LMFDB - Embedded newform 252.4.k.e.37.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [252,4,Mod(37,252)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(252, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("252.37");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 252 = 2^{2} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	252.k (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$14.8684813214$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{385})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 97x^{2} + 96x + 9216 $ x^4 - x^3 + 97x^2 + 96x + 9216
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			37.2
Root			$-4.65535 + 8.06331i$ of defining polynomial
Character	$\chi$	$=$	252.37
Dual form			252.4.k.e.109.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(9.81071 - 16.9926i) q^{5} +(3.50000 - 18.1865i) q^{7} +O(q^{10})$	$q+(9.81071 - 16.9926i) q^{5} +(3.50000 - 18.1865i) q^{7} +(-9.81071 - 16.9926i) q^{11} -54.0000 q^{13} +(19.6214 + 33.9853i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-98.1071 + 169.926i) q^{23} +(-130.000 - 225.167i) q^{25} +98.1071 q^{29} +(-100.500 - 174.071i) q^{31} +(-274.700 - 237.897i) q^{35} +(101.000 - 174.937i) q^{37} +470.914 q^{41} -244.000 q^{43} +(117.729 - 203.912i) q^{47} +(-318.500 - 127.306i) q^{49} +(-323.753 - 560.757i) q^{53} -385.000 q^{55} +(304.132 + 526.772i) q^{59} +(294.000 - 509.223i) q^{61} +(-529.778 + 917.603i) q^{65} +(151.000 + 261.540i) q^{67} -156.971 q^{71} +(223.000 + 386.247i) q^{73} +(-343.375 + 118.949i) q^{77} +(133.500 - 231.229i) q^{79} +725.992 q^{83} +770.000 q^{85} +(58.8643 - 101.956i) q^{89} +(-189.000 + 982.073i) q^{91} +(19.6214 + 33.9853i) q^{95} +595.000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 14 q^{7}+O(q^{10}) $ 4 * q + 14 * q^7	$ 4 q + 14 q^{7} - 216 q^{13} - 4 q^{19} - 520 q^{25} - 402 q^{31} + 404 q^{37} - 976 q^{43} - 1274 q^{49} - 1540 q^{55} + 1176 q^{61} + 604 q^{67} + 892 q^{73} + 534 q^{79} + 3080 q^{85} - 756 q^{91} + 2380 q^{97}+O(q^{100}) $ 4 * q + 14 * q^7 - 216 * q^13 - 4 * q^19 - 520 * q^25 - 402 * q^31 + 404 * q^37 - 976 * q^43 - 1274 * q^49 - 1540 * q^55 + 1176 * q^61 + 604 * q^67 + 892 * q^73 + 534 * q^79 + 3080 * q^85 - 756 * q^91 + 2380 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/252\mathbb{Z}\right)^\times$.

$n$	$29$	$73$	$127$
$\chi(n)$	$1$	$e\left(\frac{1}{3}\right)$	$1$

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
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	9.81071	−	16.9926i	0.877496	−	1.51987i	0.0234170	−	0.999726i	$-0.492545\pi$
$5$	9.81071	−	16.9926i	0.877496	−	1.51987i	0.854079	−	0.520143i	$-0.174121\pi$
$6$		0			0
$7$	3.50000	−	18.1865i	0.188982	−	0.981981i
$7$	3.50000	−	18.1865i	0.188982	−	0.981981i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.81071	−	16.9926i	−0.268913	−	0.465770i	0.699669	−	0.714468i	$-0.253331\pi$
$11$	−9.81071	−	16.9926i	−0.268913	−	0.465770i	−0.968581	+	0.248697i	$0.919998\pi$
$12$		0			0
$13$	−54.0000			−1.15207			−0.576035	−	0.817425i	$-0.695401\pi$
$13$	−54.0000			−1.15207			−0.576035	+	0.817425i	$0.695401\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	19.6214	+	33.9853i	0.279935	+	0.484861i	0.971368	−	0.237579i	$-0.0763538\pi$
$17$	19.6214	+	33.9853i	0.279935	+	0.484861i	−0.691433	+	0.722440i	$0.743020\pi$
$18$		0			0
$19$	−1.00000	+	1.73205i	−0.0120745	+	0.0209137i	−0.872000	−	0.489507i	$-0.837177\pi$
$19$	−1.00000	+	1.73205i	−0.0120745	+	0.0209137i	0.859925	+	0.510420i	$0.170510\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−98.1071	+	169.926i	−0.889424	+	1.54053i	−0.0488654	+	0.998805i	$0.515561\pi$
$23$	−98.1071	+	169.926i	−0.889424	+	1.54053i	−0.840558	+	0.541721i	$0.817773\pi$
$24$		0			0
$25$	−130.000	−	225.167i	−1.04000	−	1.80133i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	98.1071			0.628208			0.314104	−	0.949389i	$-0.398296\pi$
$29$	98.1071			0.628208			0.314104	+	0.949389i	$0.398296\pi$
$30$		0			0
$31$	−100.500	−	174.071i	−0.582269	−	1.00852i	−0.995210	−	0.0977614i	$-0.968832\pi$
$31$	−100.500	−	174.071i	−0.582269	−	1.00852i	0.412941	−	0.910758i	$-0.364502\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−274.700	−	237.897i	−1.32665	−	1.14891i
$36$		0			0
$37$	101.000	−	174.937i	0.448765	−	0.777283i	−0.549541	−	0.835467i	$-0.685197\pi$
$37$	101.000	−	174.937i	0.448765	−	0.777283i	0.998306	+	0.0581832i	$0.0185308\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	470.914			1.79377			0.896883	−	0.442268i	$-0.145826\pi$
$41$	470.914			1.79377			0.896883	+	0.442268i	$0.145826\pi$
$42$		0			0
$43$	−244.000			−0.865341			−0.432670	−	0.901552i	$-0.642429\pi$
$43$	−244.000			−0.865341			−0.432670	+	0.901552i	$0.642429\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	117.729	−	203.912i	0.365372	−	0.632842i	−0.623464	−	0.781852i	$-0.714275\pi$
$47$	117.729	−	203.912i	0.365372	−	0.632842i	0.988836	+	0.149010i	$0.0476086\pi$
$48$		0			0
$49$	−318.500	−	127.306i	−0.928571	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−323.753	−	560.757i	−0.839074	−	1.45332i	−0.890669	−	0.454652i	$-0.849764\pi$
$53$	−323.753	−	560.757i	−0.839074	−	1.45332i	0.0515949	−	0.998668i	$-0.483570\pi$
$54$		0			0
$55$	−385.000			−0.943880
$56$		0			0
$57$		0			0
$58$		0			0
$59$	304.132	+	526.772i	0.671095	+	1.16237i	0.977594	+	0.210500i	$0.0675091\pi$
$59$	304.132	+	526.772i	0.671095	+	1.16237i	−0.306499	+	0.951871i	$0.599158\pi$
$60$		0			0
$61$	294.000	−	509.223i	0.617096	−	1.06884i	−0.372917	−	0.927865i	$-0.621642\pi$
$61$	294.000	−	509.223i	0.617096	−	1.06884i	0.990013	−	0.140977i	$-0.0450242\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−529.778	+	917.603i	−1.01094	+	1.75099i
$66$		0			0
$67$	151.000	+	261.540i	0.275337	+	0.476898i	0.970220	−	0.242225i	$-0.0778772\pi$
$67$	151.000	+	261.540i	0.275337	+	0.476898i	−0.694883	+	0.719123i	$0.744544\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−156.971			−0.262381			−0.131191	−	0.991357i	$-0.541880\pi$
$71$	−156.971			−0.262381			−0.131191	+	0.991357i	$0.541880\pi$
$72$		0			0
$73$	223.000	+	386.247i	0.357537	+	0.619272i	0.987549	−	0.157314i	$-0.0502834\pi$
$73$	223.000	+	386.247i	0.357537	+	0.619272i	−0.630012	+	0.776585i	$0.716950\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−343.375	+	118.949i	−0.508197	+	0.176045i
$78$		0			0
$79$	133.500	−	231.229i	0.190126	−	0.329307i	−0.755166	−	0.655533i	$-0.772444\pi$
$79$	133.500	−	231.229i	0.190126	−	0.329307i	0.945292	+	0.326226i	$0.105777\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	725.992			0.960097			0.480048	−	0.877242i	$-0.340619\pi$
$83$	725.992			0.960097			0.480048	+	0.877242i	$0.340619\pi$
$84$		0			0
$85$	770.000			0.982567
$86$		0			0
$87$		0			0
$88$		0			0
$89$	58.8643	−	101.956i	0.0701078	−	0.121430i	−0.828841	−	0.559485i	$-0.810999\pi$
$89$	58.8643	−	101.956i	0.0701078	−	0.121430i	0.898948	+	0.438055i	$0.144332\pi$
$90$		0			0
$91$	−189.000	+	982.073i	−0.217721	+	1.13131i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	19.6214	+	33.9853i	0.0211907	+	0.0367033i
$96$		0			0
$97$	595.000			0.622815			0.311408	−	0.950276i	$-0.399200\pi$
$97$	595.000			0.622815			0.311408	+	0.950276i	$0.399200\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−608.264	−	1053.54i	−0.599253	−	1.03794i	−0.992932	−	0.118688i	$-0.962131\pi$
$101$	−608.264	−	1053.54i	−0.599253	−	1.03794i	0.393679	−	0.919248i	$-0.371202\pi$
$102$		0			0
$103$	−832.000	+	1441.07i	−0.795916	+	1.37857i	0.126339	+	0.991987i	$0.459677\pi$
$103$	−832.000	+	1441.07i	−0.795916	+	1.37857i	−0.922256	+	0.386581i	$0.873656\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	794.667	−	1376.40i	0.717976	−	1.24357i	−0.243825	−	0.969819i	$-0.578402\pi$
$107$	794.667	−	1376.40i	0.717976	−	1.24357i	0.961801	−	0.273751i	$-0.0882644\pi$
$108$		0			0
$109$	−179.000	−	310.037i	−0.157294	−	0.272442i	0.776598	−	0.629997i	$-0.216944\pi$
$109$	−179.000	−	310.037i	−0.157294	−	0.272442i	−0.933892	+	0.357555i	$0.883610\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1098.80			0.914746			0.457373	−	0.889275i	$-0.348790\pi$
$113$	1098.80			0.914746			0.457373	+	0.889275i	$0.348790\pi$
$114$		0			0
$115$	1925.00	+	3334.20i	1.56093	+	2.70361i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	686.750	−	237.897i	0.529027	−	0.183260i
$120$		0			0
$121$	473.000	−	819.260i	0.355372	−	0.615522i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2648.89			−1.89539
$126$		0			0
$127$	2127.00			1.48615			0.743074	−	0.669210i	$-0.233367\pi$
$127$	2127.00			1.48615			0.743074	+	0.669210i	$0.233367\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−755.425	+	1308.43i	−0.503830	+	0.872659i	0.496160	+	0.868231i	$0.334743\pi$
$131$	−755.425	+	1308.43i	−0.503830	+	0.872659i	−0.999990	+	0.00442833i	$0.998590\pi$
$132$		0			0
$133$	28.0000	+	24.2487i	0.0182549	+	0.0158092i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1275.39	+	2209.04i	0.795358	+	1.37760i	0.922611	+	0.385731i	$0.126051\pi$
$137$	1275.39	+	2209.04i	0.795358	+	1.37760i	−0.127253	+	0.991870i	$0.540616\pi$
$138$		0			0
$139$	−2438.00			−1.48769			−0.743843	−	0.668354i	$-0.766999\pi$
$139$	−2438.00			−1.48769			−0.743843	+	0.668354i	$0.766999\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	529.778	+	917.603i	0.309806	+	0.536600i
$144$		0			0
$145$	962.500	−	1667.10i	0.551250	−	0.954793i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	765.235	−	1325.43i	0.420742	−	0.728746i	−0.575270	−	0.817963i	$-0.695103\pi$
$149$	765.235	−	1325.43i	0.420742	−	0.728746i	0.996012	+	0.0892172i	$0.0284365\pi$
$150$		0			0
$151$	473.500	+	820.126i	0.255185	+	0.441993i	0.964946	−	0.262450i	$-0.0845304\pi$
$151$	473.500	+	820.126i	0.255185	+	0.441993i	−0.709761	+	0.704443i	$0.751197\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−3943.90			−2.04376
$156$		0			0
$157$	770.000	+	1333.68i	0.391418	+	0.677957i	0.992637	−	0.121128i	$-0.0386512\pi$
$157$	770.000	+	1333.68i	0.391418	+	0.677957i	−0.601218	+	0.799085i	$0.705318\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2747.00	+	2378.97i	1.34468	+	1.16453i
$162$		0			0
$163$	545.000	−	943.968i	0.261888	−	0.453603i	−0.704856	−	0.709351i	$-0.748988\pi$
$163$	545.000	−	943.968i	0.261888	−	0.453603i	0.966744	+	0.255748i	$0.0823217\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−2079.87			−0.963744			−0.481872	−	0.876242i	$-0.660043\pi$
$167$	−2079.87			−0.963744			−0.481872	+	0.876242i	$0.660043\pi$
$168$		0			0
$169$	719.000			0.327264
$170$		0			0
$171$		0			0
$172$		0			0
$173$	922.207	−	1597.31i	0.405284	−	0.701972i	−0.589071	−	0.808081i	$-0.700506\pi$
$173$	922.207	−	1597.31i	0.405284	−	0.701972i	0.994354	+	0.106110i	$0.0338394\pi$
$174$		0			0
$175$	−4550.00	+	1576.17i	−1.96542	+	0.680840i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	588.643	+	1019.56i	0.245794	+	0.425728i	0.962355	−	0.271797i	$-0.0876178\pi$
$179$	588.643	+	1019.56i	0.245794	+	0.425728i	−0.716560	+	0.697525i	$0.754285\pi$
$180$		0			0
$181$	−2004.00			−0.822962			−0.411481	−	0.911418i	$-0.634988\pi$
$181$	−2004.00			−0.822962			−0.411481	+	0.911418i	$0.634988\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1981.76	−	3432.51i	−0.787579	−	1.36413i
$186$		0			0
$187$	385.000	−	666.840i	0.150556	−	0.260771i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2236.84	−	3874.32i	0.847394	−	1.46773i	−0.0361326	−	0.999347i	$-0.511504\pi$
$191$	2236.84	−	3874.32i	0.847394	−	1.46773i	0.883526	−	0.468382i	$-0.155163\pi$
$192$		0			0
$193$	−2324.50	−	4026.15i	−0.866949	−	1.50160i	−0.865099	−	0.501602i	$-0.832744\pi$
$193$	−2324.50	−	4026.15i	−0.866949	−	1.50160i	−0.00185031	−	0.999998i	$-0.500589\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	4198.98			1.51860			0.759302	−	0.650738i	$-0.225540\pi$
$197$	4198.98			1.51860			0.759302	+	0.650738i	$0.225540\pi$
$198$		0			0
$199$	308.000	+	533.472i	0.109716	+	0.190034i	0.915655	−	0.401964i	$-0.131672\pi$
$199$	308.000	+	533.472i	0.109716	+	0.190034i	−0.805939	+	0.591999i	$0.798339\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	343.375	−	1784.23i	0.118720	−	0.616888i
$204$		0			0
$205$	4620.00	−	8002.07i	1.57402	−	2.72629i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	39.2428			0.0129880
$210$		0			0
$211$	−3208.00			−1.04667			−0.523336	−	0.852126i	$-0.675313\pi$
$211$	−3208.00			−1.04667			−0.523336	+	0.852126i	$0.675313\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2393.81	+	4146.21i	−0.759333	+	1.31520i
$216$		0			0
$217$	−3517.50	+	1218.50i	−1.10038	+	0.381184i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1059.56	−	1835.21i	−0.322504	−	0.558594i
$222$		0			0
$223$	−2065.00			−0.620101			−0.310051	−	0.950720i	$-0.600346\pi$
$223$	−2065.00			−0.620101			−0.310051	+	0.950720i	$0.600346\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	912.396	+	1580.32i	0.266775	+	0.462067i	0.968027	−	0.250846i	$-0.0807087\pi$
$227$	912.396	+	1580.32i	0.266775	+	0.462067i	−0.701252	+	0.712913i	$0.747375\pi$
$228$		0			0
$229$	1954.00	−	3384.43i	0.563860	−	0.976634i	−0.433295	−	0.901252i	$-0.642649\pi$
$229$	1954.00	−	3384.43i	0.563860	−	0.976634i	0.997155	−	0.0753820i	$-0.0240176\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2217.22	+	3840.34i	−0.623412	+	1.07978i	0.365434	+	0.930837i	$0.380921\pi$
$233$	−2217.22	+	3840.34i	−0.623412	+	1.07978i	−0.988846	+	0.148943i	$0.952413\pi$
$234$		0			0
$235$	−2310.00	−	4001.04i	−0.641225	−	1.11063i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1530.47			−0.414217			−0.207109	−	0.978318i	$-0.566405\pi$
$239$	−1530.47			−0.414217			−0.207109	+	0.978318i	$0.566405\pi$
$240$		0			0
$241$	1956.50	+	3388.76i	0.522943	+	0.905764i	0.999644	+	0.0266980i	$0.00849925\pi$
$241$	1956.50	+	3388.76i	0.522943	+	0.905764i	−0.476701	+	0.879066i	$0.658167\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−5287.97	+	4163.20i	−1.37892	+	1.08562i
$246$		0			0
$247$	54.0000	−	93.5307i	0.0139107	−	0.0240940i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	4650.28			1.16941			0.584707	−	0.811245i	$-0.301210\pi$
$251$	4650.28			1.16941			0.584707	+	0.811245i	$0.301210\pi$
$252$		0			0
$253$	3850.00			0.956709
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1000.69	−	1733.25i	0.242885	−	0.420689i	−0.718650	−	0.695372i	$-0.755240\pi$
$257$	1000.69	−	1733.25i	0.242885	−	0.420689i	0.961535	+	0.274683i	$0.0885729\pi$
$258$		0			0
$259$	−2828.00	−	2449.12i	−0.678469	−	0.587571i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	235.457	+	407.823i	0.0552049	+	0.0956178i	0.892307	−	0.451429i	$-0.149085\pi$
$263$	235.457	+	407.823i	0.0552049	+	0.0956178i	−0.837102	+	0.547046i	$0.815752\pi$
$264$		0			0
$265$	−12705.0			−2.94514
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1010.50	+	1750.24i	0.229039	+	0.396707i	0.957524	−	0.288355i	$-0.0931084\pi$
$269$	1010.50	+	1750.24i	0.229039	+	0.396707i	−0.728485	+	0.685062i	$0.759775\pi$
$270$		0			0
$271$	−598.500	+	1036.63i	−0.134156	+	0.232365i	−0.925275	−	0.379298i	$-0.876166\pi$
$271$	−598.500	+	1036.63i	−0.134156	+	0.232365i	0.791119	+	0.611663i	$0.209499\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2550.78	+	4418.09i	−0.559338	+	0.968803i
$276$		0			0
$277$	−766.000	−	1326.75i	−0.166153	−	0.287786i	0.770911	−	0.636943i	$-0.219801\pi$
$277$	−766.000	−	1326.75i	−0.166153	−	0.287786i	−0.937064	+	0.349157i	$0.886468\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−510.157			−0.108304			−0.0541520	−	0.998533i	$-0.517246\pi$
$281$	−510.157			−0.108304			−0.0541520	+	0.998533i	$0.517246\pi$
$282$		0			0
$283$	3142.00	+	5442.10i	0.659974	+	1.14311i	0.980622	+	0.195909i	$0.0627659\pi$
$283$	3142.00	+	5442.10i	0.659974	+	1.14311i	−0.320648	+	0.947198i	$0.603901\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1648.20	−	8564.29i	0.338990	−	1.76144i
$288$		0			0
$289$	1686.50	−	2921.10i	0.343273	−	0.594566i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3708.45			0.739419			0.369710	−	0.929147i	$-0.379457\pi$
$293$	3708.45			0.739419			0.369710	+	0.929147i	$0.379457\pi$
$294$		0			0
$295$	11935.0			2.35553
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5297.78	−	9176.03i	1.02468	−	1.77479i
$300$		0			0
$301$	−854.000	+	4437.51i	−0.163534	+	0.849748i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5768.70	−	9991.68i	−1.08300	−	1.87581i
$306$		0			0
$307$	1762.00			0.327566			0.163783	−	0.986496i	$-0.447630\pi$
$307$	1762.00			0.327566			0.163783	+	0.986496i	$0.447630\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	3335.64	+	5777.50i	0.608189	+	1.05341i	0.991539	+	0.129811i	$0.0414371\pi$
$311$	3335.64	+	5777.50i	0.608189	+	1.05341i	−0.383350	+	0.923603i	$0.625230\pi$
$312$		0			0
$313$	−660.500	+	1144.02i	−0.119277	+	0.206594i	−0.919481	−	0.393134i	$-0.871391\pi$
$313$	−660.500	+	1144.02i	−0.119277	+	0.206594i	0.800204	+	0.599727i	$0.204724\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3149.24	+	5454.64i	−0.557977	+	0.966445i	0.439688	+	0.898151i	$0.355089\pi$
$317$	−3149.24	+	5454.64i	−0.557977	+	0.966445i	−0.997665	+	0.0682944i	$0.978244\pi$
$318$		0			0
$319$	−962.500	−	1667.10i	−0.168933	−	0.292601i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−78.4857			−0.0135203
$324$		0			0
$325$	7020.00	+	12159.0i	1.19815	+	2.07526i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3296.40	−	2854.76i	−0.552390	−	0.478384i
$330$		0			0
$331$	4868.00	−	8431.62i	0.808367	−	1.40013i	−0.105627	−	0.994406i	$-0.533685\pi$
$331$	4868.00	−	8431.62i	0.808367	−	1.40013i	0.913994	−	0.405727i	$-0.132982\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5925.67			0.966429
$336$		0			0
$337$	−6747.00			−1.09060			−0.545300	−	0.838241i	$-0.683584\pi$
$337$	−6747.00			−1.09060			−0.545300	+	0.838241i	$0.683584\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1971.95	+	3415.52i	−0.313159	+	0.542407i
$342$		0			0
$343$	−3430.00	+	5346.84i	−0.539949	+	0.841698i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1059.56	+	1835.21i	0.163919	+	0.283916i	0.936271	−	0.351279i	$-0.114253\pi$
$347$	1059.56	+	1835.21i	0.163919	+	0.283916i	−0.772352	+	0.635195i	$0.780920\pi$
$348$		0			0
$349$	9646.00			1.47948			0.739740	−	0.672893i	$-0.234948\pi$
$349$	9646.00			1.47948			0.739740	+	0.672893i	$0.234948\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−235.457	−	407.823i	−0.0355017	−	0.0614908i	0.847729	−	0.530430i	$-0.177970\pi$
$353$	−235.457	−	407.823i	−0.0355017	−	0.0614908i	−0.883230	+	0.468939i	$0.844636\pi$
$354$		0			0
$355$	−1540.00	+	2667.36i	−0.230239	+	0.398785i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3198.29	+	5539.60i	−0.470193	+	0.814398i	−0.999419	−	0.0340826i	$-0.989149\pi$
$359$	−3198.29	+	5539.60i	−0.470193	+	0.814398i	0.529226	+	0.848481i	$0.322482\pi$
$360$		0			0
$361$	3427.50	+	5936.60i	0.499708	+	0.865520i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	8751.15			1.25495
$366$		0			0
$367$	3378.50	+	5851.73i	0.480535	+	0.832311i	0.999751	−	0.0223325i	$-0.00710925\pi$
$367$	3378.50	+	5851.73i	0.480535	+	0.832311i	−0.519216	+	0.854643i	$0.673776\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−11331.4	+	3925.30i	−1.58570	+	0.549303i
$372$		0			0
$373$	−4322.00	+	7485.92i	−0.599959	+	1.03916i	0.392868	+	0.919595i	$0.371483\pi$
$373$	−4322.00	+	7485.92i	−0.599959	+	1.03916i	−0.992826	+	0.119564i	$0.961850\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−5297.78			−0.723739
$378$		0			0
$379$	7372.00			0.999140			0.499570	−	0.866273i	$-0.333491\pi$
$379$	7372.00			0.999140			0.499570	+	0.866273i	$0.333491\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3747.69	+	6491.19i	−0.499995	+	0.866017i	−1.00000		5.89356e-6i	$-0.999998\pi$
$383$	−3747.69	+	6491.19i	−0.499995	+	0.866017i	0.500005	+	0.866022i	$0.333331\pi$
$384$		0			0
$385$	−1347.50	+	7001.82i	−0.178377	+	0.926872i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6808.63	+	11792.9i	0.887433	+	1.53708i	0.842900	+	0.538071i	$0.180847\pi$
$389$	6808.63	+	11792.9i	0.887433	+	1.53708i	0.0445330	+	0.999008i	$0.485820\pi$
$390$		0			0
$391$	−7700.00			−0.995923
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2619.46	−	4537.04i	−0.333669	−	0.577932i
$396$		0			0
$397$	2316.00	−	4011.43i	0.292788	−	0.507123i	−0.681680	−	0.731650i	$-0.738750\pi$
$397$	2316.00	−	4011.43i	0.292788	−	0.507123i	0.974468	+	0.224527i	$0.0720838\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2590.03	−	4486.06i	0.322543	−	0.558661i	−0.658469	−	0.752608i	$-0.728796\pi$
$401$	2590.03	−	4486.06i	0.322543	−	0.558661i	0.981012	+	0.193947i	$0.0621289\pi$
$402$		0			0
$403$	5427.00	+	9399.84i	0.670814	+	1.16188i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3963.53			−0.482714
$408$		0			0
$409$	2286.50	+	3960.33i	0.276431	+	0.478792i	0.970495	−	0.241121i	$-0.0775150\pi$
$409$	2286.50	+	3960.33i	0.276431	+	0.478792i	−0.694064	+	0.719913i	$0.744182\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	10644.6	−	3687.40i	1.26825	−	0.439335i
$414$		0			0
$415$	7122.50	−	12336.5i	0.842481	−	1.45922i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2590.03			−0.301984			−0.150992	−	0.988535i	$-0.548247\pi$
$419$	−2590.03			−0.301984			−0.150992	+	0.988535i	$0.548247\pi$
$420$		0			0
$421$	6096.00			0.705703			0.352851	−	0.935679i	$-0.385212\pi$
$421$	6096.00			0.705703			0.352851	+	0.935679i	$0.385212\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	5101.57	−	8836.18i	0.582265	−	1.00851i
$426$		0			0
$427$	−8232.00	−	7129.12i	−0.932961	−	0.807968i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8476.45	−	14681.6i	−0.947323	−	1.64081i	−0.751031	−	0.660267i	$-0.770443\pi$
$431$	−8476.45	−	14681.6i	−0.947323	−	1.64081i	−0.196292	−	0.980546i	$-0.562890\pi$
$432$		0			0
$433$	−5206.00			−0.577793			−0.288897	−	0.957360i	$-0.593288\pi$
$433$	−5206.00			−0.577793			−0.288897	+	0.957360i	$0.593288\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−196.214	−	339.853i	−0.0214787	−	0.0372022i
$438$		0			0
$439$	6205.50	−	10748.2i	0.674652	−	1.16853i	−0.301918	−	0.953334i	$-0.597627\pi$
$439$	6205.50	−	10748.2i	0.674652	−	1.16853i	0.976570	−	0.215198i	$-0.0690397\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6700.71	−	11606.0i	0.718647	−	1.24473i	−0.242889	−	0.970054i	$-0.578095\pi$
$443$	6700.71	−	11606.0i	0.718647	−	1.24473i	0.961536	−	0.274679i	$-0.0885715\pi$
$444$		0			0
$445$	−1155.00	−	2000.52i	−0.123039	−	0.213109i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−4669.90			−0.490838			−0.245419	−	0.969417i	$-0.578925\pi$
$449$	−4669.90			−0.490838			−0.245419	+	0.969417i	$0.578925\pi$
$450$		0			0
$451$	−4620.00	−	8002.07i	−0.482367	−	0.835483i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14833.8	+	12846.4i	1.52839	+	1.32363i
$456$		0			0
$457$	−7413.50	+	12840.6i	−0.758838	+	1.31435i	0.184606	+	0.982813i	$0.440899\pi$
$457$	−7413.50	+	12840.6i	−0.758838	+	1.31435i	−0.943444	+	0.331533i	$0.892434\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	1451.98			0.146693			0.0733467	−	0.997307i	$-0.476632\pi$
$461$	1451.98			0.146693			0.0733467	+	0.997307i	$0.476632\pi$
$462$		0			0
$463$	−2276.00			−0.228455			−0.114228	−	0.993455i	$-0.536439\pi$
$463$	−2276.00			−0.228455			−0.114228	+	0.993455i	$0.536439\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2786.24	+	4825.91i	−0.276085	+	0.478194i	−0.970408	−	0.241470i	$-0.922371\pi$
$467$	−2786.24	+	4825.91i	−0.276085	+	0.478194i	0.694323	+	0.719664i	$0.255704\pi$
$468$		0			0
$469$	5285.00	−	1830.78i	0.520338	−	0.180250i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	2393.81	+	4146.21i	0.232701	+	0.403050i
$474$		0			0
$475$	520.000			0.0502300
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−5572.48	−	9651.82i	−0.531552	−	0.920674i	−0.999322	−	0.0368242i	$-0.988276\pi$
$479$	−5572.48	−	9651.82i	−0.531552	−	0.920674i	0.467770	−	0.883850i	$-0.345058\pi$
$480$		0			0
$481$	−5454.00	+	9446.61i	−0.517008	+	0.895485i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5837.37	−	10110.6i	0.546518	−	0.946598i
$486$		0			0
$487$	−9233.50	−	15992.9i	−0.859158	−	1.48810i	−0.872733	−	0.488197i	$-0.837655\pi$
$487$	−9233.50	−	15992.9i	−0.859158	−	1.48810i	0.0135757	−	0.999908i	$-0.495679\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3826.18			−0.351676			−0.175838	−	0.984419i	$-0.556263\pi$
$491$	−3826.18			−0.351676			−0.175838	+	0.984419i	$0.556263\pi$
$492$		0			0
$493$	1925.00	+	3334.20i	0.175857	+	0.304594i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−549.400	+	2854.76i	−0.0495854	+	0.257653i
$498$		0			0
$499$	6961.00	−	12056.8i	0.624483	−	1.08164i	−0.364157	−	0.931337i	$-0.618643\pi$
$499$	6961.00	−	12056.8i	0.624483	−	1.08164i	0.988641	−	0.150299i	$-0.0480237\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−20837.9			−1.84715			−0.923577	−	0.383414i	$-0.874748\pi$
$503$	−20837.9			−1.84715			−0.923577	+	0.383414i	$0.874748\pi$
$504$		0			0
$505$	−23870.0			−2.10337
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7034.28	+	12183.7i	−0.612552	+	1.06097i	0.378257	+	0.925701i	$0.376524\pi$
$509$	−7034.28	+	12183.7i	−0.612552	+	1.06097i	−0.990809	+	0.135270i	$0.956810\pi$
$510$		0			0
$511$	7805.00	−	2703.73i	0.675681	−	0.234063i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	16325.0	+	28275.8i	1.39683	+	2.41938i
$516$		0			0
$517$	−4620.00			−0.393012
$518$		0			0
$519$		0			0
$520$		0			0
$521$	6357.34	+	11011.2i	0.534587	+	0.925933i	0.999183	+	0.0404098i	$0.0128663\pi$
$521$	6357.34	+	11011.2i	0.534587	+	0.925933i	−0.464596	+	0.885523i	$0.653800\pi$
$522$		0			0
$523$	−8233.00	+	14260.0i	−0.688344	+	1.19225i	0.284029	+	0.958816i	$0.408329\pi$
$523$	−8233.00	+	14260.0i	−0.688344	+	1.19225i	−0.972373	+	0.233431i	$0.925005\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	3943.90	−	6831.04i	0.325995	−	0.564639i
$528$		0			0
$529$	−13166.5	−	22805.0i	−1.08215	−	1.87434i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−25429.4			−2.06654
$534$		0			0
$535$	−15592.5	−	27007.0i	−1.26004	−	2.18246i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	961.449	+	6661.12i	0.0768322	+	0.532309i
$540$		0			0
$541$	−192.000	+	332.554i	−0.0152583	+	0.0264281i	−0.873554	−	0.486728i	$-0.838190\pi$
$541$	−192.000	+	332.554i	−0.0152583	+	0.0264281i	0.858295	+	0.513156i	$0.171524\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7024.47			−0.552101
$546$		0			0
$547$	10110.0			0.790260			0.395130	−	0.918625i	$-0.370699\pi$
$547$	10110.0			0.790260			0.395130	+	0.918625i	$0.370699\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−98.1071	+	169.926i	−0.00758530	+	0.0131381i
$552$		0			0
$553$	−3738.00	−	3237.20i	−0.287443	−	0.248933i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−814.289	−	1410.39i	−0.0619435	−	0.107289i	0.833391	−	0.552684i	$-0.186397\pi$
$557$	−814.289	−	1410.39i	−0.0619435	−	0.107289i	−0.895334	+	0.445395i	$0.853063\pi$
$558$		0			0
$559$	13176.0			0.996933
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6838.06	−	11843.9i	−0.511883	−	0.886607i	−0.999905	−	0.0137759i	$-0.995615\pi$
$563$	−6838.06	−	11843.9i	−0.511883	−	0.886607i	0.488022	−	0.872831i	$-0.337718\pi$
$564$		0			0
$565$	10780.0	−	18671.5i	0.802687	−	1.39029i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	7475.76	−	12948.4i	0.550791	−	0.953998i	−0.447427	−	0.894321i	$-0.647659\pi$
$569$	7475.76	−	12948.4i	0.550791	−	0.953998i	0.998218	−	0.0596776i	$-0.0190073\pi$
$570$		0			0
$571$	5268.00	+	9124.44i	0.386093	+	0.668732i	0.991920	−	0.126864i	$-0.0404911\pi$
$571$	5268.00	+	9124.44i	0.386093	+	0.668732i	−0.605827	+	0.795596i	$0.707158\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	51015.7			3.70000
$576$		0			0
$577$	−1143.50	−	1980.60i	−0.0825035	−	0.142900i	0.821821	−	0.569746i	$-0.192958\pi$
$577$	−1143.50	−	1980.60i	−0.0825035	−	0.142900i	−0.904325	+	0.426845i	$0.859625\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2540.97	−	13203.3i	0.181441	−	0.942796i
$582$		0			0
$583$	−6352.50	+	11002.9i	−0.451276	+	0.781632i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3904.66			0.274553			0.137277	−	0.990533i	$-0.456165\pi$
$587$	3904.66			0.274553			0.137277	+	0.990533i	$0.456165\pi$
$588$		0			0
$589$	402.000			0.0281224
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−7828.95	+	13560.1i	−0.542152	+	0.939035i	0.456628	+	0.889658i	$0.349057\pi$
$593$	−7828.95	+	13560.1i	−0.542152	+	0.939035i	−0.998780	+	0.0493774i	$0.984276\pi$
$594$		0			0
$595$	2695.00	−	14003.6i	0.185688	−	0.964862i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10772.2	−	18657.9i	−0.734789	−	1.27269i	−0.954816	−	0.297198i	$-0.903948\pi$
$599$	−10772.2	−	18657.9i	−0.734789	−	1.27269i	0.220027	−	0.975494i	$-0.429385\pi$
$600$		0			0
$601$	−17089.0			−1.15986			−0.579929	−	0.814667i	$-0.696920\pi$
$601$	−17089.0			−1.15986			−0.579929	+	0.814667i	$0.696920\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−9280.93	−	16075.0i	−0.623675	−	1.08024i
$606$		0			0
$607$	−2473.50	+	4284.23i	−0.165397	+	0.286477i	−0.936796	−	0.349875i	$-0.886224\pi$
$607$	−2473.50	+	4284.23i	−0.165397	+	0.286477i	0.771399	+	0.636352i	$0.219557\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6357.34	+	11011.2i	−0.420934	+	0.729078i
$612$		0			0
$613$	9015.00	+	15614.4i	0.593984	+	1.02881i	0.993689	+	0.112167i	$0.0357792\pi$
$613$	9015.00	+	15614.4i	0.593984	+	1.02881i	−0.399705	+	0.916644i	$0.630887\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−16403.5			−1.07031			−0.535154	−	0.844754i	$-0.679746\pi$
$617$	−16403.5			−1.07031			−0.535154	+	0.844754i	$0.679746\pi$
$618$		0			0
$619$	−13154.0	−	22783.4i	−0.854126	−	1.47939i	−0.877454	−	0.479661i	$-0.840759\pi$
$619$	−13154.0	−	22783.4i	−0.854126	−	1.47939i	0.0233278	−	0.999728i	$-0.492574\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1648.20	−	1427.38i	−0.105993	−	0.0917927i
$624$		0			0
$625$	−9737.50	+	16865.8i	−0.623200	+	1.07941i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	7927.05			0.502500
$630$		0			0
$631$	−10813.0			−0.682185			−0.341092	−	0.940030i	$-0.610797\pi$
$631$	−10813.0			−0.682185			−0.341092	+	0.940030i	$0.610797\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	20867.4	−	36143.4i	1.30409	−	2.25875i
$636$		0			0
$637$	17199.0	+	6874.51i	1.06978	+	0.427595i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6318.10	+	10943.3i	0.389313	+	0.674310i	0.992357	−	0.123398i	$-0.0393790\pi$
$641$	6318.10	+	10943.3i	0.389313	+	0.674310i	−0.603044	+	0.797708i	$0.706046\pi$
$642$		0			0
$643$	11174.0			0.685318			0.342659	−	0.939460i	$-0.388672\pi$
$643$	11174.0			0.685318			0.342659	+	0.939460i	$0.388672\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	9771.47	+	16924.7i	0.593750	+	1.02840i	0.993722	+	0.111877i	$0.0356863\pi$
$647$	9771.47	+	16924.7i	0.593750	+	1.02840i	−0.399972	+	0.916527i	$0.630980\pi$
$648$		0			0
$649$	5967.50	−	10336.0i	0.360932	−	0.625153i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−7328.60	+	12693.5i	−0.439189	+	0.760697i	−0.997627	−	0.0688488i	$-0.978067\pi$
$653$	−7328.60	+	12693.5i	−0.439189	+	0.760697i	0.558438	+	0.829546i	$0.311401\pi$
$654$		0			0
$655$	14822.5	+	25673.3i	0.884218	+	1.53151i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9810.71			−0.579926			−0.289963	−	0.957038i	$-0.593643\pi$
$659$	−9810.71			−0.579926			−0.289963	+	0.957038i	$0.593643\pi$
$660$		0			0
$661$	4012.00	+	6948.99i	0.236080	+	0.408902i	0.959586	−	0.281416i	$-0.0908040\pi$
$661$	4012.00	+	6948.99i	0.236080	+	0.408902i	−0.723506	+	0.690318i	$0.757471\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	686.750	−	237.897i	0.0400466	−	0.0138726i
$666$		0			0
$667$	−9625.00	+	16671.0i	−0.558743	+	0.967771i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−11537.4			−0.663780
$672$		0			0
$673$	−29791.0			−1.70633			−0.853164	−	0.521643i	$-0.825319\pi$
$673$	−29791.0			−1.70633			−0.853164	+	0.521643i	$0.825319\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4444.25	−	7697.67i	0.252299	−	0.436995i	−0.711859	−	0.702322i	$-0.752147\pi$
$677$	4444.25	−	7697.67i	0.252299	−	0.436995i	0.964158	+	0.265327i	$0.0854800\pi$
$678$		0			0
$679$	2082.50	−	10821.0i	0.117701	−	0.611593i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2501.73	−	4333.12i	−0.140155	−	0.242756i	0.787400	−	0.616443i	$-0.211427\pi$
$683$	−2501.73	−	4333.12i	−0.140155	−	0.242756i	−0.927555	+	0.373687i	$0.878094\pi$
$684$		0			0
$685$	50050.0			2.79170
$686$		0			0
$687$		0			0
$688$		0			0
$689$	17482.7	+	30280.9i	0.966672	+	1.67433i
$690$		0			0
$691$	−13588.0	+	23535.1i	−0.748064	+	1.29568i	0.200686	+	0.979656i	$0.435683\pi$
$691$	−13588.0	+	23535.1i	−0.748064	+	1.29568i	−0.948750	+	0.316028i	$0.897651\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−23918.5	+	41428.1i	−1.30544	+	2.26109i
$696$		0			0
$697$	9240.00	+	16004.1i	0.502138	+	0.869728i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−9320.17			−0.502166			−0.251083	−	0.967966i	$-0.580787\pi$
$701$	−9320.17			−0.502166			−0.251083	+	0.967966i	$0.580787\pi$
$702$		0			0
$703$	202.000	+	349.874i	0.0108372	+	0.0187706i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−21289.2	+	7374.81i	−1.13248	+	0.392303i
$708$		0			0
$709$	2717.00	−	4705.98i	0.143920	−	0.249276i	−0.785050	−	0.619433i	$-0.787363\pi$
$709$	2717.00	−	4705.98i	0.143920	−	0.249276i	0.928969	+	0.370157i	$0.120696\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	39439.0			2.07153
$714$		0			0
$715$	20790.0			1.08742
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9006.23	+	15599.2i	−0.467143	+	0.809115i	−0.999295	−	0.0375333i	$-0.988050\pi$
$719$	−9006.23	+	15599.2i	−0.467143	+	0.809115i	0.532153	+	0.846649i	$0.321383\pi$
$720$		0			0
$721$	23296.0	+	20174.9i	1.20331	+	1.04210i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−12753.9	−	22090.4i	−0.653336	−	1.13161i
$726$		0			0
$727$	29353.0			1.49744			0.748722	−	0.662884i	$-0.230668\pi$
$727$	29353.0			1.49744			0.748722	+	0.662884i	$0.230668\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−4787.63	−	8292.41i	−0.242239	−	0.419570i
$732$		0			0
$733$	5718.00	−	9903.87i	0.288130	−	0.499055i	−0.685234	−	0.728323i	$-0.740300\pi$
$733$	5718.00	−	9903.87i	0.288130	−	0.499055i	0.973363	+	0.229268i	$0.0736332\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	2962.83	−	5131.78i	0.148083	−	0.256488i
$738$		0			0
$739$	39.0000	+	67.5500i	0.00194132	+	0.00336247i	0.866994	−	0.498318i	$-0.166049\pi$
$739$	39.0000	+	67.5500i	0.00194132	+	0.00336247i	−0.865053	+	0.501680i	$0.832715\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	17306.1			0.854507			0.427254	−	0.904132i	$-0.359481\pi$
$743$	17306.1			0.854507			0.427254	+	0.904132i	$0.359481\pi$
$744$		0			0
$745$	−15015.0	−	26006.7i	−0.738399	−	1.27894i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−22250.7	−	19269.7i	−1.08548	−	0.940051i
$750$		0			0
$751$	−13195.5	+	22855.3i	−0.641159	+	1.11052i	0.344015	+	0.938964i	$0.388213\pi$
$751$	−13195.5	+	22855.3i	−0.641159	+	1.11052i	−0.985174	+	0.171556i	$0.945120\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18581.5			0.895695
$756$		0			0
$757$	−660.000			−0.0316884			−0.0158442	−	0.999874i	$-0.505044\pi$
$757$	−660.000			−0.0316884			−0.0158442	+	0.999874i	$0.505044\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2570.41	+	4452.07i	−0.122440	+	0.212073i	−0.920730	−	0.390201i	$-0.872405\pi$
$761$	−2570.41	+	4452.07i	−0.122440	+	0.212073i	0.798289	+	0.602274i	$0.205739\pi$
$762$		0			0
$763$	−6265.00	+	2170.26i	−0.297258	+	0.102973i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−16423.1	−	28445.7i	−0.773148	−	1.33913i
$768$		0			0
$769$	8773.00			0.411395			0.205697	−	0.978616i	$-0.434054\pi$
$769$	8773.00			0.411395			0.205697	+	0.978616i	$0.434054\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	5513.62	+	9549.87i	0.256547	+	0.444353i	0.965315	−	0.261089i	$-0.0840817\pi$
$773$	5513.62	+	9549.87i	0.256547	+	0.444353i	−0.708767	+	0.705442i	$0.750748\pi$
$774$		0			0
$775$	−26130.0	+	45258.5i	−1.21112	+	2.09772i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−470.914	+	815.647i	−0.0216589	+	0.0375142i
$780$		0			0
$781$	1540.00	+	2667.36i	0.0705577	+	0.122209i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	30217.0			1.37387
$786$		0			0
$787$	2940.00	+	5092.23i	0.133164	+	0.230646i	0.924894	−	0.380224i	$-0.124153\pi$
$787$	2940.00	+	5092.23i	0.133164	+	0.230646i	−0.791731	+	0.610870i	$0.790820\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	3845.80	−	19983.4i	0.172871	−	0.898263i
$792$		0			0
$793$	−15876.0	+	27498.0i	−0.710937	+	1.23138i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	22898.2			1.01769			0.508843	−	0.860859i	$-0.330073\pi$
$797$	22898.2			1.01769			0.508843	+	0.860859i	$0.330073\pi$
$798$		0			0
$799$	9240.00			0.409121
$800$		0			0
$801$		0			0
$802$		0			0
$803$	4375.58	−	7578.72i	0.192292	−	0.333060i
$804$		0			0
$805$	67375.0	−	23339.4i	2.94988	−	1.02187i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1608.96	−	2786.79i	−0.0699232	−	0.121111i	0.828944	−	0.559331i	$-0.188942\pi$
$809$	−1608.96	−	2786.79i	−0.0699232	−	0.121111i	−0.898867	+	0.438221i	$0.855609\pi$
$810$		0			0
$811$	−6062.00			−0.262473			−0.131237	−	0.991351i	$-0.541895\pi$
$811$	−6062.00			−0.262473			−0.131237	+	0.991351i	$0.541895\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−10693.7	−	18522.0i	−0.459611	−	0.796070i
$816$		0			0
$817$	244.000	−	422.620i	0.0104486	−	0.0180974i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3404.32	−	5896.45i	0.144716	−	0.250655i	−0.784551	−	0.620064i	$-0.787107\pi$
$821$	3404.32	−	5896.45i	0.144716	−	0.250655i	0.929267	+	0.369409i	$0.120440\pi$
$822$		0			0
$823$	16120.0	+	27920.7i	0.682756	+	1.18257i	0.974137	+	0.225960i	$0.0725520\pi$
$823$	16120.0	+	27920.7i	0.682756	+	1.18257i	−0.291381	+	0.956607i	$0.594115\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−28313.7			−1.19052			−0.595262	−	0.803531i	$-0.702952\pi$
$827$	−28313.7			−1.19052			−0.595262	+	0.803531i	$0.702952\pi$
$828$		0			0
$829$	−16001.0	−	27714.5i	−0.670371	−	1.16112i	−0.977799	−	0.209545i	$-0.932802\pi$
$829$	−16001.0	−	27714.5i	−0.670371	−	1.16112i	0.307428	−	0.951571i	$-0.400532\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1922.90	−	13322.2i	−0.0799814	−	0.554127i
$834$		0			0
$835$	−20405.0	+	35342.5i	−0.845682	+	1.46476i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	5376.27			0.221227			0.110613	−	0.993864i	$-0.464718\pi$
$839$	5376.27			0.221227			0.110613	+	0.993864i	$0.464718\pi$
$840$		0			0
$841$	−14764.0			−0.605355
$842$		0			0
$843$		0			0
$844$		0			0
$845$	7053.90	−	12217.7i	0.287173	−	0.497399i
$846$		0			0
$847$	−13244.0	−	11469.6i	−0.537272	−	0.465291i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	19817.6	+	34325.1i	0.798284	+	1.38267i
$852$		0			0
$853$	−19966.0			−0.801434			−0.400717	−	0.916202i	$-0.631239\pi$
$853$	−19966.0			−0.801434			−0.400717	+	0.916202i	$0.631239\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6945.98	+	12030.8i	0.276861	+	0.479538i	0.970603	−	0.240686i	$-0.0773724\pi$
$857$	6945.98	+	12030.8i	0.276861	+	0.479538i	−0.693742	+	0.720224i	$0.744039\pi$
$858$		0			0
$859$	16688.0	−	28904.5i	0.662849	−	1.14809i	−0.317015	−	0.948421i	$-0.602680\pi$
$859$	16688.0	−	28904.5i	0.662849	−	1.14809i	0.979864	−	0.199667i	$-0.0639862\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4493.30	+	7782.63i	−0.177235	+	0.306980i	−0.940932	−	0.338594i	$-0.890049\pi$
$863$	−4493.30	+	7782.63i	−0.177235	+	0.306980i	0.763697	+	0.645574i	$0.223382\pi$
$864$		0			0
$865$	−18095.0	−	31341.5i	−0.711270	−	1.23196i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5238.92			−0.204509
$870$		0			0
$871$	−8154.00	−	14123.1i	−0.317208	−	0.549420i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−9271.12	+	48174.1i	−0.358195	+	1.86124i
$876$		0			0
$877$	2260.00	−	3914.43i	0.0870180	−	0.150720i	−0.819231	−	0.573463i	$-0.805600\pi$
$877$	2260.00	−	3914.43i	0.0870180	−	0.150720i	0.906249	+	0.422744i	$0.138933\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−37555.4			−1.43618			−0.718089	−	0.695951i	$-0.754983\pi$
$881$	−37555.4			−1.43618			−0.718089	+	0.695951i	$0.754983\pi$
$882$		0			0
$883$	−7636.00			−0.291021			−0.145511	−	0.989357i	$-0.546483\pi$
$883$	−7636.00			−0.291021			−0.145511	+	0.989357i	$0.546483\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−12184.9	+	21104.9i	−0.461250	+	0.798909i	−0.999024	−	0.0441807i	$-0.985932\pi$
$887$	−12184.9	+	21104.9i	−0.461250	+	0.798909i	0.537773	+	0.843089i	$0.319266\pi$
$888$		0			0
$889$	7444.50	−	38682.8i	0.280855	−	1.45937i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	235.457	+	407.823i	0.00882337	+	0.0152825i
$894$		0			0
$895$	23100.0			0.862735
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9859.76	−	17077.6i	−0.365786	−	0.633560i
$900$		0			0
$901$	12705.0	−	22005.7i	0.469772	−	0.813670i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−19660.7	+	34053.3i	−0.722146	+	1.25079i
$906$		0			0
$907$	6051.00	+	10480.6i	0.221522	+	0.383687i	0.955270	−	0.295734i	$-0.0955643\pi$
$907$	6051.00	+	10480.6i	0.221522	+	0.383687i	−0.733749	+	0.679421i	$0.762231\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5180.05			0.188390			0.0941948	−	0.995554i	$-0.469972\pi$
$911$	5180.05			0.188390			0.0941948	+	0.995554i	$0.469972\pi$
$912$		0			0
$913$	−7122.50	−	12336.5i	−0.258182	−	0.447185i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	21151.9	+	18318.1i	0.761720	+	0.659669i
$918$		0			0
$919$	16894.0	−	29261.3i	0.606400	−	1.05032i	−0.385429	−	0.922738i	$-0.625946\pi$
$919$	16894.0	−	29261.3i	0.606400	−	1.05032i	0.991829	−	0.127578i	$-0.0407203\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8476.45			0.302281
$924$		0			0
$925$	−52520.0			−1.86686
$926$		0			0
$927$		0			0
$928$		0			0
$929$	706.371	−	1223.47i	0.0249465	−	0.0432086i	−0.853283	−	0.521449i	$-0.825392\pi$
$929$	706.371	−	1223.47i	0.0249465	−	0.0432086i	0.878229	+	0.478240i	$0.158725\pi$
$930$		0			0
$931$	539.000	−	424.352i	0.0189742	−	0.0149383i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−7554.25	−	13084.3i	−0.264225	−	0.457651i
$936$		0			0
$937$	17603.0			0.613730			0.306865	−	0.951753i	$-0.400720\pi$
$937$	17603.0			0.613730			0.306865	+	0.951753i	$0.400720\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	9545.82	+	16533.8i	0.330696	+	0.572782i	0.982648	−	0.185478i	$-0.0593832\pi$
$941$	9545.82	+	16533.8i	0.330696	+	0.572782i	−0.651953	+	0.758260i	$0.726050\pi$
$942$		0			0
$943$	−46200.0	+	80020.7i	−1.59542	+	2.76334i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	27823.2	−	48191.1i	0.954732	−	1.65364i	0.219753	−	0.975556i	$-0.429475\pi$
$947$	27823.2	−	48191.1i	0.954732	−	1.65364i	0.734980	−	0.678089i	$-0.237192\pi$
$948$		0			0
$949$	−12042.0	−	20857.4i	−0.411907	−	0.713444i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−17345.3			−0.589581			−0.294790	−	0.955562i	$-0.595250\pi$
$953$	−17345.3			−0.589581			−0.294790	+	0.955562i	$0.595250\pi$
$954$		0			0
$955$	−43890.0	−	76019.7i	−1.48717	−	2.57585i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	44638.7	−	15463.3i	1.50309	−	0.520684i
$960$		0			0
$961$	−5305.00	+	9188.53i	−0.178074	+	0.308433i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−91220.0			−3.04298
$966$		0			0
$967$	−4747.00			−0.157863			−0.0789313	−	0.996880i	$-0.525151\pi$
$967$	−4747.00			−0.157863			−0.0789313	+	0.996880i	$0.525151\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	16452.6	−	28496.7i	0.543757	−	0.941814i	−0.454927	−	0.890529i	$-0.650335\pi$
$971$	16452.6	−	28496.7i	0.543757	−	0.941814i	0.998684	−	0.0512856i	$-0.0163319\pi$
$972$		0			0
$973$	−8533.00	+	44338.8i	−0.281146	+	1.46088i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10477.8	+	18148.1i	0.343107	+	0.594279i	0.985008	−	0.172509i	$-0.0551873\pi$
$977$	10477.8	+	18148.1i	0.343107	+	0.594279i	−0.641901	+	0.766788i	$0.721854\pi$
$978$		0			0
$979$	−2310.00			−0.0754116
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16952.9	+	29363.3i	0.550065	+	0.952740i	0.998269	+	0.0588091i	$0.0187303\pi$
$983$	16952.9	+	29363.3i	0.550065	+	0.952740i	−0.448204	+	0.893931i	$0.647936\pi$
$984$		0			0
$985$	41195.0	−	71351.8i	1.33257	−	2.30808i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	23938.1	−	41462.1i	0.769654	−	1.33308i
$990$		0			0
$991$	−6734.50	−	11664.5i	−0.215871	−	0.373900i	0.737670	−	0.675161i	$-0.235926\pi$
$991$	−6734.50	−	11664.5i	−0.215871	−	0.373900i	−0.953542	+	0.301261i	$0.902592\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	12086.8			0.385103
$996$		0			0
$997$	−7141.00	−	12368.6i	−0.226838	−	0.392895i	0.730031	−	0.683414i	$-0.239506\pi$
$997$	−7141.00	−	12368.6i	−0.226838	−	0.392895i	−0.956869	+	0.290519i	$0.906172\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	252.4.k.e.37.2	yes	4
3.2	odd	2	inner	252.4.k.e.37.1	✓	4
7.2	even	3		1764.4.a.r.1.1		2
7.3	odd	6		1764.4.k.x.361.1		4
7.4	even	3	inner	252.4.k.e.109.2	yes	4
7.5	odd	6		1764.4.a.u.1.2		2
7.6	odd	2		1764.4.k.x.1549.1		4
21.2	odd	6		1764.4.a.r.1.2		2
21.5	even	6		1764.4.a.u.1.1		2
21.11	odd	6	inner	252.4.k.e.109.1	yes	4
21.17	even	6		1764.4.k.x.361.2		4
21.20	even	2		1764.4.k.x.1549.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.k.e.37.1	✓	4	3.2	odd	2	inner
252.4.k.e.37.2	yes	4	1.1	even	1	trivial
252.4.k.e.109.1	yes	4	21.11	odd	6	inner
252.4.k.e.109.2	yes	4	7.4	even	3	inner
1764.4.a.r.1.1		2	7.2	even	3
1764.4.a.r.1.2		2	21.2	odd	6
1764.4.a.u.1.1		2	21.5	even	6
1764.4.a.u.1.2		2	7.5	odd	6
1764.4.k.x.361.1		4	7.3	odd	6
1764.4.k.x.361.2		4	21.17	even	6
1764.4.k.x.1549.1		4	7.6	odd	2
1764.4.k.x.1549.2		4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	252.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(9.81071 - 16.9926i) q^{5} +(3.50000 - 18.1865i) q^{7} +O(q^{10})\)	\(q+(9.81071 - 16.9926i) q^{5} +(3.50000 - 18.1865i) q^{7} +(-9.81071 - 16.9926i) q^{11} -54.0000 q^{13} +(19.6214 + 33.9853i) q^{17} +(-1.00000 + 1.73205i) q^{19} +(-98.1071 + 169.926i) q^{23} +(-130.000 - 225.167i) q^{25} +98.1071 q^{29} +(-100.500 - 174.071i) q^{31} +(-274.700 - 237.897i) q^{35} +(101.000 - 174.937i) q^{37} +470.914 q^{41} -244.000 q^{43} +(117.729 - 203.912i) q^{47} +(-318.500 - 127.306i) q^{49} +(-323.753 - 560.757i) q^{53} -385.000 q^{55} +(304.132 + 526.772i) q^{59} +(294.000 - 509.223i) q^{61} +(-529.778 + 917.603i) q^{65} +(151.000 + 261.540i) q^{67} -156.971 q^{71} +(223.000 + 386.247i) q^{73} +(-343.375 + 118.949i) q^{77} +(133.500 - 231.229i) q^{79} +725.992 q^{83} +770.000 q^{85} +(58.8643 - 101.956i) q^{89} +(-189.000 + 982.073i) q^{91} +(19.6214 + 33.9853i) q^{95} +595.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 14 q^{7}+O(q^{10}) \) 4 * q + 14 * q^7	\( 4 q + 14 q^{7} - 216 q^{13} - 4 q^{19} - 520 q^{25} - 402 q^{31} + 404 q^{37} - 976 q^{43} - 1274 q^{49} - 1540 q^{55} + 1176 q^{61} + 604 q^{67} + 892 q^{73} + 534 q^{79} + 3080 q^{85} - 756 q^{91} + 2380 q^{97}+O(q^{100}) \) 4 * q + 14 * q^7 - 216 * q^13 - 4 * q^19 - 520 * q^25 - 402 * q^31 + 404 * q^37 - 976 * q^43 - 1274 * q^49 - 1540 * q^55 + 1176 * q^61 + 604 * q^67 + 892 * q^73 + 534 * q^79 + 3080 * q^85 - 756 * q^91 + 2380 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more