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Properties

Label 760.2.q.f
Level $760$
Weight $2$
Character orbit 760.q
Analytic conductor $6.069$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [760,2,Mod(121,760)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(760, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("760.121");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 760 = 2^{3} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 760.q (of order \(3\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.06863055362\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.4601315889.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} - \beta_{5} q^{5} + (\beta_{6} - \beta_{4} + \cdots - \beta_{2}) q^{7}+ \cdots + ( - \beta_{7} - \beta_{6} + \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} - \beta_{5} q^{5} + (\beta_{6} - \beta_{4} + \cdots - \beta_{2}) q^{7}+ \cdots + (6 \beta_{7} + 3 \beta_{6} - 6 \beta_{5} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - 4 q^{5} + 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - 4 q^{5} + 4 q^{7} - q^{9} - 8 q^{11} + q^{13} + q^{15} + 13 q^{17} + q^{19} + 12 q^{21} + 8 q^{23} - 4 q^{25} - 20 q^{27} - 3 q^{29} + 4 q^{31} - 15 q^{33} - 2 q^{35} + 20 q^{37} - 2 q^{39} - 8 q^{41} - 3 q^{43} + 2 q^{45} + 10 q^{47} + 12 q^{49} - 16 q^{51} - 11 q^{53} + 4 q^{55} - 29 q^{57} + q^{59} - 25 q^{63} - 2 q^{65} - 8 q^{67} + 22 q^{69} - 4 q^{71} - 20 q^{73} - 2 q^{75} - 36 q^{77} - 3 q^{79} + 12 q^{81} - 30 q^{83} + 13 q^{85} + 24 q^{87} + 17 q^{89} - 4 q^{91} + 12 q^{93} + 4 q^{95} + 11 q^{97} + 30 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{5} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} - 12\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/760\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(381\) \(401\) \(457\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{5}\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
1.07988 + 1.87040i
−1.02359 − 1.77290i
0.689667 + 1.19454i
−0.245959 − 0.426014i
1.07988 − 1.87040i
−1.02359 + 1.77290i
0.689667 − 1.19454i
−0.245959 + 0.426014i
0 −0.832272 − 1.44154i 0 −0.500000 − 0.866025i 0 1.93047 0 0.114645 − 0.198571i 0
121.2 0 −0.595455 − 1.03136i 0 −0.500000 − 0.866025i 0 −3.62891 0 0.790867 − 1.36982i 0
121.3 0 0.548719 + 0.950409i 0 −0.500000 − 0.866025i 0 −0.416295 0 0.897815 − 1.55506i 0
121.4 0 1.37901 + 2.38851i 0 −0.500000 − 0.866025i 0 4.11474 0 −2.30333 + 3.98948i 0
201.1 0 −0.832272 + 1.44154i 0 −0.500000 + 0.866025i 0 1.93047 0 0.114645 + 0.198571i 0
201.2 0 −0.595455 + 1.03136i 0 −0.500000 + 0.866025i 0 −3.62891 0 0.790867 + 1.36982i 0
201.3 0 0.548719 − 0.950409i 0 −0.500000 + 0.866025i 0 −0.416295 0 0.897815 + 1.55506i 0
201.4 0 1.37901 − 2.38851i 0 −0.500000 + 0.866025i 0 4.11474 0 −2.30333 − 3.98948i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 121.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 760.2.q.f ✓ 8
4.b odd 2 1 1520.2.q.k 8
19.c even 3 1 inner 760.2.q.f ✓ 8
76.g odd 6 1 1520.2.q.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
760.2.q.f ✓ 8 1.a even 1 1 trivial
760.2.q.f ✓ 8 19.c even 3 1 inner
1520.2.q.k 8 4.b odd 2 1
1520.2.q.k 8 76.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(760, [\chi])\):

\( T_{3}^{8} - T_{3}^{7} + 7T_{3}^{6} + 4T_{3}^{5} + 31T_{3}^{4} + 6T_{3}^{3} + 37T_{3}^{2} + 6T_{3} + 36 \) Copy content Toggle raw display
\( T_{7}^{4} - 2T_{7}^{3} - 15T_{7}^{2} + 23T_{7} + 12 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 36 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} - 15 T^{2} + \cdots + 12)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4 T^{3} - 15 T^{2} + \cdots + 27)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} - T^{7} + \cdots + 144 \) Copy content Toggle raw display
$17$ \( T^{8} - 13 T^{7} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( T^{8} - T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 8 T^{7} + \cdots + 27556 \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{7} + \cdots + 7921 \) Copy content Toggle raw display
$31$ \( (T^{4} - 2 T^{3} - 39 T^{2} + \cdots - 69)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 10 T^{3} + \cdots - 346)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 27556 \) Copy content Toggle raw display
$43$ \( T^{8} + 3 T^{7} + \cdots + 1444 \) Copy content Toggle raw display
$47$ \( T^{8} - 10 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$53$ \( T^{8} + 11 T^{7} + \cdots + 381924 \) Copy content Toggle raw display
$59$ \( T^{8} - T^{7} + \cdots + 410881 \) Copy content Toggle raw display
$61$ \( T^{8} + 74 T^{6} + \cdots + 961 \) Copy content Toggle raw display
$67$ \( T^{8} + 8 T^{7} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{8} + 4 T^{7} + \cdots + 5958481 \) Copy content Toggle raw display
$73$ \( T^{8} + 20 T^{7} + \cdots + 68644 \) Copy content Toggle raw display
$79$ \( T^{8} + 3 T^{7} + \cdots + 20720704 \) Copy content Toggle raw display
$83$ \( (T^{4} + 15 T^{3} + \cdots - 3634)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 17 T^{7} + \cdots + 76457536 \) Copy content Toggle raw display
$97$ \( T^{8} - 11 T^{7} + \cdots + 68644 \) Copy content Toggle raw display
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