# Properties

 Label 3025.2.a.o Level $3025$ Weight $2$ Character orbit 3025.a Self dual yes Analytic conductor $24.155$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3025 = 5^{2} \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3025.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$24.1547466114$$ 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.

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

## 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
−0.414214 −2.82843 −1.82843 0 1.17157 −2.00000 1.58579 5.00000 0
1.2 2.41421 2.82843 3.82843 0 6.82843 −2.00000 4.41421 5.00000 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
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 3025.2.a.o 2
5.b even 2 1 605.2.a.d 2
11.b odd 2 1 275.2.a.c 2
15.d odd 2 1 5445.2.a.y 2
20.d odd 2 1 9680.2.a.bn 2
33.d even 2 1 2475.2.a.x 2
44.c even 2 1 4400.2.a.bn 2
55.d odd 2 1 55.2.a.b 2
55.e even 4 2 275.2.b.d 4
55.h odd 10 4 605.2.g.f 8
55.j even 10 4 605.2.g.l 8
165.d even 2 1 495.2.a.b 2
165.l odd 4 2 2475.2.c.l 4
220.g even 2 1 880.2.a.m 2
220.i odd 4 2 4400.2.b.q 4
385.h even 2 1 2695.2.a.f 2
440.c even 2 1 3520.2.a.bo 2
440.o odd 2 1 3520.2.a.bn 2
660.g odd 2 1 7920.2.a.ch 2
715.c odd 2 1 9295.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 55.d odd 2 1
275.2.a.c 2 11.b odd 2 1
275.2.b.d 4 55.e even 4 2
495.2.a.b 2 165.d even 2 1
605.2.a.d 2 5.b even 2 1
605.2.g.f 8 55.h odd 10 4
605.2.g.l 8 55.j even 10 4
880.2.a.m 2 220.g even 2 1
2475.2.a.x 2 33.d even 2 1
2475.2.c.l 4 165.l odd 4 2
2695.2.a.f 2 385.h even 2 1
3025.2.a.o 2 1.a even 1 1 trivial
3520.2.a.bn 2 440.o odd 2 1
3520.2.a.bo 2 440.c even 2 1
4400.2.a.bn 2 44.c even 2 1
4400.2.b.q 4 220.i odd 4 2
5445.2.a.y 2 15.d odd 2 1
7920.2.a.ch 2 660.g odd 2 1
9295.2.a.g 2 715.c odd 2 1
9680.2.a.bn 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3025))$$:

 $$T_{2}^{2} - 2 T_{2} - 1$$ $$T_{3}^{2} - 8$$ $$T_{19}$$

## Hecke characteristic polynomials

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