# Properties

 Label 325.2.a.f Level $325$ Weight $2$ Character orbit 325.a Self dual yes Analytic conductor $2.595$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(1,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("325.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.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: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} - 2 \beta q^{3} + ( - 2 \beta + 1) q^{4} + (2 \beta - 4) q^{6} + (\beta - 1) q^{7} + (\beta - 3) q^{8} + 5 q^{9} +O(q^{10})$$ q + (b - 1) * q^2 - 2*b * q^3 + (-2*b + 1) * q^4 + (2*b - 4) * q^6 + (b - 1) * q^7 + (b - 3) * q^8 + 5 * q^9 $$q + (\beta - 1) q^{2} - 2 \beta q^{3} + ( - 2 \beta + 1) q^{4} + (2 \beta - 4) q^{6} + (\beta - 1) q^{7} + (\beta - 3) q^{8} + 5 q^{9} + ( - \beta + 5) q^{11} + ( - 2 \beta + 8) q^{12} - q^{13} + ( - 2 \beta + 3) q^{14} + 3 q^{16} + ( - 2 \beta + 1) q^{17} + (5 \beta - 5) q^{18} + (4 \beta + 2) q^{19} + (2 \beta - 4) q^{21} + (6 \beta - 7) q^{22} + (2 \beta + 6) q^{23} + (6 \beta - 4) q^{24} + ( - \beta + 1) q^{26} - 4 \beta q^{27} + (3 \beta - 5) q^{28} + ( - 2 \beta - 3) q^{29} + ( - 3 \beta - 3) q^{31} + (\beta + 3) q^{32} + ( - 10 \beta + 4) q^{33} + (3 \beta - 5) q^{34} + ( - 10 \beta + 5) q^{36} + 6 q^{37} + ( - 2 \beta + 6) q^{38} + 2 \beta q^{39} + 4 \beta q^{41} + ( - 6 \beta + 8) q^{42} + (2 \beta + 2) q^{43} + ( - 11 \beta + 9) q^{44} + (4 \beta - 2) q^{46} + ( - \beta - 5) q^{47} - 6 \beta q^{48} + ( - 2 \beta - 4) q^{49} + ( - 2 \beta + 8) q^{51} + (2 \beta - 1) q^{52} - 3 q^{53} + (4 \beta - 8) q^{54} + ( - 4 \beta + 5) q^{56} + ( - 4 \beta - 16) q^{57} + ( - \beta - 1) q^{58} + ( - 3 \beta + 9) q^{59} + q^{61} - 3 q^{62} + (5 \beta - 5) q^{63} + (2 \beta - 7) q^{64} + (14 \beta - 24) q^{66} + (3 \beta + 7) q^{67} + ( - 4 \beta + 9) q^{68} + ( - 12 \beta - 8) q^{69} + (8 \beta + 2) q^{71} + (5 \beta - 15) q^{72} + 6 q^{73} + (6 \beta - 6) q^{74} - 14 q^{76} + (6 \beta - 7) q^{77} + ( - 2 \beta + 4) q^{78} - 6 q^{79} + q^{81} + ( - 4 \beta + 8) q^{82} + (7 \beta + 3) q^{83} + (10 \beta - 12) q^{84} + 2 q^{86} + (6 \beta + 8) q^{87} + (8 \beta - 17) q^{88} + 6 \beta q^{89} + ( - \beta + 1) q^{91} + ( - 10 \beta - 2) q^{92} + (6 \beta + 12) q^{93} + ( - 4 \beta + 3) q^{94} + ( - 6 \beta - 4) q^{96} + ( - 2 \beta - 2) q^{97} - 2 \beta q^{98} + ( - 5 \beta + 25) q^{99} +O(q^{100})$$ q + (b - 1) * q^2 - 2*b * q^3 + (-2*b + 1) * q^4 + (2*b - 4) * q^6 + (b - 1) * q^7 + (b - 3) * q^8 + 5 * q^9 + (-b + 5) * q^11 + (-2*b + 8) * q^12 - q^13 + (-2*b + 3) * q^14 + 3 * q^16 + (-2*b + 1) * q^17 + (5*b - 5) * q^18 + (4*b + 2) * q^19 + (2*b - 4) * q^21 + (6*b - 7) * q^22 + (2*b + 6) * q^23 + (6*b - 4) * q^24 + (-b + 1) * q^26 - 4*b * q^27 + (3*b - 5) * q^28 + (-2*b - 3) * q^29 + (-3*b - 3) * q^31 + (b + 3) * q^32 + (-10*b + 4) * q^33 + (3*b - 5) * q^34 + (-10*b + 5) * q^36 + 6 * q^37 + (-2*b + 6) * q^38 + 2*b * q^39 + 4*b * q^41 + (-6*b + 8) * q^42 + (2*b + 2) * q^43 + (-11*b + 9) * q^44 + (4*b - 2) * q^46 + (-b - 5) * q^47 - 6*b * q^48 + (-2*b - 4) * q^49 + (-2*b + 8) * q^51 + (2*b - 1) * q^52 - 3 * q^53 + (4*b - 8) * q^54 + (-4*b + 5) * q^56 + (-4*b - 16) * q^57 + (-b - 1) * q^58 + (-3*b + 9) * q^59 + q^61 - 3 * q^62 + (5*b - 5) * q^63 + (2*b - 7) * q^64 + (14*b - 24) * q^66 + (3*b + 7) * q^67 + (-4*b + 9) * q^68 + (-12*b - 8) * q^69 + (8*b + 2) * q^71 + (5*b - 15) * q^72 + 6 * q^73 + (6*b - 6) * q^74 - 14 * q^76 + (6*b - 7) * q^77 + (-2*b + 4) * q^78 - 6 * q^79 + q^81 + (-4*b + 8) * q^82 + (7*b + 3) * q^83 + (10*b - 12) * q^84 + 2 * q^86 + (6*b + 8) * q^87 + (8*b - 17) * q^88 + 6*b * q^89 + (-b + 1) * q^91 + (-10*b - 2) * q^92 + (6*b + 12) * q^93 + (-4*b + 3) * q^94 + (-6*b - 4) * q^96 + (-2*b - 2) * q^97 - 2*b * q^98 + (-5*b + 25) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 8 q^{6} - 2 q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 8 * q^6 - 2 * q^7 - 6 * q^8 + 10 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 8 q^{6} - 2 q^{7} - 6 q^{8} + 10 q^{9} + 10 q^{11} + 16 q^{12} - 2 q^{13} + 6 q^{14} + 6 q^{16} + 2 q^{17} - 10 q^{18} + 4 q^{19} - 8 q^{21} - 14 q^{22} + 12 q^{23} - 8 q^{24} + 2 q^{26} - 10 q^{28} - 6 q^{29} - 6 q^{31} + 6 q^{32} + 8 q^{33} - 10 q^{34} + 10 q^{36} + 12 q^{37} + 12 q^{38} + 16 q^{42} + 4 q^{43} + 18 q^{44} - 4 q^{46} - 10 q^{47} - 8 q^{49} + 16 q^{51} - 2 q^{52} - 6 q^{53} - 16 q^{54} + 10 q^{56} - 32 q^{57} - 2 q^{58} + 18 q^{59} + 2 q^{61} - 6 q^{62} - 10 q^{63} - 14 q^{64} - 48 q^{66} + 14 q^{67} + 18 q^{68} - 16 q^{69} + 4 q^{71} - 30 q^{72} + 12 q^{73} - 12 q^{74} - 28 q^{76} - 14 q^{77} + 8 q^{78} - 12 q^{79} + 2 q^{81} + 16 q^{82} + 6 q^{83} - 24 q^{84} + 4 q^{86} + 16 q^{87} - 34 q^{88} + 2 q^{91} - 4 q^{92} + 24 q^{93} + 6 q^{94} - 8 q^{96} - 4 q^{97} + 50 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 8 * q^6 - 2 * q^7 - 6 * q^8 + 10 * q^9 + 10 * q^11 + 16 * q^12 - 2 * q^13 + 6 * q^14 + 6 * q^16 + 2 * q^17 - 10 * q^18 + 4 * q^19 - 8 * q^21 - 14 * q^22 + 12 * q^23 - 8 * q^24 + 2 * q^26 - 10 * q^28 - 6 * q^29 - 6 * q^31 + 6 * q^32 + 8 * q^33 - 10 * q^34 + 10 * q^36 + 12 * q^37 + 12 * q^38 + 16 * q^42 + 4 * q^43 + 18 * q^44 - 4 * q^46 - 10 * q^47 - 8 * q^49 + 16 * q^51 - 2 * q^52 - 6 * q^53 - 16 * q^54 + 10 * q^56 - 32 * q^57 - 2 * q^58 + 18 * q^59 + 2 * q^61 - 6 * q^62 - 10 * q^63 - 14 * q^64 - 48 * q^66 + 14 * q^67 + 18 * q^68 - 16 * q^69 + 4 * q^71 - 30 * q^72 + 12 * q^73 - 12 * q^74 - 28 * q^76 - 14 * q^77 + 8 * q^78 - 12 * q^79 + 2 * q^81 + 16 * q^82 + 6 * q^83 - 24 * q^84 + 4 * q^86 + 16 * q^87 - 34 * q^88 + 2 * q^91 - 4 * q^92 + 24 * q^93 + 6 * q^94 - 8 * q^96 - 4 * q^97 + 50 * q^99

## 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
 −1.41421 1.41421
−2.41421 2.82843 3.82843 0 −6.82843 −2.41421 −4.41421 5.00000 0
1.2 0.414214 −2.82843 −1.82843 0 −1.17157 0.414214 −1.58579 5.00000 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
$$5$$ $$-1$$
$$13$$ $$+1$$

## 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 325.2.a.f 2
3.b odd 2 1 2925.2.a.bd 2
4.b odd 2 1 5200.2.a.bt 2
5.b even 2 1 325.2.a.h yes 2
5.c odd 4 2 325.2.b.d 4
13.b even 2 1 4225.2.a.z 2
15.d odd 2 1 2925.2.a.w 2
15.e even 4 2 2925.2.c.q 4
20.d odd 2 1 5200.2.a.br 2
65.d even 2 1 4225.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.a.f 2 1.a even 1 1 trivial
325.2.a.h yes 2 5.b even 2 1
325.2.b.d 4 5.c odd 4 2
2925.2.a.w 2 15.d odd 2 1
2925.2.a.bd 2 3.b odd 2 1
2925.2.c.q 4 15.e even 4 2
4225.2.a.s 2 65.d even 2 1
4225.2.a.z 2 13.b even 2 1
5200.2.a.br 2 20.d odd 2 1
5200.2.a.bt 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(325))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{3}^{2} - 8$$ T3^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$T^{2} - 8$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2T - 1$$
$11$ $$T^{2} - 10T + 23$$
$13$ $$(T + 1)^{2}$$
$17$ $$T^{2} - 2T - 7$$
$19$ $$T^{2} - 4T - 28$$
$23$ $$T^{2} - 12T + 28$$
$29$ $$T^{2} + 6T + 1$$
$31$ $$T^{2} + 6T - 9$$
$37$ $$(T - 6)^{2}$$
$41$ $$T^{2} - 32$$
$43$ $$T^{2} - 4T - 4$$
$47$ $$T^{2} + 10T + 23$$
$53$ $$(T + 3)^{2}$$
$59$ $$T^{2} - 18T + 63$$
$61$ $$(T - 1)^{2}$$
$67$ $$T^{2} - 14T + 31$$
$71$ $$T^{2} - 4T - 124$$
$73$ $$(T - 6)^{2}$$
$79$ $$(T + 6)^{2}$$
$83$ $$T^{2} - 6T - 89$$
$89$ $$T^{2} - 72$$
$97$ $$T^{2} + 4T - 4$$