LMFDB - Embedded newform 2432.2.c.c.1217.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2432,2,Mod(1217,2432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("2432.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2432 = 2^{7} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2432.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$19.4196177716$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.1
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2432.1217
Dual form			2432.2.c.c.1217.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{3} +2.00000i q^{5} -4.46410 q^{7} +2.00000 q^{9} +O(q^{10})$	$q-1.00000i q^{3} +2.00000i q^{5} -4.46410 q^{7} +2.00000 q^{9} -4.00000i q^{11} +4.46410i q^{13} +2.00000 q^{15} -7.92820 q^{17} -1.00000i q^{19} +4.46410i q^{21} +6.46410 q^{23} +1.00000 q^{25} -5.00000i q^{27} -2.46410i q^{29} +4.92820 q^{31} -4.00000 q^{33} -8.92820i q^{35} +2.00000i q^{37} +4.46410 q^{39} +6.92820 q^{41} +4.92820i q^{43} +4.00000i q^{45} -6.92820 q^{47} +12.9282 q^{49} +7.92820i q^{51} -12.4641i q^{53} +8.00000 q^{55} -1.00000 q^{57} -9.92820i q^{59} -8.92820i q^{61} -8.92820 q^{63} -8.92820 q^{65} +5.92820i q^{67} -6.46410i q^{69} +14.0000 q^{71} +7.00000 q^{73} -1.00000i q^{75} +17.8564i q^{77} +2.00000 q^{79} +1.00000 q^{81} -2.92820i q^{83} -15.8564i q^{85} -2.46410 q^{87} +0.928203 q^{89} -19.9282i q^{91} -4.92820i q^{93} +2.00000 q^{95} +12.9282 q^{97} -8.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^7 + 8 * q^9	$ 4 q - 4 q^{7} + 8 q^{9} + 8 q^{15} - 4 q^{17} + 12 q^{23} + 4 q^{25} - 8 q^{31} - 16 q^{33} + 4 q^{39} + 24 q^{49} + 32 q^{55} - 4 q^{57} - 8 q^{63} - 8 q^{65} + 56 q^{71} + 28 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87} - 24 q^{89} + 8 q^{95} + 24 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 8 * q^9 + 8 * q^15 - 4 * q^17 + 12 * q^23 + 4 * q^25 - 8 * q^31 - 16 * q^33 + 4 * q^39 + 24 * q^49 + 32 * q^55 - 4 * q^57 - 8 * q^63 - 8 * q^65 + 56 * q^71 + 28 * q^73 + 8 * q^79 + 4 * q^81 + 4 * q^87 - 24 * q^89 + 8 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2432\mathbb{Z}\right)^\times$.

$n$	$1407$	$1921$	$2053$
$\chi(n)$	$1$	$1$	$-1$

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
$3$	− 1.00000i	− 0.577350i	−0.957427	−	0.288675i	$-0.906785\pi$
$3$	− 1.00000i	− 0.577350i	0.957427	−	0.288675i	$-0.0932147\pi$
$4$	0	0
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	−4.46410	−1.68727	−0.843636	−	0.536916i	$-0.819589\pi$
$7$	−4.46410	−1.68727	−0.843636	+	0.536916i	$0.819589\pi$
$8$	0	0
$9$	2.00000	0.666667
$10$	0	0
$11$	− 4.00000i	− 1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$	− 4.00000i	− 1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$	0	0
$13$	4.46410i	1.23812i	0.785344	+	0.619060i	$0.212486\pi$
$13$	4.46410i	1.23812i	−0.785344	+	0.619060i	$0.787514\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	−7.92820	−1.92287	−0.961436	−	0.275029i	$-0.911312\pi$
$17$	−7.92820	−1.92287	−0.961436	+	0.275029i	$0.911312\pi$
$18$	0	0
$19$	− 1.00000i	− 0.229416i
$19$	− 1.00000i	− 0.229416i
$20$	0	0
$21$	4.46410i	0.974147i
$22$	0	0
$23$	6.46410	1.34786	0.673929	−	0.738796i	$-0.264605\pi$
$23$	6.46410	1.34786	0.673929	+	0.738796i	$0.264605\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	− 5.00000i	− 0.962250i
$28$	0	0
$29$	− 2.46410i	− 0.457572i	−0.973477	−	0.228786i	$-0.926524\pi$
$29$	− 2.46410i	− 0.457572i	0.973477	−	0.228786i	$-0.0734756\pi$
$30$	0	0
$31$	4.92820	0.885131	0.442566	−	0.896736i	$-0.354068\pi$
$31$	4.92820	0.885131	0.442566	+	0.896736i	$0.354068\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	− 8.92820i	− 1.50914i
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	4.46410	0.714828
$40$	0	0
$41$	6.92820	1.08200	0.541002	−	0.841021i	$-0.318045\pi$
$41$	6.92820	1.08200	0.541002	+	0.841021i	$0.318045\pi$
$42$	0	0
$43$	4.92820i	0.751544i	0.926712	+	0.375772i	$0.122622\pi$
$43$	4.92820i	0.751544i	−0.926712	+	0.375772i	$0.877378\pi$
$44$	0	0
$45$	4.00000i	0.596285i
$46$	0	0
$47$	−6.92820	−1.01058	−0.505291	−	0.862949i	$-0.668615\pi$
$47$	−6.92820	−1.01058	−0.505291	+	0.862949i	$0.668615\pi$
$48$	0	0
$49$	12.9282	1.84689
$50$	0	0
$51$	7.92820i	1.11017i
$52$	0	0
$53$	− 12.4641i	− 1.71208i	−0.516913	−	0.856038i	$-0.672919\pi$
$53$	− 12.4641i	− 1.71208i	0.516913	−	0.856038i	$-0.327081\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	− 9.92820i	− 1.29254i	−0.763108	−	0.646271i	$-0.776328\pi$
$59$	− 9.92820i	− 1.29254i	0.763108	−	0.646271i	$-0.223672\pi$
$60$	0	0
$61$	− 8.92820i	− 1.14314i	−0.820554	−	0.571570i	$-0.806335\pi$
$61$	− 8.92820i	− 1.14314i	0.820554	−	0.571570i	$-0.193665\pi$
$62$	0	0
$63$	−8.92820	−1.12485
$64$	0	0
$65$	−8.92820	−1.10741
$66$	0	0
$67$	5.92820i	0.724245i	0.932130	+	0.362123i	$0.117948\pi$
$67$	5.92820i	0.724245i	−0.932130	+	0.362123i	$0.882052\pi$
$68$	0	0
$69$	− 6.46410i	− 0.778186i
$70$	0	0
$71$	14.0000	1.66149	0.830747	−	0.556650i	$-0.187914\pi$
$71$	14.0000	1.66149	0.830747	+	0.556650i	$0.187914\pi$
$72$	0	0
$73$	7.00000	0.819288	0.409644	−	0.912245i	$-0.365653\pi$
$73$	7.00000	0.819288	0.409644	+	0.912245i	$0.365653\pi$
$74$	0	0
$75$	− 1.00000i	− 0.115470i
$76$	0	0
$77$	17.8564i	2.03493i
$78$	0	0
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	− 2.92820i	− 0.321412i	−0.987002	−	0.160706i	$-0.948623\pi$
$83$	− 2.92820i	− 0.321412i	0.987002	−	0.160706i	$-0.0513771\pi$
$84$	0	0
$85$	− 15.8564i	− 1.71987i
$86$	0	0
$87$	−2.46410	−0.264179
$88$	0	0
$89$	0.928203	0.0983893	0.0491947	−	0.998789i	$-0.484335\pi$
$89$	0.928203	0.0983893	0.0491947	+	0.998789i	$0.484335\pi$
$90$	0	0
$91$	− 19.9282i	− 2.08904i
$92$	0	0
$93$	− 4.92820i	− 0.511031i
$94$	0	0
$95$	2.00000	0.205196
$96$	0	0
$97$	12.9282	1.31266	0.656330	−	0.754474i	$-0.272108\pi$
$97$	12.9282	1.31266	0.656330	+	0.754474i	$0.272108\pi$
$98$	0	0
$99$	− 8.00000i	− 0.804030i
$100$	0	0
$101$	9.85641i	0.980749i	0.871512	+	0.490375i	$0.163140\pi$
$101$	9.85641i	0.980749i	−0.871512	+	0.490375i	$0.836860\pi$
$102$	0	0
$103$	−2.92820	−0.288524	−0.144262	−	0.989539i	$-0.546081\pi$
$103$	−2.92820	−0.288524	−0.144262	+	0.989539i	$0.546081\pi$
$104$	0	0
$105$	−8.92820	−0.871303
$106$	0	0
$107$	13.9282i	1.34649i	0.739419	+	0.673245i	$0.235100\pi$
$107$	13.9282i	1.34649i	−0.739419	+	0.673245i	$0.764900\pi$
$108$	0	0
$109$	− 7.39230i	− 0.708054i	−0.935235	−	0.354027i	$-0.884812\pi$
$109$	− 7.39230i	− 0.708054i	0.935235	−	0.354027i	$-0.115188\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	0	0
$113$	12.9282	1.21618	0.608092	−	0.793867i	$-0.291935\pi$
$113$	12.9282	1.21618	0.608092	+	0.793867i	$0.291935\pi$
$114$	0	0
$115$	12.9282i	1.20556i
$116$	0	0
$117$	8.92820i	0.825413i
$118$	0	0
$119$	35.3923	3.24441
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	− 6.92820i	− 0.624695i
$124$	0	0
$125$	12.0000i	1.07331i
$126$	0	0
$127$	0.928203	0.0823647	0.0411824	−	0.999152i	$-0.486888\pi$
$127$	0.928203	0.0823647	0.0411824	+	0.999152i	$0.486888\pi$
$128$	0	0
$129$	4.92820	0.433904
$130$	0	0
$131$	− 7.07180i	− 0.617866i	−0.951084	−	0.308933i	$-0.900028\pi$
$131$	− 7.07180i	− 0.617866i	0.951084	−	0.308933i	$-0.0999718\pi$
$132$	0	0
$133$	4.46410i	0.387087i
$134$	0	0
$135$	10.0000	0.860663
$136$	0	0
$137$	−8.85641	−0.756654	−0.378327	−	0.925672i	$-0.623500\pi$
$137$	−8.85641	−0.756654	−0.378327	+	0.925672i	$0.623500\pi$
$138$	0	0
$139$	− 15.8564i	− 1.34492i	−0.740132	−	0.672461i	$-0.765237\pi$
$139$	− 15.8564i	− 1.34492i	0.740132	−	0.672461i	$-0.234763\pi$
$140$	0	0
$141$	6.92820i	0.583460i
$142$	0	0
$143$	17.8564	1.49323
$144$	0	0
$145$	4.92820	0.409265
$146$	0	0
$147$	− 12.9282i	− 1.06630i
$148$	0	0
$149$	− 10.9282i	− 0.895273i	−0.894216	−	0.447637i	$-0.852266\pi$
$149$	− 10.9282i	− 0.895273i	0.894216	−	0.447637i	$-0.147734\pi$
$150$	0	0
$151$	16.0000	1.30206	0.651031	−	0.759051i	$-0.274337\pi$
$151$	16.0000	1.30206	0.651031	+	0.759051i	$0.274337\pi$
$152$	0	0
$153$	−15.8564	−1.28191
$154$	0	0
$155$	9.85641i	0.791686i
$156$	0	0
$157$	− 10.0000i	− 0.798087i	−0.916932	−	0.399043i	$-0.869342\pi$
$157$	− 10.0000i	− 0.798087i	0.916932	−	0.399043i	$-0.130658\pi$
$158$	0	0
$159$	−12.4641	−0.988468
$160$	0	0
$161$	−28.8564	−2.27420
$162$	0	0
$163$	− 23.8564i	− 1.86858i	−0.356517	−	0.934289i	$-0.616036\pi$
$163$	− 23.8564i	− 1.86858i	0.356517	−	0.934289i	$-0.383964\pi$
$164$	0	0
$165$	− 8.00000i	− 0.622799i
$166$	0	0
$167$	5.07180	0.392467	0.196234	−	0.980557i	$-0.437129\pi$
$167$	5.07180	0.392467	0.196234	+	0.980557i	$0.437129\pi$
$168$	0	0
$169$	−6.92820	−0.532939
$170$	0	0
$171$	− 2.00000i	− 0.152944i
$172$	0	0
$173$	− 10.0000i	− 0.760286i	−0.924928	−	0.380143i	$-0.875875\pi$
$173$	− 10.0000i	− 0.760286i	0.924928	−	0.380143i	$-0.124125\pi$
$174$	0	0
$175$	−4.46410	−0.337454
$176$	0	0
$177$	−9.92820	−0.746249
$178$	0	0
$179$	− 25.8564i	− 1.93260i	−0.257421	−	0.966299i	$-0.582873\pi$
$179$	− 25.8564i	− 1.93260i	0.257421	−	0.966299i	$-0.417127\pi$
$180$	0	0
$181$	10.0000i	0.743294i	0.928374	+	0.371647i	$0.121207\pi$
$181$	10.0000i	0.743294i	−0.928374	+	0.371647i	$0.878793\pi$
$182$	0	0
$183$	−8.92820	−0.659992
$184$	0	0
$185$	−4.00000	−0.294086
$186$	0	0
$187$	31.7128i	2.31907i
$188$	0	0
$189$	22.3205i	1.62358i
$190$	0	0
$191$	−3.53590	−0.255849	−0.127924	−	0.991784i	$-0.540831\pi$
$191$	−3.53590	−0.255849	−0.127924	+	0.991784i	$0.540831\pi$
$192$	0	0
$193$	9.85641	0.709480	0.354740	−	0.934965i	$-0.384569\pi$
$193$	9.85641	0.709480	0.354740	+	0.934965i	$0.384569\pi$
$194$	0	0
$195$	8.92820i	0.639362i
$196$	0	0
$197$	6.92820i	0.493614i	0.969065	+	0.246807i	$0.0793814\pi$
$197$	6.92820i	0.493614i	−0.969065	+	0.246807i	$0.920619\pi$
$198$	0	0
$199$	−13.5359	−0.959534	−0.479767	−	0.877396i	$-0.659279\pi$
$199$	−13.5359	−0.959534	−0.479767	+	0.877396i	$0.659279\pi$
$200$	0	0
$201$	5.92820	0.418143
$202$	0	0
$203$	11.0000i	0.772049i
$204$	0	0
$205$	13.8564i	0.967773i
$206$	0	0
$207$	12.9282	0.898572
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	24.8564i	1.71119i	0.517649	+	0.855593i	$0.326807\pi$
$211$	24.8564i	1.71119i	−0.517649	+	0.855593i	$0.673193\pi$
$212$	0	0
$213$	− 14.0000i	− 0.959264i
$214$	0	0
$215$	−9.85641	−0.672201
$216$	0	0
$217$	−22.0000	−1.49346
$218$	0	0
$219$	− 7.00000i	− 0.473016i
$220$	0	0
$221$	− 35.3923i	− 2.38074i
$222$	0	0
$223$	−26.9282	−1.80325	−0.901623	−	0.432523i	$-0.857623\pi$
$223$	−26.9282	−1.80325	−0.901623	+	0.432523i	$0.857623\pi$
$224$	0	0
$225$	2.00000	0.133333
$226$	0	0
$227$	1.92820i	0.127979i	0.997951	+	0.0639897i	$0.0203825\pi$
$227$	1.92820i	0.127979i	−0.997951	+	0.0639897i	$0.979618\pi$
$228$	0	0
$229$	3.07180i	0.202990i	0.994836	+	0.101495i	$0.0323626\pi$
$229$	3.07180i	0.202990i	−0.994836	+	0.101495i	$0.967637\pi$
$230$	0	0
$231$	17.8564	1.17487
$232$	0	0
$233$	15.8564	1.03879	0.519394	−	0.854535i	$-0.326158\pi$
$233$	15.8564	1.03879	0.519394	+	0.854535i	$0.326158\pi$
$234$	0	0
$235$	− 13.8564i	− 0.903892i
$236$	0	0
$237$	− 2.00000i	− 0.129914i
$238$	0	0
$239$	−16.3205	−1.05569	−0.527843	−	0.849342i	$-0.676999\pi$
$239$	−16.3205	−1.05569	−0.527843	+	0.849342i	$0.676999\pi$
$240$	0	0
$241$	−24.7846	−1.59652	−0.798259	−	0.602315i	$-0.794245\pi$
$241$	−24.7846	−1.59652	−0.798259	+	0.602315i	$0.794245\pi$
$242$	0	0
$243$	− 16.0000i	− 1.02640i
$244$	0	0
$245$	25.8564i	1.65191i
$246$	0	0
$247$	4.46410	0.284044
$248$	0	0
$249$	−2.92820	−0.185567
$250$	0	0
$251$	12.9282i	0.816021i	0.912977	+	0.408010i	$0.133777\pi$
$251$	12.9282i	0.816021i	−0.912977	+	0.408010i	$0.866223\pi$
$252$	0	0
$253$	− 25.8564i	− 1.62558i
$254$	0	0
$255$	−15.8564	−0.992967
$256$	0	0
$257$	−0.928203	−0.0578997	−0.0289499	−	0.999581i	$-0.509216\pi$
$257$	−0.928203	−0.0578997	−0.0289499	+	0.999581i	$0.509216\pi$
$258$	0	0
$259$	− 8.92820i	− 0.554772i
$260$	0	0
$261$	− 4.92820i	− 0.305048i
$262$	0	0
$263$	2.92820	0.180561	0.0902804	−	0.995916i	$-0.471224\pi$
$263$	2.92820	0.180561	0.0902804	+	0.995916i	$0.471224\pi$
$264$	0	0
$265$	24.9282	1.53133
$266$	0	0
$267$	− 0.928203i	− 0.0568051i
$268$	0	0
$269$	− 7.85641i	− 0.479014i	−0.970895	−	0.239507i	$-0.923014\pi$
$269$	− 7.85641i	− 0.479014i	0.970895	−	0.239507i	$-0.0769857\pi$
$270$	0	0
$271$	−4.60770	−0.279898	−0.139949	−	0.990159i	$-0.544694\pi$
$271$	−4.60770	−0.279898	−0.139949	+	0.990159i	$0.544694\pi$
$272$	0	0
$273$	−19.9282	−1.20611
$274$	0	0
$275$	− 4.00000i	− 0.241209i
$276$	0	0
$277$	24.9282i	1.49779i	0.662688	+	0.748895i	$0.269415\pi$
$277$	24.9282i	1.49779i	−0.662688	+	0.748895i	$0.730585\pi$
$278$	0	0
$279$	9.85641	0.590088
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	− 18.9282i	− 1.12516i	−0.826741	−	0.562582i	$-0.809808\pi$
$283$	− 18.9282i	− 1.12516i	0.826741	−	0.562582i	$-0.190192\pi$
$284$	0	0
$285$	− 2.00000i	− 0.118470i
$286$	0	0
$287$	−30.9282	−1.82563
$288$	0	0
$289$	45.8564	2.69744
$290$	0	0
$291$	− 12.9282i	− 0.757865i
$292$	0	0
$293$	− 8.32051i	− 0.486089i	−0.970015	−	0.243045i	$-0.921854\pi$
$293$	− 8.32051i	− 0.486089i	0.970015	−	0.243045i	$-0.0781462\pi$
$294$	0	0
$295$	19.8564	1.15608
$296$	0	0
$297$	−20.0000	−1.16052
$298$	0	0
$299$	28.8564i	1.66881i
$300$	0	0
$301$	− 22.0000i	− 1.26806i
$302$	0	0
$303$	9.85641	0.566236
$304$	0	0
$305$	17.8564	1.02245
$306$	0	0
$307$	− 1.85641i	− 0.105951i	−0.998596	−	0.0529754i	$-0.983130\pi$
$307$	− 1.85641i	− 0.105951i	0.998596	−	0.0529754i	$-0.0168705\pi$
$308$	0	0
$309$	2.92820i	0.166580i
$310$	0	0
$311$	−7.39230	−0.419179	−0.209590	−	0.977789i	$-0.567213\pi$
$311$	−7.39230	−0.419179	−0.209590	+	0.977789i	$0.567213\pi$
$312$	0	0
$313$	16.8564	0.952780	0.476390	−	0.879234i	$-0.341945\pi$
$313$	16.8564	0.952780	0.476390	+	0.879234i	$0.341945\pi$
$314$	0	0
$315$	− 17.8564i	− 1.00609i
$316$	0	0
$317$	14.4641i	0.812385i	0.913788	+	0.406192i	$0.133144\pi$
$317$	14.4641i	0.812385i	−0.913788	+	0.406192i	$0.866856\pi$
$318$	0	0
$319$	−9.85641	−0.551853
$320$	0	0
$321$	13.9282	0.777396
$322$	0	0
$323$	7.92820i	0.441137i
$324$	0	0
$325$	4.46410i	0.247624i
$326$	0	0
$327$	−7.39230	−0.408795
$328$	0	0
$329$	30.9282	1.70513
$330$	0	0
$331$	− 9.78461i	− 0.537811i	−0.963167	−	0.268905i	$-0.913338\pi$
$331$	− 9.78461i	− 0.537811i	0.963167	−	0.268905i	$-0.0866619\pi$
$332$	0	0
$333$	4.00000i	0.219199i
$334$	0	0
$335$	−11.8564	−0.647785
$336$	0	0
$337$	−7.07180	−0.385225	−0.192613	−	0.981275i	$-0.561696\pi$
$337$	−7.07180	−0.385225	−0.192613	+	0.981275i	$0.561696\pi$
$338$	0	0
$339$	− 12.9282i	− 0.702164i
$340$	0	0
$341$	− 19.7128i	− 1.06751i
$342$	0	0
$343$	−26.4641	−1.42893
$344$	0	0
$345$	12.9282	0.696031
$346$	0	0
$347$	− 27.8564i	− 1.49541i	−0.664031	−	0.747705i	$-0.731156\pi$
$347$	− 27.8564i	− 1.49541i	0.664031	−	0.747705i	$-0.268844\pi$
$348$	0	0
$349$	14.0000i	0.749403i	0.927146	+	0.374701i	$0.122255\pi$
$349$	14.0000i	0.749403i	−0.927146	+	0.374701i	$0.877745\pi$
$350$	0	0
$351$	22.3205	1.19138
$352$	0	0
$353$	10.0718	0.536068	0.268034	−	0.963410i	$-0.413626\pi$
$353$	10.0718	0.536068	0.268034	+	0.963410i	$0.413626\pi$
$354$	0	0
$355$	28.0000i	1.48609i
$356$	0	0
$357$	− 35.3923i	− 1.87316i
$358$	0	0
$359$	24.4641	1.29117	0.645583	−	0.763690i	$-0.276614\pi$
$359$	24.4641	1.29117	0.645583	+	0.763690i	$0.276614\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	5.00000i	0.262432i
$364$	0	0
$365$	14.0000i	0.732793i
$366$	0	0
$367$	−13.0718	−0.682342	−0.341171	−	0.940001i	$-0.610824\pi$
$367$	−13.0718	−0.682342	−0.341171	+	0.940001i	$0.610824\pi$
$368$	0	0
$369$	13.8564	0.721336
$370$	0	0
$371$	55.6410i	2.88874i
$372$	0	0
$373$	11.3923i	0.589871i	0.955517	+	0.294936i	$0.0952982\pi$
$373$	11.3923i	0.589871i	−0.955517	+	0.294936i	$0.904702\pi$
$374$	0	0
$375$	12.0000	0.619677
$376$	0	0
$377$	11.0000	0.566529
$378$	0	0
$379$	6.85641i	0.352190i	0.984373	+	0.176095i	$0.0563466\pi$
$379$	6.85641i	0.352190i	−0.984373	+	0.176095i	$0.943653\pi$
$380$	0	0
$381$	− 0.928203i	− 0.0475533i
$382$	0	0
$383$	24.7846	1.26643	0.633217	−	0.773974i	$-0.281734\pi$
$383$	24.7846	1.26643	0.633217	+	0.773974i	$0.281734\pi$
$384$	0	0
$385$	−35.7128	−1.82009
$386$	0	0
$387$	9.85641i	0.501029i
$388$	0	0
$389$	25.8564i	1.31097i	0.755207	+	0.655486i	$0.227536\pi$
$389$	25.8564i	1.31097i	−0.755207	+	0.655486i	$0.772464\pi$
$390$	0	0
$391$	−51.2487	−2.59176
$392$	0	0
$393$	−7.07180	−0.356725
$394$	0	0
$395$	4.00000i	0.201262i
$396$	0	0
$397$	− 8.92820i	− 0.448094i	−0.974578	−	0.224047i	$-0.928073\pi$
$397$	− 8.92820i	− 0.448094i	0.974578	−	0.224047i	$-0.0719269\pi$
$398$	0	0
$399$	4.46410	0.223485
$400$	0	0
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	0	0
$403$	22.0000i	1.09590i
$404$	0	0
$405$	2.00000i	0.0993808i
$406$	0	0
$407$	8.00000	0.396545
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	8.85641i	0.436854i
$412$	0	0
$413$	44.3205i	2.18087i
$414$	0	0
$415$	5.85641	0.287480
$416$	0	0
$417$	−15.8564	−0.776492
$418$	0	0
$419$	30.7846i	1.50393i	0.659205	+	0.751963i	$0.270893\pi$
$419$	30.7846i	1.50393i	−0.659205	+	0.751963i	$0.729107\pi$
$420$	0	0
$421$	11.5359i	0.562225i	0.959675	+	0.281113i	$0.0907035\pi$
$421$	11.5359i	0.562225i	−0.959675	+	0.281113i	$0.909297\pi$
$422$	0	0
$423$	−13.8564	−0.673722
$424$	0	0
$425$	−7.92820	−0.384574
$426$	0	0
$427$	39.8564i	1.92879i
$428$	0	0
$429$	− 17.8564i	− 0.862115i
$430$	0	0
$431$	−12.9282	−0.622730	−0.311365	−	0.950290i	$-0.600786\pi$
$431$	−12.9282	−0.622730	−0.311365	+	0.950290i	$0.600786\pi$
$432$	0	0
$433$	−24.7846	−1.19107	−0.595536	−	0.803328i	$-0.703060\pi$
$433$	−24.7846	−1.19107	−0.595536	+	0.803328i	$0.703060\pi$
$434$	0	0
$435$	− 4.92820i	− 0.236289i
$436$	0	0
$437$	− 6.46410i	− 0.309220i
$438$	0	0
$439$	−4.00000	−0.190910	−0.0954548	−	0.995434i	$-0.530431\pi$
$439$	−4.00000	−0.190910	−0.0954548	+	0.995434i	$0.530431\pi$
$440$	0	0
$441$	25.8564	1.23126
$442$	0	0
$443$	− 17.8564i	− 0.848383i	−0.905572	−	0.424192i	$-0.860558\pi$
$443$	− 17.8564i	− 0.848383i	0.905572	−	0.424192i	$-0.139442\pi$
$444$	0	0
$445$	1.85641i	0.0880021i
$446$	0	0
$447$	−10.9282	−0.516886
$448$	0	0
$449$	20.0000	0.943858	0.471929	−	0.881636i	$-0.343558\pi$
$449$	20.0000	0.943858	0.471929	+	0.881636i	$0.343558\pi$
$450$	0	0
$451$	− 27.7128i	− 1.30495i
$452$	0	0
$453$	− 16.0000i	− 0.751746i
$454$	0	0
$455$	39.8564	1.86850
$456$	0	0
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	0	0
$459$	39.6410i	1.85028i
$460$	0	0
$461$	13.8564i	0.645357i	0.946509	+	0.322679i	$0.104583\pi$
$461$	13.8564i	0.645357i	−0.946509	+	0.322679i	$0.895417\pi$
$462$	0	0
$463$	−16.7846	−0.780047	−0.390023	−	0.920805i	$-0.627533\pi$
$463$	−16.7846	−0.780047	−0.390023	+	0.920805i	$0.627533\pi$
$464$	0	0
$465$	9.85641	0.457080
$466$	0	0
$467$	− 2.78461i	− 0.128856i	−0.997922	−	0.0644282i	$-0.979478\pi$
$467$	− 2.78461i	− 0.128856i	0.997922	−	0.0644282i	$-0.0205223\pi$
$468$	0	0
$469$	− 26.4641i	− 1.22200i
$470$	0	0
$471$	−10.0000	−0.460776
$472$	0	0
$473$	19.7128	0.906396
$474$	0	0
$475$	− 1.00000i	− 0.0458831i
$476$	0	0
$477$	− 24.9282i	− 1.14138i
$478$	0	0
$479$	16.7846	0.766908	0.383454	−	0.923560i	$-0.374734\pi$
$479$	16.7846	0.766908	0.383454	+	0.923560i	$0.374734\pi$
$480$	0	0
$481$	−8.92820	−0.407091
$482$	0	0
$483$	28.8564i	1.31301i
$484$	0	0
$485$	25.8564i	1.17408i
$486$	0	0
$487$	34.6410	1.56973	0.784867	−	0.619664i	$-0.212731\pi$
$487$	34.6410	1.56973	0.784867	+	0.619664i	$0.212731\pi$
$488$	0	0
$489$	−23.8564	−1.07882
$490$	0	0
$491$	14.1436i	0.638291i	0.947706	+	0.319146i	$0.103396\pi$
$491$	14.1436i	0.638291i	−0.947706	+	0.319146i	$0.896604\pi$
$492$	0	0
$493$	19.5359i	0.879853i
$494$	0	0
$495$	16.0000	0.719147
$496$	0	0
$497$	−62.4974	−2.80339
$498$	0	0
$499$	22.7846i	1.01998i	0.860181	+	0.509990i	$0.170351\pi$
$499$	22.7846i	1.01998i	−0.860181	+	0.509990i	$0.829649\pi$
$500$	0	0
$501$	− 5.07180i	− 0.226591i
$502$	0	0
$503$	−22.3205	−0.995222	−0.497611	−	0.867400i	$-0.665789\pi$
$503$	−22.3205	−0.995222	−0.497611	+	0.867400i	$0.665789\pi$
$504$	0	0
$505$	−19.7128	−0.877209
$506$	0	0
$507$	6.92820i	0.307692i
$508$	0	0
$509$	− 18.0000i	− 0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$	− 18.0000i	− 0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$	0	0
$511$	−31.2487	−1.38236
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	− 5.85641i	− 0.258064i
$516$	0	0
$517$	27.7128i	1.21881i
$518$	0	0
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−13.0718	−0.572686	−0.286343	−	0.958127i	$-0.592440\pi$
$521$	−13.0718	−0.572686	−0.286343	+	0.958127i	$0.592440\pi$
$522$	0	0
$523$	31.0000i	1.35554i	0.735276	+	0.677768i	$0.237052\pi$
$523$	31.0000i	1.35554i	−0.735276	+	0.677768i	$0.762948\pi$
$524$	0	0
$525$	4.46410i	0.194829i
$526$	0	0
$527$	−39.0718	−1.70199
$528$	0	0
$529$	18.7846	0.816722
$530$	0	0
$531$	− 19.8564i	− 0.861695i
$532$	0	0
$533$	30.9282i	1.33965i
$534$	0	0
$535$	−27.8564	−1.20434
$536$	0	0
$537$	−25.8564	−1.11579
$538$	0	0
$539$	− 51.7128i	− 2.22743i
$540$	0	0
$541$	12.7846i	0.549653i	0.961494	+	0.274827i	$0.0886205\pi$
$541$	12.7846i	0.549653i	−0.961494	+	0.274827i	$0.911380\pi$
$542$	0	0
$543$	10.0000	0.429141
$544$	0	0
$545$	14.7846	0.633303
$546$	0	0
$547$	6.14359i	0.262681i	0.991337	+	0.131341i	$0.0419281\pi$
$547$	6.14359i	0.262681i	−0.991337	+	0.131341i	$0.958072\pi$
$548$	0	0
$549$	− 17.8564i	− 0.762093i
$550$	0	0
$551$	−2.46410	−0.104974
$552$	0	0
$553$	−8.92820	−0.379666
$554$	0	0
$555$	4.00000i	0.169791i
$556$	0	0
$557$	− 22.0000i	− 0.932170i	−0.884740	−	0.466085i	$-0.845664\pi$
$557$	− 22.0000i	− 0.932170i	0.884740	−	0.466085i	$-0.154336\pi$
$558$	0	0
$559$	−22.0000	−0.930501
$560$	0	0
$561$	31.7128	1.33892
$562$	0	0
$563$	− 31.7128i	− 1.33654i	−0.743921	−	0.668268i	$-0.767036\pi$
$563$	− 31.7128i	− 1.33654i	0.743921	−	0.668268i	$-0.232964\pi$
$564$	0	0
$565$	25.8564i	1.08779i
$566$	0	0
$567$	−4.46410	−0.187475
$568$	0	0
$569$	−8.78461	−0.368270	−0.184135	−	0.982901i	$-0.558948\pi$
$569$	−8.78461	−0.368270	−0.184135	+	0.982901i	$0.558948\pi$
$570$	0	0
$571$	− 28.7846i	− 1.20460i	−0.798270	−	0.602299i	$-0.794251\pi$
$571$	− 28.7846i	− 1.20460i	0.798270	−	0.602299i	$-0.205749\pi$
$572$	0	0
$573$	3.53590i	0.147714i
$574$	0	0
$575$	6.46410	0.269572
$576$	0	0
$577$	11.7846	0.490600	0.245300	−	0.969447i	$-0.421114\pi$
$577$	11.7846	0.490600	0.245300	+	0.969447i	$0.421114\pi$
$578$	0	0
$579$	− 9.85641i	− 0.409618i
$580$	0	0
$581$	13.0718i	0.542310i
$582$	0	0
$583$	−49.8564	−2.06484
$584$	0	0
$585$	−17.8564	−0.738272
$586$	0	0
$587$	0	0	1.00000			$0$
$587$	0	0	−1.00000			$\pi$
$588$	0	0
$589$	− 4.92820i	− 0.203063i
$590$	0	0
$591$	6.92820	0.284988
$592$	0	0
$593$	16.1436	0.662938	0.331469	−	0.943466i	$-0.392456\pi$
$593$	16.1436	0.662938	0.331469	+	0.943466i	$0.392456\pi$
$594$	0	0
$595$	70.7846i	2.90189i
$596$	0	0
$597$	13.5359i	0.553987i
$598$	0	0
$599$	9.85641	0.402722	0.201361	−	0.979517i	$-0.435464\pi$
$599$	9.85641	0.402722	0.201361	+	0.979517i	$0.435464\pi$
$600$	0	0
$601$	−27.8564	−1.13629	−0.568143	−	0.822930i	$-0.692338\pi$
$601$	−27.8564	−1.13629	−0.568143	+	0.822930i	$0.692338\pi$
$602$	0	0
$603$	11.8564i	0.482830i
$604$	0	0
$605$	− 10.0000i	− 0.406558i
$606$	0	0
$607$	−39.8564	−1.61772	−0.808861	−	0.588000i	$-0.799915\pi$
$607$	−39.8564	−1.61772	−0.808861	+	0.588000i	$0.799915\pi$
$608$	0	0
$609$	11.0000	0.445742
$610$	0	0
$611$	− 30.9282i	− 1.25122i
$612$	0	0
$613$	− 34.6410i	− 1.39914i	−0.714565	−	0.699569i	$-0.753375\pi$
$613$	− 34.6410i	− 1.39914i	0.714565	−	0.699569i	$-0.246625\pi$
$614$	0	0
$615$	13.8564	0.558744
$616$	0	0
$617$	14.0000	0.563619	0.281809	−	0.959470i	$-0.409065\pi$
$617$	14.0000	0.563619	0.281809	+	0.959470i	$0.409065\pi$
$618$	0	0
$619$	27.8564i	1.11964i	0.828613	+	0.559822i	$0.189130\pi$
$619$	27.8564i	1.11964i	−0.828613	+	0.559822i	$0.810870\pi$
$620$	0	0
$621$	− 32.3205i	− 1.29698i
$622$	0	0
$623$	−4.14359	−0.166010
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	4.00000i	0.159745i
$628$	0	0
$629$	− 15.8564i	− 0.632236i
$630$	0	0
$631$	38.6410	1.53827	0.769137	−	0.639084i	$-0.220686\pi$
$631$	38.6410	1.53827	0.769137	+	0.639084i	$0.220686\pi$
$632$	0	0
$633$	24.8564	0.987953
$634$	0	0
$635$	1.85641i	0.0736692i
$636$	0	0
$637$	57.7128i	2.28666i
$638$	0	0
$639$	28.0000	1.10766
$640$	0	0
$641$	8.78461	0.346971	0.173486	−	0.984836i	$-0.444497\pi$
$641$	8.78461	0.346971	0.173486	+	0.984836i	$0.444497\pi$
$642$	0	0
$643$	32.9282i	1.29856i	0.760549	+	0.649281i	$0.224930\pi$
$643$	32.9282i	1.29856i	−0.760549	+	0.649281i	$0.775070\pi$
$644$	0	0
$645$	9.85641i	0.388096i
$646$	0	0
$647$	−13.5359	−0.532151	−0.266076	−	0.963952i	$-0.585727\pi$
$647$	−13.5359	−0.532151	−0.266076	+	0.963952i	$0.585727\pi$
$648$	0	0
$649$	−39.7128	−1.55886
$650$	0	0
$651$	22.0000i	0.862248i
$652$	0	0
$653$	− 6.78461i	− 0.265502i	−0.991149	−	0.132751i	$-0.957619\pi$
$653$	− 6.78461i	− 0.265502i	0.991149	−	0.132751i	$-0.0423811\pi$
$654$	0	0
$655$	14.1436	0.552636
$656$	0	0
$657$	14.0000	0.546192
$658$	0	0
$659$	2.85641i	0.111270i	0.998451	+	0.0556349i	$0.0177183\pi$
$659$	2.85641i	0.111270i	−0.998451	+	0.0556349i	$0.982282\pi$
$660$	0	0
$661$	− 30.4641i	− 1.18492i	−0.805601	−	0.592458i	$-0.798158\pi$
$661$	− 30.4641i	− 1.18492i	0.805601	−	0.592458i	$-0.201842\pi$
$662$	0	0
$663$	−35.3923	−1.37452
$664$	0	0
$665$	−8.92820	−0.346221
$666$	0	0
$667$	− 15.9282i	− 0.616742i
$668$	0	0
$669$	26.9282i	1.04110i
$670$	0	0
$671$	−35.7128	−1.37868
$672$	0	0
$673$	9.07180	0.349692	0.174846	−	0.984596i	$-0.444057\pi$
$673$	9.07180	0.349692	0.174846	+	0.984596i	$0.444057\pi$
$674$	0	0
$675$	− 5.00000i	− 0.192450i
$676$	0	0
$677$	7.39230i	0.284109i	0.989859	+	0.142055i	$0.0453709\pi$
$677$	7.39230i	0.284109i	−0.989859	+	0.142055i	$0.954629\pi$
$678$	0	0
$679$	−57.7128	−2.21481
$680$	0	0
$681$	1.92820	0.0738889
$682$	0	0
$683$	17.8564i	0.683256i	0.939835	+	0.341628i	$0.110978\pi$
$683$	17.8564i	0.683256i	−0.939835	+	0.341628i	$0.889022\pi$
$684$	0	0
$685$	− 17.7128i	− 0.676772i
$686$	0	0
$687$	3.07180	0.117196
$688$	0	0
$689$	55.6410	2.11975
$690$	0	0
$691$	7.85641i	0.298872i	0.988771	+	0.149436i	$0.0477458\pi$
$691$	7.85641i	0.298872i	−0.988771	+	0.149436i	$0.952254\pi$
$692$	0	0
$693$	35.7128i	1.35662i
$694$	0	0
$695$	31.7128	1.20294
$696$	0	0
$697$	−54.9282	−2.08055
$698$	0	0
$699$	− 15.8564i	− 0.599744i
$700$	0	0
$701$	7.71281i	0.291309i	0.989336	+	0.145654i	$0.0465288\pi$
$701$	7.71281i	0.291309i	−0.989336	+	0.145654i	$0.953471\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	0	0
$705$	−13.8564	−0.521862
$706$	0	0
$707$	− 44.0000i	− 1.65479i
$708$	0	0
$709$	− 10.7846i	− 0.405025i	−0.979280	−	0.202512i	$-0.935089\pi$
$709$	− 10.7846i	− 0.405025i	0.979280	−	0.202512i	$-0.0649106\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	31.8564	1.19303
$714$	0	0
$715$	35.7128i	1.33558i
$716$	0	0
$717$	16.3205i	0.609501i
$718$	0	0
$719$	−13.3923	−0.499449	−0.249724	−	0.968317i	$-0.580340\pi$
$719$	−13.3923	−0.499449	−0.249724	+	0.968317i	$0.580340\pi$
$720$	0	0
$721$	13.0718	0.486819
$722$	0	0
$723$	24.7846i	0.921750i
$724$	0	0
$725$	− 2.46410i	− 0.0915144i
$726$	0	0
$727$	0.320508	0.0118870	0.00594349	−	0.999982i	$-0.498108\pi$
$727$	0.320508	0.0118870	0.00594349	+	0.999982i	$0.498108\pi$
$728$	0	0
$729$	−13.0000	−0.481481
$730$	0	0
$731$	− 39.0718i	− 1.44512i
$732$	0	0
$733$	43.8564i	1.61987i	0.586517	+	0.809937i	$0.300499\pi$
$733$	43.8564i	1.61987i	−0.586517	+	0.809937i	$0.699501\pi$
$734$	0	0
$735$	25.8564	0.953728
$736$	0	0
$737$	23.7128	0.873473
$738$	0	0
$739$	9.85641i	0.362574i	0.983430	+	0.181287i	$0.0580263\pi$
$739$	9.85641i	0.362574i	−0.983430	+	0.181287i	$0.941974\pi$
$740$	0	0
$741$	− 4.46410i	− 0.163993i
$742$	0	0
$743$	−16.9282	−0.621036	−0.310518	−	0.950568i	$-0.600502\pi$
$743$	−16.9282	−0.621036	−0.310518	+	0.950568i	$0.600502\pi$
$744$	0	0
$745$	21.8564	0.800757
$746$	0	0
$747$	− 5.85641i	− 0.214275i
$748$	0	0
$749$	− 62.1769i	− 2.27190i
$750$	0	0
$751$	−33.0718	−1.20681	−0.603404	−	0.797436i	$-0.706189\pi$
$751$	−33.0718	−1.20681	−0.603404	+	0.797436i	$0.706189\pi$
$752$	0	0
$753$	12.9282	0.471130
$754$	0	0
$755$	32.0000i	1.16460i
$756$	0	0
$757$	− 1.21539i	− 0.0441741i	−0.999756	−	0.0220871i	$-0.992969\pi$
$757$	− 1.21539i	− 0.0441741i	0.999756	−	0.0220871i	$-0.00703110\pi$
$758$	0	0
$759$	−25.8564	−0.938528
$760$	0	0
$761$	48.7128	1.76584	0.882919	−	0.469525i	$-0.155575\pi$
$761$	48.7128	1.76584	0.882919	+	0.469525i	$0.155575\pi$
$762$	0	0
$763$	33.0000i	1.19468i
$764$	0	0
$765$	− 31.7128i	− 1.14658i
$766$	0	0
$767$	44.3205	1.60032
$768$	0	0
$769$	−47.0000	−1.69486	−0.847432	−	0.530904i	$-0.821852\pi$
$769$	−47.0000	−1.69486	−0.847432	+	0.530904i	$0.821852\pi$
$770$	0	0
$771$	0.928203i	0.0334284i
$772$	0	0
$773$	− 54.4641i	− 1.95894i	−0.201596	−	0.979469i	$-0.564613\pi$
$773$	− 54.4641i	− 1.95894i	0.201596	−	0.979469i	$-0.435387\pi$
$774$	0	0
$775$	4.92820	0.177026
$776$	0	0
$777$	−8.92820	−0.320298
$778$	0	0
$779$	− 6.92820i	− 0.248229i
$780$	0	0
$781$	− 56.0000i	− 2.00384i
$782$	0	0
$783$	−12.3205	−0.440299
$784$	0	0
$785$	20.0000	0.713831
$786$	0	0
$787$	19.7846i	0.705245i	0.935766	+	0.352623i	$0.114710\pi$
$787$	19.7846i	0.705245i	−0.935766	+	0.352623i	$0.885290\pi$
$788$	0	0
$789$	− 2.92820i	− 0.104247i
$790$	0	0
$791$	−57.7128	−2.05203
$792$	0	0
$793$	39.8564	1.41534
$794$	0	0
$795$	− 24.9282i	− 0.884112i
$796$	0	0
$797$	− 30.1769i	− 1.06892i	−0.845193	−	0.534461i	$-0.820515\pi$
$797$	− 30.1769i	− 1.06892i	0.845193	−	0.534461i	$-0.179485\pi$
$798$	0	0
$799$	54.9282	1.94322
$800$	0	0
$801$	1.85641	0.0655929
$802$	0	0
$803$	− 28.0000i	− 0.988099i
$804$	0	0
$805$	− 57.7128i	− 2.03411i
$806$	0	0
$807$	−7.85641	−0.276559
$808$	0	0
$809$	−18.0718	−0.635371	−0.317685	−	0.948196i	$-0.602906\pi$
$809$	−18.0718	−0.635371	−0.317685	+	0.948196i	$0.602906\pi$
$810$	0	0
$811$	1.14359i	0.0401570i	0.999798	+	0.0200785i	$0.00639161\pi$
$811$	1.14359i	0.0401570i	−0.999798	+	0.0200785i	$0.993608\pi$
$812$	0	0
$813$	4.60770i	0.161599i
$814$	0	0
$815$	47.7128	1.67131
$816$	0	0
$817$	4.92820	0.172416
$818$	0	0
$819$	− 39.8564i	− 1.39270i
$820$	0	0
$821$	− 44.6410i	− 1.55798i	−0.627035	−	0.778991i	$-0.715732\pi$
$821$	− 44.6410i	− 1.55798i	0.627035	−	0.778991i	$-0.284268\pi$
$822$	0	0
$823$	−23.5359	−0.820410	−0.410205	−	0.911993i	$-0.634543\pi$
$823$	−23.5359	−0.820410	−0.410205	+	0.911993i	$0.634543\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	− 14.7128i	− 0.511615i	−0.966728	−	0.255807i	$-0.917659\pi$
$827$	− 14.7128i	− 0.511615i	0.966728	−	0.255807i	$-0.0823413\pi$
$828$	0	0
$829$	12.3205i	0.427909i	0.976844	+	0.213954i	$0.0686344\pi$
$829$	12.3205i	0.427909i	−0.976844	+	0.213954i	$0.931366\pi$
$830$	0	0
$831$	24.9282	0.864750
$832$	0	0
$833$	−102.497	−3.55133
$834$	0	0
$835$	10.1436i	0.351034i
$836$	0	0
$837$	− 24.6410i	− 0.851718i
$838$	0	0
$839$	22.0000	0.759524	0.379762	−	0.925084i	$-0.376006\pi$
$839$	22.0000	0.759524	0.379762	+	0.925084i	$0.376006\pi$
$840$	0	0
$841$	22.9282	0.790628
$842$	0	0
$843$	− 4.00000i	− 0.137767i
$844$	0	0
$845$	− 13.8564i	− 0.476675i
$846$	0	0
$847$	22.3205	0.766942
$848$	0	0
$849$	−18.9282	−0.649614
$850$	0	0
$851$	12.9282i	0.443173i
$852$	0	0
$853$	− 13.0718i	− 0.447570i	−0.974639	−	0.223785i	$-0.928159\pi$
$853$	− 13.0718i	− 0.447570i	0.974639	−	0.223785i	$-0.0718413\pi$
$854$	0	0
$855$	4.00000	0.136797
$856$	0	0
$857$	−49.8564	−1.70306	−0.851531	−	0.524304i	$-0.824326\pi$
$857$	−49.8564	−1.70306	−0.851531	+	0.524304i	$0.824326\pi$
$858$	0	0
$859$	− 21.0718i	− 0.718960i	−0.933153	−	0.359480i	$-0.882954\pi$
$859$	− 21.0718i	− 0.718960i	0.933153	−	0.359480i	$-0.117046\pi$
$860$	0	0
$861$	30.9282i	1.05403i
$862$	0	0
$863$	14.7846	0.503274	0.251637	−	0.967822i	$-0.419031\pi$
$863$	14.7846	0.503274	0.251637	+	0.967822i	$0.419031\pi$
$864$	0	0
$865$	20.0000	0.680020
$866$	0	0
$867$	− 45.8564i	− 1.55737i
$868$	0	0
$869$	− 8.00000i	− 0.271381i
$870$	0	0
$871$	−26.4641	−0.896702
$872$	0	0
$873$	25.8564	0.875107
$874$	0	0
$875$	− 53.5692i	− 1.81097i
$876$	0	0
$877$	31.2487i	1.05519i	0.849495	+	0.527597i	$0.176907\pi$
$877$	31.2487i	1.05519i	−0.849495	+	0.527597i	$0.823093\pi$
$878$	0	0
$879$	−8.32051	−0.280644
$880$	0	0
$881$	−22.0000	−0.741199	−0.370599	−	0.928793i	$-0.620848\pi$
$881$	−22.0000	−0.741199	−0.370599	+	0.928793i	$0.620848\pi$
$882$	0	0
$883$	− 49.7128i	− 1.67297i	−0.547990	−	0.836485i	$-0.684607\pi$
$883$	− 49.7128i	− 1.67297i	0.547990	−	0.836485i	$-0.315393\pi$
$884$	0	0
$885$	− 19.8564i	− 0.667466i
$886$	0	0
$887$	12.9282	0.434087	0.217043	−	0.976162i	$-0.430359\pi$
$887$	12.9282	0.434087	0.217043	+	0.976162i	$0.430359\pi$
$888$	0	0
$889$	−4.14359	−0.138972
$890$	0	0
$891$	− 4.00000i	− 0.134005i
$892$	0	0
$893$	6.92820i	0.231843i
$894$	0	0
$895$	51.7128	1.72857
$896$	0	0
$897$	28.8564	0.963487
$898$	0	0
$899$	− 12.1436i	− 0.405012i
$900$	0	0
$901$	98.8179i	3.29210i
$902$	0	0
$903$	−22.0000	−0.732114
$904$	0	0
$905$	−20.0000	−0.664822
$906$	0	0
$907$	13.1436i	0.436426i	0.975901	+	0.218213i	$0.0700227\pi$
$907$	13.1436i	0.436426i	−0.975901	+	0.218213i	$0.929977\pi$
$908$	0	0
$909$	19.7128i	0.653833i
$910$	0	0
$911$	−7.85641	−0.260294	−0.130147	−	0.991495i	$-0.541545\pi$
$911$	−7.85641	−0.260294	−0.130147	+	0.991495i	$0.541545\pi$
$912$	0	0
$913$	−11.7128	−0.387638
$914$	0	0
$915$	− 17.8564i	− 0.590315i
$916$	0	0
$917$	31.5692i	1.04251i
$918$	0	0
$919$	−5.39230	−0.177876	−0.0889379	−	0.996037i	$-0.528347\pi$
$919$	−5.39230	−0.177876	−0.0889379	+	0.996037i	$0.528347\pi$
$920$	0	0
$921$	−1.85641	−0.0611707
$922$	0	0
$923$	62.4974i	2.05713i
$924$	0	0
$925$	2.00000i	0.0657596i
$926$	0	0
$927$	−5.85641	−0.192350
$928$	0	0
$929$	8.71281	0.285858	0.142929	−	0.989733i	$-0.454348\pi$
$929$	8.71281	0.285858	0.142929	+	0.989733i	$0.454348\pi$
$930$	0	0
$931$	− 12.9282i	− 0.423705i
$932$	0	0
$933$	7.39230i	0.242013i
$934$	0	0
$935$	−63.4256	−2.07424
$936$	0	0
$937$	29.9282	0.977712	0.488856	−	0.872365i	$-0.337414\pi$
$937$	29.9282	0.977712	0.488856	+	0.872365i	$0.337414\pi$
$938$	0	0
$939$	− 16.8564i	− 0.550088i
$940$	0	0
$941$	− 41.1051i	− 1.33999i	−0.742366	−	0.669994i	$-0.766297\pi$
$941$	− 41.1051i	− 1.33999i	0.742366	−	0.669994i	$-0.233703\pi$
$942$	0	0
$943$	44.7846	1.45839
$944$	0	0
$945$	−44.6410	−1.45217
$946$	0	0
$947$	− 16.9282i	− 0.550093i	−0.961431	−	0.275046i	$-0.911307\pi$
$947$	− 16.9282i	− 0.550093i	0.961431	−	0.275046i	$-0.0886932\pi$
$948$	0	0
$949$	31.2487i	1.01438i
$950$	0	0
$951$	14.4641	0.469031
$952$	0	0
$953$	7.21539	0.233729	0.116865	−	0.993148i	$-0.462716\pi$
$953$	7.21539	0.233729	0.116865	+	0.993148i	$0.462716\pi$
$954$	0	0
$955$	− 7.07180i	− 0.228838i
$956$	0	0
$957$	9.85641i	0.318612i
$958$	0	0
$959$	39.5359	1.27668
$960$	0	0
$961$	−6.71281	−0.216542
$962$	0	0
$963$	27.8564i	0.897660i
$964$	0	0
$965$	19.7128i	0.634578i
$966$	0	0
$967$	24.7846	0.797019	0.398510	−	0.917164i	$-0.369528\pi$
$967$	24.7846	0.797019	0.398510	+	0.917164i	$0.369528\pi$
$968$	0	0
$969$	7.92820	0.254691
$970$	0	0
$971$	25.8564i	0.829772i	0.909873	+	0.414886i	$0.136178\pi$
$971$	25.8564i	0.829772i	−0.909873	+	0.414886i	$0.863822\pi$
$972$	0	0
$973$	70.7846i	2.26925i
$974$	0	0
$975$	4.46410	0.142966
$976$	0	0
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	0	0
$979$	− 3.71281i	− 0.118662i
$980$	0	0
$981$	− 14.7846i	− 0.472036i
$982$	0	0
$983$	20.6410	0.658346	0.329173	−	0.944270i	$-0.393230\pi$
$983$	20.6410	0.658346	0.329173	+	0.944270i	$0.393230\pi$
$984$	0	0
$985$	−13.8564	−0.441502
$986$	0	0
$987$	− 30.9282i	− 0.984456i
$988$	0	0
$989$	31.8564i	1.01297i
$990$	0	0
$991$	−43.5692	−1.38402	−0.692011	−	0.721887i	$-0.743275\pi$
$991$	−43.5692	−1.38402	−0.692011	+	0.721887i	$0.743275\pi$
$992$	0	0
$993$	−9.78461	−0.310505
$994$	0	0
$995$	− 27.0718i	− 0.858234i
$996$	0	0
$997$	− 42.7846i	− 1.35500i	−0.735522	−	0.677501i	$-0.763063\pi$
$997$	− 42.7846i	− 1.35500i	0.735522	−	0.677501i	$-0.236937\pi$
$998$	0	0
$999$	10.0000	0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2432.2.c.c.1217.1	✓	4
4.3	odd	2		2432.2.c.d.1217.4	yes	4
8.3	odd	2		2432.2.c.d.1217.2	yes	4
8.5	even	2	inner	2432.2.c.c.1217.3	yes	4
16.3	odd	4		4864.2.a.u.1.1		2
16.5	even	4		4864.2.a.s.1.2		2
16.11	odd	4		4864.2.a.v.1.1		2
16.13	even	4		4864.2.a.x.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2432.2.c.c.1217.1	✓	4	1.1	even	1	trivial
2432.2.c.c.1217.3	yes	4	8.5	even	2	inner
2432.2.c.d.1217.2	yes	4	8.3	odd	2
2432.2.c.d.1217.4	yes	4	4.3	odd	2
4864.2.a.s.1.2		2	16.5	even	4
4864.2.a.u.1.1		2	16.3	odd	4
4864.2.a.v.1.1		2	16.11	odd	4
4864.2.a.x.1.2		2	16.13	even	4

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:	\( N \)	\(=\)	\( 2432 = 2^{7} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{3} +2.00000i q^{5} -4.46410 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{3} +2.00000i q^{5} -4.46410 q^{7} +2.00000 q^{9} -4.00000i q^{11} +4.46410i q^{13} +2.00000 q^{15} -7.92820 q^{17} -1.00000i q^{19} +4.46410i q^{21} +6.46410 q^{23} +1.00000 q^{25} -5.00000i q^{27} -2.46410i q^{29} +4.92820 q^{31} -4.00000 q^{33} -8.92820i q^{35} +2.00000i q^{37} +4.46410 q^{39} +6.92820 q^{41} +4.92820i q^{43} +4.00000i q^{45} -6.92820 q^{47} +12.9282 q^{49} +7.92820i q^{51} -12.4641i q^{53} +8.00000 q^{55} -1.00000 q^{57} -9.92820i q^{59} -8.92820i q^{61} -8.92820 q^{63} -8.92820 q^{65} +5.92820i q^{67} -6.46410i q^{69} +14.0000 q^{71} +7.00000 q^{73} -1.00000i q^{75} +17.8564i q^{77} +2.00000 q^{79} +1.00000 q^{81} -2.92820i q^{83} -15.8564i q^{85} -2.46410 q^{87} +0.928203 q^{89} -19.9282i q^{91} -4.92820i q^{93} +2.00000 q^{95} +12.9282 q^{97} -8.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^7 + 8 * q^9	\( 4 q - 4 q^{7} + 8 q^{9} + 8 q^{15} - 4 q^{17} + 12 q^{23} + 4 q^{25} - 8 q^{31} - 16 q^{33} + 4 q^{39} + 24 q^{49} + 32 q^{55} - 4 q^{57} - 8 q^{63} - 8 q^{65} + 56 q^{71} + 28 q^{73} + 8 q^{79} + 4 q^{81} + 4 q^{87} - 24 q^{89} + 8 q^{95} + 24 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 8 * q^9 + 8 * q^15 - 4 * q^17 + 12 * q^23 + 4 * q^25 - 8 * q^31 - 16 * q^33 + 4 * q^39 + 24 * q^49 + 32 * q^55 - 4 * q^57 - 8 * q^63 - 8 * q^65 + 56 * q^71 + 28 * q^73 + 8 * q^79 + 4 * q^81 + 4 * q^87 - 24 * q^89 + 8 * q^95 + 24 * q^97

Introduction

L-functions

Modular forms

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