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Properties

Label 252.6.k.g
Level $252$
Weight $6$
Character orbit 252.k
Analytic conductor $40.417$
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] = [252,6,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 252.k (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: \(40.4167225929\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 590 x^{10} - 137 x^{9} + 242606 x^{8} - 40639 x^{7} + 51190139 x^{6} + \cdots + 27882539673664 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{9}\cdot 7^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{8} q^{5} + ( - \beta_{7} + 3 \beta_{4} - 16 \beta_1 - 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{8} q^{5} + ( - \beta_{7} + 3 \beta_{4} - 16 \beta_1 - 8) q^{7} + ( - \beta_{9} - 2 \beta_{6} - \beta_{2}) q^{11} + ( - 9 \beta_{7} + 7 \beta_{5} + \cdots + 330) q^{13}+ \cdots + ( - 1481 \beta_{7} + 111 \beta_{5} + \cdots + 46723) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 182 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 182 q^{7} + 3940 q^{13} - 4542 q^{19} - 12806 q^{25} + 10666 q^{31} - 23742 q^{37} + 45036 q^{43} + 50274 q^{49} - 126112 q^{55} - 80444 q^{61} + 76310 q^{67} + 4790 q^{73} + 9406 q^{79} - 405328 q^{85} - 139454 q^{91} + 565712 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 590 x^{10} - 137 x^{9} + 242606 x^{8} - 40639 x^{7} + 51190139 x^{6} + \cdots + 27882539673664 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 64\!\cdots\!08 \nu^{11} + \cdots - 10\!\cdots\!64 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 49\!\cdots\!65 \nu^{11} + \cdots - 70\!\cdots\!80 ) / 85\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 32\!\cdots\!43 \nu^{11} + \cdots - 18\!\cdots\!16 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 33\!\cdots\!27 \nu^{11} + \cdots + 16\!\cdots\!36 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\!\cdots\!61 \nu^{11} + \cdots + 99\!\cdots\!48 ) / 21\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 61\!\cdots\!58 \nu^{11} + \cdots + 18\!\cdots\!92 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 87\!\cdots\!57 \nu^{11} + \cdots + 29\!\cdots\!04 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 29\!\cdots\!15 \nu^{11} + \cdots + 93\!\cdots\!20 ) / 85\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18\!\cdots\!39 \nu^{11} + \cdots + 40\!\cdots\!52 ) / 12\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 66\!\cdots\!00 \nu^{11} + \cdots - 50\!\cdots\!48 ) / 42\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 75\!\cdots\!47 \nu^{11} + \cdots - 22\!\cdots\!48 ) / 85\!\cdots\!72 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -7\beta_{8} - 3\beta_{7} - 3\beta_{5} - 12\beta_{4} - 12\beta_{3} - 7\beta_{2} - 45\beta _1 + 45 ) / 252 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9 \beta_{10} - 2 \beta_{9} - 4 \beta_{7} - 209 \beta_{6} - 37 \beta_{5} - 37 \beta_{4} + \cdots - 49503 \beta_1 ) / 252 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 6 \beta_{11} + 6 \beta_{10} - 1511 \beta_{9} + 1985 \beta_{8} + 8049 \beta_{7} - 1985 \beta_{6} + \cdots - 30195 ) / 252 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 1121 \beta_{11} + 27247 \beta_{8} - 481 \beta_{7} - 481 \beta_{5} + 5132 \beta_{4} + 5132 \beta_{3} + \cdots - 3866361 ) / 84 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1614 \beta_{10} + 363723 \beta_{9} - 2215412 \beta_{7} + 794649 \beta_{6} + 2671153 \beta_{5} + \cdots + 14205099 \beta_1 ) / 252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1034913 \beta_{11} - 1034913 \beta_{10} + 467038 \beta_{9} - 26595157 \beta_{8} - 4637417 \beta_{7} + \cdots + 2985516411 ) / 252 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1438878 \beta_{11} - 301711641 \beta_{8} - 123195853 \beta_{7} - 123195853 \beta_{5} + \cdots + 6010132587 ) / 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 101473101 \beta_{10} - 64486340 \beta_{9} + 1412527340 \beta_{7} - 2765584079 \beta_{6} + \cdots - 269213993097 \beta_1 ) / 84 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 596420370 \beta_{11} - 596420370 \beta_{10} - 24856193719 \beta_{9} + 106608220201 \beta_{8} + \cdots - 2356095314763 ) / 252 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 88595789661 \beta_{11} + 2560327398149 \beta_{8} - 1021623685507 \beta_{7} - 1021623685507 \beta_{5} + \cdots - 225119415320571 ) / 252 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 193652265018 \beta_{10} + 6827126767823 \beta_{9} - 63178272067116 \beta_{7} + \cdots + 873605217291435 \beta_1 ) / 252 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−4.51437 + 7.81912i
7.13866 − 12.3645i
4.75605 − 8.23773i
−8.34353 + 14.4514i
−7.33392 + 12.7027i
8.79711 − 15.2370i
−4.51437 − 7.81912i
7.13866 + 12.3645i
4.75605 + 8.23773i
−8.34353 − 14.4514i
−7.33392 − 12.7027i
8.79711 + 15.2370i
0 0 0 −54.1310 + 93.7576i 0 −97.3068 + 85.6644i 0 0 0
37.2 0 0 0 −31.6760 + 54.8644i 0 127.599 − 22.9213i 0 0 0
37.3 0 0 0 −3.31159 + 5.73583i 0 −75.7926 − 105.178i 0 0 0
37.4 0 0 0 3.31159 − 5.73583i 0 −75.7926 − 105.178i 0 0 0
37.5 0 0 0 31.6760 − 54.8644i 0 127.599 − 22.9213i 0 0 0
37.6 0 0 0 54.1310 − 93.7576i 0 −97.3068 + 85.6644i 0 0 0
109.1 0 0 0 −54.1310 − 93.7576i 0 −97.3068 − 85.6644i 0 0 0
109.2 0 0 0 −31.6760 − 54.8644i 0 127.599 + 22.9213i 0 0 0
109.3 0 0 0 −3.31159 − 5.73583i 0 −75.7926 + 105.178i 0 0 0
109.4 0 0 0 3.31159 + 5.73583i 0 −75.7926 + 105.178i 0 0 0
109.5 0 0 0 31.6760 + 54.8644i 0 127.599 + 22.9213i 0 0 0
109.6 0 0 0 54.1310 + 93.7576i 0 −97.3068 − 85.6644i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.6
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.6.k.g ✓ 12
3.b odd 2 1 inner 252.6.k.g ✓ 12
7.c even 3 1 inner 252.6.k.g ✓ 12
21.h odd 6 1 inner 252.6.k.g ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.6.k.g ✓ 12 1.a even 1 1 trivial
252.6.k.g ✓ 12 3.b odd 2 1 inner
252.6.k.g ✓ 12 7.c even 3 1 inner
252.6.k.g ✓ 12 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 15778 T_{5}^{10} + 201214531 T_{5}^{8} + 748968822882 T_{5}^{6} + \cdots + 42\!\cdots\!76 \) acting on \(S_{6}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 4747561509943)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{3} - 985 T^{2} + \cdots + 66950988)^{4} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 2378738643856)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 17\!\cdots\!61)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 12\!\cdots\!36)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 11259 T^{2} + \cdots - 100996641488)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 47\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 13\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 29\!\cdots\!84)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 86\!\cdots\!96)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 31\!\cdots\!29)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 13\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$97$ \( (T^{3} + \cdots + 617411450683874)^{4} \) Copy content Toggle raw display
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