Properties

Label 22.12.a.a
Level $22$
Weight $12$
Character orbit 22.a
Self dual yes
Analytic conductor $16.904$
Analytic rank $1$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{331}) \)
Defining polynomial: \( x^{2} - 331 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8\sqrt{331}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 32 q^{2} + (3 \beta - 213) q^{3} + 1024 q^{4} + ( - 50 \beta + 1145) q^{5} + (96 \beta - 6816) q^{6} + ( - 165 \beta - 43162) q^{7} + 32768 q^{8} + ( - 1278 \beta + 58878) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 32 q^{2} + (3 \beta - 213) q^{3} + 1024 q^{4} + ( - 50 \beta + 1145) q^{5} + (96 \beta - 6816) q^{6} + ( - 165 \beta - 43162) q^{7} + 32768 q^{8} + ( - 1278 \beta + 58878) q^{9} + ( - 1600 \beta + 36640) q^{10} - 161051 q^{11} + (3072 \beta - 218112) q^{12} + (8839 \beta - 1050092) q^{13} + ( - 5280 \beta - 1381184) q^{14} + (14085 \beta - 3421485) q^{15} + 1048576 q^{16} + (19951 \beta - 1441138) q^{17} + ( - 40896 \beta + 1884096) q^{18} + (6371 \beta - 9785856) q^{19} + ( - 51200 \beta + 1172480) q^{20} + ( - 94341 \beta - 1292574) q^{21} - 5153632 q^{22} + (106463 \beta + 6340267) q^{23} + (98304 \beta - 6979584) q^{24} + ( - 114500 \beta + 5442900) q^{25} + (282848 \beta - 33602944) q^{26} + ( - 82593 \beta - 56028159) q^{27} + ( - 168960 \beta - 44197888) q^{28} + ( - 420342 \beta + 22831248) q^{29} + (450720 \beta - 109487520) q^{30} + (190135 \beta - 253252085) q^{31} + 33554432 q^{32} + ( - 483153 \beta + 34303863) q^{33} + (638432 \beta - 46116416) q^{34} + (1969175 \beta + 125347510) q^{35} + ( - 1308672 \beta + 60291072) q^{36} + (2302978 \beta + 201336259) q^{37} + (203872 \beta - 313147392) q^{38} + ( - 5032983 \beta + 785405724) q^{39} + ( - 1638400 \beta + 37519360) q^{40} + ( - 6012597 \beta + 304432008) q^{41} + ( - 3018912 \beta - 41362368) q^{42} + (8085116 \beta + 550047282) q^{43} - 164916224 q^{44} + ( - 4407210 \beta + 1421072910) q^{45} + (3406816 \beta + 202888544) q^{46} + (14560338 \beta + 506171136) q^{47} + (3145728 \beta - 223346688) q^{48} + (14243460 \beta + 462365901) q^{49} + ( - 3664000 \beta + 174172800) q^{50} + ( - 8572977 \beta + 1574888346) q^{51} + (9051136 \beta - 1075294208) q^{52} + ( - 24738110 \beta - 34094638) q^{53} + ( - 2642976 \beta - 1792901088) q^{54} + (8052550 \beta - 184403395) q^{55} + ( - 5406720 \beta - 1414332416) q^{56} + ( - 30714591 \beta + 2489277120) q^{57} + ( - 13450944 \beta + 730599936) q^{58} + (22140099 \beta - 3395808759) q^{59} + (14423040 \beta - 3503600640) q^{60} + ( - 57000010 \beta + 2852023260) q^{61} + (6084320 \beta - 8104066720) q^{62} + (45446166 \beta + 1925777844) q^{63} + 1073741824 q^{64} + (62625255 \beta - 10564624140) q^{65} + ( - 15460896 \beta + 1097723616) q^{66} + ( - 9427609 \beta - 18257155851) q^{67} + (20429824 \beta - 1475725312) q^{68} + ( - 3655818 \beta + 5415459705) q^{69} + (63013600 \beta + 4011120320) q^{70} + ( - 115678293 \beta + 10336098297) q^{71} + ( - 41877504 \beta + 1929314304) q^{72} + (51530311 \beta + 1541435428) q^{73} + (73695296 \beta + 6442760288) q^{74} + (40717200 \beta - 8436041700) q^{75} + (6523904 \beta - 10020716544) q^{76} + (26573415 \beta + 6951283262) q^{77} + ( - 161055456 \beta + 25132983168) q^{78} + ( - 2327652 \beta - 14340691090) q^{79} + ( - 52428800 \beta + 1200619520) q^{80} + (75901698 \beta - 3745013535) q^{81} + ( - 192403104 \beta + 9741824256) q^{82} + ( - 68251162 \beta + 14766320386) q^{83} + ( - 96605184 \beta - 1323595776) q^{84} + (94900795 \beta - 22782202210) q^{85} + (258723712 \beta + 17601513024) q^{86} + (158026590 \beta - 31576630608) q^{87} - 5277319168 q^{88} + (35615342 \beta + 42531871231) q^{89} + ( - 141030720 \beta + 45474333120) q^{90} + ( - 208243738 \beta + 14428583864) q^{91} + (109018112 \beta + 6492433408) q^{92} + ( - 800255010 \beta + 66026153625) q^{93} + (465930816 \beta + 16197476352) q^{94} + (496587595 \beta - 17952968320) q^{95} + (100663296 \beta - 7147094016) q^{96} + ( - 149119060 \beta - 90416224839) q^{97} + (455790720 \beta + 14795708832) q^{98} + (205823178 \beta - 9482360778) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 64 q^{2} - 426 q^{3} + 2048 q^{4} + 2290 q^{5} - 13632 q^{6} - 86324 q^{7} + 65536 q^{8} + 117756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 64 q^{2} - 426 q^{3} + 2048 q^{4} + 2290 q^{5} - 13632 q^{6} - 86324 q^{7} + 65536 q^{8} + 117756 q^{9} + 73280 q^{10} - 322102 q^{11} - 436224 q^{12} - 2100184 q^{13} - 2762368 q^{14} - 6842970 q^{15} + 2097152 q^{16} - 2882276 q^{17} + 3768192 q^{18} - 19571712 q^{19} + 2344960 q^{20} - 2585148 q^{21} - 10307264 q^{22} + 12680534 q^{23} - 13959168 q^{24} + 10885800 q^{25} - 67205888 q^{26} - 112056318 q^{27} - 88395776 q^{28} + 45662496 q^{29} - 218975040 q^{30} - 506504170 q^{31} + 67108864 q^{32} + 68607726 q^{33} - 92232832 q^{34} + 250695020 q^{35} + 120582144 q^{36} + 402672518 q^{37} - 626294784 q^{38} + 1570811448 q^{39} + 75038720 q^{40} + 608864016 q^{41} - 82724736 q^{42} + 1100094564 q^{43} - 329832448 q^{44} + 2842145820 q^{45} + 405777088 q^{46} + 1012342272 q^{47} - 446693376 q^{48} + 924731802 q^{49} + 348345600 q^{50} + 3149776692 q^{51} - 2150588416 q^{52} - 68189276 q^{53} - 3585802176 q^{54} - 368806790 q^{55} - 2828664832 q^{56} + 4978554240 q^{57} + 1461199872 q^{58} - 6791617518 q^{59} - 7007201280 q^{60} + 5704046520 q^{61} - 16208133440 q^{62} + 3851555688 q^{63} + 2147483648 q^{64} - 21129248280 q^{65} + 2195447232 q^{66} - 36514311702 q^{67} - 2951450624 q^{68} + 10830919410 q^{69} + 8022240640 q^{70} + 20672196594 q^{71} + 3858628608 q^{72} + 3082870856 q^{73} + 12885520576 q^{74} - 16872083400 q^{75} - 20041433088 q^{76} + 13902566524 q^{77} + 50265966336 q^{78} - 28681382180 q^{79} + 2401239040 q^{80} - 7490027070 q^{81} + 19483648512 q^{82} + 29532640772 q^{83} - 2647191552 q^{84} - 45564404420 q^{85} + 35203026048 q^{86} - 63153261216 q^{87} - 10554638336 q^{88} + 85063742462 q^{89} + 90948666240 q^{90} + 28857167728 q^{91} + 12984866816 q^{92} + 132052307250 q^{93} + 32394952704 q^{94} - 35905936640 q^{95} - 14294188032 q^{96} - 180832449678 q^{97} + 29591417664 q^{98} - 18964721556 q^{99}+O(q^{100}) \) 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
−18.1934
18.1934
32.0000 −649.642 1024.00 8422.36 −20788.5 −19146.7 32768.0 244887. 269516.
1.2 32.0000 223.642 1024.00 −6132.36 7156.54 −67177.3 32768.0 −127131. −196236.
\(n\): 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.a 2
3.b odd 2 1 198.12.a.c 2
4.b odd 2 1 176.12.a.a 2
11.b odd 2 1 242.12.a.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.12.a.a 2 1.a even 1 1 trivial
176.12.a.a 2 4.b odd 2 1
198.12.a.c 2 3.b odd 2 1
242.12.a.a 2 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 426T_{3} - 145287 \) acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 32)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 426T - 145287 \) Copy content Toggle raw display
$5$ \( T^{2} - 2290 T - 51648975 \) Copy content Toggle raw display
$7$ \( T^{2} + 86324 T + 1286223844 \) Copy content Toggle raw display
$11$ \( (T + 161051)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 2100184 T - 552368670000 \) Copy content Toggle raw display
$17$ \( T^{2} + 2882276 T - 6355251487740 \) Copy content Toggle raw display
$19$ \( T^{2} + 19571712 T + 94903126697792 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 199908316265607 \) Copy content Toggle raw display
$29$ \( T^{2} - 45662496 T - 32\!\cdots\!72 \) Copy content Toggle raw display
$31$ \( T^{2} + 506504170 T + 63\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{2} - 402672518 T - 71\!\cdots\!75 \) Copy content Toggle raw display
$41$ \( T^{2} - 608864016 T - 67\!\cdots\!92 \) Copy content Toggle raw display
$43$ \( T^{2} - 1100094564 T - 10\!\cdots\!80 \) Copy content Toggle raw display
$47$ \( T^{2} - 1012342272 T - 42\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{2} + 68189276 T - 12\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( T^{2} + 6791617518 T + 11\!\cdots\!97 \) Copy content Toggle raw display
$61$ \( T^{2} - 5704046520 T - 60\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} + 36514311702 T + 33\!\cdots\!97 \) Copy content Toggle raw display
$71$ \( T^{2} - 20672196594 T - 17\!\cdots\!07 \) Copy content Toggle raw display
$73$ \( T^{2} - 3082870856 T - 53\!\cdots\!80 \) Copy content Toggle raw display
$79$ \( T^{2} + 28681382180 T + 20\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{2} - 29532640772 T + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( T^{2} - 85063742462 T + 17\!\cdots\!85 \) Copy content Toggle raw display
$97$ \( T^{2} + 180832449678 T + 77\!\cdots\!21 \) Copy content Toggle raw display
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