LMFDB - Embedded newform 32.20.b.a.17.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [32,20,Mod(17,32)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(32, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 20, names="a")

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

chi := DirichletCharacter("32.17");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 32 = 2^{5} $
Weight:	$ k $	$=$	$ 20 $
Character orbit:	$[\chi]$	$=$	32.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$73.2213428980$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 9 x^{17} + 847482358 x^{16} - 6779858660 x^{15} + \cdots + 11\!\cdots\!76 $ x^18 - 9x^17 + 847482358x^16 - 6779858660x^15 + 288755694526787238x^14 - 2021289743039984830x^13 + 51121850635988052944521232x^12 - 306731077539161966630890980x^11 + 5053028465804290954773516593713297x^10 - 25265139517319958838954290038554593x^9 + 277525242785226225995541366728053703523138x^8 - 1110100819550071174104182779845774739997744x^7 + 7843581170028346695466748185748667347430921666496x^6 - 23530739624732277774733733894667178736900394654048x^5 + 88699644118206331836031766951870758907094189945148981184x^4 - 177399249018514584235839520819873862547126510378177038080x^3 + 27738447216461514660719783835924541644179956872586393406592768x^2 - 27738358516840927193180104500114450400133860395870028292288768*x + 1155804113673024778339834102840085457131269774629483366903990104576
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	multiple of $ 2^{306}\cdot 3^{16}\cdot 5^{4}\cdot 7 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			17.9
Root			$0.500000 + 222.384i$ of defining polynomial
Character	$\chi$	$=$	32.17
Dual form			32.20.b.a.17.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-889.536i q^{3} -5.56401e6i q^{5} +4.51327e6 q^{7} +1.16147e9 q^{9} +O(q^{10})$	$q-889.536i q^{3} -5.56401e6i q^{5} +4.51327e6 q^{7} +1.16147e9 q^{9} +1.78940e9i q^{11} +3.40689e10i q^{13} -4.94939e9 q^{15} -4.37021e11 q^{17} +2.55409e12i q^{19} -4.01471e9i q^{21} +5.58979e12 q^{23} -1.18847e13 q^{25} -2.06704e12i q^{27} -9.17732e13i q^{29} +6.96656e13 q^{31} +1.59173e12 q^{33} -2.51119e13i q^{35} +7.21521e14i q^{37} +3.03055e13 q^{39} -1.74108e15 q^{41} +3.57351e15i q^{43} -6.46243e15i q^{45} +1.43582e16 q^{47} -1.13785e16 q^{49} +3.88746e14i q^{51} -1.88104e16i q^{53} +9.95622e15 q^{55} +2.27196e15 q^{57} +4.76955e16i q^{59} +1.16266e17i q^{61} +5.24203e15 q^{63} +1.89560e17 q^{65} -1.76329e17i q^{67} -4.97232e15i q^{69} -6.82001e16 q^{71} +5.94941e17 q^{73} +1.05719e16i q^{75} +8.07603e15i q^{77} +1.48803e18 q^{79} +1.34809e18 q^{81} +1.21993e18i q^{83} +2.43159e18i q^{85} -8.16356e16 q^{87} +3.29788e18 q^{89} +1.53762e17i q^{91} -6.19700e16i q^{93} +1.42110e19 q^{95} +4.52106e18 q^{97} +2.07833e18i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 80707216 q^{7} - 6198727826 q^{9}+O(q^{10}) $ 18 * q + 80707216 * q^7 - 6198727826 * q^9	$ 18 q + 80707216 q^{7} - 6198727826 q^{9} - 156097960432 q^{15} + 14121426692 q^{17} - 2177121583952 q^{23} - 44414474211734 q^{25} - 428505770260416 q^{31} - 185380269683736 q^{33} - 942830575043152 q^{39} + 12\!\cdots\!24 q^{41}+ \cdots + 82\!\cdots\!20 q^{97}+O(q^{100}) $ 18 * q + 80707216 * q^7 - 6198727826 * q^9 - 156097960432 * q^15 + 14121426692 * q^17 - 2177121583952 * q^23 - 44414474211734 * q^25 - 428505770260416 * q^31 - 185380269683736 * q^33 - 942830575043152 * q^39 + 1200260021141524 * q^41 + 1134160959992928 * q^47 + 31570092936329058 * q^49 + 51757047863478736 * q^55 - 7740744140670520 * q^57 - 430168668665078416 * q^63 - 184007586011668640 * q^65 + 1599477989199799056 * q^71 - 93004298989809068 * q^73 - 1813975801025264480 * q^79 + 1801762045144484330 * q^81 + 13124960835754102512 * q^87 + 4470509771045743540 * q^89 - 23809730986995679856 * q^95 + 8246429789464117220 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/32\mathbb{Z}\right)^\times$.

$n$	$5$	$31$
$\chi(n)$	$-1$	$1$

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
$3$	− 889.536i	− 0.0260923i	−0.999915	−	0.0130461i	$-0.995847\pi$
$3$	− 889.536i	− 0.0260923i	0.999915	−	0.0130461i	$-0.00415283\pi$
$4$	0	0
$5$	− 5.56401e6i	− 1.27401i	−0.770860	−	0.637005i	$-0.780173\pi$
$5$	− 5.56401e6i	− 1.27401i	0.770860	−	0.637005i	$-0.219827\pi$
$6$	0	0
$7$	4.51327e6	0.0422727	0.0211363	−	0.999777i	$-0.493272\pi$
$7$	4.51327e6	0.0422727	0.0211363	+	0.999777i	$0.493272\pi$
$8$	0	0
$9$	1.16147e9	0.999319
$10$	0	0
$11$	1.78940e9i	0.228811i	0.993434	+	0.114405i	$0.0364962\pi$
$11$	1.78940e9i	0.228811i	−0.993434	+	0.114405i	$0.963504\pi$
$12$	0	0
$13$	3.40689e10i	0.891037i	0.895273	+	0.445519i	$0.146981\pi$
$13$	3.40689e10i	0.891037i	−0.895273	+	0.445519i	$0.853019\pi$
$14$	0	0
$15$	−4.94939e9	−0.0332418
$16$	0	0
$17$	−4.37021e11	−0.893794	−0.446897	−	0.894586i	$-0.647471\pi$
$17$	−4.37021e11	−0.893794	−0.446897	+	0.894586i	$0.647471\pi$
$18$	0	0
$19$	2.55409e12i	1.81584i	0.419144	+	0.907920i	$0.362330\pi$
$19$	2.55409e12i	1.81584i	−0.419144	+	0.907920i	$0.637670\pi$
$20$	0	0
$21$	− 4.01471e9i	− 0.00110299i
$22$	0	0
$23$	5.58979e12	0.647114	0.323557	−	0.946209i	$-0.395121\pi$
$23$	5.58979e12	0.647114	0.323557	+	0.946209i	$0.395121\pi$
$24$	0	0
$25$	−1.18847e13	−0.623102
$26$	0	0
$27$	− 2.06704e12i	− 0.0521667i
$28$	0	0
$29$	− 9.17732e13i	− 1.17472i	−0.809325	−	0.587361i	$-0.800167\pi$
$29$	− 9.17732e13i	− 1.17472i	0.809325	−	0.587361i	$-0.199833\pi$
$30$	0	0
$31$	6.96656e13	0.473241	0.236620	−	0.971602i	$-0.423960\pi$
$31$	6.96656e13	0.473241	0.236620	+	0.971602i	$0.423960\pi$
$32$	0	0
$33$	1.59173e12	0.00597018
$34$	0	0
$35$	− 2.51119e13i	− 0.0538558i
$36$	0	0
$37$	7.21521e14i	0.912710i	0.889798	+	0.456355i	$0.150845\pi$
$37$	7.21521e14i	0.912710i	−0.889798	+	0.456355i	$0.849155\pi$
$38$	0	0
$39$	3.03055e13	0.0232492
$40$	0	0
$41$	−1.74108e15	−0.830561	−0.415281	−	0.909693i	$-0.636317\pi$
$41$	−1.74108e15	−0.830561	−0.415281	+	0.909693i	$0.636317\pi$
$42$	0	0
$43$	3.57351e15i	1.08429i	0.840285	+	0.542145i	$0.182388\pi$
$43$	3.57351e15i	1.08429i	−0.840285	+	0.542145i	$0.817612\pi$
$44$	0	0
$45$	− 6.46243e15i	− 1.27314i
$46$	0	0
$47$	1.43582e16	1.87142	0.935709	−	0.352774i	$-0.114761\pi$
$47$	1.43582e16	1.87142	0.935709	+	0.352774i	$0.114761\pi$
$48$	0	0
$49$	−1.13785e16	−0.998213
$50$	0	0
$51$	3.88746e14i	0.0233211i
$52$	0	0
$53$	− 1.88104e16i	− 0.783029i	−0.920172	−	0.391515i	$-0.871951\pi$
$53$	− 1.88104e16i	− 0.783029i	0.920172	−	0.391515i	$-0.128049\pi$
$54$	0	0
$55$	9.95622e15	0.291507
$56$	0	0
$57$	2.27196e15	0.0473794
$58$	0	0
$59$	4.76955e16i	0.716775i	0.933573	+	0.358388i	$0.116673\pi$
$59$	4.76955e16i	0.716775i	−0.933573	+	0.358388i	$0.883327\pi$
$60$	0	0
$61$	1.16266e17i	1.27297i	0.771290	+	0.636484i	$0.219612\pi$
$61$	1.16266e17i	1.27297i	−0.771290	+	0.636484i	$0.780388\pi$
$62$	0	0
$63$	5.24203e15	0.0422439
$64$	0	0
$65$	1.89560e17	1.13519
$66$	0	0
$67$	− 1.76329e17i	− 0.791793i	−0.918295	−	0.395897i	$-0.870434\pi$
$67$	− 1.76329e17i	− 0.791793i	0.918295	−	0.395897i	$-0.129566\pi$
$68$	0	0
$69$	− 4.97232e15i	− 0.0168847i
$70$	0	0
$71$	−6.82001e16	−0.176535	−0.0882675	−	0.996097i	$-0.528133\pi$
$71$	−6.82001e16	−0.176535	−0.0882675	+	0.996097i	$0.528133\pi$
$72$	0	0
$73$	5.94941e17	1.18279	0.591394	−	0.806383i	$-0.298578\pi$
$73$	5.94941e17	1.18279	0.591394	+	0.806383i	$0.298578\pi$
$74$	0	0
$75$	1.05719e16i	0.0162581i
$76$	0	0
$77$	8.07603e15i	0.00967244i
$78$	0	0
$79$	1.48803e18	1.39687	0.698433	−	0.715676i	$-0.253881\pi$
$79$	1.48803e18	1.39687	0.698433	+	0.715676i	$0.253881\pi$
$80$	0	0
$81$	1.34809e18	0.997958
$82$	0	0
$83$	1.21993e18i	0.716296i	0.933665	+	0.358148i	$0.116592\pi$
$83$	1.21993e18i	0.716296i	−0.933665	+	0.358148i	$0.883408\pi$
$84$	0	0
$85$	2.43159e18i	1.13870i
$86$	0	0
$87$	−8.16356e16	−0.0306511
$88$	0	0
$89$	3.29788e18	0.997768	0.498884	−	0.866669i	$-0.333743\pi$
$89$	3.29788e18	0.997768	0.498884	+	0.866669i	$0.333743\pi$
$90$	0	0
$91$	1.53762e17i	0.0376665i
$92$	0	0
$93$	− 6.19700e16i	− 0.0123479i
$94$	0	0
$95$	1.42110e19	2.31340
$96$	0	0
$97$	4.52106e18	0.603822	0.301911	−	0.953336i	$-0.402375\pi$
$97$	4.52106e18	0.603822	0.301911	+	0.953336i	$0.402375\pi$
$98$	0	0
$99$	2.07833e18i	0.228655i
$100$	0	0
$101$	− 1.01147e19i	− 0.920238i	−0.887857	−	0.460119i	$-0.847807\pi$
$101$	− 1.01147e19i	− 0.920238i	0.887857	−	0.460119i	$-0.152193\pi$
$102$	0	0
$103$	1.82500e19	1.37819	0.689096	−	0.724670i	$-0.258008\pi$
$103$	1.82500e19	1.37819	0.689096	+	0.724670i	$0.258008\pi$
$104$	0	0
$105$	−2.23379e16	−0.00140522
$106$	0	0
$107$	2.72881e19i	1.43492i	0.696600	+	0.717459i	$0.254695\pi$
$107$	2.72881e19i	1.43492i	−0.696600	+	0.717459i	$0.745305\pi$
$108$	0	0
$109$	1.32939e19i	0.586272i	0.956071	+	0.293136i	$0.0946989\pi$
$109$	1.32939e19i	0.586272i	−0.956071	+	0.293136i	$0.905301\pi$
$110$	0	0
$111$	6.41819e17	0.0238146
$112$	0	0
$113$	1.83425e19	0.574399	0.287199	−	0.957871i	$-0.407276\pi$
$113$	1.83425e19	0.574399	0.287199	+	0.957871i	$0.407276\pi$
$114$	0	0
$115$	− 3.11016e19i	− 0.824430i
$116$	0	0
$117$	3.95700e19i	0.890431i
$118$	0	0
$119$	−1.97239e18	−0.0377831
$120$	0	0
$121$	5.79571e19	0.947646
$122$	0	0
$123$	1.54875e18i	0.0216712i
$124$	0	0
$125$	− 3.99983e19i	− 0.480171i
$126$	0	0
$127$	−1.55505e20	−1.60550	−0.802748	−	0.596319i	$-0.796630\pi$
$127$	−1.55505e20	−1.60550	−0.802748	+	0.596319i	$0.796630\pi$
$128$	0	0
$129$	3.17877e18	0.0282916
$130$	0	0
$131$	− 2.39001e20i	− 1.83790i	−0.394376	−	0.918949i	$-0.629039\pi$
$131$	− 2.39001e20i	− 1.83790i	0.394376	−	0.918949i	$-0.370961\pi$
$132$	0	0
$133$	1.15273e19i	0.0767604i
$134$	0	0
$135$	−1.15010e19	−0.0664610
$136$	0	0
$137$	−2.33463e20	−1.17320	−0.586602	−	0.809875i	$-0.699535\pi$
$137$	−2.33463e20	−1.17320	−0.586602	+	0.809875i	$0.699535\pi$
$138$	0	0
$139$	4.01632e20i	1.75868i	0.476190	+	0.879342i	$0.342017\pi$
$139$	4.01632e20i	1.75868i	−0.476190	+	0.879342i	$0.657983\pi$
$140$	0	0
$141$	− 1.27721e19i	− 0.0488295i
$142$	0	0
$143$	−6.09627e19	−0.203879
$144$	0	0
$145$	−5.10627e20	−1.49661
$146$	0	0
$147$	1.01216e19i	0.0260456i
$148$	0	0
$149$	− 5.08909e20i	− 1.15178i	−0.817526	−	0.575892i	$-0.804655\pi$
$149$	− 5.08909e20i	− 1.15178i	0.817526	−	0.575892i	$-0.195345\pi$
$150$	0	0
$151$	−3.58863e20	−0.715563	−0.357782	−	0.933805i	$-0.616467\pi$
$151$	−3.58863e20	−0.715563	−0.357782	+	0.933805i	$0.616467\pi$
$152$	0	0
$153$	−5.07586e20	−0.893185
$154$	0	0
$155$	− 3.87620e20i	− 0.602913i
$156$	0	0
$157$	1.45807e20i	0.200786i	0.994948	+	0.100393i	$0.0320099\pi$
$157$	1.45807e20i	0.200786i	−0.994948	+	0.100393i	$0.967990\pi$
$158$	0	0
$159$	−1.67326e19	−0.0204310
$160$	0	0
$161$	2.52282e19	0.0273553
$162$	0	0
$163$	− 1.25859e21i	− 1.21367i	−0.794828	−	0.606835i	$-0.792439\pi$
$163$	− 1.25859e21i	− 1.21367i	0.794828	−	0.606835i	$-0.207561\pi$
$164$	0	0
$165$	− 8.85642e18i	− 0.00760608i
$166$	0	0
$167$	−3.81697e20	−0.292356	−0.146178	−	0.989258i	$-0.546697\pi$
$167$	−3.81697e20	−0.292356	−0.146178	+	0.989258i	$0.546697\pi$
$168$	0	0
$169$	3.01232e20	0.206053
$170$	0	0
$171$	2.96650e21i	1.81460i
$172$	0	0
$173$	− 7.20483e20i	− 0.394626i	−0.980341	−	0.197313i	$-0.936778\pi$
$173$	− 7.20483e20i	− 0.394626i	0.980341	−	0.197313i	$-0.0632216\pi$
$174$	0	0
$175$	−5.36390e19	−0.0263402
$176$	0	0
$177$	4.24268e19	0.0187023
$178$	0	0
$179$	− 1.54740e21i	− 0.613055i	−0.951862	−	0.306528i	$-0.900833\pi$
$179$	− 1.54740e21i	− 0.613055i	0.951862	−	0.306528i	$-0.0991672\pi$
$180$	0	0
$181$	3.95197e21i	1.40886i	0.709774	+	0.704429i	$0.248797\pi$
$181$	3.95197e21i	1.40886i	−0.709774	+	0.704429i	$0.751203\pi$
$182$	0	0
$183$	1.03422e20	0.0332146
$184$	0	0
$185$	4.01455e21	1.16280
$186$	0	0
$187$	− 7.82003e20i	− 0.204509i
$188$	0	0
$189$	− 9.32912e18i	− 0.00220523i
$190$	0	0
$191$	4.73400e21	1.01254	0.506269	−	0.862375i	$-0.331024\pi$
$191$	4.73400e21	1.01254	0.506269	+	0.862375i	$0.331024\pi$
$192$	0	0
$193$	7.86311e21	1.52335	0.761676	−	0.647959i	$-0.224377\pi$
$193$	7.86311e21	1.52335	0.761676	+	0.647959i	$0.224377\pi$
$194$	0	0
$195$	− 1.68620e20i	− 0.0296197i
$196$	0	0
$197$	− 5.32685e21i	− 0.849261i	−0.905367	−	0.424630i	$-0.860404\pi$
$197$	− 5.32685e21i	− 0.849261i	0.905367	−	0.424630i	$-0.139596\pi$
$198$	0	0
$199$	1.72615e21	0.250019	0.125010	−	0.992156i	$-0.460104\pi$
$199$	1.72615e21	0.250019	0.125010	+	0.992156i	$0.460104\pi$
$200$	0	0
$201$	−1.56851e20	−0.0206597
$202$	0	0
$203$	− 4.14197e20i	− 0.0496586i
$204$	0	0
$205$	9.68738e21i	1.05814i
$206$	0	0
$207$	6.49237e21	0.646674
$208$	0	0
$209$	−4.57029e21	−0.415483
$210$	0	0
$211$	1.43551e22i	1.19213i	0.802936	+	0.596065i	$0.203270\pi$
$211$	1.43551e22i	1.19213i	−0.802936	+	0.596065i	$0.796730\pi$
$212$	0	0
$213$	6.06664e19i	0.00460619i
$214$	0	0
$215$	1.98831e22	1.38140
$216$	0	0
$217$	3.14419e20	0.0200052
$218$	0	0
$219$	− 5.29221e20i	− 0.0308616i
$220$	0	0
$221$	− 1.48888e22i	− 0.796403i
$222$	0	0
$223$	−1.36844e22	−0.671938	−0.335969	−	0.941873i	$-0.609064\pi$
$223$	−1.36844e22	−0.671938	−0.335969	+	0.941873i	$0.609064\pi$
$224$	0	0
$225$	−1.38038e22	−0.622678
$226$	0	0
$227$	2.28444e22i	0.947400i	0.880686	+	0.473700i	$0.157082\pi$
$227$	2.28444e22i	0.947400i	−0.880686	+	0.473700i	$0.842918\pi$
$228$	0	0
$229$	− 1.15938e22i	− 0.442371i	−0.975232	−	0.221186i	$-0.929007\pi$
$229$	− 1.15938e22i	− 0.442371i	0.975232	−	0.221186i	$-0.0709927\pi$
$230$	0	0
$231$	7.18392e18	0.000252376 0
$232$	0	0
$233$	−1.44196e22	−0.466736	−0.233368	−	0.972388i	$-0.574975\pi$
$233$	−1.44196e22	−0.466736	−0.233368	+	0.972388i	$0.574975\pi$
$234$	0	0
$235$	− 7.98892e22i	− 2.38421i
$236$	0	0
$237$	− 1.32366e21i	− 0.0364474i
$238$	0	0
$239$	−5.79540e22	−1.47334	−0.736671	−	0.676252i	$-0.763603\pi$
$239$	−5.79540e22	−1.47334	−0.736671	+	0.676252i	$0.763603\pi$
$240$	0	0
$241$	2.53834e22	0.596195	0.298097	−	0.954535i	$-0.403648\pi$
$241$	2.53834e22	0.596195	0.298097	+	0.954535i	$0.403648\pi$
$242$	0	0
$243$	− 3.60162e21i	− 0.0782057i
$244$	0	0
$245$	6.33102e22i	1.27173i
$246$	0	0
$247$	−8.70151e22	−1.61798
$248$	0	0
$249$	1.08517e21	0.0186898
$250$	0	0
$251$	2.45272e22i	0.391514i	0.980652	+	0.195757i	$0.0627164\pi$
$251$	2.45272e22i	0.391514i	−0.980652	+	0.195757i	$0.937284\pi$
$252$	0	0
$253$	1.00023e22i	0.148067i
$254$	0	0
$255$	2.16298e21	0.0297113
$256$	0	0
$257$	−7.84682e22	−1.00076	−0.500379	−	0.865807i	$-0.666806\pi$
$257$	−7.84682e22	−1.00076	−0.500379	+	0.865807i	$0.666806\pi$
$258$	0	0
$259$	3.25642e21i	0.0385827i
$260$	0	0
$261$	− 1.06592e23i	− 1.17392i
$262$	0	0
$263$	−6.14964e22	−0.629898	−0.314949	−	0.949109i	$-0.601987\pi$
$263$	−6.14964e22	−0.629898	−0.314949	+	0.949109i	$0.601987\pi$
$264$	0	0
$265$	−1.04661e23	−0.997588
$266$	0	0
$267$	− 2.93358e21i	− 0.0260340i
$268$	0	0
$269$	− 3.37452e21i	− 0.0278975i	−0.999903	−	0.0139487i	$-0.995560\pi$
$269$	− 3.37452e21i	− 0.0278975i	0.999903	−	0.0139487i	$-0.00444016\pi$
$270$	0	0
$271$	1.70540e23	1.31407	0.657035	−	0.753860i	$-0.271811\pi$
$271$	1.70540e23	1.31407	0.657035	+	0.753860i	$0.271811\pi$
$272$	0	0
$273$	1.36777e20	0.000982805 0
$274$	0	0
$275$	− 2.12665e22i	− 0.142572i
$276$	0	0
$277$	− 9.88428e22i	− 0.618568i	−0.950970	−	0.309284i	$-0.899911\pi$
$277$	− 9.88428e22i	− 0.618568i	0.950970	−	0.309284i	$-0.100089\pi$
$278$	0	0
$279$	8.09145e22	0.472918
$280$	0	0
$281$	1.54967e23	0.846309	0.423154	−	0.906058i	$-0.360923\pi$
$281$	1.54967e23	0.846309	0.423154	+	0.906058i	$0.360923\pi$
$282$	0	0
$283$	9.10357e22i	0.464773i	0.972624	+	0.232386i	$0.0746534\pi$
$283$	9.10357e22i	0.464773i	−0.972624	+	0.232386i	$0.925347\pi$
$284$	0	0
$285$	− 1.26412e22i	− 0.0603618i
$286$	0	0
$287$	−7.85796e21	−0.0351101
$288$	0	0
$289$	−4.80854e22	−0.201133
$290$	0	0
$291$	− 4.02165e21i	− 0.0157551i
$292$	0	0
$293$	3.76601e23i	1.38242i	0.722655	+	0.691209i	$0.242922\pi$
$293$	3.76601e23i	1.38242i	−0.722655	+	0.691209i	$0.757078\pi$
$294$	0	0
$295$	2.65378e23	0.913179
$296$	0	0
$297$	3.69876e21	0.0119363
$298$	0	0
$299$	1.90438e23i	0.576603i
$300$	0	0
$301$	1.61282e22i	0.0458358i
$302$	0	0
$303$	−8.99739e21	−0.0240111
$304$	0	0
$305$	6.46903e23	1.62178
$306$	0	0
$307$	4.06724e22i	0.0958265i	0.998852	+	0.0479132i	$0.0152571\pi$
$307$	4.06724e22i	0.0958265i	−0.998852	+	0.0479132i	$0.984743\pi$
$308$	0	0
$309$	− 1.62340e22i	− 0.0359601i
$310$	0	0
$311$	2.11001e23	0.439604	0.219802	−	0.975544i	$-0.429459\pi$
$311$	2.11001e23	0.439604	0.219802	+	0.975544i	$0.429459\pi$
$312$	0	0
$313$	4.32064e23	0.846988	0.423494	−	0.905899i	$-0.360803\pi$
$313$	4.32064e23	0.846988	0.423494	+	0.905899i	$0.360803\pi$
$314$	0	0
$315$	− 2.91667e22i	− 0.0538192i
$316$	0	0
$317$	− 2.37604e22i	− 0.0412849i	−0.999787	−	0.0206424i	$-0.993429\pi$
$317$	− 2.37604e22i	− 0.0412849i	0.999787	−	0.0206424i	$-0.00657116\pi$
$318$	0	0
$319$	1.64219e23	0.268789
$320$	0	0
$321$	2.42737e22	0.0374403
$322$	0	0
$323$	− 1.11619e24i	− 1.62299i
$324$	0	0
$325$	− 4.04900e23i	− 0.555208i
$326$	0	0
$327$	1.18254e22	0.0152972
$328$	0	0
$329$	6.48025e22	0.0791098
$330$	0	0
$331$	1.23591e24i	1.42437i	0.701994	+	0.712183i	$0.252293\pi$
$331$	1.23591e24i	1.42437i	−0.701994	+	0.712183i	$0.747707\pi$
$332$	0	0
$333$	8.38025e23i	0.912088i
$334$	0	0
$335$	−9.81094e23	−1.00875
$336$	0	0
$337$	−6.39486e23	−0.621366	−0.310683	−	0.950514i	$-0.600558\pi$
$337$	−6.39486e23	−0.621366	−0.310683	+	0.950514i	$0.600558\pi$
$338$	0	0
$339$	− 1.63163e22i	− 0.0149874i
$340$	0	0
$341$	1.24659e23i	0.108282i
$342$	0	0
$343$	−1.02801e23	−0.0844698
$344$	0	0
$345$	−2.76660e22	−0.0215112
$346$	0	0
$347$	− 6.84162e23i	− 0.503534i	−0.967788	−	0.251767i	$-0.918988\pi$
$347$	− 6.84162e23i	− 0.503534i	0.967788	−	0.251767i	$-0.0810117\pi$
$348$	0	0
$349$	− 3.64379e23i	− 0.253928i	−0.991907	−	0.126964i	$-0.959477\pi$
$349$	− 3.64379e23i	− 0.253928i	0.991907	−	0.126964i	$-0.0405233\pi$
$350$	0	0
$351$	7.04218e22	0.0464825
$352$	0	0
$353$	−5.48650e23	−0.343112	−0.171556	−	0.985174i	$-0.554879\pi$
$353$	−5.48650e23	−0.343112	−0.171556	+	0.985174i	$0.554879\pi$
$354$	0	0
$355$	3.79466e23i	0.224907i
$356$	0	0
$357$	1.75451e21i	0 0.000985845i
$358$	0	0
$359$	−1.28269e24	−0.683479	−0.341739	−	0.939795i	$-0.611016\pi$
$359$	−1.28269e24	−0.683479	−0.341739	+	0.939795i	$0.611016\pi$
$360$	0	0
$361$	−4.54497e24	−2.29727
$362$	0	0
$363$	− 5.15550e22i	− 0.0247262i
$364$	0	0
$365$	− 3.31026e24i	− 1.50688i
$366$	0	0
$367$	1.23278e24	0.532793	0.266396	−	0.963864i	$-0.414167\pi$
$367$	1.23278e24	0.532793	0.266396	+	0.963864i	$0.414167\pi$
$368$	0	0
$369$	−2.02221e24	−0.829996
$370$	0	0
$371$	− 8.48965e22i	− 0.0331008i
$372$	0	0
$373$	− 1.86125e24i	− 0.689559i	−0.938684	−	0.344780i	$-0.887954\pi$
$373$	− 1.86125e24i	− 0.689559i	0.938684	−	0.344780i	$-0.112046\pi$
$374$	0	0
$375$	−3.55799e22	−0.0125288
$376$	0	0
$377$	3.12661e24	1.04672
$378$	0	0
$379$	− 1.18195e24i	− 0.376292i	−0.982141	−	0.188146i	$-0.939752\pi$
$379$	− 1.18195e24i	− 0.376292i	0.982141	−	0.188146i	$-0.0602479\pi$
$380$	0	0
$381$	1.38327e23i	0.0418910i
$382$	0	0
$383$	−3.42588e24	−0.987150	−0.493575	−	0.869703i	$-0.664310\pi$
$383$	−3.42588e24	−0.987150	−0.493575	+	0.869703i	$0.664310\pi$
$384$	0	0
$385$	4.49351e22	0.0123228
$386$	0	0
$387$	4.15053e24i	1.08355i
$388$	0	0
$389$	6.91661e24i	1.71938i	0.510815	+	0.859691i	$0.329344\pi$
$389$	6.91661e24i	1.71938i	−0.510815	+	0.859691i	$0.670656\pi$
$390$	0	0
$391$	−2.44285e24	−0.578387
$392$	0	0
$393$	−2.12600e23	−0.0479549
$394$	0	0
$395$	− 8.27942e24i	− 1.77962i
$396$	0	0
$397$	6.65245e24i	1.36293i	0.731853	+	0.681463i	$0.238656\pi$
$397$	6.65245e24i	1.36293i	−0.731853	+	0.681463i	$0.761344\pi$
$398$	0	0
$399$	1.02540e22	0.00200285
$400$	0	0
$401$	6.41010e24	1.19397	0.596984	−	0.802253i	$-0.296365\pi$
$401$	6.41010e24	1.19397	0.596984	+	0.802253i	$0.296365\pi$
$402$	0	0
$403$	2.37343e24i	0.421675i
$404$	0	0
$405$	− 7.50081e24i	− 1.27141i
$406$	0	0
$407$	−1.29109e24	−0.208838
$408$	0	0
$409$	5.52154e24	0.852490	0.426245	−	0.904608i	$-0.359836\pi$
$409$	5.52154e24	0.852490	0.426245	+	0.904608i	$0.359836\pi$
$410$	0	0
$411$	2.07674e23i	0.0306115i
$412$	0	0
$413$	2.15263e23i	0.0303000i
$414$	0	0
$415$	6.78770e24	0.912569
$416$	0	0
$417$	3.57266e23	0.0458880
$418$	0	0
$419$	3.31446e24i	0.406799i	0.979096	+	0.203400i	$0.0651990\pi$
$419$	3.31446e24i	0.406799i	−0.979096	+	0.203400i	$0.934801\pi$
$420$	0	0
$421$	− 3.93632e24i	− 0.461754i	−0.972983	−	0.230877i	$-0.925841\pi$
$421$	− 3.93632e24i	− 0.461754i	0.972983	−	0.230877i	$-0.0741595\pi$
$422$	0	0
$423$	1.66766e25	1.87014
$424$	0	0
$425$	5.19388e24	0.556925
$426$	0	0
$427$	5.24738e23i	0.0538118i
$428$	0	0
$429$	5.42285e22i	0.00531966i
$430$	0	0
$431$	−1.03414e24	−0.0970610	−0.0485305	−	0.998822i	$-0.515454\pi$
$431$	−1.03414e24	−0.0970610	−0.0485305	+	0.998822i	$0.515454\pi$
$432$	0	0
$433$	−2.24817e24	−0.201927	−0.100963	−	0.994890i	$-0.532193\pi$
$433$	−2.24817e24	−0.201927	−0.100963	+	0.994890i	$0.532193\pi$
$434$	0	0
$435$	4.54221e23i	0.0390499i
$436$	0	0
$437$	1.42768e25i	1.17506i
$438$	0	0
$439$	1.13798e25	0.896856	0.448428	−	0.893819i	$-0.351984\pi$
$439$	1.13798e25	0.896856	0.448428	+	0.893819i	$0.351984\pi$
$440$	0	0
$441$	−1.32158e25	−0.997533
$442$	0	0
$443$	2.12719e25i	1.53805i	0.639216	+	0.769027i	$0.279259\pi$
$443$	2.12719e25i	1.53805i	−0.639216	+	0.769027i	$0.720741\pi$
$444$	0	0
$445$	− 1.83494e25i	− 1.27117i
$446$	0	0
$447$	−4.52693e23	−0.0300526
$448$	0	0
$449$	5.43638e24	0.345915	0.172958	−	0.984929i	$-0.444668\pi$
$449$	5.43638e24	0.345915	0.172958	+	0.984929i	$0.444668\pi$
$450$	0	0
$451$	− 3.11548e24i	− 0.190041i
$452$	0	0
$453$	3.19222e23i	0.0186707i
$454$	0	0
$455$	8.55533e23	0.0479876
$456$	0	0
$457$	1.03816e25	0.558549	0.279274	−	0.960211i	$-0.409906\pi$
$457$	1.03816e25	0.558549	0.279274	+	0.960211i	$0.409906\pi$
$458$	0	0
$459$	9.03340e23i	0.0466263i
$460$	0	0
$461$	2.01328e25i	0.997117i	0.866856	+	0.498559i	$0.166137\pi$
$461$	2.01328e25i	0.997117i	−0.866856	+	0.498559i	$0.833863\pi$
$462$	0	0
$463$	−1.68246e25	−0.799699	−0.399849	−	0.916581i	$-0.630938\pi$
$463$	−1.68246e25	−0.799699	−0.399849	+	0.916581i	$0.630938\pi$
$464$	0	0
$465$	−3.44802e23	−0.0157314
$466$	0	0
$467$	− 2.96900e25i	− 1.30047i	−0.759734	−	0.650234i	$-0.774671\pi$
$467$	− 2.96900e25i	− 1.30047i	0.759734	−	0.650234i	$-0.225329\pi$
$468$	0	0
$469$	− 7.95818e23i	− 0.0334712i
$470$	0	0
$471$	1.29701e23	0.00523895
$472$	0	0
$473$	−6.39444e24	−0.248097
$474$	0	0
$475$	− 3.03547e25i	− 1.13145i
$476$	0	0
$477$	− 2.18478e25i	− 0.782496i
$478$	0	0
$479$	−3.48219e25	−1.19857	−0.599287	−	0.800534i	$-0.704549\pi$
$479$	−3.48219e25	−1.19857	−0.599287	+	0.800534i	$0.704549\pi$
$480$	0	0
$481$	−2.45814e25	−0.813258
$482$	0	0
$483$	− 2.24414e22i	0 0.000713760i
$484$	0	0
$485$	− 2.51552e25i	− 0.769276i
$486$	0	0
$487$	−3.81856e25	−1.12299	−0.561494	−	0.827481i	$-0.689773\pi$
$487$	−3.81856e25	−1.12299	−0.561494	+	0.827481i	$0.689773\pi$
$488$	0	0
$489$	−1.11956e24	−0.0316674
$490$	0	0
$491$	4.66960e25i	1.27059i	0.772270	+	0.635294i	$0.219121\pi$
$491$	4.66960e25i	1.27059i	−0.772270	+	0.635294i	$0.780879\pi$
$492$	0	0
$493$	4.01068e25i	1.04996i
$494$	0	0
$495$	1.15639e25	0.291309
$496$	0	0
$497$	−3.07805e23	−0.00746261
$498$	0	0
$499$	1.20220e25i	0.280558i	0.990112	+	0.140279i	$0.0447999\pi$
$499$	1.20220e25i	0.280558i	−0.990112	+	0.140279i	$0.955200\pi$
$500$	0	0
$501$	3.39533e23i	0.00762823i
$502$	0	0
$503$	−3.04737e25	−0.659218	−0.329609	−	0.944118i	$-0.606917\pi$
$503$	−3.04737e25	−0.659218	−0.329609	+	0.944118i	$0.606917\pi$
$504$	0	0
$505$	−5.62783e25	−1.17239
$506$	0	0
$507$	− 2.67957e23i	− 0.00537637i
$508$	0	0
$509$	− 1.44729e25i	− 0.279728i	−0.990171	−	0.139864i	$-0.955333\pi$
$509$	− 1.44729e25i	− 0.279728i	0.990171	−	0.139864i	$-0.0446665\pi$
$510$	0	0
$511$	2.68513e24	0.0499996
$512$	0	0
$513$	5.27942e24	0.0947264
$514$	0	0
$515$	− 1.01543e26i	− 1.75583i
$516$	0	0
$517$	2.56925e25i	0.428200i
$518$	0	0
$519$	−6.40896e23	−0.0102967
$520$	0	0
$521$	4.00543e25	0.620426	0.310213	−	0.950667i	$-0.399600\pi$
$521$	4.00543e25	0.620426	0.310213	+	0.950667i	$0.399600\pi$
$522$	0	0
$523$	2.12968e25i	0.318088i	0.987271	+	0.159044i	$0.0508412\pi$
$523$	2.12968e25i	0.318088i	−0.987271	+	0.159044i	$0.949159\pi$
$524$	0	0
$525$	4.77138e22i	0 0.000687276i
$526$	0	0
$527$	−3.04453e25	−0.422979
$528$	0	0
$529$	−4.33697e25	−0.581243
$530$	0	0
$531$	5.53969e25i	0.716287i
$532$	0	0
$533$	− 5.93166e25i	− 0.740061i
$534$	0	0
$535$	1.51831e26	1.82810
$536$	0	0
$537$	−1.37647e24	−0.0159960
$538$	0	0
$539$	− 2.03607e25i	− 0.228402i
$540$	0	0
$541$	− 1.51095e26i	− 1.63635i	−0.574971	−	0.818174i	$-0.694987\pi$
$541$	− 1.51095e26i	− 1.63635i	0.574971	−	0.818174i	$-0.305013\pi$
$542$	0	0
$543$	3.51542e24	0.0367603
$544$	0	0
$545$	7.39672e25	0.746917
$546$	0	0
$547$	1.21086e26i	1.18090i	0.807075	+	0.590449i	$0.201049\pi$
$547$	1.21086e26i	1.18090i	−0.807075	+	0.590449i	$0.798951\pi$
$548$	0	0
$549$	1.35039e26i	1.27210i
$550$	0	0
$551$	2.34397e26	2.13311
$552$	0	0
$553$	6.71589e24	0.0590493
$554$	0	0
$555$	− 3.57109e24i	− 0.0303401i
$556$	0	0
$557$	2.13063e26i	1.74938i	0.484683	+	0.874690i	$0.338935\pi$
$557$	2.13063e26i	1.74938i	−0.484683	+	0.874690i	$0.661065\pi$
$558$	0	0
$559$	−1.21746e26	−0.966142
$560$	0	0
$561$	−6.95620e23	−0.00533611
$562$	0	0
$563$	2.61624e25i	0.194020i	0.995283	+	0.0970102i	$0.0309279\pi$
$563$	2.61624e25i	0.194020i	−0.995283	+	0.0970102i	$0.969072\pi$
$564$	0	0
$565$	− 1.02058e26i	− 0.731790i
$566$	0	0
$567$	6.08431e24	0.0421864
$568$	0	0
$569$	1.22661e26	0.822508	0.411254	−	0.911521i	$-0.365091\pi$
$569$	1.22661e26	0.822508	0.411254	+	0.911521i	$0.365091\pi$
$570$	0	0
$571$	2.63692e25i	0.171023i	0.996337	+	0.0855113i	$0.0272524\pi$
$571$	2.63692e25i	0.171023i	−0.996337	+	0.0855113i	$0.972748\pi$
$572$	0	0
$573$	− 4.21107e24i	− 0.0264194i
$574$	0	0
$575$	−6.64332e25	−0.403218
$576$	0	0
$577$	1.48000e26	0.869141	0.434571	−	0.900638i	$-0.356900\pi$
$577$	1.48000e26	0.869141	0.434571	+	0.900638i	$0.356900\pi$
$578$	0	0
$579$	− 6.99452e24i	− 0.0397477i
$580$	0	0
$581$	5.50587e24i	0.0302798i
$582$	0	0
$583$	3.36593e25	0.179165
$584$	0	0
$585$	2.20168e26	1.13442
$586$	0	0
$587$	− 3.16221e26i	− 1.57735i	−0.614809	−	0.788676i	$-0.710767\pi$
$587$	− 3.16221e26i	− 1.57735i	0.614809	−	0.788676i	$-0.289233\pi$
$588$	0	0
$589$	1.77932e26i	0.859329i
$590$	0	0
$591$	−4.73842e24	−0.0221591
$592$	0	0
$593$	4.11232e25	0.186238	0.0931188	−	0.995655i	$-0.470316\pi$
$593$	4.11232e25	0.186238	0.0931188	+	0.995655i	$0.470316\pi$
$594$	0	0
$595$	1.09744e25i	0.0481360i
$596$	0	0
$597$	− 1.53547e24i	− 0.00652357i
$598$	0	0
$599$	1.03552e26	0.426191	0.213096	−	0.977031i	$-0.431645\pi$
$599$	1.03552e26	0.426191	0.213096	+	0.977031i	$0.431645\pi$
$600$	0	0
$601$	−1.98522e26	−0.791592	−0.395796	−	0.918338i	$-0.629531\pi$
$601$	−1.98522e26	−0.791592	−0.395796	+	0.918338i	$0.629531\pi$
$602$	0	0
$603$	− 2.04800e26i	− 0.791254i
$604$	0	0
$605$	− 3.22474e26i	− 1.20731i
$606$	0	0
$607$	−4.20554e26	−1.52591	−0.762956	−	0.646450i	$-0.776253\pi$
$607$	−4.20554e26	−1.52591	−0.762956	+	0.646450i	$0.776253\pi$
$608$	0	0
$609$	−3.68443e23	−0.00129571
$610$	0	0
$611$	4.89168e26i	1.66750i
$612$	0	0
$613$	2.47119e26i	0.816641i	0.912839	+	0.408320i	$0.133885\pi$
$613$	2.47119e26i	0.816641i	−0.912839	+	0.408320i	$0.866115\pi$
$614$	0	0
$615$	8.61727e24	0.0276094
$616$	0	0
$617$	−4.47234e26	−1.38939	−0.694697	−	0.719303i	$-0.744462\pi$
$617$	−4.47234e26	−1.38939	−0.694697	+	0.719303i	$0.744462\pi$
$618$	0	0
$619$	− 9.70369e25i	− 0.292331i	−0.989260	−	0.146166i	$-0.953307\pi$
$619$	− 9.70369e25i	− 0.292331i	0.989260	−	0.146166i	$-0.0466933\pi$
$620$	0	0
$621$	− 1.15543e25i	− 0.0337578i
$622$	0	0
$623$	1.48842e25	0.0421783
$624$	0	0
$625$	−4.49234e26	−1.23485
$626$	0	0
$627$	4.06543e24i	0.0108409i
$628$	0	0
$629$	− 3.15320e26i	− 0.815774i
$630$	0	0
$631$	1.42435e26	0.357551	0.178776	−	0.983890i	$-0.442786\pi$
$631$	1.42435e26	0.357551	0.178776	+	0.983890i	$0.442786\pi$
$632$	0	0
$633$	1.27694e25	0.0311054
$634$	0	0
$635$	8.65232e26i	2.04542i
$636$	0	0
$637$	− 3.87654e26i	− 0.889445i
$638$	0	0
$639$	−7.92124e25	−0.176415
$640$	0	0
$641$	−7.22851e26	−1.56278	−0.781389	−	0.624044i	$-0.785488\pi$
$641$	−7.22851e26	−1.56278	−0.781389	+	0.624044i	$0.785488\pi$
$642$	0	0
$643$	− 9.33528e26i	− 1.95940i	−0.200470	−	0.979700i	$-0.564247\pi$
$643$	− 9.33528e26i	− 1.95940i	0.200470	−	0.979700i	$-0.435753\pi$
$644$	0	0
$645$	− 1.76867e25i	− 0.0360437i
$646$	0	0
$647$	5.11281e26	1.01174	0.505870	−	0.862610i	$-0.331172\pi$
$647$	5.11281e26	1.01174	0.505870	+	0.862610i	$0.331172\pi$
$648$	0	0
$649$	−8.53461e25	−0.164006
$650$	0	0
$651$	− 2.79687e23i	0 0.000521980i
$652$	0	0
$653$	3.98338e26i	0.722065i	0.932553	+	0.361033i	$0.117576\pi$
$653$	3.98338e26i	0.722065i	−0.932553	+	0.361033i	$0.882424\pi$
$654$	0	0
$655$	−1.32980e27	−2.34150
$656$	0	0
$657$	6.91006e26	1.18198
$658$	0	0
$659$	− 4.20018e26i	− 0.698001i	−0.937123	−	0.349000i	$-0.886521\pi$
$659$	− 4.20018e26i	− 0.698001i	0.937123	−	0.349000i	$-0.113479\pi$
$660$	0	0
$661$	− 4.64134e26i	− 0.749427i	−0.927141	−	0.374713i	$-0.877741\pi$
$661$	− 4.64134e26i	− 0.749427i	0.927141	−	0.374713i	$-0.122259\pi$
$662$	0	0
$663$	−1.32441e25	−0.0207800
$664$	0	0
$665$	6.41381e25	0.0977936
$666$	0	0
$667$	− 5.12993e26i	− 0.760179i
$668$	0	0
$669$	1.21728e25i	0.0175324i
$670$	0	0
$671$	−2.08045e26	−0.291269
$672$	0	0
$673$	2.53795e26	0.345414	0.172707	−	0.984973i	$-0.444749\pi$
$673$	2.53795e26	0.345414	0.172707	+	0.984973i	$0.444749\pi$
$674$	0	0
$675$	2.45663e25i	0.0325052i
$676$	0	0
$677$	6.54796e26i	0.842391i	0.906970	+	0.421195i	$0.138389\pi$
$677$	6.54796e26i	0.842391i	−0.906970	+	0.421195i	$0.861611\pi$
$678$	0	0
$679$	2.04048e25	0.0255252
$680$	0	0
$681$	2.03209e25	0.0247198
$682$	0	0
$683$	1.26437e27i	1.49582i	0.663802	+	0.747909i	$0.268942\pi$
$683$	1.26437e27i	1.49582i	−0.663802	+	0.747909i	$0.731058\pi$
$684$	0	0
$685$	1.29899e27i	1.49467i
$686$	0	0
$687$	−1.03131e25	−0.0115425
$688$	0	0
$689$	6.40850e26	0.697708
$690$	0	0
$691$	− 8.15641e25i	− 0.0863889i	−0.999067	−	0.0431944i	$-0.986247\pi$
$691$	− 8.15641e25i	− 0.0863889i	0.999067	−	0.0431944i	$-0.0137535\pi$
$692$	0	0
$693$	9.38007e24i	0.00966585i
$694$	0	0
$695$	2.23469e27	2.24058
$696$	0	0
$697$	7.60887e26	0.742350
$698$	0	0
$699$	1.28267e25i	0.0121782i
$700$	0	0
$701$	− 1.50490e27i	− 1.39055i	−0.718744	−	0.695275i	$-0.755283\pi$
$701$	− 1.50490e27i	− 1.39055i	0.718744	−	0.695275i	$-0.244717\pi$
$702$	0	0
$703$	−1.84283e27	−1.65733
$704$	0	0
$705$	−7.10643e25	−0.0622093
$706$	0	0
$707$	− 4.56504e25i	− 0.0389009i
$708$	0	0
$709$	− 1.48850e25i	− 0.0123484i	−0.999981	−	0.00617420i	$-0.998035\pi$
$709$	− 1.48850e25i	− 0.0123484i	0.999981	−	0.00617420i	$-0.00196532\pi$
$710$	0	0
$711$	1.72830e27	1.39591
$712$	0	0
$713$	3.89416e26	0.306241
$714$	0	0
$715$	3.39197e26i	0.259744i
$716$	0	0
$717$	5.15522e25i	0.0384428i
$718$	0	0
$719$	−5.98917e25	−0.0434953	−0.0217476	−	0.999763i	$-0.506923\pi$
$719$	−5.98917e25	−0.0434953	−0.0217476	+	0.999763i	$0.506923\pi$
$720$	0	0
$721$	8.23672e25	0.0582599
$722$	0	0
$723$	− 2.25795e25i	− 0.0155561i
$724$	0	0
$725$	1.09070e27i	0.731972i
$726$	0	0
$727$	1.58883e27	1.03872	0.519362	−	0.854554i	$-0.326169\pi$
$727$	1.58883e27	1.03872	0.519362	+	0.854554i	$0.326169\pi$
$728$	0	0
$729$	1.56363e27	0.995917
$730$	0	0
$731$	− 1.56170e27i	− 0.969131i
$732$	0	0
$733$	− 7.88881e26i	− 0.477006i	−0.971142	−	0.238503i	$-0.923343\pi$
$733$	− 7.88881e26i	− 0.477006i	0.971142	−	0.238503i	$-0.0766566\pi$
$734$	0	0
$735$	5.63167e25	0.0331824
$736$	0	0
$737$	3.15522e26	0.181171
$738$	0	0
$739$	7.06002e26i	0.395079i	0.980295	+	0.197539i	$0.0632950\pi$
$739$	7.06002e26i	0.395079i	−0.980295	+	0.197539i	$0.936705\pi$
$740$	0	0
$741$	7.74030e25i	0.0422168i
$742$	0	0
$743$	3.24185e27	1.72345	0.861726	−	0.507373i	$-0.169384\pi$
$743$	3.24185e27	1.72345	0.861726	+	0.507373i	$0.169384\pi$
$744$	0	0
$745$	−2.83158e27	−1.46738
$746$	0	0
$747$	1.41691e27i	0.715808i
$748$	0	0
$749$	1.23158e26i	0.0606579i
$750$	0	0
$751$	−3.47111e27	−1.66682	−0.833411	−	0.552654i	$-0.813615\pi$
$751$	−3.47111e27	−1.66682	−0.833411	+	0.552654i	$0.813615\pi$
$752$	0	0
$753$	2.18178e25	0.0102155
$754$	0	0
$755$	1.99672e27i	0.911635i
$756$	0	0
$757$	− 8.85845e26i	− 0.394409i	−0.980362	−	0.197205i	$-0.936814\pi$
$757$	− 8.85845e26i	− 0.394409i	0.980362	−	0.197205i	$-0.0631863\pi$
$758$	0	0
$759$	8.89745e24	0.00386339
$760$	0	0
$761$	3.97430e26	0.168309	0.0841543	−	0.996453i	$-0.473181\pi$
$761$	3.97430e26	0.168309	0.0841543	+	0.996453i	$0.473181\pi$
$762$	0	0
$763$	5.99988e25i	0.0247833i
$764$	0	0
$765$	2.82422e27i	1.13793i
$766$	0	0
$767$	−1.62493e27	−0.638674
$768$	0	0
$769$	−2.90265e27	−1.11300	−0.556499	−	0.830848i	$-0.687856\pi$
$769$	−2.90265e27	−1.11300	−0.556499	+	0.830848i	$0.687856\pi$
$770$	0	0
$771$	6.98003e25i	0.0261120i
$772$	0	0
$773$	− 3.03505e27i	− 1.10780i	−0.832584	−	0.553899i	$-0.813139\pi$
$773$	− 3.03505e27i	− 1.10780i	0.832584	−	0.553899i	$-0.186861\pi$
$774$	0	0
$775$	−8.27957e26	−0.294877
$776$	0	0
$777$	2.89670e24	0.00100671
$778$	0	0
$779$	− 4.44688e27i	− 1.50817i
$780$	0	0
$781$	− 1.22037e26i	− 0.0403931i
$782$	0	0
$783$	−1.89699e26	−0.0612814
$784$	0	0
$785$	8.11273e26	0.255803
$786$	0	0
$787$	− 1.60790e27i	− 0.494879i	−0.968903	−	0.247440i	$-0.920411\pi$
$787$	− 1.60790e27i	− 0.494879i	0.968903	−	0.247440i	$-0.0795892\pi$
$788$	0	0
$789$	5.47033e25i	0.0164355i
$790$	0	0
$791$	8.27847e25	0.0242814
$792$	0	0
$793$	−3.96104e27	−1.13426
$794$	0	0
$795$	9.31001e25i	0.0260293i
$796$	0	0
$797$	− 2.46654e27i	− 0.673341i	−0.941623	−	0.336670i	$-0.890699\pi$
$797$	− 2.46654e27i	− 0.673341i	0.941623	−	0.336670i	$-0.109301\pi$
$798$	0	0
$799$	−6.27483e27	−1.67266
$800$	0	0
$801$	3.83039e27	0.997089
$802$	0	0
$803$	1.06459e27i	0.270634i
$804$	0	0
$805$	− 1.40370e26i	− 0.0348509i
$806$	0	0
$807$	−3.00175e24	−0.000727908 0
$808$	0	0
$809$	3.85670e27	0.913492	0.456746	−	0.889597i	$-0.349015\pi$
$809$	3.85670e27	0.913492	0.456746	+	0.889597i	$0.349015\pi$
$810$	0	0
$811$	− 3.23598e27i	− 0.748700i	−0.927288	−	0.374350i	$-0.877866\pi$
$811$	− 3.23598e27i	− 0.748700i	0.927288	−	0.374350i	$-0.122134\pi$
$812$	0	0
$813$	− 1.51702e26i	− 0.0342870i
$814$	0	0
$815$	−7.00279e27	−1.54623
$816$	0	0
$817$	−9.12709e27	−1.96890
$818$	0	0
$819$	1.78590e26i	0.0376409i
$820$	0	0
$821$	8.58529e27i	1.76805i	0.467439	+	0.884026i	$0.345177\pi$
$821$	8.58529e27i	1.76805i	−0.467439	+	0.884026i	$0.654823\pi$
$822$	0	0
$823$	−3.42172e25	−0.00688566	−0.00344283	−	0.999994i	$-0.501096\pi$
$823$	−3.42172e25	−0.00688566	−0.00344283	+	0.999994i	$0.501096\pi$
$824$	0	0
$825$	−1.89173e25	−0.00372004
$826$	0	0
$827$	7.32223e25i	0.0140715i	0.999975	+	0.00703576i	$0.00223957\pi$
$827$	7.32223e25i	0.0140715i	−0.999975	+	0.00703576i	$0.997760\pi$
$828$	0	0
$829$	3.74412e26i	0.0703205i	0.999382	+	0.0351602i	$0.0111942\pi$
$829$	3.74412e26i	0.0703205i	−0.999382	+	0.0351602i	$0.988806\pi$
$830$	0	0
$831$	−8.79242e25	−0.0161398
$832$	0	0
$833$	4.97265e27	0.892196
$834$	0	0
$835$	2.12377e27i	0.372465i
$836$	0	0
$837$	− 1.44002e26i	− 0.0246874i
$838$	0	0
$839$	8.83835e27	1.48126	0.740632	−	0.671911i	$-0.234526\pi$
$839$	8.83835e27	1.48126	0.740632	+	0.671911i	$0.234526\pi$
$840$	0	0
$841$	−2.31906e27	−0.379971
$842$	0	0
$843$	− 1.37849e26i	− 0.0220821i
$844$	0	0
$845$	− 1.67606e27i	− 0.262513i
$846$	0	0
$847$	2.61576e26	0.0400595
$848$	0	0
$849$	8.09795e25	0.0121270
$850$	0	0
$851$	4.03315e27i	0.590627i
$852$	0	0
$853$	− 1.18640e28i	− 1.69908i	−0.527525	−	0.849540i	$-0.676880\pi$
$853$	− 1.18640e28i	− 1.69908i	0.527525	−	0.849540i	$-0.323120\pi$
$854$	0	0
$855$	1.65057e28	2.31182
$856$	0	0
$857$	−2.48012e27	−0.339746	−0.169873	−	0.985466i	$-0.554336\pi$
$857$	−2.48012e27	−0.339746	−0.169873	+	0.985466i	$0.554336\pi$
$858$	0	0
$859$	8.04241e27i	1.07758i	0.842439	+	0.538792i	$0.181119\pi$
$859$	8.04241e27i	1.07758i	−0.842439	+	0.538792i	$0.818881\pi$
$860$	0	0
$861$	6.98993e24i	0 0.000916101i
$862$	0	0
$863$	−4.84578e27	−0.621242	−0.310621	−	0.950534i	$-0.600537\pi$
$863$	−4.84578e27	−0.621242	−0.310621	+	0.950534i	$0.600537\pi$
$864$	0	0
$865$	−4.00878e27	−0.502758
$866$	0	0
$867$	4.27737e25i	0.00524802i
$868$	0	0
$869$	2.66268e27i	0.319618i
$870$	0	0
$871$	6.00731e27	0.705517
$872$	0	0
$873$	5.25108e27	0.603411
$874$	0	0
$875$	− 1.80523e26i	− 0.0202981i
$876$	0	0
$877$	− 1.11292e28i	− 1.22452i	−0.790655	−	0.612261i	$-0.790260\pi$
$877$	− 1.11292e28i	− 1.22452i	0.790655	−	0.612261i	$-0.209740\pi$
$878$	0	0
$879$	3.35000e26	0.0360704
$880$	0	0
$881$	1.49766e28	1.57813	0.789065	−	0.614310i	$-0.210565\pi$
$881$	1.49766e28	1.57813	0.789065	+	0.614310i	$0.210565\pi$
$882$	0	0
$883$	− 1.18632e28i	− 1.22342i	−0.791082	−	0.611711i	$-0.790482\pi$
$883$	− 1.18632e28i	− 1.22342i	0.791082	−	0.611711i	$-0.209518\pi$
$884$	0	0
$885$	− 2.36063e26i	− 0.0238269i
$886$	0	0
$887$	1.21862e28	1.20391	0.601953	−	0.798532i	$-0.294389\pi$
$887$	1.21862e28	1.20391	0.601953	+	0.798532i	$0.294389\pi$
$888$	0	0
$889$	−7.01836e26	−0.0678686
$890$	0	0
$891$	2.41227e27i	0.228343i
$892$	0	0
$893$	3.66722e28i	3.39819i
$894$	0	0
$895$	−8.60976e27	−0.781039
$896$	0	0
$897$	1.69401e26	0.0150449
$898$	0	0
$899$	− 6.39343e27i	− 0.555926i
$900$	0	0
$901$	8.22055e27i	0.699867i
$902$	0	0
$903$	1.43466e25	0.00119596
$904$	0	0
$905$	2.19888e28	1.79490
$906$	0	0
$907$	2.10303e28i	1.68103i	0.541789	+	0.840515i	$0.317747\pi$
$907$	2.10303e28i	1.68103i	−0.541789	+	0.840515i	$0.682253\pi$
$908$	0	0
$909$	− 1.17479e28i	− 0.919612i
$910$	0	0
$911$	−9.47120e27	−0.726074	−0.363037	−	0.931775i	$-0.618260\pi$
$911$	−9.47120e27	−0.726074	−0.363037	+	0.931775i	$0.618260\pi$
$912$	0	0
$913$	−2.18294e27	−0.163896
$914$	0	0
$915$	− 5.75443e26i	− 0.0423158i
$916$	0	0
$917$	− 1.07867e27i	− 0.0776929i
$918$	0	0
$919$	−1.97843e28	−1.39580	−0.697901	−	0.716194i	$-0.745883\pi$
$919$	−1.97843e28	−1.39580	−0.697901	+	0.716194i	$0.745883\pi$
$920$	0	0
$921$	3.61796e25	0.00250033
$922$	0	0
$923$	− 2.32350e27i	− 0.157299i
$924$	0	0
$925$	− 8.57509e27i	− 0.568712i
$926$	0	0
$927$	2.11969e28	1.37725
$928$	0	0
$929$	2.35154e27	0.149693	0.0748467	−	0.997195i	$-0.476153\pi$
$929$	2.35154e27	0.149693	0.0748467	+	0.997195i	$0.476153\pi$
$930$	0	0
$931$	− 2.90618e28i	− 1.81259i
$932$	0	0
$933$	− 1.87693e26i	− 0.0114703i
$934$	0	0
$935$	−4.35108e27	−0.260547
$936$	0	0
$937$	−2.37072e28	−1.39109	−0.695544	−	0.718483i	$-0.744837\pi$
$937$	−2.37072e28	−1.39109	−0.695544	+	0.718483i	$0.744837\pi$
$938$	0	0
$939$	− 3.84337e26i	− 0.0220998i
$940$	0	0
$941$	− 1.63014e28i	− 0.918592i	−0.888283	−	0.459296i	$-0.848102\pi$
$941$	− 1.63014e28i	− 0.918592i	0.888283	−	0.459296i	$-0.151898\pi$
$942$	0	0
$943$	−9.73226e27	−0.537468
$944$	0	0
$945$	−5.19073e25	−0.00280948
$946$	0	0
$947$	1.67186e28i	0.886899i	0.896299	+	0.443449i	$0.146245\pi$
$947$	1.67186e28i	0.886899i	−0.896299	+	0.443449i	$0.853755\pi$
$948$	0	0
$949$	2.02690e28i	1.05391i
$950$	0	0
$951$	−2.11357e25	−0.00107722
$952$	0	0
$953$	−4.74178e27	−0.236896	−0.118448	−	0.992960i	$-0.537792\pi$
$953$	−4.74178e27	−0.236896	−0.118448	+	0.992960i	$0.537792\pi$
$954$	0	0
$955$	− 2.63400e28i	− 1.28998i
$956$	0	0
$957$	− 1.46078e26i	− 0.00701330i
$958$	0	0
$959$	−1.05368e27	−0.0495945
$960$	0	0
$961$	−1.68174e28	−0.776043
$962$	0	0
$963$	3.16943e28i	1.43394i
$964$	0	0
$965$	− 4.37504e28i	− 1.94077i
$966$	0	0
$967$	−3.60277e27	−0.156706	−0.0783529	−	0.996926i	$-0.524966\pi$
$967$	−3.60277e27	−0.156706	−0.0783529	+	0.996926i	$0.524966\pi$
$968$	0	0
$969$	−9.92892e26	−0.0423474
$970$	0	0
$971$	4.48458e27i	0.187559i	0.995593	+	0.0937797i	$0.0298949\pi$
$971$	4.48458e27i	0.187559i	−0.995593	+	0.0937797i	$0.970105\pi$
$972$	0	0
$973$	1.81267e27i	0.0743443i
$974$	0	0
$975$	−3.60173e26	−0.0144866
$976$	0	0
$977$	−3.13619e28	−1.23710	−0.618550	−	0.785746i	$-0.712279\pi$
$977$	−3.13619e28	−1.23710	−0.618550	+	0.785746i	$0.712279\pi$
$978$	0	0
$979$	5.90121e27i	0.228300i
$980$	0	0
$981$	1.54404e28i	0.585873i
$982$	0	0
$983$	−5.43983e27	−0.202454	−0.101227	−	0.994863i	$-0.532277\pi$
$983$	−5.43983e27	−0.202454	−0.101227	+	0.994863i	$0.532277\pi$
$984$	0	0
$985$	−2.96386e28	−1.08197
$986$	0	0
$987$	− 5.76441e25i	− 0.00206415i
$988$	0	0
$989$	1.99752e28i	0.701659i
$990$	0	0
$991$	−4.64392e28	−1.60024	−0.800120	−	0.599841i	$-0.795231\pi$
$991$	−4.64392e28	−1.60024	−0.800120	+	0.599841i	$0.795231\pi$
$992$	0	0
$993$	1.09939e27	0.0371649
$994$	0	0
$995$	− 9.60430e27i	− 0.318527i
$996$	0	0
$997$	− 8.60787e27i	− 0.280086i	−0.990145	−	0.140043i	$-0.955276\pi$
$997$	− 8.60787e27i	− 0.280086i	0.990145	−	0.140043i	$-0.0447241\pi$
$998$	0	0
$999$	1.49141e27	0.0476131

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	32.20.b.a.17.9		18
4.3	odd	2		8.20.b.a.5.2	yes	18
8.3	odd	2		8.20.b.a.5.1	✓	18
8.5	even	2	inner	32.20.b.a.17.10		18
12.11	even	2		72.20.d.b.37.17		18
24.11	even	2		72.20.d.b.37.18		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.20.b.a.5.1	✓	18	8.3	odd	2
8.20.b.a.5.2	yes	18	4.3	odd	2
32.20.b.a.17.9		18	1.1	even	1	trivial
32.20.b.a.17.10		18	8.5	even	2	inner
72.20.d.b.37.17		18	12.11	even	2
72.20.d.b.37.18		18	24.11	even	2

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:	\( N \)	\(=\)	\( 32 = 2^{5} \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	32.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-889.536i q^{3} -5.56401e6i q^{5} +4.51327e6 q^{7} +1.16147e9 q^{9} +O(q^{10})\)	\(q-889.536i q^{3} -5.56401e6i q^{5} +4.51327e6 q^{7} +1.16147e9 q^{9} +1.78940e9i q^{11} +3.40689e10i q^{13} -4.94939e9 q^{15} -4.37021e11 q^{17} +2.55409e12i q^{19} -4.01471e9i q^{21} +5.58979e12 q^{23} -1.18847e13 q^{25} -2.06704e12i q^{27} -9.17732e13i q^{29} +6.96656e13 q^{31} +1.59173e12 q^{33} -2.51119e13i q^{35} +7.21521e14i q^{37} +3.03055e13 q^{39} -1.74108e15 q^{41} +3.57351e15i q^{43} -6.46243e15i q^{45} +1.43582e16 q^{47} -1.13785e16 q^{49} +3.88746e14i q^{51} -1.88104e16i q^{53} +9.95622e15 q^{55} +2.27196e15 q^{57} +4.76955e16i q^{59} +1.16266e17i q^{61} +5.24203e15 q^{63} +1.89560e17 q^{65} -1.76329e17i q^{67} -4.97232e15i q^{69} -6.82001e16 q^{71} +5.94941e17 q^{73} +1.05719e16i q^{75} +8.07603e15i q^{77} +1.48803e18 q^{79} +1.34809e18 q^{81} +1.21993e18i q^{83} +2.43159e18i q^{85} -8.16356e16 q^{87} +3.29788e18 q^{89} +1.53762e17i q^{91} -6.19700e16i q^{93} +1.42110e19 q^{95} +4.52106e18 q^{97} +2.07833e18i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 80707216 q^{7} - 6198727826 q^{9}+O(q^{10}) \) 18 * q + 80707216 * q^7 - 6198727826 * q^9	\( 18 q + 80707216 q^{7} - 6198727826 q^{9} - 156097960432 q^{15} + 14121426692 q^{17} - 2177121583952 q^{23} - 44414474211734 q^{25} - 428505770260416 q^{31} - 185380269683736 q^{33} - 942830575043152 q^{39} + 12\!\cdots\!24 q^{41}+ \cdots + 82\!\cdots\!20 q^{97}+O(q^{100}) \) 18 * q + 80707216 * q^7 - 6198727826 * q^9 - 156097960432 * q^15 + 14121426692 * q^17 - 2177121583952 * q^23 - 44414474211734 * q^25 - 428505770260416 * q^31 - 185380269683736 * q^33 - 942830575043152 * q^39 + 1200260021141524 * q^41 + 1134160959992928 * q^47 + 31570092936329058 * q^49 + 51757047863478736 * q^55 - 7740744140670520 * q^57 - 430168668665078416 * q^63 - 184007586011668640 * q^65 + 1599477989199799056 * q^71 - 93004298989809068 * q^73 - 1813975801025264480 * q^79 + 1801762045144484330 * q^81 + 13124960835754102512 * q^87 + 4470509771045743540 * q^89 - 23809730986995679856 * q^95 + 8246429789464117220 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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