Properties

Label 1110.2.i.i
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: \(\Q(\sqrt{-3}, \sqrt{145})\)
Defining polynomial: \(x^{4} - x^{3} + 37 x^{2} + 36 x + 1296\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
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 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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 37 x^{2} + 36 x + 1296\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 37 \nu^{2} - 37 \nu + 1296 \)\()/1332\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 73 \)\()/37\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 36 \beta_{2} + \beta_{1} - 37\)
\(\nu^{3}\)\(=\)\(37 \beta_{3} - 73\)

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-\beta_{2}\) \(1\)

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
−2.76040 4.78115i
3.26040 + 5.64718i
−2.76040 + 4.78115i
3.26040 5.64718i
−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
121.2 −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
211.2 −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.i 4
37.c even 3 1 inner 1110.2.i.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.i 4 1.a even 1 1 trivial
1110.2.i.i 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 + T + T^{2} )^{2} \)
$5$ \( ( 1 - T + T^{2} )^{2} \)
$7$ \( ( 1 + T + T^{2} )^{2} \)
$11$ \( ( 2 + T )^{4} \)
$13$ \( ( 1 - T + T^{2} )^{2} \)
$17$ \( ( 4 - 2 T + T^{2} )^{2} \)
$19$ \( 1296 + 36 T + 37 T^{2} - T^{3} + T^{4} \)
$23$ \( T^{4} \)
$29$ \( ( -36 + T + T^{2} )^{2} \)
$31$ \( ( -24 - 7 T + T^{2} )^{2} \)
$37$ \( 1369 + 111 T + 40 T^{2} + 3 T^{3} + T^{4} \)
$41$ \( T^{4} \)
$43$ \( ( -16 + 9 T + T^{2} )^{2} \)
$47$ \( ( -144 - 2 T + T^{2} )^{2} \)
$53$ \( T^{4} \)
$59$ \( 18496 - 816 T + 172 T^{2} + 6 T^{3} + T^{4} \)
$61$ \( ( 100 + 10 T + T^{2} )^{2} \)
$67$ \( 36 + 66 T + 127 T^{2} - 11 T^{3} + T^{4} \)
$71$ \( 36 - 78 T + 163 T^{2} - 13 T^{3} + T^{4} \)
$73$ \( ( -24 - 7 T + T^{2} )^{2} \)
$79$ \( 576 + 168 T + 73 T^{2} - 7 T^{3} + T^{4} \)
$83$ \( 5476 - 1554 T + 367 T^{2} - 21 T^{3} + T^{4} \)
$89$ \( 20736 + 288 T + 148 T^{2} - 2 T^{3} + T^{4} \)
$97$ \( ( -36 + T + T^{2} )^{2} \)
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