LMFDB - Embedded newform 504.4.bl.a.17.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [504,4,Mod(17,504)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(504, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 3, 1]))

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

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

chi := DirichletCharacter("504.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 504 = 2^{3} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	504.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$29.7369626429$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			17.5
Character	$\chi$	$=$	504.17
Dual form			504.4.bl.a.89.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.29543 - 12.6361i) q^{5} +(18.4933 + 0.998739i) q^{7} +O(q^{10})$	$q+(-7.29543 - 12.6361i) q^{5} +(18.4933 + 0.998739i) q^{7} +(38.7318 + 22.3618i) q^{11} -76.8588i q^{13} +(-58.3464 + 101.059i) q^{17} +(83.5432 - 48.2337i) q^{19} +(-6.88438 + 3.97470i) q^{23} +(-43.9467 + 76.1180i) q^{25} -86.8540i q^{29} +(216.262 + 124.859i) q^{31} +(-122.297 - 240.969i) q^{35} +(-160.031 - 277.182i) q^{37} +231.290 q^{41} -413.282 q^{43} +(-235.508 - 407.911i) q^{47} +(341.005 + 36.9400i) q^{49} +(-600.339 - 346.606i) q^{53} -652.557i q^{55} +(-143.802 + 249.072i) q^{59} +(740.898 - 427.758i) q^{61} +(-971.193 + 560.718i) q^{65} +(240.341 - 416.283i) q^{67} -930.276i q^{71} +(-98.4174 - 56.8213i) q^{73} +(693.946 + 452.227i) q^{77} +(111.374 + 192.905i) q^{79} +692.424 q^{83} +1702.65 q^{85} +(258.667 + 448.025i) q^{89} +(76.7619 - 1421.37i) q^{91} +(-1218.97 - 703.771i) q^{95} -807.511i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 24 q^{7}+O(q^{10}) $ 48 * q - 24 * q^7	$ 48 q - 24 q^{7} + 540 q^{19} - 924 q^{25} + 648 q^{31} - 132 q^{37} - 792 q^{43} + 672 q^{49} - 12 q^{67} + 2412 q^{73} + 1680 q^{79} + 480 q^{85} + 1404 q^{91}+O(q^{100}) $ 48 * q - 24 * q^7 + 540 * q^19 - 924 * q^25 + 648 * q^31 - 132 * q^37 - 792 * q^43 + 672 * q^49 - 12 * q^67 + 2412 * q^73 + 1680 * q^79 + 480 * q^85 + 1404 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/504\mathbb{Z}\right)^\times$.

$n$	$73$	$127$	$253$	$281$
$\chi(n)$	$e\left(\frac{1}{6}\right)$	$1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−7.29543	−	12.6361i	−0.652523	−	1.13020i	−0.982509	−	0.186218i	$-0.940377\pi$
$5$	−7.29543	−	12.6361i	−0.652523	−	1.13020i	0.329985	−	0.943986i	$-0.392956\pi$
$6$		0			0
$7$	18.4933	+	0.998739i	0.998545	+	0.0539268i
$7$	18.4933	+	0.998739i	0.998545	+	0.0539268i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	38.7318	+	22.3618i	1.06164	+	0.612940i	0.925886	−	0.377802i	$-0.123320\pi$
$11$	38.7318	+	22.3618i	1.06164	+	0.612940i	0.135757	+	0.990742i	$0.456653\pi$
$12$		0			0
$13$		−	76.8588i		−	1.63975i	−0.572540	−	0.819877i	$-0.694042\pi$
$13$		−	76.8588i		−	1.63975i	0.572540	−	0.819877i	$-0.305958\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−58.3464	+	101.059i	−0.832416	+	1.44179i	0.0637008	+	0.997969i	$0.479710\pi$
$17$	−58.3464	+	101.059i	−0.832416	+	1.44179i	−0.896117	+	0.443818i	$0.853624\pi$
$18$		0			0
$19$	83.5432	−	48.2337i	1.00874	−	0.582398i	0.0979195	−	0.995194i	$-0.468781\pi$
$19$	83.5432	−	48.2337i	1.00874	−	0.582398i	0.910823	+	0.412796i	$0.135448\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.88438	+	3.97470i	−0.0624127	+	0.0360340i	−0.530882	−	0.847446i	$-0.678139\pi$
$23$	−6.88438	+	3.97470i	−0.0624127	+	0.0360340i	0.468469	+	0.883480i	$0.344806\pi$
$24$		0			0
$25$	−43.9467	+	76.1180i	−0.351574	+	0.608944i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	86.8540i		−	0.556151i	−0.960559	−	0.278076i	$-0.910303\pi$
$29$		−	86.8540i		−	0.556151i	0.960559	−	0.278076i	$-0.0896966\pi$
$30$		0			0
$31$	216.262	+	124.859i	1.25296	+	0.723397i	0.971696	−	0.236233i	$-0.0759128\pi$
$31$	216.262	+	124.859i	1.25296	+	0.723397i	0.281265	+	0.959630i	$0.409246\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−122.297	−	240.969i	−0.590626	−	1.16375i
$36$		0			0
$37$	−160.031	−	277.182i	−0.711053	−	1.23158i	−0.964462	−	0.264221i	$-0.914885\pi$
$37$	−160.031	−	277.182i	−0.711053	−	1.23158i	0.253409	−	0.967359i	$-0.418448\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	231.290			0.881010			0.440505	−	0.897750i	$-0.354799\pi$
$41$	231.290			0.881010			0.440505	+	0.897750i	$0.354799\pi$
$42$		0			0
$43$	−413.282			−1.46569			−0.732847	−	0.680393i	$-0.761809\pi$
$43$	−413.282			−1.46569			−0.732847	+	0.680393i	$0.761809\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−235.508	−	407.911i	−0.730901	−	1.26596i	−0.956499	−	0.291737i	$-0.905767\pi$
$47$	−235.508	−	407.911i	−0.730901	−	1.26596i	0.225598	−	0.974220i	$-0.427566\pi$
$48$		0			0
$49$	341.005	+	36.9400i	0.994184	+	0.107697i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−600.339	−	346.606i	−1.55590	−	0.898301i	−0.997642	−	0.0686360i	$-0.978135\pi$
$53$	−600.339	−	346.606i	−1.55590	−	0.898301i	−0.558261	−	0.829665i	$-0.688531\pi$
$54$		0			0
$55$		−	652.557i		−	1.59983i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−143.802	+	249.072i	−0.317312	+	0.549600i	−0.979926	−	0.199361i	$-0.936113\pi$
$59$	−143.802	+	249.072i	−0.317312	+	0.549600i	0.662615	+	0.748961i	$0.269447\pi$
$60$		0			0
$61$	740.898	−	427.758i	1.55512	−	0.897849i	0.557408	−	0.830238i	$-0.311796\pi$
$61$	740.898	−	427.758i	1.55512	−	0.897849i	0.997712	−	0.0676105i	$-0.0215375\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−971.193	+	560.718i	−1.85326	+	1.06998i
$66$		0			0
$67$	240.341	−	416.283i	0.438244	−	0.759061i	−0.559310	−	0.828958i	$-0.688934\pi$
$67$	240.341	−	416.283i	0.438244	−	0.759061i	0.997554	+	0.0698977i	$0.0222673\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	930.276i		−	1.55498i	−0.628897	−	0.777489i	$-0.716493\pi$
$71$		−	930.276i		−	1.55498i	0.628897	−	0.777489i	$-0.283507\pi$
$72$		0			0
$73$	−98.4174	−	56.8213i	−0.157793	−	0.0911018i	0.419024	−	0.907975i	$-0.362372\pi$
$73$	−98.4174	−	56.8213i	−0.157793	−	0.0911018i	−0.576817	+	0.816873i	$0.695706\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	693.946	+	452.227i	1.02704	+	0.669299i
$78$		0			0
$79$	111.374	+	192.905i	0.158614	+	0.274727i	0.934369	−	0.356307i	$-0.115964\pi$
$79$	111.374	+	192.905i	0.158614	+	0.274727i	−0.775755	+	0.631034i	$0.782631\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	692.424			0.915703			0.457852	−	0.889029i	$-0.348619\pi$
$83$	692.424			0.915703			0.457852	+	0.889029i	$0.348619\pi$
$84$		0			0
$85$	1702.65			2.17268
$86$		0			0
$87$		0			0
$88$		0			0
$89$	258.667	+	448.025i	0.308075	+	0.533602i	0.977941	−	0.208880i	$-0.0669818\pi$
$89$	258.667	+	448.025i	0.308075	+	0.533602i	−0.669866	+	0.742482i	$0.733648\pi$
$90$		0			0
$91$	76.7619	−	1421.37i	0.0884267	−	1.63737i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1218.97	−	703.771i	−1.31646	−	0.760057i
$96$		0			0
$97$		−	807.511i		−	0.845261i	−0.906302	−	0.422630i	$-0.861107\pi$
$97$		−	807.511i		−	0.845261i	0.906302	−	0.422630i	$-0.138893\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	115.748	−	200.482i	0.114033	−	0.197511i	−0.803360	−	0.595494i	$-0.796956\pi$
$101$	115.748	−	200.482i	0.114033	−	0.197511i	0.917393	+	0.397983i	$0.130290\pi$
$102$		0			0
$103$	−1028.60	+	593.864i	−0.983992	+	0.568108i	−0.903473	−	0.428645i	$-0.858991\pi$
$103$	−1028.60	+	593.864i	−0.983992	+	0.568108i	−0.0805191	+	0.996753i	$0.525658\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.77425	+	2.17906i	−0.00341000	+	0.00196877i	−0.501704	−	0.865039i	$-0.667293\pi$
$107$	−3.77425	+	2.17906i	−0.00341000	+	0.00196877i	0.498294	+	0.867008i	$0.333960\pi$
$108$		0			0
$109$	−113.726	+	196.980i	−0.0999360	+	0.173094i	−0.911658	−	0.410950i	$-0.865197\pi$
$109$	−113.726	+	196.980i	−0.0999360	+	0.173094i	0.811722	+	0.584044i	$0.198530\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	517.680i		−	0.430967i	−0.976507	−	0.215483i	$-0.930867\pi$
$113$		−	517.680i		−	0.430967i	0.976507	−	0.215483i	$-0.0691327\pi$
$114$		0			0
$115$	100.449	+	57.9943i	0.0814515	+	0.0470261i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1179.95	+	1810.64i	−0.908956	+	1.39480i
$120$		0			0
$121$	334.602	+	579.548i	0.251392	+	0.435423i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−541.417			−0.387406
$126$		0			0
$127$	−1665.46			−1.16367			−0.581833	−	0.813308i	$-0.697664\pi$
$127$	−1665.46			−1.16367			−0.581833	+	0.813308i	$0.697664\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−342.534	−	593.286i	−0.228453	−	0.395692i	0.728897	−	0.684624i	$-0.240033\pi$
$131$	−342.534	−	593.286i	−0.228453	−	0.395692i	−0.957350	+	0.288931i	$0.906700\pi$
$132$		0			0
$133$	1593.16	−	808.562i	1.03868	−	0.527152i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	910.457	+	525.653i	0.567778	+	0.327807i	0.756261	−	0.654270i	$-0.227024\pi$
$137$	910.457	+	525.653i	0.567778	+	0.327807i	−0.188483	+	0.982076i	$0.560357\pi$
$138$		0			0
$139$			26.6372i			0.0162542i	0.999967	+	0.00812712i	$0.00258697\pi$
$139$			26.6372i			0.0162542i	−0.999967	+	0.00812712i	$0.997413\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1718.70	−	2976.88i	1.00507	−	1.74083i
$144$		0			0
$145$	−1097.49	+	633.638i	−0.628564	+	0.362902i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1914.14	+	1105.13i	−1.05243	+	0.607622i	−0.923329	−	0.384010i	$-0.874543\pi$
$149$	−1914.14	+	1105.13i	−1.05243	+	0.607622i	−0.129102	+	0.991631i	$0.541210\pi$
$150$		0			0
$151$	958.101	−	1659.48i	0.516352	−	0.894348i	−0.483468	−	0.875362i	$-0.660623\pi$
$151$	958.101	−	1659.48i	0.516352	−	0.894348i	0.999820	−	0.0189855i	$-0.00604363\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	3643.60i		−	1.88814i
$156$		0			0
$157$	1859.51	+	1073.59i	0.945255	+	0.545743i	0.891604	−	0.452816i	$-0.149581\pi$
$157$	1859.51	+	1073.59i	0.945255	+	0.545743i	0.0536515	+	0.998560i	$0.482914\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−131.285	+	66.6296i	−0.0642651	+	0.0326159i
$162$		0			0
$163$	713.783	+	1236.31i	0.342993	+	0.594081i	0.984987	−	0.172628i	$-0.0552260\pi$
$163$	713.783	+	1236.31i	0.342993	+	0.594081i	−0.641994	+	0.766710i	$0.721893\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	558.327			0.258710			0.129355	−	0.991598i	$-0.458709\pi$
$167$	558.327			0.258710			0.129355	+	0.991598i	$0.458709\pi$
$168$		0			0
$169$	−3710.28			−1.68879
$170$		0			0
$171$		0			0
$172$		0			0
$173$	317.963	+	550.728i	0.139736	+	0.242029i	0.927397	−	0.374080i	$-0.122041\pi$
$173$	317.963	+	550.728i	0.139736	+	0.242029i	−0.787661	+	0.616109i	$0.788708\pi$
$174$		0			0
$175$	−888.742	+	1363.78i	−0.383901	+	0.589098i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1071.64	+	618.711i	0.447475	+	0.258350i	0.706763	−	0.707450i	$-0.250155\pi$
$179$	1071.64	+	618.711i	0.447475	+	0.258350i	−0.259288	+	0.965800i	$0.583488\pi$
$180$		0			0
$181$		−	252.299i		−	0.103609i	−0.998657	−	0.0518046i	$-0.983503\pi$
$181$		−	252.299i		−	0.103609i	0.998657	−	0.0518046i	$-0.0164973\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2334.99	+	4044.33i	−0.927958	+	1.60727i
$186$		0			0
$187$	−4519.72	+	2609.46i	−1.76746	+	1.02044i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2004.84	−	1157.50i	0.759505	−	0.438500i	−0.0696131	−	0.997574i	$-0.522176\pi$
$191$	2004.84	−	1157.50i	0.759505	−	0.438500i	0.829118	+	0.559074i	$0.188843\pi$
$192$		0			0
$193$	−1128.33	+	1954.32i	−0.420824	+	0.728888i	−0.996020	−	0.0891273i	$-0.971592\pi$
$193$	−1128.33	+	1954.32i	−0.420824	+	0.728888i	0.575197	+	0.818015i	$0.304926\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			4940.41i			1.78675i	0.449312	+	0.893375i	$0.351669\pi$
$197$			4940.41i			1.78675i	−0.449312	+	0.893375i	$0.648331\pi$
$198$		0			0
$199$	−505.394	−	291.789i	−0.180032	−	0.103942i	0.407276	−	0.913305i	$-0.366479\pi$
$199$	−505.394	−	291.789i	−0.180032	−	0.103942i	−0.587308	+	0.809364i	$0.699812\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	86.7445	−	1606.22i	0.0299915	−	0.555342i
$204$		0			0
$205$	−1687.36	−	2922.59i	−0.574879	−	0.995720i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4314.37			1.42790
$210$		0			0
$211$	3849.63			1.25602			0.628009	−	0.778206i	$-0.283870\pi$
$211$	3849.63			1.25602			0.628009	+	0.778206i	$0.283870\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3015.07	+	5222.25i	0.956400	+	1.65653i
$216$		0			0
$217$	3874.70	+	2525.04i	1.21213	+	0.789913i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	7767.27	+	4484.43i	2.36418	+	1.36496i
$222$		0			0
$223$			6244.17i			1.87507i	0.347889	+	0.937536i	$0.386898\pi$
$223$			6244.17i			1.87507i	−0.347889	+	0.937536i	$0.613102\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	703.572	−	1218.62i	0.205717	−	0.356312i	−0.744644	−	0.667462i	$-0.767381\pi$
$227$	703.572	−	1218.62i	0.205717	−	0.356312i	0.950361	+	0.311150i	$0.100714\pi$
$228$		0			0
$229$	2122.90	−	1225.66i	0.612600	−	0.353685i	−0.161382	−	0.986892i	$-0.551595\pi$
$229$	2122.90	−	1225.66i	0.612600	−	0.353685i	0.773982	+	0.633207i	$0.218262\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2021.13	+	1166.90i	−0.568278	+	0.328095i	−0.756461	−	0.654039i	$-0.773073\pi$
$233$	−2021.13	+	1166.90i	−0.568278	+	0.328095i	0.188184	+	0.982134i	$0.439740\pi$
$234$		0			0
$235$	−3436.26	+	5951.78i	−0.953860	+	1.65213i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			1158.54i			0.313555i	0.987634	+	0.156778i	$0.0501106\pi$
$239$			1158.54i			0.313555i	−0.987634	+	0.156778i	$0.949889\pi$
$240$		0			0
$241$	2667.43	+	1540.04i	0.712964	+	0.411630i	0.812158	−	0.583438i	$-0.198293\pi$
$241$	2667.43	+	1540.04i	0.712964	+	0.411630i	−0.0991935	+	0.995068i	$0.531626\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−2021.00	−	4578.45i	−0.527009	−	1.19390i
$246$		0			0
$247$	−3707.18	−	6421.03i	−0.954989	−	1.65409i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−1292.55			−0.325039			−0.162520	−	0.986705i	$-0.551962\pi$
$251$	−1292.55			−0.325039			−0.162520	+	0.986705i	$0.551962\pi$
$252$		0			0
$253$	−355.526			−0.0883468
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−393.202	−	681.045i	−0.0954368	−	0.165301i	0.814354	−	0.580368i	$-0.197091\pi$
$257$	−393.202	−	681.045i	−0.0954368	−	0.165301i	−0.909791	+	0.415067i	$0.863758\pi$
$258$		0			0
$259$	−2682.67	−	5285.85i	−0.643603	−	1.26813i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3353.52	−	1936.16i	−0.786262	−	0.453948i	0.0523831	−	0.998627i	$-0.483318\pi$
$263$	−3353.52	−	1936.16i	−0.786262	−	0.453948i	−0.838645	+	0.544679i	$0.816652\pi$
$264$		0			0
$265$			10114.6i			2.34465i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	1629.53	−	2822.43i	0.369347	−	0.639727i	−0.620117	−	0.784510i	$-0.712915\pi$
$269$	1629.53	−	2822.43i	0.369347	−	0.639727i	0.989464	+	0.144782i	$0.0462481\pi$
$270$		0			0
$271$	−2607.80	+	1505.61i	−0.584548	+	0.337489i	−0.762939	−	0.646471i	$-0.776244\pi$
$271$	−2607.80	+	1505.61i	−0.584548	+	0.337489i	0.178391	+	0.983960i	$0.442911\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3404.27	+	1965.46i	−0.746492	+	0.430987i
$276$		0			0
$277$	−2955.86	+	5119.70i	−0.641157	+	1.11052i	0.344018	+	0.938963i	$0.388212\pi$
$277$	−2955.86	+	5119.70i	−0.641157	+	1.11052i	−0.985175	+	0.171553i	$0.945122\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	594.162i		−	0.126138i	−0.998009	−	0.0630690i	$-0.979911\pi$
$281$		−	594.162i		−	0.126138i	0.998009	−	0.0630690i	$-0.0200888\pi$
$282$		0			0
$283$	−1858.80	−	1073.18i	−0.390440	−	0.225420i	0.291911	−	0.956446i	$-0.405709\pi$
$283$	−1858.80	−	1073.18i	−0.390440	−	0.225420i	−0.682351	+	0.731025i	$0.739042\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4277.31	+	230.998i	0.879728	+	0.0475101i
$288$		0			0
$289$	−4352.10	−	7538.06i	−0.885833	−	1.53431i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3508.37			0.699527			0.349763	−	0.936838i	$-0.386262\pi$
$293$	3508.37			0.699527			0.349763	+	0.936838i	$0.386262\pi$
$294$		0			0
$295$	4196.38			0.828213
$296$		0			0
$297$		0			0
$298$		0			0
$299$	305.491	+	529.125i	0.0590869	+	0.102341i
$300$		0			0
$301$	−7642.95	−	412.760i	−1.46356	−	0.0790403i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−10810.4	−	6241.36i	−2.02950	−	1.17174i
$306$		0			0
$307$		−	3932.23i		−	0.731024i	−0.930807	−	0.365512i	$-0.880894\pi$
$307$		−	3932.23i		−	0.731024i	0.930807	−	0.365512i	$-0.119106\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	2383.96	−	4129.14i	0.434669	−	0.752868i	−0.562600	−	0.826729i	$-0.690199\pi$
$311$	2383.96	−	4129.14i	0.434669	−	0.752868i	0.997269	+	0.0738611i	$0.0235321\pi$
$312$		0			0
$313$	−1322.49	+	763.537i	−0.238822	+	0.137884i	−0.614635	−	0.788811i	$-0.710697\pi$
$313$	−1322.49	+	763.537i	−0.238822	+	0.137884i	0.375813	+	0.926695i	$0.377363\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5430.41	−	3135.25i	0.962152	−	0.555499i	0.0653175	−	0.997865i	$-0.479194\pi$
$317$	5430.41	−	3135.25i	0.962152	−	0.555499i	0.896835	+	0.442366i	$0.145861\pi$
$318$		0			0
$319$	1942.21	−	3364.01i	0.340888	−	0.590435i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			11257.0i			1.93919i
$324$		0			0
$325$	5850.34	+	3377.69i	0.998518	+	0.576494i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−3947.92	−	7778.84i	−0.661568	−	1.30353i
$330$		0			0
$331$	4010.93	+	6947.14i	0.666045	+	1.15362i	0.979001	+	0.203856i	$0.0653474\pi$
$331$	4010.93	+	6947.14i	0.666045	+	1.15362i	−0.312956	+	0.949768i	$0.601319\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7013.57			−1.14386
$336$		0			0
$337$	7526.55			1.21661			0.608305	−	0.793704i	$-0.291850\pi$
$337$	7526.55			1.21661			0.608305	+	0.793704i	$0.291850\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	5584.14	+	9672.02i	0.886799	+	1.53598i
$342$		0			0
$343$	6269.42	+	1023.72i	0.986929	+	0.161153i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	9728.57	+	5616.79i	1.50506	+	0.868948i	0.999983	+	0.00587585i	$0.00187035\pi$
$347$	9728.57	+	5616.79i	1.50506	+	0.868948i	0.505080	+	0.863073i	$0.331463\pi$
$348$		0			0
$349$			88.0987i			0.0135124i	0.999977	+	0.00675618i	$0.00215058\pi$
$349$			88.0987i			0.0135124i	−0.999977	+	0.00675618i	$0.997849\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2911.54	−	5042.93i	0.438996	−	0.760363i	−0.558616	−	0.829426i	$-0.688668\pi$
$353$	2911.54	−	5042.93i	0.438996	−	0.760363i	0.997612	+	0.0690628i	$0.0220009\pi$
$354$		0			0
$355$	−11755.0	+	6786.77i	−1.75744	+	1.01466i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−3226.19	+	1862.64i	−0.474295	+	0.273834i	−0.718036	−	0.696006i	$-0.754959\pi$
$359$	−3226.19	+	1862.64i	−0.474295	+	0.273834i	0.243741	+	0.969840i	$0.421625\pi$
$360$		0			0
$361$	1223.47	−	2119.12i	0.178375	−	0.308954i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			1658.14i			0.237784i
$366$		0			0
$367$	8554.28	+	4938.81i	1.21670	+	0.702463i	0.964211	−	0.265136i	$-0.0854169\pi$
$367$	8554.28	+	4938.81i	1.21670	+	0.702463i	0.252491	+	0.967599i	$0.418750\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−10756.1	−	7009.47i	−1.50520	−	0.980899i
$372$		0			0
$373$	−6764.39	−	11716.3i	−0.938999	−	1.62639i	−0.767342	−	0.641238i	$-0.778421\pi$
$373$	−6764.39	−	11716.3i	−0.938999	−	1.62639i	−0.171657	−	0.985157i	$-0.554912\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6675.50			−0.911951
$378$		0			0
$379$	5023.35			0.680824			0.340412	−	0.940276i	$-0.389433\pi$
$379$	5023.35			0.680824			0.340412	+	0.940276i	$0.389433\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	6925.52	+	11995.4i	0.923962	+	1.60035i	0.793222	+	0.608933i	$0.208402\pi$
$383$	6925.52	+	11995.4i	0.923962	+	1.60035i	0.130741	+	0.991417i	$0.458264\pi$
$384$		0			0
$385$	651.734	−	12067.9i	0.0862739	−	1.59750i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	3180.84	+	1836.46i	0.414589	+	0.239363i	0.692760	−	0.721169i	$-0.256395\pi$
$389$	3180.84	+	1836.46i	0.414589	+	0.239363i	−0.278171	+	0.960532i	$0.589728\pi$
$390$		0			0
$391$		−	927.637i		−	0.119981i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1625.04	−	2814.65i	0.206999	−	0.358532i
$396$		0			0
$397$	−1790.92	+	1033.99i	−0.226407	+	0.130716i	−0.608913	−	0.793237i	$-0.708394\pi$
$397$	−1790.92	+	1033.99i	−0.226407	+	0.130716i	0.382506	+	0.923953i	$0.375061\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−2813.54	+	1624.40i	−0.350377	+	0.202290i	−0.664851	−	0.746976i	$-0.731505\pi$
$401$	−2813.54	+	1624.40i	−0.350377	+	0.202290i	0.314474	+	0.949266i	$0.398172\pi$
$402$		0			0
$403$	9596.51	−	16621.6i	1.18619	−	2.05455i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	14314.4i		−	1.74333i
$408$		0			0
$409$	−4769.99	−	2753.96i	−0.576677	−	0.332945i	0.183135	−	0.983088i	$-0.441376\pi$
$409$	−4769.99	−	2753.96i	−0.576677	−	0.332945i	−0.759812	+	0.650143i	$0.774709\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−2908.13	+	4462.54i	−0.346488	+	0.531688i
$414$		0			0
$415$	−5051.53	−	8749.51i	−0.597518	−	1.03493i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7281.32			−0.848964			−0.424482	−	0.905436i	$-0.639544\pi$
$419$	−7281.32			−0.848964			−0.424482	+	0.905436i	$0.639544\pi$
$420$		0			0
$421$	1524.24			0.176453			0.0882265	−	0.996100i	$-0.471880\pi$
$421$	1524.24			0.176453			0.0882265	+	0.996100i	$0.471880\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5128.26	−	8882.41i	−0.585311	−	1.01379i
$426$		0			0
$427$	14128.9	−	7170.70i	1.60128	−	0.812680i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4465.09	+	2577.92i	0.499016	+	0.288107i	0.728307	−	0.685251i	$-0.240308\pi$
$431$	4465.09	+	2577.92i	0.499016	+	0.288107i	−0.229291	+	0.973358i	$0.573641\pi$
$432$		0			0
$433$			1041.23i			0.115562i	0.998329	+	0.0577811i	$0.0184026\pi$
$433$			1041.23i			0.115562i	−0.998329	+	0.0577811i	$0.981597\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−383.429	+	664.118i	−0.0419723	+	0.0726981i
$438$		0			0
$439$	4138.98	−	2389.64i	0.449983	−	0.259798i	−0.257840	−	0.966188i	$-0.583011\pi$
$439$	4138.98	−	2389.64i	0.449983	−	0.259798i	0.707823	+	0.706390i	$0.249677\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	2397.66	−	1384.29i	0.257148	−	0.148464i	−0.365885	−	0.930660i	$-0.619234\pi$
$443$	2397.66	−	1384.29i	0.257148	−	0.148464i	0.623033	+	0.782196i	$0.285900\pi$
$444$		0			0
$445$	3774.18	−	6537.08i	0.402053	−	0.696376i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	14237.8i		−	1.49649i	−0.663422	−	0.748245i	$-0.730897\pi$
$449$		−	14237.8i		−	1.49649i	0.663422	−	0.748245i	$-0.269103\pi$
$450$		0			0
$451$	8958.27	+	5172.06i	0.935318	+	0.540006i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−18520.6	+	9399.57i	−1.90826	+	0.968481i
$456$		0			0
$457$	1764.03	+	3055.39i	0.180564	+	0.312747i	0.942073	−	0.335408i	$-0.108874\pi$
$457$	1764.03	+	3055.39i	0.180564	+	0.312747i	−0.761508	+	0.648155i	$0.775541\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5905.31			0.596611			0.298306	−	0.954470i	$-0.403579\pi$
$461$	5905.31			0.596611			0.298306	+	0.954470i	$0.403579\pi$
$462$		0			0
$463$	5242.57			0.526227			0.263113	−	0.964765i	$-0.415251\pi$
$463$	5242.57			0.526227			0.263113	+	0.964765i	$0.415251\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−5974.83	−	10348.7i	−0.592039	−	1.02544i	−0.993957	−	0.109766i	$-0.964990\pi$
$467$	−5974.83	−	10348.7i	−0.592039	−	1.02544i	0.401919	−	0.915675i	$-0.368343\pi$
$468$		0			0
$469$	4860.46	−	7458.41i	0.478540	−	0.734323i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−16007.1	−	9241.73i	−1.55605	−	0.898383i
$474$		0			0
$475$			8478.85i			0.819023i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	886.818	−	1536.01i	0.0845923	−	0.146518i	−0.820625	−	0.571467i	$-0.806375\pi$
$479$	886.818	−	1536.01i	0.0845923	−	0.146518i	0.905217	+	0.424949i	$0.139708\pi$
$480$		0			0
$481$	−21303.9	+	12299.8i	−2.01949	+	1.16595i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10203.8	+	5891.14i	−0.955317	+	0.551553i
$486$		0			0
$487$	−1340.79	+	2322.32i	−0.124758	+	0.216087i	−0.921638	−	0.388050i	$-0.873149\pi$
$487$	−1340.79	+	2322.32i	−0.124758	+	0.216087i	0.796880	+	0.604137i	$0.206482\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			5871.49i			0.539667i	0.962907	+	0.269834i	$0.0869687\pi$
$491$			5871.49i			0.539667i	−0.962907	+	0.269834i	$0.913031\pi$
$492$		0			0
$493$	8777.37	+	5067.62i	0.801852	+	0.462949i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	929.103	−	17203.9i	0.0838550	−	1.55272i
$498$		0			0
$499$	−6882.08	−	11920.1i	−0.617403	−	1.06937i	−0.989958	−	0.141363i	$-0.954851\pi$
$499$	−6882.08	−	11920.1i	−0.617403	−	1.06937i	0.372555	−	0.928010i	$-0.378482\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	9430.25			0.835933			0.417966	−	0.908462i	$-0.362743\pi$
$503$	9430.25			0.835933			0.417966	+	0.908462i	$0.362743\pi$
$504$		0			0
$505$	−3377.73			−0.297638
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3495.09	+	6053.68i	0.304356	+	0.527160i	0.977118	−	0.212699i	$-0.0682254\pi$
$509$	3495.09	+	6053.68i	0.304356	+	0.527160i	−0.672762	+	0.739859i	$0.734892\pi$
$510$		0			0
$511$	−1763.31	−	1149.11i	−0.152650	−	0.0994785i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	15008.2	+	8664.99i	1.28416	+	0.741408i
$516$		0			0
$517$		−	21065.5i		−	1.79199i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	8093.72	−	14018.7i	0.680600	−	1.17883i	−0.294199	−	0.955744i	$-0.595053\pi$
$521$	8093.72	−	14018.7i	0.680600	−	1.17883i	0.974798	−	0.223089i	$-0.0716140\pi$
$522$		0			0
$523$	12519.5	−	7228.11i	1.04673	−	0.604327i	0.124995	−	0.992157i	$-0.460109\pi$
$523$	12519.5	−	7228.11i	1.04673	−	0.604327i	0.921731	+	0.387830i	$0.126775\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−25236.2	+	14570.1i	−2.08597	+	1.20434i
$528$		0			0
$529$	−6051.90	+	10482.2i	−0.497403	+	0.861527i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	17776.7i		−	1.44464i
$534$		0			0
$535$	55.0696	+	31.7944i	0.00445021	+	0.00256933i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	12381.7	+	9056.25i	0.989457	+	0.723711i
$540$		0			0
$541$	−8083.69	−	14001.4i	−0.642412	−	1.11269i	−0.984893	−	0.173165i	$-0.944601\pi$
$541$	−8083.69	−	14001.4i	−0.642412	−	1.11269i	0.342481	−	0.939525i	$-0.388733\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	3318.74			0.260842
$546$		0			0
$547$	−15779.2			−1.23340			−0.616700	−	0.787199i	$-0.711531\pi$
$547$	−15779.2			−1.23340			−0.616700	+	0.787199i	$0.711531\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−4189.29	−	7256.06i	−0.323901	−	0.561014i
$552$		0			0
$553$	1867.00	+	3678.68i	0.143568	+	0.282881i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3537.74	−	2042.51i	−0.269118	−	0.155375i	0.359369	−	0.933196i	$-0.382992\pi$
$557$	−3537.74	−	2042.51i	−0.269118	−	0.155375i	−0.628487	+	0.777820i	$0.716325\pi$
$558$		0			0
$559$			31764.3i			2.40338i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1330.97	−	2305.30i	0.0996332	−	0.172570i	−0.811900	−	0.583797i	$-0.801566\pi$
$563$	1330.97	−	2305.30i	0.0996332	−	0.172570i	0.911533	+	0.411227i	$0.134900\pi$
$564$		0			0
$565$	−6541.44	+	3776.70i	−0.487080	+	0.281216i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11350.6	+	6553.27i	−0.836278	+	0.482825i	−0.855997	−	0.516980i	$-0.827056\pi$
$569$	−11350.6	+	6553.27i	−0.836278	+	0.482825i	0.0197197	+	0.999806i	$0.493723\pi$
$570$		0			0
$571$	−3592.46	+	6222.32i	−0.263292	+	0.456035i	−0.967115	−	0.254340i	$-0.918142\pi$
$571$	−3592.46	+	6222.32i	−0.263292	+	0.456035i	0.703823	+	0.710376i	$0.251475\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	698.700i		−	0.0506744i
$576$		0			0
$577$	3870.62	+	2234.70i	0.279265	+	0.161234i	0.633091	−	0.774078i	$-0.281786\pi$
$577$	3870.62	+	2234.70i	0.279265	+	0.161234i	−0.353826	+	0.935311i	$0.615119\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	12805.2	+	691.550i	0.914371	+	0.0493810i
$582$		0			0
$583$	−15501.5	−	26849.3i	−1.10121	−	1.90735i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4137.01			0.290891			0.145445	−	0.989366i	$-0.453539\pi$
$587$	4137.01			0.290891			0.145445	+	0.989366i	$0.453539\pi$
$588$		0			0
$589$	24089.6			1.68522
$590$		0			0
$591$		0			0
$592$		0			0
$593$	12000.7	+	20785.8i	0.831044	+	1.43941i	0.897211	+	0.441602i	$0.145590\pi$
$593$	12000.7	+	20785.8i	0.831044	+	1.43941i	−0.0661665	+	0.997809i	$0.521077\pi$
$594$		0			0
$595$	31487.6	+	1700.50i	2.16952	+	0.117166i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	1301.96	+	751.686i	0.0888089	+	0.0512739i	0.543747	−	0.839249i	$-0.317005\pi$
$599$	1301.96	+	751.686i	0.0888089	+	0.0512739i	−0.454938	+	0.890523i	$0.650339\pi$
$600$		0			0
$601$		−	3475.50i		−	0.235888i	−0.993020	−	0.117944i	$-0.962370\pi$
$601$		−	3475.50i		−	0.235888i	0.993020	−	0.117944i	$-0.0376303\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4882.14	−	8456.11i	0.328078	−	0.568247i
$606$		0			0
$607$	−1722.46	+	994.463i	−0.115177	+	0.0664975i	−0.556482	−	0.830860i	$-0.687849\pi$
$607$	−1722.46	+	994.463i	−0.115177	+	0.0664975i	0.441305	+	0.897357i	$0.354516\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−31351.6	+	18100.8i	−2.07586	+	1.19850i
$612$		0			0
$613$	−11830.3	+	20490.7i	−0.779479	+	1.35010i	0.152763	+	0.988263i	$0.451183\pi$
$613$	−11830.3	+	20490.7i	−0.779479	+	1.35010i	−0.932242	+	0.361835i	$0.882150\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15390.3i			1.00420i	0.864811	+	0.502098i	$0.167438\pi$
$617$			15390.3i			1.00420i	−0.864811	+	0.502098i	$0.832562\pi$
$618$		0			0
$619$	6817.56	+	3936.12i	0.442683	+	0.255583i	0.704735	−	0.709470i	$-0.251066\pi$
$619$	6817.56	+	3936.12i	0.442683	+	0.255583i	−0.262052	+	0.965054i	$0.584399\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4336.16	+	8543.81i	0.278852	+	0.549439i
$624$		0			0
$625$	9443.21	+	16356.1i	0.604366	+	1.04679i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	37349.0			2.36757
$630$		0			0
$631$	6083.98			0.383834			0.191917	−	0.981411i	$-0.438529\pi$
$631$	6083.98			0.383834			0.191917	+	0.981411i	$0.438529\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	12150.3	+	21044.9i	0.759320	+	1.31518i
$636$		0			0
$637$	2839.16	−	26209.2i	0.176596	−	1.63022i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−20571.8	−	11877.2i	−1.26761	−	0.731855i	−0.293075	−	0.956089i	$-0.594679\pi$
$641$	−20571.8	−	11877.2i	−1.26761	−	0.731855i	−0.974535	+	0.224234i	$0.928012\pi$
$642$		0			0
$643$		−	12895.8i		−	0.790918i	−0.918484	−	0.395459i	$-0.870585\pi$
$643$		−	12895.8i		−	0.790918i	0.918484	−	0.395459i	$-0.129415\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	552.298	−	956.608i	0.0335596	−	0.0581270i	−0.848758	−	0.528782i	$-0.822649\pi$
$647$	552.298	−	956.608i	0.0335596	−	0.0581270i	0.882317	+	0.470655i	$0.155982\pi$
$648$		0			0
$649$	−11139.4	+	6431.33i	−0.673744	+	0.388986i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	15648.4	−	9034.58i	0.937775	−	0.541425i	0.0485131	−	0.998823i	$-0.484552\pi$
$653$	15648.4	−	9034.58i	0.937775	−	0.541425i	0.889262	+	0.457398i	$0.151218\pi$
$654$		0			0
$655$	−4997.87	+	8656.56i	−0.298142	+	0.516397i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	15980.9i		−	0.944656i	−0.881423	−	0.472328i	$-0.843414\pi$
$659$		−	15980.9i		−	0.944656i	0.881423	−	0.472328i	$-0.156586\pi$
$660$		0			0
$661$	26260.0	+	15161.2i	1.54523	+	0.892138i	0.998496	+	0.0548308i	$0.0174619\pi$
$661$	26260.0	+	15161.2i	1.54523	+	0.892138i	0.546733	+	0.837307i	$0.315871\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−21839.9	−	14232.5i	−1.27355	−	0.829943i
$666$		0			0
$667$	345.219	+	597.936i	0.0200404	+	0.0347109i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	38261.8			2.20131
$672$		0			0
$673$	−17767.4			−1.01766			−0.508829	−	0.860868i	$-0.669921\pi$
$673$	−17767.4			−1.01766			−0.508829	+	0.860868i	$0.669921\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	8243.00	+	14277.3i	0.467953	+	0.810519i	0.999329	−	0.0366172i	$-0.0116582\pi$
$677$	8243.00	+	14277.3i	0.467953	+	0.810519i	−0.531376	+	0.847136i	$0.678325\pi$
$678$		0			0
$679$	806.493	−	14933.6i	0.0455822	−	0.844031i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1757.98	−	1014.97i	−0.0984877	−	0.0568619i	0.449947	−	0.893055i	$-0.351443\pi$
$683$	−1757.98	−	1014.97i	−0.0984877	−	0.0568619i	−0.548435	+	0.836193i	$0.684776\pi$
$684$		0			0
$685$		−	15339.5i		−	0.855607i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−26639.7	+	46141.3i	−1.47299	+	2.55130i
$690$		0			0
$691$	−27185.8	+	15695.7i	−1.49667	+	0.864101i	−0.999993	−	0.00383605i	$-0.998779\pi$
$691$	−27185.8	+	15695.7i	−1.49667	+	0.864101i	−0.496674	+	0.867937i	$0.665446\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	336.590	−	194.330i	0.0183706	−	0.0106063i
$696$		0			0
$697$	−13494.9	+	23373.9i	−0.733367	+	1.27023i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			7749.11i			0.417517i	0.977967	+	0.208759i	$0.0669423\pi$
$701$			7749.11i			0.417517i	−0.977967	+	0.208759i	$0.933058\pi$
$702$		0			0
$703$	−26739.0	−	15437.8i	−1.43454	−	0.828232i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	2340.79	−	3591.96i	0.124519	−	0.191075i
$708$		0			0
$709$	8907.48	+	15428.2i	0.471830	+	0.817233i	0.999481	−	0.0322280i	$-0.0102603\pi$
$709$	8907.48	+	15428.2i	0.471830	+	0.817233i	−0.527651	+	0.849462i	$0.676927\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1985.11			−0.104268
$714$		0			0
$715$	−50154.7			−2.62333
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10695.4	−	18525.0i	−0.554759	−	0.960871i	−0.997922	−	0.0644299i	$-0.979477\pi$
$719$	−10695.4	−	18525.0i	−0.554759	−	0.960871i	0.443163	−	0.896441i	$-0.353856\pi$
$720$		0			0
$721$	−19615.4	+	9955.20i	−1.01320	+	0.514218i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	6611.15	+	3816.95i	0.338665	+	0.195528i
$726$		0			0
$727$			27791.9i			1.41780i	0.705307	+	0.708902i	$0.250809\pi$
$727$			27791.9i			1.41780i	−0.705307	+	0.708902i	$0.749191\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	24113.5	−	41765.8i	1.22007	−	2.11322i
$732$		0			0
$733$	−17159.5	+	9907.02i	−0.864665	+	0.499214i	−0.865572	−	0.500785i	$-0.833045\pi$
$733$	−17159.5	+	9907.02i	−0.864665	+	0.499214i	0.000906890		1.00000i	$0.499711\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	18617.7	−	10748.9i	0.930518	−	0.537235i
$738$		0			0
$739$	−15164.2	+	26265.2i	−0.754838	+	1.30742i	0.190617	+	0.981664i	$0.438951\pi$
$739$	−15164.2	+	26265.2i	−0.754838	+	1.30742i	−0.945455	+	0.325753i	$0.894382\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	9992.06i		−	0.493369i	−0.969096	−	0.246685i	$-0.920659\pi$
$743$		−	9992.06i		−	0.493369i	0.969096	−	0.246685i	$-0.0793411\pi$
$744$		0			0
$745$	27928.9	+	16124.8i	1.37347	+	0.792975i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−71.9746	+	36.5286i	−0.00351121	+	0.00178201i
$750$		0			0
$751$	19991.9	+	34627.1i	0.971393	+	1.68250i	0.691358	+	0.722513i	$0.257013\pi$
$751$	19991.9	+	34627.1i	0.971393	+	1.68250i	0.280035	+	0.959990i	$0.409654\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−27959.0			−1.34773
$756$		0			0
$757$	−11981.7			−0.575273			−0.287637	−	0.957740i	$-0.592870\pi$
$757$	−11981.7			−0.575273			−0.287637	+	0.957740i	$0.592870\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	10395.4	+	18005.4i	0.495182	+	0.857680i	0.999985	−	0.00555440i	$-0.00176803\pi$
$761$	10395.4	+	18005.4i	0.495182	+	0.857680i	−0.504803	+	0.863235i	$0.668435\pi$
$762$		0			0
$763$	−2299.91	+	3529.23i	−0.109125	+	0.167453i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	19143.4	+	11052.4i	0.901208	+	0.520313i
$768$		0			0
$769$			24512.2i			1.14946i	0.818344	+	0.574728i	$0.194892\pi$
$769$			24512.2i			1.14946i	−0.818344	+	0.574728i	$0.805108\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−18063.2	+	31286.4i	−0.840477	+	1.45575i	0.0490158	+	0.998798i	$0.484392\pi$
$773$	−18063.2	+	31286.4i	−0.840477	+	1.45575i	−0.889492	+	0.456950i	$0.848942\pi$
$774$		0			0
$775$	−19008.0	+	10974.3i	−0.881017	+	0.508655i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	19322.7	−	11156.0i	0.888712	−	0.513098i
$780$		0			0
$781$	20802.7	−	36031.3i	0.953109	−	1.65083i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		−	31329.2i		−	1.42444i
$786$		0			0
$787$	22356.7	+	12907.6i	1.01262	+	0.584634i	0.911957	−	0.410287i	$-0.134571\pi$
$787$	22356.7	+	12907.6i	1.01262	+	0.584634i	0.100660	+	0.994921i	$0.467905\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	517.027	−	9573.62i	0.0232407	−	0.430340i
$792$		0			0
$793$	−32877.0	−	56944.6i	−1.47225	−	2.55001i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	11502.7			0.511223			0.255612	−	0.966780i	$-0.417723\pi$
$797$	11502.7			0.511223			0.255612	+	0.966780i	$0.417723\pi$
$798$		0			0
$799$	54964.1			2.43365
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−2541.26	−	4401.58i	−0.111680	−	0.193435i
$804$		0			0
$805$	1799.72	+	1172.83i	0.0787971	+	0.0513501i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	20637.7	+	11915.2i	0.896887	+	0.517818i	0.876189	−	0.481968i	$-0.160078\pi$
$809$	20637.7	+	11915.2i	0.896887	+	0.517818i	0.0206978	+	0.999786i	$0.493411\pi$
$810$		0			0
$811$		−	319.208i		−	0.0138211i	−0.999976	−	0.00691055i	$-0.997800\pi$
$811$		−	319.208i		−	0.0138211i	0.999976	−	0.00691055i	$-0.00219971\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	10414.7	−	18038.8i	0.447622	−	0.775304i
$816$		0			0
$817$	−34526.8	+	19934.1i	−1.47851	+	0.853617i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−16147.2	+	9322.60i	−0.686409	+	0.396298i	−0.802265	−	0.596968i	$-0.796372\pi$
$821$	−16147.2	+	9322.60i	−0.686409	+	0.396298i	0.115857	+	0.993266i	$0.463039\pi$
$822$		0			0
$823$	15224.2	−	26369.2i	0.644816	−	1.11685i	−0.339528	−	0.940596i	$-0.610267\pi$
$823$	15224.2	−	26369.2i	0.644816	−	1.11685i	0.984344	−	0.176259i	$-0.0563995\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			29203.6i			1.22794i	0.789328	+	0.613972i	$0.210429\pi$
$827$			29203.6i			1.22794i	−0.789328	+	0.613972i	$0.789571\pi$
$828$		0			0
$829$	9997.78	+	5772.22i	0.418863	+	0.241830i	0.694591	−	0.719405i	$-0.255586\pi$
$829$	9997.78	+	5772.22i	0.418863	+	0.241830i	−0.275728	+	0.961236i	$0.588919\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−23629.5	+	32306.3i	−0.982850	+	1.34375i
$834$		0			0
$835$	−4073.24	−	7055.05i	−0.168815	−	0.292395i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−4335.20			−0.178388			−0.0891941	−	0.996014i	$-0.528429\pi$
$839$	−4335.20			−0.178388			−0.0891941	+	0.996014i	$0.528429\pi$
$840$		0			0
$841$	16845.4			0.690696
$842$		0			0
$843$		0			0
$844$		0			0
$845$	27068.1	+	46883.3i	1.10198	+	1.90868i
$846$		0			0
$847$	5609.08	+	11051.9i	0.227545	+	0.448346i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2203.43	+	1272.15i	0.0887575	+	0.0512442i
$852$		0			0
$853$		−	14779.7i		−	0.593257i	−0.954993	−	0.296629i	$-0.904138\pi$
$853$		−	14779.7i		−	0.593257i	0.954993	−	0.296629i	$-0.0958624\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	10530.9	−	18240.1i	0.419754	−	0.727036i	−0.576160	−	0.817337i	$-0.695450\pi$
$857$	10530.9	−	18240.1i	0.419754	−	0.727036i	0.995915	+	0.0903008i	$0.0287829\pi$
$858$		0			0
$859$	−22556.0	+	13022.7i	−0.895928	+	0.517264i	−0.875877	−	0.482535i	$-0.839716\pi$
$859$	−22556.0	+	13022.7i	−0.895928	+	0.517264i	−0.0200509	+	0.999799i	$0.506383\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26253.5	+	15157.5i	−1.03555	+	0.597874i	−0.918569	−	0.395260i	$-0.870655\pi$
$863$	−26253.5	+	15157.5i	−1.03555	+	0.597874i	−0.116980	+	0.993134i	$0.537321\pi$
$864$		0			0
$865$	4639.36	−	8035.60i	0.182362	−	0.315860i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			9962.06i			0.388883i
$870$		0			0
$871$	−31995.0	−	18472.3i	−1.24467	−	0.718612i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−10012.6	−	540.734i	−0.386843	−	0.0208916i
$876$		0			0
$877$	11591.2	+	20076.5i	0.446301	+	0.773017i	0.998142	−	0.0609329i	$-0.0194076\pi$
$877$	11591.2	+	20076.5i	0.446301	+	0.773017i	−0.551840	+	0.833950i	$0.686074\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	10346.4			0.395663			0.197832	−	0.980236i	$-0.436610\pi$
$881$	10346.4			0.395663			0.197832	+	0.980236i	$0.436610\pi$
$882$		0			0
$883$	−11461.4			−0.436814			−0.218407	−	0.975858i	$-0.570086\pi$
$883$	−11461.4			−0.436814			−0.218407	+	0.975858i	$0.570086\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	2162.53	+	3745.61i	0.0818608	+	0.141787i	0.904049	−	0.427428i	$-0.140580\pi$
$887$	2162.53	+	3745.61i	0.0818608	+	0.141787i	−0.822188	+	0.569215i	$0.807247\pi$
$888$		0			0
$889$	−30799.9	−	1663.36i	−1.16197	−	0.0627529i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−39350.1	−	22718.8i	−1.47458	−	0.851350i
$894$		0			0
$895$		−	18055.1i		−	0.674317i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	10844.5	−	18783.2i	0.402319	−	0.696836i
$900$		0			0
$901$	70055.2	−	40446.4i	2.59032	−	1.49552i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3188.07	+	1840.63i	−0.117099	+	0.0676074i
$906$		0			0
$907$	7966.77	−	13798.9i	0.291656	−	0.505164i	−0.682545	−	0.730843i	$-0.739127\pi$
$907$	7966.77	−	13798.9i	0.291656	−	0.505164i	0.974202	+	0.225680i	$0.0724603\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	27990.0i		−	1.01795i	−0.860782	−	0.508974i	$-0.830025\pi$
$911$		−	27990.0i		−	1.01795i	0.860782	−	0.508974i	$-0.169975\pi$
$912$		0			0
$913$	26818.8	+	15483.9i	0.972151	+	0.561271i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−5742.05	−	11313.9i	−0.206782	−	0.407436i
$918$		0			0
$919$	−21383.1	−	37036.7i	−0.767535	−	1.32941i	−0.938896	−	0.344201i	$-0.888150\pi$
$919$	−21383.1	−	37036.7i	−0.767535	−	1.32941i	0.171361	−	0.985208i	$-0.445183\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−71499.9			−2.54978
$924$		0			0
$925$	28131.4			0.999951
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−18019.2	−	31210.2i	−0.636375	−	1.10223i	−0.986222	−	0.165427i	$-0.947100\pi$
$929$	−18019.2	−	31210.2i	−0.636375	−	1.10223i	0.349847	−	0.936807i	$-0.386234\pi$
$930$		0			0
$931$	30270.4	−	13361.8i	1.06560	−	0.470372i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	65946.7	+	38074.3i	2.30662	+	1.33173i
$936$		0			0
$937$			36919.6i			1.28720i	0.765361	+	0.643602i	$0.222561\pi$
$937$			36919.6i			1.28720i	−0.765361	+	0.643602i	$0.777439\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−7399.78	+	12816.8i	−0.256351	+	0.444013i	−0.965262	−	0.261286i	$-0.915854\pi$
$941$	−7399.78	+	12816.8i	−0.256351	+	0.444013i	0.708911	+	0.705298i	$0.249187\pi$
$942$		0			0
$943$	−1592.29	+	919.307i	−0.0549862	+	0.0317463i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	46041.6	−	26582.1i	1.57988	−	0.912147i	0.585010	−	0.811026i	$-0.301091\pi$
$947$	46041.6	−	26582.1i	1.57988	−	0.912147i	0.994874	−	0.101120i	$-0.0322427\pi$
$948$		0			0
$949$	−4367.22	+	7564.24i	−0.149384	+	0.258741i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			27940.1i			0.949704i	0.880066	+	0.474852i	$0.157498\pi$
$953$			27940.1i			0.949704i	−0.880066	+	0.474852i	$0.842502\pi$
$954$		0			0
$955$	−29252.4	−	16888.9i	−0.991189	−	0.572263i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16312.4	+	10630.4i	0.549274	+	0.357948i
$960$		0			0
$961$	16284.0	+	28204.7i	0.546608	+	0.946752i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	32926.6			1.09839
$966$		0			0
$967$	−7724.02			−0.256864			−0.128432	−	0.991718i	$-0.540994\pi$
$967$	−7724.02			−0.256864			−0.128432	+	0.991718i	$0.540994\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−12460.4	−	21582.0i	−0.411814	−	0.713283i	0.583274	−	0.812276i	$-0.301771\pi$
$971$	−12460.4	−	21582.0i	−0.411814	−	0.713283i	−0.995088	+	0.0989921i	$0.968438\pi$
$972$		0			0
$973$	−26.6036	+	492.611i	−0.000876540	+	0.0162306i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−19376.2	−	11186.9i	−0.634494	−	0.366325i	0.147997	−	0.988988i	$-0.452717\pi$
$977$	−19376.2	−	11186.9i	−0.634494	−	0.366325i	−0.782490	+	0.622663i	$0.786051\pi$
$978$		0			0
$979$			23137.1i			0.755327i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18175.0	−	31480.0i	0.589717	−	1.02142i	−0.404552	−	0.914515i	$-0.632573\pi$
$983$	18175.0	−	31480.0i	0.589717	−	1.02142i	0.994269	−	0.106905i	$-0.0340941\pi$
$984$		0			0
$985$	62427.4	−	36042.4i	2.01939	−	1.16590i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2845.19	−	1642.67i	0.0914780	−	0.0528148i
$990$		0			0
$991$	3752.26	−	6499.11i	0.120277	−	0.208326i	−0.799600	−	0.600533i	$-0.794955\pi$
$991$	3752.26	−	6499.11i	0.120277	−	0.208326i	0.919877	+	0.392207i	$0.128288\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			8514.92i			0.271298i
$996$		0			0
$997$	−40355.2	−	23299.1i	−1.28191	−	0.740110i	−0.304711	−	0.952445i	$-0.598560\pi$
$997$	−40355.2	−	23299.1i	−1.28191	−	0.740110i	−0.977197	+	0.212335i	$0.931893\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	504.4.bl.a.17.5	✓	48
3.2	odd	2	inner	504.4.bl.a.17.20	yes	48
4.3	odd	2		1008.4.bt.d.17.5		48
7.5	odd	6	inner	504.4.bl.a.89.20	yes	48
12.11	even	2		1008.4.bt.d.17.20		48
21.5	even	6	inner	504.4.bl.a.89.5	yes	48
28.19	even	6		1008.4.bt.d.593.20		48
84.47	odd	6		1008.4.bt.d.593.5		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.5	✓	48	1.1	even	1	trivial
504.4.bl.a.17.20	yes	48	3.2	odd	2	inner
504.4.bl.a.89.5	yes	48	21.5	even	6	inner
504.4.bl.a.89.20	yes	48	7.5	odd	6	inner
1008.4.bt.d.17.5		48	4.3	odd	2
1008.4.bt.d.17.20		48	12.11	even	2
1008.4.bt.d.593.5		48	84.47	odd	6
1008.4.bt.d.593.20		48	28.19	even	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:	\( N \)	\(=\)	\( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	504.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-7.29543 - 12.6361i) q^{5} +(18.4933 + 0.998739i) q^{7} +O(q^{10})\)	\(q+(-7.29543 - 12.6361i) q^{5} +(18.4933 + 0.998739i) q^{7} +(38.7318 + 22.3618i) q^{11} -76.8588i q^{13} +(-58.3464 + 101.059i) q^{17} +(83.5432 - 48.2337i) q^{19} +(-6.88438 + 3.97470i) q^{23} +(-43.9467 + 76.1180i) q^{25} -86.8540i q^{29} +(216.262 + 124.859i) q^{31} +(-122.297 - 240.969i) q^{35} +(-160.031 - 277.182i) q^{37} +231.290 q^{41} -413.282 q^{43} +(-235.508 - 407.911i) q^{47} +(341.005 + 36.9400i) q^{49} +(-600.339 - 346.606i) q^{53} -652.557i q^{55} +(-143.802 + 249.072i) q^{59} +(740.898 - 427.758i) q^{61} +(-971.193 + 560.718i) q^{65} +(240.341 - 416.283i) q^{67} -930.276i q^{71} +(-98.4174 - 56.8213i) q^{73} +(693.946 + 452.227i) q^{77} +(111.374 + 192.905i) q^{79} +692.424 q^{83} +1702.65 q^{85} +(258.667 + 448.025i) q^{89} +(76.7619 - 1421.37i) q^{91} +(-1218.97 - 703.771i) q^{95} -807.511i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 24 q^{7}+O(q^{10}) \) 48 * q - 24 * q^7	\( 48 q - 24 q^{7} + 540 q^{19} - 924 q^{25} + 648 q^{31} - 132 q^{37} - 792 q^{43} + 672 q^{49} - 12 q^{67} + 2412 q^{73} + 1680 q^{79} + 480 q^{85} + 1404 q^{91}+O(q^{100}) \) 48 * q - 24 * q^7 + 540 * q^19 - 924 * q^25 + 648 * q^31 - 132 * q^37 - 792 * q^43 + 672 * q^49 - 12 * q^67 + 2412 * q^73 + 1680 * q^79 + 480 * q^85 + 1404 * q^91

Introduction

L-functions

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