Properties

Label 40.11.i.a
Level $40$
Weight $11$
Character orbit 40.i
Analytic conductor $25.414$
Analytic rank $0$
Dimension $116$
Inner twists $4$

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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: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

The algebraic \(q\)-expansion of this newform has not been computed, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 116 q - 2 q^{2} - 4864 q^{6} - 4 q^{7} + 69124 q^{8} + 256166 q^{10} + 502036 q^{12} - 4 q^{15} + 3502536 q^{16} - 905772 q^{17} - 5688470 q^{18} + 4385256 q^{20} + 9808012 q^{22} - 4 q^{23} - 1476988 q^{25}+ \cdots - 65612488734 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1 −31.8852 − 2.70791i −181.420 + 181.420i 1009.33 + 172.685i 1980.27 − 2417.47i 6275.90 − 5293.36i 16061.1 − 16061.1i −31715.2 − 8239.28i − 6777.67i −69687.7 + 71719.1i
13.2 −31.7327 + 4.12737i −33.5937 + 33.5937i 989.930 − 261.945i −939.110 + 2980.55i 927.364 − 1204.67i 5089.54 − 5089.54i −30332.0 + 12398.0i 56791.9i 17498.7 − 98457.1i
13.3 −31.4478 − 5.91901i 299.253 − 299.253i 953.931 + 372.280i 2782.63 − 1422.17i −11182.2 + 7639.58i 6432.45 − 6432.45i −27795.5 − 17353.7i − 120056.i −95925.6 + 28253.8i
13.4 −31.2076 − 7.07732i 28.6547 − 28.6547i 923.823 + 441.732i 2754.83 + 1475.31i −1097.04 + 691.445i −18007.5 + 18007.5i −25704.0 − 20323.6i 57406.8i −75530.3 − 65537.5i
13.5 −31.1769 + 7.21110i 179.852 − 179.852i 920.000 − 449.640i −1869.56 − 2504.07i −4310.29 + 6904.15i 532.991 − 532.991i −25440.4 + 20652.6i − 5644.22i 76344.3 + 64587.6i
13.6 −30.5658 − 9.47256i 41.2659 − 41.2659i 844.541 + 579.073i −3105.70 − 346.803i −1652.22 + 870.433i 4455.29 − 4455.29i −20328.8 − 25699.8i 55643.3i 91643.1 + 40019.2i
13.7 −30.4587 + 9.81171i −258.070 + 258.070i 831.461 − 597.703i −2935.38 − 1071.99i 5328.36 − 10392.6i −12430.0 + 12430.0i −19460.7 + 26363.3i − 74151.3i 99925.8 + 3850.39i
13.8 −29.2557 + 12.9654i 267.142 − 267.142i 687.797 − 758.625i −1085.61 + 2930.37i −4351.84 + 11279.1i −2054.40 + 2054.40i −10286.1 + 31111.7i − 83681.0i −6233.15 − 99805.6i
13.9 −28.2521 + 15.0272i −257.663 + 257.663i 572.366 − 849.101i 2420.01 + 1977.17i 3407.58 − 11151.5i −1516.95 + 1516.95i −3410.93 + 32590.0i − 73731.8i −98081.7 − 19493.2i
13.10 −27.5829 − 16.2229i −281.304 + 281.304i 497.633 + 894.951i −1103.56 + 2923.66i 12322.8 − 3195.61i 5272.38 − 5272.38i 792.570 − 32758.4i − 99215.2i 77869.7 − 62740.1i
13.11 −26.9779 + 17.2103i 31.2791 − 31.2791i 431.610 − 928.595i 1815.55 − 2543.50i −305.520 + 1382.17i −13255.7 + 13255.7i 4337.49 + 32479.7i 57092.2i −5205.37 + 99864.4i
13.12 −26.7504 − 17.5617i −186.299 + 186.299i 407.172 + 939.568i −1009.16 − 2957.57i 8255.33 − 1711.85i −18272.5 + 18272.5i 5608.41 − 32284.5i − 10365.9i −24944.5 + 96838.9i
13.13 −24.3931 − 20.7117i 279.968 − 279.968i 166.050 + 1010.45i −2824.19 + 1337.75i −12627.9 + 1030.68i −19417.8 + 19417.8i 16877.6 − 28087.1i − 97714.9i 96597.9 + 25862.0i
13.14 −23.6958 + 21.5060i 119.074 − 119.074i 98.9850 − 1019.20i 2966.97 + 981.174i −260.753 + 5382.34i 20153.9 − 20153.9i 19573.5 + 26279.7i 30692.0i −91406.0 + 40557.9i
13.15 −23.6711 − 21.5332i 155.126 − 155.126i 96.6435 + 1019.43i 1092.16 + 2927.94i −7012.36 + 331.648i 15010.7 − 15010.7i 19663.9 − 26212.1i 10920.9i 37195.1 − 92825.2i
13.16 −21.5060 + 23.6958i −119.074 + 119.074i −98.9850 − 1019.20i −2966.97 − 981.174i −260.753 − 5382.34i 20153.9 − 20153.9i 26279.7 + 19573.5i 30692.0i 87057.4 − 49203.7i
13.17 −21.0650 − 24.0887i 107.979 − 107.979i −136.529 + 1014.86i −95.2624 − 3123.55i −4875.66 − 326.491i 7115.44 − 7115.44i 27322.6 − 18089.2i 35730.0i −73235.4 + 68092.4i
13.18 −18.0006 − 26.4571i −159.283 + 159.283i −375.958 + 952.487i 3109.51 + 310.741i 7081.36 + 1346.98i −794.929 + 794.929i 31967.5 − 7198.58i 8306.77i −47751.7 − 87862.2i
13.19 −17.2103 + 26.9779i −31.2791 + 31.2791i −431.610 − 928.595i −1815.55 + 2543.50i −305.520 − 1382.17i −13255.7 + 13255.7i 32479.7 + 4337.49i 57092.2i −37371.9 − 92754.2i
13.20 −15.0272 + 28.2521i 257.663 − 257.663i −572.366 − 849.101i −2420.01 − 1977.17i 3407.58 + 11151.5i −1516.95 + 1516.95i 32590.0 − 3410.93i − 73731.8i 92225.1 − 38659.1i
See next 80 embeddings (of 116 total)
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 13.58
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
8.b even 2 1 inner
40.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.11.i.a ✓ 116
5.c odd 4 1 inner 40.11.i.a ✓ 116
8.b even 2 1 inner 40.11.i.a ✓ 116
40.i odd 4 1 inner 40.11.i.a ✓ 116
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.11.i.a ✓ 116 1.a even 1 1 trivial
40.11.i.a ✓ 116 5.c odd 4 1 inner
40.11.i.a ✓ 116 8.b even 2 1 inner
40.11.i.a ✓ 116 40.i odd 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(40, [\chi])\).