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Properties

Label 84.4.f.a
Level $84$
Weight $4$
Character orbit 84.f
Analytic conductor $4.956$
Analytic rank $0$
Dimension $8$
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] = [84,4,Mod(41,84)]
 
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([0, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.f (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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 66x^{6} - 19x^{5} + 698x^{4} + 3793x^{3} - 926x^{2} + 32403x + 135121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{5} q^{5} + ( - \beta_{7} - 3) q^{7} + ( - \beta_{7} + \beta_{4} + \beta_1 - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{5} q^{5} + ( - \beta_{7} - 3) q^{7} + ( - \beta_{7} + \beta_{4} + \beta_1 - 2) q^{9} + \beta_{3} q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{4} + \cdots - 1) q^{13}+ \cdots + (36 \beta_{7} - 36 \beta_{4} + \cdots + 60) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{7} - 12 q^{9} - 60 q^{15} - 24 q^{21} + 320 q^{25} + 80 q^{37} - 732 q^{39} + 320 q^{43} - 304 q^{49} + 552 q^{51} - 780 q^{57} + 1200 q^{63} + 1840 q^{67} - 3176 q^{79} - 3456 q^{81} + 960 q^{85} - 3192 q^{91} + 3960 q^{93} + 576 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 66x^{6} - 19x^{5} + 698x^{4} + 3793x^{3} - 926x^{2} + 32403x + 135121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 939 \nu^{7} - 450567 \nu^{6} + 1131195 \nu^{5} - 28383660 \nu^{4} + 52202793 \nu^{3} + \cdots + 551903338 ) / 147346724 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 89893 \nu^{7} - 562280 \nu^{6} + 5143112 \nu^{5} - 33300789 \nu^{4} + 21588839 \nu^{3} + \cdots + 1485572809 ) / 884080344 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41033 \nu^{7} - 435678 \nu^{6} + 9530572 \nu^{5} - 27773155 \nu^{4} + 473676125 \nu^{3} + \cdots + 15805116033 ) / 294693448 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 168761 \nu^{7} + 1555886 \nu^{6} - 12286488 \nu^{5} + 86185191 \nu^{4} - 241231037 \nu^{3} + \cdots - 3838921849 ) / 884080344 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 97390 \nu^{7} - 58483 \nu^{6} + 6098381 \nu^{5} + 3802039 \nu^{4} + 48151970 \nu^{3} + \cdots - 479259963 ) / 442040172 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 43513 \nu^{7} + 14558 \nu^{6} + 2524723 \nu^{5} - 1330381 \nu^{4} + 19000752 \nu^{3} + \cdots - 490954982 ) / 147346724 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 551255 \nu^{7} + 203780 \nu^{6} - 38736198 \nu^{5} + 11727927 \nu^{4} - 526660055 \nu^{3} + \cdots - 19804263157 ) / 884080344 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -2\beta_{7} - \beta_{6} - 3\beta_{5} - 4\beta_{4} - \beta_{3} - 9\beta_{2} ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -10\beta_{7} - 19\beta_{6} - 21\beta_{5} + 4\beta_{4} + \beta_{3} + 45\beta_{2} - 8\beta _1 - 392 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16\beta_{7} + 12\beta_{6} + 9\beta_{5} + 2\beta_{4} + 8\beta_{3} + 36\beta_{2} - 9\beta _1 - 39 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 494\beta_{7} + 295\beta_{6} + 1677\beta_{5} - 452\beta_{4} - 41\beta_{3} - 2241\beta_{2} + 200\beta _1 + 17120 ) / 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 7458 \beta_{7} - 5759 \beta_{6} - 3057 \beta_{5} + 516 \beta_{4} - 2763 \beta_{3} - 17343 \beta_{2} + \cdots - 27320 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3126 \beta_{7} - 888 \beta_{6} - 10602 \beta_{5} + 4494 \beta_{4} + 480 \beta_{3} + 14076 \beta_{2} + \cdots - 109267 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 393614 \beta_{7} + 362111 \beta_{6} + 207213 \beta_{5} - 42836 \beta_{4} + 134815 \beta_{3} + \cdots + 2263792 ) / 24 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−2.79644 + 0.701417i
−2.79644 − 0.701417i
2.59921 − 2.90122i
2.59921 + 2.90122i
−0.333647 − 7.30557i
−0.333647 + 7.30557i
1.03088 + 4.35541i
1.03088 − 4.35541i
0 −4.33552 − 2.86414i 0 −6.63823 0 9.59339 + 15.8419i 0 10.5934 + 24.8351i 0
41.2 0 −4.33552 + 2.86414i 0 −6.63823 0 9.59339 − 15.8419i 0 10.5934 − 24.8351i 0
41.3 0 −2.58907 − 4.50519i 0 16.9096 0 −14.5934 − 11.4032i 0 −13.5934 + 23.3285i 0
41.4 0 −2.58907 + 4.50519i 0 16.9096 0 −14.5934 + 11.4032i 0 −13.5934 − 23.3285i 0
41.5 0 2.58907 − 4.50519i 0 −16.9096 0 −14.5934 − 11.4032i 0 −13.5934 − 23.3285i 0
41.6 0 2.58907 + 4.50519i 0 −16.9096 0 −14.5934 + 11.4032i 0 −13.5934 + 23.3285i 0
41.7 0 4.33552 − 2.86414i 0 6.63823 0 9.59339 + 15.8419i 0 10.5934 − 24.8351i 0
41.8 0 4.33552 + 2.86414i 0 6.63823 0 9.59339 − 15.8419i 0 10.5934 + 24.8351i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 84.4.f.a ✓ 8
3.b odd 2 1 inner 84.4.f.a ✓ 8
4.b odd 2 1 336.4.k.d 8
7.b odd 2 1 inner 84.4.f.a ✓ 8
7.c even 3 2 588.4.k.d 16
7.d odd 6 2 588.4.k.d 16
12.b even 2 1 336.4.k.d 8
21.c even 2 1 inner 84.4.f.a ✓ 8
21.g even 6 2 588.4.k.d 16
21.h odd 6 2 588.4.k.d 16
28.d even 2 1 336.4.k.d 8
84.h odd 2 1 336.4.k.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.4.f.a ✓ 8 1.a even 1 1 trivial
84.4.f.a ✓ 8 3.b odd 2 1 inner
84.4.f.a ✓ 8 7.b odd 2 1 inner
84.4.f.a ✓ 8 21.c even 2 1 inner
336.4.k.d 8 4.b odd 2 1
336.4.k.d 8 12.b even 2 1
336.4.k.d 8 28.d even 2 1
336.4.k.d 8 84.h odd 2 1
588.4.k.d 16 7.c even 3 2
588.4.k.d 16 7.d odd 6 2
588.4.k.d 16 21.g even 6 2
588.4.k.d 16 21.h odd 6 2

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(84, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 6 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} - 330 T^{2} + 12600)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} + 10 T^{3} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 4716 T^{2} + 5370624)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2694 T^{2} + 522144)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1068 T^{2} + 2016)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 7410 T^{2} + 11255400)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 48240 T^{2} + 411188400)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 57204 T^{2} + 65790144)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 48060 T^{2} + 345254400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 20 T - 84140)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 274380 T^{2} + 2076933600)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 80 T - 7760)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 322008 T^{2} + 285014016)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 221940 T^{2} + 2718878400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 558870 T^{2} + 31693989600)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 67350 T^{2} + 326340000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 460 T - 61760)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 1163340 T^{2} + 537062400)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 484896 T^{2} + 45748168704)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 794 T - 479456)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1944870 T^{2} + 869269413600)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 57780 T^{2} + 261273600)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 2446437556224)^{2} \) Copy content Toggle raw display
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