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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 37.31
Character \(\chi\) \(=\) 40.37
Dual form 40.11.i.a.13.31

$q$-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+(2.70791 - 31.8852i) q^{2} +(181.420 + 181.420i) q^{3} +(-1009.33 - 172.685i) q^{4} +(-1980.27 - 2417.47i) q^{5} +(6275.90 - 5293.36i) q^{6} +(16061.1 + 16061.1i) q^{7} +(-8239.28 + 31715.2i) q^{8} +6777.67i q^{9} +(-82443.9 + 56595.1i) q^{10} +9369.49i q^{11} +(-151785. - 214442. i) q^{12} +(44722.8 + 44722.8i) q^{13} +(555602. - 468618. i) q^{14} +(79316.0 - 797839. i) q^{15} +(988936. + 348593. i) q^{16} +(409687. + 409687. i) q^{17} +(216108. + 18353.4i) q^{18} +2.36738e6 q^{19} +(1.58130e6 + 2.78200e6i) q^{20} +5.82760e6i q^{21} +(298748. + 25371.8i) q^{22} +(2.24613e6 - 2.24613e6i) q^{23} +(-7.24856e6 + 4.25901e6i) q^{24} +(-1.92267e6 + 9.57449e6i) q^{25} +(1.54710e6 - 1.30489e6i) q^{26} +(9.48308e6 - 9.48308e6i) q^{27} +(-1.34375e7 - 1.89845e7i) q^{28} +2.96608e7 q^{29} +(-2.52245e7 - 4.68949e6i) q^{30} +3.83325e7 q^{31} +(1.37929e7 - 3.05885e7i) q^{32} +(-1.69982e6 + 1.69982e6i) q^{33} +(1.41724e7 - 1.19536e7i) q^{34} +(7.02181e6 - 7.06323e7i) q^{35} +(1.17040e6 - 6.84094e6i) q^{36} +(1.26503e7 - 1.26503e7i) q^{37} +(6.41065e6 - 7.54843e7i) q^{38} +1.62272e7i q^{39} +(9.29866e7 - 4.28866e7i) q^{40} -7.38613e7 q^{41} +(1.85814e8 + 1.57806e7i) q^{42} +(-1.96061e8 - 1.96061e8i) q^{43} +(1.61797e6 - 9.45695e6i) q^{44} +(1.63848e7 - 1.34216e7i) q^{45} +(-6.55359e7 - 7.77005e7i) q^{46} +(2.72087e8 + 2.72087e8i) q^{47} +(1.16171e8 + 2.42655e8i) q^{48} +2.33439e8i q^{49} +(3.00078e8 + 8.72316e7i) q^{50} +1.48651e8i q^{51} +(-3.74173e7 - 5.28632e7i) q^{52} +(-2.93142e7 - 2.93142e7i) q^{53} +(-2.76691e8 - 3.28049e8i) q^{54} +(2.26504e7 - 1.85541e7i) q^{55} +(-6.41711e8 + 3.77048e8i) q^{56} +(4.29490e8 + 4.29490e8i) q^{57} +(8.03190e7 - 9.45742e8i) q^{58} +6.79472e8 q^{59} +(-2.17831e8 + 7.91590e8i) q^{60} +1.01534e9i q^{61} +(1.03801e8 - 1.22224e9i) q^{62} +(-1.08857e8 + 1.08857e8i) q^{63} +(-9.37970e8 - 5.22622e8i) q^{64} +(1.95526e7 - 1.96679e8i) q^{65} +(4.95960e7 + 5.88019e7i) q^{66} +(3.41798e8 - 3.41798e8i) q^{67} +(-3.42765e8 - 4.84258e8i) q^{68} +8.14986e8 q^{69} +(-2.23311e9 - 4.15158e8i) q^{70} -6.24796e8 q^{71} +(-2.14956e8 - 5.58432e7i) q^{72} +(-2.00067e9 + 2.00067e9i) q^{73} +(-3.69103e8 - 4.37615e8i) q^{74} +(-2.08582e9 + 1.38820e9i) q^{75} +(-2.38947e9 - 4.08810e8i) q^{76} +(-1.50484e8 + 1.50484e8i) q^{77} +(5.17409e8 + 4.39420e7i) q^{78} +5.65597e9i q^{79} +(-1.11565e9 - 3.08103e9i) q^{80} +3.84106e9 q^{81} +(-2.00010e8 + 2.35508e9i) q^{82} +(-1.18232e9 - 1.18232e9i) q^{83} +(1.00634e9 - 5.88200e9i) q^{84} +(1.79113e8 - 1.80170e9i) q^{85} +(-6.78237e9 + 5.72054e9i) q^{86} +(5.38108e9 + 5.38108e9i) q^{87} +(-2.97156e8 - 7.71979e7i) q^{88} -1.86976e9i q^{89} +(-3.83583e8 - 5.58778e8i) q^{90} +1.43659e9i q^{91} +(-2.65496e9 + 1.87922e9i) q^{92} +(6.95429e9 + 6.95429e9i) q^{93} +(9.41235e9 - 7.93877e9i) q^{94} +(-4.68805e9 - 5.72305e9i) q^{95} +(8.05169e9 - 3.04705e9i) q^{96} +(4.01722e9 + 4.01722e9i) q^{97} +(7.44327e9 + 6.32134e8i) q^{98} -6.35033e7 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 \(\left(\mathbb{Z}/40\mathbb{Z}\right)^\times\).

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

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.70791 − 31.8852i 0.0846223 − 0.996413i
\(3\) 181.420 + 181.420i 0.746586 + 0.746586i 0.973836 − 0.227251i \(-0.0729736\pi\)
−0.227251 + 0.973836i \(0.572974\pi\)
\(4\) −1009.33 − 172.685i −0.985678 − 0.168638i
\(5\) −1980.27 − 2417.47i −0.633687 − 0.773589i
\(6\) 6275.90 − 5293.36i 0.807086 − 0.680730i
\(7\) 16061.1 + 16061.1i 0.955617 + 0.955617i 0.999056 − 0.0434393i \(-0.0138315\pi\)
−0.0434393 + 0.999056i \(0.513832\pi\)
\(8\) −8239.28 + 31715.2i −0.251443 + 0.967872i
\(9\) 6777.67i 0.114781i
\(10\) −82443.9 + 56595.1i −0.824439 + 0.565951i
\(11\) 9369.49i 0.0581771i 0.999577 + 0.0290886i \(0.00926049\pi\)
−0.999577 + 0.0290886i \(0.990740\pi\)
\(12\) −151785. − 214442.i −0.609991 − 0.861796i
\(13\) 44722.8 + 44722.8i 0.120451 + 0.120451i 0.764763 − 0.644312i \(-0.222856\pi\)
−0.644312 + 0.764763i \(0.722856\pi\)
\(14\) 555602. − 468618.i 1.03306 − 0.871323i
\(15\) 79316.0 − 797839.i 0.104449 − 1.05065i
\(16\) 988936. + 348593.i 0.943123 + 0.332445i
\(17\) 409687. + 409687.i 0.288541 + 0.288541i 0.836503 − 0.547962i \(-0.184596\pi\)
−0.547962 + 0.836503i \(0.684596\pi\)
\(18\) 216108. + 18353.4i 0.114369 + 0.00971299i
\(19\) 2.36738e6 0.956091 0.478046 − 0.878335i \(-0.341345\pi\)
0.478046 + 0.878335i \(0.341345\pi\)
\(20\) 1.58130e6 + 2.78200e6i 0.494155 + 0.869374i
\(21\) 5.82760e6i 1.42690i
\(22\) 298748. + 25371.8i 0.0579685 + 0.00492308i
\(23\) 2.24613e6 − 2.24613e6i 0.348976 − 0.348976i −0.510752 − 0.859728i \(-0.670633\pi\)
0.859728 + 0.510752i \(0.170633\pi\)
\(24\) −7.24856e6 + 4.25901e6i −0.910323 + 0.534876i
\(25\) −1.92267e6 + 9.57449e6i −0.196881 + 0.980427i
\(26\) 1.54710e6 − 1.30489e6i 0.130212 − 0.109827i
\(27\) 9.48308e6 − 9.48308e6i 0.660892 − 0.660892i
\(28\) −1.34375e7 − 1.89845e7i −0.780778 − 1.10308i
\(29\) 2.96608e7 1.44608 0.723042 − 0.690804i \(-0.242743\pi\)
0.723042 + 0.690804i \(0.242743\pi\)
\(30\) −2.52245e7 − 4.68949e6i −1.03805 − 0.192983i
\(31\) 3.83325e7 1.33893 0.669466 − 0.742843i \(-0.266523\pi\)
0.669466 + 0.742843i \(0.266523\pi\)
\(32\) 1.37929e7 − 3.05885e7i 0.411061 − 0.911608i
\(33\) −1.69982e6 + 1.69982e6i −0.0434342 + 0.0434342i
\(34\) 1.41724e7 − 1.19536e7i 0.311923 − 0.263089i
\(35\) 7.02181e6 − 7.06323e7i 0.133693 − 1.34482i
\(36\) 1.17040e6 − 6.84094e6i 0.0193563 − 0.113137i
\(37\) 1.26503e7 − 1.26503e7i 0.182429 − 0.182429i −0.609984 − 0.792413i \(-0.708824\pi\)
0.792413 + 0.609984i \(0.208824\pi\)
\(38\) 6.41065e6 − 7.54843e7i 0.0809066 − 0.952662i
\(39\) 1.62272e7i 0.179855i
\(40\) 9.29866e7 − 4.28866e7i 0.908072 − 0.418814i
\(41\) −7.38613e7 −0.637525 −0.318763 − 0.947835i \(-0.603267\pi\)
−0.318763 + 0.947835i \(0.603267\pi\)
\(42\) 1.85814e8 + 1.57806e7i 1.42178 + 0.120748i
\(43\) −1.96061e8 − 1.96061e8i −1.33367 − 1.33367i −0.902058 − 0.431614i \(-0.857944\pi\)
−0.431614 − 0.902058i \(-0.642056\pi\)
\(44\) 1.61797e6 − 9.45695e6i 0.00981085 − 0.0573439i
\(45\) 1.63848e7 − 1.34216e7i 0.0887930 − 0.0727349i
\(46\) −6.55359e7 − 7.77005e7i −0.318193 − 0.377255i
\(47\) 2.72087e8 + 2.72087e8i 1.18637 + 1.18637i 0.978065 + 0.208302i \(0.0667938\pi\)
0.208302 + 0.978065i \(0.433206\pi\)
\(48\) 1.16171e8 + 2.42655e8i 0.455924 + 0.952320i
\(49\) 2.33439e8i 0.826407i
\(50\) 3.00078e8 + 8.72316e7i 0.960250 + 0.279141i
\(51\) 1.48651e8i 0.430842i
\(52\) −3.74173e7 − 5.28632e7i −0.0984137 − 0.139039i
\(53\) −2.93142e7 − 2.93142e7i −0.0700969 − 0.0700969i 0.671189 − 0.741286i \(-0.265784\pi\)
−0.741286 + 0.671189i \(0.765784\pi\)
\(54\) −2.76691e8 − 3.28049e8i −0.602595 − 0.714448i
\(55\) 2.26504e7 − 1.85541e7i 0.0450052 − 0.0368661i
\(56\) −6.41711e8 + 3.77048e8i −1.16520 + 0.684632i
\(57\) 4.29490e8 + 4.29490e8i 0.713804 + 0.713804i
\(58\) 8.03190e7 − 9.45742e8i 0.122371 − 1.44090i
\(59\) 6.79472e8 0.950411 0.475206 − 0.879875i \(-0.342374\pi\)
0.475206 + 0.879875i \(0.342374\pi\)
\(60\) −2.17831e8 + 7.91590e8i −0.280133 + 1.01799i
\(61\) 1.01534e9i 1.20217i 0.799187 + 0.601083i \(0.205264\pi\)
−0.799187 + 0.601083i \(0.794736\pi\)
\(62\) 1.03801e8 − 1.22224e9i 0.113304 − 1.33413i
\(63\) −1.08857e8 + 1.08857e8i −0.109686 + 0.109686i
\(64\) −9.37970e8 − 5.22622e8i −0.873553 − 0.486729i
\(65\) 1.95526e7 − 1.96679e8i 0.0168514 − 0.169508i
\(66\) 4.95960e7 + 5.88019e7i 0.0396029 + 0.0469539i
\(67\) 3.41798e8 − 3.41798e8i 0.253160 − 0.253160i −0.569105 − 0.822265i \(-0.692710\pi\)
0.822265 + 0.569105i \(0.192710\pi\)
\(68\) −3.42765e8 − 4.84258e8i −0.235750 − 0.333068i
\(69\) 8.14986e8 0.521080
\(70\) −2.23311e9 − 4.15158e8i −1.32868 − 0.247015i
\(71\) −6.24796e8 −0.346295 −0.173148 − 0.984896i \(-0.555394\pi\)
−0.173148 + 0.984896i \(0.555394\pi\)
\(72\) −2.14956e8 − 5.58432e7i −0.111093 − 0.0288608i
\(73\) −2.00067e9 + 2.00067e9i −0.965075 + 0.965075i −0.999410 − 0.0343352i \(-0.989069\pi\)
0.0343352 + 0.999410i \(0.489069\pi\)
\(74\) −3.69103e8 − 4.37615e8i −0.166337 − 0.197212i
\(75\) −2.08582e9 + 1.38820e9i −0.878962 + 0.584984i
\(76\) −2.38947e9 − 4.08810e8i −0.942398 − 0.161233i
\(77\) −1.50484e8 + 1.50484e8i −0.0555951 + 0.0555951i
\(78\) 5.17409e8 + 4.39420e7i 0.179210 + 0.0152197i
\(79\) 5.65597e9i 1.83811i 0.394130 + 0.919055i \(0.371046\pi\)
−0.394130 + 0.919055i \(0.628954\pi\)
\(80\) −1.11565e9 − 3.08103e9i −0.340469 − 0.940256i
\(81\) 3.84106e9 1.10161
\(82\) −2.00010e8 + 2.35508e9i −0.0539489 + 0.635239i
\(83\) −1.18232e9 − 1.18232e9i −0.300153 − 0.300153i 0.540920 − 0.841074i \(-0.318076\pi\)
−0.841074 + 0.540920i \(0.818076\pi\)
\(84\) 1.00634e9 − 5.88200e9i 0.240629 − 1.40646i
\(85\) 1.79113e8 − 1.80170e9i 0.0403676 − 0.406057i
\(86\) −6.78237e9 + 5.72054e9i −1.44175 + 1.21603i
\(87\) 5.38108e9 + 5.38108e9i 1.07963 + 1.07963i
\(88\) −2.97156e8 − 7.71979e7i −0.0563080 − 0.0146282i
\(89\) − 1.86976e9i − 0.334840i −0.985886 − 0.167420i \(-0.946456\pi\)
0.985886 − 0.167420i \(-0.0535435\pi\)
\(90\) −3.83583e8 − 5.58778e8i −0.0649602 − 0.0946295i
\(91\) 1.43659e9i 0.230211i
\(92\) −2.65496e9 + 1.87922e9i −0.402828 + 0.285127i
\(93\) 6.95429e9 + 6.95429e9i 0.999628 + 0.999628i
\(94\) 9.41235e9 − 7.93877e9i 1.28250 − 1.08172i
\(95\) −4.68805e9 − 5.72305e9i −0.605863 − 0.739622i
\(96\) 8.05169e9 − 3.04705e9i 0.987486 − 0.373701i
\(97\) 4.01722e9 + 4.01722e9i 0.467808 + 0.467808i 0.901204 − 0.433396i \(-0.142685\pi\)
−0.433396 + 0.901204i \(0.642685\pi\)
\(98\) 7.44327e9 + 6.32134e8i 0.823442 + 0.0699324i
\(99\) −6.35033e7 −0.00667760
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.37.31 yes 116
5.3 odd 4 inner 40.11.i.a.13.1 ✓ 116
8.5 even 2 inner 40.11.i.a.37.1 yes 116
40.13 odd 4 inner 40.11.i.a.13.31 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.1 ✓ 116 5.3 odd 4 inner
40.11.i.a.13.31 yes 116 40.13 odd 4 inner
40.11.i.a.37.1 yes 116 8.5 even 2 inner
40.11.i.a.37.31 yes 116 1.1 even 1 trivial