LMFDB - Embedded newform 756.4.f.e.377.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [756,4,Mod(377,756)] 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("756.377"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$44.6054439643$
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^{15} - 36 x^{14} - 780 x^{13} + 17052 x^{12} - 100012 x^{11} + 336938 x^{10} + \cdots + 1976222698452 $ x^16 - 4x^15 - 36x^14 - 780x^13 + 17052x^12 - 100012x^11 + 336938x^10 - 6730840x^9 + 90353572x^8 - 474561804x^7 + 689078700x^6 + 9372890772x^5 - 37121562243x^4 - 109129608396x^3 + 718624177776x^2 + 62521834824*x + 1976222698452
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{8}\cdot 3^{26} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			377.9
Root			$4.61314 + 2.16386i$ of defining polynomial
Character	$\chi$	$=$	756.377
Dual form			756.4.f.e.377.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.0103268 q^{5} +(16.5266 - 8.35886i) q^{7} -60.1026i q^{11} -31.7159i q^{13} -91.2417 q^{17} -12.6579i q^{19} +86.0802i q^{23} -125.000 q^{25} -122.068i q^{29} +297.925i q^{31} +(0.170668 - 0.0863205i) q^{35} +159.053 q^{37} -182.152 q^{41} -95.7857 q^{43} -43.8453 q^{47} +(203.259 - 276.288i) q^{49} +17.6574i q^{53} -0.620670i q^{55} -317.260 q^{59} -802.687i q^{61} -0.327524i q^{65} -383.748 q^{67} -230.376i q^{71} +847.573i q^{73} +(-502.390 - 993.294i) q^{77} +411.427 q^{79} -1231.15 q^{83} -0.942237 q^{85} -1545.46 q^{89} +(-265.109 - 524.157i) q^{91} -0.130716i q^{95} +1584.03i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 14 q^{7} + 388 q^{25} - 344 q^{37} - 404 q^{43} - 14 q^{49} + 1772 q^{67} - 1456 q^{79} - 4704 q^{85} + 2436 q^{91}+O(q^{100}) $ 16 * q + 14 * q^7 + 388 * q^25 - 344 * q^37 - 404 * q^43 - 14 * q^49 + 1772 * q^67 - 1456 * q^79 - 4704 * q^85 + 2436 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$-1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.0103268			0.000923659			0.000461830	−	1.00000i	$-0.499853\pi$
$5$	0.0103268			0.000923659			0.000461830		1.00000i	$0.499853\pi$
$6$		0			0
$7$	16.5266	−	8.35886i	0.892354	−	0.451336i
$7$	16.5266	−	8.35886i	0.892354	−	0.451336i
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	60.1026i		−	1.64742i	−0.567011	−	0.823710i	$-0.691900\pi$
$11$		−	60.1026i		−	1.64742i	0.567011	−	0.823710i	$-0.308100\pi$
$12$		0			0
$13$		−	31.7159i		−	0.676646i	−0.941030	−	0.338323i	$-0.890140\pi$
$13$		−	31.7159i		−	0.676646i	0.941030	−	0.338323i	$-0.109860\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−91.2417			−1.30173			−0.650864	−	0.759195i	$-0.725593\pi$
$17$	−91.2417			−1.30173			−0.650864	+	0.759195i	$0.725593\pi$
$18$		0			0
$19$		−	12.6579i		−	0.152838i	−0.997076	−	0.0764192i	$-0.975651\pi$
$19$		−	12.6579i		−	0.152838i	0.997076	−	0.0764192i	$-0.0243487\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			86.0802i			0.780390i	0.920732	+	0.390195i	$0.127592\pi$
$23$			86.0802i			0.780390i	−0.920732	+	0.390195i	$0.872408\pi$
$24$		0			0
$25$	−125.000			−0.999999
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	122.068i		−	0.781636i	−0.920468	−	0.390818i	$-0.872192\pi$
$29$		−	122.068i		−	0.781636i	0.920468	−	0.390818i	$-0.127808\pi$
$30$		0			0
$31$			297.925i			1.72609i	0.505124	+	0.863047i	$0.331447\pi$
$31$			297.925i			1.72609i	−0.505124	+	0.863047i	$0.668553\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	0.170668	−	0.0863205i	0.000824231	−	0.000416881i
$36$		0			0
$37$	159.053			0.706707			0.353354	−	0.935490i	$-0.385041\pi$
$37$	159.053			0.706707			0.353354	+	0.935490i	$0.385041\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−182.152			−0.693840			−0.346920	−	0.937895i	$-0.612772\pi$
$41$	−182.152			−0.693840			−0.346920	+	0.937895i	$0.612772\pi$
$42$		0			0
$43$	−95.7857			−0.339702			−0.169851	−	0.985470i	$-0.554329\pi$
$43$	−95.7857			−0.339702			−0.169851	+	0.985470i	$0.554329\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−43.8453			−0.136074			−0.0680372	−	0.997683i	$-0.521674\pi$
$47$	−43.8453			−0.136074			−0.0680372	+	0.997683i	$0.521674\pi$
$48$		0			0
$49$	203.259	−	276.288i	0.592592	−	0.805503i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			17.6574i			0.0457630i	0.999738	+	0.0228815i	$0.00728404\pi$
$53$			17.6574i			0.0457630i	−0.999738	+	0.0228815i	$0.992716\pi$
$54$		0			0
$55$		−	0.620670i		−	0.00152166i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−317.260			−0.700063			−0.350032	−	0.936738i	$-0.613829\pi$
$59$	−317.260			−0.700063			−0.350032	+	0.936738i	$0.613829\pi$
$60$		0			0
$61$		−	802.687i		−	1.68481i	−0.538843	−	0.842406i	$-0.681138\pi$
$61$		−	802.687i		−	1.68481i	0.538843	−	0.842406i	$-0.318862\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	0.327524i		−	0.000624991i
$66$		0			0
$67$	−383.748			−0.699735			−0.349868	−	0.936799i	$-0.613773\pi$
$67$	−383.748			−0.699735			−0.349868	+	0.936799i	$0.613773\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	230.376i		−	0.385079i	−0.981289	−	0.192540i	$-0.938328\pi$
$71$		−	230.376i		−	0.385079i	0.981289	−	0.192540i	$-0.0616724\pi$
$72$		0			0
$73$			847.573i			1.35892i	0.733714	+	0.679458i	$0.237785\pi$
$73$			847.573i			1.35892i	−0.733714	+	0.679458i	$0.762215\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−502.390	−	993.294i	−0.743540	−	1.47008i
$78$		0			0
$79$	411.427			0.585939			0.292969	−	0.956122i	$-0.405357\pi$
$79$	411.427			0.585939			0.292969	+	0.956122i	$0.405357\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1231.15			−1.62815			−0.814073	−	0.580762i	$-0.802755\pi$
$83$	−1231.15			−1.62815			−0.814073	+	0.580762i	$0.802755\pi$
$84$		0			0
$85$	−0.942237			−0.00120235
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−1545.46			−1.84065			−0.920325	−	0.391154i	$-0.872076\pi$
$89$	−1545.46			−1.84065			−0.920325	+	0.391154i	$0.872076\pi$
$90$		0			0
$91$	−265.109	−	524.157i	−0.305395	−	0.603808i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	0.130716i		−	0.000141171i
$96$		0			0
$97$			1584.03i			1.65808i	0.559192	+	0.829038i	$0.311111\pi$
$97$			1584.03i			1.65808i	−0.559192	+	0.829038i	$0.688889\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−451.800			−0.445106			−0.222553	−	0.974921i	$-0.571439\pi$
$101$	−451.800			−0.445106			−0.222553	+	0.974921i	$0.571439\pi$
$102$		0			0
$103$		−	137.356i		−	0.131399i	−0.997839	−	0.0656996i	$-0.979072\pi$
$103$		−	137.356i		−	0.131399i	0.997839	−	0.0656996i	$-0.0209279\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1723.43i		−	1.55711i	−0.627578	−	0.778554i	$-0.715954\pi$
$107$		−	1723.43i		−	1.55711i	0.627578	−	0.778554i	$-0.284046\pi$
$108$		0			0
$109$	893.261			0.784944			0.392472	−	0.919764i	$-0.371620\pi$
$109$	893.261			0.784944			0.392472	+	0.919764i	$0.371620\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1267.25i		−	1.05498i	−0.849561	−	0.527491i	$-0.823133\pi$
$113$		−	1267.25i		−	1.05498i	0.849561	−	0.527491i	$-0.176867\pi$
$114$		0			0
$115$			0.888935i			0.000720814i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1507.92	+	762.677i	−1.16160	+	0.587516i
$120$		0			0
$121$	−2281.33			−1.71400
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.58171			−0.00184732
$126$		0			0
$127$	394.727			0.275798			0.137899	−	0.990446i	$-0.455965\pi$
$127$	394.727			0.275798			0.137899	+	0.990446i	$0.455965\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−256.278			−0.170925			−0.0854623	−	0.996341i	$-0.527237\pi$
$131$	−256.278			−0.170925			−0.0854623	+	0.996341i	$0.527237\pi$
$132$		0			0
$133$	−105.806	−	209.193i	−0.0689815	−	0.136386i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	1830.57i		−	1.14158i	−0.821097	−	0.570788i	$-0.806638\pi$
$137$		−	1830.57i		−	1.14158i	0.821097	−	0.570788i	$-0.193362\pi$
$138$		0			0
$139$		−	2172.14i		−	1.32546i	−0.748860	−	0.662728i	$-0.769398\pi$
$139$		−	2172.14i		−	1.32546i	0.748860	−	0.662728i	$-0.230602\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1906.21			−1.11472
$144$		0			0
$145$		−	1.26057i		−	0.000721965i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			2394.27i			1.31642i	0.752834	+	0.658210i	$0.228686\pi$
$149$			2394.27i			1.31642i	−0.752834	+	0.658210i	$0.771314\pi$
$150$		0			0
$151$	2063.40			1.11203			0.556017	−	0.831171i	$-0.312329\pi$
$151$	2063.40			1.11203			0.556017	+	0.831171i	$0.312329\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			3.07662i			0.00159432i
$156$		0			0
$157$		−	1443.14i		−	0.733598i	−0.930300	−	0.366799i	$-0.880454\pi$
$157$		−	1443.14i		−	0.733598i	0.930300	−	0.366799i	$-0.119546\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	719.533	+	1422.62i	0.352218	+	0.696384i
$162$		0			0
$163$	1603.09			0.770332			0.385166	−	0.922847i	$-0.374144\pi$
$163$	1603.09			0.770332			0.385166	+	0.922847i	$0.374144\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3251.45			−1.50661			−0.753307	−	0.657670i	$-0.771542\pi$
$167$	−3251.45			−1.50661			−0.753307	+	0.657670i	$0.771542\pi$
$168$		0			0
$169$	1191.10			0.542150
$170$		0			0
$171$		0			0
$172$		0			0
$173$	2305.43			1.01317			0.506585	−	0.862190i	$-0.330908\pi$
$173$	2305.43			1.01317			0.506585	+	0.862190i	$0.330908\pi$
$174$		0			0
$175$	−2065.83	+	1044.86i	−0.892353	+	0.451336i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	1566.35i		−	0.654048i	−0.945016	−	0.327024i	$-0.893954\pi$
$179$		−	1566.35i		−	0.654048i	0.945016	−	0.327024i	$-0.106046\pi$
$180$		0			0
$181$		−	17.5906i		−	0.00722375i	−0.999993	−	0.00361187i	$-0.998850\pi$
$181$		−	17.5906i		−	0.00722375i	0.999993	−	0.00361187i	$-0.00114970\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.64251			0.000652757
$186$		0			0
$187$			5483.87i			2.14449i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3605.80i		−	1.36600i	−0.730417	−	0.683002i	$-0.760674\pi$
$191$		−	3605.80i		−	1.36600i	0.730417	−	0.683002i	$-0.239326\pi$
$192$		0			0
$193$	−2394.01			−0.892874			−0.446437	−	0.894815i	$-0.647307\pi$
$193$	−2394.01			−0.892874			−0.446437	+	0.894815i	$0.647307\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4038.24i			1.46047i	0.683196	+	0.730235i	$0.260589\pi$
$197$			4038.24i			1.46047i	−0.683196	+	0.730235i	$0.739411\pi$
$198$		0			0
$199$		−	1374.65i		−	0.489680i	−0.969563	−	0.244840i	$-0.921265\pi$
$199$		−	1374.65i		−	0.489680i	0.969563	−	0.244840i	$-0.0787355\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1020.35	−	2017.37i	−0.352780	−	0.697496i
$204$		0			0
$205$	−1.88106			−0.000640871
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−760.775			−0.251789
$210$		0			0
$211$	−4556.21			−1.48655			−0.743276	−	0.668985i	$-0.766729\pi$
$211$	−4556.21			−1.48655			−0.743276	+	0.668985i	$0.766729\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.989162			−0.000313769
$216$		0			0
$217$	2490.31	+	4923.70i	0.779048	+	1.54029i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			2893.81i			0.880809i
$222$		0			0
$223$		−	2086.57i		−	0.626579i	−0.949658	−	0.313289i	$-0.898569\pi$
$223$		−	2086.57i		−	0.626579i	0.949658	−	0.313289i	$-0.101431\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	289.688			0.0847017			0.0423509	−	0.999103i	$-0.486515\pi$
$227$	289.688			0.0847017			0.0423509	+	0.999103i	$0.486515\pi$
$228$		0			0
$229$		−	4053.61i		−	1.16974i	−0.811128	−	0.584869i	$-0.801146\pi$
$229$		−	4053.61i		−	1.16974i	0.811128	−	0.584869i	$-0.198854\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	4047.96i		−	1.13816i	−0.822283	−	0.569078i	$-0.807300\pi$
$233$		−	4047.96i		−	1.13816i	0.822283	−	0.569078i	$-0.192700\pi$
$234$		0			0
$235$	−0.452783			−0.000125686
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5589.97i		−	1.51291i	−0.654047	−	0.756454i	$-0.726930\pi$
$239$		−	5589.97i		−	1.51291i	0.654047	−	0.756454i	$-0.273070\pi$
$240$		0			0
$241$		−	4226.92i		−	1.12979i	−0.825162	−	0.564896i	$-0.808916\pi$
$241$		−	4226.92i		−	1.12979i	0.825162	−	0.564896i	$-0.191084\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.09902	−	2.85317i	0.000547353	−	0.000744010i
$246$		0			0
$247$	−401.457			−0.103418
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−460.991			−0.115926			−0.0579631	−	0.998319i	$-0.518461\pi$
$251$	−460.991			−0.115926			−0.0579631	+	0.998319i	$0.518461\pi$
$252$		0			0
$253$	5173.65			1.28563
$254$		0			0
$255$		0			0
$256$		0			0
$257$	6438.45			1.56272			0.781360	−	0.624080i	$-0.214526\pi$
$257$	6438.45			1.56272			0.781360	+	0.624080i	$0.214526\pi$
$258$		0			0
$259$	2628.61	−	1329.50i	0.630633	−	0.318963i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			4049.94i			0.949543i	0.880109	+	0.474771i	$0.157469\pi$
$263$			4049.94i			0.949543i	−0.880109	+	0.474771i	$0.842531\pi$
$264$		0			0
$265$			0.182345i			4.22694e-5i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	6292.49			1.42625			0.713123	−	0.701039i	$-0.247280\pi$
$269$	6292.49			1.42625			0.713123	+	0.701039i	$0.247280\pi$
$270$		0			0
$271$			4321.82i			0.968753i	0.874860	+	0.484376i	$0.160953\pi$
$271$			4321.82i			0.968753i	−0.874860	+	0.484376i	$0.839047\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			7512.82i			1.64742i
$276$		0			0
$277$	−2244.13			−0.486775			−0.243388	−	0.969929i	$-0.578259\pi$
$277$	−2244.13			−0.486775			−0.243388	+	0.969929i	$0.578259\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			6904.91i			1.46588i	0.680294	+	0.732940i	$0.261852\pi$
$281$			6904.91i			1.46588i	−0.680294	+	0.732940i	$0.738148\pi$
$282$		0			0
$283$		−	3130.27i		−	0.657510i	−0.944415	−	0.328755i	$-0.893371\pi$
$283$		−	3130.27i		−	0.657510i	0.944415	−	0.328755i	$-0.106629\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−3010.37	+	1522.59i	−0.619151	+	0.313155i
$288$		0			0
$289$	3412.05			0.694494
$290$		0			0
$291$		0			0
$292$		0			0
$293$	6039.64			1.20423			0.602115	−	0.798409i	$-0.294325\pi$
$293$	6039.64			1.20423			0.602115	+	0.798409i	$0.294325\pi$
$294$		0			0
$295$	−3.27629			−0.000646620
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2730.11			0.528048
$300$		0			0
$301$	−1583.01	+	800.659i	−0.303134	+	0.153320i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	8.28921i		−	0.00155619i
$306$		0			0
$307$			3496.09i			0.649943i	0.945724	+	0.324972i	$0.105355\pi$
$307$			3496.09i			0.649943i	−0.945724	+	0.324972i	$0.894645\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	4823.31			0.879436			0.439718	−	0.898136i	$-0.355078\pi$
$311$	4823.31			0.879436			0.439718	+	0.898136i	$0.355078\pi$
$312$		0			0
$313$		−	2581.16i		−	0.466120i	−0.972462	−	0.233060i	$-0.925126\pi$
$313$		−	2581.16i		−	0.466120i	0.972462	−	0.233060i	$-0.0748739\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			4035.00i			0.714916i	0.933929	+	0.357458i	$0.116356\pi$
$317$			4035.00i			0.714916i	−0.933929	+	0.357458i	$0.883644\pi$
$318$		0			0
$319$	−7336.60			−1.28768
$320$		0			0
$321$		0			0
$322$		0			0
$323$			1154.93i			0.198954i
$324$		0			0
$325$			3964.48i			0.676646i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−724.615	+	366.497i	−0.121426	+	0.0614152i
$330$		0			0
$331$	2937.17			0.487739			0.243870	−	0.969808i	$-0.421583\pi$
$331$	2937.17			0.487739			0.243870	+	0.969808i	$0.421583\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.96290			−0.000646317
$336$		0			0
$337$	−10203.9			−1.64938			−0.824688	−	0.565588i	$-0.808649\pi$
$337$	−10203.9			−1.64938			−0.824688	+	0.565588i	$0.808649\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	17906.1			2.84360
$342$		0			0
$343$	1049.74	−	6265.11i	0.165249	−	0.986252i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	2620.73i		−	0.405441i	−0.979237	−	0.202720i	$-0.935022\pi$
$347$		−	2620.73i		−	0.405441i	0.979237	−	0.202720i	$-0.0649783\pi$
$348$		0			0
$349$			5984.10i			0.917826i	0.888481	+	0.458913i	$0.151761\pi$
$349$			5984.10i			0.917826i	−0.888481	+	0.458913i	$0.848239\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9769.53			1.47303			0.736515	−	0.676421i	$-0.236470\pi$
$353$	9769.53			1.47303			0.736515	+	0.676421i	$0.236470\pi$
$354$		0			0
$355$		−	2.37906i		−	0.000355682i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			6538.05i			0.961184i	0.876944	+	0.480592i	$0.159578\pi$
$359$			6538.05i			0.961184i	−0.876944	+	0.480592i	$0.840422\pi$
$360$		0			0
$361$	6698.78			0.976640
$362$		0			0
$363$		0			0
$364$		0			0
$365$			8.75274i			0.00125518i
$366$		0			0
$367$			9794.95i			1.39317i	0.717476	+	0.696583i	$0.245297\pi$
$367$			9794.95i			1.39317i	−0.717476	+	0.696583i	$0.754703\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	147.596	+	291.818i	0.0206545	+	0.0408368i
$372$		0			0
$373$	−6955.96			−0.965592			−0.482796	−	0.875733i	$-0.660379\pi$
$373$	−6955.96			−0.965592			−0.482796	+	0.875733i	$0.660379\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−3871.49			−0.528891
$378$		0			0
$379$	9763.91			1.32332			0.661660	−	0.749804i	$-0.269852\pi$
$379$	9763.91			1.32332			0.661660	+	0.749804i	$0.269852\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2864.19			0.382123			0.191062	−	0.981578i	$-0.438807\pi$
$383$	2864.19			0.382123			0.191062	+	0.981578i	$0.438807\pi$
$384$		0			0
$385$	−5.18809	−	10.2576i	−0.000686778	−	0.00135786i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			1215.80i			0.158467i	0.996856	+	0.0792335i	$0.0252473\pi$
$389$			1215.80i			0.158467i	−0.996856	+	0.0792335i	$0.974753\pi$
$390$		0			0
$391$		−	7854.11i		−	1.01585i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	4.24874			0.000541208
$396$		0			0
$397$		−	3466.06i		−	0.438178i	−0.975705	−	0.219089i	$-0.929691\pi$
$397$		−	3466.06i		−	0.438178i	0.975705	−	0.219089i	$-0.0703085\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			4506.84i			0.561249i	0.959818	+	0.280625i	$0.0905416\pi$
$401$			4506.84i			0.561249i	−0.959818	+	0.280625i	$0.909458\pi$
$402$		0			0
$403$	9448.95			1.16796
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	9559.52i		−	1.16424i
$408$		0			0
$409$			305.452i			0.0369282i	0.999830	+	0.0184641i	$0.00587764\pi$
$409$			305.452i			0.0369282i	−0.999830	+	0.0184641i	$0.994122\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5243.24	+	2651.93i	−0.624704	+	0.315964i
$414$		0			0
$415$	−12.7139			−0.00150385
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−10429.5			−1.21603			−0.608014	−	0.793927i	$-0.708033\pi$
$419$	−10429.5			−1.21603			−0.608014	+	0.793927i	$0.708033\pi$
$420$		0			0
$421$	2287.85			0.264853			0.132426	−	0.991193i	$-0.457723\pi$
$421$	2287.85			0.264853			0.132426	+	0.991193i	$0.457723\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	11405.2			1.30173
$426$		0			0
$427$	−6709.55	−	13265.7i	−0.760417	−	1.50345i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			2822.09i			0.315395i	0.987487	+	0.157698i	$0.0504071\pi$
$431$			2822.09i			0.315395i	−0.987487	+	0.157698i	$0.949593\pi$
$432$		0			0
$433$			705.523i			0.0783031i	0.999233	+	0.0391516i	$0.0124655\pi$
$433$			705.523i			0.0783031i	−0.999233	+	0.0391516i	$0.987534\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1089.60			0.119274
$438$		0			0
$439$		−	9350.90i		−	1.01661i	−0.861176	−	0.508307i	$-0.830271\pi$
$439$		−	9350.90i		−	1.01661i	0.861176	−	0.508307i	$-0.169729\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	6496.83i		−	0.696780i	−0.937350	−	0.348390i	$-0.886728\pi$
$443$		−	6496.83i		−	0.696780i	0.937350	−	0.348390i	$-0.113272\pi$
$444$		0			0
$445$	−15.9596			−0.00170013
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3587.89i		−	0.377112i	−0.982062	−	0.188556i	$-0.939619\pi$
$449$		−	3587.89i		−	0.377112i	0.982062	−	0.188556i	$-0.0603807\pi$
$450$		0			0
$451$			10947.8i			1.14305i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.73773	−	5.41287i	−0.000282081	−	0.000557713i
$456$		0			0
$457$	1237.42			0.126661			0.0633307	−	0.997993i	$-0.479828\pi$
$457$	1237.42			0.126661			0.0633307	+	0.997993i	$0.479828\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	947.776			0.0957534			0.0478767	−	0.998853i	$-0.484755\pi$
$461$	947.776			0.0957534			0.0478767	+	0.998853i	$0.484755\pi$
$462$		0			0
$463$	−2636.57			−0.264648			−0.132324	−	0.991207i	$-0.542244\pi$
$463$	−2636.57			−0.264648			−0.132324	+	0.991207i	$0.542244\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	14030.8			1.39029			0.695147	−	0.718868i	$-0.255339\pi$
$467$	14030.8			1.39029			0.695147	+	0.718868i	$0.255339\pi$
$468$		0			0
$469$	−6342.06	+	3207.70i	−0.624412	+	0.315816i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			5756.97i			0.559632i
$474$		0			0
$475$			1582.24i			0.152838i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9035.22			−0.861858			−0.430929	−	0.902386i	$-0.641814\pi$
$479$	−9035.22			−0.861858			−0.430929	+	0.902386i	$0.641814\pi$
$480$		0			0
$481$		−	5044.51i		−	0.478191i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			16.3580i			0.00153150i
$486$		0			0
$487$	2481.39			0.230888			0.115444	−	0.993314i	$-0.463171\pi$
$487$	2481.39			0.230888			0.115444	+	0.993314i	$0.463171\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			20226.4i			1.85907i	0.368734	+	0.929535i	$0.379791\pi$
$491$			20226.4i			1.85907i	−0.368734	+	0.929535i	$0.620209\pi$
$492$		0			0
$493$			11137.7i			1.01748i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−1925.68	−	3807.34i	−0.173800	−	0.343627i
$498$		0			0
$499$	−19290.9			−1.73062			−0.865311	−	0.501236i	$-0.832879\pi$
$499$	−19290.9			−1.73062			−0.865311	+	0.501236i	$0.832879\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−19768.7			−1.75238			−0.876188	−	0.481970i	$-0.839921\pi$
$503$	−19768.7			−1.75238			−0.876188	+	0.481970i	$0.839921\pi$
$504$		0			0
$505$	−4.66566			−0.000411127
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15648.7			1.36270			0.681352	−	0.731956i	$-0.261392\pi$
$509$	15648.7			1.36270			0.681352	+	0.731956i	$0.261392\pi$
$510$		0			0
$511$	7084.74	+	14007.5i	0.613328	+	1.21263i
$512$		0			0
$513$		0			0
$514$		0			0
$515$		−	1.41846i		−	0.000121368i
$516$		0			0
$517$			2635.22i			0.224172i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	10365.3			0.871613			0.435806	−	0.900040i	$-0.356463\pi$
$521$	10365.3			0.871613			0.435806	+	0.900040i	$0.356463\pi$
$522$		0			0
$523$			3227.64i			0.269857i	0.990855	+	0.134928i	$0.0430804\pi$
$523$			3227.64i			0.269857i	−0.990855	+	0.134928i	$0.956920\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	27183.2i		−	2.24690i
$528$		0			0
$529$	4757.20			0.390992
$530$		0			0
$531$		0			0
$532$		0			0
$533$			5777.12i			0.469484i
$534$		0			0
$535$		−	17.7976i		−	0.00143824i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−16605.6	−	12216.4i	−1.32700	−	0.976248i
$540$		0			0
$541$	13814.7			1.09785			0.548927	−	0.835870i	$-0.315037\pi$
$541$	13814.7			1.09785			0.548927	+	0.835870i	$0.315037\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	9.22455			0.000725021
$546$		0			0
$547$	−951.129			−0.0743461			−0.0371731	−	0.999309i	$-0.511835\pi$
$547$	−951.129			−0.0743461			−0.0371731	+	0.999309i	$0.511835\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1545.13			−0.119464
$552$		0			0
$553$	6799.50	−	3439.06i	0.522865	−	0.264455i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			8523.08i			0.648356i	0.945996	+	0.324178i	$0.105088\pi$
$557$			8523.08i			0.648356i	−0.945996	+	0.324178i	$0.894912\pi$
$558$		0			0
$559$			3037.93i			0.229858i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−582.531			−0.0436070			−0.0218035	−	0.999762i	$-0.506941\pi$
$563$	−582.531			−0.0436070			−0.0218035	+	0.999762i	$0.506941\pi$
$564$		0			0
$565$		−	13.0867i		−	0.000974443i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			18065.8i			1.33103i	0.746384	+	0.665515i	$0.231788\pi$
$569$			18065.8i			1.33103i	−0.746384	+	0.665515i	$0.768212\pi$
$570$		0			0
$571$	−11363.7			−0.832848			−0.416424	−	0.909170i	$-0.636717\pi$
$571$	−11363.7			−0.832848			−0.416424	+	0.909170i	$0.636717\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	10760.0i		−	0.780389i
$576$		0			0
$577$		−	19896.3i		−	1.43552i	−0.696292	−	0.717759i	$-0.745168\pi$
$577$		−	19896.3i		−	1.43552i	0.696292	−	0.717759i	$-0.254832\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−20346.7	+	10291.0i	−1.45288	+	0.734841i
$582$		0			0
$583$	1061.26			0.0753908
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−20560.6			−1.44570			−0.722849	−	0.691006i	$-0.757168\pi$
$587$	−20560.6			−1.44570			−0.722849	+	0.691006i	$0.757168\pi$
$588$		0			0
$589$	3771.11			0.263813
$590$		0			0
$591$		0			0
$592$		0			0
$593$	5879.34			0.407142			0.203571	−	0.979060i	$-0.434745\pi$
$593$	5879.34			0.407142			0.203571	+	0.979060i	$0.434745\pi$
$594$		0			0
$595$	−15.5720	+	7.87603i	−0.00107292	+	0.000542665i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	24447.6i		−	1.66761i	−0.552057	−	0.833806i	$-0.686157\pi$
$599$		−	24447.6i		−	1.66761i	0.552057	−	0.833806i	$-0.313843\pi$
$600$		0			0
$601$		−	14478.4i		−	0.982675i	−0.870969	−	0.491337i	$-0.836508\pi$
$601$		−	14478.4i		−	0.982675i	0.870969	−	0.491337i	$-0.163492\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−23.5589			−0.00158315
$606$		0			0
$607$		−	1608.41i		−	0.107551i	−0.998553	−	0.0537755i	$-0.982874\pi$
$607$		−	1608.41i		−	0.107551i	0.998553	−	0.0537755i	$-0.0171255\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			1390.59i			0.0920742i
$612$		0			0
$613$	−16309.3			−1.07459			−0.537297	−	0.843393i	$-0.680555\pi$
$613$	−16309.3			−1.07459			−0.537297	+	0.843393i	$0.680555\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	3904.69i		−	0.254776i	−0.991853	−	0.127388i	$-0.959341\pi$
$617$		−	3904.69i		−	0.254776i	0.991853	−	0.127388i	$-0.0406594\pi$
$618$		0			0
$619$		−	21633.3i		−	1.40471i	−0.711828	−	0.702354i	$-0.752132\pi$
$619$		−	21633.3i		−	1.40471i	0.711828	−	0.702354i	$-0.247868\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−25541.2	+	12918.2i	−1.64251	+	0.830752i
$624$		0			0
$625$	15625.0			0.999997
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−14512.3			−0.919940
$630$		0			0
$631$	5500.79			0.347041			0.173521	−	0.984830i	$-0.444486\pi$
$631$	5500.79			0.347041			0.173521	+	0.984830i	$0.444486\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.07628			0.000254744
$636$		0			0
$637$	−8762.70	−	6446.54i	−0.545041	−	0.400975i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	18979.9i		−	1.16952i	−0.811207	−	0.584759i	$-0.801189\pi$
$641$		−	18979.9i		−	1.16952i	0.811207	−	0.584759i	$-0.198811\pi$
$642$		0			0
$643$			10414.3i			0.638727i	0.947632	+	0.319364i	$0.103469\pi$
$643$			10414.3i			0.638727i	−0.947632	+	0.319364i	$0.896531\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−10013.5			−0.608457			−0.304228	−	0.952599i	$-0.598399\pi$
$647$	−10013.5			−0.608457			−0.304228	+	0.952599i	$0.598399\pi$
$648$		0			0
$649$			19068.2i			1.15330i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	31055.2i		−	1.86108i	−0.366190	−	0.930540i	$-0.619338\pi$
$653$		−	31055.2i		−	1.86108i	0.366190	−	0.930540i	$-0.380662\pi$
$654$		0			0
$655$	−2.64654			−0.000157876
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	17504.5i		−	1.03472i	−0.855768	−	0.517359i	$-0.826915\pi$
$659$		−	17504.5i		−	1.03472i	0.855768	−	0.517359i	$-0.173085\pi$
$660$		0			0
$661$		−	10617.0i		−	0.624742i	−0.949960	−	0.312371i	$-0.898877\pi$
$661$		−	10617.0i		−	0.624742i	0.949960	−	0.312371i	$-0.101123\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1.09264	−	2.16030i	−6.37154e−5	−	0.000125974i
$666$		0			0
$667$	10507.6			0.609981
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−48243.6			−2.77560
$672$		0			0
$673$	−28804.2			−1.64981			−0.824904	−	0.565273i	$-0.808771\pi$
$673$	−28804.2			−1.64981			−0.824904	+	0.565273i	$0.808771\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	16170.4			0.917991			0.458996	−	0.888439i	$-0.348209\pi$
$677$	16170.4			0.917991			0.458996	+	0.888439i	$0.348209\pi$
$678$		0			0
$679$	13240.6	+	26178.6i	0.748349	+	1.47959i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	21560.7i		−	1.20790i	−0.797021	−	0.603951i	$-0.793592\pi$
$683$		−	21560.7i		−	1.20790i	0.797021	−	0.603951i	$-0.206408\pi$
$684$		0			0
$685$		−	18.9040i		−	0.00105443i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	560.021			0.0309653
$690$		0			0
$691$			25964.2i			1.42942i	0.699423	+	0.714708i	$0.253440\pi$
$691$			25964.2i			1.42942i	−0.699423	+	0.714708i	$0.746560\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	22.4313i		−	0.00122427i
$696$		0			0
$697$	16619.9			0.903190
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	12967.1i		−	0.698661i	−0.937000	−	0.349330i	$-0.886409\pi$
$701$		−	12967.1i		−	0.698661i	0.937000	−	0.349330i	$-0.113591\pi$
$702$		0			0
$703$		−	2013.28i		−	0.108012i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7466.73	+	3776.53i	−0.397193	+	0.200893i
$708$		0			0
$709$	−7541.85			−0.399493			−0.199746	−	0.979848i	$-0.564012\pi$
$709$	−7541.85			−0.399493			−0.199746	+	0.979848i	$0.564012\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−25645.5			−1.34703
$714$		0			0
$715$	−19.6851			−0.00102962
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1242.50			−0.0644470			−0.0322235	−	0.999481i	$-0.510259\pi$
$719$	−1242.50			−0.0644470			−0.0322235	+	0.999481i	$0.510259\pi$
$720$		0			0
$721$	−1148.14	−	2270.04i	−0.0593052	−	0.117255i
$722$		0			0
$723$		0			0
$724$		0			0
$725$			15258.5i			0.781635i
$726$		0			0
$727$		−	14004.0i		−	0.714414i	−0.934025	−	0.357207i	$-0.883729\pi$
$727$		−	14004.0i		−	0.714414i	0.934025	−	0.357207i	$-0.116271\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	8739.65			0.442199
$732$		0			0
$733$			31060.7i			1.56515i	0.622558	+	0.782574i	$0.286094\pi$
$733$			31060.7i			1.56515i	−0.622558	+	0.782574i	$0.713906\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			23064.3i			1.15276i
$738$		0			0
$739$	−25552.9			−1.27196			−0.635981	−	0.771705i	$-0.719404\pi$
$739$	−25552.9			−1.27196			−0.635981	+	0.771705i	$0.719404\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			18684.5i			0.922567i	0.887253	+	0.461283i	$0.152611\pi$
$743$			18684.5i			0.922567i	−0.887253	+	0.461283i	$0.847389\pi$
$744$		0			0
$745$			24.7253i			0.00121592i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14405.9	−	28482.5i	−0.702779	−	1.38949i
$750$		0			0
$751$	−2508.63			−0.121893			−0.0609463	−	0.998141i	$-0.519412\pi$
$751$	−2508.63			−0.121893			−0.0609463	+	0.998141i	$0.519412\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	21.3084			0.00102714
$756$		0			0
$757$	20319.8			0.975606			0.487803	−	0.872954i	$-0.337798\pi$
$757$	20319.8			0.975606			0.487803	+	0.872954i	$0.337798\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	26376.4			1.25643			0.628216	−	0.778039i	$-0.283785\pi$
$761$	26376.4			1.25643			0.628216	+	0.778039i	$0.283785\pi$
$762$		0			0
$763$	14762.6	−	7466.64i	0.700448	−	0.354274i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			10062.2i			0.473695i
$768$		0			0
$769$			11786.3i			0.552698i	0.961057	+	0.276349i	$0.0891246\pi$
$769$			11786.3i			0.552698i	−0.961057	+	0.276349i	$0.910875\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−20883.4			−0.971697			−0.485849	−	0.874043i	$-0.661489\pi$
$773$	−20883.4			−0.971697			−0.485849	+	0.874043i	$0.661489\pi$
$774$		0			0
$775$		−	37240.6i		−	1.72609i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			2305.67i			0.106045i
$780$		0			0
$781$	−13846.2			−0.634388
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	14.9030i		−	0.000677595i
$786$		0			0
$787$		−	28611.0i		−	1.29590i	−0.761683	−	0.647950i	$-0.775626\pi$
$787$		−	28611.0i		−	1.29590i	0.761683	−	0.647950i	$-0.224374\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−10592.8	−	20943.4i	−0.476151	−	0.941417i
$792$		0			0
$793$	−25457.9			−1.14002
$794$		0			0
$795$		0			0
$796$		0			0
$797$	2768.30			0.123034			0.0615171	−	0.998106i	$-0.480406\pi$
$797$	2768.30			0.123034			0.0615171	+	0.998106i	$0.480406\pi$
$798$		0			0
$799$	4000.52			0.177132
$800$		0			0
$801$		0			0
$802$		0			0
$803$	50941.4			2.23871
$804$		0			0
$805$	7.43049	+	14.6911i	0.000325329	+	0.000643222i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			23164.2i			1.00669i	0.864087	+	0.503343i	$0.167897\pi$
$809$			23164.2i			1.00669i	−0.864087	+	0.503343i	$0.832103\pi$
$810$		0			0
$811$		−	1221.35i		−	0.0528823i	−0.999650	−	0.0264411i	$-0.991583\pi$
$811$		−	1221.35i		−	0.0528823i	0.999650	−	0.0264411i	$-0.00841746\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16.5549			0.000711524
$816$		0			0
$817$			1212.45i			0.0519195i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	13659.0i		−	0.580635i	−0.956930	−	0.290318i	$-0.906239\pi$
$821$		−	13659.0i		−	0.580635i	0.956930	−	0.290318i	$-0.0937610\pi$
$822$		0			0
$823$	34680.4			1.46887			0.734437	−	0.678677i	$-0.237447\pi$
$823$	34680.4			1.46887			0.734437	+	0.678677i	$0.237447\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			12343.1i			0.519000i	0.965743	+	0.259500i	$0.0835578\pi$
$827$			12343.1i			0.519000i	−0.965743	+	0.259500i	$0.916442\pi$
$828$		0			0
$829$			4449.70i			0.186423i	0.995646	+	0.0932114i	$0.0297133\pi$
$829$			4449.70i			0.186423i	−0.995646	+	0.0932114i	$0.970287\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−18545.7	+	25208.9i	−0.771393	+	1.04855i
$834$		0			0
$835$	−33.5771			−0.00139160
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−35305.1			−1.45276			−0.726382	−	0.687291i	$-0.758800\pi$
$839$	−35305.1			−1.45276			−0.726382	+	0.687291i	$0.758800\pi$
$840$		0			0
$841$	9488.42			0.389045
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.3003			0.000500762
$846$		0			0
$847$	−37702.7	+	19069.3i	−1.52949	+	0.773588i
$848$		0			0
$849$		0			0
$850$		0			0
$851$			13691.3i			0.551507i
$852$		0			0
$853$		−	23176.0i		−	0.930281i	−0.885237	−	0.465141i	$-0.846004\pi$
$853$		−	23176.0i		−	0.930281i	0.885237	−	0.465141i	$-0.153996\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1735.54			0.0691773			0.0345886	−	0.999402i	$-0.488988\pi$
$857$	1735.54			0.0691773			0.0345886	+	0.999402i	$0.488988\pi$
$858$		0			0
$859$		−	25706.3i		−	1.02106i	−0.859861	−	0.510528i	$-0.829450\pi$
$859$		−	25706.3i		−	1.02106i	0.859861	−	0.510528i	$-0.170550\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			1907.12i			0.0752248i	0.999292	+	0.0376124i	$0.0119752\pi$
$863$			1907.12i			0.0752248i	−0.999292	+	0.0376124i	$0.988025\pi$
$864$		0			0
$865$	23.8077			0.000935824
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	24727.9i		−	0.965288i
$870$		0			0
$871$			12170.9i			0.473473i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−42.6669	+	21.5801i	−0.00164846	+	0.000833761i
$876$		0			0
$877$	43537.3			1.67634			0.838170	−	0.545410i	$-0.183626\pi$
$877$	43537.3			1.67634			0.838170	+	0.545410i	$0.183626\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−3570.54			−0.136543			−0.0682715	−	0.997667i	$-0.521748\pi$
$881$	−3570.54			−0.136543			−0.0682715	+	0.997667i	$0.521748\pi$
$882$		0			0
$883$	1843.70			0.0702668			0.0351334	−	0.999383i	$-0.488814\pi$
$883$	1843.70			0.0702668			0.0351334	+	0.999383i	$0.488814\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	46507.5			1.76050			0.880252	−	0.474506i	$-0.157373\pi$
$887$	46507.5			1.76050			0.880252	+	0.474506i	$0.157373\pi$
$888$		0			0
$889$	6523.51	−	3299.47i	0.246110	−	0.124478i
$890$		0			0
$891$		0			0
$892$		0			0
$893$			554.991i			0.0207974i
$894$		0			0
$895$		−	16.1754i		−	0.000604118i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	36367.1			1.34918
$900$		0			0
$901$		−	1611.10i		−	0.0595709i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	0.181655i		−	6.67228e-6i
$906$		0			0
$907$	37899.9			1.38748			0.693740	−	0.720225i	$-0.255962\pi$
$907$	37899.9			1.38748			0.693740	+	0.720225i	$0.255962\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	13460.6i		−	0.489538i	−0.969581	−	0.244769i	$-0.921288\pi$
$911$		−	13460.6i		−	0.489538i	0.969581	−	0.244769i	$-0.0787121\pi$
$912$		0			0
$913$			73995.3i			2.68224i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−4235.41	+	2142.19i	−0.152525	+	0.0771444i
$918$		0			0
$919$	−31176.5			−1.11906			−0.559532	−	0.828809i	$-0.689019\pi$
$919$	−31176.5			−1.11906			−0.559532	+	0.828809i	$0.689019\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−7306.59			−0.260563
$924$		0			0
$925$	−19881.6			−0.706707
$926$		0			0
$927$		0			0
$928$		0			0
$929$	31780.1			1.12236			0.561180	−	0.827694i	$-0.310348\pi$
$929$	31780.1			1.12236			0.561180	+	0.827694i	$0.310348\pi$
$930$		0			0
$931$	−3497.23	−	2572.84i	−0.123112	−	0.0905707i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			56.6309i			0.00198078i
$936$		0			0
$937$		−	8578.03i		−	0.299074i	−0.988756	−	0.149537i	$-0.952222\pi$
$937$		−	8578.03i		−	0.299074i	0.988756	−	0.149537i	$-0.0477783\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−8658.59			−0.299960			−0.149980	−	0.988689i	$-0.547921\pi$
$941$	−8658.59			−0.299960			−0.149980	+	0.988689i	$0.547921\pi$
$942$		0			0
$943$		−	15679.7i		−	0.541465i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			28404.4i			0.974678i	0.873213	+	0.487339i	$0.162032\pi$
$947$			28404.4i			0.974678i	−0.873213	+	0.487339i	$0.837968\pi$
$948$		0			0
$949$	26881.5			0.919506
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	13517.5i		−	0.459470i	−0.973253	−	0.229735i	$-0.926214\pi$
$953$		−	13517.5i		−	0.459470i	0.973253	−	0.229735i	$-0.0737860\pi$
$954$		0			0
$955$		−	37.2365i		−	0.00126172i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−15301.5	−	30253.1i	−0.515235	−	1.01869i
$960$		0			0
$961$	−58968.3			−1.97940
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−24.7225			−0.000824711
$966$		0			0
$967$	−22083.5			−0.734392			−0.367196	−	0.930144i	$-0.619682\pi$
$967$	−22083.5			−0.734392			−0.367196	+	0.930144i	$0.619682\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	32038.4			1.05887			0.529435	−	0.848351i	$-0.322404\pi$
$971$	32038.4			1.05887			0.529435	+	0.848351i	$0.322404\pi$
$972$		0			0
$973$	−18156.6	−	35898.1i	−0.598226	−	1.18278i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	34202.0i		−	1.11998i	−0.828500	−	0.559988i	$-0.810806\pi$
$977$		−	34202.0i		−	1.11998i	0.828500	−	0.559988i	$-0.189194\pi$
$978$		0			0
$979$			92885.9i			3.03233i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	29965.2			0.972271			0.486135	−	0.873883i	$-0.338406\pi$
$983$	29965.2			0.972271			0.486135	+	0.873883i	$0.338406\pi$
$984$		0			0
$985$			41.7022i			0.00134898i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	8245.25i		−	0.265100i
$990$		0			0
$991$	−38593.5			−1.23710			−0.618549	−	0.785747i	$-0.712279\pi$
$991$	−38593.5			−1.23710			−0.618549	+	0.785747i	$0.712279\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	14.1958i		−	0.000452298i
$996$		0			0
$997$		−	7132.00i		−	0.226552i	−0.993564	−	0.113276i	$-0.963865\pi$
$997$		−	7132.00i		−	0.226552i	0.993564	−	0.113276i	$-0.0361345\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.f.e.377.9	yes	16
3.2	odd	2	inner	756.4.f.e.377.7	✓	16
7.6	odd	2	inner	756.4.f.e.377.8	yes	16
21.20	even	2	inner	756.4.f.e.377.10	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.4.f.e.377.7	✓	16	3.2	odd	2	inner
756.4.f.e.377.8	yes	16	7.6	odd	2	inner
756.4.f.e.377.9	yes	16	1.1	even	1	trivial
756.4.f.e.377.10	yes	16	21.20	even	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

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.0103268 q^{5} +(16.5266 - 8.35886i) q^{7} -60.1026i q^{11} -31.7159i q^{13} -91.2417 q^{17} -12.6579i q^{19} +86.0802i q^{23} -125.000 q^{25} -122.068i q^{29} +297.925i q^{31} +(0.170668 - 0.0863205i) q^{35} +159.053 q^{37} -182.152 q^{41} -95.7857 q^{43} -43.8453 q^{47} +(203.259 - 276.288i) q^{49} +17.6574i q^{53} -0.620670i q^{55} -317.260 q^{59} -802.687i q^{61} -0.327524i q^{65} -383.748 q^{67} -230.376i q^{71} +847.573i q^{73} +(-502.390 - 993.294i) q^{77} +411.427 q^{79} -1231.15 q^{83} -0.942237 q^{85} -1545.46 q^{89} +(-265.109 - 524.157i) q^{91} -0.130716i q^{95} +1584.03i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 14 q^{7} + 388 q^{25} - 344 q^{37} - 404 q^{43} - 14 q^{49} + 1772 q^{67} - 1456 q^{79} - 4704 q^{85} + 2436 q^{91}+O(q^{100}) \) 16 * q + 14 * q^7 + 388 * q^25 - 344 * q^37 - 404 * q^43 - 14 * q^49 + 1772 * q^67 - 1456 * q^79 - 4704 * q^85 + 2436 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more