Properties

Label 1110.2.d.i
Level $1110$
Weight $2$
Character orbit 1110.d
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.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.5161984.1
Defining polynomial: \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - 4 x^{3} + 25 x^{2} - 20 x + 8\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -5 \nu^{5} - 2 \nu^{4} - 25 \nu^{3} + 10 \nu^{2} - 121 \nu + 100 \)\()/121\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{5} + 27 \nu^{4} + 35 \nu^{3} - 14 \nu^{2} + 223 \)\()/121\)
\(\beta_{4}\)\(=\)\((\)\( -25 \nu^{5} - 10 \nu^{4} - 4 \nu^{3} + 50 \nu^{2} - 605 \nu + 258 \)\()/242\)
\(\beta_{5}\)\(=\)\((\)\( -65 \nu^{5} - 26 \nu^{4} + 38 \nu^{3} + 372 \nu^{2} - 1331 \nu + 574 \)\()/242\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{4} - 5 \beta_{2} + 2\)
\(\nu^{4}\)\(=\)\(5 \beta_{3} + 7 \beta_{2} + 7 \beta_{1} - 15\)
\(\nu^{5}\)\(=\)\(2 \beta_{5} - 16 \beta_{4} - 2 \beta_{3} - 29 \beta_{1} + 16\)

Character values

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

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
889.1
0.432320 0.432320i
−1.75233 + 1.75233i
1.32001 1.32001i
0.432320 + 0.432320i
−1.75233 1.75233i
1.32001 + 1.32001i
1.00000i 1.00000i −1.00000 −2.19388 + 0.432320i −1.00000 1.86464i 1.00000i −1.00000 0.432320 + 2.19388i
889.2 1.00000i 1.00000i −1.00000 1.38900 1.75233i −1.00000 2.50466i 1.00000i −1.00000 −1.75233 1.38900i
889.3 1.00000i 1.00000i −1.00000 1.80487 + 1.32001i −1.00000 3.64002i 1.00000i −1.00000 1.32001 1.80487i
889.4 1.00000i 1.00000i −1.00000 −2.19388 0.432320i −1.00000 1.86464i 1.00000i −1.00000 0.432320 2.19388i
889.5 1.00000i 1.00000i −1.00000 1.38900 + 1.75233i −1.00000 2.50466i 1.00000i −1.00000 −1.75233 + 1.38900i
889.6 1.00000i 1.00000i −1.00000 1.80487 1.32001i −1.00000 3.64002i 1.00000i −1.00000 1.32001 + 1.80487i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 889.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.d.i 6
3.b odd 2 1 3330.2.d.n 6
5.b even 2 1 inner 1110.2.d.i 6
5.c odd 4 1 5550.2.a.cf 3
5.c odd 4 1 5550.2.a.cg 3
15.d odd 2 1 3330.2.d.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.i 6 1.a even 1 1 trivial
1110.2.d.i 6 5.b even 2 1 inner
3330.2.d.n 6 3.b odd 2 1
3330.2.d.n 6 15.d odd 2 1
5550.2.a.cf 3 5.c odd 4 1
5550.2.a.cg 3 5.c odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{6} + 23 T_{7}^{4} + 151 T_{7}^{2} + 289 \)
\( T_{11}^{3} + 5 T_{11}^{2} - 11 T_{11} - 59 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( ( 1 + T^{2} )^{3} \)
$5$ \( 125 - 50 T - 15 T^{2} + 24 T^{3} - 3 T^{4} - 2 T^{5} + T^{6} \)
$7$ \( 289 + 151 T^{2} + 23 T^{4} + T^{6} \)
$11$ \( ( -59 - 11 T + 5 T^{2} + T^{3} )^{2} \)
$13$ \( 64 + 57 T^{2} + 14 T^{4} + T^{6} \)
$17$ \( 1 + 23 T^{2} + 55 T^{4} + T^{6} \)
$19$ \( ( 8 + 25 T + 10 T^{2} + T^{3} )^{2} \)
$23$ \( 256 + 401 T^{2} + 82 T^{4} + T^{6} \)
$29$ \( ( 50 + 66 T - 17 T^{2} + T^{3} )^{2} \)
$31$ \( ( 20 - 24 T + T^{2} + T^{3} )^{2} \)
$37$ \( ( 1 + T^{2} )^{3} \)
$41$ \( ( -34 - 30 T + 9 T^{2} + T^{3} )^{2} \)
$43$ \( 59536 + 6280 T^{2} + 185 T^{4} + T^{6} \)
$47$ \( 64 + 132 T^{2} + 44 T^{4} + T^{6} \)
$53$ \( 121 + 467 T^{2} + 91 T^{4} + T^{6} \)
$59$ \( ( 232 - 62 T - 4 T^{2} + T^{3} )^{2} \)
$61$ \( ( 108 - 72 T - 3 T^{2} + T^{3} )^{2} \)
$67$ \( 295936 + 25232 T^{2} + 312 T^{4} + T^{6} \)
$71$ \( ( -148 - 162 T - 2 T^{2} + T^{3} )^{2} \)
$73$ \( 4 + 377 T^{2} + 54 T^{4} + T^{6} \)
$79$ \( ( -16 + 38 T - 12 T^{2} + T^{3} )^{2} \)
$83$ \( 1936 + 1897 T^{2} + 206 T^{4} + T^{6} \)
$89$ \( ( 4 - 45 T + 6 T^{2} + T^{3} )^{2} \)
$97$ \( 1085764 + 63872 T^{2} + 493 T^{4} + T^{6} \)
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