Properties

Label 11.6.a.a
Level 11
Weight 6
Character orbit 11.a
Self dual yes
Analytic conductor 1.764
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 11.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.76422201794\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q - 4q^{2} - 15q^{3} - 16q^{4} - 19q^{5} + 60q^{6} + 10q^{7} + 192q^{8} - 18q^{9} + O(q^{10}) \) \( q - 4q^{2} - 15q^{3} - 16q^{4} - 19q^{5} + 60q^{6} + 10q^{7} + 192q^{8} - 18q^{9} + 76q^{10} - 121q^{11} + 240q^{12} - 1148q^{13} - 40q^{14} + 285q^{15} - 256q^{16} + 686q^{17} + 72q^{18} - 384q^{19} + 304q^{20} - 150q^{21} + 484q^{22} + 3709q^{23} - 2880q^{24} - 2764q^{25} + 4592q^{26} + 3915q^{27} - 160q^{28} - 5424q^{29} - 1140q^{30} - 6443q^{31} - 5120q^{32} + 1815q^{33} - 2744q^{34} - 190q^{35} + 288q^{36} + 12063q^{37} + 1536q^{38} + 17220q^{39} - 3648q^{40} - 1528q^{41} + 600q^{42} - 4026q^{43} + 1936q^{44} + 342q^{45} - 14836q^{46} + 7168q^{47} + 3840q^{48} - 16707q^{49} + 11056q^{50} - 10290q^{51} + 18368q^{52} - 29862q^{53} - 15660q^{54} + 2299q^{55} + 1920q^{56} + 5760q^{57} + 21696q^{58} - 6461q^{59} - 4560q^{60} - 16980q^{61} + 25772q^{62} - 180q^{63} + 28672q^{64} + 21812q^{65} - 7260q^{66} + 29999q^{67} - 10976q^{68} - 55635q^{69} + 760q^{70} + 31023q^{71} - 3456q^{72} + 1924q^{73} - 48252q^{74} + 41460q^{75} + 6144q^{76} - 1210q^{77} - 68880q^{78} + 65138q^{79} + 4864q^{80} - 54351q^{81} + 6112q^{82} - 102714q^{83} + 2400q^{84} - 13034q^{85} + 16104q^{86} + 81360q^{87} - 23232q^{88} + 17415q^{89} - 1368q^{90} - 11480q^{91} - 59344q^{92} + 96645q^{93} - 28672q^{94} + 7296q^{95} + 76800q^{96} + 66905q^{97} + 66828q^{98} + 2178q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−4.00000 −15.0000 −16.0000 −19.0000 60.0000 10.0000 192.000 −18.0000 76.0000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.6.a.a 1
3.b odd 2 1 99.6.a.c 1
4.b odd 2 1 176.6.a.c 1
5.b even 2 1 275.6.a.a 1
5.c odd 4 2 275.6.b.a 2
7.b odd 2 1 539.6.a.c 1
8.b even 2 1 704.6.a.h 1
8.d odd 2 1 704.6.a.c 1
11.b odd 2 1 121.6.a.b 1
33.d even 2 1 1089.6.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.6.a.a 1 1.a even 1 1 trivial
99.6.a.c 1 3.b odd 2 1
121.6.a.b 1 11.b odd 2 1
176.6.a.c 1 4.b odd 2 1
275.6.a.a 1 5.b even 2 1
275.6.b.a 2 5.c odd 4 2
539.6.a.c 1 7.b odd 2 1
704.6.a.c 1 8.d odd 2 1
704.6.a.h 1 8.b even 2 1
1089.6.a.c 1 33.d even 2 1

Atkin-Lehner signs

\( p \) Sign
\(11\) \(1\)

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 4 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(11))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 4 T + 32 T^{2} \)
$3$ \( 1 + 15 T + 243 T^{2} \)
$5$ \( 1 + 19 T + 3125 T^{2} \)
$7$ \( 1 - 10 T + 16807 T^{2} \)
$11$ \( 1 + 121 T \)
$13$ \( 1 + 1148 T + 371293 T^{2} \)
$17$ \( 1 - 686 T + 1419857 T^{2} \)
$19$ \( 1 + 384 T + 2476099 T^{2} \)
$23$ \( 1 - 3709 T + 6436343 T^{2} \)
$29$ \( 1 + 5424 T + 20511149 T^{2} \)
$31$ \( 1 + 6443 T + 28629151 T^{2} \)
$37$ \( 1 - 12063 T + 69343957 T^{2} \)
$41$ \( 1 + 1528 T + 115856201 T^{2} \)
$43$ \( 1 + 4026 T + 147008443 T^{2} \)
$47$ \( 1 - 7168 T + 229345007 T^{2} \)
$53$ \( 1 + 29862 T + 418195493 T^{2} \)
$59$ \( 1 + 6461 T + 714924299 T^{2} \)
$61$ \( 1 + 16980 T + 844596301 T^{2} \)
$67$ \( 1 - 29999 T + 1350125107 T^{2} \)
$71$ \( 1 - 31023 T + 1804229351 T^{2} \)
$73$ \( 1 - 1924 T + 2073071593 T^{2} \)
$79$ \( 1 - 65138 T + 3077056399 T^{2} \)
$83$ \( 1 + 102714 T + 3939040643 T^{2} \)
$89$ \( 1 - 17415 T + 5584059449 T^{2} \)
$97$ \( 1 - 66905 T + 8587340257 T^{2} \)
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