LMFDB - Embedded newform 432.9.e.c.161.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,9,Mod(161,432)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("432.161"); S:= CuspForms(chi, 9); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(432, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	432.e (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,0,0,-5278] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$175.987559546$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 108)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$2.44949i$ of defining polynomial
Character	$\chi$	$=$	432.161
Dual form			432.9.e.c.161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+705.453i q^{5} -2639.00 q^{7} +24690.9i q^{11} +13919.0 q^{13} -81127.1i q^{17} +77041.0 q^{19} -447963. i q^{23} -107039. q^{25} -585526. i q^{29} -1.67424e6 q^{31} -1.86169e6i q^{35} +27359.0 q^{37} -1.30509e6i q^{41} -1.99920e6 q^{43} +7.24853e6i q^{47} +1.19952e6 q^{49} -2.40559e6i q^{53} -1.74182e7 q^{55} -2.23311e7i q^{59} +2.49283e7 q^{61} +9.81920e6i q^{65} -1.19983e7 q^{67} +2.18690e6i q^{71} +3.80304e7 q^{73} -6.51592e7i q^{77} +2.00184e6 q^{79} -3.44332e7i q^{83} +5.72314e7 q^{85} -462072. i q^{89} -3.67322e7 q^{91} +5.43488e7i q^{95} +5.24165e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 5278 q^{7} + 27838 q^{13} + 154082 q^{19} - 214078 q^{25} - 3348484 q^{31} + 54718 q^{37} - 3998404 q^{43} + 2399040 q^{49} - 34836480 q^{55} + 49856638 q^{61} - 23996638 q^{67} + 76060798 q^{73}+ \cdots + 104832958 q^{97}+O(q^{100}) $ 2 * q - 5278 * q^7 + 27838 * q^13 + 154082 * q^19 - 214078 * q^25 - 3348484 * q^31 + 54718 * q^37 - 3998404 * q^43 + 2399040 * q^49 - 34836480 * q^55 + 49856638 * q^61 - 23996638 * q^67 + 76060798 * q^73 + 4003682 * q^79 + 114462720 * q^85 - 73464482 * q^91 + 104832958 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	705.453i	1.12872i	0.825527	+	0.564362i	$0.190878\pi$
$5$	705.453i	1.12872i	−0.825527	+	0.564362i	$0.809122\pi$
$6$	0	0
$7$	−2639.00	−1.09913	−0.549563	−	0.835452i	$-0.685206\pi$
$7$	−2639.00	−1.09913	−0.549563	+	0.835452i	$0.685206\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	24690.9i	1.68642i	0.537585	+	0.843209i	$0.319337\pi$
$11$	24690.9i	1.68642i	−0.537585	+	0.843209i	$0.680663\pi$
$12$	0	0
$13$	13919.0	0.487343	0.243671	−	0.969858i	$-0.421648\pi$
$13$	13919.0	0.487343	0.243671	+	0.969858i	$0.421648\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 81127.1i	− 0.971338i	−0.874143	−	0.485669i	$-0.838576\pi$
$17$	− 81127.1i	− 0.971338i	0.874143	−	0.485669i	$-0.161424\pi$
$18$	0	0
$19$	77041.0	0.591163	0.295582	−	0.955317i	$-0.404487\pi$
$19$	77041.0	0.591163	0.295582	+	0.955317i	$0.404487\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 447963.i	− 1.60078i	−0.599482	−	0.800388i	$-0.704627\pi$
$23$	− 447963.i	− 1.60078i	0.599482	−	0.800388i	$-0.295373\pi$
$24$	0	0
$25$	−107039.	−0.274020
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 585526.i	− 0.827855i	−0.910310	−	0.413927i	$-0.864157\pi$
$29$	− 585526.i	− 0.827855i	0.910310	−	0.413927i	$-0.135843\pi$
$30$	0	0
$31$	−1.67424e6	−1.81289	−0.906445	−	0.422324i	$-0.861214\pi$
$31$	−1.67424e6	−1.81289	−0.906445	+	0.422324i	$0.861214\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.86169e6i	− 1.24061i
$36$	0	0
$37$	27359.0	0.0145980	0.00729900	−	0.999973i	$-0.497677\pi$
$37$	27359.0	0.0145980	0.00729900	+	0.999973i	$0.497677\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.30509e6i	− 0.461854i	−0.972971	−	0.230927i	$-0.925824\pi$
$41$	− 1.30509e6i	− 0.461854i	0.972971	−	0.230927i	$-0.0741758\pi$
$42$	0	0
$43$	−1.99920e6	−0.584767	−0.292383	−	0.956301i	$-0.594448\pi$
$43$	−1.99920e6	−0.584767	−0.292383	+	0.956301i	$0.594448\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.24853e6i	1.48545i	0.669596	+	0.742726i	$0.266467\pi$
$47$	7.24853e6i	1.48545i	−0.669596	+	0.742726i	$0.733533\pi$
$48$	0	0
$49$	1.19952e6	0.208077
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.40559e6i	− 0.304873i	−0.988313	−	0.152437i	$-0.951288\pi$
$53$	− 2.40559e6i	− 0.304873i	0.988313	−	0.152437i	$-0.0487120\pi$
$54$	0	0
$55$	−1.74182e7	−1.90350
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.23311e7i	− 1.84290i	−0.388494	−	0.921451i	$-0.627005\pi$
$59$	− 2.23311e7i	− 1.84290i	0.388494	−	0.921451i	$-0.372995\pi$
$60$	0	0
$61$	2.49283e7	1.80042	0.900210	−	0.435457i	$-0.143413\pi$
$61$	2.49283e7	1.80042	0.900210	+	0.435457i	$0.143413\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.81920e6i	0.550076i
$66$	0	0
$67$	−1.19983e7	−0.595417	−0.297708	−	0.954657i	$-0.596222\pi$
$67$	−1.19983e7	−0.595417	−0.297708	+	0.954657i	$0.596222\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.18690e6i	0.0860590i	0.999074	+	0.0430295i	$0.0137009\pi$
$71$	2.18690e6i	0.0860590i	−0.999074	+	0.0430295i	$0.986299\pi$
$72$	0	0
$73$	3.80304e7	1.33918	0.669591	−	0.742730i	$-0.266470\pi$
$73$	3.80304e7	1.33918	0.669591	+	0.742730i	$0.266470\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 6.51592e7i	− 1.85359i
$78$	0	0
$79$	2.00184e6	0.0513950	0.0256975	−	0.999670i	$-0.491819\pi$
$79$	2.00184e6	0.0513950	0.0256975	+	0.999670i	$0.491819\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 3.44332e7i	− 0.725545i	−0.931878	−	0.362773i	$-0.881830\pi$
$83$	− 3.44332e7i	− 0.725545i	0.931878	−	0.362773i	$-0.118170\pi$
$84$	0	0
$85$	5.72314e7	1.09637
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 462072.i	− 0.00736460i	−0.999993	−	0.00368230i	$-0.998828\pi$
$89$	− 462072.i	− 0.00736460i	0.999993	−	0.00368230i	$-0.00117212\pi$
$90$	0	0
$91$	−3.67322e7	−0.535651
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.43488e7i	0.667261i
$96$	0	0
$97$	5.24165e7	0.592081	0.296040	−	0.955175i	$-0.404334\pi$
$97$	5.24165e7	0.592081	0.296040	+	0.955175i	$0.404334\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	9.10599e7i	0.875068i	0.899202	+	0.437534i	$0.144148\pi$
$101$	9.10599e7i	0.875068i	−0.899202	+	0.437534i	$0.855852\pi$
$102$	0	0
$103$	−6.91980e7	−0.614815	−0.307408	−	0.951578i	$-0.599462\pi$
$103$	−6.91980e7	−0.614815	−0.307408	+	0.951578i	$0.599462\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.35366e7i	0.484718i	0.970187	+	0.242359i	$0.0779212\pi$
$107$	6.35366e7i	0.484718i	−0.970187	+	0.242359i	$0.922079\pi$
$108$	0	0
$109$	−1.23077e7	−0.0871907	−0.0435953	−	0.999049i	$-0.513881\pi$
$109$	−1.23077e7	−0.0871907	−0.0435953	+	0.999049i	$0.513881\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	2.52725e8i	1.55001i	0.631955	+	0.775005i	$0.282253\pi$
$113$	2.52725e8i	1.55001i	−0.631955	+	0.775005i	$0.717747\pi$
$114$	0	0
$115$	3.16017e8	1.80684
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.14094e8i	1.06762i
$120$	0	0
$121$	−3.95280e8	−1.84401
$122$	0	0
$123$	0	0
$124$	0	0
$125$	2.00057e8i	0.819432i
$126$	0	0
$127$	−1.96635e8	−0.755868	−0.377934	−	0.925833i	$-0.623365\pi$
$127$	−1.96635e8	−0.755868	−0.377934	+	0.925833i	$0.623365\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 5.84609e7i	− 0.198509i	−0.995062	−	0.0992545i	$-0.968354\pi$
$131$	− 5.84609e7i	− 0.198509i	0.995062	−	0.0992545i	$-0.0316458\pi$
$132$	0	0
$133$	−2.03311e8	−0.649763
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 5.91484e8i	− 1.67904i	−0.543331	−	0.839519i	$-0.682837\pi$
$137$	− 5.91484e8i	− 1.67904i	0.543331	−	0.839519i	$-0.317163\pi$
$138$	0	0
$139$	5.22321e8	1.39919	0.699597	−	0.714537i	$-0.253363\pi$
$139$	5.22321e8	1.39919	0.699597	+	0.714537i	$0.253363\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.43672e8i	0.821864i
$144$	0	0
$145$	4.13061e8	0.934420
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 4.91574e8i	− 0.997341i	−0.866792	−	0.498670i	$-0.833822\pi$
$149$	− 4.91574e8i	− 0.997341i	0.866792	−	0.498670i	$-0.166178\pi$
$150$	0	0
$151$	8.44622e8	1.62463	0.812315	−	0.583218i	$-0.198207\pi$
$151$	8.44622e8	1.62463	0.812315	+	0.583218i	$0.198207\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 1.18110e9i	− 2.04625i
$156$	0	0
$157$	−6.72002e8	−1.10604	−0.553022	−	0.833167i	$-0.686525\pi$
$157$	−6.72002e8	−1.10604	−0.553022	+	0.833167i	$0.686525\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.18217e9i	1.75945i
$162$	0	0
$163$	2.35500e8	0.333611	0.166805	−	0.985990i	$-0.446655\pi$
$163$	2.35500e8	0.333611	0.166805	+	0.985990i	$0.446655\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 5.87131e8i	− 0.754865i	−0.926037	−	0.377432i	$-0.876807\pi$
$167$	− 5.87131e8i	− 0.754865i	0.926037	−	0.377432i	$-0.123193\pi$
$168$	0	0
$169$	−6.21992e8	−0.762497
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.13993e9i	− 1.27261i	−0.771437	−	0.636305i	$-0.780462\pi$
$173$	− 1.13993e9i	− 1.27261i	0.771437	−	0.636305i	$-0.219538\pi$
$174$	0	0
$175$	2.82476e8	0.301182
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1.40519e9i	1.36875i	0.729131	+	0.684374i	$0.239924\pi$
$179$	1.40519e9i	1.36875i	−0.729131	+	0.684374i	$0.760076\pi$
$180$	0	0
$181$	2.00063e9	1.86402	0.932012	−	0.362426i	$-0.118051\pi$
$181$	2.00063e9	1.86402	0.932012	+	0.362426i	$0.118051\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.93005e7i	0.0164771i
$186$	0	0
$187$	2.00310e9	1.63808
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.50002e9i	− 1.12710i	−0.826082	−	0.563550i	$-0.809435\pi$
$191$	− 1.50002e9i	− 1.12710i	0.826082	−	0.563550i	$-0.190565\pi$
$192$	0	0
$193$	2.53788e9	1.82912	0.914559	−	0.404452i	$-0.132538\pi$
$193$	2.53788e9	1.82912	0.914559	+	0.404452i	$0.132538\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 1.55082e9i	− 1.02967i	−0.857290	−	0.514834i	$-0.827854\pi$
$197$	− 1.55082e9i	− 1.02967i	0.857290	−	0.514834i	$-0.172146\pi$
$198$	0	0
$199$	−2.67114e9	−1.70328	−0.851638	−	0.524130i	$-0.824390\pi$
$199$	−2.67114e9	−1.70328	−0.851638	+	0.524130i	$0.824390\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1.54520e9i	0.909916i
$204$	0	0
$205$	9.20678e8	0.521306
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.90221e9i	0.996949i
$210$	0	0
$211$	2.85862e9	1.44220	0.721101	−	0.692830i	$-0.243636\pi$
$211$	2.85862e9	1.44220	0.721101	+	0.692830i	$0.243636\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.41034e9i	− 0.660041i
$216$	0	0
$217$	4.41832e9	1.99259
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 1.12921e9i	− 0.473375i
$222$	0	0
$223$	−2.67732e8	−0.108263	−0.0541317	−	0.998534i	$-0.517239\pi$
$223$	−2.67732e8	−0.108263	−0.0541317	+	0.998534i	$0.517239\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.15769e9i	1.18923i	0.804010	+	0.594616i	$0.202696\pi$
$227$	3.15769e9i	1.18923i	−0.804010	+	0.594616i	$0.797304\pi$
$228$	0	0
$229$	−1.99790e7	−0.00726495	−0.00363248	−	0.999993i	$-0.501156\pi$
$229$	−1.99790e7	−0.00726495	−0.00363248	+	0.999993i	$0.501156\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	3.55234e9i	1.20529i	0.798011	+	0.602644i	$0.205886\pi$
$233$	3.55234e9i	1.20529i	−0.798011	+	0.602644i	$0.794114\pi$
$234$	0	0
$235$	−5.11350e9	−1.67667
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 3.10051e9i	− 0.950258i	−0.879916	−	0.475129i	$-0.842401\pi$
$239$	− 3.10051e9i	− 0.950258i	0.879916	−	0.475129i	$-0.157599\pi$
$240$	0	0
$241$	−1.41544e9	−0.419587	−0.209794	−	0.977746i	$-0.567279\pi$
$241$	−1.41544e9	−0.419587	−0.209794	+	0.977746i	$0.567279\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	8.46205e8i	0.234861i
$246$	0	0
$247$	1.07233e9	0.288099
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 6.36947e8i	− 0.160475i	−0.996776	−	0.0802376i	$-0.974432\pi$
$251$	− 6.36947e8i	− 0.160475i	0.996776	−	0.0802376i	$-0.0255679\pi$
$252$	0	0
$253$	1.10606e10	2.69958
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.39390e9i	0.319520i	0.987156	+	0.159760i	$0.0510721\pi$
$257$	1.39390e9i	0.319520i	−0.987156	+	0.159760i	$0.948928\pi$
$258$	0	0
$259$	−7.22004e7	−0.0160450
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.27753e9i	1.31210i	0.754719	+	0.656048i	$0.227773\pi$
$263$	6.27753e9i	1.31210i	−0.754719	+	0.656048i	$0.772227\pi$
$264$	0	0
$265$	1.69703e9	0.344118
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 7.75652e9i	− 1.48135i	−0.671863	−	0.740676i	$-0.734506\pi$
$269$	− 7.75652e9i	− 1.48135i	0.671863	−	0.740676i	$-0.265494\pi$
$270$	0	0
$271$	8.57427e9	1.58972	0.794859	−	0.606794i	$-0.207545\pi$
$271$	8.57427e9	1.58972	0.794859	+	0.606794i	$0.207545\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 2.64288e9i	− 0.462112i
$276$	0	0
$277$	−5.55250e9	−0.943125	−0.471562	−	0.881833i	$-0.656310\pi$
$277$	−5.55250e9	−0.943125	−0.471562	+	0.881833i	$0.656310\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3.32759e9i	− 0.533708i	−0.963737	−	0.266854i	$-0.914016\pi$
$281$	− 3.32759e9i	− 0.533708i	0.963737	−	0.266854i	$-0.0859842\pi$
$282$	0	0
$283$	4.46395e9	0.695942	0.347971	−	0.937505i	$-0.386871\pi$
$283$	4.46395e9	0.695942	0.347971	+	0.937505i	$0.386871\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.44413e9i	0.507635i
$288$	0	0
$289$	3.94151e8	0.0565030
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.95567e9i	0.672407i	0.941789	+	0.336203i	$0.109143\pi$
$293$	4.95567e9i	0.672407i	−0.941789	+	0.336203i	$0.890857\pi$
$294$	0	0
$295$	1.57536e10	2.08013
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 6.23519e9i	− 0.780127i
$300$	0	0
$301$	5.27589e9	0.642732
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1.75858e10i	2.03218i
$306$	0	0
$307$	9.36938e9	1.05477	0.527385	−	0.849627i	$-0.323173\pi$
$307$	9.36938e9	1.05477	0.527385	+	0.849627i	$0.323173\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.46419e10i	1.56515i	0.622555	+	0.782576i	$0.286095\pi$
$311$	1.46419e10i	1.56515i	−0.622555	+	0.782576i	$0.713905\pi$
$312$	0	0
$313$	1.17993e10	1.22936	0.614681	−	0.788776i	$-0.289285\pi$
$313$	1.17993e10	1.22936	0.614681	+	0.788776i	$0.289285\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.51923e10i	1.50448i	0.658890	+	0.752239i	$0.271026\pi$
$317$	1.51923e10i	1.50448i	−0.658890	+	0.752239i	$0.728974\pi$
$318$	0	0
$319$	1.44571e10	1.39611
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 6.25011e9i	− 0.574219i
$324$	0	0
$325$	−1.48988e9	−0.133542
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 1.91289e10i	− 1.63270i
$330$	0	0
$331$	9.30582e9	0.775252	0.387626	−	0.921817i	$-0.373295\pi$
$331$	9.30582e9	0.775252	0.387626	+	0.921817i	$0.373295\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 8.46425e9i	− 0.672062i
$336$	0	0
$337$	−8.54025e9	−0.662142	−0.331071	−	0.943606i	$-0.607410\pi$
$337$	−8.54025e9	−0.662142	−0.331071	+	0.943606i	$0.607410\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 4.13385e10i	− 3.05729i
$342$	0	0
$343$	1.20478e10	0.870423
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 1.30473e10i	− 0.899918i	−0.893049	−	0.449959i	$-0.851439\pi$
$347$	− 1.30473e10i	− 0.899918i	0.893049	−	0.449959i	$-0.148561\pi$
$348$	0	0
$349$	1.89904e9	0.128007	0.0640034	−	0.997950i	$-0.479613\pi$
$349$	1.89904e9	0.128007	0.0640034	+	0.997950i	$0.479613\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.53302e10i	− 0.987302i	−0.869660	−	0.493651i	$-0.835662\pi$
$353$	− 1.53302e10i	− 0.987302i	0.869660	−	0.493651i	$-0.164338\pi$
$354$	0	0
$355$	−1.54276e9	−0.0971370
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 2.20067e10i	− 1.32488i	−0.749113	−	0.662442i	$-0.769520\pi$
$359$	− 2.20067e10i	− 1.32488i	0.749113	−	0.662442i	$-0.230480\pi$
$360$	0	0
$361$	−1.10482e10	−0.650526
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.68287e10i	1.51157i
$366$	0	0
$367$	−2.74499e9	−0.151313	−0.0756565	−	0.997134i	$-0.524105\pi$
$367$	−2.74499e9	−0.151313	−0.0756565	+	0.997134i	$0.524105\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	6.34836e9i	0.335094i
$372$	0	0
$373$	−6.25799e9	−0.323295	−0.161648	−	0.986849i	$-0.551681\pi$
$373$	−6.25799e9	−0.323295	−0.161648	+	0.986849i	$0.551681\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 8.14994e9i	− 0.403449i
$378$	0	0
$379$	7.38075e9	0.357720	0.178860	−	0.983875i	$-0.442759\pi$
$379$	7.38075e9	0.357720	0.178860	+	0.983875i	$0.442759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	2.01521e10i	0.936538i	0.883586	+	0.468269i	$0.155122\pi$
$383$	2.01521e10i	0.936538i	−0.883586	+	0.468269i	$0.844878\pi$
$384$	0	0
$385$	4.59667e10	2.09219
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9.41663e9i	− 0.411242i	−0.978632	−	0.205621i	$-0.934079\pi$
$389$	− 9.41663e9i	− 0.411242i	0.978632	−	0.205621i	$-0.0659214\pi$
$390$	0	0
$391$	−3.63419e10	−1.55489
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.41220e9i	0.0580109i
$396$	0	0
$397$	−3.66441e9	−0.147517	−0.0737585	−	0.997276i	$-0.523499\pi$
$397$	−3.66441e9	−0.147517	−0.0737585	+	0.997276i	$0.523499\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.03474e10i	1.17367i	0.809708	+	0.586833i	$0.199625\pi$
$401$	3.03474e10i	1.17367i	−0.809708	+	0.586833i	$0.800375\pi$
$402$	0	0
$403$	−2.33038e10	−0.883499
$404$	0	0
$405$	0	0
$406$	0	0
$407$	6.75517e8i	0.0246183i
$408$	0	0
$409$	2.91939e10	1.04328	0.521638	−	0.853167i	$-0.325321\pi$
$409$	2.91939e10	1.04328	0.521638	+	0.853167i	$0.325321\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5.89318e10i	2.02558i
$414$	0	0
$415$	2.42910e10	0.818941
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.62166e9i	0.279727i	0.990171	+	0.139864i	$0.0446664\pi$
$419$	8.62166e9i	0.279727i	−0.990171	+	0.139864i	$0.955334\pi$
$420$	0	0
$421$	2.35374e10	0.749257	0.374628	−	0.927175i	$-0.377770\pi$
$421$	2.35374e10	0.749257	0.374628	+	0.927175i	$0.377770\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	8.68376e9i	0.266166i
$426$	0	0
$427$	−6.57858e10	−1.97889
$428$	0	0
$429$	0	0
$430$	0	0
$431$	9.81034e9i	0.284299i	0.989845	+	0.142149i	$0.0454013\pi$
$431$	9.81034e9i	0.284299i	−0.989845	+	0.142149i	$0.954599\pi$
$432$	0	0
$433$	−4.87335e10	−1.38636	−0.693180	−	0.720764i	$-0.743791\pi$
$433$	−4.87335e10	−1.38636	−0.693180	+	0.720764i	$0.743791\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 3.45115e10i	− 0.946320i
$438$	0	0
$439$	−1.89161e10	−0.509300	−0.254650	−	0.967033i	$-0.581960\pi$
$439$	−1.89161e10	−0.509300	−0.254650	+	0.967033i	$0.581960\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	7.14896e10i	1.85621i	0.372315	+	0.928106i	$0.378564\pi$
$443$	7.14896e10i	1.85621i	−0.372315	+	0.928106i	$0.621436\pi$
$444$	0	0
$445$	3.25970e8	0.00831261
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 3.44073e10i	− 0.846574i	−0.905996	−	0.423287i	$-0.860876\pi$
$449$	− 3.44073e10i	− 0.846574i	0.905996	−	0.423287i	$-0.139124\pi$
$450$	0	0
$451$	3.22237e10	0.778879
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 2.59129e10i	− 0.604603i
$456$	0	0
$457$	6.84276e10	1.56880	0.784398	−	0.620258i	$-0.212972\pi$
$457$	6.84276e10	1.56880	0.784398	+	0.620258i	$0.212972\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.91110e10i	1.30877i	0.756160	+	0.654387i	$0.227073\pi$
$461$	5.91110e10i	1.30877i	−0.756160	+	0.654387i	$0.772927\pi$
$462$	0	0
$463$	−2.39209e10	−0.520540	−0.260270	−	0.965536i	$-0.583812\pi$
$463$	−2.39209e10	−0.520540	−0.260270	+	0.965536i	$0.583812\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	5.73549e10i	1.20588i	0.797787	+	0.602939i	$0.206004\pi$
$467$	5.73549e10i	1.20588i	−0.797787	+	0.602939i	$0.793996\pi$
$468$	0	0
$469$	3.16636e10	0.654438
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 4.93620e10i	− 0.986162i
$474$	0	0
$475$	−8.24639e9	−0.161990
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 5.08983e10i	− 0.966855i	−0.875385	−	0.483427i	$-0.839392\pi$
$479$	− 5.08983e10i	− 0.966855i	0.875385	−	0.483427i	$-0.160608\pi$
$480$	0	0
$481$	3.80810e8	0.00711423
$482$	0	0
$483$	0	0
$484$	0	0
$485$	3.69774e10i	0.668296i
$486$	0	0
$487$	−7.54226e9	−0.134087	−0.0670433	−	0.997750i	$-0.521357\pi$
$487$	−7.54226e9	−0.134087	−0.0670433	+	0.997750i	$0.521357\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 1.05897e11i	− 1.82204i	−0.412360	−	0.911021i	$-0.635295\pi$
$491$	− 1.05897e11i	− 1.82204i	0.412360	−	0.911021i	$-0.364705\pi$
$492$	0	0
$493$	−4.75020e10	−0.804127
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.77124e9i	− 0.0945897i
$498$	0	0
$499$	3.06212e10	0.493879	0.246939	−	0.969031i	$-0.420575\pi$
$499$	3.06212e10	0.493879	0.246939	+	0.969031i	$0.420575\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 5.46233e10i	− 0.853308i	−0.904415	−	0.426654i	$-0.859692\pi$
$503$	− 5.46233e10i	− 0.853308i	0.904415	−	0.426654i	$-0.140308\pi$
$504$	0	0
$505$	−6.42385e10	−0.987710
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 2.27411e10i	− 0.338797i	−0.985548	−	0.169399i	$-0.945817\pi$
$509$	− 2.27411e10i	− 0.338797i	0.985548	−	0.169399i	$-0.0541826\pi$
$510$	0	0
$511$	−1.00362e11	−1.47193
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 4.88159e10i	− 0.693957i
$516$	0	0
$517$	−1.78972e11	−2.50509
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 7.23199e10i	− 0.981538i	−0.871290	−	0.490769i	$-0.836716\pi$
$521$	− 7.23199e10i	− 0.981538i	0.871290	−	0.490769i	$-0.163284\pi$
$522$	0	0
$523$	1.34399e11	1.79634	0.898168	−	0.439652i	$-0.144898\pi$
$523$	1.34399e11	1.79634	0.898168	+	0.439652i	$0.144898\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.35826e11i	1.76093i
$528$	0	0
$529$	−1.22360e11	−1.56248
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 1.81655e10i	− 0.225081i
$534$	0	0
$535$	−4.48221e10	−0.547113
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.96172e10i	0.350904i
$540$	0	0
$541$	2.43319e10	0.284044	0.142022	−	0.989863i	$-0.454640\pi$
$541$	2.43319e10	0.284044	0.142022	+	0.989863i	$0.454640\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.68249e9i	− 0.0984143i
$546$	0	0
$547$	8.61341e9	0.0962113	0.0481056	−	0.998842i	$-0.484682\pi$
$547$	8.61341e9	0.0962113	0.0481056	+	0.998842i	$0.484682\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 4.51095e10i	− 0.489397i
$552$	0	0
$553$	−5.28286e9	−0.0564896
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.83454e10i	1.02172i	0.859663	+	0.510862i	$0.170674\pi$
$557$	9.83454e10i	1.02172i	−0.859663	+	0.510862i	$0.829326\pi$
$558$	0	0
$559$	−2.78269e10	−0.284982
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.84563e10i	0.183701i	0.995773	+	0.0918503i	$0.0292781\pi$
$563$	1.84563e10i	0.183701i	−0.995773	+	0.0918503i	$0.970722\pi$
$564$	0	0
$565$	−1.78286e11	−1.74953
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.42645e11i	− 1.36085i	−0.732820	−	0.680423i	$-0.761796\pi$
$569$	− 1.42645e11i	− 1.36085i	0.732820	−	0.680423i	$-0.238204\pi$
$570$	0	0
$571$	−9.46568e10	−0.890445	−0.445223	−	0.895420i	$-0.646875\pi$
$571$	−9.46568e10	−0.890445	−0.445223	+	0.895420i	$0.646875\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.79495e10i	0.438644i
$576$	0	0
$577$	−6.51933e10	−0.588165	−0.294083	−	0.955780i	$-0.595014\pi$
$577$	−6.51933e10	−0.588165	−0.294083	+	0.955780i	$0.595014\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.08691e10i	0.797465i
$582$	0	0
$583$	5.93962e10	0.514144
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 3.95169e10i	− 0.332836i	−0.986055	−	0.166418i	$-0.946780\pi$
$587$	− 3.95169e10i	− 0.332836i	0.986055	−	0.166418i	$-0.0532201\pi$
$588$	0	0
$589$	−1.28985e11	−1.07171
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.40557e11i	1.13667i	0.822798	+	0.568334i	$0.192412\pi$
$593$	1.40557e11i	1.13667i	−0.822798	+	0.568334i	$0.807588\pi$
$594$	0	0
$595$	−1.51034e11	−1.20505
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 1.30685e11i	− 1.01512i	−0.861615	−	0.507562i	$-0.830547\pi$
$599$	− 1.30685e11i	− 1.01512i	0.861615	−	0.507562i	$-0.169453\pi$
$600$	0	0
$601$	−1.22569e10	−0.0939468	−0.0469734	−	0.998896i	$-0.514958\pi$
$601$	−1.22569e10	−0.0939468	−0.0469734	+	0.998896i	$0.514958\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.78851e11i	− 2.08138i
$606$	0	0
$607$	−7.91136e10	−0.582769	−0.291385	−	0.956606i	$-0.594116\pi$
$607$	−7.91136e10	−0.582769	−0.291385	+	0.956606i	$0.594116\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.00892e11i	0.723924i
$612$	0	0
$613$	−1.81518e10	−0.128552	−0.0642760	−	0.997932i	$-0.520474\pi$
$613$	−1.81518e10	−0.128552	−0.0642760	+	0.997932i	$0.520474\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.73990e11i	1.20056i	0.799791	+	0.600278i	$0.204943\pi$
$617$	1.73990e11i	1.20056i	−0.799791	+	0.600278i	$0.795057\pi$
$618$	0	0
$619$	5.88216e10	0.400658	0.200329	−	0.979729i	$-0.435799\pi$
$619$	5.88216e10	0.400658	0.200329	+	0.979729i	$0.435799\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	1.21941e9i	0.00809462i
$624$	0	0
$625$	−1.82943e11	−1.19893
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 2.21956e9i	− 0.0141796i
$630$	0	0
$631$	−1.13829e11	−0.718020	−0.359010	−	0.933334i	$-0.616886\pi$
$631$	−1.13829e11	−0.718020	−0.359010	+	0.933334i	$0.616886\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 1.38717e11i	− 0.853166i
$636$	0	0
$637$	1.66961e10	0.101405
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2.36215e11i	1.39918i	0.714542	+	0.699592i	$0.246635\pi$
$641$	2.36215e11i	1.39918i	−0.714542	+	0.699592i	$0.753365\pi$
$642$	0	0
$643$	−1.51704e11	−0.887469	−0.443735	−	0.896158i	$-0.646347\pi$
$643$	−1.51704e11	−0.887469	−0.443735	+	0.896158i	$0.646347\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	8.87726e9i	0.0506596i	0.999679	+	0.0253298i	$0.00806359\pi$
$647$	8.87726e9i	0.0506596i	−0.999679	+	0.0253298i	$0.991936\pi$
$648$	0	0
$649$	5.51374e11	3.10791
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.30726e11i	− 1.26895i	−0.772944	−	0.634475i	$-0.781217\pi$
$653$	− 2.30726e11i	− 1.26895i	0.772944	−	0.634475i	$-0.218783\pi$
$654$	0	0
$655$	4.12414e10	0.224062
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.18370e11i	− 0.627624i	−0.949485	−	0.313812i	$-0.898394\pi$
$659$	− 1.18370e11i	− 0.627624i	0.949485	−	0.313812i	$-0.101606\pi$
$660$	0	0
$661$	−2.26005e11	−1.18389	−0.591946	−	0.805977i	$-0.701640\pi$
$661$	−2.26005e11	−1.18389	−0.591946	+	0.805977i	$0.701640\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.43427e11i	− 0.733403i
$666$	0	0
$667$	−2.62294e11	−1.32521
$668$	0	0
$669$	0	0
$670$	0	0
$671$	6.15502e11i	3.03626i
$672$	0	0
$673$	2.90839e11	1.41773	0.708864	−	0.705345i	$-0.249208\pi$
$673$	2.90839e11	1.41773	0.708864	+	0.705345i	$0.249208\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	6.30605e10i	0.300194i	0.988671	+	0.150097i	$0.0479586\pi$
$677$	6.30605e10i	0.300194i	−0.988671	+	0.150097i	$0.952041\pi$
$678$	0	0
$679$	−1.38327e11	−0.650771
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 6.54747e10i	− 0.300878i	−0.988619	−	0.150439i	$-0.951931\pi$
$683$	− 6.54747e10i	− 0.300878i	0.988619	−	0.150439i	$-0.0480687\pi$
$684$	0	0
$685$	4.17264e11	1.89517
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.34835e10i	− 0.148578i
$690$	0	0
$691$	7.28879e10	0.319700	0.159850	−	0.987141i	$-0.448899\pi$
$691$	7.28879e10	0.319700	0.159850	+	0.987141i	$0.448899\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.68473e11i	1.57931i
$696$	0	0
$697$	−1.05878e11	−0.448616
$698$	0	0
$699$	0	0
$700$	0	0
$701$	2.18377e10i	0.0904347i	0.998977	+	0.0452173i	$0.0143980\pi$
$701$	2.18377e10i	0.0904347i	−0.998977	+	0.0452173i	$0.985602\pi$
$702$	0	0
$703$	2.10776e9	0.00862980
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 2.40307e11i	− 0.961809i
$708$	0	0
$709$	−2.96192e11	−1.17216	−0.586082	−	0.810252i	$-0.699330\pi$
$709$	−2.96192e11	−1.17216	−0.586082	+	0.810252i	$0.699330\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	7.49998e11i	2.90203i
$714$	0	0
$715$	−2.42444e11	−0.927659
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 4.64816e11i	− 1.73927i	−0.493699	−	0.869633i	$-0.664356\pi$
$719$	− 4.64816e11i	− 1.73927i	0.493699	−	0.869633i	$-0.335644\pi$
$720$	0	0
$721$	1.82614e11	0.675759
$722$	0	0
$723$	0	0
$724$	0	0
$725$	6.26741e10i	0.226849i
$726$	0	0
$727$	3.87014e11	1.38544	0.692722	−	0.721204i	$-0.256411\pi$
$727$	3.87014e11	1.38544	0.692722	+	0.721204i	$0.256411\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	1.62189e11i	0.568006i
$732$	0	0
$733$	−1.14514e11	−0.396684	−0.198342	−	0.980133i	$-0.563556\pi$
$733$	−1.14514e11	−0.396684	−0.198342	+	0.980133i	$0.563556\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.96249e11i	− 1.00412i
$738$	0	0
$739$	−9.88171e10	−0.331325	−0.165663	−	0.986183i	$-0.552976\pi$
$739$	−9.88171e10	−0.331325	−0.165663	+	0.986183i	$0.552976\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 3.23957e11i	− 1.06300i	−0.847060	−	0.531498i	$-0.821629\pi$
$743$	− 3.23957e11i	− 1.06300i	0.847060	−	0.531498i	$-0.178371\pi$
$744$	0	0
$745$	3.46782e11	1.12572
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 1.67673e11i	− 0.532766i
$750$	0	0
$751$	−2.57803e11	−0.810454	−0.405227	−	0.914216i	$-0.632808\pi$
$751$	−2.57803e11	−0.810454	−0.405227	+	0.914216i	$0.632808\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.95841e11i	1.83376i
$756$	0	0
$757$	−2.74781e11	−0.836765	−0.418382	−	0.908271i	$-0.637403\pi$
$757$	−2.74781e11	−0.836765	−0.418382	+	0.908271i	$0.637403\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 7.89821e10i	− 0.235499i	−0.993043	−	0.117750i	$-0.962432\pi$
$761$	− 7.89821e10i	− 0.235499i	0.993043	−	0.117750i	$-0.0375681\pi$
$762$	0	0
$763$	3.24800e10	0.0958335
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.10827e11i	− 0.898125i
$768$	0	0
$769$	−2.57601e11	−0.736619	−0.368309	−	0.929703i	$-0.620063\pi$
$769$	−2.57601e11	−0.736619	−0.368309	+	0.929703i	$0.620063\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 2.55605e11i	− 0.715898i	−0.933741	−	0.357949i	$-0.883476\pi$
$773$	− 2.55605e11i	− 0.715898i	0.933741	−	0.357949i	$-0.116524\pi$
$774$	0	0
$775$	1.79209e11	0.496768
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.00545e11i	− 0.273031i
$780$	0	0
$781$	−5.39965e10	−0.145132
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.74066e11i	− 1.24842i
$786$	0	0
$787$	−3.59939e11	−0.938274	−0.469137	−	0.883125i	$-0.655435\pi$
$787$	−3.59939e11	−0.938274	−0.469137	+	0.883125i	$0.655435\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 6.66941e11i	− 1.70366i
$792$	0	0
$793$	3.46977e11	0.877422
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.68262e10i	0.165620i	0.996565	+	0.0828101i	$0.0263895\pi$
$797$	6.68262e10i	0.165620i	−0.996565	+	0.0828101i	$0.973610\pi$
$798$	0	0
$799$	5.88052e11	1.44288
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.39003e11i	2.25842i
$804$	0	0
$805$	−8.33968e11	−1.98594
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.52798e11i	0.356717i	0.983966	+	0.178358i	$0.0570786\pi$
$809$	1.52798e11i	0.356717i	−0.983966	+	0.178358i	$0.942921\pi$
$810$	0	0
$811$	7.47417e11	1.72774	0.863872	−	0.503712i	$-0.168033\pi$
$811$	7.47417e11	1.72774	0.863872	+	0.503712i	$0.168033\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.66134e11i	0.376555i
$816$	0	0
$817$	−1.54021e11	−0.345693
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.16174e10i	− 0.0475807i	−0.999717	−	0.0237903i	$-0.992427\pi$
$821$	− 2.16174e10i	− 0.0475807i	0.999717	−	0.0237903i	$-0.00757341\pi$
$822$	0	0
$823$	2.96832e11	0.647011	0.323505	−	0.946226i	$-0.395139\pi$
$823$	2.96832e11	0.647011	0.323505	+	0.946226i	$0.395139\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.77124e11i	− 1.66138i	−0.556737	−	0.830689i	$-0.687947\pi$
$827$	− 7.77124e11i	− 1.66138i	0.556737	−	0.830689i	$-0.312053\pi$
$828$	0	0
$829$	−6.11841e10	−0.129545	−0.0647725	−	0.997900i	$-0.520632\pi$
$829$	−6.11841e10	−0.129545	−0.0647725	+	0.997900i	$0.520632\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 9.73136e10i	− 0.202113i
$834$	0	0
$835$	4.14193e11	0.852034
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 6.90886e11i	− 1.39431i	−0.716922	−	0.697153i	$-0.754450\pi$
$839$	− 6.90886e11i	− 1.39431i	0.716922	−	0.697153i	$-0.245550\pi$
$840$	0	0
$841$	1.57406e11	0.314656
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 4.38786e11i	− 0.860649i
$846$	0	0
$847$	1.04314e12	2.02680
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.22558e10i	− 0.0233681i
$852$	0	0
$853$	8.06772e11	1.52389	0.761947	−	0.647640i	$-0.224244\pi$
$853$	8.06772e11	1.52389	0.761947	+	0.647640i	$0.224244\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.13922e11i	− 0.396581i	−0.980143	−	0.198290i	$-0.936461\pi$
$857$	− 2.13922e11i	− 0.396581i	0.980143	−	0.198290i	$-0.0635389\pi$
$858$	0	0
$859$	6.27657e11	1.15279	0.576395	−	0.817171i	$-0.304459\pi$
$859$	6.27657e11	1.15279	0.576395	+	0.817171i	$0.304459\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	2.94626e11i	0.531163i	0.964088	+	0.265581i	$0.0855639\pi$
$863$	2.94626e11i	0.531163i	−0.964088	+	0.265581i	$0.914436\pi$
$864$	0	0
$865$	8.04170e11	1.43643
$866$	0	0
$867$	0	0
$868$	0	0
$869$	4.94272e10i	0.0866736i
$870$	0	0
$871$	−1.67005e11	−0.290172
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 5.27949e11i	− 0.900658i
$876$	0	0
$877$	−4.69118e11	−0.793019	−0.396510	−	0.918031i	$-0.629779\pi$
$877$	−4.69118e11	−0.793019	−0.396510	+	0.918031i	$0.629779\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.66926e11i	0.941072i	0.882381	+	0.470536i	$0.155939\pi$
$881$	5.66926e11i	0.941072i	−0.882381	+	0.470536i	$0.844061\pi$
$882$	0	0
$883$	9.08701e11	1.49478	0.747392	−	0.664384i	$-0.231306\pi$
$883$	9.08701e11	1.49478	0.747392	+	0.664384i	$0.231306\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.97506e11i	0.803718i	0.915701	+	0.401859i	$0.131636\pi$
$887$	4.97506e11i	0.803718i	−0.915701	+	0.401859i	$0.868364\pi$
$888$	0	0
$889$	5.18919e11	0.830793
$890$	0	0
$891$	0	0
$892$	0	0
$893$	5.58434e11i	0.878145i
$894$	0	0
$895$	−9.91297e11	−1.54494
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.80312e11i	1.50081i
$900$	0	0
$901$	−1.95159e11	−0.296135
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.41135e12i	2.10397i
$906$	0	0
$907$	−1.35983e11	−0.200935	−0.100468	−	0.994940i	$-0.532034\pi$
$907$	−1.35983e11	−0.200935	−0.100468	+	0.994940i	$0.532034\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	2.18939e11i	0.317870i	0.987289	+	0.158935i	$0.0508060\pi$
$911$	2.18939e11i	0.317870i	−0.987289	+	0.158935i	$0.949194\pi$
$912$	0	0
$913$	8.50184e11	1.22357
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.54278e11i	0.218186i
$918$	0	0
$919$	8.19281e11	1.14860	0.574302	−	0.818643i	$-0.305273\pi$
$919$	8.19281e11	1.14860	0.574302	+	0.818643i	$0.305273\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	3.04395e10i	0.0419403i
$924$	0	0
$925$	−2.92848e9	−0.00400014
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 9.73932e11i	− 1.30757i	−0.756679	−	0.653786i	$-0.773180\pi$
$929$	− 9.73932e11i	− 1.30757i	0.756679	−	0.653786i	$-0.226820\pi$
$930$	0	0
$931$	9.24122e10	0.123007
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.41309e12i	1.84894i
$936$	0	0
$937$	2.32658e11	0.301828	0.150914	−	0.988547i	$-0.451778\pi$
$937$	2.32658e11	0.301828	0.150914	+	0.988547i	$0.451778\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 5.47479e11i	− 0.698247i	−0.937077	−	0.349124i	$-0.886479\pi$
$941$	− 5.47479e11i	− 0.698247i	0.937077	−	0.349124i	$-0.113521\pi$
$942$	0	0
$943$	−5.84631e11	−0.739324
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.39829e11i	0.173859i	0.996214	+	0.0869293i	$0.0277054\pi$
$947$	1.39829e11i	0.173859i	−0.996214	+	0.0869293i	$0.972295\pi$
$948$	0	0
$949$	5.29345e11	0.652641
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.90654e11i	0.231140i	0.993299	+	0.115570i	$0.0368694\pi$
$953$	1.90654e11i	0.231140i	−0.993299	+	0.115570i	$0.963131\pi$
$954$	0	0
$955$	1.05819e12	1.27219
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.56093e12i	1.84547i
$960$	0	0
$961$	1.95020e12	2.28657
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.79036e12i	2.06457i
$966$	0	0
$967$	−1.03836e12	−1.18752	−0.593760	−	0.804642i	$-0.702357\pi$
$967$	−1.03836e12	−1.18752	−0.593760	+	0.804642i	$0.702357\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 1.21983e12i	− 1.37222i	−0.727500	−	0.686108i	$-0.759318\pi$
$971$	− 1.21983e12i	− 1.37222i	0.727500	−	0.686108i	$-0.240682\pi$
$972$	0	0
$973$	−1.37840e12	−1.53789
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.46272e12i	− 1.60539i	−0.596387	−	0.802697i	$-0.703398\pi$
$977$	− 1.46272e12i	− 1.60539i	0.596387	−	0.802697i	$-0.296602\pi$
$978$	0	0
$979$	1.14089e10	0.0124198
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 8.69146e11i	− 0.930847i	−0.885088	−	0.465424i	$-0.845902\pi$
$983$	− 8.69146e11i	− 0.930847i	0.885088	−	0.465424i	$-0.154098\pi$
$984$	0	0
$985$	1.09403e12	1.16221
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.95568e11i	0.936081i
$990$	0	0
$991$	−1.22632e11	−0.127148	−0.0635738	−	0.997977i	$-0.520250\pi$
$991$	−1.22632e11	−0.127148	−0.0635738	+	0.997977i	$0.520250\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1.88437e12i	− 1.92253i
$996$	0	0
$997$	1.15490e11	0.116886	0.0584431	−	0.998291i	$-0.481386\pi$
$997$	1.15490e11	0.116886	0.0584431	+	0.998291i	$0.481386\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.9.e.c.161.2		2
3.2	odd	2	inner	432.9.e.c.161.1		2
4.3	odd	2		108.9.c.c.53.2	yes	2
12.11	even	2		108.9.c.c.53.1	✓	2
36.7	odd	6		324.9.g.c.53.1		4
36.11	even	6		324.9.g.c.53.2		4
36.23	even	6		324.9.g.c.269.1		4
36.31	odd	6		324.9.g.c.269.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
108.9.c.c.53.1	✓	2	12.11	even	2
108.9.c.c.53.2	yes	2	4.3	odd	2
324.9.g.c.53.1		4	36.7	odd	6
324.9.g.c.53.2		4	36.11	even	6
324.9.g.c.269.1		4	36.23	even	6
324.9.g.c.269.2		4	36.31	odd	6
432.9.e.c.161.1		2	3.2	odd	2	inner
432.9.e.c.161.2		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	432.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+705.453i q^{5} -2639.00 q^{7} +24690.9i q^{11} +13919.0 q^{13} -81127.1i q^{17} +77041.0 q^{19} -447963. i q^{23} -107039. q^{25} -585526. i q^{29} -1.67424e6 q^{31} -1.86169e6i q^{35} +27359.0 q^{37} -1.30509e6i q^{41} -1.99920e6 q^{43} +7.24853e6i q^{47} +1.19952e6 q^{49} -2.40559e6i q^{53} -1.74182e7 q^{55} -2.23311e7i q^{59} +2.49283e7 q^{61} +9.81920e6i q^{65} -1.19983e7 q^{67} +2.18690e6i q^{71} +3.80304e7 q^{73} -6.51592e7i q^{77} +2.00184e6 q^{79} -3.44332e7i q^{83} +5.72314e7 q^{85} -462072. i q^{89} -3.67322e7 q^{91} +5.43488e7i q^{95} +5.24165e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 5278 q^{7} + 27838 q^{13} + 154082 q^{19} - 214078 q^{25} - 3348484 q^{31} + 54718 q^{37} - 3998404 q^{43} + 2399040 q^{49} - 34836480 q^{55} + 49856638 q^{61} - 23996638 q^{67} + 76060798 q^{73}+ \cdots + 104832958 q^{97}+O(q^{100}) \) 2 * q - 5278 * q^7 + 27838 * q^13 + 154082 * q^19 - 214078 * q^25 - 3348484 * q^31 + 54718 * q^37 - 3998404 * q^43 + 2399040 * q^49 - 34836480 * q^55 + 49856638 * q^61 - 23996638 * q^67 + 76060798 * q^73 + 4003682 * q^79 + 114462720 * q^85 - 73464482 * q^91 + 104832958 * q^97

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