# Properties

 Label 3630.2.a.w Level $3630$ Weight $2$ Character orbit 3630.a Self dual yes Analytic conductor $28.986$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3630 = 2 \cdot 3 \cdot 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3630.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$28.9856959337$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 30) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + 4q^{7} + q^{8} + q^{9} - q^{10} + q^{12} - 2q^{13} + 4q^{14} - q^{15} + q^{16} - 6q^{17} + q^{18} + 4q^{19} - q^{20} + 4q^{21} + q^{24} + q^{25} - 2q^{26} + q^{27} + 4q^{28} + 6q^{29} - q^{30} + 8q^{31} + q^{32} - 6q^{34} - 4q^{35} + q^{36} + 2q^{37} + 4q^{38} - 2q^{39} - q^{40} + 6q^{41} + 4q^{42} + 4q^{43} - q^{45} + q^{48} + 9q^{49} + q^{50} - 6q^{51} - 2q^{52} - 6q^{53} + q^{54} + 4q^{56} + 4q^{57} + 6q^{58} - q^{60} + 10q^{61} + 8q^{62} + 4q^{63} + q^{64} + 2q^{65} - 4q^{67} - 6q^{68} - 4q^{70} + q^{72} - 2q^{73} + 2q^{74} + q^{75} + 4q^{76} - 2q^{78} - 8q^{79} - q^{80} + q^{81} + 6q^{82} - 12q^{83} + 4q^{84} + 6q^{85} + 4q^{86} + 6q^{87} + 18q^{89} - q^{90} - 8q^{91} + 8q^{93} - 4q^{95} + q^{96} + 2q^{97} + 9q^{98} + O(q^{100})$$

## 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 1.00000 1.00000 −1.00000 1.00000 4.00000 1.00000 1.00000 −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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$

## 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 3630.2.a.w 1
11.b odd 2 1 30.2.a.a 1
33.d even 2 1 90.2.a.c 1
44.c even 2 1 240.2.a.b 1
55.d odd 2 1 150.2.a.b 1
55.e even 4 2 150.2.c.a 2
77.b even 2 1 1470.2.a.d 1
77.h odd 6 2 1470.2.i.o 2
77.i even 6 2 1470.2.i.q 2
88.b odd 2 1 960.2.a.e 1
88.g even 2 1 960.2.a.p 1
99.g even 6 2 810.2.e.b 2
99.h odd 6 2 810.2.e.l 2
132.d odd 2 1 720.2.a.j 1
143.d odd 2 1 5070.2.a.w 1
143.g even 4 2 5070.2.b.k 2
165.d even 2 1 450.2.a.d 1
165.l odd 4 2 450.2.c.b 2
176.i even 4 2 3840.2.k.f 2
176.l odd 4 2 3840.2.k.y 2
187.b odd 2 1 8670.2.a.g 1
220.g even 2 1 1200.2.a.k 1
220.i odd 4 2 1200.2.f.e 2
231.h odd 2 1 4410.2.a.z 1
264.m even 2 1 2880.2.a.a 1
264.p odd 2 1 2880.2.a.q 1
385.h even 2 1 7350.2.a.ct 1
440.c even 2 1 4800.2.a.d 1
440.o odd 2 1 4800.2.a.cq 1
440.t even 4 2 4800.2.f.p 2
440.w odd 4 2 4800.2.f.w 2
660.g odd 2 1 3600.2.a.f 1
660.q even 4 2 3600.2.f.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.a.a 1 11.b odd 2 1
90.2.a.c 1 33.d even 2 1
150.2.a.b 1 55.d odd 2 1
150.2.c.a 2 55.e even 4 2
240.2.a.b 1 44.c even 2 1
450.2.a.d 1 165.d even 2 1
450.2.c.b 2 165.l odd 4 2
720.2.a.j 1 132.d odd 2 1
810.2.e.b 2 99.g even 6 2
810.2.e.l 2 99.h odd 6 2
960.2.a.e 1 88.b odd 2 1
960.2.a.p 1 88.g even 2 1
1200.2.a.k 1 220.g even 2 1
1200.2.f.e 2 220.i odd 4 2
1470.2.a.d 1 77.b even 2 1
1470.2.i.o 2 77.h odd 6 2
1470.2.i.q 2 77.i even 6 2
2880.2.a.a 1 264.m even 2 1
2880.2.a.q 1 264.p odd 2 1
3600.2.a.f 1 660.g odd 2 1
3600.2.f.i 2 660.q even 4 2
3630.2.a.w 1 1.a even 1 1 trivial
3840.2.k.f 2 176.i even 4 2
3840.2.k.y 2 176.l odd 4 2
4410.2.a.z 1 231.h odd 2 1
4800.2.a.d 1 440.c even 2 1
4800.2.a.cq 1 440.o odd 2 1
4800.2.f.p 2 440.t even 4 2
4800.2.f.w 2 440.w odd 4 2
5070.2.a.w 1 143.d odd 2 1
5070.2.b.k 2 143.g even 4 2
7350.2.a.ct 1 385.h even 2 1
8670.2.a.g 1 187.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3630))$$:

 $$T_{7} - 4$$ $$T_{13} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$-4 + T$$
$11$ $$T$$
$13$ $$2 + T$$
$17$ $$6 + T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$-6 + T$$
$31$ $$-8 + T$$
$37$ $$-2 + T$$
$41$ $$-6 + T$$
$43$ $$-4 + T$$
$47$ $$T$$
$53$ $$6 + T$$
$59$ $$T$$
$61$ $$-10 + T$$
$67$ $$4 + T$$
$71$ $$T$$
$73$ $$2 + T$$
$79$ $$8 + T$$
$83$ $$12 + T$$
$89$ $$-18 + T$$
$97$ $$-2 + T$$