Properties

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

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 252.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.01223013094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.2312.1
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 2 + \beta_{3} ) q^{8} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{2} q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{7} + ( 2 + \beta_{3} ) q^{8} + ( -2 + \beta_{2} + \beta_{3} ) q^{10} + ( -\beta_{1} + \beta_{2} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{14} + ( -2 + 2 \beta_{1} + \beta_{3} ) q^{16} + ( \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{17} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{19} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{20} + ( 2 - \beta_{2} + \beta_{3} ) q^{22} + ( \beta_{1} - \beta_{2} ) q^{23} + ( -1 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{25} + ( -4 + 2 \beta_{2} - 2 \beta_{3} ) q^{26} + ( -2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{28} + 2 q^{29} + ( -2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{32} + ( -2 + \beta_{2} - 3 \beta_{3} ) q^{34} + ( 2 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{35} + ( -4 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} + ( 4 + 2 \beta_{1} + 2 \beta_{2} ) q^{38} + ( -4 - 2 \beta_{2} + 2 \beta_{3} ) q^{40} + ( -3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{41} + ( -4 \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{43} + ( -4 + 2 \beta_{1} ) q^{44} + ( -2 + \beta_{2} - \beta_{3} ) q^{46} + ( 4 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 3 - \beta_{2} + 3 \beta_{3} ) q^{49} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{50} + ( 8 - 4 \beta_{1} ) q^{52} + ( -2 - 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{53} + ( 2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{55} + ( 6 - 2 \beta_{2} - \beta_{3} ) q^{56} + 2 \beta_{1} q^{58} -4 q^{59} + ( -4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{61} + ( 2 - 2 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} ) q^{64} + ( -4 + 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{65} + ( 2 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{67} + ( 8 - 2 \beta_{1} - 2 \beta_{3} ) q^{68} + ( 2 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{70} + ( -5 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{71} + ( -2 \beta_{1} - 2 \beta_{3} ) q^{73} + ( 4 - 4 \beta_{1} + 2 \beta_{2} ) q^{74} + ( 4 + 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 2 - 3 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{77} + ( 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{79} + ( -8 - 4 \beta_{1} ) q^{80} + ( 6 - 3 \beta_{2} + \beta_{3} ) q^{82} + ( 4 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{83} + ( 2 + 6 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{85} + ( 8 - 4 \beta_{2} - 2 \beta_{3} ) q^{86} + ( -4 \beta_{1} + 2 \beta_{2} ) q^{88} + ( 3 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{89} + ( -4 + 6 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{91} + ( 4 - 2 \beta_{1} ) q^{92} + ( -8 + 4 \beta_{1} - 4 \beta_{2} ) q^{94} + ( 2 \beta_{1} + 4 \beta_{2} + 6 \beta_{3} ) q^{95} + ( -2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} ) q^{97} + ( -8 + 3 \beta_{1} + 2 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + q^{2} + q^{4} + 2q^{7} + 7q^{8} + O(q^{10}) \) \( 4q + q^{2} + q^{4} + 2q^{7} + 7q^{8} - 8q^{10} - 7q^{14} - 7q^{16} + 12q^{19} - 4q^{20} + 6q^{22} - 8q^{25} - 12q^{26} + q^{28} + 8q^{29} - 9q^{32} - 4q^{34} + 12q^{35} - 12q^{37} + 20q^{38} - 20q^{40} - 14q^{44} - 6q^{46} + 8q^{47} + 8q^{49} - 19q^{50} + 28q^{52} - 16q^{53} + 4q^{55} + 23q^{56} + 2q^{58} - 16q^{59} + q^{64} - 8q^{65} + 32q^{68} + 16q^{70} + 14q^{74} + 20q^{76} + 8q^{77} - 36q^{80} + 20q^{82} + 8q^{83} + 20q^{85} + 30q^{86} - 2q^{88} - 16q^{91} + 14q^{92} - 32q^{94} - 31q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} - 2 x + 4\):

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
−0.780776 1.17915i
−0.780776 + 1.17915i
1.28078 0.599676i
1.28078 + 0.599676i
−0.780776 1.17915i 0 −0.780776 + 1.84130i 1.69614i 0 2.56155 0.662153i 2.78078 0.516994i 0 −2.00000 + 1.32431i
55.2 −0.780776 + 1.17915i 0 −0.780776 1.84130i 1.69614i 0 2.56155 + 0.662153i 2.78078 + 0.516994i 0 −2.00000 1.32431i
55.3 1.28078 0.599676i 0 1.28078 1.53610i 3.33513i 0 −1.56155 + 2.13578i 0.719224 2.73546i 0 −2.00000 4.27156i
55.4 1.28078 + 0.599676i 0 1.28078 + 1.53610i 3.33513i 0 −1.56155 2.13578i 0.719224 + 2.73546i 0 −2.00000 + 4.27156i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.2.b.e 4
3.b odd 2 1 84.2.b.a 4
4.b odd 2 1 252.2.b.d 4
7.b odd 2 1 252.2.b.d 4
8.b even 2 1 4032.2.b.n 4
8.d odd 2 1 4032.2.b.j 4
12.b even 2 1 84.2.b.b yes 4
21.c even 2 1 84.2.b.b yes 4
21.g even 6 2 588.2.o.a 8
21.h odd 6 2 588.2.o.c 8
24.f even 2 1 1344.2.b.e 4
24.h odd 2 1 1344.2.b.f 4
28.d even 2 1 inner 252.2.b.e 4
56.e even 2 1 4032.2.b.n 4
56.h odd 2 1 4032.2.b.j 4
84.h odd 2 1 84.2.b.a 4
84.j odd 6 2 588.2.o.c 8
84.n even 6 2 588.2.o.a 8
168.e odd 2 1 1344.2.b.f 4
168.i even 2 1 1344.2.b.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
84.2.b.a 4 3.b odd 2 1
84.2.b.a 4 84.h odd 2 1
84.2.b.b yes 4 12.b even 2 1
84.2.b.b yes 4 21.c even 2 1
252.2.b.d 4 4.b odd 2 1
252.2.b.d 4 7.b odd 2 1
252.2.b.e 4 1.a even 1 1 trivial
252.2.b.e 4 28.d even 2 1 inner
588.2.o.a 8 21.g even 6 2
588.2.o.a 8 84.n even 6 2
588.2.o.c 8 21.h odd 6 2
588.2.o.c 8 84.j odd 6 2
1344.2.b.e 4 24.f even 2 1
1344.2.b.e 4 168.i even 2 1
1344.2.b.f 4 24.h odd 2 1
1344.2.b.f 4 168.e odd 2 1
4032.2.b.j 4 8.d odd 2 1
4032.2.b.j 4 56.h odd 2 1
4032.2.b.n 4 8.b even 2 1
4032.2.b.n 4 56.e even 2 1

Hecke kernels

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

\( T_{5}^{4} + 14 T_{5}^{2} + 32 \)
\( T_{11}^{4} + 10 T_{11}^{2} + 8 \)
\( T_{19}^{2} - 6 T_{19} - 8 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T - 2 T^{3} + 4 T^{4} \)
$3$ \( \)
$5$ \( 1 - 6 T^{2} + 42 T^{4} - 150 T^{6} + 625 T^{8} \)
$7$ \( 1 - 2 T - 2 T^{2} - 14 T^{3} + 49 T^{4} \)
$11$ \( 1 - 34 T^{2} + 514 T^{4} - 4114 T^{6} + 14641 T^{8} \)
$13$ \( 1 - 12 T^{2} + 102 T^{4} - 2028 T^{6} + 28561 T^{8} \)
$17$ \( 1 - 22 T^{2} + 682 T^{4} - 6358 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 6 T + 30 T^{2} - 114 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 82 T^{2} + 2722 T^{4} - 43378 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 2 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 31 T^{2} )^{4} \)
$37$ \( ( 1 + 6 T + 66 T^{2} + 222 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( 1 - 102 T^{2} + 5130 T^{4} - 171462 T^{6} + 2825761 T^{8} \)
$43$ \( 1 - 24 T^{2} + 3774 T^{4} - 44376 T^{6} + 3418801 T^{8} \)
$47$ \( ( 1 - 4 T + 30 T^{2} - 188 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( ( 1 + 8 T + 54 T^{2} + 424 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( ( 1 + 4 T + 59 T^{2} )^{4} \)
$61$ \( 1 - 132 T^{2} + 10710 T^{4} - 491172 T^{6} + 13845841 T^{8} \)
$67$ \( 1 - 144 T^{2} + 10830 T^{4} - 646416 T^{6} + 20151121 T^{8} \)
$71$ \( 1 - 114 T^{2} + 8418 T^{4} - 574674 T^{6} + 25411681 T^{8} \)
$73$ \( 1 - 236 T^{2} + 24310 T^{4} - 1257644 T^{6} + 28398241 T^{8} \)
$79$ \( 1 - 288 T^{2} + 33150 T^{4} - 1797408 T^{6} + 38950081 T^{8} \)
$83$ \( ( 1 - 4 T + 102 T^{2} - 332 T^{3} + 6889 T^{4} )^{2} \)
$89$ \( 1 - 294 T^{2} + 36618 T^{4} - 2328774 T^{6} + 62742241 T^{8} \)
$97$ \( 1 - 204 T^{2} + 28950 T^{4} - 1919436 T^{6} + 88529281 T^{8} \)
show more
show less