LMFDB - Embedded newform 4600.2.a.bj.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$0$
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} - 3x^{7} - 13x^{6} + 38x^{5} + 41x^{4} - 123x^{3} + 15x^{2} + 32x - 8 $ x^8 - 3x^7 - 13x^6 + 38x^5 + 41x^4 - 123x^3 + 15x^2 + 32*x - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.296848$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.296848 q^{3} +3.46037 q^{7} -2.91188 q^{9} +O(q^{10})$	$q-0.296848 q^{3} +3.46037 q^{7} -2.91188 q^{9} -3.11377 q^{11} +4.60394 q^{13} +5.49355 q^{17} +4.48919 q^{19} -1.02720 q^{21} +1.00000 q^{23} +1.75493 q^{27} +9.19670 q^{29} -5.89980 q^{31} +0.924314 q^{33} -6.95324 q^{37} -1.36667 q^{39} -9.03617 q^{41} -5.55051 q^{43} +5.48499 q^{47} +4.97418 q^{49} -1.63075 q^{51} -2.74411 q^{53} -1.33261 q^{57} +9.33659 q^{59} -1.40206 q^{61} -10.0762 q^{63} +3.49779 q^{67} -0.296848 q^{69} +4.28869 q^{71} +4.92048 q^{73} -10.7748 q^{77} +2.12284 q^{79} +8.21470 q^{81} -16.0159 q^{83} -2.73002 q^{87} +11.9821 q^{89} +15.9314 q^{91} +1.75134 q^{93} -4.37324 q^{97} +9.06692 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	$ 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} + 7 q^{11} + 11 q^{13} - 7 q^{17} + 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} + 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} - 19 q^{51} + 4 q^{53} + 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} + 15 q^{71} + 6 q^{73} + 18 q^{77} - 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} - 35 q^{91} + 20 q^{93} + 3 q^{97} + 61 q^{99}+O(q^{100}) $ 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 + 7 * q^11 + 11 * q^13 - 7 * q^17 + 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 + 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 - 19 * q^51 + 4 * q^53 + 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 + 15 * q^71 + 6 * q^73 + 18 * q^77 - 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 - 35 * q^91 + 20 * q^93 + 3 * q^97 + 61 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.296848	−0.171385	−0.0856925	−	0.996322i	$-0.527310\pi$
$3$	−0.296848	−0.171385	−0.0856925	+	0.996322i	$0.527310\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.46037	1.30790	0.653949	−	0.756539i	$-0.273111\pi$
$7$	3.46037	1.30790	0.653949	+	0.756539i	$0.273111\pi$
$8$	0	0
$9$	−2.91188	−0.970627
$10$	0	0
$11$	−3.11377	−0.938836	−0.469418	−	0.882976i	$-0.655536\pi$
$11$	−3.11377	−0.938836	−0.469418	+	0.882976i	$0.655536\pi$
$12$	0	0
$13$	4.60394	1.27690	0.638452	−	0.769662i	$-0.279575\pi$
$13$	4.60394	1.27690	0.638452	+	0.769662i	$0.279575\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.49355	1.33238	0.666190	−	0.745782i	$-0.267924\pi$
$17$	5.49355	1.33238	0.666190	+	0.745782i	$0.267924\pi$
$18$	0	0
$19$	4.48919	1.02989	0.514946	−	0.857223i	$-0.327812\pi$
$19$	4.48919	1.02989	0.514946	+	0.857223i	$0.327812\pi$
$20$	0	0
$21$	−1.02720	−0.224154
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.75493	0.337736
$28$	0	0
$29$	9.19670	1.70778	0.853892	−	0.520450i	$-0.174236\pi$
$29$	9.19670	1.70778	0.853892	+	0.520450i	$0.174236\pi$
$30$	0	0
$31$	−5.89980	−1.05964	−0.529818	−	0.848111i	$-0.677740\pi$
$31$	−5.89980	−1.05964	−0.529818	+	0.848111i	$0.677740\pi$
$32$	0	0
$33$	0.924314	0.160902
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−6.95324	−1.14311	−0.571553	−	0.820565i	$-0.693659\pi$
$37$	−6.95324	−1.14311	−0.571553	+	0.820565i	$0.693659\pi$
$38$	0	0
$39$	−1.36667	−0.218842
$40$	0	0
$41$	−9.03617	−1.41121	−0.705607	−	0.708604i	$-0.749325\pi$
$41$	−9.03617	−1.41121	−0.705607	+	0.708604i	$0.749325\pi$
$42$	0	0
$43$	−5.55051	−0.846444	−0.423222	−	0.906026i	$-0.639101\pi$
$43$	−5.55051	−0.846444	−0.423222	+	0.906026i	$0.639101\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	5.48499	0.800068	0.400034	−	0.916500i	$-0.368998\pi$
$47$	5.48499	0.800068	0.400034	+	0.916500i	$0.368998\pi$
$48$	0	0
$49$	4.97418	0.710598
$50$	0	0
$51$	−1.63075	−0.228350
$52$	0	0
$53$	−2.74411	−0.376932	−0.188466	−	0.982080i	$-0.560352\pi$
$53$	−2.74411	−0.376932	−0.188466	+	0.982080i	$0.560352\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.33261	−0.176508
$58$	0	0
$59$	9.33659	1.21552	0.607760	−	0.794120i	$-0.292068\pi$
$59$	9.33659	1.21552	0.607760	+	0.794120i	$0.292068\pi$
$60$	0	0
$61$	−1.40206	−0.179515	−0.0897576	−	0.995964i	$-0.528609\pi$
$61$	−1.40206	−0.179515	−0.0897576	+	0.995964i	$0.528609\pi$
$62$	0	0
$63$	−10.0762	−1.26948
$64$	0	0
$65$	0	0
$66$	0	0
$67$	3.49779	0.427323	0.213662	−	0.976908i	$-0.431461\pi$
$67$	3.49779	0.427323	0.213662	+	0.976908i	$0.431461\pi$
$68$	0	0
$69$	−0.296848	−0.0357362
$70$	0	0
$71$	4.28869	0.508974	0.254487	−	0.967076i	$-0.418093\pi$
$71$	4.28869	0.508974	0.254487	+	0.967076i	$0.418093\pi$
$72$	0	0
$73$	4.92048	0.575899	0.287949	−	0.957646i	$-0.407026\pi$
$73$	4.92048	0.575899	0.287949	+	0.957646i	$0.407026\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−10.7748	−1.22790
$78$	0	0
$79$	2.12284	0.238838	0.119419	−	0.992844i	$-0.461897\pi$
$79$	2.12284	0.238838	0.119419	+	0.992844i	$0.461897\pi$
$80$	0	0
$81$	8.21470	0.912744
$82$	0	0
$83$	−16.0159	−1.75797	−0.878987	−	0.476845i	$-0.841780\pi$
$83$	−16.0159	−1.75797	−0.878987	+	0.476845i	$0.841780\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−2.73002	−0.292689
$88$	0	0
$89$	11.9821	1.27010	0.635048	−	0.772473i	$-0.280980\pi$
$89$	11.9821	1.27010	0.635048	+	0.772473i	$0.280980\pi$
$90$	0	0
$91$	15.9314	1.67006
$92$	0	0
$93$	1.75134	0.181606
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−4.37324	−0.444035	−0.222018	−	0.975043i	$-0.571264\pi$
$97$	−4.37324	−0.444035	−0.222018	+	0.975043i	$0.571264\pi$
$98$	0	0
$99$	9.06692	0.911259
$100$	0	0
$101$	10.4175	1.03658	0.518288	−	0.855206i	$-0.326570\pi$
$101$	10.4175	1.03658	0.518288	+	0.855206i	$0.326570\pi$
$102$	0	0
$103$	7.21376	0.710793	0.355396	−	0.934716i	$-0.384346\pi$
$103$	7.21376	0.710793	0.355396	+	0.934716i	$0.384346\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.07230	0.683705	0.341853	−	0.939754i	$-0.388946\pi$
$107$	7.07230	0.683705	0.341853	+	0.939754i	$0.388946\pi$
$108$	0	0
$109$	−9.92058	−0.950219	−0.475110	−	0.879927i	$-0.657592\pi$
$109$	−9.92058	−0.950219	−0.475110	+	0.879927i	$0.657592\pi$
$110$	0	0
$111$	2.06405	0.195911
$112$	0	0
$113$	12.9797	1.22103	0.610513	−	0.792007i	$-0.290964\pi$
$113$	12.9797	1.22103	0.610513	+	0.792007i	$0.290964\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−13.4061	−1.23940
$118$	0	0
$119$	19.0097	1.74262
$120$	0	0
$121$	−1.30446	−0.118587
$122$	0	0
$123$	2.68237	0.241861
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−14.9664	−1.32805	−0.664027	−	0.747709i	$-0.731154\pi$
$127$	−14.9664	−1.32805	−0.664027	+	0.747709i	$0.731154\pi$
$128$	0	0
$129$	1.64765	0.145068
$130$	0	0
$131$	18.5945	1.62461	0.812305	−	0.583233i	$-0.198212\pi$
$131$	18.5945	1.62461	0.812305	+	0.583233i	$0.198212\pi$
$132$	0	0
$133$	15.5343	1.34699
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.9298	1.70272	0.851359	−	0.524583i	$-0.175779\pi$
$137$	19.9298	1.70272	0.851359	+	0.524583i	$0.175779\pi$
$138$	0	0
$139$	−0.346731	−0.0294094	−0.0147047	−	0.999892i	$-0.504681\pi$
$139$	−0.346731	−0.0294094	−0.0147047	+	0.999892i	$0.504681\pi$
$140$	0	0
$141$	−1.62821	−0.137120
$142$	0	0
$143$	−14.3356	−1.19880
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.47657	−0.121786
$148$	0	0
$149$	−1.87953	−0.153977	−0.0769884	−	0.997032i	$-0.524530\pi$
$149$	−1.87953	−0.153977	−0.0769884	+	0.997032i	$0.524530\pi$
$150$	0	0
$151$	9.84743	0.801373	0.400686	−	0.916215i	$-0.368772\pi$
$151$	9.84743	0.801373	0.400686	+	0.916215i	$0.368772\pi$
$152$	0	0
$153$	−15.9966	−1.29324
$154$	0	0
$155$	0	0
$156$	0	0
$157$	16.5630	1.32187	0.660936	−	0.750442i	$-0.270159\pi$
$157$	16.5630	1.32187	0.660936	+	0.750442i	$0.270159\pi$
$158$	0	0
$159$	0.814582	0.0646006
$160$	0	0
$161$	3.46037	0.272716
$162$	0	0
$163$	−9.48888	−0.743226	−0.371613	−	0.928388i	$-0.621195\pi$
$163$	−9.48888	−0.743226	−0.371613	+	0.928388i	$0.621195\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−9.28434	−0.718444	−0.359222	−	0.933252i	$-0.616958\pi$
$167$	−9.28434	−0.718444	−0.359222	+	0.933252i	$0.616958\pi$
$168$	0	0
$169$	8.19629	0.630484
$170$	0	0
$171$	−13.0720	−0.999640
$172$	0	0
$173$	9.34023	0.710124	0.355062	−	0.934843i	$-0.384460\pi$
$173$	9.34023	0.710124	0.355062	+	0.934843i	$0.384460\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−2.77154	−0.208322
$178$	0	0
$179$	−0.255383	−0.0190882	−0.00954411	−	0.999954i	$-0.503038\pi$
$179$	−0.255383	−0.0190882	−0.00954411	+	0.999954i	$0.503038\pi$
$180$	0	0
$181$	−4.02663	−0.299297	−0.149649	−	0.988739i	$-0.547814\pi$
$181$	−4.02663	−0.299297	−0.149649	+	0.988739i	$0.547814\pi$
$182$	0	0
$183$	0.416198	0.0307662
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−17.1056	−1.25089
$188$	0	0
$189$	6.07270	0.441724
$190$	0	0
$191$	−24.7376	−1.78995	−0.894974	−	0.446118i	$-0.852806\pi$
$191$	−24.7376	−1.78995	−0.894974	+	0.446118i	$0.852806\pi$
$192$	0	0
$193$	−7.34062	−0.528390	−0.264195	−	0.964469i	$-0.585106\pi$
$193$	−7.34062	−0.528390	−0.264195	+	0.964469i	$0.585106\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	16.2033	1.15444	0.577220	−	0.816589i	$-0.304138\pi$
$197$	16.2033	1.15444	0.577220	+	0.816589i	$0.304138\pi$
$198$	0	0
$199$	−13.8498	−0.981786	−0.490893	−	0.871220i	$-0.663329\pi$
$199$	−13.8498	−0.981786	−0.490893	+	0.871220i	$0.663329\pi$
$200$	0	0
$201$	−1.03831	−0.0732368
$202$	0	0
$203$	31.8240	2.23361
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.91188	−0.202390
$208$	0	0
$209$	−13.9783	−0.966899
$210$	0	0
$211$	19.8811	1.36867	0.684335	−	0.729168i	$-0.260093\pi$
$211$	19.8811	1.36867	0.684335	+	0.729168i	$0.260093\pi$
$212$	0	0
$213$	−1.27309	−0.0872305
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−20.4155	−1.38590
$218$	0	0
$219$	−1.46063	−0.0987004
$220$	0	0
$221$	25.2920	1.70132
$222$	0	0
$223$	−14.3441	−0.960550	−0.480275	−	0.877118i	$-0.659463\pi$
$223$	−14.3441	−0.960550	−0.480275	+	0.877118i	$0.659463\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.219192	−0.0145483	−0.00727413	−	0.999974i	$-0.502315\pi$
$227$	−0.219192	−0.0145483	−0.00727413	+	0.999974i	$0.502315\pi$
$228$	0	0
$229$	0.919300	0.0607491	0.0303745	−	0.999539i	$-0.490330\pi$
$229$	0.919300	0.0607491	0.0303745	+	0.999539i	$0.490330\pi$
$230$	0	0
$231$	3.19847	0.210444
$232$	0	0
$233$	22.5267	1.47578	0.737888	−	0.674923i	$-0.235823\pi$
$233$	22.5267	1.47578	0.737888	+	0.674923i	$0.235823\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−0.630160	−0.0409333
$238$	0	0
$239$	−11.4614	−0.741376	−0.370688	−	0.928758i	$-0.620878\pi$
$239$	−11.4614	−0.741376	−0.370688	+	0.928758i	$0.620878\pi$
$240$	0	0
$241$	15.3035	0.985784	0.492892	−	0.870090i	$-0.335940\pi$
$241$	15.3035	0.985784	0.492892	+	0.870090i	$0.335940\pi$
$242$	0	0
$243$	−7.70330	−0.494167
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.6680	1.31507
$248$	0	0
$249$	4.75428	0.301290
$250$	0	0
$251$	21.1450	1.33466	0.667329	−	0.744763i	$-0.267437\pi$
$251$	21.1450	1.33466	0.667329	+	0.744763i	$0.267437\pi$
$252$	0	0
$253$	−3.11377	−0.195761
$254$	0	0
$255$	0	0
$256$	0	0
$257$	2.81665	0.175698	0.0878489	−	0.996134i	$-0.472001\pi$
$257$	2.81665	0.175698	0.0878489	+	0.996134i	$0.472001\pi$
$258$	0	0
$259$	−24.0608	−1.49507
$260$	0	0
$261$	−26.7797	−1.65762
$262$	0	0
$263$	16.8627	1.03980	0.519899	−	0.854228i	$-0.325970\pi$
$263$	16.8627	1.03980	0.519899	+	0.854228i	$0.325970\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−3.55684	−0.217675
$268$	0	0
$269$	27.8171	1.69604	0.848019	−	0.529966i	$-0.177795\pi$
$269$	27.8171	1.69604	0.848019	+	0.529966i	$0.177795\pi$
$270$	0	0
$271$	−4.73892	−0.287869	−0.143934	−	0.989587i	$-0.545975\pi$
$271$	−4.73892	−0.287869	−0.143934	+	0.989587i	$0.545975\pi$
$272$	0	0
$273$	−4.72919	−0.286223
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.983771	−0.0591091	−0.0295545	−	0.999563i	$-0.509409\pi$
$277$	−0.983771	−0.0591091	−0.0295545	+	0.999563i	$0.509409\pi$
$278$	0	0
$279$	17.1795	1.02851
$280$	0	0
$281$	24.1462	1.44044	0.720219	−	0.693746i	$-0.244041\pi$
$281$	24.1462	1.44044	0.720219	+	0.693746i	$0.244041\pi$
$282$	0	0
$283$	−22.6523	−1.34654	−0.673269	−	0.739397i	$-0.735110\pi$
$283$	−22.6523	−1.34654	−0.673269	+	0.739397i	$0.735110\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−31.2685	−1.84572
$288$	0	0
$289$	13.1791	0.775238
$290$	0	0
$291$	1.29819	0.0761010
$292$	0	0
$293$	12.9668	0.757529	0.378764	−	0.925493i	$-0.376349\pi$
$293$	12.9668	0.757529	0.378764	+	0.925493i	$0.376349\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.46443	−0.317079
$298$	0	0
$299$	4.60394	0.266253
$300$	0	0
$301$	−19.2068	−1.10706
$302$	0	0
$303$	−3.09240	−0.177654
$304$	0	0
$305$	0	0
$306$	0	0
$307$	27.4380	1.56597	0.782984	−	0.622041i	$-0.213696\pi$
$307$	27.4380	1.56597	0.782984	+	0.622041i	$0.213696\pi$
$308$	0	0
$309$	−2.14139	−0.121819
$310$	0	0
$311$	−30.8655	−1.75022	−0.875112	−	0.483920i	$-0.839212\pi$
$311$	−30.8655	−1.75022	−0.875112	+	0.483920i	$0.839212\pi$
$312$	0	0
$313$	−3.64274	−0.205900	−0.102950	−	0.994687i	$-0.532828\pi$
$313$	−3.64274	−0.205900	−0.102950	+	0.994687i	$0.532828\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	24.0344	1.34990	0.674952	−	0.737862i	$-0.264164\pi$
$317$	24.0344	1.34990	0.674952	+	0.737862i	$0.264164\pi$
$318$	0	0
$319$	−28.6364	−1.60333
$320$	0	0
$321$	−2.09940	−0.117177
$322$	0	0
$323$	24.6616	1.37221
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.94490	0.162853
$328$	0	0
$329$	18.9801	1.04641
$330$	0	0
$331$	−21.7999	−1.19823	−0.599115	−	0.800663i	$-0.704481\pi$
$331$	−21.7999	−1.19823	−0.599115	+	0.800663i	$0.704481\pi$
$332$	0	0
$333$	20.2470	1.10953
$334$	0	0
$335$	0	0
$336$	0	0
$337$	2.82526	0.153902	0.0769510	−	0.997035i	$-0.475482\pi$
$337$	2.82526	0.153902	0.0769510	+	0.997035i	$0.475482\pi$
$338$	0	0
$339$	−3.85298	−0.209265
$340$	0	0
$341$	18.3706	0.994824
$342$	0	0
$343$	−7.01008	−0.378509
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.5727	−0.674939	−0.337469	−	0.941337i	$-0.609571\pi$
$347$	−12.5727	−0.674939	−0.337469	+	0.941337i	$0.609571\pi$
$348$	0	0
$349$	7.15382	0.382935	0.191468	−	0.981499i	$-0.438675\pi$
$349$	7.15382	0.382935	0.191468	+	0.981499i	$0.438675\pi$
$350$	0	0
$351$	8.07959	0.431256
$352$	0	0
$353$	−20.7013	−1.10182	−0.550909	−	0.834565i	$-0.685719\pi$
$353$	−20.7013	−1.10182	−0.550909	+	0.834565i	$0.685719\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.64299	−0.298659
$358$	0	0
$359$	28.6077	1.50985	0.754927	−	0.655808i	$-0.227672\pi$
$359$	28.6077	1.50985	0.754927	+	0.655808i	$0.227672\pi$
$360$	0	0
$361$	1.15284	0.0606756
$362$	0	0
$363$	0.387226	0.0203241
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−20.4541	−1.06769	−0.533847	−	0.845581i	$-0.679254\pi$
$367$	−20.4541	−1.06769	−0.533847	+	0.845581i	$0.679254\pi$
$368$	0	0
$369$	26.3123	1.36976
$370$	0	0
$371$	−9.49564	−0.492989
$372$	0	0
$373$	−3.75830	−0.194598	−0.0972988	−	0.995255i	$-0.531020\pi$
$373$	−3.75830	−0.194598	−0.0972988	+	0.995255i	$0.531020\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	42.3411	2.18068
$378$	0	0
$379$	28.9884	1.48903	0.744516	−	0.667605i	$-0.232680\pi$
$379$	28.9884	1.48903	0.744516	+	0.667605i	$0.232680\pi$
$380$	0	0
$381$	4.44274	0.227608
$382$	0	0
$383$	−29.8354	−1.52452	−0.762258	−	0.647273i	$-0.775909\pi$
$383$	−29.8354	−1.52452	−0.762258	+	0.647273i	$0.775909\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	16.1624	0.821582
$388$	0	0
$389$	−4.02138	−0.203892	−0.101946	−	0.994790i	$-0.532507\pi$
$389$	−4.02138	−0.203892	−0.101946	+	0.994790i	$0.532507\pi$
$390$	0	0
$391$	5.49355	0.277821
$392$	0	0
$393$	−5.51974	−0.278434
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.86060	0.294135	0.147068	−	0.989126i	$-0.453017\pi$
$397$	5.86060	0.294135	0.147068	+	0.989126i	$0.453017\pi$
$398$	0	0
$399$	−4.61131	−0.230854
$400$	0	0
$401$	22.3492	1.11607	0.558034	−	0.829818i	$-0.311556\pi$
$401$	22.3492	1.11607	0.558034	+	0.829818i	$0.311556\pi$
$402$	0	0
$403$	−27.1624	−1.35305
$404$	0	0
$405$	0	0
$406$	0	0
$407$	21.6508	1.07319
$408$	0	0
$409$	−36.4740	−1.80352	−0.901762	−	0.432234i	$-0.857725\pi$
$409$	−36.4740	−1.80352	−0.901762	+	0.432234i	$0.857725\pi$
$410$	0	0
$411$	−5.91611	−0.291820
$412$	0	0
$413$	32.3081	1.58978
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0.102926	0.00504032
$418$	0	0
$419$	29.0555	1.41945	0.709727	−	0.704477i	$-0.248818\pi$
$419$	29.0555	1.41945	0.709727	+	0.704477i	$0.248818\pi$
$420$	0	0
$421$	28.3055	1.37952	0.689762	−	0.724036i	$-0.257715\pi$
$421$	28.3055	1.37952	0.689762	+	0.724036i	$0.257715\pi$
$422$	0	0
$423$	−15.9716	−0.776568
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.85165	−0.234788
$428$	0	0
$429$	4.25549	0.205457
$430$	0	0
$431$	−26.4640	−1.27472	−0.637362	−	0.770564i	$-0.719974\pi$
$431$	−26.4640	−1.27472	−0.637362	+	0.770564i	$0.719974\pi$
$432$	0	0
$433$	−7.95631	−0.382356	−0.191178	−	0.981555i	$-0.561231\pi$
$433$	−7.95631	−0.382356	−0.191178	+	0.981555i	$0.561231\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.48919	0.214747
$438$	0	0
$439$	20.0799	0.958361	0.479181	−	0.877716i	$-0.340934\pi$
$439$	20.0799	0.958361	0.479181	+	0.877716i	$0.340934\pi$
$440$	0	0
$441$	−14.4842	−0.689725
$442$	0	0
$443$	12.8355	0.609834	0.304917	−	0.952379i	$-0.401371\pi$
$443$	12.8355	0.609834	0.304917	+	0.952379i	$0.401371\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0.557933	0.0263893
$448$	0	0
$449$	4.18739	0.197615	0.0988075	−	0.995107i	$-0.468497\pi$
$449$	4.18739	0.197615	0.0988075	+	0.995107i	$0.468497\pi$
$450$	0	0
$451$	28.1365	1.32490
$452$	0	0
$453$	−2.92319	−0.137343
$454$	0	0
$455$	0	0
$456$	0	0
$457$	42.1962	1.97386	0.986929	−	0.161158i	$-0.0515228\pi$
$457$	42.1962	1.97386	0.986929	+	0.161158i	$0.0515228\pi$
$458$	0	0
$459$	9.64078	0.449993
$460$	0	0
$461$	−10.4930	−0.488709	−0.244355	−	0.969686i	$-0.578576\pi$
$461$	−10.4930	−0.488709	−0.244355	+	0.969686i	$0.578576\pi$
$462$	0	0
$463$	39.5643	1.83871	0.919354	−	0.393432i	$-0.128712\pi$
$463$	39.5643	1.83871	0.919354	+	0.393432i	$0.128712\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−27.2819	−1.26246	−0.631228	−	0.775598i	$-0.717449\pi$
$467$	−27.2819	−1.26246	−0.631228	+	0.775598i	$0.717449\pi$
$468$	0	0
$469$	12.1037	0.558895
$470$	0	0
$471$	−4.91669	−0.226549
$472$	0	0
$473$	17.2830	0.794672
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.99052	0.365861
$478$	0	0
$479$	2.88476	0.131808	0.0659041	−	0.997826i	$-0.479007\pi$
$479$	2.88476	0.131808	0.0659041	+	0.997826i	$0.479007\pi$
$480$	0	0
$481$	−32.0123	−1.45964
$482$	0	0
$483$	−1.02720	−0.0467394
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.55229	−0.0703409	−0.0351704	−	0.999381i	$-0.511197\pi$
$487$	−1.55229	−0.0703409	−0.0351704	+	0.999381i	$0.511197\pi$
$488$	0	0
$489$	2.81675	0.127378
$490$	0	0
$491$	0.415377	0.0187457	0.00937286	−	0.999956i	$-0.497016\pi$
$491$	0.415377	0.0187457	0.00937286	+	0.999956i	$0.497016\pi$
$492$	0	0
$493$	50.5225	2.27542
$494$	0	0
$495$	0	0
$496$	0	0
$497$	14.8405	0.665686
$498$	0	0
$499$	13.8674	0.620789	0.310394	−	0.950608i	$-0.399539\pi$
$499$	13.8674	0.620789	0.310394	+	0.950608i	$0.399539\pi$
$500$	0	0
$501$	2.75603	0.123130
$502$	0	0
$503$	−34.5409	−1.54010	−0.770051	−	0.637982i	$-0.779769\pi$
$503$	−34.5409	−1.54010	−0.770051	+	0.637982i	$0.779769\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−2.43305	−0.108056
$508$	0	0
$509$	15.5389	0.688748	0.344374	−	0.938833i	$-0.388091\pi$
$509$	15.5389	0.688748	0.344374	+	0.938833i	$0.388091\pi$
$510$	0	0
$511$	17.0267	0.753217
$512$	0	0
$513$	7.87821	0.347831
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−17.0790	−0.751133
$518$	0	0
$519$	−2.77262	−0.121705
$520$	0	0
$521$	−5.67959	−0.248827	−0.124414	−	0.992230i	$-0.539705\pi$
$521$	−5.67959	−0.248827	−0.124414	+	0.992230i	$0.539705\pi$
$522$	0	0
$523$	18.4095	0.804992	0.402496	−	0.915422i	$-0.368143\pi$
$523$	18.4095	0.804992	0.402496	+	0.915422i	$0.368143\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−32.4108	−1.41184
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−27.1871	−1.17982
$532$	0	0
$533$	−41.6020	−1.80198
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0.0758098	0.00327143
$538$	0	0
$539$	−15.4884	−0.667134
$540$	0	0
$541$	−17.5169	−0.753110	−0.376555	−	0.926394i	$-0.622891\pi$
$541$	−17.5169	−0.753110	−0.376555	+	0.926394i	$0.622891\pi$
$542$	0	0
$543$	1.19530	0.0512951
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−38.1355	−1.63056	−0.815278	−	0.579070i	$-0.803416\pi$
$547$	−38.1355	−1.63056	−0.815278	+	0.579070i	$0.803416\pi$
$548$	0	0
$549$	4.08263	0.174242
$550$	0	0
$551$	41.2857	1.75883
$552$	0	0
$553$	7.34582	0.312376
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−9.09458	−0.385350	−0.192675	−	0.981263i	$-0.561716\pi$
$557$	−9.09458	−0.385350	−0.192675	+	0.981263i	$0.561716\pi$
$558$	0	0
$559$	−25.5542	−1.08083
$560$	0	0
$561$	5.07776	0.214383
$562$	0	0
$563$	−37.7282	−1.59005	−0.795027	−	0.606574i	$-0.792543\pi$
$563$	−37.7282	−1.59005	−0.795027	+	0.606574i	$0.792543\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	28.4259	1.19378
$568$	0	0
$569$	−34.0456	−1.42727	−0.713633	−	0.700520i	$-0.752952\pi$
$569$	−34.0456	−1.42727	−0.713633	+	0.700520i	$0.752952\pi$
$570$	0	0
$571$	−11.4449	−0.478953	−0.239477	−	0.970902i	$-0.576976\pi$
$571$	−11.4449	−0.478953	−0.239477	+	0.970902i	$0.576976\pi$
$572$	0	0
$573$	7.34329	0.306770
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−40.8683	−1.70137	−0.850686	−	0.525675i	$-0.823813\pi$
$577$	−40.8683	−1.70137	−0.850686	+	0.525675i	$0.823813\pi$
$578$	0	0
$579$	2.17905	0.0905581
$580$	0	0
$581$	−55.4210	−2.29925
$582$	0	0
$583$	8.54452	0.353878
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.3630	−1.17067	−0.585334	−	0.810793i	$-0.699036\pi$
$587$	−28.3630	−1.17067	−0.585334	+	0.810793i	$0.699036\pi$
$588$	0	0
$589$	−26.4853	−1.09131
$590$	0	0
$591$	−4.80992	−0.197854
$592$	0	0
$593$	−25.2847	−1.03832	−0.519159	−	0.854678i	$-0.673755\pi$
$593$	−25.2847	−1.03832	−0.519159	+	0.854678i	$0.673755\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	4.11128	0.168263
$598$	0	0
$599$	−25.9826	−1.06162	−0.530811	−	0.847490i	$-0.678112\pi$
$599$	−25.9826	−1.06162	−0.530811	+	0.847490i	$0.678112\pi$
$600$	0	0
$601$	−10.8318	−0.441840	−0.220920	−	0.975292i	$-0.570906\pi$
$601$	−10.8318	−0.441840	−0.220920	+	0.975292i	$0.570906\pi$
$602$	0	0
$603$	−10.1852	−0.414772
$604$	0	0
$605$	0	0
$606$	0	0
$607$	14.5105	0.588962	0.294481	−	0.955657i	$-0.404853\pi$
$607$	14.5105	0.588962	0.294481	+	0.955657i	$0.404853\pi$
$608$	0	0
$609$	−9.44688	−0.382807
$610$	0	0
$611$	25.2526	1.02161
$612$	0	0
$613$	3.87278	0.156420	0.0782101	−	0.996937i	$-0.475080\pi$
$613$	3.87278	0.156420	0.0782101	+	0.996937i	$0.475080\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−26.3999	−1.06282	−0.531410	−	0.847114i	$-0.678338\pi$
$617$	−26.3999	−1.06282	−0.531410	+	0.847114i	$0.678338\pi$
$618$	0	0
$619$	46.0115	1.84936	0.924679	−	0.380748i	$-0.124334\pi$
$619$	46.0115	1.84936	0.924679	+	0.380748i	$0.124334\pi$
$620$	0	0
$621$	1.75493	0.0704228
$622$	0	0
$623$	41.4624	1.66116
$624$	0	0
$625$	0	0
$626$	0	0
$627$	4.14942	0.165712
$628$	0	0
$629$	−38.1979	−1.52305
$630$	0	0
$631$	1.15900	0.0461389	0.0230695	−	0.999734i	$-0.492656\pi$
$631$	1.15900	0.0461389	0.0230695	+	0.999734i	$0.492656\pi$
$632$	0	0
$633$	−5.90165	−0.234569
$634$	0	0
$635$	0	0
$636$	0	0
$637$	22.9009	0.907365
$638$	0	0
$639$	−12.4882	−0.494024
$640$	0	0
$641$	−5.15529	−0.203622	−0.101811	−	0.994804i	$-0.532464\pi$
$641$	−5.15529	−0.203622	−0.101811	+	0.994804i	$0.532464\pi$
$642$	0	0
$643$	−8.72811	−0.344203	−0.172102	−	0.985079i	$-0.555056\pi$
$643$	−8.72811	−0.344203	−0.172102	+	0.985079i	$0.555056\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10.4205	−0.409672	−0.204836	−	0.978796i	$-0.565666\pi$
$647$	−10.4205	−0.409672	−0.204836	+	0.978796i	$0.565666\pi$
$648$	0	0
$649$	−29.0720	−1.14117
$650$	0	0
$651$	6.06030	0.237522
$652$	0	0
$653$	−36.2315	−1.41785	−0.708924	−	0.705285i	$-0.750819\pi$
$653$	−36.2315	−1.41785	−0.708924	+	0.705285i	$0.750819\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−14.3279	−0.558983
$658$	0	0
$659$	13.4717	0.524783	0.262392	−	0.964961i	$-0.415489\pi$
$659$	13.4717	0.524783	0.262392	+	0.964961i	$0.415489\pi$
$660$	0	0
$661$	−18.9448	−0.736867	−0.368434	−	0.929654i	$-0.620106\pi$
$661$	−18.9448	−0.736867	−0.368434	+	0.929654i	$0.620106\pi$
$662$	0	0
$663$	−7.50786	−0.291581
$664$	0	0
$665$	0	0
$666$	0	0
$667$	9.19670	0.356098
$668$	0	0
$669$	4.25800	0.164624
$670$	0	0
$671$	4.36568	0.168535
$672$	0	0
$673$	−27.3807	−1.05545	−0.527725	−	0.849415i	$-0.676955\pi$
$673$	−27.3807	−1.05545	−0.527725	+	0.849415i	$0.676955\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−23.9986	−0.922342	−0.461171	−	0.887311i	$-0.652571\pi$
$677$	−23.9986	−0.922342	−0.461171	+	0.887311i	$0.652571\pi$
$678$	0	0
$679$	−15.1330	−0.580753
$680$	0	0
$681$	0.0650665	0.00249335
$682$	0	0
$683$	28.6152	1.09493	0.547464	−	0.836829i	$-0.315593\pi$
$683$	28.6152	1.09493	0.547464	+	0.836829i	$0.315593\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−0.272892	−0.0104115
$688$	0	0
$689$	−12.6337	−0.481307
$690$	0	0
$691$	−24.0645	−0.915457	−0.457728	−	0.889092i	$-0.651337\pi$
$691$	−24.0645	−0.915457	−0.457728	+	0.889092i	$0.651337\pi$
$692$	0	0
$693$	31.3749	1.19183
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−49.6406	−1.88027
$698$	0	0
$699$	−6.68701	−0.252926
$700$	0	0
$701$	41.5376	1.56885	0.784427	−	0.620221i	$-0.212957\pi$
$701$	41.5376	1.56885	0.784427	+	0.620221i	$0.212957\pi$
$702$	0	0
$703$	−31.2144	−1.17727
$704$	0	0
$705$	0	0
$706$	0	0
$707$	36.0483	1.35574
$708$	0	0
$709$	−0.239337	−0.00898849	−0.00449424	−	0.999990i	$-0.501431\pi$
$709$	−0.239337	−0.00898849	−0.00449424	+	0.999990i	$0.501431\pi$
$710$	0	0
$711$	−6.18146	−0.231823
$712$	0	0
$713$	−5.89980	−0.220949
$714$	0	0
$715$	0	0
$716$	0	0
$717$	3.40229	0.127061
$718$	0	0
$719$	18.2322	0.679948	0.339974	−	0.940435i	$-0.389582\pi$
$719$	18.2322	0.679948	0.339974	+	0.940435i	$0.389582\pi$
$720$	0	0
$721$	24.9623	0.929645
$722$	0	0
$723$	−4.54280	−0.168949
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.16149	0.0430772	0.0215386	−	0.999768i	$-0.493144\pi$
$727$	1.16149	0.0430772	0.0215386	+	0.999768i	$0.493144\pi$
$728$	0	0
$729$	−22.3574	−0.828052
$730$	0	0
$731$	−30.4920	−1.12779
$732$	0	0
$733$	16.9474	0.625967	0.312984	−	0.949759i	$-0.398671\pi$
$733$	16.9474	0.625967	0.312984	+	0.949759i	$0.398671\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−10.8913	−0.401186
$738$	0	0
$739$	−30.7566	−1.13140	−0.565700	−	0.824611i	$-0.691394\pi$
$739$	−30.7566	−1.13140	−0.565700	+	0.824611i	$0.691394\pi$
$740$	0	0
$741$	−6.13524	−0.225384
$742$	0	0
$743$	6.66705	0.244591	0.122295	−	0.992494i	$-0.460975\pi$
$743$	6.66705	0.244591	0.122295	+	0.992494i	$0.460975\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	46.6364	1.70634
$748$	0	0
$749$	24.4728	0.894217
$750$	0	0
$751$	−8.32271	−0.303700	−0.151850	−	0.988404i	$-0.548523\pi$
$751$	−8.32271	−0.303700	−0.151850	+	0.988404i	$0.548523\pi$
$752$	0	0
$753$	−6.27683	−0.228740
$754$	0	0
$755$	0	0
$756$	0	0
$757$	34.5436	1.25551	0.627754	−	0.778412i	$-0.283974\pi$
$757$	34.5436	1.25551	0.627754	+	0.778412i	$0.283974\pi$
$758$	0	0
$759$	0.924314	0.0335505
$760$	0	0
$761$	−42.1781	−1.52896	−0.764478	−	0.644650i	$-0.777003\pi$
$761$	−42.1781	−1.52896	−0.764478	+	0.644650i	$0.777003\pi$
$762$	0	0
$763$	−34.3289	−1.24279
$764$	0	0
$765$	0	0
$766$	0	0
$767$	42.9851	1.55210
$768$	0	0
$769$	13.0704	0.471332	0.235666	−	0.971834i	$-0.424273\pi$
$769$	13.0704	0.471332	0.235666	+	0.971834i	$0.424273\pi$
$770$	0	0
$771$	−0.836115	−0.0301120
$772$	0	0
$773$	−7.14112	−0.256848	−0.128424	−	0.991719i	$-0.540992\pi$
$773$	−7.14112	−0.256848	−0.128424	+	0.991719i	$0.540992\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	7.14239	0.256232
$778$	0	0
$779$	−40.5651	−1.45340
$780$	0	0
$781$	−13.3540	−0.477843
$782$	0	0
$783$	16.1395	0.576780
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−30.5980	−1.09070	−0.545351	−	0.838208i	$-0.683604\pi$
$787$	−30.5980	−1.09070	−0.545351	+	0.838208i	$0.683604\pi$
$788$	0	0
$789$	−5.00565	−0.178206
$790$	0	0
$791$	44.9145	1.59698
$792$	0	0
$793$	−6.45500	−0.229224
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−32.8338	−1.16303	−0.581516	−	0.813535i	$-0.697540\pi$
$797$	−32.8338	−1.16303	−0.581516	+	0.813535i	$0.697540\pi$
$798$	0	0
$799$	30.1321	1.06600
$800$	0	0
$801$	−34.8903	−1.23279
$802$	0	0
$803$	−15.3212	−0.540674
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−8.25743	−0.290675
$808$	0	0
$809$	−43.7461	−1.53803	−0.769015	−	0.639230i	$-0.779253\pi$
$809$	−43.7461	−1.53803	−0.769015	+	0.639230i	$0.779253\pi$
$810$	0	0
$811$	38.0475	1.33603	0.668015	−	0.744148i	$-0.267144\pi$
$811$	38.0475	1.33603	0.668015	+	0.744148i	$0.267144\pi$
$812$	0	0
$813$	1.40674	0.0493364
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−24.9173	−0.871746
$818$	0	0
$819$	−46.3902	−1.62101
$820$	0	0
$821$	9.79962	0.342009	0.171004	−	0.985270i	$-0.445299\pi$
$821$	9.79962	0.342009	0.171004	+	0.985270i	$0.445299\pi$
$822$	0	0
$823$	−33.2526	−1.15911	−0.579557	−	0.814932i	$-0.696774\pi$
$823$	−33.2526	−1.15911	−0.579557	+	0.814932i	$0.696774\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−11.9711	−0.416275	−0.208138	−	0.978100i	$-0.566740\pi$
$827$	−11.9711	−0.416275	−0.208138	+	0.978100i	$0.566740\pi$
$828$	0	0
$829$	−7.99628	−0.277722	−0.138861	−	0.990312i	$-0.544344\pi$
$829$	−7.99628	−0.277722	−0.138861	+	0.990312i	$0.544344\pi$
$830$	0	0
$831$	0.292030	0.0101304
$832$	0	0
$833$	27.3259	0.946787
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−10.3537	−0.357877
$838$	0	0
$839$	8.94284	0.308741	0.154371	−	0.988013i	$-0.450665\pi$
$839$	8.94284	0.308741	0.154371	+	0.988013i	$0.450665\pi$
$840$	0	0
$841$	55.5793	1.91653
$842$	0	0
$843$	−7.16773	−0.246870
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−4.51393	−0.155100
$848$	0	0
$849$	6.72427	0.230777
$850$	0	0
$851$	−6.95324	−0.238354
$852$	0	0
$853$	44.8144	1.53442	0.767208	−	0.641398i	$-0.221645\pi$
$853$	44.8144	1.53442	0.767208	+	0.641398i	$0.221645\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−10.2025	−0.348510	−0.174255	−	0.984701i	$-0.555752\pi$
$857$	−10.2025	−0.348510	−0.174255	+	0.984701i	$0.555752\pi$
$858$	0	0
$859$	−30.0587	−1.02559	−0.512796	−	0.858511i	$-0.671390\pi$
$859$	−30.0587	−1.02559	−0.512796	+	0.858511i	$0.671390\pi$
$860$	0	0
$861$	9.28199	0.316329
$862$	0	0
$863$	19.9013	0.677449	0.338724	−	0.940886i	$-0.390005\pi$
$863$	19.9013	0.677449	0.338724	+	0.940886i	$0.390005\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−3.91217	−0.132864
$868$	0	0
$869$	−6.61003	−0.224230
$870$	0	0
$871$	16.1036	0.545651
$872$	0	0
$873$	12.7344	0.430993
$874$	0	0
$875$	0	0
$876$	0	0
$877$	49.0835	1.65743	0.828717	−	0.559668i	$-0.189071\pi$
$877$	49.0835	1.65743	0.828717	+	0.559668i	$0.189071\pi$
$878$	0	0
$879$	−3.84917	−0.129829
$880$	0	0
$881$	25.0658	0.844489	0.422245	−	0.906482i	$-0.361242\pi$
$881$	25.0658	0.844489	0.422245	+	0.906482i	$0.361242\pi$
$882$	0	0
$883$	−35.7237	−1.20220	−0.601100	−	0.799174i	$-0.705271\pi$
$883$	−35.7237	−1.20220	−0.601100	+	0.799174i	$0.705271\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	27.4000	0.920001	0.460001	−	0.887919i	$-0.347849\pi$
$887$	27.4000	0.920001	0.460001	+	0.887919i	$0.347849\pi$
$888$	0	0
$889$	−51.7893	−1.73696
$890$	0	0
$891$	−25.5786	−0.856917
$892$	0	0
$893$	24.6232	0.823983
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.36667	−0.0456318
$898$	0	0
$899$	−54.2587	−1.80963
$900$	0	0
$901$	−15.0749	−0.502218
$902$	0	0
$903$	5.70150	0.189734
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−53.9804	−1.79239	−0.896195	−	0.443661i	$-0.853679\pi$
$907$	−53.9804	−1.79239	−0.896195	+	0.443661i	$0.853679\pi$
$908$	0	0
$909$	−30.3344	−1.00613
$910$	0	0
$911$	8.29792	0.274922	0.137461	−	0.990507i	$-0.456106\pi$
$911$	8.29792	0.274922	0.137461	+	0.990507i	$0.456106\pi$
$912$	0	0
$913$	49.8698	1.65045
$914$	0	0
$915$	0	0
$916$	0	0
$917$	64.3440	2.12482
$918$	0	0
$919$	−42.3852	−1.39816	−0.699080	−	0.715043i	$-0.746407\pi$
$919$	−42.3852	−1.39816	−0.699080	+	0.715043i	$0.746407\pi$
$920$	0	0
$921$	−8.14490	−0.268384
$922$	0	0
$923$	19.7449	0.649911
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−21.0056	−0.689915
$928$	0	0
$929$	22.9844	0.754092	0.377046	−	0.926194i	$-0.376940\pi$
$929$	22.9844	0.754092	0.377046	+	0.926194i	$0.376940\pi$
$930$	0	0
$931$	22.3301	0.731838
$932$	0	0
$933$	9.16236	0.299962
$934$	0	0
$935$	0	0
$936$	0	0
$937$	37.8592	1.23681	0.618403	−	0.785861i	$-0.287780\pi$
$937$	37.8592	1.23681	0.618403	+	0.785861i	$0.287780\pi$
$938$	0	0
$939$	1.08134	0.0352881
$940$	0	0
$941$	−7.13773	−0.232683	−0.116342	−	0.993209i	$-0.537117\pi$
$941$	−7.13773	−0.232683	−0.116342	+	0.993209i	$0.537117\pi$
$942$	0	0
$943$	−9.03617	−0.294258
$944$	0	0
$945$	0	0
$946$	0	0
$947$	23.1645	0.752745	0.376372	−	0.926468i	$-0.377171\pi$
$947$	23.1645	0.752745	0.376372	+	0.926468i	$0.377171\pi$
$948$	0	0
$949$	22.6536	0.735368
$950$	0	0
$951$	−7.13454	−0.231353
$952$	0	0
$953$	−37.0608	−1.20052	−0.600258	−	0.799806i	$-0.704935\pi$
$953$	−37.0608	−1.20052	−0.600258	+	0.799806i	$0.704935\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.50064	0.274787
$958$	0	0
$959$	68.9646	2.22698
$960$	0	0
$961$	3.80769	0.122829
$962$	0	0
$963$	−20.5937	−0.663623
$964$	0	0
$965$	0	0
$966$	0	0
$967$	47.2555	1.51964	0.759818	−	0.650136i	$-0.225288\pi$
$967$	47.2555	1.51964	0.759818	+	0.650136i	$0.225288\pi$
$968$	0	0
$969$	−7.32073	−0.235176
$970$	0	0
$971$	−53.8987	−1.72969	−0.864845	−	0.502038i	$-0.832584\pi$
$971$	−53.8987	−1.72969	−0.864845	+	0.502038i	$0.832584\pi$
$972$	0	0
$973$	−1.19982	−0.0384644
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−61.0840	−1.95425	−0.977126	−	0.212662i	$-0.931787\pi$
$977$	−61.0840	−1.95425	−0.977126	+	0.212662i	$0.931787\pi$
$978$	0	0
$979$	−37.3093	−1.19241
$980$	0	0
$981$	28.8876	0.922308
$982$	0	0
$983$	−53.4981	−1.70632	−0.853162	−	0.521645i	$-0.825318\pi$
$983$	−53.4981	−1.70632	−0.853162	+	0.521645i	$0.825318\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−5.63420	−0.179339
$988$	0	0
$989$	−5.55051	−0.176496
$990$	0	0
$991$	18.2274	0.579013	0.289506	−	0.957176i	$-0.406509\pi$
$991$	18.2274	0.579013	0.289506	+	0.957176i	$0.406509\pi$
$992$	0	0
$993$	6.47125	0.205359
$994$	0	0
$995$	0	0
$996$	0	0
$997$	55.7823	1.76664	0.883322	−	0.468767i	$-0.155302\pi$
$997$	55.7823	1.76664	0.883322	+	0.468767i	$0.155302\pi$
$998$	0	0
$999$	−12.2024	−0.386068

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.bj.1.5		8
4.3	odd	2		9200.2.a.de.1.4		8
5.2	odd	4		920.2.e.c.369.9	yes	16
5.3	odd	4		920.2.e.c.369.8	✓	16
5.4	even	2		4600.2.a.bk.1.4		8
20.3	even	4		1840.2.e.h.369.9		16
20.7	even	4		1840.2.e.h.369.8		16
20.19	odd	2		9200.2.a.dd.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.e.c.369.8	✓	16	5.3	odd	4
920.2.e.c.369.9	yes	16	5.2	odd	4
1840.2.e.h.369.8		16	20.7	even	4
1840.2.e.h.369.9		16	20.3	even	4
4600.2.a.bj.1.5		8	1.1	even	1	trivial
4600.2.a.bk.1.4		8	5.4	even	2
9200.2.a.dd.1.5		8	20.19	odd	2
9200.2.a.de.1.4		8	4.3	odd	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.296848 q^{3} +3.46037 q^{7} -2.91188 q^{9} +O(q^{10})\)	\(q-0.296848 q^{3} +3.46037 q^{7} -2.91188 q^{9} -3.11377 q^{11} +4.60394 q^{13} +5.49355 q^{17} +4.48919 q^{19} -1.02720 q^{21} +1.00000 q^{23} +1.75493 q^{27} +9.19670 q^{29} -5.89980 q^{31} +0.924314 q^{33} -6.95324 q^{37} -1.36667 q^{39} -9.03617 q^{41} -5.55051 q^{43} +5.48499 q^{47} +4.97418 q^{49} -1.63075 q^{51} -2.74411 q^{53} -1.33261 q^{57} +9.33659 q^{59} -1.40206 q^{61} -10.0762 q^{63} +3.49779 q^{67} -0.296848 q^{69} +4.28869 q^{71} +4.92048 q^{73} -10.7748 q^{77} +2.12284 q^{79} +8.21470 q^{81} -16.0159 q^{83} -2.73002 q^{87} +11.9821 q^{89} +15.9314 q^{91} +1.75134 q^{93} -4.37324 q^{97} +9.06692 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9}+O(q^{10}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9	\( 8 q - 3 q^{3} - 7 q^{7} + 11 q^{9} + 7 q^{11} + 11 q^{13} - 7 q^{17} + 11 q^{19} + 8 q^{23} - 12 q^{27} + 22 q^{29} + 9 q^{31} - 9 q^{33} + 4 q^{37} + 7 q^{41} - 22 q^{43} - 4 q^{47} + 39 q^{49} - 19 q^{51} + 4 q^{53} + 32 q^{59} + 17 q^{61} - 44 q^{63} + 4 q^{67} - 3 q^{69} + 15 q^{71} + 6 q^{73} + 18 q^{77} - 2 q^{79} + 24 q^{81} - 36 q^{83} + 4 q^{87} + 46 q^{89} - 35 q^{91} + 20 q^{93} + 3 q^{97} + 61 q^{99}+O(q^{100}) \) 8 * q - 3 * q^3 - 7 * q^7 + 11 * q^9 + 7 * q^11 + 11 * q^13 - 7 * q^17 + 11 * q^19 + 8 * q^23 - 12 * q^27 + 22 * q^29 + 9 * q^31 - 9 * q^33 + 4 * q^37 + 7 * q^41 - 22 * q^43 - 4 * q^47 + 39 * q^49 - 19 * q^51 + 4 * q^53 + 32 * q^59 + 17 * q^61 - 44 * q^63 + 4 * q^67 - 3 * q^69 + 15 * q^71 + 6 * q^73 + 18 * q^77 - 2 * q^79 + 24 * q^81 - 36 * q^83 + 4 * q^87 + 46 * q^89 - 35 * q^91 + 20 * q^93 + 3 * q^97 + 61 * q^99

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