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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 13.48
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.48

$q$-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+(26.4571 + 18.0006i) 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} +(-7198.58 + 31967.5i) q^{8} +8306.77i q^{9} +(-76675.2 - 64194.4i) q^{10} +91848.8i q^{11} +(211599. + 91831.4i) 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} +(-149527. + 219773. i) q^{18} +1.00673e6 q^{19} +(-873068. - 3.07860e6i) q^{20} +253238. i q^{21} +(-1.65333e6 + 2.43005e6i) 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} +(-1.05602e6 - 458300. i) q^{28} -2.38485e7 q^{29} +(-2.24381e7 + 1.98798e6i) q^{30} +3.19606e6 q^{31} +(-3.31550e7 + 5.16188e6i) 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} +(2.66352e7 + 1.81217e7i) q^{38} +7.15466e7i q^{39} +(3.23177e7 - 9.71665e7i) q^{40} -6.83894e7 q^{41} +(-4.55843e6 + 6.69994e6i) 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} +(-7.91605e6 + 2.36070e8i) q^{48} +2.81211e8i q^{49} +(2.18475e8 + 2.23439e8i) q^{50} +2.56294e8i q^{51} +(-2.98354e8 - 1.29482e8i) 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} +(-6.30963e8 - 4.29287e8i) q^{58} +8.88892e8 q^{59} +(-6.29433e8 - 3.51303e8i) q^{60} -3.74674e8i q^{61} +(8.45586e7 + 5.75310e7i) 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} +(-1.06876e9 - 4.63831e8i) q^{68} +1.59772e9 q^{69} +(1.11981e8 - 9.92137e6i) q^{70} +1.49503e9 q^{71} +(-2.65547e8 - 5.97969e7i) 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} +(-1.28788e9 + 1.89292e9i) q^{78} -1.47005e9i q^{79} +(2.60409e9 - 1.98901e9i) q^{80} +2.92728e9 q^{81} +(-1.80939e9 - 1.23105e9i) 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} +(-2.93618e9 - 6.61180e8i) q^{88} +7.21320e8i q^{89} +(5.33248e8 - 6.36923e8i) q^{90} -3.57065e8i q^{91} +(-2.89150e9 + 6.66262e9i) 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} +(-5.06197e9 + 7.44004e9i) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 26.4571 + 18.0006i 0.826785 + 0.562518i
\(3\) 159.283 − 159.283i 0.655486 − 0.655486i −0.298823 − 0.954309i \(-0.596594\pi\)
0.954309 + 0.298823i \(0.0965939\pi\)
\(4\) 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 \(0.260648\pi\)
\(8\) −7198.58 + 31967.5i −0.219683 + 0.975571i
\(9\) 8306.77i 0.140676i
\(10\) −76675.2 − 64194.4i −0.766752 − 0.641944i
\(11\) 91848.8i 0.570308i 0.958482 + 0.285154i \(0.0920448\pi\)
−0.958482 + 0.285154i \(0.907955\pi\)
\(12\) 211599. + 91831.4i 0.850368 + 0.369050i
\(13\) −224589. + 224589.i −0.604884 + 0.604884i −0.941605 − 0.336720i \(-0.890682\pi\)
0.336720 + 0.941605i \(0.390682\pi\)
\(14\) −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 \(0.381220\pi\)
\(18\) −149527. + 219773.i −0.0791328 + 0.116309i
\(19\) 1.00673e6 0.406579 0.203289 − 0.979119i \(-0.434837\pi\)
0.203289 + 0.979119i \(0.434837\pi\)
\(20\) −873068. − 3.07860e6i −0.272834 − 0.962061i
\(21\) 253238.i 0.0620057i
\(22\) −1.65333e6 + 2.43005e6i −0.320809 + 0.471522i
\(23\) 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
\(28\) −1.05602e6 − 458300.i −0.0613595 − 0.0266293i
\(29\) −2.38485e7 −1.16271 −0.581355 − 0.813650i \(-0.697477\pi\)
−0.581355 + 0.813650i \(0.697477\pi\)
\(30\) −2.24381e7 + 1.98798e6i −0.923380 + 0.0818101i
\(31\) 3.19606e6 0.111637 0.0558183 − 0.998441i \(-0.482223\pi\)
0.0558183 + 0.998441i \(0.482223\pi\)
\(32\) −3.31550e7 + 5.16188e6i −0.988096 + 0.153836i
\(33\) 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 \(-0.952978\pi\)
−0.147187 − 0.989109i \(-0.547022\pi\)
\(38\) 2.66352e7 + 1.81217e7i 0.336153 + 0.228708i
\(39\) 7.15466e7i 0.792986i
\(40\) 3.23177e7 − 9.71665e7i 0.315602 − 0.948892i
\(41\) −6.83894e7 −0.590296 −0.295148 − 0.955452i \(-0.595369\pi\)
−0.295148 + 0.955452i \(0.595369\pi\)
\(42\) −4.55843e6 + 6.69994e6i −0.0348794 + 0.0512654i
\(43\) 6.09026e7 − 6.09026e7i 0.414279 − 0.414279i −0.468947 − 0.883226i \(-0.655366\pi\)
0.883226 + 0.468947i \(0.155366\pi\)
\(44\) −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 \(0.456991\pi\)
\(48\) −7.91605e6 + 2.36070e8i −0.0310672 + 0.926477i
\(49\) 2.81211e8i 0.995526i
\(50\) 2.18475e8 + 2.23439e8i 0.699119 + 0.715006i
\(51\) 2.56294e8i 0.742827i
\(52\) −2.98354e8 − 1.29482e8i −0.784722 − 0.340560i
\(53\) 4.23378e8 − 4.23378e8i 1.01239 − 1.01239i 0.0124715 − 0.999922i \(-0.496030\pi\)
0.999922 − 0.0124715i \(-0.00396990\pi\)
\(54\) 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
\(58\) −6.30963e8 − 4.29287e8i −0.961311 − 0.654046i
\(59\) 8.88892e8 1.24334 0.621669 − 0.783280i \(-0.286455\pi\)
0.621669 + 0.783280i \(0.286455\pi\)
\(60\) −6.29433e8 − 3.51303e8i −0.809456 − 0.451779i
\(61\) − 3.74674e8i − 0.443613i −0.975091 − 0.221807i \(-0.928805\pi\)
0.975091 − 0.221807i \(-0.0711954\pi\)
\(62\) 8.45586e7 + 5.75310e7i 0.0922995 + 0.0627977i
\(63\) −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 0.572303 − 0.820042i \(-0.306050\pi\)
−0.820042 + 0.572303i \(0.806050\pi\)
\(68\) −1.06876e9 − 4.63831e8i −0.735085 − 0.319018i
\(69\) 1.59772e9 1.02154
\(70\) 1.11981e8 − 9.92137e6i 0.0666278 − 0.00590312i
\(71\) 1.49503e9 0.828625 0.414313 − 0.910135i \(-0.364022\pi\)
0.414313 + 0.910135i \(0.364022\pi\)
\(72\) −2.65547e8 − 5.97969e7i −0.137239 − 0.0309041i
\(73\) 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
\(78\) −1.28788e9 + 1.89292e9i −0.446069 + 0.655629i
\(79\) − 1.47005e9i − 0.477747i −0.971051 − 0.238873i \(-0.923222\pi\)
0.971051 − 0.238873i \(-0.0767781\pi\)
\(80\) 2.60409e9 − 1.98901e9i 0.794704 − 0.606997i
\(81\) 2.92728e9 0.839534
\(82\) −1.80939e9 − 1.23105e9i −0.488048 − 0.332052i
\(83\) 4.12465e9 − 4.12465e9i 1.04712 − 1.04712i 0.0482858 − 0.998834i \(-0.484624\pi\)
0.998834 − 0.0482858i \(-0.0153758\pi\)
\(84\) −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
\(88\) −2.93618e9 − 6.61180e8i −0.556377 − 0.125287i
\(89\) 7.21320e8i 0.129175i 0.997912 + 0.0645874i \(0.0205731\pi\)
−0.997912 + 0.0645874i \(0.979427\pi\)
\(90\) 5.33248e8 − 6.36923e8i 0.0903060 − 0.107864i
\(91\) − 3.57065e8i − 0.0572190i
\(92\) −2.89150e9 + 6.66262e9i −0.438716 + 1.01089i
\(93\) 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 \(0.320013\pi\)
\(98\) −5.06197e9 + 7.44004e9i −0.560002 + 0.823086i
\(99\) −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.13.48 yes 116
5.2 odd 4 inner 40.11.i.a.37.18 yes 116
8.5 even 2 inner 40.11.i.a.13.18 ✓ 116
40.37 odd 4 inner 40.11.i.a.37.48 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.18 ✓ 116 8.5 even 2 inner
40.11.i.a.13.48 yes 116 1.1 even 1 trivial
40.11.i.a.37.18 yes 116 5.2 odd 4 inner
40.11.i.a.37.48 yes 116 40.37 odd 4 inner