# 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 $2$

# Related objects

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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

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