Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
Defining polynomial: \(x^{4} + 17 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 17 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 9 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 9 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 9\)
\(\nu^{3}\)\(=\)\(8 \beta_{2} - 9 \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
961.1
2.37228i
3.37228i
2.37228i
3.37228i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −2.37228 1.00000i 1.00000 −1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.37228 1.00000i 1.00000 −1.00000
961.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −2.37228 1.00000i 1.00000 −1.00000
961.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i 3.37228 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.e 4
3.b odd 2 1 3330.2.h.k 4
37.b even 2 1 inner 1110.2.h.e 4
111.d odd 2 1 3330.2.h.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.e 4 1.a even 1 1 trivial
1110.2.h.e 4 37.b even 2 1 inner
3330.2.h.k 4 3.b odd 2 1
3330.2.h.k 4 111.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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{2} - T_{7} - 8 \)
\( T_{13}^{4} + 68 T_{13}^{2} + 1024 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( -8 - T + T^{2} )^{2} \)
$11$ \( ( 4 - 7 T + T^{2} )^{2} \)
$13$ \( 1024 + 68 T^{2} + T^{4} \)
$17$ \( 4 + 29 T^{2} + T^{4} \)
$19$ \( ( 36 + T^{2} )^{2} \)
$23$ \( 576 + 84 T^{2} + T^{4} \)
$29$ \( 1156 + 101 T^{2} + T^{4} \)
$31$ \( 484 + 77 T^{2} + T^{4} \)
$37$ \( 1369 - 58 T^{2} + T^{4} \)
$41$ \( ( -74 + T + T^{2} )^{2} \)
$43$ \( 36 + 21 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( 22 + 11 T + T^{2} )^{2} \)
$59$ \( ( 64 + T^{2} )^{2} \)
$61$ \( 4096 + 161 T^{2} + T^{4} \)
$67$ \( ( -24 - 6 T + T^{2} )^{2} \)
$71$ \( ( -24 + 6 T + T^{2} )^{2} \)
$73$ \( ( -14 + T )^{4} \)
$79$ \( 1024 + 68 T^{2} + T^{4} \)
$83$ \( ( -96 - 12 T + T^{2} )^{2} \)
$89$ \( 576 + 84 T^{2} + T^{4} \)
$97$ \( 5184 + 153 T^{2} + T^{4} \)
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