# Properties

 Label 1305.2.d.e Level $1305$ Weight $2$ Character orbit 1305.d Analytic conductor $10.420$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(811,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("1305.811");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.d (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$10.4204774638$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 13x^{8} + 56x^{6} + 89x^{4} + 41x^{2} + 4$$ x^10 + 13*x^8 + 56*x^6 + 89*x^4 + 41*x^2 + 4 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} + \beta_{7} q^{7} + \beta_{3} q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 - 1) * q^4 - q^5 + b7 * q^7 + b3 * q^8 $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} - q^{5} + \beta_{7} q^{7} + \beta_{3} q^{8} - \beta_1 q^{10} + ( - \beta_{9} - \beta_{6}) q^{11} + (\beta_{4} - \beta_{2} + 1) q^{13} - \beta_{6} q^{14} + \beta_{4} q^{16} + ( - \beta_{9} + \beta_{8} - \beta_{3}) q^{17} + (\beta_{9} - \beta_{8} - \beta_{6} + \beta_1) q^{19} + ( - \beta_{2} + 1) q^{20} + ( - \beta_{7} + \beta_{5} + 1) q^{22} + (\beta_{7} + \beta_{5} - 2 \beta_{4} + \cdots - 1) q^{23}+ \cdots + (\beta_{9} + \beta_{8} + \cdots - 4 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 - 1) * q^4 - q^5 + b7 * q^7 + b3 * q^8 - b1 * q^10 + (-b9 - b6) * q^11 + (b4 - b2 + 1) * q^13 - b6 * q^14 + b4 * q^16 + (-b9 + b8 - b3) * q^17 + (b9 - b8 - b6 + b1) * q^19 + (-b2 + 1) * q^20 + (-b7 + b5 + 1) * q^22 + (b7 + b5 - 2*b4 + b2 - 1) * q^23 + q^25 + (b9 + b8 - 2*b3 + b1) * q^26 + (b7 + b5 - b4 + b2) * q^28 + (b8 + b5 + b2 + b1) * q^29 + (-b9 - b8 + b6 - 2*b3 - b1) * q^31 + (b9 + b8 + b3 - b1) * q^32 + (b5 - b4 + b2) * q^34 - b7 * q^35 + (-b9 - 2*b6 - b3 - b1) * q^37 + (-b7 - b4 + 3*b2 - 5) * q^38 - b3 * q^40 + (-b9 - 2*b8 - 2*b6 + b3 + b1) * q^41 + (-b9 - 2*b8 - 2*b6 + 3*b3 - b1) * q^43 + (-b9 - b8 + b6 - b3) * q^44 + (-b9 - 3*b8 + b6 + 2*b3 - b1) * q^46 + (b9 + b8 - 2*b6 + b3 + 2*b1) * q^47 + (-2*b7 + b4 + b2 - 2) * q^49 + b1 * q^50 + (b5 - 2*b4 + 4*b2 - 5) * q^52 + (-2*b7 + b5 + b4 - 2*b2 + 2) * q^53 + (b9 + b6) * q^55 + (-2*b8 - b6 + b3 - b1) * q^56 + (b9 - b8 + 2*b6 + b5 - b4 + b2 - 2*b1 - 2) * q^58 + (2*b7 + 2*b4 - 2) * q^59 + (4*b9 + 2*b8 + 2*b6 - 2*b1) * q^61 + (b7 - 2*b5 + b4 + b2 - 1) * q^62 + (b5 + b4 - 2*b2 + 5) * q^64 + (-b4 + b2 - 1) * q^65 + (b7 + 2*b5 - 2*b4) * q^67 + (-2*b9 + 2*b6 - b3 - b1) * q^68 + b6 * q^70 + (4*b7 - 2*b4 + 2) * q^71 + (-3*b9 - 3*b3 + 3*b1) * q^73 + (-2*b7 + 2*b5 - 2*b4 + 2*b2 + 2) * q^74 + (b9 - 3*b8 - b6 + 4*b3 - 5*b1) * q^76 + (-b9 + b8 + 2*b6 - b3 + 2*b1) * q^77 + (-b9 + b8 + b6 - b1) * q^79 - b4 * q^80 + (-2*b7 + 2*b4 - 2) * q^82 + (-3*b7 + b5 + b2 - 3) * q^83 + (b9 - b8 + b3) * q^85 + (-2*b7 + 4*b4 - 6*b2 + 8) * q^86 + (-b7 + 2*b4) * q^88 + (3*b9 - b8 - 2*b6 + 3*b3 - 2*b1) * q^89 + (b4 - b2 + 3) * q^91 + (3*b7 - 2*b5 + 3*b4 - 5*b2 + 3) * q^92 + (-2*b7 + 3*b5 - 3*b4 + 3*b2 - 4) * q^94 + (-b9 + b8 + b6 - b1) * q^95 + (-3*b9 + 2*b8 - b3 + 3*b1) * q^97 + (b9 + b8 + 2*b6 - 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 6 q^{4} - 10 q^{5} - 4 q^{7}+O(q^{10})$$ 10 * q - 6 * q^4 - 10 * q^5 - 4 * q^7 $$10 q - 6 q^{4} - 10 q^{5} - 4 q^{7} + 4 q^{13} - 2 q^{16} + 6 q^{20} + 10 q^{22} - 10 q^{23} + 10 q^{25} - 2 q^{28} + 2 q^{34} + 4 q^{35} - 32 q^{38} - 10 q^{49} - 34 q^{52} + 14 q^{53} - 18 q^{58} - 32 q^{59} - 4 q^{62} + 36 q^{64} - 4 q^{65} - 8 q^{67} + 8 q^{71} + 32 q^{74} + 2 q^{80} - 16 q^{82} - 18 q^{83} + 56 q^{86} + 24 q^{91} - 26 q^{94}+O(q^{100})$$ 10 * q - 6 * q^4 - 10 * q^5 - 4 * q^7 + 4 * q^13 - 2 * q^16 + 6 * q^20 + 10 * q^22 - 10 * q^23 + 10 * q^25 - 2 * q^28 + 2 * q^34 + 4 * q^35 - 32 * q^38 - 10 * q^49 - 34 * q^52 + 14 * q^53 - 18 * q^58 - 32 * q^59 - 4 * q^62 + 36 * q^64 - 4 * q^65 - 8 * q^67 + 8 * q^71 + 32 * q^74 + 2 * q^80 - 16 * q^82 - 18 * q^83 + 56 * q^86 + 24 * q^91 - 26 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 13x^{8} + 56x^{6} + 89x^{4} + 41x^{2} + 4$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 4\nu$$ v^3 + 4*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 6\nu^{2} + 4$$ v^4 + 6*v^2 + 4 $$\beta_{5}$$ $$=$$ $$\nu^{6} + 9\nu^{4} + 20\nu^{2} + 5$$ v^6 + 9*v^4 + 20*v^2 + 5 $$\beta_{6}$$ $$=$$ $$( -\nu^{9} - 12\nu^{7} - 46\nu^{5} - 59\nu^{3} - 12\nu ) / 2$$ (-v^9 - 12*v^7 - 46*v^5 - 59*v^3 - 12*v) / 2 $$\beta_{7}$$ $$=$$ $$( \nu^{8} + 12\nu^{6} + 46\nu^{4} + 59\nu^{2} + 12 ) / 2$$ (v^8 + 12*v^6 + 46*v^4 + 59*v^2 + 12) / 2 $$\beta_{8}$$ $$=$$ $$( -\nu^{9} - 13\nu^{7} - 54\nu^{5} - 73\nu^{3} - 13\nu ) / 2$$ (-v^9 - 13*v^7 - 54*v^5 - 73*v^3 - 13*v) / 2 $$\beta_{9}$$ $$=$$ $$( \nu^{9} + 13\nu^{7} + 56\nu^{5} + 87\nu^{3} + 31\nu ) / 2$$ (v^9 + 13*v^7 + 56*v^5 + 87*v^3 + 31*v) / 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 4\beta_1$$ b3 - 4*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 6\beta_{2} + 14$$ b4 - 6*b2 + 14 $$\nu^{5}$$ $$=$$ $$\beta_{9} + \beta_{8} - 7\beta_{3} + 19\beta_1$$ b9 + b8 - 7*b3 + 19*b1 $$\nu^{6}$$ $$=$$ $$\beta_{5} - 9\beta_{4} + 34\beta_{2} - 71$$ b5 - 9*b4 + 34*b2 - 71 $$\nu^{7}$$ $$=$$ $$-8\beta_{9} - 10\beta_{8} + 2\beta_{6} + 42\beta_{3} - 97\beta_1$$ -8*b9 - 10*b8 + 2*b6 + 42*b3 - 97*b1 $$\nu^{8}$$ $$=$$ $$2\beta_{7} - 12\beta_{5} + 62\beta_{4} - 191\beta_{2} + 373$$ 2*b7 - 12*b5 + 62*b4 - 191*b2 + 373 $$\nu^{9}$$ $$=$$ $$50\beta_{9} + 74\beta_{8} - 26\beta_{6} - 241\beta_{3} + 514\beta_1$$ 50*b9 + 74*b8 - 26*b6 - 241*b3 + 514*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times$$.

 $$n$$ $$146$$ $$262$$ $$901$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
811.1
 − 2.38254i − 2.09920i − 1.49936i − 0.732188i − 0.364257i 0.364257i 0.732188i 1.49936i 2.09920i 2.38254i
2.38254i 0 −3.67649 −1.00000 0 1.34246 3.99431i 0 2.38254i
811.2 2.09920i 0 −2.40666 −1.00000 0 −2.25380 0.853652i 0 2.09920i
811.3 1.49936i 0 −0.248070 −1.00000 0 0.522278 2.62677i 0 1.49936i
811.4 0.732188i 0 1.46390 −1.00000 0 −4.08783 2.53623i 0 0.732188i
811.5 0.364257i 0 1.86732 −1.00000 0 2.47690 1.40870i 0 0.364257i
811.6 0.364257i 0 1.86732 −1.00000 0 2.47690 1.40870i 0 0.364257i
811.7 0.732188i 0 1.46390 −1.00000 0 −4.08783 2.53623i 0 0.732188i
811.8 1.49936i 0 −0.248070 −1.00000 0 0.522278 2.62677i 0 1.49936i
811.9 2.09920i 0 −2.40666 −1.00000 0 −2.25380 0.853652i 0 2.09920i
811.10 2.38254i 0 −3.67649 −1.00000 0 1.34246 3.99431i 0 2.38254i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 811.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1305.2.d.e 10
3.b odd 2 1 1305.2.d.f yes 10
29.b even 2 1 inner 1305.2.d.e 10
87.d odd 2 1 1305.2.d.f yes 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1305.2.d.e 10 1.a even 1 1 trivial
1305.2.d.e 10 29.b even 2 1 inner
1305.2.d.f yes 10 3.b odd 2 1
1305.2.d.f yes 10 87.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1305, [\chi])$$:

 $$T_{2}^{10} + 13T_{2}^{8} + 56T_{2}^{6} + 89T_{2}^{4} + 41T_{2}^{2} + 4$$ T2^10 + 13*T2^8 + 56*T2^6 + 89*T2^4 + 41*T2^2 + 4 $$T_{23}^{5} + 5T_{23}^{4} - 86T_{23}^{3} - 270T_{23}^{2} + 1728T_{23} - 648$$ T23^5 + 5*T23^4 - 86*T23^3 - 270*T23^2 + 1728*T23 - 648

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 13 T^{8} + \cdots + 4$$
$3$ $$T^{10}$$
$5$ $$(T + 1)^{10}$$
$7$ $$(T^{5} + 2 T^{4} - 13 T^{3} + \cdots - 16)^{2}$$
$11$ $$T^{10} + 47 T^{8} + \cdots + 6724$$
$13$ $$(T^{5} - 2 T^{4} - 25 T^{3} + \cdots + 8)^{2}$$
$17$ $$T^{10} + 62 T^{8} + \cdots + 4096$$
$19$ $$T^{10} + 140 T^{8} + \cdots + 186624$$
$23$ $$(T^{5} + 5 T^{4} + \cdots - 648)^{2}$$
$29$ $$T^{10} - 43 T^{8} + \cdots + 20511149$$
$31$ $$T^{10} + 212 T^{8} + \cdots + 6718464$$
$37$ $$T^{10} + 229 T^{8} + \cdots + 186624$$
$41$ $$T^{10} + 197 T^{8} + \cdots + 1024$$
$43$ $$T^{10} + \cdots + 215737344$$
$47$ $$T^{10} + 314 T^{8} + \cdots + 63744256$$
$53$ $$(T^{5} - 7 T^{4} + \cdots - 6480)^{2}$$
$59$ $$(T^{5} + 16 T^{4} + \cdots + 46656)^{2}$$
$61$ $$T^{10} + 512 T^{8} + \cdots + 74649600$$
$67$ $$(T^{5} + 4 T^{4} + \cdots + 2836)^{2}$$
$71$ $$(T^{5} - 4 T^{4} + \cdots - 3456)^{2}$$
$73$ $$T^{10} + \cdots + 967458816$$
$79$ $$T^{10} + 140 T^{8} + \cdots + 186624$$
$83$ $$(T^{5} + 9 T^{4} + \cdots + 38664)^{2}$$
$89$ $$T^{10} + \cdots + 132342016$$
$97$ $$T^{10} + \cdots + 191102976$$