Properties

 Label 3380.2.a.i Level $3380$ Weight $2$ Character orbit 3380.a Self dual yes Analytic conductor $26.989$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3380 = 2^{2} \cdot 5 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3380.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$26.9894358832$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 260) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} + q^{7} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 + q^7 - 2 * q^9 $$q + q^{3} + q^{5} + q^{7} - 2 q^{9} - 3 q^{11} + q^{15} - 3 q^{17} - 5 q^{19} + q^{21} + 9 q^{23} + q^{25} - 5 q^{27} - 9 q^{29} - 8 q^{31} - 3 q^{33} + q^{35} + 7 q^{37} - 3 q^{41} - q^{43} - 2 q^{45} - 6 q^{49} - 3 q^{51} + 6 q^{53} - 3 q^{55} - 5 q^{57} - 9 q^{59} - q^{61} - 2 q^{63} - 5 q^{67} + 9 q^{69} - 9 q^{71} - 2 q^{73} + q^{75} - 3 q^{77} + 8 q^{79} + q^{81} - 3 q^{85} - 9 q^{87} - 3 q^{89} - 8 q^{93} - 5 q^{95} - 17 q^{97} + 6 q^{99}+O(q^{100})$$ q + q^3 + q^5 + q^7 - 2 * q^9 - 3 * q^11 + q^15 - 3 * q^17 - 5 * q^19 + q^21 + 9 * q^23 + q^25 - 5 * q^27 - 9 * q^29 - 8 * q^31 - 3 * q^33 + q^35 + 7 * q^37 - 3 * q^41 - q^43 - 2 * q^45 - 6 * q^49 - 3 * q^51 + 6 * q^53 - 3 * q^55 - 5 * q^57 - 9 * q^59 - q^61 - 2 * q^63 - 5 * q^67 + 9 * q^69 - 9 * q^71 - 2 * q^73 + q^75 - 3 * q^77 + 8 * q^79 + q^81 - 3 * q^85 - 9 * q^87 - 3 * q^89 - 8 * q^93 - 5 * q^95 - 17 * q^97 + 6 * q^99

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 1.00000 0 1.00000 0 1.00000 0 −2.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$13$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3380.2.a.i 1
13.b even 2 1 3380.2.a.f 1
13.d odd 4 2 3380.2.f.d 2
13.e even 6 2 260.2.i.a 2
39.h odd 6 2 2340.2.q.f 2
52.i odd 6 2 1040.2.q.i 2
65.l even 6 2 1300.2.i.d 2
65.r odd 12 4 1300.2.bb.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
260.2.i.a 2 13.e even 6 2
1040.2.q.i 2 52.i odd 6 2
1300.2.i.d 2 65.l even 6 2
1300.2.bb.b 4 65.r odd 12 4
2340.2.q.f 2 39.h odd 6 2
3380.2.a.f 1 13.b even 2 1
3380.2.a.i 1 1.a even 1 1 trivial
3380.2.f.d 2 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3380))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{7} - 1$$ T7 - 1 $$T_{19} + 5$$ T19 + 5

Hecke characteristic polynomials

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