Properties

Label 6.7.b.a
Level 6
Weight 7
Character orbit 6.b
Analytic conductor 1.380
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 6 = 2 \cdot 3 \)
Weight: \( k \) = \( 7 \)
Character orbit: \([\chi]\) = 6.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.38032450172\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( + ( 21 + 3 \beta ) q^{3} \) \( -32 q^{4} \) \( -30 \beta q^{5} \) \( + ( -96 + 21 \beta ) q^{6} \) \( + 2 q^{7} \) \( -32 \beta q^{8} \) \( + ( 153 + 126 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( + ( 21 + 3 \beta ) q^{3} \) \( -32 q^{4} \) \( -30 \beta q^{5} \) \( + ( -96 + 21 \beta ) q^{6} \) \( + 2 q^{7} \) \( -32 \beta q^{8} \) \( + ( 153 + 126 \beta ) q^{9} \) \( + 960 q^{10} \) \( -6 \beta q^{11} \) \( + ( -672 - 96 \beta ) q^{12} \) \( -2950 q^{13} \) \( + 2 \beta q^{14} \) \( + ( 2880 - 630 \beta ) q^{15} \) \( + 1024 q^{16} \) \( + 792 \beta q^{17} \) \( + ( -4032 + 153 \beta ) q^{18} \) \( + 5258 q^{19} \) \( + 960 \beta q^{20} \) \( + ( 42 + 6 \beta ) q^{21} \) \( + 192 q^{22} \) \( -1812 \beta q^{23} \) \( + ( 3072 - 672 \beta ) q^{24} \) \( -13175 q^{25} \) \( -2950 \beta q^{26} \) \( + ( -8883 + 3105 \beta ) q^{27} \) \( -64 q^{28} \) \( + 390 \beta q^{29} \) \( + ( 20160 + 2880 \beta ) q^{30} \) \( + 22898 q^{31} \) \( + 1024 \beta q^{32} \) \( + ( 576 - 126 \beta ) q^{33} \) \( -25344 q^{34} \) \( -60 \beta q^{35} \) \( + ( -4896 - 4032 \beta ) q^{36} \) \( + 34058 q^{37} \) \( + 5258 \beta q^{38} \) \( + ( -61950 - 8850 \beta ) q^{39} \) \( -30720 q^{40} \) \( -2964 \beta q^{41} \) \( + ( -192 + 42 \beta ) q^{42} \) \( -6406 q^{43} \) \( + 192 \beta q^{44} \) \( + ( 120960 - 4590 \beta ) q^{45} \) \( + 57984 q^{46} \) \( + 31800 \beta q^{47} \) \( + ( 21504 + 3072 \beta ) q^{48} \) \( -117645 q^{49} \) \( -13175 \beta q^{50} \) \( + ( -76032 + 16632 \beta ) q^{51} \) \( + 94400 q^{52} \) \( -34038 \beta q^{53} \) \( + ( -99360 - 8883 \beta ) q^{54} \) \( -5760 q^{55} \) \( -64 \beta q^{56} \) \( + ( 110418 + 15774 \beta ) q^{57} \) \( -12480 q^{58} \) \( -57774 \beta q^{59} \) \( + ( -92160 + 20160 \beta ) q^{60} \) \( -62566 q^{61} \) \( + 22898 \beta q^{62} \) \( + ( 306 + 252 \beta ) q^{63} \) \( -32768 q^{64} \) \( + 88500 \beta q^{65} \) \( + ( 4032 + 576 \beta ) q^{66} \) \( + 438698 q^{67} \) \( -25344 \beta q^{68} \) \( + ( 173952 - 38052 \beta ) q^{69} \) \( + 1920 q^{70} \) \( -12060 \beta q^{71} \) \( + ( 129024 - 4896 \beta ) q^{72} \) \( -730510 q^{73} \) \( + 34058 \beta q^{74} \) \( + ( -276675 - 39525 \beta ) q^{75} \) \( -168256 q^{76} \) \( -12 \beta q^{77} \) \( + ( 283200 - 61950 \beta ) q^{78} \) \( + 340562 q^{79} \) \( -30720 \beta q^{80} \) \( + ( -484623 + 38556 \beta ) q^{81} \) \( + 94848 q^{82} \) \( + 87726 \beta q^{83} \) \( + ( -1344 - 192 \beta ) q^{84} \) \( + 760320 q^{85} \) \( -6406 \beta q^{86} \) \( + ( -37440 + 8190 \beta ) q^{87} \) \( -6144 q^{88} \) \( -68364 \beta q^{89} \) \( + ( 146880 + 120960 \beta ) q^{90} \) \( -5900 q^{91} \) \( + 57984 \beta q^{92} \) \( + ( 480858 + 68694 \beta ) q^{93} \) \( -1017600 q^{94} \) \( -157740 \beta q^{95} \) \( + ( -98304 + 21504 \beta ) q^{96} \) \( -281086 q^{97} \) \( -117645 \beta q^{98} \) \( + ( 24192 - 918 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 42q^{3} \) \(\mathstrut -\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 192q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 306q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 42q^{3} \) \(\mathstrut -\mathstrut 64q^{4} \) \(\mathstrut -\mathstrut 192q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut +\mathstrut 306q^{9} \) \(\mathstrut +\mathstrut 1920q^{10} \) \(\mathstrut -\mathstrut 1344q^{12} \) \(\mathstrut -\mathstrut 5900q^{13} \) \(\mathstrut +\mathstrut 5760q^{15} \) \(\mathstrut +\mathstrut 2048q^{16} \) \(\mathstrut -\mathstrut 8064q^{18} \) \(\mathstrut +\mathstrut 10516q^{19} \) \(\mathstrut +\mathstrut 84q^{21} \) \(\mathstrut +\mathstrut 384q^{22} \) \(\mathstrut +\mathstrut 6144q^{24} \) \(\mathstrut -\mathstrut 26350q^{25} \) \(\mathstrut -\mathstrut 17766q^{27} \) \(\mathstrut -\mathstrut 128q^{28} \) \(\mathstrut +\mathstrut 40320q^{30} \) \(\mathstrut +\mathstrut 45796q^{31} \) \(\mathstrut +\mathstrut 1152q^{33} \) \(\mathstrut -\mathstrut 50688q^{34} \) \(\mathstrut -\mathstrut 9792q^{36} \) \(\mathstrut +\mathstrut 68116q^{37} \) \(\mathstrut -\mathstrut 123900q^{39} \) \(\mathstrut -\mathstrut 61440q^{40} \) \(\mathstrut -\mathstrut 384q^{42} \) \(\mathstrut -\mathstrut 12812q^{43} \) \(\mathstrut +\mathstrut 241920q^{45} \) \(\mathstrut +\mathstrut 115968q^{46} \) \(\mathstrut +\mathstrut 43008q^{48} \) \(\mathstrut -\mathstrut 235290q^{49} \) \(\mathstrut -\mathstrut 152064q^{51} \) \(\mathstrut +\mathstrut 188800q^{52} \) \(\mathstrut -\mathstrut 198720q^{54} \) \(\mathstrut -\mathstrut 11520q^{55} \) \(\mathstrut +\mathstrut 220836q^{57} \) \(\mathstrut -\mathstrut 24960q^{58} \) \(\mathstrut -\mathstrut 184320q^{60} \) \(\mathstrut -\mathstrut 125132q^{61} \) \(\mathstrut +\mathstrut 612q^{63} \) \(\mathstrut -\mathstrut 65536q^{64} \) \(\mathstrut +\mathstrut 8064q^{66} \) \(\mathstrut +\mathstrut 877396q^{67} \) \(\mathstrut +\mathstrut 347904q^{69} \) \(\mathstrut +\mathstrut 3840q^{70} \) \(\mathstrut +\mathstrut 258048q^{72} \) \(\mathstrut -\mathstrut 1461020q^{73} \) \(\mathstrut -\mathstrut 553350q^{75} \) \(\mathstrut -\mathstrut 336512q^{76} \) \(\mathstrut +\mathstrut 566400q^{78} \) \(\mathstrut +\mathstrut 681124q^{79} \) \(\mathstrut -\mathstrut 969246q^{81} \) \(\mathstrut +\mathstrut 189696q^{82} \) \(\mathstrut -\mathstrut 2688q^{84} \) \(\mathstrut +\mathstrut 1520640q^{85} \) \(\mathstrut -\mathstrut 74880q^{87} \) \(\mathstrut -\mathstrut 12288q^{88} \) \(\mathstrut +\mathstrut 293760q^{90} \) \(\mathstrut -\mathstrut 11800q^{91} \) \(\mathstrut +\mathstrut 961716q^{93} \) \(\mathstrut -\mathstrut 2035200q^{94} \) \(\mathstrut -\mathstrut 196608q^{96} \) \(\mathstrut -\mathstrut 562172q^{97} \) \(\mathstrut +\mathstrut 48384q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
5.65685i 21.0000 16.9706i −32.0000 169.706i −96.0000 118.794i 2.00000 181.019i 153.000 712.764i 960.000
5.2 5.65685i 21.0000 + 16.9706i −32.0000 169.706i −96.0000 + 118.794i 2.00000 181.019i 153.000 + 712.764i 960.000
\(n\): 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 \(S_{7}^{\mathrm{new}}(6, [\chi])\).