LMFDB - Embedded newform 400.6.a.t.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(1,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, 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("400.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.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:	$64.1535279252$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{11}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 11 $ x^2 - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.31662$ of defining polynomial
Character	$\chi$	$=$	400.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+19.8997 q^{3} -59.6992 q^{7} +153.000 q^{9} +O(q^{10})$	$q+19.8997 q^{3} -59.6992 q^{7} +153.000 q^{9} -252.000 q^{11} -119.398 q^{13} +689.858 q^{17} -220.000 q^{19} -1188.00 q^{21} -2434.40 q^{23} -1790.98 q^{27} +6930.00 q^{29} -6752.00 q^{31} -5014.74 q^{33} -13969.6 q^{37} -2376.00 q^{39} -198.000 q^{41} +417.895 q^{43} -10540.2 q^{47} -13243.0 q^{49} +13728.0 q^{51} -5823.99 q^{53} -4377.94 q^{57} -24660.0 q^{59} -5698.00 q^{61} -9133.98 q^{63} -43640.1 q^{67} -48444.0 q^{69} -53352.0 q^{71} +70922.7 q^{73} +15044.2 q^{77} +51920.0 q^{79} -72819.0 q^{81} +61841.8 q^{83} +137905. q^{87} +9990.00 q^{89} +7128.00 q^{91} -134363. q^{93} +101250. q^{97} -38556.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 306 q^{9}+O(q^{10}) $ 2 * q + 306 * q^9	$ 2 q + 306 q^{9} - 504 q^{11} - 440 q^{19} - 2376 q^{21} + 13860 q^{29} - 13504 q^{31} - 4752 q^{39} - 396 q^{41} - 26486 q^{49} + 27456 q^{51} - 49320 q^{59} - 11396 q^{61} - 96888 q^{69} - 106704 q^{71} + 103840 q^{79} - 145638 q^{81} + 19980 q^{89} + 14256 q^{91} - 77112 q^{99}+O(q^{100}) $ 2 * q + 306 * q^9 - 504 * q^11 - 440 * q^19 - 2376 * q^21 + 13860 * q^29 - 13504 * q^31 - 4752 * q^39 - 396 * q^41 - 26486 * q^49 + 27456 * q^51 - 49320 * q^59 - 11396 * q^61 - 96888 * q^69 - 106704 * q^71 + 103840 * q^79 - 145638 * q^81 + 19980 * q^89 + 14256 * q^91 - 77112 * 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
$3$	19.8997	1.27657	0.638285	−	0.769800i	$-0.279644\pi$
$3$	19.8997	1.27657	0.638285	+	0.769800i	$0.279644\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−59.6992	−0.460494	−0.230247	−	0.973132i	$-0.573953\pi$
$7$	−59.6992	−0.460494	−0.230247	+	0.973132i	$0.573953\pi$
$8$	0	0
$9$	153.000	0.629630
$10$	0	0
$11$	−252.000	−0.627941	−0.313970	−	0.949433i	$-0.601659\pi$
$11$	−252.000	−0.627941	−0.313970	+	0.949433i	$0.601659\pi$
$12$	0	0
$13$	−119.398	−0.195948	−0.0979739	−	0.995189i	$-0.531236\pi$
$13$	−119.398	−0.195948	−0.0979739	+	0.995189i	$0.531236\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	689.858	0.578945	0.289473	−	0.957186i	$-0.406520\pi$
$17$	689.858	0.578945	0.289473	+	0.957186i	$0.406520\pi$
$18$	0	0
$19$	−220.000	−0.139810	−0.0699051	−	0.997554i	$-0.522270\pi$
$19$	−220.000	−0.139810	−0.0699051	+	0.997554i	$0.522270\pi$
$20$	0	0
$21$	−1188.00	−0.587852
$22$	0	0
$23$	−2434.40	−0.959561	−0.479781	−	0.877388i	$-0.659284\pi$
$23$	−2434.40	−0.959561	−0.479781	+	0.877388i	$0.659284\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1790.98	−0.472804
$28$	0	0
$29$	6930.00	1.53016	0.765082	−	0.643932i	$-0.222698\pi$
$29$	6930.00	1.53016	0.765082	+	0.643932i	$0.222698\pi$
$30$	0	0
$31$	−6752.00	−1.26191	−0.630955	−	0.775820i	$-0.717337\pi$
$31$	−6752.00	−1.26191	−0.630955	+	0.775820i	$0.717337\pi$
$32$	0	0
$33$	−5014.74	−0.801610
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−13969.6	−1.67757	−0.838785	−	0.544464i	$-0.816733\pi$
$37$	−13969.6	−1.67757	−0.838785	+	0.544464i	$0.816733\pi$
$38$	0	0
$39$	−2376.00	−0.250141
$40$	0	0
$41$	−198.000	−0.0183952	−0.00919762	−	0.999958i	$-0.502928\pi$
$41$	−198.000	−0.0183952	−0.00919762	+	0.999958i	$0.502928\pi$
$42$	0	0
$43$	417.895	0.0344664	0.0172332	−	0.999851i	$-0.494514\pi$
$43$	417.895	0.0344664	0.0172332	+	0.999851i	$0.494514\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−10540.2	−0.695994	−0.347997	−	0.937496i	$-0.613138\pi$
$47$	−10540.2	−0.695994	−0.347997	+	0.937496i	$0.613138\pi$
$48$	0	0
$49$	−13243.0	−0.787945
$50$	0	0
$51$	13728.0	0.739064
$52$	0	0
$53$	−5823.99	−0.284794	−0.142397	−	0.989810i	$-0.545481\pi$
$53$	−5823.99	−0.284794	−0.142397	+	0.989810i	$0.545481\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−4377.94	−0.178477
$58$	0	0
$59$	−24660.0	−0.922281	−0.461140	−	0.887327i	$-0.652560\pi$
$59$	−24660.0	−0.922281	−0.461140	+	0.887327i	$0.652560\pi$
$60$	0	0
$61$	−5698.00	−0.196064	−0.0980320	−	0.995183i	$-0.531255\pi$
$61$	−5698.00	−0.196064	−0.0980320	+	0.995183i	$0.531255\pi$
$62$	0	0
$63$	−9133.98	−0.289941
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−43640.1	−1.18768	−0.593840	−	0.804583i	$-0.702389\pi$
$67$	−43640.1	−1.18768	−0.593840	+	0.804583i	$0.702389\pi$
$68$	0	0
$69$	−48444.0	−1.22495
$70$	0	0
$71$	−53352.0	−1.25604	−0.628022	−	0.778196i	$-0.716135\pi$
$71$	−53352.0	−1.25604	−0.628022	+	0.778196i	$0.716135\pi$
$72$	0	0
$73$	70922.7	1.55768	0.778840	−	0.627223i	$-0.215808\pi$
$73$	70922.7	1.55768	0.778840	+	0.627223i	$0.215808\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15044.2	0.289163
$78$	0	0
$79$	51920.0	0.935981	0.467990	−	0.883734i	$-0.344978\pi$
$79$	51920.0	0.935981	0.467990	+	0.883734i	$0.344978\pi$
$80$	0	0
$81$	−72819.0	−1.23320
$82$	0	0
$83$	61841.8	0.985342	0.492671	−	0.870216i	$-0.336021\pi$
$83$	61841.8	0.985342	0.492671	+	0.870216i	$0.336021\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	137905.	1.95336
$88$	0	0
$89$	9990.00	0.133687	0.0668437	−	0.997763i	$-0.478707\pi$
$89$	9990.00	0.133687	0.0668437	+	0.997763i	$0.478707\pi$
$90$	0	0
$91$	7128.00	0.0902328
$92$	0	0
$93$	−134363.	−1.61092
$94$	0	0
$95$	0	0
$96$	0	0
$97$	101250.	1.09261	0.546305	−	0.837586i	$-0.316034\pi$
$97$	101250.	1.09261	0.546305	+	0.837586i	$0.316034\pi$
$98$	0	0
$99$	−38556.0	−0.395370
$100$	0	0
$101$	−109098.	−1.06418	−0.532088	−	0.846689i	$-0.678592\pi$
$101$	−109098.	−1.06418	−0.532088	+	0.846689i	$0.678592\pi$
$102$	0	0
$103$	−70624.2	−0.655935	−0.327967	−	0.944689i	$-0.606364\pi$
$103$	−70624.2	−0.655935	−0.327967	+	0.944689i	$0.606364\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	97117.4	0.820045	0.410022	−	0.912075i	$-0.365521\pi$
$107$	97117.4	0.820045	0.410022	+	0.912075i	$0.365521\pi$
$108$	0	0
$109$	21010.0	0.169379	0.0846895	−	0.996407i	$-0.473010\pi$
$109$	21010.0	0.169379	0.0846895	+	0.996407i	$0.473010\pi$
$110$	0	0
$111$	−277992.	−2.14153
$112$	0	0
$113$	105018.	0.773688	0.386844	−	0.922145i	$-0.373565\pi$
$113$	105018.	0.773688	0.386844	+	0.922145i	$0.373565\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−18268.0	−0.123375
$118$	0	0
$119$	−41184.0	−0.266601
$120$	0	0
$121$	−97547.0	−0.605690
$122$	0	0
$123$	−3940.15	−0.0234828
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−87220.6	−0.479855	−0.239927	−	0.970791i	$-0.577124\pi$
$127$	−87220.6	−0.479855	−0.239927	+	0.970791i	$0.577124\pi$
$128$	0	0
$129$	8316.00	0.0439987
$130$	0	0
$131$	−192852.	−0.981852	−0.490926	−	0.871201i	$-0.663341\pi$
$131$	−192852.	−0.981852	−0.490926	+	0.871201i	$0.663341\pi$
$132$	0	0
$133$	13133.8	0.0643817
$134$	0	0
$135$	0	0
$136$	0	0
$137$	143570.	0.653525	0.326763	−	0.945106i	$-0.394042\pi$
$137$	143570.	0.653525	0.326763	+	0.945106i	$0.394042\pi$
$138$	0	0
$139$	−318340.	−1.39751	−0.698754	−	0.715362i	$-0.746262\pi$
$139$	−318340.	−1.39751	−0.698754	+	0.715362i	$0.746262\pi$
$140$	0	0
$141$	−209748.	−0.888485
$142$	0	0
$143$	30088.4	0.123044
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−263532.	−1.00587
$148$	0	0
$149$	−84150.0	−0.310519	−0.155260	−	0.987874i	$-0.549621\pi$
$149$	−84150.0	−0.310519	−0.155260	+	0.987874i	$0.549621\pi$
$150$	0	0
$151$	155848.	0.556236	0.278118	−	0.960547i	$-0.410289\pi$
$151$	155848.	0.556236	0.278118	+	0.960547i	$0.410289\pi$
$152$	0	0
$153$	105548.	0.364521
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−356643.	−1.15474	−0.577371	−	0.816482i	$-0.695921\pi$
$157$	−356643.	−1.15474	−0.577371	+	0.816482i	$0.695921\pi$
$158$	0	0
$159$	−115896.	−0.363560
$160$	0	0
$161$	145332.	0.441872
$162$	0	0
$163$	144890.	0.427139	0.213570	−	0.976928i	$-0.431491\pi$
$163$	144890.	0.427139	0.213570	+	0.976928i	$0.431491\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18102.1	−0.0502272	−0.0251136	−	0.999685i	$-0.507995\pi$
$167$	−18102.1	−0.0502272	−0.0251136	+	0.999685i	$0.507995\pi$
$168$	0	0
$169$	−357037.	−0.961604
$170$	0	0
$171$	−33660.0	−0.0880286
$172$	0	0
$173$	−492572.	−1.25128	−0.625640	−	0.780112i	$-0.715162\pi$
$173$	−492572.	−1.25128	−0.625640	+	0.780112i	$0.715162\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−490728.	−1.17736
$178$	0	0
$179$	444420.	1.03672	0.518359	−	0.855163i	$-0.326543\pi$
$179$	444420.	1.03672	0.518359	+	0.855163i	$0.326543\pi$
$180$	0	0
$181$	156902.	0.355985	0.177993	−	0.984032i	$-0.443040\pi$
$181$	156902.	0.355985	0.177993	+	0.984032i	$0.443040\pi$
$182$	0	0
$183$	−113389.	−0.250289
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−173844.	−0.363543
$188$	0	0
$189$	106920.	0.217723
$190$	0	0
$191$	−332352.	−0.659196	−0.329598	−	0.944121i	$-0.606913\pi$
$191$	−332352.	−0.659196	−0.329598	+	0.944121i	$0.606913\pi$
$192$	0	0
$193$	786120.	1.51913	0.759566	−	0.650430i	$-0.225411\pi$
$193$	786120.	1.51913	0.759566	+	0.650430i	$0.225411\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−59606.4	−0.109428	−0.0547138	−	0.998502i	$-0.517425\pi$
$197$	−59606.4	−0.109428	−0.0547138	+	0.998502i	$0.517425\pi$
$198$	0	0
$199$	−395800.	−0.708505	−0.354253	−	0.935150i	$-0.615265\pi$
$199$	−395800.	−0.708505	−0.354253	+	0.935150i	$0.615265\pi$
$200$	0	0
$201$	−868428.	−1.51616
$202$	0	0
$203$	−413716.	−0.704631
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−372464.	−0.604168
$208$	0	0
$209$	55440.0	0.0877925
$210$	0	0
$211$	251548.	0.388969	0.194484	−	0.980906i	$-0.437697\pi$
$211$	251548.	0.388969	0.194484	+	0.980906i	$0.437697\pi$
$212$	0	0
$213$	−1.06169e6	−1.60343
$214$	0	0
$215$	0	0
$216$	0	0
$217$	403089.	0.581101
$218$	0	0
$219$	1.41134e6	1.98849
$220$	0	0
$221$	−82368.0	−0.113443
$222$	0	0
$223$	288765.	0.388851	0.194425	−	0.980917i	$-0.437716\pi$
$223$	288765.	0.388851	0.194425	+	0.980917i	$0.437716\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.16414e6	1.49948	0.749741	−	0.661731i	$-0.230178\pi$
$227$	1.16414e6	1.49948	0.749741	+	0.661731i	$0.230178\pi$
$228$	0	0
$229$	−547670.	−0.690129	−0.345064	−	0.938579i	$-0.612143\pi$
$229$	−547670.	−0.690129	−0.345064	+	0.938579i	$0.612143\pi$
$230$	0	0
$231$	299376.	0.369137
$232$	0	0
$233$	48104.3	0.0580489	0.0290245	−	0.999579i	$-0.490760\pi$
$233$	48104.3	0.0580489	0.0290245	+	0.999579i	$0.490760\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1.03319e6	1.19484
$238$	0	0
$239$	−1.00584e6	−1.13903	−0.569514	−	0.821982i	$-0.692868\pi$
$239$	−1.00584e6	−1.13903	−0.569514	+	0.821982i	$0.692868\pi$
$240$	0	0
$241$	895202.	0.992838	0.496419	−	0.868083i	$-0.334648\pi$
$241$	895202.	0.992838	0.496419	+	0.868083i	$0.334648\pi$
$242$	0	0
$243$	−1.01387e6	−1.10146
$244$	0	0
$245$	0	0
$246$	0	0
$247$	26267.7	0.0273955
$248$	0	0
$249$	1.23064e6	1.25786
$250$	0	0
$251$	−558252.	−0.559301	−0.279651	−	0.960102i	$-0.590219\pi$
$251$	−558252.	−0.559301	−0.279651	+	0.960102i	$0.590219\pi$
$252$	0	0
$253$	613469.	0.602548
$254$	0	0
$255$	0	0
$256$	0	0
$257$	787924.	0.744135	0.372067	−	0.928206i	$-0.378649\pi$
$257$	787924.	0.744135	0.372067	+	0.928206i	$0.378649\pi$
$258$	0	0
$259$	833976.	0.772510
$260$	0	0
$261$	1.06029e6	0.963437
$262$	0	0
$263$	1.63173e6	1.45465	0.727327	−	0.686291i	$-0.240762\pi$
$263$	1.63173e6	1.45465	0.727327	+	0.686291i	$0.240762\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	198798.	0.170661
$268$	0	0
$269$	1.73637e6	1.46306	0.731529	−	0.681810i	$-0.238807\pi$
$269$	1.73637e6	1.46306	0.731529	+	0.681810i	$0.238807\pi$
$270$	0	0
$271$	1.72005e6	1.42271	0.711357	−	0.702831i	$-0.248081\pi$
$271$	1.72005e6	1.42271	0.711357	+	0.702831i	$0.248081\pi$
$272$	0	0
$273$	141845.	0.115188
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−1.27243e6	−0.996402	−0.498201	−	0.867062i	$-0.666006\pi$
$277$	−1.27243e6	−0.996402	−0.498201	+	0.867062i	$0.666006\pi$
$278$	0	0
$279$	−1.03306e6	−0.794536
$280$	0	0
$281$	1.46500e6	1.10681	0.553404	−	0.832913i	$-0.313329\pi$
$281$	1.46500e6	1.10681	0.553404	+	0.832913i	$0.313329\pi$
$282$	0	0
$283$	1.65051e6	1.22504	0.612521	−	0.790455i	$-0.290156\pi$
$283$	1.65051e6	1.22504	0.612521	+	0.790455i	$0.290156\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	11820.5	0.00847089
$288$	0	0
$289$	−943953.	−0.664823
$290$	0	0
$291$	2.01485e6	1.39479
$292$	0	0
$293$	2.38772e6	1.62485	0.812426	−	0.583064i	$-0.198146\pi$
$293$	2.38772e6	1.62485	0.812426	+	0.583064i	$0.198146\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	451326.	0.296893
$298$	0	0
$299$	290664.	0.188024
$300$	0	0
$301$	−24948.0	−0.0158716
$302$	0	0
$303$	−2.17102e6	−1.35849
$304$	0	0
$305$	0	0
$306$	0	0
$307$	928264.	0.562115	0.281058	−	0.959691i	$-0.409315\pi$
$307$	928264.	0.562115	0.281058	+	0.959691i	$0.409315\pi$
$308$	0	0
$309$	−1.40540e6	−0.837346
$310$	0	0
$311$	−568152.	−0.333092	−0.166546	−	0.986034i	$-0.553261\pi$
$311$	−568152.	−0.333092	−0.166546	+	0.986034i	$0.553261\pi$
$312$	0	0
$313$	−1.72244e6	−0.993766	−0.496883	−	0.867818i	$-0.665522\pi$
$313$	−1.72244e6	−0.993766	−0.496883	+	0.867818i	$0.665522\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−131643.	−0.0735785	−0.0367893	−	0.999323i	$-0.511713\pi$
$317$	−131643.	−0.0735785	−0.0367893	+	0.999323i	$0.511713\pi$
$318$	0	0
$319$	−1.74636e6	−0.960853
$320$	0	0
$321$	1.93261e6	1.04684
$322$	0	0
$323$	−151769.	−0.0809424
$324$	0	0
$325$	0	0
$326$	0	0
$327$	418094.	0.216224
$328$	0	0
$329$	629244.	0.320501
$330$	0	0
$331$	1.58055e6	0.792935	0.396468	−	0.918049i	$-0.370236\pi$
$331$	1.58055e6	0.792935	0.396468	+	0.918049i	$0.370236\pi$
$332$	0	0
$333$	−2.13735e6	−1.05625
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−1.22885e6	−0.589419	−0.294709	−	0.955587i	$-0.595223\pi$
$337$	−1.22885e6	−0.589419	−0.294709	+	0.955587i	$0.595223\pi$
$338$	0	0
$339$	2.08982e6	0.987667
$340$	0	0
$341$	1.70150e6	0.792405
$342$	0	0
$343$	1.79396e6	0.823338
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.84224e6	−1.71301	−0.856506	−	0.516137i	$-0.827370\pi$
$347$	−3.84224e6	−1.71301	−0.856506	+	0.516137i	$0.827370\pi$
$348$	0	0
$349$	1.59445e6	0.700725	0.350362	−	0.936614i	$-0.386058\pi$
$349$	1.59445e6	0.700725	0.350362	+	0.936614i	$0.386058\pi$
$350$	0	0
$351$	213840.	0.0926448
$352$	0	0
$353$	−295365.	−0.126160	−0.0630802	−	0.998008i	$-0.520092\pi$
$353$	−295365.	−0.126160	−0.0630802	+	0.998008i	$0.520092\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−819551.	−0.340334
$358$	0	0
$359$	1.10484e6	0.452442	0.226221	−	0.974076i	$-0.427363\pi$
$359$	1.10484e6	0.452442	0.226221	+	0.974076i	$0.427363\pi$
$360$	0	0
$361$	−2.42770e6	−0.980453
$362$	0	0
$363$	−1.94116e6	−0.773206
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−1.83760e6	−0.712174	−0.356087	−	0.934453i	$-0.615889\pi$
$367$	−1.83760e6	−0.712174	−0.356087	+	0.934453i	$0.615889\pi$
$368$	0	0
$369$	−30294.0	−0.0115822
$370$	0	0
$371$	347688.	0.131146
$372$	0	0
$373$	2.93350e6	1.09173	0.545864	−	0.837874i	$-0.316202\pi$
$373$	2.93350e6	1.09173	0.545864	+	0.837874i	$0.316202\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−827432.	−0.299832
$378$	0	0
$379$	5.09342e6	1.82143	0.910713	−	0.413040i	$-0.135533\pi$
$379$	5.09342e6	1.82143	0.910713	+	0.413040i	$0.135533\pi$
$380$	0	0
$381$	−1.73567e6	−0.612568
$382$	0	0
$383$	−3.17485e6	−1.10593	−0.552964	−	0.833205i	$-0.686503\pi$
$383$	−3.17485e6	−1.10593	−0.552964	+	0.833205i	$0.686503\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	63937.9	0.0217011
$388$	0	0
$389$	−1.79991e6	−0.603083	−0.301541	−	0.953453i	$-0.597501\pi$
$389$	−1.79991e6	−0.603083	−0.301541	+	0.953453i	$0.597501\pi$
$390$	0	0
$391$	−1.67939e6	−0.555533
$392$	0	0
$393$	−3.83771e6	−1.25340
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.90405e6	1.56163	0.780817	−	0.624760i	$-0.214803\pi$
$397$	4.90405e6	1.56163	0.780817	+	0.624760i	$0.214803\pi$
$398$	0	0
$399$	261360.	0.0821877
$400$	0	0
$401$	−642798.	−0.199624	−0.0998122	−	0.995006i	$-0.531824\pi$
$401$	−642798.	−0.199624	−0.0998122	+	0.995006i	$0.531824\pi$
$402$	0	0
$403$	806179.	0.247268
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.52035e6	1.05341
$408$	0	0
$409$	2.05711e6	0.608064	0.304032	−	0.952662i	$-0.401667\pi$
$409$	2.05711e6	0.608064	0.304032	+	0.952662i	$0.401667\pi$
$410$	0	0
$411$	2.85701e6	0.834271
$412$	0	0
$413$	1.47218e6	0.424704
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−6.33489e6	−1.78402
$418$	0	0
$419$	−2.93742e6	−0.817393	−0.408697	−	0.912670i	$-0.634017\pi$
$419$	−2.93742e6	−0.817393	−0.408697	+	0.912670i	$0.634017\pi$
$420$	0	0
$421$	2.71770e6	0.747303	0.373651	−	0.927569i	$-0.378106\pi$
$421$	2.71770e6	0.747303	0.373651	+	0.927569i	$0.378106\pi$
$422$	0	0
$423$	−1.61266e6	−0.438219
$424$	0	0
$425$	0	0
$426$	0	0
$427$	340166.	0.0902862
$428$	0	0
$429$	598752.	0.157074
$430$	0	0
$431$	−4.99435e6	−1.29505	−0.647524	−	0.762045i	$-0.724196\pi$
$431$	−4.99435e6	−1.29505	−0.647524	+	0.762045i	$0.724196\pi$
$432$	0	0
$433$	−2.08183e6	−0.533612	−0.266806	−	0.963750i	$-0.585968\pi$
$433$	−2.08183e6	−0.533612	−0.266806	+	0.963750i	$0.585968\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	535569.	0.134156
$438$	0	0
$439$	−4.70404e6	−1.16496	−0.582478	−	0.812846i	$-0.697917\pi$
$439$	−4.70404e6	−1.16496	−0.582478	+	0.812846i	$0.697917\pi$
$440$	0	0
$441$	−2.02618e6	−0.496114
$442$	0	0
$443$	5.70103e6	1.38021	0.690103	−	0.723711i	$-0.257565\pi$
$443$	5.70103e6	1.38021	0.690103	+	0.723711i	$0.257565\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−1.67456e6	−0.396399
$448$	0	0
$449$	−6.20325e6	−1.45212	−0.726062	−	0.687630i	$-0.758651\pi$
$449$	−6.20325e6	−1.45212	−0.726062	+	0.687630i	$0.758651\pi$
$450$	0	0
$451$	49896.0	0.0115511
$452$	0	0
$453$	3.10134e6	0.710074
$454$	0	0
$455$	0	0
$456$	0	0
$457$	2.15371e6	0.482388	0.241194	−	0.970477i	$-0.422461\pi$
$457$	2.15371e6	0.482388	0.241194	+	0.970477i	$0.422461\pi$
$458$	0	0
$459$	−1.23552e6	−0.273727
$460$	0	0
$461$	−3.85130e6	−0.844024	−0.422012	−	0.906590i	$-0.638676\pi$
$461$	−3.85130e6	−0.844024	−0.422012	+	0.906590i	$0.638676\pi$
$462$	0	0
$463$	2.08213e6	0.451394	0.225697	−	0.974198i	$-0.427534\pi$
$463$	2.08213e6	0.451394	0.225697	+	0.974198i	$0.427534\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.30822e6	0.277579	0.138790	−	0.990322i	$-0.455679\pi$
$467$	1.30822e6	0.277579	0.138790	+	0.990322i	$0.455679\pi$
$468$	0	0
$469$	2.60528e6	0.546919
$470$	0	0
$471$	−7.09711e6	−1.47411
$472$	0	0
$473$	−105309.	−0.0216429
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−891071.	−0.179315
$478$	0	0
$479$	−6.76368e6	−1.34693	−0.673464	−	0.739220i	$-0.735194\pi$
$479$	−6.76368e6	−1.34693	−0.673464	+	0.739220i	$0.735194\pi$
$480$	0	0
$481$	1.66795e6	0.328716
$482$	0	0
$483$	2.89207e6	0.564080
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.67193e6	1.27476	0.637381	−	0.770549i	$-0.280018\pi$
$487$	6.67193e6	1.27476	0.637381	+	0.770549i	$0.280018\pi$
$488$	0	0
$489$	2.88328e6	0.545273
$490$	0	0
$491$	6.87575e6	1.28711	0.643556	−	0.765399i	$-0.277458\pi$
$491$	6.87575e6	1.28711	0.643556	+	0.765399i	$0.277458\pi$
$492$	0	0
$493$	4.78072e6	0.885881
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.18507e6	0.578400
$498$	0	0
$499$	6.94010e6	1.24771	0.623856	−	0.781539i	$-0.285565\pi$
$499$	6.94010e6	1.24771	0.623856	+	0.781539i	$0.285565\pi$
$500$	0	0
$501$	−360228.	−0.0641185
$502$	0	0
$503$	−921007.	−0.162309	−0.0811546	−	0.996702i	$-0.525861\pi$
$503$	−921007.	−0.162309	−0.0811546	+	0.996702i	$0.525861\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−7.10495e6	−1.22755
$508$	0	0
$509$	−4.97979e6	−0.851955	−0.425977	−	0.904734i	$-0.640070\pi$
$509$	−4.97979e6	−0.851955	−0.425977	+	0.904734i	$0.640070\pi$
$510$	0	0
$511$	−4.23403e6	−0.717302
$512$	0	0
$513$	394015.	0.0661027
$514$	0	0
$515$	0	0
$516$	0	0
$517$	2.65614e6	0.437043
$518$	0	0
$519$	−9.80206e6	−1.59735
$520$	0	0
$521$	−147798.	−0.0238547	−0.0119274	−	0.999929i	$-0.503797\pi$
$521$	−147798.	−0.0238547	−0.0119274	+	0.999929i	$0.503797\pi$
$522$	0	0
$523$	−1.23884e7	−1.98043	−0.990216	−	0.139543i	$-0.955437\pi$
$523$	−1.23884e7	−1.98043	−0.990216	+	0.139543i	$0.955437\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−4.65792e6	−0.730576
$528$	0	0
$529$	−510027.	−0.0792417
$530$	0	0
$531$	−3.77298e6	−0.580695
$532$	0	0
$533$	23640.9	0.00360451
$534$	0	0
$535$	0	0
$536$	0	0
$537$	8.84385e6	1.32344
$538$	0	0
$539$	3.33724e6	0.494783
$540$	0	0
$541$	−9.99810e6	−1.46867	−0.734335	−	0.678787i	$-0.762506\pi$
$541$	−9.99810e6	−1.46867	−0.734335	+	0.678787i	$0.762506\pi$
$542$	0	0
$543$	3.12231e6	0.454440
$544$	0	0
$545$	0	0
$546$	0	0
$547$	1.18580e7	1.69451	0.847253	−	0.531189i	$-0.178255\pi$
$547$	1.18580e7	1.69451	0.847253	+	0.531189i	$0.178255\pi$
$548$	0	0
$549$	−871794.	−0.123448
$550$	0	0
$551$	−1.52460e6	−0.213933
$552$	0	0
$553$	−3.09958e6	−0.431013
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−904550.	−0.123536	−0.0617681	−	0.998091i	$-0.519674\pi$
$557$	−904550.	−0.123536	−0.0617681	+	0.998091i	$0.519674\pi$
$558$	0	0
$559$	−49896.0	−0.00675361
$560$	0	0
$561$	−3.45946e6	−0.464088
$562$	0	0
$563$	8.68719e6	1.15507	0.577535	−	0.816366i	$-0.304015\pi$
$563$	8.68719e6	1.15507	0.577535	+	0.816366i	$0.304015\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	4.34724e6	0.567879
$568$	0	0
$569$	2.27007e6	0.293940	0.146970	−	0.989141i	$-0.453048\pi$
$569$	2.27007e6	0.293940	0.146970	+	0.989141i	$0.453048\pi$
$570$	0	0
$571$	−1.43807e7	−1.84582	−0.922908	−	0.385021i	$-0.874194\pi$
$571$	−1.43807e7	−1.84582	−0.922908	+	0.385021i	$0.874194\pi$
$572$	0	0
$573$	−6.61372e6	−0.841510
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−5.63943e6	−0.705173	−0.352586	−	0.935779i	$-0.614698\pi$
$577$	−5.63943e6	−0.705173	−0.352586	+	0.935779i	$0.614698\pi$
$578$	0	0
$579$	1.56436e7	1.93928
$580$	0	0
$581$	−3.69191e6	−0.453744
$582$	0	0
$583$	1.46765e6	0.178834
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.28473e6	0.153893	0.0769464	−	0.997035i	$-0.475483\pi$
$587$	1.28473e6	0.153893	0.0769464	+	0.997035i	$0.475483\pi$
$588$	0	0
$589$	1.48544e6	0.176428
$590$	0	0
$591$	−1.18615e6	−0.139692
$592$	0	0
$593$	7.00943e6	0.818552	0.409276	−	0.912411i	$-0.365781\pi$
$593$	7.00943e6	0.818552	0.409276	+	0.912411i	$0.365781\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−7.87632e6	−0.904456
$598$	0	0
$599$	−8.80020e6	−1.00213	−0.501067	−	0.865409i	$-0.667059\pi$
$599$	−8.80020e6	−1.00213	−0.501067	+	0.865409i	$0.667059\pi$
$600$	0	0
$601$	−1.07670e7	−1.21593	−0.607965	−	0.793964i	$-0.708014\pi$
$601$	−1.07670e7	−1.21593	−0.607965	+	0.793964i	$0.708014\pi$
$602$	0	0
$603$	−6.67694e6	−0.747798
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.51219e7	−1.66584	−0.832921	−	0.553391i	$-0.813333\pi$
$607$	−1.51219e7	−1.66584	−0.832921	+	0.553391i	$0.813333\pi$
$608$	0	0
$609$	−8.23284e6	−0.899511
$610$	0	0
$611$	1.25849e6	0.136379
$612$	0	0
$613$	−8.31622e6	−0.893871	−0.446936	−	0.894566i	$-0.647485\pi$
$613$	−8.31622e6	−0.893871	−0.446936	+	0.894566i	$0.647485\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−1.21083e7	−1.28047	−0.640237	−	0.768178i	$-0.721164\pi$
$617$	−1.21083e7	−1.28047	−0.640237	+	0.768178i	$0.721164\pi$
$618$	0	0
$619$	9.73238e6	1.02092	0.510461	−	0.859901i	$-0.329475\pi$
$619$	9.73238e6	1.02092	0.510461	+	0.859901i	$0.329475\pi$
$620$	0	0
$621$	4.35996e6	0.453684
$622$	0	0
$623$	−596395.	−0.0615622
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.10324e6	0.112073
$628$	0	0
$629$	−9.63706e6	−0.971220
$630$	0	0
$631$	8.60145e6	0.859999	0.430000	−	0.902829i	$-0.358514\pi$
$631$	8.60145e6	0.859999	0.430000	+	0.902829i	$0.358514\pi$
$632$	0	0
$633$	5.00574e6	0.496546
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1.58119e6	0.154396
$638$	0	0
$639$	−8.16286e6	−0.790842
$640$	0	0
$641$	−6.42440e6	−0.617572	−0.308786	−	0.951132i	$-0.599923\pi$
$641$	−6.42440e6	−0.617572	−0.308786	+	0.951132i	$0.599923\pi$
$642$	0	0
$643$	−3.64721e6	−0.347883	−0.173941	−	0.984756i	$-0.555650\pi$
$643$	−3.64721e6	−0.347883	−0.173941	+	0.984756i	$0.555650\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	3.78036e6	0.355036	0.177518	−	0.984118i	$-0.443193\pi$
$647$	3.78036e6	0.355036	0.177518	+	0.984118i	$0.443193\pi$
$648$	0	0
$649$	6.21432e6	0.579138
$650$	0	0
$651$	8.02138e6	0.741816
$652$	0	0
$653$	−1.66957e7	−1.53223	−0.766113	−	0.642706i	$-0.777812\pi$
$653$	−1.66957e7	−1.53223	−0.766113	+	0.642706i	$0.777812\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.08512e7	0.980761
$658$	0	0
$659$	−1.22166e6	−0.109581	−0.0547907	−	0.998498i	$-0.517449\pi$
$659$	−1.22166e6	−0.109581	−0.0547907	+	0.998498i	$0.517449\pi$
$660$	0	0
$661$	1.62789e7	1.44918	0.724589	−	0.689182i	$-0.242030\pi$
$661$	1.62789e7	1.44918	0.724589	+	0.689182i	$0.242030\pi$
$662$	0	0
$663$	−1.63910e6	−0.144818
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.68704e7	−1.46829
$668$	0	0
$669$	5.74636e6	0.496395
$670$	0	0
$671$	1.43590e6	0.123117
$672$	0	0
$673$	1.43928e7	1.22492	0.612459	−	0.790503i	$-0.290181\pi$
$673$	1.43928e7	1.22492	0.612459	+	0.790503i	$0.290181\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−2.62429e6	−0.220059	−0.110030	−	0.993928i	$-0.535095\pi$
$677$	−2.62429e6	−0.220059	−0.110030	+	0.993928i	$0.535095\pi$
$678$	0	0
$679$	−6.04454e6	−0.503140
$680$	0	0
$681$	2.31661e7	1.91419
$682$	0	0
$683$	−1.03039e7	−0.845184	−0.422592	−	0.906320i	$-0.638880\pi$
$683$	−1.03039e7	−0.845184	−0.422592	+	0.906320i	$0.638880\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−1.08985e7	−0.880998
$688$	0	0
$689$	695376.	0.0558048
$690$	0	0
$691$	−4.50285e6	−0.358751	−0.179375	−	0.983781i	$-0.557408\pi$
$691$	−4.50285e6	−0.358751	−0.179375	+	0.983781i	$0.557408\pi$
$692$	0	0
$693$	2.30176e6	0.182066
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−136592.	−0.0106498
$698$	0	0
$699$	957264.	0.0741035
$700$	0	0
$701$	−4.88090e6	−0.375150	−0.187575	−	0.982250i	$-0.560063\pi$
$701$	−4.88090e6	−0.375150	−0.187575	+	0.982250i	$0.560063\pi$
$702$	0	0
$703$	3.07332e6	0.234541
$704$	0	0
$705$	0	0
$706$	0	0
$707$	6.51307e6	0.490046
$708$	0	0
$709$	9.96961e6	0.744839	0.372420	−	0.928064i	$-0.378528\pi$
$709$	9.96961e6	0.744839	0.372420	+	0.928064i	$0.378528\pi$
$710$	0	0
$711$	7.94376e6	0.589321
$712$	0	0
$713$	1.64371e7	1.21088
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−2.00160e7	−1.45405
$718$	0	0
$719$	−1.19167e7	−0.859675	−0.429838	−	0.902906i	$-0.641429\pi$
$719$	−1.19167e7	−0.859675	−0.429838	+	0.902906i	$0.641429\pi$
$720$	0	0
$721$	4.21621e6	0.302054
$722$	0	0
$723$	1.78143e7	1.26743
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−1.38269e6	−0.0970264	−0.0485132	−	0.998823i	$-0.515448\pi$
$727$	−1.38269e6	−0.0970264	−0.0485132	+	0.998823i	$0.515448\pi$
$728$	0	0
$729$	−2.48079e6	−0.172890
$730$	0	0
$731$	288288.	0.0199541
$732$	0	0
$733$	−6.09661e6	−0.419110	−0.209555	−	0.977797i	$-0.567202\pi$
$733$	−6.09661e6	−0.419110	−0.209555	+	0.977797i	$0.567202\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	1.09973e7	0.745793
$738$	0	0
$739$	6.16946e6	0.415562	0.207781	−	0.978175i	$-0.433376\pi$
$739$	6.16946e6	0.415562	0.207781	+	0.978175i	$0.433376\pi$
$740$	0	0
$741$	522720.	0.0349723
$742$	0	0
$743$	1.57574e7	1.04716	0.523578	−	0.851978i	$-0.324597\pi$
$743$	1.57574e7	1.04716	0.523578	+	0.851978i	$0.324597\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	9.46179e6	0.620400
$748$	0	0
$749$	−5.79784e6	−0.377626
$750$	0	0
$751$	1.51816e7	0.982243	0.491122	−	0.871091i	$-0.336587\pi$
$751$	1.51816e7	0.982243	0.491122	+	0.871091i	$0.336587\pi$
$752$	0	0
$753$	−1.11091e7	−0.713987
$754$	0	0
$755$	0	0
$756$	0	0
$757$	652274.	0.0413705	0.0206852	−	0.999786i	$-0.493415\pi$
$757$	652274.	0.0413705	0.0206852	+	0.999786i	$0.493415\pi$
$758$	0	0
$759$	1.22079e7	0.769194
$760$	0	0
$761$	4.51420e6	0.282566	0.141283	−	0.989969i	$-0.454877\pi$
$761$	4.51420e6	0.282566	0.141283	+	0.989969i	$0.454877\pi$
$762$	0	0
$763$	−1.25428e6	−0.0779980
$764$	0	0
$765$	0	0
$766$	0	0
$767$	2.94437e6	0.180719
$768$	0	0
$769$	1.20799e7	0.736625	0.368312	−	0.929702i	$-0.379936\pi$
$769$	1.20799e7	0.736625	0.368312	+	0.929702i	$0.379936\pi$
$770$	0	0
$771$	1.56795e7	0.949939
$772$	0	0
$773$	−1.04245e7	−0.627492	−0.313746	−	0.949507i	$-0.601584\pi$
$773$	−1.04245e7	−0.627492	−0.313746	+	0.949507i	$0.601584\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.65959e7	0.986163
$778$	0	0
$779$	43560.0	0.00257184
$780$	0	0
$781$	1.34447e7	0.788721
$782$	0	0
$783$	−1.24115e7	−0.723467
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.45366e7	1.98766	0.993830	−	0.110913i	$-0.0353776\pi$
$787$	3.45366e7	1.98766	0.993830	+	0.110913i	$0.0353776\pi$
$788$	0	0
$789$	3.24711e7	1.85697
$790$	0	0
$791$	−6.26947e6	−0.356279
$792$	0	0
$793$	680333.	0.0384183
$794$	0	0
$795$	0	0
$796$	0	0
$797$	2.09287e7	1.16707	0.583533	−	0.812089i	$-0.301670\pi$
$797$	2.09287e7	1.16707	0.583533	+	0.812089i	$0.301670\pi$
$798$	0	0
$799$	−7.27126e6	−0.402942
$800$	0	0
$801$	1.52847e6	0.0841735
$802$	0	0
$803$	−1.78725e7	−0.978131
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.45533e7	1.86770
$808$	0	0
$809$	−2.48797e7	−1.33651	−0.668257	−	0.743930i	$-0.732959\pi$
$809$	−2.48797e7	−1.33651	−0.668257	+	0.743930i	$0.732959\pi$
$810$	0	0
$811$	3.95415e6	0.211106	0.105553	−	0.994414i	$-0.466339\pi$
$811$	3.95415e6	0.211106	0.105553	+	0.994414i	$0.466339\pi$
$812$	0	0
$813$	3.42285e7	1.81619
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−91936.8	−0.00481875
$818$	0	0
$819$	1.09058e6	0.0568132
$820$	0	0
$821$	−3.43550e6	−0.177882	−0.0889410	−	0.996037i	$-0.528348\pi$
$821$	−3.43550e6	−0.177882	−0.0889410	+	0.996037i	$0.528348\pi$
$822$	0	0
$823$	3.94833e6	0.203195	0.101598	−	0.994826i	$-0.467605\pi$
$823$	3.94833e6	0.203195	0.101598	+	0.994826i	$0.467605\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.38176e7	−1.71941	−0.859705	−	0.510791i	$-0.829353\pi$
$827$	−3.38176e7	−1.71941	−0.859705	+	0.510791i	$0.829353\pi$
$828$	0	0
$829$	−1.52015e7	−0.768244	−0.384122	−	0.923282i	$-0.625496\pi$
$829$	−1.52015e7	−0.768244	−0.384122	+	0.923282i	$0.625496\pi$
$830$	0	0
$831$	−2.53210e7	−1.27198
$832$	0	0
$833$	−9.13579e6	−0.456177
$834$	0	0
$835$	0	0
$836$	0	0
$837$	1.20927e7	0.596635
$838$	0	0
$839$	2.89012e7	1.41746	0.708729	−	0.705481i	$-0.249269\pi$
$839$	2.89012e7	1.41746	0.708729	+	0.705481i	$0.249269\pi$
$840$	0	0
$841$	2.75138e7	1.34140
$842$	0	0
$843$	2.91532e7	1.41292
$844$	0	0
$845$	0	0
$846$	0	0
$847$	5.82348e6	0.278917
$848$	0	0
$849$	3.28446e7	1.56385
$850$	0	0
$851$	3.40077e7	1.60973
$852$	0	0
$853$	−2.02107e7	−0.951062	−0.475531	−	0.879699i	$-0.657744\pi$
$853$	−2.02107e7	−0.951062	−0.475531	+	0.879699i	$0.657744\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.70522e7	−0.793101	−0.396550	−	0.918013i	$-0.629793\pi$
$857$	−1.70522e7	−0.793101	−0.396550	+	0.918013i	$0.629793\pi$
$858$	0	0
$859$	1.95505e7	0.904015	0.452008	−	0.892014i	$-0.350708\pi$
$859$	1.95505e7	0.904015	0.452008	+	0.892014i	$0.350708\pi$
$860$	0	0
$861$	235224.	0.0108137
$862$	0	0
$863$	−2.70896e7	−1.23816	−0.619078	−	0.785330i	$-0.712493\pi$
$863$	−2.70896e7	−1.23816	−0.619078	+	0.785330i	$0.712493\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.87844e7	−0.848692
$868$	0	0
$869$	−1.30838e7	−0.587741
$870$	0	0
$871$	5.21057e6	0.232723
$872$	0	0
$873$	1.54912e7	0.687940
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−1.98285e6	−0.0870545	−0.0435272	−	0.999052i	$-0.513860\pi$
$877$	−1.98285e6	−0.0870545	−0.0435272	+	0.999052i	$0.513860\pi$
$878$	0	0
$879$	4.75150e7	2.07424
$880$	0	0
$881$	−4.22840e7	−1.83542	−0.917712	−	0.397247i	$-0.869966\pi$
$881$	−4.22840e7	−1.83542	−0.917712	+	0.397247i	$0.869966\pi$
$882$	0	0
$883$	−134502.	−0.00580535	−0.00290267	−	0.999996i	$-0.500924\pi$
$883$	−134502.	−0.00580535	−0.00290267	+	0.999996i	$0.500924\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	3.87668e6	0.165444	0.0827219	−	0.996573i	$-0.473639\pi$
$887$	3.87668e6	0.165444	0.0827219	+	0.996573i	$0.473639\pi$
$888$	0	0
$889$	5.20700e6	0.220970
$890$	0	0
$891$	1.83504e7	0.774374
$892$	0	0
$893$	2.31885e6	0.0973070
$894$	0	0
$895$	0	0
$896$	0	0
$897$	5.78414e6	0.240026
$898$	0	0
$899$	−4.67914e7	−1.93093
$900$	0	0
$901$	−4.01773e6	−0.164880
$902$	0	0
$903$	−496459.	−0.0202611
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−2.87363e7	−1.15988	−0.579939	−	0.814660i	$-0.696924\pi$
$907$	−2.87363e7	−1.15988	−0.579939	+	0.814660i	$0.696924\pi$
$908$	0	0
$909$	−1.66920e7	−0.670037
$910$	0	0
$911$	−1.87675e6	−0.0749223	−0.0374611	−	0.999298i	$-0.511927\pi$
$911$	−1.87675e6	−0.0749223	−0.0374611	+	0.999298i	$0.511927\pi$
$912$	0	0
$913$	−1.55841e7	−0.618736
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.15131e7	0.452137
$918$	0	0
$919$	−6.76852e6	−0.264366	−0.132183	−	0.991225i	$-0.542199\pi$
$919$	−6.76852e6	−0.264366	−0.132183	+	0.991225i	$0.542199\pi$
$920$	0	0
$921$	1.84722e7	0.717579
$922$	0	0
$923$	6.37015e6	0.246119
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.08055e7	−0.412996
$928$	0	0
$929$	−1.15356e7	−0.438530	−0.219265	−	0.975665i	$-0.570366\pi$
$929$	−1.15356e7	−0.438530	−0.219265	+	0.975665i	$0.570366\pi$
$930$	0	0
$931$	2.91346e6	0.110163
$932$	0	0
$933$	−1.13061e7	−0.425214
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−3.92632e7	−1.46096	−0.730478	−	0.682936i	$-0.760703\pi$
$937$	−3.92632e7	−1.46096	−0.730478	+	0.682936i	$0.760703\pi$
$938$	0	0
$939$	−3.42762e7	−1.26861
$940$	0	0
$941$	2.94919e7	1.08575	0.542874	−	0.839814i	$-0.317336\pi$
$941$	2.94919e7	1.08575	0.542874	+	0.839814i	$0.317336\pi$
$942$	0	0
$943$	482012.	0.0176514
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.09628e7	0.759581	0.379791	−	0.925072i	$-0.375996\pi$
$947$	2.09628e7	0.759581	0.379791	+	0.925072i	$0.375996\pi$
$948$	0	0
$949$	−8.46806e6	−0.305224
$950$	0	0
$951$	−2.61967e6	−0.0939281
$952$	0	0
$953$	−1.64122e7	−0.585375	−0.292687	−	0.956208i	$-0.594549\pi$
$953$	−1.64122e7	−0.585375	−0.292687	+	0.956208i	$0.594549\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3.47521e7	−1.22660
$958$	0	0
$959$	−8.57102e6	−0.300944
$960$	0	0
$961$	1.69604e7	0.592416
$962$	0	0
$963$	1.48590e7	0.516325
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−4.71911e7	−1.62291	−0.811454	−	0.584416i	$-0.801324\pi$
$967$	−4.71911e7	−1.62291	−0.811454	+	0.584416i	$0.801324\pi$
$968$	0	0
$969$	−3.02016e6	−0.103329
$970$	0	0
$971$	−3.84771e7	−1.30965	−0.654823	−	0.755783i	$-0.727257\pi$
$971$	−3.84771e7	−1.30965	−0.654823	+	0.755783i	$0.727257\pi$
$972$	0	0
$973$	1.90047e7	0.643544
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.70184e7	−0.905572	−0.452786	−	0.891619i	$-0.649570\pi$
$977$	−2.70184e7	−0.905572	−0.452786	+	0.891619i	$0.649570\pi$
$978$	0	0
$979$	−2.51748e6	−0.0839478
$980$	0	0
$981$	3.21453e6	0.106646
$982$	0	0
$983$	−2.88475e7	−0.952192	−0.476096	−	0.879393i	$-0.657949\pi$
$983$	−2.88475e7	−0.952192	−0.476096	+	0.879393i	$0.657949\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	1.25218e7	0.409142
$988$	0	0
$989$	−1.01732e6	−0.0330726
$990$	0	0
$991$	5.21596e7	1.68714	0.843569	−	0.537021i	$-0.180450\pi$
$991$	5.21596e7	1.68714	0.843569	+	0.537021i	$0.180450\pi$
$992$	0	0
$993$	3.14525e7	1.01224
$994$	0	0
$995$	0	0
$996$	0	0
$997$	9.78148e6	0.311650	0.155825	−	0.987785i	$-0.450196\pi$
$997$	9.78148e6	0.311650	0.155825	+	0.987785i	$0.450196\pi$
$998$	0	0
$999$	2.50193e7	0.793161

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.t.1.2		2
4.3	odd	2		25.6.a.c.1.1		2
5.2	odd	4		80.6.c.a.49.1		2
5.3	odd	4		80.6.c.a.49.2		2
5.4	even	2	inner	400.6.a.t.1.1		2
12.11	even	2		225.6.a.n.1.2		2
15.2	even	4		720.6.f.f.289.2		2
15.8	even	4		720.6.f.f.289.1		2
20.3	even	4		5.6.b.a.4.2	yes	2
20.7	even	4		5.6.b.a.4.1	✓	2
20.19	odd	2		25.6.a.c.1.2		2
40.3	even	4		320.6.c.f.129.2		2
40.13	odd	4		320.6.c.g.129.1		2
40.27	even	4		320.6.c.f.129.1		2
40.37	odd	4		320.6.c.g.129.2		2
60.23	odd	4		45.6.b.b.19.1		2
60.47	odd	4		45.6.b.b.19.2		2
60.59	even	2		225.6.a.n.1.1		2
140.27	odd	4		245.6.b.a.99.1		2
140.83	odd	4		245.6.b.a.99.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.6.b.a.4.1	✓	2	20.7	even	4
5.6.b.a.4.2	yes	2	20.3	even	4
25.6.a.c.1.1		2	4.3	odd	2
25.6.a.c.1.2		2	20.19	odd	2
45.6.b.b.19.1		2	60.23	odd	4
45.6.b.b.19.2		2	60.47	odd	4
80.6.c.a.49.1		2	5.2	odd	4
80.6.c.a.49.2		2	5.3	odd	4
225.6.a.n.1.1		2	60.59	even	2
225.6.a.n.1.2		2	12.11	even	2
245.6.b.a.99.1		2	140.27	odd	4
245.6.b.a.99.2		2	140.83	odd	4
320.6.c.f.129.1		2	40.27	even	4
320.6.c.f.129.2		2	40.3	even	4
320.6.c.g.129.1		2	40.13	odd	4
320.6.c.g.129.2		2	40.37	odd	4
400.6.a.t.1.1		2	5.4	even	2	inner
400.6.a.t.1.2		2	1.1	even	1	trivial
720.6.f.f.289.1		2	15.8	even	4
720.6.f.f.289.2		2	15.2	even	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q+19.8997 q^{3} -59.6992 q^{7} +153.000 q^{9} +O(q^{10})\)	\(q+19.8997 q^{3} -59.6992 q^{7} +153.000 q^{9} -252.000 q^{11} -119.398 q^{13} +689.858 q^{17} -220.000 q^{19} -1188.00 q^{21} -2434.40 q^{23} -1790.98 q^{27} +6930.00 q^{29} -6752.00 q^{31} -5014.74 q^{33} -13969.6 q^{37} -2376.00 q^{39} -198.000 q^{41} +417.895 q^{43} -10540.2 q^{47} -13243.0 q^{49} +13728.0 q^{51} -5823.99 q^{53} -4377.94 q^{57} -24660.0 q^{59} -5698.00 q^{61} -9133.98 q^{63} -43640.1 q^{67} -48444.0 q^{69} -53352.0 q^{71} +70922.7 q^{73} +15044.2 q^{77} +51920.0 q^{79} -72819.0 q^{81} +61841.8 q^{83} +137905. q^{87} +9990.00 q^{89} +7128.00 q^{91} -134363. q^{93} +101250. q^{97} -38556.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 306 q^{9}+O(q^{10}) \) 2 * q + 306 * q^9	\( 2 q + 306 q^{9} - 504 q^{11} - 440 q^{19} - 2376 q^{21} + 13860 q^{29} - 13504 q^{31} - 4752 q^{39} - 396 q^{41} - 26486 q^{49} + 27456 q^{51} - 49320 q^{59} - 11396 q^{61} - 96888 q^{69} - 106704 q^{71} + 103840 q^{79} - 145638 q^{81} + 19980 q^{89} + 14256 q^{91} - 77112 q^{99}+O(q^{100}) \) 2 * q + 306 * q^9 - 504 * q^11 - 440 * q^19 - 2376 * q^21 + 13860 * q^29 - 13504 * q^31 - 4752 * q^39 - 396 * q^41 - 26486 * q^49 + 27456 * q^51 - 49320 * q^59 - 11396 * q^61 - 96888 * q^69 - 106704 * q^71 + 103840 * q^79 - 145638 * q^81 + 19980 * q^89 + 14256 * q^91 - 77112 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more