# Properties

 Label 6600.2.d.n Level 6600 Weight 2 Character orbit 6600.d Analytic conductor 52.701 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6600 = 2^{3} \cdot 3 \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6600.d (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$52.7012653340$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 264) 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.

 $$f(q)$$ $$=$$ $$q -i q^{3} + 4 i q^{7} - q^{9} +O(q^{10})$$ $$q -i q^{3} + 4 i q^{7} - q^{9} - q^{11} -6 i q^{13} + 6 i q^{17} + 8 q^{19} + 4 q^{21} + i q^{27} + 6 q^{29} + i q^{33} + 6 i q^{37} -6 q^{39} -10 q^{41} + 8 i q^{43} -9 q^{49} + 6 q^{51} -6 i q^{53} -8 i q^{57} -4 q^{59} -2 q^{61} -4 i q^{63} -12 i q^{67} -8 q^{71} -2 i q^{73} -4 i q^{77} + 4 q^{79} + q^{81} + 12 i q^{83} -6 i q^{87} + 6 q^{89} + 24 q^{91} + 2 i q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{9} + O(q^{10})$$ $$2q - 2q^{9} - 2q^{11} + 16q^{19} + 8q^{21} + 12q^{29} - 12q^{39} - 20q^{41} - 18q^{49} + 12q^{51} - 8q^{59} - 4q^{61} - 16q^{71} + 8q^{79} + 2q^{81} + 12q^{89} + 48q^{91} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1201$$ $$2201$$ $$2377$$ $$3301$$ $$4951$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 1.00000i − 1.00000i
0 1.00000i 0 0 0 4.00000i 0 −1.00000 0
1849.2 0 1.00000i 0 0 0 4.00000i 0 −1.00000 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 6600.2.d.n 2
5.b even 2 1 inner 6600.2.d.n 2
5.c odd 4 1 264.2.a.b 1
5.c odd 4 1 6600.2.a.a 1
15.e even 4 1 792.2.a.f 1
20.e even 4 1 528.2.a.b 1
40.i odd 4 1 2112.2.a.m 1
40.k even 4 1 2112.2.a.y 1
55.e even 4 1 2904.2.a.i 1
60.l odd 4 1 1584.2.a.n 1
120.q odd 4 1 6336.2.a.o 1
120.w even 4 1 6336.2.a.v 1
165.l odd 4 1 8712.2.a.r 1
220.i odd 4 1 5808.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
264.2.a.b 1 5.c odd 4 1
528.2.a.b 1 20.e even 4 1
792.2.a.f 1 15.e even 4 1
1584.2.a.n 1 60.l odd 4 1
2112.2.a.m 1 40.i odd 4 1
2112.2.a.y 1 40.k even 4 1
2904.2.a.i 1 55.e even 4 1
5808.2.a.f 1 220.i odd 4 1
6336.2.a.o 1 120.q odd 4 1
6336.2.a.v 1 120.w even 4 1
6600.2.a.a 1 5.c odd 4 1
6600.2.d.n 2 1.a even 1 1 trivial
6600.2.d.n 2 5.b even 2 1 inner
8712.2.a.r 1 165.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(6600, [\chi])$$:

 $$T_{7}^{2} + 16$$ $$T_{13}^{2} + 36$$ $$T_{17}^{2} + 36$$ $$T_{19} - 8$$ $$T_{29} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T^{2}$$
$5$ 1
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + T )^{2}$$
$13$ $$( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} )$$
$17$ $$1 + 2 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 31 T^{2} )^{2}$$
$37$ $$1 - 38 T^{2} + 1369 T^{4}$$
$41$ $$( 1 + 10 T + 41 T^{2} )^{2}$$
$43$ $$1 - 22 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 47 T^{2} )^{2}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 + 4 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{2}$$
$67$ $$1 + 10 T^{2} + 4489 T^{4}$$
$71$ $$( 1 + 8 T + 71 T^{2} )^{2}$$
$73$ $$1 - 142 T^{2} + 5329 T^{4}$$
$79$ $$( 1 - 4 T + 79 T^{2} )^{2}$$
$83$ $$1 - 22 T^{2} + 6889 T^{4}$$
$89$ $$( 1 - 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$