LMFDB - Embedded newform 450.6.a.u.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [450,6,Mod(1,450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(450, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("450.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 450 = 2 \cdot 3^{2} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	450.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:	$72.1727189158$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 10)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{2} +16.0000 q^{4} +118.000 q^{7} +64.0000 q^{8} +O(q^{10})$

$q+4.00000 q^{2} +16.0000 q^{4} +118.000 q^{7} +64.0000 q^{8} -192.000 q^{11} -1106.00 q^{13} +472.000 q^{14} +256.000 q^{16} +762.000 q^{17} -2740.00 q^{19} -768.000 q^{22} +1566.00 q^{23} -4424.00 q^{26} +1888.00 q^{28} -5910.00 q^{29} -6868.00 q^{31} +1024.00 q^{32} +3048.00 q^{34} +5518.00 q^{37} -10960.0 q^{38} +378.000 q^{41} +2434.00 q^{43} -3072.00 q^{44} +6264.00 q^{46} +13122.0 q^{47} -2883.00 q^{49} -17696.0 q^{52} -9174.00 q^{53} +7552.00 q^{56} -23640.0 q^{58} +34980.0 q^{59} -9838.00 q^{61} -27472.0 q^{62} +4096.00 q^{64} -33722.0 q^{67} +12192.0 q^{68} -70212.0 q^{71} -21986.0 q^{73} +22072.0 q^{74} -43840.0 q^{76} -22656.0 q^{77} +4520.00 q^{79} +1512.00 q^{82} -109074. q^{83} +9736.00 q^{86} -12288.0 q^{88} -38490.0 q^{89} -130508. q^{91} +25056.0 q^{92} +52488.0 q^{94} +1918.00 q^{97} -11532.0 q^{98} +O(q^{100})$

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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	118.000	0.910200	0.455100	−	0.890440i	$-0.349603\pi$
$7$	118.000	0.910200	0.455100	+	0.890440i	$0.349603\pi$
$8$	64.0000	0.353553
$9$	0	0
$10$	0	0
$11$	−192.000	−0.478431	−0.239216	−	0.970966i	$-0.576890\pi$
$11$	−192.000	−0.478431	−0.239216	+	0.970966i	$0.576890\pi$
$12$	0	0
$13$	−1106.00	−1.81508	−0.907542	−	0.419961i	$-0.862044\pi$
$13$	−1106.00	−1.81508	−0.907542	+	0.419961i	$0.862044\pi$
$14$	472.000	0.643609
$15$	0	0
$16$	256.000	0.250000
$17$	762.000	0.639488	0.319744	−	0.947504i	$-0.396403\pi$
$17$	762.000	0.639488	0.319744	+	0.947504i	$0.396403\pi$
$18$	0	0
$19$	−2740.00	−1.74127	−0.870636	−	0.491928i	$-0.836292\pi$
$19$	−2740.00	−1.74127	−0.870636	+	0.491928i	$0.836292\pi$
$20$	0	0
$21$	0	0
$22$	−768.000	−0.338302
$23$	1566.00	0.617266	0.308633	−	0.951181i	$-0.400129\pi$
$23$	1566.00	0.617266	0.308633	+	0.951181i	$0.400129\pi$
$24$	0	0
$25$	0	0
$26$	−4424.00	−1.28346
$27$	0	0
$28$	1888.00	0.455100
$29$	−5910.00	−1.30495	−0.652473	−	0.757812i	$-0.726268\pi$
$29$	−5910.00	−1.30495	−0.652473	+	0.757812i	$0.726268\pi$
$30$	0	0
$31$	−6868.00	−1.28359	−0.641795	−	0.766877i	$-0.721810\pi$
$31$	−6868.00	−1.28359	−0.641795	+	0.766877i	$0.721810\pi$
$32$	1024.00	0.176777
$33$	0	0
$34$	3048.00	0.452187
$35$	0	0
$36$	0	0
$37$	5518.00	0.662640	0.331320	−	0.943519i	$-0.392506\pi$
$37$	5518.00	0.662640	0.331320	+	0.943519i	$0.392506\pi$
$38$	−10960.0	−1.23127
$39$	0	0
$40$	0	0
$41$	378.000	0.0351182	0.0175591	−	0.999846i	$-0.494410\pi$
$41$	378.000	0.0351182	0.0175591	+	0.999846i	$0.494410\pi$
$42$	0	0
$43$	2434.00	0.200747	0.100374	−	0.994950i	$-0.467996\pi$
$43$	2434.00	0.200747	0.100374	+	0.994950i	$0.467996\pi$
$44$	−3072.00	−0.239216
$45$	0	0
$46$	6264.00	0.436473
$47$	13122.0	0.866474	0.433237	−	0.901280i	$-0.357371\pi$
$47$	13122.0	0.866474	0.433237	+	0.901280i	$0.357371\pi$
$48$	0	0
$49$	−2883.00	−0.171536
$50$	0	0
$51$	0	0
$52$	−17696.0	−0.907542
$53$	−9174.00	−0.448610	−0.224305	−	0.974519i	$-0.572011\pi$
$53$	−9174.00	−0.448610	−0.224305	+	0.974519i	$0.572011\pi$
$54$	0	0
$55$	0	0
$56$	7552.00	0.321804
$57$	0	0
$58$	−23640.0	−0.922736
$59$	34980.0	1.30825	0.654124	−	0.756388i	$-0.273038\pi$
$59$	34980.0	1.30825	0.654124	+	0.756388i	$0.273038\pi$
$60$	0	0
$61$	−9838.00	−0.338518	−0.169259	−	0.985572i	$-0.554137\pi$
$61$	−9838.00	−0.338518	−0.169259	+	0.985572i	$0.554137\pi$
$62$	−27472.0	−0.907635
$63$	0	0
$64$	4096.00	0.125000
$65$	0	0
$66$	0	0
$67$	−33722.0	−0.917754	−0.458877	−	0.888500i	$-0.651748\pi$
$67$	−33722.0	−0.917754	−0.458877	+	0.888500i	$0.651748\pi$
$68$	12192.0	0.319744
$69$	0	0
$70$	0	0
$71$	−70212.0	−1.65297	−0.826486	−	0.562957i	$-0.809664\pi$
$71$	−70212.0	−1.65297	−0.826486	+	0.562957i	$0.809664\pi$
$72$	0	0
$73$	−21986.0	−0.482880	−0.241440	−	0.970416i	$-0.577620\pi$
$73$	−21986.0	−0.482880	−0.241440	+	0.970416i	$0.577620\pi$
$74$	22072.0	0.468557
$75$	0	0
$76$	−43840.0	−0.870636
$77$	−22656.0	−0.435468
$78$	0	0
$79$	4520.00	0.0814837	0.0407418	−	0.999170i	$-0.487028\pi$
$79$	4520.00	0.0814837	0.0407418	+	0.999170i	$0.487028\pi$
$80$	0	0
$81$	0	0
$82$	1512.00	0.0248323
$83$	−109074.	−1.73790	−0.868952	−	0.494896i	$-0.835206\pi$
$83$	−109074.	−1.73790	−0.868952	+	0.494896i	$0.835206\pi$
$84$	0	0
$85$	0	0
$86$	9736.00	0.141950
$87$	0	0
$88$	−12288.0	−0.169151
$89$	−38490.0	−0.515078	−0.257539	−	0.966268i	$-0.582912\pi$
$89$	−38490.0	−0.515078	−0.257539	+	0.966268i	$0.582912\pi$
$90$	0	0
$91$	−130508.	−1.65209
$92$	25056.0	0.308633
$93$	0	0
$94$	52488.0	0.612689
$95$	0	0
$96$	0	0
$97$	1918.00	0.0206976	0.0103488	−	0.999946i	$-0.496706\pi$
$97$	1918.00	0.0206976	0.0103488	+	0.999946i	$0.496706\pi$
$98$	−11532.0	−0.121294
$99$	0	0
$100$	0	0
$101$	−77622.0	−0.757149	−0.378575	−	0.925571i	$-0.623586\pi$
$101$	−77622.0	−0.757149	−0.378575	+	0.925571i	$0.623586\pi$
$102$	0	0
$103$	46714.0	0.433864	0.216932	−	0.976187i	$-0.430395\pi$
$103$	46714.0	0.433864	0.216932	+	0.976187i	$0.430395\pi$
$104$	−70784.0	−0.641729
$105$	0	0
$106$	−36696.0	−0.317215
$107$	−1038.00	−0.00876472	−0.00438236	−	0.999990i	$-0.501395\pi$
$107$	−1038.00	−0.00876472	−0.00438236	+	0.999990i	$0.501395\pi$
$108$	0	0
$109$	206930.	1.66823	0.834117	−	0.551587i	$-0.185977\pi$
$109$	206930.	1.66823	0.834117	+	0.551587i	$0.185977\pi$
$110$	0	0
$111$	0	0
$112$	30208.0	0.227550
$113$	139386.	1.02689	0.513444	−	0.858123i	$-0.328369\pi$
$113$	139386.	1.02689	0.513444	+	0.858123i	$0.328369\pi$
$114$	0	0
$115$	0	0
$116$	−94560.0	−0.652473
$117$	0	0
$118$	139920.	0.925070
$119$	89916.0	0.582062
$120$	0	0
$121$	−124187.	−0.771104
$122$	−39352.0	−0.239369
$123$	0	0
$124$	−109888.	−0.641795
$125$	0	0
$126$	0	0
$127$	−299882.	−1.64984	−0.824919	−	0.565252i	$-0.808779\pi$
$127$	−299882.	−1.64984	−0.824919	+	0.565252i	$0.808779\pi$
$128$	16384.0	0.0883883
$129$	0	0
$130$	0	0
$131$	−7872.00	−0.0400781	−0.0200390	−	0.999799i	$-0.506379\pi$
$131$	−7872.00	−0.0400781	−0.0200390	+	0.999799i	$0.506379\pi$
$132$	0	0
$133$	−323320.	−1.58491
$134$	−134888.	−0.648950
$135$	0	0
$136$	48768.0	0.226093
$137$	−164238.	−0.747605	−0.373803	−	0.927508i	$-0.621946\pi$
$137$	−164238.	−0.747605	−0.373803	+	0.927508i	$0.621946\pi$
$138$	0	0
$139$	−282100.	−1.23841	−0.619207	−	0.785228i	$-0.712546\pi$
$139$	−282100.	−1.23841	−0.619207	+	0.785228i	$0.712546\pi$
$140$	0	0
$141$	0	0
$142$	−280848.	−1.16883
$143$	212352.	0.868393
$144$	0	0
$145$	0	0
$146$	−87944.0	−0.341448
$147$	0	0
$148$	88288.0	0.331320
$149$	388950.	1.43525	0.717626	−	0.696429i	$-0.245229\pi$
$149$	388950.	1.43525	0.717626	+	0.696429i	$0.245229\pi$
$150$	0	0
$151$	−97948.0	−0.349585	−0.174793	−	0.984605i	$-0.555926\pi$
$151$	−97948.0	−0.349585	−0.174793	+	0.984605i	$0.555926\pi$
$152$	−175360.	−0.615633
$153$	0	0
$154$	−90624.0	−0.307923
$155$	0	0
$156$	0	0
$157$	3718.00	0.0120382	0.00601908	−	0.999982i	$-0.498084\pi$
$157$	3718.00	0.0120382	0.00601908	+	0.999982i	$0.498084\pi$
$158$	18080.0	0.0576177
$159$	0	0
$160$	0	0
$161$	184788.	0.561835
$162$	0	0
$163$	43234.0	0.127455	0.0637274	−	0.997967i	$-0.479701\pi$
$163$	43234.0	0.127455	0.0637274	+	0.997967i	$0.479701\pi$
$164$	6048.00	0.0175591
$165$	0	0
$166$	−436296.	−1.22888
$167$	186522.	0.517534	0.258767	−	0.965940i	$-0.416684\pi$
$167$	186522.	0.517534	0.258767	+	0.965940i	$0.416684\pi$
$168$	0	0
$169$	851943.	2.29453
$170$	0	0
$171$	0	0
$172$	38944.0	0.100374
$173$	−374454.	−0.951225	−0.475612	−	0.879655i	$-0.657774\pi$
$173$	−374454.	−0.951225	−0.475612	+	0.879655i	$0.657774\pi$
$174$	0	0
$175$	0	0
$176$	−49152.0	−0.119608
$177$	0	0
$178$	−153960.	−0.364215
$179$	−272100.	−0.634740	−0.317370	−	0.948302i	$-0.602800\pi$
$179$	−272100.	−0.634740	−0.317370	+	0.948302i	$0.602800\pi$
$180$	0	0
$181$	−75418.0	−0.171111	−0.0855556	−	0.996333i	$-0.527267\pi$
$181$	−75418.0	−0.171111	−0.0855556	+	0.996333i	$0.527267\pi$
$182$	−522032.	−1.16820
$183$	0	0
$184$	100224.	0.218236
$185$	0	0
$186$	0	0
$187$	−146304.	−0.305951
$188$	209952.	0.433237
$189$	0	0
$190$	0	0
$191$	356988.	0.708060	0.354030	−	0.935234i	$-0.384811\pi$
$191$	356988.	0.708060	0.354030	+	0.935234i	$0.384811\pi$
$192$	0	0
$193$	438694.	0.847751	0.423876	−	0.905720i	$-0.360669\pi$
$193$	438694.	0.847751	0.423876	+	0.905720i	$0.360669\pi$
$194$	7672.00	0.0146354
$195$	0	0
$196$	−46128.0	−0.0857678
$197$	−156798.	−0.287856	−0.143928	−	0.989588i	$-0.545973\pi$
$197$	−156798.	−0.287856	−0.143928	+	0.989588i	$0.545973\pi$
$198$	0	0
$199$	−162520.	−0.290920	−0.145460	−	0.989364i	$-0.546466\pi$
$199$	−162520.	−0.290920	−0.145460	+	0.989364i	$0.546466\pi$
$200$	0	0
$201$	0	0
$202$	−310488.	−0.535385
$203$	−697380.	−1.18776
$204$	0	0
$205$	0	0
$206$	186856.	0.306788
$207$	0	0
$208$	−283136.	−0.453771
$209$	526080.	0.833079
$210$	0	0
$211$	−181648.	−0.280882	−0.140441	−	0.990089i	$-0.544852\pi$
$211$	−181648.	−0.280882	−0.140441	+	0.990089i	$0.544852\pi$
$212$	−146784.	−0.224305
$213$	0	0
$214$	−4152.00	−0.00619759
$215$	0	0
$216$	0	0
$217$	−810424.	−1.16832
$218$	827720.	1.17962
$219$	0	0
$220$	0	0
$221$	−842772.	−1.16073
$222$	0	0
$223$	288274.	0.388189	0.194095	−	0.980983i	$-0.437823\pi$
$223$	288274.	0.388189	0.194095	+	0.980983i	$0.437823\pi$
$224$	120832.	0.160902
$225$	0	0
$226$	557544.	0.726119
$227$	1.12552e6	1.44974	0.724869	−	0.688887i	$-0.241900\pi$
$227$	1.12552e6	1.44974	0.724869	+	0.688887i	$0.241900\pi$
$228$	0	0
$229$	−415810.	−0.523970	−0.261985	−	0.965072i	$-0.584377\pi$
$229$	−415810.	−0.523970	−0.261985	+	0.965072i	$0.584377\pi$
$230$	0	0
$231$	0	0
$232$	−378240.	−0.461368
$233$	770586.	0.929889	0.464945	−	0.885340i	$-0.346074\pi$
$233$	770586.	0.929889	0.464945	+	0.885340i	$0.346074\pi$
$234$	0	0
$235$	0	0
$236$	559680.	0.654124
$237$	0	0
$238$	359664.	0.411580
$239$	595320.	0.674149	0.337074	−	0.941478i	$-0.390563\pi$
$239$	595320.	0.674149	0.337074	+	0.941478i	$0.390563\pi$
$240$	0	0
$241$	273902.	0.303775	0.151888	−	0.988398i	$-0.451465\pi$
$241$	273902.	0.303775	0.151888	+	0.988398i	$0.451465\pi$
$242$	−496748.	−0.545253
$243$	0	0
$244$	−157408.	−0.169259
$245$	0	0
$246$	0	0
$247$	3.03044e6	3.16055
$248$	−439552.	−0.453817
$249$	0	0
$250$	0	0
$251$	−850752.	−0.852351	−0.426176	−	0.904640i	$-0.640139\pi$
$251$	−850752.	−0.852351	−0.426176	+	0.904640i	$0.640139\pi$
$252$	0	0
$253$	−300672.	−0.295319
$254$	−1.19953e6	−1.16661
$255$	0	0
$256$	65536.0	0.0625000
$257$	825402.	0.779530	0.389765	−	0.920914i	$-0.372556\pi$
$257$	825402.	0.779530	0.389765	+	0.920914i	$0.372556\pi$
$258$	0	0
$259$	651124.	0.603135
$260$	0	0
$261$	0	0
$262$	−31488.0	−0.0283395
$263$	1.36465e6	1.21655	0.608276	−	0.793726i	$-0.291861\pi$
$263$	1.36465e6	1.21655	0.608276	+	0.793726i	$0.291861\pi$
$264$	0	0
$265$	0	0
$266$	−1.29328e6	−1.12070
$267$	0	0
$268$	−539552.	−0.458877
$269$	113310.	0.0954745	0.0477373	−	0.998860i	$-0.484799\pi$
$269$	113310.	0.0954745	0.0477373	+	0.998860i	$0.484799\pi$
$270$	0	0
$271$	−849628.	−0.702758	−0.351379	−	0.936233i	$-0.614287\pi$
$271$	−849628.	−0.702758	−0.351379	+	0.936233i	$0.614287\pi$
$272$	195072.	0.159872
$273$	0	0
$274$	−656952.	−0.528637
$275$	0	0
$276$	0	0
$277$	−438602.	−0.343456	−0.171728	−	0.985144i	$-0.554935\pi$
$277$	−438602.	−0.343456	−0.171728	+	0.985144i	$0.554935\pi$
$278$	−1.12840e6	−0.875691
$279$	0	0
$280$	0	0
$281$	1.45670e6	1.10053	0.550267	−	0.834989i	$-0.314526\pi$
$281$	1.45670e6	1.10053	0.550267	+	0.834989i	$0.314526\pi$
$282$	0	0
$283$	120394.	0.0893591	0.0446795	−	0.999001i	$-0.485773\pi$
$283$	120394.	0.0893591	0.0446795	+	0.999001i	$0.485773\pi$
$284$	−1.12339e6	−0.826486
$285$	0	0
$286$	849408.	0.614047
$287$	44604.0	0.0319646
$288$	0	0
$289$	−839213.	−0.591055
$290$	0	0
$291$	0	0
$292$	−351776.	−0.241440
$293$	−2.64209e6	−1.79796	−0.898978	−	0.437993i	$-0.855689\pi$
$293$	−2.64209e6	−1.79796	−0.898978	+	0.437993i	$0.855689\pi$
$294$	0	0
$295$	0	0
$296$	353152.	0.234278
$297$	0	0
$298$	1.55580e6	1.01488
$299$	−1.73200e6	−1.12039
$300$	0	0
$301$	287212.	0.182720
$302$	−391792.	−0.247194
$303$	0	0
$304$	−701440.	−0.435318
$305$	0	0
$306$	0	0
$307$	1.44756e6	0.876577	0.438288	−	0.898834i	$-0.355585\pi$
$307$	1.44756e6	0.876577	0.438288	+	0.898834i	$0.355585\pi$
$308$	−362496.	−0.217734
$309$	0	0
$310$	0	0
$311$	928068.	0.544100	0.272050	−	0.962283i	$-0.412298\pi$
$311$	928068.	0.544100	0.272050	+	0.962283i	$0.412298\pi$
$312$	0	0
$313$	−2.29563e6	−1.32446	−0.662232	−	0.749299i	$-0.730391\pi$
$313$	−2.29563e6	−1.32446	−0.662232	+	0.749299i	$0.730391\pi$
$314$	14872.0	0.00851227
$315$	0	0
$316$	72320.0	0.0407418
$317$	2.73652e6	1.52950	0.764752	−	0.644324i	$-0.222861\pi$
$317$	2.73652e6	1.52950	0.764752	+	0.644324i	$0.222861\pi$
$318$	0	0
$319$	1.13472e6	0.624327
$320$	0	0
$321$	0	0
$322$	739152.	0.397278
$323$	−2.08788e6	−1.11352
$324$	0	0
$325$	0	0
$326$	172936.	0.0901242
$327$	0	0
$328$	24192.0	0.0124162
$329$	1.54840e6	0.788665
$330$	0	0
$331$	3.81879e6	1.91583	0.957913	−	0.287059i	$-0.0926776\pi$
$331$	3.81879e6	1.91583	0.957913	+	0.287059i	$0.0926776\pi$
$332$	−1.74518e6	−0.868952
$333$	0	0
$334$	746088.	0.365952
$335$	0	0
$336$	0	0
$337$	2.21088e6	1.06045	0.530225	−	0.847857i	$-0.322108\pi$
$337$	2.21088e6	1.06045	0.530225	+	0.847857i	$0.322108\pi$
$338$	3.40777e6	1.62248
$339$	0	0
$340$	0	0
$341$	1.31866e6	0.614109
$342$	0	0
$343$	−2.32342e6	−1.06633
$344$	155776.	0.0709748
$345$	0	0
$346$	−1.49782e6	−0.672618
$347$	−2.32724e6	−1.03757	−0.518785	−	0.854905i	$-0.673615\pi$
$347$	−2.32724e6	−1.03757	−0.518785	+	0.854905i	$0.673615\pi$
$348$	0	0
$349$	−311290.	−0.136805	−0.0684024	−	0.997658i	$-0.521790\pi$
$349$	−311290.	−0.136805	−0.0684024	+	0.997658i	$0.521790\pi$
$350$	0	0
$351$	0	0
$352$	−196608.	−0.0845755
$353$	−3.08657e6	−1.31838	−0.659189	−	0.751977i	$-0.729100\pi$
$353$	−3.08657e6	−1.31838	−0.659189	+	0.751977i	$0.729100\pi$
$354$	0	0
$355$	0	0
$356$	−615840.	−0.257539
$357$	0	0
$358$	−1.08840e6	−0.448829
$359$	3.53076e6	1.44588	0.722940	−	0.690911i	$-0.242790\pi$
$359$	3.53076e6	1.44588	0.722940	+	0.690911i	$0.242790\pi$
$360$	0	0
$361$	5.03150e6	2.03203
$362$	−301672.	−0.120994
$363$	0	0
$364$	−2.08813e6	−0.826045
$365$	0	0
$366$	0	0
$367$	−35762.0	−0.0138598	−0.00692989	−	0.999976i	$-0.502206\pi$
$367$	−35762.0	−0.0138598	−0.00692989	+	0.999976i	$0.502206\pi$
$368$	400896.	0.154316
$369$	0	0
$370$	0	0
$371$	−1.08253e6	−0.408325
$372$	0	0
$373$	1.71525e6	0.638346	0.319173	−	0.947696i	$-0.396595\pi$
$373$	1.71525e6	0.638346	0.319173	+	0.947696i	$0.396595\pi$
$374$	−585216.	−0.216340
$375$	0	0
$376$	839808.	0.306345
$377$	6.53646e6	2.36859
$378$	0	0
$379$	−3.10174e6	−1.10919	−0.554597	−	0.832119i	$-0.687127\pi$
$379$	−3.10174e6	−1.10919	−0.554597	+	0.832119i	$0.687127\pi$
$380$	0	0
$381$	0	0
$382$	1.42795e6	0.500674
$383$	5.31949e6	1.85299	0.926494	−	0.376309i	$-0.122807\pi$
$383$	5.31949e6	1.85299	0.926494	+	0.376309i	$0.122807\pi$
$384$	0	0
$385$	0	0
$386$	1.75478e6	0.599451
$387$	0	0
$388$	30688.0	0.0103488
$389$	−1.16145e6	−0.389158	−0.194579	−	0.980887i	$-0.562334\pi$
$389$	−1.16145e6	−0.389158	−0.194579	+	0.980887i	$0.562334\pi$
$390$	0	0
$391$	1.19329e6	0.394734
$392$	−184512.	−0.0606470
$393$	0	0
$394$	−627192.	−0.203545
$395$	0	0
$396$	0	0
$397$	−628562.	−0.200157	−0.100079	−	0.994980i	$-0.531909\pi$
$397$	−628562.	−0.200157	−0.100079	+	0.994980i	$0.531909\pi$
$398$	−650080.	−0.205712
$399$	0	0
$400$	0	0
$401$	2.72432e6	0.846052	0.423026	−	0.906118i	$-0.360968\pi$
$401$	2.72432e6	0.846052	0.423026	+	0.906118i	$0.360968\pi$
$402$	0	0
$403$	7.59601e6	2.32982
$404$	−1.24195e6	−0.378575
$405$	0	0
$406$	−2.78952e6	−0.839875
$407$	−1.05946e6	−0.317027
$408$	0	0
$409$	1.78019e6	0.526209	0.263104	−	0.964767i	$-0.415254\pi$
$409$	1.78019e6	0.526209	0.263104	+	0.964767i	$0.415254\pi$
$410$	0	0
$411$	0	0
$412$	747424.	0.216932
$413$	4.12764e6	1.19077
$414$	0	0
$415$	0	0
$416$	−1.13254e6	−0.320865
$417$	0	0
$418$	2.10432e6	0.589076
$419$	−650580.	−0.181036	−0.0905181	−	0.995895i	$-0.528852\pi$
$419$	−650580.	−0.181036	−0.0905181	+	0.995895i	$0.528852\pi$
$420$	0	0
$421$	−3.54060e6	−0.973579	−0.486790	−	0.873519i	$-0.661832\pi$
$421$	−3.54060e6	−0.973579	−0.486790	+	0.873519i	$0.661832\pi$
$422$	−726592.	−0.198614
$423$	0	0
$424$	−587136.	−0.158608
$425$	0	0
$426$	0	0
$427$	−1.16088e6	−0.308119
$428$	−16608.0	−0.00438236
$429$	0	0
$430$	0	0
$431$	548748.	0.142292	0.0711459	−	0.997466i	$-0.477334\pi$
$431$	548748.	0.142292	0.0711459	+	0.997466i	$0.477334\pi$
$432$	0	0
$433$	1.49241e6	0.382534	0.191267	−	0.981538i	$-0.438740\pi$
$433$	1.49241e6	0.382534	0.191267	+	0.981538i	$0.438740\pi$
$434$	−3.24170e6	−0.826129
$435$	0	0
$436$	3.31088e6	0.834117
$437$	−4.29084e6	−1.07483
$438$	0	0
$439$	4.86212e6	1.20411	0.602053	−	0.798456i	$-0.294350\pi$
$439$	4.86212e6	1.20411	0.602053	+	0.798456i	$0.294350\pi$
$440$	0	0
$441$	0	0
$442$	−3.37109e6	−0.820757
$443$	−1.86155e6	−0.450678	−0.225339	−	0.974280i	$-0.572349\pi$
$443$	−1.86155e6	−0.450678	−0.225339	+	0.974280i	$0.572349\pi$
$444$	0	0
$445$	0	0
$446$	1.15310e6	0.274491
$447$	0	0
$448$	483328.	0.113775
$449$	−3.73719e6	−0.874841	−0.437421	−	0.899257i	$-0.644108\pi$
$449$	−3.73719e6	−0.874841	−0.437421	+	0.899257i	$0.644108\pi$
$450$	0	0
$451$	−72576.0	−0.0168016
$452$	2.23018e6	0.513444
$453$	0	0
$454$	4.50209e6	1.02512
$455$	0	0
$456$	0	0
$457$	6.48276e6	1.45201	0.726005	−	0.687690i	$-0.241375\pi$
$457$	6.48276e6	1.45201	0.726005	+	0.687690i	$0.241375\pi$
$458$	−1.66324e6	−0.370503
$459$	0	0
$460$	0	0
$461$	−1.50910e6	−0.330724	−0.165362	−	0.986233i	$-0.552879\pi$
$461$	−1.50910e6	−0.330724	−0.165362	+	0.986233i	$0.552879\pi$
$462$	0	0
$463$	−8.68401e6	−1.88264	−0.941321	−	0.337513i	$-0.890414\pi$
$463$	−8.68401e6	−1.88264	−0.941321	+	0.337513i	$0.890414\pi$
$464$	−1.51296e6	−0.326236
$465$	0	0
$466$	3.08234e6	0.657531
$467$	6.96412e6	1.47766	0.738829	−	0.673893i	$-0.235379\pi$
$467$	6.96412e6	1.47766	0.738829	+	0.673893i	$0.235379\pi$
$468$	0	0
$469$	−3.97920e6	−0.835340
$470$	0	0
$471$	0	0
$472$	2.23872e6	0.462535
$473$	−467328.	−0.0960437
$474$	0	0
$475$	0	0
$476$	1.43866e6	0.291031
$477$	0	0
$478$	2.38128e6	0.476695
$479$	5.51052e6	1.09737	0.548686	−	0.836029i	$-0.315128\pi$
$479$	5.51052e6	1.09737	0.548686	+	0.836029i	$0.315128\pi$
$480$	0	0
$481$	−6.10291e6	−1.20275
$482$	1.09561e6	0.214802
$483$	0	0
$484$	−1.98699e6	−0.385552
$485$	0	0
$486$	0	0
$487$	−5.51808e6	−1.05430	−0.527152	−	0.849771i	$-0.676740\pi$
$487$	−5.51808e6	−1.05430	−0.527152	+	0.849771i	$0.676740\pi$
$488$	−629632.	−0.119684
$489$	0	0
$490$	0	0
$491$	1.51277e6	0.283184	0.141592	−	0.989925i	$-0.454778\pi$
$491$	1.51277e6	0.283184	0.141592	+	0.989925i	$0.454778\pi$
$492$	0	0
$493$	−4.50342e6	−0.834498
$494$	1.21218e7	2.23485
$495$	0	0
$496$	−1.75821e6	−0.320897
$497$	−8.28502e6	−1.50454
$498$	0	0
$499$	−1.93042e6	−0.347057	−0.173528	−	0.984829i	$-0.555517\pi$
$499$	−1.93042e6	−0.347057	−0.173528	+	0.984829i	$0.555517\pi$
$500$	0	0
$501$	0	0
$502$	−3.40301e6	−0.602703
$503$	6.73105e6	1.18621	0.593106	−	0.805124i	$-0.297901\pi$
$503$	6.73105e6	1.18621	0.593106	+	0.805124i	$0.297901\pi$
$504$	0	0
$505$	0	0
$506$	−1.20269e6	−0.208822
$507$	0	0
$508$	−4.79811e6	−0.824919
$509$	556650.	0.0952331	0.0476165	−	0.998866i	$-0.484837\pi$
$509$	556650.	0.0952331	0.0476165	+	0.998866i	$0.484837\pi$
$510$	0	0
$511$	−2.59435e6	−0.439517
$512$	262144.	0.0441942
$513$	0	0
$514$	3.30161e6	0.551211
$515$	0	0
$516$	0	0
$517$	−2.51942e6	−0.414548
$518$	2.60450e6	0.426481
$519$	0	0
$520$	0	0
$521$	−1.01110e7	−1.63192	−0.815962	−	0.578106i	$-0.803792\pi$
$521$	−1.01110e7	−1.63192	−0.815962	+	0.578106i	$0.803792\pi$
$522$	0	0
$523$	7.03719e6	1.12498	0.562491	−	0.826804i	$-0.309843\pi$
$523$	7.03719e6	1.12498	0.562491	+	0.826804i	$0.309843\pi$
$524$	−125952.	−0.0200390
$525$	0	0
$526$	5.45858e6	0.860232
$527$	−5.23342e6	−0.820840
$528$	0	0
$529$	−3.98399e6	−0.618983
$530$	0	0
$531$	0	0
$532$	−5.17312e6	−0.792453
$533$	−418068.	−0.0637425
$534$	0	0
$535$	0	0
$536$	−2.15821e6	−0.324475
$537$	0	0
$538$	453240.	0.0675107
$539$	553536.	0.0820680
$540$	0	0
$541$	−4.23114e6	−0.621533	−0.310766	−	0.950486i	$-0.600586\pi$
$541$	−4.23114e6	−0.621533	−0.310766	+	0.950486i	$0.600586\pi$
$542$	−3.39851e6	−0.496925
$543$	0	0
$544$	780288.	0.113047
$545$	0	0
$546$	0	0
$547$	−4.44024e6	−0.634510	−0.317255	−	0.948340i	$-0.602761\pi$
$547$	−4.44024e6	−0.634510	−0.317255	+	0.948340i	$0.602761\pi$
$548$	−2.62781e6	−0.373803
$549$	0	0
$550$	0	0
$551$	1.61934e7	2.27227
$552$	0	0
$553$	533360.	0.0741665
$554$	−1.75441e6	−0.242860
$555$	0	0
$556$	−4.51360e6	−0.619207
$557$	−9.01448e6	−1.23113	−0.615563	−	0.788088i	$-0.711071\pi$
$557$	−9.01448e6	−1.23113	−0.615563	+	0.788088i	$0.711071\pi$
$558$	0	0
$559$	−2.69200e6	−0.364373
$560$	0	0
$561$	0	0
$562$	5.82679e6	0.778196
$563$	−9.81287e6	−1.30474	−0.652372	−	0.757899i	$-0.726226\pi$
$563$	−9.81287e6	−1.30474	−0.652372	+	0.757899i	$0.726226\pi$
$564$	0	0
$565$	0	0
$566$	481576.	0.0631864
$567$	0	0
$568$	−4.49357e6	−0.584414
$569$	−1.33152e7	−1.72412	−0.862061	−	0.506804i	$-0.830827\pi$
$569$	−1.33152e7	−1.72412	−0.862061	+	0.506804i	$0.830827\pi$
$570$	0	0
$571$	9.95895e6	1.27827	0.639136	−	0.769094i	$-0.279292\pi$
$571$	9.95895e6	1.27827	0.639136	+	0.769094i	$0.279292\pi$
$572$	3.39763e6	0.434196
$573$	0	0
$574$	178416.	0.0226024
$575$	0	0
$576$	0	0
$577$	−4.50372e6	−0.563160	−0.281580	−	0.959538i	$-0.590859\pi$
$577$	−4.50372e6	−0.563160	−0.281580	+	0.959538i	$0.590859\pi$
$578$	−3.35685e6	−0.417939
$579$	0	0
$580$	0	0
$581$	−1.28707e7	−1.58184
$582$	0	0
$583$	1.76141e6	0.214629
$584$	−1.40710e6	−0.170724
$585$	0	0
$586$	−1.05684e7	−1.27135
$587$	625842.	0.0749669	0.0374834	−	0.999297i	$-0.488066\pi$
$587$	625842.	0.0749669	0.0374834	+	0.999297i	$0.488066\pi$
$588$	0	0
$589$	1.88183e7	2.23508
$590$	0	0
$591$	0	0
$592$	1.41261e6	0.165660
$593$	−2.50385e6	−0.292397	−0.146198	−	0.989255i	$-0.546704\pi$
$593$	−2.50385e6	−0.292397	−0.146198	+	0.989255i	$0.546704\pi$
$594$	0	0
$595$	0	0
$596$	6.22320e6	0.717626
$597$	0	0
$598$	−6.92798e6	−0.792235
$599$	756480.	0.0861451	0.0430725	−	0.999072i	$-0.486285\pi$
$599$	756480.	0.0861451	0.0430725	+	0.999072i	$0.486285\pi$
$600$	0	0
$601$	−1.38565e7	−1.56483	−0.782413	−	0.622760i	$-0.786011\pi$
$601$	−1.38565e7	−1.56483	−0.782413	+	0.622760i	$0.786011\pi$
$602$	1.14885e6	0.129203
$603$	0	0
$604$	−1.56717e6	−0.174793
$605$	0	0
$606$	0	0
$607$	−1.13772e7	−1.25333	−0.626663	−	0.779291i	$-0.715580\pi$
$607$	−1.13772e7	−1.25333	−0.626663	+	0.779291i	$0.715580\pi$
$608$	−2.80576e6	−0.307816
$609$	0	0
$610$	0	0
$611$	−1.45129e7	−1.57272
$612$	0	0
$613$	7.00161e6	0.752570	0.376285	−	0.926504i	$-0.377201\pi$
$613$	7.00161e6	0.752570	0.376285	+	0.926504i	$0.377201\pi$
$614$	5.79023e6	0.619833
$615$	0	0
$616$	−1.44998e6	−0.153961
$617$	7.90300e6	0.835755	0.417878	−	0.908503i	$-0.362774\pi$
$617$	7.90300e6	0.835755	0.417878	+	0.908503i	$0.362774\pi$
$618$	0	0
$619$	4.02362e6	0.422076	0.211038	−	0.977478i	$-0.432316\pi$
$619$	4.02362e6	0.422076	0.211038	+	0.977478i	$0.432316\pi$
$620$	0	0
$621$	0	0
$622$	3.71227e6	0.384737
$623$	−4.54182e6	−0.468824
$624$	0	0
$625$	0	0
$626$	−9.18250e6	−0.936538
$627$	0	0
$628$	59488.0	0.00601908
$629$	4.20472e6	0.423750
$630$	0	0
$631$	−1.00227e7	−1.00210	−0.501049	−	0.865419i	$-0.667052\pi$
$631$	−1.00227e7	−1.00210	−0.501049	+	0.865419i	$0.667052\pi$
$632$	289280.	0.0288088
$633$	0	0
$634$	1.09461e7	1.08152
$635$	0	0
$636$	0	0
$637$	3.18860e6	0.311352
$638$	4.53888e6	0.441466
$639$	0	0
$640$	0	0
$641$	−6.37390e6	−0.612718	−0.306359	−	0.951916i	$-0.599111\pi$
$641$	−6.37390e6	−0.612718	−0.306359	+	0.951916i	$0.599111\pi$
$642$	0	0
$643$	−5.00457e6	−0.477352	−0.238676	−	0.971099i	$-0.576713\pi$
$643$	−5.00457e6	−0.477352	−0.238676	+	0.971099i	$0.576713\pi$
$644$	2.95661e6	0.280918
$645$	0	0
$646$	−8.35152e6	−0.787380
$647$	−8.71928e6	−0.818879	−0.409440	−	0.912337i	$-0.634276\pi$
$647$	−8.71928e6	−0.818879	−0.409440	+	0.912337i	$0.634276\pi$
$648$	0	0
$649$	−6.71616e6	−0.625906
$650$	0	0
$651$	0	0
$652$	691744.	0.0637274
$653$	−1.58477e6	−0.145440	−0.0727201	−	0.997352i	$-0.523168\pi$
$653$	−1.58477e6	−0.145440	−0.0727201	+	0.997352i	$0.523168\pi$
$654$	0	0
$655$	0	0
$656$	96768.0	0.00877955
$657$	0	0
$658$	6.19358e6	0.557670
$659$	−1.26410e7	−1.13388	−0.566940	−	0.823759i	$-0.691873\pi$
$659$	−1.26410e7	−1.13388	−0.566940	+	0.823759i	$0.691873\pi$
$660$	0	0
$661$	−3.61572e6	−0.321878	−0.160939	−	0.986964i	$-0.551452\pi$
$661$	−3.61572e6	−0.321878	−0.160939	+	0.986964i	$0.551452\pi$
$662$	1.52752e7	1.35469
$663$	0	0
$664$	−6.98074e6	−0.614442
$665$	0	0
$666$	0	0
$667$	−9.25506e6	−0.805498
$668$	2.98435e6	0.258767
$669$	0	0
$670$	0	0
$671$	1.88890e6	0.161958
$672$	0	0
$673$	−1.11313e7	−0.947349	−0.473675	−	0.880700i	$-0.657073\pi$
$673$	−1.11313e7	−0.947349	−0.473675	+	0.880700i	$0.657073\pi$
$674$	8.84351e6	0.749851
$675$	0	0
$676$	1.36311e7	1.14727
$677$	−235518.	−0.0197493	−0.00987467	−	0.999951i	$-0.503143\pi$
$677$	−235518.	−0.0197493	−0.00987467	+	0.999951i	$0.503143\pi$
$678$	0	0
$679$	226324.	0.0188389
$680$	0	0
$681$	0	0
$682$	5.27462e6	0.434241
$683$	2.05830e7	1.68833	0.844164	−	0.536084i	$-0.180097\pi$
$683$	2.05830e7	1.68833	0.844164	+	0.536084i	$0.180097\pi$
$684$	0	0
$685$	0	0
$686$	−9.29368e6	−0.754011
$687$	0	0
$688$	623104.	0.0501868
$689$	1.01464e7	0.814265
$690$	0	0
$691$	−9.54825e6	−0.760727	−0.380363	−	0.924837i	$-0.624201\pi$
$691$	−9.54825e6	−0.760727	−0.380363	+	0.924837i	$0.624201\pi$
$692$	−5.99126e6	−0.475612
$693$	0	0
$694$	−9.30895e6	−0.733672
$695$	0	0
$696$	0	0
$697$	288036.	0.0224577
$698$	−1.24516e6	−0.0967357
$699$	0	0
$700$	0	0
$701$	−1.29304e6	−0.0993843	−0.0496921	−	0.998765i	$-0.515824\pi$
$701$	−1.29304e6	−0.0993843	−0.0496921	+	0.998765i	$0.515824\pi$
$702$	0	0
$703$	−1.51193e7	−1.15384
$704$	−786432.	−0.0598039
$705$	0	0
$706$	−1.23463e7	−0.932234
$707$	−9.15940e6	−0.689157
$708$	0	0
$709$	−2.12720e7	−1.58926	−0.794628	−	0.607097i	$-0.792334\pi$
$709$	−2.12720e7	−1.58926	−0.794628	+	0.607097i	$0.792334\pi$
$710$	0	0
$711$	0	0
$712$	−2.46336e6	−0.182108
$713$	−1.07553e7	−0.792316
$714$	0	0
$715$	0	0
$716$	−4.35360e6	−0.317370
$717$	0	0
$718$	1.41230e7	1.02239
$719$	−8.31732e6	−0.600014	−0.300007	−	0.953937i	$-0.596989\pi$
$719$	−8.31732e6	−0.600014	−0.300007	+	0.953937i	$0.596989\pi$
$720$	0	0
$721$	5.51225e6	0.394903
$722$	2.01260e7	1.43686
$723$	0	0
$724$	−1.20669e6	−0.0855556
$725$	0	0
$726$	0	0
$727$	4.36740e6	0.306469	0.153235	−	0.988190i	$-0.451031\pi$
$727$	4.36740e6	0.306469	0.153235	+	0.988190i	$0.451031\pi$
$728$	−8.35251e6	−0.584102
$729$	0	0
$730$	0	0
$731$	1.85471e6	0.128375
$732$	0	0
$733$	4.05645e6	0.278860	0.139430	−	0.990232i	$-0.455473\pi$
$733$	4.05645e6	0.278860	0.139430	+	0.990232i	$0.455473\pi$
$734$	−143048.	−0.00980035
$735$	0	0
$736$	1.60358e6	0.109118
$737$	6.47462e6	0.439082
$738$	0	0
$739$	768260.	0.0517484	0.0258742	−	0.999665i	$-0.491763\pi$
$739$	768260.	0.0517484	0.0258742	+	0.999665i	$0.491763\pi$
$740$	0	0
$741$	0	0
$742$	−4.33013e6	−0.288729
$743$	6.18781e6	0.411211	0.205605	−	0.978635i	$-0.434084\pi$
$743$	6.18781e6	0.411211	0.205605	+	0.978635i	$0.434084\pi$
$744$	0	0
$745$	0	0
$746$	6.86102e6	0.451379
$747$	0	0
$748$	−2.34086e6	−0.152976
$749$	−122484.	−0.00797765
$750$	0	0
$751$	1.81698e7	1.17557	0.587787	−	0.809016i	$-0.299999\pi$
$751$	1.81698e7	1.17557	0.587787	+	0.809016i	$0.299999\pi$
$752$	3.35923e6	0.216618
$753$	0	0
$754$	2.61458e7	1.67484
$755$	0	0
$756$	0	0
$757$	−1.93494e7	−1.22724	−0.613618	−	0.789603i	$-0.710286\pi$
$757$	−1.93494e7	−1.22724	−0.613618	+	0.789603i	$0.710286\pi$
$758$	−1.24070e7	−0.784318
$759$	0	0
$760$	0	0
$761$	3.01992e7	1.89031	0.945155	−	0.326621i	$-0.105910\pi$
$761$	3.01992e7	1.89031	0.945155	+	0.326621i	$0.105910\pi$
$762$	0	0
$763$	2.44177e7	1.51843
$764$	5.71181e6	0.354030
$765$	0	0
$766$	2.12779e7	1.31026
$767$	−3.86879e7	−2.37458
$768$	0	0
$769$	2.15854e7	1.31627	0.658134	−	0.752901i	$-0.271346\pi$
$769$	2.15854e7	1.31627	0.658134	+	0.752901i	$0.271346\pi$
$770$	0	0
$771$	0	0
$772$	7.01910e6	0.423876
$773$	3.90895e6	0.235294	0.117647	−	0.993055i	$-0.462465\pi$
$773$	3.90895e6	0.235294	0.117647	+	0.993055i	$0.462465\pi$
$774$	0	0
$775$	0	0
$776$	122752.	0.00731769
$777$	0	0
$778$	−4.64580e6	−0.275177
$779$	−1.03572e6	−0.0611503
$780$	0	0
$781$	1.34807e7	0.790833
$782$	4.77317e6	0.279119
$783$	0	0
$784$	−738048.	−0.0428839
$785$	0	0
$786$	0	0
$787$	2.65082e7	1.52561	0.762806	−	0.646628i	$-0.223821\pi$
$787$	2.65082e7	1.52561	0.762806	+	0.646628i	$0.223821\pi$
$788$	−2.50877e6	−0.143928
$789$	0	0
$790$	0	0
$791$	1.64475e7	0.934674
$792$	0	0
$793$	1.08808e7	0.614439
$794$	−2.51425e6	−0.141533
$795$	0	0
$796$	−2.60032e6	−0.145460
$797$	1.07940e7	0.601919	0.300960	−	0.953637i	$-0.402693\pi$
$797$	1.07940e7	0.601919	0.300960	+	0.953637i	$0.402693\pi$
$798$	0	0
$799$	9.99896e6	0.554100
$800$	0	0
$801$	0	0
$802$	1.08973e7	0.598249
$803$	4.22131e6	0.231025
$804$	0	0
$805$	0	0
$806$	3.03840e7	1.64743
$807$	0	0
$808$	−4.96781e6	−0.267693
$809$	1.11446e7	0.598675	0.299338	−	0.954147i	$-0.403234\pi$
$809$	1.11446e7	0.598675	0.299338	+	0.954147i	$0.403234\pi$
$810$	0	0
$811$	−1.14866e7	−0.613253	−0.306626	−	0.951830i	$-0.599200\pi$
$811$	−1.14866e7	−0.613253	−0.306626	+	0.951830i	$0.599200\pi$
$812$	−1.11581e7	−0.593881
$813$	0	0
$814$	−4.23782e6	−0.224172
$815$	0	0
$816$	0	0
$817$	−6.66916e6	−0.349555
$818$	7.12076e6	0.372086
$819$	0	0
$820$	0	0
$821$	−3.04347e7	−1.57584	−0.787918	−	0.615781i	$-0.788841\pi$
$821$	−3.04347e7	−1.57584	−0.787918	+	0.615781i	$0.788841\pi$
$822$	0	0
$823$	−4.09773e6	−0.210884	−0.105442	−	0.994425i	$-0.533626\pi$
$823$	−4.09773e6	−0.210884	−0.105442	+	0.994425i	$0.533626\pi$
$824$	2.98970e6	0.153394
$825$	0	0
$826$	1.65106e7	0.841999
$827$	−1.70652e7	−0.867654	−0.433827	−	0.900996i	$-0.642837\pi$
$827$	−1.70652e7	−0.867654	−0.433827	+	0.900996i	$0.642837\pi$
$828$	0	0
$829$	−2.47617e7	−1.25139	−0.625697	−	0.780066i	$-0.715185\pi$
$829$	−2.47617e7	−1.25139	−0.625697	+	0.780066i	$0.715185\pi$
$830$	0	0
$831$	0	0
$832$	−4.53018e6	−0.226886
$833$	−2.19685e6	−0.109695
$834$	0	0
$835$	0	0
$836$	8.41728e6	0.416539
$837$	0	0
$838$	−2.60232e6	−0.128012
$839$	−3.16529e7	−1.55242	−0.776208	−	0.630476i	$-0.782860\pi$
$839$	−3.16529e7	−1.55242	−0.776208	+	0.630476i	$0.782860\pi$
$840$	0	0
$841$	1.44170e7	0.702884
$842$	−1.41624e7	−0.688425
$843$	0	0
$844$	−2.90637e6	−0.140441
$845$	0	0
$846$	0	0
$847$	−1.46541e7	−0.701859
$848$	−2.34854e6	−0.112153
$849$	0	0
$850$	0	0
$851$	8.64119e6	0.409025
$852$	0	0
$853$	−2.82671e7	−1.33017	−0.665087	−	0.746765i	$-0.731606\pi$
$853$	−2.82671e7	−1.33017	−0.665087	+	0.746765i	$0.731606\pi$
$854$	−4.64354e6	−0.217873
$855$	0	0
$856$	−66432.0	−0.00309880
$857$	2.60870e7	1.21331	0.606655	−	0.794966i	$-0.292511\pi$
$857$	2.60870e7	1.21331	0.606655	+	0.794966i	$0.292511\pi$
$858$	0	0
$859$	−3.38111e7	−1.56342	−0.781710	−	0.623642i	$-0.785652\pi$
$859$	−3.38111e7	−1.56342	−0.781710	+	0.623642i	$0.785652\pi$
$860$	0	0
$861$	0	0
$862$	2.19499e6	0.100615
$863$	2.22817e7	1.01841	0.509204	−	0.860646i	$-0.329940\pi$
$863$	2.22817e7	1.01841	0.509204	+	0.860646i	$0.329940\pi$
$864$	0	0
$865$	0	0
$866$	5.96966e6	0.270492
$867$	0	0
$868$	−1.29668e7	−0.584162
$869$	−867840.	−0.0389843
$870$	0	0
$871$	3.72965e7	1.66580
$872$	1.32435e7	0.589810
$873$	0	0
$874$	−1.71634e7	−0.760018
$875$	0	0
$876$	0	0
$877$	3.46748e7	1.52235	0.761177	−	0.648545i	$-0.224622\pi$
$877$	3.46748e7	1.52235	0.761177	+	0.648545i	$0.224622\pi$
$878$	1.94485e7	0.851431
$879$	0	0
$880$	0	0
$881$	−1.42603e7	−0.618998	−0.309499	−	0.950900i	$-0.600161\pi$
$881$	−1.42603e7	−0.618998	−0.309499	+	0.950900i	$0.600161\pi$
$882$	0	0
$883$	3.75177e7	1.61933	0.809663	−	0.586895i	$-0.199650\pi$
$883$	3.75177e7	1.61933	0.809663	+	0.586895i	$0.199650\pi$
$884$	−1.34844e7	−0.580363
$885$	0	0
$886$	−7.44622e6	−0.318677
$887$	4.07657e7	1.73975	0.869873	−	0.493275i	$-0.164200\pi$
$887$	4.07657e7	1.73975	0.869873	+	0.493275i	$0.164200\pi$
$888$	0	0
$889$	−3.53861e7	−1.50168
$890$	0	0
$891$	0	0
$892$	4.61238e6	0.194095
$893$	−3.59543e7	−1.50877
$894$	0	0
$895$	0	0
$896$	1.93331e6	0.0804511
$897$	0	0
$898$	−1.49488e7	−0.618606
$899$	4.05899e7	1.67501
$900$	0	0
$901$	−6.99059e6	−0.286881
$902$	−290304.	−0.0118806
$903$	0	0
$904$	8.92070e6	0.363060
$905$	0	0
$906$	0	0
$907$	3.57116e7	1.44142	0.720712	−	0.693235i	$-0.243815\pi$
$907$	3.57116e7	1.44142	0.720712	+	0.693235i	$0.243815\pi$
$908$	1.80084e7	0.724869
$909$	0	0
$910$	0	0
$911$	2.11389e7	0.843893	0.421947	−	0.906621i	$-0.361347\pi$
$911$	2.11389e7	0.843893	0.421947	+	0.906621i	$0.361347\pi$
$912$	0	0
$913$	2.09422e7	0.831468
$914$	2.59310e7	1.02673
$915$	0	0
$916$	−6.65296e6	−0.261985
$917$	−928896.	−0.0364791
$918$	0	0
$919$	1.85996e7	0.726465	0.363233	−	0.931698i	$-0.381673\pi$
$919$	1.85996e7	0.726465	0.363233	+	0.931698i	$0.381673\pi$
$920$	0	0
$921$	0	0
$922$	−6.03641e6	−0.233857
$923$	7.76545e7	3.00028
$924$	0	0
$925$	0	0
$926$	−3.47360e7	−1.33123
$927$	0	0
$928$	−6.05184e6	−0.230684
$929$	−4.45110e7	−1.69211	−0.846055	−	0.533096i	$-0.821028\pi$
$929$	−4.45110e7	−1.69211	−0.846055	+	0.533096i	$0.821028\pi$
$930$	0	0
$931$	7.89942e6	0.298690
$932$	1.23294e7	0.464945
$933$	0	0
$934$	2.78565e7	1.04486
$935$	0	0
$936$	0	0
$937$	2.19419e7	0.816441	0.408221	−	0.912883i	$-0.366149\pi$
$937$	2.19419e7	0.816441	0.408221	+	0.912883i	$0.366149\pi$
$938$	−1.59168e7	−0.590675
$939$	0	0
$940$	0	0
$941$	7.77722e6	0.286319	0.143160	−	0.989700i	$-0.454274\pi$
$941$	7.77722e6	0.286319	0.143160	+	0.989700i	$0.454274\pi$
$942$	0	0
$943$	591948.	0.0216773
$944$	8.95488e6	0.327062
$945$	0	0
$946$	−1.86931e6	−0.0679132
$947$	3.17199e7	1.14936	0.574681	−	0.818378i	$-0.305126\pi$
$947$	3.17199e7	1.14936	0.574681	+	0.818378i	$0.305126\pi$
$948$	0	0
$949$	2.43165e7	0.876468
$950$	0	0
$951$	0	0
$952$	5.75462e6	0.205790
$953$	−5.60285e6	−0.199838	−0.0999188	−	0.994996i	$-0.531858\pi$
$953$	−5.60285e6	−0.199838	−0.0999188	+	0.994996i	$0.531858\pi$
$954$	0	0
$955$	0	0
$956$	9.52512e6	0.337074
$957$	0	0
$958$	2.20421e7	0.775959
$959$	−1.93801e7	−0.680470
$960$	0	0
$961$	1.85403e7	0.647601
$962$	−2.44116e7	−0.850470
$963$	0	0
$964$	4.38243e6	0.151888
$965$	0	0
$966$	0	0
$967$	2.03532e7	0.699949	0.349975	−	0.936759i	$-0.386190\pi$
$967$	2.03532e7	0.699949	0.349975	+	0.936759i	$0.386190\pi$
$968$	−7.94797e6	−0.272626
$969$	0	0
$970$	0	0
$971$	2.34306e7	0.797510	0.398755	−	0.917057i	$-0.369442\pi$
$971$	2.34306e7	0.797510	0.398755	+	0.917057i	$0.369442\pi$
$972$	0	0
$973$	−3.32878e7	−1.12721
$974$	−2.20723e7	−0.745505
$975$	0	0
$976$	−2.51853e6	−0.0846296
$977$	−4.30412e7	−1.44261	−0.721303	−	0.692619i	$-0.756457\pi$
$977$	−4.30412e7	−1.44261	−0.721303	+	0.692619i	$0.756457\pi$
$978$	0	0
$979$	7.39008e6	0.246429
$980$	0	0
$981$	0	0
$982$	6.05107e6	0.200241
$983$	−4.75003e7	−1.56788	−0.783940	−	0.620837i	$-0.786793\pi$
$983$	−4.75003e7	−1.56788	−0.783940	+	0.620837i	$0.786793\pi$
$984$	0	0
$985$	0	0
$986$	−1.80137e7	−0.590079
$987$	0	0
$988$	4.84870e7	1.58028
$989$	3.81164e6	0.123914
$990$	0	0
$991$	2.09231e7	0.676770	0.338385	−	0.941008i	$-0.390119\pi$
$991$	2.09231e7	0.676770	0.338385	+	0.941008i	$0.390119\pi$
$992$	−7.03283e6	−0.226909
$993$	0	0
$994$	−3.31401e7	−1.06387
$995$	0	0
$996$	0	0
$997$	−2.96332e7	−0.944148	−0.472074	−	0.881559i	$-0.656495\pi$
$997$	−2.96332e7	−0.944148	−0.472074	+	0.881559i	$0.656495\pi$
$998$	−7.72168e6	−0.245406
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	450.6.a.u.1.1		1
3.2	odd	2		50.6.a.b.1.1		1
5.2	odd	4		450.6.c.f.199.2		2
5.3	odd	4		450.6.c.f.199.1		2
5.4	even	2		90.6.a.b.1.1		1
12.11	even	2		400.6.a.i.1.1		1
15.2	even	4		50.6.b.b.49.1		2
15.8	even	4		50.6.b.b.49.2		2
15.14	odd	2		10.6.a.c.1.1	✓	1
20.19	odd	2		720.6.a.v.1.1		1
60.23	odd	4		400.6.c.i.49.2		2
60.47	odd	4		400.6.c.i.49.1		2
60.59	even	2		80.6.a.c.1.1		1
105.104	even	2		490.6.a.k.1.1		1
120.29	odd	2		320.6.a.f.1.1		1
120.59	even	2		320.6.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
10.6.a.c.1.1	✓	1	15.14	odd	2
50.6.a.b.1.1		1	3.2	odd	2
50.6.b.b.49.1		2	15.2	even	4
50.6.b.b.49.2		2	15.8	even	4
80.6.a.c.1.1		1	60.59	even	2
90.6.a.b.1.1		1	5.4	even	2
320.6.a.f.1.1		1	120.29	odd	2
320.6.a.k.1.1		1	120.59	even	2
400.6.a.i.1.1		1	12.11	even	2
400.6.c.i.49.1		2	60.47	odd	4
400.6.c.i.49.2		2	60.23	odd	4
450.6.a.u.1.1		1	1.1	even	1	trivial
450.6.c.f.199.1		2	5.3	odd	4
450.6.c.f.199.2		2	5.2	odd	4
490.6.a.k.1.1		1	105.104	even	2
720.6.a.v.1.1		1	20.19	odd	2

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:	\( N \)	\(=\)	\( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	450.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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