# Properties

 Label 2205.2.d.f Level $2205$ Weight $2$ Character orbit 2205.d Analytic conductor $17.607$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2205 = 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2205.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.6070136457$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{4} + ( 1 + 2 i ) q^{5} + 3 i q^{8} +O(q^{10})$$ $$q + i q^{2} + q^{4} + ( 1 + 2 i ) q^{5} + 3 i q^{8} + ( -2 + i ) q^{10} + 6 q^{11} -2 i q^{13} - q^{16} + 4 i q^{17} -6 q^{19} + ( 1 + 2 i ) q^{20} + 6 i q^{22} + ( -3 + 4 i ) q^{25} + 2 q^{26} -2 q^{29} + 10 q^{31} + 5 i q^{32} -4 q^{34} -4 i q^{37} -6 i q^{38} + ( -6 + 3 i ) q^{40} + 2 q^{41} + 4 i q^{43} + 6 q^{44} + ( -4 - 3 i ) q^{50} -2 i q^{52} + 6 i q^{53} + ( 6 + 12 i ) q^{55} -2 i q^{58} + 8 q^{59} + 2 q^{61} + 10 i q^{62} -7 q^{64} + ( 4 - 2 i ) q^{65} -16 i q^{67} + 4 i q^{68} -10 q^{71} -6 i q^{73} + 4 q^{74} -6 q^{76} -4 q^{79} + ( -1 - 2 i ) q^{80} + 2 i q^{82} -8 i q^{83} + ( -8 + 4 i ) q^{85} -4 q^{86} + 18 i q^{88} -6 q^{89} + ( -6 - 12 i ) q^{95} + 2 i q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{4} + 2 q^{5} + O(q^{10})$$ $$2 q + 2 q^{4} + 2 q^{5} - 4 q^{10} + 12 q^{11} - 2 q^{16} - 12 q^{19} + 2 q^{20} - 6 q^{25} + 4 q^{26} - 4 q^{29} + 20 q^{31} - 8 q^{34} - 12 q^{40} + 4 q^{41} + 12 q^{44} - 8 q^{50} + 12 q^{55} + 16 q^{59} + 4 q^{61} - 14 q^{64} + 8 q^{65} - 20 q^{71} + 8 q^{74} - 12 q^{76} - 8 q^{79} - 2 q^{80} - 16 q^{85} - 8 q^{86} - 12 q^{89} - 12 q^{95} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2205\mathbb{Z}\right)^\times$$.

 $$n$$ $$442$$ $$1081$$ $$1226$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1324.1
 − 1.00000i 1.00000i
1.00000i 0 1.00000 1.00000 2.00000i 0 0 3.00000i 0 −2.00000 1.00000i
1324.2 1.00000i 0 1.00000 1.00000 + 2.00000i 0 0 3.00000i 0 −2.00000 + 1.00000i
 $$n$$: 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 2205.2.d.f 2
3.b odd 2 1 735.2.d.a 2
5.b even 2 1 inner 2205.2.d.f 2
7.b odd 2 1 315.2.d.c 2
15.d odd 2 1 735.2.d.a 2
15.e even 4 1 3675.2.a.d 1
15.e even 4 1 3675.2.a.l 1
21.c even 2 1 105.2.d.a 2
21.g even 6 2 735.2.q.a 4
21.h odd 6 2 735.2.q.b 4
28.d even 2 1 5040.2.t.e 2
35.c odd 2 1 315.2.d.c 2
35.f even 4 1 1575.2.a.e 1
35.f even 4 1 1575.2.a.i 1
84.h odd 2 1 1680.2.t.f 2
105.g even 2 1 105.2.d.a 2
105.k odd 4 1 525.2.a.b 1
105.k odd 4 1 525.2.a.c 1
105.o odd 6 2 735.2.q.b 4
105.p even 6 2 735.2.q.a 4
140.c even 2 1 5040.2.t.e 2
420.o odd 2 1 1680.2.t.f 2
420.w even 4 1 8400.2.a.bj 1
420.w even 4 1 8400.2.a.ch 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 21.c even 2 1
105.2.d.a 2 105.g even 2 1
315.2.d.c 2 7.b odd 2 1
315.2.d.c 2 35.c odd 2 1
525.2.a.b 1 105.k odd 4 1
525.2.a.c 1 105.k odd 4 1
735.2.d.a 2 3.b odd 2 1
735.2.d.a 2 15.d odd 2 1
735.2.q.a 4 21.g even 6 2
735.2.q.a 4 105.p even 6 2
735.2.q.b 4 21.h odd 6 2
735.2.q.b 4 105.o odd 6 2
1575.2.a.e 1 35.f even 4 1
1575.2.a.i 1 35.f even 4 1
1680.2.t.f 2 84.h odd 2 1
1680.2.t.f 2 420.o odd 2 1
2205.2.d.f 2 1.a even 1 1 trivial
2205.2.d.f 2 5.b even 2 1 inner
3675.2.a.d 1 15.e even 4 1
3675.2.a.l 1 15.e even 4 1
5040.2.t.e 2 28.d even 2 1
5040.2.t.e 2 140.c even 2 1
8400.2.a.bj 1 420.w even 4 1
8400.2.a.ch 1 420.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2205, [\chi])$$:

 $$T_{2}^{2} + 1$$ $$T_{11} - 6$$ $$T_{13}^{2} + 4$$ $$T_{19} + 6$$ $$T_{29} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$5 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -6 + T )^{2}$$
$13$ $$4 + T^{2}$$
$17$ $$16 + T^{2}$$
$19$ $$( 6 + T )^{2}$$
$23$ $$T^{2}$$
$29$ $$( 2 + T )^{2}$$
$31$ $$( -10 + T )^{2}$$
$37$ $$16 + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$16 + T^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + T^{2}$$
$59$ $$( -8 + T )^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$256 + T^{2}$$
$71$ $$( 10 + T )^{2}$$
$73$ $$36 + T^{2}$$
$79$ $$( 4 + T )^{2}$$
$83$ $$64 + T^{2}$$
$89$ $$( 6 + T )^{2}$$
$97$ $$4 + T^{2}$$