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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 13.18
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.18

$q$-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+(-18.0006 - 26.4571i) q^{2} +(-159.283 + 159.283i) q^{3} +(-375.958 + 952.487i) q^{4} +(3109.51 + 310.741i) q^{5} +(7081.36 + 1346.98i) q^{6} +(-794.929 + 794.929i) q^{7} +(31967.5 - 7198.58i) q^{8} +8306.77i q^{9} +(-47751.7 - 87862.2i) q^{10} -91848.8i q^{11} +(-91831.4 - 211599. i) q^{12} +(224589. - 224589. i) q^{13} +(35340.7 + 6722.34i) q^{14} +(-544789. + 445797. i) q^{15} +(-765888. - 716190. i) q^{16} +(-804523. + 804523. i) q^{17} +(219773. - 149527. i) q^{18} -1.00673e6 q^{19} +(-1.46502e6 + 2.84494e6i) q^{20} -253238. i q^{21} +(-2.43005e6 + 1.65333e6i) q^{22} +(5.01535e6 + 5.01535e6i) q^{23} +(-3.94528e6 + 6.23850e6i) q^{24} +(9.57250e6 + 1.93251e6i) q^{25} +(-9.98472e6 - 1.89924e6i) q^{26} +(-1.07286e7 - 1.07286e7i) q^{27} +(-458300. - 1.05602e6i) q^{28} +2.38485e7 q^{29} +(2.16010e7 + 6.38893e6i) q^{30} +3.19606e6 q^{31} +(-5.16188e6 + 3.31550e7i) q^{32} +(1.46300e7 + 1.46300e7i) q^{33} +(3.57672e7 + 6.80347e6i) q^{34} +(-2.71886e6 + 2.22482e6i) q^{35} +(-7.91209e6 - 3.12299e6i) q^{36} +(7.87953e7 + 7.87953e7i) q^{37} +(1.81217e7 + 2.66352e7i) q^{38} +7.15466e7i q^{39} +(1.01640e8 - 1.24504e7i) q^{40} -6.83894e7 q^{41} +(-6.69994e6 + 4.55843e6i) q^{42} +(-6.09026e7 + 6.09026e7i) q^{43} +(8.74848e7 + 3.45312e7i) q^{44} +(-2.58126e6 + 2.58300e7i) q^{45} +(4.24124e7 - 2.22971e8i) q^{46} +(-1.96360e8 + 1.96360e8i) q^{47} +(2.36070e8 - 7.91605e6i) q^{48} +2.81211e8i q^{49} +(-1.21182e8 - 2.88047e8i) q^{50} -2.56294e8i q^{51} +(1.29482e8 + 2.98354e8i) q^{52} +(-4.23378e8 + 4.23378e8i) q^{53} +(-9.07270e7 + 4.76971e8i) q^{54} +(2.85412e7 - 2.85605e8i) q^{55} +(-1.96896e7 + 3.11343e7i) q^{56} +(1.60355e8 - 1.60355e8i) q^{57} +(-4.29287e8 - 6.30963e8i) q^{58} -8.88892e8 q^{59} +(-2.19798e8 - 6.86505e8i) q^{60} +3.74674e8i q^{61} +(-5.75310e7 - 8.45586e7i) q^{62} +(-6.60329e6 - 6.60329e6i) q^{63} +(9.70103e8 - 4.60241e8i) q^{64} +(7.68152e8 - 6.28574e8i) q^{65} +(1.23719e8 - 6.50414e8i) q^{66} +(3.34479e8 + 3.34479e8i) q^{67} +(-4.63831e8 - 1.06876e9i) q^{68} -1.59772e9 q^{69} +(1.07803e8 + 3.18850e7i) q^{70} +1.49503e9 q^{71} +(5.97969e7 + 2.65547e8i) q^{72} +(7.08288e8 + 7.08288e8i) q^{73} +(6.66334e8 - 3.50306e9i) q^{74} +(-1.83255e9 + 1.21692e9i) q^{75} +(3.78488e8 - 9.58897e8i) q^{76} +(7.30132e7 + 7.30132e7i) q^{77} +(1.89292e9 - 1.28788e9i) q^{78} -1.47005e9i q^{79} +(-2.15899e9 - 2.46499e9i) q^{80} +2.92728e9 q^{81} +(1.23105e9 + 1.80939e9i) q^{82} +(-4.12465e9 + 4.12465e9i) q^{83} +(2.41206e8 + 9.52066e7i) q^{84} +(-2.75167e9 + 2.25168e9i) q^{85} +(2.70759e9 + 5.15024e8i) q^{86} +(-3.79867e9 + 3.79867e9i) q^{87} +(-6.61180e8 - 2.93618e9i) q^{88} +7.21320e8i q^{89} +(7.29851e8 - 3.96663e8i) q^{90} +3.57065e8i q^{91} +(-6.66262e9 + 2.89150e9i) q^{92} +(-5.09079e8 + 5.09079e8i) q^{93} +(8.72973e9 + 1.66053e9i) q^{94} +(-3.13044e9 - 3.12832e8i) q^{95} +(-4.45883e9 - 6.10324e9i) q^{96} +(-2.64968e9 + 2.64968e9i) q^{97} +(7.44004e9 - 5.06197e9i) q^{98} +7.62967e8 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −18.0006 − 26.4571i −0.562518 − 0.826785i
\(3\) −159.283 + 159.283i −0.655486 + 0.655486i −0.954309 − 0.298823i \(-0.903406\pi\)
0.298823 + 0.954309i \(0.403406\pi\)
\(4\) −375.958 + 952.487i −0.367146 + 0.930163i
\(5\) 3109.51 + 310.741i 0.995044 + 0.0994372i
\(6\) 7081.36 + 1346.98i 0.910669 + 0.173223i
\(7\) −794.929 + 794.929i −0.0472975 + 0.0472975i −0.730360 − 0.683062i \(-0.760648\pi\)
0.683062 + 0.730360i \(0.260648\pi\)
\(8\) 31967.5 − 7198.58i 0.975571 − 0.219683i
\(9\) 8306.77i 0.140676i
\(10\) −47751.7 − 87862.2i −0.477517 − 0.878622i
\(11\) − 91848.8i − 0.570308i −0.958482 − 0.285154i \(-0.907955\pi\)
0.958482 − 0.285154i \(-0.0920448\pi\)
\(12\) −91831.4 − 211599.i −0.369050 − 0.850368i
\(13\) 224589. − 224589.i 0.604884 − 0.604884i −0.336720 − 0.941605i \(-0.609318\pi\)
0.941605 + 0.336720i \(0.109318\pi\)
\(14\) 35340.7 + 6722.34i 0.0657106 + 0.0124991i
\(15\) −544789. + 445797.i −0.717417 + 0.587058i
\(16\) −765888. − 716190.i −0.730407 − 0.683012i
\(17\) −804523. + 804523.i −0.566623 + 0.566623i −0.931181 − 0.364558i \(-0.881220\pi\)
0.364558 + 0.931181i \(0.381220\pi\)
\(18\) 219773. − 149527.i 0.116309 − 0.0791328i
\(19\) −1.00673e6 −0.406579 −0.203289 − 0.979119i \(-0.565163\pi\)
−0.203289 + 0.979119i \(0.565163\pi\)
\(20\) −1.46502e6 + 2.84494e6i −0.457819 + 0.889045i
\(21\) − 253238.i − 0.0620057i
\(22\) −2.43005e6 + 1.65333e6i −0.471522 + 0.320809i
\(23\) 5.01535e6 + 5.01535e6i 0.779224 + 0.779224i 0.979699 − 0.200475i \(-0.0642486\pi\)
−0.200475 + 0.979699i \(0.564249\pi\)
\(24\) −3.94528e6 + 6.23850e6i −0.495474 + 0.783473i
\(25\) 9.57250e6 + 1.93251e6i 0.980224 + 0.197889i
\(26\) −9.98472e6 − 1.89924e6i −0.840368 − 0.159851i
\(27\) −1.07286e7 − 1.07286e7i −0.747697 − 0.747697i
\(28\) −458300. − 1.05602e6i −0.0266293 − 0.0613595i
\(29\) 2.38485e7 1.16271 0.581355 − 0.813650i \(-0.302523\pi\)
0.581355 + 0.813650i \(0.302523\pi\)
\(30\) 2.16010e7 + 6.38893e6i 0.888931 + 0.262919i
\(31\) 3.19606e6 0.111637 0.0558183 − 0.998441i \(-0.482223\pi\)
0.0558183 + 0.998441i \(0.482223\pi\)
\(32\) −5.16188e6 + 3.31550e7i −0.153836 + 0.988096i
\(33\) 1.46300e7 + 1.46300e7i 0.373829 + 0.373829i
\(34\) 3.57672e7 + 6.80347e6i 0.787211 + 0.149739i
\(35\) −2.71886e6 + 2.22482e6i −0.0517662 + 0.0423600i
\(36\) −7.91209e6 − 3.12299e6i −0.130852 − 0.0516486i
\(37\) 7.87953e7 + 7.87953e7i 1.13630 + 1.13630i 0.989109 + 0.147187i \(0.0470220\pi\)
0.147187 + 0.989109i \(0.452978\pi\)
\(38\) 1.81217e7 + 2.66352e7i 0.228708 + 0.336153i
\(39\) 7.15466e7i 0.792986i
\(40\) 1.01640e8 − 1.24504e7i 0.992581 − 0.121586i
\(41\) −6.83894e7 −0.590296 −0.295148 − 0.955452i \(-0.595369\pi\)
−0.295148 + 0.955452i \(0.595369\pi\)
\(42\) −6.69994e6 + 4.55843e6i −0.0512654 + 0.0348794i
\(43\) −6.09026e7 + 6.09026e7i −0.414279 + 0.414279i −0.883226 − 0.468947i \(-0.844634\pi\)
0.468947 + 0.883226i \(0.344634\pi\)
\(44\) 8.74848e7 + 3.45312e7i 0.530480 + 0.209387i
\(45\) −2.58126e6 + 2.58300e7i −0.0139884 + 0.139979i
\(46\) 4.24124e7 − 2.22971e8i 0.205923 − 1.08258i
\(47\) −1.96360e8 + 1.96360e8i −0.856179 + 0.856179i −0.990886 − 0.134707i \(-0.956991\pi\)
0.134707 + 0.990886i \(0.456991\pi\)
\(48\) 2.36070e8 − 7.91605e6i 0.926477 − 0.0310672i
\(49\) 2.81211e8i 0.995526i
\(50\) −1.21182e8 − 2.88047e8i −0.387783 − 0.921751i
\(51\) − 2.56294e8i − 0.742827i
\(52\) 1.29482e8 + 2.98354e8i 0.340560 + 0.784722i
\(53\) −4.23378e8 + 4.23378e8i −1.01239 + 1.01239i −0.0124715 + 0.999922i \(0.503970\pi\)
−0.999922 + 0.0124715i \(0.996030\pi\)
\(54\) −9.07270e7 + 4.76971e8i −0.197591 + 1.03878i
\(55\) 2.85412e7 − 2.85605e8i 0.0567099 − 0.567482i
\(56\) −1.96896e7 + 3.11343e7i −0.0357516 + 0.0565325i
\(57\) 1.60355e8 − 1.60355e8i 0.266507 − 0.266507i
\(58\) −4.29287e8 − 6.30963e8i −0.654046 − 0.961311i
\(59\) −8.88892e8 −1.24334 −0.621669 − 0.783280i \(-0.713545\pi\)
−0.621669 + 0.783280i \(0.713545\pi\)
\(60\) −2.19798e8 − 6.86505e8i −0.282663 − 0.882851i
\(61\) 3.74674e8i 0.443613i 0.975091 + 0.221807i \(0.0711954\pi\)
−0.975091 + 0.221807i \(0.928805\pi\)
\(62\) −5.75310e7 − 8.45586e7i −0.0627977 − 0.0922995i
\(63\) −6.60329e6 − 6.60329e6i −0.00665362 − 0.00665362i
\(64\) 9.70103e8 − 4.60241e8i 0.903479 − 0.428633i
\(65\) 7.68152e8 − 6.28574e8i 0.662034 − 0.541738i
\(66\) 1.23719e8 − 6.50414e8i 0.0987905 − 0.519362i
\(67\) 3.34479e8 + 3.34479e8i 0.247739 + 0.247739i 0.820042 − 0.572303i \(-0.193950\pi\)
−0.572303 + 0.820042i \(0.693950\pi\)
\(68\) −4.63831e8 − 1.06876e9i −0.319018 − 0.735085i
\(69\) −1.59772e9 −1.02154
\(70\) 1.07803e8 + 3.18850e7i 0.0641420 + 0.0189713i
\(71\) 1.49503e9 0.828625 0.414313 − 0.910135i \(-0.364022\pi\)
0.414313 + 0.910135i \(0.364022\pi\)
\(72\) 5.97969e7 + 2.65547e8i 0.0309041 + 0.137239i
\(73\) 7.08288e8 + 7.08288e8i 0.341661 + 0.341661i 0.856992 − 0.515330i \(-0.172331\pi\)
−0.515330 + 0.856992i \(0.672331\pi\)
\(74\) 6.66334e8 − 3.50306e9i 0.300285 − 1.57866i
\(75\) −1.83255e9 + 1.21692e9i −0.772237 + 0.512810i
\(76\) 3.78488e8 − 9.58897e8i 0.149274 − 0.378185i
\(77\) 7.30132e7 + 7.30132e7i 0.0269742 + 0.0269742i
\(78\) 1.89292e9 − 1.28788e9i 0.655629 − 0.446069i
\(79\) − 1.47005e9i − 0.477747i −0.971051 − 0.238873i \(-0.923222\pi\)
0.971051 − 0.238873i \(-0.0767781\pi\)
\(80\) −2.15899e9 − 2.46499e9i −0.658871 − 0.752256i
\(81\) 2.92728e9 0.839534
\(82\) 1.23105e9 + 1.80939e9i 0.332052 + 0.488048i
\(83\) −4.12465e9 + 4.12465e9i −1.04712 + 1.04712i −0.0482858 + 0.998834i \(0.515376\pi\)
−0.998834 + 0.0482858i \(0.984624\pi\)
\(84\) 2.41206e8 + 9.52066e7i 0.0576754 + 0.0227652i
\(85\) −2.75167e9 + 2.25168e9i −0.620158 + 0.507471i
\(86\) 2.70759e9 + 5.15024e8i 0.575560 + 0.109480i
\(87\) −3.79867e9 + 3.79867e9i −0.762140 + 0.762140i
\(88\) −6.61180e8 − 2.93618e9i −0.125287 − 0.556377i
\(89\) 7.21320e8i 0.129175i 0.997912 + 0.0645874i \(0.0205731\pi\)
−0.997912 + 0.0645874i \(0.979427\pi\)
\(90\) 7.29851e8 − 3.96663e8i 0.123601 − 0.0671752i
\(91\) 3.57065e8i 0.0572190i
\(92\) −6.66262e9 + 2.89150e9i −1.01089 + 0.438716i
\(93\) −5.09079e8 + 5.09079e8i −0.0731763 + 0.0731763i
\(94\) 8.72973e9 + 1.66053e9i 1.18949 + 0.226259i
\(95\) −3.13044e9 − 3.12832e8i −0.404564 − 0.0404291i
\(96\) −4.45883e9 − 6.10324e9i −0.546846 − 0.748521i
\(97\) −2.64968e9 + 2.64968e9i −0.308556 + 0.308556i −0.844349 − 0.535793i \(-0.820013\pi\)
0.535793 + 0.844349i \(0.320013\pi\)
\(98\) 7.44004e9 − 5.06197e9i 0.823086 − 0.560002i
\(99\) 7.62967e8 0.0802287
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.13.18 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.48 yes 116
8.5 even 2 inner 40.11.i.a.13.48 yes 116
40.37 odd 4 inner 40.11.i.a.37.18 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.18 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.48 yes 116 8.5 even 2 inner
40.11.i.a.37.18 yes 116 40.37 odd 4 inner
40.11.i.a.37.48 yes 116 5.2 odd 4 inner