# Properties

 Label 3675.2.a.b Level $3675$ Weight $2$ Character orbit 3675.a Self dual yes Analytic conductor $29.345$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3675 = 3 \cdot 5^{2} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3675.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$29.3450227428$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 75) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{6} + q^{9}+O(q^{10})$$ q - 2 * q^2 - q^3 + 2 * q^4 + 2 * q^6 + q^9 $$q - 2 q^{2} - q^{3} + 2 q^{4} + 2 q^{6} + q^{9} + 2 q^{11} - 2 q^{12} + q^{13} - 4 q^{16} + 2 q^{17} - 2 q^{18} + 5 q^{19} - 4 q^{22} - 6 q^{23} - 2 q^{26} - q^{27} + 10 q^{29} + 3 q^{31} + 8 q^{32} - 2 q^{33} - 4 q^{34} + 2 q^{36} - 2 q^{37} - 10 q^{38} - q^{39} + 8 q^{41} - q^{43} + 4 q^{44} + 12 q^{46} + 2 q^{47} + 4 q^{48} - 2 q^{51} + 2 q^{52} + 4 q^{53} + 2 q^{54} - 5 q^{57} - 20 q^{58} + 10 q^{59} - 7 q^{61} - 6 q^{62} - 8 q^{64} + 4 q^{66} + 3 q^{67} + 4 q^{68} + 6 q^{69} - 8 q^{71} - 14 q^{73} + 4 q^{74} + 10 q^{76} + 2 q^{78} + q^{81} - 16 q^{82} + 6 q^{83} + 2 q^{86} - 10 q^{87} - 12 q^{92} - 3 q^{93} - 4 q^{94} - 8 q^{96} + 17 q^{97} + 2 q^{99}+O(q^{100})$$ q - 2 * q^2 - q^3 + 2 * q^4 + 2 * q^6 + q^9 + 2 * q^11 - 2 * q^12 + q^13 - 4 * q^16 + 2 * q^17 - 2 * q^18 + 5 * q^19 - 4 * q^22 - 6 * q^23 - 2 * q^26 - q^27 + 10 * q^29 + 3 * q^31 + 8 * q^32 - 2 * q^33 - 4 * q^34 + 2 * q^36 - 2 * q^37 - 10 * q^38 - q^39 + 8 * q^41 - q^43 + 4 * q^44 + 12 * q^46 + 2 * q^47 + 4 * q^48 - 2 * q^51 + 2 * q^52 + 4 * q^53 + 2 * q^54 - 5 * q^57 - 20 * q^58 + 10 * q^59 - 7 * q^61 - 6 * q^62 - 8 * q^64 + 4 * q^66 + 3 * q^67 + 4 * q^68 + 6 * q^69 - 8 * q^71 - 14 * q^73 + 4 * q^74 + 10 * q^76 + 2 * q^78 + q^81 - 16 * q^82 + 6 * q^83 + 2 * q^86 - 10 * q^87 - 12 * q^92 - 3 * q^93 - 4 * q^94 - 8 * q^96 + 17 * q^97 + 2 * 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
−2.00000 −1.00000 2.00000 0 2.00000 0 0 1.00000 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
$$3$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$

## 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 3675.2.a.b 1
5.b even 2 1 3675.2.a.q 1
7.b odd 2 1 75.2.a.a 1
21.c even 2 1 225.2.a.e 1
28.d even 2 1 1200.2.a.c 1
35.c odd 2 1 75.2.a.c yes 1
35.f even 4 2 75.2.b.a 2
56.e even 2 1 4800.2.a.br 1
56.h odd 2 1 4800.2.a.bb 1
77.b even 2 1 9075.2.a.s 1
84.h odd 2 1 3600.2.a.j 1
105.g even 2 1 225.2.a.a 1
105.k odd 4 2 225.2.b.a 2
140.c even 2 1 1200.2.a.p 1
140.j odd 4 2 1200.2.f.d 2
280.c odd 2 1 4800.2.a.bq 1
280.n even 2 1 4800.2.a.be 1
280.s even 4 2 4800.2.f.l 2
280.y odd 4 2 4800.2.f.y 2
385.h even 2 1 9075.2.a.a 1
420.o odd 2 1 3600.2.a.bk 1
420.w even 4 2 3600.2.f.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 7.b odd 2 1
75.2.a.c yes 1 35.c odd 2 1
75.2.b.a 2 35.f even 4 2
225.2.a.a 1 105.g even 2 1
225.2.a.e 1 21.c even 2 1
225.2.b.a 2 105.k odd 4 2
1200.2.a.c 1 28.d even 2 1
1200.2.a.p 1 140.c even 2 1
1200.2.f.d 2 140.j odd 4 2
3600.2.a.j 1 84.h odd 2 1
3600.2.a.bk 1 420.o odd 2 1
3600.2.f.p 2 420.w even 4 2
3675.2.a.b 1 1.a even 1 1 trivial
3675.2.a.q 1 5.b even 2 1
4800.2.a.bb 1 56.h odd 2 1
4800.2.a.be 1 280.n even 2 1
4800.2.a.bq 1 280.c odd 2 1
4800.2.a.br 1 56.e even 2 1
4800.2.f.l 2 280.s even 4 2
4800.2.f.y 2 280.y odd 4 2
9075.2.a.a 1 385.h even 2 1
9075.2.a.s 1 77.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3675))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{11} - 2$$ T11 - 2 $$T_{13} - 1$$ T13 - 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T$$
$11$ $$T - 2$$
$13$ $$T - 1$$
$17$ $$T - 2$$
$19$ $$T - 5$$
$23$ $$T + 6$$
$29$ $$T - 10$$
$31$ $$T - 3$$
$37$ $$T + 2$$
$41$ $$T - 8$$
$43$ $$T + 1$$
$47$ $$T - 2$$
$53$ $$T - 4$$
$59$ $$T - 10$$
$61$ $$T + 7$$
$67$ $$T - 3$$
$71$ $$T + 8$$
$73$ $$T + 14$$
$79$ $$T$$
$83$ $$T - 6$$
$89$ $$T$$
$97$ $$T - 17$$