Properties

Label 1150.2.b.f
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: \(\Q(i, \sqrt{13})\)
Defining polynomial: \(x^{4} + 7 x^{2} + 9\)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 7 x^{2} + 9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 4 \nu \)\()/3\)
\(\beta_{3}\)\(=\)\( \nu^{2} + 4 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} - 4\)
\(\nu^{3}\)\(=\)\(3 \beta_{2} - 4 \beta_{1}\)

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
2.30278i
1.30278i
1.30278i
2.30278i
1.00000i 3.30278i −1.00000 0 −3.30278 0.302776i 1.00000i −7.90833 0
599.2 1.00000i 0.302776i −1.00000 0 0.302776 3.30278i 1.00000i 2.90833 0
599.3 1.00000i 0.302776i −1.00000 0 0.302776 3.30278i 1.00000i 2.90833 0
599.4 1.00000i 3.30278i −1.00000 0 −3.30278 0.302776i 1.00000i −7.90833 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.f 4
5.b even 2 1 inner 1150.2.b.f 4
5.c odd 4 1 230.2.a.b 2
5.c odd 4 1 1150.2.a.m 2
15.e even 4 1 2070.2.a.w 2
20.e even 4 1 1840.2.a.j 2
20.e even 4 1 9200.2.a.ca 2
40.i odd 4 1 7360.2.a.bc 2
40.k even 4 1 7360.2.a.bu 2
115.e even 4 1 5290.2.a.j 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 5.c odd 4 1
1150.2.a.m 2 5.c odd 4 1
1150.2.b.f 4 1.a even 1 1 trivial
1150.2.b.f 4 5.b even 2 1 inner
1840.2.a.j 2 20.e even 4 1
2070.2.a.w 2 15.e even 4 1
5290.2.a.j 2 115.e even 4 1
7360.2.a.bc 2 40.i odd 4 1
7360.2.a.bu 2 40.k even 4 1
9200.2.a.ca 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 \(S_{2}^{\mathrm{new}}(1150, [\chi])\):

\( T_{3}^{4} + 11 T_{3}^{2} + 1 \)
\( T_{7}^{4} + 11 T_{7}^{2} + 1 \)
\( T_{11}^{2} + 7 T_{11} + 9 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T^{2} )^{2} \)
$3$ \( 1 + 11 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 1 + 11 T^{2} + T^{4} \)
$11$ \( ( 9 + 7 T + T^{2} )^{2} \)
$13$ \( 1 + 11 T^{2} + T^{4} \)
$17$ \( 729 + 63 T^{2} + T^{4} \)
$19$ \( ( -29 + T + T^{2} )^{2} \)
$23$ \( ( 1 + T^{2} )^{2} \)
$29$ \( ( -12 + 2 T + T^{2} )^{2} \)
$31$ \( ( -23 + 5 T + T^{2} )^{2} \)
$37$ \( ( 64 + T^{2} )^{2} \)
$41$ \( ( -9 + 9 T + T^{2} )^{2} \)
$43$ \( 2304 + 112 T^{2} + T^{4} \)
$47$ \( 144 + 28 T^{2} + T^{4} \)
$53$ \( 1296 + 136 T^{2} + T^{4} \)
$59$ \( ( 36 - 14 T + T^{2} )^{2} \)
$61$ \( ( -75 - 5 T + T^{2} )^{2} \)
$67$ \( ( 16 + T^{2} )^{2} \)
$71$ \( ( 207 + 29 T + T^{2} )^{2} \)
$73$ \( 8464 + 284 T^{2} + T^{4} \)
$79$ \( ( -208 + T^{2} )^{2} \)
$83$ \( 1296 + 136 T^{2} + T^{4} \)
$89$ \( T^{4} \)
$97$ \( 289 + 47 T^{2} + T^{4} \)
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