LMFDB - Embedded newform 384.4.j.a.289.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [384,4,Mod(97,384)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("384.97"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(384, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.j (of order $4$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [24,0,0,0,0,0,0,0,0,0,-40] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$22.6567334422$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 48)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			289.7
Character	$\chi$	$=$	384.289
Dual form			384.4.j.a.97.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(2.12132 + 2.12132i) q^{3} +(11.8955 - 11.8955i) q^{5} -0.485059i q^{7} +9.00000i q^{9} +(-30.9469 + 30.9469i) q^{11} +(18.4511 + 18.4511i) q^{13} +50.4682 q^{15} +135.964 q^{17} +(65.8832 + 65.8832i) q^{19} +(1.02897 - 1.02897i) q^{21} -128.108i q^{23} -158.004i q^{25} +(-19.0919 + 19.0919i) q^{27} +(-6.64817 - 6.64817i) q^{29} +15.1323 q^{31} -131.297 q^{33} +(-5.77001 - 5.77001i) q^{35} +(-51.4363 + 51.4363i) q^{37} +78.2813i q^{39} -410.253i q^{41} +(69.9911 - 69.9911i) q^{43} +(107.059 + 107.059i) q^{45} +487.269 q^{47} +342.765 q^{49} +(288.423 + 288.423i) q^{51} +(217.138 - 217.138i) q^{53} +736.257i q^{55} +279.519i q^{57} +(-293.944 + 293.944i) q^{59} +(-207.076 - 207.076i) q^{61} +4.36553 q^{63} +438.968 q^{65} +(284.693 + 284.693i) q^{67} +(271.759 - 271.759i) q^{69} +614.701i q^{71} -486.171i q^{73} +(335.178 - 335.178i) q^{75} +(15.0111 + 15.0111i) q^{77} -960.347 q^{79} -81.0000 q^{81} +(463.472 + 463.472i) q^{83} +(1617.36 - 1617.36i) q^{85} -28.2058i q^{87} +1278.38i q^{89} +(8.94987 - 8.94987i) q^{91} +(32.1004 + 32.1004i) q^{93} +1567.42 q^{95} -994.918 q^{97} +(-278.523 - 278.523i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} - 16 q^{37} + 1240 q^{43} - 1176 q^{49} + 744 q^{51} - 752 q^{53} - 1376 q^{59} + 912 q^{61} + 504 q^{63} + 976 q^{65} - 2256 q^{67}+ \cdots - 360 q^{99}+O(q^{100}) $ 24 * q - 40 * q^11 - 120 * q^15 + 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 - 16 * q^37 + 1240 * q^43 - 1176 * q^49 + 744 * q^51 - 752 * q^53 - 1376 * q^59 + 912 * q^61 + 504 * q^63 + 976 * q^65 - 2256 * q^67 + 528 * q^69 + 1104 * q^75 - 1904 * q^77 - 5992 * q^79 - 1944 * q^81 + 2680 * q^83 + 240 * q^85 - 3496 * q^91 + 7728 * q^95 - 360 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/384\mathbb{Z}\right)^\times$.

$n$	$127$	$133$	$257$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$

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
$3$	2.12132	+	2.12132i	0.408248	+	0.408248i
$3$	2.12132	+	2.12132i	0.408248	+	0.408248i
$4$		0			0
$5$	11.8955	−	11.8955i	1.06396	−	1.06396i	0.0661534	−	0.997809i	$-0.478927\pi$
$5$	11.8955	−	11.8955i	1.06396	−	1.06396i	0.997809	−	0.0661534i	$-0.0210727\pi$
$6$		0			0
$7$		−	0.485059i		−	0.0261907i	−0.999914	−	0.0130954i	$-0.995831\pi$
$7$		−	0.485059i		−	0.0261907i	0.999914	−	0.0130954i	$-0.00416851\pi$
$8$		0			0
$9$			9.00000i			0.333333i
$10$		0			0
$11$	−30.9469	+	30.9469i	−0.848260	+	0.848260i	−0.989916	−	0.141656i	$-0.954757\pi$
$11$	−30.9469	+	30.9469i	−0.848260	+	0.848260i	0.141656	+	0.989916i	$0.454757\pi$
$12$		0			0
$13$	18.4511	+	18.4511i	0.393647	+	0.393647i	0.875985	−	0.482338i	$-0.160212\pi$
$13$	18.4511	+	18.4511i	0.393647	+	0.393647i	−0.482338	+	0.875985i	$0.660212\pi$
$14$		0			0
$15$	50.4682			0.868722
$16$		0			0
$17$	135.964			1.93977			0.969886	−	0.243558i	$-0.0783148\pi$
$17$	135.964			1.93977			0.969886	+	0.243558i	$0.0783148\pi$
$18$		0			0
$19$	65.8832	+	65.8832i	0.795508	+	0.795508i	0.982384	−	0.186876i	$-0.0598362\pi$
$19$	65.8832	+	65.8832i	0.795508	+	0.795508i	−0.186876	+	0.982384i	$0.559836\pi$
$20$		0			0
$21$	1.02897	−	1.02897i	0.0106923	−	0.0106923i
$22$		0			0
$23$		−	128.108i		−	1.16141i	−0.814114	−	0.580705i	$-0.802777\pi$
$23$		−	128.108i		−	1.16141i	0.814114	−	0.580705i	$-0.197223\pi$
$24$		0			0
$25$		−	158.004i		−	1.26403i
$26$		0			0
$27$	−19.0919	+	19.0919i	−0.136083	+	0.136083i
$28$		0			0
$29$	−6.64817	−	6.64817i	−0.0425702	−	0.0425702i	0.685501	−	0.728071i	$-0.259583\pi$
$29$	−6.64817	−	6.64817i	−0.0425702	−	0.0425702i	−0.728071	+	0.685501i	$0.759583\pi$
$30$		0			0
$31$	15.1323			0.0876722			0.0438361	−	0.999039i	$-0.486042\pi$
$31$	15.1323			0.0876722			0.0438361	+	0.999039i	$0.486042\pi$
$32$		0			0
$33$	−131.297			−0.692601
$34$		0			0
$35$	−5.77001	−	5.77001i	−0.0278660	−	0.0278660i
$36$		0			0
$37$	−51.4363	+	51.4363i	−0.228542	+	0.228542i	−0.812084	−	0.583541i	$-0.801667\pi$
$37$	−51.4363	+	51.4363i	−0.228542	+	0.228542i	0.583541	+	0.812084i	$0.301667\pi$
$38$		0			0
$39$			78.2813i			0.321411i
$40$		0			0
$41$		−	410.253i		−	1.56270i	−0.624092	−	0.781350i	$-0.714531\pi$
$41$		−	410.253i		−	1.56270i	0.624092	−	0.781350i	$-0.285469\pi$
$42$		0			0
$43$	69.9911	−	69.9911i	0.248222	−	0.248222i	−0.572019	−	0.820241i	$-0.693839\pi$
$43$	69.9911	−	69.9911i	0.248222	−	0.248222i	0.820241	+	0.572019i	$0.193839\pi$
$44$		0			0
$45$	107.059	+	107.059i	0.354654	+	0.354654i
$46$		0			0
$47$	487.269			1.51225			0.756123	−	0.654430i	$-0.227091\pi$
$47$	487.269			1.51225			0.756123	+	0.654430i	$0.227091\pi$
$48$		0			0
$49$	342.765			0.999314
$50$		0			0
$51$	288.423	+	288.423i	0.791909	+	0.791909i
$52$		0			0
$53$	217.138	−	217.138i	0.562757	−	0.562757i	−0.367333	−	0.930090i	$-0.619729\pi$
$53$	217.138	−	217.138i	0.562757	−	0.562757i	0.930090	+	0.367333i	$0.119729\pi$
$54$		0			0
$55$			736.257i			1.80503i
$56$		0			0
$57$			279.519i			0.649529i
$58$		0			0
$59$	−293.944	+	293.944i	−0.648613	+	0.648613i	−0.952658	−	0.304044i	$-0.901663\pi$
$59$	−293.944	+	293.944i	−0.648613	+	0.648613i	0.304044	+	0.952658i	$0.401663\pi$
$60$		0			0
$61$	−207.076	−	207.076i	−0.434646	−	0.434646i	0.455559	−	0.890205i	$-0.349439\pi$
$61$	−207.076	−	207.076i	−0.434646	−	0.434646i	−0.890205	+	0.455559i	$0.849439\pi$
$62$		0			0
$63$	4.36553			0.00873025
$64$		0			0
$65$	438.968			0.837651
$66$		0			0
$67$	284.693	+	284.693i	0.519115	+	0.519115i	0.917304	−	0.398188i	$-0.130361\pi$
$67$	284.693	+	284.693i	0.519115	+	0.519115i	−0.398188	+	0.917304i	$0.630361\pi$
$68$		0			0
$69$	271.759	−	271.759i	0.474144	−	0.474144i
$70$		0			0
$71$			614.701i			1.02749i	0.857944	+	0.513744i	$0.171742\pi$
$71$			614.701i			1.02749i	−0.857944	+	0.513744i	$0.828258\pi$
$72$		0			0
$73$		−	486.171i		−	0.779479i	−0.920925	−	0.389740i	$-0.872565\pi$
$73$		−	486.171i		−	0.779479i	0.920925	−	0.389740i	$-0.127435\pi$
$74$		0			0
$75$	335.178	−	335.178i	0.516040	−	0.516040i
$76$		0			0
$77$	15.0111	+	15.0111i	0.0222166	+	0.0222166i
$78$		0			0
$79$	−960.347			−1.36769			−0.683845	−	0.729628i	$-0.739693\pi$
$79$	−960.347			−1.36769			−0.683845	+	0.729628i	$0.739693\pi$
$80$		0			0
$81$	−81.0000			−0.111111
$82$		0			0
$83$	463.472	+	463.472i	0.612923	+	0.612923i	0.943707	−	0.330784i	$-0.107313\pi$
$83$	463.472	+	463.472i	0.612923	+	0.612923i	−0.330784	+	0.943707i	$0.607313\pi$
$84$		0			0
$85$	1617.36	−	1617.36i	2.06385	−	2.06385i
$86$		0			0
$87$		−	28.2058i		−	0.0347584i
$88$		0			0
$89$			1278.38i			1.52257i	0.648420	+	0.761283i	$0.275430\pi$
$89$			1278.38i			1.52257i	−0.648420	+	0.761283i	$0.724570\pi$
$90$		0			0
$91$	8.94987	−	8.94987i	0.0103099	−	0.0103099i
$92$		0			0
$93$	32.1004	+	32.1004i	0.0357920	+	0.0357920i
$94$		0			0
$95$	1567.42			1.69278
$96$		0			0
$97$	−994.918			−1.04143			−0.520714	−	0.853731i	$-0.674334\pi$
$97$	−994.918			−1.04143			−0.520714	+	0.853731i	$0.674334\pi$
$98$		0			0
$99$	−278.523	−	278.523i	−0.282753	−	0.282753i
$100$		0			0
$101$	−1317.80	+	1317.80i	−1.29828	+	1.29828i	−0.368750	+	0.929529i	$0.620214\pi$
$101$	−1317.80	+	1317.80i	−1.29828	+	1.29828i	−0.929529	+	0.368750i	$0.879786\pi$
$102$		0			0
$103$		−	1345.19i		−	1.28685i	−0.765511	−	0.643423i	$-0.777514\pi$
$103$		−	1345.19i		−	1.28685i	0.765511	−	0.643423i	$-0.222486\pi$
$104$		0			0
$105$		−	24.4801i		−	0.0227525i
$106$		0			0
$107$	437.734	−	437.734i	0.395489	−	0.395489i	−0.481149	−	0.876639i	$-0.659781\pi$
$107$	437.734	−	437.734i	0.395489	−	0.395489i	0.876639	+	0.481149i	$0.159781\pi$
$108$		0			0
$109$	−173.959	−	173.959i	−0.152864	−	0.152864i	0.626532	−	0.779396i	$-0.284474\pi$
$109$	−173.959	−	173.959i	−0.152864	−	0.152864i	−0.779396	+	0.626532i	$0.784474\pi$
$110$		0			0
$111$	−218.226			−0.186604
$112$		0			0
$113$	312.932			0.260514			0.130257	−	0.991480i	$-0.458420\pi$
$113$	312.932			0.260514			0.130257	+	0.991480i	$0.458420\pi$
$114$		0			0
$115$	−1523.91	−	1523.91i	−1.23570	−	1.23570i
$116$		0			0
$117$	−166.060	+	166.060i	−0.131216	+	0.131216i
$118$		0			0
$119$		−	65.9507i		−	0.0508041i
$120$		0			0
$121$		−	584.427i		−	0.439089i
$122$		0			0
$123$	870.278	−	870.278i	0.637970	−	0.637970i
$124$		0			0
$125$	−392.601	−	392.601i	−0.280922	−	0.280922i
$126$		0			0
$127$	−1457.94			−1.01867			−0.509335	−	0.860569i	$-0.670108\pi$
$127$	−1457.94			−1.01867			−0.509335	+	0.860569i	$0.670108\pi$
$128$		0			0
$129$	296.947			0.202672
$130$		0			0
$131$	−203.507	−	203.507i	−0.135729	−	0.135729i	0.635978	−	0.771707i	$-0.280597\pi$
$131$	−203.507	−	203.507i	−0.135729	−	0.135729i	−0.771707	+	0.635978i	$0.780597\pi$
$132$		0			0
$133$	31.9573	−	31.9573i	0.0208349	−	0.0208349i
$134$		0			0
$135$			454.214i			0.289574i
$136$		0			0
$137$		−	432.979i		−	0.270014i	−0.990845	−	0.135007i	$-0.956894\pi$
$137$		−	432.979i		−	0.270014i	0.990845	−	0.135007i	$-0.0431056\pi$
$138$		0			0
$139$	−1058.62	+	1058.62i	−0.645975	+	0.645975i	−0.952018	−	0.306043i	$-0.900995\pi$
$139$	−1058.62	+	1058.62i	−0.645975	+	0.645975i	0.306043	+	0.952018i	$0.400995\pi$
$140$		0			0
$141$	1033.65	+	1033.65i	0.617372	+	0.617372i
$142$		0			0
$143$	−1142.01			−0.667829
$144$		0			0
$145$	−158.166			−0.0905861
$146$		0			0
$147$	727.114	+	727.114i	0.407968	+	0.407968i
$148$		0			0
$149$	−1703.13	+	1703.13i	−0.936414	+	0.936414i	−0.998096	−	0.0616815i	$-0.980354\pi$
$149$	−1703.13	+	1703.13i	−0.936414	+	0.936414i	0.0616815	+	0.998096i	$0.480354\pi$
$150$		0			0
$151$			541.560i			0.291864i	0.989295	+	0.145932i	$0.0466182\pi$
$151$			541.560i			0.291864i	−0.989295	+	0.145932i	$0.953382\pi$
$152$		0			0
$153$			1223.68i			0.646591i
$154$		0			0
$155$	180.006	−	180.006i	0.0932800	−	0.0932800i
$156$		0			0
$157$	8.94805	+	8.94805i	0.00454861	+	0.00454861i	0.709377	−	0.704829i	$-0.248976\pi$
$157$	8.94805	+	8.94805i	0.00454861	+	0.00454861i	−0.704829	+	0.709377i	$0.748976\pi$
$158$		0			0
$159$	921.236			0.459489
$160$		0			0
$161$	−62.1402			−0.0304182
$162$		0			0
$163$	−1308.41	−	1308.41i	−0.628727	−	0.628727i	0.319021	−	0.947748i	$-0.396646\pi$
$163$	−1308.41	−	1308.41i	−0.628727	−	0.628727i	−0.947748	+	0.319021i	$0.896646\pi$
$164$		0			0
$165$	−1561.84	+	1561.84i	−0.736902	+	0.736902i
$166$		0			0
$167$		−	2374.12i		−	1.10009i	−0.835135	−	0.550045i	$-0.814610\pi$
$167$		−	2374.12i		−	1.10009i	0.835135	−	0.550045i	$-0.185390\pi$
$168$		0			0
$169$		−	1516.12i		−	0.690084i
$170$		0			0
$171$	−592.949	+	592.949i	−0.265169	+	0.265169i
$172$		0			0
$173$	1310.49	+	1310.49i	0.575922	+	0.575922i	0.933777	−	0.357855i	$-0.116492\pi$
$173$	1310.49	+	1310.49i	0.575922	+	0.575922i	−0.357855	+	0.933777i	$0.616492\pi$
$174$		0			0
$175$	−76.6414			−0.0331060
$176$		0			0
$177$	−1247.10			−0.529591
$178$		0			0
$179$	248.652	+	248.652i	0.103828	+	0.103828i	0.757112	−	0.653285i	$-0.226610\pi$
$179$	248.652	+	248.652i	0.103828	+	0.103828i	−0.653285	+	0.757112i	$0.726610\pi$
$180$		0			0
$181$	−1152.86	+	1152.86i	−0.473434	+	0.473434i	−0.903024	−	0.429590i	$-0.858658\pi$
$181$	−1152.86	+	1152.86i	−0.473434	+	0.473434i	0.429590	+	0.903024i	$0.358658\pi$
$182$		0			0
$183$		−	878.550i		−	0.354887i
$184$		0			0
$185$			1223.72i			0.486321i
$186$		0			0
$187$	−4207.67	+	4207.67i	−1.64543	+	1.64543i
$188$		0			0
$189$	9.26070	+	9.26070i	0.00356411	+	0.00356411i
$190$		0			0
$191$	457.697			0.173392			0.0866959	−	0.996235i	$-0.472369\pi$
$191$	457.697			0.173392			0.0866959	+	0.996235i	$0.472369\pi$
$192$		0			0
$193$	−61.7567			−0.0230329			−0.0115164	−	0.999934i	$-0.503666\pi$
$193$	−61.7567			−0.0230329			−0.0115164	+	0.999934i	$0.503666\pi$
$194$		0			0
$195$	931.192	+	931.192i	0.341970	+	0.341970i
$196$		0			0
$197$	613.568	−	613.568i	0.221903	−	0.221903i	−0.587396	−	0.809299i	$-0.699847\pi$
$197$	613.568	−	613.568i	0.221903	−	0.221903i	0.809299	+	0.587396i	$0.199847\pi$
$198$		0			0
$199$		−	3343.30i		−	1.19095i	−0.803372	−	0.595477i	$-0.796963\pi$
$199$		−	3343.30i		−	1.19095i	0.803372	−	0.595477i	$-0.203037\pi$
$200$		0			0
$201$			1207.85i			0.423856i
$202$		0			0
$203$	−3.22476	+	3.22476i	−0.00111494	+	0.00111494i
$204$		0			0
$205$	−4880.15	−	4880.15i	−1.66266	−	1.66266i
$206$		0			0
$207$	1152.98			0.387137
$208$		0			0
$209$	−4077.77			−1.34959
$210$		0			0
$211$	−445.986	−	445.986i	−0.145512	−	0.145512i	0.630598	−	0.776110i	$-0.282810\pi$
$211$	−445.986	−	445.986i	−0.145512	−	0.145512i	−0.776110	+	0.630598i	$0.782810\pi$
$212$		0			0
$213$	−1303.98	+	1303.98i	−0.419470	+	0.419470i
$214$		0			0
$215$		−	1665.15i		−	0.528198i
$216$		0			0
$217$		−	7.34006i		−	0.00229620i
$218$		0			0
$219$	1031.32	−	1031.32i	0.318221	−	0.318221i
$220$		0			0
$221$	2508.68	+	2508.68i	0.763585	+	0.763585i
$222$		0			0
$223$	−6100.84			−1.83203			−0.916014	−	0.401146i	$-0.868612\pi$
$223$	−6100.84			−1.83203			−0.916014	+	0.401146i	$0.868612\pi$
$224$		0			0
$225$	1422.04			0.421345
$226$		0			0
$227$	−3652.90	−	3652.90i	−1.06807	−	1.06807i	−0.997507	−	0.0705613i	$-0.977521\pi$
$227$	−3652.90	−	3652.90i	−1.06807	−	1.06807i	−0.0705613	−	0.997507i	$-0.522479\pi$
$228$		0			0
$229$	−1707.86	+	1707.86i	−0.492832	+	0.492832i	−0.909197	−	0.416365i	$-0.863304\pi$
$229$	−1707.86	+	1707.86i	−0.492832	+	0.492832i	0.416365	+	0.909197i	$0.363304\pi$
$230$		0			0
$231$			63.6867i			0.0181397i
$232$		0			0
$233$			3019.27i			0.848924i	0.905446	+	0.424462i	$0.139537\pi$
$233$			3019.27i			0.848924i	−0.905446	+	0.424462i	$0.860463\pi$
$234$		0			0
$235$	5796.29	−	5796.29i	1.60897	−	1.60897i
$236$		0			0
$237$	−2037.20	−	2037.20i	−0.558357	−	0.558357i
$238$		0			0
$239$	−2964.22			−0.802256			−0.401128	−	0.916022i	$-0.631382\pi$
$239$	−2964.22			−0.802256			−0.401128	+	0.916022i	$0.631382\pi$
$240$		0			0
$241$	−2606.84			−0.696769			−0.348384	−	0.937352i	$-0.613270\pi$
$241$	−2606.84			−0.696769			−0.348384	+	0.937352i	$0.613270\pi$
$242$		0			0
$243$	−171.827	−	171.827i	−0.0453609	−	0.0453609i
$244$		0			0
$245$	4077.35	−	4077.35i	1.06323	−	1.06323i
$246$		0			0
$247$			2431.23i			0.626298i
$248$		0			0
$249$			1966.34i			0.500450i
$250$		0			0
$251$	3018.73	−	3018.73i	0.759126	−	0.759126i	−0.217037	−	0.976163i	$-0.569639\pi$
$251$	3018.73	−	3018.73i	0.759126	−	0.759126i	0.976163	+	0.217037i	$0.0696394\pi$
$252$		0			0
$253$	3964.56	+	3964.56i	0.985177	+	0.985177i
$254$		0			0
$255$	6861.86			1.68512
$256$		0			0
$257$	1988.50			0.482644			0.241322	−	0.970445i	$-0.422419\pi$
$257$	1988.50			0.482644			0.241322	+	0.970445i	$0.422419\pi$
$258$		0			0
$259$	24.9496	+	24.9496i	0.00598570	+	0.00598570i
$260$		0			0
$261$	59.8336	−	59.8336i	0.0141901	−	0.0141901i
$262$		0			0
$263$			6049.85i			1.41844i	0.704987	+	0.709220i	$0.250953\pi$
$263$			6049.85i			1.41844i	−0.704987	+	0.709220i	$0.749047\pi$
$264$		0			0
$265$		−	5165.90i		−	1.19751i
$266$		0			0
$267$	−2711.86	+	2711.86i	−0.621585	+	0.621585i
$268$		0			0
$269$	−3883.56	−	3883.56i	−0.880242	−	0.880242i	0.113317	−	0.993559i	$-0.463852\pi$
$269$	−3883.56	−	3883.56i	−0.880242	−	0.880242i	−0.993559	+	0.113317i	$0.963852\pi$
$270$		0			0
$271$	3321.60			0.744549			0.372275	−	0.928123i	$-0.378578\pi$
$271$	3321.60			0.744549			0.372275	+	0.928123i	$0.378578\pi$
$272$		0			0
$273$	37.9711			0.00841800
$274$		0			0
$275$	4889.75	+	4889.75i	1.07223	+	1.07223i
$276$		0			0
$277$	3603.29	−	3603.29i	0.781591	−	0.781591i	−0.198508	−	0.980099i	$-0.563610\pi$
$277$	3603.29	−	3603.29i	0.781591	−	0.781591i	0.980099	+	0.198508i	$0.0636096\pi$
$278$		0			0
$279$			136.191i			0.0292241i
$280$		0			0
$281$			823.661i			0.174859i	0.996171	+	0.0874297i	$0.0278653\pi$
$281$			823.661i			0.174859i	−0.996171	+	0.0874297i	$0.972135\pi$
$282$		0			0
$283$	−2003.74	+	2003.74i	−0.420883	+	0.420883i	−0.885508	−	0.464624i	$-0.846189\pi$
$283$	−2003.74	+	2003.74i	−0.420883	+	0.420883i	0.464624	+	0.885508i	$0.346189\pi$
$284$		0			0
$285$	3325.01	+	3325.01i	0.691075	+	0.691075i
$286$		0			0
$287$	−198.997			−0.0409283
$288$		0			0
$289$	13573.2			2.76272
$290$		0			0
$291$	−2110.54	−	2110.54i	−0.425162	−	0.425162i
$292$		0			0
$293$	1900.08	−	1900.08i	0.378852	−	0.378852i	−0.491836	−	0.870688i	$-0.663674\pi$
$293$	1900.08	−	1900.08i	0.378852	−	0.378852i	0.870688	+	0.491836i	$0.163674\pi$
$294$		0			0
$295$			6993.19i			1.38020i
$296$		0			0
$297$		−	1181.67i		−	0.230867i
$298$		0			0
$299$	2363.74	−	2363.74i	0.457185	−	0.457185i
$300$		0			0
$301$	−33.9499	−	33.9499i	−0.00650112	−	0.00650112i
$302$		0			0
$303$	−5590.96			−1.06004
$304$		0			0
$305$	−4926.54			−0.924894
$306$		0			0
$307$	−3489.52	−	3489.52i	−0.648721	−	0.648721i	0.303963	−	0.952684i	$-0.401690\pi$
$307$	−3489.52	−	3489.52i	−0.648721	−	0.648721i	−0.952684	+	0.303963i	$0.901690\pi$
$308$		0			0
$309$	2853.57	−	2853.57i	0.525352	−	0.525352i
$310$		0			0
$311$			1471.68i			0.268332i	0.990959	+	0.134166i	$0.0428356\pi$
$311$			1471.68i			0.268332i	−0.990959	+	0.134166i	$0.957164\pi$
$312$		0			0
$313$		−	1258.85i		−	0.227331i	−0.993519	−	0.113665i	$-0.963741\pi$
$313$		−	1258.85i		−	0.227331i	0.993519	−	0.113665i	$-0.0362592\pi$
$314$		0			0
$315$	51.9301	−	51.9301i	0.00928866	−	0.00928866i
$316$		0			0
$317$	5902.41	+	5902.41i	1.04578	+	1.04578i	0.998901	+	0.0468795i	$0.0149277\pi$
$317$	5902.41	+	5902.41i	1.04578	+	1.04578i	0.0468795	+	0.998901i	$0.485072\pi$
$318$		0			0
$319$	411.481			0.0722211
$320$		0			0
$321$	1857.15			0.322916
$322$		0			0
$323$	8957.75	+	8957.75i	1.54310	+	1.54310i
$324$		0			0
$325$	2915.35	−	2915.35i	0.497583	−	0.497583i
$326$		0			0
$327$		−	738.044i		−	0.124813i
$328$		0			0
$329$		−	236.355i		−	0.0396068i
$330$		0			0
$331$	−3256.70	+	3256.70i	−0.540798	+	0.540798i	−0.923763	−	0.382965i	$-0.874903\pi$
$331$	−3256.70	+	3256.70i	−0.540798	+	0.540798i	0.382965	+	0.923763i	$0.374903\pi$
$332$		0			0
$333$	−462.926	−	462.926i	−0.0761808	−	0.0761808i
$334$		0			0
$335$	6773.10			1.10464
$336$		0			0
$337$	−4882.53			−0.789224			−0.394612	−	0.918848i	$-0.629121\pi$
$337$	−4882.53			−0.789224			−0.394612	+	0.918848i	$0.629121\pi$
$338$		0			0
$339$	663.828	+	663.828i	0.106355	+	0.106355i
$340$		0			0
$341$	−468.298	+	468.298i	−0.0743688	+	0.0743688i
$342$		0			0
$343$		−	332.637i		−	0.0523635i
$344$		0			0
$345$		−	6465.40i		−	1.00894i
$346$		0			0
$347$	1744.70	−	1744.70i	0.269914	−	0.269914i	−0.559151	−	0.829066i	$-0.688873\pi$
$347$	1744.70	−	1744.70i	0.269914	−	0.269914i	0.829066	+	0.559151i	$0.188873\pi$
$348$		0			0
$349$	−6212.48	−	6212.48i	−0.952855	−	0.952855i	0.0460825	−	0.998938i	$-0.485326\pi$
$349$	−6212.48	−	6212.48i	−0.952855	−	0.952855i	−0.998938	+	0.0460825i	$0.985326\pi$
$350$		0			0
$351$	−704.532			−0.107137
$352$		0			0
$353$	683.045			0.102988			0.0514941	−	0.998673i	$-0.483602\pi$
$353$	683.045			0.102988			0.0514941	+	0.998673i	$0.483602\pi$
$354$		0			0
$355$	7312.16	+	7312.16i	1.09321	+	1.09321i
$356$		0			0
$357$	139.902	−	139.902i	0.0207407	−	0.0207407i
$358$		0			0
$359$			4019.67i			0.590947i	0.955351	+	0.295473i	$0.0954773\pi$
$359$			4019.67i			0.590947i	−0.955351	+	0.295473i	$0.904523\pi$
$360$		0			0
$361$			1822.20i			0.265665i
$362$		0			0
$363$	1239.76	−	1239.76i	0.179257	−	0.179257i
$364$		0			0
$365$	−5783.23	−	5783.23i	−0.829337	−	0.829337i
$366$		0			0
$367$	8041.99			1.14384			0.571919	−	0.820310i	$-0.306199\pi$
$367$	8041.99			1.14384			0.571919	+	0.820310i	$0.306199\pi$
$368$		0			0
$369$	3692.28			0.520900
$370$		0			0
$371$	−105.325	−	105.325i	−0.0147390	−	0.0147390i
$372$		0			0
$373$	−805.266	+	805.266i	−0.111783	+	0.111783i	−0.760786	−	0.649003i	$-0.775186\pi$
$373$	−805.266	+	805.266i	−0.111783	+	0.111783i	0.649003	+	0.760786i	$0.275186\pi$
$374$		0			0
$375$		−	1665.66i		−	0.229372i
$376$		0			0
$377$		−	245.332i		−	0.0335152i
$378$		0			0
$379$	519.461	−	519.461i	0.0704034	−	0.0704034i	−0.671028	−	0.741432i	$-0.734147\pi$
$379$	519.461	−	519.461i	0.0704034	−	0.0704034i	0.741432	+	0.671028i	$0.234147\pi$
$380$		0			0
$381$	−3092.75	−	3092.75i	−0.415870	−	0.415870i
$382$		0			0
$383$	−2907.38			−0.387886			−0.193943	−	0.981013i	$-0.562128\pi$
$383$	−2907.38			−0.387886			−0.193943	+	0.981013i	$0.562128\pi$
$384$		0			0
$385$	357.128			0.0472752
$386$		0			0
$387$	629.920	+	629.920i	0.0827407	+	0.0827407i
$388$		0			0
$389$	−6046.78	+	6046.78i	−0.788133	+	0.788133i	−0.981188	−	0.193055i	$-0.938160\pi$
$389$	−6046.78	+	6046.78i	−0.788133	+	0.788133i	0.193055	+	0.981188i	$0.438160\pi$
$390$		0			0
$391$		−	17418.1i		−	2.25287i
$392$		0			0
$393$		−	863.406i		−	0.110822i
$394$		0			0
$395$	−11423.8	+	11423.8i	−1.45517	+	1.45517i
$396$		0			0
$397$	4121.17	+	4121.17i	0.520996	+	0.520996i	0.917872	−	0.396876i	$-0.129906\pi$
$397$	4121.17	+	4121.17i	0.520996	+	0.520996i	−0.396876	+	0.917872i	$0.629906\pi$
$398$		0			0
$399$	135.583			0.0170117
$400$		0			0
$401$	3991.60			0.497084			0.248542	−	0.968621i	$-0.420049\pi$
$401$	3991.60			0.497084			0.248542	+	0.968621i	$0.420049\pi$
$402$		0			0
$403$	279.207	+	279.207i	0.0345119	+	0.0345119i
$404$		0			0
$405$	−963.533	+	963.533i	−0.118218	+	0.118218i
$406$		0			0
$407$		−	3183.59i		−	0.387727i
$408$		0			0
$409$			7470.32i			0.903138i	0.892236	+	0.451569i	$0.149136\pi$
$409$			7470.32i			0.903138i	−0.892236	+	0.451569i	$0.850864\pi$
$410$		0			0
$411$	918.486	−	918.486i	0.110233	−	0.110233i
$412$		0			0
$413$	142.580	+	142.580i	0.0169877	+	0.0169877i
$414$		0			0
$415$	11026.4			1.30425
$416$		0			0
$417$	−4491.32			−0.527437
$418$		0			0
$419$	−11052.5	−	11052.5i	−1.28866	−	1.28866i	−0.935601	−	0.353059i	$-0.885142\pi$
$419$	−11052.5	−	11052.5i	−1.28866	−	1.28866i	−0.353059	−	0.935601i	$-0.614858\pi$
$420$		0			0
$421$	−2747.59	+	2747.59i	−0.318075	+	0.318075i	−0.848027	−	0.529952i	$-0.822210\pi$
$421$	−2747.59	+	2747.59i	−0.318075	+	0.318075i	0.529952	+	0.848027i	$0.322210\pi$
$422$		0			0
$423$			4385.42i			0.504082i
$424$		0			0
$425$		−	21482.9i		−	2.45194i
$426$		0			0
$427$	−100.444	+	100.444i	−0.0113837	+	0.0113837i
$428$		0			0
$429$	−2422.57	−	2422.57i	−0.272640	−	0.272640i
$430$		0			0
$431$	−11.7027			−0.00130789			−0.000653945	−	1.00000i	$-0.500208\pi$
$431$	−11.7027			−0.00130789			−0.000653945		1.00000i	$0.500208\pi$
$432$		0			0
$433$	−8291.16			−0.920202			−0.460101	−	0.887866i	$-0.652187\pi$
$433$	−8291.16			−0.920202			−0.460101	+	0.887866i	$0.652187\pi$
$434$		0			0
$435$	−335.521	−	335.521i	−0.0369816	−	0.0369816i
$436$		0			0
$437$	8440.19	−	8440.19i	0.923911	−	0.923911i
$438$		0			0
$439$			14287.6i			1.55332i	0.629918	+	0.776661i	$0.283088\pi$
$439$			14287.6i			1.55332i	−0.629918	+	0.776661i	$0.716912\pi$
$440$		0			0
$441$			3084.88i			0.333105i
$442$		0			0
$443$	−5070.34	+	5070.34i	−0.543790	+	0.543790i	−0.924638	−	0.380847i	$-0.875632\pi$
$443$	−5070.34	+	5070.34i	−0.543790	+	0.543790i	0.380847	+	0.924638i	$0.375632\pi$
$444$		0			0
$445$	15207.0	+	15207.0i	1.61995	+	1.61995i
$446$		0			0
$447$	−7225.76			−0.764579
$448$		0			0
$449$	16521.3			1.73650			0.868249	−	0.496128i	$-0.165245\pi$
$449$	16521.3			1.73650			0.868249	+	0.496128i	$0.165245\pi$
$450$		0			0
$451$	12696.1	+	12696.1i	1.32558	+	1.32558i
$452$		0			0
$453$	−1148.82	+	1148.82i	−0.119153	+	0.119153i
$454$		0			0
$455$		−	212.926i		−	0.0219387i
$456$		0			0
$457$		−	3323.22i		−	0.340161i	−0.985430	−	0.170080i	$-0.945597\pi$
$457$		−	3323.22i		−	0.340161i	0.985430	−	0.170080i	$-0.0544028\pi$
$458$		0			0
$459$	−2595.81	+	2595.81i	−0.263970	+	0.263970i
$460$		0			0
$461$	1193.80	+	1193.80i	0.120609	+	0.120609i	0.764835	−	0.644226i	$-0.222820\pi$
$461$	1193.80	+	1193.80i	0.120609	+	0.120609i	−0.644226	+	0.764835i	$0.722820\pi$
$462$		0			0
$463$	−6199.48			−0.622277			−0.311139	−	0.950365i	$-0.600710\pi$
$463$	−6199.48			−0.622277			−0.311139	+	0.950365i	$0.600710\pi$
$464$		0			0
$465$	763.699			0.0761628
$466$		0			0
$467$	−2566.04	−	2566.04i	−0.254265	−	0.254265i	0.568451	−	0.822717i	$-0.307543\pi$
$467$	−2566.04	−	2566.04i	−0.254265	−	0.254265i	−0.822717	+	0.568451i	$0.807543\pi$
$468$		0			0
$469$	138.093	−	138.093i	0.0135960	−	0.0135960i
$470$		0			0
$471$			37.9634i			0.00371393i
$472$		0			0
$473$			4332.02i			0.421113i
$474$		0			0
$475$	10409.8	−	10409.8i	1.00555	−	1.00555i
$476$		0			0
$477$	1954.24	+	1954.24i	0.187586	+	0.187586i
$478$		0			0
$479$	−4563.85			−0.435339			−0.217670	−	0.976023i	$-0.569846\pi$
$479$	−4563.85			−0.435339			−0.217670	+	0.976023i	$0.569846\pi$
$480$		0			0
$481$	−1898.11			−0.179930
$482$		0			0
$483$	−131.819	−	131.819i	−0.0124182	−	0.0124182i
$484$		0			0
$485$	−11835.0	+	11835.0i	−1.10804	+	1.10804i
$486$		0			0
$487$			1099.80i			0.102334i	0.998690	+	0.0511672i	$0.0162941\pi$
$487$			1099.80i			0.102334i	−0.998690	+	0.0511672i	$0.983706\pi$
$488$		0			0
$489$		−	5551.11i		−	0.513353i
$490$		0			0
$491$	11017.7	−	11017.7i	1.01267	−	1.01267i	0.0127523	−	0.999919i	$-0.495941\pi$
$491$	11017.7	−	11017.7i	1.01267	−	1.01267i	0.999919	−	0.0127523i	$-0.00405929\pi$
$492$		0			0
$493$	−903.913	−	903.913i	−0.0825764	−	0.0825764i
$494$		0			0
$495$	−6626.31			−0.601678
$496$		0			0
$497$	298.167			0.0269107
$498$		0			0
$499$	−3603.08	−	3603.08i	−0.323238	−	0.323238i	0.526770	−	0.850008i	$-0.323403\pi$
$499$	−3603.08	−	3603.08i	−0.323238	−	0.323238i	−0.850008	+	0.526770i	$0.823403\pi$
$500$		0			0
$501$	5036.28	−	5036.28i	0.449110	−	0.449110i
$502$		0			0
$503$		−	17947.0i		−	1.59089i	−0.606028	−	0.795443i	$-0.707238\pi$
$503$		−	17947.0i		−	1.59089i	0.606028	−	0.795443i	$-0.292762\pi$
$504$		0			0
$505$			31351.7i			2.76264i
$506$		0			0
$507$	3216.17	−	3216.17i	0.281726	−	0.281726i
$508$		0			0
$509$	−1844.54	−	1844.54i	−0.160625	−	0.160625i	0.622219	−	0.782843i	$-0.286231\pi$
$509$	−1844.54	−	1844.54i	−0.160625	−	0.160625i	−0.782843	+	0.622219i	$0.786231\pi$
$510$		0			0
$511$	−235.822			−0.0204151
$512$		0			0
$513$	−2515.67			−0.216510
$514$		0			0
$515$	−16001.6	−	16001.6i	−1.36916	−	1.36916i
$516$		0			0
$517$	−15079.5	+	15079.5i	−1.28278	+	1.28278i
$518$		0			0
$519$			5559.93i			0.470239i
$520$		0			0
$521$		−	5221.57i		−	0.439081i	−0.975603	−	0.219541i	$-0.929544\pi$
$521$		−	5221.57i		−	0.439081i	0.975603	−	0.219541i	$-0.0704558\pi$
$522$		0			0
$523$	−581.412	+	581.412i	−0.0486106	+	0.0486106i	−0.730994	−	0.682384i	$-0.760943\pi$
$523$	−581.412	+	581.412i	−0.0486106	+	0.0486106i	0.682384	+	0.730994i	$0.260943\pi$
$524$		0			0
$525$	−162.581	−	162.581i	−0.0135155	−	0.0135155i
$526$		0			0
$527$	2057.45			0.170064
$528$		0			0
$529$	−4244.75			−0.348874
$530$		0			0
$531$	−2645.49	−	2645.49i	−0.216204	−	0.216204i
$532$		0			0
$533$	7569.61	−	7569.61i	0.615152	−	0.615152i
$534$		0			0
$535$		−	10414.1i		−	0.841572i
$536$		0			0
$537$			1054.94i			0.0847749i
$538$		0			0
$539$	−10607.5	+	10607.5i	−0.847678	+	0.847678i
$540$		0			0
$541$	−4813.94	−	4813.94i	−0.382564	−	0.382564i	0.489461	−	0.872025i	$-0.337194\pi$
$541$	−4813.94	−	4813.94i	−0.382564	−	0.382564i	−0.872025	+	0.489461i	$0.837194\pi$
$542$		0			0
$543$	−4891.18			−0.386557
$544$		0			0
$545$	−4138.64			−0.325284
$546$		0			0
$547$	−10287.7	−	10287.7i	−0.804154	−	0.804154i	0.179588	−	0.983742i	$-0.442524\pi$
$547$	−10287.7	−	10287.7i	−0.804154	−	0.804154i	−0.983742	+	0.179588i	$0.942524\pi$
$548$		0			0
$549$	1863.69	−	1863.69i	0.144882	−	0.144882i
$550$		0			0
$551$		−	876.006i		−	0.0677298i
$552$		0			0
$553$			465.825i			0.0358208i
$554$		0			0
$555$	−2595.89	+	2595.89i	−0.198540	+	0.198540i
$556$		0			0
$557$	−16473.4	−	16473.4i	−1.25315	−	1.25315i	−0.954302	−	0.298843i	$-0.903399\pi$
$557$	−16473.4	−	16473.4i	−1.25315	−	1.25315i	−0.298843	−	0.954302i	$-0.596601\pi$
$558$		0			0
$559$	2582.82			0.195424
$560$		0			0
$561$	−17851.6			−1.34349
$562$		0			0
$563$	−3846.62	−	3846.62i	−0.287950	−	0.287950i	0.548319	−	0.836269i	$-0.315268\pi$
$563$	−3846.62	−	3846.62i	−0.287950	−	0.287950i	−0.836269	+	0.548319i	$0.815268\pi$
$564$		0			0
$565$	3722.47	−	3722.47i	0.277178	−	0.277178i
$566$		0			0
$567$			39.2898i			0.00291008i
$568$		0			0
$569$			16206.0i			1.19401i	0.802239	+	0.597003i	$0.203642\pi$
$569$			16206.0i			1.19401i	−0.802239	+	0.597003i	$0.796358\pi$
$570$		0			0
$571$	15842.4	−	15842.4i	1.16109	−	1.16109i	0.176858	−	0.984236i	$-0.443407\pi$
$571$	15842.4	−	15842.4i	1.16109	−	1.16109i	0.984236	−	0.176858i	$-0.0565933\pi$
$572$		0			0
$573$	970.923	+	970.923i	0.0707869	+	0.0707869i
$574$		0			0
$575$	−20241.7			−1.46806
$576$		0			0
$577$	−18218.5			−1.31446			−0.657231	−	0.753689i	$-0.728272\pi$
$577$	−18218.5			−1.31446			−0.657231	+	0.753689i	$0.728272\pi$
$578$		0			0
$579$	−131.006	−	131.006i	−0.00940314	−	0.00940314i
$580$		0			0
$581$	224.811	−	224.811i	0.0160529	−	0.0160529i
$582$		0			0
$583$			13439.5i			0.954728i
$584$		0			0
$585$			3950.71i			0.279217i
$586$		0			0
$587$	3225.64	−	3225.64i	0.226808	−	0.226808i	−0.584550	−	0.811358i	$-0.698729\pi$
$587$	3225.64	−	3225.64i	0.226808	−	0.226808i	0.811358	+	0.584550i	$0.198729\pi$
$588$		0			0
$589$	996.964	+	996.964i	0.0697439	+	0.0697439i
$590$		0			0
$591$	2603.15			0.181183
$592$		0			0
$593$	−15436.7			−1.06898			−0.534492	−	0.845174i	$-0.679497\pi$
$593$	−15436.7			−1.06898			−0.534492	+	0.845174i	$0.679497\pi$
$594$		0			0
$595$	−784.514	−	784.514i	−0.0540537	−	0.0540537i
$596$		0			0
$597$	7092.20	−	7092.20i	0.486205	−	0.486205i
$598$		0			0
$599$		−	13650.7i		−	0.931142i	−0.885010	−	0.465571i	$-0.845849\pi$
$599$		−	13650.7i		−	0.931142i	0.885010	−	0.465571i	$-0.154151\pi$
$600$		0			0
$601$			27042.7i			1.83543i	0.397236	+	0.917716i	$0.369969\pi$
$601$			27042.7i			1.83543i	−0.397236	+	0.917716i	$0.630031\pi$
$602$		0			0
$603$	−2562.23	+	2562.23i	−0.173038	+	0.173038i
$604$		0			0
$605$	−6952.03	−	6952.03i	−0.467174	−	0.467174i
$606$		0			0
$607$	3562.32			0.238205			0.119102	−	0.992882i	$-0.461998\pi$
$607$	3562.32			0.238205			0.119102	+	0.992882i	$0.461998\pi$
$608$		0			0
$609$	−13.6815			−0.000910348
$610$		0			0
$611$	8990.64	+	8990.64i	0.595290	+	0.595290i
$612$		0			0
$613$	−11418.0	+	11418.0i	−0.752315	+	0.752315i	−0.974911	−	0.222595i	$-0.928547\pi$
$613$	−11418.0	+	11418.0i	−0.752315	+	0.752315i	0.222595	+	0.974911i	$0.428547\pi$
$614$		0			0
$615$		−	20704.7i		−	1.35755i
$616$		0			0
$617$			3981.62i			0.259796i	0.991527	+	0.129898i	$0.0414649\pi$
$617$			3981.62i			0.259796i	−0.991527	+	0.129898i	$0.958535\pi$
$618$		0			0
$619$	6160.23	−	6160.23i	0.400001	−	0.400001i	−0.478232	−	0.878233i	$-0.658722\pi$
$619$	6160.23	−	6160.23i	0.400001	−	0.400001i	0.878233	+	0.478232i	$0.158722\pi$
$620$		0			0
$621$	2445.83	+	2445.83i	0.158048	+	0.158048i
$622$		0			0
$623$	620.092			0.0398771
$624$		0			0
$625$	10410.2			0.666252
$626$		0			0
$627$	−8650.26	−	8650.26i	−0.550970	−	0.550970i
$628$		0			0
$629$	−6993.48	+	6993.48i	−0.443320	+	0.443320i
$630$		0			0
$631$			670.100i			0.0422761i	0.999777	+	0.0211381i	$0.00672896\pi$
$631$			670.100i			0.0422761i	−0.999777	+	0.0211381i	$0.993271\pi$
$632$		0			0
$633$		−	1892.16i		−	0.118810i
$634$		0			0
$635$	−17342.8	+	17342.8i	−1.08383	+	1.08383i
$636$		0			0
$637$	6324.38	+	6324.38i	0.393377	+	0.393377i
$638$		0			0
$639$	−5532.31			−0.342496
$640$		0			0
$641$	−3038.11			−0.187204			−0.0936022	−	0.995610i	$-0.529838\pi$
$641$	−3038.11			−0.187204			−0.0936022	+	0.995610i	$0.529838\pi$
$642$		0			0
$643$	11363.4	+	11363.4i	0.696934	+	0.696934i	0.963748	−	0.266814i	$-0.0859710\pi$
$643$	11363.4	+	11363.4i	0.696934	+	0.696934i	−0.266814	+	0.963748i	$0.585971\pi$
$644$		0			0
$645$	3532.33	−	3532.33i	0.215636	−	0.215636i
$646$		0			0
$647$			25695.9i			1.56137i	0.624923	+	0.780687i	$0.285131\pi$
$647$			25695.9i			1.56137i	−0.624923	+	0.780687i	$0.714869\pi$
$648$		0			0
$649$		−	18193.3i		−	1.10039i
$650$		0			0
$651$	15.5706	−	15.5706i	0.000937420	−	0.000937420i
$652$		0			0
$653$	20933.8	+	20933.8i	1.25452	+	1.25452i	0.953673	+	0.300847i	$0.0972692\pi$
$653$	20933.8	+	20933.8i	1.25452	+	1.25452i	0.300847	+	0.953673i	$0.402731\pi$
$654$		0			0
$655$	−4841.61			−0.288821
$656$		0			0
$657$	4375.54			0.259826
$658$		0			0
$659$	5253.27	+	5253.27i	0.310529	+	0.310529i	0.845114	−	0.534586i	$-0.179532\pi$
$659$	5253.27	+	5253.27i	0.310529	+	0.310529i	−0.534586	+	0.845114i	$0.679532\pi$
$660$		0			0
$661$	−3182.14	+	3182.14i	−0.187248	+	0.187248i	−0.794505	−	0.607257i	$-0.792270\pi$
$661$	−3182.14	+	3182.14i	−0.187248	+	0.187248i	0.607257	+	0.794505i	$0.292270\pi$
$662$		0			0
$663$			10643.4i			0.623465i
$664$		0			0
$665$		−	760.293i		−	0.0443352i
$666$		0			0
$667$	−851.686	+	851.686i	−0.0494414	+	0.0494414i
$668$		0			0
$669$	−12941.8	−	12941.8i	−0.747922	−	0.747922i
$670$		0			0
$671$	12816.8			0.737385
$672$		0			0
$673$	27736.4			1.58865			0.794325	−	0.607493i	$-0.207825\pi$
$673$	27736.4			1.58865			0.794325	+	0.607493i	$0.207825\pi$
$674$		0			0
$675$	3016.60	+	3016.60i	0.172013	+	0.172013i
$676$		0			0
$677$	−6076.45	+	6076.45i	−0.344959	+	0.344959i	−0.858228	−	0.513269i	$-0.828434\pi$
$677$	−6076.45	+	6076.45i	−0.344959	+	0.344959i	0.513269	+	0.858228i	$0.328434\pi$
$678$		0			0
$679$			482.594i			0.0272758i
$680$		0			0
$681$		−	15498.0i		−	0.872075i
$682$		0			0
$683$	−2360.09	+	2360.09i	−0.132220	+	0.132220i	−0.770120	−	0.637900i	$-0.779803\pi$
$683$	−2360.09	+	2360.09i	−0.132220	+	0.132220i	0.637900	+	0.770120i	$0.279803\pi$
$684$		0			0
$685$	−5150.48	−	5150.48i	−0.287284	−	0.287284i
$686$		0			0
$687$	−7245.84			−0.402396
$688$		0			0
$689$	8012.84			0.443055
$690$		0			0
$691$	10248.5	+	10248.5i	0.564211	+	0.564211i	0.930501	−	0.366290i	$-0.119372\pi$
$691$	10248.5	+	10248.5i	0.564211	+	0.564211i	−0.366290	+	0.930501i	$0.619372\pi$
$692$		0			0
$693$	−135.100	+	135.100i	−0.00740552	+	0.00740552i
$694$		0			0
$695$			25185.4i			1.37459i
$696$		0			0
$697$		−	55779.7i		−	3.03128i
$698$		0			0
$699$	−6404.85	+	6404.85i	−0.346572	+	0.346572i
$700$		0			0
$701$	5637.61	+	5637.61i	0.303751	+	0.303751i	0.842480	−	0.538728i	$-0.181095\pi$
$701$	5637.61	+	5637.61i	0.303751	+	0.303751i	−0.538728	+	0.842480i	$0.681095\pi$
$702$		0			0
$703$	−6777.57			−0.363614
$704$		0			0
$705$	24591.6			1.31372
$706$		0			0
$707$	639.212	+	639.212i	0.0340029	+	0.0340029i
$708$		0			0
$709$	1798.00	−	1798.00i	0.0952402	−	0.0952402i	−0.657881	−	0.753122i	$-0.728547\pi$
$709$	1798.00	−	1798.00i	0.0952402	−	0.0952402i	0.753122	+	0.657881i	$0.228547\pi$
$710$		0			0
$711$		−	8643.12i		−	0.455897i
$712$		0			0
$713$		−	1938.57i		−	0.101823i
$714$		0			0
$715$	−13584.7	+	13584.7i	−0.710545	+	0.710545i
$716$		0			0
$717$	−6288.05	−	6288.05i	−0.327520	−	0.327520i
$718$		0			0
$719$	35035.3			1.81724			0.908622	−	0.417620i	$-0.137136\pi$
$719$	35035.3			1.81724			0.908622	+	0.417620i	$0.137136\pi$
$720$		0			0
$721$	−652.495			−0.0337034
$722$		0			0
$723$	−5529.94	−	5529.94i	−0.284455	−	0.284455i
$724$		0			0
$725$	−1050.44	+	1050.44i	−0.0538101	+	0.0538101i
$726$		0			0
$727$		−	24515.4i		−	1.25066i	−0.780362	−	0.625328i	$-0.784965\pi$
$727$		−	24515.4i		−	1.25066i	0.780362	−	0.625328i	$-0.215035\pi$
$728$		0			0
$729$		−	729.000i		−	0.0370370i
$730$		0			0
$731$	9516.28	−	9516.28i	0.481494	−	0.481494i
$732$		0			0
$733$	23583.9	+	23583.9i	1.18839	+	1.18839i	0.977512	+	0.210878i	$0.0676322\pi$
$733$	23583.9	+	23583.9i	1.18839	+	1.18839i	0.210878	+	0.977512i	$0.432368\pi$
$734$		0			0
$735$	17298.7			0.868126
$736$		0			0
$737$	−17620.7			−0.880689
$738$		0			0
$739$	−17246.2	−	17246.2i	−0.858472	−	0.858472i	0.132686	−	0.991158i	$-0.457640\pi$
$739$	−17246.2	−	17246.2i	−0.858472	−	0.858472i	−0.991158	+	0.132686i	$0.957640\pi$
$740$		0			0
$741$	−5157.42	+	5157.42i	−0.255685	+	0.255685i
$742$		0			0
$743$		−	14249.5i		−	0.703587i	−0.936078	−	0.351793i	$-0.885572\pi$
$743$		−	14249.5i		−	0.703587i	0.936078	−	0.351793i	$-0.114428\pi$
$744$		0			0
$745$			40519.0i			1.99262i
$746$		0			0
$747$	−4171.24	+	4171.24i	−0.204308	+	0.204308i
$748$		0			0
$749$	−212.327	−	212.327i	−0.0103582	−	0.0103582i
$750$		0			0
$751$	−10260.0			−0.498524			−0.249262	−	0.968436i	$-0.580188\pi$
$751$	−10260.0			−0.498524			−0.249262	+	0.968436i	$0.580188\pi$
$752$		0			0
$753$	12807.4			0.619824
$754$		0			0
$755$	6442.11	+	6442.11i	0.310533	+	0.310533i
$756$		0			0
$757$	18953.4	−	18953.4i	0.910005	−	0.910005i	−0.0862671	−	0.996272i	$-0.527494\pi$
$757$	18953.4	−	18953.4i	0.910005	−	0.910005i	0.996272	+	0.0862671i	$0.0274939\pi$
$758$		0			0
$759$			16820.2i			0.804394i
$760$		0			0
$761$		−	24434.3i		−	1.16392i	−0.813218	−	0.581960i	$-0.802286\pi$
$761$		−	24434.3i		−	1.16392i	0.813218	−	0.581960i	$-0.197714\pi$
$762$		0			0
$763$	−84.3802	+	84.3802i	−0.00400363	+	0.00400363i
$764$		0			0
$765$	14556.2	+	14556.2i	0.687949	+	0.687949i
$766$		0			0
$767$	−10847.1			−0.510649
$768$		0			0
$769$	−6752.03			−0.316625			−0.158312	−	0.987389i	$-0.550605\pi$
$769$	−6752.03			−0.316625			−0.158312	+	0.987389i	$0.550605\pi$
$770$		0			0
$771$	4218.26	+	4218.26i	0.197039	+	0.197039i
$772$		0			0
$773$	18608.6	−	18608.6i	0.865854	−	0.865854i	−0.126157	−	0.992010i	$-0.540264\pi$
$773$	18608.6	−	18608.6i	0.865854	−	0.865854i	0.992010	+	0.126157i	$0.0402642\pi$
$774$		0			0
$775$		−	2390.96i		−	0.110821i
$776$		0			0
$777$			105.852i			0.00488730i
$778$		0			0
$779$	27028.8	−	27028.8i	1.24314	−	1.24314i
$780$		0			0
$781$	−19023.1	−	19023.1i	−0.871576	−	0.871576i
$782$		0			0
$783$	253.852			0.0115861
$784$		0			0
$785$	212.883			0.00967911
$786$		0			0
$787$	23406.2	+	23406.2i	1.06016	+	1.06016i	0.998071	+	0.0620850i	$0.0197750\pi$
$787$	23406.2	+	23406.2i	1.06016	+	1.06016i	0.0620850	+	0.998071i	$0.480225\pi$
$788$		0			0
$789$	−12833.7	+	12833.7i	−0.579076	+	0.579076i
$790$		0			0
$791$		−	151.790i		−	0.00682307i
$792$		0			0
$793$		−	7641.56i		−	0.342194i
$794$		0			0
$795$	10958.5	−	10958.5i	0.488880	−	0.488880i
$796$		0			0
$797$	−1487.10	−	1487.10i	−0.0660925	−	0.0660925i	0.673288	−	0.739380i	$-0.264881\pi$
$797$	−1487.10	−	1487.10i	−0.0660925	−	0.0660925i	−0.739380	+	0.673288i	$0.764881\pi$
$798$		0			0
$799$	66251.1			2.93341
$800$		0			0
$801$	−11505.5			−0.507522
$802$		0			0
$803$	15045.5	+	15045.5i	0.661201	+	0.661201i
$804$		0			0
$805$	−739.186	+	739.186i	−0.0323638	+	0.0323638i
$806$		0			0
$807$		−	16476.6i		−	0.718715i
$808$		0			0
$809$		−	10505.2i		−	0.456542i	−0.973598	−	0.228271i	$-0.926693\pi$
$809$		−	10505.2i		−	0.456542i	0.973598	−	0.228271i	$-0.0733072\pi$
$810$		0			0
$811$	−27073.1	+	27073.1i	−1.17221	+	1.17221i	−0.190531	+	0.981681i	$0.561021\pi$
$811$	−27073.1	+	27073.1i	−1.17221	+	1.17221i	−0.981681	+	0.190531i	$0.938979\pi$
$812$		0			0
$813$	7046.18	+	7046.18i	0.303961	+	0.303961i
$814$		0			0
$815$	−31128.3			−1.33788
$816$		0			0
$817$	9222.48			0.394925
$818$		0			0
$819$	80.5488	+	80.5488i	0.00343663	+	0.00343663i
$820$		0			0
$821$	9907.96	−	9907.96i	0.421182	−	0.421182i	−0.464429	−	0.885610i	$-0.653740\pi$
$821$	9907.96	−	9907.96i	0.421182	−	0.421182i	0.885610	+	0.464429i	$0.153740\pi$
$822$		0			0
$823$			17501.8i			0.741281i	0.928776	+	0.370640i	$0.120862\pi$
$823$			17501.8i			0.741281i	−0.928776	+	0.370640i	$0.879138\pi$
$824$		0			0
$825$			20745.4i			0.875471i
$826$		0			0
$827$	7074.02	−	7074.02i	0.297446	−	0.297446i	−0.542567	−	0.840013i	$-0.682547\pi$
$827$	7074.02	−	7074.02i	0.297446	−	0.297446i	0.840013	+	0.542567i	$0.182547\pi$
$828$		0			0
$829$	−21066.1	−	21066.1i	−0.882576	−	0.882576i	0.111220	−	0.993796i	$-0.464524\pi$
$829$	−21066.1	−	21066.1i	−0.882576	−	0.882576i	−0.993796	+	0.111220i	$0.964524\pi$
$830$		0			0
$831$	15287.5			0.638167
$832$		0			0
$833$	46603.7			1.93844
$834$		0			0
$835$	−28241.3	−	28241.3i	−1.17046	−	1.17046i
$836$		0			0
$837$	−288.904	+	288.904i	−0.0119307	+	0.0119307i
$838$		0			0
$839$		−	30444.6i		−	1.25276i	−0.779519	−	0.626379i	$-0.784536\pi$
$839$		−	30444.6i		−	1.25276i	0.779519	−	0.626379i	$-0.215464\pi$
$840$		0			0
$841$		−	24300.6i		−	0.996376i
$842$		0			0
$843$	−1747.25	+	1747.25i	−0.0713861	+	0.0713861i
$844$		0			0
$845$	−18034.9	−	18034.9i	−0.734224	−	0.734224i
$846$		0			0
$847$	−283.482			−0.0115001
$848$		0			0
$849$	−8501.15			−0.343650
$850$		0			0
$851$	6589.41	+	6589.41i	0.265431	+	0.265431i
$852$		0			0
$853$	882.218	−	882.218i	0.0354122	−	0.0354122i	−0.689179	−	0.724591i	$-0.742029\pi$
$853$	882.218	−	882.218i	0.0354122	−	0.0354122i	0.724591	+	0.689179i	$0.242029\pi$
$854$		0			0
$855$			14106.8i			0.564261i
$856$		0			0
$857$			8006.81i			0.319145i	0.987186	+	0.159572i	$0.0510116\pi$
$857$			8006.81i			0.319145i	−0.987186	+	0.159572i	$0.948988\pi$
$858$		0			0
$859$	−20266.0	+	20266.0i	−0.804967	+	0.804967i	−0.983867	−	0.178900i	$-0.942746\pi$
$859$	−20266.0	+	20266.0i	−0.804967	+	0.804967i	0.178900	+	0.983867i	$0.442746\pi$
$860$		0			0
$861$	−422.136	−	422.136i	−0.0167089	−	0.0167089i
$862$		0			0
$863$	−41478.0			−1.63607			−0.818034	−	0.575169i	$-0.804936\pi$
$863$	−41478.0			−1.63607			−0.818034	+	0.575169i	$0.804936\pi$
$864$		0			0
$865$	31177.7			1.22552
$866$		0			0
$867$	28793.2	+	28793.2i	1.12787	+	1.12787i
$868$		0			0
$869$	29719.8	−	29719.8i	1.16016	−	1.16016i
$870$		0			0
$871$			10505.8i			0.408696i
$872$		0			0
$873$		−	8954.26i		−	0.347143i
$874$		0			0
$875$	−190.435	+	190.435i	−0.00735756	+	0.00735756i
$876$		0			0
$877$	−6027.15	−	6027.15i	−0.232067	−	0.232067i	0.581488	−	0.813555i	$-0.302471\pi$
$877$	−6027.15	−	6027.15i	−0.232067	−	0.232067i	−0.813555	+	0.581488i	$0.802471\pi$
$878$		0			0
$879$	8061.34			0.309331
$880$		0			0
$881$	−40154.3			−1.53556			−0.767782	−	0.640712i	$-0.778639\pi$
$881$	−40154.3			−1.53556			−0.767782	+	0.640712i	$0.778639\pi$
$882$		0			0
$883$	4435.93	+	4435.93i	0.169061	+	0.169061i	0.786567	−	0.617505i	$-0.211857\pi$
$883$	4435.93	+	4435.93i	0.169061	+	0.169061i	−0.617505	+	0.786567i	$0.711857\pi$
$884$		0			0
$885$	−14834.8	+	14834.8i	−0.563465	+	0.563465i
$886$		0			0
$887$			3275.31i			0.123984i	0.998077	+	0.0619921i	$0.0197454\pi$
$887$			3275.31i			0.123984i	−0.998077	+	0.0619921i	$0.980255\pi$
$888$		0			0
$889$			707.186i			0.0266797i
$890$		0			0
$891$	2506.70	−	2506.70i	0.0942511	−	0.0942511i
$892$		0			0
$893$	32102.9	+	32102.9i	1.20300	+	1.20300i
$894$		0			0
$895$	5915.67			0.220937
$896$		0			0
$897$	10028.5			0.373290
$898$		0			0
$899$	−100.602	−	100.602i	−0.00373222	−	0.00373222i
$900$		0			0
$901$	29522.9	−	29522.9i	1.09162	−	1.09162i
$902$		0			0
$903$		−	144.037i		−	0.00530814i
$904$		0			0
$905$			27427.7i			1.00743i
$906$		0			0
$907$	18595.8	−	18595.8i	0.680775	−	0.680775i	−0.279400	−	0.960175i	$-0.590135\pi$
$907$	18595.8	−	18595.8i	0.680775	−	0.680775i	0.960175	+	0.279400i	$0.0901354\pi$
$908$		0			0
$909$	−11860.2	−	11860.2i	−0.432759	−	0.432759i
$910$		0			0
$911$	25144.8			0.914472			0.457236	−	0.889345i	$-0.348839\pi$
$911$	25144.8			0.914472			0.457236	+	0.889345i	$0.348839\pi$
$912$		0			0
$913$	−28686.1			−1.03984
$914$		0			0
$915$	−10450.8	−	10450.8i	−0.377586	−	0.377586i
$916$		0			0
$917$	−98.7128	+	98.7128i	−0.00355484	+	0.00355484i
$918$		0			0
$919$			18292.8i			0.656608i	0.944572	+	0.328304i	$0.106477\pi$
$919$			18292.8i			0.656608i	−0.944572	+	0.328304i	$0.893523\pi$
$920$		0			0
$921$		−	14804.8i		−	0.529679i
$922$		0			0
$923$	−11341.9	+	11341.9i	−0.404467	+	0.404467i
$924$		0			0
$925$	8127.14	+	8127.14i	0.288885	+	0.288885i
$926$		0			0
$927$	12106.7			0.428948
$928$		0			0
$929$	16680.0			0.589078			0.294539	−	0.955640i	$-0.404834\pi$
$929$	16680.0			0.589078			0.294539	+	0.955640i	$0.404834\pi$
$930$		0			0
$931$	22582.4	+	22582.4i	0.794962	+	0.794962i
$932$		0			0
$933$	−3121.91	+	3121.91i	−0.109546	+	0.109546i
$934$		0			0
$935$			100104.i			3.50135i
$936$		0			0
$937$		−	2053.17i		−	0.0715839i	−0.999359	−	0.0357920i	$-0.988605\pi$
$937$		−	2053.17i		−	0.0715839i	0.999359	−	0.0357920i	$-0.0113954\pi$
$938$		0			0
$939$	2670.43	−	2670.43i	0.0928074	−	0.0928074i
$940$		0			0
$941$	17468.5	+	17468.5i	0.605160	+	0.605160i	0.941677	−	0.336517i	$-0.109249\pi$
$941$	17468.5	+	17468.5i	0.605160	+	0.605160i	−0.336517	+	0.941677i	$0.609249\pi$
$942$		0			0
$943$	−52556.8			−1.81494
$944$		0			0
$945$	220.321			0.00758416
$946$		0			0
$947$	22999.8	+	22999.8i	0.789223	+	0.789223i	0.981367	−	0.192144i	$-0.0615441\pi$
$947$	22999.8	+	22999.8i	0.789223	+	0.789223i	−0.192144	+	0.981367i	$0.561544\pi$
$948$		0			0
$949$	8970.37	−	8970.37i	0.306839	−	0.306839i
$950$		0			0
$951$			25041.8i			0.853876i
$952$		0			0
$953$			29341.4i			0.997336i	0.866793	+	0.498668i	$0.166177\pi$
$953$			29341.4i			0.997336i	−0.866793	+	0.498668i	$0.833823\pi$
$954$		0			0
$955$	5444.52	−	5444.52i	0.184482	−	0.184482i
$956$		0			0
$957$	872.884	+	872.884i	0.0294841	+	0.0294841i
$958$		0			0
$959$	−210.020			−0.00707186
$960$		0			0
$961$	−29562.0			−0.992314
$962$		0			0
$963$	3939.61	+	3939.61i	0.131830	+	0.131830i
$964$		0			0
$965$	−734.625	+	734.625i	−0.0245061	+	0.0245061i
$966$		0			0
$967$		−	8861.99i		−	0.294708i	−0.989084	−	0.147354i	$-0.952924\pi$
$967$		−	8861.99i		−	0.294708i	0.989084	−	0.147354i	$-0.0470756\pi$
$968$		0			0
$969$			38004.5i			1.25994i
$970$		0			0
$971$	38037.4	−	38037.4i	1.25714	−	1.25714i	0.304681	−	0.952454i	$-0.401450\pi$
$971$	38037.4	−	38037.4i	1.25714	−	1.25714i	0.952454	−	0.304681i	$-0.0985500\pi$
$972$		0			0
$973$	513.491	+	513.491i	0.0169186	+	0.0169186i
$974$		0			0
$975$	12368.8			0.406275
$976$		0			0
$977$	26081.9			0.854078			0.427039	−	0.904233i	$-0.359557\pi$
$977$	26081.9			0.854078			0.427039	+	0.904233i	$0.359557\pi$
$978$		0			0
$979$	−39562.1	−	39562.1i	−1.29153	−	1.29153i
$980$		0			0
$981$	1565.63	−	1565.63i	0.0509548	−	0.0509548i
$982$		0			0
$983$		−	28987.6i		−	0.940552i	−0.882520	−	0.470276i	$-0.844154\pi$
$983$		−	28987.6i		−	0.940552i	0.882520	−	0.470276i	$-0.155846\pi$
$984$		0			0
$985$		−	14597.3i		−	0.472193i
$986$		0			0
$987$	501.384	−	501.384i	0.0161694	−	0.0161694i
$988$		0			0
$989$	−8966.45	−	8966.45i	−0.288288	−	0.288288i
$990$		0			0
$991$	−33373.6			−1.06977			−0.534887	−	0.844923i	$-0.679646\pi$
$991$	−33373.6			−1.06977			−0.534887	+	0.844923i	$0.679646\pi$
$992$		0			0
$993$	−13817.0			−0.441560
$994$		0			0
$995$	−39770.1	−	39770.1i	−1.26713	−	1.26713i
$996$		0			0
$997$	5283.79	−	5283.79i	0.167843	−	0.167843i	−0.618188	−	0.786031i	$-0.712133\pi$
$997$	5283.79	−	5283.79i	0.167843	−	0.167843i	0.786031	+	0.618188i	$0.212133\pi$
$998$		0			0
$999$		−	1964.03i		−	0.0622014i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.j.a.289.7		24
4.3	odd	2		384.4.j.b.289.6		24
8.3	odd	2		48.4.j.a.13.5	✓	24
8.5	even	2		192.4.j.a.145.1		24
16.3	odd	4		48.4.j.a.37.5	yes	24
16.5	even	4	inner	384.4.j.a.97.7		24
16.11	odd	4		384.4.j.b.97.6		24
16.13	even	4		192.4.j.a.49.1		24
24.5	odd	2		576.4.k.b.145.12		24
24.11	even	2		144.4.k.b.109.8		24
48.29	odd	4		576.4.k.b.433.12		24
48.35	even	4		144.4.k.b.37.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
48.4.j.a.13.5	✓	24	8.3	odd	2
48.4.j.a.37.5	yes	24	16.3	odd	4
144.4.k.b.37.8		24	48.35	even	4
144.4.k.b.109.8		24	24.11	even	2
192.4.j.a.49.1		24	16.13	even	4
192.4.j.a.145.1		24	8.5	even	2
384.4.j.a.97.7		24	16.5	even	4	inner
384.4.j.a.289.7		24	1.1	even	1	trivial
384.4.j.b.97.6		24	16.11	odd	4
384.4.j.b.289.6		24	4.3	odd	2
576.4.k.b.145.12		24	24.5	odd	2
576.4.k.b.433.12		24	48.29	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.j (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(2.12132 + 2.12132i) q^{3} +(11.8955 - 11.8955i) q^{5} -0.485059i q^{7} +9.00000i q^{9} +(-30.9469 + 30.9469i) q^{11} +(18.4511 + 18.4511i) q^{13} +50.4682 q^{15} +135.964 q^{17} +(65.8832 + 65.8832i) q^{19} +(1.02897 - 1.02897i) q^{21} -128.108i q^{23} -158.004i q^{25} +(-19.0919 + 19.0919i) q^{27} +(-6.64817 - 6.64817i) q^{29} +15.1323 q^{31} -131.297 q^{33} +(-5.77001 - 5.77001i) q^{35} +(-51.4363 + 51.4363i) q^{37} +78.2813i q^{39} -410.253i q^{41} +(69.9911 - 69.9911i) q^{43} +(107.059 + 107.059i) q^{45} +487.269 q^{47} +342.765 q^{49} +(288.423 + 288.423i) q^{51} +(217.138 - 217.138i) q^{53} +736.257i q^{55} +279.519i q^{57} +(-293.944 + 293.944i) q^{59} +(-207.076 - 207.076i) q^{61} +4.36553 q^{63} +438.968 q^{65} +(284.693 + 284.693i) q^{67} +(271.759 - 271.759i) q^{69} +614.701i q^{71} -486.171i q^{73} +(335.178 - 335.178i) q^{75} +(15.0111 + 15.0111i) q^{77} -960.347 q^{79} -81.0000 q^{81} +(463.472 + 463.472i) q^{83} +(1617.36 - 1617.36i) q^{85} -28.2058i q^{87} +1278.38i q^{89} +(8.94987 - 8.94987i) q^{91} +(32.1004 + 32.1004i) q^{93} +1567.42 q^{95} -994.918 q^{97} +(-278.523 - 278.523i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 40 q^{11} - 120 q^{15} + 24 q^{19} - 400 q^{29} + 744 q^{31} - 456 q^{35} - 16 q^{37} + 1240 q^{43} - 1176 q^{49} + 744 q^{51} - 752 q^{53} - 1376 q^{59} + 912 q^{61} + 504 q^{63} + 976 q^{65} - 2256 q^{67}+ \cdots - 360 q^{99}+O(q^{100}) \) 24 * q - 40 * q^11 - 120 * q^15 + 24 * q^19 - 400 * q^29 + 744 * q^31 - 456 * q^35 - 16 * q^37 + 1240 * q^43 - 1176 * q^49 + 744 * q^51 - 752 * q^53 - 1376 * q^59 + 912 * q^61 + 504 * q^63 + 976 * q^65 - 2256 * q^67 + 528 * q^69 + 1104 * q^75 - 1904 * q^77 - 5992 * q^79 - 1944 * q^81 + 2680 * q^83 + 240 * q^85 - 3496 * q^91 + 7728 * q^95 - 360 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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