LMFDB - Embedded newform 5000.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5000,2,Mod(1,5000)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5000, 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("5000.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5000 = 2^{3} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5000.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:	$39.9252010106$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 26 x^{14} + 110 x^{13} + 250 x^{12} - 1154 x^{11} - 1074 x^{10} + 5784 x^{9} + \cdots + 80 $ x^16 - 4x^15 - 26x^14 + 110x^13 + 250x^12 - 1154x^11 - 1074x^10 + 5784x^9 + 1890x^8 - 14210x^7 - 726x^6 + 15974x^5 + 61x^4 - 7820x^3 - 180x^2 + 1360*x + 80
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 5^{3} $
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.71029$ of defining polynomial
Character	$\chi$	$=$	5000.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.71029 q^{3} -0.760910 q^{7} +4.34569 q^{9} -6.48670 q^{11} +5.73162 q^{13} -2.72868 q^{17} -5.57409 q^{19} +2.06229 q^{21} -2.90834 q^{23} -3.64722 q^{27} -1.51092 q^{29} -2.16780 q^{31} +17.5809 q^{33} -1.49315 q^{37} -15.5344 q^{39} +4.49950 q^{41} -0.0184418 q^{43} +8.54996 q^{47} -6.42102 q^{49} +7.39552 q^{51} -5.28679 q^{53} +15.1074 q^{57} -7.91593 q^{59} +3.63893 q^{61} -3.30668 q^{63} +4.87160 q^{67} +7.88246 q^{69} -7.29823 q^{71} -11.2129 q^{73} +4.93580 q^{77} +4.77881 q^{79} -3.15204 q^{81} -4.71005 q^{83} +4.09502 q^{87} -9.86350 q^{89} -4.36124 q^{91} +5.87537 q^{93} +0.763267 q^{97} -28.1892 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 4 q^{3} + 8 q^{7} + 20 q^{9} - 12 q^{11} + 10 q^{13} + 8 q^{17} - 12 q^{19} + 8 q^{21} + 12 q^{23} + 22 q^{27} + 16 q^{29} - 2 q^{31} + 24 q^{33} + 22 q^{37} - 4 q^{39} + 20 q^{41} + 26 q^{43} + 24 q^{47}+ \cdots - 46 q^{99}+O(q^{100}) $ 16 * q + 4 * q^3 + 8 * q^7 + 20 * q^9 - 12 * q^11 + 10 * q^13 + 8 * q^17 - 12 * q^19 + 8 * q^21 + 12 * q^23 + 22 * q^27 + 16 * q^29 - 2 * q^31 + 24 * q^33 + 22 * q^37 - 4 * q^39 + 20 * q^41 + 26 * q^43 + 24 * q^47 + 30 * q^49 - 30 * q^51 + 16 * q^53 + 24 * q^57 - 40 * q^59 + 22 * q^61 + 52 * q^63 + 50 * q^67 + 22 * q^69 - 4 * q^71 + 32 * q^73 + 30 * q^77 - 8 * q^79 + 44 * q^81 + 30 * q^83 + 48 * q^87 + 28 * q^89 - 32 * q^91 + 40 * q^93 + 36 * q^97 - 46 * 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
$3$	−2.71029	−1.56479	−0.782394	−	0.622783i	$-0.786002\pi$
$3$	−2.71029	−1.56479	−0.782394	+	0.622783i	$0.786002\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.760910	−0.287597	−0.143798	−	0.989607i	$-0.545932\pi$
$7$	−0.760910	−0.287597	−0.143798	+	0.989607i	$0.545932\pi$
$8$	0	0
$9$	4.34569	1.44856
$10$	0	0
$11$	−6.48670	−1.95581	−0.977907	−	0.209040i	$-0.932966\pi$
$11$	−6.48670	−1.95581	−0.977907	+	0.209040i	$0.932966\pi$
$12$	0	0
$13$	5.73162	1.58966	0.794832	−	0.606829i	$-0.207559\pi$
$13$	5.73162	1.58966	0.794832	+	0.606829i	$0.207559\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.72868	−0.661802	−0.330901	−	0.943666i	$-0.607353\pi$
$17$	−2.72868	−0.661802	−0.330901	+	0.943666i	$0.607353\pi$
$18$	0	0
$19$	−5.57409	−1.27878	−0.639392	−	0.768881i	$-0.720814\pi$
$19$	−5.57409	−1.27878	−0.639392	+	0.768881i	$0.720814\pi$
$20$	0	0
$21$	2.06229	0.450028
$22$	0	0
$23$	−2.90834	−0.606431	−0.303216	−	0.952922i	$-0.598060\pi$
$23$	−2.90834	−0.606431	−0.303216	+	0.952922i	$0.598060\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−3.64722	−0.701908
$28$	0	0
$29$	−1.51092	−0.280570	−0.140285	−	0.990111i	$-0.544802\pi$
$29$	−1.51092	−0.280570	−0.140285	+	0.990111i	$0.544802\pi$
$30$	0	0
$31$	−2.16780	−0.389348	−0.194674	−	0.980868i	$-0.562365\pi$
$31$	−2.16780	−0.389348	−0.194674	+	0.980868i	$0.562365\pi$
$32$	0	0
$33$	17.5809	3.06044
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−1.49315	−0.245473	−0.122737	−	0.992439i	$-0.539167\pi$
$37$	−1.49315	−0.245473	−0.122737	+	0.992439i	$0.539167\pi$
$38$	0	0
$39$	−15.5344	−2.48749
$40$	0	0
$41$	4.49950	0.702704	0.351352	−	0.936243i	$-0.385722\pi$
$41$	4.49950	0.702704	0.351352	+	0.936243i	$0.385722\pi$
$42$	0	0
$43$	−0.0184418	−0.00281235	−0.00140617	−	0.999999i	$-0.500448\pi$
$43$	−0.0184418	−0.00281235	−0.00140617	+	0.999999i	$0.500448\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.54996	1.24714	0.623570	−	0.781767i	$-0.285682\pi$
$47$	8.54996	1.24714	0.623570	+	0.781767i	$0.285682\pi$
$48$	0	0
$49$	−6.42102	−0.917288
$50$	0	0
$51$	7.39552	1.03558
$52$	0	0
$53$	−5.28679	−0.726196	−0.363098	−	0.931751i	$-0.618281\pi$
$53$	−5.28679	−0.726196	−0.363098	+	0.931751i	$0.618281\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	15.1074	2.00103
$58$	0	0
$59$	−7.91593	−1.03057	−0.515283	−	0.857020i	$-0.672313\pi$
$59$	−7.91593	−1.03057	−0.515283	+	0.857020i	$0.672313\pi$
$60$	0	0
$61$	3.63893	0.465918	0.232959	−	0.972487i	$-0.425159\pi$
$61$	3.63893	0.465918	0.232959	+	0.972487i	$0.425159\pi$
$62$	0	0
$63$	−3.30668	−0.416602
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.87160	0.595161	0.297581	−	0.954697i	$-0.403820\pi$
$67$	4.87160	0.595161	0.297581	+	0.954697i	$0.403820\pi$
$68$	0	0
$69$	7.88246	0.948937
$70$	0	0
$71$	−7.29823	−0.866141	−0.433070	−	0.901360i	$-0.642570\pi$
$71$	−7.29823	−0.866141	−0.433070	+	0.901360i	$0.642570\pi$
$72$	0	0
$73$	−11.2129	−1.31237	−0.656184	−	0.754600i	$-0.727831\pi$
$73$	−11.2129	−1.31237	−0.656184	+	0.754600i	$0.727831\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.93580	0.562486
$78$	0	0
$79$	4.77881	0.537659	0.268829	−	0.963188i	$-0.413363\pi$
$79$	4.77881	0.537659	0.268829	+	0.963188i	$0.413363\pi$
$80$	0	0
$81$	−3.15204	−0.350227
$82$	0	0
$83$	−4.71005	−0.516995	−0.258498	−	0.966012i	$-0.583227\pi$
$83$	−4.71005	−0.516995	−0.258498	+	0.966012i	$0.583227\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.09502	0.439033
$88$	0	0
$89$	−9.86350	−1.04553	−0.522764	−	0.852477i	$-0.675099\pi$
$89$	−9.86350	−1.04553	−0.522764	+	0.852477i	$0.675099\pi$
$90$	0	0
$91$	−4.36124	−0.457183
$92$	0	0
$93$	5.87537	0.609247
$94$	0	0
$95$	0	0
$96$	0	0
$97$	0.763267	0.0774981	0.0387490	−	0.999249i	$-0.487663\pi$
$97$	0.763267	0.0774981	0.0387490	+	0.999249i	$0.487663\pi$
$98$	0	0
$99$	−28.1892	−2.83312
$100$	0	0
$101$	−16.8574	−1.67737	−0.838685	−	0.544617i	$-0.816675\pi$
$101$	−16.8574	−1.67737	−0.838685	+	0.544617i	$0.816675\pi$
$102$	0	0
$103$	3.01118	0.296701	0.148350	−	0.988935i	$-0.452604\pi$
$103$	3.01118	0.296701	0.148350	+	0.988935i	$0.452604\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.8596	1.14651	0.573255	−	0.819377i	$-0.305680\pi$
$107$	11.8596	1.14651	0.573255	+	0.819377i	$0.305680\pi$
$108$	0	0
$109$	−14.0958	−1.35014	−0.675068	−	0.737756i	$-0.735886\pi$
$109$	−14.0958	−1.35014	−0.675068	+	0.737756i	$0.735886\pi$
$110$	0	0
$111$	4.04689	0.384113
$112$	0	0
$113$	5.92783	0.557643	0.278822	−	0.960343i	$-0.410056\pi$
$113$	5.92783	0.557643	0.278822	+	0.960343i	$0.410056\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	24.9078	2.30273
$118$	0	0
$119$	2.07628	0.190332
$120$	0	0
$121$	31.0773	2.82521
$122$	0	0
$123$	−12.1950	−1.09958
$124$	0	0
$125$	0	0
$126$	0	0
$127$	9.08189	0.805888	0.402944	−	0.915225i	$-0.367987\pi$
$127$	9.08189	0.805888	0.402944	+	0.915225i	$0.367987\pi$
$128$	0	0
$129$	0.0499827	0.00440073
$130$	0	0
$131$	6.22054	0.543491	0.271745	−	0.962369i	$-0.412399\pi$
$131$	6.22054	0.543491	0.271745	+	0.962369i	$0.412399\pi$
$132$	0	0
$133$	4.24138	0.367774
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.2460	1.38799	0.693995	−	0.719980i	$-0.255849\pi$
$137$	16.2460	1.38799	0.693995	+	0.719980i	$0.255849\pi$
$138$	0	0
$139$	−10.1343	−0.859578	−0.429789	−	0.902929i	$-0.641412\pi$
$139$	−10.1343	−0.859578	−0.429789	+	0.902929i	$0.641412\pi$
$140$	0	0
$141$	−23.1729	−1.95151
$142$	0	0
$143$	−37.1793	−3.10909
$144$	0	0
$145$	0	0
$146$	0	0
$147$	17.4028	1.43536
$148$	0	0
$149$	−12.0280	−0.985371	−0.492686	−	0.870207i	$-0.663985\pi$
$149$	−12.0280	−0.985371	−0.492686	+	0.870207i	$0.663985\pi$
$150$	0	0
$151$	12.4820	1.01577	0.507884	−	0.861425i	$-0.330428\pi$
$151$	12.4820	1.01577	0.507884	+	0.861425i	$0.330428\pi$
$152$	0	0
$153$	−11.8580	−0.958662
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−21.7736	−1.73772	−0.868861	−	0.495056i	$-0.835148\pi$
$157$	−21.7736	−1.73772	−0.868861	+	0.495056i	$0.835148\pi$
$158$	0	0
$159$	14.3287	1.13634
$160$	0	0
$161$	2.21299	0.174408
$162$	0	0
$163$	6.47354	0.507047	0.253524	−	0.967329i	$-0.418410\pi$
$163$	6.47354	0.507047	0.253524	+	0.967329i	$0.418410\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.2070	−1.17675	−0.588377	−	0.808587i	$-0.700233\pi$
$167$	−15.2070	−1.17675	−0.588377	+	0.808587i	$0.700233\pi$
$168$	0	0
$169$	19.8514	1.52703
$170$	0	0
$171$	−24.2233	−1.85240
$172$	0	0
$173$	−0.110462	−0.00839828	−0.00419914	−	0.999991i	$-0.501337\pi$
$173$	−0.110462	−0.00839828	−0.00419914	+	0.999991i	$0.501337\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	21.4545	1.61262
$178$	0	0
$179$	−8.39526	−0.627491	−0.313746	−	0.949507i	$-0.601584\pi$
$179$	−8.39526	−0.627491	−0.313746	+	0.949507i	$0.601584\pi$
$180$	0	0
$181$	5.18226	0.385194	0.192597	−	0.981278i	$-0.438309\pi$
$181$	5.18226	0.385194	0.192597	+	0.981278i	$0.438309\pi$
$182$	0	0
$183$	−9.86258	−0.729063
$184$	0	0
$185$	0	0
$186$	0	0
$187$	17.7001	1.29436
$188$	0	0
$189$	2.77520	0.201866
$190$	0	0
$191$	26.6183	1.92603	0.963017	−	0.269441i	$-0.0868388\pi$
$191$	26.6183	1.92603	0.963017	+	0.269441i	$0.0868388\pi$
$192$	0	0
$193$	21.8300	1.57136	0.785678	−	0.618635i	$-0.212314\pi$
$193$	21.8300	1.57136	0.785678	+	0.618635i	$0.212314\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−4.53464	−0.323080	−0.161540	−	0.986866i	$-0.551646\pi$
$197$	−4.53464	−0.323080	−0.161540	+	0.986866i	$0.551646\pi$
$198$	0	0
$199$	15.5338	1.10116	0.550582	−	0.834781i	$-0.314406\pi$
$199$	15.5338	1.10116	0.550582	+	0.834781i	$0.314406\pi$
$200$	0	0
$201$	−13.2035	−0.931302
$202$	0	0
$203$	1.14967	0.0806910
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−12.6388	−0.878455
$208$	0	0
$209$	36.1575	2.50106
$210$	0	0
$211$	10.2158	0.703284	0.351642	−	0.936135i	$-0.385623\pi$
$211$	10.2158	0.703284	0.351642	+	0.936135i	$0.385623\pi$
$212$	0	0
$213$	19.7803	1.35533
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.64950	0.111975
$218$	0	0
$219$	30.3902	2.05358
$220$	0	0
$221$	−15.6397	−1.05204
$222$	0	0
$223$	16.8138	1.12593	0.562967	−	0.826479i	$-0.309660\pi$
$223$	16.8138	1.12593	0.562967	+	0.826479i	$0.309660\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1.59655	−0.105966	−0.0529832	−	0.998595i	$-0.516873\pi$
$227$	−1.59655	−0.105966	−0.0529832	+	0.998595i	$0.516873\pi$
$228$	0	0
$229$	6.63184	0.438244	0.219122	−	0.975697i	$-0.429681\pi$
$229$	6.63184	0.438244	0.219122	+	0.975697i	$0.429681\pi$
$230$	0	0
$231$	−13.3775	−0.880172
$232$	0	0
$233$	−21.5364	−1.41090	−0.705449	−	0.708761i	$-0.749255\pi$
$233$	−21.5364	−1.41090	−0.705449	+	0.708761i	$0.749255\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−12.9520	−0.841322
$238$	0	0
$239$	3.52263	0.227860	0.113930	−	0.993489i	$-0.463656\pi$
$239$	3.52263	0.227860	0.113930	+	0.993489i	$0.463656\pi$
$240$	0	0
$241$	5.20231	0.335110	0.167555	−	0.985863i	$-0.446413\pi$
$241$	5.20231	0.335110	0.167555	+	0.985863i	$0.446413\pi$
$242$	0	0
$243$	19.4846	1.24994
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−31.9486	−2.03284
$248$	0	0
$249$	12.7656	0.808988
$250$	0	0
$251$	10.9106	0.688669	0.344335	−	0.938847i	$-0.388105\pi$
$251$	10.9106	0.688669	0.344335	+	0.938847i	$0.388105\pi$
$252$	0	0
$253$	18.8656	1.18607
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.2429	0.826069	0.413034	−	0.910715i	$-0.364469\pi$
$257$	13.2429	0.826069	0.413034	+	0.910715i	$0.364469\pi$
$258$	0	0
$259$	1.13616	0.0705973
$260$	0	0
$261$	−6.56597	−0.406423
$262$	0	0
$263$	−15.7116	−0.968820	−0.484410	−	0.874841i	$-0.660966\pi$
$263$	−15.7116	−0.968820	−0.484410	+	0.874841i	$0.660966\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	26.7330	1.63603
$268$	0	0
$269$	21.8907	1.33470	0.667348	−	0.744746i	$-0.267429\pi$
$269$	21.8907	1.33470	0.667348	+	0.744746i	$0.267429\pi$
$270$	0	0
$271$	3.58346	0.217680	0.108840	−	0.994059i	$-0.465286\pi$
$271$	3.58346	0.217680	0.108840	+	0.994059i	$0.465286\pi$
$272$	0	0
$273$	11.8202	0.715394
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10.3940	−0.624516	−0.312258	−	0.949997i	$-0.601085\pi$
$277$	−10.3940	−0.624516	−0.312258	+	0.949997i	$0.601085\pi$
$278$	0	0
$279$	−9.42058	−0.563995
$280$	0	0
$281$	23.9989	1.43166	0.715828	−	0.698276i	$-0.246049\pi$
$281$	23.9989	1.43166	0.715828	+	0.698276i	$0.246049\pi$
$282$	0	0
$283$	3.39388	0.201745	0.100873	−	0.994899i	$-0.467837\pi$
$283$	3.39388	0.201745	0.100873	+	0.994899i	$0.467837\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.42372	−0.202096
$288$	0	0
$289$	−9.55432	−0.562019
$290$	0	0
$291$	−2.06868	−0.121268
$292$	0	0
$293$	0.853181	0.0498434	0.0249217	−	0.999689i	$-0.492066\pi$
$293$	0.853181	0.0498434	0.0249217	+	0.999689i	$0.492066\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	23.6584	1.37280
$298$	0	0
$299$	−16.6695	−0.964022
$300$	0	0
$301$	0.0140325	0.000808822 0
$302$	0	0
$303$	45.6884	2.62473
$304$	0	0
$305$	0	0
$306$	0	0
$307$	32.9825	1.88241	0.941207	−	0.337832i	$-0.109693\pi$
$307$	32.9825	1.88241	0.941207	+	0.337832i	$0.109693\pi$
$308$	0	0
$309$	−8.16119	−0.464274
$310$	0	0
$311$	3.26183	0.184961	0.0924807	−	0.995714i	$-0.470520\pi$
$311$	3.26183	0.184961	0.0924807	+	0.995714i	$0.470520\pi$
$312$	0	0
$313$	5.23551	0.295928	0.147964	−	0.988993i	$-0.452728\pi$
$313$	5.23551	0.295928	0.147964	+	0.988993i	$0.452728\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−21.5986	−1.21310	−0.606548	−	0.795047i	$-0.707446\pi$
$317$	−21.5986	−1.21310	−0.606548	+	0.795047i	$0.707446\pi$
$318$	0	0
$319$	9.80086	0.548743
$320$	0	0
$321$	−32.1430	−1.79405
$322$	0	0
$323$	15.2099	0.846301
$324$	0	0
$325$	0	0
$326$	0	0
$327$	38.2038	2.11268
$328$	0	0
$329$	−6.50575	−0.358674
$330$	0	0
$331$	6.08386	0.334399	0.167200	−	0.985923i	$-0.446528\pi$
$331$	6.08386	0.334399	0.167200	+	0.985923i	$0.446528\pi$
$332$	0	0
$333$	−6.48879	−0.355583
$334$	0	0
$335$	0	0
$336$	0	0
$337$	9.70193	0.528498	0.264249	−	0.964455i	$-0.414876\pi$
$337$	9.70193	0.528498	0.264249	+	0.964455i	$0.414876\pi$
$338$	0	0
$339$	−16.0662	−0.872594
$340$	0	0
$341$	14.0619	0.761492
$342$	0	0
$343$	10.2122	0.551406
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.90768	−0.102409	−0.0512047	−	0.998688i	$-0.516306\pi$
$347$	−1.90768	−0.102409	−0.0512047	+	0.998688i	$0.516306\pi$
$348$	0	0
$349$	−1.02648	−0.0549462	−0.0274731	−	0.999623i	$-0.508746\pi$
$349$	−1.02648	−0.0549462	−0.0274731	+	0.999623i	$0.508746\pi$
$350$	0	0
$351$	−20.9045	−1.11580
$352$	0	0
$353$	1.92893	0.102666	0.0513332	−	0.998682i	$-0.483653\pi$
$353$	1.92893	0.102666	0.0513332	+	0.998682i	$0.483653\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−5.62732	−0.297829
$358$	0	0
$359$	−21.9452	−1.15822	−0.579111	−	0.815249i	$-0.696600\pi$
$359$	−21.9452	−1.15822	−0.579111	+	0.815249i	$0.696600\pi$
$360$	0	0
$361$	12.0705	0.635289
$362$	0	0
$363$	−84.2286	−4.42086
$364$	0	0
$365$	0	0
$366$	0	0
$367$	18.9632	0.989871	0.494936	−	0.868930i	$-0.335192\pi$
$367$	18.9632	0.989871	0.494936	+	0.868930i	$0.335192\pi$
$368$	0	0
$369$	19.5534	1.01791
$370$	0	0
$371$	4.02277	0.208852
$372$	0	0
$373$	11.0240	0.570801	0.285401	−	0.958408i	$-0.407873\pi$
$373$	11.0240	0.570801	0.285401	+	0.958408i	$0.407873\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.65999	−0.446012
$378$	0	0
$379$	1.96246	0.100805	0.0504024	−	0.998729i	$-0.483950\pi$
$379$	1.96246	0.100805	0.0504024	+	0.998729i	$0.483950\pi$
$380$	0	0
$381$	−24.6146	−1.26104
$382$	0	0
$383$	−27.4873	−1.40454	−0.702268	−	0.711913i	$-0.747829\pi$
$383$	−27.4873	−1.40454	−0.702268	+	0.711913i	$0.747829\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.0801423	−0.00407386
$388$	0	0
$389$	32.2764	1.63648	0.818239	−	0.574878i	$-0.194951\pi$
$389$	32.2764	1.63648	0.818239	+	0.574878i	$0.194951\pi$
$390$	0	0
$391$	7.93593	0.401337
$392$	0	0
$393$	−16.8595	−0.850448
$394$	0	0
$395$	0	0
$396$	0	0
$397$	24.6152	1.23540	0.617701	−	0.786413i	$-0.288064\pi$
$397$	24.6152	1.23540	0.617701	+	0.786413i	$0.288064\pi$
$398$	0	0
$399$	−11.4954	−0.575489
$400$	0	0
$401$	18.5748	0.927581	0.463791	−	0.885945i	$-0.346489\pi$
$401$	18.5748	0.927581	0.463791	+	0.885945i	$0.346489\pi$
$402$	0	0
$403$	−12.4250	−0.618933
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.68565	0.480100
$408$	0	0
$409$	25.1825	1.24520	0.622598	−	0.782542i	$-0.286077\pi$
$409$	25.1825	1.24520	0.622598	+	0.782542i	$0.286077\pi$
$410$	0	0
$411$	−44.0314	−2.17191
$412$	0	0
$413$	6.02331	0.296388
$414$	0	0
$415$	0	0
$416$	0	0
$417$	27.4669	1.34506
$418$	0	0
$419$	−11.5811	−0.565772	−0.282886	−	0.959154i	$-0.591292\pi$
$419$	−11.5811	−0.565772	−0.282886	+	0.959154i	$0.591292\pi$
$420$	0	0
$421$	23.3236	1.13672	0.568362	−	0.822778i	$-0.307577\pi$
$421$	23.3236	1.13672	0.568362	+	0.822778i	$0.307577\pi$
$422$	0	0
$423$	37.1555	1.80656
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−2.76890	−0.133997
$428$	0	0
$429$	100.767	4.86507
$430$	0	0
$431$	9.97368	0.480415	0.240208	−	0.970722i	$-0.422784\pi$
$431$	9.97368	0.480415	0.240208	+	0.970722i	$0.422784\pi$
$432$	0	0
$433$	1.21527	0.0584021	0.0292010	−	0.999574i	$-0.490704\pi$
$433$	1.21527	0.0584021	0.0292010	+	0.999574i	$0.490704\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	16.2114	0.775495
$438$	0	0
$439$	−22.3661	−1.06748	−0.533739	−	0.845650i	$-0.679213\pi$
$439$	−22.3661	−1.06748	−0.533739	+	0.845650i	$0.679213\pi$
$440$	0	0
$441$	−27.9038	−1.32875
$442$	0	0
$443$	7.44696	0.353816	0.176908	−	0.984227i	$-0.443390\pi$
$443$	7.44696	0.353816	0.176908	+	0.984227i	$0.443390\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	32.5994	1.54190
$448$	0	0
$449$	7.86162	0.371013	0.185506	−	0.982643i	$-0.440607\pi$
$449$	7.86162	0.371013	0.185506	+	0.982643i	$0.440607\pi$
$450$	0	0
$451$	−29.1869	−1.37436
$452$	0	0
$453$	−33.8298	−1.58946
$454$	0	0
$455$	0	0
$456$	0	0
$457$	29.4630	1.37822	0.689109	−	0.724657i	$-0.258002\pi$
$457$	29.4630	1.37822	0.689109	+	0.724657i	$0.258002\pi$
$458$	0	0
$459$	9.95208	0.464523
$460$	0	0
$461$	11.3422	0.528260	0.264130	−	0.964487i	$-0.414915\pi$
$461$	11.3422	0.528260	0.264130	+	0.964487i	$0.414915\pi$
$462$	0	0
$463$	−18.5740	−0.863206	−0.431603	−	0.902064i	$-0.642052\pi$
$463$	−18.5740	−0.863206	−0.431603	+	0.902064i	$0.642052\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−9.20296	−0.425862	−0.212931	−	0.977067i	$-0.568301\pi$
$467$	−9.20296	−0.425862	−0.212931	+	0.977067i	$0.568301\pi$
$468$	0	0
$469$	−3.70685	−0.171166
$470$	0	0
$471$	59.0128	2.71917
$472$	0	0
$473$	0.119626	0.00550043
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−22.9747	−1.05194
$478$	0	0
$479$	27.5266	1.25772	0.628862	−	0.777517i	$-0.283521\pi$
$479$	27.5266	1.25772	0.628862	+	0.777517i	$0.283521\pi$
$480$	0	0
$481$	−8.55819	−0.390220
$482$	0	0
$483$	−5.99784	−0.272911
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−4.52343	−0.204976	−0.102488	−	0.994734i	$-0.532680\pi$
$487$	−4.52343	−0.204976	−0.102488	+	0.994734i	$0.532680\pi$
$488$	0	0
$489$	−17.5452	−0.793422
$490$	0	0
$491$	−30.1404	−1.36022	−0.680108	−	0.733112i	$-0.738067\pi$
$491$	−30.1404	−1.36022	−0.680108	+	0.733112i	$0.738067\pi$
$492$	0	0
$493$	4.12280	0.185682
$494$	0	0
$495$	0	0
$496$	0	0
$497$	5.55329	0.249099
$498$	0	0
$499$	−15.3588	−0.687556	−0.343778	−	0.939051i	$-0.611707\pi$
$499$	−15.3588	−0.687556	−0.343778	+	0.939051i	$0.611707\pi$
$500$	0	0
$501$	41.2154	1.84137
$502$	0	0
$503$	−24.2687	−1.08209	−0.541043	−	0.840995i	$-0.681970\pi$
$503$	−24.2687	−1.08209	−0.541043	+	0.840995i	$0.681970\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−53.8032	−2.38948
$508$	0	0
$509$	33.4276	1.48165	0.740827	−	0.671696i	$-0.234434\pi$
$509$	33.4276	1.48165	0.740827	+	0.671696i	$0.234434\pi$
$510$	0	0
$511$	8.53199	0.377433
$512$	0	0
$513$	20.3299	0.897588
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−55.4611	−2.43918
$518$	0	0
$519$	0.299385	0.0131415
$520$	0	0
$521$	−12.1739	−0.533347	−0.266674	−	0.963787i	$-0.585925\pi$
$521$	−12.1739	−0.533347	−0.266674	+	0.963787i	$0.585925\pi$
$522$	0	0
$523$	33.7025	1.47371	0.736854	−	0.676052i	$-0.236310\pi$
$523$	33.7025	1.47371	0.736854	+	0.676052i	$0.236310\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.91522	0.257671
$528$	0	0
$529$	−14.5415	−0.632241
$530$	0	0
$531$	−34.4002	−1.49284
$532$	0	0
$533$	25.7894	1.11706
$534$	0	0
$535$	0	0
$536$	0	0
$537$	22.7536	0.981892
$538$	0	0
$539$	41.6512	1.79405
$540$	0	0
$541$	−14.1486	−0.608296	−0.304148	−	0.952625i	$-0.598372\pi$
$541$	−14.1486	−0.608296	−0.304148	+	0.952625i	$0.598372\pi$
$542$	0	0
$543$	−14.0454	−0.602748
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15.3479	−0.656229	−0.328114	−	0.944638i	$-0.606413\pi$
$547$	−15.3479	−0.656229	−0.328114	+	0.944638i	$0.606413\pi$
$548$	0	0
$549$	15.8137	0.674912
$550$	0	0
$551$	8.42198	0.358788
$552$	0	0
$553$	−3.63625	−0.154629
$554$	0	0
$555$	0	0
$556$	0	0
$557$	27.6553	1.17179	0.585896	−	0.810386i	$-0.300743\pi$
$557$	27.6553	1.17179	0.585896	+	0.810386i	$0.300743\pi$
$558$	0	0
$559$	−0.105701	−0.00447069
$560$	0	0
$561$	−47.9725	−2.02540
$562$	0	0
$563$	17.5714	0.740547	0.370273	−	0.928923i	$-0.379264\pi$
$563$	17.5714	0.740547	0.370273	+	0.928923i	$0.379264\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2.39842	0.100724
$568$	0	0
$569$	−8.20625	−0.344024	−0.172012	−	0.985095i	$-0.555027\pi$
$569$	−8.20625	−0.344024	−0.172012	+	0.985095i	$0.555027\pi$
$570$	0	0
$571$	13.0085	0.544387	0.272193	−	0.962243i	$-0.412251\pi$
$571$	13.0085	0.544387	0.272193	+	0.962243i	$0.412251\pi$
$572$	0	0
$573$	−72.1435	−3.01384
$574$	0	0
$575$	0	0
$576$	0	0
$577$	41.7137	1.73656	0.868281	−	0.496073i	$-0.165225\pi$
$577$	41.7137	1.73656	0.868281	+	0.496073i	$0.165225\pi$
$578$	0	0
$579$	−59.1657	−2.45884
$580$	0	0
$581$	3.58392	0.148686
$582$	0	0
$583$	34.2938	1.42030
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.2971	−0.796478	−0.398239	−	0.917282i	$-0.630379\pi$
$587$	−19.2971	−0.796478	−0.398239	+	0.917282i	$0.630379\pi$
$588$	0	0
$589$	12.0835	0.497892
$590$	0	0
$591$	12.2902	0.505552
$592$	0	0
$593$	−22.3316	−0.917049	−0.458525	−	0.888682i	$-0.651622\pi$
$593$	−22.3316	−0.917049	−0.458525	+	0.888682i	$0.651622\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−42.1012	−1.72309
$598$	0	0
$599$	−23.3783	−0.955212	−0.477606	−	0.878574i	$-0.658495\pi$
$599$	−23.3783	−0.955212	−0.477606	+	0.878574i	$0.658495\pi$
$600$	0	0
$601$	21.1044	0.860866	0.430433	−	0.902623i	$-0.358361\pi$
$601$	21.1044	0.860866	0.430433	+	0.902623i	$0.358361\pi$
$602$	0	0
$603$	21.1705	0.862129
$604$	0	0
$605$	0	0
$606$	0	0
$607$	27.5146	1.11678	0.558391	−	0.829578i	$-0.311419\pi$
$607$	27.5146	1.11678	0.558391	+	0.829578i	$0.311419\pi$
$608$	0	0
$609$	−3.11594	−0.126264
$610$	0	0
$611$	49.0051	1.98253
$612$	0	0
$613$	−22.8425	−0.922602	−0.461301	−	0.887244i	$-0.652617\pi$
$613$	−22.8425	−0.922602	−0.461301	+	0.887244i	$0.652617\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	26.3212	1.05965	0.529826	−	0.848106i	$-0.322257\pi$
$617$	26.3212	1.05965	0.529826	+	0.848106i	$0.322257\pi$
$618$	0	0
$619$	−39.0155	−1.56817	−0.784083	−	0.620656i	$-0.786866\pi$
$619$	−39.0155	−1.56817	−0.784083	+	0.620656i	$0.786866\pi$
$620$	0	0
$621$	10.6074	0.425659
$622$	0	0
$623$	7.50523	0.300691
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−97.9974	−3.91364
$628$	0	0
$629$	4.07434	0.162454
$630$	0	0
$631$	6.04499	0.240647	0.120324	−	0.992735i	$-0.461607\pi$
$631$	6.04499	0.240647	0.120324	+	0.992735i	$0.461607\pi$
$632$	0	0
$633$	−27.6878	−1.10049
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−36.8028	−1.45818
$638$	0	0
$639$	−31.7159	−1.25466
$640$	0	0
$641$	31.9890	1.26349	0.631744	−	0.775177i	$-0.282339\pi$
$641$	31.9890	1.26349	0.631744	+	0.775177i	$0.282339\pi$
$642$	0	0
$643$	9.81465	0.387052	0.193526	−	0.981095i	$-0.438008\pi$
$643$	9.81465	0.387052	0.193526	+	0.981095i	$0.438008\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	26.0186	1.02290	0.511448	−	0.859314i	$-0.329109\pi$
$647$	26.0186	1.02290	0.511448	+	0.859314i	$0.329109\pi$
$648$	0	0
$649$	51.3483	2.01560
$650$	0	0
$651$	−4.47063	−0.175218
$652$	0	0
$653$	−38.0226	−1.48794	−0.743970	−	0.668213i	$-0.767060\pi$
$653$	−38.0226	−1.48794	−0.743970	+	0.668213i	$0.767060\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−48.7277	−1.90105
$658$	0	0
$659$	8.33138	0.324544	0.162272	−	0.986746i	$-0.448118\pi$
$659$	8.33138	0.324544	0.162272	+	0.986746i	$0.448118\pi$
$660$	0	0
$661$	−26.1090	−1.01552	−0.507762	−	0.861497i	$-0.669527\pi$
$661$	−26.1090	−1.01552	−0.507762	+	0.861497i	$0.669527\pi$
$662$	0	0
$663$	42.3883	1.64622
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.39426	0.170146
$668$	0	0
$669$	−45.5703	−1.76185
$670$	0	0
$671$	−23.6047	−0.911249
$672$	0	0
$673$	−48.6360	−1.87478	−0.937390	−	0.348283i	$-0.886765\pi$
$673$	−48.6360	−1.87478	−0.937390	+	0.348283i	$0.886765\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−31.4149	−1.20737	−0.603687	−	0.797221i	$-0.706302\pi$
$677$	−31.4149	−1.20737	−0.603687	+	0.797221i	$0.706302\pi$
$678$	0	0
$679$	−0.580778	−0.0222882
$680$	0	0
$681$	4.32711	0.165815
$682$	0	0
$683$	33.8482	1.29516	0.647582	−	0.761996i	$-0.275780\pi$
$683$	33.8482	1.29516	0.647582	+	0.761996i	$0.275780\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−17.9742	−0.685760
$688$	0	0
$689$	−30.3018	−1.15441
$690$	0	0
$691$	−45.3013	−1.72334	−0.861672	−	0.507466i	$-0.830582\pi$
$691$	−45.3013	−1.72334	−0.861672	+	0.507466i	$0.830582\pi$
$692$	0	0
$693$	21.4494	0.814797
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−12.2777	−0.465051
$698$	0	0
$699$	58.3700	2.20776
$700$	0	0
$701$	3.00762	0.113596	0.0567982	−	0.998386i	$-0.481911\pi$
$701$	3.00762	0.113596	0.0567982	+	0.998386i	$0.481911\pi$
$702$	0	0
$703$	8.32298	0.313907
$704$	0	0
$705$	0	0
$706$	0	0
$707$	12.8269	0.482406
$708$	0	0
$709$	−27.3379	−1.02670	−0.513349	−	0.858180i	$-0.671595\pi$
$709$	−27.3379	−1.02670	−0.513349	+	0.858180i	$0.671595\pi$
$710$	0	0
$711$	20.7673	0.778833
$712$	0	0
$713$	6.30470	0.236113
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−9.54736	−0.356553
$718$	0	0
$719$	−6.96333	−0.259688	−0.129844	−	0.991534i	$-0.541448\pi$
$719$	−6.96333	−0.259688	−0.129844	+	0.991534i	$0.541448\pi$
$720$	0	0
$721$	−2.29124	−0.0853302
$722$	0	0
$723$	−14.0998	−0.524377
$724$	0	0
$725$	0	0
$726$	0	0
$727$	39.4766	1.46411	0.732054	−	0.681247i	$-0.238562\pi$
$727$	39.4766	1.46411	0.732054	+	0.681247i	$0.238562\pi$
$728$	0	0
$729$	−43.3529	−1.60566
$730$	0	0
$731$	0.0503217	0.00186121
$732$	0	0
$733$	−1.45690	−0.0538119	−0.0269059	−	0.999638i	$-0.508565\pi$
$733$	−1.45690	−0.0538119	−0.0269059	+	0.999638i	$0.508565\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−31.6006	−1.16402
$738$	0	0
$739$	11.3945	0.419154	0.209577	−	0.977792i	$-0.432791\pi$
$739$	11.3945	0.419154	0.209577	+	0.977792i	$0.432791\pi$
$740$	0	0
$741$	86.5900	3.18096
$742$	0	0
$743$	38.1536	1.39972	0.699860	−	0.714280i	$-0.253246\pi$
$743$	38.1536	1.39972	0.699860	+	0.714280i	$0.253246\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−20.4684	−0.748900
$748$	0	0
$749$	−9.02408	−0.329733
$750$	0	0
$751$	−47.6723	−1.73959	−0.869794	−	0.493415i	$-0.835749\pi$
$751$	−47.6723	−1.73959	−0.869794	+	0.493415i	$0.835749\pi$
$752$	0	0
$753$	−29.5709	−1.07762
$754$	0	0
$755$	0	0
$756$	0	0
$757$	11.0241	0.400676	0.200338	−	0.979727i	$-0.435796\pi$
$757$	11.0241	0.400676	0.200338	+	0.979727i	$0.435796\pi$
$758$	0	0
$759$	−51.1312	−1.85594
$760$	0	0
$761$	50.9538	1.84707	0.923537	−	0.383509i	$-0.125284\pi$
$761$	50.9538	1.84707	0.923537	+	0.383509i	$0.125284\pi$
$762$	0	0
$763$	10.7257	0.388295
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−45.3711	−1.63826
$768$	0	0
$769$	31.8537	1.14867	0.574337	−	0.818619i	$-0.305260\pi$
$769$	31.8537	1.14867	0.574337	+	0.818619i	$0.305260\pi$
$770$	0	0
$771$	−35.8921	−1.29262
$772$	0	0
$773$	15.2598	0.548857	0.274428	−	0.961608i	$-0.411511\pi$
$773$	15.2598	0.548857	0.274428	+	0.961608i	$0.411511\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3.07932	−0.110470
$778$	0	0
$779$	−25.0806	−0.898607
$780$	0	0
$781$	47.3414	1.69401
$782$	0	0
$783$	5.51064	0.196934
$784$	0	0
$785$	0	0
$786$	0	0
$787$	6.42204	0.228921	0.114461	−	0.993428i	$-0.463486\pi$
$787$	6.42204	0.228921	0.114461	+	0.993428i	$0.463486\pi$
$788$	0	0
$789$	42.5831	1.51600
$790$	0	0
$791$	−4.51054	−0.160376
$792$	0	0
$793$	20.8570	0.740653
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−19.7061	−0.698027	−0.349014	−	0.937118i	$-0.613483\pi$
$797$	−19.7061	−0.698027	−0.349014	+	0.937118i	$0.613483\pi$
$798$	0	0
$799$	−23.3301	−0.825360
$800$	0	0
$801$	−42.8637	−1.51451
$802$	0	0
$803$	72.7346	2.56675
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−59.3301	−2.08852
$808$	0	0
$809$	−17.2003	−0.604732	−0.302366	−	0.953192i	$-0.597776\pi$
$809$	−17.2003	−0.604732	−0.302366	+	0.953192i	$0.597776\pi$
$810$	0	0
$811$	−40.4792	−1.42142	−0.710709	−	0.703486i	$-0.751626\pi$
$811$	−40.4792	−1.42142	−0.710709	+	0.703486i	$0.751626\pi$
$812$	0	0
$813$	−9.71223	−0.340623
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0.102796	0.00359638
$818$	0	0
$819$	−18.9526	−0.662258
$820$	0	0
$821$	−14.0464	−0.490222	−0.245111	−	0.969495i	$-0.578824\pi$
$821$	−14.0464	−0.490222	−0.245111	+	0.969495i	$0.578824\pi$
$822$	0	0
$823$	−15.5496	−0.542026	−0.271013	−	0.962576i	$-0.587359\pi$
$823$	−15.5496	−0.542026	−0.271013	+	0.962576i	$0.587359\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−5.65592	−0.196676	−0.0983378	−	0.995153i	$-0.531353\pi$
$827$	−5.65592	−0.196676	−0.0983378	+	0.995153i	$0.531353\pi$
$828$	0	0
$829$	−50.2986	−1.74694	−0.873472	−	0.486874i	$-0.838137\pi$
$829$	−50.2986	−1.74694	−0.873472	+	0.486874i	$0.838137\pi$
$830$	0	0
$831$	28.1709	0.977236
$832$	0	0
$833$	17.5209	0.607063
$834$	0	0
$835$	0	0
$836$	0	0
$837$	7.90643	0.273286
$838$	0	0
$839$	14.3841	0.496596	0.248298	−	0.968684i	$-0.420129\pi$
$839$	14.3841	0.496596	0.248298	+	0.968684i	$0.420129\pi$
$840$	0	0
$841$	−26.7171	−0.921281
$842$	0	0
$843$	−65.0442	−2.24024
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−23.6470	−0.812521
$848$	0	0
$849$	−9.19842	−0.315689
$850$	0	0
$851$	4.34261	0.148863
$852$	0	0
$853$	19.7582	0.676507	0.338254	−	0.941055i	$-0.390164\pi$
$853$	19.7582	0.676507	0.338254	+	0.941055i	$0.390164\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.9746	−0.784798	−0.392399	−	0.919795i	$-0.628355\pi$
$857$	−22.9746	−0.784798	−0.392399	+	0.919795i	$0.628355\pi$
$858$	0	0
$859$	−16.1697	−0.551704	−0.275852	−	0.961200i	$-0.588960\pi$
$859$	−16.1697	−0.551704	−0.275852	+	0.961200i	$0.588960\pi$
$860$	0	0
$861$	9.27927	0.316237
$862$	0	0
$863$	18.0901	0.615793	0.307896	−	0.951420i	$-0.400375\pi$
$863$	18.0901	0.615793	0.307896	+	0.951420i	$0.400375\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	25.8950	0.879441
$868$	0	0
$869$	−30.9987	−1.05156
$870$	0	0
$871$	27.9222	0.946107
$872$	0	0
$873$	3.31692	0.112261
$874$	0	0
$875$	0	0
$876$	0	0
$877$	5.26062	0.177639	0.0888193	−	0.996048i	$-0.471691\pi$
$877$	5.26062	0.177639	0.0888193	+	0.996048i	$0.471691\pi$
$878$	0	0
$879$	−2.31237	−0.0779944
$880$	0	0
$881$	−40.8212	−1.37530	−0.687651	−	0.726042i	$-0.741358\pi$
$881$	−40.8212	−1.37530	−0.687651	+	0.726042i	$0.741358\pi$
$882$	0	0
$883$	24.4931	0.824259	0.412130	−	0.911125i	$-0.364785\pi$
$883$	24.4931	0.824259	0.412130	+	0.911125i	$0.364785\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	26.6217	0.893869	0.446935	−	0.894567i	$-0.352516\pi$
$887$	26.6217	0.893869	0.446935	+	0.894567i	$0.352516\pi$
$888$	0	0
$889$	−6.91050	−0.231771
$890$	0	0
$891$	20.4463	0.684978
$892$	0	0
$893$	−47.6583	−1.59482
$894$	0	0
$895$	0	0
$896$	0	0
$897$	45.1793	1.50849
$898$	0	0
$899$	3.27536	0.109239
$900$	0	0
$901$	14.4259	0.480598
$902$	0	0
$903$	−0.0380323	−0.00126564
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−12.8213	−0.425723	−0.212862	−	0.977082i	$-0.568278\pi$
$907$	−12.8213	−0.425723	−0.212862	+	0.977082i	$0.568278\pi$
$908$	0	0
$909$	−73.2569	−2.42978
$910$	0	0
$911$	−22.4036	−0.742265	−0.371133	−	0.928580i	$-0.621030\pi$
$911$	−22.4036	−0.742265	−0.371133	+	0.928580i	$0.621030\pi$
$912$	0	0
$913$	30.5527	1.01115
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4.73327	−0.156306
$918$	0	0
$919$	−60.3521	−1.99083	−0.995416	−	0.0956372i	$-0.969511\pi$
$919$	−60.3521	−1.99083	−0.995416	+	0.0956372i	$0.969511\pi$
$920$	0	0
$921$	−89.3924	−2.94558
$922$	0	0
$923$	−41.8307	−1.37687
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13.0857	0.429790
$928$	0	0
$929$	−29.8340	−0.978821	−0.489411	−	0.872053i	$-0.662788\pi$
$929$	−29.8340	−0.978821	−0.489411	+	0.872053i	$0.662788\pi$
$930$	0	0
$931$	35.7913	1.17301
$932$	0	0
$933$	−8.84051	−0.289425
$934$	0	0
$935$	0	0
$936$	0	0
$937$	2.57817	0.0842252	0.0421126	−	0.999113i	$-0.486591\pi$
$937$	2.57817	0.0842252	0.0421126	+	0.999113i	$0.486591\pi$
$938$	0	0
$939$	−14.1898	−0.463065
$940$	0	0
$941$	−8.38161	−0.273233	−0.136616	−	0.990624i	$-0.543623\pi$
$941$	−8.38161	−0.273233	−0.136616	+	0.990624i	$0.543623\pi$
$942$	0	0
$943$	−13.0861	−0.426142
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32.6284	−1.06028	−0.530140	−	0.847910i	$-0.677861\pi$
$947$	−32.6284	−1.06028	−0.530140	+	0.847910i	$0.677861\pi$
$948$	0	0
$949$	−64.2680	−2.08623
$950$	0	0
$951$	58.5384	1.89824
$952$	0	0
$953$	−4.80673	−0.155705	−0.0778527	−	0.996965i	$-0.524806\pi$
$953$	−4.80673	−0.155705	−0.0778527	+	0.996965i	$0.524806\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−26.5632	−0.858666
$958$	0	0
$959$	−12.3617	−0.399181
$960$	0	0
$961$	−26.3007	−0.848408
$962$	0	0
$963$	51.5381	1.66079
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−21.8099	−0.701360	−0.350680	−	0.936495i	$-0.614049\pi$
$967$	−21.8099	−0.701360	−0.350680	+	0.936495i	$0.614049\pi$
$968$	0	0
$969$	−41.2233	−1.32428
$970$	0	0
$971$	24.0451	0.771643	0.385821	−	0.922574i	$-0.373918\pi$
$971$	24.0451	0.771643	0.385821	+	0.922574i	$0.373918\pi$
$972$	0	0
$973$	7.71127	0.247212
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.92161	−0.0934705	−0.0467352	−	0.998907i	$-0.514882\pi$
$977$	−2.92161	−0.0934705	−0.0467352	+	0.998907i	$0.514882\pi$
$978$	0	0
$979$	63.9816	2.04486
$980$	0	0
$981$	−61.2561	−1.95576
$982$	0	0
$983$	0.843348	0.0268986	0.0134493	−	0.999910i	$-0.495719\pi$
$983$	0.843348	0.0268986	0.0134493	+	0.999910i	$0.495719\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	17.6325	0.561249
$988$	0	0
$989$	0.0536350	0.00170549
$990$	0	0
$991$	6.22823	0.197846	0.0989231	−	0.995095i	$-0.468460\pi$
$991$	6.22823	0.197846	0.0989231	+	0.995095i	$0.468460\pi$
$992$	0	0
$993$	−16.4891	−0.523265
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−52.4910	−1.66241	−0.831203	−	0.555969i	$-0.812347\pi$
$997$	−52.4910	−1.66241	−0.831203	+	0.555969i	$0.812347\pi$
$998$	0	0
$999$	5.44586	0.172299

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5000.2.a.r.1.2		16
4.3	odd	2		10000.2.a.bq.1.15		16
5.4	even	2		5000.2.a.q.1.15		16
20.19	odd	2		10000.2.a.br.1.2		16
25.3	odd	20		1000.2.q.c.49.7		32
25.4	even	10		1000.2.m.e.201.8		32
25.6	even	5		1000.2.m.d.801.1		32
25.8	odd	20		200.2.q.a.89.2	yes	32
25.17	odd	20		1000.2.q.c.449.7		32
25.19	even	10		1000.2.m.e.801.8		32
25.21	even	5		1000.2.m.d.201.1		32
25.22	odd	20		200.2.q.a.9.2	✓	32
100.47	even	20		400.2.y.d.209.7		32
100.83	even	20		400.2.y.d.289.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.2.q.a.9.2	✓	32	25.22	odd	20
200.2.q.a.89.2	yes	32	25.8	odd	20
400.2.y.d.209.7		32	100.47	even	20
400.2.y.d.289.7		32	100.83	even	20
1000.2.m.d.201.1		32	25.21	even	5
1000.2.m.d.801.1		32	25.6	even	5
1000.2.m.e.201.8		32	25.4	even	10
1000.2.m.e.801.8		32	25.19	even	10
1000.2.q.c.49.7		32	25.3	odd	20
1000.2.q.c.449.7		32	25.17	odd	20
5000.2.a.q.1.15		16	5.4	even	2
5000.2.a.r.1.2		16	1.1	even	1	trivial
10000.2.a.bq.1.15		16	4.3	odd	2
10000.2.a.br.1.2		16	20.19	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:	\( N \)	\(=\)	\( 5000 = 2^{3} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5000.a (trivial)

\(f(q)\)	\(=\)	\(q-2.71029 q^{3} -0.760910 q^{7} +4.34569 q^{9} -6.48670 q^{11} +5.73162 q^{13} -2.72868 q^{17} -5.57409 q^{19} +2.06229 q^{21} -2.90834 q^{23} -3.64722 q^{27} -1.51092 q^{29} -2.16780 q^{31} +17.5809 q^{33} -1.49315 q^{37} -15.5344 q^{39} +4.49950 q^{41} -0.0184418 q^{43} +8.54996 q^{47} -6.42102 q^{49} +7.39552 q^{51} -5.28679 q^{53} +15.1074 q^{57} -7.91593 q^{59} +3.63893 q^{61} -3.30668 q^{63} +4.87160 q^{67} +7.88246 q^{69} -7.29823 q^{71} -11.2129 q^{73} +4.93580 q^{77} +4.77881 q^{79} -3.15204 q^{81} -4.71005 q^{83} +4.09502 q^{87} -9.86350 q^{89} -4.36124 q^{91} +5.87537 q^{93} +0.763267 q^{97} -28.1892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 4 q^{3} + 8 q^{7} + 20 q^{9} - 12 q^{11} + 10 q^{13} + 8 q^{17} - 12 q^{19} + 8 q^{21} + 12 q^{23} + 22 q^{27} + 16 q^{29} - 2 q^{31} + 24 q^{33} + 22 q^{37} - 4 q^{39} + 20 q^{41} + 26 q^{43} + 24 q^{47}+ \cdots - 46 q^{99}+O(q^{100}) \) 16 * q + 4 * q^3 + 8 * q^7 + 20 * q^9 - 12 * q^11 + 10 * q^13 + 8 * q^17 - 12 * q^19 + 8 * q^21 + 12 * q^23 + 22 * q^27 + 16 * q^29 - 2 * q^31 + 24 * q^33 + 22 * q^37 - 4 * q^39 + 20 * q^41 + 26 * q^43 + 24 * q^47 + 30 * q^49 - 30 * q^51 + 16 * q^53 + 24 * q^57 - 40 * q^59 + 22 * q^61 + 52 * q^63 + 50 * q^67 + 22 * q^69 - 4 * q^71 + 32 * q^73 + 30 * q^77 - 8 * q^79 + 44 * q^81 + 30 * q^83 + 48 * q^87 + 28 * q^89 - 32 * q^91 + 40 * q^93 + 36 * q^97 - 46 * q^99

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