Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{12}^{2} ) q^{2} + \zeta_{12}^{2} q^{3} -\zeta_{12}^{2} q^{4} -\zeta_{12}^{2} q^{5} + q^{6} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} - q^{8} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{12}^{2} ) q^{2} + \zeta_{12}^{2} q^{3} -\zeta_{12}^{2} q^{4} -\zeta_{12}^{2} q^{5} + q^{6} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{7} - q^{8} + ( -1 + \zeta_{12}^{2} ) q^{9} - q^{10} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{11} + ( 1 - \zeta_{12}^{2} ) q^{12} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{13} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{14} + ( 1 - \zeta_{12}^{2} ) q^{15} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -6 + \zeta_{12} + 6 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{17} + \zeta_{12}^{2} q^{18} + ( -2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{19} + ( -1 + \zeta_{12}^{2} ) q^{20} + ( 1 - \zeta_{12} - \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{21} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{22} + ( 2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{2} q^{24} + ( -1 + \zeta_{12}^{2} ) q^{25} + ( -3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{26} - q^{27} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{28} + ( -6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{29} -\zeta_{12}^{2} q^{30} + ( 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{31} + \zeta_{12}^{2} q^{32} + ( 2 \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{33} + ( -\zeta_{12} + 6 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{34} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{35} + q^{36} + ( 6 \zeta_{12} + \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{37} + ( -4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{38} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{39} + \zeta_{12}^{2} q^{40} + ( \zeta_{12} - 9 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{41} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{42} + ( 2 + 12 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{43} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{44} + q^{45} + ( 2 + 4 \zeta_{12} - 2 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{46} + ( 1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{47} - q^{48} + ( 3 + 2 \zeta_{12} - 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{49} + \zeta_{12}^{2} q^{50} + ( -6 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{51} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{52} + ( 5 - \zeta_{12} - 5 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{53} + ( -1 + \zeta_{12}^{2} ) q^{54} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{55} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{56} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{57} + ( -3 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{58} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{59} - q^{60} + ( 5 \zeta_{12} + 3 \zeta_{12}^{2} + 5 \zeta_{12}^{3} ) q^{61} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{62} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{63} + q^{64} + ( -3 + 2 \zeta_{12} + 3 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{65} + ( 3 + 4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{66} + ( \zeta_{12} - 6 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{67} + ( 6 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{68} + ( 4 \zeta_{12} + 2 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{69} + ( -\zeta_{12} + \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{70} + ( -\zeta_{12} + 5 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{71} + ( 1 - \zeta_{12}^{2} ) q^{72} + ( 1 - 6 \zeta_{12}^{3} ) q^{74} - q^{75} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{76} + ( \zeta_{12} + 3 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{77} + ( 2 \zeta_{12} - 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{78} -4 \zeta_{12}^{2} q^{79} + q^{80} -\zeta_{12}^{2} q^{81} + ( -9 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{82} + ( -5 - 5 \zeta_{12} + 5 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{83} + ( -1 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{84} + ( 6 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{85} + ( 2 + 6 \zeta_{12} - 2 \zeta_{12}^{2} - 12 \zeta_{12}^{3} ) q^{86} + ( -3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{87} + ( -3 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{88} + ( 2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{89} + ( 1 - \zeta_{12}^{2} ) q^{90} + ( -9 + 5 \zeta_{12} + 9 \zeta_{12}^{2} - 10 \zeta_{12}^{3} ) q^{91} + ( -4 \zeta_{12} - 2 \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{92} + ( 2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{93} + ( 1 - 2 \zeta_{12} - \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{94} + ( -2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{95} + ( -1 + \zeta_{12}^{2} ) q^{96} + ( 2 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{97} + ( -2 \zeta_{12} - 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{98} + ( -3 - 2 \zeta_{12} + 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} + 2q^{3} - 2q^{4} - 2q^{5} + 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} - 4q^{10} + 12q^{11} + 2q^{12} - 6q^{13} - 4q^{14} + 2q^{15} - 2q^{16} - 12q^{17} + 2q^{18} - 2q^{20} + 2q^{21} + 6q^{22} + 8q^{23} - 2q^{24} - 2q^{25} - 12q^{26} - 4q^{27} - 2q^{28} - 2q^{30} + 2q^{32} + 6q^{33} + 12q^{34} - 2q^{35} + 4q^{36} + 2q^{37} + 6q^{39} + 2q^{40} - 18q^{41} - 2q^{42} + 8q^{43} - 6q^{44} + 4q^{45} + 4q^{46} + 4q^{47} - 4q^{48} + 6q^{49} + 2q^{50} - 24q^{51} - 6q^{52} + 10q^{53} - 2q^{54} - 6q^{55} + 2q^{56} + 6q^{59} - 4q^{60} + 6q^{61} + 4q^{63} + 4q^{64} - 6q^{65} + 12q^{66} - 12q^{67} + 24q^{68} + 4q^{69} + 2q^{70} + 10q^{71} + 2q^{72} + 4q^{74} - 4q^{75} + 6q^{77} - 6q^{78} - 8q^{79} + 4q^{80} - 2q^{81} - 36q^{82} - 10q^{83} - 4q^{84} + 24q^{85} + 4q^{86} - 12q^{88} + 2q^{90} - 18q^{91} - 4q^{92} + 2q^{94} - 2q^{96} + 8q^{97} - 6q^{98} - 6q^{99} + O(q^{100}) \)

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
121.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 −1.36603 2.36603i −1.00000 −0.500000 + 0.866025i −1.00000
121.2 0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 0.366025 + 0.633975i −1.00000 −0.500000 + 0.866025i −1.00000
211.1 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 −1.36603 + 2.36603i −1.00000 −0.500000 0.866025i −1.00000
211.2 0.500000 + 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0.366025 0.633975i −1.00000 −0.500000 0.866025i −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.i.n 4
37.c even 3 1 inner 1110.2.i.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.n 4 1.a even 1 1 trivial
1110.2.i.n 4 37.c even 3 1 inner

Hecke kernels

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

\( T_{7}^{4} + 2 T_{7}^{3} + 6 T_{7}^{2} - 4 T_{7} + 4 \)
\( T_{11}^{2} - 6 T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( 4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( -3 - 6 T + T^{2} )^{2} \)
$13$ \( 9 - 18 T + 39 T^{2} + 6 T^{3} + T^{4} \)
$17$ \( 1089 + 396 T + 111 T^{2} + 12 T^{3} + T^{4} \)
$19$ \( 144 + 12 T^{2} + T^{4} \)
$23$ \( ( -44 - 4 T + T^{2} )^{2} \)
$29$ \( ( -27 + T^{2} )^{2} \)
$31$ \( ( -12 + T^{2} )^{2} \)
$37$ \( 1369 - 74 T - 33 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( 6084 + 1404 T + 246 T^{2} + 18 T^{3} + T^{4} \)
$43$ \( ( -104 - 4 T + T^{2} )^{2} \)
$47$ \( ( -11 - 2 T + T^{2} )^{2} \)
$53$ \( 484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4} \)
$61$ \( 4356 + 396 T + 102 T^{2} - 6 T^{3} + T^{4} \)
$67$ \( 1089 + 396 T + 111 T^{2} + 12 T^{3} + T^{4} \)
$71$ \( 484 - 220 T + 78 T^{2} - 10 T^{3} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( ( 16 + 4 T + T^{2} )^{2} \)
$83$ \( 2500 - 500 T + 150 T^{2} + 10 T^{3} + T^{4} \)
$89$ \( 144 + 12 T^{2} + T^{4} \)
$97$ \( ( -44 - 4 T + T^{2} )^{2} \)
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