LMFDB - Embedded newform 2160.4.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,4,Mod(1,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2160.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2160.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:	$127.444125612$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 8x + 9 $ x^3 - x^2 - 8*x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3}\cdot 3 $
Twist minimal:	no (minimal twist has level 1080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.14974$ of defining polynomial
Character	$\chi$	$=$	2160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} +3.85875 q^{7} +O(q^{10})$	$q-5.00000 q^{5} +3.85875 q^{7} -42.2783 q^{11} +4.96700 q^{13} -25.8876 q^{17} -28.9958 q^{19} +191.469 q^{23} +25.0000 q^{25} +287.595 q^{29} -52.6915 q^{31} -19.2937 q^{35} -225.550 q^{37} +73.9907 q^{41} +275.370 q^{43} +192.271 q^{47} -328.110 q^{49} +275.204 q^{53} +211.392 q^{55} -497.084 q^{59} +44.0473 q^{61} -24.8350 q^{65} -761.667 q^{67} -264.284 q^{71} +728.781 q^{73} -163.141 q^{77} -664.347 q^{79} -1491.33 q^{83} +129.438 q^{85} +106.783 q^{89} +19.1664 q^{91} +144.979 q^{95} -924.560 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 15 q^{5} + 10 q^{7}+O(q^{10}) $ 3 * q - 15 * q^5 + 10 * q^7	$ 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53} - 140 q^{55} + 234 q^{59} - 673 q^{61} + 390 q^{65} - 386 q^{67} + 16 q^{71} - 892 q^{73} + 1800 q^{77} - 1263 q^{79} - 1815 q^{83} + 55 q^{85} + 1800 q^{89} - 1284 q^{91} - 355 q^{95} - 840 q^{97}+O(q^{100}) $ 3 * q - 15 * q^5 + 10 * q^7 + 28 * q^11 - 78 * q^13 - 11 * q^17 + 71 * q^19 - 25 * q^23 + 75 * q^25 + 118 * q^29 + 107 * q^31 - 50 * q^35 - 410 * q^37 + 592 * q^41 - 52 * q^43 - 580 * q^47 + 479 * q^49 + 169 * q^53 - 140 * q^55 + 234 * q^59 - 673 * q^61 + 390 * q^65 - 386 * q^67 + 16 * q^71 - 892 * q^73 + 1800 * q^77 - 1263 * q^79 - 1815 * q^83 + 55 * q^85 + 1800 * q^89 - 1284 * q^91 - 355 * q^95 - 840 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	3.85875	0.208353	0.104176	−	0.994559i	$-0.466779\pi$
$7$	3.85875	0.208353	0.104176	+	0.994559i	$0.466779\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−42.2783	−1.15885	−0.579427	−	0.815024i	$-0.696724\pi$
$11$	−42.2783	−1.15885	−0.579427	+	0.815024i	$0.696724\pi$
$12$	0	0
$13$	4.96700	0.105969	0.0529845	−	0.998595i	$-0.483127\pi$
$13$	4.96700	0.105969	0.0529845	+	0.998595i	$0.483127\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−25.8876	−0.369333	−0.184666	−	0.982801i	$-0.559120\pi$
$17$	−25.8876	−0.369333	−0.184666	+	0.982801i	$0.559120\pi$
$18$	0	0
$19$	−28.9958	−0.350110	−0.175055	−	0.984559i	$-0.556010\pi$
$19$	−28.9958	−0.350110	−0.175055	+	0.984559i	$0.556010\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	191.469	1.73583	0.867914	−	0.496715i	$-0.165461\pi$
$23$	191.469	1.73583	0.867914	+	0.496715i	$0.165461\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	287.595	1.84155	0.920776	−	0.390092i	$-0.127557\pi$
$29$	287.595	1.84155	0.920776	+	0.390092i	$0.127557\pi$
$30$	0	0
$31$	−52.6915	−0.305280	−0.152640	−	0.988282i	$-0.548777\pi$
$31$	−52.6915	−0.305280	−0.152640	+	0.988282i	$0.548777\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−19.2937	−0.0931782
$36$	0	0
$37$	−225.550	−1.00217	−0.501084	−	0.865398i	$-0.667065\pi$
$37$	−225.550	−1.00217	−0.501084	+	0.865398i	$0.667065\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	73.9907	0.281839	0.140920	−	0.990021i	$-0.454994\pi$
$41$	73.9907	0.281839	0.140920	+	0.990021i	$0.454994\pi$
$42$	0	0
$43$	275.370	0.976593	0.488297	−	0.872678i	$-0.337618\pi$
$43$	275.370	0.976593	0.488297	+	0.872678i	$0.337618\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	192.271	0.596715	0.298357	−	0.954454i	$-0.403561\pi$
$47$	192.271	0.596715	0.298357	+	0.954454i	$0.403561\pi$
$48$	0	0
$49$	−328.110	−0.956589
$50$	0	0
$51$	0	0
$52$	0	0
$53$	275.204	0.713249	0.356624	−	0.934248i	$-0.383928\pi$
$53$	275.204	0.713249	0.356624	+	0.934248i	$0.383928\pi$
$54$	0	0
$55$	211.392	0.518255
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−497.084	−1.09686	−0.548431	−	0.836196i	$-0.684775\pi$
$59$	−497.084	−1.09686	−0.548431	+	0.836196i	$0.684775\pi$
$60$	0	0
$61$	44.0473	0.0924538	0.0462269	−	0.998931i	$-0.485280\pi$
$61$	44.0473	0.0924538	0.0462269	+	0.998931i	$0.485280\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−24.8350	−0.0473908
$66$	0	0
$67$	−761.667	−1.38884	−0.694422	−	0.719568i	$-0.744340\pi$
$67$	−761.667	−1.38884	−0.694422	+	0.719568i	$0.744340\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−264.284	−0.441757	−0.220879	−	0.975301i	$-0.570892\pi$
$71$	−264.284	−0.441757	−0.220879	+	0.975301i	$0.570892\pi$
$72$	0	0
$73$	728.781	1.16846	0.584229	−	0.811589i	$-0.301397\pi$
$73$	728.781	1.16846	0.584229	+	0.811589i	$0.301397\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−163.141	−0.241450
$78$	0	0
$79$	−664.347	−0.946137	−0.473069	−	0.881026i	$-0.656854\pi$
$79$	−664.347	−0.946137	−0.473069	+	0.881026i	$0.656854\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1491.33	−1.97223	−0.986113	−	0.166075i	$-0.946891\pi$
$83$	−1491.33	−1.97223	−0.986113	+	0.166075i	$0.946891\pi$
$84$	0	0
$85$	129.438	0.165171
$86$	0	0
$87$	0	0
$88$	0	0
$89$	106.783	0.127180	0.0635899	−	0.997976i	$-0.479745\pi$
$89$	106.783	0.127180	0.0635899	+	0.997976i	$0.479745\pi$
$90$	0	0
$91$	19.1664	0.0220789
$92$	0	0
$93$	0	0
$94$	0	0
$95$	144.979	0.156574
$96$	0	0
$97$	−924.560	−0.967782	−0.483891	−	0.875128i	$-0.660777\pi$
$97$	−924.560	−0.967782	−0.483891	+	0.875128i	$0.660777\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	627.973	0.618670	0.309335	−	0.950953i	$-0.399894\pi$
$101$	627.973	0.618670	0.309335	+	0.950953i	$0.399894\pi$
$102$	0	0
$103$	224.878	0.215125	0.107562	−	0.994198i	$-0.465695\pi$
$103$	224.878	0.215125	0.107562	+	0.994198i	$0.465695\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	70.1076	0.0633417	0.0316708	−	0.999498i	$-0.489917\pi$
$107$	70.1076	0.0633417	0.0316708	+	0.999498i	$0.489917\pi$
$108$	0	0
$109$	−235.022	−0.206523	−0.103261	−	0.994654i	$-0.532928\pi$
$109$	−235.022	−0.206523	−0.103261	+	0.994654i	$0.532928\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1647.70	1.37170	0.685850	−	0.727743i	$-0.259431\pi$
$113$	1647.70	1.37170	0.685850	+	0.727743i	$0.259431\pi$
$114$	0	0
$115$	−957.344	−0.776286
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−99.8936	−0.0769515
$120$	0	0
$121$	456.458	0.342943
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	1282.64	0.896189	0.448095	−	0.893986i	$-0.352103\pi$
$127$	1282.64	0.896189	0.448095	+	0.893986i	$0.352103\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1070.64	−0.714066	−0.357033	−	0.934092i	$-0.616212\pi$
$131$	−1070.64	−0.714066	−0.357033	+	0.934092i	$0.616212\pi$
$132$	0	0
$133$	−111.888	−0.0729465
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2045.70	1.27574	0.637869	−	0.770145i	$-0.279816\pi$
$137$	2045.70	1.27574	0.637869	+	0.770145i	$0.279816\pi$
$138$	0	0
$139$	−784.256	−0.478559	−0.239280	−	0.970951i	$-0.576911\pi$
$139$	−784.256	−0.478559	−0.239280	+	0.970951i	$0.576911\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−209.997	−0.122803
$144$	0	0
$145$	−1437.97	−0.823567
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1300.68	0.715140	0.357570	−	0.933886i	$-0.383605\pi$
$149$	1300.68	0.715140	0.357570	+	0.933886i	$0.383605\pi$
$150$	0	0
$151$	−1511.72	−0.814718	−0.407359	−	0.913268i	$-0.633550\pi$
$151$	−1511.72	−0.814718	−0.407359	+	0.913268i	$0.633550\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	263.458	0.136525
$156$	0	0
$157$	167.967	0.0853838	0.0426919	−	0.999088i	$-0.486407\pi$
$157$	167.967	0.0853838	0.0426919	+	0.999088i	$0.486407\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	738.830	0.361664
$162$	0	0
$163$	67.1490	0.0322670	0.0161335	−	0.999870i	$-0.494864\pi$
$163$	67.1490	0.0322670	0.0161335	+	0.999870i	$0.494864\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2568.36	−1.19009	−0.595046	−	0.803692i	$-0.702866\pi$
$167$	−2568.36	−1.19009	−0.595046	+	0.803692i	$0.702866\pi$
$168$	0	0
$169$	−2172.33	−0.988771
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3012.46	−1.32389	−0.661945	−	0.749553i	$-0.730269\pi$
$173$	−3012.46	−1.32389	−0.661945	+	0.749553i	$0.730269\pi$
$174$	0	0
$175$	96.4687	0.0416705
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−2462.80	−1.02837	−0.514185	−	0.857679i	$-0.671905\pi$
$179$	−2462.80	−1.02837	−0.514185	+	0.857679i	$0.671905\pi$
$180$	0	0
$181$	−3691.10	−1.51578	−0.757892	−	0.652380i	$-0.773771\pi$
$181$	−3691.10	−1.51578	−0.757892	+	0.652380i	$0.773771\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1127.75	0.448183
$186$	0	0
$187$	1094.48	0.428003
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−1969.02	−0.745932	−0.372966	−	0.927845i	$-0.621659\pi$
$191$	−1969.02	−0.745932	−0.372966	+	0.927845i	$0.621659\pi$
$192$	0	0
$193$	519.119	0.193612	0.0968058	−	0.995303i	$-0.469137\pi$
$193$	519.119	0.193612	0.0968058	+	0.995303i	$0.469137\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3286.33	1.18853	0.594267	−	0.804268i	$-0.297442\pi$
$197$	3286.33	1.18853	0.594267	+	0.804268i	$0.297442\pi$
$198$	0	0
$199$	−3626.47	−1.29183	−0.645914	−	0.763410i	$-0.723524\pi$
$199$	−3626.47	−1.29183	−0.645914	+	0.763410i	$0.723524\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	1109.76	0.383692
$204$	0	0
$205$	−369.954	−0.126042
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1225.90	0.405727
$210$	0	0
$211$	−1410.39	−0.460166	−0.230083	−	0.973171i	$-0.573900\pi$
$211$	−1410.39	−0.460166	−0.230083	+	0.973171i	$0.573900\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1376.85	−0.436746
$216$	0	0
$217$	−203.323	−0.0636059
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−128.584	−0.0391379
$222$	0	0
$223$	−4026.25	−1.20905	−0.604524	−	0.796587i	$-0.706637\pi$
$223$	−4026.25	−1.20905	−0.604524	+	0.796587i	$0.706637\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−2288.33	−0.669083	−0.334542	−	0.942381i	$-0.608581\pi$
$227$	−2288.33	−0.669083	−0.334542	+	0.942381i	$0.608581\pi$
$228$	0	0
$229$	2003.69	0.578200	0.289100	−	0.957299i	$-0.406644\pi$
$229$	2003.69	0.578200	0.289100	+	0.957299i	$0.406644\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	2837.72	0.797877	0.398939	−	0.916978i	$-0.369379\pi$
$233$	2837.72	0.797877	0.398939	+	0.916978i	$0.369379\pi$
$234$	0	0
$235$	−961.355	−0.266859
$236$	0	0
$237$	0	0
$238$	0	0
$239$	1749.50	0.473496	0.236748	−	0.971571i	$-0.423918\pi$
$239$	1749.50	0.473496	0.236748	+	0.971571i	$0.423918\pi$
$240$	0	0
$241$	3645.10	0.974279	0.487140	−	0.873324i	$-0.338040\pi$
$241$	3645.10	0.974279	0.487140	+	0.873324i	$0.338040\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1640.55	0.427800
$246$	0	0
$247$	−144.022	−0.0371009
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1286.39	0.323491	0.161745	−	0.986833i	$-0.448288\pi$
$251$	1286.39	0.323491	0.161745	+	0.986833i	$0.448288\pi$
$252$	0	0
$253$	−8094.99	−2.01157
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2860.83	−0.694372	−0.347186	−	0.937796i	$-0.612863\pi$
$257$	−2860.83	−0.694372	−0.347186	+	0.937796i	$0.612863\pi$
$258$	0	0
$259$	−870.341	−0.208805
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−4615.68	−1.08219	−0.541093	−	0.840963i	$-0.681989\pi$
$263$	−4615.68	−1.08219	−0.541093	+	0.840963i	$0.681989\pi$
$264$	0	0
$265$	−1376.02	−0.318974
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−3053.54	−0.692111	−0.346055	−	0.938214i	$-0.612479\pi$
$269$	−3053.54	−0.692111	−0.346055	+	0.938214i	$0.612479\pi$
$270$	0	0
$271$	5867.51	1.31523	0.657613	−	0.753356i	$-0.271566\pi$
$271$	5867.51	1.31523	0.657613	+	0.753356i	$0.271566\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1056.96	−0.231771
$276$	0	0
$277$	−7256.41	−1.57399	−0.786995	−	0.616959i	$-0.788364\pi$
$277$	−7256.41	−1.57399	−0.786995	+	0.616959i	$0.788364\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−1318.05	−0.279816	−0.139908	−	0.990165i	$-0.544681\pi$
$281$	−1318.05	−0.279816	−0.139908	+	0.990165i	$0.544681\pi$
$282$	0	0
$283$	−4810.74	−1.01049	−0.505245	−	0.862976i	$-0.668598\pi$
$283$	−4810.74	−1.01049	−0.505245	+	0.862976i	$0.668598\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	285.511	0.0587220
$288$	0	0
$289$	−4242.83	−0.863593
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1512.59	0.301593	0.150796	−	0.988565i	$-0.451816\pi$
$293$	1512.59	0.301593	0.150796	+	0.988565i	$0.451816\pi$
$294$	0	0
$295$	2485.42	0.490531
$296$	0	0
$297$	0	0
$298$	0	0
$299$	951.026	0.183944
$300$	0	0
$301$	1062.58	0.203476
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−220.237	−0.0413466
$306$	0	0
$307$	−3092.67	−0.574945	−0.287472	−	0.957789i	$-0.592815\pi$
$307$	−3092.67	−0.574945	−0.287472	+	0.957789i	$0.592815\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.7251	−0.00396115	−0.00198058	−	0.999998i	$-0.500630\pi$
$311$	−21.7251	−0.00396115	−0.00198058	+	0.999998i	$0.500630\pi$
$312$	0	0
$313$	758.733	0.137016	0.0685082	−	0.997651i	$-0.478176\pi$
$313$	758.733	0.137016	0.0685082	+	0.997651i	$0.478176\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1764.70	−0.312668	−0.156334	−	0.987704i	$-0.549968\pi$
$317$	−1764.70	−0.312668	−0.156334	+	0.987704i	$0.549968\pi$
$318$	0	0
$319$	−12159.0	−2.13409
$320$	0	0
$321$	0	0
$322$	0	0
$323$	750.632	0.129307
$324$	0	0
$325$	124.175	0.0211938
$326$	0	0
$327$	0	0
$328$	0	0
$329$	741.925	0.124327
$330$	0	0
$331$	−10098.1	−1.67687	−0.838433	−	0.545005i	$-0.816528\pi$
$331$	−10098.1	−1.67687	−0.838433	+	0.545005i	$0.816528\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3808.34	0.621109
$336$	0	0
$337$	−4485.28	−0.725012	−0.362506	−	0.931981i	$-0.618079\pi$
$337$	−4485.28	−0.725012	−0.362506	+	0.931981i	$0.618079\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2227.71	0.353775
$342$	0	0
$343$	−2589.64	−0.407661
$344$	0	0
$345$	0	0
$346$	0	0
$347$	585.137	0.0905239	0.0452620	−	0.998975i	$-0.485588\pi$
$347$	585.137	0.0905239	0.0452620	+	0.998975i	$0.485588\pi$
$348$	0	0
$349$	3461.97	0.530989	0.265494	−	0.964112i	$-0.414465\pi$
$349$	3461.97	0.530989	0.265494	+	0.964112i	$0.414465\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−6797.01	−1.02484	−0.512420	−	0.858735i	$-0.671251\pi$
$353$	−6797.01	−1.02484	−0.512420	+	0.858735i	$0.671251\pi$
$354$	0	0
$355$	1321.42	0.197560
$356$	0	0
$357$	0	0
$358$	0	0
$359$	4622.13	0.679518	0.339759	−	0.940513i	$-0.389655\pi$
$359$	4622.13	0.679518	0.339759	+	0.940513i	$0.389655\pi$
$360$	0	0
$361$	−6018.24	−0.877423
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3643.91	−0.522550
$366$	0	0
$367$	147.843	0.0210281	0.0105141	−	0.999945i	$-0.496653\pi$
$367$	147.843	0.0210281	0.0105141	+	0.999945i	$0.496653\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1061.94	0.148607
$372$	0	0
$373$	8840.27	1.22716	0.613582	−	0.789631i	$-0.289728\pi$
$373$	8840.27	1.22716	0.613582	+	0.789631i	$0.289728\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1428.48	0.195148
$378$	0	0
$379$	6063.51	0.821799	0.410899	−	0.911681i	$-0.365215\pi$
$379$	6063.51	0.821799	0.410899	+	0.911681i	$0.365215\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−496.589	−0.0662520	−0.0331260	−	0.999451i	$-0.510546\pi$
$383$	−496.589	−0.0662520	−0.0331260	+	0.999451i	$0.510546\pi$
$384$	0	0
$385$	815.707	0.107980
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−3432.37	−0.447373	−0.223686	−	0.974661i	$-0.571809\pi$
$389$	−3432.37	−0.447373	−0.223686	+	0.974661i	$0.571809\pi$
$390$	0	0
$391$	−4956.66	−0.641098
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3321.73	0.423125
$396$	0	0
$397$	−15311.8	−1.93571	−0.967856	−	0.251506i	$-0.919074\pi$
$397$	−15311.8	−1.93571	−0.967856	+	0.251506i	$0.919074\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	7088.63	0.882766	0.441383	−	0.897319i	$-0.354488\pi$
$401$	7088.63	0.882766	0.441383	+	0.897319i	$0.354488\pi$
$402$	0	0
$403$	−261.719	−0.0323502
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9535.89	1.16137
$408$	0	0
$409$	9991.23	1.20791	0.603954	−	0.797019i	$-0.293591\pi$
$409$	9991.23	1.20791	0.603954	+	0.797019i	$0.293591\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−1918.12	−0.228534
$414$	0	0
$415$	7456.65	0.882006
$416$	0	0
$417$	0	0
$418$	0	0
$419$	8172.83	0.952909	0.476455	−	0.879199i	$-0.341922\pi$
$419$	8172.83	0.952909	0.476455	+	0.879199i	$0.341922\pi$
$420$	0	0
$421$	−15475.1	−1.79147	−0.895734	−	0.444590i	$-0.853350\pi$
$421$	−15475.1	−1.79147	−0.895734	+	0.444590i	$0.853350\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−647.189	−0.0738666
$426$	0	0
$427$	169.968	0.0192630
$428$	0	0
$429$	0	0
$430$	0	0
$431$	3426.00	0.382888	0.191444	−	0.981504i	$-0.438683\pi$
$431$	3426.00	0.382888	0.191444	+	0.981504i	$0.438683\pi$
$432$	0	0
$433$	−1806.95	−0.200546	−0.100273	−	0.994960i	$-0.531972\pi$
$433$	−1806.95	−0.200546	−0.100273	+	0.994960i	$0.531972\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5551.80	−0.607731
$438$	0	0
$439$	−7974.34	−0.866957	−0.433479	−	0.901164i	$-0.642714\pi$
$439$	−7974.34	−0.866957	−0.433479	+	0.901164i	$0.642714\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3566.86	0.382544	0.191272	−	0.981537i	$-0.438739\pi$
$443$	3566.86	0.382544	0.191272	+	0.981537i	$0.438739\pi$
$444$	0	0
$445$	−533.916	−0.0568765
$446$	0	0
$447$	0	0
$448$	0	0
$449$	13687.2	1.43862	0.719308	−	0.694692i	$-0.244459\pi$
$449$	13687.2	1.43862	0.719308	+	0.694692i	$0.244459\pi$
$450$	0	0
$451$	−3128.21	−0.326611
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−95.8320	−0.00987401
$456$	0	0
$457$	1691.89	0.173180	0.0865902	−	0.996244i	$-0.472403\pi$
$457$	1691.89	0.173180	0.0865902	+	0.996244i	$0.472403\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−17881.6	−1.80657	−0.903285	−	0.429041i	$-0.858851\pi$
$461$	−17881.6	−1.80657	−0.903285	+	0.429041i	$0.858851\pi$
$462$	0	0
$463$	−11459.6	−1.15027	−0.575135	−	0.818059i	$-0.695050\pi$
$463$	−11459.6	−1.15027	−0.575135	+	0.818059i	$0.695050\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18329.4	1.81624	0.908119	−	0.418711i	$-0.137518\pi$
$467$	18329.4	1.81624	0.908119	+	0.418711i	$0.137518\pi$
$468$	0	0
$469$	−2939.08	−0.289369
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−11642.2	−1.13173
$474$	0	0
$475$	−724.896	−0.0700221
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15475.4	−1.47618	−0.738090	−	0.674703i	$-0.764272\pi$
$479$	−15475.4	−1.47618	−0.738090	+	0.674703i	$0.764272\pi$
$480$	0	0
$481$	−1120.31	−0.106199
$482$	0	0
$483$	0	0
$484$	0	0
$485$	4622.80	0.432805
$486$	0	0
$487$	16320.0	1.51854	0.759270	−	0.650776i	$-0.225556\pi$
$487$	16320.0	1.51854	0.759270	+	0.650776i	$0.225556\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	6138.33	0.564193	0.282097	−	0.959386i	$-0.408970\pi$
$491$	6138.33	0.564193	0.282097	+	0.959386i	$0.408970\pi$
$492$	0	0
$493$	−7445.13	−0.680146
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1019.81	−0.0920413
$498$	0	0
$499$	−3272.20	−0.293555	−0.146777	−	0.989170i	$-0.546890\pi$
$499$	−3272.20	−0.293555	−0.146777	+	0.989170i	$0.546890\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	4696.96	0.416356	0.208178	−	0.978091i	$-0.433247\pi$
$503$	4696.96	0.416356	0.208178	+	0.978091i	$0.433247\pi$
$504$	0	0
$505$	−3139.87	−0.276678
$506$	0	0
$507$	0	0
$508$	0	0
$509$	10011.5	0.871815	0.435907	−	0.899992i	$-0.356428\pi$
$509$	10011.5	0.871815	0.435907	+	0.899992i	$0.356428\pi$
$510$	0	0
$511$	2812.18	0.243451
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1124.39	−0.0962067
$516$	0	0
$517$	−8128.89	−0.691505
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19483.2	−1.63833	−0.819167	−	0.573554i	$-0.805564\pi$
$521$	−19483.2	−1.63833	−0.819167	+	0.573554i	$0.805564\pi$
$522$	0	0
$523$	5181.82	0.433241	0.216621	−	0.976256i	$-0.430497\pi$
$523$	5181.82	0.433241	0.216621	+	0.976256i	$0.430497\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1364.06	0.112750
$528$	0	0
$529$	24493.3	2.01310
$530$	0	0
$531$	0	0
$532$	0	0
$533$	367.512	0.0298663
$534$	0	0
$535$	−350.538	−0.0283273
$536$	0	0
$537$	0	0
$538$	0	0
$539$	13871.9	1.10855
$540$	0	0
$541$	3178.24	0.252576	0.126288	−	0.991994i	$-0.459694\pi$
$541$	3178.24	0.252576	0.126288	+	0.991994i	$0.459694\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	1175.11	0.0923599
$546$	0	0
$547$	13887.9	1.08556	0.542781	−	0.839874i	$-0.317371\pi$
$547$	13887.9	1.08556	0.542781	+	0.839874i	$0.317371\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−8339.05	−0.644747
$552$	0	0
$553$	−2563.55	−0.197130
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10574.5	−0.804406	−0.402203	−	0.915551i	$-0.631755\pi$
$557$	−10574.5	−0.804406	−0.402203	+	0.915551i	$0.631755\pi$
$558$	0	0
$559$	1367.76	0.103489
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−7815.91	−0.585083	−0.292541	−	0.956253i	$-0.594501\pi$
$563$	−7815.91	−0.585083	−0.292541	+	0.956253i	$0.594501\pi$
$564$	0	0
$565$	−8238.48	−0.613443
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20024.0	−1.47531	−0.737655	−	0.675178i	$-0.764067\pi$
$569$	−20024.0	−1.47531	−0.737655	+	0.675178i	$0.764067\pi$
$570$	0	0
$571$	−23886.5	−1.75065	−0.875323	−	0.483538i	$-0.839351\pi$
$571$	−23886.5	−1.75065	−0.875323	+	0.483538i	$0.839351\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4786.72	0.347165
$576$	0	0
$577$	−18777.2	−1.35478	−0.677389	−	0.735625i	$-0.736888\pi$
$577$	−18777.2	−1.35478	−0.677389	+	0.735625i	$0.736888\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5754.67	−0.410919
$582$	0	0
$583$	−11635.2	−0.826551
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−23209.8	−1.63198	−0.815988	−	0.578069i	$-0.803806\pi$
$587$	−23209.8	−1.63198	−0.815988	+	0.578069i	$0.803806\pi$
$588$	0	0
$589$	1527.83	0.106882
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−4204.44	−0.291157	−0.145578	−	0.989347i	$-0.546504\pi$
$593$	−4204.44	−0.291157	−0.145578	+	0.989347i	$0.546504\pi$
$594$	0	0
$595$	499.468	0.0344138
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26962.9	1.83919	0.919593	−	0.392872i	$-0.128518\pi$
$599$	26962.9	1.83919	0.919593	+	0.392872i	$0.128518\pi$
$600$	0	0
$601$	13595.2	0.922727	0.461363	−	0.887211i	$-0.347360\pi$
$601$	13595.2	0.922727	0.461363	+	0.887211i	$0.347360\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2282.29	−0.153369
$606$	0	0
$607$	−14337.5	−0.958720	−0.479360	−	0.877618i	$-0.659131\pi$
$607$	−14337.5	−0.958720	−0.479360	+	0.877618i	$0.659131\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	955.010	0.0632333
$612$	0	0
$613$	7963.62	0.524710	0.262355	−	0.964971i	$-0.415501\pi$
$613$	7963.62	0.524710	0.262355	+	0.964971i	$0.415501\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−23594.2	−1.53949	−0.769745	−	0.638352i	$-0.779617\pi$
$617$	−23594.2	−1.53949	−0.769745	+	0.638352i	$0.779617\pi$
$618$	0	0
$619$	−12132.6	−0.787803	−0.393901	−	0.919153i	$-0.628875\pi$
$619$	−12132.6	−0.787803	−0.393901	+	0.919153i	$0.628875\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	412.049	0.0264982
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5838.95	0.370134
$630$	0	0
$631$	614.270	0.0387539	0.0193769	−	0.999812i	$-0.493832\pi$
$631$	614.270	0.0387539	0.0193769	+	0.999812i	$0.493832\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−6413.21	−0.400788
$636$	0	0
$637$	−1629.72	−0.101369
$638$	0	0
$639$	0	0
$640$	0	0
$641$	2057.06	0.126754	0.0633769	−	0.997990i	$-0.479813\pi$
$641$	2057.06	0.126754	0.0633769	+	0.997990i	$0.479813\pi$
$642$	0	0
$643$	−17827.2	−1.09337	−0.546684	−	0.837339i	$-0.684110\pi$
$643$	−17827.2	−1.09337	−0.546684	+	0.837339i	$0.684110\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17499.4	−1.06333	−0.531664	−	0.846955i	$-0.678433\pi$
$647$	−17499.4	−1.06333	−0.531664	+	0.846955i	$0.678433\pi$
$648$	0	0
$649$	21015.9	1.27110
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6049.52	−0.362536	−0.181268	−	0.983434i	$-0.558020\pi$
$653$	−6049.52	−0.362536	−0.181268	+	0.983434i	$0.558020\pi$
$654$	0	0
$655$	5353.22	0.319340
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1436.32	0.0849027	0.0424514	−	0.999099i	$-0.486483\pi$
$659$	1436.32	0.0849027	0.0424514	+	0.999099i	$0.486483\pi$
$660$	0	0
$661$	9287.18	0.546489	0.273245	−	0.961945i	$-0.411903\pi$
$661$	9287.18	0.546489	0.273245	+	0.961945i	$0.411903\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	559.438	0.0326226
$666$	0	0
$667$	55065.4	3.19662
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1862.25	−0.107141
$672$	0	0
$673$	15014.9	0.860000	0.430000	−	0.902829i	$-0.358514\pi$
$673$	15014.9	0.860000	0.430000	+	0.902829i	$0.358514\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	9147.19	0.519284	0.259642	−	0.965705i	$-0.416396\pi$
$677$	9147.19	0.519284	0.259642	+	0.965705i	$0.416396\pi$
$678$	0	0
$679$	−3567.64	−0.201640
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−10388.4	−0.581992	−0.290996	−	0.956724i	$-0.593987\pi$
$683$	−10388.4	−0.581992	−0.290996	+	0.956724i	$0.593987\pi$
$684$	0	0
$685$	−10228.5	−0.570527
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1366.94	0.0755823
$690$	0	0
$691$	25955.9	1.42895	0.714477	−	0.699659i	$-0.246665\pi$
$691$	25955.9	1.42895	0.714477	+	0.699659i	$0.246665\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3921.28	0.214018
$696$	0	0
$697$	−1915.44	−0.104093
$698$	0	0
$699$	0	0
$700$	0	0
$701$	31771.2	1.71181	0.855906	−	0.517131i	$-0.173000\pi$
$701$	31771.2	1.71181	0.855906	+	0.517131i	$0.173000\pi$
$702$	0	0
$703$	6540.02	0.350870
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2423.19	0.128902
$708$	0	0
$709$	−7002.47	−0.370921	−0.185461	−	0.982652i	$-0.559378\pi$
$709$	−7002.47	−0.370921	−0.185461	+	0.982652i	$0.559378\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−10088.8	−0.529913
$714$	0	0
$715$	1049.98	0.0549191
$716$	0	0
$717$	0	0
$718$	0	0
$719$	34634.3	1.79644	0.898221	−	0.439545i	$-0.144860\pi$
$719$	34634.3	1.79644	0.898221	+	0.439545i	$0.144860\pi$
$720$	0	0
$721$	867.745	0.0448218
$722$	0	0
$723$	0	0
$724$	0	0
$725$	7189.87	0.368310
$726$	0	0
$727$	−10821.6	−0.552064	−0.276032	−	0.961148i	$-0.589020\pi$
$727$	−10821.6	−0.552064	−0.276032	+	0.961148i	$0.589020\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7128.66	−0.360688
$732$	0	0
$733$	−5576.04	−0.280977	−0.140488	−	0.990082i	$-0.544867\pi$
$733$	−5576.04	−0.280977	−0.140488	+	0.990082i	$0.544867\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	32202.0	1.60947
$738$	0	0
$739$	−29317.7	−1.45936	−0.729681	−	0.683788i	$-0.760331\pi$
$739$	−29317.7	−1.45936	−0.729681	+	0.683788i	$0.760331\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7521.08	0.371361	0.185681	−	0.982610i	$-0.440551\pi$
$743$	7521.08	0.371361	0.185681	+	0.982610i	$0.440551\pi$
$744$	0	0
$745$	−6503.40	−0.319820
$746$	0	0
$747$	0	0
$748$	0	0
$749$	270.528	0.0131974
$750$	0	0
$751$	37300.7	1.81241	0.906206	−	0.422837i	$-0.138966\pi$
$751$	37300.7	1.81241	0.906206	+	0.422837i	$0.138966\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	7558.62	0.364353
$756$	0	0
$757$	−26432.1	−1.26908	−0.634538	−	0.772891i	$-0.718810\pi$
$757$	−26432.1	−1.26908	−0.634538	+	0.772891i	$0.718810\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23440.4	1.11658	0.558288	−	0.829647i	$-0.311458\pi$
$761$	23440.4	1.11658	0.558288	+	0.829647i	$0.311458\pi$
$762$	0	0
$763$	−906.889	−0.0430296
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−2469.02	−0.116233
$768$	0	0
$769$	34075.2	1.59790	0.798948	−	0.601401i	$-0.205390\pi$
$769$	34075.2	1.59790	0.798948	+	0.601401i	$0.205390\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−14623.5	−0.680427	−0.340214	−	0.940348i	$-0.610499\pi$
$773$	−14623.5	−0.680427	−0.340214	+	0.940348i	$0.610499\pi$
$774$	0	0
$775$	−1317.29	−0.0610560
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−2145.42	−0.0986749
$780$	0	0
$781$	11173.5	0.511932
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−839.837	−0.0381848
$786$	0	0
$787$	33523.1	1.51839	0.759193	−	0.650865i	$-0.225594\pi$
$787$	33523.1	1.51839	0.759193	+	0.650865i	$0.225594\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	6358.04	0.285797
$792$	0	0
$793$	218.783	0.00979725
$794$	0	0
$795$	0	0
$796$	0	0
$797$	29447.5	1.30876	0.654381	−	0.756165i	$-0.272929\pi$
$797$	29447.5	1.30876	0.654381	+	0.756165i	$0.272929\pi$
$798$	0	0
$799$	−4977.43	−0.220386
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−30811.6	−1.35407
$804$	0	0
$805$	−3694.15	−0.161741
$806$	0	0
$807$	0	0
$808$	0	0
$809$	26891.1	1.16866	0.584328	−	0.811518i	$-0.301358\pi$
$809$	26891.1	1.16866	0.584328	+	0.811518i	$0.301358\pi$
$810$	0	0
$811$	−14912.6	−0.645687	−0.322844	−	0.946452i	$-0.604639\pi$
$811$	−14912.6	−0.645687	−0.322844	+	0.946452i	$0.604639\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−335.745	−0.0144302
$816$	0	0
$817$	−7984.58	−0.341916
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−42644.2	−1.81278	−0.906391	−	0.422439i	$-0.861174\pi$
$821$	−42644.2	−1.81278	−0.906391	+	0.422439i	$0.861174\pi$
$822$	0	0
$823$	25121.7	1.06402	0.532010	−	0.846738i	$-0.321437\pi$
$823$	25121.7	1.06402	0.532010	+	0.846738i	$0.321437\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	4918.93	0.206830	0.103415	−	0.994638i	$-0.467023\pi$
$827$	4918.93	0.206830	0.103415	+	0.994638i	$0.467023\pi$
$828$	0	0
$829$	−22807.6	−0.955538	−0.477769	−	0.878486i	$-0.658554\pi$
$829$	−22807.6	−0.955538	−0.477769	+	0.878486i	$0.658554\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	8493.97	0.353300
$834$	0	0
$835$	12841.8	0.532225
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25772.5	−1.06051	−0.530255	−	0.847839i	$-0.677904\pi$
$839$	−25772.5	−1.06051	−0.530255	+	0.847839i	$0.677904\pi$
$840$	0	0
$841$	58321.7	2.39131
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10861.6	0.442192
$846$	0	0
$847$	1761.35	0.0714532
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−43185.9	−1.73959
$852$	0	0
$853$	41636.0	1.67126	0.835632	−	0.549289i	$-0.185101\pi$
$853$	41636.0	1.67126	0.835632	+	0.549289i	$0.185101\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−17941.6	−0.715137	−0.357569	−	0.933887i	$-0.616394\pi$
$857$	−17941.6	−0.715137	−0.357569	+	0.933887i	$0.616394\pi$
$858$	0	0
$859$	−23917.0	−0.949985	−0.474993	−	0.879990i	$-0.657549\pi$
$859$	−23917.0	−0.949985	−0.474993	+	0.879990i	$0.657549\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−9786.53	−0.386022	−0.193011	−	0.981197i	$-0.561825\pi$
$863$	−9786.53	−0.386022	−0.193011	+	0.981197i	$0.561825\pi$
$864$	0	0
$865$	15062.3	0.592061
$866$	0	0
$867$	0	0
$868$	0	0
$869$	28087.5	1.09644
$870$	0	0
$871$	−3783.20	−0.147174
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−482.343	−0.0186356
$876$	0	0
$877$	−29186.8	−1.12379	−0.561897	−	0.827207i	$-0.689928\pi$
$877$	−29186.8	−1.12379	−0.561897	+	0.827207i	$0.689928\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−18426.6	−0.704664	−0.352332	−	0.935875i	$-0.614611\pi$
$881$	−18426.6	−0.704664	−0.352332	+	0.935875i	$0.614611\pi$
$882$	0	0
$883$	−3942.63	−0.150261	−0.0751303	−	0.997174i	$-0.523937\pi$
$883$	−3942.63	−0.150261	−0.0751303	+	0.997174i	$0.523937\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	436.876	0.0165376	0.00826881	−	0.999966i	$-0.497368\pi$
$887$	436.876	0.0165376	0.00826881	+	0.999966i	$0.497368\pi$
$888$	0	0
$889$	4949.39	0.186723
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−5575.05	−0.208916
$894$	0	0
$895$	12314.0	0.459901
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−15153.8	−0.562189
$900$	0	0
$901$	−7124.36	−0.263426
$902$	0	0
$903$	0	0
$904$	0	0
$905$	18455.5	0.677879
$906$	0	0
$907$	43243.2	1.58309	0.791547	−	0.611108i	$-0.209276\pi$
$907$	43243.2	1.58309	0.791547	+	0.611108i	$0.209276\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	14141.5	0.514301	0.257150	−	0.966371i	$-0.417216\pi$
$911$	14141.5	0.514301	0.257150	+	0.966371i	$0.417216\pi$
$912$	0	0
$913$	63051.0	2.28552
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−4131.34	−0.148778
$918$	0	0
$919$	−1506.36	−0.0540697	−0.0270349	−	0.999634i	$-0.508607\pi$
$919$	−1506.36	−0.0540697	−0.0270349	+	0.999634i	$0.508607\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−1312.70	−0.0468126
$924$	0	0
$925$	−5638.76	−0.200434
$926$	0	0
$927$	0	0
$928$	0	0
$929$	32173.3	1.13624	0.568122	−	0.822944i	$-0.307670\pi$
$929$	32173.3	1.13624	0.568122	+	0.822944i	$0.307670\pi$
$930$	0	0
$931$	9513.82	0.334912
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−5472.42	−0.191409
$936$	0	0
$937$	34833.5	1.21447	0.607236	−	0.794522i	$-0.292278\pi$
$937$	34833.5	1.21447	0.607236	+	0.794522i	$0.292278\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3709.66	−0.128514	−0.0642569	−	0.997933i	$-0.520468\pi$
$941$	−3709.66	−0.128514	−0.0642569	+	0.997933i	$0.520468\pi$
$942$	0	0
$943$	14166.9	0.489224
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−32085.0	−1.10097	−0.550487	−	0.834844i	$-0.685558\pi$
$947$	−32085.0	−1.10097	−0.550487	+	0.834844i	$0.685558\pi$
$948$	0	0
$949$	3619.86	0.123820
$950$	0	0
$951$	0	0
$952$	0	0
$953$	35654.2	1.21191	0.605956	−	0.795498i	$-0.292791\pi$
$953$	35654.2	1.21191	0.605956	+	0.795498i	$0.292791\pi$
$954$	0	0
$955$	9845.08	0.333591
$956$	0	0
$957$	0	0
$958$	0	0
$959$	7893.84	0.265803
$960$	0	0
$961$	−27014.6	−0.906804
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−2595.60	−0.0865857
$966$	0	0
$967$	32345.4	1.07565	0.537827	−	0.843055i	$-0.319245\pi$
$967$	32345.4	1.07565	0.537827	+	0.843055i	$0.319245\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	20137.4	0.665542	0.332771	−	0.943008i	$-0.392016\pi$
$971$	20137.4	0.665542	0.332771	+	0.943008i	$0.392016\pi$
$972$	0	0
$973$	−3026.25	−0.0997091
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30087.1	−0.985231	−0.492615	−	0.870247i	$-0.663959\pi$
$977$	−30087.1	−0.985231	−0.492615	+	0.870247i	$0.663959\pi$
$978$	0	0
$979$	−4514.62	−0.147383
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−59417.5	−1.92790	−0.963950	−	0.266084i	$-0.914270\pi$
$983$	−59417.5	−1.92790	−0.963950	+	0.266084i	$0.914270\pi$
$984$	0	0
$985$	−16431.6	−0.531529
$986$	0	0
$987$	0	0
$988$	0	0
$989$	52724.8	1.69520
$990$	0	0
$991$	20153.5	0.646011	0.323006	−	0.946397i	$-0.395307\pi$
$991$	20153.5	0.646011	0.323006	+	0.946397i	$0.395307\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	18132.4	0.577723
$996$	0	0
$997$	−20772.3	−0.659846	−0.329923	−	0.944008i	$-0.607023\pi$
$997$	−20772.3	−0.659846	−0.329923	+	0.944008i	$0.607023\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.4.a.bk.1.2		3
3.2	odd	2		2160.4.a.bs.1.2		3
4.3	odd	2		1080.4.a.d.1.2	✓	3
12.11	even	2		1080.4.a.j.1.2	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.4.a.d.1.2	✓	3	4.3	odd	2
1080.4.a.j.1.2	yes	3	12.11	even	2
2160.4.a.bk.1.2		3	1.1	even	1	trivial
2160.4.a.bs.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2160.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} +3.85875 q^{7} +O(q^{10})\)	\(q-5.00000 q^{5} +3.85875 q^{7} -42.2783 q^{11} +4.96700 q^{13} -25.8876 q^{17} -28.9958 q^{19} +191.469 q^{23} +25.0000 q^{25} +287.595 q^{29} -52.6915 q^{31} -19.2937 q^{35} -225.550 q^{37} +73.9907 q^{41} +275.370 q^{43} +192.271 q^{47} -328.110 q^{49} +275.204 q^{53} +211.392 q^{55} -497.084 q^{59} +44.0473 q^{61} -24.8350 q^{65} -761.667 q^{67} -264.284 q^{71} +728.781 q^{73} -163.141 q^{77} -664.347 q^{79} -1491.33 q^{83} +129.438 q^{85} +106.783 q^{89} +19.1664 q^{91} +144.979 q^{95} -924.560 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 15 q^{5} + 10 q^{7}+O(q^{10}) \) 3 * q - 15 * q^5 + 10 * q^7	\( 3 q - 15 q^{5} + 10 q^{7} + 28 q^{11} - 78 q^{13} - 11 q^{17} + 71 q^{19} - 25 q^{23} + 75 q^{25} + 118 q^{29} + 107 q^{31} - 50 q^{35} - 410 q^{37} + 592 q^{41} - 52 q^{43} - 580 q^{47} + 479 q^{49} + 169 q^{53} - 140 q^{55} + 234 q^{59} - 673 q^{61} + 390 q^{65} - 386 q^{67} + 16 q^{71} - 892 q^{73} + 1800 q^{77} - 1263 q^{79} - 1815 q^{83} + 55 q^{85} + 1800 q^{89} - 1284 q^{91} - 355 q^{95} - 840 q^{97}+O(q^{100}) \) 3 * q - 15 * q^5 + 10 * q^7 + 28 * q^11 - 78 * q^13 - 11 * q^17 + 71 * q^19 - 25 * q^23 + 75 * q^25 + 118 * q^29 + 107 * q^31 - 50 * q^35 - 410 * q^37 + 592 * q^41 - 52 * q^43 - 580 * q^47 + 479 * q^49 + 169 * q^53 - 140 * q^55 + 234 * q^59 - 673 * q^61 + 390 * q^65 - 386 * q^67 + 16 * q^71 - 892 * q^73 + 1800 * q^77 - 1263 * q^79 - 1815 * q^83 + 55 * q^85 + 1800 * q^89 - 1284 * q^91 - 355 * q^95 - 840 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more