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Properties

Label 931.2.g.e
Level $931$
Weight $2$
Character orbit 931.g
Analytic conductor $7.434$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [931,2,Mod(30,931)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(931, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("931.30");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 931 = 7^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 931.g (of order \(3\), degree \(2\), not minimal)

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: no
Analytic conductor: \(7.43407242818\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 32x^{6} - 9x^{5} + 63x^{4} + 20x^{3} + 68x^{2} - 8x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 133)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + (\beta_{9} + \beta_{6} - 1) q^{4} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{5}+ \cdots + ( - \beta_{5} - \beta_{4} + \cdots + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{3} - 1) q^{3} + (\beta_{9} + \beta_{6} - 1) q^{4} + ( - \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{5}+ \cdots + ( - 2 \beta_{8} - 3 \beta_{7} + \cdots + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{2} - 6 q^{3} - 3 q^{4} - q^{5} + 3 q^{6} - 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{2} - 6 q^{3} - 3 q^{4} - q^{5} + 3 q^{6} - 6 q^{8} - 4 q^{10} - q^{11} + 2 q^{12} + 6 q^{15} + 9 q^{16} + 14 q^{17} - 3 q^{18} - 4 q^{19} + 14 q^{20} + 5 q^{22} + 4 q^{23} + 30 q^{24} - 2 q^{25} + 12 q^{26} - 24 q^{27} - 6 q^{29} - 22 q^{30} + 3 q^{31} - 8 q^{32} + 10 q^{33} + 15 q^{34} + 6 q^{36} + 5 q^{37} + 28 q^{38} - 17 q^{39} + 6 q^{40} + 19 q^{41} - 16 q^{43} - 20 q^{44} + 16 q^{45} + 10 q^{46} - 20 q^{47} - 4 q^{48} - 34 q^{50} - 14 q^{51} - 22 q^{52} + 7 q^{53} + 10 q^{54} + 5 q^{55} + 6 q^{57} + 5 q^{58} - 32 q^{59} - 46 q^{60} + 54 q^{61} - 21 q^{62} - 38 q^{64} + 16 q^{65} + q^{66} - 11 q^{67} - 22 q^{68} - 54 q^{69} - 42 q^{72} - 10 q^{73} + 16 q^{74} - 17 q^{75} + 3 q^{76} + 4 q^{78} + 21 q^{79} - 8 q^{80} + 26 q^{81} - 22 q^{82} - 16 q^{83} + 34 q^{85} + 96 q^{86} - 6 q^{87} + 10 q^{88} - 44 q^{89} + 26 q^{90} + 18 q^{92} + 12 q^{93} - 21 q^{94} + 5 q^{95} + 15 q^{96} + 28 q^{97} - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} + 7x^{8} - 2x^{7} + 32x^{6} - 9x^{5} + 63x^{4} + 20x^{3} + 68x^{2} - 8x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1315 \nu^{9} + 10212 \nu^{8} - 29785 \nu^{7} + 62655 \nu^{6} - 116587 \nu^{5} + 246790 \nu^{4} + \cdots + 94555 ) / 759446 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2630 \nu^{9} + 20424 \nu^{8} - 59570 \nu^{7} + 125310 \nu^{6} - 233174 \nu^{5} + 493580 \nu^{4} + \cdots + 568833 ) / 379723 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8897 \nu^{9} - 20580 \nu^{8} + 60025 \nu^{7} - 74507 \nu^{6} + 234955 \nu^{5} - 497350 \nu^{4} + \cdots - 2277023 ) / 759446 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 15274 \nu^{9} + 38916 \nu^{8} - 113505 \nu^{7} + 207978 \nu^{6} - 444291 \nu^{5} + \cdots + 1130040 ) / 379723 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 94555 \nu^{9} - 93240 \nu^{8} + 651673 \nu^{7} - 159325 \nu^{6} + 2963105 \nu^{5} - 734408 \nu^{4} + \cdots + 44705 ) / 759446 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 53119 \nu^{9} + 45493 \nu^{8} - 354193 \nu^{7} + 54788 \nu^{6} - 1690191 \nu^{5} + 276681 \nu^{4} + \cdots - 26801 ) / 379723 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 134864 \nu^{9} + 127326 \nu^{8} - 940952 \nu^{7} + 260698 \nu^{6} - 4301561 \nu^{5} + \cdots + 1066270 ) / 379723 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 283665 \nu^{9} + 279720 \nu^{8} - 1955019 \nu^{7} + 477975 \nu^{6} - 8889315 \nu^{5} + \cdots + 2144223 ) / 759446 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 3\beta_{6} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{9} + \beta_{8} - 12\beta_{6} - \beta_{5} - 5\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{9} + \beta_{8} - 6\beta_{7} - 7\beta_{6} - 6\beta_{3} + 16\beta_{2} - 16\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{5} + 23\beta_{4} - 2\beta_{3} + \beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10\beta_{9} - 9\beta_{8} + 30\beta_{7} + 40\beta_{6} + 9\beta_{5} + 10\beta_{4} + 65\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 104\beta_{9} - 39\beta_{8} + 19\beta_{7} + 223\beta_{6} + 19\beta_{3} - 11\beta_{2} + 11\beta _1 - 223 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -58\beta_{5} - 69\beta_{4} + 143\beta_{3} - 269\beta_{2} - 215 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/931\mathbb{Z}\right)^\times\).

\(n\) \(248\) \(344\)
\(\chi(n)\) \(-1 + \beta_{6}\) \(-\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
−0.983281 − 1.70309i
−0.552442 − 0.956858i
0.0595946 + 0.103221i
0.883493 + 1.53025i
1.09264 + 1.89250i
−0.983281 + 1.70309i
−0.552442 + 0.956858i
0.0595946 − 0.103221i
0.883493 − 1.53025i
1.09264 − 1.89250i
−0.983281 − 1.70309i −0.260829 −0.933684 + 1.61719i −1.22681 − 2.12490i 0.256468 + 0.444216i 0 −0.260829 −2.93197 −2.41260 + 4.17875i
30.2 −0.552442 − 0.956858i −3.07073 0.389615 − 0.674832i 0.643958 + 1.11537i 1.69640 + 2.93825i 0 −3.07073 6.42937 0.711500 − 1.23235i
30.3 0.0595946 + 0.103221i 0.475063 0.992897 − 1.71975i −0.412094 − 0.713768i 0.0283112 + 0.0490364i 0 0.475063 −2.77431 0.0491171 − 0.0850734i
30.4 0.883493 + 1.53025i 1.55099 −0.561119 + 0.971886i 1.75378 + 3.03764i 1.37029 + 2.37341i 0 1.55099 −0.594419 −3.09891 + 5.36747i
30.5 1.09264 + 1.89250i −1.69450 −1.38771 + 2.40358i −1.25884 − 2.18037i −1.85147 − 3.20684i 0 −1.69450 −0.128672 2.75090 − 4.76470i
900.1 −0.983281 + 1.70309i −0.260829 −0.933684 − 1.61719i −1.22681 + 2.12490i 0.256468 − 0.444216i 0 −0.260829 −2.93197 −2.41260 − 4.17875i
900.2 −0.552442 + 0.956858i −3.07073 0.389615 + 0.674832i 0.643958 − 1.11537i 1.69640 − 2.93825i 0 −3.07073 6.42937 0.711500 + 1.23235i
900.3 0.0595946 − 0.103221i 0.475063 0.992897 + 1.71975i −0.412094 + 0.713768i 0.0283112 − 0.0490364i 0 0.475063 −2.77431 0.0491171 + 0.0850734i
900.4 0.883493 − 1.53025i 1.55099 −0.561119 − 0.971886i 1.75378 − 3.03764i 1.37029 − 2.37341i 0 1.55099 −0.594419 −3.09891 − 5.36747i
900.5 1.09264 − 1.89250i −1.69450 −1.38771 − 2.40358i −1.25884 + 2.18037i −1.85147 + 3.20684i 0 −1.69450 −0.128672 2.75090 + 4.76470i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
133.g even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 931.2.g.e 10
7.b odd 2 1 931.2.g.f 10
7.c even 3 1 133.2.e.c ✓ 10
7.c even 3 1 931.2.h.f 10
7.d odd 6 1 931.2.e.c 10
7.d odd 6 1 931.2.h.e 10
19.c even 3 1 931.2.h.f 10
21.h odd 6 1 1197.2.k.g 10
133.g even 3 1 inner 931.2.g.e 10
133.g even 3 1 2527.2.a.j 5
133.h even 3 1 133.2.e.c ✓ 10
133.k odd 6 1 931.2.g.f 10
133.m odd 6 1 931.2.h.e 10
133.n odd 6 1 2527.2.a.k 5
133.t odd 6 1 931.2.e.c 10
399.n odd 6 1 1197.2.k.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
133.2.e.c ✓ 10 7.c even 3 1
133.2.e.c ✓ 10 133.h even 3 1
931.2.e.c 10 7.d odd 6 1
931.2.e.c 10 133.t odd 6 1
931.2.g.e 10 1.a even 1 1 trivial
931.2.g.e 10 133.g even 3 1 inner
931.2.g.f 10 7.b odd 2 1
931.2.g.f 10 133.k odd 6 1
931.2.h.e 10 7.d odd 6 1
931.2.h.e 10 133.m odd 6 1
931.2.h.f 10 7.c even 3 1
931.2.h.f 10 19.c even 3 1
1197.2.k.g 10 21.h odd 6 1
1197.2.k.g 10 399.n odd 6 1
2527.2.a.j 5 133.g even 3 1
2527.2.a.k 5 133.n odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(931, [\chi])\):

\( T_{2}^{10} - T_{2}^{9} + 7T_{2}^{8} - 2T_{2}^{7} + 32T_{2}^{6} - 9T_{2}^{5} + 63T_{2}^{4} + 20T_{2}^{3} + 68T_{2}^{2} - 8T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{5} + 3T_{3}^{4} - 3T_{3}^{3} - 8T_{3}^{2} + 2T_{3} + 1 \) Copy content Toggle raw display
\( T_{5}^{10} + T_{5}^{9} + 14 T_{5}^{8} + 23 T_{5}^{7} + 165 T_{5}^{6} + 213 T_{5}^{5} + 587 T_{5}^{4} + \cdots + 529 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{5} + 3 T^{4} - 3 T^{3} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + T^{9} + \cdots + 529 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{10} + 20 T^{8} + \cdots + 11881 \) Copy content Toggle raw display
$17$ \( (T^{5} - 7 T^{4} + \cdots - 387)^{2} \) Copy content Toggle raw display
$19$ \( T^{10} + 4 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( (T^{5} - 2 T^{4} + \cdots - 6003)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + 6 T^{9} + \cdots + 78943225 \) Copy content Toggle raw display
$31$ \( T^{10} - 3 T^{9} + \cdots + 1638400 \) Copy content Toggle raw display
$37$ \( T^{10} - 5 T^{9} + \cdots + 8549776 \) Copy content Toggle raw display
$41$ \( T^{10} - 19 T^{9} + \cdots + 2627641 \) Copy content Toggle raw display
$43$ \( T^{10} + 16 T^{9} + \cdots + 727609 \) Copy content Toggle raw display
$47$ \( (T^{5} + 10 T^{4} + \cdots + 1928)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 7 T^{9} + \cdots + 25230529 \) Copy content Toggle raw display
$59$ \( (T^{5} + 16 T^{4} + \cdots + 1773)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 27 T^{4} + \cdots - 639)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 11 T^{9} + \cdots + 3003289 \) Copy content Toggle raw display
$71$ \( T^{10} + 42 T^{8} + \cdots + 109561 \) Copy content Toggle raw display
$73$ \( (T^{5} + 5 T^{4} + \cdots + 26181)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 5068158481 \) Copy content Toggle raw display
$83$ \( (T^{5} + 8 T^{4} + \cdots + 144)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + 22 T^{4} + \cdots - 19512)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 201554809 \) Copy content Toggle raw display
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