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Properties

Label 1568.4.a.bm
Level $1568$
Weight $4$
Character orbit 1568.a
Self dual yes
Analytic conductor $92.515$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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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] = [1568,4,Mod(1,1568)] 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("1568.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1568, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1568 = 2^{5} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1568.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,0,0,0,0,126,0,44,0,0,0,256,0,0,0,0,0,0,0,304,0,422] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(92.5149948890\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 104x^{8} - 12x^{7} + 3241x^{6} + 572x^{5} - 36090x^{4} - 6576x^{3} + 107292x^{2} + 10832x - 95032 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14}\cdot 7^{2} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{2} - \beta_1) q^{5} + (\beta_{4} + 13) q^{9} + ( - \beta_{8} + 4) q^{11} + ( - \beta_{5} + \beta_{2} - 2 \beta_1) q^{13} + (\beta_{8} - \beta_{7} + 2 \beta_{4} + 27) q^{15}+ \cdots + (17 \beta_{8} + 12 \beta_{6} + \cdots + 124) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 126 q^{9} + 44 q^{11} + 256 q^{15} + 304 q^{23} + 422 q^{25} - 348 q^{29} - 364 q^{37} + 760 q^{39} + 772 q^{43} + 580 q^{51} - 996 q^{53} - 1220 q^{57} + 1592 q^{65} + 2344 q^{67} - 1400 q^{71}+ \cdots + 1140 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 104x^{8} - 12x^{7} + 3241x^{6} + 572x^{5} - 36090x^{4} - 6576x^{3} + 107292x^{2} + 10832x - 95032 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6789 \nu^{9} - 10397 \nu^{8} - 691990 \nu^{7} + 982522 \nu^{6} + 20684493 \nu^{5} + \cdots - 477686356 ) / 1234632 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 62712 \nu^{9} - 118295 \nu^{8} - 6385342 \nu^{7} + 11318470 \nu^{6} + 190741842 \nu^{5} + \cdots - 5448833068 ) / 3703896 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 32090 \nu^{9} - 29011 \nu^{8} - 3272798 \nu^{7} + 2570032 \nu^{6} + 97731024 \nu^{5} + \cdots - 1227376948 ) / 1587384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6789 \nu^{9} + 10397 \nu^{8} + 691990 \nu^{7} - 982522 \nu^{6} - 20684493 \nu^{5} + \cdots + 464722720 ) / 308658 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 340985 \nu^{9} - 358741 \nu^{8} - 34688642 \nu^{7} + 32401990 \nu^{6} + 1031219493 \nu^{5} + \cdots - 15419497828 ) / 11111688 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 236162 \nu^{9} + 198892 \nu^{8} + 24068813 \nu^{7} - 17385103 \nu^{6} - 717749349 \nu^{5} + \cdots + 8264477956 ) / 5555844 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 247694 \nu^{9} + 194707 \nu^{8} + 25228040 \nu^{7} - 16779982 \nu^{6} - 751381530 \nu^{5} + \cdots + 7942873120 ) / 5555844 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 314395 \nu^{9} - 757532 \nu^{8} - 31987936 \nu^{7} + 73369526 \nu^{6} + 955929189 \nu^{5} + \cdots - 34572972728 ) / 5555844 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 773800 \nu^{9} - 507425 \nu^{8} - 78877510 \nu^{7} + 42235202 \nu^{6} + 2351523738 \nu^{5} + \cdots - 19870370324 ) / 11111688 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 2\beta_{6} - 2\beta_{3} - 1 ) / 14 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 4\beta_{6} + 7\beta_{4} - 4\beta_{3} + 28\beta _1 + 292 ) / 14 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 21\beta_{9} + 62\beta_{7} - 82\beta_{6} + 7\beta_{4} - 118\beta_{3} + 21\beta_{2} + 8 ) / 14 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 28 \beta_{9} + 42 \beta_{8} + 114 \beta_{7} - 228 \beta_{6} + 385 \beta_{4} - 344 \beta_{3} + \cdots + 12192 ) / 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1645 \beta_{9} + 3476 \beta_{7} - 4138 \beta_{6} - 210 \beta_{5} + 511 \beta_{4} - 7338 \beta_{3} + \cdots + 2586 ) / 14 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2506 \beta_{9} + 3234 \beta_{8} + 6754 \beta_{7} - 13256 \beta_{6} + 21483 \beta_{4} - 23932 \beta_{3} + \cdots + 625276 ) / 14 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 105889 \beta_{9} + 252 \beta_{8} + 199060 \beta_{7} - 229490 \beta_{6} - 16758 \beta_{5} + \cdots + 254190 ) / 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 178920 \beta_{9} + 202146 \beta_{8} + 416250 \beta_{7} - 795372 \beta_{6} - 2016 \beta_{5} + \cdots + 34871240 ) / 14 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6449877 \beta_{9} + 41160 \beta_{8} + 11590984 \beta_{7} - 13264130 \beta_{6} - 1071378 \beta_{5} + \cdots + 20760062 ) / 14 \) 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
−7.57755
7.75666
−4.28635
4.15915
−1.46613
1.36230
1.33073
−1.45792
4.92824
−4.74913
0 −10.1305 0 −18.6704 0 0 0 75.6269 0
1.2 0 −7.55537 0 −7.69228 0 0 0 30.0836 0
1.3 0 −5.47603 0 14.6659 0 0 0 2.98688 0
1.4 0 −2.46772 0 14.4251 0 0 0 −20.9104 0
1.5 0 −1.48764 0 2.25206 0 0 0 −24.7869 0
1.6 0 1.48764 0 −2.25206 0 0 0 −24.7869 0
1.7 0 2.46772 0 −14.4251 0 0 0 −20.9104 0
1.8 0 5.47603 0 −14.6659 0 0 0 2.98688 0
1.9 0 7.55537 0 7.69228 0 0 0 30.0836 0
1.10 0 10.1305 0 18.6704 0 0 0 75.6269 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

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

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1568.4.a.bm yes 10
4.b odd 2 1 1568.4.a.bl ✓ 10
7.b odd 2 1 inner 1568.4.a.bm yes 10
28.d even 2 1 1568.4.a.bl ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1568.4.a.bl ✓ 10 4.b odd 2 1
1568.4.a.bl ✓ 10 28.d even 2 1
1568.4.a.bm yes 10 1.a even 1 1 trivial
1568.4.a.bm yes 10 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1568))\):

\( T_{3}^{10} - 198T_{3}^{8} + 12236T_{3}^{6} - 266632T_{3}^{4} + 1602048T_{3}^{2} - 2367488 \) Copy content Toggle raw display
\( T_{5}^{10} - 836T_{5}^{8} + 242148T_{5}^{6} - 28184896T_{5}^{4} + 1059981568T_{5}^{2} - 4682022912 \) Copy content Toggle raw display
\( T_{11}^{5} - 22T_{11}^{4} - 2740T_{11}^{3} + 71416T_{11}^{2} + 412672T_{11} - 1502720 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - 198 T^{8} + \cdots - 2367488 \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots - 4682022912 \) Copy content Toggle raw display
$7$ \( T^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} - 22 T^{4} + \cdots - 1502720)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots - 55\!\cdots\!68 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots - 68\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots - 12\!\cdots\!08 \) Copy content Toggle raw display
$23$ \( (T^{5} - 152 T^{4} + \cdots + 7363665920)^{2} \) Copy content Toggle raw display
$29$ \( (T^{5} + 174 T^{4} + \cdots + 82109900160)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots - 73\!\cdots\!68 \) Copy content Toggle raw display
$37$ \( (T^{5} + 182 T^{4} + \cdots - 55233499264)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots - 51\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( (T^{5} - 386 T^{4} + \cdots - 870907046400)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots - 36\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( (T^{5} + \cdots + 15432593499936)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots - 42\!\cdots\!28 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots - 42\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( (T^{5} - 1172 T^{4} + \cdots - 335954313216)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 5085314220032)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots - 46\!\cdots\!28 \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots + 40827290370048)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 10\!\cdots\!52 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 50\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots - 27\!\cdots\!68 \) Copy content Toggle raw display
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