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Properties

Label 40.8.d.a
Level $40$
Weight $8$
Character orbit 40.d
Analytic conductor $12.495$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,8,Mod(21,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.21");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 40.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: \(12.4954010194\)
Analytic rank: \(0\)
Dimension: \(28\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} - 208 q^{4} - 228 q^{6} + 1372 q^{7} - 2024 q^{8} - 20412 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 28 q - 14 q^{2} - 208 q^{4} - 228 q^{6} + 1372 q^{7} - 2024 q^{8} - 20412 q^{9} - 3250 q^{10} - 2788 q^{12} - 26620 q^{14} - 13500 q^{15} - 48824 q^{16} + 126562 q^{18} + 20500 q^{20} - 113892 q^{22} + 2588 q^{23} - 312408 q^{24} - 437500 q^{25} + 325932 q^{26} + 482476 q^{28} - 239000 q^{30} + 268024 q^{31} - 1445944 q^{32} - 99016 q^{33} + 651372 q^{34} + 1544216 q^{36} - 443660 q^{38} - 283944 q^{39} - 533000 q^{40} - 601208 q^{41} + 2692532 q^{42} + 239528 q^{44} - 142196 q^{46} - 2076460 q^{47} - 1158872 q^{48} + 4316268 q^{49} + 218750 q^{50} + 1591040 q^{52} + 4926168 q^{54} + 1331000 q^{55} + 2368232 q^{56} + 3788536 q^{57} - 5872008 q^{58} + 435500 q^{60} + 4008592 q^{62} - 14839060 q^{63} + 10179584 q^{64} - 9178320 q^{66} - 11198568 q^{68} - 2491000 q^{70} - 1001656 q^{71} - 8674904 q^{72} + 2534128 q^{73} - 21884804 q^{74} - 13848304 q^{76} + 25352888 q^{78} - 22040096 q^{79} - 7560000 q^{80} + 11166300 q^{81} + 20089464 q^{82} - 38088296 q^{84} + 17672012 q^{86} + 34127184 q^{87} + 11943152 q^{88} + 3807224 q^{89} + 13443750 q^{90} + 31988508 q^{92} + 17925028 q^{94} - 13718000 q^{95} - 30275488 q^{96} + 15198608 q^{97} + 11064654 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1 −11.2955 − 0.641370i 42.7235i 127.177 + 14.4892i 125.000i 27.4016 − 482.584i 1344.00 −1427.24 − 245.231i 361.702 80.1713 − 1411.94i
21.2 −11.2955 + 0.641370i − 42.7235i 127.177 − 14.4892i − 125.000i 27.4016 + 482.584i 1344.00 −1427.24 + 245.231i 361.702 80.1713 + 1411.94i
21.3 −11.1368 − 1.99270i − 69.9529i 120.058 + 44.3848i 125.000i −139.395 + 779.055i −814.432 −1248.62 − 733.546i −2706.41 249.088 − 1392.10i
21.4 −11.1368 + 1.99270i 69.9529i 120.058 − 44.3848i − 125.000i −139.395 − 779.055i −814.432 −1248.62 + 733.546i −2706.41 249.088 + 1392.10i
21.5 −9.65923 − 5.89062i − 15.3942i 58.6013 + 113.798i − 125.000i −90.6811 + 148.696i −206.895 104.295 − 1444.39i 1950.02 −736.327 + 1207.40i
21.6 −9.65923 + 5.89062i 15.3942i 58.6013 − 113.798i 125.000i −90.6811 − 148.696i −206.895 104.295 + 1444.39i 1950.02 −736.327 − 1207.40i
21.7 −6.69744 − 9.11835i 81.3925i −38.2885 + 122.139i − 125.000i 742.165 − 545.121i 1696.29 1370.14 − 468.892i −4437.73 −1139.79 + 837.180i
21.8 −6.69744 + 9.11835i − 81.3925i −38.2885 − 122.139i 125.000i 742.165 + 545.121i 1696.29 1370.14 + 468.892i −4437.73 −1139.79 − 837.180i
21.9 −6.11309 − 9.51999i − 28.9351i −53.2604 + 116.393i 125.000i −275.462 + 176.883i 724.987 1433.65 − 204.483i 1349.76 1190.00 − 764.136i
21.10 −6.11309 + 9.51999i 28.9351i −53.2604 − 116.393i − 125.000i −275.462 − 176.883i 724.987 1433.65 + 204.483i 1349.76 1190.00 + 764.136i
21.11 −4.74304 − 10.2715i − 91.4325i −83.0072 + 97.4362i − 125.000i −939.148 + 433.668i −386.271 1394.52 + 390.465i −6172.90 −1283.94 + 592.880i
21.12 −4.74304 + 10.2715i 91.4325i −83.0072 − 97.4362i 125.000i −939.148 − 433.668i −386.271 1394.52 − 390.465i −6172.90 −1283.94 − 592.880i
21.13 −1.95163 − 11.1441i 63.3835i −120.382 + 43.4983i 125.000i 706.353 − 123.701i −633.994 719.691 + 1256.66i −1830.47 1393.01 − 243.953i
21.14 −1.95163 + 11.1441i − 63.3835i −120.382 − 43.4983i − 125.000i 706.353 + 123.701i −633.994 719.691 − 1256.66i −1830.47 1393.01 + 243.953i
21.15 −1.71424 − 11.1831i 13.4707i −122.123 + 38.3411i − 125.000i 150.644 − 23.0921i −1011.70 638.119 + 1299.98i 2005.54 −1397.89 + 214.280i
21.16 −1.71424 + 11.1831i − 13.4707i −122.123 − 38.3411i 125.000i 150.644 + 23.0921i −1011.70 638.119 − 1299.98i 2005.54 −1397.89 − 214.280i
21.17 4.26315 − 10.4798i − 58.0742i −91.6512 − 89.3536i 125.000i −608.605 − 247.579i −772.054 −1327.13 + 579.556i −1185.62 1309.97 + 532.893i
21.18 4.26315 + 10.4798i 58.0742i −91.6512 + 89.3536i − 125.000i −608.605 + 247.579i −772.054 −1327.13 − 579.556i −1185.62 1309.97 − 532.893i
21.19 5.94445 − 9.62619i 47.3200i −57.3270 − 114.445i − 125.000i 455.511 + 281.291i 608.992 −1442.44 − 128.471i −52.1788 −1203.27 − 743.056i
21.20 5.94445 + 9.62619i − 47.3200i −57.3270 + 114.445i 125.000i 455.511 − 281.291i 608.992 −1442.44 + 128.471i −52.1788 −1203.27 + 743.056i
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.28
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.8.d.a ✓ 28
4.b odd 2 1 160.8.d.a 28
8.b even 2 1 inner 40.8.d.a ✓ 28
8.d odd 2 1 160.8.d.a 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.8.d.a ✓ 28 1.a even 1 1 trivial
40.8.d.a ✓ 28 8.b even 2 1 inner
160.8.d.a 28 4.b odd 2 1
160.8.d.a 28 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{8}^{\mathrm{new}}(40, [\chi])\).