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Properties

Label 325.6.b.i
Level $325$
Weight $6$
Character orbit 325.b
Analytic conductor $52.125$
Analytic rank $0$
Dimension $22$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [325,6,Mod(274,325)] 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("325.274"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [22,0,0,-374,0,702] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(22\)
Twist minimal: yes
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.

\(\operatorname{Tr}(f)(q) = \) \( 22 q - 374 q^{4} + 702 q^{6} - 2744 q^{9} + 2552 q^{11} - 1156 q^{14} + 11414 q^{16} - 7040 q^{19} + 3412 q^{21} - 15446 q^{24} + 1690 q^{26} - 36850 q^{29} + 18136 q^{31} + 28910 q^{34} + 75368 q^{36}+ \cdots - 853396 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1 − 11.2252i 19.1897i −94.0044 0 215.408 − 70.9909i 696.009i −125.246 0
274.2 − 10.1023i 24.5176i −70.0572 0 247.685 − 72.1186i 384.466i −358.110 0
274.3 − 9.60154i − 25.9222i −60.1895 0 −248.893 181.554i 270.662i −428.963 0
274.4 − 9.27946i − 5.85765i −54.1083 0 −54.3558 90.4001i 205.153i 208.688 0
274.5 − 6.07859i 25.6203i −4.94922 0 155.735 − 137.301i − 164.431i −413.398 0
274.6 − 5.84001i 12.3562i −2.10573 0 72.1603 − 135.041i − 174.583i 90.3246 0
274.7 − 5.83767i − 12.7661i −2.07840 0 −74.5243 − 231.930i − 174.672i 80.0268 0
274.8 − 3.78432i − 8.20788i 17.6789 0 −31.0612 88.9969i − 188.001i 175.631 0
274.9 − 2.89241i − 3.45730i 23.6340 0 −9.99993 148.288i − 160.916i 231.047 0
274.10 − 1.96479i 29.7675i 28.1396 0 58.4868 176.972i − 118.162i −643.103 0
274.11 − 0.979695i 20.7822i 31.0402 0 20.3602 233.896i − 61.7601i −188.898 0
274.12 0.979695i − 20.7822i 31.0402 0 20.3602 − 233.896i 61.7601i −188.898 0
274.13 1.96479i − 29.7675i 28.1396 0 58.4868 − 176.972i 118.162i −643.103 0
274.14 2.89241i 3.45730i 23.6340 0 −9.99993 − 148.288i 160.916i 231.047 0
274.15 3.78432i 8.20788i 17.6789 0 −31.0612 − 88.9969i 188.001i 175.631 0
274.16 5.83767i 12.7661i −2.07840 0 −74.5243 231.930i 174.672i 80.0268 0
274.17 5.84001i − 12.3562i −2.10573 0 72.1603 135.041i 174.583i 90.3246 0
274.18 6.07859i − 25.6203i −4.94922 0 155.735 137.301i 164.431i −413.398 0
274.19 9.27946i 5.85765i −54.1083 0 −54.3558 − 90.4001i − 205.153i 208.688 0
274.20 9.60154i 25.9222i −60.1895 0 −248.893 − 181.554i − 270.662i −428.963 0
See all 22 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.22
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.6.b.i 22
5.b even 2 1 inner 325.6.b.i 22
5.c odd 4 1 325.6.a.j ✓ 11
5.c odd 4 1 325.6.a.k yes 11
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.6.a.j ✓ 11 5.c odd 4 1
325.6.a.k yes 11 5.c odd 4 1
325.6.b.i 22 1.a even 1 1 trivial
325.6.b.i 22 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{22} + 539 T_{2}^{20} + 122167 T_{2}^{18} + 15190681 T_{2}^{16} + 1135601504 T_{2}^{14} + \cdots + 19\!\cdots\!00 \) acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display