Properties

Label 1150.2.b.g
Level $1150$
Weight $2$
Character orbit 1150.b
Analytic conductor $9.183$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{21})\)
Defining polynomial: \(x^{4} + 11 x^{2} + 25\)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 11 x^{2} + 25\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 6 \nu \)\()/5\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 6 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 6\)
\(\nu^{3}\)\(=\)\(5 \beta_{2} - 6 \beta_{1}\)

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
1.79129i
2.79129i
2.79129i
1.79129i
1.00000i 1.79129i −1.00000 0 −1.79129 2.79129i 1.00000i −0.208712 0
599.2 1.00000i 2.79129i −1.00000 0 2.79129 1.79129i 1.00000i −4.79129 0
599.3 1.00000i 2.79129i −1.00000 0 2.79129 1.79129i 1.00000i −4.79129 0
599.4 1.00000i 1.79129i −1.00000 0 −1.79129 2.79129i 1.00000i −0.208712 0
\(n\): 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 1150.2.b.g 4
5.b even 2 1 inner 1150.2.b.g 4
5.c odd 4 1 230.2.a.a 2
5.c odd 4 1 1150.2.a.o 2
15.e even 4 1 2070.2.a.x 2
20.e even 4 1 1840.2.a.n 2
20.e even 4 1 9200.2.a.bs 2
40.i odd 4 1 7360.2.a.bq 2
40.k even 4 1 7360.2.a.bk 2
115.e even 4 1 5290.2.a.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 5.c odd 4 1
1150.2.a.o 2 5.c odd 4 1
1150.2.b.g 4 1.a even 1 1 trivial
1150.2.b.g 4 5.b even 2 1 inner
1840.2.a.n 2 20.e even 4 1
2070.2.a.x 2 15.e even 4 1
5290.2.a.e 2 115.e even 4 1
7360.2.a.bk 2 40.k even 4 1
7360.2.a.bq 2 40.i odd 4 1
9200.2.a.bs 2 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 \(S_{2}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{4} + 11 T_{3}^{2} + 25 \)
\( T_{7}^{4} + 11 T_{7}^{2} + 25 \)
\( T_{11}^{2} - 3 T_{11} - 3 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 25 + 11 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 25 + 11 T^{2} + T^{4} \)
$11$ \( ( -3 - 3 T + T^{2} )^{2} \)
$13$ \( 49 + 35 T^{2} + T^{4} \)
$17$ \( 9 + 15 T^{2} + T^{4} \)
$19$ \( ( 7 + 7 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -12 + 6 T + T^{2} )^{2} \)
$31$ \( ( -35 - 7 T + T^{2} )^{2} \)
$37$ \( ( 16 + T^{2} )^{2} \)
$41$ \( ( 15 + 9 T + T^{2} )^{2} \)
$43$ \( 6400 + 176 T^{2} + T^{4} \)
$47$ \( 3600 + 204 T^{2} + T^{4} \)
$53$ \( ( 36 + T^{2} )^{2} \)
$59$ \( ( 60 - 18 T + T^{2} )^{2} \)
$61$ \( ( -35 - 7 T + T^{2} )^{2} \)
$67$ \( 6400 + 176 T^{2} + T^{4} \)
$71$ \( ( -45 - 3 T + T^{2} )^{2} \)
$73$ \( 35344 + 380 T^{2} + T^{4} \)
$79$ \( ( 8 + T )^{4} \)
$83$ \( ( 36 + T^{2} )^{2} \)
$89$ \( ( -48 + 12 T + T^{2} )^{2} \)
$97$ \( 14161 + 287 T^{2} + T^{4} \)
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