# Properties

 Label 42.3.g.a Level $42$ Weight $3$ Character orbit 42.g 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.g (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14441711031$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( -5 - 2 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 2 + \beta_{2} ) q^{3} + 2 \beta_{2} q^{4} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{5} + ( 2 \beta_{1} + \beta_{3} ) q^{6} + ( -5 - 2 \beta_{1} - 5 \beta_{2} - 4 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + ( 3 + 3 \beta_{2} ) q^{9} + ( -8 + 2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{10} + 6 \beta_{2} q^{11} + ( -2 + 2 \beta_{2} ) q^{12} + ( -1 - 16 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} ) q^{13} + ( 8 - 5 \beta_{1} + 4 \beta_{2} - 5 \beta_{3} ) q^{14} + ( 6 + 6 \beta_{3} ) q^{15} + ( -4 - 4 \beta_{2} ) q^{16} + ( -16 + 2 \beta_{1} - 8 \beta_{2} - 2 \beta_{3} ) q^{17} + ( 3 \beta_{1} + 3 \beta_{3} ) q^{18} + ( -7 - 2 \beta_{1} + 7 \beta_{2} - 4 \beta_{3} ) q^{19} + ( 4 - 8 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{20} + ( -5 - 10 \beta_{2} - 6 \beta_{3} ) q^{21} + 6 \beta_{3} q^{22} + ( 12 + 18 \beta_{1} + 12 \beta_{2} ) q^{23} + ( -2 \beta_{1} + 2 \beta_{3} ) q^{24} + ( 24 \beta_{1} - 11 \beta_{2} + 24 \beta_{3} ) q^{25} + ( 16 - \beta_{1} - 16 \beta_{2} - 2 \beta_{3} ) q^{26} + ( 3 + 6 \beta_{2} ) q^{27} + ( 10 + 8 \beta_{1} + 4 \beta_{3} ) q^{28} + 24 \beta_{3} q^{29} + ( -12 + 6 \beta_{1} - 12 \beta_{2} ) q^{30} + ( 34 + 6 \beta_{1} + 17 \beta_{2} - 6 \beta_{3} ) q^{31} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -6 + 6 \beta_{2} ) q^{33} + ( 4 - 16 \beta_{1} + 8 \beta_{2} - 8 \beta_{3} ) q^{34} + ( -20 - 2 \beta_{1} + 14 \beta_{2} - 22 \beta_{3} ) q^{35} -6 q^{36} + ( 11 + 12 \beta_{1} + 11 \beta_{2} ) q^{37} + ( 8 - 7 \beta_{1} + 4 \beta_{2} + 7 \beta_{3} ) q^{38} + ( -24 \beta_{1} - 3 \beta_{2} - 24 \beta_{3} ) q^{39} + ( 8 + 4 \beta_{1} - 8 \beta_{2} + 8 \beta_{3} ) q^{40} + ( -26 - 8 \beta_{1} - 52 \beta_{2} - 4 \beta_{3} ) q^{41} + ( 12 - 5 \beta_{1} + 12 \beta_{2} - 10 \beta_{3} ) q^{42} + ( 7 + 6 \beta_{3} ) q^{43} + ( -12 - 12 \beta_{2} ) q^{44} + ( 12 - 6 \beta_{1} + 6 \beta_{2} + 6 \beta_{3} ) q^{45} + ( 12 \beta_{1} + 36 \beta_{2} + 12 \beta_{3} ) q^{46} + ( -22 + 2 \beta_{1} + 22 \beta_{2} + 4 \beta_{3} ) q^{47} + ( -4 - 8 \beta_{2} ) q^{48} + ( -20 \beta_{1} + \beta_{2} + 20 \beta_{3} ) q^{49} + ( -48 - 11 \beta_{3} ) q^{50} + ( -24 + 6 \beta_{1} - 24 \beta_{2} ) q^{51} + ( 4 + 16 \beta_{1} + 2 \beta_{2} - 16 \beta_{3} ) q^{52} + ( -18 \beta_{1} - 60 \beta_{2} - 18 \beta_{3} ) q^{53} + ( 3 \beta_{1} + 6 \beta_{3} ) q^{54} + ( 12 - 24 \beta_{1} + 24 \beta_{2} - 12 \beta_{3} ) q^{55} + ( -8 + 10 \beta_{1} + 8 \beta_{2} ) q^{56} + ( -21 - 6 \beta_{3} ) q^{57} + ( -48 - 48 \beta_{2} ) q^{58} + ( -8 - 14 \beta_{1} - 4 \beta_{2} + 14 \beta_{3} ) q^{59} + ( -12 \beta_{1} + 12 \beta_{2} - 12 \beta_{3} ) q^{60} + ( -12 - 8 \beta_{1} + 12 \beta_{2} - 16 \beta_{3} ) q^{61} + ( 12 + 34 \beta_{1} + 24 \beta_{2} + 17 \beta_{3} ) q^{62} + ( 6 \beta_{1} - 15 \beta_{2} - 6 \beta_{3} ) q^{63} + 8 q^{64} + ( 90 - 42 \beta_{1} + 90 \beta_{2} ) q^{65} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{66} + ( 42 \beta_{1} - 55 \beta_{2} + 42 \beta_{3} ) q^{67} + ( 16 + 4 \beta_{1} - 16 \beta_{2} + 8 \beta_{3} ) q^{68} + ( 12 + 36 \beta_{1} + 24 \beta_{2} + 18 \beta_{3} ) q^{69} + ( 44 - 20 \beta_{1} + 40 \beta_{2} + 14 \beta_{3} ) q^{70} + ( 78 - 42 \beta_{3} ) q^{71} -6 \beta_{1} q^{72} + ( -22 + 40 \beta_{1} - 11 \beta_{2} - 40 \beta_{3} ) q^{73} + ( 11 \beta_{1} + 24 \beta_{2} + 11 \beta_{3} ) q^{74} + ( 11 + 24 \beta_{1} - 11 \beta_{2} + 48 \beta_{3} ) q^{75} + ( -14 + 8 \beta_{1} - 28 \beta_{2} + 4 \beta_{3} ) q^{76} + ( 30 + 24 \beta_{1} + 12 \beta_{3} ) q^{77} + ( 48 - 3 \beta_{3} ) q^{78} + ( -5 - 66 \beta_{1} - 5 \beta_{2} ) q^{79} + ( -16 + 8 \beta_{1} - 8 \beta_{2} - 8 \beta_{3} ) q^{80} + 9 \beta_{2} q^{81} + ( 8 - 26 \beta_{1} - 8 \beta_{2} - 52 \beta_{3} ) q^{82} + ( 10 + 76 \beta_{1} + 20 \beta_{2} + 38 \beta_{3} ) q^{83} + ( 20 + 12 \beta_{1} + 10 \beta_{2} + 12 \beta_{3} ) q^{84} + ( -72 - 60 \beta_{3} ) q^{85} + ( -12 + 7 \beta_{1} - 12 \beta_{2} ) q^{86} + ( -24 \beta_{1} + 24 \beta_{3} ) q^{87} + ( -12 \beta_{1} - 12 \beta_{3} ) q^{88} + ( -12 + 12 \beta_{2} ) q^{89} + ( -12 + 12 \beta_{1} - 24 \beta_{2} + 6 \beta_{3} ) q^{90} + ( -101 + 34 \beta_{1} - 91 \beta_{2} + 80 \beta_{3} ) q^{91} + ( -24 + 36 \beta_{3} ) q^{92} + ( 51 + 18 \beta_{1} + 51 \beta_{2} ) q^{93} + ( -8 - 22 \beta_{1} - 4 \beta_{2} + 22 \beta_{3} ) q^{94} + ( -54 \beta_{1} + 66 \beta_{2} - 54 \beta_{3} ) q^{95} + ( -4 \beta_{1} - 8 \beta_{3} ) q^{96} + ( -12 + 8 \beta_{1} - 24 \beta_{2} + 4 \beta_{3} ) q^{97} + ( -40 - 80 \beta_{2} + \beta_{3} ) q^{98} -18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 6q^{3} - 4q^{4} + 12q^{5} - 10q^{7} + 6q^{9} + O(q^{10})$$ $$4q + 6q^{3} - 4q^{4} + 12q^{5} - 10q^{7} + 6q^{9} - 24q^{10} - 12q^{11} - 12q^{12} + 24q^{14} + 24q^{15} - 8q^{16} - 48q^{17} - 42q^{19} + 24q^{23} + 22q^{25} + 96q^{26} + 40q^{28} - 24q^{30} + 102q^{31} - 36q^{33} - 108q^{35} - 24q^{36} + 22q^{37} + 24q^{38} + 6q^{39} + 48q^{40} + 24q^{42} + 28q^{43} - 24q^{44} + 36q^{45} - 72q^{46} - 132q^{47} - 2q^{49} - 192q^{50} - 48q^{51} + 12q^{52} + 120q^{53} - 48q^{56} - 84q^{57} - 96q^{58} - 24q^{59} - 24q^{60} - 72q^{61} + 30q^{63} + 32q^{64} + 180q^{65} + 110q^{67} + 96q^{68} + 96q^{70} + 312q^{71} - 66q^{73} - 48q^{74} + 66q^{75} + 120q^{77} + 192q^{78} - 10q^{79} - 48q^{80} - 18q^{81} + 48q^{82} + 60q^{84} - 288q^{85} - 24q^{86} - 72q^{89} - 222q^{91} - 96q^{92} + 102q^{93} - 24q^{94} - 132q^{95} - 72q^{99} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3}$$

## Character values

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

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$1$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 −0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i 0.707107 + 1.22474i
−0.707107 + 1.22474i 1.50000 0.866025i −1.00000 1.73205i 7.24264 + 4.18154i 2.44949i −6.74264 + 1.88064i 2.82843 1.50000 2.59808i −10.2426 + 5.91359i
19.2 0.707107 1.22474i 1.50000 0.866025i −1.00000 1.73205i −1.24264 0.717439i 2.44949i 1.74264 + 6.77962i −2.82843 1.50000 2.59808i −1.75736 + 1.01461i
31.1 −0.707107 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i 7.24264 4.18154i 2.44949i −6.74264 1.88064i 2.82843 1.50000 + 2.59808i −10.2426 5.91359i
31.2 0.707107 + 1.22474i 1.50000 + 0.866025i −1.00000 + 1.73205i −1.24264 + 0.717439i 2.44949i 1.74264 6.77962i −2.82843 1.50000 + 2.59808i −1.75736 1.01461i
 $$n$$: 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
7.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.3.g.a 4
3.b odd 2 1 126.3.n.a 4
4.b odd 2 1 336.3.bh.e 4
5.b even 2 1 1050.3.p.a 4
5.c odd 4 2 1050.3.q.a 8
7.b odd 2 1 294.3.g.a 4
7.c even 3 1 294.3.c.a 4
7.c even 3 1 294.3.g.a 4
7.d odd 6 1 inner 42.3.g.a 4
7.d odd 6 1 294.3.c.a 4
12.b even 2 1 1008.3.cg.h 4
21.c even 2 1 882.3.n.e 4
21.g even 6 1 126.3.n.a 4
21.g even 6 1 882.3.c.b 4
21.h odd 6 1 882.3.c.b 4
21.h odd 6 1 882.3.n.e 4
28.f even 6 1 336.3.bh.e 4
28.f even 6 1 2352.3.f.e 4
28.g odd 6 1 2352.3.f.e 4
35.i odd 6 1 1050.3.p.a 4
35.k even 12 2 1050.3.q.a 8
84.j odd 6 1 1008.3.cg.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.g.a 4 1.a even 1 1 trivial
42.3.g.a 4 7.d odd 6 1 inner
126.3.n.a 4 3.b odd 2 1
126.3.n.a 4 21.g even 6 1
294.3.c.a 4 7.c even 3 1
294.3.c.a 4 7.d odd 6 1
294.3.g.a 4 7.b odd 2 1
294.3.g.a 4 7.c even 3 1
336.3.bh.e 4 4.b odd 2 1
336.3.bh.e 4 28.f even 6 1
882.3.c.b 4 21.g even 6 1
882.3.c.b 4 21.h odd 6 1
882.3.n.e 4 21.c even 2 1
882.3.n.e 4 21.h odd 6 1
1008.3.cg.h 4 12.b even 2 1
1008.3.cg.h 4 84.j odd 6 1
1050.3.p.a 4 5.b even 2 1
1050.3.p.a 4 35.i odd 6 1
1050.3.q.a 8 5.c odd 4 2
1050.3.q.a 8 35.k even 12 2
2352.3.f.e 4 28.f even 6 1
2352.3.f.e 4 28.g odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} + 4 T^{4}$$
$3$ $$( 1 - 3 T + 3 T^{2} )^{2}$$
$5$ $$1 - 12 T + 86 T^{2} - 456 T^{3} + 2019 T^{4} - 11400 T^{5} + 53750 T^{6} - 187500 T^{7} + 390625 T^{8}$$
$7$ $$1 + 10 T + 51 T^{2} + 490 T^{3} + 2401 T^{4}$$
$11$ $$( 1 + 6 T - 85 T^{2} + 726 T^{3} + 14641 T^{4} )^{2}$$
$13$ $$1 + 98 T^{2} + 54915 T^{4} + 2798978 T^{6} + 815730721 T^{8}$$
$17$ $$1 + 48 T + 1514 T^{2} + 35808 T^{3} + 694947 T^{4} + 10348512 T^{5} + 126450794 T^{6} + 1158603312 T^{7} + 6975757441 T^{8}$$
$19$ $$1 + 42 T + 1433 T^{2} + 35490 T^{3} + 795972 T^{4} + 12811890 T^{5} + 186749993 T^{6} + 1975927002 T^{7} + 16983563041 T^{8}$$
$23$ $$1 - 24 T + 22 T^{2} + 12096 T^{3} - 277629 T^{4} + 6398784 T^{5} + 6156502 T^{6} - 3552861336 T^{7} + 78310985281 T^{8}$$
$29$ $$( 1 + 530 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$1 - 102 T + 6041 T^{2} - 262446 T^{3} + 9029556 T^{4} - 252210606 T^{5} + 5578990361 T^{6} - 90525375462 T^{7} + 852891037441 T^{8}$$
$37$ $$1 - 22 T - 2087 T^{2} + 3674 T^{3} + 4073284 T^{4} + 5029706 T^{5} - 3911374007 T^{6} - 56445980998 T^{7} + 3512479453921 T^{8}$$
$41$ $$1 - 2476 T^{2} + 6405414 T^{4} - 6996584236 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 - 14 T + 3675 T^{2} - 25886 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 + 132 T + 11654 T^{2} + 771672 T^{3} + 42125907 T^{4} + 1704623448 T^{5} + 56867802374 T^{6} + 1422856423428 T^{7} + 23811286661761 T^{8}$$
$53$ $$1 - 120 T + 5830 T^{2} - 354240 T^{3} + 25104819 T^{4} - 995060160 T^{5} + 46001504230 T^{6} - 2659723335480 T^{7} + 62259690411361 T^{8}$$
$59$ $$1 + 24 T + 6026 T^{2} + 140016 T^{3} + 22586547 T^{4} + 487395696 T^{5} + 73019217386 T^{6} + 1012332807384 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 + 72 T + 9218 T^{2} + 539280 T^{3} + 48684147 T^{4} + 2006660880 T^{5} + 127630962338 T^{6} + 3709466953992 T^{7} + 191707312997281 T^{8}$$
$67$ $$1 - 110 T + 3625 T^{2} + 55330 T^{3} - 2642396 T^{4} + 248376370 T^{5} + 73047813625 T^{6} - 9950422038590 T^{7} + 406067677556641 T^{8}$$
$71$ $$( 1 - 156 T + 12638 T^{2} - 786396 T^{3} + 25411681 T^{4} )^{2}$$
$73$ $$1 + 66 T + 2873 T^{2} + 93786 T^{3} - 18641292 T^{4} + 499785594 T^{5} + 81588146393 T^{6} + 9988058935074 T^{7} + 806460091894081 T^{8}$$
$79$ $$1 + 10 T - 3695 T^{2} - 86870 T^{3} - 25172156 T^{4} - 542155670 T^{5} - 143920549295 T^{6} + 2430874555210 T^{7} + 1517108809906561 T^{8}$$
$83$ $$1 - 9628 T^{2} + 107694438 T^{4} - 456928714588 T^{6} + 2252292232139041 T^{8}$$
$89$ $$( 1 + 36 T + 8353 T^{2} + 285156 T^{3} + 62742241 T^{4} )^{2}$$
$97$ $$1 - 36580 T^{2} + 511416774 T^{4} - 3238401098980 T^{6} + 7837433594376961 T^{8}$$