# Properties

 Label 1150.2.a.m Level $1150$ Weight $2$ Character orbit 1150.a Self dual yes Analytic conductor $9.183$ Analytic rank $1$ Dimension $2$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} + (\beta - 2) q^{7} + q^{8} + (3 \beta + 1) q^{9}+O(q^{10})$$ q + q^2 + (-b - 1) * q^3 + q^4 + (-b - 1) * q^6 + (b - 2) * q^7 + q^8 + (3*b + 1) * q^9 $$q + q^{2} + ( - \beta - 1) q^{3} + q^{4} + ( - \beta - 1) q^{6} + (\beta - 2) q^{7} + q^{8} + (3 \beta + 1) q^{9} + ( - \beta - 3) q^{11} + ( - \beta - 1) q^{12} + (\beta - 2) q^{13} + (\beta - 2) q^{14} + q^{16} + (3 \beta - 3) q^{17} + (3 \beta + 1) q^{18} + ( - 3 \beta + 2) q^{19} - q^{21} + ( - \beta - 3) q^{22} + q^{23} + ( - \beta - 1) q^{24} + (\beta - 2) q^{26} + ( - 4 \beta - 7) q^{27} + (\beta - 2) q^{28} + 2 \beta q^{29} + (3 \beta - 4) q^{31} + q^{32} + (5 \beta + 6) q^{33} + (3 \beta - 3) q^{34} + (3 \beta + 1) q^{36} - 8 q^{37} + ( - 3 \beta + 2) q^{38} - q^{39} + ( - 3 \beta - 3) q^{41} - q^{42} + ( - 4 \beta + 4) q^{43} + ( - \beta - 3) q^{44} + q^{46} - 2 \beta q^{47} + ( - \beta - 1) q^{48} - 3 \beta q^{49} + ( - 3 \beta - 6) q^{51} + (\beta - 2) q^{52} + ( - 4 \beta + 6) q^{53} + ( - 4 \beta - 7) q^{54} + (\beta - 2) q^{56} + (4 \beta + 7) q^{57} + 2 \beta q^{58} + ( - 2 \beta - 6) q^{59} + ( - 5 \beta + 5) q^{61} + (3 \beta - 4) q^{62} + ( - 2 \beta + 7) q^{63} + q^{64} + (5 \beta + 6) q^{66} + 4 q^{67} + (3 \beta - 3) q^{68} + ( - \beta - 1) q^{69} + (\beta - 15) q^{71} + (3 \beta + 1) q^{72} + ( - 6 \beta - 2) q^{73} - 8 q^{74} + ( - 3 \beta + 2) q^{76} + ( - 2 \beta + 3) q^{77} - q^{78} + (8 \beta - 4) q^{79} + (6 \beta + 16) q^{81} + ( - 3 \beta - 3) q^{82} + (4 \beta - 6) q^{83} - q^{84} + ( - 4 \beta + 4) q^{86} + ( - 4 \beta - 6) q^{87} + ( - \beta - 3) q^{88} + ( - 3 \beta + 7) q^{91} + q^{92} + ( - 2 \beta - 5) q^{93} - 2 \beta q^{94} + ( - \beta - 1) q^{96} + (\beta - 5) q^{97} - 3 \beta q^{98} + ( - 13 \beta - 12) q^{99} +O(q^{100})$$ q + q^2 + (-b - 1) * q^3 + q^4 + (-b - 1) * q^6 + (b - 2) * q^7 + q^8 + (3*b + 1) * q^9 + (-b - 3) * q^11 + (-b - 1) * q^12 + (b - 2) * q^13 + (b - 2) * q^14 + q^16 + (3*b - 3) * q^17 + (3*b + 1) * q^18 + (-3*b + 2) * q^19 - q^21 + (-b - 3) * q^22 + q^23 + (-b - 1) * q^24 + (b - 2) * q^26 + (-4*b - 7) * q^27 + (b - 2) * q^28 + 2*b * q^29 + (3*b - 4) * q^31 + q^32 + (5*b + 6) * q^33 + (3*b - 3) * q^34 + (3*b + 1) * q^36 - 8 * q^37 + (-3*b + 2) * q^38 - q^39 + (-3*b - 3) * q^41 - q^42 + (-4*b + 4) * q^43 + (-b - 3) * q^44 + q^46 - 2*b * q^47 + (-b - 1) * q^48 - 3*b * q^49 + (-3*b - 6) * q^51 + (b - 2) * q^52 + (-4*b + 6) * q^53 + (-4*b - 7) * q^54 + (b - 2) * q^56 + (4*b + 7) * q^57 + 2*b * q^58 + (-2*b - 6) * q^59 + (-5*b + 5) * q^61 + (3*b - 4) * q^62 + (-2*b + 7) * q^63 + q^64 + (5*b + 6) * q^66 + 4 * q^67 + (3*b - 3) * q^68 + (-b - 1) * q^69 + (b - 15) * q^71 + (3*b + 1) * q^72 + (-6*b - 2) * q^73 - 8 * q^74 + (-3*b + 2) * q^76 + (-2*b + 3) * q^77 - q^78 + (8*b - 4) * q^79 + (6*b + 16) * q^81 + (-3*b - 3) * q^82 + (4*b - 6) * q^83 - q^84 + (-4*b + 4) * q^86 + (-4*b - 6) * q^87 + (-b - 3) * q^88 + (-3*b + 7) * q^91 + q^92 + (-2*b - 5) * q^93 - 2*b * q^94 + (-b - 1) * q^96 + (b - 5) * q^97 - 3*b * q^98 + (-13*b - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 3 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 3 * q^7 + 2 * q^8 + 5 * q^9 $$2 q + 2 q^{2} - 3 q^{3} + 2 q^{4} - 3 q^{6} - 3 q^{7} + 2 q^{8} + 5 q^{9} - 7 q^{11} - 3 q^{12} - 3 q^{13} - 3 q^{14} + 2 q^{16} - 3 q^{17} + 5 q^{18} + q^{19} - 2 q^{21} - 7 q^{22} + 2 q^{23} - 3 q^{24} - 3 q^{26} - 18 q^{27} - 3 q^{28} + 2 q^{29} - 5 q^{31} + 2 q^{32} + 17 q^{33} - 3 q^{34} + 5 q^{36} - 16 q^{37} + q^{38} - 2 q^{39} - 9 q^{41} - 2 q^{42} + 4 q^{43} - 7 q^{44} + 2 q^{46} - 2 q^{47} - 3 q^{48} - 3 q^{49} - 15 q^{51} - 3 q^{52} + 8 q^{53} - 18 q^{54} - 3 q^{56} + 18 q^{57} + 2 q^{58} - 14 q^{59} + 5 q^{61} - 5 q^{62} + 12 q^{63} + 2 q^{64} + 17 q^{66} + 8 q^{67} - 3 q^{68} - 3 q^{69} - 29 q^{71} + 5 q^{72} - 10 q^{73} - 16 q^{74} + q^{76} + 4 q^{77} - 2 q^{78} + 38 q^{81} - 9 q^{82} - 8 q^{83} - 2 q^{84} + 4 q^{86} - 16 q^{87} - 7 q^{88} + 11 q^{91} + 2 q^{92} - 12 q^{93} - 2 q^{94} - 3 q^{96} - 9 q^{97} - 3 q^{98} - 37 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 - 3 * q^3 + 2 * q^4 - 3 * q^6 - 3 * q^7 + 2 * q^8 + 5 * q^9 - 7 * q^11 - 3 * q^12 - 3 * q^13 - 3 * q^14 + 2 * q^16 - 3 * q^17 + 5 * q^18 + q^19 - 2 * q^21 - 7 * q^22 + 2 * q^23 - 3 * q^24 - 3 * q^26 - 18 * q^27 - 3 * q^28 + 2 * q^29 - 5 * q^31 + 2 * q^32 + 17 * q^33 - 3 * q^34 + 5 * q^36 - 16 * q^37 + q^38 - 2 * q^39 - 9 * q^41 - 2 * q^42 + 4 * q^43 - 7 * q^44 + 2 * q^46 - 2 * q^47 - 3 * q^48 - 3 * q^49 - 15 * q^51 - 3 * q^52 + 8 * q^53 - 18 * q^54 - 3 * q^56 + 18 * q^57 + 2 * q^58 - 14 * q^59 + 5 * q^61 - 5 * q^62 + 12 * q^63 + 2 * q^64 + 17 * q^66 + 8 * q^67 - 3 * q^68 - 3 * q^69 - 29 * q^71 + 5 * q^72 - 10 * q^73 - 16 * q^74 + q^76 + 4 * q^77 - 2 * q^78 + 38 * q^81 - 9 * q^82 - 8 * q^83 - 2 * q^84 + 4 * q^86 - 16 * q^87 - 7 * q^88 + 11 * q^91 + 2 * q^92 - 12 * q^93 - 2 * q^94 - 3 * q^96 - 9 * q^97 - 3 * q^98 - 37 * q^99

## 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
 2.30278 −1.30278
1.00000 −3.30278 1.00000 0 −3.30278 0.302776 1.00000 7.90833 0
1.2 1.00000 0.302776 1.00000 0 0.302776 −3.30278 1.00000 −2.90833 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.m 2
4.b odd 2 1 9200.2.a.ca 2
5.b even 2 1 230.2.a.b 2
5.c odd 4 2 1150.2.b.f 4
15.d odd 2 1 2070.2.a.w 2
20.d odd 2 1 1840.2.a.j 2
40.e odd 2 1 7360.2.a.bu 2
40.f even 2 1 7360.2.a.bc 2
115.c odd 2 1 5290.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 5.b even 2 1
1150.2.a.m 2 1.a even 1 1 trivial
1150.2.b.f 4 5.c odd 4 2
1840.2.a.j 2 20.d odd 2 1
2070.2.a.w 2 15.d odd 2 1
5290.2.a.j 2 115.c odd 2 1
7360.2.a.bc 2 40.f even 2 1
7360.2.a.bu 2 40.e odd 2 1
9200.2.a.ca 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1150))$$:

 $$T_{3}^{2} + 3T_{3} - 1$$ T3^2 + 3*T3 - 1 $$T_{7}^{2} + 3T_{7} - 1$$ T7^2 + 3*T7 - 1 $$T_{11}^{2} + 7T_{11} + 9$$ T11^2 + 7*T11 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{2}$$
$3$ $$T^{2} + 3T - 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 3T - 1$$
$11$ $$T^{2} + 7T + 9$$
$13$ $$T^{2} + 3T - 1$$
$17$ $$T^{2} + 3T - 27$$
$19$ $$T^{2} - T - 29$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} - 2T - 12$$
$31$ $$T^{2} + 5T - 23$$
$37$ $$(T + 8)^{2}$$
$41$ $$T^{2} + 9T - 9$$
$43$ $$T^{2} - 4T - 48$$
$47$ $$T^{2} + 2T - 12$$
$53$ $$T^{2} - 8T - 36$$
$59$ $$T^{2} + 14T + 36$$
$61$ $$T^{2} - 5T - 75$$
$67$ $$(T - 4)^{2}$$
$71$ $$T^{2} + 29T + 207$$
$73$ $$T^{2} + 10T - 92$$
$79$ $$T^{2} - 208$$
$83$ $$T^{2} + 8T - 36$$
$89$ $$T^{2}$$
$97$ $$T^{2} + 9T + 17$$