LMFDB - Embedded newform 4232.2.a.bb.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4232,2,Mod(1,4232)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4232, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4232.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4232 = 2^{3} \cdot 23^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4232.a (trivial)

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:	yes
Analytic conductor:	$33.7926901354$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 30 x^{13} + 28 x^{12} + 354 x^{11} - 302 x^{10} - 2111 x^{9} + 1596 x^{8} + 6777 x^{7} + \cdots - 419 $ x^15 - x^14 - 30x^13 + 28x^12 + 354x^11 - 302x^10 - 2111x^9 + 1596x^8 + 6777x^7 - 4292x^6 - 11506x^5 + 5297x^4 + 9480x^3 - 1825x^2 - 3136*x - 419
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.98867$ of defining polynomial
Character	$\chi$	$=$	4232.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.98867 q^{3} +3.85556 q^{5} +0.534429 q^{7} +0.954809 q^{9} +0.710045 q^{11} +4.18982 q^{13} +7.66744 q^{15} -5.05384 q^{17} +6.82427 q^{19} +1.06280 q^{21} +9.86535 q^{25} -4.06721 q^{27} +4.85806 q^{29} +3.07319 q^{31} +1.41205 q^{33} +2.06052 q^{35} -1.38911 q^{37} +8.33217 q^{39} -4.75690 q^{41} -0.295423 q^{43} +3.68133 q^{45} -10.8335 q^{47} -6.71439 q^{49} -10.0504 q^{51} -12.0191 q^{53} +2.73762 q^{55} +13.5712 q^{57} -8.62219 q^{59} +12.9129 q^{61} +0.510278 q^{63} +16.1541 q^{65} +13.5268 q^{67} +3.76034 q^{71} -6.77557 q^{73} +19.6189 q^{75} +0.379469 q^{77} +2.15068 q^{79} -10.9528 q^{81} +3.97828 q^{83} -19.4854 q^{85} +9.66109 q^{87} -2.13176 q^{89} +2.23916 q^{91} +6.11156 q^{93} +26.3114 q^{95} +7.88215 q^{97} +0.677958 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) $ 15 * q + q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 + q^21 + 23 * q^25 + q^27 - 2 * q^29 + 20 * q^31 + 18 * q^33 - 18 * q^35 + 24 * q^37 - 19 * q^39 + 9 * q^41 + 48 * q^43 + 4 * q^45 - 36 * q^47 + 25 * q^49 + 35 * q^51 - 5 * q^53 - 10 * q^55 + 23 * q^57 - 22 * q^59 + 12 * q^61 + 35 * q^63 - 26 * q^65 + 58 * q^67 + 2 * q^71 + 5 * q^73 - 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 + 19 * q^87 - 6 * q^89 + 71 * q^91 - 55 * q^93 - 12 * q^95 + 40 * q^97 + 63 * q^99

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
$3$	1.98867	1.14816	0.574080	−	0.818799i	$-0.305360\pi$
$3$	1.98867	1.14816	0.574080	+	0.818799i	$0.305360\pi$
$4$	0	0
$5$	3.85556	1.72426	0.862130	−	0.506688i	$-0.169130\pi$
$5$	3.85556	1.72426	0.862130	+	0.506688i	$0.169130\pi$
$6$	0	0
$7$	0.534429	0.201995	0.100998	−	0.994887i	$-0.467797\pi$
$7$	0.534429	0.201995	0.100998	+	0.994887i	$0.467797\pi$
$8$	0	0
$9$	0.954809	0.318270
$10$	0	0
$11$	0.710045	0.214087	0.107043	−	0.994254i	$-0.465862\pi$
$11$	0.710045	0.214087	0.107043	+	0.994254i	$0.465862\pi$
$12$	0	0
$13$	4.18982	1.16205	0.581024	−	0.813887i	$-0.302652\pi$
$13$	4.18982	1.16205	0.581024	+	0.813887i	$0.302652\pi$
$14$	0	0
$15$	7.66744	1.97972
$16$	0	0
$17$	−5.05384	−1.22574	−0.612869	−	0.790185i	$-0.709984\pi$
$17$	−5.05384	−1.22574	−0.612869	+	0.790185i	$0.709984\pi$
$18$	0	0
$19$	6.82427	1.56559	0.782797	−	0.622277i	$-0.213792\pi$
$19$	6.82427	1.56559	0.782797	+	0.622277i	$0.213792\pi$
$20$	0	0
$21$	1.06280	0.231923
$22$	0	0
$23$	0	0
$23$	0	0
$24$	0	0
$25$	9.86535	1.97307
$26$	0	0
$27$	−4.06721	−0.782735
$28$	0	0
$29$	4.85806	0.902120	0.451060	−	0.892494i	$-0.351046\pi$
$29$	4.85806	0.902120	0.451060	+	0.892494i	$0.351046\pi$
$30$	0	0
$31$	3.07319	0.551961	0.275980	−	0.961163i	$-0.410997\pi$
$31$	3.07319	0.551961	0.275980	+	0.961163i	$0.410997\pi$
$32$	0	0
$33$	1.41205	0.245806
$34$	0	0
$35$	2.06052	0.348292
$36$	0	0
$37$	−1.38911	−0.228368	−0.114184	−	0.993460i	$-0.536425\pi$
$37$	−1.38911	−0.228368	−0.114184	+	0.993460i	$0.536425\pi$
$38$	0	0
$39$	8.33217	1.33422
$40$	0	0
$41$	−4.75690	−0.742903	−0.371451	−	0.928452i	$-0.621140\pi$
$41$	−4.75690	−0.742903	−0.371451	+	0.928452i	$0.621140\pi$
$42$	0	0
$43$	−0.295423	−0.0450516	−0.0225258	−	0.999746i	$-0.507171\pi$
$43$	−0.295423	−0.0450516	−0.0225258	+	0.999746i	$0.507171\pi$
$44$	0	0
$45$	3.68133	0.548780
$46$	0	0
$47$	−10.8335	−1.58024	−0.790118	−	0.612955i	$-0.789981\pi$
$47$	−10.8335	−1.58024	−0.790118	+	0.612955i	$0.789981\pi$
$48$	0	0
$49$	−6.71439	−0.959198
$50$	0	0
$51$	−10.0504	−1.40734
$52$	0	0
$53$	−12.0191	−1.65095	−0.825477	−	0.564436i	$-0.809094\pi$
$53$	−12.0191	−1.65095	−0.825477	+	0.564436i	$0.809094\pi$
$54$	0	0
$55$	2.73762	0.369141
$56$	0	0
$57$	13.5712	1.79755
$58$	0	0
$59$	−8.62219	−1.12251	−0.561257	−	0.827642i	$-0.689682\pi$
$59$	−8.62219	−1.12251	−0.561257	+	0.827642i	$0.689682\pi$
$60$	0	0
$61$	12.9129	1.65332	0.826661	−	0.562701i	$-0.190238\pi$
$61$	12.9129	1.65332	0.826661	+	0.562701i	$0.190238\pi$
$62$	0	0
$63$	0.510278	0.0642890
$64$	0	0
$65$	16.1541	2.00367
$66$	0	0
$67$	13.5268	1.65257	0.826283	−	0.563255i	$-0.190451\pi$
$67$	13.5268	1.65257	0.826283	+	0.563255i	$0.190451\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.76034	0.446270	0.223135	−	0.974788i	$-0.428371\pi$
$71$	3.76034	0.446270	0.223135	+	0.974788i	$0.428371\pi$
$72$	0	0
$73$	−6.77557	−0.793021	−0.396510	−	0.918030i	$-0.629779\pi$
$73$	−6.77557	−0.793021	−0.396510	+	0.918030i	$0.629779\pi$
$74$	0	0
$75$	19.6189	2.26540
$76$	0	0
$77$	0.379469	0.0432445
$78$	0	0
$79$	2.15068	0.241971	0.120985	−	0.992654i	$-0.461395\pi$
$79$	2.15068	0.241971	0.120985	+	0.992654i	$0.461395\pi$
$80$	0	0
$81$	−10.9528	−1.21697
$82$	0	0
$83$	3.97828	0.436673	0.218337	−	0.975873i	$-0.429937\pi$
$83$	3.97828	0.436673	0.218337	+	0.975873i	$0.429937\pi$
$84$	0	0
$85$	−19.4854	−2.11349
$86$	0	0
$87$	9.66109	1.03578
$88$	0	0
$89$	−2.13176	−0.225966	−0.112983	−	0.993597i	$-0.536041\pi$
$89$	−2.13176	−0.225966	−0.112983	+	0.993597i	$0.536041\pi$
$90$	0	0
$91$	2.23916	0.234728
$92$	0	0
$93$	6.11156	0.633739
$94$	0	0
$95$	26.3114	2.69949
$96$	0	0
$97$	7.88215	0.800311	0.400155	−	0.916447i	$-0.368956\pi$
$97$	7.88215	0.800311	0.400155	+	0.916447i	$0.368956\pi$
$98$	0	0
$99$	0.677958	0.0681373
$100$	0	0
$101$	−8.51529	−0.847303	−0.423651	−	0.905825i	$-0.639252\pi$
$101$	−8.51529	−0.847303	−0.423651	+	0.905825i	$0.639252\pi$
$102$	0	0
$103$	9.93191	0.978620	0.489310	−	0.872110i	$-0.337249\pi$
$103$	9.93191	0.978620	0.489310	+	0.872110i	$0.337249\pi$
$104$	0	0
$105$	4.09770	0.399895
$106$	0	0
$107$	−3.12803	−0.302398	−0.151199	−	0.988503i	$-0.548313\pi$
$107$	−3.12803	−0.302398	−0.151199	+	0.988503i	$0.548313\pi$
$108$	0	0
$109$	8.56535	0.820411	0.410206	−	0.911993i	$-0.365457\pi$
$109$	8.56535	0.820411	0.410206	+	0.911993i	$0.365457\pi$
$110$	0	0
$111$	−2.76248	−0.262203
$112$	0	0
$113$	−19.0207	−1.78932	−0.894660	−	0.446748i	$-0.852582\pi$
$113$	−19.0207	−1.78932	−0.894660	+	0.446748i	$0.852582\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4.00048	0.369845
$118$	0	0
$119$	−2.70092	−0.247593
$120$	0	0
$121$	−10.4958	−0.954167
$122$	0	0
$123$	−9.45990	−0.852971
$124$	0	0
$125$	18.7587	1.67783
$126$	0	0
$127$	12.4647	1.10606	0.553032	−	0.833160i	$-0.313471\pi$
$127$	12.4647	1.10606	0.553032	+	0.833160i	$0.313471\pi$
$128$	0	0
$129$	−0.587499	−0.0517264
$130$	0	0
$131$	−16.3784	−1.43099	−0.715494	−	0.698619i	$-0.753798\pi$
$131$	−16.3784	−1.43099	−0.715494	+	0.698619i	$0.753798\pi$
$132$	0	0
$133$	3.64709	0.316243
$134$	0	0
$135$	−15.6814	−1.34964
$136$	0	0
$137$	4.29346	0.366815	0.183407	−	0.983037i	$-0.441287\pi$
$137$	4.29346	0.366815	0.183407	+	0.983037i	$0.441287\pi$
$138$	0	0
$139$	−6.93180	−0.587948	−0.293974	−	0.955813i	$-0.594978\pi$
$139$	−6.93180	−0.587948	−0.293974	+	0.955813i	$0.594978\pi$
$140$	0	0
$141$	−21.5444	−1.81436
$142$	0	0
$143$	2.97496	0.248779
$144$	0	0
$145$	18.7306	1.55549
$146$	0	0
$147$	−13.3527	−1.10131
$148$	0	0
$149$	−5.39418	−0.441909	−0.220954	−	0.975284i	$-0.570917\pi$
$149$	−5.39418	−0.441909	−0.220954	+	0.975284i	$0.570917\pi$
$150$	0	0
$151$	15.2173	1.23836	0.619181	−	0.785248i	$-0.287465\pi$
$151$	15.2173	1.23836	0.619181	+	0.785248i	$0.287465\pi$
$152$	0	0
$153$	−4.82546	−0.390115
$154$	0	0
$155$	11.8489	0.951724
$156$	0	0
$157$	9.02350	0.720153	0.360077	−	0.932923i	$-0.382751\pi$
$157$	9.02350	0.720153	0.360077	+	0.932923i	$0.382751\pi$
$158$	0	0
$159$	−23.9021	−1.89556
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−10.9531	−0.857910	−0.428955	−	0.903326i	$-0.641118\pi$
$163$	−10.9531	−0.857910	−0.428955	+	0.903326i	$0.641118\pi$
$164$	0	0
$165$	5.44423	0.423833
$166$	0	0
$167$	12.3568	0.956199	0.478099	−	0.878306i	$-0.341326\pi$
$167$	12.3568	0.956199	0.478099	+	0.878306i	$0.341326\pi$
$168$	0	0
$169$	4.55460	0.350354
$170$	0	0
$171$	6.51588	0.498281
$172$	0	0
$173$	6.93906	0.527567	0.263784	−	0.964582i	$-0.415030\pi$
$173$	6.93906	0.527567	0.263784	+	0.964582i	$0.415030\pi$
$174$	0	0
$175$	5.27233	0.398551
$176$	0	0
$177$	−17.1467	−1.28882
$178$	0	0
$179$	3.81343	0.285029	0.142515	−	0.989793i	$-0.454481\pi$
$179$	3.81343	0.285029	0.142515	+	0.989793i	$0.454481\pi$
$180$	0	0
$181$	−11.6133	−0.863209	−0.431604	−	0.902063i	$-0.642052\pi$
$181$	−11.6133	−0.863209	−0.431604	+	0.902063i	$0.642052\pi$
$182$	0	0
$183$	25.6794	1.89828
$184$	0	0
$185$	−5.35580	−0.393766
$186$	0	0
$187$	−3.58846	−0.262414
$188$	0	0
$189$	−2.17363	−0.158109
$190$	0	0
$191$	18.4213	1.33292	0.666458	−	0.745543i	$-0.267810\pi$
$191$	18.4213	1.33292	0.666458	+	0.745543i	$0.267810\pi$
$192$	0	0
$193$	−1.06315	−0.0765272	−0.0382636	−	0.999268i	$-0.512183\pi$
$193$	−1.06315	−0.0765272	−0.0382636	+	0.999268i	$0.512183\pi$
$194$	0	0
$195$	32.1252	2.30053
$196$	0	0
$197$	−2.49678	−0.177888	−0.0889440	−	0.996037i	$-0.528349\pi$
$197$	−2.49678	−0.177888	−0.0889440	+	0.996037i	$0.528349\pi$
$198$	0	0
$199$	−1.51195	−0.107180	−0.0535898	−	0.998563i	$-0.517066\pi$
$199$	−1.51195	−0.107180	−0.0535898	+	0.998563i	$0.517066\pi$
$200$	0	0
$201$	26.9004	1.89741
$202$	0	0
$203$	2.59629	0.182224
$204$	0	0
$205$	−18.3405	−1.28096
$206$	0	0
$207$	0	0
$208$	0	0
$209$	4.84554	0.335173
$210$	0	0
$211$	−1.61170	−0.110954	−0.0554771	−	0.998460i	$-0.517668\pi$
$211$	−1.61170	−0.110954	−0.0554771	+	0.998460i	$0.517668\pi$
$212$	0	0
$213$	7.47808	0.512390
$214$	0	0
$215$	−1.13902	−0.0776806
$216$	0	0
$217$	1.64240	0.111493
$218$	0	0
$219$	−13.4744	−0.910514
$220$	0	0
$221$	−21.1747	−1.42436
$222$	0	0
$223$	−8.91383	−0.596914	−0.298457	−	0.954423i	$-0.596472\pi$
$223$	−8.91383	−0.596914	−0.298457	+	0.954423i	$0.596472\pi$
$224$	0	0
$225$	9.41953	0.627969
$226$	0	0
$227$	23.9738	1.59119	0.795597	−	0.605826i	$-0.207157\pi$
$227$	23.9738	1.59119	0.795597	+	0.605826i	$0.207157\pi$
$228$	0	0
$229$	−6.40509	−0.423260	−0.211630	−	0.977350i	$-0.567877\pi$
$229$	−6.40509	−0.423260	−0.211630	+	0.977350i	$0.567877\pi$
$230$	0	0
$231$	0.754638	0.0496516
$232$	0	0
$233$	1.25898	0.0824784	0.0412392	−	0.999149i	$-0.486869\pi$
$233$	1.25898	0.0824784	0.0412392	+	0.999149i	$0.486869\pi$
$234$	0	0
$235$	−41.7694	−2.72474
$236$	0	0
$237$	4.27700	0.277821
$238$	0	0
$239$	−25.9367	−1.67771	−0.838853	−	0.544359i	$-0.816773\pi$
$239$	−25.9367	−1.67771	−0.838853	+	0.544359i	$0.816773\pi$
$240$	0	0
$241$	−0.962695	−0.0620127	−0.0310063	−	0.999519i	$-0.509871\pi$
$241$	−0.962695	−0.0620127	−0.0310063	+	0.999519i	$0.509871\pi$
$242$	0	0
$243$	−9.57981	−0.614545
$244$	0	0
$245$	−25.8877	−1.65391
$246$	0	0
$247$	28.5925	1.81930
$248$	0	0
$249$	7.91150	0.501371
$250$	0	0
$251$	6.60729	0.417049	0.208524	−	0.978017i	$-0.433134\pi$
$251$	6.60729	0.417049	0.208524	+	0.978017i	$0.433134\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−38.7500	−2.42662
$256$	0	0
$257$	25.5637	1.59462	0.797311	−	0.603569i	$-0.206255\pi$
$257$	25.5637	1.59462	0.797311	+	0.603569i	$0.206255\pi$
$258$	0	0
$259$	−0.742381	−0.0461293
$260$	0	0
$261$	4.63852	0.287117
$262$	0	0
$263$	−23.2049	−1.43087	−0.715437	−	0.698677i	$-0.753772\pi$
$263$	−23.2049	−1.43087	−0.715437	+	0.698677i	$0.753772\pi$
$264$	0	0
$265$	−46.3405	−2.84667
$266$	0	0
$267$	−4.23936	−0.259445
$268$	0	0
$269$	−26.2680	−1.60159	−0.800793	−	0.598942i	$-0.795588\pi$
$269$	−26.2680	−1.60159	−0.800793	+	0.598942i	$0.795588\pi$
$270$	0	0
$271$	−16.2798	−0.988926	−0.494463	−	0.869199i	$-0.664635\pi$
$271$	−16.2798	−0.988926	−0.494463	+	0.869199i	$0.664635\pi$
$272$	0	0
$273$	4.45295	0.269505
$274$	0	0
$275$	7.00485	0.422408
$276$	0	0
$277$	−6.35841	−0.382040	−0.191020	−	0.981586i	$-0.561180\pi$
$277$	−6.35841	−0.382040	−0.191020	+	0.981586i	$0.561180\pi$
$278$	0	0
$279$	2.93431	0.175672
$280$	0	0
$281$	−2.13599	−0.127422	−0.0637112	−	0.997968i	$-0.520294\pi$
$281$	−2.13599	−0.127422	−0.0637112	+	0.997968i	$0.520294\pi$
$282$	0	0
$283$	−19.6196	−1.16626	−0.583131	−	0.812378i	$-0.698173\pi$
$283$	−19.6196	−1.16626	−0.583131	+	0.812378i	$0.698173\pi$
$284$	0	0
$285$	52.3247	3.09945
$286$	0	0
$287$	−2.54222	−0.150063
$288$	0	0
$289$	8.54133	0.502431
$290$	0	0
$291$	15.6750	0.918884
$292$	0	0
$293$	−3.87628	−0.226455	−0.113227	−	0.993569i	$-0.536119\pi$
$293$	−3.87628	−0.226455	−0.113227	+	0.993569i	$0.536119\pi$
$294$	0	0
$295$	−33.2434	−1.93550
$296$	0	0
$297$	−2.88790	−0.167573
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−0.157883	−0.00910020
$302$	0	0
$303$	−16.9341	−0.972839
$304$	0	0
$305$	49.7863	2.85076
$306$	0	0
$307$	28.5115	1.62724	0.813619	−	0.581398i	$-0.197494\pi$
$307$	28.5115	1.62724	0.813619	+	0.581398i	$0.197494\pi$
$308$	0	0
$309$	19.7513	1.12361
$310$	0	0
$311$	9.83963	0.557954	0.278977	−	0.960298i	$-0.410005\pi$
$311$	9.83963	0.557954	0.278977	+	0.960298i	$0.410005\pi$
$312$	0	0
$313$	9.69844	0.548188	0.274094	−	0.961703i	$-0.411622\pi$
$313$	9.69844	0.548188	0.274094	+	0.961703i	$0.411622\pi$
$314$	0	0
$315$	1.96741	0.110851
$316$	0	0
$317$	10.3179	0.579513	0.289757	−	0.957100i	$-0.406426\pi$
$317$	10.3179	0.579513	0.289757	+	0.957100i	$0.406426\pi$
$318$	0	0
$319$	3.44945	0.193132
$320$	0	0
$321$	−6.22062	−0.347201
$322$	0	0
$323$	−34.4888	−1.91901
$324$	0	0
$325$	41.3341	2.29280
$326$	0	0
$327$	17.0336	0.941963
$328$	0	0
$329$	−5.78976	−0.319200
$330$	0	0
$331$	−23.0484	−1.26686	−0.633429	−	0.773801i	$-0.718353\pi$
$331$	−23.0484	−1.26686	−0.633429	+	0.773801i	$0.718353\pi$
$332$	0	0
$333$	−1.32634	−0.0726828
$334$	0	0
$335$	52.1536	2.84945
$336$	0	0
$337$	3.66812	0.199815	0.0999075	−	0.994997i	$-0.468145\pi$
$337$	3.66812	0.199815	0.0999075	+	0.994997i	$0.468145\pi$
$338$	0	0
$339$	−37.8260	−2.05442
$340$	0	0
$341$	2.18210	0.118168
$342$	0	0
$343$	−7.32936	−0.395748
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.7765	1.49112	0.745560	−	0.666439i	$-0.232182\pi$
$347$	27.7765	1.49112	0.745560	+	0.666439i	$0.232182\pi$
$348$	0	0
$349$	−25.8029	−1.38120	−0.690599	−	0.723238i	$-0.742653\pi$
$349$	−25.8029	−1.38120	−0.690599	+	0.723238i	$0.742653\pi$
$350$	0	0
$351$	−17.0409	−0.909575
$352$	0	0
$353$	16.6311	0.885182	0.442591	−	0.896724i	$-0.354059\pi$
$353$	16.6311	0.885182	0.442591	+	0.896724i	$0.354059\pi$
$354$	0	0
$355$	14.4982	0.769486
$356$	0	0
$357$	−5.37124	−0.284276
$358$	0	0
$359$	−4.81708	−0.254236	−0.127118	−	0.991888i	$-0.540573\pi$
$359$	−4.81708	−0.254236	−0.127118	+	0.991888i	$0.540573\pi$
$360$	0	0
$361$	27.5706	1.45109
$362$	0	0
$363$	−20.8728	−1.09554
$364$	0	0
$365$	−26.1236	−1.36737
$366$	0	0
$367$	14.2168	0.742111	0.371056	−	0.928611i	$-0.378996\pi$
$367$	14.2168	0.742111	0.371056	+	0.928611i	$0.378996\pi$
$368$	0	0
$369$	−4.54193	−0.236444
$370$	0	0
$371$	−6.42337	−0.333485
$372$	0	0
$373$	−32.5150	−1.68356	−0.841782	−	0.539818i	$-0.818493\pi$
$373$	−32.5150	−1.68356	−0.841782	+	0.539818i	$0.818493\pi$
$374$	0	0
$375$	37.3048	1.92641
$376$	0	0
$377$	20.3544	1.04831
$378$	0	0
$379$	4.97464	0.255530	0.127765	−	0.991804i	$-0.459220\pi$
$379$	4.97464	0.255530	0.127765	+	0.991804i	$0.459220\pi$
$380$	0	0
$381$	24.7882	1.26994
$382$	0	0
$383$	−9.88227	−0.504960	−0.252480	−	0.967602i	$-0.581246\pi$
$383$	−9.88227	−0.504960	−0.252480	+	0.967602i	$0.581246\pi$
$384$	0	0
$385$	1.46307	0.0745647
$386$	0	0
$387$	−0.282073	−0.0143386
$388$	0	0
$389$	−14.2872	−0.724389	−0.362194	−	0.932103i	$-0.617972\pi$
$389$	−14.2872	−0.724389	−0.362194	+	0.932103i	$0.617972\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−32.5713	−1.64300
$394$	0	0
$395$	8.29209	0.417220
$396$	0	0
$397$	−28.8376	−1.44732	−0.723658	−	0.690158i	$-0.757541\pi$
$397$	−28.8376	−1.44732	−0.723658	+	0.690158i	$0.757541\pi$
$398$	0	0
$399$	7.25285	0.363097
$400$	0	0
$401$	−16.9139	−0.844642	−0.422321	−	0.906446i	$-0.638784\pi$
$401$	−16.9139	−0.844642	−0.422321	+	0.906446i	$0.638784\pi$
$402$	0	0
$403$	12.8761	0.641405
$404$	0	0
$405$	−42.2291	−2.09838
$406$	0	0
$407$	−0.986332	−0.0488907
$408$	0	0
$409$	−19.2430	−0.951506	−0.475753	−	0.879579i	$-0.657824\pi$
$409$	−19.2430	−0.951506	−0.475753	+	0.879579i	$0.657824\pi$
$410$	0	0
$411$	8.53827	0.421162
$412$	0	0
$413$	−4.60795	−0.226742
$414$	0	0
$415$	15.3385	0.752938
$416$	0	0
$417$	−13.7851	−0.675058
$418$	0	0
$419$	37.8868	1.85089	0.925446	−	0.378878i	$-0.123690\pi$
$419$	37.8868	1.85089	0.925446	+	0.378878i	$0.123690\pi$
$420$	0	0
$421$	18.8280	0.917620	0.458810	−	0.888534i	$-0.348276\pi$
$421$	18.8280	0.917620	0.458810	+	0.888534i	$0.348276\pi$
$422$	0	0
$423$	−10.3440	−0.502941
$424$	0	0
$425$	−49.8580	−2.41847
$426$	0	0
$427$	6.90100	0.333963
$428$	0	0
$429$	5.91622	0.285638
$430$	0	0
$431$	−18.0926	−0.871490	−0.435745	−	0.900070i	$-0.643515\pi$
$431$	−18.0926	−0.871490	−0.435745	+	0.900070i	$0.643515\pi$
$432$	0	0
$433$	14.8629	0.714267	0.357134	−	0.934053i	$-0.383754\pi$
$433$	14.8629	0.714267	0.357134	+	0.934053i	$0.383754\pi$
$434$	0	0
$435$	37.2489	1.78595
$436$	0	0
$437$	0	0
$438$	0	0
$439$	10.2767	0.490478	0.245239	−	0.969463i	$-0.421134\pi$
$439$	10.2767	0.490478	0.245239	+	0.969463i	$0.421134\pi$
$440$	0	0
$441$	−6.41096	−0.305284
$442$	0	0
$443$	23.7504	1.12841	0.564207	−	0.825633i	$-0.309182\pi$
$443$	23.7504	1.12841	0.564207	+	0.825633i	$0.309182\pi$
$444$	0	0
$445$	−8.21912	−0.389624
$446$	0	0
$447$	−10.7272	−0.507381
$448$	0	0
$449$	8.41894	0.397314	0.198657	−	0.980069i	$-0.436342\pi$
$449$	8.41894	0.397314	0.198657	+	0.980069i	$0.436342\pi$
$450$	0	0
$451$	−3.37761	−0.159046
$452$	0	0
$453$	30.2621	1.42184
$454$	0	0
$455$	8.63323	0.404732
$456$	0	0
$457$	−5.10007	−0.238571	−0.119286	−	0.992860i	$-0.538060\pi$
$457$	−5.10007	−0.238571	−0.119286	+	0.992860i	$0.538060\pi$
$458$	0	0
$459$	20.5550	0.959427
$460$	0	0
$461$	1.41542	0.0659225	0.0329613	−	0.999457i	$-0.489506\pi$
$461$	1.41542	0.0659225	0.0329613	+	0.999457i	$0.489506\pi$
$462$	0	0
$463$	18.6855	0.868390	0.434195	−	0.900819i	$-0.357033\pi$
$463$	18.6855	0.868390	0.434195	+	0.900819i	$0.357033\pi$
$464$	0	0
$465$	23.5635	1.09273
$466$	0	0
$467$	−27.8164	−1.28719	−0.643594	−	0.765367i	$-0.722558\pi$
$467$	−27.8164	−1.28719	−0.643594	+	0.765367i	$0.722558\pi$
$468$	0	0
$469$	7.22914	0.333810
$470$	0	0
$471$	17.9448	0.826851
$472$	0	0
$473$	−0.209764	−0.00964495
$474$	0	0
$475$	67.3238	3.08903
$476$	0	0
$477$	−11.4760	−0.525449
$478$	0	0
$479$	4.99856	0.228390	0.114195	−	0.993458i	$-0.463571\pi$
$479$	4.99856	0.228390	0.114195	+	0.993458i	$0.463571\pi$
$480$	0	0
$481$	−5.82013	−0.265375
$482$	0	0
$483$	0	0
$484$	0	0
$485$	30.3901	1.37994
$486$	0	0
$487$	33.2628	1.50728	0.753640	−	0.657287i	$-0.228296\pi$
$487$	33.2628	1.50728	0.753640	+	0.657287i	$0.228296\pi$
$488$	0	0
$489$	−21.7820	−0.985018
$490$	0	0
$491$	−17.3141	−0.781376	−0.390688	−	0.920523i	$-0.627763\pi$
$491$	−17.3141	−0.781376	−0.390688	+	0.920523i	$0.627763\pi$
$492$	0	0
$493$	−24.5519	−1.10576
$494$	0	0
$495$	2.61391	0.117486
$496$	0	0
$497$	2.00964	0.0901445
$498$	0	0
$499$	−6.69668	−0.299784	−0.149892	−	0.988702i	$-0.547893\pi$
$499$	−6.69668	−0.299784	−0.149892	+	0.988702i	$0.547893\pi$
$500$	0	0
$501$	24.5736	1.09787
$502$	0	0
$503$	29.0572	1.29559	0.647797	−	0.761813i	$-0.275690\pi$
$503$	29.0572	1.29559	0.647797	+	0.761813i	$0.275690\pi$
$504$	0	0
$505$	−32.8312	−1.46097
$506$	0	0
$507$	9.05760	0.402262
$508$	0	0
$509$	0.893727	0.0396138	0.0198069	−	0.999804i	$-0.493695\pi$
$509$	0.893727	0.0396138	0.0198069	+	0.999804i	$0.493695\pi$
$510$	0	0
$511$	−3.62106	−0.160186
$512$	0	0
$513$	−27.7557	−1.22545
$514$	0	0
$515$	38.2931	1.68739
$516$	0	0
$517$	−7.69231	−0.338307
$518$	0	0
$519$	13.7995	0.605731
$520$	0	0
$521$	4.51721	0.197903	0.0989513	−	0.995092i	$-0.468451\pi$
$521$	4.51721	0.197903	0.0989513	+	0.995092i	$0.468451\pi$
$522$	0	0
$523$	3.22710	0.141111	0.0705556	−	0.997508i	$-0.477523\pi$
$523$	3.22710	0.141111	0.0705556	+	0.997508i	$0.477523\pi$
$524$	0	0
$525$	10.4849	0.457600
$526$	0	0
$527$	−15.5314	−0.676559
$528$	0	0
$529$	0	0
$530$	0	0
$531$	−8.23255	−0.357262
$532$	0	0
$533$	−19.9306	−0.863288
$534$	0	0
$535$	−12.0603	−0.521413
$536$	0	0
$537$	7.58366	0.327259
$538$	0	0
$539$	−4.76752	−0.205352
$540$	0	0
$541$	−19.9367	−0.857148	−0.428574	−	0.903507i	$-0.640984\pi$
$541$	−19.9367	−0.857148	−0.428574	+	0.903507i	$0.640984\pi$
$542$	0	0
$543$	−23.0950	−0.991101
$544$	0	0
$545$	33.0242	1.41460
$546$	0	0
$547$	−34.0842	−1.45734	−0.728668	−	0.684867i	$-0.759860\pi$
$547$	−34.0842	−1.45734	−0.728668	+	0.684867i	$0.759860\pi$
$548$	0	0
$549$	12.3293	0.526202
$550$	0	0
$551$	33.1527	1.41235
$552$	0	0
$553$	1.14939	0.0488769
$554$	0	0
$555$	−10.6509	−0.452107
$556$	0	0
$557$	−24.7799	−1.04996	−0.524980	−	0.851115i	$-0.675927\pi$
$557$	−24.7799	−1.04996	−0.524980	+	0.851115i	$0.675927\pi$
$558$	0	0
$559$	−1.23777	−0.0523521
$560$	0	0
$561$	−7.13626	−0.301293
$562$	0	0
$563$	23.1730	0.976627	0.488314	−	0.872668i	$-0.337612\pi$
$563$	23.1730	0.976627	0.488314	+	0.872668i	$0.337612\pi$
$564$	0	0
$565$	−73.3356	−3.08525
$566$	0	0
$567$	−5.85348	−0.245823
$568$	0	0
$569$	23.2289	0.973807	0.486904	−	0.873456i	$-0.338126\pi$
$569$	23.2289	0.973807	0.486904	+	0.873456i	$0.338126\pi$
$570$	0	0
$571$	6.88339	0.288061	0.144030	−	0.989573i	$-0.453994\pi$
$571$	6.88339	0.288061	0.144030	+	0.989573i	$0.453994\pi$
$572$	0	0
$573$	36.6338	1.53040
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.7329	1.44595	0.722975	−	0.690875i	$-0.242774\pi$
$577$	34.7329	1.44595	0.722975	+	0.690875i	$0.242774\pi$
$578$	0	0
$579$	−2.11426	−0.0878655
$580$	0	0
$581$	2.12611	0.0882059
$582$	0	0
$583$	−8.53412	−0.353447
$584$	0	0
$585$	15.4241	0.637708
$586$	0	0
$587$	11.3568	0.468746	0.234373	−	0.972147i	$-0.424696\pi$
$587$	11.3568	0.468746	0.234373	+	0.972147i	$0.424696\pi$
$588$	0	0
$589$	20.9723	0.864147
$590$	0	0
$591$	−4.96527	−0.204244
$592$	0	0
$593$	−2.92966	−0.120307	−0.0601533	−	0.998189i	$-0.519159\pi$
$593$	−2.92966	−0.120307	−0.0601533	+	0.998189i	$0.519159\pi$
$594$	0	0
$595$	−10.4136	−0.426914
$596$	0	0
$597$	−3.00678	−0.123059
$598$	0	0
$599$	10.1828	0.416058	0.208029	−	0.978123i	$-0.433295\pi$
$599$	10.1828	0.416058	0.208029	+	0.978123i	$0.433295\pi$
$600$	0	0
$601$	−17.4749	−0.712816	−0.356408	−	0.934330i	$-0.615999\pi$
$601$	−17.4749	−0.712816	−0.356408	+	0.934330i	$0.615999\pi$
$602$	0	0
$603$	12.9156	0.525962
$604$	0	0
$605$	−40.4673	−1.64523
$606$	0	0
$607$	−14.4097	−0.584870	−0.292435	−	0.956285i	$-0.594466\pi$
$607$	−14.4097	−0.584870	−0.292435	+	0.956285i	$0.594466\pi$
$608$	0	0
$609$	5.16316	0.209222
$610$	0	0
$611$	−45.3906	−1.83631
$612$	0	0
$613$	−19.5013	−0.787652	−0.393826	−	0.919185i	$-0.628849\pi$
$613$	−19.5013	−0.787652	−0.393826	+	0.919185i	$0.628849\pi$
$614$	0	0
$615$	−36.4732	−1.47074
$616$	0	0
$617$	−12.9401	−0.520950	−0.260475	−	0.965481i	$-0.583879\pi$
$617$	−12.9401	−0.520950	−0.260475	+	0.965481i	$0.583879\pi$
$618$	0	0
$619$	35.0776	1.40989	0.704944	−	0.709263i	$-0.250972\pi$
$619$	35.0776	1.40989	0.704944	+	0.709263i	$0.250972\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.13927	−0.0456440
$624$	0	0
$625$	22.9984	0.919937
$626$	0	0
$627$	9.63618	0.384832
$628$	0	0
$629$	7.02035	0.279920
$630$	0	0
$631$	26.1279	1.04014	0.520068	−	0.854125i	$-0.325906\pi$
$631$	26.1279	1.04014	0.520068	+	0.854125i	$0.325906\pi$
$632$	0	0
$633$	−3.20515	−0.127393
$634$	0	0
$635$	48.0585	1.90714
$636$	0	0
$637$	−28.1321	−1.11463
$638$	0	0
$639$	3.59041	0.142034
$640$	0	0
$641$	22.4262	0.885782	0.442891	−	0.896576i	$-0.353953\pi$
$641$	22.4262	0.885782	0.442891	+	0.896576i	$0.353953\pi$
$642$	0	0
$643$	−27.6829	−1.09171	−0.545853	−	0.837881i	$-0.683794\pi$
$643$	−27.6829	−1.09171	−0.545853	+	0.837881i	$0.683794\pi$
$644$	0	0
$645$	−2.26514	−0.0891897
$646$	0	0
$647$	−20.9762	−0.824660	−0.412330	−	0.911035i	$-0.635285\pi$
$647$	−20.9762	−0.824660	−0.412330	+	0.911035i	$0.635285\pi$
$648$	0	0
$649$	−6.12215	−0.240315
$650$	0	0
$651$	3.26619	0.128012
$652$	0	0
$653$	−16.5699	−0.648429	−0.324215	−	0.945984i	$-0.605100\pi$
$653$	−16.5699	−0.648429	−0.324215	+	0.945984i	$0.605100\pi$
$654$	0	0
$655$	−63.1480	−2.46740
$656$	0	0
$657$	−6.46938	−0.252394
$658$	0	0
$659$	−20.4367	−0.796101	−0.398050	−	0.917364i	$-0.630313\pi$
$659$	−20.4367	−0.796101	−0.398050	+	0.917364i	$0.630313\pi$
$660$	0	0
$661$	−35.1706	−1.36798	−0.683990	−	0.729492i	$-0.739757\pi$
$661$	−35.1706	−1.36798	−0.683990	+	0.729492i	$0.739757\pi$
$662$	0	0
$663$	−42.1095	−1.63540
$664$	0	0
$665$	14.0616	0.545284
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−17.7267	−0.685353
$670$	0	0
$671$	9.16871	0.353954
$672$	0	0
$673$	−3.59843	−0.138709	−0.0693547	−	0.997592i	$-0.522094\pi$
$673$	−3.59843	−0.138709	−0.0693547	+	0.997592i	$0.522094\pi$
$674$	0	0
$675$	−40.1245	−1.54439
$676$	0	0
$677$	7.60114	0.292135	0.146068	−	0.989275i	$-0.453338\pi$
$677$	7.60114	0.292135	0.146068	+	0.989275i	$0.453338\pi$
$678$	0	0
$679$	4.21245	0.161659
$680$	0	0
$681$	47.6759	1.82694
$682$	0	0
$683$	39.6610	1.51759	0.758793	−	0.651332i	$-0.225789\pi$
$683$	39.6610	1.51759	0.758793	+	0.651332i	$0.225789\pi$
$684$	0	0
$685$	16.5537	0.632484
$686$	0	0
$687$	−12.7376	−0.485970
$688$	0	0
$689$	−50.3580	−1.91849
$690$	0	0
$691$	45.2958	1.72313	0.861566	−	0.507645i	$-0.169484\pi$
$691$	45.2958	1.72313	0.861566	+	0.507645i	$0.169484\pi$
$692$	0	0
$693$	0.362320	0.0137634
$694$	0	0
$695$	−26.7260	−1.01377
$696$	0	0
$697$	24.0406	0.910604
$698$	0	0
$699$	2.50369	0.0946983
$700$	0	0
$701$	20.9064	0.789625	0.394813	−	0.918762i	$-0.370810\pi$
$701$	20.9064	0.789625	0.394813	+	0.918762i	$0.370810\pi$
$702$	0	0
$703$	−9.47967	−0.357532
$704$	0	0
$705$	−83.0656	−3.12843
$706$	0	0
$707$	−4.55082	−0.171151
$708$	0	0
$709$	−18.7483	−0.704109	−0.352054	−	0.935980i	$-0.614517\pi$
$709$	−18.7483	−0.704109	−0.352054	+	0.935980i	$0.614517\pi$
$710$	0	0
$711$	2.05349	0.0770120
$712$	0	0
$713$	0	0
$714$	0	0
$715$	11.4702	0.428959
$716$	0	0
$717$	−51.5795	−1.92627
$718$	0	0
$719$	−20.0265	−0.746863	−0.373431	−	0.927658i	$-0.621819\pi$
$719$	−20.0265	−0.746863	−0.373431	+	0.927658i	$0.621819\pi$
$720$	0	0
$721$	5.30790	0.197677
$722$	0	0
$723$	−1.91448	−0.0712004
$724$	0	0
$725$	47.9265	1.77995
$726$	0	0
$727$	−1.23830	−0.0459261	−0.0229631	−	0.999736i	$-0.507310\pi$
$727$	−1.23830	−0.0459261	−0.0229631	+	0.999736i	$0.507310\pi$
$728$	0	0
$729$	13.8072	0.511378
$730$	0	0
$731$	1.49302	0.0552214
$732$	0	0
$733$	−48.3323	−1.78519	−0.892597	−	0.450856i	$-0.851119\pi$
$733$	−48.3323	−1.78519	−0.892597	+	0.450856i	$0.851119\pi$
$734$	0	0
$735$	−51.4821	−1.89895
$736$	0	0
$737$	9.60467	0.353793
$738$	0	0
$739$	−25.4977	−0.937948	−0.468974	−	0.883212i	$-0.655376\pi$
$739$	−25.4977	−0.937948	−0.468974	+	0.883212i	$0.655376\pi$
$740$	0	0
$741$	56.8610	2.08884
$742$	0	0
$743$	8.34447	0.306129	0.153065	−	0.988216i	$-0.451086\pi$
$743$	8.34447	0.306129	0.153065	+	0.988216i	$0.451086\pi$
$744$	0	0
$745$	−20.7976	−0.761965
$746$	0	0
$747$	3.79850	0.138980
$748$	0	0
$749$	−1.67171	−0.0610829
$750$	0	0
$751$	22.1894	0.809704	0.404852	−	0.914382i	$-0.367323\pi$
$751$	22.1894	0.809704	0.404852	+	0.914382i	$0.367323\pi$
$752$	0	0
$753$	13.1397	0.478838
$754$	0	0
$755$	58.6711	2.13526
$756$	0	0
$757$	11.3813	0.413660	0.206830	−	0.978377i	$-0.433685\pi$
$757$	11.3813	0.413660	0.206830	+	0.978377i	$0.433685\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	9.63212	0.349164	0.174582	−	0.984643i	$-0.444143\pi$
$761$	9.63212	0.349164	0.174582	+	0.984643i	$0.444143\pi$
$762$	0	0
$763$	4.57757	0.165719
$764$	0	0
$765$	−18.6048	−0.672660
$766$	0	0
$767$	−36.1254	−1.30441
$768$	0	0
$769$	−18.0353	−0.650368	−0.325184	−	0.945651i	$-0.605426\pi$
$769$	−18.0353	−0.650368	−0.325184	+	0.945651i	$0.605426\pi$
$770$	0	0
$771$	50.8378	1.83088
$772$	0	0
$773$	39.8157	1.43207	0.716036	−	0.698063i	$-0.245955\pi$
$773$	39.8157	1.43207	0.716036	+	0.698063i	$0.245955\pi$
$774$	0	0
$775$	30.3181	1.08906
$776$	0	0
$777$	−1.47635	−0.0529638
$778$	0	0
$779$	−32.4624	−1.16308
$780$	0	0
$781$	2.67001	0.0955406
$782$	0	0
$783$	−19.7588	−0.706121
$784$	0	0
$785$	34.7906	1.24173
$786$	0	0
$787$	35.0705	1.25013	0.625064	−	0.780574i	$-0.285073\pi$
$787$	35.0705	1.25013	0.625064	+	0.780574i	$0.285073\pi$
$788$	0	0
$789$	−46.1468	−1.64287
$790$	0	0
$791$	−10.1652	−0.361434
$792$	0	0
$793$	54.1025	1.92124
$794$	0	0
$795$	−92.1559	−3.26843
$796$	0	0
$797$	5.39536	0.191114	0.0955568	−	0.995424i	$-0.469537\pi$
$797$	5.39536	0.191114	0.0955568	+	0.995424i	$0.469537\pi$
$798$	0	0
$799$	54.7511	1.93695
$800$	0	0
$801$	−2.03542	−0.0719181
$802$	0	0
$803$	−4.81096	−0.169775
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−52.2383	−1.83888
$808$	0	0
$809$	−2.49262	−0.0876360	−0.0438180	−	0.999040i	$-0.513952\pi$
$809$	−2.49262	−0.0876360	−0.0438180	+	0.999040i	$0.513952\pi$
$810$	0	0
$811$	−48.0562	−1.68748	−0.843741	−	0.536751i	$-0.819651\pi$
$811$	−48.0562	−1.68748	−0.843741	+	0.536751i	$0.819651\pi$
$812$	0	0
$813$	−32.3751	−1.13544
$814$	0	0
$815$	−42.2302	−1.47926
$816$	0	0
$817$	−2.01605	−0.0705325
$818$	0	0
$819$	2.13797	0.0747068
$820$	0	0
$821$	−53.0263	−1.85063	−0.925314	−	0.379201i	$-0.876199\pi$
$821$	−53.0263	−1.85063	−0.925314	+	0.379201i	$0.876199\pi$
$822$	0	0
$823$	43.6633	1.52201	0.761004	−	0.648747i	$-0.224707\pi$
$823$	43.6633	1.52201	0.761004	+	0.648747i	$0.224707\pi$
$824$	0	0
$825$	13.9303	0.484992
$826$	0	0
$827$	−1.53424	−0.0533508	−0.0266754	−	0.999644i	$-0.508492\pi$
$827$	−1.53424	−0.0533508	−0.0266754	+	0.999644i	$0.508492\pi$
$828$	0	0
$829$	30.4738	1.05840	0.529200	−	0.848497i	$-0.322492\pi$
$829$	30.4738	1.05840	0.529200	+	0.848497i	$0.322492\pi$
$830$	0	0
$831$	−12.6448	−0.438643
$832$	0	0
$833$	33.9335	1.17572
$834$	0	0
$835$	47.6424	1.64873
$836$	0	0
$837$	−12.4993	−0.432039
$838$	0	0
$839$	−33.0054	−1.13947	−0.569737	−	0.821827i	$-0.692955\pi$
$839$	−33.0054	−1.13947	−0.569737	+	0.821827i	$0.692955\pi$
$840$	0	0
$841$	−5.39922	−0.186180
$842$	0	0
$843$	−4.24778	−0.146301
$844$	0	0
$845$	17.5605	0.604101
$846$	0	0
$847$	−5.60928	−0.192737
$848$	0	0
$849$	−39.0169	−1.33906
$850$	0	0
$851$	0	0
$852$	0	0
$853$	13.3225	0.456155	0.228077	−	0.973643i	$-0.426756\pi$
$853$	13.3225	0.456155	0.228077	+	0.973643i	$0.426756\pi$
$854$	0	0
$855$	25.1224	0.859167
$856$	0	0
$857$	−57.9051	−1.97800	−0.989000	−	0.147912i	$-0.952745\pi$
$857$	−57.9051	−1.97800	−0.989000	+	0.147912i	$0.952745\pi$
$858$	0	0
$859$	−19.8890	−0.678603	−0.339302	−	0.940678i	$-0.610191\pi$
$859$	−19.8890	−0.678603	−0.339302	+	0.940678i	$0.610191\pi$
$860$	0	0
$861$	−5.05565	−0.172296
$862$	0	0
$863$	−4.79313	−0.163160	−0.0815800	−	0.996667i	$-0.525997\pi$
$863$	−4.79313	−0.163160	−0.0815800	+	0.996667i	$0.525997\pi$
$864$	0	0
$865$	26.7540	0.909663
$866$	0	0
$867$	16.9859	0.576871
$868$	0	0
$869$	1.52708	0.0518027
$870$	0	0
$871$	56.6751	1.92036
$872$	0	0
$873$	7.52595	0.254715
$874$	0	0
$875$	10.0252	0.338913
$876$	0	0
$877$	−37.1764	−1.25536	−0.627680	−	0.778472i	$-0.715995\pi$
$877$	−37.1764	−1.25536	−0.627680	+	0.778472i	$0.715995\pi$
$878$	0	0
$879$	−7.70865	−0.260006
$880$	0	0
$881$	−39.1904	−1.32036	−0.660179	−	0.751108i	$-0.729520\pi$
$881$	−39.1904	−1.32036	−0.660179	+	0.751108i	$0.729520\pi$
$882$	0	0
$883$	−52.0333	−1.75106	−0.875530	−	0.483165i	$-0.839487\pi$
$883$	−52.0333	−1.75106	−0.875530	+	0.483165i	$0.839487\pi$
$884$	0	0
$885$	−66.1101	−2.22227
$886$	0	0
$887$	−3.16609	−0.106307	−0.0531535	−	0.998586i	$-0.516927\pi$
$887$	−3.16609	−0.106307	−0.0531535	+	0.998586i	$0.516927\pi$
$888$	0	0
$889$	6.66151	0.223420
$890$	0	0
$891$	−7.77696	−0.260538
$892$	0	0
$893$	−73.9310	−2.47401
$894$	0	0
$895$	14.7029	0.491465
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.9297	0.497935
$900$	0	0
$901$	60.7428	2.02364
$902$	0	0
$903$	−0.313976	−0.0104485
$904$	0	0
$905$	−44.7757	−1.48840
$906$	0	0
$907$	2.09188	0.0694598	0.0347299	−	0.999397i	$-0.488943\pi$
$907$	2.09188	0.0694598	0.0347299	+	0.999397i	$0.488943\pi$
$908$	0	0
$909$	−8.13048	−0.269671
$910$	0	0
$911$	−11.8491	−0.392579	−0.196289	−	0.980546i	$-0.562889\pi$
$911$	−11.8491	−0.392579	−0.196289	+	0.980546i	$0.562889\pi$
$912$	0	0
$913$	2.82476	0.0934860
$914$	0	0
$915$	99.0085	3.27312
$916$	0	0
$917$	−8.75310	−0.289053
$918$	0	0
$919$	−0.658676	−0.0217277	−0.0108639	−	0.999941i	$-0.503458\pi$
$919$	−0.658676	−0.0217277	−0.0108639	+	0.999941i	$0.503458\pi$
$920$	0	0
$921$	56.7000	1.86833
$922$	0	0
$923$	15.7552	0.518587
$924$	0	0
$925$	−13.7041	−0.450587
$926$	0	0
$927$	9.48308	0.311465
$928$	0	0
$929$	19.3078	0.633468	0.316734	−	0.948514i	$-0.397414\pi$
$929$	19.3078	0.633468	0.316734	+	0.948514i	$0.397414\pi$
$930$	0	0
$931$	−45.8208	−1.50172
$932$	0	0
$933$	19.5678	0.640621
$934$	0	0
$935$	−13.8355	−0.452470
$936$	0	0
$937$	−42.6193	−1.39231	−0.696156	−	0.717890i	$-0.745108\pi$
$937$	−42.6193	−1.39231	−0.696156	+	0.717890i	$0.745108\pi$
$938$	0	0
$939$	19.2870	0.629407
$940$	0	0
$941$	2.68258	0.0874495	0.0437248	−	0.999044i	$-0.486078\pi$
$941$	2.68258	0.0874495	0.0437248	+	0.999044i	$0.486078\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	−8.38058	−0.272620
$946$	0	0
$947$	−7.74852	−0.251793	−0.125897	−	0.992043i	$-0.540181\pi$
$947$	−7.74852	−0.251793	−0.125897	+	0.992043i	$0.540181\pi$
$948$	0	0
$949$	−28.3884	−0.921527
$950$	0	0
$951$	20.5190	0.665373
$952$	0	0
$953$	−29.8038	−0.965439	−0.482720	−	0.875775i	$-0.660351\pi$
$953$	−29.8038	−0.965439	−0.482720	+	0.875775i	$0.660351\pi$
$954$	0	0
$955$	71.0243	2.29829
$956$	0	0
$957$	6.85981	0.221746
$958$	0	0
$959$	2.29455	0.0740948
$960$	0	0
$961$	−21.5555	−0.695339
$962$	0	0
$963$	−2.98667	−0.0962442
$964$	0	0
$965$	−4.09904	−0.131953
$966$	0	0
$967$	1.80855	0.0581592	0.0290796	−	0.999577i	$-0.490742\pi$
$967$	1.80855	0.0581592	0.0290796	+	0.999577i	$0.490742\pi$
$968$	0	0
$969$	−68.5868	−2.20333
$970$	0	0
$971$	5.36217	0.172080	0.0860400	−	0.996292i	$-0.472579\pi$
$971$	5.36217	0.172080	0.0860400	+	0.996292i	$0.472579\pi$
$972$	0	0
$973$	−3.70455	−0.118763
$974$	0	0
$975$	82.1998	2.63250
$976$	0	0
$977$	16.6059	0.531269	0.265634	−	0.964074i	$-0.414419\pi$
$977$	16.6059	0.531269	0.265634	+	0.964074i	$0.414419\pi$
$978$	0	0
$979$	−1.51365	−0.0483763
$980$	0	0
$981$	8.17827	0.261112
$982$	0	0
$983$	−21.3525	−0.681041	−0.340520	−	0.940237i	$-0.610603\pi$
$983$	−21.3525	−0.681041	−0.340520	+	0.940237i	$0.610603\pi$
$984$	0	0
$985$	−9.62648	−0.306725
$986$	0	0
$987$	−11.5139	−0.366492
$988$	0	0
$989$	0	0
$990$	0	0
$991$	35.2155	1.11866	0.559329	−	0.828946i	$-0.311059\pi$
$991$	35.2155	1.11866	0.559329	+	0.828946i	$0.311059\pi$
$992$	0	0
$993$	−45.8358	−1.45455
$994$	0	0
$995$	−5.82943	−0.184805
$996$	0	0
$997$	35.3369	1.11913	0.559566	−	0.828786i	$-0.310968\pi$
$997$	35.3369	1.11913	0.559566	+	0.828786i	$0.310968\pi$
$998$	0	0
$999$	5.64981	0.178752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4232.2.a.bb.1.12		15
4.3	odd	2		8464.2.a.cg.1.4		15
23.4	even	11		184.2.i.b.177.3	yes	30
23.6	even	11		184.2.i.b.105.3	✓	30
23.22	odd	2		4232.2.a.ba.1.12		15
92.27	odd	22		368.2.m.e.177.1		30
92.75	odd	22		368.2.m.e.289.1		30
92.91	even	2		8464.2.a.ch.1.4		15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.2.i.b.105.3	✓	30	23.6	even	11
184.2.i.b.177.3	yes	30	23.4	even	11
368.2.m.e.177.1		30	92.27	odd	22
368.2.m.e.289.1		30	92.75	odd	22
4232.2.a.ba.1.12		15	23.22	odd	2
4232.2.a.bb.1.12		15	1.1	even	1	trivial
8464.2.a.cg.1.4		15	4.3	odd	2
8464.2.a.ch.1.4		15	92.91	even	2

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:	\( N \)	\(=\)	\( 4232 = 2^{3} \cdot 23^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4232.a (trivial)

\(f(q)\)	\(=\)	\(q+1.98867 q^{3} +3.85556 q^{5} +0.534429 q^{7} +0.954809 q^{9} +0.710045 q^{11} +4.18982 q^{13} +7.66744 q^{15} -5.05384 q^{17} +6.82427 q^{19} +1.06280 q^{21} +9.86535 q^{25} -4.06721 q^{27} +4.85806 q^{29} +3.07319 q^{31} +1.41205 q^{33} +2.06052 q^{35} -1.38911 q^{37} +8.33217 q^{39} -4.75690 q^{41} -0.295423 q^{43} +3.68133 q^{45} -10.8335 q^{47} -6.71439 q^{49} -10.0504 q^{51} -12.0191 q^{53} +2.73762 q^{55} +13.5712 q^{57} -8.62219 q^{59} +12.9129 q^{61} +0.510278 q^{63} +16.1541 q^{65} +13.5268 q^{67} +3.76034 q^{71} -6.77557 q^{73} +19.6189 q^{75} +0.379469 q^{77} +2.15068 q^{79} -10.9528 q^{81} +3.97828 q^{83} -19.4854 q^{85} +9.66109 q^{87} -2.13176 q^{89} +2.23916 q^{91} +6.11156 q^{93} +26.3114 q^{95} +7.88215 q^{97} +0.677958 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + q^{3} + 10 q^{7} + 16 q^{9} + 23 q^{11} + 10 q^{15} + 29 q^{19} + q^{21} + 23 q^{25} + q^{27} - 2 q^{29} + 20 q^{31} + 18 q^{33} - 18 q^{35} + 24 q^{37} - 19 q^{39} + 9 q^{41} + 48 q^{43} + 4 q^{45}+ \cdots + 63 q^{99}+O(q^{100}) \) 15 * q + q^3 + 10 * q^7 + 16 * q^9 + 23 * q^11 + 10 * q^15 + 29 * q^19 + q^21 + 23 * q^25 + q^27 - 2 * q^29 + 20 * q^31 + 18 * q^33 - 18 * q^35 + 24 * q^37 - 19 * q^39 + 9 * q^41 + 48 * q^43 + 4 * q^45 - 36 * q^47 + 25 * q^49 + 35 * q^51 - 5 * q^53 - 10 * q^55 + 23 * q^57 - 22 * q^59 + 12 * q^61 + 35 * q^63 - 26 * q^65 + 58 * q^67 + 2 * q^71 + 5 * q^73 - 17 * q^75 + 26 * q^77 + 26 * q^79 - 21 * q^81 + 68 * q^83 - 72 * q^85 + 19 * q^87 - 6 * q^89 + 71 * q^91 - 55 * q^93 - 12 * q^95 + 40 * q^97 + 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more