# Properties

 Label 3150.2.g.e 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 210) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} -i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} -i q^{7} -i q^{8} -4 q^{11} -2 i q^{13} + q^{14} + q^{16} + 2 i q^{17} + 4 q^{19} -4 i q^{22} + 8 i q^{23} + 2 q^{26} + i q^{28} + 6 q^{29} -8 q^{31} + i q^{32} -2 q^{34} + 2 i q^{37} + 4 i q^{38} -2 q^{41} -12 i q^{43} + 4 q^{44} -8 q^{46} -8 i q^{47} - q^{49} + 2 i q^{52} -6 i q^{53} - q^{56} + 6 i q^{58} + 4 q^{59} -2 q^{61} -8 i q^{62} - q^{64} -12 i q^{67} -2 i q^{68} -8 q^{71} -14 i q^{73} -2 q^{74} -4 q^{76} + 4 i q^{77} -2 i q^{82} -12 i q^{83} + 12 q^{86} + 4 i q^{88} + 2 q^{89} -2 q^{91} -8 i q^{92} + 8 q^{94} -10 i q^{97} -i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 8q^{11} + 2q^{14} + 2q^{16} + 8q^{19} + 4q^{26} + 12q^{29} - 16q^{31} - 4q^{34} - 4q^{41} + 8q^{44} - 16q^{46} - 2q^{49} - 2q^{56} + 8q^{59} - 4q^{61} - 2q^{64} - 16q^{71} - 4q^{74} - 8q^{76} + 24q^{86} + 4q^{89} - 4q^{91} + 16q^{94} + O(q^{100})$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.c 1 15.e even 4 1
630.2.a.b 1 5.c odd 4 1
1050.2.a.h 1 15.e even 4 1
1050.2.g.d 2 3.b odd 2 1
1050.2.g.d 2 15.d odd 2 1
1470.2.a.q 1 105.k odd 4 1
1470.2.i.b 2 105.w odd 12 2
1470.2.i.f 2 105.x even 12 2
1680.2.a.q 1 60.l odd 4 1
3150.2.a.w 1 5.c odd 4 1
3150.2.g.e 2 1.a even 1 1 trivial
3150.2.g.e 2 5.b even 2 1 inner
4410.2.a.l 1 35.f even 4 1
5040.2.a.i 1 20.e even 4 1
6720.2.a.k 1 120.q odd 4 1
6720.2.a.bp 1 120.w even 4 1
7350.2.a.p 1 105.k odd 4 1
8400.2.a.p 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 $$S_{2}^{\mathrm{new}}(3150, [\chi])$$:

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 1
$5$ 1
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 4 T + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} )$$
$19$ $$( 1 - 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + 18 T^{2} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 8 T + 31 T^{2} )^{2}$$
$37$ $$( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} )$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 + 58 T^{2} + 1849 T^{4}$$
$47$ $$1 - 30 T^{2} + 2209 T^{4}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{2}$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$1 + 50 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$