LMFDB - Embedded newform 245.3.i.b.129.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [245,3,Mod(19,245)] 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("245.19"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(245, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 245 = 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	245.i (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$6.67576647683$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 35)
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			129.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	245.129
Dual form			245.3.i.b.19.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(4.00000 + 6.92820i) q^{9} +(6.50000 - 11.2583i) q^{11} +(2.00000 + 3.46410i) q^{12} +19.0000 q^{13} -5.00000 q^{15} +(-8.00000 - 13.8564i) q^{16} +(14.5000 - 25.1147i) q^{17} +20.0000 q^{20} +(-12.5000 + 21.6506i) q^{25} +17.0000 q^{27} +23.0000 q^{29} +(-6.50000 - 11.2583i) q^{33} -32.0000 q^{36} +(9.50000 - 16.4545i) q^{39} +(26.0000 + 45.0333i) q^{44} +(20.0000 - 34.6410i) q^{45} +(-15.5000 - 26.8468i) q^{47} -16.0000 q^{48} +(-14.5000 - 25.1147i) q^{51} +(-38.0000 + 65.8179i) q^{52} -65.0000 q^{55} +(10.0000 - 17.3205i) q^{60} +64.0000 q^{64} +(-47.5000 - 82.2724i) q^{65} +(58.0000 + 100.459i) q^{68} +2.00000 q^{71} +(-17.0000 + 29.4449i) q^{73} +(12.5000 + 21.6506i) q^{75} +(78.5000 + 135.966i) q^{79} +(-40.0000 + 69.2820i) q^{80} +(-27.5000 + 47.6314i) q^{81} -86.0000 q^{83} -145.000 q^{85} +(11.5000 - 19.9186i) q^{87} -149.000 q^{97} +104.000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 4 q^{4} - 5 q^{5} + 8 q^{9} + 13 q^{11} + 4 q^{12} + 38 q^{13} - 10 q^{15} - 16 q^{16} + 29 q^{17} + 40 q^{20} - 25 q^{25} + 34 q^{27} + 46 q^{29} - 13 q^{33} - 64 q^{36} + 19 q^{39} + 52 q^{44}+ \cdots + 208 q^{99}+O(q^{100}) $ 2 * q + q^3 - 4 * q^4 - 5 * q^5 + 8 * q^9 + 13 * q^11 + 4 * q^12 + 38 * q^13 - 10 * q^15 - 16 * q^16 + 29 * q^17 + 40 * q^20 - 25 * q^25 + 34 * q^27 + 46 * q^29 - 13 * q^33 - 64 * q^36 + 19 * q^39 + 52 * q^44 + 40 * q^45 - 31 * q^47 - 32 * q^48 - 29 * q^51 - 76 * q^52 - 130 * q^55 + 20 * q^60 + 128 * q^64 - 95 * q^65 + 116 * q^68 + 4 * q^71 - 34 * q^73 + 25 * q^75 + 157 * q^79 - 80 * q^80 - 55 * q^81 - 172 * q^83 - 290 * q^85 + 23 * q^87 - 298 * q^97 + 208 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/245\mathbb{Z}\right)^\times$.

$n$	$101$	$197$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$-1$

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		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	0.500000	−	0.866025i	0.166667	−	0.288675i	−0.770579	−	0.637344i	$-0.780033\pi$
$3$	0.500000	−	0.866025i	0.166667	−	0.288675i	0.937246	+	0.348669i	$0.113366\pi$
$4$	−2.00000	+	3.46410i	−0.500000	+	0.866025i
$5$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$5$	−2.50000	−	4.33013i	−0.500000	−	0.866025i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	4.00000	+	6.92820i	0.444444	+	0.769800i
$10$		0			0
$11$	6.50000	−	11.2583i	0.590909	−	1.02348i	−0.403201	−	0.915111i	$-0.632103\pi$
$11$	6.50000	−	11.2583i	0.590909	−	1.02348i	0.994110	−	0.108373i	$-0.0345641\pi$
$12$	2.00000	+	3.46410i	0.166667	+	0.288675i
$13$	19.0000			1.46154			0.730769	−	0.682625i	$-0.239162\pi$
$13$	19.0000			1.46154			0.730769	+	0.682625i	$0.239162\pi$
$14$		0			0
$15$	−5.00000			−0.333333
$16$	−8.00000	−	13.8564i	−0.500000	−	0.866025i
$17$	14.5000	−	25.1147i	0.852941	−	1.47734i	−0.0256008	−	0.999672i	$-0.508150\pi$
$17$	14.5000	−	25.1147i	0.852941	−	1.47734i	0.878542	−	0.477665i	$-0.158517\pi$
$18$		0			0
$19$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$19$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$20$	20.0000			1.00000
$21$		0			0
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	−12.5000	+	21.6506i	−0.500000	+	0.866025i
$26$		0			0
$27$	17.0000			0.629630
$28$		0			0
$29$	23.0000			0.793103			0.396552	−	0.918012i	$-0.370207\pi$
$29$	23.0000			0.793103			0.396552	+	0.918012i	$0.370207\pi$
$30$		0			0
$31$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$31$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$32$		0			0
$33$	−6.50000	−	11.2583i	−0.196970	−	0.341162i
$34$		0			0
$35$		0			0
$36$	−32.0000			−0.888889
$37$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$37$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$38$		0			0
$39$	9.50000	−	16.4545i	0.243590	−	0.421910i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$		0			0		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$	26.0000	+	45.0333i	0.590909	+	1.02348i
$45$	20.0000	−	34.6410i	0.444444	−	0.769800i
$46$		0			0
$47$	−15.5000	−	26.8468i	−0.329787	−	0.571208i	0.652682	−	0.757632i	$-0.273644\pi$
$47$	−15.5000	−	26.8468i	−0.329787	−	0.571208i	−0.982469	+	0.186424i	$0.940310\pi$
$48$	−16.0000			−0.333333
$49$		0			0
$50$		0			0
$51$	−14.5000	−	25.1147i	−0.284314	−	0.492446i
$52$	−38.0000	+	65.8179i	−0.730769	+	1.26573i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$	−65.0000			−1.18182
$56$		0			0
$57$		0			0
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$	10.0000	−	17.3205i	0.166667	−	0.288675i
$61$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$61$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$62$		0			0
$63$		0			0
$64$	64.0000			1.00000
$65$	−47.5000	−	82.2724i	−0.730769	−	1.26573i
$66$		0			0
$67$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$67$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$68$	58.0000	+	100.459i	0.852941	+	1.47734i
$69$		0			0
$70$		0			0
$71$	2.00000			0.0281690			0.0140845	−	0.999901i	$-0.495517\pi$
$71$	2.00000			0.0281690			0.0140845	+	0.999901i	$0.495517\pi$
$72$		0			0
$73$	−17.0000	+	29.4449i	−0.232877	+	0.403354i	−0.958653	−	0.284576i	$-0.908147\pi$
$73$	−17.0000	+	29.4449i	−0.232877	+	0.403354i	0.725777	+	0.687930i	$0.241481\pi$
$74$		0			0
$75$	12.5000	+	21.6506i	0.166667	+	0.288675i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	78.5000	+	135.966i	0.993671	+	1.72109i	0.594117	+	0.804379i	$0.297502\pi$
$79$	78.5000	+	135.966i	0.993671	+	1.72109i	0.399554	+	0.916710i	$0.369165\pi$
$80$	−40.0000	+	69.2820i	−0.500000	+	0.866025i
$81$	−27.5000	+	47.6314i	−0.339506	+	0.588042i
$82$		0			0
$83$	−86.0000			−1.03614			−0.518072	−	0.855337i	$-0.673350\pi$
$83$	−86.0000			−1.03614			−0.518072	+	0.855337i	$0.673350\pi$
$84$		0			0
$85$	−145.000			−1.70588
$86$		0			0
$87$	11.5000	−	19.9186i	0.132184	−	0.228949i
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−149.000			−1.53608			−0.768041	−	0.640400i	$-0.778768\pi$
$97$	−149.000			−1.53608			−0.768041	+	0.640400i	$0.778768\pi$
$98$		0			0
$99$	104.000			1.05051
$100$	−50.0000	−	86.6025i	−0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−99.5000	−	172.339i	−0.966019	−	1.67319i	−0.706851	−	0.707363i	$-0.749885\pi$
$103$	−99.5000	−	172.339i	−0.966019	−	1.67319i	−0.259169	−	0.965832i	$-0.583449\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−34.0000	+	58.8897i	−0.314815	+	0.545275i
$109$	48.5000	−	84.0045i	0.444954	−	0.770683i	−0.553095	−	0.833118i	$-0.686553\pi$
$109$	48.5000	−	84.0045i	0.444954	−	0.770683i	0.998049	+	0.0624351i	$0.0198866\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$	−46.0000	+	79.6743i	−0.396552	+	0.686848i
$117$	76.0000	+	131.636i	0.649573	+	1.12509i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−24.0000	−	41.5692i	−0.198347	−	0.343547i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	125.000			1.00000
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$	52.0000			0.393939
$133$		0			0
$134$		0			0
$135$	−42.5000	−	73.6122i	−0.314815	−	0.545275i
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$		0			0		1.00000			$0$
$139$		0			0		−1.00000			$\pi$
$140$		0			0
$141$	−31.0000			−0.219858
$142$		0			0
$143$	123.500	−	213.908i	0.863636	−	1.49586i
$144$	64.0000	−	110.851i	0.444444	−	0.769800i
$145$	−57.5000	−	99.5929i	−0.396552	−	0.686848i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	131.000	+	226.899i	0.879195	+	1.52281i	0.852226	+	0.523173i	$0.175252\pi$
$149$	131.000	+	226.899i	0.879195	+	1.52281i	0.0269684	+	0.999636i	$0.491415\pi$
$150$		0			0
$151$	6.50000	−	11.2583i	0.0430464	−	0.0745585i	−0.843699	−	0.536816i	$-0.819627\pi$
$151$	6.50000	−	11.2583i	0.0430464	−	0.0745585i	0.886746	+	0.462257i	$0.152960\pi$
$152$		0			0
$153$	232.000			1.51634
$154$		0			0
$155$		0			0
$156$	38.0000	+	65.8179i	0.243590	+	0.421910i
$157$	67.0000	−	116.047i	0.426752	−	0.739155i	−0.569831	−	0.821762i	$-0.692991\pi$
$157$	67.0000	−	116.047i	0.426752	−	0.739155i	0.996582	+	0.0826067i	$0.0263245\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$163$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$164$		0			0
$165$	−32.5000	+	56.2917i	−0.196970	+	0.341162i
$166$		0			0
$167$	271.000			1.62275			0.811377	−	0.584523i	$-0.198718\pi$
$167$	271.000			1.62275			0.811377	+	0.584523i	$0.198718\pi$
$168$		0			0
$169$	192.000			1.13609
$170$		0			0
$171$		0			0
$172$		0			0
$173$	110.500	+	191.392i	0.638728	+	1.10631i	0.985712	+	0.168439i	$0.0538726\pi$
$173$	110.500	+	191.392i	0.638728	+	1.10631i	−0.346984	+	0.937871i	$0.612794\pi$
$174$		0			0
$175$		0			0
$176$	−208.000			−1.18182
$177$		0			0
$178$		0			0
$179$	−109.000	+	188.794i	−0.608939	+	1.05471i	0.382477	+	0.923965i	$0.375071\pi$
$179$	−109.000	+	188.794i	−0.608939	+	1.05471i	−0.991416	+	0.130748i	$0.958262\pi$
$180$	80.0000	+	138.564i	0.444444	+	0.769800i
$181$		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
$187$	−188.500	−	326.492i	−1.00802	−	1.74594i
$188$	124.000			0.659574
$189$		0			0
$190$		0			0
$191$	−173.500	−	300.511i	−0.908377	−	1.57336i	−0.816319	−	0.577601i	$-0.803989\pi$
$191$	−173.500	−	300.511i	−0.908377	−	1.57336i	−0.0920580	−	0.995754i	$-0.529345\pi$
$192$	32.0000	−	55.4256i	0.166667	−	0.288675i
$193$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$193$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$194$		0			0
$195$	−95.0000			−0.487179
$196$		0			0
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$199$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$	116.000			0.568627
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−152.000	−	263.272i	−0.730769	−	1.26573i
$209$		0			0
$210$		0			0
$211$	107.000			0.507109			0.253555	−	0.967321i	$-0.418400\pi$
$211$	107.000			0.507109			0.253555	+	0.967321i	$0.418400\pi$
$212$		0			0
$213$	1.00000	−	1.73205i	0.00469484	−	0.00813169i
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	17.0000	+	29.4449i	0.0776256	+	0.134451i
$220$	130.000	−	225.167i	0.590909	−	1.02348i
$221$	275.500	−	477.180i	1.24661	−	2.15919i
$222$		0			0
$223$	−401.000			−1.79821			−0.899103	−	0.437737i	$-0.855780\pi$
$223$	−401.000			−1.79821			−0.899103	+	0.437737i	$0.855780\pi$
$224$		0			0
$225$	−200.000			−0.888889
$226$		0			0
$227$	−195.500	+	338.616i	−0.861233	+	1.49170i	0.00950559	+	0.999955i	$0.496974\pi$
$227$	−195.500	+	338.616i	−0.861233	+	1.49170i	−0.870739	+	0.491745i	$0.836359\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$	−77.5000	+	134.234i	−0.329787	+	0.571208i
$236$		0			0
$237$	157.000			0.662447
$238$		0			0
$239$	−397.000			−1.66109			−0.830544	−	0.556953i	$-0.811970\pi$
$239$	−397.000			−1.66109			−0.830544	+	0.556953i	$0.811970\pi$
$240$	40.0000	+	69.2820i	0.166667	+	0.288675i
$241$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$241$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$242$		0			0
$243$	104.000	+	180.133i	0.427984	+	0.741289i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−43.0000	+	74.4782i	−0.172691	+	0.299109i
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	−72.5000	+	125.574i	−0.284314	+	0.492446i
$256$	−128.000	+	221.703i	−0.500000	+	0.866025i
$257$	247.000	+	427.817i	0.961089	+	1.66466i	0.719773	+	0.694209i	$0.244246\pi$
$257$	247.000	+	427.817i	0.961089	+	1.66466i	0.241316	+	0.970447i	$0.422421\pi$
$258$		0			0
$259$		0			0
$260$	380.000			1.46154
$261$	92.0000	+	159.349i	0.352490	+	0.610531i
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$271$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$272$	−464.000			−1.70588
$273$		0			0
$274$		0			0
$275$	162.500	+	281.458i	0.590909	+	1.02348i
$276$		0			0
$277$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$277$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	527.000			1.87544			0.937722	−	0.347385i	$-0.112930\pi$
$281$	527.000			1.87544			0.937722	+	0.347385i	$0.112930\pi$
$282$		0			0
$283$	−279.500	+	484.108i	−0.987633	+	1.71063i	−0.358035	+	0.933708i	$0.616553\pi$
$283$	−279.500	+	484.108i	−0.987633	+	1.71063i	−0.629597	+	0.776922i	$0.716780\pi$
$284$	−4.00000	+	6.92820i	−0.0140845	+	0.0243951i
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−276.000	−	478.046i	−0.955017	−	1.65414i
$290$		0			0
$291$	−74.5000	+	129.038i	−0.256014	+	0.443429i
$292$	−68.0000	−	117.779i	−0.232877	−	0.403354i
$293$	19.0000			0.0648464			0.0324232	−	0.999474i	$-0.489678\pi$
$293$	19.0000			0.0648464			0.0324232	+	0.999474i	$0.489678\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	110.500	−	191.392i	0.372054	−	0.644416i
$298$		0			0
$299$		0			0
$300$	−100.000			−0.333333
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−569.000			−1.85342			−0.926710	−	0.375777i	$-0.877376\pi$
$307$	−569.000			−1.85342			−0.926710	+	0.375777i	$0.877376\pi$
$308$		0			0
$309$	−199.000			−0.644013
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	110.500	+	191.392i	0.353035	+	0.611475i	0.986780	−	0.162068i	$-0.0518163\pi$
$313$	110.500	+	191.392i	0.353035	+	0.611475i	−0.633745	+	0.773542i	$0.718483\pi$
$314$		0			0
$315$		0			0
$316$	−628.000			−1.98734
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$	149.500	−	258.942i	0.468652	−	0.811729i
$320$	−160.000	−	277.128i	−0.500000	−	0.866025i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−110.000	−	190.526i	−0.339506	−	0.588042i
$325$	−237.500	+	411.362i	−0.730769	+	1.26573i
$326$		0			0
$327$	−48.5000	−	84.0045i	−0.148318	−	0.256894i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	299.000	+	517.883i	0.903323	+	1.56460i	0.823152	+	0.567821i	$0.192213\pi$
$331$	299.000	+	517.883i	0.903323	+	1.56460i	0.0801710	+	0.996781i	$0.474453\pi$
$332$	172.000	−	297.913i	0.518072	−	0.897328i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$		0			0
$340$	290.000	−	502.295i	0.852941	−	1.47734i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$	46.0000	+	79.6743i	0.132184	+	0.228949i
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$	323.000			0.920228
$352$		0			0
$353$	−69.5000	+	120.378i	−0.196884	+	0.341013i	−0.947516	−	0.319707i	$-0.896416\pi$
$353$	−69.5000	+	120.378i	−0.196884	+	0.341013i	0.750633	+	0.660720i	$0.229749\pi$
$354$		0			0
$355$	−5.00000	−	8.66025i	−0.0140845	−	0.0243951i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−289.000	−	500.563i	−0.805014	−	1.39433i	−0.916282	−	0.400535i	$-0.868824\pi$
$359$	−289.000	−	500.563i	−0.805014	−	1.39433i	0.111268	−	0.993790i	$-0.464509\pi$
$360$		0			0
$361$	−180.500	+	312.635i	−0.500000	+	0.866025i
$362$		0			0
$363$	−48.0000			−0.132231
$364$		0			0
$365$	170.000			0.465753
$366$		0			0
$367$	−195.500	+	338.616i	−0.532698	+	0.922659i	0.466573	+	0.884482i	$0.345488\pi$
$367$	−195.500	+	338.616i	−0.532698	+	0.922659i	−0.999271	+	0.0381768i	$0.987845\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$373$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$374$		0			0
$375$	62.5000	−	108.253i	0.166667	−	0.288675i
$376$		0			0
$377$	437.000			1.15915
$378$		0			0
$379$	−502.000			−1.32454			−0.662269	−	0.749266i	$-0.730406\pi$
$379$	−502.000			−1.32454			−0.662269	+	0.749266i	$0.730406\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−257.000	−	445.137i	−0.671018	−	1.16224i	−0.977616	−	0.210398i	$-0.932524\pi$
$383$	−257.000	−	445.137i	−0.671018	−	1.16224i	0.306597	−	0.951839i	$-0.400809\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$	298.000	−	516.151i	0.768041	−	1.33029i
$389$	−371.500	+	643.457i	−0.955013	+	1.65413i	−0.220674	+	0.975348i	$0.570826\pi$
$389$	−371.500	+	643.457i	−0.955013	+	1.65413i	−0.734339	+	0.678783i	$0.762508\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$	392.500	−	679.830i	0.993671	−	1.72109i
$396$	−208.000	+	360.267i	−0.525253	+	0.909764i
$397$	194.500	+	336.884i	0.489924	+	0.848574i	0.999933	−	0.0115954i	$-0.00369103\pi$
$397$	194.500	+	336.884i	0.489924	+	0.848574i	−0.510008	+	0.860169i	$0.670358\pi$
$398$		0			0
$399$		0			0
$400$	400.000			1.00000
$401$	36.5000	+	63.2199i	0.0910224	+	0.157655i	0.907942	−	0.419097i	$-0.137653\pi$
$401$	36.5000	+	63.2199i	0.0910224	+	0.157655i	−0.816919	+	0.576752i	$0.804320\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$	275.000			0.679012
$406$		0			0
$407$		0			0
$408$		0			0
$409$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$409$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$410$		0			0
$411$		0			0
$412$	796.000			1.93204
$413$		0			0
$414$		0			0
$415$	215.000	+	372.391i	0.518072	+	0.897328i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	527.000			1.25178			0.625891	−	0.779911i	$-0.284736\pi$
$421$	527.000			1.25178			0.625891	+	0.779911i	$0.284736\pi$
$422$		0			0
$423$	124.000	−	214.774i	0.293144	−	0.507741i
$424$		0			0
$425$	362.500	+	627.868i	0.852941	+	1.47734i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−123.500	−	213.908i	−0.287879	−	0.498621i
$430$		0			0
$431$	426.500	−	738.720i	0.989559	−	1.71397i	0.369962	−	0.929047i	$-0.379371\pi$
$431$	426.500	−	738.720i	0.989559	−	1.71397i	0.619597	−	0.784920i	$-0.287296\pi$
$432$	−136.000	−	235.559i	−0.314815	−	0.545275i
$433$	754.000			1.74134			0.870670	−	0.491868i	$-0.163686\pi$
$433$	754.000			1.74134			0.870670	+	0.491868i	$0.163686\pi$
$434$		0			0
$435$	−115.000			−0.264368
$436$	194.000	+	336.018i	0.444954	+	0.770683i
$437$		0			0
$438$		0			0
$439$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$439$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	262.000			0.586130
$448$		0			0
$449$	−817.000			−1.81960			−0.909800	−	0.415048i	$-0.863765\pi$
$449$	−817.000			−1.81960			−0.909800	+	0.415048i	$0.863765\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−6.50000	−	11.2583i	−0.0143488	−	0.0248528i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$457$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$458$		0			0
$459$	246.500	−	426.951i	0.537037	−	0.930175i
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$	−184.000	−	318.697i	−0.396552	−	0.686848i
$465$		0			0
$466$		0			0
$467$	−435.500	−	754.308i	−0.932548	−	1.61522i	−0.778949	−	0.627088i	$-0.784247\pi$
$467$	−435.500	−	754.308i	−0.932548	−	1.61522i	−0.153599	−	0.988133i	$-0.549087\pi$
$468$	−608.000			−1.29915
$469$		0			0
$470$		0			0
$471$	−67.0000	−	116.047i	−0.142251	−	0.246385i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$	192.000			0.396694
$485$	372.500	+	645.189i	0.768041	+	1.33029i
$486$		0			0
$487$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$487$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	107.000			0.217923			0.108961	−	0.994046i	$-0.465248\pi$
$491$	107.000			0.217923			0.108961	+	0.994046i	$0.465248\pi$
$492$		0			0
$493$	333.500	−	577.639i	0.676471	−	1.17168i
$494$		0			0
$495$	−260.000	−	450.333i	−0.525253	−	0.909764i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−341.500	−	591.495i	−0.684369	−	1.18536i	−0.973635	−	0.228113i	$-0.926745\pi$
$499$	−341.500	−	591.495i	−0.684369	−	1.18536i	0.289266	−	0.957249i	$-0.406589\pi$
$500$	−250.000	+	433.013i	−0.500000	+	0.866025i
$501$	135.500	−	234.693i	0.270459	−	0.468449i
$502$		0			0
$503$	439.000			0.872763			0.436382	−	0.899762i	$-0.356260\pi$
$503$	439.000			0.872763			0.436382	+	0.899762i	$0.356260\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	96.0000	−	166.277i	0.189349	−	0.327962i
$508$		0			0
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−497.500	+	861.695i	−0.966019	+	1.67319i
$516$		0			0
$517$	−403.000			−0.779497
$518$		0			0
$519$	221.000			0.425819
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	163.000	+	282.324i	0.311663	+	0.539817i	0.978723	−	0.205188i	$-0.0657805\pi$
$523$	163.000	+	282.324i	0.311663	+	0.539817i	−0.667059	+	0.745005i	$0.732447\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−104.000	+	180.133i	−0.196970	+	0.341162i
$529$	−264.500	+	458.127i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	109.000	+	188.794i	0.202980	+	0.351571i
$538$		0			0
$539$		0			0
$540$	340.000			0.629630
$541$	−383.500	−	664.241i	−0.708872	−	1.22780i	−0.965276	−	0.261233i	$-0.915871\pi$
$541$	−383.500	−	664.241i	−0.708872	−	1.22780i	0.256403	−	0.966570i	$-0.417462\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−485.000			−0.889908
$546$		0			0
$547$		0			0		1.00000			$0$
$547$		0			0		−1.00000			$\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−377.000			−0.672014
$562$		0			0
$563$	−437.000	+	756.906i	−0.776199	+	1.34442i	0.157919	+	0.987452i	$0.449521\pi$
$563$	−437.000	+	756.906i	−0.776199	+	1.34442i	−0.934118	+	0.356964i	$0.883812\pi$
$564$	62.0000	−	107.387i	0.109929	−	0.190403i
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	551.000	+	954.360i	0.968366	+	1.67726i	0.700287	+	0.713861i	$0.253055\pi$
$569$	551.000	+	954.360i	0.968366	+	1.67726i	0.268078	+	0.963397i	$0.413611\pi$
$570$		0			0
$571$	59.0000	−	102.191i	0.103327	−	0.178968i	−0.809726	−	0.586808i	$-0.800384\pi$
$571$	59.0000	−	102.191i	0.103327	−	0.178968i	0.913054	+	0.407839i	$0.133718\pi$
$572$	494.000	+	855.633i	0.863636	+	1.49586i
$573$	−347.000			−0.605585
$574$		0			0
$575$		0			0
$576$	256.000	+	443.405i	0.444444	+	0.769800i
$577$	14.5000	−	25.1147i	0.0251300	−	0.0435264i	−0.853187	−	0.521605i	$-0.825333\pi$
$577$	14.5000	−	25.1147i	0.0251300	−	0.0435264i	0.878317	+	0.478079i	$0.158667\pi$
$578$		0			0
$579$		0			0
$580$	460.000			0.793103
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$	380.000	−	658.179i	0.649573	−	1.12509i
$586$		0			0
$587$	−1094.00			−1.86371			−0.931857	−	0.362826i	$-0.881812\pi$
$587$	−1094.00			−1.86371			−0.931857	+	0.362826i	$0.881812\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−309.500	−	536.070i	−0.521922	−	0.903996i	−0.999675	−	0.0255016i	$-0.991882\pi$
$593$	−309.500	−	536.070i	−0.521922	−	0.903996i	0.477752	−	0.878495i	$-0.341452\pi$
$594$		0			0
$595$		0			0
$596$	−1048.00			−1.75839
$597$		0			0
$598$		0			0
$599$	−161.500	+	279.726i	−0.269616	+	0.466989i	−0.968763	−	0.247990i	$-0.920230\pi$
$599$	−161.500	+	279.726i	−0.269616	+	0.466989i	0.699147	+	0.714978i	$0.253563\pi$
$600$		0			0
$601$		0			0		1.00000			$0$
$601$		0			0		−1.00000			$\pi$
$602$		0			0
$603$		0			0
$604$	26.0000	+	45.0333i	0.0430464	+	0.0745585i
$605$	−120.000	+	207.846i	−0.198347	+	0.343547i
$606$		0			0
$607$	404.500	+	700.615i	0.666392	+	1.15422i	0.978906	+	0.204312i	$0.0654956\pi$
$607$	404.500	+	700.615i	0.666392	+	1.15422i	−0.312514	+	0.949913i	$0.601171\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−294.500	−	510.089i	−0.481997	−	0.834843i
$612$	−464.000	+	803.672i	−0.758170	+	1.31319i
$613$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$613$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$619$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−304.000			−0.487179
$625$	−312.500	−	541.266i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	268.000	+	464.190i	0.426752	+	0.739155i
$629$		0			0
$630$		0			0
$631$	947.000			1.50079			0.750396	−	0.660988i	$-0.229863\pi$
$631$	947.000			1.50079			0.750396	+	0.660988i	$0.229863\pi$
$632$		0			0
$633$	53.5000	−	92.6647i	0.0845182	−	0.146390i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$	8.00000	+	13.8564i	0.0125196	+	0.0216845i
$640$		0			0
$641$	479.000	−	829.652i	0.747270	−	1.29431i	−0.201857	−	0.979415i	$-0.564698\pi$
$641$	479.000	−	829.652i	0.747270	−	1.29431i	0.949127	−	0.314894i	$-0.101969\pi$
$642$		0			0
$643$	−1241.00			−1.93002			−0.965008	−	0.262221i	$-0.915545\pi$
$643$	−1241.00			−1.93002			−0.965008	+	0.262221i	$0.915545\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	487.000	−	843.509i	0.752705	−	1.30372i	−0.193803	−	0.981041i	$-0.562082\pi$
$647$	487.000	−	843.509i	0.752705	−	1.30372i	0.946507	−	0.322682i	$-0.104585\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−272.000			−0.414003
$658$		0			0
$659$	1283.00			1.94689			0.973445	−	0.228923i	$-0.0735203\pi$
$659$	1283.00			1.94689			0.973445	+	0.228923i	$0.0735203\pi$
$660$	−130.000	−	225.167i	−0.196970	−	0.341162i
$661$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$661$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$662$		0			0
$663$	−275.500	−	477.180i	−0.415535	−	0.719729i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$	−542.000	+	938.772i	−0.811377	+	1.40535i
$669$	−200.500	+	347.276i	−0.299701	+	0.519097i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		0			0		1.00000			$0$
$673$		0			0		−1.00000			$\pi$
$674$		0			0
$675$	−212.500	+	368.061i	−0.314815	+	0.545275i
$676$	−384.000	+	665.108i	−0.568047	+	0.983887i
$677$	−645.500	−	1118.04i	−0.953471	−	1.65146i	−0.737829	−	0.674988i	$-0.764149\pi$
$677$	−645.500	−	1118.04i	−0.953471	−	1.65146i	−0.215643	−	0.976472i	$-0.569185\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	195.500	+	338.616i	0.287078	+	0.497233i
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$691$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$692$	−884.000			−1.27746
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−313.000			−0.446505			−0.223252	−	0.974761i	$-0.571667\pi$
$701$	−313.000			−0.446505			−0.223252	+	0.974761i	$0.571667\pi$
$702$		0			0
$703$		0			0
$704$	416.000	−	720.533i	0.590909	−	1.02348i
$705$	77.5000	+	134.234i	0.109929	+	0.190403i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	708.500	+	1227.16i	0.999295	+	1.73083i	0.532166	+	0.846640i	$0.321378\pi$
$709$	708.500	+	1227.16i	0.999295	+	1.73083i	0.467129	+	0.884189i	$0.345288\pi$
$710$		0			0
$711$	−628.000	+	1087.73i	−0.883263	+	1.52986i
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−1235.00			−1.72727
$716$	−436.000	−	755.174i	−0.608939	−	1.05471i
$717$	−198.500	+	343.812i	−0.276848	+	0.479515i
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$	−640.000			−0.888889
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−287.500	+	497.965i	−0.396552	+	0.686848i
$726$		0			0
$727$	1426.00			1.96149			0.980743	−	0.195304i	$-0.0625693\pi$
$727$	1426.00			1.96149			0.980743	+	0.195304i	$0.0625693\pi$
$728$		0			0
$729$	−287.000			−0.393690
$730$		0			0
$731$		0			0
$732$		0			0
$733$	530.500	+	918.853i	0.723738	+	1.25355i	0.959491	+	0.281738i	$0.0909109\pi$
$733$	530.500	+	918.853i	0.723738	+	1.25355i	−0.235753	+	0.971813i	$0.575756\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	678.500	−	1175.20i	0.918133	−	1.59025i	0.115884	−	0.993263i	$-0.463030\pi$
$739$	678.500	−	1175.20i	0.918133	−	1.59025i	0.802249	−	0.596990i	$-0.203637\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$	655.000	−	1134.49i	0.879195	−	1.52281i
$746$		0			0
$747$	−344.000	−	595.825i	−0.460509	−	0.797624i
$748$	1508.00			2.01604
$749$		0			0
$750$		0			0
$751$	666.500	+	1154.41i	0.887483	+	1.53717i	0.842841	+	0.538163i	$0.180882\pi$
$751$	666.500	+	1154.41i	0.887483	+	1.53717i	0.0446427	+	0.999003i	$0.485785\pi$
$752$	−248.000	+	429.549i	−0.329787	+	0.571208i
$753$		0			0
$754$		0			0
$755$	−65.0000			−0.0860927
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$		0			0
$764$	1388.00			1.81675
$765$	−580.000	−	1004.59i	−0.758170	−	1.31319i
$766$		0			0
$767$		0			0
$768$	128.000	+	221.703i	0.166667	+	0.288675i
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$	494.000			0.640726
$772$		0			0
$773$	770.500	−	1334.55i	0.996766	−	1.72645i	0.428788	−	0.903405i	$-0.358941\pi$
$773$	770.500	−	1334.55i	0.996766	−	1.72645i	0.567977	−	0.823044i	$-0.307726\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$	190.000	−	329.090i	0.243590	−	0.421910i
$781$	13.0000	−	22.5167i	0.0166453	−	0.0288306i
$782$		0			0
$783$	391.000			0.499361
$784$		0			0
$785$	−670.000			−0.853503
$786$		0			0
$787$	224.500	−	388.845i	0.285260	−	0.494086i	−0.687412	−	0.726268i	$-0.741253\pi$
$787$	224.500	−	388.845i	0.285260	−	0.494086i	0.972672	+	0.232182i	$0.0745866\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$	1531.00			1.92095			0.960477	−	0.278360i	$-0.0897909\pi$
$797$	1531.00			1.92095			0.960477	+	0.278360i	$0.0897909\pi$
$798$		0			0
$799$	−899.000			−1.12516
$800$		0			0
$801$		0			0
$802$		0			0
$803$	221.000	+	382.783i	0.275218	+	0.476691i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	48.5000	−	84.0045i	0.0599506	−	0.103837i	−0.834492	−	0.551019i	$-0.814239\pi$
$809$	48.5000	−	84.0045i	0.0599506	−	0.103837i	0.894443	+	0.447182i	$0.147572\pi$
$810$		0			0
$811$		0			0		1.00000			$0$
$811$		0			0		−1.00000			$\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−232.000	+	401.836i	−0.284314	+	0.492446i
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−803.500	−	1391.70i	−0.978685	−	1.69513i	−0.667197	−	0.744881i	$-0.732506\pi$
$821$	−803.500	−	1391.70i	−0.978685	−	1.69513i	−0.311487	−	0.950250i	$-0.600827\pi$
$822$		0			0
$823$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$823$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$824$		0			0
$825$	325.000			0.393939
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$829$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$830$		0			0
$831$		0			0
$832$	1216.00			1.46154
$833$		0			0
$834$		0			0
$835$	−677.500	−	1173.46i	−0.811377	−	1.40535i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−312.000			−0.370987
$842$		0			0
$843$	263.500	−	456.395i	0.312574	−	0.541394i
$844$	−214.000	+	370.659i	−0.253555	+	0.439169i
$845$	−480.000	−	831.384i	−0.568047	−	0.983887i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	279.500	+	484.108i	0.329211	+	0.570210i
$850$		0			0
$851$		0			0
$852$	4.00000	+	6.92820i	0.00469484	+	0.00813169i
$853$	−86.0000			−0.100821			−0.0504103	−	0.998729i	$-0.516053\pi$
$853$	−86.0000			−0.100821			−0.0504103	+	0.998729i	$0.516053\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−353.000	+	611.414i	−0.411902	+	0.713435i	−0.995098	−	0.0988965i	$-0.968469\pi$
$857$	−353.000	+	611.414i	−0.411902	+	0.713435i	0.583196	+	0.812332i	$0.301802\pi$
$858$		0			0
$859$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$859$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$	552.500	−	956.958i	0.638728	−	1.10631i
$866$		0			0
$867$	−552.000			−0.636678
$868$		0			0
$869$	2041.00			2.34868
$870$		0			0
$871$		0			0
$872$		0			0
$873$	−596.000	−	1032.30i	−0.682703	−	1.18248i
$874$		0			0
$875$		0			0
$876$	−136.000			−0.155251
$877$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$877$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$878$		0			0
$879$	9.50000	−	16.4545i	0.0108077	−	0.0187195i
$880$	520.000	+	900.666i	0.590909	+	1.02348i
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$		0			0		1.00000			$0$
$883$		0			0		−1.00000			$\pi$
$884$	1102.00	+	1908.72i	1.24661	+	2.15919i
$885$		0			0
$886$		0			0
$887$	247.000	+	427.817i	0.278467	+	0.482319i	0.971004	−	0.239064i	$-0.0768405\pi$
$887$	247.000	+	427.817i	0.278467	+	0.482319i	−0.692537	+	0.721382i	$0.743507\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	357.500	+	619.208i	0.401235	+	0.694959i
$892$	802.000	−	1389.10i	0.899103	−	1.55729i
$893$		0			0
$894$		0			0
$895$	1090.00			1.21788
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	400.000	−	692.820i	0.444444	−	0.769800i
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$907$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$908$	−782.000	−	1354.46i	−0.861233	−	1.49170i
$909$		0			0
$910$		0			0
$911$	−1678.00			−1.84193			−0.920966	−	0.389643i	$-0.872598\pi$
$911$	−1678.00			−1.84193			−0.920966	+	0.389643i	$0.872598\pi$
$912$		0			0
$913$	−559.000	+	968.216i	−0.612267	+	1.06048i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	498.500	+	863.427i	0.542437	+	0.939529i	0.998763	+	0.0497165i	$0.0158318\pi$
$919$	498.500	+	863.427i	0.542437	+	0.939529i	−0.456326	+	0.889813i	$0.650835\pi$
$920$		0			0
$921$	−284.500	+	492.768i	−0.308903	+	0.535036i
$922$		0			0
$923$	38.0000			0.0411701
$924$		0			0
$925$		0			0
$926$		0			0
$927$	796.000	−	1378.71i	0.858684	−	1.48728i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−942.500	+	1632.46i	−1.00802	+	1.74594i
$936$		0			0
$937$	−1829.00			−1.95197			−0.975987	−	0.217828i	$-0.930103\pi$
$937$	−1829.00			−1.95197			−0.975987	+	0.217828i	$0.930103\pi$
$938$		0			0
$939$	221.000			0.235357
$940$	−310.000	−	536.936i	−0.329787	−	0.571208i
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	−314.000	+	543.864i	−0.331224	+	0.573696i
$949$	−323.000	+	559.452i	−0.340358	+	0.589518i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$	−867.500	+	1502.55i	−0.908377	+	1.57336i
$956$	794.000	−	1375.25i	0.830544	−	1.43854i
$957$	−149.500	−	258.942i	−0.156217	−	0.270576i
$958$		0			0
$959$		0			0
$960$	−320.000			−0.333333
$961$	−480.500	−	832.250i	−0.500000	−	0.866025i
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−832.000			−0.855967
$973$		0			0
$974$		0			0
$975$	237.500	+	411.362i	0.243590	+	0.421910i
$976$		0			0
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	776.000			0.791030
$982$		0			0
$983$	560.500	−	970.814i	0.570193	−	0.987604i	−0.426352	−	0.904557i	$-0.640202\pi$
$983$	560.500	−	970.814i	0.570193	−	0.987604i	0.996546	−	0.0830466i	$-0.0264650\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$	−361.000	+	625.270i	−0.364279	+	0.630949i	−0.988660	−	0.150171i	$-0.952018\pi$
$991$	−361.000	+	625.270i	−0.364279	+	0.630949i	0.624382	+	0.781119i	$0.285351\pi$
$992$		0			0
$993$	598.000			0.602216
$994$		0			0
$995$		0			0
$996$	−172.000	−	297.913i	−0.172691	−	0.299109i
$997$	−825.500	+	1429.81i	−0.827984	+	1.43411i	0.0716333	+	0.997431i	$0.477179\pi$
$997$	−825.500	+	1429.81i	−0.827984	+	1.43411i	−0.899617	+	0.436679i	$0.856154\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	245.3.i.b.129.1		2
5.4	even	2		245.3.i.a.129.1		2
7.2	even	3	inner	245.3.i.b.19.1		2
7.3	odd	6		35.3.c.b.34.1	yes	1
7.4	even	3		35.3.c.a.34.1	✓	1
7.5	odd	6		245.3.i.a.19.1		2
7.6	odd	2		245.3.i.a.129.1		2
21.11	odd	6		315.3.e.a.244.1		1
21.17	even	6		315.3.e.b.244.1		1
28.3	even	6		560.3.p.a.209.1		1
28.11	odd	6		560.3.p.b.209.1		1
35.3	even	12		175.3.d.e.76.2		2
35.4	even	6		35.3.c.b.34.1	yes	1
35.9	even	6		245.3.i.a.19.1		2
35.17	even	12		175.3.d.e.76.1		2
35.18	odd	12		175.3.d.e.76.1		2
35.19	odd	6	inner	245.3.i.b.19.1		2
35.24	odd	6		35.3.c.a.34.1	✓	1
35.32	odd	12		175.3.d.e.76.2		2
35.34	odd	2	CM	245.3.i.b.129.1		2
105.59	even	6		315.3.e.a.244.1		1
105.74	odd	6		315.3.e.b.244.1		1
140.39	odd	6		560.3.p.a.209.1		1
140.59	even	6		560.3.p.b.209.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
35.3.c.a.34.1	✓	1	7.4	even	3
35.3.c.a.34.1	✓	1	35.24	odd	6
35.3.c.b.34.1	yes	1	7.3	odd	6
35.3.c.b.34.1	yes	1	35.4	even	6
175.3.d.e.76.1		2	35.17	even	12
175.3.d.e.76.1		2	35.18	odd	12
175.3.d.e.76.2		2	35.3	even	12
175.3.d.e.76.2		2	35.32	odd	12
245.3.i.a.19.1		2	7.5	odd	6
245.3.i.a.19.1		2	35.9	even	6
245.3.i.a.129.1		2	5.4	even	2
245.3.i.a.129.1		2	7.6	odd	2
245.3.i.b.19.1		2	7.2	even	3	inner
245.3.i.b.19.1		2	35.19	odd	6	inner
245.3.i.b.129.1		2	1.1	even	1	trivial
245.3.i.b.129.1		2	35.34	odd	2	CM
315.3.e.a.244.1		1	21.11	odd	6
315.3.e.a.244.1		1	105.59	even	6
315.3.e.b.244.1		1	21.17	even	6
315.3.e.b.244.1		1	105.74	odd	6
560.3.p.a.209.1		1	28.3	even	6
560.3.p.a.209.1		1	140.39	odd	6
560.3.p.b.209.1		1	28.11	odd	6
560.3.p.b.209.1		1	140.59	even	6

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

Level:	\( N \)	\(=\)	\( 245 = 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	245.i (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 - 0.866025i) q^{3} +(-2.00000 + 3.46410i) q^{4} +(-2.50000 - 4.33013i) q^{5} +(4.00000 + 6.92820i) q^{9} +(6.50000 - 11.2583i) q^{11} +(2.00000 + 3.46410i) q^{12} +19.0000 q^{13} -5.00000 q^{15} +(-8.00000 - 13.8564i) q^{16} +(14.5000 - 25.1147i) q^{17} +20.0000 q^{20} +(-12.5000 + 21.6506i) q^{25} +17.0000 q^{27} +23.0000 q^{29} +(-6.50000 - 11.2583i) q^{33} -32.0000 q^{36} +(9.50000 - 16.4545i) q^{39} +(26.0000 + 45.0333i) q^{44} +(20.0000 - 34.6410i) q^{45} +(-15.5000 - 26.8468i) q^{47} -16.0000 q^{48} +(-14.5000 - 25.1147i) q^{51} +(-38.0000 + 65.8179i) q^{52} -65.0000 q^{55} +(10.0000 - 17.3205i) q^{60} +64.0000 q^{64} +(-47.5000 - 82.2724i) q^{65} +(58.0000 + 100.459i) q^{68} +2.00000 q^{71} +(-17.0000 + 29.4449i) q^{73} +(12.5000 + 21.6506i) q^{75} +(78.5000 + 135.966i) q^{79} +(-40.0000 + 69.2820i) q^{80} +(-27.5000 + 47.6314i) q^{81} -86.0000 q^{83} -145.000 q^{85} +(11.5000 - 19.9186i) q^{87} -149.000 q^{97} +104.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 4 q^{4} - 5 q^{5} + 8 q^{9} + 13 q^{11} + 4 q^{12} + 38 q^{13} - 10 q^{15} - 16 q^{16} + 29 q^{17} + 40 q^{20} - 25 q^{25} + 34 q^{27} + 46 q^{29} - 13 q^{33} - 64 q^{36} + 19 q^{39} + 52 q^{44}+ \cdots + 208 q^{99}+O(q^{100}) \) 2 * q + q^3 - 4 * q^4 - 5 * q^5 + 8 * q^9 + 13 * q^11 + 4 * q^12 + 38 * q^13 - 10 * q^15 - 16 * q^16 + 29 * q^17 + 40 * q^20 - 25 * q^25 + 34 * q^27 + 46 * q^29 - 13 * q^33 - 64 * q^36 + 19 * q^39 + 52 * q^44 + 40 * q^45 - 31 * q^47 - 32 * q^48 - 29 * q^51 - 76 * q^52 - 130 * q^55 + 20 * q^60 + 128 * q^64 - 95 * q^65 + 116 * q^68 + 4 * q^71 - 34 * q^73 + 25 * q^75 + 157 * q^79 - 80 * q^80 - 55 * q^81 - 172 * q^83 - 290 * q^85 + 23 * q^87 - 298 * q^97 + 208 * q^99

Introduction

L-functions

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