# Properties

 Label 675.2.a.i Level $675$ Weight $2$ Character orbit 675.a Self dual yes Analytic conductor $5.390$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.38990213644$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 135) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 2 q^{2} + 2 q^{4} + 3 q^{7}+O(q^{10})$$ q + 2 * q^2 + 2 * q^4 + 3 * q^7 $$q + 2 q^{2} + 2 q^{4} + 3 q^{7} - 2 q^{11} + 5 q^{13} + 6 q^{14} - 4 q^{16} + 8 q^{17} + q^{19} - 4 q^{22} - 6 q^{23} + 10 q^{26} + 6 q^{28} + 2 q^{29} - 8 q^{32} + 16 q^{34} - 5 q^{37} + 2 q^{38} - 10 q^{41} - 4 q^{43} - 4 q^{44} - 12 q^{46} - 4 q^{47} + 2 q^{49} + 10 q^{52} + 2 q^{53} + 4 q^{58} - 8 q^{59} + 7 q^{61} - 8 q^{64} + 9 q^{67} + 16 q^{68} + 2 q^{71} + 5 q^{73} - 10 q^{74} + 2 q^{76} - 6 q^{77} - 3 q^{79} - 20 q^{82} - 6 q^{83} - 8 q^{86} - 12 q^{89} + 15 q^{91} - 12 q^{92} - 8 q^{94} + 13 q^{97} + 4 q^{98}+O(q^{100})$$ q + 2 * q^2 + 2 * q^4 + 3 * q^7 - 2 * q^11 + 5 * q^13 + 6 * q^14 - 4 * q^16 + 8 * q^17 + q^19 - 4 * q^22 - 6 * q^23 + 10 * q^26 + 6 * q^28 + 2 * q^29 - 8 * q^32 + 16 * q^34 - 5 * q^37 + 2 * q^38 - 10 * q^41 - 4 * q^43 - 4 * q^44 - 12 * q^46 - 4 * q^47 + 2 * q^49 + 10 * q^52 + 2 * q^53 + 4 * q^58 - 8 * q^59 + 7 * q^61 - 8 * q^64 + 9 * q^67 + 16 * q^68 + 2 * q^71 + 5 * q^73 - 10 * q^74 + 2 * q^76 - 6 * q^77 - 3 * q^79 - 20 * q^82 - 6 * q^83 - 8 * q^86 - 12 * q^89 + 15 * q^91 - 12 * q^92 - 8 * q^94 + 13 * q^97 + 4 * q^98

## 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
2.00000 0 2.00000 0 0 3.00000 0 0 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
$$3$$ $$-1$$
$$5$$ $$1$$

## 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 675.2.a.i 1
3.b odd 2 1 675.2.a.a 1
5.b even 2 1 135.2.a.a 1
5.c odd 4 2 675.2.b.a 2
15.d odd 2 1 135.2.a.b yes 1
15.e even 4 2 675.2.b.b 2
20.d odd 2 1 2160.2.a.j 1
35.c odd 2 1 6615.2.a.a 1
40.e odd 2 1 8640.2.a.ce 1
40.f even 2 1 8640.2.a.bh 1
45.h odd 6 2 405.2.e.b 2
45.j even 6 2 405.2.e.h 2
60.h even 2 1 2160.2.a.v 1
105.g even 2 1 6615.2.a.j 1
120.i odd 2 1 8640.2.a.c 1
120.m even 2 1 8640.2.a.bb 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.a.a 1 5.b even 2 1
135.2.a.b yes 1 15.d odd 2 1
405.2.e.b 2 45.h odd 6 2
405.2.e.h 2 45.j even 6 2
675.2.a.a 1 3.b odd 2 1
675.2.a.i 1 1.a even 1 1 trivial
675.2.b.a 2 5.c odd 4 2
675.2.b.b 2 15.e even 4 2
2160.2.a.j 1 20.d odd 2 1
2160.2.a.v 1 60.h even 2 1
6615.2.a.a 1 35.c odd 2 1
6615.2.a.j 1 105.g even 2 1
8640.2.a.c 1 120.i odd 2 1
8640.2.a.bb 1 120.m even 2 1
8640.2.a.bh 1 40.f even 2 1
8640.2.a.ce 1 40.e odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(675))$$:

 $$T_{2} - 2$$ T2 - 2 $$T_{7} - 3$$ T7 - 3 $$T_{11} + 2$$ T11 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 3$$
$11$ $$T + 2$$
$13$ $$T - 5$$
$17$ $$T - 8$$
$19$ $$T - 1$$
$23$ $$T + 6$$
$29$ $$T - 2$$
$31$ $$T$$
$37$ $$T + 5$$
$41$ $$T + 10$$
$43$ $$T + 4$$
$47$ $$T + 4$$
$53$ $$T - 2$$
$59$ $$T + 8$$
$61$ $$T - 7$$
$67$ $$T - 9$$
$71$ $$T - 2$$
$73$ $$T - 5$$
$79$ $$T + 3$$
$83$ $$T + 6$$
$89$ $$T + 12$$
$97$ $$T - 13$$