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

Downloads

Learn more about

Newspace parameters

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

Newform invariants

Self dual: No
Analytic conductor: \(2.44427166037\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

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

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:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
3.b Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{9}^{\mathrm{new}}(6, [\chi])\).