Properties

Label 252.2.bi.a
Level 252
Weight 2
Character orbit 252.bi
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.bi (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: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

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

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
139.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.366025 + 1.36603i −1.73205 −1.73205 1.00000i −1.50000 0.866025i 0.633975 2.36603i 2.59808 0.500000i 2.00000 2.00000i 3.00000 1.73205 1.73205i
139.2 1.36603 + 0.366025i 1.73205 1.73205 + 1.00000i −1.50000 0.866025i 2.36603 + 0.633975i −2.59808 + 0.500000i 2.00000 + 2.00000i 3.00000 −1.73205 1.73205i
223.1 −0.366025 1.36603i −1.73205 −1.73205 + 1.00000i −1.50000 + 0.866025i 0.633975 + 2.36603i 2.59808 + 0.500000i 2.00000 + 2.00000i 3.00000 1.73205 + 1.73205i
223.2 1.36603 0.366025i 1.73205 1.73205 1.00000i −1.50000 + 0.866025i 2.36603 0.633975i −2.59808 0.500000i 2.00000 2.00000i 3.00000 −1.73205 + 1.73205i
\(n\): 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.l odd 6 1 inner
252.bi 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.bi.a 4
3.b odd 2 1 756.2.bi.b 4
4.b odd 2 1 inner 252.2.bi.a 4
7.b odd 2 1 252.2.bi.b yes 4
9.c even 3 1 252.2.bi.b yes 4
9.d odd 6 1 756.2.bi.a 4
12.b even 2 1 756.2.bi.b 4
21.c even 2 1 756.2.bi.a 4
28.d even 2 1 252.2.bi.b yes 4
36.f odd 6 1 252.2.bi.b yes 4
36.h even 6 1 756.2.bi.a 4
63.l odd 6 1 inner 252.2.bi.a 4
63.o even 6 1 756.2.bi.b 4
84.h odd 2 1 756.2.bi.a 4
252.s odd 6 1 756.2.bi.b 4
252.bi even 6 1 inner 252.2.bi.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.bi.a 4 1.a even 1 1 trivial
252.2.bi.a 4 4.b odd 2 1 inner
252.2.bi.a 4 63.l odd 6 1 inner
252.2.bi.a 4 252.bi even 6 1 inner
252.2.bi.b yes 4 7.b odd 2 1
252.2.bi.b yes 4 9.c even 3 1
252.2.bi.b yes 4 28.d even 2 1
252.2.bi.b yes 4 36.f odd 6 1
756.2.bi.a 4 9.d odd 6 1
756.2.bi.a 4 21.c even 2 1
756.2.bi.a 4 36.h even 6 1
756.2.bi.a 4 84.h odd 2 1
756.2.bi.b 4 3.b odd 2 1
756.2.bi.b 4 12.b even 2 1
756.2.bi.b 4 63.o even 6 1
756.2.bi.b 4 252.s odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 3 T_{5} + 3 \) acting on \(S_{2}^{\mathrm{new}}(252, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 2 T^{2} - 4 T^{3} + 4 T^{4} \)
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ \( ( 1 + 3 T + 8 T^{2} + 15 T^{3} + 25 T^{4} )^{2} \)
$7$ \( 1 - 13 T^{2} + 49 T^{4} \)
$11$ \( 1 + 21 T^{2} + 320 T^{4} + 2541 T^{6} + 14641 T^{8} \)
$13$ \( ( 1 - 5 T + 13 T^{2} )^{2}( 1 + 2 T + 13 T^{2} )^{2} \)
$17$ \( ( 1 - 22 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 10 T^{2} + 361 T^{4} )^{2} \)
$23$ \( 1 + 21 T^{2} - 88 T^{4} + 11109 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 5 T - 4 T^{2} + 145 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 46 T^{2} + 961 T^{4} )( 1 + 59 T^{2} + 961 T^{4} ) \)
$37$ \( ( 1 + 37 T^{2} )^{4} \)
$41$ \( ( 1 - 15 T + 116 T^{2} - 615 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 + 85 T^{2} + 5376 T^{4} + 157165 T^{6} + 3418801 T^{8} \)
$47$ \( 1 - 67 T^{2} + 2280 T^{4} - 148003 T^{6} + 4879681 T^{8} \)
$53$ \( ( 1 - 4 T + 53 T^{2} )^{4} \)
$59$ \( 1 - 43 T^{2} - 1632 T^{4} - 149683 T^{6} + 12117361 T^{8} \)
$61$ \( ( 1 + 9 T + 88 T^{2} + 549 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 - 91 T^{2} + 3792 T^{4} - 408499 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 126 T^{2} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 - 38 T^{2} + 5329 T^{4} )^{2} \)
$79$ \( 1 + 149 T^{2} + 15960 T^{4} + 929909 T^{6} + 38950081 T^{8} \)
$83$ \( 1 - 163 T^{2} + 19680 T^{4} - 1122907 T^{6} + 47458321 T^{8} \)
$89$ \( ( 1 + 122 T^{2} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 5 T + 97 T^{2} )^{2}( 1 + 14 T + 97 T^{2} )^{2} \)
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