Properties

Label 1110.2.i.e
Level $1110$
Weight $2$
Character orbit 1110.i
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\) \(1\)

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
0.500000 + 0.866025i
0.500000 0.866025i
0.500000 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i 1.00000 1.50000 + 2.59808i −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 1.50000 2.59808i −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.e 2
37.c even 3 1 inner 1110.2.i.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.e 2 1.a even 1 1 trivial
1110.2.i.e 2 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} - 3 T_{7} + 9 \)
\( T_{11} + 2 \)

Hecke characteristic polynomials

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