LMFDB - Embedded newform 1176.3.d.f.785.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1176,3,Mod(785,1176)] 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("1176.785"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1176, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1176 = 2^{3} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1176.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,8,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$32.0436790888$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{14} - 4 x^{13} - 33 x^{12} + 220 x^{11} + 516 x^{10} - 2136 x^{9} - 3744 x^{8} + \cdots + 43046721 $ x^16 - 4x^14 - 4x^13 - 33x^12 + 220x^11 + 516x^10 - 2136x^9 - 3744x^8 - 19224x^7 + 41796x^6 + 160380x^5 - 216513x^4 - 236196x^3 - 2125764*x^2 + 43046721
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{14}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			785.2
Root			$2.97573 - 0.380818i$ of defining polynomial
Character	$\chi$	$=$	1176.785
Dual form			1176.3.d.f.785.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.97573 + 0.380818i) q^{3} -9.40596i q^{5} +(8.70995 - 2.26643i) q^{9} +9.96283i q^{11} -15.4775 q^{13} +(3.58196 + 27.9896i) q^{15} -14.4450i q^{17} -14.5554 q^{19} -14.2215i q^{23} -63.4720 q^{25} +(-25.0554 + 10.0612i) q^{27} -14.0780i q^{29} +23.9271 q^{31} +(-3.79403 - 29.6467i) q^{33} +4.41570 q^{37} +(46.0568 - 5.89410i) q^{39} -17.7797i q^{41} -20.0097 q^{43} +(-21.3179 - 81.9254i) q^{45} +49.5674i q^{47} +(5.50090 + 42.9843i) q^{51} +95.9683i q^{53} +93.7100 q^{55} +(43.3128 - 5.54294i) q^{57} -46.5367i q^{59} -33.5425 q^{61} +145.580i q^{65} +58.9220 q^{67} +(5.41580 + 42.3193i) q^{69} -23.5956i q^{71} +54.7924 q^{73} +(188.876 - 24.1713i) q^{75} -87.4365 q^{79} +(70.7266 - 39.4809i) q^{81} +141.820i q^{83} -135.869 q^{85} +(5.36116 + 41.8924i) q^{87} -107.353i q^{89} +(-71.2006 + 9.11188i) q^{93} +136.907i q^{95} +22.3461 q^{97} +(22.5800 + 86.7758i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 8 q^{9} - 4 q^{13} + 8 q^{15} - 16 q^{19} - 76 q^{25} - 12 q^{27} + 36 q^{31} - 32 q^{33} - 28 q^{37} - 4 q^{39} - 120 q^{43} - 28 q^{45} - 36 q^{51} - 28 q^{55} - 88 q^{57} + 152 q^{61} + 88 q^{67}+ \cdots + 368 q^{99}+O(q^{100}) $ 16 * q + 8 * q^9 - 4 * q^13 + 8 * q^15 - 16 * q^19 - 76 * q^25 - 12 * q^27 + 36 * q^31 - 32 * q^33 - 28 * q^37 - 4 * q^39 - 120 * q^43 - 28 * q^45 - 36 * q^51 - 28 * q^55 - 88 * q^57 + 152 * q^61 + 88 * q^67 - 112 * q^69 + 156 * q^73 + 312 * q^75 - 84 * q^79 + 164 * q^81 + 152 * q^85 + 176 * q^87 - 8 * q^93 - 20 * q^97 + 368 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1176\mathbb{Z}\right)^\times$.

$n$	$295$	$589$	$785$	$1081$
$\chi(n)$	$1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−2.97573	+	0.380818i	−0.991910	+	0.126939i
$3$	−2.97573	+	0.380818i	−0.991910	+	0.126939i
$4$		0			0
$5$		−	9.40596i		−	1.88119i	−0.339529	−	0.940596i	$-0.610268\pi$
$5$		−	9.40596i		−	1.88119i	0.339529	−	0.940596i	$-0.389732\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	8.70995	−	2.26643i	0.967773	−	0.251825i
$10$		0			0
$11$			9.96283i			0.905712i	0.891584	+	0.452856i	$0.149595\pi$
$11$			9.96283i			0.905712i	−0.891584	+	0.452856i	$0.850405\pi$
$12$		0			0
$13$	−15.4775			−1.19057			−0.595287	−	0.803513i	$-0.702962\pi$
$13$	−15.4775			−1.19057			−0.595287	+	0.803513i	$0.702962\pi$
$14$		0			0
$15$	3.58196	+	27.9896i	0.238797	+	1.86597i
$16$		0			0
$17$		−	14.4450i		−	0.849704i	−0.905263	−	0.424852i	$-0.860326\pi$
$17$		−	14.4450i		−	0.849704i	0.905263	−	0.424852i	$-0.139674\pi$
$18$		0			0
$19$	−14.5554			−0.766071			−0.383036	−	0.923734i	$-0.625121\pi$
$19$	−14.5554			−0.766071			−0.383036	+	0.923734i	$0.625121\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	14.2215i		−	0.618326i	−0.951009	−	0.309163i	$-0.899951\pi$
$23$		−	14.2215i		−	0.618326i	0.951009	−	0.309163i	$-0.100049\pi$
$24$		0			0
$25$	−63.4720			−2.53888
$26$		0			0
$27$	−25.0554	+	10.0612i	−0.927977	+	0.372636i
$28$		0			0
$29$		−	14.0780i		−	0.485448i	−0.970095	−	0.242724i	$-0.921959\pi$
$29$		−	14.0780i		−	0.485448i	0.970095	−	0.242724i	$-0.0780410\pi$
$30$		0			0
$31$	23.9271			0.771842			0.385921	−	0.922532i	$-0.373884\pi$
$31$	23.9271			0.771842			0.385921	+	0.922532i	$0.373884\pi$
$32$		0			0
$33$	−3.79403	−	29.6467i	−0.114971	−	0.898385i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.41570			0.119343			0.0596716	−	0.998218i	$-0.480995\pi$
$37$	4.41570			0.119343			0.0596716	+	0.998218i	$0.480995\pi$
$38$		0			0
$39$	46.0568	−	5.89410i	1.18094	−	0.151131i
$40$		0			0
$41$		−	17.7797i		−	0.433652i	−0.976210	−	0.216826i	$-0.930430\pi$
$41$		−	17.7797i		−	0.433652i	0.976210	−	0.216826i	$-0.0695704\pi$
$42$		0			0
$43$	−20.0097			−0.465342			−0.232671	−	0.972556i	$-0.574746\pi$
$43$	−20.0097			−0.465342			−0.232671	+	0.972556i	$0.574746\pi$
$44$		0			0
$45$	−21.3179	−	81.9254i	−0.473731	−	1.82057i
$46$		0			0
$47$			49.5674i			1.05462i	0.849671	+	0.527312i	$0.176800\pi$
$47$			49.5674i			1.05462i	−0.849671	+	0.527312i	$0.823200\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	5.50090	+	42.9843i	0.107861	+	0.842830i
$52$		0			0
$53$			95.9683i			1.81072i	0.424643	+	0.905361i	$0.360400\pi$
$53$			95.9683i			1.81072i	−0.424643	+	0.905361i	$0.639600\pi$
$54$		0			0
$55$	93.7100			1.70382
$56$		0			0
$57$	43.3128	−	5.54294i	0.759874	−	0.0972446i
$58$		0			0
$59$		−	46.5367i		−	0.788758i	−0.918948	−	0.394379i	$-0.870960\pi$
$59$		−	46.5367i		−	0.788758i	0.918948	−	0.394379i	$-0.129040\pi$
$60$		0			0
$61$	−33.5425			−0.549877			−0.274939	−	0.961462i	$-0.588658\pi$
$61$	−33.5425			−0.549877			−0.274939	+	0.961462i	$0.588658\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			145.580i			2.23970i
$66$		0			0
$67$	58.9220			0.879433			0.439717	−	0.898137i	$-0.355079\pi$
$67$	58.9220			0.879433			0.439717	+	0.898137i	$0.355079\pi$
$68$		0			0
$69$	5.41580	+	42.3193i	0.0784899	+	0.613324i
$70$		0			0
$71$		−	23.5956i		−	0.332333i	−0.986098	−	0.166166i	$-0.946861\pi$
$71$		−	23.5956i		−	0.332333i	0.986098	−	0.166166i	$-0.0531389\pi$
$72$		0			0
$73$	54.7924			0.750581			0.375291	−	0.926907i	$-0.377543\pi$
$73$	54.7924			0.750581			0.375291	+	0.926907i	$0.377543\pi$
$74$		0			0
$75$	188.876	−	24.1713i	2.51834	−	0.322284i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−87.4365			−1.10679			−0.553396	−	0.832918i	$-0.686668\pi$
$79$	−87.4365			−1.10679			−0.553396	+	0.832918i	$0.686668\pi$
$80$		0			0
$81$	70.7266	−	39.4809i	0.873168	−	0.487419i
$82$		0			0
$83$			141.820i			1.70867i	0.519724	+	0.854334i	$0.326035\pi$
$83$			141.820i			1.70867i	−0.519724	+	0.854334i	$0.673965\pi$
$84$		0			0
$85$	−135.869			−1.59845
$86$		0			0
$87$	5.36116	+	41.8924i	0.0616225	+	0.481521i
$88$		0			0
$89$		−	107.353i		−	1.20621i	−0.797660	−	0.603107i	$-0.793929\pi$
$89$		−	107.353i		−	1.20621i	0.797660	−	0.603107i	$-0.206071\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−71.2006	+	9.11188i	−0.765598	+	0.0979772i
$94$		0			0
$95$			136.907i			1.44113i
$96$		0			0
$97$	22.3461			0.230372			0.115186	−	0.993344i	$-0.463254\pi$
$97$	22.3461			0.230372			0.115186	+	0.993344i	$0.463254\pi$
$98$		0			0
$99$	22.5800	+	86.7758i	0.228081	+	0.876524i
$100$		0			0
$101$			131.714i			1.30410i	0.758178	+	0.652048i	$0.226090\pi$
$101$			131.714i			1.30410i	−0.758178	+	0.652048i	$0.773910\pi$
$102$		0			0
$103$	11.9272			0.115798			0.0578988	−	0.998322i	$-0.481560\pi$
$103$	11.9272			0.115798			0.0578988	+	0.998322i	$0.481560\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			87.6005i			0.818696i	0.912378	+	0.409348i	$0.134244\pi$
$107$			87.6005i			0.818696i	−0.912378	+	0.409348i	$0.865756\pi$
$108$		0			0
$109$	−144.971			−1.33001			−0.665004	−	0.746840i	$-0.731570\pi$
$109$	−144.971			−1.33001			−0.665004	+	0.746840i	$0.731570\pi$
$110$		0			0
$111$	−13.1399	+	1.68158i	−0.118378	+	0.0151494i
$112$		0			0
$113$			156.993i			1.38932i	0.719339	+	0.694659i	$0.244445\pi$
$113$			156.993i			1.38932i	−0.719339	+	0.694659i	$0.755555\pi$
$114$		0			0
$115$	−133.767			−1.16319
$116$		0			0
$117$	−134.808	+	35.0785i	−1.15221	+	0.299817i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	21.7420			0.179686
$122$		0			0
$123$	6.77084	+	52.9077i	0.0550475	+	0.430144i
$124$		0			0
$125$			361.866i			2.89493i
$126$		0			0
$127$	115.410			0.908740			0.454370	−	0.890813i	$-0.349864\pi$
$127$	115.410			0.908740			0.454370	+	0.890813i	$0.349864\pi$
$128$		0			0
$129$	59.5435	−	7.62005i	0.461577	−	0.0590702i
$130$		0			0
$131$		−	66.8133i		−	0.510025i	−0.966938	−	0.255013i	$-0.917920\pi$
$131$		−	66.8133i		−	0.510025i	0.966938	−	0.255013i	$-0.0820796\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	94.6350	+	235.670i	0.701000	+	1.74570i
$136$		0			0
$137$		−	65.1596i		−	0.475617i	−0.971312	−	0.237809i	$-0.923571\pi$
$137$		−	65.1596i		−	0.475617i	0.971312	−	0.237809i	$-0.0764291\pi$
$138$		0			0
$139$	−220.413			−1.58571			−0.792853	−	0.609413i	$-0.791405\pi$
$139$	−220.413			−1.58571			−0.792853	+	0.609413i	$0.791405\pi$
$140$		0			0
$141$	−18.8762	−	147.499i	−0.133873	−	1.04609i
$142$		0			0
$143$		−	154.199i		−	1.07832i
$144$		0			0
$145$	−132.417			−0.913221
$146$		0			0
$147$		0			0
$148$		0			0
$149$			5.25956i			0.0352991i	0.999844	+	0.0176495i	$0.00561832\pi$
$149$			5.25956i			0.0352991i	−0.999844	+	0.0176495i	$0.994382\pi$
$150$		0			0
$151$	74.0446			0.490362			0.245181	−	0.969477i	$-0.421153\pi$
$151$	74.0446			0.490362			0.245181	+	0.969477i	$0.421153\pi$
$152$		0			0
$153$	−32.7384	−	125.815i	−0.213977	−	0.822320i
$154$		0			0
$155$		−	225.057i		−	1.45198i
$156$		0			0
$157$	46.6823			0.297340			0.148670	−	0.988887i	$-0.452501\pi$
$157$	46.6823			0.297340			0.148670	+	0.988887i	$0.452501\pi$
$158$		0			0
$159$	−36.5465	−	285.576i	−0.229852	−	1.79607i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	110.564			0.678306			0.339153	−	0.940731i	$-0.389860\pi$
$163$	110.564			0.678306			0.339153	+	0.940731i	$0.389860\pi$
$164$		0			0
$165$	−278.856	+	35.6865i	−1.69003	+	0.216282i
$166$		0			0
$167$			264.745i			1.58530i	0.609679	+	0.792649i	$0.291298\pi$
$167$			264.745i			1.58530i	−0.609679	+	0.792649i	$0.708702\pi$
$168$		0			0
$169$	70.5521			0.417468
$170$		0			0
$171$	−126.776	+	32.9886i	−0.741383	+	0.192916i
$172$		0			0
$173$			321.756i			1.85986i	0.367731	+	0.929932i	$0.380135\pi$
$173$			321.756i			1.85986i	−0.367731	+	0.929932i	$0.619865\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	17.7220	+	138.481i	0.100125	+	0.782378i
$178$		0			0
$179$			43.2370i			0.241547i	0.992680	+	0.120774i	$0.0385375\pi$
$179$			43.2370i			0.241547i	−0.992680	+	0.120774i	$0.961462\pi$
$180$		0			0
$181$	−142.957			−0.789819			−0.394909	−	0.918720i	$-0.629224\pi$
$181$	−142.957			−0.789819			−0.394909	+	0.918720i	$0.629224\pi$
$182$		0			0
$183$	99.8135	−	12.7736i	0.545429	−	0.0698011i
$184$		0			0
$185$		−	41.5339i		−	0.224507i
$186$		0			0
$187$	143.913			0.769587
$188$		0			0
$189$		0			0
$190$		0			0
$191$			54.2280i			0.283916i	0.989873	+	0.141958i	$0.0453398\pi$
$191$			54.2280i			0.283916i	−0.989873	+	0.141958i	$0.954660\pi$
$192$		0			0
$193$	−57.3720			−0.297264			−0.148632	−	0.988893i	$-0.547487\pi$
$193$	−57.3720			−0.297264			−0.148632	+	0.988893i	$0.547487\pi$
$194$		0			0
$195$	−55.4397	−	433.208i	−0.284306	−	2.22158i
$196$		0			0
$197$			114.074i			0.579058i	0.957169	+	0.289529i	$0.0934987\pi$
$197$			114.074i			0.579058i	−0.957169	+	0.289529i	$0.906501\pi$
$198$		0			0
$199$	95.2651			0.478719			0.239360	−	0.970931i	$-0.423063\pi$
$199$	95.2651			0.478719			0.239360	+	0.970931i	$0.423063\pi$
$200$		0			0
$201$	−175.336	+	22.4386i	−0.872319	+	0.111635i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−167.235			−0.815782
$206$		0			0
$207$	−32.2320	−	123.869i	−0.155710	−	0.598399i
$208$		0			0
$209$		−	145.013i		−	0.693840i
$210$		0			0
$211$	198.164			0.939167			0.469584	−	0.882888i	$-0.344404\pi$
$211$	198.164			0.939167			0.469584	+	0.882888i	$0.344404\pi$
$212$		0			0
$213$	8.98565	+	70.2143i	0.0421861	+	0.329644i
$214$		0			0
$215$			188.210i			0.875396i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−163.048	+	20.8660i	−0.744509	+	0.0952783i
$220$		0			0
$221$			223.571i			1.01164i
$222$		0			0
$223$	−128.499			−0.576228			−0.288114	−	0.957596i	$-0.593028\pi$
$223$	−128.499			−0.576228			−0.288114	+	0.957596i	$0.593028\pi$
$224$		0			0
$225$	−552.838	+	143.855i	−2.45706	+	0.639353i
$226$		0			0
$227$		−	98.5416i		−	0.434104i	−0.976160	−	0.217052i	$-0.930356\pi$
$227$		−	98.5416i		−	0.434104i	0.976160	−	0.217052i	$-0.0696441\pi$
$228$		0			0
$229$	364.468			1.59156			0.795782	−	0.605583i	$-0.207060\pi$
$229$	364.468			1.59156			0.795782	+	0.605583i	$0.207060\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	156.235i		−	0.670536i	−0.942123	−	0.335268i	$-0.891173\pi$
$233$		−	156.235i		−	0.670536i	0.942123	−	0.335268i	$-0.108827\pi$
$234$		0			0
$235$	466.228			1.98395
$236$		0			0
$237$	260.188	−	33.2974i	1.09784	−	0.140495i
$238$		0			0
$239$			68.6050i			0.287050i	0.989647	+	0.143525i	$0.0458438\pi$
$239$			68.6050i			0.287050i	−0.989647	+	0.143525i	$0.954156\pi$
$240$		0			0
$241$	−193.013			−0.800882			−0.400441	−	0.916322i	$-0.631143\pi$
$241$	−193.013			−0.800882			−0.400441	+	0.916322i	$0.631143\pi$
$242$		0			0
$243$	−195.428	+	144.419i	−0.804232	+	0.594315i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	225.280			0.912065
$248$		0			0
$249$	−54.0075	−	422.017i	−0.216897	−	1.69485i
$250$		0			0
$251$			64.5979i			0.257362i	0.991686	+	0.128681i	$0.0410743\pi$
$251$			64.5979i			0.257362i	−0.991686	+	0.128681i	$0.958926\pi$
$252$		0			0
$253$	141.686			0.560025
$254$		0			0
$255$	404.309	−	51.7413i	1.58552	−	0.202907i
$256$		0			0
$257$		−	309.065i		−	1.20259i	−0.799028	−	0.601293i	$-0.794652\pi$
$257$		−	309.065i		−	1.20259i	0.799028	−	0.601293i	$-0.205348\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−31.9067	−	122.619i	−0.122248	−	0.469804i
$262$		0			0
$263$			159.621i			0.606922i	0.952844	+	0.303461i	$0.0981423\pi$
$263$			159.621i			0.606922i	−0.952844	+	0.303461i	$0.901858\pi$
$264$		0			0
$265$	902.673			3.40631
$266$		0			0
$267$	40.8820	+	319.454i	0.153116	+	1.19646i
$268$		0			0
$269$		−	64.5418i		−	0.239933i	−0.992778	−	0.119966i	$-0.961721\pi$
$269$		−	64.5418i		−	0.239933i	0.992778	−	0.119966i	$-0.0382786\pi$
$270$		0			0
$271$	178.244			0.657728			0.328864	−	0.944377i	$-0.393334\pi$
$271$	178.244			0.657728			0.328864	+	0.944377i	$0.393334\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	632.361i		−	2.29949i
$276$		0			0
$277$	168.288			0.607536			0.303768	−	0.952746i	$-0.401755\pi$
$277$	168.288			0.607536			0.303768	+	0.952746i	$0.401755\pi$
$278$		0			0
$279$	208.404	−	54.2290i	0.746968	−	0.194369i
$280$		0			0
$281$		−	384.258i		−	1.36747i	−0.729732	−	0.683733i	$-0.760355\pi$
$281$		−	384.258i		−	1.36747i	0.729732	−	0.683733i	$-0.239645\pi$
$282$		0			0
$283$	−315.530			−1.11495			−0.557474	−	0.830194i	$-0.688229\pi$
$283$	−315.530			−1.11495			−0.557474	+	0.830194i	$0.688229\pi$
$284$		0			0
$285$	−52.1367	−	407.398i	−0.182936	−	1.42947i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	80.3431			0.278004
$290$		0			0
$291$	−66.4960	+	8.50981i	−0.228509	+	0.0292433i
$292$		0			0
$293$			31.0572i			0.105997i	0.998595	+	0.0529986i	$0.0168779\pi$
$293$			31.0572i			0.105997i	−0.998595	+	0.0529986i	$0.983122\pi$
$294$		0			0
$295$	−437.722			−1.48381
$296$		0			0
$297$	−100.238	−	249.623i	−0.337501	−	0.840480i
$298$		0			0
$299$			220.113i			0.736163i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−50.1589	−	391.944i	−0.165541	−	1.29355i
$304$		0			0
$305$			315.499i			1.03442i
$306$		0			0
$307$	−305.651			−0.995606			−0.497803	−	0.867290i	$-0.665860\pi$
$307$	−305.651			−0.995606			−0.497803	+	0.867290i	$0.665860\pi$
$308$		0			0
$309$	−35.4920	+	4.54208i	−0.114861	+	0.0146993i
$310$		0			0
$311$			548.441i			1.76348i	0.471739	+	0.881738i	$0.343626\pi$
$311$			548.441i			1.76348i	−0.471739	+	0.881738i	$0.656374\pi$
$312$		0			0
$313$	38.6850			0.123594			0.0617972	−	0.998089i	$-0.480317\pi$
$313$	38.6850			0.123594			0.0617972	+	0.998089i	$0.480317\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	508.322i		−	1.60354i	−0.597633	−	0.801770i	$-0.703892\pi$
$317$		−	508.322i		−	1.60354i	0.597633	−	0.801770i	$-0.296108\pi$
$318$		0			0
$319$	140.257			0.439676
$320$		0			0
$321$	−33.3598	−	260.675i	−0.103925	−	0.812073i
$322$		0			0
$323$			210.251i			0.650933i
$324$		0			0
$325$	982.386			3.02273
$326$		0			0
$327$	431.394	−	55.2075i	1.31925	−	0.168830i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−187.307			−0.565882			−0.282941	−	0.959137i	$-0.591310\pi$
$331$	−187.307			−0.565882			−0.282941	+	0.959137i	$0.591310\pi$
$332$		0			0
$333$	38.4605	−	10.0079i	0.115497	−	0.0300536i
$334$		0			0
$335$		−	554.218i		−	1.65438i
$336$		0			0
$337$	61.2773			0.181832			0.0909158	−	0.995859i	$-0.471021\pi$
$337$	61.2773			0.181832			0.0909158	+	0.995859i	$0.471021\pi$
$338$		0			0
$339$	−59.7858	−	467.169i	−0.176359	−	1.37808i
$340$		0			0
$341$			238.382i			0.699067i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	398.054	−	50.9408i	1.15378	−	0.147654i
$346$		0			0
$347$		−	277.637i		−	0.800106i	−0.916492	−	0.400053i	$-0.868992\pi$
$347$		−	277.637i		−	0.800106i	0.916492	−	0.400053i	$-0.131008\pi$
$348$		0			0
$349$	−297.676			−0.852939			−0.426470	−	0.904502i	$-0.640243\pi$
$349$	−297.676			−0.852939			−0.426470	+	0.904502i	$0.640243\pi$
$350$		0			0
$351$	387.794	−	155.722i	1.10483	−	0.443651i
$352$		0			0
$353$			61.9169i			0.175402i	0.996147	+	0.0877010i	$0.0279520\pi$
$353$			61.9169i			0.175402i	−0.996147	+	0.0877010i	$0.972048\pi$
$354$		0			0
$355$	−221.939			−0.625182
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	134.449i		−	0.374509i	−0.982311	−	0.187254i	$-0.940041\pi$
$359$		−	134.449i		−	0.374509i	0.982311	−	0.187254i	$-0.0599588\pi$
$360$		0			0
$361$	−149.142			−0.413135
$362$		0			0
$363$	−64.6982	+	8.27973i	−0.178232	+	0.0228092i
$364$		0			0
$365$		−	515.375i		−	1.41199i
$366$		0			0
$367$	−634.900			−1.72997			−0.864987	−	0.501795i	$-0.832673\pi$
$367$	−634.900			−1.72997			−0.864987	+	0.501795i	$0.832673\pi$
$368$		0			0
$369$	−40.2964	−	154.861i	−0.109204	−	0.419676i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−501.254			−1.34385			−0.671923	−	0.740621i	$-0.734531\pi$
$373$	−501.254			−1.34385			−0.671923	+	0.740621i	$0.734531\pi$
$374$		0			0
$375$	−137.805	−	1076.82i	−0.367480	−	2.87151i
$376$		0			0
$377$			217.892i			0.577963i
$378$		0			0
$379$	413.826			1.09189			0.545945	−	0.837821i	$-0.316171\pi$
$379$	413.826			1.09189			0.545945	+	0.837821i	$0.316171\pi$
$380$		0			0
$381$	−343.429	+	43.9502i	−0.901389	+	0.115355i
$382$		0			0
$383$			525.680i			1.37253i	0.727350	+	0.686267i	$0.240752\pi$
$383$			525.680i			1.37253i	−0.727350	+	0.686267i	$0.759248\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−174.283	+	45.3505i	−0.450345	+	0.117185i
$388$		0			0
$389$		−	308.331i		−	0.792625i	−0.918116	−	0.396313i	$-0.870290\pi$
$389$		−	308.331i		−	0.792625i	0.918116	−	0.396313i	$-0.129710\pi$
$390$		0			0
$391$	−205.429			−0.525394
$392$		0			0
$393$	25.4437	+	198.818i	0.0647423	+	0.505899i
$394$		0			0
$395$			822.424i			2.08209i
$396$		0			0
$397$	−642.361			−1.61804			−0.809019	−	0.587783i	$-0.800001\pi$
$397$	−642.361			−1.61804			−0.809019	+	0.587783i	$0.800001\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	134.639i		−	0.335759i	−0.985808	−	0.167879i	$-0.946308\pi$
$401$		−	134.639i		−	0.335759i	0.985808	−	0.167879i	$-0.0536919\pi$
$402$		0			0
$403$	−370.331			−0.918936
$404$		0			0
$405$	−371.356	−	665.252i	−0.916928	−	1.64260i
$406$		0			0
$407$			43.9929i			0.108091i
$408$		0			0
$409$	−328.930			−0.804229			−0.402114	−	0.915589i	$-0.631725\pi$
$409$	−328.930			−0.804229			−0.402114	+	0.915589i	$0.631725\pi$
$410$		0			0
$411$	24.8140	+	193.897i	0.0603746	+	0.471770i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	1333.95			3.21433
$416$		0			0
$417$	655.890	−	83.9373i	1.57288	−	0.201289i
$418$		0			0
$419$			30.9017i			0.0737510i	0.999320	+	0.0368755i	$0.0117405\pi$
$419$			30.9017i			0.0737510i	−0.999320	+	0.0368755i	$0.988260\pi$
$420$		0			0
$421$	−384.449			−0.913180			−0.456590	−	0.889677i	$-0.650929\pi$
$421$	−384.449			−0.913180			−0.456590	+	0.889677i	$0.650929\pi$
$422$		0			0
$423$	112.341	+	431.730i	0.265581	+	1.02064i
$424$		0			0
$425$			916.850i			2.15729i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	58.7220	+	458.856i	0.136881	+	1.06959i
$430$		0			0
$431$			373.738i			0.867142i	0.901119	+	0.433571i	$0.142747\pi$
$431$			373.738i			0.867142i	−0.901119	+	0.433571i	$0.857253\pi$
$432$		0			0
$433$	−229.345			−0.529664			−0.264832	−	0.964295i	$-0.585317\pi$
$433$	−229.345			−0.529664			−0.264832	+	0.964295i	$0.585317\pi$
$434$		0			0
$435$	394.038	−	50.4268i	0.905834	−	0.115924i
$436$		0			0
$437$			206.999i			0.473681i
$438$		0			0
$439$	−450.321			−1.02579			−0.512894	−	0.858452i	$-0.671427\pi$
$439$	−450.321			−1.02579			−0.512894	+	0.858452i	$0.671427\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			485.506i			1.09595i	0.836494	+	0.547976i	$0.184601\pi$
$443$			485.506i			1.09595i	−0.836494	+	0.547976i	$0.815399\pi$
$444$		0			0
$445$	−1009.76			−2.26912
$446$		0			0
$447$	−2.00294	−	15.6511i	−0.00448085	−	0.0350135i
$448$		0			0
$449$		−	613.661i		−	1.36673i	−0.730078	−	0.683364i	$-0.760516\pi$
$449$		−	613.661i		−	1.36673i	0.730078	−	0.683364i	$-0.239484\pi$
$450$		0			0
$451$	177.136			0.392764
$452$		0			0
$453$	−220.337	+	28.1975i	−0.486395	+	0.0622462i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−183.432			−0.401384			−0.200692	−	0.979654i	$-0.564319\pi$
$457$	−183.432			−0.401384			−0.200692	+	0.979654i	$0.564319\pi$
$458$		0			0
$459$	145.333	+	361.924i	0.316630	+	0.788506i
$460$		0			0
$461$		−	560.504i		−	1.21584i	−0.793997	−	0.607922i	$-0.792003\pi$
$461$		−	560.504i		−	1.21584i	0.793997	−	0.607922i	$-0.207997\pi$
$462$		0			0
$463$	−844.836			−1.82470			−0.912350	−	0.409411i	$-0.865734\pi$
$463$	−844.836			−1.82470			−0.912350	+	0.409411i	$0.865734\pi$
$464$		0			0
$465$	85.7059	+	669.710i	0.184314	+	1.44024i
$466$		0			0
$467$		−	549.740i		−	1.17717i	−0.808434	−	0.588586i	$-0.799685\pi$
$467$		−	549.740i		−	1.17717i	0.808434	−	0.588586i	$-0.200315\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−138.914	+	17.7775i	−0.294934	+	0.0377441i
$472$		0			0
$473$		−	199.353i		−	0.421466i
$474$		0			0
$475$	923.857			1.94496
$476$		0			0
$477$	217.505	+	835.879i	0.455985	+	1.75237i
$478$		0			0
$479$		−	616.018i		−	1.28605i	−0.765845	−	0.643025i	$-0.777679\pi$
$479$		−	616.018i		−	1.28605i	0.765845	−	0.643025i	$-0.222321\pi$
$480$		0			0
$481$	−68.3438			−0.142087
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	210.187i		−	0.433374i
$486$		0			0
$487$	−291.654			−0.598878			−0.299439	−	0.954115i	$-0.596800\pi$
$487$	−291.654			−0.598878			−0.299439	+	0.954115i	$0.596800\pi$
$488$		0			0
$489$	−329.008	+	42.1047i	−0.672819	+	0.0861038i
$490$		0			0
$491$		−	577.104i		−	1.17536i	−0.809092	−	0.587682i	$-0.800040\pi$
$491$		−	577.104i		−	1.17536i	0.809092	−	0.587682i	$-0.199960\pi$
$492$		0			0
$493$	−203.356			−0.412487
$494$		0			0
$495$	816.210	−	212.387i	1.64891	−	0.429064i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−803.403			−1.61003			−0.805013	−	0.593258i	$-0.797842\pi$
$499$	−803.403			−1.61003			−0.805013	+	0.593258i	$0.797842\pi$
$500$		0			0
$501$	−100.820	−	787.809i	−0.201237	−	1.57247i
$502$		0			0
$503$			252.591i			0.502168i	0.967965	+	0.251084i	$0.0807871\pi$
$503$			252.591i			0.502168i	−0.967965	+	0.251084i	$0.919213\pi$
$504$		0			0
$505$	1238.89			2.45325
$506$		0			0
$507$	−209.944	+	26.8675i	−0.414091	+	0.0529931i
$508$		0			0
$509$		−	463.647i		−	0.910899i	−0.890262	−	0.455449i	$-0.849479\pi$
$509$		−	463.647i		−	0.910899i	0.890262	−	0.455449i	$-0.150521\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	364.690	−	146.444i	0.710897	−	0.285466i
$514$		0			0
$515$		−	112.186i		−	0.217838i
$516$		0			0
$517$	−493.831			−0.955187
$518$		0			0
$519$	−122.531	−	957.461i	−0.236090	−	1.84482i
$520$		0			0
$521$			350.148i			0.672069i	0.941850	+	0.336035i	$0.109086\pi$
$521$			350.148i			0.672069i	−0.941850	+	0.336035i	$0.890914\pi$
$522$		0			0
$523$	−547.527			−1.04690			−0.523449	−	0.852057i	$-0.675355\pi$
$523$	−547.527			−1.04690			−0.523449	+	0.852057i	$0.675355\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	345.626i		−	0.655837i
$528$		0			0
$529$	326.749			0.617673
$530$		0			0
$531$	−105.472	−	405.333i	−0.198629	−	0.763339i
$532$		0			0
$533$			275.185i			0.516295i
$534$		0			0
$535$	823.966			1.54012
$536$		0			0
$537$	−16.4654	−	128.662i	−0.0306619	−	0.239593i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−144.947			−0.267925			−0.133962	−	0.990986i	$-0.542770\pi$
$541$	−144.947			−0.267925			−0.133962	+	0.990986i	$0.542770\pi$
$542$		0			0
$543$	425.402	−	54.4407i	0.783429	−	0.100259i
$544$		0			0
$545$			1363.59i			2.50200i
$546$		0			0
$547$	−945.541			−1.72859			−0.864297	−	0.502981i	$-0.832236\pi$
$547$	−945.541			−1.72859			−0.864297	+	0.502981i	$0.832236\pi$
$548$		0			0
$549$	−292.154	+	76.0216i	−0.532156	+	0.138473i
$550$		0			0
$551$			204.910i			0.371888i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	15.8169	+	123.594i	0.0284988	+	0.222691i
$556$		0			0
$557$		−	651.981i		−	1.17052i	−0.810845	−	0.585262i	$-0.800992\pi$
$557$		−	651.981i		−	1.17052i	0.810845	−	0.585262i	$-0.199008\pi$
$558$		0			0
$559$	309.699			0.554024
$560$		0			0
$561$	−428.246	+	54.8046i	−0.763361	+	0.0976909i
$562$		0			0
$563$		−	327.184i		−	0.581144i	−0.956853	−	0.290572i	$-0.906154\pi$
$563$		−	327.184i		−	0.581144i	0.956853	−	0.290572i	$-0.0938456\pi$
$564$		0			0
$565$	1476.67			2.61357
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	612.607i		−	1.07664i	−0.842741	−	0.538319i	$-0.819059\pi$
$569$		−	612.607i		−	1.07664i	0.842741	−	0.538319i	$-0.180941\pi$
$570$		0			0
$571$	853.872			1.49540			0.747698	−	0.664038i	$-0.231159\pi$
$571$	853.872			1.49540			0.747698	+	0.664038i	$0.231159\pi$
$572$		0			0
$573$	−20.6510	−	161.368i	−0.0360401	−	0.281619i
$574$		0			0
$575$			902.666i			1.56985i
$576$		0			0
$577$	170.852			0.296104			0.148052	−	0.988980i	$-0.452700\pi$
$577$	170.852			0.296104			0.148052	+	0.988980i	$0.452700\pi$
$578$		0			0
$579$	170.724	−	21.8483i	0.294860	−	0.0377346i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−956.116			−1.63999
$584$		0			0
$585$	329.947	+	1268.00i	0.564012	+	2.16752i
$586$		0			0
$587$			744.390i			1.26813i	0.773281	+	0.634063i	$0.218614\pi$
$587$			744.390i			1.26813i	−0.773281	+	0.634063i	$0.781386\pi$
$588$		0			0
$589$	−348.267			−0.591286
$590$		0			0
$591$	−43.4416	−	339.455i	−0.0735053	−	0.574374i
$592$		0			0
$593$			806.226i			1.35957i	0.733411	+	0.679786i	$0.237927\pi$
$593$			806.226i			1.35957i	−0.733411	+	0.679786i	$0.762073\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−283.483	+	36.2787i	−0.474847	+	0.0607683i
$598$		0			0
$599$			889.093i			1.48430i	0.670236	+	0.742148i	$0.266193\pi$
$599$			889.093i			1.48430i	−0.670236	+	0.742148i	$0.733807\pi$
$600$		0			0
$601$	1005.75			1.67345			0.836727	−	0.547621i	$-0.184466\pi$
$601$	1005.75			1.67345			0.836727	+	0.547621i	$0.184466\pi$
$602$		0			0
$603$	513.208	−	133.542i	0.851092	−	0.221463i
$604$		0			0
$605$		−	204.504i		−	0.338023i
$606$		0			0
$607$	−343.014			−0.565097			−0.282548	−	0.959253i	$-0.591180\pi$
$607$	−343.014			−0.565097			−0.282548	+	0.959253i	$0.591180\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	767.177i		−	1.25561i
$612$		0			0
$613$	466.818			0.761530			0.380765	−	0.924672i	$-0.375661\pi$
$613$	466.818			0.761530			0.380765	+	0.924672i	$0.375661\pi$
$614$		0			0
$615$	497.647	−	63.6862i	0.809182	−	0.103555i
$616$		0			0
$617$			1028.46i			1.66686i	0.552622	+	0.833432i	$0.313627\pi$
$617$			1028.46i			1.66686i	−0.552622	+	0.833432i	$0.686373\pi$
$618$		0			0
$619$	−139.505			−0.225372			−0.112686	−	0.993631i	$-0.535945\pi$
$619$	−139.505			−0.225372			−0.112686	+	0.993631i	$0.535945\pi$
$620$		0			0
$621$	143.085	+	356.325i	0.230411	+	0.573792i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	1816.89			2.90703
$626$		0			0
$627$	55.2234	+	431.518i	0.0880756	+	0.688227i
$628$		0			0
$629$		−	63.7846i		−	0.101406i
$630$		0			0
$631$	801.221			1.26976			0.634882	−	0.772609i	$-0.281049\pi$
$631$	801.221			1.26976			0.634882	+	0.772609i	$0.281049\pi$
$632$		0			0
$633$	−589.684	+	75.4646i	−0.931570	+	0.119217i
$634$		0			0
$635$		−	1085.54i		−	1.70951i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−53.4777	−	205.517i	−0.0836897	−	0.321623i
$640$		0			0
$641$			424.115i			0.661646i	0.943693	+	0.330823i	$0.107326\pi$
$641$			424.115i			0.661646i	−0.943693	+	0.330823i	$0.892674\pi$
$642$		0			0
$643$	552.561			0.859348			0.429674	−	0.902984i	$-0.358628\pi$
$643$	552.561			0.859348			0.429674	+	0.902984i	$0.358628\pi$
$644$		0			0
$645$	−71.6739	−	560.063i	−0.111122	−	0.868315i
$646$		0			0
$647$			211.382i			0.326711i	0.986567	+	0.163356i	$0.0522318\pi$
$647$			211.382i			0.326711i	−0.986567	+	0.163356i	$0.947768\pi$
$648$		0			0
$649$	463.638			0.714388
$650$		0			0
$651$		0			0
$652$		0			0
$653$			660.756i			1.01188i	0.862570	+	0.505938i	$0.168854\pi$
$653$			660.756i			1.01188i	−0.862570	+	0.505938i	$0.831146\pi$
$654$		0			0
$655$	−628.443			−0.959454
$656$		0			0
$657$	477.239	−	124.183i	0.726392	−	0.189015i
$658$		0			0
$659$		−	160.732i		−	0.243903i	−0.992536	−	0.121951i	$-0.961085\pi$
$659$		−	160.732i		−	0.243903i	0.992536	−	0.121951i	$-0.0389152\pi$
$660$		0			0
$661$	−15.3511			−0.0232241			−0.0116120	−	0.999933i	$-0.503696\pi$
$661$	−15.3511			−0.0232241			−0.0116120	+	0.999933i	$0.503696\pi$
$662$		0			0
$663$	−85.1401	−	665.289i	−0.128416	−	1.00345i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−200.210			−0.300165
$668$		0			0
$669$	382.378	−	48.9347i	0.571566	−	0.0731460i
$670$		0			0
$671$		−	334.178i		−	0.498031i
$672$		0			0
$673$	155.122			0.230493			0.115246	−	0.993337i	$-0.463234\pi$
$673$	155.122			0.230493			0.115246	+	0.993337i	$0.463234\pi$
$674$		0			0
$675$	1590.32	−	638.603i	2.35602	−	0.946079i
$676$		0			0
$677$		−	383.160i		−	0.565968i	−0.959125	−	0.282984i	$-0.908676\pi$
$677$		−	383.160i		−	0.565968i	0.959125	−	0.282984i	$-0.0913243\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	37.5265	+	293.233i	0.0551049	+	0.430592i
$682$		0			0
$683$		−	1253.71i		−	1.83560i	−0.397043	−	0.917800i	$-0.629964\pi$
$683$		−	1253.71i		−	1.83560i	0.397043	−	0.917800i	$-0.370036\pi$
$684$		0			0
$685$	−612.888			−0.894727
$686$		0			0
$687$	−1084.56	+	138.796i	−1.57869	+	0.202032i
$688$		0			0
$689$		−	1485.35i		−	2.15580i
$690$		0			0
$691$	−141.090			−0.204183			−0.102091	−	0.994775i	$-0.532553\pi$
$691$	−141.090			−0.204183			−0.102091	+	0.994775i	$0.532553\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			2073.20i			2.98302i
$696$		0			0
$697$	−256.827			−0.368475
$698$		0			0
$699$	59.4971	+	464.913i	0.0851175	+	0.665112i
$700$		0			0
$701$			549.277i			0.783562i	0.920058	+	0.391781i	$0.128141\pi$
$701$			549.277i			0.783562i	−0.920058	+	0.391781i	$0.871859\pi$
$702$		0			0
$703$	−64.2720			−0.0914254
$704$		0			0
$705$	−1387.37	+	177.548i	−1.96790	+	0.251842i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−566.855			−0.799513			−0.399757	−	0.916621i	$-0.630905\pi$
$709$	−566.855			−0.799513			−0.399757	+	0.916621i	$0.630905\pi$
$710$		0			0
$711$	−761.568	+	198.168i	−1.07112	+	0.278718i
$712$		0			0
$713$		−	340.279i		−	0.477250i
$714$		0			0
$715$	−1450.39			−2.02852
$716$		0			0
$717$	−26.1260	−	204.150i	−0.0364380	−	0.284728i
$718$		0			0
$719$			62.4969i			0.0869220i	0.999055	+	0.0434610i	$0.0138384\pi$
$719$			62.4969i			0.0869220i	−0.999055	+	0.0434610i	$0.986162\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	574.354	−	73.5027i	0.794404	−	0.101664i
$724$		0			0
$725$			893.559i			1.23249i
$726$		0			0
$727$	−1039.29			−1.42956			−0.714781	−	0.699348i	$-0.753474\pi$
$727$	−1039.29			−1.42956			−0.714781	+	0.699348i	$0.753474\pi$
$728$		0			0
$729$	526.545	−	504.174i	0.722284	−	0.691596i
$730$		0			0
$731$			289.039i			0.395402i
$732$		0			0
$733$	956.992			1.30558			0.652791	−	0.757538i	$-0.273598\pi$
$733$	956.992			1.30558			0.652791	+	0.757538i	$0.273598\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			587.030i			0.796513i
$738$		0			0
$739$	−686.705			−0.929236			−0.464618	−	0.885511i	$-0.653808\pi$
$739$	−686.705			−0.929236			−0.464618	+	0.885511i	$0.653808\pi$
$740$		0			0
$741$	−670.373	+	85.7907i	−0.904687	+	0.115777i
$742$		0			0
$743$		−	128.814i		−	0.173370i	−0.996236	−	0.0866848i	$-0.972373\pi$
$743$		−	128.814i		−	0.173370i	0.996236	−	0.0866848i	$-0.0276273\pi$
$744$		0			0
$745$	49.4712			0.0664043
$746$		0			0
$747$	321.423	+	1235.24i	0.430286	+	1.65360i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−961.035			−1.27967			−0.639837	−	0.768511i	$-0.720998\pi$
$751$	−961.035			−1.27967			−0.639837	+	0.768511i	$0.720998\pi$
$752$		0			0
$753$	−24.6000	−	192.226i	−0.0326694	−	0.255280i
$754$		0			0
$755$		−	696.460i		−	0.922464i
$756$		0			0
$757$	1187.08			1.56813			0.784066	−	0.620678i	$-0.213143\pi$
$757$	1187.08			1.56813			0.784066	+	0.620678i	$0.213143\pi$
$758$		0			0
$759$	−421.621	+	53.9567i	−0.555495	+	0.0710893i
$760$		0			0
$761$		−	1076.04i		−	1.41398i	−0.707221	−	0.706992i	$-0.750052\pi$
$761$		−	1076.04i		−	1.41398i	0.707221	−	0.706992i	$-0.249948\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	−1183.41	+	307.936i	−1.54694	+	0.402531i
$766$		0			0
$767$			720.271i			0.939076i
$768$		0			0
$769$	−878.937			−1.14296			−0.571480	−	0.820616i	$-0.693631\pi$
$769$	−878.937			−1.14296			−0.571480	+	0.820616i	$0.693631\pi$
$770$		0			0
$771$	117.697	+	919.694i	0.152656	+	1.19286i
$772$		0			0
$773$		−	855.482i		−	1.10670i	−0.832947	−	0.553352i	$-0.813348\pi$
$773$		−	855.482i		−	1.10670i	0.832947	−	0.553352i	$-0.186652\pi$
$774$		0			0
$775$	−1518.70			−1.95961
$776$		0			0
$777$		0			0
$778$		0			0
$779$			258.790i			0.332208i
$780$		0			0
$781$	235.079			0.300998
$782$		0			0
$783$	141.641	+	352.730i	0.180896	+	0.450485i
$784$		0			0
$785$		−	439.092i		−	0.559353i
$786$		0			0
$787$	−610.933			−0.776281			−0.388141	−	0.921600i	$-0.626883\pi$
$787$	−610.933			−0.776281			−0.388141	+	0.921600i	$0.626883\pi$
$788$		0			0
$789$	−60.7864	−	474.988i	−0.0770424	−	0.602013i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	519.153			0.654670
$794$		0			0
$795$	−2686.11	+	343.754i	−3.37876	+	0.432395i
$796$		0			0
$797$		−	1035.91i		−	1.29976i	−0.760036	−	0.649882i	$-0.774818\pi$
$797$		−	1035.91i		−	1.29976i	0.760036	−	0.649882i	$-0.225182\pi$
$798$		0			0
$799$	715.999			0.896119
$800$		0			0
$801$	−243.308	−	935.040i	−0.303755	−	1.16734i
$802$		0			0
$803$			545.888i			0.679810i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	24.5787	+	192.059i	0.0304569	+	0.237992i
$808$		0			0
$809$			1195.90i			1.47825i	0.673570	+	0.739123i	$0.264760\pi$
$809$			1195.90i			1.47825i	−0.673570	+	0.739123i	$0.735240\pi$
$810$		0			0
$811$	−715.026			−0.881660			−0.440830	−	0.897591i	$-0.645316\pi$
$811$	−715.026			−0.881660			−0.440830	+	0.897591i	$0.645316\pi$
$812$		0			0
$813$	−530.407	+	67.8787i	−0.652407	+	0.0834916i
$814$		0			0
$815$		−	1039.96i		−	1.27602i
$816$		0			0
$817$	291.248			0.356485
$818$		0			0
$819$		0			0
$820$		0			0
$821$			819.409i			0.998063i	0.866584	+	0.499031i	$0.166311\pi$
$821$			819.409i			0.998063i	−0.866584	+	0.499031i	$0.833689\pi$
$822$		0			0
$823$	398.683			0.484426			0.242213	−	0.970223i	$-0.422127\pi$
$823$	398.683			0.484426			0.242213	+	0.970223i	$0.422127\pi$
$824$		0			0
$825$	240.815	+	1881.74i	0.291896	+	2.28089i
$826$		0			0
$827$		−	28.4443i		−	0.0343945i	−0.999852	−	0.0171973i	$-0.994526\pi$
$827$		−	28.4443i		−	0.0343945i	0.999852	−	0.0171973i	$-0.00547433\pi$
$828$		0			0
$829$	608.175			0.733625			0.366812	−	0.930295i	$-0.380449\pi$
$829$	608.175			0.733625			0.366812	+	0.930295i	$0.380449\pi$
$830$		0			0
$831$	−500.778	+	64.0870i	−0.602621	+	0.0771203i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	2490.18			2.98225
$836$		0			0
$837$	−599.503	+	240.735i	−0.716252	+	0.287616i
$838$		0			0
$839$			1434.22i			1.70943i	0.519094	+	0.854717i	$0.326269\pi$
$839$			1434.22i			1.70943i	−0.519094	+	0.854717i	$0.673731\pi$
$840$		0			0
$841$	642.810			0.764340
$842$		0			0
$843$	146.332	+	1143.45i	0.173585	+	1.35640i
$844$		0			0
$845$		−	663.610i		−	0.785337i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	938.934	−	120.160i	1.10593	−	0.141531i
$850$		0			0
$851$		−	62.7978i		−	0.0737930i
$852$		0			0
$853$	−908.389			−1.06493			−0.532467	−	0.846451i	$-0.678735\pi$
$853$	−908.389			−1.06493			−0.532467	+	0.846451i	$0.678735\pi$
$854$		0			0
$855$	310.289	+	1192.45i	0.362912	+	1.39468i
$856$		0			0
$857$			1510.48i			1.76252i	0.472634	+	0.881259i	$0.343303\pi$
$857$			1510.48i			1.76252i	−0.472634	+	0.881259i	$0.656697\pi$
$858$		0			0
$859$	−512.050			−0.596100			−0.298050	−	0.954550i	$-0.596336\pi$
$859$	−512.050			−0.596100			−0.298050	+	0.954550i	$0.596336\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1062.02i		−	1.23062i	−0.788286	−	0.615308i	$-0.789031\pi$
$863$		−	1062.02i		−	1.23062i	0.788286	−	0.615308i	$-0.210969\pi$
$864$		0			0
$865$	3026.43			3.49876
$866$		0			0
$867$	−239.080	+	30.5961i	−0.275755	+	0.0352896i
$868$		0			0
$869$		−	871.116i		−	1.00243i
$870$		0			0
$871$	−911.964			−1.04703
$872$		0			0
$873$	194.634	−	50.6458i	0.222948	−	0.0580135i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−1546.34			−1.76321			−0.881607	−	0.471984i	$-0.843538\pi$
$877$	−1546.34			−1.76321			−0.881607	+	0.471984i	$0.843538\pi$
$878$		0			0
$879$	−11.8271	−	92.4179i	−0.0134552	−	0.105140i
$880$		0			0
$881$			249.840i			0.283587i	0.989896	+	0.141793i	$0.0452869\pi$
$881$			249.840i			0.283587i	−0.989896	+	0.141793i	$0.954713\pi$
$882$		0			0
$883$	−86.1477			−0.0975625			−0.0487812	−	0.998809i	$-0.515534\pi$
$883$	−86.1477			−0.0975625			−0.0487812	+	0.998809i	$0.515534\pi$
$884$		0			0
$885$	1302.54	−	166.693i	1.47180	−	0.188353i
$886$		0			0
$887$			4.84664i			0.00546408i	0.999996	+	0.00273204i	$0.000869636\pi$
$887$			4.84664i			0.00546408i	−0.999996	+	0.00273204i	$0.999130\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	393.342	+	704.638i	0.441461	+	0.790839i
$892$		0			0
$893$		−	721.470i		−	0.807918i
$894$		0			0
$895$	406.685			0.454397
$896$		0			0
$897$	−83.8229	−	654.996i	−0.0934481	−	0.730208i
$898$		0			0
$899$		−	336.846i		−	0.374689i
$900$		0			0
$901$	1386.26			1.53858
$902$		0			0
$903$		0			0
$904$		0			0
$905$			1344.65i			1.48580i
$906$		0			0
$907$	−480.454			−0.529717			−0.264859	−	0.964287i	$-0.585325\pi$
$907$	−480.454			−0.529717			−0.264859	+	0.964287i	$0.585325\pi$
$908$		0			0
$909$	298.519	+	1147.22i	0.328404	+	1.26207i
$910$		0			0
$911$			477.015i			0.523617i	0.965120	+	0.261809i	$0.0843189\pi$
$911$			477.015i			0.523617i	−0.965120	+	0.261809i	$0.915681\pi$
$912$		0			0
$913$	−1412.92			−1.54756
$914$		0			0
$915$	−120.148	−	938.841i	−0.131309	−	1.02606i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	951.417			1.03527			0.517637	−	0.855600i	$-0.326812\pi$
$919$	951.417			1.03527			0.517637	+	0.855600i	$0.326812\pi$
$920$		0			0
$921$	909.535	−	116.397i	0.987552	−	0.126382i
$922$		0			0
$923$			365.201i			0.395667i
$924$		0			0
$925$	−280.273			−0.302998
$926$		0			0
$927$	103.885	−	27.0320i	0.112066	−	0.0291608i
$928$		0			0
$929$		−	627.120i		−	0.675048i	−0.941317	−	0.337524i	$-0.890411\pi$
$929$		−	627.120i		−	0.675048i	0.941317	−	0.337524i	$-0.109589\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−208.856	−	1632.01i	−0.223855	−	1.74921i
$934$		0			0
$935$		−	1353.64i		−	1.44774i
$936$		0			0
$937$	545.810			0.582508			0.291254	−	0.956646i	$-0.405927\pi$
$937$	545.810			0.582508			0.291254	+	0.956646i	$0.405927\pi$
$938$		0			0
$939$	−115.116	+	14.7320i	−0.122595	+	0.0156890i
$940$		0			0
$941$			326.914i			0.347411i	0.984798	+	0.173706i	$0.0555741\pi$
$941$			326.914i			0.347411i	−0.984798	+	0.173706i	$0.944426\pi$
$942$		0			0
$943$	−252.854			−0.268138
$944$		0			0
$945$		0			0
$946$		0			0
$947$			1147.96i			1.21221i	0.795386	+	0.606103i	$0.207268\pi$
$947$			1147.96i			1.21221i	−0.795386	+	0.606103i	$0.792732\pi$
$948$		0			0
$949$	−848.048			−0.893623
$950$		0			0
$951$	193.578	+	1512.63i	0.203552	+	1.59057i
$952$		0			0
$953$			932.135i			0.978106i	0.872254	+	0.489053i	$0.162658\pi$
$953$			932.135i			0.978106i	−0.872254	+	0.489053i	$0.837342\pi$
$954$		0			0
$955$	510.066			0.534100
$956$		0			0
$957$	−417.367	+	53.4123i	−0.436120	+	0.0558123i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−388.494			−0.404260
$962$		0			0
$963$	198.540	+	762.996i	0.206168	+	0.792312i
$964$		0			0
$965$			539.639i			0.559211i
$966$		0			0
$967$	770.649			0.796949			0.398474	−	0.917179i	$-0.369540\pi$
$967$	770.649			0.796949			0.398474	+	0.917179i	$0.369540\pi$
$968$		0			0
$969$	−80.0676	−	625.652i	−0.0826291	−	0.645668i
$970$		0			0
$971$			942.110i			0.970248i	0.874445	+	0.485124i	$0.161225\pi$
$971$			942.110i			0.970248i	−0.874445	+	0.485124i	$0.838775\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−2923.32	+	374.110i	−2.99827	+	0.383703i
$976$		0			0
$977$			490.175i			0.501714i	0.968024	+	0.250857i	$0.0807125\pi$
$977$			490.175i			0.501714i	−0.968024	+	0.250857i	$0.919288\pi$
$978$		0			0
$979$	1069.54			1.09248
$980$		0			0
$981$	−1262.69	+	328.566i	−1.28715	+	0.334929i
$982$		0			0
$983$			396.738i			0.403599i	0.979427	+	0.201799i	$0.0646789\pi$
$983$			396.738i			0.403599i	−0.979427	+	0.201799i	$0.935321\pi$
$984$		0			0
$985$	1072.98			1.08932
$986$		0			0
$987$		0			0
$988$		0			0
$989$			284.568i			0.287733i
$990$		0			0
$991$	−736.035			−0.742719			−0.371360	−	0.928489i	$-0.621108\pi$
$991$	−736.035			−0.742719			−0.371360	+	0.928489i	$0.621108\pi$
$992$		0			0
$993$	557.375	−	71.3299i	0.561304	−	0.0718327i
$994$		0			0
$995$		−	896.060i		−	0.900562i
$996$		0			0
$997$	1151.69			1.15515			0.577576	−	0.816337i	$-0.303999\pi$
$997$	1151.69			1.15515			0.577576	+	0.816337i	$0.303999\pi$
$998$		0			0
$999$	−110.637	+	44.4272i	−0.110748	+	0.0444716i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1176.3.d.f.785.2		16
3.2	odd	2	inner	1176.3.d.f.785.1		16
7.2	even	3		168.3.bf.a.137.10	yes	32
7.4	even	3		168.3.bf.a.65.12	yes	32
7.6	odd	2		1176.3.d.g.785.15		16
21.2	odd	6		168.3.bf.a.137.12	yes	32
21.11	odd	6		168.3.bf.a.65.10	✓	32
21.20	even	2		1176.3.d.g.785.16		16
28.11	odd	6		336.3.bn.h.65.5		32
28.23	odd	6		336.3.bn.h.305.7		32
84.11	even	6		336.3.bn.h.65.7		32
84.23	even	6		336.3.bn.h.305.5		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.3.bf.a.65.10	✓	32	21.11	odd	6
168.3.bf.a.65.12	yes	32	7.4	even	3
168.3.bf.a.137.10	yes	32	7.2	even	3
168.3.bf.a.137.12	yes	32	21.2	odd	6
336.3.bn.h.65.5		32	28.11	odd	6
336.3.bn.h.65.7		32	84.11	even	6
336.3.bn.h.305.5		32	84.23	even	6
336.3.bn.h.305.7		32	28.23	odd	6
1176.3.d.f.785.1		16	3.2	odd	2	inner
1176.3.d.f.785.2		16	1.1	even	1	trivial
1176.3.d.g.785.15		16	7.6	odd	2
1176.3.d.g.785.16		16	21.20	even	2

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:	\( N \)	\(=\)	\( 1176 = 2^{3} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1176.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.97573 + 0.380818i) q^{3} -9.40596i q^{5} +(8.70995 - 2.26643i) q^{9} +9.96283i q^{11} -15.4775 q^{13} +(3.58196 + 27.9896i) q^{15} -14.4450i q^{17} -14.5554 q^{19} -14.2215i q^{23} -63.4720 q^{25} +(-25.0554 + 10.0612i) q^{27} -14.0780i q^{29} +23.9271 q^{31} +(-3.79403 - 29.6467i) q^{33} +4.41570 q^{37} +(46.0568 - 5.89410i) q^{39} -17.7797i q^{41} -20.0097 q^{43} +(-21.3179 - 81.9254i) q^{45} +49.5674i q^{47} +(5.50090 + 42.9843i) q^{51} +95.9683i q^{53} +93.7100 q^{55} +(43.3128 - 5.54294i) q^{57} -46.5367i q^{59} -33.5425 q^{61} +145.580i q^{65} +58.9220 q^{67} +(5.41580 + 42.3193i) q^{69} -23.5956i q^{71} +54.7924 q^{73} +(188.876 - 24.1713i) q^{75} -87.4365 q^{79} +(70.7266 - 39.4809i) q^{81} +141.820i q^{83} -135.869 q^{85} +(5.36116 + 41.8924i) q^{87} -107.353i q^{89} +(-71.2006 + 9.11188i) q^{93} +136.907i q^{95} +22.3461 q^{97} +(22.5800 + 86.7758i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 8 q^{9} - 4 q^{13} + 8 q^{15} - 16 q^{19} - 76 q^{25} - 12 q^{27} + 36 q^{31} - 32 q^{33} - 28 q^{37} - 4 q^{39} - 120 q^{43} - 28 q^{45} - 36 q^{51} - 28 q^{55} - 88 q^{57} + 152 q^{61} + 88 q^{67}+ \cdots + 368 q^{99}+O(q^{100}) \) 16 * q + 8 * q^9 - 4 * q^13 + 8 * q^15 - 16 * q^19 - 76 * q^25 - 12 * q^27 + 36 * q^31 - 32 * q^33 - 28 * q^37 - 4 * q^39 - 120 * q^43 - 28 * q^45 - 36 * q^51 - 28 * q^55 - 88 * q^57 + 152 * q^61 + 88 * q^67 - 112 * q^69 + 156 * q^73 + 312 * q^75 - 84 * q^79 + 164 * q^81 + 152 * q^85 + 176 * q^87 - 8 * q^93 - 20 * q^97 + 368 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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