# Properties

 Label 225.4.a.o Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ Defining polynomial: $$x^{2} - x - 10$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 + \beta ) q^{2} + ( 3 + 3 \beta ) q^{4} + ( -6 + 6 \beta ) q^{7} + ( 25 + \beta ) q^{8} +O(q^{10})$$ $$q + ( 1 + \beta ) q^{2} + ( 3 + 3 \beta ) q^{4} + ( -6 + 6 \beta ) q^{7} + ( 25 + \beta ) q^{8} + ( 18 + 6 \beta ) q^{11} + ( -42 + 6 \beta ) q^{13} + ( 54 + 6 \beta ) q^{14} + ( 11 + 3 \beta ) q^{16} + ( 46 + 10 \beta ) q^{17} + ( 40 - 24 \beta ) q^{19} + ( 78 + 30 \beta ) q^{22} + ( -28 + 8 \beta ) q^{23} + ( 18 - 30 \beta ) q^{26} + ( 162 + 18 \beta ) q^{28} + ( 180 - 42 \beta ) q^{29} + ( 32 - 12 \beta ) q^{31} + ( -159 + 9 \beta ) q^{32} + ( 146 + 66 \beta ) q^{34} + ( -126 - 54 \beta ) q^{37} + ( -200 - 8 \beta ) q^{38} + ( 198 + 12 \beta ) q^{41} + ( -12 - 96 \beta ) q^{43} + ( 234 + 90 \beta ) q^{44} + ( 52 - 12 \beta ) q^{46} + ( 136 - 92 \beta ) q^{47} + ( 53 - 36 \beta ) q^{49} + ( 54 - 90 \beta ) q^{52} + ( 242 - 82 \beta ) q^{53} + ( -90 + 150 \beta ) q^{56} + ( -240 + 96 \beta ) q^{58} + ( 90 + 6 \beta ) q^{59} + ( 122 + 96 \beta ) q^{61} + ( -88 + 8 \beta ) q^{62} + ( -157 - 165 \beta ) q^{64} + ( -336 - 60 \beta ) q^{67} + ( 438 + 198 \beta ) q^{68} + ( 108 - 180 \beta ) q^{71} + ( -612 - 108 \beta ) q^{73} + ( -666 - 234 \beta ) q^{74} + ( -600 - 24 \beta ) q^{76} + ( 252 + 108 \beta ) q^{77} + ( 40 + 300 \beta ) q^{79} + ( 318 + 222 \beta ) q^{82} + ( -388 - 208 \beta ) q^{83} + ( -972 - 204 \beta ) q^{86} + ( 510 + 174 \beta ) q^{88} + ( -630 + 144 \beta ) q^{89} + ( 612 - 252 \beta ) q^{91} + ( 156 - 36 \beta ) q^{92} + ( -784 - 48 \beta ) q^{94} + ( 264 + 240 \beta ) q^{97} + ( -307 - 19 \beta ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 3 q^{2} + 9 q^{4} - 6 q^{7} + 51 q^{8} + O(q^{10})$$ $$2 q + 3 q^{2} + 9 q^{4} - 6 q^{7} + 51 q^{8} + 42 q^{11} - 78 q^{13} + 114 q^{14} + 25 q^{16} + 102 q^{17} + 56 q^{19} + 186 q^{22} - 48 q^{23} + 6 q^{26} + 342 q^{28} + 318 q^{29} + 52 q^{31} - 309 q^{32} + 358 q^{34} - 306 q^{37} - 408 q^{38} + 408 q^{41} - 120 q^{43} + 558 q^{44} + 92 q^{46} + 180 q^{47} + 70 q^{49} + 18 q^{52} + 402 q^{53} - 30 q^{56} - 384 q^{58} + 186 q^{59} + 340 q^{61} - 168 q^{62} - 479 q^{64} - 732 q^{67} + 1074 q^{68} + 36 q^{71} - 1332 q^{73} - 1566 q^{74} - 1224 q^{76} + 612 q^{77} + 380 q^{79} + 858 q^{82} - 984 q^{83} - 2148 q^{86} + 1194 q^{88} - 1116 q^{89} + 972 q^{91} + 276 q^{92} - 1616 q^{94} + 768 q^{97} - 633 q^{98} + O(q^{100})$$

## 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
 −2.70156 3.70156
−1.70156 0 −5.10469 0 0 −22.2094 22.2984 0 0
1.2 4.70156 0 14.1047 0 0 16.2094 28.7016 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 225.4.a.o 2
3.b odd 2 1 75.4.a.c 2
5.b even 2 1 225.4.a.i 2
5.c odd 4 2 45.4.b.b 4
12.b even 2 1 1200.4.a.bt 2
15.d odd 2 1 75.4.a.f 2
15.e even 4 2 15.4.b.a 4
20.e even 4 2 720.4.f.j 4
60.h even 2 1 1200.4.a.bn 2
60.l odd 4 2 240.4.f.f 4
120.q odd 4 2 960.4.f.p 4
120.w even 4 2 960.4.f.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 15.e even 4 2
45.4.b.b 4 5.c odd 4 2
75.4.a.c 2 3.b odd 2 1
75.4.a.f 2 15.d odd 2 1
225.4.a.i 2 5.b even 2 1
225.4.a.o 2 1.a even 1 1 trivial
240.4.f.f 4 60.l odd 4 2
720.4.f.j 4 20.e even 4 2
960.4.f.p 4 120.q odd 4 2
960.4.f.q 4 120.w even 4 2
1200.4.a.bn 2 60.h even 2 1
1200.4.a.bt 2 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2}^{2} - 3 T_{2} - 8$$ $$T_{7}^{2} + 6 T_{7} - 360$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-8 - 3 T + T^{2}$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$-360 + 6 T + T^{2}$$
$11$ $$72 - 42 T + T^{2}$$
$13$ $$1152 + 78 T + T^{2}$$
$17$ $$1576 - 102 T + T^{2}$$
$19$ $$-5120 - 56 T + T^{2}$$
$23$ $$-80 + 48 T + T^{2}$$
$29$ $$7200 - 318 T + T^{2}$$
$31$ $$-800 - 52 T + T^{2}$$
$37$ $$-6480 + 306 T + T^{2}$$
$41$ $$40140 - 408 T + T^{2}$$
$43$ $$-90864 + 120 T + T^{2}$$
$47$ $$-78656 - 180 T + T^{2}$$
$53$ $$-28520 - 402 T + T^{2}$$
$59$ $$8280 - 186 T + T^{2}$$
$61$ $$-65564 - 340 T + T^{2}$$
$67$ $$97056 + 732 T + T^{2}$$
$71$ $$-331776 - 36 T + T^{2}$$
$73$ $$324000 + 1332 T + T^{2}$$
$79$ $$-886400 - 380 T + T^{2}$$
$83$ $$-201392 + 984 T + T^{2}$$
$89$ $$98820 + 1116 T + T^{2}$$
$97$ $$-442944 - 768 T + T^{2}$$