# Properties

 Label 225.4.a.g Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + 3 q^{2} + q^{4} - 20 q^{7} - 21 q^{8}+O(q^{10})$$ q + 3 * q^2 + q^4 - 20 * q^7 - 21 * q^8 $$q + 3 q^{2} + q^{4} - 20 q^{7} - 21 q^{8} + 24 q^{11} - 74 q^{13} - 60 q^{14} - 71 q^{16} + 54 q^{17} - 124 q^{19} + 72 q^{22} - 120 q^{23} - 222 q^{26} - 20 q^{28} + 78 q^{29} + 200 q^{31} - 45 q^{32} + 162 q^{34} + 70 q^{37} - 372 q^{38} - 330 q^{41} - 92 q^{43} + 24 q^{44} - 360 q^{46} - 24 q^{47} + 57 q^{49} - 74 q^{52} + 450 q^{53} + 420 q^{56} + 234 q^{58} - 24 q^{59} - 322 q^{61} + 600 q^{62} + 433 q^{64} + 196 q^{67} + 54 q^{68} + 288 q^{71} + 430 q^{73} + 210 q^{74} - 124 q^{76} - 480 q^{77} - 520 q^{79} - 990 q^{82} + 156 q^{83} - 276 q^{86} - 504 q^{88} - 1026 q^{89} + 1480 q^{91} - 120 q^{92} - 72 q^{94} + 286 q^{97} + 171 q^{98}+O(q^{100})$$ q + 3 * q^2 + q^4 - 20 * q^7 - 21 * q^8 + 24 * q^11 - 74 * q^13 - 60 * q^14 - 71 * q^16 + 54 * q^17 - 124 * q^19 + 72 * q^22 - 120 * q^23 - 222 * q^26 - 20 * q^28 + 78 * q^29 + 200 * q^31 - 45 * q^32 + 162 * q^34 + 70 * q^37 - 372 * q^38 - 330 * q^41 - 92 * q^43 + 24 * q^44 - 360 * q^46 - 24 * q^47 + 57 * q^49 - 74 * q^52 + 450 * q^53 + 420 * q^56 + 234 * q^58 - 24 * q^59 - 322 * q^61 + 600 * q^62 + 433 * q^64 + 196 * q^67 + 54 * q^68 + 288 * q^71 + 430 * q^73 + 210 * q^74 - 124 * q^76 - 480 * q^77 - 520 * q^79 - 990 * q^82 + 156 * q^83 - 276 * q^86 - 504 * q^88 - 1026 * q^89 + 1480 * q^91 - 120 * q^92 - 72 * q^94 + 286 * q^97 + 171 * 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
 0
3.00000 0 1.00000 0 0 −20.0000 −21.0000 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.g 1
3.b odd 2 1 75.4.a.a 1
5.b even 2 1 45.4.a.b 1
5.c odd 4 2 225.4.b.d 2
12.b even 2 1 1200.4.a.o 1
15.d odd 2 1 15.4.a.b 1
15.e even 4 2 75.4.b.a 2
20.d odd 2 1 720.4.a.r 1
35.c odd 2 1 2205.4.a.c 1
45.h odd 6 2 405.4.e.d 2
45.j even 6 2 405.4.e.k 2
60.h even 2 1 240.4.a.f 1
60.l odd 4 2 1200.4.f.m 2
105.g even 2 1 735.4.a.i 1
120.i odd 2 1 960.4.a.bi 1
120.m even 2 1 960.4.a.l 1
165.d even 2 1 1815.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.a.b 1 15.d odd 2 1
45.4.a.b 1 5.b even 2 1
75.4.a.a 1 3.b odd 2 1
75.4.b.a 2 15.e even 4 2
225.4.a.g 1 1.a even 1 1 trivial
225.4.b.d 2 5.c odd 4 2
240.4.a.f 1 60.h even 2 1
405.4.e.d 2 45.h odd 6 2
405.4.e.k 2 45.j even 6 2
720.4.a.r 1 20.d odd 2 1
735.4.a.i 1 105.g even 2 1
960.4.a.l 1 120.m even 2 1
960.4.a.bi 1 120.i odd 2 1
1200.4.a.o 1 12.b even 2 1
1200.4.f.m 2 60.l odd 4 2
1815.4.a.a 1 165.d even 2 1
2205.4.a.c 1 35.c odd 2 1

## Hecke kernels

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

 $$T_{2} - 3$$ T2 - 3 $$T_{7} + 20$$ T7 + 20

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 3$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 20$$
$11$ $$T - 24$$
$13$ $$T + 74$$
$17$ $$T - 54$$
$19$ $$T + 124$$
$23$ $$T + 120$$
$29$ $$T - 78$$
$31$ $$T - 200$$
$37$ $$T - 70$$
$41$ $$T + 330$$
$43$ $$T + 92$$
$47$ $$T + 24$$
$53$ $$T - 450$$
$59$ $$T + 24$$
$61$ $$T + 322$$
$67$ $$T - 196$$
$71$ $$T - 288$$
$73$ $$T - 430$$
$79$ $$T + 520$$
$83$ $$T - 156$$
$89$ $$T + 1026$$
$97$ $$T - 286$$