Properties

 Label 450.6.c.d 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 90) 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.

 $$f(q)$$ $$=$$ $$q -4 i q^{2} -16 q^{4} + 98 i q^{7} + 64 i q^{8} +O(q^{10})$$ $$q -4 i q^{2} -16 q^{4} + 98 i q^{7} + 64 i q^{8} -354 q^{11} -404 i q^{13} + 392 q^{14} + 256 q^{16} -654 i q^{17} -1796 q^{19} + 1416 i q^{22} -1080 i q^{23} -1616 q^{26} -1568 i q^{28} -5754 q^{29} + 10196 q^{31} -1024 i q^{32} -2616 q^{34} + 5552 i q^{37} + 7184 i q^{38} + 12960 q^{41} + 8968 i q^{43} + 5664 q^{44} -4320 q^{46} + 5400 i q^{47} + 7203 q^{49} + 6464 i q^{52} + 8214 i q^{53} -6272 q^{56} + 23016 i q^{58} + 3954 q^{59} + 962 q^{61} -40784 i q^{62} -4096 q^{64} -17956 i q^{67} + 10464 i q^{68} + 56148 q^{71} + 85690 i q^{73} + 22208 q^{74} + 28736 q^{76} -34692 i q^{77} + 26044 q^{79} -51840 i q^{82} -93468 i q^{83} + 35872 q^{86} -22656 i q^{88} + 73428 q^{89} + 39592 q^{91} + 17280 i q^{92} + 21600 q^{94} + 128978 i q^{97} -28812 i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} + O(q^{10})$$ $$2q - 32q^{4} - 708q^{11} + 784q^{14} + 512q^{16} - 3592q^{19} - 3232q^{26} - 11508q^{29} + 20392q^{31} - 5232q^{34} + 25920q^{41} + 11328q^{44} - 8640q^{46} + 14406q^{49} - 12544q^{56} + 7908q^{59} + 1924q^{61} - 8192q^{64} + 112296q^{71} + 44416q^{74} + 57472q^{76} + 52088q^{79} + 71744q^{86} + 146856q^{89} + 79184q^{91} + 43200q^{94} + O(q^{100})$$

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 98.0000i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 98.0000i 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.d 2
3.b odd 2 1 450.6.c.l 2
5.b even 2 1 inner 450.6.c.d 2
5.c odd 4 1 90.6.a.c 1
5.c odd 4 1 450.6.a.o 1
15.d odd 2 1 450.6.c.l 2
15.e even 4 1 90.6.a.e yes 1
15.e even 4 1 450.6.a.d 1
20.e even 4 1 720.6.a.o 1
60.l odd 4 1 720.6.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.6.a.c 1 5.c odd 4 1
90.6.a.e yes 1 15.e even 4 1
450.6.a.d 1 15.e even 4 1
450.6.a.o 1 5.c odd 4 1
450.6.c.d 2 1.a even 1 1 trivial
450.6.c.d 2 5.b even 2 1 inner
450.6.c.l 2 3.b odd 2 1
450.6.c.l 2 15.d odd 2 1
720.6.a.c 1 60.l odd 4 1
720.6.a.o 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 $$S_{6}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 9604$$ $$T_{11} + 354$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$9604 + T^{2}$$
$11$ $$( 354 + T )^{2}$$
$13$ $$163216 + T^{2}$$
$17$ $$427716 + T^{2}$$
$19$ $$( 1796 + T )^{2}$$
$23$ $$1166400 + T^{2}$$
$29$ $$( 5754 + T )^{2}$$
$31$ $$( -10196 + T )^{2}$$
$37$ $$30824704 + T^{2}$$
$41$ $$( -12960 + T )^{2}$$
$43$ $$80425024 + T^{2}$$
$47$ $$29160000 + T^{2}$$
$53$ $$67469796 + T^{2}$$
$59$ $$( -3954 + T )^{2}$$
$61$ $$( -962 + T )^{2}$$
$67$ $$322417936 + T^{2}$$
$71$ $$( -56148 + T )^{2}$$
$73$ $$7342776100 + T^{2}$$
$79$ $$( -26044 + T )^{2}$$
$83$ $$8736267024 + T^{2}$$
$89$ $$( -73428 + T )^{2}$$
$97$ $$16635324484 + T^{2}$$