Properties

Label 47.7.b.a
Level $47$
Weight $7$
Character orbit 47.b
Self dual yes
Analytic conductor $10.813$
Analytic rank $0$
Dimension $5$
CM discriminant -47
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [47,7,Mod(46,47)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(47, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("47.46");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 47 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 47.b (of order \(2\), degree \(1\), minimal)

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: \(10.8125419301\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.6903125.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 10x^{3} + 20x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{3} + \beta_1) q^{2} + ( - 4 \beta_{4} + 5 \beta_{2}) q^{3} + (17 \beta_{4} - 2 \beta_{2} + 64) q^{4} + (3 \beta_{4} + 89 \beta_{3} + 47 \beta_{2} - 6 \beta_1) q^{6} + ( - 137 \beta_{3} + 19 \beta_1) q^{7} + ( - 109 \beta_{4} - 128 \beta_{3} - 65 \beta_{2} + 64 \beta_1) q^{8} + ( - 289 \beta_{3} + 43 \beta_1 + 729) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{3} + \beta_1) q^{2} + ( - 4 \beta_{4} + 5 \beta_{2}) q^{3} + (17 \beta_{4} - 2 \beta_{2} + 64) q^{4} + (3 \beta_{4} + 89 \beta_{3} + 47 \beta_{2} - 6 \beta_1) q^{6} + ( - 137 \beta_{3} + 19 \beta_1) q^{7} + ( - 109 \beta_{4} - 128 \beta_{3} - 65 \beta_{2} + 64 \beta_1) q^{8} + ( - 289 \beta_{3} + 43 \beta_1 + 729) q^{9} + ( - 256 \beta_{4} + 593 \beta_{3} + 320 \beta_{2} + 106 \beta_1 - 2231) q^{12} + (521 \beta_{4} - 434 \beta_{2} + 4313) q^{14} + (1088 \beta_{4} - 191 \beta_{3} - 128 \beta_{2} - 566 \beta_1 + 4096) q^{16} + ( - 1180 \beta_{4} - 443 \beta_{2}) q^{17} + (1137 \beta_{4} - 1458 \beta_{3} - 898 \beta_{2} + 729 \beta_1 + 9361) q^{18} + ( - 1428 \beta_{4} + 2879 \beta_{3} + 1685 \beta_{2} - 1005 \beta_1) q^{21} + (192 \beta_{4} + 4462 \beta_{3} + 3008 \beta_{2} - 2231 \beta_1 - 1727) q^{24} + 15625 q^{25} + ( - 2916 \beta_{4} + 6263 \beta_{3} + 3645 \beta_{2} - 2085 \beta_1) q^{27} + ( - 1477 \beta_{4} - 8626 \beta_{3} - 4601 \beta_{2} + 4313 \beta_1) q^{28} + ( - 6976 \beta_{4} - 8192 \beta_{3} - 4160 \beta_{2} + \cdots - 47311) q^{32}+ \cdots + (100835 \beta_{4} - 590881 \beta_{3} - 190577 \beta_{2} + 81947 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 320 q^{4} + 3645 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 320 q^{4} + 3645 q^{9} - 11155 q^{12} + 21565 q^{14} + 20480 q^{16} + 46805 q^{18} - 8635 q^{24} + 78125 q^{25} - 236555 q^{32} + 233280 q^{36} - 725755 q^{42} - 519115 q^{47} - 713920 q^{48} + 588245 q^{49} + 175010 q^{51} - 1533235 q^{54} + 1380160 q^{56} + 2499530 q^{63} + 1310720 q^{64} - 2648195 q^{68} + 2995520 q^{72} + 2657205 q^{81} + 2220130 q^{83} - 3948835 q^{84} - 552640 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 10x^{3} + 20x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 5\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} + \nu^{2} - 6\nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - 8\nu^{2} - 2\nu + 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -2\nu^{3} + 3\nu^{2} + 12\nu - 12 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 2\beta_{2} + 20 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{4} + 3\beta_{2} + 6\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 8\beta_{4} + 5\beta_{3} + 16\beta_{2} + 2\beta _1 + 120 ) / 5 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/47\mathbb{Z}\right)^\times\).

\(n\) \(5\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
46.1
−2.48472
0.517401
−2.05304
2.80449
1.21587
−15.8135 −20.8314 186.066 0 329.417 −468.255 −1930.28 −295.051 0
46.2 −7.20346 −12.4306 −12.1102 0 89.5432 −621.494 548.257 −574.481 0
46.3 −2.56980 46.1366 −57.3961 0 −118.562 332.096 311.964 1399.58 0
46.4 11.3615 40.9445 65.0833 0 465.191 84.1505 12.3078 947.456 0
46.5 14.2252 −53.8191 138.357 0 −765.589 673.501 1057.75 2167.49 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 46.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
47.b odd 2 1 CM by \(\Q(\sqrt{-47}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 47.7.b.a 5
47.b odd 2 1 CM 47.7.b.a 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
47.7.b.a 5 1.a even 1 1 trivial
47.7.b.a 5 47.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{5} - 320T_{2}^{3} + 20480T_{2} + 47311 \) acting on \(S_{7}^{\mathrm{new}}(47, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} - 320 T^{3} + 20480 T + 47311 \) Copy content Toggle raw display
$3$ \( T^{5} - 3645 T^{3} + \cdots + 26326214 \) Copy content Toggle raw display
$5$ \( T^{5} \) Copy content Toggle raw display
$7$ \( T^{5} - 588245 T^{3} + \cdots - 5477442893714 \) Copy content Toggle raw display
$11$ \( T^{5} \) Copy content Toggle raw display
$13$ \( T^{5} \) Copy content Toggle raw display
$17$ \( T^{5} - 120687845 T^{3} + \cdots + 14\!\cdots\!86 \) Copy content Toggle raw display
$19$ \( T^{5} \) Copy content Toggle raw display
$23$ \( T^{5} \) Copy content Toggle raw display
$29$ \( T^{5} \) Copy content Toggle raw display
$31$ \( T^{5} \) Copy content Toggle raw display
$37$ \( T^{5} - 12828632045 T^{3} + \cdots - 63\!\cdots\!14 \) Copy content Toggle raw display
$41$ \( T^{5} \) Copy content Toggle raw display
$43$ \( T^{5} \) Copy content Toggle raw display
$47$ \( (T + 103823)^{5} \) Copy content Toggle raw display
$53$ \( T^{5} - 110821805645 T^{3} + \cdots - 11\!\cdots\!86 \) Copy content Toggle raw display
$59$ \( T^{5} - 210902668205 T^{3} + \cdots + 27\!\cdots\!02 \) Copy content Toggle raw display
$61$ \( T^{5} - 257601871805 T^{3} + \cdots - 90\!\cdots\!02 \) Copy content Toggle raw display
$67$ \( T^{5} \) Copy content Toggle raw display
$71$ \( T^{5} - 640501419605 T^{3} + \cdots + 11\!\cdots\!98 \) Copy content Toggle raw display
$73$ \( T^{5} \) Copy content Toggle raw display
$79$ \( T^{5} - 1215437277605 T^{3} + \cdots - 21\!\cdots\!98 \) Copy content Toggle raw display
$83$ \( (T - 444026)^{5} \) Copy content Toggle raw display
$89$ \( T^{5} - 2484906454805 T^{3} + \cdots - 15\!\cdots\!98 \) Copy content Toggle raw display
$97$ \( T^{5} - 4164860024645 T^{3} + \cdots - 45\!\cdots\!14 \) Copy content Toggle raw display
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