Properties

Label 1110.2.d.f
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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{3} + 2 \nu \)
\(\beta_{2}\)\(=\)\( \nu^{3} + 4 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\(-\beta_{2} + 2 \beta_{1}\)

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
1.61803i
0.618034i
0.618034i
1.61803i
1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 −2.23607
889.2 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 2.23607
889.3 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 2.23607
889.4 1.00000i 1.00000i −1.00000 2.23607i −1.00000 2.00000i 1.00000i −1.00000 −2.23607
\(n\): 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.f 4
3.b odd 2 1 3330.2.d.j 4
5.b even 2 1 inner 1110.2.d.f 4
5.c odd 4 1 5550.2.a.bu 2
5.c odd 4 1 5550.2.a.bz 2
15.d odd 2 1 3330.2.d.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.f 4 1.a even 1 1 trivial
1110.2.d.f 4 5.b even 2 1 inner
3330.2.d.j 4 3.b odd 2 1
3330.2.d.j 4 15.d odd 2 1
5550.2.a.bu 2 5.c odd 4 1
5550.2.a.bz 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} + 4 \)
\( T_{11}^{2} - 6 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( ( 5 + T^{2} )^{2} \)
$7$ \( ( 4 + T^{2} )^{2} \)
$11$ \( ( 4 - 6 T + T^{2} )^{2} \)
$13$ \( ( 20 + T^{2} )^{2} \)
$17$ \( ( 20 + T^{2} )^{2} \)
$19$ \( ( 20 - 10 T + T^{2} )^{2} \)
$23$ \( ( 16 + T^{2} )^{2} \)
$29$ \( ( 4 + T )^{4} \)
$31$ \( ( -16 - 4 T + T^{2} )^{2} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( -20 + T^{2} )^{2} \)
$43$ \( 256 + 112 T^{2} + T^{4} \)
$47$ \( 1296 + 108 T^{2} + T^{4} \)
$53$ \( 1936 + 168 T^{2} + T^{4} \)
$59$ \( ( -76 - 4 T + T^{2} )^{2} \)
$61$ \( ( 4 - 6 T + T^{2} )^{2} \)
$67$ \( 256 + 112 T^{2} + T^{4} \)
$71$ \( ( -16 - 4 T + T^{2} )^{2} \)
$73$ \( 256 + 112 T^{2} + T^{4} \)
$79$ \( ( -16 + 4 T + T^{2} )^{2} \)
$83$ \( 30976 + 368 T^{2} + T^{4} \)
$89$ \( ( 124 - 24 T + T^{2} )^{2} \)
$97$ \( 13456 + 268 T^{2} + T^{4} \)
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