# Properties

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

# Learn more about

## 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: $$2$$ Twist minimal: no (minimal twist has level 360) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{7} +O(q^{10})$$ $$q + 2 i q^{7} + 2 q^{11} + 4 i q^{13} -2 i q^{17} + 4 q^{19} + 8 i q^{23} -10 q^{29} -4 q^{31} + 8 i q^{43} -8 i q^{47} + 3 q^{49} -6 i q^{53} + 14 q^{59} -14 q^{61} -4 i q^{67} + 12 q^{71} + 6 i q^{73} + 4 i q^{77} -12 q^{79} + 4 i q^{83} -12 q^{89} -8 q^{91} + 14 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + O(q^{10})$$ $$2q + 4q^{11} + 8q^{19} - 20q^{29} - 8q^{31} + 6q^{49} + 28q^{59} - 28q^{61} + 24q^{71} - 24q^{79} - 24q^{89} - 16q^{91} + O(q^{100})$$

## 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 2.00000i 0 0 0
2449.2 0 0 0 0 0 2.00000i 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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.q 2
3.b odd 2 1 3600.2.f.g 2
4.b odd 2 1 1800.2.f.d 2
5.b even 2 1 inner 3600.2.f.q 2
5.c odd 4 1 720.2.a.i 1
5.c odd 4 1 3600.2.a.bh 1
12.b even 2 1 1800.2.f.h 2
15.d odd 2 1 3600.2.f.g 2
15.e even 4 1 720.2.a.a 1
15.e even 4 1 3600.2.a.bd 1
20.d odd 2 1 1800.2.f.d 2
20.e even 4 1 360.2.a.d yes 1
20.e even 4 1 1800.2.a.f 1
40.i odd 4 1 2880.2.a.e 1
40.k even 4 1 2880.2.a.n 1
60.h even 2 1 1800.2.f.h 2
60.l odd 4 1 360.2.a.c 1
60.l odd 4 1 1800.2.a.i 1
120.q odd 4 1 2880.2.a.bd 1
120.w even 4 1 2880.2.a.w 1
180.v odd 12 2 3240.2.q.n 2
180.x even 12 2 3240.2.q.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
360.2.a.c 1 60.l odd 4 1
360.2.a.d yes 1 20.e even 4 1
720.2.a.a 1 15.e even 4 1
720.2.a.i 1 5.c odd 4 1
1800.2.a.f 1 20.e even 4 1
1800.2.a.i 1 60.l odd 4 1
1800.2.f.d 2 4.b odd 2 1
1800.2.f.d 2 20.d odd 2 1
1800.2.f.h 2 12.b even 2 1
1800.2.f.h 2 60.h even 2 1
2880.2.a.e 1 40.i odd 4 1
2880.2.a.n 1 40.k even 4 1
2880.2.a.w 1 120.w even 4 1
2880.2.a.bd 1 120.q odd 4 1
3240.2.q.d 2 180.x even 12 2
3240.2.q.n 2 180.v odd 12 2
3600.2.a.bd 1 15.e even 4 1
3600.2.a.bh 1 5.c odd 4 1
3600.2.f.g 2 3.b odd 2 1
3600.2.f.g 2 15.d odd 2 1
3600.2.f.q 2 1.a even 1 1 trivial
3600.2.f.q 2 5.b even 2 1 inner

## Hecke kernels

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

 $$T_{7}^{2} + 4$$ $$T_{11} - 2$$ $$T_{13}^{2} + 16$$ $$T_{17}^{2} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$4 + T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$16 + T^{2}$$
$17$ $$4 + T^{2}$$
$19$ $$( -4 + T )^{2}$$
$23$ $$64 + T^{2}$$
$29$ $$( 10 + T )^{2}$$
$31$ $$( 4 + T )^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$64 + T^{2}$$
$47$ $$64 + T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -14 + T )^{2}$$
$61$ $$( 14 + T )^{2}$$
$67$ $$16 + T^{2}$$
$71$ $$( -12 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( 12 + T )^{2}$$
$83$ $$16 + T^{2}$$
$89$ $$( 12 + T )^{2}$$
$97$ $$196 + T^{2}$$
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