LMFDB - Embedded newform 1560.4.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1560,4,Mod(1,1560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1560.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1560.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:	$92.0429796090$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\mathbb{Q}[x]/(x^{4} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 41x^{2} - 30x - 4 $ x^4 - x^3 - 41x^2 - 30x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{23}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$7.24301$ of defining polynomial
Character	$\chi$	$=$	1560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{3} +5.00000 q^{5} +4.92462 q^{7} +9.00000 q^{9} +O(q^{10})$	$q+3.00000 q^{3} +5.00000 q^{5} +4.92462 q^{7} +9.00000 q^{9} +33.0195 q^{11} +13.0000 q^{13} +15.0000 q^{15} -119.438 q^{17} -127.793 q^{19} +14.7738 q^{21} -30.2485 q^{23} +25.0000 q^{25} +27.0000 q^{27} -309.449 q^{29} -57.9486 q^{31} +99.0584 q^{33} +24.6231 q^{35} -191.930 q^{37} +39.0000 q^{39} +78.4634 q^{41} -524.538 q^{43} +45.0000 q^{45} +183.052 q^{47} -318.748 q^{49} -358.313 q^{51} -241.053 q^{53} +165.097 q^{55} -383.380 q^{57} +103.095 q^{59} +778.610 q^{61} +44.3215 q^{63} +65.0000 q^{65} +230.431 q^{67} -90.7454 q^{69} +668.821 q^{71} +706.428 q^{73} +75.0000 q^{75} +162.608 q^{77} -283.228 q^{79} +81.0000 q^{81} -564.393 q^{83} -597.188 q^{85} -928.346 q^{87} -1561.14 q^{89} +64.0200 q^{91} -173.846 q^{93} -638.967 q^{95} +1444.89 q^{97} +297.175 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	$ 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) $ 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * q^99

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
$3$	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
$7$	4.92462	0.265904	0.132952	−	0.991122i	$-0.457554\pi$
$7$	4.92462	0.265904	0.132952	+	0.991122i	$0.457554\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	33.0195	0.905067	0.452534	−	0.891747i	$-0.350520\pi$
$11$	33.0195	0.905067	0.452534	+	0.891747i	$0.350520\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	−119.438	−1.70399	−0.851996	−	0.523549i	$-0.824608\pi$
$17$	−119.438	−1.70399	−0.851996	+	0.523549i	$0.824608\pi$
$18$	0	0
$19$	−127.793	−1.54304	−0.771521	−	0.636204i	$-0.780504\pi$
$19$	−127.793	−1.54304	−0.771521	+	0.636204i	$0.780504\pi$
$20$	0	0
$21$	14.7738	0.153520
$22$	0	0
$23$	−30.2485	−0.274228	−0.137114	−	0.990555i	$-0.543783\pi$
$23$	−30.2485	−0.274228	−0.137114	+	0.990555i	$0.543783\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−309.449	−1.98149	−0.990744	−	0.135743i	$-0.956658\pi$
$29$	−309.449	−1.98149	−0.990744	+	0.135743i	$0.956658\pi$
$30$	0	0
$31$	−57.9486	−0.335738	−0.167869	−	0.985809i	$-0.553689\pi$
$31$	−57.9486	−0.335738	−0.167869	+	0.985809i	$0.553689\pi$
$32$	0	0
$33$	99.0584	0.522541
$34$	0	0
$35$	24.6231	0.118916
$36$	0	0
$37$	−191.930	−0.852786	−0.426393	−	0.904538i	$-0.640216\pi$
$37$	−191.930	−0.852786	−0.426393	+	0.904538i	$0.640216\pi$
$38$	0	0
$39$	39.0000	0.160128
$40$	0	0
$41$	78.4634	0.298876	0.149438	−	0.988771i	$-0.452254\pi$
$41$	78.4634	0.298876	0.149438	+	0.988771i	$0.452254\pi$
$42$	0	0
$43$	−524.538	−1.86026	−0.930131	−	0.367227i	$-0.880307\pi$
$43$	−524.538	−1.86026	−0.930131	+	0.367227i	$0.880307\pi$
$44$	0	0
$45$	45.0000	0.149071
$46$	0	0
$47$	183.052	0.568104	0.284052	−	0.958809i	$-0.408321\pi$
$47$	183.052	0.568104	0.284052	+	0.958809i	$0.408321\pi$
$48$	0	0
$49$	−318.748	−0.929295
$50$	0	0
$51$	−358.313	−0.983800
$52$	0	0
$53$	−241.053	−0.624738	−0.312369	−	0.949961i	$-0.601123\pi$
$53$	−241.053	−0.624738	−0.312369	+	0.949961i	$0.601123\pi$
$54$	0	0
$55$	165.097	0.404758
$56$	0	0
$57$	−383.380	−0.890876
$58$	0	0
$59$	103.095	0.227489	0.113745	−	0.993510i	$-0.463715\pi$
$59$	103.095	0.227489	0.113745	+	0.993510i	$0.463715\pi$
$60$	0	0
$61$	778.610	1.63427	0.817137	−	0.576443i	$-0.195560\pi$
$61$	778.610	1.63427	0.817137	+	0.576443i	$0.195560\pi$
$62$	0	0
$63$	44.3215	0.0886348
$64$	0	0
$65$	65.0000	0.124035
$66$	0	0
$67$	230.431	0.420173	0.210087	−	0.977683i	$-0.432625\pi$
$67$	230.431	0.420173	0.210087	+	0.977683i	$0.432625\pi$
$68$	0	0
$69$	−90.7454	−0.158325
$70$	0	0
$71$	668.821	1.11795	0.558975	−	0.829184i	$-0.311195\pi$
$71$	668.821	1.11795	0.558975	+	0.829184i	$0.311195\pi$
$72$	0	0
$73$	706.428	1.13262	0.566309	−	0.824193i	$-0.308371\pi$
$73$	706.428	1.13262	0.566309	+	0.824193i	$0.308371\pi$
$74$	0	0
$75$	75.0000	0.115470
$76$	0	0
$77$	162.608	0.240661
$78$	0	0
$79$	−283.228	−0.403362	−0.201681	−	0.979451i	$-0.564640\pi$
$79$	−283.228	−0.403362	−0.201681	+	0.979451i	$0.564640\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−564.393	−0.746388	−0.373194	−	0.927753i	$-0.621737\pi$
$83$	−564.393	−0.746388	−0.373194	+	0.927753i	$0.621737\pi$
$84$	0	0
$85$	−597.188	−0.762048
$86$	0	0
$87$	−928.346	−1.14401
$88$	0	0
$89$	−1561.14	−1.85933	−0.929665	−	0.368405i	$-0.879904\pi$
$89$	−1561.14	−1.85933	−0.929665	+	0.368405i	$0.879904\pi$
$90$	0	0
$91$	64.0200	0.0737486
$92$	0	0
$93$	−173.846	−0.193838
$94$	0	0
$95$	−638.967	−0.690069
$96$	0	0
$97$	1444.89	1.51244	0.756220	−	0.654318i	$-0.227044\pi$
$97$	1444.89	1.51244	0.756220	+	0.654318i	$0.227044\pi$
$98$	0	0
$99$	297.175	0.301689
$100$	0	0
$101$	−693.814	−0.683535	−0.341768	−	0.939784i	$-0.611026\pi$
$101$	−693.814	−0.683535	−0.341768	+	0.939784i	$0.611026\pi$
$102$	0	0
$103$	−278.392	−0.266318	−0.133159	−	0.991095i	$-0.542512\pi$
$103$	−278.392	−0.266318	−0.133159	+	0.991095i	$0.542512\pi$
$104$	0	0
$105$	73.8692	0.0686562
$106$	0	0
$107$	1392.53	1.25814	0.629070	−	0.777349i	$-0.283436\pi$
$107$	1392.53	1.25814	0.629070	+	0.777349i	$0.283436\pi$
$108$	0	0
$109$	−462.126	−0.406089	−0.203044	−	0.979170i	$-0.565084\pi$
$109$	−462.126	−0.406089	−0.203044	+	0.979170i	$0.565084\pi$
$110$	0	0
$111$	−575.790	−0.492356
$112$	0	0
$113$	−112.029	−0.0932635	−0.0466317	−	0.998912i	$-0.514849\pi$
$113$	−112.029	−0.0932635	−0.0466317	+	0.998912i	$0.514849\pi$
$114$	0	0
$115$	−151.242	−0.122638
$116$	0	0
$117$	117.000	0.0924500
$118$	0	0
$119$	−588.184	−0.453099
$120$	0	0
$121$	−240.715	−0.180853
$122$	0	0
$123$	235.390	0.172556
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	−1986.87	−1.38824	−0.694119	−	0.719860i	$-0.744206\pi$
$127$	−1986.87	−1.38824	−0.694119	+	0.719860i	$0.744206\pi$
$128$	0	0
$129$	−1573.61	−1.07402
$130$	0	0
$131$	−2941.06	−1.96154	−0.980771	−	0.195163i	$-0.937476\pi$
$131$	−2941.06	−1.96154	−0.980771	+	0.195163i	$0.937476\pi$
$132$	0	0
$133$	−629.333	−0.410301
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−473.482	−0.295272	−0.147636	−	0.989042i	$-0.547166\pi$
$137$	−473.482	−0.295272	−0.147636	+	0.989042i	$0.547166\pi$
$138$	0	0
$139$	2.57545	0.00157156	0.000785780	−	1.00000i	$-0.499750\pi$
$139$	2.57545	0.00157156	0.000785780		1.00000i	$0.499750\pi$
$140$	0	0
$141$	549.157	0.327995
$142$	0	0
$143$	429.253	0.251021
$144$	0	0
$145$	−1547.24	−0.886148
$146$	0	0
$147$	−956.244	−0.536529
$148$	0	0
$149$	3276.52	1.80150	0.900749	−	0.434340i	$-0.143018\pi$
$149$	3276.52	1.80150	0.900749	+	0.434340i	$0.143018\pi$
$150$	0	0
$151$	−1431.27	−0.771360	−0.385680	−	0.922633i	$-0.626033\pi$
$151$	−1431.27	−0.771360	−0.385680	+	0.922633i	$0.626033\pi$
$152$	0	0
$153$	−1074.94	−0.567997
$154$	0	0
$155$	−289.743	−0.150147
$156$	0	0
$157$	297.943	0.151455	0.0757275	−	0.997129i	$-0.475872\pi$
$157$	297.943	0.151455	0.0757275	+	0.997129i	$0.475872\pi$
$158$	0	0
$159$	−723.158	−0.360693
$160$	0	0
$161$	−148.962	−0.0729183
$162$	0	0
$163$	1548.45	0.744074	0.372037	−	0.928218i	$-0.378659\pi$
$163$	1548.45	0.744074	0.372037	+	0.928218i	$0.378659\pi$
$164$	0	0
$165$	495.292	0.233687
$166$	0	0
$167$	585.292	0.271205	0.135602	−	0.990763i	$-0.456703\pi$
$167$	585.292	0.271205	0.135602	+	0.990763i	$0.456703\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−1150.14	−0.514347
$172$	0	0
$173$	794.258	0.349054	0.174527	−	0.984652i	$-0.444160\pi$
$173$	794.258	0.349054	0.174527	+	0.984652i	$0.444160\pi$
$174$	0	0
$175$	123.115	0.0531809
$176$	0	0
$177$	309.286	0.131341
$178$	0	0
$179$	1889.39	0.788935	0.394467	−	0.918910i	$-0.370929\pi$
$179$	1889.39	0.788935	0.394467	+	0.918910i	$0.370929\pi$
$180$	0	0
$181$	−743.010	−0.305124	−0.152562	−	0.988294i	$-0.548752\pi$
$181$	−743.010	−0.305124	−0.152562	+	0.988294i	$0.548752\pi$
$182$	0	0
$183$	2335.83	0.943549
$184$	0	0
$185$	−959.650	−0.381377
$186$	0	0
$187$	−3943.76	−1.54223
$188$	0	0
$189$	132.965	0.0511733
$190$	0	0
$191$	1598.20	0.605454	0.302727	−	0.953077i	$-0.402103\pi$
$191$	1598.20	0.605454	0.302727	+	0.953077i	$0.402103\pi$
$192$	0	0
$193$	−3079.75	−1.14863	−0.574314	−	0.818635i	$-0.694731\pi$
$193$	−3079.75	−1.14863	−0.574314	+	0.818635i	$0.694731\pi$
$194$	0	0
$195$	195.000	0.0716115
$196$	0	0
$197$	3809.78	1.37785	0.688923	−	0.724835i	$-0.258084\pi$
$197$	3809.78	1.37785	0.688923	+	0.724835i	$0.258084\pi$
$198$	0	0
$199$	3923.17	1.39752	0.698760	−	0.715357i	$-0.253736\pi$
$199$	3923.17	1.39752	0.698760	+	0.715357i	$0.253736\pi$
$200$	0	0
$201$	691.293	0.242587
$202$	0	0
$203$	−1523.92	−0.526886
$204$	0	0
$205$	392.317	0.133661
$206$	0	0
$207$	−272.236	−0.0914092
$208$	0	0
$209$	−4219.67	−1.39656
$210$	0	0
$211$	5000.96	1.63166	0.815830	−	0.578292i	$-0.196280\pi$
$211$	5000.96	1.63166	0.815830	+	0.578292i	$0.196280\pi$
$212$	0	0
$213$	2006.46	0.645449
$214$	0	0
$215$	−2622.69	−0.831935
$216$	0	0
$217$	−285.375	−0.0892741
$218$	0	0
$219$	2119.28	0.653917
$220$	0	0
$221$	−1552.69	−0.472602
$222$	0	0
$223$	−3007.61	−0.903158	−0.451579	−	0.892231i	$-0.649139\pi$
$223$	−3007.61	−0.903158	−0.451579	+	0.892231i	$0.649139\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−1368.56	−0.400152	−0.200076	−	0.979780i	$-0.564119\pi$
$227$	−1368.56	−0.400152	−0.200076	+	0.979780i	$0.564119\pi$
$228$	0	0
$229$	−3877.06	−1.11879	−0.559396	−	0.828901i	$-0.688967\pi$
$229$	−3877.06	−1.11879	−0.559396	+	0.828901i	$0.688967\pi$
$230$	0	0
$231$	487.824	0.138946
$232$	0	0
$233$	−1785.04	−0.501898	−0.250949	−	0.968000i	$-0.580743\pi$
$233$	−1785.04	−0.501898	−0.250949	+	0.968000i	$0.580743\pi$
$234$	0	0
$235$	915.261	0.254064
$236$	0	0
$237$	−849.683	−0.232881
$238$	0	0
$239$	−2595.79	−0.702543	−0.351272	−	0.936274i	$-0.614251\pi$
$239$	−2595.79	−0.702543	−0.351272	+	0.936274i	$0.614251\pi$
$240$	0	0
$241$	1357.57	0.362858	0.181429	−	0.983404i	$-0.441928\pi$
$241$	1357.57	0.362858	0.181429	+	0.983404i	$0.441928\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−1593.74	−0.415593
$246$	0	0
$247$	−1661.31	−0.427963
$248$	0	0
$249$	−1693.18	−0.430927
$250$	0	0
$251$	−4350.81	−1.09411	−0.547053	−	0.837098i	$-0.684250\pi$
$251$	−4350.81	−1.09411	−0.547053	+	0.837098i	$0.684250\pi$
$252$	0	0
$253$	−998.787	−0.248195
$254$	0	0
$255$	−1791.56	−0.439969
$256$	0	0
$257$	1521.29	0.369242	0.184621	−	0.982810i	$-0.440894\pi$
$257$	1521.29	0.369242	0.184621	+	0.982810i	$0.440894\pi$
$258$	0	0
$259$	−945.181	−0.226759
$260$	0	0
$261$	−2785.04	−0.660496
$262$	0	0
$263$	−4402.93	−1.03230	−0.516152	−	0.856497i	$-0.672636\pi$
$263$	−4402.93	−1.03230	−0.516152	+	0.856497i	$0.672636\pi$
$264$	0	0
$265$	−1205.26	−0.279391
$266$	0	0
$267$	−4683.42	−1.07348
$268$	0	0
$269$	507.808	0.115099	0.0575494	−	0.998343i	$-0.481671\pi$
$269$	507.808	0.115099	0.0575494	+	0.998343i	$0.481671\pi$
$270$	0	0
$271$	1851.72	0.415069	0.207535	−	0.978228i	$-0.433456\pi$
$271$	1851.72	0.415069	0.207535	+	0.978228i	$0.433456\pi$
$272$	0	0
$273$	192.060	0.0425788
$274$	0	0
$275$	825.486	0.181013
$276$	0	0
$277$	−7984.20	−1.73186	−0.865928	−	0.500169i	$-0.833271\pi$
$277$	−7984.20	−1.73186	−0.865928	+	0.500169i	$0.833271\pi$
$278$	0	0
$279$	−521.537	−0.111913
$280$	0	0
$281$	6594.79	1.40004	0.700021	−	0.714122i	$-0.253174\pi$
$281$	6594.79	1.40004	0.700021	+	0.714122i	$0.253174\pi$
$282$	0	0
$283$	−8838.66	−1.85655	−0.928276	−	0.371893i	$-0.878709\pi$
$283$	−8838.66	−1.85655	−0.928276	+	0.371893i	$0.878709\pi$
$284$	0	0
$285$	−1916.90	−0.398412
$286$	0	0
$287$	386.402	0.0794724
$288$	0	0
$289$	9352.32	1.90359
$290$	0	0
$291$	4334.68	0.873207
$292$	0	0
$293$	4720.35	0.941180	0.470590	−	0.882352i	$-0.344041\pi$
$293$	4720.35	0.941180	0.470590	+	0.882352i	$0.344041\pi$
$294$	0	0
$295$	515.477	0.101736
$296$	0	0
$297$	891.525	0.174180
$298$	0	0
$299$	−393.230	−0.0760571
$300$	0	0
$301$	−2583.15	−0.494652
$302$	0	0
$303$	−2081.44	−0.394639
$304$	0	0
$305$	3893.05	0.730870
$306$	0	0
$307$	−5097.82	−0.947714	−0.473857	−	0.880602i	$-0.657139\pi$
$307$	−5097.82	−0.947714	−0.473857	+	0.880602i	$0.657139\pi$
$308$	0	0
$309$	−835.175	−0.153759
$310$	0	0
$311$	−8865.03	−1.61637	−0.808183	−	0.588932i	$-0.799549\pi$
$311$	−8865.03	−1.61637	−0.808183	+	0.588932i	$0.799549\pi$
$312$	0	0
$313$	−2904.17	−0.524452	−0.262226	−	0.965006i	$-0.584457\pi$
$313$	−2904.17	−0.524452	−0.262226	+	0.965006i	$0.584457\pi$
$314$	0	0
$315$	221.608	0.0396387
$316$	0	0
$317$	3599.19	0.637699	0.318850	−	0.947805i	$-0.396704\pi$
$317$	3599.19	0.637699	0.318850	+	0.947805i	$0.396704\pi$
$318$	0	0
$319$	−10217.8	−1.79338
$320$	0	0
$321$	4177.59	0.726388
$322$	0	0
$323$	15263.3	2.62933
$324$	0	0
$325$	325.000	0.0554700
$326$	0	0
$327$	−1386.38	−0.234455
$328$	0	0
$329$	901.462	0.151061
$330$	0	0
$331$	−1313.36	−0.218092	−0.109046	−	0.994037i	$-0.534780\pi$
$331$	−1313.36	−0.218092	−0.109046	+	0.994037i	$0.534780\pi$
$332$	0	0
$333$	−1727.37	−0.284262
$334$	0	0
$335$	1152.15	0.187907
$336$	0	0
$337$	532.634	0.0860963	0.0430481	−	0.999073i	$-0.486293\pi$
$337$	532.634	0.0860963	0.0430481	+	0.999073i	$0.486293\pi$
$338$	0	0
$339$	−336.086	−0.0538457
$340$	0	0
$341$	−1913.43	−0.303865
$342$	0	0
$343$	−3258.86	−0.513008
$344$	0	0
$345$	−453.727	−0.0708053
$346$	0	0
$347$	5191.65	0.803177	0.401588	−	0.915820i	$-0.368458\pi$
$347$	5191.65	0.803177	0.401588	+	0.915820i	$0.368458\pi$
$348$	0	0
$349$	3081.50	0.472633	0.236316	−	0.971676i	$-0.424060\pi$
$349$	3081.50	0.472633	0.236316	+	0.971676i	$0.424060\pi$
$350$	0	0
$351$	351.000	0.0533761
$352$	0	0
$353$	5601.39	0.844566	0.422283	−	0.906464i	$-0.361229\pi$
$353$	5601.39	0.844566	0.422283	+	0.906464i	$0.361229\pi$
$354$	0	0
$355$	3344.11	0.499962
$356$	0	0
$357$	−1764.55	−0.261597
$358$	0	0
$359$	58.2648	0.00856573	0.00428286	−	0.999991i	$-0.498637\pi$
$359$	58.2648	0.00856573	0.00428286	+	0.999991i	$0.498637\pi$
$360$	0	0
$361$	9472.13	1.38098
$362$	0	0
$363$	−722.146	−0.104416
$364$	0	0
$365$	3532.14	0.506522
$366$	0	0
$367$	10571.6	1.50363	0.751813	−	0.659377i	$-0.229180\pi$
$367$	10571.6	1.50363	0.751813	+	0.659377i	$0.229180\pi$
$368$	0	0
$369$	706.170	0.0996254
$370$	0	0
$371$	−1187.09	−0.166121
$372$	0	0
$373$	−5244.76	−0.728052	−0.364026	−	0.931389i	$-0.618598\pi$
$373$	−5244.76	−0.728052	−0.364026	+	0.931389i	$0.618598\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−4022.83	−0.549566
$378$	0	0
$379$	−11078.0	−1.50142	−0.750708	−	0.660634i	$-0.770288\pi$
$379$	−11078.0	−1.50142	−0.750708	+	0.660634i	$0.770288\pi$
$380$	0	0
$381$	−5960.61	−0.801500
$382$	0	0
$383$	−12771.9	−1.70395	−0.851974	−	0.523585i	$-0.824594\pi$
$383$	−12771.9	−1.70395	−0.851974	+	0.523585i	$0.824594\pi$
$384$	0	0
$385$	813.041	0.107627
$386$	0	0
$387$	−4720.84	−0.620088
$388$	0	0
$389$	−7046.01	−0.918372	−0.459186	−	0.888340i	$-0.651859\pi$
$389$	−7046.01	−0.918372	−0.459186	+	0.888340i	$0.651859\pi$
$390$	0	0
$391$	3612.80	0.467282
$392$	0	0
$393$	−8823.19	−1.13250
$394$	0	0
$395$	−1416.14	−0.180389
$396$	0	0
$397$	4637.68	0.586293	0.293146	−	0.956068i	$-0.405298\pi$
$397$	4637.68	0.586293	0.293146	+	0.956068i	$0.405298\pi$
$398$	0	0
$399$	−1888.00	−0.236888
$400$	0	0
$401$	4933.11	0.614333	0.307167	−	0.951656i	$-0.400619\pi$
$401$	4933.11	0.614333	0.307167	+	0.951656i	$0.400619\pi$
$402$	0	0
$403$	−753.332	−0.0931169
$404$	0	0
$405$	405.000	0.0496904
$406$	0	0
$407$	−6337.42	−0.771829
$408$	0	0
$409$	−3363.64	−0.406653	−0.203327	−	0.979111i	$-0.565175\pi$
$409$	−3363.64	−0.406653	−0.203327	+	0.979111i	$0.565175\pi$
$410$	0	0
$411$	−1420.45	−0.170475
$412$	0	0
$413$	507.705	0.0604904
$414$	0	0
$415$	−2821.96	−0.333795
$416$	0	0
$417$	7.72635	0.000907340 0
$418$	0	0
$419$	−3927.92	−0.457974	−0.228987	−	0.973429i	$-0.573541\pi$
$419$	−3927.92	−0.457974	−0.228987	+	0.973429i	$0.573541\pi$
$420$	0	0
$421$	−7755.35	−0.897797	−0.448899	−	0.893583i	$-0.648184\pi$
$421$	−7755.35	−0.897797	−0.448899	+	0.893583i	$0.648184\pi$
$422$	0	0
$423$	1647.47	0.189368
$424$	0	0
$425$	−2985.94	−0.340798
$426$	0	0
$427$	3834.35	0.434561
$428$	0	0
$429$	1287.76	0.144927
$430$	0	0
$431$	937.848	0.104813	0.0524067	−	0.998626i	$-0.483311\pi$
$431$	937.848	0.104813	0.0524067	+	0.998626i	$0.483311\pi$
$432$	0	0
$433$	13242.5	1.46973	0.734865	−	0.678214i	$-0.237246\pi$
$433$	13242.5	1.46973	0.734865	+	0.678214i	$0.237246\pi$
$434$	0	0
$435$	−4641.73	−0.511618
$436$	0	0
$437$	3865.55	0.423145
$438$	0	0
$439$	−17088.1	−1.85779	−0.928897	−	0.370337i	$-0.879242\pi$
$439$	−17088.1	−1.85779	−0.928897	+	0.370337i	$0.879242\pi$
$440$	0	0
$441$	−2868.73	−0.309765
$442$	0	0
$443$	10963.6	1.17584	0.587920	−	0.808919i	$-0.299947\pi$
$443$	10963.6	1.17584	0.587920	+	0.808919i	$0.299947\pi$
$444$	0	0
$445$	−7805.70	−0.831518
$446$	0	0
$447$	9829.57	1.04010
$448$	0	0
$449$	13803.9	1.45088	0.725440	−	0.688286i	$-0.241636\pi$
$449$	13803.9	1.45088	0.725440	+	0.688286i	$0.241636\pi$
$450$	0	0
$451$	2590.82	0.270503
$452$	0	0
$453$	−4293.82	−0.445345
$454$	0	0
$455$	320.100	0.0329814
$456$	0	0
$457$	10184.9	1.04252	0.521260	−	0.853398i	$-0.325462\pi$
$457$	10184.9	1.04252	0.521260	+	0.853398i	$0.325462\pi$
$458$	0	0
$459$	−3224.81	−0.327933
$460$	0	0
$461$	16798.2	1.69712	0.848559	−	0.529101i	$-0.177471\pi$
$461$	16798.2	1.69712	0.848559	+	0.529101i	$0.177471\pi$
$462$	0	0
$463$	−8215.56	−0.824643	−0.412321	−	0.911038i	$-0.635282\pi$
$463$	−8215.56	−0.824643	−0.412321	+	0.911038i	$0.635282\pi$
$464$	0	0
$465$	−869.229	−0.0866871
$466$	0	0
$467$	10706.2	1.06086	0.530431	−	0.847728i	$-0.322030\pi$
$467$	10706.2	1.06086	0.530431	+	0.847728i	$0.322030\pi$
$468$	0	0
$469$	1134.78	0.111726
$470$	0	0
$471$	893.829	0.0874426
$472$	0	0
$473$	−17320.0	−1.68366
$474$	0	0
$475$	−3194.83	−0.308608
$476$	0	0
$477$	−2169.47	−0.208246
$478$	0	0
$479$	−9145.64	−0.872390	−0.436195	−	0.899852i	$-0.643674\pi$
$479$	−9145.64	−0.872390	−0.436195	+	0.899852i	$0.643674\pi$
$480$	0	0
$481$	−2495.09	−0.236520
$482$	0	0
$483$	−446.886	−0.0420994
$484$	0	0
$485$	7224.46	0.676383
$486$	0	0
$487$	−1996.38	−0.185759	−0.0928797	−	0.995677i	$-0.529607\pi$
$487$	−1996.38	−0.185759	−0.0928797	+	0.995677i	$0.529607\pi$
$488$	0	0
$489$	4645.35	0.429591
$490$	0	0
$491$	1733.44	0.159326	0.0796628	−	0.996822i	$-0.474616\pi$
$491$	1733.44	0.159326	0.0796628	+	0.996822i	$0.474616\pi$
$492$	0	0
$493$	36959.8	3.37644
$494$	0	0
$495$	1485.88	0.134919
$496$	0	0
$497$	3293.69	0.297268
$498$	0	0
$499$	1391.46	0.124830	0.0624152	−	0.998050i	$-0.480120\pi$
$499$	1391.46	0.124830	0.0624152	+	0.998050i	$0.480120\pi$
$500$	0	0
$501$	1755.88	0.156580
$502$	0	0
$503$	−10970.4	−0.972460	−0.486230	−	0.873831i	$-0.661628\pi$
$503$	−10970.4	−0.972460	−0.486230	+	0.873831i	$0.661628\pi$
$504$	0	0
$505$	−3469.07	−0.305686
$506$	0	0
$507$	507.000	0.0444116
$508$	0	0
$509$	−8556.24	−0.745086	−0.372543	−	0.928015i	$-0.621514\pi$
$509$	−8556.24	−0.745086	−0.372543	+	0.928015i	$0.621514\pi$
$510$	0	0
$511$	3478.89	0.301168
$512$	0	0
$513$	−3450.42	−0.296959
$514$	0	0
$515$	−1391.96	−0.119101
$516$	0	0
$517$	6044.28	0.514173
$518$	0	0
$519$	2382.77	0.201526
$520$	0	0
$521$	7973.37	0.670479	0.335240	−	0.942133i	$-0.391183\pi$
$521$	7973.37	0.670479	0.335240	+	0.942133i	$0.391183\pi$
$522$	0	0
$523$	−19676.9	−1.64514	−0.822572	−	0.568660i	$-0.807462\pi$
$523$	−19676.9	−1.64514	−0.822572	+	0.568660i	$0.807462\pi$
$524$	0	0
$525$	369.346	0.0307040
$526$	0	0
$527$	6921.23	0.572094
$528$	0	0
$529$	−11252.0	−0.924799
$530$	0	0
$531$	927.859	0.0758298
$532$	0	0
$533$	1020.02	0.0828933
$534$	0	0
$535$	6962.65	0.562657
$536$	0	0
$537$	5668.16	0.455492
$538$	0	0
$539$	−10524.9	−0.841074
$540$	0	0
$541$	−2186.81	−0.173786	−0.0868932	−	0.996218i	$-0.527694\pi$
$541$	−2186.81	−0.173786	−0.0868932	+	0.996218i	$0.527694\pi$
$542$	0	0
$543$	−2229.03	−0.176164
$544$	0	0
$545$	−2310.63	−0.181608
$546$	0	0
$547$	−23647.4	−1.84843	−0.924213	−	0.381878i	$-0.875278\pi$
$547$	−23647.4	−1.84843	−0.924213	+	0.381878i	$0.875278\pi$
$548$	0	0
$549$	7007.49	0.544758
$550$	0	0
$551$	39545.5	3.05752
$552$	0	0
$553$	−1394.79	−0.107256
$554$	0	0
$555$	−2878.95	−0.220188
$556$	0	0
$557$	21859.1	1.66283	0.831416	−	0.555650i	$-0.187530\pi$
$557$	21859.1	1.66283	0.831416	+	0.555650i	$0.187530\pi$
$558$	0	0
$559$	−6818.99	−0.515944
$560$	0	0
$561$	−11831.3	−0.890405
$562$	0	0
$563$	4854.00	0.363360	0.181680	−	0.983358i	$-0.441847\pi$
$563$	4854.00	0.363360	0.181680	+	0.983358i	$0.441847\pi$
$564$	0	0
$565$	−560.143	−0.0417087
$566$	0	0
$567$	398.894	0.0295449
$568$	0	0
$569$	5896.70	0.434451	0.217225	−	0.976121i	$-0.430299\pi$
$569$	5896.70	0.434451	0.217225	+	0.976121i	$0.430299\pi$
$570$	0	0
$571$	−4297.28	−0.314949	−0.157474	−	0.987523i	$-0.550335\pi$
$571$	−4297.28	−0.314949	−0.157474	+	0.987523i	$0.550335\pi$
$572$	0	0
$573$	4794.60	0.349559
$574$	0	0
$575$	−756.211	−0.0548455
$576$	0	0
$577$	−5148.34	−0.371453	−0.185726	−	0.982602i	$-0.559464\pi$
$577$	−5148.34	−0.371453	−0.185726	+	0.982602i	$0.559464\pi$
$578$	0	0
$579$	−9239.25	−0.663161
$580$	0	0
$581$	−2779.42	−0.198468
$582$	0	0
$583$	−7959.43	−0.565430
$584$	0	0
$585$	585.000	0.0413449
$586$	0	0
$587$	−9766.94	−0.686754	−0.343377	−	0.939198i	$-0.611571\pi$
$587$	−9766.94	−0.686754	−0.343377	+	0.939198i	$0.611571\pi$
$588$	0	0
$589$	7405.44	0.518058
$590$	0	0
$591$	11429.3	0.795500
$592$	0	0
$593$	27428.8	1.89943	0.949717	−	0.313111i	$-0.101371\pi$
$593$	27428.8	1.89943	0.949717	+	0.313111i	$0.101371\pi$
$594$	0	0
$595$	−2940.92	−0.202632
$596$	0	0
$597$	11769.5	0.806858
$598$	0	0
$599$	14371.6	0.980311	0.490156	−	0.871635i	$-0.336940\pi$
$599$	14371.6	0.980311	0.490156	+	0.871635i	$0.336940\pi$
$600$	0	0
$601$	12103.3	0.821470	0.410735	−	0.911755i	$-0.365272\pi$
$601$	12103.3	0.821470	0.410735	+	0.911755i	$0.365272\pi$
$602$	0	0
$603$	2073.88	0.140058
$604$	0	0
$605$	−1203.58	−0.0808800
$606$	0	0
$607$	−14922.6	−0.997839	−0.498920	−	0.866648i	$-0.666270\pi$
$607$	−14922.6	−0.997839	−0.498920	+	0.866648i	$0.666270\pi$
$608$	0	0
$609$	−4571.75	−0.304198
$610$	0	0
$611$	2379.68	0.157564
$612$	0	0
$613$	16112.5	1.06163	0.530815	−	0.847488i	$-0.321886\pi$
$613$	16112.5	1.06163	0.530815	+	0.847488i	$0.321886\pi$
$614$	0	0
$615$	1176.95	0.0771695
$616$	0	0
$617$	−22097.7	−1.44185	−0.720923	−	0.693015i	$-0.756282\pi$
$617$	−22097.7	−1.44185	−0.720923	+	0.693015i	$0.756282\pi$
$618$	0	0
$619$	−5307.99	−0.344663	−0.172331	−	0.985039i	$-0.555130\pi$
$619$	−5307.99	−0.344663	−0.172331	+	0.985039i	$0.555130\pi$
$620$	0	0
$621$	−816.708	−0.0527752
$622$	0	0
$623$	−7688.01	−0.494404
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−12659.0	−0.806302
$628$	0	0
$629$	22923.6	1.45314
$630$	0	0
$631$	−21024.5	−1.32642	−0.663211	−	0.748433i	$-0.730807\pi$
$631$	−21024.5	−1.32642	−0.663211	+	0.748433i	$0.730807\pi$
$632$	0	0
$633$	15002.9	0.942039
$634$	0	0
$635$	−9934.35	−0.620839
$636$	0	0
$637$	−4143.73	−0.257740
$638$	0	0
$639$	6019.39	0.372650
$640$	0	0
$641$	12148.9	0.748599	0.374300	−	0.927308i	$-0.377883\pi$
$641$	12148.9	0.748599	0.374300	+	0.927308i	$0.377883\pi$
$642$	0	0
$643$	15272.5	0.936687	0.468344	−	0.883546i	$-0.344851\pi$
$643$	15272.5	0.936687	0.468344	+	0.883546i	$0.344851\pi$
$644$	0	0
$645$	−7868.07	−0.480318
$646$	0	0
$647$	17144.3	1.04175	0.520875	−	0.853633i	$-0.325606\pi$
$647$	17144.3	1.04175	0.520875	+	0.853633i	$0.325606\pi$
$648$	0	0
$649$	3404.15	0.205893
$650$	0	0
$651$	−856.124	−0.0515424
$652$	0	0
$653$	3379.46	0.202524	0.101262	−	0.994860i	$-0.467712\pi$
$653$	3379.46	0.202524	0.101262	+	0.994860i	$0.467712\pi$
$654$	0	0
$655$	−14705.3	−0.877228
$656$	0	0
$657$	6357.85	0.377539
$658$	0	0
$659$	12684.6	0.749803	0.374902	−	0.927065i	$-0.377677\pi$
$659$	12684.6	0.749803	0.374902	+	0.927065i	$0.377677\pi$
$660$	0	0
$661$	−22946.2	−1.35023	−0.675116	−	0.737712i	$-0.735906\pi$
$661$	−22946.2	−1.35023	−0.675116	+	0.737712i	$0.735906\pi$
$662$	0	0
$663$	−4658.06	−0.272857
$664$	0	0
$665$	−3146.67	−0.183492
$666$	0	0
$667$	9360.34	0.543379
$668$	0	0
$669$	−9022.82	−0.521438
$670$	0	0
$671$	25709.3	1.47913
$672$	0	0
$673$	16725.6	0.957986	0.478993	−	0.877819i	$-0.341002\pi$
$673$	16725.6	0.957986	0.478993	+	0.877819i	$0.341002\pi$
$674$	0	0
$675$	675.000	0.0384900
$676$	0	0
$677$	14220.8	0.807313	0.403657	−	0.914911i	$-0.367739\pi$
$677$	14220.8	0.807313	0.403657	+	0.914911i	$0.367739\pi$
$678$	0	0
$679$	7115.54	0.402164
$680$	0	0
$681$	−4105.68	−0.231028
$682$	0	0
$683$	32769.8	1.83587	0.917936	−	0.396728i	$-0.129854\pi$
$683$	32769.8	1.83587	0.917936	+	0.396728i	$0.129854\pi$
$684$	0	0
$685$	−2367.41	−0.132050
$686$	0	0
$687$	−11631.2	−0.645935
$688$	0	0
$689$	−3133.68	−0.173271
$690$	0	0
$691$	−6824.82	−0.375729	−0.187864	−	0.982195i	$-0.560157\pi$
$691$	−6824.82	−0.375729	−0.187864	+	0.982195i	$0.560157\pi$
$692$	0	0
$693$	1463.47	0.0802204
$694$	0	0
$695$	12.8772	0.000702823 0
$696$	0	0
$697$	−9371.47	−0.509282
$698$	0	0
$699$	−5355.13	−0.289771
$700$	0	0
$701$	−2989.72	−0.161085	−0.0805423	−	0.996751i	$-0.525665\pi$
$701$	−2989.72	−0.161085	−0.0805423	+	0.996751i	$0.525665\pi$
$702$	0	0
$703$	24527.4	1.31588
$704$	0	0
$705$	2745.78	0.146684
$706$	0	0
$707$	−3416.77	−0.181755
$708$	0	0
$709$	−27605.9	−1.46229	−0.731143	−	0.682224i	$-0.761013\pi$
$709$	−27605.9	−1.46229	−0.731143	+	0.682224i	$0.761013\pi$
$710$	0	0
$711$	−2549.05	−0.134454
$712$	0	0
$713$	1752.85	0.0920686
$714$	0	0
$715$	2146.26	0.112260
$716$	0	0
$717$	−7787.38	−0.405614
$718$	0	0
$719$	−15643.3	−0.811401	−0.405701	−	0.914006i	$-0.632972\pi$
$719$	−15643.3	−0.811401	−0.405701	+	0.914006i	$0.632972\pi$
$720$	0	0
$721$	−1370.97	−0.0708150
$722$	0	0
$723$	4072.71	0.209496
$724$	0	0
$725$	−7736.21	−0.396298
$726$	0	0
$727$	27676.5	1.41192	0.705958	−	0.708253i	$-0.250517\pi$
$727$	27676.5	1.41192	0.705958	+	0.708253i	$0.250517\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	62649.5	3.16987
$732$	0	0
$733$	26683.5	1.34458	0.672290	−	0.740288i	$-0.265311\pi$
$733$	26683.5	1.34458	0.672290	+	0.740288i	$0.265311\pi$
$734$	0	0
$735$	−4781.22	−0.239943
$736$	0	0
$737$	7608.70	0.380285
$738$	0	0
$739$	21455.4	1.06800	0.533998	−	0.845486i	$-0.320689\pi$
$739$	21455.4	1.06800	0.533998	+	0.845486i	$0.320689\pi$
$740$	0	0
$741$	−4983.94	−0.247084
$742$	0	0
$743$	9493.05	0.468730	0.234365	−	0.972149i	$-0.424699\pi$
$743$	9493.05	0.468730	0.234365	+	0.972149i	$0.424699\pi$
$744$	0	0
$745$	16382.6	0.805654
$746$	0	0
$747$	−5079.54	−0.248796
$748$	0	0
$749$	6857.68	0.334545
$750$	0	0
$751$	−21585.4	−1.04882	−0.524410	−	0.851466i	$-0.675714\pi$
$751$	−21585.4	−1.04882	−0.524410	+	0.851466i	$0.675714\pi$
$752$	0	0
$753$	−13052.4	−0.631682
$754$	0	0
$755$	−7156.36	−0.344963
$756$	0	0
$757$	8706.02	0.418000	0.209000	−	0.977916i	$-0.432979\pi$
$757$	8706.02	0.418000	0.209000	+	0.977916i	$0.432979\pi$
$758$	0	0
$759$	−2996.36	−0.143295
$760$	0	0
$761$	−2484.28	−0.118338	−0.0591688	−	0.998248i	$-0.518845\pi$
$761$	−2484.28	−0.118338	−0.0591688	+	0.998248i	$0.518845\pi$
$762$	0	0
$763$	−2275.80	−0.107981
$764$	0	0
$765$	−5374.69	−0.254016
$766$	0	0
$767$	1340.24	0.0630942
$768$	0	0
$769$	−6852.69	−0.321345	−0.160673	−	0.987008i	$-0.551366\pi$
$769$	−6852.69	−0.321345	−0.160673	+	0.987008i	$0.551366\pi$
$770$	0	0
$771$	4563.86	0.213182
$772$	0	0
$773$	32276.9	1.50184	0.750919	−	0.660394i	$-0.229611\pi$
$773$	32276.9	1.50184	0.750919	+	0.660394i	$0.229611\pi$
$774$	0	0
$775$	−1448.71	−0.0671476
$776$	0	0
$777$	−2835.54	−0.130920
$778$	0	0
$779$	−10027.1	−0.461178
$780$	0	0
$781$	22084.1	1.01182
$782$	0	0
$783$	−8355.11	−0.381338
$784$	0	0
$785$	1489.72	0.0677328
$786$	0	0
$787$	13011.8	0.589355	0.294677	−	0.955597i	$-0.404788\pi$
$787$	13011.8	0.589355	0.294677	+	0.955597i	$0.404788\pi$
$788$	0	0
$789$	−13208.8	−0.596001
$790$	0	0
$791$	−551.698	−0.0247992
$792$	0	0
$793$	10121.9	0.453266
$794$	0	0
$795$	−3615.79	−0.161307
$796$	0	0
$797$	25309.1	1.12484	0.562419	−	0.826852i	$-0.309871\pi$
$797$	25309.1	1.12484	0.562419	+	0.826852i	$0.309871\pi$
$798$	0	0
$799$	−21863.3	−0.968045
$800$	0	0
$801$	−14050.3	−0.619777
$802$	0	0
$803$	23325.9	1.02510
$804$	0	0
$805$	−744.810	−0.0326101
$806$	0	0
$807$	1523.42	0.0664524
$808$	0	0
$809$	−37180.8	−1.61583	−0.807914	−	0.589300i	$-0.799404\pi$
$809$	−37180.8	−1.61583	−0.807914	+	0.589300i	$0.799404\pi$
$810$	0	0
$811$	−18270.7	−0.791085	−0.395543	−	0.918448i	$-0.629443\pi$
$811$	−18270.7	−0.791085	−0.395543	+	0.918448i	$0.629443\pi$
$812$	0	0
$813$	5555.15	0.239640
$814$	0	0
$815$	7742.25	0.332760
$816$	0	0
$817$	67032.5	2.87046
$818$	0	0
$819$	576.180	0.0245829
$820$	0	0
$821$	−25619.8	−1.08908	−0.544542	−	0.838733i	$-0.683297\pi$
$821$	−25619.8	−1.08908	−0.544542	+	0.838733i	$0.683297\pi$
$822$	0	0
$823$	20112.4	0.851854	0.425927	−	0.904758i	$-0.359948\pi$
$823$	20112.4	0.851854	0.425927	+	0.904758i	$0.359948\pi$
$824$	0	0
$825$	2476.46	0.104508
$826$	0	0
$827$	−3220.54	−0.135416	−0.0677080	−	0.997705i	$-0.521569\pi$
$827$	−3220.54	−0.135416	−0.0677080	+	0.997705i	$0.521569\pi$
$828$	0	0
$829$	−20396.5	−0.854524	−0.427262	−	0.904128i	$-0.640522\pi$
$829$	−20396.5	−0.854524	−0.427262	+	0.904128i	$0.640522\pi$
$830$	0	0
$831$	−23952.6	−0.999888
$832$	0	0
$833$	38070.5	1.58351
$834$	0	0
$835$	2926.46	0.121287
$836$	0	0
$837$	−1564.61	−0.0646128
$838$	0	0
$839$	36682.5	1.50944	0.754720	−	0.656047i	$-0.227773\pi$
$839$	36682.5	1.50944	0.754720	+	0.656047i	$0.227773\pi$
$840$	0	0
$841$	71369.4	2.92630
$842$	0	0
$843$	19784.4	0.808315
$844$	0	0
$845$	845.000	0.0344010
$846$	0	0
$847$	−1185.43	−0.0480896
$848$	0	0
$849$	−26516.0	−1.07188
$850$	0	0
$851$	5805.58	0.233858
$852$	0	0
$853$	−22759.0	−0.913544	−0.456772	−	0.889584i	$-0.650994\pi$
$853$	−22759.0	−0.913544	−0.456772	+	0.889584i	$0.650994\pi$
$854$	0	0
$855$	−5750.70	−0.230023
$856$	0	0
$857$	4632.17	0.184635	0.0923173	−	0.995730i	$-0.470573\pi$
$857$	4632.17	0.184635	0.0923173	+	0.995730i	$0.470573\pi$
$858$	0	0
$859$	25367.9	1.00762	0.503808	−	0.863816i	$-0.331932\pi$
$859$	25367.9	1.00762	0.503808	+	0.863816i	$0.331932\pi$
$860$	0	0
$861$	1159.21	0.0458834
$862$	0	0
$863$	−40737.3	−1.60685	−0.803426	−	0.595404i	$-0.796992\pi$
$863$	−40737.3	−1.60685	−0.803426	+	0.595404i	$0.796992\pi$
$864$	0	0
$865$	3971.29	0.156102
$866$	0	0
$867$	28057.0	1.09904
$868$	0	0
$869$	−9352.02	−0.365070
$870$	0	0
$871$	2995.60	0.116535
$872$	0	0
$873$	13004.0	0.504146
$874$	0	0
$875$	615.577	0.0237832
$876$	0	0
$877$	804.453	0.0309743	0.0154871	−	0.999880i	$-0.495070\pi$
$877$	804.453	0.0309743	0.0154871	+	0.999880i	$0.495070\pi$
$878$	0	0
$879$	14161.0	0.543391
$880$	0	0
$881$	10971.4	0.419564	0.209782	−	0.977748i	$-0.432725\pi$
$881$	10971.4	0.419564	0.209782	+	0.977748i	$0.432725\pi$
$882$	0	0
$883$	−40822.6	−1.55582	−0.777910	−	0.628375i	$-0.783720\pi$
$883$	−40822.6	−1.55582	−0.777910	+	0.628375i	$0.783720\pi$
$884$	0	0
$885$	1546.43	0.0587375
$886$	0	0
$887$	−34628.9	−1.31085	−0.655425	−	0.755260i	$-0.727510\pi$
$887$	−34628.9	−1.31085	−0.655425	+	0.755260i	$0.727510\pi$
$888$	0	0
$889$	−9784.58	−0.369138
$890$	0	0
$891$	2674.58	0.100563
$892$	0	0
$893$	−23392.8	−0.876609
$894$	0	0
$895$	9446.93	0.352822
$896$	0	0
$897$	−1179.69	−0.0439116
$898$	0	0
$899$	17932.1	0.665261
$900$	0	0
$901$	28790.7	1.06455
$902$	0	0
$903$	−7749.45	−0.285587
$904$	0	0
$905$	−3715.05	−0.136456
$906$	0	0
$907$	15188.6	0.556040	0.278020	−	0.960575i	$-0.410322\pi$
$907$	15188.6	0.556040	0.278020	+	0.960575i	$0.410322\pi$
$908$	0	0
$909$	−6244.33	−0.227845
$910$	0	0
$911$	−22243.2	−0.808947	−0.404474	−	0.914550i	$-0.632545\pi$
$911$	−22243.2	−0.808947	−0.404474	+	0.914550i	$0.632545\pi$
$912$	0	0
$913$	−18635.9	−0.675531
$914$	0	0
$915$	11679.1	0.421968
$916$	0	0
$917$	−14483.6	−0.521582
$918$	0	0
$919$	−22385.9	−0.803529	−0.401765	−	0.915743i	$-0.631603\pi$
$919$	−22385.9	−0.803529	−0.401765	+	0.915743i	$0.631603\pi$
$920$	0	0
$921$	−15293.5	−0.547163
$922$	0	0
$923$	8694.67	0.310064
$924$	0	0
$925$	−4798.25	−0.170557
$926$	0	0
$927$	−2505.52	−0.0887726
$928$	0	0
$929$	2508.24	0.0885819	0.0442910	−	0.999019i	$-0.485897\pi$
$929$	2508.24	0.0885819	0.0442910	+	0.999019i	$0.485897\pi$
$930$	0	0
$931$	40733.9	1.43394
$932$	0	0
$933$	−26595.1	−0.933209
$934$	0	0
$935$	−19718.8	−0.689705
$936$	0	0
$937$	−47700.9	−1.66310	−0.831548	−	0.555452i	$-0.812545\pi$
$937$	−47700.9	−1.66310	−0.831548	+	0.555452i	$0.812545\pi$
$938$	0	0
$939$	−8712.51	−0.302793
$940$	0	0
$941$	−39333.0	−1.36261	−0.681306	−	0.731999i	$-0.738588\pi$
$941$	−39333.0	−1.36261	−0.681306	+	0.731999i	$0.738588\pi$
$942$	0	0
$943$	−2373.40	−0.0819601
$944$	0	0
$945$	664.823	0.0228854
$946$	0	0
$947$	−19389.4	−0.665333	−0.332667	−	0.943044i	$-0.607948\pi$
$947$	−19389.4	−0.665333	−0.332667	+	0.943044i	$0.607948\pi$
$948$	0	0
$949$	9183.56	0.314132
$950$	0	0
$951$	10797.6	0.368176
$952$	0	0
$953$	48296.3	1.64163	0.820814	−	0.571195i	$-0.193520\pi$
$953$	48296.3	1.64163	0.820814	+	0.571195i	$0.193520\pi$
$954$	0	0
$955$	7991.00	0.270767
$956$	0	0
$957$	−30653.5	−1.03541
$958$	0	0
$959$	−2331.72	−0.0785141
$960$	0	0
$961$	−26433.0	−0.887280
$962$	0	0
$963$	12532.8	0.419380
$964$	0	0
$965$	−15398.7	−0.513682
$966$	0	0
$967$	−38596.2	−1.28353	−0.641763	−	0.766903i	$-0.721797\pi$
$967$	−38596.2	−1.28353	−0.641763	+	0.766903i	$0.721797\pi$
$968$	0	0
$969$	45789.9	1.51804
$970$	0	0
$971$	48306.0	1.59651	0.798256	−	0.602319i	$-0.205756\pi$
$971$	48306.0	1.59651	0.798256	+	0.602319i	$0.205756\pi$
$972$	0	0
$973$	12.6831	0.000417884 0
$974$	0	0
$975$	975.000	0.0320256
$976$	0	0
$977$	5886.02	0.192744	0.0963718	−	0.995345i	$-0.469276\pi$
$977$	5886.02	0.192744	0.0963718	+	0.995345i	$0.469276\pi$
$978$	0	0
$979$	−51548.0	−1.68282
$980$	0	0
$981$	−4159.14	−0.135363
$982$	0	0
$983$	6014.75	0.195158	0.0975792	−	0.995228i	$-0.468890\pi$
$983$	6014.75	0.195158	0.0975792	+	0.995228i	$0.468890\pi$
$984$	0	0
$985$	19048.9	0.616191
$986$	0	0
$987$	2704.39	0.0872153
$988$	0	0
$989$	15866.5	0.510136
$990$	0	0
$991$	55642.6	1.78360	0.891799	−	0.452432i	$-0.149443\pi$
$991$	55642.6	1.78360	0.891799	+	0.452432i	$0.149443\pi$
$992$	0	0
$993$	−3940.07	−0.125916
$994$	0	0
$995$	19615.9	0.624990
$996$	0	0
$997$	42674.5	1.35558	0.677791	−	0.735255i	$-0.262937\pi$
$997$	42674.5	1.35558	0.677791	+	0.735255i	$0.262937\pi$
$998$	0	0
$999$	−5182.11	−0.164119

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1560.4.a.n.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1560.4.a.n.1.3	✓	4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1560 = 2^{3} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1560.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{3} +5.00000 q^{5} +4.92462 q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q+3.00000 q^{3} +5.00000 q^{5} +4.92462 q^{7} +9.00000 q^{9} +33.0195 q^{11} +13.0000 q^{13} +15.0000 q^{15} -119.438 q^{17} -127.793 q^{19} +14.7738 q^{21} -30.2485 q^{23} +25.0000 q^{25} +27.0000 q^{27} -309.449 q^{29} -57.9486 q^{31} +99.0584 q^{33} +24.6231 q^{35} -191.930 q^{37} +39.0000 q^{39} +78.4634 q^{41} -524.538 q^{43} +45.0000 q^{45} +183.052 q^{47} -318.748 q^{49} -358.313 q^{51} -241.053 q^{53} +165.097 q^{55} -383.380 q^{57} +103.095 q^{59} +778.610 q^{61} +44.3215 q^{63} +65.0000 q^{65} +230.431 q^{67} -90.7454 q^{69} +668.821 q^{71} +706.428 q^{73} +75.0000 q^{75} +162.608 q^{77} -283.228 q^{79} +81.0000 q^{81} -564.393 q^{83} -597.188 q^{85} -928.346 q^{87} -1561.14 q^{89} +64.0200 q^{91} -173.846 q^{93} -638.967 q^{95} +1444.89 q^{97} +297.175 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9}+O(q^{10}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9	\( 4 q + 12 q^{3} + 20 q^{5} - 34 q^{7} + 36 q^{9} - 38 q^{11} + 52 q^{13} + 60 q^{15} - 50 q^{17} - 180 q^{19} - 102 q^{21} - 170 q^{23} + 100 q^{25} + 108 q^{27} - 84 q^{29} - 408 q^{31} - 114 q^{33} - 170 q^{35} - 38 q^{37} + 156 q^{39} - 90 q^{41} - 732 q^{43} + 180 q^{45} - 520 q^{47} + 230 q^{49} - 150 q^{51} - 338 q^{53} - 190 q^{55} - 540 q^{57} - 232 q^{59} + 326 q^{61} - 306 q^{63} + 260 q^{65} - 1040 q^{67} - 510 q^{69} - 702 q^{71} - 368 q^{73} + 300 q^{75} + 14 q^{77} - 678 q^{79} + 324 q^{81} - 1888 q^{83} - 250 q^{85} - 252 q^{87} - 2070 q^{89} - 442 q^{91} - 1224 q^{93} - 900 q^{95} + 1126 q^{97} - 342 q^{99}+O(q^{100}) \) 4 * q + 12 * q^3 + 20 * q^5 - 34 * q^7 + 36 * q^9 - 38 * q^11 + 52 * q^13 + 60 * q^15 - 50 * q^17 - 180 * q^19 - 102 * q^21 - 170 * q^23 + 100 * q^25 + 108 * q^27 - 84 * q^29 - 408 * q^31 - 114 * q^33 - 170 * q^35 - 38 * q^37 + 156 * q^39 - 90 * q^41 - 732 * q^43 + 180 * q^45 - 520 * q^47 + 230 * q^49 - 150 * q^51 - 338 * q^53 - 190 * q^55 - 540 * q^57 - 232 * q^59 + 326 * q^61 - 306 * q^63 + 260 * q^65 - 1040 * q^67 - 510 * q^69 - 702 * q^71 - 368 * q^73 + 300 * q^75 + 14 * q^77 - 678 * q^79 + 324 * q^81 - 1888 * q^83 - 250 * q^85 - 252 * q^87 - 2070 * q^89 - 442 * q^91 - 1224 * q^93 - 900 * q^95 + 1126 * q^97 - 342 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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