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Properties

Label 49.10.a.g
Level $49$
Weight $10$
Character orbit 49.a
Self dual yes
Analytic conductor $25.237$
Analytic rank $1$
Dimension $8$
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] = [49,10,Mod(1,49)] 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("49.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 49.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-66,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(25.2367559720\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2 x^{7} - 2635 x^{6} + 6784 x^{5} + 1834350 x^{4} - 5545800 x^{3} - 141427156 x^{2} + \cdots + 2560703544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3}\cdot 7^{6} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 8) q^{2} + ( - \beta_{5} - \beta_{3}) q^{3} + (\beta_{2} - 15 \beta_1 + 209) q^{4} + ( - \beta_{7} - 2 \beta_{5} + \cdots + 2 \beta_{3}) q^{5} + (2 \beta_{7} + 13 \beta_{5} + \cdots + 6 \beta_{3}) q^{6}+ \cdots + ( - 461849 \beta_{6} + \cdots - 391729289) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 66 q^{2} + 1706 q^{4} - 58542 q^{8} + 27940 q^{9} - 82092 q^{11} - 82336 q^{15} + 1569570 q^{16} - 307774 q^{18} - 1300076 q^{22} - 2388480 q^{23} - 6191476 q^{25} - 18443892 q^{29} - 16703832 q^{30}+ \cdots - 3129730532 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2 x^{7} - 2635 x^{6} + 6784 x^{5} + 1834350 x^{4} - 5545800 x^{3} - 141427156 x^{2} + \cdots + 2560703544 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 49274683 \nu^{7} - 44711660 \nu^{6} - 125962207375 \nu^{5} + 197200820962 \nu^{4} + \cdots + 25\!\cdots\!04 ) / 58\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 31390019 \nu^{7} - 1541690600 \nu^{6} + 80023892879 \nu^{5} + 2987518494754 \nu^{4} + \cdots - 71\!\cdots\!40 ) / 11\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 716486374 \nu^{7} - 105637852963 \nu^{6} + 1788590423572 \nu^{5} + 280751654882687 \nu^{4} + \cdots + 78\!\cdots\!88 ) / 16\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 94108819 \nu^{7} - 1746475430 \nu^{6} + 240317116963 \nu^{5} + 3255370619424 \nu^{4} + \cdots + 60\!\cdots\!36 ) / 388530217644816 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2414459467 \nu^{7} - 2190871340 \nu^{6} - 6172148161375 \nu^{5} + 9662840227138 \nu^{4} + \cdots + 12\!\cdots\!96 ) / 58\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 24430655821 \nu^{7} - 30841390430 \nu^{6} - 65830850904205 \nu^{5} + 119190301919344 \nu^{4} + \cdots + 60\!\cdots\!08 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 378764652766 \nu^{7} + 94704282965 \nu^{6} + 999588533933260 \nu^{5} + \cdots + 93\!\cdots\!32 ) / 81\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - 49\beta_1 ) / 49 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -6\beta_{5} - 14\beta_{4} + 49\beta_{2} + 49\beta _1 + 32291 ) / 49 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 147\beta_{7} + 196\beta_{6} + 1919\beta_{5} + 63\beta_{4} - 147\beta_{3} + 49\beta_{2} - 60711\beta _1 - 42875 ) / 49 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 196 \beta_{7} + 196 \beta_{6} - 15252 \beta_{5} - 33628 \beta_{4} + 10780 \beta_{3} + 65023 \beta_{2} + \cdots + 40032951 ) / 49 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 323155 \beta_{7} + 261660 \beta_{6} + 3956499 \beta_{5} + 168945 \beta_{4} - 309435 \beta_{3} + \cdots - 84252707 ) / 49 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 165718 \beta_{7} + 115052 \beta_{6} - 32418714 \beta_{5} - 64326038 \beta_{4} + 21594202 \beta_{3} + \cdots + 51575914565 ) / 49 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 578382035 \beta_{7} + 338776788 \beta_{6} + 7089231871 \beta_{5} + 351619737 \beta_{4} + \cdots - 142566983377 ) / 49 \) 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
36.7406
33.9122
5.94065
8.76908
−4.28890
−7.11733
−37.3924
−34.5639
−43.3264 −99.9907 1365.18 −1890.52 4332.24 0 −36965.0 −9684.86 81909.3
1.2 −43.3264 99.9907 1365.18 1890.52 −4332.24 0 −36965.0 −9684.86 −81909.3
1.3 −15.3549 −124.067 −276.228 −246.278 1905.03 0 12103.1 −4290.46 3781.57
1.4 −15.3549 124.067 −276.228 246.278 −1905.03 0 12103.1 −4290.46 −3781.57
1.5 −2.29688 −258.122 −506.724 972.905 592.877 0 2339.89 46944.1 −2234.65
1.6 −2.29688 258.122 −506.724 −972.905 −592.877 0 2339.89 46944.1 2234.65
1.7 27.9781 −26.1570 270.776 368.111 −731.823 0 −6748.99 −18998.8 10299.1
1.8 27.9781 26.1570 270.776 −368.111 731.823 0 −6748.99 −18998.8 −10299.1
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

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

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 49.10.a.g ✓ 8
7.b odd 2 1 inner 49.10.a.g ✓ 8
7.c even 3 2 49.10.c.h 16
7.d odd 6 2 49.10.c.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.10.a.g ✓ 8 1.a even 1 1 trivial
49.10.a.g ✓ 8 7.b odd 2 1 inner
49.10.c.h 16 7.c even 3 2
49.10.c.h 16 7.d odd 6 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(49))\):

\( T_{2}^{4} + 33T_{2}^{3} - 906T_{2}^{2} - 20856T_{2} - 42752 \) Copy content Toggle raw display
\( T_{3}^{8} - 92702T_{3}^{6} + 1908561916T_{3}^{4} - 11516433342120T_{3}^{2} + 7015444108029504 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 33 T^{3} + \cdots - 42752)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 70\!\cdots\!04 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots - 40\!\cdots\!80)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 26\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 12\!\cdots\!60)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 92\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 60\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots - 12\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 25\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 68\!\cdots\!56 \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots - 37\!\cdots\!88)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 52\!\cdots\!08)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 57\!\cdots\!96 \) Copy content Toggle raw display
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