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Properties

Label 2401.4.a.d
Level $2401$
Weight $4$
Character orbit 2401.a
Self dual yes
Analytic conductor $141.664$
Analytic rank $0$
Dimension $39$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2401,4,Mod(1,2401)] 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("2401.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2401, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2401 = 7^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2401.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [39,1,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(141.663585924\)
Analytic rank: \(0\)
Dimension: \(39\)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 39 q + q^{2} + q^{3} + 145 q^{4} + 27 q^{5} + 41 q^{6} - 12 q^{8} + 312 q^{9} + 78 q^{10} + q^{11} - 91 q^{12} + 77 q^{13} - 161 q^{15} + 461 q^{16} + 211 q^{17} + 8 q^{18} + 314 q^{19} + 476 q^{20}+ \cdots + 5384 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.52124 −4.05315 22.4841 14.0923 22.3784 0 −79.9700 −10.5720 −77.8068
1.2 −5.47147 −8.31145 21.9370 −5.94042 45.4758 0 −76.2558 42.0802 32.5028
1.3 −5.14785 6.96013 18.5003 3.24701 −35.8297 0 −54.0542 21.4435 −16.7151
1.4 −4.68909 5.32877 13.9876 −6.47646 −24.9871 0 −28.0763 1.39583 30.3687
1.5 −4.44388 6.83693 11.7481 −2.20264 −30.3825 0 −16.6559 19.7437 9.78829
1.6 −4.34611 −2.32971 10.8887 −2.24811 10.1252 0 −12.5546 −21.5725 9.77055
1.7 −3.93571 −3.09696 7.48982 −18.2977 12.1887 0 2.00794 −17.4089 72.0144
1.8 −3.74761 2.12177 6.04460 10.5286 −7.95158 0 7.32810 −22.4981 −39.4572
1.9 −3.57033 −9.05210 4.74728 15.7342 32.3190 0 11.6133 54.9406 −56.1764
1.10 −2.58597 3.61260 −1.31274 3.92216 −9.34209 0 24.0825 −13.9491 −10.1426
1.11 −2.46450 −5.32585 −1.92624 15.0765 13.1256 0 24.4632 1.36464 −37.1561
1.12 −2.34370 −8.19575 −2.50708 0.508590 19.2083 0 24.6254 40.1702 −1.19198
1.13 −2.11024 −5.68890 −3.54689 −12.6417 12.0050 0 24.3667 5.36363 26.6769
1.14 −2.08878 −2.27918 −3.63702 −20.5308 4.76069 0 24.3071 −21.8054 42.8842
1.15 −2.07454 7.56379 −3.69627 −6.53935 −15.6914 0 24.2644 30.2110 13.5662
1.16 −1.58052 3.81461 −5.50196 14.4403 −6.02907 0 21.3401 −12.4487 −22.8232
1.17 −1.28751 1.89643 −6.34231 17.6887 −2.44167 0 18.4659 −23.4036 −22.7744
1.18 −0.904464 9.08614 −7.18194 −3.14137 −8.21809 0 13.7315 55.5579 2.84125
1.19 −0.275911 3.60983 −7.92387 −3.79895 −0.995991 0 4.39357 −13.9691 1.04817
1.20 0.246994 9.97745 −7.93899 −15.8609 2.46437 0 −3.93684 72.5495 −3.91755
See all 39 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.39
Significant digits:
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Atkin-Lehner signs

\( p \) Sign
\(7\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2401.4.a.d 39
7.b odd 2 1 2401.4.a.c 39
49.e even 7 2 49.4.e.a ✓ 78
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.4.e.a ✓ 78 49.e even 7 2
2401.4.a.c 39 7.b odd 2 1
2401.4.a.d 39 1.a even 1 1 trivial