LMFDB - Embedded newform 1008.4.bt.d.593.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1008,4,Mod(17,1008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1008, 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("1008.17");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1008 = 2^{4} \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1008.bt (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$59.4739252858$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 504)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			593.4
Character	$\chi$	$=$	1008.593
Dual form			1008.4.bt.d.17.4

$q$-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 $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

$n$	$127$	$577$	$757$	$785$
$\chi(n)$	$1$	$e\left(\frac{5}{6}\right)$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−7.89184	+	13.6691i	−0.705867	+	1.22260i	0.260510	+	0.965471i	$0.416109\pi$
$5$	−7.89184	+	13.6691i	−0.705867	+	1.22260i	−0.966378	+	0.257127i	$0.917224\pi$
$6$		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	$-0.519171\pi$
$11$	28.2439	−	16.3066i	0.774167	−	0.446966i	0.834359	+	0.551221i	$0.185838\pi$
$12$		0			0
$13$		−	54.3118i		−	1.15872i	−0.815071	−	0.579361i	$-0.803302\pi$
$13$		−	54.3118i		−	1.15872i	0.815071	−	0.579361i	$-0.196698\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−12.3222	−	21.3427i	−0.175798	−	0.304491i	0.764639	−	0.644459i	$-0.222917\pi$
$17$	−12.3222	−	21.3427i	−0.175798	−	0.304491i	−0.940437	+	0.339967i	$0.889584\pi$
$18$		0			0
$19$	−16.2989	−	9.41015i	−0.196801	−	0.113623i	0.398362	−	0.917228i	$-0.369579\pi$
$19$	−16.2989	−	9.41015i	−0.196801	−	0.113623i	−0.595162	+	0.803606i	$0.702912\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−46.7289	−	26.9789i	−0.423637	−	0.244587i	0.272995	−	0.962015i	$-0.411986\pi$
$23$	−46.7289	−	26.9789i	−0.423637	−	0.244587i	−0.696632	+	0.717429i	$0.745319\pi$
$24$		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	$-0.832299\pi$
$29$		−	157.048i		−	1.00562i	0.864397	−	0.502811i	$-0.167701\pi$
$30$		0			0
$31$	−41.4452	+	23.9284i	−0.240122	+	0.138634i	−0.615233	−	0.788345i	$-0.710938\pi$
$31$	−41.4452	+	23.9284i	−0.240122	+	0.138634i	0.375111	+	0.926980i	$0.377605\pi$
$32$		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	−0.953010	−	0.302940i	$-0.902032\pi$
$37$	−48.1973	+	83.4802i	−0.214151	+	0.370921i	0.738858	+	0.673861i	$0.235365\pi$
$38$		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	$-0.651877\pi$
$43$	−258.982			−0.918474			−0.459237	+	0.888314i	$0.651877\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−62.5951	+	108.418i	−0.194264	+	0.336476i	−0.946659	−	0.322237i	$-0.895565\pi$
$47$	−62.5951	+	108.418i	−0.194264	+	0.336476i	0.752395	+	0.658713i	$0.228899\pi$
$48$		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	0.256810	−	0.966462i	$-0.417329\pi$
$53$	471.579	−	272.266i	1.22220	−	0.705635i	0.965386	+	0.260827i	$0.0839952\pi$
$54$		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	$-0.303900\pi$
$59$	−189.386	−	328.027i	−0.417899	−	0.723822i	−0.995728	+	0.0923360i	$0.970567\pi$
$60$		0			0
$61$	587.204	+	339.022i	1.23252	+	0.711596i	0.967555	−	0.252662i	$-0.0813061\pi$
$61$	587.204	+	339.022i	1.23252	+	0.711596i	0.264966	+	0.964258i	$0.414639\pi$
$62$		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	0.987128	+	0.159929i	$0.0511266\pi$
$67$	346.637	+	600.394i	0.632067	+	1.09477i	−0.355061	+	0.934843i	$0.615540\pi$
$68$		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	0.100058	−	0.994982i	$-0.468097\pi$
$73$	631.051	−	364.337i	1.01177	−	0.584143i	0.911708	+	0.410838i	$0.134764\pi$
$74$		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	0.361856	+	0.932234i	$0.382143\pi$
$79$	−439.845	+	761.833i	−0.626410	+	1.08497i	−0.988266	+	0.152740i	$0.951190\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1222.11			1.61619			0.808096	−	0.589050i	$-0.200498\pi$
$83$	1222.11			1.61619			0.808096	+	0.589050i	$0.200498\pi$
$84$		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	−0.817244	−	0.576292i	$-0.804499\pi$
$89$	75.9540	−	131.556i	0.0904619	−	0.156685i	0.907706	+	0.419608i	$0.137832\pi$
$90$		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	$-0.686672\pi$
$97$		−	1591.43i		−	1.66582i	0.553406	−	0.832912i	$-0.313328\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−28.6940	−	49.6994i	−0.0282689	−	0.0489631i	0.851545	−	0.524282i	$-0.175666\pi$
$101$	−28.6940	−	49.6994i	−0.0282689	−	0.0489631i	−0.879814	+	0.475319i	$0.842333\pi$
$102$		0			0
$103$	1674.25	+	966.626i	1.60163	+	0.924704i	0.991161	+	0.132667i	$0.0423541\pi$
$103$	1674.25	+	966.626i	1.60163	+	0.924704i	0.610473	+	0.792037i	$0.290979\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1729.67	−	998.625i	−1.56274	−	0.902249i	−0.996978	−	0.0776819i	$-0.975248\pi$
$107$	−1729.67	−	998.625i	−1.56274	−	0.902249i	−0.565764	−	0.824567i	$-0.691419\pi$
$108$		0			0
$109$	698.958	+	1210.63i	0.614202	+	1.06383i	0.990524	+	0.137340i	$0.0438554\pi$
$109$	698.958	+	1210.63i	0.614202	+	1.06383i	−0.376322	+	0.926489i	$0.622811\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	1636.31i		−	1.36222i	−0.732181	−	0.681110i	$-0.761498\pi$
$113$		−	1636.31i		−	1.36222i	0.732181	−	0.681110i	$-0.238502\pi$
$114$		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	$-0.706468\pi$
$127$	−1729.20			−1.20820			−0.604101	+	0.796908i	$0.706468\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	89.9310	−	155.765i	0.0599795	−	0.103887i	−0.834476	−	0.551044i	$-0.814230\pi$
$131$	89.9310	−	155.765i	0.0599795	−	0.103887i	0.894456	+	0.447156i	$0.147563\pi$
$132$		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	0.539205	−	0.842174i	$-0.318725\pi$
$137$	2466.49	−	1424.03i	1.53815	−	0.888053i	0.998947	−	0.0458784i	$-0.0146087\pi$
$138$		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	0.680337	−	0.732900i	$-0.261834\pi$
$149$	701.675	+	405.112i	0.385795	+	0.222739i	−0.294542	+	0.955639i	$0.595167\pi$
$150$		0			0
$151$	−1271.69	−	2202.64i	−0.685357	−	1.18707i	−0.973325	−	0.229433i	$-0.926313\pi$
$151$	−1271.69	−	2202.64i	−0.685357	−	1.18707i	0.287968	−	0.957640i	$-0.407020\pi$
$152$		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	−0.938566	−	0.345101i	$-0.887845\pi$
$157$	−2181.59	+	1259.54i	−1.10898	+	0.640271i	−0.170417	+	0.985372i	$0.554511\pi$
$158$		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	0.484631	+	0.874718i	$0.338954\pi$
$163$	−1072.18	+	1857.07i	−0.515213	+	0.892375i	−0.999844	+	0.0176561i	$0.994380\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3381.59			1.56692			0.783458	−	0.621445i	$-0.213454\pi$
$167$	3381.59			1.56692			0.783458	+	0.621445i	$0.213454\pi$
$168$		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	−0.843454	−	0.537202i	$-0.819481\pi$
$173$	98.9912	−	171.458i	0.0435038	−	0.0753509i	0.886957	+	0.461851i	$0.152815\pi$
$174$		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	$-0.408482\pi$
$179$	3007.52	−	1736.40i	1.25583	−	0.725052i	0.972260	+	0.233901i	$0.0751491\pi$
$180$		0			0
$181$			2872.95i			1.17980i	0.807475	+	0.589902i	$0.200834\pi$
$181$			2872.95i			1.17980i	−0.807475	+	0.589902i	$0.799166\pi$
$182$		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	$-0.257770\pi$
$191$	1075.20	+	620.765i	0.407322	+	0.235167i	−0.282317	+	0.959321i	$0.591103\pi$
$192$		0			0
$193$	−1341.31	−	2323.21i	−0.500256	−	0.866468i	−1.00000		0.000295140i	$-0.999906\pi$
$193$	−1341.31	−	2323.21i	−0.500256	−	0.866468i	0.499744	−	0.866173i	$-0.333427\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3056.57i		−	1.10544i	−0.833367	−	0.552719i	$-0.813590\pi$
$197$		−	3056.57i		−	1.10544i	0.833367	−	0.552719i	$-0.186410\pi$
$198$		0			0
$199$	1864.33	−	1076.37i	0.664115	−	0.383427i	−0.129728	−	0.991550i	$-0.541411\pi$
$199$	1864.33	−	1076.37i	0.664115	−	0.383427i	0.793843	+	0.608123i	$0.208077\pi$
$200$		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	$-0.279661\pi$
$211$	3912.38			1.27649			0.638245	+	0.769834i	$0.279661\pi$
$212$		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	0.928607	+	0.371064i	$0.121007\pi$
$227$	2687.02	+	4654.05i	0.785654	+	1.36079i	−0.142953	+	0.989729i	$0.545660\pi$
$228$		0			0
$229$	5385.90	+	3109.55i	1.55419	+	0.897314i	0.997793	+	0.0663998i	$0.0211513\pi$
$229$	5385.90	+	3109.55i	1.55419	+	0.897314i	0.556400	+	0.830914i	$0.312182\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5180.55	−	2990.99i	−1.45661	−	0.840972i	−0.457764	−	0.889074i	$-0.651350\pi$
$233$	−5180.55	−	2990.99i	−1.45661	−	0.840972i	−0.998842	+	0.0481015i	$0.984683\pi$
$234$		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	−0.648001	−	0.761640i	$-0.724395\pi$
$241$	−1168.80	+	674.804i	−0.312401	+	0.180365i	0.335599	+	0.942005i	$0.391061\pi$
$242$		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	$-0.470697\pi$
$251$	731.129			0.183858			0.0919292	+	0.995766i	$0.470697\pi$
$252$		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	0.265972	+	0.963981i	$0.414307\pi$
$257$	−2891.62	+	5008.43i	−0.701845	+	1.21563i	−0.967818	+	0.251651i	$0.919026\pi$
$258$		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	$-0.721827\pi$
$263$	−1273.79	+	735.424i	−0.298651	+	0.172426i	0.343185	+	0.939268i	$0.388494\pi$
$264$		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	−0.999703	−	0.0243516i	$-0.992248\pi$
$269$	−2298.35	−	3980.86i	−0.520941	−	0.902296i	0.478763	−	0.877944i	$-0.341085\pi$
$270$		0			0
$271$	352.954	+	203.778i	0.0791160	+	0.0456777i	0.539036	−	0.842283i	$-0.318789\pi$
$271$	352.954	+	203.778i	0.0791160	+	0.0456777i	−0.459920	+	0.887960i	$0.652122\pi$
$272$		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	0.768086	−	0.640346i	$-0.221209\pi$
$277$	−786.099	−	1361.56i	−0.170513	−	0.295337i	−0.938599	+	0.345009i	$0.887876\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			391.409i			0.0830943i	0.999137	+	0.0415471i	$0.0132287\pi$
$281$			391.409i			0.0830943i	−0.999137	+	0.0415471i	$0.986771\pi$
$282$		0			0
$283$	994.194	−	573.998i	0.208829	−	0.120568i	−0.391938	−	0.919992i	$-0.628195\pi$
$283$	994.194	−	573.998i	0.208829	−	0.120568i	0.600767	+	0.799424i	$0.294862\pi$
$284$		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	$-0.0419733\pi$
$311$	2093.96	+	3626.85i	0.381793	+	0.661285i	−0.609525	+	0.792767i	$0.708640\pi$
$312$		0			0
$313$	6183.56	+	3570.08i	1.11666	+	0.644706i	0.940547	−	0.339663i	$-0.110313\pi$
$313$	6183.56	+	3570.08i	1.11666	+	0.644706i	0.176117	+	0.984369i	$0.443646\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5241.84	+	3026.38i	0.928742	+	0.536210i	0.886414	−	0.462894i	$-0.153189\pi$
$317$	5241.84	+	3026.38i	0.928742	+	0.536210i	0.0423286	+	0.999104i	$0.486522\pi$
$318$		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	−0.0280795	+	0.999606i	$0.508939\pi$
$331$	−5297.71	+	9175.90i	−0.879724	+	1.52373i	−0.851644	+	0.524120i	$0.824394\pi$
$332$		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	$-0.656571\pi$
$347$	355.046	−	204.986i	0.0549276	−	0.0317125i	0.527212	+	0.849734i	$0.323237\pi$
$348$		0			0
$349$		−	1777.77i		−	0.272670i	−0.990663	−	0.136335i	$-0.956468\pi$
$349$		−	1777.77i		−	0.272670i	0.990663	−	0.136335i	$-0.0435324\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4076.99	−	7061.56i	−0.614721	−	1.06473i	−0.990433	−	0.137991i	$-0.955935\pi$
$353$	−4076.99	−	7061.56i	−0.614721	−	1.06473i	0.375713	−	0.926736i	$-0.377398\pi$
$354$		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	$-0.266619\pi$
$359$	2451.26	+	1415.24i	0.360370	+	0.208060i	−0.308873	+	0.951103i	$0.599952\pi$
$360$		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	$-0.660569\pi$
$367$	233.258	−	134.671i	0.0331770	−	0.0191547i	0.516497	+	0.856289i	$0.327236\pi$
$368$		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	−0.0321693	−	0.999482i	$-0.510242\pi$
$373$	6119.60	−	10599.5i	0.849493	−	1.47136i	0.881662	−	0.471882i	$-0.156425\pi$
$374$		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	$-0.514909\pi$
$379$	−690.914			−0.0936409			−0.0468204	+	0.998903i	$0.514909\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	4423.65	−	7661.98i	0.590177	−	1.02222i	−0.404031	−	0.914745i	$-0.632391\pi$
$383$	4423.65	−	7661.98i	0.590177	−	1.02222i	0.994208	−	0.107472i	$-0.0342755\pi$
$384$		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	−0.305665	−	0.952139i	$-0.598879\pi$
$389$	2808.67	−	1621.58i	0.366080	−	0.211356i	0.671744	+	0.740783i	$0.265545\pi$
$390$		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	−0.226173	−	0.974087i	$-0.572621\pi$
$397$	−9356.49	−	5401.97i	−1.18284	−	0.682915i	−0.956671	+	0.291172i	$0.905955\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−595.785	−	343.976i	−0.0741947	−	0.0428363i	0.462444	−	0.886649i	$-0.346973\pi$
$401$	−595.785	−	343.976i	−0.0741947	−	0.0428363i	−0.536638	+	0.843812i	$0.680306\pi$
$402$		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	0.0149935	−	0.999888i	$-0.495227\pi$
$409$	7348.56	−	4242.70i	0.888418	−	0.512929i	0.873425	+	0.486959i	$0.161894\pi$
$410$		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	$-0.277799\pi$
$419$	11025.1			1.28547			0.642736	+	0.766088i	$0.277799\pi$
$420$		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	$-0.763975\pi$
$431$	−4664.33	+	2692.95i	−0.521283	+	0.300963i	0.216176	+	0.976354i	$0.430641\pi$
$432$		0			0
$433$		−	4428.30i		−	0.491480i	−0.969336	−	0.245740i	$-0.920969\pi$
$433$		−	4428.30i		−	0.491480i	0.969336	−	0.245740i	$-0.0790309\pi$
$434$		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	−0.992342	−	0.123523i	$-0.960581\pi$
$439$	−14675.4	−	8472.85i	−1.59549	−	0.921155i	−0.603145	−	0.797632i	$-0.706086\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−15157.8	−	8751.39i	−1.62567	−	0.938580i	−0.985365	−	0.170458i	$-0.945475\pi$
$443$	−15157.8	−	8751.39i	−1.62567	−	0.938580i	−0.640304	−	0.768122i	$-0.721191\pi$
$444$		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	$0.0331024\pi$
$449$			1975.26i			0.207614i	−0.994597	+	0.103807i	$0.966898\pi$
$450$		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	−0.371927	−	0.928262i	$-0.621303\pi$
$457$	6036.94	−	10456.3i	0.617935	−	1.07029i	0.989862	−	0.142032i	$-0.0453637\pi$
$458$		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	$-0.789387\pi$
$463$	−15720.4			−1.57795			−0.788973	+	0.614428i	$0.789387\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	682.156	−	1181.53i	0.0675940	−	0.117076i	−0.830248	−	0.557395i	$-0.811801\pi$
$467$	682.156	−	1181.53i	0.0675940	−	0.117076i	0.897842	+	0.440318i	$0.145134\pi$
$468$		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	$-0.103139\pi$
$479$	2078.39	+	3599.88i	0.198255	+	0.343388i	−0.749708	+	0.661769i	$0.769806\pi$
$480$		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	−0.995723	−	0.0923918i	$-0.970549\pi$
$487$	−6210.51	−	10756.9i	−0.577875	−	1.00091i	0.417848	−	0.908517i	$-0.362785\pi$
$488$		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	$-0.790698\pi$
$499$	1488.56	−	2578.26i	0.133541	−	0.231300i	0.925039	+	0.379871i	$0.124032\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	537.349			0.0476327			0.0238163	−	0.999716i	$-0.492418\pi$
$503$	537.349			0.0476327			0.0238163	+	0.999716i	$0.492418\pi$
$504$		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	−0.996362	−	0.0852256i	$-0.972839\pi$
$509$	−4873.32	+	8440.83i	−0.424373	+	0.735036i	0.571988	+	0.820262i	$0.306172\pi$
$510$		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	0.588204	−	0.808713i	$-0.299835\pi$
$521$	−4831.31	−	8368.07i	−0.406264	−	0.703670i	−0.994468	+	0.105043i	$0.966502\pi$
$522$		0			0
$523$	−5153.64	−	2975.46i	−0.430886	−	0.248772i	0.268838	−	0.963185i	$-0.413360\pi$
$523$	−5153.64	−	2975.46i	−0.430886	−	0.248772i	−0.699724	+	0.714413i	$0.746694\pi$
$524$		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	−0.631473	−	0.775398i	$-0.717549\pi$
$541$	4476.87	−	7754.17i	0.355778	−	0.616225i	0.987251	+	0.159173i	$0.0508826\pi$
$542$		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	$-0.199507\pi$
$547$	20723.2			1.61985			0.809926	+	0.586532i	$0.199507\pi$
$548$		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	−0.863453	−	0.504430i	$-0.831703\pi$
$557$	−11283.3	+	6514.43i	−0.858330	+	0.495557i	0.00512296	+	0.999987i	$0.498369\pi$
$558$		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	$-0.189451\pi$
$563$	−955.378	−	1654.76i	−0.0715176	−	0.123872i	−0.899567	+	0.436784i	$0.856118\pi$
$564$		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	0.689049	−	0.724715i	$-0.258029\pi$
$569$	5509.88	+	3181.13i	0.405951	+	0.234376i	−0.283098	+	0.959091i	$0.591362\pi$
$570$		0			0
$571$	3394.59	+	5879.61i	0.248790	+	0.430918i	0.963190	−	0.268820i	$-0.0866336\pi$
$571$	3394.59	+	5879.61i	0.248790	+	0.430918i	−0.714400	+	0.699737i	$0.753300\pi$
$572$		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	−0.263154	−	0.964754i	$-0.584763\pi$
$577$	6109.08	−	3527.08i	0.440770	−	0.254479i	0.703924	+	0.710275i	$0.251429\pi$
$578$		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	$-0.326569\pi$
$587$	14742.1			1.03658			0.518290	+	0.855205i	$0.326569\pi$
$588$		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	−0.999845	−	0.0176216i	$-0.994391\pi$
$593$	−6998.75	+	12122.2i	−0.484662	+	0.839458i	0.515183	+	0.857080i	$0.327724\pi$
$594$		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	$-0.663515\pi$
$599$	251.377	−	145.133i	0.0171469	−	0.00989977i	0.508549	+	0.861033i	$0.330182\pi$
$600$		0			0
$601$		−	3671.51i		−	0.249191i	−0.992208	−	0.124596i	$-0.960237\pi$
$601$		−	3671.51i		−	0.249191i	0.992208	−	0.124596i	$-0.0397634\pi$
$602$		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	$-0.518492\pi$
$607$	−14231.9	−	8216.78i	−0.951655	−	0.549438i	−0.893595	+	0.448875i	$0.851825\pi$
$608$		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	0.945574	+	0.325408i	$0.105502\pi$
$613$	11452.7	+	19836.6i	0.754599	+	1.30700i	−0.190975	+	0.981595i	$0.561165\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	10613.9i		−	0.692541i	−0.938135	−	0.346271i	$-0.887448\pi$
$617$		−	10613.9i		−	0.692541i	0.938135	−	0.346271i	$-0.112552\pi$
$618$		0			0
$619$	5261.57	−	3037.77i	0.341648	−	0.197251i	−0.319352	−	0.947636i	$-0.603465\pi$
$619$	5261.57	−	3037.77i	0.341648	−	0.197251i	0.661001	+	0.750385i	$0.270132\pi$
$620$		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	$-0.679582\pi$
$631$	−16951.1			−1.06944			−0.534718	+	0.845031i	$0.679582\pi$
$632$		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	−0.789461	−	0.613800i	$-0.789640\pi$
$641$	−10591.3	+	6114.91i	−0.652625	+	0.376793i	0.136836	+	0.990594i	$0.456307\pi$
$642$		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	$-0.103630\pi$
$647$	3237.83	+	5608.08i	0.196742	+	0.340767i	−0.750728	+	0.660611i	$0.770297\pi$
$648$		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	0.903409	−	0.428781i	$-0.141057\pi$
$653$	16416.0	+	9477.77i	0.983778	+	0.567984i	0.0803694	+	0.996765i	$0.474390\pi$
$654$		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	−0.851940	−	0.523640i	$-0.824574\pi$
$661$	−14010.5	+	8088.96i	−0.824425	+	0.475982i	0.0275153	+	0.999621i	$0.491240\pi$
$662$		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	−0.294163	−	0.955755i	$-0.595041\pi$
$677$	11989.3	−	20766.0i	0.680627	−	1.17888i	0.974790	−	0.223125i	$-0.0716258\pi$
$678$		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	$-0.686488\pi$
$683$	−1924.00	+	1110.82i	−0.107789	+	0.0622319i	0.445136	+	0.895463i	$0.353155\pi$
$684$		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	$-0.554667\pi$
$691$	−20155.6	−	11636.9i	−1.10963	−	0.640647i	−0.938734	+	0.344642i	$0.888000\pi$
$692$		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	$-0.956959\pi$
$701$		−	5003.94i		−	0.269610i	0.990872	−	0.134805i	$-0.0430407\pi$
$702$		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	0.128381	+	0.991725i	$0.459022\pi$
$709$	−15002.2	+	25984.6i	−0.794669	+	1.37641i	−0.923049	+	0.384682i	$0.874311\pi$
$710$		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	$-0.711587\pi$
$719$	7195.48	−	12462.9i	0.373221	−	0.646438i	0.990059	+	0.140652i	$0.0449198\pi$
$720$		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	0.876422	−	0.481543i	$-0.159924\pi$
$733$	17813.2	+	10284.4i	0.897604	+	0.518232i	0.0211822	+	0.999776i	$0.493257\pi$
$734$		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	−0.926170	−	0.377106i	$-0.876919\pi$
$739$	−15864.0	−	27477.2i	−0.789669	−	1.36775i	0.136501	−	0.990640i	$-0.456414\pi$
$740$		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	−0.0736070	−	0.997287i	$-0.523451\pi$
$751$	17017.6	−	29475.4i	0.826873	−	1.43219i	0.900480	−	0.434898i	$-0.143216\pi$
$752$		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	−0.289742	−	0.957105i	$-0.593570\pi$
$761$	14359.4	−	24871.2i	0.684006	−	1.18473i	0.973748	−	0.227628i	$-0.0730971\pi$
$762$		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	$-0.993176\pi$
$769$		−	914.245i		−	0.0428719i	0.999770	−	0.0214360i	$-0.00682380\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−6138.69	−	10632.5i	−0.285632	−	0.494729i	0.687131	−	0.726534i	$-0.258870\pi$
$773$	−6138.69	−	10632.5i	−0.285632	−	0.494729i	−0.972762	+	0.231805i	$0.925537\pi$
$774$		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	$-0.520618\pi$
$787$	16936.5	−	9778.31i	0.767119	−	0.442896i	0.831846	+	0.555007i	$0.187284\pi$
$788$		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	−0.962261	−	0.272126i	$-0.912273\pi$
$809$	−27790.1	+	16044.6i	−1.20772	+	0.697280i	−0.245462	+	0.969406i	$0.578940\pi$
$810$		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	0.579385	−	0.815054i	$-0.303293\pi$
$821$	3839.60	+	2216.79i	0.163219	+	0.0942347i	−0.416165	+	0.909289i	$0.636626\pi$
$822$		0			0
$823$	−6029.71	−	10443.8i	−0.255386	−	0.442341i	0.709614	−	0.704590i	$-0.248869\pi$
$823$	−6029.71	−	10443.8i	−0.255386	−	0.442341i	−0.965000	+	0.262249i	$0.915536\pi$
$824$		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	−0.417034	−	0.908891i	$-0.636930\pi$
$829$	3856.52	−	2226.56i	0.161571	−	0.0932832i	0.578605	+	0.815608i	$0.303597\pi$
$830$		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	$-0.654742\pi$
$839$	−22708.5			−0.934429			−0.467214	+	0.884144i	$0.654742\pi$
$840$		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	$0.0875499\pi$
$853$			13532.2i			0.543183i	−0.962413	+	0.271591i	$0.912450\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−7847.69	−	13592.6i	−0.312803	−	0.541790i	0.666165	−	0.745804i	$-0.267935\pi$
$857$	−7847.69	−	13592.6i	−0.312803	−	0.541790i	−0.978968	+	0.204014i	$0.934601\pi$
$858$		0			0
$859$	31542.4	+	18211.0i	1.25287	+	0.723342i	0.971678	−	0.236311i	$-0.0759383\pi$
$859$	31542.4	+	18211.0i	1.25287	+	0.723342i	0.281188	+	0.959653i	$0.409272\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	26424.8	+	15256.4i	1.04231	+	0.601777i	0.920486	−	0.390775i	$-0.127793\pi$
$863$	26424.8	+	15256.4i	1.04231	+	0.601777i	0.121822	+	0.992552i	$0.461126\pi$
$864$		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	−0.837612	−	0.546266i	$-0.816049\pi$
$877$	1409.59	−	2441.48i	0.0542742	−	0.0940057i	0.891886	+	0.452260i	$0.149382\pi$
$878$		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	$-0.484617\pi$
$883$	2535.09			0.0966168			0.0483084	+	0.998832i	$0.484617\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	21522.4	−	37278.0i	0.814716	−	1.41113i	−0.0948157	−	0.995495i	$-0.530226\pi$
$887$	21522.4	−	37278.0i	0.814716	−	1.41113i	0.909532	−	0.415635i	$-0.136440\pi$
$888$		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	$-0.0860992\pi$
$907$	6840.28	+	11847.7i	0.250416	+	0.433734i	−0.713224	+	0.700936i	$0.752766\pi$
$908$		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	0.375694	+	0.926744i	$0.377404\pi$
$919$	−17126.3	+	29663.5i	−0.614737	+	1.06476i	−0.990431	+	0.138011i	$0.955929\pi$
$920$		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	−0.561642	−	0.827380i	$-0.689830\pi$
$929$	12337.4	−	21369.0i	0.435712	−	0.754675i	0.997353	+	0.0727058i	$0.0231634\pi$
$930$		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	$0.194325\pi$
$937$			32885.0i			1.14654i	−0.819367	+	0.573269i	$0.805675\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	4016.54	+	6956.85i	0.139145	+	0.241006i	0.927173	−	0.374633i	$-0.122231\pi$
$941$	4016.54	+	6956.85i	0.139145	+	0.241006i	−0.788028	+	0.615639i	$0.788898\pi$
$942$		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	$-0.430646\pi$
$947$	−15192.1	−	8771.14i	−0.521305	−	0.300975i	−0.737468	+	0.675382i	$0.763979\pi$
$948$		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	$0.243915\pi$
$953$			40802.9i			1.38692i	−0.720494	+	0.693461i	$0.756085\pi$
$954$		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	$-0.0864169\pi$
$967$	57938.1			1.92675			0.963373	+	0.268164i	$0.0864169\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	7580.65	−	13130.1i	0.250540	−	0.433948i	−0.713134	−	0.701027i	$-0.752725\pi$
$971$	7580.65	−	13130.1i	0.250540	−	0.433948i	0.963675	+	0.267079i	$0.0860584\pi$
$972$		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	−0.0144714	−	0.999895i	$-0.504607\pi$
$977$	25781.1	−	14884.7i	0.844228	−	0.487415i	0.858699	+	0.512480i	$0.171273\pi$
$978$		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	−0.999117	−	0.0420055i	$-0.986625\pi$
$983$	−16517.5	−	28609.1i	−0.535937	−	0.928269i	0.463181	−	0.886264i	$-0.346708\pi$
$984$		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	$-0.215343\pi$
$991$	−4752.11	−	8230.89i	−0.152327	−	0.263837i	−0.932082	+	0.362246i	$0.882010\pi$
$992$		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	−0.232769	−	0.972532i	$-0.574779\pi$
$997$	15522.5	−	8961.95i	0.493083	−	0.284682i	0.725853	+	0.687850i	$0.241445\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1008.4.bt.d.593.4		48
3.2	odd	2	inner	1008.4.bt.d.593.21		48
4.3	odd	2		504.4.bl.a.89.4	yes	48
7.3	odd	6	inner	1008.4.bt.d.17.21		48
12.11	even	2		504.4.bl.a.89.21	yes	48
21.17	even	6	inner	1008.4.bt.d.17.4		48
28.3	even	6		504.4.bl.a.17.21	yes	48
84.59	odd	6		504.4.bl.a.17.4	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.4	✓	48	84.59	odd	6
504.4.bl.a.17.21	yes	48	28.3	even	6
504.4.bl.a.89.4	yes	48	4.3	odd	2
504.4.bl.a.89.21	yes	48	12.11	even	2
1008.4.bt.d.17.4		48	21.17	even	6	inner
1008.4.bt.d.17.21		48	7.3	odd	6	inner
1008.4.bt.d.593.4		48	1.1	even	1	trivial
1008.4.bt.d.593.21		48	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1008 = 2^{4} \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1008.bt (of order \(6\), degree \(2\), not 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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