Properties

Label 76.3.g.b
Level $76$
Weight $3$
Character orbit 76.g
Analytic conductor $2.071$
Analytic rank $0$
Dimension $4$
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(7,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 76.g (of order \(6\), 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: \(2.07085000914\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 10x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + 4 q^{4} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{5} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{6} + (\beta_{3} - 8 \beta_{2} + 4) q^{7} + 8 q^{8} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + 4 q^{4} + (2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{5} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{6} + (\beta_{3} - 8 \beta_{2} + 4) q^{7} + 8 q^{8} + ( - 2 \beta_{3} + 4 \beta_{2} - 2 \beta_1) q^{9} + (4 \beta_{3} + 4 \beta_{2} - 2 \beta_1 - 4) q^{10} + ( - 5 \beta_{3} + 2 \beta_{2} - 1) q^{11} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{12} - 6 \beta_{2} q^{13} + (2 \beta_{3} - 16 \beta_{2} + 8) q^{14} + ( - \beta_{3} + 8 \beta_{2} + \cdots - 16) q^{15}+ \cdots + ( - 26 \beta_{3} + 104 \beta_{2} + \cdots - 208) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} - 6 q^{3} + 16 q^{4} - 4 q^{5} - 12 q^{6} + 32 q^{8} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} - 6 q^{3} + 16 q^{4} - 4 q^{5} - 12 q^{6} + 32 q^{8} + 8 q^{9} - 8 q^{10} - 24 q^{12} - 12 q^{13} - 48 q^{15} + 64 q^{16} - 32 q^{17} + 16 q^{18} + 42 q^{19} - 16 q^{20} - 44 q^{21} + 12 q^{23} - 48 q^{24} - 18 q^{25} - 24 q^{26} - 4 q^{29} - 96 q^{30} + 128 q^{32} + 106 q^{33} - 64 q^{34} - 12 q^{35} + 32 q^{36} - 96 q^{37} + 84 q^{38} - 32 q^{40} + 26 q^{41} - 88 q^{42} - 72 q^{43} + 208 q^{45} + 24 q^{46} - 96 q^{48} - 36 q^{49} - 36 q^{50} - 24 q^{51} - 48 q^{52} + 124 q^{53} + 288 q^{55} + 124 q^{57} - 8 q^{58} - 78 q^{59} - 192 q^{60} - 44 q^{61} + 216 q^{63} + 256 q^{64} + 48 q^{65} + 212 q^{66} - 102 q^{67} - 128 q^{68} - 384 q^{69} - 24 q^{70} + 204 q^{71} + 64 q^{72} - 26 q^{73} - 192 q^{74} + 168 q^{76} + 248 q^{77} - 360 q^{79} - 64 q^{80} - 38 q^{81} + 52 q^{82} - 176 q^{84} - 184 q^{85} - 144 q^{86} - 16 q^{89} + 416 q^{90} - 144 q^{91} + 48 q^{92} - 80 q^{93} + 24 q^{95} - 192 q^{96} - 234 q^{97} - 72 q^{98} - 624 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 10x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 10 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 10\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 10\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-\beta_{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−2.73861 + 1.58114i
2.73861 1.58114i
−2.73861 1.58114i
2.73861 + 1.58114i
2.00000 −4.23861 + 2.44716i 4.00000 1.73861 + 3.01137i −8.47723 + 4.89433i 10.0905i 8.00000 7.47723 12.9509i 3.47723 + 6.02273i
7.2 2.00000 1.23861 0.715113i 4.00000 −3.73861 6.47547i 2.47723 1.43023i 3.76593i 8.00000 −3.47723 + 6.02273i −7.47723 12.9509i
11.1 2.00000 −4.23861 2.44716i 4.00000 1.73861 3.01137i −8.47723 4.89433i 10.0905i 8.00000 7.47723 + 12.9509i 3.47723 6.02273i
11.2 2.00000 1.23861 + 0.715113i 4.00000 −3.73861 + 6.47547i 2.47723 + 1.43023i 3.76593i 8.00000 −3.47723 6.02273i −7.47723 + 12.9509i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
76.g odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 76.3.g.b yes 4
4.b odd 2 1 76.3.g.a 4
19.c even 3 1 76.3.g.a 4
76.g odd 6 1 inner 76.3.g.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.3.g.a 4 4.b odd 2 1
76.3.g.a 4 19.c even 3 1
76.3.g.b yes 4 1.a even 1 1 trivial
76.3.g.b yes 4 76.g odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 6T_{3}^{3} + 5T_{3}^{2} - 42T_{3} + 49 \) acting on \(S_{3}^{\mathrm{new}}(76, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 6 T^{3} + \cdots + 49 \) Copy content Toggle raw display
$5$ \( T^{4} + 4 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$7$ \( T^{4} + 116T^{2} + 1444 \) Copy content Toggle raw display
$11$ \( T^{4} + 506 T^{2} + 61009 \) Copy content Toggle raw display
$13$ \( (T^{2} + 6 T + 36)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 32 T^{3} + \cdots + 18496 \) Copy content Toggle raw display
$19$ \( T^{4} - 42 T^{3} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{4} - 12 T^{3} + \cdots + 636804 \) Copy content Toggle raw display
$29$ \( T^{4} + 4 T^{3} + \cdots + 676 \) Copy content Toggle raw display
$31$ \( T^{4} + 1580 T^{2} + 36100 \) Copy content Toggle raw display
$37$ \( (T^{2} + 48 T - 174)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 26 T^{3} + \cdots + 96721 \) Copy content Toggle raw display
$43$ \( T^{4} + 72 T^{3} + \cdots + 322624 \) Copy content Toggle raw display
$47$ \( T^{4} - 490 T^{2} + 240100 \) Copy content Toggle raw display
$53$ \( T^{4} - 124 T^{3} + \cdots + 11316496 \) Copy content Toggle raw display
$59$ \( T^{4} + 78 T^{3} + \cdots + 247009 \) Copy content Toggle raw display
$61$ \( T^{4} + 44 T^{3} + \cdots + 206116 \) Copy content Toggle raw display
$67$ \( T^{4} + 102 T^{3} + \cdots + 12552849 \) Copy content Toggle raw display
$71$ \( T^{4} - 204 T^{3} + \cdots + 2274064 \) Copy content Toggle raw display
$73$ \( T^{4} + 26 T^{3} + \cdots + 96721 \) Copy content Toggle raw display
$79$ \( T^{4} + 360 T^{3} + \cdots + 103225600 \) Copy content Toggle raw display
$83$ \( T^{4} + 9386 T^{2} + 6385729 \) Copy content Toggle raw display
$89$ \( T^{4} + 16 T^{3} + \cdots + 208975936 \) Copy content Toggle raw display
$97$ \( T^{4} + 234 T^{3} + \cdots + 184117761 \) Copy content Toggle raw display
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