Properties

Label 1110.2.h.d
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: \(\Q(i, \sqrt{73})\)
Defining polynomial: \(x^{4} + 37 x^{2} + 324\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 37 x^{2} + 324\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 19 \nu \)\()/18\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 19 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 19\)
\(\nu^{3}\)\(=\)\(18 \beta_{2} - 19 \beta_{1}\)

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
4.77200i
3.77200i
3.77200i
4.77200i
1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.2 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.3 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 1.00000i 1.00000 1.00000
961.4 1.00000i −1.00000 −1.00000 1.00000i 1.00000i −1.00000 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.d 4
3.b odd 2 1 3330.2.h.j 4
37.b even 2 1 inner 1110.2.h.d 4
111.d odd 2 1 3330.2.h.j 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.d 4 1.a even 1 1 trivial
1110.2.h.d 4 37.b even 2 1 inner
3330.2.h.j 4 3.b odd 2 1
3330.2.h.j 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7} + 1 \)
\( T_{13}^{4} + 37 T_{13}^{2} + 324 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T )^{4} \)
$5$ \( ( 1 + T^{2} )^{2} \)
$7$ \( ( 1 + T )^{4} \)
$11$ \( ( 1 + T )^{4} \)
$13$ \( 324 + 37 T^{2} + T^{4} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( 36 + 61 T^{2} + T^{4} \)
$23$ \( 36 + 61 T^{2} + T^{4} \)
$29$ \( 36 + 61 T^{2} + T^{4} \)
$31$ \( 36 + 61 T^{2} + T^{4} \)
$37$ \( 1369 - 74 T + 2 T^{2} - 2 T^{3} + T^{4} \)
$41$ \( ( -6 - 7 T + T^{2} )^{2} \)
$43$ \( 144 + 97 T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( ( -69 + 4 T + T^{2} )^{2} \)
$59$ \( 5184 + 148 T^{2} + T^{4} \)
$61$ \( 2916 + 181 T^{2} + T^{4} \)
$67$ \( ( -72 - 2 T + T^{2} )^{2} \)
$71$ \( ( 6 + T )^{4} \)
$73$ \( ( 2 - 9 T + T^{2} )^{2} \)
$79$ \( 4096 + 164 T^{2} + T^{4} \)
$83$ \( ( -12 - 5 T + T^{2} )^{2} \)
$89$ \( 20736 + 369 T^{2} + T^{4} \)
$97$ \( 576 + 121 T^{2} + T^{4} \)
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