Properties

Label 22.6.a.d
Level $22$
Weight $6$
Character orbit 22.a
Self dual yes
Analytic conductor $3.528$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.52844403589\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{793}) \)
Defining polynomial: \( x^{2} - x - 198 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{793})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 4 q^{2} + ( - \beta + 15) q^{3} + 16 q^{4} + (5 \beta - 9) q^{5} + ( - 4 \beta + 60) q^{6} + (6 \beta - 10) q^{7} + 64 q^{8} + ( - 29 \beta + 180) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 q^{2} + ( - \beta + 15) q^{3} + 16 q^{4} + (5 \beta - 9) q^{5} + ( - 4 \beta + 60) q^{6} + (6 \beta - 10) q^{7} + 64 q^{8} + ( - 29 \beta + 180) q^{9} + (20 \beta - 36) q^{10} - 121 q^{11} + ( - 16 \beta + 240) q^{12} + (10 \beta - 328) q^{13} + (24 \beta - 40) q^{14} + (79 \beta - 1125) q^{15} + 256 q^{16} + ( - 124 \beta - 42) q^{17} + ( - 116 \beta + 720) q^{18} + (44 \beta - 1096) q^{19} + (80 \beta - 144) q^{20} + (94 \beta - 1338) q^{21} - 484 q^{22} + (7 \beta + 171) q^{23} + ( - 64 \beta + 960) q^{24} + ( - 65 \beta + 1906) q^{25} + (40 \beta - 1312) q^{26} + ( - 343 \beta + 4797) q^{27} + (96 \beta - 160) q^{28} + (366 \beta + 2028) q^{29} + (316 \beta - 4500) q^{30} + ( - 89 \beta + 7235) q^{31} + 1024 q^{32} + (121 \beta - 1815) q^{33} + ( - 496 \beta - 168) q^{34} + ( - 74 \beta + 6030) q^{35} + ( - 464 \beta + 2880) q^{36} + ( - 149 \beta - 2059) q^{37} + (176 \beta - 4384) q^{38} + (468 \beta - 6900) q^{39} + (320 \beta - 576) q^{40} + ( - 1014 \beta - 4548) q^{41} + (376 \beta - 5352) q^{42} + (470 \beta - 5134) q^{43} - 1936 q^{44} + (1016 \beta - 30330) q^{45} + (28 \beta + 684) q^{46} + (312 \beta + 10416) q^{47} + ( - 256 \beta + 3840) q^{48} + ( - 84 \beta - 9579) q^{49} + ( - 260 \beta + 7624) q^{50} + ( - 1694 \beta + 23922) q^{51} + (160 \beta - 5248) q^{52} + ( - 388 \beta + 19986) q^{53} + ( - 1372 \beta + 19188) q^{54} + ( - 605 \beta + 1089) q^{55} + (384 \beta - 640) q^{56} + (1712 \beta - 25152) q^{57} + (1464 \beta + 8112) q^{58} + (333 \beta + 45309) q^{59} + (1264 \beta - 18000) q^{60} + ( - 526 \beta - 14512) q^{61} + ( - 356 \beta + 28940) q^{62} + (1196 \beta - 36252) q^{63} + 4096 q^{64} + ( - 1680 \beta + 12852) q^{65} + (484 \beta - 7260) q^{66} + ( - 1495 \beta - 31327) q^{67} + ( - 1984 \beta - 672) q^{68} + ( - 73 \beta + 1179) q^{69} + ( - 296 \beta + 24120) q^{70} + (4173 \beta + 9705) q^{71} + ( - 1856 \beta + 11520) q^{72} + (2926 \beta - 20992) q^{73} + ( - 596 \beta - 8236) q^{74} + ( - 2816 \beta + 41460) q^{75} + (704 \beta - 17536) q^{76} + ( - 726 \beta + 1210) q^{77} + (1872 \beta - 27600) q^{78} + ( - 3258 \beta - 25858) q^{79} + (1280 \beta - 2304) q^{80} + ( - 2552 \beta + 96129) q^{81} + ( - 4056 \beta - 18192) q^{82} + (5278 \beta + 12354) q^{83} + (1504 \beta - 21408) q^{84} + (286 \beta - 122382) q^{85} + (1880 \beta - 20536) q^{86} + (3096 \beta - 42048) q^{87} - 7744 q^{88} + ( - 1001 \beta - 8523) q^{89} + (4064 \beta - 121320) q^{90} + ( - 2008 \beta + 15160) q^{91} + (112 \beta + 2736) q^{92} + ( - 8481 \beta + 126147) q^{93} + (1248 \beta + 41664) q^{94} + ( - 5656 \beta + 53424) q^{95} + ( - 1024 \beta + 15360) q^{96} + (8213 \beta - 19261) q^{97} + ( - 336 \beta - 38316) q^{98} + (3509 \beta - 21780) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 29 q^{3} + 32 q^{4} - 13 q^{5} + 116 q^{6} - 14 q^{7} + 128 q^{8} + 331 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 29 q^{3} + 32 q^{4} - 13 q^{5} + 116 q^{6} - 14 q^{7} + 128 q^{8} + 331 q^{9} - 52 q^{10} - 242 q^{11} + 464 q^{12} - 646 q^{13} - 56 q^{14} - 2171 q^{15} + 512 q^{16} - 208 q^{17} + 1324 q^{18} - 2148 q^{19} - 208 q^{20} - 2582 q^{21} - 968 q^{22} + 349 q^{23} + 1856 q^{24} + 3747 q^{25} - 2584 q^{26} + 9251 q^{27} - 224 q^{28} + 4422 q^{29} - 8684 q^{30} + 14381 q^{31} + 2048 q^{32} - 3509 q^{33} - 832 q^{34} + 11986 q^{35} + 5296 q^{36} - 4267 q^{37} - 8592 q^{38} - 13332 q^{39} - 832 q^{40} - 10110 q^{41} - 10328 q^{42} - 9798 q^{43} - 3872 q^{44} - 59644 q^{45} + 1396 q^{46} + 21144 q^{47} + 7424 q^{48} - 19242 q^{49} + 14988 q^{50} + 46150 q^{51} - 10336 q^{52} + 39584 q^{53} + 37004 q^{54} + 1573 q^{55} - 896 q^{56} - 48592 q^{57} + 17688 q^{58} + 90951 q^{59} - 34736 q^{60} - 29550 q^{61} + 57524 q^{62} - 71308 q^{63} + 8192 q^{64} + 24024 q^{65} - 14036 q^{66} - 64149 q^{67} - 3328 q^{68} + 2285 q^{69} + 47944 q^{70} + 23583 q^{71} + 21184 q^{72} - 39058 q^{73} - 17068 q^{74} + 80104 q^{75} - 34368 q^{76} + 1694 q^{77} - 53328 q^{78} - 54974 q^{79} - 3328 q^{80} + 189706 q^{81} - 40440 q^{82} + 29986 q^{83} - 41312 q^{84} - 244478 q^{85} - 39192 q^{86} - 81000 q^{87} - 15488 q^{88} - 18047 q^{89} - 238576 q^{90} + 28312 q^{91} + 5584 q^{92} + 243813 q^{93} + 84576 q^{94} + 101192 q^{95} + 29696 q^{96} - 30309 q^{97} - 76968 q^{98} - 40051 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
14.5801
−13.5801
4.00000 0.419872 16.0000 63.9006 1.67949 77.4808 64.0000 −242.824 255.603
1.2 4.00000 28.5801 16.0000 −76.9006 114.321 −91.4808 64.0000 573.824 −307.603
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(11\) \(1\)

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 22.6.a.d 2
3.b odd 2 1 198.6.a.k 2
4.b odd 2 1 176.6.a.f 2
5.b even 2 1 550.6.a.h 2
5.c odd 4 2 550.6.b.j 4
7.b odd 2 1 1078.6.a.h 2
8.b even 2 1 704.6.a.k 2
8.d odd 2 1 704.6.a.p 2
11.b odd 2 1 242.6.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.6.a.d 2 1.a even 1 1 trivial
176.6.a.f 2 4.b odd 2 1
198.6.a.k 2 3.b odd 2 1
242.6.a.g 2 11.b odd 2 1
550.6.a.h 2 5.b even 2 1
550.6.b.j 4 5.c odd 4 2
704.6.a.k 2 8.b even 2 1
704.6.a.p 2 8.d odd 2 1
1078.6.a.h 2 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 29T_{3} + 12 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 29T + 12 \) Copy content Toggle raw display
$5$ \( T^{2} + 13T - 4914 \) Copy content Toggle raw display
$7$ \( T^{2} + 14T - 7088 \) Copy content Toggle raw display
$11$ \( (T + 121)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 646T + 84504 \) Copy content Toggle raw display
$17$ \( T^{2} + 208 T - 3037476 \) Copy content Toggle raw display
$19$ \( T^{2} + 2148 T + 769664 \) Copy content Toggle raw display
$23$ \( T^{2} - 349T + 20736 \) Copy content Toggle raw display
$29$ \( T^{2} - 4422 T - 21668256 \) Copy content Toggle raw display
$31$ \( T^{2} - 14381 T + 50132952 \) Copy content Toggle raw display
$37$ \( T^{2} + 4267 T + 150474 \) Copy content Toggle raw display
$41$ \( T^{2} + 10110 T - 178286832 \) Copy content Toggle raw display
$43$ \( T^{2} + 9798 T - 19793224 \) Copy content Toggle raw display
$47$ \( T^{2} - 21144 T + 92468736 \) Copy content Toggle raw display
$53$ \( T^{2} - 39584 T + 361877916 \) Copy content Toggle raw display
$59$ \( T^{2} - 90951 T + 2046037356 \) Copy content Toggle raw display
$61$ \( T^{2} + 29550 T + 163449608 \) Copy content Toggle raw display
$67$ \( T^{2} + 64149 T + 585679844 \) Copy content Toggle raw display
$71$ \( T^{2} - 23583 T - 3313271952 \) Copy content Toggle raw display
$73$ \( T^{2} + 39058 T - 1315930776 \) Copy content Toggle raw display
$79$ \( T^{2} + 54974 T - 1348802144 \) Copy content Toggle raw display
$83$ \( T^{2} - 29986 T - 5297916504 \) Copy content Toggle raw display
$89$ \( T^{2} + 18047 T - 117223146 \) Copy content Toggle raw display
$97$ \( T^{2} + 30309 T - 13142971534 \) Copy content Toggle raw display
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