LMFDB - Embedded newform 547.3.b.b.546.2

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			546.2
Character	$\chi$	$=$	547.546
Dual form			547.3.b.b.546.87

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.88566i q^{2} -2.17695i q^{3} -11.0983 q^{4} -9.52039i q^{5} -8.45889 q^{6} -6.36562i q^{7} +27.5816i q^{8} +4.26087 q^{9} +O(q^{10})$	\(q-3.88566i q^{2} -2.17695i q^{3} -11.0983 q^{4} -9.52039i q^{5} -8.45889 q^{6} -6.36562i q^{7} +27.5816i q^{8} +4.26087 q^{9} -36.9930 q^{10} -13.8171 q^{11} +24.1605i q^{12} +7.82071 q^{13} -24.7346 q^{14} -20.7255 q^{15} +62.7793 q^{16} -9.95994i q^{17} -16.5563i q^{18} +10.7082 q^{19} +105.660i q^{20} -13.8576 q^{21} +53.6884i q^{22} -8.14939i q^{23} +60.0439 q^{24} -65.6379 q^{25} -30.3886i q^{26} -28.8683i q^{27} +70.6476i q^{28} +42.9911 q^{29} +80.5320i q^{30} +4.30660i q^{31} -133.612i q^{32} +30.0791i q^{33} -38.7009 q^{34} -60.6032 q^{35} -47.2885 q^{36} -1.98043i q^{37} -41.6082i q^{38} -17.0253i q^{39} +262.588 q^{40} -6.08011i q^{41} +53.8460i q^{42} -58.3595i q^{43} +153.346 q^{44} -40.5652i q^{45} -31.6657 q^{46} +46.1669 q^{47} -136.668i q^{48} +8.47893 q^{49} +255.046i q^{50} -21.6823 q^{51} -86.7967 q^{52} +78.0473 q^{53} -112.172 q^{54} +131.544i q^{55} +175.574 q^{56} -23.3112i q^{57} -167.049i q^{58} +6.78347i q^{59} +230.018 q^{60} +59.9948i q^{61} +16.7340 q^{62} -27.1231i q^{63} -268.054 q^{64} -74.4563i q^{65} +116.877 q^{66} -52.0804 q^{67} +110.539i q^{68} -17.7408 q^{69} +235.483i q^{70} +70.5799i q^{71} +117.522i q^{72} +110.003 q^{73} -7.69526 q^{74} +142.891i q^{75} -118.843 q^{76} +87.9542i q^{77} -66.1545 q^{78} -141.443i q^{79} -597.684i q^{80} -24.4971 q^{81} -23.6252 q^{82} -82.4877i q^{83} +153.797 q^{84} -94.8225 q^{85} -226.765 q^{86} -93.5896i q^{87} -381.097i q^{88} +96.6837i q^{89} -157.622 q^{90} -49.7836i q^{91} +90.4445i q^{92} +9.37527 q^{93} -179.389i q^{94} -101.946i q^{95} -290.868 q^{96} -145.501 q^{97} -32.9462i q^{98} -58.8728 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.88566i		−	1.94283i	−0.237393	−	0.971414i	$-0.576293\pi$
$2$		−	3.88566i		−	1.94283i	0.237393	−	0.971414i	$-0.423707\pi$
$3$		−	2.17695i		−	0.725651i	−0.931857	−	0.362826i	$-0.881812\pi$
$3$		−	2.17695i		−	0.725651i	0.931857	−	0.362826i	$-0.118188\pi$
$4$	−11.0983			−2.77458
$5$		−	9.52039i		−	1.90408i	−0.305975	−	0.952039i	$-0.598982\pi$
$5$		−	9.52039i		−	1.90408i	0.305975	−	0.952039i	$-0.401018\pi$
$6$	−8.45889			−1.40981
$7$		−	6.36562i		−	0.909374i	−0.890651	−	0.454687i	$-0.849751\pi$
$7$		−	6.36562i		−	0.909374i	0.890651	−	0.454687i	$-0.150249\pi$
$8$			27.5816i			3.44770i
$9$	4.26087			0.473431
$10$	−36.9930			−3.69930
$11$	−13.8171			−1.25610			−0.628049	−	0.778174i	$-0.716146\pi$
$11$	−13.8171			−1.25610			−0.628049	+	0.778174i	$0.716146\pi$
$12$			24.1605i			2.01338i
$13$	7.82071			0.601593			0.300797	−	0.953688i	$-0.402747\pi$
$13$	7.82071			0.601593			0.300797	+	0.953688i	$0.402747\pi$
$14$	−24.7346			−1.76676
$15$	−20.7255			−1.38170
$16$	62.7793			3.92371
$17$		−	9.95994i		−	0.585879i	−0.956131	−	0.292939i	$-0.905367\pi$
$17$		−	9.95994i		−	0.585879i	0.956131	−	0.292939i	$-0.0946334\pi$
$18$		−	16.5563i		−	0.919794i
$19$	10.7082			0.563587			0.281794	−	0.959475i	$-0.409071\pi$
$19$	10.7082			0.563587			0.281794	+	0.959475i	$0.409071\pi$
$20$			105.660i			5.28302i
$21$	−13.8576			−0.659888
$22$			53.6884i			2.44038i
$23$		−	8.14939i		−	0.354321i	−0.984182	−	0.177161i	$-0.943309\pi$
$23$		−	8.14939i		−	0.354321i	0.984182	−	0.177161i	$-0.0566912\pi$
$24$	60.0439			2.50183
$25$	−65.6379			−2.62552
$26$		−	30.3886i		−	1.16879i
$27$		−	28.8683i		−	1.06920i
$28$			70.6476i			2.52313i
$29$	42.9911			1.48245			0.741226	−	0.671256i	$-0.234245\pi$
$29$	42.9911			1.48245			0.741226	+	0.671256i	$0.234245\pi$
$30$			80.5320i			2.68440i
$31$			4.30660i			0.138923i	0.997585	+	0.0694613i	$0.0221280\pi$
$31$			4.30660i			0.138923i	−0.997585	+	0.0694613i	$0.977872\pi$
$32$		−	133.612i		−	4.17539i
$33$			30.0791i			0.911488i
$34$	−38.7009			−1.13826
$35$	−60.6032			−1.73152
$36$	−47.2885			−1.31357
$37$		−	1.98043i		−	0.0535251i	−0.999642	−	0.0267625i	$-0.991480\pi$
$37$		−	1.98043i		−	0.0535251i	0.999642	−	0.0267625i	$-0.00851979\pi$
$38$		−	41.6082i		−	1.09495i
$39$		−	17.0253i		−	0.436547i
$40$	262.588			6.56469
$41$		−	6.08011i		−	0.148295i	−0.997247	−	0.0741476i	$-0.976376\pi$
$41$		−	6.08011i		−	0.148295i	0.997247	−	0.0741476i	$-0.0236236\pi$
$42$			53.8460i			1.28205i
$43$		−	58.3595i		−	1.35720i	−0.734509	−	0.678599i	$-0.762588\pi$
$43$		−	58.3595i		−	1.35720i	0.734509	−	0.678599i	$-0.237412\pi$
$44$	153.346			3.48514
$45$		−	40.5652i		−	0.901449i
$46$	−31.6657			−0.688386
$47$	46.1669			0.982275			0.491138	−	0.871082i	$-0.336581\pi$
$47$	46.1669			0.982275			0.491138	+	0.871082i	$0.336581\pi$
$48$		−	136.668i		−	2.84724i
$49$	8.47893			0.173039
$50$			255.046i			5.10093i
$51$	−21.6823			−0.425143
$52$	−86.7967			−1.66917
$53$	78.0473			1.47259			0.736296	−	0.676660i	$-0.236573\pi$
$53$	78.0473			1.47259			0.736296	+	0.676660i	$0.236573\pi$
$54$	−112.172			−2.07726
$55$			131.544i			2.39171i
$56$	175.574			3.13525
$57$		−	23.3112i		−	0.408968i
$58$		−	167.049i		−	2.88015i
$59$			6.78347i			0.114974i	0.998346	+	0.0574871i	$0.0183088\pi$
$59$			6.78347i			0.114974i	−0.998346	+	0.0574871i	$0.981691\pi$
$60$	230.018			3.83363
$61$			59.9948i			0.983521i	0.870730	+	0.491761i	$0.163647\pi$
$61$			59.9948i			0.983521i	−0.870730	+	0.491761i	$0.836353\pi$
$62$	16.7340			0.269903
$63$		−	27.1231i		−	0.430525i
$64$	−268.054			−4.18835
$65$		−	74.4563i		−	1.14548i
$66$	116.877			1.77086
$67$	−52.0804			−0.777319			−0.388660	−	0.921381i	$-0.627062\pi$
$67$	−52.0804			−0.777319			−0.388660	+	0.921381i	$0.627062\pi$
$68$			110.539i			1.62557i
$69$	−17.7408			−0.257114
$70$			235.483i			3.36404i
$71$			70.5799i			0.994083i	0.867727	+	0.497041i	$0.165580\pi$
$71$			70.5799i			0.994083i	−0.867727	+	0.497041i	$0.834420\pi$
$72$			117.522i			1.63225i
$73$	110.003			1.50689			0.753445	−	0.657510i	$-0.228390\pi$
$73$	110.003			1.50689			0.753445	+	0.657510i	$0.228390\pi$
$74$	−7.69526			−0.103990
$75$			142.891i			1.90521i
$76$	−118.843			−1.56372
$77$			87.9542i			1.14226i
$78$	−66.1545			−0.848135
$79$		−	141.443i		−	1.79042i	−0.445643	−	0.895211i	$-0.647025\pi$
$79$		−	141.443i		−	1.79042i	0.445643	−	0.895211i	$-0.352975\pi$
$80$		−	597.684i		−	7.47105i
$81$	−24.4971			−0.302433
$82$	−23.6252			−0.288112
$83$		−	82.4877i		−	0.993828i	−0.867800	−	0.496914i	$-0.834466\pi$
$83$		−	82.4877i		−	0.993828i	0.867800	−	0.496914i	$-0.165534\pi$
$84$	153.797			1.83091
$85$	−94.8225			−1.11556
$86$	−226.765			−2.63680
$87$		−	93.5896i		−	1.07574i
$88$		−	381.097i		−	4.33065i
$89$			96.6837i			1.08633i	0.839625	+	0.543167i	$0.182775\pi$
$89$			96.6837i			1.08633i	−0.839625	+	0.543167i	$0.817225\pi$
$90$	−157.622			−1.75136
$91$		−	49.7836i		−	0.547073i
$92$			90.4445i			0.983093i
$93$	9.37527			0.100809
$94$		−	179.389i		−	1.90839i
$95$		−	101.946i		−	1.07311i
$96$	−290.868			−3.02987
$97$	−145.501			−1.50002			−0.750008	−	0.661429i	$-0.769950\pi$
$97$	−145.501			−1.50002			−0.750008	+	0.661429i	$0.769950\pi$
$98$		−	32.9462i		−	0.336186i
$99$	−58.8728			−0.594675
$100$	728.470			7.28470
$101$			190.658i			1.88770i	0.330369	+	0.943852i	$0.392827\pi$
$101$			190.658i			1.88770i	−0.330369	+	0.943852i	$0.607173\pi$
$102$			84.2500i			0.825980i
$103$		−	101.232i		−	0.982840i	−0.870923	−	0.491420i	$-0.836478\pi$
$103$		−	101.232i		−	0.982840i	0.870923	−	0.491420i	$-0.163522\pi$
$104$			215.708i			2.07411i
$105$			131.930i			1.25648i
$106$		−	303.265i		−	2.86099i
$107$			135.594i			1.26724i	0.773646	+	0.633618i	$0.218431\pi$
$107$			135.594i			1.26724i	−0.773646	+	0.633618i	$0.781569\pi$
$108$			320.390i			2.96657i
$109$			81.5591i			0.748248i	0.927379	+	0.374124i	$0.122057\pi$
$109$			81.5591i			0.748248i	−0.927379	+	0.374124i	$0.877943\pi$
$110$	511.134			4.64668
$111$	−4.31130			−0.0388405
$112$		−	399.629i		−	3.56812i
$113$	48.4233			0.428525			0.214262	−	0.976776i	$-0.431265\pi$
$113$	48.4233			0.428525			0.214262	+	0.976776i	$0.431265\pi$
$114$	−90.5791			−0.794554
$115$	−77.5854			−0.674656
$116$	−477.129			−4.11318
$117$	33.3231			0.284813
$118$	26.3582			0.223375
$119$	−63.4011			−0.532783
$120$		−	571.641i		−	4.76368i
$121$	69.9114			0.577780
$122$	233.119			1.91081
$123$	−13.2361			−0.107611
$124$		−	47.7960i		−	0.385452i
$125$			386.889i			3.09511i
$126$	−105.391			−0.836436
$127$	173.283			1.36443			0.682217	−	0.731150i	$-0.261016\pi$
$127$	173.283			1.36443			0.682217	+	0.731150i	$0.261016\pi$
$128$			507.117i			3.96185i
$129$	−127.046			−0.984852
$130$	−289.311			−2.22547
$131$	−81.5463			−0.622491			−0.311246	−	0.950330i	$-0.600746\pi$
$131$	−81.5463			−0.622491			−0.311246	+	0.950330i	$0.600746\pi$
$132$		−	333.827i		−	2.52900i
$133$		−	68.1640i		−	0.512511i
$134$			202.366i			1.51020i
$135$	−274.838			−2.03583
$136$	274.711			2.01993
$137$	−113.665			−0.829674			−0.414837	−	0.909896i	$-0.636161\pi$
$137$	−113.665			−0.829674			−0.414837	+	0.909896i	$0.636161\pi$
$138$			68.9348i			0.499528i
$139$	−188.050			−1.35288			−0.676438	−	0.736500i	$-0.736477\pi$
$139$	−188.050			−1.35288			−0.676438	+	0.736500i	$0.736477\pi$
$140$	672.593			4.80424
$141$		−	100.503i		−	0.712789i
$142$	274.249			1.93133
$143$	−108.059			−0.755660
$144$	267.495			1.85760
$145$		−	409.292i		−	2.82270i
$146$		−	427.434i		−	2.92763i
$147$		−	18.4582i		−	0.125566i
$148$			21.9794i			0.148509i
$149$	250.465			1.68097			0.840486	−	0.541834i	$-0.182270\pi$
$149$	250.465			1.68097			0.840486	+	0.541834i	$0.182270\pi$
$150$	555.224			3.70149
$151$			138.942i			0.920147i	0.887881	+	0.460073i	$0.152177\pi$
$151$			138.942i			0.920147i	−0.887881	+	0.460073i	$0.847823\pi$
$152$			295.348i			1.94308i
$153$		−	42.4380i		−	0.277373i
$154$	341.760			2.21922
$155$	41.0005			0.264520
$156$			188.952i			1.21123i
$157$	−92.0301			−0.586179			−0.293090	−	0.956085i	$-0.594683\pi$
$157$	−92.0301			−0.586179			−0.293090	+	0.956085i	$0.594683\pi$
$158$	−549.600			−3.47848
$159$		−	169.905i		−	1.06859i
$160$	−1272.04			−7.95027
$161$	−51.8759			−0.322211
$162$			95.1872i			0.587575i
$163$			193.517i			1.18722i	0.804752	+	0.593612i	$0.202299\pi$
$163$			193.517i			1.18722i	−0.804752	+	0.593612i	$0.797701\pi$
$164$			67.4789i			0.411457i
$165$	286.365			1.73555
$166$	−320.519			−1.93084
$167$	6.19685			0.0371069			0.0185534	−	0.999828i	$-0.494094\pi$
$167$	6.19685			0.0371069			0.0185534	+	0.999828i	$0.494094\pi$
$168$		−	382.216i		−	2.27510i
$169$	−107.836			−0.638086
$170$			368.448i			2.16734i
$171$	45.6261			0.266819
$172$			647.692i			3.76565i
$173$		−	79.2156i		−	0.457894i	−0.973439	−	0.228947i	$-0.926472\pi$
$173$		−	79.2156i		−	0.457894i	0.973439	−	0.228947i	$-0.0735282\pi$
$174$	−363.657			−2.08998
$175$			417.826i			2.38758i
$176$	−867.426			−4.92856
$177$	14.7673			0.0834311
$178$	375.680			2.11056
$179$	344.565			1.92494			0.962471	−	0.271383i	$-0.0874811\pi$
$179$	344.565			1.92494			0.962471	+	0.271383i	$0.0874811\pi$
$180$			450.205i			2.50114i
$181$	−274.003			−1.51383			−0.756915	−	0.653513i	$-0.773294\pi$
$181$	−274.003			−1.51383			−0.756915	+	0.653513i	$0.773294\pi$
$182$	−193.442			−1.06287
$183$	130.606			0.713693
$184$	224.773			1.22159
$185$	−18.8544			−0.101916
$186$		−	36.4291i		−	0.195855i
$187$			137.617i			0.735921i
$188$	−512.375			−2.72540
$189$	−183.765			−0.972299
$190$	−396.127			−2.08488
$191$	−6.31849			−0.0330811			−0.0165405	−	0.999863i	$-0.505265\pi$
$191$	−6.31849			−0.0330811			−0.0165405	+	0.999863i	$0.505265\pi$
$192$			583.542i			3.03928i
$193$	194.707			1.00885			0.504423	−	0.863457i	$-0.331705\pi$
$193$	194.707			1.00885			0.504423	+	0.863457i	$0.331705\pi$
$194$			565.369i			2.91427i
$195$	−162.088			−0.831219
$196$	−94.1018			−0.480111
$197$			162.945i			0.827134i	0.910474	+	0.413567i	$0.135717\pi$
$197$			162.945i			0.827134i	−0.910474	+	0.413567i	$0.864283\pi$
$198$			228.759i			1.15535i
$199$	−144.592			−0.726595			−0.363298	−	0.931673i	$-0.618349\pi$
$199$	−144.592			−0.726595			−0.363298	+	0.931673i	$0.618349\pi$
$200$		−	1810.40i		−	9.05199i
$201$			113.377i			0.564062i
$202$	740.832			3.66748
$203$		−	273.665i		−	1.34810i
$204$	240.637			1.17959
$205$	−57.8850			−0.282366
$206$	−393.355			−1.90949
$207$		−	34.7235i		−	0.167747i
$208$	490.979			2.36048
$209$	−147.955			−0.707920
$210$	512.636			2.44112
$211$			181.070i			0.858152i	0.903268	+	0.429076i	$0.141161\pi$
$211$			181.070i			0.858152i	−0.903268	+	0.429076i	$0.858839\pi$
$212$	−866.194			−4.08582
$213$	153.649			0.721357
$214$	526.873			2.46202
$215$	−555.605			−2.58421
$216$	796.234			3.68627
$217$	27.4142			0.126333
$218$	316.910			1.45372
$219$		−	239.471i		−	1.09348i
$220$		−	1459.92i		−	6.63598i
$221$		−	77.8938i		−	0.352461i
$222$			16.7522i			0.0754604i
$223$			135.040i			0.605563i	0.953060	+	0.302781i	$0.0979152\pi$
$223$			135.040i			0.605563i	−0.953060	+	0.302781i	$0.902085\pi$
$224$	−850.525			−3.79699
$225$	−279.675			−1.24300
$226$		−	188.156i		−	0.832550i
$227$	183.758			0.809508			0.404754	−	0.914426i	$-0.367357\pi$
$227$	183.758			0.809508			0.404754	+	0.914426i	$0.367357\pi$
$228$			258.715i			1.13471i
$229$		−	420.201i		−	1.83494i	−0.397805	−	0.917470i	$-0.630228\pi$
$229$		−	420.201i		−	1.83494i	0.397805	−	0.917470i	$-0.369772\pi$
$230$			301.470i			1.31074i
$231$	191.472			0.828884
$232$			1185.76i			5.11105i
$233$	304.272			1.30589			0.652945	−	0.757405i	$-0.273533\pi$
$233$	304.272			1.30589			0.652945	+	0.757405i	$0.273533\pi$
$234$		−	129.482i		−	0.553342i
$235$		−	439.527i		−	1.87033i
$236$		−	75.2851i		−	0.319005i
$237$	−307.915			−1.29922
$238$			246.355i			1.03510i
$239$	−43.4717			−0.181890			−0.0909450	−	0.995856i	$-0.528989\pi$
$239$	−43.4717			−0.181890			−0.0909450	+	0.995856i	$0.528989\pi$
$240$	−1301.13			−5.42138
$241$			191.255i			0.793588i	0.917908	+	0.396794i	$0.129877\pi$
$241$			191.255i			0.793588i	−0.917908	+	0.396794i	$0.870123\pi$
$242$		−	271.652i		−	1.12253i
$243$		−	206.486i		−	0.849736i
$244$		−	665.841i		−	2.72886i
$245$		−	80.7228i		−	0.329481i
$246$			51.4309i			0.209069i
$247$	83.7454			0.339050
$248$	−118.783			−0.478964
$249$	−179.572			−0.721172
$250$	1503.32			6.01327
$251$			70.6208i			0.281358i	0.990055	+	0.140679i	$0.0449285\pi$
$251$			70.6208i			0.281358i	−0.990055	+	0.140679i	$0.955071\pi$
$252$			301.021i			1.19453i
$253$			112.601i			0.445062i
$254$		−	673.318i		−	2.65086i
$255$			206.424i			0.809507i
$256$	898.265			3.50885
$257$		−	67.6759i		−	0.263330i	−0.991294	−	0.131665i	$-0.957968\pi$
$257$		−	67.6759i		−	0.263330i	0.991294	−	0.131665i	$-0.0420324\pi$
$258$			493.656i			1.91340i
$259$	−12.6066			−0.0486743
$260$			826.339i			3.17823i
$261$	183.180			0.701838
$262$			316.861i			1.20939i
$263$	207.319			0.788285			0.394143	−	0.919049i	$-0.371042\pi$
$263$	207.319			0.788285			0.394143	+	0.919049i	$0.371042\pi$
$264$	−829.630			−3.14254
$265$		−	743.041i		−	2.80393i
$266$	−264.862			−0.995721
$267$	210.476			0.788300
$268$	578.004			2.15673
$269$	−82.1496			−0.305389			−0.152694	−	0.988273i	$-0.548795\pi$
$269$	−82.1496			−0.305389			−0.152694	+	0.988273i	$0.548795\pi$
$270$			1067.92i			3.95528i
$271$		−	339.860i		−	1.25410i	−0.778981	−	0.627048i	$-0.784263\pi$
$271$		−	339.860i		−	1.25410i	0.778981	−	0.627048i	$-0.215737\pi$
$272$		−	625.278i		−	2.29882i
$273$	−108.377			−0.396984
$274$			441.664i			1.61191i
$275$	906.924			3.29790
$276$	196.894			0.713382
$277$	−313.530			−1.13188			−0.565939	−	0.824447i	$-0.691486\pi$
$277$	−313.530			−1.13188			−0.565939	+	0.824447i	$0.691486\pi$
$278$			730.696i			2.62840i
$279$			18.3499i			0.0657702i
$280$		−	1671.53i		−	5.96976i
$281$			69.3864i			0.246927i	0.992349	+	0.123463i	$0.0394002\pi$
$281$			69.3864i			0.246927i	−0.992349	+	0.123463i	$0.960600\pi$
$282$	−390.521			−1.38483
$283$		−	7.47275i		−	0.0264055i	−0.999913	−	0.0132027i	$-0.995797\pi$
$283$		−	7.47275i		−	0.0264055i	0.999913	−	0.0132027i	$-0.00420268\pi$
$284$		−	783.318i		−	2.75816i
$285$	−221.931			−0.778707
$286$			419.881i			1.46812i
$287$	−38.7036			−0.134856
$288$		−	569.306i		−	1.97676i
$289$	189.800			0.656746
$290$	−1590.37			−5.48403
$291$			316.750i			1.08849i
$292$	−1220.85			−4.18099
$293$	102.108			0.348493			0.174246	−	0.984702i	$-0.444251\pi$
$293$	102.108			0.348493			0.174246	+	0.984702i	$0.444251\pi$
$294$	−71.7223			−0.243953
$295$	64.5813			0.218920
$296$	54.6234			0.184538
$297$			398.875i			1.34301i
$298$		−	973.219i		−	3.26584i
$299$		−	63.7341i		−	0.213157i
$300$		−	1585.85i		−	5.28615i
$301$	−371.494			−1.23420
$302$	539.881			1.78769
$303$	415.054			1.36981
$304$	672.251			2.21135
$305$	571.174			1.87270
$306$	−164.900			−0.538888
$307$		−	111.754i		−	0.364019i	−0.983297	−	0.182009i	$-0.941740\pi$
$307$		−	111.754i		−	0.364019i	0.983297	−	0.182009i	$-0.0582601\pi$
$308$		−	976.143i		−	3.16930i
$309$	−220.378			−0.713199
$310$		−	159.314i		−	0.513916i
$311$	−393.530			−1.26537			−0.632685	−	0.774410i	$-0.718047\pi$
$311$	−393.530			−1.26537			−0.632685	+	0.774410i	$0.718047\pi$
$312$	469.586			1.50508
$313$	−143.346			−0.457973			−0.228987	−	0.973430i	$-0.573541\pi$
$313$	−143.346			−0.457973			−0.228987	+	0.973430i	$0.573541\pi$
$314$			357.597i			1.13884i
$315$	−258.223			−0.819754
$316$			1569.78i			4.96766i
$317$	295.585			0.932445			0.466223	−	0.884667i	$-0.345615\pi$
$317$	295.585			0.932445			0.466223	+	0.884667i	$0.345615\pi$
$318$	−660.194			−2.07608
$319$	−594.011			−1.86210
$320$			2551.98i			7.97495i
$321$	295.182			0.919571
$322$			201.572i			0.626000i
$323$		−	106.653i		−	0.330194i
$324$	271.876			0.839124
$325$	−513.335			−1.57949
$326$	751.942			2.30657
$327$	177.550			0.542967
$328$	167.699			0.511278
$329$		−	293.881i		−	0.893255i
$330$		−	1112.72i		−	3.37187i
$331$			49.2613i			0.148826i	0.997228	+	0.0744128i	$0.0237082\pi$
$331$			49.2613i			0.148826i	−0.997228	+	0.0744128i	$0.976292\pi$
$332$			915.474i			2.75745i
$333$		−	8.43835i		−	0.0253404i
$334$		−	24.0788i		−	0.0720923i
$335$			495.826i			1.48008i
$336$	−869.974			−2.58921
$337$		−	284.948i		−	0.845543i	−0.906236	−	0.422772i	$-0.861057\pi$
$337$		−	284.948i		−	0.845543i	0.906236	−	0.422772i	$-0.138943\pi$
$338$			419.015i			1.23969i
$339$		−	105.415i		−	0.310960i
$340$	1052.37			3.09521
$341$		−	59.5046i		−	0.174500i
$342$		−	177.287i		−	0.518384i
$343$		−	365.889i		−	1.06673i
$344$	1609.65			4.67921
$345$			168.900i			0.489565i
$346$	−307.805			−0.889609
$347$	−454.137			−1.30875			−0.654376	−	0.756170i	$-0.727068\pi$
$347$	−454.137			−1.30875			−0.654376	+	0.756170i	$0.727068\pi$
$348$			1038.69i			2.98473i
$349$	123.225			0.353081			0.176540	−	0.984293i	$-0.443509\pi$
$349$	123.225			0.353081			0.176540	+	0.984293i	$0.443509\pi$
$350$	1623.53			4.63865
$351$		−	225.771i		−	0.643221i
$352$			1846.13i			5.24469i
$353$	532.412			1.50825			0.754125	−	0.656731i	$-0.228061\pi$
$353$	532.412			1.50825			0.754125	+	0.656731i	$0.228061\pi$
$354$		−	57.3806i		−	0.162092i
$355$	671.948			1.89281
$356$		−	1073.03i		−	3.01412i
$357$			138.021i			0.386614i
$358$		−	1338.86i		−	3.73983i
$359$		−	330.282i		−	0.920005i	−0.887918	−	0.460002i	$-0.847849\pi$
$359$		−	330.282i		−	0.920005i	0.887918	−	0.460002i	$-0.152151\pi$
$360$	1118.85			3.10793
$361$	−246.335			−0.682369
$362$			1064.68i			2.94111i
$363$		−	152.194i		−	0.419267i
$364$			552.515i			1.51790i
$365$		−	1047.27i		−	2.86924i
$366$		−	507.489i		−	1.38658i
$367$	97.6017			0.265945			0.132972	−	0.991120i	$-0.457548\pi$
$367$	97.6017			0.265945			0.132972	+	0.991120i	$0.457548\pi$
$368$		−	511.614i		−	1.39025i
$369$		−	25.9066i		−	0.0702075i
$370$			73.2619i			0.198005i
$371$		−	496.819i		−	1.33914i
$372$	−104.050			−0.279704
$373$		−	400.414i		−	1.07350i	−0.843743	−	0.536748i	$-0.819653\pi$
$373$		−	400.414i		−	1.07350i	0.843743	−	0.536748i	$-0.180347\pi$
$374$	534.733			1.42977
$375$	842.239			2.24597
$376$			1273.36i			3.38659i
$377$	336.221			0.891833
$378$			714.046i			1.88901i
$379$	337.227			0.889781			0.444891	−	0.895585i	$-0.353243\pi$
$379$	337.227			0.889781			0.444891	+	0.895585i	$0.353243\pi$
$380$			1131.43i			2.97744i
$381$		−	377.229i		−	0.990103i
$382$			24.5515i			0.0642708i
$383$	−410.459			−1.07169			−0.535847	−	0.844315i	$-0.680008\pi$
$383$	−410.459			−1.07169			−0.535847	+	0.844315i	$0.680008\pi$
$384$	1103.97			2.87492
$385$	837.358			2.17496
$386$		−	756.565i		−	1.96001i
$387$		−	248.662i		−	0.642539i
$388$	1614.82			4.16191
$389$			688.239i			1.76925i	0.466302	+	0.884625i	$0.345586\pi$
$389$			688.239i			1.76925i	−0.466302	+	0.884625i	$0.654414\pi$
$390$			629.817i			1.61492i
$391$	−81.1674			−0.207589
$392$			233.862i			0.596588i
$393$			177.523i			0.451711i
$394$	633.150			1.60698
$395$	−1346.60			−3.40910
$396$	653.389			1.64997
$397$		−	260.358i		−	0.655812i	−0.944710	−	0.327906i	$-0.893657\pi$
$397$		−	260.358i		−	0.655812i	0.944710	−	0.327906i	$-0.106343\pi$
$398$			561.836i			1.41165i
$399$	−148.390			−0.371904
$400$	−4120.70			−10.3018
$401$	−106.407			−0.265355			−0.132678	−	0.991159i	$-0.542358\pi$
$401$	−106.407			−0.265355			−0.132678	+	0.991159i	$0.542358\pi$
$402$	440.542			1.09588
$403$			33.6807i			0.0835749i
$404$		−	2115.98i		−	5.23758i
$405$			233.222i			0.575856i
$406$	−1063.37			−2.61913
$407$			27.3637i			0.0672327i
$408$		−	598.033i		−	1.46577i
$409$	27.9266			0.0682801			0.0341400	−	0.999417i	$-0.489131\pi$
$409$	27.9266			0.0682801			0.0341400	+	0.999417i	$0.489131\pi$
$410$			224.921i			0.548588i
$411$			247.444i			0.602053i
$412$			1123.51i			2.72697i
$413$	43.1810			0.104554
$414$	−134.924			−0.325903
$415$	−785.315			−1.89233
$416$		−	1044.94i		−	2.51189i
$417$			409.375i			0.981716i
$418$			574.903i			1.37537i
$419$	309.258			0.738085			0.369043	−	0.929412i	$-0.379686\pi$
$419$	309.258			0.738085			0.369043	+	0.929412i	$0.379686\pi$
$420$		−	1464.20i		−	3.48620i
$421$			277.730i			0.659690i	0.944035	+	0.329845i	$0.106997\pi$
$421$			277.730i			0.659690i	−0.944035	+	0.329845i	$0.893003\pi$
$422$	703.576			1.66724
$423$	196.712			0.465039
$424$			2152.67i			5.07705i
$425$			653.749i			1.53823i
$426$		−	597.027i		−	1.40147i
$427$	381.904			0.894389
$428$		−	1504.87i		−	3.51605i
$429$			235.240i			0.548345i
$430$			2158.89i			5.02068i
$431$		−	619.000i		−	1.43619i	−0.695943	−	0.718097i	$-0.745013\pi$
$431$		−	619.000i		−	1.43619i	0.695943	−	0.718097i	$-0.254987\pi$
$432$		−	1812.33i		−	4.19521i
$433$		−	282.898i		−	0.653344i	−0.945138	−	0.326672i	$-0.894073\pi$
$433$		−	282.898i		−	0.653344i	0.945138	−	0.326672i	$-0.105927\pi$
$434$		−	106.522i		−	0.245442i
$435$	−891.010			−2.04830
$436$		−	905.168i		−	2.07607i
$437$		−	87.2650i		−	0.199691i
$438$	−930.504			−2.12444
$439$	−422.621			−0.962691			−0.481346	−	0.876531i	$-0.659852\pi$
$439$	−422.621			−0.962691			−0.481346	+	0.876531i	$0.659852\pi$
$440$	−3628.19			−8.24589
$441$	36.1277			0.0819221
$442$	−302.668			−0.684770
$443$	−619.016			−1.39733			−0.698664	−	0.715450i	$-0.746222\pi$
$443$	−619.016			−1.39733			−0.698664	+	0.715450i	$0.746222\pi$
$444$	47.8481			0.107766
$445$	920.467			2.06847
$446$	524.721			1.17650
$447$		−	545.250i		−	1.21980i
$448$			1706.33i			3.80878i
$449$	−380.662			−0.847800			−0.423900	−	0.905709i	$-0.639339\pi$
$449$	−380.662			−0.847800			−0.423900	+	0.905709i	$0.639339\pi$
$450$			1086.72i			2.41493i
$451$			84.0093i			0.186273i
$452$	−537.417			−1.18898
$453$	302.471			0.667706
$454$		−	714.021i		−	1.57273i
$455$	−473.960			−1.04167
$456$	642.959			1.41000
$457$		−	406.273i		−	0.889000i	−0.895779	−	0.444500i	$-0.853381\pi$
$457$		−	406.273i		−	0.889000i	0.895779	−	0.444500i	$-0.146619\pi$
$458$	−1632.76			−3.56497
$459$	−287.526			−0.626419
$460$	861.068			1.87189
$461$			196.747i			0.426782i	0.976967	+	0.213391i	$0.0684509\pi$
$461$			196.747i			0.426782i	−0.976967	+	0.213391i	$0.931549\pi$
$462$		−	743.995i		−	1.61038i
$463$			393.885i			0.850723i	0.905024	+	0.425361i	$0.139853\pi$
$463$			393.885i			0.850723i	−0.905024	+	0.425361i	$0.860147\pi$
$464$	2698.95			5.81671
$465$		−	89.2563i		−	0.191949i
$466$		−	1182.30i		−	2.53712i
$467$	−67.7629			−0.145103			−0.0725513	−	0.997365i	$-0.523114\pi$
$467$	−67.7629			−0.145103			−0.0725513	+	0.997365i	$0.523114\pi$
$468$	−369.830			−0.790235
$469$			331.524i			0.706874i
$470$	−1707.85			−3.63373
$471$			200.345i			0.425361i
$472$	−187.099			−0.396396
$473$			806.357i			1.70477i
$474$			1196.45i			2.52416i
$475$	−702.861			−1.47971
$476$	703.646			1.47825
$477$	332.550			0.697170
$478$			168.916i			0.353381i
$479$	675.523			1.41028			0.705139	−	0.709070i	$-0.250885\pi$
$479$	675.523			1.41028			0.705139	+	0.709070i	$0.250885\pi$
$480$			2769.18i			5.76912i
$481$		−	15.4883i		−	0.0322003i
$482$	743.149			1.54180
$483$			112.931i			0.233813i
$484$	−775.899			−1.60310
$485$			1385.23i			2.85615i
$486$	−802.332			−1.65089
$487$		−	493.058i		−	1.01244i	−0.862405	−	0.506219i	$-0.831043\pi$
$487$		−	493.058i		−	1.01244i	0.862405	−	0.506219i	$-0.168957\pi$
$488$	−1654.75			−3.39089
$489$	421.278			0.861510
$490$	−313.661			−0.640124
$491$		−	783.006i		−	1.59472i	−0.603506	−	0.797358i	$-0.706230\pi$
$491$		−	783.006i		−	1.59472i	0.603506	−	0.797358i	$-0.293770\pi$
$492$	146.898			0.298574
$493$		−	428.189i		−	0.868537i
$494$		−	325.406i		−	0.658716i
$495$			560.492i			1.13231i
$496$			270.366i			0.545092i
$497$	449.284			0.903993
$498$			697.754i			1.40111i
$499$	235.371			0.471686			0.235843	−	0.971791i	$-0.424215\pi$
$499$	235.371			0.471686			0.235843	+	0.971791i	$0.424215\pi$
$500$		−	4293.82i		−	8.58763i
$501$		−	13.4903i		−	0.0269267i
$502$	274.408			0.546630
$503$			392.662i			0.780640i	0.920679	+	0.390320i	$0.127636\pi$
$503$			392.662i			0.780640i	−0.920679	+	0.390320i	$0.872364\pi$
$504$	748.098			1.48432
$505$	1815.14			3.59434
$506$	437.528			0.864679
$507$			234.755i			0.463028i
$508$	−1923.15			−3.78573
$509$	−148.123			−0.291007			−0.145504	−	0.989358i	$-0.546480\pi$
$509$	−148.123			−0.291007			−0.145504	+	0.989358i	$0.546480\pi$
$510$	802.093			1.57273
$511$		−	700.237i		−	1.37033i
$512$		−	1461.88i		−	2.85523i
$513$		−	309.126i		−	0.602585i
$514$	−262.965			−0.511606
$515$	−963.773			−1.87140
$516$	1410.00			2.73255
$517$	−637.892			−1.23383
$518$			48.9851i			0.0945657i
$519$	−172.449			−0.332271
$520$	2053.62			3.94927
$521$	156.783			0.300928			0.150464	−	0.988616i	$-0.451923\pi$
$521$	156.783			0.300928			0.150464	+	0.988616i	$0.451923\pi$
$522$		−	711.773i		−	1.36355i
$523$			755.353i			1.44427i	0.691753	+	0.722135i	$0.256839\pi$
$523$			755.353i			1.44427i	−0.691753	+	0.722135i	$0.743161\pi$
$524$	905.027			1.72715
$525$	909.587			1.73255
$526$		−	805.570i		−	1.53150i
$527$	42.8935			0.0813918
$528$			1888.35i			3.57641i
$529$	462.587			0.874456
$530$	−2887.20			−5.44755
$531$			28.9035i			0.0544323i
$532$			756.506i			1.42200i
$533$		−	47.5508i		−	0.0892134i
$534$		−	817.837i		−	1.53153i
$535$	1290.91			2.41292
$536$		−	1436.46i		−	2.67996i
$537$		−	750.101i		−	1.39684i
$538$			319.205i			0.593318i
$539$	−117.154			−0.217354
$540$	3050.23			5.64858
$541$			572.573i			1.05836i	0.848509	+	0.529180i	$0.177501\pi$
$541$			572.573i			1.05836i	−0.848509	+	0.529180i	$0.822499\pi$
$542$	−1320.58			−2.43649
$543$			596.492i			1.09851i
$544$	−1330.77			−2.44627
$545$	776.474			1.42472
$546$			421.114i			0.771272i
$547$	−461.883	+	293.041i	−0.844394	+	0.535723i
$547$	−461.883	+	293.041i	−0.844394	+	0.535723i
$548$	1261.49			2.30199
$549$			255.630i			0.465629i
$550$		−	3523.99i		−	6.40726i
$551$	460.355			0.835491
$552$		−	489.321i		−	0.886451i
$553$	−900.374			−1.62816
$554$			1218.27i			2.19904i
$555$			41.0453i			0.0739554i
$556$	2087.04			3.75366
$557$	−626.705			−1.12514			−0.562571	−	0.826749i	$-0.690188\pi$
$557$	−626.705			−1.12514			−0.562571	+	0.826749i	$0.690188\pi$
$558$	71.3013			0.127780
$559$		−	456.413i		−	0.816481i
$560$	−3804.63			−6.79398
$561$	299.586			0.534022
$562$	269.612			0.479736
$563$	−186.265			−0.330844			−0.165422	−	0.986223i	$-0.552899\pi$
$563$	−186.265			−0.330844			−0.165422	+	0.986223i	$0.552899\pi$
$564$			1115.42i			1.97769i
$565$		−	461.009i		−	0.815945i
$566$	−29.0365			−0.0513013
$567$			155.939i			0.275025i
$568$	−1946.71			−3.42730
$569$		−	977.023i		−	1.71709i	−0.512740	−	0.858544i	$-0.671369\pi$
$569$		−	977.023i		−	1.71709i	0.512740	−	0.858544i	$-0.328631\pi$
$570$			862.349i			1.51289i
$571$	157.513			0.275854			0.137927	−	0.990442i	$-0.455956\pi$
$571$	157.513			0.275854			0.137927	+	0.990442i	$0.455956\pi$
$572$	1199.28			2.09664
$573$			13.7550i			0.0240053i
$574$			150.389i			0.262002i
$575$			534.909i			0.930277i
$576$	−1142.15			−1.98289
$577$			588.957i			1.02072i	0.859960	+	0.510361i	$0.170488\pi$
$577$			588.957i			1.02072i	−0.859960	+	0.510361i	$0.829512\pi$
$578$		−	737.496i		−	1.27594i
$579$		−	423.868i		−	0.732070i
$580$			4542.45i			7.83182i
$581$	−525.085			−0.903761
$582$	1230.78			2.11474
$583$	−1078.39			−1.84972
$584$			3034.06i			5.19531i
$585$		−	317.249i		−	0.542306i
$586$		−	396.758i		−	0.677061i
$587$	−185.681			−0.316322			−0.158161	−	0.987413i	$-0.550556\pi$
$587$	−185.681			−0.316322			−0.158161	+	0.987413i	$0.550556\pi$
$588$			204.855i			0.348393i
$589$			46.1158i			0.0782950i
$590$		−	250.941i		−	0.425323i
$591$	354.724			0.600211
$592$		−	124.330i		−	0.210017i
$593$	65.9384			0.111195			0.0555973	−	0.998453i	$-0.482294\pi$
$593$	65.9384			0.111195			0.0555973	+	0.998453i	$0.482294\pi$
$594$	1549.89			2.60925
$595$			603.604i			1.01446i
$596$	−2779.74			−4.66399
$597$			314.771i			0.527255i
$598$	−247.649			−0.414128
$599$	200.594			0.334881			0.167440	−	0.985882i	$-0.446450\pi$
$599$	200.594			0.334881			0.167440	+	0.985882i	$0.446450\pi$
$600$	−3941.15			−6.56859
$601$	479.066			0.797115			0.398558	−	0.917143i	$-0.369511\pi$
$601$	479.066			0.797115			0.398558	+	0.917143i	$0.369511\pi$
$602$			1443.50i			2.39784i
$603$	−221.908			−0.368007
$604$		−	1542.02i		−	2.55302i
$605$		−	665.584i		−	1.10014i
$606$		−	1612.76i		−	2.66131i
$607$	−751.127			−1.23744			−0.618721	−	0.785611i	$-0.712349\pi$
$607$	−751.127			−1.23744			−0.618721	+	0.785611i	$0.712349\pi$
$608$		−	1430.74i		−	2.35320i
$609$	−595.755			−0.978252
$610$		−	2219.39i		−	3.63834i
$611$	361.058			0.590930
$612$			470.991i			0.769593i
$613$	306.143			0.499418			0.249709	−	0.968321i	$-0.419665\pi$
$613$	306.143			0.499418			0.249709	+	0.968321i	$0.419665\pi$
$614$	−434.236			−0.707225
$615$			126.013i			0.204899i
$616$	−2425.92			−3.93818
$617$		−	498.504i		−	0.807947i	−0.914771	−	0.403974i	$-0.867629\pi$
$617$		−	498.504i		−	0.807947i	0.914771	−	0.403974i	$-0.132371\pi$
$618$			856.314i			1.38562i
$619$		−	58.1552i		−	0.0939502i	−0.998896	−	0.0469751i	$-0.985042\pi$
$619$		−	58.1552i		−	0.0939502i	0.998896	−	0.0469751i	$-0.0149581\pi$
$620$	−455.037			−0.733931
$621$	−235.259			−0.378839
$622$			1529.12i			2.45839i
$623$	615.452			0.987884
$624$		−	1068.84i		−	1.71288i
$625$	2042.39			3.26782
$626$			556.991i			0.889763i
$627$			322.092i			0.513703i
$628$	1021.38			1.62640
$629$	−19.7249			−0.0313592
$630$			1003.36i			1.59264i
$631$	234.513			0.371652			0.185826	−	0.982583i	$-0.440504\pi$
$631$	234.513			0.371652			0.185826	+	0.982583i	$0.440504\pi$
$632$	3901.23			6.17284
$633$	394.181			0.622719
$634$		−	1148.54i		−	1.81158i
$635$		−	1649.72i		−	2.59799i
$636$			1885.66i			2.96488i
$637$	66.3113			0.104099
$638$			2308.12i			3.61775i
$639$			300.732i			0.470629i
$640$	4827.96			7.54368
$641$		−	122.527i		−	0.191150i	−0.995422	−	0.0955752i	$-0.969531\pi$
$641$		−	122.527i		−	0.191150i	0.995422	−	0.0955752i	$-0.0304690\pi$
$642$		−	1146.98i		−	1.78657i
$643$	509.247			0.791986			0.395993	−	0.918253i	$-0.370400\pi$
$643$	509.247			0.791986			0.395993	+	0.918253i	$0.370400\pi$
$644$	575.735			0.893999
$645$			1209.53i			1.87524i
$646$	−414.415			−0.641509
$647$	303.715			0.469421			0.234710	−	0.972065i	$-0.424586\pi$
$647$	303.715			0.469421			0.234710	+	0.972065i	$0.424586\pi$
$648$		−	675.669i		−	1.04270i
$649$		−	93.7277i		−	0.144419i
$650$			1994.64i			3.06868i
$651$		−	59.6794i		−	0.0916734i
$652$		−	2147.72i		−	3.29405i
$653$			564.055i			0.863791i	0.901924	+	0.431895i	$0.142155\pi$
$653$			564.055i			0.863791i	−0.901924	+	0.431895i	$0.857845\pi$
$654$		−	689.899i		−	1.05489i
$655$			776.353i			1.18527i
$656$		−	381.705i		−	0.581867i
$657$	468.709			0.713408
$658$	−1141.92			−1.73544
$659$		−	91.7082i		−	0.139163i	−0.997576	−	0.0695813i	$-0.977834\pi$
$659$		−	91.7082i		−	0.139163i	0.997576	−	0.0695813i	$-0.0221663\pi$
$660$	−3178.17			−4.81541
$661$	−144.826			−0.219101			−0.109551	−	0.993981i	$-0.534941\pi$
$661$	−144.826			−0.219101			−0.109551	+	0.993981i	$0.534941\pi$
$662$	191.412			0.289142
$663$	−169.571			−0.255763
$664$	2275.14			3.42642
$665$	−648.948			−0.975862
$666$	−32.7885			−0.0492320
$667$		−	350.351i		−	0.525264i
$668$	−68.7746			−0.102956
$669$	293.977			0.439427
$670$	1926.61			2.87553
$671$		−	828.952i		−	1.23540i
$672$			1851.55i			2.75529i
$673$	−87.5377			−0.130071			−0.0650354	−	0.997883i	$-0.520716\pi$
$673$	−87.5377			−0.130071			−0.0650354	+	0.997883i	$0.520716\pi$
$674$	−1107.21			−1.64275
$675$			1894.86i			2.80719i
$676$	1196.80			1.77042
$677$	−886.521			−1.30948			−0.654742	−	0.755852i	$-0.727223\pi$
$677$	−886.521			−1.30948			−0.654742	+	0.755852i	$0.727223\pi$
$678$	−409.608			−0.604141
$679$			926.207i			1.36407i
$680$		−	2615.36i		−	3.84611i
$681$		−	400.033i		−	0.587420i
$682$	−231.214			−0.339024
$683$	−1053.42			−1.54235			−0.771173	−	0.636625i	$-0.780330\pi$
$683$	−1053.42			−1.54235			−0.771173	+	0.636625i	$0.780330\pi$
$684$	−506.373			−0.740311
$685$			1082.14i			1.57976i
$686$	−1421.72			−2.07247
$687$	−914.758			−1.33153
$688$		−	3663.77i		−	5.32525i
$689$	610.386			0.885901
$690$	656.287			0.951140
$691$	−251.746			−0.364322			−0.182161	−	0.983269i	$-0.558309\pi$
$691$	−251.746			−0.364322			−0.182161	+	0.983269i	$0.558309\pi$
$692$			879.160i			1.27046i
$693$			374.762i			0.540782i
$694$			1764.62i			2.54268i
$695$			1790.31i			2.57598i
$696$	2581.35			3.70884
$697$	−60.5575			−0.0868830
$698$		−	478.811i		−	0.685975i
$699$		−	662.387i		−	0.947620i
$700$		−	4637.16i		−	6.62452i
$701$	−291.272			−0.415509			−0.207754	−	0.978181i	$-0.566615\pi$
$701$	−291.272			−0.415509			−0.207754	+	0.978181i	$0.566615\pi$
$702$	−877.267			−1.24967
$703$		−	21.2067i		−	0.0301660i
$704$	3703.73			5.26098
$705$	−956.831			−1.35721
$706$		−	2068.77i		−	2.93027i
$707$	1213.66			1.71663
$708$	−163.892			−0.231486
$709$			386.215i			0.544732i	0.962194	+	0.272366i	$0.0878061\pi$
$709$			386.215i			0.544732i	−0.962194	+	0.272366i	$0.912194\pi$
$710$		−	2610.96i		−	3.67741i
$711$		−	602.672i		−	0.847640i
$712$	−2666.69			−3.74535
$713$	35.0962			0.0492233
$714$	536.303			0.751125
$715$			1028.77i			1.43884i
$716$	−3824.09			−5.34090
$717$			94.6359i			0.131989i
$718$	−1283.36			−1.78741
$719$			460.951i			0.641100i	0.947232	+	0.320550i	$0.103868\pi$
$719$			460.951i			0.641100i	−0.947232	+	0.320550i	$0.896132\pi$
$720$		−	2546.66i		−	3.53702i
$721$	−644.407			−0.893769
$722$			957.174i			1.32573i
$723$	416.352			0.575868
$724$	3040.98			4.20024
$725$	−2821.85			−3.89220
$726$	−591.373			−0.814563
$727$			1102.89i			1.51704i	0.651650	+	0.758520i	$0.274077\pi$
$727$			1102.89i			1.51704i	−0.651650	+	0.758520i	$0.725923\pi$
$728$	1373.11			1.88614
$729$	−669.983			−0.919045
$730$	−4069.34			−5.57444
$731$	−581.257			−0.795153
$732$	−1449.51			−1.98020
$733$		−	397.940i		−	0.542893i	−0.962454	−	0.271446i	$-0.912498\pi$
$733$		−	397.940i		−	0.542893i	0.962454	−	0.271446i	$-0.0875019\pi$
$734$		−	379.247i		−	0.516685i
$735$	−175.730			−0.239088
$736$	−1088.86			−1.47943
$737$	719.598			0.976388
$738$	−100.664			−0.136401
$739$		−	26.6248i		−	0.0360281i	−0.999838	−	0.0180140i	$-0.994266\pi$
$739$		−	26.6248i		−	0.0360281i	0.999838	−	0.0180140i	$-0.00573436\pi$
$740$	209.253			0.282774
$741$		−	182.310i		−	0.246032i
$742$	−1930.47			−2.60171
$743$	168.974			0.227421			0.113710	−	0.993514i	$-0.463726\pi$
$743$	168.974			0.227421			0.113710	+	0.993514i	$0.463726\pi$
$744$			258.585i			0.347560i
$745$		−	2384.52i		−	3.20070i
$746$	−1555.87			−2.08562
$747$		−	351.470i		−	0.470508i
$748$		−	1527.32i		−	2.04187i
$749$	863.141			1.15239
$750$		−	3272.65i		−	4.36353i
$751$	218.624			0.291111			0.145556	−	0.989350i	$-0.453503\pi$
$751$	218.624			0.291111			0.145556	+	0.989350i	$0.453503\pi$
$752$	2898.33			3.85416
$753$	153.738			0.204168
$754$		−	1306.44i		−	1.73268i
$755$	1322.78			1.75203
$756$	2039.48			2.69772
$757$	1430.22			1.88933			0.944663	−	0.328041i	$-0.106389\pi$
$757$	1430.22			1.88933			0.944663	+	0.328041i	$0.106389\pi$
$758$		−	1310.35i		−	1.72869i
$759$	245.127			0.322960
$760$	2811.83			3.69978
$761$	697.318			0.916318			0.458159	−	0.888870i	$-0.348509\pi$
$761$	697.318			0.916318			0.458159	+	0.888870i	$0.348509\pi$
$762$	−1465.78			−1.92360
$763$	519.174			0.680437
$764$	70.1245			0.0917861
$765$	−404.027			−0.528140
$766$			1594.90i			2.08212i
$767$			53.0516i			0.0691676i
$768$		−	1955.48i		−	2.54620i
$769$			990.704i			1.28830i	0.764898	+	0.644151i	$0.222789\pi$
$769$			990.704i			1.28830i	−0.764898	+	0.644151i	$0.777211\pi$
$770$		−	3253.69i		−	4.22557i
$771$	−147.327			−0.191086
$772$	−2160.92			−2.79912
$773$			293.001i			0.379044i	0.981876	+	0.189522i	$0.0606939\pi$
$773$			293.001i			0.379044i	−0.981876	+	0.189522i	$0.939306\pi$
$774$	−966.217			−1.24834
$775$		−	282.676i		−	0.364744i
$776$		−	4013.16i		−	5.17160i
$777$			27.4441i			0.0353205i
$778$	2674.26			3.43735
$779$		−	65.1067i		−	0.0835773i
$780$	1798.90			2.30628
$781$		−	975.207i		−	1.24866i
$782$			315.389i			0.403310i
$783$		−	1241.08i		−	1.58503i
$784$	532.302			0.678956
$785$			876.163i			1.11613i
$786$	689.791			0.877597
$787$	503.433			0.639686			0.319843	−	0.947471i	$-0.396370\pi$
$787$	503.433			0.639686			0.319843	+	0.947471i	$0.396370\pi$
$788$		−	1808.42i		−	2.29495i
$789$		−	451.324i		−	0.572020i
$790$			5232.41i			6.62330i
$791$		−	308.244i		−	0.389689i
$792$		−	1623.81i		−	2.05026i
$793$			469.202i			0.591680i
$794$	−1011.66			−1.27413
$795$	−1617.57			−2.03467
$796$	1604.73			2.01600
$797$	848.755			1.06494			0.532469	−	0.846450i	$-0.321264\pi$
$797$	848.755			1.06494			0.532469	+	0.846450i	$0.321264\pi$
$798$			576.592i			0.722546i
$799$		−	459.820i		−	0.575494i
$800$			8770.04i			10.9626i
$801$			411.957i			0.514304i
$802$			413.463i			0.515540i
$803$	−1519.92			−1.89280
$804$		−	1258.29i		−	1.56504i
$805$			493.879i			0.613515i
$806$	130.872			0.162372
$807$			178.836i			0.221606i
$808$	−5258.66			−6.50824
$809$		−	790.278i		−	0.976858i	−0.872604	−	0.488429i	$-0.837570\pi$
$809$		−	790.278i		−	0.976858i	0.872604	−	0.488429i	$-0.162430\pi$
$810$	906.219			1.11879
$811$	−1469.54			−1.81201			−0.906005	−	0.423267i	$-0.860883\pi$
$811$	−1469.54			−1.81201			−0.906005	+	0.423267i	$0.860883\pi$
$812$			3037.22i			3.74042i
$813$	−739.859			−0.910035
$814$	106.326			0.130621
$815$	1842.36			2.26057
$816$	−1361.20			−1.66814
$817$		−	624.923i		−	0.764899i
$818$		−	108.513i		−	0.132656i
$819$		−	212.122i		−	0.259001i
$820$	642.426			0.783446
$821$			20.8871i			0.0254410i	0.999919	+	0.0127205i	$0.00404917\pi$
$821$			20.8871i			0.0254410i	−0.999919	+	0.0127205i	$0.995951\pi$
$822$	961.482			1.16969
$823$	−1190.97			−1.44710			−0.723552	−	0.690270i	$-0.757492\pi$
$823$	−1190.97			−1.44710			−0.723552	+	0.690270i	$0.757492\pi$
$824$	2792.15			3.38854
$825$		−	1974.33i		−	2.39313i
$826$		−	167.786i		−	0.203131i
$827$			841.451i			1.01747i	0.860922	+	0.508737i	$0.169887\pi$
$827$			841.451i			1.01747i	−0.860922	+	0.508737i	$0.830113\pi$
$828$			385.373i			0.465426i
$829$	−54.0214			−0.0651645			−0.0325823	−	0.999469i	$-0.510373\pi$
$829$	−54.0214			−0.0651645			−0.0325823	+	0.999469i	$0.510373\pi$
$830$			3051.46i			3.67646i
$831$			682.541i			0.821349i
$832$	−2096.38			−2.51968
$833$		−	84.4496i		−	0.101380i
$834$	1590.69			1.90730
$835$		−	58.9965i		−	0.0706545i
$836$	1642.06			1.96418
$837$	124.324			0.148536
$838$		−	1201.67i		−	1.43397i
$839$	−419.580			−0.500096			−0.250048	−	0.968233i	$-0.580446\pi$
$839$	−419.580			−0.500096			−0.250048	+	0.968233i	$0.580446\pi$
$840$	−3638.85			−4.33196
$841$	1007.23			1.19766
$842$	1079.16			1.28166
$843$	151.051			0.179183
$844$		−	2009.57i		−	2.38101i
$845$			1026.65i			1.21497i
$846$		−	764.353i		−	0.903491i
$847$		−	445.029i		−	0.525418i
$848$	4899.76			5.77802
$849$	−16.2678			−0.0191612
$850$	2540.24			2.98852
$851$	−16.1393			−0.0189651
$852$	−1705.25			−2.00146
$853$	755.304			0.885468			0.442734	−	0.896653i	$-0.354009\pi$
$853$	755.304			0.885468			0.442734	+	0.896653i	$0.354009\pi$
$854$		−	1483.95i		−	1.73764i
$855$		−	434.379i		−	0.508045i
$856$	−3739.91			−4.36905
$857$			341.005i			0.397905i	0.980009	+	0.198953i	$0.0637540\pi$
$857$			341.005i			0.397905i	−0.980009	+	0.198953i	$0.936246\pi$
$858$	914.062			1.06534
$859$	−778.203			−0.905941			−0.452971	−	0.891525i	$-0.649636\pi$
$859$	−778.203			−0.905941			−0.452971	+	0.891525i	$0.649636\pi$
$860$	6166.28			7.17010
$861$			84.2560i			0.0978583i
$862$	−2405.22			−2.79028
$863$			1136.65i			1.31709i	0.752541	+	0.658546i	$0.228828\pi$
$863$			1136.65i			1.31709i	−0.752541	+	0.658546i	$0.771172\pi$
$864$	−3857.16			−4.46431
$865$	−754.164			−0.871866
$866$	−1099.24			−1.26933
$867$		−	413.185i		−	0.476569i
$868$	−304.251			−0.350520
$869$			1954.33i			2.24894i
$870$			3462.16i			3.97949i
$871$	−407.306			−0.467630
$872$	−2249.53			−2.57974
$873$	−619.964			−0.710153
$874$	−339.082			−0.387965
$875$	2462.79			2.81461
$876$			2657.73i			3.03394i
$877$			202.562i			0.230972i	0.993309	+	0.115486i	$0.0368425\pi$
$877$			202.562i			0.230972i	−0.993309	+	0.115486i	$0.963157\pi$
$878$			1642.16i			1.87034i
$879$		−	222.285i		−	0.252884i
$880$			8258.24i			9.38437i
$881$			888.636i			1.00867i	0.863509	+	0.504334i	$0.168262\pi$
$881$			888.636i			1.00867i	−0.863509	+	0.504334i	$0.831738\pi$
$882$		−	140.380i		−	0.159161i
$883$	−16.1835			−0.0183278			−0.00916391	−	0.999958i	$-0.502917\pi$
$883$	−16.1835			−0.0183278			−0.00916391	+	0.999958i	$0.502917\pi$
$884$			864.490i			0.977930i
$885$		−	140.591i		−	0.158859i
$886$			2405.28i			2.71477i
$887$	−1526.66			−1.72115			−0.860576	−	0.509321i	$-0.829896\pi$
$887$	−1526.66			−1.72115			−0.860576	+	0.509321i	$0.829896\pi$
$888$		−	118.912i		−	0.133910i
$889$		−	1103.05i		−	1.24078i
$890$		−	3576.62i		−	4.01867i
$891$	338.478			0.379885
$892$		−	1498.72i		−	1.68018i
$893$	494.363			0.553598
$894$	−2118.65			−2.36986
$895$		−	3280.39i		−	3.66524i
$896$	3228.11			3.60281
$897$	−138.746			−0.154678
$898$			1479.12i			1.64713i
$899$			185.146i			0.205946i
$900$	3103.92			3.44880
$901$		−	777.347i		−	0.862760i
$902$	326.431			0.361897
$903$			808.725i			0.895598i
$904$			1335.59i			1.47743i
$905$			2608.62i			2.88245i
$906$		−	1175.30i		−	1.29724i
$907$	−584.024			−0.643907			−0.321953	−	0.946755i	$-0.604339\pi$
$907$	−584.024			−0.643907			−0.321953	+	0.946755i	$0.604339\pi$
$908$	−2039.41			−2.24604
$909$			812.370i			0.893697i
$910$			1841.64i			2.02379i
$911$		−	221.489i		−	0.243127i	−0.992584	−	0.121564i	$-0.961209\pi$
$911$		−	221.489i		−	0.243127i	0.992584	−	0.121564i	$-0.0387909\pi$
$912$		−	1463.46i		−	1.60467i
$913$			1139.74i			1.24834i
$914$	−1578.64			−1.72717
$915$		−	1243.42i		−	1.35893i
$916$			4663.52i			5.09118i
$917$			519.093i			0.566077i
$918$			1117.23i			1.21702i
$919$	−425.859			−0.463394			−0.231697	−	0.972788i	$-0.574428\pi$
$919$	−425.859			−0.463394			−0.231697	+	0.972788i	$0.574428\pi$
$920$		−	2139.93i		−	2.32601i
$921$	−243.283			−0.264150
$922$	764.490			0.829165
$923$			551.985i			0.598033i
$924$	−2125.02			−2.29980
$925$			129.991i			0.140531i
$926$	1530.50			1.65281
$927$		−	431.339i		−	0.465306i
$928$		−	5744.14i		−	6.18981i
$929$			928.599i			0.999569i	0.866150	+	0.499784i	$0.166587\pi$
$929$			928.599i			0.999569i	−0.866150	+	0.499784i	$0.833413\pi$
$930$	−346.819			−0.372924
$931$	90.7937			0.0975228
$932$	−3376.91			−3.62329
$933$			856.696i			0.918216i
$934$			263.303i			0.281909i
$935$	1310.17			1.40125
$936$			919.104i			0.981948i
$937$			1414.10i			1.50918i	0.656198	+	0.754589i	$0.272164\pi$
$937$			1414.10i			1.50918i	−0.656198	+	0.754589i	$0.727836\pi$
$938$	1288.19			1.37333
$939$			312.057i			0.332329i
$940$			4878.01i			5.18938i
$941$	−650.452			−0.691235			−0.345618	−	0.938375i	$-0.612331\pi$
$941$	−650.452			−0.691235			−0.345618	+	0.938375i	$0.612331\pi$
$942$	778.473			0.826404
$943$	−49.5492			−0.0525442
$944$			425.862i			0.451125i
$945$			1749.51i			1.85133i
$946$	3133.23			3.31208
$947$	858.474			0.906520			0.453260	−	0.891378i	$-0.350261\pi$
$947$	858.474			0.906520			0.453260	+	0.891378i	$0.350261\pi$
$948$	3417.34			3.60479
$949$	860.302			0.906535
$950$			2731.08i			2.87482i
$951$		−	643.475i		−	0.676630i
$952$		−	1748.70i		−	1.83687i
$953$	232.834			0.244317			0.122158	−	0.992511i	$-0.461018\pi$
$953$	232.834			0.244317			0.122158	+	0.992511i	$0.461018\pi$
$954$		−	1292.17i		−	1.35448i
$955$			60.1545i			0.0629890i
$956$	482.463			0.504668
$957$			1293.13i			1.35124i
$958$		−	2624.85i		−	2.73992i
$959$			723.549i			0.754483i
$960$	5555.55			5.78703
$961$	942.453			0.980700
$962$	−60.1824			−0.0625597
$963$			577.750i			0.599948i
$964$		−	2122.60i		−	2.20187i
$965$		−	1853.69i		−	1.92092i
$966$	438.813			0.454257
$967$		−	218.714i		−	0.226178i	−0.993585	−	0.113089i	$-0.963925\pi$
$967$		−	218.714i		−	0.226178i	0.993585	−	0.113089i	$-0.0360746\pi$
$968$			1928.27i			1.99201i
$969$	−232.178			−0.239605
$970$	5382.53			5.54900
$971$			1139.36i			1.17339i	0.809807	+	0.586697i	$0.199572\pi$
$971$			1139.36i			1.17339i	−0.809807	+	0.586697i	$0.800428\pi$
$972$			2291.64i			2.35766i
$973$			1197.05i			1.23027i
$974$	−1915.85			−1.96699
$975$			1117.51i			1.14616i
$976$			3766.43i			3.85905i
$977$		−	1720.05i		−	1.76054i	−0.474473	−	0.880270i	$-0.657361\pi$
$977$		−	1720.05i		−	1.76054i	0.474473	−	0.880270i	$-0.342639\pi$
$978$		−	1636.94i		−	1.67377i
$979$		−	1335.89i		−	1.36454i
$980$			895.887i			0.914170i
$981$			347.513i			0.354244i
$982$	−3042.49			−3.09826
$983$		−	1502.97i		−	1.52896i	−0.644649	−	0.764479i	$-0.722996\pi$
$983$		−	1502.97i		−	1.52896i	0.644649	−	0.764479i	$-0.277004\pi$
$984$		−	365.073i		−	0.371009i
$985$	1551.30			1.57493
$986$	−1663.79			−1.68742
$987$	−639.765			−0.648192
$988$	−929.433			−0.940722
$989$	−475.594			−0.480884
$990$	2177.88			2.19988
$991$	821.527			0.828988			0.414494	−	0.910052i	$-0.363959\pi$
$991$	821.527			0.828988			0.414494	+	0.910052i	$0.363959\pi$
$992$	575.415			0.580056
$993$	107.239			0.107995
$994$		−	1745.76i		−	1.75630i
$995$			1376.58i			1.38349i
$996$	1992.95			2.00095
$997$		−	916.568i		−	0.919326i	−0.888093	−	0.459663i	$-0.847970\pi$
$997$		−	916.568i		−	0.919326i	0.888093	−	0.459663i	$-0.152030\pi$
$998$		−	914.572i		−	0.916405i
$999$	−57.1716			−0.0572288

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.3.b.b.546.2	✓	88
547.546	odd	2	inner	547.3.b.b.546.87	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.3.b.b.546.2	✓	88	1.1	even	1	trivial
547.3.b.b.546.87	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)

\(f(q)\)	\(=\)	\(q-3.88566i q^{2} -2.17695i q^{3} -11.0983 q^{4} -9.52039i q^{5} -8.45889 q^{6} -6.36562i q^{7} +27.5816i q^{8} +4.26087 q^{9} +O(q^{10})\)	\(q-3.88566i q^{2} -2.17695i q^{3} -11.0983 q^{4} -9.52039i q^{5} -8.45889 q^{6} -6.36562i q^{7} +27.5816i q^{8} +4.26087 q^{9} -36.9930 q^{10} -13.8171 q^{11} +24.1605i q^{12} +7.82071 q^{13} -24.7346 q^{14} -20.7255 q^{15} +62.7793 q^{16} -9.95994i q^{17} -16.5563i q^{18} +10.7082 q^{19} +105.660i q^{20} -13.8576 q^{21} +53.6884i q^{22} -8.14939i q^{23} +60.0439 q^{24} -65.6379 q^{25} -30.3886i q^{26} -28.8683i q^{27} +70.6476i q^{28} +42.9911 q^{29} +80.5320i q^{30} +4.30660i q^{31} -133.612i q^{32} +30.0791i q^{33} -38.7009 q^{34} -60.6032 q^{35} -47.2885 q^{36} -1.98043i q^{37} -41.6082i q^{38} -17.0253i q^{39} +262.588 q^{40} -6.08011i q^{41} +53.8460i q^{42} -58.3595i q^{43} +153.346 q^{44} -40.5652i q^{45} -31.6657 q^{46} +46.1669 q^{47} -136.668i q^{48} +8.47893 q^{49} +255.046i q^{50} -21.6823 q^{51} -86.7967 q^{52} +78.0473 q^{53} -112.172 q^{54} +131.544i q^{55} +175.574 q^{56} -23.3112i q^{57} -167.049i q^{58} +6.78347i q^{59} +230.018 q^{60} +59.9948i q^{61} +16.7340 q^{62} -27.1231i q^{63} -268.054 q^{64} -74.4563i q^{65} +116.877 q^{66} -52.0804 q^{67} +110.539i q^{68} -17.7408 q^{69} +235.483i q^{70} +70.5799i q^{71} +117.522i q^{72} +110.003 q^{73} -7.69526 q^{74} +142.891i q^{75} -118.843 q^{76} +87.9542i q^{77} -66.1545 q^{78} -141.443i q^{79} -597.684i q^{80} -24.4971 q^{81} -23.6252 q^{82} -82.4877i q^{83} +153.797 q^{84} -94.8225 q^{85} -226.765 q^{86} -93.5896i q^{87} -381.097i q^{88} +96.6837i q^{89} -157.622 q^{90} -49.7836i q^{91} +90.4445i q^{92} +9.37527 q^{93} -179.389i q^{94} -101.946i q^{95} -290.868 q^{96} -145.501 q^{97} -32.9462i q^{98} -58.8728 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) 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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