# 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 $2$

# 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: $$6$$ 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: 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.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

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