LMFDB - Embedded newform 6042.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6042,2,Mod(1,6042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("6042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6042 = 2 \cdot 3 \cdot 19 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6042.a (trivial)

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:	yes
Analytic conductor:	$48.2456129013$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	6042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.44504 q^{5} +1.00000 q^{6} -3.80194 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.44504 q^{5} +1.00000 q^{6} -3.80194 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.44504 q^{10} +0.198062 q^{11} +1.00000 q^{12} +3.15883 q^{13} -3.80194 q^{14} -2.44504 q^{15} +1.00000 q^{16} -2.91185 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.44504 q^{20} -3.80194 q^{21} +0.198062 q^{22} +8.23490 q^{23} +1.00000 q^{24} +0.978230 q^{25} +3.15883 q^{26} +1.00000 q^{27} -3.80194 q^{28} -5.93900 q^{29} -2.44504 q^{30} +6.59179 q^{31} +1.00000 q^{32} +0.198062 q^{33} -2.91185 q^{34} +9.29590 q^{35} +1.00000 q^{36} -11.8998 q^{37} +1.00000 q^{38} +3.15883 q^{39} -2.44504 q^{40} +10.8388 q^{41} -3.80194 q^{42} +1.64310 q^{43} +0.198062 q^{44} -2.44504 q^{45} +8.23490 q^{46} -10.8780 q^{47} +1.00000 q^{48} +7.45473 q^{49} +0.978230 q^{50} -2.91185 q^{51} +3.15883 q^{52} +1.00000 q^{53} +1.00000 q^{54} -0.484271 q^{55} -3.80194 q^{56} +1.00000 q^{57} -5.93900 q^{58} -4.33513 q^{59} -2.44504 q^{60} -9.04892 q^{61} +6.59179 q^{62} -3.80194 q^{63} +1.00000 q^{64} -7.72348 q^{65} +0.198062 q^{66} -2.88471 q^{67} -2.91185 q^{68} +8.23490 q^{69} +9.29590 q^{70} -15.7017 q^{71} +1.00000 q^{72} -2.96615 q^{73} -11.8998 q^{74} +0.978230 q^{75} +1.00000 q^{76} -0.753020 q^{77} +3.15883 q^{78} -11.2228 q^{79} -2.44504 q^{80} +1.00000 q^{81} +10.8388 q^{82} -10.9487 q^{83} -3.80194 q^{84} +7.11960 q^{85} +1.64310 q^{86} -5.93900 q^{87} +0.198062 q^{88} -2.62565 q^{89} -2.44504 q^{90} -12.0097 q^{91} +8.23490 q^{92} +6.59179 q^{93} -10.8780 q^{94} -2.44504 q^{95} +1.00000 q^{96} -7.86294 q^{97} +7.45473 q^{98} +0.198062 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 7 q^{5} + 3 q^{6} - 7 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 7 * q^5 + 3 * q^6 - 7 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 7 q^{5} + 3 q^{6} - 7 q^{7} + 3 q^{8} + 3 q^{9} - 7 q^{10} + 5 q^{11} + 3 q^{12} + q^{13} - 7 q^{14} - 7 q^{15} + 3 q^{16} - 5 q^{17} + 3 q^{18} + 3 q^{19} - 7 q^{20} - 7 q^{21} + 5 q^{22} + q^{23} + 3 q^{24} + 6 q^{25} + q^{26} + 3 q^{27} - 7 q^{28} - 8 q^{29} - 7 q^{30} - 8 q^{31} + 3 q^{32} + 5 q^{33} - 5 q^{34} + 14 q^{35} + 3 q^{36} - 13 q^{37} + 3 q^{38} + q^{39} - 7 q^{40} - 7 q^{42} + 9 q^{43} + 5 q^{44} - 7 q^{45} + q^{46} - 13 q^{47} + 3 q^{48} + 6 q^{50} - 5 q^{51} + q^{52} + 3 q^{53} + 3 q^{54} - 14 q^{55} - 7 q^{56} + 3 q^{57} - 8 q^{58} - 12 q^{59} - 7 q^{60} - 18 q^{61} - 8 q^{62} - 7 q^{63} + 3 q^{64} + 7 q^{65} + 5 q^{66} - 11 q^{67} - 5 q^{68} + q^{69} + 14 q^{70} - 20 q^{71} + 3 q^{72} + 7 q^{73} - 13 q^{74} + 6 q^{75} + 3 q^{76} - 7 q^{77} + q^{78} + 9 q^{79} - 7 q^{80} + 3 q^{81} - q^{83} - 7 q^{84} + 9 q^{86} - 8 q^{87} + 5 q^{88} + 4 q^{89} - 7 q^{90} - 14 q^{91} + q^{92} - 8 q^{93} - 13 q^{94} - 7 q^{95} + 3 q^{96} - 29 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 7 * q^5 + 3 * q^6 - 7 * q^7 + 3 * q^8 + 3 * q^9 - 7 * q^10 + 5 * q^11 + 3 * q^12 + q^13 - 7 * q^14 - 7 * q^15 + 3 * q^16 - 5 * q^17 + 3 * q^18 + 3 * q^19 - 7 * q^20 - 7 * q^21 + 5 * q^22 + q^23 + 3 * q^24 + 6 * q^25 + q^26 + 3 * q^27 - 7 * q^28 - 8 * q^29 - 7 * q^30 - 8 * q^31 + 3 * q^32 + 5 * q^33 - 5 * q^34 + 14 * q^35 + 3 * q^36 - 13 * q^37 + 3 * q^38 + q^39 - 7 * q^40 - 7 * q^42 + 9 * q^43 + 5 * q^44 - 7 * q^45 + q^46 - 13 * q^47 + 3 * q^48 + 6 * q^50 - 5 * q^51 + q^52 + 3 * q^53 + 3 * q^54 - 14 * q^55 - 7 * q^56 + 3 * q^57 - 8 * q^58 - 12 * q^59 - 7 * q^60 - 18 * q^61 - 8 * q^62 - 7 * q^63 + 3 * q^64 + 7 * q^65 + 5 * q^66 - 11 * q^67 - 5 * q^68 + q^69 + 14 * q^70 - 20 * q^71 + 3 * q^72 + 7 * q^73 - 13 * q^74 + 6 * q^75 + 3 * q^76 - 7 * q^77 + q^78 + 9 * q^79 - 7 * q^80 + 3 * q^81 - q^83 - 7 * q^84 + 9 * q^86 - 8 * q^87 + 5 * q^88 + 4 * q^89 - 7 * q^90 - 14 * q^91 + q^92 - 8 * q^93 - 13 * q^94 - 7 * q^95 + 3 * q^96 - 29 * q^97 + 5 * q^99

Display 100 coefficients 10 coefficients

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−2.44504	−1.09346	−0.546728	−	0.837310i	$-0.684127\pi$
$5$	−2.44504	−1.09346	−0.546728	+	0.837310i	$0.684127\pi$
$6$	1.00000	0.408248
$7$	−3.80194	−1.43700	−0.718499	−	0.695528i	$-0.755170\pi$
$7$	−3.80194	−1.43700	−0.718499	+	0.695528i	$0.755170\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.44504	−0.773190
$11$	0.198062	0.0597180	0.0298590	−	0.999554i	$-0.490494\pi$
$11$	0.198062	0.0597180	0.0298590	+	0.999554i	$0.490494\pi$
$12$	1.00000	0.288675
$13$	3.15883	0.876103	0.438051	−	0.898950i	$-0.355669\pi$
$13$	3.15883	0.876103	0.438051	+	0.898950i	$0.355669\pi$
$14$	−3.80194	−1.01611
$15$	−2.44504	−0.631307
$16$	1.00000	0.250000
$17$	−2.91185	−0.706228	−0.353114	−	0.935580i	$-0.614877\pi$
$17$	−2.91185	−0.706228	−0.353114	+	0.935580i	$0.614877\pi$
$18$	1.00000	0.235702
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−2.44504	−0.546728
$21$	−3.80194	−0.829651
$22$	0.198062	0.0422270
$23$	8.23490	1.71709	0.858547	−	0.512734i	$-0.171367\pi$
$23$	8.23490	1.71709	0.858547	+	0.512734i	$0.171367\pi$
$24$	1.00000	0.204124
$25$	0.978230	0.195646
$26$	3.15883	0.619498
$27$	1.00000	0.192450
$28$	−3.80194	−0.718499
$29$	−5.93900	−1.10284	−0.551422	−	0.834226i	$-0.685915\pi$
$29$	−5.93900	−1.10284	−0.551422	+	0.834226i	$0.685915\pi$
$30$	−2.44504	−0.446402
$31$	6.59179	1.18392	0.591961	−	0.805967i	$-0.298354\pi$
$31$	6.59179	1.18392	0.591961	+	0.805967i	$0.298354\pi$
$32$	1.00000	0.176777
$33$	0.198062	0.0344782
$34$	−2.91185	−0.499379
$35$	9.29590	1.57129
$36$	1.00000	0.166667
$37$	−11.8998	−1.95631	−0.978155	−	0.207876i	$-0.933345\pi$
$37$	−11.8998	−1.95631	−0.978155	+	0.207876i	$0.933345\pi$
$38$	1.00000	0.162221
$39$	3.15883	0.505818
$40$	−2.44504	−0.386595
$41$	10.8388	1.69273	0.846366	−	0.532602i	$-0.178786\pi$
$41$	10.8388	1.69273	0.846366	+	0.532602i	$0.178786\pi$
$42$	−3.80194	−0.586652
$43$	1.64310	0.250571	0.125286	−	0.992121i	$-0.460015\pi$
$43$	1.64310	0.250571	0.125286	+	0.992121i	$0.460015\pi$
$44$	0.198062	0.0298590
$45$	−2.44504	−0.364485
$46$	8.23490	1.21417
$47$	−10.8780	−1.58672	−0.793360	−	0.608753i	$-0.791670\pi$
$47$	−10.8780	−1.58672	−0.793360	+	0.608753i	$0.791670\pi$
$48$	1.00000	0.144338
$49$	7.45473	1.06496
$50$	0.978230	0.138343
$51$	−2.91185	−0.407741
$52$	3.15883	0.438051
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	1.00000	0.136083
$55$	−0.484271	−0.0652990
$56$	−3.80194	−0.508055
$57$	1.00000	0.132453
$58$	−5.93900	−0.779829
$59$	−4.33513	−0.564385	−0.282193	−	0.959358i	$-0.591062\pi$
$59$	−4.33513	−0.564385	−0.282193	+	0.959358i	$0.591062\pi$
$60$	−2.44504	−0.315654
$61$	−9.04892	−1.15860	−0.579298	−	0.815116i	$-0.696673\pi$
$61$	−9.04892	−1.15860	−0.579298	+	0.815116i	$0.696673\pi$
$62$	6.59179	0.837159
$63$	−3.80194	−0.478999
$64$	1.00000	0.125000
$65$	−7.72348	−0.957980
$66$	0.198062	0.0243798
$67$	−2.88471	−0.352423	−0.176212	−	0.984352i	$-0.556384\pi$
$67$	−2.88471	−0.352423	−0.176212	+	0.984352i	$0.556384\pi$
$68$	−2.91185	−0.353114
$69$	8.23490	0.991365
$70$	9.29590	1.11107
$71$	−15.7017	−1.86345	−0.931725	−	0.363164i	$-0.881696\pi$
$71$	−15.7017	−1.86345	−0.931725	+	0.363164i	$0.881696\pi$
$72$	1.00000	0.117851
$73$	−2.96615	−0.347161	−0.173581	−	0.984820i	$-0.555534\pi$
$73$	−2.96615	−0.347161	−0.173581	+	0.984820i	$0.555534\pi$
$74$	−11.8998	−1.38332
$75$	0.978230	0.112956
$76$	1.00000	0.114708
$77$	−0.753020	−0.0858146
$78$	3.15883	0.357667
$79$	−11.2228	−1.26267	−0.631333	−	0.775512i	$-0.717492\pi$
$79$	−11.2228	−1.26267	−0.631333	+	0.775512i	$0.717492\pi$
$80$	−2.44504	−0.273364
$81$	1.00000	0.111111
$82$	10.8388	1.19694
$83$	−10.9487	−1.20177	−0.600887	−	0.799334i	$-0.705186\pi$
$83$	−10.9487	−1.20177	−0.600887	+	0.799334i	$0.705186\pi$
$84$	−3.80194	−0.414825
$85$	7.11960	0.772230
$86$	1.64310	0.177180
$87$	−5.93900	−0.636728
$88$	0.198062	0.0211135
$89$	−2.62565	−0.278318	−0.139159	−	0.990270i	$-0.544440\pi$
$89$	−2.62565	−0.278318	−0.139159	+	0.990270i	$0.544440\pi$
$90$	−2.44504	−0.257730
$91$	−12.0097	−1.25896
$92$	8.23490	0.858547
$93$	6.59179	0.683537
$94$	−10.8780	−1.12198
$95$	−2.44504	−0.250856
$96$	1.00000	0.102062
$97$	−7.86294	−0.798360	−0.399180	−	0.916873i	$-0.630705\pi$
$97$	−7.86294	−0.798360	−0.399180	+	0.916873i	$0.630705\pi$
$98$	7.45473	0.753042
$99$	0.198062	0.0199060
$100$	0.978230	0.0978230
$101$	8.20237	0.816167	0.408083	−	0.912945i	$-0.366197\pi$
$101$	8.20237	0.816167	0.408083	+	0.912945i	$0.366197\pi$
$102$	−2.91185	−0.288317
$103$	−13.2959	−1.31008	−0.655042	−	0.755593i	$-0.727349\pi$
$103$	−13.2959	−1.31008	−0.655042	+	0.755593i	$0.727349\pi$
$104$	3.15883	0.309749
$105$	9.29590	0.907187
$106$	1.00000	0.0971286
$107$	−16.5429	−1.59926	−0.799630	−	0.600493i	$-0.794971\pi$
$107$	−16.5429	−1.59926	−0.799630	+	0.600493i	$0.794971\pi$
$108$	1.00000	0.0962250
$109$	−2.81402	−0.269534	−0.134767	−	0.990877i	$-0.543029\pi$
$109$	−2.81402	−0.269534	−0.134767	+	0.990877i	$0.543029\pi$
$110$	−0.484271	−0.0461734
$111$	−11.8998	−1.12948
$112$	−3.80194	−0.359249
$113$	8.03146	0.755536	0.377768	−	0.925900i	$-0.376692\pi$
$113$	8.03146	0.755536	0.377768	+	0.925900i	$0.376692\pi$
$114$	1.00000	0.0936586
$115$	−20.1347	−1.87757
$116$	−5.93900	−0.551422
$117$	3.15883	0.292034
$118$	−4.33513	−0.399081
$119$	11.0707	1.01485
$120$	−2.44504	−0.223201
$121$	−10.9608	−0.996434
$122$	−9.04892	−0.819250
$123$	10.8388	0.977299
$124$	6.59179	0.591961
$125$	9.83340	0.879526
$126$	−3.80194	−0.338704
$127$	2.18598	0.193974	0.0969872	−	0.995286i	$-0.469079\pi$
$127$	2.18598	0.193974	0.0969872	+	0.995286i	$0.469079\pi$
$128$	1.00000	0.0883883
$129$	1.64310	0.144667
$130$	−7.72348	−0.677394
$131$	−5.24698	−0.458431	−0.229215	−	0.973376i	$-0.573616\pi$
$131$	−5.24698	−0.458431	−0.229215	+	0.973376i	$0.573616\pi$
$132$	0.198062	0.0172391
$133$	−3.80194	−0.329670
$134$	−2.88471	−0.249201
$135$	−2.44504	−0.210436
$136$	−2.91185	−0.249689
$137$	5.73556	0.490022	0.245011	−	0.969520i	$-0.421208\pi$
$137$	5.73556	0.490022	0.245011	+	0.969520i	$0.421208\pi$
$138$	8.23490	0.701001
$139$	2.51573	0.213381	0.106691	−	0.994292i	$-0.465975\pi$
$139$	2.51573	0.213381	0.106691	+	0.994292i	$0.465975\pi$
$140$	9.29590	0.785647
$141$	−10.8780	−0.916093
$142$	−15.7017	−1.31766
$143$	0.625646	0.0523191
$144$	1.00000	0.0833333
$145$	14.5211	1.20591
$146$	−2.96615	−0.245480
$147$	7.45473	0.614856
$148$	−11.8998	−0.978155
$149$	20.7560	1.70040	0.850199	−	0.526461i	$-0.176482\pi$
$149$	20.7560	1.70040	0.850199	+	0.526461i	$0.176482\pi$
$150$	0.978230	0.0798721
$151$	−15.3599	−1.24997	−0.624985	−	0.780637i	$-0.714895\pi$
$151$	−15.3599	−1.24997	−0.624985	+	0.780637i	$0.714895\pi$
$152$	1.00000	0.0811107
$153$	−2.91185	−0.235409
$154$	−0.753020	−0.0606801
$155$	−16.1172	−1.29457
$156$	3.15883	0.252909
$157$	10.1347	0.808835	0.404417	−	0.914575i	$-0.367474\pi$
$157$	10.1347	0.808835	0.404417	+	0.914575i	$0.367474\pi$
$158$	−11.2228	−0.892839
$159$	1.00000	0.0793052
$160$	−2.44504	−0.193298
$161$	−31.3086	−2.46746
$162$	1.00000	0.0785674
$163$	13.6474	1.06895	0.534474	−	0.845185i	$-0.320510\pi$
$163$	13.6474	1.06895	0.534474	+	0.845185i	$0.320510\pi$
$164$	10.8388	0.846366
$165$	−0.484271	−0.0377004
$166$	−10.9487	−0.849783
$167$	−4.72587	−0.365699	−0.182850	−	0.983141i	$-0.558532\pi$
$167$	−4.72587	−0.365699	−0.182850	+	0.983141i	$0.558532\pi$
$168$	−3.80194	−0.293326
$169$	−3.02177	−0.232444
$170$	7.11960	0.546049
$171$	1.00000	0.0764719
$172$	1.64310	0.125286
$173$	−25.2446	−1.91931	−0.959655	−	0.281179i	$-0.909274\pi$
$173$	−25.2446	−1.91931	−0.959655	+	0.281179i	$0.909274\pi$
$174$	−5.93900	−0.450235
$175$	−3.71917	−0.281143
$176$	0.198062	0.0149295
$177$	−4.33513	−0.325848
$178$	−2.62565	−0.196800
$179$	25.7754	1.92654	0.963271	−	0.268530i	$-0.0865379\pi$
$179$	25.7754	1.92654	0.963271	+	0.268530i	$0.0865379\pi$
$180$	−2.44504	−0.182243
$181$	−6.10752	−0.453969	−0.226984	−	0.973898i	$-0.572887\pi$
$181$	−6.10752	−0.453969	−0.226984	+	0.973898i	$0.572887\pi$
$182$	−12.0097	−0.890217
$183$	−9.04892	−0.668915
$184$	8.23490	0.607085
$185$	29.0954	2.13914
$186$	6.59179	0.483334
$187$	−0.576728	−0.0421746
$188$	−10.8780	−0.793360
$189$	−3.80194	−0.276550
$190$	−2.44504	−0.177382
$191$	−11.0435	−0.799083	−0.399541	−	0.916715i	$-0.630831\pi$
$191$	−11.0435	−0.799083	−0.399541	+	0.916715i	$0.630831\pi$
$192$	1.00000	0.0721688
$193$	5.87800	0.423108	0.211554	−	0.977366i	$-0.432148\pi$
$193$	5.87800	0.423108	0.211554	+	0.977366i	$0.432148\pi$
$194$	−7.86294	−0.564526
$195$	−7.72348	−0.553090
$196$	7.45473	0.532481
$197$	4.36227	0.310799	0.155400	−	0.987852i	$-0.450334\pi$
$197$	4.36227	0.310799	0.155400	+	0.987852i	$0.450334\pi$
$198$	0.198062	0.0140757
$199$	−2.48427	−0.176105	−0.0880526	−	0.996116i	$-0.528064\pi$
$199$	−2.48427	−0.176105	−0.0880526	+	0.996116i	$0.528064\pi$
$200$	0.978230	0.0691713
$201$	−2.88471	−0.203472
$202$	8.20237	0.577117
$203$	22.5797	1.58479
$204$	−2.91185	−0.203871
$205$	−26.5013	−1.85093
$206$	−13.2959	−0.926369
$207$	8.23490	0.572365
$208$	3.15883	0.219026
$209$	0.198062	0.0137003
$210$	9.29590	0.641478
$211$	−0.660563	−0.0454750	−0.0227375	−	0.999741i	$-0.507238\pi$
$211$	−0.660563	−0.0454750	−0.0227375	+	0.999741i	$0.507238\pi$
$212$	1.00000	0.0686803
$213$	−15.7017	−1.07586
$214$	−16.5429	−1.13085
$215$	−4.01746	−0.273988
$216$	1.00000	0.0680414
$217$	−25.0616	−1.70129
$218$	−2.81402	−0.190589
$219$	−2.96615	−0.200434
$220$	−0.484271	−0.0326495
$221$	−9.19806	−0.618729
$222$	−11.8998	−0.798660
$223$	12.1685	0.814865	0.407432	−	0.913235i	$-0.366424\pi$
$223$	12.1685	0.814865	0.407432	+	0.913235i	$0.366424\pi$
$224$	−3.80194	−0.254028
$225$	0.978230	0.0652153
$226$	8.03146	0.534245
$227$	6.39373	0.424367	0.212183	−	0.977230i	$-0.431943\pi$
$227$	6.39373	0.424367	0.212183	+	0.977230i	$0.431943\pi$
$228$	1.00000	0.0662266
$229$	8.52542	0.563375	0.281688	−	0.959506i	$-0.409106\pi$
$229$	8.52542	0.563375	0.281688	+	0.959506i	$0.409106\pi$
$230$	−20.1347	−1.32764
$231$	−0.753020	−0.0495451
$232$	−5.93900	−0.389915
$233$	−23.1879	−1.51909	−0.759545	−	0.650455i	$-0.774578\pi$
$233$	−23.1879	−1.51909	−0.759545	+	0.650455i	$0.774578\pi$
$234$	3.15883	0.206499
$235$	26.5972	1.73501
$236$	−4.33513	−0.282193
$237$	−11.2228	−0.729000
$238$	11.0707	0.717606
$239$	16.9584	1.09695	0.548473	−	0.836168i	$-0.315209\pi$
$239$	16.9584	1.09695	0.548473	+	0.836168i	$0.315209\pi$
$240$	−2.44504	−0.157827
$241$	1.04892	0.0675667	0.0337834	−	0.999429i	$-0.489244\pi$
$241$	1.04892	0.0675667	0.0337834	+	0.999429i	$0.489244\pi$
$242$	−10.9608	−0.704585
$243$	1.00000	0.0641500
$244$	−9.04892	−0.579298
$245$	−18.2271	−1.16449
$246$	10.8388	0.691055
$247$	3.15883	0.200992
$248$	6.59179	0.418579
$249$	−10.9487	−0.693845
$250$	9.83340	0.621919
$251$	−0.351519	−0.0221877	−0.0110938	−	0.999938i	$-0.503531\pi$
$251$	−0.351519	−0.0221877	−0.0110938	+	0.999938i	$0.503531\pi$
$252$	−3.80194	−0.239500
$253$	1.63102	0.102542
$254$	2.18598	0.137161
$255$	7.11960	0.445847
$256$	1.00000	0.0625000
$257$	−6.08038	−0.379283	−0.189642	−	0.981853i	$-0.560733\pi$
$257$	−6.08038	−0.379283	−0.189642	+	0.981853i	$0.560733\pi$
$258$	1.64310	0.102295
$259$	45.2422	2.81121
$260$	−7.72348	−0.478990
$261$	−5.93900	−0.367615
$262$	−5.24698	−0.324159
$263$	−17.2989	−1.06669	−0.533347	−	0.845896i	$-0.679066\pi$
$263$	−17.2989	−1.06669	−0.533347	+	0.845896i	$0.679066\pi$
$264$	0.198062	0.0121899
$265$	−2.44504	−0.150198
$266$	−3.80194	−0.233112
$267$	−2.62565	−0.160687
$268$	−2.88471	−0.176212
$269$	9.82132	0.598816	0.299408	−	0.954125i	$-0.403211\pi$
$269$	9.82132	0.598816	0.299408	+	0.954125i	$0.403211\pi$
$270$	−2.44504	−0.148801
$271$	18.0344	1.09551	0.547757	−	0.836637i	$-0.315482\pi$
$271$	18.0344	1.09551	0.547757	+	0.836637i	$0.315482\pi$
$272$	−2.91185	−0.176557
$273$	−12.0097	−0.726859
$274$	5.73556	0.346498
$275$	0.193750	0.0116836
$276$	8.23490	0.495683
$277$	24.1987	1.45396	0.726978	−	0.686661i	$-0.240924\pi$
$277$	24.1987	1.45396	0.726978	+	0.686661i	$0.240924\pi$
$278$	2.51573	0.150883
$279$	6.59179	0.394640
$280$	9.29590	0.555536
$281$	23.3884	1.39523	0.697616	−	0.716472i	$-0.254244\pi$
$281$	23.3884	1.39523	0.697616	+	0.716472i	$0.254244\pi$
$282$	−10.8780	−0.647776
$283$	−14.4993	−0.861896	−0.430948	−	0.902377i	$-0.641821\pi$
$283$	−14.4993	−0.861896	−0.430948	+	0.902377i	$0.641821\pi$
$284$	−15.7017	−0.931725
$285$	−2.44504	−0.144832
$286$	0.625646	0.0369952
$287$	−41.2083	−2.43245
$288$	1.00000	0.0589256
$289$	−8.52111	−0.501242
$290$	14.5211	0.852709
$291$	−7.86294	−0.460934
$292$	−2.96615	−0.173581
$293$	0.653858	0.0381988	0.0190994	−	0.999818i	$-0.493920\pi$
$293$	0.653858	0.0381988	0.0190994	+	0.999818i	$0.493920\pi$
$294$	7.45473	0.434769
$295$	10.5996	0.617130
$296$	−11.8998	−0.691660
$297$	0.198062	0.0114927
$298$	20.7560	1.20236
$299$	26.0127	1.50435
$300$	0.978230	0.0564781
$301$	−6.24698	−0.360070
$302$	−15.3599	−0.883862
$303$	8.20237	0.471214
$304$	1.00000	0.0573539
$305$	22.1250	1.26687
$306$	−2.91185	−0.166460
$307$	2.78687	0.159055	0.0795276	−	0.996833i	$-0.474659\pi$
$307$	2.78687	0.159055	0.0795276	+	0.996833i	$0.474659\pi$
$308$	−0.753020	−0.0429073
$309$	−13.2959	−0.756377
$310$	−16.1172	−0.915396
$311$	−9.03684	−0.512432	−0.256216	−	0.966620i	$-0.582476\pi$
$311$	−9.03684	−0.512432	−0.256216	+	0.966620i	$0.582476\pi$
$312$	3.15883	0.178834
$313$	−4.84548	−0.273883	−0.136941	−	0.990579i	$-0.543727\pi$
$313$	−4.84548	−0.273883	−0.136941	+	0.990579i	$0.543727\pi$
$314$	10.1347	0.571933
$315$	9.29590	0.523764
$316$	−11.2228	−0.631333
$317$	−20.0441	−1.12579	−0.562895	−	0.826528i	$-0.690313\pi$
$317$	−20.0441	−1.12579	−0.562895	+	0.826528i	$0.690313\pi$
$318$	1.00000	0.0560772
$319$	−1.17629	−0.0658597
$320$	−2.44504	−0.136682
$321$	−16.5429	−0.923333
$322$	−31.3086	−1.74476
$323$	−2.91185	−0.162020
$324$	1.00000	0.0555556
$325$	3.09006	0.171406
$326$	13.6474	0.755860
$327$	−2.81402	−0.155616
$328$	10.8388	0.598471
$329$	41.3575	2.28011
$330$	−0.484271	−0.0266582
$331$	−28.3327	−1.55731	−0.778654	−	0.627454i	$-0.784097\pi$
$331$	−28.3327	−1.55731	−0.778654	+	0.627454i	$0.784097\pi$
$332$	−10.9487	−0.600887
$333$	−11.8998	−0.652104
$334$	−4.72587	−0.258588
$335$	7.05323	0.385359
$336$	−3.80194	−0.207413
$337$	−25.8213	−1.40658	−0.703288	−	0.710905i	$-0.748286\pi$
$337$	−25.8213	−1.40658	−0.703288	+	0.710905i	$0.748286\pi$
$338$	−3.02177	−0.164363
$339$	8.03146	0.436209
$340$	7.11960	0.386115
$341$	1.30559	0.0707014
$342$	1.00000	0.0540738
$343$	−1.72886	−0.0933495
$344$	1.64310	0.0885902
$345$	−20.1347	−1.08401
$346$	−25.2446	−1.35716
$347$	11.6987	0.628021	0.314010	−	0.949420i	$-0.398327\pi$
$347$	11.6987	0.628021	0.314010	+	0.949420i	$0.398327\pi$
$348$	−5.93900	−0.318364
$349$	−33.6993	−1.80388	−0.901942	−	0.431858i	$-0.857858\pi$
$349$	−33.6993	−1.80388	−0.901942	+	0.431858i	$0.857858\pi$
$350$	−3.71917	−0.198798
$351$	3.15883	0.168606
$352$	0.198062	0.0105568
$353$	19.0610	1.01451	0.507257	−	0.861795i	$-0.330659\pi$
$353$	19.0610	1.01451	0.507257	+	0.861795i	$0.330659\pi$
$354$	−4.33513	−0.230409
$355$	38.3913	2.03760
$356$	−2.62565	−0.139159
$357$	11.0707	0.585923
$358$	25.7754	1.36227
$359$	29.5023	1.55707	0.778536	−	0.627599i	$-0.215962\pi$
$359$	29.5023	1.55707	0.778536	+	0.627599i	$0.215962\pi$
$360$	−2.44504	−0.128865
$361$	1.00000	0.0526316
$362$	−6.10752	−0.321004
$363$	−10.9608	−0.575291
$364$	−12.0097	−0.629479
$365$	7.25236	0.379606
$366$	−9.04892	−0.472994
$367$	−34.4198	−1.79670	−0.898350	−	0.439281i	$-0.855233\pi$
$367$	−34.4198	−1.79670	−0.898350	+	0.439281i	$0.855233\pi$
$368$	8.23490	0.429274
$369$	10.8388	0.564244
$370$	29.0954	1.51260
$371$	−3.80194	−0.197387
$372$	6.59179	0.341769
$373$	−15.6635	−0.811028	−0.405514	−	0.914089i	$-0.632907\pi$
$373$	−15.6635	−0.811028	−0.405514	+	0.914089i	$0.632907\pi$
$374$	−0.576728	−0.0298219
$375$	9.83340	0.507794
$376$	−10.8780	−0.560990
$377$	−18.7603	−0.966205
$378$	−3.80194	−0.195551
$379$	13.5308	0.695030	0.347515	−	0.937674i	$-0.387026\pi$
$379$	13.5308	0.695030	0.347515	+	0.937674i	$0.387026\pi$
$380$	−2.44504	−0.125428
$381$	2.18598	0.111991
$382$	−11.0435	−0.565037
$383$	−3.87369	−0.197936	−0.0989682	−	0.995091i	$-0.531554\pi$
$383$	−3.87369	−0.197936	−0.0989682	+	0.995091i	$0.531554\pi$
$384$	1.00000	0.0510310
$385$	1.84117	0.0938345
$386$	5.87800	0.299182
$387$	1.64310	0.0835237
$388$	−7.86294	−0.399180
$389$	−1.22414	−0.0620666	−0.0310333	−	0.999518i	$-0.509880\pi$
$389$	−1.22414	−0.0620666	−0.0310333	+	0.999518i	$0.509880\pi$
$390$	−7.72348	−0.391094
$391$	−23.9788	−1.21266
$392$	7.45473	0.376521
$393$	−5.24698	−0.264675
$394$	4.36227	0.219768
$395$	27.4403	1.38067
$396$	0.198062	0.00995300
$397$	20.1420	1.01090	0.505448	−	0.862857i	$-0.331327\pi$
$397$	20.1420	1.01090	0.505448	+	0.862857i	$0.331327\pi$
$398$	−2.48427	−0.124525
$399$	−3.80194	−0.190335
$400$	0.978230	0.0489115
$401$	19.9379	0.995653	0.497827	−	0.867277i	$-0.334132\pi$
$401$	19.9379	0.995653	0.497827	+	0.867277i	$0.334132\pi$
$402$	−2.88471	−0.143876
$403$	20.8224	1.03724
$404$	8.20237	0.408083
$405$	−2.44504	−0.121495
$406$	22.5797	1.12061
$407$	−2.35690	−0.116827
$408$	−2.91185	−0.144158
$409$	−6.51871	−0.322330	−0.161165	−	0.986928i	$-0.551525\pi$
$409$	−6.51871	−0.322330	−0.161165	+	0.986928i	$0.551525\pi$
$410$	−26.5013	−1.30880
$411$	5.73556	0.282914
$412$	−13.2959	−0.655042
$413$	16.4819	0.811020
$414$	8.23490	0.404723
$415$	26.7700	1.31409
$416$	3.15883	0.154875
$417$	2.51573	0.123196
$418$	0.198062	0.00968754
$419$	−10.1105	−0.493931	−0.246965	−	0.969024i	$-0.579433\pi$
$419$	−10.1105	−0.493931	−0.246965	+	0.969024i	$0.579433\pi$
$420$	9.29590	0.453593
$421$	−36.1051	−1.75966	−0.879828	−	0.475292i	$-0.842342\pi$
$421$	−36.1051	−1.75966	−0.879828	+	0.475292i	$0.842342\pi$
$422$	−0.660563	−0.0321557
$423$	−10.8780	−0.528907
$424$	1.00000	0.0485643
$425$	−2.84846	−0.138171
$426$	−15.7017	−0.760750
$427$	34.4034	1.66490
$428$	−16.5429	−0.799630
$429$	0.625646	0.0302065
$430$	−4.01746	−0.193739
$431$	4.43535	0.213643	0.106822	−	0.994278i	$-0.465933\pi$
$431$	4.43535	0.213643	0.106822	+	0.994278i	$0.465933\pi$
$432$	1.00000	0.0481125
$433$	9.67217	0.464815	0.232407	−	0.972619i	$-0.425340\pi$
$433$	9.67217	0.464815	0.232407	+	0.972619i	$0.425340\pi$
$434$	−25.0616	−1.20299
$435$	14.5211	0.696234
$436$	−2.81402	−0.134767
$437$	8.23490	0.393929
$438$	−2.96615	−0.141728
$439$	−28.7885	−1.37400	−0.687001	−	0.726657i	$-0.741073\pi$
$439$	−28.7885	−1.37400	−0.687001	+	0.726657i	$0.741073\pi$
$440$	−0.484271	−0.0230867
$441$	7.45473	0.354987
$442$	−9.19806	−0.437507
$443$	−33.4969	−1.59149	−0.795744	−	0.605633i	$-0.792920\pi$
$443$	−33.4969	−1.59149	−0.795744	+	0.605633i	$0.792920\pi$
$444$	−11.8998	−0.564738
$445$	6.41981	0.304328
$446$	12.1685	0.576196
$447$	20.7560	0.981725
$448$	−3.80194	−0.179625
$449$	−5.88769	−0.277857	−0.138929	−	0.990302i	$-0.544366\pi$
$449$	−5.88769	−0.277857	−0.138929	+	0.990302i	$0.544366\pi$
$450$	0.978230	0.0461142
$451$	2.14675	0.101087
$452$	8.03146	0.377768
$453$	−15.3599	−0.721670
$454$	6.39373	0.300073
$455$	29.3642	1.37661
$456$	1.00000	0.0468293
$457$	−5.10859	−0.238970	−0.119485	−	0.992836i	$-0.538124\pi$
$457$	−5.10859	−0.238970	−0.119485	+	0.992836i	$0.538124\pi$
$458$	8.52542	0.398367
$459$	−2.91185	−0.135914
$460$	−20.1347	−0.938784
$461$	35.4873	1.65281	0.826403	−	0.563079i	$-0.190383\pi$
$461$	35.4873	1.65281	0.826403	+	0.563079i	$0.190383\pi$
$462$	−0.753020	−0.0350337
$463$	30.0030	1.39436	0.697178	−	0.716898i	$-0.254439\pi$
$463$	30.0030	1.39436	0.697178	+	0.716898i	$0.254439\pi$
$464$	−5.93900	−0.275711
$465$	−16.1172	−0.747418
$466$	−23.1879	−1.07416
$467$	35.2010	1.62891	0.814455	−	0.580227i	$-0.197036\pi$
$467$	35.2010	1.62891	0.814455	+	0.580227i	$0.197036\pi$
$468$	3.15883	0.146017
$469$	10.9675	0.506431
$470$	26.5972	1.22684
$471$	10.1347	0.466981
$472$	−4.33513	−0.199540
$473$	0.325437	0.0149636
$474$	−11.2228	−0.515481
$475$	0.978230	0.0448843
$476$	11.0707	0.507424
$477$	1.00000	0.0457869
$478$	16.9584	0.775658
$479$	−38.6088	−1.76408	−0.882040	−	0.471174i	$-0.843830\pi$
$479$	−38.6088	−1.76408	−0.882040	+	0.471174i	$0.843830\pi$
$480$	−2.44504	−0.111600
$481$	−37.5894	−1.71393
$482$	1.04892	0.0477769
$483$	−31.3086	−1.42459
$484$	−10.9608	−0.498217
$485$	19.2252	0.872972
$486$	1.00000	0.0453609
$487$	13.9511	0.632184	0.316092	−	0.948729i	$-0.397629\pi$
$487$	13.9511	0.632184	0.316092	+	0.948729i	$0.397629\pi$
$488$	−9.04892	−0.409625
$489$	13.6474	0.617157
$490$	−18.2271	−0.823418
$491$	41.8364	1.88805	0.944025	−	0.329875i	$-0.107007\pi$
$491$	41.8364	1.88805	0.944025	+	0.329875i	$0.107007\pi$
$492$	10.8388	0.488650
$493$	17.2935	0.778860
$494$	3.15883	0.142123
$495$	−0.484271	−0.0217663
$496$	6.59179	0.295980
$497$	59.6969	2.67777
$498$	−10.9487	−0.490623
$499$	38.9366	1.74304	0.871521	−	0.490358i	$-0.163134\pi$
$499$	38.9366	1.74304	0.871521	+	0.490358i	$0.163134\pi$
$500$	9.83340	0.439763
$501$	−4.72587	−0.211136
$502$	−0.351519	−0.0156891
$503$	−8.08815	−0.360633	−0.180316	−	0.983609i	$-0.557712\pi$
$503$	−8.08815	−0.360633	−0.180316	+	0.983609i	$0.557712\pi$
$504$	−3.80194	−0.169352
$505$	−20.0551	−0.892442
$506$	1.63102	0.0725078
$507$	−3.02177	−0.134202
$508$	2.18598	0.0969872
$509$	−23.6504	−1.04829	−0.524143	−	0.851630i	$-0.675614\pi$
$509$	−23.6504	−1.04829	−0.524143	+	0.851630i	$0.675614\pi$
$510$	7.11960	0.315261
$511$	11.2771	0.498870
$512$	1.00000	0.0441942
$513$	1.00000	0.0441511
$514$	−6.08038	−0.268194
$515$	32.5090	1.43252
$516$	1.64310	0.0723336
$517$	−2.15452	−0.0947558
$518$	45.2422	1.98783
$519$	−25.2446	−1.10811
$520$	−7.72348	−0.338697
$521$	−7.89200	−0.345755	−0.172877	−	0.984943i	$-0.555306\pi$
$521$	−7.89200	−0.345755	−0.172877	+	0.984943i	$0.555306\pi$
$522$	−5.93900	−0.259943
$523$	25.3394	1.10802	0.554008	−	0.832511i	$-0.313098\pi$
$523$	25.3394	1.10802	0.554008	+	0.832511i	$0.313098\pi$
$524$	−5.24698	−0.229215
$525$	−3.71917	−0.162318
$526$	−17.2989	−0.754267
$527$	−19.1943	−0.836119
$528$	0.198062	0.00861955
$529$	44.8135	1.94842
$530$	−2.44504	−0.106206
$531$	−4.33513	−0.188128
$532$	−3.80194	−0.164835
$533$	34.2379	1.48301
$534$	−2.62565	−0.113623
$535$	40.4480	1.74872
$536$	−2.88471	−0.124600
$537$	25.7754	1.11229
$538$	9.82132	0.423427
$539$	1.47650	0.0635974
$540$	−2.44504	−0.105218
$541$	−25.5297	−1.09761	−0.548804	−	0.835951i	$-0.684917\pi$
$541$	−25.5297	−1.09761	−0.548804	+	0.835951i	$0.684917\pi$
$542$	18.0344	0.774646
$543$	−6.10752	−0.262099
$544$	−2.91185	−0.124845
$545$	6.88040	0.294724
$546$	−12.0097	−0.513967
$547$	−37.5918	−1.60731	−0.803655	−	0.595096i	$-0.797114\pi$
$547$	−37.5918	−1.60731	−0.803655	+	0.595096i	$0.797114\pi$
$548$	5.73556	0.245011
$549$	−9.04892	−0.386198
$550$	0.193750	0.00826154
$551$	−5.93900	−0.253010
$552$	8.23490	0.350501
$553$	42.6684	1.81445
$554$	24.1987	1.02810
$555$	29.0954	1.23503
$556$	2.51573	0.106691
$557$	−35.3250	−1.49677	−0.748383	−	0.663266i	$-0.769170\pi$
$557$	−35.3250	−1.49677	−0.748383	+	0.663266i	$0.769170\pi$
$558$	6.59179	0.279053
$559$	5.19029	0.219526
$560$	9.29590	0.392823
$561$	−0.576728	−0.0243495
$562$	23.3884	0.986578
$563$	−35.1105	−1.47973	−0.739866	−	0.672755i	$-0.765111\pi$
$563$	−35.1105	−1.47973	−0.739866	+	0.672755i	$0.765111\pi$
$564$	−10.8780	−0.458047
$565$	−19.6373	−0.826145
$566$	−14.4993	−0.609453
$567$	−3.80194	−0.159666
$568$	−15.7017	−0.658829
$569$	1.93064	0.0809367	0.0404683	−	0.999181i	$-0.487115\pi$
$569$	1.93064	0.0809367	0.0404683	+	0.999181i	$0.487115\pi$
$570$	−2.44504	−0.102412
$571$	33.7657	1.41305	0.706525	−	0.707688i	$-0.250262\pi$
$571$	33.7657	1.41305	0.706525	+	0.707688i	$0.250262\pi$
$572$	0.625646	0.0261596
$573$	−11.0435	−0.461351
$574$	−41.2083	−1.72000
$575$	8.05562	0.335943
$576$	1.00000	0.0416667
$577$	−39.5532	−1.64662	−0.823310	−	0.567592i	$-0.807875\pi$
$577$	−39.5532	−1.64662	−0.823310	+	0.567592i	$0.807875\pi$
$578$	−8.52111	−0.354431
$579$	5.87800	0.244281
$580$	14.5211	0.602956
$581$	41.6262	1.72695
$582$	−7.86294	−0.325929
$583$	0.198062	0.00820290
$584$	−2.96615	−0.122740
$585$	−7.72348	−0.319327
$586$	0.653858	0.0270106
$587$	25.9571	1.07136	0.535681	−	0.844420i	$-0.320055\pi$
$587$	25.9571	1.07136	0.535681	+	0.844420i	$0.320055\pi$
$588$	7.45473	0.307428
$589$	6.59179	0.271610
$590$	10.5996	0.436377
$591$	4.36227	0.179440
$592$	−11.8998	−0.489078
$593$	36.0049	1.47854	0.739272	−	0.673407i	$-0.235170\pi$
$593$	36.0049	1.47854	0.739272	+	0.673407i	$0.235170\pi$
$594$	0.198062	0.00812659
$595$	−27.0683	−1.10969
$596$	20.7560	0.850199
$597$	−2.48427	−0.101674
$598$	26.0127	1.06374
$599$	8.08947	0.330527	0.165263	−	0.986249i	$-0.447153\pi$
$599$	8.08947	0.330527	0.165263	+	0.986249i	$0.447153\pi$
$600$	0.978230	0.0399361
$601$	−0.554958	−0.0226372	−0.0113186	−	0.999936i	$-0.503603\pi$
$601$	−0.554958	−0.0226372	−0.0113186	+	0.999936i	$0.503603\pi$
$602$	−6.24698	−0.254608
$603$	−2.88471	−0.117474
$604$	−15.3599	−0.624985
$605$	26.7995	1.08956
$606$	8.20237	0.333199
$607$	−8.00862	−0.325060	−0.162530	−	0.986704i	$-0.551965\pi$
$607$	−8.00862	−0.325060	−0.162530	+	0.986704i	$0.551965\pi$
$608$	1.00000	0.0405554
$609$	22.5797	0.914976
$610$	22.1250	0.895814
$611$	−34.3618	−1.39013
$612$	−2.91185	−0.117705
$613$	−12.5459	−0.506723	−0.253361	−	0.967372i	$-0.581536\pi$
$613$	−12.5459	−0.506723	−0.253361	+	0.967372i	$0.581536\pi$
$614$	2.78687	0.112469
$615$	−26.5013	−1.06863
$616$	−0.753020	−0.0303401
$617$	23.9487	0.964138	0.482069	−	0.876133i	$-0.339885\pi$
$617$	23.9487	0.964138	0.482069	+	0.876133i	$0.339885\pi$
$618$	−13.2959	−0.534839
$619$	14.3840	0.578143	0.289072	−	0.957307i	$-0.406653\pi$
$619$	14.3840	0.578143	0.289072	+	0.957307i	$0.406653\pi$
$620$	−16.1172	−0.647283
$621$	8.23490	0.330455
$622$	−9.03684	−0.362344
$623$	9.98254	0.399942
$624$	3.15883	0.126455
$625$	−28.9342	−1.15737
$626$	−4.84548	−0.193664
$627$	0.198062	0.00790984
$628$	10.1347	0.404417
$629$	34.6504	1.38160
$630$	9.29590	0.370357
$631$	−8.25129	−0.328479	−0.164239	−	0.986421i	$-0.552517\pi$
$631$	−8.25129	−0.328479	−0.164239	+	0.986421i	$0.552517\pi$
$632$	−11.2228	−0.446420
$633$	−0.660563	−0.0262550
$634$	−20.0441	−0.796054
$635$	−5.34481	−0.212102
$636$	1.00000	0.0396526
$637$	23.5483	0.933016
$638$	−1.17629	−0.0465698
$639$	−15.7017	−0.621150
$640$	−2.44504	−0.0966488
$641$	20.9554	0.827688	0.413844	−	0.910348i	$-0.364186\pi$
$641$	20.9554	0.827688	0.413844	+	0.910348i	$0.364186\pi$
$642$	−16.5429	−0.652895
$643$	24.8364	0.979451	0.489726	−	0.871877i	$-0.337097\pi$
$643$	24.8364	0.979451	0.489726	+	0.871877i	$0.337097\pi$
$644$	−31.3086	−1.23373
$645$	−4.01746	−0.158187
$646$	−2.91185	−0.114565
$647$	33.1215	1.30214	0.651071	−	0.759017i	$-0.274320\pi$
$647$	33.1215	1.30214	0.651071	+	0.759017i	$0.274320\pi$
$648$	1.00000	0.0392837
$649$	−0.858625	−0.0337040
$650$	3.09006	0.121202
$651$	−25.0616	−0.982241
$652$	13.6474	0.534474
$653$	−13.9970	−0.547746	−0.273873	−	0.961766i	$-0.588305\pi$
$653$	−13.9970	−0.547746	−0.273873	+	0.961766i	$0.588305\pi$
$654$	−2.81402	−0.110037
$655$	12.8291	0.501274
$656$	10.8388	0.423183
$657$	−2.96615	−0.115720
$658$	41.3575	1.61228
$659$	14.3907	0.560584	0.280292	−	0.959915i	$-0.409569\pi$
$659$	14.3907	0.560584	0.280292	+	0.959915i	$0.409569\pi$
$660$	−0.484271	−0.0188502
$661$	5.00836	0.194803	0.0974013	−	0.995245i	$-0.468947\pi$
$661$	5.00836	0.194803	0.0974013	+	0.995245i	$0.468947\pi$
$662$	−28.3327	−1.10118
$663$	−9.19806	−0.357223
$664$	−10.9487	−0.424892
$665$	9.29590	0.360479
$666$	−11.8998	−0.461107
$667$	−48.9071	−1.89369
$668$	−4.72587	−0.182850
$669$	12.1685	0.470462
$670$	7.05323	0.272490
$671$	−1.79225	−0.0691890
$672$	−3.80194	−0.146663
$673$	21.7536	0.838540	0.419270	−	0.907862i	$-0.362286\pi$
$673$	21.7536	0.838540	0.419270	+	0.907862i	$0.362286\pi$
$674$	−25.8213	−0.994600
$675$	0.978230	0.0376521
$676$	−3.02177	−0.116222
$677$	−9.18119	−0.352862	−0.176431	−	0.984313i	$-0.556455\pi$
$677$	−9.18119	−0.352862	−0.176431	+	0.984313i	$0.556455\pi$
$678$	8.03146	0.308446
$679$	29.8944	1.14724
$680$	7.11960	0.273024
$681$	6.39373	0.245008
$682$	1.30559	0.0499935
$683$	−0.477566	−0.0182735	−0.00913677	−	0.999958i	$-0.502908\pi$
$683$	−0.477566	−0.0182735	−0.00913677	+	0.999958i	$0.502908\pi$
$684$	1.00000	0.0382360
$685$	−14.0237	−0.535818
$686$	−1.72886	−0.0660081
$687$	8.52542	0.325265
$688$	1.64310	0.0626428
$689$	3.15883	0.120342
$690$	−20.1347	−0.766514
$691$	−5.90515	−0.224642	−0.112321	−	0.993672i	$-0.535829\pi$
$691$	−5.90515	−0.224642	−0.112321	+	0.993672i	$0.535829\pi$
$692$	−25.2446	−0.959655
$693$	−0.753020	−0.0286049
$694$	11.6987	0.444078
$695$	−6.15106	−0.233323
$696$	−5.93900	−0.225117
$697$	−31.5609	−1.19546
$698$	−33.6993	−1.27554
$699$	−23.1879	−0.877047
$700$	−3.71917	−0.140571
$701$	−13.0441	−0.492670	−0.246335	−	0.969185i	$-0.579226\pi$
$701$	−13.0441	−0.492670	−0.246335	+	0.969185i	$0.579226\pi$
$702$	3.15883	0.119222
$703$	−11.8998	−0.448808
$704$	0.198062	0.00746475
$705$	26.5972	1.00171
$706$	19.0610	0.717370
$707$	−31.1849	−1.17283
$708$	−4.33513	−0.162924
$709$	−37.5351	−1.40966	−0.704830	−	0.709376i	$-0.748977\pi$
$709$	−37.5351	−1.40966	−0.704830	+	0.709376i	$0.748977\pi$
$710$	38.3913	1.44080
$711$	−11.2228	−0.420888
$712$	−2.62565	−0.0984002
$713$	54.2828	2.03290
$714$	11.0707	0.414310
$715$	−1.52973	−0.0572087
$716$	25.7754	0.963271
$717$	16.9584	0.633322
$718$	29.5023	1.10102
$719$	31.4392	1.17248	0.586242	−	0.810136i	$-0.300607\pi$
$719$	31.4392	1.17248	0.586242	+	0.810136i	$0.300607\pi$
$720$	−2.44504	−0.0911213
$721$	50.5502	1.88259
$722$	1.00000	0.0372161
$723$	1.04892	0.0390097
$724$	−6.10752	−0.226984
$725$	−5.80971	−0.215767
$726$	−10.9608	−0.406792
$727$	−19.7399	−0.732111	−0.366056	−	0.930593i	$-0.619292\pi$
$727$	−19.7399	−0.732111	−0.366056	+	0.930593i	$0.619292\pi$
$728$	−12.0097	−0.445109
$729$	1.00000	0.0370370
$730$	7.25236	0.268422
$731$	−4.78448	−0.176960
$732$	−9.04892	−0.334458
$733$	14.6093	0.539605	0.269803	−	0.962916i	$-0.413042\pi$
$733$	14.6093	0.539605	0.269803	+	0.962916i	$0.413042\pi$
$734$	−34.4198	−1.27046
$735$	−18.2271	−0.672318
$736$	8.23490	0.303542
$737$	−0.571352	−0.0210460
$738$	10.8388	0.398981
$739$	−8.60148	−0.316411	−0.158205	−	0.987406i	$-0.550571\pi$
$739$	−8.60148	−0.316411	−0.158205	+	0.987406i	$0.550571\pi$
$740$	29.0954	1.06957
$741$	3.15883	0.116043
$742$	−3.80194	−0.139574
$743$	33.9536	1.24564	0.622818	−	0.782367i	$-0.285988\pi$
$743$	33.9536	1.24564	0.622818	+	0.782367i	$0.285988\pi$
$744$	6.59179	0.241667
$745$	−50.7493	−1.85931
$746$	−15.6635	−0.573483
$747$	−10.9487	−0.400592
$748$	−0.576728	−0.0210873
$749$	62.8950	2.29813
$750$	9.83340	0.359065
$751$	−0.348272	−0.0127086	−0.00635432	−	0.999980i	$-0.502023\pi$
$751$	−0.348272	−0.0127086	−0.00635432	+	0.999980i	$0.502023\pi$
$752$	−10.8780	−0.396680
$753$	−0.351519	−0.0128101
$754$	−18.7603	−0.683210
$755$	37.5555	1.36679
$756$	−3.80194	−0.138275
$757$	−39.9355	−1.45148	−0.725741	−	0.687968i	$-0.758503\pi$
$757$	−39.9355	−1.45148	−0.725741	+	0.687968i	$0.758503\pi$
$758$	13.5308	0.491461
$759$	1.63102	0.0592024
$760$	−2.44504	−0.0886910
$761$	−48.5260	−1.75907	−0.879533	−	0.475837i	$-0.842145\pi$
$761$	−48.5260	−1.75907	−0.879533	+	0.475837i	$0.842145\pi$
$762$	2.18598	0.0791897
$763$	10.6987	0.387320
$764$	−11.0435	−0.399541
$765$	7.11960	0.257410
$766$	−3.87369	−0.139962
$767$	−13.6939	−0.494460
$768$	1.00000	0.0360844
$769$	−37.0780	−1.33707	−0.668533	−	0.743682i	$-0.733078\pi$
$769$	−37.0780	−1.33707	−0.668533	+	0.743682i	$0.733078\pi$
$770$	1.84117	0.0663510
$771$	−6.08038	−0.218979
$772$	5.87800	0.211554
$773$	−10.4638	−0.376358	−0.188179	−	0.982135i	$-0.560258\pi$
$773$	−10.4638	−0.376358	−0.188179	+	0.982135i	$0.560258\pi$
$774$	1.64310	0.0590602
$775$	6.44829	0.231629
$776$	−7.86294	−0.282263
$777$	45.2422	1.62305
$778$	−1.22414	−0.0438877
$779$	10.8388	0.388339
$780$	−7.72348	−0.276545
$781$	−3.10992	−0.111282
$782$	−23.9788	−0.857481
$783$	−5.93900	−0.212243
$784$	7.45473	0.266240
$785$	−24.7797	−0.884425
$786$	−5.24698	−0.187154
$787$	−9.44935	−0.336833	−0.168417	−	0.985716i	$-0.553865\pi$
$787$	−9.44935	−0.336833	−0.168417	+	0.985716i	$0.553865\pi$
$788$	4.36227	0.155400
$789$	−17.2989	−0.615856
$790$	27.4403	0.976280
$791$	−30.5351	−1.08570
$792$	0.198062	0.00703784
$793$	−28.5840	−1.01505
$794$	20.1420	0.714812
$795$	−2.44504	−0.0867167
$796$	−2.48427	−0.0880526
$797$	−34.5593	−1.22415	−0.612076	−	0.790799i	$-0.709665\pi$
$797$	−34.5593	−1.22415	−0.612076	+	0.790799i	$0.709665\pi$
$798$	−3.80194	−0.134587
$799$	31.6752	1.12059
$800$	0.978230	0.0345856
$801$	−2.62565	−0.0927726
$802$	19.9379	0.704033
$803$	−0.587482	−0.0207318
$804$	−2.88471	−0.101736
$805$	76.5508	2.69806
$806$	20.8224	0.733437
$807$	9.82132	0.345727
$808$	8.20237	0.288559
$809$	−43.4016	−1.52592	−0.762960	−	0.646446i	$-0.776255\pi$
$809$	−43.4016	−1.52592	−0.762960	+	0.646446i	$0.776255\pi$
$810$	−2.44504	−0.0859100
$811$	−11.7453	−0.412432	−0.206216	−	0.978507i	$-0.566115\pi$
$811$	−11.7453	−0.412432	−0.206216	+	0.978507i	$0.566115\pi$
$812$	22.5797	0.792393
$813$	18.0344	0.632496
$814$	−2.35690	−0.0826092
$815$	−33.3685	−1.16885
$816$	−2.91185	−0.101935
$817$	1.64310	0.0574849
$818$	−6.51871	−0.227921
$819$	−12.0097	−0.419652
$820$	−26.5013	−0.925464
$821$	38.5230	1.34446	0.672231	−	0.740341i	$-0.265336\pi$
$821$	38.5230	1.34446	0.672231	+	0.740341i	$0.265336\pi$
$822$	5.73556	0.200051
$823$	−35.1836	−1.22642	−0.613211	−	0.789919i	$-0.710123\pi$
$823$	−35.1836	−1.22642	−0.613211	+	0.789919i	$0.710123\pi$
$824$	−13.2959	−0.463185
$825$	0.193750	0.00674552
$826$	16.4819	0.573478
$827$	22.4862	0.781921	0.390961	−	0.920407i	$-0.372143\pi$
$827$	22.4862	0.781921	0.390961	+	0.920407i	$0.372143\pi$
$828$	8.23490	0.286182
$829$	−13.2218	−0.459210	−0.229605	−	0.973284i	$-0.573743\pi$
$829$	−13.2218	−0.459210	−0.229605	+	0.973284i	$0.573743\pi$
$830$	26.7700	0.929200
$831$	24.1987	0.839442
$832$	3.15883	0.109513
$833$	−21.7071	−0.752106
$834$	2.51573	0.0871126
$835$	11.5550	0.399876
$836$	0.198062	0.00685013
$837$	6.59179	0.227846
$838$	−10.1105	−0.349262
$839$	14.1400	0.488169	0.244084	−	0.969754i	$-0.421513\pi$
$839$	14.1400	0.488169	0.244084	+	0.969754i	$0.421513\pi$
$840$	9.29590	0.320739
$841$	6.27173	0.216267
$842$	−36.1051	−1.24426
$843$	23.3884	0.805538
$844$	−0.660563	−0.0227375
$845$	7.38835	0.254167
$846$	−10.8780	−0.373993
$847$	41.6722	1.43187
$848$	1.00000	0.0343401
$849$	−14.4993	−0.497616
$850$	−2.84846	−0.0977014
$851$	−97.9934	−3.35917
$852$	−15.7017	−0.537932
$853$	−42.5435	−1.45666	−0.728330	−	0.685226i	$-0.759703\pi$
$853$	−42.5435	−1.45666	−0.728330	+	0.685226i	$0.759703\pi$
$854$	34.4034	1.17726
$855$	−2.44504	−0.0836187
$856$	−16.5429	−0.565424
$857$	−3.79895	−0.129770	−0.0648849	−	0.997893i	$-0.520668\pi$
$857$	−3.79895	−0.129770	−0.0648849	+	0.997893i	$0.520668\pi$
$858$	0.625646	0.0213592
$859$	−13.2174	−0.450973	−0.225487	−	0.974246i	$-0.572397\pi$
$859$	−13.2174	−0.450973	−0.225487	+	0.974246i	$0.572397\pi$
$860$	−4.01746	−0.136994
$861$	−41.2083	−1.40438
$862$	4.43535	0.151069
$863$	10.2301	0.348237	0.174119	−	0.984725i	$-0.444292\pi$
$863$	10.2301	0.348237	0.174119	+	0.984725i	$0.444292\pi$
$864$	1.00000	0.0340207
$865$	61.7241	2.09868
$866$	9.67217	0.328674
$867$	−8.52111	−0.289392
$868$	−25.0616	−0.850646
$869$	−2.22282	−0.0754039
$870$	14.5211	0.492312
$871$	−9.11231	−0.308759
$872$	−2.81402	−0.0952947
$873$	−7.86294	−0.266120
$874$	8.23490	0.278550
$875$	−37.3860	−1.26388
$876$	−2.96615	−0.100217
$877$	51.1909	1.72859	0.864297	−	0.502982i	$-0.167764\pi$
$877$	51.1909	1.72859	0.864297	+	0.502982i	$0.167764\pi$
$878$	−28.7885	−0.971566
$879$	0.653858	0.0220541
$880$	−0.484271	−0.0163248
$881$	−8.89248	−0.299595	−0.149798	−	0.988717i	$-0.547862\pi$
$881$	−8.89248	−0.299595	−0.149798	+	0.988717i	$0.547862\pi$
$882$	7.45473	0.251014
$883$	10.2024	0.343337	0.171669	−	0.985155i	$-0.445084\pi$
$883$	10.2024	0.343337	0.171669	+	0.985155i	$0.445084\pi$
$884$	−9.19806	−0.309364
$885$	10.5996	0.356300
$886$	−33.4969	−1.12535
$887$	−13.3502	−0.448256	−0.224128	−	0.974560i	$-0.571953\pi$
$887$	−13.3502	−0.448256	−0.224128	+	0.974560i	$0.571953\pi$
$888$	−11.8998	−0.399330
$889$	−8.31096	−0.278741
$890$	6.41981	0.215193
$891$	0.198062	0.00663534
$892$	12.1685	0.407432
$893$	−10.8780	−0.364018
$894$	20.7560	0.694185
$895$	−63.0219	−2.10659
$896$	−3.80194	−0.127014
$897$	26.0127	0.868538
$898$	−5.88769	−0.196475
$899$	−39.1487	−1.30568
$900$	0.978230	0.0326077
$901$	−2.91185	−0.0970079
$902$	2.14675	0.0714790
$903$	−6.24698	−0.207886
$904$	8.03146	0.267122
$905$	14.9332	0.496395
$906$	−15.3599	−0.510298
$907$	−5.96316	−0.198004	−0.0990018	−	0.995087i	$-0.531565\pi$
$907$	−5.96316	−0.198004	−0.0990018	+	0.995087i	$0.531565\pi$
$908$	6.39373	0.212183
$909$	8.20237	0.272056
$910$	29.3642	0.973413
$911$	44.9667	1.48981	0.744907	−	0.667168i	$-0.232494\pi$
$911$	44.9667	1.48981	0.744907	+	0.667168i	$0.232494\pi$
$912$	1.00000	0.0331133
$913$	−2.16852	−0.0717676
$914$	−5.10859	−0.168977
$915$	22.1250	0.731429
$916$	8.52542	0.281688
$917$	19.9487	0.658764
$918$	−2.91185	−0.0961055
$919$	−33.3357	−1.09964	−0.549822	−	0.835282i	$-0.685304\pi$
$919$	−33.3357	−1.09964	−0.549822	+	0.835282i	$0.685304\pi$
$920$	−20.1347	−0.663820
$921$	2.78687	0.0918306
$922$	35.4873	1.16871
$923$	−49.5991	−1.63257
$924$	−0.753020	−0.0247726
$925$	−11.6407	−0.382744
$926$	30.0030	0.985959
$927$	−13.2959	−0.436695
$928$	−5.93900	−0.194957
$929$	31.1667	1.02255	0.511273	−	0.859418i	$-0.329174\pi$
$929$	31.1667	1.02255	0.511273	+	0.859418i	$0.329174\pi$
$930$	−16.1172	−0.528504
$931$	7.45473	0.244319
$932$	−23.1879	−0.759545
$933$	−9.03684	−0.295853
$934$	35.2010	1.15181
$935$	1.41013	0.0461160
$936$	3.15883	0.103250
$937$	11.9571	0.390620	0.195310	−	0.980742i	$-0.437429\pi$
$937$	11.9571	0.390620	0.195310	+	0.980742i	$0.437429\pi$
$938$	10.9675	0.358101
$939$	−4.84548	−0.158126
$940$	26.5972	0.867504
$941$	5.94139	0.193684	0.0968420	−	0.995300i	$-0.469126\pi$
$941$	5.94139	0.193684	0.0968420	+	0.995300i	$0.469126\pi$
$942$	10.1347	0.330205
$943$	89.2562	2.90658
$944$	−4.33513	−0.141096
$945$	9.29590	0.302396
$946$	0.325437	0.0105809
$947$	7.77346	0.252604	0.126302	−	0.991992i	$-0.459689\pi$
$947$	7.77346	0.252604	0.126302	+	0.991992i	$0.459689\pi$
$948$	−11.2228	−0.364500
$949$	−9.36957	−0.304149
$950$	0.978230	0.0317380
$951$	−20.0441	−0.649975
$952$	11.0707	0.358803
$953$	29.1812	0.945272	0.472636	−	0.881258i	$-0.343303\pi$
$953$	29.1812	0.945272	0.472636	+	0.881258i	$0.343303\pi$
$954$	1.00000	0.0323762
$955$	27.0019	0.873762
$956$	16.9584	0.548473
$957$	−1.17629	−0.0380241
$958$	−38.6088	−1.24739
$959$	−21.8062	−0.704161
$960$	−2.44504	−0.0789134
$961$	12.4517	0.401669
$962$	−37.5894	−1.21193
$963$	−16.5429	−0.533087
$964$	1.04892	0.0337834
$965$	−14.3720	−0.462650
$966$	−31.3086	−1.00734
$967$	22.2513	0.715553	0.357777	−	0.933807i	$-0.383535\pi$
$967$	22.2513	0.715553	0.357777	+	0.933807i	$0.383535\pi$
$968$	−10.9608	−0.352293
$969$	−2.91185	−0.0935422
$970$	19.2252	0.617284
$971$	−17.0989	−0.548730	−0.274365	−	0.961626i	$-0.588468\pi$
$971$	−17.0989	−0.548730	−0.274365	+	0.961626i	$0.588468\pi$
$972$	1.00000	0.0320750
$973$	−9.56465	−0.306628
$974$	13.9511	0.447022
$975$	3.09006	0.0989613
$976$	−9.04892	−0.289649
$977$	−6.10992	−0.195474	−0.0977368	−	0.995212i	$-0.531160\pi$
$977$	−6.10992	−0.195474	−0.0977368	+	0.995212i	$0.531160\pi$
$978$	13.6474	0.436396
$979$	−0.520041	−0.0166206
$980$	−18.2271	−0.582244
$981$	−2.81402	−0.0898447
$982$	41.8364	1.33505
$983$	−57.3919	−1.83052	−0.915259	−	0.402866i	$-0.868014\pi$
$983$	−57.3919	−1.83052	−0.915259	+	0.402866i	$0.868014\pi$
$984$	10.8388	0.345527
$985$	−10.6659	−0.339845
$986$	17.2935	0.550737
$987$	41.3575	1.31642
$988$	3.15883	0.100496
$989$	13.5308	0.430254
$990$	−0.484271	−0.0153911
$991$	−2.74632	−0.0872396	−0.0436198	−	0.999048i	$-0.513889\pi$
$991$	−2.74632	−0.0872396	−0.0436198	+	0.999048i	$0.513889\pi$
$992$	6.59179	0.209290
$993$	−28.3327	−0.899112
$994$	59.6969	1.89347
$995$	6.07415	0.192563
$996$	−10.9487	−0.346923
$997$	2.29649	0.0727305	0.0363653	−	0.999339i	$-0.488422\pi$
$997$	2.29649	0.0727305	0.0363653	+	0.999339i	$0.488422\pi$
$998$	38.9366	1.23252
$999$	−11.8998	−0.376492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6042.2.a.s.1.2	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6042.2.a.s.1.2	✓	3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6042.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.44504 q^{5} +1.00000 q^{6} -3.80194 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.44504 q^{5} +1.00000 q^{6} -3.80194 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.44504 q^{10} +0.198062 q^{11} +1.00000 q^{12} +3.15883 q^{13} -3.80194 q^{14} -2.44504 q^{15} +1.00000 q^{16} -2.91185 q^{17} +1.00000 q^{18} +1.00000 q^{19} -2.44504 q^{20} -3.80194 q^{21} +0.198062 q^{22} +8.23490 q^{23} +1.00000 q^{24} +0.978230 q^{25} +3.15883 q^{26} +1.00000 q^{27} -3.80194 q^{28} -5.93900 q^{29} -2.44504 q^{30} +6.59179 q^{31} +1.00000 q^{32} +0.198062 q^{33} -2.91185 q^{34} +9.29590 q^{35} +1.00000 q^{36} -11.8998 q^{37} +1.00000 q^{38} +3.15883 q^{39} -2.44504 q^{40} +10.8388 q^{41} -3.80194 q^{42} +1.64310 q^{43} +0.198062 q^{44} -2.44504 q^{45} +8.23490 q^{46} -10.8780 q^{47} +1.00000 q^{48} +7.45473 q^{49} +0.978230 q^{50} -2.91185 q^{51} +3.15883 q^{52} +1.00000 q^{53} +1.00000 q^{54} -0.484271 q^{55} -3.80194 q^{56} +1.00000 q^{57} -5.93900 q^{58} -4.33513 q^{59} -2.44504 q^{60} -9.04892 q^{61} +6.59179 q^{62} -3.80194 q^{63} +1.00000 q^{64} -7.72348 q^{65} +0.198062 q^{66} -2.88471 q^{67} -2.91185 q^{68} +8.23490 q^{69} +9.29590 q^{70} -15.7017 q^{71} +1.00000 q^{72} -2.96615 q^{73} -11.8998 q^{74} +0.978230 q^{75} +1.00000 q^{76} -0.753020 q^{77} +3.15883 q^{78} -11.2228 q^{79} -2.44504 q^{80} +1.00000 q^{81} +10.8388 q^{82} -10.9487 q^{83} -3.80194 q^{84} +7.11960 q^{85} +1.64310 q^{86} -5.93900 q^{87} +0.198062 q^{88} -2.62565 q^{89} -2.44504 q^{90} -12.0097 q^{91} +8.23490 q^{92} +6.59179 q^{93} -10.8780 q^{94} -2.44504 q^{95} +1.00000 q^{96} -7.86294 q^{97} +7.45473 q^{98} +0.198062 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 7 q^{5} + 3 q^{6} - 7 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 7 * q^5 + 3 * q^6 - 7 * q^7 + 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{2} + 3 q^{3} + 3 q^{4} - 7 q^{5} + 3 q^{6} - 7 q^{7} + 3 q^{8} + 3 q^{9} - 7 q^{10} + 5 q^{11} + 3 q^{12} + q^{13} - 7 q^{14} - 7 q^{15} + 3 q^{16} - 5 q^{17} + 3 q^{18} + 3 q^{19} - 7 q^{20} - 7 q^{21} + 5 q^{22} + q^{23} + 3 q^{24} + 6 q^{25} + q^{26} + 3 q^{27} - 7 q^{28} - 8 q^{29} - 7 q^{30} - 8 q^{31} + 3 q^{32} + 5 q^{33} - 5 q^{34} + 14 q^{35} + 3 q^{36} - 13 q^{37} + 3 q^{38} + q^{39} - 7 q^{40} - 7 q^{42} + 9 q^{43} + 5 q^{44} - 7 q^{45} + q^{46} - 13 q^{47} + 3 q^{48} + 6 q^{50} - 5 q^{51} + q^{52} + 3 q^{53} + 3 q^{54} - 14 q^{55} - 7 q^{56} + 3 q^{57} - 8 q^{58} - 12 q^{59} - 7 q^{60} - 18 q^{61} - 8 q^{62} - 7 q^{63} + 3 q^{64} + 7 q^{65} + 5 q^{66} - 11 q^{67} - 5 q^{68} + q^{69} + 14 q^{70} - 20 q^{71} + 3 q^{72} + 7 q^{73} - 13 q^{74} + 6 q^{75} + 3 q^{76} - 7 q^{77} + q^{78} + 9 q^{79} - 7 q^{80} + 3 q^{81} - q^{83} - 7 q^{84} + 9 q^{86} - 8 q^{87} + 5 q^{88} + 4 q^{89} - 7 q^{90} - 14 q^{91} + q^{92} - 8 q^{93} - 13 q^{94} - 7 q^{95} + 3 q^{96} - 29 q^{97} + 5 q^{99}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 - 7 * q^5 + 3 * q^6 - 7 * q^7 + 3 * q^8 + 3 * q^9 - 7 * q^10 + 5 * q^11 + 3 * q^12 + q^13 - 7 * q^14 - 7 * q^15 + 3 * q^16 - 5 * q^17 + 3 * q^18 + 3 * q^19 - 7 * q^20 - 7 * q^21 + 5 * q^22 + q^23 + 3 * q^24 + 6 * q^25 + q^26 + 3 * q^27 - 7 * q^28 - 8 * q^29 - 7 * q^30 - 8 * q^31 + 3 * q^32 + 5 * q^33 - 5 * q^34 + 14 * q^35 + 3 * q^36 - 13 * q^37 + 3 * q^38 + q^39 - 7 * q^40 - 7 * q^42 + 9 * q^43 + 5 * q^44 - 7 * q^45 + q^46 - 13 * q^47 + 3 * q^48 + 6 * q^50 - 5 * q^51 + q^52 + 3 * q^53 + 3 * q^54 - 14 * q^55 - 7 * q^56 + 3 * q^57 - 8 * q^58 - 12 * q^59 - 7 * q^60 - 18 * q^61 - 8 * q^62 - 7 * q^63 + 3 * q^64 + 7 * q^65 + 5 * q^66 - 11 * q^67 - 5 * q^68 + q^69 + 14 * q^70 - 20 * q^71 + 3 * q^72 + 7 * q^73 - 13 * q^74 + 6 * q^75 + 3 * q^76 - 7 * q^77 + q^78 + 9 * q^79 - 7 * q^80 + 3 * q^81 - q^83 - 7 * q^84 + 9 * q^86 - 8 * q^87 + 5 * q^88 + 4 * q^89 - 7 * q^90 - 14 * q^91 + q^92 - 8 * q^93 - 13 * q^94 - 7 * q^95 + 3 * q^96 - 29 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more