# Properties

 Label 1045.2.a.f Level $1045$ Weight $2$ Character orbit 1045.a Self dual yes Analytic conductor $8.344$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1045,2,Mod(1,1045)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1045, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1045.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1045 = 5 \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1045.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: $$8.34436701122$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.7281497.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ x^6 - 2*x^5 - 5*x^4 + 7*x^3 + 6*x^2 - 2*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{4} q^{2} + \beta_{3} q^{3} + (\beta_{5} + \beta_{4}) q^{4} - q^{5} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{9}+O(q^{10})$$ q - b4 * q^2 + b3 * q^3 + (b5 + b4) * q^4 - q^5 + b2 * q^6 + (-b3 - b1 + 1) * q^7 + (-b5 - b4 - b1 - 1) * q^8 + (-b5 - b3 + b1) * q^9 $$q - \beta_{4} q^{2} + \beta_{3} q^{3} + (\beta_{5} + \beta_{4}) q^{4} - q^{5} + \beta_{2} q^{6} + ( - \beta_{3} - \beta_1 + 1) q^{7} + ( - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{8} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{9} + \beta_{4} q^{10} - q^{11} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots - 1) q^{12}+ \cdots + (\beta_{5} + \beta_{3} - \beta_1) q^{99}+O(q^{100})$$ q - b4 * q^2 + b3 * q^3 + (b5 + b4) * q^4 - q^5 + b2 * q^6 + (-b3 - b1 + 1) * q^7 + (-b5 - b4 - b1 - 1) * q^8 + (-b5 - b3 + b1) * q^9 + b4 * q^10 - q^11 + (b4 - b3 - b2 + 2*b1 - 1) * q^12 + (b4 + b3 - 1) * q^13 + (-b4 - 2*b2 - 1) * q^14 - b3 * q^15 + (-b5 + 2*b4 - b2 + b1) * q^16 + (b4 - b3 - 2*b2 + 2*b1 - 1) * q^17 + (2*b4 + b1) * q^18 - q^19 + (-b5 - b4) * q^20 + (b3 - b2 - b1 - 3) * q^21 + b4 * q^22 + (-b5 + b4 - 2*b3 + 2*b2 + b1) * q^23 + (-b5 - b4 - b3 - 2*b1 + 1) * q^24 + q^25 + (-b5 + b2 - 2) * q^26 + (2*b5 - b4 + b2 - 3*b1 - 2) * q^27 + (b5 + 2*b2 - 2*b1 + 2) * q^28 + (b5 - b3 + b2 + 3*b1 - 3) * q^29 - b2 * q^30 + (-b4 + b3 - 3*b1 - 2) * q^31 + (b4 - b3 + 2*b2 + b1 - 1) * q^32 - b3 * q^33 + (-b5 - 2*b4 - 2*b3 + 3*b2 - 4*b1 + 2) * q^34 + (b3 + b1 - 1) * q^35 + (-2*b4 + 2*b3 + b2 - 2*b1 - 3) * q^36 + (b5 + 3*b4 + b3 - b2 + b1 - 2) * q^37 + b4 * q^38 + (-b5 - 2*b3 - b2 + b1 + 3) * q^39 + (b5 + b4 + b1 + 1) * q^40 + (-b5 - 2*b4 + b3 - 2*b1 - 2) * q^41 + (2*b4 - b3 + b2 - 2*b1) * q^42 + (b5 + b4 - b3 + 2*b2 + b1) * q^43 + (-b5 - b4) * q^44 + (b5 + b3 - b1) * q^45 + (-b5 + 3*b4 + 2*b3 - 3*b2 + 5*b1 - 4) * q^46 + (b5 - 2*b4 + 2*b2 - 2*b1 - 2) * q^47 + (b5 + 2*b3 - b2 - 3*b1 + 1) * q^48 + (b5 - b3 + 3*b2 - 1) * q^49 - b4 * q^50 + (3*b5 + 2*b4 + 2*b3 + b2 - 3*b1 - 3) * q^51 + (3*b4 - b3 - b2 + 3*b1) * q^52 + (-b4 + b3 + 2*b1 - 3) * q^53 + (b5 + b3 - 4*b2) * q^54 + q^55 + (2*b3 + 3*b1 - 1) * q^56 - b3 * q^57 + (2*b4 + b3 + b2 + b1 + 3) * q^58 + (-2*b5 - b4 + b3 - 2*b2 - 2*b1 - 3) * q^59 + (-b4 + b3 + b2 - 2*b1 + 1) * q^60 + (3*b4 - b3 - 3*b2 + 2*b1 + 1) * q^61 + (b5 + 3*b4 - 2*b2 - 1) * q^62 + (-2*b5 + b4 - 2*b3 - b2 + 3*b1) * q^63 + (b5 - 2*b4 + 2*b3 + 2*b1 - 3) * q^64 + (-b4 - b3 + 1) * q^65 - b2 * q^66 + (b5 + 2*b4 - 2*b3 - b2 + 6*b1 - 2) * q^67 + (2*b5 + 3*b4 + 5*b3 - 5*b2 + 3*b1 - 2) * q^68 + (3*b5 - 3*b4 + 4*b3 - 2*b1 - 5) * q^69 + (b4 + 2*b2 + 1) * q^70 + (2*b5 - 2*b4 - b2 - 3*b1 - 1) * q^71 + (2*b5 + 2*b4 + b3 - b2 + 1) * q^72 + (b5 + 2*b4 + 2*b2 + b1 - 1) * q^73 + (-3*b5 - 4*b4 - b3 + 3*b2 - 3*b1 - 3) * q^74 + b3 * q^75 + (-b5 - b4) * q^76 + (b3 + b1 - 1) * q^77 + (-2*b4 - b3 - b1 + 1) * q^78 + (b5 - 3*b4 + 3*b3 + 3*b2 - b1 + 2) * q^79 + (b5 - 2*b4 + b2 - b1) * q^80 + (b4 - 4*b3 - 2*b2 + 2*b1 - 2) * q^81 + (2*b5 + 6*b4 - b2 + b1 + 1) * q^82 + (2*b5 - 5*b4 + 3*b3 - 4*b1 + 1) * q^83 + (-2*b5 - b4 - b3 - 2*b2 + 4*b1 - 1) * q^84 + (-b4 + b3 + 2*b2 - 2*b1 + 1) * q^85 + (-b5 - b4 + 2*b3 - 2*b2 + 3*b1 - 2) * q^86 + (4*b5 + 3*b2 + 2*b1 - 4) * q^87 + (b5 + b4 + b1 + 1) * q^88 + (-3*b5 - 2*b4 + b2 - 3*b1 - 1) * q^89 + (-2*b4 - b1) * q^90 + (b4 + 2*b3 + b2 - 3) * q^91 + (-b5 - 2*b4 + b3 + 6*b2 - 7*b1 + 1) * q^92 + (-4*b5 - 6*b3 - 2*b2 + b1 + 3) * q^93 + (2*b5 + 4*b4 + 2*b3 - 4*b2 + 3*b1 + 1) * q^94 + q^95 + (2*b5 - 2*b4 + b3 + b1 - 3) * q^96 + (-4*b5 - 5*b4 - 2*b3 - 2*b2 - 2*b1 + 3) * q^97 + (2*b4 + 3*b3 - 4*b2 + 5*b1 - 2) * q^98 + (b5 + b3 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9}+O(q^{10})$$ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 $$6 q - 2 q^{2} - q^{3} + 4 q^{4} - 6 q^{5} + 5 q^{7} - 12 q^{8} + q^{9} + 2 q^{10} - 6 q^{11} + q^{12} - 5 q^{13} - 8 q^{14} + q^{15} + 4 q^{16} + q^{17} + 6 q^{18} - 6 q^{19} - 4 q^{20} - 21 q^{21} + 2 q^{22} + 4 q^{23} - q^{24} + 6 q^{25} - 14 q^{26} - 16 q^{27} + 10 q^{28} - 9 q^{29} - 21 q^{31} - q^{32} + q^{33} - 5 q^{35} - 28 q^{36} - 3 q^{37} + 2 q^{38} + 20 q^{39} + 12 q^{40} - 23 q^{41} + q^{42} + 7 q^{43} - 4 q^{44} - q^{45} - 12 q^{46} - 18 q^{47} - 3 q^{49} - 2 q^{50} - 16 q^{51} + 13 q^{52} - 17 q^{53} + q^{54} + 6 q^{55} - 2 q^{56} + q^{57} + 23 q^{58} - 29 q^{59} - q^{60} + 17 q^{61} + 2 q^{62} + 6 q^{63} - 18 q^{64} + 5 q^{65} + 8 q^{67} - q^{68} - 38 q^{69} + 8 q^{70} - 12 q^{71} + 13 q^{72} + 2 q^{73} - 37 q^{74} - q^{75} - 4 q^{76} - 5 q^{77} + q^{78} + 3 q^{79} - 4 q^{80} - 2 q^{81} + 24 q^{82} - 11 q^{83} - 3 q^{84} - q^{85} - 12 q^{86} - 12 q^{87} + 12 q^{88} - 22 q^{89} - 6 q^{90} - 18 q^{91} - 15 q^{92} + 18 q^{93} + 22 q^{94} + 6 q^{95} - 17 q^{96} - 2 q^{97} - q^{98} - q^{99}+O(q^{100})$$ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - 6 * q^5 + 5 * q^7 - 12 * q^8 + q^9 + 2 * q^10 - 6 * q^11 + q^12 - 5 * q^13 - 8 * q^14 + q^15 + 4 * q^16 + q^17 + 6 * q^18 - 6 * q^19 - 4 * q^20 - 21 * q^21 + 2 * q^22 + 4 * q^23 - q^24 + 6 * q^25 - 14 * q^26 - 16 * q^27 + 10 * q^28 - 9 * q^29 - 21 * q^31 - q^32 + q^33 - 5 * q^35 - 28 * q^36 - 3 * q^37 + 2 * q^38 + 20 * q^39 + 12 * q^40 - 23 * q^41 + q^42 + 7 * q^43 - 4 * q^44 - q^45 - 12 * q^46 - 18 * q^47 - 3 * q^49 - 2 * q^50 - 16 * q^51 + 13 * q^52 - 17 * q^53 + q^54 + 6 * q^55 - 2 * q^56 + q^57 + 23 * q^58 - 29 * q^59 - q^60 + 17 * q^61 + 2 * q^62 + 6 * q^63 - 18 * q^64 + 5 * q^65 + 8 * q^67 - q^68 - 38 * q^69 + 8 * q^70 - 12 * q^71 + 13 * q^72 + 2 * q^73 - 37 * q^74 - q^75 - 4 * q^76 - 5 * q^77 + q^78 + 3 * q^79 - 4 * q^80 - 2 * q^81 + 24 * q^82 - 11 * q^83 - 3 * q^84 - q^85 - 12 * q^86 - 12 * q^87 + 12 * q^88 - 22 * q^89 - 6 * q^90 - 18 * q^91 - 15 * q^92 + 18 * q^93 + 22 * q^94 + 6 * q^95 - 17 * q^96 - 2 * q^97 - q^98 - q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2 $$\beta_{3}$$ $$=$$ $$\nu^{5} - 2\nu^{4} - 4\nu^{3} + 6\nu^{2} + 2\nu - 1$$ v^5 - 2*v^4 - 4*v^3 + 6*v^2 + 2*v - 1 $$\beta_{4}$$ $$=$$ $$\nu^{5} - 2\nu^{4} - 5\nu^{3} + 7\nu^{2} + 5\nu - 1$$ v^5 - 2*v^4 - 5*v^3 + 7*v^2 + 5*v - 1 $$\beta_{5}$$ $$=$$ $$-\nu^{5} + 3\nu^{4} + 3\nu^{3} - 11\nu^{2} + 4$$ -v^5 + 3*v^4 + 3*v^3 - 11*v^2 + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 2$$ -b4 + b3 + b2 + 4*b1 + 2 $$\nu^{4}$$ $$=$$ $$\beta_{5} - \beta_{4} + 2\beta_{3} + 6\beta_{2} + 7\beta _1 + 9$$ b5 - b4 + 2*b3 + 6*b2 + 7*b1 + 9 $$\nu^{5}$$ $$=$$ $$2\beta_{5} - 6\beta_{4} + 9\beta_{3} + 10\beta_{2} + 22\beta _1 + 15$$ 2*b5 - 6*b4 + 9*b3 + 10*b2 + 22*b1 + 15

## 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
 0.497517 −1.79049 −0.748369 1.77015 2.59744 −0.326248
−2.51246 0.895533 4.31247 −1.00000 −2.24999 −0.393051 −5.80998 −2.19802 2.51246
1.2 −2.23198 −1.34246 2.98176 −1.00000 2.99635 4.13295 −2.19127 −1.19780 2.23198
1.3 −0.412130 1.67805 −1.83015 −1.00000 −0.691575 0.0703171 1.57852 −0.184142 0.412130
1.4 0.205229 −3.10246 −1.95788 −1.00000 −0.636714 2.33231 −0.812271 6.62527 −0.205229
1.5 1.21244 1.77266 −0.529980 −1.00000 2.14925 −3.37010 −3.06746 0.142317 −1.21244
1.6 1.73890 −0.901323 1.02379 −1.00000 −1.56731 2.22757 −1.69754 −2.18762 −1.73890
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$11$$ $$+1$$
$$19$$ $$+1$$

## 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 1045.2.a.f 6
3.b odd 2 1 9405.2.a.z 6
5.b even 2 1 5225.2.a.l 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1045.2.a.f 6 1.a even 1 1 trivial
5225.2.a.l 6 5.b even 2 1
9405.2.a.z 6 3.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 2T_{2}^{5} - 6T_{2}^{4} - 8T_{2}^{3} + 11T_{2}^{2} + 3T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1045))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6} + 2 T^{5} + \cdots - 1$$
$3$ $$T^{6} + T^{5} + \cdots - 10$$
$5$ $$(T + 1)^{6}$$
$7$ $$T^{6} - 5 T^{5} + \cdots + 2$$
$11$ $$(T + 1)^{6}$$
$13$ $$T^{6} + 5 T^{5} + \cdots + 2$$
$17$ $$T^{6} - T^{5} + \cdots - 1360$$
$19$ $$(T + 1)^{6}$$
$23$ $$T^{6} - 4 T^{5} + \cdots - 8912$$
$29$ $$T^{6} + 9 T^{5} + \cdots + 3074$$
$31$ $$T^{6} + 21 T^{5} + \cdots - 20$$
$37$ $$T^{6} + 3 T^{5} + \cdots - 592$$
$41$ $$T^{6} + 23 T^{5} + \cdots - 4210$$
$43$ $$T^{6} - 7 T^{5} + \cdots - 326$$
$47$ $$T^{6} + 18 T^{5} + \cdots - 620$$
$53$ $$T^{6} + 17 T^{5} + \cdots - 4084$$
$59$ $$T^{6} + 29 T^{5} + \cdots - 54968$$
$61$ $$T^{6} - 17 T^{5} + \cdots + 2294$$
$67$ $$T^{6} - 8 T^{5} + \cdots - 73006$$
$71$ $$T^{6} + 12 T^{5} + \cdots - 5912$$
$73$ $$T^{6} - 2 T^{5} + \cdots - 4000$$
$79$ $$T^{6} - 3 T^{5} + \cdots + 28264$$
$83$ $$T^{6} + 11 T^{5} + \cdots - 82582$$
$89$ $$T^{6} + 22 T^{5} + \cdots + 3286$$
$97$ $$T^{6} + 2 T^{5} + \cdots - 967268$$