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 $2$

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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:

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

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