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Properties

Label 84.4.k.c
Level $84$
Weight $4$
Character orbit 84.k
Analytic conductor $4.956$
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] = [84,4,Mod(5,84)]
 
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, 3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("84.5");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 84 = 2^{2} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 84.k (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: \(4.95616044048\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} + \cdots + 531441 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{7} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} - \beta_{9} q^{5} + ( - \beta_{6} - 2 \beta_{4} + 2 \beta_1 - 1) q^{7} + ( - \beta_{10} + \beta_{7} + \cdots + 14 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} - \beta_{9} q^{5} + ( - \beta_{6} - 2 \beta_{4} + 2 \beta_1 - 1) q^{7} + ( - \beta_{10} + \beta_{7} + \cdots + 14 \beta_1) q^{9}+ \cdots + (12 \beta_{11} - 12 \beta_{10} + \cdots + 774) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 42 q^{7} - 84 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 42 q^{7} - 84 q^{9} + 132 q^{15} + 204 q^{19} - 378 q^{21} - 444 q^{25} + 1458 q^{31} - 108 q^{33} + 240 q^{37} - 432 q^{39} - 342 q^{45} - 1218 q^{49} - 300 q^{51} + 180 q^{57} + 2148 q^{61} + 1596 q^{63} + 1980 q^{67} - 3084 q^{73} - 3384 q^{75} - 438 q^{79} + 1008 q^{81} - 6144 q^{85} - 2898 q^{87} + 3780 q^{91} + 882 q^{93} + 9216 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 14 x^{10} - 32 x^{9} + 70 x^{8} + 224 x^{7} - 50 x^{6} + 2016 x^{5} + 5670 x^{4} + \cdots + 531441 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 36 \nu^{11} - 109 \nu^{10} + 180 \nu^{9} + 1949 \nu^{8} + 2588 \nu^{7} - 1681 \nu^{6} + \cdots + 6436341 ) / 2571912 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 589 \nu^{11} - 2916 \nu^{10} - 583 \nu^{9} + 33428 \nu^{8} + 116639 \nu^{7} + 77692 \nu^{6} + \cdots + 198640836 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 871 \nu^{11} - 2952 \nu^{10} - 7489 \nu^{9} + 31292 \nu^{8} + 131909 \nu^{7} + \cdots + 192499740 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1643 \nu^{11} + 4941 \nu^{10} - 403 \nu^{9} - 56869 \nu^{8} - 156097 \nu^{7} - 149963 \nu^{6} + \cdots - 250426809 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1765 \nu^{11} + 2916 \nu^{10} - 15881 \nu^{9} - 71060 \nu^{8} - 34319 \nu^{7} + 185732 \nu^{6} + \cdots - 198640836 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 74 \nu^{11} - 207 \nu^{10} + 793 \nu^{9} + 3808 \nu^{8} + 4846 \nu^{7} - 2878 \nu^{6} + \cdots + 14407956 ) / 826686 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2819 \nu^{11} + 4941 \nu^{10} - 10063 \nu^{9} - 94501 \nu^{8} - 169033 \nu^{7} - 104267 \nu^{6} + \cdots - 329788665 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 125 \nu^{11} - 477 \nu^{10} + 454 \nu^{9} + 9949 \nu^{8} + 18097 \nu^{7} - 9712 \nu^{6} + \cdots + 40743810 ) / 826686 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3554 \nu^{11} + 16407 \nu^{10} - 12982 \nu^{9} - 145867 \nu^{8} - 174346 \nu^{7} + \cdots - 264480471 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4378 \nu^{11} + 17505 \nu^{10} - 34238 \nu^{9} - 291125 \nu^{8} - 409382 \nu^{7} + \cdots - 1060106697 ) / 23147208 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 2633 \nu^{11} - 3987 \nu^{10} + 11023 \nu^{9} + 78109 \nu^{8} + 62263 \nu^{7} + \cdots + 251962083 ) / 11573604 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + \beta_{3} - 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{10} - \beta_{7} - \beta_{6} - \beta_{2} - 14\beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + 2\beta_{10} - 2\beta_{9} + \beta_{8} - 6\beta_{7} + \beta_{6} + 7\beta_{4} + \beta_{3} + \beta _1 + 25 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{11} - 9\beta_{7} + 7\beta_{6} + 16\beta_{5} - 14\beta_{4} - 2\beta_{3} - 37\beta_{2} + 47\beta _1 + 61 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 12 \beta_{10} - 14 \beta_{9} + 28 \beta_{8} + 12 \beta_{7} + 12 \beta_{6} - 45 \beta_{5} - 14 \beta_{4} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24 \beta_{11} - 24 \beta_{10} - 32 \beta_{9} + 16 \beta_{8} - 250 \beta_{7} + 86 \beta_{6} + 448 \beta_{5} + \cdots + 635 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 164 \beta_{11} + 70 \beta_{9} + 70 \beta_{8} - 214 \beta_{7} + 587 \beta_{6} + 157 \beta_{5} + \cdots - 3756 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 45 \beta_{10} - 448 \beta_{9} + 896 \beta_{8} - 1921 \beta_{7} - 1921 \beta_{6} + 400 \beta_{5} + \cdots - 137 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1966 \beta_{11} + 1966 \beta_{10} - 622 \beta_{9} + 311 \beta_{8} - 2272 \beta_{7} + 5687 \beta_{6} + \cdots + 6383 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 3415 \beta_{11} + 5376 \beta_{9} + 5376 \beta_{8} - 8175 \beta_{7} + 3953 \beta_{6} + 13472 \beta_{5} + \cdots - 42881 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2608 \beta_{10} - 10136 \beta_{9} + 20272 \beta_{8} + 7136 \beta_{7} + 7136 \beta_{6} + 20991 \beta_{5} + \cdots - 35640 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
−2.92030 − 0.686929i
−2.14770 + 2.09461i
0.865250 − 2.87252i
−1.67777 + 2.48698i
2.88784 − 0.812653i
2.99268 − 0.209499i
−2.92030 + 0.686929i
−2.14770 − 2.09461i
0.865250 + 2.87252i
−1.67777 − 2.48698i
2.88784 + 0.812653i
2.99268 + 0.209499i
0 −4.97534 − 1.49866i 0 −0.619556 + 1.07310i 0 8.82127 + 16.2845i 0 22.5081 + 14.9127i 0
5.2 0 −1.40756 − 5.00188i 0 −9.68891 + 16.7817i 0 −2.66851 − 18.3270i 0 −23.0376 + 14.0809i 0
5.3 0 −1.18980 + 5.05810i 0 0.619556 − 1.07310i 0 8.82127 + 16.2845i 0 −24.1688 − 12.0362i 0
5.4 0 −0.362863 − 5.18347i 0 7.41560 − 12.8442i 0 −16.6528 + 8.10467i 0 −26.7367 + 3.76177i 0
5.5 0 3.62798 + 3.71992i 0 9.68891 − 16.7817i 0 −2.66851 − 18.3270i 0 −0.675594 + 26.9915i 0
5.6 0 4.30758 + 2.90598i 0 −7.41560 + 12.8442i 0 −16.6528 + 8.10467i 0 10.1105 + 25.0355i 0
17.1 0 −4.97534 + 1.49866i 0 −0.619556 − 1.07310i 0 8.82127 − 16.2845i 0 22.5081 − 14.9127i 0
17.2 0 −1.40756 + 5.00188i 0 −9.68891 − 16.7817i 0 −2.66851 + 18.3270i 0 −23.0376 − 14.0809i 0
17.3 0 −1.18980 − 5.05810i 0 0.619556 + 1.07310i 0 8.82127 − 16.2845i 0 −24.1688 + 12.0362i 0
17.4 0 −0.362863 + 5.18347i 0 7.41560 + 12.8442i 0 −16.6528 − 8.10467i 0 −26.7367 − 3.76177i 0
17.5 0 3.62798 − 3.71992i 0 9.68891 + 16.7817i 0 −2.66851 + 18.3270i 0 −0.675594 − 26.9915i 0
17.6 0 4.30758 − 2.90598i 0 −7.41560 − 12.8442i 0 −16.6528 − 8.10467i 0 10.1105 − 25.0355i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

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

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(84, [\chi])\):

\( T_{5}^{12} + 597T_{5}^{10} + 272898T_{5}^{8} + 49602429T_{5}^{6} + 6898376178T_{5}^{4} + 10590781509T_{5}^{2} + 16083058761 \) Copy content Toggle raw display
\( T_{13}^{6} + 11724T_{13}^{4} + 42650496T_{13}^{2} + 49422707712 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 387420489 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 16083058761 \) Copy content Toggle raw display
$7$ \( (T^{6} + 21 T^{5} + \cdots + 40353607)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 11724549836769 \) Copy content Toggle raw display
$13$ \( (T^{6} + 11724 T^{4} + \cdots + 49422707712)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 12\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( (T^{6} - 102 T^{5} + \cdots + 33716904588)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 45\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + 55377 T^{4} + \cdots + 42952073472)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 2248345829547)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 120 T^{5} + \cdots + 236777613604)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 18213269969664)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 100848 T + 12209024)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 17\!\cdots\!89 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 36\!\cdots\!01 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 687780167594688)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 11\!\cdots\!44)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 15\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 13\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 23\!\cdots\!01)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 18\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 14\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 59\!\cdots\!52)^{2} \) Copy content Toggle raw display
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