# Properties

 Label 3150.2.a.bk Level $3150$ Weight $2$ Character orbit 3150.a Self dual yes Analytic conductor $25.153$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.1528766367$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + q^{7} + q^{8} - 2q^{11} - 6q^{13} + q^{14} + q^{16} + 4q^{17} - 6q^{19} - 2q^{22} - 8q^{23} - 6q^{26} + q^{28} - 6q^{29} - 2q^{31} + q^{32} + 4q^{34} - 4q^{37} - 6q^{38} - 2q^{41} - 4q^{43} - 2q^{44} - 8q^{46} + 8q^{47} + q^{49} - 6q^{52} + 6q^{53} + q^{56} - 6q^{58} + 8q^{59} - 10q^{61} - 2q^{62} + q^{64} - 8q^{67} + 4q^{68} + 6q^{71} + 14q^{73} - 4q^{74} - 6q^{76} - 2q^{77} - 12q^{79} - 2q^{82} - 8q^{83} - 4q^{86} - 2q^{88} + 10q^{89} - 6q^{91} - 8q^{92} + 8q^{94} + 10q^{97} + q^{98} + O(q^{100})$$

## 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
 0
1.00000 0 1.00000 0 0 1.00000 1.00000 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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$-1$$

## 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 3150.2.a.bk 1
3.b odd 2 1 1050.2.a.d 1
5.b even 2 1 3150.2.a.d 1
5.c odd 4 2 630.2.g.c 2
12.b even 2 1 8400.2.a.bp 1
15.d odd 2 1 1050.2.a.p 1
15.e even 4 2 210.2.g.b 2
20.e even 4 2 5040.2.t.h 2
21.c even 2 1 7350.2.a.bk 1
60.h even 2 1 8400.2.a.w 1
60.l odd 4 2 1680.2.t.e 2
105.g even 2 1 7350.2.a.bz 1
105.k odd 4 2 1470.2.g.b 2
105.w odd 12 4 1470.2.n.f 4
105.x even 12 4 1470.2.n.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.g.b 2 15.e even 4 2
630.2.g.c 2 5.c odd 4 2
1050.2.a.d 1 3.b odd 2 1
1050.2.a.p 1 15.d odd 2 1
1470.2.g.b 2 105.k odd 4 2
1470.2.n.b 4 105.x even 12 4
1470.2.n.f 4 105.w odd 12 4
1680.2.t.e 2 60.l odd 4 2
3150.2.a.d 1 5.b even 2 1
3150.2.a.bk 1 1.a even 1 1 trivial
5040.2.t.h 2 20.e even 4 2
7350.2.a.bk 1 21.c even 2 1
7350.2.a.bz 1 105.g even 2 1
8400.2.a.w 1 60.h even 2 1
8400.2.a.bp 1 12.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3150))$$:

 $$T_{11} + 2$$ $$T_{13} + 6$$ $$T_{17} - 4$$ $$T_{19} + 6$$ $$T_{29} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$T$$
$5$ $$T$$
$7$ $$-1 + T$$
$11$ $$2 + T$$
$13$ $$6 + T$$
$17$ $$-4 + T$$
$19$ $$6 + T$$
$23$ $$8 + T$$
$29$ $$6 + T$$
$31$ $$2 + T$$
$37$ $$4 + T$$
$41$ $$2 + T$$
$43$ $$4 + T$$
$47$ $$-8 + T$$
$53$ $$-6 + T$$
$59$ $$-8 + T$$
$61$ $$10 + T$$
$67$ $$8 + T$$
$71$ $$-6 + T$$
$73$ $$-14 + T$$
$79$ $$12 + T$$
$83$ $$8 + T$$
$89$ $$-10 + T$$
$97$ $$-10 + T$$