# Properties

 Label 3042.2.b.a Level $3042$ Weight $2$ Character orbit 3042.b Analytic conductor $24.290$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3042 = 2 \cdot 3^{2} \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3042.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.2904922949$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 26) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} - q^{4} + 3 i q^{5} + i q^{7} - i q^{8} +O(q^{10})$$ q + i * q^2 - q^4 + 3*i * q^5 + i * q^7 - i * q^8 $$q + i q^{2} - q^{4} + 3 i q^{5} + i q^{7} - i q^{8} - 3 q^{10} + 6 i q^{11} - q^{14} + q^{16} - 3 q^{17} + 2 i q^{19} - 3 i q^{20} - 6 q^{22} - 4 q^{25} - i q^{28} - 6 q^{29} - 4 i q^{31} + i q^{32} - 3 i q^{34} - 3 q^{35} + 7 i q^{37} - 2 q^{38} + 3 q^{40} + q^{43} - 6 i q^{44} + 3 i q^{47} + 6 q^{49} - 4 i q^{50} - 18 q^{55} + q^{56} - 6 i q^{58} - 6 i q^{59} + 8 q^{61} + 4 q^{62} - q^{64} + 14 i q^{67} + 3 q^{68} - 3 i q^{70} + 3 i q^{71} - 2 i q^{73} - 7 q^{74} - 2 i q^{76} - 6 q^{77} + 8 q^{79} + 3 i q^{80} - 12 i q^{83} - 9 i q^{85} + i q^{86} + 6 q^{88} - 6 i q^{89} - 3 q^{94} - 6 q^{95} - 10 i q^{97} + 6 i q^{98} +O(q^{100})$$ q + i * q^2 - q^4 + 3*i * q^5 + i * q^7 - i * q^8 - 3 * q^10 + 6*i * q^11 - q^14 + q^16 - 3 * q^17 + 2*i * q^19 - 3*i * q^20 - 6 * q^22 - 4 * q^25 - i * q^28 - 6 * q^29 - 4*i * q^31 + i * q^32 - 3*i * q^34 - 3 * q^35 + 7*i * q^37 - 2 * q^38 + 3 * q^40 + q^43 - 6*i * q^44 + 3*i * q^47 + 6 * q^49 - 4*i * q^50 - 18 * q^55 + q^56 - 6*i * q^58 - 6*i * q^59 + 8 * q^61 + 4 * q^62 - q^64 + 14*i * q^67 + 3 * q^68 - 3*i * q^70 + 3*i * q^71 - 2*i * q^73 - 7 * q^74 - 2*i * q^76 - 6 * q^77 + 8 * q^79 + 3*i * q^80 - 12*i * q^83 - 9*i * q^85 + i * q^86 + 6 * q^88 - 6*i * q^89 - 3 * q^94 - 6 * q^95 - 10*i * q^97 + 6*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} - 6 q^{10} - 2 q^{14} + 2 q^{16} - 6 q^{17} - 12 q^{22} - 8 q^{25} - 12 q^{29} - 6 q^{35} - 4 q^{38} + 6 q^{40} + 2 q^{43} + 12 q^{49} - 36 q^{55} + 2 q^{56} + 16 q^{61} + 8 q^{62} - 2 q^{64} + 6 q^{68} - 14 q^{74} - 12 q^{77} + 16 q^{79} + 12 q^{88} - 6 q^{94} - 12 q^{95}+O(q^{100})$$ 2 * q - 2 * q^4 - 6 * q^10 - 2 * q^14 + 2 * q^16 - 6 * q^17 - 12 * q^22 - 8 * q^25 - 12 * q^29 - 6 * q^35 - 4 * q^38 + 6 * q^40 + 2 * q^43 + 12 * q^49 - 36 * q^55 + 2 * q^56 + 16 * q^61 + 8 * q^62 - 2 * q^64 + 6 * q^68 - 14 * q^74 - 12 * q^77 + 16 * q^79 + 12 * q^88 - 6 * q^94 - 12 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3042\mathbb{Z}\right)^\times$$.

 $$n$$ $$677$$ $$847$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1351.1
 − 1.00000i 1.00000i
1.00000i 0 −1.00000 3.00000i 0 1.00000i 1.00000i 0 −3.00000
1351.2 1.00000i 0 −1.00000 3.00000i 0 1.00000i 1.00000i 0 −3.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3042.2.b.a 2
3.b odd 2 1 338.2.b.c 2
12.b even 2 1 2704.2.f.d 2
13.b even 2 1 inner 3042.2.b.a 2
13.d odd 4 1 234.2.a.e 1
13.d odd 4 1 3042.2.a.a 1
39.d odd 2 1 338.2.b.c 2
39.f even 4 1 26.2.a.a 1
39.f even 4 1 338.2.a.f 1
39.h odd 6 2 338.2.e.a 4
39.i odd 6 2 338.2.e.a 4
39.k even 12 2 338.2.c.a 2
39.k even 12 2 338.2.c.d 2
52.f even 4 1 1872.2.a.q 1
65.f even 4 1 5850.2.e.a 2
65.g odd 4 1 5850.2.a.p 1
65.k even 4 1 5850.2.e.a 2
104.j odd 4 1 7488.2.a.g 1
104.m even 4 1 7488.2.a.h 1
117.y odd 12 2 2106.2.e.b 2
117.z even 12 2 2106.2.e.ba 2
156.h even 2 1 2704.2.f.d 2
156.l odd 4 1 208.2.a.a 1
156.l odd 4 1 2704.2.a.f 1
195.j odd 4 1 650.2.b.d 2
195.n even 4 1 650.2.a.j 1
195.n even 4 1 8450.2.a.c 1
195.u odd 4 1 650.2.b.d 2
273.o odd 4 1 1274.2.a.d 1
273.cb odd 12 2 1274.2.f.r 2
273.cd even 12 2 1274.2.f.p 2
312.w odd 4 1 832.2.a.i 1
312.y even 4 1 832.2.a.d 1
429.l odd 4 1 3146.2.a.n 1
624.s odd 4 1 3328.2.b.j 2
624.u even 4 1 3328.2.b.m 2
624.bm even 4 1 3328.2.b.m 2
624.bo odd 4 1 3328.2.b.j 2
663.q even 4 1 7514.2.a.c 1
741.p odd 4 1 9386.2.a.j 1
780.bb odd 4 1 5200.2.a.x 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.2.a.a 1 39.f even 4 1
208.2.a.a 1 156.l odd 4 1
234.2.a.e 1 13.d odd 4 1
338.2.a.f 1 39.f even 4 1
338.2.b.c 2 3.b odd 2 1
338.2.b.c 2 39.d odd 2 1
338.2.c.a 2 39.k even 12 2
338.2.c.d 2 39.k even 12 2
338.2.e.a 4 39.h odd 6 2
338.2.e.a 4 39.i odd 6 2
650.2.a.j 1 195.n even 4 1
650.2.b.d 2 195.j odd 4 1
650.2.b.d 2 195.u odd 4 1
832.2.a.d 1 312.y even 4 1
832.2.a.i 1 312.w odd 4 1
1274.2.a.d 1 273.o odd 4 1
1274.2.f.p 2 273.cd even 12 2
1274.2.f.r 2 273.cb odd 12 2
1872.2.a.q 1 52.f even 4 1
2106.2.e.b 2 117.y odd 12 2
2106.2.e.ba 2 117.z even 12 2
2704.2.a.f 1 156.l odd 4 1
2704.2.f.d 2 12.b even 2 1
2704.2.f.d 2 156.h even 2 1
3042.2.a.a 1 13.d odd 4 1
3042.2.b.a 2 1.a even 1 1 trivial
3042.2.b.a 2 13.b even 2 1 inner
3146.2.a.n 1 429.l odd 4 1
3328.2.b.j 2 624.s odd 4 1
3328.2.b.j 2 624.bo odd 4 1
3328.2.b.m 2 624.u even 4 1
3328.2.b.m 2 624.bm even 4 1
5200.2.a.x 1 780.bb odd 4 1
5850.2.a.p 1 65.g odd 4 1
5850.2.e.a 2 65.f even 4 1
5850.2.e.a 2 65.k even 4 1
7488.2.a.g 1 104.j odd 4 1
7488.2.a.h 1 104.m even 4 1
7514.2.a.c 1 663.q even 4 1
8450.2.a.c 1 195.n even 4 1
9386.2.a.j 1 741.p odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3042, [\chi])$$:

 $$T_{5}^{2} + 9$$ T5^2 + 9 $$T_{7}^{2} + 1$$ T7^2 + 1 $$T_{17} + 3$$ T17 + 3 $$T_{23}$$ T23

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 9$$
$7$ $$T^{2} + 1$$
$11$ $$T^{2} + 36$$
$13$ $$T^{2}$$
$17$ $$(T + 3)^{2}$$
$19$ $$T^{2} + 4$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 49$$
$41$ $$T^{2}$$
$43$ $$(T - 1)^{2}$$
$47$ $$T^{2} + 9$$
$53$ $$T^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$(T - 8)^{2}$$
$67$ $$T^{2} + 196$$
$71$ $$T^{2} + 9$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 36$$
$97$ $$T^{2} + 100$$