# Properties

 Label 1305.2.c.a Level $1305$ Weight $2$ Character orbit 1305.c Analytic conductor $10.420$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(784,1305)]

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, 1, 0]))

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

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

chi := DirichletCharacter("1305.784");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1305.c (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: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 i q^{2} - 2 q^{4} + ( - i + 2) q^{5} - 2 i q^{7} +O(q^{10})$$ q + 2*i * q^2 - 2 * q^4 + (-i + 2) * q^5 - 2*i * q^7 $$q + 2 i q^{2} - 2 q^{4} + ( - i + 2) q^{5} - 2 i q^{7} + (4 i + 2) q^{10} + 3 q^{11} - 4 i q^{13} + 4 q^{14} - 4 q^{16} - 8 i q^{17} + (2 i - 4) q^{20} + 6 i q^{22} - i q^{23} + ( - 4 i + 3) q^{25} + 8 q^{26} + 4 i q^{28} + q^{29} - 8 q^{31} - 8 i q^{32} + 16 q^{34} + ( - 4 i - 2) q^{35} - 7 i q^{37} - 7 q^{41} - 9 i q^{43} - 6 q^{44} + 2 q^{46} + 12 i q^{47} + 3 q^{49} + (6 i + 8) q^{50} + 8 i q^{52} + 9 i q^{53} + ( - 3 i + 6) q^{55} + 2 i q^{58} + 10 q^{59} + 2 q^{61} - 16 i q^{62} + 8 q^{64} + ( - 8 i - 4) q^{65} + 8 i q^{67} + 16 i q^{68} + ( - 4 i + 8) q^{70} + 8 q^{71} + i q^{73} + 14 q^{74} - 6 i q^{77} + 10 q^{79} + (4 i - 8) q^{80} - 14 i q^{82} + 9 i q^{83} + ( - 16 i - 8) q^{85} + 18 q^{86} + 10 q^{89} - 8 q^{91} + 2 i q^{92} - 24 q^{94} + 13 i q^{97} + 6 i q^{98} +O(q^{100})$$ q + 2*i * q^2 - 2 * q^4 + (-i + 2) * q^5 - 2*i * q^7 + (4*i + 2) * q^10 + 3 * q^11 - 4*i * q^13 + 4 * q^14 - 4 * q^16 - 8*i * q^17 + (2*i - 4) * q^20 + 6*i * q^22 - i * q^23 + (-4*i + 3) * q^25 + 8 * q^26 + 4*i * q^28 + q^29 - 8 * q^31 - 8*i * q^32 + 16 * q^34 + (-4*i - 2) * q^35 - 7*i * q^37 - 7 * q^41 - 9*i * q^43 - 6 * q^44 + 2 * q^46 + 12*i * q^47 + 3 * q^49 + (6*i + 8) * q^50 + 8*i * q^52 + 9*i * q^53 + (-3*i + 6) * q^55 + 2*i * q^58 + 10 * q^59 + 2 * q^61 - 16*i * q^62 + 8 * q^64 + (-8*i - 4) * q^65 + 8*i * q^67 + 16*i * q^68 + (-4*i + 8) * q^70 + 8 * q^71 + i * q^73 + 14 * q^74 - 6*i * q^77 + 10 * q^79 + (4*i - 8) * q^80 - 14*i * q^82 + 9*i * q^83 + (-16*i - 8) * q^85 + 18 * q^86 + 10 * q^89 - 8 * q^91 + 2*i * q^92 - 24 * q^94 + 13*i * q^97 + 6*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 4 q^{4} + 4 q^{5}+O(q^{10})$$ 2 * q - 4 * q^4 + 4 * q^5 $$2 q - 4 q^{4} + 4 q^{5} + 4 q^{10} + 6 q^{11} + 8 q^{14} - 8 q^{16} - 8 q^{20} + 6 q^{25} + 16 q^{26} + 2 q^{29} - 16 q^{31} + 32 q^{34} - 4 q^{35} - 14 q^{41} - 12 q^{44} + 4 q^{46} + 6 q^{49} + 16 q^{50} + 12 q^{55} + 20 q^{59} + 4 q^{61} + 16 q^{64} - 8 q^{65} + 16 q^{70} + 16 q^{71} + 28 q^{74} + 20 q^{79} - 16 q^{80} - 16 q^{85} + 36 q^{86} + 20 q^{89} - 16 q^{91} - 48 q^{94}+O(q^{100})$$ 2 * q - 4 * q^4 + 4 * q^5 + 4 * q^10 + 6 * q^11 + 8 * q^14 - 8 * q^16 - 8 * q^20 + 6 * q^25 + 16 * q^26 + 2 * q^29 - 16 * q^31 + 32 * q^34 - 4 * q^35 - 14 * q^41 - 12 * q^44 + 4 * q^46 + 6 * q^49 + 16 * q^50 + 12 * q^55 + 20 * q^59 + 4 * q^61 + 16 * q^64 - 8 * q^65 + 16 * q^70 + 16 * q^71 + 28 * q^74 + 20 * q^79 - 16 * q^80 - 16 * q^85 + 36 * q^86 + 20 * q^89 - 16 * q^91 - 48 * q^94

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
784.1
 − 1.00000i 1.00000i
2.00000i 0 −2.00000 2.00000 + 1.00000i 0 2.00000i 0 0 2.00000 4.00000i
784.2 2.00000i 0 −2.00000 2.00000 1.00000i 0 2.00000i 0 0 2.00000 + 4.00000i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.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.c.a 2
3.b odd 2 1 435.2.c.a 2
5.b even 2 1 inner 1305.2.c.a 2
5.c odd 4 1 6525.2.a.b 1
5.c odd 4 1 6525.2.a.l 1
15.d odd 2 1 435.2.c.a 2
15.e even 4 1 2175.2.a.a 1
15.e even 4 1 2175.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.a 2 3.b odd 2 1
435.2.c.a 2 15.d odd 2 1
1305.2.c.a 2 1.a even 1 1 trivial
1305.2.c.a 2 5.b even 2 1 inner
2175.2.a.a 1 15.e even 4 1
2175.2.a.j 1 15.e even 4 1
6525.2.a.b 1 5.c odd 4 1
6525.2.a.l 1 5.c odd 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 4$$ T2^2 + 4 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11} - 3$$ T11 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2} - 4T + 5$$
$7$ $$T^{2} + 4$$
$11$ $$(T - 3)^{2}$$
$13$ $$T^{2} + 16$$
$17$ $$T^{2} + 64$$
$19$ $$T^{2}$$
$23$ $$T^{2} + 1$$
$29$ $$(T - 1)^{2}$$
$31$ $$(T + 8)^{2}$$
$37$ $$T^{2} + 49$$
$41$ $$(T + 7)^{2}$$
$43$ $$T^{2} + 81$$
$47$ $$T^{2} + 144$$
$53$ $$T^{2} + 81$$
$59$ $$(T - 10)^{2}$$
$61$ $$(T - 2)^{2}$$
$67$ $$T^{2} + 64$$
$71$ $$(T - 8)^{2}$$
$73$ $$T^{2} + 1$$
$79$ $$(T - 10)^{2}$$
$83$ $$T^{2} + 81$$
$89$ $$(T - 10)^{2}$$
$97$ $$T^{2} + 169$$