# Properties

 Label 42.3.b.a Level $42$ Weight $3$ Character orbit 42.b Analytic conductor $1.144$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 42.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14441711031$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{7})$$ Defining polynomial: $$x^{4} + 8 x^{2} + 9$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} -2 q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{5} + ( -2 - \beta_{2} ) q^{6} -\beta_{3} q^{7} + 2 \beta_{1} q^{8} + ( 5 - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} -2 q^{4} + ( -2 \beta_{1} + \beta_{2} ) q^{5} + ( -2 - \beta_{2} ) q^{6} -\beta_{3} q^{7} + 2 \beta_{1} q^{8} + ( 5 - 2 \beta_{2} ) q^{9} + ( -4 + 2 \beta_{3} ) q^{10} + ( 5 \beta_{1} + 2 \beta_{2} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{12} + ( 10 - 4 \beta_{3} ) q^{13} + \beta_{2} q^{14} + ( -4 + 7 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{15} + 4 q^{16} + ( 2 \beta_{1} + 5 \beta_{2} ) q^{17} + ( -5 \beta_{1} - 4 \beta_{3} ) q^{18} -16 q^{19} + ( 4 \beta_{1} - 2 \beta_{2} ) q^{20} + ( -7 + \beta_{2} ) q^{21} + ( 10 + 4 \beta_{3} ) q^{22} + ( -\beta_{1} - 10 \beta_{2} ) q^{23} + ( 4 + 2 \beta_{2} ) q^{24} + ( 3 + 8 \beta_{3} ) q^{25} + ( -10 \beta_{1} + 4 \beta_{2} ) q^{26} + ( -19 \beta_{1} + \beta_{3} ) q^{27} + 2 \beta_{3} q^{28} + ( -20 \beta_{1} - 2 \beta_{2} ) q^{29} + ( 14 + 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{30} + ( -32 - 10 \beta_{3} ) q^{31} -4 \beta_{1} q^{32} + ( 10 + 14 \beta_{1} + 5 \beta_{2} + 4 \beta_{3} ) q^{33} + ( 4 + 10 \beta_{3} ) q^{34} + ( -7 \beta_{1} + 2 \beta_{2} ) q^{35} + ( -10 + 4 \beta_{2} ) q^{36} + 20 q^{37} + 16 \beta_{1} q^{38} + ( -28 - 10 \beta_{1} + 4 \beta_{2} + 10 \beta_{3} ) q^{39} + ( 8 - 4 \beta_{3} ) q^{40} + ( 30 \beta_{1} - 9 \beta_{2} ) q^{41} + ( 7 \beta_{1} + 2 \beta_{3} ) q^{42} + ( 20 - 12 \beta_{3} ) q^{43} + ( -10 \beta_{1} - 4 \beta_{2} ) q^{44} + ( 28 - 10 \beta_{1} + 5 \beta_{2} - 8 \beta_{3} ) q^{45} + ( -2 - 20 \beta_{3} ) q^{46} + 6 \beta_{1} q^{47} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{48} + 7 q^{49} + ( -3 \beta_{1} - 8 \beta_{2} ) q^{50} + ( 4 + 35 \beta_{1} + 2 \beta_{2} + 10 \beta_{3} ) q^{51} + ( -20 + 8 \beta_{3} ) q^{52} + 36 \beta_{1} q^{53} + ( -38 - \beta_{2} ) q^{54} + ( -8 - 2 \beta_{3} ) q^{55} -2 \beta_{2} q^{56} + ( 16 \beta_{1} - 16 \beta_{3} ) q^{57} + ( -40 - 4 \beta_{3} ) q^{58} + ( -20 \beta_{1} - 8 \beta_{2} ) q^{59} + ( 8 - 14 \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{60} + ( -14 + 20 \beta_{3} ) q^{61} + ( 32 \beta_{1} + 10 \beta_{2} ) q^{62} + ( 14 \beta_{1} - 5 \beta_{3} ) q^{63} -8 q^{64} + ( -48 \beta_{1} + 18 \beta_{2} ) q^{65} + ( 28 - 10 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{66} + ( 60 + 4 \beta_{3} ) q^{67} + ( -4 \beta_{1} - 10 \beta_{2} ) q^{68} + ( -2 - 70 \beta_{1} - \beta_{2} - 20 \beta_{3} ) q^{69} + ( -14 + 4 \beta_{3} ) q^{70} + ( -25 \beta_{1} + 14 \beta_{2} ) q^{71} + ( 10 \beta_{1} + 8 \beta_{3} ) q^{72} + ( 30 + 16 \beta_{3} ) q^{73} -20 \beta_{1} q^{74} + ( 56 - 3 \beta_{1} - 8 \beta_{2} + 3 \beta_{3} ) q^{75} + 32 q^{76} + ( -14 \beta_{1} - 5 \beta_{2} ) q^{77} + ( -20 + 28 \beta_{1} - 10 \beta_{2} + 8 \beta_{3} ) q^{78} + ( -32 + 20 \beta_{3} ) q^{79} + ( -8 \beta_{1} + 4 \beta_{2} ) q^{80} + ( -31 - 20 \beta_{2} ) q^{81} + ( 60 - 18 \beta_{3} ) q^{82} + ( 50 \beta_{1} + 20 \beta_{2} ) q^{83} + ( 14 - 2 \beta_{2} ) q^{84} + ( -62 + 16 \beta_{3} ) q^{85} + ( -20 \beta_{1} + 12 \beta_{2} ) q^{86} + ( -40 - 14 \beta_{1} - 20 \beta_{2} - 4 \beta_{3} ) q^{87} + ( -20 - 8 \beta_{3} ) q^{88} + ( -30 \beta_{1} - 3 \beta_{2} ) q^{89} + ( -20 - 28 \beta_{1} + 8 \beta_{2} + 10 \beta_{3} ) q^{90} + ( 28 - 10 \beta_{3} ) q^{91} + ( 2 \beta_{1} + 20 \beta_{2} ) q^{92} + ( -70 + 32 \beta_{1} + 10 \beta_{2} - 32 \beta_{3} ) q^{93} + 12 q^{94} + ( 32 \beta_{1} - 16 \beta_{2} ) q^{95} + ( -8 - 4 \beta_{2} ) q^{96} + ( -90 - 8 \beta_{3} ) q^{97} -7 \beta_{1} q^{98} + ( 56 + 25 \beta_{1} + 10 \beta_{2} + 20 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{4} - 8q^{6} + 20q^{9} + O(q^{10})$$ $$4q - 8q^{4} - 8q^{6} + 20q^{9} - 16q^{10} + 40q^{13} - 16q^{15} + 16q^{16} - 64q^{19} - 28q^{21} + 40q^{22} + 16q^{24} + 12q^{25} + 56q^{30} - 128q^{31} + 40q^{33} + 16q^{34} - 40q^{36} + 80q^{37} - 112q^{39} + 32q^{40} + 80q^{43} + 112q^{45} - 8q^{46} + 28q^{49} + 16q^{51} - 80q^{52} - 152q^{54} - 32q^{55} - 160q^{58} + 32q^{60} - 56q^{61} - 32q^{64} + 112q^{66} + 240q^{67} - 8q^{69} - 56q^{70} + 120q^{73} + 224q^{75} + 128q^{76} - 80q^{78} - 128q^{79} - 124q^{81} + 240q^{82} + 56q^{84} - 248q^{85} - 160q^{87} - 80q^{88} - 80q^{90} + 112q^{91} - 280q^{93} + 48q^{94} - 32q^{96} - 360q^{97} + 224q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 8 x^{2} + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 5 \nu$$$$)/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 11 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ $$\nu^{3}$$ $$=$$ $$($$$$-5 \beta_{2} + 11 \beta_{1}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/42\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1
 − 2.57794i 1.16372i 2.57794i − 1.16372i
1.41421i −2.64575 1.41421i −2.00000 6.57008i −2.00000 + 3.74166i 2.64575 2.82843i 5.00000 + 7.48331i −9.29150
29.2 1.41421i 2.64575 1.41421i −2.00000 0.913230i −2.00000 3.74166i −2.64575 2.82843i 5.00000 7.48331i 1.29150
29.3 1.41421i −2.64575 + 1.41421i −2.00000 6.57008i −2.00000 3.74166i 2.64575 2.82843i 5.00000 7.48331i −9.29150
29.4 1.41421i 2.64575 + 1.41421i −2.00000 0.913230i −2.00000 + 3.74166i −2.64575 2.82843i 5.00000 + 7.48331i 1.29150
 $$n$$: 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 42.3.b.a 4
3.b odd 2 1 inner 42.3.b.a 4
4.b odd 2 1 336.3.d.b 4
5.b even 2 1 1050.3.e.a 4
5.c odd 4 2 1050.3.c.a 8
7.b odd 2 1 294.3.b.h 4
7.c even 3 2 294.3.h.d 8
7.d odd 6 2 294.3.h.g 8
8.b even 2 1 1344.3.d.c 4
8.d odd 2 1 1344.3.d.e 4
9.c even 3 2 1134.3.q.a 8
9.d odd 6 2 1134.3.q.a 8
12.b even 2 1 336.3.d.b 4
15.d odd 2 1 1050.3.e.a 4
15.e even 4 2 1050.3.c.a 8
21.c even 2 1 294.3.b.h 4
21.g even 6 2 294.3.h.g 8
21.h odd 6 2 294.3.h.d 8
24.f even 2 1 1344.3.d.e 4
24.h odd 2 1 1344.3.d.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.b.a 4 1.a even 1 1 trivial
42.3.b.a 4 3.b odd 2 1 inner
294.3.b.h 4 7.b odd 2 1
294.3.b.h 4 21.c even 2 1
294.3.h.d 8 7.c even 3 2
294.3.h.d 8 21.h odd 6 2
294.3.h.g 8 7.d odd 6 2
294.3.h.g 8 21.g even 6 2
336.3.d.b 4 4.b odd 2 1
336.3.d.b 4 12.b even 2 1
1050.3.c.a 8 5.c odd 4 2
1050.3.c.a 8 15.e even 4 2
1050.3.e.a 4 5.b even 2 1
1050.3.e.a 4 15.d odd 2 1
1134.3.q.a 8 9.c even 3 2
1134.3.q.a 8 9.d odd 6 2
1344.3.d.c 4 8.b even 2 1
1344.3.d.c 4 24.h odd 2 1
1344.3.d.e 4 8.d odd 2 1
1344.3.d.e 4 24.f even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ $$1 - 10 T^{2} + 81 T^{4}$$
$5$ $$1 - 56 T^{2} + 1586 T^{4} - 35000 T^{6} + 390625 T^{8}$$
$7$ $$( 1 - 7 T^{2} )^{2}$$
$11$ $$1 - 272 T^{2} + 36578 T^{4} - 3982352 T^{6} + 214358881 T^{8}$$
$13$ $$( 1 - 20 T + 326 T^{2} - 3380 T^{3} + 28561 T^{4} )^{2}$$
$17$ $$1 - 440 T^{2} + 204242 T^{4} - 36749240 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 + 16 T + 361 T^{2} )^{4}$$
$23$ $$1 + 688 T^{2} + 666818 T^{4} + 192530608 T^{6} + 78310985281 T^{8}$$
$29$ $$1 - 1652 T^{2} + 1917638 T^{4} - 1168428212 T^{6} + 500246412961 T^{8}$$
$31$ $$( 1 + 64 T + 2246 T^{2} + 61504 T^{3} + 923521 T^{4} )^{2}$$
$37$ $$( 1 - 20 T + 1369 T^{2} )^{4}$$
$41$ $$1 - 856 T^{2} - 2330094 T^{4} - 2418851416 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 - 40 T + 3090 T^{2} - 73960 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$( 1 - 4346 T^{2} + 4879681 T^{4} )^{2}$$
$53$ $$( 1 - 3026 T^{2} + 7890481 T^{4} )^{2}$$
$59$ $$1 - 10532 T^{2} + 49098278 T^{4} - 127620046052 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 + 28 T + 4838 T^{2} + 104188 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 - 120 T + 12466 T^{2} - 538680 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$1 - 12176 T^{2} + 74167106 T^{4} - 309412627856 T^{6} + 645753531245761 T^{8}$$
$73$ $$( 1 - 60 T + 9766 T^{2} - 319740 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 + 64 T + 10706 T^{2} + 399424 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 6356 T^{2} - 6983674 T^{4} - 301645088276 T^{6} + 2252292232139041 T^{8}$$
$89$ $$1 - 27832 T^{2} + 318232338 T^{4} - 1746242051512 T^{6} + 3936588805702081 T^{8}$$
$97$ $$( 1 + 180 T + 26470 T^{2} + 1693620 T^{3} + 88529281 T^{4} )^{2}$$