# Properties

 Label 1300.2.i.e Level $1300$ Weight $2$ Character orbit 1300.i Analytic conductor $10.381$ 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] = [1300,2,Mod(601,1300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1300, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1300.601");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1300 = 2^{2} \cdot 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1300.i (of order $$3$$, degree $$2$$, 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: $$10.3805522628$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} +O(q^{10})$$ q + (-z + 1) * q^3 - z * q^7 + 2*z * q^9 $$q + ( - \zeta_{6} + 1) q^{3} - \zeta_{6} q^{7} + 2 \zeta_{6} q^{9} + (3 \zeta_{6} - 3) q^{11} + (4 \zeta_{6} - 3) q^{13} - 3 \zeta_{6} q^{17} + 7 \zeta_{6} q^{19} - q^{21} + (3 \zeta_{6} - 3) q^{23} + 5 q^{27} + (3 \zeta_{6} - 3) q^{29} - 4 q^{31} + 3 \zeta_{6} q^{33} + (7 \zeta_{6} - 7) q^{37} + (3 \zeta_{6} + 1) q^{39} + ( - 9 \zeta_{6} + 9) q^{41} + 11 \zeta_{6} q^{43} + ( - 6 \zeta_{6} + 6) q^{49} - 3 q^{51} + 6 q^{53} + 7 q^{57} + 3 \zeta_{6} q^{59} - 11 \zeta_{6} q^{61} + ( - 2 \zeta_{6} + 2) q^{63} + (7 \zeta_{6} - 7) q^{67} + 3 \zeta_{6} q^{69} + 3 \zeta_{6} q^{71} - 2 q^{73} + 3 q^{77} + 8 q^{79} + (\zeta_{6} - 1) q^{81} + 12 q^{83} + 3 \zeta_{6} q^{87} + (15 \zeta_{6} - 15) q^{89} + ( - \zeta_{6} + 4) q^{91} + (4 \zeta_{6} - 4) q^{93} - 7 \zeta_{6} q^{97} - 6 q^{99} +O(q^{100})$$ q + (-z + 1) * q^3 - z * q^7 + 2*z * q^9 + (3*z - 3) * q^11 + (4*z - 3) * q^13 - 3*z * q^17 + 7*z * q^19 - q^21 + (3*z - 3) * q^23 + 5 * q^27 + (3*z - 3) * q^29 - 4 * q^31 + 3*z * q^33 + (7*z - 7) * q^37 + (3*z + 1) * q^39 + (-9*z + 9) * q^41 + 11*z * q^43 + (-6*z + 6) * q^49 - 3 * q^51 + 6 * q^53 + 7 * q^57 + 3*z * q^59 - 11*z * q^61 + (-2*z + 2) * q^63 + (7*z - 7) * q^67 + 3*z * q^69 + 3*z * q^71 - 2 * q^73 + 3 * q^77 + 8 * q^79 + (z - 1) * q^81 + 12 * q^83 + 3*z * q^87 + (15*z - 15) * q^89 + (-z + 4) * q^91 + (4*z - 4) * q^93 - 7*z * q^97 - 6 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{7} + 2 q^{9}+O(q^{10})$$ 2 * q + q^3 - q^7 + 2 * q^9 $$2 q + q^{3} - q^{7} + 2 q^{9} - 3 q^{11} - 2 q^{13} - 3 q^{17} + 7 q^{19} - 2 q^{21} - 3 q^{23} + 10 q^{27} - 3 q^{29} - 8 q^{31} + 3 q^{33} - 7 q^{37} + 5 q^{39} + 9 q^{41} + 11 q^{43} + 6 q^{49} - 6 q^{51} + 12 q^{53} + 14 q^{57} + 3 q^{59} - 11 q^{61} + 2 q^{63} - 7 q^{67} + 3 q^{69} + 3 q^{71} - 4 q^{73} + 6 q^{77} + 16 q^{79} - q^{81} + 24 q^{83} + 3 q^{87} - 15 q^{89} + 7 q^{91} - 4 q^{93} - 7 q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + q^3 - q^7 + 2 * q^9 - 3 * q^11 - 2 * q^13 - 3 * q^17 + 7 * q^19 - 2 * q^21 - 3 * q^23 + 10 * q^27 - 3 * q^29 - 8 * q^31 + 3 * q^33 - 7 * q^37 + 5 * q^39 + 9 * q^41 + 11 * q^43 + 6 * q^49 - 6 * q^51 + 12 * q^53 + 14 * q^57 + 3 * q^59 - 11 * q^61 + 2 * q^63 - 7 * q^67 + 3 * q^69 + 3 * q^71 - 4 * q^73 + 6 * q^77 + 16 * q^79 - q^81 + 24 * q^83 + 3 * q^87 - 15 * q^89 + 7 * q^91 - 4 * q^93 - 7 * q^97 - 12 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times$$.

 $$n$$ $$301$$ $$651$$ $$677$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
601.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 0 0 −0.500000 + 0.866025i 0 1.00000 1.73205i 0
1101.1 0 0.500000 0.866025i 0 0 0 −0.500000 0.866025i 0 1.00000 + 1.73205i 0
 $$n$$: 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
13.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1300.2.i.e 2
5.b even 2 1 260.2.i.b 2
5.c odd 4 2 1300.2.bb.a 4
13.c even 3 1 inner 1300.2.i.e 2
15.d odd 2 1 2340.2.q.b 2
20.d odd 2 1 1040.2.q.j 2
65.l even 6 1 3380.2.a.g 1
65.n even 6 1 260.2.i.b 2
65.n even 6 1 3380.2.a.h 1
65.q odd 12 2 1300.2.bb.a 4
65.s odd 12 2 3380.2.f.e 2
195.x odd 6 1 2340.2.q.b 2
260.v odd 6 1 1040.2.q.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.b 2 5.b even 2 1
260.2.i.b 2 65.n even 6 1
1040.2.q.j 2 20.d odd 2 1
1040.2.q.j 2 260.v odd 6 1
1300.2.i.e 2 1.a even 1 1 trivial
1300.2.i.e 2 13.c even 3 1 inner
1300.2.bb.a 4 5.c odd 4 2
1300.2.bb.a 4 65.q odd 12 2
2340.2.q.b 2 15.d odd 2 1
2340.2.q.b 2 195.x odd 6 1
3380.2.a.g 1 65.l even 6 1
3380.2.a.h 1 65.n even 6 1
3380.2.f.e 2 65.s odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1300, [\chi])$$:

 $$T_{3}^{2} - T_{3} + 1$$ T3^2 - T3 + 1 $$T_{19}^{2} - 7T_{19} + 49$$ T19^2 - 7*T19 + 49

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - T + 1$$
$5$ $$T^{2}$$
$7$ $$T^{2} + T + 1$$
$11$ $$T^{2} + 3T + 9$$
$13$ $$T^{2} + 2T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} - 7T + 49$$
$23$ $$T^{2} + 3T + 9$$
$29$ $$T^{2} + 3T + 9$$
$31$ $$(T + 4)^{2}$$
$37$ $$T^{2} + 7T + 49$$
$41$ $$T^{2} - 9T + 81$$
$43$ $$T^{2} - 11T + 121$$
$47$ $$T^{2}$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} - 3T + 9$$
$61$ $$T^{2} + 11T + 121$$
$67$ $$T^{2} + 7T + 49$$
$71$ $$T^{2} - 3T + 9$$
$73$ $$(T + 2)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$(T - 12)^{2}$$
$89$ $$T^{2} + 15T + 225$$
$97$ $$T^{2} + 7T + 49$$