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Properties

Label 182.4.d.b
Level $182$
Weight $4$
Character orbit 182.d
Analytic conductor $10.738$
Analytic rank $0$
Dimension $6$
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] = [182,4,Mod(155,182)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(182, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("182.155"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 182 = 2 \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 182.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6] 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: \(10.7383476210\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.337971456.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 23x^{4} + 115x^{2} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_{3} q^{2} + ( - \beta_{4} - \beta_{2} - 1) q^{3} - 4 q^{4} + (\beta_{5} - 10 \beta_{3} - 3 \beta_1) q^{5} + (2 \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{6} - 7 \beta_{3} q^{7} - 8 \beta_{3} q^{8}+ \cdots + ( - 162 \beta_{5} + 477 \beta_{3} + 37 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{3} - 24 q^{4} + 78 q^{9} + 108 q^{10} + 32 q^{12} + 136 q^{13} + 84 q^{14} + 96 q^{16} + 264 q^{17} - 128 q^{22} - 30 q^{23} - 168 q^{25} + 164 q^{26} + 340 q^{27} - 102 q^{29} - 80 q^{30} - 378 q^{35}+ \cdots - 858 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 23x^{4} + 115x^{2} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 12\nu^{2} - 3 ) / 14 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 26\nu^{3} + 151\nu ) / 42 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} + 19\nu^{2} + 46 ) / 7 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -5\nu^{5} - 109\nu^{3} - 482\nu ) / 21 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - 2\beta_{2} - 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + 10\beta_{3} - 13\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -12\beta_{4} + 38\beta_{2} + 87 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -26\beta_{5} - 218\beta_{3} + 187\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/182\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(157\)
\(\chi(n)\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
155.1
− 0.281995i
3.96506i
− 2.68306i
0.281995i
− 3.96506i
2.68306i
− 2.00000i −7.07449 −4.00000 17.2025i 14.1490i 7.00000i 8.00000i 23.0485 34.4049
155.2 − 2.00000i −4.17350 −4.00000 − 2.68672i 8.34700i 7.00000i 8.00000i −9.58189 −5.37344
155.3 − 2.00000i 7.24799 −4.00000 12.4842i − 14.4960i 7.00000i 8.00000i 25.5334 24.9685
155.4 2.00000i −7.07449 −4.00000 − 17.2025i − 14.1490i − 7.00000i − 8.00000i 23.0485 34.4049
155.5 2.00000i −4.17350 −4.00000 2.68672i − 8.34700i − 7.00000i − 8.00000i −9.58189 −5.37344
155.6 2.00000i 7.24799 −4.00000 − 12.4842i 14.4960i − 7.00000i − 8.00000i 25.5334 24.9685
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 155.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 182.4.d.b ✓ 6
13.b even 2 1 inner 182.4.d.b ✓ 6
13.d odd 4 1 2366.4.a.m 3
13.d odd 4 1 2366.4.a.o 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.4.d.b ✓ 6 1.a even 1 1 trivial
182.4.d.b ✓ 6 13.b even 2 1 inner
2366.4.a.m 3 13.d odd 4 1
2366.4.a.o 3 13.d odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 4T_{3}^{2} - 52T_{3} - 214 \) acting on \(S_{4}^{\mathrm{new}}(182, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$3$ \( (T^{3} + 4 T^{2} + \cdots - 214)^{2} \) Copy content Toggle raw display
$5$ \( T^{6} + 459 T^{4} + \cdots + 332929 \) Copy content Toggle raw display
$7$ \( (T^{2} + 49)^{3} \) Copy content Toggle raw display
$11$ \( T^{6} + 4940 T^{4} + \cdots + 312653124 \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 10604499373 \) Copy content Toggle raw display
$17$ \( (T^{3} - 132 T^{2} + \cdots - 40102)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + \cdots + 2583993889 \) Copy content Toggle raw display
$23$ \( (T^{3} + 15 T^{2} + \cdots + 479241)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 51 T^{2} + \cdots - 250047)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + \cdots + 5179798190889 \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 38197368881604 \) Copy content Toggle raw display
$41$ \( T^{6} + \cdots + 18\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( (T^{3} - 681 T^{2} + \cdots - 545731)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 205758684292681 \) Copy content Toggle raw display
$53$ \( (T^{3} - 317 T^{2} + \cdots + 3443153)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 48525040656 \) Copy content Toggle raw display
$61$ \( (T^{3} + 1394 T^{2} + \cdots + 93701016)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 66\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 249290436856356 \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 531353994947689 \) Copy content Toggle raw display
$79$ \( (T^{3} + 189 T^{2} + \cdots - 235809153)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 10\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 20\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 414518076320089 \) Copy content Toggle raw display
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