# Properties

 Label 325.2.a.k Level $325$ Weight $2$ Character orbit 325.a Self dual yes Analytic conductor $2.595$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{3} - x^{2} - 3x + 1$$ x^3 - x^2 - 3*x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta_{2} + 1) q^{2} + (\beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} + 2 \beta_1) q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9}+O(q^{10})$$ q + (b2 + 1) * q^2 + (b1 + 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 + 2*b1) * q^6 + (-b2 - b1 + 1) * q^7 + (-3*b1 + 4) * q^8 + (b2 + 3*b1) * q^9 $$q + (\beta_{2} + 1) q^{2} + (\beta_1 + 1) q^{3} + (\beta_{2} - \beta_1 + 2) q^{4} + (\beta_{2} + 2 \beta_1) q^{6} + ( - \beta_{2} - \beta_1 + 1) q^{7} + ( - 3 \beta_1 + 4) q^{8} + (\beta_{2} + 3 \beta_1) q^{9} + ( - \beta_{2} - 2) q^{11} + (\beta_1 - 1) q^{12} - q^{13} + (\beta_{2} - \beta_1 - 1) q^{14} + (2 \beta_{2} - 4 \beta_1 + 3) q^{16} + ( - 2 \beta_{2} + 2) q^{17} + 5 \beta_1 q^{18} - \beta_{2} q^{19} + ( - 2 \beta_{2} - 2 \beta_1) q^{21} + ( - 2 \beta_{2} + \beta_1 - 5) q^{22} + ( - \beta_1 + 5) q^{23} + ( - 3 \beta_{2} - 2 \beta_1 - 2) q^{24} + ( - \beta_{2} - 1) q^{26} + (4 \beta_{2} + 4 \beta_1 + 2) q^{27} + (\beta_{2} - \beta_1 + 1) q^{28} + ( - 3 \beta_{2} - 3 \beta_1 + 3) q^{29} + (\beta_{2} + 2 \beta_1 - 4) q^{31} + (3 \beta_{2} - 4 \beta_1 + 5) q^{32} + ( - \beta_{2} - 3 \beta_1 - 1) q^{33} + (2 \beta_{2} + 2 \beta_1 - 4) q^{34} + ( - 2 \beta_{2} + 4 \beta_1 - 5) q^{36} + ( - \beta_{2} - 3 \beta_1 + 1) q^{37} + (\beta_1 - 3) q^{38} + ( - \beta_1 - 1) q^{39} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + ( - 2 \beta_1 - 4) q^{42} + ( - 2 \beta_{2} + 3 \beta_1 + 1) q^{43} + ( - 3 \beta_{2} + 4 \beta_1 - 8) q^{44} + (5 \beta_{2} - 2 \beta_1 + 6) q^{46} + (\beta_{2} + \beta_1 + 3) q^{47} + ( - 2 \beta_{2} - 3 \beta_1 - 7) q^{48} + ( - 2 \beta_{2} - 3) q^{49} + ( - 2 \beta_{2} + 4) q^{51} + ( - \beta_{2} + \beta_1 - 2) q^{52} + (4 \beta_{2} + 2 \beta_1 + 2) q^{53} + (2 \beta_{2} + 4 \beta_1 + 10) q^{54} + ( - \beta_{2} - \beta_1 + 7) q^{56} + ( - \beta_{2} - \beta_1 + 1) q^{57} + (3 \beta_{2} - 3 \beta_1 - 3) q^{58} + (3 \beta_{2} - 2 \beta_1 - 2) q^{59} + (\beta_{2} + 3 \beta_1 + 1) q^{61} + ( - 4 \beta_{2} + 3 \beta_1 - 3) q^{62} + ( - \beta_{2} - 3 \beta_1 - 5) q^{63} + (\beta_{2} - 3 \beta_1 + 12) q^{64} + ( - \beta_{2} - 5 \beta_1 - 1) q^{66} + (5 \beta_{2} + \beta_1 + 3) q^{67} + (2 \beta_1 - 4) q^{68} + ( - \beta_{2} + 3 \beta_1 + 3) q^{69} + (\beta_{2} - 6 \beta_1 - 2) q^{71} + ( - 5 \beta_{2} - 15) q^{72} + (\beta_{2} + 3 \beta_1 - 9) q^{73} + (\beta_{2} - 5 \beta_1 + 1) q^{74} + ( - \beta_{2} + 2 \beta_1 - 4) q^{76} + 2 \beta_1 q^{77} + ( - \beta_{2} - 2 \beta_1) q^{78} + (4 \beta_{2} + 2 \beta_1 - 6) q^{79} + (5 \beta_{2} + 5 \beta_1 + 6) q^{81} + ( - 2 \beta_{2} + 6 \beta_1 - 10) q^{82} + ( - \beta_{2} + \beta_1 + 7) q^{83} - 2 q^{84} + (\beta_{2} + 8 \beta_1 - 8) q^{86} + ( - 6 \beta_{2} - 6 \beta_1) q^{87} + ( - 4 \beta_{2} + 9 \beta_1 - 11) q^{88} + (4 \beta_{2} + 4 \beta_1 + 2) q^{89} + (\beta_{2} + \beta_1 - 1) q^{91} + (6 \beta_{2} - 7 \beta_1 + 13) q^{92} + (3 \beta_{2} + \beta_1 - 1) q^{93} + (3 \beta_{2} + \beta_1 + 5) q^{94} + ( - \beta_{2} - 6) q^{96} + (6 \beta_{2} + 4 \beta_1 - 6) q^{97} + ( - 3 \beta_{2} + 2 \beta_1 - 9) q^{98} + ( - \beta_{2} - 8 \beta_1) q^{99}+O(q^{100})$$ q + (b2 + 1) * q^2 + (b1 + 1) * q^3 + (b2 - b1 + 2) * q^4 + (b2 + 2*b1) * q^6 + (-b2 - b1 + 1) * q^7 + (-3*b1 + 4) * q^8 + (b2 + 3*b1) * q^9 + (-b2 - 2) * q^11 + (b1 - 1) * q^12 - q^13 + (b2 - b1 - 1) * q^14 + (2*b2 - 4*b1 + 3) * q^16 + (-2*b2 + 2) * q^17 + 5*b1 * q^18 - b2 * q^19 + (-2*b2 - 2*b1) * q^21 + (-2*b2 + b1 - 5) * q^22 + (-b1 + 5) * q^23 + (-3*b2 - 2*b1 - 2) * q^24 + (-b2 - 1) * q^26 + (4*b2 + 4*b1 + 2) * q^27 + (b2 - b1 + 1) * q^28 + (-3*b2 - 3*b1 + 3) * q^29 + (b2 + 2*b1 - 4) * q^31 + (3*b2 - 4*b1 + 5) * q^32 + (-b2 - 3*b1 - 1) * q^33 + (2*b2 + 2*b1 - 4) * q^34 + (-2*b2 + 4*b1 - 5) * q^36 + (-b2 - 3*b1 + 1) * q^37 + (b1 - 3) * q^38 + (-b1 - 1) * q^39 + (-2*b2 + 2*b1 - 2) * q^41 + (-2*b1 - 4) * q^42 + (-2*b2 + 3*b1 + 1) * q^43 + (-3*b2 + 4*b1 - 8) * q^44 + (5*b2 - 2*b1 + 6) * q^46 + (b2 + b1 + 3) * q^47 + (-2*b2 - 3*b1 - 7) * q^48 + (-2*b2 - 3) * q^49 + (-2*b2 + 4) * q^51 + (-b2 + b1 - 2) * q^52 + (4*b2 + 2*b1 + 2) * q^53 + (2*b2 + 4*b1 + 10) * q^54 + (-b2 - b1 + 7) * q^56 + (-b2 - b1 + 1) * q^57 + (3*b2 - 3*b1 - 3) * q^58 + (3*b2 - 2*b1 - 2) * q^59 + (b2 + 3*b1 + 1) * q^61 + (-4*b2 + 3*b1 - 3) * q^62 + (-b2 - 3*b1 - 5) * q^63 + (b2 - 3*b1 + 12) * q^64 + (-b2 - 5*b1 - 1) * q^66 + (5*b2 + b1 + 3) * q^67 + (2*b1 - 4) * q^68 + (-b2 + 3*b1 + 3) * q^69 + (b2 - 6*b1 - 2) * q^71 + (-5*b2 - 15) * q^72 + (b2 + 3*b1 - 9) * q^73 + (b2 - 5*b1 + 1) * q^74 + (-b2 + 2*b1 - 4) * q^76 + 2*b1 * q^77 + (-b2 - 2*b1) * q^78 + (4*b2 + 2*b1 - 6) * q^79 + (5*b2 + 5*b1 + 6) * q^81 + (-2*b2 + 6*b1 - 10) * q^82 + (-b2 + b1 + 7) * q^83 - 2 * q^84 + (b2 + 8*b1 - 8) * q^86 + (-6*b2 - 6*b1) * q^87 + (-4*b2 + 9*b1 - 11) * q^88 + (4*b2 + 4*b1 + 2) * q^89 + (b2 + b1 - 1) * q^91 + (6*b2 - 7*b1 + 13) * q^92 + (3*b2 + b1 - 1) * q^93 + (3*b2 + b1 + 5) * q^94 + (-b2 - 6) * q^96 + (6*b2 + 4*b1 - 6) * q^97 + (-3*b2 + 2*b1 - 9) * q^98 + (-b2 - 8*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^2 + 4 * q^3 + 5 * q^4 + 2 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 $$3 q + 3 q^{2} + 4 q^{3} + 5 q^{4} + 2 q^{6} + 2 q^{7} + 9 q^{8} + 3 q^{9} - 6 q^{11} - 2 q^{12} - 3 q^{13} - 4 q^{14} + 5 q^{16} + 6 q^{17} + 5 q^{18} - 2 q^{21} - 14 q^{22} + 14 q^{23} - 8 q^{24} - 3 q^{26} + 10 q^{27} + 2 q^{28} + 6 q^{29} - 10 q^{31} + 11 q^{32} - 6 q^{33} - 10 q^{34} - 11 q^{36} - 8 q^{38} - 4 q^{39} - 4 q^{41} - 14 q^{42} + 6 q^{43} - 20 q^{44} + 16 q^{46} + 10 q^{47} - 24 q^{48} - 9 q^{49} + 12 q^{51} - 5 q^{52} + 8 q^{53} + 34 q^{54} + 20 q^{56} + 2 q^{57} - 12 q^{58} - 8 q^{59} + 6 q^{61} - 6 q^{62} - 18 q^{63} + 33 q^{64} - 8 q^{66} + 10 q^{67} - 10 q^{68} + 12 q^{69} - 12 q^{71} - 45 q^{72} - 24 q^{73} - 2 q^{74} - 10 q^{76} + 2 q^{77} - 2 q^{78} - 16 q^{79} + 23 q^{81} - 24 q^{82} + 22 q^{83} - 6 q^{84} - 16 q^{86} - 6 q^{87} - 24 q^{88} + 10 q^{89} - 2 q^{91} + 32 q^{92} - 2 q^{93} + 16 q^{94} - 18 q^{96} - 14 q^{97} - 25 q^{98} - 8 q^{99}+O(q^{100})$$ 3 * q + 3 * q^2 + 4 * q^3 + 5 * q^4 + 2 * q^6 + 2 * q^7 + 9 * q^8 + 3 * q^9 - 6 * q^11 - 2 * q^12 - 3 * q^13 - 4 * q^14 + 5 * q^16 + 6 * q^17 + 5 * q^18 - 2 * q^21 - 14 * q^22 + 14 * q^23 - 8 * q^24 - 3 * q^26 + 10 * q^27 + 2 * q^28 + 6 * q^29 - 10 * q^31 + 11 * q^32 - 6 * q^33 - 10 * q^34 - 11 * q^36 - 8 * q^38 - 4 * q^39 - 4 * q^41 - 14 * q^42 + 6 * q^43 - 20 * q^44 + 16 * q^46 + 10 * q^47 - 24 * q^48 - 9 * q^49 + 12 * q^51 - 5 * q^52 + 8 * q^53 + 34 * q^54 + 20 * q^56 + 2 * q^57 - 12 * q^58 - 8 * q^59 + 6 * q^61 - 6 * q^62 - 18 * q^63 + 33 * q^64 - 8 * q^66 + 10 * q^67 - 10 * q^68 + 12 * q^69 - 12 * q^71 - 45 * q^72 - 24 * q^73 - 2 * q^74 - 10 * q^76 + 2 * q^77 - 2 * q^78 - 16 * q^79 + 23 * q^81 - 24 * q^82 + 22 * q^83 - 6 * q^84 - 16 * q^86 - 6 * q^87 - 24 * q^88 + 10 * q^89 - 2 * q^91 + 32 * q^92 - 2 * q^93 + 16 * q^94 - 18 * q^96 - 14 * q^97 - 25 * q^98 - 8 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ v^2 - v - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 2$$ b2 + b1 + 2

## 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
 0.311108 2.17009 −1.48119
−1.21432 1.31111 −0.525428 0 −1.59210 2.90321 3.06668 −1.28100 0
1.2 1.53919 3.17009 0.369102 0 4.87936 −1.70928 −2.51026 7.04945 0
1.3 2.67513 −0.481194 5.15633 0 −1.28726 0.806063 8.44358 −2.76845 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.k 3
3.b odd 2 1 2925.2.a.bf 3
4.b odd 2 1 5200.2.a.cb 3
5.b even 2 1 325.2.a.j 3
5.c odd 4 2 65.2.b.a 6
13.b even 2 1 4225.2.a.ba 3
15.d odd 2 1 2925.2.a.bj 3
15.e even 4 2 585.2.c.b 6
20.d odd 2 1 5200.2.a.cj 3
20.e even 4 2 1040.2.d.c 6
65.d even 2 1 4225.2.a.bh 3
65.f even 4 2 845.2.d.a 6
65.h odd 4 2 845.2.b.c 6
65.k even 4 2 845.2.d.b 6
65.o even 12 4 845.2.l.d 12
65.q odd 12 4 845.2.n.f 12
65.r odd 12 4 845.2.n.g 12
65.t even 12 4 845.2.l.e 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.b.a 6 5.c odd 4 2
325.2.a.j 3 5.b even 2 1
325.2.a.k 3 1.a even 1 1 trivial
585.2.c.b 6 15.e even 4 2
845.2.b.c 6 65.h odd 4 2
845.2.d.a 6 65.f even 4 2
845.2.d.b 6 65.k even 4 2
845.2.l.d 12 65.o even 12 4
845.2.l.e 12 65.t even 12 4
845.2.n.f 12 65.q odd 12 4
845.2.n.g 12 65.r odd 12 4
1040.2.d.c 6 20.e even 4 2
2925.2.a.bf 3 3.b odd 2 1
2925.2.a.bj 3 15.d odd 2 1
4225.2.a.ba 3 13.b even 2 1
4225.2.a.bh 3 65.d even 2 1
5200.2.a.cb 3 4.b odd 2 1
5200.2.a.cj 3 20.d odd 2 1

## Hecke kernels

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

 $$T_{2}^{3} - 3T_{2}^{2} - T_{2} + 5$$ T2^3 - 3*T2^2 - T2 + 5 $$T_{3}^{3} - 4T_{3}^{2} + 2T_{3} + 2$$ T3^3 - 4*T3^2 + 2*T3 + 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3} - 3T^{2} - T + 5$$
$3$ $$T^{3} - 4 T^{2} + \cdots + 2$$
$5$ $$T^{3}$$
$7$ $$T^{3} - 2 T^{2} + \cdots + 4$$
$11$ $$T^{3} + 6 T^{2} + \cdots - 2$$
$13$ $$(T + 1)^{3}$$
$17$ $$T^{3} - 6 T^{2} + \cdots + 8$$
$19$ $$T^{3} - 4T - 2$$
$23$ $$T^{3} - 14 T^{2} + \cdots - 86$$
$29$ $$T^{3} - 6 T^{2} + \cdots + 108$$
$31$ $$T^{3} + 10 T^{2} + \cdots - 26$$
$37$ $$T^{3} - 28T + 52$$
$41$ $$T^{3} + 4 T^{2} + \cdots + 32$$
$43$ $$T^{3} - 6 T^{2} + \cdots + 278$$
$47$ $$T^{3} - 10 T^{2} + \cdots - 20$$
$53$ $$T^{3} - 8 T^{2} + \cdots + 304$$
$59$ $$T^{3} + 8 T^{2} + \cdots - 262$$
$61$ $$T^{3} - 6 T^{2} + \cdots - 4$$
$67$ $$T^{3} - 10 T^{2} + \cdots + 604$$
$71$ $$T^{3} + 12 T^{2} + \cdots - 754$$
$73$ $$T^{3} + 24 T^{2} + \cdots + 236$$
$79$ $$T^{3} + 16 T^{2} + \cdots - 16$$
$83$ $$T^{3} - 22 T^{2} + \cdots - 316$$
$89$ $$T^{3} - 10 T^{2} + \cdots + 200$$
$97$ $$T^{3} + 14 T^{2} + \cdots - 200$$