Properties

Label 22.12.a.d
Level $22$
Weight $12$
Character orbit 22.a
Self dual yes
Analytic conductor $16.904$
Analytic rank $0$
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: \(0\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
Defining polynomial: \( x^{3} - x^{2} - 206434x + 34594984 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2}\cdot 5 \)
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.

\(f(q)\) \(=\) \( q + 32 q^{2} + ( - \beta_1 + 123) q^{3} + 1024 q^{4} + ( - \beta_{2} - 5 \beta_1 - 644) q^{5} + ( - 32 \beta_1 + 3936) q^{6} + (\beta_{2} - 4 \beta_1 + 19489) q^{7} + 32768 q^{8} + (27 \beta_{2} - 199 \beta_1 + 120669) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 q^{2} + ( - \beta_1 + 123) q^{3} + 1024 q^{4} + ( - \beta_{2} - 5 \beta_1 - 644) q^{5} + ( - 32 \beta_1 + 3936) q^{6} + (\beta_{2} - 4 \beta_1 + 19489) q^{7} + 32768 q^{8} + (27 \beta_{2} - 199 \beta_1 + 120669) q^{9} + ( - 32 \beta_{2} - 160 \beta_1 - 20608) q^{10} + 161051 q^{11} + ( - 1024 \beta_1 + 125952) q^{12} + ( - 53 \beta_{2} + 2036 \beta_1 + 450409) q^{13} + (32 \beta_{2} - 128 \beta_1 + 623648) q^{14} + ( - 36 \beta_{2} + 5415 \beta_1 + 1320651) q^{15} + 1048576 q^{16} + ( - 323 \beta_{2} + 7970 \beta_1 + 1484909) q^{17} + (864 \beta_{2} - 6368 \beta_1 + 3861408) q^{18} + (755 \beta_{2} - 8594 \beta_1 + 4358529) q^{19} + ( - 1024 \beta_{2} - 5120 \beta_1 - 659456) q^{20} + (279 \beta_{2} - 24944 \beta_1 + 3541467) q^{21} + 5153632 q^{22} + ( - 2214 \beta_{2} - 1581 \beta_1 + 3295437) q^{23} + ( - 32768 \beta_1 + 4030464) q^{24} + ( - 3667 \beta_{2} + 49465 \beta_1 + 12496217) q^{25} + ( - 1696 \beta_{2} + 65152 \beta_1 + 14413088) q^{26} + (9990 \beta_{2} - 97723 \beta_1 + 49674363) q^{27} + (1024 \beta_{2} - 4096 \beta_1 + 19956736) q^{28} + (14920 \beta_{2} + 43430 \beta_1 + 94400420) q^{29} + ( - 1152 \beta_{2} + 173280 \beta_1 + 42260832) q^{30} + ( - 23642 \beta_{2} + 139811 \beta_1 - 7172687) q^{31} + 33554432 q^{32} + ( - 161051 \beta_1 + 19809273) q^{33} + ( - 10336 \beta_{2} + 255040 \beta_1 + 47517088) q^{34} + ( - 14395 \beta_{2} - 89420 \beta_1 - 60861755) q^{35} + (27648 \beta_{2} - 203776 \beta_1 + 123565056) q^{36} + (18841 \beta_{2} - 573043 \beta_1 - 133595072) q^{37} + (24160 \beta_{2} - 275008 \beta_1 + 139472928) q^{38} + ( - 64035 \beta_{2} - 22670 \beta_1 - 520869741) q^{39} + ( - 32768 \beta_{2} - 163840 \beta_1 - 21102592) q^{40} + (79259 \beta_{2} + 600484 \beta_1 - 294176291) q^{41} + (8928 \beta_{2} - 798208 \beta_1 + 113326944) q^{42} + ( - 23978 \beta_{2} + 23882 \beta_1 - 1063019732) q^{43} + 164916224 q^{44} + (24786 \beta_{2} + 162060 \beta_1 - 1254715956) q^{45} + ( - 70848 \beta_{2} - 50592 \beta_1 + 105453984) q^{46} + (156526 \beta_{2} + 4339820 \beta_1 - 624498910) q^{47} + ( - 1048576 \beta_1 + 128974848) q^{48} + (34644 \beta_{2} - 206028 \beta_1 - 1538895903) q^{49} + ( - 117344 \beta_{2} + 1582880 \beta_1 + 399878944) q^{50} + ( - 270423 \beta_{2} + 784584 \beta_1 - 2074755339) q^{51} + ( - 54272 \beta_{2} + 2084864 \beta_1 + 461218816) q^{52} + ( - 164402 \beta_{2} - 3070816 \beta_1 - 1136020540) q^{53} + (319680 \beta_{2} - 3127136 \beta_1 + 1589579616) q^{54} + ( - 161051 \beta_{2} - 805255 \beta_1 - 103716844) q^{55} + (32768 \beta_{2} - 131072 \beta_1 + 638615552) q^{56} + (361143 \beta_{2} - 8900678 \beta_1 + 2975758005) q^{57} + (477440 \beta_{2} + 1389760 \beta_1 + 3020813440) q^{58} + (674178 \beta_{2} - 3126087 \beta_1 + 4707033879) q^{59} + ( - 36864 \beta_{2} + 5544960 \beta_1 + 1352346624) q^{60} + ( - 969920 \beta_{2} + 4851578 \beta_1 + 2604744784) q^{61} + ( - 756544 \beta_{2} + 4473952 \beta_1 - 229525984) q^{62} + (544050 \beta_{2} - 6165752 \beta_1 + 4038313674) q^{63} + 1073741824 q^{64} + ( - 879079 \beta_{2} - 13676090 \beta_1 - 282939041) q^{65} + ( - 5153632 \beta_1 + 633896736) q^{66} + (413510 \beta_{2} + 15320197 \beta_1 + 295380879) q^{67} + ( - 330752 \beta_{2} + 8161280 \beta_1 + 1520546816) q^{68} + ( - 335907 \beta_{2} + 7988721 \beta_1 + 822218490) q^{69} + ( - 460640 \beta_{2} - 2861440 \beta_1 - 1947576160) q^{70} + (453248 \beta_{2} + 39496549 \beta_1 - 3901505795) q^{71} + (884736 \beta_{2} - 6520832 \beta_1 + 3954081792) q^{72} + (2739611 \beta_{2} - 24196712 \beta_1 + 4179790353) q^{73} + (602912 \beta_{2} - 18337376 \beta_1 - 4275042304) q^{74} + ( - 1962612 \beta_{2} + 10151840 \beta_1 - 12495846288) q^{75} + (773120 \beta_{2} - 8800256 \beta_1 + 4463133696) q^{76} + (161051 \beta_{2} - 644204 \beta_1 + 3138722939) q^{77} + ( - 2049120 \beta_{2} - 725440 \beta_1 - 16667831712) q^{78} + ( - 980870 \beta_{2} + 67824338 \beta_1 + 5617347412) q^{79} + ( - 1048576 \beta_{2} - 5242880 \beta_1 - 675282944) q^{80} + ( - 436158 \beta_{2} - 73307548 \beta_1 + 12494401287) q^{81} + (2536288 \beta_{2} + 19215488 \beta_1 - 9413641312) q^{82} + (1255528 \beta_{2} + 14138462 \beta_1 + 11401561442) q^{83} + (285696 \beta_{2} - 25542656 \beta_1 + 3626462208) q^{84} + ( - 3711257 \beta_{2} - 50284960 \beta_1 + 5299834487) q^{85} + ( - 767296 \beta_{2} + 764224 \beta_1 - 34016631424) q^{86} + (1378710 \beta_{2} - 167952660 \beta_1 - 463350510) q^{87} + 5277319168 q^{88} + (2307715 \beta_{2} + 87269957 \beta_1 + 4389451238) q^{89} + (793152 \beta_{2} + 5185920 \beta_1 - 40150910592) q^{90} + ( - 637350 \beta_{2} + 49586628 \beta_1 + 3584466214) q^{91} + ( - 2267136 \beta_{2} - 1618944 \beta_1 + 3374527488) q^{92} + ( - 7817679 \beta_{2} + 139578265 \beta_1 - 40725861882) q^{93} + (5008832 \beta_{2} + 138874240 \beta_1 - 19983965120) q^{94} + ( - 27621 \beta_{2} + 17877450 \beta_1 - 31478739759) q^{95} + ( - 33554432 \beta_1 + 4127195136) q^{96} + (14048191 \beta_{2} + 35970227 \beta_1 + 17215407448) q^{97} + (1108608 \beta_{2} - 6592896 \beta_1 - 49244668896) q^{98} + (4348377 \beta_{2} - 32049149 \beta_1 + 19433863119) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 96 q^{2} + 370 q^{3} + 3072 q^{4} - 1928 q^{5} + 11840 q^{6} + 58472 q^{7} + 98304 q^{8} + 362233 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 96 q^{2} + 370 q^{3} + 3072 q^{4} - 1928 q^{5} + 11840 q^{6} + 58472 q^{7} + 98304 q^{8} + 362233 q^{9} - 61696 q^{10} + 483153 q^{11} + 378880 q^{12} + 1349138 q^{13} + 1871104 q^{14} + 3956502 q^{15} + 3145728 q^{16} + 4446434 q^{17} + 11591456 q^{18} + 13084936 q^{19} - 1974272 q^{20} + 10649624 q^{21} + 15460896 q^{22} + 9885678 q^{23} + 12124160 q^{24} + 37435519 q^{25} + 43172416 q^{26} + 149130802 q^{27} + 59875328 q^{28} + 283172750 q^{29} + 126608064 q^{30} - 21681514 q^{31} + 100663296 q^{32} + 59588870 q^{33} + 142285888 q^{34} - 182510240 q^{35} + 370926592 q^{36} - 400193332 q^{37} + 418717952 q^{38} - 1562650588 q^{39} - 63176704 q^{40} - 883050098 q^{41} + 340787968 q^{42} - 3189107056 q^{43} + 494748672 q^{44} - 3764285142 q^{45} + 316341696 q^{46} - 1877680024 q^{47} + 387973120 q^{48} - 4616447037 q^{49} + 1197936608 q^{50} - 6225321024 q^{51} + 1381517312 q^{52} - 3405155206 q^{53} + 4772185664 q^{54} - 310506328 q^{55} + 1916010496 q^{56} + 8936535836 q^{57} + 9061528000 q^{58} + 14124901902 q^{59} + 4051458048 q^{60} + 7808412854 q^{61} - 693808448 q^{62} + 12121650824 q^{63} + 3221225472 q^{64} - 836020112 q^{65} + 1906843840 q^{66} + 871235950 q^{67} + 4553148416 q^{68} + 2458330842 q^{69} - 5840327680 q^{70} - 11743560686 q^{71} + 11869650944 q^{72} + 12566307382 q^{73} - 12806186624 q^{74} - 37499653316 q^{75} + 13398974464 q^{76} + 9416974072 q^{77} - 50004818816 q^{78} + 16783237028 q^{79} - 2021654528 q^{80} + 37556075251 q^{81} - 28257603136 q^{82} + 34191801392 q^{83} + 10905214976 q^{84} + 15946077164 q^{85} - 102051425792 q^{86} - 1220720160 q^{87} + 15831957504 q^{88} + 13083391472 q^{89} - 120457124544 q^{90} + 10703174664 q^{91} + 10122934272 q^{92} - 122324981590 q^{93} - 60085760768 q^{94} - 94454124348 q^{95} + 12415139840 q^{96} + 51624300308 q^{97} - 147726305184 q^{98} + 58337986883 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 206434x + 34594984 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} + 261\nu - 137728 ) / 54 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{2} + 279\nu + 137539 ) / 27 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + 2\beta _1 + 7 ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -261\beta_{2} + 558\beta _1 + 2752733 ) / 20 \) 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
306.200
−521.746
216.545
32.0000 −542.722 1024.00 −8758.18 −17367.1 21611.7 32768.0 117400. −280262.
1.2 32.0000 154.206 1024.00 9891.53 4934.59 9234.32 32768.0 −153368. 316529.
1.3 32.0000 758.516 1024.00 −3061.35 24272.5 27626.0 32768.0 398200. −97963.4
\(n\): 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.d 3
3.b odd 2 1 198.12.a.j 3
4.b odd 2 1 176.12.a.b 3
11.b odd 2 1 242.12.a.b 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.12.a.d 3 1.a even 1 1 trivial
176.12.a.b 3 4.b odd 2 1
198.12.a.j 3 3.b odd 2 1
242.12.a.b 3 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{3} - 370T_{3}^{2} - 378387T_{3} + 63480996 \) 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
$3$ \( T^{3} - 370 T^{2} + \cdots + 63480996 \) Copy content Toggle raw display
$5$ \( T^{3} + 1928 T^{2} + \cdots - 265210541550 \) Copy content Toggle raw display
$7$ \( T^{3} - 58472 T^{2} + \cdots - 5513298935200 \) Copy content Toggle raw display
$11$ \( (T - 161051)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 1349138 T^{2} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( T^{3} - 4446434 T^{2} + \cdots + 12\!\cdots\!28 \) Copy content Toggle raw display
$19$ \( T^{3} - 13084936 T^{2} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{3} - 9885678 T^{2} + \cdots - 17\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{3} - 283172750 T^{2} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{3} + 21681514 T^{2} + \cdots - 14\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{3} + 400193332 T^{2} + \cdots - 44\!\cdots\!66 \) Copy content Toggle raw display
$41$ \( T^{3} + 883050098 T^{2} + \cdots - 12\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{3} + 3189107056 T^{2} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{3} + 1877680024 T^{2} + \cdots - 18\!\cdots\!04 \) Copy content Toggle raw display
$53$ \( T^{3} + 3405155206 T^{2} + \cdots - 27\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{3} - 14124901902 T^{2} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{3} - 7808412854 T^{2} + \cdots + 88\!\cdots\!40 \) Copy content Toggle raw display
$67$ \( T^{3} - 871235950 T^{2} + \cdots - 39\!\cdots\!28 \) Copy content Toggle raw display
$71$ \( T^{3} + 11743560686 T^{2} + \cdots - 63\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{3} - 12566307382 T^{2} + \cdots + 96\!\cdots\!48 \) Copy content Toggle raw display
$79$ \( T^{3} - 16783237028 T^{2} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{3} - 34191801392 T^{2} + \cdots + 52\!\cdots\!48 \) Copy content Toggle raw display
$89$ \( T^{3} - 13083391472 T^{2} + \cdots - 62\!\cdots\!50 \) Copy content Toggle raw display
$97$ \( T^{3} - 51624300308 T^{2} + \cdots + 10\!\cdots\!94 \) Copy content Toggle raw display
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