LMFDB - Embedded newform 547.3.b.b.546.16

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			546.16
Character	$\chi$	$=$	547.546
Dual form			547.3.b.b.546.73

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.83323i q^{2} -2.06299i q^{3} -4.02719 q^{4} -2.31552i q^{5} -5.84492 q^{6} -4.45862i q^{7} +0.0770388i q^{8} +4.74408 q^{9} +O(q^{10})$	\(q-2.83323i q^{2} -2.06299i q^{3} -4.02719 q^{4} -2.31552i q^{5} -5.84492 q^{6} -4.45862i q^{7} +0.0770388i q^{8} +4.74408 q^{9} -6.56039 q^{10} -8.35960 q^{11} +8.30805i q^{12} -0.489884 q^{13} -12.6323 q^{14} -4.77688 q^{15} -15.8905 q^{16} -3.86050i q^{17} -13.4411i q^{18} -13.8370 q^{19} +9.32502i q^{20} -9.19807 q^{21} +23.6847i q^{22} +0.638370i q^{23} +0.158930 q^{24} +19.6384 q^{25} +1.38795i q^{26} -28.3539i q^{27} +17.9557i q^{28} -29.0104 q^{29} +13.5340i q^{30} -43.8137i q^{31} +45.3296i q^{32} +17.2457i q^{33} -10.9377 q^{34} -10.3240 q^{35} -19.1053 q^{36} +17.8463i q^{37} +39.2033i q^{38} +1.01062i q^{39} +0.178385 q^{40} +51.2553i q^{41} +26.0602i q^{42} +62.5101i q^{43} +33.6657 q^{44} -10.9850i q^{45} +1.80865 q^{46} -2.65351 q^{47} +32.7819i q^{48} +29.1208 q^{49} -55.6401i q^{50} -7.96416 q^{51} +1.97286 q^{52} +30.1592 q^{53} -80.3330 q^{54} +19.3568i q^{55} +0.343486 q^{56} +28.5455i q^{57} +82.1932i q^{58} -58.4033i q^{59} +19.2374 q^{60} -72.2687i q^{61} -124.134 q^{62} -21.1520i q^{63} +64.8671 q^{64} +1.13433i q^{65} +48.8611 q^{66} +63.6871 q^{67} +15.5470i q^{68} +1.31695 q^{69} +29.2502i q^{70} +113.594i q^{71} +0.365479i q^{72} -24.4669 q^{73} +50.5627 q^{74} -40.5137i q^{75} +55.7241 q^{76} +37.2722i q^{77} +2.86333 q^{78} -115.503i q^{79} +36.7947i q^{80} -15.7969 q^{81} +145.218 q^{82} -111.272i q^{83} +37.0424 q^{84} -8.93905 q^{85} +177.106 q^{86} +59.8482i q^{87} -0.644013i q^{88} +75.9989i q^{89} -31.1230 q^{90} +2.18420i q^{91} -2.57084i q^{92} -90.3871 q^{93} +7.51799i q^{94} +32.0397i q^{95} +93.5144 q^{96} +22.5264 q^{97} -82.5058i q^{98} -39.6586 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	2.83323i		−	1.41661i	−0.705904	−	0.708307i	$-0.749459\pi$
$2$		−	2.83323i		−	1.41661i	0.705904	−	0.708307i	$-0.250541\pi$
$3$		−	2.06299i		−	0.687662i	−0.939031	−	0.343831i	$-0.888275\pi$
$3$		−	2.06299i		−	0.687662i	0.939031	−	0.343831i	$-0.111725\pi$
$4$	−4.02719			−1.00680
$5$		−	2.31552i		−	0.463103i	−0.972823	−	0.231552i	$-0.925620\pi$
$5$		−	2.31552i		−	0.463103i	0.972823	−	0.231552i	$-0.0743802\pi$
$6$	−5.84492			−0.974153
$7$		−	4.45862i		−	0.636945i	−0.947932	−	0.318473i	$-0.896830\pi$
$7$		−	4.45862i		−	0.636945i	0.947932	−	0.318473i	$-0.103170\pi$
$8$			0.0770388i			0.00962985i
$9$	4.74408			0.527120
$10$	−6.56039			−0.656039
$11$	−8.35960			−0.759963			−0.379982	−	0.924994i	$-0.624070\pi$
$11$	−8.35960			−0.759963			−0.379982	+	0.924994i	$0.624070\pi$
$12$			8.30805i			0.692337i
$13$	−0.489884			−0.0376834			−0.0188417	−	0.999822i	$-0.505998\pi$
$13$	−0.489884			−0.0376834			−0.0188417	+	0.999822i	$0.505998\pi$
$14$	−12.6323			−0.902306
$15$	−4.77688			−0.318459
$16$	−15.8905			−0.993156
$17$		−	3.86050i		−	0.227088i	−0.993533	−	0.113544i	$-0.963780\pi$
$17$		−	3.86050i		−	0.227088i	0.993533	−	0.113544i	$-0.0362203\pi$
$18$		−	13.4411i		−	0.746726i
$19$	−13.8370			−0.728262			−0.364131	−	0.931348i	$-0.618634\pi$
$19$	−13.8370			−0.728262			−0.364131	+	0.931348i	$0.618634\pi$
$20$			9.32502i			0.466251i
$21$	−9.19807			−0.438003
$22$			23.6847i			1.07658i
$23$			0.638370i			0.0277552i	0.999904	+	0.0138776i	$0.00441752\pi$
$23$			0.638370i			0.0277552i	−0.999904	+	0.0138776i	$0.995582\pi$
$24$	0.158930			0.00662209
$25$	19.6384			0.785536
$26$			1.38795i			0.0533828i
$27$		−	28.3539i		−	1.05014i
$28$			17.9557i			0.641275i
$29$	−29.0104			−1.00036			−0.500180	−	0.865921i	$-0.666733\pi$
$29$	−29.0104			−1.00036			−0.500180	+	0.865921i	$0.666733\pi$
$30$			13.5340i			0.451133i
$31$		−	43.8137i		−	1.41334i	−0.707541	−	0.706672i	$-0.750195\pi$
$31$		−	43.8137i		−	1.41334i	0.707541	−	0.706672i	$-0.249805\pi$
$32$			45.3296i			1.41655i
$33$			17.2457i			0.522598i
$34$	−10.9377			−0.321697
$35$	−10.3240			−0.294971
$36$	−19.1053			−0.530704
$37$			17.8463i			0.482333i	0.970484	+	0.241166i	$0.0775300\pi$
$37$			17.8463i			0.482333i	−0.970484	+	0.241166i	$0.922470\pi$
$38$			39.2033i			1.03167i
$39$			1.01062i			0.0259134i
$40$	0.178385			0.00445961
$41$			51.2553i			1.25013i	0.780573	+	0.625065i	$0.214927\pi$
$41$			51.2553i			1.25013i	−0.780573	+	0.625065i	$0.785073\pi$
$42$			26.0602i			0.620482i
$43$			62.5101i			1.45372i	0.686784	+	0.726862i	$0.259022\pi$
$43$			62.5101i			1.45372i	−0.686784	+	0.726862i	$0.740978\pi$
$44$	33.6657			0.765129
$45$		−	10.9850i		−	0.244111i
$46$	1.80865			0.0393185
$47$	−2.65351			−0.0564576			−0.0282288	−	0.999601i	$-0.508987\pi$
$47$	−2.65351			−0.0564576			−0.0282288	+	0.999601i	$0.508987\pi$
$48$			32.7819i			0.682956i
$49$	29.1208			0.594301
$50$		−	55.6401i		−	1.11280i
$51$	−7.96416			−0.156160
$52$	1.97286			0.0379395
$53$	30.1592			0.569042			0.284521	−	0.958670i	$-0.408165\pi$
$53$	30.1592			0.569042			0.284521	+	0.958670i	$0.408165\pi$
$54$	−80.3330			−1.48765
$55$			19.3568i			0.351941i
$56$	0.343486			0.00613369
$57$			28.5455i			0.500798i
$58$			82.1932i			1.41712i
$59$		−	58.4033i		−	0.989887i	−0.868925	−	0.494944i	$-0.835189\pi$
$59$		−	58.4033i		−	0.989887i	0.868925	−	0.494944i	$-0.164811\pi$
$60$	19.2374			0.320623
$61$		−	72.2687i		−	1.18473i	−0.805669	−	0.592366i	$-0.798194\pi$
$61$		−	72.2687i		−	1.18473i	0.805669	−	0.592366i	$-0.201806\pi$
$62$	−124.134			−2.00217
$63$		−	21.1520i		−	0.335747i
$64$	64.8671			1.01355
$65$			1.13433i			0.0174513i
$66$	48.8611			0.740320
$67$	63.6871			0.950554			0.475277	−	0.879836i	$-0.342348\pi$
$67$	63.6871			0.950554			0.475277	+	0.879836i	$0.342348\pi$
$68$			15.5470i			0.228632i
$69$	1.31695			0.0190862
$70$			29.2502i			0.417861i
$71$			113.594i			1.59992i	0.600056	+	0.799958i	$0.295145\pi$
$71$			113.594i			1.59992i	−0.600056	+	0.799958i	$0.704855\pi$
$72$			0.365479i			0.00507609i
$73$	−24.4669			−0.335162			−0.167581	−	0.985858i	$-0.553596\pi$
$73$	−24.4669			−0.335162			−0.167581	+	0.985858i	$0.553596\pi$
$74$	50.5627			0.683280
$75$		−	40.5137i		−	0.540183i
$76$	55.7241			0.733212
$77$			37.2722i			0.484055i
$78$	2.86333			0.0367094
$79$		−	115.503i		−	1.46206i	−0.682345	−	0.731031i	$-0.739040\pi$
$79$		−	115.503i		−	1.46206i	0.682345	−	0.731031i	$-0.260960\pi$
$80$			36.7947i			0.459934i
$81$	−15.7969			−0.195024
$82$	145.218			1.77095
$83$		−	111.272i		−	1.34062i	−0.742079	−	0.670312i	$-0.766160\pi$
$83$		−	111.272i		−	1.34062i	0.742079	−	0.670312i	$-0.233840\pi$
$84$	37.0424			0.440981
$85$	−8.93905			−0.105165
$86$	177.106			2.05937
$87$			59.8482i			0.687910i
$88$		−	0.644013i		−	0.00731833i
$89$			75.9989i			0.853920i	0.904271	+	0.426960i	$0.140415\pi$
$89$			75.9989i			0.853920i	−0.904271	+	0.426960i	$0.859585\pi$
$90$	−31.1230			−0.345811
$91$			2.18420i			0.0240022i
$92$		−	2.57084i		−	0.0279439i
$93$	−90.3871			−0.971904
$94$			7.51799i			0.0799786i
$95$			32.0397i			0.337260i
$96$	93.5144			0.974108
$97$	22.5264			0.232231			0.116115	−	0.993236i	$-0.462956\pi$
$97$	22.5264			0.232231			0.116115	+	0.993236i	$0.462956\pi$
$98$		−	82.5058i		−	0.841896i
$99$	−39.6586			−0.400592
$100$	−79.0875			−0.790875
$101$		−	18.7470i		−	0.185614i	−0.995684	−	0.0928070i	$-0.970416\pi$
$101$		−	18.7470i		−	0.185614i	0.995684	−	0.0928070i	$-0.0295840\pi$
$102$			22.5643i			0.221219i
$103$		−	35.9030i		−	0.348573i	−0.984695	−	0.174286i	$-0.944238\pi$
$103$		−	35.9030i		−	0.348573i	0.984695	−	0.174286i	$-0.0557618\pi$
$104$		−	0.0377401i		−	0.000362885i
$105$			21.2983i			0.202841i
$106$		−	85.4480i		−	0.806113i
$107$		−	54.8857i		−	0.512951i	−0.966551	−	0.256475i	$-0.917439\pi$
$107$		−	54.8857i		−	0.512951i	0.966551	−	0.256475i	$-0.0825613\pi$
$108$			114.186i			1.05728i
$109$		−	66.6023i		−	0.611031i	−0.952187	−	0.305515i	$-0.901171\pi$
$109$		−	66.6023i		−	0.611031i	0.952187	−	0.305515i	$-0.0988287\pi$
$110$	54.8422			0.498565
$111$	36.8167			0.331682
$112$			70.8496i			0.632586i
$113$	−48.0728			−0.425423			−0.212712	−	0.977115i	$-0.568229\pi$
$113$	−48.0728			−0.425423			−0.212712	+	0.977115i	$0.568229\pi$
$114$	80.8760			0.709438
$115$	1.47816			0.0128535
$116$	116.831			1.00716
$117$	−2.32405			−0.0198637
$118$	−165.470			−1.40229
$119$	−17.2125			−0.144643
$120$		−	0.368005i		−	0.00306671i
$121$	−51.1172			−0.422456
$122$	−204.754			−1.67831
$123$	105.739			0.859667
$124$			176.446i			1.42295i
$125$		−	103.361i		−	0.826887i
$126$	−59.9286			−0.475624
$127$	−110.702			−0.871667			−0.435833	−	0.900027i	$-0.643546\pi$
$127$	−110.702			−0.871667			−0.435833	+	0.900027i	$0.643546\pi$
$128$		−	2.46519i		−	0.0192593i
$129$	128.958			0.999671
$130$	3.21383			0.0247217
$131$	−99.3685			−0.758538			−0.379269	−	0.925286i	$-0.623825\pi$
$131$	−99.3685			−0.758538			−0.379269	+	0.925286i	$0.623825\pi$
$132$		−	69.4519i		−	0.526151i
$133$			61.6937i			0.463863i
$134$		−	180.440i		−	1.34657i
$135$	−65.6538			−0.486325
$136$	0.297408			0.00218683
$137$	−92.9827			−0.678706			−0.339353	−	0.940659i	$-0.610208\pi$
$137$	−92.9827			−0.678706			−0.339353	+	0.940659i	$0.610208\pi$
$138$		−	3.73122i		−	0.0270378i
$139$	263.879			1.89841			0.949205	−	0.314657i	$-0.101890\pi$
$139$	263.879			1.89841			0.949205	+	0.314657i	$0.101890\pi$
$140$	41.5767			0.296976
$141$			5.47415i			0.0388237i
$142$	321.838			2.26646
$143$	4.09523			0.0286380
$144$	−75.3858			−0.523513
$145$			67.1741i			0.463270i
$146$			69.3202i			0.474796i
$147$		−	60.0757i		−	0.408679i
$148$		−	71.8705i		−	0.485612i
$149$	−4.54270			−0.0304879			−0.0152440	−	0.999884i	$-0.504852\pi$
$149$	−4.54270			−0.0304879			−0.0152440	+	0.999884i	$0.504852\pi$
$150$	−114.785			−0.765232
$151$		−	191.455i		−	1.26791i	−0.773368	−	0.633957i	$-0.781429\pi$
$151$		−	191.455i		−	1.26791i	0.773368	−	0.633957i	$-0.218571\pi$
$152$		−	1.06598i		−	0.00701305i
$153$		−	18.3145i		−	0.119703i
$154$	105.601			0.685719
$155$	−101.451			−0.654524
$156$		−	4.06998i		−	0.0260896i
$157$	−25.7119			−0.163770			−0.0818851	−	0.996642i	$-0.526094\pi$
$157$	−25.7119			−0.163770			−0.0818851	+	0.996642i	$0.526094\pi$
$158$	−327.246			−2.07118
$159$		−	62.2181i		−	0.391309i
$160$	104.961			0.656008
$161$	2.84625			0.0176786
$162$			44.7564i			0.276274i
$163$			15.0068i			0.0920661i	0.998940	+	0.0460331i	$0.0146580\pi$
$163$			15.0068i			0.0920661i	−0.998940	+	0.0460331i	$0.985342\pi$
$164$		−	206.415i		−	1.25863i
$165$	39.9328			0.242017
$166$	−315.259			−1.89915
$167$	−255.994			−1.53290			−0.766451	−	0.642303i	$-0.777979\pi$
$167$	−255.994			−1.53290			−0.766451	+	0.642303i	$0.777979\pi$
$168$		−	0.708608i		−	0.00421791i
$169$	−168.760			−0.998580
$170$			25.3264i			0.148979i
$171$	−65.6437			−0.383882
$172$		−	251.740i		−	1.46361i
$173$		−	100.934i		−	0.583435i	−0.956505	−	0.291717i	$-0.905773\pi$
$173$		−	100.934i		−	0.583435i	0.956505	−	0.291717i	$-0.0942266\pi$
$174$	169.564			0.974504
$175$		−	87.5600i		−	0.500343i
$176$	132.838			0.754762
$177$	−120.485			−0.680708
$178$	215.322			1.20968
$179$	−147.499			−0.824016			−0.412008	−	0.911180i	$-0.635172\pi$
$179$	−147.499			−0.824016			−0.412008	+	0.911180i	$0.635172\pi$
$180$			44.2387i			0.245770i
$181$	292.676			1.61699			0.808497	−	0.588500i	$-0.200281\pi$
$181$	292.676			1.61699			0.808497	+	0.588500i	$0.200281\pi$
$182$	6.18835			0.0340019
$183$	−149.089			−0.814696
$184$	−0.0491793			−0.000267279
$185$	41.3234			0.223370
$186$			256.087i			1.37681i
$187$			32.2722i			0.172579i
$188$	10.6862			0.0568414
$189$	−126.419			−0.668884
$190$	90.7759			0.477768
$191$	−276.732			−1.44886			−0.724428	−	0.689350i	$-0.757896\pi$
$191$	−276.732			−1.44886			−0.724428	+	0.689350i	$0.757896\pi$
$192$		−	133.820i		−	0.696980i
$193$	293.684			1.52168			0.760839	−	0.648940i	$-0.224787\pi$
$193$	293.684			1.52168			0.760839	+	0.648940i	$0.224787\pi$
$194$		−	63.8224i		−	0.328981i
$195$	2.34012			0.0120006
$196$	−117.275			−0.598341
$197$		−	326.323i		−	1.65646i	−0.560388	−	0.828230i	$-0.689348\pi$
$197$		−	326.323i		−	1.65646i	0.560388	−	0.828230i	$-0.310652\pi$
$198$			112.362i			0.567485i
$199$	−25.2411			−0.126840			−0.0634199	−	0.997987i	$-0.520201\pi$
$199$	−25.2411			−0.126840			−0.0634199	+	0.997987i	$0.520201\pi$
$200$			1.51292i			0.00756459i
$201$		−	131.386i		−	0.653661i
$202$	−53.1146			−0.262944
$203$			129.346i			0.637174i
$204$	32.0732			0.157222
$205$	118.682			0.578939
$206$	−101.721			−0.493793
$207$			3.02848i			0.0146303i
$208$	7.78450			0.0374255
$209$	115.671			0.553452
$210$	60.3429			0.287347
$211$		−	361.683i		−	1.71414i	−0.515202	−	0.857069i	$-0.672283\pi$
$211$		−	361.683i		−	1.71414i	0.515202	−	0.857069i	$-0.327717\pi$
$212$	−121.457			−0.572910
$213$	234.343			1.10020
$214$	−155.504			−0.726654
$215$	144.743			0.673224
$216$	2.18435			0.0101127
$217$	−195.348			−0.900223
$218$	−188.700			−0.865595
$219$			50.4748i			0.230479i
$220$		−	77.9534i		−	0.354334i
$221$			1.89120i			0.00855745i
$222$		−	104.310i		−	0.469866i
$223$			323.030i			1.44857i	0.689502	+	0.724283i	$0.257829\pi$
$223$			323.030i			1.44857i	−0.689502	+	0.724283i	$0.742171\pi$
$224$	202.107			0.902264
$225$	93.1661			0.414072
$226$			136.201i			0.602661i
$227$	−256.744			−1.13103			−0.565515	−	0.824738i	$-0.691322\pi$
$227$	−256.744			−1.13103			−0.565515	+	0.824738i	$0.691322\pi$
$228$		−	114.958i		−	0.504203i
$229$		−	95.3686i		−	0.416457i	−0.978080	−	0.208228i	$-0.933230\pi$
$229$		−	95.3686i		−	0.416457i	0.978080	−	0.208228i	$-0.0667697\pi$
$230$		−	4.18796i		−	0.0182085i
$231$	76.8921			0.332866
$232$		−	2.23493i		−	0.00963332i
$233$	95.3362			0.409168			0.204584	−	0.978849i	$-0.434416\pi$
$233$	95.3362			0.409168			0.204584	+	0.978849i	$0.434416\pi$
$234$			6.58457i			0.0281392i
$235$			6.14423i			0.0261457i
$236$			235.201i			0.996616i
$237$	−238.281			−1.00540
$238$			48.7669i			0.204903i
$239$	247.795			1.03680			0.518399	−	0.855139i	$-0.326528\pi$
$239$	247.795			1.03680			0.518399	+	0.855139i	$0.326528\pi$
$240$	75.9070			0.316279
$241$			180.141i			0.747474i	0.927535	+	0.373737i	$0.121924\pi$
$241$			180.141i			0.747474i	−0.927535	+	0.373737i	$0.878076\pi$
$242$			144.827i			0.598457i
$243$		−	222.596i		−	0.916033i
$244$			291.040i			1.19279i
$245$		−	67.4296i		−	0.275223i
$246$		−	299.583i		−	1.21782i
$247$	6.77851			0.0274434
$248$	3.37536			0.0136103
$249$	−229.552			−0.921897
$250$	−292.845			−1.17138
$251$		−	453.717i		−	1.80764i	−0.427916	−	0.903818i	$-0.640752\pi$
$251$		−	453.717i		−	1.80764i	0.427916	−	0.903818i	$-0.359248\pi$
$252$			85.1833i			0.338029i
$253$		−	5.33652i		−	0.0210930i
$254$			313.643i			1.23482i
$255$			18.4411i			0.0723182i
$256$	252.484			0.986266
$257$			90.3059i			0.351385i	0.984445	+	0.175692i	$0.0562164\pi$
$257$			90.3059i			0.351385i	−0.984445	+	0.175692i	$0.943784\pi$
$258$		−	365.367i		−	1.41615i
$259$	79.5698			0.307219
$260$		−	4.56818i		−	0.0175699i
$261$	−137.628			−0.527310
$262$			281.534i			1.07456i
$263$	389.443			1.48077			0.740386	−	0.672182i	$-0.234643\pi$
$263$	389.443			1.48077			0.740386	+	0.672182i	$0.234643\pi$
$264$	−1.32859			−0.00503254
$265$		−	69.8341i		−	0.263525i
$266$	174.793			0.657115
$267$	156.785			0.587209
$268$	−256.480			−0.957016
$269$	83.0941			0.308900			0.154450	−	0.988001i	$-0.450639\pi$
$269$	83.0941			0.308900			0.154450	+	0.988001i	$0.450639\pi$
$270$			186.012i			0.688935i
$271$			124.589i			0.459739i	0.973222	+	0.229869i	$0.0738299\pi$
$271$			124.589i			0.459739i	−0.973222	+	0.229869i	$0.926170\pi$
$272$			61.3453i			0.225534i
$273$	4.50598			0.0165054
$274$			263.441i			0.961465i
$275$	−164.169			−0.596978
$276$	−5.30361			−0.0192160
$277$	275.059			0.992992			0.496496	−	0.868039i	$-0.334620\pi$
$277$	275.059			0.992992			0.496496	+	0.868039i	$0.334620\pi$
$278$		−	747.630i		−	2.68932i
$279$		−	207.856i		−	0.745003i
$280$		−	0.795348i		−	0.00284053i
$281$		−	330.640i		−	1.17666i	−0.808622	−	0.588328i	$-0.799786\pi$
$281$		−	330.640i		−	1.17666i	0.808622	−	0.588328i	$-0.200214\pi$
$282$	15.5095			0.0549983
$283$			119.162i			0.421068i	0.977587	+	0.210534i	$0.0675203\pi$
$283$			119.162i			0.421068i	−0.977587	+	0.210534i	$0.932480\pi$
$284$		−	457.465i		−	1.61079i
$285$	66.0975			0.231921
$286$		−	11.6027i		−	0.0405690i
$287$	228.528			0.796264
$288$			215.047i			0.746692i
$289$	274.097			0.948431
$290$	190.320			0.656275
$291$		−	46.4716i		−	0.159696i
$292$	98.5327			0.337441
$293$	−25.3824			−0.0866294			−0.0433147	−	0.999061i	$-0.513792\pi$
$293$	−25.3824			−0.0866294			−0.0433147	+	0.999061i	$0.513792\pi$
$294$	−170.208			−0.578940
$295$	−135.234			−0.458420
$296$	−1.37486			−0.00464479
$297$			237.027i			0.798070i
$298$			12.8705i			0.0431897i
$299$		−	0.312727i		−	0.00104591i
$300$			163.157i			0.543855i
$301$	278.709			0.925942
$302$	−542.436			−1.79615
$303$	−38.6749			−0.127640
$304$	219.876			0.723277
$305$	−167.339			−0.548653
$306$	−51.8893			−0.169573
$307$		−	145.718i		−	0.474651i	−0.971430	−	0.237326i	$-0.923729\pi$
$307$		−	145.718i		−	0.474651i	0.971430	−	0.237326i	$-0.0762708\pi$
$308$		−	150.102i		−	0.487345i
$309$	−74.0674			−0.239700
$310$			287.435i			0.927209i
$311$	563.711			1.81258			0.906288	−	0.422661i	$-0.138904\pi$
$311$	563.711			1.81258			0.906288	+	0.422661i	$0.138904\pi$
$312$	−0.0778573			−0.000249543
$313$	190.491			0.608597			0.304298	−	0.952577i	$-0.401578\pi$
$313$	190.491			0.608597			0.304298	+	0.952577i	$0.401578\pi$
$314$			72.8477i			0.231999i
$315$	−48.9779			−0.155485
$316$			465.152i			1.47200i
$317$	−245.383			−0.774079			−0.387040	−	0.922063i	$-0.626502\pi$
$317$	−245.383			−0.774079			−0.387040	+	0.922063i	$0.626502\pi$
$318$	−176.278			−0.554334
$319$	242.516			0.760237
$320$		−	150.201i		−	0.469378i
$321$	−113.229			−0.352737
$322$		−	8.06407i		−	0.0250437i
$323$			53.4176i			0.165380i
$324$	63.6173			0.196350
$325$	−9.62053			−0.0296016
$326$	42.5176			0.130422
$327$	−137.400			−0.420183
$328$	−3.94865			−0.0120386
$329$			11.8310i			0.0359604i
$330$		−	113.139i		−	0.342845i
$331$			137.765i			0.416210i	0.978107	+	0.208105i	$0.0667296\pi$
$331$			137.765i			0.416210i	−0.978107	+	0.208105i	$0.933270\pi$
$332$			448.113i			1.34974i
$333$			84.6644i			0.254247i
$334$			725.291i			2.17153i
$335$		−	147.469i		−	0.440205i
$336$	146.162			0.435006
$337$			313.962i			0.931638i	0.884880	+	0.465819i	$0.154240\pi$
$337$			313.962i			0.931638i	−0.884880	+	0.465819i	$0.845760\pi$
$338$			478.136i			1.41460i
$339$			99.1736i			0.292548i
$340$	35.9993			0.105880
$341$			366.265i			1.07409i
$342$			185.984i			0.543812i
$343$		−	348.310i		−	1.01548i
$344$	−4.81571			−0.0139991
$345$		−	3.04942i		−	0.00883889i
$346$	−285.970			−0.826502
$347$	−32.3147			−0.0931259			−0.0465629	−	0.998915i	$-0.514827\pi$
$347$	−32.3147			−0.0931259			−0.0465629	+	0.998915i	$0.514827\pi$
$348$		−	241.020i		−	0.692586i
$349$	−26.1264			−0.0748608			−0.0374304	−	0.999299i	$-0.511917\pi$
$349$	−26.1264			−0.0748608			−0.0374304	+	0.999299i	$0.511917\pi$
$350$	−248.078			−0.708793
$351$			13.8901i			0.0395729i
$352$		−	378.937i		−	1.07653i
$353$	164.800			0.466855			0.233428	−	0.972374i	$-0.425006\pi$
$353$	164.800			0.466855			0.233428	+	0.972374i	$0.425006\pi$
$354$			341.363i			0.964302i
$355$	263.029			0.740926
$356$		−	306.062i		−	0.859725i
$357$			35.5091i			0.0994654i
$358$			417.898i			1.16731i
$359$			586.388i			1.63339i	0.577067	+	0.816697i	$0.304197\pi$
$359$			586.388i			1.63339i	−0.577067	+	0.816697i	$0.695803\pi$
$360$	0.846271			0.00235075
$361$	−169.538			−0.469635
$362$		−	829.218i		−	2.29066i
$363$			105.454i			0.290507i
$364$		−	8.79620i		−	0.0241654i
$365$			56.6534i			0.155215i
$366$			422.405i			1.15411i
$367$	−313.742			−0.854882			−0.427441	−	0.904043i	$-0.640585\pi$
$367$	−313.742			−0.854882			−0.427441	+	0.904043i	$0.640585\pi$
$368$		−	10.1440i		−	0.0275653i
$369$			243.159i			0.658969i
$370$		−	117.079i		−	0.316429i
$371$		−	134.468i		−	0.362448i
$372$	364.006			0.978511
$373$			296.410i			0.794666i	0.917675	+	0.397333i	$0.130064\pi$
$373$			296.410i			0.794666i	−0.917675	+	0.397333i	$0.869936\pi$
$374$	91.4346			0.244478
$375$	−213.232			−0.568619
$376$		−	0.204423i		−	0.000543678i
$377$	14.2117			0.0376969
$378$			358.174i			0.947550i
$379$	620.552			1.63734			0.818670	−	0.574264i	$-0.194712\pi$
$379$	620.552			1.63734			0.818670	+	0.574264i	$0.194712\pi$
$380$		−	129.030i		−	0.339553i
$381$			228.376i			0.599413i
$382$			784.044i			2.05247i
$383$	557.014			1.45435			0.727173	−	0.686455i	$-0.240834\pi$
$383$	557.014			1.45435			0.727173	+	0.686455i	$0.240834\pi$
$384$	−5.08565			−0.0132439
$385$	86.3044			0.224167
$386$		−	832.074i		−	2.15563i
$387$			296.553i			0.766287i
$388$	−90.7180			−0.233809
$389$			259.706i			0.667624i	0.942640	+	0.333812i	$0.108335\pi$
$389$			259.706i			0.667624i	−0.942640	+	0.333812i	$0.891665\pi$
$390$		−	6.63009i		−	0.0170002i
$391$	2.46443			0.00630289
$392$			2.24343i			0.00572303i
$393$			204.996i			0.521618i
$394$	−924.547			−2.34657
$395$	−267.449			−0.677085
$396$	159.713			0.403315
$397$			6.56252i			0.0165303i	0.999966	+	0.00826514i	$0.00263091\pi$
$397$			6.56252i			0.0165303i	−0.999966	+	0.00826514i	$0.997369\pi$
$398$			71.5139i			0.179683i
$399$	127.273			0.318981
$400$	−312.064			−0.780159
$401$	12.2942			0.0306588			0.0153294	−	0.999882i	$-0.495120\pi$
$401$	12.2942			0.0306588			0.0153294	+	0.999882i	$0.495120\pi$
$402$	−372.246			−0.925985
$403$			21.4636i			0.0532596i
$404$			75.4978i			0.186876i
$405$			36.5781i			0.0903162i
$406$	366.468			0.902631
$407$		−	149.188i		−	0.366555i
$408$		−	0.613550i		−	0.00150380i
$409$	496.490			1.21391			0.606956	−	0.794735i	$-0.292390\pi$
$409$	496.490			1.21391			0.606956	+	0.794735i	$0.292390\pi$
$410$		−	336.255i		−	0.820133i
$411$			191.822i			0.466721i
$412$			144.588i			0.350942i
$413$	−260.398			−0.630504
$414$	8.58038			0.0207256
$415$	−257.652			−0.620848
$416$		−	22.2062i		−	0.0533804i
$417$		−	544.379i		−	1.30547i
$418$		−	327.724i		−	0.784028i
$419$	351.030			0.837781			0.418891	−	0.908037i	$-0.362419\pi$
$419$	351.030			0.837781			0.418891	+	0.908037i	$0.362419\pi$
$420$		−	85.7722i		−	0.204220i
$421$			88.9933i			0.211386i	0.994399	+	0.105693i	$0.0337060\pi$
$421$			88.9933i			0.211386i	−0.994399	+	0.105693i	$0.966294\pi$
$422$	−1024.73			−2.42827
$423$	−12.5884			−0.0297599
$424$			2.32343i			0.00547979i
$425$		−	75.8140i		−	0.178386i
$426$		−	663.948i		−	1.55856i
$427$	−322.218			−0.754610
$428$			221.035i			0.516438i
$429$		−	8.44841i		−	0.0196933i
$430$		−	410.091i		−	0.953699i
$431$		−	217.389i		−	0.504383i	−0.967677	−	0.252192i	$-0.918849\pi$
$431$		−	217.389i		−	0.504383i	0.967677	−	0.252192i	$-0.0811514\pi$
$432$			450.557i			1.04296i
$433$		−	43.7445i		−	0.101027i	−0.998723	−	0.0505133i	$-0.983914\pi$
$433$		−	43.7445i		−	0.101027i	0.998723	−	0.0505133i	$-0.0160857\pi$
$434$			553.467i			1.27527i
$435$	138.579			0.318573
$436$			268.220i			0.615184i
$437$		−	8.83311i		−	0.0202131i
$438$	143.007			0.326499
$439$	−293.658			−0.668925			−0.334463	−	0.942409i	$-0.608555\pi$
$439$	−293.658			−0.668925			−0.334463	+	0.942409i	$0.608555\pi$
$440$	−1.49122			−0.00338914
$441$	138.151			0.313268
$442$	5.35819			0.0121226
$443$	−319.964			−0.722267			−0.361134	−	0.932514i	$-0.617610\pi$
$443$	−319.964			−0.722267			−0.361134	+	0.932514i	$0.617610\pi$
$444$	−148.268			−0.333937
$445$	175.977			0.395453
$446$	915.219			2.05206
$447$			9.37154i			0.0209654i
$448$		−	289.218i		−	0.645575i
$449$	−727.267			−1.61975			−0.809874	−	0.586604i	$-0.800464\pi$
$449$	−727.267			−1.61975			−0.809874	+	0.586604i	$0.800464\pi$
$450$		−	263.961i		−	0.586580i
$451$		−	428.474i		−	0.950052i
$452$	193.598			0.428315
$453$	−394.970			−0.871897
$454$			727.414i			1.60223i
$455$	5.05756			0.0111155
$456$	−2.19911			−0.00482261
$457$			521.595i			1.14135i	0.821177	+	0.570673i	$0.193318\pi$
$457$			521.595i			1.14135i	−0.821177	+	0.570673i	$0.806682\pi$
$458$	−270.201			−0.589959
$459$	−109.460			−0.238475
$460$	−5.95282			−0.0129409
$461$		−	219.089i		−	0.475247i	−0.971357	−	0.237623i	$-0.923632\pi$
$461$		−	219.089i		−	0.475247i	0.971357	−	0.237623i	$-0.0763684\pi$
$462$		−	217.853i		−	0.471543i
$463$		−	336.885i		−	0.727613i	−0.931475	−	0.363807i	$-0.881477\pi$
$463$		−	336.885i		−	0.727613i	0.931475	−	0.363807i	$-0.118523\pi$
$464$	460.990			0.993514
$465$			209.293i			0.450092i
$466$		−	270.109i		−	0.579634i
$467$	105.992			0.226964			0.113482	−	0.993540i	$-0.463800\pi$
$467$	105.992			0.226964			0.113482	+	0.993540i	$0.463800\pi$
$468$	9.35939			0.0199987
$469$		−	283.956i		−	0.605451i
$470$	17.4080			0.0370383
$471$			53.0433i			0.112619i
$472$	4.49932			0.00953247
$473$		−	522.559i		−	1.10478i
$474$			675.105i			1.42427i
$475$	−271.736			−0.572075
$476$	69.3180			0.145626
$477$	143.078			0.299953
$478$		−	702.059i		−	1.46874i
$479$	−4.57962			−0.00956080			−0.00478040	−	0.999989i	$-0.501522\pi$
$479$	−4.57962			−0.00956080			−0.00478040	+	0.999989i	$0.501522\pi$
$480$		−	216.534i		−	0.451112i
$481$		−	8.74262i		−	0.0181759i
$482$	510.382			1.05888
$483$		−	5.87177i		−	0.0121569i
$484$	205.859			0.425328
$485$		−	52.1602i		−	0.107547i
$486$	−630.665			−1.29767
$487$		−	355.111i		−	0.729182i	−0.931168	−	0.364591i	$-0.881209\pi$
$487$		−	355.111i		−	0.729182i	0.931168	−	0.364591i	$-0.118791\pi$
$488$	5.56749			0.0114088
$489$	30.9588			0.0633104
$490$	−191.043			−0.389885
$491$		−	472.368i		−	0.962054i	−0.876706	−	0.481027i	$-0.840264\pi$
$491$		−	472.368i		−	0.962054i	0.876706	−	0.481027i	$-0.159736\pi$
$492$	−425.831			−0.865511
$493$			111.995i			0.227170i
$494$		−	19.2051i		−	0.0388767i
$495$			91.8301i			0.185515i
$496$			696.221i			1.40367i
$497$	506.472			1.01906
$498$			650.375i			1.30597i
$499$	341.206			0.683779			0.341890	−	0.939740i	$-0.388933\pi$
$499$	341.206			0.683779			0.341890	+	0.939740i	$0.388933\pi$
$500$			416.254i			0.832508i
$501$			528.113i			1.05412i
$502$	−1285.48			−2.56073
$503$			11.2281i			0.0223222i	0.999938	+	0.0111611i	$0.00355277\pi$
$503$			11.2281i			0.0223222i	−0.999938	+	0.0111611i	$0.996447\pi$
$504$	1.62953			0.00323319
$505$	−43.4090			−0.0859584
$506$	−15.1196			−0.0298806
$507$			348.150i			0.686686i
$508$	445.817			0.877592
$509$	44.7032			0.0878256			0.0439128	−	0.999035i	$-0.486018\pi$
$509$	44.7032			0.0878256			0.0439128	+	0.999035i	$0.486018\pi$
$510$	52.2480			0.102447
$511$			109.088i			0.213480i
$512$		−	725.206i		−	1.41642i
$513$			392.332i			0.764779i
$514$	255.857			0.497777
$515$	−83.1339			−0.161425
$516$	−519.337			−1.00647
$517$	22.1822			0.0429057
$518$		−	225.440i		−	0.435212i
$519$	−208.226			−0.401206
$520$	−0.0873877			−0.000168053
$521$	902.156			1.73158			0.865792	−	0.500403i	$-0.166815\pi$
$521$	902.156			1.73158			0.865792	+	0.500403i	$0.166815\pi$
$522$			389.932i			0.746995i
$523$		−	387.046i		−	0.740050i	−0.929022	−	0.370025i	$-0.879349\pi$
$523$		−	387.046i		−	0.740050i	0.929022	−	0.370025i	$-0.120651\pi$
$524$	400.176			0.763695
$525$	−180.635			−0.344067
$526$		−	1103.38i		−	2.09768i
$527$	−169.143			−0.320954
$528$		−	274.043i		−	0.519022i
$529$	528.592			0.999230
$530$	−197.856			−0.373313
$531$		−	277.070i		−	0.521790i
$532$		−	248.452i		−	0.467016i
$533$		−	25.1091i		−	0.0471091i
$534$		−	444.207i		−	0.831849i
$535$	−127.089			−0.237549
$536$			4.90638i			0.00915370i
$537$			304.288i			0.566645i
$538$		−	235.425i		−	0.437593i
$539$	−243.438			−0.451647
$540$	264.401			0.489631
$541$		−	274.805i		−	0.507957i	−0.967210	−	0.253978i	$-0.918261\pi$
$541$		−	274.805i		−	0.507957i	0.967210	−	0.253978i	$-0.0817392\pi$
$542$	352.990			0.651273
$543$		−	603.787i		−	1.11195i
$544$	174.995			0.321682
$545$	−154.219			−0.282970
$546$		−	12.7665i		−	0.0233818i
$547$	101.055	+	537.584i	0.184744	+	0.982787i
$547$	101.055	+	537.584i	0.184744	+	0.982787i
$548$	374.459			0.683320
$549$		−	342.849i		−	0.624497i
$550$			465.128i			0.845688i
$551$	401.417			0.728524
$552$			0.101456i			0.000183798i
$553$	−514.983			−0.931253
$554$		−	779.305i		−	1.40669i
$555$		−	85.2497i		−	0.153603i
$556$	−1062.69			−1.91132
$557$	−29.8063			−0.0535123			−0.0267561	−	0.999642i	$-0.508518\pi$
$557$	−29.8063			−0.0535123			−0.0267561	+	0.999642i	$0.508518\pi$
$558$	−588.903			−1.05538
$559$		−	30.6227i		−	0.0547812i
$560$	164.053			0.292952
$561$	66.5772			0.118676
$562$	−936.780			−1.66687
$563$	−649.908			−1.15437			−0.577183	−	0.816615i	$-0.695848\pi$
$563$	−649.908			−1.15437			−0.577183	+	0.816615i	$0.695848\pi$
$564$		−	22.0454i		−	0.0390877i
$565$			111.313i			0.197015i
$566$	337.614			0.596492
$567$			70.4325i			0.124220i
$568$	−8.75115			−0.0154070
$569$			972.022i			1.70830i	0.520027	+	0.854150i	$0.325922\pi$
$569$			972.022i			1.70830i	−0.520027	+	0.854150i	$0.674078\pi$
$570$		−	187.270i		−	0.328543i
$571$	803.537			1.40725			0.703623	−	0.710574i	$-0.251565\pi$
$571$	803.537			1.40725			0.703623	+	0.710574i	$0.251565\pi$
$572$	−16.4923			−0.0288326
$573$			570.894i			0.996324i
$574$		−	647.471i		−	1.12800i
$575$			12.5366i			0.0218027i
$576$	307.735			0.534262
$577$			810.727i			1.40507i	0.711648	+	0.702536i	$0.247949\pi$
$577$			810.727i			1.40507i	−0.711648	+	0.702536i	$0.752051\pi$
$578$		−	776.578i		−	1.34356i
$579$		−	605.866i		−	1.04640i
$580$		−	270.523i		−	0.466419i
$581$	−496.118			−0.853904
$582$	−131.665			−0.226228
$583$	−252.119			−0.432451
$584$		−	1.88490i		−	0.00322756i
$585$			5.38137i			0.00919893i
$586$			71.9142i			0.122721i
$587$	−81.1586			−0.138260			−0.0691300	−	0.997608i	$-0.522022\pi$
$587$	−81.1586			−0.138260			−0.0691300	+	0.997608i	$0.522022\pi$
$588$			241.937i			0.411457i
$589$			606.249i			1.02929i
$590$			383.149i			0.649404i
$591$	−673.200			−1.13909
$592$		−	283.587i		−	0.479032i
$593$	−649.734			−1.09567			−0.547836	−	0.836585i	$-0.684548\pi$
$593$	−649.734			−1.09567			−0.547836	+	0.836585i	$0.684548\pi$
$594$	671.552			1.13056
$595$			39.8558i			0.0669845i
$596$	18.2943			0.0306952
$597$			52.0721i			0.0872230i
$598$	−0.886028			−0.00148165
$599$	−34.8478			−0.0581767			−0.0290883	−	0.999577i	$-0.509260\pi$
$599$	−34.8478			−0.0581767			−0.0290883	+	0.999577i	$0.509260\pi$
$600$	3.12113			0.00520189
$601$	734.721			1.22250			0.611249	−	0.791438i	$-0.290667\pi$
$601$	734.721			1.22250			0.611249	+	0.791438i	$0.290667\pi$
$602$		−	789.645i		−	1.31170i
$603$	302.137			0.501056
$604$			771.026i			1.27653i
$605$			118.363i			0.195641i
$606$			109.575i			0.180816i
$607$	−282.435			−0.465296			−0.232648	−	0.972561i	$-0.574739\pi$
$607$	−282.435			−0.465296			−0.232648	+	0.972561i	$0.574739\pi$
$608$		−	627.224i		−	1.03162i
$609$	266.840			0.438161
$610$			474.111i			0.777231i
$611$	1.29991			0.00212751
$612$			73.7561i			0.120517i
$613$	−492.022			−0.802646			−0.401323	−	0.915937i	$-0.631449\pi$
$613$	−492.022			−0.802646			−0.401323	+	0.915937i	$0.631449\pi$
$614$	−412.852			−0.672398
$615$		−	244.840i		−	0.398115i
$616$	−2.87141			−0.00466138
$617$		−	46.6214i		−	0.0755614i	−0.999286	−	0.0377807i	$-0.987971\pi$
$617$		−	46.6214i		−	0.0755614i	0.999286	−	0.0377807i	$-0.0120288\pi$
$618$			209.850i			0.339563i
$619$			208.155i			0.336276i	0.985763	+	0.168138i	$0.0537755\pi$
$619$			208.155i			0.336276i	−0.985763	+	0.168138i	$0.946225\pi$
$620$	408.564			0.658974
$621$	18.1003			0.0291470
$622$		−	1597.12i		−	2.56772i
$623$	338.850			0.543900
$624$		−	16.0593i		−	0.0257361i
$625$	251.626			0.402602
$626$		−	539.704i		−	0.862147i
$627$		−	238.629i		−	0.380588i
$628$	103.547			0.164883
$629$	68.8957			0.109532
$630$			138.766i			0.220263i
$631$	−700.706			−1.11047			−0.555235	−	0.831694i	$-0.687372\pi$
$631$	−700.706			−1.11047			−0.555235	+	0.831694i	$0.687372\pi$
$632$	8.89820			0.0140794
$633$	−746.148			−1.17875
$634$			695.227i			1.09657i
$635$			256.331i			0.403672i
$636$			250.564i			0.393969i
$637$	−14.2658			−0.0223953
$638$		−	687.102i		−	1.07696i
$639$			538.899i			0.843348i
$640$	−5.70817			−0.00891902
$641$		−	215.053i		−	0.335496i	−0.985830	−	0.167748i	$-0.946351\pi$
$641$		−	215.053i		−	0.335496i	0.985830	−	0.167748i	$-0.0536495\pi$
$642$			320.803i			0.499693i
$643$	104.988			0.163279			0.0816393	−	0.996662i	$-0.473984\pi$
$643$	104.988			0.163279			0.0816393	+	0.996662i	$0.473984\pi$
$644$	−11.4624			−0.0177987
$645$		−	298.603i		−	0.462951i
$646$	151.344			0.234279
$647$	453.740			0.701298			0.350649	−	0.936507i	$-0.385961\pi$
$647$	453.740			0.701298			0.350649	+	0.936507i	$0.385961\pi$
$648$		−	1.21698i		−	0.00187805i
$649$			488.228i			0.752278i
$650$			27.2572i			0.0419341i
$651$			403.001i			0.619050i
$652$		−	60.4352i		−	0.0926920i
$653$			89.2244i			0.136638i	0.997664	+	0.0683188i	$0.0217635\pi$
$653$			89.2244i			0.136638i	−0.997664	+	0.0683188i	$0.978236\pi$
$654$			389.285i			0.595237i
$655$			230.089i			0.351282i
$656$		−	814.472i		−	1.24157i
$657$	−116.073			−0.176671
$658$	33.5198			0.0509420
$659$		−	272.765i		−	0.413908i	−0.978351	−	0.206954i	$-0.933645\pi$
$659$		−	272.765i		−	0.413908i	0.978351	−	0.206954i	$-0.0663550\pi$
$660$	−160.817			−0.243662
$661$	−387.021			−0.585508			−0.292754	−	0.956188i	$-0.594572\pi$
$661$	−387.021			−0.585508			−0.292754	+	0.956188i	$0.594572\pi$
$662$	390.321			0.589609
$663$	3.90151			0.00588464
$664$	8.57225			0.0129100
$665$	142.853			0.214816
$666$	239.874			0.360171
$667$		−	18.5194i		−	0.0277652i
$668$	1030.94			1.54332
$669$	666.408			0.996125
$670$	−417.812			−0.623600
$671$			604.137i			0.900353i
$672$		−	416.945i		−	0.620453i
$673$	−31.3408			−0.0465688			−0.0232844	−	0.999729i	$-0.507412\pi$
$673$	−31.3408			−0.0465688			−0.0232844	+	0.999729i	$0.507412\pi$
$674$	889.526			1.31977
$675$		−	556.824i		−	0.824925i
$676$	679.629			1.00537
$677$	124.102			0.183312			0.0916559	−	0.995791i	$-0.470784\pi$
$677$	124.102			0.183312			0.0916559	+	0.995791i	$0.470784\pi$
$678$	280.982			0.414427
$679$		−	100.436i		−	0.147918i
$680$		−	0.688654i		−	0.00101273i
$681$			529.659i			0.777767i
$682$	1037.71			1.52157
$683$	522.733			0.765348			0.382674	−	0.923883i	$-0.375003\pi$
$683$	522.733			0.765348			0.382674	+	0.923883i	$0.375003\pi$
$684$	264.360			0.386491
$685$			215.303i			0.314311i
$686$	−986.843			−1.43855
$687$	−196.744			−0.286382
$688$		−	993.317i		−	1.44377i
$689$	−14.7745			−0.0214434
$690$	−8.63970			−0.0125213
$691$	737.069			1.06667			0.533335	−	0.845904i	$-0.320939\pi$
$691$	737.069			1.06667			0.533335	+	0.845904i	$0.320939\pi$
$692$			406.481i			0.587401i
$693$			176.822i			0.255155i
$694$			91.5549i			0.131924i
$695$		−	611.016i		−	0.879160i
$696$	−4.61063			−0.00662447
$697$	197.871			0.283890
$698$			74.0221i			0.106049i
$699$		−	196.677i		−	0.281370i
$700$			352.621i			0.503744i
$701$	−857.092			−1.22267			−0.611336	−	0.791372i	$-0.709367\pi$
$701$	−857.092			−1.22267			−0.611336	+	0.791372i	$0.709367\pi$
$702$	39.3538			0.0560596
$703$		−	246.939i		−	0.351265i
$704$	−542.263			−0.770260
$705$	12.6755			0.0179794
$706$		−	466.916i		−	0.661354i
$707$	−83.5857			−0.118226
$708$	485.218			0.685336
$709$			169.070i			0.238462i	0.992867	+	0.119231i	$0.0380430\pi$
$709$			169.070i			0.238462i	−0.992867	+	0.119231i	$0.961957\pi$
$710$		−	745.221i		−	1.04961i
$711$		−	547.955i		−	0.770682i
$712$	−5.85486			−0.00822312
$713$	27.9694			0.0392277
$714$	100.606			0.140904
$715$		−	9.48257i		−	0.0132623i
$716$	594.006			0.829618
$717$		−	511.197i		−	0.712967i
$718$	1661.37			2.31389
$719$		−	1306.10i		−	1.81655i	−0.418368	−	0.908277i	$-0.637398\pi$
$719$		−	1306.10i		−	1.81655i	0.418368	−	0.908277i	$-0.362602\pi$
$720$			174.557i			0.242440i
$721$	−160.078			−0.222022
$722$			480.341i			0.665292i
$723$	371.629			0.514010
$724$	−1178.66			−1.62799
$725$	−569.718			−0.785818
$726$	298.776			0.411537
$727$			538.615i			0.740873i	0.928858	+	0.370436i	$0.120792\pi$
$727$			538.615i			0.740873i	−0.928858	+	0.370436i	$0.879208\pi$
$728$	−0.168268			−0.000231138
$729$	−601.385			−0.824945
$730$	160.512			0.219879
$731$	241.320			0.330124
$732$	600.412			0.820234
$733$		−	808.030i		−	1.10236i	−0.834386	−	0.551180i	$-0.814178\pi$
$733$		−	808.030i		−	1.10236i	0.834386	−	0.551180i	$-0.185822\pi$
$734$			888.902i			1.21104i
$735$	−139.106			−0.189260
$736$	−28.9371			−0.0393167
$737$	−532.399			−0.722386
$738$	688.927			0.933505
$739$			217.463i			0.294266i	0.989117	+	0.147133i	$0.0470046\pi$
$739$			217.463i			0.294266i	−0.989117	+	0.147133i	$0.952995\pi$
$740$	−166.417			−0.224888
$741$		−	13.9840i		−	0.0188718i
$742$	−380.980			−0.513450
$743$	1014.76			1.36576			0.682878	−	0.730533i	$-0.260728\pi$
$743$	1014.76			1.36576			0.682878	+	0.730533i	$0.260728\pi$
$744$		−	6.96332i		−	0.00935930i
$745$			10.5187i			0.0141191i
$746$	839.799			1.12574
$747$		−	527.883i		−	0.706671i
$748$		−	129.966i		−	0.173752i
$749$	−244.714			−0.326721
$750$			604.136i			0.805514i
$751$	514.283			0.684797			0.342399	−	0.939555i	$-0.388761\pi$
$751$	514.283			0.684797			0.342399	+	0.939555i	$0.388761\pi$
$752$	42.1655			0.0560712
$753$	−936.012			−1.24304
$754$		−	40.2651i		−	0.0534020i
$755$	−443.317			−0.587175
$756$	509.113			0.673431
$757$	450.262			0.594798			0.297399	−	0.954753i	$-0.403881\pi$
$757$	450.262			0.594798			0.297399	+	0.954753i	$0.403881\pi$
$758$		−	1758.17i		−	2.31948i
$759$	−11.0092			−0.0145048
$760$	−2.46830			−0.00324777
$761$	361.470			0.474993			0.237497	−	0.971388i	$-0.423673\pi$
$761$	361.470			0.474993			0.237497	+	0.971388i	$0.423673\pi$
$762$	647.042			0.849137
$763$	−296.954			−0.389193
$764$	1114.45			1.45871
$765$	−42.4076			−0.0554347
$766$		−	1578.15i		−	2.06025i
$767$			28.6109i			0.0373023i
$768$		−	520.872i		−	0.678218i
$769$			144.776i			0.188265i	0.995560	+	0.0941326i	$0.0300078\pi$
$769$			144.776i			0.188265i	−0.995560	+	0.0941326i	$0.969992\pi$
$770$		−	244.520i		−	0.317559i
$771$	186.300			0.241634
$772$	−1182.72			−1.53202
$773$			1390.13i			1.79836i	0.437576	+	0.899182i	$0.355837\pi$
$773$			1390.13i			1.79836i	−0.437576	+	0.899182i	$0.644163\pi$
$774$	840.203			1.08553
$775$		−	860.430i		−	1.11023i
$776$			1.73541i			0.00223635i
$777$		−	164.152i		−	0.211263i
$778$	735.805			0.945765
$779$		−	709.218i		−	0.910421i
$780$	−9.42409			−0.0120822
$781$		−	949.600i		−	1.21588i
$782$		−	6.98229i		−	0.00892876i
$783$			822.558i			1.05052i
$784$	−462.743			−0.590234
$785$			59.5363i			0.0758424i
$786$	580.801			0.738932
$787$	−634.679			−0.806453			−0.403227	−	0.915100i	$-0.632111\pi$
$787$	−634.679			−0.806453			−0.403227	+	0.915100i	$0.632111\pi$
$788$			1314.16i			1.66772i
$789$		−	803.416i		−	1.01827i
$790$			757.743i			0.959169i
$791$			214.338i			0.270971i
$792$		−	3.05525i		−	0.00385764i
$793$			35.4033i			0.0446447i
$794$	18.5931			0.0234170
$795$	−144.067			−0.181216
$796$	101.651			0.127702
$797$	1073.92			1.34746			0.673728	−	0.738980i	$-0.264692\pi$
$797$	1073.92			1.34746			0.673728	+	0.738980i	$0.264692\pi$
$798$		−	360.595i		−	0.451873i
$799$			10.2439i			0.0128208i
$800$			890.200i			1.11275i
$801$			360.545i			0.450118i
$802$		−	34.8322i		−	0.0434317i
$803$	204.533			0.254711
$804$			529.116i			0.658104i
$805$		−	6.59053i		−	0.00818699i
$806$	60.8114			0.0754483
$807$		−	171.422i		−	0.212419i
$808$	1.44425			0.00178744
$809$			1311.87i			1.62160i	0.585325	+	0.810799i	$0.300967\pi$
$809$			1311.87i			1.62160i	−0.585325	+	0.810799i	$0.699033\pi$
$810$	103.634			0.127943
$811$	−612.190			−0.754858			−0.377429	−	0.926038i	$-0.623192\pi$
$811$	−612.190			−0.754858			−0.377429	+	0.926038i	$0.623192\pi$
$812$		−	520.903i		−	0.641506i
$813$	257.026			0.316145
$814$	−422.684			−0.519268
$815$	34.7484			0.0426361
$816$	126.555			0.155091
$817$		−	864.951i		−	1.05869i
$818$		−	1406.67i		−	1.71965i
$819$			10.3620i			0.0126521i
$820$	−477.957			−0.582874
$821$		−	423.360i		−	0.515663i	−0.966190	−	0.257832i	$-0.916992\pi$
$821$		−	423.360i		−	0.515663i	0.966190	−	0.257832i	$-0.0830080\pi$
$822$	543.476			0.661163
$823$	994.521			1.20841			0.604205	−	0.796829i	$-0.293491\pi$
$823$	994.521			1.20841			0.604205	+	0.796829i	$0.293491\pi$
$824$	2.76592			0.00335670
$825$			338.679i			0.410519i
$826$			737.767i			0.893181i
$827$			176.930i			0.213942i	0.994262	+	0.106971i	$0.0341153\pi$
$827$			176.930i			0.213942i	−0.994262	+	0.106971i	$0.965885\pi$
$828$		−	12.1963i		−	0.0147298i
$829$	607.487			0.732795			0.366398	−	0.930458i	$-0.380591\pi$
$829$	607.487			0.732795			0.366398	+	0.930458i	$0.380591\pi$
$830$			729.987i			0.879502i
$831$		−	567.443i		−	0.682843i
$832$	−31.7774			−0.0381939
$833$		−	112.421i		−	0.134959i
$834$	−1542.35			−1.84934
$835$			592.759i			0.709891i
$836$	−465.831			−0.557214
$837$	−1242.29			−1.48421
$838$		−	994.550i		−	1.18681i
$839$	−824.419			−0.982621			−0.491310	−	0.870985i	$-0.663482\pi$
$839$	−824.419			−0.982621			−0.491310	+	0.870985i	$0.663482\pi$
$840$	−1.64079			−0.00195333
$841$	0.605705			0.000720220
$842$	252.139			0.299452
$843$	−682.107			−0.809142
$844$			1456.57i			1.72579i
$845$			390.766i			0.462445i
$846$			35.6660i			0.0421584i
$847$			227.912i			0.269081i
$848$	−479.245			−0.565147
$849$	245.830			0.289553
$850$	−214.798			−0.252704
$851$	−11.3926			−0.0133873
$852$	−943.744			−1.10768
$853$	−976.828			−1.14517			−0.572584	−	0.819846i	$-0.694059\pi$
$853$	−976.828			−1.14517			−0.572584	+	0.819846i	$0.694059\pi$
$854$			912.918i			1.06899i
$855$			151.999i			0.177777i
$856$	4.22833			0.00493964
$857$			1083.80i			1.26465i	0.774705	+	0.632323i	$0.217898\pi$
$857$			1083.80i			1.26465i	−0.774705	+	0.632323i	$0.782102\pi$
$858$	−23.9363			−0.0278978
$859$	−1373.74			−1.59923			−0.799616	−	0.600512i	$-0.794963\pi$
$859$	−1373.74			−1.59923			−0.799616	+	0.600512i	$0.794963\pi$
$860$	−582.908			−0.677800
$861$		−	471.450i		−	0.547561i
$862$	−615.914			−0.714517
$863$			162.805i			0.188650i	0.995541	+	0.0943249i	$0.0300692\pi$
$863$			162.805i			0.188650i	−0.995541	+	0.0943249i	$0.969931\pi$
$864$	1285.27			1.48758
$865$	−233.715			−0.270190
$866$	−123.938			−0.143116
$867$		−	565.458i		−	0.652200i
$868$	786.705			0.906343
$869$			965.557i			1.11111i
$870$		−	392.627i		−	0.451296i
$871$	−31.1993			−0.0358201
$872$	5.13096			0.00588413
$873$	106.867			0.122414
$874$	−25.0262			−0.0286341
$875$	−460.846			−0.526682
$876$		−	203.272i		−	0.232045i
$877$			1578.64i			1.80005i	0.435840	+	0.900024i	$0.356451\pi$
$877$			1578.64i			1.80005i	−0.435840	+	0.900024i	$0.643549\pi$
$878$			832.001i			0.947609i
$879$			52.3636i			0.0595718i
$880$		−	307.589i		−	0.349533i
$881$			494.444i			0.561230i	0.959820	+	0.280615i	$0.0905384\pi$
$881$			494.444i			0.561230i	−0.959820	+	0.280615i	$0.909462\pi$
$882$		−	391.414i		−	0.443780i
$883$	−679.712			−0.769775			−0.384888	−	0.922963i	$-0.625760\pi$
$883$	−679.712			−0.769775			−0.384888	+	0.922963i	$0.625760\pi$
$884$		−	7.61621i		−	0.00861562i
$885$			278.986i			0.315238i
$886$			906.532i			1.02317i
$887$	403.544			0.454954			0.227477	−	0.973783i	$-0.426952\pi$
$887$	403.544			0.454954			0.227477	+	0.973783i	$0.426952\pi$
$888$			2.83632i			0.00319405i
$889$			493.576i			0.555204i
$890$		−	498.582i		−	0.560205i
$891$	132.056			0.148211
$892$		−	1300.90i		−	1.45841i
$893$	36.7165			0.0411159
$894$	26.5517			0.0296999
$895$			341.536i			0.381605i
$896$	−10.9913			−0.0122671
$897$	−0.645152			−0.000719233
$898$			2060.51i			2.29456i
$899$			1271.05i			1.41385i
$900$	−375.198			−0.416886
$901$		−	116.430i		−	0.129223i
$902$	−1213.96			−1.34586
$903$		−	574.972i		−	0.636736i
$904$		−	3.70347i		−	0.00409676i
$905$		−	677.696i		−	0.748835i
$906$			1119.04i			1.23514i
$907$	−128.322			−0.141480			−0.0707399	−	0.997495i	$-0.522536\pi$
$907$	−128.322			−0.141480			−0.0707399	+	0.997495i	$0.522536\pi$
$908$	1033.96			1.13872
$909$		−	88.9374i		−	0.0978409i
$910$		−	14.3292i		−	0.0157464i
$911$			427.142i			0.468872i	0.972132	+	0.234436i	$0.0753243\pi$
$911$			427.142i			0.468872i	−0.972132	+	0.234436i	$0.924676\pi$
$912$		−	453.602i		−	0.497371i
$913$			930.188i			1.01883i
$914$	1477.80			1.61685
$915$			345.219i			0.377288i
$916$			384.068i			0.419288i
$917$			443.046i			0.483147i
$918$			310.126i			0.337828i
$919$	−1283.31			−1.39642			−0.698208	−	0.715895i	$-0.746019\pi$
$919$	−1283.31			−1.39642			−0.698208	+	0.715895i	$0.746019\pi$
$920$			0.113875i			0.000123778i
$921$	−300.614			−0.326400
$922$	−620.729			−0.673241
$923$		−	55.6479i		−	0.0602902i
$924$	−309.659			−0.335129
$925$			350.473i			0.378890i
$926$	−954.472			−1.03075
$927$		−	170.327i		−	0.183740i
$928$		−	1315.03i		−	1.41706i
$929$			942.526i			1.01456i	0.861781	+	0.507280i	$0.169349\pi$
$929$			942.526i			1.01456i	−0.861781	+	0.507280i	$0.830651\pi$
$930$	592.974			0.637607
$931$	−402.943			−0.432807
$932$	−383.937			−0.411950
$933$		−	1162.93i		−	1.24644i
$934$		−	300.300i		−	0.321520i
$935$	74.7268			0.0799217
$936$		−	0.179042i		−	0.000191284i
$937$		−	802.961i		−	0.856948i	−0.903554	−	0.428474i	$-0.859051\pi$
$937$		−	802.961i		−	0.856948i	0.903554	−	0.428474i	$-0.140949\pi$
$938$	−804.514			−0.857691
$939$		−	392.980i		−	0.418509i
$940$		−	24.7440i		−	0.0263234i
$941$	−452.379			−0.480743			−0.240372	−	0.970681i	$-0.577269\pi$
$941$	−452.379			−0.480743			−0.240372	+	0.970681i	$0.577269\pi$
$942$	150.284			0.159537
$943$	−32.7199			−0.0346976
$944$			928.058i			0.983112i
$945$			292.725i			0.309762i
$946$	−1480.53			−1.56504
$947$	−1099.85			−1.16141			−0.580704	−	0.814115i	$-0.697223\pi$
$947$	−1099.85			−1.16141			−0.580704	+	0.814115i	$0.697223\pi$
$948$	959.603			1.01224
$949$	11.9859			0.0126300
$950$			769.890i			0.810411i
$951$			506.222i			0.532305i
$952$		−	1.32603i		−	0.00139289i
$953$	−1482.38			−1.55549			−0.777744	−	0.628581i	$-0.783636\pi$
$953$	−1482.38			−1.55549			−0.777744	+	0.628581i	$0.783636\pi$
$954$		−	405.372i		−	0.424919i
$955$			640.776i			0.670970i
$956$	−997.917			−1.04385
$957$		−	500.307i		−	0.522786i
$958$			12.9751i			0.0135440i
$959$			414.574i			0.432298i
$960$	−309.863			−0.322773
$961$	−958.640			−0.997544
$962$	−24.7699			−0.0257483
$963$		−	260.382i		−	0.270387i
$964$		−	725.463i		−	0.752555i
$965$		−	680.030i		−	0.704694i
$966$	−16.6361			−0.0172216
$967$		−	1401.24i		−	1.44905i	−0.689246	−	0.724527i	$-0.742058\pi$
$967$		−	1401.24i		−	1.44905i	0.689246	−	0.724527i	$-0.257942\pi$
$968$		−	3.93801i		−	0.00406819i
$969$	110.200			0.113725
$970$	−147.782			−0.152352
$971$			752.471i			0.774944i	0.921881	+	0.387472i	$0.126652\pi$
$971$			752.471i			0.774944i	−0.921881	+	0.387472i	$0.873348\pi$
$972$			896.436i			0.922260i
$973$		−	1176.54i		−	1.20918i
$974$	−1006.11			−1.03297
$975$			19.8470i			0.0203559i
$976$			1148.39i			1.17662i
$977$			959.748i			0.982341i	0.871063	+	0.491171i	$0.163431\pi$
$977$			959.748i			0.982341i	−0.871063	+	0.491171i	$0.836569\pi$
$978$		−	87.7134i		−	0.0896865i
$979$		−	635.320i		−	0.648948i
$980$			271.552i			0.277094i
$981$		−	315.967i		−	0.322087i
$982$	−1338.33			−1.36286
$983$		−	392.221i		−	0.399004i	−0.979897	−	0.199502i	$-0.936068\pi$
$983$		−	392.221i		−	0.399004i	0.979897	−	0.199502i	$-0.0639324\pi$
$984$			8.14601i			0.00827847i
$985$	−755.605			−0.767112
$986$	317.307			0.321812
$987$	24.4071			0.0247286
$988$	−27.2983			−0.0276299
$989$	−39.9046			−0.0403484
$990$	260.176			0.262804
$991$	583.905			0.589207			0.294604	−	0.955620i	$-0.404812\pi$
$991$	583.905			0.589207			0.294604	+	0.955620i	$0.404812\pi$
$992$	1986.06			2.00207
$993$	284.208			0.286212
$994$		−	1434.95i		−	1.44361i
$995$			58.4462i			0.0587399i
$996$	924.452			0.928164
$997$			1479.01i			1.48346i	0.670698	+	0.741731i	$0.265995\pi$
$997$			1479.01i			1.48346i	−0.670698	+	0.741731i	$0.734005\pi$
$998$		−	966.715i		−	0.968652i
$999$	506.012			0.506519

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.3.b.b.546.16	✓	88
547.546	odd	2	inner	547.3.b.b.546.73	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.3.b.b.546.16	✓	88	1.1	even	1	trivial
547.3.b.b.546.73	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)

\(f(q)\)	\(=\)	\(q-2.83323i q^{2} -2.06299i q^{3} -4.02719 q^{4} -2.31552i q^{5} -5.84492 q^{6} -4.45862i q^{7} +0.0770388i q^{8} +4.74408 q^{9} +O(q^{10})\)	\(q-2.83323i q^{2} -2.06299i q^{3} -4.02719 q^{4} -2.31552i q^{5} -5.84492 q^{6} -4.45862i q^{7} +0.0770388i q^{8} +4.74408 q^{9} -6.56039 q^{10} -8.35960 q^{11} +8.30805i q^{12} -0.489884 q^{13} -12.6323 q^{14} -4.77688 q^{15} -15.8905 q^{16} -3.86050i q^{17} -13.4411i q^{18} -13.8370 q^{19} +9.32502i q^{20} -9.19807 q^{21} +23.6847i q^{22} +0.638370i q^{23} +0.158930 q^{24} +19.6384 q^{25} +1.38795i q^{26} -28.3539i q^{27} +17.9557i q^{28} -29.0104 q^{29} +13.5340i q^{30} -43.8137i q^{31} +45.3296i q^{32} +17.2457i q^{33} -10.9377 q^{34} -10.3240 q^{35} -19.1053 q^{36} +17.8463i q^{37} +39.2033i q^{38} +1.01062i q^{39} +0.178385 q^{40} +51.2553i q^{41} +26.0602i q^{42} +62.5101i q^{43} +33.6657 q^{44} -10.9850i q^{45} +1.80865 q^{46} -2.65351 q^{47} +32.7819i q^{48} +29.1208 q^{49} -55.6401i q^{50} -7.96416 q^{51} +1.97286 q^{52} +30.1592 q^{53} -80.3330 q^{54} +19.3568i q^{55} +0.343486 q^{56} +28.5455i q^{57} +82.1932i q^{58} -58.4033i q^{59} +19.2374 q^{60} -72.2687i q^{61} -124.134 q^{62} -21.1520i q^{63} +64.8671 q^{64} +1.13433i q^{65} +48.8611 q^{66} +63.6871 q^{67} +15.5470i q^{68} +1.31695 q^{69} +29.2502i q^{70} +113.594i q^{71} +0.365479i q^{72} -24.4669 q^{73} +50.5627 q^{74} -40.5137i q^{75} +55.7241 q^{76} +37.2722i q^{77} +2.86333 q^{78} -115.503i q^{79} +36.7947i q^{80} -15.7969 q^{81} +145.218 q^{82} -111.272i q^{83} +37.0424 q^{84} -8.93905 q^{85} +177.106 q^{86} +59.8482i q^{87} -0.644013i q^{88} +75.9989i q^{89} -31.1230 q^{90} +2.18420i q^{91} -2.57084i q^{92} -90.3871 q^{93} +7.51799i q^{94} +32.0397i q^{95} +93.5144 q^{96} +22.5264 q^{97} -82.5058i q^{98} -39.6586 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) 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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