# Properties

 Label 1200.3.l.l Level $1200$ Weight $3$ Character orbit 1200.l Analytic conductor $32.698$ Analytic rank $0$ Dimension $2$ CM discriminant -15 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1200 = 2^{4} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1200.l (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$32.6976317232$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{17}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 - 9 * q^9 $$q + 3 i q^{3} - 9 q^{9} + 14 i q^{17} - 22 q^{19} - 34 i q^{23} - 27 i q^{27} - 2 q^{31} - 14 i q^{47} - 49 q^{49} - 42 q^{51} - 86 i q^{53} - 66 i q^{57} - 118 q^{61} + 102 q^{69} + 98 q^{79} + 81 q^{81} - 154 i q^{83} - 6 i q^{93} +O(q^{100})$$ q + 3*i * q^3 - 9 * q^9 + 14*i * q^17 - 22 * q^19 - 34*i * q^23 - 27*i * q^27 - 2 * q^31 - 14*i * q^47 - 49 * q^49 - 42 * q^51 - 86*i * q^53 - 66*i * q^57 - 118 * q^61 + 102 * q^69 + 98 * q^79 + 81 * q^81 - 154*i * q^83 - 6*i * q^93 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 44 q^{19} - 4 q^{31} - 98 q^{49} - 84 q^{51} - 236 q^{61} + 204 q^{69} + 196 q^{79} + 162 q^{81}+O(q^{100})$$ 2 * q - 18 * q^9 - 44 * q^19 - 4 * q^31 - 98 * q^49 - 84 * q^51 - 236 * q^61 + 204 * q^69 + 196 * q^79 + 162 * q^81

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$577$$ $$751$$ $$901$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 0 0 −9.00000 0
401.2 0 3.00000i 0 0 0 0 0 −9.00000 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
15.d odd 2 1 CM by $$\Q(\sqrt{-15})$$
3.b odd 2 1 inner
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 1200.3.l.l 2
3.b odd 2 1 inner 1200.3.l.l 2
4.b odd 2 1 75.3.c.d 2
5.b even 2 1 inner 1200.3.l.l 2
5.c odd 4 1 240.3.c.a 1
5.c odd 4 1 240.3.c.b 1
12.b even 2 1 75.3.c.d 2
15.d odd 2 1 CM 1200.3.l.l 2
15.e even 4 1 240.3.c.a 1
15.e even 4 1 240.3.c.b 1
20.d odd 2 1 75.3.c.d 2
20.e even 4 1 15.3.d.a 1
20.e even 4 1 15.3.d.b yes 1
40.i odd 4 1 960.3.c.a 1
40.i odd 4 1 960.3.c.d 1
40.k even 4 1 960.3.c.b 1
40.k even 4 1 960.3.c.c 1
60.h even 2 1 75.3.c.d 2
60.l odd 4 1 15.3.d.a 1
60.l odd 4 1 15.3.d.b yes 1
120.q odd 4 1 960.3.c.b 1
120.q odd 4 1 960.3.c.c 1
120.w even 4 1 960.3.c.a 1
120.w even 4 1 960.3.c.d 1
180.v odd 12 2 405.3.h.a 2
180.v odd 12 2 405.3.h.b 2
180.x even 12 2 405.3.h.a 2
180.x even 12 2 405.3.h.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.d.a 1 20.e even 4 1
15.3.d.a 1 60.l odd 4 1
15.3.d.b yes 1 20.e even 4 1
15.3.d.b yes 1 60.l odd 4 1
75.3.c.d 2 4.b odd 2 1
75.3.c.d 2 12.b even 2 1
75.3.c.d 2 20.d odd 2 1
75.3.c.d 2 60.h even 2 1
240.3.c.a 1 5.c odd 4 1
240.3.c.a 1 15.e even 4 1
240.3.c.b 1 5.c odd 4 1
240.3.c.b 1 15.e even 4 1
405.3.h.a 2 180.v odd 12 2
405.3.h.a 2 180.x even 12 2
405.3.h.b 2 180.v odd 12 2
405.3.h.b 2 180.x even 12 2
960.3.c.a 1 40.i odd 4 1
960.3.c.a 1 120.w even 4 1
960.3.c.b 1 40.k even 4 1
960.3.c.b 1 120.q odd 4 1
960.3.c.c 1 40.k even 4 1
960.3.c.c 1 120.q odd 4 1
960.3.c.d 1 40.i odd 4 1
960.3.c.d 1 120.w even 4 1
1200.3.l.l 2 1.a even 1 1 trivial
1200.3.l.l 2 3.b odd 2 1 inner
1200.3.l.l 2 5.b even 2 1 inner
1200.3.l.l 2 15.d odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}$$ T7 $$T_{11}$$ T11 $$T_{13}$$ T13

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2}$$
$17$ $$T^{2} + 196$$
$19$ $$(T + 22)^{2}$$
$23$ $$T^{2} + 1156$$
$29$ $$T^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2}$$
$41$ $$T^{2}$$
$43$ $$T^{2}$$
$47$ $$T^{2} + 196$$
$53$ $$T^{2} + 7396$$
$59$ $$T^{2}$$
$61$ $$(T + 118)^{2}$$
$67$ $$T^{2}$$
$71$ $$T^{2}$$
$73$ $$T^{2}$$
$79$ $$(T - 98)^{2}$$
$83$ $$T^{2} + 23716$$
$89$ $$T^{2}$$
$97$ $$T^{2}$$