Properties

Label 3600.2.f.v
Level $3600$
Weight $2$
Character orbit 3600.f
Analytic conductor $28.746$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3600.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.7461447277\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 300)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{7} +O(q^{10})\) \( q + i q^{7} + 6 q^{11} -5 i q^{13} + 6 i q^{17} + 5 q^{19} + 6 i q^{23} -6 q^{29} + q^{31} + 2 i q^{37} -i q^{43} + 6 i q^{47} + 6 q^{49} -12 i q^{53} + 6 q^{59} -13 q^{61} -11 i q^{67} -2 i q^{73} + 6 i q^{77} + 8 q^{79} + 6 i q^{83} + 5 q^{91} -7 i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + O(q^{10}) \) \( 2q + 12q^{11} + 10q^{19} - 12q^{29} + 2q^{31} + 12q^{49} + 12q^{59} - 26q^{61} + 16q^{79} + 10q^{91} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2449.1
1.00000i
1.00000i
0 0 0 0 0 1.00000i 0 0 0
2449.2 0 0 0 0 0 1.00000i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3600.2.f.v 2
3.b odd 2 1 1200.2.f.a 2
4.b odd 2 1 900.2.d.a 2
5.b even 2 1 inner 3600.2.f.v 2
5.c odd 4 1 3600.2.a.s 1
5.c odd 4 1 3600.2.a.z 1
12.b even 2 1 300.2.d.a 2
15.d odd 2 1 1200.2.f.a 2
15.e even 4 1 1200.2.a.f 1
15.e even 4 1 1200.2.a.n 1
20.d odd 2 1 900.2.d.a 2
20.e even 4 1 900.2.a.c 1
20.e even 4 1 900.2.a.e 1
24.f even 2 1 4800.2.f.b 2
24.h odd 2 1 4800.2.f.bi 2
60.h even 2 1 300.2.d.a 2
60.l odd 4 1 300.2.a.b 1
60.l odd 4 1 300.2.a.c yes 1
120.i odd 2 1 4800.2.f.bi 2
120.m even 2 1 4800.2.f.b 2
120.q odd 4 1 4800.2.a.o 1
120.q odd 4 1 4800.2.a.ce 1
120.w even 4 1 4800.2.a.p 1
120.w even 4 1 4800.2.a.cf 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.a.b 1 60.l odd 4 1
300.2.a.c yes 1 60.l odd 4 1
300.2.d.a 2 12.b even 2 1
300.2.d.a 2 60.h even 2 1
900.2.a.c 1 20.e even 4 1
900.2.a.e 1 20.e even 4 1
900.2.d.a 2 4.b odd 2 1
900.2.d.a 2 20.d odd 2 1
1200.2.a.f 1 15.e even 4 1
1200.2.a.n 1 15.e even 4 1
1200.2.f.a 2 3.b odd 2 1
1200.2.f.a 2 15.d odd 2 1
3600.2.a.s 1 5.c odd 4 1
3600.2.a.z 1 5.c odd 4 1
3600.2.f.v 2 1.a even 1 1 trivial
3600.2.f.v 2 5.b even 2 1 inner
4800.2.a.o 1 120.q odd 4 1
4800.2.a.p 1 120.w even 4 1
4800.2.a.ce 1 120.q odd 4 1
4800.2.a.cf 1 120.w even 4 1
4800.2.f.b 2 24.f even 2 1
4800.2.f.b 2 120.m even 2 1
4800.2.f.bi 2 24.h odd 2 1
4800.2.f.bi 2 120.i odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3600, [\chi])\):

\( T_{7}^{2} + 1 \)
\( T_{11} - 6 \)
\( T_{13}^{2} + 25 \)
\( T_{17}^{2} + 36 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 + T^{2} \)
$11$ \( ( -6 + T )^{2} \)
$13$ \( 25 + T^{2} \)
$17$ \( 36 + T^{2} \)
$19$ \( ( -5 + T )^{2} \)
$23$ \( 36 + T^{2} \)
$29$ \( ( 6 + T )^{2} \)
$31$ \( ( -1 + T )^{2} \)
$37$ \( 4 + T^{2} \)
$41$ \( T^{2} \)
$43$ \( 1 + T^{2} \)
$47$ \( 36 + T^{2} \)
$53$ \( 144 + T^{2} \)
$59$ \( ( -6 + T )^{2} \)
$61$ \( ( 13 + T )^{2} \)
$67$ \( 121 + T^{2} \)
$71$ \( T^{2} \)
$73$ \( 4 + T^{2} \)
$79$ \( ( -8 + T )^{2} \)
$83$ \( 36 + T^{2} \)
$89$ \( T^{2} \)
$97$ \( 49 + T^{2} \)
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