# Properties

 Label 1800.3.v.n Level $1800$ Weight $3$ Character orbit 1800.v Analytic conductor $49.046$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1800 = 2^{3} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1800.v (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$49.0464475849$$ 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: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} +O(q^{10})$$ $$q + ( 3 - 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{7} + ( 4 - \beta_{3} ) q^{11} + ( -12 - 12 \beta_{1} + \beta_{2} + \beta_{3} ) q^{13} + ( -6 + 6 \beta_{1} - 7 \beta_{2} + 7 \beta_{3} ) q^{17} + ( -2 \beta_{1} + 6 \beta_{2} ) q^{19} + ( -14 - 14 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{23} + ( 22 \beta_{1} + 9 \beta_{2} ) q^{29} + ( -20 - 10 \beta_{3} ) q^{31} + ( -28 + 28 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{37} + ( 14 + 14 \beta_{3} ) q^{41} + ( 2 + 2 \beta_{1} + 14 \beta_{2} + 14 \beta_{3} ) q^{43} + ( -4 + 4 \beta_{1} - 19 \beta_{2} + 19 \beta_{3} ) q^{47} + ( 19 \beta_{1} + 12 \beta_{2} ) q^{49} + ( -30 - 30 \beta_{1} - 10 \beta_{2} - 10 \beta_{3} ) q^{53} + ( 4 \beta_{1} - 21 \beta_{2} ) q^{59} + ( -6 - 22 \beta_{3} ) q^{61} + ( 2 - 2 \beta_{1} + 34 \beta_{2} - 34 \beta_{3} ) q^{67} + ( -68 - 8 \beta_{3} ) q^{71} + ( -27 - 27 \beta_{1} - 20 \beta_{2} - 20 \beta_{3} ) q^{73} + ( 18 - 18 \beta_{1} + 7 \beta_{2} - 7 \beta_{3} ) q^{77} + ( 112 \beta_{1} - 6 \beta_{2} ) q^{79} + ( 68 + 68 \beta_{1} - 11 \beta_{2} - 11 \beta_{3} ) q^{83} + ( -62 \beta_{1} + 4 \beta_{2} ) q^{89} + ( -84 + 30 \beta_{3} ) q^{91} + ( 37 - 37 \beta_{1} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 12 q^{7} + O(q^{10})$$ $$4 q + 12 q^{7} + 16 q^{11} - 48 q^{13} - 24 q^{17} - 56 q^{23} - 80 q^{31} - 112 q^{37} + 56 q^{41} + 8 q^{43} - 16 q^{47} - 120 q^{53} - 24 q^{61} + 8 q^{67} - 272 q^{71} - 108 q^{73} + 72 q^{77} + 272 q^{83} - 336 q^{91} + 148 q^{97} + O(q^{100})$$

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2}$$$$/3$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 3 \nu$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 3 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$3 \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$($$$$-3 \beta_{3} + 3 \beta_{2}$$$$)/2$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times$$.

 $$n$$ $$577$$ $$901$$ $$1001$$ $$1351$$ $$\chi(n)$$ $$-\beta_{1}$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
793.1
 1.22474 + 1.22474i −1.22474 − 1.22474i 1.22474 − 1.22474i −1.22474 + 1.22474i
0 0 0 0 0 0.550510 0.550510i 0 0 0
793.2 0 0 0 0 0 5.44949 5.44949i 0 0 0
1657.1 0 0 0 0 0 0.550510 + 0.550510i 0 0 0
1657.2 0 0 0 0 0 5.44949 + 5.44949i 0 0 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 1800.3.v.n 4
3.b odd 2 1 600.3.u.e 4
5.b even 2 1 360.3.v.b 4
5.c odd 4 1 360.3.v.b 4
5.c odd 4 1 inner 1800.3.v.n 4
12.b even 2 1 1200.3.bg.e 4
15.d odd 2 1 120.3.u.a 4
15.e even 4 1 120.3.u.a 4
15.e even 4 1 600.3.u.e 4
20.d odd 2 1 720.3.bh.g 4
20.e even 4 1 720.3.bh.g 4
60.h even 2 1 240.3.bg.c 4
60.l odd 4 1 240.3.bg.c 4
60.l odd 4 1 1200.3.bg.e 4
120.i odd 2 1 960.3.bg.c 4
120.m even 2 1 960.3.bg.d 4
120.q odd 4 1 960.3.bg.d 4
120.w even 4 1 960.3.bg.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.u.a 4 15.d odd 2 1
120.3.u.a 4 15.e even 4 1
240.3.bg.c 4 60.h even 2 1
240.3.bg.c 4 60.l odd 4 1
360.3.v.b 4 5.b even 2 1
360.3.v.b 4 5.c odd 4 1
600.3.u.e 4 3.b odd 2 1
600.3.u.e 4 15.e even 4 1
720.3.bh.g 4 20.d odd 2 1
720.3.bh.g 4 20.e even 4 1
960.3.bg.c 4 120.i odd 2 1
960.3.bg.c 4 120.w even 4 1
960.3.bg.d 4 120.m even 2 1
960.3.bg.d 4 120.q odd 4 1
1200.3.bg.e 4 12.b even 2 1
1200.3.bg.e 4 60.l odd 4 1
1800.3.v.n 4 1.a even 1 1 trivial
1800.3.v.n 4 5.c odd 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1800, [\chi])$$:

 $$T_{7}^{4} - 12 T_{7}^{3} + 72 T_{7}^{2} - 72 T_{7} + 36$$ $$T_{11}^{2} - 8 T_{11} + 10$$ $$T_{17}^{4} + 24 T_{17}^{3} + 288 T_{17}^{2} - 12384 T_{17} + 266256$$

## Hecke characteristic polynomials

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