# Properties

 Label 225.6.a.i Level $225$ Weight $6$ Character orbit 225.a Self dual yes Analytic conductor $36.086$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$36.0863594579$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{89})$$ Defining polynomial: $$x^{2} - x - 22$$ x^2 - x - 22 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{89})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 4) q^{2} + (9 \beta + 6) q^{4} + (24 \beta + 42) q^{7} + ( - 19 \beta - 94) q^{8}+O(q^{10})$$ q + (-b - 4) * q^2 + (9*b + 6) * q^4 + (24*b + 42) * q^7 + (-19*b - 94) * q^8 $$q + ( - \beta - 4) q^{2} + (9 \beta + 6) q^{4} + (24 \beta + 42) q^{7} + ( - 19 \beta - 94) q^{8} + ( - 108 \beta - 30) q^{11} + ( - 84 \beta + 690) q^{13} + ( - 162 \beta - 696) q^{14} + ( - 99 \beta + 602) q^{16} + (140 \beta - 358) q^{17} + ( - 72 \beta - 632) q^{19} + (570 \beta + 2496) q^{22} + (88 \beta - 2996) q^{23} + ( - 270 \beta - 912) q^{26} + (738 \beta + 5004) q^{28} + (864 \beta + 1344) q^{29} + ( - 144 \beta - 5752) q^{31} + (501 \beta + 2778) q^{32} + ( - 342 \beta - 1648) q^{34} + ( - 396 \beta + 7542) q^{37} + (992 \beta + 4112) q^{38} + (3024 \beta - 2418) q^{41} + (1344 \beta - 4452) q^{43} + ( - 1890 \beta - 21564) q^{44} + (2556 \beta + 10048) q^{46} + ( - 1288 \beta - 976) q^{47} + (2592 \beta - 2371) q^{49} + (4950 \beta - 12492) q^{52} + ( - 2012 \beta + 12094) q^{53} + ( - 3510 \beta - 13980) q^{56} + ( - 5664 \beta - 24384) q^{58} + (3348 \beta - 30342) q^{59} + ( - 3168 \beta - 13486) q^{61} + (6472 \beta + 26176) q^{62} + ( - 2115 \beta - 41398) q^{64} + ( - 4560 \beta - 312) q^{67} + ( - 1122 \beta + 25572) q^{68} + ( - 3240 \beta - 7092) q^{71} + (792 \beta - 2124) q^{73} + ( - 5562 \beta - 21456) q^{74} + ( - 6768 \beta - 18048) q^{76} + ( - 7848 \beta - 58284) q^{77} + ( - 9360 \beta - 24080) q^{79} + ( - 12702 \beta - 56856) q^{82} + (6832 \beta - 70412) q^{83} + ( - 2268 \beta - 11760) q^{86} + (12774 \beta + 47964) q^{88} + (4752 \beta - 70758) q^{89} + (11016 \beta - 15372) q^{91} + ( - 25644 \beta - 552) q^{92} + (7416 \beta + 32240) q^{94} + ( - 15600 \beta + 50568) q^{97} + ( - 10589 \beta - 47540) q^{98}+O(q^{100})$$ q + (-b - 4) * q^2 + (9*b + 6) * q^4 + (24*b + 42) * q^7 + (-19*b - 94) * q^8 + (-108*b - 30) * q^11 + (-84*b + 690) * q^13 + (-162*b - 696) * q^14 + (-99*b + 602) * q^16 + (140*b - 358) * q^17 + (-72*b - 632) * q^19 + (570*b + 2496) * q^22 + (88*b - 2996) * q^23 + (-270*b - 912) * q^26 + (738*b + 5004) * q^28 + (864*b + 1344) * q^29 + (-144*b - 5752) * q^31 + (501*b + 2778) * q^32 + (-342*b - 1648) * q^34 + (-396*b + 7542) * q^37 + (992*b + 4112) * q^38 + (3024*b - 2418) * q^41 + (1344*b - 4452) * q^43 + (-1890*b - 21564) * q^44 + (2556*b + 10048) * q^46 + (-1288*b - 976) * q^47 + (2592*b - 2371) * q^49 + (4950*b - 12492) * q^52 + (-2012*b + 12094) * q^53 + (-3510*b - 13980) * q^56 + (-5664*b - 24384) * q^58 + (3348*b - 30342) * q^59 + (-3168*b - 13486) * q^61 + (6472*b + 26176) * q^62 + (-2115*b - 41398) * q^64 + (-4560*b - 312) * q^67 + (-1122*b + 25572) * q^68 + (-3240*b - 7092) * q^71 + (792*b - 2124) * q^73 + (-5562*b - 21456) * q^74 + (-6768*b - 18048) * q^76 + (-7848*b - 58284) * q^77 + (-9360*b - 24080) * q^79 + (-12702*b - 56856) * q^82 + (6832*b - 70412) * q^83 + (-2268*b - 11760) * q^86 + (12774*b + 47964) * q^88 + (4752*b - 70758) * q^89 + (11016*b - 15372) * q^91 + (-25644*b - 552) * q^92 + (7416*b + 32240) * q^94 + (-15600*b + 50568) * q^97 + (-10589*b - 47540) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8}+O(q^{10})$$ 2 * q - 9 * q^2 + 21 * q^4 + 108 * q^7 - 207 * q^8 $$2 q - 9 q^{2} + 21 q^{4} + 108 q^{7} - 207 q^{8} - 168 q^{11} + 1296 q^{13} - 1554 q^{14} + 1105 q^{16} - 576 q^{17} - 1336 q^{19} + 5562 q^{22} - 5904 q^{23} - 2094 q^{26} + 10746 q^{28} + 3552 q^{29} - 11648 q^{31} + 6057 q^{32} - 3638 q^{34} + 14688 q^{37} + 9216 q^{38} - 1812 q^{41} - 7560 q^{43} - 45018 q^{44} + 22652 q^{46} - 3240 q^{47} - 2150 q^{49} - 20034 q^{52} + 22176 q^{53} - 31470 q^{56} - 54432 q^{58} - 57336 q^{59} - 30140 q^{61} + 58824 q^{62} - 84911 q^{64} - 5184 q^{67} + 50022 q^{68} - 17424 q^{71} - 3456 q^{73} - 48474 q^{74} - 42864 q^{76} - 124416 q^{77} - 57520 q^{79} - 126414 q^{82} - 133992 q^{83} - 25788 q^{86} + 108702 q^{88} - 136764 q^{89} - 19728 q^{91} - 26748 q^{92} + 71896 q^{94} + 85536 q^{97} - 105669 q^{98}+O(q^{100})$$ 2 * q - 9 * q^2 + 21 * q^4 + 108 * q^7 - 207 * q^8 - 168 * q^11 + 1296 * q^13 - 1554 * q^14 + 1105 * q^16 - 576 * q^17 - 1336 * q^19 + 5562 * q^22 - 5904 * q^23 - 2094 * q^26 + 10746 * q^28 + 3552 * q^29 - 11648 * q^31 + 6057 * q^32 - 3638 * q^34 + 14688 * q^37 + 9216 * q^38 - 1812 * q^41 - 7560 * q^43 - 45018 * q^44 + 22652 * q^46 - 3240 * q^47 - 2150 * q^49 - 20034 * q^52 + 22176 * q^53 - 31470 * q^56 - 54432 * q^58 - 57336 * q^59 - 30140 * q^61 + 58824 * q^62 - 84911 * q^64 - 5184 * q^67 + 50022 * q^68 - 17424 * q^71 - 3456 * q^73 - 48474 * q^74 - 42864 * q^76 - 124416 * q^77 - 57520 * q^79 - 126414 * q^82 - 133992 * q^83 - 25788 * q^86 + 108702 * q^88 - 136764 * q^89 - 19728 * q^91 - 26748 * q^92 + 71896 * q^94 + 85536 * q^97 - 105669 * q^98

## 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
 5.21699 −4.21699
−9.21699 0 52.9529 0 0 167.208 −193.123 0 0
1.2 0.216991 0 −31.9529 0 0 −59.2078 −13.8772 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
$$3$$ $$-1$$
$$5$$ $$-1$$

## 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 225.6.a.i 2
3.b odd 2 1 75.6.a.j 2
5.b even 2 1 225.6.a.u 2
5.c odd 4 2 45.6.b.c 4
15.d odd 2 1 75.6.a.f 2
15.e even 4 2 15.6.b.a 4
20.e even 4 2 720.6.f.h 4
60.l odd 4 2 240.6.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.b.a 4 15.e even 4 2
45.6.b.c 4 5.c odd 4 2
75.6.a.f 2 15.d odd 2 1
75.6.a.j 2 3.b odd 2 1
225.6.a.i 2 1.a even 1 1 trivial
225.6.a.u 2 5.b even 2 1
240.6.f.c 4 60.l odd 4 2
720.6.f.h 4 20.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 $$S_{6}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} + 9T_{2} - 2$$ T2^2 + 9*T2 - 2 $$T_{7}^{2} - 108T_{7} - 9900$$ T7^2 - 108*T7 - 9900

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9T - 2$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 108T - 9900$$
$11$ $$T^{2} + 168T - 252468$$
$13$ $$T^{2} - 1296 T + 262908$$
$17$ $$T^{2} + 576T - 353156$$
$19$ $$T^{2} + 1336 T + 330880$$
$23$ $$T^{2} + 5904 T + 8542000$$
$29$ $$T^{2} - 3552 T - 13455360$$
$31$ $$T^{2} + 11648 T + 33457600$$
$37$ $$T^{2} - 14688 T + 50445180$$
$41$ $$T^{2} + 1812 T - 202645980$$
$43$ $$T^{2} + 7560 T - 25902576$$
$47$ $$T^{2} + 3240 T - 34287104$$
$53$ $$T^{2} - 22176 T + 32872540$$
$59$ $$T^{2} + 57336 T + 572451660$$
$61$ $$T^{2} + 30140 T + 3798916$$
$67$ $$T^{2} + 5184 T - 455939136$$
$71$ $$T^{2} + 17424 T - 157672656$$
$73$ $$T^{2} + 3456 T - 10970640$$
$79$ $$T^{2} + 57520 T - 1122176000$$
$83$ $$T^{2} + 133992 T + 3449918032$$
$89$ $$T^{2} + 136764 T + 4173659460$$
$97$ $$T^{2} - 85536 T - 3585658176$$