LMFDB - Embedded newform 4840.2.a.ba.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4840,2,Mod(1,4840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4840, 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("4840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4840 = 2^{3} \cdot 5 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4840.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:	$38.6475945783$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.25903625.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 7x^{4} + 17x^{3} + 16x^{2} - 20x - 5 $ x^6 - 3x^5 - 7x^4 + 17x^3 + 16x^2 - 20*x - 5
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 440)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.03795$ of defining polynomial
Character	$\chi$	$=$	4840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.03795 q^{3} -1.00000 q^{5} +2.60210 q^{7} -1.92266 q^{9} +2.87756 q^{13} +1.03795 q^{15} -0.810293 q^{17} -5.15069 q^{19} -2.70085 q^{21} +4.98262 q^{23} +1.00000 q^{25} +5.10948 q^{27} -6.72933 q^{29} -4.14084 q^{31} -2.60210 q^{35} +4.05133 q^{37} -2.98677 q^{39} -9.76082 q^{41} +5.92912 q^{43} +1.92266 q^{45} -0.967904 q^{47} -0.229090 q^{49} +0.841046 q^{51} +4.47445 q^{53} +5.34617 q^{57} +6.80756 q^{59} +2.61307 q^{61} -5.00294 q^{63} -2.87756 q^{65} -15.8408 q^{67} -5.17172 q^{69} -1.41346 q^{71} +13.1429 q^{73} -1.03795 q^{75} -5.69123 q^{79} +0.464568 q^{81} -7.96967 q^{83} +0.810293 q^{85} +6.98472 q^{87} -14.5196 q^{89} +7.48768 q^{91} +4.29799 q^{93} +5.15069 q^{95} -2.99325 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) $ 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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
$3$	−1.03795	−0.599262	−0.299631	−	0.954055i	$-0.596864\pi$
$3$	−1.03795	−0.599262	−0.299631	+	0.954055i	$0.596864\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.60210	0.983500	0.491750	−	0.870736i	$-0.336357\pi$
$7$	2.60210	0.983500	0.491750	+	0.870736i	$0.336357\pi$
$8$	0	0
$9$	−1.92266	−0.640885
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.87756	0.798091	0.399045	−	0.916931i	$-0.369342\pi$
$13$	2.87756	0.798091	0.399045	+	0.916931i	$0.369342\pi$
$14$	0	0
$15$	1.03795	0.267998
$16$	0	0
$17$	−0.810293	−0.196525	−0.0982625	−	0.995161i	$-0.531328\pi$
$17$	−0.810293	−0.196525	−0.0982625	+	0.995161i	$0.531328\pi$
$18$	0	0
$19$	−5.15069	−1.18165	−0.590824	−	0.806800i	$-0.701197\pi$
$19$	−5.15069	−1.18165	−0.590824	+	0.806800i	$0.701197\pi$
$20$	0	0
$21$	−2.70085	−0.589374
$22$	0	0
$23$	4.98262	1.03895	0.519474	−	0.854486i	$-0.326128\pi$
$23$	4.98262	1.03895	0.519474	+	0.854486i	$0.326128\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.10948	0.983320
$28$	0	0
$29$	−6.72933	−1.24961	−0.624803	−	0.780783i	$-0.714821\pi$
$29$	−6.72933	−1.24961	−0.624803	+	0.780783i	$0.714821\pi$
$30$	0	0
$31$	−4.14084	−0.743717	−0.371858	−	0.928290i	$-0.621279\pi$
$31$	−4.14084	−0.743717	−0.371858	+	0.928290i	$0.621279\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.60210	−0.439835
$36$	0	0
$37$	4.05133	0.666034	0.333017	−	0.942921i	$-0.391933\pi$
$37$	4.05133	0.666034	0.333017	+	0.942921i	$0.391933\pi$
$38$	0	0
$39$	−2.98677	−0.478266
$40$	0	0
$41$	−9.76082	−1.52438	−0.762192	−	0.647351i	$-0.775877\pi$
$41$	−9.76082	−1.52438	−0.762192	+	0.647351i	$0.775877\pi$
$42$	0	0
$43$	5.92912	0.904182	0.452091	−	0.891972i	$-0.350678\pi$
$43$	5.92912	0.904182	0.452091	+	0.891972i	$0.350678\pi$
$44$	0	0
$45$	1.92266	0.286613
$46$	0	0
$47$	−0.967904	−0.141183	−0.0705917	−	0.997505i	$-0.522489\pi$
$47$	−0.967904	−0.141183	−0.0705917	+	0.997505i	$0.522489\pi$
$48$	0	0
$49$	−0.229090	−0.0327272
$50$	0	0
$51$	0.841046	0.117770
$52$	0	0
$53$	4.47445	0.614613	0.307307	−	0.951611i	$-0.400572\pi$
$53$	4.47445	0.614613	0.307307	+	0.951611i	$0.400572\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	5.34617	0.708117
$58$	0	0
$59$	6.80756	0.886269	0.443135	−	0.896455i	$-0.353866\pi$
$59$	6.80756	0.886269	0.443135	+	0.896455i	$0.353866\pi$
$60$	0	0
$61$	2.61307	0.334569	0.167285	−	0.985909i	$-0.446500\pi$
$61$	2.61307	0.334569	0.167285	+	0.985909i	$0.446500\pi$
$62$	0	0
$63$	−5.00294	−0.630311
$64$	0	0
$65$	−2.87756	−0.356917
$66$	0	0
$67$	−15.8408	−1.93526	−0.967629	−	0.252378i	$-0.918788\pi$
$67$	−15.8408	−1.93526	−0.967629	+	0.252378i	$0.918788\pi$
$68$	0	0
$69$	−5.17172	−0.622602
$70$	0	0
$71$	−1.41346	−0.167746	−0.0838732	−	0.996476i	$-0.526729\pi$
$71$	−1.41346	−0.167746	−0.0838732	+	0.996476i	$0.526729\pi$
$72$	0	0
$73$	13.1429	1.53826	0.769131	−	0.639091i	$-0.220689\pi$
$73$	13.1429	1.53826	0.769131	+	0.639091i	$0.220689\pi$
$74$	0	0
$75$	−1.03795	−0.119852
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−5.69123	−0.640314	−0.320157	−	0.947365i	$-0.603736\pi$
$79$	−5.69123	−0.640314	−0.320157	+	0.947365i	$0.603736\pi$
$80$	0	0
$81$	0.464568	0.0516187
$82$	0	0
$83$	−7.96967	−0.874785	−0.437392	−	0.899271i	$-0.644098\pi$
$83$	−7.96967	−0.874785	−0.437392	+	0.899271i	$0.644098\pi$
$84$	0	0
$85$	0.810293	0.0878887
$86$	0	0
$87$	6.98472	0.748841
$88$	0	0
$89$	−14.5196	−1.53907	−0.769535	−	0.638605i	$-0.779512\pi$
$89$	−14.5196	−1.53907	−0.769535	+	0.638605i	$0.779512\pi$
$90$	0	0
$91$	7.48768	0.784923
$92$	0	0
$93$	4.29799	0.445681
$94$	0	0
$95$	5.15069	0.528449
$96$	0	0
$97$	−2.99325	−0.303919	−0.151959	−	0.988387i	$-0.548558\pi$
$97$	−2.99325	−0.303919	−0.151959	+	0.988387i	$0.548558\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	15.3700	1.52938	0.764688	−	0.644401i	$-0.222893\pi$
$101$	15.3700	1.52938	0.764688	+	0.644401i	$0.222893\pi$
$102$	0	0
$103$	5.49309	0.541251	0.270625	−	0.962685i	$-0.412770\pi$
$103$	5.49309	0.541251	0.270625	+	0.962685i	$0.412770\pi$
$104$	0	0
$105$	2.70085	0.263576
$106$	0	0
$107$	6.23672	0.602927	0.301463	−	0.953478i	$-0.402525\pi$
$107$	6.23672	0.602927	0.301463	+	0.953478i	$0.402525\pi$
$108$	0	0
$109$	19.1460	1.83385	0.916927	−	0.399056i	$-0.130662\pi$
$109$	19.1460	1.83385	0.916927	+	0.399056i	$0.130662\pi$
$110$	0	0
$111$	−4.20508	−0.399129
$112$	0	0
$113$	−10.8681	−1.02238	−0.511192	−	0.859467i	$-0.670796\pi$
$113$	−10.8681	−1.02238	−0.511192	+	0.859467i	$0.670796\pi$
$114$	0	0
$115$	−4.98262	−0.464632
$116$	0	0
$117$	−5.53255	−0.511485
$118$	0	0
$119$	−2.10846	−0.193282
$120$	0	0
$121$	0	0
$122$	0	0
$123$	10.1313	0.913505
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−18.9400	−1.68065	−0.840325	−	0.542083i	$-0.817636\pi$
$127$	−18.9400	−1.68065	−0.840325	+	0.542083i	$0.817636\pi$
$128$	0	0
$129$	−6.15414	−0.541842
$130$	0	0
$131$	−14.5409	−1.27045	−0.635224	−	0.772328i	$-0.719092\pi$
$131$	−14.5409	−1.27045	−0.635224	+	0.772328i	$0.719092\pi$
$132$	0	0
$133$	−13.4026	−1.16215
$134$	0	0
$135$	−5.10948	−0.439754
$136$	0	0
$137$	−19.9840	−1.70735	−0.853675	−	0.520806i	$-0.825632\pi$
$137$	−19.9840	−1.70735	−0.853675	+	0.520806i	$0.825632\pi$
$138$	0	0
$139$	2.07071	0.175635	0.0878176	−	0.996137i	$-0.472011\pi$
$139$	2.07071	0.175635	0.0878176	+	0.996137i	$0.472011\pi$
$140$	0	0
$141$	1.00464	0.0846058
$142$	0	0
$143$	0	0
$144$	0	0
$145$	6.72933	0.558840
$146$	0	0
$147$	0.237785	0.0196122
$148$	0	0
$149$	−0.504694	−0.0413461	−0.0206731	−	0.999786i	$-0.506581\pi$
$149$	−0.504694	−0.0413461	−0.0206731	+	0.999786i	$0.506581\pi$
$150$	0	0
$151$	−12.1188	−0.986216	−0.493108	−	0.869968i	$-0.664139\pi$
$151$	−12.1188	−0.986216	−0.493108	+	0.869968i	$0.664139\pi$
$152$	0	0
$153$	1.55791	0.125950
$154$	0	0
$155$	4.14084	0.332600
$156$	0	0
$157$	−11.8853	−0.948548	−0.474274	−	0.880377i	$-0.657289\pi$
$157$	−11.8853	−0.948548	−0.474274	+	0.880377i	$0.657289\pi$
$158$	0	0
$159$	−4.64427	−0.368314
$160$	0	0
$161$	12.9653	1.02181
$162$	0	0
$163$	0.0631196	0.00494391	0.00247195	−	0.999997i	$-0.499213\pi$
$163$	0.0631196	0.00494391	0.00247195	+	0.999997i	$0.499213\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	7.54352	0.583735	0.291868	−	0.956459i	$-0.405723\pi$
$167$	7.54352	0.583735	0.291868	+	0.956459i	$0.405723\pi$
$168$	0	0
$169$	−4.71966	−0.363051
$170$	0	0
$171$	9.90300	0.757301
$172$	0	0
$173$	1.41262	0.107400	0.0536998	−	0.998557i	$-0.482899\pi$
$173$	1.41262	0.107400	0.0536998	+	0.998557i	$0.482899\pi$
$174$	0	0
$175$	2.60210	0.196700
$176$	0	0
$177$	−7.06593	−0.531108
$178$	0	0
$179$	−15.8217	−1.18257	−0.591285	−	0.806463i	$-0.701379\pi$
$179$	−15.8217	−1.18257	−0.591285	+	0.806463i	$0.701379\pi$
$180$	0	0
$181$	18.9773	1.41057	0.705287	−	0.708922i	$-0.250818\pi$
$181$	18.9773	1.41057	0.705287	+	0.708922i	$0.250818\pi$
$182$	0	0
$183$	−2.71224	−0.200495
$184$	0	0
$185$	−4.05133	−0.297860
$186$	0	0
$187$	0	0
$188$	0	0
$189$	13.2954	0.967096
$190$	0	0
$191$	10.3666	0.750098	0.375049	−	0.927005i	$-0.377626\pi$
$191$	10.3666	0.750098	0.375049	+	0.927005i	$0.377626\pi$
$192$	0	0
$193$	0.608652	0.0438117	0.0219059	−	0.999760i	$-0.493027\pi$
$193$	0.608652	0.0438117	0.0219059	+	0.999760i	$0.493027\pi$
$194$	0	0
$195$	2.98677	0.213887
$196$	0	0
$197$	−11.1521	−0.794553	−0.397277	−	0.917699i	$-0.630045\pi$
$197$	−11.1521	−0.794553	−0.397277	+	0.917699i	$0.630045\pi$
$198$	0	0
$199$	8.91481	0.631954	0.315977	−	0.948767i	$-0.397668\pi$
$199$	8.91481	0.631954	0.315977	+	0.948767i	$0.397668\pi$
$200$	0	0
$201$	16.4420	1.15973
$202$	0	0
$203$	−17.5104	−1.22899
$204$	0	0
$205$	9.76082	0.681725
$206$	0	0
$207$	−9.57987	−0.665847
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−17.4046	−1.19818	−0.599092	−	0.800680i	$-0.704472\pi$
$211$	−17.4046	−1.19818	−0.599092	+	0.800680i	$0.704472\pi$
$212$	0	0
$213$	1.46710	0.100524
$214$	0	0
$215$	−5.92912	−0.404362
$216$	0	0
$217$	−10.7749	−0.731445
$218$	0	0
$219$	−13.6417	−0.921822
$220$	0	0
$221$	−2.33167	−0.156845
$222$	0	0
$223$	−1.53670	−0.102905	−0.0514524	−	0.998675i	$-0.516385\pi$
$223$	−1.53670	−0.102905	−0.0514524	+	0.998675i	$0.516385\pi$
$224$	0	0
$225$	−1.92266	−0.128177
$226$	0	0
$227$	−10.0981	−0.670231	−0.335116	−	0.942177i	$-0.608775\pi$
$227$	−10.0981	−0.670231	−0.335116	+	0.942177i	$0.608775\pi$
$228$	0	0
$229$	2.39276	0.158118	0.0790590	−	0.996870i	$-0.474808\pi$
$229$	2.39276	0.158118	0.0790590	+	0.996870i	$0.474808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.28726	0.542916	0.271458	−	0.962450i	$-0.412494\pi$
$233$	8.28726	0.542916	0.271458	+	0.962450i	$0.412494\pi$
$234$	0	0
$235$	0.967904	0.0631391
$236$	0	0
$237$	5.90723	0.383716
$238$	0	0
$239$	−4.56206	−0.295095	−0.147548	−	0.989055i	$-0.547138\pi$
$239$	−4.56206	−0.295095	−0.147548	+	0.989055i	$0.547138\pi$
$240$	0	0
$241$	−23.4148	−1.50828	−0.754139	−	0.656715i	$-0.771946\pi$
$241$	−23.4148	−1.50828	−0.754139	+	0.656715i	$0.771946\pi$
$242$	0	0
$243$	−15.8106	−1.01425
$244$	0	0
$245$	0.229090	0.0146361
$246$	0	0
$247$	−14.8214	−0.943063
$248$	0	0
$249$	8.27214	0.524225
$250$	0	0
$251$	−27.7446	−1.75122	−0.875611	−	0.483017i	$-0.839541\pi$
$251$	−27.7446	−1.75122	−0.875611	+	0.483017i	$0.839541\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−0.841046	−0.0526683
$256$	0	0
$257$	−25.4843	−1.58966	−0.794832	−	0.606829i	$-0.792441\pi$
$257$	−25.4843	−1.58966	−0.794832	+	0.606829i	$0.792441\pi$
$258$	0	0
$259$	10.5419	0.655045
$260$	0	0
$261$	12.9382	0.800853
$262$	0	0
$263$	−16.1271	−0.994442	−0.497221	−	0.867624i	$-0.665646\pi$
$263$	−16.1271	−0.994442	−0.497221	+	0.867624i	$0.665646\pi$
$264$	0	0
$265$	−4.47445	−0.274863
$266$	0	0
$267$	15.0706	0.922306
$268$	0	0
$269$	17.5236	1.06843	0.534216	−	0.845348i	$-0.320607\pi$
$269$	17.5236	1.06843	0.534216	+	0.845348i	$0.320607\pi$
$270$	0	0
$271$	−7.33746	−0.445719	−0.222860	−	0.974851i	$-0.571539\pi$
$271$	−7.33746	−0.445719	−0.222860	+	0.974851i	$0.571539\pi$
$272$	0	0
$273$	−7.77186	−0.470374
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.4068	1.04588	0.522938	−	0.852371i	$-0.324836\pi$
$277$	17.4068	1.04588	0.522938	+	0.852371i	$0.324836\pi$
$278$	0	0
$279$	7.96140	0.476637
$280$	0	0
$281$	−13.7827	−0.822206	−0.411103	−	0.911589i	$-0.634856\pi$
$281$	−13.7827	−0.822206	−0.411103	+	0.911589i	$0.634856\pi$
$282$	0	0
$283$	−11.5904	−0.688980	−0.344490	−	0.938790i	$-0.611948\pi$
$283$	−11.5904	−0.688980	−0.344490	+	0.938790i	$0.611948\pi$
$284$	0	0
$285$	−5.34617	−0.316680
$286$	0	0
$287$	−25.3986	−1.49923
$288$	0	0
$289$	−16.3434	−0.961378
$290$	0	0
$291$	3.10685	0.182127
$292$	0	0
$293$	16.1449	0.943197	0.471598	−	0.881813i	$-0.343677\pi$
$293$	16.1449	0.943197	0.471598	+	0.881813i	$0.343677\pi$
$294$	0	0
$295$	−6.80756	−0.396352
$296$	0	0
$297$	0	0
$298$	0	0
$299$	14.3378	0.829176
$300$	0	0
$301$	15.4281	0.889263
$302$	0	0
$303$	−15.9534	−0.916497
$304$	0	0
$305$	−2.61307	−0.149624
$306$	0	0
$307$	14.3107	0.816753	0.408377	−	0.912814i	$-0.366095\pi$
$307$	14.3107	0.816753	0.408377	+	0.912814i	$0.366095\pi$
$308$	0	0
$309$	−5.70157	−0.324351
$310$	0	0
$311$	−22.3744	−1.26873	−0.634367	−	0.773032i	$-0.718739\pi$
$311$	−22.3744	−1.26873	−0.634367	+	0.773032i	$0.718739\pi$
$312$	0	0
$313$	11.5368	0.652100	0.326050	−	0.945352i	$-0.394282\pi$
$313$	11.5368	0.652100	0.326050	+	0.945352i	$0.394282\pi$
$314$	0	0
$315$	5.00294	0.281883
$316$	0	0
$317$	−25.3314	−1.42275	−0.711376	−	0.702812i	$-0.751928\pi$
$317$	−25.3314	−1.42275	−0.711376	+	0.702812i	$0.751928\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−6.47342	−0.361311
$322$	0	0
$323$	4.17357	0.232224
$324$	0	0
$325$	2.87756	0.159618
$326$	0	0
$327$	−19.8726	−1.09896
$328$	0	0
$329$	−2.51858	−0.138854
$330$	0	0
$331$	−20.9669	−1.15244	−0.576221	−	0.817294i	$-0.695473\pi$
$331$	−20.9669	−1.15244	−0.576221	+	0.817294i	$0.695473\pi$
$332$	0	0
$333$	−7.78931	−0.426851
$334$	0	0
$335$	15.8408	0.865474
$336$	0	0
$337$	18.1430	0.988311	0.494156	−	0.869373i	$-0.335477\pi$
$337$	18.1430	0.988311	0.494156	+	0.869373i	$0.335477\pi$
$338$	0	0
$339$	11.2806	0.612675
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−18.8108	−1.01569
$344$	0	0
$345$	5.17172	0.278436
$346$	0	0
$347$	−18.9900	−1.01944	−0.509718	−	0.860342i	$-0.670250\pi$
$347$	−18.9900	−1.01944	−0.509718	+	0.860342i	$0.670250\pi$
$348$	0	0
$349$	−16.1172	−0.862732	−0.431366	−	0.902177i	$-0.641968\pi$
$349$	−16.1172	−0.862732	−0.431366	+	0.902177i	$0.641968\pi$
$350$	0	0
$351$	14.7028	0.784779
$352$	0	0
$353$	28.0252	1.49163	0.745816	−	0.666152i	$-0.232060\pi$
$353$	28.0252	1.49163	0.745816	+	0.666152i	$0.232060\pi$
$354$	0	0
$355$	1.41346	0.0750185
$356$	0	0
$357$	2.18848	0.115827
$358$	0	0
$359$	−5.23896	−0.276502	−0.138251	−	0.990397i	$-0.544148\pi$
$359$	−5.23896	−0.276502	−0.138251	+	0.990397i	$0.544148\pi$
$360$	0	0
$361$	7.52958	0.396294
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−13.1429	−0.687932
$366$	0	0
$367$	−26.3079	−1.37326	−0.686632	−	0.727005i	$-0.740911\pi$
$367$	−26.3079	−1.37326	−0.686632	+	0.727005i	$0.740911\pi$
$368$	0	0
$369$	18.7667	0.976955
$370$	0	0
$371$	11.6430	0.604472
$372$	0	0
$373$	36.0081	1.86443	0.932215	−	0.361905i	$-0.117873\pi$
$373$	36.0081	1.86443	0.932215	+	0.361905i	$0.117873\pi$
$374$	0	0
$375$	1.03795	0.0535996
$376$	0	0
$377$	−19.3640	−0.997298
$378$	0	0
$379$	−14.0945	−0.723986	−0.361993	−	0.932181i	$-0.617904\pi$
$379$	−14.0945	−0.723986	−0.361993	+	0.932181i	$0.617904\pi$
$380$	0	0
$381$	19.6588	1.00715
$382$	0	0
$383$	2.14120	0.109410	0.0547052	−	0.998503i	$-0.482578\pi$
$383$	2.14120	0.109410	0.0547052	+	0.998503i	$0.482578\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−11.3996	−0.579477
$388$	0	0
$389$	−29.2825	−1.48468	−0.742340	−	0.670023i	$-0.766284\pi$
$389$	−29.2825	−1.48468	−0.742340	+	0.670023i	$0.766284\pi$
$390$	0	0
$391$	−4.03739	−0.204179
$392$	0	0
$393$	15.0928	0.761331
$394$	0	0
$395$	5.69123	0.286357
$396$	0	0
$397$	9.09620	0.456525	0.228263	−	0.973600i	$-0.426696\pi$
$397$	9.09620	0.456525	0.228263	+	0.973600i	$0.426696\pi$
$398$	0	0
$399$	13.9112	0.696434
$400$	0	0
$401$	32.3614	1.61605	0.808027	−	0.589146i	$-0.200536\pi$
$401$	32.3614	1.61605	0.808027	+	0.589146i	$0.200536\pi$
$402$	0	0
$403$	−11.9155	−0.593553
$404$	0	0
$405$	−0.464568	−0.0230846
$406$	0	0
$407$	0	0
$408$	0	0
$409$	20.8620	1.03156	0.515779	−	0.856722i	$-0.327502\pi$
$409$	20.8620	1.03156	0.515779	+	0.856722i	$0.327502\pi$
$410$	0	0
$411$	20.7425	1.02315
$412$	0	0
$413$	17.7139	0.871646
$414$	0	0
$415$	7.96967	0.391216
$416$	0	0
$417$	−2.14930	−0.105251
$418$	0	0
$419$	−36.4444	−1.78043	−0.890213	−	0.455545i	$-0.849444\pi$
$419$	−36.4444	−1.78043	−0.890213	+	0.455545i	$0.849444\pi$
$420$	0	0
$421$	−32.4285	−1.58047	−0.790235	−	0.612804i	$-0.790041\pi$
$421$	−32.4285	−1.58047	−0.790235	+	0.612804i	$0.790041\pi$
$422$	0	0
$423$	1.86095	0.0904823
$424$	0	0
$425$	−0.810293	−0.0393050
$426$	0	0
$427$	6.79946	0.329049
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−28.8732	−1.39077	−0.695387	−	0.718636i	$-0.744767\pi$
$431$	−28.8732	−1.39077	−0.695387	+	0.718636i	$0.744767\pi$
$432$	0	0
$433$	7.55542	0.363090	0.181545	−	0.983383i	$-0.441890\pi$
$433$	7.55542	0.363090	0.181545	+	0.983383i	$0.441890\pi$
$434$	0	0
$435$	−6.98472	−0.334892
$436$	0	0
$437$	−25.6639	−1.22767
$438$	0	0
$439$	1.96072	0.0935801	0.0467900	−	0.998905i	$-0.485101\pi$
$439$	1.96072	0.0935801	0.0467900	+	0.998905i	$0.485101\pi$
$440$	0	0
$441$	0.440462	0.0209744
$442$	0	0
$443$	6.03304	0.286638	0.143319	−	0.989677i	$-0.454222\pi$
$443$	6.03304	0.286638	0.143319	+	0.989677i	$0.454222\pi$
$444$	0	0
$445$	14.5196	0.688293
$446$	0	0
$447$	0.523848	0.0247772
$448$	0	0
$449$	9.82399	0.463623	0.231811	−	0.972761i	$-0.425535\pi$
$449$	9.82399	0.463623	0.231811	+	0.972761i	$0.425535\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	12.5788	0.591002
$454$	0	0
$455$	−7.48768	−0.351028
$456$	0	0
$457$	7.82880	0.366216	0.183108	−	0.983093i	$-0.441384\pi$
$457$	7.82880	0.366216	0.183108	+	0.983093i	$0.441384\pi$
$458$	0	0
$459$	−4.14018	−0.193247
$460$	0	0
$461$	3.19972	0.149026	0.0745130	−	0.997220i	$-0.476260\pi$
$461$	3.19972	0.149026	0.0745130	+	0.997220i	$0.476260\pi$
$462$	0	0
$463$	−21.2096	−0.985696	−0.492848	−	0.870115i	$-0.664044\pi$
$463$	−21.2096	−0.985696	−0.492848	+	0.870115i	$0.664044\pi$
$464$	0	0
$465$	−4.29799	−0.199315
$466$	0	0
$467$	−31.2967	−1.44824	−0.724118	−	0.689676i	$-0.757753\pi$
$467$	−31.2967	−1.44824	−0.724118	+	0.689676i	$0.757753\pi$
$468$	0	0
$469$	−41.2192	−1.90333
$470$	0	0
$471$	12.3363	0.568429
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−5.15069	−0.236330
$476$	0	0
$477$	−8.60283	−0.393896
$478$	0	0
$479$	−18.0022	−0.822542	−0.411271	−	0.911513i	$-0.634915\pi$
$479$	−18.0022	−0.822542	−0.411271	+	0.911513i	$0.634915\pi$
$480$	0	0
$481$	11.6579	0.531556
$482$	0	0
$483$	−13.4573	−0.612330
$484$	0	0
$485$	2.99325	0.135917
$486$	0	0
$487$	21.0873	0.955558	0.477779	−	0.878480i	$-0.341442\pi$
$487$	21.0873	0.955558	0.477779	+	0.878480i	$0.341442\pi$
$488$	0	0
$489$	−0.0655151	−0.00296270
$490$	0	0
$491$	22.0057	0.993104	0.496552	−	0.868007i	$-0.334599\pi$
$491$	22.0057	0.993104	0.496552	+	0.868007i	$0.334599\pi$
$492$	0	0
$493$	5.45273	0.245579
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−3.67795	−0.164979
$498$	0	0
$499$	16.1255	0.721877	0.360939	−	0.932590i	$-0.382456\pi$
$499$	16.1255	0.721877	0.360939	+	0.932590i	$0.382456\pi$
$500$	0	0
$501$	−7.82982	−0.349810
$502$	0	0
$503$	−3.16694	−0.141207	−0.0706035	−	0.997504i	$-0.522493\pi$
$503$	−3.16694	−0.141207	−0.0706035	+	0.997504i	$0.522493\pi$
$504$	0	0
$505$	−15.3700	−0.683957
$506$	0	0
$507$	4.89878	0.217563
$508$	0	0
$509$	28.9843	1.28470	0.642352	−	0.766409i	$-0.277959\pi$
$509$	28.9843	1.28470	0.642352	+	0.766409i	$0.277959\pi$
$510$	0	0
$511$	34.1992	1.51288
$512$	0	0
$513$	−26.3173	−1.16194
$514$	0	0
$515$	−5.49309	−0.242055
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−1.46623	−0.0643605
$520$	0	0
$521$	−15.8497	−0.694388	−0.347194	−	0.937793i	$-0.612865\pi$
$521$	−15.8497	−0.694388	−0.347194	+	0.937793i	$0.612865\pi$
$522$	0	0
$523$	24.0929	1.05351	0.526754	−	0.850018i	$-0.323409\pi$
$523$	24.0929	1.05351	0.526754	+	0.850018i	$0.323409\pi$
$524$	0	0
$525$	−2.70085	−0.117875
$526$	0	0
$527$	3.35529	0.146159
$528$	0	0
$529$	1.82653	0.0794143
$530$	0	0
$531$	−13.0886	−0.567997
$532$	0	0
$533$	−28.0873	−1.21660
$534$	0	0
$535$	−6.23672	−0.269637
$536$	0	0
$537$	16.4222	0.708669
$538$	0	0
$539$	0	0
$540$	0	0
$541$	21.3207	0.916648	0.458324	−	0.888785i	$-0.348450\pi$
$541$	21.3207	0.916648	0.458324	+	0.888785i	$0.348450\pi$
$542$	0	0
$543$	−19.6976	−0.845303
$544$	0	0
$545$	−19.1460	−0.820124
$546$	0	0
$547$	29.3180	1.25355	0.626775	−	0.779201i	$-0.284375\pi$
$547$	29.3180	1.25355	0.626775	+	0.779201i	$0.284375\pi$
$548$	0	0
$549$	−5.02403	−0.214420
$550$	0	0
$551$	34.6607	1.47659
$552$	0	0
$553$	−14.8091	−0.629749
$554$	0	0
$555$	4.20508	0.178496
$556$	0	0
$557$	1.68279	0.0713019	0.0356510	−	0.999364i	$-0.488650\pi$
$557$	1.68279	0.0713019	0.0356510	+	0.999364i	$0.488650\pi$
$558$	0	0
$559$	17.0614	0.721619
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.14833	−0.0483963	−0.0241981	−	0.999707i	$-0.507703\pi$
$563$	−1.14833	−0.0483963	−0.0241981	+	0.999707i	$0.507703\pi$
$564$	0	0
$565$	10.8681	0.457224
$566$	0	0
$567$	1.20885	0.0507670
$568$	0	0
$569$	6.68468	0.280237	0.140118	−	0.990135i	$-0.455252\pi$
$569$	6.68468	0.280237	0.140118	+	0.990135i	$0.455252\pi$
$570$	0	0
$571$	−17.7964	−0.744756	−0.372378	−	0.928081i	$-0.621458\pi$
$571$	−17.7964	−0.744756	−0.372378	+	0.928081i	$0.621458\pi$
$572$	0	0
$573$	−10.7600	−0.449505
$574$	0	0
$575$	4.98262	0.207790
$576$	0	0
$577$	−45.9807	−1.91420	−0.957102	−	0.289752i	$-0.906427\pi$
$577$	−45.9807	−1.91420	−0.957102	+	0.289752i	$0.906427\pi$
$578$	0	0
$579$	−0.631752	−0.0262547
$580$	0	0
$581$	−20.7379	−0.860351
$582$	0	0
$583$	0	0
$584$	0	0
$585$	5.53255	0.228743
$586$	0	0
$587$	4.72240	0.194914	0.0974571	−	0.995240i	$-0.468929\pi$
$587$	4.72240	0.194914	0.0974571	+	0.995240i	$0.468929\pi$
$588$	0	0
$589$	21.3282	0.878812
$590$	0	0
$591$	11.5753	0.476146
$592$	0	0
$593$	−3.67545	−0.150932	−0.0754662	−	0.997148i	$-0.524045\pi$
$593$	−3.67545	−0.150932	−0.0754662	+	0.997148i	$0.524045\pi$
$594$	0	0
$595$	2.10846	0.0864385
$596$	0	0
$597$	−9.25315	−0.378706
$598$	0	0
$599$	−39.8714	−1.62910	−0.814550	−	0.580093i	$-0.803016\pi$
$599$	−39.8714	−1.62910	−0.814550	+	0.580093i	$0.803016\pi$
$600$	0	0
$601$	12.3476	0.503670	0.251835	−	0.967770i	$-0.418966\pi$
$601$	12.3476	0.503670	0.251835	+	0.967770i	$0.418966\pi$
$602$	0	0
$603$	30.4563	1.24028
$604$	0	0
$605$	0	0
$606$	0	0
$607$	19.7067	0.799869	0.399935	−	0.916544i	$-0.369033\pi$
$607$	19.7067	0.799869	0.399935	+	0.916544i	$0.369033\pi$
$608$	0	0
$609$	18.1749	0.736485
$610$	0	0
$611$	−2.78520	−0.112677
$612$	0	0
$613$	−37.7646	−1.52530	−0.762650	−	0.646812i	$-0.776102\pi$
$613$	−37.7646	−1.52530	−0.762650	+	0.646812i	$0.776102\pi$
$614$	0	0
$615$	−10.1313	−0.408532
$616$	0	0
$617$	−3.49353	−0.140644	−0.0703222	−	0.997524i	$-0.522403\pi$
$617$	−3.49353	−0.140644	−0.0703222	+	0.997524i	$0.522403\pi$
$618$	0	0
$619$	6.48840	0.260791	0.130395	−	0.991462i	$-0.458375\pi$
$619$	6.48840	0.260791	0.130395	+	0.991462i	$0.458375\pi$
$620$	0	0
$621$	25.4586	1.02162
$622$	0	0
$623$	−37.7813	−1.51368
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.28276	−0.130892
$630$	0	0
$631$	32.7333	1.30309	0.651547	−	0.758609i	$-0.274120\pi$
$631$	32.7333	1.30309	0.651547	+	0.758609i	$0.274120\pi$
$632$	0	0
$633$	18.0652	0.718026
$634$	0	0
$635$	18.9400	0.751609
$636$	0	0
$637$	−0.659221	−0.0261193
$638$	0	0
$639$	2.71759	0.107506
$640$	0	0
$641$	−7.19375	−0.284136	−0.142068	−	0.989857i	$-0.545375\pi$
$641$	−7.19375	−0.284136	−0.142068	+	0.989857i	$0.545375\pi$
$642$	0	0
$643$	8.21093	0.323808	0.161904	−	0.986807i	$-0.448237\pi$
$643$	8.21093	0.323808	0.161904	+	0.986807i	$0.448237\pi$
$644$	0	0
$645$	6.15414	0.242319
$646$	0	0
$647$	−49.0008	−1.92642	−0.963210	−	0.268750i	$-0.913389\pi$
$647$	−49.0008	−1.92642	−0.963210	+	0.268750i	$0.913389\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	11.1838	0.438327
$652$	0	0
$653$	−43.8455	−1.71581	−0.857903	−	0.513811i	$-0.828233\pi$
$653$	−43.8455	−1.71581	−0.857903	+	0.513811i	$0.828233\pi$
$654$	0	0
$655$	14.5409	0.568161
$656$	0	0
$657$	−25.2693	−0.985850
$658$	0	0
$659$	37.2905	1.45263	0.726315	−	0.687362i	$-0.241231\pi$
$659$	37.2905	1.45263	0.726315	+	0.687362i	$0.241231\pi$
$660$	0	0
$661$	35.5579	1.38304	0.691520	−	0.722357i	$-0.256941\pi$
$661$	35.5579	1.38304	0.691520	+	0.722357i	$0.256941\pi$
$662$	0	0
$663$	2.42016	0.0939912
$664$	0	0
$665$	13.4026	0.519730
$666$	0	0
$667$	−33.5297	−1.29828
$668$	0	0
$669$	1.59502	0.0616669
$670$	0	0
$671$	0	0
$672$	0	0
$673$	13.6219	0.525084	0.262542	−	0.964921i	$-0.415439\pi$
$673$	13.6219	0.525084	0.262542	+	0.964921i	$0.415439\pi$
$674$	0	0
$675$	5.10948	0.196664
$676$	0	0
$677$	−6.89247	−0.264899	−0.132449	−	0.991190i	$-0.542284\pi$
$677$	−6.89247	−0.264899	−0.132449	+	0.991190i	$0.542284\pi$
$678$	0	0
$679$	−7.78874	−0.298904
$680$	0	0
$681$	10.4813	0.401644
$682$	0	0
$683$	8.29642	0.317454	0.158727	−	0.987323i	$-0.449261\pi$
$683$	8.29642	0.317454	0.158727	+	0.987323i	$0.449261\pi$
$684$	0	0
$685$	19.9840	0.763550
$686$	0	0
$687$	−2.48357	−0.0947541
$688$	0	0
$689$	12.8755	0.490517
$690$	0	0
$691$	18.3562	0.698303	0.349151	−	0.937066i	$-0.386470\pi$
$691$	18.3562	0.698303	0.349151	+	0.937066i	$0.386470\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2.07071	−0.0785464
$696$	0	0
$697$	7.90913	0.299580
$698$	0	0
$699$	−8.60178	−0.325349
$700$	0	0
$701$	28.0624	1.05990	0.529951	−	0.848028i	$-0.322210\pi$
$701$	28.0624	1.05990	0.529951	+	0.848028i	$0.322210\pi$
$702$	0	0
$703$	−20.8671	−0.787018
$704$	0	0
$705$	−1.00464	−0.0378369
$706$	0	0
$707$	39.9943	1.50414
$708$	0	0
$709$	−16.2588	−0.610612	−0.305306	−	0.952254i	$-0.598759\pi$
$709$	−16.2588	−0.610612	−0.305306	+	0.952254i	$0.598759\pi$
$710$	0	0
$711$	10.9423	0.410368
$712$	0	0
$713$	−20.6322	−0.772683
$714$	0	0
$715$	0	0
$716$	0	0
$717$	4.73520	0.176839
$718$	0	0
$719$	−3.75469	−0.140026	−0.0700132	−	0.997546i	$-0.522304\pi$
$719$	−3.75469	−0.140026	−0.0700132	+	0.997546i	$0.522304\pi$
$720$	0	0
$721$	14.2936	0.532320
$722$	0	0
$723$	24.3034	0.903854
$724$	0	0
$725$	−6.72933	−0.249921
$726$	0	0
$727$	35.1985	1.30544	0.652720	−	0.757599i	$-0.273628\pi$
$727$	35.1985	1.30544	0.652720	+	0.757599i	$0.273628\pi$
$728$	0	0
$729$	15.0170	0.556185
$730$	0	0
$731$	−4.80432	−0.177694
$732$	0	0
$733$	20.2869	0.749313	0.374656	−	0.927164i	$-0.377761\pi$
$733$	20.2869	0.749313	0.374656	+	0.927164i	$0.377761\pi$
$734$	0	0
$735$	−0.237785	−0.00877083
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−15.6336	−0.575093	−0.287546	−	0.957767i	$-0.592840\pi$
$739$	−15.6336	−0.575093	−0.287546	+	0.957767i	$0.592840\pi$
$740$	0	0
$741$	15.3839	0.565142
$742$	0	0
$743$	19.9836	0.733129	0.366564	−	0.930393i	$-0.380534\pi$
$743$	19.9836	0.733129	0.366564	+	0.930393i	$0.380534\pi$
$744$	0	0
$745$	0.504694	0.0184905
$746$	0	0
$747$	15.3229	0.560637
$748$	0	0
$749$	16.2286	0.592979
$750$	0	0
$751$	31.4457	1.14747	0.573734	−	0.819041i	$-0.305494\pi$
$751$	31.4457	1.14747	0.573734	+	0.819041i	$0.305494\pi$
$752$	0	0
$753$	28.7975	1.04944
$754$	0	0
$755$	12.1188	0.441049
$756$	0	0
$757$	45.3850	1.64955	0.824773	−	0.565463i	$-0.191303\pi$
$757$	45.3850	1.64955	0.824773	+	0.565463i	$0.191303\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−35.6102	−1.29087	−0.645435	−	0.763816i	$-0.723324\pi$
$761$	−35.6102	−1.29087	−0.645435	+	0.763816i	$0.723324\pi$
$762$	0	0
$763$	49.8197	1.80360
$764$	0	0
$765$	−1.55791	−0.0563265
$766$	0	0
$767$	19.5892	0.707324
$768$	0	0
$769$	18.9907	0.684821	0.342411	−	0.939550i	$-0.388757\pi$
$769$	18.9907	0.684821	0.342411	+	0.939550i	$0.388757\pi$
$770$	0	0
$771$	26.4514	0.952625
$772$	0	0
$773$	43.6808	1.57109	0.785544	−	0.618805i	$-0.212383\pi$
$773$	43.6808	1.57109	0.785544	+	0.618805i	$0.212383\pi$
$774$	0	0
$775$	−4.14084	−0.148743
$776$	0	0
$777$	−10.9420	−0.392543
$778$	0	0
$779$	50.2749	1.80129
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−34.3834	−1.22876
$784$	0	0
$785$	11.8853	0.424204
$786$	0	0
$787$	50.8854	1.81387	0.906934	−	0.421272i	$-0.138416\pi$
$787$	50.8854	1.81387	0.906934	+	0.421272i	$0.138416\pi$
$788$	0	0
$789$	16.7392	0.595931
$790$	0	0
$791$	−28.2798	−1.00551
$792$	0	0
$793$	7.51925	0.267017
$794$	0	0
$795$	4.64427	0.164715
$796$	0	0
$797$	−18.9156	−0.670025	−0.335013	−	0.942214i	$-0.608741\pi$
$797$	−18.9156	−0.670025	−0.335013	+	0.942214i	$0.608741\pi$
$798$	0	0
$799$	0.784286	0.0277461
$800$	0	0
$801$	27.9161	0.986367
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−12.9653	−0.456966
$806$	0	0
$807$	−18.1887	−0.640271
$808$	0	0
$809$	−20.8204	−0.732005	−0.366002	−	0.930614i	$-0.619274\pi$
$809$	−20.8204	−0.732005	−0.366002	+	0.930614i	$0.619274\pi$
$810$	0	0
$811$	36.0642	1.26639	0.633193	−	0.773994i	$-0.281744\pi$
$811$	36.0642	1.26639	0.633193	+	0.773994i	$0.281744\pi$
$812$	0	0
$813$	7.61594	0.267103
$814$	0	0
$815$	−0.0631196	−0.00221098
$816$	0	0
$817$	−30.5390	−1.06843
$818$	0	0
$819$	−14.3962	−0.503045
$820$	0	0
$821$	37.9278	1.32369	0.661845	−	0.749641i	$-0.269774\pi$
$821$	37.9278	1.32369	0.661845	+	0.749641i	$0.269774\pi$
$822$	0	0
$823$	−47.3524	−1.65060	−0.825300	−	0.564695i	$-0.808994\pi$
$823$	−47.3524	−1.65060	−0.825300	+	0.564695i	$0.808994\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	32.7566	1.13906	0.569530	−	0.821970i	$-0.307125\pi$
$827$	32.7566	1.13906	0.569530	+	0.821970i	$0.307125\pi$
$828$	0	0
$829$	−24.3065	−0.844199	−0.422099	−	0.906550i	$-0.638707\pi$
$829$	−24.3065	−0.844199	−0.422099	+	0.906550i	$0.638707\pi$
$830$	0	0
$831$	−18.0675	−0.626753
$832$	0	0
$833$	0.185630	0.00643171
$834$	0	0
$835$	−7.54352	−0.261054
$836$	0	0
$837$	−21.1575	−0.731311
$838$	0	0
$839$	−38.6436	−1.33413	−0.667063	−	0.745001i	$-0.732449\pi$
$839$	−38.6436	−1.33413	−0.667063	+	0.745001i	$0.732449\pi$
$840$	0	0
$841$	16.2839	0.561513
$842$	0	0
$843$	14.3058	0.492717
$844$	0	0
$845$	4.71966	0.162361
$846$	0	0
$847$	0	0
$848$	0	0
$849$	12.0303	0.412879
$850$	0	0
$851$	20.1862	0.691975
$852$	0	0
$853$	−28.7982	−0.986033	−0.493017	−	0.870020i	$-0.664106\pi$
$853$	−28.7982	−0.986033	−0.493017	+	0.870020i	$0.664106\pi$
$854$	0	0
$855$	−9.90300	−0.338675
$856$	0	0
$857$	44.6116	1.52390	0.761952	−	0.647633i	$-0.224241\pi$
$857$	44.6116	1.52390	0.761952	+	0.647633i	$0.224241\pi$
$858$	0	0
$859$	46.4941	1.58636	0.793180	−	0.608988i	$-0.208424\pi$
$859$	46.4941	1.58636	0.793180	+	0.608988i	$0.208424\pi$
$860$	0	0
$861$	26.3625	0.898433
$862$	0	0
$863$	15.6647	0.533231	0.266616	−	0.963803i	$-0.414095\pi$
$863$	15.6647	0.533231	0.266616	+	0.963803i	$0.414095\pi$
$864$	0	0
$865$	−1.41262	−0.0480306
$866$	0	0
$867$	16.9637	0.576117
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−45.5827	−1.54451
$872$	0	0
$873$	5.75499	0.194777
$874$	0	0
$875$	−2.60210	−0.0879669
$876$	0	0
$877$	33.6285	1.13555	0.567777	−	0.823182i	$-0.307804\pi$
$877$	33.6285	1.13555	0.567777	+	0.823182i	$0.307804\pi$
$878$	0	0
$879$	−16.7577	−0.565222
$880$	0	0
$881$	42.4530	1.43028	0.715139	−	0.698982i	$-0.246363\pi$
$881$	42.4530	1.43028	0.715139	+	0.698982i	$0.246363\pi$
$882$	0	0
$883$	−8.63358	−0.290543	−0.145272	−	0.989392i	$-0.546406\pi$
$883$	−8.63358	−0.290543	−0.145272	+	0.989392i	$0.546406\pi$
$884$	0	0
$885$	7.06593	0.237519
$886$	0	0
$887$	4.32578	0.145246	0.0726228	−	0.997359i	$-0.476863\pi$
$887$	4.32578	0.145246	0.0726228	+	0.997359i	$0.476863\pi$
$888$	0	0
$889$	−49.2836	−1.65292
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.98537	0.166829
$894$	0	0
$895$	15.8217	0.528862
$896$	0	0
$897$	−14.8819	−0.496893
$898$	0	0
$899$	27.8651	0.929352
$900$	0	0
$901$	−3.62562	−0.120787
$902$	0	0
$903$	−16.0137	−0.532902
$904$	0	0
$905$	−18.9773	−0.630828
$906$	0	0
$907$	−47.4842	−1.57669	−0.788344	−	0.615234i	$-0.789061\pi$
$907$	−47.4842	−1.57669	−0.788344	+	0.615234i	$0.789061\pi$
$908$	0	0
$909$	−29.5513	−0.980154
$910$	0	0
$911$	−22.7476	−0.753662	−0.376831	−	0.926282i	$-0.622986\pi$
$911$	−22.7476	−0.753662	−0.376831	+	0.926282i	$0.622986\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	2.71224	0.0896639
$916$	0	0
$917$	−37.8369	−1.24949
$918$	0	0
$919$	−37.0964	−1.22370	−0.611849	−	0.790975i	$-0.709574\pi$
$919$	−37.0964	−1.22370	−0.611849	+	0.790975i	$0.709574\pi$
$920$	0	0
$921$	−14.8538	−0.489449
$922$	0	0
$923$	−4.06730	−0.133877
$924$	0	0
$925$	4.05133	0.133207
$926$	0	0
$927$	−10.5613	−0.346879
$928$	0	0
$929$	43.0548	1.41258	0.706290	−	0.707922i	$-0.250367\pi$
$929$	43.0548	1.41258	0.706290	+	0.707922i	$0.250367\pi$
$930$	0	0
$931$	1.17997	0.0386721
$932$	0	0
$933$	23.2235	0.760304
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.5075	−0.767956	−0.383978	−	0.923342i	$-0.625446\pi$
$937$	−23.5075	−0.767956	−0.383978	+	0.923342i	$0.625446\pi$
$938$	0	0
$939$	−11.9747	−0.390779
$940$	0	0
$941$	37.1410	1.21076	0.605382	−	0.795935i	$-0.293021\pi$
$941$	37.1410	1.21076	0.605382	+	0.795935i	$0.293021\pi$
$942$	0	0
$943$	−48.6345	−1.58376
$944$	0	0
$945$	−13.2954	−0.432498
$946$	0	0
$947$	−16.3128	−0.530094	−0.265047	−	0.964235i	$-0.585388\pi$
$947$	−16.3128	−0.530094	−0.265047	+	0.964235i	$0.585388\pi$
$948$	0	0
$949$	37.8195	1.22767
$950$	0	0
$951$	26.2928	0.852601
$952$	0	0
$953$	−58.7967	−1.90461	−0.952306	−	0.305146i	$-0.901295\pi$
$953$	−58.7967	−1.90461	−0.952306	+	0.305146i	$0.901295\pi$
$954$	0	0
$955$	−10.3666	−0.335454
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−52.0004	−1.67918
$960$	0	0
$961$	−13.8535	−0.446886
$962$	0	0
$963$	−11.9911	−0.386407
$964$	0	0
$965$	−0.608652	−0.0195932
$966$	0	0
$967$	60.4542	1.94408	0.972039	−	0.234821i	$-0.0754502\pi$
$967$	60.4542	1.94408	0.972039	+	0.234821i	$0.0754502\pi$
$968$	0	0
$969$	−4.33196	−0.139163
$970$	0	0
$971$	41.6267	1.33586	0.667932	−	0.744222i	$-0.267180\pi$
$971$	41.6267	1.33586	0.667932	+	0.744222i	$0.267180\pi$
$972$	0	0
$973$	5.38818	0.172737
$974$	0	0
$975$	−2.98677	−0.0956531
$976$	0	0
$977$	47.7290	1.52699	0.763493	−	0.645816i	$-0.223483\pi$
$977$	47.7290	1.52699	0.763493	+	0.645816i	$0.223483\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−36.8111	−1.17529
$982$	0	0
$983$	−5.92744	−0.189056	−0.0945280	−	0.995522i	$-0.530134\pi$
$983$	−5.92744	−0.189056	−0.0945280	+	0.995522i	$0.530134\pi$
$984$	0	0
$985$	11.1521	0.355335
$986$	0	0
$987$	2.61417	0.0832098
$988$	0	0
$989$	29.5425	0.939398
$990$	0	0
$991$	34.5440	1.09733	0.548663	−	0.836044i	$-0.315137\pi$
$991$	34.5440	1.09733	0.548663	+	0.836044i	$0.315137\pi$
$992$	0	0
$993$	21.7626	0.690615
$994$	0	0
$995$	−8.91481	−0.282618
$996$	0	0
$997$	−34.4542	−1.09117	−0.545587	−	0.838054i	$-0.683693\pi$
$997$	−34.4542	−1.09117	−0.545587	+	0.838054i	$0.683693\pi$
$998$	0	0
$999$	20.7002	0.654925

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4840.2.a.ba.1.3		6
4.3	odd	2		9680.2.a.dd.1.4		6
11.2	odd	10		440.2.y.c.81.2	✓	12
11.6	odd	10		440.2.y.c.201.2	yes	12
11.10	odd	2		4840.2.a.bb.1.3		6
44.35	even	10		880.2.bo.i.81.2		12
44.39	even	10		880.2.bo.i.641.2		12
44.43	even	2		9680.2.a.dc.1.4		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.y.c.81.2	✓	12	11.2	odd	10
440.2.y.c.201.2	yes	12	11.6	odd	10
880.2.bo.i.81.2		12	44.35	even	10
880.2.bo.i.641.2		12	44.39	even	10
4840.2.a.ba.1.3		6	1.1	even	1	trivial
4840.2.a.bb.1.3		6	11.10	odd	2
9680.2.a.dc.1.4		6	44.43	even	2
9680.2.a.dd.1.4		6	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4840 = 2^{3} \cdot 5 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4840.a (trivial)

\(f(q)\)	\(=\)	\(q-1.03795 q^{3} -1.00000 q^{5} +2.60210 q^{7} -1.92266 q^{9} +2.87756 q^{13} +1.03795 q^{15} -0.810293 q^{17} -5.15069 q^{19} -2.70085 q^{21} +4.98262 q^{23} +1.00000 q^{25} +5.10948 q^{27} -6.72933 q^{29} -4.14084 q^{31} -2.60210 q^{35} +4.05133 q^{37} -2.98677 q^{39} -9.76082 q^{41} +5.92912 q^{43} +1.92266 q^{45} -0.967904 q^{47} -0.229090 q^{49} +0.841046 q^{51} +4.47445 q^{53} +5.34617 q^{57} +6.80756 q^{59} +2.61307 q^{61} -5.00294 q^{63} -2.87756 q^{65} -15.8408 q^{67} -5.17172 q^{69} -1.41346 q^{71} +13.1429 q^{73} -1.03795 q^{75} -5.69123 q^{79} +0.464568 q^{81} -7.96967 q^{83} +0.810293 q^{85} +6.98472 q^{87} -14.5196 q^{89} +7.48768 q^{91} +4.29799 q^{93} +5.15069 q^{95} -2.99325 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 3 q^{3} - 6 q^{5} - 7 q^{7} + 5 q^{9} + q^{13} + 3 q^{15} + 6 q^{17} - 7 q^{19} + 14 q^{21} - 9 q^{23} + 6 q^{25} - 21 q^{27} + 10 q^{29} + q^{31} + 7 q^{35} + 3 q^{37} - q^{39} - 6 q^{41} + 18 q^{43}+ \cdots - 33 q^{97}+O(q^{100}) \) 6 * q - 3 * q^3 - 6 * q^5 - 7 * q^7 + 5 * q^9 + q^13 + 3 * q^15 + 6 * q^17 - 7 * q^19 + 14 * q^21 - 9 * q^23 + 6 * q^25 - 21 * q^27 + 10 * q^29 + q^31 + 7 * q^35 + 3 * q^37 - q^39 - 6 * q^41 + 18 * q^43 - 5 * q^45 - 3 * q^47 + 17 * q^49 + 15 * q^51 - 23 * q^53 + 9 * q^57 - 2 * q^59 + 6 * q^61 - 49 * q^63 - q^65 - 22 * q^67 + 2 * q^69 - 13 * q^71 + 10 * q^73 - 3 * q^75 - 22 * q^79 + 10 * q^81 + 10 * q^83 - 6 * q^85 - 3 * q^87 - 25 * q^89 + 12 * q^91 + 19 * q^93 + 7 * q^95 - 33 * q^97

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