Properties

Label 252.2.bj.a
Level $252$
Weight $2$
Character orbit 252.bj
Analytic conductor $2.012$
Analytic rank $1$
Dimension $4$
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] = [252,2,Mod(103,252)]
 
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([3, 2, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.103");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.bj (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: \(2.01223013094\)
Analytic rank: \(1\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - 1) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{2} - 2) q^{5} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{6} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - 1) q^{2} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{3} - 2 \zeta_{12}^{3} q^{4} + ( - 2 \zeta_{12}^{2} - 2) q^{5} + (\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} + 2) q^{6} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + (3 \zeta_{12}^{2} - 3) q^{9} + ( - 4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} + 2) q^{10} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{11} + (2 \zeta_{12}^{2} - 4) q^{12} + (3 \zeta_{12}^{2} - 6) q^{13} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{14} + 6 \zeta_{12}^{3} q^{15} - 4 q^{16} + ( - 3 \zeta_{12}^{2} - 3) q^{17} + ( - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{18} + (\zeta_{12}^{3} + \zeta_{12}) q^{19} + (8 \zeta_{12}^{3} - 4 \zeta_{12}) q^{20} + ( - 5 \zeta_{12}^{2} + 1) q^{21} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12}) q^{22} + 4 \zeta_{12} q^{23} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{24} + 7 \zeta_{12}^{2} q^{25} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{26} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{27} + ( - 6 \zeta_{12}^{2} + 4) q^{28} + (5 \zeta_{12}^{2} - 5) q^{29} + ( - 6 \zeta_{12}^{3} - 6) q^{30} + (3 \zeta_{12}^{3} - 6 \zeta_{12}) q^{31} + ( - 4 \zeta_{12}^{3} + 4) q^{32} + (2 \zeta_{12}^{2} + 2) q^{33} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{34} + ( - 2 \zeta_{12}^{3} - 8 \zeta_{12}) q^{35} + 6 \zeta_{12} q^{36} - 3 \zeta_{12}^{2} q^{37} + ( - \zeta_{12}^{3} + \zeta_{12}^{2} - \zeta_{12} - 2) q^{38} + 9 \zeta_{12} q^{39} + ( - 8 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{40} + (\zeta_{12}^{2} - 2) q^{41} + ( - 4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 1) q^{42} - 11 \zeta_{12} q^{43} + 4 \zeta_{12}^{2} q^{44} + ( - 6 \zeta_{12}^{2} + 12) q^{45} + (4 \zeta_{12}^{2} - 4 \zeta_{12} - 4) q^{46} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{47} + (4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{48} + (3 \zeta_{12}^{2} + 5) q^{49} + (7 \zeta_{12}^{3} - 7 \zeta_{12}^{2} - 7 \zeta_{12}) q^{50} + 9 \zeta_{12}^{3} q^{51} + (6 \zeta_{12}^{3} + 6 \zeta_{12}) q^{52} + (\zeta_{12}^{2} - 1) q^{53} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 6 \zeta_{12} - 3) q^{54} + ( - 4 \zeta_{12}^{3} + 8 \zeta_{12}) q^{55} + ( - 2 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12} - 4) q^{56} + ( - 3 \zeta_{12}^{2} + 3) q^{57} + ( - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 5) q^{58} + ( - \zeta_{12}^{3} + 2 \zeta_{12}) q^{59} + 12 q^{60} + ( - 6 \zeta_{12}^{2} + 3) q^{61} + ( - 3 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 6 \zeta_{12} + 3) q^{62} + (9 \zeta_{12}^{3} - 6 \zeta_{12}) q^{63} + 8 \zeta_{12}^{3} q^{64} + 18 q^{65} + (4 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} - 2) q^{66} - 9 \zeta_{12}^{3} q^{67} + (12 \zeta_{12}^{3} - 6 \zeta_{12}) q^{68} + ( - 8 \zeta_{12}^{2} + 4) q^{69} + (2 \zeta_{12}^{3} - 8 \zeta_{12}^{2} + 8 \zeta_{12} + 10) q^{70} + 2 \zeta_{12}^{3} q^{71} + (6 \zeta_{12}^{2} - 6 \zeta_{12} - 6) q^{72} + ( - 7 \zeta_{12}^{2} - 7) q^{73} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12}) q^{74} + ( - 14 \zeta_{12}^{3} + 7 \zeta_{12}) q^{75} + ( - 2 \zeta_{12}^{2} + 4) q^{76} + (2 \zeta_{12}^{2} - 6) q^{77} + (9 \zeta_{12}^{2} - 9 \zeta_{12} - 9) q^{78} + 3 \zeta_{12}^{3} q^{79} + (8 \zeta_{12}^{2} + 8) q^{80} - 9 \zeta_{12}^{2} q^{81} + ( - \zeta_{12}^{3} - \zeta_{12}^{2} - \zeta_{12} + 2) q^{82} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{83} + (8 \zeta_{12}^{3} - 10 \zeta_{12}) q^{84} + 18 \zeta_{12}^{2} q^{85} + ( - 11 \zeta_{12}^{2} + 11 \zeta_{12} + 11) q^{86} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12}) q^{87} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{88} + ( - 5 \zeta_{12}^{2} + 10) q^{89} + (6 \zeta_{12}^{3} + 6 \zeta_{12}^{2} + 6 \zeta_{12} - 12) q^{90} + (12 \zeta_{12}^{3} - 15 \zeta_{12}) q^{91} + ( - 8 \zeta_{12}^{2} + 8) q^{92} + 9 \zeta_{12}^{2} q^{93} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{94} - 6 \zeta_{12}^{3} q^{95} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} - 8) q^{96} + ( - \zeta_{12}^{2} - 1) q^{97} + (8 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} - 5) q^{98} - 6 \zeta_{12}^{3} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{2} - 12 q^{5} + 6 q^{6} + 8 q^{8} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{2} - 12 q^{5} + 6 q^{6} + 8 q^{8} - 6 q^{9} + 12 q^{10} - 12 q^{12} - 18 q^{13} - 2 q^{14} - 16 q^{16} - 18 q^{17} + 6 q^{18} - 6 q^{21} - 4 q^{22} + 12 q^{24} + 14 q^{25} + 18 q^{26} + 4 q^{28} - 10 q^{29} - 24 q^{30} + 16 q^{32} + 12 q^{33} + 18 q^{34} - 6 q^{37} - 6 q^{38} - 24 q^{40} - 6 q^{41} + 6 q^{42} + 8 q^{44} + 36 q^{45} - 8 q^{46} + 26 q^{49} - 14 q^{50} - 2 q^{53} - 4 q^{56} + 6 q^{57} + 10 q^{58} + 48 q^{60} + 72 q^{65} - 12 q^{66} + 24 q^{70} - 12 q^{72} - 42 q^{73} + 6 q^{74} + 12 q^{76} - 20 q^{77} - 18 q^{78} + 48 q^{80} - 18 q^{81} + 6 q^{82} + 36 q^{85} + 22 q^{86} - 8 q^{88} + 30 q^{89} - 36 q^{90} + 16 q^{92} + 18 q^{93} - 24 q^{96} - 6 q^{97} - 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-\zeta_{12}^{2}\) \(\zeta_{12}^{2}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
103.1
0.866025 0.500000i
−0.866025 + 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−1.00000 1.00000i −0.866025 + 1.50000i 2.00000i −3.00000 + 1.73205i 2.36603 0.633975i 2.59808 0.500000i 2.00000 2.00000i −1.50000 2.59808i 4.73205 + 1.26795i
103.2 −1.00000 + 1.00000i 0.866025 1.50000i 2.00000i −3.00000 + 1.73205i 0.633975 + 2.36603i −2.59808 + 0.500000i 2.00000 + 2.00000i −1.50000 2.59808i 1.26795 4.73205i
115.1 −1.00000 1.00000i 0.866025 + 1.50000i 2.00000i −3.00000 1.73205i 0.633975 2.36603i −2.59808 0.500000i 2.00000 2.00000i −1.50000 + 2.59808i 1.26795 + 4.73205i
115.2 −1.00000 + 1.00000i −0.866025 1.50000i 2.00000i −3.00000 1.73205i 2.36603 + 0.633975i 2.59808 + 0.500000i 2.00000 + 2.00000i −1.50000 + 2.59808i 4.73205 1.26795i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
63.t odd 6 1 inner
252.bj even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.bj.a yes 4
3.b odd 2 1 756.2.bj.a 4
4.b odd 2 1 inner 252.2.bj.a yes 4
7.d odd 6 1 252.2.n.a 4
9.c even 3 1 252.2.n.a 4
9.d odd 6 1 756.2.n.a 4
12.b even 2 1 756.2.bj.a 4
21.g even 6 1 756.2.n.a 4
28.f even 6 1 252.2.n.a 4
36.f odd 6 1 252.2.n.a 4
36.h even 6 1 756.2.n.a 4
63.i even 6 1 756.2.bj.a 4
63.t odd 6 1 inner 252.2.bj.a yes 4
84.j odd 6 1 756.2.n.a 4
252.r odd 6 1 756.2.bj.a 4
252.bj even 6 1 inner 252.2.bj.a yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.n.a 4 7.d odd 6 1
252.2.n.a 4 9.c even 3 1
252.2.n.a 4 28.f even 6 1
252.2.n.a 4 36.f odd 6 1
252.2.bj.a yes 4 1.a even 1 1 trivial
252.2.bj.a yes 4 4.b odd 2 1 inner
252.2.bj.a yes 4 63.t odd 6 1 inner
252.2.bj.a yes 4 252.bj even 6 1 inner
756.2.n.a 4 9.d odd 6 1
756.2.n.a 4 21.g even 6 1
756.2.n.a 4 36.h even 6 1
756.2.n.a 4 84.j odd 6 1
756.2.bj.a 4 3.b odd 2 1
756.2.bj.a 4 12.b even 2 1
756.2.bj.a 4 63.i even 6 1
756.2.bj.a 4 252.r odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 6T_{5} + 12 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$5$ \( (T^{2} + 6 T + 12)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 13T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$13$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$23$ \( T^{4} - 16T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 27)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$47$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 27)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 81)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 21 T + 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$89$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3 T + 3)^{2} \) Copy content Toggle raw display
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