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Properties

Label 242.3.d.b
Level $242$
Weight $3$
Character orbit 242.d
Analytic conductor $6.594$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [242,3,Mod(161,242)] 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("242.161"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(242, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([7])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 242 = 2 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 242.d (of order \(10\), degree \(4\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,-2,4,2,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.59402239752\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.64000000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) 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 22)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{4} + \beta_{2} - 1) q^{3} + 2 \beta_{2} q^{4} - \beta_{4} q^{5} + (\beta_{7} - \beta_{5} + \beta_{3} - \beta_1) q^{6} + 6 \beta_{7} q^{7} + 2 \beta_{3} q^{8} + 8 \beta_{6} q^{9}+ \cdots - 23 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{3} + 4 q^{4} + 2 q^{5} + 16 q^{9} - 16 q^{12} - 24 q^{14} + 2 q^{15} - 8 q^{16} - 4 q^{20} + 136 q^{23} + 48 q^{25} + 24 q^{26} + 34 q^{27} - 34 q^{31} - 288 q^{34} - 32 q^{36} - 94 q^{37} - 72 q^{38}+ \cdots + 242 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{6} + 4x^{4} - 8x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).

\(n\) \(123\)
\(\chi(n)\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
−0.831254 + 1.14412i
0.831254 − 1.14412i
−1.34500 + 0.437016i
1.34500 − 0.437016i
−1.34500 − 0.437016i
1.34500 + 0.437016i
−0.831254 − 1.14412i
0.831254 + 1.14412i
−0.831254 + 1.14412i 0.309017 − 0.951057i −0.618034 − 1.90211i 0.809017 − 0.587785i 0.831254 + 1.14412i −8.06998 + 2.62210i 2.68999 + 0.874032i 6.47214 + 4.70228i 1.41421i
161.2 0.831254 − 1.14412i 0.309017 − 0.951057i −0.618034 − 1.90211i 0.809017 − 0.587785i −0.831254 − 1.14412i 8.06998 − 2.62210i −2.68999 − 0.874032i 6.47214 + 4.70228i − 1.41421i
215.1 −1.34500 + 0.437016i −0.809017 − 0.587785i 1.61803 − 1.17557i −0.309017 + 0.951057i 1.34500 + 0.437016i 4.98752 + 6.86474i −1.66251 + 2.28825i −2.47214 − 7.60845i − 1.41421i
215.2 1.34500 − 0.437016i −0.809017 − 0.587785i 1.61803 − 1.17557i −0.309017 + 0.951057i −1.34500 − 0.437016i −4.98752 − 6.86474i 1.66251 − 2.28825i −2.47214 − 7.60845i 1.41421i
233.1 −1.34500 − 0.437016i −0.809017 + 0.587785i 1.61803 + 1.17557i −0.309017 − 0.951057i 1.34500 − 0.437016i 4.98752 − 6.86474i −1.66251 − 2.28825i −2.47214 + 7.60845i 1.41421i
233.2 1.34500 + 0.437016i −0.809017 + 0.587785i 1.61803 + 1.17557i −0.309017 − 0.951057i −1.34500 + 0.437016i −4.98752 + 6.86474i 1.66251 + 2.28825i −2.47214 + 7.60845i − 1.41421i
239.1 −0.831254 − 1.14412i 0.309017 + 0.951057i −0.618034 + 1.90211i 0.809017 + 0.587785i 0.831254 − 1.14412i −8.06998 − 2.62210i 2.68999 − 0.874032i 6.47214 − 4.70228i − 1.41421i
239.2 0.831254 + 1.14412i 0.309017 + 0.951057i −0.618034 + 1.90211i 0.809017 + 0.587785i −0.831254 + 1.14412i 8.06998 + 2.62210i −2.68999 + 0.874032i 6.47214 − 4.70228i 1.41421i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 242.3.d.b 8
11.b odd 2 1 inner 242.3.d.b 8
11.c even 5 1 22.3.b.a ✓ 2
11.c even 5 3 inner 242.3.d.b 8
11.d odd 10 1 22.3.b.a ✓ 2
11.d odd 10 3 inner 242.3.d.b 8
33.f even 10 1 198.3.d.b 2
33.h odd 10 1 198.3.d.b 2
44.g even 10 1 176.3.h.c 2
44.h odd 10 1 176.3.h.c 2
55.h odd 10 1 550.3.d.a 2
55.j even 10 1 550.3.d.a 2
55.k odd 20 2 550.3.c.a 4
55.l even 20 2 550.3.c.a 4
77.j odd 10 1 1078.3.d.a 2
77.l even 10 1 1078.3.d.a 2
88.k even 10 1 704.3.h.e 2
88.l odd 10 1 704.3.h.e 2
88.o even 10 1 704.3.h.d 2
88.p odd 10 1 704.3.h.d 2
132.n odd 10 1 1584.3.j.d 2
132.o even 10 1 1584.3.j.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.3.b.a ✓ 2 11.c even 5 1
22.3.b.a ✓ 2 11.d odd 10 1
176.3.h.c 2 44.g even 10 1
176.3.h.c 2 44.h odd 10 1
198.3.d.b 2 33.f even 10 1
198.3.d.b 2 33.h odd 10 1
242.3.d.b 8 1.a even 1 1 trivial
242.3.d.b 8 11.b odd 2 1 inner
242.3.d.b 8 11.c even 5 3 inner
242.3.d.b 8 11.d odd 10 3 inner
550.3.c.a 4 55.k odd 20 2
550.3.c.a 4 55.l even 20 2
550.3.d.a 2 55.h odd 10 1
550.3.d.a 2 55.j even 10 1
704.3.h.d 2 88.o even 10 1
704.3.h.d 2 88.p odd 10 1
704.3.h.e 2 88.k even 10 1
704.3.h.e 2 88.l odd 10 1
1078.3.d.a 2 77.j odd 10 1
1078.3.d.a 2 77.l even 10 1
1584.3.j.d 2 132.n odd 10 1
1584.3.j.d 2 132.o even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(242, [\chi])\):

\( T_{3}^{4} + T_{3}^{3} + T_{3}^{2} + T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{8} - 72T_{7}^{6} + 5184T_{7}^{4} - 373248T_{7}^{2} + 26873856 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 72 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 72 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 176319369216 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 176319369216 \) Copy content Toggle raw display
$23$ \( (T - 17)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 1761205026816 \) Copy content Toggle raw display
$31$ \( (T^{4} + 17 T^{3} + \cdots + 83521)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 47 T^{3} + \cdots + 4879681)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 72 T^{6} + \cdots + 26873856 \) Copy content Toggle raw display
$43$ \( (T^{2} + 288)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 58 T^{3} + \cdots + 11316496)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 2 T^{3} + 4 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 55 T^{3} + \cdots + 9150625)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T - 89)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 7 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 1761205026816 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 1761205026816 \) Copy content Toggle raw display
$89$ \( (T + 97)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 121 T^{3} + \cdots + 214358881)^{2} \) Copy content Toggle raw display
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