LMFDB - Embedded newform 200.6.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [200,6,Mod(1,200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("200.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 200 = 2^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	200.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:	$32.0767639626$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.1595208.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 20x^{2} + 33x - 3 $ x^4 - x^3 - 20x^2 + 33x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8}\cdot 5^{2} $
Twist minimal:	no (minimal twist has level 40)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.0965878$ of defining polynomial
Character	$\chi$	$=$	200.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-24.1383 q^{3} +179.876 q^{7} +339.657 q^{9} +O(q^{10})$	$q-24.1383 q^{3} +179.876 q^{7} +339.657 q^{9} -653.681 q^{11} -284.851 q^{13} +383.672 q^{17} -2563.29 q^{19} -4341.90 q^{21} +948.124 q^{23} -2333.13 q^{27} -1524.26 q^{29} +3103.88 q^{31} +15778.7 q^{33} +9991.75 q^{37} +6875.83 q^{39} +15120.7 q^{41} -1754.92 q^{43} -14760.7 q^{47} +15548.5 q^{49} -9261.18 q^{51} +8704.12 q^{53} +61873.3 q^{57} -12632.1 q^{59} +43098.3 q^{61} +61096.2 q^{63} +26125.5 q^{67} -22886.1 q^{69} -46285.8 q^{71} +51303.2 q^{73} -117582. q^{77} +39314.0 q^{79} -26218.8 q^{81} +69544.8 q^{83} +36793.1 q^{87} -13092.2 q^{89} -51238.0 q^{91} -74922.5 q^{93} +26258.1 q^{97} -222027. q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9	$ 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} - 368 q^{11} + 440 q^{13} - 672 q^{17} - 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} + 2112 q^{31} + 26864 q^{33} + 8792 q^{37} + 1504 q^{39} + 11800 q^{41} + 48276 q^{43} + 14724 q^{47} + 22500 q^{49} - 62400 q^{51} + 84296 q^{53} + 71024 q^{57} - 45840 q^{59} + 61928 q^{61} + 186292 q^{63} + 72700 q^{67} + 38368 q^{69} - 62816 q^{71} + 133072 q^{73} + 11440 q^{77} - 21632 q^{79} + 204836 q^{81} + 74660 q^{83} - 12472 q^{87} + 20952 q^{89} - 243808 q^{91} + 105600 q^{93} + 59456 q^{97} - 133424 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 - 368 * q^11 + 440 * q^13 - 672 * q^17 - 688 * q^19 + 992 * q^21 + 4492 * q^23 + 8152 * q^27 - 2936 * q^29 + 2112 * q^31 + 26864 * q^33 + 8792 * q^37 + 1504 * q^39 + 11800 * q^41 + 48276 * q^43 + 14724 * q^47 + 22500 * q^49 - 62400 * q^51 + 84296 * q^53 + 71024 * q^57 - 45840 * q^59 + 61928 * q^61 + 186292 * q^63 + 72700 * q^67 + 38368 * q^69 - 62816 * q^71 + 133072 * q^73 + 11440 * q^77 - 21632 * q^79 + 204836 * q^81 + 74660 * q^83 - 12472 * q^87 + 20952 * q^89 - 243808 * q^91 + 105600 * q^93 + 59456 * q^97 - 133424 * 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
$3$	−24.1383	−1.54847	−0.774236	−	0.632897i	$-0.781866\pi$
$3$	−24.1383	−1.54847	−0.774236	+	0.632897i	$0.781866\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	179.876	1.38749	0.693743	−	0.720223i	$-0.255960\pi$
$7$	179.876	1.38749	0.693743	+	0.720223i	$0.255960\pi$
$8$	0	0
$9$	339.657	1.39777
$10$	0	0
$11$	−653.681	−1.62886	−0.814431	−	0.580260i	$-0.802951\pi$
$11$	−653.681	−1.62886	−0.814431	+	0.580260i	$0.802951\pi$
$12$	0	0
$13$	−284.851	−0.467477	−0.233738	−	0.972300i	$-0.575096\pi$
$13$	−284.851	−0.467477	−0.233738	+	0.972300i	$0.575096\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	383.672	0.321986	0.160993	−	0.986956i	$-0.448530\pi$
$17$	383.672	0.321986	0.160993	+	0.986956i	$0.448530\pi$
$18$	0	0
$19$	−2563.29	−1.62897	−0.814485	−	0.580185i	$-0.802980\pi$
$19$	−2563.29	−1.62897	−0.814485	+	0.580185i	$0.802980\pi$
$20$	0	0
$21$	−4341.90	−2.14848
$22$	0	0
$23$	948.124	0.373719	0.186860	−	0.982387i	$-0.440169\pi$
$23$	948.124	0.373719	0.186860	+	0.982387i	$0.440169\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−2333.13	−0.615929
$28$	0	0
$29$	−1524.26	−0.336561	−0.168281	−	0.985739i	$-0.553822\pi$
$29$	−1524.26	−0.336561	−0.168281	+	0.985739i	$0.553822\pi$
$30$	0	0
$31$	3103.88	0.580098	0.290049	−	0.957012i	$-0.406328\pi$
$31$	3103.88	0.580098	0.290049	+	0.957012i	$0.406328\pi$
$32$	0	0
$33$	15778.7	2.52225
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9991.75	1.19988	0.599939	−	0.800046i	$-0.295191\pi$
$37$	9991.75	1.19988	0.599939	+	0.800046i	$0.295191\pi$
$38$	0	0
$39$	6875.83	0.723875
$40$	0	0
$41$	15120.7	1.40479	0.702394	−	0.711788i	$-0.252114\pi$
$41$	15120.7	1.40479	0.702394	+	0.711788i	$0.252114\pi$
$42$	0	0
$43$	−1754.92	−0.144739	−0.0723696	−	0.997378i	$-0.523056\pi$
$43$	−1754.92	−0.144739	−0.0723696	+	0.997378i	$0.523056\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−14760.7	−0.974682	−0.487341	−	0.873212i	$-0.662033\pi$
$47$	−14760.7	−0.974682	−0.487341	+	0.873212i	$0.662033\pi$
$48$	0	0
$49$	15548.5	0.925118
$50$	0	0
$51$	−9261.18	−0.498587
$52$	0	0
$53$	8704.12	0.425633	0.212816	−	0.977092i	$-0.431736\pi$
$53$	8704.12	0.425633	0.212816	+	0.977092i	$0.431736\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	61873.3	2.52241
$58$	0	0
$59$	−12632.1	−0.472438	−0.236219	−	0.971700i	$-0.575908\pi$
$59$	−12632.1	−0.472438	−0.236219	+	0.971700i	$0.575908\pi$
$60$	0	0
$61$	43098.3	1.48298	0.741490	−	0.670964i	$-0.234119\pi$
$61$	43098.3	1.48298	0.741490	+	0.670964i	$0.234119\pi$
$62$	0	0
$63$	61096.2	1.93938
$64$	0	0
$65$	0	0
$66$	0	0
$67$	26125.5	0.711014	0.355507	−	0.934674i	$-0.384308\pi$
$67$	26125.5	0.711014	0.355507	+	0.934674i	$0.384308\pi$
$68$	0	0
$69$	−22886.1	−0.578694
$70$	0	0
$71$	−46285.8	−1.08969	−0.544844	−	0.838538i	$-0.683411\pi$
$71$	−46285.8	−1.08969	−0.544844	+	0.838538i	$0.683411\pi$
$72$	0	0
$73$	51303.2	1.12678	0.563388	−	0.826192i	$-0.309498\pi$
$73$	51303.2	1.12678	0.563388	+	0.826192i	$0.309498\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−117582.	−2.26002
$78$	0	0
$79$	39314.0	0.708729	0.354364	−	0.935107i	$-0.384697\pi$
$79$	39314.0	0.708729	0.354364	+	0.935107i	$0.384697\pi$
$80$	0	0
$81$	−26218.8	−0.444017
$82$	0	0
$83$	69544.8	1.10808	0.554038	−	0.832492i	$-0.313086\pi$
$83$	69544.8	1.10808	0.554038	+	0.832492i	$0.313086\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	36793.1	0.521156
$88$	0	0
$89$	−13092.2	−0.175201	−0.0876007	−	0.996156i	$-0.527920\pi$
$89$	−13092.2	−0.175201	−0.0876007	+	0.996156i	$0.527920\pi$
$90$	0	0
$91$	−51238.0	−0.648618
$92$	0	0
$93$	−74922.5	−0.898265
$94$	0	0
$95$	0	0
$96$	0	0
$97$	26258.1	0.283357	0.141678	−	0.989913i	$-0.454750\pi$
$97$	26258.1	0.283357	0.141678	+	0.989913i	$0.454750\pi$
$98$	0	0
$99$	−222027.	−2.27677
$100$	0	0
$101$	7382.54	0.0720116	0.0360058	−	0.999352i	$-0.488537\pi$
$101$	7382.54	0.0720116	0.0360058	+	0.999352i	$0.488537\pi$
$102$	0	0
$103$	−44518.9	−0.413477	−0.206738	−	0.978396i	$-0.566285\pi$
$103$	−44518.9	−0.413477	−0.206738	+	0.978396i	$0.566285\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	194929.	1.64595	0.822974	−	0.568079i	$-0.192313\pi$
$107$	194929.	1.64595	0.822974	+	0.568079i	$0.192313\pi$
$108$	0	0
$109$	213783.	1.72348	0.861739	−	0.507351i	$-0.169375\pi$
$109$	213783.	1.72348	0.861739	+	0.507351i	$0.169375\pi$
$110$	0	0
$111$	−241184.	−1.85798
$112$	0	0
$113$	148946.	1.09732	0.548660	−	0.836046i	$-0.315138\pi$
$113$	148946.	1.09732	0.548660	+	0.836046i	$0.315138\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−96751.8	−0.653423
$118$	0	0
$119$	69013.4	0.446751
$120$	0	0
$121$	266248.	1.65319
$122$	0	0
$123$	−364987.	−2.17528
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−116757.	−0.642351	−0.321175	−	0.947020i	$-0.604078\pi$
$127$	−116757.	−0.642351	−0.321175	+	0.947020i	$0.604078\pi$
$128$	0	0
$129$	42360.8	0.224125
$130$	0	0
$131$	349111.	1.77740	0.888700	−	0.458490i	$-0.151610\pi$
$131$	349111.	1.77740	0.888700	+	0.458490i	$0.151610\pi$
$132$	0	0
$133$	−461074.	−2.26017
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−219776.	−1.00041	−0.500205	−	0.865907i	$-0.666742\pi$
$137$	−219776.	−1.00041	−0.500205	+	0.865907i	$0.666742\pi$
$138$	0	0
$139$	126594.	0.555744	0.277872	−	0.960618i	$-0.410371\pi$
$139$	126594.	0.555744	0.277872	+	0.960618i	$0.410371\pi$
$140$	0	0
$141$	356299.	1.50927
$142$	0	0
$143$	186202.	0.761455
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−375313.	−1.43252
$148$	0	0
$149$	−25809.9	−0.0952403	−0.0476202	−	0.998866i	$-0.515164\pi$
$149$	−25809.9	−0.0952403	−0.0476202	+	0.998866i	$0.515164\pi$
$150$	0	0
$151$	−311270.	−1.11095	−0.555475	−	0.831533i	$-0.687464\pi$
$151$	−311270.	−1.11095	−0.555475	+	0.831533i	$0.687464\pi$
$152$	0	0
$153$	130317.	0.450061
$154$	0	0
$155$	0	0
$156$	0	0
$157$	429151.	1.38951	0.694755	−	0.719247i	$-0.255513\pi$
$157$	429151.	1.38951	0.694755	+	0.719247i	$0.255513\pi$
$158$	0	0
$159$	−210102.	−0.659080
$160$	0	0
$161$	170545.	0.518530
$162$	0	0
$163$	162467.	0.478957	0.239479	−	0.970902i	$-0.423023\pi$
$163$	162467.	0.478957	0.239479	+	0.970902i	$0.423023\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−280057.	−0.777060	−0.388530	−	0.921436i	$-0.627017\pi$
$167$	−280057.	−0.777060	−0.388530	+	0.921436i	$0.627017\pi$
$168$	0	0
$169$	−290153.	−0.781465
$170$	0	0
$171$	−870638.	−2.27692
$172$	0	0
$173$	−742577.	−1.88637	−0.943184	−	0.332271i	$-0.892185\pi$
$173$	−742577.	−1.88637	−0.943184	+	0.332271i	$0.892185\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	304917.	0.731557
$178$	0	0
$179$	−14556.0	−0.0339554	−0.0169777	−	0.999856i	$-0.505404\pi$
$179$	−14556.0	−0.0339554	−0.0169777	+	0.999856i	$0.505404\pi$
$180$	0	0
$181$	−243243.	−0.551878	−0.275939	−	0.961175i	$-0.588989\pi$
$181$	−243243.	−0.551878	−0.275939	+	0.961175i	$0.588989\pi$
$182$	0	0
$183$	−1.04032e6	−2.29635
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−250799.	−0.524471
$188$	0	0
$189$	−419675.	−0.854592
$190$	0	0
$191$	75871.1	0.150485	0.0752425	−	0.997165i	$-0.476027\pi$
$191$	75871.1	0.150485	0.0752425	+	0.997165i	$0.476027\pi$
$192$	0	0
$193$	370161.	0.715314	0.357657	−	0.933853i	$-0.383576\pi$
$193$	370161.	0.715314	0.357657	+	0.933853i	$0.383576\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−213848.	−0.392590	−0.196295	−	0.980545i	$-0.562891\pi$
$197$	−213848.	−0.392590	−0.196295	+	0.980545i	$0.562891\pi$
$198$	0	0
$199$	466075.	0.834302	0.417151	−	0.908837i	$-0.363029\pi$
$199$	466075.	0.834302	0.417151	+	0.908837i	$0.363029\pi$
$200$	0	0
$201$	−630626.	−1.10099
$202$	0	0
$203$	−274178.	−0.466974
$204$	0	0
$205$	0	0
$206$	0	0
$207$	322037.	0.522372
$208$	0	0
$209$	1.67557e6	2.65337
$210$	0	0
$211$	−228211.	−0.352883	−0.176441	−	0.984311i	$-0.556459\pi$
$211$	−228211.	−0.352883	−0.176441	+	0.984311i	$0.556459\pi$
$212$	0	0
$213$	1.11726e6	1.68735
$214$	0	0
$215$	0	0
$216$	0	0
$217$	558315.	0.804878
$218$	0	0
$219$	−1.23837e6	−1.74478
$220$	0	0
$221$	−109289.	−0.150521
$222$	0	0
$223$	972750.	1.30990	0.654951	−	0.755671i	$-0.272689\pi$
$223$	972750.	1.30990	0.654951	+	0.755671i	$0.272689\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	357063.	0.459917	0.229959	−	0.973200i	$-0.426141\pi$
$227$	357063.	0.459917	0.229959	+	0.973200i	$0.426141\pi$
$228$	0	0
$229$	−295196.	−0.371982	−0.185991	−	0.982551i	$-0.559549\pi$
$229$	−295196.	−0.371982	−0.185991	+	0.982551i	$0.559549\pi$
$230$	0	0
$231$	2.83822e6	3.49958
$232$	0	0
$233$	492373.	0.594162	0.297081	−	0.954852i	$-0.403987\pi$
$233$	492373.	0.594162	0.297081	+	0.954852i	$0.403987\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−948974.	−1.09745
$238$	0	0
$239$	−71796.0	−0.0813028	−0.0406514	−	0.999173i	$-0.512943\pi$
$239$	−71796.0	−0.0813028	−0.0406514	+	0.999173i	$0.512943\pi$
$240$	0	0
$241$	614304.	0.681304	0.340652	−	0.940190i	$-0.389352\pi$
$241$	614304.	0.681304	0.340652	+	0.940190i	$0.389352\pi$
$242$	0	0
$243$	1.19983e6	1.30348
$244$	0	0
$245$	0	0
$246$	0	0
$247$	730156.	0.761505
$248$	0	0
$249$	−1.67869e6	−1.71582
$250$	0	0
$251$	−244276.	−0.244735	−0.122368	−	0.992485i	$-0.539049\pi$
$251$	−244276.	−0.244735	−0.122368	+	0.992485i	$0.539049\pi$
$252$	0	0
$253$	−619771.	−0.608737
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−951728.	−0.898835	−0.449418	−	0.893322i	$-0.648368\pi$
$257$	−951728.	−0.898835	−0.449418	+	0.893322i	$0.648368\pi$
$258$	0	0
$259$	1.79728e6	1.66481
$260$	0	0
$261$	−517726.	−0.470434
$262$	0	0
$263$	258512.	0.230458	0.115229	−	0.993339i	$-0.463240\pi$
$263$	258512.	0.230458	0.115229	+	0.993339i	$0.463240\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	316023.	0.271295
$268$	0	0
$269$	894716.	0.753884	0.376942	−	0.926237i	$-0.376976\pi$
$269$	894716.	0.753884	0.376942	+	0.926237i	$0.376976\pi$
$270$	0	0
$271$	474149.	0.392186	0.196093	−	0.980585i	$-0.437175\pi$
$271$	474149.	0.392186	0.196093	+	0.980585i	$0.437175\pi$
$272$	0	0
$273$	1.23680e6	1.00437
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−142752.	−0.111785	−0.0558923	−	0.998437i	$-0.517800\pi$
$277$	−142752.	−0.111785	−0.0558923	+	0.998437i	$0.517800\pi$
$278$	0	0
$279$	1.05426e6	0.810841
$280$	0	0
$281$	−427838.	−0.323231	−0.161616	−	0.986854i	$-0.551670\pi$
$281$	−427838.	−0.323231	−0.161616	+	0.986854i	$0.551670\pi$
$282$	0	0
$283$	2.30233e6	1.70884	0.854419	−	0.519585i	$-0.173913\pi$
$283$	2.30233e6	1.70884	0.854419	+	0.519585i	$0.173913\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.71985e6	1.94913
$288$	0	0
$289$	−1.27265e6	−0.896325
$290$	0	0
$291$	−633825.	−0.438770
$292$	0	0
$293$	−1.59390e6	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$293$	−1.59390e6	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.52513e6	1.00326
$298$	0	0
$299$	−270074.	−0.174705
$300$	0	0
$301$	−315668.	−0.200824
$302$	0	0
$303$	−178202.	−0.111508
$304$	0	0
$305$	0	0
$306$	0	0
$307$	322680.	0.195401	0.0977005	−	0.995216i	$-0.468851\pi$
$307$	322680.	0.195401	0.0977005	+	0.995216i	$0.468851\pi$
$308$	0	0
$309$	1.07461e6	0.640257
$310$	0	0
$311$	−660204.	−0.387059	−0.193529	−	0.981094i	$-0.561994\pi$
$311$	−660204.	−0.387059	−0.193529	+	0.981094i	$0.561994\pi$
$312$	0	0
$313$	−1.15378e6	−0.665677	−0.332838	−	0.942984i	$-0.608006\pi$
$313$	−1.15378e6	−0.665677	−0.332838	+	0.942984i	$0.608006\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.20296e6	0.672364	0.336182	−	0.941797i	$-0.390864\pi$
$317$	1.20296e6	0.672364	0.336182	+	0.941797i	$0.390864\pi$
$318$	0	0
$319$	996381.	0.548212
$320$	0	0
$321$	−4.70524e6	−2.54870
$322$	0	0
$323$	−983460.	−0.524506
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−5.16034e6	−2.66876
$328$	0	0
$329$	−2.65510e6	−1.35236
$330$	0	0
$331$	1.82424e6	0.915193	0.457597	−	0.889160i	$-0.348710\pi$
$331$	1.82424e6	0.915193	0.457597	+	0.889160i	$0.348710\pi$
$332$	0	0
$333$	3.39377e6	1.67715
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−485867.	−0.233047	−0.116523	−	0.993188i	$-0.537175\pi$
$337$	−485867.	−0.233047	−0.116523	+	0.993188i	$0.537175\pi$
$338$	0	0
$339$	−3.59530e6	−1.69917
$340$	0	0
$341$	−2.02895e6	−0.944899
$342$	0	0
$343$	−226383.	−0.103898
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−2.35066e6	−1.04801	−0.524006	−	0.851715i	$-0.675563\pi$
$347$	−2.35066e6	−1.04801	−0.524006	+	0.851715i	$0.675563\pi$
$348$	0	0
$349$	−2.62295e6	−1.15273	−0.576364	−	0.817193i	$-0.695529\pi$
$349$	−2.62295e6	−1.15273	−0.576364	+	0.817193i	$0.695529\pi$
$350$	0	0
$351$	664597.	0.287932
$352$	0	0
$353$	−583638.	−0.249291	−0.124646	−	0.992201i	$-0.539779\pi$
$353$	−583638.	−0.249291	−0.124646	+	0.992201i	$0.539779\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−1.66587e6	−0.691782
$358$	0	0
$359$	1.64503e6	0.673654	0.336827	−	0.941567i	$-0.390646\pi$
$359$	1.64503e6	0.673654	0.336827	+	0.941567i	$0.390646\pi$
$360$	0	0
$361$	4.09433e6	1.65354
$362$	0	0
$363$	−6.42678e6	−2.55992
$364$	0	0
$365$	0	0
$366$	0	0
$367$	2.62251e6	1.01637	0.508186	−	0.861247i	$-0.330316\pi$
$367$	2.62251e6	1.01637	0.508186	+	0.861247i	$0.330316\pi$
$368$	0	0
$369$	5.13584e6	1.96357
$370$	0	0
$371$	1.56566e6	0.590559
$372$	0	0
$373$	3.73068e6	1.38840	0.694202	−	0.719780i	$-0.255757\pi$
$373$	3.73068e6	1.38840	0.694202	+	0.719780i	$0.255757\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	434188.	0.157335
$378$	0	0
$379$	−3.00539e6	−1.07474	−0.537368	−	0.843348i	$-0.680582\pi$
$379$	−3.00539e6	−1.07474	−0.537368	+	0.843348i	$0.680582\pi$
$380$	0	0
$381$	2.81831e6	0.994662
$382$	0	0
$383$	343090.	0.119512	0.0597560	−	0.998213i	$-0.480968\pi$
$383$	343090.	0.119512	0.0597560	+	0.998213i	$0.480968\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−596071.	−0.202311
$388$	0	0
$389$	190583.	0.0638572	0.0319286	−	0.999490i	$-0.489835\pi$
$389$	190583.	0.0638572	0.0319286	+	0.999490i	$0.489835\pi$
$390$	0	0
$391$	363768.	0.120332
$392$	0	0
$393$	−8.42694e6	−2.75225
$394$	0	0
$395$	0	0
$396$	0	0
$397$	5.22551e6	1.66400	0.831999	−	0.554778i	$-0.187197\pi$
$397$	5.22551e6	1.66400	0.831999	+	0.554778i	$0.187197\pi$
$398$	0	0
$399$	1.11295e7	3.49981
$400$	0	0
$401$	−1.60312e6	−0.497859	−0.248929	−	0.968522i	$-0.580079\pi$
$401$	−1.60312e6	−0.497859	−0.248929	+	0.968522i	$0.580079\pi$
$402$	0	0
$403$	−884146.	−0.271182
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.53142e6	−1.95444
$408$	0	0
$409$	4.58732e6	1.35597	0.677986	−	0.735075i	$-0.262853\pi$
$409$	4.58732e6	1.35597	0.677986	+	0.735075i	$0.262853\pi$
$410$	0	0
$411$	5.30501e6	1.54911
$412$	0	0
$413$	−2.27221e6	−0.655501
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−3.05575e6	−0.860554
$418$	0	0
$419$	−6.16854e6	−1.71651	−0.858256	−	0.513221i	$-0.828452\pi$
$419$	−6.16854e6	−1.71651	−0.858256	+	0.513221i	$0.828452\pi$
$420$	0	0
$421$	−6.15589e6	−1.69272	−0.846361	−	0.532610i	$-0.821211\pi$
$421$	−6.15589e6	−1.69272	−0.846361	+	0.532610i	$0.821211\pi$
$422$	0	0
$423$	−5.01358e6	−1.36238
$424$	0	0
$425$	0	0
$426$	0	0
$427$	7.75236e6	2.05761
$428$	0	0
$429$	−4.49460e6	−1.17909
$430$	0	0
$431$	1.15900e6	0.300532	0.150266	−	0.988646i	$-0.451987\pi$
$431$	1.15900e6	0.300532	0.150266	+	0.988646i	$0.451987\pi$
$432$	0	0
$433$	−6.73906e6	−1.72735	−0.863673	−	0.504052i	$-0.831842\pi$
$433$	−6.73906e6	−1.72735	−0.863673	+	0.504052i	$0.831842\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.43031e6	−0.608777
$438$	0	0
$439$	−3.69068e6	−0.913998	−0.456999	−	0.889467i	$-0.651076\pi$
$439$	−3.69068e6	−0.913998	−0.456999	+	0.889467i	$0.651076\pi$
$440$	0	0
$441$	5.28114e6	1.29310
$442$	0	0
$443$	4.55124e6	1.10185	0.550923	−	0.834556i	$-0.314276\pi$
$443$	4.55124e6	1.10185	0.550923	+	0.834556i	$0.314276\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	623007.	0.147477
$448$	0	0
$449$	−3.31419e6	−0.775822	−0.387911	−	0.921697i	$-0.626803\pi$
$449$	−3.31419e6	−0.775822	−0.387911	+	0.921697i	$0.626803\pi$
$450$	0	0
$451$	−9.88409e6	−2.28821
$452$	0	0
$453$	7.51352e6	1.72028
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.10489e6	−0.919414	−0.459707	−	0.888071i	$-0.652046\pi$
$457$	−4.10489e6	−0.919414	−0.459707	+	0.888071i	$0.652046\pi$
$458$	0	0
$459$	−895158.	−0.198321
$460$	0	0
$461$	8.16525e6	1.78944	0.894720	−	0.446628i	$-0.147375\pi$
$461$	8.16525e6	1.78944	0.894720	+	0.446628i	$0.147375\pi$
$462$	0	0
$463$	5.65614e6	1.22622	0.613109	−	0.789998i	$-0.289919\pi$
$463$	5.65614e6	1.22622	0.613109	+	0.789998i	$0.289919\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−8.77216e6	−1.86129	−0.930646	−	0.365921i	$-0.880754\pi$
$467$	−8.77216e6	−1.86129	−0.930646	+	0.365921i	$0.880754\pi$
$468$	0	0
$469$	4.69936e6	0.986522
$470$	0	0
$471$	−1.03590e7	−2.15162
$472$	0	0
$473$	1.14716e6	0.235760
$474$	0	0
$475$	0	0
$476$	0	0
$477$	2.95641e6	0.594935
$478$	0	0
$479$	417978.	0.0832367	0.0416184	−	0.999134i	$-0.486749\pi$
$479$	417978.	0.0832367	0.0416184	+	0.999134i	$0.486749\pi$
$480$	0	0
$481$	−2.84616e6	−0.560915
$482$	0	0
$483$	−4.11666e6	−0.802929
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7.07076e6	1.35096	0.675482	−	0.737376i	$-0.263936\pi$
$487$	7.07076e6	1.35096	0.675482	+	0.737376i	$0.263936\pi$
$488$	0	0
$489$	−3.92168e6	−0.741652
$490$	0	0
$491$	−1.00127e7	−1.87433	−0.937166	−	0.348883i	$-0.886561\pi$
$491$	−1.00127e7	−1.87433	−0.937166	+	0.348883i	$0.886561\pi$
$492$	0	0
$493$	−584816.	−0.108368
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−8.32571e6	−1.51193
$498$	0	0
$499$	−7.74452e6	−1.39233	−0.696166	−	0.717881i	$-0.745112\pi$
$499$	−7.74452e6	−1.39233	−0.696166	+	0.717881i	$0.745112\pi$
$500$	0	0
$501$	6.76009e6	1.20326
$502$	0	0
$503$	2.20376e6	0.388370	0.194185	−	0.980965i	$-0.437794\pi$
$503$	2.20376e6	0.388370	0.194185	+	0.980965i	$0.437794\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	7.00379e6	1.21008
$508$	0	0
$509$	1.40053e6	0.239607	0.119803	−	0.992798i	$-0.461774\pi$
$509$	1.40053e6	0.239607	0.119803	+	0.992798i	$0.461774\pi$
$510$	0	0
$511$	9.22823e6	1.56339
$512$	0	0
$513$	5.98049e6	1.00333
$514$	0	0
$515$	0	0
$516$	0	0
$517$	9.64881e6	1.58762
$518$	0	0
$519$	1.79245e7	2.92099
$520$	0	0
$521$	1.60571e6	0.259162	0.129581	−	0.991569i	$-0.458637\pi$
$521$	1.60571e6	0.259162	0.129581	+	0.991569i	$0.458637\pi$
$522$	0	0
$523$	5.80485e6	0.927977	0.463988	−	0.885841i	$-0.346418\pi$
$523$	5.80485e6	0.927977	0.463988	+	0.885841i	$0.346418\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.19087e6	0.186784
$528$	0	0
$529$	−5.53740e6	−0.860334
$530$	0	0
$531$	−4.29057e6	−0.660357
$532$	0	0
$533$	−4.30714e6	−0.656706
$534$	0	0
$535$	0	0
$536$	0	0
$537$	351356.	0.0525790
$538$	0	0
$539$	−1.01637e7	−1.50689
$540$	0	0
$541$	−9.37930e6	−1.37777	−0.688886	−	0.724870i	$-0.741900\pi$
$541$	−9.37930e6	−1.37777	−0.688886	+	0.724870i	$0.741900\pi$
$542$	0	0
$543$	5.87146e6	0.854568
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.33685e6	0.333936	0.166968	−	0.985962i	$-0.446602\pi$
$547$	2.33685e6	0.333936	0.166968	+	0.985962i	$0.446602\pi$
$548$	0	0
$549$	1.46386e7	2.07286
$550$	0	0
$551$	3.90712e6	0.548248
$552$	0	0
$553$	7.07166e6	0.983351
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−7.02238e6	−0.959061	−0.479530	−	0.877525i	$-0.659193\pi$
$557$	−7.02238e6	−0.959061	−0.479530	+	0.877525i	$0.659193\pi$
$558$	0	0
$559$	499891.	0.0676622
$560$	0	0
$561$	6.05386e6	0.812129
$562$	0	0
$563$	1.81010e6	0.240675	0.120338	−	0.992733i	$-0.461602\pi$
$563$	1.81010e6	0.240675	0.120338	+	0.992733i	$0.461602\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−4.71613e6	−0.616068
$568$	0	0
$569$	5.62751e6	0.728678	0.364339	−	0.931266i	$-0.381295\pi$
$569$	5.62751e6	0.728678	0.364339	+	0.931266i	$0.381295\pi$
$570$	0	0
$571$	−2.16702e6	−0.278145	−0.139073	−	0.990282i	$-0.544412\pi$
$571$	−2.16702e6	−0.278145	−0.139073	+	0.990282i	$0.544412\pi$
$572$	0	0
$573$	−1.83140e6	−0.233022
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.04992e7	1.31286	0.656430	−	0.754387i	$-0.272066\pi$
$577$	1.04992e7	1.31286	0.656430	+	0.754387i	$0.272066\pi$
$578$	0	0
$579$	−8.93504e6	−1.10764
$580$	0	0
$581$	1.25095e7	1.53744
$582$	0	0
$583$	−5.68972e6	−0.693297
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.47459e7	1.76634	0.883172	−	0.469050i	$-0.155404\pi$
$587$	1.47459e7	1.76634	0.883172	+	0.469050i	$0.155404\pi$
$588$	0	0
$589$	−7.95614e6	−0.944962
$590$	0	0
$591$	5.16192e6	0.607915
$592$	0	0
$593$	4.85519e6	0.566982	0.283491	−	0.958975i	$-0.408507\pi$
$593$	4.85519e6	0.566982	0.283491	+	0.958975i	$0.408507\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.12503e7	−1.29189
$598$	0	0
$599$	−4.48204e6	−0.510398	−0.255199	−	0.966889i	$-0.582141\pi$
$599$	−4.48204e6	−0.510398	−0.255199	+	0.966889i	$0.582141\pi$
$600$	0	0
$601$	4.29562e6	0.485110	0.242555	−	0.970138i	$-0.422015\pi$
$601$	4.29562e6	0.485110	0.242555	+	0.970138i	$0.422015\pi$
$602$	0	0
$603$	8.87372e6	0.993831
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.05706e7	1.16447	0.582235	−	0.813020i	$-0.302178\pi$
$607$	1.05706e7	1.16447	0.582235	+	0.813020i	$0.302178\pi$
$608$	0	0
$609$	6.61820e6	0.723097
$610$	0	0
$611$	4.20461e6	0.455641
$612$	0	0
$613$	1.30960e7	1.40763	0.703816	−	0.710383i	$-0.251478\pi$
$613$	1.30960e7	1.40763	0.703816	+	0.710383i	$0.251478\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.69519e7	1.79270	0.896348	−	0.443351i	$-0.146211\pi$
$617$	1.69519e7	1.79270	0.896348	+	0.443351i	$0.146211\pi$
$618$	0	0
$619$	1.27214e7	1.33447	0.667236	−	0.744846i	$-0.267477\pi$
$619$	1.27214e7	1.33447	0.667236	+	0.744846i	$0.267477\pi$
$620$	0	0
$621$	−2.21210e6	−0.230184
$622$	0	0
$623$	−2.35498e6	−0.243090
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−4.04454e7	−4.10866
$628$	0	0
$629$	3.83355e6	0.386344
$630$	0	0
$631$	8.32823e6	0.832682	0.416341	−	0.909209i	$-0.363312\pi$
$631$	8.32823e6	0.832682	0.416341	+	0.909209i	$0.363312\pi$
$632$	0	0
$633$	5.50862e6	0.546429
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−4.42900e6	−0.432471
$638$	0	0
$639$	−1.57213e7	−1.52313
$640$	0	0
$641$	−4.18862e6	−0.402648	−0.201324	−	0.979525i	$-0.564524\pi$
$641$	−4.18862e6	−0.402648	−0.201324	+	0.979525i	$0.564524\pi$
$642$	0	0
$643$	271198.	0.0258677	0.0129339	−	0.999916i	$-0.495883\pi$
$643$	271198.	0.0258677	0.0129339	+	0.999916i	$0.495883\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.01692e7	−0.955052	−0.477526	−	0.878618i	$-0.658466\pi$
$647$	−1.01692e7	−0.955052	−0.477526	+	0.878618i	$0.658466\pi$
$648$	0	0
$649$	8.25735e6	0.769536
$650$	0	0
$651$	−1.34768e7	−1.24633
$652$	0	0
$653$	−8.95062e6	−0.821429	−0.410714	−	0.911764i	$-0.634721\pi$
$653$	−8.95062e6	−0.821429	−0.410714	+	0.911764i	$0.634721\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.74255e7	1.57497
$658$	0	0
$659$	1.07182e7	0.961409	0.480705	−	0.876883i	$-0.340381\pi$
$659$	1.07182e7	0.961409	0.480705	+	0.876883i	$0.340381\pi$
$660$	0	0
$661$	4.65131e6	0.414068	0.207034	−	0.978334i	$-0.433619\pi$
$661$	4.65131e6	0.414068	0.207034	+	0.978334i	$0.433619\pi$
$662$	0	0
$663$	2.63806e6	0.233078
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.44519e6	−0.125779
$668$	0	0
$669$	−2.34805e7	−2.02835
$670$	0	0
$671$	−2.81725e7	−2.41557
$672$	0	0
$673$	9.92139e6	0.844374	0.422187	−	0.906509i	$-0.361263\pi$
$673$	9.92139e6	0.844374	0.422187	+	0.906509i	$0.361263\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−6.45936e6	−0.541649	−0.270824	−	0.962629i	$-0.587296\pi$
$677$	−6.45936e6	−0.541649	−0.270824	+	0.962629i	$0.587296\pi$
$678$	0	0
$679$	4.72321e6	0.393154
$680$	0	0
$681$	−8.61888e6	−0.712169
$682$	0	0
$683$	−1.26684e7	−1.03913	−0.519564	−	0.854432i	$-0.673905\pi$
$683$	−1.26684e7	−1.03913	−0.519564	+	0.854432i	$0.673905\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7.12552e6	0.576003
$688$	0	0
$689$	−2.47938e6	−0.198973
$690$	0	0
$691$	4.07160e6	0.324392	0.162196	−	0.986759i	$-0.448142\pi$
$691$	4.07160e6	0.324392	0.162196	+	0.986759i	$0.448142\pi$
$692$	0	0
$693$	−3.99375e7	−3.15898
$694$	0	0
$695$	0	0
$696$	0	0
$697$	5.80137e6	0.452323
$698$	0	0
$699$	−1.18850e7	−0.920043
$700$	0	0
$701$	−2.11804e7	−1.62795	−0.813973	−	0.580903i	$-0.802700\pi$
$701$	−2.11804e7	−1.62795	−0.813973	+	0.580903i	$0.802700\pi$
$702$	0	0
$703$	−2.56117e7	−1.95457
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.32794e6	0.0999150
$708$	0	0
$709$	−1.87573e7	−1.40138	−0.700689	−	0.713467i	$-0.747124\pi$
$709$	−1.87573e7	−1.40138	−0.700689	+	0.713467i	$0.747124\pi$
$710$	0	0
$711$	1.33533e7	0.990637
$712$	0	0
$713$	2.94287e6	0.216794
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.73303e6	0.125895
$718$	0	0
$719$	1.82800e7	1.31872	0.659362	−	0.751826i	$-0.270826\pi$
$719$	1.82800e7	1.31872	0.659362	+	0.751826i	$0.270826\pi$
$720$	0	0
$721$	−8.00788e6	−0.573693
$722$	0	0
$723$	−1.48282e7	−1.05498
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−6.45855e6	−0.453210	−0.226605	−	0.973987i	$-0.572763\pi$
$727$	−6.45855e6	−0.453210	−0.226605	+	0.973987i	$0.572763\pi$
$728$	0	0
$729$	−2.25906e7	−1.57438
$730$	0	0
$731$	−673313.	−0.0466040
$732$	0	0
$733$	−1.88365e6	−0.129491	−0.0647454	−	0.997902i	$-0.520624\pi$
$733$	−1.88365e6	−0.129491	−0.0647454	+	0.997902i	$0.520624\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.70778e7	−1.15814
$738$	0	0
$739$	2.24002e7	1.50883	0.754417	−	0.656395i	$-0.227920\pi$
$739$	2.24002e7	1.50883	0.754417	+	0.656395i	$0.227920\pi$
$740$	0	0
$741$	−1.76247e7	−1.17917
$742$	0	0
$743$	2.22827e6	0.148080	0.0740400	−	0.997255i	$-0.476411\pi$
$743$	2.22827e6	0.148080	0.0740400	+	0.997255i	$0.476411\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.36214e7	1.54883
$748$	0	0
$749$	3.50630e7	2.28373
$750$	0	0
$751$	1.92986e7	1.24861	0.624303	−	0.781183i	$-0.285383\pi$
$751$	1.92986e7	1.24861	0.624303	+	0.781183i	$0.285383\pi$
$752$	0	0
$753$	5.89641e6	0.378966
$754$	0	0
$755$	0	0
$756$	0	0
$757$	7.22211e6	0.458062	0.229031	−	0.973419i	$-0.426444\pi$
$757$	7.22211e6	0.458062	0.229031	+	0.973419i	$0.426444\pi$
$758$	0	0
$759$	1.49602e7	0.942612
$760$	0	0
$761$	1.08001e7	0.676028	0.338014	−	0.941141i	$-0.390245\pi$
$761$	1.08001e7	0.676028	0.338014	+	0.941141i	$0.390245\pi$
$762$	0	0
$763$	3.84544e7	2.39130
$764$	0	0
$765$	0	0
$766$	0	0
$767$	3.59827e6	0.220854
$768$	0	0
$769$	5.50354e6	0.335603	0.167802	−	0.985821i	$-0.446333\pi$
$769$	5.50354e6	0.335603	0.167802	+	0.985821i	$0.446333\pi$
$770$	0	0
$771$	2.29731e7	1.39182
$772$	0	0
$773$	−2.91304e7	−1.75347	−0.876733	−	0.480977i	$-0.840282\pi$
$773$	−2.91304e7	−1.75347	−0.876733	+	0.480977i	$0.840282\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−4.33832e7	−2.57792
$778$	0	0
$779$	−3.87586e7	−2.28836
$780$	0	0
$781$	3.02562e7	1.77495
$782$	0	0
$783$	3.55631e6	0.207298
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−5.40251e6	−0.310927	−0.155464	−	0.987842i	$-0.549687\pi$
$787$	−5.40251e6	−0.310927	−0.155464	+	0.987842i	$0.549687\pi$
$788$	0	0
$789$	−6.24005e6	−0.356858
$790$	0	0
$791$	2.67919e7	1.52251
$792$	0	0
$793$	−1.22766e7	−0.693259
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2.19820e7	−1.22580	−0.612902	−	0.790159i	$-0.709998\pi$
$797$	−2.19820e7	−1.22580	−0.612902	+	0.790159i	$0.709998\pi$
$798$	0	0
$799$	−5.66327e6	−0.313834
$800$	0	0
$801$	−4.44686e6	−0.244891
$802$	0	0
$803$	−3.35360e7	−1.83536
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−2.15969e7	−1.16737
$808$	0	0
$809$	3.37377e7	1.81236	0.906178	−	0.422896i	$-0.138986\pi$
$809$	3.37377e7	1.81236	0.906178	+	0.422896i	$0.138986\pi$
$810$	0	0
$811$	3.30907e6	0.176666	0.0883331	−	0.996091i	$-0.471846\pi$
$811$	3.30907e6	0.176666	0.0883331	+	0.996091i	$0.471846\pi$
$812$	0	0
$813$	−1.14451e7	−0.607289
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.49836e6	0.235776
$818$	0	0
$819$	−1.74033e7	−0.906615
$820$	0	0
$821$	4.00294e6	0.207263	0.103631	−	0.994616i	$-0.466954\pi$
$821$	4.00294e6	0.207263	0.103631	+	0.994616i	$0.466954\pi$
$822$	0	0
$823$	1.17703e7	0.605740	0.302870	−	0.953032i	$-0.402055\pi$
$823$	1.17703e7	0.605740	0.302870	+	0.953032i	$0.402055\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.12347e7	0.571214	0.285607	−	0.958347i	$-0.407805\pi$
$827$	1.12347e7	0.571214	0.285607	+	0.958347i	$0.407805\pi$
$828$	0	0
$829$	2.44928e7	1.23781	0.618903	−	0.785468i	$-0.287577\pi$
$829$	2.44928e7	1.23781	0.618903	+	0.785468i	$0.287577\pi$
$830$	0	0
$831$	3.44578e6	0.173095
$832$	0	0
$833$	5.96550e6	0.297875
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−7.24178e6	−0.357299
$838$	0	0
$839$	−1.20271e6	−0.0589868	−0.0294934	−	0.999565i	$-0.509389\pi$
$839$	−1.20271e6	−0.0589868	−0.0294934	+	0.999565i	$0.509389\pi$
$840$	0	0
$841$	−1.81878e7	−0.886726
$842$	0	0
$843$	1.03273e7	0.500515
$844$	0	0
$845$	0	0
$846$	0	0
$847$	4.78917e7	2.29378
$848$	0	0
$849$	−5.55742e7	−2.64609
$850$	0	0
$851$	9.47341e6	0.448418
$852$	0	0
$853$	3.92510e7	1.84705	0.923523	−	0.383543i	$-0.125296\pi$
$853$	3.92510e7	1.84705	0.923523	+	0.383543i	$0.125296\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.29324e6	−0.0601488	−0.0300744	−	0.999548i	$-0.509574\pi$
$857$	−1.29324e6	−0.0601488	−0.0300744	+	0.999548i	$0.509574\pi$
$858$	0	0
$859$	2.87256e7	1.32827	0.664134	−	0.747613i	$-0.268800\pi$
$859$	2.87256e7	1.32827	0.664134	+	0.747613i	$0.268800\pi$
$860$	0	0
$861$	−6.56525e7	−3.01817
$862$	0	0
$863$	8.57964e6	0.392141	0.196070	−	0.980590i	$-0.437182\pi$
$863$	8.57964e6	0.392141	0.196070	+	0.980590i	$0.437182\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	3.07197e7	1.38793
$868$	0	0
$869$	−2.56989e7	−1.15442
$870$	0	0
$871$	−7.44190e6	−0.332383
$872$	0	0
$873$	8.91875e6	0.396067
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.79556e7	1.22735	0.613676	−	0.789558i	$-0.289690\pi$
$877$	2.79556e7	1.22735	0.613676	+	0.789558i	$0.289690\pi$
$878$	0	0
$879$	3.84739e7	1.67955
$880$	0	0
$881$	−5.36409e6	−0.232839	−0.116420	−	0.993200i	$-0.537142\pi$
$881$	−5.36409e6	−0.232839	−0.116420	+	0.993200i	$0.537142\pi$
$882$	0	0
$883$	−4.40226e7	−1.90009	−0.950044	−	0.312117i	$-0.898962\pi$
$883$	−4.40226e7	−1.90009	−0.950044	+	0.312117i	$0.898962\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1.88424e7	−0.804133	−0.402067	−	0.915610i	$-0.631708\pi$
$887$	−1.88424e7	−0.804133	−0.402067	+	0.915610i	$0.631708\pi$
$888$	0	0
$889$	−2.10017e7	−0.891253
$890$	0	0
$891$	1.71387e7	0.723243
$892$	0	0
$893$	3.78359e7	1.58773
$894$	0	0
$895$	0	0
$896$	0	0
$897$	6.51913e6	0.270526
$898$	0	0
$899$	−4.73113e6	−0.195239
$900$	0	0
$901$	3.33952e6	0.137048
$902$	0	0
$903$	7.61969e6	0.310970
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.40217e7	−1.77684	−0.888421	−	0.459029i	$-0.848197\pi$
$907$	−4.40217e7	−1.77684	−0.888421	+	0.459029i	$0.848197\pi$
$908$	0	0
$909$	2.50753e6	0.100655
$910$	0	0
$911$	−3.02447e7	−1.20741	−0.603703	−	0.797209i	$-0.706309\pi$
$911$	−3.02447e7	−1.20741	−0.603703	+	0.797209i	$0.706309\pi$
$912$	0	0
$913$	−4.54601e7	−1.80490
$914$	0	0
$915$	0	0
$916$	0	0
$917$	6.27967e7	2.46612
$918$	0	0
$919$	−2.35173e7	−0.918543	−0.459272	−	0.888296i	$-0.651890\pi$
$919$	−2.35173e7	−0.918543	−0.459272	+	0.888296i	$0.651890\pi$
$920$	0	0
$921$	−7.78895e6	−0.302573
$922$	0	0
$923$	1.31846e7	0.509404
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.51211e7	−0.577943
$928$	0	0
$929$	4.07876e7	1.55056	0.775281	−	0.631616i	$-0.217608\pi$
$929$	4.07876e7	1.55056	0.775281	+	0.631616i	$0.217608\pi$
$930$	0	0
$931$	−3.98551e7	−1.50699
$932$	0	0
$933$	1.59362e7	0.599350
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.42613e7	0.530654	0.265327	−	0.964159i	$-0.414520\pi$
$937$	1.42613e7	0.530654	0.265327	+	0.964159i	$0.414520\pi$
$938$	0	0
$939$	2.78504e7	1.03078
$940$	0	0
$941$	2.24934e7	0.828095	0.414048	−	0.910255i	$-0.364115\pi$
$941$	2.24934e7	0.828095	0.414048	+	0.910255i	$0.364115\pi$
$942$	0	0
$943$	1.43363e7	0.524997
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.45401e7	−0.889205	−0.444603	−	0.895728i	$-0.646655\pi$
$947$	−2.45401e7	−0.889205	−0.444603	+	0.895728i	$0.646655\pi$
$948$	0	0
$949$	−1.46138e7	−0.526742
$950$	0	0
$951$	−2.90375e7	−1.04114
$952$	0	0
$953$	4.06466e7	1.44975	0.724873	−	0.688882i	$-0.241898\pi$
$953$	4.06466e7	1.44975	0.724873	+	0.688882i	$0.241898\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.40509e7	−0.848891
$958$	0	0
$959$	−3.95324e7	−1.38806
$960$	0	0
$961$	−1.89951e7	−0.663486
$962$	0	0
$963$	6.62089e7	2.30065
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.22263e7	−1.45217	−0.726084	−	0.687606i	$-0.758662\pi$
$967$	−4.22263e7	−1.45217	−0.726084	+	0.687606i	$0.758662\pi$
$968$	0	0
$969$	2.37390e7	0.812183
$970$	0	0
$971$	4.00999e7	1.36488	0.682442	−	0.730940i	$-0.260918\pi$
$971$	4.00999e7	1.36488	0.682442	+	0.730940i	$0.260918\pi$
$972$	0	0
$973$	2.27712e7	0.771087
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.75634e7	0.588671	0.294336	−	0.955702i	$-0.404902\pi$
$977$	1.75634e7	0.588671	0.294336	+	0.955702i	$0.404902\pi$
$978$	0	0
$979$	8.55813e6	0.285379
$980$	0	0
$981$	7.26127e7	2.40902
$982$	0	0
$983$	3.46862e6	0.114491	0.0572456	−	0.998360i	$-0.481768\pi$
$983$	3.46862e6	0.114491	0.0572456	+	0.998360i	$0.481768\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6.40896e7	2.09409
$988$	0	0
$989$	−1.66388e6	−0.0540918
$990$	0	0
$991$	−1.88098e7	−0.608414	−0.304207	−	0.952606i	$-0.598391\pi$
$991$	−1.88098e7	−0.608414	−0.304207	+	0.952606i	$0.598391\pi$
$992$	0	0
$993$	−4.40341e7	−1.41715
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.29454e7	−0.412455	−0.206227	−	0.978504i	$-0.566119\pi$
$997$	−1.29454e7	−0.412455	−0.206227	+	0.978504i	$0.566119\pi$
$998$	0	0
$999$	−2.33121e7	−0.739039

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	200.6.a.k.1.1		4
4.3	odd	2		400.6.a.z.1.4		4
5.2	odd	4		40.6.c.a.9.7	yes	8
5.3	odd	4		40.6.c.a.9.2	✓	8
5.4	even	2		200.6.a.j.1.4		4
15.2	even	4		360.6.f.b.289.2		8
15.8	even	4		360.6.f.b.289.1		8
20.3	even	4		80.6.c.d.49.7		8
20.7	even	4		80.6.c.d.49.2		8
20.19	odd	2		400.6.a.ba.1.1		4
40.3	even	4		320.6.c.i.129.2		8
40.13	odd	4		320.6.c.j.129.7		8
40.27	even	4		320.6.c.i.129.7		8
40.37	odd	4		320.6.c.j.129.2		8
60.23	odd	4		720.6.f.n.289.1		8
60.47	odd	4		720.6.f.n.289.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.c.a.9.2	✓	8	5.3	odd	4
40.6.c.a.9.7	yes	8	5.2	odd	4
80.6.c.d.49.2		8	20.7	even	4
80.6.c.d.49.7		8	20.3	even	4
200.6.a.j.1.4		4	5.4	even	2
200.6.a.k.1.1		4	1.1	even	1	trivial
320.6.c.i.129.2		8	40.3	even	4
320.6.c.i.129.7		8	40.27	even	4
320.6.c.j.129.2		8	40.37	odd	4
320.6.c.j.129.7		8	40.13	odd	4
360.6.f.b.289.1		8	15.8	even	4
360.6.f.b.289.2		8	15.2	even	4
400.6.a.z.1.4		4	4.3	odd	2
400.6.a.ba.1.1		4	20.19	odd	2
720.6.f.n.289.1		8	60.23	odd	4
720.6.f.n.289.2		8	60.47	odd	4

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:	\( N \)	\(=\)	\( 200 = 2^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	200.a (trivial)

\(f(q)\)	\(=\)	\(q-24.1383 q^{3} +179.876 q^{7} +339.657 q^{9} +O(q^{10})\)	\(q-24.1383 q^{3} +179.876 q^{7} +339.657 q^{9} -653.681 q^{11} -284.851 q^{13} +383.672 q^{17} -2563.29 q^{19} -4341.90 q^{21} +948.124 q^{23} -2333.13 q^{27} -1524.26 q^{29} +3103.88 q^{31} +15778.7 q^{33} +9991.75 q^{37} +6875.83 q^{39} +15120.7 q^{41} -1754.92 q^{43} -14760.7 q^{47} +15548.5 q^{49} -9261.18 q^{51} +8704.12 q^{53} +61873.3 q^{57} -12632.1 q^{59} +43098.3 q^{61} +61096.2 q^{63} +26125.5 q^{67} -22886.1 q^{69} -46285.8 q^{71} +51303.2 q^{73} -117582. q^{77} +39314.0 q^{79} -26218.8 q^{81} +69544.8 q^{83} +36793.1 q^{87} -13092.2 q^{89} -51238.0 q^{91} -74922.5 q^{93} +26258.1 q^{97} -222027. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9	\( 4 q + 4 q^{3} + 148 q^{7} + 500 q^{9} - 368 q^{11} + 440 q^{13} - 672 q^{17} - 688 q^{19} + 992 q^{21} + 4492 q^{23} + 8152 q^{27} - 2936 q^{29} + 2112 q^{31} + 26864 q^{33} + 8792 q^{37} + 1504 q^{39} + 11800 q^{41} + 48276 q^{43} + 14724 q^{47} + 22500 q^{49} - 62400 q^{51} + 84296 q^{53} + 71024 q^{57} - 45840 q^{59} + 61928 q^{61} + 186292 q^{63} + 72700 q^{67} + 38368 q^{69} - 62816 q^{71} + 133072 q^{73} + 11440 q^{77} - 21632 q^{79} + 204836 q^{81} + 74660 q^{83} - 12472 q^{87} + 20952 q^{89} - 243808 q^{91} + 105600 q^{93} + 59456 q^{97} - 133424 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 148 * q^7 + 500 * q^9 - 368 * q^11 + 440 * q^13 - 672 * q^17 - 688 * q^19 + 992 * q^21 + 4492 * q^23 + 8152 * q^27 - 2936 * q^29 + 2112 * q^31 + 26864 * q^33 + 8792 * q^37 + 1504 * q^39 + 11800 * q^41 + 48276 * q^43 + 14724 * q^47 + 22500 * q^49 - 62400 * q^51 + 84296 * q^53 + 71024 * q^57 - 45840 * q^59 + 61928 * q^61 + 186292 * q^63 + 72700 * q^67 + 38368 * q^69 - 62816 * q^71 + 133072 * q^73 + 11440 * q^77 - 21632 * q^79 + 204836 * q^81 + 74660 * q^83 - 12472 * q^87 + 20952 * q^89 - 243808 * q^91 + 105600 * q^93 + 59456 * q^97 - 133424 * q^99

Introduction

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