↑

Properties

Label 325.10.a.k
Level $325$
Weight $10$
Character orbit 325.a
Self dual yes
Analytic conductor $167.387$
Analytic rank $1$
Dimension $27$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [325,10,Mod(1,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [27,-48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(167.386646753\)
Analytic rank: \(1\)
Dimension: \(27\)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 27 q - 48 q^{2} - 324 q^{3} + 6570 q^{4} - 1786 q^{6} - 3736 q^{7} - 36864 q^{8} + 214173 q^{9} - 66096 q^{11} - 114398 q^{12} + 771147 q^{13} - 359458 q^{14} + 998622 q^{16} - 779040 q^{17} - 1709648 q^{18}+ \cdots + 2142297632 q^{99}+O(q^{100}) \) 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   \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −44.6081 −211.776 1477.88 0 9446.90 2776.39 −43086.1 25165.9 0
1.2 −40.6502 264.880 1140.44 0 −10767.4 −1410.33 −25546.3 50478.3 0
1.3 −40.6286 113.166 1138.68 0 −4597.79 10738.1 −25461.2 −6876.35 0
1.4 −37.2225 −132.249 873.511 0 4922.65 −6816.48 −13456.3 −2193.09 0
1.5 −31.3739 61.3825 472.322 0 −1925.81 −6404.18 1244.87 −15915.2 0
1.6 −30.3783 124.713 410.841 0 −3788.56 2698.71 3073.04 −4129.74 0
1.7 −29.2796 −270.897 345.298 0 7931.76 −6108.11 4880.99 53702.0 0
1.8 −22.9877 −40.8325 16.4339 0 938.645 7844.21 11391.9 −18015.7 0
1.9 −20.8068 −72.3996 −79.0763 0 1506.40 5461.35 12298.4 −14441.3 0
1.10 −16.7669 206.316 −230.871 0 −3459.29 −7929.76 12455.6 22883.5 0
1.11 −16.3195 −195.494 −245.673 0 3190.36 4181.36 12364.9 18534.7 0
1.12 −11.5026 −50.8152 −379.691 0 584.504 −11334.3 10256.7 −17100.8 0
1.13 −9.94975 186.339 −413.003 0 −1854.02 2543.77 9203.54 15039.1 0
1.14 −3.55822 −263.202 −499.339 0 936.531 8747.23 3598.56 49592.5 0
1.15 5.61283 −84.3580 −480.496 0 −473.487 1730.99 −5570.71 −12566.7 0
1.16 7.88453 −18.3933 −449.834 0 −145.022 −3534.00 −7583.61 −19344.7 0
1.17 9.01531 96.0448 −430.724 0 865.874 −6689.08 −8498.95 −10458.4 0
1.18 9.91262 218.926 −413.740 0 2170.13 1909.27 −9176.51 28245.4 0
1.19 18.1415 10.3900 −182.888 0 188.489 11293.6 −12606.3 −19575.0 0
1.20 24.0061 −127.628 64.2915 0 −3063.86 −1056.77 −10747.7 −3394.00 0
See all 27 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.27
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(13\) \( -1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.10.a.k 27
5.b even 2 1 325.10.a.l 27
5.c odd 4 2 65.10.b.a ✓ 54
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.10.b.a ✓ 54 5.c odd 4 2
325.10.a.k 27 1.a even 1 1 trivial
325.10.a.l 27 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{27} + 48 T_{2}^{26} - 9045 T_{2}^{25} - 442352 T_{2}^{24} + 35528421 T_{2}^{23} + \cdots - 28\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(325))\). Copy content Toggle raw display