Properties

Label 1150.2.b.j
Level $1150$
Weight $2$
Character orbit 1150.b
Analytic conductor $9.183$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.77580864.1
Defining polynomial: \(x^{6} + 19 x^{4} + 105 x^{2} + 144\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} -\beta_{3} q^{6} + ( \beta_{1} + \beta_{2} - \beta_{4} ) q^{7} -\beta_{2} q^{8} + ( -4 - \beta_{3} + \beta_{5} ) q^{9} +O(q^{10})\) \( q + \beta_{2} q^{2} + \beta_{1} q^{3} - q^{4} -\beta_{3} q^{6} + ( \beta_{1} + \beta_{2} - \beta_{4} ) q^{7} -\beta_{2} q^{8} + ( -4 - \beta_{3} + \beta_{5} ) q^{9} + ( 2 + \beta_{3} - 2 \beta_{5} ) q^{11} -\beta_{1} q^{12} + ( \beta_{1} + \beta_{2} + \beta_{4} ) q^{13} + ( -1 - \beta_{3} - \beta_{5} ) q^{14} + q^{16} + ( \beta_{1} - 2 \beta_{2} ) q^{17} + ( -\beta_{1} - 4 \beta_{2} - \beta_{4} ) q^{18} + ( -1 - \beta_{3} - \beta_{5} ) q^{19} + ( -9 - 2 \beta_{3} + 3 \beta_{5} ) q^{21} + ( \beta_{1} + 2 \beta_{2} + 2 \beta_{4} ) q^{22} + \beta_{2} q^{23} + \beta_{3} q^{24} + ( -1 - \beta_{3} + \beta_{5} ) q^{26} + ( -2 \beta_{1} - 5 \beta_{2} + \beta_{4} ) q^{27} + ( -\beta_{1} - \beta_{2} + \beta_{4} ) q^{28} + ( 2 + 2 \beta_{3} ) q^{29} + ( -1 + \beta_{3} - \beta_{5} ) q^{31} + \beta_{2} q^{32} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{4} ) q^{33} + ( 2 - \beta_{3} ) q^{34} + ( 4 + \beta_{3} - \beta_{5} ) q^{36} + 2 \beta_{4} q^{37} + ( -\beta_{1} - \beta_{2} + \beta_{4} ) q^{38} + ( -5 - 2 \beta_{3} - \beta_{5} ) q^{39} + ( \beta_{3} + 2 \beta_{5} ) q^{41} + ( -2 \beta_{1} - 9 \beta_{2} - 3 \beta_{4} ) q^{42} -8 \beta_{2} q^{43} + ( -2 - \beta_{3} + 2 \beta_{5} ) q^{44} - q^{46} + ( -2 \beta_{1} - 6 \beta_{2} - 2 \beta_{4} ) q^{47} + \beta_{1} q^{48} + ( -11 - \beta_{3} + 2 \beta_{5} ) q^{49} + ( -7 + \beta_{3} + \beta_{5} ) q^{51} + ( -\beta_{1} - \beta_{2} - \beta_{4} ) q^{52} + 6 \beta_{2} q^{53} + ( 5 + 2 \beta_{3} + \beta_{5} ) q^{54} + ( 1 + \beta_{3} + \beta_{5} ) q^{56} + ( -2 \beta_{1} - 9 \beta_{2} - 3 \beta_{4} ) q^{57} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{58} + ( -4 + 2 \beta_{3} ) q^{59} + ( 2 + \beta_{3} - 4 \beta_{5} ) q^{61} + ( \beta_{1} - \beta_{2} + \beta_{4} ) q^{62} + ( -8 \beta_{1} - 5 \beta_{2} + \beta_{4} ) q^{63} - q^{64} + ( -3 - 3 \beta_{3} - 3 \beta_{5} ) q^{66} + ( 4 \beta_{2} + 4 \beta_{4} ) q^{67} + ( -\beta_{1} + 2 \beta_{2} ) q^{68} -\beta_{3} q^{69} + ( 4 + \beta_{3} ) q^{71} + ( \beta_{1} + 4 \beta_{2} + \beta_{4} ) q^{72} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{4} ) q^{73} + 2 \beta_{5} q^{74} + ( 1 + \beta_{3} + \beta_{5} ) q^{76} + ( 8 \beta_{1} - 5 \beta_{2} + \beta_{4} ) q^{77} + ( -2 \beta_{1} - 5 \beta_{2} + \beta_{4} ) q^{78} + ( 4 + 4 \beta_{3} - 4 \beta_{5} ) q^{79} + ( 4 + 4 \beta_{3} - \beta_{5} ) q^{81} + ( \beta_{1} - 2 \beta_{4} ) q^{82} + ( -2 \beta_{2} + 2 \beta_{4} ) q^{83} + ( 9 + 2 \beta_{3} - 3 \beta_{5} ) q^{84} + 8 q^{86} + ( 4 \beta_{1} + 14 \beta_{2} + 2 \beta_{4} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} - 2 \beta_{4} ) q^{88} + ( -4 + 4 \beta_{3} - 2 \beta_{5} ) q^{89} + ( -2 - 5 \beta_{3} + 2 \beta_{5} ) q^{91} -\beta_{2} q^{92} + ( 5 \beta_{2} - \beta_{4} ) q^{93} + ( 6 + 2 \beta_{3} - 2 \beta_{5} ) q^{94} -\beta_{3} q^{96} + ( 3 \beta_{1} - 10 \beta_{2} ) q^{97} + ( -\beta_{1} - 11 \beta_{2} - 2 \beta_{4} ) q^{98} + ( -21 - 3 \beta_{3} + 3 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + O(q^{10}) \) \( 6q - 6q^{4} + 2q^{6} - 20q^{9} + 6q^{11} - 6q^{14} + 6q^{16} - 6q^{19} - 44q^{21} - 2q^{24} - 2q^{26} + 8q^{29} - 10q^{31} + 14q^{34} + 20q^{36} - 28q^{39} + 2q^{41} - 6q^{44} - 6q^{46} - 60q^{49} - 42q^{51} + 28q^{54} + 6q^{56} - 28q^{59} + 2q^{61} - 6q^{64} - 18q^{66} + 2q^{69} + 22q^{71} + 4q^{74} + 6q^{76} + 8q^{79} + 14q^{81} + 44q^{84} + 48q^{86} - 36q^{89} + 2q^{91} + 28q^{94} + 2q^{96} - 114q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 19 x^{4} + 105 x^{2} + 144\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{5} + 7 \nu^{3} - 15 \nu \)\()/36\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{4} + 10 \nu^{2} + 12 \)\()/3\)
\(\beta_{4}\)\(=\)\((\)\( 5 \nu^{5} + 71 \nu^{3} + 213 \nu \)\()/36\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{4} + 13 \nu^{2} + 33 \)\()/3\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{5} - \beta_{3} - 7\)
\(\nu^{3}\)\(=\)\(\beta_{4} - 5 \beta_{2} - 8 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-10 \beta_{5} + 13 \beta_{3} + 58\)
\(\nu^{5}\)\(=\)\(-7 \beta_{4} + 71 \beta_{2} + 71 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1150\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(277\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
3.11903i
1.43163i
2.68740i
2.68740i
1.43163i
3.11903i
1.00000i 3.11903i −1.00000 0 −3.11903 4.50973i 1.00000i −6.72833 0
599.2 1.00000i 1.43163i −1.00000 0 1.43163 3.08719i 1.00000i 0.950444 0
599.3 1.00000i 2.68740i −1.00000 0 2.68740 4.59692i 1.00000i −4.22212 0
599.4 1.00000i 2.68740i −1.00000 0 2.68740 4.59692i 1.00000i −4.22212 0
599.5 1.00000i 1.43163i −1.00000 0 1.43163 3.08719i 1.00000i 0.950444 0
599.6 1.00000i 3.11903i −1.00000 0 −3.11903 4.50973i 1.00000i −6.72833 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 599.6
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 1150.2.b.j 6
5.b even 2 1 inner 1150.2.b.j 6
5.c odd 4 1 230.2.a.d 3
5.c odd 4 1 1150.2.a.q 3
15.e even 4 1 2070.2.a.z 3
20.e even 4 1 1840.2.a.r 3
20.e even 4 1 9200.2.a.cf 3
40.i odd 4 1 7360.2.a.bz 3
40.k even 4 1 7360.2.a.ce 3
115.e even 4 1 5290.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 5.c odd 4 1
1150.2.a.q 3 5.c odd 4 1
1150.2.b.j 6 1.a even 1 1 trivial
1150.2.b.j 6 5.b even 2 1 inner
1840.2.a.r 3 20.e even 4 1
2070.2.a.z 3 15.e even 4 1
5290.2.a.r 3 115.e even 4 1
7360.2.a.bz 3 40.i odd 4 1
7360.2.a.ce 3 40.k even 4 1
9200.2.a.cf 3 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{6} + 19 T_{3}^{4} + 105 T_{3}^{2} + 144 \)
\( T_{7}^{6} + 51 T_{7}^{4} + 825 T_{7}^{2} + 4096 \)
\( T_{11}^{3} - 3 T_{11}^{2} - 39 T_{11} + 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{3} \)
$3$ \( 144 + 105 T^{2} + 19 T^{4} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 4096 + 825 T^{2} + 51 T^{4} + T^{6} \)
$11$ \( ( 144 - 39 T - 3 T^{2} + T^{3} )^{2} \)
$13$ \( 324 + 261 T^{2} + 31 T^{4} + T^{6} \)
$17$ \( 324 + 301 T^{2} + 35 T^{4} + T^{6} \)
$19$ \( ( -64 - 21 T + 3 T^{2} + T^{3} )^{2} \)
$23$ \( ( 1 + T^{2} )^{3} \)
$29$ \( ( -24 - 32 T - 4 T^{2} + T^{3} )^{2} \)
$31$ \( ( -8 - 7 T + 5 T^{2} + T^{3} )^{2} \)
$37$ \( 1024 + 1728 T^{2} + 84 T^{4} + T^{6} \)
$41$ \( ( 186 - 59 T - T^{2} + T^{3} )^{2} \)
$43$ \( ( 64 + T^{2} )^{3} \)
$47$ \( 82944 + 8080 T^{2} + 188 T^{4} + T^{6} \)
$53$ \( ( 36 + T^{2} )^{3} \)
$59$ \( ( -144 + 28 T + 14 T^{2} + T^{3} )^{2} \)
$61$ \( ( 526 - 157 T - T^{2} + T^{3} )^{2} \)
$67$ \( 147456 + 26880 T^{2} + 352 T^{4} + T^{6} \)
$71$ \( ( -24 + 31 T - 11 T^{2} + T^{3} )^{2} \)
$73$ \( 61504 + 5568 T^{2} + 144 T^{4} + T^{6} \)
$79$ \( ( 1152 - 240 T - 4 T^{2} + T^{3} )^{2} \)
$83$ \( 9216 + 1936 T^{2} + 104 T^{4} + T^{6} \)
$89$ \( ( -1152 - 48 T + 18 T^{2} + T^{3} )^{2} \)
$97$ \( 27556 + 66885 T^{2} + 531 T^{4} + T^{6} \)
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