LMFDB - Embedded newform 624.4.bc.d.31.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(31,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("624.31");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.bc (of order $4$, degree $2$, 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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$28$
Relative dimension:	$14$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			31.14
Character	$\chi$	$=$	624.31
Dual form			624.4.bc.d.463.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.00000i q^{3} +(13.7178 - 13.7178i) q^{5} +(-9.00439 + 9.00439i) q^{7} -9.00000 q^{9} +(5.13681 - 5.13681i) q^{11} +(-29.7474 - 36.2228i) q^{13} +(41.1534 + 41.1534i) q^{15} -41.0920i q^{17} +(28.6094 + 28.6094i) q^{19} +(-27.0132 - 27.0132i) q^{21} +58.8194 q^{23} -251.357i q^{25} -27.0000i q^{27} -13.6058 q^{29} +(-163.913 - 163.913i) q^{31} +(15.4104 + 15.4104i) q^{33} +247.041i q^{35} +(-129.152 - 129.152i) q^{37} +(108.669 - 89.2421i) q^{39} +(30.4179 - 30.4179i) q^{41} -342.340 q^{43} +(-123.460 + 123.460i) q^{45} +(328.489 - 328.489i) q^{47} +180.842i q^{49} +123.276 q^{51} +109.796 q^{53} -140.932i q^{55} +(-85.8283 + 85.8283i) q^{57} +(-262.191 + 262.191i) q^{59} +69.0168 q^{61} +(81.0396 - 81.0396i) q^{63} +(-904.967 - 88.8295i) q^{65} +(-138.791 - 138.791i) q^{67} +176.458i q^{69} +(-185.237 - 185.237i) q^{71} +(-597.825 - 597.825i) q^{73} +754.071 q^{75} +92.5078i q^{77} -627.547i q^{79} +81.0000 q^{81} +(542.734 + 542.734i) q^{83} +(-563.692 - 563.692i) q^{85} -40.8173i q^{87} +(-306.102 - 306.102i) q^{89} +(594.022 + 58.3078i) q^{91} +(491.740 - 491.740i) q^{93} +784.917 q^{95} +(1229.81 - 1229.81i) q^{97} +(-46.2313 + 46.2313i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) $ 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * q^99

Character values

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

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$-1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$			3.00000i			0.577350i
$3$			3.00000i			0.577350i
$4$		0			0
$5$	13.7178	−	13.7178i	1.22696	−	1.22696i	0.261850	−	0.965109i	$-0.415667\pi$
$5$	13.7178	−	13.7178i	1.22696	−	1.22696i	0.965109	−	0.261850i	$-0.0843327\pi$
$6$		0			0
$7$	−9.00439	+	9.00439i	−0.486192	+	0.486192i	−0.907102	−	0.420911i	$-0.861711\pi$
$7$	−9.00439	+	9.00439i	−0.486192	+	0.486192i	0.420911	+	0.907102i	$0.361711\pi$
$8$		0			0
$9$	−9.00000			−0.333333
$10$		0			0
$11$	5.13681	−	5.13681i	0.140801	−	0.140801i	−0.633193	−	0.773994i	$-0.718256\pi$
$11$	5.13681	−	5.13681i	0.140801	−	0.140801i	0.773994	+	0.633193i	$0.218256\pi$
$12$		0			0
$13$	−29.7474	−	36.2228i	−0.634649	−	0.772801i
$13$	−29.7474	−	36.2228i	−0.634649	−	0.772801i
$14$		0			0
$15$	41.1534	+	41.1534i	0.708385	+	0.708385i
$16$		0			0
$17$		−	41.0920i		−	0.586251i	−0.956074	−	0.293125i	$-0.905305\pi$
$17$		−	41.0920i		−	0.586251i	0.956074	−	0.293125i	$-0.0946954\pi$
$18$		0			0
$19$	28.6094	+	28.6094i	0.345445	+	0.345445i	0.858410	−	0.512965i	$-0.171453\pi$
$19$	28.6094	+	28.6094i	0.345445	+	0.345445i	−0.512965	+	0.858410i	$0.671453\pi$
$20$		0			0
$21$	−27.0132	−	27.0132i	−0.280703	−	0.280703i
$22$		0			0
$23$	58.8194			0.533247			0.266624	−	0.963801i	$-0.414092\pi$
$23$	58.8194			0.533247			0.266624	+	0.963801i	$0.414092\pi$
$24$		0			0
$25$		−	251.357i		−	2.01086i
$26$		0			0
$27$		−	27.0000i		−	0.192450i
$28$		0			0
$29$	−13.6058			−0.0871216			−0.0435608	−	0.999051i	$-0.513870\pi$
$29$	−13.6058			−0.0871216			−0.0435608	+	0.999051i	$0.513870\pi$
$30$		0			0
$31$	−163.913	−	163.913i	−0.949668	−	0.949668i	0.0491243	−	0.998793i	$-0.484357\pi$
$31$	−163.913	−	163.913i	−0.949668	−	0.949668i	−0.998793	+	0.0491243i	$0.984357\pi$
$32$		0			0
$33$	15.4104	+	15.4104i	0.0812913	+	0.0812913i
$34$		0			0
$35$			247.041i			1.19307i
$36$		0			0
$37$	−129.152	−	129.152i	−0.573851	−	0.573851i	0.359351	−	0.933202i	$-0.382998\pi$
$37$	−129.152	−	129.152i	−0.573851	−	0.573851i	−0.933202	+	0.359351i	$0.882998\pi$
$38$		0			0
$39$	108.669	−	89.2421i	0.446177	−	0.366415i
$40$		0			0
$41$	30.4179	−	30.4179i	0.115865	−	0.115865i	−0.646797	−	0.762662i	$-0.723892\pi$
$41$	30.4179	−	30.4179i	0.115865	−	0.115865i	0.762662	+	0.646797i	$0.223892\pi$
$42$		0			0
$43$	−342.340			−1.21410			−0.607051	−	0.794663i	$-0.707648\pi$
$43$	−342.340			−1.21410			−0.607051	+	0.794663i	$0.707648\pi$
$44$		0			0
$45$	−123.460	+	123.460i	−0.408986	+	0.408986i
$46$		0			0
$47$	328.489	−	328.489i	1.01947	−	1.01947i	0.0196632	−	0.999807i	$-0.493741\pi$
$47$	328.489	−	328.489i	1.01947	−	1.01947i	0.999807	−	0.0196632i	$-0.00625940\pi$
$48$		0			0
$49$			180.842i			0.527235i
$50$		0			0
$51$	123.276			0.338472
$52$		0			0
$53$	109.796			0.284559			0.142279	−	0.989827i	$-0.454557\pi$
$53$	109.796			0.284559			0.142279	+	0.989827i	$0.454557\pi$
$54$		0			0
$55$		−	140.932i		−	0.345513i
$56$		0			0
$57$	−85.8283	+	85.8283i	−0.199443	+	0.199443i
$58$		0			0
$59$	−262.191	+	262.191i	−0.578549	+	0.578549i	−0.934503	−	0.355954i	$-0.884156\pi$
$59$	−262.191	+	262.191i	−0.578549	+	0.578549i	0.355954	+	0.934503i	$0.384156\pi$
$60$		0			0
$61$	69.0168			0.144864			0.0724320	−	0.997373i	$-0.476924\pi$
$61$	69.0168			0.144864			0.0724320	+	0.997373i	$0.476924\pi$
$62$		0			0
$63$	81.0396	−	81.0396i	0.162064	−	0.162064i
$64$		0			0
$65$	−904.967	−	88.8295i	−1.72688	−	0.169507i
$66$		0			0
$67$	−138.791	−	138.791i	−0.253075	−	0.253075i	0.569155	−	0.822230i	$-0.307270\pi$
$67$	−138.791	−	138.791i	−0.253075	−	0.253075i	−0.822230	+	0.569155i	$0.807270\pi$
$68$		0			0
$69$			176.458i			0.307870i
$70$		0			0
$71$	−185.237	−	185.237i	−0.309629	−	0.309629i	0.535137	−	0.844765i	$-0.320260\pi$
$71$	−185.237	−	185.237i	−0.309629	−	0.309629i	−0.844765	+	0.535137i	$0.820260\pi$
$72$		0			0
$73$	−597.825	−	597.825i	−0.958494	−	0.958494i	0.0406781	−	0.999172i	$-0.487048\pi$
$73$	−597.825	−	597.825i	−0.958494	−	0.958494i	−0.999172	+	0.0406781i	$0.987048\pi$
$74$		0			0
$75$	754.071			1.16097
$76$		0			0
$77$			92.5078i			0.136912i
$78$		0			0
$79$		−	627.547i		−	0.893728i	−0.894602	−	0.446864i	$-0.852541\pi$
$79$		−	627.547i		−	0.893728i	0.894602	−	0.446864i	$-0.147459\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$	542.734	+	542.734i	0.717745	+	0.717745i	0.968143	−	0.250398i	$-0.0805615\pi$
$83$	542.734	+	542.734i	0.717745	+	0.717745i	−0.250398	+	0.968143i	$0.580561\pi$
$84$		0			0
$85$	−563.692	−	563.692i	−0.719306	−	0.719306i
$86$		0			0
$87$		−	40.8173i		−	0.0502997i
$88$		0			0
$89$	−306.102	−	306.102i	−0.364570	−	0.364570i	0.500922	−	0.865492i	$-0.332994\pi$
$89$	−306.102	−	306.102i	−0.364570	−	0.364570i	−0.865492	+	0.500922i	$0.832994\pi$
$90$		0			0
$91$	594.022	+	58.3078i	0.684290	+	0.0671683i
$92$		0			0
$93$	491.740	−	491.740i	0.548291	−	0.548291i
$94$		0			0
$95$	784.917			0.847693
$96$		0			0
$97$	1229.81	−	1229.81i	1.28730	−	1.28730i	0.350885	−	0.936419i	$-0.385881\pi$
$97$	1229.81	−	1229.81i	1.28730	−	1.28730i	0.936419	−	0.350885i	$-0.114119\pi$
$98$		0			0
$99$	−46.2313	+	46.2313i	−0.0469336	+	0.0469336i
$100$		0			0
$101$		−	1764.63i		−	1.73848i	−0.494386	−	0.869242i	$-0.664607\pi$
$101$		−	1764.63i		−	1.73848i	0.494386	−	0.869242i	$-0.335393\pi$
$102$		0			0
$103$	465.459			0.445272			0.222636	−	0.974902i	$-0.428534\pi$
$103$	465.459			0.445272			0.222636	+	0.974902i	$0.428534\pi$
$104$		0			0
$105$	−741.124			−0.688822
$106$		0			0
$107$		−	790.881i		−	0.714554i	−0.933998	−	0.357277i	$-0.883705\pi$
$107$		−	790.881i		−	0.714554i	0.933998	−	0.357277i	$-0.116295\pi$
$108$		0			0
$109$	425.633	−	425.633i	0.374020	−	0.374020i	−0.494919	−	0.868939i	$-0.664802\pi$
$109$	425.633	−	425.633i	0.374020	−	0.374020i	0.868939	+	0.494919i	$0.164802\pi$
$110$		0			0
$111$	387.457	−	387.457i	0.331313	−	0.331313i
$112$		0			0
$113$	−540.166			−0.449686			−0.224843	−	0.974395i	$-0.572187\pi$
$113$	−540.166			−0.449686			−0.224843	+	0.974395i	$0.572187\pi$
$114$		0			0
$115$	806.873	−	806.873i	0.654272	−	0.654272i
$116$		0			0
$117$	267.726	+	326.006i	0.211550	+	0.257600i
$118$		0			0
$119$	370.008	+	370.008i	0.285030	+	0.285030i
$120$		0			0
$121$			1278.23i			0.960350i
$122$		0			0
$123$	91.2538	+	91.2538i	0.0668950	+	0.0668950i
$124$		0			0
$125$	−1733.34	−	1733.34i	−1.24028	−	1.24028i
$126$		0			0
$127$	2282.24			1.59462			0.797309	−	0.603571i	$-0.206256\pi$
$127$	2282.24			1.59462			0.797309	+	0.603571i	$0.206256\pi$
$128$		0			0
$129$		−	1027.02i		−	0.700962i
$130$		0			0
$131$		−	1635.72i		−	1.09094i	−0.838129	−	0.545472i	$-0.816350\pi$
$131$		−	1635.72i		−	1.09094i	0.838129	−	0.545472i	$-0.183650\pi$
$132$		0			0
$133$	−515.221			−0.335905
$134$		0			0
$135$	−370.381	−	370.381i	−0.236128	−	0.236128i
$136$		0			0
$137$	−267.758	−	267.758i	−0.166979	−	0.166979i	0.618671	−	0.785650i	$-0.287671\pi$
$137$	−267.758	−	267.758i	−0.166979	−	0.166979i	−0.785650	+	0.618671i	$0.787671\pi$
$138$		0			0
$139$			1191.06i			0.726794i	0.931634	+	0.363397i	$0.118383\pi$
$139$			1191.06i			0.726794i	−0.931634	+	0.363397i	$0.881617\pi$
$140$		0			0
$141$	985.468	+	985.468i	0.588591	+	0.588591i
$142$		0			0
$143$	−338.877	−	33.2633i	−0.198170	−	0.0194519i
$144$		0			0
$145$	−186.641	+	186.641i	−0.106895	+	0.106895i
$146$		0			0
$147$	−542.525			−0.304400
$148$		0			0
$149$	2481.47	−	2481.47i	1.36436	−	1.36436i	0.496098	−	0.868267i	$-0.334766\pi$
$149$	2481.47	−	2481.47i	1.36436	−	1.36436i	0.868267	−	0.496098i	$-0.165234\pi$
$150$		0			0
$151$	182.274	−	182.274i	0.0982334	−	0.0982334i	−0.656282	−	0.754516i	$-0.727872\pi$
$151$	182.274	−	182.274i	0.0982334	−	0.0982334i	0.754516	+	0.656282i	$0.227872\pi$
$152$		0			0
$153$			369.828i			0.195417i
$154$		0			0
$155$	−4497.07			−2.33041
$156$		0			0
$157$	−2513.06			−1.27748			−0.638738	−	0.769424i	$-0.720543\pi$
$157$	−2513.06			−1.27748			−0.638738	+	0.769424i	$0.720543\pi$
$158$		0			0
$159$			329.388i			0.164290i
$160$		0			0
$161$	−529.633	+	529.633i	−0.259260	+	0.259260i
$162$		0			0
$163$	1650.41	−	1650.41i	0.793069	−	0.793069i	−0.188923	−	0.981992i	$-0.560500\pi$
$163$	1650.41	−	1650.41i	0.793069	−	0.793069i	0.981992	+	0.188923i	$0.0604996\pi$
$164$		0			0
$165$	422.795			0.199482
$166$		0			0
$167$	−2428.77	+	2428.77i	−1.12541	+	1.12541i	−0.134496	+	0.990914i	$0.542942\pi$
$167$	−2428.77	+	2428.77i	−1.12541	+	1.12541i	−0.990914	+	0.134496i	$0.957058\pi$
$168$		0			0
$169$	−427.189	+	2155.07i	−0.194442	+	0.980914i
$170$		0			0
$171$	−257.485	−	257.485i	−0.115148	−	0.115148i
$172$		0			0
$173$			2121.86i			0.932496i	0.884654	+	0.466248i	$0.154395\pi$
$173$			2121.86i			0.932496i	−0.884654	+	0.466248i	$0.845605\pi$
$174$		0			0
$175$	2263.32	+	2263.32i	0.977661	+	0.977661i
$176$		0			0
$177$	−786.574	−	786.574i	−0.334025	−	0.334025i
$178$		0			0
$179$	−3736.03			−1.56002			−0.780011	−	0.625766i	$-0.784787\pi$
$179$	−3736.03			−1.56002			−0.780011	+	0.625766i	$0.784787\pi$
$180$		0			0
$181$			3377.75i			1.38711i	0.720406	+	0.693553i	$0.243956\pi$
$181$			3377.75i			1.38711i	−0.720406	+	0.693553i	$0.756044\pi$
$182$		0			0
$183$			207.051i			0.0836372i
$184$		0			0
$185$	−3543.37			−1.40818
$186$		0			0
$187$	−211.082	−	211.082i	−0.0825445	−	0.0825445i
$188$		0			0
$189$	243.119	+	243.119i	0.0935676	+	0.0935676i
$190$		0			0
$191$			2547.01i			0.964895i	0.875925	+	0.482448i	$0.160252\pi$
$191$			2547.01i			0.964895i	−0.875925	+	0.482448i	$0.839748\pi$
$192$		0			0
$193$	2766.47	+	2766.47i	1.03179	+	1.03179i	0.999478	+	0.0323099i	$0.0102864\pi$
$193$	2766.47	+	2766.47i	1.03179	+	1.03179i	0.0323099	+	0.999478i	$0.489714\pi$
$194$		0			0
$195$	266.488	−	2714.90i	0.0978648	−	0.997016i
$196$		0			0
$197$	581.717	−	581.717i	0.210384	−	0.210384i	−0.594047	−	0.804431i	$-0.702471\pi$
$197$	581.717	−	581.717i	0.210384	−	0.210384i	0.804431	+	0.594047i	$0.202471\pi$
$198$		0			0
$199$	−2241.01			−0.798297			−0.399148	−	0.916886i	$-0.630694\pi$
$199$	−2241.01			−0.798297			−0.399148	+	0.916886i	$0.630694\pi$
$200$		0			0
$201$	416.373	−	416.373i	0.146113	−	0.146113i
$202$		0			0
$203$	122.512	−	122.512i	0.0423578	−	0.0423578i
$204$		0			0
$205$		−	834.535i		−	0.284324i
$206$		0			0
$207$	−529.374			−0.177749
$208$		0			0
$209$	293.922			0.0972777
$210$		0			0
$211$			236.747i			0.0772434i	0.999254	+	0.0386217i	$0.0122967\pi$
$211$			236.747i			0.0772434i	−0.999254	+	0.0386217i	$0.987703\pi$
$212$		0			0
$213$	555.712	−	555.712i	0.178764	−	0.178764i
$214$		0			0
$215$	−4696.16	+	4696.16i	−1.48965	+	1.48965i
$216$		0			0
$217$	2951.88			0.923442
$218$		0			0
$219$	1793.47	−	1793.47i	0.553387	−	0.553387i
$220$		0			0
$221$	−1488.47	+	1222.38i	−0.453055	+	0.372063i
$222$		0			0
$223$	2934.75	+	2934.75i	0.881279	+	0.881279i	0.993665	−	0.112386i	$-0.0358492\pi$
$223$	2934.75	+	2934.75i	0.881279	+	0.881279i	−0.112386	+	0.993665i	$0.535849\pi$
$224$		0			0
$225$			2262.21i			0.670285i
$226$		0			0
$227$	1861.03	+	1861.03i	0.544144	+	0.544144i	0.924741	−	0.380597i	$-0.124281\pi$
$227$	1861.03	+	1861.03i	0.544144	+	0.544144i	−0.380597	+	0.924741i	$0.624281\pi$
$228$		0			0
$229$	−2720.40	−	2720.40i	−0.785019	−	0.785019i	0.195654	−	0.980673i	$-0.437317\pi$
$229$	−2720.40	−	2720.40i	−0.785019	−	0.785019i	−0.980673	+	0.195654i	$0.937317\pi$
$230$		0			0
$231$	−277.523			−0.0790463
$232$		0			0
$233$		−	4481.19i		−	1.25997i	−0.776608	−	0.629984i	$-0.783061\pi$
$233$		−	4481.19i		−	1.25997i	0.776608	−	0.629984i	$-0.216939\pi$
$234$		0			0
$235$		−	9012.31i		−	2.50170i
$236$		0			0
$237$	1882.64			0.515994
$238$		0			0
$239$	3103.05	+	3103.05i	0.839830	+	0.839830i	0.988836	−	0.149006i	$-0.0476075\pi$
$239$	3103.05	+	3103.05i	0.839830	+	0.839830i	−0.149006	+	0.988836i	$0.547608\pi$
$240$		0			0
$241$	3522.37	+	3522.37i	0.941476	+	0.941476i	0.998380	−	0.0569037i	$-0.0181228\pi$
$241$	3522.37	+	3522.37i	0.941476	+	0.941476i	−0.0569037	+	0.998380i	$0.518123\pi$
$242$		0			0
$243$			243.000i			0.0641500i
$244$		0			0
$245$	2480.75	+	2480.75i	0.646896	+	0.646896i
$246$		0			0
$247$	185.260	−	1887.37i	0.0477239	−	0.486196i
$248$		0			0
$249$	−1628.20	+	1628.20i	−0.414390	+	0.414390i
$250$		0			0
$251$	−7437.96			−1.87044			−0.935219	−	0.354071i	$-0.884797\pi$
$251$	−7437.96			−1.87044			−0.935219	+	0.354071i	$0.884797\pi$
$252$		0			0
$253$	302.144	−	302.144i	0.0750816	−	0.0750816i
$254$		0			0
$255$	1691.08	−	1691.08i	0.415291	−	0.415291i
$256$		0			0
$257$		−	900.226i		−	0.218500i	−0.994014	−	0.109250i	$-0.965155\pi$
$257$		−	900.226i		−	0.218500i	0.994014	−	0.109250i	$-0.0348449\pi$
$258$		0			0
$259$	2325.87			0.558003
$260$		0			0
$261$	122.452			0.0290405
$262$		0			0
$263$			489.304i			0.114722i	0.998354	+	0.0573608i	$0.0182685\pi$
$263$			489.304i			0.114722i	−0.998354	+	0.0573608i	$0.981731\pi$
$264$		0			0
$265$	1506.16	−	1506.16i	0.349142	−	0.349142i
$266$		0			0
$267$	918.306	−	918.306i	0.210485	−	0.210485i
$268$		0			0
$269$	8008.84			1.81527			0.907635	−	0.419759i	$-0.137885\pi$
$269$	8008.84			1.81527			0.907635	+	0.419759i	$0.137885\pi$
$270$		0			0
$271$	−2524.78	+	2524.78i	−0.565938	+	0.565938i	−0.930988	−	0.365050i	$-0.881052\pi$
$271$	−2524.78	+	2524.78i	−0.565938	+	0.565938i	0.365050	+	0.930988i	$0.381052\pi$
$272$		0			0
$273$	−174.923	+	1782.07i	−0.0387797	+	0.395075i
$274$		0			0
$275$	−1291.17	−	1291.17i	−0.283130	−	0.283130i
$276$		0			0
$277$			3548.82i			0.769776i	0.922963	+	0.384888i	$0.125760\pi$
$277$			3548.82i			0.769776i	−0.922963	+	0.384888i	$0.874240\pi$
$278$		0			0
$279$	1475.22	+	1475.22i	0.316556	+	0.316556i
$280$		0			0
$281$	−3456.24	−	3456.24i	−0.733743	−	0.733743i	0.237616	−	0.971359i	$-0.423634\pi$
$281$	−3456.24	−	3456.24i	−0.733743	−	0.733743i	−0.971359	+	0.237616i	$0.923634\pi$
$282$		0			0
$283$	−2211.69			−0.464563			−0.232282	−	0.972649i	$-0.574619\pi$
$283$	−2211.69			−0.464563			−0.232282	+	0.972649i	$0.574619\pi$
$284$		0			0
$285$			2354.75i			0.489416i
$286$		0			0
$287$			547.790i			0.112666i
$288$		0			0
$289$	3224.45			0.656310
$290$		0			0
$291$	3689.43	+	3689.43i	0.743225	+	0.743225i
$292$		0			0
$293$	4706.80	+	4706.80i	0.938478	+	0.938478i	0.998214	−	0.0597364i	$-0.0190260\pi$
$293$	4706.80	+	4706.80i	0.938478	+	0.938478i	−0.0597364	+	0.998214i	$0.519026\pi$
$294$		0			0
$295$			7193.38i			1.41971i
$296$		0			0
$297$	−138.694	−	138.694i	−0.0270971	−	0.0270971i
$298$		0			0
$299$	−1749.72	−	2130.60i	−0.338425	−	0.412094i
$300$		0			0
$301$	3082.57	−	3082.57i	0.590286	−	0.590286i
$302$		0			0
$303$	5293.88			1.00371
$304$		0			0
$305$	946.760	−	946.760i	0.177742	−	0.177742i
$306$		0			0
$307$	252.125	−	252.125i	0.0468714	−	0.0468714i	−0.683283	−	0.730154i	$-0.739448\pi$
$307$	252.125	−	252.125i	0.0468714	−	0.0468714i	0.730154	+	0.683283i	$0.239448\pi$
$308$		0			0
$309$			1396.38i			0.257078i
$310$		0			0
$311$	1704.62			0.310804			0.155402	−	0.987851i	$-0.450333\pi$
$311$	1704.62			0.310804			0.155402	+	0.987851i	$0.450333\pi$
$312$		0			0
$313$	5415.34			0.977933			0.488967	−	0.872303i	$-0.337374\pi$
$313$	5415.34			0.977933			0.488967	+	0.872303i	$0.337374\pi$
$314$		0			0
$315$		−	2223.37i		−	0.397691i
$316$		0			0
$317$	−2676.97	+	2676.97i	−0.474302	+	0.474302i	−0.903304	−	0.429002i	$-0.858865\pi$
$317$	−2676.97	+	2676.97i	−0.474302	+	0.474302i	0.429002	+	0.903304i	$0.358865\pi$
$318$		0			0
$319$	−69.8903	+	69.8903i	−0.0122668	+	0.0122668i
$320$		0			0
$321$	2372.64			0.412548
$322$		0			0
$323$	1175.62	−	1175.62i	0.202517	−	0.202517i
$324$		0			0
$325$	−9104.86	+	7477.21i	−1.55399	+	1.27619i
$326$		0			0
$327$	1276.90	+	1276.90i	0.215941	+	0.215941i
$328$		0			0
$329$			5915.69i			0.991315i
$330$		0			0
$331$	−3653.62	−	3653.62i	−0.606710	−	0.606710i	0.335375	−	0.942085i	$-0.391137\pi$
$331$	−3653.62	−	3653.62i	−0.606710	−	0.606710i	−0.942085	+	0.335375i	$0.891137\pi$
$332$		0			0
$333$	1162.37	+	1162.37i	0.191284	+	0.191284i
$334$		0			0
$335$	−3807.82			−0.621025
$336$		0			0
$337$			5365.74i			0.867330i	0.901074	+	0.433665i	$0.142780\pi$
$337$			5365.74i			0.867330i	−0.901074	+	0.433665i	$0.857220\pi$
$338$		0			0
$339$		−	1620.50i		−	0.259626i
$340$		0			0
$341$	−1683.99			−0.267428
$342$		0			0
$343$	−4716.88	−	4716.88i	−0.742529	−	0.742529i
$344$		0			0
$345$	2420.62	+	2420.62i	0.377744	+	0.377744i
$346$		0			0
$347$			5576.25i			0.862677i	0.902190	+	0.431339i	$0.141959\pi$
$347$			5576.25i			0.862677i	−0.902190	+	0.431339i	$0.858041\pi$
$348$		0			0
$349$	1806.21	+	1806.21i	0.277032	+	0.277032i	0.831923	−	0.554891i	$-0.187240\pi$
$349$	1806.21	+	1806.21i	0.277032	+	0.277032i	−0.554891	+	0.831923i	$0.687240\pi$
$350$		0			0
$351$	−978.017	+	803.179i	−0.148726	+	0.122138i
$352$		0			0
$353$	−1813.36	+	1813.36i	−0.273415	+	0.273415i	−0.830473	−	0.557058i	$-0.811930\pi$
$353$	−1813.36	+	1813.36i	−0.273415	+	0.273415i	0.557058	+	0.830473i	$0.311930\pi$
$354$		0			0
$355$	−5082.11			−0.759804
$356$		0			0
$357$	−1110.02	+	1110.02i	−0.164562	+	0.164562i
$358$		0			0
$359$	−8955.68	+	8955.68i	−1.31661	+	1.31661i	−0.400168	+	0.916442i	$0.631048\pi$
$359$	−8955.68	+	8955.68i	−1.31661	+	1.31661i	−0.916442	+	0.400168i	$0.868952\pi$
$360$		0			0
$361$		−	5222.00i		−	0.761336i
$362$		0			0
$363$	−3834.68			−0.554459
$364$		0			0
$365$	−16401.7			−2.35207
$366$		0			0
$367$		−	94.3385i		−	0.0134181i	−0.999977	−	0.00670903i	$-0.997864\pi$
$367$		−	94.3385i		−	0.0134181i	0.999977	−	0.00670903i	$-0.00213557\pi$
$368$		0			0
$369$	−273.761	+	273.761i	−0.0386218	+	0.0386218i
$370$		0			0
$371$	−988.645	+	988.645i	−0.138350	+	0.138350i
$372$		0			0
$373$	−1045.50			−0.145131			−0.0725654	−	0.997364i	$-0.523119\pi$
$373$	−1045.50			−0.145131			−0.0725654	+	0.997364i	$0.523119\pi$
$374$		0			0
$375$	5200.03	−	5200.03i	0.716075	−	0.716075i
$376$		0			0
$377$	404.736	+	492.840i	0.0552916	+	0.0673277i
$378$		0			0
$379$	−8530.89	−	8530.89i	−1.15621	−	1.15621i	−0.985285	−	0.170922i	$-0.945325\pi$
$379$	−8530.89	−	8530.89i	−1.15621	−	1.15621i	−0.170922	−	0.985285i	$-0.554675\pi$
$380$		0			0
$381$			6846.73i			0.920653i
$382$		0			0
$383$	9103.57	+	9103.57i	1.21454	+	1.21454i	0.969516	+	0.245029i	$0.0787975\pi$
$383$	9103.57	+	9103.57i	1.21454	+	1.21454i	0.245029	+	0.969516i	$0.421203\pi$
$384$		0			0
$385$	1269.01	+	1269.01i	0.167986	+	0.167986i
$386$		0			0
$387$	3081.06			0.404701
$388$		0			0
$389$		−	6371.12i		−	0.830408i	−0.909728	−	0.415204i	$-0.863710\pi$
$389$		−	6371.12i		−	0.830408i	0.909728	−	0.415204i	$-0.136290\pi$
$390$		0			0
$391$		−	2417.00i		−	0.312617i
$392$		0			0
$393$	4907.17			0.629857
$394$		0			0
$395$	−8608.57	−	8608.57i	−1.09657	−	1.09657i
$396$		0			0
$397$	4784.83	+	4784.83i	0.604896	+	0.604896i	0.941608	−	0.336712i	$-0.109315\pi$
$397$	4784.83	+	4784.83i	0.604896	+	0.604896i	−0.336712	+	0.941608i	$0.609315\pi$
$398$		0			0
$399$		−	1545.66i		−	0.193935i
$400$		0			0
$401$	8438.09	+	8438.09i	1.05082	+	1.05082i	0.998638	+	0.0521812i	$0.0166173\pi$
$401$	8438.09	+	8438.09i	1.05082	+	1.05082i	0.0521812	+	0.998638i	$0.483383\pi$
$402$		0			0
$403$	−1061.42	+	10813.4i	−0.131199	+	1.33661i
$404$		0			0
$405$	1111.14	−	1111.14i	0.136329	−	0.136329i
$406$		0			0
$407$	−1326.86			−0.161597
$408$		0			0
$409$	5772.49	−	5772.49i	0.697876	−	0.697876i	−0.266076	−	0.963952i	$-0.585727\pi$
$409$	5772.49	−	5772.49i	0.697876	−	0.697876i	0.963952	+	0.266076i	$0.0857272\pi$
$410$		0			0
$411$	803.274	−	803.274i	0.0964053	−	0.0964053i
$412$		0			0
$413$		−	4721.75i		−	0.562571i
$414$		0			0
$415$	14890.3			1.76129
$416$		0			0
$417$	−3573.18			−0.419614
$418$		0			0
$419$			3381.06i			0.394213i	0.980382	+	0.197107i	$0.0631545\pi$
$419$			3381.06i			0.394213i	−0.980382	+	0.197107i	$0.936845\pi$
$420$		0			0
$421$	2293.30	−	2293.30i	0.265484	−	0.265484i	−0.561794	−	0.827277i	$-0.689888\pi$
$421$	2293.30	−	2293.30i	0.265484	−	0.265484i	0.827277	+	0.561794i	$0.189888\pi$
$422$		0			0
$423$	−2956.40	+	2956.40i	−0.339823	+	0.339823i
$424$		0			0
$425$	−10328.8			−1.17887
$426$		0			0
$427$	−621.455	+	621.455i	−0.0704316	+	0.0704316i
$428$		0			0
$429$	99.7900	−	1016.63i	0.0112306	−	0.114413i
$430$		0			0
$431$	−6460.43	−	6460.43i	−0.722014	−	0.722014i	0.247001	−	0.969015i	$-0.420555\pi$
$431$	−6460.43	−	6460.43i	−0.722014	−	0.722014i	−0.969015	+	0.247001i	$0.920555\pi$
$432$		0			0
$433$		−	8343.41i		−	0.926001i	−0.886358	−	0.463001i	$-0.846773\pi$
$433$		−	8343.41i		−	0.926001i	0.886358	−	0.463001i	$-0.153227\pi$
$434$		0			0
$435$	−559.924	−	559.924i	−0.0617157	−	0.0617157i
$436$		0			0
$437$	1682.79	+	1682.79i	0.184207	+	0.184207i
$438$		0			0
$439$	434.752			0.0472655			0.0236328	−	0.999721i	$-0.492477\pi$
$439$	434.752			0.0472655			0.0236328	+	0.999721i	$0.492477\pi$
$440$		0			0
$441$		−	1627.58i		−	0.175745i
$442$		0			0
$443$			8889.00i			0.953339i	0.879083	+	0.476669i	$0.158156\pi$
$443$			8889.00i			0.953339i	−0.879083	+	0.476669i	$0.841844\pi$
$444$		0			0
$445$	−8398.11			−0.894626
$446$		0			0
$447$	7444.42	+	7444.42i	0.787716	+	0.787716i
$448$		0			0
$449$	−10551.8	−	10551.8i	−1.10907	−	1.10907i	−0.993273	−	0.115793i	$-0.963059\pi$
$449$	−10551.8	−	10551.8i	−1.10907	−	1.10907i	−0.115793	−	0.993273i	$-0.536941\pi$
$450$		0			0
$451$		−	312.503i		−	0.0326279i
$452$		0			0
$453$	546.822	+	546.822i	0.0567151	+	0.0567151i
$454$		0			0
$455$	8948.54	−	7348.83i	0.922009	−	0.757183i
$456$		0			0
$457$	188.466	−	188.466i	0.0192912	−	0.0192912i	−0.697395	−	0.716687i	$-0.745658\pi$
$457$	188.466	−	188.466i	0.0192912	−	0.0192912i	0.716687	+	0.697395i	$0.245658\pi$
$458$		0			0
$459$	−1109.48			−0.112824
$460$		0			0
$461$	10294.2	−	10294.2i	1.04002	−	1.04002i	0.0408504	−	0.999165i	$-0.486993\pi$
$461$	10294.2	−	10294.2i	1.04002	−	1.04002i	0.999165	−	0.0408504i	$-0.0130067\pi$
$462$		0			0
$463$	8477.15	−	8477.15i	0.850900	−	0.850900i	−0.139344	−	0.990244i	$-0.544499\pi$
$463$	8477.15	−	8477.15i	0.850900	−	0.850900i	0.990244	+	0.139344i	$0.0444994\pi$
$464$		0			0
$465$		−	13491.2i		−	1.34546i
$466$		0			0
$467$	−15708.8			−1.55656			−0.778281	−	0.627916i	$-0.783908\pi$
$467$	−15708.8			−1.55656			−0.778281	+	0.627916i	$0.783908\pi$
$468$		0			0
$469$	2499.46			0.246086
$470$		0			0
$471$		−	7539.17i		−	0.737551i
$472$		0			0
$473$	−1758.54	+	1758.54i	−0.170946	+	0.170946i
$474$		0			0
$475$	7191.18	−	7191.18i	0.694640	−	0.694640i
$476$		0			0
$477$	−988.163			−0.0948530
$478$		0			0
$479$	5409.75	−	5409.75i	0.516028	−	0.516028i	−0.400339	−	0.916367i	$-0.631108\pi$
$479$	5409.75	−	5409.75i	0.516028	−	0.516028i	0.916367	+	0.400339i	$0.131108\pi$
$480$		0			0
$481$	−836.323	+	8520.20i	−0.0792787	+	0.807667i
$482$		0			0
$483$	−1588.90	−	1588.90i	−0.149684	−	0.149684i
$484$		0			0
$485$		−	33740.7i		−	3.15894i
$486$		0			0
$487$	11371.6	+	11371.6i	1.05810	+	1.05810i	0.998205	+	0.0598975i	$0.0190774\pi$
$487$	11371.6	+	11371.6i	1.05810	+	1.05810i	0.0598975	+	0.998205i	$0.480923\pi$
$488$		0			0
$489$	4951.24	+	4951.24i	0.457879	+	0.457879i
$490$		0			0
$491$	8193.44			0.753085			0.376543	−	0.926399i	$-0.377113\pi$
$491$	8193.44			0.753085			0.376543	+	0.926399i	$0.377113\pi$
$492$		0			0
$493$			559.088i			0.0510751i
$494$		0			0
$495$			1268.39i			0.115171i
$496$		0			0
$497$	3335.90			0.301078
$498$		0			0
$499$	4086.38	+	4086.38i	0.366596	+	0.366596i	0.866234	−	0.499638i	$-0.166534\pi$
$499$	4086.38	+	4086.38i	0.366596	+	0.366596i	−0.499638	+	0.866234i	$0.666534\pi$
$500$		0			0
$501$	−7286.30	−	7286.30i	−0.649756	−	0.649756i
$502$		0			0
$503$			149.395i			0.0132430i	0.999978	+	0.00662148i	$0.00210770\pi$
$503$			149.395i			0.0132430i	−0.999978	+	0.00662148i	$0.997892\pi$
$504$		0			0
$505$	−24206.8	−	24206.8i	−2.13305	−	2.13305i
$506$		0			0
$507$	−6465.20	−	1281.57i	−0.566331	−	0.112261i
$508$		0			0
$509$	574.591	−	574.591i	0.0500359	−	0.0500359i	−0.681646	−	0.731682i	$-0.738736\pi$
$509$	574.591	−	574.591i	0.0500359	−	0.0500359i	0.731682	+	0.681646i	$0.238736\pi$
$510$		0			0
$511$	10766.1			0.932024
$512$		0			0
$513$	772.454	−	772.454i	0.0664809	−	0.0664809i
$514$		0			0
$515$	6385.08	−	6385.08i	0.546331	−	0.546331i
$516$		0			0
$517$		−	3374.78i		−	0.287084i
$518$		0			0
$519$	−6365.57			−0.538377
$520$		0			0
$521$	−2636.90			−0.221736			−0.110868	−	0.993835i	$-0.535363\pi$
$521$	−2636.90			−0.221736			−0.110868	+	0.993835i	$0.535363\pi$
$522$		0			0
$523$			16807.1i			1.40520i	0.711583	+	0.702602i	$0.247978\pi$
$523$			16807.1i			1.40520i	−0.711583	+	0.702602i	$0.752022\pi$
$524$		0			0
$525$	−6789.95	+	6789.95i	−0.564453	+	0.564453i
$526$		0			0
$527$	−6735.52	+	6735.52i	−0.556744	+	0.556744i
$528$		0			0
$529$	−8707.28			−0.715648
$530$		0			0
$531$	2359.72	−	2359.72i	0.192850	−	0.192850i
$532$		0			0
$533$	−2006.68	−	196.971i	−0.163075	−	0.0160070i
$534$		0			0
$535$	−10849.2	−	10849.2i	−0.876729	−	0.876729i
$536$		0			0
$537$		−	11208.1i		−	0.900679i
$538$		0			0
$539$	928.950	+	928.950i	0.0742351	+	0.0742351i
$540$		0			0
$541$	4408.78	+	4408.78i	0.350367	+	0.350367i	0.860246	−	0.509879i	$-0.170310\pi$
$541$	4408.78	+	4408.78i	0.350367	+	0.350367i	−0.509879	+	0.860246i	$0.670310\pi$
$542$		0			0
$543$	−10133.2			−0.800846
$544$		0			0
$545$		−	11677.5i		−	0.917815i
$546$		0			0
$547$		−	23307.3i		−	1.82184i	−0.412582	−	0.910920i	$-0.635373\pi$
$547$		−	23307.3i		−	1.82184i	0.412582	−	0.910920i	$-0.364627\pi$
$548$		0			0
$549$	−621.152			−0.0482880
$550$		0			0
$551$	−389.253	−	389.253i	−0.0300957	−	0.0300957i
$552$		0			0
$553$	5650.68	+	5650.68i	0.434523	+	0.434523i
$554$		0			0
$555$		−	10630.1i		−	0.813015i
$556$		0			0
$557$	1815.00	+	1815.00i	0.138069	+	0.138069i	0.772763	−	0.634695i	$-0.218874\pi$
$557$	1815.00	+	1815.00i	0.138069	+	0.138069i	−0.634695	+	0.772763i	$0.718874\pi$
$558$		0			0
$559$	10183.7	+	12400.5i	0.770528	+	0.938259i
$560$		0			0
$561$	633.245	−	633.245i	0.0476571	−	0.0476571i
$562$		0			0
$563$	16303.7			1.22046			0.610229	−	0.792225i	$-0.291077\pi$
$563$	16303.7			1.22046			0.610229	+	0.792225i	$0.291077\pi$
$564$		0			0
$565$	−7409.89	+	7409.89i	−0.551746	+	0.551746i
$566$		0			0
$567$	−729.356	+	729.356i	−0.0540213	+	0.0540213i
$568$		0			0
$569$			10877.8i			0.801443i	0.916200	+	0.400722i	$0.131241\pi$
$569$			10877.8i			0.801443i	−0.916200	+	0.400722i	$0.868759\pi$
$570$		0			0
$571$	21731.5			1.59270			0.796352	−	0.604833i	$-0.206760\pi$
$571$	21731.5			1.59270			0.796352	+	0.604833i	$0.206760\pi$
$572$		0			0
$573$	−7641.02			−0.557083
$574$		0			0
$575$		−	14784.7i		−	1.07228i
$576$		0			0
$577$	−12694.3	+	12694.3i	−0.915895	+	0.915895i	−0.996728	−	0.0808325i	$-0.974242\pi$
$577$	−12694.3	+	12694.3i	−0.915895	+	0.915895i	0.0808325	+	0.996728i	$0.474242\pi$
$578$		0			0
$579$	−8299.42	+	8299.42i	−0.595703	+	0.595703i
$580$		0			0
$581$	−9773.99			−0.697923
$582$		0			0
$583$	564.001	−	564.001i	0.0400661	−	0.0400661i
$584$		0			0
$585$	8144.70	+	799.465i	0.575628	+	0.0565023i
$586$		0			0
$587$	1253.09	+	1253.09i	0.0881103	+	0.0881103i	0.749788	−	0.661678i	$-0.230155\pi$
$587$	1253.09	+	1253.09i	0.0881103	+	0.0881103i	−0.661678	+	0.749788i	$0.730155\pi$
$588$		0			0
$589$		−	9378.93i		−	0.656116i
$590$		0			0
$591$	1745.15	+	1745.15i	0.121465	+	0.121465i
$592$		0			0
$593$	13411.8	+	13411.8i	0.928762	+	0.928762i	0.997626	−	0.0688639i	$-0.0219374\pi$
$593$	13411.8	+	13411.8i	0.928762	+	0.928762i	−0.0688639	+	0.997626i	$0.521937\pi$
$594$		0			0
$595$	10151.4			0.699441
$596$		0			0
$597$		−	6723.03i		−	0.460897i
$598$		0			0
$599$			7464.13i			0.509142i	0.967054	+	0.254571i	$0.0819342\pi$
$599$			7464.13i			0.509142i	−0.967054	+	0.254571i	$0.918066\pi$
$600$		0			0
$601$	17257.4			1.17128			0.585642	−	0.810570i	$-0.300842\pi$
$601$	17257.4			1.17128			0.585642	+	0.810570i	$0.300842\pi$
$602$		0			0
$603$	1249.12	+	1249.12i	0.0843584	+	0.0843584i
$604$		0			0
$605$	17534.5	+	17534.5i	1.17831	+	1.17831i
$606$		0			0
$607$			21613.0i			1.44521i	0.691259	+	0.722607i	$0.257057\pi$
$607$			21613.0i			1.44521i	−0.691259	+	0.722607i	$0.742943\pi$
$608$		0			0
$609$	367.535	+	367.535i	0.0244553	+	0.0244553i
$610$		0			0
$611$	−21670.5	−	2127.13i	−1.43485	−	0.140842i
$612$		0			0
$613$	−645.520	+	645.520i	−0.0425323	+	0.0425323i	−0.728053	−	0.685521i	$-0.759575\pi$
$613$	−645.520	+	645.520i	−0.0425323	+	0.0425323i	0.685521	+	0.728053i	$0.259575\pi$
$614$		0			0
$615$	2503.61			0.164155
$616$		0			0
$617$	−14221.1	+	14221.1i	−0.927910	+	0.927910i	−0.997571	−	0.0696608i	$-0.977808\pi$
$617$	−14221.1	+	14221.1i	−0.927910	+	0.927910i	0.0696608	+	0.997571i	$0.477808\pi$
$618$		0			0
$619$	−2498.37	+	2498.37i	−0.162226	+	0.162226i	−0.783552	−	0.621326i	$-0.786594\pi$
$619$	−2498.37	+	2498.37i	−0.162226	+	0.162226i	0.621326	+	0.783552i	$0.286594\pi$
$620$		0			0
$621$		−	1588.12i		−	0.102623i
$622$		0			0
$623$	5512.53			0.354502
$624$		0			0
$625$	−16135.7			−1.03269
$626$		0			0
$627$			881.767i			0.0561633i
$628$		0			0
$629$	−5307.12	+	5307.12i	−0.336421	+	0.336421i
$630$		0			0
$631$	4879.60	−	4879.60i	0.307850	−	0.307850i	−0.536225	−	0.844075i	$-0.680150\pi$
$631$	4879.60	−	4879.60i	0.307850	−	0.307850i	0.844075	+	0.536225i	$0.180150\pi$
$632$		0			0
$633$	−710.242			−0.0445965
$634$		0			0
$635$	31307.4	−	31307.4i	1.95653	−	1.95653i
$636$		0			0
$637$	6550.60	−	5379.57i	0.407448	−	0.334609i
$638$		0			0
$639$	1667.14	+	1667.14i	0.103210	+	0.103210i
$640$		0			0
$641$		−	18009.6i		−	1.10973i	−0.831940	−	0.554866i	$-0.812770\pi$
$641$		−	18009.6i		−	1.10973i	0.831940	−	0.554866i	$-0.187230\pi$
$642$		0			0
$643$	−20402.6	−	20402.6i	−1.25132	−	1.25132i	−0.955127	−	0.296198i	$-0.904281\pi$
$643$	−20402.6	−	20402.6i	−1.25132	−	1.25132i	−0.296198	−	0.955127i	$-0.595719\pi$
$644$		0			0
$645$	−14088.5	−	14088.5i	−0.860051	−	0.860051i
$646$		0			0
$647$	−22975.7			−1.39609			−0.698044	−	0.716055i	$-0.745946\pi$
$647$	−22975.7			−1.39609			−0.698044	+	0.716055i	$0.745946\pi$
$648$		0			0
$649$			2693.66i			0.162920i
$650$		0			0
$651$			8855.65i			0.533149i
$652$		0			0
$653$	12090.4			0.724554			0.362277	−	0.932071i	$-0.382000\pi$
$653$	12090.4			0.724554			0.362277	+	0.932071i	$0.382000\pi$
$654$		0			0
$655$	−22438.5	−	22438.5i	−1.33854	−	1.33854i
$656$		0			0
$657$	5380.42	+	5380.42i	0.319498	+	0.319498i
$658$		0			0
$659$		−	29644.6i		−	1.75233i	−0.482007	−	0.876167i	$-0.660092\pi$
$659$		−	29644.6i		−	1.75233i	0.482007	−	0.876167i	$-0.339908\pi$
$660$		0			0
$661$	−12887.9	−	12887.9i	−0.758366	−	0.758366i	0.217659	−	0.976025i	$-0.430158\pi$
$661$	−12887.9	−	12887.9i	−0.758366	−	0.758366i	−0.976025	+	0.217659i	$0.930158\pi$
$662$		0			0
$663$	−3667.13	−	4465.40i	−0.214811	−	0.261572i
$664$		0			0
$665$	−7067.71	+	7067.71i	−0.412141	+	0.412141i
$666$		0			0
$667$	−800.283			−0.0464574
$668$		0			0
$669$	−8804.24	+	8804.24i	−0.508807	+	0.508807i
$670$		0			0
$671$	354.527	−	354.527i	0.0203969	−	0.0203969i
$672$		0			0
$673$			4371.82i			0.250403i	0.992131	+	0.125202i	$0.0399578\pi$
$673$			4371.82i			0.250403i	−0.992131	+	0.125202i	$0.960042\pi$
$674$		0			0
$675$	−6786.64			−0.386989
$676$		0			0
$677$	6590.00			0.374112			0.187056	−	0.982349i	$-0.440105\pi$
$677$	6590.00			0.374112			0.187056	+	0.982349i	$0.440105\pi$
$678$		0			0
$679$			22147.4i			1.25175i
$680$		0			0
$681$	−5583.08	+	5583.08i	−0.314162	+	0.314162i
$682$		0			0
$683$	2427.54	−	2427.54i	0.135999	−	0.135999i	−0.635830	−	0.771829i	$-0.719342\pi$
$683$	2427.54	−	2427.54i	0.135999	−	0.135999i	0.771829	+	0.635830i	$0.219342\pi$
$684$		0			0
$685$	−7346.11			−0.409753
$686$		0			0
$687$	8161.21	−	8161.21i	0.453231	−	0.453231i
$688$		0			0
$689$	−3266.14	−	3977.12i	−0.180595	−	0.219907i
$690$		0			0
$691$	−13332.7	−	13332.7i	−0.734008	−	0.734008i	0.237403	−	0.971411i	$-0.423704\pi$
$691$	−13332.7	−	13332.7i	−0.734008	−	0.734008i	−0.971411	+	0.237403i	$0.923704\pi$
$692$		0			0
$693$		−	832.570i		−	0.0456374i
$694$		0			0
$695$	16338.7	+	16338.7i	0.891746	+	0.891746i
$696$		0			0
$697$	−1249.93	−	1249.93i	−0.0679262	−	0.0679262i
$698$		0			0
$699$	13443.6			0.727443
$700$		0			0
$701$		−	17271.4i		−	0.930572i	−0.885160	−	0.465286i	$-0.845951\pi$
$701$		−	17271.4i		−	0.930572i	0.885160	−	0.465286i	$-0.154049\pi$
$702$		0			0
$703$		−	7389.94i		−	0.396468i
$704$		0			0
$705$	27036.9			1.44435
$706$		0			0
$707$	15889.4	+	15889.4i	0.845237	+	0.845237i
$708$		0			0
$709$	18290.8	+	18290.8i	0.968867	+	0.968867i	0.999530	−	0.0306626i	$-0.00976173\pi$
$709$	18290.8	+	18290.8i	0.968867	+	0.968867i	−0.0306626	+	0.999530i	$0.509762\pi$
$710$		0			0
$711$			5647.92i			0.297909i
$712$		0			0
$713$	−9641.28	−	9641.28i	−0.506408	−	0.506408i
$714$		0			0
$715$	−5104.95	+	4192.35i	−0.267013	+	0.219280i
$716$		0			0
$717$	−9309.14	+	9309.14i	−0.484876	+	0.484876i
$718$		0			0
$719$	36123.2			1.87367			0.936833	−	0.349776i	$-0.113742\pi$
$719$	36123.2			1.87367			0.936833	+	0.349776i	$0.113742\pi$
$720$		0			0
$721$	−4191.18	+	4191.18i	−0.216488	+	0.216488i
$722$		0			0
$723$	−10567.1	+	10567.1i	−0.543561	+	0.543561i
$724$		0			0
$725$			3419.90i			0.175189i
$726$		0			0
$727$	31610.5			1.61261			0.806306	−	0.591499i	$-0.201464\pi$
$727$	31610.5			1.61261			0.806306	+	0.591499i	$0.201464\pi$
$728$		0			0
$729$	−729.000			−0.0370370
$730$		0			0
$731$			14067.4i			0.711768i
$732$		0			0
$733$	20533.6	−	20533.6i	1.03469	−	1.03469i	0.0353108	−	0.999376i	$-0.488758\pi$
$733$	20533.6	−	20533.6i	1.03469	−	1.03469i	0.999376	−	0.0353108i	$-0.0112421\pi$
$734$		0			0
$735$	−7442.26	+	7442.26i	−0.373486	+	0.373486i
$736$		0			0
$737$	−1425.89			−0.0712663
$738$		0			0
$739$	−11889.5	+	11889.5i	−0.591832	+	0.591832i	−0.938126	−	0.346294i	$-0.887440\pi$
$739$	−11889.5	+	11889.5i	−0.591832	+	0.591832i	0.346294	+	0.938126i	$0.387440\pi$
$740$		0			0
$741$	5662.11	+	555.779i	0.280705	+	0.0275534i
$742$		0			0
$743$	18983.2	+	18983.2i	0.937315	+	0.937315i	0.998148	−	0.0608329i	$-0.0193757\pi$
$743$	18983.2	+	18983.2i	0.937315	+	0.937315i	−0.0608329	+	0.998148i	$0.519376\pi$
$744$		0			0
$745$		−	68080.8i		−	3.34804i
$746$		0			0
$747$	−4884.61	−	4884.61i	−0.239248	−	0.239248i
$748$		0			0
$749$	7121.40	+	7121.40i	0.347410	+	0.347410i
$750$		0			0
$751$	−32739.5			−1.59079			−0.795394	−	0.606092i	$-0.792736\pi$
$751$	−32739.5			−1.59079			−0.795394	+	0.606092i	$0.792736\pi$
$752$		0			0
$753$		−	22313.9i		−	1.07990i
$754$		0			0
$755$		−	5000.80i		−	0.241057i
$756$		0			0
$757$	10925.0			0.524538			0.262269	−	0.964995i	$-0.415529\pi$
$757$	10925.0			0.524538			0.262269	+	0.964995i	$0.415529\pi$
$758$		0			0
$759$	906.432	+	906.432i	0.0433484	+	0.0433484i
$760$		0			0
$761$	22481.9	+	22481.9i	1.07092	+	1.07092i	0.997285	+	0.0736331i	$0.0234594\pi$
$761$	22481.9	+	22481.9i	1.07092	+	1.07092i	0.0736331	+	0.997285i	$0.476541\pi$
$762$		0			0
$763$			7665.13i			0.363691i
$764$		0			0
$765$	5073.23	+	5073.23i	0.239769	+	0.239769i
$766$		0			0
$767$	17296.8	+	1697.82i	0.814279	+	0.0799277i
$768$		0			0
$769$	3212.83	−	3212.83i	0.150660	−	0.150660i	−0.627753	−	0.778413i	$-0.716025\pi$
$769$	3212.83	−	3212.83i	0.150660	−	0.150660i	0.778413	+	0.627753i	$0.216025\pi$
$770$		0			0
$771$	2700.68			0.126151
$772$		0			0
$773$	11867.0	−	11867.0i	0.552168	−	0.552168i	−0.374898	−	0.927066i	$-0.622322\pi$
$773$	11867.0	−	11867.0i	0.552168	−	0.552168i	0.927066	+	0.374898i	$0.122322\pi$
$774$		0			0
$775$	−41200.8	+	41200.8i	−1.90965	+	1.90965i
$776$		0			0
$777$			6977.62i			0.322163i
$778$		0			0
$779$	1740.48			0.0800502
$780$		0			0
$781$	−1903.06			−0.0871919
$782$		0			0
$783$			367.356i			0.0167666i
$784$		0			0
$785$	−34473.6	+	34473.6i	−1.56741	+	1.56741i
$786$		0			0
$787$	−9364.89	+	9364.89i	−0.424171	+	0.424171i	−0.886637	−	0.462466i	$-0.846965\pi$
$787$	−9364.89	+	9364.89i	−0.424171	+	0.424171i	0.462466	+	0.886637i	$0.346965\pi$
$788$		0			0
$789$	−1467.91			−0.0662346
$790$		0			0
$791$	4863.86	−	4863.86i	0.218634	−	0.218634i
$792$		0			0
$793$	−2053.07	−	2499.99i	−0.0919377	−	0.111951i
$794$		0			0
$795$	4518.48	+	4518.48i	0.201577	+	0.201577i
$796$		0			0
$797$		−	5540.68i		−	0.246249i	−0.992391	−	0.123125i	$-0.960708\pi$
$797$		−	5540.68i		−	0.246249i	0.992391	−	0.123125i	$-0.0392915\pi$
$798$		0			0
$799$	−13498.3	−	13498.3i	−0.597665	−	0.597665i
$800$		0			0
$801$	2754.92	+	2754.92i	0.121523	+	0.121523i
$802$		0			0
$803$	−6141.83			−0.269913
$804$		0			0
$805$			14530.8i			0.636203i
$806$		0			0
$807$			24026.5i			1.04805i
$808$		0			0
$809$	756.806			0.0328898			0.0164449	−	0.999865i	$-0.494765\pi$
$809$	756.806			0.0328898			0.0164449	+	0.999865i	$0.494765\pi$
$810$		0			0
$811$	−23212.8	−	23212.8i	−1.00507	−	1.00507i	−0.999987	−	0.00508197i	$-0.998382\pi$
$811$	−23212.8	−	23212.8i	−1.00507	−	1.00507i	−0.00508197	−	0.999987i	$-0.501618\pi$
$812$		0			0
$813$	−7574.33	−	7574.33i	−0.326745	−	0.326745i
$814$		0			0
$815$		−	45280.1i		−	1.94613i
$816$		0			0
$817$	−9794.15	−	9794.15i	−0.419405	−	0.419405i
$818$		0			0
$819$	−5346.20	−	524.770i	−0.228097	−	0.0223894i
$820$		0			0
$821$	−11727.2	+	11727.2i	−0.498518	+	0.498518i	−0.910977	−	0.412458i	$-0.864670\pi$
$821$	−11727.2	+	11727.2i	−0.498518	+	0.498518i	0.412458	+	0.910977i	$0.364670\pi$
$822$		0			0
$823$	−4551.94			−0.192796			−0.0963978	−	0.995343i	$-0.530732\pi$
$823$	−4551.94			−0.192796			−0.0963978	+	0.995343i	$0.530732\pi$
$824$		0			0
$825$	3873.52	−	3873.52i	0.163465	−	0.163465i
$826$		0			0
$827$	10079.8	−	10079.8i	0.423833	−	0.423833i	−0.462688	−	0.886521i	$-0.653115\pi$
$827$	10079.8	−	10079.8i	0.423833	−	0.423833i	0.886521	+	0.462688i	$0.153115\pi$
$828$		0			0
$829$			2208.99i			0.0925468i	0.998929	+	0.0462734i	$0.0147345\pi$
$829$			2208.99i			0.0925468i	−0.998929	+	0.0462734i	$0.985265\pi$
$830$		0			0
$831$	−10646.5			−0.444430
$832$		0			0
$833$	7431.14			0.309092
$834$		0			0
$835$			66634.7i			2.76166i
$836$		0			0
$837$	−4425.66	+	4425.66i	−0.182764	+	0.182764i
$838$		0			0
$839$	−24447.0	+	24447.0i	−1.00597	+	1.00597i	−0.00598415	+	0.999982i	$0.501905\pi$
$839$	−24447.0	+	24447.0i	−1.00597	+	1.00597i	−0.999982	+	0.00598415i	$0.998095\pi$
$840$		0			0
$841$	−24203.9			−0.992410
$842$		0			0
$843$	10368.7	−	10368.7i	0.423627	−	0.423627i
$844$		0			0
$845$	23702.7	+	35422.9i	0.964969	+	1.44211i
$846$		0			0
$847$	−11509.7	−	11509.7i	−0.466914	−	0.466914i
$848$		0			0
$849$		−	6635.07i		−	0.268216i
$850$		0			0
$851$	−7596.65	−	7596.65i	−0.306004	−	0.306004i
$852$		0			0
$853$	20833.5	+	20833.5i	0.836253	+	0.836253i	0.988364	−	0.152110i	$-0.0486069\pi$
$853$	20833.5	+	20833.5i	0.836253	+	0.836253i	−0.152110	+	0.988364i	$0.548607\pi$
$854$		0			0
$855$	−7064.26			−0.282564
$856$		0			0
$857$			21453.2i			0.855108i	0.903990	+	0.427554i	$0.140625\pi$
$857$			21453.2i			0.855108i	−0.903990	+	0.427554i	$0.859375\pi$
$858$		0			0
$859$		−	25866.6i		−	1.02742i	−0.857962	−	0.513712i	$-0.828270\pi$
$859$		−	25866.6i		−	1.02742i	0.857962	−	0.513712i	$-0.171730\pi$
$860$		0			0
$861$	−1643.37			−0.0650475
$862$		0			0
$863$	−10638.3	−	10638.3i	−0.419620	−	0.419620i	0.465453	−	0.885073i	$-0.345891\pi$
$863$	−10638.3	−	10638.3i	−0.419620	−	0.419620i	−0.885073	+	0.465453i	$0.845891\pi$
$864$		0			0
$865$	29107.2	+	29107.2i	1.14413	+	1.14413i
$866$		0			0
$867$			9673.35i			0.378921i
$868$		0			0
$869$	−3223.59	−	3223.59i	−0.125838	−	0.125838i
$870$		0			0
$871$	−898.739	+	9156.08i	−0.0349628	+	0.356190i
$872$		0			0
$873$	−11068.3	+	11068.3i	−0.429101	+	0.429101i
$874$		0			0
$875$	31215.4			1.20603
$876$		0			0
$877$	27081.2	−	27081.2i	1.04272	−	1.04272i	0.0436767	−	0.999046i	$-0.486093\pi$
$877$	27081.2	−	27081.2i	1.04272	−	1.04272i	0.999046	−	0.0436767i	$-0.0139072\pi$
$878$		0			0
$879$	−14120.4	+	14120.4i	−0.541830	+	0.541830i
$880$		0			0
$881$			8767.54i			0.335285i	0.985848	+	0.167642i	$0.0536154\pi$
$881$			8767.54i			0.335285i	−0.985848	+	0.167642i	$0.946385\pi$
$882$		0			0
$883$	−28566.9			−1.08873			−0.544367	−	0.838847i	$-0.683230\pi$
$883$	−28566.9			−1.08873			−0.544367	+	0.838847i	$0.683230\pi$
$884$		0			0
$885$	−21580.2			−0.819671
$886$		0			0
$887$		−	20405.9i		−	0.772449i	−0.922405	−	0.386224i	$-0.873779\pi$
$887$		−	20405.9i		−	0.772449i	0.922405	−	0.386224i	$-0.126221\pi$
$888$		0			0
$889$	−20550.2	+	20550.2i	−0.775290	+	0.775290i
$890$		0			0
$891$	416.082	−	416.082i	0.0156445	−	0.0156445i
$892$		0			0
$893$	18795.8			0.704341
$894$		0			0
$895$	−51250.2	+	51250.2i	−1.91408	+	1.91408i
$896$		0			0
$897$	6391.81	−	5249.16i	0.237922	−	0.195390i
$898$		0			0
$899$	2230.17	+	2230.17i	0.0827367	+	0.0827367i
$900$		0			0
$901$		−	4511.73i		−	0.166823i
$902$		0			0
$903$	9247.70	+	9247.70i	0.340802	+	0.340802i
$904$		0			0
$905$	46335.3	+	46335.3i	1.70192	+	1.70192i
$906$		0			0
$907$	43074.1			1.57690			0.788452	−	0.615096i	$-0.210883\pi$
$907$	43074.1			1.57690			0.788452	+	0.615096i	$0.210883\pi$
$908$		0			0
$909$			15881.6i			0.579495i
$910$		0			0
$911$		−	45705.5i		−	1.66223i	−0.556101	−	0.831115i	$-0.687703\pi$
$911$		−	45705.5i		−	1.66223i	0.556101	−	0.831115i	$-0.312297\pi$
$912$		0			0
$913$	5575.85			0.202118
$914$		0			0
$915$	2840.28	+	2840.28i	0.102619	+	0.102619i
$916$		0			0
$917$	14728.7	+	14728.7i	0.530408	+	0.530408i
$918$		0			0
$919$		−	37095.3i		−	1.33151i	−0.746168	−	0.665757i	$-0.768109\pi$
$919$		−	37095.3i		−	1.33151i	0.746168	−	0.665757i	$-0.231891\pi$
$920$		0			0
$921$	756.375	+	756.375i	0.0270612	+	0.0270612i
$922$		0			0
$923$	−1199.50	+	12220.2i	−0.0427758	+	0.435787i
$924$		0			0
$925$	−32463.3	+	32463.3i	−1.15393	+	1.15393i
$926$		0			0
$927$	−4189.13			−0.148424
$928$		0			0
$929$	3260.29	−	3260.29i	0.115142	−	0.115142i	−0.647188	−	0.762330i	$-0.724055\pi$
$929$	3260.29	−	3260.29i	0.115142	−	0.115142i	0.762330	+	0.647188i	$0.224055\pi$
$930$		0			0
$931$	−5173.78	+	5173.78i	−0.182131	+	0.182131i
$932$		0			0
$933$			5113.86i			0.179443i
$934$		0			0
$935$	−5791.16			−0.202557
$936$		0			0
$937$	32258.5			1.12469			0.562347	−	0.826901i	$-0.309898\pi$
$937$	32258.5			1.12469			0.562347	+	0.826901i	$0.309898\pi$
$938$		0			0
$939$			16246.0i			0.564610i
$940$		0			0
$941$	−13982.5	+	13982.5i	−0.484394	+	0.484394i	−0.906532	−	0.422137i	$-0.861280\pi$
$941$	−13982.5	+	13982.5i	−0.484394	+	0.484394i	0.422137	+	0.906532i	$0.361280\pi$
$942$		0			0
$943$	1789.16	−	1789.16i	0.0617849	−	0.0617849i
$944$		0			0
$945$	6670.11			0.229607
$946$		0			0
$947$	2521.02	−	2521.02i	0.0865071	−	0.0865071i	−0.662529	−	0.749036i	$-0.730517\pi$
$947$	2521.02	−	2521.02i	0.0865071	−	0.0865071i	0.749036	+	0.662529i	$0.230517\pi$
$948$		0			0
$949$	−3871.20	+	39438.6i	−0.132418	+	1.34903i
$950$		0			0
$951$	−8030.92	−	8030.92i	−0.273838	−	0.273838i
$952$		0			0
$953$			7398.75i			0.251489i	0.992063	+	0.125745i	$0.0401320\pi$
$953$			7398.75i			0.251489i	−0.992063	+	0.125745i	$0.959868\pi$
$954$		0			0
$955$	34939.4	+	34939.4i	1.18389	+	1.18389i
$956$		0			0
$957$	−209.671	−	209.671i	−0.00708223	−	0.00708223i
$958$		0			0
$959$	4822.00			0.162368
$960$		0			0
$961$			23944.2i			0.803740i
$962$		0			0
$963$			7117.93i			0.238185i
$964$		0			0
$965$	75899.9			2.53192
$966$		0			0
$967$	−35721.3	−	35721.3i	−1.18792	−	1.18792i	−0.977642	−	0.210278i	$-0.932563\pi$
$967$	−35721.3	−	35721.3i	−1.18792	−	1.18792i	−0.210278	−	0.977642i	$-0.567437\pi$
$968$		0			0
$969$	3526.85	+	3526.85i	0.116923	+	0.116923i
$970$		0			0
$971$			11414.2i			0.377239i	0.982050	+	0.188620i	$0.0604013\pi$
$971$			11414.2i			0.377239i	−0.982050	+	0.188620i	$0.939599\pi$
$972$		0			0
$973$	−10724.8	−	10724.8i	−0.353361	−	0.353361i
$974$		0			0
$975$	−22431.6	−	27314.6i	−0.736807	−	0.897197i
$976$		0			0
$977$	−34570.4	+	34570.4i	−1.13204	+	1.13204i	−0.142205	+	0.989837i	$0.545419\pi$
$977$	−34570.4	+	34570.4i	−1.13204	+	1.13204i	−0.989837	+	0.142205i	$0.954581\pi$
$978$		0			0
$979$	−3144.78			−0.102664
$980$		0			0
$981$	−3830.69	+	3830.69i	−0.124673	+	0.124673i
$982$		0			0
$983$	23504.8	−	23504.8i	0.762650	−	0.762650i	−0.214150	−	0.976801i	$-0.568698\pi$
$983$	23504.8	−	23504.8i	0.762650	−	0.762650i	0.976801	+	0.214150i	$0.0686982\pi$
$984$		0			0
$985$		−	15959.8i		−	0.516264i
$986$		0			0
$987$	−17747.1			−0.572336
$988$		0			0
$989$	−20136.2			−0.647416
$990$		0			0
$991$		−	52539.1i		−	1.68412i	−0.539386	−	0.842058i	$-0.681344\pi$
$991$		−	52539.1i		−	1.68412i	0.539386	−	0.842058i	$-0.318656\pi$
$992$		0			0
$993$	10960.8	−	10960.8i	0.350284	−	0.350284i
$994$		0			0
$995$	−30741.8	+	30741.8i	−0.979477	+	0.979477i
$996$		0			0
$997$	−59778.9			−1.89891			−0.949457	−	0.313898i	$-0.898365\pi$
$997$	−59778.9			−1.89891			−0.949457	+	0.313898i	$0.898365\pi$
$998$		0			0
$999$	−3487.11	+	3487.11i	−0.110438	+	0.110438i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.bc.d.31.14	yes	28
4.3	odd	2		624.4.bc.c.31.14	✓	28
13.8	odd	4		624.4.bc.c.463.14	yes	28
52.47	even	4	inner	624.4.bc.d.463.14	yes	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
624.4.bc.c.31.14	✓	28	4.3	odd	2
624.4.bc.c.463.14	yes	28	13.8	odd	4
624.4.bc.d.31.14	yes	28	1.1	even	1	trivial
624.4.bc.d.463.14	yes	28	52.47	even	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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.bc (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{3} +(13.7178 - 13.7178i) q^{5} +(-9.00439 + 9.00439i) q^{7} -9.00000 q^{9} +(5.13681 - 5.13681i) q^{11} +(-29.7474 - 36.2228i) q^{13} +(41.1534 + 41.1534i) q^{15} -41.0920i q^{17} +(28.6094 + 28.6094i) q^{19} +(-27.0132 - 27.0132i) q^{21} +58.8194 q^{23} -251.357i q^{25} -27.0000i q^{27} -13.6058 q^{29} +(-163.913 - 163.913i) q^{31} +(15.4104 + 15.4104i) q^{33} +247.041i q^{35} +(-129.152 - 129.152i) q^{37} +(108.669 - 89.2421i) q^{39} +(30.4179 - 30.4179i) q^{41} -342.340 q^{43} +(-123.460 + 123.460i) q^{45} +(328.489 - 328.489i) q^{47} +180.842i q^{49} +123.276 q^{51} +109.796 q^{53} -140.932i q^{55} +(-85.8283 + 85.8283i) q^{57} +(-262.191 + 262.191i) q^{59} +69.0168 q^{61} +(81.0396 - 81.0396i) q^{63} +(-904.967 - 88.8295i) q^{65} +(-138.791 - 138.791i) q^{67} +176.458i q^{69} +(-185.237 - 185.237i) q^{71} +(-597.825 - 597.825i) q^{73} +754.071 q^{75} +92.5078i q^{77} -627.547i q^{79} +81.0000 q^{81} +(542.734 + 542.734i) q^{83} +(-563.692 - 563.692i) q^{85} -40.8173i q^{87} +(-306.102 - 306.102i) q^{89} +(594.022 + 58.3078i) q^{91} +(491.740 - 491.740i) q^{93} +784.917 q^{95} +(1229.81 - 1229.81i) q^{97} +(-46.2313 + 46.2313i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 4 q^{5} + 8 q^{7} - 252 q^{9} - 64 q^{11} - 32 q^{13} + 12 q^{15} - 56 q^{19} + 24 q^{21} - 384 q^{23} - 32 q^{29} + 168 q^{31} - 192 q^{33} + 412 q^{37} - 252 q^{39} + 1340 q^{41} - 624 q^{43} - 36 q^{45}+ \cdots + 576 q^{99}+O(q^{100}) \) 28 * q + 4 * q^5 + 8 * q^7 - 252 * q^9 - 64 * q^11 - 32 * q^13 + 12 * q^15 - 56 * q^19 + 24 * q^21 - 384 * q^23 - 32 * q^29 + 168 * q^31 - 192 * q^33 + 412 * q^37 - 252 * q^39 + 1340 * q^41 - 624 * q^43 - 36 * q^45 + 688 * q^47 - 312 * q^51 + 2152 * q^53 + 168 * q^57 - 624 * q^59 - 216 * q^61 - 72 * q^63 + 60 * q^65 + 2024 * q^67 - 448 * q^71 - 724 * q^73 + 1716 * q^75 + 2268 * q^81 - 2288 * q^83 - 1528 * q^85 - 1876 * q^89 + 3448 * q^91 - 504 * q^93 - 192 * q^95 + 76 * q^97 + 576 * q^99

Introduction

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