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Properties

Label 84.4.i.b
Level $84$
Weight $4$
Character orbit 84.i
Analytic conductor $4.956$
Analytic rank $0$
Dimension $4$
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] = [84,4,Mod(25,84)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(84, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.25");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.i (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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-19})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{5} + ( - \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 3) q^{7} + ( - 9 \beta_{2} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 \beta_{2} q^{3} + ( - \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 1) q^{5} + ( - \beta_{3} + 6 \beta_{2} + 3 \beta_1 - 3) q^{7} + ( - 9 \beta_{2} - 9) q^{9} + (2 \beta_{3} - 25 \beta_{2} - \beta_1 + 1) q^{11} + (5 \beta_{3} + 5 \beta_{2} + 5 \beta_1 + 33) q^{13} + (3 \beta_{3} + 3 \beta_{2} + 3 \beta_1 + 6) q^{15} + ( - 16 \beta_{3} + 8 \beta_{2} + \cdots - 8) q^{17}+ \cdots + ( - 9 \beta_{3} - 9 \beta_{2} + \cdots - 234) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{3} + 3 q^{5} - 20 q^{7} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{3} + 3 q^{5} - 20 q^{7} - 18 q^{9} + 51 q^{11} + 122 q^{13} + 18 q^{15} - 24 q^{17} - 169 q^{19} + 15 q^{21} + 192 q^{23} - 11 q^{25} - 108 q^{27} - 78 q^{29} + 92 q^{31} - 153 q^{33} - 294 q^{35} + 173 q^{37} + 183 q^{39} + 348 q^{41} - 994 q^{43} + 27 q^{45} - 180 q^{47} - 146 q^{49} + 72 q^{51} - 285 q^{53} + 666 q^{55} - 1014 q^{57} + 1269 q^{59} - 328 q^{61} + 225 q^{63} + 1374 q^{65} + 875 q^{67} + 1152 q^{69} - 2808 q^{71} + 1361 q^{73} + 33 q^{75} + 897 q^{77} + 182 q^{79} - 162 q^{81} - 798 q^{83} - 4176 q^{85} - 117 q^{87} - 822 q^{89} + 1955 q^{91} - 276 q^{93} + 510 q^{95} + 1682 q^{97} - 918 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 4\nu^{2} + 56\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 4\nu^{2} - 4\nu - 25 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{3} + 2\nu^{2} + 28\nu + 35 ) / 10 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} + 14\beta_{2} + 14 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -4\beta_{3} + 4\beta _1 + 19 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(43\) \(73\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
−1.63746 − 1.52274i
2.13746 + 0.656712i
−1.63746 + 1.52274i
2.13746 − 0.656712i
0 1.50000 − 2.59808i 0 −4.91238 − 8.50848i 0 −16.3248 − 8.74657i 0 −4.50000 − 7.79423i 0
25.2 0 1.50000 − 2.59808i 0 6.41238 + 11.1066i 0 6.32475 + 17.4068i 0 −4.50000 − 7.79423i 0
37.1 0 1.50000 + 2.59808i 0 −4.91238 + 8.50848i 0 −16.3248 + 8.74657i 0 −4.50000 + 7.79423i 0
37.2 0 1.50000 + 2.59808i 0 6.41238 − 11.1066i 0 6.32475 − 17.4068i 0 −4.50000 + 7.79423i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.4.i.b ✓ 4
3.b odd 2 1 252.4.k.d 4
4.b odd 2 1 336.4.q.h 4
7.b odd 2 1 588.4.i.i 4
7.c even 3 1 inner 84.4.i.b ✓ 4
7.c even 3 1 588.4.a.g 2
7.d odd 6 1 588.4.a.h 2
7.d odd 6 1 588.4.i.i 4
21.c even 2 1 1764.4.k.z 4
21.g even 6 1 1764.4.a.p 2
21.g even 6 1 1764.4.k.z 4
21.h odd 6 1 252.4.k.d 4
21.h odd 6 1 1764.4.a.x 2
28.f even 6 1 2352.4.a.bp 2
28.g odd 6 1 336.4.q.h 4
28.g odd 6 1 2352.4.a.cb 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.i.b ✓ 4 1.a even 1 1 trivial
84.4.i.b ✓ 4 7.c even 3 1 inner
252.4.k.d 4 3.b odd 2 1
252.4.k.d 4 21.h odd 6 1
336.4.q.h 4 4.b odd 2 1
336.4.q.h 4 28.g odd 6 1
588.4.a.g 2 7.c even 3 1
588.4.a.h 2 7.d odd 6 1
588.4.i.i 4 7.b odd 2 1
588.4.i.i 4 7.d odd 6 1
1764.4.a.p 2 21.g even 6 1
1764.4.a.x 2 21.h odd 6 1
1764.4.k.z 4 21.c even 2 1
1764.4.k.z 4 21.g even 6 1
2352.4.a.bp 2 28.f even 6 1
2352.4.a.cb 2 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 3T_{5}^{3} + 135T_{5}^{2} + 378T_{5} + 15876 \) acting on \(S_{4}^{\mathrm{new}}(84, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} - 3 T^{3} + \cdots + 15876 \) Copy content Toggle raw display
$7$ \( T^{4} + 20 T^{3} + \cdots + 117649 \) Copy content Toggle raw display
$11$ \( T^{4} - 51 T^{3} + \cdots + 272484 \) Copy content Toggle raw display
$13$ \( (T^{2} - 61 T - 2276)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 24 T^{3} + \cdots + 65028096 \) Copy content Toggle raw display
$19$ \( T^{4} + 169 T^{3} + \cdots + 49168144 \) Copy content Toggle raw display
$23$ \( (T^{2} - 96 T + 9216)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 39 T - 36684)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 92 T^{3} + \cdots + 114682681 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 1506681856 \) Copy content Toggle raw display
$41$ \( (T^{2} - 174 T - 140688)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 497 T + 15454)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 180 T^{3} + \cdots + 611671824 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 5356483344 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 161150862096 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 19408947856 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 32767516324 \) Copy content Toggle raw display
$71$ \( (T^{2} + 1404 T + 361476)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 165260136484 \) Copy content Toggle raw display
$79$ \( T^{4} - 182 T^{3} + \cdots + 38800441 \) Copy content Toggle raw display
$83$ \( (T^{2} + 399 T - 197334)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 11416495104 \) Copy content Toggle raw display
$97$ \( (T^{2} - 841 T + 120262)^{2} \) Copy content Toggle raw display
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