LMFDB - Embedded newform 432.7.e.k.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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 75x^{4} - 90x^{3} + 1861x^{2} + 7864x + 10098 $ x^6 - 2x^5 - 75x^4 - 90x^3 + 1861x^2 + 7864*x + 10098
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{23}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-4.12061 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	432.161
Dual form			432.7.e.k.161.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-146.431i q^{5} -472.768 q^{7} +O(q^{10})$	$q-146.431i q^{5} -472.768 q^{7} +1221.87i q^{11} +949.511 q^{13} -3068.90i q^{17} -7964.04 q^{19} -1396.08i q^{23} -5817.02 q^{25} +37594.9i q^{29} -22099.6 q^{31} +69227.9i q^{35} -62468.2 q^{37} +2792.72i q^{41} +100025. q^{43} +123775. i q^{47} +105861. q^{49} -253157. i q^{53} +178920. q^{55} +113481. i q^{59} +342125. q^{61} -139038. i q^{65} -466251. q^{67} -394842. i q^{71} +50308.2 q^{73} -577663. i q^{77} +688552. q^{79} +986952. i q^{83} -449382. q^{85} -676556. i q^{89} -448899. q^{91} +1.16618e6i q^{95} +131262. q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 114 q^{7}+O(q^{10}) $ 6 * q + 114 * q^7	$ 6 q + 114 q^{7} + 4026 q^{13} - 1854 q^{19} + 7686 q^{25} - 708 q^{31} - 66198 q^{37} + 116172 q^{43} + 237072 q^{49} + 158544 q^{55} + 170154 q^{61} + 84162 q^{67} + 748986 q^{73} + 560130 q^{79} + 1073040 q^{85} + 2795982 q^{91} + 1967418 q^{97}+O(q^{100}) $ 6 * q + 114 * q^7 + 4026 * q^13 - 1854 * q^19 + 7686 * q^25 - 708 * q^31 - 66198 * q^37 + 116172 * q^43 + 237072 * q^49 + 158544 * q^55 + 170154 * q^61 + 84162 * q^67 + 748986 * q^73 + 560130 * q^79 + 1073040 * q^85 + 2795982 * q^91 + 1967418 * 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
$5$	− 146.431i	− 1.17145i	−0.810511	−	0.585724i	$-0.800810\pi$
$5$	− 146.431i	− 1.17145i	0.810511	−	0.585724i	$-0.199190\pi$
$6$	0	0
$7$	−472.768	−1.37833	−0.689167	−	0.724603i	$-0.742023\pi$
$7$	−472.768	−1.37833	−0.689167	+	0.724603i	$0.742023\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1221.87i	0.918011i	0.888433	+	0.459006i	$0.151794\pi$
$11$	1221.87i	0.918011i	−0.888433	+	0.459006i	$0.848206\pi$
$12$	0	0
$13$	949.511	0.432185	0.216093	−	0.976373i	$-0.430669\pi$
$13$	949.511	0.432185	0.216093	+	0.976373i	$0.430669\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 3068.90i	− 0.624650i	−0.949975	−	0.312325i	$-0.898892\pi$
$17$	− 3068.90i	− 0.624650i	0.949975	−	0.312325i	$-0.101108\pi$
$18$	0	0
$19$	−7964.04	−1.16111	−0.580554	−	0.814222i	$-0.697164\pi$
$19$	−7964.04	−1.16111	−0.580554	+	0.814222i	$0.697164\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 1396.08i	− 0.114743i	−0.998353	−	0.0573716i	$-0.981728\pi$
$23$	− 1396.08i	− 0.114743i	0.998353	−	0.0573716i	$-0.0182720\pi$
$24$	0	0
$25$	−5817.02	−0.372289
$26$	0	0
$27$	0	0
$28$	0	0
$29$	37594.9i	1.54147i	0.637157	+	0.770734i	$0.280110\pi$
$29$	37594.9i	1.54147i	−0.637157	+	0.770734i	$0.719890\pi$
$30$	0	0
$31$	−22099.6	−0.741821	−0.370910	−	0.928669i	$-0.620954\pi$
$31$	−22099.6	−0.741821	−0.370910	+	0.928669i	$0.620954\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	69227.9i	1.61465i
$36$	0	0
$37$	−62468.2	−1.23326	−0.616629	−	0.787254i	$-0.711502\pi$
$37$	−62468.2	−1.23326	−0.616629	+	0.787254i	$0.711502\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2792.72i	0.0405206i	0.999795	+	0.0202603i	$0.00644949\pi$
$41$	2792.72i	0.0405206i	−0.999795	+	0.0202603i	$0.993551\pi$
$42$	0	0
$43$	100025.	1.25806	0.629031	−	0.777380i	$-0.283452\pi$
$43$	100025.	1.25806	0.629031	+	0.777380i	$0.283452\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	123775.i	1.19217i	0.802922	+	0.596085i	$0.203278\pi$
$47$	123775.i	1.19217i	−0.802922	+	0.596085i	$0.796722\pi$
$48$	0	0
$49$	105861.	0.899804
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 253157.i	− 1.70044i	−0.526427	−	0.850221i	$-0.676468\pi$
$53$	− 253157.i	− 1.70044i	0.526427	−	0.850221i	$-0.323532\pi$
$54$	0	0
$55$	178920.	1.07540
$56$	0	0
$57$	0	0
$58$	0	0
$59$	113481.i	0.552542i	0.961080	+	0.276271i	$0.0890988\pi$
$59$	113481.i	0.552542i	−0.961080	+	0.276271i	$0.910901\pi$
$60$	0	0
$61$	342125.	1.50728	0.753642	−	0.657285i	$-0.228295\pi$
$61$	342125.	1.50728	0.753642	+	0.657285i	$0.228295\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 139038.i	− 0.506282i
$66$	0	0
$67$	−466251.	−1.55023	−0.775114	−	0.631822i	$-0.782308\pi$
$67$	−466251.	−1.55023	−0.775114	+	0.631822i	$0.782308\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 394842.i	− 1.10319i	−0.834113	−	0.551593i	$-0.814020\pi$
$71$	− 394842.i	− 1.10319i	0.834113	−	0.551593i	$-0.185980\pi$
$72$	0	0
$73$	50308.2	0.129321	0.0646607	−	0.997907i	$-0.479403\pi$
$73$	50308.2	0.129321	0.0646607	+	0.997907i	$0.479403\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 577663.i	− 1.26533i
$78$	0	0
$79$	688552.	1.39655	0.698274	−	0.715831i	$-0.253952\pi$
$79$	688552.	1.39655	0.698274	+	0.715831i	$0.253952\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	986952.i	1.72608i	0.505133	+	0.863042i	$0.331444\pi$
$83$	986952.i	1.72608i	−0.505133	+	0.863042i	$0.668556\pi$
$84$	0	0
$85$	−449382.	−0.731744
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 676556.i	− 0.959696i	−0.877352	−	0.479848i	$-0.840692\pi$
$89$	− 676556.i	− 0.959696i	0.877352	−	0.479848i	$-0.159308\pi$
$90$	0	0
$91$	−448899.	−0.595696
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.16618e6i	1.36018i
$96$	0	0
$97$	131262.	0.143822	0.0719109	−	0.997411i	$-0.477090\pi$
$97$	131262.	0.143822	0.0719109	+	0.997411i	$0.477090\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.92628e6i	− 1.86963i	−0.355135	−	0.934815i	$-0.615565\pi$
$101$	− 1.92628e6i	− 1.86963i	0.355135	−	0.934815i	$-0.384435\pi$
$102$	0	0
$103$	1.54921e6	1.41774	0.708872	−	0.705337i	$-0.249204\pi$
$103$	1.54921e6	1.41774	0.708872	+	0.705337i	$0.249204\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 146790.i	− 0.119824i	−0.998204	−	0.0599122i	$-0.980918\pi$
$107$	− 146790.i	− 0.119824i	0.998204	−	0.0599122i	$-0.0190821\pi$
$108$	0	0
$109$	609442.	0.470601	0.235300	−	0.971923i	$-0.424393\pi$
$109$	609442.	0.470601	0.235300	+	0.971923i	$0.424393\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.31847e6i	0.913765i	0.889527	+	0.456883i	$0.151034\pi$
$113$	1.31847e6i	0.913765i	−0.889527	+	0.456883i	$0.848966\pi$
$114$	0	0
$115$	−204429.	−0.134416
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.45088e6i	0.860976i
$120$	0	0
$121$	278588.	0.157255
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 1.43619e6i	− 0.735330i
$126$	0	0
$127$	1.42111e6	0.693769	0.346885	−	0.937908i	$-0.387240\pi$
$127$	1.42111e6	0.693769	0.346885	+	0.937908i	$0.387240\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 890155.i	− 0.395961i	−0.980206	−	0.197980i	$-0.936562\pi$
$131$	− 890155.i	− 0.395961i	0.980206	−	0.197980i	$-0.0634382\pi$
$132$	0	0
$133$	3.76515e6	1.60039
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.62366e6i	1.40924i	0.709584	+	0.704621i	$0.248883\pi$
$137$	3.62366e6i	1.40924i	−0.709584	+	0.704621i	$0.751117\pi$
$138$	0	0
$139$	2.23399e6	0.831835	0.415917	−	0.909402i	$-0.363461\pi$
$139$	2.23399e6	0.831835	0.415917	+	0.909402i	$0.363461\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.16018e6i	0.396751i
$144$	0	0
$145$	5.50505e6	1.80575
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.64931e6i	0.498591i	0.968427	+	0.249295i	$0.0801990\pi$
$149$	1.64931e6i	0.498591i	−0.968427	+	0.249295i	$0.919801\pi$
$150$	0	0
$151$	1.71652e6	0.498560	0.249280	−	0.968431i	$-0.419806\pi$
$151$	1.71652e6	0.498560	0.249280	+	0.968431i	$0.419806\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.23606e6i	0.869004i
$156$	0	0
$157$	4.26511e6	1.10213	0.551064	−	0.834463i	$-0.314222\pi$
$157$	4.26511e6	1.10213	0.551064	+	0.834463i	$0.314222\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	660023.i	0.158155i
$162$	0	0
$163$	5.71583e6	1.31983	0.659913	−	0.751342i	$-0.270593\pi$
$163$	5.71583e6	1.31983	0.659913	+	0.751342i	$0.270593\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.80831e6i	− 0.817679i	−0.912606	−	0.408839i	$-0.865934\pi$
$167$	− 3.80831e6i	− 0.817679i	0.912606	−	0.408839i	$-0.134066\pi$
$168$	0	0
$169$	−3.92524e6	−0.813216
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 6.01753e6i	− 1.16220i	−0.813833	−	0.581099i	$-0.802623\pi$
$173$	− 6.01753e6i	− 1.16220i	0.813833	−	0.581099i	$-0.197377\pi$
$174$	0	0
$175$	2.75010e6	0.513139
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 8.67107e6i	− 1.51187i	−0.654649	−	0.755933i	$-0.727183\pi$
$179$	− 8.67107e6i	− 1.51187i	0.654649	−	0.755933i	$-0.272817\pi$
$180$	0	0
$181$	−2.28705e6	−0.385691	−0.192845	−	0.981229i	$-0.561772\pi$
$181$	−2.28705e6	−0.385691	−0.192845	+	0.981229i	$0.561772\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.14728e6i	1.44470i
$186$	0	0
$187$	3.74981e6	0.573435
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.69718e6i	− 0.243573i	−0.992556	−	0.121786i	$-0.961138\pi$
$191$	− 1.69718e6i	− 0.243573i	0.992556	−	0.121786i	$-0.0388623\pi$
$192$	0	0
$193$	4.05686e6	0.564310	0.282155	−	0.959369i	$-0.408951\pi$
$193$	4.05686e6	0.564310	0.282155	+	0.959369i	$0.408951\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.33394e7i	1.74477i	0.488819	+	0.872385i	$0.337428\pi$
$197$	1.33394e7i	1.74477i	−0.488819	+	0.872385i	$0.662572\pi$
$198$	0	0
$199$	2.61779e6	0.332182	0.166091	−	0.986110i	$-0.446886\pi$
$199$	2.61779e6	0.332182	0.166091	+	0.986110i	$0.446886\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.77737e7i	− 2.12466i
$204$	0	0
$205$	408941.	0.0474677
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 9.73104e6i	− 1.06591i
$210$	0	0
$211$	−3.58068e6	−0.381170	−0.190585	−	0.981671i	$-0.561038\pi$
$211$	−3.58068e6	−0.381170	−0.190585	+	0.981671i	$0.561038\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 1.46467e7i	− 1.47375i
$216$	0	0
$217$	1.04480e7	1.02248
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.91396e6i	− 0.269964i
$222$	0	0
$223$	−5.10469e6	−0.460315	−0.230157	−	0.973153i	$-0.573924\pi$
$223$	−5.10469e6	−0.460315	−0.230157	+	0.973153i	$0.573924\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.04708e7i	0.895166i	0.894242	+	0.447583i	$0.147715\pi$
$227$	1.04708e7i	0.895166i	−0.894242	+	0.447583i	$0.852285\pi$
$228$	0	0
$229$	1.49147e7	1.24196	0.620982	−	0.783825i	$-0.286734\pi$
$229$	1.49147e7	1.24196	0.620982	+	0.783825i	$0.286734\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1.72665e7i	1.36501i	0.730881	+	0.682505i	$0.239109\pi$
$233$	1.72665e7i	1.36501i	−0.730881	+	0.682505i	$0.760891\pi$
$234$	0	0
$235$	1.81244e7	1.39656
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 7.99682e6i	− 0.585765i	−0.956148	−	0.292883i	$-0.905385\pi$
$239$	− 7.99682e6i	− 0.585765i	0.956148	−	0.292883i	$-0.0946145\pi$
$240$	0	0
$241$	1.13822e7	0.813158	0.406579	−	0.913616i	$-0.366722\pi$
$241$	1.13822e7	0.813158	0.406579	+	0.913616i	$0.366722\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.55013e7i	− 1.05407i
$246$	0	0
$247$	−7.56194e6	−0.501814
$248$	0	0
$249$	0	0
$250$	0	0
$251$	2.92977e7i	1.85273i	0.376626	+	0.926365i	$0.377084\pi$
$251$	2.92977e7i	1.85273i	−0.376626	+	0.926365i	$0.622916\pi$
$252$	0	0
$253$	1.70583e6	0.105336
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1.66872e7i	0.983068i	0.870858	+	0.491534i	$0.163564\pi$
$257$	1.66872e7i	0.983068i	−0.870858	+	0.491534i	$0.836436\pi$
$258$	0	0
$259$	2.95330e7	1.69984
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.45293e7i	0.798690i	0.916801	+	0.399345i	$0.130762\pi$
$263$	1.45293e7i	0.798690i	−0.916801	+	0.399345i	$0.869238\pi$
$264$	0	0
$265$	−3.70700e7	−1.99198
$266$	0	0
$267$	0	0
$268$	0	0
$269$	2.46847e7i	1.26815i	0.773271	+	0.634076i	$0.218619\pi$
$269$	2.46847e7i	1.26815i	−0.773271	+	0.634076i	$0.781381\pi$
$270$	0	0
$271$	−2.36672e6	−0.118916	−0.0594579	−	0.998231i	$-0.518937\pi$
$271$	−2.36672e6	−0.118916	−0.0594579	+	0.998231i	$0.518937\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 7.10765e6i	− 0.341765i
$276$	0	0
$277$	3.76419e7	1.77106	0.885528	−	0.464585i	$-0.153797\pi$
$277$	3.76419e7	1.77106	0.885528	+	0.464585i	$0.153797\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 3.23780e7i	− 1.45926i	−0.683844	−	0.729628i	$-0.739693\pi$
$281$	− 3.23780e7i	− 1.45926i	0.683844	−	0.729628i	$-0.260307\pi$
$282$	0	0
$283$	5.87574e6	0.259241	0.129620	−	0.991564i	$-0.458624\pi$
$283$	5.87574e6	0.259241	0.129620	+	0.991564i	$0.458624\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.32031e6i	− 0.0558509i
$288$	0	0
$289$	1.47194e7	0.609813
$290$	0	0
$291$	0	0
$292$	0	0
$293$	4.38235e7i	1.74222i	0.491084	+	0.871112i	$0.336601\pi$
$293$	4.38235e7i	1.74222i	−0.491084	+	0.871112i	$0.663399\pi$
$294$	0	0
$295$	1.66171e7	0.647274
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.32559e6i	− 0.0495904i
$300$	0	0
$301$	−4.72886e7	−1.73403
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 5.00976e7i	− 1.76570i
$306$	0	0
$307$	−3.49381e7	−1.20749	−0.603746	−	0.797177i	$-0.706326\pi$
$307$	−3.49381e7	−1.20749	−0.603746	+	0.797177i	$0.706326\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	2.37426e7i	0.789309i	0.918830	+	0.394654i	$0.129136\pi$
$311$	2.37426e7i	0.789309i	−0.918830	+	0.394654i	$0.870864\pi$
$312$	0	0
$313$	−2.99332e7	−0.976157	−0.488079	−	0.872800i	$-0.662302\pi$
$313$	−2.99332e7	−0.976157	−0.488079	+	0.872800i	$0.662302\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 2.61248e7i	− 0.820116i	−0.912060	−	0.410058i	$-0.865508\pi$
$317$	− 2.61248e7i	− 0.820116i	0.912060	−	0.410058i	$-0.134492\pi$
$318$	0	0
$319$	−4.59361e7	−1.41508
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.44409e7i	0.725286i
$324$	0	0
$325$	−5.52332e6	−0.160898
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 5.85167e7i	− 1.64321i
$330$	0	0
$331$	−5.33614e7	−1.47144	−0.735721	−	0.677284i	$-0.763157\pi$
$331$	−5.33614e7	−1.47144	−0.735721	+	0.677284i	$0.763157\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	6.82736e7i	1.81601i
$336$	0	0
$337$	−5.40614e7	−1.41253	−0.706265	−	0.707947i	$-0.749621\pi$
$337$	−5.40614e7	−1.41253	−0.706265	+	0.707947i	$0.749621\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.70029e7i	− 0.681000i
$342$	0	0
$343$	5.57298e6	0.138104
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 6.04628e7i	− 1.44710i	−0.690270	−	0.723552i	$-0.742508\pi$
$347$	− 6.04628e7i	− 1.44710i	0.690270	−	0.723552i	$-0.257492\pi$
$348$	0	0
$349$	2.36082e7	0.555375	0.277687	−	0.960671i	$-0.410432\pi$
$349$	2.36082e7	0.555375	0.277687	+	0.960671i	$0.410432\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4.52084e7i	− 1.02777i	−0.857860	−	0.513883i	$-0.828206\pi$
$353$	− 4.52084e7i	− 1.02777i	0.857860	−	0.513883i	$-0.171794\pi$
$354$	0	0
$355$	−5.78171e7	−1.29232
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 5.56998e7i	− 1.20384i	−0.798555	−	0.601922i	$-0.794402\pi$
$359$	− 5.56998e7i	− 1.20384i	0.798555	−	0.601922i	$-0.205598\pi$
$360$	0	0
$361$	1.63800e7	0.348171
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 7.36668e6i	− 0.151493i
$366$	0	0
$367$	2.23289e7	0.451720	0.225860	−	0.974160i	$-0.427481\pi$
$367$	2.23289e7	0.451720	0.225860	+	0.974160i	$0.427481\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.19684e8i	2.34378i
$372$	0	0
$373$	6.55588e7	1.26329	0.631647	−	0.775256i	$-0.282379\pi$
$373$	6.55588e7	1.26329	0.631647	+	0.775256i	$0.282379\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.56967e7i	0.666200i
$378$	0	0
$379$	1.34639e7	0.247317	0.123659	−	0.992325i	$-0.460537\pi$
$379$	1.34639e7	0.247317	0.123659	+	0.992325i	$0.460537\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	6.40566e7i	1.14017i	0.821587	+	0.570083i	$0.193089\pi$
$383$	6.40566e7i	1.14017i	−0.821587	+	0.570083i	$0.806911\pi$
$384$	0	0
$385$	−8.45877e7	−1.48226
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.40481e7i	0.408537i	0.978915	+	0.204269i	$0.0654816\pi$
$389$	2.40481e7i	0.408537i	−0.978915	+	0.204269i	$0.934518\pi$
$390$	0	0
$391$	−4.28444e6	−0.0716743
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.00825e8i	− 1.63598i
$396$	0	0
$397$	−9.17149e7	−1.46578	−0.732890	−	0.680347i	$-0.761829\pi$
$397$	−9.17149e7	−1.46578	−0.732890	+	0.680347i	$0.761829\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.25388e7i	0.349540i	0.984609	+	0.174770i	$0.0559182\pi$
$401$	2.25388e7i	0.349540i	−0.984609	+	0.174770i	$0.944082\pi$
$402$	0	0
$403$	−2.09838e7	−0.320604
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 7.63282e7i	− 1.13214i
$408$	0	0
$409$	−9.44843e7	−1.38099	−0.690494	−	0.723338i	$-0.742607\pi$
$409$	−9.44843e7	−1.38099	−0.690494	+	0.723338i	$0.742607\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 5.36500e7i	− 0.761588i
$414$	0	0
$415$	1.44520e8	2.02202
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.49683e7i	− 0.339427i	−0.985493	−	0.169714i	$-0.945716\pi$
$419$	− 2.49683e7i	− 0.339427i	0.985493	−	0.169714i	$-0.0542842\pi$
$420$	0	0
$421$	2.40805e7	0.322715	0.161358	−	0.986896i	$-0.448413\pi$
$421$	2.40805e7	0.322715	0.161358	+	0.986896i	$0.448413\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	1.78519e7i	0.232550i
$426$	0	0
$427$	−1.61746e8	−2.07754
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.79955e7i	0.224766i	0.993665	+	0.112383i	$0.0358484\pi$
$431$	1.79955e7i	0.224766i	−0.993665	+	0.112383i	$0.964152\pi$
$432$	0	0
$433$	8.92306e7	1.09913	0.549566	−	0.835450i	$-0.314793\pi$
$433$	8.92306e7	1.09913	0.549566	+	0.835450i	$0.314793\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.11184e7i	0.133229i
$438$	0	0
$439$	−8.22690e7	−0.972395	−0.486198	−	0.873849i	$-0.661616\pi$
$439$	−8.22690e7	−0.972395	−0.486198	+	0.873849i	$0.661616\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	9.40001e6i	0.108123i	0.998538	+	0.0540614i	$0.0172167\pi$
$443$	9.40001e6i	0.108123i	−0.998538	+	0.0540614i	$0.982783\pi$
$444$	0	0
$445$	−9.90687e7	−1.12423
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.54614e7i	0.281283i	0.990061	+	0.140641i	$0.0449165\pi$
$449$	2.54614e7i	0.281283i	−0.990061	+	0.140641i	$0.955084\pi$
$450$	0	0
$451$	−3.41235e6	−0.0371984
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.57327e7i	0.697826i
$456$	0	0
$457$	−1.59641e8	−1.67261	−0.836306	−	0.548262i	$-0.815290\pi$
$457$	−1.59641e8	−1.67261	−0.836306	+	0.548262i	$0.815290\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.38576e8i	1.41444i	0.706994	+	0.707219i	$0.250051\pi$
$461$	1.38576e8i	1.41444i	−0.706994	+	0.707219i	$0.749949\pi$
$462$	0	0
$463$	−1.41995e8	−1.43064	−0.715320	−	0.698797i	$-0.753719\pi$
$463$	−1.41995e8	−1.43064	−0.715320	+	0.698797i	$0.753719\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.45214e8i	1.42580i	0.701265	+	0.712901i	$0.252619\pi$
$467$	1.45214e8i	1.42580i	−0.701265	+	0.712901i	$0.747381\pi$
$468$	0	0
$469$	2.20429e8	2.13673
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.22218e8i	1.15492i
$474$	0	0
$475$	4.63269e7	0.432268
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.34347e8i	− 1.22242i	−0.791467	−	0.611212i	$-0.790682\pi$
$479$	− 1.34347e8i	− 1.22242i	0.791467	−	0.611212i	$-0.209318\pi$
$480$	0	0
$481$	−5.93143e7	−0.532996
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.92209e7i	− 0.168480i
$486$	0	0
$487$	−1.67976e8	−1.45432	−0.727160	−	0.686468i	$-0.759160\pi$
$487$	−1.67976e8	−1.45432	−0.727160	+	0.686468i	$0.759160\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 1.15045e8i	− 0.971903i	−0.873986	−	0.485951i	$-0.838473\pi$
$491$	− 1.15045e8i	− 0.971903i	0.873986	−	0.485951i	$-0.161527\pi$
$492$	0	0
$493$	1.15375e8	0.962877
$494$	0	0
$495$	0	0
$496$	0	0
$497$	1.86669e8i	1.52056i
$498$	0	0
$499$	4.11447e7	0.331141	0.165570	−	0.986198i	$-0.447054\pi$
$499$	4.11447e7	0.331141	0.165570	+	0.986198i	$0.447054\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 1.59099e8i	− 1.25015i	−0.780563	−	0.625077i	$-0.785067\pi$
$503$	− 1.59099e8i	− 1.25015i	0.780563	−	0.625077i	$-0.214933\pi$
$504$	0	0
$505$	−2.82067e8	−2.19017
$506$	0	0
$507$	0	0
$508$	0	0
$509$	3.22616e7i	0.244643i	0.992491	+	0.122321i	$0.0390338\pi$
$509$	3.22616e7i	0.244643i	−0.992491	+	0.122321i	$0.960966\pi$
$510$	0	0
$511$	−2.37841e7	−0.178248
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 2.26852e8i	− 1.66081i
$516$	0	0
$517$	−1.51237e8	−1.09442
$518$	0	0
$519$	0	0
$520$	0	0
$521$	9.53582e7i	0.674287i	0.941453	+	0.337144i	$0.109461\pi$
$521$	9.53582e7i	0.674287i	−0.941453	+	0.337144i	$0.890539\pi$
$522$	0	0
$523$	1.35938e8	0.950244	0.475122	−	0.879920i	$-0.342404\pi$
$523$	1.35938e8	0.950244	0.475122	+	0.879920i	$0.342404\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	6.78215e7i	0.463378i
$528$	0	0
$529$	1.46087e8	0.986834
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.65172e6i	0.0175124i
$534$	0	0
$535$	−2.14946e7	−0.140368
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.29349e8i	0.826030i
$540$	0	0
$541$	2.49641e8	1.57661	0.788305	−	0.615285i	$-0.210959\pi$
$541$	2.49641e8	1.57661	0.788305	+	0.615285i	$0.210959\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.92411e7i	− 0.551284i
$546$	0	0
$547$	1.25835e8	0.768844	0.384422	−	0.923157i	$-0.374401\pi$
$547$	1.25835e8	0.768844	0.384422	+	0.923157i	$0.374401\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 2.99407e8i	− 1.78981i
$552$	0	0
$553$	−3.25526e8	−1.92491
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.17755e8i	− 0.681415i	−0.940169	−	0.340708i	$-0.889333\pi$
$557$	− 1.17755e8i	− 0.681415i	0.940169	−	0.340708i	$-0.110667\pi$
$558$	0	0
$559$	9.49747e7	0.543716
$560$	0	0
$561$	0	0
$562$	0	0
$563$	6.01348e7i	0.336977i	0.985704	+	0.168489i	$0.0538886\pi$
$563$	6.01348e7i	0.336977i	−0.985704	+	0.168489i	$0.946111\pi$
$564$	0	0
$565$	1.93065e8	1.07043
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 5.11493e7i	− 0.277653i	−0.990317	−	0.138827i	$-0.955667\pi$
$569$	− 5.11493e7i	− 0.277653i	0.990317	−	0.138827i	$-0.0443330\pi$
$570$	0	0
$571$	−5815.98	−3.12402e−5 0	−1.56201e−5	−	1.00000i	$-0.500005\pi$
$571$	−5815.98	−3.12402e−5 0	−1.56201e−5		1.00000i	$0.500005\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	8.12103e6i	0.0427177i
$576$	0	0
$577$	7.49481e7	0.390152	0.195076	−	0.980788i	$-0.437505\pi$
$577$	7.49481e7	0.390152	0.195076	+	0.980788i	$0.437505\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.66600e8i	− 2.37912i
$582$	0	0
$583$	3.09325e8	1.56102
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.33156e8i	− 0.658332i	−0.944272	−	0.329166i	$-0.893232\pi$
$587$	− 1.33156e8i	− 0.658332i	0.944272	−	0.329166i	$-0.106768\pi$
$588$	0	0
$589$	1.76002e8	0.861334
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.05881e8i	0.987307i	0.869659	+	0.493654i	$0.164339\pi$
$593$	2.05881e8i	0.987307i	−0.869659	+	0.493654i	$0.835661\pi$
$594$	0	0
$595$	2.12454e8	1.00859
$596$	0	0
$597$	0	0
$598$	0	0
$599$	1.42986e8i	0.665293i	0.943052	+	0.332647i	$0.107942\pi$
$599$	1.42986e8i	0.665293i	−0.943052	+	0.332647i	$0.892058\pi$
$600$	0	0
$601$	3.02341e8	1.39275	0.696377	−	0.717677i	$-0.254794\pi$
$601$	3.02341e8	1.39275	0.696377	+	0.717677i	$0.254794\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.07938e7i	− 0.184216i
$606$	0	0
$607$	−1.95000e8	−0.871903	−0.435951	−	0.899970i	$-0.643588\pi$
$607$	−1.95000e8	−0.871903	−0.435951	+	0.899970i	$0.643588\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	1.17525e8i	0.515238i
$612$	0	0
$613$	4.31389e8	1.87279	0.936393	−	0.350954i	$-0.114142\pi$
$613$	4.31389e8	1.87279	0.936393	+	0.350954i	$0.114142\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 2.64193e8i	− 1.12478i	−0.826873	−	0.562389i	$-0.809882\pi$
$617$	− 2.64193e8i	− 1.12478i	0.826873	−	0.562389i	$-0.190118\pi$
$618$	0	0
$619$	2.51249e8	1.05933	0.529667	−	0.848206i	$-0.322317\pi$
$619$	2.51249e8	1.05933	0.529667	+	0.848206i	$0.322317\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.19854e8i	1.32278i
$624$	0	0
$625$	−3.01194e8	−1.23369
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.91709e8i	0.770354i
$630$	0	0
$631$	1.48143e8	0.589647	0.294823	−	0.955552i	$-0.404739\pi$
$631$	1.48143e8	0.589647	0.294823	+	0.955552i	$0.404739\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 2.08094e8i	− 0.812714i
$636$	0	0
$637$	1.00516e8	0.388882
$638$	0	0
$639$	0	0
$640$	0	0
$641$	4.39982e8i	1.67056i	0.549828	+	0.835278i	$0.314693\pi$
$641$	4.39982e8i	1.67056i	−0.549828	+	0.835278i	$0.685307\pi$
$642$	0	0
$643$	−4.25984e8	−1.60236	−0.801180	−	0.598423i	$-0.795794\pi$
$643$	−4.25984e8	−1.60236	−0.801180	+	0.598423i	$0.795794\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	2.56892e8i	0.948499i	0.880390	+	0.474250i	$0.157281\pi$
$647$	2.56892e8i	0.948499i	−0.880390	+	0.474250i	$0.842719\pi$
$648$	0	0
$649$	−1.38659e8	−0.507240
$650$	0	0
$651$	0	0
$652$	0	0
$653$	1.69558e8i	0.608947i	0.952521	+	0.304473i	$0.0984805\pi$
$653$	1.69558e8i	0.608947i	−0.952521	+	0.304473i	$0.901520\pi$
$654$	0	0
$655$	−1.30346e8	−0.463847
$656$	0	0
$657$	0	0
$658$	0	0
$659$	7.20906e7i	0.251897i	0.992037	+	0.125948i	$0.0401974\pi$
$659$	7.20906e7i	0.251897i	−0.992037	+	0.125948i	$0.959803\pi$
$660$	0	0
$661$	2.43846e8	0.844327	0.422163	−	0.906520i	$-0.361271\pi$
$661$	2.43846e8	0.844327	0.422163	+	0.906520i	$0.361271\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 5.51334e8i	− 1.87478i
$666$	0	0
$667$	5.24855e7	0.176873
$668$	0	0
$669$	0	0
$670$	0	0
$671$	4.18033e8i	1.38370i
$672$	0	0
$673$	−1.97825e8	−0.648987	−0.324493	−	0.945888i	$-0.605194\pi$
$673$	−1.97825e8	−0.648987	−0.324493	+	0.945888i	$0.605194\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 7.18878e7i	− 0.231680i	−0.993268	−	0.115840i	$-0.963044\pi$
$677$	− 7.18878e7i	− 0.231680i	0.993268	−	0.115840i	$-0.0369560\pi$
$678$	0	0
$679$	−6.20567e7	−0.198234
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.85099e8i	0.580955i	0.956882	+	0.290477i	$0.0938141\pi$
$683$	1.85099e8i	0.580955i	−0.956882	+	0.290477i	$0.906186\pi$
$684$	0	0
$685$	5.30616e8	1.65085
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 2.40375e8i	− 0.734906i
$690$	0	0
$691$	4.01905e8	1.21812	0.609059	−	0.793125i	$-0.291547\pi$
$691$	4.01905e8	1.21812	0.609059	+	0.793125i	$0.291547\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 3.27125e8i	− 0.974450i
$696$	0	0
$697$	8.57059e6	0.0253112
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4.46213e7i	− 0.129535i	−0.997900	−	0.0647677i	$-0.979369\pi$
$701$	− 4.46213e7i	− 0.129535i	0.997900	−	0.0647677i	$-0.0206306\pi$
$702$	0	0
$703$	4.97499e8	1.43195
$704$	0	0
$705$	0	0
$706$	0	0
$707$	9.10685e8i	2.57697i
$708$	0	0
$709$	−1.64821e8	−0.462460	−0.231230	−	0.972899i	$-0.574275\pi$
$709$	−1.64821e8	−0.462460	−0.231230	+	0.972899i	$0.574275\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	3.08528e7i	0.0851189i
$714$	0	0
$715$	1.69887e8	0.464773
$716$	0	0
$717$	0	0
$718$	0	0
$719$	6.06761e8i	1.63242i	0.577758	+	0.816208i	$0.303928\pi$
$719$	6.06761e8i	1.63242i	−0.577758	+	0.816208i	$0.696072\pi$
$720$	0	0
$721$	−7.32417e8	−1.95413
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 2.18690e8i	− 0.573872i
$726$	0	0
$727$	−2.01946e8	−0.525572	−0.262786	−	0.964854i	$-0.584641\pi$
$727$	−2.01946e8	−0.525572	−0.262786	+	0.964854i	$0.584641\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3.06966e8i	− 0.785848i
$732$	0	0
$733$	1.55453e8	0.394718	0.197359	−	0.980331i	$-0.436764\pi$
$733$	1.55453e8	0.394718	0.197359	+	0.980331i	$0.436764\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 5.69700e8i	− 1.42313i
$738$	0	0
$739$	3.61039e8	0.894582	0.447291	−	0.894388i	$-0.352389\pi$
$739$	3.61039e8	0.894582	0.447291	+	0.894388i	$0.352389\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 4.16298e8i	− 1.01493i	−0.861671	−	0.507467i	$-0.830582\pi$
$743$	− 4.16298e8i	− 1.01493i	0.861671	−	0.507467i	$-0.169418\pi$
$744$	0	0
$745$	2.41510e8	0.584073
$746$	0	0
$747$	0	0
$748$	0	0
$749$	6.93977e7i	0.165158i
$750$	0	0
$751$	6.21455e8	1.46720	0.733601	−	0.679581i	$-0.237838\pi$
$751$	6.21455e8	1.46720	0.733601	+	0.679581i	$0.237838\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.51351e8i	− 0.584037i
$756$	0	0
$757$	4.73200e8	1.09083	0.545415	−	0.838166i	$-0.316372\pi$
$757$	4.73200e8	1.09083	0.545415	+	0.838166i	$0.316372\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	4.85185e8i	1.10091i	0.834863	+	0.550457i	$0.185547\pi$
$761$	4.85185e8i	1.10091i	−0.834863	+	0.550457i	$0.814453\pi$
$762$	0	0
$763$	−2.88125e8	−0.648645
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.07751e8i	0.238801i
$768$	0	0
$769$	4.15701e8	0.914117	0.457058	−	0.889437i	$-0.348903\pi$
$769$	4.15701e8	0.914117	0.457058	+	0.889437i	$0.348903\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 7.86397e7i	− 0.170256i	−0.996370	−	0.0851282i	$-0.972870\pi$
$773$	− 7.86397e7i	− 0.170256i	0.996370	−	0.0851282i	$-0.0271300\pi$
$774$	0	0
$775$	1.28554e8	0.276172
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 2.22413e7i	− 0.0470488i
$780$	0	0
$781$	4.82447e8	1.01274
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 6.24545e8i	− 1.29108i
$786$	0	0
$787$	−5.91336e8	−1.21314	−0.606569	−	0.795031i	$-0.707454\pi$
$787$	−5.91336e8	−1.21314	−0.606569	+	0.795031i	$0.707454\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 6.23331e8i	− 1.25947i
$792$	0	0
$793$	3.24851e8	0.651426
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.40202e8i	0.276935i	0.990367	+	0.138468i	$0.0442177\pi$
$797$	1.40202e8i	0.276935i	−0.990367	+	0.138468i	$0.955782\pi$
$798$	0	0
$799$	3.79852e8	0.744688
$800$	0	0
$801$	0	0
$802$	0	0
$803$	6.14703e7i	0.118718i
$804$	0	0
$805$	9.66478e7	0.185270
$806$	0	0
$807$	0	0
$808$	0	0
$809$	9.06071e7i	0.171126i	0.996333	+	0.0855631i	$0.0272689\pi$
$809$	9.06071e7i	0.171126i	−0.996333	+	0.0855631i	$0.972731\pi$
$810$	0	0
$811$	−7.26353e8	−1.36171	−0.680856	−	0.732417i	$-0.738392\pi$
$811$	−7.26353e8	−1.36171	−0.680856	+	0.732417i	$0.738392\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 8.36975e8i	− 1.54611i
$816$	0	0
$817$	−7.96601e8	−1.46075
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1.07623e9i	1.94481i	0.233308	+	0.972403i	$0.425045\pi$
$821$	1.07623e9i	1.94481i	−0.233308	+	0.972403i	$0.574955\pi$
$822$	0	0
$823$	6.53142e8	1.17168	0.585839	−	0.810428i	$-0.300765\pi$
$823$	6.53142e8	1.17168	0.585839	+	0.810428i	$0.300765\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 7.54827e8i	− 1.33454i	−0.744817	−	0.667269i	$-0.767463\pi$
$827$	− 7.54827e8i	− 1.33454i	0.744817	−	0.667269i	$-0.232537\pi$
$828$	0	0
$829$	9.54226e8	1.67489	0.837447	−	0.546518i	$-0.184047\pi$
$829$	9.54226e8	1.67489	0.837447	+	0.546518i	$0.184047\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 3.24877e8i	− 0.562062i
$834$	0	0
$835$	−5.57654e8	−0.957867
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5.59573e8i	0.947482i	0.880664	+	0.473741i	$0.157097\pi$
$839$	5.59573e8i	0.947482i	−0.880664	+	0.473741i	$0.842903\pi$
$840$	0	0
$841$	−8.18550e8	−1.37612
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.74776e8i	0.952640i
$846$	0	0
$847$	−1.31707e8	−0.216750
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.72107e7i	0.141508i
$852$	0	0
$853$	2.92749e8	0.471680	0.235840	−	0.971792i	$-0.424216\pi$
$853$	2.92749e8	0.471680	0.235840	+	0.971792i	$0.424216\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 3.93247e8i	− 0.624774i	−0.949955	−	0.312387i	$-0.898871\pi$
$857$	− 3.93247e8i	− 0.624774i	0.949955	−	0.312387i	$-0.101129\pi$
$858$	0	0
$859$	8.63345e8	1.36209	0.681043	−	0.732243i	$-0.261526\pi$
$859$	8.63345e8	1.36209	0.681043	+	0.732243i	$0.261526\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 4.05821e8i	− 0.631396i	−0.948860	−	0.315698i	$-0.897761\pi$
$863$	− 4.05821e8i	− 0.631396i	0.948860	−	0.315698i	$-0.102239\pi$
$864$	0	0
$865$	−8.81152e8	−1.36145
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.41323e8i	1.28205i
$870$	0	0
$871$	−4.42711e8	−0.669986
$872$	0	0
$873$	0	0
$874$	0	0
$875$	6.78986e8i	1.01353i
$876$	0	0
$877$	−3.40017e8	−0.504083	−0.252041	−	0.967716i	$-0.581102\pi$
$877$	−3.40017e8	−0.504083	−0.252041	+	0.967716i	$0.581102\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1.60099e7i	− 0.0234132i	−0.999931	−	0.0117066i	$-0.996274\pi$
$881$	− 1.60099e7i	− 0.0234132i	0.999931	−	0.0117066i	$-0.00372641\pi$
$882$	0	0
$883$	−4.53127e8	−0.658170	−0.329085	−	0.944300i	$-0.606740\pi$
$883$	−4.53127e8	−0.658170	−0.329085	+	0.944300i	$0.606740\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	9.76437e7i	0.139918i	0.997550	+	0.0699590i	$0.0222868\pi$
$887$	9.76437e7i	0.139918i	−0.997550	+	0.0699590i	$0.977713\pi$
$888$	0	0
$889$	−6.71854e8	−0.956246
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 9.85746e8i	− 1.38424i
$894$	0	0
$895$	−1.26971e9	−1.77107
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 8.30831e8i	− 1.14349i
$900$	0	0
$901$	−7.76913e8	−1.06218
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.34894e8i	0.451816i
$906$	0	0
$907$	2.12463e8	0.284748	0.142374	−	0.989813i	$-0.454526\pi$
$907$	2.12463e8	0.284748	0.142374	+	0.989813i	$0.454526\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 1.83181e8i	− 0.242284i	−0.992635	−	0.121142i	$-0.961344\pi$
$911$	− 1.83181e8i	− 0.242284i	0.992635	−	0.121142i	$-0.0386556\pi$
$912$	0	0
$913$	−1.20593e9	−1.58456
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.20837e8i	0.545766i
$918$	0	0
$919$	−1.87858e8	−0.242038	−0.121019	−	0.992650i	$-0.538616\pi$
$919$	−1.87858e8	−0.242038	−0.121019	+	0.992650i	$0.538616\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 3.74907e8i	− 0.476781i
$924$	0	0
$925$	3.63379e8	0.459128
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.23781e9i	1.54385i	0.635713	+	0.771926i	$0.280706\pi$
$929$	1.23781e9i	1.54385i	−0.635713	+	0.771926i	$0.719294\pi$
$930$	0	0
$931$	−8.43081e8	−1.04477
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 5.49088e8i	− 0.671749i
$936$	0	0
$937$	−2.22909e8	−0.270962	−0.135481	−	0.990780i	$-0.543258\pi$
$937$	−2.22909e8	−0.270962	−0.135481	+	0.990780i	$0.543258\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	5.13522e8i	0.616298i	0.951338	+	0.308149i	$0.0997095\pi$
$941$	5.13522e8i	0.616298i	−0.951338	+	0.308149i	$0.900291\pi$
$942$	0	0
$943$	3.89886e6	0.00464947
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.71875e8i	0.202377i	0.994867	+	0.101189i	$0.0322646\pi$
$947$	1.71875e8i	0.202377i	−0.994867	+	0.101189i	$0.967735\pi$
$948$	0	0
$949$	4.77682e7	0.0558908
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.38294e9i	1.59781i	0.601459	+	0.798903i	$0.294586\pi$
$953$	1.38294e9i	1.59781i	−0.601459	+	0.798903i	$0.705414\pi$
$954$	0	0
$955$	−2.48520e8	−0.285333
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 1.71315e9i	− 1.94241i
$960$	0	0
$961$	−3.99112e8	−0.449702
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 5.94049e8i	− 0.661059i
$966$	0	0
$967$	2.03204e8	0.224726	0.112363	−	0.993667i	$-0.464158\pi$
$967$	2.03204e8	0.224726	0.112363	+	0.993667i	$0.464158\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.91854e8i	0.646483i	0.946317	+	0.323241i	$0.104773\pi$
$971$	5.91854e8i	0.646483i	−0.946317	+	0.323241i	$0.895227\pi$
$972$	0	0
$973$	−1.05616e9	−1.14655
$974$	0	0
$975$	0	0
$976$	0	0
$977$	9.81300e8i	1.05225i	0.850408	+	0.526124i	$0.176355\pi$
$977$	9.81300e8i	1.05225i	−0.850408	+	0.526124i	$0.823645\pi$
$978$	0	0
$979$	8.26666e8	0.881012
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.34077e9i	1.41154i	0.708440	+	0.705771i	$0.249399\pi$
$983$	1.34077e9i	1.41154i	−0.708440	+	0.705771i	$0.750601\pi$
$984$	0	0
$985$	1.95330e9	2.04391
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 1.39643e8i	− 0.144354i
$990$	0	0
$991$	−9.74527e8	−1.00132	−0.500660	−	0.865644i	$-0.666909\pi$
$991$	−9.74527e8	−1.00132	−0.500660	+	0.865644i	$0.666909\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 3.83326e8i	− 0.389133i
$996$	0	0
$997$	2.64896e8	0.267295	0.133647	−	0.991029i	$-0.457331\pi$
$997$	2.64896e8	0.267295	0.133647	+	0.991029i	$0.457331\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.7.e.k.161.1		6
3.2	odd	2	inner	432.7.e.k.161.6		6
4.3	odd	2		216.7.e.b.161.1	✓	6
12.11	even	2		216.7.e.b.161.6	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.7.e.b.161.1	✓	6	4.3	odd	2
216.7.e.b.161.6	yes	6	12.11	even	2
432.7.e.k.161.1		6	1.1	even	1	trivial
432.7.e.k.161.6		6	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)

\(f(q)\)	\(=\)	\(q-146.431i q^{5} -472.768 q^{7} +O(q^{10})\)	\(q-146.431i q^{5} -472.768 q^{7} +1221.87i q^{11} +949.511 q^{13} -3068.90i q^{17} -7964.04 q^{19} -1396.08i q^{23} -5817.02 q^{25} +37594.9i q^{29} -22099.6 q^{31} +69227.9i q^{35} -62468.2 q^{37} +2792.72i q^{41} +100025. q^{43} +123775. i q^{47} +105861. q^{49} -253157. i q^{53} +178920. q^{55} +113481. i q^{59} +342125. q^{61} -139038. i q^{65} -466251. q^{67} -394842. i q^{71} +50308.2 q^{73} -577663. i q^{77} +688552. q^{79} +986952. i q^{83} -449382. q^{85} -676556. i q^{89} -448899. q^{91} +1.16618e6i q^{95} +131262. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 114 q^{7}+O(q^{10}) \) 6 * q + 114 * q^7	\( 6 q + 114 q^{7} + 4026 q^{13} - 1854 q^{19} + 7686 q^{25} - 708 q^{31} - 66198 q^{37} + 116172 q^{43} + 237072 q^{49} + 158544 q^{55} + 170154 q^{61} + 84162 q^{67} + 748986 q^{73} + 560130 q^{79} + 1073040 q^{85} + 2795982 q^{91} + 1967418 q^{97}+O(q^{100}) \) 6 * q + 114 * q^7 + 4026 * q^13 - 1854 * q^19 + 7686 * q^25 - 708 * q^31 - 66198 * q^37 + 116172 * q^43 + 237072 * q^49 + 158544 * q^55 + 170154 * q^61 + 84162 * q^67 + 748986 * q^73 + 560130 * q^79 + 1073040 * q^85 + 2795982 * q^91 + 1967418 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more