LMFDB - Embedded newform 547.3.b.b.546.20

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			546.20
Character	$\chi$	$=$	547.546
Dual form			547.3.b.b.546.69

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64774i q^{2} +5.22856i q^{3} -3.01055 q^{4} -9.23817i q^{5} +13.8439 q^{6} -6.29191i q^{7} -2.61981i q^{8} -18.3379 q^{9} +O(q^{10})$	\(q-2.64774i q^{2} +5.22856i q^{3} -3.01055 q^{4} -9.23817i q^{5} +13.8439 q^{6} -6.29191i q^{7} -2.61981i q^{8} -18.3379 q^{9} -24.4603 q^{10} -16.6612 q^{11} -15.7408i q^{12} +10.1553 q^{13} -16.6594 q^{14} +48.3023 q^{15} -18.9788 q^{16} +21.8066i q^{17} +48.5540i q^{18} -0.551816 q^{19} +27.8120i q^{20} +32.8976 q^{21} +44.1145i q^{22} +43.5332i q^{23} +13.6978 q^{24} -60.3438 q^{25} -26.8887i q^{26} -48.8236i q^{27} +18.9421i q^{28} -23.6532 q^{29} -127.892i q^{30} +6.34072i q^{31} +39.7717i q^{32} -87.1139i q^{33} +57.7383 q^{34} -58.1257 q^{35} +55.2070 q^{36} -24.1141i q^{37} +1.46107i q^{38} +53.0978i q^{39} -24.2022 q^{40} -43.9968i q^{41} -87.1045i q^{42} +13.7995i q^{43} +50.1593 q^{44} +169.408i q^{45} +115.265 q^{46} -80.9321 q^{47} -99.2318i q^{48} +9.41193 q^{49} +159.775i q^{50} -114.017 q^{51} -30.5731 q^{52} +44.2297 q^{53} -129.272 q^{54} +153.919i q^{55} -16.4836 q^{56} -2.88520i q^{57} +62.6276i q^{58} -52.8725i q^{59} -145.417 q^{60} -106.596i q^{61} +16.7886 q^{62} +115.380i q^{63} +29.3903 q^{64} -93.8167i q^{65} -230.655 q^{66} -59.6078 q^{67} -65.6499i q^{68} -227.616 q^{69} +153.902i q^{70} +7.16549i q^{71} +48.0417i q^{72} +70.8639 q^{73} -63.8479 q^{74} -315.511i q^{75} +1.66127 q^{76} +104.830i q^{77} +140.589 q^{78} +29.8677i q^{79} +175.329i q^{80} +90.2363 q^{81} -116.492 q^{82} +56.4622i q^{83} -99.0399 q^{84} +201.453 q^{85} +36.5377 q^{86} -123.672i q^{87} +43.6491i q^{88} -80.6631i q^{89} +448.550 q^{90} -63.8964i q^{91} -131.059i q^{92} -33.1529 q^{93} +214.288i q^{94} +5.09777i q^{95} -207.949 q^{96} +30.2541 q^{97} -24.9204i q^{98} +305.530 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) $ 88 * q - 192 * q^4 - 306 * q^9	$ 88 q - 192 q^{4} - 306 q^{9} - 4 q^{10} - 32 q^{11} + 26 q^{13} - 26 q^{14} + 22 q^{15} + 236 q^{16} - 12 q^{19} - 16 q^{21} - 2 q^{24} - 544 q^{25} - 96 q^{29} + 26 q^{34} + 10 q^{35} + 364 q^{36} + 44 q^{40} + 124 q^{44} - 288 q^{46} - 310 q^{47} - 694 q^{49} + 86 q^{51} - 316 q^{52} + 24 q^{53} - 266 q^{54} + 158 q^{56} - 80 q^{60} + 40 q^{62} - 652 q^{64} + 528 q^{66} + 28 q^{67} + 16 q^{69} + 94 q^{73} - 614 q^{74} - 28 q^{76} - 98 q^{78} + 928 q^{81} - 772 q^{82} + 358 q^{84} + 74 q^{85} - 410 q^{86} - 214 q^{90} + 656 q^{93} - 724 q^{96} + 346 q^{97} + 50 q^{99}+O(q^{100}) $ 88 * q - 192 * q^4 - 306 * q^9 - 4 * q^10 - 32 * q^11 + 26 * q^13 - 26 * q^14 + 22 * q^15 + 236 * q^16 - 12 * q^19 - 16 * q^21 - 2 * q^24 - 544 * q^25 - 96 * q^29 + 26 * q^34 + 10 * q^35 + 364 * q^36 + 44 * q^40 + 124 * q^44 - 288 * q^46 - 310 * q^47 - 694 * q^49 + 86 * q^51 - 316 * q^52 + 24 * q^53 - 266 * q^54 + 158 * q^56 - 80 * q^60 + 40 * q^62 - 652 * q^64 + 528 * q^66 + 28 * q^67 + 16 * q^69 + 94 * q^73 - 614 * q^74 - 28 * q^76 - 98 * q^78 + 928 * q^81 - 772 * q^82 + 358 * q^84 + 74 * q^85 - 410 * q^86 - 214 * q^90 + 656 * q^93 - 724 * q^96 + 346 * q^97 + 50 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	2.64774i		−	1.32387i	−0.749560	−	0.661936i	$-0.769735\pi$
$2$		−	2.64774i		−	1.32387i	0.749560	−	0.661936i	$-0.230265\pi$
$3$			5.22856i			1.74285i	0.490525	+	0.871427i	$0.336805\pi$
$3$			5.22856i			1.74285i	−0.490525	+	0.871427i	$0.663195\pi$
$4$	−3.01055			−0.752638
$5$		−	9.23817i		−	1.84763i	−0.382834	−	0.923817i	$-0.625052\pi$
$5$		−	9.23817i		−	1.84763i	0.382834	−	0.923817i	$-0.374948\pi$
$6$	13.8439			2.30732
$7$		−	6.29191i		−	0.898844i	−0.893320	−	0.449422i	$-0.851630\pi$
$7$		−	6.29191i		−	0.898844i	0.893320	−	0.449422i	$-0.148370\pi$
$8$		−	2.61981i		−	0.327476i
$9$	−18.3379			−2.03754
$10$	−24.4603			−2.44603
$11$	−16.6612			−1.51465			−0.757326	−	0.653038i	$-0.773494\pi$
$11$	−16.6612			−1.51465			−0.757326	+	0.653038i	$0.773494\pi$
$12$		−	15.7408i		−	1.31174i
$13$	10.1553			0.781180			0.390590	−	0.920565i	$-0.372271\pi$
$13$	10.1553			0.781180			0.390590	+	0.920565i	$0.372271\pi$
$14$	−16.6594			−1.18995
$15$	48.3023			3.22016
$16$	−18.9788			−1.18617
$17$			21.8066i			1.28274i	0.767231	+	0.641371i	$0.221634\pi$
$17$			21.8066i			1.28274i	−0.767231	+	0.641371i	$0.778366\pi$
$18$			48.5540i			2.69744i
$19$	−0.551816			−0.0290430			−0.0145215	−	0.999895i	$-0.504622\pi$
$19$	−0.551816			−0.0290430			−0.0145215	+	0.999895i	$0.504622\pi$
$20$			27.8120i			1.39060i
$21$	32.8976			1.56655
$22$			44.1145i			2.00520i
$23$			43.5332i			1.89275i	0.323074	+	0.946374i	$0.395284\pi$
$23$			43.5332i			1.89275i	−0.323074	+	0.946374i	$0.604716\pi$
$24$	13.6978			0.570743
$25$	−60.3438			−2.41375
$26$		−	26.8887i		−	1.03418i
$27$		−	48.8236i		−	1.80828i
$28$			18.9421i			0.676504i
$29$	−23.6532			−0.815627			−0.407813	−	0.913065i	$-0.633709\pi$
$29$	−23.6532			−0.815627			−0.407813	+	0.913065i	$0.633709\pi$
$30$		−	127.892i		−	4.26307i
$31$			6.34072i			0.204539i	0.994757	+	0.102270i	$0.0326105\pi$
$31$			6.34072i			0.204539i	−0.994757	+	0.102270i	$0.967390\pi$
$32$			39.7717i			1.24287i
$33$		−	87.1139i		−	2.63982i
$34$	57.7383			1.69819
$35$	−58.1257			−1.66073
$36$	55.2070			1.53353
$37$		−	24.1141i		−	0.651731i	−0.945416	−	0.325866i	$-0.894344\pi$
$37$		−	24.1141i		−	0.651731i	0.945416	−	0.325866i	$-0.105656\pi$
$38$			1.46107i			0.0384492i
$39$			53.0978i			1.36148i
$40$	−24.2022			−0.605056
$41$		−	43.9968i		−	1.07309i	−0.843871	−	0.536546i	$-0.819729\pi$
$41$		−	43.9968i		−	1.07309i	0.843871	−	0.536546i	$-0.180271\pi$
$42$		−	87.1045i		−	2.07392i
$43$			13.7995i			0.320920i	0.987042	+	0.160460i	$0.0512977\pi$
$43$			13.7995i			0.320920i	−0.987042	+	0.160460i	$0.948702\pi$
$44$	50.1593			1.13998
$45$			169.408i			3.76463i
$46$	115.265			2.50576
$47$	−80.9321			−1.72196			−0.860980	−	0.508638i	$-0.830149\pi$
$47$	−80.9321			−1.72196			−0.860980	+	0.508638i	$0.830149\pi$
$48$		−	99.2318i		−	2.06733i
$49$	9.41193			0.192080
$50$			159.775i			3.19550i
$51$	−114.017			−2.23563
$52$	−30.5731			−0.587945
$53$	44.2297			0.834523			0.417262	−	0.908786i	$-0.362990\pi$
$53$	44.2297			0.834523			0.417262	+	0.908786i	$0.362990\pi$
$54$	−129.272			−2.39393
$55$			153.919i			2.79852i
$56$	−16.4836			−0.294350
$57$		−	2.88520i		−	0.0506176i
$58$			62.6276i			1.07979i
$59$		−	52.8725i		−	0.896143i	−0.893998	−	0.448072i	$-0.852111\pi$
$59$		−	52.8725i		−	0.896143i	0.893998	−	0.448072i	$-0.147889\pi$
$60$	−145.417			−2.42361
$61$		−	106.596i		−	1.74748i	−0.486396	−	0.873738i	$-0.661689\pi$
$61$		−	106.596i		−	1.74748i	0.486396	−	0.873738i	$-0.338311\pi$
$62$	16.7886			0.270784
$63$			115.380i			1.83143i
$64$	29.3903			0.459223
$65$		−	93.8167i		−	1.44333i
$66$	−230.655			−3.49478
$67$	−59.6078			−0.889669			−0.444834	−	0.895613i	$-0.646737\pi$
$67$	−59.6078			−0.889669			−0.444834	+	0.895613i	$0.646737\pi$
$68$		−	65.6499i		−	0.965439i
$69$	−227.616			−3.29878
$70$			153.902i			2.19860i
$71$			7.16549i			0.100922i	0.998726	+	0.0504612i	$0.0160691\pi$
$71$			7.16549i			0.100922i	−0.998726	+	0.0504612i	$0.983931\pi$
$72$			48.0417i			0.667246i
$73$	70.8639			0.970738			0.485369	−	0.874309i	$-0.338685\pi$
$73$	70.8639			0.970738			0.485369	+	0.874309i	$0.338685\pi$
$74$	−63.8479			−0.862809
$75$		−	315.511i		−	4.20682i
$76$	1.66127			0.0218588
$77$			104.830i			1.36143i
$78$	140.589			1.80243
$79$			29.8677i			0.378072i	0.981970	+	0.189036i	$0.0605363\pi$
$79$			29.8677i			0.378072i	−0.981970	+	0.189036i	$0.939464\pi$
$80$			175.329i			2.19162i
$81$	90.2363			1.11403
$82$	−116.492			−1.42064
$83$			56.4622i			0.680267i	0.940377	+	0.340134i	$0.110472\pi$
$83$			56.4622i			0.680267i	−0.940377	+	0.340134i	$0.889528\pi$
$84$	−99.0399			−1.17905
$85$	201.453			2.37004
$86$	36.5377			0.424857
$87$		−	123.672i		−	1.42152i
$88$			43.6491i			0.496012i
$89$		−	80.6631i		−	0.906327i	−0.891428	−	0.453163i	$-0.850295\pi$
$89$		−	80.6631i		−	0.906327i	0.891428	−	0.453163i	$-0.149705\pi$
$90$	448.550			4.98389
$91$		−	63.8964i		−	0.702158i
$92$		−	131.059i		−	1.42455i
$93$	−33.1529			−0.356482
$94$			214.288i			2.27966i
$95$			5.09777i			0.0536608i
$96$	−207.949			−2.16614
$97$	30.2541			0.311898			0.155949	−	0.987765i	$-0.450156\pi$
$97$	30.2541			0.311898			0.155949	+	0.987765i	$0.450156\pi$
$98$		−	24.9204i		−	0.254290i
$99$	305.530			3.08616
$100$	181.668			1.81668
$101$		−	187.553i		−	1.85696i	−0.371384	−	0.928479i	$-0.621117\pi$
$101$		−	187.553i		−	1.85696i	0.371384	−	0.928479i	$-0.378883\pi$
$102$			301.888i			2.95969i
$103$			32.7505i			0.317966i	0.987281	+	0.158983i	$0.0508215\pi$
$103$			32.7505i			0.317966i	−0.987281	+	0.158983i	$0.949179\pi$
$104$		−	26.6050i		−	0.255818i
$105$		−	303.914i		−	2.89442i
$106$		−	117.109i		−	1.10480i
$107$		−	148.506i		−	1.38790i	−0.720022	−	0.693951i	$-0.755868\pi$
$107$		−	148.506i		−	1.38790i	0.720022	−	0.693951i	$-0.244132\pi$
$108$			146.986i			1.36098i
$109$		−	13.4201i		−	0.123121i	−0.998103	−	0.0615603i	$-0.980392\pi$
$109$		−	13.4201i		−	0.123121i	0.998103	−	0.0615603i	$-0.0196076\pi$
$110$	407.537			3.70488
$111$	126.082			1.13587
$112$			119.413i			1.06619i
$113$	−146.448			−1.29600			−0.647999	−	0.761642i	$-0.724394\pi$
$113$	−146.448			−1.29600			−0.647999	+	0.761642i	$0.724394\pi$
$114$	−7.63929			−0.0670113
$115$	402.167			3.49710
$116$	71.2091			0.613871
$117$	−186.227			−1.59168
$118$	−139.993			−1.18638
$119$	137.205			1.15298
$120$		−	126.543i		−	1.05452i
$121$	156.594			1.29417
$122$	−282.239			−2.31344
$123$	230.040			1.87024
$124$		−	19.0891i		−	0.153944i
$125$			326.512i			2.61209i
$126$	305.497			2.42458
$127$	37.1061			0.292174			0.146087	−	0.989272i	$-0.453332\pi$
$127$	37.1061			0.292174			0.146087	+	0.989272i	$0.453332\pi$
$128$			81.2691i			0.634915i
$129$	−72.1518			−0.559316
$130$	−248.403			−1.91079
$131$	−144.981			−1.10672			−0.553361	−	0.832942i	$-0.686655\pi$
$131$	−144.981			−1.10672			−0.553361	+	0.832942i	$0.686655\pi$
$132$			262.261i			1.98682i
$133$			3.47198i			0.0261051i
$134$			157.826i			1.17781i
$135$	−451.040			−3.34104
$136$	57.1291			0.420067
$137$	−240.727			−1.75713			−0.878565	−	0.477623i	$-0.841498\pi$
$137$	−240.727			−1.75713			−0.878565	+	0.477623i	$0.841498\pi$
$138$			602.669i			4.36717i
$139$	−86.2861			−0.620763			−0.310381	−	0.950612i	$-0.600457\pi$
$139$	−86.2861			−0.620763			−0.310381	+	0.950612i	$0.600457\pi$
$140$	174.990			1.24993
$141$		−	423.159i		−	3.00113i
$142$	18.9724			0.133608
$143$	−169.200			−1.18321
$144$	348.030			2.41688
$145$			218.512i			1.50698i
$146$		−	187.629i		−	1.28513i
$147$			49.2108i			0.334768i
$148$			72.5966i			0.490518i
$149$	−139.156			−0.933934			−0.466967	−	0.884275i	$-0.654653\pi$
$149$	−139.156			−0.933934			−0.466967	+	0.884275i	$0.654653\pi$
$150$	−835.393			−5.56929
$151$			2.94185i			0.0194825i	0.999953	+	0.00974124i	$0.00310078\pi$
$151$			2.94185i			0.0194825i	−0.999953	+	0.00974124i	$0.996899\pi$
$152$			1.44565i			0.00951088i
$153$		−	399.886i		−	2.61364i
$154$	277.564			1.80237
$155$	58.5767			0.377914
$156$		−	159.854i		−	1.02470i
$157$	244.912			1.55995			0.779975	−	0.625811i	$-0.215232\pi$
$157$	244.912			1.55995			0.779975	+	0.625811i	$0.215232\pi$
$158$	79.0821			0.500520
$159$			231.258i			1.45445i
$160$	367.418			2.29636
$161$	273.907			1.70128
$162$		−	238.923i		−	1.47483i
$163$		−	28.4083i		−	0.174284i	−0.996196	−	0.0871422i	$-0.972227\pi$
$163$		−	28.4083i		−	0.174284i	0.996196	−	0.0871422i	$-0.0277734\pi$
$164$			132.455i			0.807650i
$165$	−804.773			−4.87741
$166$	149.497			0.900587
$167$	−157.968			−0.945915			−0.472957	−	0.881085i	$-0.656814\pi$
$167$	−157.968			−0.945915			−0.472957	+	0.881085i	$0.656814\pi$
$168$		−	86.1855i		−	0.513009i
$169$	−65.8692			−0.389759
$170$		−	533.396i		−	3.13763i
$171$	10.1191			0.0591762
$172$		−	41.5442i		−	0.241536i
$173$			54.3487i			0.314154i	0.987586	+	0.157077i	$0.0502071\pi$
$173$			54.3487i			0.314154i	−0.987586	+	0.157077i	$0.949793\pi$
$174$	−327.452			−1.88191
$175$			379.677i			2.16958i
$176$	316.209			1.79664
$177$	276.447			1.56185
$178$	−213.575			−1.19986
$179$	−10.4730			−0.0585081			−0.0292541	−	0.999572i	$-0.509313\pi$
$179$	−10.4730			−0.0585081			−0.0292541	+	0.999572i	$0.509313\pi$
$180$		−	510.012i		−	2.83340i
$181$	−227.772			−1.25841			−0.629206	−	0.777239i	$-0.716620\pi$
$181$	−227.772			−1.25841			−0.629206	+	0.777239i	$0.716620\pi$
$182$	−169.181			−0.929568
$183$	557.344			3.04560
$184$	114.049			0.619830
$185$	−222.770			−1.20416
$186$			87.7803i			0.471937i
$187$		−	363.323i		−	1.94291i
$188$	243.650			1.29601
$189$	−307.193			−1.62536
$190$	13.4976			0.0710400
$191$	100.853			0.528028			0.264014	−	0.964519i	$-0.414953\pi$
$191$	100.853			0.528028			0.264014	+	0.964519i	$0.414953\pi$
$192$			153.669i			0.800358i
$193$	275.300			1.42643			0.713213	−	0.700947i	$-0.247239\pi$
$193$	275.300			1.42643			0.713213	+	0.700947i	$0.247239\pi$
$194$		−	80.1052i		−	0.412913i
$195$	490.526			2.51552
$196$	−28.3351			−0.144567
$197$			89.1568i			0.452573i	0.974061	+	0.226286i	$0.0726585\pi$
$197$			89.1568i			0.452573i	−0.974061	+	0.226286i	$0.927341\pi$
$198$		−	808.965i		−	4.08568i
$199$	24.9584			0.125419			0.0627094	−	0.998032i	$-0.480026\pi$
$199$	24.9584			0.125419			0.0627094	+	0.998032i	$0.480026\pi$
$200$			158.089i			0.790446i
$201$		−	311.663i		−	1.55056i
$202$	−496.592			−2.45838
$203$			148.824i			0.733121i
$204$	343.254			1.68262
$205$	−406.450			−1.98268
$206$	86.7149			0.420946
$207$		−	798.306i		−	3.85655i
$208$	−192.736			−0.926615
$209$	9.19390			0.0439899
$210$	−804.686			−3.83184
$211$			238.907i			1.13226i	0.824316	+	0.566129i	$0.191560\pi$
$211$			238.907i			1.13226i	−0.824316	+	0.566129i	$0.808440\pi$
$212$	−133.156			−0.628094
$213$	−37.4652			−0.175893
$214$	−393.205			−1.83741
$215$	127.483			0.592942
$216$	−127.908			−0.592169
$217$	39.8952			0.183849
$218$	−35.5331			−0.162996
$219$			370.516i			1.69185i
$220$		−	463.380i		−	2.10627i
$221$			221.453i			1.00205i
$222$		−	333.833i		−	1.50375i
$223$		−	312.885i		−	1.40307i	−0.712634	−	0.701536i	$-0.752498\pi$
$223$		−	312.885i		−	1.40307i	0.712634	−	0.701536i	$-0.247502\pi$
$224$	250.240			1.11714
$225$	1106.58			4.91811
$226$			387.756i			1.71573i
$227$	292.834			1.29002			0.645008	−	0.764176i	$-0.276854\pi$
$227$	292.834			1.29002			0.645008	+	0.764176i	$0.276854\pi$
$228$			8.68606i			0.0380967i
$229$		−	33.6078i		−	0.146759i	−0.997304	−	0.0733795i	$-0.976622\pi$
$229$		−	33.6078i		−	0.146759i	0.997304	−	0.0733795i	$-0.0233784\pi$
$230$		−	1064.84i		−	4.62972i
$231$	−548.112			−2.37278
$232$			61.9668i			0.267098i
$233$	20.7195			0.0889250			0.0444625	−	0.999011i	$-0.485842\pi$
$233$	20.7195			0.0889250			0.0444625	+	0.999011i	$0.485842\pi$
$234$			493.082i			2.10719i
$235$			747.665i			3.18155i
$236$			159.175i			0.674471i
$237$	−156.165			−0.658925
$238$		−	363.284i		−	1.52640i
$239$	340.122			1.42310			0.711552	−	0.702634i	$-0.247993\pi$
$239$	340.122			1.42310			0.711552	+	0.702634i	$0.247993\pi$
$240$	−916.720			−3.81967
$241$			88.8812i			0.368802i	0.982851	+	0.184401i	$0.0590345\pi$
$241$			88.8812i			0.368802i	−0.982851	+	0.184401i	$0.940966\pi$
$242$		−	414.622i		−	1.71331i
$243$			32.3940i			0.133308i
$244$			320.913i			1.31522i
$245$		−	86.9490i		−	0.354894i
$246$		−	609.087i		−	2.47596i
$247$	−5.60388			−0.0226878
$248$	16.6115			0.0669818
$249$	−295.216			−1.18561
$250$	864.520			3.45808
$251$		−	379.531i		−	1.51207i	−0.654529	−	0.756037i	$-0.727133\pi$
$251$		−	379.531i		−	1.51207i	0.654529	−	0.756037i	$-0.272867\pi$
$252$		−	347.358i		−	1.37840i
$253$		−	725.314i		−	2.86685i
$254$		−	98.2475i		−	0.386801i
$255$			1053.31i			4.13063i
$256$	332.741			1.29977
$257$		−	214.109i		−	0.833109i	−0.909111	−	0.416554i	$-0.863238\pi$
$257$		−	214.109i		−	0.833109i	0.909111	−	0.416554i	$-0.136762\pi$
$258$			191.039i			0.740463i
$259$	−151.723			−0.585805
$260$			282.440i			1.08631i
$261$	433.749			1.66187
$262$			383.871i			1.46516i
$263$	−413.772			−1.57328			−0.786639	−	0.617413i	$-0.788181\pi$
$263$	−413.772			−1.57328			−0.786639	+	0.617413i	$0.788181\pi$
$264$	−228.222			−0.864477
$265$		−	408.602i		−	1.54189i
$266$	9.19290			0.0345598
$267$	421.752			1.57959
$268$	179.452			0.669598
$269$	−343.174			−1.27574			−0.637870	−	0.770144i	$-0.720184\pi$
$269$	−343.174			−1.27574			−0.637870	+	0.770144i	$0.720184\pi$
$270$			1194.24i			4.42311i
$271$			244.073i			0.900639i	0.892868	+	0.450319i	$0.148690\pi$
$271$			244.073i			0.900639i	−0.892868	+	0.450319i	$0.851310\pi$
$272$		−	413.863i		−	1.52155i
$273$	334.086			1.22376
$274$			637.383i			2.32622i
$275$	1005.40			3.65599
$276$	685.249			2.48279
$277$	−163.515			−0.590305			−0.295153	−	0.955450i	$-0.595371\pi$
$277$	−163.515			−0.590305			−0.295153	+	0.955450i	$0.595371\pi$
$278$			228.463i			0.821811i
$279$		−	116.275i		−	0.416757i
$280$			152.278i			0.543851i
$281$			110.741i			0.394095i	0.980394	+	0.197047i	$0.0631353\pi$
$281$			110.741i			0.394095i	−0.980394	+	0.197047i	$0.936865\pi$
$282$	−1120.42			−3.97311
$283$		−	54.7673i		−	0.193524i	−0.995308	−	0.0967619i	$-0.969151\pi$
$283$		−	54.7673i		−	0.193524i	0.995308	−	0.0967619i	$-0.0308485\pi$
$284$		−	21.5721i		−	0.0759580i
$285$	−26.6540			−0.0935228
$286$			447.997i			1.56642i
$287$	−276.824			−0.964542
$288$		−	729.329i		−	2.53239i
$289$	−186.528			−0.645425
$290$	578.564			1.99505
$291$			158.185i			0.543593i
$292$	−213.339			−0.730614
$293$	77.3963			0.264151			0.132076	−	0.991240i	$-0.457836\pi$
$293$	77.3963			0.264151			0.132076	+	0.991240i	$0.457836\pi$
$294$	130.298			0.443189
$295$	−488.445			−1.65575
$296$	−63.1742			−0.213426
$297$			813.457i			2.73891i
$298$			368.450i			1.23641i
$299$			442.094i			1.47858i
$300$			949.862i			3.16621i
$301$	86.8255			0.288457
$302$	7.78928			0.0257923
$303$	980.632			3.23641
$304$	10.4728			0.0344500
$305$	−984.753			−3.22870
$306$	−1058.80			−3.46012
$307$			10.6993i			0.0348511i	0.999848	+	0.0174255i	$0.00554700\pi$
$307$			10.6993i			0.0348511i	−0.999848	+	0.0174255i	$0.994453\pi$
$308$		−	315.597i		−	1.02467i
$309$	−171.238			−0.554168
$310$		−	155.096i		−	0.500310i
$311$	−15.0613			−0.0484287			−0.0242143	−	0.999707i	$-0.507708\pi$
$311$	−15.0613			−0.0484287			−0.0242143	+	0.999707i	$0.507708\pi$
$312$	139.106			0.445853
$313$	313.632			1.00202			0.501010	−	0.865441i	$-0.332962\pi$
$313$	313.632			1.00202			0.501010	+	0.865441i	$0.332962\pi$
$314$		−	648.464i		−	2.06517i
$315$	1065.90			3.38381
$316$		−	89.9183i		−	0.284551i
$317$	180.434			0.569194			0.284597	−	0.958647i	$-0.408140\pi$
$317$	180.434			0.569194			0.284597	+	0.958647i	$0.408140\pi$
$318$	612.312			1.92551
$319$	394.089			1.23539
$320$		−	271.512i		−	0.848476i
$321$	776.471			2.41891
$322$		−	725.235i		−	2.25228i
$323$		−	12.0332i		−	0.0372546i
$324$	−271.661			−0.838460
$325$	−612.811			−1.88557
$326$	−75.2180			−0.230730
$327$	70.1680			0.214581
$328$	−115.263			−0.351412
$329$			509.217i			1.54777i
$330$			2130.83i			6.45707i
$331$		−	481.197i		−	1.45377i	−0.686761	−	0.726884i	$-0.740968\pi$
$331$		−	481.197i		−	1.45377i	0.686761	−	0.726884i	$-0.259032\pi$
$332$		−	169.982i		−	0.511995i
$333$			442.200i			1.32793i
$334$			418.258i			1.25227i
$335$			550.667i			1.64378i
$336$	−624.357			−1.85820
$337$		−	59.3310i		−	0.176056i	−0.996118	−	0.0880281i	$-0.971943\pi$
$337$		−	59.3310i		−	0.176056i	0.996118	−	0.0880281i	$-0.0280565\pi$
$338$			174.405i			0.515990i
$339$		−	765.711i		−	2.25873i
$340$	−606.485			−1.78378
$341$		−	105.644i		−	0.309806i
$342$		−	26.7929i		−	0.0783417i
$343$		−	367.522i		−	1.07149i
$344$	36.1522			0.105094
$345$			2102.76i			6.09494i
$346$	143.901			0.415900
$347$	394.790			1.13772			0.568862	−	0.822433i	$-0.307384\pi$
$347$	394.790			1.13772			0.568862	+	0.822433i	$0.307384\pi$
$348$			372.321i			1.06989i
$349$	168.137			0.481767			0.240884	−	0.970554i	$-0.422563\pi$
$349$	168.137			0.481767			0.240884	+	0.970554i	$0.422563\pi$
$350$	1005.29			2.87225
$351$		−	495.820i		−	1.41259i
$352$		−	662.643i		−	1.88251i
$353$	−141.054			−0.399588			−0.199794	−	0.979838i	$-0.564027\pi$
$353$	−141.054			−0.399588			−0.199794	+	0.979838i	$0.564027\pi$
$354$		−	731.961i		−	2.06769i
$355$	66.1960			0.186468
$356$			242.840i			0.682136i
$357$			717.385i			2.00948i
$358$			27.7297i			0.0774573i
$359$			116.013i			0.323157i	0.986860	+	0.161578i	$0.0516585\pi$
$359$			116.013i			0.323157i	−0.986860	+	0.161578i	$0.948342\pi$
$360$	443.817			1.23283
$361$	−360.695			−0.999157
$362$			603.083i			1.66598i
$363$			818.763i			2.25555i
$364$			192.363i			0.528471i
$365$		−	654.653i		−	1.79357i
$366$		−	1475.70i		−	4.03198i
$367$	236.383			0.644097			0.322048	−	0.946723i	$-0.395629\pi$
$367$	236.383			0.644097			0.322048	+	0.946723i	$0.395629\pi$
$368$		−	826.207i		−	2.24513i
$369$			806.807i			2.18647i
$370$			589.837i			1.59416i
$371$		−	278.289i		−	0.750106i
$372$	99.8084			0.268302
$373$			89.6220i			0.240274i	0.992757	+	0.120137i	$0.0383333\pi$
$373$			89.6220i			0.240274i	−0.992757	+	0.120137i	$0.961667\pi$
$374$	−961.987			−2.57216
$375$	−1707.19			−4.55250
$376$			212.027i			0.563901i
$377$	−240.206			−0.637151
$378$			813.369i			2.15177i
$379$	−482.016			−1.27181			−0.635906	−	0.771767i	$-0.719373\pi$
$379$	−482.016			−1.27181			−0.635906	+	0.771767i	$0.719373\pi$
$380$		−	15.3471i		−	0.0403871i
$381$			194.012i			0.509217i
$382$		−	267.034i		−	0.699042i
$383$	−55.9686			−0.146132			−0.0730661	−	0.997327i	$-0.523278\pi$
$383$	−55.9686			−0.146132			−0.0730661	+	0.997327i	$0.523278\pi$
$384$	−424.920			−1.10656
$385$	968.441			2.51543
$386$		−	728.925i		−	1.88841i
$387$		−	253.054i		−	0.653887i
$388$	−91.0815			−0.234746
$389$		−	29.2447i		−	0.0751793i	−0.999293	−	0.0375896i	$-0.988032\pi$
$389$		−	29.2447i		−	0.0751793i	0.999293	−	0.0375896i	$-0.0119680\pi$
$390$		−	1298.79i		−	3.33023i
$391$	−949.311			−2.42791
$392$		−	24.6574i		−	0.0629017i
$393$		−	758.040i		−	1.92885i
$394$	236.064			0.599148
$395$	275.923			0.698539
$396$	−919.814			−2.32276
$397$			53.6950i			0.135252i	0.997711	+	0.0676260i	$0.0215425\pi$
$397$			53.6950i			0.135252i	−0.997711	+	0.0676260i	$0.978458\pi$
$398$		−	66.0833i		−	0.166039i
$399$	−18.1534			−0.0454973
$400$	1145.25			2.86313
$401$	−362.388			−0.903711			−0.451856	−	0.892091i	$-0.649238\pi$
$401$	−362.388			−0.903711			−0.451856	+	0.892091i	$0.649238\pi$
$402$	−825.204			−2.05275
$403$			64.3922i			0.159782i
$404$			564.637i			1.39762i
$405$		−	833.618i		−	2.05832i
$406$	394.047			0.970559
$407$			401.768i			0.987146i
$408$			298.703i			0.732116i
$409$	478.336			1.16953			0.584763	−	0.811204i	$-0.301188\pi$
$409$	478.336			1.16953			0.584763	+	0.811204i	$0.301188\pi$
$410$			1076.17i			2.62482i
$411$		−	1258.65i		−	3.06242i
$412$		−	98.5969i		−	0.239313i
$413$	−332.669			−0.805493
$414$	−2113.71			−5.10558
$415$	521.607			1.25688
$416$			403.895i			0.970902i
$417$		−	451.152i		−	1.08190i
$418$		−	24.3431i		−	0.0582371i
$419$	358.382			0.855326			0.427663	−	0.903938i	$-0.359337\pi$
$419$	358.382			0.855326			0.427663	+	0.903938i	$0.359337\pi$
$420$			914.948i			2.17845i
$421$		−	60.8665i		−	0.144576i	−0.997384	−	0.0722880i	$-0.976970\pi$
$421$		−	60.8665i		−	0.144576i	0.997384	−	0.0722880i	$-0.0230301\pi$
$422$	632.564			1.49897
$423$	1484.12			3.50856
$424$		−	115.873i		−	0.273286i
$425$		−	1315.89i		−	3.09622i
$426$			99.1983i			0.232860i
$427$	−670.692			−1.57071
$428$			447.084i			1.04459i
$429$		−	884.671i		−	2.06217i
$430$		−	337.541i		−	0.784980i
$431$		−	308.337i		−	0.715398i	−0.933837	−	0.357699i	$-0.883561\pi$
$431$		−	308.337i		−	0.715398i	0.933837	−	0.357699i	$-0.116439\pi$
$432$			926.612i			2.14494i
$433$			660.190i			1.52469i	0.647172	+	0.762344i	$0.275952\pi$
$433$			660.190i			1.52469i	−0.647172	+	0.762344i	$0.724048\pi$
$434$		−	105.632i		−	0.243393i
$435$	−1142.50			−2.62645
$436$			40.4020i			0.0926652i
$437$		−	24.0223i		−	0.0549710i
$438$	981.032			2.23980
$439$	55.3048			0.125979			0.0629895	−	0.998014i	$-0.479937\pi$
$439$	55.3048			0.125979			0.0629895	+	0.998014i	$0.479937\pi$
$440$	403.237			0.916449
$441$	−172.595			−0.391371
$442$	586.352			1.32659
$443$	794.833			1.79421			0.897103	−	0.441821i	$-0.145667\pi$
$443$	794.833			1.79421			0.897103	+	0.441821i	$0.145667\pi$
$444$	−379.576			−0.854900
$445$	−745.179			−1.67456
$446$	−828.440			−1.85749
$447$		−	727.586i		−	1.62771i
$448$		−	184.921i		−	0.412770i
$449$	251.606			0.560370			0.280185	−	0.959946i	$-0.409604\pi$
$449$	251.606			0.560370			0.280185	+	0.959946i	$0.409604\pi$
$450$		−	2929.93i		−	6.51095i
$451$			733.037i			1.62536i
$452$	440.888			0.975416
$453$	−15.3817			−0.0339551
$454$		−	775.348i		−	1.70782i
$455$	−590.286			−1.29733
$456$	−7.55869			−0.0165761
$457$		−	519.912i		−	1.13766i	−0.822454	−	0.568831i	$-0.807396\pi$
$457$		−	519.912i		−	1.13766i	0.822454	−	0.568831i	$-0.192604\pi$
$458$	−88.9849			−0.194290
$459$	1064.68			2.31956
$460$	−1210.74			−2.63205
$461$			79.7604i			0.173016i	0.996251	+	0.0865080i	$0.0275708\pi$
$461$			79.7604i			0.173016i	−0.996251	+	0.0865080i	$0.972429\pi$
$462$			1451.26i			3.14126i
$463$			701.708i			1.51557i	0.652506	+	0.757784i	$0.273718\pi$
$463$			701.708i			1.51557i	−0.652506	+	0.757784i	$0.726282\pi$
$464$	448.909			0.967476
$465$			306.272i			0.658649i
$466$		−	54.8600i		−	0.117725i
$467$	−571.154			−1.22303			−0.611514	−	0.791234i	$-0.709439\pi$
$467$	−571.154			−1.22303			−0.611514	+	0.791234i	$0.709439\pi$
$468$	560.646			1.19796
$469$			375.047i			0.799673i
$470$	1979.63			4.21197
$471$			1280.54i			2.71876i
$472$	−138.516			−0.293466
$473$		−	229.916i		−	0.486081i
$474$			413.486i			0.872332i
$475$	33.2987			0.0701025
$476$	−413.063			−0.867779
$477$	−811.079			−1.70037
$478$		−	900.555i		−	1.88401i
$479$	66.5614			0.138959			0.0694796	−	0.997583i	$-0.477866\pi$
$479$	66.5614			0.138959			0.0694796	+	0.997583i	$0.477866\pi$
$480$			1921.07i			4.00223i
$481$		−	244.886i		−	0.509119i
$482$	235.335			0.488246
$483$			1432.14i			2.96509i
$484$	−471.435			−0.974039
$485$		−	279.493i		−	0.576273i
$486$	85.7709			0.176483
$487$		−	513.999i		−	1.05544i	−0.849419	−	0.527719i	$-0.823047\pi$
$487$		−	513.999i		−	1.05544i	0.849419	−	0.527719i	$-0.176953\pi$
$488$	−279.261			−0.572257
$489$	148.535			0.303752
$490$	−230.219			−0.469834
$491$			743.174i			1.51359i	0.653650	+	0.756797i	$0.273237\pi$
$491$			743.174i			1.51359i	−0.653650	+	0.756797i	$0.726763\pi$
$492$	−692.547			−1.40762
$493$		−	515.795i		−	1.04624i
$494$			14.8376i			0.0300357i
$495$		−	2822.54i		−	5.70210i
$496$		−	120.339i		−	0.242619i
$497$	45.0846			0.0907134
$498$			781.656i			1.56959i
$499$	584.784			1.17191			0.585956	−	0.810343i	$-0.300719\pi$
$499$	584.784			1.17191			0.585956	+	0.810343i	$0.300719\pi$
$500$		−	982.980i		−	1.96596i
$501$		−	825.944i		−	1.64859i
$502$	−1004.90			−2.00179
$503$			336.382i			0.668752i	0.942440	+	0.334376i	$0.108526\pi$
$503$			336.382i			0.668752i	−0.942440	+	0.334376i	$0.891474\pi$
$504$	302.274			0.599749
$505$	−1732.64			−3.43098
$506$	−1920.44			−3.79535
$507$		−	344.401i		−	0.679292i
$508$	−111.710			−0.219901
$509$	427.742			0.840357			0.420179	−	0.907441i	$-0.361967\pi$
$509$	427.742			0.840357			0.420179	+	0.907441i	$0.361967\pi$
$510$	2788.90			5.46842
$511$		−	445.869i		−	0.872542i
$512$		−	555.936i		−	1.08581i
$513$			26.9416i			0.0525178i
$514$	−566.906			−1.10293
$515$	302.554			0.587484
$516$	217.217			0.420962
$517$	1348.42			2.60817
$518$			401.725i			0.775530i
$519$	−284.165			−0.547525
$520$	−245.782			−0.472657
$521$	−728.025			−1.39736			−0.698680	−	0.715434i	$-0.746229\pi$
$521$	−728.025			−1.39736			−0.698680	+	0.715434i	$0.746229\pi$
$522$		−	1148.46i		−	2.20011i
$523$			106.802i			0.204211i	0.994774	+	0.102105i	$0.0325579\pi$
$523$			106.802i			0.204211i	−0.994774	+	0.102105i	$0.967442\pi$
$524$	436.471			0.832960
$525$	−1985.17			−3.78127
$526$			1095.56i			2.08282i
$527$	−138.270			−0.262371
$528$			1653.32i			3.13128i
$529$	−1366.14			−2.58249
$530$	−1081.87			−2.04127
$531$			969.568i			1.82593i
$532$		−	10.4526i		−	0.0196477i
$533$		−	446.802i		−	0.838278i
$534$		−	1116.69i		−	2.09118i
$535$	−1371.92			−2.56434
$536$			156.161i			0.291345i
$537$		−	54.7585i		−	0.101971i
$538$			908.637i			1.68892i
$539$	−156.814			−0.290934
$540$	1357.88			2.51459
$541$			478.720i			0.884880i	0.896798	+	0.442440i	$0.145887\pi$
$541$			478.720i			0.884880i	−0.896798	+	0.442440i	$0.854113\pi$
$542$	646.243			1.19233
$543$		−	1190.92i		−	2.19323i
$544$	−867.287			−1.59428
$545$	−123.978			−0.227482
$546$		−	884.575i		−	1.62010i
$547$	389.350	+	384.208i	0.711792	+	0.702390i
$547$	389.350	+	384.208i	0.711792	+	0.702390i
$548$	724.720			1.32248
$549$			1954.74i			3.56055i
$550$		−	2662.04i		−	4.84006i
$551$	13.0522			0.0236882
$552$			596.310i			1.08027i
$553$	187.925			0.339828
$554$			432.945i			0.781489i
$555$		−	1164.77i		−	2.09868i
$556$	259.769			0.467210
$557$	181.484			0.325824			0.162912	−	0.986641i	$-0.447911\pi$
$557$	181.484			0.325824			0.162912	+	0.986641i	$0.447911\pi$
$558$	−307.867			−0.551733
$559$			140.139i			0.250696i
$560$	1103.16			1.96992
$561$	1899.66			3.38620
$562$	293.213			0.521731
$563$	525.657			0.933672			0.466836	−	0.884344i	$-0.345394\pi$
$563$	525.657			0.933672			0.466836	+	0.884344i	$0.345394\pi$
$564$			1273.94i			2.25876i
$565$			1352.91i			2.39453i
$566$	−145.010			−0.256201
$567$		−	567.758i		−	1.00134i
$568$	18.7722			0.0330497
$569$		−	114.910i		−	0.201951i	−0.994889	−	0.100975i	$-0.967804\pi$
$569$		−	114.910i		−	0.201951i	0.994889	−	0.100975i	$-0.0321963\pi$
$570$			70.5730i			0.123812i
$571$	175.929			0.308108			0.154054	−	0.988062i	$-0.450767\pi$
$571$	175.929			0.308108			0.154054	+	0.988062i	$0.450767\pi$
$572$	509.384			0.890532
$573$			527.318i			0.920276i
$574$			732.958i			1.27693i
$575$		−	2626.96i		−	4.56862i
$576$	−538.954			−0.935685
$577$		−	6.09144i		−	0.0105571i	−0.999986	−	0.00527855i	$-0.998320\pi$
$577$		−	6.09144i		−	0.0105571i	0.999986	−	0.00527855i	$-0.00168022\pi$
$578$			493.878i			0.854460i
$579$			1439.42i			2.48605i
$580$		−	657.842i		−	1.13421i
$581$	355.255			0.611454
$582$	418.835			0.719647
$583$	−736.919			−1.26401
$584$		−	185.650i		−	0.317894i
$585$			1720.40i			2.94085i
$586$		−	204.926i		−	0.349703i
$587$	−1076.30			−1.83355			−0.916777	−	0.399400i	$-0.869218\pi$
$587$	−1076.30			−1.83355			−0.916777	+	0.399400i	$0.869218\pi$
$588$		−	148.152i		−	0.251959i
$589$		−	3.49891i		−	0.00594043i
$590$			1293.28i			2.19199i
$591$	−466.162			−0.788768
$592$			457.656i			0.773067i
$593$	294.921			0.497337			0.248668	−	0.968589i	$-0.420007\pi$
$593$	294.921			0.497337			0.248668	+	0.968589i	$0.420007\pi$
$594$	2153.83			3.62597
$595$		−	1267.52i		−	2.13029i
$596$	418.937			0.702914
$597$			130.496i			0.218587i
$598$	1170.55			1.95745
$599$	209.829			0.350298			0.175149	−	0.984542i	$-0.443959\pi$
$599$	209.829			0.350298			0.175149	+	0.984542i	$0.443959\pi$
$600$	−826.579			−1.37763
$601$	736.803			1.22596			0.612981	−	0.790098i	$-0.289970\pi$
$601$	736.803			1.22596			0.612981	+	0.790098i	$0.289970\pi$
$602$		−	229.892i		−	0.381880i
$603$	1093.08			1.81274
$604$		−	8.85660i		−	0.0146632i
$605$		−	1446.64i		−	2.39115i
$606$		−	2596.46i		−	4.28459i
$607$	−450.191			−0.741665			−0.370832	−	0.928700i	$-0.620928\pi$
$607$	−450.191			−0.741665			−0.370832	+	0.928700i	$0.620928\pi$
$608$		−	21.9467i		−	0.0360965i
$609$	−778.133			−1.27772
$610$			2607.37i			4.27438i
$611$	−821.893			−1.34516
$612$			1203.88i			1.96712i
$613$	−169.096			−0.275850			−0.137925	−	0.990443i	$-0.544043\pi$
$613$	−169.096			−0.275850			−0.137925	+	0.990443i	$0.544043\pi$
$614$	28.3290			0.0461384
$615$		−	2125.15i		−	3.45552i
$616$	274.636			0.445837
$617$		−	775.318i		−	1.25659i	−0.777974	−	0.628297i	$-0.783752\pi$
$617$		−	775.318i		−	1.25659i	0.777974	−	0.628297i	$-0.216248\pi$
$618$			453.394i			0.733647i
$619$		−	170.450i		−	0.275363i	−0.990477	−	0.137682i	$-0.956035\pi$
$619$		−	170.450i		−	0.275363i	0.990477	−	0.137682i	$-0.0439651\pi$
$620$	−176.348			−0.284432
$621$	2125.45			3.42262
$622$			39.8785i			0.0641134i
$623$	−507.524			−0.814646
$624$		−	1007.73i		−	1.61495i
$625$	1507.78			2.41244
$626$		−	830.418i		−	1.32655i
$627$			48.0709i			0.0766680i
$628$	−737.320			−1.17408
$629$	525.846			0.836003
$630$		−	2822.23i		−	4.47973i
$631$	−612.860			−0.971252			−0.485626	−	0.874167i	$-0.661408\pi$
$631$	−612.860			−0.971252			−0.485626	+	0.874167i	$0.661408\pi$
$632$	78.2477			0.123810
$633$	−1249.14			−1.97336
$634$		−	477.744i		−	0.753540i
$635$		−	342.792i		−	0.539831i
$636$		−	696.214i		−	1.09468i
$637$	95.5812			0.150049
$638$		−	1043.45i		−	1.63550i
$639$		−	131.400i		−	0.205633i
$640$	750.777			1.17309
$641$			828.764i			1.29292i	0.762947	+	0.646461i	$0.223752\pi$
$641$			828.764i			1.29292i	−0.762947	+	0.646461i	$0.776248\pi$
$642$		−	2055.90i		−	3.20233i
$643$	153.697			0.239032			0.119516	−	0.992832i	$-0.461866\pi$
$643$	153.697			0.239032			0.119516	+	0.992832i	$0.461866\pi$
$644$	−824.610			−1.28045
$645$			666.550i			1.03341i
$646$	−31.8609			−0.0493203
$647$	1104.20			1.70665			0.853323	−	0.521383i	$-0.174584\pi$
$647$	1104.20			1.70665			0.853323	+	0.521383i	$0.174584\pi$
$648$		−	236.402i		−	0.364818i
$649$			880.917i			1.35734i
$650$			1622.57i			2.49626i
$651$			208.595i			0.320422i
$652$			85.5248i			0.131173i
$653$			868.524i			1.33005i	0.746820	+	0.665026i	$0.231579\pi$
$653$			868.524i			1.33005i	−0.746820	+	0.665026i	$0.768421\pi$
$654$		−	185.787i		−	0.284078i
$655$			1339.35i			2.04482i
$656$			835.005i			1.27287i
$657$	−1299.49			−1.97792
$658$	1348.28			2.04905
$659$		−	176.628i		−	0.268024i	−0.990980	−	0.134012i	$-0.957214\pi$
$659$		−	176.628i		−	0.268024i	0.990980	−	0.134012i	$-0.0427861\pi$
$660$	2422.81			3.67092
$661$	−1105.45			−1.67239			−0.836197	−	0.548429i	$-0.815226\pi$
$661$	−1105.45			−1.67239			−0.836197	+	0.548429i	$0.815226\pi$
$662$	−1274.09			−1.92460
$663$	−1157.88			−1.74643
$664$	147.920			0.222771
$665$	32.0747			0.0482326
$666$	1170.83			1.75801
$667$		−	1029.70i		−	1.54378i
$668$	475.570			0.711931
$669$	1635.94			2.44535
$670$	1458.03			2.17616
$671$			1776.01i			2.64682i
$672$			1308.40i			1.94702i
$673$	−602.366			−0.895046			−0.447523	−	0.894272i	$-0.647694\pi$
$673$	−602.366			−0.895046			−0.447523	+	0.894272i	$0.647694\pi$
$674$	−157.093			−0.233076
$675$			2946.20i			4.36474i
$676$	198.303			0.293347
$677$	1090.73			1.61113			0.805563	−	0.592510i	$-0.201863\pi$
$677$	1090.73			1.61113			0.805563	+	0.592510i	$0.201863\pi$
$678$	−2027.41			−2.99027
$679$		−	190.356i		−	0.280348i
$680$		−	527.769i		−	0.776130i
$681$			1531.10i			2.24831i
$682$	−279.718			−0.410143
$683$	−764.931			−1.11996			−0.559979	−	0.828507i	$-0.689191\pi$
$683$	−764.931			−1.11996			−0.559979	+	0.828507i	$0.689191\pi$
$684$	−30.4641			−0.0445382
$685$			2223.87i			3.24653i
$686$	−973.105			−1.41852
$687$	175.721			0.255780
$688$		−	261.899i		−	0.380667i
$689$	449.168			0.651913
$690$	5567.56			8.06892
$691$	−232.642			−0.336675			−0.168337	−	0.985729i	$-0.553840\pi$
$691$	−232.642			−0.336675			−0.168337	+	0.985729i	$0.553840\pi$
$692$		−	163.619i		−	0.236444i
$693$		−	1922.37i		−	2.77398i
$694$		−	1045.30i		−	1.50620i
$695$			797.125i			1.14694i
$696$	−323.997			−0.465513
$697$	959.420			1.37650
$698$		−	445.183i		−	0.637798i
$699$			108.333i			0.154983i
$700$		−	1143.04i		−	1.63291i
$701$	−855.682			−1.22066			−0.610330	−	0.792148i	$-0.708963\pi$
$701$	−855.682			−1.22066			−0.610330	+	0.792148i	$0.708963\pi$
$702$	−1312.80			−1.87009
$703$			13.3065i			0.0189282i
$704$	−489.676			−0.695562
$705$	−3909.21			−5.54498
$706$			373.476i			0.529003i
$707$	−1180.06			−1.66912
$708$	−832.258			−1.17550
$709$		−	583.367i		−	0.822802i	−0.911454	−	0.411401i	$-0.865040\pi$
$709$		−	583.367i		−	0.822802i	0.911454	−	0.411401i	$-0.134960\pi$
$710$		−	175.270i		−	0.246859i
$711$		−	547.710i		−	0.770337i
$712$	−211.322			−0.296800
$713$	−276.032			−0.387142
$714$	1899.45			2.66030
$715$			1563.10i			2.18615i
$716$	31.5294			0.0440354
$717$			1778.35i			2.48026i
$718$	307.174			0.427818
$719$		−	521.403i		−	0.725178i	−0.931949	−	0.362589i	$-0.881893\pi$
$719$		−	521.403i		−	0.725178i	0.931949	−	0.362589i	$-0.118107\pi$
$720$		−	3215.16i		−	4.46550i
$721$	206.063			0.285801
$722$			955.030i			1.32276i
$723$	−464.721			−0.642768
$724$	685.720			0.947128
$725$	1427.32			1.96872
$726$	2167.88			2.98605
$727$		−	1080.09i		−	1.48568i	−0.669470	−	0.742839i	$-0.733479\pi$
$727$		−	1080.09i		−	1.48568i	0.669470	−	0.742839i	$-0.266521\pi$
$728$	−167.396			−0.229940
$729$	642.753			0.881691
$730$	−1733.35			−2.37446
$731$	−300.921			−0.411657
$732$	−1677.91			−2.29223
$733$			1163.92i			1.58789i	0.607992	+	0.793943i	$0.291975\pi$
$733$			1163.92i			1.58789i	−0.607992	+	0.793943i	$0.708025\pi$
$734$		−	625.883i		−	0.852702i
$735$	454.618			0.618528
$736$	−1731.39			−2.35243
$737$	993.135			1.34754
$738$	2136.22			2.89460
$739$		−	933.507i		−	1.26320i	−0.775293	−	0.631601i	$-0.782398\pi$
$739$		−	933.507i		−	1.26320i	0.775293	−	0.631601i	$-0.217602\pi$
$740$	670.660			0.906297
$741$		−	29.3002i		−	0.0395415i
$742$	−736.839			−0.993045
$743$	655.904			0.882778			0.441389	−	0.897316i	$-0.354486\pi$
$743$	655.904			0.882778			0.441389	+	0.897316i	$0.354486\pi$
$744$			86.8542i			0.116739i
$745$			1285.55i			1.72557i
$746$	237.296			0.318091
$747$		−	1035.40i		−	1.38607i
$748$			1093.80i			1.46230i
$749$	−934.383			−1.24751
$750$			4520.20i			6.02693i
$751$	−629.684			−0.838460			−0.419230	−	0.907880i	$-0.637700\pi$
$751$	−629.684			−0.838460			−0.419230	+	0.907880i	$0.637700\pi$
$752$	1535.99			2.04255
$753$	1984.40			2.63532
$754$			636.004i			0.843507i
$755$	27.1773			0.0359965
$756$	924.821			1.22331
$757$	667.491			0.881758			0.440879	−	0.897566i	$-0.354667\pi$
$757$	667.491			0.881758			0.440879	+	0.897566i	$0.354667\pi$
$758$			1276.26i			1.68372i
$759$	3792.35			4.99650
$760$	13.3552			0.0175726
$761$	−196.493			−0.258204			−0.129102	−	0.991631i	$-0.541209\pi$
$761$	−196.493			−0.258204			−0.129102	+	0.991631i	$0.541209\pi$
$762$	513.693			0.674138
$763$	−84.4383			−0.110666
$764$	−303.624			−0.397414
$765$	−3694.22			−4.82904
$766$			148.191i			0.193460i
$767$		−	536.938i		−	0.700049i
$768$			1739.76i			2.26531i
$769$		−	583.266i		−	0.758473i	−0.925300	−	0.379236i	$-0.876187\pi$
$769$		−	583.266i		−	0.758473i	0.925300	−	0.379236i	$-0.123813\pi$
$770$		−	2564.19i		−	3.33011i
$771$	1119.48			1.45199
$772$	−828.806			−1.07358
$773$		−	250.504i		−	0.324067i	−0.986785	−	0.162033i	$-0.948195\pi$
$773$		−	250.504i		−	0.324067i	0.986785	−	0.162033i	$-0.0518052\pi$
$774$	−670.023			−0.865662
$775$		−	382.623i		−	0.493707i
$776$		−	79.2600i		−	0.102139i
$777$		−	793.295i		−	1.02097i
$778$	−77.4326			−0.0995277
$779$			24.2781i			0.0311658i
$780$	−1476.75			−1.89328
$781$		−	119.385i		−	0.152862i
$782$			2513.53i			3.21424i
$783$			1154.83i			1.47488i
$784$	−178.627			−0.227840
$785$		−	2262.54i		−	2.88221i
$786$	−2007.09			−2.55356
$787$	−251.652			−0.319761			−0.159881	−	0.987136i	$-0.551111\pi$
$787$	−251.652			−0.319761			−0.159881	+	0.987136i	$0.551111\pi$
$788$		−	268.411i		−	0.340623i
$789$		−	2163.43i		−	2.74199i
$790$		−	730.574i		−	0.924777i
$791$			921.435i			1.16490i
$792$		−	800.430i		−	1.01064i
$793$		−	1082.52i		−	1.36509i
$794$	142.171			0.179056
$795$	2136.40			2.68730
$796$	−75.1384			−0.0943949
$797$	−147.890			−0.185558			−0.0927789	−	0.995687i	$-0.529575\pi$
$797$	−147.890			−0.185558			−0.0927789	+	0.995687i	$0.529575\pi$
$798$			48.0657i			0.0602327i
$799$		−	1764.86i		−	2.20883i
$800$		−	2399.98i		−	2.99997i
$801$			1479.19i			1.84668i
$802$			959.512i			1.19640i
$803$	−1180.67			−1.47033
$804$			938.277i			1.16701i
$805$		−	2530.40i		−	3.14335i
$806$	170.494			0.211531
$807$		−	1794.31i		−	2.22343i
$808$	−491.353			−0.608110
$809$		−	801.275i		−	0.990451i	−0.868764	−	0.495226i	$-0.835085\pi$
$809$		−	801.275i		−	0.990451i	0.868764	−	0.495226i	$-0.164915\pi$
$810$	−2207.21			−2.72495
$811$	−552.245			−0.680943			−0.340471	−	0.940255i	$-0.610587\pi$
$811$	−552.245			−0.680943			−0.340471	+	0.940255i	$0.610587\pi$
$812$		−	448.041i		−	0.551774i
$813$	−1276.15			−1.56968
$814$	1063.78			1.30685
$815$	−262.441			−0.322014
$816$	2163.91			2.65185
$817$		−	7.61481i		−	0.00932046i
$818$		−	1266.51i		−	1.54830i
$819$			1171.72i			1.43068i
$820$	1223.64			1.49224
$821$			433.227i			0.527683i	0.964566	+	0.263841i	$0.0849895\pi$
$821$			433.227i			0.527683i	−0.964566	+	0.263841i	$0.915010\pi$
$822$	−3332.60			−4.05425
$823$	−319.529			−0.388249			−0.194125	−	0.980977i	$-0.562187\pi$
$823$	−319.529			−0.388249			−0.194125	+	0.980977i	$0.562187\pi$
$824$	85.8000			0.104126
$825$			5256.78i			6.37186i
$826$			880.821i			1.06637i
$827$			589.205i			0.712461i	0.934398	+	0.356230i	$0.115938\pi$
$827$			589.205i			0.712461i	−0.934398	+	0.356230i	$0.884062\pi$
$828$			2403.34i			2.90258i
$829$	1588.54			1.91621			0.958104	−	0.286419i	$-0.0924649\pi$
$829$	1588.54			1.91621			0.958104	+	0.286419i	$0.0924649\pi$
$830$		−	1381.08i		−	1.66395i
$831$		−	854.946i		−	1.02882i
$832$	298.468			0.358735
$833$			205.242i			0.246389i
$834$	−1194.54			−1.43230
$835$			1459.33i			1.74770i
$836$	−27.6787			−0.0331085
$837$	309.577			0.369865
$838$		−	948.903i		−	1.13234i
$839$	−853.023			−1.01671			−0.508357	−	0.861146i	$-0.669747\pi$
$839$	−853.023			−1.01671			−0.508357	+	0.861146i	$0.669747\pi$
$840$	−796.196			−0.947852
$841$	−281.527			−0.334753
$842$	−161.159			−0.191400
$843$	−579.015			−0.686850
$844$		−	719.240i		−	0.852181i
$845$			608.511i			0.720131i
$846$		−	3929.58i		−	4.64489i
$847$		−	985.277i		−	1.16325i
$848$	−839.427			−0.989890
$849$	286.354			0.337284
$850$	−3484.15			−4.09900
$851$	1049.76			1.23356
$852$	112.791			0.132384
$853$	741.683			0.869499			0.434749	−	0.900551i	$-0.356837\pi$
$853$	741.683			0.869499			0.434749	+	0.900551i	$0.356837\pi$
$854$			1775.82i			2.07942i
$855$		−	93.4822i		−	0.109336i
$856$	−389.056			−0.454505
$857$		−	832.618i		−	0.971550i	−0.874084	−	0.485775i	$-0.838537\pi$
$857$		−	832.618i		−	0.971550i	0.874084	−	0.485775i	$-0.161463\pi$
$858$	−2342.38			−2.73005
$859$	256.648			0.298775			0.149388	−	0.988779i	$-0.452270\pi$
$859$	256.648			0.298775			0.149388	+	0.988779i	$0.452270\pi$
$860$	−383.793			−0.446271
$861$		−	1447.39i		−	1.68106i
$862$	−816.397			−0.947096
$863$		−	245.163i		−	0.284082i	−0.989861	−	0.142041i	$-0.954633\pi$
$863$		−	245.163i		−	0.284082i	0.989861	−	0.142041i	$-0.0453665\pi$
$864$	1941.80			2.24745
$865$	502.082			0.580442
$866$	1748.01			2.01849
$867$		−	975.272i		−	1.12488i
$868$	−120.107			−0.138372
$869$		−	497.631i		−	0.572648i
$870$			3025.06i			3.47708i
$871$	−605.337			−0.694991
$872$	−35.1582			−0.0403191
$873$	−554.796			−0.635505
$874$	−63.6050			−0.0727746
$875$	2054.38			2.34786
$876$		−	1115.46i		−	1.27335i
$877$		−	1405.84i		−	1.60301i	−0.597990	−	0.801504i	$-0.704034\pi$
$877$		−	1405.84i		−	1.60301i	0.597990	−	0.801504i	$-0.295966\pi$
$878$		−	146.433i		−	0.166780i
$879$			404.672i			0.460377i
$880$		−	2921.19i		−	3.31953i
$881$			1080.06i			1.22595i	0.790103	+	0.612974i	$0.210027\pi$
$881$			1080.06i			1.22595i	−0.790103	+	0.612974i	$0.789973\pi$
$882$			456.986i			0.518125i
$883$	577.162			0.653637			0.326819	−	0.945087i	$-0.394023\pi$
$883$	577.162			0.653637			0.326819	+	0.945087i	$0.394023\pi$
$884$		−	666.696i		−	0.754181i
$885$		−	2553.86i		−	2.88572i
$886$		−	2104.52i		−	2.37530i
$887$	603.092			0.679924			0.339962	−	0.940439i	$-0.389586\pi$
$887$	603.092			0.679924			0.339962	+	0.940439i	$0.389586\pi$
$888$		−	330.310i		−	0.371971i
$889$		−	233.468i		−	0.262619i
$890$			1973.04i			2.21690i
$891$	−1503.44			−1.68736
$892$			941.956i			1.05600i
$893$	44.6597			0.0500108
$894$	−1926.46			−2.15488
$895$			96.7509i			0.108102i
$896$	511.337			0.570689
$897$	−2311.52			−2.57694
$898$		−	666.189i		−	0.741858i
$899$		−	149.978i		−	0.166828i
$900$	−3331.40			−3.70156
$901$			964.500i			1.07048i
$902$	1940.90			2.15177
$903$			453.972i			0.502738i
$904$			383.665i			0.424408i
$905$			2104.20i			2.32508i
$906$			40.7267i			0.0449522i
$907$	−1038.67			−1.14517			−0.572585	−	0.819845i	$-0.694059\pi$
$907$	−1038.67			−1.14517			−0.572585	+	0.819845i	$0.694059\pi$
$908$	−881.590			−0.970914
$909$			3439.32i			3.78363i
$910$			1562.93i			1.71750i
$911$		−	762.766i		−	0.837284i	−0.908151	−	0.418642i	$-0.862506\pi$
$911$		−	762.766i		−	0.837284i	0.908151	−	0.418642i	$-0.137494\pi$
$912$			54.7577i			0.0600413i
$913$		−	940.725i		−	1.03037i
$914$	−1376.59			−1.50612
$915$		−	5148.84i		−	5.62715i
$916$			101.178i			0.110456i
$917$			912.204i			0.994770i
$918$		−	2818.99i		−	3.07079i
$919$	369.663			0.402245			0.201122	−	0.979566i	$-0.435541\pi$
$919$	369.663			0.402245			0.201122	+	0.979566i	$0.435541\pi$
$920$		−	1053.60i		−	1.14522i
$921$	−55.9418			−0.0607403
$922$	211.185			0.229051
$923$			72.7679i			0.0788385i
$924$	1650.12			1.78584
$925$			1455.13i			1.57312i
$926$	1857.94			2.00642
$927$		−	600.573i		−	0.647868i
$928$		−	940.728i		−	1.01372i
$929$			1102.69i			1.18696i	0.804847	+	0.593482i	$0.202247\pi$
$929$			1102.69i			1.18696i	−0.804847	+	0.593482i	$0.797753\pi$
$930$	810.929			0.871967
$931$	−5.19365			−0.00557857
$932$	−62.3772			−0.0669283
$933$		−	78.7491i		−	0.0844041i
$934$			1512.27i			1.61913i
$935$	−3356.44			−3.58978
$936$			487.879i			0.521239i
$937$		−	1546.21i		−	1.65017i	−0.565008	−	0.825086i	$-0.691127\pi$
$937$		−	1546.21i		−	1.65017i	0.565008	−	0.825086i	$-0.308873\pi$
$938$	993.028			1.05866
$939$			1639.85i			1.74637i
$940$		−	2250.88i		−	2.39456i
$941$	548.734			0.583140			0.291570	−	0.956550i	$-0.405822\pi$
$941$	548.734			0.583140			0.291570	+	0.956550i	$0.405822\pi$
$942$	3390.54			3.59930
$943$	1915.32			2.03109
$944$			1003.46i			1.06298i
$945$			2837.90i			3.00307i
$946$	−608.760			−0.643510
$947$	−678.039			−0.715987			−0.357993	−	0.933724i	$-0.616539\pi$
$947$	−678.039			−0.715987			−0.357993	+	0.933724i	$0.616539\pi$
$948$	470.143			0.495932
$949$	719.646			0.758321
$950$		−	88.1664i		−	0.0928067i
$951$			943.413i			0.992022i
$952$		−	359.451i		−	0.377575i
$953$	−1759.84			−1.84663			−0.923314	−	0.384046i	$-0.874531\pi$
$953$	−1759.84			−1.84663			−0.923314	+	0.384046i	$0.874531\pi$
$954$			2147.53i			2.25108i
$955$		−	931.701i		−	0.975603i
$956$	−1023.95			−1.07108
$957$			2060.52i			2.15310i
$958$		−	176.238i		−	0.183964i
$959$			1514.63i			1.57938i
$960$	1419.62			1.47877
$961$	920.795			0.958164
$962$	−648.396			−0.674009
$963$			2723.27i			2.82791i
$964$		−	267.581i		−	0.277574i
$965$		−	2543.27i		−	2.63551i
$966$	3791.94			3.92540
$967$		−	348.172i		−	0.360053i	−0.983662	−	0.180027i	$-0.942382\pi$
$967$		−	348.172i		−	0.360053i	0.983662	−	0.180027i	$-0.0576184\pi$
$968$		−	410.247i		−	0.423809i
$969$	62.9165			0.0649293
$970$	−740.025			−0.762912
$971$			446.211i			0.459537i	0.973245	+	0.229769i	$0.0737969\pi$
$971$			446.211i			0.459537i	−0.973245	+	0.229769i	$0.926203\pi$
$972$		−	97.5237i		−	0.100333i
$973$			542.904i			0.557969i
$974$	−1360.94			−1.39727
$975$		−	3204.12i		−	3.28628i
$976$			2023.06i			2.07281i
$977$		−	1118.61i		−	1.14494i	−0.819924	−	0.572472i	$-0.805985\pi$
$977$		−	1118.61i		−	1.14494i	0.819924	−	0.572472i	$-0.194015\pi$
$978$		−	393.282i		−	0.402129i
$979$			1343.94i			1.37277i
$980$			261.764i			0.267106i
$981$			246.097i			0.250863i
$982$	1967.74			2.00380
$983$			812.065i			0.826109i	0.910706	+	0.413055i	$0.135538\pi$
$983$			812.065i			0.826109i	−0.910706	+	0.413055i	$0.864462\pi$
$984$		−	602.661i		−	0.612460i
$985$	823.646			0.836189
$986$	−1365.69			−1.38509
$987$	−2662.47			−2.69754
$988$	16.8708			0.0170757
$989$	−600.738			−0.607420
$990$	−7473.36			−7.54885
$991$	669.633			0.675714			0.337857	−	0.941197i	$-0.390298\pi$
$991$	669.633			0.675714			0.337857	+	0.941197i	$0.390298\pi$
$992$	−252.182			−0.254215
$993$	2515.97			2.53370
$994$		−	119.372i		−	0.120093i
$995$		−	230.569i		−	0.231728i
$996$	888.762			0.892332
$997$		−	272.643i		−	0.273463i	−0.990608	−	0.136732i	$-0.956340\pi$
$997$		−	272.643i		−	0.273463i	0.990608	−	0.136732i	$-0.0436598\pi$
$998$		−	1548.36i		−	1.55146i
$999$	−1177.33			−1.17851

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	547.3.b.b.546.20	✓	88
547.546	odd	2	inner	547.3.b.b.546.69	yes	88

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.3.b.b.546.20	✓	88	1.1	even	1	trivial
547.3.b.b.546.69	yes	88	547.546	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 547 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	547.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.64774i q^{2} +5.22856i q^{3} -3.01055 q^{4} -9.23817i q^{5} +13.8439 q^{6} -6.29191i q^{7} -2.61981i q^{8} -18.3379 q^{9} +O(q^{10})\)	\(q-2.64774i q^{2} +5.22856i q^{3} -3.01055 q^{4} -9.23817i q^{5} +13.8439 q^{6} -6.29191i q^{7} -2.61981i q^{8} -18.3379 q^{9} -24.4603 q^{10} -16.6612 q^{11} -15.7408i q^{12} +10.1553 q^{13} -16.6594 q^{14} +48.3023 q^{15} -18.9788 q^{16} +21.8066i q^{17} +48.5540i q^{18} -0.551816 q^{19} +27.8120i q^{20} +32.8976 q^{21} +44.1145i q^{22} +43.5332i q^{23} +13.6978 q^{24} -60.3438 q^{25} -26.8887i q^{26} -48.8236i q^{27} +18.9421i q^{28} -23.6532 q^{29} -127.892i q^{30} +6.34072i q^{31} +39.7717i q^{32} -87.1139i q^{33} +57.7383 q^{34} -58.1257 q^{35} +55.2070 q^{36} -24.1141i q^{37} +1.46107i q^{38} +53.0978i q^{39} -24.2022 q^{40} -43.9968i q^{41} -87.1045i q^{42} +13.7995i q^{43} +50.1593 q^{44} +169.408i q^{45} +115.265 q^{46} -80.9321 q^{47} -99.2318i q^{48} +9.41193 q^{49} +159.775i q^{50} -114.017 q^{51} -30.5731 q^{52} +44.2297 q^{53} -129.272 q^{54} +153.919i q^{55} -16.4836 q^{56} -2.88520i q^{57} +62.6276i q^{58} -52.8725i q^{59} -145.417 q^{60} -106.596i q^{61} +16.7886 q^{62} +115.380i q^{63} +29.3903 q^{64} -93.8167i q^{65} -230.655 q^{66} -59.6078 q^{67} -65.6499i q^{68} -227.616 q^{69} +153.902i q^{70} +7.16549i q^{71} +48.0417i q^{72} +70.8639 q^{73} -63.8479 q^{74} -315.511i q^{75} +1.66127 q^{76} +104.830i q^{77} +140.589 q^{78} +29.8677i q^{79} +175.329i q^{80} +90.2363 q^{81} -116.492 q^{82} +56.4622i q^{83} -99.0399 q^{84} +201.453 q^{85} +36.5377 q^{86} -123.672i q^{87} +43.6491i q^{88} -80.6631i q^{89} +448.550 q^{90} -63.8964i q^{91} -131.059i q^{92} -33.1529 q^{93} +214.288i q^{94} +5.09777i q^{95} -207.949 q^{96} +30.2541 q^{97} -24.9204i q^{98} +305.530 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 88 q - 192 q^{4} - 306 q^{9}+O(q^{10}) \) 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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