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Properties

Label 40.10.k.a
Level $40$
Weight $10$
Character orbit 40.k
Analytic conductor $20.601$
Analytic rank $0$
Dimension $104$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [40,10,Mod(3,40)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(40, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 2, 3]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.3");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 40.k (of order \(4\), degree \(2\), 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: \(20.6014334466\)
Analytic rank: \(0\)
Dimension: \(104\)
Relative dimension: \(52\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 4 q^{3} - 6160 q^{6} - 716 q^{8}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 104 q - 2 q^{2} - 4 q^{3} - 6160 q^{6} - 716 q^{8} - 11450 q^{10} - 8 q^{11} - 76628 q^{12} - 1071416 q^{16} + 203992 q^{17} + 681130 q^{18} + 334320 q^{20} - 1659308 q^{22} + 1721760 q^{25} - 2525788 q^{26} + 78728 q^{27} - 13780420 q^{28} + 23311180 q^{30} + 3431968 q^{32} - 78736 q^{33} - 38240620 q^{35} - 6774436 q^{36} + 22634156 q^{38} + 26791580 q^{40} - 8 q^{41} + 22309740 q^{42} - 107824612 q^{43} + 64618352 q^{46} - 48846064 q^{48} + 87428390 q^{50} - 180028664 q^{51} + 37430420 q^{52} + 3216712 q^{56} + 78728 q^{57} + 209147380 q^{58} + 332718340 q^{60} - 26060960 q^{62} - 143800800 q^{65} - 228216424 q^{66} - 308787868 q^{67} - 479482324 q^{68} + 1044459540 q^{70} + 977425180 q^{72} - 604274384 q^{73} + 915277380 q^{75} + 1490487992 q^{76} - 2304036640 q^{78} - 1555422720 q^{80} - 2726120896 q^{81} + 2025770452 q^{82} + 795523156 q^{83} + 1491849216 q^{86} + 3168953936 q^{88} - 3830453530 q^{90} - 2786075176 q^{91} - 1234597180 q^{92} - 2885969760 q^{96} + 359465136 q^{97} + 4292569674 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1 −22.5775 − 1.50277i 59.1433 + 59.1433i 507.483 + 67.8576i −864.937 + 1097.73i −1246.43 − 1424.18i −1695.71 − 1695.71i −11355.7 − 2294.69i − 12687.2i 21177.7 − 23484.1i
3.2 −22.5481 + 1.89343i 165.073 + 165.073i 504.830 − 85.3863i −51.2057 − 1396.60i −4034.63 − 3409.52i −835.012 − 835.012i −11221.3 + 2881.15i 34815.1i 3798.96 + 31393.8i
3.3 −22.4994 + 2.40357i −92.4862 − 92.4862i 500.446 − 108.158i 1219.48 + 682.642i 2303.18 + 1858.59i 2448.27 + 2448.27i −10999.8 + 3636.33i − 2575.61i −29078.3 − 12427.9i
3.4 −21.8923 − 5.72063i −42.7444 − 42.7444i 446.549 + 250.476i −1060.46 − 910.248i 691.250 + 1180.30i 3617.58 + 3617.58i −8343.12 − 8038.04i − 16028.8i 18008.7 + 25994.0i
3.5 −21.7361 + 6.28840i −38.0341 − 38.0341i 432.912 − 273.370i 416.245 − 1334.12i 1065.89 + 587.538i −8584.33 − 8584.33i −7690.74 + 8664.31i − 16789.8i −658.072 + 31615.9i
3.6 −20.7838 + 8.94614i −180.909 − 180.909i 351.933 − 371.870i −1397.19 − 31.2248i 5378.43 + 2141.55i 601.225 + 601.225i −3987.72 + 10877.3i 45773.4i 29318.3 − 11850.5i
3.7 −20.3121 + 9.97093i 135.158 + 135.158i 313.161 − 405.060i 1088.75 + 876.217i −4092.98 − 1397.69i 3965.97 + 3965.97i −2322.13 + 11350.1i 16852.2i −30851.4 − 6941.99i
3.8 −19.6856 − 11.1570i 112.603 + 112.603i 263.043 + 439.263i 1150.24 + 793.778i −960.344 − 3472.96i −8267.06 − 8267.06i −277.308 − 11581.9i 5675.84i −13786.9 − 28459.1i
3.9 −19.3427 − 11.7415i 61.6474 + 61.6474i 236.277 + 454.222i 1162.25 − 776.085i −468.594 − 1916.25i 5579.17 + 5579.17i 763.005 − 11560.1i − 12082.2i −31593.3 + 1365.10i
3.10 −19.1942 − 11.9826i −159.790 − 159.790i 224.835 + 459.993i 739.560 − 1185.82i 1152.34 + 4981.73i −5107.08 − 5107.08i 1196.38 − 11523.3i 31382.6i −28404.5 + 13899.1i
3.11 −18.7026 + 12.7363i 75.3528 + 75.3528i 187.571 − 476.404i −1388.81 + 155.946i −2369.01 − 449.571i 2853.02 + 2853.02i 2559.59 + 11298.9i − 8326.92i 23988.2 − 20605.0i
3.12 −18.5221 − 12.9974i −133.895 − 133.895i 174.136 + 481.478i −490.067 + 1308.80i 739.732 + 4220.31i 211.711 + 211.711i 3032.59 − 11181.3i 16172.9i 26088.1 − 17872.1i
3.13 −16.3239 + 15.6694i −55.4736 − 55.4736i 20.9418 − 511.572i 285.771 − 1368.01i 1774.78 + 36.3112i 6020.70 + 6020.70i 7674.15 + 8679.01i − 13528.4i 16771.0 + 26809.2i
3.14 −15.6694 + 16.3239i −55.4736 − 55.4736i −20.9418 − 511.572i −285.771 + 1368.01i 1774.78 − 36.3112i −6020.70 − 6020.70i 8679.01 + 7674.15i − 13528.4i −17853.5 − 26100.8i
3.15 −15.5446 − 16.4428i 192.723 + 192.723i −28.7297 + 511.193i −1250.95 + 623.105i 173.095 − 6164.70i 4391.82 + 4391.82i 8852.03 − 7473.91i 54601.1i 29691.0 + 10883.1i
3.16 −13.0251 − 18.5026i 20.3361 + 20.3361i −172.696 + 481.996i −1223.49 − 675.417i 111.393 − 641.151i −4822.27 − 4822.27i 11167.6 − 3082.70i − 18855.9i 3439.08 + 31435.2i
3.17 −12.7363 + 18.7026i 75.3528 + 75.3528i −187.571 − 476.404i 1388.81 − 155.946i −2369.01 + 449.571i −2853.02 − 2853.02i 11298.9 + 2559.59i − 8326.92i −14771.8 + 27960.6i
3.18 −9.97093 + 20.3121i 135.158 + 135.158i −313.161 − 405.060i −1088.75 − 876.217i −4092.98 + 1397.69i −3965.97 − 3965.97i 11350.1 − 2322.13i 16852.2i 28653.6 − 13378.0i
3.19 −9.89967 − 20.3469i 12.5094 + 12.5094i −315.993 + 402.855i 330.244 + 1357.96i 130.688 − 378.365i 4552.53 + 4552.53i 11325.1 + 2441.34i − 19370.0i 24361.0 − 20162.8i
3.20 −8.94614 + 20.7838i −180.909 − 180.909i −351.933 − 371.870i 1397.19 + 31.2248i 5378.43 − 2141.55i −601.225 − 601.225i 10877.3 − 3987.72i 45773.4i −13148.5 + 28759.7i
See next 80 embeddings (of 104 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.52
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.10.k.a ✓ 104
5.c odd 4 1 inner 40.10.k.a ✓ 104
8.d odd 2 1 inner 40.10.k.a ✓ 104
40.k even 4 1 inner 40.10.k.a ✓ 104
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.10.k.a ✓ 104 1.a even 1 1 trivial
40.10.k.a ✓ 104 5.c odd 4 1 inner
40.10.k.a ✓ 104 8.d odd 2 1 inner
40.10.k.a ✓ 104 40.k even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(40, [\chi])\).