# Properties

 Label 450.3.b.b Level $450$ Weight $3$ Character orbit 450.b Analytic conductor $12.262$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 450.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$12.2616118962$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{29}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 18) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} +O(q^{10})$$ $$q + ( \zeta_{8} - \zeta_{8}^{3} ) q^{2} + 2 q^{4} + 4 \zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{8} + ( 12 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{11} + 8 \zeta_{8}^{2} q^{13} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{14} + 4 q^{16} + ( -9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{17} + 16 q^{19} + 24 \zeta_{8}^{2} q^{22} + ( 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{23} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{26} + 8 \zeta_{8}^{2} q^{28} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{29} + 44 q^{31} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{32} -18 q^{34} + 34 \zeta_{8}^{2} q^{37} + ( 16 \zeta_{8} - 16 \zeta_{8}^{3} ) q^{38} + ( 33 \zeta_{8} + 33 \zeta_{8}^{3} ) q^{41} -40 \zeta_{8}^{2} q^{43} + ( 24 \zeta_{8} + 24 \zeta_{8}^{3} ) q^{44} + 24 q^{46} + ( -60 \zeta_{8} + 60 \zeta_{8}^{3} ) q^{47} + 33 q^{49} + 16 \zeta_{8}^{2} q^{52} + ( -27 \zeta_{8} + 27 \zeta_{8}^{3} ) q^{53} + ( 8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{56} -6 \zeta_{8}^{2} q^{58} + ( -24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{59} + 50 q^{61} + ( 44 \zeta_{8} - 44 \zeta_{8}^{3} ) q^{62} + 8 q^{64} -8 \zeta_{8}^{2} q^{67} + ( -18 \zeta_{8} + 18 \zeta_{8}^{3} ) q^{68} + ( -36 \zeta_{8} - 36 \zeta_{8}^{3} ) q^{71} -16 \zeta_{8}^{2} q^{73} + ( 34 \zeta_{8} + 34 \zeta_{8}^{3} ) q^{74} + 32 q^{76} + ( -48 \zeta_{8} + 48 \zeta_{8}^{3} ) q^{77} + 76 q^{79} + 66 \zeta_{8}^{2} q^{82} + ( -84 \zeta_{8} + 84 \zeta_{8}^{3} ) q^{83} + ( -40 \zeta_{8} - 40 \zeta_{8}^{3} ) q^{86} + 48 \zeta_{8}^{2} q^{88} + ( -9 \zeta_{8} - 9 \zeta_{8}^{3} ) q^{89} -32 q^{91} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{92} -120 q^{94} -176 \zeta_{8}^{2} q^{97} + ( 33 \zeta_{8} - 33 \zeta_{8}^{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 8q^{4} + O(q^{10})$$ $$4q + 8q^{4} + 16q^{16} + 64q^{19} + 176q^{31} - 72q^{34} + 96q^{46} + 132q^{49} + 200q^{61} + 32q^{64} + 128q^{76} + 304q^{79} - 128q^{91} - 480q^{94} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/450\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 −0.707107 + 0.707107i −0.707107 − 0.707107i 0.707107 − 0.707107i 0.707107 + 0.707107i
−1.41421 0 2.00000 0 0 4.00000i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 4.00000i −2.82843 0 0
449.3 1.41421 0 2.00000 0 0 4.00000i 2.82843 0 0
449.4 1.41421 0 2.00000 0 0 4.00000i 2.82843 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
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.3.b.b 4
3.b odd 2 1 inner 450.3.b.b 4
4.b odd 2 1 3600.3.c.b 4
5.b even 2 1 inner 450.3.b.b 4
5.c odd 4 1 18.3.b.a 2
5.c odd 4 1 450.3.d.f 2
12.b even 2 1 3600.3.c.b 4
15.d odd 2 1 inner 450.3.b.b 4
15.e even 4 1 18.3.b.a 2
15.e even 4 1 450.3.d.f 2
20.d odd 2 1 3600.3.c.b 4
20.e even 4 1 144.3.e.b 2
20.e even 4 1 3600.3.l.d 2
35.f even 4 1 882.3.b.a 2
35.k even 12 2 882.3.s.d 4
35.l odd 12 2 882.3.s.b 4
40.i odd 4 1 576.3.e.c 2
40.k even 4 1 576.3.e.f 2
45.k odd 12 2 162.3.d.b 4
45.l even 12 2 162.3.d.b 4
55.e even 4 1 2178.3.c.d 2
60.h even 2 1 3600.3.c.b 4
60.l odd 4 1 144.3.e.b 2
60.l odd 4 1 3600.3.l.d 2
65.f even 4 1 3042.3.d.a 4
65.h odd 4 1 3042.3.c.e 2
65.k even 4 1 3042.3.d.a 4
80.i odd 4 1 2304.3.h.f 4
80.j even 4 1 2304.3.h.c 4
80.s even 4 1 2304.3.h.c 4
80.t odd 4 1 2304.3.h.f 4
105.k odd 4 1 882.3.b.a 2
105.w odd 12 2 882.3.s.d 4
105.x even 12 2 882.3.s.b 4
120.q odd 4 1 576.3.e.f 2
120.w even 4 1 576.3.e.c 2
165.l odd 4 1 2178.3.c.d 2
180.v odd 12 2 1296.3.q.f 4
180.x even 12 2 1296.3.q.f 4
195.j odd 4 1 3042.3.d.a 4
195.s even 4 1 3042.3.c.e 2
195.u odd 4 1 3042.3.d.a 4
240.z odd 4 1 2304.3.h.c 4
240.bb even 4 1 2304.3.h.f 4
240.bd odd 4 1 2304.3.h.c 4
240.bf even 4 1 2304.3.h.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
18.3.b.a 2 5.c odd 4 1
18.3.b.a 2 15.e even 4 1
144.3.e.b 2 20.e even 4 1
144.3.e.b 2 60.l odd 4 1
162.3.d.b 4 45.k odd 12 2
162.3.d.b 4 45.l even 12 2
450.3.b.b 4 1.a even 1 1 trivial
450.3.b.b 4 3.b odd 2 1 inner
450.3.b.b 4 5.b even 2 1 inner
450.3.b.b 4 15.d odd 2 1 inner
450.3.d.f 2 5.c odd 4 1
450.3.d.f 2 15.e even 4 1
576.3.e.c 2 40.i odd 4 1
576.3.e.c 2 120.w even 4 1
576.3.e.f 2 40.k even 4 1
576.3.e.f 2 120.q odd 4 1
882.3.b.a 2 35.f even 4 1
882.3.b.a 2 105.k odd 4 1
882.3.s.b 4 35.l odd 12 2
882.3.s.b 4 105.x even 12 2
882.3.s.d 4 35.k even 12 2
882.3.s.d 4 105.w odd 12 2
1296.3.q.f 4 180.v odd 12 2
1296.3.q.f 4 180.x even 12 2
2178.3.c.d 2 55.e even 4 1
2178.3.c.d 2 165.l odd 4 1
2304.3.h.c 4 80.j even 4 1
2304.3.h.c 4 80.s even 4 1
2304.3.h.c 4 240.z odd 4 1
2304.3.h.c 4 240.bd odd 4 1
2304.3.h.f 4 80.i odd 4 1
2304.3.h.f 4 80.t odd 4 1
2304.3.h.f 4 240.bb even 4 1
2304.3.h.f 4 240.bf even 4 1
3042.3.c.e 2 65.h odd 4 1
3042.3.c.e 2 195.s even 4 1
3042.3.d.a 4 65.f even 4 1
3042.3.d.a 4 65.k even 4 1
3042.3.d.a 4 195.j odd 4 1
3042.3.d.a 4 195.u odd 4 1
3600.3.c.b 4 4.b odd 2 1
3600.3.c.b 4 12.b even 2 1
3600.3.c.b 4 20.d odd 2 1
3600.3.c.b 4 60.h even 2 1
3600.3.l.d 2 20.e even 4 1
3600.3.l.d 2 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 16$$ acting on $$S_{3}^{\mathrm{new}}(450, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -2 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$( 16 + T^{2} )^{2}$$
$11$ $$( 288 + T^{2} )^{2}$$
$13$ $$( 64 + T^{2} )^{2}$$
$17$ $$( -162 + T^{2} )^{2}$$
$19$ $$( -16 + T )^{4}$$
$23$ $$( -288 + T^{2} )^{2}$$
$29$ $$( 18 + T^{2} )^{2}$$
$31$ $$( -44 + T )^{4}$$
$37$ $$( 1156 + T^{2} )^{2}$$
$41$ $$( 2178 + T^{2} )^{2}$$
$43$ $$( 1600 + T^{2} )^{2}$$
$47$ $$( -7200 + T^{2} )^{2}$$
$53$ $$( -1458 + T^{2} )^{2}$$
$59$ $$( 1152 + T^{2} )^{2}$$
$61$ $$( -50 + T )^{4}$$
$67$ $$( 64 + T^{2} )^{2}$$
$71$ $$( 2592 + T^{2} )^{2}$$
$73$ $$( 256 + T^{2} )^{2}$$
$79$ $$( -76 + T )^{4}$$
$83$ $$( -14112 + T^{2} )^{2}$$
$89$ $$( 162 + T^{2} )^{2}$$
$97$ $$( 30976 + T^{2} )^{2}$$