# 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: $$x^{4} + 2x^{2} + 4$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{2} q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + \beta_{3} q^{6} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8} - 3 q^{9}+O(q^{10})$$ q + b1 * q^2 - b2 * q^3 + 2 * q^4 + (-b3 - 2*b2) * q^5 + b3 * q^6 + (-2*b3 + b2 - 3*b1 + 2) * q^7 + 2*b1 * q^8 - 3 * q^9 $$q + \beta_1 q^{2} - \beta_{2} q^{3} + 2 q^{4} + ( - \beta_{3} - 2 \beta_{2}) q^{5} + \beta_{3} q^{6} + ( - 2 \beta_{3} + \beta_{2} - 3 \beta_1 + 2) q^{7} + 2 \beta_1 q^{8} - 3 q^{9} + (2 \beta_{3} + 2 \beta_{2}) q^{10} + ( - 3 \beta_1 - 6) q^{11} - 2 \beta_{2} q^{12} + (2 \beta_{3} + 8 \beta_{2}) q^{13} + ( - \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 6) q^{14} + (3 \beta_1 - 6) q^{15} + 4 q^{16} + (11 \beta_{3} - 2 \beta_{2}) q^{17} - 3 \beta_1 q^{18} + ( - 8 \beta_{3} - 2 \beta_{2}) q^{19} + ( - 2 \beta_{3} - 4 \beta_{2}) q^{20} + ( - 3 \beta_{3} - 2 \beta_{2} + 6 \beta_1 + 3) q^{21} + ( - 6 \beta_1 - 6) q^{22} + ( - 9 \beta_1 + 6) q^{23} + 2 \beta_{3} q^{24} + (12 \beta_1 + 7) q^{25} + ( - 8 \beta_{3} - 4 \beta_{2}) q^{26} + 3 \beta_{2} q^{27} + ( - 4 \beta_{3} + 2 \beta_{2} - 6 \beta_1 + 4) q^{28} + 30 q^{29} + ( - 6 \beta_1 + 6) q^{30} + (12 \beta_{3} - 12 \beta_{2}) q^{31} + 4 \beta_1 q^{32} + ( - 3 \beta_{3} + 6 \beta_{2}) q^{33} + (2 \beta_{3} - 22 \beta_{2}) q^{34} + ( - 8 \beta_{3} - 10 \beta_{2} + 9 \beta_1 - 6) q^{35} - 6 q^{36} + (36 \beta_1 - 20) q^{37} + (2 \beta_{3} + 16 \beta_{2}) q^{38} + ( - 6 \beta_1 + 24) q^{39} + (4 \beta_{3} + 4 \beta_{2}) q^{40} + (7 \beta_{3} + 14 \beta_{2}) q^{41} + (2 \beta_{3} + 6 \beta_{2} + 3 \beta_1 + 12) q^{42} + ( - 30 \beta_1 - 32) q^{43} + ( - 6 \beta_1 - 12) q^{44} + (3 \beta_{3} + 6 \beta_{2}) q^{45} + (6 \beta_1 - 18) q^{46} + ( - 4 \beta_{3} + 28 \beta_{2}) q^{47} - 4 \beta_{2} q^{48} + ( - 2 \beta_{3} - 20 \beta_{2} - 24 \beta_1 - 5) q^{49} + (7 \beta_1 + 24) q^{50} + ( - 33 \beta_1 - 6) q^{51} + (4 \beta_{3} + 16 \beta_{2}) q^{52} + ( - 12 \beta_1 - 54) q^{53} - 3 \beta_{3} q^{54} + 6 \beta_{2} q^{55} + ( - 2 \beta_{3} + 8 \beta_{2} + 4 \beta_1 - 12) q^{56} + (24 \beta_1 - 6) q^{57} + 30 \beta_1 q^{58} + ( - 20 \beta_{3} - 28 \beta_{2}) q^{59} + (6 \beta_1 - 12) q^{60} + (4 \beta_{3} + 4 \beta_{2}) q^{61} + (12 \beta_{3} - 24 \beta_{2}) q^{62} + (6 \beta_{3} - 3 \beta_{2} + 9 \beta_1 - 6) q^{63} + 8 q^{64} + ( - 36 \beta_1 + 60) q^{65} + ( - 6 \beta_{3} + 6 \beta_{2}) q^{66} + ( - 12 \beta_1 + 44) q^{67} + (22 \beta_{3} - 4 \beta_{2}) q^{68} + ( - 9 \beta_{3} - 6 \beta_{2}) q^{69} + (10 \beta_{3} + 16 \beta_{2} - 6 \beta_1 + 18) q^{70} + (57 \beta_1 - 30) q^{71} - 6 \beta_1 q^{72} + ( - 26 \beta_{3} + 4 \beta_{2}) q^{73} + ( - 20 \beta_1 + 72) q^{74} + (12 \beta_{3} - 7 \beta_{2}) q^{75} + ( - 16 \beta_{3} - 4 \beta_{2}) q^{76} + (15 \beta_{3} - 18 \beta_{2} + 12 \beta_1 + 6) q^{77} + (24 \beta_1 - 12) q^{78} + (72 \beta_1 + 32) q^{79} + ( - 4 \beta_{3} - 8 \beta_{2}) q^{80} + 9 q^{81} + ( - 14 \beta_{3} - 14 \beta_{2}) q^{82} + ( - 20 \beta_{3} + 32 \beta_{2}) q^{83} + ( - 6 \beta_{3} - 4 \beta_{2} + 12 \beta_1 + 6) q^{84} + ( - 60 \beta_1 + 54) q^{85} + ( - 32 \beta_1 - 60) q^{86} - 30 \beta_{2} q^{87} + ( - 12 \beta_1 - 12) q^{88} + (21 \beta_{3} - 54 \beta_{2}) q^{89} + ( - 6 \beta_{3} - 6 \beta_{2}) q^{90} + (28 \beta_{3} + 28 \beta_{2} - 42 \beta_1) q^{91} + ( - 18 \beta_1 + 12) q^{92} + ( - 36 \beta_1 - 36) q^{93} + ( - 28 \beta_{3} + 8 \beta_{2}) q^{94} + (54 \beta_1 - 60) q^{95} + 4 \beta_{3} q^{96} + ( - 10 \beta_{3} + 44 \beta_{2}) q^{97} + (20 \beta_{3} + 4 \beta_{2} - 5 \beta_1 - 48) q^{98} + (9 \beta_1 + 18) q^{99}+O(q^{100})$$ q + b1 * q^2 - b2 * q^3 + 2 * q^4 + (-b3 - 2*b2) * q^5 + b3 * q^6 + (-2*b3 + b2 - 3*b1 + 2) * q^7 + 2*b1 * q^8 - 3 * q^9 + (2*b3 + 2*b2) * q^10 + (-3*b1 - 6) * q^11 - 2*b2 * q^12 + (2*b3 + 8*b2) * q^13 + (-b3 + 4*b2 + 2*b1 - 6) * q^14 + (3*b1 - 6) * q^15 + 4 * q^16 + (11*b3 - 2*b2) * q^17 - 3*b1 * q^18 + (-8*b3 - 2*b2) * q^19 + (-2*b3 - 4*b2) * q^20 + (-3*b3 - 2*b2 + 6*b1 + 3) * q^21 + (-6*b1 - 6) * q^22 + (-9*b1 + 6) * q^23 + 2*b3 * q^24 + (12*b1 + 7) * q^25 + (-8*b3 - 4*b2) * q^26 + 3*b2 * q^27 + (-4*b3 + 2*b2 - 6*b1 + 4) * q^28 + 30 * q^29 + (-6*b1 + 6) * q^30 + (12*b3 - 12*b2) * q^31 + 4*b1 * q^32 + (-3*b3 + 6*b2) * q^33 + (2*b3 - 22*b2) * q^34 + (-8*b3 - 10*b2 + 9*b1 - 6) * q^35 - 6 * q^36 + (36*b1 - 20) * q^37 + (2*b3 + 16*b2) * q^38 + (-6*b1 + 24) * q^39 + (4*b3 + 4*b2) * q^40 + (7*b3 + 14*b2) * q^41 + (2*b3 + 6*b2 + 3*b1 + 12) * q^42 + (-30*b1 - 32) * q^43 + (-6*b1 - 12) * q^44 + (3*b3 + 6*b2) * q^45 + (6*b1 - 18) * q^46 + (-4*b3 + 28*b2) * q^47 - 4*b2 * q^48 + (-2*b3 - 20*b2 - 24*b1 - 5) * q^49 + (7*b1 + 24) * q^50 + (-33*b1 - 6) * q^51 + (4*b3 + 16*b2) * q^52 + (-12*b1 - 54) * q^53 - 3*b3 * q^54 + 6*b2 * q^55 + (-2*b3 + 8*b2 + 4*b1 - 12) * q^56 + (24*b1 - 6) * q^57 + 30*b1 * q^58 + (-20*b3 - 28*b2) * q^59 + (6*b1 - 12) * q^60 + (4*b3 + 4*b2) * q^61 + (12*b3 - 24*b2) * q^62 + (6*b3 - 3*b2 + 9*b1 - 6) * q^63 + 8 * q^64 + (-36*b1 + 60) * q^65 + (-6*b3 + 6*b2) * q^66 + (-12*b1 + 44) * q^67 + (22*b3 - 4*b2) * q^68 + (-9*b3 - 6*b2) * q^69 + (10*b3 + 16*b2 - 6*b1 + 18) * q^70 + (57*b1 - 30) * q^71 - 6*b1 * q^72 + (-26*b3 + 4*b2) * q^73 + (-20*b1 + 72) * q^74 + (12*b3 - 7*b2) * q^75 + (-16*b3 - 4*b2) * q^76 + (15*b3 - 18*b2 + 12*b1 + 6) * q^77 + (24*b1 - 12) * q^78 + (72*b1 + 32) * q^79 + (-4*b3 - 8*b2) * q^80 + 9 * q^81 + (-14*b3 - 14*b2) * q^82 + (-20*b3 + 32*b2) * q^83 + (-6*b3 - 4*b2 + 12*b1 + 6) * q^84 + (-60*b1 + 54) * q^85 + (-32*b1 - 60) * q^86 - 30*b2 * q^87 + (-12*b1 - 12) * q^88 + (21*b3 - 54*b2) * q^89 + (-6*b3 - 6*b2) * q^90 + (28*b3 + 28*b2 - 42*b1) * q^91 + (-18*b1 + 12) * q^92 + (-36*b1 - 36) * q^93 + (-28*b3 + 8*b2) * q^94 + (54*b1 - 60) * q^95 + 4*b3 * q^96 + (-10*b3 + 44*b2) * q^97 + (20*b3 + 4*b2 - 5*b1 - 48) * q^98 + (9*b1 + 18) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 8 q^{4} + 8 q^{7} - 12 q^{9}+O(q^{10})$$ 4 * q + 8 * q^4 + 8 * q^7 - 12 * q^9 $$4 q + 8 q^{4} + 8 q^{7} - 12 q^{9} - 24 q^{11} - 24 q^{14} - 24 q^{15} + 16 q^{16} + 12 q^{21} - 24 q^{22} + 24 q^{23} + 28 q^{25} + 16 q^{28} + 120 q^{29} + 24 q^{30} - 24 q^{35} - 24 q^{36} - 80 q^{37} + 96 q^{39} + 48 q^{42} - 128 q^{43} - 48 q^{44} - 72 q^{46} - 20 q^{49} + 96 q^{50} - 24 q^{51} - 216 q^{53} - 48 q^{56} - 24 q^{57} - 48 q^{60} - 24 q^{63} + 32 q^{64} + 240 q^{65} + 176 q^{67} + 72 q^{70} - 120 q^{71} + 288 q^{74} + 24 q^{77} - 48 q^{78} + 128 q^{79} + 36 q^{81} + 24 q^{84} + 216 q^{85} - 240 q^{86} - 48 q^{88} + 48 q^{92} - 144 q^{93} - 240 q^{95} - 192 q^{98} + 72 q^{99}+O(q^{100})$$ 4 * q + 8 * q^4 + 8 * q^7 - 12 * q^9 - 24 * q^11 - 24 * q^14 - 24 * q^15 + 16 * q^16 + 12 * q^21 - 24 * q^22 + 24 * q^23 + 28 * q^25 + 16 * q^28 + 120 * q^29 + 24 * q^30 - 24 * q^35 - 24 * q^36 - 80 * q^37 + 96 * q^39 + 48 * q^42 - 128 * q^43 - 48 * q^44 - 72 * q^46 - 20 * q^49 + 96 * q^50 - 24 * q^51 - 216 * q^53 - 48 * q^56 - 24 * q^57 - 48 * q^60 - 24 * q^63 + 32 * q^64 + 240 * q^65 + 176 * q^67 + 72 * q^70 - 120 * q^71 + 288 * q^74 + 24 * q^77 - 48 * q^78 + 128 * q^79 + 36 * q^81 + 24 * q^84 + 216 * q^85 - 240 * q^86 - 48 * q^88 + 48 * q^92 - 144 * q^93 - 240 * q^95 - 192 * q^98 + 72 * q^99

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

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} ) / 2$$ (v^3) / 2 $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 2$$ (v^3 + 4*v) / 2
 $$\nu$$ $$=$$ $$( \beta_{3} - \beta_1 ) / 2$$ (b3 - b1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$2\beta_1$$ 2*b1

## 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

$p$ $F_p(T)$
$2$ $$(T^{2} - 2)^{2}$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$T^{4} + 36T^{2} + 36$$
$7$ $$T^{4} - 8 T^{3} + 42 T^{2} + \cdots + 2401$$
$11$ $$(T^{2} + 12 T + 18)^{2}$$
$13$ $$T^{4} + 432 T^{2} + 28224$$
$17$ $$T^{4} + 1476 T^{2} + 509796$$
$19$ $$T^{4} + 792 T^{2} + 138384$$
$23$ $$(T^{2} - 12 T - 126)^{2}$$
$29$ $$(T - 30)^{4}$$
$31$ $$T^{4} + 2592 T^{2} + 186624$$
$37$ $$(T^{2} + 40 T - 2192)^{2}$$
$41$ $$T^{4} + 1764 T^{2} + 86436$$
$43$ $$(T^{2} + 64 T - 776)^{2}$$
$47$ $$T^{4} + 4896 T^{2} + \cdots + 5089536$$
$53$ $$(T^{2} + 108 T + 2628)^{2}$$
$59$ $$T^{4} + 9504 T^{2} + 2304$$
$61$ $$T^{4} + 288T^{2} + 2304$$
$67$ $$(T^{2} - 88 T + 1648)^{2}$$
$71$ $$(T^{2} + 60 T - 5598)^{2}$$
$73$ $$T^{4} + 8208 T^{2} + \cdots + 16064064$$
$79$ $$(T^{2} - 64 T - 9344)^{2}$$
$83$ $$T^{4} + 10944 T^{2} + \cdots + 451584$$
$89$ $$T^{4} + 22788 T^{2} + \cdots + 37234404$$
$97$ $$T^{4} + 12816 T^{2} + \cdots + 27123264$$