# Properties

 Label 225.12.a.b 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

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,12,Mod(1,225)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(225, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 12, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 12);

N := Newforms(S);

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

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 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 1) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

 $$f(q)$$ $$=$$ $$q - 24 q^{2} - 1472 q^{4} + 16744 q^{7} + 84480 q^{8}+O(q^{10})$$ q - 24 * q^2 - 1472 * q^4 + 16744 * q^7 + 84480 * q^8 $$q - 24 q^{2} - 1472 q^{4} + 16744 q^{7} + 84480 q^{8} - 534612 q^{11} + 577738 q^{13} - 401856 q^{14} + 987136 q^{16} - 6905934 q^{17} + 10661420 q^{19} + 12830688 q^{22} + 18643272 q^{23} - 13865712 q^{26} - 24647168 q^{28} - 128406630 q^{29} - 52843168 q^{31} - 196706304 q^{32} + 165742416 q^{34} + 182213314 q^{37} - 255874080 q^{38} - 308120442 q^{41} + 17125708 q^{43} + 786948864 q^{44} - 447438528 q^{46} + 2687348496 q^{47} - 1696965207 q^{49} - 850430336 q^{52} - 1596055698 q^{53} + 1414533120 q^{56} + 3081759120 q^{58} + 5189203740 q^{59} + 6956478662 q^{61} + 1268236032 q^{62} + 2699296768 q^{64} + 15481826884 q^{67} + 10165534848 q^{68} - 9791485272 q^{71} - 1463791322 q^{73} - 4373119536 q^{74} - 15693610240 q^{76} - 8951543328 q^{77} + 38116845680 q^{79} + 7394890608 q^{82} - 29335099668 q^{83} - 411016992 q^{86} - 45164021760 q^{88} + 24992917110 q^{89} + 9673645072 q^{91} - 27442896384 q^{92} - 64496363904 q^{94} - 75013568546 q^{97} + 40727164968 q^{98}+O(q^{100})$$ q - 24 * q^2 - 1472 * q^4 + 16744 * q^7 + 84480 * q^8 - 534612 * q^11 + 577738 * q^13 - 401856 * q^14 + 987136 * q^16 - 6905934 * q^17 + 10661420 * q^19 + 12830688 * q^22 + 18643272 * q^23 - 13865712 * q^26 - 24647168 * q^28 - 128406630 * q^29 - 52843168 * q^31 - 196706304 * q^32 + 165742416 * q^34 + 182213314 * q^37 - 255874080 * q^38 - 308120442 * q^41 + 17125708 * q^43 + 786948864 * q^44 - 447438528 * q^46 + 2687348496 * q^47 - 1696965207 * q^49 - 850430336 * q^52 - 1596055698 * q^53 + 1414533120 * q^56 + 3081759120 * q^58 + 5189203740 * q^59 + 6956478662 * q^61 + 1268236032 * q^62 + 2699296768 * q^64 + 15481826884 * q^67 + 10165534848 * q^68 - 9791485272 * q^71 - 1463791322 * q^73 - 4373119536 * q^74 - 15693610240 * q^76 - 8951543328 * q^77 + 38116845680 * q^79 + 7394890608 * q^82 - 29335099668 * q^83 - 411016992 * q^86 - 45164021760 * q^88 + 24992917110 * q^89 + 9673645072 * q^91 - 27442896384 * q^92 - 64496363904 * q^94 - 75013568546 * q^97 + 40727164968 * 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

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
−24.0000 0 −1472.00 0 0 16744.0 84480.0 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.b 1
3.b odd 2 1 25.12.a.b 1
5.b even 2 1 9.12.a.b 1
5.c odd 4 2 225.12.b.d 2
15.d odd 2 1 1.12.a.a 1
15.e even 4 2 25.12.b.b 2
20.d odd 2 1 144.12.a.d 1
45.h odd 6 2 81.12.c.d 2
45.j even 6 2 81.12.c.b 2
60.h even 2 1 16.12.a.a 1
105.g even 2 1 49.12.a.a 1
105.o odd 6 2 49.12.c.b 2
105.p even 6 2 49.12.c.c 2
120.i odd 2 1 64.12.a.b 1
120.m even 2 1 64.12.a.f 1
165.d even 2 1 121.12.a.b 1
195.e odd 2 1 169.12.a.a 1
240.t even 4 2 256.12.b.c 2
240.bm odd 4 2 256.12.b.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 15.d odd 2 1
9.12.a.b 1 5.b even 2 1
16.12.a.a 1 60.h even 2 1
25.12.a.b 1 3.b odd 2 1
25.12.b.b 2 15.e even 4 2
49.12.a.a 1 105.g even 2 1
49.12.c.b 2 105.o odd 6 2
49.12.c.c 2 105.p even 6 2
64.12.a.b 1 120.i odd 2 1
64.12.a.f 1 120.m even 2 1
81.12.c.b 2 45.j even 6 2
81.12.c.d 2 45.h odd 6 2
121.12.a.b 1 165.d even 2 1
144.12.a.d 1 20.d odd 2 1
169.12.a.a 1 195.e odd 2 1
225.12.a.b 1 1.a even 1 1 trivial
225.12.b.d 2 5.c odd 4 2
256.12.b.c 2 240.t even 4 2
256.12.b.e 2 240.bm odd 4 2

## Hecke kernels

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

 $$T_{2} + 24$$ T2 + 24 $$T_{7} - 16744$$ T7 - 16744

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 24$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 16744$$
$11$ $$T + 534612$$
$13$ $$T - 577738$$
$17$ $$T + 6905934$$
$19$ $$T - 10661420$$
$23$ $$T - 18643272$$
$29$ $$T + 128406630$$
$31$ $$T + 52843168$$
$37$ $$T - 182213314$$
$41$ $$T + 308120442$$
$43$ $$T - 17125708$$
$47$ $$T - 2687348496$$
$53$ $$T + 1596055698$$
$59$ $$T - 5189203740$$
$61$ $$T - 6956478662$$
$67$ $$T - 15481826884$$
$71$ $$T + 9791485272$$
$73$ $$T + 1463791322$$
$79$ $$T - 38116845680$$
$83$ $$T + 29335099668$$
$89$ $$T - 24992917110$$
$97$ $$T + 75013568546$$