# Properties

 Label 6.9.b.a Level $6$ Weight $9$ Character orbit 6.b Analytic conductor $2.444$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6 = 2 \cdot 3$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 6.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.44427166037$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-2})$$ Defining polynomial: $$x^{2} + 2$$ 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 $$\beta = 4\sqrt{-2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \beta q^{2} + ( -63 + 9 \beta ) q^{3} -128 q^{4} + 102 \beta q^{5} + ( -576 - 126 \beta ) q^{6} + 2786 q^{7} -256 \beta q^{8} + ( 1377 - 1134 \beta ) q^{9} +O(q^{10})$$ $$q + 2 \beta q^{2} + ( -63 + 9 \beta ) q^{3} -128 q^{4} + 102 \beta q^{5} + ( -576 - 126 \beta ) q^{6} + 2786 q^{7} -256 \beta q^{8} + ( 1377 - 1134 \beta ) q^{9} -6528 q^{10} + 3966 \beta q^{11} + ( 8064 - 1152 \beta ) q^{12} -13150 q^{13} + 5572 \beta q^{14} + ( -29376 - 6426 \beta ) q^{15} + 16384 q^{16} -11736 \beta q^{17} + ( 72576 + 2754 \beta ) q^{18} + 144002 q^{19} -13056 \beta q^{20} + ( -175518 + 25074 \beta ) q^{21} -253824 q^{22} + 8724 \beta q^{23} + ( 73728 + 16128 \beta ) q^{24} + 57697 q^{25} -26300 \beta q^{26} + ( 239841 + 83835 \beta ) q^{27} -356608 q^{28} -110910 \beta q^{29} + ( 411264 - 58752 \beta ) q^{30} + 728738 q^{31} + 32768 \beta q^{32} + ( -1142208 - 249858 \beta ) q^{33} + 751104 q^{34} + 284172 \beta q^{35} + ( -176256 + 145152 \beta ) q^{36} -1964446 q^{37} + 288004 \beta q^{38} + ( 828450 - 118350 \beta ) q^{39} + 835584 q^{40} + 174324 \beta q^{41} + ( -1604736 - 351036 \beta ) q^{42} -78142 q^{43} -507648 \beta q^{44} + ( 3701376 + 140454 \beta ) q^{45} -558336 q^{46} -622200 \beta q^{47} + ( -1032192 + 147456 \beta ) q^{48} + 1996995 q^{49} + 115394 \beta q^{50} + ( 3379968 + 739368 \beta ) q^{51} + 1683200 q^{52} + 92286 \beta q^{53} + ( -5365440 + 479682 \beta ) q^{54} -12945024 q^{55} -713216 \beta q^{56} + ( -9072126 + 1296018 \beta ) q^{57} + 7098240 q^{58} -884634 \beta q^{59} + ( 3760128 + 822528 \beta ) q^{60} + 17578274 q^{61} + 1457476 \beta q^{62} + ( 3836322 - 3159324 \beta ) q^{63} -2097152 q^{64} -1341300 \beta q^{65} + ( 15990912 - 2284416 \beta ) q^{66} -17136766 q^{67} + 1502208 \beta q^{68} + ( -2512512 - 549612 \beta ) q^{69} -18187008 q^{70} + 4576860 \beta q^{71} + ( -9289728 - 352512 \beta ) q^{72} + 28139330 q^{73} -3928892 \beta q^{74} + ( -3634911 + 519273 \beta ) q^{75} -18432256 q^{76} + 11049276 \beta q^{77} + ( 7574400 + 1656900 \beta ) q^{78} + 9182498 q^{79} + 1671168 \beta q^{80} + ( -39254463 - 3123036 \beta ) q^{81} -11156736 q^{82} -15398742 \beta q^{83} + ( 22466304 - 3209472 \beta ) q^{84} + 38306304 q^{85} -156284 \beta q^{86} + ( 31942080 + 6987330 \beta ) q^{87} + 32489472 q^{88} -14363604 \beta q^{89} + ( -8989056 + 7402752 \beta ) q^{90} -36635900 q^{91} -1116672 \beta q^{92} + ( -45910494 + 6558642 \beta ) q^{93} + 39820800 q^{94} + 14688204 \beta q^{95} + ( -9437184 - 2064384 \beta ) q^{96} -128722558 q^{97} + 3993990 \beta q^{98} + ( 143918208 + 5461182 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 126q^{3} - 256q^{4} - 1152q^{6} + 5572q^{7} + 2754q^{9} + O(q^{10})$$ $$2q - 126q^{3} - 256q^{4} - 1152q^{6} + 5572q^{7} + 2754q^{9} - 13056q^{10} + 16128q^{12} - 26300q^{13} - 58752q^{15} + 32768q^{16} + 145152q^{18} + 288004q^{19} - 351036q^{21} - 507648q^{22} + 147456q^{24} + 115394q^{25} + 479682q^{27} - 713216q^{28} + 822528q^{30} + 1457476q^{31} - 2284416q^{33} + 1502208q^{34} - 352512q^{36} - 3928892q^{37} + 1656900q^{39} + 1671168q^{40} - 3209472q^{42} - 156284q^{43} + 7402752q^{45} - 1116672q^{46} - 2064384q^{48} + 3993990q^{49} + 6759936q^{51} + 3366400q^{52} - 10730880q^{54} - 25890048q^{55} - 18144252q^{57} + 14196480q^{58} + 7520256q^{60} + 35156548q^{61} + 7672644q^{63} - 4194304q^{64} + 31981824q^{66} - 34273532q^{67} - 5025024q^{69} - 36374016q^{70} - 18579456q^{72} + 56278660q^{73} - 7269822q^{75} - 36864512q^{76} + 15148800q^{78} + 18364996q^{79} - 78508926q^{81} - 22313472q^{82} + 44932608q^{84} + 76612608q^{85} + 63884160q^{87} + 64978944q^{88} - 17978112q^{90} - 73271800q^{91} - 91820988q^{93} + 79641600q^{94} - 18874368q^{96} - 257445116q^{97} + 287836416q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 − 1.41421i 1.41421i
11.3137i −63.0000 50.9117i −128.000 576.999i −576.000 + 712.764i 2786.00 1448.15i 1377.00 + 6414.87i −6528.00
5.2 11.3137i −63.0000 + 50.9117i −128.000 576.999i −576.000 712.764i 2786.00 1448.15i 1377.00 6414.87i −6528.00
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 6.9.b.a 2
3.b odd 2 1 inner 6.9.b.a 2
4.b odd 2 1 48.9.e.d 2
5.b even 2 1 150.9.d.a 2
5.c odd 4 2 150.9.b.a 4
8.b even 2 1 192.9.e.h 2
8.d odd 2 1 192.9.e.c 2
9.c even 3 2 162.9.d.a 4
9.d odd 6 2 162.9.d.a 4
12.b even 2 1 48.9.e.d 2
15.d odd 2 1 150.9.d.a 2
15.e even 4 2 150.9.b.a 4
24.f even 2 1 192.9.e.c 2
24.h odd 2 1 192.9.e.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.9.b.a 2 1.a even 1 1 trivial
6.9.b.a 2 3.b odd 2 1 inner
48.9.e.d 2 4.b odd 2 1
48.9.e.d 2 12.b even 2 1
150.9.b.a 4 5.c odd 4 2
150.9.b.a 4 15.e even 4 2
150.9.d.a 2 5.b even 2 1
150.9.d.a 2 15.d odd 2 1
162.9.d.a 4 9.c even 3 2
162.9.d.a 4 9.d odd 6 2
192.9.e.c 2 8.d odd 2 1
192.9.e.c 2 24.f even 2 1
192.9.e.h 2 8.b even 2 1
192.9.e.h 2 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(6, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$128 + T^{2}$$
$3$ $$6561 + 126 T + T^{2}$$
$5$ $$332928 + T^{2}$$
$7$ $$( -2786 + T )^{2}$$
$11$ $$503332992 + T^{2}$$
$13$ $$( 13150 + T )^{2}$$
$17$ $$4407478272 + T^{2}$$
$19$ $$( -144002 + T )^{2}$$
$23$ $$2435461632 + T^{2}$$
$29$ $$393632899200 + T^{2}$$
$31$ $$( -728738 + T )^{2}$$
$37$ $$( 1964446 + T )^{2}$$
$41$ $$972443423232 + T^{2}$$
$43$ $$( 78142 + T )^{2}$$
$47$ $$12388250880000 + T^{2}$$
$53$ $$272534585472 + T^{2}$$
$59$ $$25042474046592 + T^{2}$$
$61$ $$( -17578274 + T )^{2}$$
$67$ $$( 17136766 + T )^{2}$$
$71$ $$670324718707200 + T^{2}$$
$73$ $$( -28139330 + T )^{2}$$
$79$ $$( -9182498 + T )^{2}$$
$83$ $$7587880165842048 + T^{2}$$
$89$ $$6602019835802112 + T^{2}$$
$97$ $$( 128722558 + T )^{2}$$