Properties

Label 8.22.a.a
Level 8
Weight 22
Character orbit 8.a
Self dual yes
Analytic conductor 22.358
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 8.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.3581875430\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{358549}) \)
Defining polynomial: \(x^{2} - x - 89637\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 192\sqrt{358549}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -52716 - \beta ) q^{3} + ( 1054070 + 20 \beta ) q^{5} + ( 222385896 + 4222 \beta ) q^{7} + ( 5536173789 + 105432 \beta ) q^{9} +O(q^{10})\) \( q + ( -52716 - \beta ) q^{3} + ( 1054070 + 20 \beta ) q^{5} + ( 222385896 + 4222 \beta ) q^{7} + ( 5536173789 + 105432 \beta ) q^{9} + ( 26903201660 + 527181 \beta ) q^{11} + ( -245183338466 - 4314508 \beta ) q^{13} + ( -319917360840 - 2108390 \beta ) q^{15} + ( -3296932336046 - 30718184 \beta ) q^{17} + ( 9651198962660 + 366379451 \beta ) q^{19} + ( -67527792412128 - 444952848 \beta ) q^{21} + ( -204868932888136 - 944258822 \beta ) q^{23} + ( -470439074503825 + 42162800 \beta ) q^{25} + ( -1133969725036728 - 633773898 \beta ) q^{27} + ( -1202393761072530 + 23192603812 \beta ) q^{29} + ( -4344953585279584 - 18061983176 \beta ) q^{31} + ( -8386270582391376 - 54694075256 \beta ) q^{33} + ( 1350500251768560 + 8898001460 \beta ) q^{35} + ( -1093102048125930 - 50855746684 \beta ) q^{37} + ( 69952311535648344 + 472626942194 \beta ) q^{39} + ( 34089019286779338 - 13526799376 \beta ) q^{41} + ( 132264826133002012 - 769879839131 \beta ) q^{43} + ( 33706570046274270 + 221856184020 \beta ) q^{45} + ( 213247205779311216 - 2319694970380 \beta ) q^{47} + ( -273483788820065767 + 1877826505824 \beta ) q^{49} + ( 579820228277510760 + 4916272123790 \beta ) q^{51} + ( -1527990137794759066 + 2918184305860 \beta ) q^{53} + ( 167718685847412520 + 1093749709870 \beta ) q^{55} + ( -5351411440184130096 - 28965258101576 \beta ) q^{57} + ( 391712498761407212 - 7304311733439 \beta ) q^{59} + ( -3588639524539298546 + 23337258872596 \beta ) q^{61} + ( 7114746750858671688 + 46820315524230 \beta ) q^{63} + ( -1398984934878350380 - 9451460216880 \beta ) q^{65} + ( 8337061587005769044 - 77794800439793 \beta ) q^{67} + ( 23280659176128041568 + 254646480948688 \beta ) q^{69} + ( 4724131574924358184 - 231241233933906 \beta ) q^{71} + ( -5793070167251503766 - 528252850485192 \beta ) q^{73} + ( 24242377320236937900 + 468216420339025 \beta ) q^{75} + ( 35401963412776636512 + 230822936447696 \beta ) q^{77} + ( -42640351109357948912 + 355065026452268 \beta ) q^{79} + ( 10244953197363286809 + 64523790945000 \beta ) q^{81} + ( -190907311020043122908 - 591846679323509 \beta ) q^{83} + ( -11595580332466203700 - 98317762929800 \beta ) q^{85} + ( -243164018799315989352 - 20227541480862 \beta ) q^{87} + ( -29871466215347989830 - 1314255056990664 \beta ) q^{89} + ( -295294267388978008272 - 1994649782382620 \beta ) q^{91} + ( 467783744998363697280 + 5297109090385600 \beta ) q^{93} + ( 107025816003941936920 + 579213567168770 \beta ) q^{95} + ( 391697330463355975394 + 156001529627000 \beta ) q^{97} + ( 883595341143359946252 + 5755023991675929 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 105432q^{3} + 2108140q^{5} + 444771792q^{7} + 11072347578q^{9} + O(q^{10}) \) \( 2q - 105432q^{3} + 2108140q^{5} + 444771792q^{7} + 11072347578q^{9} + 53806403320q^{11} - 490366676932q^{13} - 639834721680q^{15} - 6593864672092q^{17} + 19302397925320q^{19} - 135055584824256q^{21} - 409737865776272q^{23} - 940878149007650q^{25} - 2267939450073456q^{27} - 2404787522145060q^{29} - 8689907170559168q^{31} - 16772541164782752q^{33} + 2701000503537120q^{35} - 2186204096251860q^{37} + 139904623071296688q^{39} + 68178038573558676q^{41} + 264529652266004024q^{43} + 67413140092548540q^{45} + 426494411558622432q^{47} - 546967577640131534q^{49} + 1159640456555021520q^{51} - 3055980275589518132q^{53} + 335437371694825040q^{55} - 10702822880368260192q^{57} + 783424997522814424q^{59} - 7177279049078597092q^{61} + 14229493501717343376q^{63} - 2797969869756700760q^{65} + 16674123174011538088q^{67} + 46561318352256083136q^{69} + 9448263149848716368q^{71} - 11586140334503007532q^{73} + 48484754640473875800q^{75} + 70803926825553273024q^{77} - 85280702218715897824q^{79} + 20489906394726573618q^{81} - 381814622040086245816q^{83} - 23191160664932407400q^{85} - 486328037598631978704q^{87} - 59742932430695979660q^{89} - 590588534777956016544q^{91} + 935567489996727394560q^{93} + 214051632007883873840q^{95} + 783394660926711950788q^{97} + 1767190682286719892504q^{99} + O(q^{100}) \)

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
299.895
−298.895
0 −167684. 0 3.35342e6 0 7.07779e8 0 1.76574e10 0
1.2 0 62251.6 0 −1.24528e6 0 −2.63007e8 0 −6.58509e9 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)

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 8.22.a.a 2
3.b odd 2 1 72.22.a.b 2
4.b odd 2 1 16.22.a.e 2
8.b even 2 1 64.22.a.k 2
8.d odd 2 1 64.22.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.a.a 2 1.a even 1 1 trivial
16.22.a.e 2 4.b odd 2 1
64.22.a.h 2 8.d odd 2 1
64.22.a.k 2 8.b even 2 1
72.22.a.b 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 105432 T_{3} - 10438573680 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(8))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 105432 T + 10482132726 T^{2} + 1102855958898696 T^{3} + \)\(10\!\cdots\!09\)\( T^{4} \)
$5$ \( 1 - 2108140 T + 949498359836750 T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(22\!\cdots\!25\)\( T^{4} \)
$7$ \( 1 - 444771792 T + 930940626382795406 T^{2} - \)\(24\!\cdots\!44\)\( T^{3} + \)\(31\!\cdots\!49\)\( T^{4} \)
$11$ \( 1 - 53806403320 T + \)\(11\!\cdots\!26\)\( T^{2} - \)\(39\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!21\)\( T^{4} \)
$13$ \( 1 + 490366676932 T + \)\(30\!\cdots\!78\)\( T^{2} + \)\(12\!\cdots\!16\)\( T^{3} + \)\(61\!\cdots\!69\)\( T^{4} \)
$17$ \( 1 + 6593864672092 T + \)\(13\!\cdots\!34\)\( T^{2} + \)\(45\!\cdots\!64\)\( T^{3} + \)\(47\!\cdots\!89\)\( T^{4} \)
$19$ \( 1 - 19302397925320 T - \)\(25\!\cdots\!98\)\( T^{2} - \)\(13\!\cdots\!80\)\( T^{3} + \)\(51\!\cdots\!61\)\( T^{4} \)
$23$ \( 1 + 409737865776272 T + \)\(10\!\cdots\!18\)\( T^{2} + \)\(16\!\cdots\!56\)\( T^{3} + \)\(15\!\cdots\!29\)\( T^{4} \)
$29$ \( 1 + 2404787522145060 T + \)\(46\!\cdots\!74\)\( T^{2} + \)\(12\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!41\)\( T^{4} \)
$31$ \( 1 + 8689907170559168 T + \)\(56\!\cdots\!82\)\( T^{2} + \)\(18\!\cdots\!08\)\( T^{3} + \)\(43\!\cdots\!61\)\( T^{4} \)
$37$ \( 1 + 2186204096251860 T + \)\(16\!\cdots\!58\)\( T^{2} + \)\(18\!\cdots\!20\)\( T^{3} + \)\(73\!\cdots\!69\)\( T^{4} \)
$41$ \( 1 - 68178038573558676 T + \)\(15\!\cdots\!90\)\( T^{2} - \)\(50\!\cdots\!16\)\( T^{3} + \)\(54\!\cdots\!81\)\( T^{4} \)
$43$ \( 1 - 264529652266004024 T + \)\(49\!\cdots\!34\)\( T^{2} - \)\(53\!\cdots\!32\)\( T^{3} + \)\(40\!\cdots\!49\)\( T^{4} \)
$47$ \( 1 - 426494411558622432 T + \)\(23\!\cdots\!50\)\( T^{2} - \)\(55\!\cdots\!04\)\( T^{3} + \)\(16\!\cdots\!09\)\( T^{4} \)
$53$ \( 1 + 3055980275589518132 T + \)\(54\!\cdots\!62\)\( T^{2} + \)\(49\!\cdots\!96\)\( T^{3} + \)\(26\!\cdots\!09\)\( T^{4} \)
$59$ \( 1 - 783424997522814424 T + \)\(30\!\cdots\!06\)\( T^{2} - \)\(12\!\cdots\!16\)\( T^{3} + \)\(23\!\cdots\!81\)\( T^{4} \)
$61$ \( 1 + 7177279049078597092 T + \)\(67\!\cdots\!62\)\( T^{2} + \)\(22\!\cdots\!12\)\( T^{3} + \)\(96\!\cdots\!21\)\( T^{4} \)
$67$ \( 1 - 16674123174011538088 T + \)\(43\!\cdots\!06\)\( T^{2} - \)\(37\!\cdots\!96\)\( T^{3} + \)\(49\!\cdots\!89\)\( T^{4} \)
$71$ \( 1 - 9448263149848716368 T + \)\(82\!\cdots\!02\)\( T^{2} - \)\(71\!\cdots\!28\)\( T^{3} + \)\(56\!\cdots\!41\)\( T^{4} \)
$73$ \( 1 + 11586140334503007532 T - \)\(95\!\cdots\!02\)\( T^{2} + \)\(15\!\cdots\!36\)\( T^{3} + \)\(18\!\cdots\!29\)\( T^{4} \)
$79$ \( 1 + 85280702218715897824 T + \)\(14\!\cdots\!38\)\( T^{2} + \)\(60\!\cdots\!96\)\( T^{3} + \)\(50\!\cdots\!41\)\( T^{4} \)
$83$ \( 1 + \)\(38\!\cdots\!16\)\( T + \)\(71\!\cdots\!14\)\( T^{2} + \)\(76\!\cdots\!28\)\( T^{3} + \)\(39\!\cdots\!89\)\( T^{4} \)
$89$ \( 1 + 59742932430695979660 T + \)\(15\!\cdots\!22\)\( T^{2} + \)\(51\!\cdots\!40\)\( T^{3} + \)\(74\!\cdots\!21\)\( T^{4} \)
$97$ \( 1 - \)\(78\!\cdots\!88\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(41\!\cdots\!36\)\( T^{3} + \)\(27\!\cdots\!09\)\( T^{4} \)
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