LMFDB - Embedded newform 588.4.f.c.293.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,4,Mod(293,588)]

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

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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			293.11
Root			$2.88784 + 0.812653i$ of defining polynomial
Character	$\chi$	$=$	588.293
Dual form			588.4.f.c.293.12

$q$-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	$-0.166299\pi$
$5$	19.3778			1.73320			0.866602	+	0.498999i	$0.166299\pi$
$6$		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	$0.194923\pi$
$11$			41.9412i			1.14961i	−0.818290	+	0.574806i	$0.805077\pi$
$12$		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	$-0.486444\pi$
$17$	5.96831			0.0851487			0.0425743	+	0.999093i	$0.486444\pi$
$18$		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	$-0.948444\pi$
$23$		−	35.5752i		−	0.322519i	0.986912	−	0.161260i	$-0.0515557\pi$
$24$		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	$0.261253\pi$
$29$			228.525i			1.46331i	−0.681673	+	0.731657i	$0.738747\pi$
$30$		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	$-0.418614\pi$
$41$	132.789			0.505810			0.252905	+	0.967491i	$0.418614\pi$
$42$		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	$-0.408910\pi$
$47$	181.908			0.564555			0.282277	+	0.959333i	$0.408910\pi$
$48$		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	$0.00729232\pi$
$53$			17.6775i			0.0458150i	−0.999738	+	0.0229075i	$0.992708\pi$
$54$		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	$-0.683636\pi$
$59$	−494.370			−1.09087			−0.545437	+	0.838152i	$0.683636\pi$
$60$		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	$0.134246\pi$
$71$			489.797i			0.818707i	−0.912376	+	0.409353i	$0.865754\pi$
$72$		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	$-0.575830\pi$
$83$	−356.881			−0.471961			−0.235981	+	0.971758i	$0.575830\pi$
$84$		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	$-0.178109\pi$
$89$	1423.16			1.69499			0.847496	+	0.530802i	$0.178109\pi$
$90$		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	$-0.502405\pi$
$101$	−15.3407			−0.0151134			−0.00755671	+	0.999971i	$0.502405\pi$
$102$		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	$0.00740887\pi$
$107$			51.5191i			0.0465471i	−0.999729	+	0.0232736i	$0.992591\pi$
$108$		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	$0.0615893\pi$
$113$			461.944i			0.384567i	−0.981339	+	0.192283i	$0.938411\pi$
$114$		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	$-0.432877\pi$
$131$	627.675			0.418628			0.209314	+	0.977849i	$0.432877\pi$
$132$		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	$-0.772200\pi$
$137$		−	2104.21i		−	1.31222i	0.754664	−	0.656111i	$-0.227800\pi$
$138$		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	$-0.982388\pi$
$149$		−	201.165i		−	0.110605i	0.998470	−	0.0553024i	$-0.0176123\pi$
$150$		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	$-0.830079\pi$
$167$	−3715.71			−1.72174			−0.860869	+	0.508827i	$0.830079\pi$
$168$		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	$-0.404981\pi$
$173$	1338.41			0.588193			0.294096	+	0.955776i	$0.404981\pi$
$174$		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	$0.200278\pi$
$179$			2818.71i			1.17698i	−0.808503	+	0.588492i	$0.799722\pi$
$180$		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	$0.325898\pi$
$191$			4509.14i			1.70822i	−0.520092	+	0.854111i	$0.674102\pi$
$192$		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	$-0.727460\pi$
$197$		−	4177.21i		−	1.51073i	0.655304	−	0.755365i	$-0.272540\pi$
$198$		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	$-0.751720\pi$
$227$	−4862.82			−1.42184			−0.710918	+	0.703274i	$0.751720\pi$
$228$		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	$-0.848772\pi$
$233$		−	3253.73i		−	0.914846i	0.889249	−	0.457423i	$-0.151228\pi$
$234$		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	$0.288926\pi$
$239$			5823.69i			1.57616i	−0.615571	+	0.788082i	$0.711074\pi$
$240$		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	$-0.433545\pi$
$251$	1648.38			0.414522			0.207261	+	0.978286i	$0.433545\pi$
$252$		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	$-0.627144\pi$
$257$	−3204.54			−0.777797			−0.388899	+	0.921281i	$0.627144\pi$
$258$		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	$0.132919\pi$
$263$			3459.42i			0.811092i	−0.914075	+	0.405546i	$0.867081\pi$
$264$		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	$-0.548183\pi$
$269$	−1330.60			−0.301591			−0.150795	+	0.988565i	$0.548183\pi$
$270$		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	$-0.839601\pi$
$281$		−	4548.87i		−	0.965704i	0.875702	−	0.482852i	$-0.160399\pi$
$282$		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	$-0.898514\pi$
$293$	−9525.19			−1.89921			−0.949603	+	0.313454i	$0.898514\pi$
$294$		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	$-0.897292\pi$
$311$	−10403.0			−1.89679			−0.948393	+	0.317098i	$0.897292\pi$
$312$		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	$-0.822321\pi$
$317$		−	5978.80i		−	1.05931i	0.848212	−	0.529657i	$-0.177679\pi$
$318$		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	$-0.972729\pi$
$347$		−	1106.21i		−	0.171136i	0.996332	−	0.0855681i	$-0.0272705\pi$
$348$		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	$-0.312375\pi$
$353$	7373.73			1.11180			0.555898	+	0.831251i	$0.312375\pi$
$354$		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	$0.0223773\pi$
$359$			955.590i			0.140485i	−0.997530	+	0.0702425i	$0.977623\pi$
$360$		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	$-0.594123\pi$
$383$	−4368.43			−0.582810			−0.291405	+	0.956600i	$0.594123\pi$
$384$		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	$0.0510828\pi$
$389$			2451.96i			0.319586i	−0.987151	+	0.159793i	$0.948917\pi$
$390$		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	$0.166841\pi$
$401$			8037.63i			1.00095i	−0.865751	+	0.500474i	$0.833159\pi$
$402$		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	$-0.588613\pi$
$419$	−4713.84			−0.549608			−0.274804	+	0.961500i	$0.588613\pi$
$420$		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	$-0.751673\pi$
$431$		−	12587.4i		−	1.40676i	0.710814	−	0.703380i	$-0.248327\pi$
$432$		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	$-0.689476\pi$
$443$		−	15440.8i		−	1.65601i	0.560720	−	0.828005i	$-0.310524\pi$
$444$		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	$-0.670166\pi$
$449$		−	16373.4i		−	1.72095i	0.509491	−	0.860476i	$-0.329834\pi$
$450$		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	$-0.222784\pi$
$461$	15142.3			1.52982			0.764909	+	0.644139i	$0.222784\pi$
$462$		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	$-0.204776\pi$
$467$	16149.3			1.60021			0.800106	+	0.599858i	$0.204776\pi$
$468$		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	$-0.368462\pi$
$479$	8419.81			0.803154			0.401577	+	0.915825i	$0.368462\pi$
$480$		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	$0.306289\pi$
$491$			17853.2i			1.64094i	−0.571688	+	0.820471i	$0.693711\pi$
$492$		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	$-0.607920\pi$
$503$	−7503.79			−0.665164			−0.332582	+	0.943074i	$0.607920\pi$
$504$		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	$-0.349185\pi$
$509$	10479.2			0.912541			0.456270	+	0.889841i	$0.349185\pi$
$510$		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	$-0.400759\pi$
$521$	7295.75			0.613498			0.306749	+	0.951790i	$0.400759\pi$
$522$		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	$-0.936499\pi$
$557$		−	5210.25i		−	0.396347i	0.980167	−	0.198174i	$-0.0635010\pi$
$558$		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	$-0.375459\pi$
$563$	10188.7			0.762703			0.381351	+	0.924430i	$0.375459\pi$
$564$		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	$0.0219189\pi$
$569$			1867.77i			0.137612i	−0.997630	+	0.0688059i	$0.978081\pi$
$570$		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	$-0.200601\pi$
$587$	22979.9			1.61581			0.807906	+	0.589312i	$0.200601\pi$
$588$		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	$-0.453315\pi$
$593$	4220.66			0.292279			0.146140	+	0.989264i	$0.453315\pi$
$594$		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	$-0.720520\pi$
$599$		−	22561.2i		−	1.53894i	0.638682	−	0.769471i	$-0.279480\pi$
$600$		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	$0.142938\pi$
$617$			13306.4i			0.868223i	−0.900859	+	0.434112i	$0.857062\pi$
$618$		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	$0.0495882\pi$
$641$			5036.02i			0.310313i	−0.987890	+	0.155157i	$0.950412\pi$
$642$		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	$-0.603112\pi$
$647$	−10476.6			−0.636599			−0.318299	+	0.947990i	$0.603112\pi$
$648$		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	$0.123397\pi$
$653$			12615.9i			0.756049i	−0.925796	+	0.378025i	$0.876603\pi$
$654$		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	$0.0126774\pi$
$659$			1347.17i			0.0796334i	−0.999207	+	0.0398167i	$0.987323\pi$
$660$		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	$-0.695968\pi$
$677$	−20345.0			−1.15498			−0.577492	+	0.816397i	$0.695968\pi$
$678$		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	$0.114788\pi$
$683$			12596.6i			0.705703i	−0.935679	+	0.352852i	$0.885212\pi$
$684$		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	$0.201166\pi$
$701$			21928.4i			1.18149i	−0.806858	+	0.590746i	$0.798834\pi$
$702$		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	$-0.394726\pi$
$719$	12521.2			0.649462			0.324731	+	0.945806i	$0.394726\pi$
$720$		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	$0.0505797\pi$
$743$			6409.29i			0.316466i	−0.987402	+	0.158233i	$0.949420\pi$
$744$		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	$-0.670584\pi$
$761$	−21439.1			−1.02124			−0.510621	+	0.859806i	$0.670584\pi$
$762$		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	$-0.918223\pi$
$773$	−41572.5			−1.93436			−0.967180	+	0.254094i	$0.918223\pi$
$774$		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	$-0.568271\pi$
$797$	−9577.83			−0.425676			−0.212838	+	0.977087i	$0.568271\pi$
$798$		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	$-0.901000\pi$
$809$		−	14083.6i		−	0.612054i	0.952023	−	0.306027i	$-0.0989998\pi$
$810$		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	$0.123403\pi$
$821$			17786.4i			0.756088i	−0.925788	+	0.378044i	$0.876597\pi$
$822$		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	$-0.710929\pi$
$827$		−	37498.5i		−	1.57672i	0.615212	−	0.788362i	$-0.289071\pi$
$828$		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	$-0.513365\pi$
$839$	−2040.23			−0.0839528			−0.0419764	+	0.999119i	$0.513365\pi$
$840$		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	$-0.315953\pi$
$857$	27422.4			1.09303			0.546517	+	0.837448i	$0.315953\pi$
$858$		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	$0.177656\pi$
$863$			26852.8i			1.05919i	−0.848250	+	0.529595i	$0.822344\pi$
$864$		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	$-0.774225\pi$
$881$	−39685.7			−1.51764			−0.758822	+	0.651299i	$0.774225\pi$
$882$		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	$-0.589530\pi$
$887$	−14665.4			−0.555148			−0.277574	+	0.960704i	$0.589530\pi$
$888$		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	$0.269638\pi$
$911$			41209.5i			1.49872i	−0.662165	+	0.749358i	$0.730362\pi$
$912$		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	$-0.255740\pi$
$929$	39315.6			1.38849			0.694243	+	0.719741i	$0.255740\pi$
$930$		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	$-0.601950\pi$
$941$	−18176.1			−0.629676			−0.314838	+	0.949145i	$0.601950\pi$
$942$		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	$-0.742997\pi$
$947$		−	42110.2i		−	1.44498i	0.691380	−	0.722491i	$-0.257003\pi$
$948$		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	$-0.898071\pi$
$953$		−	18521.2i		−	0.629551i	0.949166	−	0.314775i	$-0.101929\pi$
$954$		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	$-0.315411\pi$
$971$	33158.4			1.09589			0.547943	+	0.836516i	$0.315411\pi$
$972$		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	$-0.811223\pi$
$977$		−	34135.5i		−	1.11780i	0.829234	−	0.558901i	$-0.188777\pi$
$978$		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	$-0.498247\pi$
$983$	339.463			0.0110144			0.00550721	+	0.999985i	$0.498247\pi$
$984$		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

Twists

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

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

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

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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