LMFDB - Embedded newform 588.4.f.c.293.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [588,4,Mod(293,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.293"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 4, names="a")

Level:	$N$	$=$	$588 = 2^{2} \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	588.f (of order $2$ , degree $1$ , minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,0,0,0,0,0,0,168,0,0,0,0,0,132,0,0,0,0,0,0,0,0,0,888,0,0, 0,0,0,0,0,0,0,0,0,-480] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(37)] == traces)

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

Self dual:	no
Analytic conductor:	$34.6931230834$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} + \cdots + 531441$ x^12 - 14x^10 - 32x^9 + 70x^8 + 224x^7 - 50x^6 + 2016x^5 + 5670x^4 - 23328x^3 - 91854*x^2 + 531441
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$2^{10}\cdot 3^{7}$
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			293.1
Root			$-2.14770 - 2.09461i$ of defining polynomial
Character	$\chi$	$=$	588.293
Dual form			588.4.f.c.293.2

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-5.03553 - 1.28196i) q^{3} -19.3778 q^{5} +(23.7132 + 12.9107i) q^{9} -41.9412i q^{11} -77.1820i q^{13} +(97.5776 + 24.8416i) q^{15} -5.96831 q^{17} -68.1654i q^{19} +35.5752i q^{23} +250.500 q^{25} +(-102.857 - 95.4115i) q^{27} -228.525i q^{29} -72.0520i q^{31} +(-53.7668 + 211.196i) q^{33} +69.3392 q^{37} +(-98.9442 + 388.652i) q^{39} -132.789 q^{41} -366.304 q^{43} +(-459.509 - 250.181i) q^{45} -181.908 q^{47} +(30.0536 + 7.65113i) q^{51} -17.6775i q^{53} +812.728i q^{55} +(-87.3853 + 343.249i) q^{57} +494.370 q^{59} +575.525i q^{61} +1495.62i q^{65} -404.323 q^{67} +(45.6060 - 179.140i) q^{69} -489.797i q^{71} +388.753i q^{73} +(-1261.40 - 321.131i) q^{75} +253.411 q^{79} +(395.628 + 612.307i) q^{81} +356.881 q^{83} +115.653 q^{85} +(-292.960 + 1150.75i) q^{87} -1423.16 q^{89} +(-92.3677 + 362.820i) q^{93} +1320.90i q^{95} -694.168i q^{97} +(541.489 - 994.557i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 168 q^{9} + 132 q^{15} + 888 q^{25} - 480 q^{37} + 864 q^{39} + 600 q^{51} + 180 q^{57} - 3960 q^{67} + 876 q^{79} - 2016 q^{81} - 6144 q^{85} - 1764 q^{93} + 9216 q^{99}+O(q^{100})$ 12 * q + 168 * q^9 + 132 * q^15 + 888 * q^25 - 480 * q^37 + 864 * q^39 + 600 * q^51 + 180 * q^57 - 3960 * q^67 + 876 * q^79 - 2016 * q^81 - 6144 * q^85 - 1764 * q^93 + 9216 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$ .

$n$	$197$	$295$	$493$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	−5.03553	−	1.28196i	−0.969089	−	0.246713i
$3$	−5.03553	−	1.28196i	−0.969089	−	0.246713i
$4$		0			0
$5$	−19.3778			−1.73320			−0.866602	−	0.498999i	$-0.833701\pi$
$5$	−19.3778			−1.73320			−0.866602	+	0.498999i	$0.833701\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	23.7132	+	12.9107i	0.878265	+	0.478174i
$10$		0			0
$11$		−	41.9412i		−	1.14961i	−0.818290	−	0.574806i	$-0.805077\pi$
$11$		−	41.9412i		−	1.14961i	0.818290	−	0.574806i	$-0.194923\pi$
$12$		0			0
$13$		−	77.1820i		−	1.64665i	−0.567571	−	0.823325i	$-0.692117\pi$
$13$		−	77.1820i		−	1.64665i	0.567571	−	0.823325i	$-0.307883\pi$
$14$		0			0
$15$	97.5776	+	24.8416i	1.67963	+	0.427604i
$16$		0			0
$17$	−5.96831			−0.0851487			−0.0425743	−	0.999093i	$-0.513556\pi$
$17$	−5.96831			−0.0851487			−0.0425743	+	0.999093i	$0.513556\pi$
$18$		0			0
$19$		−	68.1654i		−	0.823064i	−0.911395	−	0.411532i	$-0.864994\pi$
$19$		−	68.1654i		−	0.823064i	0.911395	−	0.411532i	$-0.135006\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			35.5752i			0.322519i	0.986912	+	0.161260i	$0.0515557\pi$
$23$			35.5752i			0.322519i	−0.986912	+	0.161260i	$0.948444\pi$
$24$		0			0
$25$	250.500			2.00400
$26$		0			0
$27$	−102.857	−	95.4115i	−0.733145	−	0.680072i
$28$		0			0
$29$		−	228.525i		−	1.46331i	−0.681673	−	0.731657i	$-0.738747\pi$
$29$		−	228.525i		−	1.46331i	0.681673	−	0.731657i	$-0.261253\pi$
$30$		0			0
$31$		−	72.0520i		−	0.417449i	−0.977974	−	0.208725i	$-0.933069\pi$
$31$		−	72.0520i		−	0.417449i	0.977974	−	0.208725i	$-0.0669312\pi$
$32$		0			0
$33$	−53.7668	+	211.196i	−0.283624	+	1.11408i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	69.3392			0.308089			0.154045	−	0.988064i	$-0.450770\pi$
$37$	69.3392			0.308089			0.154045	+	0.988064i	$0.450770\pi$
$38$		0			0
$39$	−98.9442	+	388.652i	−0.406250	+	1.59575i
$40$		0			0
$41$	−132.789			−0.505810			−0.252905	−	0.967491i	$-0.581386\pi$
$41$	−132.789			−0.505810			−0.252905	+	0.967491i	$0.581386\pi$
$42$		0			0
$43$	−366.304			−1.29909			−0.649544	−	0.760324i	$-0.725040\pi$
$43$	−366.304			−1.29909			−0.649544	+	0.760324i	$0.725040\pi$
$44$		0			0
$45$	−459.509	−	250.181i	−1.52221	−	0.828773i
$46$		0			0
$47$	−181.908			−0.564555			−0.282277	−	0.959333i	$-0.591090\pi$
$47$	−181.908			−0.564555			−0.282277	+	0.959333i	$0.591090\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	30.0536	+	7.65113i	0.0825166	+	0.0210073i
$52$		0			0
$53$		−	17.6775i		−	0.0458150i	−0.999738	−	0.0229075i	$-0.992708\pi$
$53$		−	17.6775i		−	0.0458150i	0.999738	−	0.0229075i	$-0.00729232\pi$
$54$		0			0
$55$			812.728i			1.99251i
$56$		0			0
$57$	−87.3853	+	343.249i	−0.203061	+	0.797622i
$58$		0			0
$59$	494.370			1.09087			0.545437	−	0.838152i	$-0.316364\pi$
$59$	494.370			1.09087			0.545437	+	0.838152i	$0.316364\pi$
$60$		0			0
$61$			575.525i			1.20801i	0.796981	+	0.604004i	$0.206429\pi$
$61$			575.525i			1.20801i	−0.796981	+	0.604004i	$0.793571\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			1495.62i			2.85398i
$66$		0			0
$67$	−404.323			−0.737253			−0.368627	−	0.929578i	$-0.620172\pi$
$67$	−404.323			−0.737253			−0.368627	+	0.929578i	$0.620172\pi$
$68$		0			0
$69$	45.6060	−	179.140i	0.0795697	−	0.312550i
$70$		0			0
$71$		−	489.797i		−	0.818707i	−0.912376	−	0.409353i	$-0.865754\pi$
$71$		−	489.797i		−	0.818707i	0.912376	−	0.409353i	$-0.134246\pi$
$72$		0			0
$73$			388.753i			0.623290i	0.950199	+	0.311645i	$0.100880\pi$
$73$			388.753i			0.623290i	−0.950199	+	0.311645i	$0.899120\pi$
$74$		0			0
$75$	−1261.40	−	321.131i	−1.94205	−	0.494413i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	253.411			0.360899			0.180449	−	0.983584i	$-0.442245\pi$
$79$	253.411			0.360899			0.180449	+	0.983584i	$0.442245\pi$
$80$		0			0
$81$	395.628	+	612.307i	0.542700	+	0.839927i
$82$		0			0
$83$	356.881			0.471961			0.235981	−	0.971758i	$-0.424170\pi$
$83$	356.881			0.471961			0.235981	+	0.971758i	$0.424170\pi$
$84$		0			0
$85$	115.653			0.147580
$86$		0			0
$87$	−292.960	+	1150.75i	−0.361019	+	1.41808i
$88$		0			0
$89$	−1423.16			−1.69499			−0.847496	−	0.530802i	$-0.821891\pi$
$89$	−1423.16			−1.69499			−0.847496	+	0.530802i	$0.821891\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−92.3677	+	362.820i	−0.102990	+	0.404545i
$94$		0			0
$95$			1320.90i			1.42654i
$96$		0			0
$97$		−	694.168i		−	0.726619i	−0.931668	−	0.363310i	$-0.881647\pi$
$97$		−	694.168i		−	0.726619i	0.931668	−	0.363310i	$-0.118353\pi$
$98$		0			0
$99$	541.489	−	994.557i	0.549714	−	1.00966i
$100$		0			0
$101$	15.3407			0.0151134			0.00755671	−	0.999971i	$-0.497595\pi$
$101$	15.3407			0.0151134			0.00755671	+	0.999971i	$0.497595\pi$
$102$		0			0
$103$		−	353.147i		−	0.337831i	−0.985630	−	0.168916i	$-0.945973\pi$
$103$		−	353.147i		−	0.337831i	0.985630	−	0.168916i	$-0.0540266\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	51.5191i		−	0.0465471i	−0.999729	−	0.0232736i	$-0.992591\pi$
$107$		−	51.5191i		−	0.0465471i	0.999729	−	0.0232736i	$-0.00740887\pi$
$108$		0			0
$109$	−670.628			−0.589308			−0.294654	−	0.955604i	$-0.595204\pi$
$109$	−670.628			−0.589308			−0.294654	+	0.955604i	$0.595204\pi$
$110$		0			0
$111$	−349.160	−	88.8901i	−0.298566	−	0.0760097i
$112$		0			0
$113$		−	461.944i		−	0.384567i	−0.981339	−	0.192283i	$-0.938411\pi$
$113$		−	461.944i		−	0.384567i	0.981339	−	0.192283i	$-0.0615893\pi$
$114$		0			0
$115$		−	689.370i		−	0.558992i
$116$		0			0
$117$	996.473	−	1830.23i	0.787384	−	1.44619i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−428.061			−0.321608
$122$		0			0
$123$	668.665	+	170.231i	0.490175	+	0.124790i
$124$		0			0
$125$	−2431.91			−1.74014
$126$		0			0
$127$	−388.699			−0.271587			−0.135793	−	0.990737i	$-0.543358\pi$
$127$	−388.699			−0.271587			−0.135793	+	0.990737i	$0.543358\pi$
$128$		0			0
$129$	1844.53	+	469.586i	1.25893	+	0.320502i
$130$		0			0
$131$	−627.675			−0.418628			−0.209314	−	0.977849i	$-0.567123\pi$
$131$	−627.675			−0.418628			−0.209314	+	0.977849i	$0.567123\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	1993.15	+	1848.87i	1.27069	+	1.17870i
$136$		0			0
$137$			2104.21i			1.31222i	0.754664	+	0.656111i	$0.227800\pi$
$137$			2104.21i			1.31222i	−0.754664	+	0.656111i	$0.772200\pi$
$138$		0			0
$139$			1030.03i			0.628533i	0.949335	+	0.314266i	$0.101759\pi$
$139$			1030.03i			0.628533i	−0.949335	+	0.314266i	$0.898241\pi$
$140$		0			0
$141$	916.005	+	233.199i	0.547103	+	0.139283i
$142$		0			0
$143$	−3237.10			−1.89301
$144$		0			0
$145$			4428.32i			2.53622i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			201.165i			0.110605i	0.998470	+	0.0553024i	$0.0176123\pi$
$149$			201.165i			0.110605i	−0.998470	+	0.0553024i	$0.982388\pi$
$150$		0			0
$151$	1192.48			0.642664			0.321332	−	0.946967i	$-0.395869\pi$
$151$	1192.48			0.642664			0.321332	+	0.946967i	$0.395869\pi$
$152$		0			0
$153$	−141.527	−	77.0550i	−0.0747831	−	0.0407158i
$154$		0			0
$155$			1396.21i			0.723525i
$156$		0			0
$157$		−	1092.22i		−	0.555215i	−0.960695	−	0.277608i	$-0.910459\pi$
$157$		−	1092.22i		−	0.555215i	0.960695	−	0.277608i	$-0.0895415\pi$
$158$		0			0
$159$	−22.6619	+	89.0158i	−0.0113032	+	0.0443988i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−579.604			−0.278516			−0.139258	−	0.990256i	$-0.544472\pi$
$163$	−579.604			−0.278516			−0.139258	+	0.990256i	$0.544472\pi$
$164$		0			0
$165$	1041.88	−	4092.52i	0.491579	−	1.93092i
$166$		0			0
$167$	3715.71			1.72174			0.860869	−	0.508827i	$-0.169921\pi$
$167$	3715.71			1.72174			0.860869	+	0.508827i	$0.169921\pi$
$168$		0			0
$169$	−3760.06			−1.71145
$170$		0			0
$171$	880.063	−	1616.42i	0.393568	−	0.722869i
$172$		0			0
$173$	−1338.41			−0.588193			−0.294096	−	0.955776i	$-0.595019\pi$
$173$	−1338.41			−0.588193			−0.294096	+	0.955776i	$0.595019\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2489.42	−	633.763i	−1.05715	−	0.269133i
$178$		0			0
$179$		−	2818.71i		−	1.17698i	−0.808503	−	0.588492i	$-0.799722\pi$
$179$		−	2818.71i		−	1.17698i	0.808503	−	0.588492i	$-0.200278\pi$
$180$		0			0
$181$			3688.28i			1.51463i	0.653050	+	0.757315i	$0.273489\pi$
$181$			3688.28i			1.51463i	−0.653050	+	0.757315i	$0.726511\pi$
$182$		0			0
$183$	737.800	−	2898.08i	0.298031	−	1.17067i
$184$		0			0
$185$	−1343.64			−0.533982
$186$		0			0
$187$			250.318i			0.0978879i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	4509.14i		−	1.70822i	−0.520092	−	0.854111i	$-0.674102\pi$
$191$		−	4509.14i		−	1.70822i	0.520092	−	0.854111i	$-0.325898\pi$
$192$		0			0
$193$	−2211.49			−0.824802			−0.412401	−	0.911002i	$-0.635310\pi$
$193$	−2211.49			−0.824802			−0.412401	+	0.911002i	$0.635310\pi$
$194$		0			0
$195$	1917.32	−	7531.24i	0.704114	−	2.76576i
$196$		0			0
$197$			4177.21i			1.51073i	0.655304	+	0.755365i	$0.272540\pi$
$197$			4177.21i			1.51073i	−0.655304	+	0.755365i	$0.727460\pi$
$198$		0			0
$199$		−	4415.17i		−	1.57278i	−0.617730	−	0.786391i	$-0.711947\pi$
$199$		−	4415.17i		−	1.57278i	0.617730	−	0.786391i	$-0.288053\pi$
$200$		0			0
$201$	2035.98	+	518.326i	0.714463	+	0.181890i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2573.17			0.876673
$206$		0			0
$207$	−459.300	+	843.601i	−0.154220	+	0.283257i
$208$		0			0
$209$	−2858.94			−0.946205
$210$		0			0
$211$	−2002.19			−0.653252			−0.326626	−	0.945154i	$-0.605912\pi$
$211$	−2002.19			−0.653252			−0.326626	+	0.945154i	$0.605912\pi$
$212$		0			0
$213$	−627.899	+	2466.39i	−0.201986	+	0.793399i
$214$		0			0
$215$	7098.17			2.25159
$216$		0			0
$217$		0			0
$218$		0			0
$219$	498.366	−	1957.58i	0.153774	−	0.604023i
$220$		0			0
$221$			460.646i			0.140210i
$222$		0			0
$223$			2122.04i			0.637230i	0.947884	+	0.318615i	$0.103218\pi$
$223$			2122.04i			0.637230i	−0.947884	+	0.318615i	$0.896782\pi$
$224$		0			0
$225$	5940.14	+	3234.13i	1.76004	+	0.958260i
$226$		0			0
$227$	4862.82			1.42184			0.710918	−	0.703274i	$-0.248280\pi$
$227$	4862.82			1.42184			0.710918	+	0.703274i	$0.248280\pi$
$228$		0			0
$229$		−	5662.03i		−	1.63387i	−0.576726	−	0.816937i	$-0.695670\pi$
$229$		−	5662.03i		−	1.63387i	0.576726	−	0.816937i	$-0.304330\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			3253.73i			0.914846i	0.889249	+	0.457423i	$0.151228\pi$
$233$			3253.73i			0.914846i	−0.889249	+	0.457423i	$0.848772\pi$
$234$		0			0
$235$	3524.99			0.978489
$236$		0			0
$237$	−1276.06	−	324.863i	−0.349743	−	0.0890385i
$238$		0			0
$239$		−	5823.69i		−	1.57616i	−0.615571	−	0.788082i	$-0.711074\pi$
$239$		−	5823.69i		−	1.57616i	0.615571	−	0.788082i	$-0.288926\pi$
$240$		0			0
$241$			6176.56i			1.65090i	0.564474	+	0.825451i	$0.309079\pi$
$241$			6176.56i			1.65090i	−0.564474	+	0.825451i	$0.690921\pi$
$242$		0			0
$243$	−1207.25	−	3590.47i	−0.318703	−	0.947855i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−5261.14			−1.35530
$248$		0			0
$249$	−1797.09	−	457.507i	−0.457372	−	0.116439i
$250$		0			0
$251$	−1648.38			−0.414522			−0.207261	−	0.978286i	$-0.566455\pi$
$251$	−1648.38			−0.414522			−0.207261	+	0.978286i	$0.566455\pi$
$252$		0			0
$253$	1492.07			0.370772
$254$		0			0
$255$	−582.373	−	148.262i	−0.143018	−	0.0364099i
$256$		0			0
$257$	3204.54			0.777797			0.388899	−	0.921281i	$-0.372856\pi$
$257$	3204.54			0.777797			0.388899	+	0.921281i	$0.372856\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	2950.42	−	5419.06i	0.699718	−	1.28518i
$262$		0			0
$263$		−	3459.42i		−	0.811092i	−0.914075	−	0.405546i	$-0.867081\pi$
$263$		−	3459.42i		−	0.811092i	0.914075	−	0.405546i	$-0.132919\pi$
$264$		0			0
$265$			342.552i			0.0794068i
$266$		0			0
$267$	7166.35	+	1824.43i	1.64260	+	0.418177i
$268$		0			0
$269$	1330.60			0.301591			0.150795	−	0.988565i	$-0.451817\pi$
$269$	1330.60			0.301591			0.150795	+	0.988565i	$0.451817\pi$
$270$		0			0
$271$			3342.15i			0.749157i	0.927195	+	0.374578i	$0.122212\pi$
$271$			3342.15i			0.749157i	−0.927195	+	0.374578i	$0.877788\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	10506.3i		−	2.30382i
$276$		0			0
$277$	−4801.14			−1.04142			−0.520709	−	0.853734i	$-0.674332\pi$
$277$	−4801.14			−1.04142			−0.520709	+	0.853734i	$0.674332\pi$
$278$		0			0
$279$	930.241	−	1708.58i	0.199613	−	0.366631i
$280$		0			0
$281$			4548.87i			0.965704i	0.875702	+	0.482852i	$0.160399\pi$
$281$			4548.87i			0.965704i	−0.875702	+	0.482852i	$0.839601\pi$
$282$		0			0
$283$			4529.79i			0.951477i	0.879587	+	0.475739i	$0.157819\pi$
$283$			4529.79i			0.951477i	−0.879587	+	0.475739i	$0.842181\pi$
$284$		0			0
$285$	1693.34	−	6651.42i	0.351946	−	1.38244i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4877.38			−0.992750
$290$		0			0
$291$	−889.895	+	3495.50i	−0.179266	+	0.704158i
$292$		0			0
$293$	9525.19			1.89921			0.949603	−	0.313454i	$-0.101486\pi$
$293$	9525.19			1.89921			0.949603	+	0.313454i	$0.101486\pi$
$294$		0			0
$295$	−9579.82			−1.89071
$296$		0			0
$297$	−4001.67	+	4313.96i	−0.781819	+	0.842833i
$298$		0			0
$299$	2745.77			0.531076
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−77.2485	−	19.6661i	−0.0146462	−	0.00372868i
$304$		0			0
$305$		−	11152.4i		−	2.09372i
$306$		0			0
$307$			3743.83i			0.696000i	0.937495	+	0.348000i	$0.113139\pi$
$307$			3743.83i			0.696000i	−0.937495	+	0.348000i	$0.886861\pi$
$308$		0			0
$309$	−452.720	+	1778.28i	−0.0833475	+	0.327389i
$310$		0			0
$311$	10403.0			1.89679			0.948393	−	0.317098i	$-0.102708\pi$
$311$	10403.0			1.89679			0.948393	+	0.317098i	$0.102708\pi$
$312$		0			0
$313$			3291.03i			0.594313i	0.954829	+	0.297157i	$0.0960384\pi$
$313$			3291.03i			0.594313i	−0.954829	+	0.297157i	$0.903962\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			5978.80i			1.05931i	0.848212	+	0.529657i	$0.177679\pi$
$317$			5978.80i			1.05931i	−0.848212	+	0.529657i	$0.822321\pi$
$318$		0			0
$319$	−9584.62			−1.68224
$320$		0			0
$321$	−66.0454	+	259.426i	−0.0114838	+	0.0451083i
$322$		0			0
$323$			406.832i			0.0700828i
$324$		0			0
$325$		−	19334.1i		−	3.29988i
$326$		0			0
$327$	3376.97	+	859.718i	0.571091	+	0.145390i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	41.8338			0.00694680			0.00347340	−	0.999994i	$-0.498894\pi$
$331$	41.8338			0.00694680			0.00347340	+	0.999994i	$0.498894\pi$
$332$		0			0
$333$	1644.25	+	895.218i	0.270584	+	0.147320i
$334$		0			0
$335$	7834.90			1.27781
$336$		0			0
$337$	−5477.82			−0.885447			−0.442723	−	0.896658i	$-0.645988\pi$
$337$	−5477.82			−0.885447			−0.442723	+	0.896658i	$0.645988\pi$
$338$		0			0
$339$	−592.194	+	2326.13i	−0.0948777	+	0.372679i
$340$		0			0
$341$	−3021.95			−0.479905
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−883.744	+	3471.34i	−0.137911	+	0.541713i
$346$		0			0
$347$			1106.21i			0.171136i	0.996332	+	0.0855681i	$0.0272705\pi$
$347$			1106.21i			0.171136i	−0.996332	+	0.0855681i	$0.972729\pi$
$348$		0			0
$349$			7172.24i			1.10006i	0.835145	+	0.550030i	$0.185384\pi$
$349$			7172.24i			1.10006i	−0.835145	+	0.550030i	$0.814616\pi$
$350$		0			0
$351$	−7364.05	+	7938.74i	−1.11984	+	1.20723i
$352$		0			0
$353$	−7373.73			−1.11180			−0.555898	−	0.831251i	$-0.687625\pi$
$353$	−7373.73			−1.11180			−0.555898	+	0.831251i	$0.687625\pi$
$354$		0			0
$355$			9491.19i			1.41899i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	955.590i		−	0.140485i	−0.997530	−	0.0702425i	$-0.977623\pi$
$359$		−	955.590i		−	0.140485i	0.997530	−	0.0702425i	$-0.0223773\pi$
$360$		0			0
$361$	2212.48			0.322566
$362$		0			0
$363$	2155.51	+	548.756i	0.311667	+	0.0793450i
$364$		0			0
$365$		−	7533.19i		−	1.08029i
$366$		0			0
$367$			5699.26i			0.810624i	0.914178	+	0.405312i	$0.132837\pi$
$367$			5699.26i			0.810624i	−0.914178	+	0.405312i	$0.867163\pi$
$368$		0			0
$369$	−3148.86	−	1714.40i	−0.444235	−	0.241865i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	9914.06			1.37622			0.688111	−	0.725606i	$-0.258440\pi$
$373$	9914.06			1.37622			0.688111	+	0.725606i	$0.258440\pi$
$374$		0			0
$375$	12246.0	+	3117.61i	1.68635	+	0.429314i
$376$		0			0
$377$	−17638.1			−2.40956
$378$		0			0
$379$	8051.92			1.09129			0.545646	−	0.838016i	$-0.316284\pi$
$379$	8051.92			1.09129			0.545646	+	0.838016i	$0.316284\pi$
$380$		0			0
$381$	1957.31	+	498.297i	0.263191	+	0.0670040i
$382$		0			0
$383$	4368.43			0.582810			0.291405	−	0.956600i	$-0.405877\pi$
$383$	4368.43			0.582810			0.291405	+	0.956600i	$0.405877\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−8686.22	−	4729.23i	−1.14094	−	0.621190i
$388$		0			0
$389$		−	2451.96i		−	0.319586i	−0.987151	−	0.159793i	$-0.948917\pi$
$389$		−	2451.96i		−	0.319586i	0.987151	−	0.159793i	$-0.0510828\pi$
$390$		0			0
$391$		−	212.324i		−	0.0274621i
$392$		0			0
$393$	3160.68	+	804.654i	0.405687	+	0.103281i
$394$		0			0
$395$	−4910.56			−0.625511
$396$		0			0
$397$			8408.87i			1.06305i	0.847044	+	0.531523i	$0.178380\pi$
$397$			8408.87i			1.06305i	−0.847044	+	0.531523i	$0.821620\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8037.63i		−	1.00095i	−0.865751	−	0.500474i	$-0.833159\pi$
$401$		−	8037.63i		−	1.00095i	0.865751	−	0.500474i	$-0.166841\pi$
$402$		0			0
$403$	−5561.12			−0.687392
$404$		0			0
$405$	−7666.41	−	11865.2i	−0.940610	−	1.45577i
$406$		0			0
$407$		−	2908.17i		−	0.354183i
$408$		0			0
$409$		−	4916.56i		−	0.594396i	−0.954816	−	0.297198i	$-0.903948\pi$
$409$		−	4916.56i		−	0.594396i	0.954816	−	0.297198i	$-0.0960522\pi$
$410$		0			0
$411$	2697.51	−	10595.8i	0.323743	−	1.27166i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−6915.58			−0.818006
$416$		0			0
$417$	1320.46	−	5186.75i	0.155067	−	0.609104i
$418$		0			0
$419$	4713.84			0.549608			0.274804	−	0.961500i	$-0.411387\pi$
$419$	4713.84			0.549608			0.274804	+	0.961500i	$0.411387\pi$
$420$		0			0
$421$	9921.92			1.14861			0.574305	−	0.818641i	$-0.305272\pi$
$421$	9921.92			1.14861			0.574305	+	0.818641i	$0.305272\pi$
$422$		0			0
$423$	−4313.62	−	2348.56i	−0.495829	−	0.269955i
$424$		0			0
$425$	−1495.06			−0.170638
$426$		0			0
$427$		0			0
$428$		0			0
$429$	16300.5	+	4149.83i	1.83449	+	0.467030i
$430$		0			0
$431$			12587.4i			1.40676i	0.710814	+	0.703380i	$0.248327\pi$
$431$			12587.4i			1.40676i	−0.710814	+	0.703380i	$0.751673\pi$
$432$		0			0
$433$		−	3715.19i		−	0.412334i	−0.978517	−	0.206167i	$-0.933901\pi$
$433$		−	3715.19i		−	0.412334i	0.978517	−	0.206167i	$-0.0660989\pi$
$434$		0			0
$435$	5676.93	−	22299.0i	0.625719	−	2.45782i
$436$		0			0
$437$	2425.00			0.265454
$438$		0			0
$439$		−	5865.27i		−	0.637663i	−0.947811	−	0.318831i	$-0.896710\pi$
$439$		−	5865.27i		−	0.637663i	0.947811	−	0.318831i	$-0.103290\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			15440.8i			1.65601i	0.560720	+	0.828005i	$0.310524\pi$
$443$			15440.8i			1.65601i	−0.560720	+	0.828005i	$0.689476\pi$
$444$		0			0
$445$	27577.7			2.93777
$446$		0			0
$447$	257.886	−	1012.98i	0.0272877	−	0.107186i
$448$		0			0
$449$			16373.4i			1.72095i	0.509491	+	0.860476i	$0.329834\pi$
$449$			16373.4i			1.72095i	−0.509491	+	0.860476i	$0.670166\pi$
$450$		0			0
$451$			5569.34i			0.581486i
$452$		0			0
$453$	−6004.75	−	1528.70i	−0.622798	−	0.158554i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2754.42			0.281940			0.140970	−	0.990014i	$-0.454978\pi$
$457$	2754.42			0.281940			0.140970	+	0.990014i	$0.454978\pi$
$458$		0			0
$459$	613.885	+	569.445i	0.0624263	+	0.0579072i
$460$		0			0
$461$	−15142.3			−1.52982			−0.764909	−	0.644139i	$-0.777216\pi$
$461$	−15142.3			−1.52982			−0.764909	+	0.644139i	$0.777216\pi$
$462$		0			0
$463$	−10138.9			−1.01770			−0.508851	−	0.860855i	$-0.669930\pi$
$463$	−10138.9			−1.01770			−0.508851	+	0.860855i	$0.669930\pi$
$464$		0			0
$465$	1789.89	−	7030.67i	0.178503	−	0.701160i
$466$		0			0
$467$	−16149.3			−1.60021			−0.800106	−	0.599858i	$-0.795224\pi$
$467$	−16149.3			−1.60021			−0.800106	+	0.599858i	$0.795224\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−1400.18	+	5499.92i	−0.136979	+	0.538053i
$472$		0			0
$473$			15363.2i			1.49345i
$474$		0			0
$475$		−	17075.4i		−	1.64942i
$476$		0			0
$477$	228.229	−	419.190i	0.0219075	−	0.0402377i
$478$		0			0
$479$	−8419.81			−0.803154			−0.401577	−	0.915825i	$-0.631538\pi$
$479$	−8419.81			−0.803154			−0.401577	+	0.915825i	$0.631538\pi$
$480$		0			0
$481$		−	5351.74i		−	0.507315i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			13451.5i			1.25938i
$486$		0			0
$487$	−14078.2			−1.30995			−0.654975	−	0.755651i	$-0.727321\pi$
$487$	−14078.2			−1.30995			−0.654975	+	0.755651i	$0.727321\pi$
$488$		0			0
$489$	2918.61	+	743.029i	0.269907	+	0.0687135i
$490$		0			0
$491$		−	17853.2i		−	1.64094i	−0.571688	−	0.820471i	$-0.693711\pi$
$491$		−	17853.2i		−	1.64094i	0.571688	−	0.820471i	$-0.306289\pi$
$492$		0			0
$493$			1363.91i			0.124599i
$494$		0			0
$495$	−10492.9	+	19272.4i	−0.952768	+	1.74996i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	13605.8			1.22060			0.610298	−	0.792172i	$-0.291050\pi$
$499$	13605.8			1.22060			0.610298	+	0.792172i	$0.291050\pi$
$500$		0			0
$501$	−18710.6	−	4763.38i	−1.66852	−	0.424775i
$502$		0			0
$503$	7503.79			0.665164			0.332582	−	0.943074i	$-0.392080\pi$
$503$	7503.79			0.665164			0.332582	+	0.943074i	$0.392080\pi$
$504$		0			0
$505$	−297.269			−0.0261946
$506$		0			0
$507$	18933.9	+	4820.25i	1.65855	+	0.422238i
$508$		0			0
$509$	−10479.2			−0.912541			−0.456270	−	0.889841i	$-0.650815\pi$
$509$	−10479.2			−0.912541			−0.456270	+	0.889841i	$0.650815\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−6503.76	+	7011.32i	−0.559743	+	0.603425i
$514$		0			0
$515$			6843.22i			0.585531i
$516$		0			0
$517$			7629.45i			0.649019i
$518$		0			0
$519$	6739.60	+	1715.79i	0.570011	+	0.145115i
$520$		0			0
$521$	−7295.75			−0.613498			−0.306749	−	0.951790i	$-0.599241\pi$
$521$	−7295.75			−0.613498			−0.306749	+	0.951790i	$0.599241\pi$
$522$		0			0
$523$		−	15195.0i		−	1.27043i	−0.772337	−	0.635213i	$-0.780912\pi$
$523$		−	15195.0i		−	1.27043i	0.772337	−	0.635213i	$-0.219088\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			430.029i			0.0355452i
$528$		0			0
$529$	10901.4			0.895981
$530$		0			0
$531$	11723.1	+	6382.66i	0.958076	+	0.521627i
$532$		0			0
$533$			10249.0i			0.832892i
$534$		0			0
$535$			998.328i			0.0806757i
$536$		0			0
$537$	−3613.47	+	14193.7i	−0.290377	+	1.14060i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−18373.2			−1.46012			−0.730061	−	0.683382i	$-0.760508\pi$
$541$	−18373.2			−1.46012			−0.730061	+	0.683382i	$0.760508\pi$
$542$		0			0
$543$	4728.23	−	18572.5i	0.373679	−	1.46781i
$544$		0			0
$545$	12995.3			1.02139
$546$		0			0
$547$	−20764.3			−1.62307			−0.811534	−	0.584305i	$-0.801367\pi$
$547$	−20764.3			−1.62307			−0.811534	+	0.584305i	$0.801367\pi$
$548$		0			0
$549$	−7430.43	+	13647.5i	−0.577637	+	1.06095i
$550$		0			0
$551$	−15577.5			−1.20440
$552$		0			0
$553$		0			0
$554$		0			0
$555$	6765.96	+	1722.50i	0.517476	+	0.131740i
$556$		0			0
$557$			5210.25i			0.396347i	0.980167	+	0.198174i	$0.0635010\pi$
$557$			5210.25i			0.396347i	−0.980167	+	0.198174i	$0.936499\pi$
$558$		0			0
$559$			28272.1i			2.13914i
$560$		0			0
$561$	320.897	−	1260.48i	0.0241502	−	0.0948621i
$562$		0			0
$563$	−10188.7			−0.762703			−0.381351	−	0.924430i	$-0.624541\pi$
$563$	−10188.7			−0.762703			−0.381351	+	0.924430i	$0.624541\pi$
$564$		0			0
$565$			8951.47i			0.666533i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	1867.77i		−	0.137612i	−0.997630	−	0.0688059i	$-0.978081\pi$
$569$		−	1867.77i		−	0.137612i	0.997630	−	0.0688059i	$-0.0219189\pi$
$570$		0			0
$571$	−908.844			−0.0666094			−0.0333047	−	0.999445i	$-0.510603\pi$
$571$	−908.844			−0.0666094			−0.0333047	+	0.999445i	$0.510603\pi$
$572$		0			0
$573$	−5780.54	+	22705.9i	−0.421441	+	1.65542i
$574$		0			0
$575$			8911.59i			0.646328i
$576$		0			0
$577$			15044.2i			1.08544i	0.839914	+	0.542720i	$0.182605\pi$
$577$			15044.2i			1.08544i	−0.839914	+	0.542720i	$0.817395\pi$
$578$		0			0
$579$	11136.1	+	2835.05i	0.799307	+	0.203490i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−741.416			−0.0526695
$584$		0			0
$585$	−19309.5	+	35465.9i	−1.36470	+	2.50655i
$586$		0			0
$587$	−22979.9			−1.61581			−0.807906	−	0.589312i	$-0.799399\pi$
$587$	−22979.9			−1.61581			−0.807906	+	0.589312i	$0.799399\pi$
$588$		0			0
$589$	−4911.46			−0.343587
$590$		0			0
$591$	5355.01	−	21034.5i	0.372717	−	1.46403i
$592$		0			0
$593$	−4220.66			−0.292279			−0.146140	−	0.989264i	$-0.546685\pi$
$593$	−4220.66			−0.292279			−0.146140	+	0.989264i	$0.546685\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−5660.07	+	22232.8i	−0.388026	+	1.52416i
$598$		0			0
$599$			22561.2i			1.53894i	0.638682	+	0.769471i	$0.279480\pi$
$599$			22561.2i			1.53894i	−0.638682	+	0.769471i	$0.720520\pi$
$600$		0			0
$601$		−	2004.06i		−	0.136019i	−0.997685	−	0.0680095i	$-0.978335\pi$
$601$		−	2004.06i		−	0.136019i	0.997685	−	0.0680095i	$-0.0216648\pi$
$602$		0			0
$603$	−9587.78	−	5220.09i	−0.647504	−	0.352535i
$604$		0			0
$605$	8294.88			0.557413
$606$		0			0
$607$			9819.23i			0.656590i	0.944575	+	0.328295i	$0.106474\pi$
$607$			9819.23i			0.656590i	−0.944575	+	0.328295i	$0.893526\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			14040.1i			0.929623i
$612$		0			0
$613$	−18993.4			−1.25145			−0.625724	−	0.780044i	$-0.715196\pi$
$613$	−18993.4			−1.25145			−0.625724	+	0.780044i	$0.715196\pi$
$614$		0			0
$615$	−12957.3	−	3298.70i	−0.849573	−	0.216287i
$616$		0			0
$617$		−	13306.4i		−	0.868223i	−0.900859	−	0.434112i	$-0.857062\pi$
$617$		−	13306.4i		−	0.868223i	0.900859	−	0.434112i	$-0.142938\pi$
$618$		0			0
$619$			1860.24i			0.120791i	0.998175	+	0.0603953i	$0.0192361\pi$
$619$			1860.24i			0.120791i	−0.998175	+	0.0603953i	$0.980764\pi$
$620$		0			0
$621$	3394.28	−	3659.17i	0.219336	−	0.236453i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	15812.7			1.01201
$626$		0			0
$627$	14396.3	+	3665.04i	0.916956	+	0.233441i
$628$		0			0
$629$	−413.838			−0.0262334
$630$		0			0
$631$	16908.2			1.06673			0.533364	−	0.845886i	$-0.320928\pi$
$631$	16908.2			1.06673			0.533364	+	0.845886i	$0.320928\pi$
$632$		0			0
$633$	10082.1	+	2566.72i	0.633059	+	0.161166i
$634$		0			0
$635$	7532.15			0.470715
$636$		0			0
$637$		0			0
$638$		0			0
$639$	6323.61	−	11614.6i	0.391484	−	0.719042i
$640$		0			0
$641$		−	5036.02i		−	0.310313i	−0.987890	−	0.155157i	$-0.950412\pi$
$641$		−	5036.02i		−	0.310313i	0.987890	−	0.155157i	$-0.0495882\pi$
$642$		0			0
$643$		−	7842.19i		−	0.480973i	−0.970652	−	0.240487i	$-0.922693\pi$
$643$		−	7842.19i		−	0.480973i	0.970652	−	0.240487i	$-0.0773070\pi$
$644$		0			0
$645$	−35743.0	−	9099.56i	−2.18199	−	0.555496i
$646$		0			0
$647$	10476.6			0.636599			0.318299	−	0.947990i	$-0.396888\pi$
$647$	10476.6			0.636599			0.318299	+	0.947990i	$0.396888\pi$
$648$		0			0
$649$		−	20734.5i		−	1.25408i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	12615.9i		−	0.756049i	−0.925796	−	0.378025i	$-0.876603\pi$
$653$		−	12615.9i		−	0.756049i	0.925796	−	0.378025i	$-0.123397\pi$
$654$		0			0
$655$	12163.0			0.725567
$656$		0			0
$657$	−5019.08	+	9218.57i	−0.298041	+	0.547414i
$658$		0			0
$659$		−	1347.17i		−	0.0796334i	−0.999207	−	0.0398167i	$-0.987323\pi$
$659$		−	1347.17i		−	0.0796334i	0.999207	−	0.0398167i	$-0.0126774\pi$
$660$		0			0
$661$			3775.45i			0.222161i	0.993811	+	0.111080i	$0.0354311\pi$
$661$			3775.45i			0.222161i	−0.993811	+	0.111080i	$0.964569\pi$
$662$		0			0
$663$	590.529	−	2319.60i	0.0345916	−	0.135876i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8129.84			0.471947
$668$		0			0
$669$	2720.37	−	10685.6i	0.157213	−	0.617532i
$670$		0			0
$671$	24138.2			1.38874
$672$		0			0
$673$	27877.7			1.59674			0.798370	−	0.602167i	$-0.205696\pi$
$673$	27877.7			1.59674			0.798370	+	0.602167i	$0.205696\pi$
$674$		0			0
$675$	−25765.8	−	23900.6i	−1.46922	−	1.36286i
$676$		0			0
$677$	20345.0			1.15498			0.577492	−	0.816397i	$-0.304032\pi$
$677$	20345.0			1.15498			0.577492	+	0.816397i	$0.304032\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−24486.9	−	6233.94i	−1.37789	−	0.350786i
$682$		0			0
$683$		−	12596.6i		−	0.705703i	−0.935679	−	0.352852i	$-0.885212\pi$
$683$		−	12596.6i		−	0.705703i	0.935679	−	0.352852i	$-0.114788\pi$
$684$		0			0
$685$		−	40774.9i		−	2.27435i
$686$		0			0
$687$	−7258.49	+	28511.3i	−0.403098	+	1.58337i
$688$		0			0
$689$	−1364.39			−0.0754412
$690$		0			0
$691$		−	9810.01i		−	0.540073i	−0.962850	−	0.270037i	$-0.912964\pi$
$691$		−	9810.01i		−	0.540073i	0.962850	−	0.270037i	$-0.0870358\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	19959.7i		−	1.08938i
$696$		0			0
$697$	792.528			0.0430691
$698$		0			0
$699$	4171.15	−	16384.3i	0.225705	−	0.886567i
$700$		0			0
$701$		−	21928.4i		−	1.18149i	−0.806858	−	0.590746i	$-0.798834\pi$
$701$		−	21928.4i		−	1.18149i	0.806858	−	0.590746i	$-0.201166\pi$
$702$		0			0
$703$		−	4726.54i		−	0.253577i
$704$		0			0
$705$	−17750.2	−	4518.89i	−0.948242	−	0.241406i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−20021.8			−1.06056			−0.530279	−	0.847823i	$-0.677913\pi$
$709$	−20021.8			−1.06056			−0.530279	+	0.847823i	$0.677913\pi$
$710$		0			0
$711$	6009.18	+	3271.72i	0.316965	+	0.172572i
$712$		0			0
$713$	2563.27			0.134635
$714$		0			0
$715$	62728.0			3.28097
$716$		0			0
$717$	−7465.73	+	29325.4i	−0.388860	+	1.52744i
$718$		0			0
$719$	−12521.2			−0.649462			−0.324731	−	0.945806i	$-0.605274\pi$
$719$	−12521.2			−0.649462			−0.324731	+	0.945806i	$0.605274\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	7918.10	−	31102.3i	0.407299	−	1.59987i
$724$		0			0
$725$		−	57245.6i		−	2.93248i
$726$		0			0
$727$		−	27158.2i		−	1.38547i	−0.721190	−	0.692737i	$-0.756405\pi$
$727$		−	27158.2i		−	1.38547i	0.721190	−	0.692737i	$-0.243595\pi$
$728$		0			0
$729$	1476.29	+	19627.6i	0.0750034	+	0.997183i
$730$		0			0
$731$	2186.21			0.110616
$732$		0			0
$733$			5443.32i			0.274289i	0.990551	+	0.137144i	$0.0437924\pi$
$733$			5443.32i			0.274289i	−0.990551	+	0.137144i	$0.956208\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			16957.8i			0.847555i
$738$		0			0
$739$	26773.2			1.33270			0.666351	−	0.745638i	$-0.267855\pi$
$739$	26773.2			1.33270			0.666351	+	0.745638i	$0.267855\pi$
$740$		0			0
$741$	26492.7	+	6744.57i	1.31340	+	0.334370i
$742$		0			0
$743$		−	6409.29i		−	0.316466i	−0.987402	−	0.158233i	$-0.949420\pi$
$743$		−	6409.29i		−	0.316466i	0.987402	−	0.158233i	$-0.0505797\pi$
$744$		0			0
$745$		−	3898.15i		−	0.191701i
$746$		0			0
$747$	8462.78	+	4607.58i	0.414507	+	0.225680i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	18015.4			0.875353			0.437676	−	0.899133i	$-0.355802\pi$
$751$	18015.4			0.875353			0.437676	+	0.899133i	$0.355802\pi$
$752$		0			0
$753$	8300.49	+	2113.16i	0.401709	+	0.102268i
$754$		0			0
$755$	−23107.6			−1.11387
$756$		0			0
$757$	4064.93			0.195168			0.0975840	−	0.995227i	$-0.468889\pi$
$757$	4064.93			0.195168			0.0975840	+	0.995227i	$0.468889\pi$
$758$		0			0
$759$	−7513.34	−	1912.77i	−0.359311	−	0.0914743i
$760$		0			0
$761$	21439.1			1.02124			0.510621	−	0.859806i	$-0.329416\pi$
$761$	21439.1			1.02124			0.510621	+	0.859806i	$0.329416\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	2742.49	+	1493.16i	0.129614	+	0.0705689i
$766$		0			0
$767$		−	38156.5i		−	1.79629i
$768$		0			0
$769$		−	18622.9i		−	0.873286i	−0.899635	−	0.436643i	$-0.856167\pi$
$769$		−	18622.9i		−	0.873286i	0.899635	−	0.436643i	$-0.143833\pi$
$770$		0			0
$771$	−16136.6	−	4108.09i	−0.753754	−	0.191893i
$772$		0			0
$773$	41572.5			1.93436			0.967180	−	0.254094i	$-0.0817771\pi$
$773$	41572.5			1.93436			0.967180	+	0.254094i	$0.0817771\pi$
$774$		0			0
$775$		−	18049.0i		−	0.836568i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			9051.64i			0.416314i
$780$		0			0
$781$	−20542.6			−0.941195
$782$		0			0
$783$	−21804.0	+	23505.5i	−0.995159	+	1.07282i
$784$		0			0
$785$			21164.9i			0.962301i
$786$		0			0
$787$		−	3432.99i		−	0.155493i	−0.996973	−	0.0777463i	$-0.975228\pi$
$787$		−	3432.99i		−	0.155493i	0.996973	−	0.0777463i	$-0.0247724\pi$
$788$		0			0
$789$	−4434.84	+	17420.0i	−0.200107	+	0.786020i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	44420.2			1.98916
$794$		0			0
$795$	439.138	−	1724.93i	0.0195907	−	0.0769522i
$796$		0			0
$797$	9577.83			0.425676			0.212838	−	0.977087i	$-0.431729\pi$
$797$	9577.83			0.425676			0.212838	+	0.977087i	$0.431729\pi$
$798$		0			0
$799$	1085.69			0.0480711
$800$		0			0
$801$	−33747.5	−	18373.9i	−1.48865	−	0.810501i
$802$		0			0
$803$	16304.8			0.716541
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−6700.26	−	1705.77i	−0.292268	−	0.0744063i
$808$		0			0
$809$			14083.6i			0.612054i	0.952023	+	0.306027i	$0.0989998\pi$
$809$			14083.6i			0.612054i	−0.952023	+	0.306027i	$0.901000\pi$
$810$		0			0
$811$		−	8029.38i		−	0.347657i	−0.984776	−	0.173828i	$-0.944386\pi$
$811$		−	8029.38i		−	0.347657i	0.984776	−	0.173828i	$-0.0556138\pi$
$812$		0			0
$813$	4284.50	−	16829.5i	0.184827	−	0.725999i
$814$		0			0
$815$	11231.5			0.482725
$816$		0			0
$817$			24969.2i			1.06923i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	17786.4i		−	0.756088i	−0.925788	−	0.378044i	$-0.876597\pi$
$821$		−	17786.4i		−	0.756088i	0.925788	−	0.378044i	$-0.123403\pi$
$822$		0			0
$823$	−9518.53			−0.403153			−0.201577	−	0.979473i	$-0.564607\pi$
$823$	−9518.53			−0.403153			−0.201577	+	0.979473i	$0.564607\pi$
$824$		0			0
$825$	−13468.6	+	52904.6i	−0.568383	+	2.23261i
$826$		0			0
$827$			37498.5i			1.57672i	0.615212	+	0.788362i	$0.289071\pi$
$827$			37498.5i			1.57672i	−0.615212	+	0.788362i	$0.710929\pi$
$828$		0			0
$829$		−	27505.4i		−	1.15235i	−0.817325	−	0.576176i	$-0.804544\pi$
$829$		−	27505.4i		−	1.15235i	0.817325	−	0.576176i	$-0.195456\pi$
$830$		0			0
$831$	24176.3	+	6154.87i	1.00923	+	0.256931i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−72002.3			−2.98412
$836$		0			0
$837$	−6874.59	+	7411.08i	−0.283896	+	0.306051i
$838$		0			0
$839$	2040.23			0.0839528			0.0419764	−	0.999119i	$-0.486635\pi$
$839$	2040.23			0.0839528			0.0419764	+	0.999119i	$0.486635\pi$
$840$		0			0
$841$	−27834.9			−1.14129
$842$		0			0
$843$	5831.47	−	22906.0i	0.238252	−	0.935853i
$844$		0			0
$845$	72861.8			2.96630
$846$		0			0
$847$		0			0
$848$		0			0
$849$	5807.00	−	22809.9i	0.234742	−	0.922066i
$850$		0			0
$851$			2466.76i			0.0993647i
$852$		0			0
$853$		−	22267.7i		−	0.893825i	−0.894578	−	0.446913i	$-0.852524\pi$
$853$		−	22267.7i		−	0.893825i	0.894578	−	0.446913i	$-0.147476\pi$
$854$		0			0
$855$	−17053.7	+	31322.6i	−0.682133	+	1.25288i
$856$		0			0
$857$	−27422.4			−1.09303			−0.546517	−	0.837448i	$-0.684047\pi$
$857$	−27422.4			−1.09303			−0.546517	+	0.837448i	$0.684047\pi$
$858$		0			0
$859$		−	43825.7i		−	1.74076i	−0.492381	−	0.870380i	$-0.663873\pi$
$859$		−	43825.7i		−	1.74076i	0.492381	−	0.870380i	$-0.336127\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	26852.8i		−	1.05919i	−0.848250	−	0.529595i	$-0.822344\pi$
$863$		−	26852.8i		−	1.05919i	0.848250	−	0.529595i	$-0.177656\pi$
$864$		0			0
$865$	25935.4			1.01946
$866$		0			0
$867$	24560.2	+	6252.60i	0.962062	+	0.244924i
$868$		0			0
$869$		−	10628.4i		−	0.414894i
$870$		0			0
$871$			31206.5i			1.21400i
$872$		0			0
$873$	8962.18	−	16460.9i	0.347450	−	0.638164i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−39115.3			−1.50608			−0.753039	−	0.657976i	$-0.771413\pi$
$877$	−39115.3			−1.50608			−0.753039	+	0.657976i	$0.771413\pi$
$878$		0			0
$879$	−47964.4	−	12210.9i	−1.84050	−	0.468559i
$880$		0			0
$881$	39685.7			1.51764			0.758822	−	0.651299i	$-0.225775\pi$
$881$	39685.7			1.51764			0.758822	+	0.651299i	$0.225775\pi$
$882$		0			0
$883$	−37785.0			−1.44005			−0.720026	−	0.693947i	$-0.755870\pi$
$883$	−37785.0			−1.44005			−0.720026	+	0.693947i	$0.755870\pi$
$884$		0			0
$885$	48239.5	+	12280.9i	1.83226	+	0.466462i
$886$		0			0
$887$	14665.4			0.555148			0.277574	−	0.960704i	$-0.410470\pi$
$887$	14665.4			0.555148			0.277574	+	0.960704i	$0.410470\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	25680.8	−	16593.1i	0.965590	−	0.623894i
$892$		0			0
$893$			12399.9i			0.464665i
$894$		0			0
$895$			54620.4i			2.03995i
$896$		0			0
$897$	−13826.4	−	3519.96i	−0.514660	−	0.131023i
$898$		0			0
$899$	−16465.7			−0.610859
$900$		0			0
$901$			105.505i			0.00390109i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	71470.9i		−	2.62516i
$906$		0			0
$907$	−38199.7			−1.39846			−0.699229	−	0.714898i	$-0.746473\pi$
$907$	−38199.7			−1.39846			−0.699229	+	0.714898i	$0.746473\pi$
$908$		0			0
$909$	363.776	+	198.059i	0.0132736	+	0.00722684i
$910$		0			0
$911$		−	41209.5i		−	1.49872i	−0.662165	−	0.749358i	$-0.730362\pi$
$911$		−	41209.5i		−	1.49872i	0.662165	−	0.749358i	$-0.269638\pi$
$912$		0			0
$913$		−	14968.0i		−	0.542573i
$914$		0			0
$915$	−14297.0	+	56158.4i	−0.516549	+	2.02900i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	35851.3			1.28686			0.643431	−	0.765504i	$-0.277511\pi$
$919$	35851.3			1.28686			0.643431	+	0.765504i	$0.277511\pi$
$920$		0			0
$921$	4799.44	−	18852.2i	0.171712	−	0.674485i
$922$		0			0
$923$	−37803.5			−1.34812
$924$		0			0
$925$	17369.5			0.617411
$926$		0			0
$927$	4559.38	−	8374.24i	0.161542	−	0.296706i
$928$		0			0
$929$	−39315.6			−1.38849			−0.694243	−	0.719741i	$-0.744260\pi$
$929$	−39315.6			−1.38849			−0.694243	+	0.719741i	$0.744260\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	−52384.7	−	13336.2i	−1.83815	−	0.467962i
$934$		0			0
$935$		−	4850.61i		−	0.169660i
$936$		0			0
$937$		−	25396.6i		−	0.885455i	−0.896656	−	0.442728i	$-0.854011\pi$
$937$		−	25396.6i		−	0.885455i	0.896656	−	0.442728i	$-0.145989\pi$
$938$		0			0
$939$	4218.97	−	16572.1i	0.146625	−	0.575942i
$940$		0			0
$941$	18176.1			0.629676			0.314838	−	0.949145i	$-0.398050\pi$
$941$	18176.1			0.629676			0.314838	+	0.949145i	$0.398050\pi$
$942$		0			0
$943$		−	4724.01i		−	0.163134i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			42110.2i			1.44498i	0.691380	+	0.722491i	$0.257003\pi$
$947$			42110.2i			1.44498i	−0.691380	+	0.722491i	$0.742997\pi$
$948$		0			0
$949$	30004.8			1.02634
$950$		0			0
$951$	7664.57	−	30106.4i	0.261347	−	1.02657i
$952$		0			0
$953$			18521.2i			0.629551i	0.949166	+	0.314775i	$0.101929\pi$
$953$			18521.2i			0.629551i	−0.949166	+	0.314775i	$0.898071\pi$
$954$		0			0
$955$			87377.4i			2.96070i
$956$		0			0
$957$	48263.7	+	12287.1i	1.63024	+	0.415032i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	24599.5			0.825736
$962$		0			0
$963$	665.147	−	1221.68i	0.0222576	−	0.0408807i
$964$		0			0
$965$	42853.9			1.42955
$966$		0			0
$967$	48504.3			1.61302			0.806512	−	0.591218i	$-0.201353\pi$
$967$	48504.3			1.61302			0.806512	+	0.591218i	$0.201353\pi$
$968$		0			0
$969$	521.542	−	2048.62i	0.0172903	−	0.0679164i
$970$		0			0
$971$	−33158.4			−1.09589			−0.547943	−	0.836516i	$-0.684589\pi$
$971$	−33158.4			−1.09589			−0.547943	+	0.836516i	$0.684589\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−24785.5	+	97357.4i	−0.814125	+	3.19788i
$976$		0			0
$977$			34135.5i			1.11780i	0.829234	+	0.558901i	$0.188777\pi$
$977$			34135.5i			1.11780i	−0.829234	+	0.558901i	$0.811223\pi$
$978$		0			0
$979$			59688.8i			1.94858i
$980$		0			0
$981$	−15902.7	−	8658.27i	−0.517568	−	0.281791i
$982$		0			0
$983$	−339.463			−0.0110144			−0.00550721	−	0.999985i	$-0.501753\pi$
$983$	−339.463			−0.0110144			−0.00550721	+	0.999985i	$0.501753\pi$
$984$		0			0
$985$		−	80945.2i		−	2.61841i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	13031.3i		−	0.418981i
$990$		0			0
$991$	−559.940			−0.0179486			−0.00897431	−	0.999960i	$-0.502857\pi$
$991$	−559.940			−0.0179486			−0.00897431	+	0.999960i	$0.502857\pi$
$992$		0			0
$993$	−210.655	−	53.6292i	−0.00673207	−	0.00171387i
$994$		0			0
$995$			85556.5i			2.72595i
$996$		0			0
$997$			12840.4i			0.407884i	0.978983	+	0.203942i	$0.0653754\pi$
$997$			12840.4i			0.407884i	−0.978983	+	0.203942i	$0.934625\pi$
$998$		0			0
$999$	−7132.06	−	6615.76i	−0.225874	−	0.209523i

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	588.4.f.c.293.1		12
3.2	odd	2	inner	588.4.f.c.293.11		12
7.2	even	3		84.4.k.c.17.5	yes	12
7.3	odd	6		84.4.k.c.5.2	✓	12
7.4	even	3		588.4.k.c.509.5		12
7.5	odd	6		588.4.k.c.521.2		12
7.6	odd	2	inner	588.4.f.c.293.12		12
21.2	odd	6		84.4.k.c.17.2	yes	12
21.5	even	6		588.4.k.c.521.5		12
21.11	odd	6		588.4.k.c.509.2		12
21.17	even	6		84.4.k.c.5.5	yes	12
21.20	even	2	inner	588.4.f.c.293.2		12
28.3	even	6		336.4.bc.c.257.5		12
28.23	odd	6		336.4.bc.c.17.2		12
84.23	even	6		336.4.bc.c.17.5		12
84.59	odd	6		336.4.bc.c.257.2		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.4.k.c.5.2	✓	12	7.3	odd	6
84.4.k.c.5.5	yes	12	21.17	even	6
84.4.k.c.17.2	yes	12	21.2	odd	6
84.4.k.c.17.5	yes	12	7.2	even	3
336.4.bc.c.17.2		12	28.23	odd	6
336.4.bc.c.17.5		12	84.23	even	6
336.4.bc.c.257.2		12	84.59	odd	6
336.4.bc.c.257.5		12	28.3	even	6
588.4.f.c.293.1		12	1.1	even	1	trivial
588.4.f.c.293.2		12	21.20	even	2	inner
588.4.f.c.293.11		12	3.2	odd	2	inner
588.4.f.c.293.12		12	7.6	odd	2	inner
588.4.k.c.509.2		12	21.11	odd	6
588.4.k.c.509.5		12	7.4	even	3
588.4.k.c.521.2		12	7.5	odd	6
588.4.k.c.521.5		12	21.5	even	6

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	588.4.f.c.293.1

Level	$588$
Weight	$4$
Character	588.293
Analytic conductor	$34.693$
Analytic rank	$0$
Dimension	$12$
Inner twists	$4$