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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 13.3
Character \(\chi\) \(=\) 40.13
Dual form 40.11.i.a.37.3

$q$-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+(-31.4478 - 5.91901i) q^{2} +(299.253 - 299.253i) q^{3} +(953.931 + 372.280i) q^{4} +(2782.63 - 1422.17i) q^{5} +(-11182.2 + 7639.58i) q^{6} +(6432.45 - 6432.45i) q^{7} +(-27795.5 - 17353.7i) q^{8} -120056. i q^{9} +(-95925.6 + 28253.8i) q^{10} -72564.4i q^{11} +(396873. - 174061. i) q^{12} +(205709. - 205709. i) q^{13} +(-240360. + 164213. i) q^{14} +(407123. - 1.25830e6i) q^{15} +(771391. + 710258. i) q^{16} +(-1.42579e6 + 1.42579e6i) q^{17} +(-710614. + 3.77551e6i) q^{18} +3.08608e6 q^{19} +(3.18389e6 - 320735. i) q^{20} -3.84986e6i q^{21} +(-429509. + 2.28199e6i) q^{22} +(3.33931e6 + 3.33931e6i) q^{23} +(-1.35111e7 + 3.12474e6i) q^{24} +(5.72048e6 - 7.91477e6i) q^{25} +(-7.68670e6 + 5.25151e6i) q^{26} +(-1.82566e7 - 1.82566e7i) q^{27} +(8.53078e6 - 3.74144e6i) q^{28} +1.87887e7 q^{29} +(-2.02510e7 + 3.71611e7i) q^{30} -3.06307e7 q^{31} +(-2.00546e7 - 2.69019e7i) q^{32} +(-2.17151e7 - 2.17151e7i) q^{33} +(5.32771e7 - 3.63986e7i) q^{34} +(8.75110e6 - 2.70472e7i) q^{35} +(4.46945e7 - 1.14525e8i) q^{36} +(-6.88725e7 - 6.88725e7i) q^{37} +(-9.70506e7 - 1.82666e7i) q^{38} -1.23118e8i q^{39} +(-1.02025e8 - 8.75903e6i) q^{40} +1.50909e8 q^{41} +(-2.27874e7 + 1.21070e8i) q^{42} +(-1.31374e8 + 1.31374e8i) q^{43} +(2.70143e7 - 6.92214e7i) q^{44} +(-1.70741e8 - 3.34073e8i) q^{45} +(-8.52485e7 - 1.24779e8i) q^{46} +(-1.24947e8 + 1.24947e8i) q^{47} +(4.43389e8 - 1.82943e7i) q^{48} +1.99722e8i q^{49} +(-2.26744e8 + 2.15043e8i) q^{50} +8.53343e8i q^{51} +(2.72814e8 - 1.19651e8i) q^{52} +(-3.48610e8 + 3.48610e8i) q^{53} +(4.66070e8 + 6.82192e8i) q^{54} +(-1.03199e8 - 2.01920e8i) q^{55} +(-2.90420e8 + 6.71663e7i) q^{56} +(9.23521e8 - 9.23521e8i) q^{57} +(-5.90862e8 - 1.11210e8i) q^{58} -1.94393e8 q^{59} +(8.56808e8 - 1.04877e9i) q^{60} -1.06474e9i q^{61} +(9.63270e8 + 1.81304e8i) q^{62} +(-7.72256e8 - 7.72256e8i) q^{63} +(4.71439e8 + 9.64711e8i) q^{64} +(2.79860e8 - 8.64968e8i) q^{65} +(5.54362e8 + 8.11426e8i) q^{66} +(1.19181e9 + 1.19181e9i) q^{67} +(-1.89089e9 + 8.29310e8i) q^{68} +1.99860e9 q^{69} +(-4.35296e8 + 7.98778e8i) q^{70} -1.95084e9 q^{71} +(-2.08342e9 + 3.33702e9i) q^{72} +(-1.13995e9 - 1.13995e9i) q^{73} +(1.75823e9 + 2.57355e9i) q^{74} +(-6.56649e8 - 4.08039e9i) q^{75} +(2.94391e9 + 1.14889e9i) q^{76} +(-4.66767e8 - 4.66767e8i) q^{77} +(-7.28739e8 + 3.87181e9i) q^{78} -3.15934e9i q^{79} +(3.15661e9 + 8.79337e8i) q^{80} -3.83752e9 q^{81} +(-4.74577e9 - 8.93234e8i) q^{82} +(1.44998e9 - 1.44998e9i) q^{83} +(1.43323e9 - 3.67250e9i) q^{84} +(-1.93973e9 + 5.99516e9i) q^{85} +(4.90904e9 - 3.35383e9i) q^{86} +(5.62257e9 - 5.62257e9i) q^{87} +(-1.25926e9 + 2.01696e9i) q^{88} +6.71757e9i q^{89} +(3.39204e9 + 1.15165e10i) q^{90} -2.64643e9i q^{91} +(1.94231e9 + 4.42862e9i) q^{92} +(-9.16636e9 + 9.16636e9i) q^{93} +(4.66888e9 - 3.18976e9i) q^{94} +(8.58744e9 - 4.38894e9i) q^{95} +(-1.40519e10 - 2.04911e9i) q^{96} +(6.45977e9 - 6.45977e9i) q^{97} +(1.18216e9 - 6.28084e9i) q^{98} -8.71181e9 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −31.4478 − 5.91901i −0.982744 − 0.184969i
\(3\) 299.253 − 299.253i 1.23150 − 1.23150i 0.268106 − 0.963389i \(-0.413602\pi\)
0.963389 − 0.268106i \(-0.0863978\pi\)
\(4\) 953.931 + 372.280i 0.931573 + 0.363555i
\(5\) 2782.63 − 1422.17i 0.890443 − 0.455095i
\(6\) −11182.2 + 7639.58i −1.43803 + 0.982457i
\(7\) 6432.45 − 6432.45i 0.382724 − 0.382724i −0.489358 − 0.872083i \(-0.662769\pi\)
0.872083 + 0.489358i \(0.162769\pi\)
\(8\) −27795.5 − 17353.7i −0.848252 − 0.529593i
\(9\) − 120056.i − 2.03316i
\(10\) −95925.6 + 28253.8i −0.959256 + 0.282538i
\(11\) − 72564.4i − 0.450568i −0.974293 − 0.225284i \(-0.927669\pi\)
0.974293 − 0.225284i \(-0.0723309\pi\)
\(12\) 396873. − 174061.i 1.59494 − 0.699512i
\(13\) 205709. − 205709.i 0.554035 − 0.554035i −0.373568 − 0.927603i \(-0.621866\pi\)
0.927603 + 0.373568i \(0.121866\pi\)
\(14\) −240360. + 164213.i −0.446912 + 0.305328i
\(15\) 407123. − 1.25830e6i 0.536129 − 1.65702i
\(16\) 771391. + 710258.i 0.735656 + 0.677355i
\(17\) −1.42579e6 + 1.42579e6i −1.00418 + 1.00418i −0.00418492 + 0.999991i \(0.501332\pi\)
−0.999991 + 0.00418492i \(0.998668\pi\)
\(18\) −710614. + 3.77551e6i −0.376072 + 1.99808i
\(19\) 3.08608e6 1.24635 0.623175 − 0.782083i \(-0.285843\pi\)
0.623175 + 0.782083i \(0.285843\pi\)
\(20\) 3.18389e6 − 320735.i 0.994964 − 0.100230i
\(21\) − 3.84986e6i − 0.942647i
\(22\) −429509. + 2.28199e6i −0.0833411 + 0.442793i
\(23\) 3.33931e6 + 3.33931e6i 0.518820 + 0.518820i 0.917214 − 0.398394i \(-0.130432\pi\)
−0.398394 + 0.917214i \(0.630432\pi\)
\(24\) −1.35111e7 + 3.12474e6i −1.69681 + 0.392426i
\(25\) 5.72048e6 − 7.91477e6i 0.585777 − 0.810472i
\(26\) −7.68670e6 + 5.25151e6i −0.646954 + 0.441995i
\(27\) −1.82566e7 − 1.82566e7i −1.27234 − 1.27234i
\(28\) 8.53078e6 − 3.74144e6i 0.495677 − 0.217394i
\(29\) 1.87887e7 0.916022 0.458011 − 0.888947i \(-0.348562\pi\)
0.458011 + 0.888947i \(0.348562\pi\)
\(30\) −2.02510e7 + 3.71611e7i −0.833376 + 1.52926i
\(31\) −3.06307e7 −1.06991 −0.534957 − 0.844879i \(-0.679672\pi\)
−0.534957 + 0.844879i \(0.679672\pi\)
\(32\) −2.00546e7 − 2.69019e7i −0.597672 − 0.801741i
\(33\) −2.17151e7 − 2.17151e7i −0.554872 − 0.554872i
\(34\) 5.32771e7 − 3.63986e7i 1.17259 − 0.801107i
\(35\) 8.75110e6 − 2.70472e7i 0.166618 − 0.514970i
\(36\) 4.46945e7 − 1.14525e8i 0.739166 − 1.89404i
\(37\) −6.88725e7 − 6.88725e7i −0.993202 − 0.993202i 0.00677521 − 0.999977i \(-0.497843\pi\)
−0.999977 + 0.00677521i \(0.997843\pi\)
\(38\) −9.70506e7 − 1.82666e7i −1.22484 − 0.230536i
\(39\) − 1.23118e8i − 1.36458i
\(40\) −1.02025e8 − 8.75903e6i −0.996335 − 0.0855374i
\(41\) 1.50909e8 1.30256 0.651279 − 0.758839i \(-0.274233\pi\)
0.651279 + 0.758839i \(0.274233\pi\)
\(42\) −2.27874e7 + 1.21070e8i −0.174360 + 0.926381i
\(43\) −1.31374e8 + 1.31374e8i −0.893652 + 0.893652i −0.994865 − 0.101213i \(-0.967728\pi\)
0.101213 + 0.994865i \(0.467728\pi\)
\(44\) 2.70143e7 − 6.92214e7i 0.163806 − 0.419737i
\(45\) −1.70741e8 − 3.34073e8i −0.925283 − 1.81042i
\(46\) −8.52485e7 − 1.24779e8i −0.413902 − 0.605834i
\(47\) −1.24947e8 + 1.24947e8i −0.544801 + 0.544801i −0.924932 − 0.380132i \(-0.875879\pi\)
0.380132 + 0.924932i \(0.375879\pi\)
\(48\) 4.43389e8 − 1.82943e7i 1.74012 − 0.0717975i
\(49\) 1.99722e8i 0.707044i
\(50\) −2.26744e8 + 2.15043e8i −0.725581 + 0.688137i
\(51\) 8.53343e8i 2.47328i
\(52\) 2.72814e8 − 1.19651e8i 0.717546 − 0.314702i
\(53\) −3.48610e8 + 3.48610e8i −0.833605 + 0.833605i −0.988008 − 0.154403i \(-0.950655\pi\)
0.154403 + 0.988008i \(0.450655\pi\)
\(54\) 4.66070e8 + 6.82192e8i 1.01504 + 1.48572i
\(55\) −1.03199e8 − 2.01920e8i −0.205051 − 0.401205i
\(56\) −2.90420e8 + 6.71663e7i −0.527335 + 0.121958i
\(57\) 9.23521e8 − 9.23521e8i 1.53487 − 1.53487i
\(58\) −5.90862e8 − 1.11210e8i −0.900215 − 0.169436i
\(59\) −1.94393e8 −0.271907 −0.135953 − 0.990715i \(-0.543410\pi\)
−0.135953 + 0.990715i \(0.543410\pi\)
\(60\) 8.56808e8 − 1.04877e9i 1.10186 − 1.34873i
\(61\) − 1.06474e9i − 1.26064i −0.776334 − 0.630322i \(-0.782923\pi\)
0.776334 − 0.630322i \(-0.217077\pi\)
\(62\) 9.63270e8 + 1.81304e8i 1.05145 + 0.197901i
\(63\) −7.72256e8 − 7.72256e8i −0.778141 − 0.778141i
\(64\) 4.71439e8 + 9.64711e8i 0.439062 + 0.898457i
\(65\) 2.79860e8 − 8.64968e8i 0.241198 − 0.745475i
\(66\) 5.54362e8 + 8.11426e8i 0.442663 + 0.647932i
\(67\) 1.19181e9 + 1.19181e9i 0.882743 + 0.882743i 0.993813 − 0.111069i \(-0.0354276\pi\)
−0.111069 + 0.993813i \(0.535428\pi\)
\(68\) −1.89089e9 + 8.29310e8i −1.30054 + 0.570391i
\(69\) 1.99860e9 1.27785
\(70\) −4.35296e8 + 7.98778e8i −0.258997 + 0.475265i
\(71\) −1.95084e9 −1.08126 −0.540630 − 0.841260i \(-0.681814\pi\)
−0.540630 + 0.841260i \(0.681814\pi\)
\(72\) −2.08342e9 + 3.33702e9i −1.07675 + 1.72463i
\(73\) −1.13995e9 − 1.13995e9i −0.549882 − 0.549882i 0.376524 − 0.926407i \(-0.377119\pi\)
−0.926407 + 0.376524i \(0.877119\pi\)
\(74\) 1.75823e9 + 2.57355e9i 0.792352 + 1.15978i
\(75\) −6.56649e8 − 4.08039e9i −0.276712 − 1.71947i
\(76\) 2.94391e9 + 1.14889e9i 1.16107 + 0.453116i
\(77\) −4.66767e8 − 4.66767e8i −0.172443 − 0.172443i
\(78\) −7.28739e8 + 3.87181e9i −0.252406 + 1.34104i
\(79\) − 3.15934e9i − 1.02674i −0.858167 − 0.513371i \(-0.828397\pi\)
0.858167 − 0.513371i \(-0.171603\pi\)
\(80\) 3.15661e9 + 8.79337e8i 0.963321 + 0.268352i
\(81\) −3.83752e9 −1.10059
\(82\) −4.74577e9 − 8.93234e8i −1.28008 − 0.240933i
\(83\) 1.44998e9 − 1.44998e9i 0.368105 − 0.368105i −0.498681 − 0.866786i \(-0.666182\pi\)
0.866786 + 0.498681i \(0.166182\pi\)
\(84\) 1.43323e9 − 3.67250e9i 0.342704 − 0.878144i
\(85\) −1.93973e9 + 5.99516e9i −0.437166 + 1.35116i
\(86\) 4.90904e9 − 3.35383e9i 1.04353 − 0.712934i
\(87\) 5.62257e9 − 5.62257e9i 1.12808 − 1.12808i
\(88\) −1.25926e9 + 2.01696e9i −0.238618 + 0.382195i
\(89\) 6.71757e9i 1.20299i 0.798877 + 0.601495i \(0.205428\pi\)
−0.798877 + 0.601495i \(0.794572\pi\)
\(90\) 3.39204e9 + 1.15165e10i 0.574445 + 1.95032i
\(91\) − 2.64643e9i − 0.424085i
\(92\) 1.94231e9 + 4.42862e9i 0.294700 + 0.671939i
\(93\) −9.16636e9 + 9.16636e9i −1.31760 + 1.31760i
\(94\) 4.66888e9 − 3.18976e9i 0.636171 − 0.434628i
\(95\) 8.58744e9 − 4.38894e9i 1.10980 − 0.567208i
\(96\) −1.40519e10 − 2.04911e9i −1.72337 − 0.251309i
\(97\) 6.45977e9 − 6.45977e9i 0.752243 − 0.752243i −0.222654 − 0.974898i \(-0.571472\pi\)
0.974898 + 0.222654i \(0.0714721\pi\)
\(98\) 1.18216e9 − 6.28084e9i 0.130781 − 0.694844i
\(99\) −8.71181e9 −0.916078
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.13.3 ✓ 116
5.2 odd 4 inner 40.11.i.a.37.33 yes 116
8.5 even 2 inner 40.11.i.a.13.33 yes 116
40.37 odd 4 inner 40.11.i.a.37.3 yes 116
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.11.i.a.13.3 ✓ 116 1.1 even 1 trivial
40.11.i.a.13.33 yes 116 8.5 even 2 inner
40.11.i.a.37.3 yes 116 40.37 odd 4 inner
40.11.i.a.37.33 yes 116 5.2 odd 4 inner