LMFDB - Embedded newform 756.4.x.a.125.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,4,Mod(125,756)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 3]))

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

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

chi := DirichletCharacter("756.125");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	756.x (of order $6$, degree $2$, not 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:	$44.6054439643$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			125.18
Character	$\chi$	$=$	756.125
Dual form			756.4.x.a.629.18

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.16485 - 8.94579i) q^{5} +(14.4986 - 11.5234i) q^{7} +O(q^{10})$	$q+(5.16485 - 8.94579i) q^{5} +(14.4986 - 11.5234i) q^{7} +(-27.1307 + 15.6639i) q^{11} +(39.0052 + 22.5196i) q^{13} +62.4900 q^{17} +132.928i q^{19} +(58.8009 + 33.9487i) q^{23} +(9.14862 + 15.8459i) q^{25} +(116.665 - 67.3568i) q^{29} +(25.9914 + 15.0061i) q^{31} +(-28.2028 - 189.219i) q^{35} +40.4778 q^{37} +(39.7514 - 68.8515i) q^{41} +(161.452 + 279.643i) q^{43} +(-171.268 - 296.644i) q^{47} +(77.4212 - 334.148i) q^{49} -64.9125i q^{53} +323.607i q^{55} +(79.3800 - 137.490i) q^{59} +(-493.640 + 285.003i) q^{61} +(402.912 - 232.621i) q^{65} +(150.833 - 261.251i) q^{67} -719.100i q^{71} -558.706i q^{73} +(-212.856 + 539.744i) q^{77} +(456.676 + 790.986i) q^{79} +(-352.348 - 610.285i) q^{83} +(322.751 - 559.022i) q^{85} +700.133 q^{89} +(825.026 - 122.969i) q^{91} +(1189.15 + 686.554i) q^{95} +(202.787 - 117.079i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q + 6 q^{7}+O(q^{10}) $ 48 * q + 6 * q^7	$ 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) $ 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$29$	$325$	$379$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	5.16485	−	8.94579i	0.461958	−	0.800135i	−0.537100	−	0.843518i	$-0.680480\pi$
$5$	5.16485	−	8.94579i	0.461958	−	0.800135i	0.999059	+	0.0433831i	$0.0138136\pi$
$6$		0			0
$7$	14.4986	−	11.5234i	0.782853	−	0.622207i
$7$	14.4986	−	11.5234i	0.782853	−	0.622207i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−27.1307	+	15.6639i	−0.743656	+	0.429350i	−0.823397	−	0.567466i	$-0.807924\pi$
$11$	−27.1307	+	15.6639i	−0.743656	+	0.429350i	0.0797412	+	0.996816i	$0.474591\pi$
$12$		0			0
$13$	39.0052	+	22.5196i	0.832161	+	0.480448i	0.854592	−	0.519300i	$-0.173807\pi$
$13$	39.0052	+	22.5196i	0.832161	+	0.480448i	−0.0224312	+	0.999748i	$0.507141\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	62.4900			0.891532			0.445766	−	0.895150i	$-0.352931\pi$
$17$	62.4900			0.891532			0.445766	+	0.895150i	$0.352931\pi$
$18$		0			0
$19$			132.928i			1.60504i	0.596624	+	0.802521i	$0.296508\pi$
$19$			132.928i			1.60504i	−0.596624	+	0.802521i	$0.703492\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	58.8009	+	33.9487i	0.533080	+	0.307774i	0.742270	−	0.670101i	$-0.233749\pi$
$23$	58.8009	+	33.9487i	0.533080	+	0.307774i	−0.209190	+	0.977875i	$0.567083\pi$
$24$		0			0
$25$	9.14862	+	15.8459i	0.0731889	+	0.126767i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	116.665	−	67.3568i	0.747043	−	0.431305i	−0.0775817	−	0.996986i	$-0.524720\pi$
$29$	116.665	−	67.3568i	0.747043	−	0.431305i	0.824624	+	0.565681i	$0.191387\pi$
$30$		0			0
$31$	25.9914	+	15.0061i	0.150587	+	0.0869413i	0.573400	−	0.819275i	$-0.305624\pi$
$31$	25.9914	+	15.0061i	0.150587	+	0.0869413i	−0.422813	+	0.906217i	$0.638957\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−28.2028	−	189.219i	−0.136204	−	0.913822i
$36$		0			0
$37$	40.4778			0.179852			0.0899259	−	0.995948i	$-0.471337\pi$
$37$	40.4778			0.179852			0.0899259	+	0.995948i	$0.471337\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	39.7514	−	68.8515i	0.151418	−	0.262263i	−0.780331	−	0.625367i	$-0.784949\pi$
$41$	39.7514	−	68.8515i	0.151418	−	0.262263i	0.931749	+	0.363103i	$0.118283\pi$
$42$		0			0
$43$	161.452	+	279.643i	0.572586	+	0.991749i	0.996299	+	0.0859521i	$0.0273932\pi$
$43$	161.452	+	279.643i	0.572586	+	0.991749i	−0.423713	+	0.905797i	$0.639273\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−171.268	−	296.644i	−0.531531	−	0.920638i	−0.999323	−	0.0367994i	$-0.988284\pi$
$47$	−171.268	−	296.644i	−0.531531	−	0.920638i	0.467792	−	0.883839i	$-0.345050\pi$
$48$		0			0
$49$	77.4212	−	334.148i	0.225718	−	0.974193i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	64.9125i		−	0.168234i	−0.996456	−	0.0841172i	$-0.973193\pi$
$53$		−	64.9125i		−	0.168234i	0.996456	−	0.0841172i	$-0.0268070\pi$
$54$		0			0
$55$			323.607i			0.793367i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	79.3800	−	137.490i	0.175159	−	0.303385i	−0.765057	−	0.643962i	$-0.777289\pi$
$59$	79.3800	−	137.490i	0.175159	−	0.303385i	0.940216	+	0.340578i	$0.110623\pi$
$60$		0			0
$61$	−493.640	+	285.003i	−1.03613	+	0.598212i	−0.918736	−	0.394873i	$-0.870788\pi$
$61$	−493.640	+	285.003i	−1.03613	+	0.598212i	−0.117397	+	0.993085i	$0.537455\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	402.912	−	232.621i	0.768847	−	0.443894i
$66$		0			0
$67$	150.833	−	261.251i	0.275033	−	0.476372i	−0.695110	−	0.718903i	$-0.744644\pi$
$67$	150.833	−	261.251i	0.275033	−	0.476372i	0.970144	+	0.242531i	$0.0779778\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	719.100i		−	1.20199i	−0.799252	−	0.600996i	$-0.794771\pi$
$71$		−	719.100i		−	1.20199i	0.799252	−	0.600996i	$-0.205229\pi$
$72$		0			0
$73$		−	558.706i		−	0.895775i	−0.894090	−	0.447888i	$-0.852176\pi$
$73$		−	558.706i		−	0.895775i	0.894090	−	0.447888i	$-0.147824\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−212.856	+	539.744i	−0.315029	+	0.798825i
$78$		0			0
$79$	456.676	+	790.986i	0.650381	+	1.12649i	0.983031	+	0.183442i	$0.0587239\pi$
$79$	456.676	+	790.986i	0.650381	+	1.12649i	−0.332650	+	0.943050i	$0.607943\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−352.348	−	610.285i	−0.465967	−	0.807078i	0.533278	−	0.845940i	$-0.320960\pi$
$83$	−352.348	−	610.285i	−0.465967	−	0.807078i	−0.999245	+	0.0388620i	$0.987627\pi$
$84$		0			0
$85$	322.751	−	559.022i	0.411851	−	0.713346i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	700.133			0.833865			0.416932	−	0.908938i	$-0.363105\pi$
$89$	700.133			0.833865			0.416932	+	0.908938i	$0.363105\pi$
$90$		0			0
$91$	825.026	−	122.969i	0.950398	−	0.141656i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1189.15	+	686.554i	1.28425	+	0.741463i
$96$		0			0
$97$	202.787	−	117.079i	0.212267	−	0.122552i	−0.390098	−	0.920773i	$-0.627559\pi$
$97$	202.787	−	117.079i	0.212267	−	0.122552i	0.602364	+	0.798221i	$0.294225\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	938.098	+	1624.83i	0.924200	+	1.60076i	0.792843	+	0.609427i	$0.208600\pi$
$101$	938.098	+	1624.83i	0.924200	+	1.60076i	0.131358	+	0.991335i	$0.458066\pi$
$102$		0			0
$103$	524.507	+	302.824i	0.501759	+	0.289691i	0.729440	−	0.684045i	$-0.239781\pi$
$103$	524.507	+	302.824i	0.501759	+	0.289691i	−0.227681	+	0.973736i	$0.573114\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	1299.63i		−	1.17420i	−0.809513	−	0.587102i	$-0.800269\pi$
$107$		−	1299.63i		−	1.17420i	0.809513	−	0.587102i	$-0.199731\pi$
$108$		0			0
$109$	2079.24			1.82711			0.913554	−	0.406717i	$-0.133326\pi$
$109$	2079.24			1.82711			0.913554	+	0.406717i	$0.133326\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1090.53	−	629.616i	−0.907860	−	0.524153i	−0.0281181	−	0.999605i	$-0.508951\pi$
$113$	−1090.53	−	629.616i	−0.907860	−	0.524153i	−0.879742	+	0.475451i	$0.842285\pi$
$114$		0			0
$115$	607.396	−	350.680i	0.492522	−	0.284358i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	906.020	−	720.099i	0.697939	−	0.554717i
$120$		0			0
$121$	−174.783	+	302.734i	−0.131317	+	0.227448i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1480.22			1.05916
$126$		0			0
$127$	−1015.62			−0.709623			−0.354811	−	0.934938i	$-0.615455\pi$
$127$	−1015.62			−0.709623			−0.354811	+	0.934938i	$0.615455\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1353.80	+	2344.85i	−0.902915	+	1.56390i	−0.0792242	+	0.996857i	$0.525244\pi$
$131$	−1353.80	+	2344.85i	−0.902915	+	1.56390i	−0.823691	+	0.567039i	$0.808089\pi$
$132$		0			0
$133$	1531.79	+	1927.28i	0.998668	+	1.25651i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2713.91	−	1566.88i	1.69245	−	0.977135i	0.739915	−	0.672700i	$-0.234866\pi$
$137$	2713.91	−	1566.88i	1.69245	−	0.977135i	0.952533	−	0.304435i	$-0.0984677\pi$
$138$		0			0
$139$	−86.5871	−	49.9911i	−0.0528361	−	0.0305049i	0.473349	−	0.880875i	$-0.343045\pi$
$139$	−86.5871	−	49.9911i	−0.0528361	−	0.0305049i	−0.526185	+	0.850370i	$0.676378\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1410.98			−0.825121
$144$		0			0
$145$		−	1391.55i		−	0.796980i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−2280.20	−	1316.47i	−1.25370	−	0.723823i	−0.281856	−	0.959457i	$-0.590950\pi$
$149$	−2280.20	−	1316.47i	−1.25370	−	0.723823i	−0.971842	+	0.235634i	$0.924283\pi$
$150$		0			0
$151$	−944.432	−	1635.80i	−0.508985	−	0.881588i	−0.999946	−	0.0104066i	$-0.996687\pi$
$151$	−944.432	−	1635.80i	−0.508985	−	0.881588i	0.490961	−	0.871182i	$-0.336646\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	268.483	−	155.009i	0.139130	−	0.0803266i
$156$		0			0
$157$	−1110.41	−	641.097i	−0.564462	−	0.325892i	0.190472	−	0.981693i	$-0.438998\pi$
$157$	−1110.41	−	641.097i	−0.564462	−	0.325892i	−0.754935	+	0.655800i	$0.772331\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1243.74	−	185.378i	0.608822	−	0.0907442i
$162$		0			0
$163$	3558.63			1.71002			0.855011	−	0.518610i	$-0.173550\pi$
$163$	3558.63			1.71002			0.855011	+	0.518610i	$0.173550\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−471.603	+	816.841i	−0.218526	+	0.378497i	−0.954357	−	0.298667i	$-0.903458\pi$
$167$	−471.603	+	816.841i	−0.218526	+	0.378497i	0.735832	+	0.677164i	$0.236791\pi$
$168$		0			0
$169$	−84.2309	−	145.892i	−0.0383390	−	0.0664052i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−489.793	−	848.347i	−0.215250	−	0.372824i	0.738100	−	0.674692i	$-0.235723\pi$
$173$	−489.793	−	848.347i	−0.215250	−	0.372824i	−0.953350	+	0.301867i	$0.902390\pi$
$174$		0			0
$175$	315.241	+	124.320i	0.136171	+	0.0537013i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			3739.96i			1.56166i	0.624742	+	0.780831i	$0.285204\pi$
$179$			3739.96i			1.56166i	−0.624742	+	0.780831i	$0.714796\pi$
$180$		0			0
$181$			1540.52i			0.632631i	0.948654	+	0.316316i	$0.102446\pi$
$181$			1540.52i			0.632631i	−0.948654	+	0.316316i	$0.897554\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	209.062	−	362.106i	0.0830840	−	0.143906i
$186$		0			0
$187$	−1695.40	+	978.838i	−0.662993	+	0.382779i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	649.672	−	375.088i	0.246118	−	0.142097i	−0.371867	−	0.928286i	$-0.621282\pi$
$191$	649.672	−	375.088i	0.246118	−	0.142097i	0.617986	+	0.786189i	$0.287949\pi$
$192$		0			0
$193$	2026.03	−	3509.19i	0.755632	−	1.30879i	−0.189428	−	0.981895i	$-0.560663\pi$
$193$	2026.03	−	3509.19i	0.755632	−	1.30879i	0.945060	−	0.326898i	$-0.106003\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	807.631i		−	0.292088i	−0.989278	−	0.146044i	$-0.953346\pi$
$197$		−	807.631i		−	0.292088i	0.989278	−	0.146044i	$-0.0466541\pi$
$198$		0			0
$199$			4386.10i			1.56243i	0.624265	+	0.781213i	$0.285399\pi$
$199$			4386.10i			1.56243i	−0.624265	+	0.781213i	$0.714601\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	915.309	−	2320.97i	0.316464	−	0.802463i
$204$		0			0
$205$	−410.621	−	711.216i	−0.139897	−	0.242310i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2082.17	−	3606.43i	−0.689124	−	1.19360i
$210$		0			0
$211$	−2132.63	+	3693.82i	−0.695811	+	1.20518i	0.274095	+	0.961702i	$0.411622\pi$
$211$	−2132.63	+	3693.82i	−0.695811	+	1.20518i	−0.969907	+	0.243478i	$0.921712\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3335.50			1.05804
$216$		0			0
$217$	549.762	−	81.9413i	0.171983	−	0.0256338i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	2437.43	+	1407.25i	0.741898	+	0.428335i
$222$		0			0
$223$	1771.94	−	1023.03i	0.532097	−	0.307206i	−0.209773	−	0.977750i	$-0.567273\pi$
$223$	1771.94	−	1023.03i	0.532097	−	0.307206i	0.741870	+	0.670544i	$0.233939\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2797.06	−	4844.65i	−0.817830	−	1.41652i	−0.907278	−	0.420532i	$-0.861843\pi$
$227$	−2797.06	−	4844.65i	−0.817830	−	1.41652i	0.0894471	−	0.995992i	$-0.471490\pi$
$228$		0			0
$229$	318.820	+	184.071i	0.0920009	+	0.0531168i	0.545295	−	0.838244i	$-0.316418\pi$
$229$	318.820	+	184.071i	0.0920009	+	0.0531168i	−0.453294	+	0.891361i	$0.649751\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	3900.42i		−	1.09667i	−0.836258	−	0.548336i	$-0.815261\pi$
$233$		−	3900.42i		−	1.09667i	0.836258	−	0.548336i	$-0.184739\pi$
$234$		0			0
$235$	−3538.29			−0.982180
$236$		0			0
$237$		0			0
$238$		0			0
$239$	2327.57	+	1343.82i	0.629949	+	0.363701i	0.780732	−	0.624866i	$-0.214846\pi$
$239$	2327.57	+	1343.82i	0.629949	+	0.363701i	−0.150783	+	0.988567i	$0.548180\pi$
$240$		0			0
$241$	−2125.83	+	1227.35i	−0.568203	+	0.328052i	−0.756431	−	0.654073i	$-0.773059\pi$
$241$	−2125.83	+	1227.35i	−0.568203	+	0.328052i	0.188228	+	0.982125i	$0.439726\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2589.35	−	2418.42i	−0.675214	−	0.630641i
$246$		0			0
$247$	−2993.49	+	5184.88i	−0.771139	+	1.33565i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	3096.36			0.778648			0.389324	−	0.921101i	$-0.372709\pi$
$251$	3096.36			0.778648			0.389324	+	0.921101i	$0.372709\pi$
$252$		0			0
$253$	−2127.08			−0.528571
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1311.10	+	2270.89i	−0.318227	+	0.551185i	−0.980118	−	0.198415i	$-0.936421\pi$
$257$	−1311.10	+	2270.89i	−0.318227	+	0.551185i	0.661892	+	0.749600i	$0.269754\pi$
$258$		0			0
$259$	586.874	−	466.443i	0.140798	−	0.111905i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6179.60	+	3567.79i	−1.44886	+	0.836500i	−0.998414	−	0.0563032i	$-0.982069\pi$
$263$	−6179.60	+	3567.79i	−1.44886	+	0.836500i	−0.450447	+	0.892803i	$0.648735\pi$
$264$		0			0
$265$	−580.693	−	335.263i	−0.134610	−	0.0777173i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	3846.62			0.871869			0.435934	−	0.899978i	$-0.356418\pi$
$269$	3846.62			0.871869			0.435934	+	0.899978i	$0.356418\pi$
$270$		0			0
$271$			446.364i			0.100054i	0.998748	+	0.0500271i	$0.0159308\pi$
$271$			446.364i			0.100054i	−0.998748	+	0.0500271i	$0.984069\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−496.417	−	286.606i	−0.108855	−	0.0628473i
$276$		0			0
$277$	−3405.06	−	5897.74i	−0.738593	−	1.27928i	−0.953129	−	0.302565i	$-0.902157\pi$
$277$	−3405.06	−	5897.74i	−0.738593	−	1.27928i	0.214535	−	0.976716i	$-0.431176\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2970.96	+	1715.29i	−0.630722	+	0.364147i	−0.781031	−	0.624492i	$-0.785306\pi$
$281$	−2970.96	+	1715.29i	−0.630722	+	0.364147i	0.150310	+	0.988639i	$0.451973\pi$
$282$		0			0
$283$	−275.171	−	158.870i	−0.0577993	−	0.0333705i	0.470822	−	0.882228i	$-0.343957\pi$
$283$	−275.171	−	158.870i	−0.0577993	−	0.0333705i	−0.528621	+	0.848858i	$0.677291\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−217.064	−	1456.33i	−0.0446441	−	0.299527i
$288$		0			0
$289$	−1008.00			−0.205171
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3701.79	+	6411.69i	−0.738091	+	1.27841i	0.215262	+	0.976556i	$0.430939\pi$
$293$	−3701.79	+	6411.69i	−0.738091	+	1.27841i	−0.953354	+	0.301856i	$0.902394\pi$
$294$		0			0
$295$	−819.972	−	1420.23i	−0.161833	−	0.280302i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1529.03	+	2648.35i	0.295739	+	0.512235i
$300$		0			0
$301$	5563.28	+	2193.97i	1.06532	+	0.420126i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			5887.99i			1.10540i
$306$		0			0
$307$		−	9692.73i		−	1.80193i	−0.433888	−	0.900967i	$-0.642859\pi$
$307$		−	9692.73i		−	1.80193i	0.433888	−	0.900967i	$-0.357141\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3957.20	+	6854.07i	−0.721518	+	1.24971i	0.238873	+	0.971051i	$0.423222\pi$
$311$	−3957.20	+	6854.07i	−0.721518	+	1.24971i	−0.960391	+	0.278655i	$0.910111\pi$
$312$		0			0
$313$	−4644.88	+	2681.72i	−0.838799	+	0.484281i	−0.856856	−	0.515556i	$-0.827585\pi$
$313$	−4644.88	+	2681.72i	−0.838799	+	0.484281i	0.0180566	+	0.999837i	$0.494252\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3741.29	+	2160.03i	−0.662876	+	0.382712i	−0.793372	−	0.608737i	$-0.791676\pi$
$317$	−3741.29	+	2160.03i	−0.662876	+	0.382712i	0.130496	+	0.991449i	$0.458343\pi$
$318$		0			0
$319$	−2110.14	+	3654.88i	−0.370362	+	0.641485i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			8306.67i			1.43095i
$324$		0			0
$325$			824.095i			0.140654i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−5901.50	−	2327.35i	−0.988937	−	0.390002i
$330$		0			0
$331$	371.714	+	643.828i	0.0617259	+	0.106912i	0.895237	−	0.445590i	$-0.147006\pi$
$331$	371.714	+	643.828i	0.0617259	+	0.106912i	−0.833511	+	0.552503i	$0.813673\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1558.06	−	2698.65i	−0.254108	−	0.440128i
$336$		0			0
$337$	4874.39	−	8442.69i	0.787908	−	1.36470i	−0.139338	−	0.990245i	$-0.544497\pi$
$337$	4874.39	−	8442.69i	0.787908	−	1.36470i	0.927246	−	0.374452i	$-0.122169\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−940.219			−0.149313
$342$		0			0
$343$	−2728.03	−	5736.85i	−0.429445	−	0.903093i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2953.16	+	1705.01i	0.456870	+	0.263774i	0.710727	−	0.703468i	$-0.248366\pi$
$347$	2953.16	+	1705.01i	0.456870	+	0.263774i	−0.253857	+	0.967242i	$0.581699\pi$
$348$		0			0
$349$	8157.38	−	4709.67i	1.25116	−	0.722357i	0.279820	−	0.960053i	$-0.409725\pi$
$349$	8157.38	−	4709.67i	1.25116	−	0.722357i	0.971340	+	0.237695i	$0.0763919\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1363.14	−	2361.03i	−0.205532	−	0.355992i	0.744770	−	0.667321i	$-0.232559\pi$
$353$	−1363.14	−	2361.03i	−0.205532	−	0.355992i	−0.950302	+	0.311329i	$0.899226\pi$
$354$		0			0
$355$	−6432.91	−	3714.04i	−0.961756	−	0.555270i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			8978.19i			1.31992i	0.751302	+	0.659959i	$0.229426\pi$
$359$			8978.19i			1.31992i	−0.751302	+	0.659959i	$0.770574\pi$
$360$		0			0
$361$	−10810.9			−1.57616
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4998.06	−	2885.63i	−0.716742	−	0.413811i
$366$		0			0
$367$	6601.74	−	3811.52i	0.938987	−	0.542124i	0.0493444	−	0.998782i	$-0.484287\pi$
$367$	6601.74	−	3811.52i	0.938987	−	0.542124i	0.889643	+	0.456657i	$0.150953\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−748.015	−	941.143i	−0.104677	−	0.131703i
$372$		0			0
$373$	−1255.32	+	2174.28i	−0.174258	+	0.301823i	−0.939904	−	0.341438i	$-0.889086\pi$
$373$	−1255.32	+	2174.28i	−0.174258	+	0.301823i	0.765646	+	0.643262i	$0.222419\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	6067.41			0.828879
$378$		0			0
$379$	−7234.68			−0.980529			−0.490265	−	0.871574i	$-0.663100\pi$
$379$	−7234.68			−0.980529			−0.490265	+	0.871574i	$0.663100\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−6219.05	+	10771.7i	−0.829710	+	1.43710i	0.0685562	+	0.997647i	$0.478161\pi$
$383$	−6219.05	+	10771.7i	−0.829710	+	1.43710i	−0.898266	+	0.439452i	$0.855173\pi$
$384$		0			0
$385$	3729.06	+	4691.86i	0.493638	+	0.621090i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2593.05	−	1497.10i	0.337976	−	0.195131i	−0.321400	−	0.946943i	$-0.604154\pi$
$389$	2593.05	−	1497.10i	0.337976	−	0.195131i	0.659377	+	0.751813i	$0.270820\pi$
$390$		0			0
$391$	3674.47	+	2121.46i	0.475258	+	0.274390i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	9434.66			1.20180
$396$		0			0
$397$			3404.67i			0.430417i	0.976568	+	0.215208i	$0.0690430\pi$
$397$			3404.67i			0.430417i	−0.976568	+	0.215208i	$0.930957\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2674.32	−	1544.02i	−0.333040	−	0.192281i	0.324150	−	0.946006i	$-0.394922\pi$
$401$	−2674.32	−	1544.02i	−0.333040	−	0.192281i	−0.657190	+	0.753725i	$0.728255\pi$
$402$		0			0
$403$	675.866	+	1170.63i	0.0835416	+	0.144698i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1098.19	+	634.041i	−0.133748	+	0.0772193i
$408$		0			0
$409$	−8924.66	−	5152.66i	−1.07896	−	0.622940i	−0.148347	−	0.988935i	$-0.547395\pi$
$409$	−8924.66	−	5152.66i	−1.07896	−	0.622940i	−0.930617	+	0.365995i	$0.880729\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−433.456	−	2908.15i	−0.0516440	−	0.346491i
$414$		0			0
$415$	−7279.31			−0.861029
$416$		0			0
$417$		0			0
$418$		0			0
$419$	5734.36	−	9932.20i	0.668596	−	1.15804i	−0.309701	−	0.950834i	$-0.600229\pi$
$419$	5734.36	−	9932.20i	0.668596	−	1.15804i	0.978297	−	0.207208i	$-0.0664378\pi$
$420$		0			0
$421$	−3002.26	−	5200.07i	−0.347557	−	0.601986i	0.638258	−	0.769822i	$-0.279655\pi$
$421$	−3002.26	−	5200.07i	−0.347557	−	0.601986i	−0.985815	+	0.167837i	$0.946322\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	571.697	+	990.208i	0.0652503	+	0.113017i
$426$		0			0
$427$	−3872.89	+	9820.58i	−0.438929	+	1.11300i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			8312.17i			0.928963i	0.885583	+	0.464481i	$0.153759\pi$
$431$			8312.17i			0.928963i	−0.885583	+	0.464481i	$0.846241\pi$
$432$		0			0
$433$			4419.65i			0.490519i	0.969458	+	0.245259i	$0.0788731\pi$
$433$			4419.65i			0.490519i	−0.969458	+	0.245259i	$0.921127\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−4512.74	+	7816.30i	−0.493990	+	0.855616i
$438$		0			0
$439$	−14013.5	+	8090.72i	−1.52353	+	0.879610i	−0.523918	+	0.851769i	$0.675530\pi$
$439$	−14013.5	+	8090.72i	−1.52353	+	0.879610i	−0.999612	+	0.0278416i	$0.991137\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2983.47	+	1722.51i	−0.319976	+	0.184738i	−0.651382	−	0.758750i	$-0.725810\pi$
$443$	−2983.47	+	1722.51i	−0.319976	+	0.184738i	0.331406	+	0.943488i	$0.392477\pi$
$444$		0			0
$445$	3616.08	−	6263.24i	0.385211	−	0.667205i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	3345.89i		−	0.351675i	−0.984419	−	0.175838i	$-0.943737\pi$
$449$		−	3345.89i		−	0.351675i	0.984419	−	0.175838i	$-0.0562634\pi$
$450$		0			0
$451$			2490.65i			0.260045i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	3161.08	−	8015.62i	0.325701	−	0.825886i
$456$		0			0
$457$	2817.23	+	4879.58i	0.288368	+	0.499469i	0.973420	−	0.229025i	$-0.0735539\pi$
$457$	2817.23	+	4879.58i	0.288368	+	0.499469i	−0.685052	+	0.728494i	$0.740221\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−8810.53	−	15260.3i	−0.890124	−	1.54174i	−0.839726	−	0.543011i	$-0.817284\pi$
$461$	−8810.53	−	15260.3i	−0.890124	−	1.54174i	−0.0503981	−	0.998729i	$-0.516049\pi$
$462$		0			0
$463$	1050.86	−	1820.15i	0.105481	−	0.182699i	−0.808454	−	0.588560i	$-0.799695\pi$
$463$	1050.86	−	1820.15i	0.105481	−	0.182699i	0.913935	+	0.405861i	$0.133028\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4439.19			−0.439874			−0.219937	−	0.975514i	$-0.570585\pi$
$467$	−4439.19			−0.439874			−0.219937	+	0.975514i	$0.570585\pi$
$468$		0			0
$469$	−823.629	−	5525.91i	−0.0810910	−	0.544057i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−8760.62	−	5057.94i	−0.851614	−	0.491680i
$474$		0			0
$475$	−2106.36	+	1216.11i	−0.203466	+	0.117471i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1192.77	−	2065.94i	−0.113777	−	0.197067i	0.803513	−	0.595287i	$-0.202961\pi$
$479$	−1192.77	−	2065.94i	−0.113777	−	0.197067i	−0.917290	+	0.398220i	$0.869628\pi$
$480$		0			0
$481$	1578.85	+	911.547i	0.149666	+	0.0864095i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	2418.78i		−	0.226456i
$486$		0			0
$487$	−1032.17			−0.0960409			−0.0480205	−	0.998846i	$-0.515291\pi$
$487$	−1032.17			−0.0960409			−0.0480205	+	0.998846i	$0.515291\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18322.5	+	10578.5i	1.68408	+	0.972305i	0.958899	+	0.283748i	$0.0915781\pi$
$491$	18322.5	+	10578.5i	1.68408	+	0.972305i	0.725183	+	0.688556i	$0.241755\pi$
$492$		0			0
$493$	7290.42	−	4209.13i	0.666012	−	0.384522i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8286.49	−	10426.0i	−0.747887	−	0.940983i
$498$		0			0
$499$	5302.31	−	9183.86i	0.475679	−	0.823900i	−0.523933	−	0.851760i	$-0.675536\pi$
$499$	5302.31	−	9183.86i	0.475679	−	0.823900i	0.999612	+	0.0278593i	$0.00886904\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4721.95			−0.418572			−0.209286	−	0.977855i	$-0.567114\pi$
$503$	−4721.95			−0.418572			−0.209286	+	0.977855i	$0.567114\pi$
$504$		0			0
$505$	19380.5			1.70777
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6295.97	+	10904.9i	−0.548259	+	0.949613i	0.450135	+	0.892961i	$0.351376\pi$
$509$	−6295.97	+	10904.9i	−0.548259	+	0.949613i	−0.998394	+	0.0566522i	$0.981957\pi$
$510$		0			0
$511$	−6438.21	−	8100.48i	−0.557357	−	0.701261i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	5418.00	−	3128.08i	0.463583	−	0.267650i
$516$		0			0
$517$	9293.22	+	5365.44i	0.790552	+	0.456425i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	11584.6			0.974145			0.487072	−	0.873362i	$-0.338065\pi$
$521$	11584.6			0.974145			0.487072	+	0.873362i	$0.338065\pi$
$522$		0			0
$523$			3526.37i			0.294833i	0.989075	+	0.147416i	$0.0470957\pi$
$523$			3526.37i			0.294833i	−0.989075	+	0.147416i	$0.952904\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1624.20	+	937.733i	0.134253	+	0.0775110i
$528$		0			0
$529$	−3778.47	−	6544.50i	−0.310550	−	0.537889i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3101.02	−	1790.38i	0.252008	−	0.145497i
$534$		0			0
$535$	−11626.2	−	6712.39i	−0.939523	−	0.542434i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3133.58	+	10278.4i	0.250413	+	0.821376i
$540$		0			0
$541$	7849.84			0.623828			0.311914	−	0.950110i	$-0.399030\pi$
$541$	7849.84			0.623828			0.311914	+	0.950110i	$0.399030\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	10739.0	−	18600.4i	0.844048	−	1.46193i
$546$		0			0
$547$	7842.27	+	13583.2i	0.613001	+	1.06175i	0.990732	+	0.135833i	$0.0433710\pi$
$547$	7842.27	+	13583.2i	0.613001	+	1.06175i	−0.377731	+	0.925915i	$0.623296\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	8953.62	+	15508.1i	0.692263	+	1.19903i
$552$		0			0
$553$	15736.1	+	6205.75i	1.21006	+	0.477207i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	21036.4i		−	1.60025i	−0.599833	−	0.800126i	$-0.704766\pi$
$557$		−	21036.4i		−	1.60025i	0.599833	−	0.800126i	$-0.295234\pi$
$558$		0			0
$559$			14543.4i			1.10039i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6347.02	+	10993.4i	−0.475124	+	0.822939i	−0.999594	−	0.0284899i	$-0.990930\pi$
$563$	−6347.02	+	10993.4i	−0.475124	+	0.822939i	0.524470	+	0.851429i	$0.324264\pi$
$564$		0			0
$565$	−11264.8	+	6503.75i	−0.838787	+	0.484274i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	15753.0	−	9095.03i	1.16064	−	0.670094i	0.209180	−	0.977877i	$-0.432921\pi$
$569$	15753.0	−	9095.03i	1.16064	−	0.670094i	0.951456	+	0.307783i	$0.0995872\pi$
$570$		0			0
$571$	−10146.0	+	17573.3i	−0.743599	+	1.28795i	0.207247	+	0.978289i	$0.433550\pi$
$571$	−10146.0	+	17573.3i	−0.743599	+	1.28795i	−0.950846	+	0.309663i	$0.899784\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1242.34i			0.0901026i
$576$		0			0
$577$			12356.1i			0.891494i	0.895159	+	0.445747i	$0.147062\pi$
$577$			12356.1i			0.891494i	−0.895159	+	0.445747i	$0.852938\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12141.1	−	4788.05i	−0.866953	−	0.341896i
$582$		0			0
$583$	1016.78	+	1761.12i	0.0722314	+	0.125108i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1434.42	−	2484.50i	−0.100860	−	0.174695i	0.811179	−	0.584798i	$-0.198826\pi$
$587$	−1434.42	−	2484.50i	−0.100860	−	0.174695i	−0.912039	+	0.410103i	$0.865493\pi$
$588$		0			0
$589$	−1994.74	+	3454.99i	−0.139544	+	0.241698i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	3259.99			0.225754			0.112877	−	0.993609i	$-0.463993\pi$
$593$	3259.99			0.225754			0.112877	+	0.993609i	$0.463993\pi$
$594$		0			0
$595$	−1762.39	−	11824.3i	−0.121430	−	0.814702i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19971.2	−	11530.4i	−1.36227	−	0.786509i	−0.372348	−	0.928093i	$-0.621447\pi$
$599$	−19971.2	−	11530.4i	−1.36227	−	0.786509i	−0.989926	+	0.141584i	$0.954780\pi$
$600$		0			0
$601$	10827.8	−	6251.41i	0.734897	−	0.424293i	−0.0853138	−	0.996354i	$-0.527189\pi$
$601$	10827.8	−	6251.41i	0.734897	−	0.424293i	0.820211	+	0.572061i	$0.193856\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	1805.46	+	3127.15i	0.121326	+	0.210143i
$606$		0			0
$607$	−6638.81	−	3832.92i	−0.443923	−	0.256299i	0.261338	−	0.965247i	$-0.415836\pi$
$607$	−6638.81	−	3832.92i	−0.443923	−	0.256299i	−0.705260	+	0.708949i	$0.749170\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	15427.5i		−	1.02149i
$612$		0			0
$613$	20585.4			1.35634			0.678170	−	0.734905i	$-0.262773\pi$
$613$	20585.4			1.35634			0.678170	+	0.734905i	$0.262773\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	14174.2	+	8183.50i	0.924852	+	0.533963i	0.885180	−	0.465249i	$-0.154035\pi$
$617$	14174.2	+	8183.50i	0.924852	+	0.533963i	0.0396720	+	0.999213i	$0.487369\pi$
$618$		0			0
$619$	−6056.80	+	3496.90i	−0.393285	+	0.227063i	−0.683583	−	0.729873i	$-0.739579\pi$
$619$	−6056.80	+	3496.90i	−0.393285	+	0.227063i	0.290298	+	0.956936i	$0.406246\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	10151.0	−	8067.93i	0.652793	−	0.518836i
$624$		0			0
$625$	6501.53	−	11261.0i	0.416098	−	0.720703i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2529.46			0.160344
$630$		0			0
$631$	8776.78			0.553721			0.276861	−	0.960910i	$-0.410706\pi$
$631$	8776.78			0.553721			0.276861	+	0.960910i	$0.410706\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5245.55	+	9085.56i	−0.327816	+	0.567794i
$636$		0			0
$637$	10544.7	−	11290.0i	0.655883	−	0.702239i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	2562.09	−	1479.23i	0.157873	−	0.0911481i	−0.418982	−	0.907994i	$-0.637613\pi$
$641$	2562.09	−	1479.23i	0.157873	−	0.0911481i	0.576855	+	0.816846i	$0.304280\pi$
$642$		0			0
$643$	11590.6	+	6691.85i	0.710870	+	0.410421i	0.811383	−	0.584515i	$-0.198715\pi$
$643$	11590.6	+	6691.85i	0.710870	+	0.410421i	−0.100513	+	0.994936i	$0.532048\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−20353.0			−1.23672			−0.618362	−	0.785893i	$-0.712203\pi$
$647$	−20353.0			−1.23672			−0.618362	+	0.785893i	$0.712203\pi$
$648$		0			0
$649$			4973.61i			0.300818i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13525.8	+	7809.11i	0.810574	+	0.467985i	0.847155	−	0.531346i	$-0.178313\pi$
$653$	13525.8	+	7809.11i	0.810574	+	0.467985i	−0.0365814	+	0.999331i	$0.511647\pi$
$654$		0			0
$655$	13984.3	+	24221.6i	0.834219	+	1.44491i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	5847.81	−	3376.24i	0.345673	−	0.199574i	−0.317105	−	0.948390i	$-0.602711\pi$
$659$	5847.81	−	3376.24i	0.345673	−	0.199574i	0.662778	+	0.748816i	$0.269377\pi$
$660$		0			0
$661$	8658.65	+	4999.07i	0.509504	+	0.294162i	0.732630	−	0.680627i	$-0.238293\pi$
$661$	8658.65	+	4999.07i	0.509504	+	0.294162i	−0.223126	+	0.974790i	$0.571626\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	25152.5	−	3748.94i	1.46672	−	0.218613i
$666$		0			0
$667$	9146.72			0.530978
$668$		0			0
$669$		0			0
$670$		0			0
$671$	8928.53	−	15464.7i	0.513684	−	0.889727i
$672$		0			0
$673$	−31.7088	−	54.9213i	−0.00181617	−	0.00314570i	0.865116	−	0.501572i	$-0.167245\pi$
$673$	−31.7088	−	54.9213i	−0.00181617	−	0.00314570i	−0.866932	+	0.498426i	$0.833911\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8874.33	+	15370.8i	0.503793	+	0.872596i	0.999990	+	0.00438565i	$0.00139600\pi$
$677$	8874.33	+	15370.8i	0.503793	+	0.872596i	−0.496197	+	0.868210i	$0.665271\pi$
$678$		0			0
$679$	1590.98	−	4034.28i	0.0899208	−	0.228014i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	8356.42i		−	0.468155i	−0.972218	−	0.234077i	$-0.924793\pi$
$683$		−	8356.42i		−	0.468155i	0.972218	−	0.234077i	$-0.0752069\pi$
$684$		0			0
$685$		−	32370.8i		−	1.80558i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1461.81	−	2531.92i	0.0808279	−	0.139998i
$690$		0			0
$691$	−25237.8	+	14571.1i	−1.38942	+	0.802185i	−0.993250	−	0.115991i	$-0.962996\pi$
$691$	−25237.8	+	14571.1i	−1.38942	+	0.802185i	−0.396174	+	0.918175i	$0.629662\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−894.419	+	516.393i	−0.0488162	+	0.0281840i
$696$		0			0
$697$	2484.07	−	4302.53i	0.134994	−	0.233816i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	30359.4i		−	1.63574i	−0.575400	−	0.817872i	$-0.695154\pi$
$701$		−	30359.4i		−	1.63574i	0.575400	−	0.817872i	$-0.304846\pi$
$702$		0			0
$703$			5380.64i			0.288670i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	32324.8	+	12747.8i	1.71952	+	0.678118i
$708$		0			0
$709$	−6063.83	−	10502.9i	−0.321202	−	0.556338i	0.659535	−	0.751674i	$-0.270753\pi$
$709$	−6063.83	−	10502.9i	−0.321202	−	0.556338i	−0.980736	+	0.195337i	$0.937420\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1018.88	+	1764.75i	0.0535166	+	0.0926934i
$714$		0			0
$715$	−7287.52	+	12622.4i	−0.381172	+	0.660209i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	9928.20			0.514964			0.257482	−	0.966283i	$-0.417107\pi$
$719$	9928.20			0.514964			0.257482	+	0.966283i	$0.417107\pi$
$720$		0			0
$721$	11094.2	−	1653.58i	0.573051	−	0.0854125i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	2134.66	+	1232.44i	0.109351	+	0.0631335i
$726$		0			0
$727$	26251.7	−	15156.4i	1.33923	−	0.773204i	0.352536	−	0.935798i	$-0.385319\pi$
$727$	26251.7	−	15156.4i	1.33923	−	0.773204i	0.986693	+	0.162594i	$0.0519860\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	10089.1	+	17474.9i	0.510479	+	0.884176i
$732$		0			0
$733$	−16344.8	−	9436.67i	−0.823613	−	0.475513i	0.0280476	−	0.999607i	$-0.491071\pi$
$733$	−16344.8	−	9436.67i	−0.823613	−	0.475513i	−0.851661	+	0.524093i	$0.824404\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			9450.57i			0.472342i
$738$		0			0
$739$	24825.4			1.23575			0.617873	−	0.786278i	$-0.287995\pi$
$739$	24825.4			1.23575			0.617873	+	0.786278i	$0.287995\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−22599.1	−	13047.6i	−1.11586	−	0.644241i	−0.175517	−	0.984476i	$-0.556160\pi$
$743$	−22599.1	−	13047.6i	−1.11586	−	0.644241i	−0.940340	+	0.340236i	$0.889493\pi$
$744$		0			0
$745$	−23553.7	+	13598.8i	−1.15831	+	0.668752i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14976.2	−	18842.9i	−0.730598	−	0.919230i
$750$		0			0
$751$	13211.2	−	22882.5i	0.641922	−	1.11184i	−0.343081	−	0.939306i	$-0.611470\pi$
$751$	13211.2	−	22882.5i	0.641922	−	1.11184i	0.985003	−	0.172536i	$-0.0551962\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−19511.4			−0.940520
$756$		0			0
$757$	28263.9			1.35703			0.678513	−	0.734588i	$-0.262625\pi$
$757$	28263.9			1.35703			0.678513	+	0.734588i	$0.262625\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19195.4	+	33247.5i	−0.914369	+	1.58373i	−0.106545	+	0.994308i	$0.533979\pi$
$761$	−19195.4	+	33247.5i	−0.914369	+	1.58373i	−0.807823	+	0.589425i	$0.799354\pi$
$762$		0			0
$763$	30146.1	−	23959.9i	1.43036	−	1.13684i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	6192.46	−	3575.22i	0.291521	−	0.168310i
$768$		0			0
$769$	−26820.3	−	15484.7i	−1.25769	−	0.726129i	−0.285067	−	0.958508i	$-0.592016\pi$
$769$	−26820.3	−	15484.7i	−1.25769	−	0.726129i	−0.972625	+	0.232378i	$0.925349\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−28691.6			−1.33501			−0.667507	−	0.744604i	$-0.732638\pi$
$773$	−28691.6			−1.33501			−0.667507	+	0.744604i	$0.732638\pi$
$774$		0			0
$775$			549.142i			0.0254526i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	9152.30	+	5284.08i	0.420944	+	0.243032i
$780$		0			0
$781$	11263.9	+	19509.7i	0.516075	+	0.893868i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11470.2	+	6622.34i	−0.521516	+	0.301097i
$786$		0			0
$787$	−19658.2	−	11349.7i	−0.890392	−	0.514068i	−0.0163216	−	0.999867i	$-0.505196\pi$
$787$	−19658.2	−	11349.7i	−0.890392	−	0.514068i	−0.874071	+	0.485798i	$0.838529\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−23066.5	+	3438.03i	−1.03685	+	0.154542i
$792$		0			0
$793$	−25672.7			−1.14964
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4945.60	+	8566.03i	−0.219802	+	0.380708i	−0.954747	−	0.297418i	$-0.903874\pi$
$797$	−4945.60	+	8566.03i	−0.219802	+	0.380708i	0.734945	+	0.678126i	$0.237208\pi$
$798$		0			0
$799$	−10702.5	−	18537.3i	−0.473877	−	0.820778i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8751.53	+	15158.1i	0.384601	+	0.666149i
$804$		0			0
$805$	4765.38	−	12083.7i	0.208643	−	0.529060i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	15328.4i		−	0.666155i	−0.942900	−	0.333077i	$-0.891913\pi$
$809$		−	15328.4i		−	0.666155i	0.942900	−	0.333077i	$-0.108087\pi$
$810$		0			0
$811$		−	4463.96i		−	0.193281i	−0.995319	−	0.0966406i	$-0.969190\pi$
$811$		−	4463.96i		−	0.193281i	0.995319	−	0.0966406i	$-0.0308097\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	18379.8	−	31834.8i	0.789959	−	1.36825i
$816$		0			0
$817$	−37172.4	+	21461.5i	−1.59180	+	0.919025i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−13660.6	+	7886.94i	−0.580704	+	0.335269i	−0.761413	−	0.648267i	$-0.775494\pi$
$821$	−13660.6	+	7886.94i	−0.580704	+	0.335269i	0.180709	+	0.983537i	$0.442161\pi$
$822$		0			0
$823$	−960.810	+	1664.17i	−0.0406947	+	0.0704853i	−0.885655	−	0.464343i	$-0.846290\pi$
$823$	−960.810	+	1664.17i	−0.0406947	+	0.0704853i	0.844961	+	0.534828i	$0.179624\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	28140.7i		−	1.18325i	−0.806214	−	0.591624i	$-0.798487\pi$
$827$		−	28140.7i		−	1.18325i	0.806214	−	0.591624i	$-0.201513\pi$
$828$		0			0
$829$			13305.9i			0.557460i	0.960370	+	0.278730i	$0.0899134\pi$
$829$			13305.9i			0.557460i	−0.960370	+	0.278730i	$0.910087\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	4838.05	−	20880.9i	0.201235	−	0.868524i
$834$		0			0
$835$	4871.52	+	8437.73i	0.201899	+	0.349700i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	8913.20	+	15438.1i	0.366767	+	0.635260i	0.989058	−	0.147526i	$-0.0471311\pi$
$839$	8913.20	+	15438.1i	0.366767	+	0.635260i	−0.622291	+	0.782786i	$0.713798\pi$
$840$		0			0
$841$	−3120.61	+	5405.06i	−0.127952	+	0.221619i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−1740.16			−0.0708442
$846$		0			0
$847$	954.409	+	6403.33i	0.0387177	+	0.259765i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2380.14	+	1374.17i	0.0958754	+	0.0553537i
$852$		0			0
$853$	35437.7	−	20459.9i	1.42247	−	0.821261i	0.425956	−	0.904744i	$-0.359938\pi$
$853$	35437.7	−	20459.9i	1.42247	−	0.821261i	0.996509	+	0.0834834i	$0.0266046\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14081.4	+	24389.8i	0.561275	+	0.972157i	0.997386	+	0.0722641i	$0.0230225\pi$
$857$	14081.4	+	24389.8i	0.561275	+	0.972157i	−0.436110	+	0.899893i	$0.643644\pi$
$858$		0			0
$859$	3110.46	+	1795.83i	0.123548	+	0.0713304i	0.560500	−	0.828154i	$-0.310609\pi$
$859$	3110.46	+	1795.83i	0.123548	+	0.0713304i	−0.436952	+	0.899485i	$0.643942\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			18091.2i			0.713593i	0.934182	+	0.356796i	$0.116131\pi$
$863$			18091.2i			0.713593i	−0.934182	+	0.356796i	$0.883869\pi$
$864$		0			0
$865$	−10118.8			−0.397747
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−24779.9	−	14306.7i	−0.967319	−	0.558482i
$870$		0			0
$871$	11766.6	−	6793.43i	0.457744	−	0.264279i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	21461.1	−	17057.2i	0.829165	−	0.659015i
$876$		0			0
$877$	−23215.3	+	40210.1i	−0.893871	+	1.54823i	−0.0586762	+	0.998277i	$0.518688\pi$
$877$	−23215.3	+	40210.1i	−0.893871	+	1.54823i	−0.835195	+	0.549954i	$0.814645\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	5007.33			0.191488			0.0957441	−	0.995406i	$-0.469477\pi$
$881$	5007.33			0.191488			0.0957441	+	0.995406i	$0.469477\pi$
$882$		0			0
$883$	−49999.0			−1.90555			−0.952775	−	0.303677i	$-0.901786\pi$
$883$	−49999.0			−1.90555			−0.952775	+	0.303677i	$0.901786\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1730.41	−	2997.16i	0.0655034	−	0.113455i	−0.831414	−	0.555654i	$-0.812468\pi$
$887$	1730.41	−	2997.16i	0.0655034	−	0.113455i	0.896917	+	0.442199i	$0.145801\pi$
$888$		0			0
$889$	−14725.2	+	11703.5i	−0.555531	+	0.441532i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	39432.3	−	22766.3i	1.47766	−	0.853129i
$894$		0			0
$895$	33456.9	+	19316.3i	1.24954	+	0.721423i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	4043.06			0.149993
$900$		0			0
$901$		−	4056.38i		−	0.149986i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	13781.2	+	7956.58i	0.506191	+	0.292249i
$906$		0			0
$907$	14546.9	+	25196.0i	0.532550	+	0.922404i	0.999278	+	0.0380031i	$0.0120997\pi$
$907$	14546.9	+	25196.0i	0.532550	+	0.922404i	−0.466727	+	0.884401i	$0.654567\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−28776.2	+	16613.9i	−1.04654	+	0.604220i	−0.921679	−	0.387954i	$-0.873182\pi$
$911$	−28776.2	+	16613.9i	−1.04654	+	0.604220i	−0.124861	+	0.992174i	$0.539849\pi$
$912$		0			0
$913$	19118.9	+	11038.3i	0.693038	+	0.400126i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	7392.44	+	49597.5i	0.266216	+	1.78610i
$918$		0			0
$919$	−11642.5			−0.417902			−0.208951	−	0.977926i	$-0.567005\pi$
$919$	−11642.5			−0.417902			−0.208951	+	0.977926i	$0.567005\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	16193.9	−	28048.6i	0.577495	−	1.00025i
$924$		0			0
$925$	370.316	+	641.407i	0.0131632	+	0.0227993i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	19057.0	+	33007.6i	0.673024	+	1.16571i	0.977042	+	0.213045i	$0.0683381\pi$
$929$	19057.0	+	33007.6i	0.673024	+	1.16571i	−0.304019	+	0.952666i	$0.598329\pi$
$930$		0			0
$931$	44417.7	+	10291.5i	1.56362	+	0.362287i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			20222.2i			0.707312i
$936$		0			0
$937$			22454.2i			0.782869i	0.920206	+	0.391435i	$0.128021\pi$
$937$			22454.2i			0.782869i	−0.920206	+	0.391435i	$0.871979\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6159.92	−	10669.3i	0.213398	−	0.369616i	−0.739378	−	0.673291i	$-0.764880\pi$
$941$	6159.92	−	10669.3i	0.213398	−	0.369616i	0.952776	+	0.303675i	$0.0982135\pi$
$942$		0			0
$943$	4674.84	−	2699.02i	0.161436	−	0.0932049i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−13396.5	+	7734.46i	−0.459691	+	0.265403i	−0.711914	−	0.702266i	$-0.752172\pi$
$947$	−13396.5	+	7734.46i	−0.459691	+	0.265403i	0.252223	+	0.967669i	$0.418838\pi$
$948$		0			0
$949$	12581.9	−	21792.4i	0.430374	−	0.745429i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			5267.67i			0.179052i	0.995984	+	0.0895260i	$0.0285352\pi$
$953$			5267.67i			0.179052i	−0.995984	+	0.0895260i	$0.971465\pi$
$954$		0			0
$955$		−	7749.10i		−	0.262571i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	21292.3	−	53991.2i	0.716958	−	1.81801i
$960$		0			0
$961$	−14445.1	−	25019.7i	−0.484882	−	0.839841i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−20928.3	−	36248.9i	−0.698141	−	1.20922i
$966$		0			0
$967$	6237.09	−	10803.0i	0.207416	−	0.359255i	−0.743484	−	0.668754i	$-0.766828\pi$
$967$	6237.09	−	10803.0i	0.207416	−	0.359255i	0.950900	+	0.309499i	$0.100161\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−57914.6			−1.91407			−0.957037	−	0.289964i	$-0.906357\pi$
$971$	−57914.6			−1.91407			−0.957037	+	0.289964i	$0.906357\pi$
$972$		0			0
$973$	−1831.46	+	272.977i	−0.0603433	+	0.00899409i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−15954.7	−	9211.48i	−0.522454	−	0.301639i	0.215484	−	0.976507i	$-0.430867\pi$
$977$	−15954.7	−	9211.48i	−0.522454	−	0.301639i	−0.737938	+	0.674868i	$0.764200\pi$
$978$		0			0
$979$	−18995.1	+	10966.8i	−0.620108	+	0.358020i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−4293.70	−	7436.90i	−0.139316	−	0.241302i	0.787922	−	0.615775i	$-0.211157\pi$
$983$	−4293.70	−	7436.90i	−0.139316	−	0.241302i	−0.927238	+	0.374473i	$0.877824\pi$
$984$		0			0
$985$	−7224.90	−	4171.30i	−0.233710	−	0.134933i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			21924.4i			0.704909i
$990$		0			0
$991$	−17639.5			−0.565426			−0.282713	−	0.959205i	$-0.591234\pi$
$991$	−17639.5			−0.565426			−0.282713	+	0.959205i	$0.591234\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	39237.1	+	22653.6i	1.25015	+	0.721776i
$996$		0			0
$997$	−4180.85	+	2413.82i	−0.132807	+	0.0766764i	−0.564932	−	0.825138i	$-0.691097\pi$
$997$	−4180.85	+	2413.82i	−0.132807	+	0.0766764i	0.432124	+	0.901814i	$0.357764\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.4.x.a.125.18		48
3.2	odd	2		252.4.x.a.41.4	✓	48
7.6	odd	2	inner	756.4.x.a.125.7		48
9.2	odd	6	inner	756.4.x.a.629.7		48
9.4	even	3		2268.4.f.a.1133.13		48
9.5	odd	6		2268.4.f.a.1133.36		48
9.7	even	3		252.4.x.a.209.21	yes	48
21.20	even	2		252.4.x.a.41.21	yes	48
63.13	odd	6		2268.4.f.a.1133.35		48
63.20	even	6	inner	756.4.x.a.629.18		48
63.34	odd	6		252.4.x.a.209.4	yes	48
63.41	even	6		2268.4.f.a.1133.14		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.4.x.a.41.4	✓	48	3.2	odd	2
252.4.x.a.41.21	yes	48	21.20	even	2
252.4.x.a.209.4	yes	48	63.34	odd	6
252.4.x.a.209.21	yes	48	9.7	even	3
756.4.x.a.125.7		48	7.6	odd	2	inner
756.4.x.a.125.18		48	1.1	even	1	trivial
756.4.x.a.629.7		48	9.2	odd	6	inner
756.4.x.a.629.18		48	63.20	even	6	inner
2268.4.f.a.1133.13		48	9.4	even	3
2268.4.f.a.1133.14		48	63.41	even	6
2268.4.f.a.1133.35		48	63.13	odd	6
2268.4.f.a.1133.36		48	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	756.x (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(5.16485 - 8.94579i) q^{5} +(14.4986 - 11.5234i) q^{7} +O(q^{10})\)	\(q+(5.16485 - 8.94579i) q^{5} +(14.4986 - 11.5234i) q^{7} +(-27.1307 + 15.6639i) q^{11} +(39.0052 + 22.5196i) q^{13} +62.4900 q^{17} +132.928i q^{19} +(58.8009 + 33.9487i) q^{23} +(9.14862 + 15.8459i) q^{25} +(116.665 - 67.3568i) q^{29} +(25.9914 + 15.0061i) q^{31} +(-28.2028 - 189.219i) q^{35} +40.4778 q^{37} +(39.7514 - 68.8515i) q^{41} +(161.452 + 279.643i) q^{43} +(-171.268 - 296.644i) q^{47} +(77.4212 - 334.148i) q^{49} -64.9125i q^{53} +323.607i q^{55} +(79.3800 - 137.490i) q^{59} +(-493.640 + 285.003i) q^{61} +(402.912 - 232.621i) q^{65} +(150.833 - 261.251i) q^{67} -719.100i q^{71} -558.706i q^{73} +(-212.856 + 539.744i) q^{77} +(456.676 + 790.986i) q^{79} +(-352.348 - 610.285i) q^{83} +(322.751 - 559.022i) q^{85} +700.133 q^{89} +(825.026 - 122.969i) q^{91} +(1189.15 + 686.554i) q^{95} +(202.787 - 117.079i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 6 q^{7}+O(q^{10}) \) 48 * q + 6 * q^7	\( 48 q + 6 q^{7} + 12 q^{11} + 408 q^{23} - 600 q^{25} + 84 q^{29} + 336 q^{37} + 84 q^{43} + 318 q^{49} - 2964 q^{65} - 588 q^{67} - 2400 q^{77} + 204 q^{79} - 360 q^{85} - 1080 q^{91} - 300 q^{95}+O(q^{100}) \) 48 * q + 6 * q^7 + 12 * q^11 + 408 * q^23 - 600 * q^25 + 84 * q^29 + 336 * q^37 + 84 * q^43 + 318 * q^49 - 2964 * q^65 - 588 * q^67 - 2400 * q^77 + 204 * q^79 - 360 * q^85 - 1080 * q^91 - 300 * q^95

Introduction

L-functions

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