# Properties

 Label 450.6.c.h Level 450 Weight 6 Character orbit 450.c Analytic conductor 72.173 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$72.1727189158$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 10) 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.

 $$f(q)$$ $$=$$ $$q -4 i q^{2} -16 q^{4} + 172 i q^{7} + 64 i q^{8} +O(q^{10})$$ $$q -4 i q^{2} -16 q^{4} + 172 i q^{7} + 64 i q^{8} -132 q^{11} -946 i q^{13} + 688 q^{14} + 256 q^{16} -222 i q^{17} -500 q^{19} + 528 i q^{22} -3564 i q^{23} -3784 q^{26} -2752 i q^{28} + 2190 q^{29} + 2312 q^{31} -1024 i q^{32} -888 q^{34} + 11242 i q^{37} + 2000 i q^{38} -1242 q^{41} + 20624 i q^{43} + 2112 q^{44} -14256 q^{46} + 6588 i q^{47} -12777 q^{49} + 15136 i q^{52} + 21066 i q^{53} -11008 q^{56} -8760 i q^{58} + 7980 q^{59} + 16622 q^{61} -9248 i q^{62} -4096 q^{64} -1808 i q^{67} + 3552 i q^{68} + 24528 q^{71} + 20474 i q^{73} + 44968 q^{74} + 8000 q^{76} -22704 i q^{77} + 46240 q^{79} + 4968 i q^{82} + 51576 i q^{83} + 82496 q^{86} -8448 i q^{88} -110310 q^{89} + 162712 q^{91} + 57024 i q^{92} + 26352 q^{94} + 78382 i q^{97} + 51108 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} - 264q^{11} + 1376q^{14} + 512q^{16} - 1000q^{19} - 7568q^{26} + 4380q^{29} + 4624q^{31} - 1776q^{34} - 2484q^{41} + 4224q^{44} - 28512q^{46} - 25554q^{49} - 22016q^{56} + 15960q^{59} + 33244q^{61} - 8192q^{64} + 49056q^{71} + 89936q^{74} + 16000q^{76} + 92480q^{79} + 164992q^{86} - 220620q^{89} + 325424q^{91} + 52704q^{94} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 1.00000i − 1.00000i
4.00000i 0 −16.0000 0 0 172.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 172.000i 64.0000i 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.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.c.h 2
3.b odd 2 1 50.6.b.a 2
5.b even 2 1 inner 450.6.c.h 2
5.c odd 4 1 90.6.a.d 1
5.c odd 4 1 450.6.a.l 1
12.b even 2 1 400.6.c.b 2
15.d odd 2 1 50.6.b.a 2
15.e even 4 1 10.6.a.b 1
15.e even 4 1 50.6.a.d 1
20.e even 4 1 720.6.a.j 1
60.h even 2 1 400.6.c.b 2
60.l odd 4 1 80.6.a.a 1
60.l odd 4 1 400.6.a.n 1
105.k odd 4 1 490.6.a.a 1
120.q odd 4 1 320.6.a.o 1
120.w even 4 1 320.6.a.b 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.b 1 15.e even 4 1
50.6.a.d 1 15.e even 4 1
50.6.b.a 2 3.b odd 2 1
50.6.b.a 2 15.d odd 2 1
80.6.a.a 1 60.l odd 4 1
90.6.a.d 1 5.c odd 4 1
320.6.a.b 1 120.w even 4 1
320.6.a.o 1 120.q odd 4 1
400.6.a.n 1 60.l odd 4 1
400.6.c.b 2 12.b even 2 1
400.6.c.b 2 60.h even 2 1
450.6.a.l 1 5.c odd 4 1
450.6.c.h 2 1.a even 1 1 trivial
450.6.c.h 2 5.b even 2 1 inner
490.6.a.a 1 105.k odd 4 1
720.6.a.j 1 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 29584$$ $$T_{11} + 132$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 16 T^{2}$$
$3$ 1
$5$ 1
$7$ $$1 - 4030 T^{2} + 282475249 T^{4}$$
$11$ $$( 1 + 132 T + 161051 T^{2} )^{2}$$
$13$ $$1 + 152330 T^{2} + 137858491849 T^{4}$$
$17$ $$1 - 2790430 T^{2} + 2015993900449 T^{4}$$
$19$ $$( 1 + 500 T + 2476099 T^{2} )^{2}$$
$23$ $$1 - 170590 T^{2} + 41426511213649 T^{4}$$
$29$ $$( 1 - 2190 T + 20511149 T^{2} )^{2}$$
$31$ $$( 1 - 2312 T + 28629151 T^{2} )^{2}$$
$37$ $$1 - 12305350 T^{2} + 4808584372417849 T^{4}$$
$41$ $$( 1 + 1242 T + 115856201 T^{2} )^{2}$$
$43$ $$1 + 131332490 T^{2} + 21611482313284249 T^{4}$$
$47$ $$1 - 415288270 T^{2} + 52599132235830049 T^{4}$$
$53$ $$1 - 392614630 T^{2} + 174887470365513049 T^{4}$$
$59$ $$( 1 - 7980 T + 714924299 T^{2} )^{2}$$
$61$ $$( 1 - 16622 T + 844596301 T^{2} )^{2}$$
$67$ $$1 - 2696981350 T^{2} + 1822837804551761449 T^{4}$$
$71$ $$( 1 - 24528 T + 1804229351 T^{2} )^{2}$$
$73$ $$1 - 3726958510 T^{2} + 4297625829703557649 T^{4}$$
$79$ $$( 1 - 46240 T + 3077056399 T^{2} )^{2}$$
$83$ $$1 - 5217997510 T^{2} + 15516041187205853449 T^{4}$$
$89$ $$( 1 + 110310 T + 5584059449 T^{2} )^{2}$$
$97$ $$1 - 11030942590 T^{2} + 73742412689492826049 T^{4}$$