Properties

Label 1110.2.i.j
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: \(\Q(\sqrt{-3}, \sqrt{73})\)
Defining polynomial: \(x^{4} - x^{3} + 19 x^{2} + 18 x + 324\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 19 x^{2} + 18 x + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} + 19 \nu^{2} - 19 \nu + 324 \)\()/342\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{3} + 37 \)\()/19\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + 18 \beta_{2} + \beta_{1} - 19\)
\(\nu^{3}\)\(=\)\(19 \beta_{3} - 37\)

Character values

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

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

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
−1.88600 + 3.26665i
2.38600 4.13267i
−1.88600 3.26665i
2.38600 + 4.13267i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −1.00000 0 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 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 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 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.j 4
37.c even 3 1 inner 1110.2.i.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.j 4 1.a even 1 1 trivial
1110.2.i.j 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7} \)
\( T_{11} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( ( 1 - T + T^{2} )^{2} \)
$5$ \( ( 1 + T + T^{2} )^{2} \)
$7$ \( T^{4} \)
$11$ \( ( -1 + T )^{4} \)
$13$ \( ( 1 + T + T^{2} )^{2} \)
$17$ \( 36 + 42 T + 55 T^{2} - 7 T^{3} + T^{4} \)
$19$ \( 324 - 18 T + 19 T^{2} + T^{3} + T^{4} \)
$23$ \( ( -16 - 3 T + T^{2} )^{2} \)
$29$ \( ( 2 - 9 T + T^{2} )^{2} \)
$31$ \( ( -72 - 2 T + T^{2} )^{2} \)
$37$ \( 1369 + T^{2} + T^{4} \)
$41$ \( 4096 + 384 T + 100 T^{2} - 6 T^{3} + T^{4} \)
$43$ \( T^{4} \)
$47$ \( ( -3 + T )^{4} \)
$53$ \( 2304 + 480 T + 148 T^{2} - 10 T^{3} + T^{4} \)
$59$ \( 3249 + 456 T + 121 T^{2} - 8 T^{3} + T^{4} \)
$61$ \( ( 100 - 10 T + T^{2} )^{2} \)
$67$ \( 24964 + 790 T + 183 T^{2} - 5 T^{3} + T^{4} \)
$71$ \( 2304 + 480 T + 148 T^{2} - 10 T^{3} + T^{4} \)
$73$ \( T^{4} \)
$79$ \( 4096 + 384 T + 100 T^{2} - 6 T^{3} + T^{4} \)
$83$ \( 5184 - 144 T + 76 T^{2} + 2 T^{3} + T^{4} \)
$89$ \( 12996 - 2622 T + 415 T^{2} - 23 T^{3} + T^{4} \)
$97$ \( ( 8 + 18 T + T^{2} )^{2} \)
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