# Properties

 Label 1305.2.d.d 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: $$x^{10} + 17x^{8} + 96x^{6} + 201x^{4} + 121x^{2} + 16$$ x^10 + 17*x^8 + 96*x^6 + 201*x^4 + 121*x^2 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 435) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} + \cdots - 2 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 - 1) * q^4 + q^5 - b5 * q^7 + (b9 - b8 - b4 - 2*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} - 1) q^{4} + q^{5} - \beta_{5} q^{7} + (\beta_{9} - \beta_{8} + \cdots - 2 \beta_1) q^{8}+ \cdots + (\beta_{9} - 4 \beta_{8} + \cdots - 4 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b2 - 1) * q^4 + q^5 - b5 * q^7 + (b9 - b8 - b4 - 2*b1) * q^8 + b1 * q^10 + (-b8 - b4 - b1) * q^11 + (-b5 + b3) * q^13 + (b9 - b8 - b6 - b4 - b1) * q^14 + (b5 + b3 - b2 + 3) * q^16 + (-b8 + b6) * q^17 + b8 * q^19 + (b2 - 1) * q^20 + (-b7 + b5 - b2 + 1) * q^22 + (b7 - b5 - 2*b2 - 1) * q^23 + q^25 + (-b9 + 2*b8 - b6 - b4 - b1) * q^26 + (-b7 + b3 - b2 + 1) * q^28 + (-b7 + b4 - b1) * q^29 + b8 * q^31 + (-2*b9 + 3*b8 + b6 + 3*b1) * q^32 + (b7 - b5 + b3 - b2 + 1) * q^34 - b5 * q^35 + (-2*b9 - b8 - b6 - b4 - b1) * q^37 - b3 * q^38 + (b9 - b8 - b4 - 2*b1) * q^40 + (-b8 - b6 - b4 - 3*b1) * q^41 + (-b8 + b6 - b4 - 3*b1) * q^43 + (-b9 + 2*b6 + b4 + 5*b1) * q^44 + (-2*b9 + b8 - 2*b6 + 2*b1) * q^46 + (b8 + b6 + 2*b1) * q^47 + (2*b7 - b5 - b3 + 1) * q^49 + b1 * q^50 + (-3*b7 - 2*b3 + b2 - 1) * q^52 + (b7 + b5 + b3 + 3) * q^53 + (-b8 - b4 - b1) * q^55 + (2*b8 - b6 + 4*b1) * q^56 + (b9 + b7 + b6 - b5 + b4 + b3 - b2 + 2*b1 + 5) * q^58 + (-2*b7 + 2*b3 + 2) * q^59 - 2*b6 * q^61 - b3 * q^62 + (-b7 + b5 - 3*b3 + 2*b2 - 4) * q^64 + (-b5 + b3) * q^65 + (2*b7 + b5 - 2) * q^67 + (-3*b9 + b8 - b4 + b1) * q^68 + (b9 - b8 - b6 - b4 - b1) * q^70 + 8 * q^71 + (2*b9 + b8 - b6 + b4 + b1) * q^73 + (-4*b7 + 2*b5 - 2*b3 + 2*b2 - 2) * q^74 + (2*b9 - b8) * q^76 + (2*b9 + b8 - 3*b6 + 2*b1) * q^77 + (-4*b9 + b8 + 2*b6) * q^79 + (b5 + b3 - b2 + 3) * q^80 + (-2*b7 + 2*b5 - 2*b2 + 6) * q^82 + (-b7 + b5 + 2*b2 - 7) * q^83 + (-b8 + b6) * q^85 + (-4*b2 + 8) * q^86 + (-b5 + 2*b2 - 10) * q^88 + (-2*b9 - b8 - b6 + 2*b4) * q^89 + (b5 - b3 + 2*b2 + 6) * q^91 + (-2*b7 - 3*b3 + 2*b2 - 12) * q^92 + (b7 - b5 - b3 + b2 - 5) * q^94 + b8 * q^95 + (2*b9 - 3*b8 + 3*b6 - b4 + 3*b1) * q^97 + (b9 - 4*b8 - 3*b6 - 3*b4 - 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 14 q^{4} + 10 q^{5}+O(q^{10})$$ 10 * q - 14 * q^4 + 10 * q^5 $$10 q - 14 q^{4} + 10 q^{5} - 4 q^{13} + 30 q^{16} - 14 q^{20} + 14 q^{22} - 2 q^{23} + 10 q^{25} + 10 q^{28} + 10 q^{34} + 4 q^{38} + 14 q^{49} - 6 q^{52} + 26 q^{53} + 50 q^{58} + 12 q^{59} + 4 q^{62} - 36 q^{64} - 4 q^{65} - 20 q^{67} + 80 q^{71} - 20 q^{74} + 30 q^{80} + 68 q^{82} - 78 q^{83} + 96 q^{86} - 108 q^{88} + 56 q^{91} - 116 q^{92} - 50 q^{94}+O(q^{100})$$ 10 * q - 14 * q^4 + 10 * q^5 - 4 * q^13 + 30 * q^16 - 14 * q^20 + 14 * q^22 - 2 * q^23 + 10 * q^25 + 10 * q^28 + 10 * q^34 + 4 * q^38 + 14 * q^49 - 6 * q^52 + 26 * q^53 + 50 * q^58 + 12 * q^59 + 4 * q^62 - 36 * q^64 - 4 * q^65 - 20 * q^67 + 80 * q^71 - 20 * q^74 + 30 * q^80 + 68 * q^82 - 78 * q^83 + 96 * q^86 - 108 * q^88 + 56 * q^91 - 116 * q^92 - 50 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 17x^{8} + 96x^{6} + 201x^{4} + 121x^{2} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 3$$ v^2 + 3 $$\beta_{3}$$ $$=$$ $$( -\nu^{8} - 20\nu^{6} - 116\nu^{4} - 189\nu^{2} - 48 ) / 20$$ (-v^8 - 20*v^6 - 116*v^4 - 189*v^2 - 48) / 20 $$\beta_{4}$$ $$=$$ $$( -\nu^{9} - 15\nu^{7} - 76\nu^{5} - 169\nu^{3} - 183\nu ) / 20$$ (-v^9 - 15*v^7 - 76*v^5 - 169*v^3 - 183*v) / 20 $$\beta_{5}$$ $$=$$ $$( \nu^{8} + 20\nu^{6} + 136\nu^{4} + 329\nu^{2} + 128 ) / 20$$ (v^8 + 20*v^6 + 136*v^4 + 329*v^2 + 128) / 20 $$\beta_{6}$$ $$=$$ $$( \nu^{9} + 20\nu^{7} + 136\nu^{5} + 349\nu^{3} + 228\nu ) / 20$$ (v^9 + 20*v^7 + 136*v^5 + 349*v^3 + 228*v) / 20 $$\beta_{7}$$ $$=$$ $$( \nu^{8} + 15\nu^{6} + 71\nu^{4} + 114\nu^{2} + 38 ) / 5$$ (v^8 + 15*v^6 + 71*v^4 + 114*v^2 + 38) / 5 $$\beta_{8}$$ $$=$$ $$( -3\nu^{9} - 50\nu^{7} - 268\nu^{5} - 487\nu^{3} - 174\nu ) / 20$$ (-3*v^9 - 50*v^7 - 268*v^5 - 487*v^3 - 174*v) / 20 $$\beta_{9}$$ $$=$$ $$( -4\nu^{9} - 65\nu^{7} - 344\nu^{5} - 636\nu^{3} - 237\nu ) / 20$$ (-4*v^9 - 65*v^7 - 344*v^5 - 636*v^3 - 237*v) / 20
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 3$$ b2 - 3 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{8} - \beta_{4} - 6\beta_1$$ b9 - b8 - b4 - 6*b1 $$\nu^{4}$$ $$=$$ $$\beta_{5} + \beta_{3} - 7\beta_{2} + 17$$ b5 + b3 - 7*b2 + 17 $$\nu^{5}$$ $$=$$ $$-10\beta_{9} + 11\beta_{8} + \beta_{6} + 8\beta_{4} + 39\beta_1$$ -10*b9 + 11*b8 + b6 + 8*b4 + 39*b1 $$\nu^{6}$$ $$=$$ $$-\beta_{7} - 9\beta_{5} - 13\beta_{3} + 48\beta_{2} - 110$$ -b7 - 9*b5 - 13*b3 + 48*b2 - 110 $$\nu^{7}$$ $$=$$ $$84\beta_{9} - 96\beta_{8} - 8\beta_{6} - 56\beta_{4} - 261\beta_1$$ 84*b9 - 96*b8 - 8*b6 - 56*b4 - 261*b1 $$\nu^{8}$$ $$=$$ $$20\beta_{7} + 64\beta_{5} + 124\beta_{3} - 337\beta_{2} + 747$$ 20*b7 + 64*b5 + 124*b3 - 337*b2 + 747 $$\nu^{9}$$ $$=$$ $$-669\beta_{9} + 773\beta_{8} + 44\beta_{6} + 381\beta_{4} + 1782\beta_1$$ -669*b9 + 773*b8 + 44*b6 + 381*b4 + 1782*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
 − 2.72674i − 2.37667i − 1.74703i − 0.825137i − 0.428173i 0.428173i 0.825137i 1.74703i 2.37667i 2.72674i
2.72674i 0 −5.43512 1.00000 0 −1.78170 9.36669i 0 2.72674i
811.2 2.37667i 0 −3.64857 1.00000 0 −1.11968 3.91810i 0 2.37667i
811.3 1.74703i 0 −1.05213 1.00000 0 4.55534 1.65596i 0 1.74703i
811.4 0.825137i 0 1.31915 1.00000 0 1.95267 2.73875i 0 0.825137i
811.5 0.428173i 0 1.81667 1.00000 0 −3.60663 1.63419i 0 0.428173i
811.6 0.428173i 0 1.81667 1.00000 0 −3.60663 1.63419i 0 0.428173i
811.7 0.825137i 0 1.31915 1.00000 0 1.95267 2.73875i 0 0.825137i
811.8 1.74703i 0 −1.05213 1.00000 0 4.55534 1.65596i 0 1.74703i
811.9 2.37667i 0 −3.64857 1.00000 0 −1.11968 3.91810i 0 2.37667i
811.10 2.72674i 0 −5.43512 1.00000 0 −1.78170 9.36669i 0 2.72674i
 $$n$$: 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.d 10
3.b odd 2 1 435.2.d.a 10
15.d odd 2 1 2175.2.d.f 10
15.e even 4 1 2175.2.f.d 10
15.e even 4 1 2175.2.f.e 10
29.b even 2 1 inner 1305.2.d.d 10
87.d odd 2 1 435.2.d.a 10
435.b odd 2 1 2175.2.d.f 10
435.p even 4 1 2175.2.f.d 10
435.p even 4 1 2175.2.f.e 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.d.a 10 3.b odd 2 1
435.2.d.a 10 87.d odd 2 1
1305.2.d.d 10 1.a even 1 1 trivial
1305.2.d.d 10 29.b even 2 1 inner
2175.2.d.f 10 15.d odd 2 1
2175.2.d.f 10 435.b odd 2 1
2175.2.f.d 10 15.e even 4 1
2175.2.f.d 10 435.p even 4 1
2175.2.f.e 10 15.e even 4 1
2175.2.f.e 10 435.p even 4 1

## Hecke kernels

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

 $$T_{2}^{10} + 17T_{2}^{8} + 96T_{2}^{6} + 201T_{2}^{4} + 121T_{2}^{2} + 16$$ T2^10 + 17*T2^8 + 96*T2^6 + 201*T2^4 + 121*T2^2 + 16 $$T_{23}^{5} + T_{23}^{4} - 84T_{23}^{3} - 128T_{23}^{2} + 1728T_{23} + 3328$$ T23^5 + T23^4 - 84*T23^3 - 128*T23^2 + 1728*T23 + 3328

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 17 T^{8} + \cdots + 16$$
$3$ $$T^{10}$$
$5$ $$(T - 1)^{10}$$
$7$ $$(T^{5} - 21 T^{3} + \cdots + 64)^{2}$$
$11$ $$T^{10} + 59 T^{8} + \cdots + 35344$$
$13$ $$(T^{5} + 2 T^{4} + \cdots - 368)^{2}$$
$17$ $$T^{10} + 62 T^{8} + \cdots + 64$$
$19$ $$(T^{2} + 4)^{5}$$
$23$ $$(T^{5} + T^{4} - 84 T^{3} + \cdots + 3328)^{2}$$
$29$ $$T^{10} + 37 T^{8} + \cdots + 20511149$$
$31$ $$(T^{2} + 4)^{5}$$
$37$ $$T^{10} + \cdots + 666052864$$
$41$ $$T^{10} + 161 T^{8} + \cdots + 65536$$
$43$ $$T^{10} + 233 T^{8} + \cdots + 16777216$$
$47$ $$T^{10} + 94 T^{8} + \cdots + 861184$$
$53$ $$(T^{5} - 13 T^{4} + \cdots - 784)^{2}$$
$59$ $$(T^{5} - 6 T^{4} + \cdots - 21376)^{2}$$
$61$ $$T^{10} + 168 T^{8} + \cdots + 4194304$$
$67$ $$(T^{5} + 10 T^{4} + \cdots + 3616)^{2}$$
$71$ $$(T - 8)^{10}$$
$73$ $$T^{10} + 297 T^{8} + \cdots + 1183744$$
$79$ $$T^{10} + \cdots + 11097358336$$
$83$ $$(T^{5} + 39 T^{4} + \cdots + 4352)^{2}$$
$89$ $$T^{10} + 310 T^{8} + \cdots + 16384$$
$97$ $$T^{10} + \cdots + 536015104$$