LMFDB - Embedded newform 400.6.a.v.1.1

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{409}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 102 $ x^2 - x - 102
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 100)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$10.6119$ 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)

$f(q)$	$=$	$q-10.2237 q^{3} +101.342 q^{7} -138.475 q^{9} +O(q^{10})$	$q-10.2237 q^{3} +101.342 q^{7} -138.475 q^{9} +333.356 q^{11} +25.3700 q^{13} -1697.68 q^{17} -1956.07 q^{19} -1036.10 q^{21} +2850.88 q^{23} +3900.10 q^{27} +6629.70 q^{29} +7477.56 q^{31} -3408.15 q^{33} -15292.0 q^{37} -259.376 q^{39} +1061.55 q^{41} -4273.70 q^{43} +2069.53 q^{47} -6536.70 q^{49} +17356.7 q^{51} -4918.71 q^{53} +19998.4 q^{57} -4707.07 q^{59} +43577.9 q^{61} -14033.4 q^{63} -58153.0 q^{67} -29146.7 q^{69} -41411.0 q^{71} -4686.21 q^{73} +33783.2 q^{77} +204.738 q^{79} -6224.27 q^{81} -78794.1 q^{83} -67780.4 q^{87} -136510. q^{89} +2571.06 q^{91} -76448.7 q^{93} -53842.2 q^{97} -46161.5 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9}+O(q^{10}) $ 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9	$ 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9} + 60 q^{11} - 920 q^{13} - 2910 q^{17} - 2092 q^{19} - 5308 q^{21} + 120 q^{23} + 16820 q^{27} + 3552 q^{29} + 8888 q^{31} - 11670 q^{33} - 12140 q^{37} - 28832 q^{39} - 12438 q^{41} + 1160 q^{43} - 1200 q^{47} - 3366 q^{49} - 19284 q^{51} - 26340 q^{53} + 15890 q^{57} + 36696 q^{59} + 19204 q^{61} - 108800 q^{63} - 90460 q^{67} - 111684 q^{69} - 2736 q^{71} - 12770 q^{73} + 72420 q^{77} + 16184 q^{79} + 221338 q^{81} - 30300 q^{83} - 160800 q^{87} - 47322 q^{89} + 136192 q^{91} - 33820 q^{93} + 2980 q^{97} - 229440 q^{99}+O(q^{100}) $ 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9 + 60 * q^11 - 920 * q^13 - 2910 * q^17 - 2092 * q^19 - 5308 * q^21 + 120 * q^23 + 16820 * q^27 + 3552 * q^29 + 8888 * q^31 - 11670 * q^33 - 12140 * q^37 - 28832 * q^39 - 12438 * q^41 + 1160 * q^43 - 1200 * q^47 - 3366 * q^49 - 19284 * q^51 - 26340 * q^53 + 15890 * q^57 + 36696 * q^59 + 19204 * q^61 - 108800 * q^63 - 90460 * q^67 - 111684 * q^69 - 2736 * q^71 - 12770 * q^73 + 72420 * q^77 + 16184 * q^79 + 221338 * q^81 - 30300 * q^83 - 160800 * q^87 - 47322 * q^89 + 136192 * q^91 - 33820 * q^93 + 2980 * q^97 - 229440 * 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
$3$	−10.2237	−0.655854	−0.327927	−	0.944703i	$-0.606350\pi$
$3$	−10.2237	−0.655854	−0.327927	+	0.944703i	$0.606350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	101.342	0.781711	0.390856	−	0.920452i	$-0.372179\pi$
$7$	101.342	0.781711	0.390856	+	0.920452i	$0.372179\pi$
$8$	0	0
$9$	−138.475	−0.569856
$10$	0	0
$11$	333.356	0.830667	0.415333	−	0.909669i	$-0.363665\pi$
$11$	333.356	0.830667	0.415333	+	0.909669i	$0.363665\pi$
$12$	0	0
$13$	25.3700	0.0416353	0.0208176	−	0.999783i	$-0.493373\pi$
$13$	25.3700	0.0416353	0.0208176	+	0.999783i	$0.493373\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1697.68	−1.42474	−0.712369	−	0.701805i	$-0.752378\pi$
$17$	−1697.68	−1.42474	−0.712369	+	0.701805i	$0.752378\pi$
$18$	0	0
$19$	−1956.07	−1.24308	−0.621541	−	0.783381i	$-0.713493\pi$
$19$	−1956.07	−1.24308	−0.621541	+	0.783381i	$0.713493\pi$
$20$	0	0
$21$	−1036.10	−0.512688
$22$	0	0
$23$	2850.88	1.12372	0.561861	−	0.827232i	$-0.310086\pi$
$23$	2850.88	1.12372	0.561861	+	0.827232i	$0.310086\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	3900.10	1.02960
$28$	0	0
$29$	6629.70	1.46386	0.731929	−	0.681381i	$-0.238620\pi$
$29$	6629.70	1.46386	0.731929	+	0.681381i	$0.238620\pi$
$30$	0	0
$31$	7477.56	1.39751	0.698756	−	0.715360i	$-0.253737\pi$
$31$	7477.56	1.39751	0.698756	+	0.715360i	$0.253737\pi$
$32$	0	0
$33$	−3408.15	−0.544796
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−15292.0	−1.83637	−0.918186	−	0.396149i	$-0.870346\pi$
$37$	−15292.0	−1.83637	−0.918186	+	0.396149i	$0.870346\pi$
$38$	0	0
$39$	−259.376	−0.0273067
$40$	0	0
$41$	1061.55	0.0986235	0.0493118	−	0.998783i	$-0.484297\pi$
$41$	1061.55	0.0986235	0.0493118	+	0.998783i	$0.484297\pi$
$42$	0	0
$43$	−4273.70	−0.352479	−0.176239	−	0.984347i	$-0.556393\pi$
$43$	−4273.70	−0.352479	−0.176239	+	0.984347i	$0.556393\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2069.53	0.136656	0.0683279	−	0.997663i	$-0.478234\pi$
$47$	2069.53	0.136656	0.0683279	+	0.997663i	$0.478234\pi$
$48$	0	0
$49$	−6536.70	−0.388927
$50$	0	0
$51$	17356.7	0.934419
$52$	0	0
$53$	−4918.71	−0.240526	−0.120263	−	0.992742i	$-0.538374\pi$
$53$	−4918.71	−0.240526	−0.120263	+	0.992742i	$0.538374\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	19998.4	0.815281
$58$	0	0
$59$	−4707.07	−0.176044	−0.0880219	−	0.996119i	$-0.528055\pi$
$59$	−4707.07	−0.176044	−0.0880219	+	0.996119i	$0.528055\pi$
$60$	0	0
$61$	43577.9	1.49948	0.749742	−	0.661731i	$-0.230178\pi$
$61$	43577.9	1.49948	0.749742	+	0.661731i	$0.230178\pi$
$62$	0	0
$63$	−14033.4	−0.445463
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−58153.0	−1.58265	−0.791325	−	0.611396i	$-0.790608\pi$
$67$	−58153.0	−1.58265	−0.791325	+	0.611396i	$0.790608\pi$
$68$	0	0
$69$	−29146.7	−0.736997
$70$	0	0
$71$	−41411.0	−0.974922	−0.487461	−	0.873145i	$-0.662077\pi$
$71$	−41411.0	−0.974922	−0.487461	+	0.873145i	$0.662077\pi$
$72$	0	0
$73$	−4686.21	−0.102923	−0.0514617	−	0.998675i	$-0.516388\pi$
$73$	−4686.21	−0.102923	−0.0514617	+	0.998675i	$0.516388\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	33783.2	0.649342
$78$	0	0
$79$	204.738	0.00369089	0.00184544	−	0.999998i	$-0.499413\pi$
$79$	204.738	0.00369089	0.00184544	+	0.999998i	$0.499413\pi$
$80$	0	0
$81$	−6224.27	−0.105408
$82$	0	0
$83$	−78794.1	−1.25545	−0.627724	−	0.778436i	$-0.716013\pi$
$83$	−78794.1	−1.25545	−0.627724	+	0.778436i	$0.716013\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−67780.4	−0.960077
$88$	0	0
$89$	−136510.	−1.82679	−0.913393	−	0.407078i	$-0.866548\pi$
$89$	−136510.	−1.82679	−0.913393	+	0.407078i	$0.866548\pi$
$90$	0	0
$91$	2571.06	0.0325468
$92$	0	0
$93$	−76448.7	−0.916564
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−53842.2	−0.581023	−0.290511	−	0.956872i	$-0.593825\pi$
$97$	−53842.2	−0.581023	−0.290511	+	0.956872i	$0.593825\pi$
$98$	0	0
$99$	−46161.5	−0.473360
$100$	0	0
$101$	72272.8	0.704972	0.352486	−	0.935817i	$-0.385336\pi$
$101$	72272.8	0.704972	0.352486	+	0.935817i	$0.385336\pi$
$102$	0	0
$103$	−165859.	−1.54045	−0.770223	−	0.637774i	$-0.779855\pi$
$103$	−165859.	−1.54045	−0.770223	+	0.637774i	$0.779855\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	123370.	1.04172	0.520860	−	0.853642i	$-0.325611\pi$
$107$	123370.	1.04172	0.520860	+	0.853642i	$0.325611\pi$
$108$	0	0
$109$	80205.0	0.646600	0.323300	−	0.946297i	$-0.395208\pi$
$109$	80205.0	0.646600	0.323300	+	0.946297i	$0.395208\pi$
$110$	0	0
$111$	156342.	1.20439
$112$	0	0
$113$	−248670.	−1.83200	−0.916002	−	0.401173i	$-0.868603\pi$
$113$	−248670.	−1.83200	−0.916002	+	0.401173i	$0.868603\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−3513.10	−0.0237261
$118$	0	0
$119$	−172048.	−1.11373
$120$	0	0
$121$	−49924.6	−0.309993
$122$	0	0
$123$	−10853.0	−0.0646826
$124$	0	0
$125$	0	0
$126$	0	0
$127$	66527.6	0.366010	0.183005	−	0.983112i	$-0.441418\pi$
$127$	66527.6	0.366010	0.183005	+	0.983112i	$0.441418\pi$
$128$	0	0
$129$	43693.2	0.231174
$130$	0	0
$131$	98559.2	0.501786	0.250893	−	0.968015i	$-0.419276\pi$
$131$	98559.2	0.501786	0.250893	+	0.968015i	$0.419276\pi$
$132$	0	0
$133$	−198233.	−0.971732
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−383359.	−1.74504	−0.872518	−	0.488581i	$-0.837515\pi$
$137$	−383359.	−1.74504	−0.872518	+	0.488581i	$0.837515\pi$
$138$	0	0
$139$	−267882.	−1.17600	−0.587998	−	0.808862i	$-0.700084\pi$
$139$	−267882.	−1.17600	−0.587998	+	0.808862i	$0.700084\pi$
$140$	0	0
$141$	−21158.4	−0.0896262
$142$	0	0
$143$	8457.23	0.0345850
$144$	0	0
$145$	0	0
$146$	0	0
$147$	66829.6	0.255079
$148$	0	0
$149$	−155452.	−0.573627	−0.286813	−	0.957986i	$-0.592596\pi$
$149$	−155452.	−0.573627	−0.286813	+	0.957986i	$0.592596\pi$
$150$	0	0
$151$	235361.	0.840025	0.420012	−	0.907518i	$-0.362026\pi$
$151$	235361.	0.840025	0.420012	+	0.907518i	$0.362026\pi$
$152$	0	0
$153$	235087.	0.811895
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−127589.	−0.413108	−0.206554	−	0.978435i	$-0.566225\pi$
$157$	−127589.	−0.413108	−0.206554	+	0.978435i	$0.566225\pi$
$158$	0	0
$159$	50287.7	0.157750
$160$	0	0
$161$	288915.	0.878426
$162$	0	0
$163$	−222097.	−0.654748	−0.327374	−	0.944895i	$-0.606164\pi$
$163$	−222097.	−0.654748	−0.327374	+	0.944895i	$0.606164\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−389927.	−1.08191	−0.540956	−	0.841051i	$-0.681938\pi$
$167$	−389927.	−1.08191	−0.540956	+	0.841051i	$0.681938\pi$
$168$	0	0
$169$	−370649.	−0.998267
$170$	0	0
$171$	270867.	0.708378
$172$	0	0
$173$	−606977.	−1.54190	−0.770952	−	0.636894i	$-0.780219\pi$
$173$	−606977.	−1.54190	−0.770952	+	0.636894i	$0.780219\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	48123.9	0.115459
$178$	0	0
$179$	228152.	0.532222	0.266111	−	0.963942i	$-0.414261\pi$
$179$	228152.	0.532222	0.266111	+	0.963942i	$0.414261\pi$
$180$	0	0
$181$	242380.	0.549921	0.274960	−	0.961456i	$-0.411335\pi$
$181$	242380.	0.549921	0.274960	+	0.961456i	$0.411335\pi$
$182$	0	0
$183$	−445529.	−0.983442
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−565934.	−1.18348
$188$	0	0
$189$	395246.	0.804847
$190$	0	0
$191$	−722451.	−1.43293	−0.716465	−	0.697623i	$-0.754241\pi$
$191$	−722451.	−1.43293	−0.716465	+	0.697623i	$0.754241\pi$
$192$	0	0
$193$	659404.	1.27426	0.637131	−	0.770756i	$-0.280121\pi$
$193$	659404.	1.27426	0.637131	+	0.770756i	$0.280121\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	651087.	1.19529	0.597645	−	0.801760i	$-0.296103\pi$
$197$	651087.	1.19529	0.597645	+	0.801760i	$0.296103\pi$
$198$	0	0
$199$	128035.	0.229190	0.114595	−	0.993412i	$-0.463443\pi$
$199$	128035.	0.229190	0.114595	+	0.993412i	$0.463443\pi$
$200$	0	0
$201$	594541.	1.03799
$202$	0	0
$203$	671870.	1.14431
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−394775.	−0.640360
$208$	0	0
$209$	−652068.	−1.03259
$210$	0	0
$211$	324232.	0.501360	0.250680	−	0.968070i	$-0.419346\pi$
$211$	324232.	0.501360	0.250680	+	0.968070i	$0.419346\pi$
$212$	0	0
$213$	423376.	0.639407
$214$	0	0
$215$	0	0
$216$	0	0
$217$	757795.	1.09245
$218$	0	0
$219$	47910.6	0.0675027
$220$	0	0
$221$	−43070.2	−0.0593193
$222$	0	0
$223$	−62660.4	−0.0843784	−0.0421892	−	0.999110i	$-0.513433\pi$
$223$	−62660.4	−0.0843784	−0.0421892	+	0.999110i	$0.513433\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.08716e6	1.40033	0.700163	−	0.713983i	$-0.253111\pi$
$227$	1.08716e6	1.40033	0.700163	+	0.713983i	$0.253111\pi$
$228$	0	0
$229$	400555.	0.504747	0.252373	−	0.967630i	$-0.418789\pi$
$229$	400555.	0.504747	0.252373	+	0.967630i	$0.418789\pi$
$230$	0	0
$231$	−345390.	−0.425873
$232$	0	0
$233$	−597691.	−0.721252	−0.360626	−	0.932710i	$-0.617437\pi$
$233$	−597691.	−0.721252	−0.360626	+	0.932710i	$0.617437\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−2093.19	−0.00242068
$238$	0	0
$239$	1.47149e6	1.66634	0.833168	−	0.553020i	$-0.186525\pi$
$239$	1.47149e6	1.66634	0.833168	+	0.553020i	$0.186525\pi$
$240$	0	0
$241$	−1.26632e6	−1.40444	−0.702219	−	0.711961i	$-0.747807\pi$
$241$	−1.26632e6	−1.40444	−0.702219	+	0.711961i	$0.747807\pi$
$242$	0	0
$243$	−884090.	−0.960463
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−49625.4	−0.0517561
$248$	0	0
$249$	805571.	0.823390
$250$	0	0
$251$	−767189.	−0.768631	−0.384316	−	0.923202i	$-0.625563\pi$
$251$	−767189.	−0.768631	−0.384316	+	0.923202i	$0.625563\pi$
$252$	0	0
$253$	950358.	0.933439
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−1.63288e6	−1.54214	−0.771068	−	0.636753i	$-0.780277\pi$
$257$	−1.63288e6	−1.54214	−0.771068	+	0.636753i	$0.780277\pi$
$258$	0	0
$259$	−1.54973e6	−1.43551
$260$	0	0
$261$	−918047.	−0.834188
$262$	0	0
$263$	2.09710e6	1.86952	0.934759	−	0.355284i	$-0.115616\pi$
$263$	2.09710e6	1.86952	0.934759	+	0.355284i	$0.115616\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	1.39564e6	1.19811
$268$	0	0
$269$	−365904.	−0.308309	−0.154155	−	0.988047i	$-0.549265\pi$
$269$	−365904.	−0.308309	−0.154155	+	0.988047i	$0.549265\pi$
$270$	0	0
$271$	84748.2	0.0700982	0.0350491	−	0.999386i	$-0.488841\pi$
$271$	84748.2	0.0700982	0.0350491	+	0.999386i	$0.488841\pi$
$272$	0	0
$273$	−26285.8	−0.0213459
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−288788.	−0.226141	−0.113070	−	0.993587i	$-0.536069\pi$
$277$	−288788.	−0.226141	−0.113070	+	0.993587i	$0.536069\pi$
$278$	0	0
$279$	−1.03546e6	−0.796381
$280$	0	0
$281$	−1.41707e6	−1.07059	−0.535296	−	0.844664i	$-0.679800\pi$
$281$	−1.41707e6	−1.07059	−0.535296	+	0.844664i	$0.679800\pi$
$282$	0	0
$283$	1.67947e6	1.24654	0.623270	−	0.782007i	$-0.285804\pi$
$283$	1.67947e6	1.24654	0.623270	+	0.782007i	$0.285804\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	107580.	0.0770951
$288$	0	0
$289$	1.46228e6	1.02988
$290$	0	0
$291$	550469.	0.381066
$292$	0	0
$293$	−237042.	−0.161308	−0.0806541	−	0.996742i	$-0.525701\pi$
$293$	−237042.	−0.161308	−0.0806541	+	0.996742i	$0.525701\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.30012e6	0.855251
$298$	0	0
$299$	72326.6	0.0467865
$300$	0	0
$301$	−433107.	−0.275537
$302$	0	0
$303$	−738899.	−0.462358
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.51040e6	1.52019	0.760093	−	0.649815i	$-0.225153\pi$
$307$	2.51040e6	1.52019	0.760093	+	0.649815i	$0.225153\pi$
$308$	0	0
$309$	1.69570e6	1.01031
$310$	0	0
$311$	−2.65549e6	−1.55684	−0.778420	−	0.627743i	$-0.783979\pi$
$311$	−2.65549e6	−1.55684	−0.778420	+	0.627743i	$0.783979\pi$
$312$	0	0
$313$	1.82562e6	1.05329	0.526647	−	0.850084i	$-0.323449\pi$
$313$	1.82562e6	1.05329	0.526647	+	0.850084i	$0.323449\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−718332.	−0.401492	−0.200746	−	0.979643i	$-0.564337\pi$
$317$	−718332.	−0.401492	−0.200746	+	0.979643i	$0.564337\pi$
$318$	0	0
$319$	2.21005e6	1.21598
$320$	0	0
$321$	−1.26131e6	−0.683216
$322$	0	0
$323$	3.32079e6	1.77107
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−819996.	−0.424075
$328$	0	0
$329$	209732.	0.106825
$330$	0	0
$331$	1.34968e6	0.677114	0.338557	−	0.940946i	$-0.390061\pi$
$331$	1.34968e6	0.677114	0.338557	+	0.940946i	$0.390061\pi$
$332$	0	0
$333$	2.11756e6	1.04647
$334$	0	0
$335$	0	0
$336$	0	0
$337$	1.95491e6	0.937675	0.468838	−	0.883284i	$-0.344673\pi$
$337$	1.95491e6	0.937675	0.468838	+	0.883284i	$0.344673\pi$
$338$	0	0
$339$	2.54234e6	1.20153
$340$	0	0
$341$	2.49269e6	1.16087
$342$	0	0
$343$	−2.36571e6	−1.08574
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−186431.	−0.0831178	−0.0415589	−	0.999136i	$-0.513232\pi$
$347$	−186431.	−0.0831178	−0.0415589	+	0.999136i	$0.513232\pi$
$348$	0	0
$349$	−1.52510e6	−0.670248	−0.335124	−	0.942174i	$-0.608778\pi$
$349$	−1.52510e6	−0.670248	−0.335124	+	0.942174i	$0.608778\pi$
$350$	0	0
$351$	98945.5	0.0428675
$352$	0	0
$353$	−267531.	−0.114271	−0.0571357	−	0.998366i	$-0.518197\pi$
$353$	−267531.	−0.114271	−0.0571357	+	0.998366i	$0.518197\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	1.75897e6	0.730446
$358$	0	0
$359$	2.43272e6	0.996220	0.498110	−	0.867114i	$-0.334028\pi$
$359$	2.43272e6	0.996220	0.498110	+	0.867114i	$0.334028\pi$
$360$	0	0
$361$	1.35011e6	0.545255
$362$	0	0
$363$	510417.	0.203310
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.10175e6	−1.97722	−0.988609	−	0.150510i	$-0.951908\pi$
$367$	−5.10175e6	−1.97722	−0.988609	+	0.150510i	$0.951908\pi$
$368$	0	0
$369$	−146998.	−0.0562012
$370$	0	0
$371$	−498474.	−0.188022
$372$	0	0
$373$	2.66513e6	0.991849	0.495924	−	0.868366i	$-0.334829\pi$
$373$	2.66513e6	0.991849	0.495924	+	0.868366i	$0.334829\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	168195.	0.0609481
$378$	0	0
$379$	−3.35682e6	−1.20041	−0.600206	−	0.799846i	$-0.704915\pi$
$379$	−3.35682e6	−1.20041	−0.600206	+	0.799846i	$0.704915\pi$
$380$	0	0
$381$	−680162.	−0.240049
$382$	0	0
$383$	453926.	0.158121	0.0790603	−	0.996870i	$-0.474808\pi$
$383$	453926.	0.158121	0.0790603	+	0.996870i	$0.474808\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	591800.	0.200862
$388$	0	0
$389$	−1.64690e6	−0.551814	−0.275907	−	0.961184i	$-0.588978\pi$
$389$	−1.64690e6	−0.551814	−0.275907	+	0.961184i	$0.588978\pi$
$390$	0	0
$391$	−4.83989e6	−1.60101
$392$	0	0
$393$	−1.00764e6	−0.329098
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−468551.	−0.149204	−0.0746020	−	0.997213i	$-0.523769\pi$
$397$	−468551.	−0.149204	−0.0746020	+	0.997213i	$0.523769\pi$
$398$	0	0
$399$	2.02668e6	0.637314
$400$	0	0
$401$	5.49649e6	1.70696	0.853482	−	0.521122i	$-0.174486\pi$
$401$	5.49649e6	1.70696	0.853482	+	0.521122i	$0.174486\pi$
$402$	0	0
$403$	189705.	0.0581858
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−5.09769e6	−1.52541
$408$	0	0
$409$	2.00548e6	0.592803	0.296401	−	0.955063i	$-0.404213\pi$
$409$	2.00548e6	0.592803	0.296401	+	0.955063i	$0.404213\pi$
$410$	0	0
$411$	3.91937e6	1.14449
$412$	0	0
$413$	−477027.	−0.137616
$414$	0	0
$415$	0	0
$416$	0	0
$417$	2.73875e6	0.771281
$418$	0	0
$419$	−891312.	−0.248025	−0.124012	−	0.992281i	$-0.539576\pi$
$419$	−891312.	−0.248025	−0.124012	+	0.992281i	$0.539576\pi$
$420$	0	0
$421$	−5.49957e6	−1.51225	−0.756125	−	0.654427i	$-0.772910\pi$
$421$	−5.49957e6	−1.51225	−0.756125	+	0.654427i	$0.772910\pi$
$422$	0	0
$423$	−286579.	−0.0778741
$424$	0	0
$425$	0	0
$426$	0	0
$427$	4.41629e6	1.17216
$428$	0	0
$429$	−86464.6	−0.0226827
$430$	0	0
$431$	2.01042e6	0.521307	0.260653	−	0.965432i	$-0.416062\pi$
$431$	2.01042e6	0.521307	0.260653	+	0.965432i	$0.416062\pi$
$432$	0	0
$433$	−455522.	−0.116759	−0.0583794	−	0.998294i	$-0.518593\pi$
$433$	−455522.	−0.116759	−0.0583794	+	0.998294i	$0.518593\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−5.57651e6	−1.39688
$438$	0	0
$439$	231348.	0.0572935	0.0286468	−	0.999590i	$-0.490880\pi$
$439$	231348.	0.0572935	0.0286468	+	0.999590i	$0.490880\pi$
$440$	0	0
$441$	905169.	0.221632
$442$	0	0
$443$	−277485.	−0.0671785	−0.0335892	−	0.999436i	$-0.510694\pi$
$443$	−277485.	−0.0671785	−0.0335892	+	0.999436i	$0.510694\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	1.58930e6	0.376215
$448$	0	0
$449$	−624424.	−0.146172	−0.0730859	−	0.997326i	$-0.523285\pi$
$449$	−624424.	−0.146172	−0.0730859	+	0.997326i	$0.523285\pi$
$450$	0	0
$451$	353874.	0.0819233
$452$	0	0
$453$	−2.40627e6	−0.550933
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−3.95400e6	−0.885618	−0.442809	−	0.896616i	$-0.646018\pi$
$457$	−3.95400e6	−0.885618	−0.442809	+	0.896616i	$0.646018\pi$
$458$	0	0
$459$	−6.62115e6	−1.46690
$460$	0	0
$461$	−2.29630e6	−0.503241	−0.251620	−	0.967826i	$-0.580963\pi$
$461$	−2.29630e6	−0.503241	−0.251620	+	0.967826i	$0.580963\pi$
$462$	0	0
$463$	−2.02751e6	−0.439552	−0.219776	−	0.975550i	$-0.570533\pi$
$463$	−2.02751e6	−0.439552	−0.219776	+	0.975550i	$0.570533\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	8.24530e6	1.74950	0.874750	−	0.484574i	$-0.161025\pi$
$467$	8.24530e6	1.74950	0.874750	+	0.484574i	$0.161025\pi$
$468$	0	0
$469$	−5.89337e6	−1.23718
$470$	0	0
$471$	1.30444e6	0.270938
$472$	0	0
$473$	−1.42466e6	−0.292792
$474$	0	0
$475$	0	0
$476$	0	0
$477$	681118.	0.137065
$478$	0	0
$479$	−2.53107e6	−0.504041	−0.252021	−	0.967722i	$-0.581095\pi$
$479$	−2.53107e6	−0.504041	−0.252021	+	0.967722i	$0.581095\pi$
$480$	0	0
$481$	−387958.	−0.0764579
$482$	0	0
$483$	−2.95379e6	−0.576119
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−1.06777e6	−0.204013	−0.102006	−	0.994784i	$-0.532526\pi$
$487$	−1.06777e6	−0.204013	−0.102006	+	0.994784i	$0.532526\pi$
$488$	0	0
$489$	2.27067e6	0.429419
$490$	0	0
$491$	7.70189e6	1.44176	0.720881	−	0.693059i	$-0.243738\pi$
$491$	7.70189e6	1.44176	0.720881	+	0.693059i	$0.243738\pi$
$492$	0	0
$493$	−1.12551e7	−2.08561
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.19670e6	−0.762108
$498$	0	0
$499$	−7.04750e6	−1.26702	−0.633510	−	0.773734i	$-0.718387\pi$
$499$	−7.04750e6	−1.26702	−0.633510	+	0.773734i	$0.718387\pi$
$500$	0	0
$501$	3.98652e6	0.709576
$502$	0	0
$503$	−7.83892e6	−1.38145	−0.690726	−	0.723116i	$-0.742709\pi$
$503$	−7.83892e6	−1.38145	−0.690726	+	0.723116i	$0.742709\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	3.78943e6	0.654717
$508$	0	0
$509$	5.32651e6	0.911273	0.455636	−	0.890166i	$-0.349412\pi$
$509$	5.32651e6	0.911273	0.455636	+	0.890166i	$0.349412\pi$
$510$	0	0
$511$	−474912.	−0.0804564
$512$	0	0
$513$	−7.62887e6	−1.27987
$514$	0	0
$515$	0	0
$516$	0	0
$517$	689892.	0.113515
$518$	0	0
$519$	6.20558e6	1.01126
$520$	0	0
$521$	−7.82275e6	−1.26260	−0.631299	−	0.775539i	$-0.717478\pi$
$521$	−7.82275e6	−1.26260	−0.631299	+	0.775539i	$0.717478\pi$
$522$	0	0
$523$	515235.	0.0823667	0.0411833	−	0.999152i	$-0.486887\pi$
$523$	515235.	0.0823667	0.0411833	+	0.999152i	$0.486887\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.26945e7	−1.99109
$528$	0	0
$529$	1.69116e6	0.262751
$530$	0	0
$531$	651812.	0.100320
$532$	0	0
$533$	26931.5	0.00410622
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−2.33257e6	−0.349060
$538$	0	0
$539$	−2.17905e6	−0.323069
$540$	0	0
$541$	1.08113e7	1.58813	0.794065	−	0.607833i	$-0.207961\pi$
$541$	1.08113e7	1.58813	0.794065	+	0.607833i	$0.207961\pi$
$542$	0	0
$543$	−2.47803e6	−0.360668
$544$	0	0
$545$	0	0
$546$	0	0
$547$	2.10445e6	0.300726	0.150363	−	0.988631i	$-0.451956\pi$
$547$	2.10445e6	0.300726	0.150363	+	0.988631i	$0.451956\pi$
$548$	0	0
$549$	−6.03445e6	−0.854489
$550$	0	0
$551$	−1.29681e7	−1.81970
$552$	0	0
$553$	20748.7	0.00288521
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.30806e6	0.724933	0.362467	−	0.931997i	$-0.381935\pi$
$557$	5.30806e6	0.724933	0.362467	+	0.931997i	$0.381935\pi$
$558$	0	0
$559$	−108424.	−0.0146755
$560$	0	0
$561$	5.78597e6	0.776191
$562$	0	0
$563$	−5.66196e6	−0.752828	−0.376414	−	0.926452i	$-0.622843\pi$
$563$	−5.66196e6	−0.752828	−0.376414	+	0.926452i	$0.622843\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−630783.	−0.0823990
$568$	0	0
$569$	−1.05656e7	−1.36809	−0.684044	−	0.729440i	$-0.739781\pi$
$569$	−1.05656e7	−1.36809	−0.684044	+	0.729440i	$0.739781\pi$
$570$	0	0
$571$	1.43051e7	1.83611	0.918057	−	0.396448i	$-0.129757\pi$
$571$	1.43051e7	1.83611	0.918057	+	0.396448i	$0.129757\pi$
$572$	0	0
$573$	7.38616e6	0.939793
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.43466e6	0.554524	0.277262	−	0.960794i	$-0.410573\pi$
$577$	4.43466e6	0.554524	0.277262	+	0.960794i	$0.410573\pi$
$578$	0	0
$579$	−6.74158e6	−0.835729
$580$	0	0
$581$	−7.98519e6	−0.981398
$582$	0	0
$583$	−1.63968e6	−0.199797
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.91240e6	−1.18736	−0.593681	−	0.804700i	$-0.702326\pi$
$587$	−9.91240e6	−1.18736	−0.593681	+	0.804700i	$0.702326\pi$
$588$	0	0
$589$	−1.46266e7	−1.73722
$590$	0	0
$591$	−6.65655e6	−0.783936
$592$	0	0
$593$	2.48091e6	0.289717	0.144859	−	0.989452i	$-0.453727\pi$
$593$	2.48091e6	0.289717	0.144859	+	0.989452i	$0.453727\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−1.30900e6	−0.150315
$598$	0	0
$599$	3.95174e6	0.450009	0.225004	−	0.974358i	$-0.427760\pi$
$599$	3.95174e6	0.450009	0.225004	+	0.974358i	$0.427760\pi$
$600$	0	0
$601$	1.16668e7	1.31754	0.658772	−	0.752343i	$-0.271076\pi$
$601$	1.16668e7	1.31754	0.658772	+	0.752343i	$0.271076\pi$
$602$	0	0
$603$	8.05273e6	0.901882
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−1.28877e7	−1.41973	−0.709863	−	0.704340i	$-0.751243\pi$
$607$	−1.28877e7	−1.41973	−0.709863	+	0.704340i	$0.751243\pi$
$608$	0	0
$609$	−6.86903e6	−0.750503
$610$	0	0
$611$	52504.0	0.00568970
$612$	0	0
$613$	2.92567e6	0.314466	0.157233	−	0.987562i	$-0.449743\pi$
$613$	2.92567e6	0.314466	0.157233	+	0.987562i	$0.449743\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	2.74828e6	0.290635	0.145318	−	0.989385i	$-0.453580\pi$
$617$	2.74828e6	0.290635	0.145318	+	0.989385i	$0.453580\pi$
$618$	0	0
$619$	1.49290e7	1.56605	0.783023	−	0.621993i	$-0.213677\pi$
$619$	1.49290e7	1.56605	0.783023	+	0.621993i	$0.213677\pi$
$620$	0	0
$621$	1.11187e7	1.15698
$622$	0	0
$623$	−1.38342e7	−1.42802
$624$	0	0
$625$	0	0
$626$	0	0
$627$	6.66658e6	0.677227
$628$	0	0
$629$	2.59610e7	2.61635
$630$	0	0
$631$	6.95119e6	0.695002	0.347501	−	0.937680i	$-0.387030\pi$
$631$	6.95119e6	0.695002	0.347501	+	0.937680i	$0.387030\pi$
$632$	0	0
$633$	−3.31486e6	−0.328819
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−165836.	−0.0161931
$638$	0	0
$639$	5.73439e6	0.555565
$640$	0	0
$641$	−3.85240e6	−0.370328	−0.185164	−	0.982708i	$-0.559282\pi$
$641$	−3.85240e6	−0.370328	−0.185164	+	0.982708i	$0.559282\pi$
$642$	0	0
$643$	7.28010e6	0.694400	0.347200	−	0.937791i	$-0.387132\pi$
$643$	7.28010e6	0.694400	0.347200	+	0.937791i	$0.387132\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−1.03038e7	−0.967690	−0.483845	−	0.875154i	$-0.660760\pi$
$647$	−1.03038e7	−0.967690	−0.483845	+	0.875154i	$0.660760\pi$
$648$	0	0
$649$	−1.56913e6	−0.146234
$650$	0	0
$651$	−7.74750e6	−0.716489
$652$	0	0
$653$	−3.81704e6	−0.350303	−0.175152	−	0.984541i	$-0.556042\pi$
$653$	−3.81704e6	−0.350303	−0.175152	+	0.984541i	$0.556042\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	648922.	0.0586515
$658$	0	0
$659$	−1.81884e7	−1.63148	−0.815740	−	0.578419i	$-0.803670\pi$
$659$	−1.81884e7	−1.63148	−0.815740	+	0.578419i	$0.803670\pi$
$660$	0	0
$661$	8.91265e6	0.793420	0.396710	−	0.917944i	$-0.370152\pi$
$661$	8.91265e6	0.793420	0.396710	+	0.917944i	$0.370152\pi$
$662$	0	0
$663$	440339.	0.0389048
$664$	0	0
$665$	0	0
$666$	0	0
$667$	1.89005e7	1.64497
$668$	0	0
$669$	640624.	0.0553399
$670$	0	0
$671$	1.45270e7	1.24557
$672$	0	0
$673$	−627941.	−0.0534419	−0.0267209	−	0.999643i	$-0.508507\pi$
$673$	−627941.	−0.0534419	−0.0267209	+	0.999643i	$0.508507\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−815855.	−0.0684134	−0.0342067	−	0.999415i	$-0.510890\pi$
$677$	−815855.	−0.0684134	−0.0342067	+	0.999415i	$0.510890\pi$
$678$	0	0
$679$	−5.45650e6	−0.454192
$680$	0	0
$681$	−1.11149e7	−0.918409
$682$	0	0
$683$	−3.55890e6	−0.291920	−0.145960	−	0.989290i	$-0.546627\pi$
$683$	−3.55890e6	−0.291920	−0.145960	+	0.989290i	$0.546627\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−4.09517e6	−0.331040
$688$	0	0
$689$	−124788.	−0.0100144
$690$	0	0
$691$	−6.94762e6	−0.553530	−0.276765	−	0.960938i	$-0.589262\pi$
$691$	−6.94762e6	−0.553530	−0.276765	+	0.960938i	$0.589262\pi$
$692$	0	0
$693$	−4.67812e6	−0.370031
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−1.80218e6	−0.140513
$698$	0	0
$699$	6.11065e6	0.473036
$700$	0	0
$701$	−1.30671e7	−1.00434	−0.502172	−	0.864768i	$-0.667466\pi$
$701$	−1.30671e7	−1.00434	−0.502172	+	0.864768i	$0.667466\pi$
$702$	0	0
$703$	2.99123e7	2.28276
$704$	0	0
$705$	0	0
$706$	0	0
$707$	7.32431e6	0.551085
$708$	0	0
$709$	1.83106e7	1.36800	0.684000	−	0.729482i	$-0.260239\pi$
$709$	1.83106e7	1.36800	0.684000	+	0.729482i	$0.260239\pi$
$710$	0	0
$711$	−28351.1	−0.00210327
$712$	0	0
$713$	2.13176e7	1.57042
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−1.50441e7	−1.09287
$718$	0	0
$719$	1.72922e7	1.24747	0.623734	−	0.781637i	$-0.285615\pi$
$719$	1.72922e7	1.24747	0.623734	+	0.781637i	$0.285615\pi$
$720$	0	0
$721$	−1.68086e7	−1.20418
$722$	0	0
$723$	1.29466e7	0.921106
$724$	0	0
$725$	0	0
$726$	0	0
$727$	2.37243e7	1.66479	0.832393	−	0.554186i	$-0.186970\pi$
$727$	2.37243e7	1.66479	0.832393	+	0.554186i	$0.186970\pi$
$728$	0	0
$729$	1.05512e7	0.735332
$730$	0	0
$731$	7.25540e6	0.502190
$732$	0	0
$733$	−1.10926e6	−0.0762556	−0.0381278	−	0.999273i	$-0.512139\pi$
$733$	−1.10926e6	−0.0762556	−0.0381278	+	0.999273i	$0.512139\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−1.93857e7	−1.31465
$738$	0	0
$739$	−1.23339e6	−0.0830784	−0.0415392	−	0.999137i	$-0.513226\pi$
$739$	−1.23339e6	−0.0830784	−0.0415392	+	0.999137i	$0.513226\pi$
$740$	0	0
$741$	507357.	0.0339444
$742$	0	0
$743$	−2.09365e7	−1.39134	−0.695668	−	0.718364i	$-0.744891\pi$
$743$	−2.09365e7	−1.39134	−0.695668	+	0.718364i	$0.744891\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.09110e7	0.715424
$748$	0	0
$749$	1.25027e7	0.814325
$750$	0	0
$751$	−2.37979e7	−1.53971	−0.769853	−	0.638221i	$-0.779671\pi$
$751$	−2.37979e7	−1.53971	−0.769853	+	0.638221i	$0.779671\pi$
$752$	0	0
$753$	7.84354e6	0.504110
$754$	0	0
$755$	0	0
$756$	0	0
$757$	1.28737e7	0.816516	0.408258	−	0.912867i	$-0.366136\pi$
$757$	1.28737e7	0.816516	0.408258	+	0.912867i	$0.366136\pi$
$758$	0	0
$759$	−9.71622e6	−0.612199
$760$	0	0
$761$	−1.68019e7	−1.05171	−0.525857	−	0.850573i	$-0.676255\pi$
$761$	−1.68019e7	−1.05171	−0.525857	+	0.850573i	$0.676255\pi$
$762$	0	0
$763$	8.12818e6	0.505454
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−119418.	−0.00732964
$768$	0	0
$769$	−1.74155e7	−1.06199	−0.530994	−	0.847375i	$-0.678181\pi$
$769$	−1.74155e7	−1.06199	−0.530994	+	0.847375i	$0.678181\pi$
$770$	0	0
$771$	1.66942e7	1.01142
$772$	0	0
$773$	1.22956e6	0.0740117	0.0370058	−	0.999315i	$-0.488218\pi$
$773$	1.22956e6	0.0740117	0.0370058	+	0.999315i	$0.488218\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.58441e7	0.941487
$778$	0	0
$779$	−2.07646e6	−0.122597
$780$	0	0
$781$	−1.38046e7	−0.809836
$782$	0	0
$783$	2.58565e7	1.50718
$784$	0	0
$785$	0	0
$786$	0	0
$787$	4.64517e6	0.267341	0.133670	−	0.991026i	$-0.457324\pi$
$787$	4.64517e6	0.267341	0.133670	+	0.991026i	$0.457324\pi$
$788$	0	0
$789$	−2.14402e7	−1.22613
$790$	0	0
$791$	−2.52008e7	−1.43210
$792$	0	0
$793$	1.10557e6	0.0624314
$794$	0	0
$795$	0	0
$796$	0	0
$797$	6.71612e6	0.374518	0.187259	−	0.982311i	$-0.440040\pi$
$797$	6.71612e6	0.374518	0.187259	+	0.982311i	$0.440040\pi$
$798$	0	0
$799$	−3.51342e6	−0.194699
$800$	0	0
$801$	1.89032e7	1.04101
$802$	0	0
$803$	−1.56218e6	−0.0854951
$804$	0	0
$805$	0	0
$806$	0	0
$807$	3.74091e6	0.202206
$808$	0	0
$809$	2.85994e7	1.53633	0.768166	−	0.640251i	$-0.221170\pi$
$809$	2.85994e7	1.53633	0.768166	+	0.640251i	$0.221170\pi$
$810$	0	0
$811$	−6.00657e6	−0.320682	−0.160341	−	0.987062i	$-0.551259\pi$
$811$	−6.00657e6	−0.320682	−0.160341	+	0.987062i	$0.551259\pi$
$812$	0	0
$813$	−866444.	−0.0459742
$814$	0	0
$815$	0	0
$816$	0	0
$817$	8.35965e6	0.438160
$818$	0	0
$819$	−356027.	−0.0185470
$820$	0	0
$821$	−2.02199e7	−1.04694	−0.523470	−	0.852044i	$-0.675363\pi$
$821$	−2.02199e7	−1.04694	−0.523470	+	0.852044i	$0.675363\pi$
$822$	0	0
$823$	−7.90245e6	−0.406689	−0.203344	−	0.979107i	$-0.565181\pi$
$823$	−7.90245e6	−0.406689	−0.203344	+	0.979107i	$0.565181\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.57263e7	−1.30802	−0.654009	−	0.756487i	$-0.726914\pi$
$827$	−2.57263e7	−1.30802	−0.654009	+	0.756487i	$0.726914\pi$
$828$	0	0
$829$	2.87057e6	0.145071	0.0725356	−	0.997366i	$-0.476891\pi$
$829$	2.87057e6	0.145071	0.0725356	+	0.997366i	$0.476891\pi$
$830$	0	0
$831$	2.95249e6	0.148315
$832$	0	0
$833$	1.10973e7	0.554119
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.91633e7	1.43887
$838$	0	0
$839$	1.77236e7	0.869255	0.434627	−	0.900610i	$-0.356880\pi$
$839$	1.77236e7	0.869255	0.434627	+	0.900610i	$0.356880\pi$
$840$	0	0
$841$	2.34418e7	1.14288
$842$	0	0
$843$	1.44877e7	0.702152
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.05949e6	−0.242325
$848$	0	0
$849$	−1.71705e7	−0.817548
$850$	0	0
$851$	−4.35957e7	−2.06357
$852$	0	0
$853$	3.15211e7	1.48330	0.741650	−	0.670787i	$-0.234043\pi$
$853$	3.15211e7	1.48330	0.741650	+	0.670787i	$0.234043\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−6.15617e6	−0.286324	−0.143162	−	0.989699i	$-0.545727\pi$
$857$	−6.15617e6	−0.286324	−0.143162	+	0.989699i	$0.545727\pi$
$858$	0	0
$859$	1.42705e7	0.659865	0.329932	−	0.944005i	$-0.392974\pi$
$859$	1.42705e7	0.659865	0.329932	+	0.944005i	$0.392974\pi$
$860$	0	0
$861$	−1.09987e6	−0.0505631
$862$	0	0
$863$	3.50300e6	0.160108	0.0800541	−	0.996791i	$-0.474491\pi$
$863$	3.50300e6	0.160108	0.0800541	+	0.996791i	$0.474491\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−1.49500e7	−0.675448
$868$	0	0
$869$	68250.7	0.00306590
$870$	0	0
$871$	−1.47534e6	−0.0658941
$872$	0	0
$873$	7.45579e6	0.331099
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.54690e7	0.679148	0.339574	−	0.940579i	$-0.389717\pi$
$877$	1.54690e7	0.679148	0.339574	+	0.940579i	$0.389717\pi$
$878$	0	0
$879$	2.42346e6	0.105795
$880$	0	0
$881$	−5.49023e6	−0.238315	−0.119157	−	0.992875i	$-0.538019\pi$
$881$	−5.49023e6	−0.238315	−0.119157	+	0.992875i	$0.538019\pi$
$882$	0	0
$883$	−2.24202e7	−0.967695	−0.483848	−	0.875152i	$-0.660761\pi$
$883$	−2.24202e7	−0.967695	−0.483848	+	0.875152i	$0.660761\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−5.35867e6	−0.228691	−0.114345	−	0.993441i	$-0.536477\pi$
$887$	−5.35867e6	−0.228691	−0.114345	+	0.993441i	$0.536477\pi$
$888$	0	0
$889$	6.74208e6	0.286114
$890$	0	0
$891$	−2.07490e6	−0.0875593
$892$	0	0
$893$	−4.04815e6	−0.169874
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−739449.	−0.0306851
$898$	0	0
$899$	4.95740e7	2.04576
$900$	0	0
$901$	8.35042e6	0.342686
$902$	0	0
$903$	4.42798e6	0.180712
$904$	0	0
$905$	0	0
$906$	0	0
$907$	3.30460e6	0.133383	0.0666916	−	0.997774i	$-0.478756\pi$
$907$	3.30460e6	0.133383	0.0666916	+	0.997774i	$0.478756\pi$
$908$	0	0
$909$	−1.00080e7	−0.401732
$910$	0	0
$911$	1.34757e7	0.537967	0.268983	−	0.963145i	$-0.413312\pi$
$911$	1.34757e7	0.537967	0.268983	+	0.963145i	$0.413312\pi$
$912$	0	0
$913$	−2.62665e7	−1.04286
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.98823e6	0.392252
$918$	0	0
$919$	2.90104e7	1.13309	0.566545	−	0.824030i	$-0.308280\pi$
$919$	2.90104e7	1.13309	0.566545	+	0.824030i	$0.308280\pi$
$920$	0	0
$921$	−2.56657e7	−0.997019
$922$	0	0
$923$	−1.05060e6	−0.0405912
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.29674e7	0.877832
$928$	0	0
$929$	3.39231e7	1.28960	0.644802	−	0.764349i	$-0.276940\pi$
$929$	3.39231e7	1.28960	0.644802	+	0.764349i	$0.276940\pi$
$930$	0	0
$931$	1.27862e7	0.483469
$932$	0	0
$933$	2.71491e7	1.02106
$934$	0	0
$935$	0	0
$936$	0	0
$937$	7.28797e6	0.271180	0.135590	−	0.990765i	$-0.456707\pi$
$937$	7.28797e6	0.271180	0.135590	+	0.990765i	$0.456707\pi$
$938$	0	0
$939$	−1.86647e7	−0.690807
$940$	0	0
$941$	3.87899e7	1.42805	0.714027	−	0.700118i	$-0.246869\pi$
$941$	3.87899e7	1.42805	0.714027	+	0.700118i	$0.246869\pi$
$942$	0	0
$943$	3.02635e6	0.110825
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−2.26010e7	−0.818942	−0.409471	−	0.912323i	$-0.634287\pi$
$947$	−2.26010e7	−0.818942	−0.409471	+	0.912323i	$0.634287\pi$
$948$	0	0
$949$	−118889.	−0.00428524
$950$	0	0
$951$	7.34404e6	0.263320
$952$	0	0
$953$	2.01614e7	0.719100	0.359550	−	0.933126i	$-0.382930\pi$
$953$	2.01614e7	0.719100	0.359550	+	0.933126i	$0.382930\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−2.25950e7	−0.797504
$958$	0	0
$959$	−3.88506e7	−1.36412
$960$	0	0
$961$	2.72848e7	0.953042
$962$	0	0
$963$	−1.70837e7	−0.593631
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.02547e7	0.352662	0.176331	−	0.984331i	$-0.443577\pi$
$967$	1.02547e7	0.352662	0.176331	+	0.984331i	$0.443577\pi$
$968$	0	0
$969$	−3.39509e7	−1.16156
$970$	0	0
$971$	9.59974e6	0.326747	0.163373	−	0.986564i	$-0.447762\pi$
$971$	9.59974e6	0.326747	0.163373	+	0.986564i	$0.447762\pi$
$972$	0	0
$973$	−2.71478e7	−0.919289
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−3.74964e7	−1.25676	−0.628382	−	0.777905i	$-0.716282\pi$
$977$	−3.74964e7	−1.25676	−0.628382	+	0.777905i	$0.716282\pi$
$978$	0	0
$979$	−4.55063e7	−1.51745
$980$	0	0
$981$	−1.11064e7	−0.368469
$982$	0	0
$983$	5.17751e7	1.70898	0.854491	−	0.519466i	$-0.173869\pi$
$983$	5.17751e7	1.70898	0.854491	+	0.519466i	$0.173869\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.14425e6	−0.0700618
$988$	0	0
$989$	−1.21838e7	−0.396088
$990$	0	0
$991$	5.36892e7	1.73661	0.868306	−	0.496029i	$-0.165209\pi$
$991$	5.36892e7	1.73661	0.868306	+	0.496029i	$0.165209\pi$
$992$	0	0
$993$	−1.37988e7	−0.444088
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−4.39771e7	−1.40116	−0.700582	−	0.713572i	$-0.747076\pi$
$997$	−4.39771e7	−1.40116	−0.700582	+	0.713572i	$0.747076\pi$
$998$	0	0
$999$	−5.96405e7	−1.89072

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	400.6.a.v.1.1		2
4.3	odd	2		100.6.a.c.1.2	✓	2
5.2	odd	4		400.6.c.m.49.3		4
5.3	odd	4		400.6.c.m.49.2		4
5.4	even	2		400.6.a.p.1.2		2
12.11	even	2		900.6.a.s.1.1		2
20.3	even	4		100.6.c.c.49.3		4
20.7	even	4		100.6.c.c.49.2		4
20.19	odd	2		100.6.a.e.1.1	yes	2
60.23	odd	4		900.6.d.m.649.3		4
60.47	odd	4		900.6.d.m.649.2		4
60.59	even	2		900.6.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
100.6.a.c.1.2	✓	2	4.3	odd	2
100.6.a.e.1.1	yes	2	20.19	odd	2
100.6.c.c.49.2		4	20.7	even	4
100.6.c.c.49.3		4	20.3	even	4
400.6.a.p.1.2		2	5.4	even	2
400.6.a.v.1.1		2	1.1	even	1	trivial
400.6.c.m.49.2		4	5.3	odd	4
400.6.c.m.49.3		4	5.2	odd	4
900.6.a.m.1.2		2	60.59	even	2
900.6.a.s.1.1		2	12.11	even	2
900.6.d.m.649.2		4	60.47	odd	4
900.6.d.m.649.3		4	60.23	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:	\( N \)	\(=\)	\( 400 = 2^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	400.a (trivial)

\(f(q)\)	\(=\)	\(q-10.2237 q^{3} +101.342 q^{7} -138.475 q^{9} +O(q^{10})\)	\(q-10.2237 q^{3} +101.342 q^{7} -138.475 q^{9} +333.356 q^{11} +25.3700 q^{13} -1697.68 q^{17} -1956.07 q^{19} -1036.10 q^{21} +2850.88 q^{23} +3900.10 q^{27} +6629.70 q^{29} +7477.56 q^{31} -3408.15 q^{33} -15292.0 q^{37} -259.376 q^{39} +1061.55 q^{41} -4273.70 q^{43} +2069.53 q^{47} -6536.70 q^{49} +17356.7 q^{51} -4918.71 q^{53} +19998.4 q^{57} -4707.07 q^{59} +43577.9 q^{61} -14033.4 q^{63} -58153.0 q^{67} -29146.7 q^{69} -41411.0 q^{71} -4686.21 q^{73} +33783.2 q^{77} +204.738 q^{79} -6224.27 q^{81} -78794.1 q^{83} -67780.4 q^{87} -136510. q^{89} +2571.06 q^{91} -76448.7 q^{93} -53842.2 q^{97} -46161.5 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9}+O(q^{10}) \) 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9	\( 2 q + 20 q^{3} - 40 q^{7} + 532 q^{9} + 60 q^{11} - 920 q^{13} - 2910 q^{17} - 2092 q^{19} - 5308 q^{21} + 120 q^{23} + 16820 q^{27} + 3552 q^{29} + 8888 q^{31} - 11670 q^{33} - 12140 q^{37} - 28832 q^{39} - 12438 q^{41} + 1160 q^{43} - 1200 q^{47} - 3366 q^{49} - 19284 q^{51} - 26340 q^{53} + 15890 q^{57} + 36696 q^{59} + 19204 q^{61} - 108800 q^{63} - 90460 q^{67} - 111684 q^{69} - 2736 q^{71} - 12770 q^{73} + 72420 q^{77} + 16184 q^{79} + 221338 q^{81} - 30300 q^{83} - 160800 q^{87} - 47322 q^{89} + 136192 q^{91} - 33820 q^{93} + 2980 q^{97} - 229440 q^{99}+O(q^{100}) \) 2 * q + 20 * q^3 - 40 * q^7 + 532 * q^9 + 60 * q^11 - 920 * q^13 - 2910 * q^17 - 2092 * q^19 - 5308 * q^21 + 120 * q^23 + 16820 * q^27 + 3552 * q^29 + 8888 * q^31 - 11670 * q^33 - 12140 * q^37 - 28832 * q^39 - 12438 * q^41 + 1160 * q^43 - 1200 * q^47 - 3366 * q^49 - 19284 * q^51 - 26340 * q^53 + 15890 * q^57 + 36696 * q^59 + 19204 * q^61 - 108800 * q^63 - 90460 * q^67 - 111684 * q^69 - 2736 * q^71 - 12770 * q^73 + 72420 * q^77 + 16184 * q^79 + 221338 * q^81 - 30300 * q^83 - 160800 * q^87 - 47322 * q^89 + 136192 * q^91 - 33820 * q^93 + 2980 * q^97 - 229440 * q^99

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