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Properties

Label 7742.2.a.bk
Level $7742$
Weight $2$
Character orbit 7742.a
Self dual yes
Analytic conductor $61.820$
Analytic rank $1$
Dimension $12$
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] = [7742,2,Mod(1,7742)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7742, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7742.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 7742 = 2 \cdot 7^{2} \cdot 79 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7742.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-12,-3,12,3,3,0,-12,17,-3,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(61.8201812449\)
Analytic rank: \(1\)
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: \( x^{12} - 3 x^{11} - 22 x^{10} + 67 x^{9} + 148 x^{8} - 471 x^{7} - 306 x^{6} + 1117 x^{5} + 207 x^{4} + \cdots - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1106)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} - \beta_1 q^{3} + q^{4} - \beta_{2} q^{5} + \beta_1 q^{6} - q^{8} + (\beta_{11} + \beta_{10} + \beta_{7} + \cdots + 2) q^{9} + \beta_{2} q^{10} - \beta_{6} q^{11} - \beta_1 q^{12} + ( - \beta_{3} - 1) q^{13}+ \cdots + (2 \beta_{10} - \beta_{9} + 2 \beta_{5} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 3 q^{3} + 12 q^{4} + 3 q^{5} + 3 q^{6} - 12 q^{8} + 17 q^{9} - 3 q^{10} - 2 q^{11} - 3 q^{12} - 13 q^{13} - 2 q^{15} + 12 q^{16} - 5 q^{17} - 17 q^{18} - 14 q^{19} + 3 q^{20} + 2 q^{22}+ \cdots - 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} - 22 x^{10} + 67 x^{9} + 148 x^{8} - 471 x^{7} - 306 x^{6} + 1117 x^{5} + 207 x^{4} + \cdots - 7 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11297 \nu^{11} + 9171 \nu^{10} - 413349 \nu^{9} - 91619 \nu^{8} + 5167141 \nu^{7} - 885606 \nu^{6} + \cdots - 4430405 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 18613 \nu^{11} - 9460 \nu^{10} + 609327 \nu^{9} + 146331 \nu^{8} - 6988909 \nu^{7} + \cdots - 1313515 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 27487 \nu^{11} - 87686 \nu^{10} - 529706 \nu^{9} + 1776806 \nu^{8} + 2449904 \nu^{7} + \cdots - 5943728 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 32023 \nu^{11} + 9795 \nu^{10} + 900048 \nu^{9} - 150635 \nu^{8} - 8959439 \nu^{7} + \cdots + 2419669 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38445 \nu^{11} - 87318 \nu^{10} - 886815 \nu^{9} + 1942783 \nu^{8} + 6457768 \nu^{7} + \cdots + 1801387 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47829 \nu^{11} - 108224 \nu^{10} - 1108091 \nu^{9} + 2277252 \nu^{8} + 8372850 \nu^{7} + \cdots - 8221262 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 58385 \nu^{11} + 153993 \nu^{10} + 1331312 \nu^{9} - 3366063 \nu^{8} - 9657870 \nu^{7} + \cdots - 1938139 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62894 \nu^{11} - 261593 \nu^{10} - 1126006 \nu^{9} + 5584299 \nu^{8} + 3904140 \nu^{7} + \cdots + 566951 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 82223 \nu^{11} - 219541 \nu^{10} - 1875816 \nu^{9} + 4837904 \nu^{8} + 13623636 \nu^{7} + \cdots + 3059245 ) / 1661009 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 122397 \nu^{11} + 336857 \nu^{10} + 2731689 \nu^{9} - 7427464 \nu^{8} - 19015441 \nu^{7} + \cdots - 9574628 ) / 1661009 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} - \beta_{4} + \beta_{3} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{9} - \beta_{8} + \beta_{7} + \beta_{5} - \beta_{2} + 8\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12 \beta_{11} + 14 \beta_{10} - \beta_{9} + 11 \beta_{7} + 11 \beta_{6} - 12 \beta_{4} + 11 \beta_{3} + \cdots + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 7 \beta_{10} - 15 \beta_{9} - 8 \beta_{8} + 17 \beta_{7} + \beta_{6} + 16 \beta_{5} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 127 \beta_{11} + 162 \beta_{10} - 22 \beta_{9} + 116 \beta_{7} + 111 \beta_{6} + 3 \beta_{5} + \cdots + 428 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 32 \beta_{11} + 123 \beta_{10} - 182 \beta_{9} - 50 \beta_{8} + 228 \beta_{7} + 15 \beta_{6} + 200 \beta_{5} + \cdots + 41 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1305 \beta_{11} + 1770 \beta_{10} - 328 \beta_{9} - 2 \beta_{8} + 1211 \beta_{7} + 1118 \beta_{6} + \cdots + 4297 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 384 \beta_{11} + 1628 \beta_{10} - 2094 \beta_{9} - 220 \beta_{8} + 2805 \beta_{7} + 202 \beta_{6} + \cdots + 812 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 13316 \beta_{11} + 18941 \beta_{10} - 4301 \beta_{9} - 39 \beta_{8} + 12668 \beta_{7} + 11376 \beta_{6} + \cdots + 43799 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4167 \beta_{11} + 19580 \beta_{10} - 23678 \beta_{9} + 332 \beta_{8} + 33093 \beta_{7} + 2677 \beta_{6} + \cdots + 12494 \) 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
3.31126
3.15740
2.29039
1.79112
0.932671
0.644050
0.0351312
−0.540089
−1.13662
−1.33693
−2.90942
−3.23895
−1.00000 −3.31126 1.00000 1.82584 3.31126 0 −1.00000 7.96441 −1.82584
1.2 −1.00000 −3.15740 1.00000 −2.79738 3.15740 0 −1.00000 6.96915 2.79738
1.3 −1.00000 −2.29039 1.00000 1.11476 2.29039 0 −1.00000 2.24588 −1.11476
1.4 −1.00000 −1.79112 1.00000 −3.27530 1.79112 0 −1.00000 0.208114 3.27530
1.5 −1.00000 −0.932671 1.00000 −0.909357 0.932671 0 −1.00000 −2.13012 0.909357
1.6 −1.00000 −0.644050 1.00000 3.51758 0.644050 0 −1.00000 −2.58520 −3.51758
1.7 −1.00000 −0.0351312 1.00000 2.93878 0.0351312 0 −1.00000 −2.99877 −2.93878
1.8 −1.00000 0.540089 1.00000 2.34575 −0.540089 0 −1.00000 −2.70830 −2.34575
1.9 −1.00000 1.13662 1.00000 3.73696 −1.13662 0 −1.00000 −1.70810 −3.73696
1.10 −1.00000 1.33693 1.00000 −3.04483 −1.33693 0 −1.00000 −1.21261 3.04483
1.11 −1.00000 2.90942 1.00000 0.237744 −2.90942 0 −1.00000 5.46473 −0.237744
1.12 −1.00000 3.23895 1.00000 −2.69055 −3.23895 0 −1.00000 7.49081 2.69055
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(7\) \( +1 \)
\(79\) \( +1 \)

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 7742.2.a.bk 12
7.b odd 2 1 7742.2.a.bn 12
7.c even 3 2 1106.2.f.f ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1106.2.f.f ✓ 24 7.c even 3 2
7742.2.a.bk 12 1.a even 1 1 trivial
7742.2.a.bn 12 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7742))\):

\( T_{3}^{12} + 3 T_{3}^{11} - 22 T_{3}^{10} - 67 T_{3}^{9} + 148 T_{3}^{8} + 471 T_{3}^{7} - 306 T_{3}^{6} + \cdots - 7 \) Copy content Toggle raw display
\( T_{5}^{12} - 3 T_{5}^{11} - 36 T_{5}^{10} + 105 T_{5}^{9} + 482 T_{5}^{8} - 1388 T_{5}^{7} - 2836 T_{5}^{6} + \cdots - 2993 \) Copy content Toggle raw display
\( T_{11}^{12} + 2 T_{11}^{11} - 48 T_{11}^{10} - 108 T_{11}^{9} + 787 T_{11}^{8} + 1955 T_{11}^{7} + \cdots - 36643 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + \cdots - 7 \) Copy content Toggle raw display
$5$ \( T^{12} - 3 T^{11} + \cdots - 2993 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 2 T^{11} + \cdots - 36643 \) Copy content Toggle raw display
$13$ \( T^{12} + 13 T^{11} + \cdots - 10775 \) Copy content Toggle raw display
$17$ \( T^{12} + 5 T^{11} + \cdots + 2045889 \) Copy content Toggle raw display
$19$ \( T^{12} + 14 T^{11} + \cdots - 15573 \) Copy content Toggle raw display
$23$ \( T^{12} - 2 T^{11} + \cdots + 461475 \) Copy content Toggle raw display
$29$ \( T^{12} - 7 T^{11} + \cdots - 1001600 \) Copy content Toggle raw display
$31$ \( T^{12} + 15 T^{11} + \cdots + 2033955 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 449660161 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 164806283 \) Copy content Toggle raw display
$43$ \( T^{12} - 8 T^{11} + \cdots + 64966267 \) Copy content Toggle raw display
$47$ \( T^{12} - 13 T^{11} + \cdots - 60831 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 183358647 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 3282912995 \) Copy content Toggle raw display
$61$ \( T^{12} - 6 T^{11} + \cdots + 13323 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 8253309824 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 16230940189 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 315265833 \) Copy content Toggle raw display
$79$ \( (T + 1)^{12} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 180578283 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots - 168163119 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 2756739013 \) Copy content Toggle raw display
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