Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [90,11,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, 11); 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, 11, names="a")
 
Level: \( N \) \(=\) \( 90 = 2 \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 90.k (of order \(12\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [120,960] 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: \(57.1821527406\)
Analytic rank: \(0\)
Dimension: \(120\)
Relative dimension: \(30\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

Embedding invariants

Embedding label 43.28
Character \(\chi\) \(=\) 90.43
Dual form 90.11.k.b.67.28

$q$-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+(-5.85641 - 21.8564i) q^{2} +(235.426 - 60.1960i) q^{3} +(-443.405 + 256.000i) q^{4} +(-2694.26 + 1583.22i) q^{5} +(-2694.42 - 4793.04i) q^{6} +(-3016.33 - 11257.1i) q^{7} +(8192.00 + 8192.00i) q^{8} +(51801.9 - 28343.4i) q^{9} +(50382.2 + 49614.9i) q^{10} +(-102571. + 177659. i) q^{11} +(-88978.9 + 86960.3i) q^{12} +(132555. - 494704. i) q^{13} +(-228375. + 131852. i) q^{14} +(-538996. + 534915. i) q^{15} +(131072. - 227023. i) q^{16} +(-1.49151e6 + 1.49151e6i) q^{17} +(-922858. - 966212. i) q^{18} -554669. i q^{19} +(789345. - 1.39174e6i) q^{20} +(-1.38776e6 - 2.46865e6i) q^{21} +(4.48369e6 + 1.20140e6i) q^{22} +(-611943. + 2.28380e6i) q^{23} +(2.42174e6 + 1.43548e6i) q^{24} +(4.75246e6 - 8.53121e6i) q^{25} -1.15887e7 q^{26} +(1.04894e7 - 9.79105e6i) q^{27} +(4.21928e6 + 4.21928e6i) q^{28} +(3.03160e7 + 1.75030e7i) q^{29} +(1.48479e7 + 8.64783e6i) q^{30} +(6.88201e6 + 1.19200e7i) q^{31} +(-5.72953e6 - 1.53522e6i) q^{32} +(-1.34536e7 + 4.79999e7i) q^{33} +(4.13339e7 + 2.38641e7i) q^{34} +(2.59493e7 + 2.55541e7i) q^{35} +(-1.57133e7 + 2.58289e7i) q^{36} +(-5.95665e7 + 5.95665e7i) q^{37} +(-1.21231e7 + 3.24837e6i) q^{38} +(1.42782e6 - 1.24445e8i) q^{39} +(-3.50411e7 - 9.10165e6i) q^{40} +(6.98036e6 + 1.20903e7i) q^{41} +(-4.58285e7 + 4.47888e7i) q^{42} +(1.52314e8 - 4.08123e7i) q^{43} -1.05033e8i q^{44} +(-9.46939e7 + 1.58378e8i) q^{45} +5.34995e7 q^{46} +(4.57197e7 + 1.70628e8i) q^{47} +(1.71919e7 - 6.13372e7i) q^{48} +(1.27006e8 - 7.33272e7i) q^{49} +(-2.14294e8 - 5.39094e7i) q^{50} +(-2.61357e8 + 4.40923e8i) q^{51} +(6.78684e7 + 2.53288e8i) q^{52} +(1.53388e8 + 1.53388e8i) q^{53} +(-2.75427e8 - 1.71919e8i) q^{54} +(-4.91884e6 - 6.41052e8i) q^{55} +(6.75084e7 - 1.16928e8i) q^{56} +(-3.33889e7 - 1.30584e8i) q^{57} +(2.05009e8 - 7.65104e8i) q^{58} +(7.44145e7 - 4.29632e7i) q^{59} +(1.02055e8 - 3.75167e8i) q^{60} +(-6.03944e8 + 1.04606e9i) q^{61} +(2.20224e8 - 2.20224e8i) q^{62} +(-4.75317e8 - 4.97646e8i) q^{63} +1.34218e8i q^{64} +(4.26085e8 + 1.54272e9i) q^{65} +(1.12790e9 + 1.29408e7i) q^{66} +(1.90999e8 + 5.11779e7i) q^{67} +(2.79516e8 - 1.04317e9i) q^{68} +(-6.59152e6 + 5.74503e8i) q^{69} +(4.06551e8 - 7.16813e8i) q^{70} +2.29741e9 q^{71} +(6.56550e8 + 1.92172e8i) q^{72} +(2.89872e9 + 2.89872e9i) q^{73} +(1.65075e9 + 9.53063e8i) q^{74} +(6.05308e8 - 2.29455e9i) q^{75} +(1.41995e8 + 2.45943e8i) q^{76} +(2.30932e9 + 6.18779e8i) q^{77} +(-2.72829e9 + 6.97596e8i) q^{78} +(-6.81886e8 - 3.93687e8i) q^{79} +(6.28559e6 + 8.19176e8i) q^{80} +(1.88008e9 - 2.93649e9i) q^{81} +(2.23372e8 - 2.23372e8i) q^{82} +(4.49873e9 - 1.20543e9i) q^{83} +(1.24731e9 + 7.39344e8i) q^{84} +(1.65713e9 - 6.37990e9i) q^{85} +(-1.78402e9 - 3.09001e9i) q^{86} +(8.19079e9 + 2.29575e9i) q^{87} +(-2.29565e9 + 6.15117e8i) q^{88} -8.02416e9i q^{89} +(4.01615e9 + 1.14214e9i) q^{90} -5.96876e9 q^{91} +(-3.13315e8 - 1.16931e9i) q^{92} +(2.33774e9 + 2.39201e9i) q^{93} +(3.46157e9 - 1.99854e9i) q^{94} +(8.78163e8 + 1.49442e9i) q^{95} +(-1.44129e9 - 1.65366e7i) q^{96} +(2.46805e9 + 9.21088e9i) q^{97} +(-2.34647e9 - 2.34647e9i) q^{98} +(-2.77930e8 + 1.21103e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 120 q + 960 q^{2} - 128 q^{3} - 8192 q^{6} + 1830 q^{7} + 983040 q^{8} + 105792 q^{10} + 217740 q^{11} - 65536 q^{12} - 3385658 q^{15} + 15728640 q^{16} + 2438244 q^{17} + 2534464 q^{18} + 1692672 q^{20}+ \cdots + 76966514304 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(11\) \(37\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.85641 − 21.8564i −0.183013 − 0.683013i
\(3\) 235.426 − 60.1960i 0.968832 − 0.247720i
\(4\) −443.405 + 256.000i −0.433013 + 0.250000i
\(5\) −2694.26 + 1583.22i −0.862163 + 0.506630i
\(6\) −2694.42 − 4793.04i −0.346505 − 0.616388i
\(7\) −3016.33 − 11257.1i −0.179469 − 0.669787i −0.995747 − 0.0921281i \(-0.970633\pi\)
0.816278 − 0.577659i \(-0.196034\pi\)
\(8\) 8192.00 + 8192.00i 0.250000 + 0.250000i
\(9\) 51801.9 − 28343.4i 0.877269 − 0.479998i
\(10\) 50382.2 + 49614.9i 0.503822 + 0.496149i
\(11\) −102571. + 177659.i −0.636888 + 1.10312i 0.349224 + 0.937039i \(0.386445\pi\)
−0.986112 + 0.166083i \(0.946888\pi\)
\(12\) −88978.9 + 86960.3i −0.357586 + 0.349474i
\(13\) 132555. − 494704.i 0.357010 − 1.33238i −0.520925 − 0.853602i \(-0.674413\pi\)
0.877936 − 0.478778i \(-0.158920\pi\)
\(14\) −228375. + 131852.i −0.424628 + 0.245159i
\(15\) −538996. + 534915.i −0.709789 + 0.704415i
\(16\) 131072. − 227023.i 0.125000 − 0.216506i
\(17\) −1.49151e6 + 1.49151e6i −1.05046 + 1.05046i −0.0518074 + 0.998657i \(0.516498\pi\)
−0.998657 + 0.0518074i \(0.983502\pi\)
\(18\) −922858. − 966212.i −0.488396 − 0.511340i
\(19\) − 554669.i − 0.224009i −0.993708 − 0.112005i \(-0.964273\pi\)
0.993708 − 0.112005i \(-0.0357272\pi\)
\(20\) 789345. − 1.39174e6i 0.246670 − 0.434918i
\(21\) −1.38776e6 − 2.46865e6i −0.339795 − 0.604453i
\(22\) 4.48369e6 + 1.20140e6i 0.870005 + 0.233117i
\(23\) −611943. + 2.28380e6i −0.0950762 + 0.354829i −0.997031 − 0.0770012i \(-0.975465\pi\)
0.901955 + 0.431830i \(0.142132\pi\)
\(24\) 2.42174e6 + 1.43548e6i 0.304138 + 0.180278i
\(25\) 4.75246e6 − 8.53121e6i 0.486652 − 0.873596i
\(26\) −1.15887e7 −0.975370
\(27\) 1.04894e7 − 9.79105e6i 0.731021 − 0.682355i
\(28\) 4.21928e6 + 4.21928e6i 0.245159 + 0.245159i
\(29\) 3.03160e7 + 1.75030e7i 1.47803 + 0.853339i 0.999691 − 0.0248381i \(-0.00790701\pi\)
0.478335 + 0.878177i \(0.341240\pi\)
\(30\) 1.48479e7 + 8.64783e6i 0.611025 + 0.355878i
\(31\) 6.88201e6 + 1.19200e7i 0.240385 + 0.416358i 0.960824 − 0.277160i \(-0.0893931\pi\)
−0.720439 + 0.693518i \(0.756060\pi\)
\(32\) −5.72953e6 − 1.53522e6i −0.170753 − 0.0457532i
\(33\) −1.34536e7 + 4.79999e7i −0.343771 + 1.22651i
\(34\) 4.13339e7 + 2.38641e7i 0.909729 + 0.525232i
\(35\) 2.59493e7 + 2.55541e7i 0.494066 + 0.486542i
\(36\) −1.57133e7 + 2.58289e7i −0.259869 + 0.427163i
\(37\) −5.95665e7 + 5.95665e7i −0.859000 + 0.859000i −0.991220 − 0.132220i \(-0.957789\pi\)
0.132220 + 0.991220i \(0.457789\pi\)
\(38\) −1.21231e7 + 3.24837e6i −0.153001 + 0.0409966i
\(39\) 1.42782e6 − 1.24445e8i 0.0158252 − 1.37929i
\(40\) −3.50411e7 − 9.10165e6i −0.342198 − 0.0888833i
\(41\) 6.98036e6 + 1.20903e7i 0.0602502 + 0.104356i 0.894577 − 0.446913i \(-0.147477\pi\)
−0.834327 + 0.551270i \(0.814143\pi\)
\(42\) −4.58285e7 + 4.47888e7i −0.350662 + 0.342707i
\(43\) 1.52314e8 − 4.08123e7i 1.03609 − 0.277619i 0.299596 − 0.954066i \(-0.403148\pi\)
0.736491 + 0.676447i \(0.236481\pi\)
\(44\) − 1.05033e8i − 0.636888i
\(45\) −9.46939e7 + 1.58378e8i −0.513168 + 0.858288i
\(46\) 5.34995e7 0.259753
\(47\) 4.57197e7 + 1.70628e8i 0.199349 + 0.743980i 0.991098 + 0.133134i \(0.0425040\pi\)
−0.791749 + 0.610846i \(0.790829\pi\)
\(48\) 1.71919e7 − 6.13372e7i 0.0674709 − 0.240723i
\(49\) 1.27006e8 − 7.33272e7i 0.449620 − 0.259588i
\(50\) −2.14294e8 − 5.39094e7i −0.685741 − 0.172510i
\(51\) −2.61357e8 + 4.40923e8i −0.757502 + 1.27794i
\(52\) 6.78684e7 + 2.53288e8i 0.178505 + 0.666190i
\(53\) 1.53388e8 + 1.53388e8i 0.366785 + 0.366785i 0.866303 − 0.499518i \(-0.166490\pi\)
−0.499518 + 0.866303i \(0.666490\pi\)
\(54\) −2.75427e8 − 1.71919e8i −0.599843 − 0.374417i
\(55\) −4.91884e6 − 6.41052e8i −0.00977348 − 1.27374i
\(56\) 6.75084e7 − 1.16928e8i 0.122580 − 0.212314i
\(57\) −3.33889e7 − 1.30584e8i −0.0554917 − 0.217027i
\(58\) 2.05009e8 − 7.65104e8i 0.312344 − 1.16568i
\(59\) 7.44145e7 − 4.29632e7i 0.104087 − 0.0600948i −0.447053 − 0.894508i \(-0.647526\pi\)
0.551140 + 0.834413i \(0.314193\pi\)
\(60\) 1.02055e8 − 3.75167e8i 0.131244 − 0.482468i
\(61\) −6.03944e8 + 1.04606e9i −0.715068 + 1.23853i 0.247865 + 0.968795i \(0.420271\pi\)
−0.962933 + 0.269740i \(0.913062\pi\)
\(62\) 2.20224e8 − 2.20224e8i 0.240385 − 0.240385i
\(63\) −4.75317e8 − 4.97646e8i −0.478939 − 0.501439i
\(64\) 1.34218e8i 0.125000i
\(65\) 4.26085e8 + 1.54272e9i 0.367223 + 1.32960i
\(66\) 1.12790e9 + 1.29408e7i 0.900636 + 0.0103334i
\(67\) 1.90999e8 + 5.11779e7i 0.141467 + 0.0379061i 0.328858 − 0.944379i \(-0.393336\pi\)
−0.187390 + 0.982286i \(0.560003\pi\)
\(68\) 2.79516e8 − 1.04317e9i 0.192248 − 0.717481i
\(69\) −6.59152e6 + 5.74503e8i −0.00421444 + 0.367322i
\(70\) 4.06551e8 − 7.16813e8i 0.241894 − 0.426497i
\(71\) 2.29741e9 1.27335 0.636674 − 0.771133i \(-0.280310\pi\)
0.636674 + 0.771133i \(0.280310\pi\)
\(72\) 6.56550e8 + 1.92172e8i 0.339317 + 0.0993177i
\(73\) 2.89872e9 + 2.89872e9i 1.39827 + 1.39827i 0.805004 + 0.593270i \(0.202163\pi\)
0.593270 + 0.805004i \(0.297837\pi\)
\(74\) 1.65075e9 + 9.53063e8i 0.743916 + 0.429500i
\(75\) 6.05308e8 − 2.29455e9i 0.255076 − 0.966921i
\(76\) 1.41995e8 + 2.45943e8i 0.0560023 + 0.0969989i
\(77\) 2.30932e9 + 6.18779e8i 0.853158 + 0.228603i
\(78\) −2.72829e9 + 6.97596e8i −0.944970 + 0.241619i
\(79\) −6.81886e8 − 3.93687e8i −0.221603 − 0.127943i 0.385089 − 0.922879i \(-0.374171\pi\)
−0.606692 + 0.794937i \(0.707504\pi\)
\(80\) 6.28559e6 + 8.19176e8i 0.00191821 + 0.249993i
\(81\) 1.88008e9 − 2.93649e9i 0.539203 − 0.842176i
\(82\) 2.23372e8 − 2.23372e8i 0.0602502 − 0.0602502i
\(83\) 4.49873e9 − 1.20543e9i 1.14209 − 0.306021i 0.362297 − 0.932063i \(-0.381992\pi\)
0.779790 + 0.626041i \(0.215326\pi\)
\(84\) 1.24731e9 + 7.39344e8i 0.298249 + 0.176787i
\(85\) 1.65713e9 − 6.37990e9i 0.373475 − 1.43787i
\(86\) −1.78402e9 − 3.09001e9i −0.379234 − 0.656853i
\(87\) 8.19079e9 + 2.29575e9i 1.64335 + 0.460605i
\(88\) −2.29565e9 + 6.15117e8i −0.435002 + 0.116559i
\(89\) − 8.02416e9i − 1.43698i −0.695539 − 0.718488i \(-0.744835\pi\)
0.695539 − 0.718488i \(-0.255165\pi\)
\(90\) 4.01615e9 + 1.14214e9i 0.680138 + 0.193423i
\(91\) −5.96876e9 −0.956483
\(92\) −3.13315e8 − 1.16931e9i −0.0475381 − 0.177415i
\(93\) 2.33774e9 + 2.39201e9i 0.336033 + 0.343833i
\(94\) 3.46157e9 − 1.99854e9i 0.471665 − 0.272316i
\(95\) 8.78163e8 + 1.49442e9i 0.113490 + 0.193133i
\(96\) −1.44129e9 − 1.65366e7i −0.176765 − 0.00202810i
\(97\) 2.46805e9 + 9.21088e9i 0.287405 + 1.07261i 0.947064 + 0.321046i \(0.104034\pi\)
−0.659658 + 0.751566i \(0.729299\pi\)
\(98\) −2.34647e9 − 2.34647e9i −0.259588 − 0.259588i
\(99\) −2.77930e8 + 1.21103e10i −0.0292253 + 1.27344i
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 90.11.k.b.43.28 yes 120
5.2 odd 4 inner 90.11.k.b.7.13 ✓ 120
9.4 even 3 inner 90.11.k.b.13.13 yes 120
45.22 odd 12 inner 90.11.k.b.67.28 yes 120
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.11.k.b.7.13 ✓ 120 5.2 odd 4 inner
90.11.k.b.13.13 yes 120 9.4 even 3 inner
90.11.k.b.43.28 yes 120 1.1 even 1 trivial
90.11.k.b.67.28 yes 120 45.22 odd 12 inner