Properties

Label 36.2.e.a
Level 36
Weight 2
Character orbit 36.e
Analytic conductor 0.287
Analytic rank 0
Dimension 2
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 36 = 2^{2} \cdot 3^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 36.e (of order \(3\) and degree \(2\))

Newform invariants

Self dual: No
Analytic conductor: \(0.287461447277\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -3 + 3 \zeta_{6} ) q^{5} + \zeta_{6} q^{7} -3 q^{9} +O(q^{10})\) \( q + ( 1 - 2 \zeta_{6} ) q^{3} + ( -3 + 3 \zeta_{6} ) q^{5} + \zeta_{6} q^{7} -3 q^{9} -3 \zeta_{6} q^{11} + ( 1 - \zeta_{6} ) q^{13} + ( 3 + 3 \zeta_{6} ) q^{15} + 6 q^{17} -4 q^{19} + ( 2 - \zeta_{6} ) q^{21} + ( 3 - 3 \zeta_{6} ) q^{23} -4 \zeta_{6} q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} -3 \zeta_{6} q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( -6 + 3 \zeta_{6} ) q^{33} -3 q^{35} + 2 q^{37} + ( -1 - \zeta_{6} ) q^{39} + ( -3 + 3 \zeta_{6} ) q^{41} + \zeta_{6} q^{43} + ( 9 - 9 \zeta_{6} ) q^{45} + 9 \zeta_{6} q^{47} + ( 6 - 6 \zeta_{6} ) q^{49} + ( 6 - 12 \zeta_{6} ) q^{51} -6 q^{53} + 9 q^{55} + ( -4 + 8 \zeta_{6} ) q^{57} + ( 3 - 3 \zeta_{6} ) q^{59} + 13 \zeta_{6} q^{61} -3 \zeta_{6} q^{63} + 3 \zeta_{6} q^{65} + ( 7 - 7 \zeta_{6} ) q^{67} + ( -3 - 3 \zeta_{6} ) q^{69} -12 q^{71} -10 q^{73} + ( -8 + 4 \zeta_{6} ) q^{75} + ( 3 - 3 \zeta_{6} ) q^{77} -11 \zeta_{6} q^{79} + 9 q^{81} + 9 \zeta_{6} q^{83} + ( -18 + 18 \zeta_{6} ) q^{85} + ( -6 + 3 \zeta_{6} ) q^{87} + 6 q^{89} + q^{91} + ( 5 + 5 \zeta_{6} ) q^{93} + ( 12 - 12 \zeta_{6} ) q^{95} -11 \zeta_{6} q^{97} + 9 \zeta_{6} q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 3q^{5} + q^{7} - 6q^{9} + O(q^{10}) \) \( 2q - 3q^{5} + q^{7} - 6q^{9} - 3q^{11} + q^{13} + 9q^{15} + 12q^{17} - 8q^{19} + 3q^{21} + 3q^{23} - 4q^{25} - 3q^{29} - 5q^{31} - 9q^{33} - 6q^{35} + 4q^{37} - 3q^{39} - 3q^{41} + q^{43} + 9q^{45} + 9q^{47} + 6q^{49} - 12q^{53} + 18q^{55} + 3q^{59} + 13q^{61} - 3q^{63} + 3q^{65} + 7q^{67} - 9q^{69} - 24q^{71} - 20q^{73} - 12q^{75} + 3q^{77} - 11q^{79} + 18q^{81} + 9q^{83} - 18q^{85} - 9q^{87} + 12q^{89} + 2q^{91} + 15q^{93} + 12q^{95} - 11q^{97} + 9q^{99} + O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/36\mathbb{Z}\right)^\times\).

\(n\) \(19\) \(29\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
13.1
0.500000 + 0.866025i
0.500000 0.866025i
0 1.73205i 0 −1.50000 + 2.59808i 0 0.500000 + 0.866025i 0 −3.00000 0
25.1 0 1.73205i 0 −1.50000 2.59808i 0 0.500000 0.866025i 0 −3.00000 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
9.c Even 1 yes

Hecke kernels

There are no other newforms in \(S_{2}^{\mathrm{new}}(36, [\chi])\).