LMFDB - Embedded newform 1856.2.e.h.1217.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1856,2,Mod(1217,1856)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1856, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1856.1217");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1856 = 2^{6} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1856.e (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:	$14.8202346151$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 928)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1217.4
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	1856.1217
Dual form			1856.2.e.h.1217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -1.00000 q^{5} +4.47214 q^{7} +2.00000 q^{9} +O(q^{10})$	$q+1.00000i q^{3} -1.00000 q^{5} +4.47214 q^{7} +2.00000 q^{9} +3.00000i q^{11} +1.00000 q^{13} -1.00000i q^{15} +4.47214i q^{17} -4.00000i q^{19} +4.47214i q^{21} +4.47214 q^{23} -4.00000 q^{25} +5.00000i q^{27} +(3.00000 - 4.47214i) q^{29} -5.00000i q^{31} -3.00000 q^{33} -4.47214 q^{35} +1.00000i q^{39} +4.47214i q^{41} -1.00000i q^{43} -2.00000 q^{45} -3.00000i q^{47} +13.0000 q^{49} -4.47214 q^{51} +9.00000 q^{53} -3.00000i q^{55} +4.00000 q^{57} -4.47214 q^{59} +13.4164i q^{61} +8.94427 q^{63} -1.00000 q^{65} -8.94427 q^{67} +4.47214i q^{69} -8.94427 q^{71} -8.94427i q^{73} -4.00000i q^{75} +13.4164i q^{77} -15.0000i q^{79} +1.00000 q^{81} +4.47214 q^{83} -4.47214i q^{85} +(4.47214 + 3.00000i) q^{87} +13.4164i q^{89} +4.47214 q^{91} +5.00000 q^{93} +4.00000i q^{95} +13.4164i q^{97} +6.00000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 8 q^{9}+O(q^{10}) $ 4 * q - 4 * q^5 + 8 * q^9	$ 4 q - 4 q^{5} + 8 q^{9} + 4 q^{13} - 16 q^{25} + 12 q^{29} - 12 q^{33} - 8 q^{45} + 52 q^{49} + 36 q^{53} + 16 q^{57} - 4 q^{65} + 4 q^{81} + 20 q^{93}+O(q^{100}) $ 4 * q - 4 * q^5 + 8 * q^9 + 4 * q^13 - 16 * q^25 + 12 * q^29 - 12 * q^33 - 8 * q^45 + 52 * q^49 + 36 * q^53 + 16 * q^57 - 4 * q^65 + 4 * q^81 + 20 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1856\mathbb{Z}\right)^\times$.

$n$	$321$	$581$	$639$
$\chi(n)$	$-1$	$1$	$1$

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
$3$			1.00000i			0.577350i	0.957427	+	0.288675i	$0.0932147\pi$
$3$			1.00000i			0.577350i	−0.957427	+	0.288675i	$0.906785\pi$
$4$		0			0
$5$	−1.00000			−0.447214			−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000			−0.447214			−0.223607	+	0.974679i	$0.571783\pi$
$6$		0			0
$7$	4.47214			1.69031			0.845154	−	0.534522i	$-0.179509\pi$
$7$	4.47214			1.69031			0.845154	+	0.534522i	$0.179509\pi$
$8$		0			0
$9$	2.00000			0.666667
$10$		0			0
$11$			3.00000i			0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$			3.00000i			0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$		0			0
$13$	1.00000			0.277350			0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000			0.277350			0.138675	+	0.990338i	$0.455716\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$			4.47214i			1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$			4.47214i			1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$		0			0
$19$		−	4.00000i		−	0.917663i	−0.888523	−	0.458831i	$-0.848268\pi$
$19$		−	4.00000i		−	0.917663i	0.888523	−	0.458831i	$-0.151732\pi$
$20$		0			0
$21$			4.47214i			0.975900i
$22$		0			0
$23$	4.47214			0.932505			0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214			0.932505			0.466252	+	0.884652i	$0.345604\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$			5.00000i			0.962250i
$28$		0			0
$29$	3.00000	−	4.47214i	0.557086	−	0.830455i
$29$	3.00000	−	4.47214i	0.557086	−	0.830455i
$30$		0			0
$31$		−	5.00000i		−	0.898027i	−0.893525	−	0.449013i	$-0.851776\pi$
$31$		−	5.00000i		−	0.898027i	0.893525	−	0.449013i	$-0.148224\pi$
$32$		0			0
$33$	−3.00000			−0.522233
$34$		0			0
$35$	−4.47214			−0.755929
$36$		0			0
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$		0			0
$39$			1.00000i			0.160128i
$40$		0			0
$41$			4.47214i			0.698430i	0.937043	+	0.349215i	$0.113552\pi$
$41$			4.47214i			0.698430i	−0.937043	+	0.349215i	$0.886448\pi$
$42$		0			0
$43$		−	1.00000i		−	0.152499i	−0.997089	−	0.0762493i	$-0.975706\pi$
$43$		−	1.00000i		−	0.152499i	0.997089	−	0.0762493i	$-0.0242945\pi$
$44$		0			0
$45$	−2.00000			−0.298142
$46$		0			0
$47$		−	3.00000i		−	0.437595i	−0.975770	−	0.218797i	$-0.929787\pi$
$47$		−	3.00000i		−	0.437595i	0.975770	−	0.218797i	$-0.0702134\pi$
$48$		0			0
$49$	13.0000			1.85714
$50$		0			0
$51$	−4.47214			−0.626224
$52$		0			0
$53$	9.00000			1.23625			0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000			1.23625			0.618123	+	0.786082i	$0.287894\pi$
$54$		0			0
$55$		−	3.00000i		−	0.404520i
$56$		0			0
$57$	4.00000			0.529813
$58$		0			0
$59$	−4.47214			−0.582223			−0.291111	−	0.956689i	$-0.594025\pi$
$59$	−4.47214			−0.582223			−0.291111	+	0.956689i	$0.594025\pi$
$60$		0			0
$61$			13.4164i			1.71780i	0.512148	+	0.858898i	$0.328850\pi$
$61$			13.4164i			1.71780i	−0.512148	+	0.858898i	$0.671150\pi$
$62$		0			0
$63$	8.94427			1.12687
$64$		0			0
$65$	−1.00000			−0.124035
$66$		0			0
$67$	−8.94427			−1.09272			−0.546358	−	0.837552i	$-0.683986\pi$
$67$	−8.94427			−1.09272			−0.546358	+	0.837552i	$0.683986\pi$
$68$		0			0
$69$			4.47214i			0.538382i
$70$		0			0
$71$	−8.94427			−1.06149			−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427			−1.06149			−0.530745	+	0.847532i	$0.678088\pi$
$72$		0			0
$73$		−	8.94427i		−	1.04685i	−0.852072	−	0.523424i	$-0.824654\pi$
$73$		−	8.94427i		−	1.04685i	0.852072	−	0.523424i	$-0.175346\pi$
$74$		0			0
$75$		−	4.00000i		−	0.461880i
$76$		0			0
$77$			13.4164i			1.52894i
$78$		0			0
$79$		−	15.0000i		−	1.68763i	−0.536633	−	0.843816i	$-0.680304\pi$
$79$		−	15.0000i		−	1.68763i	0.536633	−	0.843816i	$-0.319696\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$	4.47214			0.490881			0.245440	−	0.969412i	$-0.421067\pi$
$83$	4.47214			0.490881			0.245440	+	0.969412i	$0.421067\pi$
$84$		0			0
$85$		−	4.47214i		−	0.485071i
$86$		0			0
$87$	4.47214	+	3.00000i	0.479463	+	0.321634i
$88$		0			0
$89$			13.4164i			1.42214i	0.703123	+	0.711068i	$0.251788\pi$
$89$			13.4164i			1.42214i	−0.703123	+	0.711068i	$0.748212\pi$
$90$		0			0
$91$	4.47214			0.468807
$92$		0			0
$93$	5.00000			0.518476
$94$		0			0
$95$			4.00000i			0.410391i
$96$		0			0
$97$			13.4164i			1.36223i	0.732177	+	0.681115i	$0.238505\pi$
$97$			13.4164i			1.36223i	−0.732177	+	0.681115i	$0.761495\pi$
$98$		0			0
$99$			6.00000i			0.603023i
$100$		0			0
$101$			8.94427i			0.889988i	0.895533	+	0.444994i	$0.146794\pi$
$101$			8.94427i			0.889988i	−0.895533	+	0.444994i	$0.853206\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		−	4.47214i		−	0.436436i
$106$		0			0
$107$	13.4164			1.29701			0.648507	−	0.761209i	$-0.275394\pi$
$107$	13.4164			1.29701			0.648507	+	0.761209i	$0.275394\pi$
$108$		0			0
$109$	−1.00000			−0.0957826			−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000			−0.0957826			−0.0478913	+	0.998853i	$0.515250\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			17.8885i			1.68281i	0.540403	+	0.841406i	$0.318272\pi$
$113$			17.8885i			1.68281i	−0.540403	+	0.841406i	$0.681728\pi$
$114$		0			0
$115$	−4.47214			−0.417029
$116$		0			0
$117$	2.00000			0.184900
$118$		0			0
$119$			20.0000i			1.83340i
$120$		0			0
$121$	2.00000			0.181818
$122$		0			0
$123$	−4.47214			−0.403239
$124$		0			0
$125$	9.00000			0.804984
$126$		0			0
$127$			12.0000i			1.06483i	0.846484	+	0.532414i	$0.178715\pi$
$127$			12.0000i			1.06483i	−0.846484	+	0.532414i	$0.821285\pi$
$128$		0			0
$129$	1.00000			0.0880451
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$		−	17.8885i		−	1.55113i
$134$		0			0
$135$		−	5.00000i		−	0.430331i
$136$		0			0
$137$		−	17.8885i		−	1.52832i	−0.645026	−	0.764161i	$-0.723153\pi$
$137$		−	17.8885i		−	1.52832i	0.645026	−	0.764161i	$-0.276847\pi$
$138$		0			0
$139$	4.47214			0.379322			0.189661	−	0.981850i	$-0.439261\pi$
$139$	4.47214			0.379322			0.189661	+	0.981850i	$0.439261\pi$
$140$		0			0
$141$	3.00000			0.252646
$142$		0			0
$143$			3.00000i			0.250873i
$144$		0			0
$145$	−3.00000	+	4.47214i	−0.249136	+	0.371391i
$146$		0			0
$147$			13.0000i			1.07222i
$148$		0			0
$149$	1.00000			0.0819232			0.0409616	−	0.999161i	$-0.486958\pi$
$149$	1.00000			0.0819232			0.0409616	+	0.999161i	$0.486958\pi$
$150$		0			0
$151$	−13.4164			−1.09181			−0.545906	−	0.837846i	$-0.683814\pi$
$151$	−13.4164			−1.09181			−0.545906	+	0.837846i	$0.683814\pi$
$152$		0			0
$153$			8.94427i			0.723102i
$154$		0			0
$155$			5.00000i			0.401610i
$156$		0			0
$157$		−	4.47214i		−	0.356915i	−0.983948	−	0.178458i	$-0.942889\pi$
$157$		−	4.47214i		−	0.356915i	0.983948	−	0.178458i	$-0.0571108\pi$
$158$		0			0
$159$			9.00000i			0.713746i
$160$		0			0
$161$	20.0000			1.57622
$162$		0			0
$163$			1.00000i			0.0783260i	0.999233	+	0.0391630i	$0.0124692\pi$
$163$			1.00000i			0.0783260i	−0.999233	+	0.0391630i	$0.987531\pi$
$164$		0			0
$165$	3.00000			0.233550
$166$		0			0
$167$	−8.94427			−0.692129			−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427			−0.692129			−0.346064	+	0.938211i	$0.612482\pi$
$168$		0			0
$169$	−12.0000			−0.923077
$170$		0			0
$171$		−	8.00000i		−	0.611775i
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$	−17.8885			−1.35225
$176$		0			0
$177$		−	4.47214i		−	0.336146i
$178$		0			0
$179$	17.8885			1.33705			0.668526	−	0.743689i	$-0.266925\pi$
$179$	17.8885			1.33705			0.668526	+	0.743689i	$0.266925\pi$
$180$		0			0
$181$	−25.0000			−1.85824			−0.929118	−	0.369784i	$-0.879432\pi$
$181$	−25.0000			−1.85824			−0.929118	+	0.369784i	$0.879432\pi$
$182$		0			0
$183$	−13.4164			−0.991769
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−13.4164			−0.981105
$188$		0			0
$189$			22.3607i			1.62650i
$190$		0			0
$191$			8.00000i			0.578860i	0.957199	+	0.289430i	$0.0934657\pi$
$191$			8.00000i			0.578860i	−0.957199	+	0.289430i	$0.906534\pi$
$192$		0			0
$193$			4.47214i			0.321911i	0.986962	+	0.160956i	$0.0514576\pi$
$193$			4.47214i			0.321911i	−0.986962	+	0.160956i	$0.948542\pi$
$194$		0			0
$195$		−	1.00000i		−	0.0716115i
$196$		0			0
$197$	−22.0000			−1.56744			−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000			−1.56744			−0.783718	+	0.621117i	$0.786679\pi$
$198$		0			0
$199$	4.47214			0.317021			0.158511	−	0.987357i	$-0.449331\pi$
$199$	4.47214			0.317021			0.158511	+	0.987357i	$0.449331\pi$
$200$		0			0
$201$		−	8.94427i		−	0.630880i
$202$		0			0
$203$	13.4164	−	20.0000i	0.941647	−	1.40372i
$204$		0			0
$205$		−	4.47214i		−	0.312348i
$206$		0			0
$207$	8.94427			0.621670
$208$		0			0
$209$	12.0000			0.830057
$210$		0			0
$211$		−	27.0000i		−	1.85876i	−0.369129	−	0.929378i	$-0.620344\pi$
$211$		−	27.0000i		−	1.85876i	0.369129	−	0.929378i	$-0.379656\pi$
$212$		0			0
$213$		−	8.94427i		−	0.612851i
$214$		0			0
$215$			1.00000i			0.0681994i
$216$		0			0
$217$		−	22.3607i		−	1.51794i
$218$		0			0
$219$	8.94427			0.604398
$220$		0			0
$221$			4.47214i			0.300828i
$222$		0			0
$223$	17.8885			1.19791			0.598953	−	0.800784i	$-0.295584\pi$
$223$	17.8885			1.19791			0.598953	+	0.800784i	$0.295584\pi$
$224$		0			0
$225$	−8.00000			−0.533333
$226$		0			0
$227$	17.8885			1.18730			0.593652	−	0.804722i	$-0.297686\pi$
$227$	17.8885			1.18730			0.593652	+	0.804722i	$0.297686\pi$
$228$		0			0
$229$		−	22.3607i		−	1.47764i	−0.673905	−	0.738818i	$-0.735384\pi$
$229$		−	22.3607i		−	1.47764i	0.673905	−	0.738818i	$-0.264616\pi$
$230$		0			0
$231$	−13.4164			−0.882735
$232$		0			0
$233$	−9.00000			−0.589610			−0.294805	−	0.955557i	$-0.595255\pi$
$233$	−9.00000			−0.589610			−0.294805	+	0.955557i	$0.595255\pi$
$234$		0			0
$235$			3.00000i			0.195698i
$236$		0			0
$237$	15.0000			0.974355
$238$		0			0
$239$	−22.3607			−1.44639			−0.723196	−	0.690643i	$-0.757328\pi$
$239$	−22.3607			−1.44639			−0.723196	+	0.690643i	$0.757328\pi$
$240$		0			0
$241$	−17.0000			−1.09507			−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000			−1.09507			−0.547533	+	0.836784i	$0.684433\pi$
$242$		0			0
$243$			16.0000i			1.02640i
$244$		0			0
$245$	−13.0000			−0.830540
$246$		0			0
$247$		−	4.00000i		−	0.254514i
$248$		0			0
$249$			4.47214i			0.283410i
$250$		0			0
$251$		−	23.0000i		−	1.45175i	−0.687828	−	0.725874i	$-0.741436\pi$
$251$		−	23.0000i		−	1.45175i	0.687828	−	0.725874i	$-0.258564\pi$
$252$		0			0
$253$			13.4164i			0.843482i
$254$		0			0
$255$	4.47214			0.280056
$256$		0			0
$257$	17.0000			1.06043			0.530215	−	0.847863i	$-0.322111\pi$
$257$	17.0000			1.06043			0.530215	+	0.847863i	$0.322111\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	6.00000	−	8.94427i	0.371391	−	0.553637i
$262$		0			0
$263$		−	21.0000i		−	1.29492i	−0.762101	−	0.647458i	$-0.775832\pi$
$263$		−	21.0000i		−	1.29492i	0.762101	−	0.647458i	$-0.224168\pi$
$264$		0			0
$265$	−9.00000			−0.552866
$266$		0			0
$267$	−13.4164			−0.821071
$268$		0			0
$269$		−	4.47214i		−	0.272671i	−0.990663	−	0.136335i	$-0.956467\pi$
$269$		−	4.47214i		−	0.272671i	0.990663	−	0.136335i	$-0.0435325\pi$
$270$		0			0
$271$		−	17.0000i		−	1.03268i	−0.856385	−	0.516338i	$-0.827295\pi$
$271$		−	17.0000i		−	1.03268i	0.856385	−	0.516338i	$-0.172705\pi$
$272$		0			0
$273$			4.47214i			0.270666i
$274$		0			0
$275$		−	12.0000i		−	0.723627i
$276$		0			0
$277$	22.0000			1.32185			0.660926	−	0.750451i	$-0.270164\pi$
$277$	22.0000			1.32185			0.660926	+	0.750451i	$0.270164\pi$
$278$		0			0
$279$		−	10.0000i		−	0.598684i
$280$		0			0
$281$	−5.00000			−0.298275			−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000			−0.298275			−0.149137	+	0.988816i	$0.547650\pi$
$282$		0			0
$283$	4.47214			0.265841			0.132920	−	0.991127i	$-0.457565\pi$
$283$	4.47214			0.265841			0.132920	+	0.991127i	$0.457565\pi$
$284$		0			0
$285$	−4.00000			−0.236940
$286$		0			0
$287$			20.0000i			1.18056i
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$	−13.4164			−0.786484
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$	4.47214			0.260378
$296$		0			0
$297$	−15.0000			−0.870388
$298$		0			0
$299$	4.47214			0.258630
$300$		0			0
$301$		−	4.47214i		−	0.257770i
$302$		0			0
$303$	−8.94427			−0.513835
$304$		0			0
$305$		−	13.4164i		−	0.768221i
$306$		0			0
$307$			27.0000i			1.54097i	0.637457	+	0.770486i	$0.279986\pi$
$307$			27.0000i			1.54097i	−0.637457	+	0.770486i	$0.720014\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	1.00000			0.0565233			0.0282617	−	0.999601i	$-0.491003\pi$
$313$	1.00000			0.0565233			0.0282617	+	0.999601i	$0.491003\pi$
$314$		0			0
$315$	−8.94427			−0.503953
$316$		0			0
$317$		−	4.47214i		−	0.251180i	−0.992082	−	0.125590i	$-0.959918\pi$
$317$		−	4.47214i		−	0.251180i	0.992082	−	0.125590i	$-0.0400824\pi$
$318$		0			0
$319$	13.4164	+	9.00000i	0.751175	+	0.503903i
$320$		0			0
$321$			13.4164i			0.748831i
$322$		0			0
$323$	17.8885			0.995345
$324$		0			0
$325$	−4.00000			−0.221880
$326$		0			0
$327$		−	1.00000i		−	0.0553001i
$328$		0			0
$329$		−	13.4164i		−	0.739671i
$330$		0			0
$331$		−	15.0000i		−	0.824475i	−0.911077	−	0.412237i	$-0.864747\pi$
$331$		−	15.0000i		−	0.824475i	0.911077	−	0.412237i	$-0.135253\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	8.94427			0.488678
$336$		0			0
$337$		−	22.3607i		−	1.21806i	−0.793146	−	0.609032i	$-0.791558\pi$
$337$		−	22.3607i		−	1.21806i	0.793146	−	0.609032i	$-0.208442\pi$
$338$		0			0
$339$	−17.8885			−0.971572
$340$		0			0
$341$	15.0000			0.812296
$342$		0			0
$343$	26.8328			1.44884
$344$		0			0
$345$		−	4.47214i		−	0.240772i
$346$		0			0
$347$	−17.8885			−0.960307			−0.480154	−	0.877184i	$-0.659419\pi$
$347$	−17.8885			−0.960307			−0.480154	+	0.877184i	$0.659419\pi$
$348$		0			0
$349$	9.00000			0.481759			0.240879	−	0.970555i	$-0.422564\pi$
$349$	9.00000			0.481759			0.240879	+	0.970555i	$0.422564\pi$
$350$		0			0
$351$			5.00000i			0.266880i
$352$		0			0
$353$	−14.0000			−0.745145			−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000			−0.745145			−0.372572	+	0.928003i	$0.621524\pi$
$354$		0			0
$355$	8.94427			0.474713
$356$		0			0
$357$	−20.0000			−1.05851
$358$		0			0
$359$		−	11.0000i		−	0.580558i	−0.956942	−	0.290279i	$-0.906252\pi$
$359$		−	11.0000i		−	0.580558i	0.956942	−	0.290279i	$-0.0937481\pi$
$360$		0			0
$361$	3.00000			0.157895
$362$		0			0
$363$			2.00000i			0.104973i
$364$		0			0
$365$			8.94427i			0.468165i
$366$		0			0
$367$		−	8.00000i		−	0.417597i	−0.977959	−	0.208798i	$-0.933045\pi$
$367$		−	8.00000i		−	0.417597i	0.977959	−	0.208798i	$-0.0669552\pi$
$368$		0			0
$369$			8.94427i			0.465620i
$370$		0			0
$371$	40.2492			2.08964
$372$		0			0
$373$	−21.0000			−1.08734			−0.543669	−	0.839299i	$-0.682965\pi$
$373$	−21.0000			−1.08734			−0.543669	+	0.839299i	$0.682965\pi$
$374$		0			0
$375$			9.00000i			0.464758i
$376$		0			0
$377$	3.00000	−	4.47214i	0.154508	−	0.230327i
$378$		0			0
$379$			20.0000i			1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$			20.0000i			1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$		0			0
$381$	−12.0000			−0.614779
$382$		0			0
$383$	−13.4164			−0.685546			−0.342773	−	0.939418i	$-0.611366\pi$
$383$	−13.4164			−0.685546			−0.342773	+	0.939418i	$0.611366\pi$
$384$		0			0
$385$		−	13.4164i		−	0.683763i
$386$		0			0
$387$		−	2.00000i		−	0.101666i
$388$		0			0
$389$		−	8.94427i		−	0.453493i	−0.973954	−	0.226746i	$-0.927191\pi$
$389$		−	8.94427i		−	0.453493i	0.973954	−	0.226746i	$-0.0728088\pi$
$390$		0			0
$391$			20.0000i			1.01144i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			15.0000i			0.754732i
$396$		0			0
$397$	−13.0000			−0.652451			−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000			−0.652451			−0.326226	+	0.945292i	$0.605777\pi$
$398$		0			0
$399$	17.8885			0.895547
$400$		0			0
$401$	15.0000			0.749064			0.374532	−	0.927214i	$-0.377803\pi$
$401$	15.0000			0.749064			0.374532	+	0.927214i	$0.377803\pi$
$402$		0			0
$403$		−	5.00000i		−	0.249068i
$404$		0			0
$405$	−1.00000			−0.0496904
$406$		0			0
$407$		0			0
$408$		0			0
$409$		−	17.8885i		−	0.884532i	−0.896884	−	0.442266i	$-0.854175\pi$
$409$		−	17.8885i		−	0.884532i	0.896884	−	0.442266i	$-0.145825\pi$
$410$		0			0
$411$	17.8885			0.882377
$412$		0			0
$413$	−20.0000			−0.984136
$414$		0			0
$415$	−4.47214			−0.219529
$416$		0			0
$417$			4.47214i			0.219001i
$418$		0			0
$419$	−35.7771			−1.74783			−0.873913	−	0.486083i	$-0.838425\pi$
$419$	−35.7771			−1.74783			−0.873913	+	0.486083i	$0.838425\pi$
$420$		0			0
$421$		−	31.3050i		−	1.52571i	−0.646570	−	0.762855i	$-0.723797\pi$
$421$		−	31.3050i		−	1.52571i	0.646570	−	0.762855i	$-0.276203\pi$
$422$		0			0
$423$		−	6.00000i		−	0.291730i
$424$		0			0
$425$		−	17.8885i		−	0.867722i
$426$		0			0
$427$			60.0000i			2.90360i
$428$		0			0
$429$	−3.00000			−0.144841
$430$		0			0
$431$	−4.47214			−0.215415			−0.107708	−	0.994183i	$-0.534351\pi$
$431$	−4.47214			−0.215415			−0.107708	+	0.994183i	$0.534351\pi$
$432$		0			0
$433$			8.94427i			0.429834i	0.976632	+	0.214917i	$0.0689481\pi$
$433$			8.94427i			0.429834i	−0.976632	+	0.214917i	$0.931052\pi$
$434$		0			0
$435$	−4.47214	−	3.00000i	−0.214423	−	0.143839i
$436$		0			0
$437$		−	17.8885i		−	0.855725i
$438$		0			0
$439$	40.2492			1.92099			0.960495	−	0.278296i	$-0.0897697\pi$
$439$	40.2492			1.92099			0.960495	+	0.278296i	$0.0897697\pi$
$440$		0			0
$441$	26.0000			1.23810
$442$		0			0
$443$		−	4.00000i		−	0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$		−	4.00000i		−	0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$		0			0
$445$		−	13.4164i		−	0.635999i
$446$		0			0
$447$			1.00000i			0.0472984i
$448$		0			0
$449$			17.8885i			0.844213i	0.906546	+	0.422106i	$0.138709\pi$
$449$			17.8885i			0.844213i	−0.906546	+	0.422106i	$0.861291\pi$
$450$		0			0
$451$	−13.4164			−0.631754
$452$		0			0
$453$		−	13.4164i		−	0.630358i
$454$		0			0
$455$	−4.47214			−0.209657
$456$		0			0
$457$	−38.0000			−1.77757			−0.888783	−	0.458329i	$-0.848448\pi$
$457$	−38.0000			−1.77757			−0.888783	+	0.458329i	$0.848448\pi$
$458$		0			0
$459$	−22.3607			−1.04371
$460$		0			0
$461$			17.8885i			0.833153i	0.909101	+	0.416576i	$0.136770\pi$
$461$			17.8885i			0.833153i	−0.909101	+	0.416576i	$0.863230\pi$
$462$		0			0
$463$	−13.4164			−0.623513			−0.311757	−	0.950162i	$-0.600917\pi$
$463$	−13.4164			−0.623513			−0.311757	+	0.950162i	$0.600917\pi$
$464$		0			0
$465$	−5.00000			−0.231869
$466$		0			0
$467$		−	37.0000i		−	1.71216i	−0.516847	−	0.856078i	$-0.672894\pi$
$467$		−	37.0000i		−	1.71216i	0.516847	−	0.856078i	$-0.327106\pi$
$468$		0			0
$469$	−40.0000			−1.84703
$470$		0			0
$471$	4.47214			0.206065
$472$		0			0
$473$	3.00000			0.137940
$474$		0			0
$475$			16.0000i			0.734130i
$476$		0			0
$477$	18.0000			0.824163
$478$		0			0
$479$		−	5.00000i		−	0.228456i	−0.993455	−	0.114228i	$-0.963561\pi$
$479$		−	5.00000i		−	0.228456i	0.993455	−	0.114228i	$-0.0364394\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$			20.0000i			0.910032i
$484$		0			0
$485$		−	13.4164i		−	0.609208i
$486$		0			0
$487$	40.2492			1.82387			0.911933	−	0.410339i	$-0.134590\pi$
$487$	40.2492			1.82387			0.911933	+	0.410339i	$0.134590\pi$
$488$		0			0
$489$	−1.00000			−0.0452216
$490$		0			0
$491$		−	25.0000i		−	1.12823i	−0.825695	−	0.564117i	$-0.809217\pi$
$491$		−	25.0000i		−	1.12823i	0.825695	−	0.564117i	$-0.190783\pi$
$492$		0			0
$493$	20.0000	+	13.4164i	0.900755	+	0.604245i
$494$		0			0
$495$		−	6.00000i		−	0.269680i
$496$		0			0
$497$	−40.0000			−1.79425
$498$		0			0
$499$	−13.4164			−0.600601			−0.300300	−	0.953845i	$-0.597087\pi$
$499$	−13.4164			−0.600601			−0.300300	+	0.953845i	$0.597087\pi$
$500$		0			0
$501$		−	8.94427i		−	0.399601i
$502$		0			0
$503$		−	9.00000i		−	0.401290i	−0.979664	−	0.200645i	$-0.935696\pi$
$503$		−	9.00000i		−	0.401290i	0.979664	−	0.200645i	$-0.0643038\pi$
$504$		0			0
$505$		−	8.94427i		−	0.398015i
$506$		0			0
$507$		−	12.0000i		−	0.532939i
$508$		0			0
$509$	25.0000			1.10811			0.554053	−	0.832482i	$-0.313081\pi$
$509$	25.0000			1.10811			0.554053	+	0.832482i	$0.313081\pi$
$510$		0			0
$511$		−	40.0000i		−	1.76950i
$512$		0			0
$513$	20.0000			0.883022
$514$		0			0
$515$		0			0
$516$		0			0
$517$	9.00000			0.395820
$518$		0			0
$519$			6.00000i			0.263371i
$520$		0			0
$521$	−23.0000			−1.00765			−0.503824	−	0.863806i	$-0.668074\pi$
$521$	−23.0000			−1.00765			−0.503824	+	0.863806i	$0.668074\pi$
$522$		0			0
$523$	−26.8328			−1.17332			−0.586659	−	0.809834i	$-0.699557\pi$
$523$	−26.8328			−1.17332			−0.586659	+	0.809834i	$0.699557\pi$
$524$		0			0
$525$		−	17.8885i		−	0.780720i
$526$		0			0
$527$	22.3607			0.974047
$528$		0			0
$529$	−3.00000			−0.130435
$530$		0			0
$531$	−8.94427			−0.388148
$532$		0			0
$533$			4.47214i			0.193710i
$534$		0			0
$535$	−13.4164			−0.580042
$536$		0			0
$537$			17.8885i			0.771948i
$538$		0			0
$539$			39.0000i			1.67985i
$540$		0			0
$541$		−	35.7771i		−	1.53818i	−0.639142	−	0.769089i	$-0.720710\pi$
$541$		−	35.7771i		−	1.53818i	0.639142	−	0.769089i	$-0.279290\pi$
$542$		0			0
$543$		−	25.0000i		−	1.07285i
$544$		0			0
$545$	1.00000			0.0428353
$546$		0			0
$547$	−4.47214			−0.191215			−0.0956074	−	0.995419i	$-0.530479\pi$
$547$	−4.47214			−0.191215			−0.0956074	+	0.995419i	$0.530479\pi$
$548$		0			0
$549$			26.8328i			1.14520i
$550$		0			0
$551$	−17.8885	−	12.0000i	−0.762078	−	0.511217i
$552$		0			0
$553$		−	67.0820i		−	2.85262i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−18.0000			−0.762684			−0.381342	−	0.924434i	$-0.624538\pi$
$557$	−18.0000			−0.762684			−0.381342	+	0.924434i	$0.624538\pi$
$558$		0			0
$559$		−	1.00000i		−	0.0422955i
$560$		0			0
$561$		−	13.4164i		−	0.566441i
$562$		0			0
$563$		−	11.0000i		−	0.463595i	−0.972764	−	0.231797i	$-0.925539\pi$
$563$		−	11.0000i		−	0.463595i	0.972764	−	0.231797i	$-0.0744606\pi$
$564$		0			0
$565$		−	17.8885i		−	0.752577i
$566$		0			0
$567$	4.47214			0.187812
$568$		0			0
$569$			44.7214i			1.87482i	0.348232	+	0.937408i	$0.386782\pi$
$569$			44.7214i			1.87482i	−0.348232	+	0.937408i	$0.613218\pi$
$570$		0			0
$571$	26.8328			1.12292			0.561459	−	0.827504i	$-0.310240\pi$
$571$	26.8328			1.12292			0.561459	+	0.827504i	$0.310240\pi$
$572$		0			0
$573$	−8.00000			−0.334205
$574$		0			0
$575$	−17.8885			−0.746004
$576$		0			0
$577$		−	22.3607i		−	0.930887i	−0.885078	−	0.465444i	$-0.845895\pi$
$577$		−	22.3607i		−	0.930887i	0.885078	−	0.465444i	$-0.154105\pi$
$578$		0			0
$579$	−4.47214			−0.185856
$580$		0			0
$581$	20.0000			0.829740
$582$		0			0
$583$			27.0000i			1.11823i
$584$		0			0
$585$	−2.00000			−0.0826898
$586$		0			0
$587$	−40.2492			−1.66126			−0.830632	−	0.556822i	$-0.812020\pi$
$587$	−40.2492			−1.66126			−0.830632	+	0.556822i	$0.812020\pi$
$588$		0			0
$589$	−20.0000			−0.824086
$590$		0			0
$591$		−	22.0000i		−	0.904959i
$592$		0			0
$593$	−11.0000			−0.451716			−0.225858	−	0.974160i	$-0.572519\pi$
$593$	−11.0000			−0.451716			−0.225858	+	0.974160i	$0.572519\pi$
$594$		0			0
$595$		−	20.0000i		−	0.819920i
$596$		0			0
$597$			4.47214i			0.183032i
$598$		0			0
$599$		−	15.0000i		−	0.612883i	−0.951889	−	0.306442i	$-0.900862\pi$
$599$		−	15.0000i		−	0.612883i	0.951889	−	0.306442i	$-0.0991384\pi$
$600$		0			0
$601$			4.47214i			0.182422i	0.995832	+	0.0912111i	$0.0290738\pi$
$601$			4.47214i			0.182422i	−0.995832	+	0.0912111i	$0.970926\pi$
$602$		0			0
$603$	−17.8885			−0.728478
$604$		0			0
$605$	−2.00000			−0.0813116
$606$		0			0
$607$		−	13.0000i		−	0.527654i	−0.964570	−	0.263827i	$-0.915015\pi$
$607$		−	13.0000i		−	0.527654i	0.964570	−	0.263827i	$-0.0849848\pi$
$608$		0			0
$609$	20.0000	+	13.4164i	0.810441	+	0.543660i
$610$		0			0
$611$		−	3.00000i		−	0.121367i
$612$		0			0
$613$	−29.0000			−1.17130			−0.585649	−	0.810564i	$-0.699160\pi$
$613$	−29.0000			−1.17130			−0.585649	+	0.810564i	$0.699160\pi$
$614$		0			0
$615$	4.47214			0.180334
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$			25.0000i			1.00483i	0.864625	+	0.502417i	$0.167556\pi$
$619$			25.0000i			1.00483i	−0.864625	+	0.502417i	$0.832444\pi$
$620$		0			0
$621$			22.3607i			0.897303i
$622$		0			0
$623$			60.0000i			2.40385i
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$			12.0000i			0.479234i
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−17.8885			−0.712132			−0.356066	−	0.934461i	$-0.615882\pi$
$631$	−17.8885			−0.712132			−0.356066	+	0.934461i	$0.615882\pi$
$632$		0			0
$633$	27.0000			1.07315
$634$		0			0
$635$		−	12.0000i		−	0.476205i
$636$		0			0
$637$	13.0000			0.515079
$638$		0			0
$639$	−17.8885			−0.707660
$640$		0			0
$641$			22.3607i			0.883194i	0.897214	+	0.441597i	$0.145588\pi$
$641$			22.3607i			0.883194i	−0.897214	+	0.441597i	$0.854412\pi$
$642$		0			0
$643$	−40.2492			−1.58727			−0.793637	−	0.608391i	$-0.791815\pi$
$643$	−40.2492			−1.58727			−0.793637	+	0.608391i	$0.791815\pi$
$644$		0			0
$645$	−1.00000			−0.0393750
$646$		0			0
$647$	13.4164			0.527453			0.263727	−	0.964597i	$-0.415048\pi$
$647$	13.4164			0.527453			0.263727	+	0.964597i	$0.415048\pi$
$648$		0			0
$649$		−	13.4164i		−	0.526640i
$650$		0			0
$651$	22.3607			0.876384
$652$		0			0
$653$			4.47214i			0.175008i	0.996164	+	0.0875041i	$0.0278891\pi$
$653$			4.47214i			0.175008i	−0.996164	+	0.0875041i	$0.972111\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	17.8885i		−	0.697899i
$658$		0			0
$659$			9.00000i			0.350590i	0.984516	+	0.175295i	$0.0560880\pi$
$659$			9.00000i			0.350590i	−0.984516	+	0.175295i	$0.943912\pi$
$660$		0			0
$661$	42.0000			1.63361			0.816805	−	0.576913i	$-0.195743\pi$
$661$	42.0000			1.63361			0.816805	+	0.576913i	$0.195743\pi$
$662$		0			0
$663$	−4.47214			−0.173683
$664$		0			0
$665$			17.8885i			0.693688i
$666$		0			0
$667$	13.4164	−	20.0000i	0.519485	−	0.774403i
$668$		0			0
$669$			17.8885i			0.691611i
$670$		0			0
$671$	−40.2492			−1.55380
$672$		0			0
$673$	19.0000			0.732396			0.366198	−	0.930537i	$-0.380659\pi$
$673$	19.0000			0.732396			0.366198	+	0.930537i	$0.380659\pi$
$674$		0			0
$675$		−	20.0000i		−	0.769800i
$676$		0			0
$677$			35.7771i			1.37503i	0.726172	+	0.687513i	$0.241297\pi$
$677$			35.7771i			1.37503i	−0.726172	+	0.687513i	$0.758703\pi$
$678$		0			0
$679$			60.0000i			2.30259i
$680$		0			0
$681$			17.8885i			0.685490i
$682$		0			0
$683$	17.8885			0.684486			0.342243	−	0.939611i	$-0.388813\pi$
$683$	17.8885			0.684486			0.342243	+	0.939611i	$0.388813\pi$
$684$		0			0
$685$			17.8885i			0.683486i
$686$		0			0
$687$	22.3607			0.853113
$688$		0			0
$689$	9.00000			0.342873
$690$		0			0
$691$	26.8328			1.02077			0.510384	−	0.859946i	$-0.329503\pi$
$691$	26.8328			1.02077			0.510384	+	0.859946i	$0.329503\pi$
$692$		0			0
$693$			26.8328i			1.01929i
$694$		0			0
$695$	−4.47214			−0.169638
$696$		0			0
$697$	−20.0000			−0.757554
$698$		0			0
$699$		−	9.00000i		−	0.340411i
$700$		0			0
$701$	−3.00000			−0.113308			−0.0566542	−	0.998394i	$-0.518043\pi$
$701$	−3.00000			−0.113308			−0.0566542	+	0.998394i	$0.518043\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$	−3.00000			−0.112987
$706$		0			0
$707$			40.0000i			1.50435i
$708$		0			0
$709$	−35.0000			−1.31445			−0.657226	−	0.753693i	$-0.728270\pi$
$709$	−35.0000			−1.31445			−0.657226	+	0.753693i	$0.728270\pi$
$710$		0			0
$711$		−	30.0000i		−	1.12509i
$712$		0			0
$713$		−	22.3607i		−	0.837414i
$714$		0			0
$715$		−	3.00000i		−	0.112194i
$716$		0			0
$717$		−	22.3607i		−	0.835075i
$718$		0			0
$719$	−17.8885			−0.667130			−0.333565	−	0.942727i	$-0.608252\pi$
$719$	−17.8885			−0.667130			−0.333565	+	0.942727i	$0.608252\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	17.0000i		−	0.632237i
$724$		0			0
$725$	−12.0000	+	17.8885i	−0.445669	+	0.664364i
$726$		0			0
$727$		−	32.0000i		−	1.18681i	−0.804902	−	0.593407i	$-0.797782\pi$
$727$		−	32.0000i		−	1.18681i	0.804902	−	0.593407i	$-0.202218\pi$
$728$		0			0
$729$	−13.0000			−0.481481
$730$		0			0
$731$	4.47214			0.165408
$732$		0			0
$733$			35.7771i			1.32146i	0.750625	+	0.660728i	$0.229753\pi$
$733$			35.7771i			1.32146i	−0.750625	+	0.660728i	$0.770247\pi$
$734$		0			0
$735$		−	13.0000i		−	0.479512i
$736$		0			0
$737$		−	26.8328i		−	0.988399i
$738$		0			0
$739$			1.00000i			0.0367856i	0.999831	+	0.0183928i	$0.00585494\pi$
$739$			1.00000i			0.0367856i	−0.999831	+	0.0183928i	$0.994145\pi$
$740$		0			0
$741$	4.00000			0.146944
$742$		0			0
$743$		−	16.0000i		−	0.586983i	−0.955962	−	0.293492i	$-0.905183\pi$
$743$		−	16.0000i		−	0.586983i	0.955962	−	0.293492i	$-0.0948173\pi$
$744$		0			0
$745$	−1.00000			−0.0366372
$746$		0			0
$747$	8.94427			0.327254
$748$		0			0
$749$	60.0000			2.19235
$750$		0			0
$751$			40.0000i			1.45962i	0.683650	+	0.729810i	$0.260392\pi$
$751$			40.0000i			1.45962i	−0.683650	+	0.729810i	$0.739608\pi$
$752$		0			0
$753$	23.0000			0.838167
$754$		0			0
$755$	13.4164			0.488273
$756$		0			0
$757$		−	31.3050i		−	1.13780i	−0.822407	−	0.568899i	$-0.807370\pi$
$757$		−	31.3050i		−	1.13780i	0.822407	−	0.568899i	$-0.192630\pi$
$758$		0			0
$759$	−13.4164			−0.486985
$760$		0			0
$761$	−30.0000			−1.08750			−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000			−1.08750			−0.543750	+	0.839248i	$0.682996\pi$
$762$		0			0
$763$	−4.47214			−0.161902
$764$		0			0
$765$		−	8.94427i		−	0.323381i
$766$		0			0
$767$	−4.47214			−0.161479
$768$		0			0
$769$			49.1935i			1.77396i	0.461805	+	0.886981i	$0.347202\pi$
$769$			49.1935i			1.77396i	−0.461805	+	0.886981i	$0.652798\pi$
$770$		0			0
$771$			17.0000i			0.612240i
$772$		0			0
$773$		−	4.47214i		−	0.160852i	−0.996761	−	0.0804258i	$-0.974372\pi$
$773$		−	4.47214i		−	0.160852i	0.996761	−	0.0804258i	$-0.0256280\pi$
$774$		0			0
$775$			20.0000i			0.718421i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	17.8885			0.640924
$780$		0			0
$781$		−	26.8328i		−	0.960154i
$782$		0			0
$783$	22.3607	+	15.0000i	0.799106	+	0.536056i
$784$		0			0
$785$			4.47214i			0.159617i
$786$		0			0
$787$	26.8328			0.956487			0.478243	−	0.878227i	$-0.341274\pi$
$787$	26.8328			0.956487			0.478243	+	0.878227i	$0.341274\pi$
$788$		0			0
$789$	21.0000			0.747620
$790$		0			0
$791$			80.0000i			2.84447i
$792$		0			0
$793$			13.4164i			0.476431i
$794$		0			0
$795$		−	9.00000i		−	0.319197i
$796$		0			0
$797$		−	22.3607i		−	0.792056i	−0.918238	−	0.396028i	$-0.870388\pi$
$797$		−	22.3607i		−	0.792056i	0.918238	−	0.396028i	$-0.129612\pi$
$798$		0			0
$799$	13.4164			0.474638
$800$		0			0
$801$			26.8328i			0.948091i
$802$		0			0
$803$	26.8328			0.946910
$804$		0			0
$805$	−20.0000			−0.704907
$806$		0			0
$807$	4.47214			0.157427
$808$		0			0
$809$		−	26.8328i		−	0.943392i	−0.881761	−	0.471696i	$-0.843642\pi$
$809$		−	26.8328i		−	0.943392i	0.881761	−	0.471696i	$-0.156358\pi$
$810$		0			0
$811$	4.47214			0.157038			0.0785190	−	0.996913i	$-0.474981\pi$
$811$	4.47214			0.157038			0.0785190	+	0.996913i	$0.474981\pi$
$812$		0			0
$813$	17.0000			0.596216
$814$		0			0
$815$		−	1.00000i		−	0.0350285i
$816$		0			0
$817$	−4.00000			−0.139942
$818$		0			0
$819$	8.94427			0.312538
$820$		0			0
$821$	25.0000			0.872506			0.436253	−	0.899824i	$-0.356305\pi$
$821$	25.0000			0.872506			0.436253	+	0.899824i	$0.356305\pi$
$822$		0			0
$823$		−	44.0000i		−	1.53374i	−0.641800	−	0.766872i	$-0.721812\pi$
$823$		−	44.0000i		−	1.53374i	0.641800	−	0.766872i	$-0.278188\pi$
$824$		0			0
$825$	12.0000			0.417786
$826$		0			0
$827$		−	23.0000i		−	0.799788i	−0.916561	−	0.399894i	$-0.869047\pi$
$827$		−	23.0000i		−	0.799788i	0.916561	−	0.399894i	$-0.130953\pi$
$828$		0			0
$829$			40.2492i			1.39791i	0.715164	+	0.698957i	$0.246352\pi$
$829$			40.2492i			1.39791i	−0.715164	+	0.698957i	$0.753648\pi$
$830$		0			0
$831$			22.0000i			0.763172i
$832$		0			0
$833$			58.1378i			2.01435i
$834$		0			0
$835$	8.94427			0.309529
$836$		0			0
$837$	25.0000			0.864126
$838$		0			0
$839$			5.00000i			0.172619i	0.996268	+	0.0863096i	$0.0275074\pi$
$839$			5.00000i			0.172619i	−0.996268	+	0.0863096i	$0.972493\pi$
$840$		0			0
$841$	−11.0000	−	26.8328i	−0.379310	−	0.925270i
$842$		0			0
$843$		−	5.00000i		−	0.172209i
$844$		0			0
$845$	12.0000			0.412813
$846$		0			0
$847$	8.94427			0.307329
$848$		0			0
$849$			4.47214i			0.153483i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	8.94427i		−	0.306246i	−0.988207	−	0.153123i	$-0.951067\pi$
$853$		−	8.94427i		−	0.306246i	0.988207	−	0.153123i	$-0.0489331\pi$
$854$		0			0
$855$			8.00000i			0.273594i
$856$		0			0
$857$	53.0000			1.81045			0.905223	−	0.424937i	$-0.139704\pi$
$857$	53.0000			1.81045			0.905223	+	0.424937i	$0.139704\pi$
$858$		0			0
$859$			15.0000i			0.511793i	0.966704	+	0.255897i	$0.0823707\pi$
$859$			15.0000i			0.511793i	−0.966704	+	0.255897i	$0.917629\pi$
$860$		0			0
$861$	−20.0000			−0.681598
$862$		0			0
$863$	8.94427			0.304467			0.152233	−	0.988345i	$-0.451353\pi$
$863$	8.94427			0.304467			0.152233	+	0.988345i	$0.451353\pi$
$864$		0			0
$865$	−6.00000			−0.204006
$866$		0			0
$867$		−	3.00000i		−	0.101885i
$868$		0			0
$869$	45.0000			1.52652
$870$		0			0
$871$	−8.94427			−0.303065
$872$		0			0
$873$			26.8328i			0.908153i
$874$		0			0
$875$	40.2492			1.36067
$876$		0			0
$877$	−43.0000			−1.45201			−0.726003	−	0.687691i	$-0.758624\pi$
$877$	−43.0000			−1.45201			−0.726003	+	0.687691i	$0.758624\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	8.94427i		−	0.301340i	−0.988584	−	0.150670i	$-0.951857\pi$
$881$		−	8.94427i		−	0.301340i	0.988584	−	0.150670i	$-0.0481431\pi$
$882$		0			0
$883$	40.2492			1.35449			0.677247	−	0.735756i	$-0.263173\pi$
$883$	40.2492			1.35449			0.677247	+	0.735756i	$0.263173\pi$
$884$		0			0
$885$			4.47214i			0.150329i
$886$		0			0
$887$		−	53.0000i		−	1.77957i	−0.456384	−	0.889783i	$-0.650856\pi$
$887$		−	53.0000i		−	1.77957i	0.456384	−	0.889783i	$-0.349144\pi$
$888$		0			0
$889$			53.6656i			1.79989i
$890$		0			0
$891$			3.00000i			0.100504i
$892$		0			0
$893$	−12.0000			−0.401565
$894$		0			0
$895$	−17.8885			−0.597948
$896$		0			0
$897$			4.47214i			0.149320i
$898$		0			0
$899$	−22.3607	−	15.0000i	−0.745770	−	0.500278i
$900$		0			0
$901$			40.2492i			1.34090i
$902$		0			0
$903$	4.47214			0.148823
$904$		0			0
$905$	25.0000			0.831028
$906$		0			0
$907$		−	28.0000i		−	0.929725i	−0.885383	−	0.464862i	$-0.846104\pi$
$907$		−	28.0000i		−	0.929725i	0.885383	−	0.464862i	$-0.153896\pi$
$908$		0			0
$909$			17.8885i			0.593326i
$910$		0			0
$911$			55.0000i			1.82223i	0.412151	+	0.911116i	$0.364778\pi$
$911$			55.0000i			1.82223i	−0.412151	+	0.911116i	$0.635222\pi$
$912$		0			0
$913$			13.4164i			0.444018i
$914$		0			0
$915$	13.4164			0.443533
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−53.6656			−1.77027			−0.885133	−	0.465338i	$-0.845933\pi$
$919$	−53.6656			−1.77027			−0.885133	+	0.465338i	$0.845933\pi$
$920$		0			0
$921$	−27.0000			−0.889680
$922$		0			0
$923$	−8.94427			−0.294404
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	46.0000			1.50921			0.754606	−	0.656179i	$-0.227828\pi$
$929$	46.0000			1.50921			0.754606	+	0.656179i	$0.227828\pi$
$930$		0			0
$931$		−	52.0000i		−	1.70423i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	13.4164			0.438763
$936$		0			0
$937$	2.00000			0.0653372			0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000			0.0653372			0.0326686	+	0.999466i	$0.489599\pi$
$938$		0			0
$939$			1.00000i			0.0326338i
$940$		0			0
$941$	−33.0000			−1.07577			−0.537885	−	0.843018i	$-0.680776\pi$
$941$	−33.0000			−1.07577			−0.537885	+	0.843018i	$0.680776\pi$
$942$		0			0
$943$			20.0000i			0.651290i
$944$		0			0
$945$		−	22.3607i		−	0.727393i
$946$		0			0
$947$			27.0000i			0.877382i	0.898638	+	0.438691i	$0.144558\pi$
$947$			27.0000i			0.877382i	−0.898638	+	0.438691i	$0.855442\pi$
$948$		0			0
$949$		−	8.94427i		−	0.290343i
$950$		0			0
$951$	4.47214			0.145019
$952$		0			0
$953$	39.0000			1.26333			0.631667	−	0.775240i	$-0.282371\pi$
$953$	39.0000			1.26333			0.631667	+	0.775240i	$0.282371\pi$
$954$		0			0
$955$		−	8.00000i		−	0.258874i
$956$		0			0
$957$	−9.00000	+	13.4164i	−0.290929	+	0.433691i
$958$		0			0
$959$		−	80.0000i		−	2.58333i
$960$		0			0
$961$	6.00000			0.193548
$962$		0			0
$963$	26.8328			0.864675
$964$		0			0
$965$		−	4.47214i		−	0.143963i
$966$		0			0
$967$		−	7.00000i		−	0.225105i	−0.993646	−	0.112552i	$-0.964097\pi$
$967$		−	7.00000i		−	0.225105i	0.993646	−	0.112552i	$-0.0359026\pi$
$968$		0			0
$969$			17.8885i			0.574663i
$970$		0			0
$971$		−	20.0000i		−	0.641831i	−0.947108	−	0.320915i	$-0.896010\pi$
$971$		−	20.0000i		−	0.641831i	0.947108	−	0.320915i	$-0.103990\pi$
$972$		0			0
$973$	20.0000			0.641171
$974$		0			0
$975$		−	4.00000i		−	0.128103i
$976$		0			0
$977$	−43.0000			−1.37569			−0.687846	−	0.725857i	$-0.741444\pi$
$977$	−43.0000			−1.37569			−0.687846	+	0.725857i	$0.741444\pi$
$978$		0			0
$979$	−40.2492			−1.28637
$980$		0			0
$981$	−2.00000			−0.0638551
$982$		0			0
$983$			9.00000i			0.287055i	0.989646	+	0.143528i	$0.0458446\pi$
$983$			9.00000i			0.287055i	−0.989646	+	0.143528i	$0.954155\pi$
$984$		0			0
$985$	22.0000			0.700978
$986$		0			0
$987$	13.4164			0.427049
$988$		0			0
$989$		−	4.47214i		−	0.142206i
$990$		0			0
$991$		0			0			−	1.00000i	$-0.5\pi$
$991$		0			0				1.00000i	$0.5\pi$
$992$		0			0
$993$	15.0000			0.476011
$994$		0			0
$995$	−4.47214			−0.141776
$996$		0			0
$997$		−	17.8885i		−	0.566536i	−0.959041	−	0.283268i	$-0.908581\pi$
$997$		−	17.8885i		−	0.566536i	0.959041	−	0.283268i	$-0.0914186\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1856.2.e.h.1217.4		4
4.3	odd	2	inner	1856.2.e.h.1217.1		4
8.3	odd	2		928.2.e.b.289.3	yes	4
8.5	even	2		928.2.e.b.289.2	yes	4
29.28	even	2	inner	1856.2.e.h.1217.2		4
116.115	odd	2	inner	1856.2.e.h.1217.3		4
232.115	odd	2		928.2.e.b.289.1	✓	4
232.173	even	2		928.2.e.b.289.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
928.2.e.b.289.1	✓	4	232.115	odd	2
928.2.e.b.289.2	yes	4	8.5	even	2
928.2.e.b.289.3	yes	4	8.3	odd	2
928.2.e.b.289.4	yes	4	232.173	even	2
1856.2.e.h.1217.1		4	4.3	odd	2	inner
1856.2.e.h.1217.2		4	29.28	even	2	inner
1856.2.e.h.1217.3		4	116.115	odd	2	inner
1856.2.e.h.1217.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1856 = 2^{6} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1856.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -1.00000 q^{5} +4.47214 q^{7} +2.00000 q^{9} +O(q^{10})\)	\(q+1.00000i q^{3} -1.00000 q^{5} +4.47214 q^{7} +2.00000 q^{9} +3.00000i q^{11} +1.00000 q^{13} -1.00000i q^{15} +4.47214i q^{17} -4.00000i q^{19} +4.47214i q^{21} +4.47214 q^{23} -4.00000 q^{25} +5.00000i q^{27} +(3.00000 - 4.47214i) q^{29} -5.00000i q^{31} -3.00000 q^{33} -4.47214 q^{35} +1.00000i q^{39} +4.47214i q^{41} -1.00000i q^{43} -2.00000 q^{45} -3.00000i q^{47} +13.0000 q^{49} -4.47214 q^{51} +9.00000 q^{53} -3.00000i q^{55} +4.00000 q^{57} -4.47214 q^{59} +13.4164i q^{61} +8.94427 q^{63} -1.00000 q^{65} -8.94427 q^{67} +4.47214i q^{69} -8.94427 q^{71} -8.94427i q^{73} -4.00000i q^{75} +13.4164i q^{77} -15.0000i q^{79} +1.00000 q^{81} +4.47214 q^{83} -4.47214i q^{85} +(4.47214 + 3.00000i) q^{87} +13.4164i q^{89} +4.47214 q^{91} +5.00000 q^{93} +4.00000i q^{95} +13.4164i q^{97} +6.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 8 q^{9}+O(q^{10}) \) 4 * q - 4 * q^5 + 8 * q^9	\( 4 q - 4 q^{5} + 8 q^{9} + 4 q^{13} - 16 q^{25} + 12 q^{29} - 12 q^{33} - 8 q^{45} + 52 q^{49} + 36 q^{53} + 16 q^{57} - 4 q^{65} + 4 q^{81} + 20 q^{93}+O(q^{100}) \) 4 * q - 4 * q^5 + 8 * q^9 + 4 * q^13 - 16 * q^25 + 12 * q^29 - 12 * q^33 - 8 * q^45 + 52 * q^49 + 36 * q^53 + 16 * q^57 - 4 * q^65 + 4 * q^81 + 20 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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