Properties

Label 8.23.d.a
Level 8
Weight 23
Character orbit 8.d
Self dual Yes
Analytic conductor 24.537
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 23 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(24.5365947873\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 2048q^{2} \) \(\mathstrut -\mathstrut 199058q^{3} \) \(\mathstrut +\mathstrut 4194304q^{4} \) \(\mathstrut +\mathstrut 407670784q^{6} \) \(\mathstrut -\mathstrut 8589934592q^{8} \) \(\mathstrut +\mathstrut 8243027755q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 2048q^{2} \) \(\mathstrut -\mathstrut 199058q^{3} \) \(\mathstrut +\mathstrut 4194304q^{4} \) \(\mathstrut +\mathstrut 407670784q^{6} \) \(\mathstrut -\mathstrut 8589934592q^{8} \) \(\mathstrut +\mathstrut 8243027755q^{9} \) \(\mathstrut -\mathstrut 550486028386q^{11} \) \(\mathstrut -\mathstrut 834909765632q^{12} \) \(\mathstrut +\mathstrut 17592186044416q^{16} \) \(\mathstrut -\mathstrut 41341538741182q^{17} \) \(\mathstrut -\mathstrut 16881720842240q^{18} \) \(\mathstrut -\mathstrut 86441697072754q^{19} \) \(\mathstrut +\mathstrut 1127395386134528q^{22} \) \(\mathstrut +\mathstrut 1709895200014336q^{24} \) \(\mathstrut +\mathstrut 2384185791015625q^{25} \) \(\mathstrut +\mathstrut 4605810344793532q^{27} \) \(\mathstrut -\mathstrut 36028797018963968q^{32} \) \(\mathstrut +\mathstrut 109578647838460388q^{33} \) \(\mathstrut +\mathstrut 84667471341940736q^{34} \) \(\mathstrut +\mathstrut 34573764284907520q^{36} \) \(\mathstrut +\mathstrut 177032595605000192q^{38} \) \(\mathstrut +\mathstrut 291475944231948914q^{41} \) \(\mathstrut -\mathstrut 1810527321484332514q^{43} \) \(\mathstrut -\mathstrut 2308905750803513344q^{44} \) \(\mathstrut -\mathstrut 3501865369629360128q^{48} \) \(\mathstrut +\mathstrut 3909821048582988049q^{49} \) \(\mathstrut -\mathstrut 4882812500000000000q^{50} \) \(\mathstrut +\mathstrut 8229364018742206556q^{51} \) \(\mathstrut -\mathstrut 9432699586137153536q^{54} \) \(\mathstrut +\mathstrut 17206911335908265732q^{57} \) \(\mathstrut +\mathstrut 49867838021809381118q^{59} \) \(\mathstrut +\mathstrut 73786976294838206464q^{64} \) \(\mathstrut -\mathstrut 224417070773166874624q^{66} \) \(\mathstrut +\mathstrut 204466126247920757678q^{67} \) \(\mathstrut -\mathstrut 173398981308294627328q^{68} \) \(\mathstrut -\mathstrut 70807069255490600960q^{72} \) \(\mathstrut +\mathstrut 531355580825828150162q^{73} \) \(\mathstrut -\mathstrut 474591255187988281250q^{75} \) \(\mathstrut -\mathstrut 362562755799040393216q^{76} \) \(\mathstrut -\mathstrut 1175498340952207340651q^{81} \) \(\mathstrut -\mathstrut 596942733787031375872q^{82} \) \(\mathstrut -\mathstrut 2470066709866614036658q^{83} \) \(\mathstrut +\mathstrut 3707959954399912988672q^{86} \) \(\mathstrut +\mathstrut 4728638977645595328512q^{88} \) \(\mathstrut +\mathstrut 5077570440143736010226q^{89} \) \(\mathstrut +\mathstrut 7171820277000929542144q^{96} \) \(\mathstrut -\mathstrut 9422684141117979760606q^{97} \) \(\mathstrut -\mathstrut 8007313507497959524352q^{98} \) \(\mathstrut -\mathstrut 4537671610725515853430q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−2048.00 −199058. 4.19430e6 0 4.07671e8 0 −8.58993e9 8.24303e9 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 \(T_{3} \) \(\mathstrut +\mathstrut 199058 \) acting on \(S_{23}^{\mathrm{new}}(8, [\chi])\).