# Properties

 Label 3150.2.g.f 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 350) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} -i q^{7} -i q^{8} +O(q^{10})$$ $$q + i q^{2} - q^{4} -i q^{7} -i q^{8} -3 q^{11} + 2 i q^{13} + q^{14} + q^{16} + 3 i q^{17} + 7 q^{19} -3 i q^{22} -2 q^{26} + i q^{28} -6 q^{29} -4 q^{31} + i q^{32} -3 q^{34} -8 i q^{37} + 7 i q^{38} + 9 q^{41} + 8 i q^{43} + 3 q^{44} -6 i q^{47} - q^{49} -2 i q^{52} + 12 i q^{53} - q^{56} -6 i q^{58} + 12 q^{59} -10 q^{61} -4 i q^{62} - q^{64} + 7 i q^{67} -3 i q^{68} -6 q^{71} + 5 i q^{73} + 8 q^{74} -7 q^{76} + 3 i q^{77} -14 q^{79} + 9 i q^{82} + 9 i q^{83} -8 q^{86} + 3 i q^{88} -15 q^{89} + 2 q^{91} + 6 q^{94} + 10 i q^{97} -i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{4} + O(q^{10})$$ $$2q - 2q^{4} - 6q^{11} + 2q^{14} + 2q^{16} + 14q^{19} - 4q^{26} - 12q^{29} - 8q^{31} - 6q^{34} + 18q^{41} + 6q^{44} - 2q^{49} - 2q^{56} + 24q^{59} - 20q^{61} - 2q^{64} - 12q^{71} + 16q^{74} - 14q^{76} - 28q^{79} - 16q^{86} - 30q^{89} + 4q^{91} + 12q^{94} + O(q^{100})$$

## 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.f 2
3.b odd 2 1 350.2.c.c 2
5.b even 2 1 inner 3150.2.g.f 2
5.c odd 4 1 3150.2.a.m 1
5.c odd 4 1 3150.2.a.x 1
12.b even 2 1 2800.2.g.i 2
15.d odd 2 1 350.2.c.c 2
15.e even 4 1 350.2.a.a 1
15.e even 4 1 350.2.a.e yes 1
21.c even 2 1 2450.2.c.h 2
60.h even 2 1 2800.2.g.i 2
60.l odd 4 1 2800.2.a.h 1
60.l odd 4 1 2800.2.a.x 1
105.g even 2 1 2450.2.c.h 2
105.k odd 4 1 2450.2.a.m 1
105.k odd 4 1 2450.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.a.a 1 15.e even 4 1
350.2.a.e yes 1 15.e even 4 1
350.2.c.c 2 3.b odd 2 1
350.2.c.c 2 15.d odd 2 1
2450.2.a.m 1 105.k odd 4 1
2450.2.a.x 1 105.k odd 4 1
2450.2.c.h 2 21.c even 2 1
2450.2.c.h 2 105.g even 2 1
2800.2.a.h 1 60.l odd 4 1
2800.2.a.x 1 60.l odd 4 1
2800.2.g.i 2 12.b even 2 1
2800.2.g.i 2 60.h even 2 1
3150.2.a.m 1 5.c odd 4 1
3150.2.a.x 1 5.c odd 4 1
3150.2.g.f 2 1.a even 1 1 trivial
3150.2.g.f 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{11} + 3$$ $$T_{13}^{2} + 4$$ $$T_{17}^{2} + 9$$ $$T_{19} - 7$$ $$T_{29} + 6$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ 
$5$ 
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 3 T + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$1 - 25 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 7 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 + 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 4 T + 31 T^{2} )^{2}$$
$37$ $$1 - 10 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 9 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$1 - 58 T^{2} + 2209 T^{4}$$
$53$ $$1 + 38 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 12 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 10 T + 61 T^{2} )^{2}$$
$67$ $$1 - 85 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 121 T^{2} + 5329 T^{4}$$
$79$ $$( 1 + 14 T + 79 T^{2} )^{2}$$
$83$ $$1 - 85 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 15 T + 89 T^{2} )^{2}$$
$97$ $$1 - 94 T^{2} + 9409 T^{4}$$