# Properties

 Label 42.3.h.a Level $42$ Weight $3$ Character orbit 42.h Analytic conductor $1.144$ Analytic rank $0$ Dimension $4$ 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: $$4$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} -6 \beta_{1} q^{5} + ( 4 + \beta_{3} ) q^{6} + ( -7 + 7 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + ( 7 + 4 \beta_{1} - 7 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 2 \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} -6 \beta_{1} q^{5} + ( 4 + \beta_{3} ) q^{6} + ( -7 + 7 \beta_{2} ) q^{7} + 2 \beta_{3} q^{8} + ( 7 + 4 \beta_{1} - 7 \beta_{2} ) q^{9} -12 \beta_{2} q^{10} + ( -2 + 4 \beta_{1} + 2 \beta_{2} ) q^{12} - q^{13} + ( -7 \beta_{1} + 7 \beta_{3} ) q^{14} + ( -24 - 6 \beta_{3} ) q^{15} + ( -4 + 4 \beta_{2} ) q^{16} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{17} + ( 7 \beta_{1} + 8 \beta_{2} - 7 \beta_{3} ) q^{18} + ( 31 - 31 \beta_{2} ) q^{19} -12 \beta_{3} q^{20} + ( -7 + 14 \beta_{3} ) q^{21} + 6 \beta_{1} q^{23} + ( -2 \beta_{1} + 8 \beta_{2} + 2 \beta_{3} ) q^{24} + 47 \beta_{2} q^{25} -\beta_{1} q^{26} + ( 23 - 10 \beta_{3} ) q^{27} -14 q^{28} + 12 \beta_{3} q^{29} + ( 12 - 24 \beta_{1} - 12 \beta_{2} ) q^{30} + 7 \beta_{2} q^{31} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{32} -12 q^{34} + ( 42 \beta_{1} - 42 \beta_{3} ) q^{35} + ( 14 + 8 \beta_{3} ) q^{36} + ( 1 - \beta_{2} ) q^{37} + ( 31 \beta_{1} - 31 \beta_{3} ) q^{38} + ( -2 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{39} + ( 24 - 24 \beta_{2} ) q^{40} + 24 \beta_{3} q^{41} + ( -28 - 7 \beta_{1} + 28 \beta_{2} ) q^{42} -31 q^{43} + ( -42 \beta_{1} - 48 \beta_{2} + 42 \beta_{3} ) q^{45} + 12 \beta_{2} q^{46} -30 \beta_{1} q^{47} + ( -4 + 8 \beta_{3} ) q^{48} -49 \beta_{2} q^{49} + 47 \beta_{3} q^{50} + ( -24 - 6 \beta_{1} + 24 \beta_{2} ) q^{51} -2 \beta_{2} q^{52} + ( 18 \beta_{1} - 18 \beta_{3} ) q^{53} + ( 20 + 23 \beta_{1} - 20 \beta_{2} ) q^{54} -14 \beta_{1} q^{56} + ( 31 - 62 \beta_{3} ) q^{57} + ( -24 + 24 \beta_{2} ) q^{58} + ( -6 \beta_{1} + 6 \beta_{3} ) q^{59} + ( 12 \beta_{1} - 48 \beta_{2} - 12 \beta_{3} ) q^{60} + ( -50 + 50 \beta_{2} ) q^{61} + 7 \beta_{3} q^{62} + ( -28 \beta_{1} + 49 \beta_{2} + 28 \beta_{3} ) q^{63} -8 q^{64} + 6 \beta_{1} q^{65} -65 \beta_{2} q^{67} -12 \beta_{1} q^{68} + ( 24 + 6 \beta_{3} ) q^{69} + 84 q^{70} -42 \beta_{3} q^{71} + ( -16 + 14 \beta_{1} + 16 \beta_{2} ) q^{72} + 97 \beta_{2} q^{73} + ( \beta_{1} - \beta_{3} ) q^{74} + ( -47 + 94 \beta_{1} + 47 \beta_{2} ) q^{75} + 62 q^{76} + ( -4 - \beta_{3} ) q^{78} + ( 103 - 103 \beta_{2} ) q^{79} + ( 24 \beta_{1} - 24 \beta_{3} ) q^{80} + ( 56 \beta_{1} - 17 \beta_{2} - 56 \beta_{3} ) q^{81} + ( -48 + 48 \beta_{2} ) q^{82} + 30 \beta_{3} q^{83} + ( -28 \beta_{1} - 14 \beta_{2} + 28 \beta_{3} ) q^{84} + 72 q^{85} -31 \beta_{1} q^{86} + ( -12 \beta_{1} + 48 \beta_{2} + 12 \beta_{3} ) q^{87} -84 \beta_{1} q^{89} + ( -84 - 48 \beta_{3} ) q^{90} + ( 7 - 7 \beta_{2} ) q^{91} + 12 \beta_{3} q^{92} + ( -7 + 14 \beta_{1} + 7 \beta_{2} ) q^{93} -60 \beta_{2} q^{94} + ( -186 \beta_{1} + 186 \beta_{3} ) q^{95} + ( -16 - 4 \beta_{1} + 16 \beta_{2} ) q^{96} -166 q^{97} -49 \beta_{3} q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 2q^{3} + 4q^{4} + 16q^{6} - 14q^{7} + 14q^{9} + O(q^{10})$$ $$4q + 2q^{3} + 4q^{4} + 16q^{6} - 14q^{7} + 14q^{9} - 24q^{10} - 4q^{12} - 4q^{13} - 96q^{15} - 8q^{16} + 16q^{18} + 62q^{19} - 28q^{21} + 16q^{24} + 94q^{25} + 92q^{27} - 56q^{28} + 24q^{30} + 14q^{31} - 48q^{34} + 56q^{36} + 2q^{37} - 2q^{39} + 48q^{40} - 56q^{42} - 124q^{43} - 96q^{45} + 24q^{46} - 16q^{48} - 98q^{49} - 48q^{51} - 4q^{52} + 40q^{54} + 124q^{57} - 48q^{58} - 96q^{60} - 100q^{61} + 98q^{63} - 32q^{64} - 130q^{67} + 96q^{69} + 336q^{70} - 32q^{72} + 194q^{73} - 94q^{75} + 248q^{76} - 16q^{78} + 206q^{79} - 34q^{81} - 96q^{82} - 28q^{84} + 288q^{85} + 96q^{87} - 336q^{90} + 14q^{91} - 14q^{93} - 120q^{94} - 32q^{96} - 664q^{97} + O(q^{100})$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −1.22474 + 0.707107i 1.22474 − 0.707107i −1.22474 − 0.707107i 1.22474 + 0.707107i
−1.22474 + 0.707107i −1.94949 2.28024i 1.00000 1.73205i 7.34847 4.24264i 4.00000 + 1.41421i −3.50000 6.06218i 2.82843i −1.39898 + 8.89060i −6.00000 + 10.3923i
11.2 1.22474 0.707107i 2.94949 + 0.548188i 1.00000 1.73205i −7.34847 + 4.24264i 4.00000 1.41421i −3.50000 6.06218i 2.82843i 8.39898 + 3.23375i −6.00000 + 10.3923i
23.1 −1.22474 0.707107i −1.94949 + 2.28024i 1.00000 + 1.73205i 7.34847 + 4.24264i 4.00000 1.41421i −3.50000 + 6.06218i 2.82843i −1.39898 8.89060i −6.00000 10.3923i
23.2 1.22474 + 0.707107i 2.94949 0.548188i 1.00000 + 1.73205i −7.34847 4.24264i 4.00000 + 1.41421i −3.50000 + 6.06218i 2.82843i 8.39898 3.23375i −6.00000 10.3923i
 $$n$$: 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
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.a 4
3.b odd 2 1 inner 42.3.h.a 4
4.b odd 2 1 336.3.bn.c 4
7.b odd 2 1 294.3.h.b 4
7.c even 3 1 inner 42.3.h.a 4
7.c even 3 1 294.3.b.b 2
7.d odd 6 1 294.3.b.c 2
7.d odd 6 1 294.3.h.b 4
12.b even 2 1 336.3.bn.c 4
21.c even 2 1 294.3.h.b 4
21.g even 6 1 294.3.b.c 2
21.g even 6 1 294.3.h.b 4
21.h odd 6 1 inner 42.3.h.a 4
21.h odd 6 1 294.3.b.b 2
28.g odd 6 1 336.3.bn.c 4
84.n even 6 1 336.3.bn.c 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{4} - 72 T_{5}^{2} + 5184$$ acting on $$S_{3}^{\mathrm{new}}(42, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 2 T^{2} + 4 T^{4}$$
$3$ $$1 - 2 T - 5 T^{2} - 18 T^{3} + 81 T^{4}$$
$5$ $$1 - 22 T^{2} - 141 T^{4} - 13750 T^{6} + 390625 T^{8}$$
$7$ $$( 1 + 7 T + 49 T^{2} )^{2}$$
$11$ $$( 1 - 11 T + 121 T^{2} )^{2}( 1 + 11 T + 121 T^{2} )^{2}$$
$13$ $$( 1 + T + 169 T^{2} )^{4}$$
$17$ $$1 + 506 T^{2} + 172515 T^{4} + 42261626 T^{6} + 6975757441 T^{8}$$
$19$ $$( 1 - 31 T + 600 T^{2} - 11191 T^{3} + 130321 T^{4} )^{2}$$
$23$ $$1 + 986 T^{2} + 692355 T^{4} + 275923226 T^{6} + 78310985281 T^{8}$$
$29$ $$( 1 - 1394 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$( 1 - 7 T - 912 T^{2} - 6727 T^{3} + 923521 T^{4} )^{2}$$
$37$ $$( 1 - T - 1368 T^{2} - 1369 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 - 2210 T^{2} + 2825761 T^{4} )^{2}$$
$43$ $$( 1 + 31 T + 1849 T^{2} )^{4}$$
$47$ $$1 + 2618 T^{2} + 1974243 T^{4} + 12775004858 T^{6} + 23811286661761 T^{8}$$
$53$ $$1 + 4970 T^{2} + 16810419 T^{4} + 39215690570 T^{6} + 62259690411361 T^{8}$$
$59$ $$1 + 6890 T^{2} + 35354739 T^{4} + 83488617290 T^{6} + 146830437604321 T^{8}$$
$61$ $$( 1 + 50 T - 1221 T^{2} + 186050 T^{3} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 + 65 T - 264 T^{2} + 291785 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 - 6554 T^{2} + 25411681 T^{4} )^{2}$$
$73$ $$( 1 - 143 T + 5329 T^{2} )^{2}( 1 + 46 T + 5329 T^{2} )^{2}$$
$79$ $$( 1 - 103 T + 4368 T^{2} - 642823 T^{3} + 38950081 T^{4} )^{2}$$
$83$ $$( 1 - 11978 T^{2} + 47458321 T^{4} )^{2}$$
$89$ $$1 + 1730 T^{2} - 59749341 T^{4} + 108544076930 T^{6} + 3936588805702081 T^{8}$$
$97$ $$( 1 + 166 T + 9409 T^{2} )^{4}$$