# Properties

 Label 1200.2.o.i Level $1200$ Weight $2$ Character orbit 1200.o Analytic conductor $9.582$ Analytic rank $0$ Dimension $4$ CM discriminant -3 Inner twists $8$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.58204824255$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: no (minimal twist has level 48) Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + 2 \beta_1 q^{7} + 3 q^{9}+O(q^{10})$$ q + b1 * q^3 + 2*b1 * q^7 + 3 * q^9 $$q + \beta_1 q^{3} + 2 \beta_1 q^{7} + 3 q^{9} - \beta_{3} q^{13} + \beta_{2} q^{19} + 6 q^{21} + 3 \beta_1 q^{27} - 3 \beta_{2} q^{31} + 5 \beta_{3} q^{37} - \beta_{2} q^{39} - 6 \beta_1 q^{43} + 5 q^{49} + 3 \beta_{3} q^{57} + 14 q^{61} + 6 \beta_1 q^{63} - 2 \beta_1 q^{67} + 5 \beta_{3} q^{73} - 5 \beta_{2} q^{79} + 9 q^{81} - 2 \beta_{2} q^{91} - 9 \beta_{3} q^{93} + 7 \beta_{3} q^{97}+O(q^{100})$$ q + b1 * q^3 + 2*b1 * q^7 + 3 * q^9 - b3 * q^13 + b2 * q^19 + 6 * q^21 + 3*b1 * q^27 - 3*b2 * q^31 + 5*b3 * q^37 - b2 * q^39 - 6*b1 * q^43 + 5 * q^49 + 3*b3 * q^57 + 14 * q^61 + 6*b1 * q^63 - 2*b1 * q^67 + 5*b3 * q^73 - 5*b2 * q^79 + 9 * q^81 - 2*b2 * q^91 - 9*b3 * q^93 + 7*b3 * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{9}+O(q^{10})$$ 4 * q + 12 * q^9 $$4 q + 12 q^{9} + 24 q^{21} + 20 q^{49} + 56 q^{61} + 36 q^{81}+O(q^{100})$$ 4 * q + 12 * q^9 + 24 * q^21 + 20 * q^49 + 56 * q^61 + 36 * q^81

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$-\zeta_{12}^{3} + 2\zeta_{12}$$ -v^3 + 2*v $$\beta_{2}$$ $$=$$ $$4\zeta_{12}^{2} - 2$$ 4*v^2 - 2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_{3} + 2\beta_1 ) / 4$$ (b3 + 2*b1) / 4 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{2} + 2 ) / 4$$ (b2 + 2) / 4 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1199.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i
0 −1.73205 0 0 0 −3.46410 0 3.00000 0
1199.2 0 −1.73205 0 0 0 −3.46410 0 3.00000 0
1199.3 0 1.73205 0 0 0 3.46410 0 3.00000 0
1199.4 0 1.73205 0 0 0 3.46410 0 3.00000 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
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
4.b odd 2 1 inner
5.b even 2 1 inner
12.b even 2 1 inner
15.d odd 2 1 inner
20.d odd 2 1 inner
60.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.2.o.i 4
3.b odd 2 1 CM 1200.2.o.i 4
4.b odd 2 1 inner 1200.2.o.i 4
5.b even 2 1 inner 1200.2.o.i 4
5.c odd 4 1 48.2.c.a 2
5.c odd 4 1 1200.2.h.e 2
12.b even 2 1 inner 1200.2.o.i 4
15.d odd 2 1 inner 1200.2.o.i 4
15.e even 4 1 48.2.c.a 2
15.e even 4 1 1200.2.h.e 2
20.d odd 2 1 inner 1200.2.o.i 4
20.e even 4 1 48.2.c.a 2
20.e even 4 1 1200.2.h.e 2
35.f even 4 1 2352.2.h.c 2
40.i odd 4 1 192.2.c.a 2
40.k even 4 1 192.2.c.a 2
45.k odd 12 1 1296.2.s.b 2
45.k odd 12 1 1296.2.s.e 2
45.l even 12 1 1296.2.s.b 2
45.l even 12 1 1296.2.s.e 2
60.h even 2 1 inner 1200.2.o.i 4
60.l odd 4 1 48.2.c.a 2
60.l odd 4 1 1200.2.h.e 2
80.i odd 4 1 768.2.f.d 4
80.j even 4 1 768.2.f.d 4
80.s even 4 1 768.2.f.d 4
80.t odd 4 1 768.2.f.d 4
105.k odd 4 1 2352.2.h.c 2
120.q odd 4 1 192.2.c.a 2
120.w even 4 1 192.2.c.a 2
140.j odd 4 1 2352.2.h.c 2
180.v odd 12 1 1296.2.s.b 2
180.v odd 12 1 1296.2.s.e 2
180.x even 12 1 1296.2.s.b 2
180.x even 12 1 1296.2.s.e 2
240.z odd 4 1 768.2.f.d 4
240.bb even 4 1 768.2.f.d 4
240.bd odd 4 1 768.2.f.d 4
240.bf even 4 1 768.2.f.d 4
420.w even 4 1 2352.2.h.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
48.2.c.a 2 5.c odd 4 1
48.2.c.a 2 15.e even 4 1
48.2.c.a 2 20.e even 4 1
48.2.c.a 2 60.l odd 4 1
192.2.c.a 2 40.i odd 4 1
192.2.c.a 2 40.k even 4 1
192.2.c.a 2 120.q odd 4 1
192.2.c.a 2 120.w even 4 1
768.2.f.d 4 80.i odd 4 1
768.2.f.d 4 80.j even 4 1
768.2.f.d 4 80.s even 4 1
768.2.f.d 4 80.t odd 4 1
768.2.f.d 4 240.z odd 4 1
768.2.f.d 4 240.bb even 4 1
768.2.f.d 4 240.bd odd 4 1
768.2.f.d 4 240.bf even 4 1
1200.2.h.e 2 5.c odd 4 1
1200.2.h.e 2 15.e even 4 1
1200.2.h.e 2 20.e even 4 1
1200.2.h.e 2 60.l odd 4 1
1200.2.o.i 4 1.a even 1 1 trivial
1200.2.o.i 4 3.b odd 2 1 CM
1200.2.o.i 4 4.b odd 2 1 inner
1200.2.o.i 4 5.b even 2 1 inner
1200.2.o.i 4 12.b even 2 1 inner
1200.2.o.i 4 15.d odd 2 1 inner
1200.2.o.i 4 20.d odd 2 1 inner
1200.2.o.i 4 60.h even 2 1 inner
1296.2.s.b 2 45.k odd 12 1
1296.2.s.b 2 45.l even 12 1
1296.2.s.b 2 180.v odd 12 1
1296.2.s.b 2 180.x even 12 1
1296.2.s.e 2 45.k odd 12 1
1296.2.s.e 2 45.l even 12 1
1296.2.s.e 2 180.v odd 12 1
1296.2.s.e 2 180.x even 12 1
2352.2.h.c 2 35.f even 4 1
2352.2.h.c 2 105.k odd 4 1
2352.2.h.c 2 140.j odd 4 1
2352.2.h.c 2 420.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 $$S_{2}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} - 12$$ T7^2 - 12 $$T_{11}$$ T11 $$T_{17}$$ T17

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 12)^{2}$$
$11$ $$T^{4}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4}$$
$19$ $$(T^{2} + 12)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$(T^{2} + 108)^{2}$$
$37$ $$(T^{2} + 100)^{2}$$
$41$ $$T^{4}$$
$43$ $$(T^{2} - 108)^{2}$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$(T - 14)^{4}$$
$67$ $$(T^{2} - 12)^{2}$$
$71$ $$T^{4}$$
$73$ $$(T^{2} + 100)^{2}$$
$79$ $$(T^{2} + 300)^{2}$$
$83$ $$T^{4}$$
$89$ $$T^{4}$$
$97$ $$(T^{2} + 196)^{2}$$