Properties

Label 8.19.d.a
Level 8
Weight 19
Character orbit 8.d
Self dual Yes
Analytic conductor 16.431
Analytic rank 0
Dimension 1
CM disc. -8
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 8 = 2^{3} \)
Weight: \( k \) = \( 19 \)
Character orbit: \([\chi]\) = 8.d (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(16.4308910168\)
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

\(f(q)\) \(=\) \(q \) \(\mathstrut -\mathstrut 512q^{2} \) \(\mathstrut -\mathstrut 3266q^{3} \) \(\mathstrut +\mathstrut 262144q^{4} \) \(\mathstrut +\mathstrut 1672192q^{6} \) \(\mathstrut -\mathstrut 134217728q^{8} \) \(\mathstrut -\mathstrut 376753733q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(q \) \(\mathstrut -\mathstrut 512q^{2} \) \(\mathstrut -\mathstrut 3266q^{3} \) \(\mathstrut +\mathstrut 262144q^{4} \) \(\mathstrut +\mathstrut 1672192q^{6} \) \(\mathstrut -\mathstrut 134217728q^{8} \) \(\mathstrut -\mathstrut 376753733q^{9} \) \(\mathstrut -\mathstrut 354349618q^{11} \) \(\mathstrut -\mathstrut 856162304q^{12} \) \(\mathstrut +\mathstrut 68719476736q^{16} \) \(\mathstrut +\mathstrut 119842447106q^{17} \) \(\mathstrut +\mathstrut 192897911296q^{18} \) \(\mathstrut +\mathstrut 335013705758q^{19} \) \(\mathstrut +\mathstrut 181427004416q^{22} \) \(\mathstrut +\mathstrut 438355099648q^{24} \) \(\mathstrut +\mathstrut 3814697265625q^{25} \) \(\mathstrut +\mathstrut 2495793009052q^{27} \) \(\mathstrut -\mathstrut 35184372088832q^{32} \) \(\mathstrut +\mathstrut 1157305852388q^{33} \) \(\mathstrut -\mathstrut 61359332918272q^{34} \) \(\mathstrut -\mathstrut 98763730583552q^{36} \) \(\mathstrut -\mathstrut 171527017348096q^{38} \) \(\mathstrut +\mathstrut 522162604887122q^{41} \) \(\mathstrut +\mathstrut 1000250360894414q^{43} \) \(\mathstrut -\mathstrut 92890626260992q^{44} \) \(\mathstrut -\mathstrut 224437811019776q^{48} \) \(\mathstrut +\mathstrut 1628413597910449q^{49} \) \(\mathstrut -\mathstrut 1953125000000000q^{50} \) \(\mathstrut -\mathstrut 391405432248196q^{51} \) \(\mathstrut -\mathstrut 1277846020634624q^{54} \) \(\mathstrut -\mathstrut 1094154763005628q^{57} \) \(\mathstrut -\mathstrut 10228968070290322q^{59} \) \(\mathstrut +\mathstrut 18014398509481984q^{64} \) \(\mathstrut -\mathstrut 592540596422656q^{66} \) \(\mathstrut -\mathstrut 50467064407716994q^{67} \) \(\mathstrut +\mathstrut 31415978454155264q^{68} \) \(\mathstrut +\mathstrut 50567030058778624q^{72} \) \(\mathstrut +\mathstrut 60615173082969074q^{73} \) \(\mathstrut -\mathstrut 12458801269531250q^{75} \) \(\mathstrut +\mathstrut 87821832882225152q^{76} \) \(\mathstrut +\mathstrut 137810855503871605q^{81} \) \(\mathstrut -\mathstrut 267347253702206464q^{82} \) \(\mathstrut -\mathstrut 352960460737558306q^{83} \) \(\mathstrut -\mathstrut 512128184777939968q^{86} \) \(\mathstrut +\mathstrut 47560000645627904q^{88} \) \(\mathstrut +\mathstrut 487058048464217618q^{89} \) \(\mathstrut +\mathstrut 114912159242125312q^{96} \) \(\mathstrut +\mathstrut 1501067319528053666q^{97} \) \(\mathstrut -\mathstrut 833747762130149888q^{98} \) \(\mathstrut +\mathstrut 133502541368623994q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
−512.000 −3266.00 262144. 0 1.67219e6 0 −1.34218e8 −3.76754e8 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 \(T_{3} \) \(\mathstrut +\mathstrut 3266 \) acting on \(S_{19}^{\mathrm{new}}(8, [\chi])\).