# Properties

 Label 225.4.a.l Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2754297513$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ Defining polynomial: $$x^{2} - 10$$ x^2 - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 2 q^{4} - 15 q^{7} - 6 \beta q^{8} +O(q^{10})$$ q + b * q^2 + 2 * q^4 - 15 * q^7 - 6*b * q^8 $$q + \beta q^{2} + 2 q^{4} - 15 q^{7} - 6 \beta q^{8} - 20 \beta q^{11} - 35 q^{13} - 15 \beta q^{14} - 76 q^{16} + 28 \beta q^{17} + 91 q^{19} - 200 q^{22} - 36 \beta q^{23} - 35 \beta q^{26} - 30 q^{28} + 20 \beta q^{29} - 147 q^{31} - 28 \beta q^{32} + 280 q^{34} - 370 q^{37} + 91 \beta q^{38} + 140 \beta q^{41} - 335 q^{43} - 40 \beta q^{44} - 360 q^{46} - 56 \beta q^{47} - 118 q^{49} - 70 q^{52} + 28 \beta q^{53} + 90 \beta q^{56} + 200 q^{58} + 280 \beta q^{59} + 427 q^{61} - 147 \beta q^{62} + 328 q^{64} - 15 q^{67} + 56 \beta q^{68} + 20 \beta q^{71} + 70 q^{73} - 370 \beta q^{74} + 182 q^{76} + 300 \beta q^{77} - 876 q^{79} + 1400 q^{82} - 168 \beta q^{83} - 335 \beta q^{86} + 1200 q^{88} + 525 q^{91} - 72 \beta q^{92} - 560 q^{94} + 1085 q^{97} - 118 \beta q^{98} +O(q^{100})$$ q + b * q^2 + 2 * q^4 - 15 * q^7 - 6*b * q^8 - 20*b * q^11 - 35 * q^13 - 15*b * q^14 - 76 * q^16 + 28*b * q^17 + 91 * q^19 - 200 * q^22 - 36*b * q^23 - 35*b * q^26 - 30 * q^28 + 20*b * q^29 - 147 * q^31 - 28*b * q^32 + 280 * q^34 - 370 * q^37 + 91*b * q^38 + 140*b * q^41 - 335 * q^43 - 40*b * q^44 - 360 * q^46 - 56*b * q^47 - 118 * q^49 - 70 * q^52 + 28*b * q^53 + 90*b * q^56 + 200 * q^58 + 280*b * q^59 + 427 * q^61 - 147*b * q^62 + 328 * q^64 - 15 * q^67 + 56*b * q^68 + 20*b * q^71 + 70 * q^73 - 370*b * q^74 + 182 * q^76 + 300*b * q^77 - 876 * q^79 + 1400 * q^82 - 168*b * q^83 - 335*b * q^86 + 1200 * q^88 + 525 * q^91 - 72*b * q^92 - 560 * q^94 + 1085 * q^97 - 118*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} - 30 q^{7}+O(q^{10})$$ 2 * q + 4 * q^4 - 30 * q^7 $$2 q + 4 q^{4} - 30 q^{7} - 70 q^{13} - 152 q^{16} + 182 q^{19} - 400 q^{22} - 60 q^{28} - 294 q^{31} + 560 q^{34} - 740 q^{37} - 670 q^{43} - 720 q^{46} - 236 q^{49} - 140 q^{52} + 400 q^{58} + 854 q^{61} + 656 q^{64} - 30 q^{67} + 140 q^{73} + 364 q^{76} - 1752 q^{79} + 2800 q^{82} + 2400 q^{88} + 1050 q^{91} - 1120 q^{94} + 2170 q^{97}+O(q^{100})$$ 2 * q + 4 * q^4 - 30 * q^7 - 70 * q^13 - 152 * q^16 + 182 * q^19 - 400 * q^22 - 60 * q^28 - 294 * q^31 + 560 * q^34 - 740 * q^37 - 670 * q^43 - 720 * q^46 - 236 * q^49 - 140 * q^52 + 400 * q^58 + 854 * q^61 + 656 * q^64 - 30 * q^67 + 140 * q^73 + 364 * q^76 - 1752 * q^79 + 2800 * q^82 + 2400 * q^88 + 1050 * q^91 - 1120 * q^94 + 2170 * q^97

## 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
 −3.16228 3.16228
−3.16228 0 2.00000 0 0 −15.0000 18.9737 0 0
1.2 3.16228 0 2.00000 0 0 −15.0000 −18.9737 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 225.4.a.l 2
3.b odd 2 1 inner 225.4.a.l 2
5.b even 2 1 225.4.a.m yes 2
5.c odd 4 2 225.4.b.i 4
15.d odd 2 1 225.4.a.m yes 2
15.e even 4 2 225.4.b.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.4.a.l 2 1.a even 1 1 trivial
225.4.a.l 2 3.b odd 2 1 inner
225.4.a.m yes 2 5.b even 2 1
225.4.a.m yes 2 15.d odd 2 1
225.4.b.i 4 5.c odd 4 2
225.4.b.i 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} - 10$$ T2^2 - 10 $$T_{7} + 15$$ T7 + 15

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 10$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 15)^{2}$$
$11$ $$T^{2} - 4000$$
$13$ $$(T + 35)^{2}$$
$17$ $$T^{2} - 7840$$
$19$ $$(T - 91)^{2}$$
$23$ $$T^{2} - 12960$$
$29$ $$T^{2} - 4000$$
$31$ $$(T + 147)^{2}$$
$37$ $$(T + 370)^{2}$$
$41$ $$T^{2} - 196000$$
$43$ $$(T + 335)^{2}$$
$47$ $$T^{2} - 31360$$
$53$ $$T^{2} - 7840$$
$59$ $$T^{2} - 784000$$
$61$ $$(T - 427)^{2}$$
$67$ $$(T + 15)^{2}$$
$71$ $$T^{2} - 4000$$
$73$ $$(T - 70)^{2}$$
$79$ $$(T + 876)^{2}$$
$83$ $$T^{2} - 282240$$
$89$ $$T^{2}$$
$97$ $$(T - 1085)^{2}$$