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Properties

Label 2445.4.a.i
Level $2445$
Weight $4$
Character orbit 2445.a
Self dual yes
Analytic conductor $144.260$
Analytic rank $0$
Dimension $43$
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] = [2445,4,Mod(1,2445)] 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("2445.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2445, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 2445 = 3 \cdot 5 \cdot 163 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2445.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [43,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(144.259669964\)
Analytic rank: \(0\)
Dimension: \(43\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 43 q + 18 q^{2} + 129 q^{3} + 198 q^{4} + 215 q^{5} + 54 q^{6} + 137 q^{7} + 201 q^{8} + 387 q^{9} + 90 q^{10} + 137 q^{11} + 594 q^{12} + 212 q^{13} + 246 q^{14} + 645 q^{15} + 930 q^{16} + 547 q^{17}+ \cdots + 1233 q^{99}+O(q^{100}) \) 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 −5.35636 3.00000 20.6906 5.00000 −16.0691 1.10380 −67.9753 9.00000 −26.7818
1.2 −5.21979 3.00000 19.2462 5.00000 −15.6594 5.24462 −58.7030 9.00000 −26.0990
1.3 −5.15244 3.00000 18.5476 5.00000 −15.4573 27.2982 −54.3458 9.00000 −25.7622
1.4 −4.81383 3.00000 15.1730 5.00000 −14.4415 −18.0932 −34.5296 9.00000 −24.0692
1.5 −4.35152 3.00000 10.9357 5.00000 −13.0546 −18.0483 −12.7748 9.00000 −21.7576
1.6 −4.24931 3.00000 10.0566 5.00000 −12.7479 19.1518 −8.73932 9.00000 −21.2466
1.7 −3.77545 3.00000 6.25402 5.00000 −11.3264 32.7551 6.59184 9.00000 −18.8773
1.8 −3.73401 3.00000 5.94286 5.00000 −11.2020 −31.4152 7.68138 9.00000 −18.6701
1.9 −3.12518 3.00000 1.76674 5.00000 −9.37554 20.6956 19.4800 9.00000 −15.6259
1.10 −3.10884 3.00000 1.66487 5.00000 −9.32651 7.59013 19.6949 9.00000 −15.5442
1.11 −3.00113 3.00000 1.00679 5.00000 −9.00339 −5.88728 20.9876 9.00000 −15.0057
1.12 −2.68894 3.00000 −0.769593 5.00000 −8.06682 −16.5732 23.5809 9.00000 −13.4447
1.13 −2.59675 3.00000 −1.25690 5.00000 −7.79024 −31.1905 24.0378 9.00000 −12.9837
1.14 −2.30192 3.00000 −2.70115 5.00000 −6.90577 31.0368 24.6332 9.00000 −11.5096
1.15 −1.51577 3.00000 −5.70243 5.00000 −4.54732 −14.0193 20.7698 9.00000 −7.57887
1.16 −1.30480 3.00000 −6.29750 5.00000 −3.91440 12.1054 18.6554 9.00000 −6.52400
1.17 −1.23734 3.00000 −6.46899 5.00000 −3.71202 −14.9528 17.9031 9.00000 −6.18670
1.18 −0.624529 3.00000 −7.60996 5.00000 −1.87359 14.9776 9.74888 9.00000 −3.12265
1.19 −0.0381026 3.00000 −7.99855 5.00000 −0.114308 11.4468 0.609586 9.00000 −0.190513
1.20 0.00992648 3.00000 −7.99990 5.00000 0.0297794 2.34539 −0.158823 9.00000 0.0496324
See all 43 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.43
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(163\) \( +1 \)

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 2445.4.a.i ✓ 43
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2445.4.a.i ✓ 43 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{43} - 18 T_{2}^{42} - 109 T_{2}^{41} + 3743 T_{2}^{40} - 2670 T_{2}^{39} - 347968 T_{2}^{38} + \cdots + 97630374854656 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2445))\). Copy content Toggle raw display