LMFDB - Embedded newform 400.6.c.e.49.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,6,Mod(49,400)]

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, 1]))

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

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 400 = 2^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	400.c (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$64.1535279252$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	400.49
Dual form			400.6.c.e.49.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+18.0000i q^{3} +242.000i q^{7} -81.0000 q^{9} -656.000 q^{11} -206.000i q^{13} -1690.00i q^{17} -1364.00 q^{19} -4356.00 q^{21} -2198.00i q^{23} +2916.00i q^{27} +2218.00 q^{29} +1700.00 q^{31} -11808.0i q^{33} +846.000i q^{37} +3708.00 q^{39} -1818.00 q^{41} -10534.0i q^{43} +12074.0i q^{47} -41757.0 q^{49} +30420.0 q^{51} +32586.0i q^{53} -24552.0i q^{57} +8668.00 q^{59} -34670.0 q^{61} -19602.0i q^{63} -47566.0i q^{67} +39564.0 q^{69} -948.000 q^{71} -63102.0i q^{73} -158752. i q^{77} +46536.0 q^{79} -72171.0 q^{81} +88778.0i q^{83} +39924.0i q^{87} +104934. q^{89} +49852.0 q^{91} +30600.0i q^{93} +36254.0i q^{97} +53136.0 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 162 q^{9} - 1312 q^{11} - 2728 q^{19} - 8712 q^{21} + 4436 q^{29} + 3400 q^{31} + 7416 q^{39} - 3636 q^{41} - 83514 q^{49} + 60840 q^{51} + 17336 q^{59} - 69340 q^{61} + 79128 q^{69} - 1896 q^{71}+ \cdots + 106272 q^{99}+O(q^{100}) $ 2 * q - 162 * q^9 - 1312 * q^11 - 2728 * q^19 - 8712 * q^21 + 4436 * q^29 + 3400 * q^31 + 7416 * q^39 - 3636 * q^41 - 83514 * q^49 + 60840 * q^51 + 17336 * q^59 - 69340 * q^61 + 79128 * q^69 - 1896 * q^71 + 93072 * q^79 - 144342 * q^81 + 209868 * q^89 + 99704 * q^91 + 106272 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$.

$n$	$101$	$177$	$351$
$\chi(n)$	$1$	$-1$	$1$

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
$3$	18.0000i	1.15470i	0.816497	+	0.577350i	$0.195913\pi$
$3$	18.0000i	1.15470i	−0.816497	+	0.577350i	$0.804087\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	242.000i	1.86668i	0.358991	+	0.933341i	$0.383121\pi$
$7$	242.000i	1.86668i	−0.358991	+	0.933341i	$0.616879\pi$
$8$	0	0
$9$	−81.0000	−0.333333
$10$	0	0
$11$	−656.000	−1.63464	−0.817320	−	0.576184i	$-0.804541\pi$
$11$	−656.000	−1.63464	−0.817320	+	0.576184i	$0.804541\pi$
$12$	0	0
$13$	− 206.000i	− 0.338072i	−0.985610	−	0.169036i	$-0.945935\pi$
$13$	− 206.000i	− 0.338072i	0.985610	−	0.169036i	$-0.0540654\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 1690.00i	− 1.41829i	−0.705064	−	0.709144i	$-0.749082\pi$
$17$	− 1690.00i	− 1.41829i	0.705064	−	0.709144i	$-0.250918\pi$
$18$	0	0
$19$	−1364.00	−0.866823	−0.433411	−	0.901196i	$-0.642690\pi$
$19$	−1364.00	−0.866823	−0.433411	+	0.901196i	$0.642690\pi$
$20$	0	0
$21$	−4356.00	−2.15546
$22$	0	0
$23$	− 2198.00i	− 0.866379i	−0.901303	−	0.433190i	$-0.857388\pi$
$23$	− 2198.00i	− 0.866379i	0.901303	−	0.433190i	$-0.142612\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	2916.00i	0.769800i
$28$	0	0
$29$	2218.00	0.489741	0.244871	−	0.969556i	$-0.421255\pi$
$29$	2218.00	0.489741	0.244871	+	0.969556i	$0.421255\pi$
$30$	0	0
$31$	1700.00	0.317720	0.158860	−	0.987301i	$-0.449218\pi$
$31$	1700.00	0.317720	0.158860	+	0.987301i	$0.449218\pi$
$32$	0	0
$33$	− 11808.0i	− 1.88752i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	846.000i	0.101594i	0.998709	+	0.0507968i	$0.0161761\pi$
$37$	846.000i	0.101594i	−0.998709	+	0.0507968i	$0.983824\pi$
$38$	0	0
$39$	3708.00	0.390372
$40$	0	0
$41$	−1818.00	−0.168902	−0.0844509	−	0.996428i	$-0.526914\pi$
$41$	−1818.00	−0.168902	−0.0844509	+	0.996428i	$0.526914\pi$
$42$	0	0
$43$	− 10534.0i	− 0.868805i	−0.900719	−	0.434402i	$-0.856960\pi$
$43$	− 10534.0i	− 0.868805i	0.900719	−	0.434402i	$-0.143040\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12074.0i	0.797272i	0.917109	+	0.398636i	$0.130516\pi$
$47$	12074.0i	0.797272i	−0.917109	+	0.398636i	$0.869484\pi$
$48$	0	0
$49$	−41757.0	−2.48450
$50$	0	0
$51$	30420.0	1.63770
$52$	0	0
$53$	32586.0i	1.59346i	0.604335	+	0.796730i	$0.293439\pi$
$53$	32586.0i	1.59346i	−0.604335	+	0.796730i	$0.706561\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 24552.0i	− 1.00092i
$58$	0	0
$59$	8668.00	0.324182	0.162091	−	0.986776i	$-0.448176\pi$
$59$	8668.00	0.324182	0.162091	+	0.986776i	$0.448176\pi$
$60$	0	0
$61$	−34670.0	−1.19297	−0.596485	−	0.802624i	$-0.703436\pi$
$61$	−34670.0	−1.19297	−0.596485	+	0.802624i	$0.703436\pi$
$62$	0	0
$63$	− 19602.0i	− 0.622227i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 47566.0i	− 1.29452i	−0.762268	−	0.647261i	$-0.775914\pi$
$67$	− 47566.0i	− 1.29452i	0.762268	−	0.647261i	$-0.224086\pi$
$68$	0	0
$69$	39564.0	1.00041
$70$	0	0
$71$	−948.000	−0.0223184	−0.0111592	−	0.999938i	$-0.503552\pi$
$71$	−948.000	−0.0223184	−0.0111592	+	0.999938i	$0.503552\pi$
$72$	0	0
$73$	− 63102.0i	− 1.38591i	−0.720979	−	0.692957i	$-0.756308\pi$
$73$	− 63102.0i	− 1.38591i	0.720979	−	0.692957i	$-0.243692\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 158752.i	− 3.05135i
$78$	0	0
$79$	46536.0	0.838921	0.419461	−	0.907773i	$-0.362219\pi$
$79$	46536.0	0.838921	0.419461	+	0.907773i	$0.362219\pi$
$80$	0	0
$81$	−72171.0	−1.22222
$82$	0	0
$83$	88778.0i	1.41452i	0.706952	+	0.707262i	$0.250070\pi$
$83$	88778.0i	1.41452i	−0.706952	+	0.707262i	$0.749930\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	39924.0i	0.565504i
$88$	0	0
$89$	104934.	1.40424	0.702120	−	0.712059i	$-0.252237\pi$
$89$	104934.	1.40424	0.702120	+	0.712059i	$0.252237\pi$
$90$	0	0
$91$	49852.0	0.631072
$92$	0	0
$93$	30600.0i	0.366872i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	36254.0i	0.391225i	0.980681	+	0.195612i	$0.0626695\pi$
$97$	36254.0i	0.391225i	−0.980681	+	0.195612i	$0.937331\pi$
$98$	0	0
$99$	53136.0	0.544880
$100$	0	0
$101$	42486.0	0.414422	0.207211	−	0.978296i	$-0.433561\pi$
$101$	42486.0	0.414422	0.207211	+	0.978296i	$0.433561\pi$
$102$	0	0
$103$	− 147934.i	− 1.37396i	−0.726675	−	0.686981i	$-0.758935\pi$
$103$	− 147934.i	− 1.37396i	0.726675	−	0.686981i	$-0.241065\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 18390.0i	− 0.155282i	−0.996981	−	0.0776412i	$-0.975261\pi$
$107$	− 18390.0i	− 0.155282i	0.996981	−	0.0776412i	$-0.0247389\pi$
$108$	0	0
$109$	145006.	1.16901	0.584507	−	0.811389i	$-0.301288\pi$
$109$	145006.	1.16901	0.584507	+	0.811389i	$0.301288\pi$
$110$	0	0
$111$	−15228.0	−0.117310
$112$	0	0
$113$	82746.0i	0.609608i	0.952415	+	0.304804i	$0.0985910\pi$
$113$	82746.0i	0.609608i	−0.952415	+	0.304804i	$0.901409\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16686.0i	0.112691i
$118$	0	0
$119$	408980.	2.64749
$120$	0	0
$121$	269285.	1.67205
$122$	0	0
$123$	− 32724.0i	− 0.195031i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 274446.i	− 1.50990i	−0.655784	−	0.754949i	$-0.727662\pi$
$127$	− 274446.i	− 1.50990i	0.655784	−	0.754949i	$-0.272338\pi$
$128$	0	0
$129$	189612.	1.00321
$130$	0	0
$131$	−202608.	−1.03152	−0.515761	−	0.856733i	$-0.672491\pi$
$131$	−202608.	−1.03152	−0.515761	+	0.856733i	$0.672491\pi$
$132$	0	0
$133$	− 330088.i	− 1.61808i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	48142.0i	0.219141i	0.993979	+	0.109570i	$0.0349475\pi$
$137$	48142.0i	0.219141i	−0.993979	+	0.109570i	$0.965053\pi$
$138$	0	0
$139$	−111156.	−0.487973	−0.243987	−	0.969779i	$-0.578455\pi$
$139$	−111156.	−0.487973	−0.243987	+	0.969779i	$0.578455\pi$
$140$	0	0
$141$	−217332.	−0.920610
$142$	0	0
$143$	135136.i	0.552626i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 751626.i	− 2.86885i
$148$	0	0
$149$	−243178.	−0.897343	−0.448672	−	0.893697i	$-0.648103\pi$
$149$	−243178.	−0.897343	−0.448672	+	0.893697i	$0.648103\pi$
$150$	0	0
$151$	−368852.	−1.31647	−0.658233	−	0.752814i	$-0.728696\pi$
$151$	−368852.	−1.31647	−0.658233	+	0.752814i	$0.728696\pi$
$152$	0	0
$153$	136890.i	0.472763i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 319546.i	− 1.03463i	−0.855796	−	0.517314i	$-0.826932\pi$
$157$	− 319546.i	− 1.03463i	0.855796	−	0.517314i	$-0.173068\pi$
$158$	0	0
$159$	−586548.	−1.83997
$160$	0	0
$161$	531916.	1.61725
$162$	0	0
$163$	− 69862.0i	− 0.205955i	−0.994684	−	0.102977i	$-0.967163\pi$
$163$	− 69862.0i	− 0.205955i	0.994684	−	0.102977i	$-0.0328369\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 343422.i	− 0.952877i	−0.879208	−	0.476439i	$-0.841927\pi$
$167$	− 343422.i	− 0.952877i	0.879208	−	0.476439i	$-0.158073\pi$
$168$	0	0
$169$	328857.	0.885708
$170$	0	0
$171$	110484.	0.288941
$172$	0	0
$173$	− 1142.00i	− 0.00290102i	−0.999999	−	0.00145051i	$-0.999538\pi$
$173$	− 1142.00i	− 0.00290102i	0.999999	−	0.00145051i	$-0.000461712\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	156024.i	0.374333i
$178$	0	0
$179$	−86684.0	−0.202212	−0.101106	−	0.994876i	$-0.532238\pi$
$179$	−86684.0	−0.202212	−0.101106	+	0.994876i	$0.532238\pi$
$180$	0	0
$181$	−651418.	−1.47796	−0.738981	−	0.673726i	$-0.764693\pi$
$181$	−651418.	−1.47796	−0.738981	+	0.673726i	$0.764693\pi$
$182$	0	0
$183$	− 624060.i	− 1.37752i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	1.10864e6i	2.31839i
$188$	0	0
$189$	−705672.	−1.43697
$190$	0	0
$191$	−29140.0	−0.0577971	−0.0288986	−	0.999582i	$-0.509200\pi$
$191$	−29140.0	−0.0577971	−0.0288986	+	0.999582i	$0.509200\pi$
$192$	0	0
$193$	− 646406.i	− 1.24914i	−0.780968	−	0.624571i	$-0.785274\pi$
$193$	− 646406.i	− 1.24914i	0.780968	−	0.624571i	$-0.214726\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 431138.i	− 0.791500i	−0.918358	−	0.395750i	$-0.870485\pi$
$197$	− 431138.i	− 0.791500i	0.918358	−	0.395750i	$-0.129515\pi$
$198$	0	0
$199$	−131608.	−0.235586	−0.117793	−	0.993038i	$-0.537582\pi$
$199$	−131608.	−0.235586	−0.117793	+	0.993038i	$0.537582\pi$
$200$	0	0
$201$	856188.	1.49479
$202$	0	0
$203$	536756.i	0.914191i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	178038.i	0.288793i
$208$	0	0
$209$	894784.	1.41694
$210$	0	0
$211$	−1.21078e6	−1.87224	−0.936118	−	0.351686i	$-0.885608\pi$
$211$	−1.21078e6	−1.87224	−0.936118	+	0.351686i	$0.885608\pi$
$212$	0	0
$213$	− 17064.0i	− 0.0257710i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	411400.i	0.593082i
$218$	0	0
$219$	1.13584e6	1.60031
$220$	0	0
$221$	−348140.	−0.479483
$222$	0	0
$223$	− 34886.0i	− 0.0469774i	−0.999724	−	0.0234887i	$-0.992523\pi$
$223$	− 34886.0i	− 0.0469774i	0.999724	−	0.0234887i	$-0.00747737\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 124182.i	− 0.159954i	−0.996797	−	0.0799768i	$-0.974515\pi$
$227$	− 124182.i	− 0.159954i	0.996797	−	0.0799768i	$-0.0254846\pi$
$228$	0	0
$229$	456386.	0.575100	0.287550	−	0.957766i	$-0.407159\pi$
$229$	456386.	0.575100	0.287550	+	0.957766i	$0.407159\pi$
$230$	0	0
$231$	2.85754e6	3.52340
$232$	0	0
$233$	252666.i	0.304900i	0.988311	+	0.152450i	$0.0487163\pi$
$233$	252666.i	0.304900i	−0.988311	+	0.152450i	$0.951284\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	837648.i	0.968703i
$238$	0	0
$239$	65064.0	0.0736794	0.0368397	−	0.999321i	$-0.488271\pi$
$239$	65064.0	0.0736794	0.0368397	+	0.999321i	$0.488271\pi$
$240$	0	0
$241$	−1.40600e6	−1.55935	−0.779675	−	0.626185i	$-0.784615\pi$
$241$	−1.40600e6	−1.55935	−0.779675	+	0.626185i	$0.784615\pi$
$242$	0	0
$243$	− 590490.i	− 0.641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	280984.i	0.293048i
$248$	0	0
$249$	−1.59800e6	−1.63335
$250$	0	0
$251$	548400.	0.549431	0.274715	−	0.961526i	$-0.411416\pi$
$251$	548400.	0.549431	0.274715	+	0.961526i	$0.411416\pi$
$252$	0	0
$253$	1.44189e6i	1.41622i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	493830.i	0.466385i	0.972431	+	0.233193i	$0.0749172\pi$
$257$	493830.i	0.466385i	−0.972431	+	0.233193i	$0.925083\pi$
$258$	0	0
$259$	−204732.	−0.189643
$260$	0	0
$261$	−179658.	−0.163247
$262$	0	0
$263$	1.07181e6i	0.955495i	0.878497	+	0.477748i	$0.158547\pi$
$263$	1.07181e6i	0.955495i	−0.878497	+	0.477748i	$0.841453\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.88881e6i	1.62148i
$268$	0	0
$269$	−999394.	−0.842085	−0.421043	−	0.907041i	$-0.638336\pi$
$269$	−999394.	−0.842085	−0.421043	+	0.907041i	$0.638336\pi$
$270$	0	0
$271$	−1.00760e6	−0.833425	−0.416713	−	0.909038i	$-0.636818\pi$
$271$	−1.00760e6	−0.833425	−0.416713	+	0.909038i	$0.636818\pi$
$272$	0	0
$273$	897336.i	0.728700i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.02286e6i	− 0.800969i	−0.916303	−	0.400485i	$-0.868842\pi$
$277$	− 1.02286e6i	− 0.800969i	0.916303	−	0.400485i	$-0.131158\pi$
$278$	0	0
$279$	−137700.	−0.105907
$280$	0	0
$281$	−1.18172e6	−0.892790	−0.446395	−	0.894836i	$-0.647292\pi$
$281$	−1.18172e6	−0.892790	−0.446395	+	0.894836i	$0.647292\pi$
$282$	0	0
$283$	917506.i	0.680993i	0.940246	+	0.340497i	$0.110595\pi$
$283$	917506.i	0.680993i	−0.940246	+	0.340497i	$0.889405\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 439956.i	− 0.315286i
$288$	0	0
$289$	−1.43624e6	−1.01154
$290$	0	0
$291$	−652572.	−0.451748
$292$	0	0
$293$	− 512302.i	− 0.348624i	−0.984690	−	0.174312i	$-0.944230\pi$
$293$	− 512302.i	− 0.348624i	0.984690	−	0.174312i	$-0.0557701\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 1.91290e6i	− 1.25835i
$298$	0	0
$299$	−452788.	−0.292898
$300$	0	0
$301$	2.54923e6	1.62178
$302$	0	0
$303$	764748.i	0.478533i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	1.40946e6i	0.853505i	0.904368	+	0.426753i	$0.140342\pi$
$307$	1.40946e6i	0.853505i	−0.904368	+	0.426753i	$0.859658\pi$
$308$	0	0
$309$	2.66281e6	1.58652
$310$	0	0
$311$	2.78604e6	1.63337	0.816687	−	0.577081i	$-0.195808\pi$
$311$	2.78604e6	1.63337	0.816687	+	0.577081i	$0.195808\pi$
$312$	0	0
$313$	1.55086e6i	0.894770i	0.894342	+	0.447385i	$0.147645\pi$
$313$	1.55086e6i	0.894770i	−0.894342	+	0.447385i	$0.852355\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 377322.i	− 0.210894i	−0.994425	−	0.105447i	$-0.966373\pi$
$317$	− 377322.i	− 0.210894i	0.994425	−	0.105447i	$-0.0336273\pi$
$318$	0	0
$319$	−1.45501e6	−0.800550
$320$	0	0
$321$	331020.	0.179305
$322$	0	0
$323$	2.30516e6i	1.22940i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	2.61011e6i	1.34986i
$328$	0	0
$329$	−2.92191e6	−1.48825
$330$	0	0
$331$	1.63063e6	0.818062	0.409031	−	0.912521i	$-0.365867\pi$
$331$	1.63063e6	0.818062	0.409031	+	0.912521i	$0.365867\pi$
$332$	0	0
$333$	− 68526.0i	− 0.0338645i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	3.36717e6i	1.61506i	0.589824	+	0.807532i	$0.299197\pi$
$337$	3.36717e6i	1.61506i	−0.589824	+	0.807532i	$0.700803\pi$
$338$	0	0
$339$	−1.48943e6	−0.703915
$340$	0	0
$341$	−1.11520e6	−0.519358
$342$	0	0
$343$	− 6.03790e6i	− 2.77109i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	837202.i	0.373256i	0.982431	+	0.186628i	$0.0597559\pi$
$347$	837202.i	0.373256i	−0.982431	+	0.186628i	$0.940244\pi$
$348$	0	0
$349$	1.51910e6	0.667609	0.333805	−	0.942642i	$-0.391667\pi$
$349$	1.51910e6	0.667609	0.333805	+	0.942642i	$0.391667\pi$
$350$	0	0
$351$	600696.	0.260248
$352$	0	0
$353$	− 3.51851e6i	− 1.50287i	−0.659806	−	0.751436i	$-0.729362\pi$
$353$	− 3.51851e6i	− 1.50287i	0.659806	−	0.751436i	$-0.270638\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	7.36164e6i	3.05706i
$358$	0	0
$359$	−3.57089e6	−1.46231	−0.731156	−	0.682210i	$-0.761019\pi$
$359$	−3.57089e6	−1.46231	−0.731156	+	0.682210i	$0.761019\pi$
$360$	0	0
$361$	−615603.	−0.248618
$362$	0	0
$363$	4.84713e6i	1.93071i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 3.58231e6i	− 1.38835i	−0.719808	−	0.694173i	$-0.755770\pi$
$367$	− 3.58231e6i	− 1.38835i	0.719808	−	0.694173i	$-0.244230\pi$
$368$	0	0
$369$	147258.	0.0563006
$370$	0	0
$371$	−7.88581e6	−2.97448
$372$	0	0
$373$	635530.i	0.236518i	0.992983	+	0.118259i	$0.0377313\pi$
$373$	635530.i	0.236518i	−0.992983	+	0.118259i	$0.962269\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 456908.i	− 0.165568i
$378$	0	0
$379$	67060.0	0.0239809	0.0119905	−	0.999928i	$-0.496183\pi$
$379$	67060.0	0.0239809	0.0119905	+	0.999928i	$0.496183\pi$
$380$	0	0
$381$	4.94003e6	1.74348
$382$	0	0
$383$	− 4.45129e6i	− 1.55056i	−0.631618	−	0.775280i	$-0.717609\pi$
$383$	− 4.45129e6i	− 1.55056i	0.631618	−	0.775280i	$-0.282391\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	853254.i	0.289602i
$388$	0	0
$389$	−5.79825e6	−1.94278	−0.971388	−	0.237496i	$-0.923673\pi$
$389$	−5.79825e6	−1.94278	−0.971388	+	0.237496i	$0.923673\pi$
$390$	0	0
$391$	−3.71462e6	−1.22878
$392$	0	0
$393$	− 3.64694e6i	− 1.19110i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 333874.i	− 0.106318i	−0.998586	−	0.0531589i	$-0.983071\pi$
$397$	− 333874.i	− 0.106318i	0.998586	−	0.0531589i	$-0.0169290\pi$
$398$	0	0
$399$	5.94158e6	1.86840
$400$	0	0
$401$	−2.55689e6	−0.794057	−0.397029	−	0.917806i	$-0.629959\pi$
$401$	−2.55689e6	−0.794057	−0.397029	+	0.917806i	$0.629959\pi$
$402$	0	0
$403$	− 350200.i	− 0.107412i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 554976.i	− 0.166069i
$408$	0	0
$409$	3.05511e6	0.903063	0.451531	−	0.892255i	$-0.350878\pi$
$409$	3.05511e6	0.903063	0.451531	+	0.892255i	$0.350878\pi$
$410$	0	0
$411$	−866556.	−0.253042
$412$	0	0
$413$	2.09766e6i	0.605145i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 2.00081e6i	− 0.563463i
$418$	0	0
$419$	−3.54347e6	−0.986038	−0.493019	−	0.870019i	$-0.664107\pi$
$419$	−3.54347e6	−0.986038	−0.493019	+	0.870019i	$0.664107\pi$
$420$	0	0
$421$	1.97294e6	0.542511	0.271255	−	0.962507i	$-0.412561\pi$
$421$	1.97294e6	0.542511	0.271255	+	0.962507i	$0.412561\pi$
$422$	0	0
$423$	− 977994.i	− 0.265757i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 8.39014e6i	− 2.22689i
$428$	0	0
$429$	−2.43245e6	−0.638117
$430$	0	0
$431$	−1.37396e6	−0.356270	−0.178135	−	0.984006i	$-0.557006\pi$
$431$	−1.37396e6	−0.356270	−0.178135	+	0.984006i	$0.557006\pi$
$432$	0	0
$433$	5.18813e6i	1.32981i	0.746926	+	0.664907i	$0.231529\pi$
$433$	5.18813e6i	1.32981i	−0.746926	+	0.664907i	$0.768471\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.99807e6i	0.750997i
$438$	0	0
$439$	−2.94082e6	−0.728296	−0.364148	−	0.931341i	$-0.618640\pi$
$439$	−2.94082e6	−0.728296	−0.364148	+	0.931341i	$0.618640\pi$
$440$	0	0
$441$	3.38232e6	0.828167
$442$	0	0
$443$	1.28347e6i	0.310724i	0.987858	+	0.155362i	$0.0496544\pi$
$443$	1.28347e6i	0.310724i	−0.987858	+	0.155362i	$0.950346\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 4.37720e6i	− 1.03616i
$448$	0	0
$449$	4.95263e6	1.15937	0.579683	−	0.814842i	$-0.303176\pi$
$449$	4.95263e6	1.15937	0.579683	+	0.814842i	$0.303176\pi$
$450$	0	0
$451$	1.19261e6	0.276094
$452$	0	0
$453$	− 6.63934e6i	− 1.52012i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 7.91315e6i	− 1.77239i	−0.463315	−	0.886194i	$-0.653340\pi$
$457$	− 7.91315e6i	− 1.77239i	0.463315	−	0.886194i	$-0.346660\pi$
$458$	0	0
$459$	4.92804e6	1.09180
$460$	0	0
$461$	−6.18530e6	−1.35553	−0.677764	−	0.735280i	$-0.737051\pi$
$461$	−6.18530e6	−1.35553	−0.677764	+	0.735280i	$0.737051\pi$
$462$	0	0
$463$	− 491934.i	− 0.106648i	−0.998577	−	0.0533242i	$-0.983018\pi$
$463$	− 491934.i	− 0.106648i	0.998577	−	0.0533242i	$-0.0169817\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	447442.i	0.0949390i	0.998873	+	0.0474695i	$0.0151157\pi$
$467$	447442.i	0.0949390i	−0.998873	+	0.0474695i	$0.984884\pi$
$468$	0	0
$469$	1.15110e7	2.41646
$470$	0	0
$471$	5.75183e6	1.19469
$472$	0	0
$473$	6.91030e6i	1.42018i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 2.63947e6i	− 0.531154i
$478$	0	0
$479$	8.18487e6	1.62995	0.814973	−	0.579499i	$-0.196752\pi$
$479$	8.18487e6	1.62995	0.814973	+	0.579499i	$0.196752\pi$
$480$	0	0
$481$	174276.	0.0343459
$482$	0	0
$483$	9.57449e6i	1.86744i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	6.21524e6i	1.18751i	0.804648	+	0.593753i	$0.202354\pi$
$487$	6.21524e6i	1.18751i	−0.804648	+	0.593753i	$0.797646\pi$
$488$	0	0
$489$	1.25752e6	0.237816
$490$	0	0
$491$	−827856.	−0.154971	−0.0774856	−	0.996993i	$-0.524689\pi$
$491$	−827856.	−0.154971	−0.0774856	+	0.996993i	$0.524689\pi$
$492$	0	0
$493$	− 3.74842e6i	− 0.694594i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 229416.i	− 0.0416613i
$498$	0	0
$499$	−1.04004e7	−1.86982	−0.934908	−	0.354890i	$-0.884518\pi$
$499$	−1.04004e7	−1.86982	−0.934908	+	0.354890i	$0.884518\pi$
$500$	0	0
$501$	6.18160e6	1.10029
$502$	0	0
$503$	− 2.03821e6i	− 0.359193i	−0.983740	−	0.179597i	$-0.942521\pi$
$503$	− 2.03821e6i	− 0.359193i	0.983740	−	0.179597i	$-0.0574792\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	5.91943e6i	1.02273i
$508$	0	0
$509$	−3.66133e6	−0.626390	−0.313195	−	0.949689i	$-0.601399\pi$
$509$	−3.66133e6	−0.626390	−0.313195	+	0.949689i	$0.601399\pi$
$510$	0	0
$511$	1.52707e7	2.58706
$512$	0	0
$513$	− 3.97742e6i	− 0.667281i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 7.92054e6i	− 1.30325i
$518$	0	0
$519$	20556.0	0.00334981
$520$	0	0
$521$	3.24713e6	0.524089	0.262045	−	0.965056i	$-0.415603\pi$
$521$	3.24713e6	0.524089	0.262045	+	0.965056i	$0.415603\pi$
$522$	0	0
$523$	4.97357e6i	0.795086i	0.917584	+	0.397543i	$0.130137\pi$
$523$	4.97357e6i	0.795086i	−0.917584	+	0.397543i	$0.869863\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 2.87300e6i	− 0.450619i
$528$	0	0
$529$	1.60514e6	0.249387
$530$	0	0
$531$	−702108.	−0.108061
$532$	0	0
$533$	374508.i	0.0571009i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 1.56031e6i	− 0.233494i
$538$	0	0
$539$	2.73926e7	4.06126
$540$	0	0
$541$	2.42544e6	0.356285	0.178142	−	0.984005i	$-0.442991\pi$
$541$	2.42544e6	0.356285	0.178142	+	0.984005i	$0.442991\pi$
$542$	0	0
$543$	− 1.17255e7i	− 1.70660i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 731254.i	− 0.104496i	−0.998634	−	0.0522480i	$-0.983361\pi$
$547$	− 731254.i	− 0.104496i	0.998634	−	0.0522480i	$-0.0166386\pi$
$548$	0	0
$549$	2.80827e6	0.397656
$550$	0	0
$551$	−3.02535e6	−0.424519
$552$	0	0
$553$	1.12617e7i	1.56600i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 7.71992e6i	− 1.05433i	−0.849764	−	0.527163i	$-0.823256\pi$
$557$	− 7.71992e6i	− 1.05433i	0.849764	−	0.527163i	$-0.176744\pi$
$558$	0	0
$559$	−2.17000e6	−0.293718
$560$	0	0
$561$	−1.99555e7	−2.67705
$562$	0	0
$563$	− 3.10576e6i	− 0.412949i	−0.978452	−	0.206475i	$-0.933801\pi$
$563$	− 3.10576e6i	− 0.412949i	0.978452	−	0.206475i	$-0.0661991\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.74654e7i	− 2.28150i
$568$	0	0
$569$	482498.	0.0624762	0.0312381	−	0.999512i	$-0.490055\pi$
$569$	482498.	0.0624762	0.0312381	+	0.999512i	$0.490055\pi$
$570$	0	0
$571$	−1.38502e7	−1.77773	−0.888865	−	0.458169i	$-0.848505\pi$
$571$	−1.38502e7	−1.77773	−0.888865	+	0.458169i	$0.848505\pi$
$572$	0	0
$573$	− 524520.i	− 0.0667384i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 7.09764e6i	− 0.887513i	−0.896147	−	0.443756i	$-0.853646\pi$
$577$	− 7.09764e6i	− 0.887513i	0.896147	−	0.443756i	$-0.146354\pi$
$578$	0	0
$579$	1.16353e7	1.44239
$580$	0	0
$581$	−2.14843e7	−2.64046
$582$	0	0
$583$	− 2.13764e7i	− 2.60473i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	1.56926e6i	0.187975i	0.995573	+	0.0939873i	$0.0299613\pi$
$587$	1.56926e6i	0.187975i	−0.995573	+	0.0939873i	$0.970039\pi$
$588$	0	0
$589$	−2.31880e6	−0.275407
$590$	0	0
$591$	7.76048e6	0.913945
$592$	0	0
$593$	− 1.30477e7i	− 1.52370i	−0.647756	−	0.761848i	$-0.724293\pi$
$593$	− 1.30477e7i	− 1.52370i	0.647756	−	0.761848i	$-0.275707\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 2.36894e6i	− 0.272031i
$598$	0	0
$599$	−5.24688e6	−0.597495	−0.298747	−	0.954332i	$-0.596569\pi$
$599$	−5.24688e6	−0.597495	−0.298747	+	0.954332i	$0.596569\pi$
$600$	0	0
$601$	−5.57316e6	−0.629384	−0.314692	−	0.949194i	$-0.601901\pi$
$601$	−5.57316e6	−0.629384	−0.314692	+	0.949194i	$0.601901\pi$
$602$	0	0
$603$	3.85285e6i	0.431508i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	1.98249e6i	0.218393i	0.994020	+	0.109197i	$0.0348278\pi$
$607$	1.98249e6i	0.218393i	−0.994020	+	0.109197i	$0.965172\pi$
$608$	0	0
$609$	−9.66161e6	−1.05562
$610$	0	0
$611$	2.48724e6	0.269535
$612$	0	0
$613$	969810.i	0.104240i	0.998641	+	0.0521201i	$0.0165979\pi$
$613$	969810.i	0.104240i	−0.998641	+	0.0521201i	$0.983402\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.12946e7i	1.19442i	0.802084	+	0.597211i	$0.203725\pi$
$617$	1.12946e7i	1.19442i	−0.802084	+	0.597211i	$0.796275\pi$
$618$	0	0
$619$	−1.80728e7	−1.89583	−0.947914	−	0.318528i	$-0.896812\pi$
$619$	−1.80728e7	−1.89583	−0.947914	+	0.318528i	$0.896812\pi$
$620$	0	0
$621$	6.40937e6	0.666939
$622$	0	0
$623$	2.53940e7i	2.62127i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1.61061e7i	1.63615i
$628$	0	0
$629$	1.42974e6	0.144089
$630$	0	0
$631$	−4.62634e6	−0.462556	−0.231278	−	0.972888i	$-0.574291\pi$
$631$	−4.62634e6	−0.462556	−0.231278	+	0.972888i	$0.574291\pi$
$632$	0	0
$633$	− 2.17941e7i	− 2.16187i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	8.60194e6i	0.839939i
$638$	0	0
$639$	76788.0	0.00743946
$640$	0	0
$641$	1.99058e7	1.91352	0.956762	−	0.290871i	$-0.0939452\pi$
$641$	1.99058e7	1.91352	0.956762	+	0.290871i	$0.0939452\pi$
$642$	0	0
$643$	− 1.21078e7i	− 1.15489i	−0.816431	−	0.577443i	$-0.804051\pi$
$643$	− 1.21078e7i	− 1.15489i	0.816431	−	0.577443i	$-0.195949\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 2.53124e6i	− 0.237724i	−0.992911	−	0.118862i	$-0.962075\pi$
$647$	− 2.53124e6i	− 0.237724i	0.992911	−	0.118862i	$-0.0379245\pi$
$648$	0	0
$649$	−5.68621e6	−0.529921
$650$	0	0
$651$	−7.40520e6	−0.684832
$652$	0	0
$653$	1.37043e7i	1.25770i	0.777529	+	0.628848i	$0.216473\pi$
$653$	1.37043e7i	1.25770i	−0.777529	+	0.628848i	$0.783527\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	5.11126e6i	0.461971i
$658$	0	0
$659$	−9.83320e6	−0.882026	−0.441013	−	0.897501i	$-0.645381\pi$
$659$	−9.83320e6	−0.882026	−0.441013	+	0.897501i	$0.645381\pi$
$660$	0	0
$661$	6.68687e6	0.595278	0.297639	−	0.954679i	$-0.403801\pi$
$661$	6.68687e6	0.595278	0.297639	+	0.954679i	$0.403801\pi$
$662$	0	0
$663$	− 6.26652e6i	− 0.553659i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.87516e6i	− 0.424302i
$668$	0	0
$669$	627948.	0.0542448
$670$	0	0
$671$	2.27435e7	1.95008
$672$	0	0
$673$	− 727566.i	− 0.0619205i	−0.999521	−	0.0309603i	$-0.990143\pi$
$673$	− 727566.i	− 0.0619205i	0.999521	−	0.0309603i	$-0.00985654\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.86951e7i	− 1.56768i	−0.620964	−	0.783839i	$-0.713259\pi$
$677$	− 1.86951e7i	− 1.56768i	0.620964	−	0.783839i	$-0.286741\pi$
$678$	0	0
$679$	−8.77347e6	−0.730293
$680$	0	0
$681$	2.23528e6	0.184699
$682$	0	0
$683$	1.79850e7i	1.47523i	0.675223	+	0.737614i	$0.264048\pi$
$683$	1.79850e7i	1.47523i	−0.675223	+	0.737614i	$0.735952\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	8.21495e6i	0.664069i
$688$	0	0
$689$	6.71272e6	0.538704
$690$	0	0
$691$	−1.20006e6	−0.0956107	−0.0478053	−	0.998857i	$-0.515223\pi$
$691$	−1.20006e6	−0.0956107	−0.0478053	+	0.998857i	$0.515223\pi$
$692$	0	0
$693$	1.28589e7i	1.01712i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	3.07242e6i	0.239551i
$698$	0	0
$699$	−4.54799e6	−0.352068
$700$	0	0
$701$	−1.15194e7	−0.885393	−0.442697	−	0.896671i	$-0.645978\pi$
$701$	−1.15194e7	−0.885393	−0.442697	+	0.896671i	$0.645978\pi$
$702$	0	0
$703$	− 1.15394e6i	− 0.0880636i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.02816e7i	0.773593i
$708$	0	0
$709$	610738.	0.0456288	0.0228144	−	0.999740i	$-0.492737\pi$
$709$	610738.	0.0456288	0.0228144	+	0.999740i	$0.492737\pi$
$710$	0	0
$711$	−3.76942e6	−0.279640
$712$	0	0
$713$	− 3.73660e6i	− 0.275266i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	1.17115e6i	0.0850776i
$718$	0	0
$719$	3.97278e6	0.286597	0.143299	−	0.989680i	$-0.454229\pi$
$719$	3.97278e6	0.286597	0.143299	+	0.989680i	$0.454229\pi$
$720$	0	0
$721$	3.58000e7	2.56475
$722$	0	0
$723$	− 2.53080e7i	− 1.80058i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 1.23220e7i	− 0.864658i	−0.901716	−	0.432329i	$-0.857692\pi$
$727$	− 1.23220e7i	− 0.864658i	0.901716	−	0.432329i	$-0.142308\pi$
$728$	0	0
$729$	−6.90873e6	−0.481481
$730$	0	0
$731$	−1.78025e7	−1.23222
$732$	0	0
$733$	8.14579e6i	0.559981i	0.960003	+	0.279990i	$0.0903313\pi$
$733$	8.14579e6i	0.559981i	−0.960003	+	0.279990i	$0.909669\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.12033e7i	2.11608i
$738$	0	0
$739$	7.16653e6	0.482723	0.241361	−	0.970435i	$-0.422406\pi$
$739$	7.16653e6	0.482723	0.241361	+	0.970435i	$0.422406\pi$
$740$	0	0
$741$	−5.05771e6	−0.338383
$742$	0	0
$743$	5.65041e6i	0.375498i	0.982217	+	0.187749i	$0.0601192\pi$
$743$	5.65041e6i	0.375498i	−0.982217	+	0.187749i	$0.939881\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 7.19102e6i	− 0.471508i
$748$	0	0
$749$	4.45038e6	0.289863
$750$	0	0
$751$	−9.09500e6	−0.588441	−0.294221	−	0.955738i	$-0.595060\pi$
$751$	−9.09500e6	−0.588441	−0.294221	+	0.955738i	$0.595060\pi$
$752$	0	0
$753$	9.87120e6i	0.634428i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.12880e7i	0.715944i	0.933732	+	0.357972i	$0.116532\pi$
$757$	1.12880e7i	0.715944i	−0.933732	+	0.357972i	$0.883468\pi$
$758$	0	0
$759$	−2.59540e7	−1.63531
$760$	0	0
$761$	1.52933e7	0.957283	0.478641	−	0.878011i	$-0.341129\pi$
$761$	1.52933e7	0.957283	0.478641	+	0.878011i	$0.341129\pi$
$762$	0	0
$763$	3.50915e7i	2.18218i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.78561e6i	− 0.109597i
$768$	0	0
$769$	−1.77402e6	−0.108179	−0.0540894	−	0.998536i	$-0.517226\pi$
$769$	−1.77402e6	−0.108179	−0.0540894	+	0.998536i	$0.517226\pi$
$770$	0	0
$771$	−8.88894e6	−0.538535
$772$	0	0
$773$	− 1.46441e7i	− 0.881484i	−0.897634	−	0.440742i	$-0.854715\pi$
$773$	− 1.46441e7i	− 0.881484i	0.897634	−	0.440742i	$-0.145285\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 3.68518e6i	− 0.218981i
$778$	0	0
$779$	2.47975e6	0.146408
$780$	0	0
$781$	621888.	0.0364825
$782$	0	0
$783$	6.46769e6i	0.377003i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 5.97074e6i	− 0.343630i	−0.985129	−	0.171815i	$-0.945037\pi$
$787$	− 5.97074e6i	− 0.343630i	0.985129	−	0.171815i	$-0.0549632\pi$
$788$	0	0
$789$	−1.92926e7	−1.10331
$790$	0	0
$791$	−2.00245e7	−1.13794
$792$	0	0
$793$	7.14202e6i	0.403309i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.40500e7i	1.89876i	0.314125	+	0.949382i	$0.398289\pi$
$797$	3.40500e7i	1.89876i	−0.314125	+	0.949382i	$0.601711\pi$
$798$	0	0
$799$	2.04051e7	1.13076
$800$	0	0
$801$	−8.49965e6	−0.468080
$802$	0	0
$803$	4.13949e7i	2.26547i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 1.79891e7i	− 0.972356i
$808$	0	0
$809$	2.63540e7	1.41571	0.707857	−	0.706356i	$-0.249662\pi$
$809$	2.63540e7	1.41571	0.707857	+	0.706356i	$0.249662\pi$
$810$	0	0
$811$	9.49658e6	0.507008	0.253504	−	0.967334i	$-0.418417\pi$
$811$	9.49658e6	0.507008	0.253504	+	0.967334i	$0.418417\pi$
$812$	0	0
$813$	− 1.81369e7i	− 0.962357i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.43684e7i	0.753100i
$818$	0	0
$819$	−4.03801e6	−0.210357
$820$	0	0
$821$	−1.59887e7	−0.827856	−0.413928	−	0.910310i	$-0.635843\pi$
$821$	−1.59887e7	−0.827856	−0.413928	+	0.910310i	$0.635843\pi$
$822$	0	0
$823$	− 3.18347e7i	− 1.63833i	−0.573559	−	0.819164i	$-0.694438\pi$
$823$	− 3.18347e7i	− 1.63833i	0.573559	−	0.819164i	$-0.305562\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.27575e7i	0.648635i	0.945948	+	0.324317i	$0.105135\pi$
$827$	1.27575e7i	0.648635i	−0.945948	+	0.324317i	$0.894865\pi$
$828$	0	0
$829$	−6.18613e6	−0.312631	−0.156316	−	0.987707i	$-0.549962\pi$
$829$	−6.18613e6	−0.312631	−0.156316	+	0.987707i	$0.549962\pi$
$830$	0	0
$831$	1.84114e7	0.924880
$832$	0	0
$833$	7.05693e7i	3.52374i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	4.95720e6i	0.244581i
$838$	0	0
$839$	−5.66754e6	−0.277965	−0.138982	−	0.990295i	$-0.544383\pi$
$839$	−5.66754e6	−0.277965	−0.138982	+	0.990295i	$0.544383\pi$
$840$	0	0
$841$	−1.55916e7	−0.760154
$842$	0	0
$843$	− 2.12710e7i	− 1.03091i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.51670e7i	3.12118i
$848$	0	0
$849$	−1.65151e7	−0.786343
$850$	0	0
$851$	1.85951e6	0.0880185
$852$	0	0
$853$	− 1.76010e7i	− 0.828257i	−0.910218	−	0.414129i	$-0.864086\pi$
$853$	− 1.76010e7i	− 0.828257i	0.910218	−	0.414129i	$-0.135914\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 162162.i	− 0.00754218i	−0.999993	−	0.00377109i	$-0.998800\pi$
$857$	− 162162.i	− 0.00754218i	0.999993	−	0.00377109i	$-0.00120038\pi$
$858$	0	0
$859$	−7.10520e6	−0.328544	−0.164272	−	0.986415i	$-0.552527\pi$
$859$	−7.10520e6	−0.328544	−0.164272	+	0.986415i	$0.552527\pi$
$860$	0	0
$861$	7.91921e6	0.364061
$862$	0	0
$863$	− 4.08956e6i	− 0.186917i	−0.995623	−	0.0934586i	$-0.970208\pi$
$863$	− 4.08956e6i	− 0.186917i	0.995623	−	0.0934586i	$-0.0297923\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 2.58524e7i	− 1.16803i
$868$	0	0
$869$	−3.05276e7	−1.37133
$870$	0	0
$871$	−9.79860e6	−0.437641
$872$	0	0
$873$	− 2.93657e6i	− 0.130408i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 809194.i	− 0.0355266i	−0.999842	−	0.0177633i	$-0.994345\pi$
$877$	− 809194.i	− 0.0355266i	0.999842	−	0.0177633i	$-0.00565453\pi$
$878$	0	0
$879$	9.22144e6	0.402556
$880$	0	0
$881$	3.90411e6	0.169466	0.0847329	−	0.996404i	$-0.472996\pi$
$881$	3.90411e6	0.169466	0.0847329	+	0.996404i	$0.472996\pi$
$882$	0	0
$883$	− 3.58290e7i	− 1.54644i	−0.634138	−	0.773220i	$-0.718645\pi$
$883$	− 3.58290e7i	− 1.54644i	0.634138	−	0.773220i	$-0.281355\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.77571e7i	1.18458i	0.805725	+	0.592290i	$0.201776\pi$
$887$	2.77571e7i	1.18458i	−0.805725	+	0.592290i	$0.798224\pi$
$888$	0	0
$889$	6.64159e7	2.81850
$890$	0	0
$891$	4.73442e7	1.99789
$892$	0	0
$893$	− 1.64689e7i	− 0.691094i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 8.15018e6i	− 0.338210i
$898$	0	0
$899$	3.77060e6	0.155601
$900$	0	0
$901$	5.50703e7	2.25999
$902$	0	0
$903$	4.58861e7i	1.87267i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	2.01914e7i	0.814981i	0.913209	+	0.407490i	$0.133596\pi$
$907$	2.01914e7i	0.814981i	−0.913209	+	0.407490i	$0.866404\pi$
$908$	0	0
$909$	−3.44137e6	−0.138141
$910$	0	0
$911$	−2.75179e7	−1.09855	−0.549274	−	0.835642i	$-0.685096\pi$
$911$	−2.75179e7	−1.09855	−0.549274	+	0.835642i	$0.685096\pi$
$912$	0	0
$913$	− 5.82384e7i	− 2.31224i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 4.90311e7i	− 1.92552i
$918$	0	0
$919$	1.31786e7	0.514730	0.257365	−	0.966314i	$-0.417146\pi$
$919$	1.31786e7	0.514730	0.257365	+	0.966314i	$0.417146\pi$
$920$	0	0
$921$	−2.53702e7	−0.985543
$922$	0	0
$923$	195288.i	0.00754521i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	1.19827e7i	0.457988i
$928$	0	0
$929$	−4.00688e7	−1.52323	−0.761617	−	0.648027i	$-0.775594\pi$
$929$	−4.00688e7	−1.52323	−0.761617	+	0.648027i	$0.775594\pi$
$930$	0	0
$931$	5.69565e7	2.15362
$932$	0	0
$933$	5.01486e7i	1.88606i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 3.04258e7i	− 1.13212i	−0.824364	−	0.566060i	$-0.808467\pi$
$937$	− 3.04258e7i	− 1.13212i	0.824364	−	0.566060i	$-0.191533\pi$
$938$	0	0
$939$	−2.79154e7	−1.03319
$940$	0	0
$941$	−3.26349e7	−1.20146	−0.600729	−	0.799452i	$-0.705123\pi$
$941$	−3.26349e7	−1.20146	−0.600729	+	0.799452i	$0.705123\pi$
$942$	0	0
$943$	3.99596e6i	0.146333i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	3.01534e7i	1.09260i	0.837589	+	0.546300i	$0.183964\pi$
$947$	3.01534e7i	1.09260i	−0.837589	+	0.546300i	$0.816036\pi$
$948$	0	0
$949$	−1.29990e7	−0.468538
$950$	0	0
$951$	6.79180e6	0.243519
$952$	0	0
$953$	303066.i	0.0108095i	0.999985	+	0.00540474i	$0.00172039\pi$
$953$	303066.i	0.0108095i	−0.999985	+	0.00540474i	$0.998280\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 2.61901e7i	− 0.924396i
$958$	0	0
$959$	−1.16504e7	−0.409066
$960$	0	0
$961$	−2.57392e7	−0.899054
$962$	0	0
$963$	1.48959e6i	0.0517608i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 7.39863e6i	− 0.254440i	−0.991875	−	0.127220i	$-0.959395\pi$
$967$	− 7.39863e6i	− 0.254440i	0.991875	−	0.127220i	$-0.0406054\pi$
$968$	0	0
$969$	−4.14929e7	−1.41959
$970$	0	0
$971$	6.18414e6	0.210490	0.105245	−	0.994446i	$-0.466437\pi$
$971$	6.18414e6	0.210490	0.105245	+	0.994446i	$0.466437\pi$
$972$	0	0
$973$	− 2.68998e7i	− 0.910891i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.63928e6i	− 0.0549436i	−0.999623	−	0.0274718i	$-0.991254\pi$
$977$	− 1.63928e6i	− 0.0549436i	0.999623	−	0.0274718i	$-0.00874565\pi$
$978$	0	0
$979$	−6.88367e7	−2.29543
$980$	0	0
$981$	−1.17455e7	−0.389671
$982$	0	0
$983$	1.13020e7i	0.373052i	0.982450	+	0.186526i	$0.0597229\pi$
$983$	1.13020e7i	0.373052i	−0.982450	+	0.186526i	$0.940277\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 5.25943e7i	− 1.71849i
$988$	0	0
$989$	−2.31537e7	−0.752714
$990$	0	0
$991$	−3.12643e6	−0.101126	−0.0505632	−	0.998721i	$-0.516102\pi$
$991$	−3.12643e6	−0.101126	−0.0505632	+	0.998721i	$0.516102\pi$
$992$	0	0
$993$	2.93514e7i	0.944616i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	3.55827e7i	1.13371i	0.823818	+	0.566854i	$0.191840\pi$
$997$	3.55827e7i	1.13371i	−0.823818	+	0.566854i	$0.808160\pi$
$998$	0	0
$999$	−2.46694e6	−0.0782067

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.c.e.49.2		2
4.3	odd	2		200.6.c.b.49.1		2
5.2	odd	4		80.6.a.g.1.1		1
5.3	odd	4		400.6.a.b.1.1		1
5.4	even	2	inner	400.6.c.e.49.1		2
15.2	even	4		720.6.a.k.1.1		1
20.3	even	4		200.6.a.d.1.1		1
20.7	even	4		40.6.a.a.1.1	✓	1
20.19	odd	2		200.6.c.b.49.2		2
40.27	even	4		320.6.a.m.1.1		1
40.37	odd	4		320.6.a.d.1.1		1
60.47	odd	4		360.6.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.6.a.a.1.1	✓	1	20.7	even	4
80.6.a.g.1.1		1	5.2	odd	4
200.6.a.d.1.1		1	20.3	even	4
200.6.c.b.49.1		2	4.3	odd	2
200.6.c.b.49.2		2	20.19	odd	2
320.6.a.d.1.1		1	40.37	odd	4
320.6.a.m.1.1		1	40.27	even	4
360.6.a.i.1.1		1	60.47	odd	4
400.6.a.b.1.1		1	5.3	odd	4
400.6.c.e.49.1		2	5.4	even	2	inner
400.6.c.e.49.2		2	1.1	even	1	trivial
720.6.a.k.1.1		1	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.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+18.0000i q^{3} +242.000i q^{7} -81.0000 q^{9} -656.000 q^{11} -206.000i q^{13} -1690.00i q^{17} -1364.00 q^{19} -4356.00 q^{21} -2198.00i q^{23} +2916.00i q^{27} +2218.00 q^{29} +1700.00 q^{31} -11808.0i q^{33} +846.000i q^{37} +3708.00 q^{39} -1818.00 q^{41} -10534.0i q^{43} +12074.0i q^{47} -41757.0 q^{49} +30420.0 q^{51} +32586.0i q^{53} -24552.0i q^{57} +8668.00 q^{59} -34670.0 q^{61} -19602.0i q^{63} -47566.0i q^{67} +39564.0 q^{69} -948.000 q^{71} -63102.0i q^{73} -158752. i q^{77} +46536.0 q^{79} -72171.0 q^{81} +88778.0i q^{83} +39924.0i q^{87} +104934. q^{89} +49852.0 q^{91} +30600.0i q^{93} +36254.0i q^{97} +53136.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 162 q^{9} - 1312 q^{11} - 2728 q^{19} - 8712 q^{21} + 4436 q^{29} + 3400 q^{31} + 7416 q^{39} - 3636 q^{41} - 83514 q^{49} + 60840 q^{51} + 17336 q^{59} - 69340 q^{61} + 79128 q^{69} - 1896 q^{71}+ \cdots + 106272 q^{99}+O(q^{100}) \) 2 * q - 162 * q^9 - 1312 * q^11 - 2728 * q^19 - 8712 * q^21 + 4436 * q^29 + 3400 * q^31 + 7416 * q^39 - 3636 * q^41 - 83514 * q^49 + 60840 * q^51 + 17336 * q^59 - 69340 * q^61 + 79128 * q^69 - 1896 * q^71 + 93072 * q^79 - 144342 * q^81 + 209868 * q^89 + 99704 * q^91 + 106272 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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