# Properties

 Label 450.6.c.f 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} -16 q^{4} + 118 i q^{7} -64 i q^{8} +O(q^{10})$$ $$q + 4 i q^{2} -16 q^{4} + 118 i q^{7} -64 i q^{8} -192 q^{11} + 1106 i q^{13} -472 q^{14} + 256 q^{16} + 762 i q^{17} + 2740 q^{19} -768 i q^{22} -1566 i q^{23} -4424 q^{26} -1888 i q^{28} + 5910 q^{29} -6868 q^{31} + 1024 i q^{32} -3048 q^{34} + 5518 i q^{37} + 10960 i q^{38} + 378 q^{41} -2434 i q^{43} + 3072 q^{44} + 6264 q^{46} + 13122 i q^{47} + 2883 q^{49} -17696 i q^{52} + 9174 i q^{53} + 7552 q^{56} + 23640 i q^{58} -34980 q^{59} -9838 q^{61} -27472 i q^{62} -4096 q^{64} -33722 i q^{67} -12192 i q^{68} -70212 q^{71} + 21986 i q^{73} -22072 q^{74} -43840 q^{76} -22656 i q^{77} -4520 q^{79} + 1512 i q^{82} + 109074 i q^{83} + 9736 q^{86} + 12288 i q^{88} + 38490 q^{89} -130508 q^{91} + 25056 i q^{92} -52488 q^{94} + 1918 i q^{97} + 11532 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} - 384q^{11} - 944q^{14} + 512q^{16} + 5480q^{19} - 8848q^{26} + 11820q^{29} - 13736q^{31} - 6096q^{34} + 756q^{41} + 6144q^{44} + 12528q^{46} + 5766q^{49} + 15104q^{56} - 69960q^{59} - 19676q^{61} - 8192q^{64} - 140424q^{71} - 44144q^{74} - 87680q^{76} - 9040q^{79} + 19472q^{86} + 76980q^{89} - 261016q^{91} - 104976q^{94} + O(q^{100})$$

## 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 0 118.000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 118.000i 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.f 2
3.b odd 2 1 50.6.b.b 2
5.b even 2 1 inner 450.6.c.f 2
5.c odd 4 1 90.6.a.b 1
5.c odd 4 1 450.6.a.u 1
12.b even 2 1 400.6.c.i 2
15.d odd 2 1 50.6.b.b 2
15.e even 4 1 10.6.a.c 1
15.e even 4 1 50.6.a.b 1
20.e even 4 1 720.6.a.v 1
60.h even 2 1 400.6.c.i 2
60.l odd 4 1 80.6.a.c 1
60.l odd 4 1 400.6.a.i 1
105.k odd 4 1 490.6.a.k 1
120.q odd 4 1 320.6.a.k 1
120.w even 4 1 320.6.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.6.a.c 1 15.e even 4 1
50.6.a.b 1 15.e even 4 1
50.6.b.b 2 3.b odd 2 1
50.6.b.b 2 15.d odd 2 1
80.6.a.c 1 60.l odd 4 1
90.6.a.b 1 5.c odd 4 1
320.6.a.f 1 120.w even 4 1
320.6.a.k 1 120.q odd 4 1
400.6.a.i 1 60.l odd 4 1
400.6.c.i 2 12.b even 2 1
400.6.c.i 2 60.h even 2 1
450.6.a.u 1 5.c odd 4 1
450.6.c.f 2 1.a even 1 1 trivial
450.6.c.f 2 5.b even 2 1 inner
490.6.a.k 1 105.k odd 4 1
720.6.a.v 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 $$S_{6}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 13924$$ $$T_{11} + 192$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$13924 + T^{2}$$
$11$ $$( 192 + T )^{2}$$
$13$ $$1223236 + T^{2}$$
$17$ $$580644 + T^{2}$$
$19$ $$( -2740 + T )^{2}$$
$23$ $$2452356 + T^{2}$$
$29$ $$( -5910 + T )^{2}$$
$31$ $$( 6868 + T )^{2}$$
$37$ $$30448324 + T^{2}$$
$41$ $$( -378 + T )^{2}$$
$43$ $$5924356 + T^{2}$$
$47$ $$172186884 + T^{2}$$
$53$ $$84162276 + T^{2}$$
$59$ $$( 34980 + T )^{2}$$
$61$ $$( 9838 + T )^{2}$$
$67$ $$1137173284 + T^{2}$$
$71$ $$( 70212 + T )^{2}$$
$73$ $$483384196 + T^{2}$$
$79$ $$( 4520 + T )^{2}$$
$83$ $$11897137476 + T^{2}$$
$89$ $$( -38490 + T )^{2}$$
$97$ $$3678724 + T^{2}$$