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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [40,11,Mod(13,40)] 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("40.13"); S:= CuspForms(chi, 11); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(40, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 11, names="a")
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 40.i (of order \(4\), degree \(2\), 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: \(25.4142901069\)
Analytic rank: \(0\)
Dimension: \(116\)
Relative dimension: \(58\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.20
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.20

$q$-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+(-15.0272 + 28.2521i) q^{2} +(257.663 - 257.663i) q^{3} +(-572.366 - 849.101i) q^{4} +(-2420.01 - 1977.17i) q^{5} +(3407.58 + 11151.5i) q^{6} +(-1516.95 + 1516.95i) q^{7} +(32590.0 - 3410.93i) q^{8} -73731.8i q^{9} +(92225.1 - 38659.1i) q^{10} -52679.7i q^{11} +(-366260. - 71304.6i) q^{12} +(441502. - 441502. i) q^{13} +(-20061.6 - 65652.8i) q^{14} +(-1.13299e6 + 114104. i) q^{15} +(-393370. + 971993. i) q^{16} +(-1.43204e6 + 1.43204e6i) q^{17} +(2.08308e6 + 1.10798e6i) q^{18} -1.08572e6 q^{19} +(-293684. + 3.18649e6i) q^{20} +781727. i q^{21} +(1.48831e6 + 791629. i) q^{22} +(-3.59788e6 - 3.59788e6i) q^{23} +(7.51837e6 - 9.27612e6i) q^{24} +(1.94725e6 + 9.56952e6i) q^{25} +(5.83882e6 + 1.91079e7i) q^{26} +(-3.78323e6 - 3.78323e6i) q^{27} +(2.15630e6 + 419795. i) q^{28} -3.62641e7 q^{29} +(1.38020e7 - 3.37241e7i) q^{30} -3.58761e7 q^{31} +(-2.15496e7 - 2.57199e7i) q^{32} +(-1.35736e7 - 1.35736e7i) q^{33} +(-1.89387e7 - 6.19779e7i) q^{34} +(6.67031e6 - 671771. i) q^{35} +(-6.26058e7 + 4.22016e7i) q^{36} +(8.56642e6 + 8.56642e6i) q^{37} +(1.63154e7 - 3.06740e7i) q^{38} -2.27518e8i q^{39} +(-8.56120e7 - 5.61813e7i) q^{40} -5.50374e7 q^{41} +(-2.20854e7 - 1.17472e7i) q^{42} +(-3.02596e7 + 3.02596e7i) q^{43} +(-4.47304e7 + 3.01521e7i) q^{44} +(-1.45780e8 + 1.78432e8i) q^{45} +(1.55714e8 - 4.75817e7i) q^{46} +(2.74925e8 - 2.74925e8i) q^{47} +(1.49090e8 + 3.51804e8i) q^{48} +2.77873e8i q^{49} +(-2.99621e8 - 8.87890e7i) q^{50} +7.37971e8i q^{51} +(-6.27580e8 - 1.22179e8i) q^{52} +(9.74861e7 - 9.74861e7i) q^{53} +(1.63735e8 - 5.00329e7i) q^{54} +(-1.04157e8 + 1.27485e8i) q^{55} +(-4.42633e7 + 5.46117e7i) q^{56} +(-2.79751e8 + 2.79751e8i) q^{57} +(5.44949e8 - 1.02454e9i) q^{58} +4.28897e8 q^{59} +(7.45371e8 + 8.96715e8i) q^{60} +2.75054e8i q^{61} +(5.39118e8 - 1.01358e9i) q^{62} +(1.11848e8 + 1.11848e8i) q^{63} +(1.05047e9 - 2.22325e8i) q^{64} +(-1.94136e9 + 1.95515e8i) q^{65} +(5.87458e8 - 1.79510e8i) q^{66} +(-8.77077e8 - 8.77077e8i) q^{67} +(2.03560e9 + 3.96297e8i) q^{68} -1.85409e9 q^{69} +(-8.12572e7 + 1.98545e8i) q^{70} -1.36684e9 q^{71} +(-2.51494e8 - 2.40292e9i) q^{72} +(4.21499e8 + 4.21499e8i) q^{73} +(-3.70749e8 + 1.13290e8i) q^{74} +(2.96745e9 + 1.96398e9i) q^{75} +(6.21431e8 + 9.21889e8i) q^{76} +(7.99127e7 + 7.99127e7i) q^{77} +(6.42786e9 + 3.41895e9i) q^{78} -2.97537e9i q^{79} +(2.87375e9 - 1.57447e9i) q^{80} +2.40419e9 q^{81} +(8.27059e8 - 1.55492e9i) q^{82} +(-4.71268e9 + 4.71268e9i) q^{83} +(6.63765e8 - 4.47434e8i) q^{84} +(6.29695e9 - 6.34170e8i) q^{85} +(-4.00181e8 - 1.30961e9i) q^{86} +(-9.34394e9 + 9.34394e9i) q^{87} +(-1.79687e8 - 1.71683e9i) q^{88} -6.41481e9i q^{89} +(-2.85041e9 - 6.79993e9i) q^{90} +1.33947e9i q^{91} +(-9.95662e8 + 5.11428e9i) q^{92} +(-9.24396e9 + 9.24396e9i) q^{93} +(3.63586e9 + 1.18986e10i) q^{94} +(2.62746e9 + 2.14665e9i) q^{95} +(-1.21796e10 - 1.07453e9i) q^{96} +(6.09122e9 - 6.09122e9i) q^{97} +(-7.85050e9 - 4.17566e9i) q^{98} -3.88417e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 116 q - 2 q^{2} - 4864 q^{6} - 4 q^{7} + 69124 q^{8} + 256166 q^{10} + 502036 q^{12} - 4 q^{15} + 3502536 q^{16} - 905772 q^{17} - 5688470 q^{18} + 4385256 q^{20} + 9808012 q^{22} - 4 q^{23} - 1476988 q^{25}+ \cdots - 65612488734 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(21\) \(31\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −15.0272 + 28.2521i −0.469600 + 0.882879i
\(3\) 257.663 − 257.663i 1.06034 − 1.06034i 0.0622846 − 0.998058i \(-0.480161\pi\)
0.998058 − 0.0622846i \(-0.0198387\pi\)
\(4\) −572.366 − 849.101i −0.558951 − 0.829201i
\(5\) −2420.01 − 1977.17i −0.774403 − 0.632693i
\(6\) 3407.58 + 11151.5i 0.438217 + 1.43409i
\(7\) −1516.95 + 1516.95i −0.0902572 + 0.0902572i −0.750794 − 0.660537i \(-0.770329\pi\)
0.660537 + 0.750794i \(0.270329\pi\)
\(8\) 32590.0 − 3410.93i 0.994568 − 0.104093i
\(9\) − 73731.8i − 1.24865i
\(10\) 92225.1 − 38659.1i 0.922251 − 0.386591i
\(11\) − 52679.7i − 0.327100i −0.986535 − 0.163550i \(-0.947706\pi\)
0.986535 − 0.163550i \(-0.0522944\pi\)
\(12\) −366260. − 71304.6i −1.47192 − 0.286557i
\(13\) 441502. − 441502.i 1.18909 − 1.18909i 0.211773 − 0.977319i \(-0.432076\pi\)
0.977319 − 0.211773i \(-0.0679238\pi\)
\(14\) −20061.6 − 65652.8i −0.0373014 − 0.122071i
\(15\) −1.13299e6 + 114104.i −1.49200 + 0.150261i
\(16\) −393370. + 971993.i −0.375147 + 0.926965i
\(17\) −1.43204e6 + 1.43204e6i −1.00858 + 1.00858i −0.00862071 + 0.999963i \(0.502744\pi\)
−0.999963 + 0.00862071i \(0.997256\pi\)
\(18\) 2.08308e6 + 1.10798e6i 1.10241 + 0.586369i
\(19\) −1.08572e6 −0.438481 −0.219241 − 0.975671i \(-0.570358\pi\)
−0.219241 + 0.975671i \(0.570358\pi\)
\(20\) −293684. + 3.18649e6i −0.0917763 + 0.995780i
\(21\) 781727.i 0.191407i
\(22\) 1.48831e6 + 791629.i 0.288789 + 0.153606i
\(23\) −3.59788e6 − 3.59788e6i −0.558995 − 0.558995i 0.370026 − 0.929021i \(-0.379349\pi\)
−0.929021 + 0.370026i \(0.879349\pi\)
\(24\) 7.51837e6 − 9.27612e6i 0.944208 − 1.16496i
\(25\) 1.94725e6 + 9.56952e6i 0.199399 + 0.979918i
\(26\) 5.83882e6 + 1.91079e7i 0.491427 + 1.60822i
\(27\) −3.78323e6 − 3.78323e6i −0.263659 − 0.263659i
\(28\) 2.15630e6 + 419795.i 0.125291 + 0.0243920i
\(29\) −3.62641e7 −1.76802 −0.884010 − 0.467467i \(-0.845167\pi\)
−0.884010 + 0.467467i \(0.845167\pi\)
\(30\) 1.38020e7 − 3.37241e7i 0.567984 − 1.38782i
\(31\) −3.58761e7 −1.25313 −0.626566 − 0.779368i \(-0.715540\pi\)
−0.626566 + 0.779368i \(0.715540\pi\)
\(32\) −2.15496e7 − 2.57199e7i −0.642229 − 0.766513i
\(33\) −1.35736e7 − 1.35736e7i −0.346838 − 0.346838i
\(34\) −1.89387e7 − 6.19779e7i −0.416826 − 1.36409i
\(35\) 6.67031e6 − 671771.i 0.127001 − 0.0127903i
\(36\) −6.26058e7 + 4.22016e7i −1.03539 + 0.697937i
\(37\) 8.56642e6 + 8.56642e6i 0.123535 + 0.123535i 0.766171 − 0.642636i \(-0.222159\pi\)
−0.642636 + 0.766171i \(0.722159\pi\)
\(38\) 1.63154e7 − 3.06740e7i 0.205911 − 0.387126i
\(39\) − 2.27518e8i − 2.52169i
\(40\) −8.56120e7 − 5.61813e7i −0.836055 − 0.548646i
\(41\) −5.50374e7 −0.475049 −0.237525 − 0.971382i \(-0.576336\pi\)
−0.237525 + 0.971382i \(0.576336\pi\)
\(42\) −2.20854e7 − 1.17472e7i −0.168990 − 0.0898849i
\(43\) −3.02596e7 + 3.02596e7i −0.205836 + 0.205836i −0.802495 − 0.596659i \(-0.796494\pi\)
0.596659 + 0.802495i \(0.296494\pi\)
\(44\) −4.47304e7 + 3.01521e7i −0.271231 + 0.182833i
\(45\) −1.45780e8 + 1.78432e8i −0.790015 + 0.966962i
\(46\) 1.55714e8 − 4.75817e7i 0.756029 − 0.231021i
\(47\) 2.74925e8 − 2.74925e8i 1.19874 − 1.19874i 0.224196 − 0.974544i \(-0.428024\pi\)
0.974544 − 0.224196i \(-0.0719757\pi\)
\(48\) 1.49090e8 + 3.51804e8i 0.585116 + 1.38069i
\(49\) 2.77873e8i 0.983707i
\(50\) −2.99621e8 − 8.87890e7i −0.958787 − 0.284125i
\(51\) 7.37971e8i 2.13889i
\(52\) −6.27580e8 − 1.22179e8i −1.65064 − 0.321352i
\(53\) 9.74861e7 − 9.74861e7i 0.233111 − 0.233111i −0.580879 − 0.813990i \(-0.697291\pi\)
0.813990 + 0.580879i \(0.197291\pi\)
\(54\) 1.63735e8 − 5.00329e7i 0.356594 − 0.108965i
\(55\) −1.04157e8 + 1.27485e8i −0.206954 + 0.253307i
\(56\) −4.42633e7 + 5.46117e7i −0.0803717 + 0.0991621i
\(57\) −2.79751e8 + 2.79751e8i −0.464941 + 0.464941i
\(58\) 5.44949e8 − 1.02454e9i 0.830263 − 1.56095i
\(59\) 4.28897e8 0.599919 0.299960 − 0.953952i \(-0.403027\pi\)
0.299960 + 0.953952i \(0.403027\pi\)
\(60\) 7.45371e8 + 8.96715e8i 0.958554 + 1.15318i
\(61\) 2.75054e8i 0.325664i 0.986654 + 0.162832i \(0.0520628\pi\)
−0.986654 + 0.162832i \(0.947937\pi\)
\(62\) 5.39118e8 − 1.01358e9i 0.588471 − 1.10636i
\(63\) 1.11848e8 + 1.11848e8i 0.112700 + 0.112700i
\(64\) 1.05047e9 − 2.22325e8i 0.978329 − 0.207056i
\(65\) −1.94136e9 + 1.95515e8i −1.67317 + 0.168506i
\(66\) 5.87458e8 − 1.79510e8i 0.469091 − 0.143341i
\(67\) −8.77077e8 − 8.77077e8i −0.649626 − 0.649626i 0.303276 − 0.952903i \(-0.401919\pi\)
−0.952903 + 0.303276i \(0.901919\pi\)
\(68\) 2.03560e9 + 3.96297e8i 1.40007 + 0.272569i
\(69\) −1.85409e9 −1.18545
\(70\) −8.12572e7 + 1.98545e8i −0.0483472 + 0.118133i
\(71\) −1.36684e9 −0.757576 −0.378788 − 0.925483i \(-0.623659\pi\)
−0.378788 + 0.925483i \(0.623659\pi\)
\(72\) −2.51494e8 − 2.40292e9i −0.129977 − 1.24187i
\(73\) 4.21499e8 + 4.21499e8i 0.203321 + 0.203321i 0.801421 − 0.598100i \(-0.204078\pi\)
−0.598100 + 0.801421i \(0.704078\pi\)
\(74\) −3.70749e8 + 1.13290e8i −0.167079 + 0.0510545i
\(75\) 2.96745e9 + 1.96398e9i 1.25048 + 0.827619i
\(76\) 6.21431e8 + 9.21889e8i 0.245090 + 0.363589i
\(77\) 7.99127e7 + 7.99127e7i 0.0295231 + 0.0295231i
\(78\) 6.42786e9 + 3.41895e9i 2.22635 + 1.18419i
\(79\) − 2.97537e9i − 0.966953i −0.875357 − 0.483476i \(-0.839374\pi\)
0.875357 − 0.483476i \(-0.160626\pi\)
\(80\) 2.87375e9 − 1.57447e9i 0.877000 − 0.480491i
\(81\) 2.40419e9 0.689516
\(82\) 8.27059e8 − 1.55492e9i 0.223083 − 0.419411i
\(83\) −4.71268e9 + 4.71268e9i −1.19640 + 1.19640i −0.221166 + 0.975236i \(0.570986\pi\)
−0.975236 + 0.221166i \(0.929014\pi\)
\(84\) 6.63765e8 − 4.47434e8i 0.158715 − 0.106987i
\(85\) 6.29695e9 − 6.34170e8i 1.41917 − 0.142926i
\(86\) −4.00181e8 − 1.30961e9i −0.0850675 − 0.278388i
\(87\) −9.34394e9 + 9.34394e9i −1.87471 + 1.87471i
\(88\) −1.79687e8 − 1.71683e9i −0.0340489 − 0.325323i
\(89\) − 6.41481e9i − 1.14877i −0.818585 − 0.574386i \(-0.805241\pi\)
0.818585 − 0.574386i \(-0.194759\pi\)
\(90\) −2.85041e9 − 6.79993e9i −0.482719 − 1.15157i
\(91\) 1.33947e9i 0.214648i
\(92\) −9.95662e8 + 5.11428e9i −0.151068 + 0.775970i
\(93\) −9.24396e9 + 9.24396e9i −1.32875 + 1.32875i
\(94\) 3.63586e9 + 1.18986e10i 0.495414 + 1.62127i
\(95\) 2.62746e9 + 2.14665e9i 0.339561 + 0.277424i
\(96\) −1.21796e10 − 1.07453e9i −1.49375 − 0.131784i
\(97\) 6.09122e9 − 6.09122e9i 0.709325 − 0.709325i −0.257068 − 0.966393i \(-0.582756\pi\)
0.966393 + 0.257068i \(0.0827565\pi\)
\(98\) −7.85050e9 − 4.17566e9i −0.868495 − 0.461949i
\(99\) −3.88417e9 −0.408434
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 40.11.i.a.13.20 yes 116
5.2 odd 4 inner 40.11.i.a.37.9 yes 116
8.5 even 2 inner 40.11.i.a.13.9 ✓ 116
40.37 odd 4 inner 40.11.i.a.37.20 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.9 ✓ 116 8.5 even 2 inner
40.11.i.a.13.20 yes 116 1.1 even 1 trivial
40.11.i.a.37.9 yes 116 5.2 odd 4 inner
40.11.i.a.37.20 yes 116 40.37 odd 4 inner