Properties

Label 1110.2.u.a
Level $1110$
Weight $2$
Character orbit 1110.u
Analytic conductor $8.863$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1110.u (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(-1\) \(\zeta_{8}^{2}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
−0.707107 0.707107i 1.41421 1.00000i 1.00000i 0.707107 0.707107i −1.70711 0.292893i −2.00000 0.707107 0.707107i 1.00000 2.82843i −1.00000
191.2 0.707107 + 0.707107i −1.41421 1.00000i 1.00000i −0.707107 + 0.707107i −0.292893 1.70711i −2.00000 −0.707107 + 0.707107i 1.00000 + 2.82843i −1.00000
401.1 −0.707107 + 0.707107i 1.41421 + 1.00000i 1.00000i 0.707107 + 0.707107i −1.70711 + 0.292893i −2.00000 0.707107 + 0.707107i 1.00000 + 2.82843i −1.00000
401.2 0.707107 0.707107i −1.41421 + 1.00000i 1.00000i −0.707107 0.707107i −0.292893 + 1.70711i −2.00000 −0.707107 0.707107i 1.00000 2.82843i −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
3.b odd 2 1 inner
37.d odd 4 1 inner
111.g even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1110.2.u.a 4
3.b odd 2 1 inner 1110.2.u.a 4
37.d odd 4 1 inner 1110.2.u.a 4
111.g even 4 1 inner 1110.2.u.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.u.a 4 1.a even 1 1 trivial
1110.2.u.a 4 3.b odd 2 1 inner
1110.2.u.a 4 37.d odd 4 1 inner
1110.2.u.a 4 111.g even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7} + 2 \) acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{4} \)
$3$ \( 9 - 2 T^{2} + T^{4} \)
$5$ \( 1 + T^{4} \)
$7$ \( ( 2 + T )^{4} \)
$11$ \( ( -8 + T^{2} )^{2} \)
$13$ \( ( 18 + 6 T + T^{2} )^{2} \)
$17$ \( 16 + T^{4} \)
$19$ \( ( 2 + 2 T + T^{2} )^{2} \)
$23$ \( 4096 + T^{4} \)
$29$ \( 16 + T^{4} \)
$31$ \( ( 2 - 2 T + T^{2} )^{2} \)
$37$ \( ( 37 + 2 T + T^{2} )^{2} \)
$41$ \( T^{4} \)
$43$ \( ( 50 + 10 T + T^{2} )^{2} \)
$47$ \( T^{4} \)
$53$ \( ( 32 + T^{2} )^{2} \)
$59$ \( 256 + T^{4} \)
$61$ \( ( 162 - 18 T + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( 8 + T^{2} )^{2} \)
$73$ \( ( 100 + T^{2} )^{2} \)
$79$ \( ( 18 - 6 T + T^{2} )^{2} \)
$83$ \( ( 8 + T^{2} )^{2} \)
$89$ \( 10000 + T^{4} \)
$97$ \( ( 18 + 6 T + T^{2} )^{2} \)
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