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 \(2\) and degree \(1\))

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: \( 2 \)
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.

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

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
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. 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 \(S_{5}^{\mathrm{new}}(6, [\chi])\).