# Properties

 Label 450.6.a.y Level $450$ Weight $6$ Character orbit 450.a Self dual yes Analytic conductor $72.173$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$450 = 2 \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.1727189158$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{4081})$$ Defining polynomial: $$x^{2} - x - 1020$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2\cdot 3$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 3\sqrt{4081}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -4 q^{2} + 16 q^{4} + ( -50 - \beta ) q^{7} -64 q^{8} +O(q^{10})$$ $$q -4 q^{2} + 16 q^{4} + ( -50 - \beta ) q^{7} -64 q^{8} + ( -270 - 2 \beta ) q^{11} + ( 445 + 2 \beta ) q^{13} + ( 200 + 4 \beta ) q^{14} + 256 q^{16} + ( 246 + 10 \beta ) q^{17} + ( 296 + 5 \beta ) q^{19} + ( 1080 + 8 \beta ) q^{22} + ( 1830 + 10 \beta ) q^{23} + ( -1780 - 8 \beta ) q^{26} + ( -800 - 16 \beta ) q^{28} + ( -2850 + 2 \beta ) q^{29} + ( -2854 + 5 \beta ) q^{31} -1024 q^{32} + ( -984 - 40 \beta ) q^{34} + ( 5650 - 24 \beta ) q^{37} + ( -1184 - 20 \beta ) q^{38} + ( -7710 + 30 \beta ) q^{41} + ( 3160 + 41 \beta ) q^{43} + ( -4320 - 32 \beta ) q^{44} + ( -7320 - 40 \beta ) q^{46} + ( -3900 - 100 \beta ) q^{47} + ( 22422 + 100 \beta ) q^{49} + ( 7120 + 32 \beta ) q^{52} + ( -13914 - 70 \beta ) q^{53} + ( 3200 + 64 \beta ) q^{56} + ( 11400 - 8 \beta ) q^{58} + ( -25260 - 52 \beta ) q^{59} + ( -14563 + 90 \beta ) q^{61} + ( 11416 - 20 \beta ) q^{62} + 4096 q^{64} + ( 48700 - 3 \beta ) q^{67} + ( 3936 + 160 \beta ) q^{68} + ( -3090 - 126 \beta ) q^{71} + ( 16450 - 120 \beta ) q^{73} + ( -22600 + 96 \beta ) q^{74} + ( 4736 + 80 \beta ) q^{76} + ( 86958 + 370 \beta ) q^{77} + ( 3956 - 360 \beta ) q^{79} + ( 30840 - 120 \beta ) q^{82} + ( -81732 + 100 \beta ) q^{83} + ( -12640 - 164 \beta ) q^{86} + ( 17280 + 128 \beta ) q^{88} + ( -82320 + 336 \beta ) q^{89} + ( -95708 - 545 \beta ) q^{91} + ( 29280 + 160 \beta ) q^{92} + ( 15600 + 400 \beta ) q^{94} + ( 26215 + 364 \beta ) q^{97} + ( -89688 - 400 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 8q^{2} + 32q^{4} - 100q^{7} - 128q^{8} + O(q^{10})$$ $$2q - 8q^{2} + 32q^{4} - 100q^{7} - 128q^{8} - 540q^{11} + 890q^{13} + 400q^{14} + 512q^{16} + 492q^{17} + 592q^{19} + 2160q^{22} + 3660q^{23} - 3560q^{26} - 1600q^{28} - 5700q^{29} - 5708q^{31} - 2048q^{32} - 1968q^{34} + 11300q^{37} - 2368q^{38} - 15420q^{41} + 6320q^{43} - 8640q^{44} - 14640q^{46} - 7800q^{47} + 44844q^{49} + 14240q^{52} - 27828q^{53} + 6400q^{56} + 22800q^{58} - 50520q^{59} - 29126q^{61} + 22832q^{62} + 8192q^{64} + 97400q^{67} + 7872q^{68} - 6180q^{71} + 32900q^{73} - 45200q^{74} + 9472q^{76} + 173916q^{77} + 7912q^{79} + 61680q^{82} - 163464q^{83} - 25280q^{86} + 34560q^{88} - 164640q^{89} - 191416q^{91} + 58560q^{92} + 31200q^{94} + 52430q^{97} - 179376q^{98} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 32.4414 −31.4414
−4.00000 0 16.0000 0 0 −241.648 −64.0000 0 0
1.2 −4.00000 0 16.0000 0 0 141.648 −64.0000 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(450))$$:

 $$T_{7}^{2} + 100 T_{7} - 34229$$ $$T_{11}^{2} + 540 T_{11} - 74016$$ $$T_{17}^{2} - 492 T_{17} - 3612384$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 4 + T )^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-34229 + 100 T + T^{2}$$
$11$ $$-74016 + 540 T + T^{2}$$
$13$ $$51109 - 890 T + T^{2}$$
$17$ $$-3612384 - 492 T + T^{2}$$
$19$ $$-830609 - 592 T + T^{2}$$
$23$ $$-324000 - 3660 T + T^{2}$$
$29$ $$7975584 + 5700 T + T^{2}$$
$31$ $$7227091 + 5708 T + T^{2}$$
$37$ $$10766596 - 11300 T + T^{2}$$
$41$ $$26388000 + 15420 T + T^{2}$$
$43$ $$-51755849 - 6320 T + T^{2}$$
$47$ $$-352080000 + 7800 T + T^{2}$$
$53$ $$13627296 + 27828 T + T^{2}$$
$59$ $$538752384 + 50520 T + T^{2}$$
$61$ $$-85423931 + 29126 T + T^{2}$$
$67$ $$2371359439 - 97400 T + T^{2}$$
$71$ $$-573561504 + 6180 T + T^{2}$$
$73$ $$-258295100 - 32900 T + T^{2}$$
$79$ $$-4744428464 - 7912 T + T^{2}$$
$83$ $$6312829824 + 163464 T + T^{2}$$
$89$ $$2630025216 + 164640 T + T^{2}$$
$97$ $$-4179219359 - 52430 T + T^{2}$$