Properties

Label 1110.2.i.d
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.

\(f(q)\) \(=\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} + \zeta_{6} q^{7} - q^{8} + ( -1 + \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 1 - \zeta_{6} ) q^{2} -\zeta_{6} q^{3} -\zeta_{6} q^{4} + \zeta_{6} q^{5} - q^{6} + \zeta_{6} q^{7} - q^{8} + ( -1 + \zeta_{6} ) q^{9} + q^{10} + 6 q^{11} + ( -1 + \zeta_{6} ) q^{12} + \zeta_{6} q^{13} + q^{14} + ( 1 - \zeta_{6} ) q^{15} + ( -1 + \zeta_{6} ) q^{16} + ( -6 + 6 \zeta_{6} ) q^{17} + \zeta_{6} q^{18} + 7 \zeta_{6} q^{19} + ( 1 - \zeta_{6} ) q^{20} + ( 1 - \zeta_{6} ) q^{21} + ( 6 - 6 \zeta_{6} ) q^{22} -6 q^{23} + \zeta_{6} q^{24} + ( -1 + \zeta_{6} ) q^{25} + q^{26} + q^{27} + ( 1 - \zeta_{6} ) q^{28} + 9 q^{29} -\zeta_{6} q^{30} -4 q^{31} + \zeta_{6} q^{32} -6 \zeta_{6} q^{33} + 6 \zeta_{6} q^{34} + ( -1 + \zeta_{6} ) q^{35} + q^{36} + ( 7 - 3 \zeta_{6} ) q^{37} + 7 q^{38} + ( 1 - \zeta_{6} ) q^{39} -\zeta_{6} q^{40} -6 \zeta_{6} q^{41} -\zeta_{6} q^{42} + 2 q^{43} -6 \zeta_{6} q^{44} - q^{45} + ( -6 + 6 \zeta_{6} ) q^{46} + 12 q^{47} + q^{48} + ( 6 - 6 \zeta_{6} ) q^{49} + \zeta_{6} q^{50} + 6 q^{51} + ( 1 - \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + ( 1 - \zeta_{6} ) q^{54} + 6 \zeta_{6} q^{55} -\zeta_{6} q^{56} + ( 7 - 7 \zeta_{6} ) q^{57} + ( 9 - 9 \zeta_{6} ) q^{58} + ( 6 - 6 \zeta_{6} ) q^{59} - q^{60} -8 \zeta_{6} q^{61} + ( -4 + 4 \zeta_{6} ) q^{62} - q^{63} + q^{64} + ( -1 + \zeta_{6} ) q^{65} -6 q^{66} + 10 \zeta_{6} q^{67} + 6 q^{68} + 6 \zeta_{6} q^{69} + \zeta_{6} q^{70} + 3 \zeta_{6} q^{71} + ( 1 - \zeta_{6} ) q^{72} -10 q^{73} + ( 4 - 7 \zeta_{6} ) q^{74} + q^{75} + ( 7 - 7 \zeta_{6} ) q^{76} + 6 \zeta_{6} q^{77} -\zeta_{6} q^{78} -14 \zeta_{6} q^{79} - q^{80} -\zeta_{6} q^{81} -6 q^{82} + ( 3 - 3 \zeta_{6} ) q^{83} - q^{84} -6 q^{85} + ( 2 - 2 \zeta_{6} ) q^{86} -9 \zeta_{6} q^{87} -6 q^{88} + ( -1 + \zeta_{6} ) q^{90} + ( -1 + \zeta_{6} ) q^{91} + 6 \zeta_{6} q^{92} + 4 \zeta_{6} q^{93} + ( 12 - 12 \zeta_{6} ) q^{94} + ( -7 + 7 \zeta_{6} ) q^{95} + ( 1 - \zeta_{6} ) q^{96} -10 q^{97} -6 \zeta_{6} q^{98} + ( -6 + 6 \zeta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} - q^{3} - q^{4} + q^{5} - 2q^{6} + q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} - q^{3} - q^{4} + q^{5} - 2q^{6} + q^{7} - 2q^{8} - q^{9} + 2q^{10} + 12q^{11} - q^{12} + q^{13} + 2q^{14} + q^{15} - q^{16} - 6q^{17} + q^{18} + 7q^{19} + q^{20} + q^{21} + 6q^{22} - 12q^{23} + q^{24} - q^{25} + 2q^{26} + 2q^{27} + q^{28} + 18q^{29} - q^{30} - 8q^{31} + q^{32} - 6q^{33} + 6q^{34} - q^{35} + 2q^{36} + 11q^{37} + 14q^{38} + q^{39} - q^{40} - 6q^{41} - q^{42} + 4q^{43} - 6q^{44} - 2q^{45} - 6q^{46} + 24q^{47} + 2q^{48} + 6q^{49} + q^{50} + 12q^{51} + q^{52} - 6q^{53} + q^{54} + 6q^{55} - q^{56} + 7q^{57} + 9q^{58} + 6q^{59} - 2q^{60} - 8q^{61} - 4q^{62} - 2q^{63} + 2q^{64} - q^{65} - 12q^{66} + 10q^{67} + 12q^{68} + 6q^{69} + q^{70} + 3q^{71} + q^{72} - 20q^{73} + q^{74} + 2q^{75} + 7q^{76} + 6q^{77} - q^{78} - 14q^{79} - 2q^{80} - q^{81} - 12q^{82} + 3q^{83} - 2q^{84} - 12q^{85} + 2q^{86} - 9q^{87} - 12q^{88} - q^{90} - q^{91} + 6q^{92} + 4q^{93} + 12q^{94} - 7q^{95} + q^{96} - 20q^{97} - 6q^{98} - 6q^{99} + O(q^{100}) \)

Character values

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

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

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

\( T_{7}^{2} - T_{7} + 1 \)
\( T_{11} - 6 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 + T + T^{2} \)
$5$ \( 1 - T + T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 1 - T + T^{2} \)
$17$ \( 36 + 6 T + T^{2} \)
$19$ \( 49 - 7 T + T^{2} \)
$23$ \( ( 6 + T )^{2} \)
$29$ \( ( -9 + T )^{2} \)
$31$ \( ( 4 + T )^{2} \)
$37$ \( 37 - 11 T + T^{2} \)
$41$ \( 36 + 6 T + T^{2} \)
$43$ \( ( -2 + T )^{2} \)
$47$ \( ( -12 + T )^{2} \)
$53$ \( 36 + 6 T + T^{2} \)
$59$ \( 36 - 6 T + T^{2} \)
$61$ \( 64 + 8 T + T^{2} \)
$67$ \( 100 - 10 T + T^{2} \)
$71$ \( 9 - 3 T + T^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( 196 + 14 T + T^{2} \)
$83$ \( 9 - 3 T + T^{2} \)
$89$ \( T^{2} \)
$97$ \( ( 10 + T )^{2} \)
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