Properties

Label 1110.2.i.o
Level $1110$
Weight $2$
Character orbit 1110.i
Analytic conductor $8.863$
Analytic rank $0$
Dimension $6$
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: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.45911232.1
Defining polynomial: \(x^{6} + 11 x^{4} - 4 x^{3} + 121 x^{2} - 22 x + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 11 x^{4} - 4 x^{3} + 121 x^{2} - 22 x + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{3} - 11 \nu + 2 \)\()/11\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + \nu^{3} + 11 \nu^{2} - 2 \nu + 86 \)\()/22\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{5} + 11 \nu^{3} - 2 \nu^{2} + 121 \nu \)\()/22\)
\(\beta_{5}\)\(=\)\((\)\( -4 \nu^{5} - \nu^{4} - 44 \nu^{3} + 8 \nu^{2} - 471 \nu \)\()/22\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(2 \beta_{5} + 8 \beta_{4} + 2 \beta_{3} + \beta_{2} - 8\)
\(\nu^{3}\)\(=\)\(-11 \beta_{2} - 11 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(-22 \beta_{5} - 88 \beta_{4} + 13 \beta_{1}\)
\(\nu^{5}\)\(=\)\(4 \beta_{5} + 38 \beta_{4} + 4 \beta_{3} + 123 \beta_{2} - 38\)

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1 + \beta_{4}\) \(1\)

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
0.0911848 0.157937i
1.61084 2.79005i
−1.70202 + 2.94799i
0.0911848 + 0.157937i
1.61084 + 2.79005i
−1.70202 2.94799i
−0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −1.00000 −2.03728 3.52867i 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 −0.210618 0.364801i 1.00000 −0.500000 + 0.866025i −1.00000
121.3 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.500000 0.866025i 0.500000 + 0.866025i −1.00000 1.74790 + 3.02744i 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 −2.03728 + 3.52867i 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 −0.210618 + 0.364801i 1.00000 −0.500000 0.866025i −1.00000
211.3 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0.500000 0.866025i −1.00000 1.74790 3.02744i 1.00000 −0.500000 0.866025i −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 211.3
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.o 6
37.c even 3 1 inner 1110.2.i.o 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.i.o 6 1.a even 1 1 trivial
1110.2.i.o 6 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{6} + T_{7}^{5} + 15 T_{7}^{4} - 2 T_{7}^{3} + 202 T_{7}^{2} + 84 T_{7} + 36 \)
\( T_{11}^{3} - 11 T_{11} - 2 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{3} \)
$3$ \( ( 1 - T + T^{2} )^{3} \)
$5$ \( ( 1 - T + T^{2} )^{3} \)
$7$ \( 36 + 84 T + 202 T^{2} - 2 T^{3} + 15 T^{4} + T^{5} + T^{6} \)
$11$ \( ( -2 - 11 T + T^{3} )^{2} \)
$13$ \( ( 1 + T + T^{2} )^{3} \)
$17$ \( 324 - 126 T + 193 T^{2} + 92 T^{3} + 57 T^{4} + 8 T^{5} + T^{6} \)
$19$ \( 144 - 96 T + 100 T^{2} + 17 T^{4} - 3 T^{5} + T^{6} \)
$23$ \( ( -8 - 44 T - 2 T^{2} + T^{3} )^{2} \)
$29$ \( ( 349 - 55 T - 7 T^{2} + T^{3} )^{2} \)
$31$ \( ( 96 + 4 T - 12 T^{2} + T^{3} )^{2} \)
$37$ \( 50653 + 17797 T + 4070 T^{2} + 805 T^{3} + 110 T^{4} + 13 T^{5} + T^{6} \)
$41$ \( 21904 + 10360 T + 4604 T^{2} + 436 T^{3} + 74 T^{4} - 2 T^{5} + T^{6} \)
$43$ \( ( 48 - 56 T - 2 T^{2} + T^{3} )^{2} \)
$47$ \( ( 216 - 71 T - 2 T^{2} + T^{3} )^{2} \)
$53$ \( 4624 - 2584 T + 1308 T^{2} - 212 T^{3} + 42 T^{4} + 2 T^{5} + T^{6} \)
$59$ \( 26244 - 14742 T + 7633 T^{2} - 688 T^{3} + 107 T^{4} + 4 T^{5} + T^{6} \)
$61$ \( 64 + 48 T + 68 T^{2} - 8 T^{3} + 22 T^{4} + 4 T^{5} + T^{6} \)
$67$ \( 5476 - 4810 T + 3633 T^{2} - 668 T^{3} + 129 T^{4} + 8 T^{5} + T^{6} \)
$71$ \( 161604 - 34572 T + 8602 T^{2} - 546 T^{3} + 95 T^{4} - 3 T^{5} + T^{6} \)
$73$ \( ( -96 - 32 T + 6 T^{2} + T^{3} )^{2} \)
$79$ \( 2262016 - 96256 T + 43200 T^{2} + 4672 T^{3} + 612 T^{4} + 26 T^{5} + T^{6} \)
$83$ \( 79524 + 41172 T + 14830 T^{2} + 2794 T^{3} + 383 T^{4} + 23 T^{5} + T^{6} \)
$89$ \( 9216 - 7296 T + 7696 T^{2} + 1712 T^{3} + 324 T^{4} + 20 T^{5} + T^{6} \)
$97$ \( ( -2 + T )^{6} \)
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