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Properties

Label 45.14.f.a
Level $45$
Weight $14$
Character orbit 45.f
Analytic conductor $48.254$
Analytic rank $0$
Dimension $52$
Inner twists $4$

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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] = [45,14,Mod(8,45)] 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("45.8"); S:= CuspForms(chi, 14); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(45, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 14, names="a")
 
Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 45.f (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(48.2539180284\)
Analytic rank: \(0\)
Dimension: \(52\)
Relative dimension: \(26\) 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.

\(\operatorname{Tr}(f)(q) = \) \( 52 q - 325928 q^{7} - 1271408 q^{10} - 14021276 q^{13} - 1225161608 q^{16} + 2938227248 q^{22} + 2536151056 q^{25} + 1169060768 q^{28} + 16189871792 q^{31} - 37236688556 q^{37} - 55693852632 q^{40} + 75180385888 q^{43}+ \cdots + 8822313963316 q^{97}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
8.1 −126.074 − 126.074i 0 23597.1i −19035.8 − 29297.5i 0 −34291.7 + 34291.7i 1.94218e6 − 1.94218e6i 0 −1.29372e6 + 6.09355e6i
8.2 −120.296 − 120.296i 0 20750.4i 33048.8 + 11335.0i 0 306864. − 306864.i 1.51073e6 − 1.51073e6i 0 −2.61208e6 − 5.33921e6i
8.3 −109.356 − 109.356i 0 15725.5i −32469.0 + 12902.2i 0 −331507. + 331507.i 823831. − 823831.i 0 4.96161e6 + 2.13975e6i
8.4 −100.396 − 100.396i 0 11966.8i 15126.4 + 31494.4i 0 −183725. + 183725.i 378973. − 378973.i 0 1.64329e6 − 4.68054e6i
8.5 −85.6698 − 85.6698i 0 6486.62i −11409.2 − 33023.2i 0 165100. − 165100.i −146100. + 146100.i 0 −1.85166e6 + 3.80652e6i
8.6 −84.1671 − 84.1671i 0 5976.19i 30856.8 − 16387.8i 0 249443. − 249443.i −186498. + 186498.i 0 −3.97644e6 − 1.21781e6i
8.7 −83.1707 − 83.1707i 0 5642.73i −24838.7 + 24571.1i 0 78008.2 − 78008.2i −212024. + 212024.i 0 4.10945e6 + 22257.1i
8.8 −66.6150 − 66.6150i 0 683.109i 34925.6 + 951.343i 0 −293513. + 293513.i −500205. + 500205.i 0 −2.26319e6 − 2.38994e6i
8.9 −52.0615 − 52.0615i 0 − 2771.21i −9842.91 − 33523.4i 0 −393635. + 393635.i −570761. + 570761.i 0 −1.23284e6 + 2.25771e6i
8.10 −47.7485 − 47.7485i 0 − 3632.16i −9988.30 + 33480.4i 0 153636. − 153636.i −564586. + 564586.i 0 2.07557e6 − 1.12171e6i
8.11 −27.6344 − 27.6344i 0 − 6664.68i −33354.0 − 10402.6i 0 315557. − 315557.i −410555. + 410555.i 0 634248. + 1.20919e6i
8.12 −8.52103 − 8.52103i 0 − 8046.78i 33109.7 + 11155.9i 0 −71448.6 + 71448.6i −138371. + 138371.i 0 −187069. − 377188.i
8.13 −7.08405 − 7.08405i 0 − 8091.63i 13766.0 − 32112.3i 0 −41969.6 + 41969.6i −115354. + 115354.i 0 −325004. + 129966.i
8.14 7.08405 + 7.08405i 0 − 8091.63i −13766.0 + 32112.3i 0 −41969.6 + 41969.6i 115354. − 115354.i 0 −325004. + 129966.i
8.15 8.52103 + 8.52103i 0 − 8046.78i −33109.7 − 11155.9i 0 −71448.6 + 71448.6i 138371. − 138371.i 0 −187069. − 377188.i
8.16 27.6344 + 27.6344i 0 − 6664.68i 33354.0 + 10402.6i 0 315557. − 315557.i 410555. − 410555.i 0 634248. + 1.20919e6i
8.17 47.7485 + 47.7485i 0 − 3632.16i 9988.30 − 33480.4i 0 153636. − 153636.i 564586. − 564586.i 0 2.07557e6 − 1.12171e6i
8.18 52.0615 + 52.0615i 0 − 2771.21i 9842.91 + 33523.4i 0 −393635. + 393635.i 570761. − 570761.i 0 −1.23284e6 + 2.25771e6i
8.19 66.6150 + 66.6150i 0 683.109i −34925.6 − 951.343i 0 −293513. + 293513.i 500205. − 500205.i 0 −2.26319e6 − 2.38994e6i
8.20 83.1707 + 83.1707i 0 5642.73i 24838.7 − 24571.1i 0 78008.2 − 78008.2i 212024. − 212024.i 0 4.10945e6 + 22257.1i
See all 52 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 8.26
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 45.14.f.a ✓ 52
3.b odd 2 1 inner 45.14.f.a ✓ 52
5.c odd 4 1 inner 45.14.f.a ✓ 52
15.e even 4 1 inner 45.14.f.a ✓ 52
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.14.f.a ✓ 52 1.a even 1 1 trivial
45.14.f.a ✓ 52 3.b odd 2 1 inner
45.14.f.a ✓ 52 5.c odd 4 1 inner
45.14.f.a ✓ 52 15.e even 4 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{14}^{\mathrm{new}}(45, [\chi])\).