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Properties

Label 1275.2.b.k
Level $1275$
Weight $2$
Character orbit 1275.b
Analytic conductor $10.181$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1275,2,Mod(1174,1275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("1275.1174"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1275.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,-18,0,2,0,0,-8,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.1809262577\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12131700736.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 33x^{4} + 24x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 255)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{4} q^{3} + ( - \beta_1 - 2) q^{4} - \beta_{6} q^{6} + (\beta_{5} + \beta_{4} + \beta_{2}) q^{7} + ( - \beta_{7} + \beta_{5} + 2 \beta_{2}) q^{8} - q^{9} + (\beta_{6} + \beta_{3} + 1) q^{11}+ \cdots + ( - \beta_{6} - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 18 q^{4} + 2 q^{6} - 8 q^{9} + 4 q^{11} + 22 q^{14} + 22 q^{16} - 24 q^{19} + 8 q^{21} - 8 q^{29} + 12 q^{31} + 2 q^{34} + 18 q^{36} + 4 q^{39} - 30 q^{44} - 48 q^{46} - 44 q^{49} - 8 q^{51} - 2 q^{54}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 33x^{4} + 24x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} + 13\nu^{5} + 51\nu^{3} + 54\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} - 9\nu^{4} - 17\nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{7} + 31\nu^{5} + 81\nu^{3} + 34\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 11\nu^{5} - 31\nu^{3} - 10\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 11\nu^{4} + 31\nu^{2} + 12 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -7\nu^{7} - 75\nu^{5} - 205\nu^{3} - 90\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{5} - \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{7} - 2\beta_{5} + 4\beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{3} - 7\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{7} + 10\beta_{5} - 29\beta_{4} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -9\beta_{6} - 11\beta_{3} + 46\beta _1 - 95 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 121\beta_{7} - 55\beta_{5} + 200\beta_{4} + 35\beta_{2} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(751\) \(851\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1174.1
0.489088i
0.849256i
− 1.89761i
− 2.53744i
2.53744i
1.89761i
− 0.849256i
− 0.489088i
− 2.60015i − 1.00000i −4.76079 0 −2.60015 2.81754i 7.17848i −1.00000 0
1174.2 − 2.50575i 1.00000i −4.27876 0 2.50575 4.48302i 5.71000i −1.00000 0
1174.3 − 1.84366i 1.00000i −1.39907 0 1.84366 − 4.55250i − 1.10791i −1.00000 0
1174.4 − 0.749250i − 1.00000i 1.43862 0 −0.749250 1.11299i − 2.57639i −1.00000 0
1174.5 0.749250i 1.00000i 1.43862 0 −0.749250 − 1.11299i 2.57639i −1.00000 0
1174.6 1.84366i − 1.00000i −1.39907 0 1.84366 4.55250i 1.10791i −1.00000 0
1174.7 2.50575i − 1.00000i −4.27876 0 2.50575 − 4.48302i − 5.71000i −1.00000 0
1174.8 2.60015i 1.00000i −4.76079 0 −2.60015 − 2.81754i − 7.17848i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1174.8
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 1275.2.b.k 8
5.b even 2 1 inner 1275.2.b.k 8
5.c odd 4 1 255.2.a.d ✓ 4
5.c odd 4 1 1275.2.a.t 4
15.e even 4 1 765.2.a.m 4
15.e even 4 1 3825.2.a.bi 4
20.e even 4 1 4080.2.a.bt 4
85.g odd 4 1 4335.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
255.2.a.d ✓ 4 5.c odd 4 1
765.2.a.m 4 15.e even 4 1
1275.2.a.t 4 5.c odd 4 1
1275.2.b.k 8 1.a even 1 1 trivial
1275.2.b.k 8 5.b even 2 1 inner
3825.2.a.bi 4 15.e even 4 1
4080.2.a.bt 4 20.e even 4 1
4335.2.a.z 4 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1275, [\chi])\):

\( T_{2}^{8} + 17T_{2}^{6} + 96T_{2}^{4} + 193T_{2}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{8} + 50T_{7}^{6} + 801T_{7}^{4} + 4224T_{7}^{2} + 4096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 17 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 50 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 31 T^{2} + \cdots - 96)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 100 T^{6} + \cdots + 43264 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 12 T^{3} + \cdots - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 204 T^{6} + \cdots + 5308416 \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} - 17 T^{2} + \cdots + 12)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 6 T^{3} + \cdots + 128)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 186 T^{6} + \cdots + 1263376 \) Copy content Toggle raw display
$41$ \( (T^{4} - 109 T^{2} + \cdots + 1308)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 176 T^{6} + \cdots + 262144 \) Copy content Toggle raw display
$47$ \( T^{8} + 66 T^{6} + \cdots + 9216 \) Copy content Toggle raw display
$53$ \( T^{8} + 218 T^{6} + \cdots + 1710864 \) Copy content Toggle raw display
$59$ \( (T^{4} - 10 T^{3} + \cdots - 192)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} + \cdots + 208)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 508 T^{6} + \cdots + 173817856 \) Copy content Toggle raw display
$71$ \( (T^{4} + 20 T^{3} + \cdots - 768)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 130 T^{6} + \cdots + 11664 \) Copy content Toggle raw display
$79$ \( (T^{4} - 84 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 272 T^{6} + \cdots + 589824 \) Copy content Toggle raw display
$89$ \( (T^{4} + 10 T^{3} + \cdots + 3888)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 304 T^{6} + \cdots + 891136 \) Copy content Toggle raw display
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