LMFDB - Embedded newform 547.3.b.b.546.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [547,3,Mod(546,547)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(547, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1]))

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

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

chi := DirichletCharacter("547.546");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 547 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	547.b (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:	$14.9046704605$
Analytic rank:	$0$
Dimension:	$88$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			546.10
Character	$\chi$	$=$	547.546
Dual form			547.3.b.b.546.79

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.26612i q^{2} +1.63931i q^{3} -6.66753 q^{4} -7.05581i q^{5} +5.35418 q^{6} +7.78718i q^{7} +8.71245i q^{8} +6.31266 q^{9} +O(q^{10})$	\(q-3.26612i q^{2} +1.63931i q^{3} -6.66753 q^{4} -7.05581i q^{5} +5.35418 q^{6} +7.78718i q^{7} +8.71245i q^{8} +6.31266 q^{9} -23.0451 q^{10} -3.13419 q^{11} -10.9301i q^{12} -22.1891 q^{13} +25.4338 q^{14} +11.5667 q^{15} +1.78580 q^{16} -3.98475i q^{17} -20.6179i q^{18} -19.1182 q^{19} +47.0448i q^{20} -12.7656 q^{21} +10.2366i q^{22} +19.3134i q^{23} -14.2824 q^{24} -24.7845 q^{25} +72.4721i q^{26} +25.1022i q^{27} -51.9212i q^{28} -32.7082 q^{29} -37.7781i q^{30} -13.7696i q^{31} +29.0172i q^{32} -5.13791i q^{33} -13.0147 q^{34} +54.9449 q^{35} -42.0899 q^{36} +7.09996i q^{37} +62.4424i q^{38} -36.3747i q^{39} +61.4735 q^{40} +4.40713i q^{41} +41.6939i q^{42} -33.9397i q^{43} +20.8973 q^{44} -44.5410i q^{45} +63.0799 q^{46} +34.2642 q^{47} +2.92748i q^{48} -11.6401 q^{49} +80.9491i q^{50} +6.53223 q^{51} +147.946 q^{52} -62.1821 q^{53} +81.9867 q^{54} +22.1143i q^{55} -67.8454 q^{56} -31.3407i q^{57} +106.829i q^{58} +107.148i q^{59} -77.1210 q^{60} +67.0284i q^{61} -44.9732 q^{62} +49.1578i q^{63} +101.917 q^{64} +156.562i q^{65} -16.7810 q^{66} -37.2246 q^{67} +26.5684i q^{68} -31.6607 q^{69} -179.456i q^{70} -85.4096i q^{71} +54.9988i q^{72} +48.5373 q^{73} +23.1893 q^{74} -40.6295i q^{75} +127.471 q^{76} -24.4065i q^{77} -118.804 q^{78} +83.5765i q^{79} -12.6003i q^{80} +15.6637 q^{81} +14.3942 q^{82} -137.197i q^{83} +85.1149 q^{84} -28.1156 q^{85} -110.851 q^{86} -53.6189i q^{87} -27.3065i q^{88} +43.3646i q^{89} -145.476 q^{90} -172.790i q^{91} -128.773i q^{92} +22.5727 q^{93} -111.911i q^{94} +134.895i q^{95} -47.5682 q^{96} -119.093 q^{97} +38.0179i q^{98} -19.7851 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) $ 88 * q - 192 * q^4 - 306 * q^9	$ 88 q - 192 q^{4} - 306 q^{9} - 4 q^{10} - 32 q^{11} + 26 q^{13} - 26 q^{14} + 22 q^{15} + 236 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} - 544 q^{25} - 96 q^{29} + 26 q^{34} + 10 q^{35} + 364 q^{36} + 44 q^{40} + 124 q^{44} - 288 q^{46} - 310 q^{47} - 694 q^{49} + 86 q^{51} - 316 q^{52} + 24 q^{53} - 266 q^{54} + 158 q^{56} - 80 q^{60} + 40 q^{62} - 652 q^{64} + 528 q^{66} + 28 q^{67} + 16 q^{69} + 94 q^{73} - 614 q^{74} - 28 q^{76} - 98 q^{78} + 928 q^{81} - 772 q^{82} + 358 q^{84} + 74 q^{85} - 410 q^{86} - 214 q^{90} + 656 q^{93} - 724 q^{96} + 346 q^{97} + 50 q^{99}+O(q^{100}) $ 88 * q - 192 * q^4 - 306 * q^9 - 4 * q^10 - 32 * q^11 + 26 * q^13 - 26 * q^14 + 22 * q^15 + 236 * q^16 - 12 * q^19 - 16 * q^21 - 2 * q^24 - 544 * q^25 - 96 * q^29 + 26 * q^34 + 10 * q^35 + 364 * q^36 + 44 * q^40 + 124 * q^44 - 288 * q^46 - 310 * q^47 - 694 * q^49 + 86 * q^51 - 316 * q^52 + 24 * q^53 - 266 * q^54 + 158 * q^56 - 80 * q^60 + 40 * q^62 - 652 * q^64 + 528 * q^66 + 28 * q^67 + 16 * q^69 + 94 * q^73 - 614 * q^74 - 28 * q^76 - 98 * q^78 + 928 * q^81 - 772 * q^82 + 358 * q^84 + 74 * q^85 - 410 * q^86 - 214 * q^90 + 656 * q^93 - 724 * q^96 + 346 * q^97 + 50 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/547\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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 $

$2$		−	3.26612i		−	1.63306i	−0.577304	−	0.816529i	$-0.695895\pi$
$2$		−	3.26612i		−	1.63306i	0.577304	−	0.816529i	$-0.304105\pi$
$3$			1.63931i			0.546437i	0.961952	+	0.273218i	$0.0880881\pi$
$3$			1.63931i			0.546437i	−0.961952	+	0.273218i	$0.911912\pi$
$4$	−6.66753			−1.66688
$5$		−	7.05581i		−	1.41116i	−0.708629	−	0.705581i	$-0.750686\pi$
$5$		−	7.05581i		−	1.41116i	0.708629	−	0.705581i	$-0.249314\pi$
$6$	5.35418			0.892363
$7$			7.78718i			1.11245i	0.831031	+	0.556227i	$0.187751\pi$
$7$			7.78718i			1.11245i	−0.831031	+	0.556227i	$0.812249\pi$
$8$			8.71245i			1.08906i
$9$	6.31266			0.701407
$10$	−23.0451			−2.30451
$11$	−3.13419			−0.284927			−0.142463	−	0.989800i	$-0.545502\pi$
$11$	−3.13419			−0.284927			−0.142463	+	0.989800i	$0.545502\pi$
$12$		−	10.9301i		−	0.910845i
$13$	−22.1891			−1.70685			−0.853425	−	0.521215i	$-0.825479\pi$
$13$	−22.1891			−1.70685			−0.853425	+	0.521215i	$0.825479\pi$
$14$	25.4338			1.81670
$15$	11.5667			0.771111
$16$	1.78580			0.111612
$17$		−	3.98475i		−	0.234397i	−0.993109	−	0.117198i	$-0.962609\pi$
$17$		−	3.98475i		−	0.234397i	0.993109	−	0.117198i	$-0.0373914\pi$
$18$		−	20.6179i		−	1.14544i
$19$	−19.1182			−1.00622			−0.503112	−	0.864221i	$-0.667812\pi$
$19$	−19.1182			−1.00622			−0.503112	+	0.864221i	$0.667812\pi$
$20$			47.0448i			2.35224i
$21$	−12.7656			−0.607885
$22$			10.2366i			0.465302i
$23$			19.3134i			0.839714i	0.907590	+	0.419857i	$0.137920\pi$
$23$			19.3134i			0.839714i	−0.907590	+	0.419857i	$0.862080\pi$
$24$	−14.2824			−0.595100
$25$	−24.7845			−0.991380
$26$			72.4721i			2.78739i
$27$			25.1022i			0.929711i
$28$		−	51.9212i		−	1.85433i
$29$	−32.7082			−1.12787			−0.563934	−	0.825820i	$-0.690713\pi$
$29$	−32.7082			−1.12787			−0.563934	+	0.825820i	$0.690713\pi$
$30$		−	37.7781i		−	1.25927i
$31$		−	13.7696i		−	0.444181i	−0.975026	−	0.222091i	$-0.928712\pi$
$31$		−	13.7696i		−	0.444181i	0.975026	−	0.222091i	$-0.0712881\pi$
$32$			29.0172i			0.906787i
$33$		−	5.13791i		−	0.155694i
$34$	−13.0147			−0.382784
$35$	54.9449			1.56985
$36$	−42.0899			−1.16916
$37$			7.09996i			0.191891i	0.995387	+	0.0959454i	$0.0305874\pi$
$37$			7.09996i			0.191891i	−0.995387	+	0.0959454i	$0.969413\pi$
$38$			62.4424i			1.64322i
$39$		−	36.3747i		−	0.932686i
$40$	61.4735			1.53684
$41$			4.40713i			0.107491i	0.998555	+	0.0537455i	$0.0171160\pi$
$41$			4.40713i			0.107491i	−0.998555	+	0.0537455i	$0.982884\pi$
$42$			41.6939i			0.992712i
$43$		−	33.9397i		−	0.789296i	−0.918832	−	0.394648i	$-0.870867\pi$
$43$		−	33.9397i		−	0.789296i	0.918832	−	0.394648i	$-0.129133\pi$
$44$	20.8973			0.474939
$45$		−	44.5410i		−	0.989800i
$46$	63.0799			1.37130
$47$	34.2642			0.729026			0.364513	−	0.931198i	$-0.381235\pi$
$47$	34.2642			0.729026			0.364513	+	0.931198i	$0.381235\pi$
$48$			2.92748i			0.0609891i
$49$	−11.6401			−0.237553
$50$			80.9491i			1.61898i
$51$	6.53223			0.128083
$52$	147.946			2.84512
$53$	−62.1821			−1.17325			−0.586623	−	0.809860i	$-0.699543\pi$
$53$	−62.1821			−1.17325			−0.586623	+	0.809860i	$0.699543\pi$
$54$	81.9867			1.51827
$55$			22.1143i			0.402078i
$56$	−67.8454			−1.21153
$57$		−	31.3407i		−	0.549837i
$58$			106.829i			1.84188i
$59$			107.148i			1.81607i	0.418894	+	0.908035i	$0.362418\pi$
$59$			107.148i			1.81607i	−0.418894	+	0.908035i	$0.637582\pi$
$60$	−77.1210			−1.28535
$61$			67.0284i			1.09883i	0.835551	+	0.549413i	$0.185149\pi$
$61$			67.0284i			1.09883i	−0.835551	+	0.549413i	$0.814851\pi$
$62$	−44.9732			−0.725374
$63$			49.1578i			0.780283i
$64$	101.917			1.59245
$65$			156.562i			2.40864i
$66$	−16.7810			−0.254258
$67$	−37.2246			−0.555592			−0.277796	−	0.960640i	$-0.589604\pi$
$67$	−37.2246			−0.555592			−0.277796	+	0.960640i	$0.589604\pi$
$68$			26.5684i			0.390712i
$69$	−31.6607			−0.458850
$70$		−	179.456i		−	2.56366i
$71$		−	85.4096i		−	1.20295i	−0.798891	−	0.601476i	$-0.794579\pi$
$71$		−	85.4096i		−	1.20295i	0.798891	−	0.601476i	$-0.205421\pi$
$72$			54.9988i			0.763872i
$73$	48.5373			0.664894			0.332447	−	0.943122i	$-0.392126\pi$
$73$	48.5373			0.664894			0.332447	+	0.943122i	$0.392126\pi$
$74$	23.1893			0.313369
$75$		−	40.6295i		−	0.541726i
$76$	127.471			1.67726
$77$		−	24.4065i		−	0.316968i
$78$	−118.804			−1.52313
$79$			83.5765i			1.05793i	0.848643	+	0.528965i	$0.177420\pi$
$79$			83.5765i			1.05793i	−0.848643	+	0.528965i	$0.822580\pi$
$80$		−	12.6003i		−	0.157503i
$81$	15.6637			0.193379
$82$	14.3942			0.175539
$83$		−	137.197i		−	1.65297i	−0.562956	−	0.826487i	$-0.690336\pi$
$83$		−	137.197i		−	1.65297i	0.562956	−	0.826487i	$-0.309664\pi$
$84$	85.1149			1.01327
$85$	−28.1156			−0.330772
$86$	−110.851			−1.28897
$87$		−	53.6189i		−	0.616309i
$88$		−	27.3065i		−	0.310301i
$89$			43.3646i			0.487243i	0.969870	+	0.243621i	$0.0783355\pi$
$89$			43.3646i			0.487243i	−0.969870	+	0.243621i	$0.921665\pi$
$90$	−145.476			−1.61640
$91$		−	172.790i		−	1.89879i
$92$		−	128.773i		−	1.39970i
$93$	22.5727			0.242717
$94$		−	111.911i		−	1.19054i
$95$			134.895i			1.41995i
$96$	−47.5682			−0.495502
$97$	−119.093			−1.22776			−0.613881	−	0.789399i	$-0.710392\pi$
$97$	−119.093			−1.22776			−0.613881	+	0.789399i	$0.710392\pi$
$98$			38.0179i			0.387938i
$99$	−19.7851			−0.199850
$100$	165.251			1.65251
$101$		−	49.6531i		−	0.491614i	−0.969319	−	0.245807i	$-0.920947\pi$
$101$		−	49.6531i		−	0.491614i	0.969319	−	0.245807i	$-0.0790530\pi$
$102$		−	21.3350i		−	0.209167i
$103$		−	199.269i		−	1.93465i	−0.253534	−	0.967327i	$-0.581593\pi$
$103$		−	199.269i		−	1.93465i	0.253534	−	0.967327i	$-0.418407\pi$
$104$		−	193.321i		−	1.85886i
$105$			90.0716i			0.857825i
$106$			203.094i			1.91598i
$107$		−	2.53013i		−	0.0236461i	−0.999930	−	0.0118230i	$-0.996237\pi$
$107$		−	2.53013i		−	0.0236461i	0.999930	−	0.0118230i	$-0.00376348\pi$
$108$		−	167.370i		−	1.54972i
$109$		−	38.6783i		−	0.354847i	−0.984135	−	0.177423i	$-0.943224\pi$
$109$		−	38.6783i		−	0.354847i	0.984135	−	0.177423i	$-0.0567762\pi$
$110$	72.2279			0.656617
$111$	−11.6390			−0.104856
$112$			13.9063i			0.124164i
$113$	−102.810			−0.909819			−0.454909	−	0.890538i	$-0.650328\pi$
$113$	−102.810			−0.909819			−0.454909	+	0.890538i	$0.650328\pi$
$114$	−102.362			−0.897917
$115$	136.272			1.18497
$116$	218.083			1.88002
$117$	−140.072			−1.19720
$118$	349.958			2.96575
$119$	31.0299			0.260756
$120$			100.774i			0.839784i
$121$	−111.177			−0.918817
$122$	218.923			1.79445
$123$	−7.22465			−0.0587370
$124$			91.8093i			0.740397i
$125$		−	1.52044i		−	0.0121635i
$126$	160.555			1.27425
$127$	−212.952			−1.67679			−0.838395	−	0.545064i	$-0.816506\pi$
$127$	−212.952			−1.67679			−0.838395	+	0.545064i	$0.816506\pi$
$128$		−	216.803i		−	1.69378i
$129$	55.6377			0.431300
$130$	511.350			3.93346
$131$	79.6077			0.607692			0.303846	−	0.952721i	$-0.401729\pi$
$131$	79.6077			0.607692			0.303846	+	0.952721i	$0.401729\pi$
$132$			34.2572i			0.259524i
$133$		−	148.877i		−	1.11938i
$134$			121.580i			0.907314i
$135$	177.116			1.31197
$136$	34.7169			0.255272
$137$	12.0379			0.0878680			0.0439340	−	0.999034i	$-0.486011\pi$
$137$	12.0379			0.0878680			0.0439340	+	0.999034i	$0.486011\pi$
$138$			103.407i			0.749329i
$139$	3.37997			0.0243163			0.0121582	−	0.999926i	$-0.496130\pi$
$139$	3.37997			0.0243163			0.0121582	+	0.999926i	$0.496130\pi$
$140$	−366.346			−2.61676
$141$			56.1697i			0.398367i
$142$	−278.958			−1.96449
$143$	69.5448			0.486327
$144$	11.2732			0.0782858
$145$			230.783i			1.59161i
$146$		−	158.528i		−	1.08581i
$147$		−	19.0817i		−	0.129808i
$148$		−	47.3392i		−	0.319859i
$149$	−149.757			−1.00508			−0.502539	−	0.864554i	$-0.667601\pi$
$149$	−149.757			−1.00508			−0.502539	+	0.864554i	$0.667601\pi$
$150$	−132.701			−0.884671
$151$			136.093i			0.901278i	0.892706	+	0.450639i	$0.148804\pi$
$151$			136.093i			0.901278i	−0.892706	+	0.450639i	$0.851196\pi$
$152$		−	166.567i		−	1.09583i
$153$		−	25.1544i		−	0.164408i
$154$	−79.7145			−0.517627
$155$	−97.1559			−0.626812
$156$			242.530i			1.55468i
$157$	25.4946			0.162386			0.0811931	−	0.996698i	$-0.474127\pi$
$157$	25.4946			0.162386			0.0811931	+	0.996698i	$0.474127\pi$
$158$	272.971			1.72766
$159$		−	101.936i		−	0.641105i
$160$	204.740			1.27962
$161$	−150.397			−0.934142
$162$		−	51.1595i		−	0.315800i
$163$		−	108.942i		−	0.668354i	−0.942510	−	0.334177i	$-0.891542\pi$
$163$		−	108.942i		−	0.668354i	0.942510	−	0.334177i	$-0.108458\pi$
$164$		−	29.3846i		−	0.179175i
$165$	−36.2522			−0.219710
$166$	−448.101			−2.69940
$167$	−189.278			−1.13340			−0.566700	−	0.823924i	$-0.691780\pi$
$167$	−189.278			−1.13340			−0.566700	+	0.823924i	$0.691780\pi$
$168$		−	111.220i		−	0.662022i
$169$	323.354			1.91334
$170$			91.8290i			0.540170i
$171$	−120.687			−0.705772
$172$			226.294i			1.31566i
$173$		−	141.966i		−	0.820610i	−0.911948	−	0.410305i	$-0.865422\pi$
$173$		−	141.966i		−	0.820610i	0.911948	−	0.410305i	$-0.134578\pi$
$174$	−175.126			−1.00647
$175$		−	193.001i		−	1.10286i
$176$	−5.59704			−0.0318014
$177$	−175.649			−0.992367
$178$	141.634			0.795697
$179$	−105.911			−0.591680			−0.295840	−	0.955237i	$-0.595600\pi$
$179$	−105.911			−0.591680			−0.295840	+	0.955237i	$0.595600\pi$
$180$			296.978i			1.64988i
$181$	−241.740			−1.33558			−0.667789	−	0.744351i	$-0.732759\pi$
$181$	−241.740			−1.33558			−0.667789	+	0.744351i	$0.732759\pi$
$182$	−564.353			−3.10084
$183$	−109.880			−0.600439
$184$	−168.267			−0.914496
$185$	50.0960			0.270789
$186$		−	73.7250i		−	0.396371i
$187$			12.4890i			0.0667859i
$188$	−228.458			−1.21520
$189$	−195.475			−1.03426
$190$	440.582			2.31885
$191$	334.886			1.75333			0.876666	−	0.481100i	$-0.159763\pi$
$191$	334.886			1.75333			0.876666	+	0.481100i	$0.159763\pi$
$192$			167.073i			0.870172i
$193$	−133.685			−0.692667			−0.346333	−	0.938112i	$-0.612573\pi$
$193$	−133.685			−0.692667			−0.346333	+	0.938112i	$0.612573\pi$
$194$			388.971i			2.00501i
$195$	−256.653			−1.31617
$196$	77.6106			0.395973
$197$			287.583i			1.45981i	0.683546	+	0.729907i	$0.260437\pi$
$197$			287.583i			1.45981i	−0.683546	+	0.729907i	$0.739563\pi$
$198$			64.6205i			0.326366i
$199$	−0.814036			−0.00409063			−0.00204532	−	0.999998i	$-0.500651\pi$
$199$	−0.814036			−0.00409063			−0.00204532	+	0.999998i	$0.500651\pi$
$200$		−	215.934i		−	1.07967i
$201$		−	61.0227i		−	0.303596i
$202$	−162.173			−0.802835
$203$		−	254.704i		−	1.25470i
$204$	−43.5538			−0.213499
$205$	31.0959			0.151687
$206$	−650.837			−3.15940
$207$			121.919i			0.588981i
$208$	−39.6252			−0.190506
$209$	59.9203			0.286700
$210$	294.185			1.40088
$211$		−	253.195i		−	1.19998i	−0.800008	−	0.599989i	$-0.795172\pi$
$211$		−	253.195i		−	1.19998i	0.800008	−	0.599989i	$-0.204828\pi$
$212$	414.600			1.95566
$213$	140.013			0.657337
$214$	−8.26371			−0.0386155
$215$	−239.472			−1.11383
$216$	−218.702			−1.01251
$217$	107.226			0.494131
$218$	−126.328			−0.579486
$219$			79.5676i			0.363322i
$220$		−	147.448i		−	0.670216i
$221$			88.4178i			0.400081i
$222$			38.0144i			0.171236i
$223$		−	252.933i		−	1.13423i	−0.823639	−	0.567114i	$-0.808060\pi$
$223$		−	252.933i		−	1.13423i	0.823639	−	0.567114i	$-0.191940\pi$
$224$	−225.962			−1.00876
$225$	−156.456			−0.695361
$226$			335.788i			1.48579i
$227$	312.925			1.37852			0.689262	−	0.724512i	$-0.257935\pi$
$227$	312.925			1.37852			0.689262	+	0.724512i	$0.257935\pi$
$228$			208.965i			0.916514i
$229$		−	275.144i		−	1.20150i	−0.799437	−	0.600750i	$-0.794869\pi$
$229$		−	275.144i		−	1.20150i	0.799437	−	0.600750i	$-0.205131\pi$
$230$		−	445.080i		−	1.93513i
$231$	40.0098			0.173203
$232$		−	284.969i		−	1.22831i
$233$	−222.832			−0.956361			−0.478181	−	0.878262i	$-0.658704\pi$
$233$	−222.832			−0.956361			−0.478181	+	0.878262i	$0.658704\pi$
$234$			457.492i			1.95509i
$235$		−	241.762i		−	1.02877i
$236$		−	714.413i		−	3.02717i
$237$	−137.008			−0.578092
$238$		−	101.347i		−	0.425829i
$239$	160.596			0.671949			0.335975	−	0.941871i	$-0.390934\pi$
$239$	160.596			0.671949			0.335975	+	0.941871i	$0.390934\pi$
$240$	20.6557			0.0860656
$241$		−	69.0340i		−	0.286448i	−0.989690	−	0.143224i	$-0.954253\pi$
$241$		−	69.0340i		−	0.286448i	0.989690	−	0.143224i	$-0.0457469\pi$
$242$			363.117i			1.50048i
$243$			251.597i			1.03538i
$244$		−	446.914i		−	1.83161i
$245$			82.1304i			0.335226i
$246$			23.5966i			0.0959209i
$247$	424.216			1.71747
$248$	119.967			0.483739
$249$	224.908			0.903245
$250$	−4.96594			−0.0198638
$251$			11.6866i			0.0465603i	0.999729	+	0.0232801i	$0.00741097\pi$
$251$			11.6866i			0.0465603i	−0.999729	+	0.0232801i	$0.992589\pi$
$252$		−	327.761i		−	1.30064i
$253$		−	60.5320i		−	0.239257i
$254$			695.527i			2.73830i
$255$		−	46.0902i		−	0.180746i
$256$	−300.438			−1.17359
$257$			414.672i			1.61351i	0.590886	+	0.806755i	$0.298778\pi$
$257$			414.672i			1.61351i	−0.590886	+	0.806755i	$0.701222\pi$
$258$		−	181.719i		−	0.704339i
$259$	−55.2886			−0.213470
$260$		−	1043.88i		−	4.01492i
$261$	−206.476			−0.791095
$262$		−	260.008i		−	0.992398i
$263$	132.635			0.504316			0.252158	−	0.967686i	$-0.418860\pi$
$263$	132.635			0.504316			0.252158	+	0.967686i	$0.418860\pi$
$264$	44.7638			0.169560
$265$			438.745i			1.65564i
$266$	−486.250			−1.82801
$267$	−71.0880			−0.266247
$268$	248.196			0.926106
$269$	104.662			0.389077			0.194539	−	0.980895i	$-0.437679\pi$
$269$	104.662			0.389077			0.194539	+	0.980895i	$0.437679\pi$
$270$		−	578.483i		−	2.14253i
$271$		−	141.241i		−	0.521185i	−0.965449	−	0.260593i	$-0.916082\pi$
$271$		−	141.241i		−	0.521185i	0.965449	−	0.260593i	$-0.0839179\pi$
$272$		−	7.11596i		−	0.0261616i
$273$	283.256			1.03757
$274$		−	39.3173i		−	0.143494i
$275$	77.6794			0.282471
$276$	211.098			0.764849
$277$	−262.535			−0.947780			−0.473890	−	0.880584i	$-0.657151\pi$
$277$	−262.535			−0.947780			−0.473890	+	0.880584i	$0.657151\pi$
$278$		−	11.0394i		−	0.0397100i
$279$		−	86.9230i		−	0.311552i
$280$			478.705i			1.70966i
$281$		−	336.333i		−	1.19691i	−0.801155	−	0.598457i	$-0.795781\pi$
$281$		−	336.333i		−	1.19691i	0.801155	−	0.598457i	$-0.204219\pi$
$282$	183.457			0.650556
$283$			385.390i			1.36180i	0.732375	+	0.680901i	$0.238412\pi$
$283$			385.390i			1.36180i	−0.732375	+	0.680901i	$0.761588\pi$
$284$			569.471i			2.00518i
$285$	−221.134			−0.775910
$286$		−	227.142i		−	0.794201i
$287$	−34.3191			−0.119579
$288$			183.176i			0.636027i
$289$	273.122			0.945058
$290$	753.764			2.59919
$291$		−	195.230i		−	0.670894i
$292$	−323.624			−1.10830
$293$	30.0746			0.102644			0.0513219	−	0.998682i	$-0.483657\pi$
$293$	30.0746			0.102644			0.0513219	+	0.998682i	$0.483657\pi$
$294$	−62.3232			−0.211984
$295$	756.017			2.56277
$296$	−61.8581			−0.208980
$297$		−	78.6751i		−	0.264899i
$298$			489.123i			1.64135i
$299$		−	428.547i		−	1.43327i
$300$			270.898i			0.902994i
$301$	264.295			0.878055
$302$	444.496			1.47184
$303$	81.3967			0.268636
$304$	−34.1413			−0.112307
$305$	472.940			1.55062
$306$	−82.1571			−0.268487
$307$		−	54.8131i		−	0.178544i	−0.996007	−	0.0892721i	$-0.971546\pi$
$307$		−	54.8131i		−	0.178544i	0.996007	−	0.0892721i	$-0.0284541\pi$
$308$			162.731i			0.528348i
$309$	326.664			1.05717
$310$			317.322i			1.02362i
$311$	338.037			1.08694			0.543468	−	0.839430i	$-0.317111\pi$
$311$	338.037			1.08694			0.543468	+	0.839430i	$0.317111\pi$
$312$	316.913			1.01575
$313$	68.0576			0.217436			0.108718	−	0.994073i	$-0.465325\pi$
$313$	68.0576			0.217436			0.108718	+	0.994073i	$0.465325\pi$
$314$		−	83.2685i		−	0.265186i
$315$	346.848			1.10111
$316$		−	557.249i		−	1.76345i
$317$	−325.206			−1.02589			−0.512943	−	0.858423i	$-0.671445\pi$
$317$	−325.206			−1.02589			−0.512943	+	0.858423i	$0.671445\pi$
$318$	−332.934			−1.04696
$319$	102.514			0.321360
$320$		−	719.106i		−	2.24721i
$321$	4.14767			0.0129211
$322$			491.214i			1.52551i
$323$			76.1814i			0.235856i
$324$	−104.438			−0.322340
$325$	549.945			1.69214
$326$	−355.817			−1.09146
$327$	63.4057			0.193901
$328$	−38.3969			−0.117064
$329$			266.822i			0.811008i
$330$			118.404i			0.358799i
$331$			44.2515i			0.133690i	0.997763	+	0.0668451i	$0.0212933\pi$
$331$			44.2515i			0.133690i	−0.997763	+	0.0668451i	$0.978707\pi$
$332$			914.763i			2.75531i
$333$			44.8196i			0.134594i
$334$			618.204i			1.85091i
$335$			262.650i			0.784030i
$336$	−22.7968			−0.0678476
$337$		−	129.123i		−	0.383153i	−0.981478	−	0.191577i	$-0.938640\pi$
$337$		−	129.123i		−	0.383153i	0.981478	−	0.191577i	$-0.0613600\pi$
$338$		−	1056.11i		−	3.12460i
$339$		−	168.537i		−	0.497158i
$340$	187.462			0.551358
$341$			43.1566i			0.126559i
$342$			394.178i			1.15257i
$343$			290.928i			0.848187i
$344$	295.698			0.859588
$345$			223.392i			0.647512i
$346$	−463.676			−1.34011
$347$	−392.140			−1.13009			−0.565043	−	0.825061i	$-0.691141\pi$
$347$	−392.140			−1.13009			−0.565043	+	0.825061i	$0.691141\pi$
$348$			357.505i			1.02731i
$349$	205.966			0.590161			0.295081	−	0.955472i	$-0.404653\pi$
$349$	205.966			0.590161			0.295081	+	0.955472i	$0.404653\pi$
$350$	−630.365			−1.80104
$351$		−	556.994i		−	1.58688i
$352$		−	90.9455i		−	0.258368i
$353$	−143.336			−0.406052			−0.203026	−	0.979173i	$-0.565078\pi$
$353$	−143.336			−0.406052			−0.203026	+	0.979173i	$0.565078\pi$
$354$			573.690i			1.62059i
$355$	−602.635			−1.69756
$356$		−	289.135i		−	0.812176i
$357$			50.8677i			0.142486i
$358$			345.917i			0.966249i
$359$			486.432i			1.35496i	0.735539	+	0.677482i	$0.236929\pi$
$359$			486.432i			1.35496i	−0.735539	+	0.677482i	$0.763071\pi$
$360$	388.061			1.07795
$361$	4.50732			0.0124857
$362$			789.550i			2.18108i
$363$		−	182.253i		−	0.502075i
$364$			1152.08i			3.16506i
$365$		−	342.470i		−	0.938274i
$366$			358.882i			0.980552i
$367$	−64.7852			−0.176526			−0.0882632	−	0.996097i	$-0.528132\pi$
$367$	−64.7852			−0.176526			−0.0882632	+	0.996097i	$0.528132\pi$
$368$			34.4899i			0.0937225i
$369$			27.8207i			0.0753949i
$370$		−	163.619i		−	0.442215i
$371$		−	484.223i		−	1.30518i
$372$	−150.504			−0.404580
$373$		−	174.639i		−	0.468202i	−0.972212	−	0.234101i	$-0.924785\pi$
$373$		−	174.639i		−	0.468202i	0.972212	−	0.234101i	$-0.0752147\pi$
$374$	40.7904			0.109065
$375$	2.49247			0.00664659
$376$			298.526i			0.793951i
$377$	725.764			1.92510
$378$			638.445i			1.68901i
$379$	−528.159			−1.39356			−0.696780	−	0.717285i	$-0.745385\pi$
$379$	−528.159			−1.39356			−0.696780	+	0.717285i	$0.745385\pi$
$380$		−	899.415i		−	2.36688i
$381$		−	349.095i		−	0.916259i
$382$		−	1093.78i		−	2.86329i
$383$	41.2234			0.107633			0.0538165	−	0.998551i	$-0.482861\pi$
$383$	41.2234			0.107633			0.0538165	+	0.998551i	$0.482861\pi$
$384$	355.408			0.925541
$385$	−172.208			−0.447293
$386$			436.630i			1.13117i
$387$		−	214.250i		−	0.553618i
$388$	794.055			2.04653
$389$		−	683.375i		−	1.75675i	−0.477974	−	0.878374i	$-0.658629\pi$
$389$		−	683.375i		−	1.75675i	0.477974	−	0.878374i	$-0.341371\pi$
$390$			838.260i			2.14939i
$391$	76.9591			0.196826
$392$		−	101.414i		−	0.258709i
$393$			130.502i			0.332065i
$394$	939.281			2.38396
$395$	589.700			1.49291
$396$	131.918			0.333126
$397$			662.057i			1.66765i	0.552029	+	0.833825i	$0.313854\pi$
$397$			662.057i			1.66765i	−0.552029	+	0.833825i	$0.686146\pi$
$398$			2.65874i			0.00668024i
$399$	244.056			0.611668
$400$	−44.2602			−0.110650
$401$	735.968			1.83533			0.917666	−	0.397354i	$-0.130071\pi$
$401$	735.968			1.83533			0.917666	+	0.397354i	$0.130071\pi$
$402$	−199.307			−0.495790
$403$			305.535i			0.758151i
$404$			331.063i			0.819463i
$405$		−	110.520i		−	0.272889i
$406$	−831.895			−2.04900
$407$		−	22.2526i		−	0.0546748i
$408$			56.9118i			0.139490i
$409$	271.929			0.664863			0.332431	−	0.943127i	$-0.392131\pi$
$409$	271.929			0.664863			0.332431	+	0.943127i	$0.392131\pi$
$410$		−	101.563i		−	0.247714i
$411$			19.7339i			0.0480143i
$412$			1328.63i			3.22484i
$413$	−834.381			−2.02029
$414$	398.202			0.961841
$415$	−968.035			−2.33261
$416$		−	643.864i		−	1.54775i
$417$			5.54082i			0.0132873i
$418$		−	195.707i		−	0.468198i
$419$	253.260			0.604440			0.302220	−	0.953238i	$-0.402272\pi$
$419$	253.260			0.604440			0.302220	+	0.953238i	$0.402272\pi$
$420$		−	600.555i		−	1.42989i
$421$		−	712.147i		−	1.69156i	−0.533531	−	0.845781i	$-0.679135\pi$
$421$		−	712.147i		−	1.69156i	0.533531	−	0.845781i	$-0.320865\pi$
$422$	−826.966			−1.95964
$423$	216.299			0.511344
$424$		−	541.758i		−	1.27773i
$425$			98.7600i			0.232377i
$426$		−	457.298i		−	1.07347i
$427$	−521.962			−1.22239
$428$			16.8697i			0.0394152i
$429$			114.005i			0.265747i
$430$			782.145i			1.81894i
$431$		−	421.037i		−	0.976884i	−0.872596	−	0.488442i	$-0.837565\pi$
$431$		−	421.037i		−	0.976884i	0.872596	−	0.488442i	$-0.162435\pi$
$432$			44.8275i			0.103767i
$433$		−	594.218i		−	1.37233i	−0.727447	−	0.686164i	$-0.759293\pi$
$433$		−	594.218i		−	1.37233i	0.727447	−	0.686164i	$-0.240707\pi$
$434$		−	350.214i		−	0.806945i
$435$	−378.325			−0.869712
$436$			257.889i			0.591488i
$437$		−	369.239i		−	0.844940i
$438$	259.877			0.593327
$439$	−493.046			−1.12311			−0.561556	−	0.827439i	$-0.689797\pi$
$439$	−493.046			−1.12311			−0.561556	+	0.827439i	$0.689797\pi$
$440$	−192.670			−0.437886
$441$	−73.4800			−0.166621
$442$	288.783			0.653355
$443$	−105.683			−0.238562			−0.119281	−	0.992861i	$-0.538059\pi$
$443$	−105.683			−0.238562			−0.119281	+	0.992861i	$0.538059\pi$
$444$	77.6035			0.174783
$445$	305.973			0.687579
$446$	−826.109			−1.85226
$447$		−	245.498i		−	0.549212i
$448$			793.644i			1.77153i
$449$	556.048			1.23841			0.619207	−	0.785228i	$-0.287454\pi$
$449$	556.048			1.23841			0.619207	+	0.785228i	$0.287454\pi$
$450$			511.005i			1.13557i
$451$		−	13.8128i		−	0.0306270i
$452$	685.485			1.51656
$453$	−223.099			−0.492491
$454$		−	1022.05i		−	2.25121i
$455$	−1219.17			−2.67951
$456$	273.055			0.598804
$457$			448.001i			0.980308i	0.871636	+	0.490154i	$0.163060\pi$
$457$			448.001i			0.980308i	−0.871636	+	0.490154i	$0.836940\pi$
$458$	−898.652			−1.96212
$459$	100.026			0.217921
$460$	−908.596			−1.97521
$461$			131.210i			0.284621i	0.989822	+	0.142310i	$0.0454531\pi$
$461$			131.210i			0.284621i	−0.989822	+	0.142310i	$0.954547\pi$
$462$		−	130.677i		−	0.282850i
$463$			82.2351i			0.177614i	0.996049	+	0.0888068i	$0.0283054\pi$
$463$			82.2351i			0.177614i	−0.996049	+	0.0888068i	$0.971695\pi$
$464$	−58.4103			−0.125884
$465$		−	159.269i		−	0.342513i
$466$			727.796i			1.56179i
$467$	54.0562			0.115752			0.0578760	−	0.998324i	$-0.481567\pi$
$467$	54.0562			0.115752			0.0578760	+	0.998324i	$0.481567\pi$
$468$	933.934			1.99559
$469$		−	289.875i		−	0.618070i
$470$	−789.623			−1.68005
$471$			41.7936i			0.0887338i
$472$	−933.523			−1.97780
$473$			106.374i			0.224892i
$474$			447.484i			0.944058i
$475$	473.836			0.997550
$476$	−206.893			−0.434649
$477$	−392.534			−0.822923
$478$		−	524.525i		−	1.09733i
$479$	−567.189			−1.18411			−0.592055	−	0.805897i	$-0.701683\pi$
$479$	−567.189			−1.18411			−0.592055	+	0.805897i	$0.701683\pi$
$480$			335.632i			0.699233i
$481$		−	157.541i		−	0.327529i
$482$	−225.473			−0.467787
$483$		−	246.547i		−	0.510450i
$484$	741.274			1.53156
$485$			840.297i			1.73257i
$486$	821.747			1.69084
$487$			650.131i			1.33497i	0.744623	+	0.667485i	$0.232629\pi$
$487$			650.131i			1.33497i	−0.744623	+	0.667485i	$0.767371\pi$
$488$	−583.982			−1.19668
$489$	178.589			0.365213
$490$	268.247			0.547444
$491$		−	30.6552i		−	0.0624341i	−0.999513	−	0.0312171i	$-0.990062\pi$
$491$		−	30.6552i		−	0.0624341i	0.999513	−	0.0312171i	$-0.00993832\pi$
$492$	48.1705			0.0979076
$493$			130.334i			0.264369i
$494$		−	1385.54i		−	2.80474i
$495$			139.600i			0.282020i
$496$		−	24.5898i		−	0.0495762i
$497$	665.100			1.33823
$498$		−	734.576i		−	1.47505i
$499$	676.335			1.35538			0.677691	−	0.735347i	$-0.262981\pi$
$499$	676.335			1.35538			0.677691	+	0.735347i	$0.262981\pi$
$500$			10.1376i			0.0202752i
$501$		−	310.285i		−	0.619331i
$502$	38.1699			0.0760357
$503$		−	626.311i		−	1.24515i	−0.782560	−	0.622576i	$-0.786086\pi$
$503$		−	626.311i		−	1.24515i	0.782560	−	0.622576i	$-0.213914\pi$
$504$	−428.285			−0.849772
$505$	−350.343			−0.693748
$506$	−197.705			−0.390720
$507$			530.078i			1.04552i
$508$	1419.86			2.79501
$509$	−363.300			−0.713752			−0.356876	−	0.934152i	$-0.616158\pi$
$509$	−363.300			−0.713752			−0.356876	+	0.934152i	$0.616158\pi$
$510$	−150.536			−0.295169
$511$			377.968i			0.739664i
$512$			114.054i			0.222761i
$513$		−	479.910i		−	0.935497i
$514$	1354.37			2.63496
$515$	−1406.01			−2.73011
$516$	−370.966			−0.718926
$517$	−107.391			−0.207719
$518$			180.579i			0.348608i
$519$	232.726			0.448412
$520$	−1364.04			−2.62315
$521$	−251.014			−0.481792			−0.240896	−	0.970551i	$-0.577441\pi$
$521$	−251.014			−0.481792			−0.240896	+	0.970551i	$0.577441\pi$
$522$			674.375i			1.29191i
$523$			636.996i			1.21797i	0.793183	+	0.608983i	$0.208422\pi$
$523$			636.996i			1.21797i	−0.793183	+	0.608983i	$0.791578\pi$
$524$	−530.786			−1.01295
$525$	316.389			0.602646
$526$		−	433.202i		−	0.823578i
$527$	−54.8684			−0.104115
$528$		−	9.17528i		−	0.0173774i
$529$	155.992			0.294881
$530$	1432.99			2.70376
$531$			676.390i			1.27380i
$532$			992.642i			1.86587i
$533$		−	97.7900i		−	0.183471i
$534$			232.182i			0.434798i
$535$	−17.8521			−0.0333685
$536$		−	324.318i		−	0.605071i
$537$		−	173.621i		−	0.323316i
$538$		−	341.838i		−	0.635386i
$539$	36.4823			0.0676852
$540$	−1180.93			−2.18690
$541$			814.667i			1.50585i	0.658104	+	0.752927i	$0.271359\pi$
$541$			814.667i			1.50585i	−0.658104	+	0.752927i	$0.728641\pi$
$542$	−461.310			−0.851126
$543$		−	396.286i		−	0.729809i
$544$	115.626			0.212548
$545$	−272.907			−0.500747
$546$		−	925.149i		−	1.69441i
$547$	−213.724	+	503.519i	−0.390721	+	0.920509i
$547$	−213.724	+	503.519i	−0.390721	+	0.920509i
$548$	−80.2632			−0.146466
$549$			423.128i			0.770725i
$550$		−	253.710i		−	0.461291i
$551$	625.323			1.13489
$552$		−	275.842i		−	0.499714i
$553$	−650.825			−1.17690
$554$			857.471i			1.54778i
$555$			82.1228i			0.147969i
$556$	−22.5360			−0.0405325
$557$	−903.163			−1.62148			−0.810739	−	0.585408i	$-0.800934\pi$
$557$	−903.163			−1.62148			−0.810739	+	0.585408i	$0.800934\pi$
$558$	−283.901			−0.508783
$559$			753.091i			1.34721i
$560$	98.1205			0.175215
$561$	−20.4733			−0.0364943
$562$	−1098.50			−1.95463
$563$	543.374			0.965141			0.482571	−	0.875857i	$-0.339703\pi$
$563$	543.374			0.965141			0.482571	+	0.875857i	$0.339703\pi$
$564$		−	374.513i		−	0.664030i
$565$			725.405i			1.28390i
$566$	1258.73			2.22390
$567$			121.976i			0.215125i
$568$	744.128			1.31008
$569$			998.354i			1.75458i	0.479964	+	0.877288i	$0.340650\pi$
$569$			998.354i			1.75458i	−0.479964	+	0.877288i	$0.659350\pi$
$570$			722.251i			1.26711i
$571$	−68.0908			−0.119248			−0.0596241	−	0.998221i	$-0.518990\pi$
$571$	−68.0908			−0.119248			−0.0596241	+	0.998221i	$0.518990\pi$
$572$	−463.692			−0.810650
$573$			548.982i			0.958084i
$574$			112.090i			0.195279i
$575$		−	478.674i		−	0.832476i
$576$	643.366			1.11696
$577$		−	326.969i		−	0.566670i	−0.959021	−	0.283335i	$-0.908559\pi$
$577$		−	326.969i		−	0.566670i	0.959021	−	0.283335i	$-0.0914408\pi$
$578$		−	892.048i		−	1.54334i
$579$		−	219.151i		−	0.378498i
$580$		−	1538.75i		−	2.65302i
$581$	1068.38			1.83886
$582$	−637.644			−1.09561
$583$	194.891			0.334289
$584$			422.879i			0.724108i
$585$			988.323i			1.68944i
$586$		−	98.2273i		−	0.167623i
$587$	1104.56			1.88170			0.940851	−	0.338820i	$-0.110028\pi$
$587$	1104.56			1.88170			0.940851	+	0.338820i	$0.110028\pi$
$588$			127.228i			0.216374i
$589$			263.251i			0.446946i
$590$		−	2469.24i		−	4.18516i
$591$	−471.438			−0.797696
$592$			12.6791i			0.0214174i
$593$	488.053			0.823024			0.411512	−	0.911404i	$-0.365001\pi$
$593$	488.053			0.823024			0.411512	+	0.911404i	$0.365001\pi$
$594$	−256.962			−0.432596
$595$		−	218.941i		−	0.367969i
$596$	998.507			1.67535
$597$		−	1.33446i		−	0.00223527i
$598$	−1399.68			−2.34061
$599$	−907.158			−1.51445			−0.757227	−	0.653151i	$-0.773446\pi$
$599$	−907.158			−1.51445			−0.757227	+	0.653151i	$0.773446\pi$
$600$	353.983			0.589971
$601$	112.612			0.187375			0.0936874	−	0.995602i	$-0.470135\pi$
$601$	112.612			0.187375			0.0936874	+	0.995602i	$0.470135\pi$
$602$		−	863.218i		−	1.43392i
$603$	−234.987			−0.389696
$604$		−	907.404i		−	1.50232i
$605$			784.443i			1.29660i
$606$		−	265.851i		−	0.438698i
$607$	1014.30			1.67101			0.835506	−	0.549481i	$-0.185174\pi$
$607$	1014.30			1.67101			0.835506	+	0.549481i	$0.185174\pi$
$608$		−	554.758i		−	0.912430i
$609$	417.539			0.685615
$610$		−	1544.68i		−	2.53226i
$611$	−760.291			−1.24434
$612$			167.717i			0.274048i
$613$	516.086			0.841902			0.420951	−	0.907083i	$-0.361696\pi$
$613$	516.086			0.841902			0.420951	+	0.907083i	$0.361696\pi$
$614$	−179.026			−0.291573
$615$			50.9758i			0.0828874i
$616$	212.641			0.345196
$617$		−	659.323i		−	1.06860i	−0.845296	−	0.534298i	$-0.820576\pi$
$617$		−	659.323i		−	1.06860i	0.845296	−	0.534298i	$-0.179424\pi$
$618$		−	1066.92i		−	1.72641i
$619$			509.523i			0.823139i	0.911378	+	0.411569i	$0.135019\pi$
$619$			509.523i			0.823139i	−0.911378	+	0.411569i	$0.864981\pi$
$620$	647.789			1.04482
$621$	−484.809			−0.780691
$622$		−	1104.07i		−	1.77503i
$623$	−337.688			−0.542035
$624$		−	64.9580i		−	0.104099i
$625$	−630.341			−1.00855
$626$		−	222.284i		−	0.355086i
$627$			98.2279i			0.156663i
$628$	−169.986			−0.270679
$629$	28.2915			0.0449786
$630$		−	1132.85i		−	1.79817i
$631$	−419.421			−0.664692			−0.332346	−	0.943158i	$-0.607840\pi$
$631$	−419.421			−0.664692			−0.332346	+	0.943158i	$0.607840\pi$
$632$	−728.157			−1.15215
$633$	415.066			0.655712
$634$			1062.16i			1.67533i
$635$			1502.55i			2.36622i
$636$			679.659i			1.06865i
$637$	258.283			0.405468
$638$		−	334.822i		−	0.524800i
$639$		−	539.162i		−	0.843760i
$640$	−1529.72			−2.39019
$641$			810.462i			1.26437i	0.774817	+	0.632185i	$0.217842\pi$
$641$			810.462i			1.26437i	−0.774817	+	0.632185i	$0.782158\pi$
$642$		−	13.5468i		−	0.0211009i
$643$	444.423			0.691171			0.345586	−	0.938387i	$-0.387680\pi$
$643$	444.423			0.691171			0.345586	+	0.938387i	$0.387680\pi$
$644$	1002.78			1.55710
$645$		−	392.570i		−	0.608635i
$646$	248.817			0.385166
$647$	−665.587			−1.02873			−0.514364	−	0.857572i	$-0.671972\pi$
$647$	−665.587			−1.02873			−0.514364	+	0.857572i	$0.671972\pi$
$648$			136.469i			0.210601i
$649$		−	335.823i		−	0.517447i
$650$		−	1796.19i		−	2.76336i
$651$			175.777i			0.270011i
$652$			726.372i			1.11407i
$653$			895.412i			1.37123i	0.727966	+	0.685614i	$0.240466\pi$
$653$			895.412i			1.37123i	−0.727966	+	0.685614i	$0.759534\pi$
$654$		−	207.091i		−	0.316652i
$655$		−	561.697i		−	0.857553i
$656$			7.87025i			0.0119973i
$657$	306.400			0.466362
$658$	871.471			1.32442
$659$		−	3.48677i		−	0.00529100i	−0.999997	−	0.00264550i	$-0.999158\pi$
$659$		−	3.48677i		−	0.00529100i	0.999997	−	0.00264550i	$-0.000842090\pi$
$660$	241.712			0.366231
$661$	939.569			1.42144			0.710718	−	0.703477i	$-0.248370\pi$
$661$	939.569			1.42144			0.710718	+	0.703477i	$0.248370\pi$
$662$	144.531			0.218324
$663$	−144.944			−0.218619
$664$	1195.32			1.80018
$665$	−1050.45			−1.57962
$666$	146.386			0.219799
$667$		−	631.707i		−	0.947087i
$668$	1262.01			1.88924
$669$	414.635			0.619784
$670$	857.847			1.28037
$671$		−	210.080i		−	0.313085i
$672$		−	370.422i		−	0.551223i
$673$	−7.34355			−0.0109117			−0.00545583	−	0.999985i	$-0.501737\pi$
$673$	−7.34355			−0.0109117			−0.00545583	+	0.999985i	$0.501737\pi$
$674$	−421.730			−0.625712
$675$		−	622.146i		−	0.921697i
$676$	−2155.97			−3.18931
$677$	−979.507			−1.44683			−0.723417	−	0.690411i	$-0.757430\pi$
$677$	−979.507			−1.44683			−0.723417	+	0.690411i	$0.757430\pi$
$678$	−550.460			−0.811889
$679$		−	927.397i		−	1.36583i
$680$		−	244.956i		−	0.360230i
$681$			512.981i			0.753276i
$682$	140.955			0.206678
$683$	−1270.75			−1.86054			−0.930271	−	0.366873i	$-0.880428\pi$
$683$	−1270.75			−1.86054			−0.930271	+	0.366873i	$0.880428\pi$
$684$	804.684			1.17644
$685$		−	84.9373i		−	0.123996i
$686$	950.206			1.38514
$687$	451.046			0.656544
$688$		−	60.6096i		−	0.0880953i
$689$	1379.76			2.00256
$690$	729.624			1.05743
$691$	−338.858			−0.490387			−0.245194	−	0.969474i	$-0.578852\pi$
$691$	−338.858			−0.490387			−0.245194	+	0.969474i	$0.578852\pi$
$692$			946.559i			1.36786i
$693$		−	154.070i		−	0.222323i
$694$			1280.78i			1.84550i
$695$		−	23.8484i		−	0.0343143i
$696$	467.152			0.671195
$697$	17.5613			0.0251955
$698$		−	672.710i		−	0.963768i
$699$		−	365.291i		−	0.522591i
$700$			1286.84i			1.83834i
$701$	−1267.51			−1.80815			−0.904075	−	0.427374i	$-0.859439\pi$
$701$	−1267.51			−1.80815			−0.904075	+	0.427374i	$0.859439\pi$
$702$	−1819.21			−2.59147
$703$		−	135.739i		−	0.193085i
$704$	−319.427			−0.453731
$705$	396.323			0.562160
$706$			468.153i			0.663106i
$707$	386.657			0.546898
$708$	1171.14			1.65416
$709$		−	1394.66i		−	1.96709i	−0.180674	−	0.983543i	$-0.557828\pi$
$709$		−	1394.66i		−	1.96709i	0.180674	−	0.983543i	$-0.442172\pi$
$710$			1968.28i			2.77222i
$711$			527.591i			0.742040i
$712$	−377.812			−0.530635
$713$	265.938			0.372985
$714$	166.140			0.232689
$715$		−	490.695i		−	0.686287i
$716$	706.163			0.986261
$717$			263.266i			0.367178i
$718$	1588.74			2.21274
$719$		−	48.6971i		−	0.0677289i	−0.999426	−	0.0338645i	$-0.989219\pi$
$719$		−	48.6971i		−	0.0677289i	0.999426	−	0.0338645i	$-0.0107815\pi$
$720$		−	79.5413i		−	0.110474i
$721$	1551.74			2.15221
$722$		−	14.7214i		−	0.0203898i
$723$	113.168			0.156526
$724$	1611.81			2.22625
$725$	810.657			1.11815
$726$	−595.261			−0.819918
$727$			152.736i			0.210091i	0.994467	+	0.105046i	$0.0334989\pi$
$727$			152.736i			0.210091i	−0.994467	+	0.105046i	$0.966501\pi$
$728$	1505.43			2.06789
$729$	−271.473			−0.372391
$730$	−1118.55			−1.53226
$731$	−135.241			−0.185009
$732$	732.630			1.00086
$733$		−	19.7028i		−	0.0268797i	−0.999910	−	0.0134398i	$-0.995722\pi$
$733$		−	19.7028i		−	0.0268797i	0.999910	−	0.0134398i	$-0.00427816\pi$
$734$			211.596i			0.288278i
$735$	−134.637			−0.183180
$736$	−560.421			−0.761442
$737$	116.669			0.158303
$738$	90.8658			0.123124
$739$		−	1191.78i		−	1.61269i	−0.591444	−	0.806346i	$-0.701442\pi$
$739$		−	1191.78i		−	1.61269i	0.591444	−	0.806346i	$-0.298558\pi$
$740$	−334.016			−0.451373
$741$			695.421i			0.938490i
$742$	−1581.53			−2.13144
$743$	−968.931			−1.30408			−0.652040	−	0.758185i	$-0.726087\pi$
$743$	−968.931			−1.30408			−0.652040	+	0.758185i	$0.726087\pi$
$744$			196.663i			0.264332i
$745$			1056.66i			1.41833i
$746$	−570.393			−0.764602
$747$		−	866.077i		−	1.15941i
$748$		−	83.2705i		−	0.111324i
$749$	19.7026			0.0263052
$750$		−	8.14071i		−	0.0108543i
$751$	216.284			0.287994			0.143997	−	0.989578i	$-0.454004\pi$
$751$	216.284			0.287994			0.143997	+	0.989578i	$0.454004\pi$
$752$	61.1890			0.0813684
$753$	−19.1580			−0.0254422
$754$		−	2370.43i		−	3.14381i
$755$	960.247			1.27185
$756$	1303.34			1.72399
$757$	−386.161			−0.510120			−0.255060	−	0.966925i	$-0.582095\pi$
$757$	−386.161			−0.510120			−0.255060	+	0.966925i	$0.582095\pi$
$758$			1725.03i			2.27577i
$759$	99.2306			0.130739
$760$	−1175.26			−1.54640
$761$	1295.42			1.70227			0.851133	−	0.524950i	$-0.175916\pi$
$761$	1295.42			1.70227			0.851133	+	0.524950i	$0.175916\pi$
$762$	−1140.18			−1.49630
$763$	301.195			0.394751
$764$	−2232.86			−2.92260
$765$	−177.485			−0.232006
$766$		−	134.641i		−	0.175771i
$767$		−	2377.52i		−	3.09976i
$768$		−	492.511i		−	0.641291i
$769$			187.900i			0.244344i	0.992509	+	0.122172i	$0.0389859\pi$
$769$			187.900i			0.244344i	−0.992509	+	0.122172i	$0.961014\pi$
$770$			562.451i			0.730456i
$771$	−679.776			−0.881681
$772$	891.346			1.15459
$773$			1184.69i			1.53258i	0.642494	+	0.766290i	$0.277900\pi$
$773$			1184.69i			1.53258i	−0.642494	+	0.766290i	$0.722100\pi$
$774$	−699.766			−0.904091
$775$			341.273i			0.440353i
$776$		−	1037.59i		−	1.33710i
$777$		−	90.6351i		−	0.116648i
$778$	−2231.98			−2.86887
$779$		−	84.2566i		−	0.108160i
$780$	1711.24			2.19390
$781$			267.690i			0.342753i
$782$		−	251.357i		−	0.321429i
$783$		−	821.048i		−	1.04859i
$784$	−20.7869			−0.0265139
$785$		−	179.885i		−	0.229153i
$786$	426.234			0.542282
$787$	−1251.62			−1.59036			−0.795182	−	0.606371i	$-0.792625\pi$
$787$	−1251.62			−1.59036			−0.795182	+	0.606371i	$0.792625\pi$
$788$		−	1917.47i		−	2.43334i
$789$			217.430i			0.275577i
$790$		−	1926.03i		−	2.43801i
$791$		−	800.596i		−	1.01213i
$792$		−	172.377i		−	0.217648i
$793$		−	1487.30i		−	1.87553i
$794$	2162.36			2.72337
$795$	−719.239			−0.904703
$796$	5.42761			0.00681860
$797$	967.327			1.21371			0.606855	−	0.794813i	$-0.292431\pi$
$797$	967.327			1.21371			0.606855	+	0.794813i	$0.292431\pi$
$798$		−	797.115i		−	0.998891i
$799$		−	136.534i		−	0.170881i
$800$		−	719.177i		−	0.898971i
$801$			273.746i			0.341756i
$802$		−	2403.76i		−	2.99720i
$803$	−152.125			−0.189446
$804$			406.871i			0.506058i
$805$			1061.17i			1.31823i
$806$	997.913			1.23811
$807$			171.573i			0.212606i
$808$	432.600			0.535396
$809$		−	809.402i		−	1.00050i	−0.865882	−	0.500248i	$-0.833242\pi$
$809$		−	809.402i		−	1.00050i	0.865882	−	0.500248i	$-0.166758\pi$
$810$	−360.972			−0.445645
$811$	44.9112			0.0553775			0.0276888	−	0.999617i	$-0.491185\pi$
$811$	44.9112			0.0553775			0.0276888	+	0.999617i	$0.491185\pi$
$812$			1698.25i			2.09144i
$813$	231.538			0.284795
$814$	−72.6797			−0.0892871
$815$	−768.673			−0.943157
$816$	11.6653			0.0142957
$817$			648.868i			0.794208i
$818$		−	888.151i		−	1.08576i
$819$		−	1090.77i		−	1.33183i
$820$	−207.333			−0.252845
$821$			1246.97i			1.51885i	0.650598	+	0.759423i	$0.274518\pi$
$821$			1246.97i			1.51885i	−0.650598	+	0.759423i	$0.725482\pi$
$822$	64.4532			0.0784102
$823$	−366.592			−0.445433			−0.222717	−	0.974883i	$-0.571493\pi$
$823$	−366.592			−0.445433			−0.222717	+	0.974883i	$0.571493\pi$
$824$	1736.12			2.10695
$825$			127.341i			0.154352i
$826$			2725.19i			3.29926i
$827$			889.794i			1.07593i	0.842967	+	0.537965i	$0.180807\pi$
$827$			889.794i			1.07593i	−0.842967	+	0.537965i	$0.819193\pi$
$828$		−	812.899i		−	0.981762i
$829$	−947.718			−1.14321			−0.571603	−	0.820530i	$-0.693678\pi$
$829$	−947.718			−1.14321			−0.571603	+	0.820530i	$0.693678\pi$
$830$			3161.72i			3.80930i
$831$		−	430.376i		−	0.517902i
$832$	−2261.44			−2.71807
$833$			46.3828i			0.0556817i
$834$	18.0970			0.0216990
$835$			1335.51i			1.59941i
$836$	−399.520			−0.477895
$837$	345.648			0.412960
$838$		−	827.178i		−	0.987086i
$839$	403.241			0.480621			0.240310	−	0.970696i	$-0.422751\pi$
$839$	403.241			0.480621			0.240310	+	0.970696i	$0.422751\pi$
$840$	−784.745			−0.934220
$841$	228.826			0.272088
$842$	−2325.96			−2.76242
$843$	551.354			0.654038
$844$			1688.19i			2.00022i
$845$		−	2281.53i		−	2.70003i
$846$		−	706.457i		−	0.835055i
$847$		−	865.753i		−	1.02214i
$848$	−111.045			−0.130949
$849$	−631.774			−0.744139
$850$	322.562			0.379485
$851$	−137.124			−0.161133
$852$	−933.539			−1.09570
$853$	−306.636			−0.359480			−0.179740	−	0.983714i	$-0.557526\pi$
$853$	−306.636			−0.359480			−0.179740	+	0.983714i	$0.557526\pi$
$854$			1704.79i			1.99624i
$855$			851.546i			0.995960i
$856$	22.0437			0.0257519
$857$			443.716i			0.517755i	0.965910	+	0.258877i	$0.0833525\pi$
$857$			443.716i			0.517755i	−0.965910	+	0.258877i	$0.916647\pi$
$858$	372.355			0.433980
$859$	1589.16			1.85002			0.925008	−	0.379949i	$-0.124058\pi$
$859$	1589.16			1.85002			0.925008	+	0.379949i	$0.124058\pi$
$860$	1596.69			1.85662
$861$		−	56.2596i		−	0.0653422i
$862$	−1375.16			−1.59531
$863$			645.853i			0.748381i	0.927352	+	0.374190i	$0.122079\pi$
$863$			645.853i			0.748381i	−0.927352	+	0.374190i	$0.877921\pi$
$864$	−728.395			−0.843050
$865$	−1001.68			−1.15802
$866$	−1940.78			−2.24109
$867$			447.731i			0.516414i
$868$	−714.935			−0.823658
$869$		−	261.945i		−	0.301433i
$870$			1235.65i			1.42029i
$871$	825.980			0.948312
$872$	336.983			0.386449
$873$	−751.793			−0.861161
$874$	−1205.98			−1.37984
$875$	11.8399			0.0135314
$876$		−	530.519i		−	0.605616i
$877$			1284.87i			1.46507i	0.680727	+	0.732537i	$0.261664\pi$
$877$			1284.87i			1.46507i	−0.680727	+	0.732537i	$0.738336\pi$
$878$			1610.35i			1.83411i
$879$			49.3016i			0.0560883i
$880$			39.4917i			0.0448769i
$881$			308.886i			0.350608i	0.984514	+	0.175304i	$0.0560909\pi$
$881$			308.886i			0.350608i	−0.984514	+	0.175304i	$0.943909\pi$
$882$			239.994i			0.272103i
$883$	634.537			0.718615			0.359307	−	0.933219i	$-0.383013\pi$
$883$	634.537			0.718615			0.359307	+	0.933219i	$0.383013\pi$
$884$		−	589.528i		−	0.666887i
$885$			1239.35i			1.40039i
$886$			345.173i			0.389586i
$887$	−678.789			−0.765264			−0.382632	−	0.923901i	$-0.624982\pi$
$887$	−678.789			−0.765264			−0.382632	+	0.923901i	$0.624982\pi$
$888$		−	101.405i		−	0.114194i
$889$		−	1658.30i		−	1.86535i
$890$		−	999.343i		−	1.12286i
$891$	−49.0931			−0.0550989
$892$			1686.44i			1.89062i
$893$	−655.072			−0.733563
$894$	−801.824			−0.896895
$895$			747.287i			0.834957i
$896$	1688.29			1.88425
$897$	702.520			0.783189
$898$		−	1816.12i		−	2.02240i
$899$			450.379i			0.500978i
$900$	1043.18			1.15908
$901$			247.780i			0.275005i
$902$	−45.1142			−0.0500157
$903$			433.261i			0.479802i
$904$		−	895.723i		−	0.990844i
$905$			1705.67i			1.88472i
$906$			728.666i			0.804268i
$907$	−341.466			−0.376478			−0.188239	−	0.982123i	$-0.560278\pi$
$907$	−341.466			−0.376478			−0.188239	+	0.982123i	$0.560278\pi$
$908$	−2086.44			−2.29784
$909$		−	313.443i		−	0.344822i
$910$			3981.97i			4.37579i
$911$			462.521i			0.507707i	0.967243	+	0.253854i	$0.0816982\pi$
$911$			462.521i			0.507707i	−0.967243	+	0.253854i	$0.918302\pi$
$912$		−	55.9682i		−	0.0613687i
$913$			430.001i			0.470976i
$914$	1463.22			1.60090
$915$			775.295i			0.847317i
$916$			1834.53i			2.00276i
$917$			619.919i			0.676030i
$918$		−	326.696i		−	0.355878i
$919$	−278.904			−0.303487			−0.151743	−	0.988420i	$-0.548489\pi$
$919$	−278.904			−0.303487			−0.151743	+	0.988420i	$0.548489\pi$
$920$			1187.26i			1.29050i
$921$	89.8556			0.0975631
$922$	428.548			0.464802
$923$			1895.16i			2.05326i
$924$	−266.767			−0.288708
$925$		−	175.969i		−	0.190237i
$926$	268.590			0.290054
$927$		−	1257.92i		−	1.35698i
$928$		−	949.100i		−	1.02274i
$929$			67.0312i			0.0721542i	0.999349	+	0.0360771i	$0.0114862\pi$
$929$			67.0312i			0.0721542i	−0.999349	+	0.0360771i	$0.988514\pi$
$930$	−520.190			−0.559344
$931$	222.538			0.239031
$932$	1485.74			1.59414
$933$			554.147i			0.593942i
$934$		−	176.554i		−	0.189030i
$935$	88.1198			0.0942458
$936$		−	1220.37i		−	1.30382i
$937$			874.989i			0.933820i	0.884305	+	0.466910i	$0.154633\pi$
$937$			874.989i			0.933820i	−0.884305	+	0.466910i	$0.845367\pi$
$938$	−946.765			−1.00934
$939$			111.567i			0.118815i
$940$			1611.95i			1.71485i
$941$	1424.27			1.51358			0.756788	−	0.653661i	$-0.226768\pi$
$941$	1424.27			1.51358			0.756788	+	0.653661i	$0.226768\pi$
$942$	136.503			0.144907
$943$	−85.1167			−0.0902616
$944$			191.345i			0.202696i
$945$			1379.24i			1.45951i
$946$	347.429			0.367261
$947$	964.863			1.01886			0.509431	−	0.860511i	$-0.329856\pi$
$947$	964.863			1.01886			0.509431	+	0.860511i	$0.329856\pi$
$948$	913.503			0.963611
$949$	−1077.00			−1.13488
$950$		−	1547.61i		−	1.62906i
$951$		−	533.113i		−	0.560581i
$952$			270.347i			0.283978i
$953$	−1295.20			−1.35908			−0.679538	−	0.733640i	$-0.737820\pi$
$953$	−1295.20			−1.35908			−0.679538	+	0.733640i	$0.737820\pi$
$954$			1282.06i			1.34388i
$955$		−	2362.90i		−	2.47424i
$956$	−1070.78			−1.12006
$957$			168.052i			0.175603i
$958$			1852.51i			1.93372i
$959$			93.7414i			0.0977491i
$960$	1178.84			1.22795
$961$	771.398			0.802703
$962$	−514.549			−0.534874
$963$		−	15.9719i		−	0.0165855i
$964$			460.286i			0.477475i
$965$			943.254i			0.977465i
$966$	−805.252			−0.833594
$967$			654.324i			0.676654i	0.941029	+	0.338327i	$0.109861\pi$
$967$			654.324i			0.676654i	−0.941029	+	0.338327i	$0.890139\pi$
$968$		−	968.623i		−	1.00064i
$969$	−124.885			−0.128880
$970$	2744.51			2.82939
$971$		−	543.800i		−	0.560041i	−0.959994	−	0.280021i	$-0.909659\pi$
$971$		−	543.800i		−	0.560041i	0.959994	−	0.280021i	$-0.0903413\pi$
$972$		−	1677.53i		−	1.72586i
$973$			26.3204i			0.0270508i
$974$	2123.40			2.18009
$975$			901.530i			0.924646i
$976$			119.699i			0.122643i
$977$		−	267.088i		−	0.273376i	−0.990614	−	0.136688i	$-0.956354\pi$
$977$		−	267.088i		−	0.273376i	0.990614	−	0.136688i	$-0.0436457\pi$
$978$		−	583.294i		−	0.596415i
$979$		−	135.913i		−	0.138829i
$980$		−	547.606i		−	0.558782i
$981$		−	244.163i		−	0.248892i
$982$	−100.123			−0.101959
$983$			551.327i			0.560861i	0.959874	+	0.280431i	$0.0904773\pi$
$983$			551.327i			0.560861i	−0.959874	+	0.280431i	$0.909523\pi$
$984$		−	62.9444i		−	0.0639679i
$985$	2029.14			2.06004
$986$	425.686			0.431730
$987$	−437.403			−0.443164
$988$	−2828.47			−2.86282
$989$	655.492			0.662783
$990$	455.950			0.460556
$991$	154.911			0.156318			0.0781591	−	0.996941i	$-0.475096\pi$
$991$	154.911			0.156318			0.0781591	+	0.996941i	$0.475096\pi$
$992$	399.556			0.402778
$993$	−72.5419			−0.0730533
$994$		−	2172.29i		−	2.18541i
$995$			5.74369i			0.00577255i
$996$	−1499.58			−1.50560
$997$		−	1461.06i		−	1.46546i	−0.680522	−	0.732728i	$-0.738247\pi$
$997$		−	1461.06i		−	1.46546i	0.680522	−	0.732728i	$-0.261753\pi$
$998$		−	2208.99i		−	2.21342i
$999$	−178.225			−0.178403

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.3.b.b.546.10	✓	88
547.546	odd	2	inner	547.3.b.b.546.79	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.3.b.b.546.10	✓	88	1.1	even	1	trivial
547.3.b.b.546.79	yes	88	547.546	odd	2	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:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	547.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.26612i q^{2} +1.63931i q^{3} -6.66753 q^{4} -7.05581i q^{5} +5.35418 q^{6} +7.78718i q^{7} +8.71245i q^{8} +6.31266 q^{9} +O(q^{10})\)	\(q-3.26612i q^{2} +1.63931i q^{3} -6.66753 q^{4} -7.05581i q^{5} +5.35418 q^{6} +7.78718i q^{7} +8.71245i q^{8} +6.31266 q^{9} -23.0451 q^{10} -3.13419 q^{11} -10.9301i q^{12} -22.1891 q^{13} +25.4338 q^{14} +11.5667 q^{15} +1.78580 q^{16} -3.98475i q^{17} -20.6179i q^{18} -19.1182 q^{19} +47.0448i q^{20} -12.7656 q^{21} +10.2366i q^{22} +19.3134i q^{23} -14.2824 q^{24} -24.7845 q^{25} +72.4721i q^{26} +25.1022i q^{27} -51.9212i q^{28} -32.7082 q^{29} -37.7781i q^{30} -13.7696i q^{31} +29.0172i q^{32} -5.13791i q^{33} -13.0147 q^{34} +54.9449 q^{35} -42.0899 q^{36} +7.09996i q^{37} +62.4424i q^{38} -36.3747i q^{39} +61.4735 q^{40} +4.40713i q^{41} +41.6939i q^{42} -33.9397i q^{43} +20.8973 q^{44} -44.5410i q^{45} +63.0799 q^{46} +34.2642 q^{47} +2.92748i q^{48} -11.6401 q^{49} +80.9491i q^{50} +6.53223 q^{51} +147.946 q^{52} -62.1821 q^{53} +81.9867 q^{54} +22.1143i q^{55} -67.8454 q^{56} -31.3407i q^{57} +106.829i q^{58} +107.148i q^{59} -77.1210 q^{60} +67.0284i q^{61} -44.9732 q^{62} +49.1578i q^{63} +101.917 q^{64} +156.562i q^{65} -16.7810 q^{66} -37.2246 q^{67} +26.5684i q^{68} -31.6607 q^{69} -179.456i q^{70} -85.4096i q^{71} +54.9988i q^{72} +48.5373 q^{73} +23.1893 q^{74} -40.6295i q^{75} +127.471 q^{76} -24.4065i q^{77} -118.804 q^{78} +83.5765i q^{79} -12.6003i q^{80} +15.6637 q^{81} +14.3942 q^{82} -137.197i q^{83} +85.1149 q^{84} -28.1156 q^{85} -110.851 q^{86} -53.6189i q^{87} -27.3065i q^{88} +43.3646i q^{89} -145.476 q^{90} -172.790i q^{91} -128.773i q^{92} +22.5727 q^{93} -111.911i q^{94} +134.895i q^{95} -47.5682 q^{96} -119.093 q^{97} +38.0179i q^{98} -19.7851 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) 88 * q - 192 * q^4 - 306 * q^9	\( 88 q - 192 q^{4} - 306 q^{9} - 4 q^{10} - 32 q^{11} + 26 q^{13} - 26 q^{14} + 22 q^{15} + 236 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} - 544 q^{25} - 96 q^{29} + 26 q^{34} + 10 q^{35} + 364 q^{36} + 44 q^{40} + 124 q^{44} - 288 q^{46} - 310 q^{47} - 694 q^{49} + 86 q^{51} - 316 q^{52} + 24 q^{53} - 266 q^{54} + 158 q^{56} - 80 q^{60} + 40 q^{62} - 652 q^{64} + 528 q^{66} + 28 q^{67} + 16 q^{69} + 94 q^{73} - 614 q^{74} - 28 q^{76} - 98 q^{78} + 928 q^{81} - 772 q^{82} + 358 q^{84} + 74 q^{85} - 410 q^{86} - 214 q^{90} + 656 q^{93} - 724 q^{96} + 346 q^{97} + 50 q^{99}+O(q^{100}) \) 88 * q - 192 * q^4 - 306 * q^9 - 4 * q^10 - 32 * q^11 + 26 * q^13 - 26 * q^14 + 22 * q^15 + 236 * q^16 - 12 * q^19 - 16 * q^21 - 2 * q^24 - 544 * q^25 - 96 * q^29 + 26 * q^34 + 10 * q^35 + 364 * q^36 + 44 * q^40 + 124 * q^44 - 288 * q^46 - 310 * q^47 - 694 * q^49 + 86 * q^51 - 316 * q^52 + 24 * q^53 - 266 * q^54 + 158 * q^56 - 80 * q^60 + 40 * q^62 - 652 * q^64 + 528 * q^66 + 28 * q^67 + 16 * q^69 + 94 * q^73 - 614 * q^74 - 28 * q^76 - 98 * q^78 + 928 * q^81 - 772 * q^82 + 358 * q^84 + 74 * q^85 - 410 * q^86 - 214 * q^90 + 656 * q^93 - 724 * q^96 + 346 * q^97 + 50 * q^99

Introduction

L-functions

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