# Properties

 Label 42.3.h.b Level $42$ Weight $3$ Character orbit 42.h Analytic conductor $1.144$ Analytic rank $0$ Dimension $8$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 42.h (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.14441711031$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{6})$$ Coefficient field: 8.0.4857532416.2 Defining polynomial: $$x^{8} - 2 x^{7} - 7 x^{6} - 2 x^{5} + 98 x^{4} - 98 x^{3} + 67 x^{2} - 30 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{2} - \beta_{6} ) q^{2} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{3} + ( 2 + 2 \beta_{5} ) q^{4} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{5} + ( -2 + \beta_{3} - 2 \beta_{7} ) q^{6} + ( -3 - 2 \beta_{3} - 6 \beta_{5} + \beta_{6} ) q^{7} -2 \beta_{6} q^{8} + ( 1 + \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{2} - \beta_{6} ) q^{2} + ( \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} ) q^{3} + ( 2 + 2 \beta_{5} ) q^{4} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{5} + ( -2 + \beta_{3} - 2 \beta_{7} ) q^{6} + ( -3 - 2 \beta_{3} - 6 \beta_{5} + \beta_{6} ) q^{7} -2 \beta_{6} q^{8} + ( 1 + \beta_{1} - 4 \beta_{2} - 2 \beta_{4} + 3 \beta_{5} + 4 \beta_{6} + \beta_{7} ) q^{9} + ( 4 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - 2 \beta_{6} ) q^{10} + ( -5 - 10 \beta_{1} - \beta_{2} - 5 \beta_{5} ) q^{11} + ( 2 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + 2 \beta_{7} ) q^{12} + ( -8 + 4 \beta_{3} - 2 \beta_{6} ) q^{13} + ( -4 - 4 \beta_{1} + 3 \beta_{2} - 2 \beta_{5} + 3 \beta_{6} - 4 \beta_{7} ) q^{14} + ( 1 + \beta_{3} + 6 \beta_{6} - 5 \beta_{7} ) q^{15} + 4 \beta_{5} q^{16} + ( 1 + 2 \beta_{1} + 2 \beta_{2} + \beta_{5} ) q^{17} + ( -12 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} - \beta_{4} - 12 \beta_{5} - \beta_{6} ) q^{18} + ( -3 \beta_{2} + 6 \beta_{4} - \beta_{5} + 3 \beta_{6} ) q^{19} + ( 2 - 4 \beta_{6} + 4 \beta_{7} ) q^{20} + ( 6 - \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 12 \beta_{5} - 4 \beta_{6} - 6 \beta_{7} ) q^{21} + ( -2 - 10 \beta_{3} + 5 \beta_{6} ) q^{22} + ( 10 + 10 \beta_{1} + 7 \beta_{2} + 5 \beta_{5} - 7 \beta_{6} + 10 \beta_{7} ) q^{23} + ( 4 \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - 2 \beta_{6} ) q^{24} + ( -6 + 4 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 6 \beta_{5} - 8 \beta_{6} ) q^{25} + ( 8 + 8 \beta_{1} - 8 \beta_{2} + 4 \beta_{5} + 8 \beta_{6} + 8 \beta_{7} ) q^{26} + ( 25 - 2 \beta_{3} - 3 \beta_{6} + 10 \beta_{7} ) q^{27} + ( 6 - 2 \beta_{2} - 4 \beta_{3} + 4 \beta_{4} - 6 \beta_{5} + 4 \beta_{6} ) q^{28} + ( -10 - 10 \beta_{6} - 20 \beta_{7} ) q^{29} + ( 2 + 2 \beta_{1} + \beta_{2} + 5 \beta_{4} - 12 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{30} + ( 5 - 5 \beta_{2} - 10 \beta_{3} + 10 \beta_{4} + 5 \beta_{5} + 10 \beta_{6} ) q^{31} -4 \beta_{2} q^{32} + ( 7 + 7 \beta_{1} + 26 \beta_{2} + 4 \beta_{4} + 30 \beta_{5} - 26 \beta_{6} + 7 \beta_{7} ) q^{33} + ( 4 + 2 \beta_{3} - \beta_{6} ) q^{34} + ( -8 - 2 \beta_{1} - 5 \beta_{2} - \beta_{5} + 17 \beta_{6} - 14 \beta_{7} ) q^{35} + ( -4 - 4 \beta_{3} + 12 \beta_{6} + 2 \beta_{7} ) q^{36} -\beta_{5} q^{37} + ( 6 + 12 \beta_{1} + \beta_{2} + 6 \beta_{5} ) q^{38} + ( -24 - 12 \beta_{1} + 18 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} - 24 \beta_{5} - 6 \beta_{6} ) q^{39} + ( 2 \beta_{2} - 4 \beta_{4} + 8 \beta_{5} - 2 \beta_{6} ) q^{40} + ( 6 + 6 \beta_{6} + 12 \beta_{7} ) q^{41} + ( -16 - 12 \beta_{1} - 6 \beta_{2} - \beta_{3} + 6 \beta_{4} - 6 \beta_{6} - 4 \beta_{7} ) q^{42} + ( 26 + 12 \beta_{3} - 6 \beta_{6} ) q^{43} + ( -20 - 20 \beta_{1} - 2 \beta_{2} - 10 \beta_{5} + 2 \beta_{6} - 20 \beta_{7} ) q^{44} + ( -21 - 13 \beta_{1} - 20 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} - 21 \beta_{5} + 8 \beta_{6} ) q^{45} + ( 14 + 5 \beta_{2} + 10 \beta_{3} - 10 \beta_{4} + 14 \beta_{5} - 10 \beta_{6} ) q^{46} + ( 6 + 6 \beta_{1} - 45 \beta_{2} + 3 \beta_{5} + 45 \beta_{6} + 6 \beta_{7} ) q^{47} + ( 4 + 4 \beta_{3} + 4 \beta_{7} ) q^{48} + ( -5 + 12 \beta_{2} + 12 \beta_{3} - 24 \beta_{4} - 18 \beta_{6} ) q^{49} + ( 8 + 6 \beta_{6} + 16 \beta_{7} ) q^{50} + ( -5 - 5 \beta_{1} - 7 \beta_{2} + \beta_{4} - 6 \beta_{5} + 7 \beta_{6} - 5 \beta_{7} ) q^{51} + ( -16 + 4 \beta_{2} + 8 \beta_{3} - 8 \beta_{4} - 16 \beta_{5} - 8 \beta_{6} ) q^{52} + ( 3 + 6 \beta_{1} - 36 \beta_{2} + 3 \beta_{5} ) q^{53} + ( -4 - 4 \beta_{1} + 25 \beta_{2} - 10 \beta_{4} + 6 \beta_{5} - 25 \beta_{6} - 4 \beta_{7} ) q^{54} + ( -59 - 22 \beta_{3} + 11 \beta_{6} ) q^{55} + ( -4 + 12 \beta_{2} - 6 \beta_{6} - 8 \beta_{7} ) q^{56} + ( -31 - 4 \beta_{3} + 18 \beta_{6} + 5 \beta_{7} ) q^{57} + ( -10 \beta_{2} + 20 \beta_{4} + 20 \beta_{5} + 10 \beta_{6} ) q^{58} + ( -7 - 14 \beta_{1} + 55 \beta_{2} - 7 \beta_{5} ) q^{59} + ( 12 + 10 \beta_{1} + 14 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 12 \beta_{5} - 2 \beta_{6} ) q^{60} + ( -4 \beta_{2} + 8 \beta_{4} - 53 \beta_{5} + 4 \beta_{6} ) q^{61} + ( -10 - 5 \beta_{6} - 20 \beta_{7} ) q^{62} + ( 35 + 23 \beta_{1} - 20 \beta_{2} + 12 \beta_{3} - \beta_{4} - 3 \beta_{5} - 16 \beta_{6} + 17 \beta_{7} ) q^{63} -8 q^{64} + ( 6 \beta_{2} - 6 \beta_{6} ) q^{65} + ( 60 + 8 \beta_{1} - 23 \beta_{2} + 7 \beta_{3} - 7 \beta_{4} + 60 \beta_{5} - 7 \beta_{6} ) q^{66} + ( 39 - 13 \beta_{2} - 26 \beta_{3} + 26 \beta_{4} + 39 \beta_{5} + 26 \beta_{6} ) q^{67} + ( 4 + 4 \beta_{1} + 4 \beta_{2} + 2 \beta_{5} - 4 \beta_{6} + 4 \beta_{7} ) q^{68} + ( 11 + 2 \beta_{3} + 30 \beta_{6} - 19 \beta_{7} ) q^{69} + ( -10 - 7 \beta_{2} - 2 \beta_{3} + 14 \beta_{4} - 34 \beta_{5} + 8 \beta_{6} ) q^{70} + ( 4 - 68 \beta_{6} + 8 \beta_{7} ) q^{71} + ( -8 - 8 \beta_{1} - 4 \beta_{2} - 2 \beta_{4} - 24 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{72} + ( 33 - 4 \beta_{2} - 8 \beta_{3} + 8 \beta_{4} + 33 \beta_{5} + 8 \beta_{6} ) q^{73} + \beta_{2} q^{74} + ( -14 - 14 \beta_{1} + 26 \beta_{2} - 2 \beta_{4} - 48 \beta_{5} - 26 \beta_{6} - 14 \beta_{7} ) q^{75} + ( 2 + 12 \beta_{3} - 6 \beta_{6} ) q^{76} + ( 47 + 34 \beta_{1} + 58 \beta_{2} + 17 \beta_{5} - 6 \beta_{6} + 60 \beta_{7} ) q^{77} + ( 36 - 12 \beta_{3} + 24 \beta_{6} + 12 \beta_{7} ) q^{78} + ( 17 \beta_{2} - 34 \beta_{4} + 13 \beta_{5} - 17 \beta_{6} ) q^{79} + ( -4 - 8 \beta_{1} - 8 \beta_{2} - 4 \beta_{5} ) q^{80} + ( 42 + 35 \beta_{1} + 4 \beta_{2} - 20 \beta_{3} + 20 \beta_{4} + 42 \beta_{5} + 20 \beta_{6} ) q^{81} + ( 6 \beta_{2} - 12 \beta_{4} - 12 \beta_{5} - 6 \beta_{6} ) q^{82} + ( -8 - 44 \beta_{6} - 16 \beta_{7} ) q^{83} + ( -2 + 10 \beta_{1} - 16 \beta_{2} - 12 \beta_{3} + 4 \beta_{4} + 24 \beta_{5} + 16 \beta_{6} - 2 \beta_{7} ) q^{84} + ( 19 + 8 \beta_{3} - 4 \beta_{6} ) q^{85} + ( 24 + 24 \beta_{1} + 26 \beta_{2} + 12 \beta_{5} - 26 \beta_{6} + 24 \beta_{7} ) q^{86} + ( -60 + 10 \beta_{1} - 40 \beta_{2} + 20 \beta_{3} - 20 \beta_{4} - 60 \beta_{5} - 20 \beta_{6} ) q^{87} + ( -4 - 10 \beta_{2} - 20 \beta_{3} + 20 \beta_{4} - 4 \beta_{5} + 20 \beta_{6} ) q^{88} + ( -66 - 66 \beta_{1} - 36 \beta_{2} - 33 \beta_{5} + 36 \beta_{6} - 66 \beta_{7} ) q^{89} + ( -40 - 13 \beta_{3} + 21 \beta_{6} - 16 \beta_{7} ) q^{90} + ( -20 - 12 \beta_{2} + 4 \beta_{3} + 24 \beta_{4} + 48 \beta_{5} + 10 \beta_{6} ) q^{91} + ( 10 - 14 \beta_{6} + 20 \beta_{7} ) q^{92} + ( 15 + 15 \beta_{1} - 30 \beta_{2} + 60 \beta_{5} + 30 \beta_{6} + 15 \beta_{7} ) q^{93} + ( -90 + 3 \beta_{2} + 6 \beta_{3} - 6 \beta_{4} - 90 \beta_{5} - 6 \beta_{6} ) q^{94} + ( 13 + 26 \beta_{1} + 35 \beta_{2} + 13 \beta_{5} ) q^{95} + ( 8 + 8 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} - 4 \beta_{6} + 8 \beta_{7} ) q^{96} + ( 72 + 20 \beta_{3} - 10 \beta_{6} ) q^{97} + ( -24 \beta_{1} - 5 \beta_{2} - 12 \beta_{5} + 5 \beta_{6} + 24 \beta_{7} ) q^{98} + ( -32 + 49 \beta_{3} + 33 \beta_{6} - 29 \beta_{7} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 2q^{3} + 8q^{4} - 8q^{6} - 10q^{9} + O(q^{10})$$ $$8q - 2q^{3} + 8q^{4} - 8q^{6} - 10q^{9} + 16q^{10} + 4q^{12} - 64q^{13} + 28q^{15} - 16q^{16} - 40q^{18} + 4q^{19} + 26q^{21} - 16q^{22} - 8q^{24} - 24q^{25} + 160q^{27} + 72q^{28} + 52q^{30} + 20q^{31} - 106q^{33} + 32q^{34} - 40q^{36} + 4q^{37} - 72q^{39} - 32q^{40} - 88q^{42} + 208q^{43} - 58q^{45} + 56q^{46} + 16q^{48} - 40q^{49} + 14q^{51} - 64q^{52} - 32q^{54} - 472q^{55} - 268q^{57} - 80q^{58} + 28q^{60} + 212q^{61} + 178q^{63} - 64q^{64} + 224q^{66} + 156q^{67} + 164q^{69} + 56q^{70} + 80q^{72} + 132q^{73} + 164q^{75} + 16q^{76} + 240q^{78} - 52q^{79} + 98q^{81} + 48q^{82} - 124q^{84} + 152q^{85} - 260q^{87} - 16q^{88} - 256q^{90} - 352q^{91} - 210q^{93} - 360q^{94} + 16q^{96} + 576q^{97} - 140q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 2 x^{7} - 7 x^{6} - 2 x^{5} + 98 x^{4} - 98 x^{3} + 67 x^{2} - 30 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2272 \nu^{7} + 10231 \nu^{6} + 9128 \nu^{5} - 44054 \nu^{4} - 273322 \nu^{3} + 767308 \nu^{2} - 205992 \nu + 94527$$$$)/84870$$ $$\beta_{2}$$ $$=$$ $$($$$$3823 \nu^{7} - 12079 \nu^{6} - 22382 \nu^{5} + 30236 \nu^{4} + 414148 \nu^{3} - 781972 \nu^{2} + 298053 \nu - 135243$$$$)/84870$$ $$\beta_{3}$$ $$=$$ $$($$$$-4433 \nu^{7} + 4379 \nu^{6} + 37882 \nu^{5} + 39494 \nu^{4} - 407318 \nu^{3} + 41912 \nu^{2} - 20553 \nu - 204147$$$$)/84870$$ $$\beta_{4}$$ $$=$$ $$($$$$-4487 \nu^{7} + 6851 \nu^{6} + 30628 \nu^{5} + 27116 \nu^{4} - 392522 \nu^{3} + 276458 \nu^{2} - 506877 \nu + 39897$$$$)/84870$$ $$\beta_{5}$$ $$=$$ $$($$$$-1508 \nu^{7} + 2499 \nu^{6} + 11172 \nu^{5} + 7434 \nu^{4} - 143178 \nu^{3} + 100842 \nu^{2} - 96148 \nu + 14553$$$$)/28290$$ $$\beta_{6}$$ $$=$$ $$($$$$-209 \nu^{7} + 227 \nu^{6} + 1786 \nu^{5} + 1862 \nu^{4} - 19784 \nu^{3} + 1976 \nu^{2} - 969 \nu - 261$$$$)/2070$$ $$\beta_{7}$$ $$=$$ $$($$$$-88 \nu^{7} + 97 \nu^{6} + 752 \nu^{5} + 784 \nu^{4} - 8278 \nu^{3} + 832 \nu^{2} - 408 \nu - 477$$$$)/738$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{7} - \beta_{6} + \beta_{2} + \beta_{1} + 1$$ $$\nu^{2}$$ $$=$$ $$-2 \beta_{6} + 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_{1} + 5$$ $$\nu^{3}$$ $$=$$ $$8 \beta_{7} - 11 \beta_{6} + 3 \beta_{3} + 11$$ $$\nu^{4}$$ $$=$$ $$17 \beta_{7} - 28 \beta_{6} + 46 \beta_{5} - 16 \beta_{4} + 28 \beta_{2} + 17 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$-45 \beta_{6} + 75 \beta_{5} - 45 \beta_{4} + 45 \beta_{3} - 49 \beta_{2} - 61 \beta_{1} + 75$$ $$\nu^{6}$$ $$=$$ $$226 \beta_{7} - 210 \beta_{6} - 110 \beta_{3} - 145$$ $$\nu^{7}$$ $$=$$ $$-365 \beta_{7} + 155 \beta_{6} + 1098 \beta_{5} - 546 \beta_{4} - 155 \beta_{2} - 365 \beta_{1} - 365$$

## Character values

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

 $$n$$ $$29$$ $$31$$ $$\chi(n)$$ $$-1$$ $$\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −2.41089 + 1.96928i 0.461396 + 0.310963i 0.0386042 + 0.555062i 2.91089 − 1.10325i −2.41089 − 1.96928i 0.461396 − 0.310963i 0.0386042 − 0.555062i 2.91089 + 1.10325i
−1.22474 + 0.707107i −2.24637 + 1.98842i 1.00000 1.73205i −5.32177 + 3.07253i 1.34521 4.02373i −4.69042 + 5.19615i 2.82843i 1.09238 8.93346i 4.34521 7.52612i
11.2 −1.22474 + 0.707107i 2.97112 0.415287i 1.00000 1.73205i 0.422792 0.244099i −3.34521 + 2.60952i 4.69042 + 5.19615i 2.82843i 8.65507 2.46773i −0.345208 + 0.597918i
11.3 1.22474 0.707107i −1.12591 2.78071i 1.00000 1.73205i −0.422792 + 0.244099i −3.34521 2.60952i 4.69042 + 5.19615i 2.82843i −6.46466 + 6.26165i −0.345208 + 0.597918i
11.4 1.22474 0.707107i −0.598836 + 2.93963i 1.00000 1.73205i 5.32177 3.07253i 1.34521 + 4.02373i −4.69042 + 5.19615i 2.82843i −8.28279 3.52071i 4.34521 7.52612i
23.1 −1.22474 0.707107i −2.24637 1.98842i 1.00000 + 1.73205i −5.32177 3.07253i 1.34521 + 4.02373i −4.69042 5.19615i 2.82843i 1.09238 + 8.93346i 4.34521 + 7.52612i
23.2 −1.22474 0.707107i 2.97112 + 0.415287i 1.00000 + 1.73205i 0.422792 + 0.244099i −3.34521 2.60952i 4.69042 5.19615i 2.82843i 8.65507 + 2.46773i −0.345208 0.597918i
23.3 1.22474 + 0.707107i −1.12591 + 2.78071i 1.00000 + 1.73205i −0.422792 0.244099i −3.34521 + 2.60952i 4.69042 5.19615i 2.82843i −6.46466 6.26165i −0.345208 0.597918i
23.4 1.22474 + 0.707107i −0.598836 2.93963i 1.00000 + 1.73205i 5.32177 + 3.07253i 1.34521 4.02373i −4.69042 5.19615i 2.82843i −8.28279 + 3.52071i 4.34521 + 7.52612i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 23.4 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
7.c even 3 1 inner
21.h 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.h.b 8
3.b odd 2 1 inner 42.3.h.b 8
4.b odd 2 1 336.3.bn.g 8
7.b odd 2 1 294.3.h.h 8
7.c even 3 1 inner 42.3.h.b 8
7.c even 3 1 294.3.b.i 4
7.d odd 6 1 294.3.b.e 4
7.d odd 6 1 294.3.h.h 8
12.b even 2 1 336.3.bn.g 8
21.c even 2 1 294.3.h.h 8
21.g even 6 1 294.3.b.e 4
21.g even 6 1 294.3.h.h 8
21.h odd 6 1 inner 42.3.h.b 8
21.h odd 6 1 294.3.b.i 4
28.g odd 6 1 336.3.bn.g 8
84.n even 6 1 336.3.bn.g 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.3.h.b 8 1.a even 1 1 trivial
42.3.h.b 8 3.b odd 2 1 inner
42.3.h.b 8 7.c even 3 1 inner
42.3.h.b 8 21.h odd 6 1 inner
294.3.b.e 4 7.d odd 6 1
294.3.b.e 4 21.g even 6 1
294.3.b.i 4 7.c even 3 1
294.3.b.i 4 21.h odd 6 1
294.3.h.h 8 7.b odd 2 1
294.3.h.h 8 7.d odd 6 1
294.3.h.h 8 21.c even 2 1
294.3.h.h 8 21.g even 6 1
336.3.bn.g 8 4.b odd 2 1
336.3.bn.g 8 12.b even 2 1
336.3.bn.g 8 28.g odd 6 1
336.3.bn.g 8 84.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 38 T_{5}^{6} + 1435 T_{5}^{4} - 342 T_{5}^{2} + 81$$ acting on $$S_{3}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 2 T^{2} + 4 T^{4} )^{2}$$
$3$ $$1 + 2 T + 7 T^{2} - 42 T^{3} - 108 T^{4} - 378 T^{5} + 567 T^{6} + 1458 T^{7} + 6561 T^{8}$$
$5$ $$1 + 62 T^{2} + 1985 T^{4} + 37758 T^{6} + 662756 T^{8} + 23598750 T^{10} + 775390625 T^{12} + 15136718750 T^{14} + 152587890625 T^{16}$$
$7$ $$( 1 + 10 T^{2} + 2401 T^{4} )^{2}$$
$11$ $$1 - 70 T^{2} - 23407 T^{4} + 68250 T^{6} + 515186468 T^{8} + 999248250 T^{10} - 5017498327567 T^{12} - 219689986370470 T^{14} + 45949729863572161 T^{16}$$
$13$ $$( 1 + 16 T + 314 T^{2} + 2704 T^{3} + 28561 T^{4} )^{4}$$
$17$ $$1 + 1118 T^{2} + 770753 T^{4} + 348960222 T^{6} + 118234187396 T^{8} + 29145506701662 T^{10} + 5376585974923073 T^{12} + 651371661222872798 T^{14} + 48661191875666868481 T^{16}$$
$19$ $$( 1 - 2 T - 521 T^{2} + 394 T^{3} + 143860 T^{4} + 142234 T^{5} - 67897241 T^{6} - 94091762 T^{7} + 16983563041 T^{8} )^{2}$$
$23$ $$1 + 1370 T^{2} + 955793 T^{4} + 495152250 T^{6} + 244893547268 T^{8} + 138563900792250 T^{10} + 74849091554682833 T^{12} + 30023035471867839770 T^{14} +$$$$61\!\cdots\!61$$$$T^{16}$$
$29$ $$( 1 - 764 T^{2} + 680486 T^{4} - 540362684 T^{6} + 500246412961 T^{8} )^{2}$$
$31$ $$( 1 - 10 T - 1297 T^{2} + 5250 T^{3} + 931988 T^{4} + 5045250 T^{5} - 1197806737 T^{6} - 8875036810 T^{7} + 852891037441 T^{8} )^{2}$$
$37$ $$( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{4}$$
$41$ $$( 1 - 5788 T^{2} + 13912710 T^{4} - 16355504668 T^{6} + 7984925229121 T^{8} )^{2}$$
$43$ $$( 1 - 52 T + 3582 T^{2} - 96148 T^{3} + 3418801 T^{4} )^{4}$$
$47$ $$1 + 538 T^{2} - 7938479 T^{4} - 823914182 T^{6} + 42475035844804 T^{8} - 4020438379535942 T^{10} -$$$$18\!\cdots\!19$$$$T^{12} +$$$$62\!\cdots\!58$$$$T^{14} +$$$$56\!\cdots\!21$$$$T^{16}$$
$53$ $$1 + 5854 T^{2} + 10947457 T^{4} + 44144411038 T^{6} + 211248030023524 T^{8} + 348320636551529278 T^{10} +$$$$68\!\cdots\!77$$$$T^{12} +$$$$28\!\cdots\!14$$$$T^{14} +$$$$38\!\cdots\!21$$$$T^{16}$$
$59$ $$1 + 746 T^{2} - 10773535 T^{4} - 9626884566 T^{6} - 25203887215996 T^{8} - 116652435591550326 T^{10} -$$$$15\!\cdots\!35$$$$T^{12} +$$$$13\!\cdots\!26$$$$T^{14} +$$$$21\!\cdots\!41$$$$T^{16}$$
$61$ $$( 1 - 106 T + 1337 T^{2} - 260442 T^{3} + 42335204 T^{4} - 969104682 T^{5} + 18511889417 T^{6} - 5461159682266 T^{7} + 191707312997281 T^{8} )^{2}$$
$67$ $$( 1 - 78 T - 697 T^{2} + 171366 T^{3} - 1480236 T^{4} + 769261974 T^{5} - 14045331337 T^{6} - 7055753809182 T^{7} + 406067677556641 T^{8} )^{2}$$
$71$ $$( 1 - 1316 T^{2} + 44745734 T^{4} - 33441772196 T^{6} + 645753531245761 T^{8} )^{2}$$
$73$ $$( 1 - 66 T - 7039 T^{2} - 48642 T^{3} + 78234660 T^{4} - 259213218 T^{5} - 199895218399 T^{6} - 9988058935074 T^{7} + 806460091894081 T^{8} )^{2}$$
$79$ $$( 1 + 26 T - 5617 T^{2} - 160914 T^{3} - 3567148 T^{4} - 1004264274 T^{5} - 218782604977 T^{6} + 6320273843546 T^{7} + 1517108809906561 T^{8} )^{2}$$
$83$ $$( 1 - 18404 T^{2} + 168689894 T^{4} - 873422939684 T^{6} + 2252292232139041 T^{8} )^{2}$$
$89$ $$1 + 2542 T^{2} + 3560113 T^{4} - 311605556402 T^{6} - 4333594781349404 T^{8} - 19550830916713376882 T^{10} +$$$$14\!\cdots\!53$$$$T^{12} +$$$$62\!\cdots\!82$$$$T^{14} +$$$$15\!\cdots\!61$$$$T^{16}$$
$97$ $$( 1 - 144 T + 21802 T^{2} - 1354896 T^{3} + 88529281 T^{4} )^{4}$$