LMFDB - Embedded newform 432.7.e.j.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,7,Mod(161,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("432.161");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	432.e (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:	$99.3833641238$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{41})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 21x^{2} + 100 $ x^4 + 21*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{8} $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-3.70156i$ of defining polynomial
Character	$\chi$	$=$	432.161
Dual form			432.7.e.j.161.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-137.419i q^{5} +514.769 q^{7} -1218.26i q^{11} -170.463 q^{13} -1574.10i q^{17} +11289.1 q^{19} -10572.6i q^{23} -3258.91 q^{25} -8665.42i q^{29} +26209.1 q^{31} -70738.9i q^{35} +21866.9 q^{37} +38166.6i q^{41} -23895.1 q^{43} +26698.7i q^{47} +147338. q^{49} +213997. i q^{53} -167412. q^{55} +163689. i q^{59} +111078. q^{61} +23424.8i q^{65} +381035. q^{67} -675048. i q^{71} -762939. q^{73} -627123. i q^{77} -515708. q^{79} +903773. i q^{83} -216311. q^{85} -644278. i q^{89} -87748.8 q^{91} -1.55133e6i q^{95} -616470. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 676 q^{7} - 3448 q^{13} + 3664 q^{19} + 17392 q^{25} + 153244 q^{31} + 128960 q^{37} - 126008 q^{43} + 121872 q^{49} - 54180 q^{55} + 167696 q^{61} - 8 q^{67} - 1881676 q^{73} - 1188728 q^{79} - 176472 q^{85}+ \cdots + 975212 q^{97}+O(q^{100}) $ 4 * q + 676 * q^7 - 3448 * q^13 + 3664 * q^19 + 17392 * q^25 + 153244 * q^31 + 128960 * q^37 - 126008 * q^43 + 121872 * q^49 - 54180 * q^55 + 167696 * q^61 - 8 * q^67 - 1881676 * q^73 - 1188728 * q^79 - 176472 * q^85 + 373736 * q^91 + 975212 * 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
$5$	− 137.419i	− 1.09935i	−0.835379	−	0.549675i	$-0.814752\pi$
$5$	− 137.419i	− 1.09935i	0.835379	−	0.549675i	$-0.185248\pi$
$6$	0	0
$7$	514.769	1.50078	0.750392	−	0.660993i	$-0.229865\pi$
$7$	514.769	1.50078	0.750392	+	0.660993i	$0.229865\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 1218.26i	− 0.915298i	−0.889133	−	0.457649i	$-0.848692\pi$
$11$	− 1218.26i	− 0.915298i	0.889133	−	0.457649i	$-0.151308\pi$
$12$	0	0
$13$	−170.463	−0.0775888	−0.0387944	−	0.999247i	$-0.512352\pi$
$13$	−170.463	−0.0775888	−0.0387944	+	0.999247i	$0.512352\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1574.10i	− 0.320395i	−0.987085	−	0.160197i	$-0.948787\pi$
$17$	− 1574.10i	− 0.320395i	0.987085	−	0.160197i	$-0.0512131\pi$
$18$	0	0
$19$	11289.1	1.64588	0.822938	−	0.568131i	$-0.192333\pi$
$19$	11289.1	1.64588	0.822938	+	0.568131i	$0.192333\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 10572.6i	− 0.868960i	−0.900681	−	0.434480i	$-0.856932\pi$
$23$	− 10572.6i	− 0.868960i	0.900681	−	0.434480i	$-0.143068\pi$
$24$	0	0
$25$	−3258.91	−0.208570
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8665.42i	− 0.355300i	−0.984094	−	0.177650i	$-0.943150\pi$
$29$	− 8665.42i	− 0.355300i	0.984094	−	0.177650i	$-0.0568496\pi$
$30$	0	0
$31$	26209.1	0.879766	0.439883	−	0.898055i	$-0.355020\pi$
$31$	26209.1	0.879766	0.439883	+	0.898055i	$0.355020\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 70738.9i	− 1.64989i
$36$	0	0
$37$	21866.9	0.431701	0.215850	−	0.976426i	$-0.430748\pi$
$37$	21866.9	0.431701	0.215850	+	0.976426i	$0.430748\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	38166.6i	0.553774i	0.960903	+	0.276887i	$0.0893027\pi$
$41$	38166.6i	0.553774i	−0.960903	+	0.276887i	$0.910697\pi$
$42$	0	0
$43$	−23895.1	−0.300541	−0.150270	−	0.988645i	$-0.548014\pi$
$43$	−23895.1	−0.300541	−0.150270	+	0.988645i	$0.548014\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	26698.7i	0.257156i	0.991699	+	0.128578i	$0.0410413\pi$
$47$	26698.7i	0.257156i	−0.991699	+	0.128578i	$0.958959\pi$
$48$	0	0
$49$	147338.	1.25235
$50$	0	0
$51$	0	0
$52$	0	0
$53$	213997.i	1.43741i	0.695315	+	0.718705i	$0.255265\pi$
$53$	213997.i	1.43741i	−0.695315	+	0.718705i	$0.744735\pi$
$54$	0	0
$55$	−167412.	−1.00623
$56$	0	0
$57$	0	0
$58$	0	0
$59$	163689.i	0.797010i	0.917166	+	0.398505i	$0.130471\pi$
$59$	163689.i	0.797010i	−0.917166	+	0.398505i	$0.869529\pi$
$60$	0	0
$61$	111078.	0.489370	0.244685	−	0.969603i	$-0.421315\pi$
$61$	111078.	0.489370	0.244685	+	0.969603i	$0.421315\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	23424.8i	0.0852972i
$66$	0	0
$67$	381035.	1.26689	0.633447	−	0.773786i	$-0.281639\pi$
$67$	381035.	1.26689	0.633447	+	0.773786i	$0.281639\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 675048.i	− 1.88608i	−0.332682	−	0.943039i	$-0.607953\pi$
$71$	− 675048.i	− 1.88608i	0.332682	−	0.943039i	$-0.392047\pi$
$72$	0	0
$73$	−762939.	−1.96120	−0.980599	−	0.196025i	$-0.937197\pi$
$73$	−762939.	−1.96120	−0.980599	+	0.196025i	$0.937197\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 627123.i	− 1.37366i
$78$	0	0
$79$	−515708.	−1.04598	−0.522989	−	0.852340i	$-0.675183\pi$
$79$	−515708.	−1.04598	−0.522989	+	0.852340i	$0.675183\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	903773.i	1.58061i	0.612713	+	0.790306i	$0.290078\pi$
$83$	903773.i	1.58061i	−0.612713	+	0.790306i	$0.709922\pi$
$84$	0	0
$85$	−216311.	−0.352226
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 644278.i	− 0.913910i	−0.889490	−	0.456955i	$-0.848940\pi$
$89$	− 644278.i	− 0.913910i	0.889490	−	0.456955i	$-0.151060\pi$
$90$	0	0
$91$	−87748.8	−0.116444
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.55133e6i	− 1.80939i
$96$	0	0
$97$	−616470.	−0.675455	−0.337728	−	0.941244i	$-0.609658\pi$
$97$	−616470.	−0.675455	−0.337728	+	0.941244i	$0.609658\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 623027.i	− 0.604704i	−0.953196	−	0.302352i	$-0.902228\pi$
$101$	− 623027.i	− 0.604704i	0.953196	−	0.302352i	$-0.0977718\pi$
$102$	0	0
$103$	750695.	0.686992	0.343496	−	0.939154i	$-0.388389\pi$
$103$	750695.	0.686992	0.343496	+	0.939154i	$0.388389\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	345428.i	0.281973i	0.990012	+	0.140986i	$0.0450273\pi$
$107$	345428.i	0.281973i	−0.990012	+	0.140986i	$0.954973\pi$
$108$	0	0
$109$	1.17208e6	0.905061	0.452530	−	0.891749i	$-0.350521\pi$
$109$	1.17208e6	0.905061	0.452530	+	0.891749i	$0.350521\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 2.45331e6i	− 1.70026i	−0.526569	−	0.850132i	$-0.676522\pi$
$113$	− 2.45331e6i	− 1.70026i	0.526569	−	0.850132i	$-0.323478\pi$
$114$	0	0
$115$	−1.45288e6	−0.955291
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 810297.i	− 0.480843i
$120$	0	0
$121$	287398.	0.162229
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.69933e6i	− 0.870058i
$126$	0	0
$127$	1.53816e6	0.750913	0.375456	−	0.926840i	$-0.377486\pi$
$127$	1.53816e6	0.750913	0.375456	+	0.926840i	$0.377486\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.12723e6i	− 0.501416i	−0.968063	−	0.250708i	$-0.919337\pi$
$131$	− 1.12723e6i	− 0.501416i	0.968063	−	0.250708i	$-0.0806635\pi$
$132$	0	0
$133$	5.81126e6	2.47010
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.20599e6i	1.24681i	0.781899	+	0.623406i	$0.214251\pi$
$137$	3.20599e6i	1.24681i	−0.781899	+	0.623406i	$0.785749\pi$
$138$	0	0
$139$	66685.5	0.0248306	0.0124153	−	0.999923i	$-0.496048\pi$
$139$	66685.5	0.0248306	0.0124153	+	0.999923i	$0.496048\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	207668.i	0.0710169i
$144$	0	0
$145$	−1.19079e6	−0.390600
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.62846e6i	0.492288i	0.969233	+	0.246144i	$0.0791635\pi$
$149$	1.62846e6i	0.492288i	−0.969233	+	0.246144i	$0.920836\pi$
$150$	0	0
$151$	365620.	0.106194	0.0530968	−	0.998589i	$-0.483091\pi$
$151$	365620.	0.106194	0.0530968	+	0.998589i	$0.483091\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 3.60162e6i	− 0.967170i
$156$	0	0
$157$	−5.18926e6	−1.34093	−0.670465	−	0.741941i	$-0.733905\pi$
$157$	−5.18926e6	−1.34093	−0.670465	+	0.741941i	$0.733905\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 5.44246e6i	− 1.30412i
$162$	0	0
$163$	3.00726e6	0.694397	0.347199	−	0.937792i	$-0.387133\pi$
$163$	3.00726e6	0.694397	0.347199	+	0.937792i	$0.387133\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 4.78381e6i	− 1.02713i	−0.858051	−	0.513564i	$-0.828325\pi$
$167$	− 4.78381e6i	− 1.02713i	0.858051	−	0.513564i	$-0.171675\pi$
$168$	0	0
$169$	−4.79775e6	−0.993980
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.53596e6i	− 0.682918i	−0.939897	−	0.341459i	$-0.889079\pi$
$173$	− 3.53596e6i	− 0.682918i	0.939897	−	0.341459i	$-0.110921\pi$
$174$	0	0
$175$	−1.67759e6	−0.313019
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.57291e6i	− 1.49475i	−0.664401	−	0.747376i	$-0.731313\pi$
$179$	− 8.57291e6i	− 1.49475i	0.664401	−	0.747376i	$-0.268687\pi$
$180$	0	0
$181$	1.79208e6	0.302219	0.151110	−	0.988517i	$-0.451715\pi$
$181$	1.79208e6	0.302219	0.151110	+	0.988517i	$0.451715\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 3.00493e6i	− 0.474590i
$186$	0	0
$187$	−1.91767e6	−0.293257
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.01076e7i	− 1.45061i	−0.688429	−	0.725304i	$-0.741699\pi$
$191$	− 1.01076e7i	− 1.45061i	0.688429	−	0.725304i	$-0.258301\pi$
$192$	0	0
$193$	−7.64657e6	−1.06364	−0.531820	−	0.846857i	$-0.678492\pi$
$193$	−7.64657e6	−1.06364	−0.531820	+	0.846857i	$0.678492\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.94403e6i	− 0.646669i	−0.946285	−	0.323335i	$-0.895196\pi$
$197$	− 4.94403e6i	− 0.646669i	0.946285	−	0.323335i	$-0.104804\pi$
$198$	0	0
$199$	1.96814e6	0.249745	0.124872	−	0.992173i	$-0.460148\pi$
$199$	1.96814e6	0.249745	0.124872	+	0.992173i	$0.460148\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 4.46069e6i	− 0.533229i
$204$	0	0
$205$	5.24481e6	0.608791
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.37530e7i	− 1.50647i
$210$	0	0
$211$	−8.29966e6	−0.883513	−0.441756	−	0.897135i	$-0.645644\pi$
$211$	−8.29966e6	−0.883513	−0.441756	+	0.897135i	$0.645644\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	3.28363e6i	0.330399i
$216$	0	0
$217$	1.34916e7	1.32034
$218$	0	0
$219$	0	0
$220$	0	0
$221$	268325.i	0.0248590i
$222$	0	0
$223$	−9.37779e6	−0.845641	−0.422821	−	0.906213i	$-0.638960\pi$
$223$	−9.37779e6	−0.845641	−0.422821	+	0.906213i	$0.638960\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.66419e6i	− 0.313257i	−0.987658	−	0.156628i	$-0.949938\pi$
$227$	− 3.66419e6i	− 0.313257i	0.987658	−	0.156628i	$-0.0500625\pi$
$228$	0	0
$229$	4.98904e6	0.415442	0.207721	−	0.978188i	$-0.433395\pi$
$229$	4.98904e6	0.415442	0.207721	+	0.978188i	$0.433395\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.00370e6i	0.395570i	0.980245	+	0.197785i	$0.0633749\pi$
$233$	5.00370e6i	0.395570i	−0.980245	+	0.197785i	$0.936625\pi$
$234$	0	0
$235$	3.66890e6	0.282705
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1.89151e7i	1.38553i	0.721163	+	0.692765i	$0.243608\pi$
$239$	1.89151e7i	1.38553i	−0.721163	+	0.692765i	$0.756392\pi$
$240$	0	0
$241$	−1.52515e7	−1.08958	−0.544791	−	0.838572i	$-0.683391\pi$
$241$	−1.52515e7	−1.08958	−0.544791	+	0.838572i	$0.683391\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.02470e7i	− 1.37677i
$246$	0	0
$247$	−1.92436e6	−0.127702
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 1.44796e7i	− 0.915661i	−0.889040	−	0.457830i	$-0.848627\pi$
$251$	− 1.44796e7i	− 0.915661i	0.889040	−	0.457830i	$-0.151373\pi$
$252$	0	0
$253$	−1.28802e7	−0.795358
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.19468e7i	− 0.703807i	−0.936036	−	0.351904i	$-0.885534\pi$
$257$	− 1.19468e7i	− 0.703807i	0.936036	−	0.351904i	$-0.114466\pi$
$258$	0	0
$259$	1.12564e7	0.647889
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.51400e7i	0.832258i	0.909306	+	0.416129i	$0.136614\pi$
$263$	1.51400e7i	0.832258i	−0.909306	+	0.416129i	$0.863386\pi$
$264$	0	0
$265$	2.94073e7	1.58022
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.37864e7i	1.73574i	0.496789	+	0.867871i	$0.334512\pi$
$269$	3.37864e7i	1.73574i	−0.496789	+	0.867871i	$0.665488\pi$
$270$	0	0
$271$	292052.	0.0146741	0.00733707	−	0.999973i	$-0.497665\pi$
$271$	292052.	0.0146741	0.00733707	+	0.999973i	$0.497665\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.97021e6i	0.190904i
$276$	0	0
$277$	−1.18466e7	−0.557384	−0.278692	−	0.960381i	$-0.589901\pi$
$277$	−1.18466e7	−0.557384	−0.278692	+	0.960381i	$0.589901\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 2.67693e6i	− 0.120648i	−0.998179	−	0.0603238i	$-0.980787\pi$
$281$	− 2.67693e6i	− 0.120648i	0.998179	−	0.0603238i	$-0.0192133\pi$
$282$	0	0
$283$	3.11101e7	1.37260	0.686298	−	0.727321i	$-0.259235\pi$
$283$	3.11101e7	1.37260	0.686298	+	0.727321i	$0.259235\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.96470e7i	0.831094i
$288$	0	0
$289$	2.16598e7	0.897347
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 2.35392e6i	− 0.0935813i	−0.998905	−	0.0467906i	$-0.985101\pi$
$293$	− 2.35392e6i	− 0.0935813i	0.998905	−	0.0467906i	$-0.0148994\pi$
$294$	0	0
$295$	2.24939e7	0.876193
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1.80224e6i	0.0674216i
$300$	0	0
$301$	−1.23004e7	−0.451046
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.52642e7i	− 0.537989i
$306$	0	0
$307$	−3.99249e6	−0.137984	−0.0689920	−	0.997617i	$-0.521978\pi$
$307$	−3.99249e6	−0.137984	−0.0689920	+	0.997617i	$0.521978\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 3.01385e7i	− 1.00194i	−0.865465	−	0.500969i	$-0.832977\pi$
$311$	− 3.01385e7i	− 1.00194i	0.865465	−	0.500969i	$-0.167023\pi$
$312$	0	0
$313$	−3.78911e7	−1.23568	−0.617838	−	0.786305i	$-0.711991\pi$
$313$	−3.78911e7	−1.23568	−0.617838	+	0.786305i	$0.711991\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.47444e7i	1.09070i	0.838207	+	0.545352i	$0.183604\pi$
$317$	3.47444e7i	1.09070i	−0.838207	+	0.545352i	$0.816396\pi$
$318$	0	0
$319$	−1.05568e7	−0.325206
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.77701e7i	− 0.527330i
$324$	0	0
$325$	555522.	0.0161827
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.37437e7i	0.385936i
$330$	0	0
$331$	−3.66213e7	−1.00983	−0.504917	−	0.863168i	$-0.668477\pi$
$331$	−3.66213e7	−1.00983	−0.504917	+	0.863168i	$0.668477\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 5.23614e7i	− 1.39276i
$336$	0	0
$337$	4.53144e7	1.18398	0.591992	−	0.805944i	$-0.298342\pi$
$337$	4.53144e7	1.18398	0.591992	+	0.805944i	$0.298342\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 3.19296e7i	− 0.805248i
$342$	0	0
$343$	1.52829e7	0.378724
$344$	0	0
$345$	0	0
$346$	0	0
$347$	2.92265e7i	0.699502i	0.936843	+	0.349751i	$0.113734\pi$
$347$	2.92265e7i	0.699502i	−0.936843	+	0.349751i	$0.886266\pi$
$348$	0	0
$349$	−9.61270e6	−0.226136	−0.113068	−	0.993587i	$-0.536068\pi$
$349$	−9.61270e6	−0.226136	−0.113068	+	0.993587i	$0.536068\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.01291e7i	0.684955i	0.939526	+	0.342478i	$0.111266\pi$
$353$	3.01291e7i	0.684955i	−0.939526	+	0.342478i	$0.888734\pi$
$354$	0	0
$355$	−9.27643e7	−2.07346
$356$	0	0
$357$	0	0
$358$	0	0
$359$	3.63910e7i	0.786521i	0.919427	+	0.393260i	$0.128653\pi$
$359$	3.63910e7i	0.786521i	−0.919427	+	0.393260i	$0.871347\pi$
$360$	0	0
$361$	8.03970e7	1.70891
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.04842e8i	2.15604i
$366$	0	0
$367$	−5.05958e7	−1.02357	−0.511784	−	0.859114i	$-0.671015\pi$
$367$	−5.05958e7	−1.02357	−0.511784	+	0.859114i	$0.671015\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.10159e8i	2.15724i
$372$	0	0
$373$	−9.44996e7	−1.82097	−0.910487	−	0.413538i	$-0.864293\pi$
$373$	−9.44996e7	−1.82097	−0.910487	+	0.413538i	$0.864293\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1.47713e6i	0.0275673i
$378$	0	0
$379$	5.62274e7	1.03283	0.516417	−	0.856337i	$-0.327266\pi$
$379$	5.62274e7	1.03283	0.516417	+	0.856337i	$0.327266\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 2.12413e7i	− 0.378080i	−0.981969	−	0.189040i	$-0.939462\pi$
$383$	− 2.12413e7i	− 0.378080i	0.981969	−	0.189040i	$-0.0605376\pi$
$384$	0	0
$385$	−8.61785e7	−1.51014
$386$	0	0
$387$	0	0
$388$	0	0
$389$	9.62174e7i	1.63458i	0.576230	+	0.817288i	$0.304523\pi$
$389$	9.62174e7i	1.63458i	−0.576230	+	0.817288i	$0.695477\pi$
$390$	0	0
$391$	−1.66424e7	−0.278410
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.08679e7i	1.14990i
$396$	0	0
$397$	2.71686e7	0.434206	0.217103	−	0.976149i	$-0.430339\pi$
$397$	2.71686e7	0.434206	0.217103	+	0.976149i	$0.430339\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 1.53926e7i	− 0.238715i	−0.992851	−	0.119357i	$-0.961917\pi$
$401$	− 1.53926e7i	− 0.238715i	0.992851	−	0.119357i	$-0.0380834\pi$
$402$	0	0
$403$	−4.46767e6	−0.0682599
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 2.66397e7i	− 0.395135i
$408$	0	0
$409$	−3.26408e7	−0.477080	−0.238540	−	0.971133i	$-0.576669\pi$
$409$	−3.26408e7	−0.477080	−0.238540	+	0.971133i	$0.576669\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.42620e7i	1.19614i
$414$	0	0
$415$	1.24195e8	1.73765
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8.86249e7i	1.20480i	0.798196	+	0.602398i	$0.205788\pi$
$419$	8.86249e7i	1.20480i	−0.798196	+	0.602398i	$0.794212\pi$
$420$	0	0
$421$	9.51173e7	1.27472	0.637358	−	0.770568i	$-0.280027\pi$
$421$	9.51173e7	1.27472	0.637358	+	0.770568i	$0.280027\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	5.12985e6i	0.0668249i
$426$	0	0
$427$	5.71793e7	0.734439
$428$	0	0
$429$	0	0
$430$	0	0
$431$	8.05617e7i	1.00623i	0.864220	+	0.503115i	$0.167813\pi$
$431$	8.05617e7i	1.00623i	−0.864220	+	0.503115i	$0.832187\pi$
$432$	0	0
$433$	3.50511e7	0.431755	0.215878	−	0.976420i	$-0.430739\pi$
$433$	3.50511e7	0.431755	0.215878	+	0.976420i	$0.430739\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 1.19355e8i	− 1.43020i
$438$	0	0
$439$	4.63062e7	0.547325	0.273663	−	0.961826i	$-0.411765\pi$
$439$	4.63062e7	0.547325	0.273663	+	0.961826i	$0.411765\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.19041e8i	1.36926i	0.728893	+	0.684628i	$0.240035\pi$
$443$	1.19041e8i	1.36926i	−0.728893	+	0.684628i	$0.759965\pi$
$444$	0	0
$445$	−8.85359e7	−1.00471
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 9.75546e7i	− 1.07773i	−0.842393	−	0.538864i	$-0.818854\pi$
$449$	− 9.75546e7i	− 1.07773i	0.842393	−	0.538864i	$-0.181146\pi$
$450$	0	0
$451$	4.64970e7	0.506868
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1.20583e7i	0.128013i
$456$	0	0
$457$	−8.13740e6	−0.0852583	−0.0426292	−	0.999091i	$-0.513573\pi$
$457$	−8.13740e6	−0.0852583	−0.0426292	+	0.999091i	$0.513573\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.55318e8i	1.58533i	0.609659	+	0.792664i	$0.291306\pi$
$461$	1.55318e8i	1.58533i	−0.609659	+	0.792664i	$0.708694\pi$
$462$	0	0
$463$	−7.62984e7	−0.768728	−0.384364	−	0.923182i	$-0.625579\pi$
$463$	−7.62984e7	−0.768728	−0.384364	+	0.923182i	$0.625579\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 1.85842e8i	− 1.82470i	−0.409408	−	0.912352i	$-0.634265\pi$
$467$	− 1.85842e8i	− 1.82470i	0.409408	−	0.912352i	$-0.365735\pi$
$468$	0	0
$469$	1.96145e8	1.90133
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.91105e7i	0.275084i
$474$	0	0
$475$	−3.67901e7	−0.343281
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.23819e8i	1.12663i	0.826243	+	0.563314i	$0.190474\pi$
$479$	1.23819e8i	1.12663i	−0.826243	+	0.563314i	$0.809526\pi$
$480$	0	0
$481$	−3.72749e6	−0.0334951
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8.47145e7i	0.742561i
$486$	0	0
$487$	6.09360e7	0.527578	0.263789	−	0.964580i	$-0.415028\pi$
$487$	6.09360e7	0.527578	0.263789	+	0.964580i	$0.415028\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 6.48068e7i	− 0.547490i	−0.961802	−	0.273745i	$-0.911738\pi$
$491$	− 6.48068e7i	− 0.547490i	0.961802	−	0.273745i	$-0.0882624\pi$
$492$	0	0
$493$	−1.36402e7	−0.113836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.47494e8i	− 2.83060i
$498$	0	0
$499$	−2.87096e7	−0.231061	−0.115530	−	0.993304i	$-0.536857\pi$
$499$	−2.87096e7	−0.231061	−0.115530	+	0.993304i	$0.536857\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.12463e8i	0.883702i	0.897088	+	0.441851i	$0.145678\pi$
$503$	1.12463e8i	0.883702i	−0.897088	+	0.441851i	$0.854322\pi$
$504$	0	0
$505$	−8.56156e7	−0.664781
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.61786e7i	0.426008i	0.977051	+	0.213004i	$0.0683247\pi$
$509$	5.61786e7i	0.426008i	−0.977051	+	0.213004i	$0.931675\pi$
$510$	0	0
$511$	−3.92737e8	−2.94333
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 1.03160e8i	− 0.755245i
$516$	0	0
$517$	3.25260e7	0.235375
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.68861e8i	1.19403i	0.802230	+	0.597015i	$0.203647\pi$
$521$	1.68861e8i	1.19403i	−0.802230	+	0.597015i	$0.796353\pi$
$522$	0	0
$523$	2.39748e8	1.67591	0.837955	−	0.545739i	$-0.183751\pi$
$523$	2.39748e8	1.67591	0.837955	+	0.545739i	$0.183751\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.12557e7i	− 0.281872i
$528$	0	0
$529$	3.62552e7	0.244908
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.50598e6i	− 0.0429666i
$534$	0	0
$535$	4.74683e7	0.309986
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 1.79496e8i	− 1.14627i
$540$	0	0
$541$	1.85322e8	1.17040	0.585200	−	0.810889i	$-0.301016\pi$
$541$	1.85322e8	1.17040	0.585200	+	0.810889i	$0.301016\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.61066e8i	− 0.994978i
$546$	0	0
$547$	−2.52846e8	−1.54488	−0.772438	−	0.635090i	$-0.780963\pi$
$547$	−2.52846e8	−1.54488	−0.772438	+	0.635090i	$0.780963\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 9.78245e7i	− 0.584780i
$552$	0	0
$553$	−2.65470e8	−1.56979
$554$	0	0
$555$	0	0
$556$	0	0
$557$	1.01472e8i	0.587194i	0.955929	+	0.293597i	$0.0948524\pi$
$557$	1.01472e8i	0.587194i	−0.955929	+	0.293597i	$0.905148\pi$
$558$	0	0
$559$	4.07322e6	0.0233186
$560$	0	0
$561$	0	0
$562$	0	0
$563$	1.15852e8i	0.649202i	0.945851	+	0.324601i	$0.105230\pi$
$563$	1.15852e8i	0.649202i	−0.945851	+	0.324601i	$0.894770\pi$
$564$	0	0
$565$	−3.37130e8	−1.86919
$566$	0	0
$567$	0	0
$568$	0	0
$569$	7.30159e7i	0.396352i	0.980166	+	0.198176i	$0.0635017\pi$
$569$	7.30159e7i	0.396352i	−0.980166	+	0.198176i	$0.936498\pi$
$570$	0	0
$571$	9.75143e7	0.523793	0.261897	−	0.965096i	$-0.415652\pi$
$571$	9.75143e7	0.523793	0.261897	+	0.965096i	$0.415652\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.44553e7i	0.181239i
$576$	0	0
$577$	8.90080e7	0.463342	0.231671	−	0.972794i	$-0.425581\pi$
$577$	8.90080e7	0.463342	0.231671	+	0.972794i	$0.425581\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	4.65234e8i	2.37216i
$582$	0	0
$583$	2.60705e8	1.31566
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.26314e8i	− 0.624509i	−0.949998	−	0.312255i	$-0.898916\pi$
$587$	− 1.26314e8i	− 0.624509i	0.949998	−	0.312255i	$-0.101084\pi$
$588$	0	0
$589$	2.95876e8	1.44798
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.59275e8i	− 0.763805i	−0.924203	−	0.381902i	$-0.875269\pi$
$593$	− 1.59275e8i	− 0.763805i	0.924203	−	0.381902i	$-0.124731\pi$
$594$	0	0
$595$	−1.11350e8	−0.528615
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.42561e8i	0.663315i	0.943400	+	0.331658i	$0.107608\pi$
$599$	1.42561e8i	0.663315i	−0.943400	+	0.331658i	$0.892392\pi$
$600$	0	0
$601$	−1.18489e8	−0.545828	−0.272914	−	0.962038i	$-0.587987\pi$
$601$	−1.18489e8	−0.545828	−0.272914	+	0.962038i	$0.587987\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 3.94939e7i	− 0.178346i
$606$	0	0
$607$	8.06041e7	0.360405	0.180203	−	0.983630i	$-0.442325\pi$
$607$	8.06041e7	0.360405	0.180203	+	0.983630i	$0.442325\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 4.55113e6i	− 0.0199524i
$612$	0	0
$613$	−2.49459e8	−1.08297	−0.541486	−	0.840710i	$-0.682138\pi$
$613$	−2.49459e8	−1.08297	−0.541486	+	0.840710i	$0.682138\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.44285e8i	− 1.46576i	−0.680358	−	0.732880i	$-0.738176\pi$
$617$	− 3.44285e8i	− 1.46576i	0.680358	−	0.732880i	$-0.261824\pi$
$618$	0	0
$619$	−4.59439e7	−0.193712	−0.0968558	−	0.995298i	$-0.530879\pi$
$619$	−4.59439e7	−0.193712	−0.0968558	+	0.995298i	$0.530879\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 3.31654e8i	− 1.37158i
$624$	0	0
$625$	−2.84441e8	−1.16507
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.44207e7i	− 0.138315i
$630$	0	0
$631$	2.71320e8	1.07993	0.539963	−	0.841689i	$-0.318438\pi$
$631$	2.71320e8	1.07993	0.539963	+	0.841689i	$0.318438\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 2.11372e8i	− 0.825516i
$636$	0	0
$637$	−2.51156e7	−0.0971684
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 3.13515e8i	− 1.19037i	−0.803587	−	0.595187i	$-0.797078\pi$
$641$	− 3.13515e8i	− 1.19037i	0.803587	−	0.595187i	$-0.202922\pi$
$642$	0	0
$643$	−5.94578e7	−0.223654	−0.111827	−	0.993728i	$-0.535670\pi$
$643$	−5.94578e7	−0.223654	−0.111827	+	0.993728i	$0.535670\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.04779e8i	− 0.756089i	−0.925787	−	0.378045i	$-0.876597\pi$
$647$	− 2.04779e8i	− 0.756089i	0.925787	−	0.378045i	$-0.123403\pi$
$648$	0	0
$649$	1.99416e8	0.729502
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3.40035e7i	− 0.122119i	−0.998134	−	0.0610596i	$-0.980552\pi$
$653$	− 3.40035e7i	− 0.122119i	0.998134	−	0.0610596i	$-0.0194480\pi$
$654$	0	0
$655$	−1.54902e8	−0.551232
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 2.43145e8i	− 0.849590i	−0.905290	−	0.424795i	$-0.860346\pi$
$659$	− 2.43145e8i	− 0.849590i	0.905290	−	0.424795i	$-0.139654\pi$
$660$	0	0
$661$	4.51973e8	1.56498	0.782488	−	0.622665i	$-0.213950\pi$
$661$	4.51973e8	1.56498	0.782488	+	0.622665i	$0.213950\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 7.98575e8i	− 2.71551i
$666$	0	0
$667$	−9.16164e7	−0.308742
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.35322e8i	− 0.447920i
$672$	0	0
$673$	3.17560e8	1.04179	0.520896	−	0.853620i	$-0.325598\pi$
$673$	3.17560e8	1.04179	0.520896	+	0.853620i	$0.325598\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.70541e7i	− 0.216102i	−0.994145	−	0.108051i	$-0.965539\pi$
$677$	− 6.70541e7i	− 0.216102i	0.994145	−	0.108051i	$-0.0344610\pi$
$678$	0	0
$679$	−3.17339e8	−1.01371
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.51437e8i	0.475303i	0.971350	+	0.237652i	$0.0763777\pi$
$683$	1.51437e8i	0.475303i	−0.971350	+	0.237652i	$0.923622\pi$
$684$	0	0
$685$	4.40563e8	1.37068
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 3.64786e7i	− 0.111527i
$690$	0	0
$691$	−9.46804e7	−0.286963	−0.143482	−	0.989653i	$-0.545830\pi$
$691$	−9.46804e7	−0.286963	−0.143482	+	0.989653i	$0.545830\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 9.16384e6i	− 0.0272975i
$696$	0	0
$697$	6.00781e7	0.177426
$698$	0	0
$699$	0	0
$700$	0	0
$701$	7.92713e6i	0.0230124i	0.999934	+	0.0115062i	$0.00366262\pi$
$701$	7.92713e6i	0.0230124i	−0.999934	+	0.0115062i	$0.996337\pi$
$702$	0	0
$703$	2.46857e8	0.710526
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 3.20715e8i	− 0.907530i
$708$	0	0
$709$	−6.65474e8	−1.86721	−0.933603	−	0.358310i	$-0.883353\pi$
$709$	−6.65474e8	−1.86721	−0.933603	+	0.358310i	$0.883353\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.77099e8i	− 0.764481i
$714$	0	0
$715$	2.85375e7	0.0780724
$716$	0	0
$717$	0	0
$718$	0	0
$719$	4.90198e8i	1.31882i	0.751784	+	0.659409i	$0.229194\pi$
$719$	4.90198e8i	1.31882i	−0.751784	+	0.659409i	$0.770806\pi$
$720$	0	0
$721$	3.86434e8	1.03103
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.82398e7i	0.0741051i
$726$	0	0
$727$	2.09307e8	0.544730	0.272365	−	0.962194i	$-0.412194\pi$
$727$	2.09307e8	0.544730	0.272365	+	0.962194i	$0.412194\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	3.76133e7i	0.0962917i
$732$	0	0
$733$	−3.66672e8	−0.931034	−0.465517	−	0.885039i	$-0.654132\pi$
$733$	−3.66672e8	−0.931034	−0.465517	+	0.885039i	$0.654132\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 4.64201e8i	− 1.15959i
$738$	0	0
$739$	6.11387e8	1.51490	0.757449	−	0.652895i	$-0.226446\pi$
$739$	6.11387e8	1.51490	0.757449	+	0.652895i	$0.226446\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.10111e8i	− 0.512252i	−0.966643	−	0.256126i	$-0.917554\pi$
$743$	− 2.10111e8i	− 0.512252i	0.966643	−	0.256126i	$-0.0824461\pi$
$744$	0	0
$745$	2.23781e8	0.541196
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.77816e8i	0.423180i
$750$	0	0
$751$	−5.05697e8	−1.19391	−0.596953	−	0.802276i	$-0.703622\pi$
$751$	−5.05697e8	−1.19391	−0.596953	+	0.802276i	$0.703622\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 5.02430e7i	− 0.116744i
$756$	0	0
$757$	4.19839e8	0.967822	0.483911	−	0.875117i	$-0.339216\pi$
$757$	4.19839e8	0.967822	0.483911	+	0.875117i	$0.339216\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 7.32156e8i	− 1.66131i	−0.556791	−	0.830653i	$-0.687968\pi$
$761$	− 7.32156e8i	− 1.66131i	0.556791	−	0.830653i	$-0.312032\pi$
$762$	0	0
$763$	6.03350e8	1.35830
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 2.79029e7i	− 0.0618390i
$768$	0	0
$769$	−2.53294e8	−0.556988	−0.278494	−	0.960438i	$-0.589835\pi$
$769$	−2.53294e8	−0.556988	−0.278494	+	0.960438i	$0.589835\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	7.82197e8i	1.69347i	0.532014	+	0.846735i	$0.321435\pi$
$773$	7.82197e8i	1.69347i	−0.532014	+	0.846735i	$0.678565\pi$
$774$	0	0
$775$	−8.54131e7	−0.183493
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.30865e8i	0.911443i
$780$	0	0
$781$	−8.22386e8	−1.72632
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.13101e8i	1.47415i
$786$	0	0
$787$	−6.81837e8	−1.39880	−0.699401	−	0.714730i	$-0.746550\pi$
$787$	−6.81837e8	−1.39880	−0.699401	+	0.714730i	$0.746550\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 1.26289e9i	− 2.55173i
$792$	0	0
$793$	−1.89346e7	−0.0379696
$794$	0	0
$795$	0	0
$796$	0	0
$797$	5.08452e7i	0.100433i	0.998738	+	0.0502163i	$0.0159911\pi$
$797$	5.08452e7i	0.100433i	−0.998738	+	0.0502163i	$0.984009\pi$
$798$	0	0
$799$	4.20264e7	0.0823915
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.29460e8i	1.79508i
$804$	0	0
$805$	−7.47896e8	−1.43369
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.02851e7i	0.0194251i	0.999953	+	0.00971257i	$0.00309166\pi$
$809$	1.02851e7i	0.0194251i	−0.999953	+	0.00971257i	$0.996908\pi$
$810$	0	0
$811$	−4.84952e8	−0.909152	−0.454576	−	0.890708i	$-0.650209\pi$
$811$	−4.84952e8	−0.909152	−0.454576	+	0.890708i	$0.650209\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 4.13254e8i	− 0.763385i
$816$	0	0
$817$	−2.69753e8	−0.494653
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.96161e8i	0.896588i	0.893886	+	0.448294i	$0.147968\pi$
$821$	4.96161e8i	0.896588i	−0.893886	+	0.448294i	$0.852032\pi$
$822$	0	0
$823$	3.21483e8	0.576711	0.288356	−	0.957523i	$-0.406891\pi$
$823$	3.21483e8	0.576711	0.288356	+	0.957523i	$0.406891\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	7.54902e8i	1.33467i	0.744757	+	0.667336i	$0.232565\pi$
$827$	7.54902e8i	1.33467i	−0.744757	+	0.667336i	$0.767435\pi$
$828$	0	0
$829$	−5.82262e8	−1.02201	−0.511004	−	0.859578i	$-0.670726\pi$
$829$	−5.82262e8	−1.02201	−0.511004	+	0.859578i	$0.670726\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 2.31924e8i	− 0.401247i
$834$	0	0
$835$	−6.57385e8	−1.12917
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.64141e8i	1.12454i	0.826954	+	0.562270i	$0.190072\pi$
$839$	6.64141e8i	1.12454i	−0.826954	+	0.562270i	$0.809928\pi$
$840$	0	0
$841$	5.19734e8	0.873762
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.59301e8i	1.09273i
$846$	0	0
$847$	1.47943e8	0.243470
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 2.31191e8i	− 0.375131i
$852$	0	0
$853$	7.87779e8	1.26928	0.634640	−	0.772808i	$-0.281148\pi$
$853$	7.87779e8	1.26928	0.634640	+	0.772808i	$0.281148\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 9.65175e8i	− 1.53343i	−0.641988	−	0.766715i	$-0.721890\pi$
$857$	− 9.65175e8i	− 1.53343i	0.641988	−	0.766715i	$-0.278110\pi$
$858$	0	0
$859$	5.95457e8	0.939444	0.469722	−	0.882814i	$-0.344354\pi$
$859$	5.95457e8	0.939444	0.469722	+	0.882814i	$0.344354\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 9.85551e8i	− 1.53337i	−0.642024	−	0.766684i	$-0.721905\pi$
$863$	− 9.85551e8i	− 1.53337i	0.642024	−	0.766684i	$-0.278095\pi$
$864$	0	0
$865$	−4.85907e8	−0.750766
$866$	0	0
$867$	0	0
$868$	0	0
$869$	6.28267e8i	0.957382i
$870$	0	0
$871$	−6.49522e7	−0.0982969
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 8.74763e8i	− 1.30577i
$876$	0	0
$877$	3.68680e8	0.546577	0.273288	−	0.961932i	$-0.411889\pi$
$877$	3.68680e8	0.546577	0.273288	+	0.961932i	$0.411889\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	7.29792e8i	1.06726i	0.845717	+	0.533632i	$0.179173\pi$
$881$	7.29792e8i	1.06726i	−0.845717	+	0.533632i	$0.820827\pi$
$882$	0	0
$883$	7.12900e7	0.103549	0.0517746	−	0.998659i	$-0.483512\pi$
$883$	7.12900e7	0.103549	0.0517746	+	0.998659i	$0.483512\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.15305e8i	0.451814i	0.974149	+	0.225907i	$0.0725346\pi$
$887$	3.15305e8i	0.451814i	−0.974149	+	0.225907i	$0.927465\pi$
$888$	0	0
$889$	7.91795e8	1.12696
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.01403e8i	0.423247i
$894$	0	0
$895$	−1.17808e9	−1.64326
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 2.27113e8i	− 0.312581i
$900$	0	0
$901$	3.36853e8	0.460539
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 2.46266e8i	− 0.332245i
$906$	0	0
$907$	5.70419e8	0.764491	0.382245	−	0.924061i	$-0.375151\pi$
$907$	5.70419e8	0.764491	0.382245	+	0.924061i	$0.375151\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	3.43557e8i	0.454406i	0.973847	+	0.227203i	$0.0729581\pi$
$911$	3.43557e8i	0.454406i	−0.973847	+	0.227203i	$0.927042\pi$
$912$	0	0
$913$	1.10103e9	1.44673
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 5.80262e8i	− 0.752517i
$918$	0	0
$919$	−7.92851e8	−1.02152	−0.510758	−	0.859725i	$-0.670635\pi$
$919$	−7.92851e8	−1.02152	−0.510758	+	0.859725i	$0.670635\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.15070e8i	0.146339i
$924$	0	0
$925$	−7.12624e7	−0.0900400
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.82427e8i	− 0.227531i	−0.993508	−	0.113766i	$-0.963709\pi$
$929$	− 1.82427e8i	− 0.227531i	0.993508	−	0.113766i	$-0.0362913\pi$
$930$	0	0
$931$	1.66331e9	2.06121
$932$	0	0
$933$	0	0
$934$	0	0
$935$	2.63523e8i	0.322392i
$936$	0	0
$937$	−3.34488e8	−0.406594	−0.203297	−	0.979117i	$-0.565166\pi$
$937$	−3.34488e8	−0.406594	−0.203297	+	0.979117i	$0.565166\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	2.91160e8i	0.349433i	0.984619	+	0.174716i	$0.0559008\pi$
$941$	2.91160e8i	0.349433i	−0.984619	+	0.174716i	$0.944099\pi$
$942$	0	0
$943$	4.03522e8	0.481207
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.69634e8i	0.199739i	0.995001	+	0.0998694i	$0.0318425\pi$
$947$	1.69634e8i	0.199739i	−0.995001	+	0.0998694i	$0.968158\pi$
$948$	0	0
$949$	1.30053e8	0.152167
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 5.17958e8i	− 0.598434i	−0.954185	−	0.299217i	$-0.903275\pi$
$953$	− 5.17958e8i	− 0.598434i	0.954185	−	0.299217i	$-0.0967254\pi$
$954$	0	0
$955$	−1.38898e9	−1.59473
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.65034e9i	1.87119i
$960$	0	0
$961$	−2.00587e8	−0.226013
$962$	0	0
$963$	0	0
$964$	0	0
$965$	1.05078e9i	1.16931i
$966$	0	0
$967$	1.37422e9	1.51976	0.759881	−	0.650062i	$-0.225257\pi$
$967$	1.37422e9	1.51976	0.759881	+	0.650062i	$0.225257\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.45510e8i	0.814321i	0.913357	+	0.407160i	$0.133481\pi$
$971$	7.45510e8i	0.814321i	−0.913357	+	0.407160i	$0.866519\pi$
$972$	0	0
$973$	3.43276e7	0.0372653
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.80224e8i	1.05109i	0.850764	+	0.525547i	$0.176139\pi$
$977$	9.80224e8i	1.05109i	−0.850764	+	0.525547i	$0.823861\pi$
$978$	0	0
$979$	−7.84900e8	−0.836501
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.01328e9i	1.06676i	0.845875	+	0.533381i	$0.179079\pi$
$983$	1.01328e9i	1.06676i	−0.845875	+	0.533381i	$0.820921\pi$
$984$	0	0
$985$	−6.79402e8	−0.710916
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.52634e8i	0.261158i
$990$	0	0
$991$	1.85231e9	1.90324	0.951620	−	0.307279i	$-0.0994183\pi$
$991$	1.85231e9	1.90324	0.951620	+	0.307279i	$0.0994183\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 2.70459e8i	− 0.274557i
$996$	0	0
$997$	1.25665e9	1.26803	0.634013	−	0.773322i	$-0.281407\pi$
$997$	1.25665e9	1.26803	0.634013	+	0.773322i	$0.281407\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.7.e.j.161.1		4
3.2	odd	2	inner	432.7.e.j.161.4		4
4.3	odd	2		27.7.b.c.26.4	yes	4
12.11	even	2		27.7.b.c.26.1	✓	4
36.7	odd	6		81.7.d.e.53.4		8
36.11	even	6		81.7.d.e.53.1		8
36.23	even	6		81.7.d.e.26.4		8
36.31	odd	6		81.7.d.e.26.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.7.b.c.26.1	✓	4	12.11	even	2
27.7.b.c.26.4	yes	4	4.3	odd	2
81.7.d.e.26.1		8	36.31	odd	6
81.7.d.e.26.4		8	36.23	even	6
81.7.d.e.53.1		8	36.11	even	6
81.7.d.e.53.4		8	36.7	odd	6
432.7.e.j.161.1		4	1.1	even	1	trivial
432.7.e.j.161.4		4	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	432.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-137.419i q^{5} +514.769 q^{7} -1218.26i q^{11} -170.463 q^{13} -1574.10i q^{17} +11289.1 q^{19} -10572.6i q^{23} -3258.91 q^{25} -8665.42i q^{29} +26209.1 q^{31} -70738.9i q^{35} +21866.9 q^{37} +38166.6i q^{41} -23895.1 q^{43} +26698.7i q^{47} +147338. q^{49} +213997. i q^{53} -167412. q^{55} +163689. i q^{59} +111078. q^{61} +23424.8i q^{65} +381035. q^{67} -675048. i q^{71} -762939. q^{73} -627123. i q^{77} -515708. q^{79} +903773. i q^{83} -216311. q^{85} -644278. i q^{89} -87748.8 q^{91} -1.55133e6i q^{95} -616470. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 676 q^{7} - 3448 q^{13} + 3664 q^{19} + 17392 q^{25} + 153244 q^{31} + 128960 q^{37} - 126008 q^{43} + 121872 q^{49} - 54180 q^{55} + 167696 q^{61} - 8 q^{67} - 1881676 q^{73} - 1188728 q^{79} - 176472 q^{85}+ \cdots + 975212 q^{97}+O(q^{100}) \) 4 * q + 676 * q^7 - 3448 * q^13 + 3664 * q^19 + 17392 * q^25 + 153244 * q^31 + 128960 * q^37 - 126008 * q^43 + 121872 * q^49 - 54180 * q^55 + 167696 * q^61 - 8 * q^67 - 1881676 * q^73 - 1188728 * q^79 - 176472 * q^85 + 373736 * q^91 + 975212 * q^97

Introduction

L-functions

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