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Properties

Label 770.2.n.f
Level $770$
Weight $2$
Character orbit 770.n
Analytic conductor $6.148$
Analytic rank $0$
Dimension $8$
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] = [770,2,Mod(71,770)] 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("770.71"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(770, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 770 = 2 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 770.n (of order \(5\), degree \(4\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.14848095564\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.484000000.9
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{5} q^{2} + ( - \beta_{7} - \beta_{4} - \beta_{3}) q^{3} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{4} + \beta_{2} q^{5} + (\beta_{6} + \beta_{2}) q^{6} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{7}+ \cdots + (3 \beta_{7} + 3 \beta_{6} + 12 \beta_{5} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{7} + 2 q^{8} + 2 q^{9} - 8 q^{10} - 8 q^{12} - 2 q^{13} - 2 q^{14} + 2 q^{15} - 2 q^{16} - 6 q^{17} + 8 q^{18} + 6 q^{19} - 2 q^{20} + 8 q^{21}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{6} + 16x^{4} + 66x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 37\nu^{4} + 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} + 148\nu^{4} - 525\nu^{2} - 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} + 148\nu^{5} - 525\nu^{3} - 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} + 73\nu^{4} + 750\nu^{2} + 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -61\nu^{7} + 38\nu^{5} - 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} - 75\nu^{5} + 1275\nu^{3} + 3300\nu ) / 1991 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} + 4\beta_{6} + \beta_{4} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 7\beta_{5} + 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{7} + 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} - 75\beta_{2} - 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} - 75\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/770\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(617\) \(661\)
\(\chi(n)\) \(-1 - \beta_{2} - \beta_{3} + \beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
−1.73855 + 1.26313i
1.73855 − 1.26313i
−1.73855 − 1.26313i
1.73855 + 1.26313i
−0.476925 + 1.46782i
0.476925 − 1.46782i
−0.476925 − 1.46782i
0.476925 + 1.46782i
−0.309017 − 0.951057i −0.929529 − 0.675342i −0.809017 + 0.587785i 0.309017 − 0.951057i −0.355049 + 1.09273i 0.809017 − 0.587785i 0.809017 + 0.587785i −0.519114 − 1.59767i −1.00000
71.2 −0.309017 − 0.951057i 2.54756 + 1.85091i −0.809017 + 0.587785i 0.309017 − 0.951057i 0.973083 − 2.99484i 0.809017 − 0.587785i 0.809017 + 0.587785i 2.13715 + 6.57747i −1.00000
141.1 −0.309017 + 0.951057i −0.929529 + 0.675342i −0.809017 − 0.587785i 0.309017 + 0.951057i −0.355049 − 1.09273i 0.809017 + 0.587785i 0.809017 − 0.587785i −0.519114 + 1.59767i −1.00000
141.2 −0.309017 + 0.951057i 2.54756 − 1.85091i −0.809017 − 0.587785i 0.309017 + 0.951057i 0.973083 + 2.99484i 0.809017 + 0.587785i 0.809017 − 0.587785i 2.13715 − 6.57747i −1.00000
421.1 0.809017 − 0.587785i −0.785942 − 2.41888i 0.309017 − 0.951057i −0.809017 − 0.587785i −2.05762 − 1.49495i −0.309017 + 0.951057i −0.309017 − 0.951057i −2.80623 + 2.03884i −1.00000
421.2 0.809017 − 0.587785i 0.167908 + 0.516768i 0.309017 − 0.951057i −0.809017 − 0.587785i 0.439589 + 0.319380i −0.309017 + 0.951057i −0.309017 − 0.951057i 2.18820 − 1.58982i −1.00000
631.1 0.809017 + 0.587785i −0.785942 + 2.41888i 0.309017 + 0.951057i −0.809017 + 0.587785i −2.05762 + 1.49495i −0.309017 − 0.951057i −0.309017 + 0.951057i −2.80623 − 2.03884i −1.00000
631.2 0.809017 + 0.587785i 0.167908 − 0.516768i 0.309017 + 0.951057i −0.809017 + 0.587785i 0.439589 − 0.319380i −0.309017 − 0.951057i −0.309017 + 0.951057i 2.18820 + 1.58982i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.n.f ✓ 8
11.c even 5 1 inner 770.2.n.f ✓ 8
11.c even 5 1 8470.2.a.co 4
11.d odd 10 1 8470.2.a.cs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.n.f ✓ 8 1.a even 1 1 trivial
770.2.n.f ✓ 8 11.c even 5 1 inner
8470.2.a.co 4 11.c even 5 1
8470.2.a.cs 4 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} + 4T_{3}^{6} - 8T_{3}^{5} + 46T_{3}^{4} + 80T_{3}^{3} + 65T_{3}^{2} + 25 \) acting on \(S_{2}^{\mathrm{new}}(770, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 19 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + \cdots + 48400 \) Copy content Toggle raw display
$17$ \( T^{8} + 6 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$19$ \( T^{8} - 6 T^{7} + \cdots + 21025 \) Copy content Toggle raw display
$23$ \( (T^{4} - 32 T^{2} + 176)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 20 T^{7} + \cdots + 26896 \) Copy content Toggle raw display
$31$ \( T^{8} + 6 T^{7} + \cdots + 21939856 \) Copy content Toggle raw display
$37$ \( T^{8} - 16 T^{7} + \cdots + 1210000 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 703921 \) Copy content Toggle raw display
$43$ \( (T^{4} - 10 T^{3} + \cdots + 971)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 16 T^{7} + \cdots + 2611456 \) Copy content Toggle raw display
$53$ \( T^{8} + 30 T^{7} + \cdots + 7862416 \) Copy content Toggle raw display
$59$ \( T^{8} + 18 T^{7} + \cdots + 24025 \) Copy content Toggle raw display
$61$ \( T^{8} - 8 T^{7} + \cdots + 80656 \) Copy content Toggle raw display
$67$ \( (T^{4} - 23 T^{2} + \cdots - 29)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 22 T^{7} + \cdots + 309136 \) Copy content Toggle raw display
$73$ \( T^{8} + 50 T^{7} + \cdots + 689010001 \) Copy content Toggle raw display
$79$ \( T^{8} + 34 T^{7} + \cdots + 2016400 \) Copy content Toggle raw display
$83$ \( T^{8} + 34 T^{7} + \cdots + 1113025 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots - 2749)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 49 T^{6} + \cdots + 2042041 \) Copy content Toggle raw display
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