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Properties

Label 20.11.d.d
Level $20$
Weight $11$
Character orbit 20.d
Analytic conductor $12.707$
Analytic rank $0$
Dimension $24$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,11,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7071450535\)
Analytic rank: \(0\)
Dimension: \(24\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 608 q^{4} + 8280 q^{5} - 19584 q^{6} + 597192 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 608 q^{4} + 8280 q^{5} - 19584 q^{6} + 597192 q^{9} + 223600 q^{10} - 1706016 q^{14} - 4733376 q^{16} + 1631520 q^{20} - 13030368 q^{21} + 10190784 q^{24} + 8826200 q^{25} - 9454368 q^{26} + 121656816 q^{29} + 103616160 q^{30} - 335231168 q^{34} - 276632160 q^{36} - 559048000 q^{40} + 892843248 q^{41} + 766329600 q^{44} - 248162040 q^{45} + 433181216 q^{46} - 738102008 q^{49} + 765396000 q^{50} + 139387968 q^{54} - 2629032384 q^{56} + 296160000 q^{60} + 228563248 q^{61} - 1875284992 q^{64} - 2069145600 q^{65} - 1440259200 q^{66} - 943422432 q^{69} + 3242642240 q^{70} + 21045467232 q^{74} + 828422400 q^{76} + 11736799680 q^{80} - 5619065544 q^{81} - 28069573632 q^{84} - 18119744000 q^{85} + 8163556416 q^{86} + 4631088816 q^{89} - 16754463600 q^{90} + 63404384 q^{94} - 5617046784 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1 −31.7400 − 4.07116i 190.073 990.851 + 258.437i 101.732 + 3123.34i −6032.92 − 773.819i 14056.7 −30397.4 − 12236.7i −22921.2 9486.66 − 99549.0i
19.2 −31.7400 + 4.07116i 190.073 990.851 − 258.437i 101.732 − 3123.34i −6032.92 + 773.819i 14056.7 −30397.4 + 12236.7i −22921.2 9486.66 + 99549.0i
19.3 −28.1904 − 15.1426i −349.582 565.401 + 853.755i 1690.09 − 2628.54i 9854.87 + 5293.59i 15402.9 −3010.79 − 32629.4i 63158.6 −87447.4 + 48507.3i
19.4 −28.1904 + 15.1426i −349.582 565.401 − 853.755i 1690.09 + 2628.54i 9854.87 − 5293.59i 15402.9 −3010.79 + 32629.4i 63158.6 −87447.4 − 48507.3i
19.5 −23.9223 − 21.2538i 94.4407 120.549 + 1016.88i 3107.65 + 328.861i −2259.24 − 2007.23i −20081.9 18728.8 − 26888.2i −50129.9 −67352.4 − 73916.5i
19.6 −23.9223 + 21.2538i 94.4407 120.549 − 1016.88i 3107.65 − 328.861i −2259.24 + 2007.23i −20081.9 18728.8 + 26888.2i −50129.9 −67352.4 + 73916.5i
19.7 −21.8278 − 23.3997i 416.738 −71.0958 + 1021.53i −2532.23 − 1831.23i −9096.47 − 9751.56i −9550.48 25455.4 − 20634.1i 114622. 12422.7 + 99225.4i
19.8 −21.8278 + 23.3997i 416.738 −71.0958 − 1021.53i −2532.23 + 1831.23i −9096.47 + 9751.56i −9550.48 25455.4 + 20634.1i 114622. 12422.7 − 99225.4i
19.9 −17.2353 − 26.9619i −146.443 −429.889 + 929.393i −2538.13 + 1823.05i 2523.99 + 3948.38i 13979.7 32467.5 − 4427.76i −37603.5 92898.4 + 37012.0i
19.10 −17.2353 + 26.9619i −146.443 −429.889 − 929.393i −2538.13 − 1823.05i 2523.99 − 3948.38i 13979.7 32467.5 + 4427.76i −37603.5 92898.4 − 37012.0i
19.11 −0.302723 − 31.9986i −375.794 −1023.82 + 19.3734i 2240.90 + 2178.07i 113.762 + 12024.9i −19635.7 929.854 + 32754.8i 82172.5 69016.9 − 72364.8i
19.12 −0.302723 + 31.9986i −375.794 −1023.82 − 19.3734i 2240.90 − 2178.07i 113.762 − 12024.9i −19635.7 929.854 − 32754.8i 82172.5 69016.9 + 72364.8i
19.13 0.302723 − 31.9986i 375.794 −1023.82 − 19.3734i 2240.90 + 2178.07i 113.762 − 12024.9i 19635.7 −929.854 + 32754.8i 82172.5 70373.6 − 71046.1i
19.14 0.302723 + 31.9986i 375.794 −1023.82 + 19.3734i 2240.90 − 2178.07i 113.762 + 12024.9i 19635.7 −929.854 − 32754.8i 82172.5 70373.6 + 71046.1i
19.15 17.2353 − 26.9619i 146.443 −429.889 − 929.393i −2538.13 + 1823.05i 2523.99 − 3948.38i −13979.7 −32467.5 − 4427.76i −37603.5 5407.52 + 99853.7i
19.16 17.2353 + 26.9619i 146.443 −429.889 + 929.393i −2538.13 − 1823.05i 2523.99 + 3948.38i −13979.7 −32467.5 + 4427.76i −37603.5 5407.52 − 99853.7i
19.17 21.8278 − 23.3997i −416.738 −71.0958 − 1021.53i −2532.23 − 1831.23i −9096.47 + 9751.56i 9550.48 −25455.4 − 20634.1i 114622. −98123.4 + 19281.8i
19.18 21.8278 + 23.3997i −416.738 −71.0958 + 1021.53i −2532.23 + 1831.23i −9096.47 − 9751.56i 9550.48 −25455.4 + 20634.1i 114622. −98123.4 − 19281.8i
19.19 23.9223 − 21.2538i −94.4407 120.549 − 1016.88i 3107.65 + 328.861i −2259.24 + 2007.23i 20081.9 −18728.8 − 26888.2i −50129.9 81331.5 − 58182.3i
19.20 23.9223 + 21.2538i −94.4407 120.549 + 1016.88i 3107.65 − 328.861i −2259.24 − 2007.23i 20081.9 −18728.8 + 26888.2i −50129.9 81331.5 + 58182.3i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 20.11.d.d ✓ 24
4.b odd 2 1 inner 20.11.d.d ✓ 24
5.b even 2 1 inner 20.11.d.d ✓ 24
5.c odd 4 2 100.11.b.h 24
20.d odd 2 1 inner 20.11.d.d ✓ 24
20.e even 4 2 100.11.b.h 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.11.d.d ✓ 24 1.a even 1 1 trivial
20.11.d.d ✓ 24 4.b odd 2 1 inner
20.11.d.d ✓ 24 5.b even 2 1 inner
20.11.d.d ✓ 24 20.d odd 2 1 inner
100.11.b.h 24 5.c odd 4 2
100.11.b.h 24 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 503592 T_{3}^{10} + 93360281616 T_{3}^{8} + \cdots + 20\!\cdots\!00 \) acting on \(S_{11}^{\mathrm{new}}(20, [\chi])\). Copy content Toggle raw display