LMFDB - Embedded newform 2160.3.c.i.1889.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2160,3,Mod(1889,2160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2160, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2160.1889");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2160 = 2^{4} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2160.c (of order $2$, degree $1$, not 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:	$58.8557371018$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 270)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1889.1
Root			$0.707107 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	2160.1889
Dual form			2160.3.c.i.1889.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.53553 - 3.53553i) q^{5} -5.00000i q^{7} +O(q^{10})$	$q+(-3.53553 - 3.53553i) q^{5} -5.00000i q^{7} -1.41421i q^{11} +9.00000i q^{13} +11.3137 q^{17} -21.0000 q^{19} -1.41421 q^{23} +25.0000i q^{25} +38.1838i q^{29} -40.0000 q^{31} +(-17.6777 + 17.6777i) q^{35} -25.0000i q^{37} +52.3259i q^{41} -64.0000i q^{43} -22.6274 q^{47} +24.0000 q^{49} +72.1249 q^{53} +(-5.00000 + 5.00000i) q^{55} -90.5097i q^{59} -97.0000 q^{61} +(31.8198 - 31.8198i) q^{65} +131.000i q^{67} +89.0955i q^{71} -17.0000i q^{73} -7.07107 q^{77} +117.000 q^{79} +57.9828 q^{83} +(-40.0000 - 40.0000i) q^{85} +147.078i q^{89} +45.0000 q^{91} +(74.2462 + 74.2462i) q^{95} -41.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 84 q^{19} - 160 q^{31} + 96 q^{49} - 20 q^{55} - 388 q^{61} + 468 q^{79} - 160 q^{85} + 180 q^{91}+O(q^{100}) $ 4 * q - 84 * q^19 - 160 * q^31 + 96 * q^49 - 20 * q^55 - 388 * q^61 + 468 * q^79 - 160 * q^85 + 180 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times$.

$n$	$271$	$1297$	$1621$	$2081$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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 $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$5$	−3.53553	−	3.53553i	−0.707107	−	0.707107i
$6$		0			0
$7$		−	5.00000i		−	0.714286i	−0.934050	−	0.357143i	$-0.883751\pi$
$7$		−	5.00000i		−	0.714286i	0.934050	−	0.357143i	$-0.116249\pi$
$8$		0			0
$9$		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
$13$			9.00000i			0.692308i	0.938178	+	0.346154i	$0.112512\pi$
$13$			9.00000i			0.692308i	−0.938178	+	0.346154i	$0.887488\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	11.3137			0.665512			0.332756	−	0.943013i	$-0.392021\pi$
$17$	11.3137			0.665512			0.332756	+	0.943013i	$0.392021\pi$
$18$		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
$22$		0			0
$23$	−1.41421			−0.0614875			−0.0307438	−	0.999527i	$-0.509788\pi$
$23$	−1.41421			−0.0614875			−0.0307438	+	0.999527i	$0.509788\pi$
$24$		0			0
$25$			25.0000i			1.00000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			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	$-0.723210\pi$
$31$	−40.0000			−1.29032			−0.645161	+	0.764046i	$0.723210\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−17.6777	+	17.6777i	−0.505076	+	0.505076i
$36$		0			0
$37$		−	25.0000i		−	0.675676i	−0.941204	−	0.337838i	$-0.890304\pi$
$37$		−	25.0000i		−	0.675676i	0.941204	−	0.337838i	$-0.109696\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			52.3259i			1.27624i	0.769936	+	0.638121i	$0.220288\pi$
$41$			52.3259i			1.27624i	−0.769936	+	0.638121i	$0.779712\pi$
$42$		0			0
$43$		−	64.0000i		−	1.48837i	−0.667972	−	0.744186i	$-0.732838\pi$
$43$		−	64.0000i		−	1.48837i	0.667972	−	0.744186i	$-0.267162\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−22.6274			−0.481434			−0.240717	−	0.970595i	$-0.577383\pi$
$47$	−22.6274			−0.481434			−0.240717	+	0.970595i	$0.577383\pi$
$48$		0			0
$49$	24.0000			0.489796
$50$		0			0
$51$		0			0
$52$		0			0
$53$	72.1249			1.36085			0.680424	−	0.732819i	$-0.261796\pi$
$53$	72.1249			1.36085			0.680424	+	0.732819i	$0.261796\pi$
$54$		0			0
$55$	−5.00000	+	5.00000i	−0.0909091	+	0.0909091i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	90.5097i		−	1.53406i	−0.641610	−	0.767031i	$-0.721733\pi$
$59$		−	90.5097i		−	1.53406i	0.641610	−	0.767031i	$-0.278267\pi$
$60$		0			0
$61$	−97.0000			−1.59016			−0.795082	−	0.606502i	$-0.792572\pi$
$61$	−97.0000			−1.59016			−0.795082	+	0.606502i	$0.792572\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	31.8198	−	31.8198i	0.489535	−	0.489535i
$66$		0			0
$67$			131.000i			1.95522i	0.210416	+	0.977612i	$0.432518\pi$
$67$			131.000i			1.95522i	−0.210416	+	0.977612i	$0.567482\pi$
$68$		0			0
$69$		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
$73$		−	17.0000i		−	0.232877i	−0.993198	−	0.116438i	$-0.962852\pi$
$73$		−	17.0000i		−	0.232877i	0.993198	−	0.116438i	$-0.0371477\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.07107			−0.0918320
$78$		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
$82$		0			0
$83$	57.9828			0.698587			0.349294	−	0.937013i	$-0.386422\pi$
$83$	57.9828			0.698587			0.349294	+	0.937013i	$0.386422\pi$
$84$		0			0
$85$	−40.0000	−	40.0000i	−0.470588	−	0.470588i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			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
$92$		0			0
$93$		0			0
$94$		0			0
$95$	74.2462	+	74.2462i	0.781539	+	0.781539i
$96$		0			0
$97$		−	41.0000i		−	0.422680i	−0.977413	−	0.211340i	$-0.932217\pi$
$97$		−	41.0000i		−	0.422680i	0.977413	−	0.211340i	$-0.0677828\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	90.5097i		−	0.896135i	−0.894000	−	0.448068i	$-0.852112\pi$
$101$		−	90.5097i		−	0.896135i	0.894000	−	0.448068i	$-0.147888\pi$
$102$		0			0
$103$			13.0000i			0.126214i	0.998007	+	0.0631068i	$0.0201009\pi$
$103$			13.0000i			0.126214i	−0.998007	+	0.0631068i	$0.979899\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	123.037			1.14987			0.574937	−	0.818197i	$-0.305026\pi$
$107$	123.037			1.14987			0.574937	+	0.818197i	$0.305026\pi$
$108$		0			0
$109$	8.00000			0.0733945			0.0366972	−	0.999326i	$-0.488316\pi$
$109$	8.00000			0.0733945			0.0366972	+	0.999326i	$0.488316\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−38.1838			−0.337909			−0.168955	−	0.985624i	$-0.554039\pi$
$113$	−38.1838			−0.337909			−0.168955	+	0.985624i	$0.554039\pi$
$114$		0			0
$115$	5.00000	+	5.00000i	0.0434783	+	0.0434783i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	56.5685i		−	0.475366i
$120$		0			0
$121$	119.000			0.983471
$122$		0			0
$123$		0			0
$124$		0			0
$125$	88.3883	−	88.3883i	0.707107	−	0.707107i
$126$		0			0
$127$		−	8.00000i		−	0.0629921i	−0.999504	−	0.0314961i	$-0.989973\pi$
$127$		−	8.00000i		−	0.0629921i	0.999504	−	0.0314961i	$-0.0100272\pi$
$128$		0			0
$129$		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
$133$			105.000i			0.789474i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	267.286			1.95100			0.975498	−	0.220010i	$-0.0706089\pi$
$137$	267.286			1.95100			0.975498	+	0.220010i	$0.0706089\pi$
$138$		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
$142$		0			0
$143$	12.7279			0.0890064
$144$		0			0
$145$	135.000	−	135.000i	0.931034	−	0.931034i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			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	$-0.617540\pi$
$151$	−109.000			−0.721854			−0.360927	+	0.932594i	$0.617540\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	141.421	+	141.421i	0.912396	+	0.912396i
$156$		0			0
$157$			118.000i			0.751592i	0.926702	+	0.375796i	$0.122631\pi$
$157$			118.000i			0.751592i	−0.926702	+	0.375796i	$0.877369\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			7.07107i			0.0439197i
$162$		0			0
$163$		−	203.000i		−	1.24540i	−0.782461	−	0.622699i	$-0.786036\pi$
$163$		−	203.000i		−	1.24540i	0.782461	−	0.622699i	$-0.213964\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	101.823			0.609721			0.304860	−	0.952397i	$-0.401390\pi$
$167$	101.823			0.609721			0.304860	+	0.952397i	$0.401390\pi$
$168$		0			0
$169$	88.0000			0.520710
$170$		0			0
$171$		0			0
$172$		0			0
$173$	11.3137			0.0653972			0.0326986	−	0.999465i	$-0.489590\pi$
$173$	11.3137			0.0653972			0.0326986	+	0.999465i	$0.489590\pi$
$174$		0			0
$175$	125.000			0.714286
$176$		0			0
$177$		0			0
$178$		0			0
$179$			125.865i			0.703156i	0.936159	+	0.351578i	$0.114355\pi$
$179$			125.865i			0.703156i	−0.936159	+	0.351578i	$0.885645\pi$
$180$		0			0
$181$	−127.000			−0.701657			−0.350829	−	0.936440i	$-0.614100\pi$
$181$	−127.000			−0.701657			−0.350829	+	0.936440i	$0.614100\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−88.3883	+	88.3883i	−0.477775	+	0.477775i
$186$		0			0
$187$		−	16.0000i		−	0.0855615i
$188$		0			0
$189$		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
$193$			271.000i			1.40415i	0.712105	+	0.702073i	$0.247742\pi$
$193$			271.000i			1.40415i	−0.712105	+	0.702073i	$0.752258\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−316.784			−1.60804			−0.804020	−	0.594602i	$-0.797309\pi$
$197$	−316.784			−1.60804			−0.804020	+	0.594602i	$0.797309\pi$
$198$		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
$202$		0			0
$203$	190.919			0.940487
$204$		0			0
$205$	185.000	−	185.000i	0.902439	−	0.902439i
$206$		0			0
$207$		0			0
$208$		0			0
$209$			29.6985i			0.142098i
$210$		0			0
$211$	−141.000			−0.668246			−0.334123	−	0.942529i	$-0.608440\pi$
$211$	−141.000			−0.668246			−0.334123	+	0.942529i	$0.608440\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−226.274	+	226.274i	−1.05244	+	1.05244i
$216$		0			0
$217$			200.000i			0.921659i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			101.823i			0.460739i
$222$		0			0
$223$			8.00000i			0.0358744i	0.999839	+	0.0179372i	$0.00570990\pi$
$223$			8.00000i			0.0358744i	−0.999839	+	0.0179372i	$0.994290\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	69.2965			0.305271			0.152635	−	0.988283i	$-0.451224\pi$
$227$	69.2965			0.305271			0.152635	+	0.988283i	$0.451224\pi$
$228$		0			0
$229$	−8.00000			−0.0349345			−0.0174672	−	0.999847i	$-0.505560\pi$
$229$	−8.00000			−0.0349345			−0.0174672	+	0.999847i	$0.505560\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	316.784			1.35959			0.679794	−	0.733403i	$-0.262069\pi$
$233$	316.784			1.35959			0.679794	+	0.733403i	$0.262069\pi$
$234$		0			0
$235$	80.0000	+	80.0000i	0.340426	+	0.340426i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			205.061i			0.857996i	0.903306	+	0.428998i	$0.141133\pi$
$239$			205.061i			0.857996i	−0.903306	+	0.428998i	$0.858867\pi$
$240$		0			0
$241$	79.0000			0.327801			0.163900	−	0.986477i	$-0.447592\pi$
$241$	79.0000			0.327801			0.163900	+	0.986477i	$0.447592\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−84.8528	−	84.8528i	−0.346338	−	0.346338i
$246$		0			0
$247$		−	189.000i		−	0.765182i
$248$		0			0
$249$		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
$253$			2.00000i			0.00790514i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	169.706			0.660333			0.330167	−	0.943923i	$-0.392895\pi$
$257$	169.706			0.660333			0.330167	+	0.943923i	$0.392895\pi$
$258$		0			0
$259$	−125.000			−0.482625
$260$		0			0
$261$		0			0
$262$		0			0
$263$	282.843			1.07545			0.537724	−	0.843121i	$-0.319284\pi$
$263$	282.843			1.07545			0.537724	+	0.843121i	$0.319284\pi$
$264$		0			0
$265$	−255.000	−	255.000i	−0.962264	−	0.962264i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	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	$-0.366314\pi$
$271$	221.000			0.815498			0.407749	+	0.913094i	$0.366314\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	35.3553			0.128565
$276$		0			0
$277$		−	88.0000i		−	0.317690i	−0.987304	−	0.158845i	$-0.949223\pi$
$277$		−	88.0000i		−	0.317690i	0.987304	−	0.158845i	$-0.0507769\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			203.647i			0.724722i	0.932038	+	0.362361i	$0.118029\pi$
$281$			203.647i			0.724722i	−0.932038	+	0.362361i	$0.881971\pi$
$282$		0			0
$283$		−	32.0000i		−	0.113074i	−0.998400	−	0.0565371i	$-0.981994\pi$
$283$		−	32.0000i		−	0.113074i	0.998400	−	0.0565371i	$-0.0180059\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	261.630			0.911601
$288$		0			0
$289$	−161.000			−0.557093
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−173.948			−0.593680			−0.296840	−	0.954927i	$-0.595933\pi$
$293$	−173.948			−0.593680			−0.296840	+	0.954927i	$0.595933\pi$
$294$		0			0
$295$	−320.000	+	320.000i	−1.08475	+	1.08475i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	12.7279i		−	0.0425683i
$300$		0			0
$301$	−320.000			−1.06312
$302$		0			0
$303$		0			0
$304$		0			0
$305$	342.947	+	342.947i	1.12442	+	1.12442i
$306$		0			0
$307$			486.000i			1.58306i	0.611129	+	0.791531i	$0.290716\pi$
$307$			486.000i			1.58306i	−0.611129	+	0.791531i	$0.709284\pi$
$308$		0			0
$309$		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
$313$		−	281.000i		−	0.897764i	−0.893591	−	0.448882i	$-0.851822\pi$
$313$		−	281.000i		−	0.897764i	0.893591	−	0.448882i	$-0.148178\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	520.431			1.64174			0.820868	−	0.571117i	$-0.193490\pi$
$317$	520.431			1.64174			0.820868	+	0.571117i	$0.193490\pi$
$318$		0			0
$319$	54.0000			0.169279
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−237.588			−0.735566
$324$		0			0
$325$	−225.000			−0.692308
$326$		0			0
$327$		0			0
$328$		0			0
$329$			113.137i			0.343882i
$330$		0			0
$331$	59.0000			0.178248			0.0891239	−	0.996021i	$-0.471593\pi$
$331$	59.0000			0.178248			0.0891239	+	0.996021i	$0.471593\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	463.155	−	463.155i	1.38255	−	1.38255i
$336$		0			0
$337$		−	55.0000i		−	0.163205i	−0.996665	−	0.0816024i	$-0.973996\pi$
$337$		−	55.0000i		−	0.163205i	0.996665	−	0.0816024i	$-0.0260038\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			56.5685i			0.165890i
$342$		0			0
$343$		−	365.000i		−	1.06414i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−598.212			−1.72395			−0.861977	−	0.506947i	$-0.830774\pi$
$347$	−598.212			−1.72395			−0.861977	+	0.506947i	$0.830774\pi$
$348$		0			0
$349$	−439.000			−1.25788			−0.628940	−	0.777454i	$-0.716511\pi$
$349$	−439.000			−1.25788			−0.628940	+	0.777454i	$0.716511\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	520.431			1.47431			0.737154	−	0.675725i	$-0.236169\pi$
$353$	520.431			1.47431			0.737154	+	0.675725i	$0.236169\pi$
$354$		0			0
$355$	315.000	−	315.000i	0.887324	−	0.887324i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			55.1543i			0.153633i	0.997045	+	0.0768166i	$0.0244756\pi$
$359$			55.1543i			0.153633i	−0.997045	+	0.0768166i	$0.975524\pi$
$360$		0			0
$361$	80.0000			0.221607
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−60.1041	+	60.1041i	−0.164669	+	0.164669i
$366$		0			0
$367$			589.000i			1.60490i	0.596716	+	0.802452i	$0.296472\pi$
$367$			589.000i			1.60490i	−0.596716	+	0.802452i	$0.703528\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	360.624i		−	0.972034i
$372$		0			0
$373$			9.00000i			0.0241287i	0.999927	+	0.0120643i	$0.00384029\pi$
$373$			9.00000i			0.0241287i	−0.999927	+	0.0120643i	$0.996160\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−343.654			−0.911549
$378$		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
$382$		0			0
$383$	−282.843			−0.738493			−0.369246	−	0.929332i	$-0.620384\pi$
$383$	−282.843			−0.738493			−0.369246	+	0.929332i	$0.620384\pi$
$384$		0			0
$385$	25.0000	+	25.0000i	0.0649351	+	0.0649351i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	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
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−413.657	−	413.657i	−1.04723	−	1.04723i
$396$		0			0
$397$			296.000i			0.745592i	0.927913	+	0.372796i	$0.121601\pi$
$397$			296.000i			0.745592i	−0.927913	+	0.372796i	$0.878399\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			388.909i			0.969847i	0.874557	+	0.484924i	$0.161153\pi$
$401$			388.909i			0.969847i	−0.874557	+	0.484924i	$0.838847\pi$
$402$		0			0
$403$		−	360.000i		−	0.893300i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−35.3553			−0.0868682
$408$		0			0
$409$	145.000			0.354523			0.177262	−	0.984164i	$-0.443276\pi$
$409$	145.000			0.354523			0.177262	+	0.984164i	$0.443276\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−452.548			−1.09576
$414$		0			0
$415$	−205.000	−	205.000i	−0.493976	−	0.493976i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			701.450i			1.67410i	0.547123	+	0.837052i	$0.315723\pi$
$419$			701.450i			1.67410i	−0.547123	+	0.837052i	$0.684277\pi$
$420$		0			0
$421$	505.000			1.19952			0.599762	−	0.800178i	$-0.295262\pi$
$421$	505.000			1.19952			0.599762	+	0.800178i	$0.295262\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			282.843i			0.665512i
$426$		0			0
$427$			485.000i			1.13583i
$428$		0			0
$429$		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
$433$		−	32.0000i		−	0.0739030i	−0.999317	−	0.0369515i	$-0.988235\pi$
$433$		−	32.0000i		−	0.0739030i	0.999317	−	0.0369515i	$-0.0117647\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	29.6985			0.0679599
$438$		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
$442$		0			0
$443$	237.588			0.536316			0.268158	−	0.963375i	$-0.413585\pi$
$443$	237.588			0.536316			0.268158	+	0.963375i	$0.413585\pi$
$444$		0			0
$445$	520.000	−	520.000i	1.16854	−	1.16854i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	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
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−159.099	−	159.099i	−0.349668	−	0.349668i
$456$		0			0
$457$			752.000i			1.64551i	0.568393	+	0.822757i	$0.307565\pi$
$457$			752.000i			1.64551i	−0.568393	+	0.822757i	$0.692435\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	610.940i		−	1.32525i	−0.748951	−	0.662625i	$-0.769442\pi$
$461$		−	610.940i		−	1.32525i	0.748951	−	0.662625i	$-0.230558\pi$
$462$		0			0
$463$		−	597.000i		−	1.28942i	−0.764429	−	0.644708i	$-0.776979\pi$
$463$		−	597.000i		−	1.28942i	0.764429	−	0.644708i	$-0.223021\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	848.528			1.81698			0.908488	−	0.417910i	$-0.137237\pi$
$467$	848.528			1.81698			0.908488	+	0.417910i	$0.137237\pi$
$468$		0			0
$469$	655.000			1.39659
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−90.5097			−0.191352
$474$		0			0
$475$		−	525.000i		−	1.10526i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	168.291i		−	0.351339i	−0.984449	−	0.175670i	$-0.943791\pi$
$479$		−	168.291i		−	0.351339i	0.984449	−	0.175670i	$-0.0562090\pi$
$480$		0			0
$481$	225.000			0.467775
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−144.957	+	144.957i	−0.298880	+	0.298880i
$486$		0			0
$487$			507.000i			1.04107i	0.853841	+	0.520534i	$0.174267\pi$
$487$			507.000i			1.04107i	−0.853841	+	0.520534i	$0.825733\pi$
$488$		0			0
$489$		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
$493$			432.000i			0.876268i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	445.477			0.896333
$498$		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
$502$		0			0
$503$	−462.448			−0.919379			−0.459690	−	0.888080i	$-0.652039\pi$
$503$	−462.448			−0.919379			−0.459690	+	0.888080i	$0.652039\pi$
$504$		0			0
$505$	−320.000	+	320.000i	−0.633663	+	0.633663i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	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
$512$		0			0
$513$		0			0
$514$		0			0
$515$	45.9619	−	45.9619i	0.0892465	−	0.0892465i
$516$		0			0
$517$			32.0000i			0.0618956i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			864.084i			1.65851i	0.558869	+	0.829256i	$0.311235\pi$
$521$			864.084i			1.65851i	−0.558869	+	0.829256i	$0.688765\pi$
$522$		0			0
$523$			163.000i			0.311663i	0.987784	+	0.155832i	$0.0498058\pi$
$523$			163.000i			0.311663i	−0.987784	+	0.155832i	$0.950194\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−452.548			−0.858726
$528$		0			0
$529$	−527.000			−0.996219
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−470.933			−0.883552
$534$		0			0
$535$	−435.000	−	435.000i	−0.813084	−	0.813084i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	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	$0.277200\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−28.2843	−	28.2843i	−0.0518977	−	0.0518977i
$546$		0			0
$547$		−	389.000i		−	0.711152i	−0.934647	−	0.355576i	$-0.884285\pi$
$547$		−	389.000i		−	0.711152i	0.934647	−	0.355576i	$-0.115715\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	801.859i		−	1.45528i
$552$		0			0
$553$		−	585.000i		−	1.05787i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−1086.12			−1.94994			−0.974969	−	0.222339i	$-0.928631\pi$
$557$	−1086.12			−1.94994			−0.974969	+	0.222339i	$0.928631\pi$
$558$		0			0
$559$	576.000			1.03041
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−622.254			−1.10525			−0.552623	−	0.833431i	$-0.686373\pi$
$563$	−622.254			−1.10525			−0.552623	+	0.833431i	$0.686373\pi$
$564$		0			0
$565$	135.000	+	135.000i	0.238938	+	0.238938i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	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	$-0.200425\pi$
$571$	923.000			1.61646			0.808231	+	0.588865i	$0.200425\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	35.3553i		−	0.0614875i
$576$		0			0
$577$		−	247.000i		−	0.428076i	−0.976825	−	0.214038i	$-0.931338\pi$
$577$		−	247.000i		−	0.428076i	0.976825	−	0.214038i	$-0.0686617\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	289.914i		−	0.498991i
$582$		0			0
$583$		−	102.000i		−	0.174957i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−453.963			−0.773360			−0.386680	−	0.922214i	$-0.626378\pi$
$587$	−453.963			−0.773360			−0.386680	+	0.922214i	$0.626378\pi$
$588$		0			0
$589$	840.000			1.42615
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.24264			0.00715454			0.00357727	−	0.999994i	$-0.498861\pi$
$593$	4.24264			0.00715454			0.00357727	+	0.999994i	$0.498861\pi$
$594$		0			0
$595$	−200.000	+	200.000i	−0.336134	+	0.336134i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			224.860i			0.375392i	0.982227	+	0.187696i	$0.0601020\pi$
$599$			224.860i			0.375392i	−0.982227	+	0.187696i	$0.939898\pi$
$600$		0			0
$601$	−736.000			−1.22463			−0.612313	−	0.790616i	$-0.709761\pi$
$601$	−736.000			−1.22463			−0.612313	+	0.790616i	$0.709761\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−420.729	−	420.729i	−0.695419	−	0.695419i
$606$		0			0
$607$			437.000i			0.719934i	0.932965	+	0.359967i	$0.117212\pi$
$607$			437.000i			0.719934i	−0.932965	+	0.359967i	$0.882788\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	203.647i		−	0.333301i
$612$		0			0
$613$		−	335.000i		−	0.546493i	−0.961944	−	0.273246i	$-0.911903\pi$
$613$		−	335.000i		−	0.546493i	0.961944	−	0.273246i	$-0.0880974\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−255.973			−0.414867			−0.207433	−	0.978249i	$-0.566511\pi$
$617$	−255.973			−0.414867			−0.207433	+	0.978249i	$0.566511\pi$
$618$		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
$622$		0			0
$623$	735.391			1.18040
$624$		0			0
$625$	−625.000			−1.00000
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	282.843i		−	0.449670i
$630$		0			0
$631$	−275.000			−0.435816			−0.217908	−	0.975969i	$-0.569923\pi$
$631$	−275.000			−0.435816			−0.217908	+	0.975969i	$0.569923\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−28.2843	+	28.2843i	−0.0445422	+	0.0445422i
$636$		0			0
$637$			216.000i			0.339089i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	486.489i		−	0.758954i	−0.925201	−	0.379477i	$-0.876104\pi$
$641$		−	486.489i		−	0.758954i	0.925201	−	0.379477i	$-0.123896\pi$
$642$		0			0
$643$			1152.00i			1.79160i	0.444455	+	0.895801i	$0.353397\pi$
$643$			1152.00i			1.79160i	−0.444455	+	0.895801i	$0.646603\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−691.550			−1.06886			−0.534428	−	0.845214i	$-0.679473\pi$
$647$	−691.550			−1.06886			−0.534428	+	0.845214i	$0.679473\pi$
$648$		0			0
$649$	−128.000			−0.197227
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−350.725			−0.537098			−0.268549	−	0.963266i	$-0.586544\pi$
$653$	−350.725			−0.537098			−0.268549	+	0.963266i	$0.586544\pi$
$654$		0			0
$655$	480.000	−	480.000i	0.732824	−	0.732824i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	497.803i		−	0.755392i	−0.925930	−	0.377696i	$-0.876716\pi$
$659$		−	497.803i		−	0.755392i	0.925930	−	0.377696i	$-0.123284\pi$
$660$		0			0
$661$	−577.000			−0.872920			−0.436460	−	0.899724i	$-0.643768\pi$
$661$	−577.000			−0.872920			−0.436460	+	0.899724i	$0.643768\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	371.231	−	371.231i	0.558242	−	0.558242i
$666$		0			0
$667$		−	54.0000i		−	0.0809595i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			137.179i			0.204439i
$672$		0			0
$673$		−	489.000i		−	0.726597i	−0.931673	−	0.363299i	$-0.881650\pi$
$673$		−	489.000i		−	0.726597i	0.931673	−	0.363299i	$-0.118350\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−599.627			−0.885711			−0.442856	−	0.896593i	$-0.646035\pi$
$677$	−599.627			−0.885711			−0.442856	+	0.896593i	$0.646035\pi$
$678$		0			0
$679$	−205.000			−0.301915
$680$		0			0
$681$		0			0
$682$		0			0
$683$	236.174			0.345789			0.172894	−	0.984940i	$-0.444688\pi$
$683$	236.174			0.345789			0.172894	+	0.984940i	$0.444688\pi$
$684$		0			0
$685$	−945.000	−	945.000i	−1.37956	−	1.37956i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			649.124i			0.942125i
$690$		0			0
$691$	−640.000			−0.926194			−0.463097	−	0.886308i	$-0.653262\pi$
$691$	−640.000			−0.926194			−0.463097	+	0.886308i	$0.653262\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−130.815	−	130.815i	−0.188223	−	0.188223i
$696$		0			0
$697$			592.000i			0.849354i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1093.19i		−	1.55947i	−0.626111	−	0.779734i	$-0.715354\pi$
$701$		−	1093.19i		−	1.55947i	0.626111	−	0.779734i	$-0.284646\pi$
$702$		0			0
$703$			525.000i			0.746799i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−452.548			−0.640097
$708$		0			0
$709$	−489.000			−0.689704			−0.344852	−	0.938657i	$-0.612071\pi$
$709$	−489.000			−0.689704			−0.344852	+	0.938657i	$0.612071\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	56.5685			0.0793388
$714$		0			0
$715$	−45.0000	−	45.0000i	−0.0629371	−	0.0629371i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			620.840i			0.863477i	0.901999	+	0.431738i	$0.142100\pi$
$719$			620.840i			0.863477i	−0.901999	+	0.431738i	$0.857900\pi$
$720$		0			0
$721$	65.0000			0.0901526
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−954.594			−1.31668
$726$		0			0
$727$			1080.00i			1.48556i	0.669537	+	0.742779i	$0.266492\pi$
$727$			1080.00i			1.48556i	−0.669537	+	0.742779i	$0.733508\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	724.077i		−	0.990530i
$732$		0			0
$733$			248.000i			0.338336i	0.985587	+	0.169168i	$0.0541080\pi$
$733$			248.000i			0.338336i	−0.985587	+	0.169168i	$0.945892\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	185.262			0.251373
$738$		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
$742$		0			0
$743$	33.9411			0.0456812			0.0228406	−	0.999739i	$-0.492729\pi$
$743$	33.9411			0.0456812			0.0228406	+	0.999739i	$0.492729\pi$
$744$		0			0
$745$	920.000	−	920.000i	1.23490	−	1.23490i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	615.183i		−	0.821339i
$750$		0			0
$751$	133.000			0.177097			0.0885486	−	0.996072i	$-0.471777\pi$
$751$	133.000			0.177097			0.0885486	+	0.996072i	$0.471777\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	385.373	+	385.373i	0.510428	+	0.510428i
$756$		0			0
$757$			1271.00i			1.67900i	0.543363	+	0.839498i	$0.317151\pi$
$757$			1271.00i			1.67900i	−0.543363	+	0.839498i	$0.682849\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	538.815i		−	0.708036i	−0.935239	−	0.354018i	$-0.884815\pi$
$761$		−	538.815i		−	0.708036i	0.935239	−	0.354018i	$-0.115185\pi$
$762$		0			0
$763$		−	40.0000i		−	0.0524246i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	814.587			1.06204
$768$		0			0
$769$	−81.0000			−0.105332			−0.0526658	−	0.998612i	$-0.516772\pi$
$769$	−81.0000			−0.105332			−0.0526658	+	0.998612i	$0.516772\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−445.477			−0.576297			−0.288148	−	0.957586i	$-0.593040\pi$
$773$	−445.477			−0.576297			−0.288148	+	0.957586i	$0.593040\pi$
$774$		0			0
$775$		−	1000.00i		−	1.29032i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	1098.84i		−	1.41058i
$780$		0			0
$781$	126.000			0.161332
$782$		0			0
$783$		0			0
$784$		0			0
$785$	417.193	−	417.193i	0.531456	−	0.531456i
$786$		0			0
$787$		−	395.000i		−	0.501906i	−0.967999	−	0.250953i	$-0.919256\pi$
$787$		−	395.000i		−	0.501906i	0.967999	−	0.250953i	$-0.0807440\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			190.919i			0.241364i
$792$		0			0
$793$		−	873.000i		−	1.10088i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1074.80			−1.34856			−0.674280	−	0.738476i	$-0.735546\pi$
$797$	−1074.80			−1.34856			−0.674280	+	0.738476i	$0.735546\pi$
$798$		0			0
$799$	−256.000			−0.320401
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−24.0416			−0.0299398
$804$		0			0
$805$	25.0000	−	25.0000i	0.0310559	−	0.0310559i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			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	$-0.346548\pi$
$811$	752.000			0.927250			0.463625	+	0.886031i	$0.346548\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−717.713	+	717.713i	−0.880630	+	0.880630i
$816$		0			0
$817$			1344.00i			1.64504i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	497.803i		−	0.606338i	−0.952937	−	0.303169i	$-0.901955\pi$
$821$		−	497.803i		−	0.606338i	0.952937	−	0.303169i	$-0.0980446\pi$
$822$		0			0
$823$			531.000i			0.645200i	0.946535	+	0.322600i	$0.104557\pi$
$823$			531.000i			0.645200i	−0.946535	+	0.322600i	$0.895443\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−350.725			−0.424093			−0.212047	−	0.977260i	$-0.568013\pi$
$827$	−350.725			−0.424093			−0.212047	+	0.977260i	$0.568013\pi$
$828$		0			0
$829$	−705.000			−0.850422			−0.425211	−	0.905094i	$-0.639800\pi$
$829$	−705.000			−0.850422			−0.425211	+	0.905094i	$0.639800\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	271.529			0.325965
$834$		0			0
$835$	−360.000	−	360.000i	−0.431138	−	0.431138i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	1617.86i		−	1.92832i	−0.265322	−	0.964160i	$-0.585478\pi$
$839$		−	1617.86i		−	1.92832i	0.265322	−	0.964160i	$-0.414522\pi$
$840$		0			0
$841$	−617.000			−0.733650
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−311.127	−	311.127i	−0.368198	−	0.368198i
$846$		0			0
$847$		−	595.000i		−	0.702479i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			35.3553i			0.0415456i
$852$		0			0
$853$		−	1639.00i		−	1.92145i	−0.277496	−	0.960727i	$-0.589504\pi$
$853$		−	1639.00i		−	1.92145i	0.277496	−	0.960727i	$-0.410496\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	988.535			1.15348			0.576742	−	0.816927i	$-0.304324\pi$
$857$	988.535			1.15348			0.576742	+	0.816927i	$0.304324\pi$
$858$		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
$862$		0			0
$863$	−1459.47			−1.69116			−0.845578	−	0.533851i	$-0.820744\pi$
$863$	−1459.47			−1.69116			−0.845578	+	0.533851i	$0.820744\pi$
$864$		0			0
$865$	−40.0000	−	40.0000i	−0.0462428	−	0.0462428i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	165.463i		−	0.190406i
$870$		0			0
$871$	−1179.00			−1.35362
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−441.942	−	441.942i	−0.505076	−	0.505076i
$876$		0			0
$877$			1129.00i			1.28734i	0.765302	+	0.643672i	$0.222590\pi$
$877$			1129.00i			1.28734i	−0.765302	+	0.643672i	$0.777410\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	165.463i		−	0.187813i	−0.995581	−	0.0939063i	$-0.970065\pi$
$881$		−	165.463i		−	0.187813i	0.995581	−	0.0939063i	$-0.0299354\pi$
$882$		0			0
$883$		−	1227.00i		−	1.38958i	−0.719212	−	0.694790i	$-0.755497\pi$
$883$		−	1227.00i		−	1.38958i	0.719212	−	0.694790i	$-0.244503\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	237.588			0.267856			0.133928	−	0.990991i	$-0.457241\pi$
$887$	237.588			0.267856			0.133928	+	0.990991i	$0.457241\pi$
$888$		0			0
$889$	−40.0000			−0.0449944
$890$		0			0
$891$		0			0
$892$		0			0
$893$	475.176			0.532112
$894$		0			0
$895$	445.000	−	445.000i	0.497207	−	0.497207i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	1527.35i		−	1.69894i
$900$		0			0
$901$	816.000			0.905660
$902$		0			0
$903$		0			0
$904$		0			0
$905$	449.013	+	449.013i	0.496147	+	0.496147i
$906$		0			0
$907$			1005.00i			1.10805i	0.832501	+	0.554024i	$0.186909\pi$
$907$			1005.00i			1.10805i	−0.832501	+	0.554024i	$0.813091\pi$
$908$		0			0
$909$		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
$913$		−	82.0000i		−	0.0898138i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	678.823			0.740264
$918$		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
$922$		0			0
$923$	−801.859			−0.868753
$924$		0			0
$925$	625.000			0.675676
$926$		0			0
$927$		0			0
$928$		0			0
$929$			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
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−56.5685	+	56.5685i	−0.0605011	+	0.0605011i
$936$		0			0
$937$			1465.00i			1.56350i	0.623592	+	0.781750i	$0.285673\pi$
$937$			1465.00i			1.56350i	−0.623592	+	0.781750i	$0.714327\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			255.973i			0.272022i	0.990707	+	0.136011i	$0.0434282\pi$
$941$			255.973i			0.272022i	−0.990707	+	0.136011i	$0.956572\pi$
$942$		0			0
$943$		−	74.0000i		−	0.0784730i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	407.294			0.430088			0.215044	−	0.976604i	$-0.431010\pi$
$947$	407.294			0.430088			0.215044	+	0.976604i	$0.431010\pi$
$948$		0			0
$949$	153.000			0.161222
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−497.803			−0.522354			−0.261177	−	0.965291i	$-0.584111\pi$
$953$	−497.803			−0.522354			−0.261177	+	0.965291i	$0.584111\pi$
$954$		0			0
$955$	360.000	−	360.000i	0.376963	−	0.376963i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	1336.43i		−	1.39357i
$960$		0			0
$961$	639.000			0.664932
$962$		0			0
$963$		0			0
$964$		0			0
$965$	958.130	−	958.130i	0.992881	−	0.992881i
$966$		0			0
$967$			491.000i			0.507756i	0.967236	+	0.253878i	$0.0817062\pi$
$967$			491.000i			0.507756i	−0.967236	+	0.253878i	$0.918294\pi$
$968$		0			0
$969$		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
$973$		−	185.000i		−	0.190134i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	470.933			0.482020			0.241010	−	0.970523i	$-0.422521\pi$
$977$	470.933			0.482020			0.241010	+	0.970523i	$0.422521\pi$
$978$		0			0
$979$	208.000			0.212462
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−601.041			−0.611435			−0.305718	−	0.952122i	$-0.598896\pi$
$983$	−601.041			−0.611435			−0.305718	+	0.952122i	$0.598896\pi$
$984$		0			0
$985$	1120.00	+	1120.00i	1.13706	+	1.13706i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			90.5097i			0.0915163i
$990$		0			0
$991$	755.000			0.761857			0.380928	−	0.924605i	$-0.375604\pi$
$991$	755.000			0.761857			0.380928	+	0.924605i	$0.375604\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	519.723	+	519.723i	0.522335	+	0.522335i
$996$		0			0
$997$			24.0000i			0.0240722i	0.999928	+	0.0120361i	$0.00383130\pi$
$997$			24.0000i			0.0240722i	−0.999928	+	0.0120361i	$0.996169\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2160.3.c.i.1889.1		4
3.2	odd	2	inner	2160.3.c.i.1889.4		4
4.3	odd	2		270.3.b.c.269.1	✓	4
5.4	even	2	inner	2160.3.c.i.1889.3		4
12.11	even	2		270.3.b.c.269.4	yes	4
15.14	odd	2	inner	2160.3.c.i.1889.2		4
20.3	even	4		1350.3.d.g.701.2		2
20.7	even	4		1350.3.d.f.701.1		2
20.19	odd	2		270.3.b.c.269.3	yes	4
36.7	odd	6		810.3.j.e.539.4		8
36.11	even	6		810.3.j.e.539.1		8
36.23	even	6		810.3.j.e.269.2		8
36.31	odd	6		810.3.j.e.269.3		8
60.23	odd	4		1350.3.d.g.701.1		2
60.47	odd	4		1350.3.d.f.701.2		2
60.59	even	2		270.3.b.c.269.2	yes	4
180.59	even	6		810.3.j.e.269.4		8
180.79	odd	6		810.3.j.e.539.2		8
180.119	even	6		810.3.j.e.539.3		8
180.139	odd	6		810.3.j.e.269.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.3.b.c.269.1	✓	4	4.3	odd	2
270.3.b.c.269.2	yes	4	60.59	even	2
270.3.b.c.269.3	yes	4	20.19	odd	2
270.3.b.c.269.4	yes	4	12.11	even	2
810.3.j.e.269.1		8	180.139	odd	6
810.3.j.e.269.2		8	36.23	even	6
810.3.j.e.269.3		8	36.31	odd	6
810.3.j.e.269.4		8	180.59	even	6
810.3.j.e.539.1		8	36.11	even	6
810.3.j.e.539.2		8	180.79	odd	6
810.3.j.e.539.3		8	180.119	even	6
810.3.j.e.539.4		8	36.7	odd	6
1350.3.d.f.701.1		2	20.7	even	4
1350.3.d.f.701.2		2	60.47	odd	4
1350.3.d.g.701.1		2	60.23	odd	4
1350.3.d.g.701.2		2	20.3	even	4
2160.3.c.i.1889.1		4	1.1	even	1	trivial
2160.3.c.i.1889.2		4	15.14	odd	2	inner
2160.3.c.i.1889.3		4	5.4	even	2	inner
2160.3.c.i.1889.4		4	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2160.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.53553 - 3.53553i) q^{5} -5.00000i q^{7} +O(q^{10})\)	\(q+(-3.53553 - 3.53553i) q^{5} -5.00000i q^{7} -1.41421i q^{11} +9.00000i q^{13} +11.3137 q^{17} -21.0000 q^{19} -1.41421 q^{23} +25.0000i q^{25} +38.1838i q^{29} -40.0000 q^{31} +(-17.6777 + 17.6777i) q^{35} -25.0000i q^{37} +52.3259i q^{41} -64.0000i q^{43} -22.6274 q^{47} +24.0000 q^{49} +72.1249 q^{53} +(-5.00000 + 5.00000i) q^{55} -90.5097i q^{59} -97.0000 q^{61} +(31.8198 - 31.8198i) q^{65} +131.000i q^{67} +89.0955i q^{71} -17.0000i q^{73} -7.07107 q^{77} +117.000 q^{79} +57.9828 q^{83} +(-40.0000 - 40.0000i) q^{85} +147.078i q^{89} +45.0000 q^{91} +(74.2462 + 74.2462i) q^{95} -41.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 84 q^{19} - 160 q^{31} + 96 q^{49} - 20 q^{55} - 388 q^{61} + 468 q^{79} - 160 q^{85} + 180 q^{91}+O(q^{100}) \) 4 * q - 84 * q^19 - 160 * q^31 + 96 * q^49 - 20 * q^55 - 388 * q^61 + 468 * q^79 - 160 * q^85 + 180 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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