# Properties

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

# 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: $$\Q(\sqrt{70})$$ Defining polynomial: $$x^{2} - 70$$ x^2 - 70 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 $$\beta = \sqrt{70}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + 38 q^{4} - 45 q^{7} + 6 \beta q^{8}+O(q^{10})$$ q + b * q^2 + 38 * q^4 - 45 * q^7 + 6*b * q^8 $$q + \beta q^{2} + 38 q^{4} - 45 q^{7} + 6 \beta q^{8} - 40 \beta q^{11} - 1045 q^{13} - 45 \beta q^{14} - 796 q^{16} - 152 \beta q^{17} + 1159 q^{19} - 2800 q^{22} + 456 \beta q^{23} - 1045 \beta q^{26} - 1710 q^{28} - 440 \beta q^{29} + 3633 q^{31} - 988 \beta q^{32} - 10640 q^{34} - 3110 q^{37} + 1159 \beta q^{38} - 2120 \beta q^{41} + 10355 q^{43} - 1520 \beta q^{44} + 31920 q^{46} - 1616 \beta q^{47} - 14782 q^{49} - 39710 q^{52} + 1192 \beta q^{53} - 270 \beta q^{56} - 30800 q^{58} + 3440 \beta q^{59} - 7613 q^{61} + 3633 \beta q^{62} - 43688 q^{64} - 50445 q^{67} - 5776 \beta q^{68} + 9640 \beta q^{71} - 74710 q^{73} - 3110 \beta q^{74} + 44042 q^{76} + 1800 \beta q^{77} + 38316 q^{79} - 148400 q^{82} - 4272 \beta q^{83} + 10355 \beta q^{86} - 16800 q^{88} + 14400 \beta q^{89} + 47025 q^{91} + 17328 \beta q^{92} - 113120 q^{94} + 71755 q^{97} - 14782 \beta q^{98} +O(q^{100})$$ q + b * q^2 + 38 * q^4 - 45 * q^7 + 6*b * q^8 - 40*b * q^11 - 1045 * q^13 - 45*b * q^14 - 796 * q^16 - 152*b * q^17 + 1159 * q^19 - 2800 * q^22 + 456*b * q^23 - 1045*b * q^26 - 1710 * q^28 - 440*b * q^29 + 3633 * q^31 - 988*b * q^32 - 10640 * q^34 - 3110 * q^37 + 1159*b * q^38 - 2120*b * q^41 + 10355 * q^43 - 1520*b * q^44 + 31920 * q^46 - 1616*b * q^47 - 14782 * q^49 - 39710 * q^52 + 1192*b * q^53 - 270*b * q^56 - 30800 * q^58 + 3440*b * q^59 - 7613 * q^61 + 3633*b * q^62 - 43688 * q^64 - 50445 * q^67 - 5776*b * q^68 + 9640*b * q^71 - 74710 * q^73 - 3110*b * q^74 + 44042 * q^76 + 1800*b * q^77 + 38316 * q^79 - 148400 * q^82 - 4272*b * q^83 + 10355*b * q^86 - 16800 * q^88 + 14400*b * q^89 + 47025 * q^91 + 17328*b * q^92 - 113120 * q^94 + 71755 * q^97 - 14782*b * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 76 q^{4} - 90 q^{7}+O(q^{10})$$ 2 * q + 76 * q^4 - 90 * q^7 $$2 q + 76 q^{4} - 90 q^{7} - 2090 q^{13} - 1592 q^{16} + 2318 q^{19} - 5600 q^{22} - 3420 q^{28} + 7266 q^{31} - 21280 q^{34} - 6220 q^{37} + 20710 q^{43} + 63840 q^{46} - 29564 q^{49} - 79420 q^{52} - 61600 q^{58} - 15226 q^{61} - 87376 q^{64} - 100890 q^{67} - 149420 q^{73} + 88084 q^{76} + 76632 q^{79} - 296800 q^{82} - 33600 q^{88} + 94050 q^{91} - 226240 q^{94} + 143510 q^{97}+O(q^{100})$$ 2 * q + 76 * q^4 - 90 * q^7 - 2090 * q^13 - 1592 * q^16 + 2318 * q^19 - 5600 * q^22 - 3420 * q^28 + 7266 * q^31 - 21280 * q^34 - 6220 * q^37 + 20710 * q^43 + 63840 * q^46 - 29564 * q^49 - 79420 * q^52 - 61600 * q^58 - 15226 * q^61 - 87376 * q^64 - 100890 * q^67 - 149420 * q^73 + 88084 * q^76 + 76632 * q^79 - 296800 * q^82 - 33600 * q^88 + 94050 * q^91 - 226240 * q^94 + 143510 * 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
 −8.36660 8.36660
−8.36660 0 38.0000 0 0 −45.0000 −50.1996 0 0
1.2 8.36660 0 38.0000 0 0 −45.0000 50.1996 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 225.6.a.o 2
3.b odd 2 1 inner 225.6.a.o 2
5.b even 2 1 225.6.a.p yes 2
5.c odd 4 2 225.6.b.h 4
15.d odd 2 1 225.6.a.p yes 2
15.e even 4 2 225.6.b.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.6.a.o 2 1.a even 1 1 trivial
225.6.a.o 2 3.b odd 2 1 inner
225.6.a.p yes 2 5.b even 2 1
225.6.a.p yes 2 15.d odd 2 1
225.6.b.h 4 5.c odd 4 2
225.6.b.h 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 $$S_{6}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} - 70$$ T2^2 - 70 $$T_{7} + 45$$ T7 + 45

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 70$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$(T + 45)^{2}$$
$11$ $$T^{2} - 112000$$
$13$ $$(T + 1045)^{2}$$
$17$ $$T^{2} - 1617280$$
$19$ $$(T - 1159)^{2}$$
$23$ $$T^{2} - 14555520$$
$29$ $$T^{2} - 13552000$$
$31$ $$(T - 3633)^{2}$$
$37$ $$(T + 3110)^{2}$$
$41$ $$T^{2} - 314608000$$
$43$ $$(T - 10355)^{2}$$
$47$ $$T^{2} - 182801920$$
$53$ $$T^{2} - 99460480$$
$59$ $$T^{2} - 828352000$$
$61$ $$(T + 7613)^{2}$$
$67$ $$(T + 50445)^{2}$$
$71$ $$T^{2} - 6505072000$$
$73$ $$(T + 74710)^{2}$$
$79$ $$(T - 38316)^{2}$$
$83$ $$T^{2} - 1277498880$$
$89$ $$T^{2} - 14515200000$$
$97$ $$(T - 71755)^{2}$$