Properties

Label 1110.2.i.l
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: \(\Q(\sqrt{-3}, \sqrt{-5})\)
Defining polynomial: \(x^{4} - 5 x^{2} + 25\)
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 basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{3} -\beta_{1} q^{4} + \beta_{1} q^{5} - q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} - q^{8} + ( -1 + \beta_{1} ) q^{9} +O(q^{10})\) \( q + ( 1 - \beta_{1} ) q^{2} -\beta_{1} q^{3} -\beta_{1} q^{4} + \beta_{1} q^{5} - q^{6} + ( -\beta_{1} + \beta_{2} ) q^{7} - q^{8} + ( -1 + \beta_{1} ) q^{9} + q^{10} -5 q^{11} + ( -1 + \beta_{1} ) q^{12} + \beta_{1} q^{13} + ( -1 + \beta_{3} ) q^{14} + ( 1 - \beta_{1} ) q^{15} + ( -1 + \beta_{1} ) q^{16} + ( -4 + 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{17} + \beta_{1} q^{18} -2 \beta_{2} q^{19} + ( 1 - \beta_{1} ) q^{20} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{21} + ( -5 + 5 \beta_{1} ) q^{22} -6 q^{23} + \beta_{1} q^{24} + ( -1 + \beta_{1} ) q^{25} + q^{26} + q^{27} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{28} -\beta_{3} q^{29} -\beta_{1} q^{30} -2 \beta_{3} q^{31} + \beta_{1} q^{32} + 5 \beta_{1} q^{33} + ( 4 \beta_{1} + \beta_{2} ) q^{34} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{35} + q^{36} + ( -4 - 3 \beta_{1} ) q^{37} -2 \beta_{3} q^{38} + ( 1 - \beta_{1} ) q^{39} -\beta_{1} q^{40} + ( 3 \beta_{1} + \beta_{2} ) q^{41} + ( \beta_{1} - \beta_{2} ) q^{42} + ( -2 + 2 \beta_{3} ) q^{43} + 5 \beta_{1} q^{44} - q^{45} + ( -6 + 6 \beta_{1} ) q^{46} + ( -3 + 2 \beta_{3} ) q^{47} + q^{48} + ( -9 + 9 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{49} + \beta_{1} q^{50} + ( 4 + \beta_{3} ) q^{51} + ( 1 - \beta_{1} ) q^{52} + ( 3 - 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{53} + ( 1 - \beta_{1} ) q^{54} -5 \beta_{1} q^{55} + ( \beta_{1} - \beta_{2} ) q^{56} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{57} + ( \beta_{2} - \beta_{3} ) q^{58} + ( -5 + 5 \beta_{1} ) q^{59} - q^{60} + ( 5 \beta_{1} - \beta_{2} ) q^{61} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{62} + ( 1 - \beta_{3} ) q^{63} + q^{64} + ( -1 + \beta_{1} ) q^{65} + 5 q^{66} + ( -10 \beta_{1} - \beta_{2} ) q^{67} + ( 4 + \beta_{3} ) q^{68} + 6 \beta_{1} q^{69} + ( -\beta_{1} + \beta_{2} ) q^{70} + ( -3 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 1 - \beta_{1} ) q^{72} + 12 q^{73} + ( -7 + 4 \beta_{1} ) q^{74} + q^{75} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 5 \beta_{1} - 5 \beta_{2} ) q^{77} -\beta_{1} q^{78} + 4 \beta_{2} q^{79} - q^{80} -\beta_{1} q^{81} + ( 3 + \beta_{3} ) q^{82} + ( 1 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{83} + ( 1 - \beta_{3} ) q^{84} + ( -4 - \beta_{3} ) q^{85} + ( -2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{86} + \beta_{2} q^{87} + 5 q^{88} + ( -4 + 4 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{89} + ( -1 + \beta_{1} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + 6 \beta_{1} q^{92} + 2 \beta_{2} q^{93} + ( -3 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{94} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{95} + ( 1 - \beta_{1} ) q^{96} -10 q^{97} + ( 9 \beta_{1} - 2 \beta_{2} ) q^{98} + ( 5 - 5 \beta_{1} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{5} - 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + O(q^{10}) \) \( 4q + 2q^{2} - 2q^{3} - 2q^{4} + 2q^{5} - 4q^{6} - 2q^{7} - 4q^{8} - 2q^{9} + 4q^{10} - 20q^{11} - 2q^{12} + 2q^{13} - 4q^{14} + 2q^{15} - 2q^{16} - 8q^{17} + 2q^{18} + 2q^{20} - 2q^{21} - 10q^{22} - 24q^{23} + 2q^{24} - 2q^{25} + 4q^{26} + 4q^{27} - 2q^{28} - 2q^{30} + 2q^{32} + 10q^{33} + 8q^{34} + 2q^{35} + 4q^{36} - 22q^{37} + 2q^{39} - 2q^{40} + 6q^{41} + 2q^{42} - 8q^{43} + 10q^{44} - 4q^{45} - 12q^{46} - 12q^{47} + 4q^{48} - 18q^{49} + 2q^{50} + 16q^{51} + 2q^{52} + 6q^{53} + 2q^{54} - 10q^{55} + 2q^{56} - 10q^{59} - 4q^{60} + 10q^{61} + 4q^{63} + 4q^{64} - 2q^{65} + 20q^{66} - 20q^{67} + 16q^{68} + 12q^{69} - 2q^{70} - 6q^{71} + 2q^{72} + 48q^{73} - 20q^{74} + 4q^{75} + 10q^{77} - 2q^{78} - 4q^{80} - 2q^{81} + 12q^{82} + 2q^{83} + 4q^{84} - 16q^{85} - 4q^{86} + 20q^{88} - 8q^{89} - 2q^{90} + 2q^{91} + 12q^{92} - 6q^{94} + 2q^{96} - 40q^{97} + 18q^{98} + 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 5 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 5 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 10 \nu \)\()/5\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(5 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-5 \beta_{3} + 10 \beta_{2}\)\()/3\)

Character values

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

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

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
−1.93649 1.11803i
1.93649 + 1.11803i
−1.93649 + 1.11803i
1.93649 1.11803i
0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −1.00000 −2.43649 4.22013i −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 1.43649 + 2.48808i −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 −2.43649 + 4.22013i −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 1.43649 2.48808i −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.l 4
37.c even 3 1 inner 1110.2.i.l 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.l 4 1.a even 1 1 trivial
1110.2.i.l 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{4} + 2 T_{7}^{3} + 18 T_{7}^{2} - 28 T_{7} + 196 \)
\( T_{11} + 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( 196 - 28 T + 18 T^{2} + 2 T^{3} + T^{4} \)
$11$ \( ( 5 + T )^{4} \)
$13$ \( ( 1 - T + T^{2} )^{2} \)
$17$ \( 1 + 8 T + 63 T^{2} + 8 T^{3} + T^{4} \)
$19$ \( 3600 + 60 T^{2} + T^{4} \)
$23$ \( ( 6 + T )^{4} \)
$29$ \( ( -15 + T^{2} )^{2} \)
$31$ \( ( -60 + T^{2} )^{2} \)
$37$ \( ( 37 + 11 T + T^{2} )^{2} \)
$41$ \( 36 + 36 T + 42 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( ( -56 + 4 T + T^{2} )^{2} \)
$47$ \( ( -51 + 6 T + T^{2} )^{2} \)
$53$ \( 36 + 36 T + 42 T^{2} - 6 T^{3} + T^{4} \)
$59$ \( ( 25 + 5 T + T^{2} )^{2} \)
$61$ \( 100 - 100 T + 90 T^{2} - 10 T^{3} + T^{4} \)
$67$ \( 7225 + 1700 T + 315 T^{2} + 20 T^{3} + T^{4} \)
$71$ \( 15876 - 756 T + 162 T^{2} + 6 T^{3} + T^{4} \)
$73$ \( ( -12 + T )^{4} \)
$79$ \( 57600 + 240 T^{2} + T^{4} \)
$83$ \( 196 + 28 T + 18 T^{2} - 2 T^{3} + T^{4} \)
$89$ \( 1936 - 352 T + 108 T^{2} + 8 T^{3} + T^{4} \)
$97$ \( ( 10 + T )^{4} \)
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