Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \(x^{4} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} \)\(/3\)
\(\beta_{3}\)\(=\)\( \nu^{3} \)\(/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(3 \beta_{2}\)
\(\nu^{3}\)\(=\)\(3 \beta_{3}\)

Character values

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

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

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
1.22474 1.22474i
−1.22474 + 1.22474i
1.22474 + 1.22474i
−1.22474 1.22474i
1.00000i 1.00000i −1.00000 −0.224745 2.22474i 1.00000 3.44949i 1.00000i −1.00000 −2.22474 + 0.224745i
889.2 1.00000i 1.00000i −1.00000 2.22474 + 0.224745i 1.00000 1.44949i 1.00000i −1.00000 0.224745 2.22474i
889.3 1.00000i 1.00000i −1.00000 −0.224745 + 2.22474i 1.00000 3.44949i 1.00000i −1.00000 −2.22474 0.224745i
889.4 1.00000i 1.00000i −1.00000 2.22474 0.224745i 1.00000 1.44949i 1.00000i −1.00000 0.224745 + 2.22474i
\(n\): e.g. 2-40 or 990-1000
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.h 4
3.b odd 2 1 3330.2.d.h 4
5.b even 2 1 inner 1110.2.d.h 4
5.c odd 4 1 5550.2.a.bt 2
5.c odd 4 1 5550.2.a.cc 2
15.d odd 2 1 3330.2.d.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.h 4 1.a even 1 1 trivial
1110.2.d.h 4 5.b even 2 1 inner
3330.2.d.h 4 3.b odd 2 1
3330.2.d.h 4 15.d odd 2 1
5550.2.a.bt 2 5.c odd 4 1
5550.2.a.cc 2 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{4} + 14 T_{7}^{2} + 25 \)
\( T_{11}^{2} + 2 T_{11} - 5 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( 25 - 20 T + 8 T^{2} - 4 T^{3} + T^{4} \)
$7$ \( 25 + 14 T^{2} + T^{4} \)
$11$ \( ( -5 + 2 T + T^{2} )^{2} \)
$13$ \( 9 + 30 T^{2} + T^{4} \)
$17$ \( 25 + 14 T^{2} + T^{4} \)
$19$ \( ( 3 + T )^{4} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 30 + 12 T + T^{2} )^{2} \)
$31$ \( ( 2 + T )^{4} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( -2 - 4 T + T^{2} )^{2} \)
$43$ \( 400 + 56 T^{2} + T^{4} \)
$47$ \( 100 + 44 T^{2} + T^{4} \)
$53$ \( ( 9 + T^{2} )^{2} \)
$59$ \( ( -38 + 8 T + T^{2} )^{2} \)
$61$ \( ( 76 + 20 T + T^{2} )^{2} \)
$67$ \( 400 + 56 T^{2} + T^{4} \)
$71$ \( ( -18 - 12 T + T^{2} )^{2} \)
$73$ \( 9025 + 194 T^{2} + T^{4} \)
$79$ \( ( -18 + 12 T + T^{2} )^{2} \)
$83$ \( 25 + 14 T^{2} + T^{4} \)
$89$ \( ( -5 - 14 T + T^{2} )^{2} \)
$97$ \( 1444 + 140 T^{2} + T^{4} \)
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