# Properties

 Label 2340.2.q.f Level $2340$ Weight $2$ Character orbit 2340.q Analytic conductor $18.685$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2340 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2340.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6849940730$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

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.

 $$f(q)$$ $$=$$ $$q + q^{5} + \zeta_{6} q^{7}+O(q^{10})$$ q + q^5 + z * q^7 $$q + q^{5} + \zeta_{6} q^{7} + ( - 3 \zeta_{6} + 3) q^{11} + (4 \zeta_{6} - 1) q^{13} - 3 \zeta_{6} q^{17} - 5 \zeta_{6} q^{19} + ( - 9 \zeta_{6} + 9) q^{23} + q^{25} + (9 \zeta_{6} - 9) q^{29} + 8 q^{31} + \zeta_{6} q^{35} + ( - 7 \zeta_{6} + 7) q^{37} + ( - 3 \zeta_{6} + 3) q^{41} + \zeta_{6} q^{43} + ( - 6 \zeta_{6} + 6) q^{49} - 6 q^{53} + ( - 3 \zeta_{6} + 3) q^{55} + 9 \zeta_{6} q^{59} + \zeta_{6} q^{61} + (4 \zeta_{6} - 1) q^{65} + (5 \zeta_{6} - 5) q^{67} + 9 \zeta_{6} q^{71} + 2 q^{73} + 3 q^{77} + 8 q^{79} - 3 \zeta_{6} q^{85} + ( - 3 \zeta_{6} + 3) q^{89} + (3 \zeta_{6} - 4) q^{91} - 5 \zeta_{6} q^{95} - 17 \zeta_{6} q^{97} +O(q^{100})$$ q + q^5 + z * q^7 + (-3*z + 3) * q^11 + (4*z - 1) * q^13 - 3*z * q^17 - 5*z * q^19 + (-9*z + 9) * q^23 + q^25 + (9*z - 9) * q^29 + 8 * q^31 + z * q^35 + (-7*z + 7) * q^37 + (-3*z + 3) * q^41 + z * q^43 + (-6*z + 6) * q^49 - 6 * q^53 + (-3*z + 3) * q^55 + 9*z * q^59 + z * q^61 + (4*z - 1) * q^65 + (5*z - 5) * q^67 + 9*z * q^71 + 2 * q^73 + 3 * q^77 + 8 * q^79 - 3*z * q^85 + (-3*z + 3) * q^89 + (3*z - 4) * q^91 - 5*z * q^95 - 17*z * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{5} + q^{7}+O(q^{10})$$ 2 * q + 2 * q^5 + q^7 $$2 q + 2 q^{5} + q^{7} + 3 q^{11} + 2 q^{13} - 3 q^{17} - 5 q^{19} + 9 q^{23} + 2 q^{25} - 9 q^{29} + 16 q^{31} + q^{35} + 7 q^{37} + 3 q^{41} + q^{43} + 6 q^{49} - 12 q^{53} + 3 q^{55} + 9 q^{59} + q^{61} + 2 q^{65} - 5 q^{67} + 9 q^{71} + 4 q^{73} + 6 q^{77} + 16 q^{79} - 3 q^{85} + 3 q^{89} - 5 q^{91} - 5 q^{95} - 17 q^{97}+O(q^{100})$$ 2 * q + 2 * q^5 + q^7 + 3 * q^11 + 2 * q^13 - 3 * q^17 - 5 * q^19 + 9 * q^23 + 2 * q^25 - 9 * q^29 + 16 * q^31 + q^35 + 7 * q^37 + 3 * q^41 + q^43 + 6 * q^49 - 12 * q^53 + 3 * q^55 + 9 * q^59 + q^61 + 2 * q^65 - 5 * q^67 + 9 * q^71 + 4 * q^73 + 6 * q^77 + 16 * q^79 - 3 * q^85 + 3 * q^89 - 5 * q^91 - 5 * q^95 - 17 * q^97

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2340\mathbb{Z}\right)^\times$$.

 $$n$$ $$937$$ $$1081$$ $$1171$$ $$2081$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$ $$1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1621.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 0 0 1.00000 0 0.500000 + 0.866025i 0 0 0
2161.1 0 0 0 1.00000 0 0.500000 0.866025i 0 0 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 2340.2.q.f 2
3.b odd 2 1 260.2.i.a 2
12.b even 2 1 1040.2.q.i 2
13.c even 3 1 inner 2340.2.q.f 2
15.d odd 2 1 1300.2.i.d 2
15.e even 4 2 1300.2.bb.b 4
39.h odd 6 1 3380.2.a.i 1
39.i odd 6 1 260.2.i.a 2
39.i odd 6 1 3380.2.a.f 1
39.k even 12 2 3380.2.f.d 2
156.p even 6 1 1040.2.q.i 2
195.x odd 6 1 1300.2.i.d 2
195.bl even 12 2 1300.2.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.a 2 3.b odd 2 1
260.2.i.a 2 39.i odd 6 1
1040.2.q.i 2 12.b even 2 1
1040.2.q.i 2 156.p even 6 1
1300.2.i.d 2 15.d odd 2 1
1300.2.i.d 2 195.x odd 6 1
1300.2.bb.b 4 15.e even 4 2
1300.2.bb.b 4 195.bl even 12 2
2340.2.q.f 2 1.a even 1 1 trivial
2340.2.q.f 2 13.c even 3 1 inner
3380.2.a.f 1 39.i odd 6 1
3380.2.a.i 1 39.h odd 6 1
3380.2.f.d 2 39.k even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2340, [\chi])$$:

 $$T_{7}^{2} - T_{7} + 1$$ T7^2 - T7 + 1 $$T_{11}^{2} - 3T_{11} + 9$$ T11^2 - 3*T11 + 9 $$T_{19}^{2} + 5T_{19} + 25$$ T19^2 + 5*T19 + 25

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - T + 1$$
$11$ $$T^{2} - 3T + 9$$
$13$ $$T^{2} - 2T + 13$$
$17$ $$T^{2} + 3T + 9$$
$19$ $$T^{2} + 5T + 25$$
$23$ $$T^{2} - 9T + 81$$
$29$ $$T^{2} + 9T + 81$$
$31$ $$(T - 8)^{2}$$
$37$ $$T^{2} - 7T + 49$$
$41$ $$T^{2} - 3T + 9$$
$43$ $$T^{2} - T + 1$$
$47$ $$T^{2}$$
$53$ $$(T + 6)^{2}$$
$59$ $$T^{2} - 9T + 81$$
$61$ $$T^{2} - T + 1$$
$67$ $$T^{2} + 5T + 25$$
$71$ $$T^{2} - 9T + 81$$
$73$ $$(T - 2)^{2}$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2}$$
$89$ $$T^{2} - 3T + 9$$
$97$ $$T^{2} + 17T + 289$$