LMFDB - Embedded newform 480.4.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [480,4,Mod(1,480)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("480.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(480, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 480 = 2^{5} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	480.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-6,0,10,0,-12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$28.3209168028$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{89}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 22 $ x^2 - x - 22
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-4.21699$ of defining polynomial
Character	$\chi$	$=$	480.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +5.00000 q^{5} +12.8680 q^{7} +9.00000 q^{9} -49.7359 q^{11} -52.6039 q^{13} -15.0000 q^{15} +84.6039 q^{17} +26.6039 q^{19} -38.6039 q^{21} -136.340 q^{23} +25.0000 q^{25} -27.0000 q^{27} +6.00000 q^{29} -47.1320 q^{31} +149.208 q^{33} +64.3398 q^{35} +344.227 q^{37} +157.812 q^{39} -43.2078 q^{41} -252.000 q^{43} +45.0000 q^{45} -306.076 q^{47} -177.416 q^{49} -253.812 q^{51} -455.208 q^{53} -248.680 q^{55} -79.8117 q^{57} -708.151 q^{59} -652.039 q^{61} +115.812 q^{63} -263.019 q^{65} -704.528 q^{67} +409.019 q^{69} +531.775 q^{71} +57.6233 q^{73} -75.0000 q^{75} -640.000 q^{77} -429.699 q^{79} +81.0000 q^{81} -227.697 q^{83} +423.019 q^{85} -18.0000 q^{87} +1032.87 q^{89} -676.905 q^{91} +141.396 q^{93} +133.019 q^{95} -152.416 q^{97} -447.623 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 10 q^{5} - 12 q^{7} + 18 q^{9} - 24 q^{11} + 8 q^{13} - 30 q^{15} + 56 q^{17} - 60 q^{19} + 36 q^{21} - 84 q^{23} + 50 q^{25} - 54 q^{27} + 12 q^{29} - 132 q^{31} + 72 q^{33} - 60 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) $ 2 * q - 6 * q^3 + 10 * q^5 - 12 * q^7 + 18 * q^9 - 24 * q^11 + 8 * q^13 - 30 * q^15 + 56 * q^17 - 60 * q^19 + 36 * q^21 - 84 * q^23 + 50 * q^25 - 54 * q^27 + 12 * q^29 - 132 * q^31 + 72 * q^33 - 60 * q^35 - 104 * q^37 - 24 * q^39 + 140 * q^41 - 504 * q^43 + 90 * q^45 - 348 * q^47 + 98 * q^49 - 168 * q^51 - 684 * q^53 - 120 * q^55 + 180 * q^57 - 888 * q^59 - 172 * q^61 - 108 * q^63 + 40 * q^65 - 1560 * q^67 + 252 * q^69 - 144 * q^71 - 564 * q^73 - 150 * q^75 - 1280 * q^77 + 84 * q^79 + 162 * q^81 - 1512 * q^83 + 280 * q^85 - 36 * q^87 + 28 * q^89 - 2184 * q^91 + 396 * q^93 - 300 * q^95 + 148 * q^97 - 216 * 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
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	12.8680	0.694805	0.347402	−	0.937716i	$-0.387064\pi$
$7$	12.8680	0.694805	0.347402	+	0.937716i	$0.387064\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−49.7359	−1.36327	−0.681634	−	0.731693i	$-0.738730\pi$
$11$	−49.7359	−1.36327	−0.681634	+	0.731693i	$0.738730\pi$
$12$	0	0
$13$	−52.6039	−1.12228	−0.561142	−	0.827720i	$-0.689638\pi$
$13$	−52.6039	−1.12228	−0.561142	+	0.827720i	$0.689638\pi$
$14$	0	0
$15$	−15.0000	−0.258199
$16$	0	0
$17$	84.6039	1.20703	0.603513	−	0.797353i	$-0.293767\pi$
$17$	84.6039	1.20703	0.603513	+	0.797353i	$0.293767\pi$
$18$	0	0
$19$	26.6039	0.321229	0.160614	−	0.987017i	$-0.448652\pi$
$19$	26.6039	0.321229	0.160614	+	0.987017i	$0.448652\pi$
$20$	0	0
$21$	−38.6039	−0.401146
$22$	0	0
$23$	−136.340	−1.23604	−0.618018	−	0.786164i	$-0.712064\pi$
$23$	−136.340	−1.23604	−0.618018	+	0.786164i	$0.712064\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	6.00000	0.0384197	0.0192099	−	0.999815i	$-0.493885\pi$
$29$	6.00000	0.0384197	0.0192099	+	0.999815i	$0.493885\pi$
$30$	0	0
$31$	−47.1320	−0.273070	−0.136535	−	0.990635i	$-0.543597\pi$
$31$	−47.1320	−0.273070	−0.136535	+	0.990635i	$0.543597\pi$
$32$	0	0
$33$	149.208	0.787083
$34$	0	0
$35$	64.3398	0.310726
$36$	0	0
$37$	344.227	1.52948	0.764738	−	0.644341i	$-0.222868\pi$
$37$	344.227	1.52948	0.764738	+	0.644341i	$0.222868\pi$
$38$	0	0
$39$	157.812	0.647951
$40$	0	0
$41$	−43.2078	−0.164583	−0.0822917	−	0.996608i	$-0.526224\pi$
$41$	−43.2078	−0.164583	−0.0822917	+	0.996608i	$0.526224\pi$
$42$	0	0
$43$	−252.000	−0.893713	−0.446856	−	0.894606i	$-0.647456\pi$
$43$	−252.000	−0.893713	−0.446856	+	0.894606i	$0.647456\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	−306.076	−0.949909	−0.474955	−	0.880010i	$-0.657536\pi$
$47$	−306.076	−0.949909	−0.474955	+	0.880010i	$0.657536\pi$
$48$	0	0
$49$	−177.416	−0.517246
$50$	0	0
$51$	−253.812	−0.696877
$52$	0	0
$53$	−455.208	−1.17977	−0.589883	−	0.807489i	$-0.700826\pi$
$53$	−455.208	−1.17977	−0.589883	+	0.807489i	$0.700826\pi$
$54$	0	0
$55$	−248.680	−0.609672
$56$	0	0
$57$	−79.8117	−0.185462
$58$	0	0
$59$	−708.151	−1.56260	−0.781301	−	0.624155i	$-0.785443\pi$
$59$	−708.151	−1.56260	−0.781301	+	0.624155i	$0.785443\pi$
$60$	0	0
$61$	−652.039	−1.36861	−0.684303	−	0.729197i	$-0.739894\pi$
$61$	−652.039	−1.36861	−0.684303	+	0.729197i	$0.739894\pi$
$62$	0	0
$63$	115.812	0.231602
$64$	0	0
$65$	−263.019	−0.501901
$66$	0	0
$67$	−704.528	−1.28465	−0.642327	−	0.766431i	$-0.722031\pi$
$67$	−704.528	−1.28465	−0.642327	+	0.766431i	$0.722031\pi$
$68$	0	0
$69$	409.019	0.713625
$70$	0	0
$71$	531.775	0.888874	0.444437	−	0.895810i	$-0.353404\pi$
$71$	531.775	0.888874	0.444437	+	0.895810i	$0.353404\pi$
$72$	0	0
$73$	57.6233	0.0923877	0.0461938	−	0.998932i	$-0.485291\pi$
$73$	57.6233	0.0923877	0.0461938	+	0.998932i	$0.485291\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	−640.000	−0.947205
$78$	0	0
$79$	−429.699	−0.611961	−0.305981	−	0.952038i	$-0.598984\pi$
$79$	−429.699	−0.611961	−0.305981	+	0.952038i	$0.598984\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−227.697	−0.301120	−0.150560	−	0.988601i	$-0.548108\pi$
$83$	−227.697	−0.301120	−0.150560	+	0.988601i	$0.548108\pi$
$84$	0	0
$85$	423.019	0.539799
$86$	0	0
$87$	−18.0000	−0.0221816
$88$	0	0
$89$	1032.87	1.23016	0.615079	−	0.788466i	$-0.289124\pi$
$89$	1032.87	1.23016	0.615079	+	0.788466i	$0.289124\pi$
$90$	0	0
$91$	−676.905	−0.779768
$92$	0	0
$93$	141.396	0.157657
$94$	0	0
$95$	133.019	0.143658
$96$	0	0
$97$	−152.416	−0.159541	−0.0797704	−	0.996813i	$-0.525419\pi$
$97$	−152.416	−0.159541	−0.0797704	+	0.996813i	$0.525419\pi$
$98$	0	0
$99$	−447.623	−0.454423
$100$	0	0
$101$	−290.753	−0.286446	−0.143223	−	0.989690i	$-0.545747\pi$
$101$	−290.753	−0.286446	−0.143223	+	0.989690i	$0.545747\pi$
$102$	0	0
$103$	5.62132	0.00537753	0.00268876	−	0.999996i	$-0.499144\pi$
$103$	5.62132	0.00537753	0.00268876	+	0.999996i	$0.499144\pi$
$104$	0	0
$105$	−193.019	−0.179398
$106$	0	0
$107$	−2170.57	−1.96109	−0.980545	−	0.196294i	$-0.937109\pi$
$107$	−2170.57	−1.96109	−0.980545	+	0.196294i	$0.937109\pi$
$108$	0	0
$109$	1411.58	1.24042	0.620208	−	0.784438i	$-0.287048\pi$
$109$	1411.58	1.24042	0.620208	+	0.784438i	$0.287048\pi$
$110$	0	0
$111$	−1032.68	−0.883043
$112$	0	0
$113$	−1557.81	−1.29687	−0.648436	−	0.761269i	$-0.724577\pi$
$113$	−1557.81	−1.29687	−0.648436	+	0.761269i	$0.724577\pi$
$114$	0	0
$115$	−681.699	−0.552772
$116$	0	0
$117$	−473.435	−0.374095
$118$	0	0
$119$	1088.68	0.838648
$120$	0	0
$121$	1142.66	0.858499
$122$	0	0
$123$	129.623	0.0950223
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1920.19	1.34165	0.670825	−	0.741616i	$-0.265940\pi$
$127$	1920.19	1.34165	0.670825	+	0.741616i	$0.265940\pi$
$128$	0	0
$129$	756.000	0.515985
$130$	0	0
$131$	−600.303	−0.400372	−0.200186	−	0.979758i	$-0.564155\pi$
$131$	−600.303	−0.400372	−0.200186	+	0.979758i	$0.564155\pi$
$132$	0	0
$133$	342.338	0.223191
$134$	0	0
$135$	−135.000	−0.0860663
$136$	0	0
$137$	−1953.51	−1.21825	−0.609124	−	0.793075i	$-0.708479\pi$
$137$	−1953.51	−1.21825	−0.609124	+	0.793075i	$0.708479\pi$
$138$	0	0
$139$	2261.25	1.37983	0.689916	−	0.723890i	$-0.257648\pi$
$139$	2261.25	1.37983	0.689916	+	0.723890i	$0.257648\pi$
$140$	0	0
$141$	918.227	0.548430
$142$	0	0
$143$	2616.30	1.52997
$144$	0	0
$145$	30.0000	0.0171818
$146$	0	0
$147$	532.247	0.298632
$148$	0	0
$149$	−2938.16	−1.61546	−0.807728	−	0.589555i	$-0.799303\pi$
$149$	−2938.16	−1.61546	−0.807728	+	0.589555i	$0.799303\pi$
$150$	0	0
$151$	1053.40	0.567710	0.283855	−	0.958867i	$-0.408387\pi$
$151$	1053.40	0.567710	0.283855	+	0.958867i	$0.408387\pi$
$152$	0	0
$153$	761.435	0.402342
$154$	0	0
$155$	−235.660	−0.122121
$156$	0	0
$157$	2789.43	1.41797	0.708985	−	0.705224i	$-0.249154\pi$
$157$	2789.43	1.41797	0.708985	+	0.705224i	$0.249154\pi$
$158$	0	0
$159$	1365.62	0.681138
$160$	0	0
$161$	−1754.42	−0.858803
$162$	0	0
$163$	−2170.35	−1.04291	−0.521456	−	0.853278i	$-0.674611\pi$
$163$	−2170.35	−1.04291	−0.521456	+	0.853278i	$0.674611\pi$
$164$	0	0
$165$	746.039	0.351994
$166$	0	0
$167$	−3175.74	−1.47153	−0.735767	−	0.677234i	$-0.763178\pi$
$167$	−3175.74	−1.47153	−0.735767	+	0.677234i	$0.763178\pi$
$168$	0	0
$169$	570.169	0.259522
$170$	0	0
$171$	239.435	0.107076
$172$	0	0
$173$	947.740	0.416505	0.208252	−	0.978075i	$-0.433222\pi$
$173$	947.740	0.416505	0.208252	+	0.978075i	$0.433222\pi$
$174$	0	0
$175$	321.699	0.138961
$176$	0	0
$177$	2124.45	0.902168
$178$	0	0
$179$	3003.47	1.25413	0.627067	−	0.778965i	$-0.284255\pi$
$179$	3003.47	1.25413	0.627067	+	0.778965i	$0.284255\pi$
$180$	0	0
$181$	1502.91	0.617184	0.308592	−	0.951194i	$-0.400142\pi$
$181$	1502.91	0.617184	0.308592	+	0.951194i	$0.400142\pi$
$182$	0	0
$183$	1956.12	0.790166
$184$	0	0
$185$	1721.14	0.684002
$186$	0	0
$187$	−4207.85	−1.64550
$188$	0	0
$189$	−347.435	−0.133715
$190$	0	0
$191$	3144.00	1.19106	0.595528	−	0.803334i	$-0.296943\pi$
$191$	3144.00	1.19106	0.595528	+	0.803334i	$0.296943\pi$
$192$	0	0
$193$	−3100.87	−1.15651	−0.578253	−	0.815858i	$-0.696265\pi$
$193$	−3100.87	−1.15651	−0.578253	+	0.815858i	$0.696265\pi$
$194$	0	0
$195$	789.058	0.289773
$196$	0	0
$197$	−2563.13	−0.926982	−0.463491	−	0.886102i	$-0.653403\pi$
$197$	−2563.13	−0.926982	−0.463491	+	0.886102i	$0.653403\pi$
$198$	0	0
$199$	2929.17	1.04343	0.521717	−	0.853119i	$-0.325292\pi$
$199$	2929.17	1.04343	0.521717	+	0.853119i	$0.325292\pi$
$200$	0	0
$201$	2113.58	0.741695
$202$	0	0
$203$	77.2078	0.0266942
$204$	0	0
$205$	−216.039	−0.0736039
$206$	0	0
$207$	−1227.06	−0.412012
$208$	0	0
$209$	−1323.17	−0.437921
$210$	0	0
$211$	3876.80	1.26488	0.632440	−	0.774609i	$-0.282053\pi$
$211$	3876.80	1.26488	0.632440	+	0.774609i	$0.282053\pi$
$212$	0	0
$213$	−1595.32	−0.513192
$214$	0	0
$215$	−1260.00	−0.399680
$216$	0	0
$217$	−606.493	−0.189730
$218$	0	0
$219$	−172.870	−0.0533400
$220$	0	0
$221$	−4450.49	−1.35463
$222$	0	0
$223$	4268.94	1.28193	0.640963	−	0.767572i	$-0.278535\pi$
$223$	4268.94	1.28193	0.640963	+	0.767572i	$0.278535\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	4191.03	1.22541	0.612706	−	0.790311i	$-0.290081\pi$
$227$	4191.03	1.22541	0.612706	+	0.790311i	$0.290081\pi$
$228$	0	0
$229$	2036.12	0.587556	0.293778	−	0.955874i	$-0.405087\pi$
$229$	2036.12	0.587556	0.293778	+	0.955874i	$0.405087\pi$
$230$	0	0
$231$	1920.00	0.546869
$232$	0	0
$233$	−4544.76	−1.27784	−0.638921	−	0.769273i	$-0.720619\pi$
$233$	−4544.76	−1.27784	−0.638921	+	0.769273i	$0.720619\pi$
$234$	0	0
$235$	−1530.38	−0.424812
$236$	0	0
$237$	1289.10	0.353316
$238$	0	0
$239$	2068.00	0.559697	0.279848	−	0.960044i	$-0.409716\pi$
$239$	2068.00	0.559697	0.279848	+	0.960044i	$0.409716\pi$
$240$	0	0
$241$	−1901.25	−0.508175	−0.254087	−	0.967181i	$-0.581775\pi$
$241$	−1901.25	−0.508175	−0.254087	+	0.967181i	$0.581775\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−887.078	−0.231320
$246$	0	0
$247$	−1399.47	−0.360510
$248$	0	0
$249$	683.091	0.173852
$250$	0	0
$251$	−3794.42	−0.954191	−0.477095	−	0.878852i	$-0.658310\pi$
$251$	−3794.42	−0.954191	−0.477095	+	0.878852i	$0.658310\pi$
$252$	0	0
$253$	6780.99	1.68505
$254$	0	0
$255$	−1269.06	−0.311653
$256$	0	0
$257$	3103.33	0.753231	0.376616	−	0.926370i	$-0.377088\pi$
$257$	3103.33	0.753231	0.376616	+	0.926370i	$0.377088\pi$
$258$	0	0
$259$	4429.50	1.06269
$260$	0	0
$261$	54.0000	0.0128066
$262$	0	0
$263$	4110.83	0.963821	0.481910	−	0.876221i	$-0.339943\pi$
$263$	4110.83	0.963821	0.481910	+	0.876221i	$0.339943\pi$
$264$	0	0
$265$	−2276.04	−0.527607
$266$	0	0
$267$	−3098.61	−0.710232
$268$	0	0
$269$	−4272.34	−0.968361	−0.484180	−	0.874968i	$-0.660882\pi$
$269$	−4272.34	−0.968361	−0.484180	+	0.874968i	$0.660882\pi$
$270$	0	0
$271$	4396.64	0.985524	0.492762	−	0.870164i	$-0.335987\pi$
$271$	4396.64	0.985524	0.492762	+	0.870164i	$0.335987\pi$
$272$	0	0
$273$	2030.71	0.450199
$274$	0	0
$275$	−1243.40	−0.272654
$276$	0	0
$277$	−3068.54	−0.665598	−0.332799	−	0.942998i	$-0.607993\pi$
$277$	−3068.54	−0.665598	−0.332799	+	0.942998i	$0.607993\pi$
$278$	0	0
$279$	−424.188	−0.0910233
$280$	0	0
$281$	2672.71	0.567405	0.283702	−	0.958912i	$-0.408437\pi$
$281$	2672.71	0.567405	0.283702	+	0.958912i	$0.408437\pi$
$282$	0	0
$283$	6135.64	1.28878	0.644392	−	0.764696i	$-0.277111\pi$
$283$	6135.64	1.28878	0.644392	+	0.764696i	$0.277111\pi$
$284$	0	0
$285$	−399.058	−0.0829410
$286$	0	0
$287$	−555.996	−0.114353
$288$	0	0
$289$	2244.82	0.456914
$290$	0	0
$291$	457.247	0.0921109
$292$	0	0
$293$	40.7922	0.00813347	0.00406674	−	0.999992i	$-0.498706\pi$
$293$	40.7922	0.00813347	0.00406674	+	0.999992i	$0.498706\pi$
$294$	0	0
$295$	−3540.76	−0.698816
$296$	0	0
$297$	1342.87	0.262361
$298$	0	0
$299$	7172.00	1.38718
$300$	0	0
$301$	−3242.73	−0.620956
$302$	0	0
$303$	872.260	0.165380
$304$	0	0
$305$	−3260.19	−0.612060
$306$	0	0
$307$	−7126.80	−1.32491	−0.662456	−	0.749101i	$-0.730486\pi$
$307$	−7126.80	−1.32491	−0.662456	+	0.749101i	$0.730486\pi$
$308$	0	0
$309$	−16.8640	−0.00310472
$310$	0	0
$311$	−1704.52	−0.310787	−0.155393	−	0.987853i	$-0.549665\pi$
$311$	−1704.52	−0.310787	−0.155393	+	0.987853i	$0.549665\pi$
$312$	0	0
$313$	5343.52	0.964963	0.482482	−	0.875906i	$-0.339735\pi$
$313$	5343.52	0.964963	0.482482	+	0.875906i	$0.339735\pi$
$314$	0	0
$315$	579.058	0.103575
$316$	0	0
$317$	−5644.19	−1.00003	−0.500015	−	0.866017i	$-0.666672\pi$
$317$	−5644.19	−1.00003	−0.500015	+	0.866017i	$0.666672\pi$
$318$	0	0
$319$	−298.416	−0.0523764
$320$	0	0
$321$	6511.70	1.13224
$322$	0	0
$323$	2250.79	0.387732
$324$	0	0
$325$	−1315.10	−0.224457
$326$	0	0
$327$	−4234.75	−0.716154
$328$	0	0
$329$	−3938.57	−0.660001
$330$	0	0
$331$	−2361.55	−0.392152	−0.196076	−	0.980589i	$-0.562820\pi$
$331$	−2361.55	−0.392152	−0.196076	+	0.980589i	$0.562820\pi$
$332$	0	0
$333$	3098.04	0.509825
$334$	0	0
$335$	−3522.64	−0.574515
$336$	0	0
$337$	5117.10	0.827141	0.413570	−	0.910472i	$-0.364282\pi$
$337$	5117.10	0.827141	0.413570	+	0.910472i	$0.364282\pi$
$338$	0	0
$339$	4673.43	0.748750
$340$	0	0
$341$	2344.16	0.372267
$342$	0	0
$343$	−6696.69	−1.05419
$344$	0	0
$345$	2045.10	0.319143
$346$	0	0
$347$	−6087.39	−0.941753	−0.470876	−	0.882199i	$-0.656062\pi$
$347$	−6087.39	−0.941753	−0.470876	+	0.882199i	$0.656062\pi$
$348$	0	0
$349$	7827.52	1.20057	0.600283	−	0.799788i	$-0.295055\pi$
$349$	7827.52	1.20057	0.600283	+	0.799788i	$0.295055\pi$
$350$	0	0
$351$	1420.30	0.215984
$352$	0	0
$353$	2515.47	0.379278	0.189639	−	0.981854i	$-0.439268\pi$
$353$	2515.47	0.379278	0.189639	+	0.981854i	$0.439268\pi$
$354$	0	0
$355$	2658.87	0.397517
$356$	0	0
$357$	−3266.04	−0.484194
$358$	0	0
$359$	−915.914	−0.134652	−0.0673261	−	0.997731i	$-0.521447\pi$
$359$	−915.914	−0.134652	−0.0673261	+	0.997731i	$0.521447\pi$
$360$	0	0
$361$	−6151.23	−0.896812
$362$	0	0
$363$	−3427.99	−0.495655
$364$	0	0
$365$	288.117	0.0413170
$366$	0	0
$367$	−12237.6	−1.74060	−0.870298	−	0.492525i	$-0.836074\pi$
$367$	−12237.6	−1.74060	−0.870298	+	0.492525i	$0.836074\pi$
$368$	0	0
$369$	−388.870	−0.0548611
$370$	0	0
$371$	−5857.60	−0.819707
$372$	0	0
$373$	3825.20	0.530996	0.265498	−	0.964111i	$-0.414464\pi$
$373$	3825.20	0.530996	0.265498	+	0.964111i	$0.414464\pi$
$374$	0	0
$375$	−375.000	−0.0516398
$376$	0	0
$377$	−315.623	−0.0431178
$378$	0	0
$379$	8197.63	1.11104	0.555519	−	0.831504i	$-0.312519\pi$
$379$	8197.63	1.11104	0.555519	+	0.831504i	$0.312519\pi$
$380$	0	0
$381$	−5760.58	−0.774602
$382$	0	0
$383$	−452.196	−0.0603294	−0.0301647	−	0.999545i	$-0.509603\pi$
$383$	−452.196	−0.0603294	−0.0301647	+	0.999545i	$0.509603\pi$
$384$	0	0
$385$	−3200.00	−0.423603
$386$	0	0
$387$	−2268.00	−0.297904
$388$	0	0
$389$	6816.13	0.888410	0.444205	−	0.895925i	$-0.353486\pi$
$389$	6816.13	0.888410	0.444205	+	0.895925i	$0.353486\pi$
$390$	0	0
$391$	−11534.9	−1.49193
$392$	0	0
$393$	1800.91	0.231155
$394$	0	0
$395$	−2148.50	−0.273677
$396$	0	0
$397$	13380.3	1.69153	0.845766	−	0.533554i	$-0.179144\pi$
$397$	13380.3	1.69153	0.845766	+	0.533554i	$0.179144\pi$
$398$	0	0
$399$	−1027.01	−0.128860
$400$	0	0
$401$	6130.00	0.763386	0.381693	−	0.924289i	$-0.375341\pi$
$401$	6130.00	0.763386	0.381693	+	0.924289i	$0.375341\pi$
$402$	0	0
$403$	2479.33	0.306462
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−17120.5	−2.08509
$408$	0	0
$409$	−15003.5	−1.81387	−0.906935	−	0.421270i	$-0.861584\pi$
$409$	−15003.5	−1.81387	−0.906935	+	0.421270i	$0.861584\pi$
$410$	0	0
$411$	5860.54	0.703355
$412$	0	0
$413$	−9112.47	−1.08570
$414$	0	0
$415$	−1138.49	−0.134665
$416$	0	0
$417$	−6783.75	−0.796646
$418$	0	0
$419$	16300.7	1.90057	0.950287	−	0.311375i	$-0.100790\pi$
$419$	16300.7	1.90057	0.950287	+	0.311375i	$0.100790\pi$
$420$	0	0
$421$	7663.91	0.887211	0.443606	−	0.896222i	$-0.353699\pi$
$421$	7663.91	0.887211	0.443606	+	0.896222i	$0.353699\pi$
$422$	0	0
$423$	−2754.68	−0.316636
$424$	0	0
$425$	2115.10	0.241405
$426$	0	0
$427$	−8390.41	−0.950914
$428$	0	0
$429$	−7848.91	−0.883331
$430$	0	0
$431$	3702.20	0.413755	0.206878	−	0.978367i	$-0.433670\pi$
$431$	3702.20	0.413755	0.206878	+	0.978367i	$0.433670\pi$
$432$	0	0
$433$	−10275.8	−1.14047	−0.570237	−	0.821481i	$-0.693148\pi$
$433$	−10275.8	−1.14047	−0.570237	+	0.821481i	$0.693148\pi$
$434$	0	0
$435$	−90.0000	−0.00991993
$436$	0	0
$437$	−3627.17	−0.397050
$438$	0	0
$439$	−14178.6	−1.54148	−0.770738	−	0.637152i	$-0.780112\pi$
$439$	−14178.6	−1.54148	−0.770738	+	0.637152i	$0.780112\pi$
$440$	0	0
$441$	−1596.74	−0.172415
$442$	0	0
$443$	−13395.6	−1.43666	−0.718332	−	0.695700i	$-0.755094\pi$
$443$	−13395.6	−1.43666	−0.718332	+	0.695700i	$0.755094\pi$
$444$	0	0
$445$	5164.35	0.550143
$446$	0	0
$447$	8814.47	0.932684
$448$	0	0
$449$	−9490.39	−0.997504	−0.498752	−	0.866745i	$-0.666208\pi$
$449$	−9490.39	−0.997504	−0.498752	+	0.866745i	$0.666208\pi$
$450$	0	0
$451$	2148.98	0.224371
$452$	0	0
$453$	−3160.19	−0.327767
$454$	0	0
$455$	−3384.52	−0.348723
$456$	0	0
$457$	587.064	0.0600913	0.0300456	−	0.999549i	$-0.490435\pi$
$457$	587.064	0.0600913	0.0300456	+	0.999549i	$0.490435\pi$
$458$	0	0
$459$	−2284.30	−0.232292
$460$	0	0
$461$	2800.42	0.282925	0.141462	−	0.989944i	$-0.454820\pi$
$461$	2800.42	0.282925	0.141462	+	0.989944i	$0.454820\pi$
$462$	0	0
$463$	19745.6	1.98198	0.990988	−	0.133952i	$-0.0427668\pi$
$463$	19745.6	1.98198	0.990988	+	0.133952i	$0.0427668\pi$
$464$	0	0
$465$	706.981	0.0705063
$466$	0	0
$467$	−13552.0	−1.34285	−0.671427	−	0.741071i	$-0.734318\pi$
$467$	−13552.0	−1.34285	−0.671427	+	0.741071i	$0.734318\pi$
$468$	0	0
$469$	−9065.84	−0.892584
$470$	0	0
$471$	−8368.30	−0.818665
$472$	0	0
$473$	12533.5	1.21837
$474$	0	0
$475$	665.097	0.0642458
$476$	0	0
$477$	−4096.87	−0.393255
$478$	0	0
$479$	−14131.9	−1.34803	−0.674014	−	0.738719i	$-0.735431\pi$
$479$	−14131.9	−1.34803	−0.674014	+	0.738719i	$0.735431\pi$
$480$	0	0
$481$	−18107.7	−1.71651
$482$	0	0
$483$	5263.25	0.495830
$484$	0	0
$485$	−762.078	−0.0713488
$486$	0	0
$487$	−7958.52	−0.740524	−0.370262	−	0.928927i	$-0.620732\pi$
$487$	−7958.52	−0.740524	−0.370262	+	0.928927i	$0.620732\pi$
$488$	0	0
$489$	6511.04	0.602125
$490$	0	0
$491$	10984.3	1.00960	0.504799	−	0.863237i	$-0.331567\pi$
$491$	10984.3	1.00960	0.504799	+	0.863237i	$0.331567\pi$
$492$	0	0
$493$	507.623	0.0463736
$494$	0	0
$495$	−2238.12	−0.203224
$496$	0	0
$497$	6842.86	0.617594
$498$	0	0
$499$	17710.3	1.58882	0.794411	−	0.607381i	$-0.207780\pi$
$499$	17710.3	1.58882	0.794411	+	0.607381i	$0.207780\pi$
$500$	0	0
$501$	9527.23	0.849591
$502$	0	0
$503$	2279.89	0.202098	0.101049	−	0.994881i	$-0.467780\pi$
$503$	2279.89	0.202098	0.101049	+	0.994881i	$0.467780\pi$
$504$	0	0
$505$	−1453.77	−0.128103
$506$	0	0
$507$	−1710.51	−0.149835
$508$	0	0
$509$	−3254.99	−0.283447	−0.141724	−	0.989906i	$-0.545264\pi$
$509$	−3254.99	−0.283447	−0.141724	+	0.989906i	$0.545264\pi$
$510$	0	0
$511$	741.495	0.0641914
$512$	0	0
$513$	−718.305	−0.0618205
$514$	0	0
$515$	28.1066	0.00240490
$516$	0	0
$517$	15223.0	1.29498
$518$	0	0
$519$	−2843.22	−0.240469
$520$	0	0
$521$	21591.2	1.81560	0.907799	−	0.419406i	$-0.137762\pi$
$521$	21591.2	1.81560	0.907799	+	0.419406i	$0.137762\pi$
$522$	0	0
$523$	−16868.8	−1.41037	−0.705185	−	0.709024i	$-0.749136\pi$
$523$	−16868.8	−1.41037	−0.705185	+	0.709024i	$0.749136\pi$
$524$	0	0
$525$	−965.097	−0.0802291
$526$	0	0
$527$	−3987.55	−0.329603
$528$	0	0
$529$	6421.54	0.527784
$530$	0	0
$531$	−6373.36	−0.520867
$532$	0	0
$533$	2272.90	0.184709
$534$	0	0
$535$	−10852.8	−0.877026
$536$	0	0
$537$	−9010.42	−0.724075
$538$	0	0
$539$	8823.93	0.705145
$540$	0	0
$541$	−4366.23	−0.346985	−0.173493	−	0.984835i	$-0.555505\pi$
$541$	−4366.23	−0.346985	−0.173493	+	0.984835i	$0.555505\pi$
$542$	0	0
$543$	−4508.73	−0.356331
$544$	0	0
$545$	7057.92	0.554731
$546$	0	0
$547$	16579.7	1.29597	0.647987	−	0.761652i	$-0.275611\pi$
$547$	16579.7	1.29597	0.647987	+	0.761652i	$0.275611\pi$
$548$	0	0
$549$	−5868.35	−0.456202
$550$	0	0
$551$	159.623	0.0123415
$552$	0	0
$553$	−5529.35	−0.425193
$554$	0	0
$555$	−5163.41	−0.394909
$556$	0	0
$557$	−5971.66	−0.454268	−0.227134	−	0.973863i	$-0.572936\pi$
$557$	−5971.66	−0.454268	−0.227134	+	0.973863i	$0.572936\pi$
$558$	0	0
$559$	13256.2	1.00300
$560$	0	0
$561$	12623.6	0.950030
$562$	0	0
$563$	15561.2	1.16488	0.582441	−	0.812873i	$-0.302098\pi$
$563$	15561.2	1.16488	0.582441	+	0.812873i	$0.302098\pi$
$564$	0	0
$565$	−7789.06	−0.579979
$566$	0	0
$567$	1042.30	0.0772005
$568$	0	0
$569$	11871.6	0.874662	0.437331	−	0.899301i	$-0.355924\pi$
$569$	11871.6	0.874662	0.437331	+	0.899301i	$0.355924\pi$
$570$	0	0
$571$	−18660.0	−1.36759	−0.683797	−	0.729672i	$-0.739673\pi$
$571$	−18660.0	−1.36759	−0.683797	+	0.729672i	$0.739673\pi$
$572$	0	0
$573$	−9432.00	−0.687657
$574$	0	0
$575$	−3408.50	−0.247207
$576$	0	0
$577$	−7549.12	−0.544669	−0.272334	−	0.962203i	$-0.587796\pi$
$577$	−7549.12	−0.544669	−0.272334	+	0.962203i	$0.587796\pi$
$578$	0	0
$579$	9302.61	0.667709
$580$	0	0
$581$	−2930.00	−0.209220
$582$	0	0
$583$	22640.2	1.60834
$584$	0	0
$585$	−2367.17	−0.167300
$586$	0	0
$587$	4513.39	0.317355	0.158678	−	0.987330i	$-0.449277\pi$
$587$	4513.39	0.317355	0.158678	+	0.987330i	$0.449277\pi$
$588$	0	0
$589$	−1253.90	−0.0877179
$590$	0	0
$591$	7689.39	0.535193
$592$	0	0
$593$	8335.46	0.577228	0.288614	−	0.957445i	$-0.406806\pi$
$593$	8335.46	0.577228	0.288614	+	0.957445i	$0.406806\pi$
$594$	0	0
$595$	5443.40	0.375055
$596$	0	0
$597$	−8787.51	−0.602427
$598$	0	0
$599$	−10806.9	−0.737162	−0.368581	−	0.929596i	$-0.620156\pi$
$599$	−10806.9	−0.737162	−0.368581	+	0.929596i	$0.620156\pi$
$600$	0	0
$601$	−25653.1	−1.74112	−0.870559	−	0.492063i	$-0.836243\pi$
$601$	−25653.1	−1.74112	−0.870559	+	0.492063i	$0.836243\pi$
$602$	0	0
$603$	−6340.75	−0.428218
$604$	0	0
$605$	5713.31	0.383932
$606$	0	0
$607$	−8879.17	−0.593730	−0.296865	−	0.954919i	$-0.595941\pi$
$607$	−8879.17	−0.593730	−0.296865	+	0.954919i	$0.595941\pi$
$608$	0	0
$609$	−231.623	−0.0154119
$610$	0	0
$611$	16100.8	1.06607
$612$	0	0
$613$	3846.28	0.253425	0.126713	−	0.991939i	$-0.459557\pi$
$613$	3846.28	0.253425	0.126713	+	0.991939i	$0.459557\pi$
$614$	0	0
$615$	648.117	0.0424953
$616$	0	0
$617$	−15821.2	−1.03231	−0.516156	−	0.856495i	$-0.672637\pi$
$617$	−15821.2	−1.03231	−0.516156	+	0.856495i	$0.672637\pi$
$618$	0	0
$619$	−11518.7	−0.747942	−0.373971	−	0.927440i	$-0.622004\pi$
$619$	−11518.7	−0.747942	−0.373971	+	0.927440i	$0.622004\pi$
$620$	0	0
$621$	3681.17	0.237875
$622$	0	0
$623$	13290.9	0.854719
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	3969.51	0.252834
$628$	0	0
$629$	29123.0	1.84612
$630$	0	0
$631$	9053.33	0.571169	0.285584	−	0.958354i	$-0.407812\pi$
$631$	9053.33	0.571169	0.285584	+	0.958354i	$0.407812\pi$
$632$	0	0
$633$	−11630.4	−0.730279
$634$	0	0
$635$	9600.96	0.600004
$636$	0	0
$637$	9332.75	0.580498
$638$	0	0
$639$	4785.97	0.296291
$640$	0	0
$641$	1700.77	0.104799	0.0523996	−	0.998626i	$-0.483313\pi$
$641$	1700.77	0.104799	0.0523996	+	0.998626i	$0.483313\pi$
$642$	0	0
$643$	−17275.1	−1.05951	−0.529755	−	0.848151i	$-0.677716\pi$
$643$	−17275.1	−1.05951	−0.529755	+	0.848151i	$0.677716\pi$
$644$	0	0
$645$	3780.00	0.230756
$646$	0	0
$647$	11182.2	0.679468	0.339734	−	0.940522i	$-0.389663\pi$
$647$	11182.2	0.679468	0.339734	+	0.940522i	$0.389663\pi$
$648$	0	0
$649$	35220.6	2.13024
$650$	0	0
$651$	1819.48	0.109541
$652$	0	0
$653$	20720.5	1.24174	0.620869	−	0.783915i	$-0.286780\pi$
$653$	20720.5	1.24174	0.620869	+	0.783915i	$0.286780\pi$
$654$	0	0
$655$	−3001.51	−0.179052
$656$	0	0
$657$	518.610	0.0307959
$658$	0	0
$659$	8745.26	0.516945	0.258473	−	0.966019i	$-0.416781\pi$
$659$	8745.26	0.516945	0.258473	+	0.966019i	$0.416781\pi$
$660$	0	0
$661$	−25921.8	−1.52533	−0.762663	−	0.646796i	$-0.776108\pi$
$661$	−25921.8	−1.52533	−0.762663	+	0.646796i	$0.776108\pi$
$662$	0	0
$663$	13351.5	0.782094
$664$	0	0
$665$	1711.69	0.0998142
$666$	0	0
$667$	−818.039	−0.0474881
$668$	0	0
$669$	−12806.8	−0.740120
$670$	0	0
$671$	32429.8	1.86578
$672$	0	0
$673$	−12111.4	−0.693702	−0.346851	−	0.937920i	$-0.612749\pi$
$673$	−12111.4	−0.693702	−0.346851	+	0.937920i	$0.612749\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	4961.50	0.281663	0.140832	−	0.990034i	$-0.455022\pi$
$677$	4961.50	0.281663	0.140832	+	0.990034i	$0.455022\pi$
$678$	0	0
$679$	−1961.28	−0.110850
$680$	0	0
$681$	−12573.1	−0.707492
$682$	0	0
$683$	−24838.1	−1.39151	−0.695755	−	0.718279i	$-0.744930\pi$
$683$	−24838.1	−1.39151	−0.695755	+	0.718279i	$0.744930\pi$
$684$	0	0
$685$	−9767.56	−0.544817
$686$	0	0
$687$	−6108.35	−0.339226
$688$	0	0
$689$	23945.7	1.32403
$690$	0	0
$691$	6857.65	0.377536	0.188768	−	0.982022i	$-0.439551\pi$
$691$	6857.65	0.377536	0.188768	+	0.982022i	$0.439551\pi$
$692$	0	0
$693$	−5760.00	−0.315735
$694$	0	0
$695$	11306.2	0.617079
$696$	0	0
$697$	−3655.55	−0.198657
$698$	0	0
$699$	13634.3	0.737762
$700$	0	0
$701$	32760.0	1.76509	0.882545	−	0.470228i	$-0.155828\pi$
$701$	32760.0	1.76509	0.882545	+	0.470228i	$0.155828\pi$
$702$	0	0
$703$	9157.78	0.491312
$704$	0	0
$705$	4591.14	0.245266
$706$	0	0
$707$	−3741.40	−0.199024
$708$	0	0
$709$	8678.69	0.459711	0.229855	−	0.973225i	$-0.426175\pi$
$709$	8678.69	0.459711	0.229855	+	0.973225i	$0.426175\pi$
$710$	0	0
$711$	−3867.29	−0.203987
$712$	0	0
$713$	6425.97	0.337524
$714$	0	0
$715$	13081.5	0.684225
$716$	0	0
$717$	−6203.99	−0.323141
$718$	0	0
$719$	8160.85	0.423294	0.211647	−	0.977346i	$-0.432117\pi$
$719$	8160.85	0.423294	0.211647	+	0.977346i	$0.432117\pi$
$720$	0	0
$721$	72.3349	0.00373633
$722$	0	0
$723$	5703.74	0.293395
$724$	0	0
$725$	150.000	0.00768395
$726$	0	0
$727$	23907.3	1.21963	0.609816	−	0.792543i	$-0.291243\pi$
$727$	23907.3	1.21963	0.609816	+	0.792543i	$0.291243\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−21320.2	−1.07874
$732$	0	0
$733$	−28946.6	−1.45862	−0.729309	−	0.684185i	$-0.760158\pi$
$733$	−28946.6	−1.45862	−0.729309	+	0.684185i	$0.760158\pi$
$734$	0	0
$735$	2661.23	0.133552
$736$	0	0
$737$	35040.4	1.75133
$738$	0	0
$739$	39038.5	1.94324	0.971620	−	0.236547i	$-0.0760157\pi$
$739$	39038.5	1.94324	0.971620	+	0.236547i	$0.0760157\pi$
$740$	0	0
$741$	4198.40	0.208141
$742$	0	0
$743$	−4175.77	−0.206183	−0.103092	−	0.994672i	$-0.532874\pi$
$743$	−4175.77	−0.206183	−0.103092	+	0.994672i	$0.532874\pi$
$744$	0	0
$745$	−14690.8	−0.722454
$746$	0	0
$747$	−2049.27	−0.100373
$748$	0	0
$749$	−27930.8	−1.36257
$750$	0	0
$751$	22219.1	1.07961	0.539805	−	0.841790i	$-0.318498\pi$
$751$	22219.1	1.07961	0.539805	+	0.841790i	$0.318498\pi$
$752$	0	0
$753$	11383.3	0.550902
$754$	0	0
$755$	5266.98	0.253887
$756$	0	0
$757$	22514.1	1.08096	0.540482	−	0.841355i	$-0.318242\pi$
$757$	22514.1	1.08096	0.540482	+	0.841355i	$0.318242\pi$
$758$	0	0
$759$	−20343.0	−0.972863
$760$	0	0
$761$	−26354.3	−1.25538	−0.627689	−	0.778465i	$-0.715999\pi$
$761$	−26354.3	−1.25538	−0.627689	+	0.778465i	$0.715999\pi$
$762$	0	0
$763$	18164.2	0.861846
$764$	0	0
$765$	3807.17	0.179933
$766$	0	0
$767$	37251.5	1.75368
$768$	0	0
$769$	−26226.3	−1.22983	−0.614917	−	0.788591i	$-0.710811\pi$
$769$	−26226.3	−1.22983	−0.614917	+	0.788591i	$0.710811\pi$
$770$	0	0
$771$	−9309.99	−0.434878
$772$	0	0
$773$	−28023.6	−1.30393	−0.651965	−	0.758249i	$-0.726055\pi$
$773$	−28023.6	−1.30393	−0.651965	+	0.758249i	$0.726055\pi$
$774$	0	0
$775$	−1178.30	−0.0546140
$776$	0	0
$777$	−13288.5	−0.613543
$778$	0	0
$779$	−1149.49	−0.0528690
$780$	0	0
$781$	−26448.3	−1.21177
$782$	0	0
$783$	−162.000	−0.00739388
$784$	0	0
$785$	13947.2	0.634135
$786$	0	0
$787$	16361.2	0.741061	0.370530	−	0.928820i	$-0.379176\pi$
$787$	16361.2	0.741061	0.370530	+	0.928820i	$0.379176\pi$
$788$	0	0
$789$	−12332.5	−0.556462
$790$	0	0
$791$	−20045.9	−0.901073
$792$	0	0
$793$	34299.8	1.53597
$794$	0	0
$795$	6828.12	0.304614
$796$	0	0
$797$	−20999.9	−0.933319	−0.466660	−	0.884437i	$-0.654543\pi$
$797$	−20999.9	−0.933319	−0.466660	+	0.884437i	$0.654543\pi$
$798$	0	0
$799$	−25895.2	−1.14657
$800$	0	0
$801$	9295.83	0.410052
$802$	0	0
$803$	−2865.95	−0.125949
$804$	0	0
$805$	−8772.08	−0.384068
$806$	0	0
$807$	12817.0	0.559083
$808$	0	0
$809$	−29390.4	−1.27727	−0.638635	−	0.769510i	$-0.720501\pi$
$809$	−29390.4	−1.27727	−0.638635	+	0.769510i	$0.720501\pi$
$810$	0	0
$811$	−17296.9	−0.748924	−0.374462	−	0.927242i	$-0.622173\pi$
$811$	−17296.9	−0.748924	−0.374462	+	0.927242i	$0.622173\pi$
$812$	0	0
$813$	−13189.9	−0.568993
$814$	0	0
$815$	−10851.7	−0.466404
$816$	0	0
$817$	−6704.18	−0.287086
$818$	0	0
$819$	−6092.14	−0.259923
$820$	0	0
$821$	19989.1	0.849725	0.424862	−	0.905258i	$-0.360322\pi$
$821$	19989.1	0.849725	0.424862	+	0.905258i	$0.360322\pi$
$822$	0	0
$823$	−20226.7	−0.856694	−0.428347	−	0.903614i	$-0.640904\pi$
$823$	−20226.7	−0.856694	−0.428347	+	0.903614i	$0.640904\pi$
$824$	0	0
$825$	3730.19	0.157417
$826$	0	0
$827$	−24124.7	−1.01439	−0.507193	−	0.861833i	$-0.669317\pi$
$827$	−24124.7	−1.01439	−0.507193	+	0.861833i	$0.669317\pi$
$828$	0	0
$829$	43594.3	1.82641	0.913204	−	0.407503i	$-0.133600\pi$
$829$	43594.3	1.82641	0.913204	+	0.407503i	$0.133600\pi$
$830$	0	0
$831$	9205.61	0.384283
$832$	0	0
$833$	−15010.0	−0.624330
$834$	0	0
$835$	−15878.7	−0.658090
$836$	0	0
$837$	1272.57	0.0525523
$838$	0	0
$839$	−29937.7	−1.23190	−0.615949	−	0.787786i	$-0.711227\pi$
$839$	−29937.7	−1.23190	−0.615949	+	0.787786i	$0.711227\pi$
$840$	0	0
$841$	−24353.0	−0.998524
$842$	0	0
$843$	−8018.14	−0.327591
$844$	0	0
$845$	2850.84	0.116062
$846$	0	0
$847$	14703.7	0.596489
$848$	0	0
$849$	−18406.9	−0.744079
$850$	0	0
$851$	−46931.9	−1.89049
$852$	0	0
$853$	−2147.83	−0.0862136	−0.0431068	−	0.999070i	$-0.513726\pi$
$853$	−2147.83	−0.0862136	−0.0431068	+	0.999070i	$0.513726\pi$
$854$	0	0
$855$	1197.17	0.0478860
$856$	0	0
$857$	15771.5	0.628638	0.314319	−	0.949317i	$-0.398224\pi$
$857$	15771.5	0.628638	0.314319	+	0.949317i	$0.398224\pi$
$858$	0	0
$859$	−29364.8	−1.16637	−0.583187	−	0.812338i	$-0.698194\pi$
$859$	−29364.8	−1.16637	−0.583187	+	0.812338i	$0.698194\pi$
$860$	0	0
$861$	1667.99	0.0660219
$862$	0	0
$863$	44989.3	1.77457	0.887285	−	0.461221i	$-0.152588\pi$
$863$	44989.3	1.77457	0.887285	+	0.461221i	$0.152588\pi$
$864$	0	0
$865$	4738.70	0.186267
$866$	0	0
$867$	−6734.45	−0.263799
$868$	0	0
$869$	21371.5	0.834267
$870$	0	0
$871$	37060.9	1.44175
$872$	0	0
$873$	−1371.74	−0.0531803
$874$	0	0
$875$	1608.50	0.0621452
$876$	0	0
$877$	36547.6	1.40721	0.703606	−	0.710590i	$-0.251572\pi$
$877$	36547.6	1.40721	0.703606	+	0.710590i	$0.251572\pi$
$878$	0	0
$879$	−122.377	−0.00469586
$880$	0	0
$881$	21146.4	0.808672	0.404336	−	0.914610i	$-0.367503\pi$
$881$	21146.4	0.808672	0.404336	+	0.914610i	$0.367503\pi$
$882$	0	0
$883$	−39103.3	−1.49030	−0.745148	−	0.666899i	$-0.767621\pi$
$883$	−39103.3	−1.49030	−0.745148	+	0.666899i	$0.767621\pi$
$884$	0	0
$885$	10622.3	0.403462
$886$	0	0
$887$	32169.3	1.21775	0.608873	−	0.793268i	$-0.291622\pi$
$887$	32169.3	1.21775	0.608873	+	0.793268i	$0.291622\pi$
$888$	0	0
$889$	24709.0	0.932184
$890$	0	0
$891$	−4028.61	−0.151474
$892$	0	0
$893$	−8142.80	−0.305138
$894$	0	0
$895$	15017.4	0.560866
$896$	0	0
$897$	−21516.0	−0.800890
$898$	0	0
$899$	−282.792	−0.0104913
$900$	0	0
$901$	−38512.3	−1.42401
$902$	0	0
$903$	9728.18	0.358509
$904$	0	0
$905$	7514.54	0.276013
$906$	0	0
$907$	−10810.4	−0.395759	−0.197880	−	0.980226i	$-0.563405\pi$
$907$	−10810.4	−0.395759	−0.197880	+	0.980226i	$0.563405\pi$
$908$	0	0
$909$	−2616.78	−0.0954820
$910$	0	0
$911$	−244.183	−0.00888053	−0.00444026	−	0.999990i	$-0.501413\pi$
$911$	−244.183	−0.00888053	−0.00444026	+	0.999990i	$0.501413\pi$
$912$	0	0
$913$	11324.7	0.410508
$914$	0	0
$915$	9780.58	0.353373
$916$	0	0
$917$	−7724.68	−0.278180
$918$	0	0
$919$	−34164.1	−1.22630	−0.613151	−	0.789966i	$-0.710098\pi$
$919$	−34164.1	−1.22630	−0.613151	+	0.789966i	$0.710098\pi$
$920$	0	0
$921$	21380.4	0.764938
$922$	0	0
$923$	−27973.4	−0.997569
$924$	0	0
$925$	8605.68	0.305895
$926$	0	0
$927$	50.5919	0.00179251
$928$	0	0
$929$	−35758.6	−1.26287	−0.631433	−	0.775431i	$-0.717533\pi$
$929$	−35758.6	−1.26287	−0.631433	+	0.775431i	$0.717533\pi$
$930$	0	0
$931$	−4719.94	−0.166155
$932$	0	0
$933$	5113.57	0.179433
$934$	0	0
$935$	−21039.3	−0.735890
$936$	0	0
$937$	40301.0	1.40510	0.702549	−	0.711635i	$-0.252045\pi$
$937$	40301.0	1.40510	0.702549	+	0.711635i	$0.252045\pi$
$938$	0	0
$939$	−16030.6	−0.557122
$940$	0	0
$941$	48866.6	1.69289	0.846443	−	0.532480i	$-0.178740\pi$
$941$	48866.6	1.69289	0.846443	+	0.532480i	$0.178740\pi$
$942$	0	0
$943$	5890.94	0.203431
$944$	0	0
$945$	−1737.17	−0.0597993
$946$	0	0
$947$	42669.4	1.46417	0.732084	−	0.681214i	$-0.238548\pi$
$947$	42669.4	1.46417	0.732084	+	0.681214i	$0.238548\pi$
$948$	0	0
$949$	−3031.21	−0.103685
$950$	0	0
$951$	16932.6	0.577368
$952$	0	0
$953$	−42523.5	−1.44541	−0.722703	−	0.691158i	$-0.757101\pi$
$953$	−42523.5	−1.44541	−0.722703	+	0.691158i	$0.757101\pi$
$954$	0	0
$955$	15720.0	0.532657
$956$	0	0
$957$	895.247	0.0302395
$958$	0	0
$959$	−25137.7	−0.846444
$960$	0	0
$961$	−27569.6	−0.925433
$962$	0	0
$963$	−19535.1	−0.653697
$964$	0	0
$965$	−15504.3	−0.517205
$966$	0	0
$967$	13205.1	0.439138	0.219569	−	0.975597i	$-0.429535\pi$
$967$	13205.1	0.439138	0.219569	+	0.975597i	$0.429535\pi$
$968$	0	0
$969$	−6752.38	−0.223857
$970$	0	0
$971$	3545.55	0.117180	0.0585901	−	0.998282i	$-0.481339\pi$
$971$	3545.55	0.117180	0.0585901	+	0.998282i	$0.481339\pi$
$972$	0	0
$973$	29097.7	0.958713
$974$	0	0
$975$	3945.29	0.129590
$976$	0	0
$977$	−13070.8	−0.428015	−0.214008	−	0.976832i	$-0.568652\pi$
$977$	−13070.8	−0.428015	−0.214008	+	0.976832i	$0.568652\pi$
$978$	0	0
$979$	−51370.7	−1.67703
$980$	0	0
$981$	12704.3	0.413472
$982$	0	0
$983$	12342.3	0.400467	0.200234	−	0.979748i	$-0.435830\pi$
$983$	12342.3	0.400467	0.200234	+	0.979748i	$0.435830\pi$
$984$	0	0
$985$	−12815.7	−0.414559
$986$	0	0
$987$	11815.7	0.381052
$988$	0	0
$989$	34357.6	1.10466
$990$	0	0
$991$	−35095.4	−1.12497	−0.562484	−	0.826808i	$-0.690154\pi$
$991$	−35095.4	−1.12497	−0.562484	+	0.826808i	$0.690154\pi$
$992$	0	0
$993$	7084.64	0.226409
$994$	0	0
$995$	14645.9	0.466638
$996$	0	0
$997$	−46486.0	−1.47666	−0.738328	−	0.674442i	$-0.764384\pi$
$997$	−46486.0	−1.47666	−0.738328	+	0.674442i	$0.764384\pi$
$998$	0	0
$999$	−9294.13	−0.294348

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	480.4.a.o.1.2	✓	2
3.2	odd	2		1440.4.a.s.1.2		2
4.3	odd	2		480.4.a.r.1.1	yes	2
5.4	even	2		2400.4.a.bb.1.1		2
8.3	odd	2		960.4.a.bk.1.1		2
8.5	even	2		960.4.a.bn.1.2		2
12.11	even	2		1440.4.a.y.1.1		2
20.19	odd	2		2400.4.a.y.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
480.4.a.o.1.2	✓	2	1.1	even	1	trivial
480.4.a.r.1.1	yes	2	4.3	odd	2
960.4.a.bk.1.1		2	8.3	odd	2
960.4.a.bn.1.2		2	8.5	even	2
1440.4.a.s.1.2		2	3.2	odd	2
1440.4.a.y.1.1		2	12.11	even	2
2400.4.a.y.1.2		2	20.19	odd	2
2400.4.a.bb.1.1		2	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 480 = 2^{5} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	480.a (trivial)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +5.00000 q^{5} +12.8680 q^{7} +9.00000 q^{9} -49.7359 q^{11} -52.6039 q^{13} -15.0000 q^{15} +84.6039 q^{17} +26.6039 q^{19} -38.6039 q^{21} -136.340 q^{23} +25.0000 q^{25} -27.0000 q^{27} +6.00000 q^{29} -47.1320 q^{31} +149.208 q^{33} +64.3398 q^{35} +344.227 q^{37} +157.812 q^{39} -43.2078 q^{41} -252.000 q^{43} +45.0000 q^{45} -306.076 q^{47} -177.416 q^{49} -253.812 q^{51} -455.208 q^{53} -248.680 q^{55} -79.8117 q^{57} -708.151 q^{59} -652.039 q^{61} +115.812 q^{63} -263.019 q^{65} -704.528 q^{67} +409.019 q^{69} +531.775 q^{71} +57.6233 q^{73} -75.0000 q^{75} -640.000 q^{77} -429.699 q^{79} +81.0000 q^{81} -227.697 q^{83} +423.019 q^{85} -18.0000 q^{87} +1032.87 q^{89} -676.905 q^{91} +141.396 q^{93} +133.019 q^{95} -152.416 q^{97} -447.623 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 10 q^{5} - 12 q^{7} + 18 q^{9} - 24 q^{11} + 8 q^{13} - 30 q^{15} + 56 q^{17} - 60 q^{19} + 36 q^{21} - 84 q^{23} + 50 q^{25} - 54 q^{27} + 12 q^{29} - 132 q^{31} + 72 q^{33} - 60 q^{35}+ \cdots - 216 q^{99}+O(q^{100}) \) 2 * q - 6 * q^3 + 10 * q^5 - 12 * q^7 + 18 * q^9 - 24 * q^11 + 8 * q^13 - 30 * q^15 + 56 * q^17 - 60 * q^19 + 36 * q^21 - 84 * q^23 + 50 * q^25 - 54 * q^27 + 12 * q^29 - 132 * q^31 + 72 * q^33 - 60 * q^35 - 104 * q^37 - 24 * q^39 + 140 * q^41 - 504 * q^43 + 90 * q^45 - 348 * q^47 + 98 * q^49 - 168 * q^51 - 684 * q^53 - 120 * q^55 + 180 * q^57 - 888 * q^59 - 172 * q^61 - 108 * q^63 + 40 * q^65 - 1560 * q^67 + 252 * q^69 - 144 * q^71 - 564 * q^73 - 150 * q^75 - 1280 * q^77 + 84 * q^79 + 162 * q^81 - 1512 * q^83 + 280 * q^85 - 36 * q^87 + 28 * q^89 - 2184 * q^91 + 396 * q^93 - 300 * q^95 + 148 * q^97 - 216 * q^99

Introduction

L-functions

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