Properties

Label 8.20.a.a
Level 8
Weight 20
Character orbit 8.a
Self dual Yes
Analytic conductor 18.305
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 20 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(18.3053357245\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1453}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3\cdot 5 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 960\sqrt{1453}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( -13956 - \beta ) q^{3} \) \( + ( 613310 + 44 \beta ) q^{5} \) \( + ( 44255256 + 3190 \beta ) q^{7} \) \( + ( 371593269 + 27912 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( -13956 - \beta ) q^{3} \) \( + ( 613310 + 44 \beta ) q^{5} \) \( + ( 44255256 + 3190 \beta ) q^{7} \) \( + ( 371593269 + 27912 \beta ) q^{9} \) \( + ( -3581893804 - 223467 \beta ) q^{11} \) \( + ( -5063461802 + 71660 \beta ) q^{13} \) \( + ( -67479085560 - 1227374 \beta ) q^{15} \) \( + ( -36022539470 + 17338504 \beta ) q^{17} \) \( + ( -1560240236116 - 26137813 \beta ) q^{19} \) \( + ( -4889306864736 - 88774896 \beta ) q^{21} \) \( + ( -7379603545144 + 184291330 \beta ) q^{23} \) \( + ( -16104868999225 + 53971280 \beta ) q^{25} \) \( + ( -26341969566312 + 401128326 \beta ) q^{27} \) \( + ( -15124769622522 - 893092484 \beta ) q^{29} \) \( + ( -61694781388960 - 3864337064 \beta ) q^{31} \) \( + ( 349230172930224 + 6700599256 \beta ) q^{33} \) \( + ( 215096133585360 + 3903690164 \beta ) q^{35} \) \( + ( 1007696585087262 - 3072429508 \beta ) q^{37} \) \( + ( -25293143859288 + 4063374842 \beta ) q^{39} \) \( + ( 1270392479752122 - 47315480368 \beta ) q^{41} \) \( + ( -2816827546694732 + 16432445197 \beta ) q^{43} \) \( + ( 1872469405064790 + 33468812556 \beta ) q^{45} \) \( + ( -10974169793565168 + 13905445988 \beta ) q^{47} \) \( + ( 4186293331532393 + 282348533280 \beta ) q^{49} \) \( + ( -22714996600295880 - 205953622354 \beta ) q^{51} \) \( + ( -4709062533452338 - 810595524548 \beta ) q^{53} \) \( + ( -15363426861001640 - 294657873146 \beta ) q^{55} \) \( + ( 56775460828777296 + 1925019554344 \beta ) q^{57} \) \( + ( 49271224795203812 + 1148874921753 \beta ) q^{59} \) \( + ( 5146072688919910 - 3405518362868 \beta ) q^{61} \) \( + ( 135676101698415864 + 2420635233582 \beta ) q^{63} \) \( + ( 1116716180007380 - 178842524688 \beta ) q^{65} \) \( + ( 37876814001992252 - 6468769570097 \beta ) q^{67} \) \( + ( -143791971698754336 + 4807633743664 \beta ) q^{69} \) \( + ( 8703526283356888 - 4032598137882 \beta ) q^{71} \) \( + ( -428754127529916134 + 10723507910184 \beta ) q^{73} \) \( + ( 152487431068640100 + 15351645815545 \beta ) q^{75} \) \( + ( -1113097256235937824 - 21315830527312 \beta ) q^{77} \) \( + ( -113145960722427536 - 8826033368644 \beta ) q^{79} \) \( + ( -601404854883860151 - 11697219418248 \beta ) q^{81} \) \( + ( 383757850730492524 + 1875501918907 \beta ) q^{83} \) \( + ( 999487011407779100 + 9048886151560 \beta ) q^{85} \) \( + ( 1407007855170560232 + 27588768329226 \beta ) q^{87} \) \( + ( 3046272947217587274 + 7105196196264 \beta ) q^{89} \) \( + ( 82023827196188688 - 12981111503420 \beta ) q^{91} \) \( + ( 6035687393543352960 + 115625469454144 \beta ) q^{93} \) \( + ( -2496943855328169560 - 84681152480134 \beta ) q^{95} \) \( + ( 774074124761173538 - 186089076998104 \beta ) q^{97} \) \( + ( -9683429760739864476 - 183016652900871 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 27912q^{3} \) \(\mathstrut +\mathstrut 1226620q^{5} \) \(\mathstrut +\mathstrut 88510512q^{7} \) \(\mathstrut +\mathstrut 743186538q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 27912q^{3} \) \(\mathstrut +\mathstrut 1226620q^{5} \) \(\mathstrut +\mathstrut 88510512q^{7} \) \(\mathstrut +\mathstrut 743186538q^{9} \) \(\mathstrut -\mathstrut 7163787608q^{11} \) \(\mathstrut -\mathstrut 10126923604q^{13} \) \(\mathstrut -\mathstrut 134958171120q^{15} \) \(\mathstrut -\mathstrut 72045078940q^{17} \) \(\mathstrut -\mathstrut 3120480472232q^{19} \) \(\mathstrut -\mathstrut 9778613729472q^{21} \) \(\mathstrut -\mathstrut 14759207090288q^{23} \) \(\mathstrut -\mathstrut 32209737998450q^{25} \) \(\mathstrut -\mathstrut 52683939132624q^{27} \) \(\mathstrut -\mathstrut 30249539245044q^{29} \) \(\mathstrut -\mathstrut 123389562777920q^{31} \) \(\mathstrut +\mathstrut 698460345860448q^{33} \) \(\mathstrut +\mathstrut 430192267170720q^{35} \) \(\mathstrut +\mathstrut 2015393170174524q^{37} \) \(\mathstrut -\mathstrut 50586287718576q^{39} \) \(\mathstrut +\mathstrut 2540784959504244q^{41} \) \(\mathstrut -\mathstrut 5633655093389464q^{43} \) \(\mathstrut +\mathstrut 3744938810129580q^{45} \) \(\mathstrut -\mathstrut 21948339587130336q^{47} \) \(\mathstrut +\mathstrut 8372586663064786q^{49} \) \(\mathstrut -\mathstrut 45429993200591760q^{51} \) \(\mathstrut -\mathstrut 9418125066904676q^{53} \) \(\mathstrut -\mathstrut 30726853722003280q^{55} \) \(\mathstrut +\mathstrut 113550921657554592q^{57} \) \(\mathstrut +\mathstrut 98542449590407624q^{59} \) \(\mathstrut +\mathstrut 10292145377839820q^{61} \) \(\mathstrut +\mathstrut 271352203396831728q^{63} \) \(\mathstrut +\mathstrut 2233432360014760q^{65} \) \(\mathstrut +\mathstrut 75753628003984504q^{67} \) \(\mathstrut -\mathstrut 287583943397508672q^{69} \) \(\mathstrut +\mathstrut 17407052566713776q^{71} \) \(\mathstrut -\mathstrut 857508255059832268q^{73} \) \(\mathstrut +\mathstrut 304974862137280200q^{75} \) \(\mathstrut -\mathstrut 2226194512471875648q^{77} \) \(\mathstrut -\mathstrut 226291921444855072q^{79} \) \(\mathstrut -\mathstrut 1202809709767720302q^{81} \) \(\mathstrut +\mathstrut 767515701460985048q^{83} \) \(\mathstrut +\mathstrut 1998974022815558200q^{85} \) \(\mathstrut +\mathstrut 2814015710341120464q^{87} \) \(\mathstrut +\mathstrut 6092545894435174548q^{89} \) \(\mathstrut +\mathstrut 164047654392377376q^{91} \) \(\mathstrut +\mathstrut 12071374787086705920q^{93} \) \(\mathstrut -\mathstrut 4993887710656339120q^{95} \) \(\mathstrut +\mathstrut 1548148249522347076q^{97} \) \(\mathstrut -\mathstrut 19366859521479728952q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
19.5591
−18.5591
0 −50549.5 0 2.22342e6 0 1.60989e8 0 1.39299e9 0
1.2 0 22637.5 0 −996804. 0 −7.24780e7 0 −6.49805e8 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 \(T_{3}^{2} \) \(\mathstrut +\mathstrut 27912 T_{3} \) \(\mathstrut -\mathstrut 1144314864 \) acting on \(S_{20}^{\mathrm{new}}(\Gamma_0(8))\).