Properties

 Label 3042.2.b.g 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 78) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} + 2 i q^{5} + 4 i q^{7} + i q^{8} +O(q^{10})$$ q - i * q^2 - q^4 + 2*i * q^5 + 4*i * q^7 + i * q^8 $$q - i q^{2} - q^{4} + 2 i q^{5} + 4 i q^{7} + i q^{8} + 2 q^{10} + 4 i q^{11} + 4 q^{14} + q^{16} + 2 q^{17} + 8 i q^{19} - 2 i q^{20} + 4 q^{22} + q^{25} - 4 i q^{28} - 6 q^{29} + 4 i q^{31} - i q^{32} - 2 i q^{34} - 8 q^{35} - 2 i q^{37} + 8 q^{38} - 2 q^{40} - 10 i q^{41} - 4 q^{43} - 4 i q^{44} - 8 i q^{47} - 9 q^{49} - i q^{50} + 10 q^{53} - 8 q^{55} - 4 q^{56} + 6 i q^{58} - 4 i q^{59} - 2 q^{61} + 4 q^{62} - q^{64} + 16 i q^{67} - 2 q^{68} + 8 i q^{70} - 8 i q^{71} + 2 i q^{73} - 2 q^{74} - 8 i q^{76} - 16 q^{77} + 8 q^{79} + 2 i q^{80} - 10 q^{82} + 12 i q^{83} + 4 i q^{85} + 4 i q^{86} - 4 q^{88} - 14 i q^{89} - 8 q^{94} - 16 q^{95} - 10 i q^{97} + 9 i q^{98} +O(q^{100})$$ q - i * q^2 - q^4 + 2*i * q^5 + 4*i * q^7 + i * q^8 + 2 * q^10 + 4*i * q^11 + 4 * q^14 + q^16 + 2 * q^17 + 8*i * q^19 - 2*i * q^20 + 4 * q^22 + q^25 - 4*i * q^28 - 6 * q^29 + 4*i * q^31 - i * q^32 - 2*i * q^34 - 8 * q^35 - 2*i * q^37 + 8 * q^38 - 2 * q^40 - 10*i * q^41 - 4 * q^43 - 4*i * q^44 - 8*i * q^47 - 9 * q^49 - i * q^50 + 10 * q^53 - 8 * q^55 - 4 * q^56 + 6*i * q^58 - 4*i * q^59 - 2 * q^61 + 4 * q^62 - q^64 + 16*i * q^67 - 2 * q^68 + 8*i * q^70 - 8*i * q^71 + 2*i * q^73 - 2 * q^74 - 8*i * q^76 - 16 * q^77 + 8 * q^79 + 2*i * q^80 - 10 * q^82 + 12*i * q^83 + 4*i * q^85 + 4*i * q^86 - 4 * q^88 - 14*i * q^89 - 8 * q^94 - 16 * q^95 - 10*i * q^97 + 9*i * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4}+O(q^{10})$$ 2 * q - 2 * q^4 $$2 q - 2 q^{4} + 4 q^{10} + 8 q^{14} + 2 q^{16} + 4 q^{17} + 8 q^{22} + 2 q^{25} - 12 q^{29} - 16 q^{35} + 16 q^{38} - 4 q^{40} - 8 q^{43} - 18 q^{49} + 20 q^{53} - 16 q^{55} - 8 q^{56} - 4 q^{61} + 8 q^{62} - 2 q^{64} - 4 q^{68} - 4 q^{74} - 32 q^{77} + 16 q^{79} - 20 q^{82} - 8 q^{88} - 16 q^{94} - 32 q^{95}+O(q^{100})$$ 2 * q - 2 * q^4 + 4 * q^10 + 8 * q^14 + 2 * q^16 + 4 * q^17 + 8 * q^22 + 2 * q^25 - 12 * q^29 - 16 * q^35 + 16 * q^38 - 4 * q^40 - 8 * q^43 - 18 * q^49 + 20 * q^53 - 16 * q^55 - 8 * q^56 - 4 * q^61 + 8 * q^62 - 2 * q^64 - 4 * q^68 - 4 * q^74 - 32 * q^77 + 16 * q^79 - 20 * q^82 - 8 * q^88 - 16 * q^94 - 32 * 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 2.00000i 0 4.00000i 1.00000i 0 2.00000
1351.2 1.00000i 0 −1.00000 2.00000i 0 4.00000i 1.00000i 0 2.00000
 $$n$$: 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.g 2
3.b odd 2 1 1014.2.b.b 2
13.b even 2 1 inner 3042.2.b.g 2
13.d odd 4 1 234.2.a.c 1
13.d odd 4 1 3042.2.a.f 1
39.d odd 2 1 1014.2.b.b 2
39.f even 4 1 78.2.a.a 1
39.f even 4 1 1014.2.a.d 1
39.h odd 6 2 1014.2.i.d 4
39.i odd 6 2 1014.2.i.d 4
39.k even 12 2 1014.2.e.c 2
39.k even 12 2 1014.2.e.f 2
52.f even 4 1 1872.2.a.c 1
65.f even 4 1 5850.2.e.bb 2
65.g odd 4 1 5850.2.a.d 1
65.k even 4 1 5850.2.e.bb 2
104.j odd 4 1 7488.2.a.bz 1
104.m even 4 1 7488.2.a.bk 1
117.y odd 12 2 2106.2.e.j 2
117.z even 12 2 2106.2.e.q 2
156.l odd 4 1 624.2.a.h 1
156.l odd 4 1 8112.2.a.v 1
195.j odd 4 1 1950.2.e.i 2
195.n even 4 1 1950.2.a.w 1
195.u odd 4 1 1950.2.e.i 2
273.o odd 4 1 3822.2.a.j 1
312.w odd 4 1 2496.2.a.b 1
312.y even 4 1 2496.2.a.t 1
429.l odd 4 1 9438.2.a.t 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.a.a 1 39.f even 4 1
234.2.a.c 1 13.d odd 4 1
624.2.a.h 1 156.l odd 4 1
1014.2.a.d 1 39.f even 4 1
1014.2.b.b 2 3.b odd 2 1
1014.2.b.b 2 39.d odd 2 1
1014.2.e.c 2 39.k even 12 2
1014.2.e.f 2 39.k even 12 2
1014.2.i.d 4 39.h odd 6 2
1014.2.i.d 4 39.i odd 6 2
1872.2.a.c 1 52.f even 4 1
1950.2.a.w 1 195.n even 4 1
1950.2.e.i 2 195.j odd 4 1
1950.2.e.i 2 195.u odd 4 1
2106.2.e.j 2 117.y odd 12 2
2106.2.e.q 2 117.z even 12 2
2496.2.a.b 1 312.w odd 4 1
2496.2.a.t 1 312.y even 4 1
3042.2.a.f 1 13.d odd 4 1
3042.2.b.g 2 1.a even 1 1 trivial
3042.2.b.g 2 13.b even 2 1 inner
3822.2.a.j 1 273.o odd 4 1
5850.2.a.d 1 65.g odd 4 1
5850.2.e.bb 2 65.f even 4 1
5850.2.e.bb 2 65.k even 4 1
7488.2.a.bk 1 104.m even 4 1
7488.2.a.bz 1 104.j odd 4 1
8112.2.a.v 1 156.l odd 4 1
9438.2.a.t 1 429.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 $$S_{2}^{\mathrm{new}}(3042, [\chi])$$:

 $$T_{5}^{2} + 4$$ T5^2 + 4 $$T_{7}^{2} + 16$$ T7^2 + 16 $$T_{17} - 2$$ T17 - 2 $$T_{23}$$ T23

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2} + 4$$
$7$ $$T^{2} + 16$$
$11$ $$T^{2} + 16$$
$13$ $$T^{2}$$
$17$ $$(T - 2)^{2}$$
$19$ $$T^{2} + 64$$
$23$ $$T^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 4$$
$41$ $$T^{2} + 100$$
$43$ $$(T + 4)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T - 10)^{2}$$
$59$ $$T^{2} + 16$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 256$$
$71$ $$T^{2} + 64$$
$73$ $$T^{2} + 4$$
$79$ $$(T - 8)^{2}$$
$83$ $$T^{2} + 144$$
$89$ $$T^{2} + 196$$
$97$ $$T^{2} + 100$$