# Properties

 Label 3042.3.c.e Level $3042$ Weight $3$ Character orbit 3042.c Analytic conductor $82.888$ 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: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 3042.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$82.8884964184$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta q^{2} -2 q^{4} + 3 \beta q^{5} + 4 q^{7} + 2 \beta q^{8} +O(q^{10})$$ $$q -\beta q^{2} -2 q^{4} + 3 \beta q^{5} + 4 q^{7} + 2 \beta q^{8} + 6 q^{10} -12 \beta q^{11} -4 \beta q^{14} + 4 q^{16} -9 \beta q^{17} + 16 q^{19} -6 \beta q^{20} -24 q^{22} -12 \beta q^{23} + 7 q^{25} -8 q^{28} + 3 \beta q^{29} -44 q^{31} -4 \beta q^{32} -18 q^{34} + 12 \beta q^{35} + 34 q^{37} -16 \beta q^{38} -12 q^{40} -33 \beta q^{41} -40 q^{43} + 24 \beta q^{44} -24 q^{46} + 60 \beta q^{47} -33 q^{49} -7 \beta q^{50} + 27 \beta q^{53} + 72 q^{55} + 8 \beta q^{56} + 6 q^{58} -24 \beta q^{59} + 50 q^{61} + 44 \beta q^{62} -8 q^{64} -8 q^{67} + 18 \beta q^{68} + 24 q^{70} + 36 \beta q^{71} + 16 q^{73} -34 \beta q^{74} -32 q^{76} -48 \beta q^{77} -76 q^{79} + 12 \beta q^{80} -66 q^{82} -84 \beta q^{83} + 54 q^{85} + 40 \beta q^{86} + 48 q^{88} -9 \beta q^{89} + 24 \beta q^{92} + 120 q^{94} + 48 \beta q^{95} -176 q^{97} + 33 \beta q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{4} + 8q^{7} + O(q^{10})$$ $$2q - 4q^{4} + 8q^{7} + 12q^{10} + 8q^{16} + 32q^{19} - 48q^{22} + 14q^{25} - 16q^{28} - 88q^{31} - 36q^{34} + 68q^{37} - 24q^{40} - 80q^{43} - 48q^{46} - 66q^{49} + 144q^{55} + 12q^{58} + 100q^{61} - 16q^{64} - 16q^{67} + 48q^{70} + 32q^{73} - 64q^{76} - 152q^{79} - 132q^{82} + 108q^{85} + 96q^{88} + 240q^{94} - 352q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1691.1
 1.41421i − 1.41421i
1.41421i 0 −2.00000 4.24264i 0 4.00000 2.82843i 0 6.00000
1691.2 1.41421i 0 −2.00000 4.24264i 0 4.00000 2.82843i 0 6.00000
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3042.3.c.e 2
3.b odd 2 1 inner 3042.3.c.e 2
13.b even 2 1 18.3.b.a 2
13.d odd 4 2 3042.3.d.a 4
39.d odd 2 1 18.3.b.a 2
39.f even 4 2 3042.3.d.a 4
52.b odd 2 1 144.3.e.b 2
65.d even 2 1 450.3.d.f 2
65.h odd 4 2 450.3.b.b 4
91.b odd 2 1 882.3.b.a 2
91.r even 6 2 882.3.s.b 4
91.s odd 6 2 882.3.s.d 4
104.e even 2 1 576.3.e.c 2
104.h odd 2 1 576.3.e.f 2
117.n odd 6 2 162.3.d.b 4
117.t even 6 2 162.3.d.b 4
143.d odd 2 1 2178.3.c.d 2
156.h even 2 1 144.3.e.b 2
195.e odd 2 1 450.3.d.f 2
195.s even 4 2 450.3.b.b 4
208.o odd 4 2 2304.3.h.c 4
208.p even 4 2 2304.3.h.f 4
260.g odd 2 1 3600.3.l.d 2
260.p even 4 2 3600.3.c.b 4
273.g even 2 1 882.3.b.a 2
273.w odd 6 2 882.3.s.b 4
273.ba even 6 2 882.3.s.d 4
312.b odd 2 1 576.3.e.c 2
312.h even 2 1 576.3.e.f 2
429.e even 2 1 2178.3.c.d 2
468.x even 6 2 1296.3.q.f 4
468.bg odd 6 2 1296.3.q.f 4
624.v even 4 2 2304.3.h.c 4
624.bi odd 4 2 2304.3.h.f 4
780.d even 2 1 3600.3.l.d 2
780.w odd 4 2 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 13.b even 2 1
18.3.b.a 2 39.d odd 2 1
144.3.e.b 2 52.b odd 2 1
144.3.e.b 2 156.h even 2 1
162.3.d.b 4 117.n odd 6 2
162.3.d.b 4 117.t even 6 2
450.3.b.b 4 65.h odd 4 2
450.3.b.b 4 195.s even 4 2
450.3.d.f 2 65.d even 2 1
450.3.d.f 2 195.e odd 2 1
576.3.e.c 2 104.e even 2 1
576.3.e.c 2 312.b odd 2 1
576.3.e.f 2 104.h odd 2 1
576.3.e.f 2 312.h even 2 1
882.3.b.a 2 91.b odd 2 1
882.3.b.a 2 273.g even 2 1
882.3.s.b 4 91.r even 6 2
882.3.s.b 4 273.w odd 6 2
882.3.s.d 4 91.s odd 6 2
882.3.s.d 4 273.ba even 6 2
1296.3.q.f 4 468.x even 6 2
1296.3.q.f 4 468.bg odd 6 2
2178.3.c.d 2 143.d odd 2 1
2178.3.c.d 2 429.e even 2 1
2304.3.h.c 4 208.o odd 4 2
2304.3.h.c 4 624.v even 4 2
2304.3.h.f 4 208.p even 4 2
2304.3.h.f 4 624.bi odd 4 2
3042.3.c.e 2 1.a even 1 1 trivial
3042.3.c.e 2 3.b odd 2 1 inner
3042.3.d.a 4 13.d odd 4 2
3042.3.d.a 4 39.f even 4 2
3600.3.c.b 4 260.p even 4 2
3600.3.c.b 4 780.w odd 4 2
3600.3.l.d 2 260.g odd 2 1
3600.3.l.d 2 780.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(3042, [\chi])$$:

 $$T_{5}^{2} + 18$$ $$T_{7} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$2 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$18 + T^{2}$$
$7$ $$( -4 + T )^{2}$$
$11$ $$288 + T^{2}$$
$13$ $$T^{2}$$
$17$ $$162 + T^{2}$$
$19$ $$( -16 + T )^{2}$$
$23$ $$288 + T^{2}$$
$29$ $$18 + T^{2}$$
$31$ $$( 44 + T )^{2}$$
$37$ $$( -34 + T )^{2}$$
$41$ $$2178 + T^{2}$$
$43$ $$( 40 + T )^{2}$$
$47$ $$7200 + T^{2}$$
$53$ $$1458 + T^{2}$$
$59$ $$1152 + T^{2}$$
$61$ $$( -50 + T )^{2}$$
$67$ $$( 8 + T )^{2}$$
$71$ $$2592 + T^{2}$$
$73$ $$( -16 + T )^{2}$$
$79$ $$( 76 + T )^{2}$$
$83$ $$14112 + T^{2}$$
$89$ $$162 + T^{2}$$
$97$ $$( 176 + T )^{2}$$