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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,5,Mod(7,90)] 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("90.7"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(90, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [48,48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(9.30329667755\)
Analytic rank: \(0\)
Dimension: \(48\)
Relative dimension: \(12\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 13.7
Character \(\chi\) \(=\) 90.13
Dual form 90.5.k.b.7.7

$q$-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+(2.73205 + 0.732051i) q^{2} +(-1.50442 - 8.87337i) q^{3} +(6.92820 + 4.00000i) q^{4} +(-11.6684 - 22.1099i) q^{5} +(2.38560 - 25.3438i) q^{6} +(-19.9684 - 5.35051i) q^{7} +(16.0000 + 16.0000i) q^{8} +(-76.4734 + 26.6986i) q^{9} +(-15.6931 - 68.9473i) q^{10} +(-27.3658 - 47.3989i) q^{11} +(25.0705 - 67.4942i) q^{12} +(-187.537 + 50.2504i) q^{13} +(-50.6378 - 29.2358i) q^{14} +(-178.635 + 136.801i) q^{15} +(32.0000 + 55.4256i) q^{16} +(212.077 - 212.077i) q^{17} +(-228.474 + 16.9595i) q^{18} -423.135i q^{19} +(7.59846 - 199.856i) q^{20} +(-17.4362 + 185.236i) q^{21} +(-40.0663 - 149.529i) q^{22} +(-228.540 + 61.2370i) q^{23} +(117.903 - 166.045i) q^{24} +(-352.696 + 515.975i) q^{25} -549.147 q^{26} +(351.955 + 638.411i) q^{27} +(-116.943 - 116.943i) q^{28} +(409.130 - 236.211i) q^{29} +(-588.186 + 242.977i) q^{30} +(901.271 - 1561.05i) q^{31} +(46.8513 + 174.851i) q^{32} +(-379.418 + 314.135i) q^{33} +(734.655 - 424.153i) q^{34} +(114.700 + 503.931i) q^{35} +(-636.618 - 120.920i) q^{36} +(1276.93 - 1276.93i) q^{37} +(309.756 - 1156.03i) q^{38} +(728.026 + 1588.49i) q^{39} +(167.064 - 540.453i) q^{40} +(52.3525 - 90.6771i) q^{41} +(-183.239 + 493.311i) q^{42} +(-761.473 + 2841.86i) q^{43} -437.852i q^{44} +(1482.63 + 1379.29i) q^{45} -669.210 q^{46} +(1459.90 + 391.180i) q^{47} +(443.671 - 367.331i) q^{48} +(-1709.22 - 986.818i) q^{49} +(-1341.30 + 1151.48i) q^{50} +(-2200.89 - 1562.78i) q^{51} +(-1500.30 - 402.003i) q^{52} +(1337.06 + 1337.06i) q^{53} +(494.210 + 2001.82i) q^{54} +(-728.670 + 1158.12i) q^{55} +(-233.886 - 405.102i) q^{56} +(-3754.63 + 636.574i) q^{57} +(1290.68 - 345.837i) q^{58} +(5331.68 + 3078.25i) q^{59} +(-1784.82 + 233.243i) q^{60} +(-588.732 - 1019.71i) q^{61} +(3605.08 - 3605.08i) q^{62} +(1669.90 - 123.956i) q^{63} +512.000i q^{64} +(3299.29 + 3560.09i) q^{65} +(-1266.55 + 580.478i) q^{66} +(-439.993 - 1642.08i) q^{67} +(2317.62 - 621.004i) q^{68} +(887.199 + 1935.79i) q^{69} +(-55.5367 + 1460.73i) q^{70} +1429.77 q^{71} +(-1650.75 - 796.397i) q^{72} +(-5715.32 - 5715.32i) q^{73} +(4423.42 - 2553.86i) q^{74} +(5109.04 + 2353.36i) q^{75} +(1692.54 - 2931.57i) q^{76} +(292.842 + 1092.90i) q^{77} +(826.149 + 4872.78i) q^{78} +(1174.15 - 677.895i) q^{79} +(852.066 - 1354.25i) q^{80} +(5135.37 - 4083.47i) q^{81} +(209.410 - 209.410i) q^{82} +(496.330 - 1852.33i) q^{83} +(-861.747 + 1213.61i) q^{84} +(-7163.60 - 2214.40i) q^{85} +(-4160.77 + 7206.66i) q^{86} +(-2711.49 - 3275.00i) q^{87} +(320.530 - 1196.23i) q^{88} -2113.36i q^{89} +(3040.90 + 4853.65i) q^{90} +4013.68 q^{91} +(-1828.32 - 489.896i) q^{92} +(-15207.6 - 5648.84i) q^{93} +(3702.17 + 2137.45i) q^{94} +(-9355.48 + 4937.32i) q^{95} +(1481.04 - 678.779i) q^{96} +(-11822.8 - 3167.91i) q^{97} +(-3947.27 - 3947.27i) q^{98} +(3358.24 + 2894.13i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 48 q^{2} - 18 q^{3} - 144 q^{6} + 24 q^{7} + 768 q^{8} - 24 q^{10} + 156 q^{11} - 144 q^{12} + 630 q^{15} + 1536 q^{16} + 900 q^{17} + 216 q^{18} - 48 q^{20} + 1908 q^{21} - 312 q^{22} - 756 q^{23}+ \cdots + 17760 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/90\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.73205 + 0.732051i 0.683013 + 0.183013i
\(3\) −1.50442 − 8.87337i −0.167158 − 0.985930i
\(4\) 6.92820 + 4.00000i 0.433013 + 0.250000i
\(5\) −11.6684 − 22.1099i −0.466737 − 0.884396i
\(6\) 2.38560 − 25.3438i 0.0662666 − 0.703995i
\(7\) −19.9684 − 5.35051i −0.407518 − 0.109194i 0.0492359 − 0.998787i \(-0.484321\pi\)
−0.456754 + 0.889593i \(0.650988\pi\)
\(8\) 16.0000 + 16.0000i 0.250000 + 0.250000i
\(9\) −76.4734 + 26.6986i −0.944116 + 0.329612i
\(10\) −15.6931 − 68.9473i −0.156931 − 0.689473i
\(11\) −27.3658 − 47.3989i −0.226163 − 0.391726i 0.730504 − 0.682908i \(-0.239285\pi\)
−0.956668 + 0.291181i \(0.905952\pi\)
\(12\) 25.0705 − 67.4942i 0.174101 − 0.468710i
\(13\) −187.537 + 50.2504i −1.10969 + 0.297340i −0.766703 − 0.642001i \(-0.778104\pi\)
−0.342984 + 0.939341i \(0.611438\pi\)
\(14\) −50.6378 − 29.2358i −0.258356 − 0.149162i
\(15\) −178.635 + 136.801i −0.793934 + 0.608004i
\(16\) 32.0000 + 55.4256i 0.125000 + 0.216506i
\(17\) 212.077 − 212.077i 0.733830 − 0.733830i −0.237547 − 0.971376i \(-0.576343\pi\)
0.971376 + 0.237547i \(0.0763433\pi\)
\(18\) −228.474 + 16.9595i −0.705167 + 0.0523442i
\(19\) − 423.135i − 1.17212i −0.810268 − 0.586060i \(-0.800678\pi\)
0.810268 − 0.586060i \(-0.199322\pi\)
\(20\) 7.59846 − 199.856i 0.0189962 − 0.499639i
\(21\) −17.4362 + 185.236i −0.0395378 + 0.420037i
\(22\) −40.0663 − 149.529i −0.0827815 − 0.308945i
\(23\) −228.540 + 61.2370i −0.432022 + 0.115760i −0.468275 − 0.883583i \(-0.655124\pi\)
0.0362533 + 0.999343i \(0.488458\pi\)
\(24\) 117.903 − 166.045i 0.204693 − 0.288272i
\(25\) −352.696 + 515.975i −0.564314 + 0.825560i
\(26\) −549.147 −0.812348
\(27\) 351.955 + 638.411i 0.482792 + 0.875735i
\(28\) −116.943 − 116.943i −0.149162 − 0.149162i
\(29\) 409.130 − 236.211i 0.486480 − 0.280869i −0.236633 − 0.971599i \(-0.576044\pi\)
0.723113 + 0.690730i \(0.242711\pi\)
\(30\) −588.186 + 242.977i −0.653540 + 0.269974i
\(31\) 901.271 − 1561.05i 0.937847 − 1.62440i 0.168371 − 0.985724i \(-0.446149\pi\)
0.769476 − 0.638675i \(-0.220517\pi\)
\(32\) 46.8513 + 174.851i 0.0457532 + 0.170753i
\(33\) −379.418 + 314.135i −0.348410 + 0.288462i
\(34\) 734.655 − 424.153i 0.635515 − 0.366915i
\(35\) 114.700 + 503.931i 0.0936328 + 0.411372i
\(36\) −636.618 − 120.920i −0.491217 − 0.0933027i
\(37\) 1276.93 − 1276.93i 0.932748 − 0.932748i −0.0651290 − 0.997877i \(-0.520746\pi\)
0.997877 + 0.0651290i \(0.0207459\pi\)
\(38\) 309.756 − 1156.03i 0.214513 − 0.800572i
\(39\) 728.026 + 1588.49i 0.478650 + 1.04437i
\(40\) 167.064 − 540.453i 0.104415 − 0.337783i
\(41\) 52.3525 − 90.6771i 0.0311436 − 0.0539424i −0.850033 − 0.526729i \(-0.823418\pi\)
0.881177 + 0.472786i \(0.156752\pi\)
\(42\) −183.239 + 493.311i −0.103877 + 0.279655i
\(43\) −761.473 + 2841.86i −0.411830 + 1.53697i 0.379273 + 0.925285i \(0.376174\pi\)
−0.791102 + 0.611684i \(0.790492\pi\)
\(44\) − 437.852i − 0.226163i
\(45\) 1482.63 + 1379.29i 0.732162 + 0.681131i
\(46\) −669.210 −0.316262
\(47\) 1459.90 + 391.180i 0.660889 + 0.177085i 0.573647 − 0.819102i \(-0.305528\pi\)
0.0872419 + 0.996187i \(0.472195\pi\)
\(48\) 443.671 − 367.331i 0.192565 − 0.159432i
\(49\) −1709.22 − 986.818i −0.711878 − 0.411003i
\(50\) −1341.30 + 1151.48i −0.536521 + 0.460592i
\(51\) −2200.89 − 1562.78i −0.846170 − 0.600839i
\(52\) −1500.30 − 402.003i −0.554844 − 0.148670i
\(53\) 1337.06 + 1337.06i 0.475992 + 0.475992i 0.903847 − 0.427855i \(-0.140731\pi\)
−0.427855 + 0.903847i \(0.640731\pi\)
\(54\) 494.210 + 2001.82i 0.169482 + 0.686495i
\(55\) −728.670 + 1158.12i −0.240883 + 0.382851i
\(56\) −233.886 − 405.102i −0.0745810 − 0.129178i
\(57\) −3754.63 + 636.574i −1.15563 + 0.195929i
\(58\) 1290.68 − 345.837i 0.383675 − 0.102805i
\(59\) 5331.68 + 3078.25i 1.53165 + 0.884300i 0.999286 + 0.0377864i \(0.0120306\pi\)
0.532367 + 0.846514i \(0.321303\pi\)
\(60\) −1784.82 + 233.243i −0.495785 + 0.0647898i
\(61\) −588.732 − 1019.71i −0.158219 − 0.274043i 0.776008 − 0.630724i \(-0.217242\pi\)
−0.934226 + 0.356681i \(0.883908\pi\)
\(62\) 3605.08 − 3605.08i 0.937847 − 0.937847i
\(63\) 1669.90 − 123.956i 0.420736 − 0.0312311i
\(64\) 512.000i 0.125000i
\(65\) 3299.29 + 3560.09i 0.780898 + 0.842624i
\(66\) −1266.55 + 580.478i −0.290760 + 0.133259i
\(67\) −439.993 − 1642.08i −0.0980158 − 0.365800i 0.899444 − 0.437037i \(-0.143972\pi\)
−0.997459 + 0.0712369i \(0.977305\pi\)
\(68\) 2317.62 − 621.004i 0.501215 − 0.134300i
\(69\) 887.199 + 1935.79i 0.186347 + 0.406593i
\(70\) −55.5367 + 1460.73i −0.0113340 + 0.298109i
\(71\) 1429.77 0.283628 0.141814 − 0.989893i \(-0.454706\pi\)
0.141814 + 0.989893i \(0.454706\pi\)
\(72\) −1650.75 − 796.397i −0.318432 − 0.153626i
\(73\) −5715.32 − 5715.32i −1.07249 − 1.07249i −0.997158 − 0.0753364i \(-0.975997\pi\)
−0.0753364 − 0.997158i \(-0.524003\pi\)
\(74\) 4423.42 − 2553.86i 0.807783 − 0.466374i
\(75\) 5109.04 + 2353.36i 0.908275 + 0.418375i
\(76\) 1692.54 − 2931.57i 0.293030 − 0.507543i
\(77\) 292.842 + 1092.90i 0.0493914 + 0.184331i
\(78\) 826.149 + 4872.78i 0.135790 + 0.800918i
\(79\) 1174.15 − 677.895i 0.188135 − 0.108620i −0.402974 − 0.915211i \(-0.632024\pi\)
0.591109 + 0.806592i \(0.298690\pi\)
\(80\) 852.066 − 1354.25i 0.133135 − 0.211601i
\(81\) 5135.37 − 4083.47i 0.782711 − 0.622385i
\(82\) 209.410 − 209.410i 0.0311436 − 0.0311436i
\(83\) 496.330 − 1852.33i 0.0720467 − 0.268882i −0.920501 − 0.390741i \(-0.872219\pi\)
0.992547 + 0.121859i \(0.0388855\pi\)
\(84\) −861.747 + 1213.61i −0.122130 + 0.171997i
\(85\) −7163.60 − 2214.40i −0.991501 − 0.306491i
\(86\) −4160.77 + 7206.66i −0.562570 + 0.974399i
\(87\) −2711.49 − 3275.00i −0.358237 − 0.432686i
\(88\) 320.530 − 1196.23i 0.0413908 − 0.154472i
\(89\) − 2113.36i − 0.266804i −0.991062 − 0.133402i \(-0.957410\pi\)
0.991062 − 0.133402i \(-0.0425902\pi\)
\(90\) 3040.90 + 4853.65i 0.375420 + 0.599216i
\(91\) 4013.68 0.484686
\(92\) −1828.32 − 489.896i −0.216011 − 0.0578799i
\(93\) −15207.6 − 5648.84i −1.75831 − 0.653120i
\(94\) 3702.17 + 2137.45i 0.418987 + 0.241902i
\(95\) −9355.48 + 4937.32i −1.03662 + 0.547071i
\(96\) 1481.04 − 678.779i 0.160703 − 0.0736522i
\(97\) −11822.8 − 3167.91i −1.25654 − 0.336690i −0.431683 − 0.902026i \(-0.642080\pi\)
−0.824861 + 0.565336i \(0.808747\pi\)
\(98\) −3947.27 − 3947.27i −0.411003 − 0.411003i
\(99\) 3358.24 + 2894.13i 0.342642 + 0.295289i
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 90.5.k.b.13.7 yes 48
3.2 odd 2 270.5.l.a.253.8 48
5.2 odd 4 inner 90.5.k.b.67.1 yes 48
9.2 odd 6 270.5.l.a.73.1 48
9.7 even 3 inner 90.5.k.b.43.1 yes 48
15.2 even 4 270.5.l.a.37.1 48
45.2 even 12 270.5.l.a.127.8 48
45.7 odd 12 inner 90.5.k.b.7.7 ✓ 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.5.k.b.7.7 ✓ 48 45.7 odd 12 inner
90.5.k.b.13.7 yes 48 1.1 even 1 trivial
90.5.k.b.43.1 yes 48 9.7 even 3 inner
90.5.k.b.67.1 yes 48 5.2 odd 4 inner
270.5.l.a.37.1 48 15.2 even 4
270.5.l.a.73.1 48 9.2 odd 6
270.5.l.a.127.8 48 45.2 even 12
270.5.l.a.253.8 48 3.2 odd 2