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Properties

Label 7942.2.a.bv
Level $7942$
Weight $2$
Character orbit 7942.a
Self dual yes
Analytic conductor $63.417$
Analytic rank $1$
Dimension $12$
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] = [7942,2,Mod(1,7942)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7942, 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("7942.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7942 = 2 \cdot 11 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7942.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: \(63.4171892853\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + x^{10} + 48 x^{9} - 47 x^{8} - 138 x^{7} + 150 x^{6} + 172 x^{5} - 139 x^{4} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + \beta_{11} q^{5} + (\beta_1 - 1) q^{6} + (\beta_{11} - \beta_{6} + \cdots - \beta_{2}) q^{7}+ \cdots + (\beta_{4} + \beta_{3} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + (\beta_1 - 1) q^{3} + q^{4} + \beta_{11} q^{5} + (\beta_1 - 1) q^{6} + (\beta_{11} - \beta_{6} + \cdots - \beta_{2}) q^{7}+ \cdots + (\beta_{4} + \beta_{3} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 6 q^{3} + 12 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 12 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} - 6 q^{3} + 12 q^{4} - 6 q^{5} - 6 q^{6} - 6 q^{7} + 12 q^{8} - 2 q^{9} - 6 q^{10} + 12 q^{11} - 6 q^{12} - 26 q^{13} - 6 q^{14} + 16 q^{15} + 12 q^{16} - 2 q^{18} - 6 q^{20} - 10 q^{21} + 12 q^{22} + 4 q^{23} - 6 q^{24} - 26 q^{26} - 6 q^{27} - 6 q^{28} - 24 q^{29} + 16 q^{30} - 16 q^{31} + 12 q^{32} - 6 q^{33} + 6 q^{35} - 2 q^{36} - 8 q^{37} - 6 q^{40} - 16 q^{41} - 10 q^{42} - 32 q^{43} + 12 q^{44} - 2 q^{45} + 4 q^{46} - 2 q^{47} - 6 q^{48} - 4 q^{49} + 2 q^{51} - 26 q^{52} - 28 q^{53} - 6 q^{54} - 6 q^{55} - 6 q^{56} - 24 q^{58} - 6 q^{59} + 16 q^{60} - 16 q^{62} + 4 q^{63} + 12 q^{64} - 14 q^{65} - 6 q^{66} - 12 q^{67} - 16 q^{69} + 6 q^{70} - 4 q^{71} - 2 q^{72} - 22 q^{73} - 8 q^{74} - 20 q^{75} - 6 q^{77} - 6 q^{80} - 32 q^{81} - 16 q^{82} + 40 q^{83} - 10 q^{84} - 16 q^{85} - 32 q^{86} + 12 q^{88} - 16 q^{89} - 2 q^{90} + 60 q^{91} + 4 q^{92} - 2 q^{94} - 6 q^{96} - 20 q^{97} - 4 q^{98} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + x^{10} + 48 x^{9} - 47 x^{8} - 138 x^{7} + 150 x^{6} + 172 x^{5} - 139 x^{4} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 7 \nu^{11} + 31 \nu^{10} + 85 \nu^{9} - 491 \nu^{8} - 154 \nu^{7} + 2340 \nu^{6} - 475 \nu^{5} + \cdots - 170 ) / 101 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13 \nu^{11} + 72 \nu^{10} + 28 \nu^{9} - 681 \nu^{8} + 421 \nu^{7} + 2369 \nu^{6} - 1921 \nu^{5} + \cdots - 359 ) / 101 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13 \nu^{11} - 72 \nu^{10} - 28 \nu^{9} + 681 \nu^{8} - 421 \nu^{7} - 2369 \nu^{6} + 1921 \nu^{5} + \cdots + 157 ) / 101 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24 \nu^{11} + 164 \nu^{10} - 127 \nu^{9} - 1164 \nu^{8} + 1795 \nu^{7} + 2944 \nu^{6} - 5077 \nu^{5} + \cdots - 150 ) / 101 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 80 \nu^{11} - 513 \nu^{10} + 255 \nu^{9} + 3880 \nu^{8} - 5108 \nu^{7} - 10150 \nu^{6} + 15442 \nu^{5} + \cdots + 399 ) / 101 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( - \nu^{11} + 6 \nu^{10} - \nu^{9} - 48 \nu^{8} + 47 \nu^{7} + 138 \nu^{6} - 150 \nu^{5} - 172 \nu^{4} + \cdots - 6 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 150 \nu^{11} - 924 \nu^{10} + 314 \nu^{9} + 7073 \nu^{8} - 8214 \nu^{7} - 18905 \nu^{6} + 25444 \nu^{5} + \cdots + 786 ) / 101 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 249 \nu^{11} + 1550 \nu^{10} - 598 \nu^{9} - 11824 \nu^{8} + 14419 \nu^{7} + 31049 \nu^{6} + \cdots - 1228 ) / 101 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 279 \nu^{11} - 1755 \nu^{10} + 782 \nu^{9} + 13178 \nu^{8} - 16789 \nu^{7} - 34022 \nu^{6} + \cdots + 1062 ) / 101 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 380 \nu^{11} - 2361 \nu^{10} + 883 \nu^{9} + 18026 \nu^{8} - 21536 \nu^{7} - 47960 \nu^{6} + \cdots + 1870 ) / 101 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + \beta_{3} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} - \beta_{10} + \beta_{7} + 2\beta_{4} + 2\beta_{3} + 4\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{11} - 3\beta_{10} + 2\beta_{8} + 3\beta_{7} + \beta_{6} + 9\beta_{4} + 9\beta_{3} + 6\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10 \beta_{11} - 12 \beta_{10} + \beta_{9} + 6 \beta_{8} + 13 \beta_{7} + 3 \beta_{6} + \beta_{5} + \cdots + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27 \beta_{11} - 37 \beta_{10} + 4 \beta_{9} + 27 \beta_{8} + 40 \beta_{7} + 15 \beta_{6} + 5 \beta_{5} + \cdots + 79 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 96 \beta_{11} - 121 \beta_{10} + 20 \beta_{9} + 85 \beta_{8} + 136 \beta_{7} + 49 \beta_{6} + 23 \beta_{5} + \cdots + 204 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 283 \beta_{11} - 370 \beta_{10} + 72 \beta_{9} + 294 \beta_{8} + 419 \beta_{7} + 180 \beta_{6} + \cdots + 642 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 907 \beta_{11} - 1146 \beta_{10} + 269 \beta_{9} + 923 \beta_{8} + 1326 \beta_{7} + 584 \beta_{6} + \cdots + 1818 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2734 \beta_{11} - 3476 \beta_{10} + 909 \beta_{9} + 2957 \beta_{8} + 4060 \beta_{7} + 1940 \beta_{6} + \cdots + 5500 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 8447 \beta_{11} - 10567 \beta_{10} + 3071 \beta_{9} + 9184 \beta_{8} + 12507 \beta_{7} + 6167 \beta_{6} + \cdots + 16106 \) 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.

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
−1.73171
−1.65830
−1.24800
−0.443244
−0.220619
−0.162801
0.410422
1.26387
1.85058
2.37041
2.55692
3.01246
1.00000 −2.73171 1.00000 −3.73440 −2.73171 −3.26533 1.00000 4.46223 −3.73440
1.2 1.00000 −2.65830 1.00000 1.24304 −2.65830 1.34865 1.00000 4.06656 1.24304
1.3 1.00000 −2.24800 1.00000 −0.981787 −2.24800 4.63571 1.00000 2.05349 −0.981787
1.4 1.00000 −1.44324 1.00000 0.168593 −1.44324 −1.52457 1.00000 −0.917047 0.168593
1.5 1.00000 −1.22062 1.00000 −2.91567 −1.22062 −2.48215 1.00000 −1.51009 −2.91567
1.6 1.00000 −1.16280 1.00000 2.05915 −1.16280 1.61923 1.00000 −1.64789 2.05915
1.7 1.00000 −0.589578 1.00000 −3.50950 −0.589578 −2.22346 1.00000 −2.65240 −3.50950
1.8 1.00000 0.263871 1.00000 −2.13699 0.263871 2.93250 1.00000 −2.93037 −2.13699
1.9 1.00000 0.850582 1.00000 3.18154 0.850582 −2.86948 1.00000 −2.27651 3.18154
1.10 1.00000 1.37041 1.00000 −0.107181 1.37041 −0.953307 1.00000 −1.12197 −0.107181
1.11 1.00000 1.55692 1.00000 −0.954341 1.55692 0.269753 1.00000 −0.576007 −0.954341
1.12 1.00000 2.01246 1.00000 1.68755 2.01246 −3.48754 1.00000 1.05001 1.68755
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(-1\)
\(19\) \(1\)

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 7942.2.a.bv yes 12
19.b odd 2 1 7942.2.a.bu ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7942.2.a.bu ✓ 12 19.b odd 2 1
7942.2.a.bv yes 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7942))\):

\( T_{3}^{12} + 6 T_{3}^{11} + T_{3}^{10} - 52 T_{3}^{9} - 65 T_{3}^{8} + 146 T_{3}^{7} + 262 T_{3}^{6} + \cdots - 19 \) Copy content Toggle raw display
\( T_{5}^{12} + 6 T_{5}^{11} - 12 T_{5}^{10} - 116 T_{5}^{9} - 11 T_{5}^{8} + 718 T_{5}^{7} + 412 T_{5}^{6} + \cdots - 19 \) Copy content Toggle raw display
\( T_{13}^{12} + 26 T_{13}^{11} + 238 T_{13}^{10} + 626 T_{13}^{9} - 3367 T_{13}^{8} - 22068 T_{13}^{7} + \cdots - 14039 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 6 T^{11} + \cdots - 19 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots - 19 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + \cdots - 2099 \) Copy content Toggle raw display
$11$ \( (T - 1)^{12} \) Copy content Toggle raw display
$13$ \( T^{12} + 26 T^{11} + \cdots - 14039 \) Copy content Toggle raw display
$17$ \( T^{12} - 101 T^{10} + \cdots + 67241 \) Copy content Toggle raw display
$19$ \( T^{12} \) Copy content Toggle raw display
$23$ \( T^{12} - 4 T^{11} + \cdots + 779741 \) Copy content Toggle raw display
$29$ \( T^{12} + 24 T^{11} + \cdots - 581879 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 371736479 \) Copy content Toggle raw display
$37$ \( T^{12} + 8 T^{11} + \cdots - 34252279 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots - 830751079 \) Copy content Toggle raw display
$43$ \( T^{12} + 32 T^{11} + \cdots - 55491259 \) Copy content Toggle raw display
$47$ \( T^{12} + 2 T^{11} + \cdots + 7560841 \) Copy content Toggle raw display
$53$ \( T^{12} + 28 T^{11} + \cdots - 59875559 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 5254326481 \) Copy content Toggle raw display
$61$ \( T^{12} - 363 T^{10} + \cdots + 12707561 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 18632171741 \) Copy content Toggle raw display
$71$ \( T^{12} + 4 T^{11} + \cdots + 7382021 \) Copy content Toggle raw display
$73$ \( T^{12} + 22 T^{11} + \cdots + 6570181 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 260545559 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 70919562121 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 447105176501 \) Copy content Toggle raw display
$97$ \( T^{12} + 20 T^{11} + \cdots - 12591559 \) Copy content Toggle raw display
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