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Properties

Label 4080.2.m.q
Level $4080$
Weight $2$
Character orbit 4080.m
Analytic conductor $32.579$
Analytic rank $0$
Dimension $10$
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] = [4080,2,Mod(2449,4080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0])) 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("4080.2449"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4080.m (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,0,0,-2,0,0,0,-10,0,-10,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.5789640247\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} - 8x^{8} + 12x^{7} + 17x^{6} - 28x^{5} + 85x^{4} + 300x^{3} - 1000x^{2} - 1250x + 3125 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2040)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + \beta_{8} q^{5} + (\beta_{5} + \beta_{4}) q^{7} - q^{9} + (\beta_{8} - \beta_{7} - \beta_{6} - 1) q^{11} + ( - \beta_{9} - 2 \beta_{4} + \cdots + \beta_1) q^{13} + \beta_1 q^{15}+ \cdots + ( - \beta_{8} + \beta_{7} + \beta_{6} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{5} - 10 q^{9} - 10 q^{11} + 2 q^{15} - 6 q^{19} - 14 q^{21} - 20 q^{25} + 38 q^{29} - 12 q^{31} + 20 q^{35} + 16 q^{39} + 10 q^{41} + 2 q^{45} - 4 q^{49} + 10 q^{51} + 40 q^{55} - 20 q^{59}+ \cdots + 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} - 8x^{8} + 12x^{7} + 17x^{6} - 28x^{5} + 85x^{4} + 300x^{3} - 1000x^{2} - 1250x + 3125 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{9} - 2\nu^{8} - 8\nu^{7} + 12\nu^{6} + 17\nu^{5} - 28\nu^{4} + 85\nu^{3} + 300\nu^{2} - 1000\nu - 1250 ) / 625 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{9} + 7 \nu^{8} + 49 \nu^{7} + 5 \nu^{6} - 100 \nu^{5} - 75 \nu^{4} - 657 \nu^{3} + \cdots + 9750 ) / 250 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 79 \nu^{9} + 42 \nu^{8} - 557 \nu^{7} - 477 \nu^{6} + 193 \nu^{5} - 1487 \nu^{4} + 3315 \nu^{3} + \cdots - 95625 ) / 2500 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 129 \nu^{9} + 42 \nu^{8} - 907 \nu^{7} - 677 \nu^{6} + 493 \nu^{5} - 2187 \nu^{4} + 6265 \nu^{3} + \cdots - 158125 ) / 2500 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 153 \nu^{9} - 89 \nu^{8} + 1014 \nu^{7} + 949 \nu^{6} - 216 \nu^{5} + 3319 \nu^{4} - 5570 \nu^{3} + \cdots + 170000 ) / 2500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17 \nu^{9} - 5 \nu^{8} - 149 \nu^{7} - 68 \nu^{6} + 177 \nu^{5} - 93 \nu^{4} + 1373 \nu^{3} + \cdots - 28250 ) / 250 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 8 \nu^{9} - 3 \nu^{8} + 57 \nu^{7} + 46 \nu^{6} - 29 \nu^{5} + 136 \nu^{4} - 363 \nu^{3} + \cdots + 9875 ) / 100 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21 \nu^{9} + \nu^{8} - 174 \nu^{7} - 102 \nu^{6} + 183 \nu^{5} - 222 \nu^{4} + 1366 \nu^{3} + \cdots - 32375 ) / 250 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{8} - 2\beta_{6} + 3\beta_{5} + \beta_{4} - \beta_{3} - 3\beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{9} - 2\beta_{8} + 2\beta_{7} + 8\beta_{6} + 5\beta_{5} + 3\beta_{4} + 4\beta_{3} - 7\beta_{2} - 4\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} + 2\beta_{8} - 8\beta_{7} - 2\beta_{6} + 10\beta_{5} - 22\beta_{4} - \beta_{3} - 7\beta_{2} + 2\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23\beta_{8} + 13\beta_{7} + 11\beta_{6} + 19\beta_{5} + 30\beta_{4} + 20\beta_{3} + 13\beta_{2} + 23\beta _1 + 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9 \beta_{9} + 46 \beta_{8} + 16 \beta_{7} - 34 \beta_{6} + 82 \beta_{5} - 122 \beta_{4} + 29 \beta_{3} + \cdots - 64 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 52 \beta_{9} + 188 \beta_{8} - 12 \beta_{7} + 62 \beta_{6} + 195 \beta_{5} + 197 \beta_{4} + 49 \beta_{3} + \cdots + 279 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 5 \beta_{9} - 262 \beta_{8} - 160 \beta_{7} + 448 \beta_{6} + 533 \beta_{5} - 269 \beta_{4} + \cdots - 278 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4080\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(511\) \(817\) \(1361\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
0.440604 + 2.19223i
2.19908 + 0.405001i
−2.21950 + 0.271668i
2.21027 − 0.338667i
−1.63046 − 1.53023i
0.440604 − 2.19223i
2.19908 − 0.405001i
−2.21950 − 0.271668i
2.21027 + 0.338667i
−1.63046 + 1.53023i
0 − 1.00000i 0 −2.19223 + 0.440604i 0 − 1.54384i 0 −1.00000 0
2449.2 0 − 1.00000i 0 −0.405001 + 2.19908i 0 0.469130i 0 −1.00000 0
2449.3 0 − 1.00000i 0 −0.271668 − 2.21950i 0 1.68096i 0 −1.00000 0
2449.4 0 − 1.00000i 0 0.338667 + 2.21027i 0 − 4.95275i 0 −1.00000 0
2449.5 0 − 1.00000i 0 1.53023 − 1.63046i 0 − 2.65350i 0 −1.00000 0
2449.6 0 1.00000i 0 −2.19223 − 0.440604i 0 1.54384i 0 −1.00000 0
2449.7 0 1.00000i 0 −0.405001 − 2.19908i 0 − 0.469130i 0 −1.00000 0
2449.8 0 1.00000i 0 −0.271668 + 2.21950i 0 − 1.68096i 0 −1.00000 0
2449.9 0 1.00000i 0 0.338667 − 2.21027i 0 4.95275i 0 −1.00000 0
2449.10 0 1.00000i 0 1.53023 + 1.63046i 0 2.65350i 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2449.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4080.2.m.q 10
4.b odd 2 1 2040.2.m.h ✓ 10
5.b even 2 1 inner 4080.2.m.q 10
20.d odd 2 1 2040.2.m.h ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2040.2.m.h ✓ 10 4.b odd 2 1
2040.2.m.h ✓ 10 20.d odd 2 1
4080.2.m.q 10 1.a even 1 1 trivial
4080.2.m.q 10 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4080, [\chi])\):

\( T_{7}^{10} + 37T_{7}^{8} + 352T_{7}^{6} + 1188T_{7}^{4} + 1408T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{5} + 5T_{11}^{4} - 27T_{11}^{3} - 135T_{11}^{2} + 184T_{11} + 904 \) Copy content Toggle raw display
\( T_{23}^{10} + 110T_{23}^{8} + 3597T_{23}^{6} + 31332T_{23}^{4} + 25348T_{23}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{10} + 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 37 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{5} + 5 T^{4} + \cdots + 904)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 126 T^{8} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{5} \) Copy content Toggle raw display
$19$ \( (T^{5} + 3 T^{4} + \cdots + 1372)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 110 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( (T^{5} - 19 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + 6 T^{4} + \cdots - 208)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 269 T^{8} + \cdots + 6310144 \) Copy content Toggle raw display
$41$ \( (T^{5} - 5 T^{4} + \cdots - 7612)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 240622144 \) Copy content Toggle raw display
$47$ \( T^{10} + 265 T^{8} + \cdots + 50176 \) Copy content Toggle raw display
$53$ \( T^{10} + 273 T^{8} + \cdots + 614656 \) Copy content Toggle raw display
$59$ \( (T^{5} + 10 T^{4} + \cdots - 8320)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 16 T^{4} + \cdots - 256)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + 252 T^{8} + \cdots + 1982464 \) Copy content Toggle raw display
$71$ \( (T^{5} - 122 T^{3} + \cdots + 896)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 16183910656 \) Copy content Toggle raw display
$79$ \( (T^{5} + 6 T^{4} + \cdots - 2096)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 320 T^{8} + \cdots + 70291456 \) Copy content Toggle raw display
$89$ \( (T^{5} + 14 T^{4} + \cdots + 64000)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 3215343616 \) Copy content Toggle raw display
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