.826627	0.413313	+	0.910589i	$0.364371\pi$
$522$	0	0
$523$	9.00300e30	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$523$	9.00300e30	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$524$	0	0
$525$	7.22839e31	3.77754
$526$	0	0
$527$	3.38337e30	0.169249
$528$	0	0
$529$	−2.07968e31	−0.995991
$530$	0	0
$531$	5.20418e31	2.38652
$532$	0	0
$533$	3.32344e30	0.145957
$534$	0	0
$535$	−5.08621e31	−2.13957
$536$	0	0
$537$	6.26305e31	2.52395
$538$	0	0
$539$	−4.99432e30	−0.192844
$540$	0	0
$541$	−2.25601e31	−0.834780	−0.417390	−	0.908728i	$-0.637055\pi$
$541$	−2.25601e31	−0.834780	−0.417390	+	0.908728i	$0.637055\pi$
$542$	0	0
$543$	−1.22608e31	−0.434830
$544$	0	0
$545$	3.69540e31	1.25632
$546$	0	0
$547$	−2.88241e31	−0.939509	−0.469755	−	0.882797i	$-0.655658\pi$
$547$	−2.88241e31	−0.939509	−0.469755	+	0.882797i	$0.655658\pi$
$548$	0	0
$549$	9.42706e31	2.94642
$550$	0	0
$551$	−3.70943e30	−0.111189
$552$	0	0
$553$	4.57439e31	1.31521
$554$	0	0
$555$	1.63619e32	4.51299
$556$	0	0
$557$	−6.24521e31	−1.65277	−0.826385	−	0.563105i	$-0.809607\pi$
$557$	−6.24521e31	−1.65277	−0.826385	+	0.563105i	$0.809607\pi$
$558$	0	0
$559$	4.86009e31	1.23427
$560$	0	0
$561$	−1.52547e31	−0.371819
$562$	0	0
$563$	−5.16252e31	−1.20786	−0.603928	−	0.797039i	$-0.706399\pi$
$563$	−5.16252e31	−1.20786	−0.603928	+	0.797039i	$0.706399\pi$
$564$	0	0
$565$	−1.43891e32	−3.23205
$566$	0	0
$567$	1.71603e32	3.70100
$568$	0	0
$569$	2.14523e31	0.444307	0.222153	−	0.975012i	$-0.428691\pi$
$569$	2.14523e31	0.444307	0.222153	+	0.975012i	$0.428691\pi$
$570$	0	0
$571$	1.21454e31	0.241600	0.120800	−	0.992677i	$-0.461454\pi$
$571$	1.21454e31	0.241600	0.120800	+	0.992677i	$0.461454\pi$
$572$	0	0
$573$	1.78736e31	0.341534
$574$	0	0
$575$	7.45657e30	0.136886
$576$	0	0
$577$	−1.46874e31	−0.259074	−0.129537	−	0.991575i	$-0.541349\pi$
$577$	−1.46874e31	−0.259074	−0.129537	+	0.991575i	$0.541349\pi$
$578$	0	0
$579$	−6.98535e31	−1.18409
$580$	0	0
$581$	1.25216e30	0.0204002
$582$	0	0
$583$	−3.76654e31	−0.589867
$584$	0	0
$585$	−3.11603e32	−4.69147
$586$	0	0
$587$	−8.82904e31	−1.27813	−0.639065	−	0.769152i	$-0.720679\pi$
$587$	−8.82904e31	−1.27813	−0.639065	+	0.769152i	$0.720679\pi$
$588$	0	0
$589$	1.09438e31	0.152350
$590$	0	0
$591$	−5.96719e31	−0.798940
$592$	0	0
$593$	2.50394e31	0.322474	0.161237	−	0.986916i	$-0.448452\pi$
$593$	2.50394e31	0.322474	0.161237	+	0.986916i	$0.448452\pi$
$594$	0	0
$595$	2.54392e31	0.315180
$596$	0	0
$597$	3.84487e31	0.458328
$598$	0	0
$599$	7.24328e31	0.830858	0.415429	−	0.909626i	$-0.363631\pi$
$599$	7.24328e31	0.830858	0.415429	+	0.909626i	$0.363631\pi$
$600$	0	0
$601$	4.56317e30	0.0503744	0.0251872	−	0.999683i	$-0.491982\pi$
$601$	4.56317e30	0.0503744	0.0251872	+	0.999683i	$0.491982\pi$
$602$	0	0
$603$	−2.88071e32	−3.06090
$604$	0	0
$605$	1.39277e31	0.142460
$606$	0	0
$607$	−2.30963e31	−0.227443	−0.113722	−	0.993513i	$-0.536277\pi$
$607$	−2.30963e31	−0.227443	−0.113722	+	0.993513i	$0.536277\pi$
$608$	0	0
$609$	1.14789e32	1.08844
$610$	0	0
$611$	1.61491e31	0.147460
$612$	0	0
$613$	4.28357e31	0.376713	0.188357	−	0.982101i	$-0.439684\pi$
$613$	4.28357e31	0.376713	0.188357	+	0.982101i	$0.439684\pi$
$614$	0	0
$615$	−6.40578e31	−0.542636
$616$	0	0
$617$	−9.33284e31	−0.761613	−0.380807	−	0.924655i	$-0.624354\pi$
$617$	−9.33284e31	−0.761613	−0.380807	+	0.924655i	$0.624354\pi$
$618$	0	0
$619$	1.37163e32	1.07843	0.539217	−	0.842167i	$-0.318720\pi$
$619$	1.37163e32	1.07843	0.539217	+	0.842167i	$0.318720\pi$
$620$	0	0
$621$	2.97657e31	0.225508
$622$	0	0
$623$	7.84981e31	0.573121
$624$	0	0
$625$	2.14823e32	1.51168
$626$	0	0
$627$	−4.93427e31	−0.334693
$628$	0	0
$629$	3.93712e31	0.257452
$630$	0	0
$631$	−7.90564e31	−0.498424	−0.249212	−	0.968449i	$-0.580172\pi$
$631$	−7.90564e31	−0.498424	−0.249212	+	0.968449i	$0.580172\pi$
$632$	0	0
$633$	−1.44715e32	−0.879774
$634$	0	0
$635$	1.80424e32	1.05778
$636$	0	0
$637$	−3.32476e31	−0.188000
$638$	0	0
$639$	−6.63199e31	−0.361730
$640$	0	0
$641$	9.86969e31	0.519321	0.259661	−	0.965700i	$-0.416389\pi$
$641$	9.86969e31	0.519321	0.259661	+	0.965700i	$0.416389\pi$
$642$	0	0
$643$	−2.67901e32	−1.36003	−0.680015	−	0.733198i	$-0.738027\pi$
$643$	−2.67901e32	−1.36003	−0.680015	+	0.733198i	$0.738027\pi$
$644$	0	0
$645$	−9.36761e32	−4.58873
$646$	0	0
$647$	−3.68444e32	−1.74170	−0.870848	−	0.491553i	$-0.836430\pi$
$647$	−3.68444e32	−1.74170	−0.870848	+	0.491553i	$0.836430\pi$
$648$	0	0
$649$	1.77797e32	0.811168
$650$	0	0
$651$	−3.38659e32	−1.49136
$652$	0	0
$653$	−2.99115e32	−1.27156	−0.635780	−	0.771871i	$-0.719321\pi$
$653$	−2.99115e32	−1.27156	−0.635780	+	0.771871i	$0.719321\pi$
$654$	0	0
$655$	2.01550e32	0.827196
$656$	0	0
$657$	−1.17067e33	−4.63908
$658$	0	0
$659$	−2.30273e32	−0.881171	−0.440585	−	0.897711i	$-0.645229\pi$
$659$	−2.30273e32	−0.881171	−0.440585	+	0.897711i	$0.645229\pi$
$660$	0	0
$661$	1.78499e32	0.659656	0.329828	−	0.944041i	$-0.393009\pi$
$661$	1.78499e32	0.659656	0.329828	+	0.944041i	$0.393009\pi$
$662$	0	0
$663$	−1.01552e32	−0.362479
$664$	0	0
$665$	8.22852e31	0.283709
$666$	0	0
$667$	1.18413e31	0.0394415
$668$	0	0
$669$	4.01630e32	1.29249
$670$	0	0
$671$	3.22068e32	1.00147
$672$	0	0
$673$	1.67534e31	0.0503419	0.0251710	−	0.999683i	$-0.491987\pi$
$673$	1.67534e31	0.0503419	0.0251710	+	0.999683i	$0.491987\pi$
$674$	0	0
$675$	2.65119e33	7.69923
$676$	0	0
$677$	−6.67098e32	−1.87249	−0.936246	−	0.351346i	$-0.885724\pi$
$677$	−6.67098e32	−1.87249	−0.936246	+	0.351346i	$0.885724\pi$
$678$	0	0
$679$	−4.50781e32	−1.22310
$680$	0	0
$681$	1.20583e32	0.316295
$682$	0	0
$683$	3.68966e32	0.935725	0.467862	−	0.883801i	$-0.345024\pi$
$683$	3.68966e32	0.935725	0.467862	+	0.883801i	$0.345024\pi$
$684$	0	0
$685$	6.12871e32	1.50289
$686$	0	0
$687$	4.32448e32	1.02549
$688$	0	0
$689$	−2.50742e32	−0.575050
$690$	0	0
$691$	−3.38485e32	−0.750829	−0.375415	−	0.926857i	$-0.622500\pi$
$691$	−3.38485e32	−0.750829	−0.375415	+	0.926857i	$0.622500\pi$
$692$	0	0
$693$	1.12739e33	2.41904
$694$	0	0
$695$	−8.90557e32	−1.84857
$696$	0	0
$697$	−1.54141e31	−0.0309557
$698$	0	0
$699$	−8.81891e32	−1.71367
$700$	0	0
$701$	1.71493e31	0.0322470	0.0161235	−	0.999870i	$-0.494868\pi$
$701$	1.71493e31	0.0322470	0.0161235	+	0.999870i	$0.494868\pi$
$702$	0	0
$703$	1.27350e32	0.231745
$704$	0	0
$705$	−3.11267e32	−0.548225
$706$	0	0
$707$	2.47550e32	0.422026
$708$	0	0
$709$	4.59943e31	0.0759054	0.0379527	−	0.999280i	$-0.487916\pi$
$709$	4.59943e31	0.0759054	0.0379527	+	0.999280i	$0.487916\pi$
$710$	0	0
$711$	2.59873e33	4.15203
$712$	0	0
$713$	−3.49350e31	−0.0540420
$714$	0	0
$715$	−1.06457e33	−1.59461
$716$	0	0
$717$	−2.47615e33	−3.59176
$718$	0	0
$719$	−7.80178e32	−1.09600	−0.548001	−	0.836478i	$-0.684611\pi$
$719$	−7.80178e32	−1.09600	−0.548001	+	0.836478i	$0.684611\pi$
$720$	0	0
$721$	7.54914e32	1.02717
$722$	0	0
$723$	9.12276e32	1.20236
$724$	0	0
$725$	1.05469e33	1.34660
$726$	0	0
$727$	−1.46187e33	−1.80826	−0.904132	−	0.427253i	$-0.859481\pi$
$727$	−1.46187e33	−1.80826	−0.904132	+	0.427253i	$0.859481\pi$
$728$	0	0
$729$	3.93951e33	4.72146
$730$	0	0
$731$	−2.25410e32	−0.261772
$732$	0	0
$733$	5.52915e32	0.622247	0.311123	−	0.950370i	$-0.399295\pi$
$733$	5.52915e32	0.622247	0.311123	+	0.950370i	$0.399295\pi$
$734$	0	0
$735$	6.40834e32	0.698942
$736$	0	0
$737$	−9.84172e32	−1.04039
$738$	0	0
$739$	−1.23448e33	−1.26495	−0.632476	−	0.774580i	$-0.717961\pi$
$739$	−1.23448e33	−1.26495	−0.632476	+	0.774580i	$0.717961\pi$
$740$	0	0
$741$	−3.28479e32	−0.326286
$742$	0	0
$743$	−1.10716e33	−1.06620	−0.533102	−	0.846051i	$-0.678974\pi$
$743$	−1.10716e33	−1.06620	−0.533102	+	0.846051i	$0.678974\pi$
$744$	0	0
$745$	−1.78339e33	−1.66514
$746$	0	0
$747$	7.11357e31	0.0644022
$748$	0	0
$749$	1.22505e33	1.07550
$750$	0	0
$751$	−3.07288e32	−0.261629	−0.130815	−	0.991407i	$-0.541759\pi$
$751$	−3.07288e32	−0.261629	−0.130815	+	0.991407i	$0.541759\pi$
$752$	0	0
$753$	1.83427e33	1.51468
$754$	0	0
$755$	−2.90289e33	−2.32510
$756$	0	0
$757$	1.68066e33	1.30580	0.652900	−	0.757444i	$-0.273552\pi$
$757$	1.68066e33	1.30580	0.652900	+	0.757444i	$0.273552\pi$
$758$	0	0
$759$	1.57513e32	0.118723
$760$	0	0
$761$	1.97667e32	0.144548	0.0722739	−	0.997385i	$-0.476974\pi$
$761$	1.97667e32	0.144548	0.0722739	+	0.997385i	$0.476974\pi$
$762$	0	0
$763$	−8.90060e32	−0.631520
$764$	0	0
$765$	1.44521e33	0.995002
$766$	0	0
$767$	1.18361e33	0.790792
$768$	0	0
$769$	1.91794e33	1.24360	0.621802	−	0.783175i	$-0.286401\pi$
$769$	1.91794e33	1.24360	0.621802	+	0.783175i	$0.286401\pi$
$770$	0	0
$771$	−3.10241e33	−1.95242
$772$	0	0
$773$	1.01129e33	0.617751	0.308875	−	0.951103i	$-0.400047\pi$
$773$	1.01129e33	0.617751	0.308875	+	0.951103i	$0.400047\pi$
$774$	0	0
$775$	−3.11161e33	−1.84508
$776$	0	0
$777$	−3.94087e33	−2.26856
$778$	0	0
$779$	−4.98581e31	−0.0278647
$780$	0	0
$781$	−2.26577e32	−0.122950
$782$	0	0
$783$	4.21019e33	2.21841
$784$	0	0
$785$	−4.81473e33	−2.46361
$786$	0	0
$787$	3.49401e33	1.73626	0.868131	−	0.496335i	$-0.165321\pi$
$787$	3.49401e33	1.73626	0.868131	+	0.496335i	$0.165321\pi$
$788$	0	0
$789$	−2.07046e33	−0.999265
$790$	0	0
$791$	3.46572e33	1.62467
$792$	0	0
$793$	2.14404e33	0.976317
$794$	0	0
$795$	4.83294e33	2.13791
$796$	0	0
$797$	−1.94666e33	−0.836602	−0.418301	−	0.908308i	$-0.637374\pi$
$797$	−1.94666e33	−0.836602	−0.418301	+	0.908308i	$0.637374\pi$
$798$	0	0
$799$	−7.48994e31	−0.0312745
$800$	0	0
$801$	4.45950e33	1.80931
$802$	0	0
$803$	−3.99950e33	−1.57680
$804$	0	0
$805$	−2.62672e32	−0.100638
$806$	0	0
$807$	3.85332e33	1.43480
$808$	0	0
$809$	2.08091e33	0.753088	0.376544	−	0.926399i	$-0.377112\pi$
$809$	2.08091e33	0.753088	0.376544	+	0.926399i	$0.377112\pi$
$810$	0	0
$811$	2.34181e33	0.823781	0.411890	−	0.911233i	$-0.364869\pi$
$811$	2.34181e33	0.823781	0.411890	+	0.911233i	$0.364869\pi$
$812$	0	0
$813$	8.82764e33	3.01858
$814$	0	0
$815$	−6.27820e33	−2.08700
$816$	0	0
$817$	−7.29109e32	−0.235634
$818$	0	0
$819$	7.50515e33	2.35828
$820$	0	0
$821$	−2.96465e33	−0.905788	−0.452894	−	0.891564i	$-0.649608\pi$
$821$	−2.96465e33	−0.905788	−0.452894	+	0.891564i	$0.649608\pi$
$822$	0	0
$823$	1.46897e33	0.436430	0.218215	−	0.975901i	$-0.429977\pi$
$823$	1.46897e33	0.436430	0.218215	+	0.975901i	$0.429977\pi$
$824$	0	0
$825$	1.40295e34	4.05341
$826$	0	0
$827$	−3.54337e33	−0.995642	−0.497821	−	0.867280i	$-0.665866\pi$
$827$	−3.54337e33	−0.995642	−0.497821	+	0.867280i	$0.665866\pi$
$828$	0	0
$829$	7.22528e33	1.97459	0.987297	−	0.158888i	$-0.0507909\pi$
$829$	7.22528e33	1.97459	0.987297	+	0.158888i	$0.0507909\pi$
$830$	0	0
$831$	−9.93205e33	−2.64014
$832$	0	0
$833$	1.54202e32	0.0398724
$834$	0	0
$835$	2.36828e33	0.595716
$836$	0	0
$837$	−1.24212e34	−3.03962
$838$	0	0
$839$	4.45524e33	1.06074	0.530369	−	0.847767i	$-0.322053\pi$
$839$	4.45524e33	1.06074	0.530369	+	0.847767i	$0.322053\pi$
$840$	0	0
$841$	−2.64183e33	−0.612000
$842$	0	0
$843$	1.32395e34	2.98438
$844$	0	0
$845$	1.01936e33	0.223603
$846$	0	0
$847$	−3.35458e32	−0.0716109
$848$	0	0
$849$	−7.07664e33	−1.47024
$850$	0	0
$851$	−4.06528e32	−0.0822054
$852$	0	0
$853$	−3.02900e33	−0.596191	−0.298096	−	0.954536i	$-0.596351\pi$
$853$	−3.02900e33	−0.596191	−0.298096	+	0.954536i	$0.596351\pi$
$854$	0	0
$855$	4.67465e33	0.895651
$856$	0	0
$857$	−8.38209e32	−0.156341	−0.0781705	−	0.996940i	$-0.524908\pi$
$857$	−8.38209e32	−0.156341	−0.0781705	+	0.996940i	$0.524908\pi$
$858$	0	0
$859$	−5.16075e33	−0.937112	−0.468556	−	0.883434i	$-0.655226\pi$
$859$	−5.16075e33	−0.937112	−0.468556	+	0.883434i	$0.655226\pi$
$860$	0	0
$861$	1.54287e33	0.272768
$862$	0	0
$863$	−2.78600e33	−0.479575	−0.239787	−	0.970825i	$-0.577078\pi$
$863$	−2.78600e33	−0.479575	−0.239787	+	0.970825i	$0.577078\pi$
$864$	0	0
$865$	−1.18019e34	−1.97819
$866$	0	0
$867$	−1.15061e34	−1.87806
$868$	0	0
$869$	8.87836e33	1.41125
$870$	0	0
$871$	−6.55171e33	−1.01425
$872$	0	0
$873$	−2.56090e34	−3.86126
$874$	0	0
$875$	−1.25736e34	−1.84658
$876$	0	0
$877$	−7.56385e33	−1.08205	−0.541025	−	0.841006i	$-0.681964\pi$
$877$	−7.56385e33	−1.08205	−0.541025	+	0.841006i	$0.681964\pi$
$878$	0	0
$879$	2.18028e33	0.303837
$880$	0	0
$881$	5.42366e33	0.736326	0.368163	−	0.929761i	$-0.379987\pi$
$881$	5.42366e33	0.736326	0.368163	+	0.929761i	$0.379987\pi$
$882$	0	0
$883$	−1.67994e33	−0.222201	−0.111100	−	0.993809i	$-0.535438\pi$
$883$	−1.67994e33	−0.222201	−0.111100	+	0.993809i	$0.535438\pi$
$884$	0	0
$885$	−2.28136e34	−2.93999
$886$	0	0
$887$	−1.26014e34	−1.58232	−0.791162	−	0.611606i	$-0.790524\pi$
$887$	−1.26014e34	−1.58232	−0.791162	+	0.611606i	$0.790524\pi$
$888$	0	0
$889$	−4.34562e33	−0.531719
$890$	0	0
$891$	3.33061e34	3.97128
$892$	0	0
$893$	−2.42268e32	−0.0281517
$894$	0	0
$895$	−2.02714e34	−2.29572
$896$	0	0
$897$	1.04858e33	0.115741
$898$	0	0
$899$	−4.94136e33	−0.531630
$900$	0	0
$901$	1.16294e33	0.121961
$902$	0	0
$903$	2.25625e34	2.30663
$904$	0	0
$905$	3.96840e33	0.395509
$906$	0	0
$907$	−6.24260e33	−0.606571	−0.303285	−	0.952900i	$-0.598084\pi$
$907$	−6.24260e33	−0.606571	−0.303285	+	0.952900i	$0.598084\pi$
$908$	0	0
$909$	1.40634e34	1.33231
$910$	0	0
$911$	−9.25036e33	−0.854471	−0.427235	−	0.904140i	$-0.640512\pi$
$911$	−9.25036e33	−0.854471	−0.427235	+	0.904140i	$0.640512\pi$
$912$	0	0
$913$	2.43030e32	0.0218900
$914$	0	0
$915$	−4.13254e34	−3.62973
$916$	0	0
$917$	−4.85447e33	−0.415809
$918$	0	0
$919$	4.49930e32	0.0375852	0.0187926	−	0.999823i	$-0.494018\pi$
$919$	4.49930e32	0.0375852	0.0187926	+	0.999823i	$0.494018\pi$
$920$	0	0
$921$	3.37068e34	2.74619
$922$	0	0
$923$	−1.50834e33	−0.119862
$924$	0	0
$925$	−3.62089e34	−2.80663
$926$	0	0
$927$	4.28869e34	3.24271
$928$	0	0
$929$	2.11921e34	1.56312	0.781562	−	0.623828i	$-0.214423\pi$
$929$	2.11921e34	1.56312	0.781562	+	0.623828i	$0.214423\pi$
$930$	0	0
$931$	4.98780e32	0.0358912
$932$	0	0
$933$	3.33991e34	2.34475
$934$	0	0
$935$	4.93745e33	0.338196
$936$	0	0
$937$	1.53872e34	1.02838	0.514192	−	0.857675i	$-0.328092\pi$
$937$	1.53872e34	1.02838	0.514192	+	0.857675i	$0.328092\pi$
$938$	0	0
$939$	−3.35634e34	−2.18883
$940$	0	0
$941$	−1.44116e34	−0.917130	−0.458565	−	0.888661i	$-0.651636\pi$
$941$	−1.44116e34	−0.917130	−0.458565	+	0.888661i	$0.651636\pi$
$942$	0	0
$943$	1.59158e32	0.00988425
$944$	0	0
$945$	−9.33933e34	−5.66044
$946$	0	0
$947$	3.00127e34	1.77534	0.887669	−	0.460482i	$-0.152324\pi$
$947$	3.00127e34	1.77534	0.887669	+	0.460482i	$0.152324\pi$
$948$	0	0
$949$	−2.66250e34	−1.53719
$950$	0	0
$951$	1.36134e34	0.767171
$952$	0	0
$953$	−2.04332e33	−0.112400	−0.0562002	−	0.998420i	$-0.517899\pi$
$953$	−2.04332e33	−0.112400	−0.0562002	+	0.998420i	$0.517899\pi$
$954$	0	0
$955$	−5.78508e33	−0.310650
$956$	0	0
$957$	2.22793e34	1.16792
$958$	0	0
$959$	−1.47614e34	−0.755464
$960$	0	0
$961$	−5.43501e33	−0.271570
$962$	0	0
$963$	6.95954e34	3.39530
$964$	0	0
$965$	2.26092e34	1.07701
$966$	0	0
$967$	−1.25439e34	−0.583483	−0.291742	−	0.956497i	$-0.594235\pi$
$967$	−1.25439e34	−0.583483	−0.291742	+	0.956497i	$0.594235\pi$
$968$	0	0
$969$	1.52348e33	0.0692012
$970$	0	0
$971$	4.78021e33	0.212044	0.106022	−	0.994364i	$-0.466189\pi$
$971$	4.78021e33	0.212044	0.106022	+	0.994364i	$0.466189\pi$
$972$	0	0
$973$	2.14496e34	0.929228
$974$	0	0
$975$	9.33953e34	3.95159
$976$	0	0
$977$	5.75705e33	0.237910	0.118955	−	0.992900i	$-0.462046\pi$
$977$	5.75705e33	0.237910	0.118955	+	0.992900i	$0.462046\pi$
$978$	0	0
$979$	1.52356e34	0.614975
$980$	0	0
$981$	−5.05646e34	−1.99367
$982$	0	0
$983$	3.79059e34	1.45996	0.729981	−	0.683467i	$-0.239529\pi$
$983$	3.79059e34	1.45996	0.729981	+	0.683467i	$0.239529\pi$
$984$	0	0
$985$	1.93138e34	0.726693
$986$	0	0
$987$	7.49707e33	0.275578
$988$	0	0
$989$	2.32748e33	0.0835849
$990$	0	0
$991$	−2.27118e34	−0.796902	−0.398451	−	0.917190i	$-0.630452\pi$
$991$	−2.27118e34	−0.796902	−0.398451	+	0.917190i	$0.630452\pi$
$992$	0	0
$993$	−6.19064e34	−2.12236
$994$	0	0
$995$	−1.24445e34	−0.416882
$996$	0	0
$997$	−1.19399e34	−0.390847	−0.195424	−	0.980719i	$-0.562608\pi$
$997$	−1.19399e34	−0.390847	−0.195424	+	0.980719i	$0.562608\pi$
$998$	0	0
$999$	−1.44541e35	−4.62369

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	64.24.a.o.1.6		6
4.3	odd	2		64.24.a.m.1.1		6
8.3	odd	2		32.24.a.e.1.6	yes	6
8.5	even	2		32.24.a.c.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.24.a.c.1.1	✓	6	8.5	even	2
32.24.a.e.1.6	yes	6	8.3	odd	2
64.24.a.m.1.1		6	4.3	odd	2
64.24.a.o.1.6		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+599828. q^{3} -1.94144e8 q^{5} +4.67609e9 q^{7} +2.65650e11 q^{9} +O(q^{10})\)	\(q+599828. q^{3} -1.94144e8 q^{5} +4.67609e9 q^{7} +2.65650e11 q^{9} +9.07574e11 q^{11} +6.04180e12 q^{13} -1.16453e14 q^{15} -2.80218e13 q^{17} -9.06389e13 q^{19} +2.80485e15 q^{21} +2.89339e14 q^{23} +2.57710e16 q^{25} +1.02875e17 q^{27} +4.09254e16 q^{29} -1.20741e17 q^{31} +5.44388e17 q^{33} -9.07836e17 q^{35} -1.40502e18 q^{37} +3.62404e18 q^{39} +5.50074e17 q^{41} +8.04411e18 q^{43} -5.15744e19 q^{45} +2.67290e18 q^{47} -5.50294e18 q^{49} -1.68082e19 q^{51} -4.15012e19 q^{53} -1.76200e20 q^{55} -5.43677e19 q^{57} +1.95904e20 q^{59} +3.54867e20 q^{61} +1.24220e21 q^{63} -1.17298e21 q^{65} -1.08440e21 q^{67} +1.73554e20 q^{69} -2.49651e20 q^{71} -4.40680e21 q^{73} +1.54582e22 q^{75} +4.24390e21 q^{77} +9.78251e21 q^{79} +3.66979e22 q^{81} +2.67780e20 q^{83} +5.44027e21 q^{85} +2.45482e22 q^{87} +1.67871e22 q^{89} +2.82520e22 q^{91} -7.24236e22 q^{93} +1.75970e22 q^{95} -9.64012e22 q^{97} +2.41097e23 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9}+O(q^{10}) \) 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9	\( 6 q + 483920 q^{3} + 6100380 q^{5} + 347289696 q^{7} + 260924449726 q^{9} + 926871857520 q^{11} + 3684897167820 q^{13} - 105662245358560 q^{15} + 71064722424780 q^{17} - 453921923982960 q^{19} + 17\!\cdots\!80 q^{21}+ \cdots + 39\!\cdots\!40 q^{99}+O(q^{100}) \) 6 * q + 483920 * q^3 + 6100380 * q^5 + 347289696 * q^7 + 260924449726 * q^9 + 926871857520 * q^11 + 3684897167820 * q^13 - 105662245358560 * q^15 + 71064722424780 * q^17 - 453921923982960 * q^19 + 1712900116737280 * q^21 + 7126581767144736 * q^23 + 21617163289317450 * q^25 + 41851914072529184 * q^27 + 90718382478351852 * q^29 - 53660517982277760 * q^31 + 607667723496123520 * q^33 - 843485446441350720 * q^35 - 1178800099078290660 * q^37 - 387144388087503200 * q^39 + 1672300630976121180 * q^41 + 9707130536958833904 * q^43 - 31072675924013630900 * q^45 + 12637324971625378368 * q^47 + 20589514456941037878 * q^49 - 56125446355277416800 * q^51 + 25955354185131109500 * q^53 - 79415390320029033120 * q^55 + 169187096172727794560 * q^57 + 407713345239006043440 * q^59 + 615884547554394402156 * q^61 + 578320926136902243296 * q^63 - 1095923787012414183240 * q^65 - 523768762996317716400 * q^67 - 596596128814415845632 * q^69 - 948379808687003287200 * q^71 - 4925435605783876580100 * q^73 + 11105120520135150430000 * q^75 - 1628144191735745368320 * q^77 + 6474134783625939495360 * q^79 + 23375719407022646959702 * q^81 + 12413209217367249862032 * q^83 + 5978612007137723165880 * q^85 - 17230630950338425832032 * q^87 + 7892765755425674057628 * q^89 + 117701223538009065704640 * q^91 - 65760567831273830333440 * q^93 - 112886120912929344409440 * q^95 - 12935113600510578486420 * q^97 + 397484583772859161352240 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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