Properties

Label 22.20.a.b
Level $22$
Weight $20$
Character orbit 22.a
Self dual yes
Analytic conductor $50.340$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 22 = 2 \cdot 11 \)
Weight: \( k \) \(=\) \( 20 \)
Character orbit: \([\chi]\) \(=\) 22.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(50.3396732424\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{51151}) \)
Defining polynomial: \( x^{2} - 51151 \) 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 \(\beta = 4\sqrt{51151}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 512 q^{2} + (9 \beta + 8487) q^{3} + 262144 q^{4} + ( - 4490 \beta - 1474255) q^{5} + (4608 \beta + 4345344) q^{6} + (53625 \beta + 42464978) q^{7} + 134217728 q^{8} + (152766 \beta - 1023940602) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 512 q^{2} + (9 \beta + 8487) q^{3} + 262144 q^{4} + ( - 4490 \beta - 1474255) q^{5} + (4608 \beta + 4345344) q^{6} + (53625 \beta + 42464978) q^{7} + 134217728 q^{8} + (152766 \beta - 1023940602) q^{9} + ( - 2298880 \beta - 754818560) q^{10} - 2357947691 q^{11} + (2359296 \beta + 2224816128) q^{12} + (60013597 \beta + 6465678928) q^{13} + (27456000 \beta + 21742068736) q^{14} + ( - 51374925 \beta - 45584192745) q^{15} + 68719476736 q^{16} + ( - 184614827 \beta - 229893652438) q^{17} + (78216192 \beta - 524257588224) q^{18} + ( - 573880807 \beta - 883314463956) q^{19} + ( - 1177026560 \beta - 386467102720) q^{20} + (837300177 \beta + 755388290286) q^{21} - 1207269217792 q^{22} + ( - 12173805511 \beta - 1633300048073) q^{23} + (1207959552 \beta + 1139105857536) q^{24} + (13238809900 \beta - 400710121500) q^{25} + (30726961664 \beta + 3310427611136) q^{26} + ( - 18379293579 \beta - 17429061711699) q^{27} + (14057472000 \beta + 11131939192832) q^{28} + ( - 16593817026 \beta - 48556768210632) q^{29} + ( - 26303961600 \beta - 23339106685440) q^{30} + (58051640185 \beta - 77127101577065) q^{31} + 35184372088832 q^{32} + ( - 21221529219 \beta - 20011902053517) q^{33} + ( - 94522791424 \beta - 117705550048256) q^{34} + ( - 269724675595 \beta - 259659341561390) q^{35} + (40046690304 \beta - 268419885170688) q^{36} + (550020408874 \beta - 285696382422101) q^{37} + ( - 293826973184 \beta - 452257005545472) q^{38} + (567526508091 \beta + 496919009083104) q^{39} + ( - 602637598720 \beta - 197871156592640) q^{40} + (748621877529 \beta - 92460840021612) q^{41} + (428697690624 \beta + 386758804626432) q^{42} + ( - 432820418812 \beta + 10\!\cdots\!02) q^{43}+ \cdots + ( - 360214236963306 \beta + 24\!\cdots\!82) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 1024 q^{2} + 16974 q^{3} + 524288 q^{4} - 2948510 q^{5} + 8690688 q^{6} + 84929956 q^{7} + 268435456 q^{8} - 2047881204 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 1024 q^{2} + 16974 q^{3} + 524288 q^{4} - 2948510 q^{5} + 8690688 q^{6} + 84929956 q^{7} + 268435456 q^{8} - 2047881204 q^{9} - 1509637120 q^{10} - 4715895382 q^{11} + 4449632256 q^{12} + 12931357856 q^{13} + 43484137472 q^{14} - 91168385490 q^{15} + 137438953472 q^{16} - 459787304876 q^{17} - 1048515176448 q^{18} - 1766628927912 q^{19} - 772934205440 q^{20} + 1510776580572 q^{21} - 2414538435584 q^{22} - 3266600096146 q^{23} + 2278211715072 q^{24} - 801420243000 q^{25} + 6620855222272 q^{26} - 34858123423398 q^{27} + 22263878385664 q^{28} - 97113536421264 q^{29} - 46678213370880 q^{30} - 154254203154130 q^{31} + 70368744177664 q^{32} - 40023804107034 q^{33} - 235411100096512 q^{34} - 519318683122780 q^{35} - 536839770341376 q^{36} - 571392764844202 q^{37} - 904514011090944 q^{38} + 993838018166208 q^{39} - 395742313185280 q^{40} - 184921680043224 q^{41} + 773517609252864 q^{42} + 21\!\cdots\!04 q^{43}+ \cdots + 48\!\cdots\!64 q^{99}+O(q^{100}) \) 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
−226.166
226.166
512.000 345.029 262144. 2.58768e6 176655. −6.04760e6 1.34218e8 −1.16214e9 1.32489e9
1.2 512.000 16629.0 262144. −5.53619e6 8.51403e6 9.09776e7 1.34218e8 −8.85739e8 −2.83453e9
\(n\): 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 22.20.a.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
22.20.a.b 2 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 16974T_{3} + 5737473 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(22))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 512)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 16974 T + 5737473 \) Copy content Toggle raw display
$5$ \( T^{2} + 2948510 T - 14325920596575 \) Copy content Toggle raw display
$7$ \( T^{2} + \cdots - 550195941209516 \) Copy content Toggle raw display
$11$ \( (T + 2357947691)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 12931357856 T - 29\!\cdots\!60 \) Copy content Toggle raw display
$17$ \( T^{2} + 459787304876 T + 24\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( T^{2} + 1766628927912 T + 51\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( T^{2} + 3266600096146 T - 11\!\cdots\!07 \) Copy content Toggle raw display
$29$ \( T^{2} + 97113536421264 T + 21\!\cdots\!08 \) Copy content Toggle raw display
$31$ \( T^{2} + 154254203154130 T + 31\!\cdots\!25 \) Copy content Toggle raw display
$37$ \( T^{2} + 571392764844202 T - 16\!\cdots\!15 \) Copy content Toggle raw display
$41$ \( T^{2} + 184921680043224 T - 45\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots - 32\!\cdots\!20 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots - 73\!\cdots\!16 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 12\!\cdots\!83 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 39\!\cdots\!57 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 10\!\cdots\!33 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 32\!\cdots\!20 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{2} - 974129749376372 T - 11\!\cdots\!60 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 86\!\cdots\!75 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 56\!\cdots\!41 \) Copy content Toggle raw display
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