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Properties

Label 252.6.f.a
Level $252$
Weight $6$
Character orbit 252.f
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(125,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.125");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 252.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: \(40.4167225929\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 1584x^{10} + 918546x^{8} + 240636628x^{6} + 28535411889x^{4} + 1321520960964x^{2} + 18090373745284 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{25}\cdot 3^{15} \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{5} + (\beta_{5} - 7) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{5} + (\beta_{5} - 7) q^{7} + (\beta_{10} + \beta_{7}) q^{11} + \beta_{8} q^{13} + ( - \beta_{2} + 2 \beta_1) q^{17} + (\beta_{8} - \beta_{6} + \cdots + \beta_{4}) q^{19}+ \cdots + ( - 42 \beta_{8} + \cdots + 298 \beta_{4}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 84 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 84 q^{7} + 2676 q^{25} + 1056 q^{37} - 7656 q^{43} + 15204 q^{49} + 65472 q^{67} - 6096 q^{79} + 75888 q^{85} - 69552 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 1584x^{10} + 918546x^{8} + 240636628x^{6} + 28535411889x^{4} + 1321520960964x^{2} + 18090373745284 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 548896454 \nu^{10} + 709507148573 \nu^{8} + 269652030972203 \nu^{6} + \cdots - 13\!\cdots\!62 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1507189604 \nu^{10} - 1126790582257 \nu^{8} + \cdots - 60\!\cdots\!42 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 9337717 \nu^{10} - 13911932145 \nu^{8} - 7195978024499 \nu^{6} + \cdots + 14\!\cdots\!80 ) / 11\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 105193552671938 \nu^{11} + \cdots - 67\!\cdots\!80 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 105193552671938 \nu^{11} + \cdots + 67\!\cdots\!80 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 64353185588009 \nu^{11} + \cdots - 40\!\cdots\!42 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 16894278783 \nu^{11} + 24966702709299 \nu^{9} + \cdots + 64\!\cdots\!64 \nu ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 705182950822909 \nu^{11} + \cdots - 34\!\cdots\!42 \nu ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 33\!\cdots\!90 \nu^{11} + \cdots + 17\!\cdots\!24 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7498734843201 \nu^{11} + \cdots - 53\!\cdots\!88 \nu ) / 14\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 9217434602103 \nu^{11} + \cdots - 16\!\cdots\!96 \nu ) / 14\!\cdots\!80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 9\beta_{8} - 16\beta_{7} - 9\beta_{6} + 18\beta_{5} + 18\beta_{4} ) / 432 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + 2\beta_{9} + \beta_{7} + 162\beta_{5} - 162\beta_{4} + 18\beta_{3} + 7\beta_{2} - 59\beta _1 - 57024 ) / 216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 324 \beta_{11} + 648 \beta_{10} - 4185 \beta_{8} + 12596 \beta_{7} + 6669 \beta_{6} + \cdots - 2862 \beta_{4} ) / 432 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 257 \beta_{11} - 514 \beta_{9} - 257 \beta_{7} - 58563 \beta_{5} + 58563 \beta_{4} - 4239 \beta_{3} + \cdots + 12095352 ) / 108 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 386100 \beta_{11} - 500040 \beta_{10} + 2019105 \beta_{8} - 8736964 \beta_{7} + \cdots + 726282 \beta_{4} ) / 432 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 32755 \beta_{11} + 65510 \beta_{9} + 32755 \beta_{7} + 7822926 \beta_{5} - 7822926 \beta_{4} + \cdots - 1325295984 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 317776284 \beta_{11} + 332257464 \beta_{10} - 1043118585 \beta_{8} + 5882788172 \beta_{7} + \cdots - 236526678 \beta_{4} ) / 432 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 89323906 \beta_{11} - 178647812 \beta_{9} - 89323906 \beta_{7} - 20179303119 \beta_{5} + \cdots + 3143054457288 ) / 108 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 227001827700 \beta_{11} - 215479637640 \beta_{10} + 562479448401 \beta_{8} + \cdots + 93586675698 \beta_{4} ) / 432 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 110905616429 \beta_{11} + 221811232858 \beta_{9} + 110905616429 \beta_{7} + 22872928595310 \beta_{5} + \cdots - 34\!\cdots\!04 ) / 216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 151309556288748 \beta_{11} + 137664393939864 \beta_{10} - 311183532837225 \beta_{8} + \cdots - 43170701716062 \beta_{4} ) / 432 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
125.1
− 21.5023i
21.5023i
− 16.3091i
16.3091i
− 4.82903i
4.82903i
− 7.65746i
7.65746i
− 13.4806i
13.4806i
− 24.3307i
24.3307i
0 0 0 −74.3576 0 −129.115 − 11.6746i 0 0 0
125.2 0 0 0 −74.3576 0 −129.115 + 11.6746i 0 0 0
125.3 0 0 0 −61.1411 0 101.994 − 80.0261i 0 0 0
125.4 0 0 0 −61.1411 0 101.994 + 80.0261i 0 0 0
125.5 0 0 0 −27.8694 0 6.12081 − 129.497i 0 0 0
125.6 0 0 0 −27.8694 0 6.12081 + 129.497i 0 0 0
125.7 0 0 0 27.8694 0 6.12081 − 129.497i 0 0 0
125.8 0 0 0 27.8694 0 6.12081 + 129.497i 0 0 0
125.9 0 0 0 61.1411 0 101.994 − 80.0261i 0 0 0
125.10 0 0 0 61.1411 0 101.994 + 80.0261i 0 0 0
125.11 0 0 0 74.3576 0 −129.115 − 11.6746i 0 0 0
125.12 0 0 0 74.3576 0 −129.115 + 11.6746i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 125.12
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 252.6.f.a ✓ 12
3.b odd 2 1 inner 252.6.f.a ✓ 12
4.b odd 2 1 1008.6.k.c 12
7.b odd 2 1 inner 252.6.f.a ✓ 12
12.b even 2 1 1008.6.k.c 12
21.c even 2 1 inner 252.6.f.a ✓ 12
28.d even 2 1 1008.6.k.c 12
84.h odd 2 1 1008.6.k.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.6.f.a ✓ 12 1.a even 1 1 trivial
252.6.f.a ✓ 12 3.b odd 2 1 inner
252.6.f.a ✓ 12 7.b odd 2 1 inner
252.6.f.a ✓ 12 21.c even 2 1 inner
1008.6.k.c 12 4.b odd 2 1
1008.6.k.c 12 12.b even 2 1
1008.6.k.c 12 28.d even 2 1
1008.6.k.c 12 84.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{6}^{\mathrm{new}}(252, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 10044 T^{4} + \cdots - 16053645312)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + \cdots + 4747561509943)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 28\!\cdots\!72)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 26\!\cdots\!96)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 590697459449856)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 264 T^{2} + \cdots + 50243009408)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 66\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 1914 T^{2} + \cdots - 259352693848)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 75\!\cdots\!28)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 41\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 84\!\cdots\!72)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 21\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + \cdots - 11084941347200)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 16\!\cdots\!72)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 12972288805120)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 48\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 73\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 66\!\cdots\!36)^{2} \) Copy content Toggle raw display
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