Properties

Label 76.5.c.a
Level $76$
Weight $5$
Character orbit 76.c
Self dual yes
Analytic conductor $7.856$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,5,Mod(37,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.c (of order \(2\), degree \(1\), 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: yes
Analytic conductor: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(\beta = \frac{1}{2}(-1 + 3\sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 3 \beta - 17) q^{5} + (5 \beta + 39) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \beta - 17) q^{5} + (5 \beta + 39) q^{7} + 81 q^{9} + (5 \beta + 119) q^{11} + ( - 35 \beta + 159) q^{17} + 361 q^{19} - 158 q^{23} + (93 \beta + 816) q^{25} + ( - 187 \beta - 2583) q^{35} + (85 \beta - 1721) q^{43} + ( - 243 \beta - 1377) q^{45} + (325 \beta - 441) q^{47} + (365 \beta + 2320) q^{49} + ( - 427 \beta - 3943) q^{55} + ( - 515 \beta - 1841) q^{61} + (405 \beta + 3159) q^{63} + ( - 275 \beta + 4879) q^{73} + (765 \beta + 7841) q^{77} + 6561 q^{81} - 5678 q^{83} + (13 \beta + 10737) q^{85} + ( - 1083 \beta - 6137) q^{95} + (405 \beta + 9639) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 31 q^{5} + 73 q^{7} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 31 q^{5} + 73 q^{7} + 162 q^{9} + 233 q^{11} + 353 q^{17} + 722 q^{19} - 316 q^{23} + 1539 q^{25} - 4979 q^{35} - 3527 q^{43} - 2511 q^{45} - 1207 q^{47} + 4275 q^{49} - 7459 q^{55} - 3167 q^{61} + 5913 q^{63} + 10033 q^{73} + 14917 q^{77} + 13122 q^{81} - 11356 q^{83} + 21461 q^{85} - 11191 q^{95} + 18873 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
4.27492
−3.27492
0 0 0 −49.4743 0 93.1238 0 81.0000 0
37.2 0 0 0 18.4743 0 −20.1238 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.b odd 2 1 CM by \(\Q(\sqrt{-19}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.5.c.a 2
3.b odd 2 1 684.5.h.b 2
4.b odd 2 1 304.5.e.b 2
19.b odd 2 1 CM 76.5.c.a 2
57.d even 2 1 684.5.h.b 2
76.d even 2 1 304.5.e.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.5.c.a 2 1.a even 1 1 trivial
76.5.c.a 2 19.b odd 2 1 CM
304.5.e.b 2 4.b odd 2 1
304.5.e.b 2 76.d even 2 1
684.5.h.b 2 3.b odd 2 1
684.5.h.b 2 57.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{5}^{\mathrm{new}}(76, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 31T - 914 \) Copy content Toggle raw display
$7$ \( T^{2} - 73T - 1874 \) Copy content Toggle raw display
$11$ \( T^{2} - 233T + 10366 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} - 353T - 125954 \) Copy content Toggle raw display
$19$ \( (T - 361)^{2} \) Copy content Toggle raw display
$23$ \( (T + 158)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 3527 T + 2183326 \) Copy content Toggle raw display
$47$ \( T^{2} + 1207 T - 13182194 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 3167 T - 31507634 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 10033 T + 15466366 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T + 5678)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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