LMFDB - Embedded newform 1764.4.k.x.361.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1764,4,Mod(361,1764)] 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("1764.361"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1764, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1764.k (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,0,0,0,0,216,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$104.079369250$
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:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$5.15535 + 8.92934i$ of defining polynomial
Character	$\chi$	$=$	1764.361
Dual form			1764.4.k.x.1549.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times$.

$n$	$785$	$883$	$1081$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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	−0.854079	−	0.520143i	$-0.825879\pi$
$5$	−9.81071	−	16.9926i	−0.877496	−	1.51987i	−0.0234170	−	0.999726i	$-0.507455\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.81071	+	16.9926i	−0.268913	+	0.465770i	−0.968581	−	0.248697i	$-0.919998\pi$
$11$	−9.81071	+	16.9926i	−0.268913	+	0.465770i	0.699669	+	0.714468i	$0.253331\pi$
$12$		0			0
$13$	54.0000			1.15207			0.576035	−	0.817425i	$-0.304599\pi$
$13$	54.0000			1.15207			0.576035	+	0.817425i	$0.304599\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−19.6214	+	33.9853i	−0.279935	+	0.484861i	−0.971368	−	0.237579i	$-0.923646\pi$
$17$	−19.6214	+	33.9853i	−0.279935	+	0.484861i	0.691433	+	0.722440i	$0.256980\pi$
$18$		0			0
$19$	1.00000	+	1.73205i	0.0120745	+	0.0209137i	0.872000	−	0.489507i	$-0.162823\pi$
$19$	1.00000	+	1.73205i	0.0120745	+	0.0209137i	−0.859925	+	0.510420i	$0.829490\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−98.1071	−	169.926i	−0.889424	−	1.54053i	−0.840558	−	0.541721i	$-0.817773\pi$
$23$	−98.1071	−	169.926i	−0.889424	−	1.54053i	−0.0488654	−	0.998805i	$-0.515561\pi$
$24$		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	−0.412941	−	0.910758i	$-0.635498\pi$
$31$	100.500	−	174.071i	0.582269	−	1.00852i	0.995210	−	0.0977614i	$-0.0311682\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	101.000	+	174.937i	0.448765	+	0.777283i	0.998306	−	0.0581832i	$-0.0185308\pi$
$37$	101.000	+	174.937i	0.448765	+	0.777283i	−0.549541	+	0.835467i	$0.685197\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−470.914			−1.79377			−0.896883	−	0.442268i	$-0.854174\pi$
$41$	−470.914			−1.79377			−0.896883	+	0.442268i	$0.854174\pi$
$42$		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	$-0.285725\pi$
$47$	−117.729	−	203.912i	−0.365372	−	0.632842i	−0.988836	+	0.149010i	$0.952391\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−323.753	+	560.757i	−0.839074	+	1.45332i	0.0515949	+	0.998668i	$0.483570\pi$
$53$	−323.753	+	560.757i	−0.839074	+	1.45332i	−0.890669	+	0.454652i	$0.849764\pi$
$54$		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	0.306499	+	0.951871i	$0.400842\pi$
$59$	−304.132	+	526.772i	−0.671095	+	1.16237i	−0.977594	+	0.210500i	$0.932491\pi$
$60$		0			0
$61$	−294.000	−	509.223i	−0.617096	−	1.06884i	−0.990013	−	0.140977i	$-0.954976\pi$
$61$	−294.000	−	509.223i	−0.617096	−	1.06884i	0.372917	−	0.927865i	$-0.378358\pi$
$62$		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	−0.694883	−	0.719123i	$-0.744544\pi$
$67$	151.000	−	261.540i	0.275337	−	0.476898i	0.970220	+	0.242225i	$0.0778772\pi$
$68$		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	$-0.949717\pi$
$73$	−223.000	+	386.247i	−0.357537	+	0.619272i	0.630012	+	0.776585i	$0.283050\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	133.500	+	231.229i	0.190126	+	0.329307i	0.945292	−	0.326226i	$-0.105777\pi$
$79$	133.500	+	231.229i	0.190126	+	0.329307i	−0.755166	+	0.655533i	$0.772444\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−725.992			−0.960097			−0.480048	−	0.877242i	$-0.659381\pi$
$83$	−725.992			−0.960097			−0.480048	+	0.877242i	$0.659381\pi$
$84$		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	$-0.189001\pi$
$89$	−58.8643	−	101.956i	−0.0701078	−	0.121430i	−0.898948	+	0.438055i	$0.855668\pi$
$90$		0			0
$91$		0			0
$92$		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	$-0.600800\pi$
$97$	−595.000			−0.622815			−0.311408	+	0.950276i	$0.600800\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	608.264	−	1053.54i	0.599253	−	1.03794i	−0.393679	−	0.919248i	$-0.628798\pi$
$101$	608.264	−	1053.54i	0.599253	−	1.03794i	0.992932	−	0.118688i	$-0.0378689\pi$
$102$		0			0
$103$	832.000	+	1441.07i	0.795916	+	1.37857i	0.922256	+	0.386581i	$0.126344\pi$
$103$	832.000	+	1441.07i	0.795916	+	1.37857i	−0.126339	+	0.991987i	$0.540323\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	794.667	+	1376.40i	0.717976	+	1.24357i	0.961801	+	0.273751i	$0.0882644\pi$
$107$	794.667	+	1376.40i	0.717976	+	1.24357i	−0.243825	+	0.969819i	$0.578402\pi$
$108$		0			0
$109$	−179.000	+	310.037i	−0.157294	+	0.272442i	−0.933892	−	0.357555i	$-0.883610\pi$
$109$	−179.000	+	310.037i	−0.157294	+	0.272442i	0.776598	+	0.629997i	$0.216944\pi$
$110$		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
$119$		0			0
$120$		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	0.999990	+	0.00442833i	$0.00140959\pi$
$131$	755.425	+	1308.43i	0.503830	+	0.872659i	−0.496160	+	0.868231i	$0.665257\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1275.39	−	2209.04i	0.795358	−	1.37760i	−0.127253	−	0.991870i	$-0.540616\pi$
$137$	1275.39	−	2209.04i	0.795358	−	1.37760i	0.922611	−	0.385731i	$-0.126051\pi$
$138$		0			0
$139$	2438.00			1.48769			0.743843	−	0.668354i	$-0.233001\pi$
$139$	2438.00			1.48769			0.743843	+	0.668354i	$0.233001\pi$
$140$		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	0.996012	−	0.0892172i	$-0.0284365\pi$
$149$	765.235	+	1325.43i	0.420742	+	0.728746i	−0.575270	+	0.817963i	$0.695103\pi$
$150$		0			0
$151$	473.500	−	820.126i	0.255185	−	0.441993i	−0.709761	−	0.704443i	$-0.751197\pi$
$151$	473.500	−	820.126i	0.255185	−	0.441993i	0.964946	+	0.262450i	$0.0845304\pi$
$152$		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	$-0.961349\pi$
$157$	−770.000	+	1333.68i	−0.391418	+	0.677957i	0.601218	+	0.799085i	$0.294682\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	545.000	+	943.968i	0.261888	+	0.453603i	0.966744	−	0.255748i	$-0.0823217\pi$
$163$	545.000	+	943.968i	0.261888	+	0.453603i	−0.704856	+	0.709351i	$0.748988\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	2079.87			0.963744			0.481872	−	0.876242i	$-0.339957\pi$
$167$	2079.87			0.963744			0.481872	+	0.876242i	$0.339957\pi$
$168$		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	$-0.299494\pi$
$173$	−922.207	−	1597.31i	−0.405284	−	0.701972i	−0.994354	+	0.106110i	$0.966161\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	588.643	−	1019.56i	0.245794	−	0.425728i	−0.716560	−	0.697525i	$-0.754285\pi$
$179$	588.643	−	1019.56i	0.245794	−	0.425728i	0.962355	+	0.271797i	$0.0876178\pi$
$180$		0			0
$181$	2004.00			0.822962			0.411481	−	0.911418i	$-0.365012\pi$
$181$	2004.00			0.822962			0.411481	+	0.911418i	$0.365012\pi$
$182$		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	0.883526	+	0.468382i	$0.155163\pi$
$191$	2236.84	+	3874.32i	0.847394	+	1.46773i	−0.0361326	+	0.999347i	$0.511504\pi$
$192$		0			0
$193$	−2324.50	+	4026.15i	−0.866949	+	1.50160i	−0.00185031	+	0.999998i	$0.500589\pi$
$193$	−2324.50	+	4026.15i	−0.866949	+	1.50160i	−0.865099	+	0.501602i	$0.832744\pi$
$194$		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	$-0.868328\pi$
$199$	−308.000	+	533.472i	−0.109716	+	0.190034i	0.805939	+	0.591999i	$0.201661\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		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
$217$		0			0
$218$		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	$-0.399654\pi$
$223$	2065.00			0.620101			0.310051	+	0.950720i	$0.399654\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−912.396	+	1580.32i	−0.266775	+	0.462067i	−0.968027	−	0.250846i	$-0.919291\pi$
$227$	−912.396	+	1580.32i	−0.266775	+	0.462067i	0.701252	+	0.712913i	$0.252625\pi$
$228$		0			0
$229$	−1954.00	−	3384.43i	−0.563860	−	0.976634i	−0.997155	−	0.0753820i	$-0.975982\pi$
$229$	−1954.00	−	3384.43i	−0.563860	−	0.976634i	0.433295	−	0.901252i	$-0.357351\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2217.22	−	3840.34i	−0.623412	−	1.07978i	−0.988846	−	0.148943i	$-0.952413\pi$
$233$	−2217.22	−	3840.34i	−0.623412	−	1.07978i	0.365434	−	0.930837i	$-0.380921\pi$
$234$		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	0.476701	+	0.879066i	$0.341833\pi$
$241$	−1956.50	+	3388.76i	−0.522943	+	0.905764i	−0.999644	+	0.0266980i	$0.991501\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		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	$-0.698790\pi$
$251$	−4650.28			−1.16941			−0.584707	+	0.811245i	$0.698790\pi$
$252$		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	$-0.244760\pi$
$257$	−1000.69	−	1733.25i	−0.242885	−	0.420689i	−0.961535	+	0.274683i	$0.911427\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$	235.457	−	407.823i	0.0552049	−	0.0956178i	−0.837102	−	0.547046i	$-0.815752\pi$
$263$	235.457	−	407.823i	0.0552049	−	0.0956178i	0.892307	+	0.451429i	$0.149085\pi$
$264$		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	$-0.906892\pi$
$269$	−1010.50	+	1750.24i	−0.229039	+	0.396707i	0.728485	+	0.685062i	$0.240225\pi$
$270$		0			0
$271$	598.500	+	1036.63i	0.134156	+	0.232365i	0.925275	−	0.379298i	$-0.123834\pi$
$271$	598.500	+	1036.63i	0.134156	+	0.232365i	−0.791119	+	0.611663i	$0.790501\pi$
$272$		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	−0.937064	−	0.349157i	$-0.886468\pi$
$277$	−766.000	+	1326.75i	−0.166153	+	0.287786i	0.770911	+	0.636943i	$0.219801\pi$
$278$		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	0.320648	+	0.947198i	$0.396099\pi$
$283$	−3142.00	+	5442.10i	−0.659974	+	1.14311i	−0.980622	+	0.195909i	$0.937234\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		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	$-0.620543\pi$
$293$	−3708.45			−0.739419			−0.369710	+	0.929147i	$0.620543\pi$
$294$		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
$301$		0			0
$302$		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	$-0.552370\pi$
$307$	−1762.00			−0.327566			−0.163783	+	0.986496i	$0.552370\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3335.64	+	5777.50i	−0.608189	+	1.05341i	0.383350	+	0.923603i	$0.374770\pi$
$311$	−3335.64	+	5777.50i	−0.608189	+	1.05341i	−0.991539	+	0.129811i	$0.958563\pi$
$312$		0			0
$313$	660.500	+	1144.02i	0.119277	+	0.206594i	0.919481	−	0.393134i	$-0.128609\pi$
$313$	660.500	+	1144.02i	0.119277	+	0.206594i	−0.800204	+	0.599727i	$0.795276\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3149.24	−	5454.64i	−0.557977	−	0.966445i	−0.997665	−	0.0682944i	$-0.978244\pi$
$317$	−3149.24	−	5454.64i	−0.557977	−	0.966445i	0.439688	−	0.898151i	$-0.355089\pi$
$318$		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
$329$		0			0
$330$		0			0
$331$	4868.00	+	8431.62i	0.808367	+	1.40013i	0.913994	+	0.405727i	$0.132982\pi$
$331$	4868.00	+	8431.62i	0.808367	+	1.40013i	−0.105627	+	0.994406i	$0.533685\pi$
$332$		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
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1059.56	−	1835.21i	0.163919	−	0.283916i	−0.772352	−	0.635195i	$-0.780920\pi$
$347$	1059.56	−	1835.21i	0.163919	−	0.283916i	0.936271	+	0.351279i	$0.114253\pi$
$348$		0			0
$349$	−9646.00			−1.47948			−0.739740	−	0.672893i	$-0.765052\pi$
$349$	−9646.00			−1.47948			−0.739740	+	0.672893i	$0.765052\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	235.457	−	407.823i	0.0355017	−	0.0614908i	−0.847729	−	0.530430i	$-0.822030\pi$
$353$	235.457	−	407.823i	0.0355017	−	0.0614908i	0.883230	+	0.468939i	$0.155364\pi$
$354$		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	0.529226	−	0.848481i	$-0.322482\pi$
$359$	−3198.29	−	5539.60i	−0.470193	−	0.814398i	−0.999419	+	0.0340826i	$0.989149\pi$
$360$		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	$-0.992891\pi$
$367$	−3378.50	+	5851.73i	−0.480535	+	0.832311i	0.519216	+	0.854643i	$0.326224\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−4322.00	−	7485.92i	−0.599959	−	1.03916i	−0.992826	−	0.119564i	$-0.961850\pi$
$373$	−4322.00	−	7485.92i	−0.599959	−	1.03916i	0.392868	−	0.919595i	$-0.371483\pi$
$374$		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	$-1.87598e-6\pi$
$383$	3747.69	+	6491.19i	0.499995	+	0.866017i	−0.500005	+	0.866022i	$0.666669\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6808.63	−	11792.9i	0.887433	−	1.53708i	0.0445330	−	0.999008i	$-0.485820\pi$
$389$	6808.63	−	11792.9i	0.887433	−	1.53708i	0.842900	−	0.538071i	$-0.180847\pi$
$390$		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	$-0.261250\pi$
$397$	−2316.00	−	4011.43i	−0.292788	−	0.507123i	−0.974468	+	0.224527i	$0.927916\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	2590.03	+	4486.06i	0.322543	+	0.558661i	0.981012	−	0.193947i	$-0.0621289\pi$
$401$	2590.03	+	4486.06i	0.322543	+	0.558661i	−0.658469	+	0.752608i	$0.728796\pi$
$402$		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	$-0.922485\pi$
$409$	−2286.50	+	3960.33i	−0.276431	+	0.478792i	0.694064	+	0.719913i	$0.255818\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		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	$-0.451753\pi$
$419$	2590.03			0.301984			0.150992	+	0.988535i	$0.451753\pi$
$420$		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
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−8476.45	+	14681.6i	−0.947323	+	1.64081i	−0.196292	+	0.980546i	$0.562890\pi$
$431$	−8476.45	+	14681.6i	−0.947323	+	1.64081i	−0.751031	+	0.660267i	$0.770443\pi$
$432$		0			0
$433$	5206.00			0.577793			0.288897	−	0.957360i	$-0.406712\pi$
$433$	5206.00			0.577793			0.288897	+	0.957360i	$0.406712\pi$
$434$		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	−0.976570	−	0.215198i	$-0.930960\pi$
$439$	−6205.50	−	10748.2i	−0.674652	−	1.16853i	0.301918	−	0.953334i	$-0.402373\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6700.71	+	11606.0i	0.718647	+	1.24473i	0.961536	+	0.274679i	$0.0885715\pi$
$443$	6700.71	+	11606.0i	0.718647	+	1.24473i	−0.242889	+	0.970054i	$0.578095\pi$
$444$		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
$455$		0			0
$456$		0			0
$457$	−7413.50	−	12840.6i	−0.758838	−	1.31435i	−0.943444	−	0.331533i	$-0.892434\pi$
$457$	−7413.50	−	12840.6i	−0.758838	−	1.31435i	0.184606	−	0.982813i	$-0.440899\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1451.98			−0.146693			−0.0733467	−	0.997307i	$-0.523368\pi$
$461$	−1451.98			−0.146693			−0.0733467	+	0.997307i	$0.523368\pi$
$462$		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	$-0.0776294\pi$
$467$	2786.24	+	4825.91i	0.276085	+	0.478194i	−0.694323	+	0.719664i	$0.744296\pi$
$468$		0			0
$469$		0			0
$470$		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	−0.467770	−	0.883850i	$-0.654942\pi$
$479$	5572.48	−	9651.82i	0.531552	−	0.920674i	0.999322	−	0.0368242i	$-0.0117242\pi$
$480$		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	0.0135757	+	0.999908i	$0.495679\pi$
$487$	−9233.50	+	15992.9i	−0.859158	+	1.48810i	−0.872733	+	0.488197i	$0.837655\pi$
$488$		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
$497$		0			0
$498$		0			0
$499$	6961.00	+	12056.8i	0.624483	+	1.08164i	0.988641	+	0.150299i	$0.0480237\pi$
$499$	6961.00	+	12056.8i	0.624483	+	1.08164i	−0.364157	+	0.931337i	$0.618643\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	20837.9			1.84715			0.923577	−	0.383414i	$-0.125252\pi$
$503$	20837.9			1.84715			0.923577	+	0.383414i	$0.125252\pi$
$504$		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	0.990809	+	0.135270i	$0.0431903\pi$
$509$	7034.28	+	12183.7i	0.612552	+	1.06097i	−0.378257	+	0.925701i	$0.623476\pi$
$510$		0			0
$511$		0			0
$512$		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	0.464596	+	0.885523i	$0.346200\pi$
$521$	−6357.34	+	11011.2i	−0.534587	+	0.925933i	−0.999183	+	0.0404098i	$0.987134\pi$
$522$		0			0
$523$	8233.00	+	14260.0i	0.688344	+	1.19225i	0.972373	+	0.233431i	$0.0749954\pi$
$523$	8233.00	+	14260.0i	0.688344	+	1.19225i	−0.284029	+	0.958816i	$0.591671\pi$
$524$		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
$539$		0			0
$540$		0			0
$541$	−192.000	−	332.554i	−0.0152583	−	0.0264281i	0.858295	−	0.513156i	$-0.171524\pi$
$541$	−192.000	−	332.554i	−0.0152583	−	0.0264281i	−0.873554	+	0.486728i	$0.838190\pi$
$542$		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
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−814.289	+	1410.39i	−0.0619435	+	0.107289i	−0.895334	−	0.445395i	$-0.853063\pi$
$557$	−814.289	+	1410.39i	−0.0619435	+	0.107289i	0.833391	+	0.552684i	$0.186397\pi$
$558$		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	−0.488022	−	0.872831i	$-0.662282\pi$
$563$	6838.06	−	11843.9i	0.511883	−	0.886607i	0.999905	−	0.0137759i	$-0.00438514\pi$
$564$		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	0.998218	+	0.0596776i	$0.0190073\pi$
$569$	7475.76	+	12948.4i	0.550791	+	0.953998i	−0.447427	+	0.894321i	$0.647659\pi$
$570$		0			0
$571$	5268.00	−	9124.44i	0.386093	−	0.668732i	−0.605827	−	0.795596i	$-0.707158\pi$
$571$	5268.00	−	9124.44i	0.386093	−	0.668732i	0.991920	+	0.126864i	$0.0404911\pi$
$572$		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	$-0.807042\pi$
$577$	1143.50	−	1980.60i	0.0825035	−	0.142900i	0.904325	+	0.426845i	$0.140375\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		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	$-0.543835\pi$
$587$	−3904.66			−0.274553			−0.137277	+	0.990533i	$0.543835\pi$
$588$		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	0.998780	+	0.0493774i	$0.0157237\pi$
$593$	7828.95	+	13560.1i	0.542152	+	0.939035i	−0.456628	+	0.889658i	$0.650943\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10772.2	+	18657.9i	−0.734789	+	1.27269i	0.220027	+	0.975494i	$0.429385\pi$
$599$	−10772.2	+	18657.9i	−0.734789	+	1.27269i	−0.954816	+	0.297198i	$0.903948\pi$
$600$		0			0
$601$	17089.0			1.15986			0.579929	−	0.814667i	$-0.303080\pi$
$601$	17089.0			1.15986			0.579929	+	0.814667i	$0.303080\pi$
$602$		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	$-0.113776\pi$
$607$	2473.50	+	4284.23i	0.165397	+	0.286477i	−0.771399	+	0.636352i	$0.780443\pi$
$608$		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	−0.399705	−	0.916644i	$-0.630887\pi$
$613$	9015.00	−	15614.4i	0.593984	−	1.02881i	0.993689	−	0.112167i	$-0.0357792\pi$
$614$		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	−0.0233278	−	0.999728i	$-0.507426\pi$
$619$	13154.0	−	22783.4i	0.854126	−	1.47939i	0.877454	−	0.479661i	$-0.159241\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		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
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	6318.10	−	10943.3i	0.389313	−	0.674310i	−0.603044	−	0.797708i	$-0.706046\pi$
$641$	6318.10	−	10943.3i	0.389313	−	0.674310i	0.992357	+	0.123398i	$0.0393790\pi$
$642$		0			0
$643$	−11174.0			−0.685318			−0.342659	−	0.939460i	$-0.611328\pi$
$643$	−11174.0			−0.685318			−0.342659	+	0.939460i	$0.611328\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−9771.47	+	16924.7i	−0.593750	+	1.02840i	0.399972	+	0.916527i	$0.369020\pi$
$647$	−9771.47	+	16924.7i	−0.593750	+	1.02840i	−0.993722	+	0.111877i	$0.964314\pi$
$648$		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	0.558438	−	0.829546i	$-0.311401\pi$
$653$	−7328.60	−	12693.5i	−0.439189	−	0.760697i	−0.997627	+	0.0688488i	$0.978067\pi$
$654$		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	$-0.909196\pi$
$661$	−4012.00	+	6948.99i	−0.236080	+	0.408902i	0.723506	+	0.690318i	$0.242529\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		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	$-0.247853\pi$
$677$	−4444.25	−	7697.67i	−0.252299	−	0.436995i	−0.964158	+	0.265327i	$0.914520\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2501.73	+	4333.12i	−0.140155	+	0.242756i	−0.927555	−	0.373687i	$-0.878094\pi$
$683$	−2501.73	+	4333.12i	−0.140155	+	0.242756i	0.787400	+	0.616443i	$0.211427\pi$
$684$		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	0.948750	+	0.316028i	$0.102349\pi$
$691$	13588.0	+	23535.1i	0.748064	+	1.29568i	−0.200686	+	0.979656i	$0.564317\pi$
$692$		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
$707$		0			0
$708$		0			0
$709$	2717.00	+	4705.98i	0.143920	+	0.249276i	0.928969	−	0.370157i	$-0.120696\pi$
$709$	2717.00	+	4705.98i	0.143920	+	0.249276i	−0.785050	+	0.619433i	$0.787363\pi$
$710$		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	$-0.0119500\pi$
$719$	9006.23	+	15599.2i	0.467143	+	0.809115i	−0.532153	+	0.846649i	$0.678617\pi$
$720$		0			0
$721$		0			0
$722$		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	$-0.769332\pi$
$727$	−29353.0			−1.49744			−0.748722	+	0.662884i	$0.769332\pi$
$728$		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	$-0.259700\pi$
$733$	−5718.00	−	9903.87i	−0.288130	−	0.499055i	−0.973363	+	0.229268i	$0.926367\pi$
$734$		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	−0.865053	−	0.501680i	$-0.832715\pi$
$739$	39.0000	−	67.5500i	0.00194132	−	0.00336247i	0.866994	+	0.498318i	$0.166049\pi$
$740$		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
$749$		0			0
$750$		0			0
$751$	−13195.5	−	22855.3i	−0.641159	−	1.11052i	−0.985174	−	0.171556i	$-0.945120\pi$
$751$	−13195.5	−	22855.3i	−0.641159	−	1.11052i	0.344015	−	0.938964i	$-0.388213\pi$
$752$		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	$-0.127595\pi$
$761$	2570.41	+	4452.07i	0.122440	+	0.212073i	−0.798289	+	0.602274i	$0.794261\pi$
$762$		0			0
$763$		0			0
$764$		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	$-0.565946\pi$
$769$	−8773.00			−0.411395			−0.205697	+	0.978616i	$0.565946\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−5513.62	+	9549.87i	−0.256547	+	0.444353i	−0.965315	−	0.261089i	$-0.915918\pi$
$773$	−5513.62	+	9549.87i	−0.256547	+	0.444353i	0.708767	+	0.705442i	$0.249252\pi$
$774$		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	$-0.875847\pi$
$787$	−2940.00	+	5092.23i	−0.133164	+	0.230646i	0.791731	+	0.610870i	$0.209180\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		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	$-0.669927\pi$
$797$	−22898.2			−1.01769			−0.508843	+	0.860859i	$0.669927\pi$
$798$		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
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−1608.96	+	2786.79i	−0.0699232	+	0.121111i	−0.898867	−	0.438221i	$-0.855609\pi$
$809$	−1608.96	+	2786.79i	−0.0699232	+	0.121111i	0.828944	+	0.559331i	$0.188942\pi$
$810$		0			0
$811$	6062.00			0.262473			0.131237	−	0.991351i	$-0.458105\pi$
$811$	6062.00			0.262473			0.131237	+	0.991351i	$0.458105\pi$
$812$		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	0.929267	−	0.369409i	$-0.120440\pi$
$821$	3404.32	+	5896.45i	0.144716	+	0.250655i	−0.784551	+	0.620064i	$0.787107\pi$
$822$		0			0
$823$	16120.0	−	27920.7i	0.682756	−	1.18257i	−0.291381	−	0.956607i	$-0.594115\pi$
$823$	16120.0	−	27920.7i	0.682756	−	1.18257i	0.974137	−	0.225960i	$-0.0725520\pi$
$824$		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	−0.307428	−	0.951571i	$-0.599468\pi$
$829$	16001.0	−	27714.5i	0.670371	−	1.16112i	0.977799	−	0.209545i	$-0.0671984\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		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	$-0.535282\pi$
$839$	−5376.27			−0.221227			−0.110613	+	0.993864i	$0.535282\pi$
$840$		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
$847$		0			0
$848$		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	$-0.368761\pi$
$853$	19966.0			0.801434			0.400717	+	0.916202i	$0.368761\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6945.98	+	12030.8i	−0.276861	+	0.479538i	−0.970603	−	0.240686i	$-0.922628\pi$
$857$	−6945.98	+	12030.8i	−0.276861	+	0.479538i	0.693742	+	0.720224i	$0.255961\pi$
$858$		0			0
$859$	−16688.0	−	28904.5i	−0.662849	−	1.14809i	−0.979864	−	0.199667i	$-0.936014\pi$
$859$	−16688.0	−	28904.5i	−0.662849	−	1.14809i	0.317015	−	0.948421i	$-0.397320\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−4493.30	−	7782.63i	−0.177235	−	0.306980i	0.763697	−	0.645574i	$-0.223382\pi$
$863$	−4493.30	−	7782.63i	−0.177235	−	0.306980i	−0.940932	+	0.338594i	$0.890049\pi$
$864$		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
$875$		0			0
$876$		0			0
$877$	2260.00	+	3914.43i	0.0870180	+	0.150720i	0.906249	−	0.422744i	$-0.138933\pi$
$877$	2260.00	+	3914.43i	0.0870180	+	0.150720i	−0.819231	+	0.573463i	$0.805600\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	37555.4			1.43618			0.718089	−	0.695951i	$-0.245017\pi$
$881$	37555.4			1.43618			0.718089	+	0.695951i	$0.245017\pi$
$882$		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	$-0.0140677\pi$
$887$	12184.9	+	21104.9i	0.461250	+	0.798909i	−0.537773	+	0.843089i	$0.680734\pi$
$888$		0			0
$889$		0			0
$890$		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	−0.733749	−	0.679421i	$-0.762231\pi$
$907$	6051.00	−	10480.6i	0.221522	−	0.383687i	0.955270	+	0.295734i	$0.0955643\pi$
$908$		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
$917$		0			0
$918$		0			0
$919$	16894.0	+	29261.3i	0.606400	+	1.05032i	0.991829	+	0.127578i	$0.0407203\pi$
$919$	16894.0	+	29261.3i	0.606400	+	1.05032i	−0.385429	+	0.922738i	$0.625946\pi$
$920$		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	$-0.174608\pi$
$929$	−706.371	−	1223.47i	−0.0249465	−	0.0432086i	−0.878229	+	0.478240i	$0.841275\pi$
$930$		0			0
$931$		0			0
$932$		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	$-0.599280\pi$
$937$	−17603.0			−0.613730			−0.306865	+	0.951753i	$0.599280\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−9545.82	+	16533.8i	−0.330696	+	0.572782i	−0.982648	−	0.185478i	$-0.940617\pi$
$941$	−9545.82	+	16533.8i	−0.330696	+	0.572782i	0.651953	+	0.758260i	$0.273950\pi$
$942$		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	0.734980	+	0.678089i	$0.237192\pi$
$947$	27823.2	+	48191.1i	0.954732	+	1.65364i	0.219753	+	0.975556i	$0.429475\pi$
$948$		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
$959$		0			0
$960$		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	−0.998684	−	0.0512856i	$-0.983668\pi$
$971$	−16452.6	−	28496.7i	−0.543757	−	0.941814i	0.454927	−	0.890529i	$-0.349665\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	10477.8	−	18148.1i	0.343107	−	0.594279i	−0.641901	−	0.766788i	$-0.721854\pi$
$977$	10477.8	−	18148.1i	0.343107	−	0.594279i	0.985008	+	0.172509i	$0.0551873\pi$
$978$		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	0.448204	+	0.893931i	$0.352064\pi$
$983$	−16952.9	+	29363.3i	−0.550065	+	0.952740i	−0.998269	+	0.0588091i	$0.981270\pi$
$984$		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	−0.953542	−	0.301261i	$-0.902592\pi$
$991$	−6734.50	+	11664.5i	−0.215871	+	0.373900i	0.737670	+	0.675161i	$0.235926\pi$
$992$		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	$-0.760494\pi$
$997$	7141.00	−	12368.6i	0.226838	−	0.392895i	0.956869	+	0.290519i	$0.0938278\pi$
$998$		0			0
$999$		0			0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.k.e.37.1	✓	4	21.5	even	6
252.4.k.e.37.2	yes	4	7.5	odd	6
252.4.k.e.109.1	yes	4	21.20	even	2
252.4.k.e.109.2	yes	4	7.6	odd	2
1764.4.a.r.1.1		2	7.3	odd	6
1764.4.a.r.1.2		2	21.17	even	6
1764.4.a.u.1.1		2	21.11	odd	6
1764.4.a.u.1.2		2	7.4	even	3
1764.4.k.x.361.1		4	1.1	even	1	trivial
1764.4.k.x.361.2		4	3.2	odd	2	inner
1764.4.k.x.1549.1		4	7.2	even	3	inner
1764.4.k.x.1549.2		4	21.2	odd	6	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:	\( N \)	\(=\)	\( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1764.k (of order \(3\), degree \(2\), not minimal)

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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