Properties

Label 252.2.bf.e
Level $252$
Weight $2$
Character orbit 252.bf
Analytic conductor $2.012$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 252.bf (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 28)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} + 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{2} + 2) q^{5} + (3 \zeta_{12}^{3} - 2 \zeta_{12}) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} + ( - 2 \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} + 1) q^{10} - \zeta_{12} q^{11} + ( - 4 \zeta_{12}^{2} + 2) q^{13} + (\zeta_{12}^{3} + 3 \zeta_{12}^{2} - 3 \zeta_{12} - 2) q^{14} + 4 \zeta_{12}^{2} q^{16} + (\zeta_{12}^{2} + 1) q^{17} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{19} + ( - 2 \zeta_{12}^{3} + 4 \zeta_{12}) q^{20} + ( - \zeta_{12}^{3} - 1) q^{22} + (\zeta_{12}^{3} - \zeta_{12}) q^{23} + (2 \zeta_{12}^{2} - 2) q^{25} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{26} + (2 \zeta_{12}^{2} - 6) q^{28} - 4 q^{29} + (2 \zeta_{12}^{3} - \zeta_{12}) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{34} + (5 \zeta_{12}^{3} - \zeta_{12}) q^{35} - 3 \zeta_{12}^{2} q^{37} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{38} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 4) q^{40} + ( - 4 \zeta_{12}^{2} + 2) q^{41} + 2 \zeta_{12}^{3} q^{43} - 2 \zeta_{12}^{2} q^{44} + (\zeta_{12}^{2} - \zeta_{12} - 1) q^{46} + ( - 5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{47} + ( - 8 \zeta_{12}^{2} + 3) q^{49} + (2 \zeta_{12}^{3} - 2) q^{50} + ( - 8 \zeta_{12}^{3} + 4 \zeta_{12}) q^{52} + (\zeta_{12}^{2} - 1) q^{53} + (\zeta_{12}^{3} - 2 \zeta_{12}) q^{55} + (6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12} - 2) q^{56} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{58} + ( - 6 \zeta_{12}^{3} + 3 \zeta_{12}) q^{59} + (3 \zeta_{12}^{2} - 6) q^{61} + (\zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 1) q^{62} + 8 \zeta_{12}^{3} q^{64} - 6 \zeta_{12}^{2} q^{65} + 3 \zeta_{12} q^{67} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{68} + (4 \zeta_{12}^{3} + 5 \zeta_{12}^{2} - 5 \zeta_{12} - 1) q^{70} + 14 \zeta_{12}^{3} q^{71} + (5 \zeta_{12}^{2} + 5) q^{73} + ( - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 3) q^{74} + ( - 12 \zeta_{12}^{2} + 6) q^{76} + ( - \zeta_{12}^{2} + 3) q^{77} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{79} + (4 \zeta_{12}^{2} + 4) q^{80} + ( - 2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 4) q^{82} + ( - 8 \zeta_{12}^{3} + 16 \zeta_{12}) q^{83} + 3 q^{85} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{86} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{88} + (9 \zeta_{12}^{2} - 18) q^{89} + (2 \zeta_{12}^{3} + 8 \zeta_{12}) q^{91} - 2 q^{92} + ( - 10 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 5 \zeta_{12} - 5) q^{94} - 9 \zeta_{12} q^{95} + (20 \zeta_{12}^{2} - 10) q^{97} + ( - 3 \zeta_{12}^{3} - 5 \zeta_{12}^{2} - 5 \zeta_{12} + 8) q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} + 6 q^{5} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} + 6 q^{5} + 8 q^{8} + 6 q^{10} - 2 q^{14} + 8 q^{16} + 6 q^{17} - 4 q^{22} - 4 q^{25} + 12 q^{26} - 20 q^{28} - 16 q^{29} - 8 q^{32} - 6 q^{37} - 18 q^{38} + 12 q^{40} - 4 q^{44} - 2 q^{46} - 4 q^{49} - 8 q^{50} - 2 q^{53} - 16 q^{56} - 8 q^{58} - 18 q^{61} - 12 q^{65} + 6 q^{70} + 30 q^{73} + 6 q^{74} + 10 q^{77} + 24 q^{80} + 12 q^{82} + 12 q^{85} + 4 q^{86} + 4 q^{88} - 54 q^{89} - 8 q^{92} - 30 q^{94} + 22 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.366025 1.36603i 0 −1.73205 + 1.00000i 1.50000 + 0.866025i 0 1.73205 + 2.00000i 2.00000 + 2.00000i 0 0.633975 2.36603i
19.2 1.36603 0.366025i 0 1.73205 1.00000i 1.50000 + 0.866025i 0 −1.73205 2.00000i 2.00000 2.00000i 0 2.36603 + 0.633975i
199.1 −0.366025 + 1.36603i 0 −1.73205 1.00000i 1.50000 0.866025i 0 1.73205 2.00000i 2.00000 2.00000i 0 0.633975 + 2.36603i
199.2 1.36603 + 0.366025i 0 1.73205 + 1.00000i 1.50000 0.866025i 0 −1.73205 + 2.00000i 2.00000 + 2.00000i 0 2.36603 0.633975i
\(n\): 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
7.d odd 6 1 inner
28.f 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.bf.e 4
3.b odd 2 1 28.2.f.a 4
4.b odd 2 1 inner 252.2.bf.e 4
7.c even 3 1 1764.2.b.a 4
7.d odd 6 1 inner 252.2.bf.e 4
7.d odd 6 1 1764.2.b.a 4
12.b even 2 1 28.2.f.a 4
15.d odd 2 1 700.2.p.a 4
15.e even 4 1 700.2.t.a 4
15.e even 4 1 700.2.t.b 4
21.c even 2 1 196.2.f.a 4
21.g even 6 1 28.2.f.a 4
21.g even 6 1 196.2.d.b 4
21.h odd 6 1 196.2.d.b 4
21.h odd 6 1 196.2.f.a 4
24.f even 2 1 448.2.p.d 4
24.h odd 2 1 448.2.p.d 4
28.f even 6 1 inner 252.2.bf.e 4
28.f even 6 1 1764.2.b.a 4
28.g odd 6 1 1764.2.b.a 4
60.h even 2 1 700.2.p.a 4
60.l odd 4 1 700.2.t.a 4
60.l odd 4 1 700.2.t.b 4
84.h odd 2 1 196.2.f.a 4
84.j odd 6 1 28.2.f.a 4
84.j odd 6 1 196.2.d.b 4
84.n even 6 1 196.2.d.b 4
84.n even 6 1 196.2.f.a 4
105.p even 6 1 700.2.p.a 4
105.w odd 12 1 700.2.t.a 4
105.w odd 12 1 700.2.t.b 4
168.s odd 6 1 3136.2.f.e 4
168.v even 6 1 3136.2.f.e 4
168.ba even 6 1 448.2.p.d 4
168.ba even 6 1 3136.2.f.e 4
168.be odd 6 1 448.2.p.d 4
168.be odd 6 1 3136.2.f.e 4
420.be odd 6 1 700.2.p.a 4
420.br even 12 1 700.2.t.a 4
420.br even 12 1 700.2.t.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 3.b odd 2 1
28.2.f.a 4 12.b even 2 1
28.2.f.a 4 21.g even 6 1
28.2.f.a 4 84.j odd 6 1
196.2.d.b 4 21.g even 6 1
196.2.d.b 4 21.h odd 6 1
196.2.d.b 4 84.j odd 6 1
196.2.d.b 4 84.n even 6 1
196.2.f.a 4 21.c even 2 1
196.2.f.a 4 21.h odd 6 1
196.2.f.a 4 84.h odd 2 1
196.2.f.a 4 84.n even 6 1
252.2.bf.e 4 1.a even 1 1 trivial
252.2.bf.e 4 4.b odd 2 1 inner
252.2.bf.e 4 7.d odd 6 1 inner
252.2.bf.e 4 28.f even 6 1 inner
448.2.p.d 4 24.f even 2 1
448.2.p.d 4 24.h odd 2 1
448.2.p.d 4 168.ba even 6 1
448.2.p.d 4 168.be odd 6 1
700.2.p.a 4 15.d odd 2 1
700.2.p.a 4 60.h even 2 1
700.2.p.a 4 105.p even 6 1
700.2.p.a 4 420.be odd 6 1
700.2.t.a 4 15.e even 4 1
700.2.t.a 4 60.l odd 4 1
700.2.t.a 4 105.w odd 12 1
700.2.t.a 4 420.br even 12 1
700.2.t.b 4 15.e even 4 1
700.2.t.b 4 60.l odd 4 1
700.2.t.b 4 105.w odd 12 1
700.2.t.b 4 420.br even 12 1
1764.2.b.a 4 7.c even 3 1
1764.2.b.a 4 7.d odd 6 1
1764.2.b.a 4 28.f even 6 1
1764.2.b.a 4 28.g odd 6 1
3136.2.f.e 4 168.s odd 6 1
3136.2.f.e 4 168.v even 6 1
3136.2.f.e 4 168.ba even 6 1
3136.2.f.e 4 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\):

\( T_{5}^{2} - 3T_{5} + 3 \) Copy content Toggle raw display
\( T_{11}^{4} - T_{11}^{2} + 1 \) Copy content Toggle raw display
\( T_{19}^{4} + 27T_{19}^{2} + 729 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4 \) Copy content Toggle raw display
$3$ \( T^{4} \) Copy content Toggle raw display
$5$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} + 2T^{2} + 49 \) Copy content Toggle raw display
$11$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$13$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$17$ \( (T^{2} - 3 T + 3)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$23$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$29$ \( (T + 4)^{4} \) Copy content Toggle raw display
$31$ \( T^{4} + 3T^{2} + 9 \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} + 12)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 75T^{2} + 5625 \) Copy content Toggle raw display
$53$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 27T^{2} + 729 \) Copy content Toggle raw display
$61$ \( (T^{2} + 9 T + 27)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 9T^{2} + 81 \) Copy content Toggle raw display
$71$ \( (T^{2} + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 15 T + 75)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} - 81T^{2} + 6561 \) Copy content Toggle raw display
$83$ \( (T^{2} - 192)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 27 T + 243)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 300)^{2} \) Copy content Toggle raw display
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