# Properties

 Label 450.4.c.f Level $450$ Weight $4$ Character orbit 450.c Analytic conductor $26.551$ 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: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 450.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$26.5508595026$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} - 4 q^{4} + 7 \beta q^{7} - 4 \beta q^{8} +O(q^{10})$$ q + b * q^2 - 4 * q^4 + 7*b * q^7 - 4*b * q^8 $$q + \beta q^{2} - 4 q^{4} + 7 \beta q^{7} - 4 \beta q^{8} - 6 q^{11} - 34 \beta q^{13} - 28 q^{14} + 16 q^{16} - 39 \beta q^{17} - 44 q^{19} - 6 \beta q^{22} + 60 \beta q^{23} + 136 q^{26} - 28 \beta q^{28} + 126 q^{29} - 244 q^{31} + 16 \beta q^{32} + 156 q^{34} - 152 \beta q^{37} - 44 \beta q^{38} + 480 q^{41} - 52 \beta q^{43} + 24 q^{44} - 240 q^{46} - 300 \beta q^{47} + 147 q^{49} + 136 \beta q^{52} - 129 \beta q^{53} + 112 q^{56} + 126 \beta q^{58} + 534 q^{59} + 362 q^{61} - 244 \beta q^{62} - 64 q^{64} - 134 \beta q^{67} + 156 \beta q^{68} + 972 q^{71} - 235 \beta q^{73} + 608 q^{74} + 176 q^{76} - 42 \beta q^{77} - 1244 q^{79} + 480 \beta q^{82} + 198 \beta q^{83} + 208 q^{86} + 24 \beta q^{88} - 972 q^{89} + 952 q^{91} - 240 \beta q^{92} + 1200 q^{94} - 23 \beta q^{97} + 147 \beta q^{98} +O(q^{100})$$ q + b * q^2 - 4 * q^4 + 7*b * q^7 - 4*b * q^8 - 6 * q^11 - 34*b * q^13 - 28 * q^14 + 16 * q^16 - 39*b * q^17 - 44 * q^19 - 6*b * q^22 + 60*b * q^23 + 136 * q^26 - 28*b * q^28 + 126 * q^29 - 244 * q^31 + 16*b * q^32 + 156 * q^34 - 152*b * q^37 - 44*b * q^38 + 480 * q^41 - 52*b * q^43 + 24 * q^44 - 240 * q^46 - 300*b * q^47 + 147 * q^49 + 136*b * q^52 - 129*b * q^53 + 112 * q^56 + 126*b * q^58 + 534 * q^59 + 362 * q^61 - 244*b * q^62 - 64 * q^64 - 134*b * q^67 + 156*b * q^68 + 972 * q^71 - 235*b * q^73 + 608 * q^74 + 176 * q^76 - 42*b * q^77 - 1244 * q^79 + 480*b * q^82 + 198*b * q^83 + 208 * q^86 + 24*b * q^88 - 972 * q^89 + 952 * q^91 - 240*b * q^92 + 1200 * q^94 - 23*b * q^97 + 147*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 8 q^{4}+O(q^{10})$$ 2 * q - 8 * q^4 $$2 q - 8 q^{4} - 12 q^{11} - 56 q^{14} + 32 q^{16} - 88 q^{19} + 272 q^{26} + 252 q^{29} - 488 q^{31} + 312 q^{34} + 960 q^{41} + 48 q^{44} - 480 q^{46} + 294 q^{49} + 224 q^{56} + 1068 q^{59} + 724 q^{61} - 128 q^{64} + 1944 q^{71} + 1216 q^{74} + 352 q^{76} - 2488 q^{79} + 416 q^{86} - 1944 q^{89} + 1904 q^{91} + 2400 q^{94}+O(q^{100})$$ 2 * q - 8 * q^4 - 12 * q^11 - 56 * q^14 + 32 * q^16 - 88 * q^19 + 272 * q^26 + 252 * q^29 - 488 * q^31 + 312 * q^34 + 960 * q^41 + 48 * q^44 - 480 * q^46 + 294 * q^49 + 224 * q^56 + 1068 * q^59 + 724 * q^61 - 128 * q^64 + 1944 * q^71 + 1216 * q^74 + 352 * q^76 - 2488 * q^79 + 416 * q^86 - 1944 * q^89 + 1904 * q^91 + 2400 * q^94

## 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
2.00000i 0 −4.00000 0 0 14.0000i 8.00000i 0 0
199.2 2.00000i 0 −4.00000 0 0 14.0000i 8.00000i 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 450.4.c.f 2
3.b odd 2 1 450.4.c.g 2
5.b even 2 1 inner 450.4.c.f 2
5.c odd 4 1 90.4.a.e yes 1
5.c odd 4 1 450.4.a.c 1
15.d odd 2 1 450.4.c.g 2
15.e even 4 1 90.4.a.b 1
15.e even 4 1 450.4.a.m 1
20.e even 4 1 720.4.a.t 1
45.k odd 12 2 810.4.e.a 2
45.l even 12 2 810.4.e.u 2
60.l odd 4 1 720.4.a.e 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.4.a.b 1 15.e even 4 1
90.4.a.e yes 1 5.c odd 4 1
450.4.a.c 1 5.c odd 4 1
450.4.a.m 1 15.e even 4 1
450.4.c.f 2 1.a even 1 1 trivial
450.4.c.f 2 5.b even 2 1 inner
450.4.c.g 2 3.b odd 2 1
450.4.c.g 2 15.d odd 2 1
720.4.a.e 1 60.l odd 4 1
720.4.a.t 1 20.e even 4 1
810.4.e.a 2 45.k odd 12 2
810.4.e.u 2 45.l even 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(450, [\chi])$$:

 $$T_{7}^{2} + 196$$ T7^2 + 196 $$T_{11} + 6$$ T11 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 4$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 196$$
$11$ $$(T + 6)^{2}$$
$13$ $$T^{2} + 4624$$
$17$ $$T^{2} + 6084$$
$19$ $$(T + 44)^{2}$$
$23$ $$T^{2} + 14400$$
$29$ $$(T - 126)^{2}$$
$31$ $$(T + 244)^{2}$$
$37$ $$T^{2} + 92416$$
$41$ $$(T - 480)^{2}$$
$43$ $$T^{2} + 10816$$
$47$ $$T^{2} + 360000$$
$53$ $$T^{2} + 66564$$
$59$ $$(T - 534)^{2}$$
$61$ $$(T - 362)^{2}$$
$67$ $$T^{2} + 71824$$
$71$ $$(T - 972)^{2}$$
$73$ $$T^{2} + 220900$$
$79$ $$(T + 1244)^{2}$$
$83$ $$T^{2} + 156816$$
$89$ $$(T + 972)^{2}$$
$97$ $$T^{2} + 2116$$