# Properties

 Label 3042.2.b.o Level $3042$ Weight $2$ Character orbit 3042.b Analytic conductor $24.290$ Analytic rank $0$ Dimension $6$ 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: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5x^{4} + 6x^{2} + 1$$ x^6 + 5*x^4 + 6*x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1014) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{5} q^{2} - q^{4} + ( - \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{7} + \beta_{5} q^{8}+O(q^{10})$$ q - b5 * q^2 - q^4 + (-b3 - 2*b1) * q^5 + (-2*b5 + 2*b3 - b1) * q^7 + b5 * q^8 $$q - \beta_{5} q^{2} - q^{4} + ( - \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} + 2 \beta_{3} - \beta_1) q^{7} + \beta_{5} q^{8} + ( - 3 \beta_{4} - \beta_{2} + 1) q^{10} + (3 \beta_{5} + 3 \beta_{3} - \beta_1) q^{11} + (\beta_{4} + 2 \beta_{2} - 4) q^{14} + q^{16} + ( - 2 \beta_{4} - 2) q^{17} + ( - 4 \beta_{5} - 4 \beta_{3} + 4 \beta_1) q^{19} + (\beta_{3} + 2 \beta_1) q^{20} + (2 \beta_{4} + 3 \beta_{2}) q^{22} + ( - 2 \beta_{4} + 2 \beta_{2}) q^{23} + (5 \beta_{4} + 4 \beta_{2} - 9) q^{25} + (2 \beta_{5} - 2 \beta_{3} + \beta_1) q^{28} + (2 \beta_{4} - \beta_{2} - 4) q^{29} + (5 \beta_{5} + 5 \beta_{3} - 5 \beta_1) q^{31} - \beta_{5} q^{32} + (2 \beta_{5} + 2 \beta_1) q^{34} + ( - 11 \beta_{4} + 5) q^{35} + ( - 2 \beta_{3} + 6 \beta_1) q^{37} - 4 \beta_{2} q^{38} + (3 \beta_{4} + \beta_{2} - 1) q^{40} + (2 \beta_{5} + 6 \beta_{3} - 2 \beta_1) q^{41} + (2 \beta_{4} - 2 \beta_{2} - 4) q^{43} + ( - 3 \beta_{5} - 3 \beta_{3} + \beta_1) q^{44} + ( - 2 \beta_{5} - 2 \beta_{3} + 4 \beta_1) q^{46} + (4 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{47} + (4 \beta_{4} + 9 \beta_{2} - 11) q^{49} + (5 \beta_{5} - 4 \beta_{3} - \beta_1) q^{50} + ( - 3 \beta_{4} - 5 \beta_{2} + 1) q^{53} + (\beta_{4} + 5 \beta_{2} + 4) q^{55} + ( - \beta_{4} - 2 \beta_{2} + 4) q^{56} + (5 \beta_{5} + \beta_{3} - 3 \beta_1) q^{58} + ( - 4 \beta_{5} - 2 \beta_{3} + 5 \beta_1) q^{59} + ( - 2 \beta_{4} + 8) q^{61} + 5 \beta_{2} q^{62} - q^{64} + (2 \beta_{5} + 2 \beta_{3} + 2 \beta_1) q^{67} + (2 \beta_{4} + 2) q^{68} + ( - 5 \beta_{5} + 11 \beta_1) q^{70} + (8 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{71} + (8 \beta_{5} + 7 \beta_{3} - 4 \beta_1) q^{73} + (4 \beta_{4} - 2 \beta_{2} + 2) q^{74} + (4 \beta_{5} + 4 \beta_{3} - 4 \beta_1) q^{76} + (2 \beta_{4} + \beta_{2} - 3) q^{77} + (5 \beta_{4} + 2 \beta_{2} + 8) q^{79} + ( - \beta_{3} - 2 \beta_1) q^{80} + (4 \beta_{4} + 6 \beta_{2} - 4) q^{82} + (10 \beta_{3} - 3 \beta_1) q^{83} + (6 \beta_{5} - 2 \beta_{3} + 6 \beta_1) q^{85} + (6 \beta_{5} + 2 \beta_{3} - 4 \beta_1) q^{86} + ( - 2 \beta_{4} - 3 \beta_{2}) q^{88} + (6 \beta_{3} - 8 \beta_1) q^{89} + (2 \beta_{4} - 2 \beta_{2}) q^{92} + 4 \beta_{2} q^{94} + ( - 4 \beta_{4} - 12 \beta_{2} + 8) q^{95} + ( - 8 \beta_{5} + 3 \beta_{3} + 4 \beta_1) q^{97} + (2 \beta_{5} - 9 \beta_{3} + 5 \beta_1) q^{98}+O(q^{100})$$ q - b5 * q^2 - q^4 + (-b3 - 2*b1) * q^5 + (-2*b5 + 2*b3 - b1) * q^7 + b5 * q^8 + (-3*b4 - b2 + 1) * q^10 + (3*b5 + 3*b3 - b1) * q^11 + (b4 + 2*b2 - 4) * q^14 + q^16 + (-2*b4 - 2) * q^17 + (-4*b5 - 4*b3 + 4*b1) * q^19 + (b3 + 2*b1) * q^20 + (2*b4 + 3*b2) * q^22 + (-2*b4 + 2*b2) * q^23 + (5*b4 + 4*b2 - 9) * q^25 + (2*b5 - 2*b3 + b1) * q^28 + (2*b4 - b2 - 4) * q^29 + (5*b5 + 5*b3 - 5*b1) * q^31 - b5 * q^32 + (2*b5 + 2*b1) * q^34 + (-11*b4 + 5) * q^35 + (-2*b3 + 6*b1) * q^37 - 4*b2 * q^38 + (3*b4 + b2 - 1) * q^40 + (2*b5 + 6*b3 - 2*b1) * q^41 + (2*b4 - 2*b2 - 4) * q^43 + (-3*b5 - 3*b3 + b1) * q^44 + (-2*b5 - 2*b3 + 4*b1) * q^46 + (4*b5 + 4*b3 - 4*b1) * q^47 + (4*b4 + 9*b2 - 11) * q^49 + (5*b5 - 4*b3 - b1) * q^50 + (-3*b4 - 5*b2 + 1) * q^53 + (b4 + 5*b2 + 4) * q^55 + (-b4 - 2*b2 + 4) * q^56 + (5*b5 + b3 - 3*b1) * q^58 + (-4*b5 - 2*b3 + 5*b1) * q^59 + (-2*b4 + 8) * q^61 + 5*b2 * q^62 - q^64 + (2*b5 + 2*b3 + 2*b1) * q^67 + (2*b4 + 2) * q^68 + (-5*b5 + 11*b1) * q^70 + (8*b5 + 2*b3 - 4*b1) * q^71 + (8*b5 + 7*b3 - 4*b1) * q^73 + (4*b4 - 2*b2 + 2) * q^74 + (4*b5 + 4*b3 - 4*b1) * q^76 + (2*b4 + b2 - 3) * q^77 + (5*b4 + 2*b2 + 8) * q^79 + (-b3 - 2*b1) * q^80 + (4*b4 + 6*b2 - 4) * q^82 + (10*b3 - 3*b1) * q^83 + (6*b5 - 2*b3 + 6*b1) * q^85 + (6*b5 + 2*b3 - 4*b1) * q^86 + (-2*b4 - 3*b2) * q^88 + (6*b3 - 8*b1) * q^89 + (2*b4 - 2*b2) * q^92 + 4*b2 * q^94 + (-4*b4 - 12*b2 + 8) * q^95 + (-8*b5 + 3*b3 + 4*b1) * q^97 + (2*b5 - 9*b3 + 5*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{4}+O(q^{10})$$ 6 * q - 6 * q^4 $$6 q - 6 q^{4} - 2 q^{10} - 18 q^{14} + 6 q^{16} - 16 q^{17} + 10 q^{22} - 36 q^{25} - 22 q^{29} + 8 q^{35} - 8 q^{38} + 2 q^{40} - 24 q^{43} - 40 q^{49} - 10 q^{53} + 36 q^{55} + 18 q^{56} + 44 q^{61} + 10 q^{62} - 6 q^{64} + 16 q^{68} + 16 q^{74} - 12 q^{77} + 62 q^{79} - 4 q^{82} - 10 q^{88} + 8 q^{94} + 16 q^{95}+O(q^{100})$$ 6 * q - 6 * q^4 - 2 * q^10 - 18 * q^14 + 6 * q^16 - 16 * q^17 + 10 * q^22 - 36 * q^25 - 22 * q^29 + 8 * q^35 - 8 * q^38 + 2 * q^40 - 24 * q^43 - 40 * q^49 - 10 * q^53 + 36 * q^55 + 18 * q^56 + 44 * q^61 + 10 * q^62 - 6 * q^64 + 16 * q^68 + 16 * q^74 - 12 * q^77 + 62 * q^79 - 4 * q^82 - 10 * q^88 + 8 * q^94 + 16 * q^95

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5x^{4} + 6x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ v^2 + 2 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3\nu$$ v^3 + 3*v $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3\nu^{2} + 1$$ v^4 + 3*v^2 + 1 $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4\nu^{3} + 3\nu$$ v^5 + 4*v^3 + 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ b2 - 2 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3\beta_1$$ b3 - 3*b1 $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3\beta_{2} + 5$$ b4 - 3*b2 + 5 $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4\beta_{3} + 9\beta_1$$ b5 - 4*b3 + 9*b1

## 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
 1.80194i 0.445042i − 1.24698i 1.24698i − 0.445042i − 1.80194i
1.00000i 0 −1.00000 3.15883i 0 4.69202i 1.00000i 0 −3.15883
1351.2 1.00000i 0 −1.00000 2.13706i 0 0.0489173i 1.00000i 0 −2.13706
1351.3 1.00000i 0 −1.00000 4.29590i 0 4.35690i 1.00000i 0 4.29590
1351.4 1.00000i 0 −1.00000 4.29590i 0 4.35690i 1.00000i 0 4.29590
1351.5 1.00000i 0 −1.00000 2.13706i 0 0.0489173i 1.00000i 0 −2.13706
1351.6 1.00000i 0 −1.00000 3.15883i 0 4.69202i 1.00000i 0 −3.15883
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1351.6 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.o 6
3.b odd 2 1 1014.2.b.f 6
13.b even 2 1 inner 3042.2.b.o 6
13.d odd 4 1 3042.2.a.ba 3
13.d odd 4 1 3042.2.a.bh 3
39.d odd 2 1 1014.2.b.f 6
39.f even 4 1 1014.2.a.l 3
39.f even 4 1 1014.2.a.n yes 3
39.h odd 6 2 1014.2.i.h 12
39.i odd 6 2 1014.2.i.h 12
39.k even 12 2 1014.2.e.l 6
39.k even 12 2 1014.2.e.n 6
156.l odd 4 1 8112.2.a.cj 3
156.l odd 4 1 8112.2.a.cm 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1014.2.a.l 3 39.f even 4 1
1014.2.a.n yes 3 39.f even 4 1
1014.2.b.f 6 3.b odd 2 1
1014.2.b.f 6 39.d odd 2 1
1014.2.e.l 6 39.k even 12 2
1014.2.e.n 6 39.k even 12 2
1014.2.i.h 12 39.h odd 6 2
1014.2.i.h 12 39.i odd 6 2
3042.2.a.ba 3 13.d odd 4 1
3042.2.a.bh 3 13.d odd 4 1
3042.2.b.o 6 1.a even 1 1 trivial
3042.2.b.o 6 13.b even 2 1 inner
8112.2.a.cj 3 156.l odd 4 1
8112.2.a.cm 3 156.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 $$S_{2}^{\mathrm{new}}(3042, [\chi])$$:

 $$T_{5}^{6} + 33T_{5}^{4} + 314T_{5}^{2} + 841$$ T5^6 + 33*T5^4 + 314*T5^2 + 841 $$T_{7}^{6} + 41T_{7}^{4} + 418T_{7}^{2} + 1$$ T7^6 + 41*T7^4 + 418*T7^2 + 1 $$T_{17}^{3} + 8T_{17}^{2} + 12T_{17} - 8$$ T17^3 + 8*T17^2 + 12*T17 - 8 $$T_{23}^{3} - 28T_{23} + 56$$ T23^3 - 28*T23 + 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{3}$$
$3$ $$T^{6}$$
$5$ $$T^{6} + 33 T^{4} + 314 T^{2} + \cdots + 841$$
$7$ $$T^{6} + 41 T^{4} + 418 T^{2} + 1$$
$11$ $$T^{6} + 41 T^{4} + 54 T^{2} + 1$$
$13$ $$T^{6}$$
$17$ $$(T^{3} + 8 T^{2} + 12 T - 8)^{2}$$
$19$ $$T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096$$
$23$ $$(T^{3} - 28 T + 56)^{2}$$
$29$ $$(T^{3} + 11 T^{2} + 24 T - 29)^{2}$$
$31$ $$T^{6} + 125 T^{4} + 3750 T^{2} + \cdots + 15625$$
$37$ $$T^{6} + 152 T^{4} + 2064 T^{2} + \cdots + 64$$
$41$ $$T^{6} + 132 T^{4} + 5024 T^{2} + \cdots + 53824$$
$43$ $$(T^{3} + 12 T^{2} + 20 T - 104)^{2}$$
$47$ $$T^{6} + 80 T^{4} + 1536 T^{2} + \cdots + 4096$$
$53$ $$(T^{3} + 5 T^{2} - 36 T + 43)^{2}$$
$59$ $$T^{6} + 97 T^{4} + 2966 T^{2} + \cdots + 27889$$
$61$ $$(T^{3} - 22 T^{2} + 152 T - 328)^{2}$$
$67$ $$T^{6} + 68 T^{4} + 1504 T^{2} + \cdots + 10816$$
$71$ $$T^{6} + 164 T^{4} + 6688 T^{2} + \cdots + 64$$
$73$ $$T^{6} + 229 T^{4} + 1238 T^{2} + \cdots + 169$$
$79$ $$(T^{3} - 31 T^{2} + 276 T - 533)^{2}$$
$83$ $$T^{6} + 425 T^{4} + 57126 T^{2} + \cdots + 2455489$$
$89$ $$T^{6} + 308 T^{4} + 4704 T^{2} + \cdots + 3136$$
$97$ $$T^{6} + 349 T^{4} + 3638 T^{2} + \cdots + 9409$$