Properties

Label 29.2.b.a
Level 29
Weight 2
Character orbit 29.b
Analytic conductor 0.232
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 29 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 29.b (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(0.231566165862\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-5}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(f(q)\) \(=\) \( q\) \( + \beta q^{2} \) \( -\beta q^{3} \) \( -3 q^{4} \) \( -3 q^{5} \) \( + 5 q^{6} \) \( + 2 q^{7} \) \( -\beta q^{8} \) \( -2 q^{9} \) \(+O(q^{10})\) \( q\) \( + \beta q^{2} \) \( -\beta q^{3} \) \( -3 q^{4} \) \( -3 q^{5} \) \( + 5 q^{6} \) \( + 2 q^{7} \) \( -\beta q^{8} \) \( -2 q^{9} \) \( -3 \beta q^{10} \) \( + \beta q^{11} \) \( + 3 \beta q^{12} \) \(- q^{13}\) \( + 2 \beta q^{14} \) \( + 3 \beta q^{15} \) \(- q^{16}\) \( -2 \beta q^{17} \) \( -2 \beta q^{18} \) \( + 9 q^{20} \) \( -2 \beta q^{21} \) \( -5 q^{22} \) \( + 6 q^{23} \) \( -5 q^{24} \) \( + 4 q^{25} \) \( -\beta q^{26} \) \( -\beta q^{27} \) \( -6 q^{28} \) \( + ( -3 + 2 \beta ) q^{29} \) \( -15 q^{30} \) \( + 3 \beta q^{31} \) \( -3 \beta q^{32} \) \( + 5 q^{33} \) \( + 10 q^{34} \) \( -6 q^{35} \) \( + 6 q^{36} \) \( + \beta q^{39} \) \( + 3 \beta q^{40} \) \( -2 \beta q^{41} \) \( + 10 q^{42} \) \( -3 \beta q^{43} \) \( -3 \beta q^{44} \) \( + 6 q^{45} \) \( + 6 \beta q^{46} \) \( + \beta q^{47} \) \( + \beta q^{48} \) \( -3 q^{49} \) \( + 4 \beta q^{50} \) \( -10 q^{51} \) \( + 3 q^{52} \) \( -9 q^{53} \) \( + 5 q^{54} \) \( -3 \beta q^{55} \) \( -2 \beta q^{56} \) \( + ( -10 - 3 \beta ) q^{58} \) \( + 6 q^{59} \) \( -9 \beta q^{60} \) \( + 6 \beta q^{61} \) \( -15 q^{62} \) \( -4 q^{63} \) \( + 13 q^{64} \) \( + 3 q^{65} \) \( + 5 \beta q^{66} \) \( + 8 q^{67} \) \( + 6 \beta q^{68} \) \( -6 \beta q^{69} \) \( -6 \beta q^{70} \) \( + 2 \beta q^{72} \) \( -4 \beta q^{75} \) \( + 2 \beta q^{77} \) \( -5 q^{78} \) \( -3 \beta q^{79} \) \( + 3 q^{80} \) \( -11 q^{81} \) \( + 10 q^{82} \) \( -6 q^{83} \) \( + 6 \beta q^{84} \) \( + 6 \beta q^{85} \) \( + 15 q^{86} \) \( + ( 10 + 3 \beta ) q^{87} \) \( + 5 q^{88} \) \( -2 \beta q^{89} \) \( + 6 \beta q^{90} \) \( -2 q^{91} \) \( -18 q^{92} \) \( + 15 q^{93} \) \( -5 q^{94} \) \( -15 q^{96} \) \( + 6 \beta q^{97} \) \( -3 \beta q^{98} \) \( -2 \beta q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(2q \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(2q \) \(\mathstrut -\mathstrut 6q^{4} \) \(\mathstrut -\mathstrut 6q^{5} \) \(\mathstrut +\mathstrut 10q^{6} \) \(\mathstrut +\mathstrut 4q^{7} \) \(\mathstrut -\mathstrut 4q^{9} \) \(\mathstrut -\mathstrut 2q^{13} \) \(\mathstrut -\mathstrut 2q^{16} \) \(\mathstrut +\mathstrut 18q^{20} \) \(\mathstrut -\mathstrut 10q^{22} \) \(\mathstrut +\mathstrut 12q^{23} \) \(\mathstrut -\mathstrut 10q^{24} \) \(\mathstrut +\mathstrut 8q^{25} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut -\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 30q^{30} \) \(\mathstrut +\mathstrut 10q^{33} \) \(\mathstrut +\mathstrut 20q^{34} \) \(\mathstrut -\mathstrut 12q^{35} \) \(\mathstrut +\mathstrut 12q^{36} \) \(\mathstrut +\mathstrut 20q^{42} \) \(\mathstrut +\mathstrut 12q^{45} \) \(\mathstrut -\mathstrut 6q^{49} \) \(\mathstrut -\mathstrut 20q^{51} \) \(\mathstrut +\mathstrut 6q^{52} \) \(\mathstrut -\mathstrut 18q^{53} \) \(\mathstrut +\mathstrut 10q^{54} \) \(\mathstrut -\mathstrut 20q^{58} \) \(\mathstrut +\mathstrut 12q^{59} \) \(\mathstrut -\mathstrut 30q^{62} \) \(\mathstrut -\mathstrut 8q^{63} \) \(\mathstrut +\mathstrut 26q^{64} \) \(\mathstrut +\mathstrut 6q^{65} \) \(\mathstrut +\mathstrut 16q^{67} \) \(\mathstrut -\mathstrut 10q^{78} \) \(\mathstrut +\mathstrut 6q^{80} \) \(\mathstrut -\mathstrut 22q^{81} \) \(\mathstrut +\mathstrut 20q^{82} \) \(\mathstrut -\mathstrut 12q^{83} \) \(\mathstrut +\mathstrut 30q^{86} \) \(\mathstrut +\mathstrut 20q^{87} \) \(\mathstrut +\mathstrut 10q^{88} \) \(\mathstrut -\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 36q^{92} \) \(\mathstrut +\mathstrut 30q^{93} \) \(\mathstrut -\mathstrut 10q^{94} \) \(\mathstrut -\mathstrut 30q^{96} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/29\mathbb{Z}\right)^\times\).

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
28.1
2.23607i
2.23607i
2.23607i 2.23607i −3.00000 −3.00000 5.00000 2.00000 2.23607i −2.00000 6.70820i
28.2 2.23607i 2.23607i −3.00000 −3.00000 5.00000 2.00000 2.23607i −2.00000 6.70820i
\(n\): 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
29.b Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(29, [\chi])\).