Properties

Label 3600.1.c.a
Level 3600
Weight 1
Character orbit 3600.c
Analytic conductor 1.797
Analytic rank 0
Dimension 4
Projective image \(S_{4}\)
CM/RM No
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) = \( 1 \)
Character orbit: \([\chi]\) = 3600.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(1.79663404548\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Projective image \(S_{4}\)
Projective field Galois closure of 4.2.10800.2

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q\) \( -\zeta_{8}^{2} q^{7} \) \(+O(q^{10})\) \( q\) \( -\zeta_{8}^{2} q^{7} \) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{11} \) \( + \zeta_{8}^{2} q^{13} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{17} \) \(+ q^{19}\) \( + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{23} \) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{29} \) \(- q^{31}\) \( -\zeta_{8}^{2} q^{43} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{47} \) \( + ( \zeta_{8} + \zeta_{8}^{3} ) q^{59} \) \(+ q^{61}\) \( -\zeta_{8}^{2} q^{67} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{77} \) \( + ( \zeta_{8} - \zeta_{8}^{3} ) q^{83} \) \(+ q^{91}\) \( -\zeta_{8}^{2} q^{97} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(4q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(4q \) \(\mathstrut +\mathstrut 4q^{19} \) \(\mathstrut -\mathstrut 4q^{31} \) \(\mathstrut +\mathstrut 4q^{61} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−0.707107 0.707107i
0.707107 + 0.707107i
0.707107 0.707107i
−0.707107 + 0.707107i
0 0 0 0 0 1.00000i 0 0 0
449.2 0 0 0 0 0 1.00000i 0 0 0
449.3 0 0 0 0 0 1.00000i 0 0 0
449.4 0 0 0 0 0 1.00000i 0 0 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
3.b Odd 1 yes
5.b Even 1 yes
15.d Odd 1 yes

Hecke kernels

There are no other newforms in \(S_{1}^{\mathrm{new}}(3600, [\chi])\).