LMFDB - Embedded newform 4732.2.a.s.1.6

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			1.6
Root			$-1.75101$ 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)

$f(q)$	$=$	$q+1.75101 q^{3} +1.38536 q^{5} -1.00000 q^{7} +0.0660200 q^{9} +1.15290 q^{11} +2.42578 q^{15} -1.56249 q^{17} -8.38917 q^{19} -1.75101 q^{21} -2.22134 q^{23} -3.08077 q^{25} -5.13741 q^{27} +2.86981 q^{29} -6.65024 q^{31} +2.01874 q^{33} -1.38536 q^{35} +3.48068 q^{37} -12.5583 q^{41} -6.15682 q^{43} +0.0914616 q^{45} -7.40432 q^{47} +1.00000 q^{49} -2.73593 q^{51} +11.2301 q^{53} +1.59719 q^{55} -14.6895 q^{57} -8.05559 q^{59} +1.65841 q^{61} -0.0660200 q^{63} +3.71128 q^{67} -3.88958 q^{69} +9.93663 q^{71} -3.30200 q^{73} -5.39444 q^{75} -1.15290 q^{77} +4.64811 q^{79} -9.19370 q^{81} -1.82551 q^{83} -2.16462 q^{85} +5.02506 q^{87} +2.37681 q^{89} -11.6446 q^{93} -11.6220 q^{95} -7.54618 q^{97} +0.0761145 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.75101	1.01094	0.505472	−	0.862843i	$-0.331318\pi$
$3$	1.75101	1.01094	0.505472	+	0.862843i	$0.331318\pi$
$4$	0	0
$5$	1.38536	0.619553	0.309777	−	0.950809i	$-0.399746\pi$
$5$	1.38536	0.619553	0.309777	+	0.950809i	$0.399746\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	0.0660200	0.0220067
$10$	0	0
$11$	1.15290	0.347613	0.173806	−	0.984780i	$-0.444393\pi$
$11$	1.15290	0.347613	0.173806	+	0.984780i	$0.444393\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.42578	0.626333
$16$	0	0
$17$	−1.56249	−0.378960	−0.189480	−	0.981885i	$-0.560680\pi$
$17$	−1.56249	−0.378960	−0.189480	+	0.981885i	$0.560680\pi$
$18$	0	0
$19$	−8.38917	−1.92461	−0.962303	−	0.271978i	$-0.912322\pi$
$19$	−8.38917	−1.92461	−0.962303	+	0.271978i	$0.912322\pi$
$20$	0	0
$21$	−1.75101	−0.382101
$22$	0	0
$23$	−2.22134	−0.463182	−0.231591	−	0.972813i	$-0.574393\pi$
$23$	−2.22134	−0.463182	−0.231591	+	0.972813i	$0.574393\pi$
$24$	0	0
$25$	−3.08077	−0.616154
$26$	0	0
$27$	−5.13741	−0.988696
$28$	0	0
$29$	2.86981	0.532911	0.266455	−	0.963847i	$-0.414148\pi$
$29$	2.86981	0.532911	0.266455	+	0.963847i	$0.414148\pi$
$30$	0	0
$31$	−6.65024	−1.19442	−0.597209	−	0.802086i	$-0.703724\pi$
$31$	−6.65024	−1.19442	−0.597209	+	0.802086i	$0.703724\pi$
$32$	0	0
$33$	2.01874	0.351417
$34$	0	0
$35$	−1.38536	−0.234169
$36$	0	0
$37$	3.48068	0.572220	0.286110	−	0.958197i	$-0.407638\pi$
$37$	3.48068	0.572220	0.286110	+	0.958197i	$0.407638\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−12.5583	−1.96128	−0.980641	−	0.195814i	$-0.937265\pi$
$41$	−12.5583	−1.96128	−0.980641	+	0.195814i	$0.937265\pi$
$42$	0	0
$43$	−6.15682	−0.938906	−0.469453	−	0.882958i	$-0.655549\pi$
$43$	−6.15682	−0.938906	−0.469453	+	0.882958i	$0.655549\pi$
$44$	0	0
$45$	0.0914616	0.0136343
$46$	0	0
$47$	−7.40432	−1.08003	−0.540015	−	0.841655i	$-0.681582\pi$
$47$	−7.40432	−1.08003	−0.540015	+	0.841655i	$0.681582\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.73593	−0.383107
$52$	0	0
$53$	11.2301	1.54257	0.771285	−	0.636490i	$-0.219614\pi$
$53$	11.2301	1.54257	0.771285	+	0.636490i	$0.219614\pi$
$54$	0	0
$55$	1.59719	0.215365
$56$	0	0
$57$	−14.6895	−1.94567
$58$	0	0
$59$	−8.05559	−1.04875	−0.524374	−	0.851488i	$-0.675701\pi$
$59$	−8.05559	−1.04875	−0.524374	+	0.851488i	$0.675701\pi$
$60$	0	0
$61$	1.65841	0.212338	0.106169	−	0.994348i	$-0.466142\pi$
$61$	1.65841	0.212338	0.106169	+	0.994348i	$0.466142\pi$
$62$	0	0
$63$	−0.0660200	−0.00831774
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.71128	0.453405	0.226702	−	0.973964i	$-0.427206\pi$
$67$	3.71128	0.453405	0.226702	+	0.973964i	$0.427206\pi$
$68$	0	0
$69$	−3.88958	−0.468251
$70$	0	0
$71$	9.93663	1.17926	0.589631	−	0.807673i	$-0.299273\pi$
$71$	9.93663	1.17926	0.589631	+	0.807673i	$0.299273\pi$
$72$	0	0
$73$	−3.30200	−0.386470	−0.193235	−	0.981153i	$-0.561898\pi$
$73$	−3.30200	−0.386470	−0.193235	+	0.981153i	$0.561898\pi$
$74$	0	0
$75$	−5.39444	−0.622897
$76$	0	0
$77$	−1.15290	−0.131385
$78$	0	0
$79$	4.64811	0.522953	0.261477	−	0.965210i	$-0.415791\pi$
$79$	4.64811	0.522953	0.261477	+	0.965210i	$0.415791\pi$
$80$	0	0
$81$	−9.19370	−1.02152
$82$	0	0
$83$	−1.82551	−0.200376	−0.100188	−	0.994969i	$-0.531944\pi$
$83$	−1.82551	−0.200376	−0.100188	+	0.994969i	$0.531944\pi$
$84$	0	0
$85$	−2.16462	−0.234786
$86$	0	0
$87$	5.02506	0.538743
$88$	0	0
$89$	2.37681	0.251941	0.125970	−	0.992034i	$-0.459796\pi$
$89$	2.37681	0.251941	0.125970	+	0.992034i	$0.459796\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−11.6446	−1.20749
$94$	0	0
$95$	−11.6220	−1.19240
$96$	0	0
$97$	−7.54618	−0.766199	−0.383099	−	0.923707i	$-0.625143\pi$
$97$	−7.54618	−0.766199	−0.383099	+	0.923707i	$0.625143\pi$
$98$	0	0
$99$	0.0761145	0.00764980
$100$	0	0
$101$	13.2005	1.31350	0.656751	−	0.754108i	$-0.271930\pi$
$101$	13.2005	1.31350	0.656751	+	0.754108i	$0.271930\pi$
$102$	0	0
$103$	−5.90120	−0.581463	−0.290731	−	0.956805i	$-0.593899\pi$
$103$	−5.90120	−0.581463	−0.290731	+	0.956805i	$0.593899\pi$
$104$	0	0
$105$	−2.42578	−0.236732
$106$	0	0
$107$	10.1927	0.985363	0.492682	−	0.870210i	$-0.336017\pi$
$107$	10.1927	0.985363	0.492682	+	0.870210i	$0.336017\pi$
$108$	0	0
$109$	6.14856	0.588925	0.294463	−	0.955663i	$-0.404859\pi$
$109$	6.14856	0.588925	0.294463	+	0.955663i	$0.404859\pi$
$110$	0	0
$111$	6.09469	0.578482
$112$	0	0
$113$	14.5975	1.37322	0.686609	−	0.727027i	$-0.259098\pi$
$113$	14.5975	1.37322	0.686609	+	0.727027i	$0.259098\pi$
$114$	0	0
$115$	−3.07737	−0.286966
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.56249	0.143233
$120$	0	0
$121$	−9.67082	−0.879165
$122$	0	0
$123$	−21.9897	−1.98275
$124$	0	0
$125$	−11.1948	−1.00129
$126$	0	0
$127$	8.72229	0.773978	0.386989	−	0.922084i	$-0.373515\pi$
$127$	8.72229	0.773978	0.386989	+	0.922084i	$0.373515\pi$
$128$	0	0
$129$	−10.7806	−0.949180
$130$	0	0
$131$	19.4433	1.69877	0.849385	−	0.527773i	$-0.176973\pi$
$131$	19.4433	1.69877	0.849385	+	0.527773i	$0.176973\pi$
$132$	0	0
$133$	8.38917	0.727433
$134$	0	0
$135$	−7.11718	−0.612550
$136$	0	0
$137$	−3.35460	−0.286602	−0.143301	−	0.989679i	$-0.545772\pi$
$137$	−3.35460	−0.286602	−0.143301	+	0.989679i	$0.545772\pi$
$138$	0	0
$139$	−7.91259	−0.671137	−0.335568	−	0.942016i	$-0.608928\pi$
$139$	−7.91259	−0.671137	−0.335568	+	0.942016i	$0.608928\pi$
$140$	0	0
$141$	−12.9650	−1.09185
$142$	0	0
$143$	0	0
$144$	0	0
$145$	3.97573	0.330167
$146$	0	0
$147$	1.75101	0.144420
$148$	0	0
$149$	2.36432	0.193693	0.0968464	−	0.995299i	$-0.469124\pi$
$149$	2.36432	0.193693	0.0968464	+	0.995299i	$0.469124\pi$
$150$	0	0
$151$	−11.5883	−0.943041	−0.471520	−	0.881855i	$-0.656295\pi$
$151$	−11.5883	−0.943041	−0.471520	+	0.881855i	$0.656295\pi$
$152$	0	0
$153$	−0.103156	−0.00833965
$154$	0	0
$155$	−9.21299	−0.740005
$156$	0	0
$157$	−9.03642	−0.721184	−0.360592	−	0.932724i	$-0.617425\pi$
$157$	−9.03642	−0.721184	−0.360592	+	0.932724i	$0.617425\pi$
$158$	0	0
$159$	19.6639	1.55945
$160$	0	0
$161$	2.22134	0.175066
$162$	0	0
$163$	21.3556	1.67270	0.836351	−	0.548195i	$-0.184685\pi$
$163$	21.3556	1.67270	0.836351	+	0.548195i	$0.184685\pi$
$164$	0	0
$165$	2.79668	0.217721
$166$	0	0
$167$	−3.18129	−0.246176	−0.123088	−	0.992396i	$-0.539280\pi$
$167$	−3.18129	−0.246176	−0.123088	+	0.992396i	$0.539280\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−0.553853	−0.0423542
$172$	0	0
$173$	5.33904	0.405920	0.202960	−	0.979187i	$-0.434944\pi$
$173$	5.33904	0.405920	0.202960	+	0.979187i	$0.434944\pi$
$174$	0	0
$175$	3.08077	0.232884
$176$	0	0
$177$	−14.1054	−1.06023
$178$	0	0
$179$	9.84558	0.735893	0.367947	−	0.929847i	$-0.380061\pi$
$179$	9.84558	0.735893	0.367947	+	0.929847i	$0.380061\pi$
$180$	0	0
$181$	−15.1253	−1.12425	−0.562127	−	0.827051i	$-0.690017\pi$
$181$	−15.1253	−1.12425	−0.562127	+	0.827051i	$0.690017\pi$
$182$	0	0
$183$	2.90389	0.214662
$184$	0	0
$185$	4.82200	0.354521
$186$	0	0
$187$	−1.80140	−0.131731
$188$	0	0
$189$	5.13741	0.373692
$190$	0	0
$191$	−22.1800	−1.60489	−0.802443	−	0.596729i	$-0.796467\pi$
$191$	−22.1800	−1.60489	−0.802443	+	0.596729i	$0.796467\pi$
$192$	0	0
$193$	−6.97186	−0.501846	−0.250923	−	0.968007i	$-0.580734\pi$
$193$	−6.97186	−0.501846	−0.250923	+	0.968007i	$0.580734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.7332	−1.40593	−0.702966	−	0.711223i	$-0.748141\pi$
$197$	−19.7332	−1.40593	−0.702966	+	0.711223i	$0.748141\pi$
$198$	0	0
$199$	−13.8039	−0.978529	−0.489265	−	0.872135i	$-0.662735\pi$
$199$	−13.8039	−0.978529	−0.489265	+	0.872135i	$0.662735\pi$
$200$	0	0
$201$	6.49847	0.458367
$202$	0	0
$203$	−2.86981	−0.201421
$204$	0	0
$205$	−17.3978	−1.21512
$206$	0	0
$207$	−0.146653	−0.0101931
$208$	0	0
$209$	−9.67188	−0.669018
$210$	0	0
$211$	−9.47437	−0.652242	−0.326121	−	0.945328i	$-0.605742\pi$
$211$	−9.47437	−0.652242	−0.326121	+	0.945328i	$0.605742\pi$
$212$	0	0
$213$	17.3991	1.19217
$214$	0	0
$215$	−8.52942	−0.581702
$216$	0	0
$217$	6.65024	0.451448
$218$	0	0
$219$	−5.78182	−0.390699
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−14.5879	−0.976878	−0.488439	−	0.872598i	$-0.662434\pi$
$223$	−14.5879	−0.976878	−0.488439	+	0.872598i	$0.662434\pi$
$224$	0	0
$225$	−0.203392	−0.0135595
$226$	0	0
$227$	−7.16353	−0.475460	−0.237730	−	0.971331i	$-0.576403\pi$
$227$	−7.16353	−0.475460	−0.237730	+	0.971331i	$0.576403\pi$
$228$	0	0
$229$	−12.3005	−0.812839	−0.406420	−	0.913687i	$-0.633223\pi$
$229$	−12.3005	−0.812839	−0.406420	+	0.913687i	$0.633223\pi$
$230$	0	0
$231$	−2.01874	−0.132823
$232$	0	0
$233$	18.8037	1.23187	0.615936	−	0.787796i	$-0.288778\pi$
$233$	18.8037	1.23187	0.615936	+	0.787796i	$0.288778\pi$
$234$	0	0
$235$	−10.2577	−0.669136
$236$	0	0
$237$	8.13886	0.528676
$238$	0	0
$239$	1.60295	0.103686	0.0518432	−	0.998655i	$-0.483490\pi$
$239$	1.60295	0.103686	0.0518432	+	0.998655i	$0.483490\pi$
$240$	0	0
$241$	18.3657	1.18304	0.591519	−	0.806291i	$-0.298529\pi$
$241$	18.3657	1.18304	0.591519	+	0.806291i	$0.298529\pi$
$242$	0	0
$243$	−0.685976	−0.0440054
$244$	0	0
$245$	1.38536	0.0885076
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.19649	−0.202569
$250$	0	0
$251$	−14.3646	−0.906683	−0.453341	−	0.891337i	$-0.649768\pi$
$251$	−14.3646	−0.906683	−0.453341	+	0.891337i	$0.649768\pi$
$252$	0	0
$253$	−2.56099	−0.161008
$254$	0	0
$255$	−3.79026	−0.237355
$256$	0	0
$257$	17.3657	1.08324	0.541620	−	0.840624i	$-0.317811\pi$
$257$	17.3657	1.08324	0.541620	+	0.840624i	$0.317811\pi$
$258$	0	0
$259$	−3.48068	−0.216279
$260$	0	0
$261$	0.189465	0.0117276
$262$	0	0
$263$	16.7419	1.03235	0.516175	−	0.856483i	$-0.327355\pi$
$263$	16.7419	1.03235	0.516175	+	0.856483i	$0.327355\pi$
$264$	0	0
$265$	15.5577	0.955704
$266$	0	0
$267$	4.16180	0.254698
$268$	0	0
$269$	24.7915	1.51156	0.755781	−	0.654824i	$-0.227257\pi$
$269$	24.7915	1.51156	0.755781	+	0.654824i	$0.227257\pi$
$270$	0	0
$271$	−20.2317	−1.22899	−0.614493	−	0.788922i	$-0.710639\pi$
$271$	−20.2317	−1.22899	−0.614493	+	0.788922i	$0.710639\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.55182	−0.214183
$276$	0	0
$277$	11.0533	0.664127	0.332064	−	0.943257i	$-0.392255\pi$
$277$	11.0533	0.664127	0.332064	+	0.943257i	$0.392255\pi$
$278$	0	0
$279$	−0.439049	−0.0262852
$280$	0	0
$281$	3.18955	0.190273	0.0951363	−	0.995464i	$-0.469671\pi$
$281$	3.18955	0.190273	0.0951363	+	0.995464i	$0.469671\pi$
$282$	0	0
$283$	17.9296	1.06581	0.532903	−	0.846177i	$-0.321101\pi$
$283$	17.9296	1.06581	0.532903	+	0.846177i	$0.321101\pi$
$284$	0	0
$285$	−20.3503	−1.20545
$286$	0	0
$287$	12.5583	0.741295
$288$	0	0
$289$	−14.5586	−0.856389
$290$	0	0
$291$	−13.2134	−0.774584
$292$	0	0
$293$	16.9677	0.991264	0.495632	−	0.868533i	$-0.334936\pi$
$293$	16.9677	0.991264	0.495632	+	0.868533i	$0.334936\pi$
$294$	0	0
$295$	−11.1599	−0.649755
$296$	0	0
$297$	−5.92293	−0.343683
$298$	0	0
$299$	0	0
$300$	0	0
$301$	6.15682	0.354873
$302$	0	0
$303$	23.1142	1.32788
$304$	0	0
$305$	2.29750	0.131555
$306$	0	0
$307$	26.9884	1.54031	0.770155	−	0.637857i	$-0.220179\pi$
$307$	26.9884	1.54031	0.770155	+	0.637857i	$0.220179\pi$
$308$	0	0
$309$	−10.3330	−0.587826
$310$	0	0
$311$	−28.3781	−1.60918	−0.804588	−	0.593834i	$-0.797614\pi$
$311$	−28.3781	−1.60918	−0.804588	+	0.593834i	$0.797614\pi$
$312$	0	0
$313$	−16.6674	−0.942095	−0.471047	−	0.882108i	$-0.656124\pi$
$313$	−16.6674	−0.942095	−0.471047	+	0.882108i	$0.656124\pi$
$314$	0	0
$315$	−0.0914616	−0.00515328
$316$	0	0
$317$	−28.7427	−1.61435	−0.807175	−	0.590312i	$-0.799005\pi$
$317$	−28.7427	−1.61435	−0.807175	+	0.590312i	$0.799005\pi$
$318$	0	0
$319$	3.30861	0.185247
$320$	0	0
$321$	17.8474	0.996147
$322$	0	0
$323$	13.1080	0.729349
$324$	0	0
$325$	0	0
$326$	0	0
$327$	10.7662	0.595370
$328$	0	0
$329$	7.40432	0.408213
$330$	0	0
$331$	26.1917	1.43963	0.719814	−	0.694167i	$-0.244227\pi$
$331$	26.1917	1.43963	0.719814	+	0.694167i	$0.244227\pi$
$332$	0	0
$333$	0.229794	0.0125927
$334$	0	0
$335$	5.14147	0.280908
$336$	0	0
$337$	−4.47831	−0.243949	−0.121975	−	0.992533i	$-0.538923\pi$
$337$	−4.47831	−0.243949	−0.121975	+	0.992533i	$0.538923\pi$
$338$	0	0
$339$	25.5603	1.38825
$340$	0	0
$341$	−7.66707	−0.415195
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−5.38849	−0.290106
$346$	0	0
$347$	−26.9212	−1.44520	−0.722602	−	0.691264i	$-0.757054\pi$
$347$	−26.9212	−1.44520	−0.722602	+	0.691264i	$0.757054\pi$
$348$	0	0
$349$	20.7338	1.10986	0.554928	−	0.831898i	$-0.312746\pi$
$349$	20.7338	1.10986	0.554928	+	0.831898i	$0.312746\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	11.3041	0.601655	0.300828	−	0.953679i	$-0.402737\pi$
$353$	11.3041	0.601655	0.300828	+	0.953679i	$0.402737\pi$
$354$	0	0
$355$	13.7658	0.730615
$356$	0	0
$357$	2.73593	0.144801
$358$	0	0
$359$	−17.7453	−0.936559	−0.468279	−	0.883580i	$-0.655126\pi$
$359$	−17.7453	−0.936559	−0.468279	+	0.883580i	$0.655126\pi$
$360$	0	0
$361$	51.3781	2.70411
$362$	0	0
$363$	−16.9337	−0.888786
$364$	0	0
$365$	−4.57447	−0.239439
$366$	0	0
$367$	9.56310	0.499190	0.249595	−	0.968350i	$-0.419703\pi$
$367$	9.56310	0.499190	0.249595	+	0.968350i	$0.419703\pi$
$368$	0	0
$369$	−0.829101	−0.0431613
$370$	0	0
$371$	−11.2301	−0.583036
$372$	0	0
$373$	11.8835	0.615306	0.307653	−	0.951499i	$-0.400456\pi$
$373$	11.8835	0.615306	0.307653	+	0.951499i	$0.400456\pi$
$374$	0	0
$375$	−19.6022	−1.01225
$376$	0	0
$377$	0	0
$378$	0	0
$379$	17.2502	0.886085	0.443042	−	0.896501i	$-0.353899\pi$
$379$	17.2502	0.886085	0.443042	+	0.896501i	$0.353899\pi$
$380$	0	0
$381$	15.2728	0.782448
$382$	0	0
$383$	−13.0300	−0.665800	−0.332900	−	0.942962i	$-0.608027\pi$
$383$	−13.0300	−0.665800	−0.332900	+	0.942962i	$0.608027\pi$
$384$	0	0
$385$	−1.59719	−0.0814002
$386$	0	0
$387$	−0.406473	−0.0206622
$388$	0	0
$389$	−24.1585	−1.22488	−0.612442	−	0.790516i	$-0.709813\pi$
$389$	−24.1585	−1.22488	−0.612442	+	0.790516i	$0.709813\pi$
$390$	0	0
$391$	3.47083	0.175528
$392$	0	0
$393$	34.0454	1.71736
$394$	0	0
$395$	6.43932	0.323997
$396$	0	0
$397$	−4.81535	−0.241675	−0.120838	−	0.992672i	$-0.538558\pi$
$397$	−4.81535	−0.241675	−0.120838	+	0.992672i	$0.538558\pi$
$398$	0	0
$399$	14.6895	0.735394
$400$	0	0
$401$	−8.02173	−0.400586	−0.200293	−	0.979736i	$-0.564189\pi$
$401$	−8.02173	−0.400586	−0.200293	+	0.979736i	$0.564189\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−12.7366	−0.632887
$406$	0	0
$407$	4.01288	0.198911
$408$	0	0
$409$	−9.96745	−0.492859	−0.246429	−	0.969161i	$-0.579257\pi$
$409$	−9.96745	−0.492859	−0.246429	+	0.969161i	$0.579257\pi$
$410$	0	0
$411$	−5.87391	−0.289739
$412$	0	0
$413$	8.05559	0.396390
$414$	0	0
$415$	−2.52900	−0.124144
$416$	0	0
$417$	−13.8550	−0.678481
$418$	0	0
$419$	−29.6884	−1.45037	−0.725187	−	0.688552i	$-0.758247\pi$
$419$	−29.6884	−1.45037	−0.725187	+	0.688552i	$0.758247\pi$
$420$	0	0
$421$	13.1380	0.640305	0.320152	−	0.947366i	$-0.396266\pi$
$421$	13.1380	0.640305	0.320152	+	0.947366i	$0.396266\pi$
$422$	0	0
$423$	−0.488833	−0.0237679
$424$	0	0
$425$	4.81368	0.233498
$426$	0	0
$427$	−1.65841	−0.0802562
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.4025	0.693745	0.346872	−	0.937912i	$-0.387244\pi$
$431$	14.4025	0.693745	0.346872	+	0.937912i	$0.387244\pi$
$432$	0	0
$433$	−36.3192	−1.74539	−0.872695	−	0.488266i	$-0.837630\pi$
$433$	−36.3192	−1.74539	−0.872695	+	0.488266i	$0.837630\pi$
$434$	0	0
$435$	6.96153	0.333780
$436$	0	0
$437$	18.6352	0.891443
$438$	0	0
$439$	20.5444	0.980532	0.490266	−	0.871573i	$-0.336900\pi$
$439$	20.5444	0.980532	0.490266	+	0.871573i	$0.336900\pi$
$440$	0	0
$441$	0.0660200	0.00314381
$442$	0	0
$443$	−25.3211	−1.20304	−0.601521	−	0.798857i	$-0.705439\pi$
$443$	−25.3211	−1.20304	−0.601521	+	0.798857i	$0.705439\pi$
$444$	0	0
$445$	3.29274	0.156091
$446$	0	0
$447$	4.13994	0.195812
$448$	0	0
$449$	33.6015	1.58575	0.792876	−	0.609383i	$-0.208583\pi$
$449$	33.6015	1.58575	0.792876	+	0.609383i	$0.208583\pi$
$450$	0	0
$451$	−14.4785	−0.681767
$452$	0	0
$453$	−20.2911	−0.953361
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.9955	−1.77735	−0.888677	−	0.458534i	$-0.848375\pi$
$457$	−37.9955	−1.77735	−0.888677	+	0.458534i	$0.848375\pi$
$458$	0	0
$459$	8.02717	0.374676
$460$	0	0
$461$	−36.3513	−1.69305	−0.846525	−	0.532348i	$-0.821310\pi$
$461$	−36.3513	−1.69305	−0.846525	+	0.532348i	$0.821310\pi$
$462$	0	0
$463$	−35.3400	−1.64239	−0.821194	−	0.570649i	$-0.806692\pi$
$463$	−35.3400	−1.64239	−0.821194	+	0.570649i	$0.806692\pi$
$464$	0	0
$465$	−16.1320	−0.748104
$466$	0	0
$467$	29.5717	1.36842	0.684208	−	0.729287i	$-0.260148\pi$
$467$	29.5717	1.36842	0.684208	+	0.729287i	$0.260148\pi$
$468$	0	0
$469$	−3.71128	−0.171371
$470$	0	0
$471$	−15.8228	−0.729077
$472$	0	0
$473$	−7.09820	−0.326376
$474$	0	0
$475$	25.8451	1.18585
$476$	0	0
$477$	0.741409	0.0339468
$478$	0	0
$479$	21.7476	0.993672	0.496836	−	0.867845i	$-0.334495\pi$
$479$	21.7476	0.993672	0.496836	+	0.867845i	$0.334495\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	3.88958	0.176982
$484$	0	0
$485$	−10.4542	−0.474701
$486$	0	0
$487$	−30.0473	−1.36157	−0.680786	−	0.732482i	$-0.738362\pi$
$487$	−30.0473	−1.36157	−0.680786	+	0.732482i	$0.738362\pi$
$488$	0	0
$489$	37.3938	1.69101
$490$	0	0
$491$	−40.1797	−1.81329	−0.906643	−	0.421898i	$-0.861364\pi$
$491$	−40.1797	−1.81329	−0.906643	+	0.421898i	$0.861364\pi$
$492$	0	0
$493$	−4.48406	−0.201952
$494$	0	0
$495$	0.105446	0.00473946
$496$	0	0
$497$	−9.93663	−0.445719
$498$	0	0
$499$	−11.7400	−0.525554	−0.262777	−	0.964857i	$-0.584638\pi$
$499$	−11.7400	−0.525554	−0.262777	+	0.964857i	$0.584638\pi$
$500$	0	0
$501$	−5.57046	−0.248870
$502$	0	0
$503$	−33.3929	−1.48892	−0.744459	−	0.667669i	$-0.767292\pi$
$503$	−33.3929	−1.48892	−0.744459	+	0.667669i	$0.767292\pi$
$504$	0	0
$505$	18.2875	0.813784
$506$	0	0
$507$	0	0
$508$	0	0
$509$	20.7826	0.921172	0.460586	−	0.887615i	$-0.347639\pi$
$509$	20.7826	0.921172	0.460586	+	0.887615i	$0.347639\pi$
$510$	0	0
$511$	3.30200	0.146072
$512$	0	0
$513$	43.0986	1.90285
$514$	0	0
$515$	−8.17531	−0.360247
$516$	0	0
$517$	−8.53645	−0.375433
$518$	0	0
$519$	9.34869	0.410362
$520$	0	0
$521$	7.50239	0.328686	0.164343	−	0.986403i	$-0.447450\pi$
$521$	7.50239	0.328686	0.164343	+	0.986403i	$0.447450\pi$
$522$	0	0
$523$	7.32578	0.320334	0.160167	−	0.987090i	$-0.448797\pi$
$523$	7.32578	0.320334	0.160167	+	0.987090i	$0.448797\pi$
$524$	0	0
$525$	5.39444	0.235433
$526$	0	0
$527$	10.3909	0.452637
$528$	0	0
$529$	−18.0656	−0.785462
$530$	0	0
$531$	−0.531830	−0.0230794
$532$	0	0
$533$	0	0
$534$	0	0
$535$	14.1206	0.610485
$536$	0	0
$537$	17.2397	0.743947
$538$	0	0
$539$	1.15290	0.0496590
$540$	0	0
$541$	11.6924	0.502698	0.251349	−	0.967897i	$-0.419126\pi$
$541$	11.6924	0.502698	0.251349	+	0.967897i	$0.419126\pi$
$542$	0	0
$543$	−26.4844	−1.13656
$544$	0	0
$545$	8.51799	0.364871
$546$	0	0
$547$	5.10513	0.218280	0.109140	−	0.994026i	$-0.465190\pi$
$547$	5.10513	0.218280	0.109140	+	0.994026i	$0.465190\pi$
$548$	0	0
$549$	0.109488	0.00467285
$550$	0	0
$551$	−24.0753	−1.02564
$552$	0	0
$553$	−4.64811	−0.197658
$554$	0	0
$555$	8.44336	0.358401
$556$	0	0
$557$	12.5336	0.531065	0.265532	−	0.964102i	$-0.414452\pi$
$557$	12.5336	0.531065	0.265532	+	0.964102i	$0.414452\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−3.15426	−0.133173
$562$	0	0
$563$	3.90630	0.164631	0.0823155	−	0.996606i	$-0.473768\pi$
$563$	3.90630	0.164631	0.0823155	+	0.996606i	$0.473768\pi$
$564$	0	0
$565$	20.2229	0.850782
$566$	0	0
$567$	9.19370	0.386099
$568$	0	0
$569$	−46.1329	−1.93399	−0.966996	−	0.254793i	$-0.917993\pi$
$569$	−46.1329	−1.93399	−0.966996	+	0.254793i	$0.917993\pi$
$570$	0	0
$571$	30.1160	1.26031	0.630157	−	0.776467i	$-0.282990\pi$
$571$	30.1160	1.26031	0.630157	+	0.776467i	$0.282990\pi$
$572$	0	0
$573$	−38.8372	−1.62245
$574$	0	0
$575$	6.84345	0.285391
$576$	0	0
$577$	25.6300	1.06699	0.533496	−	0.845803i	$-0.320878\pi$
$577$	25.6300	1.06699	0.533496	+	0.845803i	$0.320878\pi$
$578$	0	0
$579$	−12.2078	−0.507338
$580$	0	0
$581$	1.82551	0.0757351
$582$	0	0
$583$	12.9472	0.536217
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−13.1805	−0.544018	−0.272009	−	0.962295i	$-0.587688\pi$
$587$	−13.1805	−0.544018	−0.272009	+	0.962295i	$0.587688\pi$
$588$	0	0
$589$	55.7900	2.29878
$590$	0	0
$591$	−34.5529	−1.42132
$592$	0	0
$593$	−6.53704	−0.268444	−0.134222	−	0.990951i	$-0.542854\pi$
$593$	−6.53704	−0.268444	−0.134222	+	0.990951i	$0.542854\pi$
$594$	0	0
$595$	2.16462	0.0887408
$596$	0	0
$597$	−24.1706	−0.989238
$598$	0	0
$599$	12.4244	0.507649	0.253824	−	0.967250i	$-0.418311\pi$
$599$	12.4244	0.507649	0.253824	+	0.967250i	$0.418311\pi$
$600$	0	0
$601$	−18.5404	−0.756278	−0.378139	−	0.925749i	$-0.623436\pi$
$601$	−18.5404	−0.756278	−0.378139	+	0.925749i	$0.623436\pi$
$602$	0	0
$603$	0.245018	0.00997792
$604$	0	0
$605$	−13.3976	−0.544690
$606$	0	0
$607$	15.7445	0.639050	0.319525	−	0.947578i	$-0.396477\pi$
$607$	15.7445	0.639050	0.319525	+	0.947578i	$0.396477\pi$
$608$	0	0
$609$	−5.02506	−0.203626
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−19.2599	−0.777901	−0.388950	−	0.921259i	$-0.627162\pi$
$613$	−19.2599	−0.777901	−0.388950	+	0.921259i	$0.627162\pi$
$614$	0	0
$615$	−30.4637	−1.22842
$616$	0	0
$617$	43.4020	1.74730	0.873649	−	0.486557i	$-0.161748\pi$
$617$	43.4020	1.74730	0.873649	+	0.486557i	$0.161748\pi$
$618$	0	0
$619$	−13.3880	−0.538109	−0.269054	−	0.963125i	$-0.586711\pi$
$619$	−13.3880	−0.538109	−0.269054	+	0.963125i	$0.586711\pi$
$620$	0	0
$621$	11.4120	0.457946
$622$	0	0
$623$	−2.37681	−0.0952247
$624$	0	0
$625$	−0.105015	−0.00420059
$626$	0	0
$627$	−16.9355	−0.676339
$628$	0	0
$629$	−5.43854	−0.216849
$630$	0	0
$631$	−15.1921	−0.604789	−0.302394	−	0.953183i	$-0.597786\pi$
$631$	−15.1921	−0.604789	−0.302394	+	0.953183i	$0.597786\pi$
$632$	0	0
$633$	−16.5897	−0.659380
$634$	0	0
$635$	12.0835	0.479520
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.656016	0.0259516
$640$	0	0
$641$	−35.9342	−1.41932	−0.709658	−	0.704546i	$-0.751151\pi$
$641$	−35.9342	−1.41932	−0.709658	+	0.704546i	$0.751151\pi$
$642$	0	0
$643$	8.91979	0.351762	0.175881	−	0.984411i	$-0.443723\pi$
$643$	8.91979	0.351762	0.175881	+	0.984411i	$0.443723\pi$
$644$	0	0
$645$	−14.9351	−0.588068
$646$	0	0
$647$	9.76721	0.383989	0.191994	−	0.981396i	$-0.438505\pi$
$647$	9.76721	0.383989	0.191994	+	0.981396i	$0.438505\pi$
$648$	0	0
$649$	−9.28730	−0.364558
$650$	0	0
$651$	11.6446	0.456388
$652$	0	0
$653$	16.5372	0.647149	0.323575	−	0.946203i	$-0.395115\pi$
$653$	16.5372	0.647149	0.323575	+	0.946203i	$0.395115\pi$
$654$	0	0
$655$	26.9361	1.05248
$656$	0	0
$657$	−0.217998	−0.00850491
$658$	0	0
$659$	15.6402	0.609256	0.304628	−	0.952471i	$-0.401468\pi$
$659$	15.6402	0.609256	0.304628	+	0.952471i	$0.401468\pi$
$660$	0	0
$661$	−25.3249	−0.985024	−0.492512	−	0.870306i	$-0.663921\pi$
$661$	−25.3249	−0.985024	−0.492512	+	0.870306i	$0.663921\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	11.6220	0.450683
$666$	0	0
$667$	−6.37484	−0.246835
$668$	0	0
$669$	−25.5435	−0.987568
$670$	0	0
$671$	1.91199	0.0738114
$672$	0	0
$673$	−8.17838	−0.315253	−0.157627	−	0.987499i	$-0.550384\pi$
$673$	−8.17838	−0.315253	−0.157627	+	0.987499i	$0.550384\pi$
$674$	0	0
$675$	15.8272	0.609189
$676$	0	0
$677$	44.9616	1.72801	0.864007	−	0.503480i	$-0.167947\pi$
$677$	44.9616	1.72801	0.864007	+	0.503480i	$0.167947\pi$
$678$	0	0
$679$	7.54618	0.289596
$680$	0	0
$681$	−12.5434	−0.480663
$682$	0	0
$683$	−20.4486	−0.782444	−0.391222	−	0.920296i	$-0.627947\pi$
$683$	−20.4486	−0.782444	−0.391222	+	0.920296i	$0.627947\pi$
$684$	0	0
$685$	−4.64733	−0.177565
$686$	0	0
$687$	−21.5382	−0.821734
$688$	0	0
$689$	0	0
$690$	0	0
$691$	33.7107	1.28242	0.641208	−	0.767367i	$-0.278433\pi$
$691$	33.7107	1.28242	0.641208	+	0.767367i	$0.278433\pi$
$692$	0	0
$693$	−0.0761145	−0.00289135
$694$	0	0
$695$	−10.9618	−0.415805
$696$	0	0
$697$	19.6223	0.743248
$698$	0	0
$699$	32.9254	1.24535
$700$	0	0
$701$	29.6172	1.11862	0.559312	−	0.828957i	$-0.311065\pi$
$701$	29.6172	1.11862	0.559312	+	0.828957i	$0.311065\pi$
$702$	0	0
$703$	−29.2000	−1.10130
$704$	0	0
$705$	−17.9612	−0.676459
$706$	0	0
$707$	−13.2005	−0.496457
$708$	0	0
$709$	−9.92641	−0.372794	−0.186397	−	0.982474i	$-0.559681\pi$
$709$	−9.92641	−0.372794	−0.186397	+	0.982474i	$0.559681\pi$
$710$	0	0
$711$	0.306868	0.0115085
$712$	0	0
$713$	14.7725	0.553233
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2.80678	0.104821
$718$	0	0
$719$	−20.6850	−0.771422	−0.385711	−	0.922620i	$-0.626044\pi$
$719$	−20.6850	−0.771422	−0.385711	+	0.922620i	$0.626044\pi$
$720$	0	0
$721$	5.90120	0.219772
$722$	0	0
$723$	32.1584	1.19598
$724$	0	0
$725$	−8.84123	−0.328355
$726$	0	0
$727$	−4.83449	−0.179301	−0.0896506	−	0.995973i	$-0.528575\pi$
$727$	−4.83449	−0.179301	−0.0896506	+	0.995973i	$0.528575\pi$
$728$	0	0
$729$	26.3800	0.977035
$730$	0	0
$731$	9.61998	0.355808
$732$	0	0
$733$	12.7037	0.469223	0.234612	−	0.972089i	$-0.424618\pi$
$733$	12.7037	0.469223	0.234612	+	0.972089i	$0.424618\pi$
$734$	0	0
$735$	2.42578	0.0894762
$736$	0	0
$737$	4.27874	0.157609
$738$	0	0
$739$	−2.58482	−0.0950840	−0.0475420	−	0.998869i	$-0.515139\pi$
$739$	−2.58482	−0.0950840	−0.0475420	+	0.998869i	$0.515139\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−11.5276	−0.422906	−0.211453	−	0.977388i	$-0.567820\pi$
$743$	−11.5276	−0.422906	−0.211453	+	0.977388i	$0.567820\pi$
$744$	0	0
$745$	3.27544	0.120003
$746$	0	0
$747$	−0.120520	−0.00440961
$748$	0	0
$749$	−10.1927	−0.372432
$750$	0	0
$751$	−7.00538	−0.255630	−0.127815	−	0.991798i	$-0.540796\pi$
$751$	−7.00538	−0.255630	−0.127815	+	0.991798i	$0.540796\pi$
$752$	0	0
$753$	−25.1524	−0.916605
$754$	0	0
$755$	−16.0540	−0.584264
$756$	0	0
$757$	−17.2018	−0.625211	−0.312605	−	0.949883i	$-0.601202\pi$
$757$	−17.2018	−0.625211	−0.312605	+	0.949883i	$0.601202\pi$
$758$	0	0
$759$	−4.48431	−0.162770
$760$	0	0
$761$	−34.5530	−1.25255	−0.626274	−	0.779603i	$-0.715421\pi$
$761$	−34.5530	−1.25255	−0.626274	+	0.779603i	$0.715421\pi$
$762$	0	0
$763$	−6.14856	−0.222593
$764$	0	0
$765$	−0.142908	−0.00516686
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−10.9495	−0.394848	−0.197424	−	0.980318i	$-0.563258\pi$
$769$	−10.9495	−0.394848	−0.197424	+	0.980318i	$0.563258\pi$
$770$	0	0
$771$	30.4074	1.09509
$772$	0	0
$773$	22.0036	0.791413	0.395707	−	0.918377i	$-0.370500\pi$
$773$	22.0036	0.791413	0.395707	+	0.918377i	$0.370500\pi$
$774$	0	0
$775$	20.4878	0.735945
$776$	0	0
$777$	−6.09469	−0.218646
$778$	0	0
$779$	105.354	3.77470
$780$	0	0
$781$	11.4560	0.409926
$782$	0	0
$783$	−14.7434	−0.526887
$784$	0	0
$785$	−12.5187	−0.446812
$786$	0	0
$787$	−39.3231	−1.40172	−0.700859	−	0.713300i	$-0.747200\pi$
$787$	−39.3231	−1.40172	−0.700859	+	0.713300i	$0.747200\pi$
$788$	0	0
$789$	29.3152	1.04365
$790$	0	0
$791$	−14.5975	−0.519028
$792$	0	0
$793$	0	0
$794$	0	0
$795$	27.2417	0.966162
$796$	0	0
$797$	−54.5240	−1.93134	−0.965669	−	0.259775i	$-0.916352\pi$
$797$	−54.5240	−1.93134	−0.965669	+	0.259775i	$0.916352\pi$
$798$	0	0
$799$	11.5692	0.409289
$800$	0	0
$801$	0.156917	0.00554438
$802$	0	0
$803$	−3.80688	−0.134342
$804$	0	0
$805$	3.07737	0.108463
$806$	0	0
$807$	43.4100	1.52810
$808$	0	0
$809$	3.12637	0.109917	0.0549586	−	0.998489i	$-0.482497\pi$
$809$	3.12637	0.109917	0.0549586	+	0.998489i	$0.482497\pi$
$810$	0	0
$811$	−4.48741	−0.157574	−0.0787872	−	0.996891i	$-0.525105\pi$
$811$	−4.48741	−0.157574	−0.0787872	+	0.996891i	$0.525105\pi$
$812$	0	0
$813$	−35.4258	−1.24244
$814$	0	0
$815$	29.5853	1.03633
$816$	0	0
$817$	51.6506	1.80702
$818$	0	0
$819$	0	0
$820$	0	0
$821$	24.6872	0.861589	0.430795	−	0.902450i	$-0.358233\pi$
$821$	24.6872	0.861589	0.430795	+	0.902450i	$0.358233\pi$
$822$	0	0
$823$	−11.0134	−0.383904	−0.191952	−	0.981404i	$-0.561482\pi$
$823$	−11.0134	−0.383904	−0.191952	+	0.981404i	$0.561482\pi$
$824$	0	0
$825$	−6.21926	−0.216527
$826$	0	0
$827$	−23.8575	−0.829605	−0.414803	−	0.909911i	$-0.636149\pi$
$827$	−23.8575	−0.829605	−0.414803	+	0.909911i	$0.636149\pi$
$828$	0	0
$829$	20.4314	0.709611	0.354806	−	0.934940i	$-0.384547\pi$
$829$	20.4314	0.709611	0.354806	+	0.934940i	$0.384547\pi$
$830$	0	0
$831$	19.3544	0.671395
$832$	0	0
$833$	−1.56249	−0.0541372
$834$	0	0
$835$	−4.40725	−0.152519
$836$	0	0
$837$	34.1650	1.18092
$838$	0	0
$839$	−24.2579	−0.837477	−0.418738	−	0.908107i	$-0.637528\pi$
$839$	−24.2579	−0.837477	−0.418738	+	0.908107i	$0.637528\pi$
$840$	0	0
$841$	−20.7642	−0.716006
$842$	0	0
$843$	5.58492	0.192355
$844$	0	0
$845$	0	0
$846$	0	0
$847$	9.67082	0.332293
$848$	0	0
$849$	31.3949	1.07747
$850$	0	0
$851$	−7.73178	−0.265042
$852$	0	0
$853$	51.8470	1.77521	0.887604	−	0.460608i	$-0.152369\pi$
$853$	51.8470	1.77521	0.887604	+	0.460608i	$0.152369\pi$
$854$	0	0
$855$	−0.767287	−0.0262407
$856$	0	0
$857$	26.9054	0.919071	0.459536	−	0.888159i	$-0.348016\pi$
$857$	26.9054	0.919071	0.459536	+	0.888159i	$0.348016\pi$
$858$	0	0
$859$	33.4509	1.14133	0.570665	−	0.821183i	$-0.306685\pi$
$859$	33.4509	1.14133	0.570665	+	0.821183i	$0.306685\pi$
$860$	0	0
$861$	21.9897	0.749407
$862$	0	0
$863$	−5.94352	−0.202320	−0.101160	−	0.994870i	$-0.532255\pi$
$863$	−5.94352	−0.202320	−0.101160	+	0.994870i	$0.532255\pi$
$864$	0	0
$865$	7.39651	0.251489
$866$	0	0
$867$	−25.4922	−0.865761
$868$	0	0
$869$	5.35881	0.181785
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−0.498199	−0.0168615
$874$	0	0
$875$	11.1948	0.378453
$876$	0	0
$877$	49.6605	1.67692	0.838458	−	0.544966i	$-0.183458\pi$
$877$	49.6605	1.67692	0.838458	+	0.544966i	$0.183458\pi$
$878$	0	0
$879$	29.7105	1.00211
$880$	0	0
$881$	−21.2905	−0.717294	−0.358647	−	0.933473i	$-0.616762\pi$
$881$	−21.2905	−0.717294	−0.358647	+	0.933473i	$0.616762\pi$
$882$	0	0
$883$	32.8053	1.10399	0.551993	−	0.833848i	$-0.313867\pi$
$883$	32.8053	1.10399	0.551993	+	0.833848i	$0.313867\pi$
$884$	0	0
$885$	−19.5411	−0.656866
$886$	0	0
$887$	−17.2358	−0.578721	−0.289361	−	0.957220i	$-0.593443\pi$
$887$	−17.2358	−0.578721	−0.289361	+	0.957220i	$0.593443\pi$
$888$	0	0
$889$	−8.72229	−0.292536
$890$	0	0
$891$	−10.5994	−0.355094
$892$	0	0
$893$	62.1161	2.07863
$894$	0	0
$895$	13.6397	0.455925
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−19.0849	−0.636518
$900$	0	0
$901$	−17.5469	−0.584572
$902$	0	0
$903$	10.7806	0.358756
$904$	0	0
$905$	−20.9540	−0.696535
$906$	0	0
$907$	−34.6356	−1.15006	−0.575028	−	0.818134i	$-0.695009\pi$
$907$	−34.6356	−1.15006	−0.575028	+	0.818134i	$0.695009\pi$
$908$	0	0
$909$	0.871499	0.0289058
$910$	0	0
$911$	−7.85153	−0.260133	−0.130066	−	0.991505i	$-0.541519\pi$
$911$	−7.85153	−0.260133	−0.130066	+	0.991505i	$0.541519\pi$
$912$	0	0
$913$	−2.10464	−0.0696533
$914$	0	0
$915$	4.02294	0.132994
$916$	0	0
$917$	−19.4433	−0.642075
$918$	0	0
$919$	−36.7317	−1.21167	−0.605834	−	0.795591i	$-0.707160\pi$
$919$	−36.7317	−1.21167	−0.605834	+	0.795591i	$0.707160\pi$
$920$	0	0
$921$	47.2568	1.55717
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−10.7232	−0.352576
$926$	0	0
$927$	−0.389597	−0.0127961
$928$	0	0
$929$	33.0655	1.08484	0.542421	−	0.840107i	$-0.317508\pi$
$929$	33.0655	1.08484	0.542421	+	0.840107i	$0.317508\pi$
$930$	0	0
$931$	−8.38917	−0.274944
$932$	0	0
$933$	−49.6902	−1.62679
$934$	0	0
$935$	−2.49559	−0.0816146
$936$	0	0
$937$	−45.8626	−1.49826	−0.749132	−	0.662421i	$-0.769529\pi$
$937$	−45.8626	−1.49826	−0.749132	+	0.662421i	$0.769529\pi$
$938$	0	0
$939$	−29.1846	−0.952404
$940$	0	0
$941$	−5.77729	−0.188334	−0.0941671	−	0.995556i	$-0.530019\pi$
$941$	−5.77729	−0.188334	−0.0941671	+	0.995556i	$0.530019\pi$
$942$	0	0
$943$	27.8964	0.908431
$944$	0	0
$945$	7.11718	0.231522
$946$	0	0
$947$	−0.0562941	−0.00182931	−0.000914656	−	1.00000i	$-0.500291\pi$
$947$	−0.0562941	−0.00182931	−0.000914656		1.00000i	$0.500291\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−50.3286	−1.63202
$952$	0	0
$953$	−40.1557	−1.30077	−0.650385	−	0.759605i	$-0.725392\pi$
$953$	−40.1557	−1.30077	−0.650385	+	0.759605i	$0.725392\pi$
$954$	0	0
$955$	−30.7273	−0.994312
$956$	0	0
$957$	5.79339	0.187274
$958$	0	0
$959$	3.35460	0.108326
$960$	0	0
$961$	13.2257	0.426634
$962$	0	0
$963$	0.672920	0.0216846
$964$	0	0
$965$	−9.65856	−0.310920
$966$	0	0
$967$	−32.3647	−1.04078	−0.520390	−	0.853929i	$-0.674213\pi$
$967$	−32.3647	−1.04078	−0.520390	+	0.853929i	$0.674213\pi$
$968$	0	0
$969$	22.9522	0.737331
$970$	0	0
$971$	−30.6918	−0.984947	−0.492473	−	0.870328i	$-0.663907\pi$
$971$	−30.6918	−0.984947	−0.492473	+	0.870328i	$0.663907\pi$
$972$	0	0
$973$	7.91259	0.253666
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−5.62992	−0.180117	−0.0900585	−	0.995936i	$-0.528705\pi$
$977$	−5.62992	−0.180117	−0.0900585	+	0.995936i	$0.528705\pi$
$978$	0	0
$979$	2.74022	0.0875779
$980$	0	0
$981$	0.405928	0.0129603
$982$	0	0
$983$	6.66363	0.212537	0.106268	−	0.994337i	$-0.466110\pi$
$983$	6.66363	0.212537	0.106268	+	0.994337i	$0.466110\pi$
$984$	0	0
$985$	−27.3376	−0.871050
$986$	0	0
$987$	12.9650	0.412681
$988$	0	0
$989$	13.6764	0.434884
$990$	0	0
$991$	−52.8312	−1.67824	−0.839120	−	0.543946i	$-0.816929\pi$
$991$	−52.8312	−1.67824	−0.839120	+	0.543946i	$0.816929\pi$
$992$	0	0
$993$	45.8618	1.45538
$994$	0	0
$995$	−19.1233	−0.606251
$996$	0	0
$997$	−14.6367	−0.463548	−0.231774	−	0.972770i	$-0.574453\pi$
$997$	−14.6367	−0.463548	−0.231774	+	0.972770i	$0.574453\pi$
$998$	0	0
$999$	−17.8817	−0.565752

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4732.2.a.s.1.6		8
13.5	odd	4		4732.2.g.k.337.11		16
13.6	odd	12		364.2.u.a.309.3	yes	16
13.8	odd	4		4732.2.g.k.337.12		16
13.11	odd	12		364.2.u.a.225.3	✓	16
13.12	even	2		4732.2.a.t.1.6		8
39.11	even	12		3276.2.cf.c.2773.3		16
39.32	even	12		3276.2.cf.c.1765.6		16
52.11	even	12		1456.2.cc.f.225.6		16
52.19	even	12		1456.2.cc.f.673.6		16
91.6	even	12		2548.2.u.c.1765.6		16
91.11	odd	12		2548.2.bq.e.1941.6		16
91.19	even	12		2548.2.bq.c.361.3		16
91.24	even	12		2548.2.bq.c.1941.3		16
91.32	odd	12		2548.2.bb.d.569.3		16
91.37	odd	12		2548.2.bb.d.1733.3		16
91.45	even	12		2548.2.bb.c.569.6		16
91.58	odd	12		2548.2.bq.e.361.6		16
91.76	even	12		2548.2.u.c.589.6		16
91.89	even	12		2548.2.bb.c.1733.6		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
364.2.u.a.225.3	✓	16	13.11	odd	12
364.2.u.a.309.3	yes	16	13.6	odd	12
1456.2.cc.f.225.6		16	52.11	even	12
1456.2.cc.f.673.6		16	52.19	even	12
2548.2.u.c.589.6		16	91.76	even	12
2548.2.u.c.1765.6		16	91.6	even	12
2548.2.bb.c.569.6		16	91.45	even	12
2548.2.bb.c.1733.6		16	91.89	even	12
2548.2.bb.d.569.3		16	91.32	odd	12
2548.2.bb.d.1733.3		16	91.37	odd	12
2548.2.bq.c.361.3		16	91.19	even	12
2548.2.bq.c.1941.3		16	91.24	even	12
2548.2.bq.e.361.6		16	91.58	odd	12
2548.2.bq.e.1941.6		16	91.11	odd	12
3276.2.cf.c.1765.6		16	39.32	even	12
3276.2.cf.c.2773.3		16	39.11	even	12
4732.2.a.s.1.6		8	1.1	even	1	trivial
4732.2.a.t.1.6		8	13.12	even	2
4732.2.g.k.337.11		16	13.5	odd	4
4732.2.g.k.337.12		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)

\(f(q)\)	\(=\)	\(q+1.75101 q^{3} +1.38536 q^{5} -1.00000 q^{7} +0.0660200 q^{9} +1.15290 q^{11} +2.42578 q^{15} -1.56249 q^{17} -8.38917 q^{19} -1.75101 q^{21} -2.22134 q^{23} -3.08077 q^{25} -5.13741 q^{27} +2.86981 q^{29} -6.65024 q^{31} +2.01874 q^{33} -1.38536 q^{35} +3.48068 q^{37} -12.5583 q^{41} -6.15682 q^{43} +0.0914616 q^{45} -7.40432 q^{47} +1.00000 q^{49} -2.73593 q^{51} +11.2301 q^{53} +1.59719 q^{55} -14.6895 q^{57} -8.05559 q^{59} +1.65841 q^{61} -0.0660200 q^{63} +3.71128 q^{67} -3.88958 q^{69} +9.93663 q^{71} -3.30200 q^{73} -5.39444 q^{75} -1.15290 q^{77} +4.64811 q^{79} -9.19370 q^{81} -1.82551 q^{83} -2.16462 q^{85} +5.02506 q^{87} +2.37681 q^{89} -11.6446 q^{93} -11.6220 q^{95} -7.54618 q^{97} +0.0761145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} - 8 q^{7} + 14 q^{9} - 12 q^{11} - 12 q^{15} + 2 q^{17} - 6 q^{19} + 22 q^{25} - 6 q^{27} + 22 q^{29} - 14 q^{31} + 28 q^{33} + 6 q^{35} - 12 q^{37} - 4 q^{41} + 6 q^{43} - 20 q^{45} - 42 q^{47}+ \cdots - 80 q^{99}+O(q^{100}) \) 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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