# Properties

 Label 43.3.b.b Level $43$ Weight $3$ Character orbit 43.b Analytic conductor $1.172$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$43$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 43.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.17166513675$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} + \cdots)$$ Defining polynomial: $$x^{6} + 20 x^{4} + 121 x^{2} + 214$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} + ( -2 + \beta_{3} + \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( 2 - \beta_{3} + 4 \beta_{4} ) q^{6} + ( -\beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{5} ) q^{8} + ( -9 - 4 \beta_{3} - 5 \beta_{4} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{5} q^{3} + ( -2 + \beta_{3} + \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( 2 - \beta_{3} + 4 \beta_{4} ) q^{6} + ( -\beta_{1} + \beta_{2} + \beta_{5} ) q^{7} + ( \beta_{2} + \beta_{5} ) q^{8} + ( -9 - 4 \beta_{3} - 5 \beta_{4} ) q^{9} + ( 2 + 8 \beta_{3} - \beta_{4} ) q^{10} + ( 7 + 3 \beta_{3} - \beta_{4} ) q^{11} + ( -\beta_{1} - \beta_{2} ) q^{12} + ( 5 - 5 \beta_{3} + 5 \beta_{4} ) q^{13} + ( 2 - 8 \beta_{3} - 4 \beta_{4} ) q^{14} + ( 2 + \beta_{3} - 9 \beta_{4} ) q^{15} + ( -12 - 3 \beta_{3} + \beta_{4} ) q^{16} + ( -5 - 5 \beta_{4} ) q^{17} + ( -4 \beta_{2} - 5 \beta_{5} ) q^{18} + ( -5 \beta_{1} + 5 \beta_{5} ) q^{19} + ( -5 \beta_{1} + 4 \beta_{2} - \beta_{5} ) q^{20} + ( 14 + 4 \beta_{3} + 10 \beta_{4} ) q^{21} + ( 5 \beta_{1} + 3 \beta_{2} - \beta_{5} ) q^{22} + ( -15 + 5 \beta_{3} - 10 \beta_{4} ) q^{23} + ( 16 + 3 \beta_{3} + 14 \beta_{4} ) q^{24} + ( -9 + 5 \beta_{3} + 10 \beta_{4} ) q^{25} + ( 5 \beta_{1} - 5 \beta_{2} + 5 \beta_{5} ) q^{26} + ( 6 \beta_{1} + 5 \beta_{2} + 9 \beta_{5} ) q^{27} + ( 10 \beta_{1} - 4 \beta_{2} ) q^{28} + ( -10 \beta_{1} - 5 \beta_{5} ) q^{29} + ( 10 \beta_{1} + \beta_{2} - 9 \beta_{5} ) q^{30} + ( -15 - 4 \beta_{3} + 15 \beta_{4} ) q^{31} + ( -10 \beta_{1} + \beta_{2} + 5 \beta_{5} ) q^{32} + ( 5 \beta_{1} + \beta_{2} - 9 \beta_{5} ) q^{33} -5 \beta_{5} q^{34} + ( 30 - 14 \beta_{3} ) q^{35} + ( -18 + 11 \beta_{3} - 4 \beta_{4} ) q^{36} + ( \beta_{1} - 5 \beta_{2} ) q^{37} + ( 20 - 25 \beta_{4} ) q^{38} + ( -15 \beta_{1} - 5 \beta_{2} - 5 \beta_{5} ) q^{39} + ( 32 - 6 \beta_{3} - \beta_{4} ) q^{40} + ( -23 + 11 \beta_{3} + 6 \beta_{4} ) q^{41} + ( 4 \beta_{2} + 10 \beta_{5} ) q^{42} + ( -5 + 10 \beta_{1} - \beta_{2} - 5 \beta_{3} - 15 \beta_{4} - 5 \beta_{5} ) q^{43} + ( -6 - 8 \beta_{3} + 8 \beta_{4} ) q^{44} + ( 19 \beta_{1} + 6 \beta_{5} ) q^{45} + ( -10 \beta_{1} + 5 \beta_{2} - 10 \beta_{5} ) q^{46} -15 \beta_{3} q^{47} + ( -5 \beta_{1} - \beta_{2} + 14 \beta_{5} ) q^{48} + ( 3 + 18 \beta_{3} - 6 \beta_{4} ) q^{49} + ( -24 \beta_{1} + 5 \beta_{2} + 10 \beta_{5} ) q^{50} + ( 10 \beta_{1} + 5 \beta_{2} + 10 \beta_{5} ) q^{51} + ( -10 + 30 \beta_{3} ) q^{52} + ( -25 - 25 \beta_{3} + 5 \beta_{4} ) q^{53} + ( -64 - 25 \beta_{3} - 25 \beta_{4} ) q^{54} + ( -19 \beta_{1} - 5 \beta_{2} + 5 \beta_{5} ) q^{55} + ( -44 + 10 \beta_{3} - 10 \beta_{4} ) q^{56} + ( 80 + 25 \beta_{3} + 5 \beta_{4} ) q^{57} + ( 70 - 15 \beta_{3} + 10 \beta_{4} ) q^{58} + ( 12 + 26 \beta_{3} + 16 \beta_{4} ) q^{59} + ( -36 - 3 \beta_{3} + 11 \beta_{4} ) q^{60} + ( -5 \beta_{1} + 5 \beta_{2} - 15 \beta_{5} ) q^{61} + ( -26 \beta_{1} - 4 \beta_{2} + 15 \beta_{5} ) q^{62} + ( -25 \beta_{1} - \beta_{2} - 19 \beta_{5} ) q^{63} + ( -25 \beta_{3} - 25 \beta_{4} ) q^{64} + ( 35 \beta_{1} - 5 \beta_{2} - 15 \beta_{5} ) q^{65} + ( -14 - 12 \beta_{3} + 42 \beta_{4} ) q^{66} + ( -25 - 15 \beta_{3} - 25 \beta_{4} ) q^{67} + ( -10 - 5 \beta_{3} ) q^{68} + ( 25 \beta_{1} + 10 \beta_{2} + 20 \beta_{5} ) q^{69} + ( 44 \beta_{1} - 14 \beta_{2} ) q^{70} + 20 \beta_{1} q^{71} + ( -25 \beta_{1} - 5 \beta_{2} - 24 \beta_{5} ) q^{72} + ( 15 \beta_{1} + 5 \beta_{2} - \beta_{5} ) q^{73} + ( 4 + 41 \beta_{3} - 4 \beta_{4} ) q^{74} + ( -15 \beta_{1} - 10 \beta_{2} - 6 \beta_{5} ) q^{75} + ( 25 \beta_{1} - 5 \beta_{5} ) q^{76} + ( 9 \beta_{1} + \beta_{2} + 5 \beta_{5} ) q^{77} + ( 110 + 20 \beta_{3} ) q^{78} + ( 32 - 2 \beta_{3} + 9 \beta_{4} ) q^{79} + ( 19 \beta_{1} + 10 \beta_{2} - 5 \beta_{5} ) q^{80} + ( 83 - 11 \beta_{3} + 69 \beta_{4} ) q^{81} + ( -40 \beta_{1} + 11 \beta_{2} + 6 \beta_{5} ) q^{82} + ( 5 + 25 \beta_{3} - 15 \beta_{4} ) q^{83} + ( 28 - 6 \beta_{3} + 4 \beta_{4} ) q^{84} + ( -5 \beta_{1} + 10 \beta_{5} ) q^{85} + ( -48 + 15 \beta_{1} - 5 \beta_{2} + 13 \beta_{3} + 29 \beta_{4} - 15 \beta_{5} ) q^{86} + ( -110 - 10 \beta_{3} - 65 \beta_{4} ) q^{87} + ( 14 \beta_{1} + 4 \beta_{2} + 4 \beta_{5} ) q^{88} + ( -15 \beta_{1} - 5 \beta_{2} + 5 \beta_{5} ) q^{89} + ( -126 + 25 \beta_{3} - 5 \beta_{4} ) q^{90} + ( -25 \beta_{1} + 15 \beta_{2} + 15 \beta_{5} ) q^{91} + ( 10 - 40 \beta_{3} - 5 \beta_{4} ) q^{92} + ( -34 \beta_{1} - 15 \beta_{2} + 4 \beta_{5} ) q^{93} + ( 15 \beta_{1} - 15 \beta_{2} ) q^{94} + ( -20 - 45 \beta_{3} + 50 \beta_{4} ) q^{95} + ( 68 + 29 \beta_{3} - 6 \beta_{4} ) q^{96} + ( -45 + 5 \beta_{3} + 50 \beta_{4} ) q^{97} + ( -9 \beta_{1} + 18 \beta_{2} - 6 \beta_{5} ) q^{98} + ( -91 - 15 \beta_{3} - 25 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 16q^{4} + 6q^{6} - 36q^{9} + O(q^{10})$$ $$6q - 16q^{4} + 6q^{6} - 36q^{9} - 2q^{10} + 38q^{11} + 30q^{13} + 36q^{14} + 28q^{15} - 68q^{16} - 20q^{17} + 56q^{21} - 80q^{23} + 62q^{24} - 84q^{25} - 112q^{31} + 208q^{35} - 122q^{36} + 170q^{38} + 206q^{40} - 172q^{41} + 10q^{43} - 36q^{44} + 30q^{47} - 6q^{49} - 120q^{52} - 110q^{53} - 284q^{54} - 264q^{56} + 420q^{57} + 430q^{58} - 12q^{59} - 232q^{60} + 100q^{64} - 144q^{66} - 70q^{67} - 50q^{68} - 50q^{74} + 620q^{78} + 178q^{79} + 382q^{81} + 10q^{83} + 172q^{84} - 372q^{86} - 510q^{87} - 796q^{90} + 150q^{92} - 130q^{95} + 362q^{96} - 380q^{97} - 466q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 20 x^{4} + 121 x^{2} + 214$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} - 11 \nu^{3} - 18 \nu$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{4} - 11 \nu^{2} - 22$$$$)/4$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 15 \nu^{2} + 46$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + 15 \nu^{3} + 50 \nu$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} + \beta_{3} - 6$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + \beta_{2} - 8 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$-11 \beta_{4} - 15 \beta_{3} + 44$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{5} - 15 \beta_{2} + 70 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/43\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
42.1
 − 3.18991i − 2.58315i − 1.77533i 1.77533i 2.58315i 3.18991i
3.18991i 0.724539i −6.17554 7.66434i 2.31122 10.1297i 6.93980i 8.47504 −24.4486
42.2 2.58315i 3.59415i −2.67267 7.02284i −9.28424 0.845536i 3.42869i −3.91793 18.1411
42.3 1.77533i 5.61757i 0.848217 2.98959i 9.97302 6.83184i 8.60717i −22.5571 5.30751
42.4 1.77533i 5.61757i 0.848217 2.98959i 9.97302 6.83184i 8.60717i −22.5571 5.30751
42.5 2.58315i 3.59415i −2.67267 7.02284i −9.28424 0.845536i 3.42869i −3.91793 18.1411
42.6 3.18991i 0.724539i −6.17554 7.66434i 2.31122 10.1297i 6.93980i 8.47504 −24.4486
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 42.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 43.3.b.b 6
3.b odd 2 1 387.3.b.c 6
4.b odd 2 1 688.3.b.e 6
43.b odd 2 1 inner 43.3.b.b 6
129.d even 2 1 387.3.b.c 6
172.d even 2 1 688.3.b.e 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.3.b.b 6 1.a even 1 1 trivial
43.3.b.b 6 43.b odd 2 1 inner
387.3.b.c 6 3.b odd 2 1
387.3.b.c 6 129.d even 2 1
688.3.b.e 6 4.b odd 2 1
688.3.b.e 6 172.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 20 T_{2}^{4} + 121 T_{2}^{2} + 214$$ acting on $$S_{3}^{\mathrm{new}}(43, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$214 + 121 T^{2} + 20 T^{4} + T^{6}$$
$3$ $$214 + 431 T^{2} + 45 T^{4} + T^{6}$$
$5$ $$25894 + 3863 T^{2} + 117 T^{4} + T^{6}$$
$7$ $$3424 + 4896 T^{2} + 150 T^{4} + T^{6}$$
$11$ $$( 244 + 48 T - 19 T^{2} + T^{3} )^{2}$$
$13$ $$( 3500 - 250 T - 15 T^{2} + T^{3} )^{2}$$
$17$ $$( -125 - 100 T + 10 T^{2} + T^{3} )^{2}$$
$19$ $$53500000 + 553125 T^{2} + 1475 T^{4} + T^{6}$$
$23$ $$( -14125 - 200 T + 40 T^{2} + T^{3} )^{2}$$
$29$ $$163843750 + 2859375 T^{2} + 3425 T^{4} + T^{6}$$
$31$ $$( -11041 - 292 T + 56 T^{2} + T^{3} )^{2}$$
$37$ $$316432384 + 2306261 T^{2} + 2955 T^{4} + T^{6}$$
$41$ $$( 2989 + 1408 T + 86 T^{2} + T^{3} )^{2}$$
$43$ $$6321363049 - 34188010 T + 271803 T^{2} + 135020 T^{3} + 147 T^{4} - 10 T^{5} + T^{6}$$
$47$ $$( -13500 - 1575 T - 15 T^{2} + T^{3} )^{2}$$
$53$ $$( -140500 - 3750 T + 55 T^{2} + T^{3} )^{2}$$
$59$ $$( 105424 - 6172 T + 6 T^{2} + T^{3} )^{2}$$
$61$ $$150281500000 + 85940000 T^{2} + 16150 T^{4} + T^{6}$$
$67$ $$( 36500 - 4450 T + 35 T^{2} + T^{3} )^{2}$$
$71$ $$13696000000 + 19360000 T^{2} + 8000 T^{4} + T^{6}$$
$73$ $$601120864 + 8323096 T^{2} + 7370 T^{4} + T^{6}$$
$79$ $$( -11236 + 2173 T - 89 T^{2} + T^{3} )^{2}$$
$83$ $$( -59500 - 5900 T - 5 T^{2} + T^{3} )^{2}$$
$89$ $$53500000 + 1835000 T^{2} + 8650 T^{4} + T^{6}$$
$97$ $$( -1046125 - 1400 T + 190 T^{2} + T^{3} )^{2}$$