LMFDB - Embedded newform 5200.2.a.cb.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5200,2,Mod(1,5200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5200, 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("5200.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5200 = 2^{4} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5200.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:	$41.5222090511$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	5200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.31111 q^{3} -2.90321 q^{7} -1.28100 q^{9} -0.214320 q^{11} -1.00000 q^{13} +6.42864 q^{17} -2.21432 q^{19} +3.80642 q^{21} -4.68889 q^{23} +5.61285 q^{27} +8.70964 q^{29} +5.59210 q^{31} +0.280996 q^{33} +2.28100 q^{37} +1.31111 q^{39} +3.05086 q^{41} -6.36196 q^{43} -1.09679 q^{47} +1.42864 q^{49} -8.42864 q^{51} -6.23506 q^{53} +2.90321 q^{57} +9.26517 q^{59} -0.280996 q^{61} +3.71900 q^{63} +7.76049 q^{67} +6.14764 q^{69} +6.08097 q^{71} -10.2810 q^{73} +0.622216 q^{77} +14.2351 q^{79} -3.51606 q^{81} -9.52543 q^{83} -11.4193 q^{87} -5.61285 q^{89} +2.90321 q^{91} -7.33185 q^{93} -18.0415 q^{97} +0.274543 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) $ 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^11 - 3 * q^13 + 6 * q^17 - 2 * q^21 - 14 * q^23 - 10 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^33 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 10 * q^47 - 9 * q^49 - 12 * q^51 + 8 * q^53 + 2 * q^57 + 8 * q^59 + 6 * q^61 + 18 * q^63 - 10 * q^67 + 12 * q^69 + 12 * q^71 - 24 * q^73 + 2 * q^77 + 16 * q^79 + 23 * q^81 - 22 * q^83 + 6 * q^87 + 10 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 8 * 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
$3$	−1.31111	−0.756968	−0.378484	−	0.925608i	$-0.623555\pi$
$3$	−1.31111	−0.756968	−0.378484	+	0.925608i	$0.623555\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.90321	−1.09731	−0.548655	−	0.836049i	$-0.684860\pi$
$7$	−2.90321	−1.09731	−0.548655	+	0.836049i	$0.684860\pi$
$8$	0	0
$9$	−1.28100	−0.426999
$10$	0	0
$11$	−0.214320	−0.0646198	−0.0323099	−	0.999478i	$-0.510286\pi$
$11$	−0.214320	−0.0646198	−0.0323099	+	0.999478i	$0.510286\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.42864	1.55917	0.779587	−	0.626294i	$-0.215429\pi$
$17$	6.42864	1.55917	0.779587	+	0.626294i	$0.215429\pi$
$18$	0	0
$19$	−2.21432	−0.508000	−0.254000	−	0.967204i	$-0.581746\pi$
$19$	−2.21432	−0.508000	−0.254000	+	0.967204i	$0.581746\pi$
$20$	0	0
$21$	3.80642	0.830630
$22$	0	0
$23$	−4.68889	−0.977702	−0.488851	−	0.872367i	$-0.662584\pi$
$23$	−4.68889	−0.977702	−0.488851	+	0.872367i	$0.662584\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.61285	1.08019
$28$	0	0
$29$	8.70964	1.61734	0.808669	−	0.588263i	$-0.200188\pi$
$29$	8.70964	1.61734	0.808669	+	0.588263i	$0.200188\pi$
$30$	0	0
$31$	5.59210	1.00437	0.502186	−	0.864760i	$-0.332529\pi$
$31$	5.59210	1.00437	0.502186	+	0.864760i	$0.332529\pi$
$32$	0	0
$33$	0.280996	0.0489152
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.28100	0.374993	0.187497	−	0.982265i	$-0.439963\pi$
$37$	2.28100	0.374993	0.187497	+	0.982265i	$0.439963\pi$
$38$	0	0
$39$	1.31111	0.209945
$40$	0	0
$41$	3.05086	0.476464	0.238232	−	0.971208i	$-0.423432\pi$
$41$	3.05086	0.476464	0.238232	+	0.971208i	$0.423432\pi$
$42$	0	0
$43$	−6.36196	−0.970190	−0.485095	−	0.874461i	$-0.661215\pi$
$43$	−6.36196	−0.970190	−0.485095	+	0.874461i	$0.661215\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.09679	−0.159983	−0.0799915	−	0.996796i	$-0.525489\pi$
$47$	−1.09679	−0.159983	−0.0799915	+	0.996796i	$0.525489\pi$
$48$	0	0
$49$	1.42864	0.204091
$50$	0	0
$51$	−8.42864	−1.18025
$52$	0	0
$53$	−6.23506	−0.856452	−0.428226	−	0.903672i	$-0.640861\pi$
$53$	−6.23506	−0.856452	−0.428226	+	0.903672i	$0.640861\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.90321	0.384540
$58$	0	0
$59$	9.26517	1.20622	0.603112	−	0.797657i	$-0.293927\pi$
$59$	9.26517	1.20622	0.603112	+	0.797657i	$0.293927\pi$
$60$	0	0
$61$	−0.280996	−0.0359779	−0.0179889	−	0.999838i	$-0.505726\pi$
$61$	−0.280996	−0.0359779	−0.0179889	+	0.999838i	$0.505726\pi$
$62$	0	0
$63$	3.71900	0.468550
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7.76049	0.948095	0.474047	−	0.880499i	$-0.342793\pi$
$67$	7.76049	0.948095	0.474047	+	0.880499i	$0.342793\pi$
$68$	0	0
$69$	6.14764	0.740089
$70$	0	0
$71$	6.08097	0.721678	0.360839	−	0.932628i	$-0.382490\pi$
$71$	6.08097	0.721678	0.360839	+	0.932628i	$0.382490\pi$
$72$	0	0
$73$	−10.2810	−1.20330	−0.601650	−	0.798760i	$-0.705490\pi$
$73$	−10.2810	−1.20330	−0.601650	+	0.798760i	$0.705490\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.622216	0.0709081
$78$	0	0
$79$	14.2351	1.60157	0.800785	−	0.598952i	$-0.204416\pi$
$79$	14.2351	1.60157	0.800785	+	0.598952i	$0.204416\pi$
$80$	0	0
$81$	−3.51606	−0.390673
$82$	0	0
$83$	−9.52543	−1.04555	−0.522776	−	0.852470i	$-0.675103\pi$
$83$	−9.52543	−1.04555	−0.522776	+	0.852470i	$0.675103\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−11.4193	−1.22427
$88$	0	0
$89$	−5.61285	−0.594961	−0.297480	−	0.954728i	$-0.596146\pi$
$89$	−5.61285	−0.594961	−0.297480	+	0.954728i	$0.596146\pi$
$90$	0	0
$91$	2.90321	0.304339
$92$	0	0
$93$	−7.33185	−0.760278
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−18.0415	−1.83184	−0.915918	−	0.401366i	$-0.868536\pi$
$97$	−18.0415	−1.83184	−0.915918	+	0.401366i	$0.868536\pi$
$98$	0	0
$99$	0.274543	0.0275926
$100$	0	0
$101$	3.93978	0.392022	0.196011	−	0.980602i	$-0.437201\pi$
$101$	3.93978	0.392022	0.196011	+	0.980602i	$0.437201\pi$
$102$	0	0
$103$	−2.82225	−0.278084	−0.139042	−	0.990286i	$-0.544402\pi$
$103$	−2.82225	−0.278084	−0.139042	+	0.990286i	$0.544402\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−17.1175	−1.65481	−0.827407	−	0.561603i	$-0.810185\pi$
$107$	−17.1175	−1.65481	−0.827407	+	0.561603i	$0.810185\pi$
$108$	0	0
$109$	−16.7239	−1.60186	−0.800931	−	0.598757i	$-0.795662\pi$
$109$	−16.7239	−1.60186	−0.800931	+	0.598757i	$0.795662\pi$
$110$	0	0
$111$	−2.99063	−0.283858
$112$	0	0
$113$	1.18421	0.111401	0.0557005	−	0.998448i	$-0.482261\pi$
$113$	1.18421	0.111401	0.0557005	+	0.998448i	$0.482261\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	1.28100	0.118428
$118$	0	0
$119$	−18.6637	−1.71090
$120$	0	0
$121$	−10.9541	−0.995824
$122$	0	0
$123$	−4.00000	−0.360668
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−2.30174	−0.204246	−0.102123	−	0.994772i	$-0.532564\pi$
$127$	−2.30174	−0.204246	−0.102123	+	0.994772i	$0.532564\pi$
$128$	0	0
$129$	8.34122	0.734403
$130$	0	0
$131$	13.4193	1.17245	0.586224	−	0.810149i	$-0.300614\pi$
$131$	13.4193	1.17245	0.586224	+	0.810149i	$0.300614\pi$
$132$	0	0
$133$	6.42864	0.557434
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19.1526	1.63631	0.818157	−	0.574995i	$-0.194996\pi$
$137$	19.1526	1.63631	0.818157	+	0.574995i	$0.194996\pi$
$138$	0	0
$139$	−19.0923	−1.61939	−0.809696	−	0.586850i	$-0.800368\pi$
$139$	−19.0923	−1.61939	−0.809696	+	0.586850i	$0.800368\pi$
$140$	0	0
$141$	1.43801	0.121102
$142$	0	0
$143$	0.214320	0.0179223
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−1.87310	−0.154491
$148$	0	0
$149$	−3.57136	−0.292577	−0.146289	−	0.989242i	$-0.546733\pi$
$149$	−3.57136	−0.292577	−0.146289	+	0.989242i	$0.546733\pi$
$150$	0	0
$151$	1.26517	0.102958	0.0514792	−	0.998674i	$-0.483606\pi$
$151$	1.26517	0.102958	0.0514792	+	0.998674i	$0.483606\pi$
$152$	0	0
$153$	−8.23506	−0.665765
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−5.61285	−0.447954	−0.223977	−	0.974594i	$-0.571904\pi$
$157$	−5.61285	−0.447954	−0.223977	+	0.974594i	$0.571904\pi$
$158$	0	0
$159$	8.17484	0.648307
$160$	0	0
$161$	13.6128	1.07284
$162$	0	0
$163$	−3.71900	−0.291295	−0.145647	−	0.989337i	$-0.546527\pi$
$163$	−3.71900	−0.291295	−0.145647	+	0.989337i	$0.546527\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.03657	0.544506	0.272253	−	0.962226i	$-0.412231\pi$
$167$	7.03657	0.544506	0.272253	+	0.962226i	$0.412231\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.83654	0.216915
$172$	0	0
$173$	0.723926	0.0550391	0.0275195	−	0.999621i	$-0.491239\pi$
$173$	0.723926	0.0550391	0.0275195	+	0.999621i	$0.491239\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−12.1476	−0.913073
$178$	0	0
$179$	4.04149	0.302075	0.151037	−	0.988528i	$-0.451739\pi$
$179$	4.04149	0.302075	0.151037	+	0.988528i	$0.451739\pi$
$180$	0	0
$181$	2.34122	0.174021	0.0870107	−	0.996207i	$-0.472269\pi$
$181$	2.34122	0.174021	0.0870107	+	0.996207i	$0.472269\pi$
$182$	0	0
$183$	0.368416	0.0272341
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−1.37778	−0.100754
$188$	0	0
$189$	−16.2953	−1.18531
$190$	0	0
$191$	2.10171	0.152074	0.0760372	−	0.997105i	$-0.475773\pi$
$191$	2.10171	0.152074	0.0760372	+	0.997105i	$0.475773\pi$
$192$	0	0
$193$	13.5210	0.973262	0.486631	−	0.873608i	$-0.338226\pi$
$193$	13.5210	0.973262	0.486631	+	0.873608i	$0.338226\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.00000	0.142494	0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000	0.142494	0.0712470	+	0.997459i	$0.477302\pi$
$198$	0	0
$199$	−22.1432	−1.56969	−0.784845	−	0.619692i	$-0.787257\pi$
$199$	−22.1432	−1.56969	−0.784845	+	0.619692i	$0.787257\pi$
$200$	0	0
$201$	−10.1748	−0.717678
$202$	0	0
$203$	−25.2859	−1.77472
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.00645	0.417477
$208$	0	0
$209$	0.474572	0.0328269
$210$	0	0
$211$	−19.6543	−1.35306	−0.676530	−	0.736415i	$-0.736517\pi$
$211$	−19.6543	−1.35306	−0.676530	+	0.736415i	$0.736517\pi$
$212$	0	0
$213$	−7.97280	−0.546287
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−16.2351	−1.10211
$218$	0	0
$219$	13.4795	0.910860
$220$	0	0
$221$	−6.42864	−0.432437
$222$	0	0
$223$	19.6686	1.31711	0.658554	−	0.752533i	$-0.271168\pi$
$223$	19.6686	1.31711	0.658554	+	0.752533i	$0.271168\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.2716	−0.880869	−0.440434	−	0.897785i	$-0.645176\pi$
$227$	−13.2716	−0.880869	−0.440434	+	0.897785i	$0.645176\pi$
$228$	0	0
$229$	−2.42864	−0.160489	−0.0802445	−	0.996775i	$-0.525570\pi$
$229$	−2.42864	−0.160489	−0.0802445	+	0.996775i	$0.525570\pi$
$230$	0	0
$231$	−0.815792	−0.0536752
$232$	0	0
$233$	−16.1748	−1.05965	−0.529825	−	0.848107i	$-0.677742\pi$
$233$	−16.1748	−1.05965	−0.529825	+	0.848107i	$0.677742\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−18.6637	−1.21234
$238$	0	0
$239$	12.7763	0.826431	0.413215	−	0.910633i	$-0.364406\pi$
$239$	12.7763	0.826431	0.413215	+	0.910633i	$0.364406\pi$
$240$	0	0
$241$	−5.89829	−0.379942	−0.189971	−	0.981790i	$-0.560839\pi$
$241$	−5.89829	−0.379942	−0.189971	+	0.981790i	$0.560839\pi$
$242$	0	0
$243$	−12.2286	−0.784466
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.21432	0.140894
$248$	0	0
$249$	12.4889	0.791450
$250$	0	0
$251$	2.07313	0.130855	0.0654274	−	0.997857i	$-0.479159\pi$
$251$	2.07313	0.130855	0.0654274	+	0.997857i	$0.479159\pi$
$252$	0	0
$253$	1.00492	0.0631789
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−18.3970	−1.14757	−0.573787	−	0.819005i	$-0.694526\pi$
$257$	−18.3970	−1.14757	−0.573787	+	0.819005i	$0.694526\pi$
$258$	0	0
$259$	−6.62222	−0.411484
$260$	0	0
$261$	−11.1570	−0.690602
$262$	0	0
$263$	11.0257	0.679872	0.339936	−	0.940449i	$-0.389595\pi$
$263$	11.0257	0.679872	0.339936	+	0.940449i	$0.389595\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	7.35905	0.450366
$268$	0	0
$269$	−16.1432	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$269$	−16.1432	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$270$	0	0
$271$	−13.0114	−0.790385	−0.395192	−	0.918598i	$-0.629322\pi$
$271$	−13.0114	−0.790385	−0.395192	+	0.918598i	$0.629322\pi$
$272$	0	0
$273$	−3.80642	−0.230375
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−7.57136	−0.454919	−0.227459	−	0.973788i	$-0.573042\pi$
$277$	−7.57136	−0.454919	−0.227459	+	0.973788i	$0.573042\pi$
$278$	0	0
$279$	−7.16346	−0.428865
$280$	0	0
$281$	6.75557	0.403003	0.201502	−	0.979488i	$-0.435418\pi$
$281$	6.75557	0.403003	0.201502	+	0.979488i	$0.435418\pi$
$282$	0	0
$283$	19.0859	1.13454	0.567269	−	0.823532i	$-0.308000\pi$
$283$	19.0859	1.13454	0.567269	+	0.823532i	$0.308000\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−8.85728	−0.522829
$288$	0	0
$289$	24.3274	1.43102
$290$	0	0
$291$	23.6543	1.38664
$292$	0	0
$293$	8.08742	0.472472	0.236236	−	0.971696i	$-0.424086\pi$
$293$	8.08742	0.472472	0.236236	+	0.971696i	$0.424086\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−1.20294	−0.0698019
$298$	0	0
$299$	4.68889	0.271166
$300$	0	0
$301$	18.4701	1.06460
$302$	0	0
$303$	−5.16547	−0.296749
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−13.4336	−0.766694	−0.383347	−	0.923604i	$-0.625229\pi$
$307$	−13.4336	−0.766694	−0.383347	+	0.923604i	$0.625229\pi$
$308$	0	0
$309$	3.70027	0.210501
$310$	0	0
$311$	−20.2034	−1.14563	−0.572815	−	0.819684i	$-0.694149\pi$
$311$	−20.2034	−1.14563	−0.572815	+	0.819684i	$0.694149\pi$
$312$	0	0
$313$	−15.1111	−0.854129	−0.427064	−	0.904221i	$-0.640452\pi$
$313$	−15.1111	−0.854129	−0.427064	+	0.904221i	$0.640452\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.2810	−1.25143	−0.625713	−	0.780054i	$-0.715192\pi$
$317$	−22.2810	−1.25143	−0.625713	+	0.780054i	$0.715192\pi$
$318$	0	0
$319$	−1.86665	−0.104512
$320$	0	0
$321$	22.4429	1.25264
$322$	0	0
$323$	−14.2351	−0.792060
$324$	0	0
$325$	0	0
$326$	0	0
$327$	21.9269	1.21256
$328$	0	0
$329$	3.18421	0.175551
$330$	0	0
$331$	−8.25581	−0.453780	−0.226890	−	0.973920i	$-0.572856\pi$
$331$	−8.25581	−0.453780	−0.226890	+	0.973920i	$0.572856\pi$
$332$	0	0
$333$	−2.92195	−0.160122
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.7462	0.748803	0.374402	−	0.927267i	$-0.377848\pi$
$337$	13.7462	0.748803	0.374402	+	0.927267i	$0.377848\pi$
$338$	0	0
$339$	−1.55262	−0.0843270
$340$	0	0
$341$	−1.19850	−0.0649023
$342$	0	0
$343$	16.1748	0.873359
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1.21924	0.0654523	0.0327262	−	0.999464i	$-0.489581\pi$
$347$	1.21924	0.0654523	0.0327262	+	0.999464i	$0.489581\pi$
$348$	0	0
$349$	22.5116	1.20502	0.602510	−	0.798112i	$-0.294168\pi$
$349$	22.5116	1.20502	0.602510	+	0.798112i	$0.294168\pi$
$350$	0	0
$351$	−5.61285	−0.299592
$352$	0	0
$353$	14.2810	0.760101	0.380050	−	0.924966i	$-0.375907\pi$
$353$	14.2810	0.760101	0.380050	+	0.924966i	$0.375907\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	24.4701	1.29510
$358$	0	0
$359$	12.1541	0.641469	0.320734	−	0.947169i	$-0.396070\pi$
$359$	12.1541	0.641469	0.320734	+	0.947169i	$0.396070\pi$
$360$	0	0
$361$	−14.0968	−0.741936
$362$	0	0
$363$	14.3620	0.753808
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−4.65725	−0.243106	−0.121553	−	0.992585i	$-0.538788\pi$
$367$	−4.65725	−0.243106	−0.121553	+	0.992585i	$0.538788\pi$
$368$	0	0
$369$	−3.90813	−0.203449
$370$	0	0
$371$	18.1017	0.939794
$372$	0	0
$373$	−34.9403	−1.80914	−0.904569	−	0.426328i	$-0.859807\pi$
$373$	−34.9403	−1.80914	−0.904569	+	0.426328i	$0.859807\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8.70964	−0.448569
$378$	0	0
$379$	17.4717	0.897459	0.448729	−	0.893668i	$-0.351877\pi$
$379$	17.4717	0.897459	0.448729	+	0.893668i	$0.351877\pi$
$380$	0	0
$381$	3.01783	0.154608
$382$	0	0
$383$	−18.6780	−0.954401	−0.477200	−	0.878794i	$-0.658348\pi$
$383$	−18.6780	−0.954401	−0.477200	+	0.878794i	$0.658348\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	8.14965	0.414270
$388$	0	0
$389$	1.61285	0.0817746	0.0408873	−	0.999164i	$-0.486982\pi$
$389$	1.61285	0.0817746	0.0408873	+	0.999164i	$0.486982\pi$
$390$	0	0
$391$	−30.1432	−1.52441
$392$	0	0
$393$	−17.5941	−0.887506
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−6.57628	−0.330054	−0.165027	−	0.986289i	$-0.552771\pi$
$397$	−6.57628	−0.330054	−0.165027	+	0.986289i	$0.552771\pi$
$398$	0	0
$399$	−8.42864	−0.421960
$400$	0	0
$401$	−21.9081	−1.09404	−0.547020	−	0.837120i	$-0.684238\pi$
$401$	−21.9081	−1.09404	−0.547020	+	0.837120i	$0.684238\pi$
$402$	0	0
$403$	−5.59210	−0.278563
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−0.488863	−0.0242320
$408$	0	0
$409$	−10.1936	−0.504040	−0.252020	−	0.967722i	$-0.581095\pi$
$409$	−10.1936	−0.504040	−0.252020	+	0.967722i	$0.581095\pi$
$410$	0	0
$411$	−25.1111	−1.23864
$412$	0	0
$413$	−26.8988	−1.32360
$414$	0	0
$415$	0	0
$416$	0	0
$417$	25.0321	1.22583
$418$	0	0
$419$	−7.31756	−0.357486	−0.178743	−	0.983896i	$-0.557203\pi$
$419$	−7.31756	−0.357486	−0.178743	+	0.983896i	$0.557203\pi$
$420$	0	0
$421$	−7.86665	−0.383397	−0.191698	−	0.981454i	$-0.561400\pi$
$421$	−7.86665	−0.383397	−0.191698	+	0.981454i	$0.561400\pi$
$422$	0	0
$423$	1.40498	0.0683125
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0.815792	0.0394789
$428$	0	0
$429$	−0.280996	−0.0135666
$430$	0	0
$431$	38.9195	1.87469	0.937343	−	0.348407i	$-0.113277\pi$
$431$	38.9195	1.87469	0.937343	+	0.348407i	$0.113277\pi$
$432$	0	0
$433$	20.2034	0.970914	0.485457	−	0.874260i	$-0.338653\pi$
$433$	20.2034	0.970914	0.485457	+	0.874260i	$0.338653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	10.3827	0.496672
$438$	0	0
$439$	−10.8889	−0.519700	−0.259850	−	0.965649i	$-0.583673\pi$
$439$	−10.8889	−0.519700	−0.259850	+	0.965649i	$0.583673\pi$
$440$	0	0
$441$	−1.83008	−0.0871468
$442$	0	0
$443$	−28.6287	−1.36019	−0.680095	−	0.733124i	$-0.738061\pi$
$443$	−28.6287	−1.36019	−0.680095	+	0.733124i	$0.738061\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4.68244	0.221472
$448$	0	0
$449$	−10.9304	−0.515838	−0.257919	−	0.966167i	$-0.583037\pi$
$449$	−10.9304	−0.515838	−0.257919	+	0.966167i	$0.583037\pi$
$450$	0	0
$451$	−0.653858	−0.0307890
$452$	0	0
$453$	−1.65878	−0.0779363
$454$	0	0
$455$	0	0
$456$	0	0
$457$	11.4064	0.533567	0.266784	−	0.963756i	$-0.414039\pi$
$457$	11.4064	0.533567	0.266784	+	0.963756i	$0.414039\pi$
$458$	0	0
$459$	36.0830	1.68421
$460$	0	0
$461$	−26.1334	−1.21715	−0.608576	−	0.793496i	$-0.708259\pi$
$461$	−26.1334	−1.21715	−0.608576	+	0.793496i	$0.708259\pi$
$462$	0	0
$463$	−7.92242	−0.368186	−0.184093	−	0.982909i	$-0.558935\pi$
$463$	−7.92242	−0.368186	−0.184093	+	0.982909i	$0.558935\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	10.8923	0.504036	0.252018	−	0.967723i	$-0.418906\pi$
$467$	10.8923	0.504036	0.252018	+	0.967723i	$0.418906\pi$
$468$	0	0
$469$	−22.5303	−1.04035
$470$	0	0
$471$	7.35905	0.339087
$472$	0	0
$473$	1.36349	0.0626935
$474$	0	0
$475$	0	0
$476$	0	0
$477$	7.98709	0.365704
$478$	0	0
$479$	9.13182	0.417244	0.208622	−	0.977996i	$-0.433102\pi$
$479$	9.13182	0.417244	0.208622	+	0.977996i	$0.433102\pi$
$480$	0	0
$481$	−2.28100	−0.104004
$482$	0	0
$483$	−17.8479	−0.812108
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−16.1891	−0.733600	−0.366800	−	0.930300i	$-0.619547\pi$
$487$	−16.1891	−0.733600	−0.366800	+	0.930300i	$0.619547\pi$
$488$	0	0
$489$	4.87601	0.220501
$490$	0	0
$491$	−26.2636	−1.18526	−0.592631	−	0.805474i	$-0.701911\pi$
$491$	−26.2636	−1.18526	−0.592631	+	0.805474i	$0.701911\pi$
$492$	0	0
$493$	55.9911	2.52171
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.6543	−0.791905
$498$	0	0
$499$	−30.0306	−1.34435	−0.672177	−	0.740391i	$-0.734641\pi$
$499$	−30.0306	−1.34435	−0.672177	+	0.740391i	$0.734641\pi$
$500$	0	0
$501$	−9.22570	−0.412174
$502$	0	0
$503$	−16.7304	−0.745971	−0.372985	−	0.927837i	$-0.621666\pi$
$503$	−16.7304	−0.745971	−0.372985	+	0.927837i	$0.621666\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−1.31111	−0.0582283
$508$	0	0
$509$	−11.9684	−0.530488	−0.265244	−	0.964181i	$-0.585453\pi$
$509$	−11.9684	−0.530488	−0.265244	+	0.964181i	$0.585453\pi$
$510$	0	0
$511$	29.8479	1.32039
$512$	0	0
$513$	−12.4286	−0.548738
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0.235063	0.0103381
$518$	0	0
$519$	−0.949145	−0.0416628
$520$	0	0
$521$	5.75065	0.251940	0.125970	−	0.992034i	$-0.459796\pi$
$521$	5.75065	0.251940	0.125970	+	0.992034i	$0.459796\pi$
$522$	0	0
$523$	−20.8035	−0.909674	−0.454837	−	0.890575i	$-0.650302\pi$
$523$	−20.8035	−0.909674	−0.454837	+	0.890575i	$0.650302\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	35.9496	1.56599
$528$	0	0
$529$	−1.01429	−0.0440996
$530$	0	0
$531$	−11.8687	−0.515056
$532$	0	0
$533$	−3.05086	−0.132147
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−5.29883	−0.228661
$538$	0	0
$539$	−0.306186	−0.0131883
$540$	0	0
$541$	16.6222	0.714645	0.357322	−	0.933981i	$-0.383690\pi$
$541$	16.6222	0.714645	0.357322	+	0.933981i	$0.383690\pi$
$542$	0	0
$543$	−3.06959	−0.131729
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−29.9748	−1.28163	−0.640815	−	0.767695i	$-0.721404\pi$
$547$	−29.9748	−1.28163	−0.640815	+	0.767695i	$0.721404\pi$
$548$	0	0
$549$	0.359955	0.0153625
$550$	0	0
$551$	−19.2859	−0.821608
$552$	0	0
$553$	−41.3274	−1.75742
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.03657	0.213406	0.106703	−	0.994291i	$-0.465971\pi$
$557$	5.03657	0.213406	0.106703	+	0.994291i	$0.465971\pi$
$558$	0	0
$559$	6.36196	0.269082
$560$	0	0
$561$	1.80642	0.0762673
$562$	0	0
$563$	2.88247	0.121482	0.0607408	−	0.998154i	$-0.480654\pi$
$563$	2.88247	0.121482	0.0607408	+	0.998154i	$0.480654\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.2079	0.428690
$568$	0	0
$569$	−4.37286	−0.183320	−0.0916600	−	0.995790i	$-0.529217\pi$
$569$	−4.37286	−0.183320	−0.0916600	+	0.995790i	$0.529217\pi$
$570$	0	0
$571$	1.58120	0.0661714	0.0330857	−	0.999453i	$-0.489467\pi$
$571$	1.58120	0.0661714	0.0330857	+	0.999453i	$0.489467\pi$
$572$	0	0
$573$	−2.75557	−0.115116
$574$	0	0
$575$	0	0
$576$	0	0
$577$	7.61729	0.317112	0.158556	−	0.987350i	$-0.449316\pi$
$577$	7.61729	0.317112	0.158556	+	0.987350i	$0.449316\pi$
$578$	0	0
$579$	−17.7275	−0.736728
$580$	0	0
$581$	27.6543	1.14730
$582$	0	0
$583$	1.33630	0.0553438
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−46.8243	−1.93264	−0.966322	−	0.257336i	$-0.917155\pi$
$587$	−46.8243	−1.93264	−0.966322	+	0.257336i	$0.917155\pi$
$588$	0	0
$589$	−12.3827	−0.510221
$590$	0	0
$591$	−2.62222	−0.107864
$592$	0	0
$593$	15.9398	0.654568	0.327284	−	0.944926i	$-0.393867\pi$
$593$	15.9398	0.654568	0.327284	+	0.944926i	$0.393867\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	29.0321	1.18821
$598$	0	0
$599$	−18.4889	−0.755434	−0.377717	−	0.925921i	$-0.623291\pi$
$599$	−18.4889	−0.755434	−0.377717	+	0.925921i	$0.623291\pi$
$600$	0	0
$601$	20.7556	0.846637	0.423319	−	0.905981i	$-0.360865\pi$
$601$	20.7556	0.846637	0.423319	+	0.905981i	$0.360865\pi$
$602$	0	0
$603$	−9.94116	−0.404835
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.0765	1.46430	0.732150	−	0.681143i	$-0.238517\pi$
$607$	36.0765	1.46430	0.732150	+	0.681143i	$0.238517\pi$
$608$	0	0
$609$	33.1526	1.34341
$610$	0	0
$611$	1.09679	0.0443713
$612$	0	0
$613$	−9.94962	−0.401861	−0.200931	−	0.979605i	$-0.564397\pi$
$613$	−9.94962	−0.401861	−0.200931	+	0.979605i	$0.564397\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−2.09187	−0.0842154	−0.0421077	−	0.999113i	$-0.513407\pi$
$617$	−2.09187	−0.0842154	−0.0421077	+	0.999113i	$0.513407\pi$
$618$	0	0
$619$	−18.4681	−0.742296	−0.371148	−	0.928574i	$-0.621036\pi$
$619$	−18.4681	−0.742296	−0.371148	+	0.928574i	$0.621036\pi$
$620$	0	0
$621$	−26.3180	−1.05611
$622$	0	0
$623$	16.2953	0.652857
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−0.622216	−0.0248489
$628$	0	0
$629$	14.6637	0.584680
$630$	0	0
$631$	38.6657	1.53926	0.769629	−	0.638492i	$-0.220441\pi$
$631$	38.6657	1.53926	0.769629	+	0.638492i	$0.220441\pi$
$632$	0	0
$633$	25.7690	1.02422
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−1.42864	−0.0566048
$638$	0	0
$639$	−7.78970	−0.308156
$640$	0	0
$641$	24.5718	0.970529	0.485265	−	0.874367i	$-0.338723\pi$
$641$	24.5718	0.970529	0.485265	+	0.874367i	$0.338723\pi$
$642$	0	0
$643$	27.4938	1.08425	0.542125	−	0.840298i	$-0.317620\pi$
$643$	27.4938	1.08425	0.542125	+	0.840298i	$0.317620\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	13.7812	0.541796	0.270898	−	0.962608i	$-0.412679\pi$
$647$	13.7812	0.541796	0.270898	+	0.962608i	$0.412679\pi$
$648$	0	0
$649$	−1.98571	−0.0779459
$650$	0	0
$651$	21.2859	0.834261
$652$	0	0
$653$	2.12045	0.0829795	0.0414897	−	0.999139i	$-0.486790\pi$
$653$	2.12045	0.0829795	0.0414897	+	0.999139i	$0.486790\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	13.1699	0.513807
$658$	0	0
$659$	−33.8894	−1.32014	−0.660072	−	0.751203i	$-0.729474\pi$
$659$	−33.8894	−1.32014	−0.660072	+	0.751203i	$0.729474\pi$
$660$	0	0
$661$	−37.3689	−1.45348	−0.726741	−	0.686912i	$-0.758966\pi$
$661$	−37.3689	−1.45348	−0.726741	+	0.686912i	$0.758966\pi$
$662$	0	0
$663$	8.42864	0.327341
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−40.8385	−1.58127
$668$	0	0
$669$	−25.7877	−0.997010
$670$	0	0
$671$	0.0602231	0.00232489
$672$	0	0
$673$	35.4608	1.36691	0.683456	−	0.729992i	$-0.260476\pi$
$673$	35.4608	1.36691	0.683456	+	0.729992i	$0.260476\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−15.3047	−0.588206	−0.294103	−	0.955774i	$-0.595021\pi$
$677$	−15.3047	−0.588206	−0.294103	+	0.955774i	$0.595021\pi$
$678$	0	0
$679$	52.3783	2.01009
$680$	0	0
$681$	17.4005	0.666790
$682$	0	0
$683$	−13.0968	−0.501135	−0.250567	−	0.968099i	$-0.580617\pi$
$683$	−13.0968	−0.501135	−0.250567	+	0.968099i	$0.580617\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	3.18421	0.121485
$688$	0	0
$689$	6.23506	0.237537
$690$	0	0
$691$	18.4079	0.700269	0.350135	−	0.936699i	$-0.386136\pi$
$691$	18.4079	0.700269	0.350135	+	0.936699i	$0.386136\pi$
$692$	0	0
$693$	−0.797056	−0.0302777
$694$	0	0
$695$	0	0
$696$	0	0
$697$	19.6128	0.742890
$698$	0	0
$699$	21.2070	0.802121
$700$	0	0
$701$	−31.3689	−1.18479	−0.592393	−	0.805649i	$-0.701817\pi$
$701$	−31.3689	−1.18479	−0.592393	+	0.805649i	$0.701817\pi$
$702$	0	0
$703$	−5.05086	−0.190497
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−11.4380	−0.430171
$708$	0	0
$709$	9.47949	0.356010	0.178005	−	0.984030i	$-0.443036\pi$
$709$	9.47949	0.356010	0.178005	+	0.984030i	$0.443036\pi$
$710$	0	0
$711$	−18.2351	−0.683868
$712$	0	0
$713$	−26.2208	−0.981976
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−16.7511	−0.625582
$718$	0	0
$719$	29.6227	1.10474	0.552370	−	0.833599i	$-0.313724\pi$
$719$	29.6227	1.10474	0.552370	+	0.833599i	$0.313724\pi$
$720$	0	0
$721$	8.19358	0.305145
$722$	0	0
$723$	7.73329	0.287604
$724$	0	0
$725$	0	0
$726$	0	0
$727$	42.6702	1.58255	0.791274	−	0.611461i	$-0.209418\pi$
$727$	42.6702	1.58255	0.791274	+	0.611461i	$0.209418\pi$
$728$	0	0
$729$	26.5812	0.984489
$730$	0	0
$731$	−40.8988	−1.51270
$732$	0	0
$733$	−26.0830	−0.963397	−0.481698	−	0.876337i	$-0.659980\pi$
$733$	−26.0830	−0.963397	−0.481698	+	0.876337i	$0.659980\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.66323	−0.0612657
$738$	0	0
$739$	28.2687	1.03988	0.519941	−	0.854202i	$-0.325954\pi$
$739$	28.2687	1.03988	0.519941	+	0.854202i	$0.325954\pi$
$740$	0	0
$741$	−2.90321	−0.106652
$742$	0	0
$743$	20.6681	0.758241	0.379120	−	0.925347i	$-0.376227\pi$
$743$	20.6681	0.758241	0.379120	+	0.925347i	$0.376227\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	12.2020	0.446449
$748$	0	0
$749$	49.6958	1.81585
$750$	0	0
$751$	−2.46028	−0.0897770	−0.0448885	−	0.998992i	$-0.514293\pi$
$751$	−2.46028	−0.0897770	−0.0448885	+	0.998992i	$0.514293\pi$
$752$	0	0
$753$	−2.71810	−0.0990530
$754$	0	0
$755$	0	0
$756$	0	0
$757$	48.6035	1.76652	0.883262	−	0.468880i	$-0.155342\pi$
$757$	48.6035	1.76652	0.883262	+	0.468880i	$0.155342\pi$
$758$	0	0
$759$	−1.31756	−0.0478244
$760$	0	0
$761$	−13.8252	−0.501162	−0.250581	−	0.968096i	$-0.580622\pi$
$761$	−13.8252	−0.501162	−0.250581	+	0.968096i	$0.580622\pi$
$762$	0	0
$763$	48.5531	1.75774
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.26517	−0.334546
$768$	0	0
$769$	−38.9688	−1.40525	−0.702626	−	0.711559i	$-0.747989\pi$
$769$	−38.9688	−1.40525	−0.702626	+	0.711559i	$0.747989\pi$
$770$	0	0
$771$	24.1204	0.868677
$772$	0	0
$773$	0.445992	0.0160412	0.00802061	−	0.999968i	$-0.497447\pi$
$773$	0.445992	0.0160412	0.00802061	+	0.999968i	$0.497447\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	8.68244	0.311481
$778$	0	0
$779$	−6.75557	−0.242043
$780$	0	0
$781$	−1.30327	−0.0466347
$782$	0	0
$783$	48.8859	1.74704
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−33.9037	−1.20854	−0.604268	−	0.796781i	$-0.706534\pi$
$787$	−33.9037	−1.20854	−0.604268	+	0.796781i	$0.706534\pi$
$788$	0	0
$789$	−14.4558	−0.514641
$790$	0	0
$791$	−3.43801	−0.122241
$792$	0	0
$793$	0.280996	0.00997847
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−10.2953	−0.364678	−0.182339	−	0.983236i	$-0.558367\pi$
$797$	−10.2953	−0.364678	−0.182339	+	0.983236i	$0.558367\pi$
$798$	0	0
$799$	−7.05086	−0.249441
$800$	0	0
$801$	7.19004	0.254047
$802$	0	0
$803$	2.20342	0.0777570
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.1655	0.745060
$808$	0	0
$809$	7.94422	0.279304	0.139652	−	0.990201i	$-0.455402\pi$
$809$	7.94422	0.279304	0.139652	+	0.990201i	$0.455402\pi$
$810$	0	0
$811$	8.12245	0.285218	0.142609	−	0.989779i	$-0.454451\pi$
$811$	8.12245	0.285218	0.142609	+	0.989779i	$0.454451\pi$
$812$	0	0
$813$	17.0593	0.598296
$814$	0	0
$815$	0	0
$816$	0	0
$817$	14.0874	0.492856
$818$	0	0
$819$	−3.71900	−0.129953
$820$	0	0
$821$	−22.2065	−0.775012	−0.387506	−	0.921867i	$-0.626663\pi$
$821$	−22.2065	−0.775012	−0.387506	+	0.921867i	$0.626663\pi$
$822$	0	0
$823$	−11.1175	−0.387533	−0.193766	−	0.981048i	$-0.562070\pi$
$823$	−11.1175	−0.387533	−0.193766	+	0.981048i	$0.562070\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−23.1570	−0.805248	−0.402624	−	0.915365i	$-0.631902\pi$
$827$	−23.1570	−0.805248	−0.402624	+	0.915365i	$0.631902\pi$
$828$	0	0
$829$	−27.1195	−0.941901	−0.470950	−	0.882160i	$-0.656089\pi$
$829$	−27.1195	−0.941901	−0.470950	+	0.882160i	$0.656089\pi$
$830$	0	0
$831$	9.92687	0.344359
$832$	0	0
$833$	9.18421	0.318214
$834$	0	0
$835$	0	0
$836$	0	0
$837$	31.3876	1.08492
$838$	0	0
$839$	−25.3955	−0.876749	−0.438374	−	0.898792i	$-0.644446\pi$
$839$	−25.3955	−0.876749	−0.438374	+	0.898792i	$0.644446\pi$
$840$	0	0
$841$	46.8578	1.61578
$842$	0	0
$843$	−8.85728	−0.305061
$844$	0	0
$845$	0	0
$846$	0	0
$847$	31.8020	1.09273
$848$	0	0
$849$	−25.0237	−0.858810
$850$	0	0
$851$	−10.6953	−0.366632
$852$	0	0
$853$	25.0651	0.858214	0.429107	−	0.903254i	$-0.358828\pi$
$853$	25.0651	0.858214	0.429107	+	0.903254i	$0.358828\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	7.61285	0.260050	0.130025	−	0.991511i	$-0.458494\pi$
$857$	7.61285	0.260050	0.130025	+	0.991511i	$0.458494\pi$
$858$	0	0
$859$	−42.1432	−1.43791	−0.718954	−	0.695058i	$-0.755379\pi$
$859$	−42.1432	−1.43791	−0.718954	+	0.695058i	$0.755379\pi$
$860$	0	0
$861$	11.6128	0.395765
$862$	0	0
$863$	−51.5768	−1.75569	−0.877847	−	0.478942i	$-0.841020\pi$
$863$	−51.5768	−1.75569	−0.877847	+	0.478942i	$0.841020\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−31.8959	−1.08324
$868$	0	0
$869$	−3.05086	−0.103493
$870$	0	0
$871$	−7.76049	−0.262954
$872$	0	0
$873$	23.1111	0.782191
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.0701	−1.15046	−0.575232	−	0.817990i	$-0.695088\pi$
$877$	−34.0701	−1.15046	−0.575232	+	0.817990i	$0.695088\pi$
$878$	0	0
$879$	−10.6035	−0.357646
$880$	0	0
$881$	3.71900	0.125296	0.0626482	−	0.998036i	$-0.480045\pi$
$881$	3.71900	0.125296	0.0626482	+	0.998036i	$0.480045\pi$
$882$	0	0
$883$	42.0163	1.41396	0.706981	−	0.707233i	$-0.250057\pi$
$883$	42.0163	1.41396	0.706981	+	0.707233i	$0.250057\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	40.3116	1.35353	0.676765	−	0.736199i	$-0.263381\pi$
$887$	40.3116	1.35353	0.676765	+	0.736199i	$0.263381\pi$
$888$	0	0
$889$	6.68244	0.224122
$890$	0	0
$891$	0.753561	0.0252452
$892$	0	0
$893$	2.42864	0.0812713
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−6.14764	−0.205264
$898$	0	0
$899$	48.7052	1.62441
$900$	0	0
$901$	−40.0830	−1.33536
$902$	0	0
$903$	−24.2163	−0.805869
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−34.8419	−1.15691	−0.578454	−	0.815715i	$-0.696344\pi$
$907$	−34.8419	−1.15691	−0.578454	+	0.815715i	$0.696344\pi$
$908$	0	0
$909$	−5.04684	−0.167393
$910$	0	0
$911$	−23.2672	−0.770876	−0.385438	−	0.922734i	$-0.625950\pi$
$911$	−23.2672	−0.770876	−0.385438	+	0.922734i	$0.625950\pi$
$912$	0	0
$913$	2.04149	0.0675634
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−38.9590	−1.28654
$918$	0	0
$919$	3.22570	0.106406	0.0532029	−	0.998584i	$-0.483057\pi$
$919$	3.22570	0.106406	0.0532029	+	0.998584i	$0.483057\pi$
$920$	0	0
$921$	17.6128	0.580363
$922$	0	0
$923$	−6.08097	−0.200157
$924$	0	0
$925$	0	0
$926$	0	0
$927$	3.61529	0.118742
$928$	0	0
$929$	−39.3461	−1.29091	−0.645453	−	0.763800i	$-0.723331\pi$
$929$	−39.3461	−1.29091	−0.645453	+	0.763800i	$0.723331\pi$
$930$	0	0
$931$	−3.16346	−0.103678
$932$	0	0
$933$	26.4889	0.867206
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51.6040	−1.68583	−0.842914	−	0.538048i	$-0.819162\pi$
$937$	−51.6040	−1.68583	−0.842914	+	0.538048i	$0.819162\pi$
$938$	0	0
$939$	19.8123	0.646548
$940$	0	0
$941$	37.5081	1.22273	0.611364	−	0.791349i	$-0.290621\pi$
$941$	37.5081	1.22273	0.611364	+	0.791349i	$0.290621\pi$
$942$	0	0
$943$	−14.3051	−0.465839
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38.1160	−1.23860	−0.619302	−	0.785153i	$-0.712584\pi$
$947$	−38.1160	−1.23860	−0.619302	+	0.785153i	$0.712584\pi$
$948$	0	0
$949$	10.2810	0.333735
$950$	0	0
$951$	29.2128	0.947290
$952$	0	0
$953$	28.7368	0.930877	0.465439	−	0.885080i	$-0.345897\pi$
$953$	28.7368	0.930877	0.465439	+	0.885080i	$0.345897\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.44738	0.0791124
$958$	0	0
$959$	−55.6040	−1.79555
$960$	0	0
$961$	0.271628	0.00876221
$962$	0	0
$963$	21.9275	0.706604
$964$	0	0
$965$	0	0
$966$	0	0
$967$	29.0593	0.934485	0.467242	−	0.884129i	$-0.345248\pi$
$967$	29.0593	0.934485	0.467242	+	0.884129i	$0.345248\pi$
$968$	0	0
$969$	18.6637	0.599565
$970$	0	0
$971$	39.8578	1.27910	0.639548	−	0.768751i	$-0.279121\pi$
$971$	39.8578	1.27910	0.639548	+	0.768751i	$0.279121\pi$
$972$	0	0
$973$	55.4291	1.77698
$974$	0	0
$975$	0	0
$976$	0	0
$977$	12.8617	0.411483	0.205742	−	0.978606i	$-0.434039\pi$
$977$	12.8617	0.411483	0.205742	+	0.978606i	$0.434039\pi$
$978$	0	0
$979$	1.20294	0.0384463
$980$	0	0
$981$	21.4233	0.683993
$982$	0	0
$983$	45.4880	1.45084	0.725420	−	0.688306i	$-0.241645\pi$
$983$	45.4880	1.45084	0.725420	+	0.688306i	$0.241645\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−4.17484	−0.132887
$988$	0	0
$989$	29.8306	0.948557
$990$	0	0
$991$	−8.07007	−0.256354	−0.128177	−	0.991751i	$-0.540913\pi$
$991$	−8.07007	−0.256354	−0.128177	+	0.991751i	$0.540913\pi$
$992$	0	0
$993$	10.8243	0.343497
$994$	0	0
$995$	0	0
$996$	0	0
$997$	32.8158	1.03929	0.519643	−	0.854383i	$-0.326065\pi$
$997$	32.8158	1.03929	0.519643	+	0.854383i	$0.326065\pi$
$998$	0	0
$999$	12.8029	0.405065

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5200.2.a.cb.1.2		3
4.3	odd	2		325.2.a.k.1.1		3
5.2	odd	4		1040.2.d.c.209.5		6
5.3	odd	4		1040.2.d.c.209.2		6
5.4	even	2		5200.2.a.cj.1.2		3
12.11	even	2		2925.2.a.bf.1.3		3
20.3	even	4		65.2.b.a.14.4	yes	6
20.7	even	4		65.2.b.a.14.3	✓	6
20.19	odd	2		325.2.a.j.1.3		3
52.51	odd	2		4225.2.a.ba.1.3		3
60.23	odd	4		585.2.c.b.469.3		6
60.47	odd	4		585.2.c.b.469.4		6
60.59	even	2		2925.2.a.bj.1.1		3
260.3	even	12		845.2.n.f.529.3		12
260.7	odd	12		845.2.l.e.699.1		12
260.23	even	12		845.2.n.g.529.4		12
260.43	even	12		845.2.n.g.484.3		12
260.47	odd	4		845.2.d.a.844.5		6
260.63	odd	12		845.2.l.d.654.5		12
260.67	odd	12		845.2.l.d.654.6		12
260.83	odd	4		845.2.d.a.844.6		6
260.87	even	12		845.2.n.f.484.3		12
260.103	even	4		845.2.b.c.339.3		6
260.107	even	12		845.2.n.f.529.4		12
260.123	odd	12		845.2.l.e.699.2		12
260.127	even	12		845.2.n.g.529.3		12
260.147	even	12		845.2.n.g.484.4		12
260.163	odd	12		845.2.l.d.699.6		12
260.167	odd	12		845.2.l.e.654.2		12
260.187	odd	4		845.2.d.b.844.1		6
260.203	odd	4		845.2.d.b.844.2		6
260.207	even	4		845.2.b.c.339.4		6
260.223	odd	12		845.2.l.e.654.1		12
260.227	odd	12		845.2.l.d.699.5		12
260.243	even	12		845.2.n.f.484.4		12
260.259	odd	2		4225.2.a.bh.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.b.a.14.3	✓	6	20.7	even	4
65.2.b.a.14.4	yes	6	20.3	even	4
325.2.a.j.1.3		3	20.19	odd	2
325.2.a.k.1.1		3	4.3	odd	2
585.2.c.b.469.3		6	60.23	odd	4
585.2.c.b.469.4		6	60.47	odd	4
845.2.b.c.339.3		6	260.103	even	4
845.2.b.c.339.4		6	260.207	even	4
845.2.d.a.844.5		6	260.47	odd	4
845.2.d.a.844.6		6	260.83	odd	4
845.2.d.b.844.1		6	260.187	odd	4
845.2.d.b.844.2		6	260.203	odd	4
845.2.l.d.654.5		12	260.63	odd	12
845.2.l.d.654.6		12	260.67	odd	12
845.2.l.d.699.5		12	260.227	odd	12
845.2.l.d.699.6		12	260.163	odd	12
845.2.l.e.654.1		12	260.223	odd	12
845.2.l.e.654.2		12	260.167	odd	12
845.2.l.e.699.1		12	260.7	odd	12
845.2.l.e.699.2		12	260.123	odd	12
845.2.n.f.484.3		12	260.87	even	12
845.2.n.f.484.4		12	260.243	even	12
845.2.n.f.529.3		12	260.3	even	12
845.2.n.f.529.4		12	260.107	even	12
845.2.n.g.484.3		12	260.43	even	12
845.2.n.g.484.4		12	260.147	even	12
845.2.n.g.529.3		12	260.127	even	12
845.2.n.g.529.4		12	260.23	even	12
1040.2.d.c.209.2		6	5.3	odd	4
1040.2.d.c.209.5		6	5.2	odd	4
2925.2.a.bf.1.3		3	12.11	even	2
2925.2.a.bj.1.1		3	60.59	even	2
4225.2.a.ba.1.3		3	52.51	odd	2
4225.2.a.bh.1.1		3	260.259	odd	2
5200.2.a.cb.1.2		3	1.1	even	1	trivial
5200.2.a.cj.1.2		3	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5200 = 2^{4} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5200.a (trivial)

\(f(q)\)	\(=\)	\(q-1.31111 q^{3} -2.90321 q^{7} -1.28100 q^{9} -0.214320 q^{11} -1.00000 q^{13} +6.42864 q^{17} -2.21432 q^{19} +3.80642 q^{21} -4.68889 q^{23} +5.61285 q^{27} +8.70964 q^{29} +5.59210 q^{31} +0.280996 q^{33} +2.28100 q^{37} +1.31111 q^{39} +3.05086 q^{41} -6.36196 q^{43} -1.09679 q^{47} +1.42864 q^{49} -8.42864 q^{51} -6.23506 q^{53} +2.90321 q^{57} +9.26517 q^{59} -0.280996 q^{61} +3.71900 q^{63} +7.76049 q^{67} +6.14764 q^{69} +6.08097 q^{71} -10.2810 q^{73} +0.622216 q^{77} +14.2351 q^{79} -3.51606 q^{81} -9.52543 q^{83} -11.4193 q^{87} -5.61285 q^{89} +2.90321 q^{91} -7.33185 q^{93} -18.0415 q^{97} +0.274543 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{3} - 2 q^{7} + 3 q^{9} + 6 q^{11} - 3 q^{13} + 6 q^{17} - 2 q^{21} - 14 q^{23} - 10 q^{27} + 6 q^{29} + 10 q^{31} - 6 q^{33} + 4 q^{39} - 4 q^{41} - 6 q^{43} - 10 q^{47} - 9 q^{49} - 12 q^{51}+ \cdots + 8 q^{99}+O(q^{100}) \) 3 * q - 4 * q^3 - 2 * q^7 + 3 * q^9 + 6 * q^11 - 3 * q^13 + 6 * q^17 - 2 * q^21 - 14 * q^23 - 10 * q^27 + 6 * q^29 + 10 * q^31 - 6 * q^33 + 4 * q^39 - 4 * q^41 - 6 * q^43 - 10 * q^47 - 9 * q^49 - 12 * q^51 + 8 * q^53 + 2 * q^57 + 8 * q^59 + 6 * q^61 + 18 * q^63 - 10 * q^67 + 12 * q^69 + 12 * q^71 - 24 * q^73 + 2 * q^77 + 16 * q^79 + 23 * q^81 - 22 * q^83 + 6 * q^87 + 10 * q^89 + 2 * q^91 - 2 * q^93 - 14 * q^97 + 8 * q^99

Introduction

L-functions

Modular forms

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