LMFDB - Embedded newform 855.3.g.a.379.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [855,3,Mod(379,855)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("855.379"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(855, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 3, names="a")

Level:	$N$	$=$	$855 = 3^{2} \cdot 5 \cdot 19$
Weight:	$k$	$=$	$3$
Character orbit:	$[\chi]$	$=$	855.g (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-8,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$23.2970626028$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - x + 5$ x^2 - x + 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 95)
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			379.1
Root			$0.500000 + 2.17945i$ of defining polynomial
Character	$\chi$	$=$	855.379
Dual form			855.3.g.a.379.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-4.00000 q^{4} +(4.50000 - 2.17945i) q^{5} -13.0767i q^{7} +3.00000 q^{11} +16.0000 q^{16} -30.5123i q^{17} -19.0000 q^{19} +(-18.0000 + 8.71780i) q^{20} +34.8712i q^{23} +(15.5000 - 19.6150i) q^{25} +52.3068i q^{28} +(-28.5000 - 58.8451i) q^{35} -13.0767i q^{43} -12.0000 q^{44} +56.6657i q^{47} -122.000 q^{49} +(13.5000 - 6.53835i) q^{55} -103.000 q^{61} -64.0000 q^{64} +122.049i q^{68} -143.844i q^{73} +76.0000 q^{76} -39.2301i q^{77} +(72.0000 - 34.8712i) q^{80} -139.485i q^{83} +(-66.5000 - 137.305i) q^{85} -139.485i q^{92} +(-85.5000 + 41.4095i) q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 8 q^{4} + 9 q^{5} + 6 q^{11} + 32 q^{16} - 38 q^{19} - 36 q^{20} + 31 q^{25} - 57 q^{35} - 24 q^{44} - 244 q^{49} + 27 q^{55} - 206 q^{61} - 128 q^{64} + 152 q^{76} + 144 q^{80} - 133 q^{85} - 171 q^{95}+O(q^{100})$ 2 * q - 8 * q^4 + 9 * q^5 + 6 * q^11 + 32 * q^16 - 38 * q^19 - 36 * q^20 + 31 * q^25 - 57 * q^35 - 24 * q^44 - 244 * q^49 + 27 * q^55 - 206 * q^61 - 128 * q^64 + 152 * q^76 + 144 * q^80 - 133 * q^85 - 171 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/855\mathbb{Z}\right)^\times$ .

$n$	$172$	$191$	$496$
$\chi(n)$	$-1$	$1$	$-1$

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			−	1.00000i	$-0.5\pi$
$2$		0			0				1.00000i	$0.5\pi$
$3$		0			0
$3$		0			0
$4$	−4.00000			−1.00000
$5$	4.50000	−	2.17945i	0.900000	−	0.435890i
$5$	4.50000	−	2.17945i	0.900000	−	0.435890i
$6$		0			0
$7$		−	13.0767i		−	1.86810i	−0.357143	−	0.934050i	$-0.616249\pi$
$7$		−	13.0767i		−	1.86810i	0.357143	−	0.934050i	$-0.383751\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.00000			0.272727			0.136364	−	0.990659i	$-0.456458\pi$
$11$	3.00000			0.272727			0.136364	+	0.990659i	$0.456458\pi$
$12$		0			0
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	16.0000			1.00000
$17$		−	30.5123i		−	1.79484i	−0.441176	−	0.897420i	$-0.645439\pi$
$17$		−	30.5123i		−	1.79484i	0.441176	−	0.897420i	$-0.354561\pi$
$18$		0			0
$19$	−19.0000			−1.00000
$19$	−19.0000			−1.00000
$20$	−18.0000	+	8.71780i	−0.900000	+	0.435890i
$21$		0			0
$22$		0			0
$23$			34.8712i			1.51614i	0.652174	+	0.758069i	$0.273857\pi$
$23$			34.8712i			1.51614i	−0.652174	+	0.758069i	$0.726143\pi$
$24$		0			0
$25$	15.5000	−	19.6150i	0.620000	−	0.784602i
$26$		0			0
$27$		0			0
$28$			52.3068i			1.86810i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−28.5000	−	58.8451i	−0.814286	−	1.68129i
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		−	13.0767i		−	0.304109i	−0.988372	−	0.152055i	$-0.951411\pi$
$43$		−	13.0767i		−	0.304109i	0.988372	−	0.152055i	$-0.0485890\pi$
$44$	−12.0000			−0.272727
$45$		0			0
$46$		0			0
$47$			56.6657i			1.20565i	0.797872	+	0.602826i	$0.205959\pi$
$47$			56.6657i			1.20565i	−0.797872	+	0.602826i	$0.794041\pi$
$48$		0			0
$49$	−122.000			−2.48980
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$	13.5000	−	6.53835i	0.245455	−	0.118879i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		1.00000			$0$
$59$		0			0		−1.00000			$\pi$
$60$		0			0
$61$	−103.000			−1.68852			−0.844262	−	0.535930i	$-0.819961\pi$
$61$	−103.000			−1.68852			−0.844262	+	0.535930i	$0.819961\pi$
$62$		0			0
$63$		0			0
$64$	−64.0000			−1.00000
$65$		0			0
$66$		0			0
$67$		0			0			−	1.00000i	$-0.5\pi$
$67$		0			0				1.00000i	$0.5\pi$
$68$			122.049i			1.79484i
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	143.844i		−	1.97046i	−0.171233	−	0.985231i	$-0.554775\pi$
$73$		−	143.844i		−	1.97046i	0.171233	−	0.985231i	$-0.445225\pi$
$74$		0			0
$75$		0			0
$76$	76.0000			1.00000
$77$		−	39.2301i		−	0.509482i
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$	72.0000	−	34.8712i	0.900000	−	0.435890i
$81$		0			0
$82$		0			0
$83$		−	139.485i		−	1.68054i	−0.542169	−	0.840270i	$-0.682397\pi$
$83$		−	139.485i		−	1.68054i	0.542169	−	0.840270i	$-0.317603\pi$
$84$		0			0
$85$	−66.5000	−	137.305i	−0.782353	−	1.61536i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		0			0
$92$		−	139.485i		−	1.51614i
$93$		0			0
$94$		0			0
$95$	−85.5000	+	41.4095i	−0.900000	+	0.435890i
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$		0			0
$100$	−62.0000	+	78.4602i	−0.620000	+	0.784602i
$101$	−102.000			−1.00990			−0.504950	−	0.863148i	$-0.668489\pi$
$101$	−102.000			−1.00990			−0.504950	+	0.863148i	$0.668489\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		−	209.227i		−	1.86810i
$113$		0			0			−	1.00000i	$-0.5\pi$
$113$		0			0				1.00000i	$0.5\pi$
$114$		0			0
$115$	76.0000	+	156.920i	0.660870	+	1.36452i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−399.000			−3.35294
$120$		0			0
$121$	−112.000			−0.925620
$122$		0			0
$123$		0			0
$124$		0			0
$125$	27.0000	−	122.049i	0.216000	−	0.976393i
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	213.000			1.62595			0.812977	−	0.582296i	$-0.197845\pi$
$131$	213.000			1.62595			0.812977	+	0.582296i	$0.197845\pi$
$132$		0			0
$133$			248.457i			1.86810i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			100.255i			0.731786i	0.930657	+	0.365893i	$0.119236\pi$
$137$			100.255i			0.731786i	−0.930657	+	0.365893i	$0.880764\pi$
$138$		0			0
$139$	197.000			1.41727			0.708633	−	0.705577i	$-0.249312\pi$
$139$	197.000			1.41727			0.708633	+	0.705577i	$0.249312\pi$
$140$	114.000	+	235.381i	0.814286	+	1.68129i
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−177.000			−1.18792			−0.593960	−	0.804495i	$-0.702436\pi$
$149$	−177.000			−1.18792			−0.593960	+	0.804495i	$0.702436\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$			313.841i			1.99899i	0.0318471	+	0.999493i	$0.489861\pi$
$157$			313.841i			1.99899i	−0.0318471	+	0.999493i	$0.510139\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	456.000			2.83230
$162$		0			0
$163$		−	209.227i		−	1.28360i	−0.766871	−	0.641801i	$-0.778187\pi$
$163$		−	209.227i		−	1.28360i	0.766871	−	0.641801i	$-0.221813\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−169.000			−1.00000
$170$		0			0
$171$		0			0
$172$			52.3068i			0.304109i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−256.500	−	202.689i	−1.46571	−	1.15822i
$176$	48.0000			0.272727
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$		−	91.5369i		−	0.489502i
$188$		−	226.663i		−	1.20565i
$189$		0			0
$190$		0			0
$191$	93.0000			0.486911			0.243455	−	0.969912i	$-0.421719\pi$
$191$	93.0000			0.486911			0.243455	+	0.969912i	$0.421719\pi$
$192$		0			0
$193$		0			0			−	1.00000i	$-0.5\pi$
$193$		0			0				1.00000i	$0.5\pi$
$194$		0			0
$195$		0			0
$196$	488.000			2.48980
$197$			383.583i			1.94712i	0.228426	+	0.973561i	$0.426642\pi$
$197$			383.583i			1.94712i	−0.228426	+	0.973561i	$0.573358\pi$
$198$		0			0
$199$	227.000			1.14070			0.570352	−	0.821401i	$-0.306807\pi$
$199$	227.000			1.14070			0.570352	+	0.821401i	$0.306807\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−57.0000			−0.272727
$210$		0			0
$211$		0			0		1.00000			$0$
$211$		0			0		−1.00000			$\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−28.5000	−	58.8451i	−0.132558	−	0.273698i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$	−54.0000	+	26.1534i	−0.245455	+	0.118879i
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$		0			0
$229$	17.0000			0.0742358			0.0371179	−	0.999311i	$-0.488182\pi$
$229$	17.0000			0.0742358			0.0371179	+	0.999311i	$0.488182\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	30.5123i		−	0.130954i	−0.997854	−	0.0654770i	$-0.979143\pi$
$233$		−	30.5123i		−	0.130954i	0.997854	−	0.0654770i	$-0.0208569\pi$
$234$		0			0
$235$	123.500	+	254.996i	0.525532	+	1.08509i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	453.000			1.89540			0.947699	−	0.319166i	$-0.103403\pi$
$239$	453.000			1.89540			0.947699	+	0.319166i	$0.103403\pi$
$240$		0			0
$241$		0			0		1.00000			$0$
$241$		0			0		−1.00000			$\pi$
$242$		0			0
$243$		0			0
$244$	412.000			1.68852
$245$	−549.000	+	265.893i	−2.24082	+	1.08528i
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−27.0000			−0.107570			−0.0537849	−	0.998553i	$-0.517129\pi$
$251$	−27.0000			−0.107570			−0.0537849	+	0.998553i	$0.517129\pi$
$252$		0			0
$253$			104.614i			0.413492i
$254$		0			0
$255$		0			0
$256$	256.000			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	335.635i		−	1.27618i	−0.769962	−	0.638090i	$-0.779725\pi$
$263$		−	335.635i		−	1.27618i	0.769962	−	0.638090i	$-0.220275\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$	142.000			0.523985			0.261993	−	0.965070i	$-0.415620\pi$
$271$	142.000			0.523985			0.261993	+	0.965070i	$0.415620\pi$
$272$		−	488.197i		−	1.79484i
$273$		0			0
$274$		0			0
$275$	46.5000	−	58.8451i	0.169091	−	0.213982i
$276$		0			0
$277$		−	143.844i		−	0.519291i	−0.965704	−	0.259646i	$-0.916394\pi$
$277$		−	143.844i		−	0.519291i	0.965704	−	0.259646i	$-0.0836057\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	405.378i		−	1.43243i	−0.697880	−	0.716215i	$-0.745873\pi$
$283$		−	405.378i		−	1.43243i	0.697880	−	0.716215i	$-0.254127\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−642.000			−2.22145
$290$		0			0
$291$		0			0
$292$			575.375i			1.97046i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$	−171.000			−0.568106
$302$		0			0
$303$		0			0
$304$	−304.000			−1.00000
$305$	−463.500	+	224.483i	−1.51967	+	0.736011i
$306$		0			0
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$			156.920i			0.509482i
$309$		0			0
$310$		0			0
$311$	603.000			1.93891			0.969453	−	0.245276i	$-0.0788785\pi$
$311$	603.000			1.93891			0.969453	+	0.245276i	$0.0788785\pi$
$312$		0			0
$313$		−	209.227i		−	0.668457i	−0.942492	−	0.334229i	$-0.891524\pi$
$313$		−	209.227i		−	0.668457i	0.942492	−	0.334229i	$-0.108476\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$	−288.000	+	139.485i	−0.900000	+	0.435890i
$321$		0			0
$322$		0			0
$323$			579.734i			1.79484i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$	741.000			2.25228
$330$		0			0
$331$		0			0		1.00000			$0$
$331$		0			0		−1.00000			$\pi$
$332$			557.939i			1.68054i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$		0			0
$340$	266.000	+	549.221i	0.782353	+	1.61536i
$341$		0			0
$342$		0			0
$343$			954.599i			2.78309i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	161.279i		−	0.464782i	−0.972622	−	0.232391i	$-0.925345\pi$
$347$		−	161.279i		−	0.464782i	0.972622	−	0.232391i	$-0.0746548\pi$
$348$		0			0
$349$	527.000			1.51003			0.755014	−	0.655708i	$-0.227630\pi$
$349$	527.000			1.51003			0.755014	+	0.655708i	$0.227630\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	488.197i		−	1.38299i	−0.722380	−	0.691497i	$-0.756952\pi$
$353$		−	488.197i		−	1.38299i	0.722380	−	0.691497i	$-0.243048\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$	243.000			0.676880			0.338440	−	0.940988i	$-0.390101\pi$
$359$	243.000			0.676880			0.338440	+	0.940988i	$0.390101\pi$
$360$		0			0
$361$	361.000			1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−313.500	−	647.296i	−0.858904	−	1.77342i
$366$		0			0
$367$		−	732.295i		−	1.99535i	−0.0681199	−	0.997677i	$-0.521700\pi$
$367$		−	732.295i		−	1.99535i	0.0681199	−	0.997677i	$-0.478300\pi$
$368$			557.939i			1.51614i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$		0			0		1.00000			$0$
$379$		0			0		−1.00000			$\pi$
$380$	342.000	−	165.638i	0.900000	−	0.435890i
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$	−85.5000	−	176.535i	−0.222078	−	0.458534i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	153.000			0.393316			0.196658	−	0.980472i	$-0.436991\pi$
$389$	153.000			0.393316			0.196658	+	0.980472i	$0.436991\pi$
$390$		0			0
$391$	1064.00			2.72123
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$		−	274.611i		−	0.691714i	−0.938287	−	0.345857i	$-0.887588\pi$
$397$		−	274.611i		−	0.691714i	0.938287	−	0.345857i	$-0.112412\pi$
$398$		0			0
$399$		0			0
$400$	248.000	−	313.841i	0.620000	−	0.784602i
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$		0			0
$404$	408.000			1.00990
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		1.00000			$0$
$409$		0			0		−1.00000			$\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−304.000	−	627.681i	−0.732530	−	1.51249i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−762.000			−1.81862			−0.909308	−	0.416124i	$-0.863388\pi$
$419$	−762.000			−1.81862			−0.909308	+	0.416124i	$0.863388\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−598.500	−	472.941i	−1.40824	−	1.11280i
$426$		0			0
$427$			1346.90i			3.15433i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		0			0
$433$		0			0			−	1.00000i	$-0.5\pi$
$433$		0			0				1.00000i	$0.5\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	662.553i		−	1.51614i
$438$		0			0
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			884.856i			1.99742i	0.0507901	+	0.998709i	$0.483826\pi$
$443$			884.856i			1.99742i	−0.0507901	+	0.998709i	$0.516174\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$			836.909i			1.86810i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	666.912i		−	1.45933i	−0.683807	−	0.729663i	$-0.739677\pi$
$457$		−	666.912i		−	1.45933i	0.683807	−	0.729663i	$-0.260323\pi$
$458$		0			0
$459$		0			0
$460$	−304.000	−	627.681i	−0.660870	−	1.36452i
$461$	−447.000			−0.969631			−0.484816	−	0.874616i	$-0.661113\pi$
$461$	−447.000			−0.969631			−0.484816	+	0.874616i	$0.661113\pi$
$462$		0			0
$463$		−	536.145i		−	1.15798i	−0.815335	−	0.578990i	$-0.803447\pi$
$463$		−	536.145i		−	1.15798i	0.815335	−	0.578990i	$-0.196553\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			187.433i			0.401355i	0.979657	+	0.200677i	$0.0643143\pi$
$467$			187.433i			0.401355i	−0.979657	+	0.200677i	$0.935686\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	39.2301i		−	0.0829389i
$474$		0			0
$475$	−294.500	+	372.686i	−0.620000	+	0.784602i
$476$	1596.00			3.35294
$477$		0			0
$478$		0			0
$479$	−942.000			−1.96660			−0.983299	−	0.182000i	$-0.941743\pi$
$479$	−942.000			−1.96660			−0.983299	+	0.182000i	$0.941743\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	448.000			0.925620
$485$		0			0
$486$		0			0
$487$		0			0			−	1.00000i	$-0.5\pi$
$487$		0			0				1.00000i	$0.5\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	918.000			1.86965			0.934827	−	0.355104i	$-0.115554\pi$
$491$	918.000			1.86965			0.934827	+	0.355104i	$0.115554\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−523.000			−1.04810			−0.524048	−	0.851689i	$-0.675579\pi$
$499$	−523.000			−1.04810			−0.524048	+	0.851689i	$0.675579\pi$
$500$	−108.000	+	488.197i	−0.216000	+	0.976393i
$501$		0			0
$502$		0			0
$503$			383.583i			0.762591i	0.924453	+	0.381295i	$0.124522\pi$
$503$			383.583i			0.762591i	−0.924453	+	0.381295i	$0.875478\pi$
$504$		0			0
$505$	−459.000	+	222.304i	−0.908911	+	0.440206i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$	−1881.00			−3.68102
$512$		0			0
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$			169.997i			0.328814i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$		0			0			−	1.00000i	$-0.5\pi$
$523$		0			0				1.00000i	$0.5\pi$
$524$	−852.000			−1.62595
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−687.000			−1.29868
$530$		0			0
$531$		0			0
$532$		−	993.829i		−	1.86810i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−366.000			−0.679035
$540$		0			0
$541$	457.000			0.844732			0.422366	−	0.906425i	$-0.361200\pi$
$541$	457.000			0.844732			0.422366	+	0.906425i	$0.361200\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$		0			0			−	1.00000i	$-0.5\pi$
$547$		0			0				1.00000i	$0.5\pi$
$548$		−	401.019i		−	0.731786i
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	−788.000			−1.41727
$557$		−	204.868i		−	0.367807i	−0.982944	−	0.183903i	$-0.941127\pi$
$557$		−	204.868i		−	0.367807i	0.982944	−	0.183903i	$-0.0588733\pi$
$558$		0			0
$559$		0			0
$560$	−456.000	−	941.522i	−0.814286	−	1.68129i
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−458.000			−0.802102			−0.401051	−	0.916056i	$-0.631355\pi$
$571$	−458.000			−0.802102			−0.401051	+	0.916056i	$0.631355\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	684.000	+	540.503i	1.18957	+	0.940006i
$576$		0			0
$577$		−	143.844i		−	0.249296i	−0.992201	−	0.124648i	$-0.960220\pi$
$577$		−	143.844i		−	0.249296i	0.992201	−	0.124648i	$-0.0397801\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1824.00			−3.13941
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	335.635i		−	0.571781i	−0.958262	−	0.285890i	$-0.907711\pi$
$587$		−	335.635i		−	0.571781i	0.958262	−	0.285890i	$-0.0922893\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	1185.62i		−	1.99936i	−0.0252951	−	0.999680i	$-0.508053\pi$
$593$		−	1185.62i		−	1.99936i	0.0252951	−	0.999680i	$-0.491947\pi$
$594$		0			0
$595$	−1795.50	+	869.600i	−3.01765	+	1.46151i
$596$	708.000			1.18792
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−504.000	+	244.098i	−0.833058	+	0.403468i
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		−	1189.98i		−	1.94124i	−0.240620	−	0.970619i	$-0.577351\pi$
$613$		−	1189.98i		−	1.94124i	0.240620	−	0.970619i	$-0.422649\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			623.323i			1.01025i	0.863047	+	0.505124i	$0.168553\pi$
$617$			623.323i			1.01025i	−0.863047	+	0.505124i	$0.831447\pi$
$618$		0			0
$619$	662.000			1.06947			0.534733	−	0.845021i	$-0.320412\pi$
$619$	662.000			1.06947			0.534733	+	0.845021i	$0.320412\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−144.500	−	608.066i	−0.231200	−	0.972906i
$626$		0			0
$627$		0			0
$628$		−	1255.36i		−	1.99899i
$629$		0			0
$630$		0			0
$631$	1037.00			1.64342			0.821712	−	0.569904i	$-0.193019\pi$
$631$	1037.00			1.64342			0.821712	+	0.569904i	$0.193019\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			640.758i			0.996513i	0.867030	+	0.498257i	$0.166026\pi$
$643$			640.758i			0.996513i	−0.867030	+	0.498257i	$0.833974\pi$
$644$	−1824.00			−2.83230
$645$		0			0
$646$		0			0
$647$		−	815.114i		−	1.25984i	−0.776662	−	0.629918i	$-0.783088\pi$
$647$		−	815.114i		−	1.25984i	0.776662	−	0.629918i	$-0.216912\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$			836.909i			1.28360i
$653$		−	1251.00i		−	1.91578i	−0.287136	−	0.957890i	$-0.592703\pi$
$653$		−	1251.00i		−	1.91578i	0.287136	−	0.957890i	$-0.407297\pi$
$654$		0			0
$655$	958.500	−	464.223i	1.46336	−	0.708737i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	541.500	+	1118.06i	0.814286	+	1.68129i
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−309.000			−0.460507
$672$		0			0
$673$		0			0			−	1.00000i	$-0.5\pi$
$673$		0			0				1.00000i	$0.5\pi$
$674$		0			0
$675$		0			0
$676$	676.000			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0			−	1.00000i	$-0.5\pi$
$683$		0			0				1.00000i	$0.5\pi$
$684$		0			0
$685$	218.500	+	451.146i	0.318978	+	0.658607i
$686$		0			0
$687$		0			0
$688$		−	209.227i		−	0.304109i
$689$		0			0
$690$		0			0
$691$	157.000			0.227207			0.113603	−	0.993526i	$-0.463761\pi$
$691$	157.000			0.227207			0.113603	+	0.993526i	$0.463761\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	886.500	−	429.352i	1.27554	−	0.617772i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1026.00	+	810.755i	1.46571	+	1.15822i
$701$	1098.00			1.56633			0.783167	−	0.621812i	$-0.213603\pi$
$701$	1098.00			1.56633			0.783167	+	0.621812i	$0.213603\pi$
$702$		0			0
$703$		0			0
$704$	−192.000			−0.272727
$705$		0			0
$706$		0			0
$707$			1333.82i			1.88660i
$708$		0			0
$709$	−1318.00			−1.85896			−0.929478	−	0.368877i	$-0.879742\pi$
$709$	−1318.00			−1.85896			−0.929478	+	0.368877i	$0.879742\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$	963.000			1.33936			0.669680	−	0.742650i	$-0.266431\pi$
$719$	963.000			1.33936			0.669680	+	0.742650i	$0.266431\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		−	1451.51i		−	1.99658i	−0.0584594	−	0.998290i	$-0.518619\pi$
$727$		−	1451.51i		−	1.99658i	0.0584594	−	0.998290i	$-0.481381\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−399.000			−0.545828
$732$		0			0
$733$		−	732.295i		−	0.999038i	−0.866303	−	0.499519i	$-0.833510\pi$
$733$		−	732.295i		−	0.999038i	0.866303	−	0.499519i	$-0.166490\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	547.000			0.740189			0.370095	−	0.928994i	$-0.379325\pi$
$739$	547.000			0.740189			0.370095	+	0.928994i	$0.379325\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$	−796.500	+	385.763i	−1.06913	+	0.517802i
$746$		0			0
$747$		0			0
$748$			366.148i			0.489502i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$			906.651i			1.20565i
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$			1294.59i			1.71016i	0.518494	+	0.855081i	$0.326493\pi$
$757$			1294.59i			1.71016i	−0.518494	+	0.855081i	$0.673507\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1503.00			1.97503			0.987516	−	0.157516i	$-0.0503486\pi$
$761$	1503.00			1.97503			0.987516	+	0.157516i	$0.0503486\pi$
$762$		0			0
$763$		0			0
$764$	−372.000			−0.486911
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−1063.00			−1.38231			−0.691157	−	0.722704i	$-0.742899\pi$
$769$	−1063.00			−1.38231			−0.691157	+	0.722704i	$0.742899\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1952.00			−2.48980
$785$	684.000	+	1412.28i	0.871338	+	1.79909i
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		−	1534.33i		−	1.94712i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$	−908.000			−1.14070
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$	1729.00			2.16395
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	431.531i		−	0.537398i
$804$		0			0
$805$	2052.00	−	993.829i	2.54907	−	1.23457i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	1593.00			1.96910			0.984549	−	0.175110i	$-0.0560282\pi$
$809$	1593.00			1.96910			0.984549	+	0.175110i	$0.0560282\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−456.000	−	941.522i	−0.559509	−	1.15524i
$816$		0			0
$817$			248.457i			0.304109i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−1167.00			−1.42144			−0.710719	−	0.703476i	$-0.751630\pi$
$821$	−1167.00			−1.42144			−0.710719	+	0.703476i	$0.751630\pi$
$822$		0			0
$823$			509.991i			0.619673i	0.950790	+	0.309837i	$0.100274\pi$
$823$			509.991i			0.619673i	−0.950790	+	0.309837i	$0.899726\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			3722.50i			4.46879i
$834$		0			0
$835$		0			0
$836$	228.000			0.272727
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	841.000			1.00000
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−760.500	+	368.327i	−0.900000	+	0.435890i
$846$		0			0
$847$			1464.59i			1.72915i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1359.98i			1.59435i	0.603751	+	0.797173i	$0.293672\pi$
$853$			1359.98i			1.59435i	−0.603751	+	0.797173i	$0.706328\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$	−1493.00			−1.73807			−0.869034	−	0.494753i	$-0.835259\pi$
$859$	−1493.00			−1.73807			−0.869034	+	0.494753i	$0.835259\pi$
$860$	114.000	+	235.381i	0.132558	+	0.273698i
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1596.00	−	353.071i	−1.82400	−	0.403510i
$876$		0			0
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$	216.000	−	104.614i	0.245455	−	0.118879i
$881$	−537.000			−0.609535			−0.304767	−	0.952427i	$-0.598579\pi$
$881$	−537.000			−0.609535			−0.304767	+	0.952427i	$0.598579\pi$
$882$		0			0
$883$			1556.13i			1.76232i	0.472820	+	0.881159i	$0.343236\pi$
$883$			1556.13i			1.76232i	−0.472820	+	0.881159i	$0.656764\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	1076.65i		−	1.20565i
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0			−	1.00000i	$-0.5\pi$
$907$		0			0				1.00000i	$0.5\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		0			0
$913$		−	418.454i		−	0.458329i
$914$		0			0
$915$		0			0
$916$	−68.0000			−0.0742358
$917$		−	2785.34i		−	3.03744i
$918$		0			0
$919$	1762.00			1.91730			0.958651	−	0.284585i	$-0.0918559\pi$
$919$	1762.00			1.91730			0.958651	+	0.284585i	$0.0918559\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−642.000			−0.691066			−0.345533	−	0.938407i	$-0.612302\pi$
$929$	−642.000			−0.691066			−0.345533	+	0.938407i	$0.612302\pi$
$930$		0			0
$931$	2318.00			2.48980
$932$			122.049i			0.130954i
$933$		0			0
$934$		0			0
$935$	−199.500	−	411.916i	−0.213369	−	0.440552i
$936$		0			0
$937$		−	1843.81i		−	1.96778i	−0.178762	−	0.983892i	$-0.557209\pi$
$937$		−	1843.81i		−	1.96778i	0.178762	−	0.983892i	$-0.442791\pi$
$938$		0			0
$939$		0			0
$940$	−494.000	−	1019.98i	−0.525532	−	1.08509i
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	488.197i		−	0.515519i	−0.966209	−	0.257760i	$-0.917016\pi$
$947$		−	488.197i		−	0.515519i	0.966209	−	0.257760i	$-0.0829843\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0			−	1.00000i	$-0.5\pi$
$953$		0			0				1.00000i	$0.5\pi$
$954$		0			0
$955$	418.500	−	202.689i	0.438220	−	0.212240i
$956$	−1812.00			−1.89540
$957$		0			0
$958$		0			0
$959$	1311.00			1.36705
$960$		0			0
$961$	961.000			1.00000
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		−	732.295i		−	0.757285i	−0.925543	−	0.378643i	$-0.876391\pi$
$967$		−	732.295i		−	0.757285i	0.925543	−	0.378643i	$-0.123609\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		1.00000			$0$
$971$		0			0		−1.00000			$\pi$
$972$		0			0
$973$		−	2576.11i		−	2.64759i
$974$		0			0
$975$		0			0
$976$	−1648.00			−1.68852
$977$		0			0			−	1.00000i	$-0.5\pi$
$977$		0			0				1.00000i	$0.5\pi$
$978$		0			0
$979$		0			0
$980$	2196.00	−	1063.57i	2.24082	−	1.08528i
$981$		0			0
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$	836.000	+	1726.12i	0.848731	+	1.75241i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	456.000			0.461072
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1021.50	−	494.735i	1.02663	−	0.497221i
$996$		0			0
$997$		−	274.611i		−	0.275437i	−0.990471	−	0.137718i	$-0.956023\pi$
$997$		−	274.611i		−	0.275437i	0.990471	−	0.137718i	$-0.0439769\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	855.3.g.a.379.1		2
3.2	odd	2		95.3.d.a.94.2	yes	2
5.4	even	2	inner	855.3.g.a.379.2		2
15.2	even	4		475.3.c.c.151.2		2
15.8	even	4		475.3.c.c.151.1		2
15.14	odd	2		95.3.d.a.94.1	✓	2
19.18	odd	2	CM	855.3.g.a.379.1		2
57.56	even	2		95.3.d.a.94.2	yes	2
95.94	odd	2	inner	855.3.g.a.379.2		2
285.113	odd	4		475.3.c.c.151.1		2
285.227	odd	4		475.3.c.c.151.2		2
285.284	even	2		95.3.d.a.94.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.3.d.a.94.1	✓	2	15.14	odd	2
95.3.d.a.94.1	✓	2	285.284	even	2
95.3.d.a.94.2	yes	2	3.2	odd	2
95.3.d.a.94.2	yes	2	57.56	even	2
475.3.c.c.151.1		2	15.8	even	4
475.3.c.c.151.1		2	285.113	odd	4
475.3.c.c.151.2		2	15.2	even	4
475.3.c.c.151.2		2	285.227	odd	4
855.3.g.a.379.1		2	1.1	even	1	trivial
855.3.g.a.379.1		2	19.18	odd	2	CM
855.3.g.a.379.2		2	5.4	even	2	inner
855.3.g.a.379.2		2	95.94	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	855.3.g.a.379.1

Level	$855$
Weight	$3$
Character	855.379
Analytic conductor	$23.297$
Analytic rank	$0$
Dimension	$2$
CM discriminant	-19
Inner twists	$4$