LMFDB - Embedded newform 1008.4.bt.d.17.19

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			17.19
Character	$\chi$	$=$	1008.17
Dual form			1008.4.bt.d.593.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(5.74088 + 9.94350i) q^{5} +(-17.2542 - 6.73007i) q^{7} +O(q^{10})$	$q+(5.74088 + 9.94350i) q^{5} +(-17.2542 - 6.73007i) q^{7} +(-49.6208 - 28.6486i) q^{11} +9.20923i q^{13} +(14.5492 - 25.1999i) q^{17} +(32.6061 - 18.8252i) q^{19} +(7.27174 - 4.19834i) q^{23} +(-3.41550 + 5.91583i) q^{25} -62.3892i q^{29} +(48.8957 + 28.2300i) q^{31} +(-32.1337 - 210.203i) q^{35} +(146.068 + 252.997i) q^{37} +54.2417 q^{41} +438.235 q^{43} +(128.386 + 222.371i) q^{47} +(252.412 + 232.243i) q^{49} +(515.128 + 297.409i) q^{53} -657.873i q^{55} +(-238.156 + 412.499i) q^{59} +(-548.703 + 316.794i) q^{61} +(-91.5720 + 52.8691i) q^{65} +(308.827 - 534.904i) q^{67} +396.155i q^{71} +(39.8998 + 23.0361i) q^{73} +(663.359 + 828.259i) q^{77} +(-344.924 - 597.425i) q^{79} -1288.72 q^{83} +334.100 q^{85} +(595.784 + 1031.93i) q^{89} +(61.9787 - 158.897i) q^{91} +(374.376 + 216.146i) q^{95} -946.768i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1008\mathbb{Z}\right)^\times$.

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	5.74088	+	9.94350i	0.513480	+	0.889374i	0.999878	+	0.0156363i	$0.00497739\pi$
$5$	5.74088	+	9.94350i	0.513480	+	0.889374i	−0.486397	+	0.873738i	$0.661689\pi$
$6$		0			0
$7$	−17.2542	−	6.73007i	−0.931637	−	0.363390i
$7$	−17.2542	−	6.73007i	−0.931637	−	0.363390i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−49.6208	−	28.6486i	−1.36011	−	0.785261i	−0.370474	−	0.928843i	$-0.620805\pi$
$11$	−49.6208	−	28.6486i	−1.36011	−	0.785261i	−0.989639	+	0.143581i	$0.954138\pi$
$12$		0			0
$13$			9.20923i			0.196475i	0.995163	+	0.0982377i	$0.0313205\pi$
$13$			9.20923i			0.196475i	−0.995163	+	0.0982377i	$0.968679\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	14.5492	−	25.1999i	0.207570	−	0.359522i	−0.743378	−	0.668871i	$-0.766778\pi$
$17$	14.5492	−	25.1999i	0.207570	−	0.359522i	0.950949	+	0.309349i	$0.100111\pi$
$18$		0			0
$19$	32.6061	−	18.8252i	0.393703	−	0.227305i	−0.290060	−	0.957008i	$-0.593675\pi$
$19$	32.6061	−	18.8252i	0.393703	−	0.227305i	0.683763	+	0.729704i	$0.260342\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	7.27174	−	4.19834i	0.0659245	−	0.0380615i	−0.466675	−	0.884429i	$-0.654548\pi$
$23$	7.27174	−	4.19834i	0.0659245	−	0.0380615i	0.532600	+	0.846367i	$0.321215\pi$
$24$		0			0
$25$	−3.41550	+	5.91583i	−0.0273240	+	0.0473266i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	62.3892i		−	0.399496i	−0.979847	−	0.199748i	$-0.935988\pi$
$29$		−	62.3892i		−	0.399496i	0.979847	−	0.199748i	$-0.0640124\pi$
$30$		0			0
$31$	48.8957	+	28.2300i	0.283288	+	0.163556i	0.634911	−	0.772585i	$-0.281037\pi$
$31$	48.8957	+	28.2300i	0.283288	+	0.163556i	−0.351623	+	0.936142i	$0.614370\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−32.1337	−	210.203i	−0.155188	−	1.01517i
$36$		0			0
$37$	146.068	+	252.997i	0.649012	+	1.12412i	0.983359	+	0.181672i	$0.0581509\pi$
$37$	146.068	+	252.997i	0.649012	+	1.12412i	−0.334347	+	0.942450i	$0.608516\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	54.2417			0.206613			0.103306	−	0.994650i	$-0.467058\pi$
$41$	54.2417			0.206613			0.103306	+	0.994650i	$0.467058\pi$
$42$		0			0
$43$	438.235			1.55419			0.777096	−	0.629382i	$-0.216692\pi$
$43$	438.235			1.55419			0.777096	+	0.629382i	$0.216692\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	128.386	+	222.371i	0.398446	+	0.690129i	0.993534	−	0.113531i	$-0.0362161\pi$
$47$	128.386	+	222.371i	0.398446	+	0.690129i	−0.595088	+	0.803661i	$0.702883\pi$
$48$		0			0
$49$	252.412	+	232.243i	0.735896	+	0.677095i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	515.128	+	297.409i	1.33506	+	0.770799i	0.986071	−	0.166327i	$-0.0531907\pi$
$53$	515.128	+	297.409i	1.33506	+	0.770799i	0.348992	+	0.937126i	$0.386524\pi$
$54$		0			0
$55$		−	657.873i		−	1.61287i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−238.156	+	412.499i	−0.525513	+	0.910216i	0.474045	+	0.880501i	$0.342793\pi$
$59$	−238.156	+	412.499i	−0.525513	+	0.910216i	−0.999558	+	0.0297153i	$0.990540\pi$
$60$		0			0
$61$	−548.703	+	316.794i	−1.15171	+	0.664939i	−0.949303	−	0.314363i	$-0.898209\pi$
$61$	−548.703	+	316.794i	−1.15171	+	0.664939i	−0.202406	+	0.979302i	$0.564876\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−91.5720	+	52.8691i	−0.174740	+	0.100886i
$66$		0			0
$67$	308.827	−	534.904i	0.563123	−	0.975357i	−0.434099	−	0.900865i	$-0.642933\pi$
$67$	308.827	−	534.904i	0.563123	−	0.975357i	0.997222	−	0.0744918i	$-0.0237335\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			396.155i			0.662183i	0.943599	+	0.331092i	$0.107417\pi$
$71$			396.155i			0.662183i	−0.943599	+	0.331092i	$0.892583\pi$
$72$		0			0
$73$	39.8998	+	23.0361i	0.0639714	+	0.0369339i	0.531645	−	0.846968i	$-0.321574\pi$
$73$	39.8998	+	23.0361i	0.0639714	+	0.0369339i	−0.467673	+	0.883901i	$0.654908\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	663.359	+	828.259i	0.981776	+	1.22583i
$78$		0			0
$79$	−344.924	−	597.425i	−0.491227	−	0.850831i	0.508722	−	0.860931i	$-0.330118\pi$
$79$	−344.924	−	597.425i	−0.491227	−	0.850831i	−0.999949	+	0.0101004i	$0.996785\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1288.72			−1.70428			−0.852138	−	0.523317i	$-0.824694\pi$
$83$	−1288.72			−1.70428			−0.852138	+	0.523317i	$0.824694\pi$
$84$		0			0
$85$	334.100			0.426333
$86$		0			0
$87$		0			0
$88$		0			0
$89$	595.784	+	1031.93i	0.709584	+	1.22904i	0.965012	+	0.262207i	$0.0844504\pi$
$89$	595.784	+	1031.93i	0.709584	+	1.22904i	−0.255428	+	0.966828i	$0.582216\pi$
$90$		0			0
$91$	61.9787	−	158.897i	0.0713971	−	0.183044i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	374.376	+	216.146i	0.404318	+	0.233433i
$96$		0			0
$97$		−	946.768i		−	0.991028i	−0.868600	−	0.495514i	$-0.834980\pi$
$97$		−	946.768i		−	0.991028i	0.868600	−	0.495514i	$-0.165020\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−636.416	+	1102.31i	−0.626988	+	1.08597i	0.361165	+	0.932502i	$0.382379\pi$
$101$	−636.416	+	1102.31i	−0.626988	+	1.08597i	−0.988153	+	0.153473i	$0.950954\pi$
$102$		0			0
$103$	−338.095	+	195.199i	−0.323431	+	0.186733i	−0.652921	−	0.757426i	$-0.726457\pi$
$103$	−338.095	+	195.199i	−0.323431	+	0.186733i	0.329490	+	0.944159i	$0.393123\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	67.4130	−	38.9209i	0.0609071	−	0.0351648i	−0.469237	−	0.883072i	$-0.655471\pi$
$107$	67.4130	−	38.9209i	0.0609071	−	0.0351648i	0.530144	+	0.847907i	$0.322138\pi$
$108$		0			0
$109$	−341.576	+	591.626i	−0.300156	+	0.519886i	−0.976171	−	0.217002i	$-0.930372\pi$
$109$	−341.576	+	591.626i	−0.300156	+	0.519886i	0.676015	+	0.736888i	$0.263705\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	286.715i		−	0.238689i	−0.992853	−	0.119345i	$-0.961921\pi$
$113$		−	286.715i		−	0.238689i	0.992853	−	0.119345i	$-0.0380793\pi$
$114$		0			0
$115$	83.4924	+	48.2044i	0.0677018	+	0.0390877i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−420.631	+	336.886i	−0.324027	+	0.259515i
$120$		0			0
$121$	975.984	+	1690.45i	0.733271	+	1.27006i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	1356.79			0.970839
$126$		0			0
$127$	2655.85			1.85566			0.927830	−	0.373002i	$-0.121672\pi$
$127$	2655.85			1.85566			0.927830	+	0.373002i	$0.121672\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1198.22	+	2075.38i	0.799151	+	1.38417i	0.920169	+	0.391521i	$0.128051\pi$
$131$	1198.22	+	2075.38i	0.799151	+	1.38417i	−0.121018	+	0.992650i	$0.538616\pi$
$132$		0			0
$133$	−689.286	+	105.371i	−0.449389	+	0.0686978i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1553.97	−	897.184i	−0.969084	−	0.559501i	−0.0701271	−	0.997538i	$-0.522340\pi$
$137$	−1553.97	−	897.184i	−0.969084	−	0.559501i	−0.898957	+	0.438037i	$0.855674\pi$
$138$		0			0
$139$			101.014i			0.0616396i	0.999525	+	0.0308198i	$0.00981180\pi$
$139$			101.014i			0.0616396i	−0.999525	+	0.0308198i	$0.990188\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	263.831	−	456.969i	0.154285	−	0.267229i
$144$		0			0
$145$	620.367	−	358.169i	0.355301	−	0.205133i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	744.159	−	429.640i	0.409154	−	0.236225i	−0.281272	−	0.959628i	$-0.590756\pi$
$149$	744.159	−	429.640i	0.409154	−	0.236225i	0.690426	+	0.723403i	$0.257423\pi$
$150$		0			0
$151$	807.459	−	1398.56i	0.435166	−	0.753730i	−0.562143	−	0.827040i	$-0.690023\pi$
$151$	807.459	−	1398.56i	0.435166	−	0.753730i	0.997309	+	0.0733102i	$0.0233563\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			648.260i			0.335932i
$156$		0			0
$157$	1622.84	+	936.947i	0.824948	+	0.476284i	0.852120	−	0.523347i	$-0.175317\pi$
$157$	1622.84	+	936.947i	0.824948	+	0.476284i	−0.0271720	+	0.999631i	$0.508650\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−153.723	+	23.4996i	−0.0752488	+	0.0115033i
$162$		0			0
$163$	664.037	+	1150.15i	0.319088	+	0.552677i	0.980298	−	0.197524i	$-0.0632900\pi$
$163$	664.037	+	1150.15i	0.319088	+	0.552677i	−0.661210	+	0.750201i	$0.729957\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	595.681			0.276019			0.138010	−	0.990431i	$-0.455930\pi$
$167$	595.681			0.276019			0.138010	+	0.990431i	$0.455930\pi$
$168$		0			0
$169$	2112.19			0.961397
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1119.44	+	1938.93i	0.491962	+	0.852103i	0.999957	−	0.00925667i	$-0.00294653\pi$
$173$	1119.44	+	1938.93i	0.491962	+	0.852103i	−0.507995	+	0.861360i	$0.669613\pi$
$174$		0			0
$175$	98.7456	−	79.0861i	0.0426541	−	0.0341620i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	1660.97	+	958.960i	0.693556	+	0.400425i	0.804943	−	0.593352i	$-0.202196\pi$
$179$	1660.97	+	958.960i	0.693556	+	0.400425i	−0.111387	+	0.993777i	$0.535529\pi$
$180$		0			0
$181$		−	4162.64i		−	1.70943i	−0.519099	−	0.854714i	$-0.673732\pi$
$181$		−	4162.64i		−	1.70943i	0.519099	−	0.854714i	$-0.326268\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1677.12	+	2904.86i	−0.666510	+	1.15443i
$186$		0			0
$187$	−1443.88	+	833.626i	−0.564637	+	0.325994i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	267.384	−	154.374i	0.101294	−	0.0584823i	−0.448497	−	0.893784i	$-0.648041\pi$
$191$	267.384	−	154.374i	0.101294	−	0.0584823i	0.549791	+	0.835302i	$0.314707\pi$
$192$		0			0
$193$	2385.32	−	4131.49i	0.889632	−	1.54089i	0.0493214	−	0.998783i	$-0.484294\pi$
$193$	2385.32	−	4131.49i	0.889632	−	1.54089i	0.840311	−	0.542105i	$-0.182373\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			1115.98i			0.403607i	0.979426	+	0.201804i	$0.0646802\pi$
$197$			1115.98i			0.403607i	−0.979426	+	0.201804i	$0.935320\pi$
$198$		0			0
$199$	2713.57	+	1566.68i	0.966634	+	0.558086i	0.898208	−	0.439570i	$-0.144869\pi$
$199$	2713.57	+	1566.68i	0.966634	+	0.558086i	0.0684255	+	0.997656i	$0.478202\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−419.884	+	1076.47i	−0.145173	+	0.372185i
$204$		0			0
$205$	311.395	+	539.352i	0.106092	+	0.183756i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2157.26			−0.713974
$210$		0			0
$211$	513.831			0.167647			0.0838237	−	0.996481i	$-0.473287\pi$
$211$	513.831			0.167647			0.0838237	+	0.996481i	$0.473287\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	2515.86	+	4357.59i	0.798047	+	1.38226i
$216$		0			0
$217$	−653.665	−	816.156i	−0.204487	−	0.255319i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	232.072	+	133.987i	0.0706372	+	0.0407824i
$222$		0			0
$223$			4529.37i			1.36013i	0.733151	+	0.680066i	$0.238049\pi$
$223$			4529.37i			1.36013i	−0.733151	+	0.680066i	$0.761951\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	146.074	−	253.008i	0.0427105	−	0.0739767i	−0.843880	−	0.536532i	$-0.819734\pi$
$227$	146.074	−	253.008i	0.0427105	−	0.0739767i	0.886590	+	0.462555i	$0.153067\pi$
$228$		0			0
$229$	5530.36	−	3192.96i	1.59588	−	0.921382i	0.603611	−	0.797279i	$-0.293728\pi$
$229$	5530.36	−	3192.96i	1.59588	−	0.921382i	0.992269	−	0.124103i	$-0.0396052\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1468.47	−	847.822i	0.412887	−	0.238381i	−0.279142	−	0.960250i	$-0.590050\pi$
$233$	1468.47	−	847.822i	0.412887	−	0.238381i	0.692030	+	0.721869i	$0.256717\pi$
$234$		0			0
$235$	−1474.10	+	2553.21i	−0.409189	+	0.708736i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			962.214i			0.260420i	0.991486	+	0.130210i	$0.0415652\pi$
$239$			962.214i			0.260420i	−0.991486	+	0.130210i	$0.958435\pi$
$240$		0			0
$241$	−4233.33	−	2444.11i	−1.13150	−	0.653274i	−0.187191	−	0.982324i	$-0.559938\pi$
$241$	−4233.33	−	2444.11i	−1.13150	−	0.653274i	−0.944313	+	0.329049i	$0.893272\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−860.243	+	3843.15i	−0.224322	+	1.00216i
$246$		0			0
$247$	173.365	+	300.277i	0.0446598	+	0.0773530i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	7473.53			1.87938			0.939691	−	0.342025i	$-0.111113\pi$
$251$	7473.53			1.87938			0.939691	+	0.342025i	$0.111113\pi$
$252$		0			0
$253$	−481.106			−0.119553
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−2720.11	−	4711.37i	−0.660217	−	1.14353i	−0.980558	−	0.196227i	$-0.937131\pi$
$257$	−2720.11	−	4711.37i	−0.660217	−	1.14353i	0.320341	−	0.947302i	$-0.396202\pi$
$258$		0			0
$259$	−817.593	−	5348.31i	−0.196150	−	1.28312i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−4868.13	−	2810.62i	−1.14138	−	0.658974i	−0.194605	−	0.980882i	$-0.562342\pi$
$263$	−4868.13	−	2810.62i	−1.14138	−	0.658974i	−0.946771	+	0.321908i	$0.895676\pi$
$264$		0			0
$265$			6829.57i			1.58316i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2259.81	−	3914.11i	0.512206	−	0.887166i	−0.487694	−	0.873014i	$-0.662162\pi$
$269$	2259.81	−	3914.11i	0.512206	−	0.887166i	0.999900	−	0.0141516i	$-0.00450473\pi$
$270$		0			0
$271$	−4739.86	+	2736.56i	−1.06246	+	0.613410i	−0.926111	−	0.377252i	$-0.876869\pi$
$271$	−4739.86	+	2736.56i	−1.06246	+	0.613410i	−0.136346	+	0.990661i	$0.543536\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	338.960	−	195.699i	0.0743275	−	0.0429130i
$276$		0			0
$277$	2082.99	−	3607.85i	0.451823	−	0.782580i	−0.546677	−	0.837344i	$-0.684107\pi$
$277$	2082.99	−	3607.85i	0.451823	−	0.782580i	0.998499	+	0.0547639i	$0.0174406\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			5979.00i			1.26931i	0.772794	+	0.634657i	$0.218858\pi$
$281$			5979.00i			1.26931i	−0.772794	+	0.634657i	$0.781142\pi$
$282$		0			0
$283$	6220.79	+	3591.57i	1.30667	+	0.754406i	0.981539	−	0.191263i	$-0.0612584\pi$
$283$	6220.79	+	3591.57i	1.30667	+	0.754406i	0.325130	+	0.945669i	$0.394592\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−935.895	−	365.050i	−0.192488	−	0.0750810i
$288$		0			0
$289$	2033.14	+	3521.51i	0.413829	+	0.716773i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3741.42			−0.745994			−0.372997	−	0.927832i	$-0.621670\pi$
$293$	−3741.42			−0.745994			−0.372997	+	0.927832i	$0.621670\pi$
$294$		0			0
$295$	−5468.91			−1.07936
$296$		0			0
$297$		0			0
$298$		0			0
$299$	38.6635	+	66.9671i	0.00747815	+	0.0129525i
$300$		0			0
$301$	−7561.38	−	2949.35i	−1.44794	−	0.564777i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−6300.08	−	3637.35i	−1.18276	−	0.682866i
$306$		0			0
$307$		−	4568.44i		−	0.849299i	−0.905358	−	0.424649i	$-0.860397\pi$
$307$		−	4568.44i		−	0.849299i	0.905358	−	0.424649i	$-0.139603\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−1633.29	+	2828.95i	−0.297799	+	0.515803i	−0.975632	−	0.219413i	$-0.929586\pi$
$311$	−1633.29	+	2828.95i	−0.297799	+	0.515803i	0.677833	+	0.735216i	$0.262919\pi$
$312$		0			0
$313$	−3415.84	+	1972.13i	−0.616852	+	0.356139i	−0.775642	−	0.631173i	$-0.782574\pi$
$313$	−3415.84	+	1972.13i	−0.616852	+	0.356139i	0.158791	+	0.987312i	$0.449241\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5604.14	−	3235.55i	0.992934	−	0.573271i	0.0867840	−	0.996227i	$-0.472341\pi$
$317$	5604.14	−	3235.55i	0.992934	−	0.573271i	0.906150	+	0.422956i	$0.139008\pi$
$318$		0			0
$319$	−1787.36	+	3095.80i	−0.313709	+	0.543360i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	1095.56i		−	0.188727i
$324$		0			0
$325$	−54.4802	−	31.4542i	−0.00929851	−	0.00536850i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−718.619	−	4700.86i	−0.120422	−	0.787742i
$330$		0			0
$331$	−902.540	−	1563.24i	−0.149873	−	0.259588i	0.781307	−	0.624147i	$-0.214553\pi$
$331$	−902.540	−	1563.24i	−0.149873	−	0.259588i	−0.931180	+	0.364559i	$0.881220\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7091.76			1.15661
$336$		0			0
$337$	−6868.31			−1.11021			−0.555105	−	0.831780i	$-0.687322\pi$
$337$	−6868.31			−1.11021			−0.555105	+	0.831780i	$0.687322\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−1617.50	−	2801.59i	−0.256869	−	0.444910i
$342$		0			0
$343$	−2792.15	−	5705.92i	−0.439539	−	0.898223i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−4458.72	−	2574.24i	−0.689788	−	0.398249i	0.113745	−	0.993510i	$-0.463715\pi$
$347$	−4458.72	−	2574.24i	−0.689788	−	0.398249i	−0.803533	+	0.595261i	$0.797049\pi$
$348$		0			0
$349$		−	9931.72i		−	1.52330i	−0.647987	−	0.761652i	$-0.724389\pi$
$349$		−	9931.72i		−	1.52330i	0.647987	−	0.761652i	$-0.275611\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−4125.94	+	7146.35i	−0.622102	+	1.07751i	0.366992	+	0.930224i	$0.380388\pi$
$353$	−4125.94	+	7146.35i	−0.622102	+	1.07751i	−0.989094	+	0.147288i	$0.952946\pi$
$354$		0			0
$355$	−3939.17	+	2274.28i	−0.588928	+	0.340018i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−7280.23	+	4203.24i	−1.07029	+	0.617935i	−0.928262	−	0.371927i	$-0.878697\pi$
$359$	−7280.23	+	4203.24i	−1.07029	+	0.617935i	−0.142033	+	0.989862i	$0.545364\pi$
$360$		0			0
$361$	−2720.73	+	4712.44i	−0.396665	+	0.687044i
$362$		0			0
$363$		0			0
$364$		0			0
$365$			528.991i			0.0758593i
$366$		0			0
$367$	4673.36	+	2698.17i	0.664707	+	0.383769i	0.794068	−	0.607829i	$-0.207959\pi$
$367$	4673.36	+	2698.17i	0.664707	+	0.383769i	−0.129361	+	0.991598i	$0.541293\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6886.52	−	8598.40i	−0.963694	−	1.20325i
$372$		0			0
$373$	3007.56	+	5209.25i	0.417495	+	0.723123i	0.995687	−	0.0927779i	$-0.0295747\pi$
$373$	3007.56	+	5209.25i	0.417495	+	0.723123i	−0.578191	+	0.815901i	$0.696241\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	574.556			0.0784911
$378$		0			0
$379$	−1456.39			−0.197388			−0.0986939	−	0.995118i	$-0.531466\pi$
$379$	−1456.39			−0.197388			−0.0986939	+	0.995118i	$0.531466\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−806.803	−	1397.42i	−0.107639	−	0.186436i	0.807174	−	0.590313i	$-0.200996\pi$
$383$	−806.803	−	1397.42i	−0.107639	−	0.186436i	−0.914813	+	0.403877i	$0.867662\pi$
$384$		0			0
$385$	−4427.53	+	11351.0i	−0.586098	+	1.50261i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	7423.22	+	4285.80i	0.967537	+	0.558608i	0.898485	−	0.439005i	$-0.144669\pi$
$389$	7423.22	+	4285.80i	0.967537	+	0.558608i	0.0690527	+	0.997613i	$0.478002\pi$
$390$		0			0
$391$		−	244.329i		−	0.0316017i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	3960.33	−	6859.50i	0.504471	−	0.873770i
$396$		0			0
$397$	11677.0	−	6741.72i	1.47620	−	0.852286i	0.476562	−	0.879141i	$-0.341883\pi$
$397$	11677.0	−	6741.72i	1.47620	−	0.852286i	0.999639	+	0.0268551i	$0.00854927\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	10042.0	−	5797.73i	1.25055	−	0.722007i	0.279333	−	0.960194i	$-0.409887\pi$
$401$	10042.0	−	5797.73i	1.25055	−	0.722007i	0.971219	+	0.238187i	$0.0765532\pi$
$402$		0			0
$403$	−259.976	+	450.292i	−0.0321348	+	0.0556591i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	16738.6i		−	2.03858i
$408$		0			0
$409$	−2081.47	−	1201.74i	−0.251643	−	0.145286i	0.368873	−	0.929480i	$-0.379744\pi$
$409$	−2081.47	−	1201.74i	−0.251643	−	0.145286i	−0.620517	+	0.784193i	$0.713077\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	6885.33	−	5514.51i	0.820351	−	0.657025i
$414$		0			0
$415$	−7398.37	−	12814.3i	−0.875112	−	1.51574i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6175.24			−0.720001			−0.360001	−	0.932952i	$-0.617223\pi$
$419$	−6175.24			−0.720001			−0.360001	+	0.932952i	$0.617223\pi$
$420$		0			0
$421$	3082.19			0.356809			0.178405	−	0.983957i	$-0.442906\pi$
$421$	3082.19			0.356809			0.178405	+	0.983957i	$0.442906\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	99.3855	+	172.141i	0.0113433	+	0.0196472i
$426$		0			0
$427$	11599.5	−	1773.20i	1.31461	−	0.200963i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1931.19	−	1114.97i	−0.215829	−	0.124609i	0.388189	−	0.921580i	$-0.373101\pi$
$431$	−1931.19	−	1114.97i	−0.215829	−	0.124609i	−0.604017	+	0.796971i	$0.706434\pi$
$432$		0			0
$433$			9238.69i			1.02536i	0.858578	+	0.512682i	$0.171348\pi$
$433$			9238.69i			1.02536i	−0.858578	+	0.512682i	$0.828652\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	158.069	−	273.783i	0.0173031	−	0.0299699i
$438$		0			0
$439$	2806.74	−	1620.47i	0.305145	−	0.176175i	−0.339607	−	0.940567i	$-0.610294\pi$
$439$	2806.74	−	1620.47i	0.305145	−	0.176175i	0.644752	+	0.764392i	$0.276961\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−6001.79	+	3465.14i	−0.643688	+	0.371633i	−0.786034	−	0.618184i	$-0.787869\pi$
$443$	−6001.79	+	3465.14i	−0.643688	+	0.371633i	0.142346	+	0.989817i	$0.454535\pi$
$444$		0			0
$445$	−6840.65	+	11848.4i	−0.728715	+	1.26217i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	6773.64i		−	0.711955i	−0.934495	−	0.355978i	$-0.884148\pi$
$449$		−	6773.64i		−	0.711955i	0.934495	−	0.355978i	$-0.115852\pi$
$450$		0			0
$451$	−2691.52	−	1553.95i	−0.281017	−	0.162245i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1935.81	−	295.926i	0.199455	−	0.0304906i
$456$		0			0
$457$	6687.33	+	11582.8i	0.684508	+	1.18560i	0.973591	+	0.228299i	$0.0733164\pi$
$457$	6687.33	+	11582.8i	0.684508	+	1.18560i	−0.289083	+	0.957304i	$0.593350\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	13126.6			1.32618			0.663090	−	0.748540i	$-0.269245\pi$
$461$	13126.6			1.32618			0.663090	+	0.748540i	$0.269245\pi$
$462$		0			0
$463$	12805.7			1.28538			0.642690	−	0.766126i	$-0.277818\pi$
$463$	12805.7			1.28538			0.642690	+	0.766126i	$0.277818\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	665.256	+	1152.26i	0.0659194	+	0.114176i	0.897101	−	0.441825i	$-0.145669\pi$
$467$	665.256	+	1152.26i	0.0659194	+	0.114176i	−0.831182	+	0.556000i	$0.812335\pi$
$468$		0			0
$469$	−8928.49	+	7150.89i	−0.879061	+	0.704046i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−21745.6	−	12554.8i	−2.11388	−	1.22045i
$474$		0			0
$475$			257.190i			0.0248435i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5608.11	−	9713.54i	0.534950	−	0.926561i	−0.464215	−	0.885722i	$-0.653664\pi$
$479$	5608.11	−	9713.54i	0.534950	−	0.926561i	0.999166	−	0.0408390i	$-0.0130031\pi$
$480$		0			0
$481$	−2329.91	+	1345.17i	−0.220862	+	0.127515i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	9414.19	−	5435.28i	0.881394	−	0.508873i
$486$		0			0
$487$	1480.99	−	2565.15i	0.137803	−	0.238682i	−0.788862	−	0.614571i	$-0.789329\pi$
$487$	1480.99	−	2565.15i	0.137803	−	0.238682i	0.926665	+	0.375889i	$0.122663\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$			4452.75i			0.409266i	0.978839	+	0.204633i	$0.0656001\pi$
$491$			4452.75i			0.409266i	−0.978839	+	0.204633i	$0.934400\pi$
$492$		0			0
$493$	−1572.20	−	907.711i	−0.143628	−	0.0829234i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	2666.15	−	6835.33i	0.240630	−	0.616914i
$498$		0			0
$499$	−6419.64	−	11119.1i	−0.575917	−	0.997518i	−0.995941	−	0.0900049i	$-0.971312\pi$
$499$	−6419.64	−	11119.1i	−0.575917	−	0.997518i	0.420024	−	0.907513i	$-0.362022\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3169.30			−0.280939			−0.140469	−	0.990085i	$-0.544861\pi$
$503$	−3169.30			−0.280939			−0.140469	+	0.990085i	$0.544861\pi$
$504$		0			0
$505$	−14614.4			−1.28778
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5393.25	−	9341.38i	−0.469649	−	0.813456i	0.529749	−	0.848155i	$-0.322286\pi$
$509$	−5393.25	−	9341.38i	−0.469649	−	0.813456i	−0.999398	+	0.0346984i	$0.988953\pi$
$510$		0			0
$511$	−533.402	−	665.997i	−0.0461768	−	0.0576556i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−3881.92	−	2241.23i	−0.332151	−	0.191768i
$516$		0			0
$517$		−	14712.3i		−	1.25154i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−8319.38	+	14409.6i	−0.699575	+	1.21170i	0.269039	+	0.963129i	$0.413294\pi$
$521$	−8319.38	+	14409.6i	−0.699575	+	1.21170i	−0.968614	+	0.248570i	$0.920039\pi$
$522$		0			0
$523$	−17007.8	+	9819.46i	−1.42199	+	0.820984i	−0.996469	−	0.0839657i	$-0.973241\pi$
$523$	−17007.8	+	9819.46i	−1.42199	+	0.820984i	−0.425518	+	0.904950i	$0.639908\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	1422.78	−	821.445i	0.117604	−	0.0678988i
$528$		0			0
$529$	−6048.25	+	10475.9i	−0.497103	+	0.861007i
$530$		0			0
$531$		0			0
$532$		0			0
$533$			499.524i			0.0405943i
$534$		0			0
$535$	774.021	+	446.881i	0.0625492	+	0.0361128i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−5871.46	−	18755.4i	−0.469205	−	1.49880i
$540$		0			0
$541$	−8795.99	−	15235.1i	−0.699019	−	1.21074i	−0.968807	−	0.247816i	$-0.920287\pi$
$541$	−8795.99	−	15235.1i	−0.699019	−	1.21074i	0.269788	−	0.962920i	$-0.413046\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−7843.78			−0.616497
$546$		0			0
$547$	−9961.21			−0.778630			−0.389315	−	0.921105i	$-0.627288\pi$
$547$	−9961.21			−0.778630			−0.389315	+	0.921105i	$0.627288\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−1174.49	−	2034.27i	−0.0908073	−	0.157283i
$552$		0			0
$553$	1930.66	+	12629.4i	0.148463	+	0.971172i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	1255.36	+	724.783i	0.0954961	+	0.0551347i	0.546988	−	0.837141i	$-0.315775\pi$
$557$	1255.36	+	724.783i	0.0954961	+	0.0551347i	−0.451491	+	0.892275i	$0.649108\pi$
$558$		0			0
$559$			4035.81i			0.305360i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6356.42	+	11009.6i	−0.475828	+	0.824159i	−0.999617	−	0.0276898i	$-0.991185\pi$
$563$	−6356.42	+	11009.6i	−0.475828	+	0.824159i	0.523788	+	0.851848i	$0.324518\pi$
$564$		0			0
$565$	2850.95	−	1646.00i	0.212284	−	0.122562i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11895.7	+	6867.98i	−0.876438	+	0.506012i	−0.869482	−	0.493964i	$-0.835547\pi$
$569$	−11895.7	+	6867.98i	−0.876438	+	0.506012i	−0.00695552	+	0.999976i	$0.502214\pi$
$570$		0			0
$571$	−11893.2	+	20599.7i	−0.871656	+	1.50975i	−0.0113732	+	0.999935i	$0.503620\pi$
$571$	−11893.2	+	20599.7i	−0.871656	+	1.50975i	−0.860283	+	0.509817i	$0.829713\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			57.3578i			0.00415998i
$576$		0			0
$577$	−16470.3	−	9509.14i	−1.18833	−	0.686085i	−0.230406	−	0.973095i	$-0.574005\pi$
$577$	−16470.3	−	9509.14i	−1.18833	−	0.686085i	−0.957928	+	0.287010i	$0.907339\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	22235.7	+	8673.14i	1.58777	+	0.619316i
$582$		0			0
$583$	−17040.7	−	29515.4i	−1.21056	−	2.09675i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−2636.09			−0.185354			−0.0926772	−	0.995696i	$-0.529542\pi$
$587$	−2636.09			−0.185354			−0.0926772	+	0.995696i	$0.529542\pi$
$588$		0			0
$589$	2125.73			0.148709
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−8294.23	−	14366.0i	−0.574373	−	0.994843i	−0.996109	−	0.0881246i	$-0.971913\pi$
$593$	−8294.23	−	14366.0i	−0.574373	−	0.994843i	0.421737	−	0.906718i	$-0.361421\pi$
$594$		0			0
$595$	−5764.62	−	2248.52i	−0.397187	−	0.154925i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5022.26	−	2899.61i	−0.342578	−	0.197787i	0.318834	−	0.947811i	$-0.396709\pi$
$599$	−5022.26	−	2899.61i	−0.342578	−	0.197787i	−0.661411	+	0.750023i	$0.730042\pi$
$600$		0			0
$601$		−	19240.0i		−	1.30585i	−0.757422	−	0.652926i	$-0.773541\pi$
$601$		−	19240.0i		−	1.30585i	0.757422	−	0.652926i	$-0.226459\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11206.0	+	19409.4i	−0.753041	+	1.30430i
$606$		0			0
$607$	−10321.5	+	5959.14i	−0.690178	+	0.398474i	−0.803679	−	0.595064i	$-0.797127\pi$
$607$	−10321.5	+	5959.14i	−0.690178	+	0.398474i	0.113501	+	0.993538i	$0.463793\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−2047.86	+	1182.33i	−0.135593	+	0.0782849i
$612$		0			0
$613$	−2031.64	+	3518.90i	−0.133862	+	0.231855i	−0.925162	−	0.379573i	$-0.876071\pi$
$613$	−2031.64	+	3518.90i	−0.133862	+	0.231855i	0.791300	+	0.611428i	$0.209404\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	15705.9i		−	1.02479i	−0.858750	−	0.512395i	$-0.828758\pi$
$617$		−	15705.9i		−	1.02479i	0.858750	−	0.512395i	$-0.171242\pi$
$618$		0			0
$619$	20914.5	+	12075.0i	1.35803	+	0.784061i	0.989359	−	0.145497i	$-0.0464781\pi$
$619$	20914.5	+	12075.0i	1.35803	+	0.784061i	0.368675	+	0.929558i	$0.379811\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−3334.81	−	21814.7i	−0.214456	−	1.40287i
$624$		0			0
$625$	8216.11	+	14230.7i	0.525831	+	0.910766i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8500.68			0.538862
$630$		0			0
$631$	23965.9			1.51199			0.755996	−	0.654576i	$-0.227153\pi$
$631$	23965.9			1.51199			0.755996	+	0.654576i	$0.227153\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	15247.0	+	26408.5i	0.952845	+	1.65038i
$636$		0			0
$637$	−2138.78	+	2324.52i	−0.133032	+	0.144585i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−8093.10	−	4672.55i	−0.498687	−	0.287917i	0.229484	−	0.973312i	$-0.426296\pi$
$641$	−8093.10	−	4672.55i	−0.498687	−	0.287917i	−0.728171	+	0.685395i	$0.759629\pi$
$642$		0			0
$643$		−	9283.22i		−	0.569354i	−0.958623	−	0.284677i	$-0.908114\pi$
$643$		−	9283.22i		−	0.569354i	0.958623	−	0.284677i	$-0.0918864\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−11327.2	+	19619.2i	−0.688280	+	1.19214i	0.284114	+	0.958791i	$0.408301\pi$
$647$	−11327.2	+	19619.2i	−0.688280	+	1.19214i	−0.972394	+	0.233345i	$0.925033\pi$
$648$		0			0
$649$	23635.0	−	13645.7i	1.42951	−	0.825331i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11545.5	−	6665.79i	0.691899	−	0.399468i	−0.112424	−	0.993660i	$-0.535862\pi$
$653$	11545.5	−	6665.79i	0.691899	−	0.399468i	0.804323	+	0.594192i	$0.202528\pi$
$654$		0			0
$655$	−13757.7	+	23829.0i	−0.820697	+	1.42149i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	5190.48i		−	0.306817i	−0.988163	−	0.153409i	$-0.950975\pi$
$659$		−	5190.48i		−	0.306817i	0.988163	−	0.153409i	$-0.0490250\pi$
$660$		0			0
$661$	2286.98	+	1320.39i	0.134574	+	0.0776963i	0.565776	−	0.824559i	$-0.308577\pi$
$661$	2286.98	+	1320.39i	0.134574	+	0.0776963i	−0.431202	+	0.902256i	$0.641910\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5004.87	−	6249.00i	−0.291850	−	0.364400i
$666$		0			0
$667$	−261.931	−	453.678i	−0.0152054	−	0.0263366i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	36302.8			2.08860
$672$		0			0
$673$	6119.56			0.350507			0.175254	−	0.984523i	$-0.443925\pi$
$673$	6119.56			0.350507			0.175254	+	0.984523i	$0.443925\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10363.3	+	17949.7i	0.588320	+	1.01900i	0.994453	+	0.105186i	$0.0335437\pi$
$677$	10363.3	+	17949.7i	0.588320	+	1.01900i	−0.406133	+	0.913814i	$0.633123\pi$
$678$		0			0
$679$	−6371.81	+	16335.7i	−0.360129	+	0.923278i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6056.22	+	3496.56i	0.339290	+	0.195889i	0.659958	−	0.751303i	$-0.270574\pi$
$683$	6056.22	+	3496.56i	0.339290	+	0.195889i	−0.320668	+	0.947192i	$0.603907\pi$
$684$		0			0
$685$		−	20602.5i		−	1.14917i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2738.91	+	4743.93i	−0.151443	+	0.262307i
$690$		0			0
$691$	18475.9	−	10667.1i	1.01716	−	0.587256i	0.103878	−	0.994590i	$-0.466875\pi$
$691$	18475.9	−	10667.1i	1.01716	−	0.587256i	0.913279	+	0.407334i	$0.133541\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1004.43	+	579.910i	−0.0548207	+	0.0316507i
$696$		0			0
$697$	789.171	−	1366.88i	0.0428867	−	0.0742819i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			7995.09i			0.430771i	0.976529	+	0.215385i	$0.0691007\pi$
$701$			7995.09i			0.430771i	−0.976529	+	0.215385i	$0.930899\pi$
$702$		0			0
$703$	9525.43	+	5499.51i	0.511036	+	0.295047i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18399.4	−	14736.2i	0.978757	−	0.783894i
$708$		0			0
$709$	−1680.51	−	2910.73i	−0.0890167	−	0.154181i	0.818079	−	0.575106i	$-0.195039\pi$
$709$	−1680.51	−	2910.73i	−0.0890167	−	0.154181i	−0.907096	+	0.420924i	$0.861706\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	474.076			0.0249008
$714$		0			0
$715$	6058.50			0.316888
$716$		0			0
$717$		0			0
$718$		0			0
$719$	11653.3	+	20184.1i	0.604444	+	1.04693i	0.992139	+	0.125139i	$0.0399378\pi$
$719$	11653.3	+	20184.1i	0.604444	+	1.04693i	−0.387696	+	0.921787i	$0.626729\pi$
$720$		0			0
$721$	7147.24	−	1092.60i	0.369178	−	0.0564360i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	369.084	+	213.091i	0.0189068	+	0.0109158i
$726$		0			0
$727$			27744.2i			1.41537i	0.706526	+	0.707687i	$0.250261\pi$
$727$			27744.2i			1.41537i	−0.706526	+	0.707687i	$0.749739\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	6375.96	−	11043.5i	0.322604	−	0.558766i
$732$		0			0
$733$	30453.8	−	17582.5i	1.53457	−	0.885983i	0.535425	−	0.844583i	$-0.320151\pi$
$733$	30453.8	−	17582.5i	1.53457	−	0.885983i	0.999143	−	0.0414005i	$-0.0131819\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−30648.5	+	17694.9i	−1.53182	+	0.884397i
$738$		0			0
$739$	−4158.66	+	7203.01i	−0.207008	+	0.358548i	−0.950771	−	0.309896i	$-0.899706\pi$
$739$	−4158.66	+	7203.01i	−0.207008	+	0.358548i	0.743763	+	0.668444i	$0.233039\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			34257.4i			1.69150i	0.533582	+	0.845748i	$0.320846\pi$
$743$			34257.4i			1.69150i	−0.533582	+	0.845748i	$0.679154\pi$
$744$		0			0
$745$	8544.26	+	4933.03i	0.420185	+	0.242594i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−1425.10	+	217.854i	−0.0695219	+	0.0106278i
$750$		0			0
$751$	−3760.57	−	6513.49i	−0.182723	−	0.316486i	0.760084	−	0.649825i	$-0.225158\pi$
$751$	−3760.57	−	6513.49i	−0.182723	−	0.316486i	−0.942807	+	0.333339i	$0.891824\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	18542.1			0.893797
$756$		0			0
$757$	−13575.7			−0.651806			−0.325903	−	0.945403i	$-0.605668\pi$
$757$	−13575.7			−0.651806			−0.325903	+	0.945403i	$0.605668\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−20338.8	−	35227.8i	−0.968830	−	1.67806i	−0.698953	−	0.715167i	$-0.746350\pi$
$761$	−20338.8	−	35227.8i	−0.968830	−	1.67806i	−0.269876	−	0.962895i	$-0.586983\pi$
$762$		0			0
$763$	9875.29	−	7909.19i	0.468558	−	0.375271i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−3798.79	−	2193.23i	−0.178835	−	0.103250i
$768$		0			0
$769$		−	5230.17i		−	0.245260i	−0.992452	−	0.122630i	$-0.960867\pi$
$769$		−	5230.17i		−	0.245260i	0.992452	−	0.122630i	$-0.0391328\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	38.3831	−	66.4815i	0.00178596	−	0.00309337i	−0.865131	−	0.501546i	$-0.832765\pi$
$773$	38.3831	−	66.4815i	0.00178596	−	0.00309337i	0.866917	+	0.498453i	$0.166098\pi$
$774$		0			0
$775$	−334.007	+	192.839i	−0.0154811	+	0.00893804i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1768.61	−	1021.11i	0.0813441	−	0.0469641i
$780$		0			0
$781$	11349.3	−	19657.6i	0.519987	−	0.900644i
$782$		0			0
$783$		0			0
$784$		0			0
$785$			21515.6i			0.978249i
$786$		0			0
$787$	26841.7	+	15497.0i	1.21576	+	0.701919i	0.964008	−	0.265873i	$-0.0856603\pi$
$787$	26841.7	+	15497.0i	1.21576	+	0.701919i	0.251751	+	0.967792i	$0.418994\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−1929.61	+	4947.03i	−0.0867372	+	0.222372i
$792$		0			0
$793$	−2917.43	−	5053.13i	−0.130644	−	0.226282i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	12755.8			0.566920			0.283460	−	0.958984i	$-0.408518\pi$
$797$	12755.8			0.566920			0.283460	+	0.958984i	$0.408518\pi$
$798$		0			0
$799$	7471.62			0.330822
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−1319.91	−	2286.14i	−0.0580056	−	0.100469i
$804$		0			0
$805$	−1116.17	−	1393.64i	−0.0488695	−	0.0610177i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−23103.5	−	13338.8i	−1.00405	−	0.579688i	−0.0946047	−	0.995515i	$-0.530159\pi$
$809$	−23103.5	−	13338.8i	−1.00405	−	0.579688i	−0.909444	+	0.415827i	$0.863492\pi$
$810$		0			0
$811$		−	10735.6i		−	0.464829i	−0.972617	−	0.232415i	$-0.925337\pi$
$811$		−	10735.6i		−	0.464829i	0.972617	−	0.232415i	$-0.0746626\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7624.32	+	13205.7i	−0.327691	+	0.567578i
$816$		0			0
$817$	14289.2	−	8249.85i	0.611890	−	0.353275i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	18107.4	−	10454.3i	0.769735	−	0.444407i	−0.0630451	−	0.998011i	$-0.520081\pi$
$821$	18107.4	−	10454.3i	0.769735	−	0.444407i	0.832780	+	0.553604i	$0.186748\pi$
$822$		0			0
$823$	−3036.16	+	5258.78i	−0.128595	+	0.222734i	−0.923133	−	0.384482i	$-0.874380\pi$
$823$	−3036.16	+	5258.78i	−0.128595	+	0.222734i	0.794537	+	0.607215i	$0.207713\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			31977.3i			1.34457i	0.740293	+	0.672285i	$0.234687\pi$
$827$			31977.3i			1.34457i	−0.740293	+	0.672285i	$0.765313\pi$
$828$		0			0
$829$	−7087.97	−	4092.24i	−0.296954	−	0.171447i	0.344119	−	0.938926i	$-0.388177\pi$
$829$	−7087.97	−	4092.24i	−0.296954	−	0.171447i	−0.641074	+	0.767479i	$0.721511\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	9524.90	−	2981.82i	0.396180	−	0.124026i
$834$		0			0
$835$	3419.74	+	5923.16i	0.141730	+	0.245484i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15870.1			0.653036			0.326518	−	0.945191i	$-0.394125\pi$
$839$	15870.1			0.653036			0.326518	+	0.945191i	$0.394125\pi$
$840$		0			0
$841$	20496.6			0.840403
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12125.8	+	21002.6i	0.493659	+	0.855042i
$846$		0			0
$847$	−5462.92	−	35735.8i	−0.221615	−	1.44970i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2124.34	+	1226.49i	0.0855716	+	0.0494048i
$852$		0			0
$853$			196.093i			0.00787116i	0.999992	+	0.00393558i	$0.00125274\pi$
$853$			196.093i			0.00787116i	−0.999992	+	0.00393558i	$0.998747\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−21147.8	+	36629.0i	−0.842934	+	1.46000i	0.0444695	+	0.999011i	$0.485840\pi$
$857$	−21147.8	+	36629.0i	−0.842934	+	1.46000i	−0.887403	+	0.460994i	$0.847493\pi$
$858$		0			0
$859$	13993.2	−	8078.97i	0.555810	−	0.320897i	−0.195652	−	0.980673i	$-0.562682\pi$
$859$	13993.2	−	8078.97i	0.555810	−	0.320897i	0.751462	+	0.659776i	$0.229349\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−29141.4	+	16824.8i	−1.14946	+	0.663642i	−0.948756	−	0.316010i	$-0.897657\pi$
$863$	−29141.4	+	16824.8i	−1.14946	+	0.663642i	−0.200705	+	0.979652i	$0.564323\pi$
$864$		0			0
$865$	−12853.1	+	22262.3i	−0.505226	+	0.875077i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			39526.3i			1.54297i
$870$		0			0
$871$	4926.05	+	2844.06i	0.191634	+	0.110640i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−23410.3	−	9131.28i	−0.904470	−	0.352793i
$876$		0			0
$877$	14172.3	+	24547.1i	0.545683	+	0.945150i	0.998564	+	0.0535793i	$0.0170630\pi$
$877$	14172.3	+	24547.1i	0.545683	+	0.945150i	−0.452881	+	0.891571i	$0.649604\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3836.42			0.146711			0.0733555	−	0.997306i	$-0.476629\pi$
$881$	3836.42			0.146711			0.0733555	+	0.997306i	$0.476629\pi$
$882$		0			0
$883$	−43488.6			−1.65743			−0.828713	−	0.559674i	$-0.810926\pi$
$883$	−43488.6			−1.65743			−0.828713	+	0.559674i	$0.810926\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−9863.09	−	17083.4i	−0.373360	−	0.646678i	0.616720	−	0.787183i	$-0.288461\pi$
$887$	−9863.09	−	17083.4i	−0.373360	−	0.646678i	−0.990080	+	0.140504i	$0.955128\pi$
$888$		0			0
$889$	−45824.5	−	17874.1i	−1.72880	−	0.674328i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8372.33	+	4833.76i	0.313739	+	0.181137i
$894$		0			0
$895$			22021.1i			0.822441i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1761.24	−	3050.56i	0.0653401	−	0.113172i
$900$		0			0
$901$	14989.4	−	8654.12i	0.554238	−	0.319990i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	41391.2	−	23897.2i	1.52032	−	0.877758i
$906$		0			0
$907$	−297.428	+	515.160i	−0.0108886	+	0.0188595i	−0.871418	−	0.490541i	$-0.836799\pi$
$907$	−297.428	+	515.160i	−0.0108886	+	0.0188595i	0.860530	+	0.509400i	$0.170133\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	38885.5i		−	1.41420i	−0.707115	−	0.707099i	$-0.750004\pi$
$911$		−	38885.5i		−	1.41420i	0.707115	−	0.707099i	$-0.249996\pi$
$912$		0			0
$913$	63947.1	+	36919.9i	2.31801	+	1.33830i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6706.84	−	43873.0i	−0.241526	−	1.57995i
$918$		0			0
$919$	6745.70	+	11683.9i	0.242133	+	0.419387i	0.961322	−	0.275428i	$-0.0888196\pi$
$919$	6745.70	+	11683.9i	0.242133	+	0.419387i	−0.719189	+	0.694815i	$0.755486\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−3648.28			−0.130103
$924$		0			0
$925$	−1995.59			−0.0709345
$926$		0			0
$927$		0			0
$928$		0			0
$929$	20401.4	+	35336.3i	0.720504	+	1.24795i	0.960798	+	0.277249i	$0.0894228\pi$
$929$	20401.4	+	35336.3i	0.720504	+	1.24795i	−0.240294	+	0.970700i	$0.577244\pi$
$930$		0			0
$931$	12602.2	+	2820.86i	0.443631	+	0.0993017i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−16578.3	−	9571.51i	−0.579860	−	0.334783i
$936$		0			0
$937$			5109.16i			0.178131i	0.996026	+	0.0890656i	$0.0283881\pi$
$937$			5109.16i			0.178131i	−0.996026	+	0.0890656i	$0.971612\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−22089.7	+	38260.5i	−0.765253	+	1.32546i	0.174859	+	0.984593i	$0.444053\pi$
$941$	−22089.7	+	38260.5i	−0.765253	+	1.32546i	−0.940113	+	0.340864i	$0.889280\pi$
$942$		0			0
$943$	394.431	−	227.725i	0.0136208	−	0.00786400i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26710.1	−	15421.1i	0.916538	−	0.529163i	0.0340090	−	0.999422i	$-0.489172\pi$
$947$	26710.1	−	15421.1i	0.916538	−	0.529163i	0.882529	+	0.470258i	$0.155839\pi$
$948$		0			0
$949$	−212.145	+	367.446i	−0.00725660	+	0.0125688i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			24444.0i			0.830870i	0.909623	+	0.415435i	$0.136371\pi$
$953$			24444.0i			0.830870i	−0.909623	+	0.415435i	$0.863629\pi$
$954$		0			0
$955$	3070.04	+	1772.49i	0.104025	+	0.0600590i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	20774.3	+	25938.5i	0.699518	+	0.873407i
$960$		0			0
$961$	−13301.6	−	23039.1i	−0.446499	−	0.773358i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	54775.4			1.82723
$966$		0			0
$967$	15106.5			0.502369			0.251185	−	0.967939i	$-0.419180\pi$
$967$	15106.5			0.502369			0.251185	+	0.967939i	$0.419180\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	22196.6	+	38445.7i	0.733598	+	1.27063i	0.955336	+	0.295523i	$0.0954938\pi$
$971$	22196.6	+	38445.7i	0.733598	+	1.27063i	−0.221738	+	0.975106i	$0.571173\pi$
$972$		0			0
$973$	679.832	−	1742.91i	0.0223992	−	0.0574258i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−22723.1	−	13119.2i	−0.744092	−	0.429602i	0.0794632	−	0.996838i	$-0.474679\pi$
$977$	−22723.1	−	13119.2i	−0.744092	−	0.429602i	−0.823555	+	0.567236i	$0.808013\pi$
$978$		0			0
$979$		−	68273.5i		−	2.22884i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−30359.2	+	52583.7i	−0.985055	+	1.70616i	−0.343361	+	0.939204i	$0.611565\pi$
$983$	−30359.2	+	52583.7i	−0.985055	+	1.70616i	−0.641694	+	0.766961i	$0.721768\pi$
$984$		0			0
$985$	−11096.8	+	6406.74i	−0.358958	+	0.207244i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	3186.73	−	1839.86i	0.102459	−	0.0591549i
$990$		0			0
$991$	21247.7	−	36802.1i	0.681085	−	1.17967i	−0.293564	−	0.955939i	$-0.594842\pi$
$991$	21247.7	−	36802.1i	0.681085	−	1.17967i	0.974650	−	0.223735i	$-0.0718251\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			35976.6i			1.14627i
$996$		0			0
$997$	−1037.94	−	599.254i	−0.0329707	−	0.0190357i	0.483424	−	0.875386i	$-0.339393\pi$
$997$	−1037.94	−	599.254i	−0.0329707	−	0.0190357i	−0.516395	+	0.856351i	$0.672726\pi$
$998$		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.17.19		48
3.2	odd	2	inner	1008.4.bt.d.17.6		48
4.3	odd	2		504.4.bl.a.17.19	yes	48
7.5	odd	6	inner	1008.4.bt.d.593.6		48
12.11	even	2		504.4.bl.a.17.6	✓	48
21.5	even	6	inner	1008.4.bt.d.593.19		48
28.19	even	6		504.4.bl.a.89.6	yes	48
84.47	odd	6		504.4.bl.a.89.19	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
504.4.bl.a.17.6	✓	48	12.11	even	2
504.4.bl.a.17.19	yes	48	4.3	odd	2
504.4.bl.a.89.6	yes	48	28.19	even	6
504.4.bl.a.89.19	yes	48	84.47	odd	6
1008.4.bt.d.17.6		48	3.2	odd	2	inner
1008.4.bt.d.17.19		48	1.1	even	1	trivial
1008.4.bt.d.593.6		48	7.5	odd	6	inner
1008.4.bt.d.593.19		48	21.5	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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)

\(f(q)\)	\(=\)	\(q+(5.74088 + 9.94350i) q^{5} +(-17.2542 - 6.73007i) q^{7} +O(q^{10})\)	\(q+(5.74088 + 9.94350i) q^{5} +(-17.2542 - 6.73007i) q^{7} +(-49.6208 - 28.6486i) q^{11} +9.20923i q^{13} +(14.5492 - 25.1999i) q^{17} +(32.6061 - 18.8252i) q^{19} +(7.27174 - 4.19834i) q^{23} +(-3.41550 + 5.91583i) q^{25} -62.3892i q^{29} +(48.8957 + 28.2300i) q^{31} +(-32.1337 - 210.203i) q^{35} +(146.068 + 252.997i) q^{37} +54.2417 q^{41} +438.235 q^{43} +(128.386 + 222.371i) q^{47} +(252.412 + 232.243i) q^{49} +(515.128 + 297.409i) q^{53} -657.873i q^{55} +(-238.156 + 412.499i) q^{59} +(-548.703 + 316.794i) q^{61} +(-91.5720 + 52.8691i) q^{65} +(308.827 - 534.904i) q^{67} +396.155i q^{71} +(39.8998 + 23.0361i) q^{73} +(663.359 + 828.259i) q^{77} +(-344.924 - 597.425i) q^{79} -1288.72 q^{83} +334.100 q^{85} +(595.784 + 1031.93i) q^{89} +(61.9787 - 158.897i) q^{91} +(374.376 + 216.146i) q^{95} -946.768i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q + 24 q^{7}+O(q^{10}) \) 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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