Properties

Label 7.17.b.a
Level 7
Weight 17
Character orbit 7.b
Self dual Yes
Analytic conductor 11.363
Analytic rank 0
Dimension 1
CM disc. -7
Inner twists 2

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 7 \)
Weight: \( k \) = \( 17 \)
Character orbit: \([\chi]\) = 7.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: Yes
Analytic conductor: \(11.36271807\)
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

\(f(q)\) \(=\) \( q + 449q^{2} + 136065q^{4} + 5764801q^{7} + 31667521q^{8} + 43046721q^{9} + O(q^{10}) \) \( q + 449q^{2} + 136065q^{4} + 5764801q^{7} + 31667521q^{8} + 43046721q^{9} - 255690046q^{11} + 2588395649q^{14} + 5301561089q^{16} + 19327977729q^{18} - 114804830654q^{22} - 156184073086q^{23} + 152587890625q^{25} + 784387648065q^{28} - 988786884286q^{29} + 305038272705q^{32} + 5857152092865q^{36} - 2723955766846q^{37} + 21863294238914q^{43} - 34790466108990q^{44} - 70126648815614q^{46} + 33232930569601q^{49} + 68511962890625q^{50} + 111956183305922q^{53} + 182556956728321q^{56} - 443965311044414q^{58} + 248155780267521q^{63} - 210480923084159q^{64} - 561251106979006q^{67} + 500488282933634q^{71} + 1363182941248641q^{72} - 1223056139313854q^{74} - 1474002232870846q^{77} + 1139826930254594q^{79} + 1853020188851841q^{81} + 9816619113272386q^{86} - 8097069901195966q^{88} - 21251185904446590q^{92} + 14921585825750849q^{98} - 11006618072639166q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0
449.000 0 136065. 0 0 5.76480e6 3.16675e7 4.30467e7 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
7.b Odd 1 CM by \(\Q(\sqrt{-7}) \) yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} - 449 \) acting on \(S_{17}^{\mathrm{new}}(7, [\chi])\).