Properties

Label 1110.2.d.e
Level $1110$
Weight $2$
Character orbit 1110.d
Analytic conductor $8.863$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 + T^{2} \)
$5$ \( 5 - 4 T + T^{2} \)
$7$ \( 4 + T^{2} \)
$11$ \( ( 2 + T )^{2} \)
$13$ \( 36 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -4 + T )^{2} \)
$23$ \( 16 + T^{2} \)
$29$ \( ( -8 + T )^{2} \)
$31$ \( T^{2} \)
$37$ \( 1 + T^{2} \)
$41$ \( ( -10 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( 144 + T^{2} \)
$53$ \( 4 + T^{2} \)
$59$ \( ( 2 + T )^{2} \)
$61$ \( ( -6 + T )^{2} \)
$67$ \( 64 + T^{2} \)
$71$ \( ( 12 + T )^{2} \)
$73$ \( 16 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 16 + T^{2} \)
$89$ \( ( 14 + T )^{2} \)
$97$ \( 144 + T^{2} \)
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