Properties

Label 349.2.b.a
Level 349
Weight 2
Character orbit 349.b
Analytic conductor 2.787
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 349 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 349.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(2.78677903054\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 2\sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + 2 q^{4} + 2 q^{5} + \beta q^{7} -2 q^{9} +O(q^{10})\) \( q - q^{3} + 2 q^{4} + 2 q^{5} + \beta q^{7} -2 q^{9} + \beta q^{11} -2 q^{12} -\beta q^{13} -2 q^{15} + 4 q^{16} + 3 q^{17} -5 q^{19} + 4 q^{20} -\beta q^{21} + q^{23} - q^{25} + 5 q^{27} + 2 \beta q^{28} + q^{29} + 7 q^{31} -\beta q^{33} + 2 \beta q^{35} -4 q^{36} -3 q^{37} + \beta q^{39} + 10 q^{41} -\beta q^{43} + 2 \beta q^{44} -4 q^{45} -2 \beta q^{47} -4 q^{48} -13 q^{49} -3 q^{51} -2 \beta q^{52} -\beta q^{53} + 2 \beta q^{55} + 5 q^{57} + \beta q^{59} -4 q^{60} -2 \beta q^{61} -2 \beta q^{63} + 8 q^{64} -2 \beta q^{65} -13 q^{67} + 6 q^{68} - q^{69} -2 \beta q^{71} -11 q^{73} + q^{75} -10 q^{76} -20 q^{77} -\beta q^{79} + 8 q^{80} + q^{81} + q^{83} -2 \beta q^{84} + 6 q^{85} - q^{87} + 2 \beta q^{89} + 20 q^{91} + 2 q^{92} -7 q^{93} -10 q^{95} -3 \beta q^{97} -2 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 4q^{4} + 4q^{5} - 4q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 4q^{4} + 4q^{5} - 4q^{9} - 4q^{12} - 4q^{15} + 8q^{16} + 6q^{17} - 10q^{19} + 8q^{20} + 2q^{23} - 2q^{25} + 10q^{27} + 2q^{29} + 14q^{31} - 8q^{36} - 6q^{37} + 20q^{41} - 8q^{45} - 8q^{48} - 26q^{49} - 6q^{51} + 10q^{57} - 8q^{60} + 16q^{64} - 26q^{67} + 12q^{68} - 2q^{69} - 22q^{73} + 2q^{75} - 20q^{76} - 40q^{77} + 16q^{80} + 2q^{81} + 2q^{83} + 12q^{85} - 2q^{87} + 40q^{91} + 4q^{92} - 14q^{93} - 20q^{95} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/349\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
348.1
2.23607i
2.23607i
0 −1.00000 2.00000 2.00000 0 4.47214i 0 −2.00000 0
348.2 0 −1.00000 2.00000 2.00000 0 4.47214i 0 −2.00000 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
349.b Even 1 yes

Hecke kernels

This newform can be constructed as the kernel of the linear operator \( T_{2} \) acting on \(S_{2}^{\mathrm{new}}(349, [\chi])\).