LMFDB - Embedded newform 4732.2.a.s.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4732,2,Mod(1,4732)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4732, 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("4732.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4732 = 2^{2} \cdot 7 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4732.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:	$37.7852102365$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 19x^{6} - 2x^{5} + 113x^{4} + 40x^{3} - 232x^{2} - 136x + 52 $ x^8 - 19x^6 - 2x^5 + 113x^4 + 40x^3 - 232x^2 - 136x + 52
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 364)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.17707$ of defining polynomial
Character	$\chi$	$=$	4732.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.17707 q^{3} +1.46614 q^{5} -1.00000 q^{7} -1.61452 q^{9} -2.75012 q^{11} +1.72574 q^{15} +4.79948 q^{17} +1.97179 q^{19} -1.17707 q^{21} -2.93781 q^{23} -2.85043 q^{25} -5.43159 q^{27} -5.46889 q^{29} -3.90020 q^{31} -3.23707 q^{33} -1.46614 q^{35} -2.16723 q^{37} -9.44655 q^{41} +11.6086 q^{43} -2.36711 q^{45} -10.3753 q^{47} +1.00000 q^{49} +5.64930 q^{51} +1.24369 q^{53} -4.03207 q^{55} +2.32092 q^{57} +4.56755 q^{59} +8.39800 q^{61} +1.61452 q^{63} -3.75711 q^{67} -3.45799 q^{69} -9.90743 q^{71} -5.93768 q^{73} -3.35514 q^{75} +2.75012 q^{77} -12.8697 q^{79} -1.54979 q^{81} +0.811244 q^{83} +7.03671 q^{85} -6.43725 q^{87} +9.67958 q^{89} -4.59079 q^{93} +2.89092 q^{95} +9.51440 q^{97} +4.44012 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5} - 8 q^{7} + 14 q^{9} - 12 q^{11} - 12 q^{15} + 2 q^{17} - 6 q^{19} + 22 q^{25} - 6 q^{27} + 22 q^{29} - 14 q^{31} + 28 q^{33} + 6 q^{35} - 12 q^{37} - 4 q^{41} + 6 q^{43} - 20 q^{45} - 42 q^{47}+ \cdots - 80 q^{99}+O(q^{100}) $ 8 * q - 6 * q^5 - 8 * q^7 + 14 * q^9 - 12 * q^11 - 12 * q^15 + 2 * q^17 - 6 * q^19 + 22 * q^25 - 6 * q^27 + 22 * q^29 - 14 * q^31 + 28 * q^33 + 6 * q^35 - 12 * q^37 - 4 * q^41 + 6 * q^43 - 20 * q^45 - 42 * q^47 + 8 * q^49 + 2 * q^51 + 4 * q^53 - 2 * q^55 + 8 * q^57 - 2 * q^59 - 4 * q^61 - 14 * q^63 - 24 * q^67 - 52 * q^69 - 28 * q^71 - 28 * q^73 - 10 * q^75 + 12 * q^77 + 4 * q^79 - 38 * q^83 + 36 * q^85 - 26 * q^87 + 22 * q^89 - 56 * q^93 - 30 * q^95 + 4 * q^97 - 80 * 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
$3$	1.17707	0.679579	0.339790	−	0.940501i	$-0.389644\pi$
$3$	1.17707	0.679579	0.339790	+	0.940501i	$0.389644\pi$
$4$	0	0
$5$	1.46614	0.655678	0.327839	−	0.944734i	$-0.393680\pi$
$5$	1.46614	0.655678	0.327839	+	0.944734i	$0.393680\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−1.61452	−0.538172
$10$	0	0
$11$	−2.75012	−0.829193	−0.414597	−	0.910005i	$-0.636077\pi$
$11$	−2.75012	−0.829193	−0.414597	+	0.910005i	$0.636077\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.72574	0.445585
$16$	0	0
$17$	4.79948	1.16404	0.582022	−	0.813173i	$-0.302262\pi$
$17$	4.79948	1.16404	0.582022	+	0.813173i	$0.302262\pi$
$18$	0	0
$19$	1.97179	0.452359	0.226179	−	0.974086i	$-0.427376\pi$
$19$	1.97179	0.452359	0.226179	+	0.974086i	$0.427376\pi$
$20$	0	0
$21$	−1.17707	−0.256857
$22$	0	0
$23$	−2.93781	−0.612575	−0.306288	−	0.951939i	$-0.599087\pi$
$23$	−2.93781	−0.612575	−0.306288	+	0.951939i	$0.599087\pi$
$24$	0	0
$25$	−2.85043	−0.570086
$26$	0	0
$27$	−5.43159	−1.04531
$28$	0	0
$29$	−5.46889	−1.01555	−0.507774	−	0.861490i	$-0.669531\pi$
$29$	−5.46889	−1.01555	−0.507774	+	0.861490i	$0.669531\pi$
$30$	0	0
$31$	−3.90020	−0.700497	−0.350248	−	0.936657i	$-0.613903\pi$
$31$	−3.90020	−0.700497	−0.350248	+	0.936657i	$0.613903\pi$
$32$	0	0
$33$	−3.23707	−0.563502
$34$	0	0
$35$	−1.46614	−0.247823
$36$	0	0
$37$	−2.16723	−0.356290	−0.178145	−	0.984004i	$-0.557010\pi$
$37$	−2.16723	−0.356290	−0.178145	+	0.984004i	$0.557010\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−9.44655	−1.47530	−0.737652	−	0.675181i	$-0.764065\pi$
$41$	−9.44655	−1.47530	−0.737652	+	0.675181i	$0.764065\pi$
$42$	0	0
$43$	11.6086	1.77029	0.885144	−	0.465316i	$-0.154059\pi$
$43$	11.6086	1.77029	0.885144	+	0.465316i	$0.154059\pi$
$44$	0	0
$45$	−2.36711	−0.352868
$46$	0	0
$47$	−10.3753	−1.51339	−0.756693	−	0.653770i	$-0.773186\pi$
$47$	−10.3753	−1.51339	−0.756693	+	0.653770i	$0.773186\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.64930	0.791061
$52$	0	0
$53$	1.24369	0.170835	0.0854173	−	0.996345i	$-0.472778\pi$
$53$	1.24369	0.170835	0.0854173	+	0.996345i	$0.472778\pi$
$54$	0	0
$55$	−4.03207	−0.543684
$56$	0	0
$57$	2.32092	0.307414
$58$	0	0
$59$	4.56755	0.594644	0.297322	−	0.954777i	$-0.403906\pi$
$59$	4.56755	0.594644	0.297322	+	0.954777i	$0.403906\pi$
$60$	0	0
$61$	8.39800	1.07525	0.537627	−	0.843183i	$-0.319321\pi$
$61$	8.39800	1.07525	0.537627	+	0.843183i	$0.319321\pi$
$62$	0	0
$63$	1.61452	0.203410
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−3.75711	−0.459005	−0.229502	−	0.973308i	$-0.573710\pi$
$67$	−3.75711	−0.459005	−0.229502	+	0.973308i	$0.573710\pi$
$68$	0	0
$69$	−3.45799	−0.416293
$70$	0	0
$71$	−9.90743	−1.17580	−0.587898	−	0.808935i	$-0.700044\pi$
$71$	−9.90743	−1.17580	−0.587898	+	0.808935i	$0.700044\pi$
$72$	0	0
$73$	−5.93768	−0.694953	−0.347477	−	0.937689i	$-0.612961\pi$
$73$	−5.93768	−0.694953	−0.347477	+	0.937689i	$0.612961\pi$
$74$	0	0
$75$	−3.35514	−0.387419
$76$	0	0
$77$	2.75012	0.313406
$78$	0	0
$79$	−12.8697	−1.44795	−0.723975	−	0.689826i	$-0.757687\pi$
$79$	−12.8697	−1.44795	−0.723975	+	0.689826i	$0.757687\pi$
$80$	0	0
$81$	−1.54979	−0.172199
$82$	0	0
$83$	0.811244	0.0890456	0.0445228	−	0.999008i	$-0.485823\pi$
$83$	0.811244	0.0890456	0.0445228	+	0.999008i	$0.485823\pi$
$84$	0	0
$85$	7.03671	0.763239
$86$	0	0
$87$	−6.43725	−0.690145
$88$	0	0
$89$	9.67958	1.02603	0.513017	−	0.858379i	$-0.328528\pi$
$89$	9.67958	1.02603	0.513017	+	0.858379i	$0.328528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−4.59079	−0.476043
$94$	0	0
$95$	2.89092	0.296602
$96$	0	0
$97$	9.51440	0.966041	0.483020	−	0.875609i	$-0.339540\pi$
$97$	9.51440	0.966041	0.483020	+	0.875609i	$0.339540\pi$
$98$	0	0
$99$	4.44012	0.446249
$100$	0	0
$101$	10.3663	1.03148	0.515742	−	0.856744i	$-0.327516\pi$
$101$	10.3663	1.03148	0.515742	+	0.856744i	$0.327516\pi$
$102$	0	0
$103$	7.81778	0.770309	0.385155	−	0.922852i	$-0.374148\pi$
$103$	7.81778	0.770309	0.385155	+	0.922852i	$0.374148\pi$
$104$	0	0
$105$	−1.72574	−0.168415
$106$	0	0
$107$	−9.76845	−0.944351	−0.472176	−	0.881505i	$-0.656531\pi$
$107$	−9.76845	−0.944351	−0.472176	+	0.881505i	$0.656531\pi$
$108$	0	0
$109$	−12.9827	−1.24352	−0.621758	−	0.783209i	$-0.713581\pi$
$109$	−12.9827	−1.24352	−0.621758	+	0.783209i	$0.713581\pi$
$110$	0	0
$111$	−2.55097	−0.242127
$112$	0	0
$113$	−11.0423	−1.03877	−0.519386	−	0.854540i	$-0.673839\pi$
$113$	−11.0423	−1.03877	−0.519386	+	0.854540i	$0.673839\pi$
$114$	0	0
$115$	−4.30724	−0.401652
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−4.79948	−0.439968
$120$	0	0
$121$	−3.43683	−0.312439
$122$	0	0
$123$	−11.1192	−1.00259
$124$	0	0
$125$	−11.5098	−1.02947
$126$	0	0
$127$	−15.6532	−1.38899	−0.694497	−	0.719496i	$-0.744373\pi$
$127$	−15.6532	−1.38899	−0.694497	+	0.719496i	$0.744373\pi$
$128$	0	0
$129$	13.6640	1.20305
$130$	0	0
$131$	−3.84949	−0.336332	−0.168166	−	0.985759i	$-0.553784\pi$
$131$	−3.84949	−0.336332	−0.168166	+	0.985759i	$0.553784\pi$
$132$	0	0
$133$	−1.97179	−0.170976
$134$	0	0
$135$	−7.96348	−0.685387
$136$	0	0
$137$	−6.34437	−0.542036	−0.271018	−	0.962574i	$-0.587360\pi$
$137$	−6.34437	−0.542036	−0.271018	+	0.962574i	$0.587360\pi$
$138$	0	0
$139$	−10.2971	−0.873386	−0.436693	−	0.899611i	$-0.643850\pi$
$139$	−10.2971	−0.873386	−0.436693	+	0.899611i	$0.643850\pi$
$140$	0	0
$141$	−12.2124	−1.02847
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−8.01817	−0.665873
$146$	0	0
$147$	1.17707	0.0970827
$148$	0	0
$149$	−20.4015	−1.67136	−0.835678	−	0.549220i	$-0.814925\pi$
$149$	−20.4015	−1.67136	−0.835678	+	0.549220i	$0.814925\pi$
$150$	0	0
$151$	−22.8050	−1.85584	−0.927922	−	0.372774i	$-0.878407\pi$
$151$	−22.8050	−1.85584	−0.927922	+	0.372774i	$0.878407\pi$
$152$	0	0
$153$	−7.74884	−0.626456
$154$	0	0
$155$	−5.71824	−0.459300
$156$	0	0
$157$	14.6229	1.16703	0.583516	−	0.812102i	$-0.301677\pi$
$157$	14.6229	1.16703	0.583516	+	0.812102i	$0.301677\pi$
$158$	0	0
$159$	1.46391	0.116096
$160$	0	0
$161$	2.93781	0.231532
$162$	0	0
$163$	−0.858968	−0.0672795	−0.0336398	−	0.999434i	$-0.510710\pi$
$163$	−0.858968	−0.0672795	−0.0336398	+	0.999434i	$0.510710\pi$
$164$	0	0
$165$	−4.74601	−0.369476
$166$	0	0
$167$	5.24059	0.405529	0.202764	−	0.979228i	$-0.435007\pi$
$167$	5.24059	0.405529	0.202764	+	0.979228i	$0.435007\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.18348	−0.243447
$172$	0	0
$173$	19.3657	1.47235	0.736174	−	0.676792i	$-0.236630\pi$
$173$	19.3657	1.47235	0.736174	+	0.676792i	$0.236630\pi$
$174$	0	0
$175$	2.85043	0.215472
$176$	0	0
$177$	5.37631	0.404108
$178$	0	0
$179$	7.91929	0.591916	0.295958	−	0.955201i	$-0.404361\pi$
$179$	7.91929	0.591916	0.295958	+	0.955201i	$0.404361\pi$
$180$	0	0
$181$	18.5469	1.37858	0.689288	−	0.724487i	$-0.257923\pi$
$181$	18.5469	1.37858	0.689288	+	0.724487i	$0.257923\pi$
$182$	0	0
$183$	9.88499	0.730720
$184$	0	0
$185$	−3.17746	−0.233612
$186$	0	0
$187$	−13.1992	−0.965218
$188$	0	0
$189$	5.43159	0.395090
$190$	0	0
$191$	15.9759	1.15598	0.577989	−	0.816045i	$-0.303838\pi$
$191$	15.9759	1.15598	0.577989	+	0.816045i	$0.303838\pi$
$192$	0	0
$193$	−13.5305	−0.973945	−0.486972	−	0.873417i	$-0.661899\pi$
$193$	−13.5305	−0.973945	−0.486972	+	0.873417i	$0.661899\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1.27550	−0.0908753	−0.0454377	−	0.998967i	$-0.514468\pi$
$197$	−1.27550	−0.0908753	−0.0454377	+	0.998967i	$0.514468\pi$
$198$	0	0
$199$	−1.69720	−0.120311	−0.0601555	−	0.998189i	$-0.519160\pi$
$199$	−1.69720	−0.120311	−0.0601555	+	0.998189i	$0.519160\pi$
$200$	0	0
$201$	−4.42237	−0.311930
$202$	0	0
$203$	5.46889	0.383841
$204$	0	0
$205$	−13.8500	−0.967325
$206$	0	0
$207$	4.74314	0.329671
$208$	0	0
$209$	−5.42266	−0.375093
$210$	0	0
$211$	0.0338871	0.00233289	0.00116644	−	0.999999i	$-0.499629\pi$
$211$	0.0338871	0.00233289	0.00116644	+	0.999999i	$0.499629\pi$
$212$	0	0
$213$	−11.6617	−0.799046
$214$	0	0
$215$	17.0198	1.16074
$216$	0	0
$217$	3.90020	0.264763
$218$	0	0
$219$	−6.98904	−0.472276
$220$	0	0
$221$	0	0
$222$	0	0
$223$	22.1213	1.48135	0.740676	−	0.671862i	$-0.234505\pi$
$223$	22.1213	1.48135	0.740676	+	0.671862i	$0.234505\pi$
$224$	0	0
$225$	4.60207	0.306804
$226$	0	0
$227$	2.07867	0.137966	0.0689832	−	0.997618i	$-0.478025\pi$
$227$	2.07867	0.137966	0.0689832	+	0.997618i	$0.478025\pi$
$228$	0	0
$229$	4.60929	0.304591	0.152295	−	0.988335i	$-0.451333\pi$
$229$	4.60929	0.304591	0.152295	+	0.988335i	$0.451333\pi$
$230$	0	0
$231$	3.23707	0.212984
$232$	0	0
$233$	27.6690	1.81266	0.906328	−	0.422576i	$-0.138874\pi$
$233$	27.6690	1.81266	0.906328	+	0.422576i	$0.138874\pi$
$234$	0	0
$235$	−15.2116	−0.992295
$236$	0	0
$237$	−15.1484	−0.983997
$238$	0	0
$239$	−4.72236	−0.305464	−0.152732	−	0.988268i	$-0.548807\pi$
$239$	−4.72236	−0.305464	−0.152732	+	0.988268i	$0.548807\pi$
$240$	0	0
$241$	6.16812	0.397323	0.198662	−	0.980068i	$-0.436341\pi$
$241$	6.16812	0.397323	0.198662	+	0.980068i	$0.436341\pi$
$242$	0	0
$243$	14.4706	0.928287
$244$	0	0
$245$	1.46614	0.0936683
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.954887	0.0605135
$250$	0	0
$251$	−9.50340	−0.599850	−0.299925	−	0.953963i	$-0.596962\pi$
$251$	−9.50340	−0.599850	−0.299925	+	0.953963i	$0.596962\pi$
$252$	0	0
$253$	8.07933	0.507943
$254$	0	0
$255$	8.28267	0.518681
$256$	0	0
$257$	−16.9509	−1.05737	−0.528684	−	0.848818i	$-0.677315\pi$
$257$	−16.9509	−1.05737	−0.528684	+	0.848818i	$0.677315\pi$
$258$	0	0
$259$	2.16723	0.134665
$260$	0	0
$261$	8.82962	0.546539
$262$	0	0
$263$	−7.66825	−0.472845	−0.236422	−	0.971650i	$-0.575975\pi$
$263$	−7.66825	−0.472845	−0.236422	+	0.971650i	$0.575975\pi$
$264$	0	0
$265$	1.82343	0.112013
$266$	0	0
$267$	11.3935	0.697271
$268$	0	0
$269$	−8.12363	−0.495307	−0.247653	−	0.968849i	$-0.579659\pi$
$269$	−8.12363	−0.495307	−0.247653	+	0.968849i	$0.579659\pi$
$270$	0	0
$271$	2.10302	0.127749	0.0638746	−	0.997958i	$-0.479654\pi$
$271$	2.10302	0.127749	0.0638746	+	0.997958i	$0.479654\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.83903	0.472711
$276$	0	0
$277$	−6.14676	−0.369323	−0.184661	−	0.982802i	$-0.559119\pi$
$277$	−6.14676	−0.369323	−0.184661	+	0.982802i	$0.559119\pi$
$278$	0	0
$279$	6.29694	0.376988
$280$	0	0
$281$	−17.5826	−1.04889	−0.524445	−	0.851444i	$-0.675727\pi$
$281$	−17.5826	−1.04889	−0.524445	+	0.851444i	$0.675727\pi$
$282$	0	0
$283$	0.349504	0.0207759	0.0103879	−	0.999946i	$-0.496693\pi$
$283$	0.349504	0.0207759	0.0103879	+	0.999946i	$0.496693\pi$
$284$	0	0
$285$	3.40280	0.201565
$286$	0	0
$287$	9.44655	0.557612
$288$	0	0
$289$	6.03500	0.355000
$290$	0	0
$291$	11.1991	0.656501
$292$	0	0
$293$	1.68707	0.0985595	0.0492798	−	0.998785i	$-0.484307\pi$
$293$	1.68707	0.0985595	0.0492798	+	0.998785i	$0.484307\pi$
$294$	0	0
$295$	6.69667	0.389895
$296$	0	0
$297$	14.9375	0.866764
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−11.6086	−0.669106
$302$	0	0
$303$	12.2018	0.700975
$304$	0	0
$305$	12.3126	0.705020
$306$	0	0
$307$	−15.8850	−0.906606	−0.453303	−	0.891357i	$-0.649754\pi$
$307$	−15.8850	−0.906606	−0.453303	+	0.891357i	$0.649754\pi$
$308$	0	0
$309$	9.20205	0.523486
$310$	0	0
$311$	5.73424	0.325159	0.162579	−	0.986695i	$-0.448019\pi$
$311$	5.73424	0.325159	0.162579	+	0.986695i	$0.448019\pi$
$312$	0	0
$313$	10.3701	0.586155	0.293078	−	0.956089i	$-0.405321\pi$
$313$	10.3701	0.586155	0.293078	+	0.956089i	$0.405321\pi$
$314$	0	0
$315$	2.36711	0.133371
$316$	0	0
$317$	20.0766	1.12761	0.563807	−	0.825907i	$-0.309336\pi$
$317$	20.0766	1.12761	0.563807	+	0.825907i	$0.309336\pi$
$318$	0	0
$319$	15.0401	0.842085
$320$	0	0
$321$	−11.4981	−0.641761
$322$	0	0
$323$	9.46355	0.526566
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−15.2815	−0.845068
$328$	0	0
$329$	10.3753	0.572006
$330$	0	0
$331$	−22.6980	−1.24759	−0.623797	−	0.781586i	$-0.714411\pi$
$331$	−22.6980	−1.24759	−0.623797	+	0.781586i	$0.714411\pi$
$332$	0	0
$333$	3.49902	0.191745
$334$	0	0
$335$	−5.50846	−0.300959
$336$	0	0
$337$	31.8134	1.73299	0.866493	−	0.499189i	$-0.166369\pi$
$337$	31.8134	1.73299	0.866493	+	0.499189i	$0.166369\pi$
$338$	0	0
$339$	−12.9975	−0.705927
$340$	0	0
$341$	10.7260	0.580847
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.06990	−0.272954
$346$	0	0
$347$	33.5527	1.80121	0.900603	−	0.434644i	$-0.143126\pi$
$347$	33.5527	1.80121	0.900603	+	0.434644i	$0.143126\pi$
$348$	0	0
$349$	−22.4496	−1.20170	−0.600850	−	0.799362i	$-0.705171\pi$
$349$	−22.4496	−1.20170	−0.600850	+	0.799362i	$0.705171\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−20.3207	−1.08156	−0.540782	−	0.841163i	$-0.681871\pi$
$353$	−20.3207	−1.08156	−0.540782	+	0.841163i	$0.681871\pi$
$354$	0	0
$355$	−14.5257	−0.770944
$356$	0	0
$357$	−5.64930	−0.298993
$358$	0	0
$359$	−32.5948	−1.72029	−0.860144	−	0.510052i	$-0.829626\pi$
$359$	−32.5948	−1.72029	−0.860144	+	0.510052i	$0.829626\pi$
$360$	0	0
$361$	−15.1121	−0.795371
$362$	0	0
$363$	−4.04537	−0.212327
$364$	0	0
$365$	−8.70548	−0.455666
$366$	0	0
$367$	−15.1385	−0.790222	−0.395111	−	0.918633i	$-0.629294\pi$
$367$	−15.1385	−0.790222	−0.395111	+	0.918633i	$0.629294\pi$
$368$	0	0
$369$	15.2516	0.793967
$370$	0	0
$371$	−1.24369	−0.0645694
$372$	0	0
$373$	−33.0340	−1.71043	−0.855217	−	0.518270i	$-0.826576\pi$
$373$	−33.0340	−1.71043	−0.855217	+	0.518270i	$0.826576\pi$
$374$	0	0
$375$	−13.5478	−0.699607
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.9858	−0.667033	−0.333517	−	0.942744i	$-0.608235\pi$
$379$	−12.9858	−0.667033	−0.333517	+	0.942744i	$0.608235\pi$
$380$	0	0
$381$	−18.4248	−0.943931
$382$	0	0
$383$	−22.6435	−1.15703	−0.578514	−	0.815673i	$-0.696367\pi$
$383$	−22.6435	−1.15703	−0.578514	+	0.815673i	$0.696367\pi$
$384$	0	0
$385$	4.03207	0.205493
$386$	0	0
$387$	−18.7422	−0.952720
$388$	0	0
$389$	−3.51171	−0.178051	−0.0890254	−	0.996029i	$-0.528375\pi$
$389$	−3.51171	−0.178051	−0.0890254	+	0.996029i	$0.528375\pi$
$390$	0	0
$391$	−14.0999	−0.713065
$392$	0	0
$393$	−4.53110	−0.228564
$394$	0	0
$395$	−18.8687	−0.949389
$396$	0	0
$397$	17.9590	0.901334	0.450667	−	0.892692i	$-0.351186\pi$
$397$	17.9590	0.901334	0.450667	+	0.892692i	$0.351186\pi$
$398$	0	0
$399$	−2.32092	−0.116191
$400$	0	0
$401$	9.18476	0.458665	0.229333	−	0.973348i	$-0.426346\pi$
$401$	9.18476	0.458665	0.229333	+	0.973348i	$0.426346\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−2.27221	−0.112907
$406$	0	0
$407$	5.96014	0.295433
$408$	0	0
$409$	33.1849	1.64089	0.820444	−	0.571727i	$-0.193727\pi$
$409$	33.1849	1.64089	0.820444	+	0.571727i	$0.193727\pi$
$410$	0	0
$411$	−7.46773	−0.368356
$412$	0	0
$413$	−4.56755	−0.224754
$414$	0	0
$415$	1.18940	0.0583853
$416$	0	0
$417$	−12.1203	−0.593535
$418$	0	0
$419$	−14.7409	−0.720142	−0.360071	−	0.932925i	$-0.617248\pi$
$419$	−14.7409	−0.720142	−0.360071	+	0.932925i	$0.617248\pi$
$420$	0	0
$421$	26.7288	1.30268	0.651341	−	0.758785i	$-0.274207\pi$
$421$	26.7288	1.30268	0.651341	+	0.758785i	$0.274207\pi$
$422$	0	0
$423$	16.7510	0.814463
$424$	0	0
$425$	−13.6806	−0.663606
$426$	0	0
$427$	−8.39800	−0.406408
$428$	0	0
$429$	0	0
$430$	0	0
$431$	10.1684	0.489793	0.244897	−	0.969549i	$-0.421246\pi$
$431$	10.1684	0.489793	0.244897	+	0.969549i	$0.421246\pi$
$432$	0	0
$433$	−13.0280	−0.626085	−0.313043	−	0.949739i	$-0.601348\pi$
$433$	−13.0280	−0.626085	−0.313043	+	0.949739i	$0.601348\pi$
$434$	0	0
$435$	−9.43791	−0.452513
$436$	0	0
$437$	−5.79273	−0.277104
$438$	0	0
$439$	21.6571	1.03364	0.516819	−	0.856095i	$-0.327116\pi$
$439$	21.6571	1.03364	0.516819	+	0.856095i	$0.327116\pi$
$440$	0	0
$441$	−1.61452	−0.0768817
$442$	0	0
$443$	26.2812	1.24866	0.624330	−	0.781161i	$-0.285372\pi$
$443$	26.2812	1.24866	0.624330	+	0.781161i	$0.285372\pi$
$444$	0	0
$445$	14.1916	0.672748
$446$	0	0
$447$	−24.0139	−1.13582
$448$	0	0
$449$	−15.1500	−0.714975	−0.357487	−	0.933918i	$-0.616366\pi$
$449$	−15.1500	−0.714975	−0.357487	+	0.933918i	$0.616366\pi$
$450$	0	0
$451$	25.9792	1.22331
$452$	0	0
$453$	−26.8430	−1.26119
$454$	0	0
$455$	0	0
$456$	0	0
$457$	30.4632	1.42501	0.712505	−	0.701667i	$-0.247560\pi$
$457$	30.4632	1.42501	0.712505	+	0.701667i	$0.247560\pi$
$458$	0	0
$459$	−26.0688	−1.21679
$460$	0	0
$461$	30.4655	1.41892	0.709461	−	0.704745i	$-0.248939\pi$
$461$	30.4655	1.41892	0.709461	+	0.704745i	$0.248939\pi$
$462$	0	0
$463$	10.4617	0.486197	0.243099	−	0.970002i	$-0.421836\pi$
$463$	10.4617	0.486197	0.243099	+	0.970002i	$0.421836\pi$
$464$	0	0
$465$	−6.73075	−0.312131
$466$	0	0
$467$	−15.2825	−0.707190	−0.353595	−	0.935399i	$-0.615041\pi$
$467$	−15.2825	−0.707190	−0.353595	+	0.935399i	$0.615041\pi$
$468$	0	0
$469$	3.75711	0.173487
$470$	0	0
$471$	17.2121	0.793090
$472$	0	0
$473$	−31.9250	−1.46791
$474$	0	0
$475$	−5.62044	−0.257884
$476$	0	0
$477$	−2.00797	−0.0919384
$478$	0	0
$479$	−19.5909	−0.895129	−0.447564	−	0.894252i	$-0.647708\pi$
$479$	−19.5909	−0.895129	−0.447564	+	0.894252i	$0.647708\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.45799	0.157344
$484$	0	0
$485$	13.9494	0.633412
$486$	0	0
$487$	6.39343	0.289714	0.144857	−	0.989453i	$-0.453728\pi$
$487$	6.39343	0.289714	0.144857	+	0.989453i	$0.453728\pi$
$488$	0	0
$489$	−1.01106	−0.0457218
$490$	0	0
$491$	−22.2520	−1.00422	−0.502109	−	0.864805i	$-0.667442\pi$
$491$	−22.2520	−1.00422	−0.502109	+	0.864805i	$0.667442\pi$
$492$	0	0
$493$	−26.2478	−1.18214
$494$	0	0
$495$	6.50984	0.292595
$496$	0	0
$497$	9.90743	0.444409
$498$	0	0
$499$	30.0165	1.34372	0.671862	−	0.740677i	$-0.265495\pi$
$499$	30.0165	1.34372	0.671862	+	0.740677i	$0.265495\pi$
$500$	0	0
$501$	6.16852	0.275589
$502$	0	0
$503$	−35.0031	−1.56071	−0.780356	−	0.625336i	$-0.784962\pi$
$503$	−35.0031	−1.56071	−0.780356	+	0.625336i	$0.784962\pi$
$504$	0	0
$505$	15.1984	0.676322
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−9.95401	−0.441204	−0.220602	−	0.975364i	$-0.570802\pi$
$509$	−9.95401	−0.441204	−0.220602	+	0.975364i	$0.570802\pi$
$510$	0	0
$511$	5.93768	0.262668
$512$	0	0
$513$	−10.7099	−0.472855
$514$	0	0
$515$	11.4620	0.505075
$516$	0	0
$517$	28.5332	1.25489
$518$	0	0
$519$	22.7947	1.00058
$520$	0	0
$521$	−7.75549	−0.339774	−0.169887	−	0.985464i	$-0.554340\pi$
$521$	−7.75549	−0.339774	−0.169887	+	0.985464i	$0.554340\pi$
$522$	0	0
$523$	4.84825	0.211999	0.105999	−	0.994366i	$-0.466196\pi$
$523$	4.84825	0.211999	0.105999	+	0.994366i	$0.466196\pi$
$524$	0	0
$525$	3.35514	0.146430
$526$	0	0
$527$	−18.7189	−0.815409
$528$	0	0
$529$	−14.3693	−0.624752
$530$	0	0
$531$	−7.37438	−0.320021
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−14.3219	−0.619191
$536$	0	0
$537$	9.32153	0.402254
$538$	0	0
$539$	−2.75012	−0.118456
$540$	0	0
$541$	28.2030	1.21254	0.606271	−	0.795258i	$-0.292665\pi$
$541$	28.2030	1.21254	0.606271	+	0.795258i	$0.292665\pi$
$542$	0	0
$543$	21.8309	0.936852
$544$	0	0
$545$	−19.0345	−0.815346
$546$	0	0
$547$	−1.10083	−0.0470680	−0.0235340	−	0.999723i	$-0.507492\pi$
$547$	−1.10083	−0.0470680	−0.0235340	+	0.999723i	$0.507492\pi$
$548$	0	0
$549$	−13.5587	−0.578671
$550$	0	0
$551$	−10.7835	−0.459392
$552$	0	0
$553$	12.8697	0.547274
$554$	0	0
$555$	−3.74008	−0.158758
$556$	0	0
$557$	20.7862	0.880738	0.440369	−	0.897817i	$-0.354848\pi$
$557$	20.7862	0.880738	0.440369	+	0.897817i	$0.354848\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−15.5363	−0.655942
$562$	0	0
$563$	33.5659	1.41463	0.707316	−	0.706897i	$-0.249906\pi$
$563$	33.5659	1.41463	0.707316	+	0.706897i	$0.249906\pi$
$564$	0	0
$565$	−16.1896	−0.681100
$566$	0	0
$567$	1.54979	0.0650850
$568$	0	0
$569$	−8.90865	−0.373470	−0.186735	−	0.982410i	$-0.559791\pi$
$569$	−8.90865	−0.373470	−0.186735	+	0.982410i	$0.559791\pi$
$570$	0	0
$571$	−29.3722	−1.22919	−0.614594	−	0.788844i	$-0.710680\pi$
$571$	−29.3722	−1.22919	−0.614594	+	0.788844i	$0.710680\pi$
$572$	0	0
$573$	18.8047	0.785578
$574$	0	0
$575$	8.37401	0.349221
$576$	0	0
$577$	34.5820	1.43967	0.719834	−	0.694146i	$-0.244218\pi$
$577$	34.5820	1.43967	0.719834	+	0.694146i	$0.244218\pi$
$578$	0	0
$579$	−15.9263	−0.661873
$580$	0	0
$581$	−0.811244	−0.0336561
$582$	0	0
$583$	−3.42031	−0.141655
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.2900	1.58040	0.790199	−	0.612850i	$-0.209977\pi$
$587$	38.2900	1.58040	0.790199	+	0.612850i	$0.209977\pi$
$588$	0	0
$589$	−7.69036	−0.316876
$590$	0	0
$591$	−1.50134	−0.0617570
$592$	0	0
$593$	44.1937	1.81482	0.907408	−	0.420250i	$-0.138058\pi$
$593$	44.1937	1.81482	0.907408	+	0.420250i	$0.138058\pi$
$594$	0	0
$595$	−7.03671	−0.288477
$596$	0	0
$597$	−1.99771	−0.0817608
$598$	0	0
$599$	−18.1168	−0.740232	−0.370116	−	0.928986i	$-0.620682\pi$
$599$	−18.1168	−0.740232	−0.370116	+	0.928986i	$0.620682\pi$
$600$	0	0
$601$	−34.7627	−1.41800	−0.709001	−	0.705208i	$-0.750854\pi$
$601$	−34.7627	−1.41800	−0.709001	+	0.705208i	$0.750854\pi$
$602$	0	0
$603$	6.06592	0.247023
$604$	0	0
$605$	−5.03887	−0.204859
$606$	0	0
$607$	17.6284	0.715514	0.357757	−	0.933815i	$-0.383542\pi$
$607$	17.6284	0.715514	0.357757	+	0.933815i	$0.383542\pi$
$608$	0	0
$609$	6.43725	0.260850
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−7.00516	−0.282936	−0.141468	−	0.989943i	$-0.545182\pi$
$613$	−7.00516	−0.282936	−0.141468	+	0.989943i	$0.545182\pi$
$614$	0	0
$615$	−16.3023	−0.657374
$616$	0	0
$617$	−25.1516	−1.01257	−0.506283	−	0.862367i	$-0.668981\pi$
$617$	−25.1516	−1.01257	−0.506283	+	0.862367i	$0.668981\pi$
$618$	0	0
$619$	5.30877	0.213377	0.106689	−	0.994292i	$-0.465975\pi$
$619$	5.30877	0.213377	0.106689	+	0.994292i	$0.465975\pi$
$620$	0	0
$621$	15.9570	0.640331
$622$	0	0
$623$	−9.67958	−0.387804
$624$	0	0
$625$	−2.62289	−0.104916
$626$	0	0
$627$	−6.38282	−0.254905
$628$	0	0
$629$	−10.4016	−0.414738
$630$	0	0
$631$	−21.9584	−0.874151	−0.437076	−	0.899425i	$-0.643986\pi$
$631$	−21.9584	−0.874151	−0.437076	+	0.899425i	$0.643986\pi$
$632$	0	0
$633$	0.0398874	0.00158538
$634$	0	0
$635$	−22.9497	−0.910732
$636$	0	0
$637$	0	0
$638$	0	0
$639$	15.9957	0.632780
$640$	0	0
$641$	29.2954	1.15710	0.578549	−	0.815648i	$-0.303619\pi$
$641$	29.2954	1.15710	0.578549	+	0.815648i	$0.303619\pi$
$642$	0	0
$643$	44.3931	1.75069	0.875346	−	0.483496i	$-0.160633\pi$
$643$	44.3931	1.75069	0.875346	+	0.483496i	$0.160633\pi$
$644$	0	0
$645$	20.0334	0.788815
$646$	0	0
$647$	2.88866	0.113565	0.0567824	−	0.998387i	$-0.481916\pi$
$647$	2.88866	0.113565	0.0567824	+	0.998387i	$0.481916\pi$
$648$	0	0
$649$	−12.5613	−0.493075
$650$	0	0
$651$	4.59079	0.179927
$652$	0	0
$653$	−18.8390	−0.737229	−0.368614	−	0.929582i	$-0.620168\pi$
$653$	−18.8390	−0.737229	−0.368614	+	0.929582i	$0.620168\pi$
$654$	0	0
$655$	−5.64390	−0.220525
$656$	0	0
$657$	9.58649	0.374004
$658$	0	0
$659$	26.2632	1.02307	0.511535	−	0.859263i	$-0.329077\pi$
$659$	26.2632	1.02307	0.511535	+	0.859263i	$0.329077\pi$
$660$	0	0
$661$	33.0289	1.28468	0.642338	−	0.766421i	$-0.277964\pi$
$661$	33.0289	1.28468	0.642338	+	0.766421i	$0.277964\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.89092	−0.112105
$666$	0	0
$667$	16.0665	0.622099
$668$	0	0
$669$	26.0382	1.00670
$670$	0	0
$671$	−23.0955	−0.891592
$672$	0	0
$673$	−4.60997	−0.177701	−0.0888507	−	0.996045i	$-0.528319\pi$
$673$	−4.60997	−0.177701	−0.0888507	+	0.996045i	$0.528319\pi$
$674$	0	0
$675$	15.4824	0.595917
$676$	0	0
$677$	−0.757520	−0.0291138	−0.0145569	−	0.999894i	$-0.504634\pi$
$677$	−0.757520	−0.0291138	−0.0145569	+	0.999894i	$0.504634\pi$
$678$	0	0
$679$	−9.51440	−0.365129
$680$	0	0
$681$	2.44674	0.0937591
$682$	0	0
$683$	−40.0568	−1.53273	−0.766366	−	0.642405i	$-0.777937\pi$
$683$	−40.0568	−1.53273	−0.766366	+	0.642405i	$0.777937\pi$
$684$	0	0
$685$	−9.30173	−0.355401
$686$	0	0
$687$	5.42544	0.206994
$688$	0	0
$689$	0	0
$690$	0	0
$691$	15.3280	0.583106	0.291553	−	0.956555i	$-0.405828\pi$
$691$	15.3280	0.583106	0.291553	+	0.956555i	$0.405828\pi$
$692$	0	0
$693$	−4.44012	−0.168666
$694$	0	0
$695$	−15.0970	−0.572660
$696$	0	0
$697$	−45.3385	−1.71732
$698$	0	0
$699$	32.5682	1.23184
$700$	0	0
$701$	15.1453	0.572032	0.286016	−	0.958225i	$-0.407669\pi$
$701$	15.1453	0.572032	0.286016	+	0.958225i	$0.407669\pi$
$702$	0	0
$703$	−4.27331	−0.161171
$704$	0	0
$705$	−17.9050	−0.674343
$706$	0	0
$707$	−10.3663	−0.389864
$708$	0	0
$709$	11.0785	0.416063	0.208032	−	0.978122i	$-0.433294\pi$
$709$	11.0785	0.416063	0.208032	+	0.978122i	$0.433294\pi$
$710$	0	0
$711$	20.7783	0.779246
$712$	0	0
$713$	11.4580	0.429107
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−5.55853	−0.207587
$718$	0	0
$719$	34.5299	1.28775	0.643874	−	0.765132i	$-0.277326\pi$
$719$	34.5299	1.28775	0.643874	+	0.765132i	$0.277326\pi$
$720$	0	0
$721$	−7.81778	−0.291149
$722$	0	0
$723$	7.26028	0.270013
$724$	0	0
$725$	15.5887	0.578950
$726$	0	0
$727$	−13.5521	−0.502621	−0.251311	−	0.967907i	$-0.580862\pi$
$727$	−13.5521	−0.502621	−0.251311	+	0.967907i	$0.580862\pi$
$728$	0	0
$729$	21.6822	0.803043
$730$	0	0
$731$	55.7150	2.06070
$732$	0	0
$733$	−12.5720	−0.464357	−0.232179	−	0.972673i	$-0.574585\pi$
$733$	−12.5720	−0.464357	−0.232179	+	0.972673i	$0.574585\pi$
$734$	0	0
$735$	1.72574	0.0636550
$736$	0	0
$737$	10.3325	0.380603
$738$	0	0
$739$	51.2157	1.88400	0.942000	−	0.335614i	$-0.108944\pi$
$739$	51.2157	1.88400	0.942000	+	0.335614i	$0.108944\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−23.6367	−0.867148	−0.433574	−	0.901118i	$-0.642748\pi$
$743$	−23.6367	−0.867148	−0.433574	+	0.901118i	$0.642748\pi$
$744$	0	0
$745$	−29.9115	−1.09587
$746$	0	0
$747$	−1.30977	−0.0479218
$748$	0	0
$749$	9.76845	0.356931
$750$	0	0
$751$	−50.3119	−1.83591	−0.917954	−	0.396686i	$-0.870160\pi$
$751$	−50.3119	−1.83591	−0.917954	+	0.396686i	$0.870160\pi$
$752$	0	0
$753$	−11.1861	−0.407645
$754$	0	0
$755$	−33.4353	−1.21684
$756$	0	0
$757$	33.7267	1.22582	0.612909	−	0.790153i	$-0.289999\pi$
$757$	33.7267	1.22582	0.612909	+	0.790153i	$0.289999\pi$
$758$	0	0
$759$	9.50990	0.345188
$760$	0	0
$761$	−27.8570	−1.00982	−0.504908	−	0.863173i	$-0.668474\pi$
$761$	−27.8570	−1.00982	−0.504908	+	0.863173i	$0.668474\pi$
$762$	0	0
$763$	12.9827	0.470005
$764$	0	0
$765$	−11.3609	−0.410754
$766$	0	0
$767$	0	0
$768$	0	0
$769$	45.9195	1.65590	0.827950	−	0.560801i	$-0.189507\pi$
$769$	45.9195	1.65590	0.827950	+	0.560801i	$0.189507\pi$
$770$	0	0
$771$	−19.9523	−0.718566
$772$	0	0
$773$	29.5683	1.06350	0.531748	−	0.846902i	$-0.321535\pi$
$773$	29.5683	1.06350	0.531748	+	0.846902i	$0.321535\pi$
$774$	0	0
$775$	11.1172	0.399343
$776$	0	0
$777$	2.55097	0.0915155
$778$	0	0
$779$	−18.6266	−0.667367
$780$	0	0
$781$	27.2466	0.974962
$782$	0	0
$783$	29.7048	1.06156
$784$	0	0
$785$	21.4392	0.765197
$786$	0	0
$787$	−26.5408	−0.946076	−0.473038	−	0.881042i	$-0.656843\pi$
$787$	−26.5408	−0.946076	−0.473038	+	0.881042i	$0.656843\pi$
$788$	0	0
$789$	−9.02604	−0.321335
$790$	0	0
$791$	11.0423	0.392619
$792$	0	0
$793$	0	0
$794$	0	0
$795$	2.14630	0.0761214
$796$	0	0
$797$	−50.4099	−1.78561	−0.892806	−	0.450442i	$-0.851267\pi$
$797$	−50.4099	−1.78561	−0.892806	+	0.450442i	$0.851267\pi$
$798$	0	0
$799$	−49.7958	−1.76165
$800$	0	0
$801$	−15.6278	−0.552183
$802$	0	0
$803$	16.3294	0.576250
$804$	0	0
$805$	4.30724	0.151810
$806$	0	0
$807$	−9.56205	−0.336600
$808$	0	0
$809$	11.1822	0.393146	0.196573	−	0.980489i	$-0.437019\pi$
$809$	11.1822	0.393146	0.196573	+	0.980489i	$0.437019\pi$
$810$	0	0
$811$	−22.8117	−0.801027	−0.400513	−	0.916291i	$-0.631168\pi$
$811$	−22.8117	−0.801027	−0.400513	+	0.916291i	$0.631168\pi$
$812$	0	0
$813$	2.47539	0.0868157
$814$	0	0
$815$	−1.25937	−0.0441137
$816$	0	0
$817$	22.8896	0.800806
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−45.7200	−1.59564	−0.797819	−	0.602897i	$-0.794013\pi$
$821$	−45.7200	−1.59564	−0.797819	+	0.602897i	$0.794013\pi$
$822$	0	0
$823$	−48.8445	−1.70261	−0.851306	−	0.524669i	$-0.824189\pi$
$823$	−48.8445	−1.70261	−0.851306	+	0.524669i	$0.824189\pi$
$824$	0	0
$825$	9.22706	0.321245
$826$	0	0
$827$	39.2720	1.36562	0.682810	−	0.730596i	$-0.260757\pi$
$827$	39.2720	1.36562	0.682810	+	0.730596i	$0.260757\pi$
$828$	0	0
$829$	37.8209	1.31358	0.656788	−	0.754075i	$-0.271915\pi$
$829$	37.8209	1.31358	0.656788	+	0.754075i	$0.271915\pi$
$830$	0	0
$831$	−7.23514	−0.250984
$832$	0	0
$833$	4.79948	0.166292
$834$	0	0
$835$	7.68344	0.265897
$836$	0	0
$837$	21.1843	0.732236
$838$	0	0
$839$	−43.5636	−1.50398	−0.751991	−	0.659174i	$-0.770906\pi$
$839$	−43.5636	−1.50398	−0.751991	+	0.659174i	$0.770906\pi$
$840$	0	0
$841$	0.908780	0.0313372
$842$	0	0
$843$	−20.6959	−0.712804
$844$	0	0
$845$	0	0
$846$	0	0
$847$	3.43683	0.118091
$848$	0	0
$849$	0.411389	0.0141188
$850$	0	0
$851$	6.36690	0.218254
$852$	0	0
$853$	−3.80505	−0.130282	−0.0651412	−	0.997876i	$-0.520750\pi$
$853$	−3.80505	−0.130282	−0.0651412	+	0.997876i	$0.520750\pi$
$854$	0	0
$855$	−4.66743	−0.159623
$856$	0	0
$857$	−48.9717	−1.67284	−0.836420	−	0.548089i	$-0.815355\pi$
$857$	−48.9717	−1.67284	−0.836420	+	0.548089i	$0.815355\pi$
$858$	0	0
$859$	11.2339	0.383296	0.191648	−	0.981464i	$-0.438617\pi$
$859$	11.2339	0.383296	0.191648	+	0.981464i	$0.438617\pi$
$860$	0	0
$861$	11.1192	0.378942
$862$	0	0
$863$	21.5744	0.734400	0.367200	−	0.930142i	$-0.380316\pi$
$863$	21.5744	0.734400	0.367200	+	0.930142i	$0.380316\pi$
$864$	0	0
$865$	28.3929	0.965387
$866$	0	0
$867$	7.10359	0.241251
$868$	0	0
$869$	35.3931	1.20063
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−15.3611	−0.519896
$874$	0	0
$875$	11.5098	0.389104
$876$	0	0
$877$	−20.1323	−0.679821	−0.339911	−	0.940458i	$-0.610397\pi$
$877$	−20.1323	−0.679821	−0.339911	+	0.940458i	$0.610397\pi$
$878$	0	0
$879$	1.98579	0.0669790
$880$	0	0
$881$	22.4408	0.756050	0.378025	−	0.925795i	$-0.376603\pi$
$881$	22.4408	0.756050	0.378025	+	0.925795i	$0.376603\pi$
$882$	0	0
$883$	−28.8016	−0.969251	−0.484625	−	0.874722i	$-0.661044\pi$
$883$	−28.8016	−0.969251	−0.484625	+	0.874722i	$0.661044\pi$
$884$	0	0
$885$	7.88242	0.264965
$886$	0	0
$887$	25.0780	0.842037	0.421018	−	0.907052i	$-0.361673\pi$
$887$	25.0780	0.842037	0.421018	+	0.907052i	$0.361673\pi$
$888$	0	0
$889$	15.6532	0.524990
$890$	0	0
$891$	4.26211	0.142786
$892$	0	0
$893$	−20.4578	−0.684594
$894$	0	0
$895$	11.6108	0.388106
$896$	0	0
$897$	0	0
$898$	0	0
$899$	21.3298	0.711388
$900$	0	0
$901$	5.96909	0.198859
$902$	0	0
$903$	−13.6640	−0.454711
$904$	0	0
$905$	27.1923	0.903903
$906$	0	0
$907$	1.14981	0.0381788	0.0190894	−	0.999818i	$-0.493923\pi$
$907$	1.14981	0.0381788	0.0190894	+	0.999818i	$0.493923\pi$
$908$	0	0
$909$	−16.7365	−0.555116
$910$	0	0
$911$	−12.7197	−0.421423	−0.210712	−	0.977548i	$-0.567578\pi$
$911$	−12.7197	−0.421423	−0.210712	+	0.977548i	$0.567578\pi$
$912$	0	0
$913$	−2.23102	−0.0738360
$914$	0	0
$915$	14.4928	0.479117
$916$	0	0
$917$	3.84949	0.127121
$918$	0	0
$919$	44.9476	1.48268	0.741342	−	0.671127i	$-0.234190\pi$
$919$	44.9476	1.48268	0.741342	+	0.671127i	$0.234190\pi$
$920$	0	0
$921$	−18.6977	−0.616111
$922$	0	0
$923$	0	0
$924$	0	0
$925$	6.17753	0.203116
$926$	0	0
$927$	−12.6219	−0.414559
$928$	0	0
$929$	25.6144	0.840381	0.420190	−	0.907436i	$-0.361963\pi$
$929$	25.6144	0.840381	0.420190	+	0.907436i	$0.361963\pi$
$930$	0	0
$931$	1.97179	0.0646227
$932$	0	0
$933$	6.74957	0.220971
$934$	0	0
$935$	−19.3518	−0.632872
$936$	0	0
$937$	−33.0018	−1.07812	−0.539061	−	0.842267i	$-0.681221\pi$
$937$	−33.0018	−1.07812	−0.539061	+	0.842267i	$0.681221\pi$
$938$	0	0
$939$	12.2063	0.398339
$940$	0	0
$941$	−34.2701	−1.11717	−0.558587	−	0.829446i	$-0.688656\pi$
$941$	−34.2701	−1.11717	−0.558587	+	0.829446i	$0.688656\pi$
$942$	0	0
$943$	27.7522	0.903734
$944$	0	0
$945$	7.96348	0.259052
$946$	0	0
$947$	26.7901	0.870560	0.435280	−	0.900295i	$-0.356649\pi$
$947$	26.7901	0.870560	0.435280	+	0.900295i	$0.356649\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	23.6315	0.766303
$952$	0	0
$953$	59.5224	1.92812	0.964061	−	0.265682i	$-0.0855973\pi$
$953$	59.5224	1.92812	0.964061	+	0.265682i	$0.0855973\pi$
$954$	0	0
$955$	23.4230	0.757949
$956$	0	0
$957$	17.7032	0.572264
$958$	0	0
$959$	6.34437	0.204870
$960$	0	0
$961$	−15.7884	−0.509304
$962$	0	0
$963$	15.7713	0.508223
$964$	0	0
$965$	−19.8376	−0.638594
$966$	0	0
$967$	−22.3740	−0.719501	−0.359750	−	0.933049i	$-0.617138\pi$
$967$	−22.3740	−0.719501	−0.359750	+	0.933049i	$0.617138\pi$
$968$	0	0
$969$	11.1392	0.357843
$970$	0	0
$971$	−24.0032	−0.770301	−0.385150	−	0.922854i	$-0.625850\pi$
$971$	−24.0032	−0.770301	−0.385150	+	0.922854i	$0.625850\pi$
$972$	0	0
$973$	10.2971	0.330109
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.7043	0.854347	0.427174	−	0.904170i	$-0.359509\pi$
$977$	26.7043	0.854347	0.427174	+	0.904170i	$0.359509\pi$
$978$	0	0
$979$	−26.6200	−0.850780
$980$	0	0
$981$	20.9608	0.669226
$982$	0	0
$983$	−25.3231	−0.807682	−0.403841	−	0.914829i	$-0.632325\pi$
$983$	−25.3231	−0.807682	−0.403841	+	0.914829i	$0.632325\pi$
$984$	0	0
$985$	−1.87006	−0.0595850
$986$	0	0
$987$	12.2124	0.388724
$988$	0	0
$989$	−34.1037	−1.08443
$990$	0	0
$991$	−14.2634	−0.453092	−0.226546	−	0.974000i	$-0.572743\pi$
$991$	−14.2634	−0.453092	−0.226546	+	0.974000i	$0.572743\pi$
$992$	0	0
$993$	−26.7170	−0.847839
$994$	0	0
$995$	−2.48833	−0.0788853
$996$	0	0
$997$	−28.9565	−0.917062	−0.458531	−	0.888678i	$-0.651624\pi$
$997$	−28.9565	−0.917062	−0.458531	+	0.888678i	$0.651624\pi$
$998$	0	0
$999$	11.7715	0.372434

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.s.1.5		8
13.2	odd	12		364.2.u.a.225.4	✓	16
13.5	odd	4		4732.2.g.k.337.9		16
13.7	odd	12		364.2.u.a.309.4	yes	16
13.8	odd	4		4732.2.g.k.337.10		16
13.12	even	2		4732.2.a.t.1.5		8
39.2	even	12		3276.2.cf.c.2773.6		16
39.20	even	12		3276.2.cf.c.1765.3		16
52.7	even	12		1456.2.cc.f.673.5		16
52.15	even	12		1456.2.cc.f.225.5		16
91.2	odd	12		2548.2.bb.d.1733.4		16
91.20	even	12		2548.2.u.c.1765.5		16
91.33	even	12		2548.2.bq.c.361.4		16
91.41	even	12		2548.2.u.c.589.5		16
91.46	odd	12		2548.2.bb.d.569.4		16
91.54	even	12		2548.2.bb.c.1733.5		16
91.59	even	12		2548.2.bb.c.569.5		16
91.67	odd	12		2548.2.bq.e.1941.5		16
91.72	odd	12		2548.2.bq.e.361.5		16
91.80	even	12		2548.2.bq.c.1941.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.u.a.225.4	✓	16	13.2	odd	12
364.2.u.a.309.4	yes	16	13.7	odd	12
1456.2.cc.f.225.5		16	52.15	even	12
1456.2.cc.f.673.5		16	52.7	even	12
2548.2.u.c.589.5		16	91.41	even	12
2548.2.u.c.1765.5		16	91.20	even	12
2548.2.bb.c.569.5		16	91.59	even	12
2548.2.bb.c.1733.5		16	91.54	even	12
2548.2.bb.d.569.4		16	91.46	odd	12
2548.2.bb.d.1733.4		16	91.2	odd	12
2548.2.bq.c.361.4		16	91.33	even	12
2548.2.bq.c.1941.4		16	91.80	even	12
2548.2.bq.e.361.5		16	91.72	odd	12
2548.2.bq.e.1941.5		16	91.67	odd	12
3276.2.cf.c.1765.3		16	39.20	even	12
3276.2.cf.c.2773.6		16	39.2	even	12
4732.2.a.s.1.5		8	1.1	even	1	trivial
4732.2.a.t.1.5		8	13.12	even	2
4732.2.g.k.337.9		16	13.5	odd	4
4732.2.g.k.337.10		16	13.8	odd	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:	\( N \)	\(=\)	\( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4732.a (trivial)

\(f(q)\)	\(=\)	\(q+1.17707 q^{3} +1.46614 q^{5} -1.00000 q^{7} -1.61452 q^{9} -2.75012 q^{11} +1.72574 q^{15} +4.79948 q^{17} +1.97179 q^{19} -1.17707 q^{21} -2.93781 q^{23} -2.85043 q^{25} -5.43159 q^{27} -5.46889 q^{29} -3.90020 q^{31} -3.23707 q^{33} -1.46614 q^{35} -2.16723 q^{37} -9.44655 q^{41} +11.6086 q^{43} -2.36711 q^{45} -10.3753 q^{47} +1.00000 q^{49} +5.64930 q^{51} +1.24369 q^{53} -4.03207 q^{55} +2.32092 q^{57} +4.56755 q^{59} +8.39800 q^{61} +1.61452 q^{63} -3.75711 q^{67} -3.45799 q^{69} -9.90743 q^{71} -5.93768 q^{73} -3.35514 q^{75} +2.75012 q^{77} -12.8697 q^{79} -1.54979 q^{81} +0.811244 q^{83} +7.03671 q^{85} -6.43725 q^{87} +9.67958 q^{89} -4.59079 q^{93} +2.89092 q^{95} +9.51440 q^{97} +4.44012 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} - 8 q^{7} + 14 q^{9} - 12 q^{11} - 12 q^{15} + 2 q^{17} - 6 q^{19} + 22 q^{25} - 6 q^{27} + 22 q^{29} - 14 q^{31} + 28 q^{33} + 6 q^{35} - 12 q^{37} - 4 q^{41} + 6 q^{43} - 20 q^{45} - 42 q^{47}+ \cdots - 80 q^{99}+O(q^{100}) \) 8 * q - 6 * q^5 - 8 * q^7 + 14 * q^9 - 12 * q^11 - 12 * q^15 + 2 * q^17 - 6 * q^19 + 22 * q^25 - 6 * q^27 + 22 * q^29 - 14 * q^31 + 28 * q^33 + 6 * q^35 - 12 * q^37 - 4 * q^41 + 6 * q^43 - 20 * q^45 - 42 * q^47 + 8 * q^49 + 2 * q^51 + 4 * q^53 - 2 * q^55 + 8 * q^57 - 2 * q^59 - 4 * q^61 - 14 * q^63 - 24 * q^67 - 52 * q^69 - 28 * q^71 - 28 * q^73 - 10 * q^75 + 12 * q^77 + 4 * q^79 - 38 * q^83 + 36 * q^85 - 26 * q^87 + 22 * q^89 - 56 * q^93 - 30 * q^95 + 4 * q^97 - 80 * q^99

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