# Properties

 Label 1200.4.f.q 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 150) 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.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 23 i q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 i q^{3} + 23 i q^{7} -9 q^{9} + 30 q^{11} + 29 i q^{13} -78 i q^{17} + 149 q^{19} -69 q^{21} -150 i q^{23} -27 i q^{27} + 234 q^{29} + 217 q^{31} + 90 i q^{33} -146 i q^{37} -87 q^{39} -156 q^{41} + 433 i q^{43} + 30 i q^{47} -186 q^{49} + 234 q^{51} -552 i q^{53} + 447 i q^{57} -270 q^{59} + 275 q^{61} -207 i q^{63} + 803 i q^{67} + 450 q^{69} -660 q^{71} -646 i q^{73} + 690 i q^{77} + 992 q^{79} + 81 q^{81} + 846 i q^{83} + 702 i q^{87} + 1488 q^{89} -667 q^{91} + 651 i q^{93} + 319 i q^{97} -270 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9} + O(q^{10})$$ $$2 q - 18 q^{9} + 60 q^{11} + 298 q^{19} - 138 q^{21} + 468 q^{29} + 434 q^{31} - 174 q^{39} - 312 q^{41} - 372 q^{49} + 468 q^{51} - 540 q^{59} + 550 q^{61} + 900 q^{69} - 1320 q^{71} + 1984 q^{79} + 162 q^{81} + 2976 q^{89} - 1334 q^{91} - 540 q^{99} + O(q^{100})$$

## 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 23.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 23.0000i 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.q 2
4.b odd 2 1 150.4.c.b 2
5.b even 2 1 inner 1200.4.f.q 2
5.c odd 4 1 1200.4.a.r 1
5.c odd 4 1 1200.4.a.v 1
12.b even 2 1 450.4.c.h 2
20.d odd 2 1 150.4.c.b 2
20.e even 4 1 150.4.a.c 1
20.e even 4 1 150.4.a.g yes 1
60.h even 2 1 450.4.c.h 2
60.l odd 4 1 450.4.a.i 1
60.l odd 4 1 450.4.a.l 1

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

## Hecke kernels

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

 $$T_{7}^{2} + 529$$ $$T_{11} - 30$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$529 + T^{2}$$
$11$ $$( -30 + T )^{2}$$
$13$ $$841 + T^{2}$$
$17$ $$6084 + T^{2}$$
$19$ $$( -149 + T )^{2}$$
$23$ $$22500 + T^{2}$$
$29$ $$( -234 + T )^{2}$$
$31$ $$( -217 + T )^{2}$$
$37$ $$21316 + T^{2}$$
$41$ $$( 156 + T )^{2}$$
$43$ $$187489 + T^{2}$$
$47$ $$900 + T^{2}$$
$53$ $$304704 + T^{2}$$
$59$ $$( 270 + T )^{2}$$
$61$ $$( -275 + T )^{2}$$
$67$ $$644809 + T^{2}$$
$71$ $$( 660 + T )^{2}$$
$73$ $$417316 + T^{2}$$
$79$ $$( -992 + T )^{2}$$
$83$ $$715716 + T^{2}$$
$89$ $$( -1488 + T )^{2}$$
$97$ $$101761 + T^{2}$$