LMFDB - Embedded newform 432.9.g.h.271.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [432,9,Mod(271,432)] 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("432.271"); 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([1, 0, 0])) N = Newforms(chi, 9, names="a")

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	432.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,0,0,0,0,57036] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$175.987559546$
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} + 3658 x^{10} + 10323672 x^{8} + 10806771976 x^{6} + 8657863222624 x^{4} + 575997114675360 x^{2} + 35\!\cdots\!00 $ x^12 + 3658x^10 + 10323672x^8 + 10806771976x^6 + 8657863222624x^4 + 575997114675360*x^2 + 35494966945166400
Coefficient ring:	$\Z[a_1, \ldots, a_{31}]$
Coefficient ring index:	$ 2^{40}\cdot 3^{25} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			271.10
Root			$24.6650 - 42.7211i$ of defining polynomial
Character	$\chi$	$=$	432.271
Dual form			432.9.g.h.271.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+612.710 q^{5} +3291.22i q^{7} +5149.41i q^{11} +26243.0 q^{13} +155584. q^{17} -121189. i q^{19} -497151. i q^{23} -15211.9 q^{25} -1.00912e6 q^{29} -1.51605e6i q^{31} +2.01656e6i q^{35} -2.43964e6 q^{37} +5.45730e6 q^{41} -2.74696e6i q^{43} -8.83959e6i q^{47} -5.06734e6 q^{49} +7.85231e6 q^{53} +3.15509e6i q^{55} -1.93978e6i q^{59} +1.59520e7 q^{61} +1.60793e7 q^{65} -1.93864e7i q^{67} -2.01068e6i q^{71} -9.41414e6 q^{73} -1.69478e7 q^{77} +6.11984e7i q^{79} +2.44137e7i q^{83} +9.53278e7 q^{85} -3.51750e7 q^{89} +8.63715e7i q^{91} -7.42535e7i q^{95} -3.81118e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 57036 q^{13} + 306996 q^{25} - 2032140 q^{37} - 6810648 q^{49} + 28116276 q^{61} + 33515148 q^{73} + 95803200 q^{85} + 90106572 q^{97}+O(q^{100}) $ 12 * q + 57036 * q^13 + 306996 * q^25 - 2032140 * q^37 - 6810648 * q^49 + 28116276 * q^61 + 33515148 * q^73 + 95803200 * q^85 + 90106572 * 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
$5$	612.710	0.980335	0.490168	−	0.871628i	$-0.336936\pi$
$5$	612.710	0.980335	0.490168	+	0.871628i	$0.336936\pi$
$6$	0	0
$7$	3291.22i	1.37077i	0.728180	+	0.685386i	$0.240366\pi$
$7$	3291.22i	1.37077i	−0.728180	+	0.685386i	$0.759634\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5149.41i	0.351712i	0.984416	+	0.175856i	$0.0562692\pi$
$11$	5149.41i	0.351712i	−0.984416	+	0.175856i	$0.943731\pi$
$12$	0	0
$13$	26243.0	0.918840	0.459420	−	0.888219i	$-0.348057\pi$
$13$	26243.0	0.918840	0.459420	+	0.888219i	$0.348057\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	155584.	1.86281	0.931406	−	0.363982i	$-0.118583\pi$
$17$	155584.	1.86281	0.931406	+	0.363982i	$0.118583\pi$
$18$	0	0
$19$	− 121189.i	− 0.929924i	−0.885331	−	0.464962i	$-0.846068\pi$
$19$	− 121189.i	− 0.929924i	0.885331	−	0.464962i	$-0.153932\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 497151.i	− 1.77655i	−0.459314	−	0.888274i	$-0.651905\pi$
$23$	− 497151.i	− 1.77655i	0.459314	−	0.888274i	$-0.348095\pi$
$24$	0	0
$25$	−15211.9	−0.0389424
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−1.00912e6	−1.42676	−0.713380	−	0.700778i	$-0.752836\pi$
$29$	−1.00912e6	−1.42676	−0.713380	+	0.700778i	$0.752836\pi$
$30$	0	0
$31$	− 1.51605e6i	− 1.64160i	−0.571214	−	0.820801i	$-0.693527\pi$
$31$	− 1.51605e6i	− 1.64160i	0.571214	−	0.820801i	$-0.306473\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.01656e6i	1.34382i
$36$	0	0
$37$	−2.43964e6	−1.30172	−0.650861	−	0.759197i	$-0.725592\pi$
$37$	−2.43964e6	−1.30172	−0.650861	+	0.759197i	$0.725592\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.45730e6	1.93127	0.965634	−	0.259904i	$-0.0836911\pi$
$41$	5.45730e6	1.93127	0.965634	+	0.259904i	$0.0836911\pi$
$42$	0	0
$43$	− 2.74696e6i	− 0.803487i	−0.915752	−	0.401743i	$-0.868404\pi$
$43$	− 2.74696e6i	− 0.803487i	0.915752	−	0.401743i	$-0.131596\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 8.83959e6i	− 1.81151i	−0.423802	−	0.905755i	$-0.639305\pi$
$47$	− 8.83959e6i	− 1.81151i	0.423802	−	0.905755i	$-0.360695\pi$
$48$	0	0
$49$	−5.06734e6	−0.879014
$50$	0	0
$51$	0	0
$52$	0	0
$53$	7.85231e6	0.995163	0.497581	−	0.867417i	$-0.334222\pi$
$53$	7.85231e6	0.995163	0.497581	+	0.867417i	$0.334222\pi$
$54$	0	0
$55$	3.15509e6i	0.344795i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1.93978e6i	− 0.160082i	−0.996792	−	0.0800412i	$-0.974495\pi$
$59$	− 1.93978e6i	− 0.160082i	0.996792	−	0.0800412i	$-0.0255052\pi$
$60$	0	0
$61$	1.59520e7	1.15212	0.576058	−	0.817409i	$-0.304590\pi$
$61$	1.59520e7	1.15212	0.576058	+	0.817409i	$0.304590\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.60793e7	0.900771
$66$	0	0
$67$	− 1.93864e7i	− 0.962050i	−0.876707	−	0.481025i	$-0.840265\pi$
$67$	− 1.93864e7i	− 0.962050i	0.876707	−	0.481025i	$-0.159735\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 2.01068e6i	− 0.0791243i	−0.999217	−	0.0395621i	$-0.987404\pi$
$71$	− 2.01068e6i	− 0.0791243i	0.999217	−	0.0395621i	$-0.0125963\pi$
$72$	0	0
$73$	−9.41414e6	−0.331504	−0.165752	−	0.986167i	$-0.553005\pi$
$73$	−9.41414e6	−0.331504	−0.165752	+	0.986167i	$0.553005\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.69478e7	−0.482116
$78$	0	0
$79$	6.11984e7i	1.57120i	0.618734	+	0.785601i	$0.287646\pi$
$79$	6.11984e7i	1.57120i	−0.618734	+	0.785601i	$0.712354\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.44137e7i	0.514424i	0.966355	+	0.257212i	$0.0828039\pi$
$83$	2.44137e7i	0.514424i	−0.966355	+	0.257212i	$0.917196\pi$
$84$	0	0
$85$	9.53278e7	1.82618
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−3.51750e7	−0.560627	−0.280313	−	0.959909i	$-0.590438\pi$
$89$	−3.51750e7	−0.560627	−0.280313	+	0.959909i	$0.590438\pi$
$90$	0	0
$91$	8.63715e7i	1.25952i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 7.42535e7i	− 0.911638i
$96$	0	0
$97$	−3.81118e7	−0.430500	−0.215250	−	0.976559i	$-0.569057\pi$
$97$	−3.81118e7	−0.430500	−0.215250	+	0.976559i	$0.569057\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	4.24287e7	0.407732	0.203866	−	0.978999i	$-0.434649\pi$
$101$	4.24287e7	0.407732	0.203866	+	0.978999i	$0.434649\pi$
$102$	0	0
$103$	− 5.30917e7i	− 0.471713i	−0.971788	−	0.235856i	$-0.924211\pi$
$103$	− 5.30917e7i	− 0.471713i	0.971788	−	0.235856i	$-0.0757895\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.31590e6i	− 0.0329258i	−0.999864	−	0.0164629i	$-0.994759\pi$
$107$	− 4.31590e6i	− 0.0329258i	0.999864	−	0.0164629i	$-0.00524054\pi$
$108$	0	0
$109$	−1.66924e8	−1.18253	−0.591267	−	0.806476i	$-0.701372\pi$
$109$	−1.66924e8	−1.18253	−0.591267	+	0.806476i	$0.701372\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.37562e7	−0.575024	−0.287512	−	0.957777i	$-0.592828\pi$
$113$	−9.37562e7	−0.575024	−0.287512	+	0.957777i	$0.592828\pi$
$114$	0	0
$115$	− 3.04609e8i	− 1.74161i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.12061e8i	2.55349i
$120$	0	0
$121$	1.87842e8	0.876299
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−2.48660e8	−1.01851
$126$	0	0
$127$	− 1.25168e8i	− 0.481148i	−0.970631	−	0.240574i	$-0.922664\pi$
$127$	− 1.25168e8i	− 0.481148i	0.970631	−	0.240574i	$-0.0773357\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 2.33675e8i	− 0.793465i	−0.917934	−	0.396732i	$-0.870144\pi$
$131$	− 2.33675e8i	− 0.793465i	0.917934	−	0.396732i	$-0.129856\pi$
$132$	0	0
$133$	3.98859e8	1.27471
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2.03282e8	−0.577054	−0.288527	−	0.957472i	$-0.593166\pi$
$137$	−2.03282e8	−0.577054	−0.288527	+	0.957472i	$0.593166\pi$
$138$	0	0
$139$	− 2.65268e8i	− 0.710600i	−0.934752	−	0.355300i	$-0.884379\pi$
$139$	− 2.65268e8i	− 0.710600i	0.934752	−	0.355300i	$-0.115621\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.35136e8i	0.323166i
$144$	0	0
$145$	−6.18297e8	−1.39870
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.75140e8	−0.558224	−0.279112	−	0.960259i	$-0.590040\pi$
$149$	−2.75140e8	−0.558224	−0.279112	+	0.960259i	$0.590040\pi$
$150$	0	0
$151$	− 3.30226e8i	− 0.635190i	−0.948227	−	0.317595i	$-0.897125\pi$
$151$	− 3.30226e8i	− 0.635190i	0.948227	−	0.317595i	$-0.102875\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9.28901e8i	− 1.60932i
$156$	0	0
$157$	1.14620e9	1.88653	0.943264	−	0.332044i	$-0.107738\pi$
$157$	1.14620e9	1.88653	0.943264	+	0.332044i	$0.107738\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.63623e9	2.43524
$162$	0	0
$163$	6.30361e8i	0.892974i	0.894790	+	0.446487i	$0.147325\pi$
$163$	6.30361e8i	0.892974i	−0.894790	+	0.446487i	$0.852675\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.67526e8i	− 0.472522i	−0.971690	−	0.236261i	$-0.924078\pi$
$167$	− 3.67526e8i	− 0.472522i	0.971690	−	0.236261i	$-0.0759220\pi$
$168$	0	0
$169$	−1.27037e8	−0.155734
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.40226e9	1.56547	0.782735	−	0.622355i	$-0.213824\pi$
$173$	1.40226e9	1.56547	0.782735	+	0.622355i	$0.213824\pi$
$174$	0	0
$175$	− 5.00656e7i	− 0.0533811i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	6.79989e8i	0.662353i	0.943569	+	0.331177i	$0.107446\pi$
$179$	6.79989e8i	0.662353i	−0.943569	+	0.331177i	$0.892554\pi$
$180$	0	0
$181$	2.02898e9	1.89044	0.945219	−	0.326437i	$-0.105848\pi$
$181$	2.02898e9	1.89044	0.945219	+	0.326437i	$0.105848\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.49479e9	−1.27612
$186$	0	0
$187$	8.01165e8i	0.655173i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	1.78138e9i	1.33851i	0.743031	+	0.669257i	$0.233387\pi$
$191$	1.78138e9i	1.33851i	−0.743031	+	0.669257i	$0.766613\pi$
$192$	0	0
$193$	−6.58917e6	−0.00474899	−0.00237450	−	0.999997i	$-0.500756\pi$
$193$	−6.58917e6	−0.00474899	−0.00237450	+	0.999997i	$0.500756\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.39585e9	−0.926777	−0.463388	−	0.886155i	$-0.653367\pi$
$197$	−1.39585e9	−0.926777	−0.463388	+	0.886155i	$0.653367\pi$
$198$	0	0
$199$	− 1.55647e9i	− 0.992493i	−0.868182	−	0.496246i	$-0.834711\pi$
$199$	− 1.55647e9i	− 0.992493i	0.868182	−	0.496246i	$-0.165289\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.32124e9i	− 1.95576i
$204$	0	0
$205$	3.34374e9	1.89329
$206$	0	0
$207$	0	0
$208$	0	0
$209$	6.24050e8	0.327065
$210$	0	0
$211$	− 1.29163e9i	− 0.651641i	−0.945432	−	0.325821i	$-0.894359\pi$
$211$	− 1.29163e9i	− 0.651641i	0.945432	−	0.325821i	$-0.105641\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.68309e9i	− 0.787686i
$216$	0	0
$217$	4.98967e9	2.25026
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.08299e9	1.71163
$222$	0	0
$223$	− 1.11728e9i	− 0.451797i	−0.974151	−	0.225899i	$-0.927468\pi$
$223$	− 1.11728e9i	− 0.451797i	0.974151	−	0.225899i	$-0.0725318\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.41579e9i	1.66305i	0.555488	+	0.831525i	$0.312532\pi$
$227$	4.41579e9i	1.66305i	−0.555488	+	0.831525i	$0.687468\pi$
$228$	0	0
$229$	−1.35159e9	−0.491478	−0.245739	−	0.969336i	$-0.579031\pi$
$229$	−1.35159e9	−0.491478	−0.245739	+	0.969336i	$0.579031\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.28517e9	−0.436050	−0.218025	−	0.975943i	$-0.569961\pi$
$233$	−1.28517e9	−0.436050	−0.218025	+	0.975943i	$0.569961\pi$
$234$	0	0
$235$	− 5.41610e9i	− 1.77589i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	2.93925e9i	0.900836i	0.892818	+	0.450418i	$0.148725\pi$
$239$	2.93925e9i	0.900836i	−0.892818	+	0.450418i	$0.851275\pi$
$240$	0	0
$241$	−7.50002e8	−0.222328	−0.111164	−	0.993802i	$-0.535458\pi$
$241$	−7.50002e8	−0.222328	−0.111164	+	0.993802i	$0.535458\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.10481e9	−0.861728
$246$	0	0
$247$	− 3.18035e9i	− 0.854451i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 2.57131e9i	− 0.647827i	−0.946087	−	0.323914i	$-0.895001\pi$
$251$	− 2.57131e9i	− 0.647827i	0.946087	−	0.323914i	$-0.104999\pi$
$252$	0	0
$253$	2.56003e9	0.624832
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−6.76328e9	−1.55033	−0.775166	−	0.631758i	$-0.782334\pi$
$257$	−6.76328e9	−1.55033	−0.775166	+	0.631758i	$0.782334\pi$
$258$	0	0
$259$	− 8.02939e9i	− 1.78436i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 3.65031e9i	− 0.762968i	−0.924375	−	0.381484i	$-0.875413\pi$
$263$	− 3.65031e9i	− 0.762968i	0.924375	−	0.381484i	$-0.124587\pi$
$264$	0	0
$265$	4.81119e9	0.975593
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3.47857e9	−0.664342	−0.332171	−	0.943219i	$-0.607781\pi$
$269$	−3.47857e9	−0.664342	−0.332171	+	0.943219i	$0.607781\pi$
$270$	0	0
$271$	− 1.28427e9i	− 0.238110i	−0.992888	−	0.119055i	$-0.962014\pi$
$271$	− 1.28427e9i	− 0.238110i	0.992888	−	0.119055i	$-0.0379865\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.83321e7i	− 0.0136965i
$276$	0	0
$277$	5.23359e9	0.888957	0.444479	−	0.895789i	$-0.353389\pi$
$277$	5.23359e9	0.888957	0.444479	+	0.895789i	$0.353389\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.05516e9	0.329625	0.164812	−	0.986325i	$-0.447298\pi$
$281$	2.05516e9	0.329625	0.164812	+	0.986325i	$0.447298\pi$
$282$	0	0
$283$	6.47858e9i	1.01003i	0.863111	+	0.505015i	$0.168513\pi$
$283$	6.47858e9i	1.01003i	−0.863111	+	0.505015i	$0.831487\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.79612e10i	2.64733i
$288$	0	0
$289$	1.72306e10	2.47007
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−2.18836e9	−0.296926	−0.148463	−	0.988918i	$-0.547433\pi$
$293$	−2.18836e9	−0.296926	−0.148463	+	0.988918i	$0.547433\pi$
$294$	0	0
$295$	− 1.18852e9i	− 0.156934i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.30467e10i	− 1.63236i
$300$	0	0
$301$	9.04086e9	1.10140
$302$	0	0
$303$	0	0
$304$	0	0
$305$	9.77396e9	1.12946
$306$	0	0
$307$	− 2.47814e9i	− 0.278980i	−0.990224	−	0.139490i	$-0.955454\pi$
$307$	− 2.47814e9i	− 0.278980i	0.990224	−	0.139490i	$-0.0445462\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	1.00671e10i	1.07613i	0.842905	+	0.538063i	$0.180844\pi$
$311$	1.00671e10i	1.07613i	−0.842905	+	0.538063i	$0.819156\pi$
$312$	0	0
$313$	1.53438e10	1.59866	0.799331	−	0.600891i	$-0.205187\pi$
$313$	1.53438e10	1.59866	0.799331	+	0.600891i	$0.205187\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.82308e9	−0.180538	−0.0902690	−	0.995917i	$-0.528773\pi$
$317$	−1.82308e9	−0.180538	−0.0902690	+	0.995917i	$0.528773\pi$
$318$	0	0
$319$	− 5.19637e9i	− 0.501808i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 1.88550e10i	− 1.73227i
$324$	0	0
$325$	−3.99205e8	−0.0357818
$326$	0	0
$327$	0	0
$328$	0	0
$329$	2.90930e10	2.48316
$330$	0	0
$331$	− 1.12382e10i	− 0.936232i	−0.883667	−	0.468116i	$-0.844933\pi$
$331$	− 1.12382e10i	− 0.936232i	0.883667	−	0.468116i	$-0.155067\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 1.18782e10i	− 0.943132i
$336$	0	0
$337$	1.78276e10	1.38220	0.691102	−	0.722757i	$-0.257125\pi$
$337$	1.78276e10	1.38220	0.691102	+	0.722757i	$0.257125\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7.80678e9	0.577370
$342$	0	0
$343$	2.29550e9i	0.165844i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.35915e10i	0.937454i	0.883343	+	0.468727i	$0.155287\pi$
$347$	1.35915e10i	0.937454i	−0.883343	+	0.468727i	$0.844713\pi$
$348$	0	0
$349$	−5.02047e9	−0.338410	−0.169205	−	0.985581i	$-0.554120\pi$
$349$	−5.02047e9	−0.338410	−0.169205	+	0.985581i	$0.554120\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−2.58668e10	−1.66588	−0.832939	−	0.553365i	$-0.813343\pi$
$353$	−2.58668e10	−1.66588	−0.832939	+	0.553365i	$0.813343\pi$
$354$	0	0
$355$	− 1.23196e9i	− 0.0775683i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 2.48530e10i	− 1.49624i	−0.663564	−	0.748119i	$-0.730957\pi$
$359$	− 2.48530e10i	− 1.49624i	0.663564	−	0.748119i	$-0.269043\pi$
$360$	0	0
$361$	2.29688e9	0.135241
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.76814e9	−0.324985
$366$	0	0
$367$	2.82782e9i	0.155879i	0.996958	+	0.0779395i	$0.0248341\pi$
$367$	2.82782e9i	0.155879i	−0.996958	+	0.0779395i	$0.975166\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.58437e10i	1.36414i
$372$	0	0
$373$	9.32666e9	0.481827	0.240913	−	0.970547i	$-0.422553\pi$
$373$	9.32666e9	0.481827	0.240913	+	0.970547i	$0.422553\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−2.64823e10	−1.31096
$378$	0	0
$379$	1.37528e10i	0.666555i	0.942829	+	0.333277i	$0.108155\pi$
$379$	1.37528e10i	0.666555i	−0.942829	+	0.333277i	$0.891845\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 1.40154e10i	− 0.651346i	−0.945483	−	0.325673i	$-0.894409\pi$
$383$	− 1.40154e10i	− 0.651346i	0.945483	−	0.325673i	$-0.105591\pi$
$384$	0	0
$385$	−1.03841e10	−0.472635
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.17057e10	1.82136	0.910682	−	0.413109i	$-0.135557\pi$
$389$	4.17057e10	1.82136	0.910682	+	0.413109i	$0.135557\pi$
$390$	0	0
$391$	− 7.73487e10i	− 3.30938i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.74969e10i	1.54030i
$396$	0	0
$397$	−1.51045e10	−0.608058	−0.304029	−	0.952663i	$-0.598332\pi$
$397$	−1.51045e10	−0.608058	−0.304029	+	0.952663i	$0.598332\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	4.94580e9	0.191275	0.0956377	−	0.995416i	$-0.469511\pi$
$401$	4.94580e9	0.191275	0.0956377	+	0.995416i	$0.469511\pi$
$402$	0	0
$403$	− 3.97858e10i	− 1.50837i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 1.25627e10i	− 0.457831i
$408$	0	0
$409$	2.39277e10	0.855084	0.427542	−	0.903996i	$-0.359380\pi$
$409$	2.39277e10	0.855084	0.427542	+	0.903996i	$0.359380\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	6.38424e9	0.219436
$414$	0	0
$415$	1.49585e10i	0.504308i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	7.98719e9i	0.259142i	0.991570	+	0.129571i	$0.0413600\pi$
$419$	7.98719e9i	0.259142i	−0.991570	+	0.129571i	$0.958640\pi$
$420$	0	0
$421$	2.96141e10	0.942691	0.471346	−	0.881949i	$-0.343768\pi$
$421$	2.96141e10	0.942691	0.471346	+	0.881949i	$0.343768\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.36672e9	−0.0725424
$426$	0	0
$427$	5.25016e10i	1.57929i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 3.32904e10i	− 0.964740i	−0.875968	−	0.482370i	$-0.839776\pi$
$431$	− 3.32904e10i	− 0.964740i	0.875968	−	0.482370i	$-0.160224\pi$
$432$	0	0
$433$	−4.61419e10	−1.31264	−0.656318	−	0.754484i	$-0.727887\pi$
$433$	−4.61419e10	−1.31264	−0.656318	+	0.754484i	$0.727887\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−6.02490e10	−1.65205
$438$	0	0
$439$	7.01864e8i	0.0188971i	0.999955	+	0.00944855i	$0.00300761\pi$
$439$	7.01864e8i	0.0188971i	−0.999955	+	0.00944855i	$0.996992\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 1.68426e10i	− 0.437315i	−0.975802	−	0.218658i	$-0.929832\pi$
$443$	− 1.68426e10i	− 0.437315i	0.975802	−	0.218658i	$-0.0701678\pi$
$444$	0	0
$445$	−2.15520e10	−0.549602
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.24578e9	0.0552564	0.0276282	−	0.999618i	$-0.491205\pi$
$449$	2.24578e9	0.0552564	0.0276282	+	0.999618i	$0.491205\pi$
$450$	0	0
$451$	2.81019e10i	0.679249i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	5.29206e10i	1.23475i
$456$	0	0
$457$	7.04798e9	0.161585	0.0807923	−	0.996731i	$-0.474255\pi$
$457$	7.04798e9	0.161585	0.0807923	+	0.996731i	$0.474255\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−6.84734e10	−1.51607	−0.758033	−	0.652216i	$-0.773839\pi$
$461$	−6.84734e10	−1.51607	−0.758033	+	0.652216i	$0.773839\pi$
$462$	0	0
$463$	4.15997e10i	0.905245i	0.891702	+	0.452622i	$0.149511\pi$
$463$	4.15997e10i	0.905245i	−0.891702	+	0.452622i	$0.850489\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	6.11460e10i	1.28558i	0.766041	+	0.642792i	$0.222224\pi$
$467$	6.11460e10i	1.28558i	−0.766041	+	0.642792i	$0.777776\pi$
$468$	0	0
$469$	6.38049e10	1.31875
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.41452e10	0.282596
$474$	0	0
$475$	1.84351e9i	0.0362135i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	2.95726e10i	0.561756i	0.959743	+	0.280878i	$0.0906256\pi$
$479$	2.95726e10i	0.561756i	−0.959743	+	0.280878i	$0.909374\pi$
$480$	0	0
$481$	−6.40233e10	−1.19607
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.33515e10	−0.422034
$486$	0	0
$487$	1.03670e11i	1.84306i	0.388308	+	0.921530i	$0.373060\pi$
$487$	1.03670e11i	1.84306i	−0.388308	+	0.921530i	$0.626940\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 5.09520e10i	− 0.876668i	−0.898812	−	0.438334i	$-0.855569\pi$
$491$	− 5.09520e10i	− 0.876668i	0.898812	−	0.438334i	$-0.144431\pi$
$492$	0	0
$493$	−1.57003e11	−2.65778
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.61760e9	0.108461
$498$	0	0
$499$	1.01548e11i	1.63784i	0.573909	+	0.818919i	$0.305426\pi$
$499$	1.01548e11i	1.63784i	−0.573909	+	0.818919i	$0.694574\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.11264e10i	0.173813i	0.996216	+	0.0869064i	$0.0276981\pi$
$503$	1.11264e10i	0.173813i	−0.996216	+	0.0869064i	$0.972302\pi$
$504$	0	0
$505$	2.59965e10	0.399714
$506$	0	0
$507$	0	0
$508$	0	0
$509$	1.07735e11	1.60504	0.802521	−	0.596623i	$-0.203491\pi$
$509$	1.07735e11	1.60504	0.802521	+	0.596623i	$0.203491\pi$
$510$	0	0
$511$	− 3.09840e10i	− 0.454417i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.25298e10i	− 0.462437i
$516$	0	0
$517$	4.55186e10	0.637129
$518$	0	0
$519$	0	0
$520$	0	0
$521$	4.84664e10	0.657794	0.328897	−	0.944366i	$-0.393323\pi$
$521$	4.84664e10	0.657794	0.328897	+	0.944366i	$0.393323\pi$
$522$	0	0
$523$	− 3.15532e10i	− 0.421732i	−0.977515	−	0.210866i	$-0.932372\pi$
$523$	− 3.15532e10i	− 0.421732i	0.977515	−	0.210866i	$-0.0676284\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.35874e11i	− 3.05800i
$528$	0	0
$529$	−1.68848e11	−2.15612
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1.43216e11	1.77453
$534$	0	0
$535$	− 2.64439e9i	− 0.0322783i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 2.60938e10i	− 0.309159i
$540$	0	0
$541$	4.36968e10	0.510107	0.255053	−	0.966927i	$-0.417907\pi$
$541$	4.36968e10	0.510107	0.255053	+	0.966927i	$0.417907\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−1.02276e11	−1.15928
$546$	0	0
$547$	− 1.09757e11i	− 1.22598i	−0.790089	−	0.612992i	$-0.789966\pi$
$547$	− 1.09757e11i	− 1.22598i	0.790089	−	0.612992i	$-0.210034\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1.22294e11i	1.32678i
$552$	0	0
$553$	−2.01418e11	−2.15376
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.22551e8	−0.00231211	−0.00115606	−	0.999999i	$-0.500368\pi$
$557$	−2.22551e8	−0.00231211	−0.00115606	+	0.999999i	$0.500368\pi$
$558$	0	0
$559$	− 7.20884e10i	− 0.738275i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7.82950e10i	0.779292i	0.920965	+	0.389646i	$0.127403\pi$
$563$	7.82950e10i	0.779292i	−0.920965	+	0.389646i	$0.872597\pi$
$564$	0	0
$565$	−5.74453e10	−0.563717
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−3.90607e9	−0.0372641	−0.0186321	−	0.999826i	$-0.505931\pi$
$569$	−3.90607e9	−0.0372641	−0.0186321	+	0.999826i	$0.505931\pi$
$570$	0	0
$571$	9.58527e10i	0.901695i	0.892601	+	0.450848i	$0.148878\pi$
$571$	9.58527e10i	0.901695i	−0.892601	+	0.450848i	$0.851122\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	7.56260e9i	0.0691830i
$576$	0	0
$577$	−5.40624e10	−0.487744	−0.243872	−	0.969807i	$-0.578418\pi$
$577$	−5.40624e10	−0.487744	−0.243872	+	0.969807i	$0.578418\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−8.03509e10	−0.705158
$582$	0	0
$583$	4.04348e10i	0.350010i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.44039e11i	1.21319i	0.795012	+	0.606594i	$0.207465\pi$
$587$	1.44039e11i	1.21319i	−0.795012	+	0.606594i	$0.792535\pi$
$588$	0	0
$589$	−1.83729e11	−1.52657
$590$	0	0
$591$	0	0
$592$	0	0
$593$	1.76145e11	1.42447	0.712233	−	0.701943i	$-0.247684\pi$
$593$	1.76145e11	1.42447	0.712233	+	0.701943i	$0.247684\pi$
$594$	0	0
$595$	3.13745e11i	2.50328i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	7.62362e10i	0.592180i	0.955160	+	0.296090i	$0.0956829\pi$
$599$	7.62362e10i	0.592180i	−0.955160	+	0.296090i	$0.904317\pi$
$600$	0	0
$601$	−9.18734e10	−0.704193	−0.352096	−	0.935964i	$-0.614531\pi$
$601$	−9.18734e10	−0.704193	−0.352096	+	0.935964i	$0.614531\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.15093e11	0.859067
$606$	0	0
$607$	− 1.40049e10i	− 0.103163i	−0.998669	−	0.0515816i	$-0.983574\pi$
$607$	− 1.40049e10i	− 0.103163i	0.998669	−	0.0515816i	$-0.0164262\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 2.31977e11i	− 1.66449i
$612$	0	0
$613$	1.95047e10	0.138133	0.0690664	−	0.997612i	$-0.477998\pi$
$613$	1.95047e10	0.138133	0.0690664	+	0.997612i	$0.477998\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.02420e11	−0.706717	−0.353359	−	0.935488i	$-0.614960\pi$
$617$	−1.02420e11	−0.706717	−0.353359	+	0.935488i	$0.614960\pi$
$618$	0	0
$619$	− 1.34870e11i	− 0.918659i	−0.888266	−	0.459329i	$-0.848090\pi$
$619$	− 1.34870e11i	− 0.918659i	0.888266	−	0.459329i	$-0.151910\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 1.15769e11i	− 0.768491i
$624$	0	0
$625$	−1.46414e11	−0.959541
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.79568e11	−2.42486
$630$	0	0
$631$	1.03098e11i	0.650331i	0.945657	+	0.325166i	$0.105420\pi$
$631$	1.03098e11i	0.650331i	−0.945657	+	0.325166i	$0.894580\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.66917e10i	− 0.471687i
$636$	0	0
$637$	−1.32982e11	−0.807673
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−1.66304e11	−0.985075	−0.492538	−	0.870291i	$-0.663931\pi$
$641$	−1.66304e11	−0.985075	−0.492538	+	0.870291i	$0.663931\pi$
$642$	0	0
$643$	1.90065e11i	1.11188i	0.831223	+	0.555940i	$0.187641\pi$
$643$	1.90065e11i	1.11188i	−0.831223	+	0.555940i	$0.812359\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1.30909e11i	0.747056i	0.927619	+	0.373528i	$0.121852\pi$
$647$	1.30909e11i	0.747056i	−0.927619	+	0.373528i	$0.878148\pi$
$648$	0	0
$649$	9.98870e9	0.0563028
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.43274e10	0.0787981	0.0393991	−	0.999224i	$-0.487456\pi$
$653$	1.43274e10	0.0787981	0.0393991	+	0.999224i	$0.487456\pi$
$654$	0	0
$655$	− 1.43175e11i	− 0.777862i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	4.63518e10i	0.245768i	0.992421	+	0.122884i	$0.0392143\pi$
$659$	4.63518e10i	0.245768i	−0.992421	+	0.122884i	$0.960786\pi$
$660$	0	0
$661$	4.00756e10	0.209930	0.104965	−	0.994476i	$-0.466527\pi$
$661$	4.00756e10	0.209930	0.104965	+	0.994476i	$0.466527\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.44385e11	1.24965
$666$	0	0
$667$	5.01685e11i	2.53471i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	8.21435e10i	0.405213i
$672$	0	0
$673$	−2.04071e10	−0.0994765	−0.0497382	−	0.998762i	$-0.515839\pi$
$673$	−2.04071e10	−0.0994765	−0.0497382	+	0.998762i	$0.515839\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.23384e11	0.587359	0.293679	−	0.955904i	$-0.405120\pi$
$677$	1.23384e11	0.587359	0.293679	+	0.955904i	$0.405120\pi$
$678$	0	0
$679$	− 1.25434e11i	− 0.590116i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	2.36674e11i	1.08760i	0.839215	+	0.543799i	$0.183015\pi$
$683$	2.36674e11i	1.08760i	−0.839215	+	0.543799i	$0.816985\pi$
$684$	0	0
$685$	−1.24553e11	−0.565707
$686$	0	0
$687$	0	0
$688$	0	0
$689$	2.06068e11	0.914395
$690$	0	0
$691$	8.80842e9i	0.0386354i	0.999813	+	0.0193177i	$0.00614940\pi$
$691$	8.80842e9i	0.0386354i	−0.999813	+	0.0193177i	$0.993851\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.62532e11i	− 0.696626i
$696$	0	0
$697$	8.49069e11	3.59759
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−1.98395e11	−0.821599	−0.410799	−	0.911726i	$-0.634750\pi$
$701$	−1.98395e11	−0.821599	−0.410799	+	0.911726i	$0.634750\pi$
$702$	0	0
$703$	2.95656e11i	1.21050i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.39642e11i	0.558907i
$708$	0	0
$709$	−3.83136e11	−1.51624	−0.758120	−	0.652116i	$-0.773882\pi$
$709$	−3.83136e11	−1.51624	−0.758120	+	0.652116i	$0.773882\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−7.53708e11	−2.91639
$714$	0	0
$715$	8.27990e10i	0.316812i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.23592e11i	0.836644i	0.908299	+	0.418322i	$0.137382\pi$
$719$	2.23592e11i	0.836644i	−0.908299	+	0.418322i	$0.862618\pi$
$720$	0	0
$721$	1.74736e11	0.646610
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.53506e10	0.0555614
$726$	0	0
$727$	− 5.07241e11i	− 1.81583i	−0.419149	−	0.907917i	$-0.637672\pi$
$727$	− 5.07241e11i	− 1.81583i	0.419149	−	0.907917i	$-0.362328\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 4.27383e11i	− 1.49674i
$732$	0	0
$733$	−2.23500e11	−0.774216	−0.387108	−	0.922034i	$-0.626526\pi$
$733$	−2.23500e11	−0.774216	−0.387108	+	0.922034i	$0.626526\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	9.98285e10	0.338364
$738$	0	0
$739$	3.46124e9i	0.0116052i	0.999983	+	0.00580261i	$0.00184704\pi$
$739$	3.46124e9i	0.0116052i	−0.999983	+	0.00580261i	$0.998153\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.33286e11i	1.42174i	0.703325	+	0.710869i	$0.251698\pi$
$743$	4.33286e11i	1.42174i	−0.703325	+	0.710869i	$0.748302\pi$
$744$	0	0
$745$	−1.68581e11	−0.547247
$746$	0	0
$747$	0	0
$748$	0	0
$749$	1.42046e10	0.0451338
$750$	0	0
$751$	− 3.23570e11i	− 1.01720i	−0.861002	−	0.508602i	$-0.830163\pi$
$751$	− 3.23570e11i	− 1.01720i	0.861002	−	0.508602i	$-0.169837\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.02333e11i	− 0.622699i
$756$	0	0
$757$	−2.00734e11	−0.611276	−0.305638	−	0.952148i	$-0.598870\pi$
$757$	−2.00734e11	−0.611276	−0.305638	+	0.952148i	$0.598870\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.48440e11	1.33710	0.668552	−	0.743665i	$-0.266914\pi$
$761$	4.48440e11	1.33710	0.668552	+	0.743665i	$0.266914\pi$
$762$	0	0
$763$	− 5.49385e11i	− 1.62098i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.09055e10i	− 0.147090i
$768$	0	0
$769$	−3.40762e11	−0.974418	−0.487209	−	0.873285i	$-0.661985\pi$
$769$	−3.40762e11	−0.974418	−0.487209	+	0.873285i	$0.661985\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	2.90199e11	0.812788	0.406394	−	0.913698i	$-0.366786\pi$
$773$	2.90199e11	0.812788	0.406394	+	0.913698i	$0.366786\pi$
$774$	0	0
$775$	2.30620e10i	0.0639279i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 6.61363e11i	− 1.79593i
$780$	0	0
$781$	1.03538e10	0.0278289
$782$	0	0
$783$	0	0
$784$	0	0
$785$	7.02290e11	1.84943
$786$	0	0
$787$	4.36682e11i	1.13832i	0.822225	+	0.569162i	$0.192732\pi$
$787$	4.36682e11i	1.13832i	−0.822225	+	0.569162i	$0.807268\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 3.08572e11i	− 0.788227i
$792$	0	0
$793$	4.18628e11	1.05861
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.76639e11	0.685614	0.342807	−	0.939406i	$-0.388622\pi$
$797$	2.76639e11	0.685614	0.342807	+	0.939406i	$0.388622\pi$
$798$	0	0
$799$	− 1.37530e12i	− 3.37450i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 4.84773e10i	− 0.116594i
$804$	0	0
$805$	1.00254e12	2.38735
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−3.22838e11	−0.753686	−0.376843	−	0.926277i	$-0.622990\pi$
$809$	−3.22838e11	−0.753686	−0.376843	+	0.926277i	$0.622990\pi$
$810$	0	0
$811$	5.78580e11i	1.33746i	0.743506	+	0.668729i	$0.233161\pi$
$811$	5.78580e11i	1.33746i	−0.743506	+	0.668729i	$0.766839\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.86228e11i	0.875414i
$816$	0	0
$817$	−3.32900e11	−0.747182
$818$	0	0
$819$	0	0
$820$	0	0
$821$	5.83753e11	1.28486	0.642432	−	0.766343i	$-0.277926\pi$
$821$	5.83753e11	1.28486	0.642432	+	0.766343i	$0.277926\pi$
$822$	0	0
$823$	3.21624e10i	0.0701049i	0.999385	+	0.0350525i	$0.0111598\pi$
$823$	3.21624e10i	0.0701049i	−0.999385	+	0.0350525i	$0.988840\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 6.95669e11i	− 1.48724i	−0.668603	−	0.743620i	$-0.733107\pi$
$827$	− 6.95669e11i	− 1.48724i	0.668603	−	0.743620i	$-0.266893\pi$
$828$	0	0
$829$	5.89574e11	1.24830	0.624152	−	0.781303i	$-0.285445\pi$
$829$	5.89574e11	1.24830	0.624152	+	0.781303i	$0.285445\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−7.88397e11	−1.63744
$834$	0	0
$835$	− 2.25187e11i	− 0.463230i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8.12279e11i	1.63930i	0.572868	+	0.819648i	$0.305831\pi$
$839$	8.12279e11i	1.63930i	−0.572868	+	0.819648i	$0.694169\pi$
$840$	0	0
$841$	5.18076e11	1.03564
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−7.78367e10	−0.152671
$846$	0	0
$847$	6.18231e11i	1.20121i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1.21287e12i	2.31257i
$852$	0	0
$853$	4.85715e11	0.917456	0.458728	−	0.888577i	$-0.348305\pi$
$853$	4.85715e11	0.917456	0.458728	+	0.888577i	$0.348305\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−2.71818e11	−0.503912	−0.251956	−	0.967739i	$-0.581074\pi$
$857$	−2.71818e11	−0.503912	−0.251956	+	0.967739i	$0.581074\pi$
$858$	0	0
$859$	− 8.01412e11i	− 1.47192i	−0.677027	−	0.735958i	$-0.736732\pi$
$859$	− 8.01412e11i	− 1.47192i	0.677027	−	0.735958i	$-0.263268\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 3.81770e11i	− 0.688269i	−0.938920	−	0.344135i	$-0.888172\pi$
$863$	− 3.81770e11i	− 0.688269i	0.938920	−	0.344135i	$-0.111828\pi$
$864$	0	0
$865$	8.59180e11	1.53469
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.15136e11	−0.552610
$870$	0	0
$871$	− 5.08757e11i	− 0.883970i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 8.18396e11i	− 1.39615i
$876$	0	0
$877$	−8.40735e10	−0.142122	−0.0710609	−	0.997472i	$-0.522638\pi$
$877$	−8.40735e10	−0.142122	−0.0710609	+	0.997472i	$0.522638\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−5.14435e11	−0.853939	−0.426970	−	0.904266i	$-0.640419\pi$
$881$	−5.14435e11	−0.853939	−0.426970	+	0.904266i	$0.640419\pi$
$882$	0	0
$883$	6.56953e10i	0.108067i	0.998539	+	0.0540333i	$0.0172077\pi$
$883$	6.56953e10i	0.108067i	−0.998539	+	0.0540333i	$0.982792\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.09769e12i	− 1.77331i	−0.462435	−	0.886653i	$-0.653024\pi$
$887$	− 1.09769e12i	− 1.77331i	0.462435	−	0.886653i	$-0.346976\pi$
$888$	0	0
$889$	4.11956e11	0.659544
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−1.07126e12	−1.68457
$894$	0	0
$895$	4.16636e11i	0.649329i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	1.52988e12i	2.34217i
$900$	0	0
$901$	1.22169e12	1.85380
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.24317e12	1.85326
$906$	0	0
$907$	9.23663e11i	1.36485i	0.730956	+	0.682424i	$0.239074\pi$
$907$	9.23663e11i	1.36485i	−0.730956	+	0.682424i	$0.760926\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 2.79308e11i	− 0.405517i	−0.979229	−	0.202759i	$-0.935009\pi$
$911$	− 2.79308e11i	− 0.405517i	0.979229	−	0.202759i	$-0.0649906\pi$
$912$	0	0
$913$	−1.25716e11	−0.180929
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.69077e11	1.08766
$918$	0	0
$919$	− 5.33058e11i	− 0.747330i	−0.927564	−	0.373665i	$-0.878101\pi$
$919$	− 5.33058e11i	− 0.747330i	0.927564	−	0.373665i	$-0.121899\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 5.27662e10i	− 0.0727025i
$924$	0	0
$925$	3.71114e10	0.0506922
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−2.77797e11	−0.372961	−0.186481	−	0.982459i	$-0.559708\pi$
$929$	−2.77797e11	−0.372961	−0.186481	+	0.982459i	$0.559708\pi$
$930$	0	0
$931$	6.14104e11i	0.817416i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	4.90882e11i	0.642289i
$936$	0	0
$937$	1.47588e12	1.91467	0.957335	−	0.288981i	$-0.0933163\pi$
$937$	1.47588e12	1.91467	0.957335	+	0.288981i	$0.0933163\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−5.85392e11	−0.746601	−0.373301	−	0.927710i	$-0.621774\pi$
$941$	−5.85392e11	−0.746601	−0.373301	+	0.927710i	$0.621774\pi$
$942$	0	0
$943$	− 2.71310e12i	− 3.43099i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.56825e12i	1.94991i	0.222406	+	0.974954i	$0.428609\pi$
$947$	1.56825e12i	1.94991i	−0.222406	+	0.974954i	$0.571391\pi$
$948$	0	0
$949$	−2.47055e11	−0.304599
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.00177e12	1.21450	0.607250	−	0.794511i	$-0.292273\pi$
$953$	1.00177e12	1.21450	0.607250	+	0.794511i	$0.292273\pi$
$954$	0	0
$955$	1.09147e12i	1.31219i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 6.69046e11i	− 0.791010i
$960$	0	0
$961$	−1.44553e12	−1.69486
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.03725e9	−0.00465560
$966$	0	0
$967$	− 3.46931e11i	− 0.396769i	−0.980124	−	0.198384i	$-0.936431\pi$
$967$	− 3.46931e11i	− 0.396769i	0.980124	−	0.198384i	$-0.0635694\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.73571e11i	0.532731i	0.963872	+	0.266366i	$0.0858229\pi$
$971$	4.73571e11i	0.532731i	−0.963872	+	0.266366i	$0.914177\pi$
$972$	0	0
$973$	8.73055e11	0.974070
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−6.30375e11	−0.691864	−0.345932	−	0.938260i	$-0.612437\pi$
$977$	−6.30375e11	−0.691864	−0.345932	+	0.938260i	$0.612437\pi$
$978$	0	0
$979$	− 1.81130e11i	− 0.197179i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 7.46792e11i	− 0.799808i	−0.916557	−	0.399904i	$-0.869043\pi$
$983$	− 7.46792e11i	− 0.799808i	0.916557	−	0.399904i	$-0.130957\pi$
$984$	0	0
$985$	−8.55254e11	−0.908552
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.36565e12	−1.42743
$990$	0	0
$991$	− 1.18922e12i	− 1.23301i	−0.787352	−	0.616504i	$-0.788548\pi$
$991$	− 1.18922e12i	− 1.23301i	0.787352	−	0.616504i	$-0.211452\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 9.53662e11i	− 0.972976i
$996$	0	0
$997$	6.58941e11	0.666907	0.333454	−	0.942766i	$-0.391786\pi$
$997$	6.58941e11	0.666907	0.333454	+	0.942766i	$0.391786\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.9.g.h.271.10	yes	12
3.2	odd	2	inner	432.9.g.h.271.4	yes	12
4.3	odd	2	inner	432.9.g.h.271.9	yes	12
12.11	even	2	inner	432.9.g.h.271.3	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.9.g.h.271.3	✓	12	12.11	even	2	inner
432.9.g.h.271.4	yes	12	3.2	odd	2	inner
432.9.g.h.271.9	yes	12	4.3	odd	2	inner
432.9.g.h.271.10	yes	12	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.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+612.710 q^{5} +3291.22i q^{7} +5149.41i q^{11} +26243.0 q^{13} +155584. q^{17} -121189. i q^{19} -497151. i q^{23} -15211.9 q^{25} -1.00912e6 q^{29} -1.51605e6i q^{31} +2.01656e6i q^{35} -2.43964e6 q^{37} +5.45730e6 q^{41} -2.74696e6i q^{43} -8.83959e6i q^{47} -5.06734e6 q^{49} +7.85231e6 q^{53} +3.15509e6i q^{55} -1.93978e6i q^{59} +1.59520e7 q^{61} +1.60793e7 q^{65} -1.93864e7i q^{67} -2.01068e6i q^{71} -9.41414e6 q^{73} -1.69478e7 q^{77} +6.11984e7i q^{79} +2.44137e7i q^{83} +9.53278e7 q^{85} -3.51750e7 q^{89} +8.63715e7i q^{91} -7.42535e7i q^{95} -3.81118e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 57036 q^{13} + 306996 q^{25} - 2032140 q^{37} - 6810648 q^{49} + 28116276 q^{61} + 33515148 q^{73} + 95803200 q^{85} + 90106572 q^{97}+O(q^{100}) \) 12 * q + 57036 * q^13 + 306996 * q^25 - 2032140 * q^37 - 6810648 * q^49 + 28116276 * q^61 + 33515148 * q^73 + 95803200 * q^85 + 90106572 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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