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Properties

Label 350.4.g.b
Level $350$
Weight $4$
Character orbit 350.g
Analytic conductor $20.651$
Analytic rank $0$
Dimension $24$
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] = [350,4,Mod(293,350)] 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("350.293"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(350, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([3, 2])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 350.g (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(20.6506685020\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 70)
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.

\(\operatorname{Tr}(f)(q) = \) \( 24 q - 4 q^{7} + 40 q^{11} - 384 q^{16} + 16 q^{18} - 432 q^{21} + 192 q^{22} - 336 q^{23} + 16 q^{28} - 896 q^{36} + 456 q^{37} - 1280 q^{42} + 1264 q^{43} + 1792 q^{46} + 2184 q^{51} + 2256 q^{53} - 384 q^{56}+ \cdots + 4064 q^{98}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
293.1 −1.41421 + 1.41421i −5.70899 + 5.70899i − 4.00000i 0 − 16.1475i −18.1966 − 3.44728i 5.65685 + 5.65685i − 38.1851i 0
293.2 −1.41421 + 1.41421i 5.70899 − 5.70899i − 4.00000i 0 16.1475i 3.44728 + 18.1966i 5.65685 + 5.65685i − 38.1851i 0
293.3 −1.41421 + 1.41421i −3.17701 + 3.17701i − 4.00000i 0 − 8.98595i −2.27797 + 18.3796i 5.65685 + 5.65685i 6.81319i 0
293.4 −1.41421 + 1.41421i 3.17701 − 3.17701i − 4.00000i 0 8.98595i −18.3796 + 2.27797i 5.65685 + 5.65685i 6.81319i 0
293.5 −1.41421 + 1.41421i −3.53853 + 3.53853i − 4.00000i 0 − 10.0085i 14.8362 − 11.0854i 5.65685 + 5.65685i 1.95766i 0
293.6 −1.41421 + 1.41421i 3.53853 − 3.53853i − 4.00000i 0 10.0085i 11.0854 − 14.8362i 5.65685 + 5.65685i 1.95766i 0
293.7 1.41421 − 1.41421i −6.82881 + 6.82881i − 4.00000i 0 19.3148i 16.1862 + 9.00041i −5.65685 − 5.65685i − 66.2652i 0
293.8 1.41421 − 1.41421i 6.82881 − 6.82881i − 4.00000i 0 − 19.3148i −9.00041 − 16.1862i −5.65685 − 5.65685i − 66.2652i 0
293.9 1.41421 − 1.41421i −1.78083 + 1.78083i − 4.00000i 0 5.03694i 18.4362 + 1.76212i −5.65685 − 5.65685i 20.6573i 0
293.10 1.41421 − 1.41421i 1.78083 − 1.78083i − 4.00000i 0 − 5.03694i −1.76212 − 18.4362i −5.65685 − 5.65685i 20.6573i 0
293.11 1.41421 − 1.41421i −1.99723 + 1.99723i − 4.00000i 0 5.64901i −18.4083 − 2.03365i −5.65685 − 5.65685i 19.0222i 0
293.12 1.41421 − 1.41421i 1.99723 − 1.99723i − 4.00000i 0 − 5.64901i 2.03365 + 18.4083i −5.65685 − 5.65685i 19.0222i 0
307.1 −1.41421 − 1.41421i −5.70899 − 5.70899i 4.00000i 0 16.1475i −18.1966 + 3.44728i 5.65685 − 5.65685i 38.1851i 0
307.2 −1.41421 − 1.41421i 5.70899 + 5.70899i 4.00000i 0 − 16.1475i 3.44728 − 18.1966i 5.65685 − 5.65685i 38.1851i 0
307.3 −1.41421 − 1.41421i −3.17701 − 3.17701i 4.00000i 0 8.98595i −2.27797 − 18.3796i 5.65685 − 5.65685i − 6.81319i 0
307.4 −1.41421 − 1.41421i 3.17701 + 3.17701i 4.00000i 0 − 8.98595i −18.3796 − 2.27797i 5.65685 − 5.65685i − 6.81319i 0
307.5 −1.41421 − 1.41421i −3.53853 − 3.53853i 4.00000i 0 10.0085i 14.8362 + 11.0854i 5.65685 − 5.65685i − 1.95766i 0
307.6 −1.41421 − 1.41421i 3.53853 + 3.53853i 4.00000i 0 − 10.0085i 11.0854 + 14.8362i 5.65685 − 5.65685i − 1.95766i 0
307.7 1.41421 + 1.41421i −6.82881 − 6.82881i 4.00000i 0 − 19.3148i 16.1862 − 9.00041i −5.65685 + 5.65685i 66.2652i 0
307.8 1.41421 + 1.41421i 6.82881 + 6.82881i 4.00000i 0 19.3148i −9.00041 + 16.1862i −5.65685 + 5.65685i 66.2652i 0
See all 24 embeddings
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 293.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.b odd 2 1 inner
35.f even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.4.g.b 24
5.b even 2 1 70.4.g.a ✓ 24
5.c odd 4 1 70.4.g.a ✓ 24
5.c odd 4 1 inner 350.4.g.b 24
7.b odd 2 1 inner 350.4.g.b 24
35.c odd 2 1 70.4.g.a ✓ 24
35.f even 4 1 70.4.g.a ✓ 24
35.f even 4 1 inner 350.4.g.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.4.g.a ✓ 24 5.b even 2 1
70.4.g.a ✓ 24 5.c odd 4 1
70.4.g.a ✓ 24 35.c odd 2 1
70.4.g.a ✓ 24 35.f even 4 1
350.4.g.b 24 1.a even 1 1 trivial
350.4.g.b 24 5.c odd 4 1 inner
350.4.g.b 24 7.b odd 2 1 inner
350.4.g.b 24 35.f even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{24} + 14086 T_{3}^{20} + 52066669 T_{3}^{16} + 46842061436 T_{3}^{12} + 13890934895476 T_{3}^{8} + \cdots + 24\!\cdots\!04 \) acting on \(S_{4}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display