# Properties

 Label 3042.3.d.a Level $3042$ Weight $3$ Character orbit 3042.d Analytic conductor $82.888$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$82.8884964184$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$2^{2}$$ 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 a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{5} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} -6 q^{10} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{11} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{17} -16 \zeta_{8}^{2} q^{19} + ( -6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{20} -24 q^{22} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{23} -7 q^{25} + 8 \zeta_{8}^{2} q^{28} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} + 44 \zeta_{8}^{2} q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} -18 \zeta_{8}^{2} q^{34} + ( -12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{35} + 34 \zeta_{8}^{2} q^{37} + ( -16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{38} -12 q^{40} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{41} + 40 q^{43} + ( -24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{44} -24 \zeta_{8}^{2} q^{46} + ( 60 \zeta_{8} - 60 \zeta_{8}^{3} ) q^{47} + 33 q^{49} + ( -7 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{50} + ( -27 \zeta_{8} - 27 \zeta_{8}^{3} ) q^{53} + 72 q^{55} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{56} -6 \zeta_{8}^{2} q^{58} + ( -24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{59} + 50 q^{61} + ( 44 \zeta_{8} + 44 \zeta_{8}^{3} ) q^{62} + 8 q^{64} + 8 \zeta_{8}^{2} q^{67} + ( -18 \zeta_{8} - 18 \zeta_{8}^{3} ) q^{68} -24 \zeta_{8}^{2} q^{70} + ( -36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{71} + 16 \zeta_{8}^{2} q^{73} + ( 34 \zeta_{8} + 34 \zeta_{8}^{3} ) q^{74} -32 \zeta_{8}^{2} q^{76} + ( -48 \zeta_{8} - 48 \zeta_{8}^{3} ) q^{77} -76 q^{79} + ( -12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{80} + 66 q^{82} + ( 84 \zeta_{8} - 84 \zeta_{8}^{3} ) q^{83} + 54 \zeta_{8}^{2} q^{85} + ( 40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{86} -48 q^{88} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{89} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{92} + 120 q^{94} + ( 48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{95} + 176 \zeta_{8}^{2} q^{97} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + O(q^{10})$$ $$4q + 8q^{4} - 24q^{10} + 16q^{16} - 96q^{22} - 28q^{25} - 48q^{40} + 160q^{43} + 132q^{49} + 288q^{55} + 200q^{61} + 32q^{64} - 304q^{79} + 264q^{82} - 192q^{88} + 480q^{94} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3041.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 0 2.00000 4.24264 0 4.00000i −2.82843 0 −6.00000
3041.2 −1.41421 0 2.00000 4.24264 0 4.00000i −2.82843 0 −6.00000
3041.3 1.41421 0 2.00000 −4.24264 0 4.00000i 2.82843 0 −6.00000
3041.4 1.41421 0 2.00000 −4.24264 0 4.00000i 2.82843 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
13.b even 2 1 inner
39.d 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.d.a 4
3.b odd 2 1 inner 3042.3.d.a 4
13.b even 2 1 inner 3042.3.d.a 4
13.d odd 4 1 18.3.b.a 2
13.d odd 4 1 3042.3.c.e 2
39.d odd 2 1 inner 3042.3.d.a 4
39.f even 4 1 18.3.b.a 2
39.f even 4 1 3042.3.c.e 2
52.f even 4 1 144.3.e.b 2
65.f even 4 1 450.3.b.b 4
65.g odd 4 1 450.3.d.f 2
65.k even 4 1 450.3.b.b 4
91.i even 4 1 882.3.b.a 2
91.z odd 12 2 882.3.s.b 4
91.bb even 12 2 882.3.s.d 4
104.j odd 4 1 576.3.e.c 2
104.m even 4 1 576.3.e.f 2
117.y odd 12 2 162.3.d.b 4
117.z even 12 2 162.3.d.b 4
143.g even 4 1 2178.3.c.d 2
156.l odd 4 1 144.3.e.b 2
195.j odd 4 1 450.3.b.b 4
195.n even 4 1 450.3.d.f 2
195.u odd 4 1 450.3.b.b 4
208.l even 4 1 2304.3.h.c 4
208.m odd 4 1 2304.3.h.f 4
208.r odd 4 1 2304.3.h.f 4
208.s even 4 1 2304.3.h.c 4
260.l odd 4 1 3600.3.c.b 4
260.s odd 4 1 3600.3.c.b 4
260.u even 4 1 3600.3.l.d 2
273.o odd 4 1 882.3.b.a 2
273.cb odd 12 2 882.3.s.d 4
273.cd even 12 2 882.3.s.b 4
312.w odd 4 1 576.3.e.f 2
312.y even 4 1 576.3.e.c 2
429.l odd 4 1 2178.3.c.d 2
468.bs even 12 2 1296.3.q.f 4
468.ch odd 12 2 1296.3.q.f 4
624.s odd 4 1 2304.3.h.c 4
624.u even 4 1 2304.3.h.f 4
624.bm even 4 1 2304.3.h.f 4
624.bo odd 4 1 2304.3.h.c 4
780.u even 4 1 3600.3.c.b 4
780.bb odd 4 1 3600.3.l.d 2
780.bn even 4 1 3600.3.c.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 13.d odd 4 1
18.3.b.a 2 39.f even 4 1
144.3.e.b 2 52.f even 4 1
144.3.e.b 2 156.l odd 4 1
162.3.d.b 4 117.y odd 12 2
162.3.d.b 4 117.z even 12 2
450.3.b.b 4 65.f even 4 1
450.3.b.b 4 65.k even 4 1
450.3.b.b 4 195.j odd 4 1
450.3.b.b 4 195.u odd 4 1
450.3.d.f 2 65.g odd 4 1
450.3.d.f 2 195.n even 4 1
576.3.e.c 2 104.j odd 4 1
576.3.e.c 2 312.y even 4 1
576.3.e.f 2 104.m even 4 1
576.3.e.f 2 312.w odd 4 1
882.3.b.a 2 91.i even 4 1
882.3.b.a 2 273.o odd 4 1
882.3.s.b 4 91.z odd 12 2
882.3.s.b 4 273.cd even 12 2
882.3.s.d 4 91.bb even 12 2
882.3.s.d 4 273.cb odd 12 2
1296.3.q.f 4 468.bs even 12 2
1296.3.q.f 4 468.ch odd 12 2
2178.3.c.d 2 143.g even 4 1
2178.3.c.d 2 429.l odd 4 1
2304.3.h.c 4 208.l even 4 1
2304.3.h.c 4 208.s even 4 1
2304.3.h.c 4 624.s odd 4 1
2304.3.h.c 4 624.bo odd 4 1
2304.3.h.f 4 208.m odd 4 1
2304.3.h.f 4 208.r odd 4 1
2304.3.h.f 4 624.u even 4 1
2304.3.h.f 4 624.bm even 4 1
3042.3.c.e 2 13.d odd 4 1
3042.3.c.e 2 39.f even 4 1
3042.3.d.a 4 1.a even 1 1 trivial
3042.3.d.a 4 3.b odd 2 1 inner
3042.3.d.a 4 13.b even 2 1 inner
3042.3.d.a 4 39.d odd 2 1 inner
3600.3.c.b 4 260.l odd 4 1
3600.3.c.b 4 260.s odd 4 1
3600.3.c.b 4 780.u even 4 1
3600.3.c.b 4 780.bn even 4 1
3600.3.l.d 2 260.u even 4 1
3600.3.l.d 2 780.bb odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 18$$ acting on $$S_{3}^{\mathrm{new}}(3042, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$( -18 + T^{2} )^{2}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$( -288 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( 162 + T^{2} )^{2}$$
$19$ $$( 256 + T^{2} )^{2}$$
$23$ $$( 288 + T^{2} )^{2}$$
$29$ $$( 18 + T^{2} )^{2}$$
$31$ $$( 1936 + T^{2} )^{2}$$
$37$ $$( 1156 + T^{2} )^{2}$$
$41$ $$( -2178 + T^{2} )^{2}$$
$43$ $$( -40 + T )^{4}$$
$47$ $$( -7200 + T^{2} )^{2}$$
$53$ $$( 1458 + T^{2} )^{2}$$
$59$ $$( -1152 + T^{2} )^{2}$$
$61$ $$( -50 + T )^{4}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( -2592 + T^{2} )^{2}$$
$73$ $$( 256 + T^{2} )^{2}$$
$79$ $$( 76 + T )^{4}$$
$83$ $$( -14112 + T^{2} )^{2}$$
$89$ $$( -162 + T^{2} )^{2}$$
$97$ $$( 30976 + T^{2} )^{2}$$