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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [75,8,Mod(2,75)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("75.2"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(20)) chi = DirichletCharacter(H, H._module([10, 1])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.l (of order \(20\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(23.4288769113\)
Analytic rank: \(0\)
Dimension: \(544\)
Relative dimension: \(68\) over \(\Q(\zeta_{20})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

Embedding invariants

Embedding label 2.12
Character \(\chi\) \(=\) 75.2
Dual form 75.8.l.a.38.12

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.55129 + 16.1082i) q^{2} +(-12.9716 + 44.9304i) q^{3} +(-131.229 - 42.6390i) q^{4} +(79.1005 - 268.082i) q^{5} +(-690.652 - 323.579i) q^{6} +(-1066.81 - 1066.81i) q^{7} +(73.9138 - 145.064i) q^{8} +(-1850.47 - 1165.64i) q^{9} +(4116.51 + 1958.12i) q^{10} +(2636.15 + 3628.35i) q^{11} +(3618.04 - 5343.09i) q^{12} +(4265.46 - 675.582i) q^{13} +(19906.1 - 14462.7i) q^{14} +(11019.0 + 7031.47i) q^{15} +(-12140.6 - 8820.63i) q^{16} +(8857.50 + 4513.12i) q^{17} +(23497.4 - 26833.9i) q^{18} +(-9145.22 + 2971.46i) q^{19} +(-21811.1 + 31807.5i) q^{20} +(61770.5 - 34094.0i) q^{21} +(-65171.8 + 33206.7i) q^{22} +(111020. + 17583.8i) q^{23} +(5559.00 + 5202.69i) q^{24} +(-65611.2 - 42410.9i) q^{25} +70432.4i q^{26} +(76376.2 - 68022.3i) q^{27} +(94509.3 + 185485. i) q^{28} +(-15784.6 + 48579.9i) q^{29} +(-141377. + 159556. i) q^{30} +(15436.6 + 47508.9i) q^{31} +(187794. - 187794. i) q^{32} +(-197218. + 71377.7i) q^{33} +(-95296.2 + 131164. i) q^{34} +(-370379. + 201608. i) q^{35} +(193135. + 231868. i) q^{36} +(-42293.7 - 267032. i) q^{37} +(-24532.8 - 154894. i) q^{38} +(-24975.7 + 200412. i) q^{39} +(-33042.5 - 31289.6i) q^{40} +(284118. - 391055. i) q^{41} +(391598. + 1.08199e6i) q^{42} +(443359. - 443359. i) q^{43} +(-191231. - 588550. i) q^{44} +(-458860. + 403877. i) q^{45} +(-566487. + 1.74347e6i) q^{46} +(-156616. - 307377. i) q^{47} +(553796. - 431062. i) q^{48} +1.45263e6i q^{49} +(850556. - 948675. i) q^{50} +(-317672. + 339428. i) q^{51} +(-588559. - 93218.7i) q^{52} +(-478814. + 243968. i) q^{53} +(900859. + 1.40383e6i) q^{54} +(1.18122e6 - 419701. i) q^{55} +(-233608. + 75903.9i) q^{56} +(-14880.6 - 449442. i) q^{57} +(-742263. - 378202. i) q^{58} +(944255. + 686042. i) q^{59} +(-1.14620e6 - 1.39257e6i) q^{60} +(1.23873e6 - 899991. i) q^{61} +(-804666. + 127446. i) q^{62} +(730592. + 3.21762e6i) q^{63} +(1.41687e6 + 1.95015e6i) q^{64} +(156288. - 1.19693e6i) q^{65} +(-646606. - 3.35894e6i) q^{66} +(1.61856e6 - 3.17661e6i) q^{67} +(-969929. - 969929. i) q^{68} +(-2.23016e6 + 4.76008e6i) q^{69} +(-2.30260e6 - 6.48049e6i) q^{70} +(-4.88387e6 - 1.58687e6i) q^{71} +(-305868. + 182281. i) q^{72} +(659422. - 4.16342e6i) q^{73} +4.40931e6 q^{74} +(2.75662e6 - 2.39780e6i) q^{75} +1.32682e6 q^{76} +(1.05849e6 - 6.68305e6i) q^{77} +(-3.16455e6 - 913621. i) q^{78} +(5.29363e6 + 1.72000e6i) q^{79} +(-3.32498e6 + 2.55695e6i) q^{80} +(2.06554e6 + 4.31397e6i) q^{81} +(5.57432e6 + 5.57432e6i) q^{82} +(-1.20224e6 + 2.35953e6i) q^{83} +(-9.55984e6 + 1.84030e6i) q^{84} +(1.91052e6 - 2.01755e6i) q^{85} +(6.01058e6 + 8.27285e6i) q^{86} +(-1.97796e6 - 1.33937e6i) q^{87} +(721192. - 114226. i) q^{88} +(8.27465e6 - 6.01189e6i) q^{89} +(-5.33504e6 - 8.42182e6i) q^{90} +(-5.27115e6 - 3.82972e6i) q^{91} +(-1.38193e7 - 7.04130e6i) q^{92} +(-2.33483e6 + 77304.2i) q^{93} +(5.35085e6 - 1.73860e6i) q^{94} +(73204.4 + 2.68671e6i) q^{95} +(6.00167e6 + 1.08737e7i) q^{96} +(-3.14338e6 + 1.60163e6i) q^{97} +(-2.33992e7 - 3.70608e6i) q^{98} +(-648788. - 9.78698e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 544 q - 34 q^{3} - 20 q^{4} - 6 q^{6} - 1364 q^{7} - 10 q^{9} + 3160 q^{10} + 2018 q^{12} - 16868 q^{13} - 37570 q^{15} + 507892 q^{16} + 106550 q^{18} + 70800 q^{19} - 6 q^{21} - 195160 q^{22} - 444200 q^{25}+ \cdots - 22089384 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{20}\right)\)

Coefficient data

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



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.55129 + 16.1082i −0.225504 + 1.42378i 0.571897 + 0.820326i \(0.306208\pi\)
−0.797401 + 0.603450i \(0.793792\pi\)
\(3\) −12.9716 + 44.9304i −0.277376 + 0.960761i
\(4\) −131.229 − 42.6390i −1.02523 − 0.333117i
\(5\) 79.1005 − 268.082i 0.282999 − 0.959120i
\(6\) −690.652 − 323.579i −1.30536 − 0.611577i
\(7\) −1066.81 − 1066.81i −1.17556 − 1.17556i −0.980864 − 0.194696i \(-0.937628\pi\)
−0.194696 − 0.980864i \(-0.562372\pi\)
\(8\) 73.9138 − 145.064i 0.0510400 − 0.100172i
\(9\) −1850.47 − 1165.64i −0.846125 − 0.532985i
\(10\) 4116.51 + 1958.12i 1.30176 + 0.619212i
\(11\) 2636.15 + 3628.35i 0.597168 + 0.821931i 0.995445 − 0.0953341i \(-0.0303919\pi\)
−0.398278 + 0.917265i \(0.630392\pi\)
\(12\) 3618.04 − 5343.09i 0.604421 − 0.892602i
\(13\) 4265.46 − 675.582i 0.538473 − 0.0852857i 0.118726 − 0.992927i \(-0.462119\pi\)
0.419747 + 0.907641i \(0.362119\pi\)
\(14\) 19906.1 − 14462.7i 1.93883 − 1.40864i
\(15\) 11019.0 + 7031.47i 0.842989 + 0.537931i
\(16\) −12140.6 − 8820.63i −0.741000 − 0.538368i
\(17\) 8857.50 + 4513.12i 0.437260 + 0.222795i 0.658741 − 0.752369i \(-0.271089\pi\)
−0.221481 + 0.975165i \(0.571089\pi\)
\(18\) 23497.4 − 26833.9i 0.949656 − 1.08450i
\(19\) −9145.22 + 2971.46i −0.305884 + 0.0993876i −0.457937 − 0.888985i \(-0.651411\pi\)
0.152053 + 0.988372i \(0.451411\pi\)
\(20\) −21811.1 + 31807.5i −0.609638 + 0.889047i
\(21\) 61770.5 − 34094.0i 1.45550 − 0.803360i
\(22\) −65171.8 + 33206.7i −1.30491 + 0.664884i
\(23\) 111020. + 17583.8i 1.90263 + 0.301346i 0.993377 − 0.114904i \(-0.0366561\pi\)
0.909250 + 0.416251i \(0.136656\pi\)
\(24\) 5559.00 + 5202.69i 0.0820838 + 0.0768225i
\(25\) −65611.2 − 42410.9i −0.839823 − 0.542860i
\(26\) 70432.4i 0.785897i
\(27\) 76376.2 − 68022.3i 0.746766 − 0.665087i
\(28\) 94509.3 + 185485.i 0.813619 + 1.59682i
\(29\) −15784.6 + 48579.9i −0.120182 + 0.369882i −0.992993 − 0.118177i \(-0.962295\pi\)
0.872811 + 0.488059i \(0.162295\pi\)
\(30\) −141377. + 159556.i −0.955991 + 1.07892i
\(31\) 15436.6 + 47508.9i 0.0930647 + 0.286424i 0.986744 − 0.162282i \(-0.0518853\pi\)
−0.893680 + 0.448705i \(0.851885\pi\)
\(32\) 187794. − 187794.i 1.01311 − 1.01311i
\(33\) −197218. + 71377.7i −0.955320 + 0.345751i
\(34\) −95296.2 + 131164.i −0.415814 + 0.572319i
\(35\) −370379. + 201608.i −1.46018 + 0.794821i
\(36\) 193135. + 231868.i 0.689926 + 0.828291i
\(37\) −42293.7 − 267032.i −0.137268 − 0.866678i −0.956185 − 0.292765i \(-0.905425\pi\)
0.818916 − 0.573913i \(-0.194575\pi\)
\(38\) −24532.8 − 154894.i −0.0725278 − 0.457922i
\(39\) −24975.7 + 200412.i −0.0674204 + 0.541000i
\(40\) −33042.5 − 31289.6i −0.0816324 − 0.0773020i
\(41\) 284118. − 391055.i 0.643806 − 0.886123i −0.355005 − 0.934864i \(-0.615521\pi\)
0.998811 + 0.0487410i \(0.0155209\pi\)
\(42\) 391598. + 1.08199e6i 0.815582 + 2.25347i
\(43\) 443359. − 443359.i 0.850386 − 0.850386i −0.139794 − 0.990181i \(-0.544644\pi\)
0.990181 + 0.139794i \(0.0446441\pi\)
\(44\) −191231. − 588550.i −0.338435 − 1.04160i
\(45\) −458860. + 403877.i −0.750649 + 0.660701i
\(46\) −566487. + 1.74347e6i −0.858100 + 2.64096i
\(47\) −156616. − 307377.i −0.220036 − 0.431845i 0.754430 − 0.656381i \(-0.227914\pi\)
−0.974466 + 0.224536i \(0.927914\pi\)
\(48\) 553796. − 431062.i 0.722779 − 0.562594i
\(49\) 1.45263e6i 1.76388i
\(50\) 850556. − 948675.i 0.962294 − 1.07330i
\(51\) −317672. + 339428.i −0.335339 + 0.358304i
\(52\) −588559. − 93218.7i −0.580468 − 0.0919372i
\(53\) −478814. + 243968.i −0.441776 + 0.225096i −0.660706 − 0.750645i \(-0.729743\pi\)
0.218931 + 0.975740i \(0.429743\pi\)
\(54\) 900859. + 1.40383e6i 0.778536 + 1.21321i
\(55\) 1.18122e6 − 419701.i 0.957328 − 0.340150i
\(56\) −233608. + 75903.9i −0.177758 + 0.0577572i
\(57\) −14880.6 − 449442.i −0.0106429 − 0.321449i
\(58\) −742263. − 378202.i −0.499528 − 0.254522i
\(59\) 944255. + 686042.i 0.598560 + 0.434879i 0.845367 − 0.534185i \(-0.179382\pi\)
−0.246808 + 0.969064i \(0.579382\pi\)
\(60\) −1.14620e6 − 1.39257e6i −0.685063 − 0.832318i
\(61\) 1.23873e6 − 899991.i 0.698752 − 0.507673i −0.180774 − 0.983525i \(-0.557860\pi\)
0.879525 + 0.475852i \(0.157860\pi\)
\(62\) −804666. + 127446.i −0.428790 + 0.0679136i
\(63\) 730592. + 3.21762e6i 0.368114 + 1.62123i
\(64\) 1.41687e6 + 1.95015e6i 0.675615 + 0.929904i
\(65\) 156288. − 1.19693e6i 0.0705878 − 0.540596i
\(66\) −646606. − 3.35894e6i −0.276844 − 1.43813i
\(67\) 1.61856e6 − 3.17661e6i 0.657457 − 1.29033i −0.285806 − 0.958287i \(-0.592261\pi\)
0.943264 − 0.332045i \(-0.107739\pi\)
\(68\) −969929. − 969929.i −0.374075 − 0.374075i
\(69\) −2.23016e6 + 4.76008e6i −0.817266 + 1.74438i
\(70\) −2.30260e6 − 6.48049e6i −0.802370 − 2.25821i
\(71\) −4.88387e6 − 1.58687e6i −1.61942 − 0.526182i −0.647616 − 0.761967i \(-0.724234\pi\)
−0.971805 + 0.235785i \(0.924234\pi\)
\(72\) −305868. + 182281.i −0.0965762 + 0.0575542i
\(73\) 659422. − 4.16342e6i 0.198396 − 1.25262i −0.664518 − 0.747273i \(-0.731363\pi\)
0.862914 − 0.505351i \(-0.168637\pi\)
\(74\) 4.40931e6 1.26491
\(75\) 2.75662e6 − 2.39780e6i 0.754506 − 0.656294i
\(76\) 1.32682e6 0.346709
\(77\) 1.05849e6 − 6.68305e6i 0.264222 − 1.66823i
\(78\) −3.16455e6 − 913621.i −0.755059 − 0.217989i
\(79\) 5.29363e6 + 1.72000e6i 1.20798 + 0.392495i 0.842690 − 0.538400i \(-0.180971\pi\)
0.365287 + 0.930895i \(0.380971\pi\)
\(80\) −3.32498e6 + 2.55695e6i −0.726062 + 0.558351i
\(81\) 2.06554e6 + 4.31397e6i 0.431854 + 0.901943i
\(82\) 5.57432e6 + 5.57432e6i 1.11646 + 1.11646i
\(83\) −1.20224e6 + 2.35953e6i −0.230790 + 0.452951i −0.977138 − 0.212605i \(-0.931805\pi\)
0.746348 + 0.665556i \(0.231805\pi\)
\(84\) −9.55984e6 + 1.84030e6i −1.75984 + 0.338775i
\(85\) 1.91052e6 − 2.01755e6i 0.337431 − 0.356334i
\(86\) 6.01058e6 + 8.27285e6i 1.01899 + 1.40253i
\(87\) −1.97796e6 − 1.33937e6i −0.322033 − 0.218063i
\(88\) 721192. − 114226.i 0.112814 − 0.0178679i
\(89\) 8.27465e6 − 6.01189e6i 1.24418 − 0.903953i 0.246315 − 0.969190i \(-0.420780\pi\)
0.997870 + 0.0652371i \(0.0207804\pi\)
\(90\) −5.33504e6 − 8.42182e6i −0.771417 − 1.21775i
\(91\) −5.27115e6 − 3.82972e6i −0.733265 − 0.532748i
\(92\) −1.38193e7 − 7.04130e6i −1.85025 − 0.942747i
\(93\) −2.33483e6 + 77304.2i −0.300999 + 0.00996581i
\(94\) 5.35085e6 − 1.73860e6i 0.664470 − 0.215899i
\(95\) 73204.4 + 2.68671e6i 0.00876001 + 0.321506i
\(96\) 6.00167e6 + 1.08737e7i 0.692345 + 1.25437i
\(97\) −3.14338e6 + 1.60163e6i −0.349701 + 0.178181i −0.620013 − 0.784591i \(-0.712873\pi\)
0.270313 + 0.962773i \(0.412873\pi\)
\(98\) −2.33992e7 − 3.70608e6i −2.51137 − 0.397762i
\(99\) −648788. − 9.78698e6i −0.0672016 − 1.01374i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 75.8.l.a.2.12 ✓ 544
3.2 odd 2 inner 75.8.l.a.2.57 yes 544
25.13 odd 20 inner 75.8.l.a.38.57 yes 544
75.38 even 20 inner 75.8.l.a.38.12 yes 544
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.8.l.a.2.12 ✓ 544 1.1 even 1 trivial
75.8.l.a.2.57 yes 544 3.2 odd 2 inner
75.8.l.a.38.12 yes 544 75.38 even 20 inner
75.8.l.a.38.57 yes 544 25.13 odd 20 inner