Properties

Label 1110.2.bb.a
Level $1110$
Weight $2$
Character orbit 1110.bb
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.bb (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1009.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i −0.866025 + 0.500000i 0.500000 + 0.866025i −0.133975 2.23205i 1.00000 3.46410 2.00000i 1.00000i 0.500000 0.866025i −1.00000 + 2.00000i
1009.2 0.866025 + 0.500000i 0.866025 0.500000i 0.500000 + 0.866025i −1.86603 1.23205i 1.00000 −3.46410 + 2.00000i 1.00000i 0.500000 0.866025i −1.00000 2.00000i
1099.1 −0.866025 + 0.500000i −0.866025 0.500000i 0.500000 0.866025i −0.133975 + 2.23205i 1.00000 3.46410 + 2.00000i 1.00000i 0.500000 + 0.866025i −1.00000 2.00000i
1099.2 0.866025 0.500000i 0.866025 + 0.500000i 0.500000 0.866025i −1.86603 + 1.23205i 1.00000 −3.46410 2.00000i 1.00000i 0.500000 + 0.866025i −1.00000 + 2.00000i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
37.c even 3 1 inner
185.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.bb.a 4
5.b even 2 1 inner 1110.2.bb.a 4
37.c even 3 1 inner 1110.2.bb.a 4
185.n even 6 1 inner 1110.2.bb.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.bb.a 4 1.a even 1 1 trivial
1110.2.bb.a 4 5.b even 2 1 inner
1110.2.bb.a 4 37.c even 3 1 inner
1110.2.bb.a 4 185.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{4} - 16 T_{7}^{2} + 256 \)
\( T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( 25 + 20 T + 11 T^{2} + 4 T^{3} + T^{4} \)
$7$ \( 256 - 16 T^{2} + T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( 81 - 9 T^{2} + T^{4} \)
$17$ \( 1296 - 36 T^{2} + T^{4} \)
$19$ \( ( 25 - 5 T + T^{2} )^{2} \)
$23$ \( ( 49 + T^{2} )^{2} \)
$29$ \( ( 10 + T )^{4} \)
$31$ \( T^{4} \)
$37$ \( 1369 - 73 T^{2} + T^{4} \)
$41$ \( ( 4 - 2 T + T^{2} )^{2} \)
$43$ \( ( 36 + T^{2} )^{2} \)
$47$ \( ( 9 + T^{2} )^{2} \)
$53$ \( 16 - 4 T^{2} + T^{4} \)
$59$ \( ( 121 - 11 T + T^{2} )^{2} \)
$61$ \( ( 144 - 12 T + T^{2} )^{2} \)
$67$ \( 10000 - 100 T^{2} + T^{4} \)
$71$ \( ( 36 + 6 T + T^{2} )^{2} \)
$73$ \( ( 196 + T^{2} )^{2} \)
$79$ \( ( 64 + 8 T + T^{2} )^{2} \)
$83$ \( 10000 - 100 T^{2} + T^{4} \)
$89$ \( ( 25 - 5 T + T^{2} )^{2} \)
$97$ \( ( 324 + T^{2} )^{2} \)
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