Show commands: Magma / Pari/GP / SageMath

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 13.4
Character \(\chi\) \(=\) 90.13
Dual form 90.5.k.b.7.4

$q$-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} +(-7.02493 - 5.62586i) q^{3} +(6.92820 + 4.00000i) q^{4} +(9.14193 + 23.2685i) q^{5} +(-15.0741 - 20.5127i) q^{6} +(-72.3024 - 19.3734i) q^{7} +(16.0000 + 16.0000i) q^{8} +(17.6994 + 79.0426i) q^{9} +(7.94247 + 70.2632i) q^{10} +(78.8083 + 136.500i) q^{11} +(-26.1667 - 67.0768i) q^{12} +(-147.650 + 39.5627i) q^{13} +(-183.352 - 105.858i) q^{14} +(66.6842 - 214.891i) q^{15} +(32.0000 + 55.4256i) q^{16} +(-152.822 + 152.822i) q^{17} +(-9.50767 + 228.905i) q^{18} +378.135i q^{19} +(-29.7370 + 197.777i) q^{20} +(398.928 + 542.860i) q^{21} +(115.383 + 430.617i) q^{22} +(111.213 - 29.7995i) q^{23} +(-22.3851 - 202.413i) q^{24} +(-457.850 + 425.439i) q^{25} -432.349 q^{26} +(320.346 - 654.843i) q^{27} +(-423.432 - 423.432i) q^{28} +(-782.716 + 451.902i) q^{29} +(339.496 - 538.277i) q^{30} +(855.273 - 1481.38i) q^{31} +(46.8513 + 174.851i) q^{32} +(214.307 - 1402.27i) q^{33} +(-529.390 + 305.643i) q^{34} +(-210.194 - 1859.48i) q^{35} +(-193.546 + 618.421i) q^{36} +(-1097.59 + 1097.59i) q^{37} +(-276.814 + 1033.09i) q^{38} +(1259.81 + 552.734i) q^{39} +(-226.026 + 518.568i) q^{40} +(1205.45 - 2087.91i) q^{41} +(692.489 + 1775.16i) q^{42} +(541.190 - 2019.75i) q^{43} +1260.93i q^{44} +(-1677.40 + 1134.44i) q^{45} +325.655 q^{46} +(1153.41 + 309.056i) q^{47} +(87.0191 - 569.389i) q^{48} +(2772.98 + 1600.98i) q^{49} +(-1562.31 + 827.151i) q^{50} +(1933.31 - 213.808i) q^{51} +(-1181.20 - 316.502i) q^{52} +(10.2234 + 10.2234i) q^{53} +(1354.58 - 1554.55i) q^{54} +(-2455.70 + 3081.63i) q^{55} +(-846.865 - 1466.81i) q^{56} +(2127.34 - 2656.38i) q^{57} +(-2469.24 + 661.630i) q^{58} +(-1998.54 - 1153.86i) q^{59} +(1321.57 - 1222.07i) q^{60} +(1359.76 + 2355.17i) q^{61} +(3421.09 - 3421.09i) q^{62} +(251.616 - 6057.87i) q^{63} +512.000i q^{64} +(-2270.37 - 3073.92i) q^{65} +(1612.03 - 3674.18i) q^{66} +(1683.57 + 6283.15i) q^{67} +(-1670.07 + 447.493i) q^{68} +(-948.915 - 416.331i) q^{69} +(786.976 - 5234.07i) q^{70} -4444.62 q^{71} +(-981.492 + 1547.87i) q^{72} +(5158.22 + 5158.22i) q^{73} +(-3802.18 + 2195.19i) q^{74} +(5609.83 - 412.878i) q^{75} +(-1512.54 + 2619.80i) q^{76} +(-3053.56 - 11396.1i) q^{77} +(3037.23 + 2432.34i) q^{78} +(30.1328 - 17.3972i) q^{79} +(-997.132 + 1251.29i) q^{80} +(-5934.47 + 2798.01i) q^{81} +(4821.81 - 4821.81i) q^{82} +(714.956 - 2668.25i) q^{83} +(592.412 + 5356.75i) q^{84} +(-4953.02 - 2158.85i) q^{85} +(2957.12 - 5121.88i) q^{86} +(8040.87 + 1228.88i) q^{87} +(-923.067 + 3444.93i) q^{88} +3431.48i q^{89} +(-5413.21 + 1871.41i) q^{90} +11441.9 q^{91} +(889.707 + 238.396i) q^{92} +(-14342.3 + 5594.92i) q^{93} +(2924.94 + 1688.71i) q^{94} +(-8798.66 + 3456.89i) q^{95} +(654.562 - 1491.90i) q^{96} +(-9284.55 - 2487.79i) q^{97} +(6403.93 + 6403.93i) q^{98} +(-9394.46 + 8645.17i) 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
\(3\) −7.02493 − 5.62586i −0.780548 − 0.625096i
\(4\) 6.92820 + 4.00000i 0.433013 + 0.250000i
\(5\) 9.14193 + 23.2685i 0.365677 + 0.930742i
\(6\) −15.0741 − 20.5127i −0.418724 − 0.569799i
\(7\) −72.3024 − 19.3734i −1.47556 − 0.395375i −0.570726 − 0.821141i \(-0.693338\pi\)
−0.904833 + 0.425766i \(0.860005\pi\)
\(8\) 16.0000 + 16.0000i 0.250000 + 0.250000i
\(9\) 17.6994 + 79.0426i 0.218511 + 0.975835i
\(10\) 7.94247 + 70.2632i 0.0794247 + 0.702632i
\(11\) 78.8083 + 136.500i 0.651308 + 1.12810i 0.982806 + 0.184643i \(0.0591128\pi\)
−0.331498 + 0.943456i \(0.607554\pi\)
\(12\) −26.1667 − 67.0768i −0.181713 − 0.465811i
\(13\) −147.650 + 39.5627i −0.873669 + 0.234099i −0.667674 − 0.744454i \(-0.732710\pi\)
−0.205995 + 0.978553i \(0.566043\pi\)
\(14\) −183.352 − 105.858i −0.935467 − 0.540092i
\(15\) 66.6842 − 214.891i 0.296374 − 0.955072i
\(16\) 32.0000 + 55.4256i 0.125000 + 0.216506i
\(17\) −152.822 + 152.822i −0.528795 + 0.528795i −0.920213 − 0.391418i \(-0.871985\pi\)
0.391418 + 0.920213i \(0.371985\pi\)
\(18\) −9.50767 + 228.905i −0.0293447 + 0.706498i
\(19\) 378.135i 1.04747i 0.851882 + 0.523733i \(0.175461\pi\)
−0.851882 + 0.523733i \(0.824539\pi\)
\(20\) −29.7370 + 197.777i −0.0743425 + 0.494442i
\(21\) 398.928 + 542.860i 0.904598 + 1.23098i
\(22\) 115.383 + 430.617i 0.238395 + 0.889704i
\(23\) 111.213 − 29.7995i 0.210233 − 0.0563318i −0.152165 − 0.988355i \(-0.548625\pi\)
0.362399 + 0.932023i \(0.381958\pi\)
\(24\) −22.3851 − 202.413i −0.0388631 − 0.351411i
\(25\) −457.850 + 425.439i −0.732560 + 0.680702i
\(26\) −432.349 −0.639570
\(27\) 320.346 − 654.843i 0.439432 − 0.898276i
\(28\) −423.432 − 423.432i −0.540092 − 0.540092i
\(29\) −782.716 + 451.902i −0.930697 + 0.537338i −0.887032 − 0.461708i \(-0.847237\pi\)
−0.0436652 + 0.999046i \(0.513903\pi\)
\(30\) 339.496 − 538.277i 0.377218 − 0.598086i
\(31\) 855.273 − 1481.38i 0.889983 − 1.54150i 0.0500888 − 0.998745i \(-0.484050\pi\)
0.839894 − 0.542751i \(-0.182617\pi\)
\(32\) 46.8513 + 174.851i 0.0457532 + 0.170753i
\(33\) 214.307 − 1402.27i 0.196793 − 1.28767i
\(34\) −529.390 + 305.643i −0.457949 + 0.264397i
\(35\) −210.194 − 1859.48i −0.171587 − 1.51794i
\(36\) −193.546 + 618.421i −0.149341 + 0.477176i
\(37\) −1097.59 + 1097.59i −0.801749 + 0.801749i −0.983369 − 0.181620i \(-0.941866\pi\)
0.181620 + 0.983369i \(0.441866\pi\)
\(38\) −276.814 + 1033.09i −0.191700 + 0.715433i
\(39\) 1259.81 + 552.734i 0.828275 + 0.363401i
\(40\) −226.026 + 518.568i −0.141266 + 0.324105i
\(41\) 1205.45 − 2087.91i 0.717104 − 1.24206i −0.245038 − 0.969514i \(-0.578800\pi\)
0.962142 − 0.272548i \(-0.0878663\pi\)
\(42\) 692.489 + 1775.16i 0.392568 + 1.00632i
\(43\) 541.190 − 2019.75i 0.292693 − 1.09235i −0.650339 − 0.759644i \(-0.725373\pi\)
0.943032 − 0.332702i \(-0.107960\pi\)
\(44\) 1260.93i 0.651308i
\(45\) −1677.40 + 1134.44i −0.828346 + 0.560217i
\(46\) 325.655 0.153901
\(47\) 1153.41 + 309.056i 0.522143 + 0.139908i 0.510258 − 0.860022i \(-0.329550\pi\)
0.0118852 + 0.999929i \(0.496217\pi\)
\(48\) 87.0191 − 569.389i 0.0377687 − 0.247131i
\(49\) 2772.98 + 1600.98i 1.15493 + 0.666798i
\(50\) −1562.31 + 827.151i −0.624925 + 0.330860i
\(51\) 1933.31 − 213.808i 0.743297 − 0.0822023i
\(52\) −1181.20 − 316.502i −0.436835 − 0.117049i
\(53\) 10.2234 + 10.2234i 0.00363950 + 0.00363950i 0.708924 − 0.705285i \(-0.249181\pi\)
−0.705285 + 0.708924i \(0.749181\pi\)
\(54\) 1354.58 − 1554.55i 0.464534 − 0.533112i
\(55\) −2455.70 + 3081.63i −0.811800 + 1.01872i
\(56\) −846.865 − 1466.81i −0.270046 − 0.467734i
\(57\) 2127.34 − 2656.38i 0.654767 − 0.817598i
\(58\) −2469.24 + 661.630i −0.734018 + 0.196679i
\(59\) −1998.54 − 1153.86i −0.574129 − 0.331474i 0.184668 − 0.982801i \(-0.440879\pi\)
−0.758797 + 0.651327i \(0.774212\pi\)
\(60\) 1321.57 − 1222.07i 0.367102 − 0.339465i
\(61\) 1359.76 + 2355.17i 0.365429 + 0.632941i 0.988845 − 0.148949i \(-0.0475891\pi\)
−0.623416 + 0.781890i \(0.714256\pi\)
\(62\) 3421.09 − 3421.09i 0.889983 − 0.889983i
\(63\) 251.616 − 6057.87i 0.0633953 − 1.52630i
\(64\) 512.000i 0.125000i
\(65\) −2270.37 − 3073.92i −0.537367 − 0.727556i
\(66\) 1612.03 − 3674.18i 0.370071 − 0.843476i
\(67\) 1683.57 + 6283.15i 0.375042 + 1.39968i 0.853283 + 0.521448i \(0.174608\pi\)
−0.478240 + 0.878229i \(0.658725\pi\)
\(68\) −1670.07 + 447.493i −0.361173 + 0.0967761i
\(69\) −948.915 − 416.331i −0.199310 − 0.0874462i
\(70\) 786.976 − 5234.07i 0.160607 − 1.06818i
\(71\) −4444.62 −0.881694 −0.440847 − 0.897582i \(-0.645322\pi\)
−0.440847 + 0.897582i \(0.645322\pi\)
\(72\) −981.492 + 1547.87i −0.189331 + 0.298586i
\(73\) 5158.22 + 5158.22i 0.967952 + 0.967952i 0.999502 − 0.0315500i \(-0.0100443\pi\)
−0.0315500 + 0.999502i \(0.510044\pi\)
\(74\) −3802.18 + 2195.19i −0.694335 + 0.400874i
\(75\) 5609.83 − 412.878i 0.997303 − 0.0734005i
\(76\) −1512.54 + 2619.80i −0.261867 + 0.453566i
\(77\) −3053.56 − 11396.1i −0.515022 − 1.92209i
\(78\) 3037.23 + 2432.34i 0.499215 + 0.399793i
\(79\) 30.1328 − 17.3972i 0.00482819 − 0.00278756i −0.497584 − 0.867416i \(-0.665779\pi\)
0.502412 + 0.864628i \(0.332446\pi\)
\(80\) −997.132 + 1251.29i −0.155802 + 0.195514i
\(81\) −5934.47 + 2798.01i −0.904506 + 0.426460i
\(82\) 4821.81 − 4821.81i 0.717104 − 0.717104i
\(83\) 714.956 − 2668.25i 0.103782 − 0.387320i −0.894422 − 0.447224i \(-0.852413\pi\)
0.998204 + 0.0599036i \(0.0190793\pi\)
\(84\) 592.412 + 5356.75i 0.0839586 + 0.759177i
\(85\) −4953.02 − 2158.85i −0.685539 − 0.298803i
\(86\) 2957.12 − 5121.88i 0.399826 − 0.692520i
\(87\) 8040.87 + 1228.88i 1.06234 + 0.162357i
\(88\) −923.067 + 3444.93i −0.119198 + 0.444852i
\(89\) 3431.48i 0.433213i 0.976259 + 0.216606i \(0.0694988\pi\)
−0.976259 + 0.216606i \(0.930501\pi\)
\(90\) −5413.21 + 1871.41i −0.668297 + 0.231038i
\(91\) 11441.9 1.38171
\(92\) 889.707 + 238.396i 0.105117 + 0.0281659i
\(93\) −14342.3 + 5594.92i −1.65826 + 0.646887i
\(94\) 2924.94 + 1688.71i 0.331025 + 0.191118i
\(95\) −8798.66 + 3456.89i −0.974921 + 0.383035i
\(96\) 654.562 − 1491.90i 0.0710245 − 0.161881i
\(97\) −9284.55 − 2487.79i −0.986773 − 0.264405i −0.270879 − 0.962613i \(-0.587314\pi\)
−0.715895 + 0.698208i \(0.753981\pi\)
\(98\) 6403.93 + 6403.93i 0.666798 + 0.666798i
\(99\) −9394.46 + 8645.17i −0.958520 + 0.882071i
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.13.4 yes 48
3.2 odd 2 270.5.l.a.253.5 48
5.2 odd 4 inner 90.5.k.b.67.4 yes 48
9.2 odd 6 270.5.l.a.73.12 48
9.7 even 3 inner 90.5.k.b.43.4 yes 48
15.2 even 4 270.5.l.a.37.12 48
45.2 even 12 270.5.l.a.127.5 48
45.7 odd 12 inner 90.5.k.b.7.4 ✓ 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.5.k.b.7.4 ✓ 48 45.7 odd 12 inner
90.5.k.b.13.4 yes 48 1.1 even 1 trivial
90.5.k.b.43.4 yes 48 9.7 even 3 inner
90.5.k.b.67.4 yes 48 5.2 odd 4 inner
270.5.l.a.37.12 48 15.2 even 4
270.5.l.a.73.12 48 9.2 odd 6
270.5.l.a.127.5 48 45.2 even 12
270.5.l.a.253.5 48 3.2 odd 2