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: \(22.358187543\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{358549}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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.

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

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:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

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

Hecke kernels

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