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Properties

Label 315.2.k.b
Level $315$
Weight $2$
Character orbit 315.k
Analytic conductor $2.515$
Analytic rank $0$
Dimension $24$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,2,Mod(16,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4, 0, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.16"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 315.k (of order \(3\), 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: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 24 q - q^{2} + q^{3} - 7 q^{4} - 24 q^{5} + 3 q^{6} + 7 q^{7} + 12 q^{8} + 9 q^{9} + q^{10} - 2 q^{11} - 16 q^{12} - 4 q^{13} - 13 q^{14} - q^{15} - 5 q^{16} - 7 q^{17} - 12 q^{18} - 2 q^{19} + 7 q^{20}+ \cdots - q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
16.1 −1.17004 − 2.02657i 1.32145 + 1.11972i −1.73798 + 3.01027i −1.00000 0.723033 − 3.98812i −1.00344 + 2.44808i 3.45385 0.492465 + 2.95930i 1.17004 + 2.02657i
16.2 −1.16277 − 2.01398i 0.434379 − 1.67670i −1.70408 + 2.95156i −1.00000 −3.88192 + 1.07479i 0.459529 − 2.60554i 3.27475 −2.62263 − 1.45664i 1.16277 + 2.01398i
16.3 −1.08176 − 1.87367i −1.55999 + 0.752618i −1.34041 + 2.32167i −1.00000 3.09769 + 2.10874i 2.14928 + 1.54292i 1.47299 1.86713 − 2.34815i 1.08176 + 1.87367i
16.4 −0.599034 − 1.03756i 1.72885 + 0.105192i 0.282317 − 0.488988i −1.00000 −0.926499 − 1.85680i 2.04342 − 1.68061i −3.07261 2.97787 + 0.363723i 0.599034 + 1.03756i
16.5 −0.259245 − 0.449026i −1.70948 + 0.278682i 0.865584 − 1.49923i −1.00000 0.568312 + 0.695356i −1.91816 + 1.82226i −1.93458 2.84467 − 0.952806i 0.259245 + 0.449026i
16.6 −0.148731 − 0.257610i 0.310785 − 1.70394i 0.955758 − 1.65542i −1.00000 −0.485175 + 0.173367i −2.64436 + 0.0857253i −1.16353 −2.80683 − 1.05912i 0.148731 + 0.257610i
16.7 0.154039 + 0.266804i 1.05264 − 1.37548i 0.952544 − 1.64985i −1.00000 0.529131 + 0.0689719i 2.20433 + 1.46319i 1.20307 −0.783880 − 2.89578i −0.154039 − 0.266804i
16.8 0.304907 + 0.528114i −1.22862 + 1.22086i 0.814064 − 1.41000i −1.00000 −1.01937 − 0.276604i 1.83653 − 1.90451i 2.21248 0.0190166 − 2.99994i −0.304907 − 0.528114i
16.9 0.517769 + 0.896802i −1.26722 − 1.18074i 0.463831 − 0.803378i −1.00000 0.402766 − 1.74780i −1.07729 − 2.41649i 3.03170 0.211690 + 2.99252i −0.517769 − 0.896802i
16.10 0.805191 + 1.39463i 1.04104 + 1.38428i −0.296664 + 0.513837i −1.00000 −1.09233 + 2.56648i 2.48389 + 0.911196i 2.26528 −0.832482 + 2.88218i −0.805191 − 1.39463i
16.11 0.859635 + 1.48893i −1.31121 − 1.13170i −0.477944 + 0.827824i −1.00000 0.557856 − 2.92514i 0.594106 + 2.57818i 1.79511 0.438533 + 2.96778i −0.859635 − 1.48893i
16.12 1.28004 + 2.21710i 1.68737 − 0.390872i −2.27701 + 3.94390i −1.00000 3.02651 + 3.24073i −1.62783 + 2.08571i −6.53853 2.69444 − 1.31909i −1.28004 − 2.21710i
256.1 −1.17004 + 2.02657i 1.32145 − 1.11972i −1.73798 − 3.01027i −1.00000 0.723033 + 3.98812i −1.00344 − 2.44808i 3.45385 0.492465 − 2.95930i 1.17004 − 2.02657i
256.2 −1.16277 + 2.01398i 0.434379 + 1.67670i −1.70408 − 2.95156i −1.00000 −3.88192 − 1.07479i 0.459529 + 2.60554i 3.27475 −2.62263 + 1.45664i 1.16277 − 2.01398i
256.3 −1.08176 + 1.87367i −1.55999 − 0.752618i −1.34041 − 2.32167i −1.00000 3.09769 − 2.10874i 2.14928 − 1.54292i 1.47299 1.86713 + 2.34815i 1.08176 − 1.87367i
256.4 −0.599034 + 1.03756i 1.72885 − 0.105192i 0.282317 + 0.488988i −1.00000 −0.926499 + 1.85680i 2.04342 + 1.68061i −3.07261 2.97787 − 0.363723i 0.599034 − 1.03756i
256.5 −0.259245 + 0.449026i −1.70948 − 0.278682i 0.865584 + 1.49923i −1.00000 0.568312 − 0.695356i −1.91816 − 1.82226i −1.93458 2.84467 + 0.952806i 0.259245 − 0.449026i
256.6 −0.148731 + 0.257610i 0.310785 + 1.70394i 0.955758 + 1.65542i −1.00000 −0.485175 − 0.173367i −2.64436 − 0.0857253i −1.16353 −2.80683 + 1.05912i 0.148731 − 0.257610i
256.7 0.154039 − 0.266804i 1.05264 + 1.37548i 0.952544 + 1.64985i −1.00000 0.529131 − 0.0689719i 2.20433 − 1.46319i 1.20307 −0.783880 + 2.89578i −0.154039 + 0.266804i
256.8 0.304907 − 0.528114i −1.22862 − 1.22086i 0.814064 + 1.41000i −1.00000 −1.01937 + 0.276604i 1.83653 + 1.90451i 2.21248 0.0190166 + 2.99994i −0.304907 + 0.528114i
See all 24 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 16.12
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.k.b ✓ 24
3.b odd 2 1 945.2.k.b 24
7.c even 3 1 315.2.l.b yes 24
9.c even 3 1 315.2.l.b yes 24
9.d odd 6 1 945.2.l.b 24
21.h odd 6 1 945.2.l.b 24
63.g even 3 1 inner 315.2.k.b ✓ 24
63.n odd 6 1 945.2.k.b 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.k.b ✓ 24 1.a even 1 1 trivial
315.2.k.b ✓ 24 63.g even 3 1 inner
315.2.l.b yes 24 7.c even 3 1
315.2.l.b yes 24 9.c even 3 1
945.2.k.b 24 3.b odd 2 1
945.2.k.b 24 63.n odd 6 1
945.2.l.b 24 9.d odd 6 1
945.2.l.b 24 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{24} + T_{2}^{23} + 16 T_{2}^{22} + 9 T_{2}^{21} + 157 T_{2}^{20} + 62 T_{2}^{19} + 930 T_{2}^{18} + \cdots + 9 \) acting on \(S_{2}^{\mathrm{new}}(315, [\chi])\). Copy content Toggle raw display