# Properties

 Label 225.6.a.v Level $225$ Weight $6$ Character orbit 225.a Self dual yes Analytic conductor $36.086$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# 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: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{5}, \sqrt{14})$$ Defining polynomial: $$x^{4} - 2x^{3} - 29x^{2} + 30x + 155$$ x^4 - 2*x^3 - 29*x^2 + 30*x + 155 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{7}\cdot 3^{2}\cdot 5^{2}$$ Twist minimal: no (minimal twist has level 45) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} - 12 q^{4} - \beta_{2} q^{7} + 44 \beta_1 q^{8}+O(q^{10})$$ q - b1 * q^2 - 12 * q^4 - b2 * q^7 + 44*b1 * q^8 $$q - \beta_1 q^{2} - 12 q^{4} - \beta_{2} q^{7} + 44 \beta_1 q^{8} - \beta_{3} q^{11} - \beta_{2} q^{13} + 2 \beta_{3} q^{14} - 496 q^{16} - 373 \beta_1 q^{17} + 484 q^{19} + 10 \beta_{2} q^{22} - 506 \beta_1 q^{23} + 2 \beta_{3} q^{26} + 12 \beta_{2} q^{28} - 11 \beta_{3} q^{29} + 3608 q^{31} - 912 \beta_1 q^{32} + 7460 q^{34} - 33 \beta_{2} q^{37} - 484 \beta_1 q^{38} + 22 \beta_{3} q^{41} - 56 \beta_{2} q^{43} + 12 \beta_{3} q^{44} + 10120 q^{46} - 2134 \beta_1 q^{47} + 33593 q^{49} + 12 \beta_{2} q^{52} - 1067 \beta_1 q^{53} - 88 \beta_{3} q^{56} + 110 \beta_{2} q^{58} + 11 \beta_{3} q^{59} + 21362 q^{61} - 3608 \beta_1 q^{62} + 34112 q^{64} + 154 \beta_{2} q^{67} + 4476 \beta_1 q^{68} + 66 \beta_{3} q^{71} + 12 \beta_{2} q^{73} + 66 \beta_{3} q^{74} - 5808 q^{76} + 25200 \beta_1 q^{77} + 99616 q^{79} - 220 \beta_{2} q^{82} + 12772 \beta_1 q^{83} + 112 \beta_{3} q^{86} - 440 \beta_{2} q^{88} - 240 \beta_{3} q^{89} + 50400 q^{91} + 6072 \beta_1 q^{92} + 42680 q^{94} - 286 \beta_{2} q^{97} - 33593 \beta_1 q^{98}+O(q^{100})$$ q - b1 * q^2 - 12 * q^4 - b2 * q^7 + 44*b1 * q^8 - b3 * q^11 - b2 * q^13 + 2*b3 * q^14 - 496 * q^16 - 373*b1 * q^17 + 484 * q^19 + 10*b2 * q^22 - 506*b1 * q^23 + 2*b3 * q^26 + 12*b2 * q^28 - 11*b3 * q^29 + 3608 * q^31 - 912*b1 * q^32 + 7460 * q^34 - 33*b2 * q^37 - 484*b1 * q^38 + 22*b3 * q^41 - 56*b2 * q^43 + 12*b3 * q^44 + 10120 * q^46 - 2134*b1 * q^47 + 33593 * q^49 + 12*b2 * q^52 - 1067*b1 * q^53 - 88*b3 * q^56 + 110*b2 * q^58 + 11*b3 * q^59 + 21362 * q^61 - 3608*b1 * q^62 + 34112 * q^64 + 154*b2 * q^67 + 4476*b1 * q^68 + 66*b3 * q^71 + 12*b2 * q^73 + 66*b3 * q^74 - 5808 * q^76 + 25200*b1 * q^77 + 99616 * q^79 - 220*b2 * q^82 + 12772*b1 * q^83 + 112*b3 * q^86 - 440*b2 * q^88 - 240*b3 * q^89 + 50400 * q^91 + 6072*b1 * q^92 + 42680 * q^94 - 286*b2 * q^97 - 33593*b1 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 48 q^{4}+O(q^{10})$$ 4 * q - 48 * q^4 $$4 q - 48 q^{4} - 1984 q^{16} + 1936 q^{19} + 14432 q^{31} + 29840 q^{34} + 40480 q^{46} + 134372 q^{49} + 85448 q^{61} + 136448 q^{64} - 23232 q^{76} + 398464 q^{79} + 201600 q^{91} + 170720 q^{94}+O(q^{100})$$ 4 * q - 48 * q^4 - 1984 * q^16 + 1936 * q^19 + 14432 * q^31 + 29840 * q^34 + 40480 * q^46 + 134372 * q^49 + 85448 * q^61 + 136448 * q^64 - 23232 * q^76 + 398464 * q^79 + 201600 * q^91 + 170720 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 29x^{2} + 30x + 155$$ :

 $$\beta_{1}$$ $$=$$ $$( 8\nu^{3} - 12\nu^{2} - 136\nu + 70 ) / 51$$ (8*v^3 - 12*v^2 - 136*v + 70) / 51 $$\beta_{2}$$ $$=$$ $$( -40\nu^{3} + 60\nu^{2} + 1700\nu - 860 ) / 17$$ (-40*v^3 + 60*v^2 + 1700*v - 860) / 17 $$\beta_{3}$$ $$=$$ $$60\nu^{2} - 60\nu - 900$$ 60*v^2 - 60*v - 900
 $$\nu$$ $$=$$ $$( \beta_{2} + 15\beta _1 + 30 ) / 60$$ (b2 + 15*b1 + 30) / 60 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + \beta_{2} + 15\beta _1 + 930 ) / 60$$ (b3 + b2 + 15*b1 + 930) / 60 $$\nu^{3}$$ $$=$$ $$( 3\beta_{3} + 37\beta_{2} + 1320\beta _1 + 2760 ) / 120$$ (3*b3 + 37*b2 + 1320*b1 + 2760) / 120

## 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
 5.35969 −2.12362 3.12362 −4.35969
−4.47214 0 −12.0000 0 0 −224.499 196.774 0 0
1.2 −4.47214 0 −12.0000 0 0 224.499 196.774 0 0
1.3 4.47214 0 −12.0000 0 0 −224.499 −196.774 0 0
1.4 4.47214 0 −12.0000 0 0 224.499 −196.774 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

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.6.a.v 4
3.b odd 2 1 inner 225.6.a.v 4
5.b even 2 1 inner 225.6.a.v 4
5.c odd 4 2 45.6.b.d 4
15.d odd 2 1 inner 225.6.a.v 4
15.e even 4 2 45.6.b.d 4
20.e even 4 2 720.6.f.k 4
60.l odd 4 2 720.6.f.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
45.6.b.d 4 5.c odd 4 2
45.6.b.d 4 15.e even 4 2
225.6.a.v 4 1.a even 1 1 trivial
225.6.a.v 4 3.b odd 2 1 inner
225.6.a.v 4 5.b even 2 1 inner
225.6.a.v 4 15.d odd 2 1 inner
720.6.f.k 4 20.e even 4 2
720.6.f.k 4 60.l odd 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} - 20$$ T2^2 - 20 $$T_{7}^{2} - 50400$$ T7^2 - 50400

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 20)^{2}$$
$3$ $$T^{4}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 50400)^{2}$$
$11$ $$(T^{2} - 252000)^{2}$$
$13$ $$(T^{2} - 50400)^{2}$$
$17$ $$(T^{2} - 2782580)^{2}$$
$19$ $$(T - 484)^{4}$$
$23$ $$(T^{2} - 5120720)^{2}$$
$29$ $$(T^{2} - 30492000)^{2}$$
$31$ $$(T - 3608)^{4}$$
$37$ $$(T^{2} - 54885600)^{2}$$
$41$ $$(T^{2} - 121968000)^{2}$$
$43$ $$(T^{2} - 158054400)^{2}$$
$47$ $$(T^{2} - 91079120)^{2}$$
$53$ $$(T^{2} - 22769780)^{2}$$
$59$ $$(T^{2} - 30492000)^{2}$$
$61$ $$(T - 21362)^{4}$$
$67$ $$(T^{2} - 1195286400)^{2}$$
$71$ $$(T^{2} - 1097712000)^{2}$$
$73$ $$(T^{2} - 7257600)^{2}$$
$79$ $$(T - 99616)^{4}$$
$83$ $$(T^{2} - 3262479680)^{2}$$
$89$ $$(T^{2} - 14515200000)^{2}$$
$97$ $$(T^{2} - 4122518400)^{2}$$