Properties

Label 6.5.b.a
Level 6
Weight 5
Character orbit 6.b
Analytic conductor 0.620
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 6.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.620219778503\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( -3 - 3 \beta ) q^{3} -8 q^{4} + 6 \beta q^{5} + ( 24 - 3 \beta ) q^{6} + 26 q^{7} -8 \beta q^{8} + ( -63 + 18 \beta ) q^{9} +O(q^{10})\) \( q + \beta q^{2} + ( -3 - 3 \beta ) q^{3} -8 q^{4} + 6 \beta q^{5} + ( 24 - 3 \beta ) q^{6} + 26 q^{7} -8 \beta q^{8} + ( -63 + 18 \beta ) q^{9} -48 q^{10} -42 \beta q^{11} + ( 24 + 24 \beta ) q^{12} + 50 q^{13} + 26 \beta q^{14} + ( 144 - 18 \beta ) q^{15} + 64 q^{16} + 72 \beta q^{17} + ( -144 - 63 \beta ) q^{18} -358 q^{19} -48 \beta q^{20} + ( -78 - 78 \beta ) q^{21} + 336 q^{22} + 132 \beta q^{23} + ( -192 + 24 \beta ) q^{24} + 337 q^{25} + 50 \beta q^{26} + ( 621 + 135 \beta ) q^{27} -208 q^{28} -510 \beta q^{29} + ( 144 + 144 \beta ) q^{30} -742 q^{31} + 64 \beta q^{32} + ( -1008 + 126 \beta ) q^{33} -576 q^{34} + 156 \beta q^{35} + ( 504 - 144 \beta ) q^{36} + 1874 q^{37} -358 \beta q^{38} + ( -150 - 150 \beta ) q^{39} + 384 q^{40} + 852 \beta q^{41} + ( 624 - 78 \beta ) q^{42} -262 q^{43} + 336 \beta q^{44} + ( -864 - 378 \beta ) q^{45} -1056 q^{46} -600 \beta q^{47} + ( -192 - 192 \beta ) q^{48} -1725 q^{49} + 337 \beta q^{50} + ( 1728 - 216 \beta ) q^{51} -400 q^{52} -162 \beta q^{53} + ( -1080 + 621 \beta ) q^{54} + 2016 q^{55} -208 \beta q^{56} + ( 1074 + 1074 \beta ) q^{57} + 4080 q^{58} -642 \beta q^{59} + ( -1152 + 144 \beta ) q^{60} -1486 q^{61} -742 \beta q^{62} + ( -1638 + 468 \beta ) q^{63} -512 q^{64} + 300 \beta q^{65} + ( -1008 - 1008 \beta ) q^{66} -4486 q^{67} -576 \beta q^{68} + ( 3168 - 396 \beta ) q^{69} -1248 q^{70} + 1260 \beta q^{71} + ( 1152 + 504 \beta ) q^{72} + 290 q^{73} + 1874 \beta q^{74} + ( -1011 - 1011 \beta ) q^{75} + 2864 q^{76} -1092 \beta q^{77} + ( 1200 - 150 \beta ) q^{78} + 9818 q^{79} + 384 \beta q^{80} + ( 1377 - 2268 \beta ) q^{81} -6816 q^{82} + 2514 \beta q^{83} + ( 624 + 624 \beta ) q^{84} -3456 q^{85} -262 \beta q^{86} + ( -12240 + 1530 \beta ) q^{87} -2688 q^{88} -2772 \beta q^{89} + ( 3024 - 864 \beta ) q^{90} + 1300 q^{91} -1056 \beta q^{92} + ( 2226 + 2226 \beta ) q^{93} + 4800 q^{94} -2148 \beta q^{95} + ( 1536 - 192 \beta ) q^{96} -478 q^{97} -1725 \beta q^{98} + ( 6048 + 2646 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 6q^{3} - 16q^{4} + 48q^{6} + 52q^{7} - 126q^{9} + O(q^{10}) \) \( 2q - 6q^{3} - 16q^{4} + 48q^{6} + 52q^{7} - 126q^{9} - 96q^{10} + 48q^{12} + 100q^{13} + 288q^{15} + 128q^{16} - 288q^{18} - 716q^{19} - 156q^{21} + 672q^{22} - 384q^{24} + 674q^{25} + 1242q^{27} - 416q^{28} + 288q^{30} - 1484q^{31} - 2016q^{33} - 1152q^{34} + 1008q^{36} + 3748q^{37} - 300q^{39} + 768q^{40} + 1248q^{42} - 524q^{43} - 1728q^{45} - 2112q^{46} - 384q^{48} - 3450q^{49} + 3456q^{51} - 800q^{52} - 2160q^{54} + 4032q^{55} + 2148q^{57} + 8160q^{58} - 2304q^{60} - 2972q^{61} - 3276q^{63} - 1024q^{64} - 2016q^{66} - 8972q^{67} + 6336q^{69} - 2496q^{70} + 2304q^{72} + 580q^{73} - 2022q^{75} + 5728q^{76} + 2400q^{78} + 19636q^{79} + 2754q^{81} - 13632q^{82} + 1248q^{84} - 6912q^{85} - 24480q^{87} - 5376q^{88} + 6048q^{90} + 2600q^{91} + 4452q^{93} + 9600q^{94} + 3072q^{96} - 956q^{97} + 12096q^{99} + O(q^{100}) \)

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
2.82843i −3.00000 + 8.48528i −8.00000 16.9706i 24.0000 + 8.48528i 26.0000 22.6274i −63.0000 50.9117i −48.0000
5.2 2.82843i −3.00000 8.48528i −8.00000 16.9706i 24.0000 8.48528i 26.0000 22.6274i −63.0000 + 50.9117i −48.0000
\(n\): 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 6.5.b.a 2
3.b odd 2 1 inner 6.5.b.a 2
4.b odd 2 1 48.5.e.b 2
5.b even 2 1 150.5.d.a 2
5.c odd 4 2 150.5.b.a 4
7.b odd 2 1 294.5.b.a 2
8.b even 2 1 192.5.e.d 2
8.d odd 2 1 192.5.e.c 2
9.c even 3 2 162.5.d.a 4
9.d odd 6 2 162.5.d.a 4
12.b even 2 1 48.5.e.b 2
15.d odd 2 1 150.5.d.a 2
15.e even 4 2 150.5.b.a 4
21.c even 2 1 294.5.b.a 2
24.f even 2 1 192.5.e.c 2
24.h odd 2 1 192.5.e.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.5.b.a 2 1.a even 1 1 trivial
6.5.b.a 2 3.b odd 2 1 inner
48.5.e.b 2 4.b odd 2 1
48.5.e.b 2 12.b even 2 1
150.5.b.a 4 5.c odd 4 2
150.5.b.a 4 15.e even 4 2
150.5.d.a 2 5.b even 2 1
150.5.d.a 2 15.d odd 2 1
162.5.d.a 4 9.c even 3 2
162.5.d.a 4 9.d odd 6 2
192.5.e.c 2 8.d odd 2 1
192.5.e.c 2 24.f even 2 1
192.5.e.d 2 8.b even 2 1
192.5.e.d 2 24.h odd 2 1
294.5.b.a 2 7.b odd 2 1
294.5.b.a 2 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 8 T^{2} \)
$3$ \( 1 + 6 T + 81 T^{2} \)
$5$ \( 1 - 962 T^{2} + 390625 T^{4} \)
$7$ \( ( 1 - 26 T + 2401 T^{2} )^{2} \)
$11$ \( 1 - 15170 T^{2} + 214358881 T^{4} \)
$13$ \( ( 1 - 50 T + 28561 T^{2} )^{2} \)
$17$ \( 1 - 125570 T^{2} + 6975757441 T^{4} \)
$19$ \( ( 1 + 358 T + 130321 T^{2} )^{2} \)
$23$ \( 1 - 420290 T^{2} + 78310985281 T^{4} \)
$29$ \( 1 + 666238 T^{2} + 500246412961 T^{4} \)
$31$ \( ( 1 + 742 T + 923521 T^{2} )^{2} \)
$37$ \( ( 1 - 1874 T + 1874161 T^{2} )^{2} \)
$41$ \( 1 + 155710 T^{2} + 7984925229121 T^{4} \)
$43$ \( ( 1 + 262 T + 3418801 T^{2} )^{2} \)
$47$ \( 1 - 6879362 T^{2} + 23811286661761 T^{4} \)
$53$ \( 1 - 15571010 T^{2} + 62259690411361 T^{4} \)
$59$ \( 1 - 20937410 T^{2} + 146830437604321 T^{4} \)
$61$ \( ( 1 + 1486 T + 13845841 T^{2} )^{2} \)
$67$ \( ( 1 + 4486 T + 20151121 T^{2} )^{2} \)
$71$ \( 1 - 38122562 T^{2} + 645753531245761 T^{4} \)
$73$ \( ( 1 - 290 T + 28398241 T^{2} )^{2} \)
$79$ \( ( 1 - 9818 T + 38950081 T^{2} )^{2} \)
$83$ \( 1 - 44355074 T^{2} + 2252292232139041 T^{4} \)
$89$ \( 1 - 64012610 T^{2} + 3936588805702081 T^{4} \)
$97$ \( ( 1 + 478 T + 88529281 T^{2} )^{2} \)
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