LMFDB - Embedded newform 432.9.e.d.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,9,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, 9, names="a")

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

chi := DirichletCharacter("432.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$175.987559546$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 6 $ x^2 + 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
Twist minimal:	no (minimal twist has level 27)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

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

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+823.029i q^{5} -1967.00 q^{7} +O(q^{10})$	$q+823.029i q^{5} -1967.00 q^{7} -12580.6i q^{11} -45505.0 q^{13} +59610.8i q^{17} -152399. q^{19} -131332. i q^{23} -286751. q^{25} +588583. i q^{29} +164350. q^{31} -1.61890e6i q^{35} -663937. q^{37} +938017. i q^{41} -575330. q^{43} -9.23426e6i q^{47} -1.89571e6 q^{49} -1.03765e7i q^{53} +1.03542e7 q^{55} -5.03987e6i q^{59} -1.92130e7 q^{61} -3.74519e7i q^{65} +598033. q^{67} +2.92721e7i q^{71} +1.28502e7 q^{73} +2.47460e7i q^{77} +2.35847e7 q^{79} +3.34509e7i q^{83} -4.90614e7 q^{85} +2.82848e7i q^{89} +8.95083e7 q^{91} -1.25429e8i q^{95} +1.36490e8 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3934 q^{7}+O(q^{10}) $ 2 * q - 3934 * q^7	$ 2 q - 3934 q^{7} - 91010 q^{13} - 304798 q^{19} - 573502 q^{25} + 328700 q^{31} - 1327874 q^{37} - 1150660 q^{43} - 3791424 q^{49} + 20708352 q^{55} - 38425922 q^{61} + 1196066 q^{67} + 25700350 q^{73} + 47169314 q^{79} - 98122752 q^{85} + 179016670 q^{91} + 272979262 q^{97}+O(q^{100}) $ 2 * q - 3934 * q^7 - 91010 * q^13 - 304798 * q^19 - 573502 * q^25 + 328700 * q^31 - 1327874 * q^37 - 1150660 * q^43 - 3791424 * q^49 + 20708352 * q^55 - 38425922 * q^61 + 1196066 * q^67 + 25700350 * q^73 + 47169314 * q^79 - 98122752 * q^85 + 179016670 * q^91 + 272979262 * 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
$5$	823.029i	1.31685i	0.752648	+	0.658423i	$0.228776\pi$
$5$	823.029i	1.31685i	−0.752648	+	0.658423i	$0.771224\pi$
$6$	0	0
$7$	−1967.00	−0.819242	−0.409621	−	0.912256i	$-0.634339\pi$
$7$	−1967.00	−0.819242	−0.409621	+	0.912256i	$0.634339\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 12580.6i	− 0.859270i	−0.903003	−	0.429635i	$-0.858642\pi$
$11$	− 12580.6i	− 0.859270i	0.903003	−	0.429635i	$-0.141358\pi$
$12$	0	0
$13$	−45505.0	−1.59326	−0.796628	−	0.604470i	$-0.793385\pi$
$13$	−45505.0	−1.59326	−0.796628	+	0.604470i	$0.793385\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	59610.8i	0.713722i	0.934157	+	0.356861i	$0.116153\pi$
$17$	59610.8i	0.713722i	−0.934157	+	0.356861i	$0.883847\pi$
$18$	0	0
$19$	−152399.	−1.16941	−0.584706	−	0.811245i	$-0.698790\pi$
$19$	−152399.	−1.16941	−0.584706	+	0.811245i	$0.698790\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 131332.i	− 0.469309i	−0.972079	−	0.234654i	$-0.924604\pi$
$23$	− 131332.i	− 0.469309i	0.972079	−	0.234654i	$-0.0753958\pi$
$24$	0	0
$25$	−286751.	−0.734083
$26$	0	0
$27$	0	0
$28$	0	0
$29$	588583.i	0.832177i	0.909324	+	0.416089i	$0.136599\pi$
$29$	588583.i	0.832177i	−0.909324	+	0.416089i	$0.863401\pi$
$30$	0	0
$31$	164350.	0.177960	0.0889801	−	0.996033i	$-0.471639\pi$
$31$	164350.	0.177960	0.0889801	+	0.996033i	$0.471639\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1.61890e6i	− 1.07882i
$36$	0	0
$37$	−663937.	−0.354258	−0.177129	−	0.984188i	$-0.556681\pi$
$37$	−663937.	−0.354258	−0.177129	+	0.984188i	$0.556681\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	938017.i	0.331952i	0.986130	+	0.165976i	$0.0530774\pi$
$41$	938017.i	0.331952i	−0.986130	+	0.165976i	$0.946923\pi$
$42$	0	0
$43$	−575330.	−0.168284	−0.0841421	−	0.996454i	$-0.526815\pi$
$43$	−575330.	−0.168284	−0.0841421	+	0.996454i	$0.526815\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 9.23426e6i	− 1.89239i	−0.323596	−	0.946195i	$-0.604892\pi$
$47$	− 9.23426e6i	− 1.89239i	0.323596	−	0.946195i	$-0.395108\pi$
$48$	0	0
$49$	−1.89571e6	−0.328843
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.03765e7i	− 1.31507i	−0.753425	−	0.657533i	$-0.771600\pi$
$53$	− 1.03765e7i	− 1.31507i	0.753425	−	0.657533i	$-0.228400\pi$
$54$	0	0
$55$	1.03542e7	1.13153
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.03987e6i	− 0.415922i	−0.978137	−	0.207961i	$-0.933317\pi$
$59$	− 5.03987e6i	− 0.415922i	0.978137	−	0.207961i	$-0.0666827\pi$
$60$	0	0
$61$	−1.92130e7	−1.38763	−0.693817	−	0.720151i	$-0.744072\pi$
$61$	−1.92130e7	−1.38763	−0.693817	+	0.720151i	$0.744072\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 3.74519e7i	− 2.09807i
$66$	0	0
$67$	598033.	0.0296774	0.0148387	−	0.999890i	$-0.495277\pi$
$67$	598033.	0.0296774	0.0148387	+	0.999890i	$0.495277\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.92721e7i	1.15191i	0.817480	+	0.575957i	$0.195370\pi$
$71$	2.92721e7i	1.15191i	−0.817480	+	0.575957i	$0.804630\pi$
$72$	0	0
$73$	1.28502e7	0.452499	0.226249	−	0.974069i	$-0.427354\pi$
$73$	1.28502e7	0.452499	0.226249	+	0.974069i	$0.427354\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.47460e7i	0.703950i
$78$	0	0
$79$	2.35847e7	0.605510	0.302755	−	0.953068i	$-0.402094\pi$
$79$	2.35847e7	0.605510	0.302755	+	0.953068i	$0.402094\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	3.34509e7i	0.704849i	0.935840	+	0.352424i	$0.114643\pi$
$83$	3.34509e7i	0.704849i	−0.935840	+	0.352424i	$0.885357\pi$
$84$	0	0
$85$	−4.90614e7	−0.939862
$86$	0	0
$87$	0	0
$88$	0	0
$89$	2.82848e7i	0.450809i	0.974265	+	0.225405i	$0.0723704\pi$
$89$	2.82848e7i	0.450809i	−0.974265	+	0.225405i	$0.927630\pi$
$90$	0	0
$91$	8.95083e7	1.30526
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1.25429e8i	− 1.53994i
$96$	0	0
$97$	1.36490e8	1.54175	0.770873	−	0.636989i	$-0.219820\pi$
$97$	1.36490e8	1.54175	0.770873	+	0.636989i	$0.219820\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4.02494e7i	− 0.386789i	−0.981121	−	0.193394i	$-0.938050\pi$
$101$	− 4.02494e7i	− 0.386789i	0.981121	−	0.193394i	$-0.0619497\pi$
$102$	0	0
$103$	−3.35907e7	−0.298449	−0.149225	−	0.988803i	$-0.547678\pi$
$103$	−3.35907e7	−0.298449	−0.149225	+	0.988803i	$0.547678\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.95483e8i	1.49133i	0.666322	+	0.745664i	$0.267868\pi$
$107$	1.95483e8i	1.49133i	−0.666322	+	0.745664i	$0.732132\pi$
$108$	0	0
$109$	−1.01182e8	−0.716800	−0.358400	−	0.933568i	$-0.616678\pi$
$109$	−1.01182e8	−0.716800	−0.358400	+	0.933568i	$0.616678\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	3.14504e8i	1.92891i	0.264238	+	0.964457i	$0.414880\pi$
$113$	3.14504e8i	1.92891i	−0.264238	+	0.964457i	$0.585120\pi$
$114$	0	0
$115$	1.08090e8	0.618007
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.17254e8i	− 0.584711i
$120$	0	0
$121$	5.60879e7	0.261654
$122$	0	0
$123$	0	0
$124$	0	0
$125$	8.54913e7i	0.350172i
$126$	0	0
$127$	−4.30869e8	−1.65627	−0.828133	−	0.560532i	$-0.810597\pi$
$127$	−4.30869e8	−1.65627	−0.828133	+	0.560532i	$0.810597\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.88933e8i	− 0.641538i	−0.947158	−	0.320769i	$-0.896059\pi$
$131$	− 1.88933e8i	− 0.641538i	0.947158	−	0.320769i	$-0.103941\pi$
$132$	0	0
$133$	2.99769e8	0.958032
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.22998e8i	0.349152i	0.984644	+	0.174576i	$0.0558555\pi$
$137$	1.22998e8i	0.349152i	−0.984644	+	0.174576i	$0.944144\pi$
$138$	0	0
$139$	−1.72072e8	−0.460947	−0.230474	−	0.973079i	$-0.574028\pi$
$139$	−1.72072e8	−0.460947	−0.230474	+	0.973079i	$0.574028\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.72479e8i	1.36904i
$144$	0	0
$145$	−4.84421e8	−1.09585
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 7.31667e8i	− 1.48446i	−0.670145	−	0.742230i	$-0.733768\pi$
$149$	− 7.31667e8i	− 1.48446i	0.670145	−	0.742230i	$-0.266232\pi$
$150$	0	0
$151$	−1.85952e8	−0.357679	−0.178840	−	0.983878i	$-0.557234\pi$
$151$	−1.85952e8	−0.357679	−0.178840	+	0.983878i	$0.557234\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1.35265e8i	0.234346i
$156$	0	0
$157$	9.74007e8	1.60311	0.801556	−	0.597920i	$-0.204006\pi$
$157$	9.74007e8	1.60311	0.801556	+	0.597920i	$0.204006\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.58330e8i	0.384477i
$162$	0	0
$163$	1.15499e9	1.63616	0.818082	−	0.575102i	$-0.195038\pi$
$163$	1.15499e9	1.63616	0.818082	+	0.575102i	$0.195038\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	1.14302e8i	0.146956i	0.997297	+	0.0734779i	$0.0234098\pi$
$167$	1.14302e8i	0.146956i	−0.997297	+	0.0734779i	$0.976590\pi$
$168$	0	0
$169$	1.25497e9	1.53847
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.65322e9i	− 1.84564i	−0.385235	−	0.922819i	$-0.625879\pi$
$173$	− 1.65322e9i	− 1.84564i	0.385235	−	0.922819i	$-0.374121\pi$
$174$	0	0
$175$	5.64039e8	0.601391
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.17229e8i	− 0.406408i	−0.979136	−	0.203204i	$-0.934865\pi$
$179$	− 4.17229e8i	− 0.406408i	0.979136	−	0.203204i	$-0.0651355\pi$
$180$	0	0
$181$	1.16424e9	1.08475	0.542374	−	0.840137i	$-0.317526\pi$
$181$	1.16424e9	1.08475	0.542374	+	0.840137i	$0.317526\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.46439e8i	− 0.466503i
$186$	0	0
$187$	7.49938e8	0.613280
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 1.61055e9i	− 1.21016i	−0.796166	−	0.605078i	$-0.793142\pi$
$191$	− 1.61055e9i	− 1.21016i	0.796166	−	0.605078i	$-0.206858\pi$
$192$	0	0
$193$	−7.10279e8	−0.511917	−0.255959	−	0.966688i	$-0.582391\pi$
$193$	−7.10279e8	−0.511917	−0.255959	+	0.966688i	$0.582391\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.41623e9i	0.940303i	0.882586	+	0.470152i	$0.155801\pi$
$197$	1.41623e9i	0.940303i	−0.882586	+	0.470152i	$0.844199\pi$
$198$	0	0
$199$	2.34324e9	1.49419	0.747093	−	0.664720i	$-0.231449\pi$
$199$	2.34324e9	1.49419	0.747093	+	0.664720i	$0.231449\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 1.15774e9i	− 0.681754i
$204$	0	0
$205$	−7.72015e8	−0.437130
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.91727e9i	1.00484i
$210$	0	0
$211$	−1.06517e9	−0.537389	−0.268695	−	0.963225i	$-0.586592\pi$
$211$	−1.06517e9	−0.537389	−0.268695	+	0.963225i	$0.586592\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 4.73513e8i	− 0.221604i
$216$	0	0
$217$	−3.23276e8	−0.145792
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 2.71259e9i	− 1.13714i
$222$	0	0
$223$	−1.99586e9	−0.807068	−0.403534	−	0.914965i	$-0.632218\pi$
$223$	−1.99586e9	−0.807068	−0.403534	+	0.914965i	$0.632218\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 1.79923e9i	− 0.677615i	−0.940856	−	0.338807i	$-0.889977\pi$
$227$	− 1.79923e9i	− 0.677615i	0.940856	−	0.338807i	$-0.110023\pi$
$228$	0	0
$229$	1.59787e9	0.581032	0.290516	−	0.956870i	$-0.406173\pi$
$229$	1.59787e9	0.581032	0.290516	+	0.956870i	$0.406173\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	4.98411e9i	1.69108i	0.533912	+	0.845540i	$0.320721\pi$
$233$	4.98411e9i	1.69108i	−0.533912	+	0.845540i	$0.679279\pi$
$234$	0	0
$235$	7.60006e9	2.49199
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5.89754e9i	1.80751i	0.428055	+	0.903753i	$0.359199\pi$
$239$	5.89754e9i	1.80751i	−0.428055	+	0.903753i	$0.640801\pi$
$240$	0	0
$241$	2.26174e9	0.670463	0.335231	−	0.942136i	$-0.391185\pi$
$241$	2.26174e9	0.670463	0.335231	+	0.942136i	$0.391185\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.56023e9i	− 0.433035i
$246$	0	0
$247$	6.93492e9	1.86317
$248$	0	0
$249$	0	0
$250$	0	0
$251$	3.31815e9i	0.835989i	0.908450	+	0.417994i	$0.137267\pi$
$251$	3.31815e9i	0.835989i	−0.908450	+	0.417994i	$0.862733\pi$
$252$	0	0
$253$	−1.65223e9	−0.403263
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3.05018e9i	0.699185i	0.936902	+	0.349593i	$0.113680\pi$
$257$	3.05018e9i	0.699185i	−0.936902	+	0.349593i	$0.886320\pi$
$258$	0	0
$259$	1.30596e9	0.290223
$260$	0	0
$261$	0	0
$262$	0	0
$263$	1.97422e9i	0.412640i	0.978485	+	0.206320i	$0.0661489\pi$
$263$	1.97422e9i	0.412640i	−0.978485	+	0.206320i	$0.933851\pi$
$264$	0	0
$265$	8.54016e9	1.73174
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 3.49334e9i	− 0.667163i	−0.942721	−	0.333581i	$-0.891743\pi$
$269$	− 3.49334e9i	− 0.667163i	0.942721	−	0.333581i	$-0.108257\pi$
$270$	0	0
$271$	−1.43226e9	−0.265549	−0.132774	−	0.991146i	$-0.542389\pi$
$271$	−1.43226e9	−0.265549	−0.132774	+	0.991146i	$0.542389\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3.60749e9i	0.630775i
$276$	0	0
$277$	3.35095e9	0.569179	0.284589	−	0.958650i	$-0.408143\pi$
$277$	3.35095e9	0.569179	0.284589	+	0.958650i	$0.408143\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	4.29040e9i	0.688133i	0.938945	+	0.344067i	$0.111805\pi$
$281$	4.29040e9i	0.688133i	−0.938945	+	0.344067i	$0.888195\pi$
$282$	0	0
$283$	9.91145e7	0.0154522	0.00772612	−	0.999970i	$-0.497541\pi$
$283$	9.91145e7	0.0154522	0.00772612	+	0.999970i	$0.497541\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.84508e9i	− 0.271949i
$288$	0	0
$289$	3.42231e9	0.490601
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 9.02972e8i	− 0.122519i	−0.998122	−	0.0612596i	$-0.980488\pi$
$293$	− 9.02972e8i	− 0.122519i	0.998122	−	0.0612596i	$-0.0195117\pi$
$294$	0	0
$295$	4.14796e9	0.547705
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.97626e9i	0.747729i
$300$	0	0
$301$	1.13167e9	0.137865
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.58128e10i	− 1.82730i
$306$	0	0
$307$	−3.36397e9	−0.378703	−0.189352	−	0.981909i	$-0.560639\pi$
$307$	−3.36397e9	−0.378703	−0.189352	+	0.981909i	$0.560639\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 1.33023e10i	− 1.42196i	−0.703215	−	0.710978i	$-0.748253\pi$
$311$	− 1.33023e10i	− 1.42196i	0.703215	−	0.710978i	$-0.251747\pi$
$312$	0	0
$313$	1.31249e10	1.36748	0.683738	−	0.729728i	$-0.260353\pi$
$313$	1.31249e10	1.36748	0.683738	+	0.729728i	$0.260353\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.98520e9i	− 0.691739i	−0.938283	−	0.345869i	$-0.887584\pi$
$317$	− 6.98520e9i	− 0.691739i	0.938283	−	0.345869i	$-0.112416\pi$
$318$	0	0
$319$	7.40472e9	0.715065
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 9.08462e9i	− 0.834635i
$324$	0	0
$325$	1.30486e10	1.16958
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.81638e10i	1.55033i
$330$	0	0
$331$	1.18733e10	0.989140	0.494570	−	0.869138i	$-0.335326\pi$
$331$	1.18733e10	0.989140	0.494570	+	0.869138i	$0.335326\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	4.92198e8i	0.0390806i
$336$	0	0
$337$	9.15419e9	0.709741	0.354871	−	0.934915i	$-0.384525\pi$
$337$	9.15419e9	0.709741	0.354871	+	0.934915i	$0.384525\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 2.06762e9i	− 0.152916i
$342$	0	0
$343$	1.50682e10	1.08864
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 2.05081e10i	− 1.41451i	−0.706957	−	0.707257i	$-0.749933\pi$
$347$	− 2.05081e10i	− 1.41451i	0.706957	−	0.707257i	$-0.250067\pi$
$348$	0	0
$349$	1.34799e10	0.908625	0.454312	−	0.890842i	$-0.349885\pi$
$349$	1.34799e10	0.908625	0.454312	+	0.890842i	$0.349885\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.78259e9i	0.630021i	0.949088	+	0.315011i	$0.102008\pi$
$353$	9.78259e9i	0.630021i	−0.949088	+	0.315011i	$0.897992\pi$
$354$	0	0
$355$	−2.40917e10	−1.51689
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 1.67338e10i	− 1.00743i	−0.863868	−	0.503717i	$-0.831965\pi$
$359$	− 1.67338e10i	− 1.00743i	0.863868	−	0.503717i	$-0.168035\pi$
$360$	0	0
$361$	6.24189e9	0.367525
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.05761e10i	0.595871i
$366$	0	0
$367$	1.49879e10	0.826184	0.413092	−	0.910689i	$-0.364449\pi$
$367$	1.49879e10	0.826184	0.413092	+	0.910689i	$0.364449\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2.04106e10i	1.07736i
$372$	0	0
$373$	−1.97964e10	−1.02271	−0.511353	−	0.859371i	$-0.670855\pi$
$373$	−1.97964e10	−1.02271	−0.511353	+	0.859371i	$0.670855\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.67835e10i	− 1.32587i
$378$	0	0
$379$	5.34899e9	0.259248	0.129624	−	0.991563i	$-0.458623\pi$
$379$	5.34899e9	0.259248	0.129624	+	0.991563i	$0.458623\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	1.44938e10i	0.673577i	0.941580	+	0.336788i	$0.109341\pi$
$383$	1.44938e10i	0.673577i	−0.941580	+	0.336788i	$0.890659\pi$
$384$	0	0
$385$	−2.03667e10	−0.926994
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.80507e10i	0.788308i	0.919044	+	0.394154i	$0.128962\pi$
$389$	1.80507e10i	0.788308i	−0.919044	+	0.394154i	$0.871038\pi$
$390$	0	0
$391$	7.82879e9	0.334956
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.94108e10i	0.797363i
$396$	0	0
$397$	3.82640e10	1.54038	0.770191	−	0.637814i	$-0.220161\pi$
$397$	3.82640e10	1.54038	0.770191	+	0.637814i	$0.220161\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 4.26602e10i	− 1.64985i	−0.565241	−	0.824926i	$-0.691217\pi$
$401$	− 4.26602e10i	− 1.64985i	0.565241	−	0.824926i	$-0.308783\pi$
$402$	0	0
$403$	−7.47875e9	−0.283536
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.35271e9i	0.304404i
$408$	0	0
$409$	−3.53953e10	−1.26489	−0.632444	−	0.774606i	$-0.717948\pi$
$409$	−3.53953e10	−1.26489	−0.632444	+	0.774606i	$0.717948\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	9.91343e9i	0.340741i
$414$	0	0
$415$	−2.75311e10	−0.928177
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1.45926e10i	0.473453i	0.971576	+	0.236726i	$0.0760745\pi$
$419$	1.45926e10i	0.473453i	−0.971576	+	0.236726i	$0.923926\pi$
$420$	0	0
$421$	−2.65099e10	−0.843879	−0.421939	−	0.906624i	$-0.638650\pi$
$421$	−2.65099e10	−0.843879	−0.421939	+	0.906624i	$0.638650\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 1.70935e10i	− 0.523931i
$426$	0	0
$427$	3.77919e10	1.13681
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.61676e9i	0.133791i	0.997760	+	0.0668957i	$0.0213095\pi$
$431$	4.61676e9i	0.133791i	−0.997760	+	0.0668957i	$0.978691\pi$
$432$	0	0
$433$	−1.64494e10	−0.467948	−0.233974	−	0.972243i	$-0.575173\pi$
$433$	−1.64494e10	−0.467948	−0.233974	+	0.972243i	$0.575173\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.00148e10i	0.548816i
$438$	0	0
$439$	2.61475e10	0.703999	0.351999	−	0.936000i	$-0.385502\pi$
$439$	2.61475e10	0.703999	0.351999	+	0.936000i	$0.385502\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 4.31455e10i	− 1.12027i	−0.828403	−	0.560133i	$-0.810750\pi$
$443$	− 4.31455e10i	− 1.12027i	0.828403	−	0.560133i	$-0.189250\pi$
$444$	0	0
$445$	−2.32792e10	−0.593646
$446$	0	0
$447$	0	0
$448$	0	0
$449$	2.18287e10i	0.537084i	0.963268	+	0.268542i	$0.0865418\pi$
$449$	2.18287e10i	0.537084i	−0.963268	+	0.268542i	$0.913458\pi$
$450$	0	0
$451$	1.18008e10	0.285237
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7.36679e10i	1.71883i
$456$	0	0
$457$	2.46905e10	0.566064	0.283032	−	0.959110i	$-0.408660\pi$
$457$	2.46905e10	0.566064	0.283032	+	0.959110i	$0.408660\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.61732e10i	0.579500i	0.957102	+	0.289750i	$0.0935722\pi$
$461$	2.61732e10i	0.579500i	−0.957102	+	0.289750i	$0.906428\pi$
$462$	0	0
$463$	−7.04102e10	−1.53219	−0.766093	−	0.642729i	$-0.777802\pi$
$463$	−7.04102e10	−1.53219	−0.766093	+	0.642729i	$0.777802\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.07220e10i	0.225428i	0.993627	+	0.112714i	$0.0359543\pi$
$467$	1.07220e10i	0.225428i	−0.993627	+	0.112714i	$0.964046\pi$
$468$	0	0
$469$	−1.17633e9	−0.0243130
$470$	0	0
$471$	0	0
$472$	0	0
$473$	7.23798e9i	0.144602i
$474$	0	0
$475$	4.37006e10	0.858445
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 1.40034e10i	− 0.266005i	−0.991116	−	0.133003i	$-0.957538\pi$
$479$	− 1.40034e10i	− 0.266005i	0.991116	−	0.133003i	$-0.0424619\pi$
$480$	0	0
$481$	3.02125e10	0.564424
$482$	0	0
$483$	0	0
$484$	0	0
$485$	1.12335e11i	2.03024i
$486$	0	0
$487$	−1.72271e9	−0.0306264	−0.0153132	−	0.999883i	$-0.504875\pi$
$487$	−1.72271e9	−0.0306264	−0.0153132	+	0.999883i	$0.504875\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 3.70419e10i	− 0.637334i	−0.947867	−	0.318667i	$-0.896765\pi$
$491$	− 3.70419e10i	− 0.637334i	0.947867	−	0.318667i	$-0.103235\pi$
$492$	0	0
$493$	−3.50859e10	−0.593943
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.75782e10i	− 0.943696i
$498$	0	0
$499$	1.33497e10	0.215312	0.107656	−	0.994188i	$-0.465665\pi$
$499$	1.33497e10	0.215312	0.107656	+	0.994188i	$0.465665\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 5.91175e10i	− 0.923516i	−0.887006	−	0.461758i	$-0.847219\pi$
$503$	− 5.91175e10i	− 0.923516i	0.887006	−	0.461758i	$-0.152781\pi$
$504$	0	0
$505$	3.31264e10	0.509341
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 3.32051e10i	− 0.494691i	−0.968927	−	0.247345i	$-0.920442\pi$
$509$	− 3.32051e10i	− 0.494691i	0.968927	−	0.247345i	$-0.0795583\pi$
$510$	0	0
$511$	−2.52763e10	−0.370706
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 2.76461e10i	− 0.393012i
$516$	0	0
$517$	−1.16172e11	−1.62608
$518$	0	0
$519$	0	0
$520$	0	0
$521$	2.53746e10i	0.344388i	0.985063	+	0.172194i	$0.0550856\pi$
$521$	2.53746e10i	0.344388i	−0.985063	+	0.172194i	$0.944914\pi$
$522$	0	0
$523$	−1.27775e11	−1.70781	−0.853903	−	0.520432i	$-0.825771\pi$
$523$	−1.27775e11	−1.70781	−0.853903	+	0.520432i	$0.825771\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	9.79703e9i	0.127014i
$528$	0	0
$529$	6.10629e10	0.779749
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 4.26845e10i	− 0.528885i
$534$	0	0
$535$	−1.60888e11	−1.96385
$536$	0	0
$537$	0	0
$538$	0	0
$539$	2.38492e10i	0.282565i
$540$	0	0
$541$	6.16835e9	0.0720078	0.0360039	−	0.999352i	$-0.488537\pi$
$541$	6.16835e9	0.0720078	0.0360039	+	0.999352i	$0.488537\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 8.32758e10i	− 0.943915i
$546$	0	0
$547$	−1.02309e11	−1.14279	−0.571394	−	0.820676i	$-0.693597\pi$
$547$	−1.02309e11	−1.14279	−0.571394	+	0.820676i	$0.693597\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 8.96995e10i	− 0.973158i
$552$	0	0
$553$	−4.63910e10	−0.496059
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 1.00553e11i	− 1.04466i	−0.852743	−	0.522331i	$-0.825063\pi$
$557$	− 1.00553e11i	− 1.04466i	0.852743	−	0.522331i	$-0.174937\pi$
$558$	0	0
$559$	2.61804e10	0.268120
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 1.29004e11i	− 1.28402i	−0.766697	−	0.642009i	$-0.778101\pi$
$563$	− 1.29004e11i	− 1.28402i	0.766697	−	0.642009i	$-0.221899\pi$
$564$	0	0
$565$	−2.58846e11	−2.54008
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 1.38131e10i	− 0.131778i	−0.997827	−	0.0658889i	$-0.979012\pi$
$569$	− 1.38131e10i	− 0.131778i	0.997827	−	0.0658889i	$-0.0209883\pi$
$570$	0	0
$571$	−7.83161e10	−0.736727	−0.368363	−	0.929682i	$-0.620082\pi$
$571$	−7.83161e10	−0.736727	−0.368363	+	0.929682i	$0.620082\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.76595e10i	0.344511i
$576$	0	0
$577$	−6.44834e10	−0.581761	−0.290880	−	0.956759i	$-0.593948\pi$
$577$	−6.44834e10	−0.581761	−0.290880	+	0.956759i	$0.593948\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.57980e10i	− 0.577442i
$582$	0	0
$583$	−1.30542e11	−1.13000
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.10003e11i	0.926512i	0.886225	+	0.463256i	$0.153319\pi$
$587$	1.10003e11i	0.926512i	−0.886225	+	0.463256i	$0.846681\pi$
$588$	0	0
$589$	−2.50468e10	−0.208109
$590$	0	0
$591$	0	0
$592$	0	0
$593$	9.33048e9i	0.0754545i	0.999288	+	0.0377273i	$0.0120118\pi$
$593$	9.33048e9i	0.0754545i	−0.999288	+	0.0377273i	$0.987988\pi$
$594$	0	0
$595$	9.65037e10	0.769974
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.12693e10i	0.631276i	0.948880	+	0.315638i	$0.102219\pi$
$599$	8.12693e10i	0.631276i	−0.948880	+	0.315638i	$0.897781\pi$
$600$	0	0
$601$	1.80583e11	1.38414	0.692069	−	0.721831i	$-0.256699\pi$
$601$	1.80583e11	1.38414	0.692069	+	0.721831i	$0.256699\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	4.61619e10i	0.344558i
$606$	0	0
$607$	−1.15750e11	−0.852644	−0.426322	−	0.904571i	$-0.640191\pi$
$607$	−1.15750e11	−0.852644	−0.426322	+	0.904571i	$0.640191\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	4.20205e11i	3.01506i
$612$	0	0
$613$	7.01257e10	0.496632	0.248316	−	0.968679i	$-0.420123\pi$
$613$	7.01257e10	0.496632	0.248316	+	0.968679i	$0.420123\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.79037e10i	0.123538i	0.998090	+	0.0617691i	$0.0196742\pi$
$617$	1.79037e10i	0.123538i	−0.998090	+	0.0617691i	$0.980326\pi$
$618$	0	0
$619$	−1.35992e11	−0.926300	−0.463150	−	0.886280i	$-0.653281\pi$
$619$	−1.35992e11	−0.926300	−0.463150	+	0.886280i	$0.653281\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 5.56362e10i	− 0.369322i
$624$	0	0
$625$	−1.82374e11	−1.19521
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.95778e10i	− 0.252842i
$630$	0	0
$631$	2.50988e11	1.58320	0.791600	−	0.611040i	$-0.209248\pi$
$631$	2.50988e11	1.58320	0.791600	+	0.611040i	$0.209248\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 3.54617e11i	− 2.18105i
$636$	0	0
$637$	8.62644e10	0.523931
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 2.57286e10i	− 0.152400i	−0.997093	−	0.0761999i	$-0.975721\pi$
$641$	− 2.57286e10i	− 0.152400i	0.997093	−	0.0761999i	$-0.0242787\pi$
$642$	0	0
$643$	−4.29706e10	−0.251378	−0.125689	−	0.992070i	$-0.540114\pi$
$643$	−4.29706e10	−0.251378	−0.125689	+	0.992070i	$0.540114\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.20975e11i	− 1.26103i	−0.776177	−	0.630516i	$-0.782843\pi$
$647$	− 2.20975e11i	− 1.26103i	0.776177	−	0.630516i	$-0.217157\pi$
$648$	0	0
$649$	−6.34045e10	−0.357389
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.99011e11i	1.64450i	0.569125	+	0.822251i	$0.307282\pi$
$653$	2.99011e11i	1.64450i	−0.569125	+	0.822251i	$0.692718\pi$
$654$	0	0
$655$	1.55497e11	0.844806
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.99806e11i	1.58964i	0.606846	+	0.794820i	$0.292435\pi$
$659$	2.99806e11i	1.58964i	−0.606846	+	0.794820i	$0.707565\pi$
$660$	0	0
$661$	−2.11448e11	−1.10764	−0.553820	−	0.832637i	$-0.686830\pi$
$661$	−2.11448e11	−1.10764	−0.553820	+	0.832637i	$0.686830\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.46718e11i	1.26158i
$666$	0	0
$667$	7.72997e10	0.390548
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2.41710e11i	1.19235i
$672$	0	0
$673$	−1.90166e11	−0.926985	−0.463493	−	0.886101i	$-0.653404\pi$
$673$	−1.90166e11	−0.926985	−0.463493	+	0.886101i	$0.653404\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.14815e11i	0.546570i	0.961933	+	0.273285i	$0.0881102\pi$
$677$	1.14815e11i	0.546570i	−0.961933	+	0.273285i	$0.911890\pi$
$678$	0	0
$679$	−2.68475e11	−1.26306
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 3.07050e11i	− 1.41100i	−0.708710	−	0.705500i	$-0.750723\pi$
$683$	− 3.07050e11i	− 1.41100i	0.708710	−	0.705500i	$-0.249277\pi$
$684$	0	0
$685$	−1.01231e11	−0.459780
$686$	0	0
$687$	0	0
$688$	0	0
$689$	4.72183e11i	2.09524i
$690$	0	0
$691$	2.20597e11	0.967582	0.483791	−	0.875184i	$-0.339259\pi$
$691$	2.20597e11	0.967582	0.483791	+	0.875184i	$0.339259\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 1.41620e11i	− 0.606996i
$696$	0	0
$697$	−5.59160e10	−0.236922
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 4.11156e11i	− 1.70269i	−0.524609	−	0.851343i	$-0.675788\pi$
$701$	− 4.11156e11i	− 1.70269i	0.524609	−	0.851343i	$-0.324212\pi$
$702$	0	0
$703$	1.01183e11	0.414274
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.91705e10i	0.316874i
$708$	0	0
$709$	1.41043e11	0.558169	0.279084	−	0.960267i	$-0.409969\pi$
$709$	1.41043e11	0.558169	0.279084	+	0.960267i	$0.409969\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 2.15844e10i	− 0.0835183i
$714$	0	0
$715$	−4.71167e11	−1.80281
$716$	0	0
$717$	0	0
$718$	0	0
$719$	1.87514e11i	0.701645i	0.936442	+	0.350823i	$0.114098\pi$
$719$	1.87514e11i	0.701645i	−0.936442	+	0.350823i	$0.885902\pi$
$720$	0	0
$721$	6.60730e10	0.244502
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.68777e11i	− 0.610887i
$726$	0	0
$727$	−6.74621e10	−0.241503	−0.120751	−	0.992683i	$-0.538530\pi$
$727$	−6.74621e10	−0.241503	−0.120751	+	0.992683i	$0.538530\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 3.42959e10i	− 0.120108i
$732$	0	0
$733$	−5.51348e11	−1.90990	−0.954948	−	0.296773i	$-0.904090\pi$
$733$	−5.51348e11	−1.90990	−0.954948	+	0.296773i	$0.904090\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 7.52360e9i	− 0.0255009i
$738$	0	0
$739$	3.53894e11	1.18658	0.593288	−	0.804990i	$-0.297829\pi$
$739$	3.53894e11	1.18658	0.593288	+	0.804990i	$0.297829\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	4.61456e10i	0.151417i	0.997130	+	0.0757086i	$0.0241219\pi$
$743$	4.61456e10i	0.151417i	−0.997130	+	0.0757086i	$0.975878\pi$
$744$	0	0
$745$	6.02183e11	1.95480
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.84515e11i	− 1.22176i
$750$	0	0
$751$	2.78555e11	0.875690	0.437845	−	0.899050i	$-0.355742\pi$
$751$	2.78555e11	0.875690	0.437845	+	0.899050i	$0.355742\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1.53044e11i	− 0.471008i
$756$	0	0
$757$	−2.54100e11	−0.773786	−0.386893	−	0.922125i	$-0.626452\pi$
$757$	−2.54100e11	−0.773786	−0.386893	+	0.922125i	$0.626452\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 5.04366e11i	− 1.50386i	−0.659243	−	0.751930i	$-0.729123\pi$
$761$	− 5.04366e11i	− 1.50386i	0.659243	−	0.751930i	$-0.270877\pi$
$762$	0	0
$763$	1.99025e11	0.587233
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.29339e11i	0.662670i
$768$	0	0
$769$	2.16447e11	0.618937	0.309468	−	0.950910i	$-0.399849\pi$
$769$	2.16447e11	0.618937	0.309468	+	0.950910i	$0.399849\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 4.05778e11i	− 1.13650i	−0.822855	−	0.568252i	$-0.807620\pi$
$773$	− 4.05778e11i	− 1.13650i	0.822855	−	0.568252i	$-0.192380\pi$
$774$	0	0
$775$	−4.71275e10	−0.130637
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.42953e11i	− 0.388189i
$780$	0	0
$781$	3.68260e11	0.989806
$782$	0	0
$783$	0	0
$784$	0	0
$785$	8.01636e11i	2.11105i
$786$	0	0
$787$	−6.28441e11	−1.63820	−0.819098	−	0.573653i	$-0.805526\pi$
$787$	−6.28441e11	−1.63820	−0.819098	+	0.573653i	$0.805526\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 6.18630e11i	− 1.58025i
$792$	0	0
$793$	8.74286e11	2.21086
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.23780e11i	− 0.802447i	−0.915980	−	0.401224i	$-0.868585\pi$
$797$	− 3.23780e11i	− 0.802447i	0.915980	−	0.401224i	$-0.131415\pi$
$798$	0	0
$799$	5.50462e11	1.35064
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.61663e11i	− 0.388819i
$804$	0	0
$805$	−2.12613e11	−0.506297
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 4.65514e10i	− 0.108677i	−0.998523	−	0.0543386i	$-0.982695\pi$
$809$	− 4.65514e10i	− 0.108677i	0.998523	−	0.0543386i	$-0.0173050\pi$
$810$	0	0
$811$	−9.90434e9	−0.0228951	−0.0114475	−	0.999934i	$-0.503644\pi$
$811$	−9.90434e9	−0.0228951	−0.0114475	+	0.999934i	$0.503644\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9.50587e11i	2.15457i
$816$	0	0
$817$	8.76797e10	0.196794
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.00799e11i	− 0.662069i	−0.943619	−	0.331034i	$-0.892602\pi$
$821$	− 3.00799e11i	− 0.662069i	0.943619	−	0.331034i	$-0.107398\pi$
$822$	0	0
$823$	−3.64988e10	−0.0795571	−0.0397785	−	0.999209i	$-0.512665\pi$
$823$	−3.64988e10	−0.0795571	−0.0397785	+	0.999209i	$0.512665\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	3.63233e11i	0.776540i	0.921546	+	0.388270i	$0.126927\pi$
$827$	3.63233e11i	0.776540i	−0.921546	+	0.388270i	$0.873073\pi$
$828$	0	0
$829$	−1.99283e11	−0.421941	−0.210971	−	0.977492i	$-0.567662\pi$
$829$	−1.99283e11	−0.421941	−0.210971	+	0.977492i	$0.567662\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 1.13005e11i	− 0.234702i
$834$	0	0
$835$	−9.40735e10	−0.193518
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 4.55688e11i	− 0.919644i	−0.888011	−	0.459822i	$-0.847913\pi$
$839$	− 4.55688e11i	− 0.919644i	0.888011	−	0.459822i	$-0.152087\pi$
$840$	0	0
$841$	1.53816e11	0.307481
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.03288e12i	2.02592i
$846$	0	0
$847$	−1.10325e11	−0.214358
$848$	0	0
$849$	0	0
$850$	0	0
$851$	8.71961e10i	0.166257i
$852$	0	0
$853$	−3.80568e11	−0.718846	−0.359423	−	0.933175i	$-0.617027\pi$
$853$	−3.80568e11	−0.718846	−0.359423	+	0.933175i	$0.617027\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.04675e11i	− 0.379438i	−0.981838	−	0.189719i	$-0.939242\pi$
$857$	− 2.04675e11i	− 0.379438i	0.981838	−	0.189719i	$-0.0607577\pi$
$858$	0	0
$859$	−2.40882e11	−0.442417	−0.221208	−	0.975227i	$-0.571000\pi$
$859$	−2.40882e11	−0.442417	−0.221208	+	0.975227i	$0.571000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 8.63484e11i	− 1.55672i	−0.627817	−	0.778361i	$-0.716051\pi$
$863$	− 8.63484e11i	− 1.55672i	0.627817	−	0.778361i	$-0.283949\pi$
$864$	0	0
$865$	1.36065e12	2.43042
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.96709e11i	− 0.520297i
$870$	0	0
$871$	−2.72135e10	−0.0472837
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 1.68161e11i	− 0.286876i
$876$	0	0
$877$	−4.79159e11	−0.809993	−0.404997	−	0.914318i	$-0.632727\pi$
$877$	−4.79159e11	−0.809993	−0.404997	+	0.914318i	$0.632727\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 6.43291e11i	− 1.06783i	−0.845537	−	0.533917i	$-0.820720\pi$
$881$	− 6.43291e11i	− 1.06783i	0.845537	−	0.533917i	$-0.179280\pi$
$882$	0	0
$883$	1.08619e12	1.78674	0.893371	−	0.449319i	$-0.148333\pi$
$883$	1.08619e12	1.78674	0.893371	+	0.449319i	$0.148333\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.68383e11i	− 0.272022i	−0.990707	−	0.136011i	$-0.956572\pi$
$887$	− 1.68383e11i	− 0.272022i	0.990707	−	0.136011i	$-0.0434283\pi$
$888$	0	0
$889$	8.47519e11	1.35688
$890$	0	0
$891$	0	0
$892$	0	0
$893$	1.40729e12i	2.21299i
$894$	0	0
$895$	3.43391e11	0.535177
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.67336e10i	0.148094i
$900$	0	0
$901$	6.18552e11	0.938592
$902$	0	0
$903$	0	0
$904$	0	0
$905$	9.58204e11i	1.42845i
$906$	0	0
$907$	6.69645e11	0.989500	0.494750	−	0.869035i	$-0.335260\pi$
$907$	6.69645e11	0.989500	0.494750	+	0.869035i	$0.335260\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.19351e12i	1.73282i	0.499337	+	0.866408i	$0.333577\pi$
$911$	1.19351e12i	1.73282i	−0.499337	+	0.866408i	$0.666423\pi$
$912$	0	0
$913$	4.20832e11	0.605656
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.71631e11i	0.525575i
$918$	0	0
$919$	7.65817e11	1.07365	0.536825	−	0.843693i	$-0.319623\pi$
$919$	7.65817e11	1.07365	0.536825	+	0.843693i	$0.319623\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 1.33203e12i	− 1.83529i
$924$	0	0
$925$	1.90385e11	0.260055
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 4.26646e11i	− 0.572803i	−0.958110	−	0.286401i	$-0.907541\pi$
$929$	− 4.26646e11i	− 0.572803i	0.958110	−	0.286401i	$-0.0924591\pi$
$930$	0	0
$931$	2.88905e11	0.384553
$932$	0	0
$933$	0	0
$934$	0	0
$935$	6.17221e11i	0.807596i
$936$	0	0
$937$	1.35431e12	1.75695	0.878477	−	0.477784i	$-0.158560\pi$
$937$	1.35431e12	1.75695	0.878477	+	0.477784i	$0.158560\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.88655e10i	− 0.0240608i	−0.999928	−	0.0120304i	$-0.996171\pi$
$941$	− 1.88655e10i	− 0.0240608i	0.999928	−	0.0120304i	$-0.00382948\pi$
$942$	0	0
$943$	1.23192e11	0.155788
$944$	0	0
$945$	0	0
$946$	0	0
$947$	6.03707e11i	0.750631i	0.926897	+	0.375316i	$0.122466\pi$
$947$	6.03707e11i	0.750631i	−0.926897	+	0.375316i	$0.877534\pi$
$948$	0	0
$949$	−5.84747e11	−0.720947
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6.59941e10i	0.0800080i	0.999200	+	0.0400040i	$0.0127371\pi$
$953$	6.59941e10i	0.0800080i	−0.999200	+	0.0400040i	$0.987263\pi$
$954$	0	0
$955$	1.32553e12	1.59359
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 2.41937e11i	− 0.286040i
$960$	0	0
$961$	−8.25880e11	−0.968330
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 5.84580e11i	− 0.674116i
$966$	0	0
$967$	−5.39935e11	−0.617498	−0.308749	−	0.951144i	$-0.599910\pi$
$967$	−5.39935e11	−0.617498	−0.308749	+	0.951144i	$0.599910\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.23031e11i	0.813355i	0.913572	+	0.406677i	$0.133313\pi$
$971$	7.23031e11i	0.813355i	−0.913572	+	0.406677i	$0.866687\pi$
$972$	0	0
$973$	3.38466e11	0.377627
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 4.79457e11i	− 0.526225i	−0.964765	−	0.263112i	$-0.915251\pi$
$977$	− 4.79457e11i	− 0.526225i	0.964765	−	0.263112i	$-0.0847490\pi$
$978$	0	0
$979$	3.55839e11	0.387367
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 9.92587e11i	− 1.06305i	−0.847042	−	0.531526i	$-0.821619\pi$
$983$	− 9.92587e11i	− 1.06305i	0.847042	−	0.531526i	$-0.178381\pi$
$984$	0	0
$985$	−1.16560e12	−1.23823
$986$	0	0
$987$	0	0
$988$	0	0
$989$	7.55591e10i	0.0789772i
$990$	0	0
$991$	−9.53660e11	−0.988778	−0.494389	−	0.869241i	$-0.664608\pi$
$991$	−9.53660e11	−0.988778	−0.494389	+	0.869241i	$0.664608\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.92855e12i	1.96761i
$996$	0	0
$997$	−1.51632e12	−1.53465	−0.767324	−	0.641259i	$-0.778412\pi$
$997$	−1.51632e12	−1.53465	−0.767324	+	0.641259i	$0.778412\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.9.e.d.161.2		2
3.2	odd	2	inner	432.9.e.d.161.1		2
4.3	odd	2		27.9.b.b.26.1	✓	2
12.11	even	2		27.9.b.b.26.2	yes	2
36.7	odd	6		81.9.d.e.53.1		4
36.11	even	6		81.9.d.e.53.2		4
36.23	even	6		81.9.d.e.26.1		4
36.31	odd	6		81.9.d.e.26.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.9.b.b.26.1	✓	2	4.3	odd	2
27.9.b.b.26.2	yes	2	12.11	even	2
81.9.d.e.26.1		4	36.23	even	6
81.9.d.e.26.2		4	36.31	odd	6
81.9.d.e.53.1		4	36.7	odd	6
81.9.d.e.53.2		4	36.11	even	6
432.9.e.d.161.1		2	3.2	odd	2	inner
432.9.e.d.161.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+823.029i q^{5} -1967.00 q^{7} +O(q^{10})\)	\(q+823.029i q^{5} -1967.00 q^{7} -12580.6i q^{11} -45505.0 q^{13} +59610.8i q^{17} -152399. q^{19} -131332. i q^{23} -286751. q^{25} +588583. i q^{29} +164350. q^{31} -1.61890e6i q^{35} -663937. q^{37} +938017. i q^{41} -575330. q^{43} -9.23426e6i q^{47} -1.89571e6 q^{49} -1.03765e7i q^{53} +1.03542e7 q^{55} -5.03987e6i q^{59} -1.92130e7 q^{61} -3.74519e7i q^{65} +598033. q^{67} +2.92721e7i q^{71} +1.28502e7 q^{73} +2.47460e7i q^{77} +2.35847e7 q^{79} +3.34509e7i q^{83} -4.90614e7 q^{85} +2.82848e7i q^{89} +8.95083e7 q^{91} -1.25429e8i q^{95} +1.36490e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3934 q^{7}+O(q^{10}) \) 2 * q - 3934 * q^7	\( 2 q - 3934 q^{7} - 91010 q^{13} - 304798 q^{19} - 573502 q^{25} + 328700 q^{31} - 1327874 q^{37} - 1150660 q^{43} - 3791424 q^{49} + 20708352 q^{55} - 38425922 q^{61} + 1196066 q^{67} + 25700350 q^{73} + 47169314 q^{79} - 98122752 q^{85} + 179016670 q^{91} + 272979262 q^{97}+O(q^{100}) \) 2 * q - 3934 * q^7 - 91010 * q^13 - 304798 * q^19 - 573502 * q^25 + 328700 * q^31 - 1327874 * q^37 - 1150660 * q^43 - 3791424 * q^49 + 20708352 * q^55 - 38425922 * q^61 + 1196066 * q^67 + 25700350 * q^73 + 47169314 * q^79 - 98122752 * q^85 + 179016670 * q^91 + 272979262 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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