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Properties

Label 315.5.e.f
Level $315$
Weight $5$
Character orbit 315.e
Analytic conductor $32.562$
Analytic rank $0$
Dimension $32$
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] = [315,5,Mod(244,315)] 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("315.244"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 315.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [32,0,0,-256,0,0,0,0,0,0,-168] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.5615383714\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 32 q - 256 q^{4} - 168 q^{11} - 588 q^{14} + 944 q^{16} - 3256 q^{25} - 864 q^{29} + 444 q^{35} - 4824 q^{44} + 25008 q^{46} - 3256 q^{49} + 8736 q^{50} + 32532 q^{56} - 14176 q^{64} - 21576 q^{65} - 7104 q^{70}+ \cdots + 34728 q^{95}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
244.1 − 3.68935i 0 2.38872 17.0477 − 18.2859i 0 44.0892 + 21.3809i − 67.8424i 0 −67.4631 − 62.8950i
244.2 3.68935i 0 2.38872 17.0477 + 18.2859i 0 44.0892 − 21.3809i 67.8424i 0 −67.4631 + 62.8950i
244.3 − 3.00009i 0 6.99947 −20.9370 − 13.6617i 0 −24.3551 − 42.5186i − 69.0004i 0 −40.9862 + 62.8129i
244.4 3.00009i 0 6.99947 −20.9370 + 13.6617i 0 −24.3551 + 42.5186i 69.0004i 0 −40.9862 − 62.8129i
244.5 − 7.63204i 0 −42.2480 0.322863 − 24.9979i 0 5.28112 − 48.7146i 200.326i 0 −190.785 − 2.46410i
244.6 7.63204i 0 −42.2480 0.322863 + 24.9979i 0 5.28112 + 48.7146i − 200.326i 0 −190.785 + 2.46410i
244.7 − 6.29647i 0 −23.6455 23.5093 + 8.50375i 0 −46.8756 + 14.2714i 48.1400i 0 53.5436 − 148.026i
244.8 6.29647i 0 −23.6455 23.5093 − 8.50375i 0 −46.8756 − 14.2714i − 48.1400i 0 53.5436 + 148.026i
244.9 − 0.522389i 0 15.7271 19.9863 + 15.0183i 0 2.38113 + 48.9421i − 16.5739i 0 7.84539 − 10.4406i
244.10 0.522389i 0 15.7271 19.9863 − 15.0183i 0 2.38113 − 48.9421i 16.5739i 0 7.84539 + 10.4406i
244.11 − 7.63204i 0 −42.2480 −0.322863 + 24.9979i 0 −5.28112 − 48.7146i 200.326i 0 190.785 + 2.46410i
244.12 7.63204i 0 −42.2480 −0.322863 − 24.9979i 0 −5.28112 + 48.7146i − 200.326i 0 190.785 − 2.46410i
244.13 − 5.52056i 0 −14.4765 −16.5868 − 18.7050i 0 0.191033 + 48.9996i − 8.41031i 0 −103.262 + 91.5682i
244.14 5.52056i 0 −14.4765 −16.5868 + 18.7050i 0 0.191033 − 48.9996i 8.41031i 0 −103.262 − 91.5682i
244.15 − 2.44594i 0 10.0174 7.22188 − 23.9342i 0 46.9408 − 14.0558i − 63.6369i 0 −58.5415 − 17.6643i
244.16 2.44594i 0 10.0174 7.22188 + 23.9342i 0 46.9408 + 14.0558i 63.6369i 0 −58.5415 + 17.6643i
244.17 − 5.89598i 0 −18.7625 9.19226 − 23.2487i 0 −47.1764 − 13.2433i 16.2879i 0 −137.074 − 54.1974i
244.18 5.89598i 0 −18.7625 9.19226 + 23.2487i 0 −47.1764 + 13.2433i − 16.2879i 0 −137.074 + 54.1974i
244.19 − 6.29647i 0 −23.6455 −23.5093 − 8.50375i 0 46.8756 + 14.2714i 48.1400i 0 −53.5436 + 148.026i
244.20 6.29647i 0 −23.6455 −23.5093 + 8.50375i 0 46.8756 − 14.2714i − 48.1400i 0 −53.5436 − 148.026i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 244.32
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.5.e.f 32
3.b odd 2 1 105.5.e.a ✓ 32
5.b even 2 1 inner 315.5.e.f 32
7.b odd 2 1 inner 315.5.e.f 32
15.d odd 2 1 105.5.e.a ✓ 32
15.e even 4 2 525.5.h.e 32
21.c even 2 1 105.5.e.a ✓ 32
35.c odd 2 1 inner 315.5.e.f 32
105.g even 2 1 105.5.e.a ✓ 32
105.k odd 4 2 525.5.h.e 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.5.e.a ✓ 32 3.b odd 2 1
105.5.e.a ✓ 32 15.d odd 2 1
105.5.e.a ✓ 32 21.c even 2 1
105.5.e.a ✓ 32 105.g even 2 1
315.5.e.f 32 1.a even 1 1 trivial
315.5.e.f 32 5.b even 2 1 inner
315.5.e.f 32 7.b odd 2 1 inner
315.5.e.f 32 35.c odd 2 1 inner
525.5.h.e 32 15.e even 4 2
525.5.h.e 32 105.k odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(315, [\chi])\):

\( T_{2}^{16} + 192 T_{2}^{14} + 14730 T_{2}^{12} + 580104 T_{2}^{10} + 12511581 T_{2}^{8} + \cdots + 489327696 \) Copy content Toggle raw display
\( T_{13}^{16} - 314528 T_{13}^{14} + 38146512148 T_{13}^{12} + \cdots + 68\!\cdots\!96 \) Copy content Toggle raw display