# Properties

 Label 225.6.a.u 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 5) q^{2} + ( - 9 \beta + 15) q^{4} + (24 \beta - 66) q^{7} + ( - 19 \beta + 113) q^{8}+O(q^{10})$$ q + (-b + 5) * q^2 + (-9*b + 15) * q^4 + (24*b - 66) * q^7 + (-19*b + 113) * q^8 $$q + ( - \beta + 5) q^{2} + ( - 9 \beta + 15) q^{4} + (24 \beta - 66) q^{7} + ( - 19 \beta + 113) q^{8} + (108 \beta - 138) q^{11} + ( - 84 \beta - 606) q^{13} + (162 \beta - 858) q^{14} + (99 \beta + 503) q^{16} + (140 \beta + 218) q^{17} + (72 \beta - 704) q^{19} + (570 \beta - 3066) q^{22} + (88 \beta + 2908) q^{23} + (270 \beta - 1182) q^{26} + (738 \beta - 5742) q^{28} + ( - 864 \beta + 2208) q^{29} + (144 \beta - 5896) q^{31} + (501 \beta - 3279) q^{32} + (342 \beta - 1990) q^{34} + ( - 396 \beta - 7146) q^{37} + (992 \beta - 5104) q^{38} + ( - 3024 \beta + 606) q^{41} + (1344 \beta + 3108) q^{43} + (1890 \beta - 23454) q^{44} + ( - 2556 \beta + 12604) q^{46} + ( - 1288 \beta + 2264) q^{47} + ( - 2592 \beta + 221) q^{49} + (4950 \beta + 7542) q^{52} + ( - 2012 \beta - 10082) q^{53} + (3510 \beta - 17490) q^{56} + ( - 5664 \beta + 30048) q^{58} + ( - 3348 \beta - 26994) q^{59} + (3168 \beta - 16654) q^{61} + (6472 \beta - 32648) q^{62} + (2115 \beta - 43513) q^{64} + ( - 4560 \beta + 4872) q^{67} + ( - 1122 \beta - 24450) q^{68} + (3240 \beta - 10332) q^{71} + (792 \beta + 1332) q^{73} + (5562 \beta - 27018) q^{74} + (6768 \beta - 24816) q^{76} + ( - 7848 \beta + 66132) q^{77} + (9360 \beta - 33440) q^{79} + ( - 12702 \beta + 69558) q^{82} + (6832 \beta + 63580) q^{83} + (2268 \beta - 14028) q^{86} + (12774 \beta - 60738) q^{88} + ( - 4752 \beta - 66006) q^{89} + ( - 11016 \beta - 4356) q^{91} + ( - 25644 \beta + 26196) q^{92} + ( - 7416 \beta + 39656) q^{94} + ( - 15600 \beta - 34968) q^{97} + ( - 10589 \beta + 58129) q^{98}+O(q^{100})$$ q + (-b + 5) * q^2 + (-9*b + 15) * q^4 + (24*b - 66) * q^7 + (-19*b + 113) * q^8 + (108*b - 138) * q^11 + (-84*b - 606) * q^13 + (162*b - 858) * q^14 + (99*b + 503) * q^16 + (140*b + 218) * q^17 + (72*b - 704) * q^19 + (570*b - 3066) * q^22 + (88*b + 2908) * q^23 + (270*b - 1182) * q^26 + (738*b - 5742) * q^28 + (-864*b + 2208) * q^29 + (144*b - 5896) * q^31 + (501*b - 3279) * q^32 + (342*b - 1990) * q^34 + (-396*b - 7146) * q^37 + (992*b - 5104) * q^38 + (-3024*b + 606) * q^41 + (1344*b + 3108) * q^43 + (1890*b - 23454) * q^44 + (-2556*b + 12604) * q^46 + (-1288*b + 2264) * q^47 + (-2592*b + 221) * q^49 + (4950*b + 7542) * q^52 + (-2012*b - 10082) * q^53 + (3510*b - 17490) * q^56 + (-5664*b + 30048) * q^58 + (-3348*b - 26994) * q^59 + (3168*b - 16654) * q^61 + (6472*b - 32648) * q^62 + (2115*b - 43513) * q^64 + (-4560*b + 4872) * q^67 + (-1122*b - 24450) * q^68 + (3240*b - 10332) * q^71 + (792*b + 1332) * q^73 + (5562*b - 27018) * q^74 + (6768*b - 24816) * q^76 + (-7848*b + 66132) * q^77 + (9360*b - 33440) * q^79 + (-12702*b + 69558) * q^82 + (6832*b + 63580) * q^83 + (2268*b - 14028) * q^86 + (12774*b - 60738) * q^88 + (-4752*b - 66006) * q^89 + (-11016*b - 4356) * q^91 + (-25644*b + 26196) * q^92 + (-7416*b + 39656) * q^94 + (-15600*b - 34968) * q^97 + (-10589*b + 58129) * 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
−0.216991 0 −31.9529 0 0 59.2078 13.8772 0 0
1.2 9.21699 0 52.9529 0 0 −167.208 193.123 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.u 2
3.b odd 2 1 75.6.a.f 2
5.b even 2 1 225.6.a.i 2
5.c odd 4 2 45.6.b.c 4
15.d odd 2 1 75.6.a.j 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 3.b odd 2 1
75.6.a.j 2 15.d odd 2 1
225.6.a.i 2 5.b even 2 1
225.6.a.u 2 1.a even 1 1 trivial
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

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