LMFDB - Embedded newform 675.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,6,Mod(1,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, 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("675.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 675 = 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	675.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:	$108.259078374$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 135)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

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

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	$-0.845080\pi$
$2$	−10.0000	−1.76777	−0.883883	+	0.467707i	$0.845080\pi$
$3$	0	0
$3$	0	0
$4$	68.0000	2.12500
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−6.00000	−0.0462814	−0.0231407	−	0.999732i	$-0.507367\pi$
$7$	−6.00000	−0.0462814	−0.0231407	+	0.999732i	$0.507367\pi$
$8$	−360.000	−1.98874
$9$	0	0
$10$	0	0
$11$	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	$-0.690001\pi$
$13$	−685.000	−1.12417	−0.562085	+	0.827079i	$0.690001\pi$
$14$	60.0000	0.0818147
$15$	0	0
$16$	1424.00	1.39062
$17$	−1045.00	−0.876989	−0.438494	−	0.898734i	$-0.644488\pi$
$17$	−1045.00	−0.876989	−0.438494	+	0.898734i	$0.644488\pi$
$18$	0	0
$19$	1108.00	0.704135	0.352067	−	0.935975i	$-0.385479\pi$
$19$	1108.00	0.704135	0.352067	+	0.935975i	$0.385479\pi$
$20$	0	0
$21$	0	0
$22$	−6850.00	−3.01741
$23$	3855.00	1.51951	0.759757	−	0.650207i	$-0.225318\pi$
$23$	3855.00	1.51951	0.759757	+	0.650207i	$0.225318\pi$
$24$	0	0
$25$	0	0
$26$	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
$35$	0	0
$36$	0	0
$37$	6826.00	0.819713	0.409857	−	0.912150i	$-0.365579\pi$
$37$	6826.00	0.819713	0.409857	+	0.912150i	$0.365579\pi$
$38$	−11080.0	−1.24475
$39$	0	0
$40$	0	0
$41$	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	$-0.622406\pi$
$43$	−9097.00	−0.750286	−0.375143	+	0.926967i	$0.622406\pi$
$44$	46580.0	3.62717
$45$	0	0
$46$	−38550.0	−2.68615
$47$	−2095.00	−0.138337	−0.0691687	−	0.997605i	$-0.522035\pi$
$47$	−2095.00	−0.138337	−0.0691687	+	0.997605i	$0.522035\pi$
$48$	0	0
$49$	−16771.0	−0.997858
$50$	0	0
$51$	0	0
$52$	−46580.0	−2.38886
$53$	−10060.0	−0.491936	−0.245968	−	0.969278i	$-0.579106\pi$
$53$	−10060.0	−0.491936	−0.245968	+	0.969278i	$0.579106\pi$
$54$	0	0
$55$	0	0
$56$	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
$65$	0	0
$66$	0	0
$67$	−13860.0	−0.377204	−0.188602	−	0.982054i	$-0.560396\pi$
$67$	−13860.0	−0.377204	−0.188602	+	0.982054i	$0.560396\pi$
$68$	−71060.0	−1.86360
$69$	0	0
$70$	0	0
$71$	−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	$-0.382943\pi$
$73$	32738.0	0.719027	0.359513	+	0.933140i	$0.382943\pi$
$74$	−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	$0.264179\pi$
$80$	0	0
$81$	0	0
$82$	−45200.0	−0.742342
$83$	−93930.0	−1.49661	−0.748306	−	0.663354i	$-0.769132\pi$
$83$	−93930.0	−1.49661	−0.748306	+	0.663354i	$0.769132\pi$
$84$	0	0
$85$	0	0
$86$	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
$95$	0	0
$96$	0	0
$97$	85966.0	0.927678	0.463839	−	0.885919i	$-0.346472\pi$
$97$	85966.0	0.927678	0.463839	+	0.885919i	$0.346472\pi$
$98$	167710.	1.76398
$99$	0	0
$100$	0	0
$101$	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	$-0.211513\pi$
$103$	169522.	1.57446	0.787232	+	0.616656i	$0.211513\pi$
$104$	246600.	2.23568
$105$	0	0
$106$	100600.	0.869628
$107$	−2850.00	−0.0240650	−0.0120325	−	0.999928i	$-0.503830\pi$
$107$	−2850.00	−0.0240650	−0.0120325	+	0.999928i	$0.503830\pi$
$108$	0	0
$109$	−210628.	−1.69805	−0.849024	−	0.528355i	$-0.822809\pi$
$109$	−210628.	−1.69805	−0.849024	+	0.528355i	$0.822809\pi$
$110$	0	0
$111$	0	0
$112$	−8544.00	−0.0643600
$113$	81335.0	0.599213	0.299607	−	0.954063i	$-0.403145\pi$
$113$	81335.0	0.599213	0.299607	+	0.954063i	$0.403145\pi$
$114$	0	0
$115$	0	0
$116$	−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
$125$	0	0
$126$	0	0
$127$	237410.	1.30614	0.653070	−	0.757298i	$-0.273481\pi$
$127$	237410.	1.30614	0.653070	+	0.757298i	$0.273481\pi$
$128$	270720.	1.46048
$129$	0	0
$130$	0	0
$131$	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	$-0.353406\pi$
$137$	195270.	0.888862	0.444431	+	0.895813i	$0.353406\pi$
$138$	0	0
$139$	13238.0	0.0581146	0.0290573	−	0.999578i	$-0.490749\pi$
$139$	13238.0	0.0581146	0.0290573	+	0.999578i	$0.490749\pi$
$140$	0	0
$141$	0	0
$142$	755800.	3.14547
$143$	−469225.	−1.91885
$144$	0	0
$145$	0	0
$146$	−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
$155$	0	0
$156$	0	0
$157$	−332411.	−1.07628	−0.538141	−	0.842855i	$-0.680873\pi$
$157$	−332411.	−1.07628	−0.538141	+	0.842855i	$0.680873\pi$
$158$	−748770.	−2.38619
$159$	0	0
$160$	0	0
$161$	−23130.0	−0.0703252
$162$	0	0
$163$	−442925.	−1.30575	−0.652877	−	0.757464i	$-0.726438\pi$
$163$	−442925.	−1.30575	−0.652877	+	0.757464i	$0.726438\pi$
$164$	307360.	0.892355
$165$	0	0
$166$	939300.	2.64566
$167$	7140.00	0.0198110	0.00990551	−	0.999951i	$-0.496847\pi$
$167$	7140.00	0.0198110	0.00990551	+	0.999951i	$0.496847\pi$
$168$	0	0
$169$	97932.0	0.263759
$170$	0	0
$171$	0	0
$172$	−618596.	−1.59436
$173$	−292650.	−0.743418	−0.371709	−	0.928349i	$-0.621228\pi$
$173$	−292650.	−0.743418	−0.371709	+	0.928349i	$0.621228\pi$
$174$	0	0
$175$	0	0
$176$	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
$185$	0	0
$186$	0	0
$187$	−715825.	−1.49693
$188$	−142460.	−0.293967
$189$	0	0
$190$	0	0
$191$	−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	$-0.552742\pi$
$193$	−170702.	−0.329872	−0.164936	+	0.986304i	$0.552742\pi$
$194$	−859660.	−1.63992
$195$	0	0
$196$	−1.14043e6	−2.12045
$197$	−333360.	−0.611995	−0.305998	−	0.952032i	$-0.598990\pi$
$197$	−333360.	−0.611995	−0.305998	+	0.952032i	$0.598990\pi$
$198$	0	0
$199$	57511.0	0.102948	0.0514740	−	0.998674i	$-0.483608\pi$
$199$	57511.0	0.102948	0.0514740	+	0.998674i	$0.483608\pi$
$200$	0	0
$201$	0	0
$202$	−624150.	−1.07624
$203$	7890.00	0.0134381
$204$	0	0
$205$	0	0
$206$	−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
$215$	0	0
$216$	0	0
$217$	59454.0	0.0857100
$218$	2.10628e6	3.00175
$219$	0	0
$220$	0	0
$221$	715825.	0.985885
$222$	0	0
$223$	−699368.	−0.941767	−0.470884	−	0.882195i	$-0.656065\pi$
$223$	−699368.	−0.941767	−0.470884	+	0.882195i	$0.656065\pi$
$224$	16320.0	0.0217320
$225$	0	0
$226$	−813350.	−1.05927
$227$	−471200.	−0.606933	−0.303466	−	0.952842i	$-0.598144\pi$
$227$	−471200.	−0.606933	−0.303466	+	0.952842i	$0.598144\pi$
$228$	0	0
$229$	1.44071e6	1.81546	0.907731	−	0.419552i	$-0.137813\pi$
$229$	1.44071e6	1.81546	0.907731	+	0.419552i	$0.137813\pi$
$230$	0	0
$231$	0	0
$232$	473400.	0.577442
$233$	294090.	0.354887	0.177444	−	0.984131i	$-0.443217\pi$
$233$	294090.	0.354887	0.177444	+	0.984131i	$0.443217\pi$
$234$	0	0
$235$	0	0
$236$	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
$245$	0	0
$246$	0	0
$247$	−758980.	−0.791567
$248$	3.56724e6	3.68301
$249$	0	0
$250$	0	0
$251$	−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	$-0.397574\pi$
$257$	669735.	0.632514	0.316257	+	0.948674i	$0.397574\pi$
$258$	0	0
$259$	−40956.0	−0.0379374
$260$	0	0
$261$	0	0
$262$	−1.58115e6	−1.42305
$263$	−1.92650e6	−1.71743	−0.858716	−	0.512451i	$-0.828737\pi$
$263$	−1.92650e6	−1.71743	−0.858716	+	0.512451i	$0.828737\pi$
$264$	0	0
$265$	0	0
$266$	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
$275$	0	0
$276$	0	0
$277$	91866.0	0.0719375	0.0359688	−	0.999353i	$-0.488548\pi$
$277$	91866.0	0.0719375	0.0359688	+	0.999353i	$0.488548\pi$
$278$	−132380.	−0.102733
$279$	0	0
$280$	0	0
$281$	−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	$-0.652487\pi$
$283$	−1.24205e6	−0.921876	−0.460938	+	0.887432i	$0.652487\pi$
$284$	−5.13944e6	−3.78112
$285$	0	0
$286$	4.69225e6	3.39208
$287$	−27120.0	−0.0194350
$288$	0	0
$289$	−327832.	−0.230891
$290$	0	0
$291$	0	0
$292$	2.22618e6	1.52793
$293$	−453690.	−0.308738	−0.154369	−	0.988013i	$-0.549334\pi$
$293$	−453690.	−0.308738	−0.154369	+	0.988013i	$0.549334\pi$
$294$	0	0
$295$	0	0
$296$	−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
$305$	0	0
$306$	0	0
$307$	2.37090e6	1.43571	0.717856	−	0.696192i	$-0.245124\pi$
$307$	2.37090e6	1.43571	0.717856	+	0.696192i	$0.245124\pi$
$308$	−279480.	−0.167870
$309$	0	0
$310$	0	0
$311$	−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	$-0.609033\pi$
$313$	−1.16432e6	−0.671756	−0.335878	+	0.941906i	$0.609033\pi$
$314$	3.32411e6	1.90262
$315$	0	0
$316$	5.09164e6	2.86840
$317$	−855800.	−0.478326	−0.239163	−	0.970979i	$-0.576873\pi$
$317$	−855800.	−0.478326	−0.239163	+	0.970979i	$0.576873\pi$
$318$	0	0
$319$	−900775.	−0.495609
$320$	0	0
$321$	0	0
$322$	231300.	0.124319
$323$	−1.15786e6	−0.617518
$324$	0	0
$325$	0	0
$326$	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
$335$	0	0
$336$	0	0
$337$	820040.	0.393333	0.196666	−	0.980470i	$-0.436988\pi$
$337$	820040.	0.393333	0.196666	+	0.980470i	$0.436988\pi$
$338$	−979320.	−0.466265
$339$	0	0
$340$	0	0
$341$	−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	$-0.439178\pi$
$347$	851960.	0.379835	0.189918	+	0.981800i	$0.439178\pi$
$348$	0	0
$349$	−1.11612e6	−0.490510	−0.245255	−	0.969459i	$-0.578872\pi$
$349$	−1.11612e6	−0.490510	−0.245255	+	0.969459i	$0.578872\pi$
$350$	0	0
$351$	0	0
$352$	−1.86320e6	−0.801499
$353$	−4.18808e6	−1.78887	−0.894433	−	0.447202i	$-0.852420\pi$
$353$	−4.18808e6	−1.78887	−0.894433	+	0.447202i	$0.852420\pi$
$354$	0	0
$355$	0	0
$356$	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
$365$	0	0
$366$	0	0
$367$	1.42773e6	0.553326	0.276663	−	0.960967i	$-0.410771\pi$
$367$	1.42773e6	0.553326	0.276663	+	0.960967i	$0.410771\pi$
$368$	5.48952e6	2.11307
$369$	0	0
$370$	0	0
$371$	60360.0	0.0227675
$372$	0	0
$373$	−821623.	−0.305774	−0.152887	−	0.988244i	$-0.548857\pi$
$373$	−821623.	−0.305774	−0.152887	+	0.988244i	$0.548857\pi$
$374$	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	$0.534760\pi$
$380$	0	0
$381$	0	0
$382$	1.50910e6	0.529127
$383$	1.08650e6	0.378469	0.189235	−	0.981932i	$-0.439399\pi$
$383$	1.08650e6	0.378469	0.189235	+	0.981932i	$0.439399\pi$
$384$	0	0
$385$	0	0
$386$	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
$395$	0	0
$396$	0	0
$397$	−3.85253e6	−1.22679	−0.613394	−	0.789777i	$-0.710196\pi$
$397$	−3.85253e6	−1.22679	−0.613394	+	0.789777i	$0.710196\pi$
$398$	−575110.	−0.181988
$399$	0	0
$400$	0	0
$401$	−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	$0.573378\pi$
$410$	0	0
$411$	0	0
$412$	1.15275e7	3.34574
$413$	−148920.	−0.0429613
$414$	0	0
$415$	0	0
$416$	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
$425$	0	0
$426$	0	0
$427$	277716.	0.0737108
$428$	−193800.	−0.0511381
$429$	0	0
$430$	0	0
$431$	−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	$-0.590452\pi$
$433$	−2.18754e6	−0.560707	−0.280353	+	0.959897i	$0.590452\pi$
$434$	−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	$0.449923\pi$
$440$	0	0
$441$	0	0
$442$	−7.15825e6	−1.74281
$443$	−672180.	−0.162733	−0.0813666	−	0.996684i	$-0.525928\pi$
$443$	−672180.	−0.162733	−0.0813666	+	0.996684i	$0.525928\pi$
$444$	0	0
$445$	0	0
$446$	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
$455$	0	0
$456$	0	0
$457$	−6.81838e6	−1.52718	−0.763591	−	0.645700i	$-0.776565\pi$
$457$	−6.81838e6	−1.52718	−0.763591	+	0.645700i	$0.776565\pi$
$458$	−1.44071e7	−3.20931
$459$	0	0
$460$	0	0
$461$	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	$-0.610973\pi$
$463$	−3.15148e6	−0.683223	−0.341611	+	0.939841i	$0.610973\pi$
$464$	−1.87256e6	−0.403776
$465$	0	0
$466$	−2.94090e6	−0.627358
$467$	7.80490e6	1.65606	0.828028	−	0.560686i	$-0.189463\pi$
$467$	7.80490e6	1.65606	0.828028	+	0.560686i	$0.189463\pi$
$468$	0	0
$469$	83160.0	0.0174575
$470$	0	0
$471$	0	0
$472$	−8.93520e6	−1.84607
$473$	−6.23145e6	−1.28067
$474$	0	0
$475$	0	0
$476$	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
$485$	0	0
$486$	0	0
$487$	−9.35510e6	−1.78742	−0.893709	−	0.448647i	$-0.851906\pi$
$487$	−9.35510e6	−1.78742	−0.893709	+	0.448647i	$0.851906\pi$
$488$	1.66630e7	3.16740
$489$	0	0
$490$	0	0
$491$	−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	$0.299791\pi$
$500$	0	0
$501$	0	0
$502$	6.76695e6	1.19849
$503$	−6.68478e6	−1.17806	−0.589030	−	0.808111i	$-0.700490\pi$
$503$	−6.68478e6	−1.17806	−0.589030	+	0.808111i	$0.700490\pi$
$504$	0	0
$505$	0	0
$506$	−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
$515$	0	0
$516$	0	0
$517$	−1.43508e6	−0.236128
$518$	409560.	0.0670646
$519$	0	0
$520$	0	0
$521$	−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	$-0.642294\pi$
$523$	−5.40826e6	−0.864577	−0.432288	+	0.901735i	$0.642294\pi$
$524$	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
$530$	0	0
$531$	0	0
$532$	−452064.	−0.0692502
$533$	−3.09620e6	−0.472075
$534$	0	0
$535$	0	0
$536$	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
$545$	0	0
$546$	0	0
$547$	−30901.0	−0.00441575	−0.00220787	−	0.999998i	$-0.500703\pi$
$547$	−30901.0	−0.00441575	−0.00220787	+	0.999998i	$0.500703\pi$
$548$	1.32784e7	1.88883
$549$	0	0
$550$	0	0
$551$	−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	$-0.641470\pi$
$557$	−6.29634e6	−0.859904	−0.429952	+	0.902852i	$0.641470\pi$
$558$	0	0
$559$	6.23144e6	0.843450
$560$	0	0
$561$	0	0
$562$	1.81323e7	2.42165
$563$	−1.23597e6	−0.164338	−0.0821688	−	0.996618i	$-0.526185\pi$
$563$	−1.23597e6	−0.164338	−0.0821688	+	0.996618i	$0.526185\pi$
$564$	0	0
$565$	0	0
$566$	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
$575$	0	0
$576$	0	0
$577$	7.58210e6	0.948090	0.474045	−	0.880501i	$-0.342793\pi$
$577$	7.58210e6	0.948090	0.474045	+	0.880501i	$0.342793\pi$
$578$	3.27832e6	0.408161
$579$	0	0
$580$	0	0
$581$	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	$-0.768424\pi$
$587$	−1.24694e7	−1.49366	−0.746828	+	0.665017i	$0.768424\pi$
$588$	0	0
$589$	−1.09792e7	−1.30401
$590$	0	0
$591$	0	0
$592$	9.72022e6	1.13991
$593$	2.94156e6	0.343511	0.171755	−	0.985140i	$-0.445056\pi$
$593$	2.94156e6	0.343511	0.171755	+	0.985140i	$0.445056\pi$
$594$	0	0
$595$	0	0
$596$	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
$605$	0	0
$606$	0	0
$607$	−5.59659e6	−0.616527	−0.308263	−	0.951301i	$-0.599748\pi$
$607$	−5.59659e6	−0.616527	−0.308263	+	0.951301i	$0.599748\pi$
$608$	−3.01376e6	−0.330636
$609$	0	0
$610$	0	0
$611$	1.43508e6	0.155515
$612$	0	0
$613$	−1.01497e7	−1.09095	−0.545473	−	0.838128i	$-0.683650\pi$
$613$	−1.01497e7	−1.09095	−0.545473	+	0.838128i	$0.683650\pi$
$614$	−2.37090e7	−2.53800
$615$	0	0
$616$	1.47960e6	0.157106
$617$	−1.53162e7	−1.61971	−0.809854	−	0.586631i	$-0.800454\pi$
$617$	−1.53162e7	−1.61971	−0.809854	+	0.586631i	$0.800454\pi$
$618$	0	0
$619$	−5.20008e6	−0.545486	−0.272743	−	0.962087i	$-0.587931\pi$
$619$	−5.20008e6	−0.545486	−0.272743	+	0.962087i	$0.587931\pi$
$620$	0	0
$621$	0	0
$622$	9.81600e6	1.01732
$623$	−741240.	−0.0765136
$624$	0	0
$625$	0	0
$626$	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
$635$	0	0
$636$	0	0
$637$	1.14881e7	1.12176
$638$	9.00775e6	0.876122
$639$	0	0
$640$	0	0
$641$	−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	$-0.131144\pi$
$643$	1.92135e7	1.83264	0.916322	+	0.400443i	$0.131144\pi$
$644$	−1.57284e6	−0.149441
$645$	0	0
$646$	1.15786e7	1.09163
$647$	8.64470e6	0.811875	0.405938	−	0.913901i	$-0.366945\pi$
$647$	8.64470e6	0.811875	0.405938	+	0.913901i	$0.366945\pi$
$648$	0	0
$649$	1.70017e7	1.58446
$650$	0	0
$651$	0	0
$652$	−3.01189e7	−2.77473
$653$	−4.50206e6	−0.413170	−0.206585	−	0.978429i	$-0.566235\pi$
$653$	−4.50206e6	−0.413170	−0.206585	+	0.978429i	$0.566235\pi$
$654$	0	0
$655$	0	0
$656$	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
$665$	0	0
$666$	0	0
$667$	−5.06932e6	−0.441200
$668$	485520.	0.0420984
$669$	0	0
$670$	0	0
$671$	−3.17059e7	−2.71853
$672$	0	0
$673$	−1.49638e7	−1.27352	−0.636758	−	0.771064i	$-0.719725\pi$
$673$	−1.49638e7	−1.27352	−0.636758	+	0.771064i	$0.719725\pi$
$674$	−8.20040e6	−0.695321
$675$	0	0
$676$	6.65938e6	0.560489
$677$	−8.83758e6	−0.741074	−0.370537	−	0.928818i	$-0.620826\pi$
$677$	−8.83758e6	−0.741074	−0.370537	+	0.928818i	$0.620826\pi$
$678$	0	0
$679$	−515796.	−0.0429342
$680$	0	0
$681$	0	0
$682$	6.78766e7	5.58804
$683$	1.09012e7	0.894172	0.447086	−	0.894491i	$-0.352462\pi$
$683$	1.09012e7	0.894172	0.447086	+	0.894491i	$0.352462\pi$
$684$	0	0
$685$	0	0
$686$	−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
$695$	0	0
$696$	0	0
$697$	−4.72340e6	−0.368275
$698$	1.11612e7	0.867108
$699$	0	0
$700$	0	0
$701$	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	$0.718208\pi$
$710$	0	0
$711$	0	0
$712$	−4.44744e7	−3.28784
$713$	−3.81992e7	−2.81404
$714$	0	0
$715$	0	0
$716$	−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
$725$	0	0
$726$	0	0
$727$	−1.66714e7	−1.16986	−0.584931	−	0.811083i	$-0.698879\pi$
$727$	−1.66714e7	−1.16986	−0.584931	+	0.811083i	$0.698879\pi$
$728$	−1.47960e6	−0.103470
$729$	0	0
$730$	0	0
$731$	9.50636e6	0.657993
$732$	0	0
$733$	1.32270e7	0.909284	0.454642	−	0.890674i	$-0.349767\pi$
$733$	1.32270e7	0.909284	0.454642	+	0.890674i	$0.349767\pi$
$734$	−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	$0.598129\pi$
$740$	0	0
$741$	0	0
$742$	−603600.	−0.0402476
$743$	2.44168e7	1.62262	0.811309	−	0.584618i	$-0.198756\pi$
$743$	2.44168e7	1.62262	0.811309	+	0.584618i	$0.198756\pi$
$744$	0	0
$745$	0	0
$746$	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
$755$	0	0
$756$	0	0
$757$	−2.76229e7	−1.75198	−0.875991	−	0.482328i	$-0.839791\pi$
$757$	−2.76229e7	−1.75198	−0.875991	+	0.482328i	$0.839791\pi$
$758$	6.09536e6	0.385324
$759$	0	0
$760$	0	0
$761$	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	$0.465983\pi$
$770$	0	0
$771$	0	0
$772$	−1.16077e7	−0.700978
$773$	−2.99115e6	−0.180048	−0.0900242	−	0.995940i	$-0.528694\pi$
$773$	−2.99115e6	−0.180048	−0.0900242	+	0.995940i	$0.528694\pi$
$774$	0	0
$775$	0	0
$776$	−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
$785$	0	0
$786$	0	0
$787$	1.38920e7	0.799520	0.399760	−	0.916620i	$-0.369094\pi$
$787$	1.38920e7	0.799520	0.399760	+	0.916620i	$0.369094\pi$
$788$	−2.26685e7	−1.30049
$789$	0	0
$790$	0	0
$791$	−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	$-0.572876\pi$
$797$	−8.13974e6	−0.453905	−0.226952	+	0.973906i	$0.572876\pi$
$798$	0	0
$799$	2.18928e6	0.121320
$800$	0	0
$801$	0	0
$802$	2.93700e6	0.161238
$803$	2.24255e7	1.22731
$804$	0	0
$805$	0	0
$806$	−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
$815$	0	0
$816$	0	0
$817$	−1.00795e7	−0.528303
$818$	1.54596e7	0.807821
$819$	0	0
$820$	0	0
$821$	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	$-0.557568\pi$
$823$	−6.99026e6	−0.359744	−0.179872	+	0.983690i	$0.557568\pi$
$824$	−6.10279e7	−3.13120
$825$	0	0
$826$	1.48920e6	0.0759457
$827$	2.68441e7	1.36485	0.682425	−	0.730955i	$-0.260925\pi$
$827$	2.68441e7	1.36485	0.682425	+	0.730955i	$0.260925\pi$
$828$	0	0
$829$	2.11268e7	1.06770	0.533848	−	0.845580i	$-0.320745\pi$
$829$	2.11268e7	1.06770	0.533848	+	0.845580i	$0.320745\pi$
$830$	0	0
$831$	0	0
$832$	1.25821e7	0.630150
$833$	1.75257e7	0.875110
$834$	0	0
$835$	0	0
$836$	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
$845$	0	0
$846$	0	0
$847$	−1.84904e6	−0.0885602
$848$	−1.43254e7	−0.684098
$849$	0	0
$850$	0	0
$851$	2.63142e7	1.24557
$852$	0	0
$853$	−5.75945e6	−0.271024	−0.135512	−	0.990776i	$-0.543268\pi$
$853$	−5.75945e6	−0.271024	−0.135512	+	0.990776i	$0.543268\pi$
$854$	−2.77716e6	−0.130304
$855$	0	0
$856$	1.02600e6	0.0478589
$857$	1.43880e7	0.669188	0.334594	−	0.942362i	$-0.391401\pi$
$857$	1.43880e7	0.669188	0.334594	+	0.942362i	$0.391401\pi$
$858$	0	0
$859$	3.28619e7	1.51953	0.759766	−	0.650197i	$-0.225314\pi$
$859$	3.28619e7	1.51953	0.759766	+	0.650197i	$0.225314\pi$
$860$	0	0
$861$	0	0
$862$	3.85980e7	1.76928
$863$	4.34132e6	0.198425	0.0992123	−	0.995066i	$-0.468368\pi$
$863$	4.34132e6	0.198425	0.0992123	+	0.995066i	$0.468368\pi$
$864$	0	0
$865$	0	0
$866$	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
$875$	0	0
$876$	0	0
$877$	−2.70727e7	−1.18859	−0.594296	−	0.804246i	$-0.702569\pi$
$877$	−2.70727e7	−1.18859	−0.594296	+	0.804246i	$0.702569\pi$
$878$	−1.26527e7	−0.553921
$879$	0	0
$880$	0	0
$881$	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	$-0.618874\pi$
$883$	−1.69054e7	−0.729667	−0.364833	+	0.931073i	$0.618874\pi$
$884$	4.86761e7	2.09501
$885$	0	0
$886$	6.72180e6	0.287674
$887$	−3.06531e7	−1.30817	−0.654087	−	0.756419i	$-0.726947\pi$
$887$	−3.06531e7	−1.30817	−0.654087	+	0.756419i	$0.726947\pi$
$888$	0	0
$889$	−1.42446e6	−0.0604499
$890$	0	0
$891$	0	0
$892$	−4.75570e7	−2.00126
$893$	−2.32126e6	−0.0974081
$894$	0	0
$895$	0	0
$896$	−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
$905$	0	0
$906$	0	0
$907$	−1.73344e7	−0.699667	−0.349833	−	0.936812i	$-0.613762\pi$
$907$	−1.73344e7	−0.699667	−0.349833	+	0.936812i	$0.613762\pi$
$908$	−3.20416e7	−1.28973
$909$	0	0
$910$	0	0
$911$	−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	$0.754973\pi$
$920$	0	0
$921$	0	0
$922$	−6.55773e7	−2.54054
$923$	5.17723e7	2.00029
$924$	0	0
$925$	0	0
$926$	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
$935$	0	0
$936$	0	0
$937$	5.26250e7	1.95814	0.979068	−	0.203533i	$-0.0652423\pi$
$937$	5.26250e7	1.95814	0.979068	+	0.203533i	$0.0652423\pi$
$938$	−831600.	−0.0308608
$939$	0	0
$940$	0	0
$941$	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	$-0.353193\pi$
$947$	2.45637e7	0.890058	0.445029	+	0.895516i	$0.353193\pi$
$948$	0	0
$949$	−2.24255e7	−0.808309
$950$	0	0
$951$	0	0
$952$	−2.25720e6	−0.0807194
$953$	−2.58428e7	−0.921737	−0.460868	−	0.887469i	$-0.652462\pi$
$953$	−2.58428e7	−0.921737	−0.460868	+	0.887469i	$0.652462\pi$
$954$	0	0
$955$	0	0
$956$	−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
$965$	0	0
$966$	0	0
$967$	1.34437e7	0.462331	0.231165	−	0.972914i	$-0.425746\pi$
$967$	1.34437e7	0.462331	0.231165	+	0.972914i	$0.425746\pi$
$968$	−1.10943e8	−3.80549
$969$	0	0
$970$	0	0
$971$	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	$-0.352913\pi$
$977$	2.66025e7	0.891634	0.445817	+	0.895124i	$0.352913\pi$
$978$	0	0
$979$	8.46249e7	2.82190
$980$	0	0
$981$	0	0
$982$	9.39394e7	3.10863
$983$	5.59346e6	0.184627	0.0923137	−	0.995730i	$-0.470574\pi$
$983$	5.59346e6	0.184627	0.0923137	+	0.995730i	$0.470574\pi$
$984$	0	0
$985$	0	0
$986$	−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
$995$	0	0
$996$	0	0
$997$	−4.48356e7	−1.42851	−0.714257	−	0.699883i	$-0.753235\pi$
$997$	−4.48356e7	−1.42851	−0.714257	+	0.699883i	$0.753235\pi$
$998$	−6.54473e7	−2.08001
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.6.a.a.1.1	✓	1	15.14	odd	2
135.6.a.b.1.1	yes	1	5.4	even	2
675.6.a.a.1.1		1	1.1	even	1	trivial
675.6.a.e.1.1		1	3.2	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:	\( N \)	\(=\)	\( 675 = 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	675.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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