# Properties

 Label 600.3.u.e Level $600$ Weight $3$ Character orbit 600.u Analytic conductor $16.349$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 600.u (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.3488158616$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + ( 3 - 3 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + ( 3 - 3 \beta_{2} + 2 \beta_{3} ) q^{7} + 3 \beta_{2} q^{9} + ( -4 + \beta_{1} - \beta_{3} ) q^{11} + ( -12 + 2 \beta_{1} - 12 \beta_{2} ) q^{13} + ( 6 - 6 \beta_{2} + 14 \beta_{3} ) q^{17} + ( 6 \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{19} + ( 6 - 3 \beta_{1} + 3 \beta_{3} ) q^{21} + ( 14 + 6 \beta_{1} + 14 \beta_{2} ) q^{23} -3 \beta_{3} q^{27} + ( -9 \beta_{1} - 22 \beta_{2} - 9 \beta_{3} ) q^{29} + ( -20 - 10 \beta_{1} + 10 \beta_{3} ) q^{31} + ( -3 + 4 \beta_{1} - 3 \beta_{2} ) q^{33} + ( -28 + 28 \beta_{2} - 6 \beta_{3} ) q^{37} + ( 12 \beta_{1} - 6 \beta_{2} + 12 \beta_{3} ) q^{39} + ( -14 - 14 \beta_{1} + 14 \beta_{3} ) q^{41} + ( 2 + 28 \beta_{1} + 2 \beta_{2} ) q^{43} + ( 4 - 4 \beta_{2} + 38 \beta_{3} ) q^{47} + ( 12 \beta_{1} + 19 \beta_{2} + 12 \beta_{3} ) q^{49} + ( 42 - 6 \beta_{1} + 6 \beta_{3} ) q^{51} + ( 30 + 20 \beta_{1} + 30 \beta_{2} ) q^{53} + ( 18 - 18 \beta_{2} + 2 \beta_{3} ) q^{57} + ( 21 \beta_{1} - 4 \beta_{2} + 21 \beta_{3} ) q^{59} + ( -6 - 22 \beta_{1} + 22 \beta_{3} ) q^{61} + ( 9 - 6 \beta_{1} + 9 \beta_{2} ) q^{63} + ( 2 - 2 \beta_{2} + 68 \beta_{3} ) q^{67} + ( -14 \beta_{1} - 18 \beta_{2} - 14 \beta_{3} ) q^{69} + ( 68 + 8 \beta_{1} - 8 \beta_{3} ) q^{71} + ( -27 - 40 \beta_{1} - 27 \beta_{2} ) q^{73} + ( -18 + 18 \beta_{2} - 14 \beta_{3} ) q^{77} + ( -6 \beta_{1} + 112 \beta_{2} - 6 \beta_{3} ) q^{79} -9 q^{81} + ( -68 + 22 \beta_{1} - 68 \beta_{2} ) q^{83} + ( -27 + 27 \beta_{2} + 22 \beta_{3} ) q^{87} + ( -4 \beta_{1} + 62 \beta_{2} - 4 \beta_{3} ) q^{89} + ( -84 + 30 \beta_{1} - 30 \beta_{3} ) q^{91} + ( 30 + 20 \beta_{1} + 30 \beta_{2} ) q^{93} + ( 37 - 37 \beta_{2} ) q^{97} + ( 3 \beta_{1} - 12 \beta_{2} + 3 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{7} + O(q^{10})$$ $$4 q + 12 q^{7} - 16 q^{11} - 48 q^{13} + 24 q^{17} + 24 q^{21} + 56 q^{23} - 80 q^{31} - 12 q^{33} - 112 q^{37} - 56 q^{41} + 8 q^{43} + 16 q^{47} + 168 q^{51} + 120 q^{53} + 72 q^{57} - 24 q^{61} + 36 q^{63} + 8 q^{67} + 272 q^{71} - 108 q^{73} - 72 q^{77} - 36 q^{81} - 272 q^{83} - 108 q^{87} - 336 q^{91} + 120 q^{93} + 148 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
193.1
 1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
0 −1.22474 1.22474i 0 0 0 0.550510 0.550510i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 5.44949 5.44949i 0 3.00000i 0
457.1 0 −1.22474 + 1.22474i 0 0 0 0.550510 + 0.550510i 0 3.00000i 0
457.2 0 1.22474 1.22474i 0 0 0 5.44949 + 5.44949i 0 3.00000i 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.u.e 4
3.b odd 2 1 1800.3.v.n 4
4.b odd 2 1 1200.3.bg.e 4
5.b even 2 1 120.3.u.a 4
5.c odd 4 1 120.3.u.a 4
5.c odd 4 1 inner 600.3.u.e 4
15.d odd 2 1 360.3.v.b 4
15.e even 4 1 360.3.v.b 4
15.e even 4 1 1800.3.v.n 4
20.d odd 2 1 240.3.bg.c 4
20.e even 4 1 240.3.bg.c 4
20.e even 4 1 1200.3.bg.e 4
40.e odd 2 1 960.3.bg.d 4
40.f even 2 1 960.3.bg.c 4
40.i odd 4 1 960.3.bg.c 4
40.k even 4 1 960.3.bg.d 4
60.h even 2 1 720.3.bh.g 4
60.l odd 4 1 720.3.bh.g 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.a 4 5.b even 2 1
120.3.u.a 4 5.c odd 4 1
240.3.bg.c 4 20.d odd 2 1
240.3.bg.c 4 20.e even 4 1
360.3.v.b 4 15.d odd 2 1
360.3.v.b 4 15.e even 4 1
600.3.u.e 4 1.a even 1 1 trivial
600.3.u.e 4 5.c odd 4 1 inner
720.3.bh.g 4 60.h even 2 1
720.3.bh.g 4 60.l odd 4 1
960.3.bg.c 4 40.f even 2 1
960.3.bg.c 4 40.i odd 4 1
960.3.bg.d 4 40.e odd 2 1
960.3.bg.d 4 40.k even 4 1
1200.3.bg.e 4 4.b odd 2 1
1200.3.bg.e 4 20.e even 4 1
1800.3.v.n 4 3.b odd 2 1
1800.3.v.n 4 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 12 T_{7}^{3} + 72 T_{7}^{2} - 72 T_{7} + 36$$ acting on $$S_{3}^{\mathrm{new}}(600, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + T^{4}$$
$5$ $$T^{4}$$
$7$ $$36 - 72 T + 72 T^{2} - 12 T^{3} + T^{4}$$
$11$ $$( 10 + 8 T + T^{2} )^{2}$$
$13$ $$76176 + 13248 T + 1152 T^{2} + 48 T^{3} + T^{4}$$
$17$ $$266256 + 12384 T + 288 T^{2} - 24 T^{3} + T^{4}$$
$19$ $$44944 + 440 T^{2} + T^{4}$$
$23$ $$80656 - 15904 T + 1568 T^{2} - 56 T^{3} + T^{4}$$
$29$ $$4 + 1940 T^{2} + T^{4}$$
$31$ $$( -200 + 40 T + T^{2} )^{2}$$
$37$ $$2131600 + 163520 T + 6272 T^{2} + 112 T^{3} + T^{4}$$
$41$ $$( -980 + 28 T + T^{2} )^{2}$$
$43$ $$5494336 + 18752 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$47$ $$18490000 + 68800 T + 128 T^{2} - 16 T^{3} + T^{4}$$
$53$ $$360000 - 72000 T + 7200 T^{2} - 120 T^{3} + T^{4}$$
$59$ $$6916900 + 5324 T^{2} + T^{4}$$
$61$ $$( -2868 + 12 T + T^{2} )^{2}$$
$67$ $$192210496 + 110912 T + 32 T^{2} - 8 T^{3} + T^{4}$$
$71$ $$( 4240 - 136 T + T^{2} )^{2}$$
$73$ $$11168964 - 360936 T + 5832 T^{2} + 108 T^{3} + T^{4}$$
$79$ $$151979584 + 25520 T^{2} + T^{4}$$
$83$ $$60777616 + 2120512 T + 36992 T^{2} + 272 T^{3} + T^{4}$$
$89$ $$14047504 + 7880 T^{2} + T^{4}$$
$97$ $$( 2738 - 74 T + T^{2} )^{2}$$