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Properties

Label 49.5.d.b
Level $49$
Weight $5$
Character orbit 49.d
Analytic conductor $5.065$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [49,5,Mod(19,49)] 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("49.19"); S:= CuspForms(chi, 5); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([5])) N = Newforms(chi, 5, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.06512819111\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{22})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 22x^{2} + 484 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} + \beta_1 - 2) q^{2} + ( - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{3} + ( - 4 \beta_{3} + 10 \beta_{2} - 4 \beta_1) q^{4} + ( - 4 \beta_{3} - 5 \beta_{2} + \cdots + 5) q^{5} + ( - 3 \beta_{3} + 48 \beta_{2} + \cdots + 24) q^{6}+ \cdots + ( - 378 \beta_{3} + 2592) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 6 q^{3} - 20 q^{4} + 30 q^{5} + 304 q^{8} - 24 q^{9} + 204 q^{10} - 58 q^{11} + 588 q^{12} + 468 q^{15} - 72 q^{16} + 246 q^{17} + 216 q^{18} - 642 q^{19} - 1264 q^{22} + 290 q^{23} - 720 q^{24}+ \cdots + 10368 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 22x^{2} + 484 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 22 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 22\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 22\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−2.34521 + 4.06202i
2.34521 − 4.06202i
−2.34521 − 4.06202i
2.34521 + 4.06202i
−3.34521 + 5.79407i −8.53562 + 4.92804i −14.3808 − 24.9083i −6.57125 − 3.79391i − 65.9413i 0 85.3808 8.07125 − 13.9798i 43.9644 − 25.3828i
19.2 1.34521 − 2.32997i 5.53562 − 3.19599i 4.38083 + 7.58782i 21.5712 + 12.4542i − 17.1971i 0 66.6192 −20.0712 + 34.7644i 58.0356 − 33.5069i
31.1 −3.34521 − 5.79407i −8.53562 − 4.92804i −14.3808 + 24.9083i −6.57125 + 3.79391i 65.9413i 0 85.3808 8.07125 + 13.9798i 43.9644 + 25.3828i
31.2 1.34521 + 2.32997i 5.53562 + 3.19599i 4.38083 − 7.58782i 21.5712 − 12.4542i 17.1971i 0 66.6192 −20.0712 − 34.7644i 58.0356 + 33.5069i
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.5.d.b 4
7.b odd 2 1 7.5.d.a ✓ 4
7.c even 3 1 7.5.d.a ✓ 4
7.c even 3 1 49.5.b.a 4
7.d odd 6 1 49.5.b.a 4
7.d odd 6 1 inner 49.5.d.b 4
21.c even 2 1 63.5.m.d 4
21.g even 6 1 441.5.d.d 4
21.h odd 6 1 63.5.m.d 4
21.h odd 6 1 441.5.d.d 4
28.d even 2 1 112.5.s.a 4
28.f even 6 1 784.5.c.c 4
28.g odd 6 1 112.5.s.a 4
28.g odd 6 1 784.5.c.c 4
35.c odd 2 1 175.5.i.a 4
35.f even 4 2 175.5.j.a 8
35.j even 6 1 175.5.i.a 4
35.l odd 12 2 175.5.j.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.5.d.a ✓ 4 7.b odd 2 1
7.5.d.a ✓ 4 7.c even 3 1
49.5.b.a 4 7.c even 3 1
49.5.b.a 4 7.d odd 6 1
49.5.d.b 4 1.a even 1 1 trivial
49.5.d.b 4 7.d odd 6 1 inner
63.5.m.d 4 21.c even 2 1
63.5.m.d 4 21.h odd 6 1
112.5.s.a 4 28.d even 2 1
112.5.s.a 4 28.g odd 6 1
175.5.i.a 4 35.c odd 2 1
175.5.i.a 4 35.j even 6 1
175.5.j.a 8 35.f even 4 2
175.5.j.a 8 35.l odd 12 2
441.5.d.d 4 21.g even 6 1
441.5.d.d 4 21.h odd 6 1
784.5.c.c 4 28.f even 6 1
784.5.c.c 4 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 4T_{2}^{3} + 34T_{2}^{2} - 72T_{2} + 324 \) acting on \(S_{5}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 4 T^{3} + \cdots + 324 \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + \cdots + 3969 \) Copy content Toggle raw display
$5$ \( T^{4} - 30 T^{3} + \cdots + 35721 \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + 58 T^{3} + \cdots + 30437289 \) Copy content Toggle raw display
$13$ \( T^{4} + 55272 T^{2} + 3111696 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 5076990009 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 3136784049 \) Copy content Toggle raw display
$23$ \( T^{4} - 290 T^{3} + \cdots + 254625849 \) Copy content Toggle raw display
$29$ \( (T^{2} + 1088 T + 188136)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 1071889573041 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 48279954529 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots + 3218392944144 \) Copy content Toggle raw display
$43$ \( (T^{2} - 1236 T - 2313076)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 4451285577249 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 6460874330625 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 163173619767249 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 14401820070729 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 151890252361 \) Copy content Toggle raw display
$71$ \( (T^{2} + 5204 T + 5524236)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 6851102086521 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 188584385155609 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 115179601694976 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 75\!\cdots\!81 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 11\!\cdots\!84 \) Copy content Toggle raw display
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