LMFDB - Embedded newform 624.3.ba.a.385.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,3,Mod(385,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("624.385");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	624.ba (of order $4$, 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:	$17.0027684961$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			385.2
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	624.385
Dual form			624.3.ba.a.577.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.73205 q^{3} +(4.73205 + 4.73205i) q^{5} +(2.73205 - 2.73205i) q^{7} +3.00000 q^{9} +(-1.73205 + 1.73205i) q^{11} +(9.92820 + 8.39230i) q^{13} +(8.19615 + 8.19615i) q^{15} -29.3205i q^{17} +(11.2679 + 11.2679i) q^{19} +(4.73205 - 4.73205i) q^{21} +29.3205i q^{23} +19.7846i q^{25} +5.19615 q^{27} +31.8564 q^{29} +(-26.9808 - 26.9808i) q^{31} +(-3.00000 + 3.00000i) q^{33} +25.8564 q^{35} +(-30.8564 + 30.8564i) q^{37} +(17.1962 + 14.5359i) q^{39} +(-14.4449 - 14.4449i) q^{41} +25.1769i q^{43} +(14.1962 + 14.1962i) q^{45} +(41.1962 - 41.1962i) q^{47} +34.0718i q^{49} -50.7846i q^{51} +2.28719 q^{53} -16.3923 q^{55} +(19.5167 + 19.5167i) q^{57} +(-54.6218 + 54.6218i) q^{59} -7.42563 q^{61} +(8.19615 - 8.19615i) q^{63} +(7.26795 + 86.6936i) q^{65} +(60.6936 + 60.6936i) q^{67} +50.7846i q^{69} +(-38.9090 - 38.9090i) q^{71} +(40.3205 - 40.3205i) q^{73} +34.2679i q^{75} +9.46410i q^{77} +148.210 q^{79} +9.00000 q^{81} +(-73.7987 - 73.7987i) q^{83} +(138.746 - 138.746i) q^{85} +55.1769 q^{87} +(25.5167 - 25.5167i) q^{89} +(50.0526 - 4.19615i) q^{91} +(-46.7321 - 46.7321i) q^{93} +106.641i q^{95} +(-86.0333 - 86.0333i) q^{97} +(-5.19615 + 5.19615i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 12 q^{5} + 4 q^{7} + 12 q^{9} + 12 q^{13} + 12 q^{15} + 52 q^{19} + 12 q^{21} + 72 q^{29} - 4 q^{31} - 12 q^{33} + 48 q^{35} - 68 q^{37} + 48 q^{39} + 60 q^{41} + 36 q^{45} + 144 q^{47} + 120 q^{53}+ \cdots - 164 q^{97}+O(q^{100}) $ 4 * q + 12 * q^5 + 4 * q^7 + 12 * q^9 + 12 * q^13 + 12 * q^15 + 52 * q^19 + 12 * q^21 + 72 * q^29 - 4 * q^31 - 12 * q^33 + 48 * q^35 - 68 * q^37 + 48 * q^39 + 60 * q^41 + 36 * q^45 + 144 * q^47 + 120 * q^53 - 24 * q^55 - 12 * q^57 + 24 * q^59 + 192 * q^61 + 12 * q^63 + 36 * q^65 + 28 * q^67 - 24 * q^71 + 92 * q^73 + 288 * q^79 + 36 * q^81 + 72 * q^83 + 264 * q^85 + 96 * q^87 + 12 * q^89 + 124 * q^91 - 180 * q^93 - 164 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$e\left(\frac{1}{4}\right)$	$1$	$1$

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
$3$	1.73205			0.577350
$3$	1.73205			0.577350
$4$		0			0
$5$	4.73205	+	4.73205i	0.946410	+	0.946410i	0.998635	−	0.0522252i	$-0.0166314\pi$
$5$	4.73205	+	4.73205i	0.946410	+	0.946410i	−0.0522252	+	0.998635i	$0.516631\pi$
$6$		0			0
$7$	2.73205	−	2.73205i	0.390293	−	0.390293i	−0.484499	−	0.874792i	$-0.660998\pi$
$7$	2.73205	−	2.73205i	0.390293	−	0.390293i	0.874792	+	0.484499i	$0.160998\pi$
$8$		0			0
$9$	3.00000			0.333333
$10$		0			0
$11$	−1.73205	+	1.73205i	−0.157459	+	0.157459i	−0.781440	−	0.623981i	$-0.785514\pi$
$11$	−1.73205	+	1.73205i	−0.157459	+	0.157459i	0.623981	+	0.781440i	$0.285514\pi$
$12$		0			0
$13$	9.92820	+	8.39230i	0.763708	+	0.645562i
$13$	9.92820	+	8.39230i	0.763708	+	0.645562i
$14$		0			0
$15$	8.19615	+	8.19615i	0.546410	+	0.546410i
$16$		0			0
$17$		−	29.3205i		−	1.72474i	−0.506282	−	0.862368i	$-0.668981\pi$
$17$		−	29.3205i		−	1.72474i	0.506282	−	0.862368i	$-0.331019\pi$
$18$		0			0
$19$	11.2679	+	11.2679i	0.593050	+	0.593050i	0.938454	−	0.345404i	$-0.112258\pi$
$19$	11.2679	+	11.2679i	0.593050	+	0.593050i	−0.345404	+	0.938454i	$0.612258\pi$
$20$		0			0
$21$	4.73205	−	4.73205i	0.225336	−	0.225336i
$22$		0			0
$23$			29.3205i			1.27480i	0.770531	+	0.637402i	$0.219991\pi$
$23$			29.3205i			1.27480i	−0.770531	+	0.637402i	$0.780009\pi$
$24$		0			0
$25$			19.7846i			0.791384i
$26$		0			0
$27$	5.19615			0.192450
$28$		0			0
$29$	31.8564			1.09850			0.549248	−	0.835659i	$-0.314914\pi$
$29$	31.8564			1.09850			0.549248	+	0.835659i	$0.314914\pi$
$30$		0			0
$31$	−26.9808	−	26.9808i	−0.870347	−	0.870347i	0.122163	−	0.992510i	$-0.461017\pi$
$31$	−26.9808	−	26.9808i	−0.870347	−	0.870347i	−0.992510	+	0.122163i	$0.961017\pi$
$32$		0			0
$33$	−3.00000	+	3.00000i	−0.0909091	+	0.0909091i
$34$		0			0
$35$	25.8564			0.738754
$36$		0			0
$37$	−30.8564	+	30.8564i	−0.833957	+	0.833957i	−0.988055	−	0.154099i	$-0.950753\pi$
$37$	−30.8564	+	30.8564i	−0.833957	+	0.833957i	0.154099	+	0.988055i	$0.450753\pi$
$38$		0			0
$39$	17.1962	+	14.5359i	0.440927	+	0.372715i
$40$		0			0
$41$	−14.4449	−	14.4449i	−0.352314	−	0.352314i	0.508656	−	0.860970i	$-0.330142\pi$
$41$	−14.4449	−	14.4449i	−0.352314	−	0.352314i	−0.860970	+	0.508656i	$0.830142\pi$
$42$		0			0
$43$			25.1769i			0.585510i	0.956188	+	0.292755i	$0.0945720\pi$
$43$			25.1769i			0.585510i	−0.956188	+	0.292755i	$0.905428\pi$
$44$		0			0
$45$	14.1962	+	14.1962i	0.315470	+	0.315470i
$46$		0			0
$47$	41.1962	−	41.1962i	0.876514	−	0.876514i	−0.116658	−	0.993172i	$-0.537218\pi$
$47$	41.1962	−	41.1962i	0.876514	−	0.876514i	0.993172	+	0.116658i	$0.0372182\pi$
$48$		0			0
$49$			34.0718i			0.695343i
$50$		0			0
$51$		−	50.7846i		−	0.995777i
$52$		0			0
$53$	2.28719			0.0431545			0.0215772	−	0.999767i	$-0.493131\pi$
$53$	2.28719			0.0431545			0.0215772	+	0.999767i	$0.493131\pi$
$54$		0			0
$55$	−16.3923			−0.298042
$56$		0			0
$57$	19.5167	+	19.5167i	0.342398	+	0.342398i
$58$		0			0
$59$	−54.6218	+	54.6218i	−0.925793	+	0.925793i	−0.997431	−	0.0716379i	$-0.977177\pi$
$59$	−54.6218	+	54.6218i	−0.925793	+	0.925793i	0.0716379	+	0.997431i	$0.477177\pi$
$60$		0			0
$61$	−7.42563			−0.121732			−0.0608658	−	0.998146i	$-0.519386\pi$
$61$	−7.42563			−0.121732			−0.0608658	+	0.998146i	$0.519386\pi$
$62$		0			0
$63$	8.19615	−	8.19615i	0.130098	−	0.130098i
$64$		0			0
$65$	7.26795	+	86.6936i	0.111815	+	1.33375i
$66$		0			0
$67$	60.6936	+	60.6936i	0.905874	+	0.905874i	0.995936	−	0.0900619i	$-0.0287065\pi$
$67$	60.6936	+	60.6936i	0.905874	+	0.905874i	−0.0900619	+	0.995936i	$0.528706\pi$
$68$		0			0
$69$			50.7846i			0.736009i
$70$		0			0
$71$	−38.9090	−	38.9090i	−0.548014	−	0.548014i	0.377852	−	0.925866i	$-0.376663\pi$
$71$	−38.9090	−	38.9090i	−0.548014	−	0.548014i	−0.925866	+	0.377852i	$0.876663\pi$
$72$		0			0
$73$	40.3205	−	40.3205i	0.552336	−	0.552336i	−0.374779	−	0.927114i	$-0.622281\pi$
$73$	40.3205	−	40.3205i	0.552336	−	0.552336i	0.927114	+	0.374779i	$0.122281\pi$
$74$		0			0
$75$			34.2679i			0.456906i
$76$		0			0
$77$			9.46410i			0.122910i
$78$		0			0
$79$	148.210			1.87608			0.938039	−	0.346528i	$-0.112640\pi$
$79$	148.210			1.87608			0.938039	+	0.346528i	$0.112640\pi$
$80$		0			0
$81$	9.00000			0.111111
$82$		0			0
$83$	−73.7987	−	73.7987i	−0.889141	−	0.889141i	0.105300	−	0.994441i	$-0.466420\pi$
$83$	−73.7987	−	73.7987i	−0.889141	−	0.889141i	−0.994441	+	0.105300i	$0.966420\pi$
$84$		0			0
$85$	138.746	−	138.746i	1.63231	−	1.63231i
$86$		0			0
$87$	55.1769			0.634217
$88$		0			0
$89$	25.5167	−	25.5167i	0.286704	−	0.286704i	−0.549071	−	0.835775i	$-0.685019\pi$
$89$	25.5167	−	25.5167i	0.286704	−	0.286704i	0.835775	+	0.549071i	$0.185019\pi$
$90$		0			0
$91$	50.0526	−	4.19615i	0.550028	−	0.0461116i
$92$		0			0
$93$	−46.7321	−	46.7321i	−0.502495	−	0.502495i
$94$		0			0
$95$			106.641i			1.12254i
$96$		0			0
$97$	−86.0333	−	86.0333i	−0.886941	−	0.886941i	0.107287	−	0.994228i	$-0.465784\pi$
$97$	−86.0333	−	86.0333i	−0.886941	−	0.886941i	−0.994228	+	0.107287i	$0.965784\pi$
$98$		0			0
$99$	−5.19615	+	5.19615i	−0.0524864	+	0.0524864i
$100$		0			0
$101$		−	104.536i		−	1.03501i	−0.855681	−	0.517504i	$-0.826861\pi$
$101$		−	104.536i		−	1.03501i	0.855681	−	0.517504i	$-0.173139\pi$
$102$		0			0
$103$			36.6795i			0.356112i	0.984020	+	0.178056i	$0.0569808\pi$
$103$			36.6795i			0.356112i	−0.984020	+	0.178056i	$0.943019\pi$
$104$		0			0
$105$	44.7846			0.426520
$106$		0			0
$107$	123.464			1.15387			0.576935	−	0.816790i	$-0.304249\pi$
$107$	123.464			1.15387			0.576935	+	0.816790i	$0.304249\pi$
$108$		0			0
$109$	119.315	+	119.315i	1.09464	+	1.09464i	0.995027	+	0.0996097i	$0.0317594\pi$
$109$	119.315	+	119.315i	1.09464	+	1.09464i	0.0996097	+	0.995027i	$0.468241\pi$
$110$		0			0
$111$	−53.4449	+	53.4449i	−0.481485	+	0.481485i
$112$		0			0
$113$	−184.277			−1.63077			−0.815384	−	0.578920i	$-0.803474\pi$
$113$	−184.277			−1.63077			−0.815384	+	0.578920i	$0.803474\pi$
$114$		0			0
$115$	−138.746	+	138.746i	−1.20649	+	1.20649i
$116$		0			0
$117$	29.7846	+	25.1769i	0.254569	+	0.215187i
$118$		0			0
$119$	−80.1051	−	80.1051i	−0.673152	−	0.673152i
$120$		0			0
$121$			115.000i			0.950413i
$122$		0			0
$123$	−25.0192	−	25.0192i	−0.203408	−	0.203408i
$124$		0			0
$125$	24.6795	−	24.6795i	0.197436	−	0.197436i
$126$		0			0
$127$		−	173.962i		−	1.36978i	−0.728648	−	0.684888i	$-0.759851\pi$
$127$		−	173.962i		−	1.36978i	0.728648	−	0.684888i	$-0.240149\pi$
$128$		0			0
$129$			43.6077i			0.338044i
$130$		0			0
$131$	−121.110			−0.924506			−0.462253	−	0.886748i	$-0.652959\pi$
$131$	−121.110			−0.924506			−0.462253	+	0.886748i	$0.652959\pi$
$132$		0			0
$133$	61.5692			0.462926
$134$		0			0
$135$	24.5885	+	24.5885i	0.182137	+	0.182137i
$136$		0			0
$137$	−130.053	+	130.053i	−0.949289	+	0.949289i	−0.998775	−	0.0494861i	$-0.984242\pi$
$137$	−130.053	+	130.053i	−0.949289	+	0.949289i	0.0494861	+	0.998775i	$0.484242\pi$
$138$		0			0
$139$	27.1384			0.195241			0.0976203	−	0.995224i	$-0.468877\pi$
$139$	27.1384			0.195241			0.0976203	+	0.995224i	$0.468877\pi$
$140$		0			0
$141$	71.3538	−	71.3538i	0.506056	−	0.506056i
$142$		0			0
$143$	−31.7321	+	2.66025i	−0.221902	+	0.0186032i
$144$		0			0
$145$	150.746	+	150.746i	1.03963	+	1.03963i
$146$		0			0
$147$			59.0141i			0.401456i
$148$		0			0
$149$	−168.224	−	168.224i	−1.12902	−	1.12902i	−0.990336	−	0.138686i	$-0.955712\pi$
$149$	−168.224	−	168.224i	−1.12902	−	1.12902i	−0.138686	−	0.990336i	$-0.544288\pi$
$150$		0			0
$151$	38.4064	−	38.4064i	0.254347	−	0.254347i	−0.568403	−	0.822750i	$-0.692439\pi$
$151$	38.4064	−	38.4064i	0.254347	−	0.254347i	0.822750	+	0.568403i	$0.192439\pi$
$152$		0			0
$153$		−	87.9615i		−	0.574912i
$154$		0			0
$155$		−	255.349i		−	1.64741i
$156$		0			0
$157$	−227.215			−1.44723			−0.723616	−	0.690203i	$-0.757521\pi$
$157$	−227.215			−1.44723			−0.723616	+	0.690203i	$0.757521\pi$
$158$		0			0
$159$	3.96152			0.0249152
$160$		0			0
$161$	80.1051	+	80.1051i	0.497547	+	0.497547i
$162$		0			0
$163$	−12.4449	+	12.4449i	−0.0763489	+	0.0763489i	−0.744250	−	0.667901i	$-0.767193\pi$
$163$	−12.4449	+	12.4449i	−0.0763489	+	0.0763489i	0.667901	+	0.744250i	$0.267193\pi$
$164$		0			0
$165$	−28.3923			−0.172075
$166$		0			0
$167$	−119.512	+	119.512i	−0.715638	+	0.715638i	−0.967709	−	0.252071i	$-0.918888\pi$
$167$	−119.512	+	119.512i	−0.715638	+	0.715638i	0.252071	+	0.967709i	$0.418888\pi$
$168$		0			0
$169$	28.1384	+	166.641i	0.166500	+	0.986042i
$170$		0			0
$171$	33.8038	+	33.8038i	0.197683	+	0.197683i
$172$		0			0
$173$		−	69.3975i		−	0.401141i	−0.979679	−	0.200571i	$-0.935720\pi$
$173$		−	69.3975i		−	0.401141i	0.979679	−	0.200571i	$-0.0642796\pi$
$174$		0			0
$175$	54.0526	+	54.0526i	0.308872	+	0.308872i
$176$		0			0
$177$	−94.6077	+	94.6077i	−0.534507	+	0.534507i
$178$		0			0
$179$		−	165.282i		−	0.923363i	−0.887046	−	0.461682i	$-0.847246\pi$
$179$		−	165.282i		−	0.923363i	0.887046	−	0.461682i	$-0.152754\pi$
$180$		0			0
$181$		−	283.856i		−	1.56827i	−0.620592	−	0.784134i	$-0.713108\pi$
$181$		−	283.856i		−	1.56827i	0.620592	−	0.784134i	$-0.286892\pi$
$182$		0			0
$183$	−12.8616			−0.0702818
$184$		0			0
$185$	−292.028			−1.57853
$186$		0			0
$187$	50.7846	+	50.7846i	0.271575	+	0.271575i
$188$		0			0
$189$	14.1962	−	14.1962i	0.0751119	−	0.0751119i
$190$		0			0
$191$	−9.28203			−0.0485970			−0.0242985	−	0.999705i	$-0.507735\pi$
$191$	−9.28203			−0.0485970			−0.0242985	+	0.999705i	$0.507735\pi$
$192$		0			0
$193$	−21.7180	+	21.7180i	−0.112528	+	0.112528i	−0.761129	−	0.648601i	$-0.775355\pi$
$193$	−21.7180	+	21.7180i	−0.112528	+	0.112528i	0.648601	+	0.761129i	$0.275355\pi$
$194$		0			0
$195$	12.5885	+	150.158i	0.0645562	+	0.770039i
$196$		0			0
$197$	−112.732	−	112.732i	−0.572244	−	0.572244i	0.360511	−	0.932755i	$-0.382602\pi$
$197$	−112.732	−	112.732i	−0.572244	−	0.572244i	−0.932755	+	0.360511i	$0.882602\pi$
$198$		0			0
$199$		−	245.100i		−	1.23166i	−0.787880	−	0.615829i	$-0.788821\pi$
$199$		−	245.100i		−	1.23166i	0.787880	−	0.615829i	$-0.211179\pi$
$200$		0			0
$201$	105.124	+	105.124i	0.523007	+	0.523007i
$202$		0			0
$203$	87.0333	−	87.0333i	0.428736	−	0.428736i
$204$		0			0
$205$		−	136.708i		−	0.666867i
$206$		0			0
$207$			87.9615i			0.424935i
$208$		0			0
$209$	−39.0333			−0.186762
$210$		0			0
$211$	−345.282			−1.63641			−0.818204	−	0.574928i	$-0.805030\pi$
$211$	−345.282			−1.63641			−0.818204	+	0.574928i	$0.805030\pi$
$212$		0			0
$213$	−67.3923	−	67.3923i	−0.316396	−	0.316396i
$214$		0			0
$215$	−119.138	+	119.138i	−0.554132	+	0.554132i
$216$		0			0
$217$	−147.426			−0.679381
$218$		0			0
$219$	69.8372	−	69.8372i	0.318891	−	0.318891i
$220$		0			0
$221$	246.067	−	291.100i	1.11342	−	1.31719i
$222$		0			0
$223$	−170.588	−	170.588i	−0.764971	−	0.764971i	0.212246	−	0.977216i	$-0.431922\pi$
$223$	−170.588	−	170.588i	−0.764971	−	0.764971i	−0.977216	+	0.212246i	$0.931922\pi$
$224$		0			0
$225$			59.3538i			0.263795i
$226$		0			0
$227$	−169.368	−	169.368i	−0.746114	−	0.746114i	0.227633	−	0.973747i	$-0.426901\pi$
$227$	−169.368	−	169.368i	−0.746114	−	0.746114i	−0.973747	+	0.227633i	$0.926901\pi$
$228$		0			0
$229$	141.823	−	141.823i	0.619315	−	0.619315i	−0.326041	−	0.945356i	$-0.605715\pi$
$229$	141.823	−	141.823i	0.619315	−	0.619315i	0.945356	+	0.326041i	$0.105715\pi$
$230$		0			0
$231$			16.3923i			0.0709624i
$232$		0			0
$233$			128.038i			0.549521i	0.961513	+	0.274761i	$0.0885986\pi$
$233$			128.038i			0.549521i	−0.961513	+	0.274761i	$0.911401\pi$
$234$		0			0
$235$	389.885			1.65908
$236$		0			0
$237$	256.708			1.08315
$238$		0			0
$239$	−88.0192	−	88.0192i	−0.368281	−	0.368281i	0.498569	−	0.866850i	$-0.333859\pi$
$239$	−88.0192	−	88.0192i	−0.368281	−	0.368281i	−0.866850	+	0.498569i	$0.833859\pi$
$240$		0			0
$241$	−147.459	+	147.459i	−0.611863	+	0.611863i	−0.943431	−	0.331568i	$-0.892422\pi$
$241$	−147.459	+	147.459i	−0.611863	+	0.611863i	0.331568	+	0.943431i	$0.392422\pi$
$242$		0			0
$243$	15.5885			0.0641500
$244$		0			0
$245$	−161.229	+	161.229i	−0.658079	+	0.658079i
$246$		0			0
$247$	17.3064	+	206.435i	0.0700665	+	0.835767i
$248$		0			0
$249$	−127.823	−	127.823i	−0.513346	−	0.513346i
$250$		0			0
$251$		−	181.377i		−	0.722617i	−0.932446	−	0.361308i	$-0.882330\pi$
$251$		−	181.377i		−	0.722617i	0.932446	−	0.361308i	$-0.117670\pi$
$252$		0			0
$253$	−50.7846	−	50.7846i	−0.200730	−	0.200730i
$254$		0			0
$255$	240.315	−	240.315i	0.942413	−	0.942413i
$256$		0			0
$257$		−	297.664i		−	1.15823i	−0.815247	−	0.579113i	$-0.803399\pi$
$257$		−	297.664i		−	1.15823i	0.815247	−	0.579113i	$-0.196601\pi$
$258$		0			0
$259$			168.603i			0.650975i
$260$		0			0
$261$	95.5692			0.366166
$262$		0			0
$263$	−22.8897			−0.0870332			−0.0435166	−	0.999053i	$-0.513856\pi$
$263$	−22.8897			−0.0870332			−0.0435166	+	0.999053i	$0.513856\pi$
$264$		0			0
$265$	10.8231	+	10.8231i	0.0408418	+	0.0408418i
$266$		0			0
$267$	44.1962	−	44.1962i	0.165529	−	0.165529i
$268$		0			0
$269$	220.774			0.820722			0.410361	−	0.911923i	$-0.365403\pi$
$269$	220.774			0.820722			0.410361	+	0.911923i	$0.365403\pi$
$270$		0			0
$271$	−148.953	+	148.953i	−0.549641	+	0.549641i	−0.926337	−	0.376696i	$-0.877060\pi$
$271$	−148.953	+	148.953i	−0.549641	+	0.549641i	0.376696	+	0.926337i	$0.377060\pi$
$272$		0			0
$273$	86.6936	−	7.26795i	0.317559	−	0.0266225i
$274$		0			0
$275$	−34.2679	−	34.2679i	−0.124611	−	0.124611i
$276$		0			0
$277$			27.2154i			0.0982505i	0.998793	+	0.0491253i	$0.0156434\pi$
$277$			27.2154i			0.0982505i	−0.998793	+	0.0491253i	$0.984357\pi$
$278$		0			0
$279$	−80.9423	−	80.9423i	−0.290116	−	0.290116i
$280$		0			0
$281$	−51.6218	+	51.6218i	−0.183707	+	0.183707i	−0.792969	−	0.609262i	$-0.791466\pi$
$281$	−51.6218	+	51.6218i	−0.183707	+	0.183707i	0.609262	+	0.792969i	$0.291466\pi$
$282$		0			0
$283$			93.8076i			0.331476i	0.986170	+	0.165738i	$0.0530006\pi$
$283$			93.8076i			0.331476i	−0.986170	+	0.165738i	$0.946999\pi$
$284$		0			0
$285$			184.708i			0.648097i
$286$		0			0
$287$	−78.9282			−0.275011
$288$		0			0
$289$	−570.692			−1.97471
$290$		0			0
$291$	−149.014	−	149.014i	−0.512076	−	0.512076i
$292$		0			0
$293$	−120.042	+	120.042i	−0.409701	+	0.409701i	−0.881634	−	0.471934i	$-0.843556\pi$
$293$	−120.042	+	120.042i	−0.409701	+	0.409701i	0.471934	+	0.881634i	$0.343556\pi$
$294$		0			0
$295$	−516.946			−1.75236
$296$		0			0
$297$	−9.00000	+	9.00000i	−0.0303030	+	0.0303030i
$298$		0			0
$299$	−246.067	+	291.100i	−0.822965	+	0.973578i
$300$		0			0
$301$	68.7846	+	68.7846i	0.228520	+	0.228520i
$302$		0			0
$303$		−	181.061i		−	0.597563i
$304$		0			0
$305$	−35.1384	−	35.1384i	−0.115208	−	0.115208i
$306$		0			0
$307$	−7.48849	+	7.48849i	−0.0243925	+	0.0243925i	−0.719198	−	0.694805i	$-0.755491\pi$
$307$	−7.48849	+	7.48849i	−0.0243925	+	0.0243925i	0.694805	+	0.719198i	$0.255491\pi$
$308$		0			0
$309$			63.5307i			0.205601i
$310$		0			0
$311$			289.377i			0.930472i	0.885187	+	0.465236i	$0.154031\pi$
$311$			289.377i			0.930472i	−0.885187	+	0.465236i	$0.845969\pi$
$312$		0			0
$313$	346.841			1.10812			0.554059	−	0.832477i	$-0.313078\pi$
$313$	346.841			1.10812			0.554059	+	0.832477i	$0.313078\pi$
$314$		0			0
$315$	77.5692			0.246251
$316$		0			0
$317$	−97.0192	−	97.0192i	−0.306054	−	0.306054i	0.537322	−	0.843377i	$-0.319436\pi$
$317$	−97.0192	−	97.0192i	−0.306054	−	0.306054i	−0.843377	+	0.537322i	$0.819436\pi$
$318$		0			0
$319$	−55.1769	+	55.1769i	−0.172968	+	0.172968i
$320$		0			0
$321$	213.846			0.666187
$322$		0			0
$323$	330.382	−	330.382i	1.02285	−	1.02285i
$324$		0			0
$325$	−166.038	+	196.426i	−0.510888	+	0.604387i
$326$		0			0
$327$	206.660	+	206.660i	0.631989	+	0.631989i
$328$		0			0
$329$		−	225.100i		−	0.684194i
$330$		0			0
$331$	126.130	+	126.130i	0.381056	+	0.381056i	0.871483	−	0.490427i	$-0.163159\pi$
$331$	126.130	+	126.130i	0.381056	+	0.381056i	−0.490427	+	0.871483i	$0.663159\pi$
$332$		0			0
$333$	−92.5692	+	92.5692i	−0.277986	+	0.277986i
$334$		0			0
$335$			574.410i			1.71466i
$336$		0			0
$337$		−	423.061i		−	1.25538i	−0.778465	−	0.627688i	$-0.784002\pi$
$337$		−	423.061i		−	1.25538i	0.778465	−	0.627688i	$-0.215998\pi$
$338$		0			0
$339$	−319.177			−0.941525
$340$		0			0
$341$	93.4641			0.274088
$342$		0			0
$343$	226.956	+	226.956i	0.661680	+	0.661680i
$344$		0			0
$345$	−240.315	+	240.315i	−0.696566	+	0.696566i
$346$		0			0
$347$	191.867			0.552930			0.276465	−	0.961024i	$-0.410837\pi$
$347$	191.867			0.552930			0.276465	+	0.961024i	$0.410837\pi$
$348$		0			0
$349$	36.7898	−	36.7898i	0.105415	−	0.105415i	−0.652432	−	0.757847i	$-0.726251\pi$
$349$	36.7898	−	36.7898i	0.105415	−	0.105415i	0.757847	+	0.652432i	$0.226251\pi$
$350$		0			0
$351$	51.5885	+	43.6077i	0.146976	+	0.124238i
$352$		0			0
$353$	117.870	+	117.870i	0.333911	+	0.333911i	0.854070	−	0.520159i	$-0.174127\pi$
$353$	117.870	+	117.870i	0.333911	+	0.333911i	−0.520159	+	0.854070i	$0.674127\pi$
$354$		0			0
$355$		−	368.238i		−	1.03729i
$356$		0			0
$357$	−138.746	−	138.746i	−0.388645	−	0.388645i
$358$		0			0
$359$	−13.1192	+	13.1192i	−0.0365437	+	0.0365437i	−0.725143	−	0.688599i	$-0.758226\pi$
$359$	−13.1192	+	13.1192i	−0.0365437	+	0.0365437i	0.688599	+	0.725143i	$0.258226\pi$
$360$		0			0
$361$		−	107.067i		−	0.296583i
$362$		0			0
$363$			199.186i			0.548721i
$364$		0			0
$365$	381.597			1.04547
$366$		0			0
$367$	600.708			1.63681			0.818403	−	0.574645i	$-0.194860\pi$
$367$	600.708			1.63681			0.818403	+	0.574645i	$0.194860\pi$
$368$		0			0
$369$	−43.3346	−	43.3346i	−0.117438	−	0.117438i
$370$		0			0
$371$	6.24871	−	6.24871i	0.0168429	−	0.0168429i
$372$		0			0
$373$	671.836			1.80117			0.900584	−	0.434682i	$-0.143139\pi$
$373$	671.836			1.80117			0.900584	+	0.434682i	$0.143139\pi$
$374$		0			0
$375$	42.7461	−	42.7461i	0.113990	−	0.113990i
$376$		0			0
$377$	316.277	+	267.349i	0.838931	+	0.709148i
$378$		0			0
$379$	−158.799	−	158.799i	−0.418994	−	0.418994i	0.465863	−	0.884857i	$-0.345744\pi$
$379$	−158.799	−	158.799i	−0.418994	−	0.418994i	−0.884857	+	0.465863i	$0.845744\pi$
$380$		0			0
$381$		−	301.310i		−	0.790840i
$382$		0			0
$383$	−233.445	−	233.445i	−0.609517	−	0.609517i	0.333303	−	0.942820i	$-0.391837\pi$
$383$	−233.445	−	233.445i	−0.609517	−	0.609517i	−0.942820	+	0.333303i	$0.891837\pi$
$384$		0			0
$385$	−44.7846	+	44.7846i	−0.116324	+	0.116324i
$386$		0			0
$387$			75.5307i			0.195170i
$388$		0			0
$389$			27.6950i			0.0711953i	0.999366	+	0.0355976i	$0.0113335\pi$
$389$			27.6950i			0.0711953i	−0.999366	+	0.0355976i	$0.988667\pi$
$390$		0			0
$391$	859.692			2.19870
$392$		0			0
$393$	−209.769			−0.533764
$394$		0			0
$395$	701.338	+	701.338i	1.77554	+	1.77554i
$396$		0			0
$397$	−398.692	+	398.692i	−1.00426	+	1.00426i	−0.00427158	+	0.999991i	$0.501360\pi$
$397$	−398.692	+	398.692i	−1.00426	+	1.00426i	−0.999991	+	0.00427158i	$0.998640\pi$
$398$		0			0
$399$	106.641			0.267271
$400$		0			0
$401$	−139.450	+	139.450i	−0.347756	+	0.347756i	−0.859273	−	0.511517i	$-0.829084\pi$
$401$	−139.450	+	139.450i	−0.347756	+	0.347756i	0.511517	+	0.859273i	$0.329084\pi$
$402$		0			0
$403$	−41.4397	−	494.301i	−0.102828	−	1.22655i
$404$		0			0
$405$	42.5885	+	42.5885i	0.105157	+	0.105157i
$406$		0			0
$407$		−	106.890i		−	0.262628i
$408$		0			0
$409$	401.813	+	401.813i	0.982427	+	0.982427i	0.999848	−	0.0174209i	$-0.00554553\pi$
$409$	401.813	+	401.813i	0.982427	+	0.982427i	−0.0174209	+	0.999848i	$0.505546\pi$
$410$		0			0
$411$	−225.258	+	225.258i	−0.548072	+	0.548072i
$412$		0			0
$413$			298.459i			0.722661i
$414$		0			0
$415$		−	698.438i		−	1.68298i
$416$		0			0
$417$	47.0052			0.112722
$418$		0			0
$419$	−139.177			−0.332164			−0.166082	−	0.986112i	$-0.553112\pi$
$419$	−139.177			−0.332164			−0.166082	+	0.986112i	$0.553112\pi$
$420$		0			0
$421$	360.244	+	360.244i	0.855685	+	0.855685i	0.990826	−	0.135141i	$-0.0431487\pi$
$421$	360.244	+	360.244i	0.855685	+	0.855685i	−0.135141	+	0.990826i	$0.543149\pi$
$422$		0			0
$423$	123.588	−	123.588i	0.292171	−	0.292171i
$424$		0			0
$425$	580.095			1.36493
$426$		0			0
$427$	−20.2872	+	20.2872i	−0.0475110	+	0.0475110i
$428$		0			0
$429$	−54.9615	+	4.60770i	−0.128115	+	0.0107405i
$430$		0			0
$431$	−213.522	−	213.522i	−0.495410	−	0.495410i	0.414596	−	0.910006i	$-0.363923\pi$
$431$	−213.522	−	213.522i	−0.495410	−	0.495410i	−0.910006	+	0.414596i	$0.863923\pi$
$432$		0			0
$433$			446.708i			1.03166i	0.856692	+	0.515829i	$0.172516\pi$
$433$			446.708i			1.03166i	−0.856692	+	0.515829i	$0.827484\pi$
$434$		0			0
$435$	261.100	+	261.100i	0.600230	+	0.600230i
$436$		0			0
$437$	−330.382	+	330.382i	−0.756023	+	0.756023i
$438$		0			0
$439$		−	164.238i		−	0.374119i	−0.982349	−	0.187060i	$-0.940104\pi$
$439$		−	164.238i		−	0.374119i	0.982349	−	0.187060i	$-0.0598958\pi$
$440$		0			0
$441$			102.215i			0.231781i
$442$		0			0
$443$	458.736			1.03552			0.517761	−	0.855526i	$-0.326766\pi$
$443$	458.736			1.03552			0.517761	+	0.855526i	$0.326766\pi$
$444$		0			0
$445$	241.492			0.542679
$446$		0			0
$447$	−291.373	−	291.373i	−0.651841	−	0.651841i
$448$		0			0
$449$	221.776	−	221.776i	0.493932	−	0.493932i	−0.415610	−	0.909543i	$-0.636432\pi$
$449$	221.776	−	221.776i	0.493932	−	0.493932i	0.909543	+	0.415610i	$0.136432\pi$
$450$		0			0
$451$	50.0385			0.110950
$452$		0			0
$453$	66.5218	−	66.5218i	0.146847	−	0.146847i
$454$		0			0
$455$	256.708	+	216.995i	0.564193	+	0.476912i
$456$		0			0
$457$	52.9615	+	52.9615i	0.115890	+	0.115890i	0.762673	−	0.646784i	$-0.223886\pi$
$457$	52.9615	+	52.9615i	0.115890	+	0.115890i	−0.646784	+	0.762673i	$0.723886\pi$
$458$		0			0
$459$		−	152.354i		−	0.331926i
$460$		0			0
$461$	67.4500	+	67.4500i	0.146312	+	0.146312i	0.776468	−	0.630156i	$-0.217009\pi$
$461$	67.4500	+	67.4500i	0.146312	+	0.146312i	−0.630156	+	0.776468i	$0.717009\pi$
$462$		0			0
$463$	−549.247	+	549.247i	−1.18628	+	1.18628i	−0.208191	+	0.978088i	$0.566758\pi$
$463$	−549.247	+	549.247i	−1.18628	+	1.18628i	−0.978088	+	0.208191i	$0.933242\pi$
$464$		0			0
$465$		−	442.277i		−	0.951133i
$466$		0			0
$467$			30.1821i			0.0646297i	0.999478	+	0.0323148i	$0.0102879\pi$
$467$			30.1821i			0.0646297i	−0.999478	+	0.0323148i	$0.989712\pi$
$468$		0			0
$469$	331.636			0.707113
$470$		0			0
$471$	−393.549			−0.835560
$472$		0			0
$473$	−43.6077	−	43.6077i	−0.0921939	−	0.0921939i
$474$		0			0
$475$	−222.932	+	222.932i	−0.469330	+	0.469330i
$476$		0			0
$477$	6.86156			0.0143848
$478$		0			0
$479$	258.870	−	258.870i	0.540439	−	0.540439i	−0.383218	−	0.923658i	$-0.625184\pi$
$479$	258.870	−	258.870i	0.540439	−	0.540439i	0.923658	+	0.383218i	$0.125184\pi$
$480$		0			0
$481$	−565.305	+	47.3923i	−1.17527	+	0.0985287i
$482$		0			0
$483$	138.746	+	138.746i	0.287259	+	0.287259i
$484$		0			0
$485$		−	814.228i		−	1.67882i
$486$		0			0
$487$	274.560	+	274.560i	0.563779	+	0.563779i	0.930379	−	0.366600i	$-0.119478\pi$
$487$	274.560	+	274.560i	0.563779	+	0.563779i	−0.366600	+	0.930379i	$0.619478\pi$
$488$		0			0
$489$	−21.5551	+	21.5551i	−0.0440800	+	0.0440800i
$490$		0			0
$491$		−	220.726i		−	0.449543i	−0.974412	−	0.224771i	$-0.927836\pi$
$491$		−	220.726i		−	0.449543i	0.974412	−	0.224771i	$-0.0721635\pi$
$492$		0			0
$493$		−	934.046i		−	1.89462i
$494$		0			0
$495$	−49.1769			−0.0993473
$496$		0			0
$497$	−212.603			−0.427772
$498$		0			0
$499$	625.065	+	625.065i	1.25264	+	1.25264i	0.954534	+	0.298102i	$0.0963534\pi$
$499$	625.065	+	625.065i	1.25264	+	1.25264i	0.298102	+	0.954534i	$0.403647\pi$
$500$		0			0
$501$	−207.000	+	207.000i	−0.413174	+	0.413174i
$502$		0			0
$503$	696.018			1.38373			0.691867	−	0.722025i	$-0.256789\pi$
$503$	696.018			1.38373			0.691867	+	0.722025i	$0.256789\pi$
$504$		0			0
$505$	494.669	−	494.669i	0.979543	−	0.979543i
$506$		0			0
$507$	48.7372	+	288.631i	0.0961286	+	0.569291i
$508$		0			0
$509$	401.678	+	401.678i	0.789151	+	0.789151i	0.981355	−	0.192204i	$-0.0615634\pi$
$509$	401.678	+	401.678i	0.789151	+	0.789151i	−0.192204	+	0.981355i	$0.561563\pi$
$510$		0			0
$511$		−	220.315i		−	0.431146i
$512$		0			0
$513$	58.5500	+	58.5500i	0.114133	+	0.114133i
$514$		0			0
$515$	−173.569	+	173.569i	−0.337028	+	0.337028i
$516$		0			0
$517$			142.708i			0.276030i
$518$		0			0
$519$		−	120.200i		−	0.231599i
$520$		0			0
$521$	−602.651			−1.15672			−0.578360	−	0.815782i	$-0.696307\pi$
$521$	−602.651			−1.15672			−0.578360	+	0.815782i	$0.696307\pi$
$522$		0			0
$523$	−854.677			−1.63418			−0.817091	−	0.576509i	$-0.804414\pi$
$523$	−854.677			−1.63418			−0.817091	+	0.576509i	$0.804414\pi$
$524$		0			0
$525$	93.6218	+	93.6218i	0.178327	+	0.178327i
$526$		0			0
$527$	−791.090	+	791.090i	−1.50112	+	1.50112i
$528$		0			0
$529$	−330.692			−0.625127
$530$		0			0
$531$	−163.865	+	163.865i	−0.308598	+	0.308598i
$532$		0			0
$533$	−22.1858	−	264.637i	−0.0416245	−	0.496505i
$534$		0			0
$535$	584.238	+	584.238i	1.09203	+	1.09203i
$536$		0			0
$537$		−	286.277i		−	0.533104i
$538$		0			0
$539$	−59.0141	−	59.0141i	−0.109488	−	0.109488i
$540$		0			0
$541$	−344.244	+	344.244i	−0.636310	+	0.636310i	−0.949643	−	0.313333i	$-0.898554\pi$
$541$	−344.244	+	344.244i	−0.636310	+	0.636310i	0.313333	+	0.949643i	$0.398554\pi$
$542$		0			0
$543$		−	491.654i		−	0.905440i
$544$		0			0
$545$			1129.21i			2.07195i
$546$		0			0
$547$	−842.200			−1.53967			−0.769835	−	0.638242i	$-0.779662\pi$
$547$	−842.200			−1.53967			−0.769835	+	0.638242i	$0.779662\pi$
$548$		0			0
$549$	−22.2769			−0.0405772
$550$		0			0
$551$	358.956	+	358.956i	0.651463	+	0.651463i
$552$		0			0
$553$	404.918	−	404.918i	0.732220	−	0.732220i
$554$		0			0
$555$	−505.808			−0.911365
$556$		0			0
$557$	99.2576	−	99.2576i	0.178200	−	0.178200i	−0.612370	−	0.790571i	$-0.709784\pi$
$557$	99.2576	−	99.2576i	0.178200	−	0.178200i	0.790571	+	0.612370i	$0.209784\pi$
$558$		0			0
$559$	−211.292	+	249.962i	−0.377983	+	0.447158i
$560$		0			0
$561$	87.9615	+	87.9615i	0.156794	+	0.156794i
$562$		0			0
$563$		−	647.174i		−	1.14951i	−0.818326	−	0.574755i	$-0.805097\pi$
$563$		−	647.174i		−	1.14951i	0.818326	−	0.574755i	$-0.194903\pi$
$564$		0			0
$565$	−872.008	−	872.008i	−1.54338	−	1.54338i
$566$		0			0
$567$	24.5885	−	24.5885i	0.0433659	−	0.0433659i
$568$		0			0
$569$			701.223i			1.23238i	0.787598	+	0.616189i	$0.211324\pi$
$569$			701.223i			1.23238i	−0.787598	+	0.616189i	$0.788676\pi$
$570$		0			0
$571$		−	218.746i		−	0.383093i	−0.981484	−	0.191547i	$-0.938650\pi$
$571$		−	218.746i		−	0.383093i	0.981484	−	0.191547i	$-0.0613503\pi$
$572$		0			0
$573$	−16.0770			−0.0280575
$574$		0			0
$575$	−580.095			−1.00886
$576$		0			0
$577$	−12.3590	−	12.3590i	−0.0214194	−	0.0214194i	0.696316	−	0.717735i	$-0.254821\pi$
$577$	−12.3590	−	12.3590i	−0.0214194	−	0.0214194i	−0.717735	+	0.696316i	$0.754821\pi$
$578$		0			0
$579$	−37.6166	+	37.6166i	−0.0649683	+	0.0649683i
$580$		0			0
$581$	−403.244			−0.694051
$582$		0			0
$583$	−3.96152	+	3.96152i	−0.00679507	+	0.00679507i
$584$		0			0
$585$	21.8038	+	260.081i	0.0372715	+	0.444582i
$586$		0			0
$587$	106.219	+	106.219i	0.180953	+	0.180953i	0.791771	−	0.610818i	$-0.209159\pi$
$587$	106.219	+	106.219i	0.180953	+	0.180953i	−0.610818	+	0.791771i	$0.709159\pi$
$588$		0			0
$589$		−	608.036i		−	1.03232i
$590$		0			0
$591$	−195.258	−	195.258i	−0.330385	−	0.330385i
$592$		0			0
$593$	770.645	−	770.645i	1.29957	−	1.29957i	0.370895	−	0.928675i	$-0.379051\pi$
$593$	770.645	−	770.645i	1.29957	−	1.29957i	0.928675	−	0.370895i	$-0.120949\pi$
$594$		0			0
$595$		−	758.123i		−	1.27416i
$596$		0			0
$597$		−	424.526i		−	0.711098i
$598$		0			0
$599$	130.392			0.217683			0.108842	−	0.994059i	$-0.465286\pi$
$599$	130.392			0.217683			0.108842	+	0.994059i	$0.465286\pi$
$600$		0			0
$601$	−551.108			−0.916984			−0.458492	−	0.888698i	$-0.651610\pi$
$601$	−551.108			−0.916984			−0.458492	+	0.888698i	$0.651610\pi$
$602$		0			0
$603$	182.081	+	182.081i	0.301958	+	0.301958i
$604$		0			0
$605$	−544.186	+	544.186i	−0.899481	+	0.899481i
$606$		0			0
$607$	−63.6001			−0.104778			−0.0523889	−	0.998627i	$-0.516684\pi$
$607$	−63.6001			−0.104778			−0.0523889	+	0.998627i	$0.516684\pi$
$608$		0			0
$609$	150.746	−	150.746i	0.247531	−	0.247531i
$610$		0			0
$611$	754.734	−	63.2731i	1.23524	−	0.103557i
$612$		0			0
$613$	−26.3487	−	26.3487i	−0.0429832	−	0.0429832i	0.685289	−	0.728272i	$-0.259676\pi$
$613$	−26.3487	−	26.3487i	−0.0429832	−	0.0429832i	−0.728272	+	0.685289i	$0.759676\pi$
$614$		0			0
$615$		−	236.785i		−	0.385016i
$616$		0			0
$617$	375.391	+	375.391i	0.608413	+	0.608413i	0.942531	−	0.334118i	$-0.108438\pi$
$617$	375.391	+	375.391i	0.608413	+	0.608413i	−0.334118	+	0.942531i	$0.608438\pi$
$618$		0			0
$619$	−264.070	+	264.070i	−0.426608	+	0.426608i	−0.887471	−	0.460863i	$-0.847540\pi$
$619$	−264.070	+	264.070i	−0.426608	+	0.426608i	0.460863	+	0.887471i	$0.347540\pi$
$620$		0			0
$621$			152.354i			0.245336i
$622$		0			0
$623$		−	139.426i		−	0.223797i
$624$		0			0
$625$	728.184			1.16510
$626$		0			0
$627$	−67.6077			−0.107827
$628$		0			0
$629$	904.726	+	904.726i	1.43836	+	1.43836i
$630$		0			0
$631$	441.594	−	441.594i	0.699831	−	0.699831i	−0.264543	−	0.964374i	$-0.585221\pi$
$631$	441.594	−	441.594i	0.699831	−	0.699831i	0.964374	+	0.264543i	$0.0852210\pi$
$632$		0			0
$633$	−598.046			−0.944780
$634$		0			0
$635$	823.195	−	823.195i	1.29637	−	1.29637i
$636$		0			0
$637$	−285.941	+	338.272i	−0.448887	+	0.531039i
$638$		0			0
$639$	−116.727	−	116.727i	−0.182671	−	0.182671i
$640$		0			0
$641$		−	829.910i		−	1.29471i	−0.762188	−	0.647356i	$-0.775875\pi$
$641$		−	829.910i		−	1.29471i	0.762188	−	0.647356i	$-0.224125\pi$
$642$		0			0
$643$	−424.196	−	424.196i	−0.659714	−	0.659714i	0.295598	−	0.955312i	$-0.404481\pi$
$643$	−424.196	−	424.196i	−0.659714	−	0.659714i	−0.955312	+	0.295598i	$0.904481\pi$
$644$		0			0
$645$	−206.354	+	206.354i	−0.319928	+	0.319928i
$646$		0			0
$647$		−	477.913i		−	0.738660i	−0.929298	−	0.369330i	$-0.879587\pi$
$647$		−	477.913i		−	0.738660i	0.929298	−	0.369330i	$-0.120413\pi$
$648$		0			0
$649$		−	189.215i		−	0.291549i
$650$		0			0
$651$	−255.349			−0.392241
$652$		0			0
$653$	−243.380			−0.372710			−0.186355	−	0.982482i	$-0.559667\pi$
$653$	−243.380			−0.372710			−0.186355	+	0.982482i	$0.559667\pi$
$654$		0			0
$655$	−573.100	−	573.100i	−0.874962	−	0.874962i
$656$		0			0
$657$	120.962	−	120.962i	0.184112	−	0.184112i
$658$		0			0
$659$	190.677			0.289343			0.144671	−	0.989480i	$-0.453788\pi$
$659$	190.677			0.289343			0.144671	+	0.989480i	$0.453788\pi$
$660$		0			0
$661$	−446.149	+	446.149i	−0.674960	+	0.674960i	−0.958855	−	0.283895i	$-0.908373\pi$
$661$	−446.149	+	446.149i	−0.674960	+	0.674960i	0.283895	+	0.958855i	$0.408373\pi$
$662$		0			0
$663$	426.200	−	504.200i	0.642835	−	0.760483i
$664$		0			0
$665$	291.349	+	291.349i	0.438118	+	0.438118i
$666$		0			0
$667$			934.046i			1.40037i
$668$		0			0
$669$	−295.468	−	295.468i	−0.441656	−	0.441656i
$670$		0			0
$671$	12.8616	−	12.8616i	0.0191678	−	0.0191678i
$672$		0			0
$673$		−	176.221i		−	0.261843i	−0.991393	−	0.130922i	$-0.958206\pi$
$673$		−	176.221i		−	0.261843i	0.991393	−	0.130922i	$-0.0417936\pi$
$674$		0			0
$675$			102.804i			0.152302i
$676$		0			0
$677$	672.600			0.993500			0.496750	−	0.867894i	$-0.334527\pi$
$677$	672.600			0.993500			0.496750	+	0.867894i	$0.334527\pi$
$678$		0			0
$679$	−470.095			−0.692334
$680$		0			0
$681$	−293.354	−	293.354i	−0.430769	−	0.430769i
$682$		0			0
$683$	−306.191	+	306.191i	−0.448303	+	0.448303i	−0.894790	−	0.446487i	$-0.852675\pi$
$683$	−306.191	+	306.191i	−0.448303	+	0.448303i	0.446487	+	0.894790i	$0.352675\pi$
$684$		0			0
$685$	−1230.83			−1.79683
$686$		0			0
$687$	245.645	−	245.645i	0.357562	−	0.357562i
$688$		0			0
$689$	22.7077	+	19.1948i	0.0329574	+	0.0278589i
$690$		0			0
$691$	474.865	+	474.865i	0.687215	+	0.687215i	0.961615	−	0.274401i	$-0.0884795\pi$
$691$	474.865	+	474.865i	0.687215	+	0.687215i	−0.274401	+	0.961615i	$0.588479\pi$
$692$		0			0
$693$			28.3923i			0.0409701i
$694$		0			0
$695$	128.420	+	128.420i	0.184778	+	0.184778i
$696$		0			0
$697$	−423.531	+	423.531i	−0.607648	+	0.607648i
$698$		0			0
$699$			221.769i			0.317266i
$700$		0			0
$701$			204.480i			0.291697i	0.989307	+	0.145848i	$0.0465912\pi$
$701$			204.480i			0.291697i	−0.989307	+	0.145848i	$0.953409\pi$
$702$		0			0
$703$	−695.377			−0.989156
$704$		0			0
$705$	675.300			0.957872
$706$		0			0
$707$	−285.597	−	285.597i	−0.403957	−	0.403957i
$708$		0			0
$709$	−629.018	+	629.018i	−0.887190	+	0.887190i	−0.994252	−	0.107062i	$-0.965856\pi$
$709$	−629.018	+	629.018i	−0.887190	+	0.887190i	0.107062	+	0.994252i	$0.465856\pi$
$710$		0			0
$711$	444.631			0.625360
$712$		0			0
$713$	791.090	−	791.090i	1.10952	−	1.10952i
$714$		0			0
$715$	−162.746	−	137.569i	−0.227617	−	0.192405i
$716$		0			0
$717$	−152.454	−	152.454i	−0.212627	−	0.212627i
$718$		0			0
$719$			863.290i			1.20068i	0.799745	+	0.600340i	$0.204968\pi$
$719$			863.290i			1.20068i	−0.799745	+	0.600340i	$0.795032\pi$
$720$		0			0
$721$	100.210	+	100.210i	0.138988	+	0.138988i
$722$		0			0
$723$	−255.406	+	255.406i	−0.353259	+	0.353259i
$724$		0			0
$725$			630.267i			0.869333i
$726$		0			0
$727$		−	605.577i		−	0.832980i	−0.909140	−	0.416490i	$-0.863260\pi$
$727$		−	605.577i		−	0.832980i	0.909140	−	0.416490i	$-0.136740\pi$
$728$		0			0
$729$	27.0000			0.0370370
$730$		0			0
$731$	738.200			1.00985
$732$		0			0
$733$	205.946	+	205.946i	0.280963	+	0.280963i	0.833493	−	0.552530i	$-0.186337\pi$
$733$	205.946	+	205.946i	0.280963	+	0.280963i	−0.552530	+	0.833493i	$0.686337\pi$
$734$		0			0
$735$	−279.258	+	279.258i	−0.379942	+	0.379942i
$736$		0			0
$737$	−210.249			−0.285276
$738$		0			0
$739$	58.7884	−	58.7884i	0.0795513	−	0.0795513i	−0.666212	−	0.745763i	$-0.732085\pi$
$739$	58.7884	−	58.7884i	0.0795513	−	0.0795513i	0.745763	+	0.666212i	$0.232085\pi$
$740$		0			0
$741$	29.9756	+	357.555i	0.0404529	+	0.482531i
$742$		0			0
$743$	885.509	+	885.509i	1.19180	+	1.19180i	0.976561	+	0.215241i	$0.0690536\pi$
$743$	885.509	+	885.509i	1.19180	+	1.19180i	0.215241	+	0.976561i	$0.430946\pi$
$744$		0			0
$745$		−	1592.09i		−	2.13704i
$746$		0			0
$747$	−221.396	−	221.396i	−0.296380	−	0.296380i
$748$		0			0
$749$	337.310	−	337.310i	0.450347	−	0.450347i
$750$		0			0
$751$			19.5720i			0.0260612i	0.999915	+	0.0130306i	$0.00414789\pi$
$751$			19.5720i			0.0260612i	−0.999915	+	0.0130306i	$0.995852\pi$
$752$		0			0
$753$		−	314.154i		−	0.417203i
$754$		0			0
$755$	363.482			0.481433
$756$		0			0
$757$	677.836			0.895424			0.447712	−	0.894178i	$-0.352239\pi$
$757$	677.836			0.895424			0.447712	+	0.894178i	$0.352239\pi$
$758$		0			0
$759$	−87.9615	−	87.9615i	−0.115891	−	0.115891i
$760$		0			0
$761$	−62.4449	+	62.4449i	−0.0820563	+	0.0820563i	−0.746944	−	0.664887i	$-0.768480\pi$
$761$	−62.4449	+	62.4449i	−0.0820563	+	0.0820563i	0.664887	+	0.746944i	$0.268480\pi$
$762$		0			0
$763$	651.951			0.854458
$764$		0			0
$765$	416.238	−	416.238i	0.544102	−	0.544102i
$766$		0			0
$767$	−1000.70	+	83.8935i	−1.30469	+	0.109379i
$768$		0			0
$769$	−71.5950	−	71.5950i	−0.0931014	−	0.0931014i	0.659022	−	0.752124i	$-0.270970\pi$
$769$	−71.5950	−	71.5950i	−0.0931014	−	0.0931014i	−0.752124	+	0.659022i	$0.770970\pi$
$770$		0			0
$771$		−	515.569i		−	0.668702i
$772$		0			0
$773$	−778.817	−	778.817i	−1.00752	−	1.00752i	−0.999971	−	0.00755317i	$-0.997596\pi$
$773$	−778.817	−	778.817i	−1.00752	−	1.00752i	−0.00755317	−	0.999971i	$-0.502404\pi$
$774$		0			0
$775$	533.804	−	533.804i	0.688779	−	0.688779i
$776$		0			0
$777$			292.028i			0.375841i
$778$		0			0
$779$		−	325.528i		−	0.417879i
$780$		0			0
$781$	134.785			0.172580
$782$		0			0
$783$	165.531			0.211406
$784$		0			0
$785$	−1075.19	−	1075.19i	−1.36967	−	1.36967i
$786$		0			0
$787$	−548.883	+	548.883i	−0.697437	+	0.697437i	−0.963857	−	0.266420i	$-0.914159\pi$
$787$	−548.883	+	548.883i	−0.697437	+	0.697437i	0.266420	+	0.963857i	$0.414159\pi$
$788$		0			0
$789$	−39.6462			−0.0502486
$790$		0			0
$791$	−503.454	+	503.454i	−0.636478	+	0.636478i
$792$		0			0
$793$	−73.7231	−	62.3181i	−0.0929674	−	0.0785853i
$794$		0			0
$795$	18.7461	+	18.7461i	0.0235800	+	0.0235800i
$796$		0			0
$797$		−	41.5871i		−	0.0521795i	−0.999660	−	0.0260898i	$-0.991694\pi$
$797$		−	41.5871i		−	0.0521795i	0.999660	−	0.0260898i	$-0.00830557\pi$
$798$		0			0
$799$	−1207.89	−	1207.89i	−1.51175	−	1.51175i
$800$		0			0
$801$	76.5500	−	76.5500i	0.0955680	−	0.0955680i
$802$		0			0
$803$			139.674i			0.173941i
$804$		0			0
$805$			758.123i			0.941768i
$806$		0			0
$807$	382.392			0.473844
$808$		0			0
$809$	−1338.40			−1.65439			−0.827194	−	0.561916i	$-0.810064\pi$
$809$	−1338.40			−1.65439			−0.827194	+	0.561916i	$0.810064\pi$
$810$		0			0
$811$	−257.258	−	257.258i	−0.317210	−	0.317210i	0.530484	−	0.847695i	$-0.322010\pi$
$811$	−257.258	−	257.258i	−0.317210	−	0.317210i	−0.847695	+	0.530484i	$0.822010\pi$
$812$		0			0
$813$	−257.993	+	257.993i	−0.317335	+	0.317335i
$814$		0			0
$815$	−117.779			−0.144515
$816$		0			0
$817$	−283.692	+	283.692i	−0.347236	+	0.347236i
$818$		0			0
$819$	150.158	−	12.5885i	0.183343	−	0.0153705i
$820$		0			0
$821$	−306.873	−	306.873i	−0.373779	−	0.373779i	0.495072	−	0.868852i	$-0.335142\pi$
$821$	−306.873	−	306.873i	−0.373779	−	0.373779i	−0.868852	+	0.495072i	$0.835142\pi$
$822$		0			0
$823$		−	1037.92i		−	1.26114i	−0.776131	−	0.630571i	$-0.782821\pi$
$823$		−	1037.92i		−	1.26114i	0.776131	−	0.630571i	$-0.217179\pi$
$824$		0			0
$825$	−59.3538	−	59.3538i	−0.0719440	−	0.0719440i
$826$		0			0
$827$	−658.153	+	658.153i	−0.795831	+	0.795831i	−0.982435	−	0.186604i	$-0.940252\pi$
$827$	−658.153	+	658.153i	−0.795831	+	0.795831i	0.186604	+	0.982435i	$0.440252\pi$
$828$		0			0
$829$			372.354i			0.449160i	0.974456	+	0.224580i	$0.0721010\pi$
$829$			372.354i			0.449160i	−0.974456	+	0.224580i	$0.927899\pi$
$830$		0			0
$831$			47.1384i			0.0567250i
$832$		0			0
$833$	999.002			1.19928
$834$		0			0
$835$	−1131.07			−1.35457
$836$		0			0
$837$	−140.196	−	140.196i	−0.167498	−	0.167498i
$838$		0			0
$839$	−1018.87	+	1018.87i	−1.21438	+	1.21438i	−0.244813	+	0.969570i	$0.578727\pi$
$839$	−1018.87	+	1018.87i	−1.21438	+	1.21438i	−0.969570	+	0.244813i	$0.921273\pi$
$840$		0			0
$841$	173.831			0.206695
$842$		0			0
$843$	−89.4115	+	89.4115i	−0.106064	+	0.106064i
$844$		0			0
$845$	−655.401	+	921.706i	−0.775623	+	1.09078i
$846$		0			0
$847$	314.186	+	314.186i	0.370940	+	0.370940i
$848$		0			0
$849$			162.480i			0.191378i
$850$		0			0
$851$	−904.726	−	904.726i	−1.06313	−	1.06313i
$852$		0			0
$853$	−366.790	+	366.790i	−0.430000	+	0.430000i	−0.888628	−	0.458628i	$-0.848341\pi$
$853$	−366.790	+	366.790i	−0.430000	+	0.430000i	0.458628	+	0.888628i	$0.348341\pi$
$854$		0			0
$855$			319.923i			0.374179i
$856$		0			0
$857$		−	469.244i		−	0.547542i	−0.961795	−	0.273771i	$-0.911729\pi$
$857$		−	469.244i		−	0.547542i	0.961795	−	0.273771i	$-0.0882711\pi$
$858$		0			0
$859$	1238.66			1.44197			0.720987	−	0.692948i	$-0.243689\pi$
$859$	1238.66			1.44197			0.720987	+	0.692948i	$0.243689\pi$
$860$		0			0
$861$	−136.708			−0.158778
$862$		0			0
$863$	5.71143	+	5.71143i	0.00661811	+	0.00661811i	0.710408	−	0.703790i	$-0.248510\pi$
$863$	5.71143	+	5.71143i	0.00661811	+	0.00661811i	−0.703790	+	0.710408i	$0.748510\pi$
$864$		0			0
$865$	328.392	−	328.392i	0.379644	−	0.379644i
$866$		0			0
$867$	−988.468			−1.14010
$868$		0			0
$869$	−256.708	+	256.708i	−0.295406	+	0.295406i
$870$		0			0
$871$	93.2192	+	1111.94i	0.107025	+	1.27662i
$872$		0			0
$873$	−258.100	−	258.100i	−0.295647	−	0.295647i
$874$		0			0
$875$		−	134.851i		−	0.154116i
$876$		0			0
$877$	591.287	+	591.287i	0.674216	+	0.674216i	0.958685	−	0.284469i	$-0.0918174\pi$
$877$	591.287	+	591.287i	0.674216	+	0.674216i	−0.284469	+	0.958685i	$0.591817\pi$
$878$		0			0
$879$	−207.919	+	207.919i	−0.236541	+	0.236541i
$880$		0			0
$881$		−	663.997i		−	0.753686i	−0.926277	−	0.376843i	$-0.877010\pi$
$881$		−	663.997i		−	0.753686i	0.926277	−	0.376843i	$-0.122990\pi$
$882$		0			0
$883$		−	570.192i		−	0.645744i	−0.946443	−	0.322872i	$-0.895352\pi$
$883$		−	570.192i		−	0.645744i	0.946443	−	0.322872i	$-0.104648\pi$
$884$		0			0
$885$	−895.377			−1.01173
$886$		0			0
$887$	−481.031			−0.542312			−0.271156	−	0.962535i	$-0.587406\pi$
$887$	−481.031			−0.542312			−0.271156	+	0.962535i	$0.587406\pi$
$888$		0			0
$889$	−475.272	−	475.272i	−0.534614	−	0.534614i
$890$		0			0
$891$	−15.5885	+	15.5885i	−0.0174955	+	0.0174955i
$892$		0			0
$893$	928.392			1.03963
$894$		0			0
$895$	782.123	−	782.123i	0.873880	−	0.873880i
$896$		0			0
$897$	−426.200	+	504.200i	−0.475139	+	0.562096i
$898$		0			0
$899$	−859.510	−	859.510i	−0.956074	−	0.956074i
$900$		0			0
$901$		−	67.0615i		−	0.0744301i
$902$		0			0
$903$	119.138	+	119.138i	0.131936	+	0.131936i
$904$		0			0
$905$	1343.22	−	1343.22i	1.48422	−	1.48422i
$906$		0			0
$907$			193.331i			0.213154i	0.994304	+	0.106577i	$0.0339891\pi$
$907$			193.331i			0.213154i	−0.994304	+	0.106577i	$0.966011\pi$
$908$		0			0
$909$		−	313.608i		−	0.345003i
$910$		0			0
$911$	868.743			0.953615			0.476808	−	0.879008i	$-0.341794\pi$
$911$	868.743			0.953615			0.476808	+	0.879008i	$0.341794\pi$
$912$		0			0
$913$	255.646			0.280007
$914$		0			0
$915$	−60.8616	−	60.8616i	−0.0665154	−	0.0665154i
$916$		0			0
$917$	−330.879	+	330.879i	−0.360828	+	0.360828i
$918$		0			0
$919$	−389.108			−0.423403			−0.211702	−	0.977334i	$-0.567900\pi$
$919$	−389.108			−0.423403			−0.211702	+	0.977334i	$0.567900\pi$
$920$		0			0
$921$	−12.9705	+	12.9705i	−0.0140830	+	0.0140830i
$922$		0			0
$923$	−59.7602	−	712.832i	−0.0647456	−	0.772299i
$924$		0			0
$925$	−610.482	−	610.482i	−0.659980	−	0.659980i
$926$		0			0
$927$			110.038i			0.118704i
$928$		0			0
$929$	−429.870	−	429.870i	−0.462724	−	0.462724i	0.436823	−	0.899547i	$-0.356103\pi$
$929$	−429.870	−	429.870i	−0.462724	−	0.462724i	−0.899547	+	0.436823i	$0.856103\pi$
$930$		0			0
$931$	−383.919	+	383.919i	−0.412373	+	0.412373i
$932$		0			0
$933$			501.215i			0.537208i
$934$		0			0
$935$			480.631i			0.514044i
$936$		0			0
$937$	916.441			0.978059			0.489029	−	0.872267i	$-0.337351\pi$
$937$	916.441			0.978059			0.489029	+	0.872267i	$0.337351\pi$
$938$		0			0
$939$	600.746			0.639772
$940$		0			0
$941$	835.601	+	835.601i	0.887993	+	0.887993i	0.994330	−	0.106337i	$-0.0339124\pi$
$941$	835.601	+	835.601i	0.887993	+	0.887993i	−0.106337	+	0.994330i	$0.533912\pi$
$942$		0			0
$943$	423.531	−	423.531i	0.449131	−	0.449131i
$944$		0			0
$945$	134.354			0.142173
$946$		0			0
$947$	378.506	−	378.506i	0.399690	−	0.399690i	−0.478434	−	0.878124i	$-0.658795\pi$
$947$	378.506	−	378.506i	0.399690	−	0.399690i	0.878124	+	0.478434i	$0.158795\pi$
$948$		0			0
$949$	738.692	−	61.9282i	0.778390	−	0.0652563i
$950$		0			0
$951$	−168.042	−	168.042i	−0.176701	−	0.176701i
$952$		0			0
$953$			839.556i			0.880961i	0.897762	+	0.440481i	$0.145192\pi$
$953$			839.556i			0.880961i	−0.897762	+	0.440481i	$0.854808\pi$
$954$		0			0
$955$	−43.9230	−	43.9230i	−0.0459927	−	0.0459927i
$956$		0			0
$957$	−95.5692	+	95.5692i	−0.0998633	+	0.0998633i
$958$		0			0
$959$			710.620i			0.741001i
$960$		0			0
$961$			494.923i			0.515008i
$962$		0			0
$963$	370.392			0.384623
$964$		0			0
$965$	−205.541			−0.212996
$966$		0			0
$967$	375.745	+	375.745i	0.388567	+	0.388567i	0.874176	−	0.485609i	$-0.161402\pi$
$967$	375.745	+	375.745i	0.388567	+	0.388567i	−0.485609	+	0.874176i	$0.661402\pi$
$968$		0			0
$969$	572.238	−	572.238i	0.590545	−	0.590545i
$970$		0			0
$971$	−134.523			−0.138541			−0.0692703	−	0.997598i	$-0.522067\pi$
$971$	−134.523			−0.138541			−0.0692703	+	0.997598i	$0.522067\pi$
$972$		0			0
$973$	74.1436	−	74.1436i	0.0762010	−	0.0762010i
$974$		0			0
$975$	−287.587	+	340.219i	−0.294961	+	0.348943i
$976$		0			0
$977$	708.060	+	708.060i	0.724729	+	0.724729i	0.969565	−	0.244836i	$-0.0787340\pi$
$977$	708.060	+	708.060i	0.724729	+	0.724729i	−0.244836	+	0.969565i	$0.578734\pi$
$978$		0			0
$979$			88.3923i			0.0902884i
$980$		0			0
$981$	357.946	+	357.946i	0.364879	+	0.364879i
$982$		0			0
$983$	904.881	−	904.881i	0.920530	−	0.920530i	−0.0765369	−	0.997067i	$-0.524386\pi$
$983$	904.881	−	904.881i	0.920530	−	0.920530i	0.997067	+	0.0765369i	$0.0243863\pi$
$984$		0			0
$985$		−	1066.91i		−	1.08315i
$986$		0			0
$987$		−	389.885i		−	0.395020i
$988$		0			0
$989$	−738.200			−0.746410
$990$		0			0
$991$	−29.0155			−0.0292790			−0.0146395	−	0.999893i	$-0.504660\pi$
$991$	−29.0155			−0.0292790			−0.0146395	+	0.999893i	$0.504660\pi$
$992$		0			0
$993$	218.463	+	218.463i	0.220003	+	0.220003i
$994$		0			0
$995$	1159.83	−	1159.83i	1.16565	−	1.16565i
$996$		0			0
$997$	336.123			0.337134			0.168567	−	0.985690i	$-0.446086\pi$
$997$	336.123			0.337134			0.168567	+	0.985690i	$0.446086\pi$
$998$		0			0
$999$	−160.335	+	160.335i	−0.160495	+	0.160495i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.3.ba.a.385.2		4
4.3	odd	2		78.3.f.a.73.1	yes	4
12.11	even	2		234.3.i.b.73.1		4
13.5	odd	4	inner	624.3.ba.a.577.2		4
52.31	even	4		78.3.f.a.31.1	✓	4
52.47	even	4		1014.3.f.a.577.1		4
52.51	odd	2		1014.3.f.a.775.1		4
156.83	odd	4		234.3.i.b.109.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.3.f.a.31.1	✓	4	52.31	even	4
78.3.f.a.73.1	yes	4	4.3	odd	2
234.3.i.b.73.1		4	12.11	even	2
234.3.i.b.109.1		4	156.83	odd	4
624.3.ba.a.385.2		4	1.1	even	1	trivial
624.3.ba.a.577.2		4	13.5	odd	4	inner
1014.3.f.a.577.1		4	52.47	even	4
1014.3.f.a.775.1		4	52.51	odd	2

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	624.ba (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+1.73205 q^{3} +(4.73205 + 4.73205i) q^{5} +(2.73205 - 2.73205i) q^{7} +3.00000 q^{9} +(-1.73205 + 1.73205i) q^{11} +(9.92820 + 8.39230i) q^{13} +(8.19615 + 8.19615i) q^{15} -29.3205i q^{17} +(11.2679 + 11.2679i) q^{19} +(4.73205 - 4.73205i) q^{21} +29.3205i q^{23} +19.7846i q^{25} +5.19615 q^{27} +31.8564 q^{29} +(-26.9808 - 26.9808i) q^{31} +(-3.00000 + 3.00000i) q^{33} +25.8564 q^{35} +(-30.8564 + 30.8564i) q^{37} +(17.1962 + 14.5359i) q^{39} +(-14.4449 - 14.4449i) q^{41} +25.1769i q^{43} +(14.1962 + 14.1962i) q^{45} +(41.1962 - 41.1962i) q^{47} +34.0718i q^{49} -50.7846i q^{51} +2.28719 q^{53} -16.3923 q^{55} +(19.5167 + 19.5167i) q^{57} +(-54.6218 + 54.6218i) q^{59} -7.42563 q^{61} +(8.19615 - 8.19615i) q^{63} +(7.26795 + 86.6936i) q^{65} +(60.6936 + 60.6936i) q^{67} +50.7846i q^{69} +(-38.9090 - 38.9090i) q^{71} +(40.3205 - 40.3205i) q^{73} +34.2679i q^{75} +9.46410i q^{77} +148.210 q^{79} +9.00000 q^{81} +(-73.7987 - 73.7987i) q^{83} +(138.746 - 138.746i) q^{85} +55.1769 q^{87} +(25.5167 - 25.5167i) q^{89} +(50.0526 - 4.19615i) q^{91} +(-46.7321 - 46.7321i) q^{93} +106.641i q^{95} +(-86.0333 - 86.0333i) q^{97} +(-5.19615 + 5.19615i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 12 q^{5} + 4 q^{7} + 12 q^{9} + 12 q^{13} + 12 q^{15} + 52 q^{19} + 12 q^{21} + 72 q^{29} - 4 q^{31} - 12 q^{33} + 48 q^{35} - 68 q^{37} + 48 q^{39} + 60 q^{41} + 36 q^{45} + 144 q^{47} + 120 q^{53}+ \cdots - 164 q^{97}+O(q^{100}) \) 4 * q + 12 * q^5 + 4 * q^7 + 12 * q^9 + 12 * q^13 + 12 * q^15 + 52 * q^19 + 12 * q^21 + 72 * q^29 - 4 * q^31 - 12 * q^33 + 48 * q^35 - 68 * q^37 + 48 * q^39 + 60 * q^41 + 36 * q^45 + 144 * q^47 + 120 * q^53 - 24 * q^55 - 12 * q^57 + 24 * q^59 + 192 * q^61 + 12 * q^63 + 36 * q^65 + 28 * q^67 - 24 * q^71 + 92 * q^73 + 288 * q^79 + 36 * q^81 + 72 * q^83 + 264 * q^85 + 96 * q^87 + 12 * q^89 + 124 * q^91 - 180 * q^93 - 164 * q^97

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