# Properties

 Label 1305.2.c.g Level $1305$ Weight $2$ Character orbit 1305.c Analytic conductor $10.420$ Analytic rank $0$ Dimension $4$ 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(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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + q^{4} + (2 \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 \beta_1 q^{8}+O(q^{10})$$ q + b1 * q^2 + q^4 + (2*b1 + 1) * q^5 + b2 * q^7 + 3*b1 * q^8 $$q + \beta_1 q^{2} + q^{4} + (2 \beta_1 + 1) q^{5} + \beta_{2} q^{7} + 3 \beta_1 q^{8} + (\beta_1 - 2) q^{10} + (\beta_{3} + 2) q^{11} - 2 \beta_{2} q^{13} - \beta_{3} q^{14} - q^{16} + (2 \beta_{2} + 2 \beta_1) q^{17} + (\beta_{3} + 2) q^{19} + (2 \beta_1 + 1) q^{20} + (\beta_{2} + 2 \beta_1) q^{22} - 3 \beta_{2} q^{23} + (4 \beta_1 - 3) q^{25} + 2 \beta_{3} q^{26} + \beta_{2} q^{28} - q^{29} + ( - \beta_{3} + 2) q^{31} + 5 \beta_1 q^{32} + ( - 2 \beta_{3} - 2) q^{34} + ( - 2 \beta_{3} + \beta_{2}) q^{35} - 6 \beta_1 q^{37} + (\beta_{2} + 2 \beta_1) q^{38} + (3 \beta_1 - 6) q^{40} + ( - 2 \beta_{3} - 2) q^{41} + ( - 2 \beta_{2} + 4 \beta_1) q^{43} + (\beta_{3} + 2) q^{44} + 3 \beta_{3} q^{46} - 2 \beta_{2} q^{47} - q^{49} + ( - 3 \beta_1 - 4) q^{50} - 2 \beta_{2} q^{52} - 4 \beta_1 q^{53} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 + 2) q^{55} - 3 \beta_{3} q^{56} - \beta_1 q^{58} - 4 q^{59} + ( - 2 \beta_{3} + 6) q^{61} + ( - \beta_{2} + 2 \beta_1) q^{62} - 7 q^{64} + (4 \beta_{3} - 2 \beta_{2}) q^{65} + (\beta_{2} - 4 \beta_1) q^{67} + (2 \beta_{2} + 2 \beta_1) q^{68} + ( - \beta_{3} - 2 \beta_{2}) q^{70} + ( - 2 \beta_{3} - 4) q^{71} + (2 \beta_{2} + 2 \beta_1) q^{73} + 6 q^{74} + (\beta_{3} + 2) q^{76} + (2 \beta_{2} + 8 \beta_1) q^{77} + ( - \beta_{3} + 6) q^{79} + ( - 2 \beta_1 - 1) q^{80} + ( - 2 \beta_{2} - 2 \beta_1) q^{82} + (3 \beta_{2} - 4 \beta_1) q^{83} + ( - 4 \beta_{3} + 2 \beta_{2} + \cdots - 4) q^{85}+ \cdots - \beta_1 q^{98}+O(q^{100})$$ q + b1 * q^2 + q^4 + (2*b1 + 1) * q^5 + b2 * q^7 + 3*b1 * q^8 + (b1 - 2) * q^10 + (b3 + 2) * q^11 - 2*b2 * q^13 - b3 * q^14 - q^16 + (2*b2 + 2*b1) * q^17 + (b3 + 2) * q^19 + (2*b1 + 1) * q^20 + (b2 + 2*b1) * q^22 - 3*b2 * q^23 + (4*b1 - 3) * q^25 + 2*b3 * q^26 + b2 * q^28 - q^29 + (-b3 + 2) * q^31 + 5*b1 * q^32 + (-2*b3 - 2) * q^34 + (-2*b3 + b2) * q^35 - 6*b1 * q^37 + (b2 + 2*b1) * q^38 + (3*b1 - 6) * q^40 + (-2*b3 - 2) * q^41 + (-2*b2 + 4*b1) * q^43 + (b3 + 2) * q^44 + 3*b3 * q^46 - 2*b2 * q^47 - q^49 + (-3*b1 - 4) * q^50 - 2*b2 * q^52 - 4*b1 * q^53 + (b3 + 2*b2 + 4*b1 + 2) * q^55 - 3*b3 * q^56 - b1 * q^58 - 4 * q^59 + (-2*b3 + 6) * q^61 + (-b2 + 2*b1) * q^62 - 7 * q^64 + (4*b3 - 2*b2) * q^65 + (b2 - 4*b1) * q^67 + (2*b2 + 2*b1) * q^68 + (-b3 - 2*b2) * q^70 + (-2*b3 - 4) * q^71 + (2*b2 + 2*b1) * q^73 + 6 * q^74 + (b3 + 2) * q^76 + (2*b2 + 8*b1) * q^77 + (-b3 + 6) * q^79 + (-2*b1 - 1) * q^80 + (-2*b2 - 2*b1) * q^82 + (3*b2 - 4*b1) * q^83 + (-4*b3 + 2*b2 + 2*b1 - 4) * q^85 + (2*b3 - 4) * q^86 + (3*b2 + 6*b1) * q^88 + (-2*b3 - 6) * q^89 + 16 * q^91 - 3*b2 * q^92 + 2*b3 * q^94 + (b3 + 2*b2 + 4*b1 + 2) * q^95 + (2*b2 - 6*b1) * q^97 - b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{4} + 4 q^{5}+O(q^{10})$$ 4 * q + 4 * q^4 + 4 * q^5 $$4 q + 4 q^{4} + 4 q^{5} - 8 q^{10} + 8 q^{11} - 4 q^{16} + 8 q^{19} + 4 q^{20} - 12 q^{25} - 4 q^{29} + 8 q^{31} - 8 q^{34} - 24 q^{40} - 8 q^{41} + 8 q^{44} - 4 q^{49} - 16 q^{50} + 8 q^{55} - 16 q^{59} + 24 q^{61} - 28 q^{64} - 16 q^{71} + 24 q^{74} + 8 q^{76} + 24 q^{79} - 4 q^{80} - 16 q^{85} - 16 q^{86} - 24 q^{89} + 64 q^{91} + 8 q^{95}+O(q^{100})$$ 4 * q + 4 * q^4 + 4 * q^5 - 8 * q^10 + 8 * q^11 - 4 * q^16 + 8 * q^19 + 4 * q^20 - 12 * q^25 - 4 * q^29 + 8 * q^31 - 8 * q^34 - 24 * q^40 - 8 * q^41 + 8 * q^44 - 4 * q^49 - 16 * q^50 + 8 * q^55 - 16 * q^59 + 24 * q^61 - 28 * q^64 - 16 * q^71 + 24 * q^74 + 8 * q^76 + 24 * q^79 - 4 * q^80 - 16 * q^85 - 16 * q^86 - 24 * q^89 + 64 * q^91 + 8 * q^95

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$2\zeta_{8}^{3} + 2\zeta_{8}$$ 2*v^3 + 2*v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 4$$ (b3 + b2) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 4$$ (-b3 + b2) / 4

## 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
 0.707107 − 0.707107i −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
1.00000i 0 1.00000 1.00000 2.00000i 0 2.82843i 3.00000i 0 −2.00000 1.00000i
784.2 1.00000i 0 1.00000 1.00000 2.00000i 0 2.82843i 3.00000i 0 −2.00000 1.00000i
784.3 1.00000i 0 1.00000 1.00000 + 2.00000i 0 2.82843i 3.00000i 0 −2.00000 + 1.00000i
784.4 1.00000i 0 1.00000 1.00000 + 2.00000i 0 2.82843i 3.00000i 0 −2.00000 + 1.00000i
 $$n$$: 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.g 4
3.b odd 2 1 435.2.c.c 4
5.b even 2 1 inner 1305.2.c.g 4
5.c odd 4 1 6525.2.a.n 2
5.c odd 4 1 6525.2.a.bd 2
15.d odd 2 1 435.2.c.c 4
15.e even 4 1 2175.2.a.k 2
15.e even 4 1 2175.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.2.c.c 4 3.b odd 2 1
435.2.c.c 4 15.d odd 2 1
1305.2.c.g 4 1.a even 1 1 trivial
1305.2.c.g 4 5.b even 2 1 inner
2175.2.a.k 2 15.e even 4 1
2175.2.a.s 2 15.e even 4 1
6525.2.a.n 2 5.c odd 4 1
6525.2.a.bd 2 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 $$S_{2}^{\mathrm{new}}(1305, [\chi])$$:

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{7}^{2} + 8$$ T7^2 + 8 $$T_{11}^{2} - 4T_{11} - 4$$ T11^2 - 4*T11 - 4

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4}$$
$5$ $$(T^{2} - 2 T + 5)^{2}$$
$7$ $$(T^{2} + 8)^{2}$$
$11$ $$(T^{2} - 4 T - 4)^{2}$$
$13$ $$(T^{2} + 32)^{2}$$
$17$ $$T^{4} + 72T^{2} + 784$$
$19$ $$(T^{2} - 4 T - 4)^{2}$$
$23$ $$(T^{2} + 72)^{2}$$
$29$ $$(T + 1)^{4}$$
$31$ $$(T^{2} - 4 T - 4)^{2}$$
$37$ $$(T^{2} + 36)^{2}$$
$41$ $$(T^{2} + 4 T - 28)^{2}$$
$43$ $$T^{4} + 96T^{2} + 256$$
$47$ $$(T^{2} + 32)^{2}$$
$53$ $$(T^{2} + 16)^{2}$$
$59$ $$(T + 4)^{4}$$
$61$ $$(T^{2} - 12 T + 4)^{2}$$
$67$ $$T^{4} + 48T^{2} + 64$$
$71$ $$(T^{2} + 8 T - 16)^{2}$$
$73$ $$T^{4} + 72T^{2} + 784$$
$79$ $$(T^{2} - 12 T + 28)^{2}$$
$83$ $$T^{4} + 176T^{2} + 3136$$
$89$ $$(T^{2} + 12 T + 4)^{2}$$
$97$ $$T^{4} + 136T^{2} + 16$$