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Properties

Label 31.7.b.c
Level $31$
Weight $7$
Character orbit 31.b
Analytic conductor $7.132$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [31,7,Mod(30,31)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(31, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("31.30"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 31 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 31.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.13167659222\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 7208 x^{10} + 19859688 x^{8} + 26566749360 x^{6} + 17884354852944 x^{4} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{2}\cdot 5 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{2} - \beta_1 q^{3} + ( - \beta_{8} + \beta_{6} + 10) q^{4} + (3 \beta_{8} + \beta_{5} - 13) q^{5} + ( - \beta_{7} - \beta_1) q^{6} + ( - 2 \beta_{10} + 3 \beta_{8} + \cdots - 1) q^{7}+ \cdots + ( - 123 \beta_{11} + \cdots + 26712 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{2} + 122 q^{4} - 146 q^{5} + 6 q^{7} + 1142 q^{8} - 5668 q^{9} - 3004 q^{10} - 1312 q^{14} - 1102 q^{16} - 16382 q^{18} + 10430 q^{19} - 8052 q^{20} + 29618 q^{25} + 136504 q^{28} + 90076 q^{31}+ \cdots + 408810 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 7208 x^{10} + 19859688 x^{8} + 26566749360 x^{6} + 17884354852944 x^{4} + \cdots + 59\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 412550224103 \nu^{10} + \cdots + 23\!\cdots\!00 ) / 59\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2673790054 \nu^{11} - 17665681213522 \nu^{9} + \cdots + 67\!\cdots\!96 \nu ) / 33\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 244094790473 \nu^{11} + \cdots - 67\!\cdots\!24 \nu ) / 23\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2232250868269 \nu^{10} + \cdots + 72\!\cdots\!20 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 62978293117 \nu^{10} - 213955610347416 \nu^{8} + \cdots + 32\!\cdots\!20 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3487692504577 \nu^{11} + \cdots - 21\!\cdots\!80 \nu ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3487692504577 \nu^{10} + \cdots + 21\!\cdots\!20 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 4312792952783 \nu^{11} + \cdots - 66\!\cdots\!00 \nu ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4450256924017 \nu^{10} + \cdots - 70\!\cdots\!80 ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4075866612067 \nu^{11} + \cdots - 70\!\cdots\!00 \nu ) / 29\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{10} + 18\beta_{8} + 5\beta_{6} - 2\beta_{5} - 3\beta_{2} - 1202 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} + \beta_{9} - 12\beta_{7} - 7\beta_{4} + 14\beta_{3} - 1704\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -23984\beta_{10} - 50052\beta_{8} - 18514\beta_{6} + 8560\beta_{5} + 8178\beta_{2} + 2042980 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8560\beta_{11} + 382\beta_{9} + 41412\beta_{7} + 27074\beta_{4} - 41104\beta_{3} + 3453828\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 55880344 \beta_{10} + 128386824 \beta_{8} + 54679868 \beta_{6} - 27125576 \beta_{5} - 17657940 \beta_{2} - 4136605400 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27125576 \beta_{11} - 9467636 \beta_{9} - 116861280 \beta_{7} - 81805444 \beta_{4} + \cdots - 7735569744 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 134634899024 \beta_{10} - 327474988464 \beta_{8} - 149178417112 \beta_{6} + 77881143088 \beta_{5} + \cdots + 9256406351344 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 77881143088 \beta_{11} + 40428918904 \beta_{9} + 314757325776 \beta_{7} + 227059560200 \beta_{4} + \cdots + 18389053293648 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 333023330805760 \beta_{10} + 840858878316768 \beta_{8} + 394128525396752 \beta_{6} + \cdots - 21\!\cdots\!44 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 213720411500288 \beta_{11} - 131256254586128 \beta_{9} - 837332413529280 \beta_{7} + \cdots - 45\!\cdots\!96 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/31\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1
30.2629i
− 30.2629i
39.6325i
− 39.6325i
14.4783i
− 14.4783i
50.8814i
− 50.8814i
23.5019i
− 23.5019i
37.0208i
− 37.0208i
−12.4307 − 30.2629i 90.5217 96.4253 376.188i 161.968 −329.682 −186.843 −1198.63
30.2 −12.4307 30.2629i 90.5217 96.4253 − 376.188i 161.968 −329.682 −186.843 −1198.63
30.3 −5.88527 − 39.6325i −29.3637 −192.768 233.248i 330.261 549.470 −841.735 1134.49
30.4 −5.88527 39.6325i −29.3637 −192.768 − 233.248i 330.261 549.470 −841.735 1134.49
30.5 −5.22681 − 14.4783i −36.6805 28.3450 75.6755i −304.878 526.238 519.378 −148.154
30.6 −5.22681 14.4783i −36.6805 28.3450 − 75.6755i −304.878 526.238 519.378 −148.154
30.7 2.80723 − 50.8814i −56.1194 198.770 − 142.836i −242.679 −337.203 −1859.92 557.994
30.8 2.80723 50.8814i −56.1194 198.770 142.836i −242.679 −337.203 −1859.92 557.994
30.9 6.26938 − 23.5019i −24.6949 −124.548 − 147.342i −222.781 −556.062 176.661 −780.837
30.10 6.26938 23.5019i −24.6949 −124.548 147.342i −222.781 −556.062 176.661 −780.837
30.11 13.4661 − 37.0208i 117.337 −79.2253 − 498.527i 281.109 718.240 −641.539 −1066.86
30.12 13.4661 37.0208i 117.337 −79.2253 498.527i 281.109 718.240 −641.539 −1066.86
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 30.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
31.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 31.7.b.c ✓ 12
3.b odd 2 1 279.7.d.f 12
4.b odd 2 1 496.7.e.c 12
31.b odd 2 1 inner 31.7.b.c ✓ 12
93.c even 2 1 279.7.d.f 12
124.d even 2 1 496.7.e.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
31.7.b.c ✓ 12 1.a even 1 1 trivial
31.7.b.c ✓ 12 31.b odd 2 1 inner
279.7.d.f 12 3.b odd 2 1
279.7.d.f 12 93.c even 2 1
496.7.e.c 12 4.b odd 2 1
496.7.e.c 12 124.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + T_{2}^{5} - 222T_{2}^{4} - 370T_{2}^{3} + 9416T_{2}^{2} + 13440T_{2} - 90624 \) acting on \(S_{7}^{\mathrm{new}}(31, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + T^{5} + \cdots - 90624)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{6} + \cdots - 1033363425000)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots - 247854254904896)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 79\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 18\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 48\!\cdots\!81 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 15\!\cdots\!08)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 26\!\cdots\!52)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 26\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 45\!\cdots\!92)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 68\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 43\!\cdots\!00)^{2} \) Copy content Toggle raw display
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