Properties

Label 252.2.t.a
Level 252
Weight 2
Character orbit 252.t
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.t (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: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{2} q^{5} + ( 2 - 3 \beta_{1} ) q^{7} +O(q^{10})\) \( q -\beta_{2} q^{5} + ( 2 - 3 \beta_{1} ) q^{7} + ( -\beta_{2} - \beta_{3} ) q^{11} + ( -1 + 2 \beta_{1} ) q^{13} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{17} + ( 6 - 3 \beta_{1} ) q^{19} + ( -1 + \beta_{1} ) q^{25} + ( 4 \beta_{2} - 2 \beta_{3} ) q^{29} + ( -1 - \beta_{1} ) q^{31} + ( \beta_{2} - 3 \beta_{3} ) q^{35} + 5 \beta_{1} q^{37} + 5 \beta_{3} q^{41} -11 q^{43} -\beta_{2} q^{47} + ( -5 - 3 \beta_{1} ) q^{49} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( -6 + 12 \beta_{1} ) q^{55} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{59} + ( -4 + 2 \beta_{1} ) q^{61} + ( -\beta_{2} + 2 \beta_{3} ) q^{65} + ( 7 - 7 \beta_{1} ) q^{67} + ( 6 \beta_{2} - 3 \beta_{3} ) q^{71} + ( 9 + 9 \beta_{1} ) q^{73} + ( 4 \beta_{2} - 5 \beta_{3} ) q^{77} -11 \beta_{1} q^{79} + 5 \beta_{3} q^{83} -12 q^{85} -6 \beta_{2} q^{89} + ( 4 + \beta_{1} ) q^{91} + ( -3 \beta_{2} - 3 \beta_{3} ) q^{95} + ( -2 + 4 \beta_{1} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{7} + O(q^{10}) \) \( 4q + 2q^{7} + 18q^{19} - 2q^{25} - 6q^{31} + 10q^{37} - 44q^{43} - 26q^{49} - 12q^{61} + 14q^{67} + 54q^{73} - 22q^{79} - 48q^{85} + 18q^{91} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
1.22474 + 0.707107i
−1.22474 0.707107i
1.22474 0.707107i
−1.22474 + 0.707107i
0 0 0 −1.22474 2.12132i 0 0.500000 2.59808i 0 0 0
17.2 0 0 0 1.22474 + 2.12132i 0 0.500000 2.59808i 0 0 0
89.1 0 0 0 −1.22474 + 2.12132i 0 0.500000 + 2.59808i 0 0 0
89.2 0 0 0 1.22474 2.12132i 0 0.500000 + 2.59808i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g 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.t.a 4
3.b odd 2 1 inner 252.2.t.a 4
4.b odd 2 1 1008.2.bt.a 4
5.b even 2 1 6300.2.ch.a 4
5.c odd 4 2 6300.2.dd.a 8
7.b odd 2 1 1764.2.t.a 4
7.c even 3 1 1764.2.f.a 4
7.c even 3 1 1764.2.t.a 4
7.d odd 6 1 inner 252.2.t.a 4
7.d odd 6 1 1764.2.f.a 4
9.c even 3 1 2268.2.w.h 4
9.c even 3 1 2268.2.bm.g 4
9.d odd 6 1 2268.2.w.h 4
9.d odd 6 1 2268.2.bm.g 4
12.b even 2 1 1008.2.bt.a 4
15.d odd 2 1 6300.2.ch.a 4
15.e even 4 2 6300.2.dd.a 8
21.c even 2 1 1764.2.t.a 4
21.g even 6 1 inner 252.2.t.a 4
21.g even 6 1 1764.2.f.a 4
21.h odd 6 1 1764.2.f.a 4
21.h odd 6 1 1764.2.t.a 4
28.f even 6 1 1008.2.bt.a 4
28.f even 6 1 7056.2.k.a 4
28.g odd 6 1 7056.2.k.a 4
35.i odd 6 1 6300.2.ch.a 4
35.k even 12 2 6300.2.dd.a 8
63.i even 6 1 2268.2.bm.g 4
63.k odd 6 1 2268.2.w.h 4
63.s even 6 1 2268.2.w.h 4
63.t odd 6 1 2268.2.bm.g 4
84.j odd 6 1 1008.2.bt.a 4
84.j odd 6 1 7056.2.k.a 4
84.n even 6 1 7056.2.k.a 4
105.p even 6 1 6300.2.ch.a 4
105.w odd 12 2 6300.2.dd.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.t.a 4 1.a even 1 1 trivial
252.2.t.a 4 3.b odd 2 1 inner
252.2.t.a 4 7.d odd 6 1 inner
252.2.t.a 4 21.g even 6 1 inner
1008.2.bt.a 4 4.b odd 2 1
1008.2.bt.a 4 12.b even 2 1
1008.2.bt.a 4 28.f even 6 1
1008.2.bt.a 4 84.j odd 6 1
1764.2.f.a 4 7.c even 3 1
1764.2.f.a 4 7.d odd 6 1
1764.2.f.a 4 21.g even 6 1
1764.2.f.a 4 21.h odd 6 1
1764.2.t.a 4 7.b odd 2 1
1764.2.t.a 4 7.c even 3 1
1764.2.t.a 4 21.c even 2 1
1764.2.t.a 4 21.h odd 6 1
2268.2.w.h 4 9.c even 3 1
2268.2.w.h 4 9.d odd 6 1
2268.2.w.h 4 63.k odd 6 1
2268.2.w.h 4 63.s even 6 1
2268.2.bm.g 4 9.c even 3 1
2268.2.bm.g 4 9.d odd 6 1
2268.2.bm.g 4 63.i even 6 1
2268.2.bm.g 4 63.t odd 6 1
6300.2.ch.a 4 5.b even 2 1
6300.2.ch.a 4 15.d odd 2 1
6300.2.ch.a 4 35.i odd 6 1
6300.2.ch.a 4 105.p even 6 1
6300.2.dd.a 8 5.c odd 4 2
6300.2.dd.a 8 15.e even 4 2
6300.2.dd.a 8 35.k even 12 2
6300.2.dd.a 8 105.w odd 12 2
7056.2.k.a 4 28.f even 6 1
7056.2.k.a 4 28.g odd 6 1
7056.2.k.a 4 84.j odd 6 1
7056.2.k.a 4 84.n even 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(252, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( \)
$3$ \( \)
$5$ \( 1 - 4 T^{2} - 9 T^{4} - 100 T^{6} + 625 T^{8} \)
$7$ \( ( 1 - T + 7 T^{2} )^{2} \)
$11$ \( 1 + 4 T^{2} - 105 T^{4} + 484 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 7 T + 13 T^{2} )^{2}( 1 + 7 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 10 T^{2} - 189 T^{4} - 2890 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 8 T + 19 T^{2} )^{2}( 1 - T + 19 T^{2} )^{2} \)
$23$ \( ( 1 + 23 T^{2} + 529 T^{4} )^{2} \)
$29$ \( ( 1 + 14 T^{2} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 4 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 5 T - 12 T^{2} - 185 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 68 T^{2} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 + 11 T + 43 T^{2} )^{4} \)
$47$ \( 1 - 88 T^{2} + 5535 T^{4} - 194392 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 34 T^{2} - 1653 T^{4} + 95506 T^{6} + 7890481 T^{8} \)
$59$ \( 1 - 94 T^{2} + 5355 T^{4} - 327214 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 6 T + 73 T^{2} + 366 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4} )^{2} \)
$71$ \( ( 1 + 20 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )^{2}( 1 - 10 T + 73 T^{2} )^{2} \)
$79$ \( ( 1 + 11 T + 42 T^{2} + 869 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 + 16 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( 1 + 38 T^{2} - 6477 T^{4} + 300998 T^{6} + 62742241 T^{8} \)
$97$ \( ( 1 - 182 T^{2} + 9409 T^{4} )^{2} \)
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