Properties

Label 22.12.a.b
Level $22$
Weight $12$
Character orbit 22.a
Self dual yes
Analytic conductor $16.904$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(16.9035499723\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - 331687x - 40657734 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 32 q^{2} + (\beta_{2} - 23) q^{3} + 1024 q^{4} + ( - \beta_{2} - \beta_1 - 1875) q^{5} + ( - 32 \beta_{2} + 736) q^{6} + ( - 54 \beta_{2} + 21 \beta_1 - 10210) q^{7} - 32768 q^{8} + ( - 343 \beta_{2} - 105 \beta_1 + 177830) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 32 q^{2} + (\beta_{2} - 23) q^{3} + 1024 q^{4} + ( - \beta_{2} - \beta_1 - 1875) q^{5} + ( - 32 \beta_{2} + 736) q^{6} + ( - 54 \beta_{2} + 21 \beta_1 - 10210) q^{7} - 32768 q^{8} + ( - 343 \beta_{2} - 105 \beta_1 + 177830) q^{9} + (32 \beta_{2} + 32 \beta_1 + 60000) q^{10} + 161051 q^{11} + (1024 \beta_{2} - 23552) q^{12} + ( - 242 \beta_{2} + 823 \beta_1 + 140936) q^{13} + (1728 \beta_{2} - 672 \beta_1 + 326720) q^{14} + ( - 703 \beta_{2} - 168 \beta_1 - 193915) q^{15} + 1048576 q^{16} + (4364 \beta_{2} - 2383 \beta_1 - 5606526) q^{17} + (10976 \beta_{2} + 3360 \beta_1 - 5690560) q^{18} + ( - 13564 \beta_{2} - 2713 \beta_1 - 4250212) q^{19} + ( - 1024 \beta_{2} - 1024 \beta_1 - 1920000) q^{20} + ( - 10822 \beta_{2} + 11403 \beta_1 - 21370930) q^{21} - 5153632 q^{22} + ( - 54647 \beta_{2} - 12026 \beta_1 - 3817227) q^{23} + ( - 32768 \beta_{2} + 753664) q^{24} + (5269 \beta_{2} + 5089 \beta_1 - 41653700) q^{25} + (7744 \beta_{2} - 26336 \beta_1 - 4509952) q^{26} + (199903 \beta_{2} + 7350 \beta_1 - 109263533) q^{27} + ( - 55296 \beta_{2} + 21504 \beta_1 - 10455040) q^{28} + (74250 \beta_{2} + 47208 \beta_1 - 45591624) q^{29} + (22496 \beta_{2} + 5376 \beta_1 + 6205280) q^{30} + (11905 \beta_{2} - 96374 \beta_1 + 80684309) q^{31} - 33554432 q^{32} + (161051 \beta_{2} - 3704173) q^{33} + ( - 139648 \beta_{2} + 76256 \beta_1 + 179408832) q^{34} + ( - 6614 \beta_{2} - 42959 \beta_1 - 39913050) q^{35} + ( - 351232 \beta_{2} - 107520 \beta_1 + 182097920) q^{36} + ( - 353015 \beta_{2} + 85897 \beta_1 + 150102743) q^{37} + (434048 \beta_{2} + 86816 \beta_1 + 136006784) q^{38} + ( - 482820 \beta_{2} + 250089 \beta_1 - 185644728) q^{39} + (32768 \beta_{2} + 32768 \beta_1 + 61440000) q^{40} + ( - 567234 \beta_{2} - 513561 \beta_1 + 32386284) q^{41} + (346304 \beta_{2} - 364896 \beta_1 + 683869760) q^{42} + ( - 647878 \beta_{2} + 336686 \beta_1 - 237709042) q^{43} + 164916224 q^{44} + (351328 \beta_{2} + 205098 \beta_1 + 107158270) q^{45} + (1748704 \beta_{2} + 384832 \beta_1 + 122151264) q^{46} + (361512 \beta_{2} + 250158 \beta_1 - 1110062592) q^{47} + (1048576 \beta_{2} - 24117248) q^{48} + (3721284 \beta_{2} - 1030596 \beta_1 + 987542157) q^{49} + ( - 168608 \beta_{2} - 162848 \beta_1 + 1332918400) q^{50} + ( - 4972690 \beta_{2} - 1108779 \beta_1 + 1955544434) q^{51} + ( - 247808 \beta_{2} + 842752 \beta_1 + 144318464) q^{52} + (4007732 \beta_{2} + 2244158 \beta_1 - 1257725718) q^{53} + ( - 6396896 \beta_{2} - 235200 \beta_1 + 3496433056) q^{54} + ( - 161051 \beta_{2} - 161051 \beta_1 - 301970625) q^{55} + (1769472 \beta_{2} - 688128 \beta_1 + 334561280) q^{56} + (2401744 \beta_{2} + 683571 \beta_1 - 4391449892) q^{57} + ( - 2376000 \beta_{2} - 1510656 \beta_1 + 1458931968) q^{58} + ( - 4601133 \beta_{2} - 844614 \beta_1 - 3099318045) q^{59} + ( - 719872 \beta_{2} - 172032 \beta_1 - 198568960) q^{60} + (6282758 \beta_{2} - 912928 \beta_1 + 884673188) q^{61} + ( - 380960 \beta_{2} + 3083968 \beta_1 - 2581897888) q^{62} + ( - 18057308 \beta_{2} + 529242 \beta_1 - 2874437420) q^{63} + 1073741824 q^{64} + ( - 2161008 \beta_{2} - 2582643 \beta_1 - 3022999800) q^{65} + ( - 5153632 \beta_{2} + 118533536) q^{66} + (13772063 \beta_{2} + 1920422 \beta_1 + 548609255) q^{67} + (4468736 \beta_{2} - 2440192 \beta_1 - 5741082624) q^{68} + (23915965 \beta_{2} + 2454837 \beta_1 - 17869775027) q^{69} + (211648 \beta_{2} + 1374688 \beta_1 + 1277217600) q^{70} + (13052763 \beta_{2} + 2282196 \beta_1 - 702164889) q^{71} + (11239424 \beta_{2} + 3440640 \beta_1 - 5827133440) q^{72} + (21271114 \beta_{2} - 5577017 \beta_1 - 1754453560) q^{73} + (11296480 \beta_{2} - 2748704 \beta_1 - 4803287776) q^{74} + ( - 47675608 \beta_{2} + 836052 \beta_1 + 2228132300) q^{75} + ( - 13889536 \beta_{2} - 2778112 \beta_1 - 4352217088) q^{76} + ( - 8696754 \beta_{2} + 3382071 \beta_1 - 1644330710) q^{77} + (15450240 \beta_{2} - 8002848 \beta_1 + 5940631296) q^{78} + ( - 26194422 \beta_{2} - 5435430 \beta_1 + 2985566042) q^{79} + ( - 1048576 \beta_{2} - 1048576 \beta_1 - 1966080000) q^{80} + ( - 118733272 \beta_{2} - 382830 \beta_1 + 41003279993) q^{81} + (18151488 \beta_{2} + 16433952 \beta_1 - 1036361088) q^{82} + (91042834 \beta_{2} - 3456056 \beta_1 + 3860084742) q^{83} + ( - 11081728 \beta_{2} + 11676672 \beta_1 - 21883832320) q^{84} + (8796106 \beta_{2} + 11976811 \beta_1 + 17631847770) q^{85} + (20732096 \beta_{2} - 10773952 \beta_1 + 7606689344) q^{86} + ( - 109572840 \beta_{2} + 5091534 \beta_1 + 21823774488) q^{87} - 5277319168 q^{88} + (58147597 \beta_{2} + 264751 \beta_1 + 60369007695) q^{89} + ( - 11242496 \beta_{2} - 6563136 \beta_1 - 3429064640) q^{90} + (94009688 \beta_{2} - 6749302 \beta_1 + 70174924240) q^{91} + ( - 55958528 \beta_{2} - 12314624 \beta_1 - 3908840448) q^{92} + (158985357 \beta_{2} - 27560127 \beta_1 + 13679042925) q^{93} + ( - 11568384 \beta_{2} - 8005056 \beta_1 + 35522002944) q^{94} + (21335960 \beta_{2} + 15042335 \beta_1 + 20467592940) q^{95} + ( - 33554432 \beta_{2} + 771751936) q^{96} + (12329735 \beta_{2} - 3199237 \beta_1 - 5715614935) q^{97} + ( - 119081088 \beta_{2} + 32979072 \beta_1 - 31601349024) q^{98} + ( - 55240493 \beta_{2} - 16910355 \beta_1 + 28639699330) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 96 q^{2} - 70 q^{3} + 3072 q^{4} - 5624 q^{5} + 2240 q^{6} - 30576 q^{7} - 98304 q^{8} + 533833 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 96 q^{2} - 70 q^{3} + 3072 q^{4} - 5624 q^{5} + 2240 q^{6} - 30576 q^{7} - 98304 q^{8} + 533833 q^{9} + 179968 q^{10} + 483153 q^{11} - 71680 q^{12} + 423050 q^{13} + 978432 q^{14} - 581042 q^{15} + 3145728 q^{16} - 16823942 q^{17} - 17082656 q^{18} - 12737072 q^{19} - 5758976 q^{20} - 64101968 q^{21} - 15460896 q^{22} - 11397034 q^{23} + 2293760 q^{24} - 124966369 q^{25} - 13537600 q^{26} - 327990502 q^{27} - 31309824 q^{28} - 136849122 q^{29} + 18593344 q^{30} + 242041022 q^{31} - 100663296 q^{32} - 11273570 q^{33} + 538366144 q^{34} - 119732536 q^{35} + 546644992 q^{36} + 450661244 q^{37} + 407586304 q^{38} - 556451364 q^{39} + 184287232 q^{40} + 97726086 q^{41} + 2051262976 q^{42} - 712479248 q^{43} + 494748672 q^{44} + 321123482 q^{45} + 364705088 q^{46} - 3330549288 q^{47} - 73400320 q^{48} + 2958905187 q^{49} + 3998923808 q^{50} + 5871605992 q^{51} + 433203200 q^{52} - 3777184886 q^{53} + 10495696064 q^{54} - 905750824 q^{55} + 1001914368 q^{56} - 13176751420 q^{57} + 4379171904 q^{58} - 9293353002 q^{59} - 594987008 q^{60} + 2647736806 q^{61} - 7745312704 q^{62} - 8605254952 q^{63} + 3221225472 q^{64} - 9066838392 q^{65} + 360754240 q^{66} + 1632055702 q^{67} - 17227716608 q^{68} - 53633241046 q^{69} + 3831441152 q^{70} - 2119547430 q^{71} - 17492639744 q^{72} - 5284631794 q^{73} - 14421159808 q^{74} + 6732072508 q^{75} - 13042761728 q^{76} - 4924295376 q^{77} + 17806443648 q^{78} + 8982892548 q^{79} - 5897191424 q^{80} + 123128573251 q^{81} - 3127234752 q^{82} + 11489211392 q^{83} - 65640415232 q^{84} + 52886747204 q^{85} + 22799335936 q^{86} + 65580896304 q^{87} - 15831957504 q^{88} + 181048875488 q^{89} - 10275951424 q^{90} + 210430763032 q^{91} - 11670562816 q^{92} + 40878143418 q^{93} + 106577577216 q^{94} + 61381442860 q^{95} + 2348810240 q^{96} - 17159174540 q^{97} - 94684965984 q^{98} + 85974338483 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - 331687x - 40657734 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 213\nu - 221198 ) / 220 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 880\beta_{2} + 213\beta _1 + 884792 ) / 4 \) 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
−129.060
629.503
−500.443
−32.0000 −827.781 1024.00 −553.980 26489.0 22407.2 −32768.0 508075. 17727.4
1.2 −32.0000 163.327 1024.00 −4579.34 −5226.46 32606.6 −32768.0 −150471. 146539.
1.3 −32.0000 594.455 1024.00 −490.681 −19022.5 −85589.8 −32768.0 176229. 15701.8
\(n\): 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 22.12.a.b 3
3.b odd 2 1 198.12.a.l 3
4.b odd 2 1 176.12.a.d 3
11.b odd 2 1 242.12.a.c 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.12.a.b 3 1.a even 1 1 trivial
176.12.a.d 3 4.b odd 2 1
198.12.a.l 3 3.b odd 2 1
242.12.a.c 3 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} + 70T_{3}^{2} - 530187T_{3} + 80369604 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 32)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + 70 T^{2} - 530187 T + 80369604 \) Copy content Toggle raw display
$5$ \( T^{3} + 5624 T^{2} + \cdots + 1244790450 \) Copy content Toggle raw display
$7$ \( T^{3} + 30576 T^{2} + \cdots + 62533813132000 \) Copy content Toggle raw display
$11$ \( (T - 161051)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 423050 T^{2} + \cdots - 32\!\cdots\!88 \) Copy content Toggle raw display
$17$ \( T^{3} + 16823942 T^{2} + \cdots - 15\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{3} + 12737072 T^{2} + \cdots - 78\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{3} + 11397034 T^{2} + \cdots - 27\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{3} + 136849122 T^{2} + \cdots - 10\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} - 242041022 T^{2} + \cdots + 54\!\cdots\!88 \) Copy content Toggle raw display
$37$ \( T^{3} - 450661244 T^{2} + \cdots + 28\!\cdots\!94 \) Copy content Toggle raw display
$41$ \( T^{3} - 97726086 T^{2} + \cdots + 73\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{3} + 712479248 T^{2} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{3} + 3330549288 T^{2} + \cdots + 87\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{3} + 3777184886 T^{2} + \cdots - 94\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{3} + 9293353002 T^{2} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} - 2647736806 T^{2} + \cdots - 59\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{3} - 1632055702 T^{2} + \cdots + 47\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{3} + 2119547430 T^{2} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} + 5284631794 T^{2} + \cdots - 42\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{3} - 8982892548 T^{2} + \cdots - 96\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} - 11489211392 T^{2} + \cdots + 54\!\cdots\!28 \) Copy content Toggle raw display
$89$ \( T^{3} - 181048875488 T^{2} + \cdots - 92\!\cdots\!30 \) Copy content Toggle raw display
$97$ \( T^{3} + 17159174540 T^{2} + \cdots - 13\!\cdots\!02 \) Copy content Toggle raw display
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