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Properties

Label 961.2.a.i
Level $961$
Weight $2$
Character orbit 961.a
Self dual yes
Analytic conductor $7.674$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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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] = [961,2,Mod(1,961)] 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("961.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
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, 2, names="a")
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 961.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,-3,8,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.67362363425\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: 8.8.2051578125.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3} + 1) q^{2} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{3} + ( - \beta_{6} + \beta_1 + 1) q^{4} + ( - \beta_{6} + \beta_{5} + \cdots - \beta_1) q^{5} + (2 \beta_{6} + \beta_{5} + \beta_{2} + \cdots - 2) q^{6}+ \cdots + ( - 2 \beta_{7} - \beta_{6} + \cdots + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 3 q^{3} + 8 q^{4} + 3 q^{5} - 11 q^{6} - 2 q^{7} - 9 q^{8} + 5 q^{9} - 13 q^{10} - 18 q^{11} - 8 q^{13} - 9 q^{14} - 18 q^{15} + 4 q^{16} - 14 q^{17} + 23 q^{18} - 6 q^{19} - 7 q^{20} + q^{21}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 9x^{6} + 19x^{5} + 14x^{4} - 28x^{3} - 11x^{2} + 6x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 7\nu^{4} + 2\nu^{3} + 4\nu^{2} + 5\nu + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 7\nu^{5} - 9\nu^{4} + \nu^{3} - \nu^{2} - 11\nu + 4 ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 17\nu^{4} - 4\nu^{3} - 26\nu^{2} - \nu + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 10\nu^{5} + 2\nu^{4} + 25\nu^{3} - 4\nu^{2} - 14\nu + 3 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 3\nu^{6} - 7\nu^{5} + 26\nu^{4} - 5\nu^{3} - 28\nu^{2} + 10\nu + 6 ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 4\nu^{6} - 7\nu^{5} + 36\nu^{4} - 4\nu^{3} - 47\nu^{2} - \nu + 2 ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{4} - \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} - 2\beta_{4} + \beta_{3} - \beta_{2} + 5\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} + 7\beta_{4} - 6\beta_{3} + 2\beta_{2} - 3\beta _1 + 17 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} + 3\beta_{6} - \beta_{5} - 19\beta_{4} + 10\beta_{3} - 7\beta_{2} + 30\beta _1 - 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{7} - 38\beta_{6} + 49\beta_{4} - 40\beta_{3} + 19\beta_{2} - 36\beta _1 + 108 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55\beta_{7} + 38\beta_{6} - 7\beta_{5} - 150\beta_{4} + 83\beta_{3} - 49\beta_{2} + 195\beta _1 - 150 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.28064
1.76152
−2.73366
−1.04940
−0.143490
0.431370
−0.662608
2.11562
−2.69016 1.41916 5.23694 0.608384 −3.81777 −1.72688 −8.70786 −0.985973 −1.63665
1.2 −1.85021 0.499354 1.42326 1.20736 −0.923909 3.73304 1.06708 −2.75065 −2.23387
1.3 −0.689493 −0.902401 −1.52460 3.70752 0.622199 0.763394 2.43019 −2.18567 −2.55631
1.4 0.351432 −2.89329 −1.87650 2.97323 −1.01680 −1.08213 −1.36233 5.37114 1.04489
1.5 1.23217 2.07497 −0.481752 −1.54562 2.55673 −3.80376 −3.05795 1.30552 −1.90447
1.6 1.26660 1.48431 −0.395721 −3.80032 1.88004 2.18899 −3.03442 −0.796809 −4.81349
1.7 2.07212 −2.13939 2.29369 2.34791 −4.43308 −3.67454 0.608557 1.57700 4.86516
1.8 2.30753 −2.54272 3.32468 −2.49846 −5.86740 1.60188 3.05673 3.46545 −5.76526
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 961.2.a.i 8
3.b odd 2 1 8649.2.a.bf 8
31.b odd 2 1 961.2.a.j 8
31.c even 3 2 961.2.c.j 16
31.d even 5 2 961.2.d.o 16
31.d even 5 2 961.2.d.p 16
31.e odd 6 2 961.2.c.i 16
31.f odd 10 2 961.2.d.n 16
31.f odd 10 2 961.2.d.q 16
31.g even 15 2 31.2.g.a ✓ 16
31.g even 15 2 961.2.g.k 16
31.g even 15 2 961.2.g.s 16
31.g even 15 2 961.2.g.t 16
31.h odd 30 2 961.2.g.j 16
31.h odd 30 2 961.2.g.l 16
31.h odd 30 2 961.2.g.m 16
31.h odd 30 2 961.2.g.n 16
93.c even 2 1 8649.2.a.be 8
93.o odd 30 2 279.2.y.c 16
124.n odd 30 2 496.2.bg.c 16
155.u even 30 2 775.2.bl.a 16
155.w odd 60 4 775.2.ck.a 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.2.g.a ✓ 16 31.g even 15 2
279.2.y.c 16 93.o odd 30 2
496.2.bg.c 16 124.n odd 30 2
775.2.bl.a 16 155.u even 30 2
775.2.ck.a 32 155.w odd 60 4
961.2.a.i 8 1.a even 1 1 trivial
961.2.a.j 8 31.b odd 2 1
961.2.c.i 16 31.e odd 6 2
961.2.c.j 16 31.c even 3 2
961.2.d.n 16 31.f odd 10 2
961.2.d.o 16 31.d even 5 2
961.2.d.p 16 31.d even 5 2
961.2.d.q 16 31.f odd 10 2
961.2.g.j 16 31.h odd 30 2
961.2.g.k 16 31.g even 15 2
961.2.g.l 16 31.h odd 30 2
961.2.g.m 16 31.h odd 30 2
961.2.g.n 16 31.h odd 30 2
961.2.g.s 16 31.g even 15 2
961.2.g.t 16 31.g even 15 2
8649.2.a.be 8 93.c even 2 1
8649.2.a.bf 8 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(961))\):

\( T_{2}^{8} - 2T_{2}^{7} - 10T_{2}^{6} + 23T_{2}^{5} + 19T_{2}^{4} - 63T_{2}^{3} + 15T_{2}^{2} + 27T_{2} - 9 \) Copy content Toggle raw display
\( T_{3}^{8} + 3T_{3}^{7} - 10T_{3}^{6} - 27T_{3}^{5} + 39T_{3}^{4} + 72T_{3}^{3} - 70T_{3}^{2} - 48T_{3} + 31 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + \cdots - 9 \) Copy content Toggle raw display
$3$ \( T^{8} + 3 T^{7} + \cdots + 31 \) Copy content Toggle raw display
$5$ \( T^{8} - 3 T^{7} + \cdots - 279 \) Copy content Toggle raw display
$7$ \( T^{8} + 2 T^{7} + \cdots + 261 \) Copy content Toggle raw display
$11$ \( T^{8} + 18 T^{7} + \cdots - 279 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + \cdots - 279 \) Copy content Toggle raw display
$17$ \( T^{8} + 14 T^{7} + \cdots - 8649 \) Copy content Toggle raw display
$19$ \( T^{8} + 6 T^{7} + \cdots + 601 \) Copy content Toggle raw display
$23$ \( T^{8} + 22 T^{7} + \cdots - 279 \) Copy content Toggle raw display
$29$ \( T^{8} + 12 T^{7} + \cdots - 279 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 8 T^{7} + \cdots - 18569 \) Copy content Toggle raw display
$41$ \( T^{8} + 22 T^{7} + \cdots - 9 \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} + \cdots - 2759 \) Copy content Toggle raw display
$47$ \( T^{8} + 18 T^{7} + \cdots + 57501 \) Copy content Toggle raw display
$53$ \( T^{8} + 6 T^{7} + \cdots + 605151 \) Copy content Toggle raw display
$59$ \( T^{8} + 4 T^{7} + \cdots + 12951 \) Copy content Toggle raw display
$61$ \( T^{8} + 30 T^{7} + \cdots + 38161 \) Copy content Toggle raw display
$67$ \( T^{8} + 13 T^{7} + \cdots + 86521 \) Copy content Toggle raw display
$71$ \( T^{8} + T^{7} + \cdots + 14661 \) Copy content Toggle raw display
$73$ \( T^{8} + 2 T^{7} + \cdots + 4176351 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} + \cdots + 9198351 \) Copy content Toggle raw display
$83$ \( T^{8} + 39 T^{7} + \cdots - 1202769 \) Copy content Toggle raw display
$89$ \( T^{8} + 27 T^{7} + \cdots - 343449 \) Copy content Toggle raw display
$97$ \( T^{8} - 34 T^{7} + \cdots - 2670579 \) Copy content Toggle raw display
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