Properties

Label 8.17.d.a
Level 8
Weight 17
Character orbit 8.d
Self dual Yes
Analytic conductor 12.986
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(12.9859635085\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

\(f(q)\) \(=\) \(q \) \(\mathstrut +\mathstrut 256q^{2} \) \(\mathstrut -\mathstrut 11966q^{3} \) \(\mathstrut +\mathstrut 65536q^{4} \) \(\mathstrut -\mathstrut 3063296q^{6} \) \(\mathstrut +\mathstrut 16777216q^{8} \) \(\mathstrut +\mathstrut 100138435q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut +\mathstrut 256q^{2} \) \(\mathstrut -\mathstrut 11966q^{3} \) \(\mathstrut +\mathstrut 65536q^{4} \) \(\mathstrut -\mathstrut 3063296q^{6} \) \(\mathstrut +\mathstrut 16777216q^{8} \) \(\mathstrut +\mathstrut 100138435q^{9} \) \(\mathstrut +\mathstrut 309273794q^{11} \) \(\mathstrut -\mathstrut 784203776q^{12} \) \(\mathstrut +\mathstrut 4294967296q^{16} \) \(\mathstrut +\mathstrut 12433289474q^{17} \) \(\mathstrut +\mathstrut 25635439360q^{18} \) \(\mathstrut -\mathstrut 28741860286q^{19} \) \(\mathstrut +\mathstrut 79174091264q^{22} \) \(\mathstrut -\mathstrut 200756166656q^{24} \) \(\mathstrut +\mathstrut 152587890625q^{25} \) \(\mathstrut -\mathstrut 683159449724q^{27} \) \(\mathstrut +\mathstrut 1099511627776q^{32} \) \(\mathstrut -\mathstrut 3700770219004q^{33} \) \(\mathstrut +\mathstrut 3182922105344q^{34} \) \(\mathstrut +\mathstrut 6562672476160q^{36} \) \(\mathstrut -\mathstrut 7357916233216q^{38} \) \(\mathstrut +\mathstrut 831999729794q^{41} \) \(\mathstrut +\mathstrut 6069438110402q^{43} \) \(\mathstrut +\mathstrut 20268567363584q^{44} \) \(\mathstrut -\mathstrut 51393578663936q^{48} \) \(\mathstrut +\mathstrut 33232930569601q^{49} \) \(\mathstrut +\mathstrut 39062500000000q^{50} \) \(\mathstrut -\mathstrut 148776741845884q^{51} \) \(\mathstrut -\mathstrut 174888819129344q^{54} \) \(\mathstrut +\mathstrut 343925100182276q^{57} \) \(\mathstrut +\mathstrut 290918580565442q^{59} \) \(\mathstrut +\mathstrut 281474976710656q^{64} \) \(\mathstrut -\mathstrut 947397176065024q^{66} \) \(\mathstrut -\mathstrut 617692243063486q^{67} \) \(\mathstrut +\mathstrut 814828058968064q^{68} \) \(\mathstrut +\mathstrut 1680044153896960q^{72} \) \(\mathstrut -\mathstrut 486139502245246q^{73} \) \(\mathstrut -\mathstrut 1825866699218750q^{75} \) \(\mathstrut -\mathstrut 1883626555703296q^{76} \) \(\mathstrut +\mathstrut 3864054702575749q^{81} \) \(\mathstrut +\mathstrut 212991930827264q^{82} \) \(\mathstrut -\mathstrut 3591943143595966q^{83} \) \(\mathstrut +\mathstrut 1553776156262912q^{86} \) \(\mathstrut +\mathstrut 5188753245077504q^{88} \) \(\mathstrut +\mathstrut 1250855726873474q^{89} \) \(\mathstrut -\mathstrut 13156756137967616q^{96} \) \(\mathstrut -\mathstrut 9681283613729278q^{97} \) \(\mathstrut +\mathstrut 8507630225817856q^{98} \) \(\mathstrut +\mathstrut 30970193717672390q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
0
256.000 −11966.0 65536.0 0 −3.06330e6 0 1.67772e7 1.00138e8 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.d Odd 1 CM by \(\Q(\sqrt{-2}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \(T_{3} \) \(\mathstrut +\mathstrut 11966 \) acting on \(S_{17}^{\mathrm{new}}(8, [\chi])\).