Properties

Label 22.8.a.d
Level $22$
Weight $8$
Character orbit 22.a
Self dual yes
Analytic conductor $6.872$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.87247056065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14881}) \)
Defining polynomial: \( x^{2} - x - 3720 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{14881})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 8 q^{2} + ( - \beta - 11) q^{3} + 64 q^{4} + (\beta + 165) q^{5} + ( - 8 \beta - 88) q^{6} + (14 \beta + 890) q^{7} + 512 q^{8} + (23 \beta + 1654) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 8 q^{2} + ( - \beta - 11) q^{3} + 64 q^{4} + (\beta + 165) q^{5} + ( - 8 \beta - 88) q^{6} + (14 \beta + 890) q^{7} + 512 q^{8} + (23 \beta + 1654) q^{9} + (8 \beta + 1320) q^{10} + 1331 q^{11} + ( - 64 \beta - 704) q^{12} + ( - 118 \beta - 2644) q^{13} + (112 \beta + 7120) q^{14} + ( - 177 \beta - 5535) q^{15} + 4096 q^{16} + (236 \beta + 7398) q^{17} + (184 \beta + 13232) q^{18} + (484 \beta + 8216) q^{19} + (64 \beta + 10560) q^{20} + ( - 1058 \beta - 61870) q^{21} + 10648 q^{22} + (567 \beta - 25959) q^{23} + ( - 512 \beta - 5632) q^{24} + (331 \beta - 47180) q^{25} + ( - 944 \beta - 21152) q^{26} + (257 \beta - 79697) q^{27} + (896 \beta + 56960) q^{28} + ( - 730 \beta - 103200) q^{29} + ( - 1416 \beta - 44280) q^{30} + (1295 \beta - 10183) q^{31} + 32768 q^{32} + ( - 1331 \beta - 14641) q^{33} + (1888 \beta + 59184) q^{34} + (3214 \beta + 198930) q^{35} + (1472 \beta + 105856) q^{36} + ( - 3385 \beta + 177359) q^{37} + (3872 \beta + 65728) q^{38} + (4060 \beta + 468044) q^{39} + (512 \beta + 84480) q^{40} + ( - 8006 \beta + 65808) q^{41} + ( - 8464 \beta - 494960) q^{42} + ( - 12682 \beta - 73570) q^{43} + 85184 q^{44} + (5472 \beta + 358470) q^{45} + (4536 \beta - 207672) q^{46} + (5768 \beta + 222696) q^{47} + ( - 4096 \beta - 45056) q^{48} + (25116 \beta + 697677) q^{49} + (2648 \beta - 377440) q^{50} + ( - 10230 \beta - 959298) q^{51} + ( - 7552 \beta - 169216) q^{52} + (9308 \beta - 635070) q^{53} + (2056 \beta - 637576) q^{54} + (1331 \beta + 219615) q^{55} + (7168 \beta + 455680) q^{56} + ( - 14024 \beta - 1890856) q^{57} + ( - 5840 \beta - 825600) q^{58} + ( - 14307 \beta + 450927) q^{59} + ( - 11328 \beta - 354240) q^{60} + ( - 12118 \beta - 292900) q^{61} + (10360 \beta - 81464) q^{62} + (43948 \beta + 2669900) q^{63} + 262144 q^{64} + ( - 22232 \beta - 875220) q^{65} + ( - 10648 \beta - 117128) q^{66} + ( - 34943 \beta + 1449827) q^{67} + (15104 \beta + 473472) q^{68} + (19155 \beta - 1823691) q^{69} + (25712 \beta + 1591440) q^{70} + ( - 40763 \beta + 673515) q^{71} + (11776 \beta + 846848) q^{72} + (29326 \beta - 2303428) q^{73} + ( - 27080 \beta + 1418872) q^{74} + (43208 \beta - 712340) q^{75} + (30976 \beta + 525824) q^{76} + (18634 \beta + 1184590) q^{77} + (32480 \beta + 3744352) q^{78} + ( - 55258 \beta - 1445542) q^{79} + (4096 \beta + 675840) q^{80} + (26312 \beta - 3696671) q^{81} + ( - 64048 \beta + 526464) q^{82} + ( - 24754 \beta + 4995102) q^{83} + ( - 67712 \beta - 3959680) q^{84} + (46574 \beta + 2098590) q^{85} + ( - 101456 \beta - 588560) q^{86} + (111960 \beta + 3850800) q^{87} + 681472 q^{88} + ( - 67837 \beta + 5126607) q^{89} + (43776 \beta + 2867760) q^{90} + ( - 143688 \beta - 8498600) q^{91} + (36288 \beta - 1661376) q^{92} + ( - 5357 \beta - 4705387) q^{93} + (46144 \beta + 1781568) q^{94} + (88560 \beta + 3156120) q^{95} + ( - 32768 \beta - 360448) q^{96} + ( - 1255 \beta - 13882111) q^{97} + (200928 \beta + 5581416) q^{98} + (30613 \beta + 2201474) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} - 23 q^{3} + 128 q^{4} + 331 q^{5} - 184 q^{6} + 1794 q^{7} + 1024 q^{8} + 3331 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} - 23 q^{3} + 128 q^{4} + 331 q^{5} - 184 q^{6} + 1794 q^{7} + 1024 q^{8} + 3331 q^{9} + 2648 q^{10} + 2662 q^{11} - 1472 q^{12} - 5406 q^{13} + 14352 q^{14} - 11247 q^{15} + 8192 q^{16} + 15032 q^{17} + 26648 q^{18} + 16916 q^{19} + 21184 q^{20} - 124798 q^{21} + 21296 q^{22} - 51351 q^{23} - 11776 q^{24} - 94029 q^{25} - 43248 q^{26} - 159137 q^{27} + 114816 q^{28} - 207130 q^{29} - 89976 q^{30} - 19071 q^{31} + 65536 q^{32} - 30613 q^{33} + 120256 q^{34} + 401074 q^{35} + 213184 q^{36} + 351333 q^{37} + 135328 q^{38} + 940148 q^{39} + 169472 q^{40} + 123610 q^{41} - 998384 q^{42} - 159822 q^{43} + 170368 q^{44} + 722412 q^{45} - 410808 q^{46} + 451160 q^{47} - 94208 q^{48} + 1420470 q^{49} - 752232 q^{50} - 1928826 q^{51} - 345984 q^{52} - 1260832 q^{53} - 1273096 q^{54} + 440561 q^{55} + 918528 q^{56} - 3795736 q^{57} - 1657040 q^{58} + 887547 q^{59} - 719808 q^{60} - 597918 q^{61} - 152568 q^{62} + 5383748 q^{63} + 524288 q^{64} - 1772672 q^{65} - 244904 q^{66} + 2864711 q^{67} + 962048 q^{68} - 3628227 q^{69} + 3208592 q^{70} + 1306267 q^{71} + 1705472 q^{72} - 4577530 q^{73} + 2810664 q^{74} - 1381472 q^{75} + 1082624 q^{76} + 2387814 q^{77} + 7521184 q^{78} - 2946342 q^{79} + 1355776 q^{80} - 7367030 q^{81} + 988880 q^{82} + 9965450 q^{83} - 7987072 q^{84} + 4243754 q^{85} - 1278576 q^{86} + 7813560 q^{87} + 1362944 q^{88} + 10185377 q^{89} + 5779296 q^{90} - 17140888 q^{91} - 3286464 q^{92} - 9416131 q^{93} + 3609280 q^{94} + 6400800 q^{95} - 753664 q^{96} - 27765477 q^{97} + 11363760 q^{98} + 4433561 q^{99}+O(q^{100}) \) 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
61.4939
−60.4939
8.00000 −72.4939 64.0000 226.494 −579.951 1750.91 512.000 3068.36 1811.95
1.2 8.00000 49.4939 64.0000 104.506 395.951 43.0861 512.000 262.641 836.049
\(n\): 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 22.8.a.d 2
3.b odd 2 1 198.8.a.f 2
4.b odd 2 1 176.8.a.e 2
5.b even 2 1 550.8.a.d 2
11.b odd 2 1 242.8.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.8.a.d 2 1.a even 1 1 trivial
176.8.a.e 2 4.b odd 2 1
198.8.a.f 2 3.b odd 2 1
242.8.a.h 2 11.b odd 2 1
550.8.a.d 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 23T_{3} - 3588 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 8)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 23T - 3588 \) Copy content Toggle raw display
$5$ \( T^{2} - 331T + 23670 \) Copy content Toggle raw display
$7$ \( T^{2} - 1794T + 75440 \) Copy content Toggle raw display
$11$ \( (T - 1331)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5406 T - 44494552 \) Copy content Toggle raw display
$17$ \( T^{2} - 15032 T - 150712788 \) Copy content Toggle raw display
$19$ \( T^{2} - 16916 T - 799953120 \) Copy content Toggle raw display
$23$ \( T^{2} + 51351 T - 536788152 \) Copy content Toggle raw display
$29$ \( T^{2} + 207130 T + 8743188000 \) Copy content Toggle raw display
$31$ \( T^{2} + 19071 T - 6148026496 \) Copy content Toggle raw display
$37$ \( T^{2} - 351333 T - 11768742334 \) Copy content Toggle raw display
$41$ \( T^{2} - 123610 T - 234633419904 \) Copy content Toggle raw display
$43$ \( T^{2} + 159822 T - 591953661640 \) Copy content Toggle raw display
$47$ \( T^{2} - 451160 T - 72885726336 \) Copy content Toggle raw display
$53$ \( T^{2} + 1260832 T + 75106099260 \) Copy content Toggle raw display
$59$ \( T^{2} - 887547 T - 564563979540 \) Copy content Toggle raw display
$61$ \( T^{2} + 597918 T - 456927065080 \) Copy content Toggle raw display
$67$ \( T^{2} - 2864711 T - 2490832261212 \) Copy content Toggle raw display
$71$ \( T^{2} - 1306267 T - 5755066505400 \) Copy content Toggle raw display
$73$ \( T^{2} + 4577530 T + 2038977114936 \) Copy content Toggle raw display
$79$ \( T^{2} + 2946342 T - 9189351784480 \) Copy content Toggle raw display
$83$ \( T^{2} - 9965450 T + 22547926115976 \) Copy content Toggle raw display
$89$ \( T^{2} - 10185377 T + 8815411816710 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 192724568772626 \) Copy content Toggle raw display
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