# Properties

 Label 605.2.a.d Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta ) q^{2} + 2 \beta q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( 4 - 2 \beta ) q^{6} + 2 q^{7} + ( -3 + \beta ) q^{8} + 5 q^{9} +O(q^{10})$$ $$q + ( -1 + \beta ) q^{2} + 2 \beta q^{3} + ( 1 - 2 \beta ) q^{4} - q^{5} + ( 4 - 2 \beta ) q^{6} + 2 q^{7} + ( -3 + \beta ) q^{8} + 5 q^{9} + ( 1 - \beta ) q^{10} + ( -8 + 2 \beta ) q^{12} + ( 4 + 2 \beta ) q^{13} + ( -2 + 2 \beta ) q^{14} -2 \beta q^{15} + 3 q^{16} + ( -4 + 2 \beta ) q^{17} + ( -5 + 5 \beta ) q^{18} + ( -1 + 2 \beta ) q^{20} + 4 \beta q^{21} + 2 \beta q^{23} + ( 4 - 6 \beta ) q^{24} + q^{25} + 2 \beta q^{26} + 4 \beta q^{27} + ( 2 - 4 \beta ) q^{28} + ( -2 - 4 \beta ) q^{29} + ( -4 + 2 \beta ) q^{30} + ( 3 + \beta ) q^{32} + ( 8 - 6 \beta ) q^{34} -2 q^{35} + ( 5 - 10 \beta ) q^{36} + ( -2 + 4 \beta ) q^{37} + ( 8 + 8 \beta ) q^{39} + ( 3 - \beta ) q^{40} -6 q^{41} + ( 8 - 4 \beta ) q^{42} + 6 q^{43} -5 q^{45} + ( 4 - 2 \beta ) q^{46} -2 \beta q^{47} + 6 \beta q^{48} -3 q^{49} + ( -1 + \beta ) q^{50} + ( 8 - 8 \beta ) q^{51} + ( -4 - 6 \beta ) q^{52} + ( 6 - 4 \beta ) q^{53} + ( 8 - 4 \beta ) q^{54} + ( -6 + 2 \beta ) q^{56} + ( -6 + 2 \beta ) q^{58} + ( -4 - 4 \beta ) q^{59} + ( 8 - 2 \beta ) q^{60} + ( -2 - 8 \beta ) q^{61} + 10 q^{63} + ( -7 + 2 \beta ) q^{64} + ( -4 - 2 \beta ) q^{65} + ( 4 - 6 \beta ) q^{67} + ( -12 + 10 \beta ) q^{68} + 8 q^{69} + ( 2 - 2 \beta ) q^{70} -8 \beta q^{71} + ( -15 + 5 \beta ) q^{72} + ( 4 + 2 \beta ) q^{73} + ( 10 - 6 \beta ) q^{74} + 2 \beta q^{75} + 8 q^{78} -4 q^{79} -3 q^{80} + q^{81} + ( 6 - 6 \beta ) q^{82} + 6 q^{83} + ( -16 + 4 \beta ) q^{84} + ( 4 - 2 \beta ) q^{85} + ( -6 + 6 \beta ) q^{86} + ( -16 - 4 \beta ) q^{87} + ( -2 + 8 \beta ) q^{89} + ( 5 - 5 \beta ) q^{90} + ( 8 + 4 \beta ) q^{91} + ( -8 + 2 \beta ) q^{92} + ( -4 + 2 \beta ) q^{94} + ( 4 + 6 \beta ) q^{96} + ( -2 - 4 \beta ) q^{97} + ( 3 - 3 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} + 8q^{6} + 4q^{7} - 6q^{8} + 10q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{4} - 2q^{5} + 8q^{6} + 4q^{7} - 6q^{8} + 10q^{9} + 2q^{10} - 16q^{12} + 8q^{13} - 4q^{14} + 6q^{16} - 8q^{17} - 10q^{18} - 2q^{20} + 8q^{24} + 2q^{25} + 4q^{28} - 4q^{29} - 8q^{30} + 6q^{32} + 16q^{34} - 4q^{35} + 10q^{36} - 4q^{37} + 16q^{39} + 6q^{40} - 12q^{41} + 16q^{42} + 12q^{43} - 10q^{45} + 8q^{46} - 6q^{49} - 2q^{50} + 16q^{51} - 8q^{52} + 12q^{53} + 16q^{54} - 12q^{56} - 12q^{58} - 8q^{59} + 16q^{60} - 4q^{61} + 20q^{63} - 14q^{64} - 8q^{65} + 8q^{67} - 24q^{68} + 16q^{69} + 4q^{70} - 30q^{72} + 8q^{73} + 20q^{74} + 16q^{78} - 8q^{79} - 6q^{80} + 2q^{81} + 12q^{82} + 12q^{83} - 32q^{84} + 8q^{85} - 12q^{86} - 32q^{87} - 4q^{89} + 10q^{90} + 16q^{91} - 16q^{92} - 8q^{94} + 8q^{96} - 4q^{97} + 6q^{98} + O(q^{100})$$

## 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
 −1.41421 1.41421
−2.41421 −2.82843 3.82843 −1.00000 6.82843 2.00000 −4.41421 5.00000 2.41421
1.2 0.414214 2.82843 −1.82843 −1.00000 1.17157 2.00000 −1.58579 5.00000 −0.414214
 $$n$$: 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.d 2
3.b odd 2 1 5445.2.a.y 2
4.b odd 2 1 9680.2.a.bn 2
5.b even 2 1 3025.2.a.o 2
11.b odd 2 1 55.2.a.b 2
11.c even 5 4 605.2.g.l 8
11.d odd 10 4 605.2.g.f 8
33.d even 2 1 495.2.a.b 2
44.c even 2 1 880.2.a.m 2
55.d odd 2 1 275.2.a.c 2
55.e even 4 2 275.2.b.d 4
77.b even 2 1 2695.2.a.f 2
88.b odd 2 1 3520.2.a.bn 2
88.g even 2 1 3520.2.a.bo 2
132.d odd 2 1 7920.2.a.ch 2
143.d odd 2 1 9295.2.a.g 2
165.d even 2 1 2475.2.a.x 2
165.l odd 4 2 2475.2.c.l 4
220.g even 2 1 4400.2.a.bn 2
220.i odd 4 2 4400.2.b.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 11.b odd 2 1
275.2.a.c 2 55.d odd 2 1
275.2.b.d 4 55.e even 4 2
495.2.a.b 2 33.d even 2 1
605.2.a.d 2 1.a even 1 1 trivial
605.2.g.f 8 11.d odd 10 4
605.2.g.l 8 11.c even 5 4
880.2.a.m 2 44.c even 2 1
2475.2.a.x 2 165.d even 2 1
2475.2.c.l 4 165.l odd 4 2
2695.2.a.f 2 77.b even 2 1
3025.2.a.o 2 5.b even 2 1
3520.2.a.bn 2 88.b odd 2 1
3520.2.a.bo 2 88.g even 2 1
4400.2.a.bn 2 220.g even 2 1
4400.2.b.q 4 220.i odd 4 2
5445.2.a.y 2 3.b odd 2 1
7920.2.a.ch 2 132.d odd 2 1
9295.2.a.g 2 143.d odd 2 1
9680.2.a.bn 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$:

 $$T_{2}^{2} + 2 T_{2} - 1$$ $$T_{3}^{2} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + 2 T + T^{2}$$
$3$ $$-8 + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( -2 + T )^{2}$$
$11$ $$T^{2}$$
$13$ $$8 - 8 T + T^{2}$$
$17$ $$8 + 8 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-8 + T^{2}$$
$29$ $$-28 + 4 T + T^{2}$$
$31$ $$T^{2}$$
$37$ $$-28 + 4 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( -6 + T )^{2}$$
$47$ $$-8 + T^{2}$$
$53$ $$4 - 12 T + T^{2}$$
$59$ $$-16 + 8 T + T^{2}$$
$61$ $$-124 + 4 T + T^{2}$$
$67$ $$-56 - 8 T + T^{2}$$
$71$ $$-128 + T^{2}$$
$73$ $$8 - 8 T + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$( -6 + T )^{2}$$
$89$ $$-124 + 4 T + T^{2}$$
$97$ $$-28 + 4 T + T^{2}$$