LMFDB - Embedded newform 432.6.c.g.431.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,6,Mod(431,432)]

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([1, 0, 1]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("432.431");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	432.c (of order $2$, degree $1$, 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:	$69.2858101592$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2 x^{11} + 43 x^{10} - 802 x^{9} + 2077 x^{8} - 26672 x^{7} + 276788 x^{6} - 792632 x^{5} + \cdots + 2123735056 $ x^12 - 2x^11 + 43x^10 - 802x^9 + 2077x^8 - 26672x^7 + 276788x^6 - 792632x^5 + 6657772x^4 - 44055712x^3 + 101881648x^2 - 498075872*x + 2123735056
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{34}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.3
Root			$6.16887 - 1.14932i$ of defining polynomial
Character	$\chi$	$=$	432.431
Dual form			432.6.c.g.431.10

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-33.9573i q^{5} -80.4860i q^{7} +O(q^{10})$	$q-33.9573i q^{5} -80.4860i q^{7} +305.470 q^{11} +443.021 q^{13} +380.175i q^{17} +434.999i q^{19} +1197.12 q^{23} +1971.90 q^{25} +1443.33i q^{29} +4928.03i q^{31} -2733.08 q^{35} -2083.88 q^{37} -14431.5i q^{41} -6007.51i q^{43} +7559.53 q^{47} +10329.0 q^{49} -9326.25i q^{53} -10372.9i q^{55} -6227.61 q^{59} -26383.6 q^{61} -15043.8i q^{65} -7382.58i q^{67} +23598.0 q^{71} -27479.9 q^{73} -24586.0i q^{77} +38992.7i q^{79} +119533. q^{83} +12909.7 q^{85} -127357. i q^{89} -35657.0i q^{91} +14771.4 q^{95} +29343.3 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q+O(q^{10}) $ 12 * q	$ 12 q - 2040 q^{13} - 7824 q^{25} + 18240 q^{37} - 69936 q^{49} + 105936 q^{61} + 200172 q^{73} - 126072 q^{85} - 282300 q^{97}+O(q^{100}) $ 12 * q - 2040 * q^13 - 7824 * q^25 + 18240 * q^37 - 69936 * q^49 + 105936 * q^61 + 200172 * q^73 - 126072 * q^85 - 282300 * q^97

Display 100 coefficients 10 coefficients

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
$5$	− 33.9573i	− 0.607446i	−0.952760	−	0.303723i	$-0.901770\pi$
$5$	− 33.9573i	− 0.607446i	0.952760	−	0.303723i	$-0.0982298\pi$
$6$	0	0
$7$	− 80.4860i	− 0.620833i	−0.950601	−	0.310417i	$-0.899531\pi$
$7$	− 80.4860i	− 0.620833i	0.950601	−	0.310417i	$-0.100469\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	305.470	0.761179	0.380589	−	0.924744i	$-0.375721\pi$
$11$	305.470	0.761179	0.380589	+	0.924744i	$0.375721\pi$
$12$	0	0
$13$	443.021	0.727053	0.363527	−	0.931584i	$-0.381573\pi$
$13$	443.021	0.727053	0.363527	+	0.931584i	$0.381573\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	380.175i	0.319052i	0.987194	+	0.159526i	$0.0509965\pi$
$17$	380.175i	0.319052i	−0.987194	+	0.159526i	$0.949003\pi$
$18$	0	0
$19$	434.999i	0.276442i	0.990401	+	0.138221i	$0.0441385\pi$
$19$	434.999i	0.276442i	−0.990401	+	0.138221i	$0.955862\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1197.12	0.471865	0.235933	−	0.971769i	$-0.424186\pi$
$23$	1197.12	0.471865	0.235933	+	0.971769i	$0.424186\pi$
$24$	0	0
$25$	1971.90	0.631009
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1443.33i	0.318691i	0.987223	+	0.159346i	$0.0509384\pi$
$29$	1443.33i	0.318691i	−0.987223	+	0.159346i	$0.949062\pi$
$30$	0	0
$31$	4928.03i	0.921020i	0.887655	+	0.460510i	$0.152333\pi$
$31$	4928.03i	0.921020i	−0.887655	+	0.460510i	$0.847667\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2733.08	−0.377123
$36$	0	0
$37$	−2083.88	−0.250247	−0.125123	−	0.992141i	$-0.539933\pi$
$37$	−2083.88	−0.250247	−0.125123	+	0.992141i	$0.539933\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 14431.5i	− 1.34076i	−0.742018	−	0.670380i	$-0.766131\pi$
$41$	− 14431.5i	− 1.34076i	0.742018	−	0.670380i	$-0.233869\pi$
$42$	0	0
$43$	− 6007.51i	− 0.495477i	−0.968827	−	0.247738i	$-0.920313\pi$
$43$	− 6007.51i	− 0.495477i	0.968827	−	0.247738i	$-0.0796873\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7559.53	0.499172	0.249586	−	0.968353i	$-0.419706\pi$
$47$	7559.53	0.499172	0.249586	+	0.968353i	$0.419706\pi$
$48$	0	0
$49$	10329.0	0.614566
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9326.25i	− 0.456055i	−0.973655	−	0.228028i	$-0.926772\pi$
$53$	− 9326.25i	− 0.456055i	0.973655	−	0.228028i	$-0.0732276\pi$
$54$	0	0
$55$	− 10372.9i	− 0.462375i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−6227.61	−0.232912	−0.116456	−	0.993196i	$-0.537153\pi$
$59$	−6227.61	−0.232912	−0.116456	+	0.993196i	$0.537153\pi$
$60$	0	0
$61$	−26383.6	−0.907840	−0.453920	−	0.891042i	$-0.649975\pi$
$61$	−26383.6	−0.907840	−0.453920	+	0.891042i	$0.649975\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 15043.8i	− 0.441646i
$66$	0	0
$67$	− 7382.58i	− 0.200919i	−0.994941	−	0.100459i	$-0.967969\pi$
$67$	− 7382.58i	− 0.200919i	0.994941	−	0.100459i	$-0.0320313\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	23598.0	0.555559	0.277779	−	0.960645i	$-0.410402\pi$
$71$	23598.0	0.555559	0.277779	+	0.960645i	$0.410402\pi$
$72$	0	0
$73$	−27479.9	−0.603542	−0.301771	−	0.953380i	$-0.597578\pi$
$73$	−27479.9	−0.603542	−0.301771	+	0.953380i	$0.597578\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 24586.0i	− 0.472565i
$78$	0	0
$79$	38992.7i	0.702935i	0.936200	+	0.351468i	$0.114317\pi$
$79$	38992.7i	0.702935i	−0.936200	+	0.351468i	$0.885683\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	119533.	1.90456	0.952278	−	0.305231i	$-0.0987336\pi$
$83$	119533.	1.90456	0.952278	+	0.305231i	$0.0987336\pi$
$84$	0	0
$85$	12909.7	0.193807
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 127357.i	− 1.70431i	−0.523293	−	0.852153i	$-0.675297\pi$
$89$	− 127357.i	− 1.70431i	0.523293	−	0.852153i	$-0.324703\pi$
$90$	0	0
$91$	− 35657.0i	− 0.451379i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	14771.4	0.167924
$96$	0	0
$97$	29343.3	0.316650	0.158325	−	0.987387i	$-0.449391\pi$
$97$	29343.3	0.316650	0.158325	+	0.987387i	$0.449391\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 96038.9i	− 0.936793i	−0.883518	−	0.468396i	$-0.844832\pi$
$101$	− 96038.9i	− 0.936793i	0.883518	−	0.468396i	$-0.155168\pi$
$102$	0	0
$103$	36810.1i	0.341880i	0.985281	+	0.170940i	$0.0546805\pi$
$103$	36810.1i	0.341880i	−0.985281	+	0.170940i	$0.945320\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	48551.6	0.409963	0.204981	−	0.978766i	$-0.434287\pi$
$107$	48551.6	0.409963	0.204981	+	0.978766i	$0.434287\pi$
$108$	0	0
$109$	128765.	1.03808	0.519040	−	0.854750i	$-0.326290\pi$
$109$	128765.	1.03808	0.519040	+	0.854750i	$0.326290\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 186175.i	− 1.37159i	−0.727793	−	0.685797i	$-0.759454\pi$
$113$	− 186175.i	− 1.37159i	0.727793	−	0.685797i	$-0.240546\pi$
$114$	0	0
$115$	− 40650.9i	− 0.286633i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	30598.7	0.198078
$120$	0	0
$121$	−67739.2	−0.420607
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 173077.i	− 0.990750i
$126$	0	0
$127$	47781.8i	0.262878i	0.991324	+	0.131439i	$0.0419597\pi$
$127$	47781.8i	0.262878i	−0.991324	+	0.131439i	$0.958040\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	113086.	0.575743	0.287872	−	0.957669i	$-0.407052\pi$
$131$	113086.	0.575743	0.287872	+	0.957669i	$0.407052\pi$
$132$	0	0
$133$	35011.3	0.171625
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 222936.i	− 1.01480i	−0.861712	−	0.507398i	$-0.830607\pi$
$137$	− 222936.i	− 1.01480i	0.861712	−	0.507398i	$-0.169393\pi$
$138$	0	0
$139$	9243.56i	0.0405791i	0.999794	+	0.0202896i	$0.00645881\pi$
$139$	9243.56i	0.0405791i	−0.999794	+	0.0202896i	$0.993541\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	135330.	0.553417
$144$	0	0
$145$	49011.5	0.193588
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 386320.i	− 1.42555i	−0.701395	−	0.712773i	$-0.747439\pi$
$149$	− 386320.i	− 1.42555i	0.701395	−	0.712773i	$-0.252561\pi$
$150$	0	0
$151$	202172.i	0.721570i	0.932649	+	0.360785i	$0.117491\pi$
$151$	202172.i	0.721570i	−0.932649	+	0.360785i	$0.882509\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	167342.	0.559470
$156$	0	0
$157$	−309682.	−1.00269	−0.501345	−	0.865247i	$-0.667161\pi$
$157$	−309682.	−1.00269	−0.501345	+	0.865247i	$0.667161\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 96351.3i	− 0.292950i
$162$	0	0
$163$	− 258018.i	− 0.760644i	−0.924854	−	0.380322i	$-0.875813\pi$
$163$	− 258018.i	− 0.760644i	0.924854	−	0.380322i	$-0.124187\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	480932.	1.33442	0.667209	−	0.744870i	$-0.267489\pi$
$167$	480932.	1.33442	0.667209	+	0.744870i	$0.267489\pi$
$168$	0	0
$169$	−175025.	−0.471394
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 434424.i	− 1.10357i	−0.833988	−	0.551783i	$-0.813948\pi$
$173$	− 434424.i	− 1.10357i	0.833988	−	0.551783i	$-0.186052\pi$
$174$	0	0
$175$	− 158711.i	− 0.391751i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	325478.	0.759257	0.379628	−	0.925139i	$-0.376052\pi$
$179$	325478.	0.759257	0.379628	+	0.925139i	$0.376052\pi$
$180$	0	0
$181$	174491.	0.395892	0.197946	−	0.980213i	$-0.436573\pi$
$181$	174491.	0.395892	0.197946	+	0.980213i	$0.436573\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	70762.8i	0.152011i
$186$	0	0
$187$	116132.i	0.242855i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−408963.	−0.811149	−0.405575	−	0.914062i	$-0.632929\pi$
$191$	−408963.	−0.811149	−0.405575	+	0.914062i	$0.632929\pi$
$192$	0	0
$193$	192413.	0.371827	0.185914	−	0.982566i	$-0.440476\pi$
$193$	192413.	0.371827	0.185914	+	0.982566i	$0.440476\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 29821.6i	− 0.0547477i	−0.999625	−	0.0273738i	$-0.991286\pi$
$197$	− 29821.6i	− 0.0547477i	0.999625	−	0.0273738i	$-0.00871445\pi$
$198$	0	0
$199$	− 941230.i	− 1.68486i	−0.538808	−	0.842428i	$-0.681125\pi$
$199$	− 941230.i	− 1.68486i	0.538808	−	0.842428i	$-0.318875\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	116168.	0.197854
$204$	0	0
$205$	−490053.	−0.814439
$206$	0	0
$207$	0	0
$208$	0	0
$209$	132879.i	0.210422i
$210$	0	0
$211$	693732.i	1.07272i	0.843990	+	0.536359i	$0.180201\pi$
$211$	693732.i	1.07272i	−0.843990	+	0.536359i	$0.819799\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−203999.	−0.300976
$216$	0	0
$217$	396637.	0.571800
$218$	0	0
$219$	0	0
$220$	0	0
$221$	168425.i	0.231967i
$222$	0	0
$223$	1.07629e6i	1.44932i	0.689105	+	0.724662i	$0.258004\pi$
$223$	1.07629e6i	1.44932i	−0.689105	+	0.724662i	$0.741996\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	74352.1	0.0957698	0.0478849	−	0.998853i	$-0.484752\pi$
$227$	74352.1	0.0957698	0.0478849	+	0.998853i	$0.484752\pi$
$228$	0	0
$229$	418858.	0.527811	0.263905	−	0.964549i	$-0.414989\pi$
$229$	418858.	0.527811	0.263905	+	0.964549i	$0.414989\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 688513.i	− 0.830849i	−0.909628	−	0.415424i	$-0.863633\pi$
$233$	− 688513.i	− 0.830849i	0.909628	−	0.415424i	$-0.136367\pi$
$234$	0	0
$235$	− 256701.i	− 0.303220i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−928417.	−1.05135	−0.525676	−	0.850685i	$-0.676188\pi$
$239$	−928417.	−1.05135	−0.525676	+	0.850685i	$0.676188\pi$
$240$	0	0
$241$	−77990.8	−0.0864970	−0.0432485	−	0.999064i	$-0.513771\pi$
$241$	−77990.8	−0.0864970	−0.0432485	+	0.999064i	$0.513771\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 350745.i	− 0.373316i
$246$	0	0
$247$	192714.i	0.200988i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	134137.	0.134389	0.0671945	−	0.997740i	$-0.478595\pi$
$251$	134137.	0.134389	0.0671945	+	0.997740i	$0.478595\pi$
$252$	0	0
$253$	365684.	0.359174
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.24365e6i	1.17453i	0.809394	+	0.587266i	$0.199796\pi$
$257$	1.24365e6i	1.17453i	−0.809394	+	0.587266i	$0.800204\pi$
$258$	0	0
$259$	167723.i	0.155361i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−779910.	−0.695273	−0.347636	−	0.937629i	$-0.613016\pi$
$263$	−779910.	−0.695273	−0.347636	+	0.937629i	$0.613016\pi$
$264$	0	0
$265$	−316694.	−0.277029
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.08789e6i	1.75925i	0.475670	+	0.879624i	$0.342206\pi$
$269$	2.08789e6i	1.75925i	−0.475670	+	0.879624i	$0.657794\pi$
$270$	0	0
$271$	− 1.82720e6i	− 1.51134i	−0.654953	−	0.755670i	$-0.727312\pi$
$271$	− 1.82720e6i	− 1.51134i	0.654953	−	0.755670i	$-0.272688\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	602357.	0.480310
$276$	0	0
$277$	−153356.	−0.120089	−0.0600443	−	0.998196i	$-0.519124\pi$
$277$	−153356.	−0.120089	−0.0600443	+	0.998196i	$0.519124\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	1.55969e6i	1.17834i	0.808009	+	0.589171i	$0.200545\pi$
$281$	1.55969e6i	1.17834i	−0.808009	+	0.589171i	$0.799455\pi$
$282$	0	0
$283$	1.48182e6i	1.09984i	0.835217	+	0.549920i	$0.185342\pi$
$283$	1.48182e6i	1.09984i	−0.835217	+	0.549920i	$0.814658\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.16153e6	−0.832388
$288$	0	0
$289$	1.27532e6	0.898206
$290$	0	0
$291$	0	0
$292$	0	0
$293$	709991.i	0.483152i	0.970382	+	0.241576i	$0.0776643\pi$
$293$	709991.i	0.483152i	−0.970382	+	0.241576i	$0.922336\pi$
$294$	0	0
$295$	211473.i	0.141481i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	530349.	0.343071
$300$	0	0
$301$	−483520.	−0.307609
$302$	0	0
$303$	0	0
$304$	0	0
$305$	895915.i	0.551464i
$306$	0	0
$307$	− 2.69157e6i	− 1.62989i	−0.579536	−	0.814946i	$-0.696766\pi$
$307$	− 2.69157e6i	− 1.62989i	0.579536	−	0.814946i	$-0.303234\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2.51406e6	−1.47392	−0.736961	−	0.675936i	$-0.763740\pi$
$311$	−2.51406e6	−1.47392	−0.736961	+	0.675936i	$0.763740\pi$
$312$	0	0
$313$	−2.21508e6	−1.27800	−0.638998	−	0.769209i	$-0.720651\pi$
$313$	−2.21508e6	−1.27800	−0.638998	+	0.769209i	$0.720651\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.92441e6i	1.63452i	0.576270	+	0.817260i	$0.304508\pi$
$317$	2.92441e6i	1.63452i	−0.576270	+	0.817260i	$0.695492\pi$
$318$	0	0
$319$	440893.i	0.242581i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−165376.	−0.0881993
$324$	0	0
$325$	873595.	0.458777
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 608436.i	− 0.309902i
$330$	0	0
$331$	2.90552e6i	1.45765i	0.684699	+	0.728826i	$0.259934\pi$
$331$	2.90552e6i	1.45765i	−0.684699	+	0.728826i	$0.740066\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−250692.	−0.122047
$336$	0	0
$337$	−2.19833e6	−1.05443	−0.527215	−	0.849732i	$-0.676764\pi$
$337$	−2.19833e6	−1.05443	−0.527215	+	0.849732i	$0.676764\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.50536e6i	0.701060i
$342$	0	0
$343$	− 2.18407e6i	− 1.00238i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	587065.	0.261736	0.130868	−	0.991400i	$-0.458224\pi$
$347$	587065.	0.261736	0.130868	+	0.991400i	$0.458224\pi$
$348$	0	0
$349$	−138574.	−0.0609001	−0.0304501	−	0.999536i	$-0.509694\pi$
$349$	−138574.	−0.0609001	−0.0304501	+	0.999536i	$0.509694\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	1.60120e6i	0.683928i	0.939713	+	0.341964i	$0.111092\pi$
$353$	1.60120e6i	0.683928i	−0.939713	+	0.341964i	$0.888908\pi$
$354$	0	0
$355$	− 801325.i	− 0.337472i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−3.96831e6	−1.62506	−0.812531	−	0.582918i	$-0.801911\pi$
$359$	−3.96831e6	−1.62506	−0.812531	+	0.582918i	$0.801911\pi$
$360$	0	0
$361$	2.28687e6	0.923580
$362$	0	0
$363$	0	0
$364$	0	0
$365$	933141.i	0.366619i
$366$	0	0
$367$	4.00727e6i	1.55304i	0.630090	+	0.776522i	$0.283018\pi$
$367$	4.00727e6i	1.55304i	−0.630090	+	0.776522i	$0.716982\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−750632.	−0.283134
$372$	0	0
$373$	−1.66145e6	−0.618324	−0.309162	−	0.951009i	$-0.600049\pi$
$373$	−1.66145e6	−0.618324	−0.309162	+	0.951009i	$0.600049\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	639425.i	0.231705i
$378$	0	0
$379$	− 3.33674e6i	− 1.19323i	−0.802527	−	0.596616i	$-0.796512\pi$
$379$	− 3.33674e6i	− 1.19323i	0.802527	−	0.596616i	$-0.203488\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.75899e6	−0.961067	−0.480533	−	0.876976i	$-0.659557\pi$
$383$	−2.75899e6	−0.961067	−0.480533	+	0.876976i	$0.659557\pi$
$384$	0	0
$385$	−834875.	−0.287058
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 251235.i	− 0.0841795i	−0.999114	−	0.0420897i	$-0.986598\pi$
$389$	− 251235.i	− 0.0841795i	0.999114	−	0.0420897i	$-0.0134015\pi$
$390$	0	0
$391$	455114.i	0.150549i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.32409e6	0.426995
$396$	0	0
$397$	5.22531e6	1.66393	0.831966	−	0.554826i	$-0.187215\pi$
$397$	5.22531e6	1.66393	0.831966	+	0.554826i	$0.187215\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.96959e6i	0.922223i	0.887342	+	0.461112i	$0.152549\pi$
$401$	2.96959e6i	0.922223i	−0.887342	+	0.461112i	$0.847451\pi$
$402$	0	0
$403$	2.18322e6i	0.669630i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−636562.	−0.190482
$408$	0	0
$409$	3.08107e6	0.910738	0.455369	−	0.890303i	$-0.349507\pi$
$409$	3.08107e6	0.910738	0.455369	+	0.890303i	$0.349507\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	501235.i	0.144599i
$414$	0	0
$415$	− 4.05903e6i	− 1.15692i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	672153.	0.187039	0.0935197	−	0.995617i	$-0.470188\pi$
$419$	672153.	0.187039	0.0935197	+	0.995617i	$0.470188\pi$
$420$	0	0
$421$	−2.64888e6	−0.728380	−0.364190	−	0.931325i	$-0.618654\pi$
$421$	−2.64888e6	−0.728380	−0.364190	+	0.931325i	$0.618654\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	749667.i	0.201324i
$426$	0	0
$427$	2.12351e6i	0.563618i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	105796.	0.0274332	0.0137166	−	0.999906i	$-0.495634\pi$
$431$	105796.	0.0274332	0.0137166	+	0.999906i	$0.495634\pi$
$432$	0	0
$433$	2.94723e6	0.755430	0.377715	−	0.925922i	$-0.376710\pi$
$433$	2.94723e6	0.755430	0.377715	+	0.925922i	$0.376710\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	520746.i	0.130443i
$438$	0	0
$439$	1.98146e6i	0.490708i	0.969434	+	0.245354i	$0.0789042\pi$
$439$	1.98146e6i	0.490708i	−0.969434	+	0.245354i	$0.921096\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.07351e6	1.22829	0.614143	−	0.789195i	$-0.289502\pi$
$443$	5.07351e6	1.22829	0.614143	+	0.789195i	$0.289502\pi$
$444$	0	0
$445$	−4.32470e6	−1.03527
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 3.47453e6i	− 0.813356i	−0.913572	−	0.406678i	$-0.866687\pi$
$449$	− 3.47453e6i	− 0.813356i	0.913572	−	0.406678i	$-0.133313\pi$
$450$	0	0
$451$	− 4.40838e6i	− 1.02056i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−1.21081e6	−0.274188
$456$	0	0
$457$	−370212.	−0.0829201	−0.0414601	−	0.999140i	$-0.513201\pi$
$457$	−370212.	−0.0829201	−0.0414601	+	0.999140i	$0.513201\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 5.17403e6i	− 1.13391i	−0.823750	−	0.566953i	$-0.808122\pi$
$461$	− 5.17403e6i	− 1.13391i	0.823750	−	0.566953i	$-0.191878\pi$
$462$	0	0
$463$	8.13232e6i	1.76304i	0.472148	+	0.881519i	$0.343479\pi$
$463$	8.13232e6i	1.76304i	−0.472148	+	0.881519i	$0.656521\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.84301e6	1.87632	0.938162	−	0.346196i	$-0.112527\pi$
$467$	8.84301e6	1.87632	0.938162	+	0.346196i	$0.112527\pi$
$468$	0	0
$469$	−594194.	−0.124737
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 1.83511e6i	− 0.377146i
$474$	0	0
$475$	857776.i	0.174438i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−7.26232e6	−1.44623	−0.723114	−	0.690729i	$-0.757290\pi$
$479$	−7.26232e6	−1.44623	−0.723114	+	0.690729i	$0.757290\pi$
$480$	0	0
$481$	−923203.	−0.181943
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 996420.i	− 0.192348i
$486$	0	0
$487$	− 4.21568e6i	− 0.805463i	−0.915318	−	0.402731i	$-0.868061\pi$
$487$	− 4.21568e6i	− 0.805463i	0.915318	−	0.402731i	$-0.131939\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	9.69372e6	1.81462	0.907312	−	0.420457i	$-0.138130\pi$
$491$	9.69372e6	1.81462	0.907312	+	0.420457i	$0.138130\pi$
$492$	0	0
$493$	−548717.	−0.101679
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 1.89931e6i	− 0.344909i
$498$	0	0
$499$	− 2.23130e6i	− 0.401149i	−0.979678	−	0.200575i	$-0.935719\pi$
$499$	− 2.23130e6i	− 0.401149i	0.979678	−	0.200575i	$-0.0642809\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−5.66002e6	−0.997466	−0.498733	−	0.866756i	$-0.666201\pi$
$503$	−5.66002e6	−0.997466	−0.498733	+	0.866756i	$0.666201\pi$
$504$	0	0
$505$	−3.26122e6	−0.569051
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 4.84707e6i	− 0.829249i	−0.909993	−	0.414625i	$-0.863913\pi$
$509$	− 4.84707e6i	− 0.829249i	0.909993	−	0.414625i	$-0.136087\pi$
$510$	0	0
$511$	2.21174e6i	0.374699i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.24997e6	0.207674
$516$	0	0
$517$	2.30921e6	0.379959
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.28208e6i	− 0.206929i	−0.994633	−	0.103464i	$-0.967007\pi$
$521$	− 1.28208e6i	− 0.206929i	0.994633	−	0.103464i	$-0.0329928\pi$
$522$	0	0
$523$	− 8.72572e6i	− 1.39491i	−0.716627	−	0.697456i	$-0.754315\pi$
$523$	− 8.72572e6i	− 1.39491i	0.716627	−	0.697456i	$-0.245685\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.87351e6	−0.293853
$528$	0	0
$529$	−5.00325e6	−0.777343
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 6.39345e6i	− 0.974803i
$534$	0	0
$535$	− 1.64868e6i	− 0.249030i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.15520e6	0.467794
$540$	0	0
$541$	6.25126e6	0.918279	0.459139	−	0.888364i	$-0.348158\pi$
$541$	6.25126e6	0.918279	0.459139	+	0.888364i	$0.348158\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 4.37250e6i	− 0.630577i
$546$	0	0
$547$	4.50121e6i	0.643221i	0.946872	+	0.321611i	$0.104224\pi$
$547$	4.50121e6i	0.643221i	−0.946872	+	0.321611i	$0.895776\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−627847.	−0.0880997
$552$	0	0
$553$	3.13836e6	0.436406
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.34552e6i	− 1.13977i	−0.821726	−	0.569883i	$-0.806989\pi$
$557$	− 8.34552e6i	− 1.13977i	0.821726	−	0.569883i	$-0.193011\pi$
$558$	0	0
$559$	− 2.66145e6i	− 0.360238i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.69650e6	0.890383	0.445191	−	0.895435i	$-0.353136\pi$
$563$	6.69650e6	0.890383	0.445191	+	0.895435i	$0.353136\pi$
$564$	0	0
$565$	−6.32200e6	−0.833170
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 7.30808e6i	− 0.946286i	−0.880986	−	0.473143i	$-0.843119\pi$
$569$	− 7.30808e6i	− 0.946286i	0.880986	−	0.473143i	$-0.156881\pi$
$570$	0	0
$571$	5.44039e6i	0.698297i	0.937067	+	0.349148i	$0.113529\pi$
$571$	5.44039e6i	0.698297i	−0.937067	+	0.349148i	$0.886471\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.36060e6	0.297751
$576$	0	0
$577$	−2.25800e6	−0.282348	−0.141174	−	0.989985i	$-0.545088\pi$
$577$	−2.25800e6	−0.282348	−0.141174	+	0.989985i	$0.545088\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 9.62076e6i	− 1.18241i
$582$	0	0
$583$	− 2.84889e6i	− 0.347139i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.21539e7	1.45587	0.727934	−	0.685648i	$-0.240481\pi$
$587$	1.21539e7	1.45587	0.727934	+	0.685648i	$0.240481\pi$
$588$	0	0
$589$	−2.14369e6	−0.254609
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 1.19519e6i	− 0.139572i	−0.997562	−	0.0697862i	$-0.977768\pi$
$593$	− 1.19519e6i	− 0.139572i	0.997562	−	0.0697862i	$-0.0222317\pi$
$594$	0	0
$595$	− 1.03905e6i	− 0.120322i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.93387e6	0.561851	0.280925	−	0.959730i	$-0.409359\pi$
$599$	4.93387e6	0.561851	0.280925	+	0.959730i	$0.409359\pi$
$600$	0	0
$601$	2.79989e6	0.316195	0.158097	−	0.987424i	$-0.449464\pi$
$601$	2.79989e6	0.316195	0.158097	+	0.987424i	$0.449464\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	2.30024e6i	0.255496i
$606$	0	0
$607$	523668.i	0.0576879i	0.999584	+	0.0288439i	$0.00918258\pi$
$607$	523668.i	0.0576879i	−0.999584	+	0.0288439i	$0.990817\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	3.34903e6	0.362924
$612$	0	0
$613$	1.27593e7	1.37143	0.685716	−	0.727869i	$-0.259489\pi$
$613$	1.27593e7	1.37143	0.685716	+	0.727869i	$0.259489\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 4.62454e6i	− 0.489053i	−0.969643	−	0.244527i	$-0.921367\pi$
$617$	− 4.62454e6i	− 0.489053i	0.969643	−	0.244527i	$-0.0786325\pi$
$618$	0	0
$619$	1.49707e7i	1.57042i	0.619232	+	0.785208i	$0.287444\pi$
$619$	1.49707e7i	1.57042i	−0.619232	+	0.785208i	$0.712556\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.02504e7	−1.05809
$624$	0	0
$625$	284973.	0.0291813
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 792238.i	− 0.0798415i
$630$	0	0
$631$	4.29018e6i	0.428946i	0.976730	+	0.214473i	$0.0688033\pi$
$631$	4.29018e6i	0.428946i	−0.976730	+	0.214473i	$0.931197\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.62254e6	0.159684
$636$	0	0
$637$	4.57597e6	0.446822
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 1.00333e7i	− 0.964493i	−0.876036	−	0.482246i	$-0.839821\pi$
$641$	− 1.00333e7i	− 0.964493i	0.876036	−	0.482246i	$-0.160179\pi$
$642$	0	0
$643$	− 3.67434e6i	− 0.350471i	−0.984527	−	0.175235i	$-0.943931\pi$
$643$	− 3.67434e6i	− 0.350471i	0.984527	−	0.175235i	$-0.0560686\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.76449e6	0.541378	0.270689	−	0.962667i	$-0.412749\pi$
$647$	5.76449e6	0.541378	0.270689	+	0.962667i	$0.412749\pi$
$648$	0	0
$649$	−1.90235e6	−0.177288
$650$	0	0
$651$	0	0
$652$	0	0
$653$	8.70697e6i	0.799068i	0.916718	+	0.399534i	$0.130828\pi$
$653$	8.70697e6i	0.799068i	−0.916718	+	0.399534i	$0.869172\pi$
$654$	0	0
$655$	− 3.84008e6i	− 0.349733i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.49366e7	1.33979	0.669895	−	0.742456i	$-0.266339\pi$
$659$	1.49366e7	1.33979	0.669895	+	0.742456i	$0.266339\pi$
$660$	0	0
$661$	−7.76645e6	−0.691383	−0.345692	−	0.938348i	$-0.612356\pi$
$661$	−7.76645e6	−0.691383	−0.345692	+	0.938348i	$0.612356\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.18889e6i	− 0.104253i
$666$	0	0
$667$	1.72784e6i	0.150379i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−8.05939e6	−0.691029
$672$	0	0
$673$	3.24346e6	0.276039	0.138020	−	0.990429i	$-0.455926\pi$
$673$	3.24346e6	0.276039	0.138020	+	0.990429i	$0.455926\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.11330e7i	0.933554i	0.884375	+	0.466777i	$0.154585\pi$
$677$	1.11330e7i	0.933554i	−0.884375	+	0.466777i	$0.845415\pi$
$678$	0	0
$679$	− 2.36173e6i	− 0.196587i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.05637e6	0.168675	0.0843374	−	0.996437i	$-0.473123\pi$
$683$	2.05637e6	0.168675	0.0843374	+	0.996437i	$0.473123\pi$
$684$	0	0
$685$	−7.57030e6	−0.616434
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.13173e6i	− 0.331576i
$690$	0	0
$691$	− 1.03166e7i	− 0.821946i	−0.911648	−	0.410973i	$-0.865189\pi$
$691$	− 1.03166e7i	− 0.821946i	0.911648	−	0.410973i	$-0.134811\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	313886.	0.0246496
$696$	0	0
$697$	5.48648e6	0.427771
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 8.42923e6i	− 0.647878i	−0.946078	−	0.323939i	$-0.894993\pi$
$701$	− 8.42923e6i	− 0.647878i	0.946078	−	0.323939i	$-0.105007\pi$
$702$	0	0
$703$	− 906486.i	− 0.0691787i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−7.72978e6	−0.581592
$708$	0	0
$709$	1.10066e7	0.822311	0.411155	−	0.911565i	$-0.365125\pi$
$709$	1.10066e7	0.822311	0.411155	+	0.911565i	$0.365125\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	5.89944e6i	0.434597i
$714$	0	0
$715$	− 4.59543e6i	− 0.336171i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−1.91055e7	−1.37828	−0.689140	−	0.724629i	$-0.742011\pi$
$719$	−1.91055e7	−1.37828	−0.689140	+	0.724629i	$0.742011\pi$
$720$	0	0
$721$	2.96270e6	0.212251
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.84610e6i	0.201097i
$726$	0	0
$727$	− 1.04952e7i	− 0.736467i	−0.929733	−	0.368234i	$-0.879963\pi$
$727$	− 1.04952e7i	− 0.736467i	0.929733	−	0.368234i	$-0.120037\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	2.28390e6	0.158083
$732$	0	0
$733$	−1.48440e7	−1.02045	−0.510225	−	0.860041i	$-0.670438\pi$
$733$	−1.48440e7	−1.02045	−0.510225	+	0.860041i	$0.670438\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 2.25515e6i	− 0.152935i
$738$	0	0
$739$	2.24036e7i	1.50906i	0.656267	+	0.754529i	$0.272135\pi$
$739$	2.24036e7i	1.50906i	−0.656267	+	0.754529i	$0.727865\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.47181e7	−0.978092	−0.489046	−	0.872258i	$-0.662655\pi$
$743$	−1.47181e7	−0.978092	−0.489046	+	0.872258i	$0.662655\pi$
$744$	0	0
$745$	−1.31184e7	−0.865943
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.90773e6i	− 0.254519i
$750$	0	0
$751$	− 6.57359e6i	− 0.425307i	−0.977128	−	0.212653i	$-0.931789\pi$
$751$	− 6.57359e6i	− 0.425307i	0.977128	−	0.212653i	$-0.0682105\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.86521e6	0.438315
$756$	0	0
$757$	−1.26137e7	−0.800024	−0.400012	−	0.916510i	$-0.630994\pi$
$757$	−1.26137e7	−0.800024	−0.400012	+	0.916510i	$0.630994\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	1.10995e7i	0.694771i	0.937722	+	0.347386i	$0.112931\pi$
$761$	1.10995e7i	0.694771i	−0.937722	+	0.347386i	$0.887069\pi$
$762$	0	0
$763$	− 1.03638e7i	− 0.644474i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2.75897e6	−0.169339
$768$	0	0
$769$	9.03762e6	0.551110	0.275555	−	0.961285i	$-0.411138\pi$
$769$	9.03762e6	0.551110	0.275555	+	0.961285i	$0.411138\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.28899e7i	0.775890i	0.921682	+	0.387945i	$0.126815\pi$
$773$	1.28899e7i	0.775890i	−0.921682	+	0.387945i	$0.873185\pi$
$774$	0	0
$775$	9.71759e6i	0.581172i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	6.27768e6	0.370643
$780$	0	0
$781$	7.20849e6	0.422879
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.05160e7i	0.609081i
$786$	0	0
$787$	5.21316e6i	0.300030i	0.988684	+	0.150015i	$0.0479321\pi$
$787$	5.21316e6i	0.300030i	−0.988684	+	0.150015i	$0.952068\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−1.49845e7	−0.851531
$792$	0	0
$793$	−1.16885e7	−0.660048
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.39819e6i	0.524081i	0.965057	+	0.262040i	$0.0843954\pi$
$797$	9.39819e6i	0.524081i	−0.965057	+	0.262040i	$0.915605\pi$
$798$	0	0
$799$	2.87394e6i	0.159261i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−8.39427e6	−0.459403
$804$	0	0
$805$	−3.27183e6	−0.177951
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.97008e7i	1.59550i	0.602987	+	0.797751i	$0.293977\pi$
$809$	2.97008e7i	1.59550i	−0.602987	+	0.797751i	$0.706023\pi$
$810$	0	0
$811$	8.12658e6i	0.433866i	0.976186	+	0.216933i	$0.0696054\pi$
$811$	8.12658e6i	0.433866i	−0.976186	+	0.216933i	$0.930395\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−8.76160e6	−0.462050
$816$	0	0
$817$	2.61326e6	0.136971
$818$	0	0
$819$	0	0
$820$	0	0
$821$	2.77322e7i	1.43591i	0.696090	+	0.717955i	$0.254922\pi$
$821$	2.77322e7i	1.43591i	−0.696090	+	0.717955i	$0.745078\pi$
$822$	0	0
$823$	21631.4i	0.00111323i	1.00000		0.000556615i	$0.000177176\pi$
$823$	21631.4i	0.00111323i	−1.00000		0.000556615i	$0.999823\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8.80967e6	−0.447915	−0.223958	−	0.974599i	$-0.571898\pi$
$827$	−8.80967e6	−0.447915	−0.223958	+	0.974599i	$0.571898\pi$
$828$	0	0
$829$	−4.58329e6	−0.231628	−0.115814	−	0.993271i	$-0.536948\pi$
$829$	−4.58329e6	−0.231628	−0.115814	+	0.993271i	$0.536948\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	3.92683e6i	0.196078i
$834$	0	0
$835$	− 1.63311e7i	− 0.810588i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	931628.	0.0456917	0.0228459	−	0.999739i	$-0.492727\pi$
$839$	931628.	0.0456917	0.0228459	+	0.999739i	$0.492727\pi$
$840$	0	0
$841$	1.84280e7	0.898436
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.94338e6i	0.286346i
$846$	0	0
$847$	5.45206e6i	0.261127i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.49465e6	−0.118083
$852$	0	0
$853$	1.75076e7	0.823862	0.411931	−	0.911215i	$-0.364855\pi$
$853$	1.75076e7	0.823862	0.411931	+	0.911215i	$0.364855\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	1.73195e7i	0.805534i	0.915303	+	0.402767i	$0.131951\pi$
$857$	1.73195e7i	0.805534i	−0.915303	+	0.402767i	$0.868049\pi$
$858$	0	0
$859$	− 1.73909e7i	− 0.804154i	−0.915606	−	0.402077i	$-0.868288\pi$
$859$	− 1.73909e7i	− 0.804154i	0.915606	−	0.402077i	$-0.131712\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1.26123e6	0.0576457	0.0288229	−	0.999585i	$-0.490824\pi$
$863$	1.26123e6	0.0576457	0.0288229	+	0.999585i	$0.490824\pi$
$864$	0	0
$865$	−1.47519e7	−0.670357
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.19111e7i	0.535059i
$870$	0	0
$871$	− 3.27064e6i	− 0.146079i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−1.39303e7	−0.615091
$876$	0	0
$877$	1.76858e7	0.776470	0.388235	−	0.921560i	$-0.373085\pi$
$877$	1.76858e7	0.776470	0.388235	+	0.921560i	$0.373085\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	2.32700e7i	1.01008i	0.863096	+	0.505040i	$0.168522\pi$
$881$	2.32700e7i	1.01008i	−0.863096	+	0.505040i	$0.831478\pi$
$882$	0	0
$883$	2.27894e7i	0.983629i	0.870700	+	0.491814i	$0.163666\pi$
$883$	2.27894e7i	0.983629i	−0.870700	+	0.491814i	$0.836334\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.72380e6	0.414980	0.207490	−	0.978237i	$-0.433471\pi$
$887$	9.72380e6	0.414980	0.207490	+	0.978237i	$0.433471\pi$
$888$	0	0
$889$	3.84577e6	0.163203
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.28839e6i	0.137992i
$894$	0	0
$895$	− 1.10523e7i	− 0.461208i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−7.11276e6	−0.293521
$900$	0	0
$901$	3.54560e6	0.145505
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 5.92524e6i	− 0.240483i
$906$	0	0
$907$	− 9.72850e6i	− 0.392670i	−0.980537	−	0.196335i	$-0.937096\pi$
$907$	− 9.72850e6i	− 0.392670i	0.980537	−	0.196335i	$-0.0629040\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−1.57689e7	−0.629514	−0.314757	−	0.949172i	$-0.601923\pi$
$911$	−1.57689e7	−0.629514	−0.314757	+	0.949172i	$0.601923\pi$
$912$	0	0
$913$	3.65138e7	1.44971
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 9.10180e6i	− 0.357441i
$918$	0	0
$919$	4.35741e7i	1.70192i	0.525230	+	0.850960i	$0.323979\pi$
$919$	4.35741e7i	1.70192i	−0.525230	+	0.850960i	$0.676021\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.04544e7	0.403921
$924$	0	0
$925$	−4.10921e6	−0.157908
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 9.09785e6i	− 0.345859i	−0.984934	−	0.172930i	$-0.944677\pi$
$929$	− 9.09785e6i	− 0.345859i	0.984934	−	0.172930i	$-0.0553233\pi$
$930$	0	0
$931$	4.49311e6i	0.169892i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	3.94352e6	0.147521
$936$	0	0
$937$	−4.70289e7	−1.74991	−0.874956	−	0.484202i	$-0.839110\pi$
$937$	−4.70289e7	−1.74991	−0.874956	+	0.484202i	$0.839110\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.69812e7i	0.625165i	0.949891	+	0.312582i	$0.101194\pi$
$941$	1.69812e7i	0.625165i	−0.949891	+	0.312582i	$0.898806\pi$
$942$	0	0
$943$	− 1.72762e7i	− 0.632657i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−5.33669e7	−1.93374	−0.966868	−	0.255279i	$-0.917833\pi$
$947$	−5.33669e7	−1.93374	−0.966868	+	0.255279i	$0.917833\pi$
$948$	0	0
$949$	−1.21742e7	−0.438807
$950$	0	0
$951$	0	0
$952$	0	0
$953$	2.48569e7i	0.886572i	0.896380	+	0.443286i	$0.146187\pi$
$953$	2.48569e7i	0.886572i	−0.896380	+	0.443286i	$0.853813\pi$
$954$	0	0
$955$	1.38873e7i	0.492730i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−1.79432e7	−0.630019
$960$	0	0
$961$	4.34369e6	0.151723
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 6.53382e6i	− 0.225865i
$966$	0	0
$967$	− 2.02145e6i	− 0.0695179i	−0.999396	−	0.0347590i	$-0.988934\pi$
$967$	− 2.02145e6i	− 0.0695179i	0.999396	−	0.0347590i	$-0.0110664\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.11340e6	0.140008	0.0700040	−	0.997547i	$-0.477699\pi$
$971$	4.11340e6	0.140008	0.0700040	+	0.997547i	$0.477699\pi$
$972$	0	0
$973$	743977.	0.0251929
$974$	0	0
$975$	0	0
$976$	0	0
$977$	4.06445e7i	1.36228i	0.732155	+	0.681138i	$0.238515\pi$
$977$	4.06445e7i	1.36228i	−0.732155	+	0.681138i	$0.761485\pi$
$978$	0	0
$979$	− 3.89037e7i	− 1.29728i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.16188e7	0.713590	0.356795	−	0.934183i	$-0.383869\pi$
$983$	2.16188e7	0.713590	0.356795	+	0.934183i	$0.383869\pi$
$984$	0	0
$985$	−1.01266e6	−0.0332563
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 7.19170e6i	− 0.233798i
$990$	0	0
$991$	2.63393e7i	0.851963i	0.904732	+	0.425981i	$0.140071\pi$
$991$	2.63393e7i	0.851963i	−0.904732	+	0.425981i	$0.859929\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−3.19616e7	−1.02346
$996$	0	0
$997$	3.25804e7	1.03805	0.519026	−	0.854759i	$-0.326295\pi$
$997$	3.25804e7	1.03805	0.519026	+	0.854759i	$0.326295\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.6.c.g.431.3	✓	12
3.2	odd	2	inner	432.6.c.g.431.9	yes	12
4.3	odd	2	inner	432.6.c.g.431.4	yes	12
12.11	even	2	inner	432.6.c.g.431.10	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.6.c.g.431.3	✓	12	1.1	even	1	trivial
432.6.c.g.431.4	yes	12	4.3	odd	2	inner
432.6.c.g.431.9	yes	12	3.2	odd	2	inner
432.6.c.g.431.10	yes	12	12.11	even	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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-33.9573i q^{5} -80.4860i q^{7} +O(q^{10})\)	\(q-33.9573i q^{5} -80.4860i q^{7} +305.470 q^{11} +443.021 q^{13} +380.175i q^{17} +434.999i q^{19} +1197.12 q^{23} +1971.90 q^{25} +1443.33i q^{29} +4928.03i q^{31} -2733.08 q^{35} -2083.88 q^{37} -14431.5i q^{41} -6007.51i q^{43} +7559.53 q^{47} +10329.0 q^{49} -9326.25i q^{53} -10372.9i q^{55} -6227.61 q^{59} -26383.6 q^{61} -15043.8i q^{65} -7382.58i q^{67} +23598.0 q^{71} -27479.9 q^{73} -24586.0i q^{77} +38992.7i q^{79} +119533. q^{83} +12909.7 q^{85} -127357. i q^{89} -35657.0i q^{91} +14771.4 q^{95} +29343.3 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q+O(q^{10}) \) 12 * q	\( 12 q - 2040 q^{13} - 7824 q^{25} + 18240 q^{37} - 69936 q^{49} + 105936 q^{61} + 200172 q^{73} - 126072 q^{85} - 282300 q^{97}+O(q^{100}) \) 12 * q - 2040 * q^13 - 7824 * q^25 + 18240 * q^37 - 69936 * q^49 + 105936 * q^61 + 200172 * q^73 - 126072 * q^85 - 282300 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more