Properties

 Label 225.12.a.f Level $225$ Weight $12$ Character orbit 225.a Self dual yes Analytic conductor $172.877$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$12$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$172.877215626$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 3) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

 $$f(q)$$ $$=$$ $$q + 78 q^{2} + 4036 q^{4} + 27760 q^{7} + 155064 q^{8}+O(q^{10})$$ q + 78 * q^2 + 4036 * q^4 + 27760 * q^7 + 155064 * q^8 $$q + 78 q^{2} + 4036 q^{4} + 27760 q^{7} + 155064 q^{8} - 637836 q^{11} - 766214 q^{13} + 2165280 q^{14} + 3829264 q^{16} + 3084354 q^{17} - 19511404 q^{19} - 49751208 q^{22} + 15312360 q^{23} - 59764692 q^{26} + 112039360 q^{28} - 10751262 q^{29} - 50937400 q^{31} - 18888480 q^{32} + 240579612 q^{34} - 664740830 q^{37} - 1521889512 q^{38} - 898833450 q^{41} + 957947188 q^{43} - 2574306096 q^{44} + 1194364080 q^{46} - 1555741344 q^{47} - 1206709143 q^{49} - 3092439704 q^{52} + 3792417030 q^{53} + 4304576640 q^{56} - 838598436 q^{58} - 555306924 q^{59} + 4950420998 q^{61} - 3973117200 q^{62} - 9315634112 q^{64} - 5292399284 q^{67} + 12448452744 q^{68} + 14831086248 q^{71} - 13971005210 q^{73} - 51849784740 q^{74} - 78748026544 q^{76} - 17706327360 q^{77} + 3720542360 q^{79} - 70109009100 q^{82} + 8768454036 q^{83} + 74719880664 q^{86} - 98905401504 q^{88} + 25472769174 q^{89} - 21270100640 q^{91} + 61800684960 q^{92} - 121347824832 q^{94} + 39092494846 q^{97} - 94123313154 q^{98}+O(q^{100})$$ q + 78 * q^2 + 4036 * q^4 + 27760 * q^7 + 155064 * q^8 - 637836 * q^11 - 766214 * q^13 + 2165280 * q^14 + 3829264 * q^16 + 3084354 * q^17 - 19511404 * q^19 - 49751208 * q^22 + 15312360 * q^23 - 59764692 * q^26 + 112039360 * q^28 - 10751262 * q^29 - 50937400 * q^31 - 18888480 * q^32 + 240579612 * q^34 - 664740830 * q^37 - 1521889512 * q^38 - 898833450 * q^41 + 957947188 * q^43 - 2574306096 * q^44 + 1194364080 * q^46 - 1555741344 * q^47 - 1206709143 * q^49 - 3092439704 * q^52 + 3792417030 * q^53 + 4304576640 * q^56 - 838598436 * q^58 - 555306924 * q^59 + 4950420998 * q^61 - 3973117200 * q^62 - 9315634112 * q^64 - 5292399284 * q^67 + 12448452744 * q^68 + 14831086248 * q^71 - 13971005210 * q^73 - 51849784740 * q^74 - 78748026544 * q^76 - 17706327360 * q^77 + 3720542360 * q^79 - 70109009100 * q^82 + 8768454036 * q^83 + 74719880664 * q^86 - 98905401504 * q^88 + 25472769174 * q^89 - 21270100640 * q^91 + 61800684960 * q^92 - 121347824832 * q^94 + 39092494846 * q^97 - 94123313154 * 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
78.0000 0 4036.00 0 0 27760.0 155064. 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 225.12.a.f 1
3.b odd 2 1 75.12.a.a 1
5.b even 2 1 9.12.a.a 1
5.c odd 4 2 225.12.b.a 2
15.d odd 2 1 3.12.a.a 1
15.e even 4 2 75.12.b.a 2
20.d odd 2 1 144.12.a.l 1
45.h odd 6 2 81.12.c.a 2
45.j even 6 2 81.12.c.e 2
60.h even 2 1 48.12.a.f 1
105.g even 2 1 147.12.a.c 1
120.i odd 2 1 192.12.a.q 1
120.m even 2 1 192.12.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.12.a.a 1 15.d odd 2 1
9.12.a.a 1 5.b even 2 1
48.12.a.f 1 60.h even 2 1
75.12.a.a 1 3.b odd 2 1
75.12.b.a 2 15.e even 4 2
81.12.c.a 2 45.h odd 6 2
81.12.c.e 2 45.j even 6 2
144.12.a.l 1 20.d odd 2 1
147.12.a.c 1 105.g even 2 1
192.12.a.g 1 120.m even 2 1
192.12.a.q 1 120.i odd 2 1
225.12.a.f 1 1.a even 1 1 trivial
225.12.b.a 2 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{12}^{\mathrm{new}}(\Gamma_0(225))$$:

 $$T_{2} - 78$$ T2 - 78 $$T_{7} - 27760$$ T7 - 27760

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 78$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 27760$$
$11$ $$T + 637836$$
$13$ $$T + 766214$$
$17$ $$T - 3084354$$
$19$ $$T + 19511404$$
$23$ $$T - 15312360$$
$29$ $$T + 10751262$$
$31$ $$T + 50937400$$
$37$ $$T + 664740830$$
$41$ $$T + 898833450$$
$43$ $$T - 957947188$$
$47$ $$T + 1555741344$$
$53$ $$T - 3792417030$$
$59$ $$T + 555306924$$
$61$ $$T - 4950420998$$
$67$ $$T + 5292399284$$
$71$ $$T - 14831086248$$
$73$ $$T + 13971005210$$
$79$ $$T - 3720542360$$
$83$ $$T - 8768454036$$
$89$ $$T - 25472769174$$
$97$ $$T - 39092494846$$