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 \(2\), degree \(1\), minimal)

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: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( 21 + 3 \beta ) q^{3} -32 q^{4} -30 \beta q^{5} + ( -96 + 21 \beta ) q^{6} + 2 q^{7} -32 \beta q^{8} + ( 153 + 126 \beta ) q^{9} +O(q^{10})\) \( 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})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 42q^{3} - 64q^{4} - 192q^{6} + 4q^{7} + 306q^{9} + O(q^{10}) \) \( 2q + 42q^{3} - 64q^{4} - 192q^{6} + 4q^{7} + 306q^{9} + 1920q^{10} - 1344q^{12} - 5900q^{13} + 5760q^{15} + 2048q^{16} - 8064q^{18} + 10516q^{19} + 84q^{21} + 384q^{22} + 6144q^{24} - 26350q^{25} - 17766q^{27} - 128q^{28} + 40320q^{30} + 45796q^{31} + 1152q^{33} - 50688q^{34} - 9792q^{36} + 68116q^{37} - 123900q^{39} - 61440q^{40} - 384q^{42} - 12812q^{43} + 241920q^{45} + 115968q^{46} + 43008q^{48} - 235290q^{49} - 152064q^{51} + 188800q^{52} - 198720q^{54} - 11520q^{55} + 220836q^{57} - 24960q^{58} - 184320q^{60} - 125132q^{61} + 612q^{63} - 65536q^{64} + 8064q^{66} + 877396q^{67} + 347904q^{69} + 3840q^{70} + 258048q^{72} - 1461020q^{73} - 553350q^{75} - 336512q^{76} + 566400q^{78} + 681124q^{79} - 969246q^{81} + 189696q^{82} - 2688q^{84} + 1520640q^{85} - 74880q^{87} - 12288q^{88} + 293760q^{90} - 11800q^{91} + 961716q^{93} - 2035200q^{94} - 196608q^{96} - 562172q^{97} + 48384q^{99} + O(q^{100}) \)

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
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 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 6.7.b.a 2
3.b odd 2 1 inner 6.7.b.a 2
4.b odd 2 1 48.7.e.b 2
5.b even 2 1 150.7.d.a 2
5.c odd 4 2 150.7.b.a 4
7.b odd 2 1 294.7.b.a 2
8.b even 2 1 192.7.e.c 2
8.d odd 2 1 192.7.e.f 2
9.c even 3 2 162.7.d.b 4
9.d odd 6 2 162.7.d.b 4
12.b even 2 1 48.7.e.b 2
15.d odd 2 1 150.7.d.a 2
15.e even 4 2 150.7.b.a 4
21.c even 2 1 294.7.b.a 2
24.f even 2 1 192.7.e.f 2
24.h odd 2 1 192.7.e.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6.7.b.a 2 1.a even 1 1 trivial
6.7.b.a 2 3.b odd 2 1 inner
48.7.e.b 2 4.b odd 2 1
48.7.e.b 2 12.b even 2 1
150.7.b.a 4 5.c odd 4 2
150.7.b.a 4 15.e even 4 2
150.7.d.a 2 5.b even 2 1
150.7.d.a 2 15.d odd 2 1
162.7.d.b 4 9.c even 3 2
162.7.d.b 4 9.d odd 6 2
192.7.e.c 2 8.b even 2 1
192.7.e.c 2 24.h odd 2 1
192.7.e.f 2 8.d odd 2 1
192.7.e.f 2 24.f even 2 1
294.7.b.a 2 7.b odd 2 1
294.7.b.a 2 21.c even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(6, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 32 T^{2} \)
$3$ \( 1 - 42 T + 729 T^{2} \)
$5$ \( 1 - 2450 T^{2} + 244140625 T^{4} \)
$7$ \( ( 1 - 2 T + 117649 T^{2} )^{2} \)
$11$ \( 1 - 3541970 T^{2} + 3138428376721 T^{4} \)
$13$ \( ( 1 + 2950 T + 4826809 T^{2} )^{2} \)
$17$ \( 1 - 28202690 T^{2} + 582622237229761 T^{4} \)
$19$ \( ( 1 - 5258 T + 47045881 T^{2} )^{2} \)
$23$ \( 1 - 191004770 T^{2} + 21914624432020321 T^{4} \)
$29$ \( 1 - 1184779442 T^{2} + 353814783205469041 T^{4} \)
$31$ \( ( 1 - 22898 T + 887503681 T^{2} )^{2} \)
$37$ \( ( 1 - 34058 T + 2565726409 T^{2} )^{2} \)
$41$ \( 1 - 9219079010 T^{2} + 22563490300366186081 T^{4} \)
$43$ \( ( 1 + 6406 T + 6321363049 T^{2} )^{2} \)
$47$ \( 1 + 10801249342 T^{2} + \)\(11\!\cdots\!41\)\( T^{4} \)
$53$ \( 1 - 7253988050 T^{2} + \)\(49\!\cdots\!41\)\( T^{4} \)
$59$ \( 1 + 22449655150 T^{2} + \)\(17\!\cdots\!81\)\( T^{4} \)
$61$ \( ( 1 + 62566 T + 51520374361 T^{2} )^{2} \)
$67$ \( ( 1 - 438698 T + 90458382169 T^{2} )^{2} \)
$71$ \( 1 - 251546372642 T^{2} + \)\(16\!\cdots\!41\)\( T^{4} \)
$73$ \( ( 1 + 730510 T + 151334226289 T^{2} )^{2} \)
$79$ \( ( 1 - 340562 T + 243087455521 T^{2} )^{2} \)
$83$ \( 1 - 407613512306 T^{2} + \)\(10\!\cdots\!61\)\( T^{4} \)
$89$ \( 1 - 844406214050 T^{2} + \)\(24\!\cdots\!21\)\( T^{4} \)
$97$ \( ( 1 + 281086 T + 832972004929 T^{2} )^{2} \)
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