Properties

Label 931.2.a.n
Level $931$
Weight $2$
Character orbit 931.a
Self dual yes
Analytic conductor $7.434$
Analytic rank $0$
Dimension $7$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(1,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 4x^{3} - 12x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 133)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{2} q^{3} + ( - \beta_{5} + \beta_{3} + 1) q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{6} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{8} + (\beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - \beta_{2} q^{3} + ( - \beta_{5} + \beta_{3} + 1) q^{4} + \beta_{6} q^{5} + ( - \beta_{6} + \beta_{4} - \beta_{3} - \beta_1) q^{6} + ( - \beta_{5} + 2 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{8} + (\beta_{6} - \beta_{5} - \beta_{4} + 2) q^{9} + (\beta_{6} - \beta_{3} + \beta_{2} + \beta_1) q^{10} + ( - \beta_{6} - \beta_1 + 1) q^{11} + ( - 2 \beta_{6} + \beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_1 - 3) q^{12} + (\beta_{6} - \beta_{5} + \beta_{3} - \beta_{2} + 1) q^{13} + (\beta_{5} - \beta_{3} - \beta_1 + 1) q^{15} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + 3) q^{16} + (\beta_{6} + 3) q^{17} + (\beta_{6} + \beta_{5} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 3) q^{18} + q^{19} + (\beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 2) q^{20} + ( - 2 \beta_{6} + \beta_{5} + 2 \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{22} + (\beta_{5} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{23} + ( - 3 \beta_{6} - 2 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 4) q^{24} + (\beta_{5} + \beta_{4}) q^{25} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3} + \beta_1 + 1) q^{26} + ( - 2 \beta_{4} - 2 \beta_{2} - 2) q^{27} + (\beta_{6} - \beta_1 + 4) q^{29} + ( - \beta_{6} + 2 \beta_{5} - 2 \beta_{3} - \beta_1 - 2) q^{30} + (2 \beta_{5} + \beta_{2}) q^{31} + ( - \beta_{5} + 4 \beta_{3} - \beta_{2} - \beta_1 + 3) q^{32} + (\beta_{6} - \beta_{5} + 3 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{33} + (\beta_{6} + 2 \beta_{3} + \beta_{2} + \beta_1) q^{34} + (3 \beta_{6} - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 + 6) q^{36} + ( - \beta_{6} - 2 \beta_{4} + \beta_1) q^{37} + \beta_{3} q^{38} + ( - \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 3) q^{39} + (3 \beta_{6} - 2 \beta_{4} - \beta_{3} + 3 \beta_{2} + \beta_1 - 2) q^{40} - 2 \beta_{3} q^{41} + (2 \beta_{5} + 2 \beta_{2} + \beta_1) q^{43} + ( - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 5) q^{44} + ( - \beta_{5} + \beta_{3} + 2 \beta_1 + 1) q^{45} + (3 \beta_{6} - 2 \beta_{4} - \beta_{3} + \beta_1 - 2) q^{46} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - \beta_1) q^{47} + ( - 4 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 3) q^{48} + ( - \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 + 3) q^{50} + (\beta_{5} - \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{51} + ( - \beta_{6} - 2 \beta_{5} + 4 \beta_{3} + \beta_1 + 4) q^{52} + (\beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + 2 \beta_1 + 3) q^{53} + ( - 2 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 6 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 2) q^{54} + ( - 2 \beta_{3} + \beta_1 - 4) q^{55} - \beta_{2} q^{57} + (\beta_{5} + 3 \beta_{3} + \beta_1 - 1) q^{58} + ( - 2 \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{59} + ( - 2 \beta_{6} + \beta_{5} - 5 \beta_{3} - \beta_1 - 5) q^{60} + (\beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_1 + 2) q^{61} + (\beta_{6} - \beta_{4} - 3 \beta_{3} + 2 \beta_{2} - \beta_1 + 4) q^{62} + ( - 2 \beta_{6} - \beta_{5} - \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 3) q^{64} + (\beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{65} + (2 \beta_{6} - 5 \beta_{5} + \beta_{4} + 2 \beta_{2} + \beta_1 + 9) q^{66} + (2 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 2 \beta_1) q^{67} + (\beta_{6} - 3 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{68} + ( - \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - 5) q^{69} + ( - 2 \beta_{5} + \beta_{2} + 4 \beta_1) q^{71} + (7 \beta_{6} - 2 \beta_{5} - 4 \beta_{4} + 5 \beta_{3} - \beta_{2} + 5 \beta_1 - 2) q^{72} + (2 \beta_{6} - \beta_{4} - \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 1) q^{73} + (\beta_{5} - \beta_{3} + 4 \beta_{2} - \beta_1 - 1) q^{74} + (\beta_{5} + 2 \beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 + 3) q^{75} + ( - \beta_{5} + \beta_{3} + 1) q^{76} + ( - 5 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - \beta_{3} - 4 \beta_1 - 1) q^{78} + ( - 2 \beta_{6} + 2 \beta_{4} - \beta_{2} - 2 \beta_1 - 2) q^{79} + (5 \beta_{6} + 2 \beta_{5} - \beta_{4} - 3 \beta_{3} + 4 \beta_{2} + 2 \beta_1) q^{80} + (\beta_{6} - 3 \beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 4) q^{81} + (2 \beta_{5} - 2 \beta_{3} - 6) q^{82} + ( - \beta_{6} + 3 \beta_{4} - \beta_{3} + \beta_1 - 1) q^{83} + (3 \beta_{6} + \beta_{5} + \beta_{4} + 5) q^{85} + (3 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3 \beta_{2} + 5) q^{86} + (\beta_{6} + \beta_{5} + \beta_{3} - 4 \beta_{2} - 1) q^{87} + ( - 4 \beta_{6} + 2 \beta_{4} + 5 \beta_{3} - 7 \beta_{2} - 2 \beta_1 - 4) q^{88} + ( - \beta_{6} + 2 \beta_{3} - \beta_1 - 2) q^{89} + (2 \beta_{6} - 3 \beta_{5} + 4 \beta_{3} + \beta_{2} + \beta_1 + 3) q^{90} + (4 \beta_{6} - 2 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} + \beta_1 - 4) q^{92} + (\beta_{6} - \beta_{5} + 3 \beta_{4} - 4 \beta_{3} + 2 \beta_{2} + 1) q^{93} + ( - 5 \beta_{6} + 2 \beta_{4} + 3 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 2) q^{94} + \beta_{6} q^{95} + ( - 3 \beta_{6} + 2 \beta_{4} - 4 \beta_{2} - 3 \beta_1) q^{96} + (2 \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{97} + ( - 2 \beta_{6} + \beta_{5} + \beta_{4} - 6 \beta_{3} + 2 \beta_{2} - 3 \beta_1 + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 2 q^{2} - 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 12 q^{8} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 2 q^{2} - 2 q^{3} + 10 q^{4} - 2 q^{5} - 4 q^{6} + 12 q^{8} + 15 q^{9} + 7 q^{11} - 22 q^{12} + 6 q^{13} + 2 q^{15} + 24 q^{16} + 19 q^{17} - 12 q^{18} + 7 q^{19} - 8 q^{20} - 6 q^{22} - q^{23} + 20 q^{24} - 3 q^{25} + 12 q^{26} - 14 q^{27} + 24 q^{29} - 20 q^{30} + 26 q^{32} - 14 q^{33} + 6 q^{34} + 46 q^{36} + 8 q^{37} + 2 q^{38} + 16 q^{39} - 10 q^{40} - 4 q^{41} + 4 q^{43} + 26 q^{44} + 14 q^{45} - 16 q^{46} + 5 q^{47} - 28 q^{48} + 16 q^{50} - 4 q^{51} + 42 q^{52} + 20 q^{53} - 24 q^{54} - 30 q^{55} - 2 q^{57} - 16 q^{59} - 44 q^{60} + 5 q^{61} + 24 q^{62} + 32 q^{64} + 26 q^{65} + 68 q^{66} - 4 q^{67} + 22 q^{68} - 36 q^{69} + 12 q^{71} + 3 q^{73} - 4 q^{74} + 18 q^{75} + 10 q^{76} - 14 q^{78} - 20 q^{79} - 4 q^{80} + 27 q^{81} - 48 q^{82} - 11 q^{83} + 26 q^{85} + 36 q^{86} - 16 q^{87} - 32 q^{88} - 10 q^{89} + 32 q^{90} - 30 q^{92} - 4 q^{93} + 16 q^{94} - 2 q^{95} - 12 q^{96} - 4 q^{97} + 15 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 6x^{5} + 18x^{4} + 4x^{3} - 12x^{2} - x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 3\nu^{5} - 5\nu^{4} + 16\nu^{3} - \nu^{2} - 3\nu - 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{6} + 3\nu^{5} + 5\nu^{4} - 16\nu^{3} + 3\nu^{2} + \nu - 3 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 4\nu^{5} - 3\nu^{4} + 23\nu^{3} - 12\nu^{2} - 11\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 11\nu^{3} - 22\nu^{2} + \nu + 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{6} + 2\nu^{5} + 9\nu^{4} - 13\nu^{3} - 20\nu^{2} + 15\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + 3\nu^{5} + 6\nu^{4} - 18\nu^{3} - 4\nu^{2} + 11\nu + 1 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{3} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -4\beta_{6} + 3\beta_{5} + \beta_{4} - 4\beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -11\beta_{6} + 9\beta_{5} + 4\beta_{4} - 13\beta_{3} + 14\beta_{2} + 18\beta _1 + 33 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -59\beta_{6} + 41\beta_{5} + 22\beta_{4} - 67\beta_{3} + 20\beta_{2} + 32\beta _1 + 69 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -54\beta_{6} + 38\beta_{5} + 27\beta_{4} - 71\beta_{3} + 50\beta_{2} + 80\beta _1 + 127 \) 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.

comment: embeddings in the coefficient field
 
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
−0.770405
0.862998
3.00704
0.273704
2.27137
−2.29398
−0.350729
−2.46342 −3.03216 4.06842 −0.527614 7.46948 0 −5.09539 6.19401 1.29973
1.2 −1.40650 3.19504 −0.0217491 0.295752 −4.49383 0 2.84360 7.20828 −0.415977
1.3 −0.812652 0.415454 −1.33960 −2.67449 −0.337619 0 2.71393 −2.82740 2.17343
1.4 0.269662 1.39869 −1.92728 3.37987 0.377172 0 −1.05904 −1.04367 0.911422
1.5 1.13506 −2.19453 −0.711632 −1.83111 −2.49093 0 −3.07787 1.81594 −2.07842
1.6 2.59421 −2.89925 4.72991 1.85806 −7.52126 0 7.08194 5.40568 4.82019
1.7 2.68364 1.11676 5.20193 −2.50047 2.99699 0 8.59284 −1.75284 −6.71038
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \(1\)
\(19\) \(-1\)

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 931.2.a.n 7
3.b odd 2 1 8379.2.a.cl 7
7.b odd 2 1 931.2.a.o 7
7.c even 3 2 133.2.f.d 14
7.d odd 6 2 931.2.f.p 14
21.c even 2 1 8379.2.a.ck 7
21.h odd 6 2 1197.2.j.l 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.f.d 14 7.c even 3 2
931.2.a.n 7 1.a even 1 1 trivial
931.2.a.o 7 7.b odd 2 1
931.2.f.p 14 7.d odd 6 2
1197.2.j.l 14 21.h odd 6 2
8379.2.a.ck 7 21.c even 2 1
8379.2.a.cl 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(931))\):

\( T_{2}^{7} - 2T_{2}^{6} - 10T_{2}^{5} + 16T_{2}^{4} + 27T_{2}^{3} - 24T_{2}^{2} - 18T_{2} + 6 \) Copy content Toggle raw display
\( T_{3}^{7} + 2T_{3}^{6} - 16T_{3}^{5} - 26T_{3}^{4} + 72T_{3}^{3} + 52T_{3}^{2} - 128T_{3} + 40 \) Copy content Toggle raw display
\( T_{5}^{7} + 2T_{5}^{6} - 14T_{5}^{5} - 32T_{5}^{4} + 33T_{5}^{3} + 90T_{5}^{2} + 12T_{5} - 12 \) Copy content Toggle raw display
\( T_{13}^{7} - 6T_{13}^{6} - 24T_{13}^{5} + 210T_{13}^{4} - 104T_{13}^{3} - 1620T_{13}^{2} + 3200T_{13} - 1208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 2 T^{6} - 10 T^{5} + 16 T^{4} + \cdots + 6 \) Copy content Toggle raw display
$3$ \( T^{7} + 2 T^{6} - 16 T^{5} - 26 T^{4} + \cdots + 40 \) Copy content Toggle raw display
$5$ \( T^{7} + 2 T^{6} - 14 T^{5} - 32 T^{4} + \cdots - 12 \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 7 T^{6} - 13 T^{5} + 131 T^{4} + \cdots - 135 \) Copy content Toggle raw display
$13$ \( T^{7} - 6 T^{6} - 24 T^{5} + \cdots - 1208 \) Copy content Toggle raw display
$17$ \( T^{7} - 19 T^{6} + 139 T^{5} + \cdots - 48 \) Copy content Toggle raw display
$19$ \( (T - 1)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + T^{6} - 85 T^{5} + 7 T^{4} + \cdots + 3165 \) Copy content Toggle raw display
$29$ \( T^{7} - 24 T^{6} + 206 T^{5} + \cdots + 12480 \) Copy content Toggle raw display
$31$ \( T^{7} - 96 T^{5} + 6 T^{4} + \cdots + 8632 \) Copy content Toggle raw display
$37$ \( T^{7} - 8 T^{6} - 154 T^{5} + \cdots + 155776 \) Copy content Toggle raw display
$41$ \( T^{7} + 4 T^{6} - 40 T^{5} - 128 T^{4} + \cdots - 768 \) Copy content Toggle raw display
$43$ \( T^{7} - 4 T^{6} - 148 T^{5} + \cdots + 303284 \) Copy content Toggle raw display
$47$ \( T^{7} - 5 T^{6} - 149 T^{5} + \cdots - 56193 \) Copy content Toggle raw display
$53$ \( T^{7} - 20 T^{6} + 56 T^{5} + \cdots - 408 \) Copy content Toggle raw display
$59$ \( T^{7} + 16 T^{6} - 2 T^{5} + \cdots + 124032 \) Copy content Toggle raw display
$61$ \( T^{7} - 5 T^{6} - 187 T^{5} + \cdots + 1281545 \) Copy content Toggle raw display
$67$ \( T^{7} + 4 T^{6} - 256 T^{5} + \cdots - 8192 \) Copy content Toggle raw display
$71$ \( T^{7} - 12 T^{6} - 256 T^{5} + \cdots + 6648 \) Copy content Toggle raw display
$73$ \( T^{7} - 3 T^{6} - 303 T^{5} + \cdots + 164299 \) Copy content Toggle raw display
$79$ \( T^{7} + 20 T^{6} - 64 T^{5} + \cdots - 151672 \) Copy content Toggle raw display
$83$ \( T^{7} + 11 T^{6} - 209 T^{5} + \cdots + 142371 \) Copy content Toggle raw display
$89$ \( T^{7} + 10 T^{6} - 22 T^{5} + \cdots + 10464 \) Copy content Toggle raw display
$97$ \( T^{7} + 4 T^{6} - 258 T^{5} - 642 T^{4} + \cdots + 32 \) Copy content Toggle raw display
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