# Properties

 Label 1150.2.a.q Level $1150$ Weight $2$ Character orbit 1150.a Self dual yes Analytic conductor $9.183$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1150,2,Mod(1,1150)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1150, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1150.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1150 = 2 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1150.a (trivial)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: yes Analytic conductor: $$9.18279623245$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.1101.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 9x + 12$$ x^3 - x^2 - 9*x + 12 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 230) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} - q^{8} + (\beta_{2} - \beta_1 + 4) q^{9}+O(q^{10})$$ q - q^2 - b1 * q^3 + q^4 + b1 * q^6 + (b2 + b1 - 1) * q^7 - q^8 + (b2 - b1 + 4) * q^9 $$q - q^{2} - \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + (\beta_{2} + \beta_1 - 1) q^{7} - q^{8} + (\beta_{2} - \beta_1 + 4) q^{9} + (2 \beta_{2} - \beta_1 + 2) q^{11} - \beta_1 q^{12} + (\beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{2} - \beta_1 + 1) q^{14} + q^{16} + (\beta_1 + 2) q^{17} + ( - \beta_{2} + \beta_1 - 4) q^{18} + ( - \beta_{2} - \beta_1 + 1) q^{19} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{21} + ( - 2 \beta_{2} + \beta_1 - 2) q^{22} + q^{23} + \beta_1 q^{24} + ( - \beta_{2} + \beta_1 - 1) q^{26} + ( - \beta_{2} - 2 \beta_1 + 5) q^{27} + (\beta_{2} + \beta_1 - 1) q^{28} + (2 \beta_1 - 2) q^{29} + (\beta_{2} - \beta_1 - 1) q^{31} - q^{32} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{33} + ( - \beta_1 - 2) q^{34} + (\beta_{2} - \beta_1 + 4) q^{36} - 2 \beta_{2} q^{37} + (\beta_{2} + \beta_1 - 1) q^{38} + ( - \beta_{2} - 2 \beta_1 + 5) q^{39} + ( - 2 \beta_{2} - \beta_1) q^{41} + (3 \beta_{2} - 2 \beta_1 + 9) q^{42} - 8 q^{43} + (2 \beta_{2} - \beta_1 + 2) q^{44} - q^{46} + (2 \beta_{2} - 2 \beta_1 + 6) q^{47} - \beta_1 q^{48} + (2 \beta_{2} - \beta_1 + 11) q^{49} + ( - \beta_{2} - \beta_1 - 7) q^{51} + (\beta_{2} - \beta_1 + 1) q^{52} + 6 q^{53} + (\beta_{2} + 2 \beta_1 - 5) q^{54} + ( - \beta_{2} - \beta_1 + 1) q^{56} + (3 \beta_{2} - 2 \beta_1 + 9) q^{57} + ( - 2 \beta_1 + 2) q^{58} + (2 \beta_1 + 4) q^{59} + (4 \beta_{2} - \beta_1 + 2) q^{61} + ( - \beta_{2} + \beta_1 + 1) q^{62} + (\beta_{2} + 8 \beta_1 - 5) q^{63} + q^{64} + (3 \beta_{2} + 3 \beta_1 - 3) q^{66} + ( - 4 \beta_{2} - 4) q^{67} + (\beta_1 + 2) q^{68} - \beta_1 q^{69} + ( - \beta_1 + 4) q^{71} + ( - \beta_{2} + \beta_1 - 4) q^{72} + (2 \beta_{2} - 2 \beta_1 + 4) q^{73} + 2 \beta_{2} q^{74} + ( - \beta_{2} - \beta_1 + 1) q^{76} + ( - \beta_{2} + 8 \beta_1 + 5) q^{77} + (\beta_{2} + 2 \beta_1 - 5) q^{78} + ( - 4 \beta_{2} + 4 \beta_1 - 4) q^{79} + (\beta_{2} - 4 \beta_1 + 4) q^{81} + (2 \beta_{2} + \beta_1) q^{82} + (2 \beta_{2} - 2) q^{83} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{84} + 8 q^{86} + ( - 2 \beta_{2} + 4 \beta_1 - 14) q^{87} + ( - 2 \beta_{2} + \beta_1 - 2) q^{88} + ( - 2 \beta_{2} + 4 \beta_1 + 4) q^{89} + ( - 2 \beta_{2} + 5 \beta_1 - 2) q^{91} + q^{92} + ( - \beta_{2} + 5) q^{93} + ( - 2 \beta_{2} + 2 \beta_1 - 6) q^{94} + \beta_1 q^{96} + (3 \beta_1 + 10) q^{97} + ( - 2 \beta_{2} + \beta_1 - 11) q^{98} + (3 \beta_{2} - 3 \beta_1 + 21) q^{99}+O(q^{100})$$ q - q^2 - b1 * q^3 + q^4 + b1 * q^6 + (b2 + b1 - 1) * q^7 - q^8 + (b2 - b1 + 4) * q^9 + (2*b2 - b1 + 2) * q^11 - b1 * q^12 + (b2 - b1 + 1) * q^13 + (-b2 - b1 + 1) * q^14 + q^16 + (b1 + 2) * q^17 + (-b2 + b1 - 4) * q^18 + (-b2 - b1 + 1) * q^19 + (-3*b2 + 2*b1 - 9) * q^21 + (-2*b2 + b1 - 2) * q^22 + q^23 + b1 * q^24 + (-b2 + b1 - 1) * q^26 + (-b2 - 2*b1 + 5) * q^27 + (b2 + b1 - 1) * q^28 + (2*b1 - 2) * q^29 + (b2 - b1 - 1) * q^31 - q^32 + (-3*b2 - 3*b1 + 3) * q^33 + (-b1 - 2) * q^34 + (b2 - b1 + 4) * q^36 - 2*b2 * q^37 + (b2 + b1 - 1) * q^38 + (-b2 - 2*b1 + 5) * q^39 + (-2*b2 - b1) * q^41 + (3*b2 - 2*b1 + 9) * q^42 - 8 * q^43 + (2*b2 - b1 + 2) * q^44 - q^46 + (2*b2 - 2*b1 + 6) * q^47 - b1 * q^48 + (2*b2 - b1 + 11) * q^49 + (-b2 - b1 - 7) * q^51 + (b2 - b1 + 1) * q^52 + 6 * q^53 + (b2 + 2*b1 - 5) * q^54 + (-b2 - b1 + 1) * q^56 + (3*b2 - 2*b1 + 9) * q^57 + (-2*b1 + 2) * q^58 + (2*b1 + 4) * q^59 + (4*b2 - b1 + 2) * q^61 + (-b2 + b1 + 1) * q^62 + (b2 + 8*b1 - 5) * q^63 + q^64 + (3*b2 + 3*b1 - 3) * q^66 + (-4*b2 - 4) * q^67 + (b1 + 2) * q^68 - b1 * q^69 + (-b1 + 4) * q^71 + (-b2 + b1 - 4) * q^72 + (2*b2 - 2*b1 + 4) * q^73 + 2*b2 * q^74 + (-b2 - b1 + 1) * q^76 + (-b2 + 8*b1 + 5) * q^77 + (b2 + 2*b1 - 5) * q^78 + (-4*b2 + 4*b1 - 4) * q^79 + (b2 - 4*b1 + 4) * q^81 + (2*b2 + b1) * q^82 + (2*b2 - 2) * q^83 + (-3*b2 + 2*b1 - 9) * q^84 + 8 * q^86 + (-2*b2 + 4*b1 - 14) * q^87 + (-2*b2 + b1 - 2) * q^88 + (-2*b2 + 4*b1 + 4) * q^89 + (-2*b2 + 5*b1 - 2) * q^91 + q^92 + (-b2 + 5) * q^93 + (-2*b2 + 2*b1 - 6) * q^94 + b1 * q^96 + (3*b1 + 10) * q^97 + (-2*b2 + b1 - 11) * q^98 + (3*b2 - 3*b1 + 21) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 3 q^{7} - 3 q^{8} + 10 q^{9}+O(q^{10})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 3 * q^7 - 3 * q^8 + 10 * q^9 $$3 q - 3 q^{2} - q^{3} + 3 q^{4} + q^{6} - 3 q^{7} - 3 q^{8} + 10 q^{9} + 3 q^{11} - q^{12} + q^{13} + 3 q^{14} + 3 q^{16} + 7 q^{17} - 10 q^{18} + 3 q^{19} - 22 q^{21} - 3 q^{22} + 3 q^{23} + q^{24} - q^{26} + 14 q^{27} - 3 q^{28} - 4 q^{29} - 5 q^{31} - 3 q^{32} + 9 q^{33} - 7 q^{34} + 10 q^{36} + 2 q^{37} - 3 q^{38} + 14 q^{39} + q^{41} + 22 q^{42} - 24 q^{43} + 3 q^{44} - 3 q^{46} + 14 q^{47} - q^{48} + 30 q^{49} - 21 q^{51} + q^{52} + 18 q^{53} - 14 q^{54} + 3 q^{56} + 22 q^{57} + 4 q^{58} + 14 q^{59} + q^{61} + 5 q^{62} - 8 q^{63} + 3 q^{64} - 9 q^{66} - 8 q^{67} + 7 q^{68} - q^{69} + 11 q^{71} - 10 q^{72} + 8 q^{73} - 2 q^{74} + 3 q^{76} + 24 q^{77} - 14 q^{78} - 4 q^{79} + 7 q^{81} - q^{82} - 8 q^{83} - 22 q^{84} + 24 q^{86} - 36 q^{87} - 3 q^{88} + 18 q^{89} + q^{91} + 3 q^{92} + 16 q^{93} - 14 q^{94} + q^{96} + 33 q^{97} - 30 q^{98} + 57 q^{99}+O(q^{100})$$ 3 * q - 3 * q^2 - q^3 + 3 * q^4 + q^6 - 3 * q^7 - 3 * q^8 + 10 * q^9 + 3 * q^11 - q^12 + q^13 + 3 * q^14 + 3 * q^16 + 7 * q^17 - 10 * q^18 + 3 * q^19 - 22 * q^21 - 3 * q^22 + 3 * q^23 + q^24 - q^26 + 14 * q^27 - 3 * q^28 - 4 * q^29 - 5 * q^31 - 3 * q^32 + 9 * q^33 - 7 * q^34 + 10 * q^36 + 2 * q^37 - 3 * q^38 + 14 * q^39 + q^41 + 22 * q^42 - 24 * q^43 + 3 * q^44 - 3 * q^46 + 14 * q^47 - q^48 + 30 * q^49 - 21 * q^51 + q^52 + 18 * q^53 - 14 * q^54 + 3 * q^56 + 22 * q^57 + 4 * q^58 + 14 * q^59 + q^61 + 5 * q^62 - 8 * q^63 + 3 * q^64 - 9 * q^66 - 8 * q^67 + 7 * q^68 - q^69 + 11 * q^71 - 10 * q^72 + 8 * q^73 - 2 * q^74 + 3 * q^76 + 24 * q^77 - 14 * q^78 - 4 * q^79 + 7 * q^81 - q^82 - 8 * q^83 - 22 * q^84 + 24 * q^86 - 36 * q^87 - 3 * q^88 + 18 * q^89 + q^91 + 3 * q^92 + 16 * q^93 - 14 * q^94 + q^96 + 33 * q^97 - 30 * q^98 + 57 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 9x + 12$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + \nu - 7$$ v^2 + v - 7
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - \beta _1 + 7$$ b2 - b1 + 7

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.68740 1.43163 −3.11903
−1.00000 −2.68740 1.00000 0 2.68740 4.59692 −1.00000 4.22212 0
1.2 −1.00000 −1.43163 1.00000 0 1.43163 −3.08719 −1.00000 −0.950444 0
1.3 −1.00000 3.11903 1.00000 0 −3.11903 −4.50973 −1.00000 6.72833 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$5$$ $$+1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1150.2.a.q 3
4.b odd 2 1 9200.2.a.cf 3
5.b even 2 1 230.2.a.d 3
5.c odd 4 2 1150.2.b.j 6
15.d odd 2 1 2070.2.a.z 3
20.d odd 2 1 1840.2.a.r 3
40.e odd 2 1 7360.2.a.ce 3
40.f even 2 1 7360.2.a.bz 3
115.c odd 2 1 5290.2.a.r 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 5.b even 2 1
1150.2.a.q 3 1.a even 1 1 trivial
1150.2.b.j 6 5.c odd 4 2
1840.2.a.r 3 20.d odd 2 1
2070.2.a.z 3 15.d odd 2 1
5290.2.a.r 3 115.c odd 2 1
7360.2.a.bz 3 40.f even 2 1
7360.2.a.ce 3 40.e odd 2 1
9200.2.a.cf 3 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1150))$$:

 $$T_{3}^{3} + T_{3}^{2} - 9T_{3} - 12$$ T3^3 + T3^2 - 9*T3 - 12 $$T_{7}^{3} + 3T_{7}^{2} - 21T_{7} - 64$$ T7^3 + 3*T7^2 - 21*T7 - 64 $$T_{11}^{3} - 3T_{11}^{2} - 39T_{11} + 144$$ T11^3 - 3*T11^2 - 39*T11 + 144

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{3}$$
$3$ $$T^{3} + T^{2} - 9T - 12$$
$5$ $$T^{3}$$
$7$ $$T^{3} + 3 T^{2} + \cdots - 64$$
$11$ $$T^{3} - 3 T^{2} + \cdots + 144$$
$13$ $$T^{3} - T^{2} + \cdots + 18$$
$17$ $$T^{3} - 7 T^{2} + \cdots + 18$$
$19$ $$T^{3} - 3 T^{2} + \cdots + 64$$
$23$ $$(T - 1)^{3}$$
$29$ $$T^{3} + 4 T^{2} + \cdots + 24$$
$31$ $$T^{3} + 5 T^{2} + \cdots - 8$$
$37$ $$T^{3} - 2 T^{2} + \cdots + 32$$
$41$ $$T^{3} - T^{2} + \cdots + 186$$
$43$ $$(T + 8)^{3}$$
$47$ $$T^{3} - 14 T^{2} + \cdots + 288$$
$53$ $$(T - 6)^{3}$$
$59$ $$T^{3} - 14 T^{2} + \cdots + 144$$
$61$ $$T^{3} - T^{2} + \cdots + 526$$
$67$ $$T^{3} + 8 T^{2} + \cdots - 384$$
$71$ $$T^{3} - 11 T^{2} + \cdots - 24$$
$73$ $$T^{3} - 8 T^{2} + \cdots + 248$$
$79$ $$T^{3} + 4 T^{2} + \cdots - 1152$$
$83$ $$T^{3} + 8 T^{2} + \cdots - 96$$
$89$ $$T^{3} - 18 T^{2} + \cdots + 1152$$
$97$ $$T^{3} - 33 T^{2} + \cdots - 166$$