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Properties

Label 84.2.e.a
Level $84$
Weight $2$
Character orbit 84.e
Analytic conductor $0.671$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [84,2,Mod(71,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.71");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 84.e (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: no
Analytic conductor: \(0.670743376979\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.2593100598870016.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} - \beta_{9} q^{3} + ( - \beta_{3} - \beta_{2}) q^{4} + ( - \beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{5}+ \cdots + ( - \beta_{9} + \beta_{8} + \beta_{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} - \beta_{9} q^{3} + ( - \beta_{3} - \beta_{2}) q^{4} + ( - \beta_{10} + \beta_{9} + \cdots - \beta_{2}) q^{5}+ \cdots + (5 \beta_{11} - \beta_{10} + 2 \beta_{9} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 4 q^{4} - 6 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 4 q^{4} - 6 q^{6} - 4 q^{9} + 4 q^{10} - 6 q^{12} + 4 q^{16} - 8 q^{18} - 16 q^{22} + 2 q^{24} - 12 q^{25} + 8 q^{28} + 20 q^{30} - 16 q^{33} + 32 q^{34} - 20 q^{36} - 16 q^{37} + 20 q^{40} + 10 q^{42} + 24 q^{45} + 46 q^{48} - 12 q^{49} - 28 q^{52} + 10 q^{54} + 16 q^{57} - 32 q^{58} + 28 q^{60} - 16 q^{61} + 20 q^{64} - 12 q^{66} - 24 q^{69} - 12 q^{70} - 32 q^{72} + 24 q^{73} - 60 q^{76} + 20 q^{78} + 28 q^{81} + 8 q^{82} - 14 q^{84} + 40 q^{85} - 56 q^{88} - 80 q^{90} + 24 q^{93} - 34 q^{96} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2x^{10} + x^{8} + 4x^{6} + 4x^{4} - 32x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{8} + \nu^{4} + 14\nu^{2} + 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} - \nu^{4} + 2\nu^{2} - 8 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - \nu^{5} + 2\nu^{3} - 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + 9 \nu^{7} - 14 \nu^{6} + 8 \nu^{5} - 16 \nu^{4} + \cdots - 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 9 \nu^{7} + 14 \nu^{6} + 8 \nu^{5} + 16 \nu^{4} + \cdots + 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{11} + 6 \nu^{10} - 2 \nu^{9} - 12 \nu^{8} - 9 \nu^{7} - 10 \nu^{6} + 24 \nu^{5} + 24 \nu^{4} + \cdots - 160 ) / 128 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{10} - \nu^{6} + 2\nu^{4} - 8\nu^{2} + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3 \nu^{11} + 2 \nu^{10} + 2 \nu^{9} - 4 \nu^{8} + 5 \nu^{7} - 14 \nu^{6} - 16 \nu^{5} + 40 \nu^{4} + \cdots - 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{11} - 2 \nu^{10} + 6 \nu^{9} + 4 \nu^{8} - 5 \nu^{7} - 18 \nu^{6} - 8 \nu^{5} - 8 \nu^{4} + \cdots - 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3 \nu^{11} - 2 \nu^{10} + 2 \nu^{9} + 4 \nu^{8} + 5 \nu^{7} + 14 \nu^{6} - 16 \nu^{5} - 40 \nu^{4} + \cdots + 96 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{11} + \beta_{9} + \beta_{6} - \beta_{5} - \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{11} - \beta_{10} - 2 \beta_{9} + 2 \beta_{8} + 3 \beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{4} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{11} - \beta_{9} - 2\beta_{8} + 3\beta_{6} - 3\beta_{5} + \beta_{3} + 3\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} - \beta_{10} + 2\beta_{9} - \beta_{7} + 5\beta_{6} + 6\beta_{5} + 3\beta_{4} - \beta_{2} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{6} + \beta_{5} - 13\beta_{3} + \beta_{2} - 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{11} + 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} - 13 \beta_{4} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -3\beta_{11} + 3\beta_{9} - 14\beta_{8} - \beta_{6} + \beta_{5} - 11\beta_{3} - 9\beta_{2} + 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 23 \beta_{11} + 7 \beta_{10} - 6 \beta_{9} - 12 \beta_{8} - 17 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + \cdots + 12 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/84\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
71.1
0.430469 − 1.34711i
0.430469 + 1.34711i
1.19877 − 0.750295i
1.19877 + 0.750295i
−1.37027 + 0.349801i
−1.37027 − 0.349801i
1.37027 + 0.349801i
1.37027 − 0.349801i
−1.19877 − 0.750295i
−1.19877 + 0.750295i
−0.430469 − 1.34711i
−0.430469 + 1.34711i
−1.34711 − 0.430469i 0.916638 − 1.46962i 1.62939 + 1.15978i − 0.348612i −1.86743 + 1.58515i − 1.00000i −1.69572 − 2.26374i −1.31955 − 2.69421i −0.150067 + 0.469617i
71.2 −1.34711 + 0.430469i 0.916638 + 1.46962i 1.62939 − 1.15978i 0.348612i −1.86743 − 1.58515i 1.00000i −1.69572 + 2.26374i −1.31955 + 2.69421i −0.150067 − 0.469617i
71.3 −0.750295 − 1.19877i −0.448478 + 1.67298i −0.874114 + 1.79887i 3.56257i 2.34202 − 0.717607i − 1.00000i 2.81228 − 0.301817i −2.59774 − 1.50059i 4.27072 − 2.67298i
71.4 −0.750295 + 1.19877i −0.448478 − 1.67298i −0.874114 − 1.79887i − 3.56257i 2.34202 + 0.717607i 1.00000i 2.81228 + 0.301817i −2.59774 + 1.50059i 4.27072 + 2.67298i
71.5 −0.349801 − 1.37027i 1.72007 + 0.203364i −1.75528 + 0.958643i − 2.27740i −0.323018 − 2.42810i 1.00000i 1.92760 + 2.06987i 2.91729 + 0.699602i −3.12065 + 0.796636i
71.6 −0.349801 + 1.37027i 1.72007 − 0.203364i −1.75528 − 0.958643i 2.27740i −0.323018 + 2.42810i − 1.00000i 1.92760 − 2.06987i 2.91729 − 0.699602i −3.12065 − 0.796636i
71.7 0.349801 − 1.37027i −1.72007 − 0.203364i −1.75528 − 0.958643i − 2.27740i −0.880346 + 2.28582i − 1.00000i −1.92760 + 2.06987i 2.91729 + 0.699602i −3.12065 − 0.796636i
71.8 0.349801 + 1.37027i −1.72007 + 0.203364i −1.75528 + 0.958643i 2.27740i −0.880346 − 2.28582i 1.00000i −1.92760 − 2.06987i 2.91729 − 0.699602i −3.12065 + 0.796636i
71.9 0.750295 − 1.19877i 0.448478 − 1.67298i −0.874114 − 1.79887i 3.56257i −1.66903 − 1.79285i 1.00000i −2.81228 − 0.301817i −2.59774 − 1.50059i 4.27072 + 2.67298i
71.10 0.750295 + 1.19877i 0.448478 + 1.67298i −0.874114 + 1.79887i − 3.56257i −1.66903 + 1.79285i − 1.00000i −2.81228 + 0.301817i −2.59774 + 1.50059i 4.27072 − 2.67298i
71.11 1.34711 − 0.430469i −0.916638 + 1.46962i 1.62939 − 1.15978i − 0.348612i −0.602184 + 2.37432i 1.00000i 1.69572 − 2.26374i −1.31955 − 2.69421i −0.150067 − 0.469617i
71.12 1.34711 + 0.430469i −0.916638 − 1.46962i 1.62939 + 1.15978i 0.348612i −0.602184 − 2.37432i − 1.00000i 1.69572 + 2.26374i −1.31955 + 2.69421i −0.150067 + 0.469617i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 71.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.2.e.a ✓ 12
3.b odd 2 1 inner 84.2.e.a ✓ 12
4.b odd 2 1 inner 84.2.e.a ✓ 12
7.b odd 2 1 588.2.e.c 12
7.c even 3 2 588.2.n.f 24
7.d odd 6 2 588.2.n.g 24
8.b even 2 1 1344.2.h.h 12
8.d odd 2 1 1344.2.h.h 12
12.b even 2 1 inner 84.2.e.a ✓ 12
21.c even 2 1 588.2.e.c 12
21.g even 6 2 588.2.n.g 24
21.h odd 6 2 588.2.n.f 24
24.f even 2 1 1344.2.h.h 12
24.h odd 2 1 1344.2.h.h 12
28.d even 2 1 588.2.e.c 12
28.f even 6 2 588.2.n.g 24
28.g odd 6 2 588.2.n.f 24
84.h odd 2 1 588.2.e.c 12
84.j odd 6 2 588.2.n.g 24
84.n even 6 2 588.2.n.f 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.e.a ✓ 12 1.a even 1 1 trivial
84.2.e.a ✓ 12 3.b odd 2 1 inner
84.2.e.a ✓ 12 4.b odd 2 1 inner
84.2.e.a ✓ 12 12.b even 2 1 inner
588.2.e.c 12 7.b odd 2 1
588.2.e.c 12 21.c even 2 1
588.2.e.c 12 28.d even 2 1
588.2.e.c 12 84.h odd 2 1
588.2.n.f 24 7.c even 3 2
588.2.n.f 24 21.h odd 6 2
588.2.n.f 24 28.g odd 6 2
588.2.n.f 24 84.n even 6 2
588.2.n.g 24 7.d odd 6 2
588.2.n.g 24 21.g even 6 2
588.2.n.g 24 28.f even 6 2
588.2.n.g 24 84.j odd 6 2
1344.2.h.h 12 8.b even 2 1
1344.2.h.h 12 8.d odd 2 1
1344.2.h.h 12 24.f even 2 1
1344.2.h.h 12 24.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 2 T^{10} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} + 2 T^{10} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{6} + 18 T^{4} + 68 T^{2} + 8)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$11$ \( (T^{6} - 34 T^{4} + \cdots - 32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{3} - 10 T - 4)^{4} \) Copy content Toggle raw display
$17$ \( (T^{6} + 34 T^{4} + \cdots + 32)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 64 T^{4} + \cdots + 3136)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 26 T^{4} + \cdots - 128)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + 144 T^{4} + \cdots + 46208)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 68 T^{4} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} + 4 T^{2} - 28 T + 16)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 66 T^{4} + \cdots + 800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 184 T^{4} + \cdots + 43264)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 104 T^{4} + \cdots - 8192)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 56 T^{4} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} - 280 T^{4} + \cdots - 204800)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 4 T^{2} - 146 T - 52)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 72 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 274 T^{4} + \cdots - 366368)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 6 T^{2} + \cdots + 104)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} + 16)^{6} \) Copy content Toggle raw display
$83$ \( (T^{6} - 128 T^{4} + \cdots - 1568)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 322 T^{4} + \cdots + 326432)^{2} \) Copy content Toggle raw display
$97$ \( (T + 2)^{12} \) Copy content Toggle raw display
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