# Properties

 Label 1200.4.f.i Level $1200$ Weight $4$ Character orbit 1200.f Analytic conductor $70.802$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$70.8022920069$$ 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: $$1$$ Twist minimal: no (minimal twist has level 600) 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.

 $$f(q)$$ $$=$$ $$q -3 i q^{3} + 5 i q^{7} -9 q^{9} +O(q^{10})$$ $$q -3 i q^{3} + 5 i q^{7} -9 q^{9} -14 q^{11} + i q^{13} -46 i q^{17} + 19 q^{19} + 15 q^{21} + 46 i q^{23} + 27 i q^{27} -14 q^{29} -133 q^{31} + 42 i q^{33} -258 i q^{37} + 3 q^{39} + 84 q^{41} + 167 i q^{43} + 410 i q^{47} + 318 q^{49} -138 q^{51} + 456 i q^{53} -57 i q^{57} -194 q^{59} -17 q^{61} -45 i q^{63} + 653 i q^{67} + 138 q^{69} -828 q^{71} + 570 i q^{73} -70 i q^{77} -552 q^{79} + 81 q^{81} -142 i q^{83} + 42 i q^{87} + 1104 q^{89} -5 q^{91} + 399 i q^{93} -841 i q^{97} + 126 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9} + O(q^{10})$$ $$2 q - 18 q^{9} - 28 q^{11} + 38 q^{19} + 30 q^{21} - 28 q^{29} - 266 q^{31} + 6 q^{39} + 168 q^{41} + 636 q^{49} - 276 q^{51} - 388 q^{59} - 34 q^{61} + 276 q^{69} - 1656 q^{71} - 1104 q^{79} + 162 q^{81} + 2208 q^{89} - 10 q^{91} + 252 q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 1.00000i − 1.00000i
0 3.00000i 0 0 0 5.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 5.00000i 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
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 1200.4.f.i 2
4.b odd 2 1 600.4.f.e 2
5.b even 2 1 inner 1200.4.f.i 2
5.c odd 4 1 1200.4.a.g 1
5.c odd 4 1 1200.4.a.bd 1
12.b even 2 1 1800.4.f.l 2
20.d odd 2 1 600.4.f.e 2
20.e even 4 1 600.4.a.e 1
20.e even 4 1 600.4.a.n yes 1
60.h even 2 1 1800.4.f.l 2
60.l odd 4 1 1800.4.a.m 1
60.l odd 4 1 1800.4.a.v 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.e 1 20.e even 4 1
600.4.a.n yes 1 20.e even 4 1
600.4.f.e 2 4.b odd 2 1
600.4.f.e 2 20.d odd 2 1
1200.4.a.g 1 5.c odd 4 1
1200.4.a.bd 1 5.c odd 4 1
1200.4.f.i 2 1.a even 1 1 trivial
1200.4.f.i 2 5.b even 2 1 inner
1800.4.a.m 1 60.l odd 4 1
1800.4.a.v 1 60.l odd 4 1
1800.4.f.l 2 12.b even 2 1
1800.4.f.l 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 25$$ $$T_{11} + 14$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$25 + T^{2}$$
$11$ $$( 14 + T )^{2}$$
$13$ $$1 + T^{2}$$
$17$ $$2116 + T^{2}$$
$19$ $$( -19 + T )^{2}$$
$23$ $$2116 + T^{2}$$
$29$ $$( 14 + T )^{2}$$
$31$ $$( 133 + T )^{2}$$
$37$ $$66564 + T^{2}$$
$41$ $$( -84 + T )^{2}$$
$43$ $$27889 + T^{2}$$
$47$ $$168100 + T^{2}$$
$53$ $$207936 + T^{2}$$
$59$ $$( 194 + T )^{2}$$
$61$ $$( 17 + T )^{2}$$
$67$ $$426409 + T^{2}$$
$71$ $$( 828 + T )^{2}$$
$73$ $$324900 + T^{2}$$
$79$ $$( 552 + T )^{2}$$
$83$ $$20164 + T^{2}$$
$89$ $$( -1104 + T )^{2}$$
$97$ $$707281 + T^{2}$$