Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
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 primitive root of unity \(\zeta_{8}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{8}^{2} q^{2} -\zeta_{8}^{2} q^{3} - q^{4} + ( -\zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + q^{6} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{2} q^{8} - q^{9} +O(q^{10})\) \( q + \zeta_{8}^{2} q^{2} -\zeta_{8}^{2} q^{3} - q^{4} + ( -\zeta_{8} - 2 \zeta_{8}^{3} ) q^{5} + q^{6} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{7} -\zeta_{8}^{2} q^{8} - q^{9} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{10} + ( -5 - \zeta_{8} + \zeta_{8}^{3} ) q^{11} + \zeta_{8}^{2} q^{12} + ( 3 \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{13} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{14} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{15} + q^{16} + ( \zeta_{8} - 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{17} -\zeta_{8}^{2} q^{18} + ( 5 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{19} + ( \zeta_{8} + 2 \zeta_{8}^{3} ) q^{20} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{21} + ( -\zeta_{8} - 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{22} + 3 \zeta_{8}^{2} q^{23} - q^{24} + ( -4 - 3 \zeta_{8}^{2} ) q^{25} + ( -1 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{26} + \zeta_{8}^{2} q^{27} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{28} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{29} + ( -\zeta_{8} - 2 \zeta_{8}^{3} ) q^{30} + ( 2 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{31} + \zeta_{8}^{2} q^{32} + ( \zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{33} + ( 5 - \zeta_{8} + \zeta_{8}^{3} ) q^{34} + ( 3 + 2 \zeta_{8} + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{35} + q^{36} + \zeta_{8}^{2} q^{37} + ( -2 \zeta_{8} + 5 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{38} + ( 1 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{39} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{40} + ( -8 + \zeta_{8} - \zeta_{8}^{3} ) q^{41} + ( \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{42} + ( 6 \zeta_{8} + 2 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{43} + ( 5 + \zeta_{8} - \zeta_{8}^{3} ) q^{44} + ( \zeta_{8} + 2 \zeta_{8}^{3} ) q^{45} -3 q^{46} + ( 7 \zeta_{8} + 2 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{47} -\zeta_{8}^{2} q^{48} + ( 4 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{49} + ( 3 - 4 \zeta_{8}^{2} ) q^{50} + ( -5 + \zeta_{8} - \zeta_{8}^{3} ) q^{51} + ( -3 \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{52} + ( -2 \zeta_{8} - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{53} - q^{54} + ( -1 + 5 \zeta_{8} + 3 \zeta_{8}^{2} + 10 \zeta_{8}^{3} ) q^{55} + ( 1 + \zeta_{8} - \zeta_{8}^{3} ) q^{56} + ( 2 \zeta_{8} - 5 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{57} + ( -5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{58} + ( 6 + 5 \zeta_{8} - 5 \zeta_{8}^{3} ) q^{59} + ( 2 \zeta_{8} - \zeta_{8}^{3} ) q^{60} + ( 2 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{61} + ( -4 \zeta_{8} + 2 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{62} + ( -\zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{63} - q^{64} + ( 9 + 2 \zeta_{8} + 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{65} + ( -5 - \zeta_{8} + \zeta_{8}^{3} ) q^{66} + ( 2 \zeta_{8} - 2 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{67} + ( -\zeta_{8} + 5 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{68} + 3 q^{69} + ( -1 + \zeta_{8} + 3 \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{70} + ( -8 - \zeta_{8} + \zeta_{8}^{3} ) q^{71} + \zeta_{8}^{2} q^{72} + ( -2 \zeta_{8} + 5 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{73} - q^{74} + ( -3 + 4 \zeta_{8}^{2} ) q^{75} + ( -5 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{76} + ( -6 \zeta_{8} - 7 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{77} + ( 3 \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{78} + ( -8 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{79} + ( -\zeta_{8} - 2 \zeta_{8}^{3} ) q^{80} + q^{81} + ( \zeta_{8} - 8 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{82} + ( 7 \zeta_{8} - 3 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{83} + ( -1 - \zeta_{8} + \zeta_{8}^{3} ) q^{84} + ( 3 - 10 \zeta_{8} + \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{85} + ( -2 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{86} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{87} + ( \zeta_{8} + 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{88} + ( 7 - 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{89} + ( -2 \zeta_{8} + \zeta_{8}^{3} ) q^{90} + ( -7 - 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{91} -3 \zeta_{8}^{2} q^{92} + ( 4 \zeta_{8} - 2 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{93} + ( -2 - 7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{94} + ( -2 - 5 \zeta_{8} + 6 \zeta_{8}^{2} - 10 \zeta_{8}^{3} ) q^{95} + q^{96} + ( 9 \zeta_{8} - 6 \zeta_{8}^{2} + 9 \zeta_{8}^{3} ) q^{97} + ( -2 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{98} + ( 5 + \zeta_{8} - \zeta_{8}^{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{6} - 4q^{9} - 20q^{11} - 4q^{14} + 4q^{16} + 20q^{19} + 4q^{21} - 4q^{24} - 16q^{25} - 4q^{26} + 8q^{31} + 20q^{34} + 12q^{35} + 4q^{36} + 4q^{39} - 32q^{41} + 20q^{44} - 12q^{46} + 16q^{49} + 12q^{50} - 20q^{51} - 4q^{54} - 4q^{55} + 4q^{56} + 24q^{59} + 8q^{61} - 4q^{64} + 36q^{65} - 20q^{66} + 12q^{69} - 4q^{70} - 32q^{71} - 4q^{74} - 12q^{75} - 20q^{76} - 32q^{79} + 4q^{81} - 4q^{84} + 12q^{85} - 8q^{86} + 28q^{89} - 28q^{91} - 8q^{94} - 8q^{95} + 4q^{96} + 20q^{99} + O(q^{100}) \)

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
−0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 0.707107i
0.707107 + 0.707107i
1.00000i 1.00000i −1.00000 −0.707107 2.12132i 1.00000 0.414214i 1.00000i −1.00000 −2.12132 + 0.707107i
889.2 1.00000i 1.00000i −1.00000 0.707107 + 2.12132i 1.00000 2.41421i 1.00000i −1.00000 2.12132 0.707107i
889.3 1.00000i 1.00000i −1.00000 −0.707107 + 2.12132i 1.00000 0.414214i 1.00000i −1.00000 −2.12132 0.707107i
889.4 1.00000i 1.00000i −1.00000 0.707107 2.12132i 1.00000 2.41421i 1.00000i −1.00000 2.12132 + 0.707107i
\(n\): 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.g 4
3.b odd 2 1 3330.2.d.k 4
5.b even 2 1 inner 1110.2.d.g 4
5.c odd 4 1 5550.2.a.bs 2
5.c odd 4 1 5550.2.a.cb 2
15.d odd 2 1 3330.2.d.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.g 4 1.a even 1 1 trivial
1110.2.d.g 4 5.b even 2 1 inner
3330.2.d.k 4 3.b odd 2 1
3330.2.d.k 4 15.d odd 2 1
5550.2.a.bs 2 5.c odd 4 1
5550.2.a.cb 2 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{4} + 6 T_{7}^{2} + 1 \)
\( T_{11}^{2} + 10 T_{11} + 23 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( ( 1 + T^{2} )^{2} \)
$5$ \( 25 + 8 T^{2} + T^{4} \)
$7$ \( 1 + 6 T^{2} + T^{4} \)
$11$ \( ( 23 + 10 T + T^{2} )^{2} \)
$13$ \( 289 + 38 T^{2} + T^{4} \)
$17$ \( 529 + 54 T^{2} + T^{4} \)
$19$ \( ( 17 - 10 T + T^{2} )^{2} \)
$23$ \( ( 9 + T^{2} )^{2} \)
$29$ \( ( -50 + T^{2} )^{2} \)
$31$ \( ( -28 - 4 T + T^{2} )^{2} \)
$37$ \( ( 1 + T^{2} )^{2} \)
$41$ \( ( 62 + 16 T + T^{2} )^{2} \)
$43$ \( 4624 + 152 T^{2} + T^{4} \)
$47$ \( 8836 + 204 T^{2} + T^{4} \)
$53$ \( 49 + 18 T^{2} + T^{4} \)
$59$ \( ( -14 - 12 T + T^{2} )^{2} \)
$61$ \( ( -68 - 4 T + T^{2} )^{2} \)
$67$ \( 16 + 24 T^{2} + T^{4} \)
$71$ \( ( 62 + 16 T + T^{2} )^{2} \)
$73$ \( 289 + 66 T^{2} + T^{4} \)
$79$ \( ( 46 + 16 T + T^{2} )^{2} \)
$83$ \( 7921 + 214 T^{2} + T^{4} \)
$89$ \( ( -49 - 14 T + T^{2} )^{2} \)
$97$ \( 15876 + 396 T^{2} + T^{4} \)
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