Properties

Label 1150.2.b.i
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: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

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

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
0.618034i
1.61803i
1.61803i
0.618034i
1.00000i 0.618034i −1.00000 0 −0.618034 1.61803i 1.00000i 2.61803 0
599.2 1.00000i 1.61803i −1.00000 0 1.61803 0.618034i 1.00000i 0.381966 0
599.3 1.00000i 1.61803i −1.00000 0 1.61803 0.618034i 1.00000i 0.381966 0
599.4 1.00000i 0.618034i −1.00000 0 −0.618034 1.61803i 1.00000i 2.61803 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.i 4
5.b even 2 1 inner 1150.2.b.i 4
5.c odd 4 1 230.2.a.c 2
5.c odd 4 1 1150.2.a.j 2
15.e even 4 1 2070.2.a.u 2
20.e even 4 1 1840.2.a.l 2
20.e even 4 1 9200.2.a.bu 2
40.i odd 4 1 7360.2.a.bh 2
40.k even 4 1 7360.2.a.bn 2
115.e even 4 1 5290.2.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.c 2 5.c odd 4 1
1150.2.a.j 2 5.c odd 4 1
1150.2.b.i 4 1.a even 1 1 trivial
1150.2.b.i 4 5.b even 2 1 inner
1840.2.a.l 2 20.e even 4 1
2070.2.a.u 2 15.e even 4 1
5290.2.a.o 2 115.e even 4 1
7360.2.a.bh 2 40.i odd 4 1
7360.2.a.bn 2 40.k even 4 1
9200.2.a.bu 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 \(S_{2}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{4} + 3 T_{3}^{2} + 1 \)
\( T_{7}^{4} + 3 T_{7}^{2} + 1 \)
\( T_{11}^{2} - T_{11} - 11 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 3 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + 3 T^{2} + T^{4} \)
$11$ \( ( -11 - T + T^{2} )^{2} \)
$13$ \( 841 + 67 T^{2} + T^{4} \)
$17$ \( 961 + 63 T^{2} + T^{4} \)
$19$ \( ( -9 - 3 T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( 44 - 14 T + T^{2} )^{2} \)
$31$ \( ( -19 - 7 T + T^{2} )^{2} \)
$37$ \( 256 + 48 T^{2} + T^{4} \)
$41$ \( ( -41 + 9 T + T^{2} )^{2} \)
$43$ \( T^{4} \)
$47$ \( 1296 + 108 T^{2} + T^{4} \)
$53$ \( 16 + 72 T^{2} + T^{4} \)
$59$ \( ( -20 - 10 T + T^{2} )^{2} \)
$61$ \( ( -59 + 3 T + T^{2} )^{2} \)
$67$ \( 6400 + 240 T^{2} + T^{4} \)
$71$ \( ( -29 - 3 T + T^{2} )^{2} \)
$73$ \( 16 + 12 T^{2} + T^{4} \)
$79$ \( ( 16 + 12 T + T^{2} )^{2} \)
$83$ \( 5776 + 168 T^{2} + T^{4} \)
$89$ \( ( 16 - 12 T + T^{2} )^{2} \)
$97$ \( 32761 + 367 T^{2} + T^{4} \)
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