LMFDB - Embedded newform 547.3.b.b.546.6

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			546.6
Character	$\chi$	$=$	547.546
Dual form			547.3.b.b.546.83

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.54864i q^{2} +2.90313i q^{3} -8.59283 q^{4} +1.02412i q^{5} +10.3021 q^{6} -3.84607i q^{7} +16.2983i q^{8} +0.571853 q^{9} +O(q^{10})$	\(q-3.54864i q^{2} +2.90313i q^{3} -8.59283 q^{4} +1.02412i q^{5} +10.3021 q^{6} -3.84607i q^{7} +16.2983i q^{8} +0.571853 q^{9} +3.63422 q^{10} -7.88798 q^{11} -24.9461i q^{12} -9.08393 q^{13} -13.6483 q^{14} -2.97315 q^{15} +23.4654 q^{16} +12.4775i q^{17} -2.02930i q^{18} +16.6325 q^{19} -8.80007i q^{20} +11.1656 q^{21} +27.9916i q^{22} +17.2618i q^{23} -47.3160 q^{24} +23.9512 q^{25} +32.2356i q^{26} +27.7883i q^{27} +33.0486i q^{28} +30.3989 q^{29} +10.5506i q^{30} +20.0465i q^{31} -18.0770i q^{32} -22.8998i q^{33} +44.2782 q^{34} +3.93883 q^{35} -4.91383 q^{36} +31.5954i q^{37} -59.0228i q^{38} -26.3718i q^{39} -16.6914 q^{40} +10.1428i q^{41} -39.6228i q^{42} +7.78099i q^{43} +67.7801 q^{44} +0.585645i q^{45} +61.2558 q^{46} +17.9012 q^{47} +68.1230i q^{48} +34.2077 q^{49} -84.9941i q^{50} -36.2238 q^{51} +78.0567 q^{52} +8.21380 q^{53} +98.6106 q^{54} -8.07823i q^{55} +62.6843 q^{56} +48.2864i q^{57} -107.875i q^{58} -93.0214i q^{59} +25.5477 q^{60} +53.2269i q^{61} +71.1377 q^{62} -2.19939i q^{63} +29.7128 q^{64} -9.30302i q^{65} -81.2631 q^{66} -0.609827 q^{67} -107.217i q^{68} -50.1132 q^{69} -13.9775i q^{70} +82.9341i q^{71} +9.32022i q^{72} -45.6883 q^{73} +112.121 q^{74} +69.5333i q^{75} -142.920 q^{76} +30.3377i q^{77} -93.5840 q^{78} -38.7556i q^{79} +24.0313i q^{80} -75.5263 q^{81} +35.9931 q^{82} +100.618i q^{83} -95.9444 q^{84} -12.7785 q^{85} +27.6119 q^{86} +88.2518i q^{87} -128.561i q^{88} +107.956i q^{89} +2.07824 q^{90} +34.9374i q^{91} -148.328i q^{92} -58.1975 q^{93} -63.5250i q^{94} +17.0337i q^{95} +52.4798 q^{96} +12.7041 q^{97} -121.391i q^{98} -4.51076 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$		−	3.54864i		−	1.77432i	−0.461463	−	0.887159i	$-0.652675\pi$
$2$		−	3.54864i		−	1.77432i	0.461463	−	0.887159i	$-0.347325\pi$
$3$			2.90313i			0.967709i	0.875148	+	0.483855i	$0.160764\pi$
$3$			2.90313i			0.967709i	−0.875148	+	0.483855i	$0.839236\pi$
$4$	−8.59283			−2.14821
$5$			1.02412i			0.204824i	0.994742	+	0.102412i	$0.0326560\pi$
$5$			1.02412i			0.204824i	−0.994742	+	0.102412i	$0.967344\pi$
$6$	10.3021			1.71702
$7$		−	3.84607i		−	0.549439i	−0.961524	−	0.274719i	$-0.911415\pi$
$7$		−	3.84607i		−	0.549439i	0.961524	−	0.274719i	$-0.0885849\pi$
$8$			16.2983i			2.03729i
$9$	0.571853			0.0635392
$10$	3.63422			0.363422
$11$	−7.88798			−0.717089			−0.358545	−	0.933513i	$-0.616727\pi$
$11$	−7.88798			−0.717089			−0.358545	+	0.933513i	$0.616727\pi$
$12$		−	24.9461i		−	2.07884i
$13$	−9.08393			−0.698764			−0.349382	−	0.936980i	$-0.613608\pi$
$13$	−9.08393			−0.698764			−0.349382	+	0.936980i	$0.613608\pi$
$14$	−13.6483			−0.974879
$15$	−2.97315			−0.198210
$16$	23.4654			1.46659
$17$			12.4775i			0.733972i	0.930226	+	0.366986i	$0.119610\pi$
$17$			12.4775i			0.733972i	−0.930226	+	0.366986i	$0.880390\pi$
$18$		−	2.02930i		−	0.112739i
$19$	16.6325			0.875396			0.437698	−	0.899122i	$-0.355794\pi$
$19$	16.6325			0.875396			0.437698	+	0.899122i	$0.355794\pi$
$20$		−	8.80007i		−	0.440004i
$21$	11.1656			0.531697
$22$			27.9916i			1.27234i
$23$			17.2618i			0.750513i	0.926921	+	0.375256i	$0.122445\pi$
$23$			17.2618i			0.750513i	−0.926921	+	0.375256i	$0.877555\pi$
$24$	−47.3160			−1.97150
$25$	23.9512			0.958047
$26$			32.2356i			1.23983i
$27$			27.7883i			1.02920i
$28$			33.0486i			1.18031i
$29$	30.3989			1.04824			0.524119	−	0.851645i	$-0.324395\pi$
$29$	30.3989			1.04824			0.524119	+	0.851645i	$0.324395\pi$
$30$			10.5506i			0.351687i
$31$			20.0465i			0.646661i	0.946286	+	0.323331i	$0.104803\pi$
$31$			20.0465i			0.646661i	−0.946286	+	0.323331i	$0.895197\pi$
$32$		−	18.0770i		−	0.564906i
$33$		−	22.8998i		−	0.693934i
$34$	44.2782			1.30230
$35$	3.93883			0.112538
$36$	−4.91383			−0.136495
$37$			31.5954i			0.853931i	0.904268	+	0.426965i	$0.140417\pi$
$37$			31.5954i			0.853931i	−0.904268	+	0.426965i	$0.859583\pi$
$38$		−	59.0228i		−	1.55323i
$39$		−	26.3718i		−	0.676200i
$40$	−16.6914			−0.417284
$41$			10.1428i			0.247386i	0.992321	+	0.123693i	$0.0394737\pi$
$41$			10.1428i			0.247386i	−0.992321	+	0.123693i	$0.960526\pi$
$42$		−	39.6228i		−	0.943400i
$43$			7.78099i			0.180953i	0.995899	+	0.0904766i	$0.0288390\pi$
$43$			7.78099i			0.180953i	−0.995899	+	0.0904766i	$0.971161\pi$
$44$	67.7801			1.54046
$45$			0.585645i			0.0130143i
$46$	61.2558			1.33165
$47$	17.9012			0.380877			0.190439	−	0.981699i	$-0.439009\pi$
$47$	17.9012			0.380877			0.190439	+	0.981699i	$0.439009\pi$
$48$			68.1230i			1.41923i
$49$	34.2077			0.698117
$50$		−	84.9941i		−	1.69988i
$51$	−36.2238			−0.710271
$52$	78.0567			1.50109
$53$	8.21380			0.154977			0.0774887	−	0.996993i	$-0.475310\pi$
$53$	8.21380			0.154977			0.0774887	+	0.996993i	$0.475310\pi$
$54$	98.6106			1.82612
$55$		−	8.07823i		−	0.146877i
$56$	62.6843			1.11936
$57$			48.2864i			0.847129i
$58$		−	107.875i		−	1.85991i
$59$		−	93.0214i		−	1.57663i	−0.615270	−	0.788317i	$-0.710953\pi$
$59$		−	93.0214i		−	1.57663i	0.615270	−	0.788317i	$-0.289047\pi$
$60$	25.5477			0.425796
$61$			53.2269i			0.872572i	0.899808	+	0.436286i	$0.143706\pi$
$61$			53.2269i			0.872572i	−0.899808	+	0.436286i	$0.856294\pi$
$62$	71.1377			1.14738
$63$		−	2.19939i		−	0.0349109i
$64$	29.7128			0.464262
$65$		−	9.30302i		−	0.143123i
$66$	−81.2631			−1.23126
$67$	−0.609827			−0.00910190			−0.00455095	−	0.999990i	$-0.501449\pi$
$67$	−0.609827			−0.00910190			−0.00455095	+	0.999990i	$0.501449\pi$
$68$		−	107.217i		−	1.57672i
$69$	−50.1132			−0.726278
$70$		−	13.9775i		−	0.199678i
$71$			82.9341i			1.16809i	0.811723	+	0.584043i	$0.198530\pi$
$71$			82.9341i			1.16809i	−0.811723	+	0.584043i	$0.801470\pi$
$72$			9.32022i			0.129447i
$73$	−45.6883			−0.625867			−0.312933	−	0.949775i	$-0.601312\pi$
$73$	−45.6883			−0.625867			−0.312933	+	0.949775i	$0.601312\pi$
$74$	112.121			1.51515
$75$			69.5333i			0.927111i
$76$	−142.920			−1.88053
$77$			30.3377i			0.393997i
$78$	−93.5840			−1.19979
$79$		−	38.7556i		−	0.490577i	−0.969450	−	0.245289i	$-0.921117\pi$
$79$		−	38.7556i		−	0.490577i	0.969450	−	0.245289i	$-0.0788827\pi$
$80$			24.0313i			0.300392i
$81$	−75.5263			−0.932424
$82$	35.9931			0.438941
$83$			100.618i			1.21227i	0.795364	+	0.606133i	$0.207280\pi$
$83$			100.618i			1.21227i	−0.795364	+	0.606133i	$0.792720\pi$
$84$	−95.9444			−1.14219
$85$	−12.7785			−0.150335
$86$	27.6119			0.321069
$87$			88.2518i			1.01439i
$88$		−	128.561i		−	1.46092i
$89$			107.956i			1.21299i	0.795086	+	0.606497i	$0.207426\pi$
$89$			107.956i			1.21299i	−0.795086	+	0.606497i	$0.792574\pi$
$90$	2.07824			0.0230916
$91$			34.9374i			0.383928i
$92$		−	148.328i		−	1.61226i
$93$	−58.1975			−0.625780
$94$		−	63.5250i		−	0.675797i
$95$			17.0337i			0.179302i
$96$	52.4798			0.546665
$97$	12.7041			0.130970			0.0654849	−	0.997854i	$-0.479141\pi$
$97$	12.7041			0.130970			0.0654849	+	0.997854i	$0.479141\pi$
$98$		−	121.391i		−	1.23868i
$99$	−4.51076			−0.0455633
$100$	−205.808			−2.05808
$101$		−	98.8214i		−	0.978430i	−0.872163	−	0.489215i	$-0.837283\pi$
$101$		−	98.8214i		−	0.978430i	0.872163	−	0.489215i	$-0.162717\pi$
$102$			128.545i			1.26025i
$103$			58.8653i			0.571508i	0.958303	+	0.285754i	$0.0922440\pi$
$103$			58.8653i			0.571508i	−0.958303	+	0.285754i	$0.907756\pi$
$104$		−	148.052i		−	1.42358i
$105$			11.4349i			0.108904i
$106$		−	29.1478i		−	0.274979i
$107$			30.0902i			0.281217i	0.990065	+	0.140609i	$0.0449059\pi$
$107$			30.0902i			0.281217i	−0.990065	+	0.140609i	$0.955094\pi$
$108$		−	238.780i		−	2.21093i
$109$		−	99.4664i		−	0.912536i	−0.889842	−	0.456268i	$-0.849186\pi$
$109$		−	99.4664i		−	0.912536i	0.889842	−	0.456268i	$-0.150814\pi$
$110$	−28.6667			−0.260606
$111$	−91.7256			−0.826357
$112$		−	90.2495i		−	0.805799i
$113$	9.07840			0.0803398			0.0401699	−	0.999193i	$-0.487210\pi$
$113$	9.07840			0.0803398			0.0401699	+	0.999193i	$0.487210\pi$
$114$	171.351			1.50308
$115$	−17.6781			−0.153723
$116$	−261.212			−2.25183
$117$	−5.19467			−0.0443989
$118$	−330.099			−2.79745
$119$	47.9894			0.403273
$120$		−	48.4572i		−	0.403810i
$121$	−58.7798			−0.485783
$122$	188.883			1.54822
$123$	−29.4459			−0.239397
$124$		−	172.256i		−	1.38916i
$125$			50.1318i			0.401054i
$126$	−7.80483			−0.0619431
$127$	−119.634			−0.942004			−0.471002	−	0.882132i	$-0.656107\pi$
$127$	−119.634			−0.942004			−0.471002	+	0.882132i	$0.656107\pi$
$128$		−	177.748i		−	1.38866i
$129$	−22.5892			−0.175110
$130$	−33.0130			−0.253947
$131$	46.7502			0.356872			0.178436	−	0.983952i	$-0.442896\pi$
$131$	46.7502			0.356872			0.178436	+	0.983952i	$0.442896\pi$
$132$			196.774i			1.49071i
$133$		−	63.9699i		−	0.480977i
$134$			2.16406i			0.0161497i
$135$	−28.4585			−0.210804
$136$	−203.362			−1.49531
$137$	184.833			1.34915			0.674573	−	0.738208i	$-0.264328\pi$
$137$	184.833			1.34915			0.674573	+	0.738208i	$0.264328\pi$
$138$			177.833i			1.28865i
$139$	−50.1999			−0.361151			−0.180575	−	0.983561i	$-0.557796\pi$
$139$	−50.1999			−0.361151			−0.180575	+	0.983561i	$0.557796\pi$
$140$	−33.8457			−0.241755
$141$			51.9695i			0.368578i
$142$	294.303			2.07256
$143$	71.6539			0.501076
$144$	13.4187			0.0931857
$145$			31.1320i			0.214704i
$146$			162.131i			1.11049i
$147$			99.3094i			0.675574i
$148$		−	271.494i		−	1.83442i
$149$	26.9398			0.180804			0.0904022	−	0.995905i	$-0.471185\pi$
$149$	26.9398			0.180804			0.0904022	+	0.995905i	$0.471185\pi$
$150$	246.749			1.64499
$151$			107.400i			0.711259i	0.934627	+	0.355629i	$0.115733\pi$
$151$			107.400i			0.711259i	−0.934627	+	0.355629i	$0.884267\pi$
$152$			271.082i			1.78343i
$153$			7.13531i			0.0466360i
$154$	107.658			0.699075
$155$	−20.5300			−0.132452
$156$			226.608i			1.45262i
$157$	−127.245			−0.810481			−0.405240	−	0.914210i	$-0.632812\pi$
$157$	−127.245			−0.810481			−0.405240	+	0.914210i	$0.632812\pi$
$158$	−137.530			−0.870440
$159$			23.8457i			0.149973i
$160$	18.5130			0.115706
$161$	66.3901			0.412361
$162$			268.015i			1.65442i
$163$		−	236.970i		−	1.45380i	−0.686741	−	0.726902i	$-0.740959\pi$
$163$		−	236.970i		−	1.45380i	0.686741	−	0.726902i	$-0.259041\pi$
$164$		−	87.1554i		−	0.531435i
$165$	23.4521			0.142134
$166$	357.057			2.15095
$167$	74.8233			0.448044			0.224022	−	0.974584i	$-0.428081\pi$
$167$	74.8233			0.448044			0.224022	+	0.974584i	$0.428081\pi$
$168$			181.981i			1.08322i
$169$	−86.4822			−0.511729
$170$			45.3461i			0.266742i
$171$	9.51136			0.0556220
$172$		−	66.8607i		−	0.388725i
$173$			313.971i			1.81486i	0.420199	+	0.907432i	$0.361960\pi$
$173$			313.971i			1.81486i	−0.420199	+	0.907432i	$0.638040\pi$
$174$	313.174			1.79985
$175$		−	92.1179i		−	0.526388i
$176$	−185.094			−1.05167
$177$	270.053			1.52572
$178$	383.098			2.15224
$179$	−176.410			−0.985532			−0.492766	−	0.870162i	$-0.664014\pi$
$179$	−176.410			−0.985532			−0.492766	+	0.870162i	$0.664014\pi$
$180$		−	5.03235i		−	0.0279575i
$181$	94.6000			0.522652			0.261326	−	0.965251i	$-0.415840\pi$
$181$	94.6000			0.522652			0.261326	+	0.965251i	$0.415840\pi$
$182$	123.980			0.681211
$183$	−154.524			−0.844396
$184$	−281.337			−1.52901
$185$	−32.3575			−0.174905
$186$			206.522i			1.11033i
$187$		−	98.4225i		−	0.526323i
$188$	−153.822			−0.818203
$189$	106.876			0.565480
$190$	60.4464			0.318139
$191$	27.2341			0.142587			0.0712934	−	0.997455i	$-0.477287\pi$
$191$	27.2341			0.142587			0.0712934	+	0.997455i	$0.477287\pi$
$192$			86.2600i			0.449271i
$193$	−59.0728			−0.306077			−0.153038	−	0.988220i	$-0.548906\pi$
$193$	−59.0728			−0.306077			−0.153038	+	0.988220i	$0.548906\pi$
$194$		−	45.0821i		−	0.232382i
$195$	27.0079			0.138502
$196$	−293.941			−1.49970
$197$			209.755i			1.06475i	0.846510	+	0.532373i	$0.178700\pi$
$197$			209.755i			1.06475i	−0.846510	+	0.532373i	$0.821300\pi$
$198$			16.0071i			0.0808438i
$199$	−57.3235			−0.288058			−0.144029	−	0.989573i	$-0.546006\pi$
$199$	−57.3235			−0.288058			−0.144029	+	0.989573i	$0.546006\pi$
$200$			390.363i			1.95182i
$201$		−	1.77041i		−	0.00880799i
$202$	−350.681			−1.73605
$203$		−	116.916i		−	0.575942i
$204$	311.265			1.52581
$205$	−10.3874			−0.0506704
$206$	208.892			1.01404
$207$			9.87120i			0.0476870i
$208$	−213.158			−1.02480
$209$	−131.197			−0.627737
$210$	40.5784			0.193231
$211$		−	85.9342i		−	0.407271i	−0.979047	−	0.203636i	$-0.934724\pi$
$211$		−	85.9342i		−	0.407271i	0.979047	−	0.203636i	$-0.0652758\pi$
$212$	−70.5798			−0.332923
$213$	−240.768			−1.13037
$214$	106.779			0.498969
$215$	−7.96866			−0.0370635
$216$	−452.902			−2.09677
$217$	77.1002			0.355301
$218$	−352.970			−1.61913
$219$		−	132.639i		−	0.605657i
$220$			69.4148i			0.315522i
$221$		−	113.345i		−	0.512873i
$222$			325.501i			1.46622i
$223$		−	215.289i		−	0.965421i	−0.875780	−	0.482711i	$-0.839652\pi$
$223$		−	215.289i		−	0.965421i	0.875780	−	0.482711i	$-0.160348\pi$
$224$	−69.5254			−0.310381
$225$	13.6966			0.0608736
$226$		−	32.2159i		−	0.142548i
$227$	−95.4662			−0.420556			−0.210278	−	0.977642i	$-0.567437\pi$
$227$	−95.4662			−0.420556			−0.210278	+	0.977642i	$0.567437\pi$
$228$		−	414.916i		−	1.81981i
$229$			225.264i			0.983688i	0.870683	+	0.491844i	$0.163677\pi$
$229$			225.264i			0.983688i	−0.870683	+	0.491844i	$0.836323\pi$
$230$			62.7332i			0.272753i
$231$	−88.0743			−0.381274
$232$			495.449i			2.13556i
$233$	127.558			0.547460			0.273730	−	0.961807i	$-0.411742\pi$
$233$	127.558			0.547460			0.273730	+	0.961807i	$0.411742\pi$
$234$			18.4340i			0.0787778i
$235$			18.3330i			0.0780126i
$236$			799.317i			3.38694i
$237$	112.512			0.474736
$238$		−	170.297i		−	0.715534i
$239$	−361.407			−1.51216			−0.756082	−	0.654477i	$-0.772889\pi$
$239$	−361.407			−1.51216			−0.756082	+	0.654477i	$0.772889\pi$
$240$	−69.7660			−0.290692
$241$		−	243.981i		−	1.01237i	−0.862425	−	0.506185i	$-0.831055\pi$
$241$		−	243.981i		−	1.01237i	0.862425	−	0.506185i	$-0.168945\pi$
$242$			208.588i			0.861934i
$243$			30.8323i			0.126882i
$244$		−	457.370i		−	1.87447i
$245$			35.0328i			0.142991i
$246$			104.493i			0.424767i
$247$	−151.089			−0.611695
$248$	−326.723			−1.31743
$249$	−292.107			−1.17312
$250$	177.900			0.711598
$251$			12.0280i			0.0479203i	0.999713	+	0.0239601i	$0.00762748\pi$
$251$			12.0280i			0.0479203i	−0.999713	+	0.0239601i	$0.992373\pi$
$252$			18.8990i			0.0749958i
$253$		−	136.161i		−	0.538184i
$254$			424.539i			1.67141i
$255$		−	37.0975i		−	0.145480i
$256$	−511.912			−1.99966
$257$			102.601i			0.399227i	0.979875	+	0.199613i	$0.0639686\pi$
$257$			102.601i			0.399227i	−0.979875	+	0.199613i	$0.936031\pi$
$258$			80.1609i			0.310701i
$259$	121.518			0.469183
$260$			79.9393i			0.307459i
$261$	17.3837			0.0666041
$262$		−	165.899i		−	0.633204i
$263$	−30.4185			−0.115660			−0.0578298	−	0.998326i	$-0.518418\pi$
$263$	−30.4185			−0.115660			−0.0578298	+	0.998326i	$0.518418\pi$
$264$	373.228			1.41374
$265$			8.41190i			0.0317430i
$266$	−227.006			−0.853406
$267$	−313.411			−1.17383
$268$	5.24014			0.0195528
$269$	388.096			1.44274			0.721369	−	0.692551i	$-0.243513\pi$
$269$	388.096			1.44274			0.721369	+	0.692551i	$0.243513\pi$
$270$			100.989i			0.374033i
$271$			264.560i			0.976237i	0.872777	+	0.488118i	$0.162317\pi$
$271$			264.560i			0.976237i	−0.872777	+	0.488118i	$0.837683\pi$
$272$			292.790i			1.07643i
$273$	−101.428			−0.371531
$274$		−	655.905i		−	2.39381i
$275$	−188.926			−0.687005
$276$	430.614			1.56020
$277$	23.7439			0.0857182			0.0428591	−	0.999081i	$-0.486353\pi$
$277$	23.7439			0.0857182			0.0428591	+	0.999081i	$0.486353\pi$
$278$			178.141i			0.640796i
$279$			11.4636i			0.0410883i
$280$			64.1962i			0.229272i
$281$		−	243.444i		−	0.866348i	−0.901310	−	0.433174i	$-0.857394\pi$
$281$		−	243.444i		−	0.866348i	0.901310	−	0.433174i	$-0.142606\pi$
$282$	184.421			0.653975
$283$		−	451.442i		−	1.59520i	−0.603186	−	0.797601i	$-0.706102\pi$
$283$		−	451.442i		−	1.59520i	0.603186	−	0.797601i	$-0.293898\pi$
$284$		−	712.639i		−	2.50929i
$285$	−49.4509			−0.173512
$286$		−	254.274i		−	0.889069i
$287$	39.0100			0.135923
$288$		−	10.3374i		−	0.0358937i
$289$	133.311			0.461285
$290$	110.476			0.380953
$291$			36.8815i			0.126741i
$292$	392.591			1.34449
$293$	189.700			0.647439			0.323719	−	0.946153i	$-0.395067\pi$
$293$	189.700			0.647439			0.323719	+	0.946153i	$0.395067\pi$
$294$	352.413			1.19868
$295$	95.2649			0.322932
$296$	−514.951			−1.73970
$297$		−	219.194i		−	0.738026i
$298$		−	95.5997i		−	0.320804i
$299$		−	156.805i		−	0.524431i
$300$		−	597.488i		−	1.99163i
$301$	29.9262			0.0994227
$302$	381.124			1.26200
$303$	286.891			0.946835
$304$	390.289			1.28384
$305$	−54.5107			−0.178723
$306$	25.3206			0.0827471
$307$		−	248.926i		−	0.810835i	−0.914132	−	0.405418i	$-0.867126\pi$
$307$		−	248.926i		−	0.810835i	0.914132	−	0.405418i	$-0.132874\pi$
$308$		−	260.687i		−	0.846386i
$309$	−170.893			−0.553053
$310$			72.8535i			0.235011i
$311$	−170.123			−0.547020			−0.273510	−	0.961869i	$-0.588185\pi$
$311$	−170.123			−0.547020			−0.273510	+	0.961869i	$0.588185\pi$
$312$	429.815			1.37761
$313$	420.662			1.34397			0.671984	−	0.740565i	$-0.265442\pi$
$313$	420.662			1.34397			0.671984	+	0.740565i	$0.265442\pi$
$314$			451.548i			1.43805i
$315$	2.25243			0.00715058
$316$			333.020i			1.05386i
$317$	−261.281			−0.824232			−0.412116	−	0.911131i	$-0.635210\pi$
$317$	−261.281			−0.824232			−0.412116	+	0.911131i	$0.635210\pi$
$318$	84.6197			0.266100
$319$	−239.786			−0.751679
$320$			30.4294i			0.0950919i
$321$	−87.3557			−0.272136
$322$		−	235.594i		−	0.731659i
$323$			207.533i			0.642516i
$324$	648.985			2.00304
$325$	−217.571			−0.669449
$326$	−840.921			−2.57951
$327$	288.764			0.883069
$328$	−165.310			−0.503995
$329$		−	68.8494i		−	0.209269i
$330$		−	83.2231i		−	0.252191i
$331$		−	61.2503i		−	0.185046i	−0.995711	−	0.0925231i	$-0.970507\pi$
$331$		−	61.2503i		−	0.185046i	0.995711	−	0.0925231i	$-0.0294932\pi$
$332$		−	864.593i		−	2.60420i
$333$			18.0679i			0.0542581i
$334$		−	265.521i		−	0.794973i
$335$		−	0.624535i		−	0.00186428i
$336$	262.006			0.779779
$337$			19.2472i			0.0571133i	0.999592	+	0.0285567i	$0.00909111\pi$
$337$			19.2472i			0.0571133i	−0.999592	+	0.0285567i	$0.990909\pi$
$338$			306.894i			0.907970i
$339$			26.3557i			0.0777455i
$340$	109.803			0.322950
$341$		−	158.126i		−	0.463714i
$342$		−	33.7524i		−	0.0986911i
$343$		−	320.023i		−	0.933011i
$344$	−126.817			−0.368653
$345$		−	51.3218i		−	0.148759i
$346$	1114.17			3.22015
$347$	178.119			0.513310			0.256655	−	0.966503i	$-0.417380\pi$
$347$	178.119			0.513310			0.256655	+	0.966503i	$0.417380\pi$
$348$		−	758.332i		−	2.17912i
$349$	−178.075			−0.510243			−0.255122	−	0.966909i	$-0.582116\pi$
$349$	−178.075			−0.510243			−0.255122	+	0.966909i	$0.582116\pi$
$350$	−326.893			−0.933981
$351$		−	252.427i		−	0.719165i
$352$			142.591i			0.405088i
$353$	659.758			1.86900			0.934502	−	0.355959i	$-0.115846\pi$
$353$	659.758			1.86900			0.934502	+	0.355959i	$0.115846\pi$
$354$		−	958.320i		−	2.70712i
$355$	−84.9344			−0.239252
$356$		−	927.651i		−	2.60576i
$357$			139.319i			0.390251i
$358$			626.016i			1.74865i
$359$		−	30.7668i		−	0.0857015i	−0.999081	−	0.0428508i	$-0.986356\pi$
$359$		−	30.7668i		−	0.0857015i	0.999081	−	0.0428508i	$-0.0136440\pi$
$360$	−9.54501			−0.0265139
$361$	−84.3589			−0.233681
$362$		−	335.701i		−	0.927351i
$363$		−	170.645i		−	0.470097i
$364$		−	300.211i		−	0.824757i
$365$		−	46.7902i		−	0.128192i
$366$			548.351i			1.49823i
$367$	497.248			1.35490			0.677450	−	0.735569i	$-0.263085\pi$
$367$	497.248			1.35490			0.677450	+	0.735569i	$0.263085\pi$
$368$			405.054i			1.10069i
$369$			5.80019i			0.0157187i
$370$			114.825i			0.310338i
$371$		−	31.5909i		−	0.0851505i
$372$	500.081			1.34430
$373$		−	149.204i		−	0.400010i	−0.979795	−	0.200005i	$-0.935904\pi$
$373$		−	149.204i		−	0.400010i	0.979795	−	0.200005i	$-0.0640958\pi$
$374$	−349.266			−0.933865
$375$	−145.539			−0.388104
$376$			291.759i			0.775955i
$377$	−276.141			−0.732470
$378$		−	379.263i		−	1.00334i
$379$	−522.535			−1.37872			−0.689360	−	0.724419i	$-0.742108\pi$
$379$	−522.535			−1.37872			−0.689360	+	0.724419i	$0.742108\pi$
$380$		−	146.368i		−	0.385178i
$381$		−	347.314i		−	0.911585i
$382$		−	96.6439i		−	0.252995i
$383$	−42.8938			−0.111994			−0.0559971	−	0.998431i	$-0.517834\pi$
$383$	−42.8938			−0.111994			−0.0559971	+	0.998431i	$0.517834\pi$
$384$	516.025			1.34381
$385$	−31.0694			−0.0806998
$386$			209.628i			0.543077i
$387$			4.44958i			0.0114976i
$388$	−109.164			−0.281350
$389$			104.283i			0.268079i	0.990976	+	0.134039i	$0.0427949\pi$
$389$			104.283i			0.268079i	−0.990976	+	0.134039i	$0.957205\pi$
$390$		−	95.8411i		−	0.245746i
$391$	−215.384			−0.550855
$392$			557.527i			1.42226i
$393$			135.722i			0.345348i
$394$	744.344			1.88920
$395$	39.6903			0.100482
$396$	38.7602			0.0978794
$397$		−	207.192i		−	0.521894i	−0.965353	−	0.260947i	$-0.915965\pi$
$397$		−	207.192i		−	0.521894i	0.965353	−	0.260947i	$-0.0840347\pi$
$398$			203.420i			0.511106i
$399$	185.713			0.465445
$400$	562.024			1.40506
$401$	−150.857			−0.376203			−0.188101	−	0.982150i	$-0.560233\pi$
$401$	−150.857			−0.376203			−0.188101	+	0.982150i	$0.560233\pi$
$402$	−6.28253			−0.0156282
$403$		−	182.101i		−	0.451863i
$404$			849.155i			2.10187i
$405$		−	77.3479i		−	0.190982i
$406$	−414.893			−1.02190
$407$		−	249.224i		−	0.612345i
$408$		−	590.386i		−	1.44703i
$409$	73.1680			0.178895			0.0894475	−	0.995992i	$-0.471490\pi$
$409$	73.1680			0.178895			0.0894475	+	0.995992i	$0.471490\pi$
$410$			36.8612i			0.0899055i
$411$			536.593i			1.30558i
$412$		−	505.819i		−	1.22772i
$413$	−357.767			−0.866264
$414$	35.0293			0.0846119
$415$	−103.045			−0.248301
$416$			164.210i			0.394736i
$417$		−	145.737i		−	0.349489i
$418$			465.571i			1.11381i
$419$	−197.251			−0.470766			−0.235383	−	0.971903i	$-0.575634\pi$
$419$	−197.251			−0.470766			−0.235383	+	0.971903i	$0.575634\pi$
$420$		−	98.2584i		−	0.233949i
$421$			92.2937i			0.219225i	0.993974	+	0.109612i	$0.0349610\pi$
$421$			92.2937i			0.219225i	−0.993974	+	0.109612i	$0.965039\pi$
$422$	−304.949			−0.722629
$423$	10.2369			0.0242006
$424$			133.871i			0.315733i
$425$			298.851i			0.703180i
$426$			854.399i			2.00563i
$427$	204.714			0.479425
$428$		−	258.560i		−	0.604112i
$429$			208.020i			0.484896i
$430$			28.2779i			0.0657625i
$431$			467.306i			1.08424i	0.840302	+	0.542118i	$0.182377\pi$
$431$			467.306i			1.08424i	−0.840302	+	0.542118i	$0.817623\pi$
$432$			652.063i			1.50941i
$433$			454.183i			1.04892i	0.851435	+	0.524460i	$0.175733\pi$
$433$			454.183i			1.04892i	−0.851435	+	0.524460i	$0.824267\pi$
$434$		−	273.601i		−	0.630417i
$435$	−90.3803			−0.207771
$436$			854.698i			1.96032i
$437$			287.107i			0.656996i
$438$	−470.687			−1.07463
$439$	214.332			0.488227			0.244113	−	0.969747i	$-0.421503\pi$
$439$	214.332			0.488227			0.244113	+	0.969747i	$0.421503\pi$
$440$	131.661			0.299230
$441$	19.5618			0.0443578
$442$	−402.220			−0.910000
$443$	−789.650			−1.78251			−0.891253	−	0.453507i	$-0.850173\pi$
$443$	−789.650			−1.78251			−0.891253	+	0.453507i	$0.850173\pi$
$444$	788.182			1.77518
$445$	−110.560			−0.248450
$446$	−763.982			−1.71297
$447$			78.2098i			0.174966i
$448$		−	114.278i		−	0.255084i
$449$	−76.4674			−0.170306			−0.0851530	−	0.996368i	$-0.527138\pi$
$449$	−76.4674			−0.170306			−0.0851530	+	0.996368i	$0.527138\pi$
$450$		−	48.6041i		−	0.108009i
$451$		−	80.0063i		−	0.177398i
$452$	−78.0091			−0.172587
$453$	−311.796			−0.688292
$454$			338.775i			0.746201i
$455$	−35.7801			−0.0786375
$456$	−786.985			−1.72584
$457$			37.4104i			0.0818609i	0.999162	+	0.0409304i	$0.0130322\pi$
$457$			37.4104i			0.0818609i	−0.999162	+	0.0409304i	$0.986968\pi$
$458$	799.382			1.74538
$459$	−346.729			−0.755401
$460$	151.905			0.330228
$461$		−	0.0658078i		−	0.000142750i	−1.00000		7.13751e-5i	$-0.999977\pi$
$461$		−	0.0658078i		−	0.000142750i	1.00000		7.13751e-5i	$-2.27194e-5\pi$
$462$			312.544i			0.676502i
$463$			288.579i			0.623280i	0.950200	+	0.311640i	$0.100878\pi$
$463$			288.579i			0.623280i	−0.950200	+	0.311640i	$0.899122\pi$
$464$	713.321			1.53733
$465$		−	59.6012i		−	0.128175i
$466$		−	452.658i		−	0.971369i
$467$	77.4511			0.165848			0.0829241	−	0.996556i	$-0.473574\pi$
$467$	77.4511			0.165848			0.0829241	+	0.996556i	$0.473574\pi$
$468$	44.6369			0.0953780
$469$			2.34544i			0.00500094i
$470$	65.0571			0.138419
$471$		−	369.410i		−	0.784309i
$472$	1516.09			3.21205
$473$		−	61.3763i		−	0.129760i
$474$		−	399.266i		−	0.842333i
$475$	398.369			0.838671
$476$	−412.365			−0.866313
$477$	4.69708			0.00984714
$478$			1282.50i			2.68306i
$479$	281.165			0.586983			0.293491	−	0.955962i	$-0.405183\pi$
$479$	281.165			0.586983			0.293491	+	0.955962i	$0.405183\pi$
$480$			53.7456i			0.111970i
$481$		−	287.011i		−	0.596696i
$482$	−865.801			−1.79627
$483$			192.739i			0.399045i
$484$	505.084			1.04356
$485$			13.0105i			0.0268257i
$486$	109.413			0.225129
$487$			712.976i			1.46402i	0.681296	+	0.732008i	$0.261417\pi$
$487$			712.976i			1.46402i	−0.681296	+	0.732008i	$0.738583\pi$
$488$	−867.507			−1.77768
$489$	687.955			1.40686
$490$	124.319			0.253711
$491$			51.6548i			0.105203i	0.998616	+	0.0526017i	$0.0167514\pi$
$491$			51.6548i			0.105203i	−0.998616	+	0.0526017i	$0.983249\pi$
$492$	253.023			0.514275
$493$			379.303i			0.769377i
$494$			536.159i			1.08534i
$495$		−	4.61956i		−	0.00933244i
$496$			470.399i			0.948384i
$497$	318.970			0.641792
$498$			1036.58i			2.08149i
$499$	−251.365			−0.503737			−0.251869	−	0.967761i	$-0.581045\pi$
$499$	−251.365			−0.503737			−0.251869	+	0.967761i	$0.581045\pi$
$500$		−	430.774i		−	0.861548i
$501$			217.222i			0.433576i
$502$	42.6830			0.0850258
$503$		−	108.396i		−	0.215499i	−0.994178	−	0.107750i	$-0.965636\pi$
$503$		−	108.396i		−	0.215499i	0.994178	−	0.107750i	$-0.0343645\pi$
$504$	35.8462			0.0711235
$505$	101.205			0.200406
$506$	−483.185			−0.954911
$507$		−	251.069i		−	0.495205i
$508$	1028.00			2.02362
$509$	863.365			1.69620			0.848099	−	0.529838i	$-0.177747\pi$
$509$	863.365			1.69620			0.848099	+	0.529838i	$0.177747\pi$
$510$	−131.646			−0.258129
$511$			175.720i			0.343875i
$512$			1105.60i			2.15937i
$513$			462.190i			0.900955i
$514$	364.095			0.708356
$515$	−60.2850			−0.117058
$516$	194.105			0.376173
$517$	−141.205			−0.273123
$518$		−	431.224i		−	0.832479i
$519$	−911.499			−1.75626
$520$	151.623			0.291583
$521$	−133.679			−0.256581			−0.128291	−	0.991737i	$-0.540949\pi$
$521$	−133.679			−0.256581			−0.128291	+	0.991737i	$0.540949\pi$
$522$		−	61.6884i		−	0.118177i
$523$		−	610.057i		−	1.16646i	−0.812308	−	0.583229i	$-0.801789\pi$
$523$		−	610.057i		−	1.16646i	0.812308	−	0.583229i	$-0.198211\pi$
$524$	−401.716			−0.766634
$525$	267.430			0.509391
$526$			107.944i			0.205217i
$527$	−250.131			−0.474631
$528$		−	537.353i		−	1.01771i
$529$	231.031			0.436731
$530$	29.8508			0.0563222
$531$		−	53.1945i		−	0.100178i
$532$			549.682i			1.03324i
$533$		−	92.1366i		−	0.172864i
$534$			1112.18i			2.08274i
$535$	−30.8160			−0.0575999
$536$		−	9.93914i		−	0.0185432i
$537$		−	512.141i		−	0.953708i
$538$		−	1377.21i		−	2.55988i
$539$	−269.830			−0.500612
$540$	244.539			0.452850
$541$		−	960.244i		−	1.77494i	−0.460863	−	0.887471i	$-0.652460\pi$
$541$		−	960.244i		−	1.77494i	0.460863	−	0.887471i	$-0.347540\pi$
$542$	938.828			1.73216
$543$			274.636i			0.505775i
$544$	225.556			0.414625
$545$	101.865			0.186909
$546$			359.931i			0.659214i
$547$	−525.375	−	152.285i	−0.960465	−	0.278400i
$547$	−525.375	−	152.285i	−0.960465	−	0.278400i
$548$	−1588.24			−2.89824
$549$			30.4380i			0.0554425i
$550$			670.432i			1.21897i
$551$	505.610			0.917623
$552$		−	816.759i		−	1.47964i
$553$	−149.057			−0.269542
$554$		−	84.2586i		−	0.152091i
$555$		−	93.9378i		−	0.169257i
$556$	431.359			0.775826
$557$	974.722			1.74995			0.874975	−	0.484169i	$-0.160878\pi$
$557$	974.722			1.74995			0.874975	+	0.484169i	$0.160878\pi$
$558$	40.6803			0.0729038
$559$		−	70.6820i		−	0.126444i
$560$	92.4262			0.165047
$561$	285.733			0.509328
$562$	−863.894			−1.53718
$563$	19.2169			0.0341331			0.0170665	−	0.999854i	$-0.494567\pi$
$563$	19.2169			0.0341331			0.0170665	+	0.999854i	$0.494567\pi$
$564$		−	446.565i		−	0.791782i
$565$			9.29735i			0.0164555i
$566$	−1602.00			−2.83040
$567$			290.480i			0.512310i
$568$	−1351.68			−2.37972
$569$		−	927.262i		−	1.62964i	−0.579717	−	0.814818i	$-0.696837\pi$
$569$		−	927.262i		−	1.62964i	0.579717	−	0.814818i	$-0.303163\pi$
$570$			175.483i			0.307866i
$571$	266.789			0.467231			0.233616	−	0.972329i	$-0.424944\pi$
$571$	266.789			0.467231			0.233616	+	0.972329i	$0.424944\pi$
$572$	−615.709			−1.07642
$573$			79.0640i			0.137983i
$574$		−	138.432i		−	0.241171i
$575$			413.440i			0.719027i
$576$	16.9913			0.0294989
$577$			756.124i			1.31044i	0.755438	+	0.655220i	$0.227424\pi$
$577$			756.124i			1.31044i	−0.755438	+	0.655220i	$0.772576\pi$
$578$		−	473.074i		−	0.818467i
$579$		−	171.496i		−	0.296193i
$580$		−	267.512i		−	0.461228i
$581$	386.984			0.666065
$582$	130.879			0.224878
$583$	−64.7903			−0.111133
$584$		−	744.640i		−	1.27507i
$585$		−	5.31996i		−	0.00909395i
$586$		−	673.175i		−	1.14876i
$587$	−125.621			−0.214005			−0.107003	−	0.994259i	$-0.534125\pi$
$587$	−125.621			−0.214005			−0.107003	+	0.994259i	$0.534125\pi$
$588$		−	853.349i		−	1.45127i
$589$			333.424i			0.566085i
$590$		−	338.061i		−	0.572984i
$591$	−608.945			−1.03036
$592$			741.399i			1.25236i
$593$	1099.11			1.85348			0.926741	−	0.375701i	$-0.122598\pi$
$593$	1099.11			1.85348			0.926741	+	0.375701i	$0.122598\pi$
$594$	−777.839			−1.30949
$595$			49.1469i			0.0825998i
$596$	−231.489			−0.388405
$597$		−	166.417i		−	0.278756i
$598$	−556.444			−0.930508
$599$	15.7358			0.0262702			0.0131351	−	0.999914i	$-0.495819\pi$
$599$	15.7358			0.0262702			0.0131351	+	0.999914i	$0.495819\pi$
$600$	−1133.27			−1.88879
$601$	499.256			0.830708			0.415354	−	0.909660i	$-0.363658\pi$
$601$	499.256			0.830708			0.415354	+	0.909660i	$0.363658\pi$
$602$		−	106.197i		−	0.176408i
$603$	−0.348731			−0.000578327
$604$		−	922.871i		−	1.52793i
$605$		−	60.1974i		−	0.0994999i
$606$		−	1018.07i		−	1.67999i
$607$	311.467			0.513125			0.256562	−	0.966528i	$-0.417410\pi$
$607$	311.467			0.513125			0.256562	+	0.966528i	$0.417410\pi$
$608$		−	300.666i		−	0.494517i
$609$	339.423			0.557344
$610$			193.439i			0.317112i
$611$	−162.613			−0.266143
$612$		−	61.3125i		−	0.100184i
$613$	−289.808			−0.472770			−0.236385	−	0.971659i	$-0.575963\pi$
$613$	−289.808			−0.472770			−0.236385	+	0.971659i	$0.575963\pi$
$614$	−883.350			−1.43868
$615$		−	30.1560i		−	0.0490342i
$616$	−494.453			−0.802683
$617$			278.692i			0.451688i	0.974163	+	0.225844i	$0.0725140\pi$
$617$			278.692i			0.451688i	−0.974163	+	0.225844i	$0.927486\pi$
$618$			606.439i			0.981293i
$619$		−	867.898i		−	1.40210i	−0.713114	−	0.701048i	$-0.752716\pi$
$619$		−	867.898i		−	1.40210i	0.713114	−	0.701048i	$-0.247284\pi$
$620$	176.411			0.284533
$621$	−479.676			−0.772425
$622$			603.706i			0.970588i
$623$	415.208			0.666466
$624$		−	618.824i		−	0.991706i
$625$	547.439			0.875902
$626$		−	1492.78i		−	2.38463i
$627$		−	380.882i		−	0.607467i
$628$	1093.40			1.74108
$629$	−394.233			−0.626761
$630$		−	7.99307i		−	0.0126874i
$631$	828.388			1.31282			0.656409	−	0.754406i	$-0.272075\pi$
$631$	828.388			1.31282			0.656409	+	0.754406i	$0.272075\pi$
$632$	631.650			0.999446
$633$	249.478			0.394120
$634$			927.193i			1.46245i
$635$		−	122.520i		−	0.192945i
$636$		−	204.902i		−	0.322173i
$637$	−310.741			−0.487819
$638$			850.913i			1.33372i
$639$			47.4261i			0.0742193i
$640$	182.035			0.284430
$641$			698.958i			1.09042i	0.838300	+	0.545209i	$0.183550\pi$
$641$			698.958i			1.09042i	−0.838300	+	0.545209i	$0.816450\pi$
$642$			309.994i			0.482856i
$643$	−875.985			−1.36234			−0.681170	−	0.732125i	$-0.738529\pi$
$643$	−875.985			−1.36234			−0.681170	+	0.732125i	$0.738529\pi$
$644$	−570.478			−0.885836
$645$		−	23.1340i		−	0.0358667i
$646$	736.459			1.14003
$647$	53.1591			0.0821625			0.0410812	−	0.999156i	$-0.486920\pi$
$647$	53.1591			0.0821625			0.0410812	+	0.999156i	$0.486920\pi$
$648$		−	1230.95i		−	1.89961i
$649$			733.751i			1.13059i
$650$			772.080i			1.18782i
$651$			223.832i			0.343828i
$652$			2036.24i			3.12307i
$653$			939.734i			1.43910i	0.694439	+	0.719551i	$0.255652\pi$
$653$			939.734i			1.43910i	−0.694439	+	0.719551i	$0.744348\pi$
$654$		−	1024.72i		−	1.56685i
$655$			47.8777i			0.0730957i
$656$			238.005i			0.362812i
$657$	−26.1270			−0.0397671
$658$	−244.321			−0.371309
$659$		−	322.953i		−	0.490065i	−0.969515	−	0.245033i	$-0.921201\pi$
$659$		−	322.953i		−	0.490065i	0.969515	−	0.245033i	$-0.0787987\pi$
$660$	−201.520			−0.305333
$661$	833.634			1.26117			0.630585	−	0.776120i	$-0.282815\pi$
$661$	833.634			1.26117			0.630585	+	0.776120i	$0.282815\pi$
$662$	−217.355			−0.328331
$663$	329.055			0.496312
$664$	−1639.90			−2.46973
$665$	65.5128			0.0985154
$666$	64.1166			0.0962711
$667$			524.739i			0.786715i
$668$	−642.944			−0.962491
$669$	625.011			0.934247
$670$	−2.21625			−0.00330784
$671$		−	419.853i		−	0.625712i
$672$		−	201.841i		−	0.300359i
$673$	577.208			0.857665			0.428832	−	0.903384i	$-0.358925\pi$
$673$	577.208			0.857665			0.428832	+	0.903384i	$0.358925\pi$
$674$	68.3013			0.101337
$675$			665.563i			0.986019i
$676$	743.127			1.09930
$677$	770.986			1.13883			0.569413	−	0.822051i	$-0.307170\pi$
$677$	770.986			1.13883			0.569413	+	0.822051i	$0.307170\pi$
$678$	93.5270			0.137945
$679$		−	48.8607i		−	0.0719598i
$680$		−	208.267i		−	0.306275i
$681$		−	277.151i		−	0.406976i
$682$	−561.133			−0.822776
$683$	1179.93			1.72757			0.863786	−	0.503859i	$-0.168087\pi$
$683$	1179.93			1.72757			0.863786	+	0.503859i	$0.168087\pi$
$684$	−81.7295			−0.119488
$685$			189.291i			0.276337i
$686$	−1135.65			−1.65546
$687$	−653.971			−0.951924
$688$			182.584i			0.265384i
$689$	−74.6136			−0.108293
$690$	−182.123			−0.263946
$691$	−816.916			−1.18222			−0.591111	−	0.806590i	$-0.701311\pi$
$691$	−816.916			−1.18222			−0.591111	+	0.806590i	$0.701311\pi$
$692$		−	2697.90i		−	3.89870i
$693$			17.3487i			0.0250342i
$694$		−	632.078i		−	0.910776i
$695$		−	51.4107i		−	0.0739722i
$696$	−1438.35			−2.06660
$697$	−126.557			−0.181574
$698$			631.923i			0.905334i
$699$			370.318i			0.529782i
$700$			791.554i			1.13079i
$701$	1362.78			1.94406			0.972029	−	0.234861i	$-0.0754635\pi$
$701$	1362.78			1.94406			0.972029	+	0.234861i	$0.0754635\pi$
$702$	−895.772			−1.27603
$703$			525.512i			0.747528i
$704$	−234.374			−0.332918
$705$	−53.2229			−0.0754935
$706$		−	2341.24i		−	3.31621i
$707$	−380.074			−0.537587
$708$	−2320.52			−3.27757
$709$		−	709.333i		−	1.00047i	−0.865890	−	0.500235i	$-0.833247\pi$
$709$		−	709.333i		−	1.00047i	0.865890	−	0.500235i	$-0.166753\pi$
$710$			301.401i			0.424509i
$711$		−	22.1625i		−	0.0311709i
$712$	−1759.50			−2.47121
$713$	−346.038			−0.485327
$714$	494.394			0.692429
$715$			73.3821i			0.102632i
$716$	1515.86			2.11713
$717$		−	1049.21i		−	1.46334i
$718$	−109.180			−0.152062
$719$			668.601i			0.929904i	0.885336	+	0.464952i	$0.153928\pi$
$719$			668.601i			0.929904i	−0.885336	+	0.464952i	$0.846072\pi$
$720$			13.7424i			0.0190866i
$721$	226.400			0.314009
$722$			299.359i			0.414625i
$723$	708.308			0.979680
$724$	−812.881			−1.12276
$725$	728.089			1.00426
$726$	−605.558			−0.834101
$727$			511.232i			0.703208i	0.936149	+	0.351604i	$0.114364\pi$
$727$			511.232i			0.703208i	−0.936149	+	0.351604i	$0.885636\pi$
$728$	−569.420			−0.782171
$729$	−769.247			−1.05521
$730$	−166.041			−0.227454
$731$	−97.0875			−0.132815
$732$	1327.80			1.81394
$733$		−	636.492i		−	0.868338i	−0.900832	−	0.434169i	$-0.857042\pi$
$733$		−	636.492i		−	0.868338i	0.900832	−	0.434169i	$-0.142958\pi$
$734$		−	1764.55i		−	2.40402i
$735$	−101.705			−0.138374
$736$	312.041			0.423969
$737$	4.81031			0.00652687
$738$	20.5828			0.0278900
$739$		−	1378.68i		−	1.86561i	−0.360386	−	0.932803i	$-0.617355\pi$
$739$		−	1378.68i		−	1.86561i	0.360386	−	0.932803i	$-0.382645\pi$
$740$	278.042			0.375733
$741$		−	438.630i		−	0.591943i
$742$	−112.104			−0.151084
$743$	345.652			0.465212			0.232606	−	0.972571i	$-0.425275\pi$
$743$	345.652			0.465212			0.232606	+	0.972571i	$0.425275\pi$
$744$		−	948.520i		−	1.27489i
$745$			27.5896i			0.0370330i
$746$	−529.470			−0.709745
$747$			57.5387i			0.0770264i
$748$			845.727i			1.13065i
$749$	115.729			0.154512
$750$			516.465i			0.688620i
$751$	284.974			0.379459			0.189730	−	0.981836i	$-0.439239\pi$
$751$	284.974			0.379459			0.189730	+	0.981836i	$0.439239\pi$
$752$	420.059			0.558589
$753$	−34.9188			−0.0463729
$754$			979.925i			1.29964i
$755$	−109.990			−0.145683
$756$	−918.365			−1.21477
$757$	−1416.74			−1.87151			−0.935756	−	0.352647i	$-0.885282\pi$
$757$	−1416.74			−1.87151			−0.935756	+	0.352647i	$0.885282\pi$
$758$			1854.29i			2.44629i
$759$	395.292			0.520806
$760$	−277.620			−0.365289
$761$	−247.063			−0.324655			−0.162328	−	0.986737i	$-0.551900\pi$
$761$	−247.063			−0.324655			−0.162328	+	0.986737i	$0.551900\pi$
$762$	−1232.49			−1.61744
$763$	−382.555			−0.501383
$764$	−234.018			−0.306306
$765$	−7.30740			−0.00955216
$766$			152.214i			0.198713i
$767$			845.000i			1.10169i
$768$		−	1486.15i		−	1.93508i
$769$		−	1314.84i		−	1.70981i	−0.518784	−	0.854905i	$-0.673615\pi$
$769$		−	1314.84i		−	1.70981i	0.518784	−	0.854905i	$-0.326385\pi$
$770$			110.254i			0.143187i
$771$	−297.865			−0.386335
$772$	507.602			0.657516
$773$			48.0285i			0.0621326i	0.999517	+	0.0310663i	$0.00989030\pi$
$773$			48.0285i			0.0621326i	−0.999517	+	0.0310663i	$0.990110\pi$
$774$	15.7900			0.0204005
$775$			480.137i			0.619532i
$776$			207.054i			0.266823i
$777$			352.783i			0.454032i
$778$	370.062			0.475658
$779$			168.701i			0.216560i
$780$	−232.074			−0.297531
$781$		−	654.183i		−	0.837622i
$782$			764.321i			0.977393i
$783$			844.733i			1.07884i
$784$	802.698			1.02385
$785$		−	130.314i		−	0.166006i
$786$	481.627			0.612757
$787$	577.561			0.733877			0.366939	−	0.930245i	$-0.380406\pi$
$787$	577.561			0.733877			0.366939	+	0.930245i	$0.380406\pi$
$788$		−	1802.39i		−	2.28730i
$789$		−	88.3087i		−	0.111925i
$790$		−	140.847i		−	0.178287i
$791$		−	34.9162i		−	0.0441418i
$792$		−	73.5177i		−	0.0928254i
$793$		−	483.510i		−	0.609722i
$794$	−735.248			−0.926006
$795$	−24.4208			−0.0307180
$796$	492.571			0.618808
$797$	247.166			0.310121			0.155060	−	0.987905i	$-0.450443\pi$
$797$	247.166			0.310121			0.155060	+	0.987905i	$0.450443\pi$
$798$		−	659.027i		−	0.825849i
$799$			223.363i			0.279553i
$800$		−	432.966i		−	0.541207i
$801$			61.7352i			0.0770727i
$802$			535.338i			0.667504i
$803$	360.388			0.448802
$804$			15.2128i			0.0189214i
$805$			67.9913i			0.0844612i
$806$	−646.210			−0.801750
$807$			1126.69i			1.39615i
$808$	1610.62			1.99334
$809$		−	420.779i		−	0.520122i	−0.965592	−	0.260061i	$-0.916257\pi$
$809$		−	420.779i		−	0.520122i	0.965592	−	0.260061i	$-0.0837427\pi$
$810$	−274.480			−0.338864
$811$	828.596			1.02170			0.510848	−	0.859671i	$-0.329331\pi$
$811$	828.596			1.02170			0.510848	+	0.859671i	$0.329331\pi$
$812$			1004.64i			1.23724i
$813$	−768.052			−0.944713
$814$	−884.406			−1.08649
$815$	242.686			0.297774
$816$	−850.006			−1.04167
$817$			129.418i			0.158406i
$818$		−	259.647i		−	0.317417i
$819$			19.9791i			0.0243945i
$820$	89.2575			0.108851
$821$		−	659.976i		−	0.803869i	−0.915668	−	0.401934i	$-0.868338\pi$
$821$		−	659.976i		−	0.803869i	0.915668	−	0.401934i	$-0.131662\pi$
$822$	1904.18			2.31652
$823$	−271.049			−0.329343			−0.164671	−	0.986349i	$-0.552656\pi$
$823$	−271.049			−0.329343			−0.164671	+	0.986349i	$0.552656\pi$
$824$	−959.403			−1.16432
$825$		−	548.478i		−	0.664821i
$826$			1269.58i			1.53703i
$827$			795.612i			0.962046i	0.876708	+	0.481023i	$0.159735\pi$
$827$			795.612i			0.962046i	−0.876708	+	0.481023i	$0.840265\pi$
$828$		−	84.8215i		−	0.102441i
$829$	−1423.45			−1.71707			−0.858537	−	0.512752i	$-0.828626\pi$
$829$	−1423.45			−1.71707			−0.858537	+	0.512752i	$0.828626\pi$
$830$			365.669i			0.440564i
$831$			68.9317i			0.0829503i
$832$	−269.909			−0.324410
$833$			426.828i			0.512398i
$834$	−517.167			−0.620104
$835$			76.6280i			0.0917700i
$836$	1127.35			1.34851
$837$	−557.058			−0.665541
$838$			699.972i			0.835289i
$839$	87.0366			0.103739			0.0518693	−	0.998654i	$-0.483482\pi$
$839$	87.0366			0.103739			0.0518693	+	0.998654i	$0.483482\pi$
$840$	−186.370			−0.221869
$841$	83.0915			0.0988008
$842$	327.517			0.388975
$843$	706.748			0.838373
$844$			738.418i			0.874903i
$845$		−	88.5680i		−	0.104814i
$846$		−	36.3269i		−	0.0429396i
$847$			226.071i			0.266908i
$848$	192.740			0.227288
$849$	1310.59			1.54369
$850$	1060.52			1.24767
$851$	−545.394			−0.640886
$852$	2068.88			2.42826
$853$	−0.553270			−0.000648616			−0.000324308	−	1.00000i	$-0.500103\pi$
$853$	−0.553270			−0.000648616			−0.000324308		1.00000i	$0.500103\pi$
$854$		−	726.457i		−	0.850653i
$855$			9.74076i			0.0113927i
$856$	−490.419			−0.572919
$857$			1357.06i			1.58350i	0.610848	+	0.791748i	$0.290829\pi$
$857$			1357.06i			1.58350i	−0.610848	+	0.791748i	$0.709171\pi$
$858$	738.189			0.860360
$859$	−14.3394			−0.0166931			−0.00834655	−	0.999965i	$-0.502657\pi$
$859$	−14.3394			−0.0166931			−0.00834655	+	0.999965i	$0.502657\pi$
$860$	68.4733			0.0796201
$861$			113.251i			0.131534i
$862$	1658.30			1.92378
$863$			723.941i			0.838866i	0.907786	+	0.419433i	$0.137771\pi$
$863$			723.941i			0.838866i	−0.907786	+	0.419433i	$0.862229\pi$
$864$	502.329			0.581400
$865$	−321.544			−0.371727
$866$	1611.73			1.86112
$867$			387.020i			0.446390i
$868$	−662.509			−0.763259
$869$			305.703i			0.351788i
$870$			320.727i			0.368652i
$871$	5.53963			0.00636008
$872$	1621.13			1.85910
$873$	7.26485			0.00832171
$874$	1018.84			1.16572
$875$	192.810			0.220355
$876$			1139.74i			1.30108i
$877$		−	1007.38i		−	1.14867i	−0.818621	−	0.574334i	$-0.805261\pi$
$877$		−	1007.38i		−	1.14867i	0.818621	−	0.574334i	$-0.194739\pi$
$878$		−	760.585i		−	0.866270i
$879$			550.722i			0.626532i
$880$		−	189.559i		−	0.215408i
$881$			560.550i			0.636266i	0.948046	+	0.318133i	$0.103056\pi$
$881$			560.550i			0.636266i	−0.948046	+	0.318133i	$0.896944\pi$
$882$		−	69.4177i		−	0.0787049i
$883$	−1514.89			−1.71562			−0.857809	−	0.513969i	$-0.828175\pi$
$883$	−1514.89			−1.71562			−0.857809	+	0.513969i	$0.828175\pi$
$884$			973.954i			1.10176i
$885$			276.566i			0.312504i
$886$			2802.18i			3.16273i
$887$	−651.122			−0.734072			−0.367036	−	0.930207i	$-0.619627\pi$
$887$	−651.122			−0.734072			−0.367036	+	0.930207i	$0.619627\pi$
$888$		−	1494.97i		−	1.68352i
$889$			460.123i			0.517573i
$890$			392.338i			0.440829i
$891$	595.750			0.668631
$892$			1849.94i			2.07392i
$893$	297.743			0.333418
$894$	277.538			0.310445
$895$		−	180.665i		−	0.201860i
$896$	−683.631			−0.762981
$897$	455.225			0.507497
$898$			271.355i			0.302177i
$899$			609.391i			0.677854i
$900$	−117.692			−0.130769
$901$			102.488i			0.113749i
$902$	−283.913			−0.314760
$903$			86.8797i			0.0962123i
$904$			147.962i			0.163675i
$905$			96.8816i			0.107051i
$906$			1106.45i			1.22125i
$907$	−802.045			−0.884283			−0.442141	−	0.896945i	$-0.645781\pi$
$907$	−802.045			−0.884283			−0.442141	+	0.896945i	$0.645781\pi$
$908$	820.325			0.903441
$909$		−	56.5113i		−	0.0621687i
$910$			126.971i			0.139528i
$911$			614.085i			0.674078i	0.941491	+	0.337039i	$0.109425\pi$
$911$			614.085i			0.674078i	−0.941491	+	0.337039i	$0.890575\pi$
$912$			1133.06i			1.24239i
$913$		−	793.673i		−	0.869302i
$914$	132.756			0.145247
$915$		−	158.251i		−	0.172952i
$916$		−	1935.66i		−	2.11316i
$917$		−	179.804i		−	0.196079i
$918$			1230.42i			1.34032i
$919$	−1638.42			−1.78283			−0.891416	−	0.453187i	$-0.850287\pi$
$919$	−1638.42			−1.78283			−0.891416	+	0.453187i	$0.850287\pi$
$920$		−	288.123i		−	0.313177i
$921$	722.665			0.784653
$922$	−0.233528			−0.000253284
$923$		−	753.368i		−	0.816216i
$924$	756.807			0.819056
$925$			756.748i			0.818106i
$926$	1024.06			1.10590
$927$			33.6623i			0.0363132i
$928$		−	549.521i		−	0.592156i
$929$		−	1500.27i		−	1.61493i	−0.589914	−	0.807466i	$-0.700839\pi$
$929$		−	1500.27i		−	1.61493i	0.589914	−	0.807466i	$-0.299161\pi$
$930$	−211.503			−0.227422
$931$	568.961			0.611129
$932$	−1096.09			−1.17606
$933$		−	493.890i		−	0.529357i
$934$		−	274.846i		−	0.294267i
$935$	100.796			0.107803
$936$		−	84.6642i		−	0.0904532i
$937$			984.366i			1.05055i	0.850932	+	0.525275i	$0.176038\pi$
$937$			984.366i			1.05055i	−0.850932	+	0.525275i	$0.823962\pi$
$938$	8.32311			0.00887325
$939$			1221.24i			1.30057i
$940$		−	157.532i		−	0.167587i
$941$	1535.85			1.63214			0.816071	−	0.577952i	$-0.196148\pi$
$941$	1535.85			1.63214			0.816071	+	0.577952i	$0.196148\pi$
$942$	−1310.90			−1.39161
$943$	−175.083			−0.185666
$944$		−	2182.78i		−	2.31227i
$945$			109.453i			0.115824i
$946$	−217.802			−0.230235
$947$	727.032			0.767721			0.383860	−	0.923391i	$-0.374594\pi$
$947$	727.032			0.767721			0.383860	+	0.923391i	$0.374594\pi$
$948$	−966.800			−1.01983
$949$	415.029			0.437333
$950$		−	1413.67i		−	1.48807i
$951$		−	758.533i		−	0.797616i
$952$			782.145i			0.821581i
$953$	−1402.88			−1.47207			−0.736035	−	0.676943i	$-0.763304\pi$
$953$	−1402.88			−1.47207			−0.736035	+	0.676943i	$0.763304\pi$
$954$		−	16.6682i		−	0.0174720i
$955$			27.8909i			0.0292052i
$956$	3105.51			3.24844
$957$		−	696.128i		−	0.727407i
$958$		−	997.752i		−	1.04149i
$959$		−	710.880i		−	0.741273i
$960$	−88.3405			−0.0920213
$961$	559.138			0.581829
$962$	−1018.50			−1.05873
$963$			17.2072i			0.0178683i
$964$			2096.49i			2.17478i
$965$		−	60.4975i		−	0.0626917i
$966$	683.960			0.708033
$967$			519.593i			0.537325i	0.963234	+	0.268662i	$0.0865816\pi$
$967$			519.593i			0.537325i	−0.963234	+	0.268662i	$0.913418\pi$
$968$		−	958.009i		−	0.989679i
$969$	−602.494			−0.621769
$970$	46.1694			0.0475973
$971$		−	980.112i		−	1.00938i	−0.863299	−	0.504692i	$-0.831606\pi$
$971$		−	980.112i		−	1.00938i	0.863299	−	0.504692i	$-0.168394\pi$
$972$		−	264.936i		−	0.272568i
$973$			193.072i			0.198430i
$974$	2530.09			2.59763
$975$		−	631.636i		−	0.647832i
$976$			1248.99i			1.27970i
$977$		−	578.975i		−	0.592605i	−0.955094	−	0.296302i	$-0.904246\pi$
$977$		−	578.975i		−	0.592605i	0.955094	−	0.296302i	$-0.0957536\pi$
$978$		−	2441.30i		−	2.49622i
$979$		−	851.559i		−	0.869825i
$980$		−	301.031i		−	0.307174i
$981$		−	56.8802i		−	0.0579818i
$982$	183.304			0.186664
$983$		−	369.458i		−	0.375848i	−0.982184	−	0.187924i	$-0.939824\pi$
$983$		−	369.458i		−	0.375848i	0.982184	−	0.187924i	$-0.0601758\pi$
$984$		−	479.917i		−	0.487720i
$985$	−214.814			−0.218085
$986$	1346.01			1.36512
$987$	199.878			0.202511
$988$	1298.28			1.31405
$989$	−134.314			−0.135808
$990$	−16.3931			−0.0165587
$991$	792.629			0.799828			0.399914	−	0.916553i	$-0.369040\pi$
$991$	792.629			0.799828			0.399914	+	0.916553i	$0.369040\pi$
$992$	362.381			0.365303
$993$	177.817			0.179071
$994$		−	1131.91i		−	1.13874i
$995$		−	58.7060i		−	0.0590010i
$996$	2510.02			2.52010
$997$			765.348i			0.767651i	0.923406	+	0.383825i	$0.125394\pi$
$997$			765.348i			0.767651i	−0.923406	+	0.383825i	$0.874606\pi$
$998$			892.003i			0.893791i
$999$	−877.984			−0.878863

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.3.b.b.546.6	✓	88
547.546	odd	2	inner	547.3.b.b.546.83	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.3.b.b.546.6	✓	88	1.1	even	1	trivial
547.3.b.b.546.83	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)

\(f(q)\)	\(=\)	\(q-3.54864i q^{2} +2.90313i q^{3} -8.59283 q^{4} +1.02412i q^{5} +10.3021 q^{6} -3.84607i q^{7} +16.2983i q^{8} +0.571853 q^{9} +O(q^{10})\)	\(q-3.54864i q^{2} +2.90313i q^{3} -8.59283 q^{4} +1.02412i q^{5} +10.3021 q^{6} -3.84607i q^{7} +16.2983i q^{8} +0.571853 q^{9} +3.63422 q^{10} -7.88798 q^{11} -24.9461i q^{12} -9.08393 q^{13} -13.6483 q^{14} -2.97315 q^{15} +23.4654 q^{16} +12.4775i q^{17} -2.02930i q^{18} +16.6325 q^{19} -8.80007i q^{20} +11.1656 q^{21} +27.9916i q^{22} +17.2618i q^{23} -47.3160 q^{24} +23.9512 q^{25} +32.2356i q^{26} +27.7883i q^{27} +33.0486i q^{28} +30.3989 q^{29} +10.5506i q^{30} +20.0465i q^{31} -18.0770i q^{32} -22.8998i q^{33} +44.2782 q^{34} +3.93883 q^{35} -4.91383 q^{36} +31.5954i q^{37} -59.0228i q^{38} -26.3718i q^{39} -16.6914 q^{40} +10.1428i q^{41} -39.6228i q^{42} +7.78099i q^{43} +67.7801 q^{44} +0.585645i q^{45} +61.2558 q^{46} +17.9012 q^{47} +68.1230i q^{48} +34.2077 q^{49} -84.9941i q^{50} -36.2238 q^{51} +78.0567 q^{52} +8.21380 q^{53} +98.6106 q^{54} -8.07823i q^{55} +62.6843 q^{56} +48.2864i q^{57} -107.875i q^{58} -93.0214i q^{59} +25.5477 q^{60} +53.2269i q^{61} +71.1377 q^{62} -2.19939i q^{63} +29.7128 q^{64} -9.30302i q^{65} -81.2631 q^{66} -0.609827 q^{67} -107.217i q^{68} -50.1132 q^{69} -13.9775i q^{70} +82.9341i q^{71} +9.32022i q^{72} -45.6883 q^{73} +112.121 q^{74} +69.5333i q^{75} -142.920 q^{76} +30.3377i q^{77} -93.5840 q^{78} -38.7556i q^{79} +24.0313i q^{80} -75.5263 q^{81} +35.9931 q^{82} +100.618i q^{83} -95.9444 q^{84} -12.7785 q^{85} +27.6119 q^{86} +88.2518i q^{87} -128.561i q^{88} +107.956i q^{89} +2.07824 q^{90} +34.9374i q^{91} -148.328i q^{92} -58.1975 q^{93} -63.5250i q^{94} +17.0337i q^{95} +52.4798 q^{96} +12.7041 q^{97} -121.391i q^{98} -4.51076 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) 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

Modular forms

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