LMFDB - Embedded newform 1350.3.d.f.701.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,3,Mod(701,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("1350.701");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1350 = 2 \cdot 3^{3} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1350.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$36.7848356886$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			701.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	1350.701
Dual form			1350.3.d.f.701.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} -2.82843i q^{8} -1.41421i q^{11} +9.00000 q^{13} -7.07107i q^{14} +4.00000 q^{16} -11.3137i q^{17} -21.0000 q^{19} +2.00000 q^{22} +1.41421i q^{23} +12.7279i q^{26} +10.0000 q^{28} +38.1838i q^{29} +40.0000 q^{31} +5.65685i q^{32} +16.0000 q^{34} +25.0000 q^{37} -29.6985i q^{38} -52.3259i q^{41} +64.0000 q^{43} +2.82843i q^{44} -2.00000 q^{46} -22.6274i q^{47} -24.0000 q^{49} -18.0000 q^{52} +72.1249i q^{53} +14.1421i q^{56} -54.0000 q^{58} +90.5097i q^{59} -97.0000 q^{61} +56.5685i q^{62} -8.00000 q^{64} +131.000 q^{67} +22.6274i q^{68} +89.0955i q^{71} -17.0000 q^{73} +35.3553i q^{74} +42.0000 q^{76} +7.07107i q^{77} +117.000 q^{79} +74.0000 q^{82} -57.9828i q^{83} +90.5097i q^{86} -4.00000 q^{88} +147.078i q^{89} -45.0000 q^{91} -2.82843i q^{92} +32.0000 q^{94} +41.0000 q^{97} -33.9411i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{4} - 10 q^{7}+O(q^{10}) $ 2 * q - 4 * q^4 - 10 * q^7	$ 2 q - 4 q^{4} - 10 q^{7} + 18 q^{13} + 8 q^{16} - 42 q^{19} + 4 q^{22} + 20 q^{28} + 80 q^{31} + 32 q^{34} + 50 q^{37} + 128 q^{43} - 4 q^{46} - 48 q^{49} - 36 q^{52} - 108 q^{58} - 194 q^{61} - 16 q^{64} + 262 q^{67} - 34 q^{73} + 84 q^{76} + 234 q^{79} + 148 q^{82} - 8 q^{88} - 90 q^{91} + 64 q^{94} + 82 q^{97}+O(q^{100}) $ 2 * q - 4 * q^4 - 10 * q^7 + 18 * q^13 + 8 * q^16 - 42 * q^19 + 4 * q^22 + 20 * q^28 + 80 * q^31 + 32 * q^34 + 50 * q^37 + 128 * q^43 - 4 * q^46 - 48 * q^49 - 36 * q^52 - 108 * q^58 - 194 * q^61 - 16 * q^64 + 262 * q^67 - 34 * q^73 + 84 * q^76 + 234 * q^79 + 148 * q^82 - 8 * q^88 - 90 * q^91 + 64 * q^94 + 82 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times$.

$n$	$1001$	$1027$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.41421i	0.707107i
$2$	1.41421i	0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−5.00000	−0.714286	−0.357143	−	0.934050i	$-0.616249\pi$
$7$	−5.00000	−0.714286	−0.357143	+	0.934050i	$0.616249\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	0	0
$11$	− 1.41421i	− 0.128565i	−0.997932	−	0.0642824i	$-0.979524\pi$
$11$	− 1.41421i	− 0.128565i	0.997932	−	0.0642824i	$-0.0204759\pi$
$12$	0	0
$13$	9.00000	0.692308	0.346154	−	0.938178i	$-0.387488\pi$
$13$	9.00000	0.692308	0.346154	+	0.938178i	$0.387488\pi$
$14$	− 7.07107i	− 0.505076i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 11.3137i	− 0.665512i	−0.943013	−	0.332756i	$-0.892021\pi$
$17$	− 11.3137i	− 0.665512i	0.943013	−	0.332756i	$-0.107979\pi$
$18$	0	0
$19$	−21.0000	−1.10526	−0.552632	−	0.833426i	$-0.686376\pi$
$19$	−21.0000	−1.10526	−0.552632	+	0.833426i	$0.686376\pi$
$20$	0	0
$21$	0	0
$22$	2.00000	0.0909091
$23$	1.41421i	0.0614875i	0.999527	+	0.0307438i	$0.00978759\pi$
$23$	1.41421i	0.0614875i	−0.999527	+	0.0307438i	$0.990212\pi$
$24$	0	0
$25$	0	0
$26$	12.7279i	0.489535i
$27$	0	0
$28$	10.0000	0.357143
$29$	38.1838i	1.31668i	0.752720	+	0.658341i	$0.228741\pi$
$29$	38.1838i	1.31668i	−0.752720	+	0.658341i	$0.771259\pi$
$30$	0	0
$31$	40.0000	1.29032	0.645161	−	0.764046i	$-0.276790\pi$
$31$	40.0000	1.29032	0.645161	+	0.764046i	$0.276790\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	16.0000	0.470588
$35$	0	0
$36$	0	0
$37$	25.0000	0.675676	0.337838	−	0.941204i	$-0.390304\pi$
$37$	25.0000	0.675676	0.337838	+	0.941204i	$0.390304\pi$
$38$	− 29.6985i	− 0.781539i
$39$	0	0
$40$	0	0
$41$	− 52.3259i	− 1.27624i	−0.769936	−	0.638121i	$-0.779712\pi$
$41$	− 52.3259i	− 1.27624i	0.769936	−	0.638121i	$-0.220288\pi$
$42$	0	0
$43$	64.0000	1.48837	0.744186	−	0.667972i	$-0.232838\pi$
$43$	64.0000	1.48837	0.744186	+	0.667972i	$0.232838\pi$
$44$	2.82843i	0.0642824i
$45$	0	0
$46$	−2.00000	−0.0434783
$47$	− 22.6274i	− 0.481434i	−0.970595	−	0.240717i	$-0.922617\pi$
$47$	− 22.6274i	− 0.481434i	0.970595	−	0.240717i	$-0.0773826\pi$
$48$	0	0
$49$	−24.0000	−0.489796
$50$	0	0
$51$	0	0
$52$	−18.0000	−0.346154
$53$	72.1249i	1.36085i	0.732819	+	0.680424i	$0.238204\pi$
$53$	72.1249i	1.36085i	−0.732819	+	0.680424i	$0.761796\pi$
$54$	0	0
$55$	0	0
$56$	14.1421i	0.252538i
$57$	0	0
$58$	−54.0000	−0.931034
$59$	90.5097i	1.53406i	0.641610	+	0.767031i	$0.278267\pi$
$59$	90.5097i	1.53406i	−0.641610	+	0.767031i	$0.721733\pi$
$60$	0	0
$61$	−97.0000	−1.59016	−0.795082	−	0.606502i	$-0.792572\pi$
$61$	−97.0000	−1.59016	−0.795082	+	0.606502i	$0.792572\pi$
$62$	56.5685i	0.912396i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	0	0
$66$	0	0
$67$	131.000	1.95522	0.977612	−	0.210416i	$-0.0674818\pi$
$67$	131.000	1.95522	0.977612	+	0.210416i	$0.0674818\pi$
$68$	22.6274i	0.332756i
$69$	0	0
$70$	0	0
$71$	89.0955i	1.25487i	0.778671	+	0.627433i	$0.215894\pi$
$71$	89.0955i	1.25487i	−0.778671	+	0.627433i	$0.784106\pi$
$72$	0	0
$73$	−17.0000	−0.232877	−0.116438	−	0.993198i	$-0.537148\pi$
$73$	−17.0000	−0.232877	−0.116438	+	0.993198i	$0.537148\pi$
$74$	35.3553i	0.477775i
$75$	0	0
$76$	42.0000	0.552632
$77$	7.07107i	0.0918320i
$78$	0	0
$79$	117.000	1.48101	0.740506	−	0.672049i	$-0.234586\pi$
$79$	117.000	1.48101	0.740506	+	0.672049i	$0.234586\pi$
$80$	0	0
$81$	0	0
$82$	74.0000	0.902439
$83$	− 57.9828i	− 0.698587i	−0.937013	−	0.349294i	$-0.886422\pi$
$83$	− 57.9828i	− 0.698587i	0.937013	−	0.349294i	$-0.113578\pi$
$84$	0	0
$85$	0	0
$86$	90.5097i	1.05244i
$87$	0	0
$88$	−4.00000	−0.0454545
$89$	147.078i	1.65256i	0.563257	+	0.826282i	$0.309548\pi$
$89$	147.078i	1.65256i	−0.563257	+	0.826282i	$0.690452\pi$
$90$	0	0
$91$	−45.0000	−0.494505
$92$	− 2.82843i	− 0.0307438i
$93$	0	0
$94$	32.0000	0.340426
$95$	0	0
$96$	0	0
$97$	41.0000	0.422680	0.211340	−	0.977413i	$-0.432217\pi$
$97$	41.0000	0.422680	0.211340	+	0.977413i	$0.432217\pi$
$98$	− 33.9411i	− 0.346338i
$99$	0	0
$100$	0	0
$101$	90.5097i	0.896135i	0.894000	+	0.448068i	$0.147888\pi$
$101$	90.5097i	0.896135i	−0.894000	+	0.448068i	$0.852112\pi$
$102$	0	0
$103$	−13.0000	−0.126214	−0.0631068	−	0.998007i	$-0.520101\pi$
$103$	−13.0000	−0.126214	−0.0631068	+	0.998007i	$0.520101\pi$
$104$	− 25.4558i	− 0.244768i
$105$	0	0
$106$	−102.000	−0.962264
$107$	123.037i	1.14987i	0.818197	+	0.574937i	$0.194974\pi$
$107$	123.037i	1.14987i	−0.818197	+	0.574937i	$0.805026\pi$
$108$	0	0
$109$	−8.00000	−0.0733945	−0.0366972	−	0.999326i	$-0.511684\pi$
$109$	−8.00000	−0.0733945	−0.0366972	+	0.999326i	$0.511684\pi$
$110$	0	0
$111$	0	0
$112$	−20.0000	−0.178571
$113$	− 38.1838i	− 0.337909i	−0.985624	−	0.168955i	$-0.945961\pi$
$113$	− 38.1838i	− 0.337909i	0.985624	−	0.168955i	$-0.0540392\pi$
$114$	0	0
$115$	0	0
$116$	− 76.3675i	− 0.658341i
$117$	0	0
$118$	−128.000	−1.08475
$119$	56.5685i	0.475366i
$120$	0	0
$121$	119.000	0.983471
$122$	− 137.179i	− 1.12442i
$123$	0	0
$124$	−80.0000	−0.645161
$125$	0	0
$126$	0	0
$127$	−8.00000	−0.0629921	−0.0314961	−	0.999504i	$-0.510027\pi$
$127$	−8.00000	−0.0629921	−0.0314961	+	0.999504i	$0.510027\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	0	0
$131$	135.765i	1.03637i	0.855268	+	0.518185i	$0.173392\pi$
$131$	135.765i	1.03637i	−0.855268	+	0.518185i	$0.826608\pi$
$132$	0	0
$133$	105.000	0.789474
$134$	185.262i	1.38255i
$135$	0	0
$136$	−32.0000	−0.235294
$137$	− 267.286i	− 1.95100i	−0.220010	−	0.975498i	$-0.570609\pi$
$137$	− 267.286i	− 1.95100i	0.220010	−	0.975498i	$-0.429391\pi$
$138$	0	0
$139$	37.0000	0.266187	0.133094	−	0.991103i	$-0.457509\pi$
$139$	37.0000	0.266187	0.133094	+	0.991103i	$0.457509\pi$
$140$	0	0
$141$	0	0
$142$	−126.000	−0.887324
$143$	− 12.7279i	− 0.0890064i
$144$	0	0
$145$	0	0
$146$	− 24.0416i	− 0.164669i
$147$	0	0
$148$	−50.0000	−0.337838
$149$	260.215i	1.74641i	0.487352	+	0.873206i	$0.337963\pi$
$149$	260.215i	1.74641i	−0.487352	+	0.873206i	$0.662037\pi$
$150$	0	0
$151$	109.000	0.721854	0.360927	−	0.932594i	$-0.382460\pi$
$151$	109.000	0.721854	0.360927	+	0.932594i	$0.382460\pi$
$152$	59.3970i	0.390770i
$153$	0	0
$154$	−10.0000	−0.0649351
$155$	0	0
$156$	0	0
$157$	−118.000	−0.751592	−0.375796	−	0.926702i	$-0.622631\pi$
$157$	−118.000	−0.751592	−0.375796	+	0.926702i	$0.622631\pi$
$158$	165.463i	1.04723i
$159$	0	0
$160$	0	0
$161$	− 7.07107i	− 0.0439197i
$162$	0	0
$163$	203.000	1.24540	0.622699	−	0.782461i	$-0.286036\pi$
$163$	203.000	1.24540	0.622699	+	0.782461i	$0.286036\pi$
$164$	104.652i	0.638121i
$165$	0	0
$166$	82.0000	0.493976
$167$	101.823i	0.609721i	0.952397	+	0.304860i	$0.0986098\pi$
$167$	101.823i	0.609721i	−0.952397	+	0.304860i	$0.901390\pi$
$168$	0	0
$169$	−88.0000	−0.520710
$170$	0	0
$171$	0	0
$172$	−128.000	−0.744186
$173$	11.3137i	0.0653972i	0.999465	+	0.0326986i	$0.0104101\pi$
$173$	11.3137i	0.0653972i	−0.999465	+	0.0326986i	$0.989590\pi$
$174$	0	0
$175$	0	0
$176$	− 5.65685i	− 0.0321412i
$177$	0	0
$178$	−208.000	−1.16854
$179$	− 125.865i	− 0.703156i	−0.936159	−	0.351578i	$-0.885645\pi$
$179$	− 125.865i	− 0.703156i	0.936159	−	0.351578i	$-0.114355\pi$
$180$	0	0
$181$	−127.000	−0.701657	−0.350829	−	0.936440i	$-0.614100\pi$
$181$	−127.000	−0.701657	−0.350829	+	0.936440i	$0.614100\pi$
$182$	− 63.6396i	− 0.349668i
$183$	0	0
$184$	4.00000	0.0217391
$185$	0	0
$186$	0	0
$187$	−16.0000	−0.0855615
$188$	45.2548i	0.240717i
$189$	0	0
$190$	0	0
$191$	101.823i	0.533107i	0.963820	+	0.266553i	$0.0858849\pi$
$191$	101.823i	0.533107i	−0.963820	+	0.266553i	$0.914115\pi$
$192$	0	0
$193$	271.000	1.40415	0.702073	−	0.712105i	$-0.252258\pi$
$193$	271.000	1.40415	0.702073	+	0.712105i	$0.252258\pi$
$194$	57.9828i	0.298880i
$195$	0	0
$196$	48.0000	0.244898
$197$	316.784i	1.60804i	0.594602	+	0.804020i	$0.297309\pi$
$197$	316.784i	1.60804i	−0.594602	+	0.804020i	$0.702691\pi$
$198$	0	0
$199$	−147.000	−0.738693	−0.369347	−	0.929292i	$-0.620419\pi$
$199$	−147.000	−0.738693	−0.369347	+	0.929292i	$0.620419\pi$
$200$	0	0
$201$	0	0
$202$	−128.000	−0.633663
$203$	− 190.919i	− 0.940487i
$204$	0	0
$205$	0	0
$206$	− 18.3848i	− 0.0892465i
$207$	0	0
$208$	36.0000	0.173077
$209$	29.6985i	0.142098i
$210$	0	0
$211$	141.000	0.668246	0.334123	−	0.942529i	$-0.391560\pi$
$211$	141.000	0.668246	0.334123	+	0.942529i	$0.391560\pi$
$212$	− 144.250i	− 0.680424i
$213$	0	0
$214$	−174.000	−0.813084
$215$	0	0
$216$	0	0
$217$	−200.000	−0.921659
$218$	− 11.3137i	− 0.0518977i
$219$	0	0
$220$	0	0
$221$	− 101.823i	− 0.460739i
$222$	0	0
$223$	−8.00000	−0.0358744	−0.0179372	−	0.999839i	$-0.505710\pi$
$223$	−8.00000	−0.0358744	−0.0179372	+	0.999839i	$0.505710\pi$
$224$	− 28.2843i	− 0.126269i
$225$	0	0
$226$	54.0000	0.238938
$227$	69.2965i	0.305271i	0.988283	+	0.152635i	$0.0487760\pi$
$227$	69.2965i	0.305271i	−0.988283	+	0.152635i	$0.951224\pi$
$228$	0	0
$229$	8.00000	0.0349345	0.0174672	−	0.999847i	$-0.494440\pi$
$229$	8.00000	0.0349345	0.0174672	+	0.999847i	$0.494440\pi$
$230$	0	0
$231$	0	0
$232$	108.000	0.465517
$233$	316.784i	1.35959i	0.733403	+	0.679794i	$0.237931\pi$
$233$	316.784i	1.35959i	−0.733403	+	0.679794i	$0.762069\pi$
$234$	0	0
$235$	0	0
$236$	− 181.019i	− 0.767031i
$237$	0	0
$238$	−80.0000	−0.336134
$239$	− 205.061i	− 0.857996i	−0.903306	−	0.428998i	$-0.858867\pi$
$239$	− 205.061i	− 0.857996i	0.903306	−	0.428998i	$-0.141133\pi$
$240$	0	0
$241$	79.0000	0.327801	0.163900	−	0.986477i	$-0.447592\pi$
$241$	79.0000	0.327801	0.163900	+	0.986477i	$0.447592\pi$
$242$	168.291i	0.695419i
$243$	0	0
$244$	194.000	0.795082
$245$	0	0
$246$	0	0
$247$	−189.000	−0.765182
$248$	− 113.137i	− 0.456198i
$249$	0	0
$250$	0	0
$251$	46.6690i	0.185932i	0.995669	+	0.0929662i	$0.0296349\pi$
$251$	46.6690i	0.185932i	−0.995669	+	0.0929662i	$0.970365\pi$
$252$	0	0
$253$	2.00000	0.00790514
$254$	− 11.3137i	− 0.0445422i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 169.706i	− 0.660333i	−0.943923	−	0.330167i	$-0.892895\pi$
$257$	− 169.706i	− 0.660333i	0.943923	−	0.330167i	$-0.107105\pi$
$258$	0	0
$259$	−125.000	−0.482625
$260$	0	0
$261$	0	0
$262$	−192.000	−0.732824
$263$	− 282.843i	− 1.07545i	−0.843121	−	0.537724i	$-0.819284\pi$
$263$	− 282.843i	− 1.07545i	0.843121	−	0.537724i	$-0.180716\pi$
$264$	0	0
$265$	0	0
$266$	148.492i	0.558242i
$267$	0	0
$268$	−262.000	−0.977612
$269$	− 101.823i	− 0.378526i	−0.981926	−	0.189263i	$-0.939390\pi$
$269$	− 101.823i	− 0.378526i	0.981926	−	0.189263i	$-0.0606098\pi$
$270$	0	0
$271$	−221.000	−0.815498	−0.407749	−	0.913094i	$-0.633686\pi$
$271$	−221.000	−0.815498	−0.407749	+	0.913094i	$0.633686\pi$
$272$	− 45.2548i	− 0.166378i
$273$	0	0
$274$	378.000	1.37956
$275$	0	0
$276$	0	0
$277$	88.0000	0.317690	0.158845	−	0.987304i	$-0.449223\pi$
$277$	88.0000	0.317690	0.158845	+	0.987304i	$0.449223\pi$
$278$	52.3259i	0.188223i
$279$	0	0
$280$	0	0
$281$	− 203.647i	− 0.724722i	−0.932038	−	0.362361i	$-0.881971\pi$
$281$	− 203.647i	− 0.724722i	0.932038	−	0.362361i	$-0.118029\pi$
$282$	0	0
$283$	32.0000	0.113074	0.0565371	−	0.998400i	$-0.481994\pi$
$283$	32.0000	0.113074	0.0565371	+	0.998400i	$0.481994\pi$
$284$	− 178.191i	− 0.627433i
$285$	0	0
$286$	18.0000	0.0629371
$287$	261.630i	0.911601i
$288$	0	0
$289$	161.000	0.557093
$290$	0	0
$291$	0	0
$292$	34.0000	0.116438
$293$	− 173.948i	− 0.593680i	−0.954927	−	0.296840i	$-0.904067\pi$
$293$	− 173.948i	− 0.593680i	0.954927	−	0.296840i	$-0.0959328\pi$
$294$	0	0
$295$	0	0
$296$	− 70.7107i	− 0.238887i
$297$	0	0
$298$	−368.000	−1.23490
$299$	12.7279i	0.0425683i
$300$	0	0
$301$	−320.000	−1.06312
$302$	154.149i	0.510428i
$303$	0	0
$304$	−84.0000	−0.276316
$305$	0	0
$306$	0	0
$307$	486.000	1.58306	0.791531	−	0.611129i	$-0.209284\pi$
$307$	486.000	1.58306	0.791531	+	0.611129i	$0.209284\pi$
$308$	− 14.1421i	− 0.0459160i
$309$	0	0
$310$	0	0
$311$	− 67.8823i	− 0.218271i	−0.994027	−	0.109135i	$-0.965192\pi$
$311$	− 67.8823i	− 0.218271i	0.994027	−	0.109135i	$-0.0348082\pi$
$312$	0	0
$313$	−281.000	−0.897764	−0.448882	−	0.893591i	$-0.648178\pi$
$313$	−281.000	−0.897764	−0.448882	+	0.893591i	$0.648178\pi$
$314$	− 166.877i	− 0.531456i
$315$	0	0
$316$	−234.000	−0.740506
$317$	− 520.431i	− 1.64174i	−0.571117	−	0.820868i	$-0.693490\pi$
$317$	− 520.431i	− 1.64174i	0.571117	−	0.820868i	$-0.306510\pi$
$318$	0	0
$319$	54.0000	0.169279
$320$	0	0
$321$	0	0
$322$	10.0000	0.0310559
$323$	237.588i	0.735566i
$324$	0	0
$325$	0	0
$326$	287.085i	0.880630i
$327$	0	0
$328$	−148.000	−0.451220
$329$	113.137i	0.343882i
$330$	0	0
$331$	−59.0000	−0.178248	−0.0891239	−	0.996021i	$-0.528407\pi$
$331$	−59.0000	−0.178248	−0.0891239	+	0.996021i	$0.528407\pi$
$332$	115.966i	0.349294i
$333$	0	0
$334$	−144.000	−0.431138
$335$	0	0
$336$	0	0
$337$	55.0000	0.163205	0.0816024	−	0.996665i	$-0.473996\pi$
$337$	55.0000	0.163205	0.0816024	+	0.996665i	$0.473996\pi$
$338$	− 124.451i	− 0.368198i
$339$	0	0
$340$	0	0
$341$	− 56.5685i	− 0.165890i
$342$	0	0
$343$	365.000	1.06414
$344$	− 181.019i	− 0.526219i
$345$	0	0
$346$	−16.0000	−0.0462428
$347$	− 598.212i	− 1.72395i	−0.506947	−	0.861977i	$-0.669226\pi$
$347$	− 598.212i	− 1.72395i	0.506947	−	0.861977i	$-0.330774\pi$
$348$	0	0
$349$	439.000	1.25788	0.628940	−	0.777454i	$-0.283489\pi$
$349$	439.000	1.25788	0.628940	+	0.777454i	$0.283489\pi$
$350$	0	0
$351$	0	0
$352$	8.00000	0.0227273
$353$	520.431i	1.47431i	0.675725	+	0.737154i	$0.263831\pi$
$353$	520.431i	1.47431i	−0.675725	+	0.737154i	$0.736169\pi$
$354$	0	0
$355$	0	0
$356$	− 294.156i	− 0.826282i
$357$	0	0
$358$	178.000	0.497207
$359$	− 55.1543i	− 0.153633i	−0.997045	−	0.0768166i	$-0.975524\pi$
$359$	− 55.1543i	− 0.153633i	0.997045	−	0.0768166i	$-0.0244756\pi$
$360$	0	0
$361$	80.0000	0.221607
$362$	− 179.605i	− 0.496147i
$363$	0	0
$364$	90.0000	0.247253
$365$	0	0
$366$	0	0
$367$	589.000	1.60490	0.802452	−	0.596716i	$-0.203528\pi$
$367$	589.000	1.60490	0.802452	+	0.596716i	$0.203528\pi$
$368$	5.65685i	0.0153719i
$369$	0	0
$370$	0	0
$371$	− 360.624i	− 0.972034i
$372$	0	0
$373$	9.00000	0.0241287	0.0120643	−	0.999927i	$-0.496160\pi$
$373$	9.00000	0.0241287	0.0120643	+	0.999927i	$0.496160\pi$
$374$	− 22.6274i	− 0.0605011i
$375$	0	0
$376$	−64.0000	−0.170213
$377$	343.654i	0.911549i
$378$	0	0
$379$	−157.000	−0.414248	−0.207124	−	0.978315i	$-0.566410\pi$
$379$	−157.000	−0.414248	−0.207124	+	0.978315i	$0.566410\pi$
$380$	0	0
$381$	0	0
$382$	−144.000	−0.376963
$383$	282.843i	0.738493i	0.929332	+	0.369246i	$0.120384\pi$
$383$	282.843i	0.738493i	−0.929332	+	0.369246i	$0.879616\pi$
$384$	0	0
$385$	0	0
$386$	383.252i	0.992881i
$387$	0	0
$388$	−82.0000	−0.211340
$389$	− 599.627i	− 1.54146i	−0.637164	−	0.770728i	$-0.719893\pi$
$389$	− 599.627i	− 1.54146i	0.637164	−	0.770728i	$-0.280107\pi$
$390$	0	0
$391$	16.0000	0.0409207
$392$	67.8823i	0.173169i
$393$	0	0
$394$	−448.000	−1.13706
$395$	0	0
$396$	0	0
$397$	−296.000	−0.745592	−0.372796	−	0.927913i	$-0.621601\pi$
$397$	−296.000	−0.745592	−0.372796	+	0.927913i	$0.621601\pi$
$398$	− 207.889i	− 0.522335i
$399$	0	0
$400$	0	0
$401$	− 388.909i	− 0.969847i	−0.874557	−	0.484924i	$-0.838847\pi$
$401$	− 388.909i	− 0.969847i	0.874557	−	0.484924i	$-0.161153\pi$
$402$	0	0
$403$	360.000	0.893300
$404$	− 181.019i	− 0.448068i
$405$	0	0
$406$	270.000	0.665025
$407$	− 35.3553i	− 0.0868682i
$408$	0	0
$409$	−145.000	−0.354523	−0.177262	−	0.984164i	$-0.556724\pi$
$409$	−145.000	−0.354523	−0.177262	+	0.984164i	$0.556724\pi$
$410$	0	0
$411$	0	0
$412$	26.0000	0.0631068
$413$	− 452.548i	− 1.09576i
$414$	0	0
$415$	0	0
$416$	50.9117i	0.122384i
$417$	0	0
$418$	−42.0000	−0.100478
$419$	− 701.450i	− 1.67410i	−0.547123	−	0.837052i	$-0.684277\pi$
$419$	− 701.450i	− 1.67410i	0.547123	−	0.837052i	$-0.315723\pi$
$420$	0	0
$421$	505.000	1.19952	0.599762	−	0.800178i	$-0.295262\pi$
$421$	505.000	1.19952	0.599762	+	0.800178i	$0.295262\pi$
$422$	199.404i	0.472522i
$423$	0	0
$424$	204.000	0.481132
$425$	0	0
$426$	0	0
$427$	485.000	1.13583
$428$	− 246.073i	− 0.574937i
$429$	0	0
$430$	0	0
$431$	− 43.8406i	− 0.101718i	−0.998706	−	0.0508592i	$-0.983804\pi$
$431$	− 43.8406i	− 0.101718i	0.998706	−	0.0508592i	$-0.0161960\pi$
$432$	0	0
$433$	−32.0000	−0.0739030	−0.0369515	−	0.999317i	$-0.511765\pi$
$433$	−32.0000	−0.0739030	−0.0369515	+	0.999317i	$0.511765\pi$
$434$	− 282.843i	− 0.651711i
$435$	0	0
$436$	16.0000	0.0366972
$437$	− 29.6985i	− 0.0679599i
$438$	0	0
$439$	504.000	1.14806	0.574032	−	0.818833i	$-0.305379\pi$
$439$	504.000	1.14806	0.574032	+	0.818833i	$0.305379\pi$
$440$	0	0
$441$	0	0
$442$	144.000	0.325792
$443$	− 237.588i	− 0.536316i	−0.963375	−	0.268158i	$-0.913585\pi$
$443$	− 237.588i	− 0.536316i	0.963375	−	0.268158i	$-0.0864149\pi$
$444$	0	0
$445$	0	0
$446$	− 11.3137i	− 0.0253671i
$447$	0	0
$448$	40.0000	0.0892857
$449$	− 67.8823i	− 0.151185i	−0.997139	−	0.0755927i	$-0.975915\pi$
$449$	− 67.8823i	− 0.151185i	0.997139	−	0.0755927i	$-0.0240849\pi$
$450$	0	0
$451$	−74.0000	−0.164080
$452$	76.3675i	0.168955i
$453$	0	0
$454$	−98.0000	−0.215859
$455$	0	0
$456$	0	0
$457$	−752.000	−1.64551	−0.822757	−	0.568393i	$-0.807565\pi$
$457$	−752.000	−1.64551	−0.822757	+	0.568393i	$0.807565\pi$
$458$	11.3137i	0.0247024i
$459$	0	0
$460$	0	0
$461$	610.940i	1.32525i	0.748951	+	0.662625i	$0.230558\pi$
$461$	610.940i	1.32525i	−0.748951	+	0.662625i	$0.769442\pi$
$462$	0	0
$463$	597.000	1.28942	0.644708	−	0.764429i	$-0.276979\pi$
$463$	597.000	1.28942	0.644708	+	0.764429i	$0.276979\pi$
$464$	152.735i	0.329170i
$465$	0	0
$466$	−448.000	−0.961373
$467$	848.528i	1.81698i	0.417910	+	0.908488i	$0.362763\pi$
$467$	848.528i	1.81698i	−0.417910	+	0.908488i	$0.637237\pi$
$468$	0	0
$469$	−655.000	−1.39659
$470$	0	0
$471$	0	0
$472$	256.000	0.542373
$473$	− 90.5097i	− 0.191352i
$474$	0	0
$475$	0	0
$476$	− 113.137i	− 0.237683i
$477$	0	0
$478$	290.000	0.606695
$479$	168.291i	0.351339i	0.984449	+	0.175670i	$0.0562090\pi$
$479$	168.291i	0.351339i	−0.984449	+	0.175670i	$0.943791\pi$
$480$	0	0
$481$	225.000	0.467775
$482$	111.723i	0.231790i
$483$	0	0
$484$	−238.000	−0.491736
$485$	0	0
$486$	0	0
$487$	507.000	1.04107	0.520534	−	0.853841i	$-0.325733\pi$
$487$	507.000	1.04107	0.520534	+	0.853841i	$0.325733\pi$
$488$	274.357i	0.562208i
$489$	0	0
$490$	0	0
$491$	428.507i	0.872722i	0.899772	+	0.436361i	$0.143733\pi$
$491$	428.507i	0.872722i	−0.899772	+	0.436361i	$0.856267\pi$
$492$	0	0
$493$	432.000	0.876268
$494$	− 267.286i	− 0.541066i
$495$	0	0
$496$	160.000	0.322581
$497$	− 445.477i	− 0.896333i
$498$	0	0
$499$	−870.000	−1.74349	−0.871743	−	0.489963i	$-0.837010\pi$
$499$	−870.000	−1.74349	−0.871743	+	0.489963i	$0.837010\pi$
$500$	0	0
$501$	0	0
$502$	−66.0000	−0.131474
$503$	462.448i	0.919379i	0.888080	+	0.459690i	$0.152039\pi$
$503$	462.448i	0.919379i	−0.888080	+	0.459690i	$0.847961\pi$
$504$	0	0
$505$	0	0
$506$	2.82843i	0.00558978i
$507$	0	0
$508$	16.0000	0.0314961
$509$	− 818.830i	− 1.60870i	−0.594154	−	0.804351i	$-0.702513\pi$
$509$	− 818.830i	− 1.60870i	0.594154	−	0.804351i	$-0.297487\pi$
$510$	0	0
$511$	85.0000	0.166341
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	240.000	0.466926
$515$	0	0
$516$	0	0
$517$	−32.0000	−0.0618956
$518$	− 176.777i	− 0.341268i
$519$	0	0
$520$	0	0
$521$	− 864.084i	− 1.65851i	−0.558869	−	0.829256i	$-0.688765\pi$
$521$	− 864.084i	− 1.65851i	0.558869	−	0.829256i	$-0.311235\pi$
$522$	0	0
$523$	−163.000	−0.311663	−0.155832	−	0.987784i	$-0.549806\pi$
$523$	−163.000	−0.311663	−0.155832	+	0.987784i	$0.549806\pi$
$524$	− 271.529i	− 0.518185i
$525$	0	0
$526$	400.000	0.760456
$527$	− 452.548i	− 0.858726i
$528$	0	0
$529$	527.000	0.996219
$530$	0	0
$531$	0	0
$532$	−210.000	−0.394737
$533$	− 470.933i	− 0.883552i
$534$	0	0
$535$	0	0
$536$	− 370.524i	− 0.691276i
$537$	0	0
$538$	144.000	0.267658
$539$	33.9411i	0.0629705i
$540$	0	0
$541$	697.000	1.28835	0.644177	−	0.764876i	$-0.277200\pi$
$541$	697.000	1.28835	0.644177	+	0.764876i	$0.277200\pi$
$542$	− 312.541i	− 0.576644i
$543$	0	0
$544$	64.0000	0.117647
$545$	0	0
$546$	0	0
$547$	−389.000	−0.711152	−0.355576	−	0.934647i	$-0.615715\pi$
$547$	−389.000	−0.711152	−0.355576	+	0.934647i	$0.615715\pi$
$548$	534.573i	0.975498i
$549$	0	0
$550$	0	0
$551$	− 801.859i	− 1.45528i
$552$	0	0
$553$	−585.000	−1.05787
$554$	124.451i	0.224640i
$555$	0	0
$556$	−74.0000	−0.133094
$557$	1086.12i	1.94994i	0.222339	+	0.974969i	$0.428631\pi$
$557$	1086.12i	1.94994i	−0.222339	+	0.974969i	$0.571369\pi$
$558$	0	0
$559$	576.000	1.03041
$560$	0	0
$561$	0	0
$562$	288.000	0.512456
$563$	622.254i	1.10525i	0.833431	+	0.552623i	$0.186373\pi$
$563$	622.254i	1.10525i	−0.833431	+	0.552623i	$0.813627\pi$
$564$	0	0
$565$	0	0
$566$	45.2548i	0.0799555i
$567$	0	0
$568$	252.000	0.443662
$569$	− 463.862i	− 0.815223i	−0.913155	−	0.407612i	$-0.866362\pi$
$569$	− 463.862i	− 0.815223i	0.913155	−	0.407612i	$-0.133638\pi$
$570$	0	0
$571$	−923.000	−1.61646	−0.808231	−	0.588865i	$-0.799575\pi$
$571$	−923.000	−1.61646	−0.808231	+	0.588865i	$0.799575\pi$
$572$	25.4558i	0.0445032i
$573$	0	0
$574$	−370.000	−0.644599
$575$	0	0
$576$	0	0
$577$	247.000	0.428076	0.214038	−	0.976825i	$-0.431338\pi$
$577$	247.000	0.428076	0.214038	+	0.976825i	$0.431338\pi$
$578$	227.688i	0.393925i
$579$	0	0
$580$	0	0
$581$	289.914i	0.498991i
$582$	0	0
$583$	102.000	0.174957
$584$	48.0833i	0.0823344i
$585$	0	0
$586$	246.000	0.419795
$587$	− 453.963i	− 0.773360i	−0.922214	−	0.386680i	$-0.873622\pi$
$587$	− 453.963i	− 0.773360i	0.922214	−	0.386680i	$-0.126378\pi$
$588$	0	0
$589$	−840.000	−1.42615
$590$	0	0
$591$	0	0
$592$	100.000	0.168919
$593$	4.24264i	0.00715454i	0.999994	+	0.00357727i	$0.00113868\pi$
$593$	4.24264i	0.00715454i	−0.999994	+	0.00357727i	$0.998861\pi$
$594$	0	0
$595$	0	0
$596$	− 520.431i	− 0.873206i
$597$	0	0
$598$	−18.0000	−0.0301003
$599$	− 224.860i	− 0.375392i	−0.982227	−	0.187696i	$-0.939898\pi$
$599$	− 224.860i	− 0.375392i	0.982227	−	0.187696i	$-0.0601020\pi$
$600$	0	0
$601$	−736.000	−1.22463	−0.612313	−	0.790616i	$-0.709761\pi$
$601$	−736.000	−1.22463	−0.612313	+	0.790616i	$0.709761\pi$
$602$	− 452.548i	− 0.751741i
$603$	0	0
$604$	−218.000	−0.360927
$605$	0	0
$606$	0	0
$607$	437.000	0.719934	0.359967	−	0.932965i	$-0.382788\pi$
$607$	437.000	0.719934	0.359967	+	0.932965i	$0.382788\pi$
$608$	− 118.794i	− 0.195385i
$609$	0	0
$610$	0	0
$611$	− 203.647i	− 0.333301i
$612$	0	0
$613$	−335.000	−0.546493	−0.273246	−	0.961944i	$-0.588097\pi$
$613$	−335.000	−0.546493	−0.273246	+	0.961944i	$0.588097\pi$
$614$	687.308i	1.11939i
$615$	0	0
$616$	20.0000	0.0324675
$617$	255.973i	0.414867i	0.978249	+	0.207433i	$0.0665110\pi$
$617$	255.973i	0.414867i	−0.978249	+	0.207433i	$0.933489\pi$
$618$	0	0
$619$	965.000	1.55897	0.779483	−	0.626423i	$-0.215482\pi$
$619$	965.000	1.55897	0.779483	+	0.626423i	$0.215482\pi$
$620$	0	0
$621$	0	0
$622$	96.0000	0.154341
$623$	− 735.391i	− 1.18040i
$624$	0	0
$625$	0	0
$626$	− 397.394i	− 0.634815i
$627$	0	0
$628$	236.000	0.375796
$629$	− 282.843i	− 0.449670i
$630$	0	0
$631$	275.000	0.435816	0.217908	−	0.975969i	$-0.430077\pi$
$631$	275.000	0.435816	0.217908	+	0.975969i	$0.430077\pi$
$632$	− 330.926i	− 0.523617i
$633$	0	0
$634$	736.000	1.16088
$635$	0	0
$636$	0	0
$637$	−216.000	−0.339089
$638$	76.3675i	0.119698i
$639$	0	0
$640$	0	0
$641$	486.489i	0.758954i	0.925201	+	0.379477i	$0.123896\pi$
$641$	486.489i	0.758954i	−0.925201	+	0.379477i	$0.876104\pi$
$642$	0	0
$643$	−1152.00	−1.79160	−0.895801	−	0.444455i	$-0.853397\pi$
$643$	−1152.00	−1.79160	−0.895801	+	0.444455i	$0.853397\pi$
$644$	14.1421i	0.0219598i
$645$	0	0
$646$	−336.000	−0.520124
$647$	− 691.550i	− 1.06886i	−0.845214	−	0.534428i	$-0.820527\pi$
$647$	− 691.550i	− 1.06886i	0.845214	−	0.534428i	$-0.179473\pi$
$648$	0	0
$649$	128.000	0.197227
$650$	0	0
$651$	0	0
$652$	−406.000	−0.622699
$653$	− 350.725i	− 0.537098i	−0.963266	−	0.268549i	$-0.913456\pi$
$653$	− 350.725i	− 0.537098i	0.963266	−	0.268549i	$-0.0865441\pi$
$654$	0	0
$655$	0	0
$656$	− 209.304i	− 0.319060i
$657$	0	0
$658$	−160.000	−0.243161
$659$	497.803i	0.755392i	0.925930	+	0.377696i	$0.123284\pi$
$659$	497.803i	0.755392i	−0.925930	+	0.377696i	$0.876716\pi$
$660$	0	0
$661$	−577.000	−0.872920	−0.436460	−	0.899724i	$-0.643768\pi$
$661$	−577.000	−0.872920	−0.436460	+	0.899724i	$0.643768\pi$
$662$	− 83.4386i	− 0.126040i
$663$	0	0
$664$	−164.000	−0.246988
$665$	0	0
$666$	0	0
$667$	−54.0000	−0.0809595
$668$	− 203.647i	− 0.304860i
$669$	0	0
$670$	0	0
$671$	137.179i	0.204439i
$672$	0	0
$673$	−489.000	−0.726597	−0.363299	−	0.931673i	$-0.618350\pi$
$673$	−489.000	−0.726597	−0.363299	+	0.931673i	$0.618350\pi$
$674$	77.7817i	0.115403i
$675$	0	0
$676$	176.000	0.260355
$677$	599.627i	0.885711i	0.896593	+	0.442856i	$0.146035\pi$
$677$	599.627i	0.885711i	−0.896593	+	0.442856i	$0.853965\pi$
$678$	0	0
$679$	−205.000	−0.301915
$680$	0	0
$681$	0	0
$682$	80.0000	0.117302
$683$	− 236.174i	− 0.345789i	−0.984940	−	0.172894i	$-0.944688\pi$
$683$	− 236.174i	− 0.345789i	0.984940	−	0.172894i	$-0.0553119\pi$
$684$	0	0
$685$	0	0
$686$	516.188i	0.752461i
$687$	0	0
$688$	256.000	0.372093
$689$	649.124i	0.942125i
$690$	0	0
$691$	640.000	0.926194	0.463097	−	0.886308i	$-0.346738\pi$
$691$	640.000	0.926194	0.463097	+	0.886308i	$0.346738\pi$
$692$	− 22.6274i	− 0.0326986i
$693$	0	0
$694$	846.000	1.21902
$695$	0	0
$696$	0	0
$697$	−592.000	−0.849354
$698$	620.840i	0.889455i
$699$	0	0
$700$	0	0
$701$	1093.19i	1.55947i	0.626111	+	0.779734i	$0.284646\pi$
$701$	1093.19i	1.55947i	−0.626111	+	0.779734i	$0.715354\pi$
$702$	0	0
$703$	−525.000	−0.746799
$704$	11.3137i	0.0160706i
$705$	0	0
$706$	−736.000	−1.04249
$707$	− 452.548i	− 0.640097i
$708$	0	0
$709$	489.000	0.689704	0.344852	−	0.938657i	$-0.387929\pi$
$709$	489.000	0.689704	0.344852	+	0.938657i	$0.387929\pi$
$710$	0	0
$711$	0	0
$712$	416.000	0.584270
$713$	56.5685i	0.0793388i
$714$	0	0
$715$	0	0
$716$	251.730i	0.351578i
$717$	0	0
$718$	78.0000	0.108635
$719$	− 620.840i	− 0.863477i	−0.901999	−	0.431738i	$-0.857900\pi$
$719$	− 620.840i	− 0.863477i	0.901999	−	0.431738i	$-0.142100\pi$
$720$	0	0
$721$	65.0000	0.0901526
$722$	113.137i	0.156700i
$723$	0	0
$724$	254.000	0.350829
$725$	0	0
$726$	0	0
$727$	1080.00	1.48556	0.742779	−	0.669537i	$-0.233508\pi$
$727$	1080.00	1.48556	0.742779	+	0.669537i	$0.233508\pi$
$728$	127.279i	0.174834i
$729$	0	0
$730$	0	0
$731$	− 724.077i	− 0.990530i
$732$	0	0
$733$	248.000	0.338336	0.169168	−	0.985587i	$-0.445892\pi$
$733$	248.000	0.338336	0.169168	+	0.985587i	$0.445892\pi$
$734$	832.972i	1.13484i
$735$	0	0
$736$	−8.00000	−0.0108696
$737$	− 185.262i	− 0.251373i
$738$	0	0
$739$	−848.000	−1.14750	−0.573748	−	0.819032i	$-0.694511\pi$
$739$	−848.000	−1.14750	−0.573748	+	0.819032i	$0.694511\pi$
$740$	0	0
$741$	0	0
$742$	510.000	0.687332
$743$	− 33.9411i	− 0.0456812i	−0.999739	−	0.0228406i	$-0.992729\pi$
$743$	− 33.9411i	− 0.0456812i	0.999739	−	0.0228406i	$-0.00727102\pi$
$744$	0	0
$745$	0	0
$746$	12.7279i	0.0170616i
$747$	0	0
$748$	32.0000	0.0427807
$749$	− 615.183i	− 0.821339i
$750$	0	0
$751$	−133.000	−0.177097	−0.0885486	−	0.996072i	$-0.528223\pi$
$751$	−133.000	−0.177097	−0.0885486	+	0.996072i	$0.528223\pi$
$752$	− 90.5097i	− 0.120359i
$753$	0	0
$754$	−486.000	−0.644562
$755$	0	0
$756$	0	0
$757$	−1271.00	−1.67900	−0.839498	−	0.543363i	$-0.817151\pi$
$757$	−1271.00	−1.67900	−0.839498	+	0.543363i	$0.817151\pi$
$758$	− 222.032i	− 0.292918i
$759$	0	0
$760$	0	0
$761$	538.815i	0.708036i	0.935239	+	0.354018i	$0.115185\pi$
$761$	538.815i	0.708036i	−0.935239	+	0.354018i	$0.884815\pi$
$762$	0	0
$763$	40.0000	0.0524246
$764$	− 203.647i	− 0.266553i
$765$	0	0
$766$	−400.000	−0.522193
$767$	814.587i	1.06204i
$768$	0	0
$769$	81.0000	0.105332	0.0526658	−	0.998612i	$-0.483228\pi$
$769$	81.0000	0.105332	0.0526658	+	0.998612i	$0.483228\pi$
$770$	0	0
$771$	0	0
$772$	−542.000	−0.702073
$773$	− 445.477i	− 0.576297i	−0.957586	−	0.288148i	$-0.906960\pi$
$773$	− 445.477i	− 0.576297i	0.957586	−	0.288148i	$-0.0930396\pi$
$774$	0	0
$775$	0	0
$776$	− 115.966i	− 0.149440i
$777$	0	0
$778$	848.000	1.08997
$779$	1098.84i	1.41058i
$780$	0	0
$781$	126.000	0.161332
$782$	22.6274i	0.0289353i
$783$	0	0
$784$	−96.0000	−0.122449
$785$	0	0
$786$	0	0
$787$	−395.000	−0.501906	−0.250953	−	0.967999i	$-0.580744\pi$
$787$	−395.000	−0.501906	−0.250953	+	0.967999i	$0.580744\pi$
$788$	− 633.568i	− 0.804020i
$789$	0	0
$790$	0	0
$791$	190.919i	0.241364i
$792$	0	0
$793$	−873.000	−1.10088
$794$	− 418.607i	− 0.527213i
$795$	0	0
$796$	294.000	0.369347
$797$	1074.80i	1.34856i	0.738476	+	0.674280i	$0.235546\pi$
$797$	1074.80i	1.34856i	−0.738476	+	0.674280i	$0.764454\pi$
$798$	0	0
$799$	−256.000	−0.320401
$800$	0	0
$801$	0	0
$802$	550.000	0.685786
$803$	24.0416i	0.0299398i
$804$	0	0
$805$	0	0
$806$	509.117i	0.631659i
$807$	0	0
$808$	256.000	0.316832
$809$	1255.82i	1.55231i	0.630540	+	0.776157i	$0.282833\pi$
$809$	1255.82i	1.55231i	−0.630540	+	0.776157i	$0.717167\pi$
$810$	0	0
$811$	−752.000	−0.927250	−0.463625	−	0.886031i	$-0.653452\pi$
$811$	−752.000	−0.927250	−0.463625	+	0.886031i	$0.653452\pi$
$812$	381.838i	0.470243i
$813$	0	0
$814$	50.0000	0.0614251
$815$	0	0
$816$	0	0
$817$	−1344.00	−1.64504
$818$	− 205.061i	− 0.250686i
$819$	0	0
$820$	0	0
$821$	497.803i	0.606338i	0.952937	+	0.303169i	$0.0980446\pi$
$821$	497.803i	0.606338i	−0.952937	+	0.303169i	$0.901955\pi$
$822$	0	0
$823$	−531.000	−0.645200	−0.322600	−	0.946535i	$-0.604557\pi$
$823$	−531.000	−0.645200	−0.322600	+	0.946535i	$0.604557\pi$
$824$	36.7696i	0.0446232i
$825$	0	0
$826$	640.000	0.774818
$827$	− 350.725i	− 0.424093i	−0.977260	−	0.212047i	$-0.931987\pi$
$827$	− 350.725i	− 0.424093i	0.977260	−	0.212047i	$-0.0680128\pi$
$828$	0	0
$829$	705.000	0.850422	0.425211	−	0.905094i	$-0.360200\pi$
$829$	705.000	0.850422	0.425211	+	0.905094i	$0.360200\pi$
$830$	0	0
$831$	0	0
$832$	−72.0000	−0.0865385
$833$	271.529i	0.325965i
$834$	0	0
$835$	0	0
$836$	− 59.3970i	− 0.0710490i
$837$	0	0
$838$	992.000	1.18377
$839$	1617.86i	1.92832i	0.265322	+	0.964160i	$0.414522\pi$
$839$	1617.86i	1.92832i	−0.265322	+	0.964160i	$0.585478\pi$
$840$	0	0
$841$	−617.000	−0.733650
$842$	714.178i	0.848192i
$843$	0	0
$844$	−282.000	−0.334123
$845$	0	0
$846$	0	0
$847$	−595.000	−0.702479
$848$	288.500i	0.340212i
$849$	0	0
$850$	0	0
$851$	35.3553i	0.0415456i
$852$	0	0
$853$	−1639.00	−1.92145	−0.960727	−	0.277496i	$-0.910496\pi$
$853$	−1639.00	−1.92145	−0.960727	+	0.277496i	$0.910496\pi$
$854$	685.894i	0.803154i
$855$	0	0
$856$	348.000	0.406542
$857$	− 988.535i	− 1.15348i	−0.816927	−	0.576742i	$-0.804324\pi$
$857$	− 988.535i	− 1.15348i	0.816927	−	0.576742i	$-0.195676\pi$
$858$	0	0
$859$	−355.000	−0.413271	−0.206636	−	0.978418i	$-0.566251\pi$
$859$	−355.000	−0.413271	−0.206636	+	0.978418i	$0.566251\pi$
$860$	0	0
$861$	0	0
$862$	62.0000	0.0719258
$863$	1459.47i	1.69116i	0.533851	+	0.845578i	$0.320744\pi$
$863$	1459.47i	1.69116i	−0.533851	+	0.845578i	$0.679256\pi$
$864$	0	0
$865$	0	0
$866$	− 45.2548i	− 0.0522573i
$867$	0	0
$868$	400.000	0.460829
$869$	− 165.463i	− 0.190406i
$870$	0	0
$871$	1179.00	1.35362
$872$	22.6274i	0.0259489i
$873$	0	0
$874$	42.0000	0.0480549
$875$	0	0
$876$	0	0
$877$	−1129.00	−1.28734	−0.643672	−	0.765302i	$-0.722590\pi$
$877$	−1129.00	−1.28734	−0.643672	+	0.765302i	$0.722590\pi$
$878$	712.764i	0.811804i
$879$	0	0
$880$	0	0
$881$	165.463i	0.187813i	0.995581	+	0.0939063i	$0.0299354\pi$
$881$	165.463i	0.187813i	−0.995581	+	0.0939063i	$0.970065\pi$
$882$	0	0
$883$	1227.00	1.38958	0.694790	−	0.719212i	$-0.255497\pi$
$883$	1227.00	1.38958	0.694790	+	0.719212i	$0.255497\pi$
$884$	203.647i	0.230370i
$885$	0	0
$886$	336.000	0.379233
$887$	237.588i	0.267856i	0.990991	+	0.133928i	$0.0427590\pi$
$887$	237.588i	0.267856i	−0.990991	+	0.133928i	$0.957241\pi$
$888$	0	0
$889$	40.0000	0.0449944
$890$	0	0
$891$	0	0
$892$	16.0000	0.0179372
$893$	475.176i	0.532112i
$894$	0	0
$895$	0	0
$896$	56.5685i	0.0631345i
$897$	0	0
$898$	96.0000	0.106904
$899$	1527.35i	1.69894i
$900$	0	0
$901$	816.000	0.905660
$902$	− 104.652i	− 0.116022i
$903$	0	0
$904$	−108.000	−0.119469
$905$	0	0
$906$	0	0
$907$	1005.00	1.10805	0.554024	−	0.832501i	$-0.313091\pi$
$907$	1005.00	1.10805	0.554024	+	0.832501i	$0.313091\pi$
$908$	− 138.593i	− 0.152635i
$909$	0	0
$910$	0	0
$911$	780.646i	0.856911i	0.903563	+	0.428455i	$0.140942\pi$
$911$	780.646i	0.856911i	−0.903563	+	0.428455i	$0.859058\pi$
$912$	0	0
$913$	−82.0000	−0.0898138
$914$	− 1063.49i	− 1.16355i
$915$	0	0
$916$	−16.0000	−0.0174672
$917$	− 678.823i	− 0.740264i
$918$	0	0
$919$	−600.000	−0.652884	−0.326442	−	0.945217i	$-0.605850\pi$
$919$	−600.000	−0.652884	−0.326442	+	0.945217i	$0.605850\pi$
$920$	0	0
$921$	0	0
$922$	−864.000	−0.937093
$923$	801.859i	0.868753i
$924$	0	0
$925$	0	0
$926$	844.285i	0.911755i
$927$	0	0
$928$	−216.000	−0.232759
$929$	1260.06i	1.35637i	0.734893	+	0.678183i	$0.237232\pi$
$929$	1260.06i	1.35637i	−0.734893	+	0.678183i	$0.762768\pi$
$930$	0	0
$931$	504.000	0.541353
$932$	− 633.568i	− 0.679794i
$933$	0	0
$934$	−1200.00	−1.28480
$935$	0	0
$936$	0	0
$937$	−1465.00	−1.56350	−0.781750	−	0.623592i	$-0.785673\pi$
$937$	−1465.00	−1.56350	−0.781750	+	0.623592i	$0.785673\pi$
$938$	− 926.310i	− 0.987537i
$939$	0	0
$940$	0	0
$941$	− 255.973i	− 0.272022i	−0.990707	−	0.136011i	$-0.956572\pi$
$941$	− 255.973i	− 0.272022i	0.990707	−	0.136011i	$-0.0434282\pi$
$942$	0	0
$943$	74.0000	0.0784730
$944$	362.039i	0.383516i
$945$	0	0
$946$	128.000	0.135307
$947$	407.294i	0.430088i	0.976604	+	0.215044i	$0.0689895\pi$
$947$	407.294i	0.430088i	−0.976604	+	0.215044i	$0.931010\pi$
$948$	0	0
$949$	−153.000	−0.161222
$950$	0	0
$951$	0	0
$952$	160.000	0.168067
$953$	− 497.803i	− 0.522354i	−0.965291	−	0.261177i	$-0.915889\pi$
$953$	− 497.803i	− 0.522354i	0.965291	−	0.261177i	$-0.0841106\pi$
$954$	0	0
$955$	0	0
$956$	410.122i	0.428998i
$957$	0	0
$958$	−238.000	−0.248434
$959$	1336.43i	1.39357i
$960$	0	0
$961$	639.000	0.664932
$962$	318.198i	0.330767i
$963$	0	0
$964$	−158.000	−0.163900
$965$	0	0
$966$	0	0
$967$	491.000	0.507756	0.253878	−	0.967236i	$-0.418294\pi$
$967$	491.000	0.507756	0.253878	+	0.967236i	$0.418294\pi$
$968$	− 336.583i	− 0.347710i
$969$	0	0
$970$	0	0
$971$	− 509.117i	− 0.524322i	−0.965024	−	0.262161i	$-0.915565\pi$
$971$	− 509.117i	− 0.524322i	0.965024	−	0.262161i	$-0.0844352\pi$
$972$	0	0
$973$	−185.000	−0.190134
$974$	717.006i	0.736146i
$975$	0	0
$976$	−388.000	−0.397541
$977$	− 470.933i	− 0.482020i	−0.970523	−	0.241010i	$-0.922521\pi$
$977$	− 470.933i	− 0.482020i	0.970523	−	0.241010i	$-0.0774786\pi$
$978$	0	0
$979$	208.000	0.212462
$980$	0	0
$981$	0	0
$982$	−606.000	−0.617108
$983$	601.041i	0.611435i	0.952122	+	0.305718i	$0.0988963\pi$
$983$	601.041i	0.611435i	−0.952122	+	0.305718i	$0.901104\pi$
$984$	0	0
$985$	0	0
$986$	610.940i	0.619615i
$987$	0	0
$988$	378.000	0.382591
$989$	90.5097i	0.0915163i
$990$	0	0
$991$	−755.000	−0.761857	−0.380928	−	0.924605i	$-0.624396\pi$
$991$	−755.000	−0.761857	−0.380928	+	0.924605i	$0.624396\pi$
$992$	226.274i	0.228099i
$993$	0	0
$994$	630.000	0.633803
$995$	0	0
$996$	0	0
$997$	−24.0000	−0.0240722	−0.0120361	−	0.999928i	$-0.503831\pi$
$997$	−24.0000	−0.0240722	−0.0120361	+	0.999928i	$0.503831\pi$
$998$	− 1230.37i	− 1.23283i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1350.3.d.f.701.2		2
3.2	odd	2	inner	1350.3.d.f.701.1		2
5.2	odd	4		270.3.b.c.269.2	yes	4
5.3	odd	4		270.3.b.c.269.4	yes	4
5.4	even	2		1350.3.d.g.701.1		2
15.2	even	4		270.3.b.c.269.3	yes	4
15.8	even	4		270.3.b.c.269.1	✓	4
15.14	odd	2		1350.3.d.g.701.2		2
20.3	even	4		2160.3.c.i.1889.4		4
20.7	even	4		2160.3.c.i.1889.2		4
45.2	even	12		810.3.j.e.539.2		8
45.7	odd	12		810.3.j.e.539.3		8
45.13	odd	12		810.3.j.e.269.2		8
45.22	odd	12		810.3.j.e.269.4		8
45.23	even	12		810.3.j.e.269.3		8
45.32	even	12		810.3.j.e.269.1		8
45.38	even	12		810.3.j.e.539.4		8
45.43	odd	12		810.3.j.e.539.1		8
60.23	odd	4		2160.3.c.i.1889.1		4
60.47	odd	4		2160.3.c.i.1889.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.b.c.269.1	✓	4	15.8	even	4
270.3.b.c.269.2	yes	4	5.2	odd	4
270.3.b.c.269.3	yes	4	15.2	even	4
270.3.b.c.269.4	yes	4	5.3	odd	4
810.3.j.e.269.1		8	45.32	even	12
810.3.j.e.269.2		8	45.13	odd	12
810.3.j.e.269.3		8	45.23	even	12
810.3.j.e.269.4		8	45.22	odd	12
810.3.j.e.539.1		8	45.43	odd	12
810.3.j.e.539.2		8	45.2	even	12
810.3.j.e.539.3		8	45.7	odd	12
810.3.j.e.539.4		8	45.38	even	12
1350.3.d.f.701.1		2	3.2	odd	2	inner
1350.3.d.f.701.2		2	1.1	even	1	trivial
1350.3.d.g.701.1		2	5.4	even	2
1350.3.d.g.701.2		2	15.14	odd	2
2160.3.c.i.1889.1		4	60.23	odd	4
2160.3.c.i.1889.2		4	20.7	even	4
2160.3.c.i.1889.3		4	60.47	odd	4
2160.3.c.i.1889.4		4	20.3	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1350.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -5.00000 q^{7} -2.82843i q^{8} -1.41421i q^{11} +9.00000 q^{13} -7.07107i q^{14} +4.00000 q^{16} -11.3137i q^{17} -21.0000 q^{19} +2.00000 q^{22} +1.41421i q^{23} +12.7279i q^{26} +10.0000 q^{28} +38.1838i q^{29} +40.0000 q^{31} +5.65685i q^{32} +16.0000 q^{34} +25.0000 q^{37} -29.6985i q^{38} -52.3259i q^{41} +64.0000 q^{43} +2.82843i q^{44} -2.00000 q^{46} -22.6274i q^{47} -24.0000 q^{49} -18.0000 q^{52} +72.1249i q^{53} +14.1421i q^{56} -54.0000 q^{58} +90.5097i q^{59} -97.0000 q^{61} +56.5685i q^{62} -8.00000 q^{64} +131.000 q^{67} +22.6274i q^{68} +89.0955i q^{71} -17.0000 q^{73} +35.3553i q^{74} +42.0000 q^{76} +7.07107i q^{77} +117.000 q^{79} +74.0000 q^{82} -57.9828i q^{83} +90.5097i q^{86} -4.00000 q^{88} +147.078i q^{89} -45.0000 q^{91} -2.82843i q^{92} +32.0000 q^{94} +41.0000 q^{97} -33.9411i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{4} - 10 q^{7}+O(q^{10}) \) 2 * q - 4 * q^4 - 10 * q^7	\( 2 q - 4 q^{4} - 10 q^{7} + 18 q^{13} + 8 q^{16} - 42 q^{19} + 4 q^{22} + 20 q^{28} + 80 q^{31} + 32 q^{34} + 50 q^{37} + 128 q^{43} - 4 q^{46} - 48 q^{49} - 36 q^{52} - 108 q^{58} - 194 q^{61} - 16 q^{64} + 262 q^{67} - 34 q^{73} + 84 q^{76} + 234 q^{79} + 148 q^{82} - 8 q^{88} - 90 q^{91} + 64 q^{94} + 82 q^{97}+O(q^{100}) \) 2 * q - 4 * q^4 - 10 * q^7 + 18 * q^13 + 8 * q^16 - 42 * q^19 + 4 * q^22 + 20 * q^28 + 80 * q^31 + 32 * q^34 + 50 * q^37 + 128 * q^43 - 4 * q^46 - 48 * q^49 - 36 * q^52 - 108 * q^58 - 194 * q^61 - 16 * q^64 + 262 * q^67 - 34 * q^73 + 84 * q^76 + 234 * q^79 + 148 * q^82 - 8 * q^88 - 90 * q^91 + 64 * q^94 + 82 * q^97

Introduction

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