Properties

Label 34.2.b.a
Level 34
Weight 2
Character orbit 34.b
Analytic conductor 0.271
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 34 = 2 \cdot 17 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 34.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.271491366872\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \( + \beta q^{3} \) \(+ q^{4}\) \( -\beta q^{5} \) \( -\beta q^{6} \) \(- q^{8}\) \( -5 q^{9} \) \(+O(q^{10})\) \( q\) \(- q^{2}\) \( + \beta q^{3} \) \(+ q^{4}\) \( -\beta q^{5} \) \( -\beta q^{6} \) \(- q^{8}\) \( -5 q^{9} \) \( + \beta q^{10} \) \( -\beta q^{11} \) \( + \beta q^{12} \) \( + 2 q^{13} \) \( + 8 q^{15} \) \(+ q^{16}\) \( + ( -3 + \beta ) q^{17} \) \( + 5 q^{18} \) \( -4 q^{19} \) \( -\beta q^{20} \) \( + \beta q^{22} \) \( + 2 \beta q^{23} \) \( -\beta q^{24} \) \( -3 q^{25} \) \( -2 q^{26} \) \( -2 \beta q^{27} \) \( -\beta q^{29} \) \( -8 q^{30} \) \(- q^{32}\) \( + 8 q^{33} \) \( + ( 3 - \beta ) q^{34} \) \( -5 q^{36} \) \( -3 \beta q^{37} \) \( + 4 q^{38} \) \( + 2 \beta q^{39} \) \( + \beta q^{40} \) \( + 2 \beta q^{41} \) \( -4 q^{43} \) \( -\beta q^{44} \) \( + 5 \beta q^{45} \) \( -2 \beta q^{46} \) \( + \beta q^{48} \) \( + 7 q^{49} \) \( + 3 q^{50} \) \( + ( -8 - 3 \beta ) q^{51} \) \( + 2 q^{52} \) \( + 6 q^{53} \) \( + 2 \beta q^{54} \) \( -8 q^{55} \) \( -4 \beta q^{57} \) \( + \beta q^{58} \) \( + 12 q^{59} \) \( + 8 q^{60} \) \( + 3 \beta q^{61} \) \(+ q^{64}\) \( -2 \beta q^{65} \) \( -8 q^{66} \) \( -4 q^{67} \) \( + ( -3 + \beta ) q^{68} \) \( -16 q^{69} \) \( + 2 \beta q^{71} \) \( + 5 q^{72} \) \( + 3 \beta q^{74} \) \( -3 \beta q^{75} \) \( -4 q^{76} \) \( -2 \beta q^{78} \) \( -6 \beta q^{79} \) \( -\beta q^{80} \) \(+ q^{81}\) \( -2 \beta q^{82} \) \( -12 q^{83} \) \( + ( 8 + 3 \beta ) q^{85} \) \( + 4 q^{86} \) \( + 8 q^{87} \) \( + \beta q^{88} \) \( + 6 q^{89} \) \( -5 \beta q^{90} \) \( + 2 \beta q^{92} \) \( + 4 \beta q^{95} \) \( -\beta q^{96} \) \( + 6 \beta q^{97} \) \( -7 q^{98} \) \( + 5 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 2q^{4} \) \(\mathstrut -\mathstrut 2q^{8} \) \(\mathstrut -\mathstrut 10q^{9} \) \(\mathstrut +\mathstrut 4q^{13} \) \(\mathstrut +\mathstrut 16q^{15} \) \(\mathstrut +\mathstrut 2q^{16} \) \(\mathstrut -\mathstrut 6q^{17} \) \(\mathstrut +\mathstrut 10q^{18} \) \(\mathstrut -\mathstrut 8q^{19} \) \(\mathstrut -\mathstrut 6q^{25} \) \(\mathstrut -\mathstrut 4q^{26} \) \(\mathstrut -\mathstrut 16q^{30} \) \(\mathstrut -\mathstrut 2q^{32} \) \(\mathstrut +\mathstrut 16q^{33} \) \(\mathstrut +\mathstrut 6q^{34} \) \(\mathstrut -\mathstrut 10q^{36} \) \(\mathstrut +\mathstrut 8q^{38} \) \(\mathstrut -\mathstrut 8q^{43} \) \(\mathstrut +\mathstrut 14q^{49} \) \(\mathstrut +\mathstrut 6q^{50} \) \(\mathstrut -\mathstrut 16q^{51} \) \(\mathstrut +\mathstrut 4q^{52} \) \(\mathstrut +\mathstrut 12q^{53} \) \(\mathstrut -\mathstrut 16q^{55} \) \(\mathstrut +\mathstrut 24q^{59} \) \(\mathstrut +\mathstrut 16q^{60} \) \(\mathstrut +\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 16q^{66} \) \(\mathstrut -\mathstrut 8q^{67} \) \(\mathstrut -\mathstrut 6q^{68} \) \(\mathstrut -\mathstrut 32q^{69} \) \(\mathstrut +\mathstrut 10q^{72} \) \(\mathstrut -\mathstrut 8q^{76} \) \(\mathstrut +\mathstrut 2q^{81} \) \(\mathstrut -\mathstrut 24q^{83} \) \(\mathstrut +\mathstrut 16q^{85} \) \(\mathstrut +\mathstrut 8q^{86} \) \(\mathstrut +\mathstrut 16q^{87} \) \(\mathstrut +\mathstrut 12q^{89} \) \(\mathstrut -\mathstrut 14q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/34\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
1.41421i
1.41421i
−1.00000 2.82843i 1.00000 2.82843i 2.82843i 0 −1.00000 −5.00000 2.82843i
33.2 −1.00000 2.82843i 1.00000 2.82843i 2.82843i 0 −1.00000 −5.00000 2.82843i
\(n\): 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
17.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(34, [\chi])\).