LMFDB - Embedded newform 1600.4.a.by.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 200)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1600.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+9.00000 q^{3} -26.0000 q^{7} +54.0000 q^{9} -59.0000 q^{11} -28.0000 q^{13} +5.00000 q^{17} +109.000 q^{19} -234.000 q^{21} +194.000 q^{23} +243.000 q^{27} +32.0000 q^{29} -10.0000 q^{31} -531.000 q^{33} +198.000 q^{37} -252.000 q^{39} +117.000 q^{41} +388.000 q^{43} +68.0000 q^{47} +333.000 q^{49} +45.0000 q^{51} +18.0000 q^{53} +981.000 q^{57} +392.000 q^{59} +710.000 q^{61} -1404.00 q^{63} -253.000 q^{67} +1746.00 q^{69} +612.000 q^{71} -549.000 q^{73} +1534.00 q^{77} -414.000 q^{79} +729.000 q^{81} -121.000 q^{83} +288.000 q^{87} -81.0000 q^{89} +728.000 q^{91} -90.0000 q^{93} -1502.00 q^{97} -3186.00 q^{99} +O(q^{100})

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
$3$	9.00000	1.73205	0.866025	−	0.500000i	$-0.166667\pi$
$3$	9.00000	1.73205	0.866025	+	0.500000i	$0.166667\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−26.0000	−1.40387	−0.701934	−	0.712242i	$-0.747680\pi$
$7$	−26.0000	−1.40387	−0.701934	+	0.712242i	$0.747680\pi$
$8$	0	0
$9$	54.0000	2.00000
$10$	0	0
$11$	−59.0000	−1.61720	−0.808599	−	0.588361i	$-0.799774\pi$
$11$	−59.0000	−1.61720	−0.808599	+	0.588361i	$0.799774\pi$
$12$	0	0
$13$	−28.0000	−0.597369	−0.298685	−	0.954352i	$-0.596548\pi$
$13$	−28.0000	−0.597369	−0.298685	+	0.954352i	$0.596548\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	5.00000	0.0713340	0.0356670	−	0.999364i	$-0.488644\pi$
$17$	5.00000	0.0713340	0.0356670	+	0.999364i	$0.488644\pi$
$18$	0	0
$19$	109.000	1.31612	0.658061	−	0.752965i	$-0.271377\pi$
$19$	109.000	1.31612	0.658061	+	0.752965i	$0.271377\pi$
$20$	0	0
$21$	−234.000	−2.43157
$22$	0	0
$23$	194.000	1.75877	0.879387	−	0.476108i	$-0.157953\pi$
$23$	194.000	1.75877	0.879387	+	0.476108i	$0.157953\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	243.000	1.73205
$28$	0	0
$29$	32.0000	0.204905	0.102453	−	0.994738i	$-0.467331\pi$
$29$	32.0000	0.204905	0.102453	+	0.994738i	$0.467331\pi$
$30$	0	0
$31$	−10.0000	−0.0579372	−0.0289686	−	0.999580i	$-0.509222\pi$
$31$	−10.0000	−0.0579372	−0.0289686	+	0.999580i	$0.509222\pi$
$32$	0	0
$33$	−531.000	−2.80107
$34$	0	0
$35$	0	0
$36$	0	0
$37$	198.000	0.879757	0.439878	−	0.898057i	$-0.355022\pi$
$37$	198.000	0.879757	0.439878	+	0.898057i	$0.355022\pi$
$38$	0	0
$39$	−252.000	−1.03467
$40$	0	0
$41$	117.000	0.445667	0.222833	−	0.974857i	$-0.428469\pi$
$41$	117.000	0.445667	0.222833	+	0.974857i	$0.428469\pi$
$42$	0	0
$43$	388.000	1.37603	0.688017	−	0.725695i	$-0.258482\pi$
$43$	388.000	1.37603	0.688017	+	0.725695i	$0.258482\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	68.0000	0.211039	0.105519	−	0.994417i	$-0.466350\pi$
$47$	68.0000	0.211039	0.105519	+	0.994417i	$0.466350\pi$
$48$	0	0
$49$	333.000	0.970845
$50$	0	0
$51$	45.0000	0.123554
$52$	0	0
$53$	18.0000	0.0466508	0.0233254	−	0.999728i	$-0.492575\pi$
$53$	18.0000	0.0466508	0.0233254	+	0.999728i	$0.492575\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	981.000	2.27959
$58$	0	0
$59$	392.000	0.864984	0.432492	−	0.901638i	$-0.357634\pi$
$59$	392.000	0.864984	0.432492	+	0.901638i	$0.357634\pi$
$60$	0	0
$61$	710.000	1.49027	0.745133	−	0.666916i	$-0.232386\pi$
$61$	710.000	1.49027	0.745133	+	0.666916i	$0.232386\pi$
$62$	0	0
$63$	−1404.00	−2.80774
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−253.000	−0.461326	−0.230663	−	0.973034i	$-0.574090\pi$
$67$	−253.000	−0.461326	−0.230663	+	0.973034i	$0.574090\pi$
$68$	0	0
$69$	1746.00	3.04629
$70$	0	0
$71$	612.000	1.02297	0.511486	−	0.859292i	$-0.329095\pi$
$71$	612.000	1.02297	0.511486	+	0.859292i	$0.329095\pi$
$72$	0	0
$73$	−549.000	−0.880214	−0.440107	−	0.897945i	$-0.645059\pi$
$73$	−549.000	−0.880214	−0.440107	+	0.897945i	$0.645059\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1534.00	2.27033
$78$	0	0
$79$	−414.000	−0.589603	−0.294802	−	0.955559i	$-0.595254\pi$
$79$	−414.000	−0.589603	−0.294802	+	0.955559i	$0.595254\pi$
$80$	0	0
$81$	729.000	1.00000
$82$	0	0
$83$	−121.000	−0.160018	−0.0800089	−	0.996794i	$-0.525495\pi$
$83$	−121.000	−0.160018	−0.0800089	+	0.996794i	$0.525495\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	288.000	0.354906
$88$	0	0
$89$	−81.0000	−0.0964717	−0.0482359	−	0.998836i	$-0.515360\pi$
$89$	−81.0000	−0.0964717	−0.0482359	+	0.998836i	$0.515360\pi$
$90$	0	0
$91$	728.000	0.838628
$92$	0	0
$93$	−90.0000	−0.100350
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−1502.00	−1.57222	−0.786108	−	0.618089i	$-0.787907\pi$
$97$	−1502.00	−1.57222	−0.786108	+	0.618089i	$0.787907\pi$
$98$	0	0
$99$	−3186.00	−3.23439
$100$	0	0
$101$	234.000	0.230533	0.115267	−	0.993335i	$-0.463228\pi$
$101$	234.000	0.230533	0.115267	+	0.993335i	$0.463228\pi$
$102$	0	0
$103$	1172.00	1.12117	0.560585	−	0.828097i	$-0.310576\pi$
$103$	1172.00	1.12117	0.560585	+	0.828097i	$0.310576\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1125.00	1.01643	0.508214	−	0.861231i	$-0.330306\pi$
$107$	1125.00	1.01643	0.508214	+	0.861231i	$0.330306\pi$
$108$	0	0
$109$	1234.00	1.08436	0.542182	−	0.840261i	$-0.317598\pi$
$109$	1234.00	1.08436	0.542182	+	0.840261i	$0.317598\pi$
$110$	0	0
$111$	1782.00	1.52378
$112$	0	0
$113$	567.000	0.472025	0.236013	−	0.971750i	$-0.424159\pi$
$113$	567.000	0.472025	0.236013	+	0.971750i	$0.424159\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−1512.00	−1.19474
$118$	0	0
$119$	−130.000	−0.100144
$120$	0	0
$121$	2150.00	1.61533
$122$	0	0
$123$	1053.00	0.771917
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2358.00	1.64755	0.823774	−	0.566918i	$-0.191864\pi$
$127$	2358.00	1.64755	0.823774	+	0.566918i	$0.191864\pi$
$128$	0	0
$129$	3492.00	2.38336
$130$	0	0
$131$	−1692.00	−1.12848	−0.564239	−	0.825611i	$-0.690831\pi$
$131$	−1692.00	−1.12848	−0.564239	+	0.825611i	$0.690831\pi$
$132$	0	0
$133$	−2834.00	−1.84766
$134$	0	0
$135$	0	0
$136$	0	0
$137$	229.000	0.142809	0.0714043	−	0.997447i	$-0.477252\pi$
$137$	229.000	0.142809	0.0714043	+	0.997447i	$0.477252\pi$
$138$	0	0
$139$	2781.00	1.69699	0.848494	−	0.529205i	$-0.177510\pi$
$139$	2781.00	1.69699	0.848494	+	0.529205i	$0.177510\pi$
$140$	0	0
$141$	612.000	0.365530
$142$	0	0
$143$	1652.00	0.966064
$144$	0	0
$145$	0	0
$146$	0	0
$147$	2997.00	1.68155
$148$	0	0
$149$	−1472.00	−0.809335	−0.404668	−	0.914464i	$-0.632613\pi$
$149$	−1472.00	−0.809335	−0.404668	+	0.914464i	$0.632613\pi$
$150$	0	0
$151$	−1322.00	−0.712469	−0.356235	−	0.934397i	$-0.615940\pi$
$151$	−1322.00	−0.712469	−0.356235	+	0.934397i	$0.615940\pi$
$152$	0	0
$153$	270.000	0.142668
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−298.000	−0.151484	−0.0757420	−	0.997127i	$-0.524133\pi$
$157$	−298.000	−0.151484	−0.0757420	+	0.997127i	$0.524133\pi$
$158$	0	0
$159$	162.000	0.0808015
$160$	0	0
$161$	−5044.00	−2.46909
$162$	0	0
$163$	−341.000	−0.163860	−0.0819300	−	0.996638i	$-0.526108\pi$
$163$	−341.000	−0.163860	−0.0819300	+	0.996638i	$0.526108\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−684.000	−0.316943	−0.158472	−	0.987364i	$-0.550657\pi$
$167$	−684.000	−0.316943	−0.158472	+	0.987364i	$0.550657\pi$
$168$	0	0
$169$	−1413.00	−0.643150
$170$	0	0
$171$	5886.00	2.63224
$172$	0	0
$173$	2344.00	1.03012	0.515061	−	0.857154i	$-0.327769\pi$
$173$	2344.00	1.03012	0.515061	+	0.857154i	$0.327769\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	3528.00	1.49820
$178$	0	0
$179$	−1111.00	−0.463911	−0.231955	−	0.972726i	$-0.574512\pi$
$179$	−1111.00	−0.463911	−0.231955	+	0.972726i	$0.574512\pi$
$180$	0	0
$181$	−2042.00	−0.838567	−0.419284	−	0.907855i	$-0.637719\pi$
$181$	−2042.00	−0.838567	−0.419284	+	0.907855i	$0.637719\pi$
$182$	0	0
$183$	6390.00	2.58122
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−295.000	−0.115361
$188$	0	0
$189$	−6318.00	−2.43157
$190$	0	0
$191$	−5270.00	−1.99646	−0.998230	−	0.0594735i	$-0.981058\pi$
$191$	−5270.00	−1.99646	−0.998230	+	0.0594735i	$0.981058\pi$
$192$	0	0
$193$	613.000	0.228625	0.114313	−	0.993445i	$-0.463533\pi$
$193$	613.000	0.228625	0.114313	+	0.993445i	$0.463533\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1174.00	−0.424589	−0.212295	−	0.977206i	$-0.568094\pi$
$197$	−1174.00	−0.424589	−0.212295	+	0.977206i	$0.568094\pi$
$198$	0	0
$199$	−3428.00	−1.22113	−0.610564	−	0.791967i	$-0.709057\pi$
$199$	−3428.00	−1.22113	−0.610564	+	0.791967i	$0.709057\pi$
$200$	0	0
$201$	−2277.00	−0.799041
$202$	0	0
$203$	−832.000	−0.287660
$204$	0	0
$205$	0	0
$206$	0	0
$207$	10476.0	3.51755
$208$	0	0
$209$	−6431.00	−2.12843
$210$	0	0
$211$	2339.00	0.763144	0.381572	−	0.924339i	$-0.375383\pi$
$211$	2339.00	0.763144	0.381572	+	0.924339i	$0.375383\pi$
$212$	0	0
$213$	5508.00	1.77184
$214$	0	0
$215$	0	0
$216$	0	0
$217$	260.000	0.0813362
$218$	0	0
$219$	−4941.00	−1.52457
$220$	0	0
$221$	−140.000	−0.0426128
$222$	0	0
$223$	−3932.00	−1.18075	−0.590373	−	0.807131i	$-0.701019\pi$
$223$	−3932.00	−1.18075	−0.590373	+	0.807131i	$0.701019\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	6084.00	1.77890	0.889448	−	0.457037i	$-0.151089\pi$
$227$	6084.00	1.77890	0.889448	+	0.457037i	$0.151089\pi$
$228$	0	0
$229$	−4996.00	−1.44168	−0.720841	−	0.693101i	$-0.756244\pi$
$229$	−4996.00	−1.44168	−0.720841	+	0.693101i	$0.756244\pi$
$230$	0	0
$231$	13806.0	3.93233
$232$	0	0
$233$	3222.00	0.905924	0.452962	−	0.891530i	$-0.350367\pi$
$233$	3222.00	0.905924	0.452962	+	0.891530i	$0.350367\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−3726.00	−1.02122
$238$	0	0
$239$	−2736.00	−0.740490	−0.370245	−	0.928934i	$-0.620726\pi$
$239$	−2736.00	−0.740490	−0.370245	+	0.928934i	$0.620726\pi$
$240$	0	0
$241$	1673.00	0.447168	0.223584	−	0.974685i	$-0.428224\pi$
$241$	1673.00	0.447168	0.223584	+	0.974685i	$0.428224\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3052.00	−0.786211
$248$	0	0
$249$	−1089.00	−0.277159
$250$	0	0
$251$	5355.00	1.34663	0.673316	−	0.739355i	$-0.264869\pi$
$251$	5355.00	1.34663	0.673316	+	0.739355i	$0.264869\pi$
$252$	0	0
$253$	−11446.0	−2.84428
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−5490.00	−1.33252	−0.666258	−	0.745721i	$-0.732105\pi$
$257$	−5490.00	−1.33252	−0.666258	+	0.745721i	$0.732105\pi$
$258$	0	0
$259$	−5148.00	−1.23506
$260$	0	0
$261$	1728.00	0.409810
$262$	0	0
$263$	−3150.00	−0.738545	−0.369272	−	0.929321i	$-0.620393\pi$
$263$	−3150.00	−0.738545	−0.369272	+	0.929321i	$0.620393\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−729.000	−0.167094
$268$	0	0
$269$	−176.000	−0.0398919	−0.0199459	−	0.999801i	$-0.506349\pi$
$269$	−176.000	−0.0398919	−0.0199459	+	0.999801i	$0.506349\pi$
$270$	0	0
$271$	−2394.00	−0.536624	−0.268312	−	0.963332i	$-0.586466\pi$
$271$	−2394.00	−0.536624	−0.268312	+	0.963332i	$0.586466\pi$
$272$	0	0
$273$	6552.00	1.45255
$274$	0	0
$275$	0	0
$276$	0	0
$277$	6256.00	1.35699	0.678496	−	0.734604i	$-0.262632\pi$
$277$	6256.00	1.35699	0.678496	+	0.734604i	$0.262632\pi$
$278$	0	0
$279$	−540.000	−0.115874
$280$	0	0
$281$	−4802.00	−1.01944	−0.509721	−	0.860340i	$-0.670251\pi$
$281$	−4802.00	−1.01944	−0.509721	+	0.860340i	$0.670251\pi$
$282$	0	0
$283$	2123.00	0.445934	0.222967	−	0.974826i	$-0.428426\pi$
$283$	2123.00	0.445934	0.222967	+	0.974826i	$0.428426\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3042.00	−0.625657
$288$	0	0
$289$	−4888.00	−0.994911
$290$	0	0
$291$	−13518.0	−2.72316
$292$	0	0
$293$	8834.00	1.76139	0.880696	−	0.473682i	$-0.157075\pi$
$293$	8834.00	1.76139	0.880696	+	0.473682i	$0.157075\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−14337.0	−2.80107
$298$	0	0
$299$	−5432.00	−1.05064
$300$	0	0
$301$	−10088.0	−1.93177
$302$	0	0
$303$	2106.00	0.399296
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1369.00	0.254505	0.127252	−	0.991870i	$-0.459384\pi$
$307$	1369.00	0.254505	0.127252	+	0.991870i	$0.459384\pi$
$308$	0	0
$309$	10548.0	1.94192
$310$	0	0
$311$	10426.0	1.90098	0.950489	−	0.310758i	$-0.100583\pi$
$311$	10426.0	1.90098	0.950489	+	0.310758i	$0.100583\pi$
$312$	0	0
$313$	−3574.00	−0.645413	−0.322707	−	0.946499i	$-0.604593\pi$
$313$	−3574.00	−0.645413	−0.322707	+	0.946499i	$0.604593\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−9036.00	−1.60099	−0.800493	−	0.599343i	$-0.795429\pi$
$317$	−9036.00	−1.60099	−0.800493	+	0.599343i	$0.795429\pi$
$318$	0	0
$319$	−1888.00	−0.331372
$320$	0	0
$321$	10125.0	1.76051
$322$	0	0
$323$	545.000	0.0938842
$324$	0	0
$325$	0	0
$326$	0	0
$327$	11106.0	1.87817
$328$	0	0
$329$	−1768.00	−0.296271
$330$	0	0
$331$	10233.0	1.69926	0.849632	−	0.527376i	$-0.176824\pi$
$331$	10233.0	1.69926	0.849632	+	0.527376i	$0.176824\pi$
$332$	0	0
$333$	10692.0	1.75951
$334$	0	0
$335$	0	0
$336$	0	0
$337$	4627.00	0.747919	0.373960	−	0.927445i	$-0.378000\pi$
$337$	4627.00	0.747919	0.373960	+	0.927445i	$0.378000\pi$
$338$	0	0
$339$	5103.00	0.817572
$340$	0	0
$341$	590.000	0.0936959
$342$	0	0
$343$	260.000	0.0409291
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4901.00	0.758212	0.379106	−	0.925353i	$-0.376232\pi$
$347$	4901.00	0.758212	0.379106	+	0.925353i	$0.376232\pi$
$348$	0	0
$349$	4482.00	0.687438	0.343719	−	0.939072i	$-0.388313\pi$
$349$	4482.00	0.687438	0.343719	+	0.939072i	$0.388313\pi$
$350$	0	0
$351$	−6804.00	−1.03467
$352$	0	0
$353$	−1210.00	−0.182441	−0.0912207	−	0.995831i	$-0.529077\pi$
$353$	−1210.00	−0.182441	−0.0912207	+	0.995831i	$0.529077\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1170.00	−0.173454
$358$	0	0
$359$	9882.00	1.45279	0.726396	−	0.687277i	$-0.241194\pi$
$359$	9882.00	1.45279	0.726396	+	0.687277i	$0.241194\pi$
$360$	0	0
$361$	5022.00	0.732177
$362$	0	0
$363$	19350.0	2.79783
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11260.0	1.60155	0.800773	−	0.598968i	$-0.204422\pi$
$367$	11260.0	1.60155	0.800773	+	0.598968i	$0.204422\pi$
$368$	0	0
$369$	6318.00	0.891333
$370$	0	0
$371$	−468.000	−0.0654915
$372$	0	0
$373$	3230.00	0.448373	0.224186	−	0.974546i	$-0.428028\pi$
$373$	3230.00	0.448373	0.224186	+	0.974546i	$0.428028\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−896.000	−0.122404
$378$	0	0
$379$	11575.0	1.56878	0.784390	−	0.620268i	$-0.212976\pi$
$379$	11575.0	1.56878	0.784390	+	0.620268i	$0.212976\pi$
$380$	0	0
$381$	21222.0	2.85364
$382$	0	0
$383$	18.0000	0.00240145	0.00120073	−	0.999999i	$-0.499618\pi$
$383$	18.0000	0.00240145	0.00120073	+	0.999999i	$0.499618\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	20952.0	2.75207
$388$	0	0
$389$	−10710.0	−1.39593	−0.697967	−	0.716130i	$-0.745912\pi$
$389$	−10710.0	−1.39593	−0.697967	+	0.716130i	$0.745912\pi$
$390$	0	0
$391$	970.000	0.125460
$392$	0	0
$393$	−15228.0	−1.95458
$394$	0	0
$395$	0	0
$396$	0	0
$397$	3788.00	0.478877	0.239439	−	0.970912i	$-0.423037\pi$
$397$	3788.00	0.478877	0.239439	+	0.970912i	$0.423037\pi$
$398$	0	0
$399$	−25506.0	−3.20024
$400$	0	0
$401$	−10539.0	−1.31245	−0.656225	−	0.754565i	$-0.727848\pi$
$401$	−10539.0	−1.31245	−0.656225	+	0.754565i	$0.727848\pi$
$402$	0	0
$403$	280.000	0.0346099
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−11682.0	−1.42274
$408$	0	0
$409$	5581.00	0.674725	0.337363	−	0.941375i	$-0.390465\pi$
$409$	5581.00	0.674725	0.337363	+	0.941375i	$0.390465\pi$
$410$	0	0
$411$	2061.00	0.247352
$412$	0	0
$413$	−10192.0	−1.21432
$414$	0	0
$415$	0	0
$416$	0	0
$417$	25029.0	2.93927
$418$	0	0
$419$	5193.00	0.605476	0.302738	−	0.953074i	$-0.402099\pi$
$419$	5193.00	0.605476	0.302738	+	0.953074i	$0.402099\pi$
$420$	0	0
$421$	−4788.00	−0.554282	−0.277141	−	0.960829i	$-0.589387\pi$
$421$	−4788.00	−0.554282	−0.277141	+	0.960829i	$0.589387\pi$
$422$	0	0
$423$	3672.00	0.422077
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−18460.0	−2.09214
$428$	0	0
$429$	14868.0	1.67327
$430$	0	0
$431$	−8006.00	−0.894746	−0.447373	−	0.894348i	$-0.647640\pi$
$431$	−8006.00	−0.894746	−0.447373	+	0.894348i	$0.647640\pi$
$432$	0	0
$433$	−2395.00	−0.265811	−0.132906	−	0.991129i	$-0.542431\pi$
$433$	−2395.00	−0.265811	−0.132906	+	0.991129i	$0.542431\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21146.0	2.31476
$438$	0	0
$439$	−1864.00	−0.202651	−0.101326	−	0.994853i	$-0.532308\pi$
$439$	−1864.00	−0.202651	−0.101326	+	0.994853i	$0.532308\pi$
$440$	0	0
$441$	17982.0	1.94169
$442$	0	0
$443$	5463.00	0.585903	0.292951	−	0.956127i	$-0.405363\pi$
$443$	5463.00	0.585903	0.292951	+	0.956127i	$0.405363\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−13248.0	−1.40181
$448$	0	0
$449$	12969.0	1.36313	0.681565	−	0.731758i	$-0.261300\pi$
$449$	12969.0	1.36313	0.681565	+	0.731758i	$0.261300\pi$
$450$	0	0
$451$	−6903.00	−0.720731
$452$	0	0
$453$	−11898.0	−1.23403
$454$	0	0
$455$	0	0
$456$	0	0
$457$	18313.0	1.87450	0.937249	−	0.348659i	$-0.113363\pi$
$457$	18313.0	1.87450	0.937249	+	0.348659i	$0.113363\pi$
$458$	0	0
$459$	1215.00	0.123554
$460$	0	0
$461$	−12492.0	−1.26206	−0.631031	−	0.775758i	$-0.717368\pi$
$461$	−12492.0	−1.26206	−0.631031	+	0.775758i	$0.717368\pi$
$462$	0	0
$463$	−4428.00	−0.444464	−0.222232	−	0.974994i	$-0.571334\pi$
$463$	−4428.00	−0.444464	−0.222232	+	0.974994i	$0.571334\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1084.00	−0.107412	−0.0537061	−	0.998557i	$-0.517103\pi$
$467$	−1084.00	−0.107412	−0.0537061	+	0.998557i	$0.517103\pi$
$468$	0	0
$469$	6578.00	0.647641
$470$	0	0
$471$	−2682.00	−0.262378
$472$	0	0
$473$	−22892.0	−2.22532
$474$	0	0
$475$	0	0
$476$	0	0
$477$	972.000	0.0933015
$478$	0	0
$479$	13082.0	1.24787	0.623937	−	0.781474i	$-0.285532\pi$
$479$	13082.0	1.24787	0.623937	+	0.781474i	$0.285532\pi$
$480$	0	0
$481$	−5544.00	−0.525540
$482$	0	0
$483$	−45396.0	−4.27658
$484$	0	0
$485$	0	0
$486$	0	0
$487$	3014.00	0.280446	0.140223	−	0.990120i	$-0.455218\pi$
$487$	3014.00	0.280446	0.140223	+	0.990120i	$0.455218\pi$
$488$	0	0
$489$	−3069.00	−0.283814
$490$	0	0
$491$	−3564.00	−0.327579	−0.163789	−	0.986495i	$-0.552372\pi$
$491$	−3564.00	−0.327579	−0.163789	+	0.986495i	$0.552372\pi$
$492$	0	0
$493$	160.000	0.0146167
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−15912.0	−1.43612
$498$	0	0
$499$	−15796.0	−1.41709	−0.708543	−	0.705667i	$-0.750647\pi$
$499$	−15796.0	−1.41709	−0.708543	+	0.705667i	$0.750647\pi$
$500$	0	0
$501$	−6156.00	−0.548962
$502$	0	0
$503$	10908.0	0.966926	0.483463	−	0.875365i	$-0.339379\pi$
$503$	10908.0	0.966926	0.483463	+	0.875365i	$0.339379\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−12717.0	−1.11397
$508$	0	0
$509$	−21946.0	−1.91108	−0.955540	−	0.294863i	$-0.904726\pi$
$509$	−21946.0	−1.91108	−0.955540	+	0.294863i	$0.904726\pi$
$510$	0	0
$511$	14274.0	1.23570
$512$	0	0
$513$	26487.0	2.27959
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4012.00	−0.341291
$518$	0	0
$519$	21096.0	1.78422
$520$	0	0
$521$	−6395.00	−0.537754	−0.268877	−	0.963174i	$-0.586653\pi$
$521$	−6395.00	−0.537754	−0.268877	+	0.963174i	$0.586653\pi$
$522$	0	0
$523$	−5615.00	−0.469459	−0.234729	−	0.972061i	$-0.575420\pi$
$523$	−5615.00	−0.469459	−0.234729	+	0.972061i	$0.575420\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−50.0000	−0.00413289
$528$	0	0
$529$	25469.0	2.09329
$530$	0	0
$531$	21168.0	1.72997
$532$	0	0
$533$	−3276.00	−0.266228
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−9999.00	−0.803517
$538$	0	0
$539$	−19647.0	−1.57005
$540$	0	0
$541$	4112.00	0.326781	0.163391	−	0.986561i	$-0.447757\pi$
$541$	4112.00	0.326781	0.163391	+	0.986561i	$0.447757\pi$
$542$	0	0
$543$	−18378.0	−1.45244
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−2167.00	−0.169386	−0.0846931	−	0.996407i	$-0.526991\pi$
$547$	−2167.00	−0.169386	−0.0846931	+	0.996407i	$0.526991\pi$
$548$	0	0
$549$	38340.0	2.98053
$550$	0	0
$551$	3488.00	0.269680
$552$	0	0
$553$	10764.0	0.827725
$554$	0	0
$555$	0	0
$556$	0	0
$557$	19444.0	1.47912	0.739559	−	0.673092i	$-0.235034\pi$
$557$	19444.0	1.47912	0.739559	+	0.673092i	$0.235034\pi$
$558$	0	0
$559$	−10864.0	−0.822000
$560$	0	0
$561$	−2655.00	−0.199811
$562$	0	0
$563$	20416.0	1.52830	0.764149	−	0.645040i	$-0.223159\pi$
$563$	20416.0	1.52830	0.764149	+	0.645040i	$0.223159\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−18954.0	−1.40387
$568$	0	0
$569$	−3127.00	−0.230388	−0.115194	−	0.993343i	$-0.536749\pi$
$569$	−3127.00	−0.230388	−0.115194	+	0.993343i	$0.536749\pi$
$570$	0	0
$571$	−22580.0	−1.65489	−0.827446	−	0.561545i	$-0.810207\pi$
$571$	−22580.0	−1.65489	−0.827446	+	0.561545i	$0.810207\pi$
$572$	0	0
$573$	−47430.0	−3.45797
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−829.000	−0.0598123	−0.0299062	−	0.999553i	$-0.509521\pi$
$577$	−829.000	−0.0598123	−0.0299062	+	0.999553i	$0.509521\pi$
$578$	0	0
$579$	5517.00	0.395991
$580$	0	0
$581$	3146.00	0.224644
$582$	0	0
$583$	−1062.00	−0.0754435
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7119.00	0.500567	0.250283	−	0.968173i	$-0.419476\pi$
$587$	7119.00	0.500567	0.250283	+	0.968173i	$0.419476\pi$
$588$	0	0
$589$	−1090.00	−0.0762524
$590$	0	0
$591$	−10566.0	−0.735410
$592$	0	0
$593$	−8217.00	−0.569025	−0.284512	−	0.958672i	$-0.591832\pi$
$593$	−8217.00	−0.569025	−0.284512	+	0.958672i	$0.591832\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−30852.0	−2.11506
$598$	0	0
$599$	90.0000	0.00613907	0.00306953	−	0.999995i	$-0.499023\pi$
$599$	90.0000	0.00613907	0.00306953	+	0.999995i	$0.499023\pi$
$600$	0	0
$601$	−17117.0	−1.16176	−0.580879	−	0.813990i	$-0.697291\pi$
$601$	−17117.0	−1.16176	−0.580879	+	0.813990i	$0.697291\pi$
$602$	0	0
$603$	−13662.0	−0.922653
$604$	0	0
$605$	0	0
$606$	0	0
$607$	15120.0	1.01104	0.505520	−	0.862815i	$-0.331301\pi$
$607$	15120.0	1.01104	0.505520	+	0.862815i	$0.331301\pi$
$608$	0	0
$609$	−7488.00	−0.498241
$610$	0	0
$611$	−1904.00	−0.126068
$612$	0	0
$613$	6570.00	0.432887	0.216444	−	0.976295i	$-0.430554\pi$
$613$	6570.00	0.432887	0.216444	+	0.976295i	$0.430554\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	18846.0	1.22968	0.614839	−	0.788653i	$-0.289221\pi$
$617$	18846.0	1.22968	0.614839	+	0.788653i	$0.289221\pi$
$618$	0	0
$619$	16316.0	1.05944	0.529722	−	0.848172i	$-0.322296\pi$
$619$	16316.0	1.05944	0.529722	+	0.848172i	$0.322296\pi$
$620$	0	0
$621$	47142.0	3.04629
$622$	0	0
$623$	2106.00	0.135434
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−57879.0	−3.68655
$628$	0	0
$629$	990.000	0.0627566
$630$	0	0
$631$	−20170.0	−1.27251	−0.636256	−	0.771478i	$-0.719518\pi$
$631$	−20170.0	−1.27251	−0.636256	+	0.771478i	$0.719518\pi$
$632$	0	0
$633$	21051.0	1.32180
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−9324.00	−0.579953
$638$	0	0
$639$	33048.0	2.04594
$640$	0	0
$641$	−12726.0	−0.784160	−0.392080	−	0.919931i	$-0.628244\pi$
$641$	−12726.0	−0.784160	−0.392080	+	0.919931i	$0.628244\pi$
$642$	0	0
$643$	2196.00	0.134684	0.0673420	−	0.997730i	$-0.478548\pi$
$643$	2196.00	0.134684	0.0673420	+	0.997730i	$0.478548\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16884.0	1.02593	0.512966	−	0.858409i	$-0.328547\pi$
$647$	16884.0	1.02593	0.512966	+	0.858409i	$0.328547\pi$
$648$	0	0
$649$	−23128.0	−1.39885
$650$	0	0
$651$	2340.00	0.140878
$652$	0	0
$653$	4018.00	0.240791	0.120395	−	0.992726i	$-0.461584\pi$
$653$	4018.00	0.240791	0.120395	+	0.992726i	$0.461584\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−29646.0	−1.76043
$658$	0	0
$659$	19071.0	1.12732	0.563658	−	0.826009i	$-0.309394\pi$
$659$	19071.0	1.12732	0.563658	+	0.826009i	$0.309394\pi$
$660$	0	0
$661$	−17424.0	−1.02529	−0.512644	−	0.858601i	$-0.671334\pi$
$661$	−17424.0	−1.02529	−0.512644	+	0.858601i	$0.671334\pi$
$662$	0	0
$663$	−1260.00	−0.0738075
$664$	0	0
$665$	0	0
$666$	0	0
$667$	6208.00	0.360382
$668$	0	0
$669$	−35388.0	−2.04511
$670$	0	0
$671$	−41890.0	−2.41005
$672$	0	0
$673$	−5382.00	−0.308263	−0.154131	−	0.988050i	$-0.549258\pi$
$673$	−5382.00	−0.308263	−0.154131	+	0.988050i	$0.549258\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	4496.00	0.255237	0.127618	−	0.991823i	$-0.459267\pi$
$677$	4496.00	0.255237	0.127618	+	0.991823i	$0.459267\pi$
$678$	0	0
$679$	39052.0	2.20718
$680$	0	0
$681$	54756.0	3.08114
$682$	0	0
$683$	3249.00	0.182020	0.0910099	−	0.995850i	$-0.470991\pi$
$683$	3249.00	0.182020	0.0910099	+	0.995850i	$0.470991\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−44964.0	−2.49706
$688$	0	0
$689$	−504.000	−0.0278677
$690$	0	0
$691$	−13399.0	−0.737658	−0.368829	−	0.929497i	$-0.620241\pi$
$691$	−13399.0	−0.737658	−0.368829	+	0.929497i	$0.620241\pi$
$692$	0	0
$693$	82836.0	4.54066
$694$	0	0
$695$	0	0
$696$	0	0
$697$	585.000	0.0317912
$698$	0	0
$699$	28998.0	1.56911
$700$	0	0
$701$	18148.0	0.977804	0.488902	−	0.872339i	$-0.337398\pi$
$701$	18148.0	0.977804	0.488902	+	0.872339i	$0.337398\pi$
$702$	0	0
$703$	21582.0	1.15787
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−6084.00	−0.323638
$708$	0	0
$709$	−4868.00	−0.257858	−0.128929	−	0.991654i	$-0.541154\pi$
$709$	−4868.00	−0.257858	−0.128929	+	0.991654i	$0.541154\pi$
$710$	0	0
$711$	−22356.0	−1.17921
$712$	0	0
$713$	−1940.00	−0.101898
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−24624.0	−1.28257
$718$	0	0
$719$	17366.0	0.900755	0.450377	−	0.892838i	$-0.351289\pi$
$719$	17366.0	0.900755	0.450377	+	0.892838i	$0.351289\pi$
$720$	0	0
$721$	−30472.0	−1.57398
$722$	0	0
$723$	15057.0	0.774517
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−21824.0	−1.11335	−0.556676	−	0.830729i	$-0.687924\pi$
$727$	−21824.0	−1.11335	−0.556676	+	0.830729i	$0.687924\pi$
$728$	0	0
$729$	−19683.0	−1.00000
$730$	0	0
$731$	1940.00	0.0981580
$732$	0	0
$733$	31428.0	1.58366	0.791828	−	0.610744i	$-0.209130\pi$
$733$	31428.0	1.58366	0.791828	+	0.610744i	$0.209130\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	14927.0	0.746056
$738$	0	0
$739$	−14292.0	−0.711420	−0.355710	−	0.934596i	$-0.615761\pi$
$739$	−14292.0	−0.711420	−0.355710	+	0.934596i	$0.615761\pi$
$740$	0	0
$741$	−27468.0	−1.36176
$742$	0	0
$743$	−13950.0	−0.688797	−0.344398	−	0.938824i	$-0.611917\pi$
$743$	−13950.0	−0.688797	−0.344398	+	0.938824i	$0.611917\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−6534.00	−0.320036
$748$	0	0
$749$	−29250.0	−1.42693
$750$	0	0
$751$	38736.0	1.88215	0.941076	−	0.338194i	$-0.109816\pi$
$751$	38736.0	1.88215	0.941076	+	0.338194i	$0.109816\pi$
$752$	0	0
$753$	48195.0	2.33243
$754$	0	0
$755$	0	0
$756$	0	0
$757$	3664.00	0.175919	0.0879593	−	0.996124i	$-0.471965\pi$
$757$	3664.00	0.175919	0.0879593	+	0.996124i	$0.471965\pi$
$758$	0	0
$759$	−103014.	−4.92644
$760$	0	0
$761$	19557.0	0.931591	0.465795	−	0.884892i	$-0.345768\pi$
$761$	19557.0	0.931591	0.465795	+	0.884892i	$0.345768\pi$
$762$	0	0
$763$	−32084.0	−1.52231
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−10976.0	−0.516715
$768$	0	0
$769$	−13283.0	−0.622883	−0.311442	−	0.950265i	$-0.600812\pi$
$769$	−13283.0	−0.622883	−0.311442	+	0.950265i	$0.600812\pi$
$770$	0	0
$771$	−49410.0	−2.30799
$772$	0	0
$773$	−24840.0	−1.15580	−0.577900	−	0.816108i	$-0.696127\pi$
$773$	−24840.0	−1.15580	−0.577900	+	0.816108i	$0.696127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−46332.0	−2.13919
$778$	0	0
$779$	12753.0	0.586552
$780$	0	0
$781$	−36108.0	−1.65435
$782$	0	0
$783$	7776.00	0.354906
$784$	0	0
$785$	0	0
$786$	0	0
$787$	18044.0	0.817280	0.408640	−	0.912696i	$-0.366003\pi$
$787$	18044.0	0.817280	0.408640	+	0.912696i	$0.366003\pi$
$788$	0	0
$789$	−28350.0	−1.27920
$790$	0	0
$791$	−14742.0	−0.662661
$792$	0	0
$793$	−19880.0	−0.890239
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6174.00	0.274397	0.137198	−	0.990544i	$-0.456190\pi$
$797$	6174.00	0.274397	0.137198	+	0.990544i	$0.456190\pi$
$798$	0	0
$799$	340.000	0.0150542
$800$	0	0
$801$	−4374.00	−0.192943
$802$	0	0
$803$	32391.0	1.42348
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−1584.00	−0.0690947
$808$	0	0
$809$	−1998.00	−0.0868306	−0.0434153	−	0.999057i	$-0.513824\pi$
$809$	−1998.00	−0.0868306	−0.0434153	+	0.999057i	$0.513824\pi$
$810$	0	0
$811$	−7156.00	−0.309841	−0.154921	−	0.987927i	$-0.549512\pi$
$811$	−7156.00	−0.309841	−0.154921	+	0.987927i	$0.549512\pi$
$812$	0	0
$813$	−21546.0	−0.929460
$814$	0	0
$815$	0	0
$816$	0	0
$817$	42292.0	1.81103
$818$	0	0
$819$	39312.0	1.67726
$820$	0	0
$821$	−27922.0	−1.18695	−0.593474	−	0.804853i	$-0.702244\pi$
$821$	−27922.0	−1.18695	−0.593474	+	0.804853i	$0.702244\pi$
$822$	0	0
$823$	−22636.0	−0.958738	−0.479369	−	0.877613i	$-0.659134\pi$
$823$	−22636.0	−0.958738	−0.479369	+	0.877613i	$0.659134\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	26559.0	1.11674	0.558372	−	0.829591i	$-0.311426\pi$
$827$	26559.0	1.11674	0.558372	+	0.829591i	$0.311426\pi$
$828$	0	0
$829$	−12580.0	−0.527046	−0.263523	−	0.964653i	$-0.584885\pi$
$829$	−12580.0	−0.527046	−0.263523	+	0.964653i	$0.584885\pi$
$830$	0	0
$831$	56304.0	2.35038
$832$	0	0
$833$	1665.00	0.0692543
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−2430.00	−0.100350
$838$	0	0
$839$	11344.0	0.466792	0.233396	−	0.972382i	$-0.425016\pi$
$839$	11344.0	0.466792	0.233396	+	0.972382i	$0.425016\pi$
$840$	0	0
$841$	−23365.0	−0.958014
$842$	0	0
$843$	−43218.0	−1.76573
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−55900.0	−2.26771
$848$	0	0
$849$	19107.0	0.772380
$850$	0	0
$851$	38412.0	1.54729
$852$	0	0
$853$	−14786.0	−0.593509	−0.296754	−	0.954954i	$-0.595904\pi$
$853$	−14786.0	−0.593509	−0.296754	+	0.954954i	$0.595904\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−29259.0	−1.16624	−0.583120	−	0.812386i	$-0.698168\pi$
$857$	−29259.0	−1.16624	−0.583120	+	0.812386i	$0.698168\pi$
$858$	0	0
$859$	−13651.0	−0.542219	−0.271109	−	0.962549i	$-0.587391\pi$
$859$	−13651.0	−0.542219	−0.271109	+	0.962549i	$0.587391\pi$
$860$	0	0
$861$	−27378.0	−1.08367
$862$	0	0
$863$	−29016.0	−1.14451	−0.572257	−	0.820074i	$-0.693932\pi$
$863$	−29016.0	−1.14451	−0.572257	+	0.820074i	$0.693932\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−43992.0	−1.72324
$868$	0	0
$869$	24426.0	0.953504
$870$	0	0
$871$	7084.00	0.275582
$872$	0	0
$873$	−81108.0	−3.14443
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−21412.0	−0.824438	−0.412219	−	0.911085i	$-0.635246\pi$
$877$	−21412.0	−0.824438	−0.412219	+	0.911085i	$0.635246\pi$
$878$	0	0
$879$	79506.0	3.05082
$880$	0	0
$881$	−1170.00	−0.0447427	−0.0223713	−	0.999750i	$-0.507122\pi$
$881$	−1170.00	−0.0447427	−0.0223713	+	0.999750i	$0.507122\pi$
$882$	0	0
$883$	−12655.0	−0.482304	−0.241152	−	0.970487i	$-0.577525\pi$
$883$	−12655.0	−0.482304	−0.241152	+	0.970487i	$0.577525\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−32764.0	−1.24026	−0.620128	−	0.784500i	$-0.712919\pi$
$887$	−32764.0	−1.24026	−0.620128	+	0.784500i	$0.712919\pi$
$888$	0	0
$889$	−61308.0	−2.31294
$890$	0	0
$891$	−43011.0	−1.61720
$892$	0	0
$893$	7412.00	0.277753
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−48888.0	−1.81976
$898$	0	0
$899$	−320.000	−0.0118716
$900$	0	0
$901$	90.0000	0.00332779
$902$	0	0
$903$	−90792.0	−3.34592
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−29844.0	−1.09256	−0.546281	−	0.837602i	$-0.683957\pi$
$907$	−29844.0	−1.09256	−0.546281	+	0.837602i	$0.683957\pi$
$908$	0	0
$909$	12636.0	0.461067
$910$	0	0
$911$	−15628.0	−0.568363	−0.284182	−	0.958770i	$-0.591722\pi$
$911$	−15628.0	−0.568363	−0.284182	+	0.958770i	$0.591722\pi$
$912$	0	0
$913$	7139.00	0.258780
$914$	0	0
$915$	0	0
$916$	0	0
$917$	43992.0	1.58424
$918$	0	0
$919$	−42974.0	−1.54253	−0.771263	−	0.636517i	$-0.780375\pi$
$919$	−42974.0	−1.54253	−0.771263	+	0.636517i	$0.780375\pi$
$920$	0	0
$921$	12321.0	0.440815
$922$	0	0
$923$	−17136.0	−0.611092
$924$	0	0
$925$	0	0
$926$	0	0
$927$	63288.0	2.24234
$928$	0	0
$929$	13342.0	0.471191	0.235596	−	0.971851i	$-0.424296\pi$
$929$	13342.0	0.471191	0.235596	+	0.971851i	$0.424296\pi$
$930$	0	0
$931$	36297.0	1.27775
$932$	0	0
$933$	93834.0	3.29259
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3005.00	−0.104770	−0.0523848	−	0.998627i	$-0.516682\pi$
$937$	−3005.00	−0.104770	−0.0523848	+	0.998627i	$0.516682\pi$
$938$	0	0
$939$	−32166.0	−1.11789
$940$	0	0
$941$	−16204.0	−0.561355	−0.280678	−	0.959802i	$-0.590559\pi$
$941$	−16204.0	−0.561355	−0.280678	+	0.959802i	$0.590559\pi$
$942$	0	0
$943$	22698.0	0.783827
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−30200.0	−1.03629	−0.518146	−	0.855292i	$-0.673378\pi$
$947$	−30200.0	−1.03629	−0.518146	+	0.855292i	$0.673378\pi$
$948$	0	0
$949$	15372.0	0.525813
$950$	0	0
$951$	−81324.0	−2.77299
$952$	0	0
$953$	−29583.0	−1.00555	−0.502774	−	0.864418i	$-0.667687\pi$
$953$	−29583.0	−1.00555	−0.502774	+	0.864418i	$0.667687\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−16992.0	−0.573953
$958$	0	0
$959$	−5954.00	−0.200485
$960$	0	0
$961$	−29691.0	−0.996643
$962$	0	0
$963$	60750.0	2.03286
$964$	0	0
$965$	0	0
$966$	0	0
$967$	6480.00	0.215494	0.107747	−	0.994178i	$-0.465636\pi$
$967$	6480.00	0.215494	0.107747	+	0.994178i	$0.465636\pi$
$968$	0	0
$969$	4905.00	0.162612
$970$	0	0
$971$	−40171.0	−1.32765	−0.663825	−	0.747888i	$-0.731068\pi$
$971$	−40171.0	−1.32765	−0.663825	+	0.747888i	$0.731068\pi$
$972$	0	0
$973$	−72306.0	−2.38235
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50801.0	1.66353	0.831765	−	0.555129i	$-0.187331\pi$
$977$	50801.0	1.66353	0.831765	+	0.555129i	$0.187331\pi$
$978$	0	0
$979$	4779.00	0.156014
$980$	0	0
$981$	66636.0	2.16873
$982$	0	0
$983$	58338.0	1.89287	0.946436	−	0.322891i	$-0.104655\pi$
$983$	58338.0	1.89287	0.946436	+	0.322891i	$0.104655\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−15912.0	−0.513156
$988$	0	0
$989$	75272.0	2.42013
$990$	0	0
$991$	51202.0	1.64126	0.820628	−	0.571462i	$-0.193624\pi$
$991$	51202.0	1.64126	0.820628	+	0.571462i	$0.193624\pi$
$992$	0	0
$993$	92097.0	2.94321
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1764.00	−0.0560345	−0.0280173	−	0.999607i	$-0.508919\pi$
$997$	−1764.00	−0.0560345	−0.0280173	+	0.999607i	$0.508919\pi$
$998$	0	0
$999$	48114.0	1.52378

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.4.a.by.1.1		1
4.3	odd	2		1600.4.a.c.1.1		1
5.4	even	2		1600.4.a.b.1.1		1
8.3	odd	2		200.4.a.j.1.1	yes	1
8.5	even	2		400.4.a.a.1.1		1
20.19	odd	2		1600.4.a.bz.1.1		1
24.11	even	2		1800.4.a.bh.1.1		1
40.3	even	4		200.4.c.b.49.2		2
40.13	odd	4		400.4.c.b.49.1		2
40.19	odd	2		200.4.a.b.1.1	✓	1
40.27	even	4		200.4.c.b.49.1		2
40.29	even	2		400.4.a.t.1.1		1
40.37	odd	4		400.4.c.b.49.2		2
120.59	even	2		1800.4.a.c.1.1		1
120.83	odd	4		1800.4.f.w.649.1		2
120.107	odd	4		1800.4.f.w.649.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.4.a.b.1.1	✓	1	40.19	odd	2
200.4.a.j.1.1	yes	1	8.3	odd	2
200.4.c.b.49.1		2	40.27	even	4
200.4.c.b.49.2		2	40.3	even	4
400.4.a.a.1.1		1	8.5	even	2
400.4.a.t.1.1		1	40.29	even	2
400.4.c.b.49.1		2	40.13	odd	4
400.4.c.b.49.2		2	40.37	odd	4
1600.4.a.b.1.1		1	5.4	even	2
1600.4.a.c.1.1		1	4.3	odd	2
1600.4.a.by.1.1		1	1.1	even	1	trivial
1600.4.a.bz.1.1		1	20.19	odd	2
1800.4.a.c.1.1		1	120.59	even	2
1800.4.a.bh.1.1		1	24.11	even	2
1800.4.f.w.649.1		2	120.83	odd	4
1800.4.f.w.649.2		2	120.107	odd	4

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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

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Newspace parameters

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$q$ -expansion

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Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1600.4.a.by.1.1

Level	$1600$
Weight	$4$
Character	1600.1
Self dual	yes
Analytic conductor	$94.403$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$