Properties

 Label 605.2.a.a Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$4.83094932229$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} - 3q^{3} - q^{4} + q^{5} + 3q^{6} - 3q^{7} + 3q^{8} + 6q^{9} + O(q^{10})$$ $$q - q^{2} - 3q^{3} - q^{4} + q^{5} + 3q^{6} - 3q^{7} + 3q^{8} + 6q^{9} - q^{10} + 3q^{12} + 4q^{13} + 3q^{14} - 3q^{15} - q^{16} - 6q^{18} + 4q^{19} - q^{20} + 9q^{21} - 8q^{23} - 9q^{24} + q^{25} - 4q^{26} - 9q^{27} + 3q^{28} + 6q^{29} + 3q^{30} - 2q^{31} - 5q^{32} - 3q^{35} - 6q^{36} - 8q^{37} - 4q^{38} - 12q^{39} + 3q^{40} - 5q^{41} - 9q^{42} + 5q^{43} + 6q^{45} + 8q^{46} - 3q^{47} + 3q^{48} + 2q^{49} - q^{50} - 4q^{52} + 4q^{53} + 9q^{54} - 9q^{56} - 12q^{57} - 6q^{58} - 2q^{59} + 3q^{60} - 11q^{61} + 2q^{62} - 18q^{63} + 7q^{64} + 4q^{65} - 13q^{67} + 24q^{69} + 3q^{70} + 2q^{71} + 18q^{72} - 8q^{73} + 8q^{74} - 3q^{75} - 4q^{76} + 12q^{78} + 10q^{79} - q^{80} + 9q^{81} + 5q^{82} + 4q^{83} - 9q^{84} - 5q^{86} - 18q^{87} + q^{89} - 6q^{90} - 12q^{91} + 8q^{92} + 6q^{93} + 3q^{94} + 4q^{95} + 15q^{96} - 8q^{97} - 2q^{98} + O(q^{100})$$

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
−1.00000 −3.00000 −1.00000 1.00000 3.00000 −3.00000 3.00000 6.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$11$$ $$-1$$

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 605.2.a.a 1
3.b odd 2 1 5445.2.a.h 1
4.b odd 2 1 9680.2.a.bf 1
5.b even 2 1 3025.2.a.g 1
11.b odd 2 1 605.2.a.c yes 1
11.c even 5 4 605.2.g.d 4
11.d odd 10 4 605.2.g.b 4
33.d even 2 1 5445.2.a.d 1
44.c even 2 1 9680.2.a.be 1
55.d odd 2 1 3025.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.a.a 1 1.a even 1 1 trivial
605.2.a.c yes 1 11.b odd 2 1
605.2.g.b 4 11.d odd 10 4
605.2.g.d 4 11.c even 5 4
3025.2.a.c 1 55.d odd 2 1
3025.2.a.g 1 5.b even 2 1
5445.2.a.d 1 33.d even 2 1
5445.2.a.h 1 3.b odd 2 1
9680.2.a.be 1 44.c even 2 1
9680.2.a.bf 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$:

 $$T_{2} + 1$$ $$T_{3} + 3$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$3 + T$$
$5$ $$-1 + T$$
$7$ $$3 + T$$
$11$ $$T$$
$13$ $$-4 + T$$
$17$ $$T$$
$19$ $$-4 + T$$
$23$ $$8 + T$$
$29$ $$-6 + T$$
$31$ $$2 + T$$
$37$ $$8 + T$$
$41$ $$5 + T$$
$43$ $$-5 + T$$
$47$ $$3 + T$$
$53$ $$-4 + T$$
$59$ $$2 + T$$
$61$ $$11 + T$$
$67$ $$13 + T$$
$71$ $$-2 + T$$
$73$ $$8 + T$$
$79$ $$-10 + T$$
$83$ $$-4 + T$$
$89$ $$-1 + T$$
$97$ $$8 + T$$