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Properties

Label 755.2.e.d
Level $755$
Weight $2$
Character orbit 755.e
Analytic conductor $6.029$
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] = [755,2,Mod(571,755)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(755, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 4])) 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("755.571"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 755 = 5 \cdot 151 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 755.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,4] 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: \(6.02870535261\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 11x^{6} + 2x^{5} + 55x^{4} - 18x^{3} + 78x^{2} + 36x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + \beta_{3} q^{3} + ( - \beta_{4} + 1) q^{4} - \beta_{4} q^{5} - \beta_1 q^{6} + (\beta_{4} - \beta_{2}) q^{7} + 3 q^{8} + (\beta_{6} + \beta_{5} - \beta_{3} + \cdots + 1) q^{9} + ( - \beta_{4} + 1) q^{10}+ \cdots + ( - 4 \beta_{7} - \beta_{4} + \cdots - 7 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 4 q^{3} + 4 q^{4} - 4 q^{5} - 2 q^{6} + 3 q^{7} + 24 q^{8} + 12 q^{9} + 4 q^{10} - 11 q^{11} - 2 q^{12} - 5 q^{13} - 3 q^{14} + 2 q^{15} + 4 q^{16} - q^{17} + 6 q^{18} + 20 q^{19} - 8 q^{20}+ \cdots - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 11x^{6} + 2x^{5} + 55x^{4} - 18x^{3} + 78x^{2} + 36x + 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 86\nu^{7} + 57\nu^{6} - 247\nu^{5} + 3074\nu^{4} - 1007\nu^{3} + 1387\nu^{2} - 33660\nu + 798 ) / 13606 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 413\nu^{7} - 280\nu^{6} + 3481\nu^{5} + 5428\nu^{4} + 29891\nu^{3} + 11328\nu^{2} + 6372\nu + 23292 ) / 40818 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -647\nu^{7} + 1707\nu^{6} - 7397\nu^{5} + 2187\nu^{4} - 30157\nu^{3} + 41537\nu^{2} - 39138\nu + 23898 ) / 40818 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 418\nu^{7} - 514\nu^{6} + 4495\nu^{5} + 3550\nu^{4} + 27222\nu^{3} + 12437\nu^{2} + 41436\nu + 20016 ) / 13606 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1645 \nu^{7} + 2153 \nu^{6} - 13865 \nu^{5} - 21620 \nu^{4} - 67516 \nu^{3} - 45120 \nu^{2} + \cdots - 187554 ) / 40818 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1784 \nu^{7} - 5937 \nu^{6} + 25727 \nu^{5} - 25146 \nu^{4} + 104887 \nu^{3} - 144467 \nu^{2} + \cdots - 83118 ) / 40818 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} + 4\beta_{4} + \beta_{3} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} - 3\beta_{5} + 7\beta_{3} - 3\beta_{2} - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -9\beta_{7} - 30\beta_{4} - 13\beta_{2} - 15\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -19\beta_{7} + 19\beta_{6} + 41\beta_{5} - 64\beta_{4} - 67\beta_{3} - 67\beta _1 + 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 89\beta_{6} + 149\beta_{5} - 191\beta_{3} + 149\beta_{2} + 290 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 251\beta_{7} + 824\beta_{4} + 489\beta_{2} + 719\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/755\mathbb{Z}\right)^\times\).

\(n\) \(6\) \(152\)
\(\chi(n)\) \(-1 + \beta_{4}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
571.1
1.69448 − 2.93492i
0.665775 − 1.15316i
−0.319339 + 0.553111i
−1.04092 + 1.80292i
1.69448 + 2.93492i
0.665775 + 1.15316i
−0.319339 − 0.553111i
−1.04092 − 1.80292i
0.500000 − 0.866025i −3.38896 0.500000 + 0.866025i −0.500000 + 0.866025i −1.69448 + 2.93492i −1.27589 + 2.20990i 3.00000 8.48505 0.500000 + 0.866025i
571.2 0.500000 − 0.866025i −1.33155 0.500000 + 0.866025i −0.500000 + 0.866025i −0.665775 + 1.15316i 2.35036 − 4.07095i 3.00000 −1.22698 0.500000 + 0.866025i
571.3 0.500000 − 0.866025i 0.638678 0.500000 + 0.866025i −0.500000 + 0.866025i 0.319339 − 0.553111i −0.290909 + 0.503869i 3.00000 −2.59209 0.500000 + 0.866025i
571.4 0.500000 − 0.866025i 2.08183 0.500000 + 0.866025i −0.500000 + 0.866025i 1.04092 − 1.80292i 0.716433 − 1.24090i 3.00000 1.33402 0.500000 + 0.866025i
636.1 0.500000 + 0.866025i −3.38896 0.500000 − 0.866025i −0.500000 − 0.866025i −1.69448 − 2.93492i −1.27589 − 2.20990i 3.00000 8.48505 0.500000 − 0.866025i
636.2 0.500000 + 0.866025i −1.33155 0.500000 − 0.866025i −0.500000 − 0.866025i −0.665775 − 1.15316i 2.35036 + 4.07095i 3.00000 −1.22698 0.500000 − 0.866025i
636.3 0.500000 + 0.866025i 0.638678 0.500000 − 0.866025i −0.500000 − 0.866025i 0.319339 + 0.553111i −0.290909 − 0.503869i 3.00000 −2.59209 0.500000 − 0.866025i
636.4 0.500000 + 0.866025i 2.08183 0.500000 − 0.866025i −0.500000 − 0.866025i 1.04092 + 1.80292i 0.716433 + 1.24090i 3.00000 1.33402 0.500000 − 0.866025i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 571.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
151.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 755.2.e.d ✓ 8
151.c even 3 1 inner 755.2.e.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
755.2.e.d ✓ 8 1.a even 1 1 trivial
755.2.e.d ✓ 8 151.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(755, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} - 7 T^{2} + \cdots + 6)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} - 3 T^{7} + \cdots + 100 \) Copy content Toggle raw display
$11$ \( T^{8} + 11 T^{7} + \cdots + 71289 \) Copy content Toggle raw display
$13$ \( T^{8} + 5 T^{7} + \cdots + 8100 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} + \cdots + 47524 \) Copy content Toggle raw display
$19$ \( (T^{4} - 10 T^{3} + 2 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( (T^{4} + 9 T^{3} + \cdots - 127)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + T^{7} + \cdots + 169744 \) Copy content Toggle raw display
$37$ \( T^{8} - 15 T^{7} + \cdots + 2102500 \) Copy content Toggle raw display
$41$ \( (T^{4} - 3 T^{3} - 26 T^{2} + \cdots + 39)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 3 T^{7} + \cdots + 399424 \) Copy content Toggle raw display
$47$ \( T^{8} + 3 T^{7} + \cdots + 161604 \) Copy content Toggle raw display
$53$ \( (T^{4} - 10 T^{3} + \cdots + 288)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 5 T^{3} + \cdots + 1651)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 18 T^{7} + \cdots + 58384881 \) Copy content Toggle raw display
$67$ \( (T^{4} + 22 T^{3} + \cdots + 288)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 17 T^{7} + \cdots + 59049 \) Copy content Toggle raw display
$73$ \( (T^{4} - 9 T^{3} + \cdots + 1124)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 6 T^{3} + \cdots - 3184)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 14 T^{3} + \cdots - 978)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + \cdots + 2322576 \) Copy content Toggle raw display
$97$ \( T^{8} + 26 T^{7} + \cdots + 760384 \) Copy content Toggle raw display
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