# Properties

 Label 1200.4.f.c 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.

 $$f(q)$$ $$=$$ $$q -3 i q^{3} + i q^{7} -9 q^{9} +O(q^{10})$$ $$q -3 i q^{3} + i q^{7} -9 q^{9} -42 q^{11} + 67 i q^{13} + 54 i q^{17} -115 q^{19} + 3 q^{21} -162 i q^{23} + 27 i q^{27} + 210 q^{29} + 193 q^{31} + 126 i q^{33} -286 i q^{37} + 201 q^{39} + 12 q^{41} + 263 i q^{43} -414 i q^{47} + 342 q^{49} + 162 q^{51} + 192 i q^{53} + 345 i q^{57} + 690 q^{59} -733 q^{61} -9 i q^{63} -299 i q^{67} -486 q^{69} + 228 q^{71} -938 i q^{73} -42 i q^{77} -160 q^{79} + 81 q^{81} -462 i q^{83} -630 i q^{87} + 240 q^{89} -67 q^{91} -579 i q^{93} -511 i q^{97} + 378 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9} + O(q^{10})$$ $$2 q - 18 q^{9} - 84 q^{11} - 230 q^{19} + 6 q^{21} + 420 q^{29} + 386 q^{31} + 402 q^{39} + 24 q^{41} + 684 q^{49} + 324 q^{51} + 1380 q^{59} - 1466 q^{61} - 972 q^{69} + 456 q^{71} - 320 q^{79} + 162 q^{81} + 480 q^{89} - 134 q^{91} + 756 q^{99} + O(q^{100})$$

## 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 1.00000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 1.00000i 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.c 2
4.b odd 2 1 150.4.c.e 2
5.b even 2 1 inner 1200.4.f.c 2
5.c odd 4 1 1200.4.a.i 1
5.c odd 4 1 1200.4.a.bb 1
12.b even 2 1 450.4.c.a 2
20.d odd 2 1 150.4.c.e 2
20.e even 4 1 150.4.a.a 1
20.e even 4 1 150.4.a.h yes 1
60.h even 2 1 450.4.c.a 2
60.l odd 4 1 450.4.a.f 1
60.l odd 4 1 450.4.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.4.a.a 1 20.e even 4 1
150.4.a.h yes 1 20.e even 4 1
150.4.c.e 2 4.b odd 2 1
150.4.c.e 2 20.d odd 2 1
450.4.a.f 1 60.l odd 4 1
450.4.a.o 1 60.l odd 4 1
450.4.c.a 2 12.b even 2 1
450.4.c.a 2 60.h even 2 1
1200.4.a.i 1 5.c odd 4 1
1200.4.a.bb 1 5.c odd 4 1
1200.4.f.c 2 1.a even 1 1 trivial
1200.4.f.c 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 $$S_{4}^{\mathrm{new}}(1200, [\chi])$$:

 $$T_{7}^{2} + 1$$ $$T_{11} + 42$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$9 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 42 + T )^{2}$$
$13$ $$4489 + T^{2}$$
$17$ $$2916 + T^{2}$$
$19$ $$( 115 + T )^{2}$$
$23$ $$26244 + T^{2}$$
$29$ $$( -210 + T )^{2}$$
$31$ $$( -193 + T )^{2}$$
$37$ $$81796 + T^{2}$$
$41$ $$( -12 + T )^{2}$$
$43$ $$69169 + T^{2}$$
$47$ $$171396 + T^{2}$$
$53$ $$36864 + T^{2}$$
$59$ $$( -690 + T )^{2}$$
$61$ $$( 733 + T )^{2}$$
$67$ $$89401 + T^{2}$$
$71$ $$( -228 + T )^{2}$$
$73$ $$879844 + T^{2}$$
$79$ $$( 160 + T )^{2}$$
$83$ $$213444 + T^{2}$$
$89$ $$( -240 + T )^{2}$$
$97$ $$261121 + T^{2}$$