↑

Properties

Label 961.4.a.l
Level $961$
Weight $4$
Character orbit 961.a
Self dual yes
Analytic conductor $56.701$
Analytic rank $1$
Dimension $28$
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] = [961,4,Mod(1,961)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("961.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 961.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [28,2,-19] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(56.7008355155\)
Analytic rank: \(1\)
Dimension: \(28\)
Twist minimal: no (minimal twist has level 31)
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.

\(\operatorname{Tr}(f)(q) = \) \( 28 q + 2 q^{2} - 19 q^{3} + 104 q^{4} - 53 q^{6} + 31 q^{7} + 99 q^{8} + 211 q^{9} - 3 q^{10} - 185 q^{11} - 266 q^{12} - 145 q^{13} - 225 q^{14} - 261 q^{15} + 284 q^{16} - 259 q^{17} + 305 q^{18} + 73 q^{19}+ \cdots - 6383 q^{99}+O(q^{100}) \) 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 −5.35407 −3.10398 20.6660 −8.73629 16.6189 −11.1861 −67.8149 −17.3653 46.7747
1.2 −4.81357 3.09607 15.1704 4.88170 −14.9031 31.7899 −34.5152 −17.4143 −23.4984
1.3 −4.48038 −8.22565 12.0738 −14.4905 36.8540 26.0637 −18.2522 40.6613 64.9229
1.4 −4.36917 5.78847 11.0896 −10.3522 −25.2908 16.8187 −13.4990 6.50634 45.2304
1.5 −4.17989 2.81358 9.47150 13.7074 −11.7605 −1.03173 −6.15072 −19.0838 −57.2953
1.6 −3.24874 6.74612 2.55430 4.69567 −21.9164 −2.20251 17.6917 18.5101 −15.2550
1.7 −3.10167 −2.93269 1.62033 −3.48617 9.09623 −26.7429 19.7876 −18.3993 10.8129
1.8 −3.02474 0.505147 1.14905 17.4734 −1.52794 −21.7088 20.7223 −26.7448 −52.8526
1.9 −2.81923 −7.72414 −0.0519391 −7.73140 21.7761 17.1626 22.7003 32.6623 21.7966
1.10 −2.02423 −8.82091 −3.90251 19.4888 17.8555 −18.7764 24.0934 50.8084 −39.4498
1.11 −1.54110 6.36283 −5.62503 −12.9956 −9.80573 6.11350 20.9975 13.4856 20.0275
1.12 −0.917157 9.23780 −7.15882 −2.60091 −8.47251 −3.67739 13.9030 58.3369 2.38544
1.13 −0.882133 −2.84427 −7.22184 −15.8441 2.50903 9.55043 13.4277 −18.9101 13.9766
1.14 −0.154880 −4.64335 −7.97601 1.44763 0.719162 −13.6137 2.47436 −5.43929 −0.224209
1.15 −0.0714956 −8.95671 −7.99489 13.8643 0.640365 19.5804 1.14356 53.2226 −0.991239
1.16 0.735972 −1.39786 −7.45835 −7.32633 −1.02878 −14.6726 −11.3769 −25.0460 −5.39197
1.17 0.923673 8.16743 −7.14683 4.12868 7.54404 13.2879 −13.9907 39.7069 3.81355
1.18 1.47570 −0.401143 −5.82232 −11.8034 −0.591966 11.0204 −20.3975 −26.8391 −17.4182
1.19 1.71255 3.61531 −5.06719 6.64692 6.19139 −2.57380 −22.3782 −13.9295 11.3832
1.20 2.83254 0.917938 0.0233066 12.0918 2.60010 32.1562 −22.5943 −26.1574 34.2506
See all 28 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.28
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(31\) \( -1 \)

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 961.4.a.l 28
31.b odd 2 1 961.4.a.m 28
31.h odd 30 2 31.4.g.a ✓ 56
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.4.g.a ✓ 56 31.h odd 30 2
961.4.a.l 28 1.a even 1 1 trivial
961.4.a.m 28 31.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(961))\):

\( T_{2}^{28} - 2 T_{2}^{27} - 162 T_{2}^{26} + 283 T_{2}^{25} + 11617 T_{2}^{24} - 17245 T_{2}^{23} + \cdots - 3201675264 \) Copy content Toggle raw display
\( T_{3}^{28} + 19 T_{3}^{27} - 303 T_{3}^{26} - 7289 T_{3}^{25} + 34648 T_{3}^{24} + 1220253 T_{3}^{23} + \cdots + 29\!\cdots\!21 \) Copy content Toggle raw display