LMFDB - Embedded newform 1440.4.a.w.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1440,4,Mod(1,1440)] 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("1440.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1440, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1440 = 2^{5} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1440.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-10,0,0,0,0,0,0,0,104] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$84.9627504083$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{85}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 21 $ x^2 - x - 21
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			1.1
Root			$5.10977$ of defining polynomial
Character	$\chi$	$=$	1440.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-5.00000 q^{5} -18.4391 q^{7} +55.3173 q^{11} +52.0000 q^{13} -66.0000 q^{17} +36.8782 q^{23} +25.0000 q^{25} +46.0000 q^{29} -258.147 q^{31} +92.1954 q^{35} -24.0000 q^{37} -72.0000 q^{41} -331.904 q^{47} -3.00000 q^{49} +62.0000 q^{53} -276.586 q^{55} -239.708 q^{59} -190.000 q^{61} -260.000 q^{65} +774.442 q^{67} -626.929 q^{71} +1078.00 q^{73} -1020.00 q^{77} +848.198 q^{79} +774.442 q^{83} +330.000 q^{85} +1116.00 q^{89} -958.833 q^{91} +834.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 10 q^{5} + 104 q^{13} - 132 q^{17} + 50 q^{25} + 92 q^{29} - 48 q^{37} - 144 q^{41} - 6 q^{49} + 124 q^{53} - 380 q^{61} - 520 q^{65} + 2156 q^{73} - 2040 q^{77} + 660 q^{85} + 2232 q^{89} + 1668 q^{97}+O(q^{100}) $ 2 * q - 10 * q^5 + 104 * q^13 - 132 * q^17 + 50 * q^25 + 92 * q^29 - 48 * q^37 - 144 * q^41 - 6 * q^49 + 124 * q^53 - 380 * q^61 - 520 * q^65 + 2156 * q^73 - 2040 * q^77 + 660 * q^85 + 2232 * q^89 + 1668 * q^97

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−18.4391	−0.995617	−0.497809	−	0.867287i	$-0.665862\pi$
$7$	−18.4391	−0.995617	−0.497809	+	0.867287i	$0.665862\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	55.3173	1.51625	0.758127	−	0.652107i	$-0.226115\pi$
$11$	55.3173	1.51625	0.758127	+	0.652107i	$0.226115\pi$
$12$	0	0
$13$	52.0000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	52.0000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−66.0000	−0.941609	−0.470804	−	0.882238i	$-0.656036\pi$
$17$	−66.0000	−0.941609	−0.470804	+	0.882238i	$0.656036\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	36.8782	0.334332	0.167166	−	0.985929i	$-0.446538\pi$
$23$	36.8782	0.334332	0.167166	+	0.985929i	$0.446538\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	46.0000	0.294551	0.147276	−	0.989095i	$-0.452950\pi$
$29$	46.0000	0.294551	0.147276	+	0.989095i	$0.452950\pi$
$30$	0	0
$31$	−258.147	−1.49563	−0.747816	−	0.663906i	$-0.768898\pi$
$31$	−258.147	−1.49563	−0.747816	+	0.663906i	$0.768898\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	92.1954	0.445254
$36$	0	0
$37$	−24.0000	−0.106637	−0.0533186	−	0.998578i	$-0.516980\pi$
$37$	−24.0000	−0.106637	−0.0533186	+	0.998578i	$0.516980\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−72.0000	−0.274256	−0.137128	−	0.990553i	$-0.543787\pi$
$41$	−72.0000	−0.274256	−0.137128	+	0.990553i	$0.543787\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−331.904	−1.03007	−0.515033	−	0.857170i	$-0.672220\pi$
$47$	−331.904	−1.03007	−0.515033	+	0.857170i	$0.672220\pi$
$48$	0	0
$49$	−3.00000	−0.00874636
$50$	0	0
$51$	0	0
$52$	0	0
$53$	62.0000	0.160686	0.0803430	−	0.996767i	$-0.474398\pi$
$53$	62.0000	0.160686	0.0803430	+	0.996767i	$0.474398\pi$
$54$	0	0
$55$	−276.586	−0.678089
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−239.708	−0.528938	−0.264469	−	0.964394i	$-0.585197\pi$
$59$	−239.708	−0.528938	−0.264469	+	0.964394i	$0.585197\pi$
$60$	0	0
$61$	−190.000	−0.398803	−0.199402	−	0.979918i	$-0.563900\pi$
$61$	−190.000	−0.398803	−0.199402	+	0.979918i	$0.563900\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−260.000	−0.496139
$66$	0	0
$67$	774.442	1.41214	0.706068	−	0.708144i	$-0.250467\pi$
$67$	774.442	1.41214	0.706068	+	0.708144i	$0.250467\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−626.929	−1.04793	−0.523963	−	0.851741i	$-0.675547\pi$
$71$	−626.929	−1.04793	−0.523963	+	0.851741i	$0.675547\pi$
$72$	0	0
$73$	1078.00	1.72836	0.864181	−	0.503182i	$-0.167837\pi$
$73$	1078.00	1.72836	0.864181	+	0.503182i	$0.167837\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1020.00	−1.50961
$78$	0	0
$79$	848.198	1.20797	0.603986	−	0.796995i	$-0.293578\pi$
$79$	848.198	1.20797	0.603986	+	0.796995i	$0.293578\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	774.442	1.02417	0.512084	−	0.858935i	$-0.328873\pi$
$83$	774.442	1.02417	0.512084	+	0.858935i	$0.328873\pi$
$84$	0	0
$85$	330.000	0.421100
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1116.00	1.32917	0.664583	−	0.747215i	$-0.268609\pi$
$89$	1116.00	1.32917	0.664583	+	0.747215i	$0.268609\pi$
$90$	0	0
$91$	−958.833	−1.10454
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	834.000	0.872988	0.436494	−	0.899707i	$-0.356220\pi$
$97$	834.000	0.872988	0.436494	+	0.899707i	$0.356220\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	270.000	0.266000	0.133000	−	0.991116i	$-0.457539\pi$
$101$	270.000	0.266000	0.133000	+	0.991116i	$0.457539\pi$
$102$	0	0
$103$	571.612	0.546821	0.273411	−	0.961897i	$-0.411848\pi$
$103$	571.612	0.546821	0.273411	+	0.961897i	$0.411848\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1069.47	0.966255	0.483128	−	0.875550i	$-0.339501\pi$
$107$	1069.47	0.966255	0.483128	+	0.875550i	$0.339501\pi$
$108$	0	0
$109$	1746.00	1.53428	0.767140	−	0.641480i	$-0.221679\pi$
$109$	1746.00	1.53428	0.767140	+	0.641480i	$0.221679\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	882.000	0.734262	0.367131	−	0.930169i	$-0.380340\pi$
$113$	882.000	0.734262	0.367131	+	0.930169i	$0.380340\pi$
$114$	0	0
$115$	−184.391	−0.149518
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1216.98	0.937482
$120$	0	0
$121$	1729.00	1.29902
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	2194.25	1.53314	0.766568	−	0.642163i	$-0.221963\pi$
$127$	2194.25	1.53314	0.766568	+	0.642163i	$0.221963\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2268.01	−1.51265	−0.756324	−	0.654198i	$-0.773006\pi$
$131$	−2268.01	−1.51265	−0.756324	+	0.654198i	$0.773006\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1474.00	0.919214	0.459607	−	0.888122i	$-0.347990\pi$
$137$	1474.00	0.919214	0.459607	+	0.888122i	$0.347990\pi$
$138$	0	0
$139$	−221.269	−0.135020	−0.0675101	−	0.997719i	$-0.521505\pi$
$139$	−221.269	−0.135020	−0.0675101	+	0.997719i	$0.521505\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2876.50	1.68213
$144$	0	0
$145$	−230.000	−0.131727
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−430.000	−0.236423	−0.118211	−	0.992988i	$-0.537716\pi$
$149$	−430.000	−0.236423	−0.118211	+	0.992988i	$0.537716\pi$
$150$	0	0
$151$	−2950.25	−1.58999	−0.794994	−	0.606617i	$-0.792526\pi$
$151$	−2950.25	−1.58999	−0.794994	+	0.606617i	$0.792526\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1290.74	0.668867
$156$	0	0
$157$	1884.00	0.957704	0.478852	−	0.877896i	$-0.341053\pi$
$157$	1884.00	0.957704	0.478852	+	0.877896i	$0.341053\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−680.000	−0.332867
$162$	0	0
$163$	−405.660	−0.194931	−0.0974655	−	0.995239i	$-0.531074\pi$
$163$	−405.660	−0.194931	−0.0974655	+	0.995239i	$0.531074\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4093.48	−1.89678	−0.948392	−	0.317101i	$-0.897290\pi$
$167$	−4093.48	−1.89678	−0.948392	+	0.317101i	$0.897290\pi$
$168$	0	0
$169$	507.000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1162.00	−0.510666	−0.255333	−	0.966853i	$-0.582185\pi$
$173$	−1162.00	−0.510666	−0.255333	+	0.966853i	$0.582185\pi$
$174$	0	0
$175$	−460.977	−0.199123
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−3927.53	−1.63998	−0.819992	−	0.572375i	$-0.806022\pi$
$179$	−3927.53	−1.63998	−0.819992	+	0.572375i	$0.806022\pi$
$180$	0	0
$181$	3538.00	1.45291	0.726457	−	0.687212i	$-0.241166\pi$
$181$	3538.00	1.45291	0.726457	+	0.687212i	$0.241166\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	120.000	0.0476896
$186$	0	0
$187$	−3650.94	−1.42772
$188$	0	0
$189$	0	0
$190$	0	0
$191$	405.660	0.153678	0.0768391	−	0.997044i	$-0.475517\pi$
$191$	405.660	0.153678	0.0768391	+	0.997044i	$0.475517\pi$
$192$	0	0
$193$	2438.00	0.909280	0.454640	−	0.890675i	$-0.349768\pi$
$193$	2438.00	0.909280	0.454640	+	0.890675i	$0.349768\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	474.000	0.171427	0.0857135	−	0.996320i	$-0.472683\pi$
$197$	474.000	0.171427	0.0857135	+	0.996320i	$0.472683\pi$
$198$	0	0
$199$	−1696.40	−0.604293	−0.302147	−	0.953261i	$-0.597703\pi$
$199$	−1696.40	−0.604293	−0.302147	+	0.953261i	$0.597703\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−848.198	−0.293260
$204$	0	0
$205$	360.000	0.122651
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	5900.51	1.92516	0.962578	−	0.271006i	$-0.0873564\pi$
$211$	5900.51	1.92516	0.962578	+	0.271006i	$0.0873564\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	4760.00	1.48908
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3432.00	−1.04462
$222$	0	0
$223$	−2452.40	−0.736434	−0.368217	−	0.929740i	$-0.620032\pi$
$223$	−2452.40	−0.736434	−0.368217	+	0.929740i	$0.620032\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	811.320	0.237221	0.118611	−	0.992941i	$-0.462156\pi$
$227$	811.320	0.237221	0.118611	+	0.992941i	$0.462156\pi$
$228$	0	0
$229$	6190.00	1.78623	0.893115	−	0.449828i	$-0.148515\pi$
$229$	6190.00	1.78623	0.893115	+	0.449828i	$0.148515\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	798.000	0.224372	0.112186	−	0.993687i	$-0.464215\pi$
$233$	798.000	0.224372	0.112186	+	0.993687i	$0.464215\pi$
$234$	0	0
$235$	1659.52	0.460660
$236$	0	0
$237$	0	0
$238$	0	0
$239$	3208.40	0.868344	0.434172	−	0.900830i	$-0.357041\pi$
$239$	3208.40	0.868344	0.434172	+	0.900830i	$0.357041\pi$
$240$	0	0
$241$	4630.00	1.23753	0.618765	−	0.785576i	$-0.287633\pi$
$241$	4630.00	1.23753	0.618765	+	0.785576i	$0.287633\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.0000	0.00391149
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1198.54	0.301399	0.150700	−	0.988580i	$-0.451847\pi$
$251$	1198.54	0.301399	0.150700	+	0.988580i	$0.451847\pi$
$252$	0	0
$253$	2040.00	0.506932
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7586.00	1.84125	0.920626	−	0.390446i	$-0.127679\pi$
$257$	7586.00	1.84125	0.920626	+	0.390446i	$0.127679\pi$
$258$	0	0
$259$	442.538	0.106170
$260$	0	0
$261$	0	0
$262$	0	0
$263$	2913.38	0.683066	0.341533	−	0.939870i	$-0.389054\pi$
$263$	2913.38	0.683066	0.341533	+	0.939870i	$0.389054\pi$
$264$	0	0
$265$	−310.000	−0.0718609
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6650.00	−1.50728	−0.753639	−	0.657289i	$-0.771703\pi$
$269$	−6650.00	−1.50728	−0.753639	+	0.657289i	$0.771703\pi$
$270$	0	0
$271$	2950.25	0.661311	0.330655	−	0.943752i	$-0.392730\pi$
$271$	2950.25	0.661311	0.330655	+	0.943752i	$0.392730\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1382.93	0.303251
$276$	0	0
$277$	2536.00	0.550085	0.275042	−	0.961432i	$-0.411308\pi$
$277$	2536.00	0.550085	0.275042	+	0.961432i	$0.411308\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	532.000	0.112941	0.0564706	−	0.998404i	$-0.482015\pi$
$281$	532.000	0.112941	0.0564706	+	0.998404i	$0.482015\pi$
$282$	0	0
$283$	2692.11	0.565474	0.282737	−	0.959197i	$-0.408758\pi$
$283$	2692.11	0.565474	0.282737	+	0.959197i	$0.408758\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1327.61	0.273054
$288$	0	0
$289$	−557.000	−0.113373
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−6918.00	−1.37936	−0.689682	−	0.724112i	$-0.742250\pi$
$293$	−6918.00	−1.37936	−0.689682	+	0.724112i	$0.742250\pi$
$294$	0	0
$295$	1198.54	0.236548
$296$	0	0
$297$	0	0
$298$	0	0
$299$	1917.67	0.370908
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	950.000	0.178350
$306$	0	0
$307$	−7449.39	−1.38488	−0.692442	−	0.721474i	$-0.743465\pi$
$307$	−7449.39	−1.38488	−0.692442	+	0.721474i	$0.743465\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3577.18	−0.652230	−0.326115	−	0.945330i	$-0.605740\pi$
$311$	−3577.18	−0.652230	−0.326115	+	0.945330i	$0.605740\pi$
$312$	0	0
$313$	3158.00	0.570290	0.285145	−	0.958484i	$-0.407958\pi$
$313$	3158.00	0.570290	0.285145	+	0.958484i	$0.407958\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2634.00	0.466688	0.233344	−	0.972394i	$-0.425033\pi$
$317$	2634.00	0.466688	0.233344	+	0.972394i	$0.425033\pi$
$318$	0	0
$319$	2544.59	0.446614
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	1300.00	0.221880
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6120.00	1.02555
$330$	0	0
$331$	7818.17	1.29826	0.649132	−	0.760675i	$-0.275132\pi$
$331$	7818.17	1.29826	0.649132	+	0.760675i	$0.275132\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3872.21	−0.631526
$336$	0	0
$337$	−2866.00	−0.463267	−0.231633	−	0.972803i	$-0.574407\pi$
$337$	−2866.00	−0.463267	−0.231633	+	0.972803i	$0.574407\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−14280.0	−2.26776
$342$	0	0
$343$	6379.92	1.00433
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5900.51	0.912841	0.456421	−	0.889764i	$-0.349131\pi$
$347$	5900.51	0.912841	0.456421	+	0.889764i	$0.349131\pi$
$348$	0	0
$349$	−390.000	−0.0598173	−0.0299086	−	0.999553i	$-0.509522\pi$
$349$	−390.000	−0.0598173	−0.0299086	+	0.999553i	$0.509522\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	8742.00	1.31810	0.659051	−	0.752099i	$-0.270958\pi$
$353$	8742.00	1.31810	0.659051	+	0.752099i	$0.270958\pi$
$354$	0	0
$355$	3134.65	0.468647
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−4867.92	−0.715652	−0.357826	−	0.933788i	$-0.616482\pi$
$359$	−4867.92	−0.715652	−0.357826	+	0.933788i	$0.616482\pi$
$360$	0	0
$361$	−6859.00	−1.00000
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5390.00	−0.772947
$366$	0	0
$367$	−2157.37	−0.306850	−0.153425	−	0.988160i	$-0.549030\pi$
$367$	−2157.37	−0.306850	−0.153425	+	0.988160i	$0.549030\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1143.22	−0.159982
$372$	0	0
$373$	8532.00	1.18437	0.592185	−	0.805802i	$-0.298265\pi$
$373$	8532.00	1.18437	0.592185	+	0.805802i	$0.298265\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2392.00	0.326775
$378$	0	0
$379$	−12022.3	−1.62940	−0.814701	−	0.579881i	$-0.803099\pi$
$379$	−12022.3	−1.62940	−0.814701	+	0.579881i	$0.803099\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−13239.3	−1.76631	−0.883153	−	0.469086i	$-0.844583\pi$
$383$	−13239.3	−1.76631	−0.883153	+	0.469086i	$0.844583\pi$
$384$	0	0
$385$	5100.00	0.675117
$386$	0	0
$387$	0	0
$388$	0	0
$389$	6250.00	0.814621	0.407311	−	0.913290i	$-0.366467\pi$
$389$	6250.00	0.814621	0.407311	+	0.913290i	$0.366467\pi$
$390$	0	0
$391$	−2433.96	−0.314810
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−4240.99	−0.540221
$396$	0	0
$397$	−3544.00	−0.448031	−0.224015	−	0.974586i	$-0.571917\pi$
$397$	−3544.00	−0.448031	−0.224015	+	0.974586i	$0.571917\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8940.00	1.11332	0.556661	−	0.830740i	$-0.312082\pi$
$401$	8940.00	1.11332	0.556661	+	0.830740i	$0.312082\pi$
$402$	0	0
$403$	−13423.7	−1.65926
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1327.61	−0.161689
$408$	0	0
$409$	6794.00	0.821373	0.410687	−	0.911777i	$-0.365289\pi$
$409$	6794.00	0.821373	0.410687	+	0.911777i	$0.365289\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4420.00	0.526620
$414$	0	0
$415$	−3872.21	−0.458022
$416$	0	0
$417$	0	0
$418$	0	0
$419$	5402.65	0.629921	0.314960	−	0.949105i	$-0.398009\pi$
$419$	5402.65	0.629921	0.314960	+	0.949105i	$0.398009\pi$
$420$	0	0
$421$	1710.00	0.197958	0.0989790	−	0.995090i	$-0.468442\pi$
$421$	1710.00	0.197958	0.0989790	+	0.995090i	$0.468442\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−1650.00	−0.188322
$426$	0	0
$427$	3503.43	0.397056
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13497.4	1.50846	0.754231	−	0.656609i	$-0.228010\pi$
$431$	13497.4	1.50846	0.754231	+	0.656609i	$0.228010\pi$
$432$	0	0
$433$	−8058.00	−0.894325	−0.447163	−	0.894453i	$-0.647565\pi$
$433$	−8058.00	−0.894325	−0.447163	+	0.894453i	$0.647565\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	5457.97	0.593382	0.296691	−	0.954974i	$-0.404117\pi$
$439$	5457.97	0.593382	0.296691	+	0.954974i	$0.404117\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−8703.25	−0.933417	−0.466709	−	0.884411i	$-0.654560\pi$
$443$	−8703.25	−0.933417	−0.466709	+	0.884411i	$0.654560\pi$
$444$	0	0
$445$	−5580.00	−0.594421
$446$	0	0
$447$	0	0
$448$	0	0
$449$	11820.0	1.24236	0.621181	−	0.783667i	$-0.286653\pi$
$449$	11820.0	1.24236	0.621181	+	0.783667i	$0.286653\pi$
$450$	0	0
$451$	−3982.84	−0.415842
$452$	0	0
$453$	0	0
$454$	0	0
$455$	4794.16	0.493964
$456$	0	0
$457$	7226.00	0.739646	0.369823	−	0.929102i	$-0.379418\pi$
$457$	7226.00	0.739646	0.369823	+	0.929102i	$0.379418\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15750.0	−1.59122	−0.795608	−	0.605812i	$-0.792848\pi$
$461$	−15750.0	−1.59122	−0.795608	+	0.605812i	$0.792848\pi$
$462$	0	0
$463$	−5882.07	−0.590417	−0.295208	−	0.955433i	$-0.595389\pi$
$463$	−5882.07	−0.590417	−0.295208	+	0.955433i	$0.595389\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6306.17	−0.624871	−0.312435	−	0.949939i	$-0.601145\pi$
$467$	−6306.17	−0.624871	−0.312435	+	0.949939i	$0.601145\pi$
$468$	0	0
$469$	−14280.0	−1.40595
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−774.442	−0.0738730	−0.0369365	−	0.999318i	$-0.511760\pi$
$479$	−774.442	−0.0738730	−0.0369365	+	0.999318i	$0.511760\pi$
$480$	0	0
$481$	−1248.00	−0.118303
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−4170.00	−0.390412
$486$	0	0
$487$	8942.96	0.832123	0.416062	−	0.909336i	$-0.363410\pi$
$487$	8942.96	0.832123	0.416062	+	0.909336i	$0.363410\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	20596.5	1.89309	0.946543	−	0.322577i	$-0.104549\pi$
$491$	20596.5	1.89309	0.946543	+	0.322577i	$0.104549\pi$
$492$	0	0
$493$	−3036.00	−0.277352
$494$	0	0
$495$	0	0
$496$	0	0
$497$	11560.0	1.04333
$498$	0	0
$499$	5900.51	0.529345	0.264672	−	0.964338i	$-0.414736\pi$
$499$	5900.51	0.529345	0.264672	+	0.964338i	$0.414736\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	110.635	0.00980706	0.00490353	−	0.999988i	$-0.498439\pi$
$503$	110.635	0.00980706	0.00490353	+	0.999988i	$0.498439\pi$
$504$	0	0
$505$	−1350.00	−0.118959
$506$	0	0
$507$	0	0
$508$	0	0
$509$	214.000	0.0186353	0.00931767	−	0.999957i	$-0.497034\pi$
$509$	214.000	0.0186353	0.00931767	+	0.999957i	$0.497034\pi$
$510$	0	0
$511$	−19877.3	−1.72079
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2858.06	−0.244546
$516$	0	0
$517$	−18360.0	−1.56184
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−16780.0	−1.41103	−0.705514	−	0.708696i	$-0.749284\pi$
$521$	−16780.0	−1.41103	−0.705514	+	0.708696i	$0.749284\pi$
$522$	0	0
$523$	17443.4	1.45840	0.729202	−	0.684298i	$-0.239891\pi$
$523$	17443.4	1.45840	0.729202	+	0.684298i	$0.239891\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	17037.7	1.40830
$528$	0	0
$529$	−10807.0	−0.888222
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−3744.00	−0.304260
$534$	0	0
$535$	−5347.34	−0.432122
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−165.952	−0.0132617
$540$	0	0
$541$	−13798.0	−1.09653	−0.548265	−	0.836305i	$-0.684711\pi$
$541$	−13798.0	−1.09653	−0.548265	+	0.836305i	$0.684711\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8730.00	−0.686151
$546$	0	0
$547$	11801.0	0.922441	0.461220	−	0.887286i	$-0.347412\pi$
$547$	11801.0	0.922441	0.461220	+	0.887286i	$0.347412\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−15640.0	−1.20268
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−8994.00	−0.684180	−0.342090	−	0.939667i	$-0.611135\pi$
$557$	−8994.00	−0.684180	−0.342090	+	0.939667i	$0.611135\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	7449.39	0.557646	0.278823	−	0.960343i	$-0.410056\pi$
$563$	7449.39	0.557646	0.278823	+	0.960343i	$0.410056\pi$
$564$	0	0
$565$	−4410.00	−0.328372
$566$	0	0
$567$	0	0
$568$	0	0
$569$	16180.0	1.19209	0.596046	−	0.802950i	$-0.296737\pi$
$569$	16180.0	1.19209	0.596046	+	0.802950i	$0.296737\pi$
$570$	0	0
$571$	−7818.17	−0.572995	−0.286498	−	0.958081i	$-0.592491\pi$
$571$	−7818.17	−0.572995	−0.286498	+	0.958081i	$0.592491\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	921.954	0.0668664
$576$	0	0
$577$	−8486.00	−0.612265	−0.306132	−	0.951989i	$-0.599035\pi$
$577$	−8486.00	−0.612265	−0.306132	+	0.951989i	$0.599035\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−14280.0	−1.01968
$582$	0	0
$583$	3429.67	0.243641
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20909.9	−1.47026	−0.735132	−	0.677924i	$-0.762880\pi$
$587$	−20909.9	−1.47026	−0.735132	+	0.677924i	$0.762880\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21818.0	1.51089	0.755445	−	0.655212i	$-0.227421\pi$
$593$	21818.0	1.51089	0.755445	+	0.655212i	$0.227421\pi$
$594$	0	0
$595$	−6084.90	−0.419255
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6933.10	−0.472919	−0.236460	−	0.971641i	$-0.575987\pi$
$599$	−6933.10	−0.472919	−0.236460	+	0.971641i	$0.575987\pi$
$600$	0	0
$601$	−9210.00	−0.625098	−0.312549	−	0.949902i	$-0.601183\pi$
$601$	−9210.00	−0.625098	−0.312549	+	0.949902i	$0.601183\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−8645.00	−0.580941
$606$	0	0
$607$	9680.52	0.647315	0.323658	−	0.946174i	$-0.395087\pi$
$607$	9680.52	0.647315	0.323658	+	0.946174i	$0.395087\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−17259.0	−1.14276
$612$	0	0
$613$	−19312.0	−1.27244	−0.636219	−	0.771509i	$-0.719502\pi$
$613$	−19312.0	−1.27244	−0.636219	+	0.771509i	$0.719502\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	20854.0	1.36070	0.680349	−	0.732888i	$-0.261828\pi$
$617$	20854.0	1.36070	0.680349	+	0.732888i	$0.261828\pi$
$618$	0	0
$619$	23823.3	1.54691	0.773457	−	0.633849i	$-0.218526\pi$
$619$	23823.3	1.54691	0.773457	+	0.633849i	$0.218526\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−20578.0	−1.32334
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1584.00	0.100411
$630$	0	0
$631$	8150.08	0.514183	0.257091	−	0.966387i	$-0.417236\pi$
$631$	8150.08	0.514183	0.257091	+	0.966387i	$0.417236\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−10971.3	−0.685640
$636$	0	0
$637$	−156.000	−0.00970321
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−14328.0	−0.882874	−0.441437	−	0.897292i	$-0.645531\pi$
$641$	−14328.0	−0.882874	−0.441437	+	0.897292i	$0.645531\pi$
$642$	0	0
$643$	6306.17	0.386767	0.193383	−	0.981123i	$-0.438054\pi$
$643$	6306.17	0.386767	0.193383	+	0.981123i	$0.438054\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	25409.1	1.54395	0.771973	−	0.635655i	$-0.219270\pi$
$647$	25409.1	1.54395	0.771973	+	0.635655i	$0.219270\pi$
$648$	0	0
$649$	−13260.0	−0.802004
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−20918.0	−1.25358	−0.626788	−	0.779190i	$-0.715631\pi$
$653$	−20918.0	−1.25358	−0.626788	+	0.779190i	$0.715631\pi$
$654$	0	0
$655$	11340.0	0.676476
$656$	0	0
$657$	0	0
$658$	0	0
$659$	14769.7	0.873060	0.436530	−	0.899690i	$-0.356207\pi$
$659$	14769.7	0.873060	0.436530	+	0.899690i	$0.356207\pi$
$660$	0	0
$661$	17330.0	1.01976	0.509878	−	0.860247i	$-0.329690\pi$
$661$	17330.0	1.01976	0.509878	+	0.860247i	$0.329690\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1696.40	0.0984778
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10510.3	−0.604687
$672$	0	0
$673$	−21938.0	−1.25653	−0.628267	−	0.777998i	$-0.716236\pi$
$673$	−21938.0	−1.25653	−0.628267	+	0.777998i	$0.716236\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	25986.0	1.47522	0.737609	−	0.675228i	$-0.235955\pi$
$677$	25986.0	1.47522	0.737609	+	0.675228i	$0.235955\pi$
$678$	0	0
$679$	−15378.2	−0.869162
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−14050.6	−0.787161	−0.393580	−	0.919290i	$-0.628764\pi$
$683$	−14050.6	−0.787161	−0.393580	+	0.919290i	$0.628764\pi$
$684$	0	0
$685$	−7370.00	−0.411085
$686$	0	0
$687$	0	0
$688$	0	0
$689$	3224.00	0.178265
$690$	0	0
$691$	−21610.6	−1.18973	−0.594867	−	0.803824i	$-0.702795\pi$
$691$	−21610.6	−1.18973	−0.594867	+	0.803824i	$0.702795\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1106.35	0.0603828
$696$	0	0
$697$	4752.00	0.258242
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−31218.0	−1.68201	−0.841004	−	0.541029i	$-0.818035\pi$
$701$	−31218.0	−1.68201	−0.841004	+	0.541029i	$0.818035\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4978.55	−0.264834
$708$	0	0
$709$	−5010.00	−0.265380	−0.132690	−	0.991158i	$-0.542361\pi$
$709$	−5010.00	−0.265380	−0.132690	+	0.991158i	$0.542361\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−9520.00	−0.500038
$714$	0	0
$715$	−14382.5	−0.752272
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−6416.80	−0.332832	−0.166416	−	0.986056i	$-0.553219\pi$
$719$	−6416.80	−0.332832	−0.166416	+	0.986056i	$0.553219\pi$
$720$	0	0
$721$	−10540.0	−0.544425
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1150.00	0.0589102
$726$	0	0
$727$	−1530.44	−0.0780757	−0.0390379	−	0.999238i	$-0.512429\pi$
$727$	−1530.44	−0.0780757	−0.0390379	+	0.999238i	$0.512429\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	8848.00	0.445850	0.222925	−	0.974836i	$-0.428439\pi$
$733$	8848.00	0.445850	0.222925	+	0.974836i	$0.428439\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	42840.0	2.14116
$738$	0	0
$739$	−2728.99	−0.135842	−0.0679211	−	0.997691i	$-0.521637\pi$
$739$	−2728.99	−0.135842	−0.0679211	+	0.997691i	$0.521637\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	38316.4	1.89192	0.945958	−	0.324290i	$-0.105125\pi$
$743$	38316.4	1.89192	0.945958	+	0.324290i	$0.105125\pi$
$744$	0	0
$745$	2150.00	0.105731
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−19720.0	−0.962020
$750$	0	0
$751$	8076.32	0.392422	0.196211	−	0.980562i	$-0.437136\pi$
$751$	8076.32	0.392422	0.196211	+	0.980562i	$0.437136\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	14751.3	0.711065
$756$	0	0
$757$	−39704.0	−1.90630	−0.953148	−	0.302504i	$-0.902178\pi$
$757$	−39704.0	−1.90630	−0.953148	+	0.302504i	$0.902178\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−39220.0	−1.86823	−0.934116	−	0.356971i	$-0.883810\pi$
$761$	−39220.0	−1.86823	−0.934116	+	0.356971i	$0.883810\pi$
$762$	0	0
$763$	−32194.6	−1.52756
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−12464.8	−0.586804
$768$	0	0
$769$	−17890.0	−0.838921	−0.419460	−	0.907774i	$-0.637781\pi$
$769$	−17890.0	−0.838921	−0.419460	+	0.907774i	$0.637781\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	23778.0	1.10638	0.553192	−	0.833054i	$-0.313409\pi$
$773$	23778.0	1.10638	0.553192	+	0.833054i	$0.313409\pi$
$774$	0	0
$775$	−6453.68	−0.299127
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−34680.0	−1.58892
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−9420.00	−0.428298
$786$	0	0
$787$	−18328.5	−0.830164	−0.415082	−	0.909784i	$-0.636247\pi$
$787$	−18328.5	−0.830164	−0.415082	+	0.909784i	$0.636247\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−16263.3	−0.731044
$792$	0	0
$793$	−9880.00	−0.442433
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34806.0	1.54692	0.773458	−	0.633848i	$-0.218525\pi$
$797$	34806.0	1.54692	0.773458	+	0.633848i	$0.218525\pi$
$798$	0	0
$799$	21905.6	0.969920
$800$	0	0
$801$	0	0
$802$	0	0
$803$	59632.0	2.62063
$804$	0	0
$805$	3400.00	0.148862
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3544.00	0.154018	0.0770089	−	0.997030i	$-0.475463\pi$
$809$	3544.00	0.154018	0.0770089	+	0.997030i	$0.475463\pi$
$810$	0	0
$811$	12759.8	0.552477	0.276239	−	0.961089i	$-0.410912\pi$
$811$	12759.8	0.552477	0.276239	+	0.961089i	$0.410912\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	2028.30	0.0871758
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	1302.00	0.0553473	0.0276736	−	0.999617i	$-0.491190\pi$
$821$	1302.00	0.0553473	0.0276736	+	0.999617i	$0.491190\pi$
$822$	0	0
$823$	−6619.63	−0.280372	−0.140186	−	0.990125i	$-0.544770\pi$
$823$	−6619.63	−0.280372	−0.140186	+	0.990125i	$0.544770\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	19840.5	0.834245	0.417122	−	0.908850i	$-0.363039\pi$
$827$	19840.5	0.834245	0.417122	+	0.908850i	$0.363039\pi$
$828$	0	0
$829$	18314.0	0.767276	0.383638	−	0.923484i	$-0.374671\pi$
$829$	18314.0	0.767276	0.383638	+	0.923484i	$0.374671\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	198.000	0.00823565
$834$	0	0
$835$	20467.4	0.848267
$836$	0	0
$837$	0	0
$838$	0	0
$839$	3466.55	0.142644	0.0713221	−	0.997453i	$-0.477278\pi$
$839$	3466.55	0.142644	0.0713221	+	0.997453i	$0.477278\pi$
$840$	0	0
$841$	−22273.0	−0.913240
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−2535.00	−0.103203
$846$	0	0
$847$	−31881.2	−1.29333
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−885.076	−0.0356522
$852$	0	0
$853$	3728.00	0.149642	0.0748208	−	0.997197i	$-0.476162\pi$
$853$	3728.00	0.149642	0.0748208	+	0.997197i	$0.476162\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23966.0	0.955266	0.477633	−	0.878559i	$-0.341495\pi$
$857$	23966.0	0.955266	0.477633	+	0.878559i	$0.341495\pi$
$858$	0	0
$859$	−47867.9	−1.90132	−0.950658	−	0.310240i	$-0.899591\pi$
$859$	−47867.9	−1.90132	−0.950658	+	0.310240i	$0.899591\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−19287.3	−0.760773	−0.380386	−	0.924828i	$-0.624209\pi$
$863$	−19287.3	−0.760773	−0.380386	+	0.924828i	$0.624209\pi$
$864$	0	0
$865$	5810.00	0.228377
$866$	0	0
$867$	0	0
$868$	0	0
$869$	46920.0	1.83159
$870$	0	0
$871$	40271.0	1.56662
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2304.89	0.0890507
$876$	0	0
$877$	−6236.00	−0.240108	−0.120054	−	0.992767i	$-0.538307\pi$
$877$	−6236.00	−0.240108	−0.120054	+	0.992767i	$0.538307\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−44532.0	−1.70298	−0.851488	−	0.524375i	$-0.824299\pi$
$881$	−44532.0	−1.70298	−0.851488	+	0.524375i	$0.824299\pi$
$882$	0	0
$883$	26957.9	1.02741	0.513707	−	0.857965i	$-0.328272\pi$
$883$	26957.9	1.02741	0.513707	+	0.857965i	$0.328272\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−51961.4	−1.96696	−0.983479	−	0.181023i	$-0.942059\pi$
$887$	−51961.4	−1.96696	−0.983479	+	0.181023i	$0.942059\pi$
$888$	0	0
$889$	−40460.0	−1.52642
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	19637.6	0.733423
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−11874.8	−0.440540
$900$	0	0
$901$	−4092.00	−0.151303
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17690.0	−0.649763
$906$	0	0
$907$	16005.1	0.585933	0.292967	−	0.956123i	$-0.405357\pi$
$907$	16005.1	0.585933	0.292967	+	0.956123i	$0.405357\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21905.6	0.796670	0.398335	−	0.917240i	$-0.369588\pi$
$911$	21905.6	0.796670	0.398335	+	0.917240i	$0.369588\pi$
$912$	0	0
$913$	42840.0	1.55290
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41820.0	1.50602
$918$	0	0
$919$	−52035.1	−1.86777	−0.933885	−	0.357575i	$-0.883604\pi$
$919$	−52035.1	−1.86777	−0.933885	+	0.357575i	$0.883604\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−32600.3	−1.16257
$924$	0	0
$925$	−600.000	−0.0213274
$926$	0	0
$927$	0	0
$928$	0	0
$929$	12460.0	0.440042	0.220021	−	0.975495i	$-0.429387\pi$
$929$	12460.0	0.440042	0.220021	+	0.975495i	$0.429387\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18254.7	0.638495
$936$	0	0
$937$	25834.0	0.900704	0.450352	−	0.892851i	$-0.351298\pi$
$937$	25834.0	0.900704	0.450352	+	0.892851i	$0.351298\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30270.0	−1.04864	−0.524322	−	0.851520i	$-0.675681\pi$
$941$	−30270.0	−1.04864	−0.524322	+	0.851520i	$0.675681\pi$
$942$	0	0
$943$	−2655.23	−0.0916926
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−23602.0	−0.809887	−0.404944	−	0.914342i	$-0.632709\pi$
$947$	−23602.0	−0.809887	−0.404944	+	0.914342i	$0.632709\pi$
$948$	0	0
$949$	56056.0	1.91744
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−6758.00	−0.229709	−0.114855	−	0.993382i	$-0.536640\pi$
$953$	−6758.00	−0.229709	−0.114855	+	0.993382i	$0.536640\pi$
$954$	0	0
$955$	−2028.30	−0.0687270
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−27179.2	−0.915185
$960$	0	0
$961$	36849.0	1.23692
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−12190.0	−0.406642
$966$	0	0
$967$	−47370.0	−1.57530	−0.787651	−	0.616122i	$-0.788703\pi$
$967$	−47370.0	−1.57530	−0.787651	+	0.616122i	$0.788703\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17314.3	−0.572237	−0.286119	−	0.958194i	$-0.592365\pi$
$971$	−17314.3	−0.572237	−0.286119	+	0.958194i	$0.592365\pi$
$972$	0	0
$973$	4080.00	0.134428
$974$	0	0
$975$	0	0
$976$	0	0
$977$	28206.0	0.923633	0.461817	−	0.886975i	$-0.347198\pi$
$977$	28206.0	0.923633	0.461817	+	0.886975i	$0.347198\pi$
$978$	0	0
$979$	61734.1	2.01535
$980$	0	0
$981$	0	0
$982$	0	0
$983$	35218.7	1.14273	0.571364	−	0.820697i	$-0.306415\pi$
$983$	35218.7	1.14273	0.571364	+	0.820697i	$0.306415\pi$
$984$	0	0
$985$	−2370.00	−0.0766644
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	36804.4	1.17975	0.589874	−	0.807495i	$-0.299177\pi$
$991$	36804.4	1.17975	0.589874	+	0.807495i	$0.299177\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	8481.98	0.270248
$996$	0	0
$997$	−8076.00	−0.256539	−0.128270	−	0.991739i	$-0.540942\pi$
$997$	−8076.00	−0.256539	−0.128270	+	0.991739i	$0.540942\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1440.4.a.w.1.1	✓	2
3.2	odd	2		1440.4.a.be.1.1	yes	2
4.3	odd	2	inner	1440.4.a.w.1.2	yes	2
12.11	even	2		1440.4.a.be.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1440.4.a.w.1.1	✓	2	1.1	even	1	trivial
1440.4.a.w.1.2	yes	2	4.3	odd	2	inner
1440.4.a.be.1.1	yes	2	3.2	odd	2
1440.4.a.be.1.2	yes	2	12.11	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:	\( N \)	\(=\)	\( 1440 = 2^{5} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1440.a (trivial)

\(f(q)\)	\(=\)	\(q-5.00000 q^{5} -18.4391 q^{7} +55.3173 q^{11} +52.0000 q^{13} -66.0000 q^{17} +36.8782 q^{23} +25.0000 q^{25} +46.0000 q^{29} -258.147 q^{31} +92.1954 q^{35} -24.0000 q^{37} -72.0000 q^{41} -331.904 q^{47} -3.00000 q^{49} +62.0000 q^{53} -276.586 q^{55} -239.708 q^{59} -190.000 q^{61} -260.000 q^{65} +774.442 q^{67} -626.929 q^{71} +1078.00 q^{73} -1020.00 q^{77} +848.198 q^{79} +774.442 q^{83} +330.000 q^{85} +1116.00 q^{89} -958.833 q^{91} +834.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 10 q^{5} + 104 q^{13} - 132 q^{17} + 50 q^{25} + 92 q^{29} - 48 q^{37} - 144 q^{41} - 6 q^{49} + 124 q^{53} - 380 q^{61} - 520 q^{65} + 2156 q^{73} - 2040 q^{77} + 660 q^{85} + 2232 q^{89} + 1668 q^{97}+O(q^{100}) \) 2 * q - 10 * q^5 + 104 * q^13 - 132 * q^17 + 50 * q^25 + 92 * q^29 - 48 * q^37 - 144 * q^41 - 6 * q^49 + 124 * q^53 - 380 * q^61 - 520 * q^65 + 2156 * q^73 - 2040 * q^77 + 660 * q^85 + 2232 * q^89 + 1668 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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