# Properties

 Label 3150.2.g.s Level 3150 Weight 2 Character orbit 3150.g Analytic conductor 25.153 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 3150.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 630) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -i q^{2} - q^{4} + i q^{7} + i q^{8} +O(q^{10})$$ $$q -i q^{2} - q^{4} + i q^{7} + i q^{8} + 4 q^{11} + 6 i q^{13} + q^{14} + q^{16} + 4 i q^{17} -6 q^{19} -4 i q^{22} + 6 q^{26} -i q^{28} -6 q^{29} -4 q^{31} -i q^{32} + 4 q^{34} -8 i q^{37} + 6 i q^{38} -10 q^{41} -2 i q^{43} -4 q^{44} + 10 i q^{47} - q^{49} -6 i q^{52} -14 i q^{53} - q^{56} + 6 i q^{58} -4 q^{59} -8 q^{61} + 4 i q^{62} - q^{64} -6 i q^{67} -4 i q^{68} + 2 q^{71} -10 i q^{73} -8 q^{74} + 6 q^{76} + 4 i q^{77} -16 q^{79} + 10 i q^{82} + 8 i q^{83} -2 q^{86} + 4 i q^{88} + 2 q^{89} -6 q^{91} + 10 q^{94} -2 i q^{97} + i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} + 8q^{11} + 2q^{14} + 2q^{16} - 12q^{19} + 12q^{26} - 12q^{29} - 8q^{31} + 8q^{34} - 20q^{41} - 8q^{44} - 2q^{49} - 2q^{56} - 8q^{59} - 16q^{61} - 2q^{64} + 4q^{71} - 16q^{74} + 12q^{76} - 32q^{79} - 4q^{86} + 4q^{89} - 12q^{91} + 20q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$451$$ $$2801$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2899.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 0
2899.2 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.2.g.s 2
3.b odd 2 1 3150.2.g.b 2
5.b even 2 1 inner 3150.2.g.s 2
5.c odd 4 1 630.2.a.g yes 1
5.c odd 4 1 3150.2.a.s 1
15.d odd 2 1 3150.2.g.b 2
15.e even 4 1 630.2.a.e 1
15.e even 4 1 3150.2.a.bh 1
20.e even 4 1 5040.2.a.n 1
35.f even 4 1 4410.2.a.bl 1
60.l odd 4 1 5040.2.a.bp 1
105.k odd 4 1 4410.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
630.2.a.e 1 15.e even 4 1
630.2.a.g yes 1 5.c odd 4 1
3150.2.a.s 1 5.c odd 4 1
3150.2.a.bh 1 15.e even 4 1
3150.2.g.b 2 3.b odd 2 1
3150.2.g.b 2 15.d odd 2 1
3150.2.g.s 2 1.a even 1 1 trivial
3150.2.g.s 2 5.b even 2 1 inner
4410.2.a.a 1 105.k odd 4 1
4410.2.a.bl 1 35.f even 4 1
5040.2.a.n 1 20.e even 4 1
5040.2.a.bp 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3150, [\chi])$$:

 $$T_{11} - 4$$ $$T_{13}^{2} + 36$$ $$T_{17}^{2} + 16$$ $$T_{19} + 6$$ $$T_{29} + 6$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 1
$5$ 1
$7$ $$1 + T^{2}$$
$11$ $$( 1 - 4 T + 11 T^{2} )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$1 - 18 T^{2} + 289 T^{4}$$
$19$ $$( 1 + 6 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 10 T + 41 T^{2} )^{2}$$
$43$ $$1 - 82 T^{2} + 1849 T^{4}$$
$47$ $$1 + 6 T^{2} + 2209 T^{4}$$
$53$ $$( 1 - 4 T + 53 T^{2} )( 1 + 4 T + 53 T^{2} )$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 8 T + 61 T^{2} )^{2}$$
$67$ $$1 - 98 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 2 T + 71 T^{2} )^{2}$$
$73$ $$1 - 46 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 16 T + 79 T^{2} )^{2}$$
$83$ $$1 - 102 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$