Properties

Label 1110.2.h.b
Level $1110$
Weight $2$
Character orbit 1110.h
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.h (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.

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
1.00000i
1.00000i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000 1.00000i 1.00000 −1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 2.00000 1.00000i 1.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.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.h.b 2
3.b odd 2 1 3330.2.h.h 2
37.b even 2 1 inner 1110.2.h.b 2
111.d odd 2 1 3330.2.h.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.b 2 1.a even 1 1 trivial
1110.2.h.b 2 37.b even 2 1 inner
3330.2.h.h 2 3.b odd 2 1
3330.2.h.h 2 111.d odd 2 1

Hecke kernels

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

\( T_{7} - 2 \)
\( T_{13}^{2} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 + T )^{2} \)
$5$ \( 1 + T^{2} \)
$7$ \( ( -2 + T )^{2} \)
$11$ \( ( 4 + T )^{2} \)
$13$ \( 4 + T^{2} \)
$17$ \( 4 + T^{2} \)
$19$ \( 36 + T^{2} \)
$23$ \( T^{2} \)
$29$ \( 4 + T^{2} \)
$31$ \( 16 + T^{2} \)
$37$ \( 37 + 12 T + T^{2} \)
$41$ \( ( 2 + T )^{2} \)
$43$ \( 144 + T^{2} \)
$47$ \( ( 6 + T )^{2} \)
$53$ \( ( 4 + T )^{2} \)
$59$ \( 100 + T^{2} \)
$61$ \( 4 + T^{2} \)
$67$ \( ( 12 + T )^{2} \)
$71$ \( ( -12 + T )^{2} \)
$73$ \( ( 10 + T )^{2} \)
$79$ \( 16 + T^{2} \)
$83$ \( T^{2} \)
$89$ \( 144 + T^{2} \)
$97$ \( 36 + T^{2} \)
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