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Properties

Label 7774.2.a.bk
Level $7774$
Weight $2$
Character orbit 7774.a
Self dual yes
Analytic conductor $62.076$
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] = [7774,2,Mod(1,7774)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
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")
 
//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);
 
Level: \( N \) \(=\) \( 7774 = 2 \cdot 13^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7774.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: \(62.0757025313\)
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: \( x^{12} - 2 x^{11} - 26 x^{10} + 46 x^{9} + 244 x^{8} - 366 x^{7} - 996 x^{6} + 1164 x^{5} + 1685 x^{4} + \cdots + 97 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 598)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{10} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{5} + \beta_1) q^{7} + q^{8} + (\beta_{11} - \beta_{9} - \beta_{5} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta_1 q^{3} + q^{4} + ( - \beta_{10} - 1) q^{5} - \beta_1 q^{6} + ( - \beta_{5} + \beta_1) q^{7} + q^{8} + (\beta_{11} - \beta_{9} - \beta_{5} + \cdots + 2) q^{9}+ \cdots + (\beta_{10} - 3 \beta_{9} + \beta_{8} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 8 q^{5} - 2 q^{6} + 12 q^{8} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} - 2 q^{3} + 12 q^{4} - 8 q^{5} - 2 q^{6} + 12 q^{8} + 20 q^{9} - 8 q^{10} - 12 q^{11} - 2 q^{12} - 18 q^{15} + 12 q^{16} - 2 q^{17} + 20 q^{18} - 14 q^{19} - 8 q^{20} - 30 q^{21} - 12 q^{22} - 12 q^{23} - 2 q^{24} + 26 q^{25} - 14 q^{27} + 2 q^{29} - 18 q^{30} - 16 q^{31} + 12 q^{32} - 54 q^{33} - 2 q^{34} + 4 q^{35} + 20 q^{36} - 12 q^{37} - 14 q^{38} - 8 q^{40} - 16 q^{41} - 30 q^{42} - 32 q^{43} - 12 q^{44} - 26 q^{45} - 12 q^{46} - 30 q^{47} - 2 q^{48} - 6 q^{49} + 26 q^{50} + 32 q^{51} - 6 q^{53} - 14 q^{54} - 32 q^{55} + 10 q^{57} + 2 q^{58} - 28 q^{59} - 18 q^{60} - 8 q^{61} - 16 q^{62} + 32 q^{63} + 12 q^{64} - 54 q^{66} - 32 q^{67} - 2 q^{68} + 2 q^{69} + 4 q^{70} - 48 q^{71} + 20 q^{72} + 24 q^{73} - 12 q^{74} - 24 q^{75} - 14 q^{76} - 16 q^{77} - 30 q^{79} - 8 q^{80} + 44 q^{81} - 16 q^{82} + 4 q^{83} - 30 q^{84} - 22 q^{85} - 32 q^{86} + 32 q^{87} - 12 q^{88} - 40 q^{89} - 26 q^{90} - 12 q^{92} - 28 q^{93} - 30 q^{94} + 4 q^{95} - 2 q^{96} - 48 q^{97} - 6 q^{98} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} - 26 x^{10} + 46 x^{9} + 244 x^{8} - 366 x^{7} - 996 x^{6} + 1164 x^{5} + 1685 x^{4} + \cdots + 97 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 110231 \nu^{11} + 246169 \nu^{10} + 2167677 \nu^{9} - 4864408 \nu^{8} - 12105720 \nu^{7} + \cdots - 65971740 ) / 30200586 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15588 \nu^{11} - 22795 \nu^{10} - 409670 \nu^{9} + 490775 \nu^{8} + 3872083 \nu^{7} + \cdots + 512246 ) / 2323122 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22412 \nu^{11} - 44796 \nu^{10} - 548078 \nu^{9} + 1072144 \nu^{8} + 4399454 \nu^{7} + \cdots + 4285161 ) / 2323122 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 344187 \nu^{11} + 585809 \nu^{10} + 8942338 \nu^{9} - 13504942 \nu^{8} - 81252050 \nu^{7} + \cdots + 66003032 ) / 30200586 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 353975 \nu^{11} - 979361 \nu^{10} - 8647461 \nu^{9} + 23157659 \nu^{8} + 73140631 \nu^{7} + \cdots + 108987247 ) / 30200586 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 385830 \nu^{11} + 1540231 \nu^{10} + 8467895 \nu^{9} - 35577710 \nu^{8} - 60556222 \nu^{7} + \cdots - 94075877 ) / 30200586 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 446752 \nu^{11} + 579285 \nu^{10} + 11269998 \nu^{9} - 10775364 \nu^{8} - 101460127 \nu^{7} + \cdots + 38987999 ) / 30200586 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 462928 \nu^{11} + 1036327 \nu^{10} + 11589010 \nu^{9} - 24049602 \nu^{8} - 102097910 \nu^{7} + \cdots - 89330779 ) / 30200586 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 625990 \nu^{11} + 1285957 \nu^{10} + 15574011 \nu^{9} - 28481080 \nu^{8} - 138262778 \nu^{7} + \cdots - 63792980 ) / 30200586 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 807115 \nu^{11} + 1622136 \nu^{10} + 20531348 \nu^{9} - 37554544 \nu^{8} - 183349960 \nu^{7} + \cdots - 174330677 ) / 30200586 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{5} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{5} - \beta_{4} - \beta_{2} + 9\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{11} + 2\beta_{10} - 11\beta_{9} + \beta_{6} - 8\beta_{5} + 5\beta_{3} - \beta_{2} + 12\beta _1 + 40 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{11} + 14 \beta_{10} - 20 \beta_{9} - \beta_{7} - 5 \beta_{6} - 13 \beta_{5} - 12 \beta_{4} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 101 \beta_{11} + 30 \beta_{10} - 115 \beta_{9} + 4 \beta_{8} - 2 \beta_{7} + 10 \beta_{6} - 73 \beta_{5} + \cdots + 360 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 46 \beta_{11} + 153 \beta_{10} - 277 \beta_{9} + 10 \beta_{8} - 16 \beta_{7} - 92 \beta_{6} - 151 \beta_{5} + \cdots + 309 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1039 \beta_{11} + 344 \beta_{10} - 1210 \beta_{9} + 106 \beta_{8} - 30 \beta_{7} + 71 \beta_{6} + \cdots + 3431 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 754 \beta_{11} + 1549 \beta_{10} - 3385 \beta_{9} + 264 \beta_{8} - 191 \beta_{7} - 1253 \beta_{6} + \cdots + 3984 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10851 \beta_{11} + 3608 \beta_{10} - 12900 \beta_{9} + 1840 \beta_{8} - 298 \beta_{7} + 259 \beta_{6} + \cdots + 33930 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 10758 \beta_{11} + 15225 \beta_{10} - 39263 \beta_{9} + 4700 \beta_{8} - 1997 \beta_{7} - 15339 \beta_{6} + \cdots + 48340 \) 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
3.37074
3.07066
2.35031
1.96609
1.20384
0.347836
−0.318878
−0.619692
−1.26231
−2.42907
−2.57377
−3.10575
1.00000 −3.37074 1.00000 −0.127004 −3.37074 1.74666 1.00000 8.36189 −0.127004
1.2 1.00000 −3.07066 1.00000 −3.88363 −3.07066 3.28470 1.00000 6.42893 −3.88363
1.3 1.00000 −2.35031 1.00000 4.02483 −2.35031 2.50718 1.00000 2.52396 4.02483
1.4 1.00000 −1.96609 1.00000 1.52254 −1.96609 −0.749241 1.00000 0.865523 1.52254
1.5 1.00000 −1.20384 1.00000 1.64757 −1.20384 1.26051 1.00000 −1.55078 1.64757
1.6 1.00000 −0.347836 1.00000 −4.00363 −0.347836 −4.05307 1.00000 −2.87901 −4.00363
1.7 1.00000 0.318878 1.00000 2.08847 0.318878 −1.41679 1.00000 −2.89832 2.08847
1.8 1.00000 0.619692 1.00000 −1.71485 0.619692 −2.12563 1.00000 −2.61598 −1.71485
1.9 1.00000 1.26231 1.00000 −2.24932 1.26231 4.16153 1.00000 −1.40657 −2.24932
1.10 1.00000 2.42907 1.00000 −4.13189 2.42907 0.659174 1.00000 2.90040 −4.13189
1.11 1.00000 2.57377 1.00000 0.739727 2.57377 −3.38079 1.00000 3.62429 0.739727
1.12 1.00000 3.10575 1.00000 −1.91281 3.10575 −1.89423 1.00000 6.64565 −1.91281
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

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

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.bk 12
13.b even 2 1 7774.2.a.bi 12
13.f odd 12 2 598.2.h.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
598.2.h.b ✓ 24 13.f odd 12 2
7774.2.a.bi 12 13.b even 2 1
7774.2.a.bk 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7774))\):

\( T_{3}^{12} + 2 T_{3}^{11} - 26 T_{3}^{10} - 46 T_{3}^{9} + 244 T_{3}^{8} + 366 T_{3}^{7} - 996 T_{3}^{6} + \cdots + 97 \) Copy content Toggle raw display
\( T_{5}^{12} + 8 T_{5}^{11} - 11 T_{5}^{10} - 214 T_{5}^{9} - 203 T_{5}^{8} + 1692 T_{5}^{7} + 2384 T_{5}^{6} + \cdots - 939 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{11} + \cdots + 97 \) Copy content Toggle raw display
$5$ \( T^{12} + 8 T^{11} + \cdots - 939 \) Copy content Toggle raw display
$7$ \( T^{12} - 39 T^{10} + \cdots + 2913 \) Copy content Toggle raw display
$11$ \( T^{12} + 12 T^{11} + \cdots + 265317 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + 2 T^{11} + \cdots - 1908351 \) Copy content Toggle raw display
$19$ \( T^{12} + 14 T^{11} + \cdots - 211923 \) Copy content Toggle raw display
$23$ \( (T + 1)^{12} \) Copy content Toggle raw display
$29$ \( T^{12} - 2 T^{11} + \cdots - 6135939 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 452001927 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 123687897 \) Copy content Toggle raw display
$41$ \( T^{12} + 16 T^{11} + \cdots - 9935679 \) Copy content Toggle raw display
$43$ \( T^{12} + 32 T^{11} + \cdots - 35601311 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1065392313 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 442137447 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots - 54356724207 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} + \cdots - 116019 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots - 3560086947 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 4544087511 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 5198181921 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 249651791953 \) Copy content Toggle raw display
$83$ \( T^{12} - 4 T^{11} + \cdots + 36153573 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 3923815389 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots - 185581467 \) Copy content Toggle raw display
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