# Properties

 Label 225.4.a.e Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $0$ 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - 7 q^{4} - 6 q^{7} - 15 q^{8}+O(q^{10})$$ q + q^2 - 7 * q^4 - 6 * q^7 - 15 * q^8 $$q + q^{2} - 7 q^{4} - 6 q^{7} - 15 q^{8} + 43 q^{11} + 28 q^{13} - 6 q^{14} + 41 q^{16} + 91 q^{17} - 35 q^{19} + 43 q^{22} + 162 q^{23} + 28 q^{26} + 42 q^{28} - 160 q^{29} + 42 q^{31} + 161 q^{32} + 91 q^{34} + 314 q^{37} - 35 q^{38} + 203 q^{41} - 92 q^{43} - 301 q^{44} + 162 q^{46} + 196 q^{47} - 307 q^{49} - 196 q^{52} + 82 q^{53} + 90 q^{56} - 160 q^{58} + 280 q^{59} - 518 q^{61} + 42 q^{62} - 167 q^{64} - 141 q^{67} - 637 q^{68} - 412 q^{71} + 763 q^{73} + 314 q^{74} + 245 q^{76} - 258 q^{77} + 510 q^{79} + 203 q^{82} + 777 q^{83} - 92 q^{86} - 645 q^{88} + 945 q^{89} - 168 q^{91} - 1134 q^{92} + 196 q^{94} - 1246 q^{97} - 307 q^{98}+O(q^{100})$$ q + q^2 - 7 * q^4 - 6 * q^7 - 15 * q^8 + 43 * q^11 + 28 * q^13 - 6 * q^14 + 41 * q^16 + 91 * q^17 - 35 * q^19 + 43 * q^22 + 162 * q^23 + 28 * q^26 + 42 * q^28 - 160 * q^29 + 42 * q^31 + 161 * q^32 + 91 * q^34 + 314 * q^37 - 35 * q^38 + 203 * q^41 - 92 * q^43 - 301 * q^44 + 162 * q^46 + 196 * q^47 - 307 * q^49 - 196 * q^52 + 82 * q^53 + 90 * q^56 - 160 * q^58 + 280 * q^59 - 518 * q^61 + 42 * q^62 - 167 * q^64 - 141 * q^67 - 637 * q^68 - 412 * q^71 + 763 * q^73 + 314 * q^74 + 245 * q^76 - 258 * q^77 + 510 * q^79 + 203 * q^82 + 777 * q^83 - 92 * q^86 - 645 * q^88 + 945 * q^89 - 168 * q^91 - 1134 * q^92 + 196 * q^94 - 1246 * q^97 - 307 * 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
1.00000 0 −7.00000 0 0 −6.00000 −15.0000 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.e 1
3.b odd 2 1 25.4.a.a 1
5.b even 2 1 225.4.a.c 1
5.c odd 4 2 225.4.b.f 2
12.b even 2 1 400.4.a.s 1
15.d odd 2 1 25.4.a.b yes 1
15.e even 4 2 25.4.b.b 2
21.c even 2 1 1225.4.a.h 1
24.f even 2 1 1600.4.a.h 1
24.h odd 2 1 1600.4.a.bt 1
60.h even 2 1 400.4.a.c 1
60.l odd 4 2 400.4.c.e 2
105.g even 2 1 1225.4.a.i 1
120.i odd 2 1 1600.4.a.i 1
120.m even 2 1 1600.4.a.bs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.4.a.a 1 3.b odd 2 1
25.4.a.b yes 1 15.d odd 2 1
25.4.b.b 2 15.e even 4 2
225.4.a.c 1 5.b even 2 1
225.4.a.e 1 1.a even 1 1 trivial
225.4.b.f 2 5.c odd 4 2
400.4.a.c 1 60.h even 2 1
400.4.a.s 1 12.b even 2 1
400.4.c.e 2 60.l odd 4 2
1225.4.a.h 1 21.c even 2 1
1225.4.a.i 1 105.g even 2 1
1600.4.a.h 1 24.f even 2 1
1600.4.a.i 1 120.i odd 2 1
1600.4.a.bs 1 120.m even 2 1
1600.4.a.bt 1 24.h odd 2 1

## Hecke kernels

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

 $$T_{2} - 1$$ T2 - 1 $$T_{7} + 6$$ T7 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T + 6$$
$11$ $$T - 43$$
$13$ $$T - 28$$
$17$ $$T - 91$$
$19$ $$T + 35$$
$23$ $$T - 162$$
$29$ $$T + 160$$
$31$ $$T - 42$$
$37$ $$T - 314$$
$41$ $$T - 203$$
$43$ $$T + 92$$
$47$ $$T - 196$$
$53$ $$T - 82$$
$59$ $$T - 280$$
$61$ $$T + 518$$
$67$ $$T + 141$$
$71$ $$T + 412$$
$73$ $$T - 763$$
$79$ $$T - 510$$
$83$ $$T - 777$$
$89$ $$T - 945$$
$97$ $$T + 1246$$