LMFDB - Embedded newform 448.3.l.b.209.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(209,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 2]))

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

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

chi := DirichletCharacter("448.209");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.l (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:	$12.2071158433$
Analytic rank:	$0$
Dimension:	$56$
Relative dimension:	$28$ over $\Q(i)$
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			209.15
Character	$\chi$	$=$	448.209
Dual form			448.3.l.b.433.15

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.746365 - 0.746365i) q^{3} +(0.822232 + 0.822232i) q^{5} +(6.88091 + 1.28574i) q^{7} +7.88588i q^{9} +O(q^{10})$	\(q+(0.746365 - 0.746365i) q^{3} +(0.822232 + 0.822232i) q^{5} +(6.88091 + 1.28574i) q^{7} +7.88588i q^{9} +(-2.57423 + 2.57423i) q^{11} +(-14.9766 + 14.9766i) q^{13} +1.22737 q^{15} +20.7103i q^{17} +(10.0977 - 10.0977i) q^{19} +(6.09530 - 4.17604i) q^{21} -19.5800i q^{23} -23.6479i q^{25} +(12.6030 + 12.6030i) q^{27} +(7.24407 + 7.24407i) q^{29} +45.5117i q^{31} +3.84263i q^{33} +(4.60052 + 6.71488i) q^{35} +(-34.5793 + 34.5793i) q^{37} +22.3561i q^{39} +61.5837 q^{41} +(16.6424 - 16.6424i) q^{43} +(-6.48402 + 6.48402i) q^{45} +55.0421i q^{47} +(45.6937 + 17.6941i) q^{49} +(15.4575 + 15.4575i) q^{51} +(38.2187 - 38.2187i) q^{53} -4.23322 q^{55} -15.0732i q^{57} +(-32.2679 - 32.2679i) q^{59} +(-32.0297 + 32.0297i) q^{61} +(-10.1392 + 54.2620i) q^{63} -24.6286 q^{65} +(-42.1559 - 42.1559i) q^{67} +(-14.6139 - 14.6139i) q^{69} -15.7491i q^{71} +65.7339 q^{73} +(-17.6500 - 17.6500i) q^{75} +(-21.0228 + 14.4032i) q^{77} +96.5103 q^{79} -52.1600 q^{81} +(31.6530 - 31.6530i) q^{83} +(-17.0287 + 17.0287i) q^{85} +10.8134 q^{87} +89.3246 q^{89} +(-122.309 + 83.7968i) q^{91} +(33.9684 + 33.9684i) q^{93} +16.6054 q^{95} -47.6196i q^{97} +(-20.3000 - 20.3000i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q+O(q^{10}) $ 56 * q	$ 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) $ 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	0.746365	−	0.746365i	0.248788	−	0.248788i	−0.571685	−	0.820473i	$-0.693710\pi$
$3$	0.746365	−	0.746365i	0.248788	−	0.248788i	0.820473	+	0.571685i	$0.193710\pi$
$4$		0			0
$5$	0.822232	+	0.822232i	0.164446	+	0.164446i	0.784533	−	0.620087i	$-0.212903\pi$
$5$	0.822232	+	0.822232i	0.164446	+	0.164446i	−0.620087	+	0.784533i	$0.712903\pi$
$6$		0			0
$7$	6.88091	+	1.28574i	0.982987	+	0.183677i
$7$	6.88091	+	1.28574i	0.982987	+	0.183677i
$8$		0			0
$9$			7.88588i			0.876209i
$10$		0			0
$11$	−2.57423	+	2.57423i	−0.234021	+	0.234021i	−0.814369	−	0.580348i	$-0.802917\pi$
$11$	−2.57423	+	2.57423i	−0.234021	+	0.234021i	0.580348	+	0.814369i	$0.302917\pi$
$12$		0			0
$13$	−14.9766	+	14.9766i	−1.15205	+	1.15205i	−0.165909	+	0.986141i	$0.553056\pi$
$13$	−14.9766	+	14.9766i	−1.15205	+	1.15205i	−0.986141	+	0.165909i	$0.946944\pi$
$14$		0			0
$15$	1.22737			0.0818247
$16$		0			0
$17$			20.7103i			1.21826i	0.793072	+	0.609128i	$0.208480\pi$
$17$			20.7103i			1.21826i	−0.793072	+	0.609128i	$0.791520\pi$
$18$		0			0
$19$	10.0977	−	10.0977i	0.531460	−	0.531460i	−0.389547	−	0.921007i	$-0.627368\pi$
$19$	10.0977	−	10.0977i	0.531460	−	0.531460i	0.921007	+	0.389547i	$0.127368\pi$
$20$		0			0
$21$	6.09530	−	4.17604i	0.290253	−	0.198859i
$22$		0			0
$23$		−	19.5800i		−	0.851305i	−0.904887	−	0.425653i	$-0.860045\pi$
$23$		−	19.5800i		−	0.851305i	0.904887	−	0.425653i	$-0.139955\pi$
$24$		0			0
$25$		−	23.6479i		−	0.945915i
$26$		0			0
$27$	12.6030	+	12.6030i	0.466779	+	0.466779i
$28$		0			0
$29$	7.24407	+	7.24407i	0.249795	+	0.249795i	0.820887	−	0.571091i	$-0.193480\pi$
$29$	7.24407	+	7.24407i	0.249795	+	0.249795i	−0.571091	+	0.820887i	$0.693480\pi$
$30$		0			0
$31$			45.5117i			1.46812i	0.679084	+	0.734060i	$0.262377\pi$
$31$			45.5117i			1.46812i	−0.679084	+	0.734060i	$0.737623\pi$
$32$		0			0
$33$			3.84263i			0.116443i
$34$		0			0
$35$	4.60052	+	6.71488i	0.131444	+	0.191854i
$36$		0			0
$37$	−34.5793	+	34.5793i	−0.934575	+	0.934575i	−0.997987	−	0.0634120i	$-0.979802\pi$
$37$	−34.5793	+	34.5793i	−0.934575	+	0.934575i	0.0634120	+	0.997987i	$0.479802\pi$
$38$		0			0
$39$			22.3561i			0.573233i
$40$		0			0
$41$	61.5837			1.50204			0.751020	−	0.660279i	$-0.229562\pi$
$41$	61.5837			1.50204			0.751020	+	0.660279i	$0.229562\pi$
$42$		0			0
$43$	16.6424	−	16.6424i	0.387034	−	0.387034i	−0.486594	−	0.873628i	$-0.661761\pi$
$43$	16.6424	−	16.6424i	0.387034	−	0.387034i	0.873628	+	0.486594i	$0.161761\pi$
$44$		0			0
$45$	−6.48402	+	6.48402i	−0.144089	+	0.144089i
$46$		0			0
$47$			55.0421i			1.17111i	0.810633	+	0.585554i	$0.199123\pi$
$47$			55.0421i			1.17111i	−0.810633	+	0.585554i	$0.800877\pi$
$48$		0			0
$49$	45.6937	+	17.6941i	0.932525	+	0.361105i
$50$		0			0
$51$	15.4575	+	15.4575i	0.303088	+	0.303088i
$52$		0			0
$53$	38.2187	−	38.2187i	0.721107	−	0.721107i	−0.247724	−	0.968831i	$-0.579683\pi$
$53$	38.2187	−	38.2187i	0.721107	−	0.721107i	0.968831	+	0.247724i	$0.0796826\pi$
$54$		0			0
$55$	−4.23322			−0.0769677
$56$		0			0
$57$		−	15.0732i		−	0.264442i
$58$		0			0
$59$	−32.2679	−	32.2679i	−0.546914	−	0.546914i	0.378633	−	0.925547i	$-0.376394\pi$
$59$	−32.2679	−	32.2679i	−0.546914	−	0.546914i	−0.925547	+	0.378633i	$0.876394\pi$
$60$		0			0
$61$	−32.0297	+	32.0297i	−0.525078	+	0.525078i	−0.919101	−	0.394023i	$-0.871083\pi$
$61$	−32.0297	+	32.0297i	−0.525078	+	0.525078i	0.394023	+	0.919101i	$0.371083\pi$
$62$		0			0
$63$	−10.1392	+	54.2620i	−0.160940	+	0.861301i
$64$		0			0
$65$	−24.6286			−0.378901
$66$		0			0
$67$	−42.1559	−	42.1559i	−0.629192	−	0.629192i	0.318673	−	0.947865i	$-0.396763\pi$
$67$	−42.1559	−	42.1559i	−0.629192	−	0.629192i	−0.947865	+	0.318673i	$0.896763\pi$
$68$		0			0
$69$	−14.6139	−	14.6139i	−0.211795	−	0.211795i
$70$		0			0
$71$		−	15.7491i		−	0.221819i	−0.993831	−	0.110909i	$-0.964624\pi$
$71$		−	15.7491i		−	0.221819i	0.993831	−	0.110909i	$-0.0353763\pi$
$72$		0			0
$73$	65.7339			0.900464			0.450232	−	0.892912i	$-0.351341\pi$
$73$	65.7339			0.900464			0.450232	+	0.892912i	$0.351341\pi$
$74$		0			0
$75$	−17.6500	−	17.6500i	−0.235333	−	0.235333i
$76$		0			0
$77$	−21.0228	+	14.4032i	−0.273023	+	0.187055i
$78$		0			0
$79$	96.5103			1.22165			0.610825	−	0.791766i	$-0.290838\pi$
$79$	96.5103			1.22165			0.610825	+	0.791766i	$0.290838\pi$
$80$		0			0
$81$	−52.1600			−0.643950
$82$		0			0
$83$	31.6530	−	31.6530i	0.381361	−	0.381361i	−0.490231	−	0.871592i	$-0.663088\pi$
$83$	31.6530	−	31.6530i	0.381361	−	0.381361i	0.871592	+	0.490231i	$0.163088\pi$
$84$		0			0
$85$	−17.0287	+	17.0287i	−0.200338	+	0.200338i
$86$		0			0
$87$	10.8134			0.124292
$88$		0			0
$89$	89.3246			1.00365			0.501824	−	0.864970i	$-0.332663\pi$
$89$	89.3246			1.00365			0.501824	+	0.864970i	$0.332663\pi$
$90$		0			0
$91$	−122.309	+	83.7968i	−1.34406	+	0.920844i
$92$		0			0
$93$	33.9684	+	33.9684i	0.365252	+	0.365252i
$94$		0			0
$95$	16.6054			0.174793
$96$		0			0
$97$		−	47.6196i		−	0.490923i	−0.969406	−	0.245462i	$-0.921060\pi$
$97$		−	47.6196i		−	0.490923i	0.969406	−	0.245462i	$-0.0789395\pi$
$98$		0			0
$99$	−20.3000	−	20.3000i	−0.205051	−	0.205051i
$100$		0			0
$101$	−57.7284	−	57.7284i	−0.571568	−	0.571568i	0.360999	−	0.932566i	$-0.382436\pi$
$101$	−57.7284	−	57.7284i	−0.571568	−	0.571568i	−0.932566	+	0.360999i	$0.882436\pi$
$102$		0			0
$103$	−54.7987			−0.532026			−0.266013	−	0.963969i	$-0.585706\pi$
$103$	−54.7987			−0.532026			−0.266013	+	0.963969i	$0.585706\pi$
$104$		0			0
$105$	8.44542	+	1.57808i	0.0804326	+	0.0150294i
$106$		0			0
$107$	53.0000	−	53.0000i	0.495327	−	0.495327i	−0.414653	−	0.909980i	$-0.636097\pi$
$107$	53.0000	−	53.0000i	0.495327	−	0.495327i	0.909980	+	0.414653i	$0.136097\pi$
$108$		0			0
$109$	−55.0097	−	55.0097i	−0.504676	−	0.504676i	0.408211	−	0.912888i	$-0.366153\pi$
$109$	−55.0097	−	55.0097i	−0.504676	−	0.504676i	−0.912888	+	0.408211i	$0.866153\pi$
$110$		0			0
$111$			51.6176i			0.465023i
$112$		0			0
$113$	6.79908			0.0601689			0.0300844	−	0.999547i	$-0.490422\pi$
$113$	6.79908			0.0601689			0.0300844	+	0.999547i	$0.490422\pi$
$114$		0			0
$115$	16.0993	−	16.0993i	0.139994	−	0.139994i
$116$		0			0
$117$	−118.104	−	118.104i	−1.00944	−	1.00944i
$118$		0			0
$119$	−26.6282	+	142.506i	−0.223766	+	1.19753i
$120$		0			0
$121$			107.747i			0.890469i
$122$		0			0
$123$	45.9639	−	45.9639i	0.373690	−	0.373690i
$124$		0			0
$125$	39.9998	−	39.9998i	0.319999	−	0.319999i
$126$		0			0
$127$	−148.279			−1.16755			−0.583775	−	0.811916i	$-0.698425\pi$
$127$	−148.279			−1.16755			−0.583775	+	0.811916i	$0.698425\pi$
$128$		0			0
$129$		−	24.8427i		−	0.192579i
$130$		0			0
$131$	104.999	−	104.999i	0.801520	−	0.801520i	−0.181813	−	0.983333i	$-0.558197\pi$
$131$	104.999	−	104.999i	0.801520	−	0.801520i	0.983333	+	0.181813i	$0.0581966\pi$
$132$		0			0
$133$	82.4647	−	56.4985i	0.620035	−	0.424801i
$134$		0			0
$135$			20.7252i			0.153520i
$136$		0			0
$137$		−	19.9653i		−	0.145732i	−0.997342	−	0.0728660i	$-0.976785\pi$
$137$		−	19.9653i		−	0.145732i	0.997342	−	0.0728660i	$-0.0232145\pi$
$138$		0			0
$139$	−166.512	−	166.512i	−1.19793	−	1.19793i	−0.974787	−	0.223139i	$-0.928370\pi$
$139$	−166.512	−	166.512i	−1.19793	−	1.19793i	−0.223139	−	0.974787i	$-0.571630\pi$
$140$		0			0
$141$	41.0815	+	41.0815i	0.291358	+	0.291358i
$142$		0			0
$143$		−	77.1066i		−	0.539207i
$144$		0			0
$145$			11.9126i			0.0821559i
$146$		0			0
$147$	47.3105	−	20.8979i	0.321840	−	0.142163i
$148$		0			0
$149$	−128.482	+	128.482i	−0.862292	+	0.862292i	−0.991604	−	0.129312i	$-0.958723\pi$
$149$	−128.482	+	128.482i	−0.862292	+	0.862292i	0.129312	+	0.991604i	$0.458723\pi$
$150$		0			0
$151$		−	254.970i		−	1.68854i	−0.535915	−	0.844272i	$-0.680033\pi$
$151$		−	254.970i		−	1.68854i	0.535915	−	0.844272i	$-0.319967\pi$
$152$		0			0
$153$	−163.319			−1.06745
$154$		0			0
$155$	−37.4212	+	37.4212i	−0.241427	+	0.241427i
$156$		0			0
$157$	−106.743	+	106.743i	−0.679893	+	0.679893i	−0.959976	−	0.280083i	$-0.909638\pi$
$157$	−106.743	+	106.743i	−0.679893	+	0.679893i	0.280083	+	0.959976i	$0.409638\pi$
$158$		0			0
$159$		−	57.0502i		−	0.358806i
$160$		0			0
$161$	25.1749	−	134.728i	0.156366	−	0.836822i
$162$		0			0
$163$	121.521	+	121.521i	0.745529	+	0.745529i	0.973636	−	0.228107i	$-0.0732537\pi$
$163$	121.521	+	121.521i	0.745529	+	0.745529i	−0.228107	+	0.973636i	$0.573254\pi$
$164$		0			0
$165$	−3.15953	+	3.15953i	−0.0191487	+	0.0191487i
$166$		0			0
$167$	168.178			1.00705			0.503527	−	0.863979i	$-0.332035\pi$
$167$	168.178			1.00705			0.503527	+	0.863979i	$0.332035\pi$
$168$		0			0
$169$		−	279.600i		−	1.65444i
$170$		0			0
$171$	79.6295	+	79.6295i	0.465670	+	0.465670i
$172$		0			0
$173$	80.3214	−	80.3214i	0.464285	−	0.464285i	−0.435772	−	0.900057i	$-0.643525\pi$
$173$	80.3214	−	80.3214i	0.464285	−	0.464285i	0.900057	+	0.435772i	$0.143525\pi$
$174$		0			0
$175$	30.4051	−	162.719i	0.173743	−	0.929822i
$176$		0			0
$177$	−48.1673			−0.272132
$178$		0			0
$179$	−186.398	−	186.398i	−1.04133	−	1.04133i	−0.999108	−	0.0422189i	$-0.986557\pi$
$179$	−186.398	−	186.398i	−1.04133	−	1.04133i	−0.0422189	−	0.999108i	$-0.513443\pi$
$180$		0			0
$181$	−14.9372	−	14.9372i	−0.0825261	−	0.0825261i	0.664639	−	0.747165i	$-0.268585\pi$
$181$	−14.9372	−	14.9372i	−0.0825261	−	0.0825261i	−0.747165	+	0.664639i	$0.768585\pi$
$182$		0			0
$183$			47.8118i			0.261267i
$184$		0			0
$185$	−56.8644			−0.307375
$186$		0			0
$187$	−53.3131	−	53.3131i	−0.285097	−	0.285097i
$188$		0			0
$189$	70.5161	+	102.925i	0.373101	+	0.544574i
$190$		0			0
$191$	−22.6772			−0.118729			−0.0593645	−	0.998236i	$-0.518907\pi$
$191$	−22.6772			−0.118729			−0.0593645	+	0.998236i	$0.518907\pi$
$192$		0			0
$193$	−13.2053			−0.0684213			−0.0342107	−	0.999415i	$-0.510892\pi$
$193$	−13.2053			−0.0684213			−0.0342107	+	0.999415i	$0.510892\pi$
$194$		0			0
$195$	−18.3819	+	18.3819i	−0.0942662	+	0.0942662i
$196$		0			0
$197$	44.3731	−	44.3731i	0.225244	−	0.225244i	−0.585458	−	0.810703i	$-0.699085\pi$
$197$	44.3731	−	44.3731i	0.225244	−	0.225244i	0.810703	+	0.585458i	$0.199085\pi$
$198$		0			0
$199$	−116.110			−0.583468			−0.291734	−	0.956499i	$-0.594232\pi$
$199$	−116.110			−0.583468			−0.291734	+	0.956499i	$0.594232\pi$
$200$		0			0
$201$	−62.9274			−0.313071
$202$		0			0
$203$	40.5317	+	59.1597i	0.199664	+	0.291427i
$204$		0			0
$205$	50.6360	+	50.6360i	0.247005	+	0.247005i
$206$		0			0
$207$	154.406			0.745921
$208$		0			0
$209$			51.9877i			0.248745i
$210$		0			0
$211$	123.830	+	123.830i	0.586873	+	0.586873i	0.936783	−	0.349910i	$-0.113788\pi$
$211$	123.830	+	123.830i	0.586873	+	0.586873i	−0.349910	+	0.936783i	$0.613788\pi$
$212$		0			0
$213$	−11.7546	−	11.7546i	−0.0551859	−	0.0551859i
$214$		0			0
$215$	27.3679			0.127293
$216$		0			0
$217$	−58.5164	+	313.162i	−0.269661	+	1.44314i
$218$		0			0
$219$	49.0615	−	49.0615i	0.224025	−	0.224025i
$220$		0			0
$221$	−310.172	−	310.172i	−1.40349	−	1.40349i
$222$		0			0
$223$		−	208.501i		−	0.934980i	−0.883998	−	0.467490i	$-0.845158\pi$
$223$		−	208.501i		−	0.934980i	0.883998	−	0.467490i	$-0.154842\pi$
$224$		0			0
$225$	186.484			0.828819
$226$		0			0
$227$	−38.5672	+	38.5672i	−0.169900	+	0.169900i	−0.786935	−	0.617036i	$-0.788333\pi$
$227$	−38.5672	+	38.5672i	−0.169900	+	0.169900i	0.617036	+	0.786935i	$0.288333\pi$
$228$		0			0
$229$	239.869	+	239.869i	1.04746	+	1.04746i	0.998816	+	0.0486463i	$0.0154907\pi$
$229$	239.869	+	239.869i	1.04746	+	1.04746i	0.0486463	+	0.998816i	$0.484509\pi$
$230$		0			0
$231$	−4.94063	+	26.4408i	−0.0213880	+	0.114462i
$232$		0			0
$233$			187.643i			0.805335i	0.915346	+	0.402667i	$0.131917\pi$
$233$			187.643i			0.805335i	−0.915346	+	0.402667i	$0.868083\pi$
$234$		0			0
$235$	−45.2573	+	45.2573i	−0.192584	+	0.192584i
$236$		0			0
$237$	72.0320	−	72.0320i	0.303932	−	0.303932i
$238$		0			0
$239$	−334.909			−1.40129			−0.700646	−	0.713509i	$-0.747105\pi$
$239$	−334.909			−1.40129			−0.700646	+	0.713509i	$0.747105\pi$
$240$		0			0
$241$			20.7818i			0.0862314i	0.999070	+	0.0431157i	$0.0137284\pi$
$241$			20.7818i			0.0862314i	−0.999070	+	0.0431157i	$0.986272\pi$
$242$		0			0
$243$	−152.358	+	152.358i	−0.626986	+	0.626986i
$244$		0			0
$245$	23.0222	+	52.1195i	0.0939680	+	0.212733i
$246$		0			0
$247$			302.461i			1.22454i
$248$		0			0
$249$		−	47.2494i		−	0.189757i
$250$		0			0
$251$	331.900	+	331.900i	1.32231	+	1.32231i	0.911901	+	0.410410i	$0.134615\pi$
$251$	331.900	+	331.900i	1.32231	+	1.32231i	0.410410	+	0.911901i	$0.365385\pi$
$252$		0			0
$253$	50.4034	+	50.4034i	0.199223	+	0.199223i
$254$		0			0
$255$			25.4193i			0.0996834i
$256$		0			0
$257$		−	4.98473i		−	0.0193958i	−0.999953	−	0.00969792i	$-0.996913\pi$
$257$		−	4.98473i		−	0.0193958i	0.999953	−	0.00969792i	$-0.00308699\pi$
$258$		0			0
$259$	−282.397	+	193.477i	−1.09034	+	0.747015i
$260$		0			0
$261$	−57.1258	+	57.1258i	−0.218873	+	0.218873i
$262$		0			0
$263$		−	300.720i		−	1.14342i	−0.820455	−	0.571711i	$-0.806280\pi$
$263$		−	300.720i		−	1.14342i	0.820455	−	0.571711i	$-0.193720\pi$
$264$		0			0
$265$	62.8492			0.237167
$266$		0			0
$267$	66.6688	−	66.6688i	0.249696	−	0.249696i
$268$		0			0
$269$	337.937	−	337.937i	1.25627	−	1.25627i	0.303411	−	0.952860i	$-0.401875\pi$
$269$	337.937	−	337.937i	1.25627	−	1.25627i	0.952860	−	0.303411i	$-0.0981253\pi$
$270$		0			0
$271$			170.823i			0.630342i	0.949035	+	0.315171i	$0.102062\pi$
$271$			170.823i			0.630342i	−0.949035	+	0.315171i	$0.897938\pi$
$272$		0			0
$273$	−28.7442	+	153.830i	−0.105290	+	0.563481i
$274$		0			0
$275$	60.8750	+	60.8750i	0.221364	+	0.221364i
$276$		0			0
$277$	177.959	−	177.959i	0.642451	−	0.642451i	−0.308707	−	0.951157i	$-0.599896\pi$
$277$	177.959	−	177.959i	0.642451	−	0.642451i	0.951157	+	0.308707i	$0.0998962\pi$
$278$		0			0
$279$	−358.900			−1.28638
$280$		0			0
$281$		−	451.491i		−	1.60673i	−0.595486	−	0.803366i	$-0.703041\pi$
$281$		−	451.491i		−	1.60673i	0.595486	−	0.803366i	$-0.296959\pi$
$282$		0			0
$283$	−29.1197	−	29.1197i	−0.102897	−	0.102897i	0.653784	−	0.756681i	$-0.273180\pi$
$283$	−29.1197	−	29.1197i	−0.102897	−	0.102897i	−0.756681	+	0.653784i	$0.773180\pi$
$284$		0			0
$285$	12.3937	−	12.3937i	0.0434866	−	0.0434866i
$286$		0			0
$287$	423.751	+	79.1807i	1.47649	+	0.275891i
$288$		0			0
$289$	−139.918			−0.484147
$290$		0			0
$291$	−35.5416	−	35.5416i	−0.122136	−	0.122136i
$292$		0			0
$293$	264.180	+	264.180i	0.901637	+	0.901637i	0.995578	−	0.0939410i	$-0.0299465\pi$
$293$	264.180	+	264.180i	0.901637	+	0.901637i	−0.0939410	+	0.995578i	$0.529947\pi$
$294$		0			0
$295$		−	53.0634i		−	0.179876i
$296$		0			0
$297$	−64.8862			−0.218472
$298$		0			0
$299$	293.243	+	293.243i	0.980746	+	0.980746i
$300$		0			0
$301$	135.913	−	93.1172i	0.451538	−	0.309360i
$302$		0			0
$303$	−86.1729			−0.284399
$304$		0			0
$305$	−52.6717			−0.172694
$306$		0			0
$307$	−402.236	+	402.236i	−1.31022	+	1.31022i	−0.388962	+	0.921254i	$0.627166\pi$
$307$	−402.236	+	402.236i	−1.31022	+	1.31022i	−0.921254	+	0.388962i	$0.872834\pi$
$308$		0			0
$309$	−40.8998	+	40.8998i	−0.132362	+	0.132362i
$310$		0			0
$311$	352.364			1.13300			0.566502	−	0.824060i	$-0.308296\pi$
$311$	352.364			1.13300			0.566502	+	0.824060i	$0.308296\pi$
$312$		0			0
$313$	−290.355			−0.927652			−0.463826	−	0.885926i	$-0.653524\pi$
$313$	−290.355			−0.927652			−0.463826	+	0.885926i	$0.653524\pi$
$314$		0			0
$315$	−52.9527	+	36.2792i	−0.168104	+	0.115172i
$316$		0			0
$317$	334.727	+	334.727i	1.05592	+	1.05592i	0.998341	+	0.0575804i	$0.0183386\pi$
$317$	334.727	+	334.727i	1.05592	+	1.05592i	0.0575804	+	0.998341i	$0.481661\pi$
$318$		0			0
$319$	−37.2957			−0.116915
$320$		0			0
$321$		−	79.1147i		−	0.246463i
$322$		0			0
$323$	209.128	+	209.128i	0.647454	+	0.647454i
$324$		0			0
$325$	354.166	+	354.166i	1.08974	+	1.08974i
$326$		0			0
$327$	−82.1147			−0.251115
$328$		0			0
$329$	−70.7699	+	378.739i	−0.215106	+	1.15118i
$330$		0			0
$331$	−33.5183	+	33.5183i	−0.101264	+	0.101264i	−0.755924	−	0.654660i	$-0.772812\pi$
$331$	−33.5183	+	33.5183i	−0.101264	+	0.101264i	0.654660	+	0.755924i	$0.272812\pi$
$332$		0			0
$333$	−272.688	−	272.688i	−0.818883	−	0.818883i
$334$		0			0
$335$		−	69.3238i		−	0.206937i
$336$		0			0
$337$	−15.4008			−0.0456997			−0.0228499	−	0.999739i	$-0.507274\pi$
$337$	−15.4008			−0.0456997			−0.0228499	+	0.999739i	$0.507274\pi$
$338$		0			0
$339$	5.07460	−	5.07460i	0.0149693	−	0.0149693i
$340$		0			0
$341$	−117.158	−	117.158i	−0.343571	−	0.343571i
$342$		0			0
$343$	291.664	+	180.502i	0.850333	+	0.526245i
$344$		0			0
$345$		−	24.0320i		−	0.0696578i
$346$		0			0
$347$	114.114	−	114.114i	0.328859	−	0.328859i	−0.523294	−	0.852153i	$-0.675297\pi$
$347$	114.114	−	114.114i	0.328859	−	0.328859i	0.852153	+	0.523294i	$0.175297\pi$
$348$		0			0
$349$	−374.354	+	374.354i	−1.07265	+	1.07265i	−0.0755012	+	0.997146i	$0.524056\pi$
$349$	−374.354	+	374.354i	−1.07265	+	1.07265i	−0.997146	+	0.0755012i	$0.975944\pi$
$350$		0			0
$351$	−377.502			−1.07551
$352$		0			0
$353$			63.2575i			0.179200i	0.995978	+	0.0895999i	$0.0285588\pi$
$353$			63.2575i			0.179200i	−0.995978	+	0.0895999i	$0.971441\pi$
$354$		0			0
$355$	12.9494	−	12.9494i	0.0364773	−	0.0364773i
$356$		0			0
$357$	86.4872	+	126.236i	0.242261	+	0.353602i
$358$		0			0
$359$			24.7880i			0.0690472i	0.999404	+	0.0345236i	$0.0109914\pi$
$359$			24.7880i			0.0690472i	−0.999404	+	0.0345236i	$0.989009\pi$
$360$		0			0
$361$			157.071i			0.435101i
$362$		0			0
$363$	80.4184	+	80.4184i	0.221538	+	0.221538i
$364$		0			0
$365$	54.0485	+	54.0485i	0.148078	+	0.148078i
$366$		0			0
$367$		−	150.362i		−	0.409705i	−0.978793	−	0.204852i	$-0.934329\pi$
$367$		−	150.362i		−	0.409705i	0.978793	−	0.204852i	$-0.0656715\pi$
$368$		0			0
$369$			485.641i			1.31610i
$370$		0			0
$371$	312.118	−	213.840i	0.841289	−	0.576387i
$372$		0			0
$373$	100.362	−	100.362i	0.269068	−	0.269068i	−0.559657	−	0.828724i	$-0.689067\pi$
$373$	100.362	−	100.362i	0.269068	−	0.269068i	0.828724	+	0.559657i	$0.189067\pi$
$374$		0			0
$375$		−	59.7090i		−	0.159224i
$376$		0			0
$377$	−216.984			−0.575553
$378$		0			0
$379$	377.568	−	377.568i	0.996222	−	0.996222i	−0.00377077	−	0.999993i	$-0.501200\pi$
$379$	377.568	−	377.568i	0.996222	−	0.996222i	0.999993	+	0.00377077i	$0.00120028\pi$
$380$		0			0
$381$	−110.670	+	110.670i	−0.290473	+	0.290473i
$382$		0			0
$383$		−	453.557i		−	1.18422i	−0.805856	−	0.592111i	$-0.798295\pi$
$383$		−	453.557i		−	1.18422i	0.805856	−	0.592111i	$-0.201705\pi$
$384$		0			0
$385$	−29.1284	−	5.44283i	−0.0756582	−	0.0141372i
$386$		0			0
$387$	131.240	+	131.240i	0.339122	+	0.339122i
$388$		0			0
$389$	406.780	−	406.780i	1.04571	−	1.04571i	0.0468024	−	0.998904i	$-0.485097\pi$
$389$	406.780	−	406.780i	1.04571	−	1.04571i	0.998904	−	0.0468024i	$-0.0149031\pi$
$390$		0			0
$391$	405.509			1.03711
$392$		0			0
$393$		−	156.735i		−	0.398818i
$394$		0			0
$395$	79.3538	+	79.3538i	0.200896	+	0.200896i
$396$		0			0
$397$	−94.3830	+	94.3830i	−0.237741	+	0.237741i	−0.815914	−	0.578173i	$-0.803766\pi$
$397$	−94.3830	+	94.3830i	−0.237741	+	0.237741i	0.578173	+	0.815914i	$0.303766\pi$
$398$		0			0
$399$	19.3802	−	103.717i	0.0485721	−	0.259943i
$400$		0			0
$401$	703.081			1.75332			0.876660	−	0.481110i	$-0.159766\pi$
$401$	703.081			1.75332			0.876660	+	0.481110i	$0.159766\pi$
$402$		0			0
$403$	−681.613	−	681.613i	−1.69135	−	1.69135i
$404$		0			0
$405$	−42.8876	−	42.8876i	−0.105895	−	0.105895i
$406$		0			0
$407$		−	178.030i		−	0.437420i
$408$		0			0
$409$	−472.432			−1.15509			−0.577546	−	0.816358i	$-0.695989\pi$
$409$	−472.432			−1.15509			−0.577546	+	0.816358i	$0.695989\pi$
$410$		0			0
$411$	−14.9014	−	14.9014i	−0.0362564	−	0.0362564i
$412$		0			0
$413$	−180.544	−	263.521i	−0.437153	−	0.638065i
$414$		0			0
$415$	52.0522			0.125427
$416$		0			0
$417$	−248.557			−0.596060
$418$		0			0
$419$	356.549	−	356.549i	0.850953	−	0.850953i	−0.139298	−	0.990251i	$-0.544485\pi$
$419$	356.549	−	356.549i	0.850953	−	0.850953i	0.990251	+	0.139298i	$0.0444845\pi$
$420$		0			0
$421$	−458.909	+	458.909i	−1.09004	+	1.09004i	−0.0945219	+	0.995523i	$0.530132\pi$
$421$	−458.909	+	458.909i	−1.09004	+	1.09004i	−0.995523	+	0.0945219i	$0.969868\pi$
$422$		0			0
$423$	−434.055			−1.02613
$424$		0			0
$425$	489.756			1.15237
$426$		0			0
$427$	−261.576	+	179.212i	−0.612589	+	0.419699i
$428$		0			0
$429$	−57.5497	−	57.5497i	−0.134148	−	0.134148i
$430$		0			0
$431$	509.215			1.18147			0.590736	−	0.806865i	$-0.298837\pi$
$431$	509.215			1.18147			0.590736	+	0.806865i	$0.298837\pi$
$432$		0			0
$433$			105.687i			0.244081i	0.992525	+	0.122041i	$0.0389438\pi$
$433$			105.687i			0.244081i	−0.992525	+	0.122041i	$0.961056\pi$
$434$		0			0
$435$	8.89116	+	8.89116i	0.0204394	+	0.0204394i
$436$		0			0
$437$	−197.714	−	197.714i	−0.452435	−	0.452435i
$438$		0			0
$439$	129.713			0.295473			0.147736	−	0.989027i	$-0.452801\pi$
$439$	129.713			0.295473			0.147736	+	0.989027i	$0.452801\pi$
$440$		0			0
$441$	−139.534	+	360.335i	−0.316403	+	0.817087i
$442$		0			0
$443$	−446.226	+	446.226i	−1.00728	+	1.00728i	−0.00730994	+	0.999973i	$0.502327\pi$
$443$	−446.226	+	446.226i	−1.00728	+	1.00728i	−0.999973	+	0.00730994i	$0.997673\pi$
$444$		0			0
$445$	73.4455	+	73.4455i	0.165046	+	0.165046i
$446$		0			0
$447$			191.788i			0.429057i
$448$		0			0
$449$	−578.506			−1.28843			−0.644216	−	0.764843i	$-0.722816\pi$
$449$	−578.506			−1.28843			−0.644216	+	0.764843i	$0.722816\pi$
$450$		0			0
$451$	−158.530	+	158.530i	−0.351509	+	0.351509i
$452$		0			0
$453$	−190.301	−	190.301i	−0.420090	−	0.420090i
$454$		0			0
$455$	−169.467	−	31.6660i	−0.372454	−	0.0695955i
$456$		0			0
$457$		−	432.070i		−	0.945450i	−0.881210	−	0.472725i	$-0.843270\pi$
$457$		−	432.070i		−	0.945450i	0.881210	−	0.472725i	$-0.156730\pi$
$458$		0			0
$459$	−261.013	+	261.013i	−0.568656	+	0.568656i
$460$		0			0
$461$	434.353	−	434.353i	0.942198	−	0.942198i	−0.0562205	−	0.998418i	$-0.517905\pi$
$461$	434.353	−	434.353i	0.942198	−	0.942198i	0.998418	+	0.0562205i	$0.0179050\pi$
$462$		0			0
$463$	−93.7911			−0.202573			−0.101286	−	0.994857i	$-0.532296\pi$
$463$	−93.7911			−0.202573			−0.101286	+	0.994857i	$0.532296\pi$
$464$		0			0
$465$			55.8598i			0.120129i
$466$		0			0
$467$	−67.8183	+	67.8183i	−0.145221	+	0.145221i	−0.775979	−	0.630758i	$-0.782744\pi$
$467$	−67.8183	+	67.8183i	−0.145221	+	0.145221i	0.630758	+	0.775979i	$0.282744\pi$
$468$		0			0
$469$	−235.869	−	344.272i	−0.502919	−	0.734056i
$470$		0			0
$471$			159.339i			0.338299i
$472$		0			0
$473$			85.6829i			0.181148i
$474$		0			0
$475$	−238.790	−	238.790i	−0.502716	−	0.502716i
$476$		0			0
$477$	301.388	+	301.388i	0.631840	+	0.631840i
$478$		0			0
$479$			113.366i			0.236673i	0.992974	+	0.118337i	$0.0377562\pi$
$479$			113.366i			0.236673i	−0.992974	+	0.118337i	$0.962244\pi$
$480$		0			0
$481$		−	1035.76i		−	2.15335i
$482$		0			0
$483$	−81.7669	−	119.346i	−0.169290	−	0.247094i
$484$		0			0
$485$	39.1543	−	39.1543i	0.0807306	−	0.0807306i
$486$		0			0
$487$			132.462i			0.271996i	0.990709	+	0.135998i	$0.0434240\pi$
$487$			132.462i			0.271996i	−0.990709	+	0.135998i	$0.956576\pi$
$488$		0			0
$489$	181.398			0.370958
$490$		0			0
$491$	614.010	−	614.010i	1.25053	−	1.25053i	0.295047	−	0.955483i	$-0.404665\pi$
$491$	614.010	−	614.010i	1.25053	−	1.25053i	0.955483	−	0.295047i	$-0.0953353\pi$
$492$		0			0
$493$	−150.027	+	150.027i	−0.304315	+	0.304315i
$494$		0			0
$495$		−	33.3827i		−	0.0674398i
$496$		0			0
$497$	20.2493	−	108.368i	0.0407431	−	0.218045i
$498$		0			0
$499$	540.237	+	540.237i	1.08264	+	1.08264i	0.996262	+	0.0863773i	$0.0275290\pi$
$499$	540.237	+	540.237i	1.08264	+	1.08264i	0.0863773	+	0.996262i	$0.472471\pi$
$500$		0			0
$501$	125.522	−	125.522i	0.250543	−	0.250543i
$502$		0			0
$503$	−417.128			−0.829280			−0.414640	−	0.909986i	$-0.636093\pi$
$503$	−417.128			−0.829280			−0.414640	+	0.909986i	$0.636093\pi$
$504$		0			0
$505$		−	94.9322i		−	0.187985i
$506$		0			0
$507$	−208.684	−	208.684i	−0.411605	−	0.411605i
$508$		0			0
$509$	−23.6510	+	23.6510i	−0.0464657	+	0.0464657i	−0.729958	−	0.683492i	$-0.760460\pi$
$509$	−23.6510	+	23.6510i	−0.0464657	+	0.0464657i	0.683492	+	0.729958i	$0.260460\pi$
$510$		0			0
$511$	452.309	+	84.5168i	0.885144	+	0.165395i
$512$		0			0
$513$	254.524			0.496149
$514$		0			0
$515$	−45.0572	−	45.0572i	−0.0874897	−	0.0874897i
$516$		0			0
$517$	−141.691	−	141.691i	−0.274063	−	0.274063i
$518$		0			0
$519$		−	119.898i		−	0.231018i
$520$		0			0
$521$	877.949			1.68512			0.842562	−	0.538600i	$-0.181047\pi$
$521$	877.949			1.68512			0.842562	+	0.538600i	$0.181047\pi$
$522$		0			0
$523$	−510.059	−	510.059i	−0.975256	−	0.975256i	0.0244456	−	0.999701i	$-0.492218\pi$
$523$	−510.059	−	510.059i	−0.975256	−	0.975256i	−0.999701	+	0.0244456i	$0.992218\pi$
$524$		0			0
$525$	−98.7544	−	144.141i	−0.188104	−	0.274554i
$526$		0			0
$527$	−942.564			−1.78855
$528$		0			0
$529$	145.623			0.275279
$530$		0			0
$531$	254.461	−	254.461i	0.479211	−	0.479211i
$532$		0			0
$533$	−922.317	+	922.317i	−1.73043	+	1.73043i
$534$		0			0
$535$	87.1565			0.162909
$536$		0			0
$537$	−278.241			−0.518140
$538$		0			0
$539$	−163.175	+	72.0773i	−0.302736	+	0.133724i
$540$		0			0
$541$	−63.2455	−	63.2455i	−0.116905	−	0.116905i	0.646234	−	0.763139i	$-0.276343\pi$
$541$	−63.2455	−	63.2455i	−0.116905	−	0.116905i	−0.763139	+	0.646234i	$0.776343\pi$
$542$		0			0
$543$	−22.2973			−0.0410631
$544$		0			0
$545$		−	90.4615i		−	0.165984i
$546$		0			0
$547$	323.315	+	323.315i	0.591069	+	0.591069i	0.937920	−	0.346851i	$-0.112749\pi$
$547$	323.315	+	323.315i	0.591069	+	0.591069i	−0.346851	+	0.937920i	$0.612749\pi$
$548$		0			0
$549$	−252.583	−	252.583i	−0.460078	−	0.460078i
$550$		0			0
$551$	146.297			0.265512
$552$		0			0
$553$	664.078	+	124.087i	1.20086	+	0.224389i
$554$		0			0
$555$	−42.4416	+	42.4416i	−0.0764714	+	0.0764714i
$556$		0			0
$557$	−234.582	−	234.582i	−0.421153	−	0.421153i	0.464448	−	0.885600i	$-0.346253\pi$
$557$	−234.582	−	234.582i	−0.421153	−	0.421153i	−0.885600	+	0.464448i	$0.846253\pi$
$558$		0			0
$559$			498.496i			0.891764i
$560$		0			0
$561$	−79.5822			−0.141858
$562$		0			0
$563$	520.051	−	520.051i	0.923714	−	0.923714i	−0.0735757	−	0.997290i	$-0.523441\pi$
$563$	520.051	−	520.051i	0.923714	−	0.923714i	0.997290	+	0.0735757i	$0.0234410\pi$
$564$		0			0
$565$	5.59042	+	5.59042i	0.00989456	+	0.00989456i
$566$		0			0
$567$	−358.908	−	67.0642i	−0.632994	−	0.118279i
$568$		0			0
$569$		−	169.180i		−	0.297328i	−0.988888	−	0.148664i	$-0.952503\pi$
$569$		−	169.180i		−	0.297328i	0.988888	−	0.148664i	$-0.0474973\pi$
$570$		0			0
$571$	−273.184	+	273.184i	−0.478432	+	0.478432i	−0.904630	−	0.426198i	$-0.859853\pi$
$571$	−273.184	+	273.184i	−0.478432	+	0.478432i	0.426198	+	0.904630i	$0.359853\pi$
$572$		0			0
$573$	−16.9255	+	16.9255i	−0.0295384	+	0.0295384i
$574$		0			0
$575$	−463.026			−0.805262
$576$		0			0
$577$		−	1152.31i		−	1.99707i	−0.0540701	−	0.998537i	$-0.517219\pi$
$577$		−	1152.31i		−	1.99707i	0.0540701	−	0.998537i	$-0.482781\pi$
$578$		0			0
$579$	−9.85599	+	9.85599i	−0.0170224	+	0.0170224i
$580$		0			0
$581$	258.499	−	177.104i	0.444921	−	0.304826i
$582$		0			0
$583$			196.767i			0.337508i
$584$		0			0
$585$		−	194.218i		−	0.331996i
$586$		0			0
$587$	−594.972	−	594.972i	−1.01358	−	1.01358i	−0.999906	−	0.0136745i	$-0.995647\pi$
$587$	−594.972	−	594.972i	−1.01358	−	1.01358i	−0.0136745	−	0.999906i	$-0.504353\pi$
$588$		0			0
$589$	459.566	+	459.566i	0.780247	+	0.780247i
$590$		0			0
$591$		−	66.2371i		−	0.112076i
$592$		0			0
$593$			567.041i			0.956225i	0.878299	+	0.478112i	$0.158679\pi$
$593$			567.041i			0.956225i	−0.878299	+	0.478112i	$0.841321\pi$
$594$		0			0
$595$	−139.067	+	95.2784i	−0.233727	+	0.160132i
$596$		0			0
$597$	−86.6606	+	86.6606i	−0.145160	+	0.145160i
$598$		0			0
$599$			353.024i			0.589355i	0.955597	+	0.294678i	$0.0952123\pi$
$599$			353.024i			0.589355i	−0.955597	+	0.294678i	$0.904788\pi$
$600$		0			0
$601$	30.5796			0.0508811			0.0254406	−	0.999676i	$-0.491901\pi$
$601$	30.5796			0.0508811			0.0254406	+	0.999676i	$0.491901\pi$
$602$		0			0
$603$	332.436	−	332.436i	0.551303	−	0.551303i
$604$		0			0
$605$	−88.5928	+	88.5928i	−0.146434	+	0.146434i
$606$		0			0
$607$			622.168i			1.02499i	0.858691	+	0.512494i	$0.171278\pi$
$607$			622.168i			1.02499i	−0.858691	+	0.512494i	$0.828722\pi$
$608$		0			0
$609$	74.4063	+	13.9033i	0.122178	+	0.0228297i
$610$		0			0
$611$	−824.346	−	824.346i	−1.34917	−	1.34917i
$612$		0			0
$613$	375.828	−	375.828i	0.613097	−	0.613097i	−0.330655	−	0.943752i	$-0.607270\pi$
$613$	375.828	−	375.828i	0.613097	−	0.613097i	0.943752	+	0.330655i	$0.107270\pi$
$614$		0			0
$615$	75.5860			0.122904
$616$		0			0
$617$		−	724.934i		−	1.17493i	−0.809248	−	0.587467i	$-0.800125\pi$
$617$		−	724.934i		−	1.17493i	0.809248	−	0.587467i	$-0.199875\pi$
$618$		0			0
$619$	−67.3282	−	67.3282i	−0.108769	−	0.108769i	0.650628	−	0.759397i	$-0.274506\pi$
$619$	−67.3282	−	67.3282i	−0.108769	−	0.108769i	−0.759397	+	0.650628i	$0.774506\pi$
$620$		0			0
$621$	246.768	−	246.768i	0.397372	−	0.397372i
$622$		0			0
$623$	614.634	+	114.848i	0.986572	+	0.184347i
$624$		0			0
$625$	−525.418			−0.840670
$626$		0			0
$627$	38.8019	+	38.8019i	0.0618849	+	0.0618849i
$628$		0			0
$629$	−716.149	−	716.149i	−1.13855	−	1.13855i
$630$		0			0
$631$		−	941.200i		−	1.49160i	−0.666169	−	0.745801i	$-0.732067\pi$
$631$		−	941.200i		−	1.49160i	0.666169	−	0.745801i	$-0.267933\pi$
$632$		0			0
$633$	184.845			0.292014
$634$		0			0
$635$	−121.919	−	121.919i	−0.191999	−	0.191999i
$636$		0			0
$637$	−949.338	+	419.340i	−1.49033	+	0.658305i
$638$		0			0
$639$	124.196			0.194359
$640$		0			0
$641$	28.5724			0.0445748			0.0222874	−	0.999752i	$-0.492905\pi$
$641$	28.5724			0.0445748			0.0222874	+	0.999752i	$0.492905\pi$
$642$		0			0
$643$	30.2697	−	30.2697i	0.0470758	−	0.0470758i	−0.683177	−	0.730253i	$-0.739402\pi$
$643$	30.2697	−	30.2697i	0.0470758	−	0.0470758i	0.730253	+	0.683177i	$0.239402\pi$
$644$		0			0
$645$	20.4265	−	20.4265i	0.0316689	−	0.0316689i
$646$		0			0
$647$	−96.1146			−0.148554			−0.0742771	−	0.997238i	$-0.523665\pi$
$647$	−96.1146			−0.148554			−0.0742771	+	0.997238i	$0.523665\pi$
$648$		0			0
$649$	166.130			0.255978
$650$		0			0
$651$	190.059	+	277.408i	0.291949	+	0.426126i
$652$		0			0
$653$	299.672	+	299.672i	0.458916	+	0.458916i	0.898300	−	0.439383i	$-0.144803\pi$
$653$	299.672	+	299.672i	0.458916	+	0.458916i	−0.439383	+	0.898300i	$0.644803\pi$
$654$		0			0
$655$	172.667			0.263614
$656$		0			0
$657$			518.369i			0.788994i
$658$		0			0
$659$	499.672	+	499.672i	0.758228	+	0.758228i	0.976000	−	0.217772i	$-0.0698789\pi$
$659$	499.672	+	499.672i	0.758228	+	0.758228i	−0.217772	+	0.976000i	$0.569879\pi$
$660$		0			0
$661$	−368.251	−	368.251i	−0.557112	−	0.557112i	0.371372	−	0.928484i	$-0.378887\pi$
$661$	−368.251	−	368.251i	−0.557112	−	0.557112i	−0.928484	+	0.371372i	$0.878887\pi$
$662$		0			0
$663$	−463.003			−0.698345
$664$		0			0
$665$	114.260	+	21.3502i	0.171819	+	0.0321056i
$666$		0			0
$667$	141.839	−	141.839i	0.212652	−	0.212652i
$668$		0			0
$669$	−155.618	−	155.618i	−0.232612	−	0.232612i
$670$		0			0
$671$		−	164.904i		−	0.245758i
$672$		0			0
$673$	360.905			0.536262			0.268131	−	0.963382i	$-0.413594\pi$
$673$	360.905			0.536262			0.268131	+	0.963382i	$0.413594\pi$
$674$		0			0
$675$	298.035	−	298.035i	0.441533	−	0.441533i
$676$		0			0
$677$	388.404	+	388.404i	0.573713	+	0.573713i	0.933164	−	0.359451i	$-0.117036\pi$
$677$	388.404	+	388.404i	0.573713	+	0.573713i	−0.359451	+	0.933164i	$0.617036\pi$
$678$		0			0
$679$	61.2264	−	327.666i	0.0901715	−	0.482571i
$680$		0			0
$681$			57.5705i			0.0845382i
$682$		0			0
$683$	−614.349	+	614.349i	−0.899486	+	0.899486i	−0.995391	−	0.0959045i	$-0.969426\pi$
$683$	−614.349	+	614.349i	−0.899486	+	0.899486i	0.0959045	+	0.995391i	$0.469426\pi$
$684$		0			0
$685$	16.4161	−	16.4161i	0.0239651	−	0.0239651i
$686$		0			0
$687$	358.060			0.521193
$688$		0			0
$689$			1144.77i			1.66150i
$690$		0			0
$691$	370.027	−	370.027i	0.535496	−	0.535496i	−0.386707	−	0.922203i	$-0.626388\pi$
$691$	370.027	−	370.027i	0.535496	−	0.535496i	0.922203	+	0.386707i	$0.126388\pi$
$692$		0			0
$693$	−113.582	−	165.783i	−0.163899	−	0.239226i
$694$		0			0
$695$		−	273.822i		−	0.393989i
$696$		0			0
$697$			1275.42i			1.82987i
$698$		0			0
$699$	140.050	+	140.050i	0.200358	+	0.200358i
$700$		0			0
$701$	12.8864	+	12.8864i	0.0183829	+	0.0183829i	0.716238	−	0.697856i	$-0.245862\pi$
$701$	12.8864	+	12.8864i	0.0183829	+	0.0183829i	−0.697856	+	0.716238i	$0.745862\pi$
$702$		0			0
$703$			698.345i			0.993379i
$704$		0			0
$705$			67.5570i			0.0958256i
$706$		0			0
$707$	−323.000	−	471.447i	−0.456859	−	0.666828i
$708$		0			0
$709$	−30.4271	+	30.4271i	−0.0429155	+	0.0429155i	−0.728239	−	0.685323i	$-0.759661\pi$
$709$	−30.4271	+	30.4271i	−0.0429155	+	0.0429155i	0.685323	+	0.728239i	$0.259661\pi$
$710$		0			0
$711$			761.068i			1.07042i
$712$		0			0
$713$	891.121			1.24982
$714$		0			0
$715$	63.3995	−	63.3995i	0.0886706	−	0.0886706i
$716$		0			0
$717$	−249.964	+	249.964i	−0.348625	+	0.348625i
$718$		0			0
$719$			317.738i			0.441916i	0.975283	+	0.220958i	$0.0709184\pi$
$719$			317.738i			0.441916i	−0.975283	+	0.220958i	$0.929082\pi$
$720$		0			0
$721$	−377.064	−	70.4569i	−0.522974	−	0.0977211i
$722$		0			0
$723$	15.5108	+	15.5108i	0.0214534	+	0.0214534i
$724$		0			0
$725$	171.307	−	171.307i	0.236285	−	0.236285i
$726$		0			0
$727$	803.482			1.10520			0.552601	−	0.833446i	$-0.313635\pi$
$727$	803.482			1.10520			0.552601	+	0.833446i	$0.313635\pi$
$728$		0			0
$729$		−	242.011i		−	0.331976i
$730$		0			0
$731$	344.671	+	344.671i	0.471506	+	0.471506i
$732$		0			0
$733$	−553.720	+	553.720i	−0.755416	+	0.755416i	−0.975484	−	0.220069i	$-0.929372\pi$
$733$	−553.720	+	553.720i	−0.755416	+	0.755416i	0.220069	+	0.975484i	$0.429372\pi$
$734$		0			0
$735$	56.0832	+	21.7173i	0.0763036	+	0.0295473i
$736$		0			0
$737$	217.038			0.294488
$738$		0			0
$739$	−82.0530	−	82.0530i	−0.111032	−	0.111032i	0.649408	−	0.760440i	$-0.275017\pi$
$739$	−82.0530	−	82.0530i	−0.111032	−	0.111032i	−0.760440	+	0.649408i	$0.775017\pi$
$740$		0			0
$741$	225.746	+	225.746i	0.304651	+	0.304651i
$742$		0			0
$743$			1174.63i			1.58093i	0.612507	+	0.790465i	$0.290161\pi$
$743$			1174.63i			1.58093i	−0.612507	+	0.790465i	$0.709839\pi$
$744$		0			0
$745$	−211.283			−0.283602
$746$		0			0
$747$	249.612	+	249.612i	0.334152	+	0.334152i
$748$		0			0
$749$	432.832	−	296.543i	0.577880	−	0.395919i
$750$		0			0
$751$	40.2707			0.0536227			0.0268114	−	0.999641i	$-0.491465\pi$
$751$	40.2707			0.0536227			0.0268114	+	0.999641i	$0.491465\pi$
$752$		0			0
$753$	495.438			0.657952
$754$		0			0
$755$	209.645	−	209.645i	0.277675	−	0.277675i
$756$		0			0
$757$	55.7299	−	55.7299i	0.0736195	−	0.0736195i	−0.669338	−	0.742958i	$-0.733422\pi$
$757$	55.7299	−	55.7299i	0.0736195	−	0.0736195i	0.742958	+	0.669338i	$0.233422\pi$
$758$		0			0
$759$	75.2388			0.0991288
$760$		0			0
$761$	−97.8619			−0.128596			−0.0642982	−	0.997931i	$-0.520481\pi$
$761$	−97.8619			−0.128596			−0.0642982	+	0.997931i	$0.520481\pi$
$762$		0			0
$763$	−307.788	−	449.245i	−0.403392	−	0.588788i
$764$		0			0
$765$	−134.286	−	134.286i	−0.175538	−	0.175538i
$766$		0			0
$767$	966.531			1.26014
$768$		0			0
$769$		−	659.680i		−	0.857841i	−0.903342	−	0.428920i	$-0.858894\pi$
$769$		−	659.680i		−	0.857841i	0.903342	−	0.428920i	$-0.141106\pi$
$770$		0			0
$771$	−3.72043	−	3.72043i	−0.00482546	−	0.00482546i
$772$		0			0
$773$	756.470	+	756.470i	0.978616	+	0.978616i	0.999776	−	0.0211604i	$-0.00673606\pi$
$773$	756.470	+	756.470i	0.978616	+	0.978616i	−0.0211604	+	0.999776i	$0.506736\pi$
$774$		0			0
$775$	1076.26			1.38872
$776$		0			0
$777$	−66.3669	+	355.176i	−0.0854142	+	0.457112i
$778$		0			0
$779$	621.856	−	621.856i	0.798274	−	0.798274i
$780$		0			0
$781$	40.5418	+	40.5418i	0.0519101	+	0.0519101i
$782$		0			0
$783$			182.594i			0.233199i
$784$		0			0
$785$	−175.535			−0.223612
$786$		0			0
$787$	−1017.48	+	1017.48i	−1.29286	+	1.29286i	−0.359854	+	0.933009i	$0.617173\pi$
$787$	−1017.48	+	1017.48i	−1.29286	+	1.29286i	−0.933009	+	0.359854i	$0.882827\pi$
$788$		0			0
$789$	−224.447	−	224.447i	−0.284470	−	0.284470i
$790$		0			0
$791$	46.7839	+	8.74187i	0.0591452	+	0.0110517i
$792$		0			0
$793$		−	959.396i		−	1.20983i
$794$		0			0
$795$	46.9085	−	46.9085i	0.0590044	−	0.0590044i
$796$		0			0
$797$	−871.234	+	871.234i	−1.09314	+	1.09314i	−0.0979499	+	0.995191i	$0.531228\pi$
$797$	−871.234	+	871.234i	−1.09314	+	1.09314i	−0.995191	+	0.0979499i	$0.968772\pi$
$798$		0			0
$799$	−1139.94			−1.42671
$800$		0			0
$801$			704.403i			0.879404i
$802$		0			0
$803$	−169.214	+	169.214i	−0.210727	+	0.210727i
$804$		0			0
$805$	131.477	−	90.0783i	0.163326	−	0.111899i
$806$		0			0
$807$		−	504.449i		−	0.625091i
$808$		0			0
$809$			678.651i			0.838876i	0.907784	+	0.419438i	$0.137773\pi$
$809$			678.651i			0.838876i	−0.907784	+	0.419438i	$0.862227\pi$
$810$		0			0
$811$	733.218	+	733.218i	0.904092	+	0.904092i	0.995787	−	0.0916954i	$-0.0292286\pi$
$811$	733.218	+	733.218i	0.904092	+	0.904092i	−0.0916954	+	0.995787i	$0.529229\pi$
$812$		0			0
$813$	127.496	+	127.496i	0.156822	+	0.156822i
$814$		0			0
$815$			199.837i			0.245199i
$816$		0			0
$817$		−	336.102i		−	0.411386i
$818$		0			0
$819$	−660.811	−	964.514i	−0.806851	−	1.17767i
$820$		0			0
$821$	219.233	−	219.233i	0.267032	−	0.267032i	−0.560871	−	0.827903i	$-0.689534\pi$
$821$	219.233	−	219.233i	0.267032	−	0.267032i	0.827903	+	0.560871i	$0.189534\pi$
$822$		0			0
$823$		−	584.619i		−	0.710351i	−0.934800	−	0.355176i	$-0.884421\pi$
$823$		−	584.619i		−	0.710351i	0.934800	−	0.355176i	$-0.115579\pi$
$824$		0			0
$825$	90.8700			0.110145
$826$		0			0
$827$	−377.082	+	377.082i	−0.455964	+	0.455964i	−0.897328	−	0.441364i	$-0.854495\pi$
$827$	−377.082	+	377.082i	−0.455964	+	0.455964i	0.441364	+	0.897328i	$0.354495\pi$
$828$		0			0
$829$	−220.663	+	220.663i	−0.266179	+	0.266179i	−0.827559	−	0.561379i	$-0.810271\pi$
$829$	−220.663	+	220.663i	−0.266179	+	0.266179i	0.561379	+	0.827559i	$0.310271\pi$
$830$		0			0
$831$		−	265.645i		−	0.319669i
$832$		0			0
$833$	−366.452	+	946.333i	−0.439918	+	1.13605i
$834$		0			0
$835$	138.281	+	138.281i	0.165606	+	0.165606i
$836$		0			0
$837$	−573.586	+	573.586i	−0.685288	+	0.685288i
$838$		0			0
$839$	−1453.30			−1.73218			−0.866090	−	0.499888i	$-0.833374\pi$
$839$	−1453.30			−1.73218			−0.866090	+	0.499888i	$0.833374\pi$
$840$		0			0
$841$		−	736.047i		−	0.875205i
$842$		0			0
$843$	−336.978	−	336.978i	−0.399736	−	0.399736i
$844$		0			0
$845$	229.896	−	229.896i	0.272066	−	0.272066i
$846$		0			0
$847$	−138.534	+	741.395i	−0.163559	+	0.875319i
$848$		0			0
$849$	−43.4679			−0.0511990
$850$		0			0
$851$	677.063	+	677.063i	0.795609	+	0.795609i
$852$		0			0
$853$	454.447	+	454.447i	0.532763	+	0.532763i	0.921394	−	0.388630i	$-0.127052\pi$
$853$	454.447	+	454.447i	0.532763	+	0.532763i	−0.388630	+	0.921394i	$0.627052\pi$
$854$		0			0
$855$			130.948i			0.153155i
$856$		0			0
$857$	−884.451			−1.03203			−0.516016	−	0.856579i	$-0.672585\pi$
$857$	−884.451			−1.03203			−0.516016	+	0.856579i	$0.672585\pi$
$858$		0			0
$859$	496.836	+	496.836i	0.578389	+	0.578389i	0.934459	−	0.356071i	$-0.115884\pi$
$859$	496.836	+	496.836i	0.578389	+	0.578389i	−0.356071	+	0.934459i	$0.615884\pi$
$860$		0			0
$861$	375.371	−	257.176i	0.435971	−	0.298694i
$862$		0			0
$863$	−209.451			−0.242701			−0.121351	−	0.992610i	$-0.538723\pi$
$863$	−209.451			−0.242701			−0.121351	+	0.992610i	$0.538723\pi$
$864$		0			0
$865$	132.086			0.152700
$866$		0			0
$867$	−104.430	+	104.430i	−0.120450	+	0.120450i
$868$		0			0
$869$	−248.439	+	248.439i	−0.285891	+	0.285891i
$870$		0			0
$871$	1262.71			1.44972
$872$		0			0
$873$	375.522			0.430151
$874$		0			0
$875$	326.665	−	223.806i	0.373331	−	0.255778i
$876$		0			0
$877$	−410.184	−	410.184i	−0.467713	−	0.467713i	0.433460	−	0.901173i	$-0.357293\pi$
$877$	−410.184	−	410.184i	−0.467713	−	0.467713i	−0.901173	+	0.433460i	$0.857293\pi$
$878$		0			0
$879$	394.349			0.448634
$880$		0			0
$881$			270.639i			0.307195i	0.988134	+	0.153597i	$0.0490859\pi$
$881$			270.639i			0.307195i	−0.988134	+	0.153597i	$0.950914\pi$
$882$		0			0
$883$	455.994	+	455.994i	0.516414	+	0.516414i	0.916484	−	0.400070i	$-0.131014\pi$
$883$	455.994	+	455.994i	0.516414	+	0.516414i	−0.400070	+	0.916484i	$0.631014\pi$
$884$		0			0
$885$	−39.6047	−	39.6047i	−0.0447511	−	0.0447511i
$886$		0			0
$887$	−270.546			−0.305013			−0.152506	−	0.988303i	$-0.548734\pi$
$887$	−270.546			−0.305013			−0.152506	+	0.988303i	$0.548734\pi$
$888$		0			0
$889$	−1020.29	−	190.648i	−1.14769	−	0.214452i
$890$		0			0
$891$	134.272	−	134.272i	0.150698	−	0.150698i
$892$		0			0
$893$	555.800	+	555.800i	0.622397	+	0.622397i
$894$		0			0
$895$		−	306.524i		−	0.342485i
$896$		0			0
$897$	437.733			0.487997
$898$		0			0
$899$	−329.690	+	329.690i	−0.366730	+	0.366730i
$900$		0			0
$901$	791.522	+	791.522i	0.878492	+	0.878492i
$902$		0			0
$903$	31.9413	−	170.940i	0.0353724	−	0.189303i
$904$		0			0
$905$		−	24.5637i		−	0.0271422i
$906$		0			0
$907$	−392.452	+	392.452i	−0.432693	+	0.432693i	−0.889543	−	0.456851i	$-0.848977\pi$
$907$	−392.452	+	392.452i	−0.432693	+	0.432693i	0.456851	+	0.889543i	$0.348977\pi$
$908$		0			0
$909$	455.239	−	455.239i	0.500813	−	0.500813i
$910$		0			0
$911$	−1096.68			−1.20382			−0.601911	−	0.798563i	$-0.705594\pi$
$911$	−1096.68			−1.20382			−0.601911	+	0.798563i	$0.705594\pi$
$912$		0			0
$913$			162.964i			0.178493i
$914$		0			0
$915$	−39.3124	+	39.3124i	−0.0429643	+	0.0429643i
$916$		0			0
$917$	857.491	−	587.487i	0.935105	−	0.640662i
$918$		0			0
$919$			141.891i			0.154397i	0.997016	+	0.0771984i	$0.0245975\pi$
$919$			141.891i			0.154397i	−0.997016	+	0.0771984i	$0.975403\pi$
$920$		0			0
$921$			600.431i			0.651933i
$922$		0			0
$923$	235.869	+	235.869i	0.255546	+	0.255546i
$924$		0			0
$925$	817.727	+	817.727i	0.884029	+	0.884029i
$926$		0			0
$927$		−	432.136i		−	0.466166i
$928$		0			0
$929$		−	173.652i		−	0.186923i	−0.995623	−	0.0934617i	$-0.970207\pi$
$929$		−	173.652i		−	0.186923i	0.995623	−	0.0934617i	$-0.0297933\pi$
$930$		0			0
$931$	640.074	−	282.733i	0.687512	−	0.303687i
$932$		0			0
$933$	262.993	−	262.993i	0.281878	−	0.281878i
$934$		0			0
$935$		−	87.6715i		−	0.0937663i
$936$		0			0
$937$	−374.767			−0.399965			−0.199982	−	0.979799i	$-0.564089\pi$
$937$	−374.767			−0.399965			−0.199982	+	0.979799i	$0.564089\pi$
$938$		0			0
$939$	−216.711	+	216.711i	−0.230789	+	0.230789i
$940$		0			0
$941$	834.839	−	834.839i	0.887182	−	0.887182i	−0.107069	−	0.994252i	$-0.534147\pi$
$941$	834.839	−	834.839i	0.887182	−	0.887182i	0.994252	+	0.107069i	$0.0341466\pi$
$942$		0			0
$943$		−	1205.81i		−	1.27870i
$944$		0			0
$945$	−26.6473	+	142.608i	−0.0281982	+	0.150908i
$946$		0			0
$947$	−208.061	−	208.061i	−0.219706	−	0.219706i	0.588669	−	0.808374i	$-0.299652\pi$
$947$	−208.061	−	208.061i	−0.219706	−	0.219706i	−0.808374	+	0.588669i	$0.799652\pi$
$948$		0			0
$949$	−984.473	+	984.473i	−1.03738	+	1.03738i
$950$		0			0
$951$	499.657			0.525402
$952$		0			0
$953$		−	226.681i		−	0.237860i	−0.992903	−	0.118930i	$-0.962054\pi$
$953$		−	226.681i		−	0.237860i	0.992903	−	0.118930i	$-0.0379464\pi$
$954$		0			0
$955$	−18.6459	−	18.6459i	−0.0195245	−	0.0195245i
$956$		0			0
$957$	−27.8363	+	27.8363i	−0.0290870	+	0.0290870i
$958$		0			0
$959$	25.6702	−	137.379i	0.0267677	−	0.143253i
$960$		0			0
$961$	−1110.32			−1.15538
$962$		0			0
$963$	417.951	+	417.951i	0.434009	+	0.434009i
$964$		0			0
$965$	−10.8578	−	10.8578i	−0.0112516	−	0.0112516i
$966$		0			0
$967$			343.065i			0.354773i	0.984141	+	0.177386i	$0.0567643\pi$
$967$			343.065i			0.354773i	−0.984141	+	0.177386i	$0.943236\pi$
$968$		0			0
$969$	312.171			0.322158
$970$		0			0
$971$	21.4880	+	21.4880i	0.0221297	+	0.0221297i	0.718085	−	0.695955i	$-0.245019\pi$
$971$	21.4880	+	21.4880i	0.0221297	+	0.0221297i	−0.695955	+	0.718085i	$0.745019\pi$
$972$		0			0
$973$	−931.660	−	1359.84i	−0.957513	−	1.39758i
$974$		0			0
$975$	528.674			0.542230
$976$		0			0
$977$	291.470			0.298332			0.149166	−	0.988812i	$-0.452341\pi$
$977$	291.470			0.298332			0.149166	+	0.988812i	$0.452341\pi$
$978$		0			0
$979$	−229.942	+	229.942i	−0.234874	+	0.234874i
$980$		0			0
$981$	433.800	−	433.800i	0.442202	−	0.442202i
$982$		0			0
$983$	−1331.33			−1.35435			−0.677177	−	0.735820i	$-0.736797\pi$
$983$	−1331.33			−1.35435			−0.677177	+	0.735820i	$0.736797\pi$
$984$		0			0
$985$	72.9700			0.0740812
$986$		0			0
$987$	229.858	+	335.498i	0.232885	+	0.339917i
$988$		0			0
$989$	−325.860	−	325.860i	−0.329484	−	0.329484i
$990$		0			0
$991$	913.947			0.922247			0.461124	−	0.887336i	$-0.347446\pi$
$991$	913.947			0.922247			0.461124	+	0.887336i	$0.347446\pi$
$992$		0			0
$993$			50.0338i			0.0503865i
$994$		0			0
$995$	−95.4695	−	95.4695i	−0.0959493	−	0.0959493i
$996$		0			0
$997$	421.536	+	421.536i	0.422805	+	0.422805i	0.886168	−	0.463364i	$-0.153358\pi$
$997$	421.536	+	421.536i	0.422805	+	0.422805i	−0.463364	+	0.886168i	$0.653358\pi$
$998$		0			0
$999$	−871.608			−0.872481

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	448.3.l.b.209.15		56
4.3	odd	2		112.3.l.b.13.5	✓	56
7.6	odd	2	inner	448.3.l.b.209.14		56
16.5	even	4	inner	448.3.l.b.433.14		56
16.11	odd	4		112.3.l.b.69.6	yes	56
28.27	even	2		112.3.l.b.13.6	yes	56
112.27	even	4		112.3.l.b.69.5	yes	56
112.69	odd	4	inner	448.3.l.b.433.15		56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.l.b.13.5	✓	56	4.3	odd	2
112.3.l.b.13.6	yes	56	28.27	even	2
112.3.l.b.69.5	yes	56	112.27	even	4
112.3.l.b.69.6	yes	56	16.11	odd	4
448.3.l.b.209.14		56	7.6	odd	2	inner
448.3.l.b.209.15		56	1.1	even	1	trivial
448.3.l.b.433.14		56	16.5	even	4	inner
448.3.l.b.433.15		56	112.69	odd	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.l (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.746365 - 0.746365i) q^{3} +(0.822232 + 0.822232i) q^{5} +(6.88091 + 1.28574i) q^{7} +7.88588i q^{9} +O(q^{10})\)	\(q+(0.746365 - 0.746365i) q^{3} +(0.822232 + 0.822232i) q^{5} +(6.88091 + 1.28574i) q^{7} +7.88588i q^{9} +(-2.57423 + 2.57423i) q^{11} +(-14.9766 + 14.9766i) q^{13} +1.22737 q^{15} +20.7103i q^{17} +(10.0977 - 10.0977i) q^{19} +(6.09530 - 4.17604i) q^{21} -19.5800i q^{23} -23.6479i q^{25} +(12.6030 + 12.6030i) q^{27} +(7.24407 + 7.24407i) q^{29} +45.5117i q^{31} +3.84263i q^{33} +(4.60052 + 6.71488i) q^{35} +(-34.5793 + 34.5793i) q^{37} +22.3561i q^{39} +61.5837 q^{41} +(16.6424 - 16.6424i) q^{43} +(-6.48402 + 6.48402i) q^{45} +55.0421i q^{47} +(45.6937 + 17.6941i) q^{49} +(15.4575 + 15.4575i) q^{51} +(38.2187 - 38.2187i) q^{53} -4.23322 q^{55} -15.0732i q^{57} +(-32.2679 - 32.2679i) q^{59} +(-32.0297 + 32.0297i) q^{61} +(-10.1392 + 54.2620i) q^{63} -24.6286 q^{65} +(-42.1559 - 42.1559i) q^{67} +(-14.6139 - 14.6139i) q^{69} -15.7491i q^{71} +65.7339 q^{73} +(-17.6500 - 17.6500i) q^{75} +(-21.0228 + 14.4032i) q^{77} +96.5103 q^{79} -52.1600 q^{81} +(31.6530 - 31.6530i) q^{83} +(-17.0287 + 17.0287i) q^{85} +10.8134 q^{87} +89.3246 q^{89} +(-122.309 + 83.7968i) q^{91} +(33.9684 + 33.9684i) q^{93} +16.6054 q^{95} -47.6196i q^{97} +(-20.3000 - 20.3000i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q+O(q^{10}) \) 56 * q	\( 56 q + 8 q^{15} - 20 q^{21} - 96 q^{29} + 100 q^{35} - 128 q^{37} + 72 q^{43} + 192 q^{49} + 128 q^{51} + 88 q^{53} - 444 q^{63} - 8 q^{65} - 440 q^{67} + 12 q^{77} + 8 q^{79} + 64 q^{81} + 96 q^{85} + 388 q^{91} + 32 q^{93} + 776 q^{95} - 8 q^{99}+O(q^{100}) \) 56 * q + 8 * q^15 - 20 * q^21 - 96 * q^29 + 100 * q^35 - 128 * q^37 + 72 * q^43 + 192 * q^49 + 128 * q^51 + 88 * q^53 - 444 * q^63 - 8 * q^65 - 440 * q^67 + 12 * q^77 + 8 * q^79 + 64 * q^81 + 96 * q^85 + 388 * q^91 + 32 * q^93 + 776 * q^95 - 8 * q^99

Introduction

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