LMFDB - Embedded newform 504.4.bl.a.17.4

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			17.4
Character	$\chi$	$=$	504.17
Dual form			504.4.bl.a.89.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-7.89184 - 13.6691i) q^{5} +(10.6185 + 15.1739i) q^{7} +O(q^{10})$	$q+(-7.89184 - 13.6691i) q^{5} +(10.6185 + 15.1739i) q^{7} +(-28.2439 - 16.3066i) q^{11} +54.3118i q^{13} +(-12.3222 + 21.3427i) q^{17} +(16.2989 - 9.41015i) q^{19} +(46.7289 - 26.9789i) q^{23} +(-62.0622 + 107.495i) q^{25} +157.048i q^{29} +(41.4452 + 23.9284i) q^{31} +(123.614 - 264.895i) q^{35} +(-48.1973 - 83.4802i) q^{37} +263.002 q^{41} +258.982 q^{43} +(62.5951 + 108.418i) q^{47} +(-117.497 + 322.248i) q^{49} +(471.579 + 272.266i) q^{53} +514.756i q^{55} +(189.386 - 328.027i) q^{59} +(587.204 - 339.022i) q^{61} +(742.392 - 428.620i) q^{65} +(-346.637 + 600.394i) q^{67} +238.831i q^{71} +(631.051 + 364.337i) q^{73} +(-52.4707 - 601.721i) q^{77} +(439.845 + 761.833i) q^{79} -1222.11 q^{83} +388.979 q^{85} +(75.9540 + 131.556i) q^{89} +(-824.125 + 576.708i) q^{91} +(-257.256 - 148.527i) q^{95} +1591.43i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−7.89184	−	13.6691i	−0.705867	−	1.22260i	−0.966378	−	0.257127i	$-0.917224\pi$
$5$	−7.89184	−	13.6691i	−0.705867	−	1.22260i	0.260510	−	0.965471i	$-0.416109\pi$
$6$		0			0
$7$	10.6185	+	15.1739i	0.573343	+	0.819316i
$7$	10.6185	+	15.1739i	0.573343	+	0.819316i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−28.2439	−	16.3066i	−0.774167	−	0.446966i	0.0601918	−	0.998187i	$-0.480829\pi$
$11$	−28.2439	−	16.3066i	−0.774167	−	0.446966i	−0.834359	+	0.551221i	$0.814162\pi$
$12$		0			0
$13$			54.3118i			1.15872i	0.815071	+	0.579361i	$0.196698\pi$
$13$			54.3118i			1.15872i	−0.815071	+	0.579361i	$0.803302\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.3222	+	21.3427i	−0.175798	+	0.304491i	−0.940437	−	0.339967i	$-0.889584\pi$
$17$	−12.3222	+	21.3427i	−0.175798	+	0.304491i	0.764639	+	0.644459i	$0.222917\pi$
$18$		0			0
$19$	16.2989	−	9.41015i	0.196801	−	0.113623i	−0.398362	−	0.917228i	$-0.630421\pi$
$19$	16.2989	−	9.41015i	0.196801	−	0.113623i	0.595162	+	0.803606i	$0.297088\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	46.7289	−	26.9789i	0.423637	−	0.244587i	−0.272995	−	0.962015i	$-0.588014\pi$
$23$	46.7289	−	26.9789i	0.423637	−	0.244587i	0.696632	+	0.717429i	$0.254681\pi$
$24$		0			0
$25$	−62.0622	+	107.495i	−0.496498	+	0.859959i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			157.048i			1.00562i	0.864397	+	0.502811i	$0.167701\pi$
$29$			157.048i			1.00562i	−0.864397	+	0.502811i	$0.832299\pi$
$30$		0			0
$31$	41.4452	+	23.9284i	0.240122	+	0.138634i	0.615233	−	0.788345i	$-0.289062\pi$
$31$	41.4452	+	23.9284i	0.240122	+	0.138634i	−0.375111	+	0.926980i	$0.622395\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	123.614	−	264.895i	0.596990	−	1.27930i
$36$		0			0
$37$	−48.1973	−	83.4802i	−0.214151	−	0.370921i	0.738858	−	0.673861i	$-0.235365\pi$
$37$	−48.1973	−	83.4802i	−0.214151	−	0.370921i	−0.953010	+	0.302940i	$0.902032\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	263.002			1.00181			0.500903	−	0.865503i	$-0.333001\pi$
$41$	263.002			1.00181			0.500903	+	0.865503i	$0.333001\pi$
$42$		0			0
$43$	258.982			0.918474			0.459237	−	0.888314i	$-0.348123\pi$
$43$	258.982			0.918474			0.459237	+	0.888314i	$0.348123\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	62.5951	+	108.418i	0.194264	+	0.336476i	0.946659	−	0.322237i	$-0.104435\pi$
$47$	62.5951	+	108.418i	0.194264	+	0.336476i	−0.752395	+	0.658713i	$0.771101\pi$
$48$		0			0
$49$	−117.497	+	322.248i	−0.342557	+	0.939497i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	471.579	+	272.266i	1.22220	+	0.705635i	0.965386	−	0.260827i	$-0.0839952\pi$
$53$	471.579	+	272.266i	1.22220	+	0.705635i	0.256810	+	0.966462i	$0.417329\pi$
$54$		0			0
$55$			514.756i			1.26199i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	189.386	−	328.027i	0.417899	−	0.723822i	−0.577829	−	0.816158i	$-0.696100\pi$
$59$	189.386	−	328.027i	0.417899	−	0.723822i	0.995728	+	0.0923360i	$0.0294334\pi$
$60$		0			0
$61$	587.204	−	339.022i	1.23252	−	0.711596i	0.264966	−	0.964258i	$-0.414639\pi$
$61$	587.204	−	339.022i	1.23252	−	0.711596i	0.967555	+	0.252662i	$0.0813061\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	742.392	−	428.620i	1.41665	−	0.817905i
$66$		0			0
$67$	−346.637	+	600.394i	−0.632067	+	1.09477i	0.355061	+	0.934843i	$0.384460\pi$
$67$	−346.637	+	600.394i	−0.632067	+	1.09477i	−0.987128	+	0.159929i	$0.948873\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			238.831i			0.399212i	0.979876	+	0.199606i	$0.0639662\pi$
$71$			238.831i			0.399212i	−0.979876	+	0.199606i	$0.936034\pi$
$72$		0			0
$73$	631.051	+	364.337i	1.01177	+	0.584143i	0.911708	−	0.410838i	$-0.134764\pi$
$73$	631.051	+	364.337i	1.01177	+	0.584143i	0.100058	+	0.994982i	$0.468097\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−52.4707	−	601.721i	−0.0776570	−	0.890552i
$78$		0			0
$79$	439.845	+	761.833i	0.626410	+	1.08497i	0.988266	+	0.152740i	$0.0488098\pi$
$79$	439.845	+	761.833i	0.626410	+	1.08497i	−0.361856	+	0.932234i	$0.617857\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1222.11			−1.61619			−0.808096	−	0.589050i	$-0.799502\pi$
$83$	−1222.11			−1.61619			−0.808096	+	0.589050i	$0.799502\pi$
$84$		0			0
$85$	388.979			0.496361
$86$		0			0
$87$		0			0
$88$		0			0
$89$	75.9540	+	131.556i	0.0904619	+	0.156685i	0.907706	−	0.419608i	$-0.137832\pi$
$89$	75.9540	+	131.556i	0.0904619	+	0.156685i	−0.817244	+	0.576292i	$0.804499\pi$
$90$		0			0
$91$	−824.125	+	576.708i	−0.949360	+	0.664345i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−257.256	−	148.527i	−0.277830	−	0.160405i
$96$		0			0
$97$			1591.43i			1.66582i	0.553406	+	0.832912i	$0.313328\pi$
$97$			1591.43i			1.66582i	−0.553406	+	0.832912i	$0.686672\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−28.6940	+	49.6994i	−0.0282689	+	0.0489631i	−0.879814	−	0.475319i	$-0.842333\pi$
$101$	−28.6940	+	49.6994i	−0.0282689	+	0.0489631i	0.851545	+	0.524282i	$0.175666\pi$
$102$		0			0
$103$	−1674.25	+	966.626i	−1.60163	+	0.924704i	−0.610473	+	0.792037i	$0.709021\pi$
$103$	−1674.25	+	966.626i	−1.60163	+	0.924704i	−0.991161	+	0.132667i	$0.957646\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1729.67	−	998.625i	1.56274	−	0.902249i	0.565764	−	0.824567i	$-0.308581\pi$
$107$	1729.67	−	998.625i	1.56274	−	0.902249i	0.996978	−	0.0776819i	$-0.0247519\pi$
$108$		0			0
$109$	698.958	−	1210.63i	0.614202	−	1.06383i	−0.376322	−	0.926489i	$-0.622811\pi$
$109$	698.958	−	1210.63i	0.614202	−	1.06383i	0.990524	−	0.137340i	$-0.0438554\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			1636.31i			1.36222i	0.732181	+	0.681110i	$0.238502\pi$
$113$			1636.31i			1.36222i	−0.732181	+	0.681110i	$0.761498\pi$
$114$		0			0
$115$	−737.553	−	425.827i	−0.598063	−	0.345292i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−454.695	+	39.6498i	−0.350267	+	0.0305436i
$120$		0			0
$121$	−133.690	−	231.558i	−0.100443	−	0.173973i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−13.8192			−0.00988823
$126$		0			0
$127$	1729.20			1.20820			0.604101	−	0.796908i	$-0.293532\pi$
$127$	1729.20			1.20820			0.604101	+	0.796908i	$0.293532\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−89.9310	−	155.765i	−0.0599795	−	0.103887i	0.834476	−	0.551044i	$-0.185770\pi$
$131$	−89.9310	−	155.765i	−0.0599795	−	0.103887i	−0.894456	+	0.447156i	$0.852437\pi$
$132$		0			0
$133$	315.858	+	147.397i	0.205927	+	0.0960970i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2466.49	+	1424.03i	1.53815	+	0.888053i	0.998947	+	0.0458784i	$0.0146087\pi$
$137$	2466.49	+	1424.03i	1.53815	+	0.888053i	0.539205	+	0.842174i	$0.318725\pi$
$138$		0			0
$139$			58.4498i			0.0356665i	0.999841	+	0.0178333i	$0.00567680\pi$
$139$			58.4498i			0.0356665i	−0.999841	+	0.0178333i	$0.994323\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	885.641	−	1533.98i	0.517909	−	0.897045i
$144$		0			0
$145$	2146.70	−	1239.40i	1.22947	−	0.709836i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	701.675	−	405.112i	0.385795	−	0.222739i	−0.294542	−	0.955639i	$-0.595167\pi$
$149$	701.675	−	405.112i	0.385795	−	0.222739i	0.680337	+	0.732900i	$0.261834\pi$
$150$		0			0
$151$	1271.69	−	2202.64i	0.685357	−	1.18707i	−0.287968	−	0.957640i	$-0.592980\pi$
$151$	1271.69	−	2202.64i	0.685357	−	1.18707i	0.973325	−	0.229433i	$-0.0736870\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		−	755.357i		−	0.391430i
$156$		0			0
$157$	−2181.59	−	1259.54i	−1.10898	−	0.640271i	−0.170417	−	0.985372i	$-0.554511\pi$
$157$	−2181.59	−	1259.54i	−1.10898	−	0.640271i	−0.938566	+	0.345101i	$0.887845\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	905.565	+	422.587i	0.443283	+	0.206860i
$162$		0			0
$163$	1072.18	+	1857.07i	0.515213	+	0.892375i	0.999844	+	0.0176561i	$0.00562041\pi$
$163$	1072.18	+	1857.07i	0.515213	+	0.892375i	−0.484631	+	0.874718i	$0.661046\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−3381.59			−1.56692			−0.783458	−	0.621445i	$-0.786546\pi$
$167$	−3381.59			−1.56692			−0.783458	+	0.621445i	$0.786546\pi$
$168$		0			0
$169$	−752.775			−0.342638
$170$		0			0
$171$		0			0
$172$		0			0
$173$	98.9912	+	171.458i	0.0435038	+	0.0753509i	0.886957	−	0.461851i	$-0.152815\pi$
$173$	98.9912	+	171.458i	0.0435038	+	0.0753509i	−0.843454	+	0.537202i	$0.819481\pi$
$174$		0			0
$175$	−2290.13	+	199.701i	−0.989242	+	0.0862628i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3007.52	−	1736.40i	−1.25583	−	0.725052i	−0.283566	−	0.958953i	$-0.591518\pi$
$179$	−3007.52	−	1736.40i	−1.25583	−	0.725052i	−0.972260	+	0.233901i	$0.924851\pi$
$180$		0			0
$181$		−	2872.95i		−	1.17980i	−0.807475	−	0.589902i	$-0.799166\pi$
$181$		−	2872.95i		−	1.17980i	0.807475	−	0.589902i	$-0.200834\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−760.731	+	1317.63i	−0.302325	+	0.523642i
$186$		0			0
$187$	696.052	−	401.866i	0.272194	−	0.157152i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1075.20	+	620.765i	−0.407322	+	0.235167i	−0.689638	−	0.724154i	$-0.742230\pi$
$191$	−1075.20	+	620.765i	−0.407322	+	0.235167i	0.282317	+	0.959321i	$0.408897\pi$
$192$		0			0
$193$	−1341.31	+	2323.21i	−0.500256	+	0.866468i	0.499744	+	0.866173i	$0.333427\pi$
$193$	−1341.31	+	2323.21i	−0.500256	+	0.866468i	−1.00000		0.000295140i	$0.999906\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			3056.57i			1.10544i	0.833367	+	0.552719i	$0.186410\pi$
$197$			3056.57i			1.10544i	−0.833367	+	0.552719i	$0.813590\pi$
$198$		0			0
$199$	−1864.33	−	1076.37i	−0.664115	−	0.383427i	0.129728	−	0.991550i	$-0.458589\pi$
$199$	−1864.33	−	1076.37i	−0.664115	−	0.383427i	−0.793843	+	0.608123i	$0.791923\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−2383.03	+	1667.60i	−0.823922	+	0.576566i
$204$		0			0
$205$	−2075.57	−	3595.00i	−0.707143	−	1.22481i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−613.790			−0.203142
$210$		0			0
$211$	−3912.38			−1.27649			−0.638245	−	0.769834i	$-0.720339\pi$
$211$	−3912.38			−1.27649			−0.638245	+	0.769834i	$0.720339\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−2043.84	−	3540.04i	−0.648321	−	1.12292i
$216$		0			0
$217$	76.9958	+	882.970i	0.0240867	+	0.276221i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1159.16	−	669.241i	−0.352821	−	0.203701i
$222$		0			0
$223$		−	885.668i		−	0.265958i	−0.991119	−	0.132979i	$-0.957546\pi$
$223$		−	885.668i		−	0.265958i	0.991119	−	0.132979i	$-0.0424544\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2687.02	+	4654.05i	−0.785654	+	1.36079i	0.142953	+	0.989729i	$0.454340\pi$
$227$	−2687.02	+	4654.05i	−0.785654	+	1.36079i	−0.928607	+	0.371064i	$0.878993\pi$
$228$		0			0
$229$	5385.90	−	3109.55i	1.55419	−	0.897314i	0.556400	−	0.830914i	$-0.312182\pi$
$229$	5385.90	−	3109.55i	1.55419	−	0.897314i	0.997793	−	0.0663998i	$-0.0211513\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5180.55	+	2990.99i	−1.45661	+	0.840972i	−0.998842	−	0.0481015i	$-0.984683\pi$
$233$	−5180.55	+	2990.99i	−1.45661	+	0.840972i	−0.457764	+	0.889074i	$0.651350\pi$
$234$		0			0
$235$	987.981	−	1711.23i	0.274250	−	0.475015i
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	3377.80i		−	0.914191i	−0.889418	−	0.457095i	$-0.848890\pi$
$239$		−	3377.80i		−	0.914191i	0.889418	−	0.457095i	$-0.151110\pi$
$240$		0			0
$241$	−1168.80	−	674.804i	−0.312401	−	0.180365i	0.335599	−	0.942005i	$-0.391061\pi$
$241$	−1168.80	−	674.804i	−0.312401	−	0.180365i	−0.648001	+	0.761640i	$0.724395\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	5332.09	−	937.052i	1.39043	−	0.244351i
$246$		0			0
$247$	511.082	+	885.221i	0.131657	+	0.228037i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−731.129			−0.183858			−0.0919292	−	0.995766i	$-0.529303\pi$
$251$	−731.129			−0.183858			−0.0919292	+	0.995766i	$0.529303\pi$
$252$		0			0
$253$	−1759.74			−0.437288
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2891.62	−	5008.43i	−0.701845	−	1.21563i	−0.967818	−	0.251651i	$-0.919026\pi$
$257$	−2891.62	−	5008.43i	−0.701845	−	1.21563i	0.265972	−	0.963981i	$-0.414307\pi$
$258$		0			0
$259$	754.943	−	1617.77i	0.181119	−	0.388122i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1273.79	+	735.424i	0.298651	+	0.172426i	0.641837	−	0.766841i	$-0.278173\pi$
$263$	1273.79	+	735.424i	0.298651	+	0.172426i	−0.343185	+	0.939268i	$0.611506\pi$
$264$		0			0
$265$		−	8594.73i		−	1.99234i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2298.35	+	3980.86i	−0.520941	+	0.902296i	0.478763	+	0.877944i	$0.341085\pi$
$269$	−2298.35	+	3980.86i	−0.520941	+	0.902296i	−0.999703	+	0.0243516i	$0.992248\pi$
$270$		0			0
$271$	−352.954	+	203.778i	−0.0791160	+	0.0456777i	−0.539036	−	0.842283i	$-0.681211\pi$
$271$	−352.954	+	203.778i	−0.0791160	+	0.0456777i	0.459920	+	0.887960i	$0.347878\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3505.75	−	2024.05i	0.768745	−	0.443835i
$276$		0			0
$277$	−786.099	+	1361.56i	−0.170513	+	0.295337i	−0.938599	−	0.345009i	$-0.887876\pi$
$277$	−786.099	+	1361.56i	−0.170513	+	0.295337i	0.768086	+	0.640346i	$0.221209\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	391.409i		−	0.0830943i	−0.999137	−	0.0415471i	$-0.986771\pi$
$281$		−	391.409i		−	0.0830943i	0.999137	−	0.0415471i	$-0.0132287\pi$
$282$		0			0
$283$	−994.194	−	573.998i	−0.208829	−	0.120568i	0.391938	−	0.919992i	$-0.371805\pi$
$283$	−994.194	−	573.998i	−0.208829	−	0.120568i	−0.600767	+	0.799424i	$0.705138\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2792.68	+	3990.78i	0.574378	+	0.820796i
$288$		0			0
$289$	2152.83	+	3728.81i	0.438190	+	0.758967i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	7106.43			1.41694			0.708468	−	0.705743i	$-0.249387\pi$
$293$	7106.43			1.41694			0.708468	+	0.705743i	$0.249387\pi$
$294$		0			0
$295$	−5978.43			−1.17992
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1465.27	+	2537.93i	0.283408	+	0.490877i
$300$		0			0
$301$	2749.99	+	3929.78i	0.526600	+	0.752520i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9268.23	−	5351.02i	−1.73999	−	1.00458i
$306$		0			0
$307$		−	8035.73i		−	1.49389i	−0.664887	−	0.746944i	$-0.731520\pi$
$307$		−	8035.73i		−	1.49389i	0.664887	−	0.746944i	$-0.268480\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2093.96	+	3626.85i	−0.381793	+	0.661285i	−0.991319	−	0.131481i	$-0.958027\pi$
$311$	−2093.96	+	3626.85i	−0.381793	+	0.661285i	0.609525	+	0.792767i	$0.291360\pi$
$312$		0			0
$313$	6183.56	−	3570.08i	1.11666	−	0.644706i	0.176117	−	0.984369i	$-0.443646\pi$
$313$	6183.56	−	3570.08i	1.11666	−	0.644706i	0.940547	+	0.339663i	$0.110313\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5241.84	−	3026.38i	0.928742	−	0.536210i	0.0423286	−	0.999104i	$-0.486522\pi$
$317$	5241.84	−	3026.38i	0.928742	−	0.536210i	0.886414	+	0.462894i	$0.153189\pi$
$318$		0			0
$319$	2560.91	−	4435.63i	0.449478	−	0.778519i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			463.814i			0.0798988i
$324$		0			0
$325$	−5838.25	−	3370.71i	−0.996454	−	0.575303i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−980.463	+	2101.04i	−0.164300	+	0.352080i
$330$		0			0
$331$	5297.71	+	9175.90i	0.879724	+	1.52373i	0.851644	+	0.524120i	$0.175606\pi$
$331$	5297.71	+	9175.90i	0.879724	+	1.52373i	0.0280795	+	0.999606i	$0.491061\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	10942.4			1.78462
$336$		0			0
$337$	−629.759			−0.101796			−0.0508978	−	0.998704i	$-0.516208\pi$
$337$	−629.759			−0.101796			−0.0508978	+	0.998704i	$0.516208\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−780.382	−	1351.66i	−0.123930	−	0.214653i
$342$		0			0
$343$	−6137.40	+	1638.88i	−0.966147	+	0.257992i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−355.046	−	204.986i	−0.0549276	−	0.0317125i	0.472285	−	0.881446i	$-0.343429\pi$
$347$	−355.046	−	204.986i	−0.0549276	−	0.0317125i	−0.527212	+	0.849734i	$0.676763\pi$
$348$		0			0
$349$			1777.77i			0.272670i	0.990663	+	0.136335i	$0.0435324\pi$
$349$			1777.77i			0.272670i	−0.990663	+	0.136335i	$0.956468\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4076.99	+	7061.56i	−0.614721	+	1.06473i	0.375713	+	0.926736i	$0.377398\pi$
$353$	−4076.99	+	7061.56i	−0.614721	+	1.06473i	−0.990433	+	0.137991i	$0.955935\pi$
$354$		0			0
$355$	3264.60	−	1884.82i	0.488076	−	0.281791i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−2451.26	+	1415.24i	−0.360370	+	0.208060i	−0.669243	−	0.743044i	$-0.733381\pi$
$359$	−2451.26	+	1415.24i	−0.360370	+	0.208060i	0.308873	+	0.951103i	$0.400048\pi$
$360$		0			0
$361$	−3252.40	+	5633.32i	−0.474180	+	0.821303i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	11501.2i		−	1.64931i
$366$		0			0
$367$	−233.258	−	134.671i	−0.0331770	−	0.0191547i	0.483320	−	0.875444i	$-0.339431\pi$
$367$	−233.258	−	134.671i	−0.0331770	−	0.0191547i	−0.516497	+	0.856289i	$0.672764\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	876.087	+	10046.8i	0.122599	+	1.40593i
$372$		0			0
$373$	6119.60	+	10599.5i	0.849493	+	1.47136i	0.881662	+	0.471882i	$0.156425\pi$
$373$	6119.60	+	10599.5i	0.849493	+	1.47136i	−0.0321693	+	0.999482i	$0.510242\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8529.55			−1.16524
$378$		0			0
$379$	690.914			0.0936409			0.0468204	−	0.998903i	$-0.485091\pi$
$379$	690.914			0.0936409			0.0468204	+	0.998903i	$0.485091\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4423.65	−	7661.98i	−0.590177	−	1.02222i	−0.994208	−	0.107472i	$-0.965725\pi$
$383$	−4423.65	−	7661.98i	−0.590177	−	1.02222i	0.404031	−	0.914745i	$-0.367609\pi$
$384$		0			0
$385$	−7810.88	+	5465.91i	−1.03397	+	0.723555i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2808.67	+	1621.58i	0.366080	+	0.211356i	0.671744	−	0.740783i	$-0.265545\pi$
$389$	2808.67	+	1621.58i	0.366080	+	0.211356i	−0.305665	+	0.952139i	$0.598879\pi$
$390$		0			0
$391$			1329.76i			0.171992i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6942.37	−	12024.5i	0.884325	−	1.53170i
$396$		0			0
$397$	−9356.49	+	5401.97i	−1.18284	+	0.682915i	−0.956671	−	0.291172i	$-0.905955\pi$
$397$	−9356.49	+	5401.97i	−1.18284	+	0.682915i	−0.226173	+	0.974087i	$0.572621\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−595.785	+	343.976i	−0.0741947	+	0.0428363i	−0.536638	−	0.843812i	$-0.680306\pi$
$401$	−595.785	+	343.976i	−0.0741947	+	0.0428363i	0.462444	+	0.886649i	$0.346973\pi$
$402$		0			0
$403$	−1299.60	+	2250.97i	−0.160639	+	0.278235i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			3143.74i			0.382873i
$408$		0			0
$409$	7348.56	+	4242.70i	0.888418	+	0.512929i	0.873425	−	0.486959i	$-0.161894\pi$
$409$	7348.56	+	4242.70i	0.888418	+	0.512929i	0.0149935	+	0.999888i	$0.495227\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6988.45	−	609.400i	0.832638	−	0.0726068i
$414$		0			0
$415$	9644.69	+	16705.1i	1.14082	+	1.97595i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11025.1			−1.28547			−0.642736	−	0.766088i	$-0.722201\pi$
$419$	−11025.1			−1.28547			−0.642736	+	0.766088i	$0.722201\pi$
$420$		0			0
$421$	−9636.91			−1.11562			−0.557808	−	0.829970i	$-0.688357\pi$
$421$	−9636.91			−1.11562			−0.557808	+	0.829970i	$0.688357\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1529.48	−	2649.15i	−0.174567	−	0.302359i
$426$		0			0
$427$	11379.5	+	5310.30i	1.28968	+	0.601835i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	4664.33	+	2692.95i	0.521283	+	0.300963i	0.737459	−	0.675391i	$-0.236025\pi$
$431$	4664.33	+	2692.95i	0.521283	+	0.300963i	−0.216176	+	0.976354i	$0.569359\pi$
$432$		0			0
$433$			4428.30i			0.491480i	0.969336	+	0.245740i	$0.0790309\pi$
$433$			4428.30i			0.491480i	−0.969336	+	0.245740i	$0.920969\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	507.751	−	879.451i	0.0555813	−	0.0962697i
$438$		0			0
$439$	14675.4	−	8472.85i	1.59549	−	0.921155i	0.603145	−	0.797632i	$-0.293914\pi$
$439$	14675.4	−	8472.85i	1.59549	−	0.921155i	0.992342	−	0.123523i	$-0.0394193\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15157.8	−	8751.39i	1.62567	−	0.938580i	0.640304	−	0.768122i	$-0.278809\pi$
$443$	15157.8	−	8751.39i	1.62567	−	0.938580i	0.985365	−	0.170458i	$-0.0545247\pi$
$444$		0			0
$445$	1198.83	−	2076.44i	0.127708	−	0.221197i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	1975.26i		−	0.207614i	−0.994597	−	0.103807i	$-0.966898\pi$
$449$		−	1975.26i		−	0.207614i	0.994597	−	0.103807i	$-0.0331024\pi$
$450$		0			0
$451$	−7428.20	−	4288.67i	−0.775566	−	0.447773i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	14386.9	+	6713.73i	1.48235	+	0.691746i
$456$		0			0
$457$	6036.94	+	10456.3i	0.617935	+	1.07029i	0.989862	+	0.142032i	$0.0453637\pi$
$457$	6036.94	+	10456.3i	0.617935	+	1.07029i	−0.371927	+	0.928262i	$0.621303\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13447.2			1.35856			0.679281	−	0.733878i	$-0.262292\pi$
$461$	13447.2			1.35856			0.679281	+	0.733878i	$0.262292\pi$
$462$		0			0
$463$	15720.4			1.57795			0.788973	−	0.614428i	$-0.210613\pi$
$463$	15720.4			1.57795			0.788973	+	0.614428i	$0.210613\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−682.156	−	1181.53i	−0.0675940	−	0.117076i	0.830248	−	0.557395i	$-0.188199\pi$
$467$	−682.156	−	1181.53i	−0.0675940	−	0.117076i	−0.897842	+	0.440318i	$0.854866\pi$
$468$		0			0
$469$	−12791.1	+	1115.40i	−1.25936	+	0.109817i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7314.65	−	4223.11i	−0.711052	−	0.410526i
$474$		0			0
$475$			2336.06i			0.225654i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−2078.39	+	3599.88i	−0.198255	+	0.343388i	−0.947963	−	0.318381i	$-0.896861\pi$
$479$	−2078.39	+	3599.88i	−0.198255	+	0.343388i	0.749708	+	0.661769i	$0.230194\pi$
$480$		0			0
$481$	4533.96	−	2617.69i	0.429794	−	0.248142i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	21753.3	−	12559.3i	2.03663	−	1.17585i
$486$		0			0
$487$	6210.51	−	10756.9i	0.577875	−	1.00091i	−0.417848	−	0.908517i	$-0.637215\pi$
$487$	6210.51	−	10756.9i	0.577875	−	1.00091i	0.995723	−	0.0923918i	$-0.0294512\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			8234.71i			0.756879i	0.925626	+	0.378439i	$0.123539\pi$
$491$			8234.71i			0.756879i	−0.925626	+	0.378439i	$0.876461\pi$
$492$		0			0
$493$	−3351.81	−	1935.17i	−0.306203	−	0.176786i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3624.01	+	2536.02i	−0.327081	+	0.228885i
$498$		0			0
$499$	−1488.56	−	2578.26i	−0.133541	−	0.231300i	0.791498	−	0.611172i	$-0.209302\pi$
$499$	−1488.56	−	2578.26i	−0.133541	−	0.231300i	−0.925039	+	0.379871i	$0.875968\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−537.349			−0.0476327			−0.0238163	−	0.999716i	$-0.507582\pi$
$503$	−537.349			−0.0476327			−0.0238163	+	0.999716i	$0.507582\pi$
$504$		0			0
$505$	905.792			0.0798163
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−4873.32	−	8440.83i	−0.424373	−	0.735036i	0.571988	−	0.820262i	$-0.306172\pi$
$509$	−4873.32	−	8440.83i	−0.424373	−	0.735036i	−0.996362	+	0.0852256i	$0.972839\pi$
$510$		0			0
$511$	1172.35	+	13444.2i	0.101491	+	1.16387i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	26425.8	+	15256.9i	2.26108	+	1.30544i
$516$		0			0
$517$		−	4082.85i		−	0.347318i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−4831.31	+	8368.07i	−0.406264	+	0.703670i	−0.994468	−	0.105043i	$-0.966502\pi$
$521$	−4831.31	+	8368.07i	−0.406264	+	0.703670i	0.588204	+	0.808713i	$0.299835\pi$
$522$		0			0
$523$	5153.64	−	2975.46i	0.430886	−	0.248772i	−0.268838	−	0.963185i	$-0.586640\pi$
$523$	5153.64	−	2975.46i	0.430886	−	0.248772i	0.699724	+	0.714413i	$0.253306\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1021.39	+	589.701i	−0.0844260	+	0.0487434i
$528$		0			0
$529$	−4627.78	+	8015.54i	−0.380355	+	0.658794i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			14284.1i			1.16082i
$534$		0			0
$535$	−27300.5	−	15762.0i	−2.20618	−	1.27374i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	8573.33	−	7185.54i	0.685119	−	0.574217i
$540$		0			0
$541$	4476.87	+	7754.17i	0.355778	+	0.616225i	0.987251	−	0.159173i	$-0.0508826\pi$
$541$	4476.87	+	7754.17i	0.355778	+	0.616225i	−0.631473	+	0.775398i	$0.717549\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−22064.3			−1.73418
$546$		0			0
$547$	−20723.2			−1.61985			−0.809926	−	0.586532i	$-0.800493\pi$
$547$	−20723.2			−1.61985			−0.809926	+	0.586532i	$0.800493\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1477.84	+	2559.70i	0.114262	+	0.197907i
$552$		0			0
$553$	−6889.54	+	14763.7i	−0.529789	+	1.13529i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−11283.3	−	6514.43i	−0.858330	−	0.495557i	0.00512296	−	0.999987i	$-0.498369\pi$
$557$	−11283.3	−	6514.43i	−0.858330	−	0.495557i	−0.863453	+	0.504430i	$0.831703\pi$
$558$		0			0
$559$			14065.8i			1.06426i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	955.378	−	1654.76i	0.0715176	−	0.123872i	−0.828049	−	0.560656i	$-0.810549\pi$
$563$	955.378	−	1654.76i	0.0715176	−	0.123872i	0.899567	+	0.436784i	$0.143882\pi$
$564$		0			0
$565$	22366.8	−	12913.5i	1.66545	−	0.961546i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	5509.88	−	3181.13i	0.405951	−	0.234376i	−0.283098	−	0.959091i	$-0.591362\pi$
$569$	5509.88	−	3181.13i	0.405951	−	0.234376i	0.689049	+	0.724715i	$0.258029\pi$
$570$		0			0
$571$	−3394.59	+	5879.61i	−0.248790	+	0.430918i	−0.963190	−	0.268820i	$-0.913366\pi$
$571$	−3394.59	+	5879.61i	−0.248790	+	0.430918i	0.714400	+	0.699737i	$0.246700\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			6697.49i			0.485747i
$576$		0			0
$577$	6109.08	+	3527.08i	0.440770	+	0.254479i	0.703924	−	0.710275i	$-0.251429\pi$
$577$	6109.08	+	3527.08i	0.440770	+	0.254479i	−0.263154	+	0.964754i	$0.584763\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−12976.9	−	18544.2i	−0.926632	−	1.32417i
$582$		0			0
$583$	−8879.47	−	15379.7i	−0.630789	−	1.09256i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−14742.1			−1.03658			−0.518290	−	0.855205i	$-0.673431\pi$
$587$	−14742.1			−1.03658			−0.518290	+	0.855205i	$0.673431\pi$
$588$		0			0
$589$	900.679			0.0630082
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−6998.75	−	12122.2i	−0.484662	−	0.839458i	0.515183	−	0.857080i	$-0.327724\pi$
$593$	−6998.75	−	12122.2i	−0.484662	−	0.839458i	−0.999845	+	0.0176216i	$0.994391\pi$
$594$		0			0
$595$	4130.35	+	5902.34i	0.284585	+	0.406676i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−251.377	−	145.133i	−0.0171469	−	0.00989977i	0.491402	−	0.870933i	$-0.336485\pi$
$599$	−251.377	−	145.133i	−0.0171469	−	0.00989977i	−0.508549	+	0.861033i	$0.669818\pi$
$600$		0			0
$601$			3671.51i			0.249191i	0.992208	+	0.124596i	$0.0397634\pi$
$601$			3671.51i			0.249191i	−0.992208	+	0.124596i	$0.960237\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2110.12	+	3654.83i	−0.141799	+	0.245603i
$606$		0			0
$607$	14231.9	−	8216.78i	0.951655	−	0.549438i	0.0580603	−	0.998313i	$-0.481508\pi$
$607$	14231.9	−	8216.78i	0.951655	−	0.549438i	0.893595	+	0.448875i	$0.148175\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−5888.37	+	3399.65i	−0.389882	+	0.225099i
$612$		0			0
$613$	11452.7	−	19836.6i	0.754599	−	1.30700i	−0.190975	−	0.981595i	$-0.561165\pi$
$613$	11452.7	−	19836.6i	0.754599	−	1.30700i	0.945574	−	0.325408i	$-0.105502\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			10613.9i			0.692541i	0.938135	+	0.346271i	$0.112552\pi$
$617$			10613.9i			0.692541i	−0.938135	+	0.346271i	$0.887448\pi$
$618$		0			0
$619$	−5261.57	−	3037.77i	−0.341648	−	0.197251i	0.319352	−	0.947636i	$-0.396535\pi$
$619$	−5261.57	−	3037.77i	−0.341648	−	0.197251i	−0.661001	+	0.750385i	$0.729868\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1189.71	+	2549.44i	−0.0765085	+	0.163951i
$624$		0			0
$625$	7866.84	+	13625.8i	0.503478	+	0.872049i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2375.59			0.150590
$630$		0			0
$631$	16951.1			1.06944			0.534718	−	0.845031i	$-0.320418\pi$
$631$	16951.1			1.06944			0.534718	+	0.845031i	$0.320418\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−13646.6	−	23636.5i	−0.852830	−	1.47715i
$636$		0			0
$637$	−17501.9	−	6381.47i	−1.08862	−	0.396928i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−10591.3	−	6114.91i	−0.652625	−	0.376793i	0.136836	−	0.990594i	$-0.456307\pi$
$641$	−10591.3	−	6114.91i	−0.652625	−	0.376793i	−0.789461	+	0.613800i	$0.789640\pi$
$642$		0			0
$643$		−	11251.9i		−	0.690095i	−0.938585	−	0.345048i	$-0.887863\pi$
$643$		−	11251.9i		−	0.690095i	0.938585	−	0.345048i	$-0.112137\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3237.83	+	5608.08i	−0.196742	+	0.340767i	−0.947470	−	0.319844i	$-0.896370\pi$
$647$	−3237.83	+	5608.08i	−0.196742	+	0.340767i	0.750728	+	0.660611i	$0.229703\pi$
$648$		0			0
$649$	−10698.0	+	6176.50i	−0.647047	+	0.373573i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	16416.0	−	9477.77i	0.983778	−	0.567984i	0.0803694	−	0.996765i	$-0.474390\pi$
$653$	16416.0	−	9477.77i	0.983778	−	0.567984i	0.903409	+	0.428781i	$0.141057\pi$
$654$		0			0
$655$	−1419.44	+	2458.55i	−0.0846751	+	0.146662i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			22148.1i			1.30921i	0.755973	+	0.654603i	$0.227164\pi$
$659$			22148.1i			1.30921i	−0.755973	+	0.654603i	$0.772836\pi$
$660$		0			0
$661$	−14010.5	−	8088.96i	−0.824425	−	0.475982i	0.0275153	−	0.999621i	$-0.491240\pi$
$661$	−14010.5	−	8088.96i	−0.824425	−	0.475982i	−0.851940	+	0.523640i	$0.824574\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−477.923	−	5480.71i	−0.0278693	−	0.319598i
$666$		0			0
$667$	4236.98	+	7338.66i	0.245962	+	0.426018i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−22113.2			−1.27224
$672$		0			0
$673$	16420.1			0.940488			0.470244	−	0.882537i	$-0.344166\pi$
$673$	16420.1			0.940488			0.470244	+	0.882537i	$0.344166\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	11989.3	+	20766.0i	0.680627	+	1.17888i	0.974790	+	0.223125i	$0.0716258\pi$
$677$	11989.3	+	20766.0i	0.680627	+	1.17888i	−0.294163	+	0.955755i	$0.595041\pi$
$678$		0			0
$679$	−24148.2	+	16898.5i	−1.36484	+	0.955088i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1924.00	+	1110.82i	0.107789	+	0.0622319i	0.552925	−	0.833231i	$-0.313512\pi$
$683$	1924.00	+	1110.82i	0.107789	+	0.0622319i	−0.445136	+	0.895463i	$0.646845\pi$
$684$		0			0
$685$		−	44952.9i		−	2.50739i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14787.3	+	25612.3i	−0.817635	+	1.41619i
$690$		0			0
$691$	20155.6	−	11636.9i	1.10963	−	0.640647i	0.170899	−	0.985289i	$-0.445333\pi$
$691$	20155.6	−	11636.9i	1.10963	−	0.640647i	0.938734	+	0.344642i	$0.112000\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	798.954	−	461.276i	0.0436058	−	0.0251758i
$696$		0			0
$697$	−3240.77	+	5613.17i	−0.176116	+	0.305042i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			5003.94i			0.269610i	0.990872	+	0.134805i	$0.0430407\pi$
$701$			5003.94i			0.269610i	−0.990872	+	0.134805i	$0.956959\pi$
$702$		0			0
$703$	−1571.12	−	907.088i	−0.0842902	−	0.0486650i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1058.82	+	92.3302i	−0.0563240	+	0.00491151i
$708$		0			0
$709$	−15002.2	−	25984.6i	−0.794669	−	1.37641i	−0.923049	−	0.384682i	$-0.874311\pi$
$709$	−15002.2	−	25984.6i	−0.794669	−	1.37641i	0.128381	−	0.991725i	$-0.459022\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2582.25			0.135633
$714$		0			0
$715$	−27957.3			−1.46230
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−7195.48	−	12462.9i	−0.373221	−	0.646438i	0.616838	−	0.787090i	$-0.288413\pi$
$719$	−7195.48	−	12462.9i	−0.373221	−	0.646438i	−0.990059	+	0.140652i	$0.955080\pi$
$720$		0			0
$721$	−32445.4	−	15140.8i	−1.67591	−	0.782072i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16881.8	−	9746.73i	−0.864794	−	0.499289i
$726$		0			0
$727$			4931.70i			0.251591i	0.992056	+	0.125795i	$0.0401483\pi$
$727$			4931.70i			0.251591i	−0.992056	+	0.125795i	$0.959852\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−3191.22	+	5527.36i	−0.161466	+	0.279667i
$732$		0			0
$733$	17813.2	−	10284.4i	0.897604	−	0.518232i	0.0211822	−	0.999776i	$-0.493257\pi$
$733$	17813.2	−	10284.4i	0.897604	−	0.518232i	0.876422	+	0.481543i	$0.159924\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	19580.7	−	11305.0i	0.978652	−	0.565025i
$738$		0			0
$739$	15864.0	−	27477.2i	0.789669	−	1.36775i	−0.136501	−	0.990640i	$-0.543586\pi$
$739$	15864.0	−	27477.2i	0.789669	−	1.36775i	0.926170	−	0.377106i	$-0.123081\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	30966.4i		−	1.52900i	−0.644623	−	0.764501i	$-0.722986\pi$
$743$		−	30966.4i		−	1.52900i	0.644623	−	0.764501i	$-0.277014\pi$
$744$		0			0
$745$	−11075.0	−	6394.16i	−0.544640	−	0.314448i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	33519.5	+	15642.0i	1.63521	+	0.763081i
$750$		0			0
$751$	−17017.6	−	29475.4i	−0.826873	−	1.43219i	−0.900480	−	0.434898i	$-0.856784\pi$
$751$	−17017.6	−	29475.4i	−0.826873	−	1.43219i	0.0736070	−	0.997287i	$-0.476549\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−40144.0			−1.93508
$756$		0			0
$757$	1216.96			0.0584293			0.0292147	−	0.999573i	$-0.490699\pi$
$757$	1216.96			0.0584293			0.0292147	+	0.999573i	$0.490699\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	14359.4	+	24871.2i	0.684006	+	1.18473i	0.973748	+	0.227628i	$0.0730971\pi$
$761$	14359.4	+	24871.2i	0.684006	+	1.18473i	−0.289742	+	0.957105i	$0.593570\pi$
$762$		0			0
$763$	25791.9	−	2249.08i	1.22376	−	0.106713i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	17815.7	+	10285.9i	0.838708	+	0.484229i
$768$		0			0
$769$			914.245i			0.0428719i	0.999770	+	0.0214360i	$0.00682380\pi$
$769$			914.245i			0.0428719i	−0.999770	+	0.0214360i	$0.993176\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6138.69	+	10632.5i	−0.285632	+	0.494729i	−0.972762	−	0.231805i	$-0.925537\pi$
$773$	−6138.69	+	10632.5i	−0.285632	+	0.494729i	0.687131	+	0.726534i	$0.258870\pi$
$774$		0			0
$775$	−5144.37	+	2970.10i	−0.238440	+	0.137663i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	4286.64	−	2474.89i	0.197156	−	0.113828i
$780$		0			0
$781$	3894.52	−	6745.51i	0.178434	−	0.309057i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			39760.5i			1.80779i
$786$		0			0
$787$	−16936.5	−	9778.31i	−0.767119	−	0.442896i	0.0647271	−	0.997903i	$-0.479382\pi$
$787$	−16936.5	−	9778.31i	−0.767119	−	0.442896i	−0.831846	+	0.555007i	$0.812716\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−24829.2	+	17375.0i	−1.11609	+	0.781018i
$792$		0			0
$793$	18412.9	+	31892.1i	0.824542	+	1.42815i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3319.70			0.147540			0.0737702	−	0.997275i	$-0.476497\pi$
$797$	3319.70			0.147540			0.0737702	+	0.997275i	$0.476497\pi$
$798$		0			0
$799$	−3085.23			−0.136605
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−11882.2	−	20580.6i	−0.522184	−	0.904450i
$804$		0			0
$805$	−1370.21	−	15713.2i	−0.0599919	−	0.687973i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−27790.1	−	16044.6i	−1.20772	−	0.697280i	−0.245462	−	0.969406i	$-0.578940\pi$
$809$	−27790.1	−	16044.6i	−1.20772	−	0.697280i	−0.962261	+	0.272126i	$0.912273\pi$
$810$		0			0
$811$		−	17924.1i		−	0.776081i	−0.921642	−	0.388040i	$-0.873152\pi$
$811$		−	17924.1i		−	0.776081i	0.921642	−	0.388040i	$-0.126848\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	16923.0	−	29311.4i	0.727344	−	1.25980i
$816$		0			0
$817$	4221.11	−	2437.06i	0.180756	−	0.104360i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3839.60	−	2216.79i	0.163219	−	0.0942347i	−0.416165	−	0.909289i	$-0.636626\pi$
$821$	3839.60	−	2216.79i	0.163219	−	0.0942347i	0.579385	+	0.815054i	$0.303293\pi$
$822$		0			0
$823$	6029.71	−	10443.8i	0.255386	−	0.442341i	−0.709614	−	0.704590i	$-0.751131\pi$
$823$	6029.71	−	10443.8i	0.255386	−	0.442341i	0.965000	+	0.262249i	$0.0844642\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1552.07i			0.0652609i	0.999467	+	0.0326304i	$0.0103884\pi$
$827$			1552.07i			0.0652609i	−0.999467	+	0.0326304i	$0.989612\pi$
$828$		0			0
$829$	3856.52	+	2226.56i	0.161571	+	0.0932832i	0.578605	−	0.815608i	$-0.303597\pi$
$829$	3856.52	+	2226.56i	0.161571	+	0.0932832i	−0.417034	+	0.908891i	$0.636930\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−5429.80	−	6478.49i	−0.225848	−	0.269467i
$834$		0			0
$835$	26686.9	+	46223.1i	1.10604	+	1.91571i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22708.5			0.934429			0.467214	−	0.884144i	$-0.345258\pi$
$839$	22708.5			0.934429			0.467214	+	0.884144i	$0.345258\pi$
$840$		0			0
$841$	−274.978			−0.0112747
$842$		0			0
$843$		0			0
$844$		0			0
$845$	5940.78	+	10289.7i	0.241857	+	0.418908i
$846$		0			0
$847$	2094.06	−	4487.39i	0.0849502	−	0.182041i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−4504.41	−	2600.62i	−0.181445	−	0.104757i
$852$		0			0
$853$		−	13532.2i		−	0.543183i	−0.962413	−	0.271591i	$-0.912450\pi$
$853$		−	13532.2i		−	0.543183i	0.962413	−	0.271591i	$-0.0875499\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7847.69	+	13592.6i	−0.312803	+	0.541790i	−0.978968	−	0.204014i	$-0.934601\pi$
$857$	−7847.69	+	13592.6i	−0.312803	+	0.541790i	0.666165	+	0.745804i	$0.267935\pi$
$858$		0			0
$859$	−31542.4	+	18211.0i	−1.25287	+	0.723342i	−0.971678	−	0.236311i	$-0.924062\pi$
$859$	−31542.4	+	18211.0i	−1.25287	+	0.723342i	−0.281188	+	0.959653i	$0.590728\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−26424.8	+	15256.4i	−1.04231	+	0.601777i	−0.920486	−	0.390775i	$-0.872207\pi$
$863$	−26424.8	+	15256.4i	−1.04231	+	0.601777i	−0.121822	+	0.992552i	$0.538874\pi$
$864$		0			0
$865$	1562.45	−	2706.24i	0.0614159	−	0.106375i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	28689.5i		−	1.11994i
$870$		0			0
$871$	−32608.5	−	18826.5i	−1.26854	−	0.732390i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−146.739	−	209.692i	−0.00566934	−	0.00810158i
$876$		0			0
$877$	1409.59	+	2441.48i	0.0542742	+	0.0940057i	0.891886	−	0.452260i	$-0.149382\pi$
$877$	1409.59	+	2441.48i	0.0542742	+	0.0940057i	−0.837612	+	0.546266i	$0.816049\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18469.6			0.706306			0.353153	−	0.935566i	$-0.385110\pi$
$881$	18469.6			0.706306			0.353153	+	0.935566i	$0.385110\pi$
$882$		0			0
$883$	−2535.09			−0.0966168			−0.0483084	−	0.998832i	$-0.515383\pi$
$883$	−2535.09			−0.0966168			−0.0483084	+	0.998832i	$0.515383\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−21522.4	−	37278.0i	−0.814716	−	1.41113i	−0.909532	−	0.415635i	$-0.863560\pi$
$887$	−21522.4	−	37278.0i	−0.814716	−	1.41113i	0.0948157	−	0.995495i	$-0.469774\pi$
$888$		0			0
$889$	18361.4	+	26238.8i	0.692714	+	0.989899i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	2040.46	+	1178.06i	0.0764628	+	0.0441458i
$894$		0			0
$895$			54813.4i			2.04716i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3757.90	+	6508.88i	−0.139414	+	0.241472i
$900$		0			0
$901$	−11621.8	+	6709.83i	−0.429720	+	0.248099i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−39270.5	+	22672.8i	−1.44243	+	0.832785i
$906$		0			0
$907$	−6840.28	+	11847.7i	−0.250416	+	0.433734i	−0.963640	−	0.267202i	$-0.913901\pi$
$907$	−6840.28	+	11847.7i	−0.250416	+	0.433734i	0.713224	+	0.700936i	$0.247234\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	33621.6i		−	1.22276i	−0.791338	−	0.611379i	$-0.790615\pi$
$911$		−	33621.6i		−	1.22276i	0.791338	−	0.611379i	$-0.209385\pi$
$912$		0			0
$913$	34517.1	+	19928.5i	1.25120	+	0.722383i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1408.64	−	3018.59i	0.0507279	−	0.108705i
$918$		0			0
$919$	17126.3	+	29663.5i	0.614737	+	1.06476i	0.990431	+	0.138011i	$0.0440711\pi$
$919$	17126.3	+	29663.5i	0.614737	+	1.06476i	−0.375694	+	0.926744i	$0.622596\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−12971.4			−0.462576
$924$		0			0
$925$	11964.9			0.425302
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12337.4	+	21369.0i	0.435712	+	0.754675i	0.997353	−	0.0727058i	$-0.0231634\pi$
$929$	12337.4	+	21369.0i	0.435712	+	0.754675i	−0.561642	+	0.827380i	$0.689830\pi$
$930$		0			0
$931$	1117.33	+	6357.93i	0.0393330	+	0.223816i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−10986.3	−	6342.92i	−0.384266	−	0.221856i
$936$		0			0
$937$		−	32885.0i		−	1.14654i	−0.819367	−	0.573269i	$-0.805675\pi$
$937$		−	32885.0i		−	1.14654i	0.819367	−	0.573269i	$-0.194325\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4016.54	−	6956.85i	0.139145	−	0.241006i	−0.788028	−	0.615639i	$-0.788898\pi$
$941$	4016.54	−	6956.85i	0.139145	−	0.241006i	0.927173	+	0.374633i	$0.122231\pi$
$942$		0			0
$943$	12289.8	−	7095.52i	0.424402	−	0.245029i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	15192.1	−	8771.14i	0.521305	−	0.300975i	−0.216164	−	0.976357i	$-0.569354\pi$
$947$	15192.1	−	8771.14i	0.521305	−	0.300975i	0.737468	+	0.675382i	$0.236021\pi$
$948$		0			0
$949$	−19787.8	+	34273.5i	−0.676860	+	1.17236i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	40802.9i		−	1.38692i	−0.720494	−	0.693461i	$-0.756085\pi$
$953$		−	40802.9i		−	1.38692i	0.720494	−	0.693461i	$-0.243915\pi$
$954$		0			0
$955$	16970.5	+	9797.95i	0.575030	+	0.331994i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4582.19	+	52547.5i	0.154293	+	1.76939i
$960$		0			0
$961$	−13750.4	−	23816.3i	−0.461561	−	0.799447i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	42341.5			1.41246
$966$		0			0
$967$	−57938.1			−1.92675			−0.963373	−	0.268164i	$-0.913583\pi$
$967$	−57938.1			−1.92675			−0.963373	+	0.268164i	$0.913583\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−7580.65	−	13130.1i	−0.250540	−	0.433948i	0.713134	−	0.701027i	$-0.247275\pi$
$971$	−7580.65	−	13130.1i	−0.250540	−	0.433948i	−0.963675	+	0.267079i	$0.913942\pi$
$972$		0			0
$973$	−886.913	+	620.646i	−0.0292221	+	0.0204491i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25781.1	+	14884.7i	0.844228	+	0.487415i	0.858699	−	0.512480i	$-0.171273\pi$
$977$	25781.1	+	14884.7i	0.844228	+	0.487415i	−0.0144714	+	0.999895i	$0.504607\pi$
$978$		0			0
$979$		−	4954.21i		−	0.161733i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	16517.5	−	28609.1i	0.535937	−	0.928269i	−0.463181	−	0.886264i	$-0.653292\pi$
$983$	16517.5	−	28609.1i	0.535937	−	0.928269i	0.999117	−	0.0420055i	$-0.0133747\pi$
$984$		0			0
$985$	41780.4	−	24121.9i	1.35151	−	0.780293i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	12101.9	−	6987.05i	0.389099	−	0.224646i
$990$		0			0
$991$	4752.11	−	8230.89i	0.152327	−	0.263837i	−0.779756	−	0.626084i	$-0.784657\pi$
$991$	4752.11	−	8230.89i	0.152327	−	0.263837i	0.932082	+	0.362246i	$0.117990\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			33978.2i			1.08259i
$996$		0			0
$997$	15522.5	+	8961.95i	0.493083	+	0.284682i	0.725853	−	0.687850i	$-0.241445\pi$
$997$	15522.5	+	8961.95i	0.493083	+	0.284682i	−0.232769	+	0.972532i	$0.574779\pi$
$998$		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.17.4	✓	48
3.2	odd	2	inner	504.4.bl.a.17.21	yes	48
4.3	odd	2		1008.4.bt.d.17.4		48
7.5	odd	6	inner	504.4.bl.a.89.21	yes	48
12.11	even	2		1008.4.bt.d.17.21		48
21.5	even	6	inner	504.4.bl.a.89.4	yes	48
28.19	even	6		1008.4.bt.d.593.21		48
84.47	odd	6		1008.4.bt.d.593.4		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.4	✓	48	1.1	even	1	trivial
504.4.bl.a.17.21	yes	48	3.2	odd	2	inner
504.4.bl.a.89.4	yes	48	21.5	even	6	inner
504.4.bl.a.89.21	yes	48	7.5	odd	6	inner
1008.4.bt.d.17.4		48	4.3	odd	2
1008.4.bt.d.17.21		48	12.11	even	2
1008.4.bt.d.593.4		48	84.47	odd	6
1008.4.bt.d.593.21		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)

\(f(q)\)	\(=\)	\(q+(-7.89184 - 13.6691i) q^{5} +(10.6185 + 15.1739i) q^{7} +O(q^{10})\)	\(q+(-7.89184 - 13.6691i) q^{5} +(10.6185 + 15.1739i) q^{7} +(-28.2439 - 16.3066i) q^{11} +54.3118i q^{13} +(-12.3222 + 21.3427i) q^{17} +(16.2989 - 9.41015i) q^{19} +(46.7289 - 26.9789i) q^{23} +(-62.0622 + 107.495i) q^{25} +157.048i q^{29} +(41.4452 + 23.9284i) q^{31} +(123.614 - 264.895i) q^{35} +(-48.1973 - 83.4802i) q^{37} +263.002 q^{41} +258.982 q^{43} +(62.5951 + 108.418i) q^{47} +(-117.497 + 322.248i) q^{49} +(471.579 + 272.266i) q^{53} +514.756i q^{55} +(189.386 - 328.027i) q^{59} +(587.204 - 339.022i) q^{61} +(742.392 - 428.620i) q^{65} +(-346.637 + 600.394i) q^{67} +238.831i q^{71} +(631.051 + 364.337i) q^{73} +(-52.4707 - 601.721i) q^{77} +(439.845 + 761.833i) q^{79} -1222.11 q^{83} +388.979 q^{85} +(75.9540 + 131.556i) q^{89} +(-824.125 + 576.708i) q^{91} +(-257.256 - 148.527i) q^{95} +1591.43i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 24 q^{7}+O(q^{10}) \) 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

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