Properties

Label 76.3.c.b
Level $76$
Weight $3$
Character orbit 76.c
Self dual yes
Analytic conductor $2.071$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,3,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, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.37");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
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: \(2.07085000914\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \frac{1}{2}(1 + \sqrt{57})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 5) q^{5} + (3 \beta + 1) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 5) q^{5} + (3 \beta + 1) q^{7} + 9 q^{9} + ( - 5 \beta + 1) q^{11} + (7 \beta - 11) q^{17} - 19 q^{19} - 30 q^{23} + ( - 9 \beta + 14) q^{25} + (11 \beta - 37) q^{35} + (3 \beta + 41) q^{43} + ( - 9 \beta + 45) q^{45} + ( - 13 \beta - 31) q^{47} + (15 \beta + 78) q^{49} + ( - 21 \beta + 75) q^{55} + (15 \beta - 59) q^{61} + (27 \beta + 9) q^{63} + ( - 33 \beta + 29) q^{73} + ( - 17 \beta - 209) q^{77} + 81 q^{81} + 90 q^{83} + (39 \beta - 153) q^{85} + (19 \beta - 95) q^{95} + ( - 45 \beta + 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 9 q^{5} + 5 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 9 q^{5} + 5 q^{7} + 18 q^{9} - 3 q^{11} - 15 q^{17} - 38 q^{19} - 60 q^{23} + 19 q^{25} - 63 q^{35} + 85 q^{43} + 81 q^{45} - 75 q^{47} + 171 q^{49} + 129 q^{55} - 103 q^{61} + 45 q^{63} + 25 q^{73} - 435 q^{77} + 162 q^{81} + 180 q^{83} - 267 q^{85} - 171 q^{95} - 27 q^{99}+O(q^{100}) \) 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 0.725083 0 13.8248 0 9.00000 0
37.2 0 0 0 8.27492 0 −8.82475 0 9.00000 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 76.3.c.b 2
3.b odd 2 1 684.3.h.a 2
4.b odd 2 1 304.3.e.e 2
5.b even 2 1 1900.3.e.a 2
5.c odd 4 2 1900.3.g.a 4
8.b even 2 1 1216.3.e.f 2
8.d odd 2 1 1216.3.e.e 2
12.b even 2 1 2736.3.o.c 2
19.b odd 2 1 CM 76.3.c.b 2
57.d even 2 1 684.3.h.a 2
76.d even 2 1 304.3.e.e 2
95.d odd 2 1 1900.3.e.a 2
95.g even 4 2 1900.3.g.a 4
152.b even 2 1 1216.3.e.e 2
152.g odd 2 1 1216.3.e.f 2
228.b odd 2 1 2736.3.o.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.c.b 2 1.a even 1 1 trivial
76.3.c.b 2 19.b odd 2 1 CM
304.3.e.e 2 4.b odd 2 1
304.3.e.e 2 76.d even 2 1
684.3.h.a 2 3.b odd 2 1
684.3.h.a 2 57.d even 2 1
1216.3.e.e 2 8.d odd 2 1
1216.3.e.e 2 152.b even 2 1
1216.3.e.f 2 8.b even 2 1
1216.3.e.f 2 152.g odd 2 1
1900.3.e.a 2 5.b even 2 1
1900.3.e.a 2 95.d odd 2 1
1900.3.g.a 4 5.c odd 4 2
1900.3.g.a 4 95.g even 4 2
2736.3.o.c 2 12.b even 2 1
2736.3.o.c 2 228.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\). 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
$5$ \( T^{2} - 9T + 6 \) Copy content Toggle raw display
$7$ \( T^{2} - 5T - 122 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T - 354 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 15T - 642 \) Copy content Toggle raw display
$19$ \( (T + 19)^{2} \) Copy content Toggle raw display
$23$ \( (T + 30)^{2} \) 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
$43$ \( T^{2} - 85T + 1678 \) Copy content Toggle raw display
$47$ \( T^{2} + 75T - 1002 \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + 103T - 554 \) Copy content Toggle raw display
$67$ \( T^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 25T - 15362 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 90)^{2} \) 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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