# Properties

 Label 225.4.a.i Level $225$ Weight $4$ Character orbit 225.a Self dual yes Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [225,4,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, 4, names="a")

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

chi := DirichletCharacter("225.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{41})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 10$$ x^2 - x - 10 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 15) 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{41})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + ( - 6 \beta + 6) q^{7} + ( - \beta - 25) q^{8}+O(q^{10})$$ q + (-b - 1) * q^2 + (3*b + 3) * q^4 + (-6*b + 6) * q^7 + (-b - 25) * q^8 $$q + ( - \beta - 1) q^{2} + (3 \beta + 3) q^{4} + ( - 6 \beta + 6) q^{7} + ( - \beta - 25) q^{8} + (6 \beta + 18) q^{11} + ( - 6 \beta + 42) q^{13} + (6 \beta + 54) q^{14} + (3 \beta + 11) q^{16} + ( - 10 \beta - 46) q^{17} + ( - 24 \beta + 40) q^{19} + ( - 30 \beta - 78) q^{22} + ( - 8 \beta + 28) q^{23} + ( - 30 \beta + 18) q^{26} + ( - 18 \beta - 162) q^{28} + ( - 42 \beta + 180) q^{29} + ( - 12 \beta + 32) q^{31} + ( - 9 \beta + 159) q^{32} + (66 \beta + 146) q^{34} + (54 \beta + 126) q^{37} + (8 \beta + 200) q^{38} + (12 \beta + 198) q^{41} + (96 \beta + 12) q^{43} + (90 \beta + 234) q^{44} + ( - 12 \beta + 52) q^{46} + (92 \beta - 136) q^{47} + ( - 36 \beta + 53) q^{49} + (90 \beta - 54) q^{52} + (82 \beta - 242) q^{53} + (150 \beta - 90) q^{56} + ( - 96 \beta + 240) q^{58} + (6 \beta + 90) q^{59} + (96 \beta + 122) q^{61} + ( - 8 \beta + 88) q^{62} + ( - 165 \beta - 157) q^{64} + (60 \beta + 336) q^{67} + ( - 198 \beta - 438) q^{68} + ( - 180 \beta + 108) q^{71} + (108 \beta + 612) q^{73} + ( - 234 \beta - 666) q^{74} + ( - 24 \beta - 600) q^{76} + ( - 108 \beta - 252) q^{77} + (300 \beta + 40) q^{79} + ( - 222 \beta - 318) q^{82} + (208 \beta + 388) q^{83} + ( - 204 \beta - 972) q^{86} + ( - 174 \beta - 510) q^{88} + (144 \beta - 630) q^{89} + ( - 252 \beta + 612) q^{91} + (36 \beta - 156) q^{92} + ( - 48 \beta - 784) q^{94} + ( - 240 \beta - 264) q^{97} + (19 \beta + 307) q^{98}+O(q^{100})$$ q + (-b - 1) * q^2 + (3*b + 3) * q^4 + (-6*b + 6) * q^7 + (-b - 25) * q^8 + (6*b + 18) * q^11 + (-6*b + 42) * q^13 + (6*b + 54) * q^14 + (3*b + 11) * q^16 + (-10*b - 46) * q^17 + (-24*b + 40) * q^19 + (-30*b - 78) * q^22 + (-8*b + 28) * q^23 + (-30*b + 18) * q^26 + (-18*b - 162) * q^28 + (-42*b + 180) * q^29 + (-12*b + 32) * q^31 + (-9*b + 159) * q^32 + (66*b + 146) * q^34 + (54*b + 126) * q^37 + (8*b + 200) * q^38 + (12*b + 198) * q^41 + (96*b + 12) * q^43 + (90*b + 234) * q^44 + (-12*b + 52) * q^46 + (92*b - 136) * q^47 + (-36*b + 53) * q^49 + (90*b - 54) * q^52 + (82*b - 242) * q^53 + (150*b - 90) * q^56 + (-96*b + 240) * q^58 + (6*b + 90) * q^59 + (96*b + 122) * q^61 + (-8*b + 88) * q^62 + (-165*b - 157) * q^64 + (60*b + 336) * q^67 + (-198*b - 438) * q^68 + (-180*b + 108) * q^71 + (108*b + 612) * q^73 + (-234*b - 666) * q^74 + (-24*b - 600) * q^76 + (-108*b - 252) * q^77 + (300*b + 40) * q^79 + (-222*b - 318) * q^82 + (208*b + 388) * q^83 + (-204*b - 972) * q^86 + (-174*b - 510) * q^88 + (144*b - 630) * q^89 + (-252*b + 612) * q^91 + (36*b - 156) * q^92 + (-48*b - 784) * q^94 + (-240*b - 264) * q^97 + (19*b + 307) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8}+O(q^{10})$$ 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8 $$2 q - 3 q^{2} + 9 q^{4} + 6 q^{7} - 51 q^{8} + 42 q^{11} + 78 q^{13} + 114 q^{14} + 25 q^{16} - 102 q^{17} + 56 q^{19} - 186 q^{22} + 48 q^{23} + 6 q^{26} - 342 q^{28} + 318 q^{29} + 52 q^{31} + 309 q^{32} + 358 q^{34} + 306 q^{37} + 408 q^{38} + 408 q^{41} + 120 q^{43} + 558 q^{44} + 92 q^{46} - 180 q^{47} + 70 q^{49} - 18 q^{52} - 402 q^{53} - 30 q^{56} + 384 q^{58} + 186 q^{59} + 340 q^{61} + 168 q^{62} - 479 q^{64} + 732 q^{67} - 1074 q^{68} + 36 q^{71} + 1332 q^{73} - 1566 q^{74} - 1224 q^{76} - 612 q^{77} + 380 q^{79} - 858 q^{82} + 984 q^{83} - 2148 q^{86} - 1194 q^{88} - 1116 q^{89} + 972 q^{91} - 276 q^{92} - 1616 q^{94} - 768 q^{97} + 633 q^{98}+O(q^{100})$$ 2 * q - 3 * q^2 + 9 * q^4 + 6 * q^7 - 51 * q^8 + 42 * q^11 + 78 * q^13 + 114 * q^14 + 25 * q^16 - 102 * q^17 + 56 * q^19 - 186 * q^22 + 48 * q^23 + 6 * q^26 - 342 * q^28 + 318 * q^29 + 52 * q^31 + 309 * q^32 + 358 * q^34 + 306 * q^37 + 408 * q^38 + 408 * q^41 + 120 * q^43 + 558 * q^44 + 92 * q^46 - 180 * q^47 + 70 * q^49 - 18 * q^52 - 402 * q^53 - 30 * q^56 + 384 * q^58 + 186 * q^59 + 340 * q^61 + 168 * q^62 - 479 * q^64 + 732 * q^67 - 1074 * q^68 + 36 * q^71 + 1332 * q^73 - 1566 * q^74 - 1224 * q^76 - 612 * q^77 + 380 * q^79 - 858 * q^82 + 984 * q^83 - 2148 * q^86 - 1194 * q^88 - 1116 * q^89 + 972 * q^91 - 276 * q^92 - 1616 * q^94 - 768 * q^97 + 633 * 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
 3.70156 −2.70156
−4.70156 0 14.1047 0 0 −16.2094 −28.7016 0 0
1.2 1.70156 0 −5.10469 0 0 22.2094 −22.2984 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.i 2
3.b odd 2 1 75.4.a.f 2
5.b even 2 1 225.4.a.o 2
5.c odd 4 2 45.4.b.b 4
12.b even 2 1 1200.4.a.bn 2
15.d odd 2 1 75.4.a.c 2
15.e even 4 2 15.4.b.a 4
20.e even 4 2 720.4.f.j 4
60.h even 2 1 1200.4.a.bt 2
60.l odd 4 2 240.4.f.f 4
120.q odd 4 2 960.4.f.p 4
120.w even 4 2 960.4.f.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.4.b.a 4 15.e even 4 2
45.4.b.b 4 5.c odd 4 2
75.4.a.c 2 15.d odd 2 1
75.4.a.f 2 3.b odd 2 1
225.4.a.i 2 1.a even 1 1 trivial
225.4.a.o 2 5.b even 2 1
240.4.f.f 4 60.l odd 4 2
720.4.f.j 4 20.e even 4 2
960.4.f.p 4 120.q odd 4 2
960.4.f.q 4 120.w even 4 2
1200.4.a.bn 2 12.b even 2 1
1200.4.a.bt 2 60.h even 2 1

## Hecke kernels

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

 $$T_{2}^{2} + 3T_{2} - 8$$ T2^2 + 3*T2 - 8 $$T_{7}^{2} - 6T_{7} - 360$$ T7^2 - 6*T7 - 360

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 3T - 8$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 6T - 360$$
$11$ $$T^{2} - 42T + 72$$
$13$ $$T^{2} - 78T + 1152$$
$17$ $$T^{2} + 102T + 1576$$
$19$ $$T^{2} - 56T - 5120$$
$23$ $$T^{2} - 48T - 80$$
$29$ $$T^{2} - 318T + 7200$$
$31$ $$T^{2} - 52T - 800$$
$37$ $$T^{2} - 306T - 6480$$
$41$ $$T^{2} - 408T + 40140$$
$43$ $$T^{2} - 120T - 90864$$
$47$ $$T^{2} + 180T - 78656$$
$53$ $$T^{2} + 402T - 28520$$
$59$ $$T^{2} - 186T + 8280$$
$61$ $$T^{2} - 340T - 65564$$
$67$ $$T^{2} - 732T + 97056$$
$71$ $$T^{2} - 36T - 331776$$
$73$ $$T^{2} - 1332 T + 324000$$
$79$ $$T^{2} - 380T - 886400$$
$83$ $$T^{2} - 984T - 201392$$
$89$ $$T^{2} + 1116T + 98820$$
$97$ $$T^{2} + 768T - 442944$$
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