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Properties

Label 7105.2.a.w
Level $7105$
Weight $2$
Character orbit 7105.a
Self dual yes
Analytic conductor $56.734$
Analytic rank $0$
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] = [7105,2,Mod(1,7105)] 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("7105.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7105, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7105.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,2,12,-14] 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: \(56.7337106361\)
Analytic rank: \(0\)
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: \( x^{14} - 20 x^{12} + 152 x^{10} - 2 x^{9} - 549 x^{8} + 18 x^{7} + 970 x^{6} - 46 x^{5} - 787 x^{4} + \cdots - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{2} + 1) q^{4} - q^{5} + (\beta_{5} + \beta_{2} + 1) q^{6} + ( - \beta_{3} - \beta_1) q^{8} + ( - \beta_{12} + \beta_{5}) q^{9} + \beta_1 q^{10} + ( - \beta_{11} + \beta_{9} + \beta_{4}) q^{11}+ \cdots + ( - \beta_{13} - \beta_{11} - \beta_{10} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{3} + 12 q^{4} - 14 q^{5} + 12 q^{6} + 6 q^{9} - 2 q^{11} + 8 q^{12} + 6 q^{13} - 2 q^{15} + 8 q^{16} + 14 q^{17} + 22 q^{18} + 2 q^{19} - 12 q^{20} - 16 q^{22} - 12 q^{23} + 28 q^{24} + 14 q^{25}+ \cdots - 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 20 x^{12} + 152 x^{10} - 2 x^{9} - 549 x^{8} + 18 x^{7} + 970 x^{6} - 46 x^{5} - 787 x^{4} + \cdots - 20 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 141 \nu^{13} - 968 \nu^{12} + 546 \nu^{11} + 17358 \nu^{10} + 18552 \nu^{9} - 113862 \nu^{8} + \cdots - 74540 ) / 11630 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 139 \nu^{13} - 118 \nu^{12} - 2889 \nu^{11} + 1323 \nu^{10} + 24272 \nu^{9} - 3187 \nu^{8} + \cdots + 16240 ) / 5815 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 461 \nu^{13} + 278 \nu^{12} - 9456 \nu^{11} - 5778 \nu^{10} + 72718 \nu^{9} + 47622 \nu^{8} + \cdots + 75280 ) / 11630 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 298 \nu^{13} + 1006 \nu^{12} + 5608 \nu^{11} - 19361 \nu^{10} - 38984 \nu^{9} + 138444 \nu^{8} + \cdots + 29985 ) / 5815 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 145 \nu^{13} - 814 \nu^{12} + 2838 \nu^{11} + 15178 \nu^{10} - 21404 \nu^{9} - 103730 \nu^{8} + \cdots - 32972 ) / 2326 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 739 \nu^{13} - 42 \nu^{12} + 15234 \nu^{11} + 3132 \nu^{10} - 121262 \nu^{9} - 41248 \nu^{8} + \cdots - 37980 ) / 11630 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 1153 \nu^{13} - 3874 \nu^{12} + 24508 \nu^{11} + 72904 \nu^{10} - 200164 \nu^{9} - 504486 \nu^{8} + \cdots - 115550 ) / 11630 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 644 \nu^{13} + 792 \nu^{12} - 13134 \nu^{11} - 14202 \nu^{10} + 102707 \nu^{9} + 89988 \nu^{8} + \cdots + 1780 ) / 5815 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 418 \nu^{13} - 189 \nu^{12} + 8077 \nu^{11} + 3627 \nu^{10} - 58382 \nu^{9} - 25122 \nu^{8} + \cdots - 12742 ) / 1163 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 3042 \nu^{13} + 346 \nu^{12} - 59042 \nu^{11} - 7526 \nu^{10} + 428821 \nu^{9} + 57639 \nu^{8} + \cdots + 65330 ) / 5815 \) 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
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} + \beta_{6} + \beta_{5} + 7\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + 8\beta_{3} + \beta_{2} + 28\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{13} + \beta_{12} + \beta_{10} + 9 \beta_{9} - \beta_{8} + \beta_{7} + 8 \beta_{6} + 8 \beta_{5} + \cdots + 84 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - \beta_{13} - 2 \beta_{12} + 9 \beta_{11} + 11 \beta_{10} + \beta_{9} - \beta_{8} + 10 \beta_{7} + \cdots + 14 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 13 \beta_{13} + 13 \beta_{12} - 2 \beta_{11} + 14 \beta_{10} + 64 \beta_{9} - 16 \beta_{8} + 11 \beta_{7} + \cdots + 491 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 12 \beta_{13} - 25 \beta_{12} + 62 \beta_{11} + 94 \beta_{10} + 12 \beta_{9} - 20 \beta_{8} + \cdots + 142 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 119 \beta_{13} + 118 \beta_{12} - 31 \beta_{11} + 139 \beta_{10} + 427 \beta_{9} - 171 \beta_{8} + \cdots + 2937 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 97 \beta_{13} - 216 \beta_{12} + 390 \beta_{11} + 737 \beta_{10} + 103 \beta_{9} - 244 \beta_{8} + \cdots + 1241 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 947 \beta_{13} + 926 \beta_{12} - 318 \beta_{11} + 1202 \beta_{10} + 2789 \beta_{9} - 1538 \beta_{8} + \cdots + 17834 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 653 \beta_{13} - 1606 \beta_{12} + 2349 \beta_{11} + 5535 \beta_{10} + 795 \beta_{9} - 2405 \beta_{8} + \cdots + 9966 \) 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
2.58936
2.41039
1.75621
1.64793
1.02070
0.604527
0.455194
−0.288646
−0.684542
−1.20243
−1.31251
−2.18587
−2.36724
−2.44307
−2.58936 −1.73283 4.70477 −1.00000 4.48691 0 −7.00360 0.00269759 2.58936
1.2 −2.41039 −0.704268 3.80996 −1.00000 1.69756 0 −4.36270 −2.50401 2.41039
1.3 −1.75621 1.70406 1.08428 −1.00000 −2.99270 0 1.60820 −0.0961640 1.75621
1.4 −1.64793 0.310137 0.715669 −1.00000 −0.511084 0 2.11649 −2.90381 1.64793
1.5 −1.02070 1.60921 −0.958179 −1.00000 −1.64251 0 3.01940 −0.410449 1.02070
1.6 −0.604527 −0.171750 −1.63455 −1.00000 0.103828 0 2.19718 −2.97050 0.604527
1.7 −0.455194 −3.07848 −1.79280 −1.00000 1.40131 0 1.72646 6.47706 0.455194
1.8 0.288646 −2.31868 −1.91668 −1.00000 −0.669278 0 −1.13054 2.37627 −0.288646
1.9 0.684542 2.76076 −1.53140 −1.00000 1.88986 0 −2.41739 4.62182 −0.684542
1.10 1.20243 1.19732 −0.554173 −1.00000 1.43969 0 −3.07120 −1.56643 −1.20243
1.11 1.31251 −0.671016 −0.277327 −1.00000 −0.880712 0 −2.98901 −2.54974 −1.31251
1.12 2.18587 −1.42582 2.77802 −1.00000 −3.11665 0 1.70064 −0.967043 −2.18587
1.13 2.36724 3.32580 3.60383 −1.00000 7.87297 0 3.79665 8.06095 −2.36724
1.14 2.44307 1.19555 3.96859 −1.00000 2.92081 0 4.80941 −1.57066 −2.44307
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.14
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(7\) \( -1 \)
\(29\) \( +1 \)

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 7105.2.a.w yes 14
7.b odd 2 1 7105.2.a.v ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7105.2.a.v ✓ 14 7.b odd 2 1
7105.2.a.w yes 14 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7105))\):

\( T_{2}^{14} - 20 T_{2}^{12} + 152 T_{2}^{10} + 2 T_{2}^{9} - 549 T_{2}^{8} - 18 T_{2}^{7} + 970 T_{2}^{6} + \cdots - 20 \) Copy content Toggle raw display
\( T_{3}^{14} - 2 T_{3}^{13} - 22 T_{3}^{12} + 40 T_{3}^{11} + 171 T_{3}^{10} - 278 T_{3}^{9} - 595 T_{3}^{8} + \cdots - 16 \) Copy content Toggle raw display
\( T_{17}^{14} - 14 T_{17}^{13} - 14 T_{17}^{12} + 1012 T_{17}^{11} - 3439 T_{17}^{10} - 17052 T_{17}^{9} + \cdots - 76672 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} - 20 T^{12} + \cdots - 20 \) Copy content Toggle raw display
$3$ \( T^{14} - 2 T^{13} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( (T + 1)^{14} \) Copy content Toggle raw display
$7$ \( T^{14} \) Copy content Toggle raw display
$11$ \( T^{14} + 2 T^{13} + \cdots - 19877888 \) Copy content Toggle raw display
$13$ \( T^{14} - 6 T^{13} + \cdots - 1908928 \) Copy content Toggle raw display
$17$ \( T^{14} - 14 T^{13} + \cdots - 76672 \) Copy content Toggle raw display
$19$ \( T^{14} - 2 T^{13} + \cdots - 1503908 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots - 1884412928 \) Copy content Toggle raw display
$29$ \( (T + 1)^{14} \) Copy content Toggle raw display
$31$ \( T^{14} - 10 T^{13} + \cdots - 667184 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 2873447600 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots - 1164140972 \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots - 2694980048 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots - 2685433040 \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots - 103340866496 \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 284678912 \) Copy content Toggle raw display
$61$ \( T^{14} + \cdots - 432814032896 \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots - 520609809664 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots - 461714128 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 6675953104 \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 159723431168 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots - 52953843200 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots - 1178925065596 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots - 8765913704560 \) Copy content Toggle raw display
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