# Properties

 Label 42.3.c.a Level $42$ Weight $3$ Character orbit 42.c 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.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14441711031$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Defining polynomial: $$x^{4} + 2 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + ( -2 \beta_{2} - \beta_{3} ) q^{5} + \beta_{3} q^{6} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{2} q^{3} + 2 q^{4} + ( -2 \beta_{2} - \beta_{3} ) q^{5} + \beta_{3} q^{6} + ( 2 - 3 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{7} + 2 \beta_{1} q^{8} -3 q^{9} + ( 2 \beta_{2} + 2 \beta_{3} ) q^{10} + ( -6 - 3 \beta_{1} ) q^{11} -2 \beta_{2} q^{12} + ( 8 \beta_{2} + 2 \beta_{3} ) q^{13} + ( -6 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{14} + ( -6 + 3 \beta_{1} ) q^{15} + 4 q^{16} + ( -2 \beta_{2} + 11 \beta_{3} ) q^{17} -3 \beta_{1} q^{18} + ( -2 \beta_{2} - 8 \beta_{3} ) q^{19} + ( -4 \beta_{2} - 2 \beta_{3} ) q^{20} + ( 3 + 6 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{21} + ( -6 - 6 \beta_{1} ) q^{22} + ( 6 - 9 \beta_{1} ) q^{23} + 2 \beta_{3} q^{24} + ( 7 + 12 \beta_{1} ) q^{25} + ( -4 \beta_{2} - 8 \beta_{3} ) q^{26} + 3 \beta_{2} q^{27} + ( 4 - 6 \beta_{1} + 2 \beta_{2} - 4 \beta_{3} ) q^{28} + 30 q^{29} + ( 6 - 6 \beta_{1} ) q^{30} + ( -12 \beta_{2} + 12 \beta_{3} ) q^{31} + 4 \beta_{1} q^{32} + ( 6 \beta_{2} - 3 \beta_{3} ) q^{33} + ( -22 \beta_{2} + 2 \beta_{3} ) q^{34} + ( -6 + 9 \beta_{1} - 10 \beta_{2} - 8 \beta_{3} ) q^{35} -6 q^{36} + ( -20 + 36 \beta_{1} ) q^{37} + ( 16 \beta_{2} + 2 \beta_{3} ) q^{38} + ( 24 - 6 \beta_{1} ) q^{39} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{40} + ( 14 \beta_{2} + 7 \beta_{3} ) q^{41} + ( 12 + 3 \beta_{1} + 6 \beta_{2} + 2 \beta_{3} ) q^{42} + ( -32 - 30 \beta_{1} ) q^{43} + ( -12 - 6 \beta_{1} ) q^{44} + ( 6 \beta_{2} + 3 \beta_{3} ) q^{45} + ( -18 + 6 \beta_{1} ) q^{46} + ( 28 \beta_{2} - 4 \beta_{3} ) q^{47} -4 \beta_{2} q^{48} + ( -5 - 24 \beta_{1} - 20 \beta_{2} - 2 \beta_{3} ) q^{49} + ( 24 + 7 \beta_{1} ) q^{50} + ( -6 - 33 \beta_{1} ) q^{51} + ( 16 \beta_{2} + 4 \beta_{3} ) q^{52} + ( -54 - 12 \beta_{1} ) q^{53} -3 \beta_{3} q^{54} + 6 \beta_{2} q^{55} + ( -12 + 4 \beta_{1} + 8 \beta_{2} - 2 \beta_{3} ) q^{56} + ( -6 + 24 \beta_{1} ) q^{57} + 30 \beta_{1} q^{58} + ( -28 \beta_{2} - 20 \beta_{3} ) q^{59} + ( -12 + 6 \beta_{1} ) q^{60} + ( 4 \beta_{2} + 4 \beta_{3} ) q^{61} + ( -24 \beta_{2} + 12 \beta_{3} ) q^{62} + ( -6 + 9 \beta_{1} - 3 \beta_{2} + 6 \beta_{3} ) q^{63} + 8 q^{64} + ( 60 - 36 \beta_{1} ) q^{65} + ( 6 \beta_{2} - 6 \beta_{3} ) q^{66} + ( 44 - 12 \beta_{1} ) q^{67} + ( -4 \beta_{2} + 22 \beta_{3} ) q^{68} + ( -6 \beta_{2} - 9 \beta_{3} ) q^{69} + ( 18 - 6 \beta_{1} + 16 \beta_{2} + 10 \beta_{3} ) q^{70} + ( -30 + 57 \beta_{1} ) q^{71} -6 \beta_{1} q^{72} + ( 4 \beta_{2} - 26 \beta_{3} ) q^{73} + ( 72 - 20 \beta_{1} ) q^{74} + ( -7 \beta_{2} + 12 \beta_{3} ) q^{75} + ( -4 \beta_{2} - 16 \beta_{3} ) q^{76} + ( 6 + 12 \beta_{1} - 18 \beta_{2} + 15 \beta_{3} ) q^{77} + ( -12 + 24 \beta_{1} ) q^{78} + ( 32 + 72 \beta_{1} ) q^{79} + ( -8 \beta_{2} - 4 \beta_{3} ) q^{80} + 9 q^{81} + ( -14 \beta_{2} - 14 \beta_{3} ) q^{82} + ( 32 \beta_{2} - 20 \beta_{3} ) q^{83} + ( 6 + 12 \beta_{1} - 4 \beta_{2} - 6 \beta_{3} ) q^{84} + ( 54 - 60 \beta_{1} ) q^{85} + ( -60 - 32 \beta_{1} ) q^{86} -30 \beta_{2} q^{87} + ( -12 - 12 \beta_{1} ) q^{88} + ( -54 \beta_{2} + 21 \beta_{3} ) q^{89} + ( -6 \beta_{2} - 6 \beta_{3} ) q^{90} + ( -42 \beta_{1} + 28 \beta_{2} + 28 \beta_{3} ) q^{91} + ( 12 - 18 \beta_{1} ) q^{92} + ( -36 - 36 \beta_{1} ) q^{93} + ( 8 \beta_{2} - 28 \beta_{3} ) q^{94} + ( -60 + 54 \beta_{1} ) q^{95} + 4 \beta_{3} q^{96} + ( 44 \beta_{2} - 10 \beta_{3} ) q^{97} + ( -48 - 5 \beta_{1} + 4 \beta_{2} + 20 \beta_{3} ) q^{98} + ( 18 + 9 \beta_{1} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + 8q^{7} - 12q^{9} + O(q^{10})$$ $$4q + 8q^{4} + 8q^{7} - 12q^{9} - 24q^{11} - 24q^{14} - 24q^{15} + 16q^{16} + 12q^{21} - 24q^{22} + 24q^{23} + 28q^{25} + 16q^{28} + 120q^{29} + 24q^{30} - 24q^{35} - 24q^{36} - 80q^{37} + 96q^{39} + 48q^{42} - 128q^{43} - 48q^{44} - 72q^{46} - 20q^{49} + 96q^{50} - 24q^{51} - 216q^{53} - 48q^{56} - 24q^{57} - 48q^{60} - 24q^{63} + 32q^{64} + 240q^{65} + 176q^{67} + 72q^{70} - 120q^{71} + 288q^{74} + 24q^{77} - 48q^{78} + 128q^{79} + 36q^{81} + 24q^{84} + 216q^{85} - 240q^{86} - 48q^{88} + 48q^{92} - 144q^{93} - 240q^{95} - 192q^{98} + 72q^{99} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 0.707107 + 1.22474i 0.707107 − 1.22474i −0.707107 − 1.22474i −0.707107 + 1.22474i
−1.41421 1.73205i 2.00000 5.91359i 2.44949i 6.24264 3.16693i −2.82843 −3.00000 8.36308i
13.2 −1.41421 1.73205i 2.00000 5.91359i 2.44949i 6.24264 + 3.16693i −2.82843 −3.00000 8.36308i
13.3 1.41421 1.73205i 2.00000 1.01461i 2.44949i −2.24264 + 6.63103i 2.82843 −3.00000 1.43488i
13.4 1.41421 1.73205i 2.00000 1.01461i 2.44949i −2.24264 6.63103i 2.82843 −3.00000 1.43488i
 $$n$$: 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.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.c.a 4
3.b odd 2 1 126.3.c.b 4
4.b odd 2 1 336.3.f.c 4
5.b even 2 1 1050.3.f.a 4
5.c odd 4 2 1050.3.h.a 8
7.b odd 2 1 inner 42.3.c.a 4
7.c even 3 1 294.3.g.b 4
7.c even 3 1 294.3.g.c 4
7.d odd 6 1 294.3.g.b 4
7.d odd 6 1 294.3.g.c 4
8.b even 2 1 1344.3.f.f 4
8.d odd 2 1 1344.3.f.e 4
12.b even 2 1 1008.3.f.g 4
21.c even 2 1 126.3.c.b 4
21.g even 6 1 882.3.n.a 4
21.g even 6 1 882.3.n.d 4
21.h odd 6 1 882.3.n.a 4
21.h odd 6 1 882.3.n.d 4
28.d even 2 1 336.3.f.c 4
35.c odd 2 1 1050.3.f.a 4
35.f even 4 2 1050.3.h.a 8
56.e even 2 1 1344.3.f.e 4
56.h odd 2 1 1344.3.f.f 4
84.h odd 2 1 1008.3.f.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.c.a 4 1.a even 1 1 trivial
42.3.c.a 4 7.b odd 2 1 inner
126.3.c.b 4 3.b odd 2 1
126.3.c.b 4 21.c even 2 1
294.3.g.b 4 7.c even 3 1
294.3.g.b 4 7.d odd 6 1
294.3.g.c 4 7.c even 3 1
294.3.g.c 4 7.d odd 6 1
336.3.f.c 4 4.b odd 2 1
336.3.f.c 4 28.d even 2 1
882.3.n.a 4 21.g even 6 1
882.3.n.a 4 21.h odd 6 1
882.3.n.d 4 21.g even 6 1
882.3.n.d 4 21.h odd 6 1
1008.3.f.g 4 12.b even 2 1
1008.3.f.g 4 84.h odd 2 1
1050.3.f.a 4 5.b even 2 1
1050.3.f.a 4 35.c odd 2 1
1050.3.h.a 8 5.c odd 4 2
1050.3.h.a 8 35.f even 4 2
1344.3.f.e 4 8.d odd 2 1
1344.3.f.e 4 56.e even 2 1
1344.3.f.f 4 8.b even 2 1
1344.3.f.f 4 56.h odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} )^{2}$$
$3$ $$( 1 + 3 T^{2} )^{2}$$
$5$ $$1 - 64 T^{2} + 1986 T^{4} - 40000 T^{6} + 390625 T^{8}$$
$7$ $$1 - 8 T + 42 T^{2} - 392 T^{3} + 2401 T^{4}$$
$11$ $$( 1 + 12 T + 260 T^{2} + 1452 T^{3} + 14641 T^{4} )^{2}$$
$13$ $$1 - 244 T^{2} + 53574 T^{4} - 6968884 T^{6} + 815730721 T^{8}$$
$17$ $$1 + 320 T^{2} + 157794 T^{4} + 26726720 T^{6} + 6975757441 T^{8}$$
$19$ $$1 - 652 T^{2} + 348486 T^{4} - 84969292 T^{6} + 16983563041 T^{8}$$
$23$ $$( 1 - 12 T + 932 T^{2} - 6348 T^{3} + 279841 T^{4} )^{2}$$
$29$ $$( 1 - 30 T + 841 T^{2} )^{4}$$
$31$ $$1 - 1252 T^{2} + 745926 T^{4} - 1156248292 T^{6} + 852891037441 T^{8}$$
$37$ $$( 1 + 40 T + 546 T^{2} + 54760 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$1 - 4960 T^{2} + 11110434 T^{4} - 14015774560 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 + 64 T + 2922 T^{2} + 118336 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 3940 T^{2} + 12737094 T^{4} - 19225943140 T^{6} + 23811286661761 T^{8}$$
$53$ $$( 1 + 108 T + 8246 T^{2} + 303372 T^{3} + 7890481 T^{4} )^{2}$$
$59$ $$1 - 4420 T^{2} + 6539622 T^{4} - 53558735620 T^{6} + 146830437604321 T^{8}$$
$61$ $$1 - 14596 T^{2} + 80934054 T^{4} - 202093895236 T^{6} + 191707312997281 T^{8}$$
$67$ $$( 1 - 88 T + 10626 T^{2} - 395032 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 + 60 T + 4484 T^{2} + 302460 T^{3} + 25411681 T^{4} )^{2}$$
$73$ $$1 - 13108 T^{2} + 98972646 T^{4} - 372244143028 T^{6} + 806460091894081 T^{8}$$
$79$ $$( 1 - 64 T + 3138 T^{2} - 399424 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$1 - 16612 T^{2} + 134415078 T^{4} - 788377628452 T^{6} + 2252292232139041 T^{8}$$
$89$ $$1 - 8896 T^{2} + 52680354 T^{4} - 558154975936 T^{6} + 3936588805702081 T^{8}$$
$97$ $$1 - 24820 T^{2} + 317127462 T^{4} - 2197296754420 T^{6} + 7837433594376961 T^{8}$$