# Properties

 Label 450.6.c.p Level $450$ Weight $6$ Character orbit 450.c Analytic conductor $72.173$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(i, \sqrt{4081})$$ Defining polynomial: $$x^{4} + 2041 x^{2} + 1040400$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 \beta_{1} q^{2} -16 q^{4} + ( -50 \beta_{1} - \beta_{2} ) q^{7} + 64 \beta_{1} q^{8} +O(q^{10})$$ $$q -4 \beta_{1} q^{2} -16 q^{4} + ( -50 \beta_{1} - \beta_{2} ) q^{7} + 64 \beta_{1} q^{8} + ( -270 - 2 \beta_{3} ) q^{11} + ( -445 \beta_{1} - 2 \beta_{2} ) q^{13} + ( -200 - 4 \beta_{3} ) q^{14} + 256 q^{16} + ( 246 \beta_{1} + 10 \beta_{2} ) q^{17} + ( -296 - 5 \beta_{3} ) q^{19} + ( 1080 \beta_{1} + 8 \beta_{2} ) q^{22} + ( -1830 \beta_{1} - 10 \beta_{2} ) q^{23} + ( -1780 - 8 \beta_{3} ) q^{26} + ( 800 \beta_{1} + 16 \beta_{2} ) q^{28} + ( 2850 - 2 \beta_{3} ) q^{29} + ( -2854 + 5 \beta_{3} ) q^{31} -1024 \beta_{1} q^{32} + ( 984 + 40 \beta_{3} ) q^{34} + ( 5650 \beta_{1} - 24 \beta_{2} ) q^{37} + ( 1184 \beta_{1} + 20 \beta_{2} ) q^{38} + ( -7710 + 30 \beta_{3} ) q^{41} + ( -3160 \beta_{1} - 41 \beta_{2} ) q^{43} + ( 4320 + 32 \beta_{3} ) q^{44} + ( -7320 - 40 \beta_{3} ) q^{46} + ( -3900 \beta_{1} - 100 \beta_{2} ) q^{47} + ( -22422 - 100 \beta_{3} ) q^{49} + ( 7120 \beta_{1} + 32 \beta_{2} ) q^{52} + ( 13914 \beta_{1} + 70 \beta_{2} ) q^{53} + ( 3200 + 64 \beta_{3} ) q^{56} + ( -11400 \beta_{1} + 8 \beta_{2} ) q^{58} + ( 25260 + 52 \beta_{3} ) q^{59} + ( -14563 + 90 \beta_{3} ) q^{61} + ( 11416 \beta_{1} - 20 \beta_{2} ) q^{62} -4096 q^{64} + ( 48700 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -3936 \beta_{1} - 160 \beta_{2} ) q^{68} + ( -3090 - 126 \beta_{3} ) q^{71} + ( -16450 \beta_{1} + 120 \beta_{2} ) q^{73} + ( 22600 - 96 \beta_{3} ) q^{74} + ( 4736 + 80 \beta_{3} ) q^{76} + ( 86958 \beta_{1} + 370 \beta_{2} ) q^{77} + ( -3956 + 360 \beta_{3} ) q^{79} + ( 30840 \beta_{1} - 120 \beta_{2} ) q^{82} + ( 81732 \beta_{1} - 100 \beta_{2} ) q^{83} + ( -12640 - 164 \beta_{3} ) q^{86} + ( -17280 \beta_{1} - 128 \beta_{2} ) q^{88} + ( 82320 - 336 \beta_{3} ) q^{89} + ( -95708 - 545 \beta_{3} ) q^{91} + ( 29280 \beta_{1} + 160 \beta_{2} ) q^{92} + ( -15600 - 400 \beta_{3} ) q^{94} + ( 26215 \beta_{1} + 364 \beta_{2} ) q^{97} + ( 89688 \beta_{1} + 400 \beta_{2} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 64q^{4} + O(q^{10})$$ $$4q - 64q^{4} - 1080q^{11} - 800q^{14} + 1024q^{16} - 1184q^{19} - 7120q^{26} + 11400q^{29} - 11416q^{31} + 3936q^{34} - 30840q^{41} + 17280q^{44} - 29280q^{46} - 89688q^{49} + 12800q^{56} + 101040q^{59} - 58252q^{61} - 16384q^{64} - 12360q^{71} + 90400q^{74} + 18944q^{76} - 15824q^{79} - 50560q^{86} + 329280q^{89} - 382832q^{91} - 62400q^{94} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 2041 x^{2} + 1040400$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} + 1021 \nu$$$$)/1020$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 3061 \nu$$$$)/340$$ $$\beta_{3}$$ $$=$$ $$6 \nu^{2} + 6123$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 3 \beta_{1}$$$$)/6$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} - 6123$$$$)/6$$ $$\nu^{3}$$ $$=$$ $$($$$$-1021 \beta_{2} + 9183 \beta_{1}$$$$)/6$$

## 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
 31.4414i − 32.4414i 32.4414i − 31.4414i
4.00000i 0 −16.0000 0 0 241.648i 64.0000i 0 0
199.2 4.00000i 0 −16.0000 0 0 141.648i 64.0000i 0 0
199.3 4.00000i 0 −16.0000 0 0 141.648i 64.0000i 0 0
199.4 4.00000i 0 −16.0000 0 0 241.648i 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.p 4
3.b odd 2 1 450.6.c.q 4
5.b even 2 1 inner 450.6.c.p 4
5.c odd 4 1 450.6.a.y 2
5.c odd 4 1 450.6.a.bf yes 2
15.d odd 2 1 450.6.c.q 4
15.e even 4 1 450.6.a.ba yes 2
15.e even 4 1 450.6.a.bd yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.6.a.y 2 5.c odd 4 1
450.6.a.ba yes 2 15.e even 4 1
450.6.a.bd yes 2 15.e even 4 1
450.6.a.bf yes 2 5.c odd 4 1
450.6.c.p 4 1.a even 1 1 trivial
450.6.c.p 4 5.b even 2 1 inner
450.6.c.q 4 3.b odd 2 1
450.6.c.q 4 15.d odd 2 1

## Hecke kernels

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

 $$T_{7}^{4} + 78458 T_{7}^{2} + 1171624441$$ $$T_{11}^{2} + 540 T_{11} - 74016$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + T^{2} )^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$1171624441 + 78458 T^{2} + T^{4}$$
$11$ $$( -74016 + 540 T + T^{2} )^{2}$$
$13$ $$2612129881 + 689882 T^{2} + T^{4}$$
$17$ $$13049318163456 + 7466832 T^{2} + T^{4}$$
$19$ $$( -830609 + 592 T + T^{2} )^{2}$$
$23$ $$104976000000 + 14043600 T^{2} + T^{4}$$
$29$ $$( 7975584 - 5700 T + T^{2} )^{2}$$
$31$ $$( 7227091 + 5708 T + T^{2} )^{2}$$
$37$ $$115919589427216 + 106156808 T^{2} + T^{4}$$
$41$ $$( 26388000 + 15420 T + T^{2} )^{2}$$
$43$ $$2678667905710801 + 143454098 T^{2} + T^{4}$$
$47$ $$123960326400000000 + 765000000 T^{2} + T^{4}$$
$53$ $$185703196271616 + 747142992 T^{2} + T^{4}$$
$59$ $$( 538752384 - 50520 T + T^{2} )^{2}$$
$61$ $$( -85423931 + 29126 T + T^{2} )^{2}$$
$67$ $$5623345588934394721 + 4744041122 T^{2} + T^{4}$$
$71$ $$( -573561504 + 6180 T + T^{2} )^{2}$$
$73$ $$66716358684010000 + 1599000200 T^{2} + T^{4}$$
$79$ $$( -4744428464 + 7912 T + T^{2} )^{2}$$
$83$ $$39851820386783870976 + 14094819648 T^{2} + T^{4}$$
$89$ $$( 2630025216 - 164640 T + T^{2} )^{2}$$
$97$ $$17465874450640370881 + 11107343618 T^{2} + T^{4}$$