# Properties

 Label 1200.4.f.v Level $1200$ Weight $4$ Character orbit 1200.f Analytic conductor $70.802$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{19})$$ Defining polynomial: $$x^{4} - 9 x^{2} + 25$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 3 \beta_{1} q^{3} + ( -13 \beta_{1} + \beta_{3} ) q^{7} -9 q^{9} +O(q^{10})$$ $$q + 3 \beta_{1} q^{3} + ( -13 \beta_{1} + \beta_{3} ) q^{7} -9 q^{9} + ( -14 + \beta_{2} ) q^{11} + ( -9 \beta_{1} - 4 \beta_{3} ) q^{13} + ( -34 \beta_{1} + 5 \beta_{3} ) q^{17} + ( 3 + \beta_{2} ) q^{19} + ( 39 - 3 \beta_{2} ) q^{21} + ( 66 \beta_{1} + 3 \beta_{3} ) q^{23} -27 \beta_{1} q^{27} + ( -46 - 7 \beta_{2} ) q^{29} + ( -61 + 7 \beta_{2} ) q^{31} + ( -42 \beta_{1} + 3 \beta_{3} ) q^{33} + ( -142 \beta_{1} - 6 \beta_{3} ) q^{37} + ( 27 + 12 \beta_{2} ) q^{39} + ( 196 + 13 \beta_{2} ) q^{41} + ( 345 \beta_{1} + \beta_{3} ) q^{43} + ( 310 \beta_{1} - 8 \beta_{3} ) q^{47} + ( -130 + 26 \beta_{2} ) q^{49} + ( 102 - 15 \beta_{2} ) q^{51} + ( 424 \beta_{1} + 7 \beta_{3} ) q^{53} + ( 9 \beta_{1} + 3 \beta_{3} ) q^{57} + ( 62 - 16 \beta_{2} ) q^{59} + ( 375 - 14 \beta_{2} ) q^{61} + ( 117 \beta_{1} - 9 \beta_{3} ) q^{63} + ( 179 \beta_{1} + 25 \beta_{3} ) q^{67} + ( -198 - 9 \beta_{2} ) q^{69} + ( -412 + 5 \beta_{2} ) q^{71} + ( 54 \beta_{1} + 2 \beta_{3} ) q^{73} + ( 486 \beta_{1} - 27 \beta_{3} ) q^{77} + ( -440 - 40 \beta_{2} ) q^{79} + 81 q^{81} + ( 78 \beta_{1} - 48 \beta_{3} ) q^{83} + ( -138 \beta_{1} - 21 \beta_{3} ) q^{87} + ( 432 - 36 \beta_{2} ) q^{89} + ( 1099 - 43 \beta_{2} ) q^{91} + ( -183 \beta_{1} + 21 \beta_{3} ) q^{93} + 521 \beta_{1} q^{97} + ( 126 - 9 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 36 q^{9} + O(q^{10})$$ $$4 q - 36 q^{9} - 56 q^{11} + 12 q^{19} + 156 q^{21} - 184 q^{29} - 244 q^{31} + 108 q^{39} + 784 q^{41} - 520 q^{49} + 408 q^{51} + 248 q^{59} + 1500 q^{61} - 792 q^{69} - 1648 q^{71} - 1760 q^{79} + 324 q^{81} + 1728 q^{89} + 4396 q^{91} + 504 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 9 x^{2} + 25$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 4 \nu$$$$)/5$$ $$\beta_{2}$$ $$=$$ $$($$$$-4 \nu^{3} + 56 \nu$$$$)/5$$ $$\beta_{3}$$ $$=$$ $$8 \nu^{2} - 36$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + 4 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 36$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 14 \beta_{1}$$$$)/2$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 2.17945 − 0.500000i −2.17945 − 0.500000i −2.17945 + 0.500000i 2.17945 + 0.500000i
0 3.00000i 0 0 0 4.43560i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 30.4356i 0 −9.00000 0
49.3 0 3.00000i 0 0 0 30.4356i 0 −9.00000 0
49.4 0 3.00000i 0 0 0 4.43560i 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.v 4
4.b odd 2 1 75.4.b.c 4
5.b even 2 1 inner 1200.4.f.v 4
5.c odd 4 1 1200.4.a.bl 2
5.c odd 4 1 1200.4.a.bu 2
12.b even 2 1 225.4.b.h 4
20.d odd 2 1 75.4.b.c 4
20.e even 4 1 75.4.a.d 2
20.e even 4 1 75.4.a.e yes 2
60.h even 2 1 225.4.b.h 4
60.l odd 4 1 225.4.a.j 2
60.l odd 4 1 225.4.a.n 2

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

 $$T_{7}^{4} + 946 T_{7}^{2} + 18225$$ $$T_{11}^{2} + 28 T_{11} - 108$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 9 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$18225 + 946 T^{2} + T^{4}$$
$11$ $$( -108 + 28 T + T^{2} )^{2}$$
$13$ $$22877089 + 9890 T^{2} + T^{4}$$
$17$ $$41525136 + 17512 T^{2} + T^{4}$$
$19$ $$( -295 - 6 T + T^{2} )^{2}$$
$23$ $$2624400 + 14184 T^{2} + T^{4}$$
$29$ $$( -12780 + 92 T + T^{2} )^{2}$$
$31$ $$( -11175 + 122 T + T^{2} )^{2}$$
$37$ $$85008400 + 62216 T^{2} + T^{4}$$
$41$ $$( -12960 - 392 T + T^{2} )^{2}$$
$43$ $$14094675841 + 238658 T^{2} + T^{4}$$
$47$ $$5874302736 + 231112 T^{2} + T^{4}$$
$53$ $$27185414400 + 389344 T^{2} + T^{4}$$
$59$ $$( -73980 - 124 T + T^{2} )^{2}$$
$61$ $$( 81041 - 750 T + T^{2} )^{2}$$
$67$ $$24951045681 + 444082 T^{2} + T^{4}$$
$71$ $$( 162144 + 824 T + T^{2} )^{2}$$
$73$ $$2890000 + 8264 T^{2} + T^{4}$$
$79$ $$( -292800 + 880 T + T^{2} )^{2}$$
$83$ $$482096926224 + 1413000 T^{2} + T^{4}$$
$89$ $$( -207360 - 864 T + T^{2} )^{2}$$
$97$ $$( 271441 + T^{2} )^{2}$$