LMFDB - Embedded newform 135.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,6,Mod(1,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, 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("135.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	135.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:	$21.6518156748$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	135.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+10.0000 q^{2} +68.0000 q^{4} -25.0000 q^{5} +6.00000 q^{7} +360.000 q^{8} +O(q^{10})$

\(q+10.0000 q^{2} +68.0000 q^{4} -25.0000 q^{5} +6.00000 q^{7} +360.000 q^{8} -250.000 q^{10} +685.000 q^{11} +685.000 q^{13} +60.0000 q^{14} +1424.00 q^{16} +1045.00 q^{17} +1108.00 q^{19} -1700.00 q^{20} +6850.00 q^{22} -3855.00 q^{23} +625.000 q^{25} +6850.00 q^{26} +408.000 q^{28} -1315.00 q^{29} -9909.00 q^{31} +2720.00 q^{32} +10450.0 q^{34} -150.000 q^{35} -6826.00 q^{37} +11080.0 q^{38} -9000.00 q^{40} +4520.00 q^{41} +9097.00 q^{43} +46580.0 q^{44} -38550.0 q^{46} +2095.00 q^{47} -16771.0 q^{49} +6250.00 q^{50} +46580.0 q^{52} +10060.0 q^{53} -17125.0 q^{55} +2160.00 q^{56} -13150.0 q^{58} +24820.0 q^{59} -46286.0 q^{61} -99090.0 q^{62} -18368.0 q^{64} -17125.0 q^{65} +13860.0 q^{67} +71060.0 q^{68} -1500.00 q^{70} -75580.0 q^{71} -32738.0 q^{73} -68260.0 q^{74} +75344.0 q^{76} +4110.00 q^{77} +74877.0 q^{79} -35600.0 q^{80} +45200.0 q^{82} +93930.0 q^{83} -26125.0 q^{85} +90970.0 q^{86} +246600. q^{88} +123540. q^{89} +4110.00 q^{91} -262140. q^{92} +20950.0 q^{94} -27700.0 q^{95} -85966.0 q^{97} -167710. q^{98} +O(q^{100})\)

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 $

$2$	10.0000	1.76777	0.883883	−	0.467707i	$-0.154920\pi$
$2$	10.0000	1.76777	0.883883	+	0.467707i	$0.154920\pi$
$3$	0	0
$3$	0	0
$4$	68.0000	2.12500
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	6.00000	0.0462814	0.0231407	−	0.999732i	$-0.492633\pi$
$7$	6.00000	0.0462814	0.0231407	+	0.999732i	$0.492633\pi$
$8$	360.000	1.98874
$9$	0	0
$10$	−250.000	−0.790569
$11$	685.000	1.70690	0.853452	−	0.521172i	$-0.174505\pi$
$11$	685.000	1.70690	0.853452	+	0.521172i	$0.174505\pi$
$12$	0	0
$13$	685.000	1.12417	0.562085	−	0.827079i	$-0.309999\pi$
$13$	685.000	1.12417	0.562085	+	0.827079i	$0.309999\pi$
$14$	60.0000	0.0818147
$15$	0	0
$16$	1424.00	1.39062
$17$	1045.00	0.876989	0.438494	−	0.898734i	$-0.355512\pi$
$17$	1045.00	0.876989	0.438494	+	0.898734i	$0.355512\pi$
$18$	0	0
$19$	1108.00	0.704135	0.352067	−	0.935975i	$-0.385479\pi$
$19$	1108.00	0.704135	0.352067	+	0.935975i	$0.385479\pi$
$20$	−1700.00	−0.950329
$21$	0	0
$22$	6850.00	3.01741
$23$	−3855.00	−1.51951	−0.759757	−	0.650207i	$-0.774682\pi$
$23$	−3855.00	−1.51951	−0.759757	+	0.650207i	$0.774682\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	6850.00	1.98727
$27$	0	0
$28$	408.000	0.0983479
$29$	−1315.00	−0.290356	−0.145178	−	0.989406i	$-0.546375\pi$
$29$	−1315.00	−0.290356	−0.145178	+	0.989406i	$0.546375\pi$
$30$	0	0
$31$	−9909.00	−1.85193	−0.925967	−	0.377604i	$-0.876748\pi$
$31$	−9909.00	−1.85193	−0.925967	+	0.377604i	$0.876748\pi$
$32$	2720.00	0.469563
$33$	0	0
$34$	10450.0	1.55031
$35$	−150.000	−0.0206977
$36$	0	0
$37$	−6826.00	−0.819713	−0.409857	−	0.912150i	$-0.634421\pi$
$37$	−6826.00	−0.819713	−0.409857	+	0.912150i	$0.634421\pi$
$38$	11080.0	1.24475
$39$	0	0
$40$	−9000.00	−0.889391
$41$	4520.00	0.419932	0.209966	−	0.977709i	$-0.432665\pi$
$41$	4520.00	0.419932	0.209966	+	0.977709i	$0.432665\pi$
$42$	0	0
$43$	9097.00	0.750286	0.375143	−	0.926967i	$-0.377594\pi$
$43$	9097.00	0.750286	0.375143	+	0.926967i	$0.377594\pi$
$44$	46580.0	3.62717
$45$	0	0
$46$	−38550.0	−2.68615
$47$	2095.00	0.138337	0.0691687	−	0.997605i	$-0.477965\pi$
$47$	2095.00	0.138337	0.0691687	+	0.997605i	$0.477965\pi$
$48$	0	0
$49$	−16771.0	−0.997858
$50$	6250.00	0.353553
$51$	0	0
$52$	46580.0	2.38886
$53$	10060.0	0.491936	0.245968	−	0.969278i	$-0.420894\pi$
$53$	10060.0	0.491936	0.245968	+	0.969278i	$0.420894\pi$
$54$	0	0
$55$	−17125.0	−0.763350
$56$	2160.00	0.0920415
$57$	0	0
$58$	−13150.0	−0.513282
$59$	24820.0	0.928265	0.464132	−	0.885766i	$-0.346366\pi$
$59$	24820.0	0.928265	0.464132	+	0.885766i	$0.346366\pi$
$60$	0	0
$61$	−46286.0	−1.59267	−0.796334	−	0.604858i	$-0.793230\pi$
$61$	−46286.0	−1.59267	−0.796334	+	0.604858i	$0.793230\pi$
$62$	−99090.0	−3.27379
$63$	0	0
$64$	−18368.0	−0.560547
$65$	−17125.0	−0.502744
$66$	0	0
$67$	13860.0	0.377204	0.188602	−	0.982054i	$-0.439604\pi$
$67$	13860.0	0.377204	0.188602	+	0.982054i	$0.439604\pi$
$68$	71060.0	1.86360
$69$	0	0
$70$	−1500.00	−0.0365886
$71$	−75580.0	−1.77935	−0.889674	−	0.456596i	$-0.849069\pi$
$71$	−75580.0	−1.77935	−0.889674	+	0.456596i	$0.849069\pi$
$72$	0	0
$73$	−32738.0	−0.719027	−0.359513	−	0.933140i	$-0.617057\pi$
$73$	−32738.0	−0.719027	−0.359513	+	0.933140i	$0.617057\pi$
$74$	−68260.0	−1.44906
$75$	0	0
$76$	75344.0	1.49629
$77$	4110.00	0.0789978
$78$	0	0
$79$	74877.0	1.34984	0.674918	−	0.737893i	$-0.264179\pi$
$79$	74877.0	1.34984	0.674918	+	0.737893i	$0.264179\pi$
$80$	−35600.0	−0.621906
$81$	0	0
$82$	45200.0	0.742342
$83$	93930.0	1.49661	0.748306	−	0.663354i	$-0.230868\pi$
$83$	93930.0	1.49661	0.748306	+	0.663354i	$0.230868\pi$
$84$	0	0
$85$	−26125.0	−0.392201
$86$	90970.0	1.32633
$87$	0	0
$88$	246600.	3.39458
$89$	123540.	1.65323	0.826614	−	0.562770i	$-0.190264\pi$
$89$	123540.	1.65323	0.826614	+	0.562770i	$0.190264\pi$
$90$	0	0
$91$	4110.00	0.0520281
$92$	−262140.	−3.22897
$93$	0	0
$94$	20950.0	0.244548
$95$	−27700.0	−0.314899
$96$	0	0
$97$	−85966.0	−0.927678	−0.463839	−	0.885919i	$-0.653528\pi$
$97$	−85966.0	−0.927678	−0.463839	+	0.885919i	$0.653528\pi$
$98$	−167710.	−1.76398
$99$	0	0
$100$	42500.0	0.425000
$101$	62415.0	0.608815	0.304408	−	0.952542i	$-0.401542\pi$
$101$	62415.0	0.608815	0.304408	+	0.952542i	$0.401542\pi$
$102$	0	0
$103$	−169522.	−1.57446	−0.787232	−	0.616656i	$-0.788487\pi$
$103$	−169522.	−1.57446	−0.787232	+	0.616656i	$0.788487\pi$
$104$	246600.	2.23568
$105$	0	0
$106$	100600.	0.869628
$107$	2850.00	0.0240650	0.0120325	−	0.999928i	$-0.496170\pi$
$107$	2850.00	0.0240650	0.0120325	+	0.999928i	$0.496170\pi$
$108$	0	0
$109$	−210628.	−1.69805	−0.849024	−	0.528355i	$-0.822809\pi$
$109$	−210628.	−1.69805	−0.849024	+	0.528355i	$0.822809\pi$
$110$	−171250.	−1.34943
$111$	0	0
$112$	8544.00	0.0643600
$113$	−81335.0	−0.599213	−0.299607	−	0.954063i	$-0.596855\pi$
$113$	−81335.0	−0.599213	−0.299607	+	0.954063i	$0.596855\pi$
$114$	0	0
$115$	96375.0	0.679547
$116$	−89420.0	−0.617006
$117$	0	0
$118$	248200.	1.64096
$119$	6270.00	0.0405882
$120$	0	0
$121$	308174.	1.91352
$122$	−462860.	−2.81546
$123$	0	0
$124$	−673812.	−3.93536
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−237410.	−1.30614	−0.653070	−	0.757298i	$-0.726519\pi$
$127$	−237410.	−1.30614	−0.653070	+	0.757298i	$0.726519\pi$
$128$	−270720.	−1.46048
$129$	0	0
$130$	−171250.	−0.888735
$131$	158115.	0.804998	0.402499	−	0.915420i	$-0.368142\pi$
$131$	158115.	0.804998	0.402499	+	0.915420i	$0.368142\pi$
$132$	0	0
$133$	6648.00	0.0325883
$134$	138600.	0.666809
$135$	0	0
$136$	376200.	1.74410
$137$	−195270.	−0.888862	−0.444431	−	0.895813i	$-0.646594\pi$
$137$	−195270.	−0.888862	−0.444431	+	0.895813i	$0.646594\pi$
$138$	0	0
$139$	13238.0	0.0581146	0.0290573	−	0.999578i	$-0.490749\pi$
$139$	13238.0	0.0581146	0.0290573	+	0.999578i	$0.490749\pi$
$140$	−10200.0	−0.0439825
$141$	0	0
$142$	−755800.	−3.14547
$143$	469225.	1.91885
$144$	0	0
$145$	32875.0	0.129851
$146$	−327380.	−1.27107
$147$	0	0
$148$	−464168.	−1.74189
$149$	242405.	0.894491	0.447245	−	0.894411i	$-0.352405\pi$
$149$	242405.	0.894491	0.447245	+	0.894411i	$0.352405\pi$
$150$	0	0
$151$	−446459.	−1.59345	−0.796726	−	0.604340i	$-0.793437\pi$
$151$	−446459.	−1.59345	−0.796726	+	0.604340i	$0.793437\pi$
$152$	398880.	1.40034
$153$	0	0
$154$	41100.0	0.139650
$155$	247725.	0.828210
$156$	0	0
$157$	332411.	1.07628	0.538141	−	0.842855i	$-0.319127\pi$
$157$	332411.	1.07628	0.538141	+	0.842855i	$0.319127\pi$
$158$	748770.	2.38619
$159$	0	0
$160$	−68000.0	−0.209995
$161$	−23130.0	−0.0703252
$162$	0	0
$163$	442925.	1.30575	0.652877	−	0.757464i	$-0.273562\pi$
$163$	442925.	1.30575	0.652877	+	0.757464i	$0.273562\pi$
$164$	307360.	0.892355
$165$	0	0
$166$	939300.	2.64566
$167$	−7140.00	−0.0198110	−0.00990551	−	0.999951i	$-0.503153\pi$
$167$	−7140.00	−0.0198110	−0.00990551	+	0.999951i	$0.503153\pi$
$168$	0	0
$169$	97932.0	0.263759
$170$	−261250.	−0.693320
$171$	0	0
$172$	618596.	1.59436
$173$	292650.	0.743418	0.371709	−	0.928349i	$-0.378772\pi$
$173$	292650.	0.743418	0.371709	+	0.928349i	$0.378772\pi$
$174$	0	0
$175$	3750.00	0.00925627
$176$	975440.	2.37366
$177$	0	0
$178$	1.23540e6	2.92252
$179$	−609980.	−1.42293	−0.711464	−	0.702722i	$-0.751968\pi$
$179$	−609980.	−1.42293	−0.711464	+	0.702722i	$0.751968\pi$
$180$	0	0
$181$	−79852.0	−0.181171	−0.0905856	−	0.995889i	$-0.528874\pi$
$181$	−79852.0	−0.181171	−0.0905856	+	0.995889i	$0.528874\pi$
$182$	41100.0	0.0919736
$183$	0	0
$184$	−1.38780e6	−3.02192
$185$	170650.	0.366587
$186$	0	0
$187$	715825.	1.49693
$188$	142460.	0.293967
$189$	0	0
$190$	−277000.	−0.556667
$191$	−150910.	−0.299319	−0.149660	−	0.988738i	$-0.547818\pi$
$191$	−150910.	−0.299319	−0.149660	+	0.988738i	$0.547818\pi$
$192$	0	0
$193$	170702.	0.329872	0.164936	−	0.986304i	$-0.447258\pi$
$193$	170702.	0.329872	0.164936	+	0.986304i	$0.447258\pi$
$194$	−859660.	−1.63992
$195$	0	0
$196$	−1.14043e6	−2.12045
$197$	333360.	0.611995	0.305998	−	0.952032i	$-0.401010\pi$
$197$	333360.	0.611995	0.305998	+	0.952032i	$0.401010\pi$
$198$	0	0
$199$	57511.0	0.102948	0.0514740	−	0.998674i	$-0.483608\pi$
$199$	57511.0	0.102948	0.0514740	+	0.998674i	$0.483608\pi$
$200$	225000.	0.397748
$201$	0	0
$202$	624150.	1.07624
$203$	−7890.00	−0.0134381
$204$	0	0
$205$	−113000.	−0.187799
$206$	−1.69522e6	−2.78329
$207$	0	0
$208$	975440.	1.56330
$209$	758980.	1.20189
$210$	0	0
$211$	428102.	0.661974	0.330987	−	0.943635i	$-0.392618\pi$
$211$	428102.	0.661974	0.330987	+	0.943635i	$0.392618\pi$
$212$	684080.	1.04536
$213$	0	0
$214$	28500.0	0.0425413
$215$	−227425.	−0.335538
$216$	0	0
$217$	−59454.0	−0.0857100
$218$	−2.10628e6	−3.00175
$219$	0	0
$220$	−1.16450e6	−1.62212
$221$	715825.	0.985885
$222$	0	0
$223$	699368.	0.941767	0.470884	−	0.882195i	$-0.343935\pi$
$223$	699368.	0.941767	0.470884	+	0.882195i	$0.343935\pi$
$224$	16320.0	0.0217320
$225$	0	0
$226$	−813350.	−1.05927
$227$	471200.	0.606933	0.303466	−	0.952842i	$-0.401856\pi$
$227$	471200.	0.606933	0.303466	+	0.952842i	$0.401856\pi$
$228$	0	0
$229$	1.44071e6	1.81546	0.907731	−	0.419552i	$-0.137813\pi$
$229$	1.44071e6	1.81546	0.907731	+	0.419552i	$0.137813\pi$
$230$	963750.	1.20128
$231$	0	0
$232$	−473400.	−0.577442
$233$	−294090.	−0.354887	−0.177444	−	0.984131i	$-0.556783\pi$
$233$	−294090.	−0.354887	−0.177444	+	0.984131i	$0.556783\pi$
$234$	0	0
$235$	−52375.0	−0.0618663
$236$	1.68776e6	1.97256
$237$	0	0
$238$	62700.0	0.0717505
$239$	−833330.	−0.943675	−0.471837	−	0.881686i	$-0.656409\pi$
$239$	−833330.	−0.943675	−0.471837	+	0.881686i	$0.656409\pi$
$240$	0	0
$241$	436477.	0.484082	0.242041	−	0.970266i	$-0.422183\pi$
$241$	436477.	0.484082	0.242041	+	0.970266i	$0.422183\pi$
$242$	3.08174e6	3.38265
$243$	0	0
$244$	−3.14745e6	−3.38442
$245$	419275.	0.446256
$246$	0	0
$247$	758980.	0.791567
$248$	−3.56724e6	−3.68301
$249$	0	0
$250$	−156250.	−0.158114
$251$	−676695.	−0.677967	−0.338984	−	0.940792i	$-0.610083\pi$
$251$	−676695.	−0.677967	−0.338984	+	0.940792i	$0.610083\pi$
$252$	0	0
$253$	−2.64068e6	−2.59366
$254$	−2.37410e6	−2.30895
$255$	0	0
$256$	−2.11942e6	−2.02124
$257$	−669735.	−0.632514	−0.316257	−	0.948674i	$-0.602426\pi$
$257$	−669735.	−0.632514	−0.316257	+	0.948674i	$0.602426\pi$
$258$	0	0
$259$	−40956.0	−0.0379374
$260$	−1.16450e6	−1.06833
$261$	0	0
$262$	1.58115e6	1.42305
$263$	1.92650e6	1.71743	0.858716	−	0.512451i	$-0.171263\pi$
$263$	1.92650e6	1.71743	0.858716	+	0.512451i	$0.171263\pi$
$264$	0	0
$265$	−251500.	−0.220000
$266$	66480.0	0.0576085
$267$	0	0
$268$	942480.	0.801558
$269$	1.06518e6	0.897512	0.448756	−	0.893654i	$-0.351867\pi$
$269$	1.06518e6	0.897512	0.448756	+	0.893654i	$0.351867\pi$
$270$	0	0
$271$	−1.39806e6	−1.15638	−0.578191	−	0.815901i	$-0.696241\pi$
$271$	−1.39806e6	−1.15638	−0.578191	+	0.815901i	$0.696241\pi$
$272$	1.48808e6	1.21956
$273$	0	0
$274$	−1.95270e6	−1.57130
$275$	428125.	0.341381
$276$	0	0
$277$	−91866.0	−0.0719375	−0.0359688	−	0.999353i	$-0.511452\pi$
$277$	−91866.0	−0.0719375	−0.0359688	+	0.999353i	$0.511452\pi$
$278$	132380.	0.102733
$279$	0	0
$280$	−54000.0	−0.0411622
$281$	−1.81323e6	−1.36989	−0.684947	−	0.728593i	$-0.740175\pi$
$281$	−1.81323e6	−1.36989	−0.684947	+	0.728593i	$0.740175\pi$
$282$	0	0
$283$	1.24205e6	0.921876	0.460938	−	0.887432i	$-0.347513\pi$
$283$	1.24205e6	0.921876	0.460938	+	0.887432i	$0.347513\pi$
$284$	−5.13944e6	−3.78112
$285$	0	0
$286$	4.69225e6	3.39208
$287$	27120.0	0.0194350
$288$	0	0
$289$	−327832.	−0.230891
$290$	328750.	0.229547
$291$	0	0
$292$	−2.22618e6	−1.52793
$293$	453690.	0.308738	0.154369	−	0.988013i	$-0.450666\pi$
$293$	453690.	0.308738	0.154369	+	0.988013i	$0.450666\pi$
$294$	0	0
$295$	−620500.	−0.415133
$296$	−2.45736e6	−1.63019
$297$	0	0
$298$	2.42405e6	1.58125
$299$	−2.64068e6	−1.70819
$300$	0	0
$301$	54582.0	0.0347243
$302$	−4.46459e6	−2.81685
$303$	0	0
$304$	1.57779e6	0.979187
$305$	1.15715e6	0.712262
$306$	0	0
$307$	−2.37090e6	−1.43571	−0.717856	−	0.696192i	$-0.754876\pi$
$307$	−2.37090e6	−1.43571	−0.717856	+	0.696192i	$0.754876\pi$
$308$	279480.	0.167870
$309$	0	0
$310$	2.47725e6	1.46408
$311$	−981600.	−0.575484	−0.287742	−	0.957708i	$-0.592905\pi$
$311$	−981600.	−0.575484	−0.287742	+	0.957708i	$0.592905\pi$
$312$	0	0
$313$	1.16432e6	0.671756	0.335878	−	0.941906i	$-0.390967\pi$
$313$	1.16432e6	0.671756	0.335878	+	0.941906i	$0.390967\pi$
$314$	3.32411e6	1.90262
$315$	0	0
$316$	5.09164e6	2.86840
$317$	855800.	0.478326	0.239163	−	0.970979i	$-0.423127\pi$
$317$	855800.	0.478326	0.239163	+	0.970979i	$0.423127\pi$
$318$	0	0
$319$	−900775.	−0.495609
$320$	459200.	0.250684
$321$	0	0
$322$	−231300.	−0.124319
$323$	1.15786e6	0.617518
$324$	0	0
$325$	428125.	0.224834
$326$	4.42925e6	2.30827
$327$	0	0
$328$	1.62720e6	0.835134
$329$	12570.0	0.00640244
$330$	0	0
$331$	65994.0	0.0331081	0.0165541	−	0.999863i	$-0.494730\pi$
$331$	65994.0	0.0331081	0.0165541	+	0.999863i	$0.494730\pi$
$332$	6.38724e6	3.18030
$333$	0	0
$334$	−71400.0	−0.0350213
$335$	−346500.	−0.168691
$336$	0	0
$337$	−820040.	−0.393333	−0.196666	−	0.980470i	$-0.563012\pi$
$337$	−820040.	−0.393333	−0.196666	+	0.980470i	$0.563012\pi$
$338$	979320.	0.466265
$339$	0	0
$340$	−1.77650e6	−0.833428
$341$	−6.78766e6	−3.16107
$342$	0	0
$343$	−201468.	−0.0924636
$344$	3.27492e6	1.49212
$345$	0	0
$346$	2.92650e6	1.31419
$347$	−851960.	−0.379835	−0.189918	−	0.981800i	$-0.560822\pi$
$347$	−851960.	−0.379835	−0.189918	+	0.981800i	$0.560822\pi$
$348$	0	0
$349$	−1.11612e6	−0.490510	−0.245255	−	0.969459i	$-0.578872\pi$
$349$	−1.11612e6	−0.490510	−0.245255	+	0.969459i	$0.578872\pi$
$350$	37500.0	0.0163629
$351$	0	0
$352$	1.86320e6	0.801499
$353$	4.18808e6	1.78887	0.894433	−	0.447202i	$-0.147580\pi$
$353$	4.18808e6	1.78887	0.894433	+	0.447202i	$0.147580\pi$
$354$	0	0
$355$	1.88950e6	0.795749
$356$	8.40072e6	3.51311
$357$	0	0
$358$	−6.09980e6	−2.51541
$359$	−1.37034e6	−0.561167	−0.280584	−	0.959830i	$-0.590528\pi$
$359$	−1.37034e6	−0.561167	−0.280584	+	0.959830i	$0.590528\pi$
$360$	0	0
$361$	−1.24843e6	−0.504194
$362$	−798520.	−0.320269
$363$	0	0
$364$	279480.	0.110560
$365$	818450.	0.321558
$366$	0	0
$367$	−1.42773e6	−0.553326	−0.276663	−	0.960967i	$-0.589229\pi$
$367$	−1.42773e6	−0.553326	−0.276663	+	0.960967i	$0.589229\pi$
$368$	−5.48952e6	−2.11307
$369$	0	0
$370$	1.70650e6	0.648040
$371$	60360.0	0.0227675
$372$	0	0
$373$	821623.	0.305774	0.152887	−	0.988244i	$-0.451143\pi$
$373$	821623.	0.305774	0.152887	+	0.988244i	$0.451143\pi$
$374$	7.15825e6	2.64623
$375$	0	0
$376$	754200.	0.275117
$377$	−900775.	−0.326410
$378$	0	0
$379$	−609536.	−0.217972	−0.108986	−	0.994043i	$-0.534760\pi$
$379$	−609536.	−0.217972	−0.108986	+	0.994043i	$0.534760\pi$
$380$	−1.88360e6	−0.669160
$381$	0	0
$382$	−1.50910e6	−0.529127
$383$	−1.08650e6	−0.378469	−0.189235	−	0.981932i	$-0.560601\pi$
$383$	−1.08650e6	−0.378469	−0.189235	+	0.981932i	$0.560601\pi$
$384$	0	0
$385$	−102750.	−0.0353289
$386$	1.70702e6	0.583137
$387$	0	0
$388$	−5.84569e6	−1.97132
$389$	2.26594e6	0.759233	0.379617	−	0.925144i	$-0.376056\pi$
$389$	2.26594e6	0.759233	0.379617	+	0.925144i	$0.376056\pi$
$390$	0	0
$391$	−4.02847e6	−1.33260
$392$	−6.03756e6	−1.98448
$393$	0	0
$394$	3.33360e6	1.08186
$395$	−1.87192e6	−0.603665
$396$	0	0
$397$	3.85253e6	1.22679	0.613394	−	0.789777i	$-0.289804\pi$
$397$	3.85253e6	1.22679	0.613394	+	0.789777i	$0.289804\pi$
$398$	575110.	0.181988
$399$	0	0
$400$	890000.	0.278125
$401$	−293700.	−0.0912101	−0.0456051	−	0.998960i	$-0.514522\pi$
$401$	−293700.	−0.0912101	−0.0456051	+	0.998960i	$0.514522\pi$
$402$	0	0
$403$	−6.78766e6	−2.08189
$404$	4.24422e6	1.29373
$405$	0	0
$406$	−78900.0	−0.0237554
$407$	−4.67581e6	−1.39917
$408$	0	0
$409$	−1.54596e6	−0.456973	−0.228486	−	0.973547i	$-0.573378\pi$
$409$	−1.54596e6	−0.456973	−0.228486	+	0.973547i	$0.573378\pi$
$410$	−1.13000e6	−0.331985
$411$	0	0
$412$	−1.15275e7	−3.34574
$413$	148920.	0.0429613
$414$	0	0
$415$	−2.34825e6	−0.669305
$416$	1.86320e6	0.527869
$417$	0	0
$418$	7.58980e6	2.12466
$419$	274355.	0.0763445	0.0381723	−	0.999271i	$-0.487846\pi$
$419$	274355.	0.0763445	0.0381723	+	0.999271i	$0.487846\pi$
$420$	0	0
$421$	122836.	0.0337769	0.0168885	−	0.999857i	$-0.494624\pi$
$421$	122836.	0.0337769	0.0168885	+	0.999857i	$0.494624\pi$
$422$	4.28102e6	1.17022
$423$	0	0
$424$	3.62160e6	0.978331
$425$	653125.	0.175398
$426$	0	0
$427$	−277716.	−0.0737108
$428$	193800.	0.0511381
$429$	0	0
$430$	−2.27425e6	−0.593153
$431$	−3.85980e6	−1.00086	−0.500428	−	0.865778i	$-0.666824\pi$
$431$	−3.85980e6	−1.00086	−0.500428	+	0.865778i	$0.666824\pi$
$432$	0	0
$433$	2.18754e6	0.560707	0.280353	−	0.959897i	$-0.409548\pi$
$433$	2.18754e6	0.560707	0.280353	+	0.959897i	$0.409548\pi$
$434$	−594540.	−0.151515
$435$	0	0
$436$	−1.43227e7	−3.60835
$437$	−4.27134e6	−1.06994
$438$	0	0
$439$	1.26527e6	0.313345	0.156672	−	0.987651i	$-0.449923\pi$
$439$	1.26527e6	0.313345	0.156672	+	0.987651i	$0.449923\pi$
$440$	−6.16500e6	−1.51810
$441$	0	0
$442$	7.15825e6	1.74281
$443$	672180.	0.162733	0.0813666	−	0.996684i	$-0.474072\pi$
$443$	672180.	0.162733	0.0813666	+	0.996684i	$0.474072\pi$
$444$	0	0
$445$	−3.08850e6	−0.739346
$446$	6.99368e6	1.66483
$447$	0	0
$448$	−110208.	−0.0259429
$449$	1.26013e6	0.294985	0.147492	−	0.989063i	$-0.452880\pi$
$449$	1.26013e6	0.294985	0.147492	+	0.989063i	$0.452880\pi$
$450$	0	0
$451$	3.09620e6	0.716783
$452$	−5.53078e6	−1.27333
$453$	0	0
$454$	4.71200e6	1.07292
$455$	−102750.	−0.0232677
$456$	0	0
$457$	6.81838e6	1.52718	0.763591	−	0.645700i	$-0.223435\pi$
$457$	6.81838e6	1.52718	0.763591	+	0.645700i	$0.223435\pi$
$458$	1.44071e7	3.20931
$459$	0	0
$460$	6.55350e6	1.44404
$461$	6.55773e6	1.43715	0.718574	−	0.695451i	$-0.244795\pi$
$461$	6.55773e6	1.43715	0.718574	+	0.695451i	$0.244795\pi$
$462$	0	0
$463$	3.15148e6	0.683223	0.341611	−	0.939841i	$-0.389027\pi$
$463$	3.15148e6	0.683223	0.341611	+	0.939841i	$0.389027\pi$
$464$	−1.87256e6	−0.403776
$465$	0	0
$466$	−2.94090e6	−0.627358
$467$	−7.80490e6	−1.65606	−0.828028	−	0.560686i	$-0.810537\pi$
$467$	−7.80490e6	−1.65606	−0.828028	+	0.560686i	$0.810537\pi$
$468$	0	0
$469$	83160.0	0.0174575
$470$	−523750.	−0.109365
$471$	0	0
$472$	8.93520e6	1.84607
$473$	6.23145e6	1.28067
$474$	0	0
$475$	692500.	0.140827
$476$	426360.	0.0862500
$477$	0	0
$478$	−8.33330e6	−1.66820
$479$	−729660.	−0.145305	−0.0726527	−	0.997357i	$-0.523146\pi$
$479$	−729660.	−0.145305	−0.0726527	+	0.997357i	$0.523146\pi$
$480$	0	0
$481$	−4.67581e6	−0.921498
$482$	4.36477e6	0.855744
$483$	0	0
$484$	2.09558e7	4.06623
$485$	2.14915e6	0.414870
$486$	0	0
$487$	9.35510e6	1.78742	0.893709	−	0.448647i	$-0.148094\pi$
$487$	9.35510e6	1.78742	0.893709	+	0.448647i	$0.148094\pi$
$488$	−1.66630e7	−3.16740
$489$	0	0
$490$	4.19275e6	0.788876
$491$	−9.39394e6	−1.75851	−0.879253	−	0.476354i	$-0.841958\pi$
$491$	−9.39394e6	−1.75851	−0.879253	+	0.476354i	$0.841958\pi$
$492$	0	0
$493$	−1.37418e6	−0.254639
$494$	7.58980e6	1.39931
$495$	0	0
$496$	−1.41104e7	−2.57535
$497$	−453480.	−0.0823507
$498$	0	0
$499$	6.54473e6	1.17663	0.588316	−	0.808631i	$-0.299791\pi$
$499$	6.54473e6	1.17663	0.588316	+	0.808631i	$0.299791\pi$
$500$	−1.06250e6	−0.190066
$501$	0	0
$502$	−6.76695e6	−1.19849
$503$	6.68478e6	1.17806	0.589030	−	0.808111i	$-0.299510\pi$
$503$	6.68478e6	1.17806	0.589030	+	0.808111i	$0.299510\pi$
$504$	0	0
$505$	−1.56037e6	−0.272270
$506$	−2.64068e7	−4.58499
$507$	0	0
$508$	−1.61439e7	−2.77555
$509$	−3.61198e6	−0.617946	−0.308973	−	0.951071i	$-0.599985\pi$
$509$	−3.61198e6	−0.617946	−0.308973	+	0.951071i	$0.599985\pi$
$510$	0	0
$511$	−196428.	−0.0332775
$512$	−1.25312e7	−2.11260
$513$	0	0
$514$	−6.69735e6	−1.11814
$515$	4.23805e6	0.704122
$516$	0	0
$517$	1.43508e6	0.236128
$518$	−409560.	−0.0670646
$519$	0	0
$520$	−6.16500e6	−0.999827
$521$	−572740.	−0.0924407	−0.0462203	−	0.998931i	$-0.514718\pi$
$521$	−572740.	−0.0924407	−0.0462203	+	0.998931i	$0.514718\pi$
$522$	0	0
$523$	5.40826e6	0.864577	0.432288	−	0.901735i	$-0.357706\pi$
$523$	5.40826e6	0.864577	0.432288	+	0.901735i	$0.357706\pi$
$524$	1.07518e7	1.71062
$525$	0	0
$526$	1.92650e7	3.03602
$527$	−1.03549e7	−1.62413
$528$	0	0
$529$	8.42468e6	1.30892
$530$	−2.51500e6	−0.388909
$531$	0	0
$532$	452064.	0.0692502
$533$	3.09620e6	0.472075
$534$	0	0
$535$	−71250.0	−0.0107622
$536$	4.98960e6	0.750160
$537$	0	0
$538$	1.06518e7	1.58659
$539$	−1.14881e7	−1.70325
$540$	0	0
$541$	−7.26253e6	−1.06683	−0.533414	−	0.845854i	$-0.679091\pi$
$541$	−7.26253e6	−1.06683	−0.533414	+	0.845854i	$0.679091\pi$
$542$	−1.39806e7	−2.04421
$543$	0	0
$544$	2.84240e6	0.411802
$545$	5.26570e6	0.759390
$546$	0	0
$547$	30901.0	0.00441575	0.00220787	−	0.999998i	$-0.499297\pi$
$547$	30901.0	0.00441575	0.00220787	+	0.999998i	$0.499297\pi$
$548$	−1.32784e7	−1.88883
$549$	0	0
$550$	4.28125e6	0.603481
$551$	−1.45702e6	−0.204450
$552$	0	0
$553$	449262.	0.0624722
$554$	−918660.	−0.127169
$555$	0	0
$556$	900184.	0.123494
$557$	6.29634e6	0.859904	0.429952	−	0.902852i	$-0.358530\pi$
$557$	6.29634e6	0.859904	0.429952	+	0.902852i	$0.358530\pi$
$558$	0	0
$559$	6.23144e6	0.843450
$560$	−213600.	−0.0287827
$561$	0	0
$562$	−1.81323e7	−2.42165
$563$	1.23597e6	0.164338	0.0821688	−	0.996618i	$-0.473815\pi$
$563$	1.23597e6	0.164338	0.0821688	+	0.996618i	$0.473815\pi$
$564$	0	0
$565$	2.03338e6	0.267976
$566$	1.24205e7	1.62966
$567$	0	0
$568$	−2.72088e7	−3.53866
$569$	6.30000e6	0.815755	0.407878	−	0.913037i	$-0.366269\pi$
$569$	6.30000e6	0.815755	0.407878	+	0.913037i	$0.366269\pi$
$570$	0	0
$571$	−2.12369e6	−0.272584	−0.136292	−	0.990669i	$-0.543519\pi$
$571$	−2.12369e6	−0.272584	−0.136292	+	0.990669i	$0.543519\pi$
$572$	3.19073e7	4.07756
$573$	0	0
$574$	271200.	0.0343566
$575$	−2.40938e6	−0.303903
$576$	0	0
$577$	−7.58210e6	−0.948090	−0.474045	−	0.880501i	$-0.657207\pi$
$577$	−7.58210e6	−0.948090	−0.474045	+	0.880501i	$0.657207\pi$
$578$	−3.27832e6	−0.408161
$579$	0	0
$580$	2.23550e6	0.275934
$581$	563580.	0.0692652
$582$	0	0
$583$	6.89110e6	0.839686
$584$	−1.17857e7	−1.42996
$585$	0	0
$586$	4.53690e6	0.545777
$587$	1.24694e7	1.49366	0.746828	−	0.665017i	$-0.231576\pi$
$587$	1.24694e7	1.49366	0.746828	+	0.665017i	$0.231576\pi$
$588$	0	0
$589$	−1.09792e7	−1.30401
$590$	−6.20500e6	−0.733858
$591$	0	0
$592$	−9.72022e6	−1.13991
$593$	−2.94156e6	−0.343511	−0.171755	−	0.985140i	$-0.554944\pi$
$593$	−2.94156e6	−0.343511	−0.171755	+	0.985140i	$0.554944\pi$
$594$	0	0
$595$	−156750.	−0.0181516
$596$	1.64835e7	1.90079
$597$	0	0
$598$	−2.64067e7	−3.01969
$599$	1.58779e7	1.80812	0.904058	−	0.427410i	$-0.140574\pi$
$599$	1.58779e7	1.80812	0.904058	+	0.427410i	$0.140574\pi$
$600$	0	0
$601$	−4.59290e6	−0.518681	−0.259341	−	0.965786i	$-0.583505\pi$
$601$	−4.59290e6	−0.518681	−0.259341	+	0.965786i	$0.583505\pi$
$602$	545820.	0.0613844
$603$	0	0
$604$	−3.03592e7	−3.38609
$605$	−7.70435e6	−0.855751
$606$	0	0
$607$	5.59659e6	0.616527	0.308263	−	0.951301i	$-0.400252\pi$
$607$	5.59659e6	0.616527	0.308263	+	0.951301i	$0.400252\pi$
$608$	3.01376e6	0.330636
$609$	0	0
$610$	1.15715e7	1.25911
$611$	1.43508e6	0.155515
$612$	0	0
$613$	1.01497e7	1.09095	0.545473	−	0.838128i	$-0.316350\pi$
$613$	1.01497e7	1.09095	0.545473	+	0.838128i	$0.316350\pi$
$614$	−2.37090e7	−2.53800
$615$	0	0
$616$	1.47960e6	0.157106
$617$	1.53162e7	1.61971	0.809854	−	0.586631i	$-0.199546\pi$
$617$	1.53162e7	1.61971	0.809854	+	0.586631i	$0.199546\pi$
$618$	0	0
$619$	−5.20008e6	−0.545486	−0.272743	−	0.962087i	$-0.587931\pi$
$619$	−5.20008e6	−0.545486	−0.272743	+	0.962087i	$0.587931\pi$
$620$	1.68453e7	1.75995
$621$	0	0
$622$	−9.81600e6	−1.01732
$623$	741240.	0.0765136
$624$	0	0
$625$	390625.	0.0400000
$626$	1.16432e7	1.18751
$627$	0	0
$628$	2.26039e7	2.28710
$629$	−7.13317e6	−0.718879
$630$	0	0
$631$	7.34675e6	0.734551	0.367276	−	0.930112i	$-0.380291\pi$
$631$	7.34675e6	0.734551	0.367276	+	0.930112i	$0.380291\pi$
$632$	2.69557e7	2.68447
$633$	0	0
$634$	8.55800e6	0.845569
$635$	5.93525e6	0.584124
$636$	0	0
$637$	−1.14881e7	−1.12176
$638$	−9.00775e6	−0.876122
$639$	0	0
$640$	6.76800e6	0.653146
$641$	−1.17297e6	−0.112757	−0.0563783	−	0.998409i	$-0.517955\pi$
$641$	−1.17297e6	−0.112757	−0.0563783	+	0.998409i	$0.517955\pi$
$642$	0	0
$643$	−1.92135e7	−1.83264	−0.916322	−	0.400443i	$-0.868856\pi$
$643$	−1.92135e7	−1.83264	−0.916322	+	0.400443i	$0.868856\pi$
$644$	−1.57284e6	−0.149441
$645$	0	0
$646$	1.15786e7	1.09163
$647$	−8.64470e6	−0.811875	−0.405938	−	0.913901i	$-0.633055\pi$
$647$	−8.64470e6	−0.811875	−0.405938	+	0.913901i	$0.633055\pi$
$648$	0	0
$649$	1.70017e7	1.58446
$650$	4.28125e6	0.397454
$651$	0	0
$652$	3.01189e7	2.77473
$653$	4.50206e6	0.413170	0.206585	−	0.978429i	$-0.433765\pi$
$653$	4.50206e6	0.413170	0.206585	+	0.978429i	$0.433765\pi$
$654$	0	0
$655$	−3.95288e6	−0.360006
$656$	6.43648e6	0.583968
$657$	0	0
$658$	125700.	0.0113180
$659$	−1.51573e7	−1.35959	−0.679795	−	0.733403i	$-0.737931\pi$
$659$	−1.51573e7	−1.35959	−0.679795	+	0.733403i	$0.737931\pi$
$660$	0	0
$661$	1.03752e7	0.923618	0.461809	−	0.886979i	$-0.347201\pi$
$661$	1.03752e7	0.923618	0.461809	+	0.886979i	$0.347201\pi$
$662$	659940.	0.0585274
$663$	0	0
$664$	3.38148e7	2.97637
$665$	−166200.	−0.0145739
$666$	0	0
$667$	5.06932e6	0.441200
$668$	−485520.	−0.0420984
$669$	0	0
$670$	−3.46500e6	−0.298206
$671$	−3.17059e7	−2.71853
$672$	0	0
$673$	1.49638e7	1.27352	0.636758	−	0.771064i	$-0.280275\pi$
$673$	1.49638e7	1.27352	0.636758	+	0.771064i	$0.280275\pi$
$674$	−8.20040e6	−0.695321
$675$	0	0
$676$	6.65938e6	0.560489
$677$	8.83758e6	0.741074	0.370537	−	0.928818i	$-0.379174\pi$
$677$	8.83758e6	0.741074	0.370537	+	0.928818i	$0.379174\pi$
$678$	0	0
$679$	−515796.	−0.0429342
$680$	−9.40500e6	−0.779985
$681$	0	0
$682$	−6.78766e7	−5.58804
$683$	−1.09012e7	−0.894172	−0.447086	−	0.894491i	$-0.647538\pi$
$683$	−1.09012e7	−0.894172	−0.447086	+	0.894491i	$0.647538\pi$
$684$	0	0
$685$	4.88175e6	0.397511
$686$	−2.01468e6	−0.163454
$687$	0	0
$688$	1.29541e7	1.04337
$689$	6.89110e6	0.553020
$690$	0	0
$691$	−9.02310e6	−0.718887	−0.359444	−	0.933167i	$-0.617033\pi$
$691$	−9.02310e6	−0.718887	−0.359444	+	0.933167i	$0.617033\pi$
$692$	1.99002e7	1.57976
$693$	0	0
$694$	−8.51960e6	−0.671461
$695$	−330950.	−0.0259896
$696$	0	0
$697$	4.72340e6	0.368275
$698$	−1.11612e7	−0.867108
$699$	0	0
$700$	255000.	0.0196696
$701$	2.15578e7	1.65695	0.828474	−	0.560028i	$-0.189210\pi$
$701$	2.15578e7	1.65695	0.828474	+	0.560028i	$0.189210\pi$
$702$	0	0
$703$	−7.56321e6	−0.577189
$704$	−1.25821e7	−0.956799
$705$	0	0
$706$	4.18808e7	3.16230
$707$	374490.	0.0281768
$708$	0	0
$709$	−1.69473e7	−1.26615	−0.633076	−	0.774090i	$-0.718208\pi$
$709$	−1.69473e7	−1.26615	−0.633076	+	0.774090i	$0.718208\pi$
$710$	1.88950e7	1.40670
$711$	0	0
$712$	4.44744e7	3.28784
$713$	3.81992e7	2.81404
$714$	0	0
$715$	−1.17306e7	−0.858136
$716$	−4.14786e7	−3.02372
$717$	0	0
$718$	−1.37034e7	−0.992013
$719$	2.05676e7	1.48375	0.741874	−	0.670539i	$-0.233937\pi$
$719$	2.05676e7	1.48375	0.741874	+	0.670539i	$0.233937\pi$
$720$	0	0
$721$	−1.01713e6	−0.0728684
$722$	−1.24844e7	−0.891298
$723$	0	0
$724$	−5.42994e6	−0.384989
$725$	−821875.	−0.0580712
$726$	0	0
$727$	1.66714e7	1.16986	0.584931	−	0.811083i	$-0.301121\pi$
$727$	1.66714e7	1.16986	0.584931	+	0.811083i	$0.301121\pi$
$728$	1.47960e6	0.103470
$729$	0	0
$730$	8.18450e6	0.568440
$731$	9.50636e6	0.657993
$732$	0	0
$733$	−1.32270e7	−0.909284	−0.454642	−	0.890674i	$-0.650233\pi$
$733$	−1.32270e7	−0.909284	−0.454642	+	0.890674i	$0.650233\pi$
$734$	−1.42773e7	−0.978151
$735$	0	0
$736$	−1.04856e7	−0.713508
$737$	9.49410e6	0.643851
$738$	0	0
$739$	−9.00924e6	−0.606844	−0.303422	−	0.952856i	$-0.598129\pi$
$739$	−9.00924e6	−0.606844	−0.303422	+	0.952856i	$0.598129\pi$
$740$	1.16042e7	0.778997
$741$	0	0
$742$	603600.	0.0402476
$743$	−2.44168e7	−1.62262	−0.811309	−	0.584618i	$-0.801244\pi$
$743$	−2.44168e7	−1.62262	−0.811309	+	0.584618i	$0.801244\pi$
$744$	0	0
$745$	−6.06012e6	−0.400028
$746$	8.21623e6	0.540537
$747$	0	0
$748$	4.86761e7	3.18099
$749$	17100.0	0.00111376
$750$	0	0
$751$	−2.35907e7	−1.52630	−0.763151	−	0.646220i	$-0.776349\pi$
$751$	−2.35907e7	−1.52630	−0.763151	+	0.646220i	$0.776349\pi$
$752$	2.98328e6	0.192375
$753$	0	0
$754$	−9.00775e6	−0.577016
$755$	1.11615e7	0.712614
$756$	0	0
$757$	2.76229e7	1.75198	0.875991	−	0.482328i	$-0.160209\pi$
$757$	2.76229e7	1.75198	0.875991	+	0.482328i	$0.160209\pi$
$758$	−6.09536e6	−0.385324
$759$	0	0
$760$	−9.97200e6	−0.626251
$761$	1.20776e7	0.755992	0.377996	−	0.925807i	$-0.376613\pi$
$761$	1.20776e7	0.755992	0.377996	+	0.925807i	$0.376613\pi$
$762$	0	0
$763$	−1.26377e6	−0.0785880
$764$	−1.02619e7	−0.636053
$765$	0	0
$766$	−1.08650e7	−0.669046
$767$	1.70017e7	1.04353
$768$	0	0
$769$	3.49831e6	0.213325	0.106663	−	0.994295i	$-0.465983\pi$
$769$	3.49831e6	0.213325	0.106663	+	0.994295i	$0.465983\pi$
$770$	−1.02750e6	−0.0624532
$771$	0	0
$772$	1.16077e7	0.700978
$773$	2.99115e6	0.180048	0.0900242	−	0.995940i	$-0.471306\pi$
$773$	2.99115e6	0.180048	0.0900242	+	0.995940i	$0.471306\pi$
$774$	0	0
$775$	−6.19312e6	−0.370387
$776$	−3.09478e7	−1.84491
$777$	0	0
$778$	2.26594e7	1.34215
$779$	5.00816e6	0.295689
$780$	0	0
$781$	−5.17723e7	−3.03718
$782$	−4.02848e7	−2.35572
$783$	0	0
$784$	−2.38819e7	−1.38765
$785$	−8.31028e6	−0.481328
$786$	0	0
$787$	−1.38920e7	−0.799520	−0.399760	−	0.916620i	$-0.630906\pi$
$787$	−1.38920e7	−0.799520	−0.399760	+	0.916620i	$0.630906\pi$
$788$	2.26685e7	1.30049
$789$	0	0
$790$	−1.87192e7	−1.06714
$791$	−488010.	−0.0277324
$792$	0	0
$793$	−3.17059e7	−1.79043
$794$	3.85253e7	2.16868
$795$	0	0
$796$	3.91075e6	0.218765
$797$	8.13974e6	0.453905	0.226952	−	0.973906i	$-0.427124\pi$
$797$	8.13974e6	0.453905	0.226952	+	0.973906i	$0.427124\pi$
$798$	0	0
$799$	2.18928e6	0.121320
$800$	1.70000e6	0.0939126
$801$	0	0
$802$	−2.93700e6	−0.161238
$803$	−2.24255e7	−1.22731
$804$	0	0
$805$	578250.	0.0314504
$806$	−6.78766e7	−3.68030
$807$	0	0
$808$	2.24694e7	1.21077
$809$	−2.68094e7	−1.44017	−0.720087	−	0.693883i	$-0.755898\pi$
$809$	−2.68094e7	−1.44017	−0.720087	+	0.693883i	$0.755898\pi$
$810$	0	0
$811$	−1.97282e7	−1.05326	−0.526629	−	0.850095i	$-0.676544\pi$
$811$	−1.97282e7	−1.05326	−0.526629	+	0.850095i	$0.676544\pi$
$812$	−536520.	−0.0285559
$813$	0	0
$814$	−4.67581e7	−2.47341
$815$	−1.10731e7	−0.583951
$816$	0	0
$817$	1.00795e7	0.528303
$818$	−1.54596e7	−0.807821
$819$	0	0
$820$	−7.68400e6	−0.399073
$821$	2.16090e7	1.11886	0.559432	−	0.828876i	$-0.311019\pi$
$821$	2.16090e7	1.11886	0.559432	+	0.828876i	$0.311019\pi$
$822$	0	0
$823$	6.99026e6	0.359744	0.179872	−	0.983690i	$-0.442432\pi$
$823$	6.99026e6	0.359744	0.179872	+	0.983690i	$0.442432\pi$
$824$	−6.10279e7	−3.13120
$825$	0	0
$826$	1.48920e6	0.0759457
$827$	−2.68441e7	−1.36485	−0.682425	−	0.730955i	$-0.739075\pi$
$827$	−2.68441e7	−1.36485	−0.682425	+	0.730955i	$0.739075\pi$
$828$	0	0
$829$	2.11268e7	1.06770	0.533848	−	0.845580i	$-0.320745\pi$
$829$	2.11268e7	1.06770	0.533848	+	0.845580i	$0.320745\pi$
$830$	−2.34825e7	−1.18318
$831$	0	0
$832$	−1.25821e7	−0.630150
$833$	−1.75257e7	−0.875110
$834$	0	0
$835$	178500.	0.00885976
$836$	5.16106e7	2.55402
$837$	0	0
$838$	2.74355e6	0.134959
$839$	−2.28583e7	−1.12108	−0.560542	−	0.828126i	$-0.689407\pi$
$839$	−2.28583e7	−1.12108	−0.560542	+	0.828126i	$0.689407\pi$
$840$	0	0
$841$	−1.87819e7	−0.915693
$842$	1.22836e6	0.0597098
$843$	0	0
$844$	2.91109e7	1.40670
$845$	−2.44830e6	−0.117957
$846$	0	0
$847$	1.84904e6	0.0885602
$848$	1.43254e7	0.684098
$849$	0	0
$850$	6.53125e6	0.310062
$851$	2.63142e7	1.24557
$852$	0	0
$853$	5.75945e6	0.271024	0.135512	−	0.990776i	$-0.456732\pi$
$853$	5.75945e6	0.271024	0.135512	+	0.990776i	$0.456732\pi$
$854$	−2.77716e6	−0.130304
$855$	0	0
$856$	1.02600e6	0.0478589
$857$	−1.43880e7	−0.669188	−0.334594	−	0.942362i	$-0.608599\pi$
$857$	−1.43880e7	−0.669188	−0.334594	+	0.942362i	$0.608599\pi$
$858$	0	0
$859$	3.28619e7	1.51953	0.759766	−	0.650197i	$-0.225314\pi$
$859$	3.28619e7	1.51953	0.759766	+	0.650197i	$0.225314\pi$
$860$	−1.54649e7	−0.713019
$861$	0	0
$862$	−3.85980e7	−1.76928
$863$	−4.34132e6	−0.198425	−0.0992123	−	0.995066i	$-0.531632\pi$
$863$	−4.34132e6	−0.198425	−0.0992123	+	0.995066i	$0.531632\pi$
$864$	0	0
$865$	−7.31625e6	−0.332467
$866$	2.18754e7	0.991199
$867$	0	0
$868$	−4.04287e6	−0.182134
$869$	5.12907e7	2.30404
$870$	0	0
$871$	9.49410e6	0.424042
$872$	−7.58261e7	−3.37697
$873$	0	0
$874$	−4.27134e7	−1.89141
$875$	−93750.0	−0.00413953
$876$	0	0
$877$	2.70727e7	1.18859	0.594296	−	0.804246i	$-0.297431\pi$
$877$	2.70727e7	1.18859	0.594296	+	0.804246i	$0.297431\pi$
$878$	1.26527e7	0.553921
$879$	0	0
$880$	−2.43860e7	−1.06153
$881$	4.98787e6	0.216509	0.108254	−	0.994123i	$-0.465474\pi$
$881$	4.98787e6	0.216509	0.108254	+	0.994123i	$0.465474\pi$
$882$	0	0
$883$	1.69054e7	0.729667	0.364833	−	0.931073i	$-0.381126\pi$
$883$	1.69054e7	0.729667	0.364833	+	0.931073i	$0.381126\pi$
$884$	4.86761e7	2.09501
$885$	0	0
$886$	6.72180e6	0.287674
$887$	3.06531e7	1.30817	0.654087	−	0.756419i	$-0.273053\pi$
$887$	3.06531e7	1.30817	0.654087	+	0.756419i	$0.273053\pi$
$888$	0	0
$889$	−1.42446e6	−0.0604499
$890$	−3.08850e7	−1.30699
$891$	0	0
$892$	4.75570e7	2.00126
$893$	2.32126e6	0.0974081
$894$	0	0
$895$	1.52495e7	0.636353
$896$	−1.62432e6	−0.0675930
$897$	0	0
$898$	1.26013e7	0.521464
$899$	1.30303e7	0.537720
$900$	0	0
$901$	1.05127e7	0.431422
$902$	3.09620e7	1.26711
$903$	0	0
$904$	−2.92806e7	−1.19168
$905$	1.99630e6	0.0810223
$906$	0	0
$907$	1.73344e7	0.699667	0.349833	−	0.936812i	$-0.386238\pi$
$907$	1.73344e7	0.699667	0.349833	+	0.936812i	$0.386238\pi$
$908$	3.20416e7	1.28973
$909$	0	0
$910$	−1.02750e6	−0.0411319
$911$	−9.85196e6	−0.393302	−0.196651	−	0.980474i	$-0.563007\pi$
$911$	−9.85196e6	−0.393302	−0.196651	+	0.980474i	$0.563007\pi$
$912$	0	0
$913$	6.43420e7	2.55457
$914$	6.81838e7	2.69970
$915$	0	0
$916$	9.79681e7	3.85786
$917$	948690.	0.0372564
$918$	0	0
$919$	−3.67691e7	−1.43613	−0.718066	−	0.695975i	$-0.754973\pi$
$919$	−3.67691e7	−1.43613	−0.718066	+	0.695975i	$0.754973\pi$
$920$	3.46950e7	1.35144
$921$	0	0
$922$	6.55773e7	2.54054
$923$	−5.17723e7	−2.00029
$924$	0	0
$925$	−4.26625e6	−0.163943
$926$	3.15148e7	1.20778
$927$	0	0
$928$	−3.57680e6	−0.136340
$929$	1.27151e7	0.483371	0.241685	−	0.970355i	$-0.422300\pi$
$929$	1.27151e7	0.483371	0.241685	+	0.970355i	$0.422300\pi$
$930$	0	0
$931$	−1.85823e7	−0.702626
$932$	−1.99981e7	−0.754136
$933$	0	0
$934$	−7.80490e7	−2.92752
$935$	−1.78956e7	−0.669450
$936$	0	0
$937$	−5.26250e7	−1.95814	−0.979068	−	0.203533i	$-0.934758\pi$
$937$	−5.26250e7	−1.95814	−0.979068	+	0.203533i	$0.934758\pi$
$938$	831600.	0.0308608
$939$	0	0
$940$	−3.56150e6	−0.131466
$941$	1.26678e6	0.0466368	0.0233184	−	0.999728i	$-0.492577\pi$
$941$	1.26678e6	0.0466368	0.0233184	+	0.999728i	$0.492577\pi$
$942$	0	0
$943$	−1.74246e7	−0.638092
$944$	3.53437e7	1.29087
$945$	0	0
$946$	6.23144e7	2.26392
$947$	−2.45637e7	−0.890058	−0.445029	−	0.895516i	$-0.646807\pi$
$947$	−2.45637e7	−0.890058	−0.445029	+	0.895516i	$0.646807\pi$
$948$	0	0
$949$	−2.24255e7	−0.808309
$950$	6.92500e6	0.248949
$951$	0	0
$952$	2.25720e6	0.0807194
$953$	2.58428e7	0.921737	0.460868	−	0.887469i	$-0.347538\pi$
$953$	2.58428e7	0.921737	0.460868	+	0.887469i	$0.347538\pi$
$954$	0	0
$955$	3.77275e6	0.133860
$956$	−5.66664e7	−2.00531
$957$	0	0
$958$	−7.29660e6	−0.256866
$959$	−1.17162e6	−0.0411377
$960$	0	0
$961$	6.95591e7	2.42966
$962$	−4.67581e7	−1.62899
$963$	0	0
$964$	2.96804e7	1.02867
$965$	−4.26755e6	−0.147523
$966$	0	0
$967$	−1.34437e7	−0.462331	−0.231165	−	0.972914i	$-0.574254\pi$
$967$	−1.34437e7	−0.462331	−0.231165	+	0.972914i	$0.574254\pi$
$968$	1.10943e8	3.80549
$969$	0	0
$970$	2.14915e7	0.733394
$971$	4.08522e7	1.39049	0.695244	−	0.718774i	$-0.255296\pi$
$971$	4.08522e7	1.39049	0.695244	+	0.718774i	$0.255296\pi$
$972$	0	0
$973$	79428.0	0.00268962
$974$	9.35510e7	3.15974
$975$	0	0
$976$	−6.59113e7	−2.21480
$977$	−2.66025e7	−0.891634	−0.445817	−	0.895124i	$-0.647087\pi$
$977$	−2.66025e7	−0.891634	−0.445817	+	0.895124i	$0.647087\pi$
$978$	0	0
$979$	8.46249e7	2.82190
$980$	2.85107e7	0.948293
$981$	0	0
$982$	−9.39394e7	−3.10863
$983$	−5.59346e6	−0.184627	−0.0923137	−	0.995730i	$-0.529426\pi$
$983$	−5.59346e6	−0.184627	−0.0923137	+	0.995730i	$0.529426\pi$
$984$	0	0
$985$	−8.33400e6	−0.273693
$986$	−1.37417e7	−0.450142
$987$	0	0
$988$	5.16106e7	1.68208
$989$	−3.50689e7	−1.14007
$990$	0	0
$991$	−1.67296e7	−0.541130	−0.270565	−	0.962702i	$-0.587211\pi$
$991$	−1.67296e7	−0.541130	−0.270565	+	0.962702i	$0.587211\pi$
$992$	−2.69525e7	−0.869600
$993$	0	0
$994$	−4.53480e6	−0.145577
$995$	−1.43778e6	−0.0460398
$996$	0	0
$997$	4.48356e7	1.42851	0.714257	−	0.699883i	$-0.246765\pi$
$997$	4.48356e7	1.42851	0.714257	+	0.699883i	$0.246765\pi$
$998$	6.54473e7	2.08001
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.6.a.b.1.1	yes	1
3.2	odd	2		135.6.a.a.1.1	✓	1
5.4	even	2		675.6.a.a.1.1		1
15.14	odd	2		675.6.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.6.a.a.1.1	✓	1	3.2	odd	2
135.6.a.b.1.1	yes	1	1.1	even	1	trivial
675.6.a.a.1.1		1	5.4	even	2
675.6.a.e.1.1		1	15.14	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	135.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more