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Properties

Label 205.2.j.d
Level $205$
Weight $2$
Character orbit 205.j
Analytic conductor $1.637$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [205,2,Mod(9,205)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(205, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([2, 3])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("205.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 205 = 5 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 205.j (of order \(4\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.63693324144\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: 16.0.2786689352468701118464.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} + 4x^{12} + 12x^{10} + 16x^{8} + 48x^{6} + 64x^{4} + 64x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - \beta_{9} q^{3} + (2 \beta_{11} + \beta_{9} + \beta_{5} + 1) q^{4} + ( - \beta_{3} + \beta_1) q^{5} + ( - \beta_{13} + \beta_{11} + 2 \beta_{8} + \cdots + 1) q^{6} + (\beta_{15} + \beta_{14} + \cdots - \beta_1) q^{7}+ \cdots + ( - \beta_{14} - \beta_{13} - \beta_{12} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 16 q^{4} + 4 q^{6} - 40 q^{10} - 16 q^{11} - 16 q^{14} - 6 q^{15} + 16 q^{16} - 20 q^{19} + 12 q^{24} + 32 q^{25} - 12 q^{26} + 16 q^{29} + 10 q^{30} + 32 q^{31} - 4 q^{34} - 10 q^{35} + 4 q^{40}+ \cdots + 68 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} + 4x^{12} + 12x^{10} + 16x^{8} + 48x^{6} + 64x^{4} + 64x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{13} + \nu^{11} + 8\nu^{7} + 16\nu^{5} + 16\nu^{3} ) / 64 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 3\nu^{13} + 10\nu^{11} + 8\nu^{9} + 8\nu^{7} + 64\nu^{5} + 64\nu ) / 384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} + \nu^{7} - 2\nu^{5} - 4\nu^{3} + 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{15} - 3\nu^{13} - 10\nu^{11} - 20\nu^{9} - 44\nu^{7} - 40\nu^{5} - 144\nu^{3} - 160\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{14} + 3 \nu^{12} - 12 \nu^{11} - 4 \nu^{10} - 12 \nu^{9} + 16 \nu^{8} - 48 \nu^{7} + \cdots + 128 ) / 384 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - \nu^{15} - \nu^{14} + 3 \nu^{13} - 3 \nu^{12} - 4 \nu^{11} + 14 \nu^{10} - 8 \nu^{9} - 8 \nu^{8} + \cdots + 128 ) / 384 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} + \nu^{13} + 8\nu^{9} + 16\nu^{7} + 16\nu^{5} ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} - \nu^{14} - 3 \nu^{13} - 3 \nu^{12} + 4 \nu^{11} + 14 \nu^{10} + 8 \nu^{9} - 8 \nu^{8} + \cdots + 128 ) / 384 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{14} - 3 \nu^{12} - 12 \nu^{11} + 4 \nu^{10} - 12 \nu^{9} - 16 \nu^{8} - 48 \nu^{7} - 16 \nu^{6} + \cdots - 128 ) / 384 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{14} - 3\nu^{12} + 4\nu^{10} + 8\nu^{8} + 8\nu^{6} + 16\nu^{4} + 96\nu^{2} + 64 ) / 192 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{14} + \nu^{12} + 4 \nu^{11} + 4 \nu^{9} + 8 \nu^{8} + 16 \nu^{7} + 16 \nu^{6} + 16 \nu^{5} + \cdots + 64 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -\nu^{14} - 3\nu^{12} - 4\nu^{10} - 14\nu^{8} - 20\nu^{6} - 16\nu^{4} - 96\nu^{2} - 64 ) / 96 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - \nu^{14} - 2 \nu^{13} - \nu^{12} - 2 \nu^{11} - 8 \nu^{9} - 8 \nu^{8} - 24 \nu^{7} + 16 \nu^{6} + \cdots + 128 ) / 128 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{14} + \nu^{10} + 2\nu^{8} - 16\nu^{6} + 16\nu^{4} - 128 ) / 96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - \nu^{15} - \nu^{14} + \nu^{13} - \nu^{12} + 2 \nu^{11} - 8 \nu^{8} + 8 \nu^{7} + 16 \nu^{6} + \cdots + 128 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - \beta_{13} + \beta_{9} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{15} + \beta_{14} + \beta_{13} - \beta_{12} - 2 \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{15} + \beta_{13} + \beta_{9} + \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \cdots + 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{15} + 3 \beta_{14} + \beta_{13} + \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} + \beta_{8} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{15} - \beta_{13} - \beta_{9} + 3 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} - \beta_{5} + \cdots + 5 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3 \beta_{15} - 3 \beta_{14} + 3 \beta_{13} - \beta_{12} + 6 \beta_{11} + \beta_{10} + 5 \beta_{9} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - \beta_{15} + \beta_{13} - 7 \beta_{9} - 3 \beta_{8} + 6 \beta_{7} + 3 \beta_{6} - 7 \beta_{5} + \cdots + 3 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 19 \beta_{15} - 21 \beta_{14} - 19 \beta_{13} + 9 \beta_{12} + 10 \beta_{11} + 7 \beta_{10} + \cdots + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 9 \beta_{15} - 9 \beta_{13} - \beta_{9} - 5 \beta_{8} + 10 \beta_{7} + 5 \beta_{6} - \beta_{5} + \cdots - 27 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 5 \beta_{15} - 3 \beta_{14} - 5 \beta_{13} - 17 \beta_{12} - 26 \beta_{11} + \beta_{10} - 3 \beta_{9} + \cdots - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 17 \beta_{15} + 17 \beta_{13} - 23 \beta_{9} - 19 \beta_{8} - 26 \beta_{7} + 19 \beta_{6} + \cdots - 13 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 13 \beta_{15} + 27 \beta_{14} + 13 \beta_{13} - 39 \beta_{12} - 22 \beta_{11} - 73 \beta_{10} + \cdots + 17 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 25 \beta_{15} - 25 \beta_{13} + 79 \beta_{9} - 21 \beta_{8} - 22 \beta_{7} + 21 \beta_{6} + \cdots - 11 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 75 \beta_{15} + 141 \beta_{14} + 75 \beta_{13} - 33 \beta_{12} + 70 \beta_{11} + 17 \beta_{10} + \cdots + 103 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 97 \beta_{15} + 97 \beta_{13} + 57 \beta_{9} + 61 \beta_{8} + 70 \beta_{7} - 61 \beta_{6} + \cdots + 99 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/205\mathbb{Z}\right)^\times\).

\(n\) \(6\) \(42\)
\(\chi(n)\) \(\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−0.842022 + 1.13622i
0.199044 − 1.40014i
−1.29041 + 0.578647i
−1.15595 − 0.814732i
1.15595 − 0.814732i
1.29041 + 0.578647i
−0.199044 − 1.40014i
0.842022 + 1.13622i
−0.842022 − 1.13622i
0.199044 + 1.40014i
−1.29041 − 0.578647i
−1.15595 + 0.814732i
1.15595 + 0.814732i
1.29041 − 0.578647i
−0.199044 + 1.40014i
0.842022 − 1.13622i
−2.53701 0.218455 − 0.218455i 4.43644 1.09024 − 1.95228i −0.554223 + 0.554223i 1.07142 − 1.07142i −6.18128 2.90455i −2.76594 + 4.95295i
9.2 −2.01029 −1.65195 + 1.65195i 2.04125 2.11391 + 0.728948i 3.32090 − 3.32090i 0.756423 − 0.756423i −0.0829304 − 2.45790i −4.24957 − 1.46539i
9.3 −1.22336 1.56044 − 1.56044i −0.503379 2.19336 + 0.434963i −1.90898 + 1.90898i 0.202973 − 0.202973i 3.06255 − 1.86993i −2.68327 − 0.532118i
9.4 −0.160274 −0.887900 + 0.887900i −1.97431 1.87935 − 1.21163i 0.142307 − 0.142307i −3.03952 + 3.03952i 0.636979 1.42327i −0.301211 + 0.194193i
9.5 0.160274 0.887900 − 0.887900i −1.97431 −1.87935 − 1.21163i 0.142307 − 0.142307i 3.03952 − 3.03952i −0.636979 1.42327i −0.301211 − 0.194193i
9.6 1.22336 −1.56044 + 1.56044i −0.503379 −2.19336 + 0.434963i −1.90898 + 1.90898i −0.202973 + 0.202973i −3.06255 − 1.86993i −2.68327 + 0.532118i
9.7 2.01029 1.65195 − 1.65195i 2.04125 −2.11391 + 0.728948i 3.32090 − 3.32090i −0.756423 + 0.756423i 0.0829304 − 2.45790i −4.24957 + 1.46539i
9.8 2.53701 −0.218455 + 0.218455i 4.43644 −1.09024 − 1.95228i −0.554223 + 0.554223i −1.07142 + 1.07142i 6.18128 2.90455i −2.76594 − 4.95295i
114.1 −2.53701 0.218455 + 0.218455i 4.43644 1.09024 + 1.95228i −0.554223 − 0.554223i 1.07142 + 1.07142i −6.18128 − 2.90455i −2.76594 − 4.95295i
114.2 −2.01029 −1.65195 − 1.65195i 2.04125 2.11391 − 0.728948i 3.32090 + 3.32090i 0.756423 + 0.756423i −0.0829304 2.45790i −4.24957 + 1.46539i
114.3 −1.22336 1.56044 + 1.56044i −0.503379 2.19336 − 0.434963i −1.90898 − 1.90898i 0.202973 + 0.202973i 3.06255 1.86993i −2.68327 + 0.532118i
114.4 −0.160274 −0.887900 − 0.887900i −1.97431 1.87935 + 1.21163i 0.142307 + 0.142307i −3.03952 − 3.03952i 0.636979 − 1.42327i −0.301211 − 0.194193i
114.5 0.160274 0.887900 + 0.887900i −1.97431 −1.87935 + 1.21163i 0.142307 + 0.142307i 3.03952 + 3.03952i −0.636979 − 1.42327i −0.301211 + 0.194193i
114.6 1.22336 −1.56044 − 1.56044i −0.503379 −2.19336 − 0.434963i −1.90898 − 1.90898i −0.202973 − 0.202973i −3.06255 1.86993i −2.68327 − 0.532118i
114.7 2.01029 1.65195 + 1.65195i 2.04125 −2.11391 − 0.728948i 3.32090 + 3.32090i −0.756423 − 0.756423i 0.0829304 2.45790i −4.24957 − 1.46539i
114.8 2.53701 −0.218455 − 0.218455i 4.43644 −1.09024 + 1.95228i −0.554223 − 0.554223i −1.07142 − 1.07142i 6.18128 − 2.90455i −2.76594 + 4.95295i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
41.c even 4 1 inner
205.j even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 205.2.j.d ✓ 16
5.b even 2 1 inner 205.2.j.d ✓ 16
41.c even 4 1 inner 205.2.j.d ✓ 16
205.j even 4 1 inner 205.2.j.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
205.2.j.d ✓ 16 1.a even 1 1 trivial
205.2.j.d ✓ 16 5.b even 2 1 inner
205.2.j.d ✓ 16 41.c even 4 1 inner
205.2.j.d ✓ 16 205.j even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 12T_{2}^{6} + 42T_{2}^{4} - 40T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(205, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - 12 T^{6} + 42 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + 56 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} - 16 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} + 348 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T^{8} + 8 T^{7} + 32 T^{6} + \cdots + 64)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 1004 T^{12} + \cdots + 1048576 \) Copy content Toggle raw display
$17$ \( T^{16} + 780 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$19$ \( (T^{8} + 10 T^{7} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 112 T^{6} + \cdots + 135424)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 8 T^{3} + \cdots + 192)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 120 T^{6} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 6 T^{7} + \cdots + 2825761)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 204 T^{6} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 55788550416 \) Copy content Toggle raw display
$53$ \( T^{16} + 6796 T^{12} + \cdots + 5308416 \) Copy content Toggle raw display
$59$ \( (T^{4} + 14 T^{3} + \cdots - 256)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + 188 T^{6} + \cdots + 3779136)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 197753906250000 \) Copy content Toggle raw display
$71$ \( (T^{8} + 2 T^{7} + \cdots + 104976)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} - 460 T^{6} + \cdots + 13366336)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 26 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 272 T^{6} + \cdots + 186624)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 44 T^{7} + \cdots + 6170256)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 32319410176 \) Copy content Toggle raw display
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