# Properties

 Label 225.6.a.c Level $225$ Weight $6$ Character orbit 225.a Self dual yes Analytic conductor $36.086$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# 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: $$0$$ 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

 $$f(q)$$ $$=$$ $$q - 2 q^{2} - 28 q^{4} + 132 q^{7} + 120 q^{8}+O(q^{10})$$ q - 2 * q^2 - 28 * q^4 + 132 * q^7 + 120 * q^8 $$q - 2 q^{2} - 28 q^{4} + 132 q^{7} + 120 q^{8} - 472 q^{11} + 686 q^{13} - 264 q^{14} + 656 q^{16} - 1562 q^{17} - 2180 q^{19} + 944 q^{22} + 264 q^{23} - 1372 q^{26} - 3696 q^{28} - 170 q^{29} + 7272 q^{31} - 5152 q^{32} + 3124 q^{34} + 142 q^{37} + 4360 q^{38} + 16198 q^{41} + 10316 q^{43} + 13216 q^{44} - 528 q^{46} + 18568 q^{47} + 617 q^{49} - 19208 q^{52} + 21514 q^{53} + 15840 q^{56} + 340 q^{58} - 34600 q^{59} - 35738 q^{61} - 14544 q^{62} - 10688 q^{64} + 5772 q^{67} + 43736 q^{68} + 69088 q^{71} + 70526 q^{73} - 284 q^{74} + 61040 q^{76} - 62304 q^{77} + 47640 q^{79} - 32396 q^{82} + 74004 q^{83} - 20632 q^{86} - 56640 q^{88} + 90030 q^{89} + 90552 q^{91} - 7392 q^{92} - 37136 q^{94} + 33502 q^{97} - 1234 q^{98}+O(q^{100})$$ q - 2 * q^2 - 28 * q^4 + 132 * q^7 + 120 * q^8 - 472 * q^11 + 686 * q^13 - 264 * q^14 + 656 * q^16 - 1562 * q^17 - 2180 * q^19 + 944 * q^22 + 264 * q^23 - 1372 * q^26 - 3696 * q^28 - 170 * q^29 + 7272 * q^31 - 5152 * q^32 + 3124 * q^34 + 142 * q^37 + 4360 * q^38 + 16198 * q^41 + 10316 * q^43 + 13216 * q^44 - 528 * q^46 + 18568 * q^47 + 617 * q^49 - 19208 * q^52 + 21514 * q^53 + 15840 * q^56 + 340 * q^58 - 34600 * q^59 - 35738 * q^61 - 14544 * q^62 - 10688 * q^64 + 5772 * q^67 + 43736 * q^68 + 69088 * q^71 + 70526 * q^73 - 284 * q^74 + 61040 * q^76 - 62304 * q^77 + 47640 * q^79 - 32396 * q^82 + 74004 * q^83 - 20632 * q^86 - 56640 * q^88 + 90030 * q^89 + 90552 * q^91 - 7392 * q^92 - 37136 * q^94 + 33502 * q^97 - 1234 * 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
−2.00000 0 −28.0000 0 0 132.000 120.000 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 225.6.a.c 1
3.b odd 2 1 75.6.a.c 1
5.b even 2 1 45.6.a.c 1
5.c odd 4 2 225.6.b.d 2
15.d odd 2 1 15.6.a.a 1
15.e even 4 2 75.6.b.d 2
20.d odd 2 1 720.6.a.w 1
60.h even 2 1 240.6.a.k 1
105.g even 2 1 735.6.a.a 1
120.i odd 2 1 960.6.a.v 1
120.m even 2 1 960.6.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.6.a.a 1 15.d odd 2 1
45.6.a.c 1 5.b even 2 1
75.6.a.c 1 3.b odd 2 1
75.6.b.d 2 15.e even 4 2
225.6.a.c 1 1.a even 1 1 trivial
225.6.b.d 2 5.c odd 4 2
240.6.a.k 1 60.h even 2 1
720.6.a.w 1 20.d odd 2 1
735.6.a.a 1 105.g even 2 1
960.6.a.m 1 120.m even 2 1
960.6.a.v 1 120.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2} + 2$$ T2 + 2 $$T_{7} - 132$$ T7 - 132

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 2$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 132$$
$11$ $$T + 472$$
$13$ $$T - 686$$
$17$ $$T + 1562$$
$19$ $$T + 2180$$
$23$ $$T - 264$$
$29$ $$T + 170$$
$31$ $$T - 7272$$
$37$ $$T - 142$$
$41$ $$T - 16198$$
$43$ $$T - 10316$$
$47$ $$T - 18568$$
$53$ $$T - 21514$$
$59$ $$T + 34600$$
$61$ $$T + 35738$$
$67$ $$T - 5772$$
$71$ $$T - 69088$$
$73$ $$T - 70526$$
$79$ $$T - 47640$$
$83$ $$T - 74004$$
$89$ $$T - 90030$$
$97$ $$T - 33502$$