Properties

Label 252.12.a.f
Level $252$
Weight $12$
Character orbit 252.a
Self dual yes
Analytic conductor $193.622$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,12,Mod(1,252)]
 
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, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

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: yes
Analytic conductor: \(193.622481501\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 522x + 2520 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5}\cdot 3\cdot 7 \)
Twist minimal: no (minimal twist has level 28)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \beta_{2} - 5 \beta_1 - 1585) q^{5} - 16807 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta_{2} - 5 \beta_1 - 1585) q^{5} - 16807 q^{7} + (297 \beta_{2} - 1199 \beta_1 - 26522) q^{11} + (2348 \beta_{2} - 167 \beta_1 + 331455) q^{13} + (9204 \beta_{2} - 986 \beta_1 + 2638624) q^{17} + (7241 \beta_{2} - 24788 \beta_1 - 5081395) q^{19} + (23451 \beta_{2} - 11903 \beta_1 + 20472268) q^{23} + (15319 \beta_{2} + 18275 \beta_1 - 36412975) q^{25} + ( - 143056 \beta_{2} - 72186 \beta_1 + 53524808) q^{29} + ( - 15432 \beta_{2} - 91254 \beta_1 - 114013926) q^{31} + (33614 \beta_{2} + 84035 \beta_1 + 26639095) q^{35} + (247498 \beta_{2} - 175312 \beta_1 - 240703792) q^{37} + ( - 293818 \beta_{2} - 864824 \beta_1 - 94709988) q^{41} + ( - 1149547 \beta_{2} + 437101 \beta_1 - 158603722) q^{43} + ( - 1037566 \beta_{2} + 2111920 \beta_1 - 408726678) q^{47} + 282475249 q^{49} + (2761202 \beta_{2} - 526010 \beta_1 - 1344268902) q^{53} + (1190290 \beta_{2} + 323570 \beta_1 + 1606862060) q^{55} + (6229197 \beta_{2} + 7563614 \beta_1 - 1745322901) q^{59} + ( - 10094050 \beta_{2} - 10085351 \beta_1 + 1901672169) q^{61} + ( - 5779781 \beta_{2} - 8435245 \beta_1 - 2712933030) q^{65} + (7776031 \beta_{2} - 12809347 \beta_1 + 947395888) q^{67} + (2836694 \beta_{2} - 22613626 \beta_1 + 5649485044) q^{71} + ( - 10051220 \beta_{2} - 30484648 \beta_1 - 4839929858) q^{73} + ( - 4991679 \beta_{2} + 20151593 \beta_1 + 445755254) q^{77} + (27865372 \beta_{2} + 55681400 \beta_1 - 10971962268) q^{79} + ( - 8290233 \beta_{2} + 31319032 \beta_1 + 38476282183) q^{83} + ( - 24833006 \beta_{2} - 39467380 \beta_1 - 12239293630) q^{85} + (17698370 \beta_{2} + 95811502 \beta_1 + 5182673670) q^{89} + ( - 39462836 \beta_{2} + 2806769 \beta_1 - 5570764185) q^{91} + (31156443 \beta_{2} + 26107345 \beta_1 + 39256979980) q^{95} + ( - 34354270 \beta_{2} + 54303700 \beta_1 + 13557821104) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4762 q^{5} - 50421 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4762 q^{5} - 50421 q^{7} - 80468 q^{11} + 996546 q^{13} + 7924090 q^{17} - 15261732 q^{19} + 61428352 q^{23} - 109205331 q^{25} + 160359182 q^{29} - 342148464 q^{31} + 80034934 q^{35} - 722039190 q^{37} - 285288606 q^{41} - 476523612 q^{43} - 1225105680 q^{47} + 847425747 q^{49} - 4030571514 q^{53} + 4822100040 q^{55} - 5222175892 q^{59} + 5684837106 q^{61} - 8153014116 q^{65} + 2837154348 q^{67} + 16928678200 q^{71} - 14560325442 q^{73} + 1352425676 q^{77} - 32832340032 q^{79} + 115451875348 q^{83} - 36782181276 q^{85} + 15661530882 q^{89} - 16748948622 q^{91} + 117828203728 q^{95} + 40693412742 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 522x + 2520 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{2} + 85\nu + 321 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\nu^{2} + 73\nu - 3855 ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 11\beta _1 + 108 ) / 336 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 85\beta_{2} - 73\beta _1 + 117036 ) / 336 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
20.4801
5.02190
−24.5020
0 0 0 −5824.83 0 −16807.0 0 0 0
1.2 0 0 0 −648.744 0 −16807.0 0 0 0
1.3 0 0 0 1711.58 0 −16807.0 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.12.a.f 3
3.b odd 2 1 28.12.a.a 3
12.b even 2 1 112.12.a.g 3
21.c even 2 1 196.12.a.c 3
21.g even 6 2 196.12.e.d 6
21.h odd 6 2 196.12.e.e 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.12.a.a 3 3.b odd 2 1
112.12.a.g 3 12.b even 2 1
196.12.a.c 3 21.c even 2 1
196.12.e.d 6 21.g even 6 2
196.12.e.e 6 21.h odd 6 2
252.12.a.f 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{3} + 4762T_{5}^{2} - 7301200T_{5} - 6467756800 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(252))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} \) Copy content Toggle raw display
$5$ \( T^{3} + 4762 T^{2} + \cdots - 6467756800 \) Copy content Toggle raw display
$7$ \( (T + 16807)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 80468 T^{2} + \cdots - 29\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{3} - 996546 T^{2} + \cdots + 54\!\cdots\!44 \) Copy content Toggle raw display
$17$ \( T^{3} - 7924090 T^{2} + \cdots + 42\!\cdots\!60 \) Copy content Toggle raw display
$19$ \( T^{3} + 15261732 T^{2} + \cdots - 41\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( T^{3} - 61428352 T^{2} + \cdots + 88\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{3} - 160359182 T^{2} + \cdots + 11\!\cdots\!12 \) Copy content Toggle raw display
$31$ \( T^{3} + 342148464 T^{2} + \cdots + 98\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{3} + 722039190 T^{2} + \cdots + 19\!\cdots\!80 \) Copy content Toggle raw display
$41$ \( T^{3} + 285288606 T^{2} + \cdots + 44\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{3} + 476523612 T^{2} + \cdots - 80\!\cdots\!20 \) Copy content Toggle raw display
$47$ \( T^{3} + 1225105680 T^{2} + \cdots - 93\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{3} + 4030571514 T^{2} + \cdots - 90\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{3} + 5222175892 T^{2} + \cdots - 26\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{3} - 5684837106 T^{2} + \cdots + 76\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{3} - 2837154348 T^{2} + \cdots - 31\!\cdots\!20 \) Copy content Toggle raw display
$71$ \( T^{3} - 16928678200 T^{2} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} + 14560325442 T^{2} + \cdots - 66\!\cdots\!60 \) Copy content Toggle raw display
$79$ \( T^{3} + 32832340032 T^{2} + \cdots - 50\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{3} - 115451875348 T^{2} + \cdots - 30\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{3} - 15661530882 T^{2} + \cdots + 42\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{3} - 40693412742 T^{2} + \cdots + 44\!\cdots\!56 \) Copy content Toggle raw display
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