Properties

Label 1110.2.d.j
Level $1110$
Weight $2$
Character orbit 1110.d
Analytic conductor $8.863$
Analytic rank $0$
Dimension $8$
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: \(8\)
Coefficient field: 8.0.57815240704.2
Defining polynomial: \(x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} + 2 x^{6} + 89 x^{4} - 170 x^{3} + 162 x^{2} - 72 x + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 198 \nu^{7} - 217 \nu^{6} + 209 \nu^{5} + 88 \nu^{4} + 18711 \nu^{3} - 17653 \nu^{2} + 16897 \nu - 7524 \)\()/9085\)
\(\beta_{3}\)\(=\)\((\)\( -1881 \nu^{7} + 2970 \nu^{6} - 2894 \nu^{5} - 836 \nu^{4} - 167761 \nu^{3} + 244926 \nu^{2} - 234110 \nu + 67844 \)\()/36340\)
\(\beta_{4}\)\(=\)\((\)\( 1969 \nu^{7} - 2057 \nu^{6} + 968 \nu^{5} + 2894 \nu^{4} + 176077 \nu^{3} - 166969 \nu^{2} + 74052 \nu + 56002 \)\()/18170\)
\(\beta_{5}\)\(=\)\((\)\( -358 \nu^{7} + 374 \nu^{6} - 176 \nu^{5} - 361 \nu^{4} - 32014 \nu^{3} + 30358 \nu^{2} - 13464 \nu - 2749 \)\()/1817\)
\(\beta_{6}\)\(=\)\((\)\( -9117 \nu^{7} + 17838 \nu^{6} - 14166 \nu^{5} - 4052 \nu^{4} - 811589 \nu^{3} + 1516102 \nu^{2} - 1096418 \nu + 328276 \)\()/36340\)
\(\beta_{7}\)\(=\)\((\)\( 9261 \nu^{7} - 16344 \nu^{6} + 14318 \nu^{5} + 4116 \nu^{4} + 825197 \nu^{3} - 1388536 \nu^{2} + 1151654 \nu - 333748 \)\()/36340\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-2 \beta_{7} - \beta_{6} - 5 \beta_{3} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(-\beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_{3} + 9 \beta_{2} + \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(11 \beta_{5} + 20 \beta_{4} - 45\)
\(\nu^{5}\)\(=\)\(11 \beta_{7} + 11 \beta_{6} + 11 \beta_{5} + 11 \beta_{4} - 18 \beta_{3} - 96 \beta_{1} + 18\)
\(\nu^{6}\)\(=\)\(192 \beta_{7} + 107 \beta_{6} + 425 \beta_{3} - 4 \beta_{2} - 111 \beta_{1}\)
\(\nu^{7}\)\(=\)\(115 \beta_{7} + 111 \beta_{6} - 111 \beta_{5} - 115 \beta_{4} - 150 \beta_{3} - 809 \beta_{2} - 111 \beta_{1} - 150\)

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
0.353624 0.353624i
−2.15569 + 2.15569i
2.20793 2.20793i
0.594137 0.594137i
0.353624 + 0.353624i
−2.15569 2.15569i
2.20793 + 2.20793i
0.594137 + 0.594137i
1.00000i 1.00000i −1.00000 −2.20793 0.353624i −1.00000 4.51003i 1.00000i −1.00000 −0.353624 + 2.20793i
889.2 1.00000i 1.00000i −1.00000 −0.594137 + 2.15569i −1.00000 1.17795i 1.00000i −1.00000 2.15569 + 0.594137i
889.3 1.00000i 1.00000i −1.00000 −0.353624 2.20793i −1.00000 3.94848i 1.00000i −1.00000 −2.20793 + 0.353624i
889.4 1.00000i 1.00000i −1.00000 2.15569 0.594137i −1.00000 2.38360i 1.00000i −1.00000 −0.594137 2.15569i
889.5 1.00000i 1.00000i −1.00000 −2.20793 + 0.353624i −1.00000 4.51003i 1.00000i −1.00000 −0.353624 2.20793i
889.6 1.00000i 1.00000i −1.00000 −0.594137 2.15569i −1.00000 1.17795i 1.00000i −1.00000 2.15569 0.594137i
889.7 1.00000i 1.00000i −1.00000 −0.353624 + 2.20793i −1.00000 3.94848i 1.00000i −1.00000 −2.20793 0.353624i
889.8 1.00000i 1.00000i −1.00000 2.15569 + 0.594137i −1.00000 2.38360i 1.00000i −1.00000 −0.594137 + 2.15569i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 889.8
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.j 8
3.b odd 2 1 3330.2.d.o 8
5.b even 2 1 inner 1110.2.d.j 8
5.c odd 4 1 5550.2.a.cl 4
5.c odd 4 1 5550.2.a.cm 4
15.d odd 2 1 3330.2.d.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.d.j 8 1.a even 1 1 trivial
1110.2.d.j 8 5.b even 2 1 inner
3330.2.d.o 8 3.b odd 2 1
3330.2.d.o 8 15.d odd 2 1
5550.2.a.cl 4 5.c odd 4 1
5550.2.a.cm 4 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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{8} + 43 T_{7}^{6} + 579 T_{7}^{4} + 2525 T_{7}^{2} + 2500 \)
\( T_{11}^{4} + 5 T_{11}^{3} - 11 T_{11}^{2} - 7 T_{11} + 4 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( 1 + T^{2} )^{4} \)
$5$ \( 625 + 250 T + 50 T^{2} - 30 T^{3} - 46 T^{4} - 6 T^{5} + 2 T^{6} + 2 T^{7} + T^{8} \)
$7$ \( 2500 + 2525 T^{2} + 579 T^{4} + 43 T^{6} + T^{8} \)
$11$ \( ( 4 - 7 T - 11 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$13$ \( 256 + 484 T^{2} + 201 T^{4} + 26 T^{6} + T^{8} \)
$17$ \( 11236 + 8285 T^{2} + 1411 T^{4} + 75 T^{6} + T^{8} \)
$19$ \( ( 128 + 138 T + 13 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$23$ \( ( 1296 + 81 T^{2} + T^{4} )^{2} \)
$29$ \( ( -16 + 46 T - 34 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$31$ \( ( -3328 - 676 T + 56 T^{2} + 19 T^{3} + T^{4} )^{2} \)
$37$ \( ( 1 + T^{2} )^{4} \)
$41$ \( ( 172 + 146 T - 8 T^{2} - 13 T^{3} + T^{4} )^{2} \)
$43$ \( 256 + 400 T^{2} + 184 T^{4} + 25 T^{6} + T^{8} \)
$47$ \( 719104 + 288720 T^{2} + 13396 T^{4} + 208 T^{6} + T^{8} \)
$53$ \( 1747684 + 921941 T^{2} + 32191 T^{4} + 335 T^{6} + T^{8} \)
$59$ \( ( -128 - 52 T + 18 T^{2} + 10 T^{3} + T^{4} )^{2} \)
$61$ \( ( 1296 - 108 T - 108 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$67$ \( 11505664 + 6672896 T^{2} + 106512 T^{4} + 568 T^{6} + T^{8} \)
$71$ \( ( -64 - 1644 T - 250 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$73$ \( 11075584 + 910884 T^{2} + 25177 T^{4} + 278 T^{6} + T^{8} \)
$79$ \( ( -1024 + 1424 T - 170 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$83$ \( 163840000 + 6534400 T^{2} + 90849 T^{4} + 518 T^{6} + T^{8} \)
$89$ \( ( 1616 + 1362 T + 339 T^{2} + 32 T^{3} + T^{4} )^{2} \)
$97$ \( 839056 + 694308 T^{2} + 32404 T^{4} + 337 T^{6} + T^{8} \)
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