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Properties

Label 40.7.g.a
Level $40$
Weight $7$
Character orbit 40.g
Analytic conductor $9.202$
Analytic rank $0$
Dimension $24$
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] = [40,7,Mod(11,40)]
 
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([1, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("40.11");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 40 = 2^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 40.g (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: \(9.20216334479\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q + 10 q^{2} + 24 q^{4} - 348 q^{6} - 440 q^{8} + 5832 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 24 q + 10 q^{2} + 24 q^{4} - 348 q^{6} - 440 q^{8} + 5832 q^{9} + 750 q^{10} + 2720 q^{11} - 8140 q^{12} - 3652 q^{14} + 9864 q^{16} + 28810 q^{18} + 3936 q^{19} - 3500 q^{20} - 38460 q^{22} + 96568 q^{24} - 75000 q^{25} + 74988 q^{26} - 68640 q^{27} - 83580 q^{28} + 13000 q^{30} - 12600 q^{32} + 44080 q^{33} + 164028 q^{34} - 145584 q^{36} - 174900 q^{38} - 57000 q^{40} + 20880 q^{41} - 291940 q^{42} + 145440 q^{43} - 31064 q^{44} + 276708 q^{46} - 271000 q^{48} - 609288 q^{49} - 31250 q^{50} + 847552 q^{51} + 655200 q^{52} + 99608 q^{54} - 302632 q^{56} - 27920 q^{57} - 412320 q^{58} - 1250912 q^{59} + 602500 q^{60} + 1670560 q^{62} + 223104 q^{64} - 401248 q^{66} + 1509600 q^{67} - 111960 q^{68} + 639000 q^{70} - 2269480 q^{72} - 514080 q^{73} + 1844268 q^{74} - 359040 q^{76} - 1364040 q^{78} - 212000 q^{80} + 1728088 q^{81} - 1280400 q^{82} + 3074400 q^{83} - 2910760 q^{84} - 1040460 q^{86} + 1242480 q^{88} - 587664 q^{89} - 704250 q^{90} - 5013504 q^{91} - 1956060 q^{92} - 2221332 q^{94} + 2834752 q^{96} - 2106720 q^{97} + 3820030 q^{98} + 6290144 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
11.1 −7.94339 − 0.950007i −37.1759 62.1950 + 15.0926i 55.9017i 295.303 + 35.3174i 609.349i −479.701 − 178.972i 653.051 53.1070 − 444.049i
11.2 −7.94339 + 0.950007i −37.1759 62.1950 − 15.0926i − 55.9017i 295.303 − 35.3174i − 609.349i −479.701 + 178.972i 653.051 53.1070 + 444.049i
11.3 −7.53564 − 2.68590i 7.36022 49.5719 + 40.4800i − 55.9017i −55.4640 − 19.7688i 283.113i −264.831 − 438.188i −674.827 −150.147 + 421.255i
11.4 −7.53564 + 2.68590i 7.36022 49.5719 − 40.4800i 55.9017i −55.4640 + 19.7688i − 283.113i −264.831 + 438.188i −674.827 −150.147 − 421.255i
11.5 −6.39296 − 4.80937i 7.20902 17.7400 + 61.4922i 55.9017i −46.0870 − 34.6708i − 329.323i 182.328 − 478.436i −677.030 268.852 − 357.377i
11.6 −6.39296 + 4.80937i 7.20902 17.7400 − 61.4922i − 55.9017i −46.0870 + 34.6708i 329.323i 182.328 + 478.436i −677.030 268.852 + 357.377i
11.7 −3.88349 − 6.99418i 40.8506 −33.8370 + 54.3236i − 55.9017i −158.643 − 285.716i − 204.762i 511.355 + 25.6964i 939.773 −390.986 + 217.094i
11.8 −3.88349 + 6.99418i 40.8506 −33.8370 − 54.3236i 55.9017i −158.643 + 285.716i 204.762i 511.355 − 25.6964i 939.773 −390.986 − 217.094i
11.9 −2.05625 − 7.73122i 26.3296 −55.5437 + 31.7947i 55.9017i −54.1402 − 203.560i 640.121i 360.023 + 364.043i −35.7518 432.189 − 114.948i
11.10 −2.05625 + 7.73122i 26.3296 −55.5437 − 31.7947i − 55.9017i −54.1402 + 203.560i − 640.121i 360.023 − 364.043i −35.7518 432.189 + 114.948i
11.11 −0.0347224 − 7.99992i −30.3557 −63.9976 + 0.555553i 55.9017i 1.05402 + 242.843i − 130.462i 6.66653 + 511.957i 192.466 447.209 − 1.94104i
11.12 −0.0347224 + 7.99992i −30.3557 −63.9976 − 0.555553i − 55.9017i 1.05402 − 242.843i 130.462i 6.66653 − 511.957i 192.466 447.209 + 1.94104i
11.13 1.57057 − 7.84432i 2.94131 −59.0666 − 24.6401i − 55.9017i 4.61952 − 23.0725i − 124.641i −286.053 + 424.638i −720.349 −438.511 − 87.7975i
11.14 1.57057 + 7.84432i 2.94131 −59.0666 + 24.6401i 55.9017i 4.61952 + 23.0725i 124.641i −286.053 − 424.638i −720.349 −438.511 + 87.7975i
11.15 3.85311 − 7.01096i 45.3926 −34.3071 − 54.0280i 55.9017i 174.903 − 318.246i − 457.976i −510.977 + 32.3496i 1331.49 391.924 + 215.395i
11.16 3.85311 + 7.01096i 45.3926 −34.3071 + 54.0280i − 55.9017i 174.903 + 318.246i 457.976i −510.977 − 32.3496i 1331.49 391.924 − 215.395i
11.17 5.23745 − 6.04724i −53.4302 −9.13830 − 63.3442i − 55.9017i −279.838 + 323.106i 358.999i −430.919 − 276.501i 2125.79 −338.051 − 292.782i
11.18 5.23745 + 6.04724i −53.4302 −9.13830 + 63.3442i 55.9017i −279.838 − 323.106i − 358.999i −430.919 + 276.501i 2125.79 −338.051 + 292.782i
11.19 6.54967 − 4.59367i −19.6320 21.7965 − 60.1740i 55.9017i −128.583 + 90.1829i − 409.691i −133.660 − 494.246i −343.585 256.794 + 366.138i
11.20 6.54967 + 4.59367i −19.6320 21.7965 + 60.1740i − 55.9017i −128.583 − 90.1829i 409.691i −133.660 + 494.246i −343.585 256.794 − 366.138i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 11.24
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 40.7.g.a ✓ 24
4.b odd 2 1 160.7.g.a 24
8.b even 2 1 160.7.g.a 24
8.d odd 2 1 inner 40.7.g.a ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
40.7.g.a ✓ 24 1.a even 1 1 trivial
40.7.g.a ✓ 24 8.d odd 2 1 inner
160.7.g.a 24 4.b odd 2 1
160.7.g.a 24 8.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(40, [\chi])\).