Properties

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

Newform invariants

Self dual: no
Analytic conductor: \(8.86339462436\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 20 x^{6} + 132 x^{4} + 297 x^{2} + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} + 20 x^{6} + 132 x^{4} + 297 x^{2} + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{6} + 10 \nu^{4} + 18 \nu^{2} - 9 \)\()/7\)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{7} + 44 \nu^{5} + 180 \nu^{3} + 155 \nu \)\()/56\)
\(\beta_{3}\)\(=\)\((\)\( -3 \nu^{7} - 44 \nu^{5} - 124 \nu^{3} + 181 \nu \)\()/56\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 164 \nu^{3} + 527 \nu \)\()/56\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{6} - 27 \nu^{4} - 85 \nu^{2} + 11 \)\()/7\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 4 \nu^{5} + 164 \nu^{3} + 639 \nu \)\()/56\)
\(\beta_{7}\)\(=\)\((\)\( 2 \nu^{6} + 27 \nu^{4} + 99 \nu^{2} + 59 \)\()/7\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{6} - \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{7} + \beta_{5} - 10\)\()/2\)
\(\nu^{3}\)\(=\)\(-3 \beta_{6} + 3 \beta_{4} + \beta_{3} + \beta_{2}\)
\(\nu^{4}\)\(=\)\((\)\(-7 \beta_{7} - 9 \beta_{5} - 4 \beta_{1} + 68\)\()/2\)
\(\nu^{5}\)\(=\)\((\)\(41 \beta_{6} - 35 \beta_{4} - 24 \beta_{3} - 22 \beta_{2}\)\()/2\)
\(\nu^{6}\)\(=\)\(26 \beta_{7} + 36 \beta_{5} + 27 \beta_{1} - 241\)
\(\nu^{7}\)\(=\)\((\)\(-293 \beta_{6} + 205 \beta_{4} + 232 \beta_{3} + 240 \beta_{2}\)\()/2\)

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
0.490113i
2.20111i
2.87151i
2.58251i
0.490113i
2.20111i
2.87151i
2.58251i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i −4.26968 1.00000i 1.00000 1.00000
961.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i −2.35622 1.00000i 1.00000 1.00000
961.3 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.374052 1.00000i 1.00000 1.00000
961.4 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 4.25184 1.00000i 1.00000 1.00000
961.5 1.00000i 1.00000 −1.00000 1.00000i 1.00000i −4.26968 1.00000i 1.00000 1.00000
961.6 1.00000i 1.00000 −1.00000 1.00000i 1.00000i −2.35622 1.00000i 1.00000 1.00000
961.7 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0.374052 1.00000i 1.00000 1.00000
961.8 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 4.25184 1.00000i 1.00000 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 961.8
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.g 8
3.b odd 2 1 3330.2.h.o 8
37.b even 2 1 inner 1110.2.h.g 8
111.d odd 2 1 3330.2.h.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1110.2.h.g 8 1.a even 1 1 trivial
1110.2.h.g 8 37.b even 2 1 inner
3330.2.h.o 8 3.b odd 2 1
3330.2.h.o 8 111.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 \(S_{2}^{\mathrm{new}}(1110, [\chi])\):

\( T_{7}^{4} + 2 T_{7}^{3} - 19 T_{7}^{2} - 36 T_{7} + 16 \)
\( T_{13}^{8} + 53 T_{13}^{6} + 852 T_{13}^{4} + 5056 T_{13}^{2} + 9216 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{4} \)
$3$ \( ( -1 + T )^{8} \)
$5$ \( ( 1 + T^{2} )^{4} \)
$7$ \( ( 16 - 36 T - 19 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$11$ \( ( 60 - 38 T - 33 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$13$ \( 9216 + 5056 T^{2} + 852 T^{4} + 53 T^{6} + T^{8} \)
$17$ \( 144 + 1192 T^{2} + 457 T^{4} + 50 T^{6} + T^{8} \)
$19$ \( 46656 + 21760 T^{2} + 2192 T^{4} + 81 T^{6} + T^{8} \)
$23$ \( 9216 + 5056 T^{2} + 852 T^{4} + 53 T^{6} + T^{8} \)
$29$ \( 3594816 + 402160 T^{2} + 14940 T^{4} + 213 T^{6} + T^{8} \)
$31$ \( 129600 + 35824 T^{2} + 3356 T^{4} + 117 T^{6} + T^{8} \)
$37$ \( 1874161 - 911754 T + 164280 T^{2} - 8510 T^{3} - 610 T^{4} - 230 T^{5} + 120 T^{6} - 18 T^{7} + T^{8} \)
$41$ \( ( 360 - 52 T - 46 T^{2} + 5 T^{3} + T^{4} )^{2} \)
$43$ \( 2304 + 11920 T^{2} + 1896 T^{4} + 89 T^{6} + T^{8} \)
$47$ \( ( -9216 + 2560 T - 124 T^{2} - 14 T^{3} + T^{4} )^{2} \)
$53$ \( ( 12 + 32 T - 7 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$59$ \( 2359296 + 323584 T^{2} + 13632 T^{4} + 212 T^{6} + T^{8} \)
$61$ \( 484416 + 121024 T^{2} + 8192 T^{4} + 177 T^{6} + T^{8} \)
$67$ \( ( 4096 + 1280 T - 208 T^{2} - 6 T^{3} + T^{4} )^{2} \)
$71$ \( ( 192 + 256 T - 28 T^{2} - 12 T^{3} + T^{4} )^{2} \)
$73$ \( ( 1880 + 228 T - 82 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$79$ \( 589824 + 4259392 T^{2} + 79664 T^{4} + 492 T^{6} + T^{8} \)
$83$ \( ( -576 + 752 T - 124 T^{2} - 3 T^{3} + T^{4} )^{2} \)
$89$ \( 36864 + 124048 T^{2} + 9912 T^{4} + 185 T^{6} + T^{8} \)
$97$ \( 191102976 + 6988048 T^{2} + 90764 T^{4} + 501 T^{6} + T^{8} \)
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