Properties

Label 1110.2.i.f
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + q^{6} -2 \zeta_{6} q^{7} - q^{8} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{5} + q^{6} -2 \zeta_{6} q^{7} - q^{8} + ( -1 + \zeta_{6} ) q^{9} + q^{10} -5 q^{11} + ( 1 - \zeta_{6} ) q^{12} -5 \zeta_{6} q^{13} -2 q^{14} + ( -1 + \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} + ( 5 - 5 \zeta_{6} ) q^{17} + \zeta_{6} q^{18} -2 \zeta_{6} q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( 2 - 2 \zeta_{6} ) q^{21} + ( -5 + 5 \zeta_{6} ) q^{22} -\zeta_{6} q^{24} + ( -1 + \zeta_{6} ) q^{25} -5 q^{26} - q^{27} + ( -2 + 2 \zeta_{6} ) q^{28} -9 q^{29} + \zeta_{6} q^{30} + 4 q^{31} + \zeta_{6} q^{32} -5 \zeta_{6} q^{33} -5 \zeta_{6} q^{34} + ( 2 - 2 \zeta_{6} ) q^{35} + q^{36} + ( 4 - 7 \zeta_{6} ) q^{37} -2 q^{38} + ( 5 - 5 \zeta_{6} ) q^{39} -\zeta_{6} q^{40} -6 \zeta_{6} q^{41} -2 \zeta_{6} q^{42} + 4 q^{43} + 5 \zeta_{6} q^{44} - q^{45} + 9 q^{47} - q^{48} + ( 3 - 3 \zeta_{6} ) q^{49} + \zeta_{6} q^{50} + 5 q^{51} + ( -5 + 5 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + ( -1 + \zeta_{6} ) q^{54} -5 \zeta_{6} q^{55} + 2 \zeta_{6} q^{56} + ( 2 - 2 \zeta_{6} ) q^{57} + ( -9 + 9 \zeta_{6} ) q^{58} + ( -3 + 3 \zeta_{6} ) q^{59} + q^{60} + ( 4 - 4 \zeta_{6} ) q^{62} + 2 q^{63} + q^{64} + ( 5 - 5 \zeta_{6} ) q^{65} -5 q^{66} + 9 \zeta_{6} q^{67} -5 q^{68} -2 \zeta_{6} q^{70} + 12 \zeta_{6} q^{71} + ( 1 - \zeta_{6} ) q^{72} -12 q^{73} + ( -3 - 4 \zeta_{6} ) q^{74} - q^{75} + ( -2 + 2 \zeta_{6} ) q^{76} + 10 \zeta_{6} q^{77} -5 \zeta_{6} q^{78} -16 \zeta_{6} q^{79} - q^{80} -\zeta_{6} q^{81} -6 q^{82} + ( 2 - 2 \zeta_{6} ) q^{83} -2 q^{84} + 5 q^{85} + ( 4 - 4 \zeta_{6} ) q^{86} -9 \zeta_{6} q^{87} + 5 q^{88} + ( 14 - 14 \zeta_{6} ) q^{89} + ( -1 + \zeta_{6} ) q^{90} + ( -10 + 10 \zeta_{6} ) q^{91} + 4 \zeta_{6} q^{93} + ( 9 - 9 \zeta_{6} ) q^{94} + ( 2 - 2 \zeta_{6} ) q^{95} + ( -1 + \zeta_{6} ) q^{96} -18 q^{97} -3 \zeta_{6} q^{98} + ( 5 - 5 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + q^{3} - q^{4} + q^{5} + 2q^{6} - 2q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} + q^{3} - q^{4} + q^{5} + 2q^{6} - 2q^{7} - 2q^{8} - q^{9} + 2q^{10} - 10q^{11} + q^{12} - 5q^{13} - 4q^{14} - q^{15} - q^{16} + 5q^{17} + q^{18} - 2q^{19} + q^{20} + 2q^{21} - 5q^{22} - q^{24} - q^{25} - 10q^{26} - 2q^{27} - 2q^{28} - 18q^{29} + q^{30} + 8q^{31} + q^{32} - 5q^{33} - 5q^{34} + 2q^{35} + 2q^{36} + q^{37} - 4q^{38} + 5q^{39} - q^{40} - 6q^{41} - 2q^{42} + 8q^{43} + 5q^{44} - 2q^{45} + 18q^{47} - 2q^{48} + 3q^{49} + q^{50} + 10q^{51} - 5q^{52} - 6q^{53} - q^{54} - 5q^{55} + 2q^{56} + 2q^{57} - 9q^{58} - 3q^{59} + 2q^{60} + 4q^{62} + 4q^{63} + 2q^{64} + 5q^{65} - 10q^{66} + 9q^{67} - 10q^{68} - 2q^{70} + 12q^{71} + q^{72} - 24q^{73} - 10q^{74} - 2q^{75} - 2q^{76} + 10q^{77} - 5q^{78} - 16q^{79} - 2q^{80} - q^{81} - 12q^{82} + 2q^{83} - 4q^{84} + 10q^{85} + 4q^{86} - 9q^{87} + 10q^{88} + 14q^{89} - q^{90} - 10q^{91} + 4q^{93} + 9q^{94} + 2q^{95} - q^{96} - 36q^{97} - 3q^{98} + 5q^{99} + O(q^{100}) \)

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 −1.00000 1.73205i −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 −1.00000 + 1.73205i −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.f 2
37.c even 3 1 inner 1110.2.i.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.f 2 1.a even 1 1 trivial
1110.2.i.f 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} + 2 T_{7} + 4 \)
\( T_{11} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 4 + 2 T + T^{2} \)
$11$ \( ( 5 + T )^{2} \)
$13$ \( 25 + 5 T + T^{2} \)
$17$ \( 25 - 5 T + T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( ( 9 + T )^{2} \)
$31$ \( ( -4 + T )^{2} \)
$37$ \( 37 - T + T^{2} \)
$41$ \( 36 + 6 T + T^{2} \)
$43$ \( ( -4 + T )^{2} \)
$47$ \( ( -9 + T )^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 9 + 3 T + T^{2} \)
$61$ \( T^{2} \)
$67$ \( 81 - 9 T + T^{2} \)
$71$ \( 144 - 12 T + T^{2} \)
$73$ \( ( 12 + T )^{2} \)
$79$ \( 256 + 16 T + T^{2} \)
$83$ \( 4 - 2 T + T^{2} \)
$89$ \( 196 - 14 T + T^{2} \)
$97$ \( ( 18 + T )^{2} \)
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