# Properties

 Label 3150.2.a.w 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

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

## 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.w 1
3.b odd 2 1 1050.2.a.h 1
5.b even 2 1 630.2.a.b 1
5.c odd 4 2 3150.2.g.e 2
12.b even 2 1 8400.2.a.p 1
15.d odd 2 1 210.2.a.c 1
15.e even 4 2 1050.2.g.d 2
20.d odd 2 1 5040.2.a.i 1
21.c even 2 1 7350.2.a.p 1
35.c odd 2 1 4410.2.a.l 1
60.h even 2 1 1680.2.a.q 1
105.g even 2 1 1470.2.a.q 1
105.o odd 6 2 1470.2.i.f 2
105.p even 6 2 1470.2.i.b 2
120.i odd 2 1 6720.2.a.bp 1
120.m even 2 1 6720.2.a.k 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.c 1 15.d odd 2 1
630.2.a.b 1 5.b even 2 1
1050.2.a.h 1 3.b odd 2 1
1050.2.g.d 2 15.e even 4 2
1470.2.a.q 1 105.g even 2 1
1470.2.i.b 2 105.p even 6 2
1470.2.i.f 2 105.o odd 6 2
1680.2.a.q 1 60.h even 2 1
3150.2.a.w 1 1.a even 1 1 trivial
3150.2.g.e 2 5.c odd 4 2
4410.2.a.l 1 35.c odd 2 1
5040.2.a.i 1 20.d odd 2 1
6720.2.a.k 1 120.m even 2 1
6720.2.a.bp 1 120.i odd 2 1
7350.2.a.p 1 21.c even 2 1
8400.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(3150))$$:

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

## Hecke characteristic polynomials

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