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Properties

Label 7774.2.a.bl
Level $7774$
Weight $2$
Character orbit 7774.a
Self dual yes
Analytic conductor $62.076$
Analytic rank $1$
Dimension $14$
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] = [7774,2,Mod(1,7774)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("7774.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7774, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7774 = 2 \cdot 13^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7774.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,-14,4,14,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.0757025313\)
Analytic rank: \(1\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 4 x^{13} - 16 x^{12} + 74 x^{11} + 76 x^{10} - 470 x^{9} - 88 x^{8} + 1200 x^{7} - 3 x^{6} + \cdots - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 598)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + \beta_1 q^{3} + q^{4} + ( - \beta_{9} - 1) q^{5} - \beta_1 q^{6} - \beta_{13} q^{7} - q^{8} + \beta_{2} q^{9} + (\beta_{9} + 1) q^{10} + (\beta_{4} - 1) q^{11} + \beta_1 q^{12} + \beta_{13} q^{14}+ \cdots + (\beta_{13} + 2 \beta_{9} - \beta_{8} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} + 4 q^{3} + 14 q^{4} - 8 q^{5} - 4 q^{6} - 2 q^{7} - 14 q^{8} + 6 q^{9} + 8 q^{10} - 12 q^{11} + 4 q^{12} + 2 q^{14} - 8 q^{15} + 14 q^{16} - 2 q^{17} - 6 q^{18} - 2 q^{19} - 8 q^{20} - 6 q^{21}+ \cdots - 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 4 x^{13} - 16 x^{12} + 74 x^{11} + 76 x^{10} - 470 x^{9} - 88 x^{8} + 1200 x^{7} - 3 x^{6} + \cdots - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14 \nu^{13} - 338 \nu^{12} + 1927 \nu^{11} + 4657 \nu^{10} - 31797 \nu^{9} - 12383 \nu^{8} + \cdots + 1384 ) / 2574 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 361 \nu^{13} - 1703 \nu^{12} - 4736 \nu^{11} + 30832 \nu^{10} + 8271 \nu^{9} - 188417 \nu^{8} + \cdots + 5374 ) / 2574 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 152 \nu^{13} + 702 \nu^{12} + 2107 \nu^{11} - 12726 \nu^{10} - 5763 \nu^{9} + 77911 \nu^{8} + \cdots + 522 ) / 858 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 261 \nu^{13} - 1196 \nu^{12} - 3474 \nu^{11} + 21421 \nu^{10} + 7110 \nu^{9} - 128433 \nu^{8} + \cdots - 1124 ) / 858 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1544 \nu^{13} + 6604 \nu^{12} + 22780 \nu^{11} - 120584 \nu^{10} - 81864 \nu^{9} + 748873 \nu^{8} + \cdots + 13762 ) / 2574 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1552 \nu^{13} + 6656 \nu^{12} + 22778 \nu^{11} - 121171 \nu^{10} - 80361 \nu^{9} + 748232 \nu^{8} + \cdots + 12224 ) / 2574 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1957 \nu^{13} - 8645 \nu^{12} - 27503 \nu^{11} + 155875 \nu^{10} + 79722 \nu^{9} - 945350 \nu^{8} + \cdots + 2896 ) / 2574 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 71 \nu^{13} - 302 \nu^{12} - 1063 \nu^{11} + 5530 \nu^{10} + 4056 \nu^{9} - 34501 \nu^{8} + 2152 \nu^{7} + \cdots - 488 ) / 66 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 1211 \nu^{13} - 5369 \nu^{12} - 17179 \nu^{11} + 97312 \nu^{10} + 52245 \nu^{9} - 596179 \nu^{8} + \cdots - 7604 ) / 858 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 445 \nu^{13} - 1898 \nu^{12} - 6587 \nu^{11} + 34597 \nu^{10} + 24138 \nu^{9} - 214154 \nu^{8} + \cdots - 2696 ) / 234 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + 7\beta_{2} + \beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{12} + \beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{5} - \beta_{4} + 10\beta_{3} + \beta_{2} + 41\beta _1 + 9 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{12} + 10 \beta_{11} - \beta_{10} + 9 \beta_{9} + 13 \beta_{8} + 12 \beta_{7} + 12 \beta_{6} + \cdots + 132 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{13} + 14 \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + 9 \beta_{9} + 22 \beta_{8} - \beta_{7} + \cdots + 73 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{13} + 16 \beta_{12} + 81 \beta_{11} - 16 \beta_{10} + 59 \beta_{9} + 139 \beta_{8} + 107 \beta_{7} + \cdots + 944 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 24 \beta_{13} + 147 \beta_{12} + 32 \beta_{11} - 40 \beta_{10} + 53 \beta_{9} + 303 \beta_{8} + \cdots + 588 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 48 \beta_{13} + 183 \beta_{12} + 621 \beta_{11} - 195 \beta_{10} + 314 \beta_{9} + 1386 \beta_{8} + \cdots + 6851 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 355 \beta_{13} + 1385 \beta_{12} + 364 \beta_{11} - 538 \beta_{10} + 174 \beta_{9} + 3453 \beta_{8} + \cdots + 4774 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 734 \beta_{13} + 1841 \beta_{12} + 4691 \beta_{11} - 2119 \beta_{10} + 1146 \beta_{9} + 13294 \beta_{8} + \cdots + 50234 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 4269 \beta_{13} + 12365 \beta_{12} + 3632 \beta_{11} - 6110 \beta_{10} - 986 \beta_{9} + 35685 \beta_{8} + \cdots + 39125 \) 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
−2.70119
−2.59019
−1.60924
−0.935010
−0.438446
−0.326982
−0.185788
0.155885
0.628502
1.72190
2.07144
2.59433
2.68866
2.92615
−1.00000 −2.70119 1.00000 −1.33585 2.70119 3.51776 −1.00000 4.29645 1.33585
1.2 −1.00000 −2.59019 1.00000 −2.06019 2.59019 −0.488916 −1.00000 3.70911 2.06019
1.3 −1.00000 −1.60924 1.00000 2.70952 1.60924 −2.39305 −1.00000 −0.410338 −2.70952
1.4 −1.00000 −0.935010 1.00000 0.444674 0.935010 −3.50397 −1.00000 −2.12576 −0.444674
1.5 −1.00000 −0.438446 1.00000 0.411787 0.438446 4.49554 −1.00000 −2.80777 −0.411787
1.6 −1.00000 −0.326982 1.00000 2.43918 0.326982 −2.45710 −1.00000 −2.89308 −2.43918
1.7 −1.00000 −0.185788 1.00000 −0.931614 0.185788 4.59769 −1.00000 −2.96548 0.931614
1.8 −1.00000 0.155885 1.00000 −3.04588 −0.155885 −4.38155 −1.00000 −2.97570 3.04588
1.9 −1.00000 0.628502 1.00000 −4.29224 −0.628502 0.325318 −1.00000 −2.60498 4.29224
1.10 −1.00000 1.72190 1.00000 3.67067 −1.72190 −0.629766 −1.00000 −0.0350623 −3.67067
1.11 −1.00000 2.07144 1.00000 −1.91535 −2.07144 −2.10411 −1.00000 1.29085 1.91535
1.12 −1.00000 2.59433 1.00000 −1.63382 −2.59433 −3.40529 −1.00000 3.73053 1.63382
1.13 −1.00000 2.68866 1.00000 −3.77055 −2.68866 4.50185 −1.00000 4.22891 3.77055
1.14 −1.00000 2.92615 1.00000 1.30968 −2.92615 −0.0744085 −1.00000 5.56233 −1.30968
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(13\) \( -1 \)
\(23\) \( -1 \)

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 7774.2.a.bl 14
13.b even 2 1 7774.2.a.bm 14
13.f odd 12 2 598.2.h.c ✓ 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
598.2.h.c ✓ 28 13.f odd 12 2
7774.2.a.bl 14 1.a even 1 1 trivial
7774.2.a.bm 14 13.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7774))\):

\( T_{3}^{14} - 4 T_{3}^{13} - 16 T_{3}^{12} + 74 T_{3}^{11} + 76 T_{3}^{10} - 470 T_{3}^{9} - 88 T_{3}^{8} + \cdots - 2 \) Copy content Toggle raw display
\( T_{5}^{14} + 8 T_{5}^{13} - 10 T_{5}^{12} - 218 T_{5}^{11} - 266 T_{5}^{10} + 1862 T_{5}^{9} + \cdots + 2301 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{14} \) Copy content Toggle raw display
$3$ \( T^{14} - 4 T^{13} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{14} + 8 T^{13} + \cdots + 2301 \) Copy content Toggle raw display
$7$ \( T^{14} + 2 T^{13} + \cdots - 1578 \) Copy content Toggle raw display
$11$ \( T^{14} + 12 T^{13} + \cdots - 83208 \) Copy content Toggle raw display
$13$ \( T^{14} \) Copy content Toggle raw display
$17$ \( T^{14} + 2 T^{13} + \cdots - 96867 \) Copy content Toggle raw display
$19$ \( T^{14} + 2 T^{13} + \cdots + 1097304 \) Copy content Toggle raw display
$23$ \( (T - 1)^{14} \) Copy content Toggle raw display
$29$ \( T^{14} - 14 T^{13} + \cdots + 933669 \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 314485776 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 351051381 \) Copy content Toggle raw display
$41$ \( T^{14} + 40 T^{13} + \cdots + 914325 \) Copy content Toggle raw display
$43$ \( T^{14} - 12 T^{13} + \cdots - 167072 \) Copy content Toggle raw display
$47$ \( T^{14} + 26 T^{13} + \cdots - 5764656 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 3794647923 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots - 12059206128 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots + 4098244525 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 319732764126 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 147765696342 \) Copy content Toggle raw display
$73$ \( T^{14} + 28 T^{13} + \cdots + 15210681 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 4131931024 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 745007838 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 375810600 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 52073089896 \) Copy content Toggle raw display
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