Properties

Label 8.18.a.b
Level 8
Weight 18
Character orbit 8.a
Self dual Yes
Analytic conductor 14.658
Analytic rank 1
Dimension 2
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 18 \)
Character orbit: \([\chi]\) = 8.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(14.6577669876\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{114}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2} \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 1152\sqrt{114}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + ( 5796 + \beta ) q^{3} \) \( + ( -395962 - 68 \beta ) q^{5} \) \( + ( -9466296 - 1582 \beta ) q^{7} \) \( + ( 55743309 + 11592 \beta ) q^{9} \) \(+O(q^{10})\) \( q\) \( + ( 5796 + \beta ) q^{3} \) \( + ( -395962 - 68 \beta ) q^{5} \) \( + ( -9466296 - 1582 \beta ) q^{7} \) \( + ( 55743309 + 11592 \beta ) q^{9} \) \( + ( -235385620 - 10269 \beta ) q^{11} \) \( + ( -1251761714 + 49948 \beta ) q^{13} \) \( + ( -12582705960 - 790090 \beta ) q^{15} \) \( + ( -24222344174 + 1366664 \beta ) q^{17} \) \( + ( -41241650380 + 6247141 \beta ) q^{19} \) \( + ( -294207203808 - 18635568 \beta ) q^{21} \) \( + ( -153548515624 - 7579978 \beta ) q^{23} \) \( + ( 93410746463 + 53850832 \beta ) q^{25} \) \( + ( 1328343844968 - 6209622 \beta ) q^{27} \) \( + ( 994787693790 + 46341932 \beta ) q^{29} \) \( + ( 5376066787616 - 263508856 \beta ) q^{31} \) \( + ( -2917890584784 - 294904744 \beta ) q^{33} \) \( + ( 20023451045808 + 1270120012 \beta ) q^{35} \) \( + ( -25587226374810 - 441997556 \beta ) q^{37} \) \( + ( 301414833144 - 962263106 \beta ) q^{39} \) \( + ( -57113645522262 + 2940525904 \beta ) q^{41} \) \( + ( -28378777101812 - 4230485749 \beta ) q^{43} \) \( + ( -141327368849394 - 8380536516 \beta ) q^{45} \) \( + ( 100776581579184 + 10149312940 \beta ) q^{47} \) \( + ( 235616999540153 + 29951360544 \beta ) q^{49} \) \( + ( 66369692927880 - 16301159630 \beta ) q^{51} \) \( + ( 108637731793526 - 56505234100 \beta ) q^{53} \) \( + ( 198848256992392 + 20072355938 \beta ) q^{55} \) \( + ( 706092456699216 - 5033221144 \beta ) q^{57} \) \( + ( -233155724445028 + 65093222991 \beta ) q^{59} \) \( + ( 1031908631813374 + 101795756156 \beta ) q^{61} \) \( + ( -3302118344023128 - 197919218070 \beta ) q^{63} \) \( + ( -18200477670316 + 65342286576 \beta ) q^{65} \) \( + ( -3318612981179164 - 53283505807 \beta ) q^{67} \) \( + ( -2036740976659872 - 197482068112 \beta ) q^{69} \) \( + ( 771842498992264 + 360055439394 \beta ) q^{71} \) \( + ( -1867053400179254 - 288694867608 \beta ) q^{73} \) \( + ( 8688493305259740 + 405530168735 \beta ) q^{75} \) \( + ( 4686018083523168 + 469589444464 \beta ) q^{77} \) \( + ( 8877689253665488 - 575220781772 \beta ) q^{79} \) \( + ( -439071903179271 - 204639893640 \beta ) q^{81} \) \( + ( 12726798779079028 - 1526245359211 \beta ) q^{83} \) \( + ( -4468715339880724 + 1105972393064 \beta ) q^{85} \) \( + ( 12776853692248632 + 1263385531662 \beta ) q^{87} \) \( + ( -33608419329601830 - 1843251450264 \beta ) q^{89} \) \( + ( -105034994694672 + 1507464478940 \beta ) q^{91} \) \( + ( -8706533777942400 + 3848769458240 \beta ) q^{93} \) \( + ( -47938649868749768 + 330801781198 \beta ) q^{95} \) \( + ( -65489512430721694 - 6403182311000 \beta ) q^{97} \) \( + ( -31130452748228868 - 3301018147161 \beta ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut +\mathstrut 11592q^{3} \) \(\mathstrut -\mathstrut 791924q^{5} \) \(\mathstrut -\mathstrut 18932592q^{7} \) \(\mathstrut +\mathstrut 111486618q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut +\mathstrut 11592q^{3} \) \(\mathstrut -\mathstrut 791924q^{5} \) \(\mathstrut -\mathstrut 18932592q^{7} \) \(\mathstrut +\mathstrut 111486618q^{9} \) \(\mathstrut -\mathstrut 470771240q^{11} \) \(\mathstrut -\mathstrut 2503523428q^{13} \) \(\mathstrut -\mathstrut 25165411920q^{15} \) \(\mathstrut -\mathstrut 48444688348q^{17} \) \(\mathstrut -\mathstrut 82483300760q^{19} \) \(\mathstrut -\mathstrut 588414407616q^{21} \) \(\mathstrut -\mathstrut 307097031248q^{23} \) \(\mathstrut +\mathstrut 186821492926q^{25} \) \(\mathstrut +\mathstrut 2656687689936q^{27} \) \(\mathstrut +\mathstrut 1989575387580q^{29} \) \(\mathstrut +\mathstrut 10752133575232q^{31} \) \(\mathstrut -\mathstrut 5835781169568q^{33} \) \(\mathstrut +\mathstrut 40046902091616q^{35} \) \(\mathstrut -\mathstrut 51174452749620q^{37} \) \(\mathstrut +\mathstrut 602829666288q^{39} \) \(\mathstrut -\mathstrut 114227291044524q^{41} \) \(\mathstrut -\mathstrut 56757554203624q^{43} \) \(\mathstrut -\mathstrut 282654737698788q^{45} \) \(\mathstrut +\mathstrut 201553163158368q^{47} \) \(\mathstrut +\mathstrut 471233999080306q^{49} \) \(\mathstrut +\mathstrut 132739385855760q^{51} \) \(\mathstrut +\mathstrut 217275463587052q^{53} \) \(\mathstrut +\mathstrut 397696513984784q^{55} \) \(\mathstrut +\mathstrut 1412184913398432q^{57} \) \(\mathstrut -\mathstrut 466311448890056q^{59} \) \(\mathstrut +\mathstrut 2063817263626748q^{61} \) \(\mathstrut -\mathstrut 6604236688046256q^{63} \) \(\mathstrut -\mathstrut 36400955340632q^{65} \) \(\mathstrut -\mathstrut 6637225962358328q^{67} \) \(\mathstrut -\mathstrut 4073481953319744q^{69} \) \(\mathstrut +\mathstrut 1543684997984528q^{71} \) \(\mathstrut -\mathstrut 3734106800358508q^{73} \) \(\mathstrut +\mathstrut 17376986610519480q^{75} \) \(\mathstrut +\mathstrut 9372036167046336q^{77} \) \(\mathstrut +\mathstrut 17755378507330976q^{79} \) \(\mathstrut -\mathstrut 878143806358542q^{81} \) \(\mathstrut +\mathstrut 25453597558158056q^{83} \) \(\mathstrut -\mathstrut 8937430679761448q^{85} \) \(\mathstrut +\mathstrut 25553707384497264q^{87} \) \(\mathstrut -\mathstrut 67216838659203660q^{89} \) \(\mathstrut -\mathstrut 210069989389344q^{91} \) \(\mathstrut -\mathstrut 17413067555884800q^{93} \) \(\mathstrut -\mathstrut 95877299737499536q^{95} \) \(\mathstrut -\mathstrut 130979024861443388q^{97} \) \(\mathstrut -\mathstrut 62260905496457736q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−10.6771
10.6771
0 −6503.99 0 440438. 0 9.99229e6 0 −8.68382e7 0
1.2 0 18096.0 0 −1.23236e6 0 −2.89249e7 0 1.98325e8 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 \(T_{3}^{2} \) \(\mathstrut -\mathstrut 11592 T_{3} \) \(\mathstrut -\mathstrut 117696240 \) acting on \(S_{18}^{\mathrm{new}}(\Gamma_0(8))\).