# Properties

 Label 325.2.c.e Level $325$ Weight $2$ Character orbit 325.c Analytic conductor $2.595$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(51,325)]

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, 1]))

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

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

chi := DirichletCharacter("325.51");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.c (of order $$2$$, degree $$1$$, minimal)

## 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: no Analytic conductor: $$2.59513806569$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 65) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
51.1
 − 1.00000i 1.00000i
1.00000i 2.00000 1.00000 0 2.00000i 0 3.00000i 1.00000 0
51.2 1.00000i 2.00000 1.00000 0 2.00000i 0 3.00000i 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.c.e 2
5.b even 2 1 325.2.c.b 2
5.c odd 4 1 65.2.d.a 2
5.c odd 4 1 65.2.d.b yes 2
13.b even 2 1 inner 325.2.c.e 2
13.d odd 4 1 4225.2.a.h 1
13.d odd 4 1 4225.2.a.m 1
15.e even 4 1 585.2.h.b 2
15.e even 4 1 585.2.h.c 2
20.e even 4 1 1040.2.f.a 2
20.e even 4 1 1040.2.f.b 2
65.d even 2 1 325.2.c.b 2
65.f even 4 1 845.2.b.a 2
65.f even 4 1 845.2.b.b 2
65.g odd 4 1 4225.2.a.e 1
65.g odd 4 1 4225.2.a.k 1
65.h odd 4 1 65.2.d.a 2
65.h odd 4 1 65.2.d.b yes 2
65.k even 4 1 845.2.b.a 2
65.k even 4 1 845.2.b.b 2
65.o even 12 2 845.2.n.a 4
65.o even 12 2 845.2.n.b 4
65.q odd 12 2 845.2.l.a 4
65.q odd 12 2 845.2.l.b 4
65.r odd 12 2 845.2.l.a 4
65.r odd 12 2 845.2.l.b 4
65.t even 12 2 845.2.n.a 4
65.t even 12 2 845.2.n.b 4
195.s even 4 1 585.2.h.b 2
195.s even 4 1 585.2.h.c 2
260.p even 4 1 1040.2.f.a 2
260.p even 4 1 1040.2.f.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.d.a 2 5.c odd 4 1
65.2.d.a 2 65.h odd 4 1
65.2.d.b yes 2 5.c odd 4 1
65.2.d.b yes 2 65.h odd 4 1
325.2.c.b 2 5.b even 2 1
325.2.c.b 2 65.d even 2 1
325.2.c.e 2 1.a even 1 1 trivial
325.2.c.e 2 13.b even 2 1 inner
585.2.h.b 2 15.e even 4 1
585.2.h.b 2 195.s even 4 1
585.2.h.c 2 15.e even 4 1
585.2.h.c 2 195.s even 4 1
845.2.b.a 2 65.f even 4 1
845.2.b.a 2 65.k even 4 1
845.2.b.b 2 65.f even 4 1
845.2.b.b 2 65.k even 4 1
845.2.l.a 4 65.q odd 12 2
845.2.l.a 4 65.r odd 12 2
845.2.l.b 4 65.q odd 12 2
845.2.l.b 4 65.r odd 12 2
845.2.n.a 4 65.o even 12 2
845.2.n.a 4 65.t even 12 2
845.2.n.b 4 65.o even 12 2
845.2.n.b 4 65.t even 12 2
1040.2.f.a 2 20.e even 4 1
1040.2.f.a 2 260.p even 4 1
1040.2.f.b 2 20.e even 4 1
1040.2.f.b 2 260.p even 4 1
4225.2.a.e 1 65.g odd 4 1
4225.2.a.h 1 13.d odd 4 1
4225.2.a.k 1 65.g odd 4 1
4225.2.a.m 1 13.d odd 4 1

## Hecke kernels

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

 $$T_{2}^{2} + 1$$ T2^2 + 1 $$T_{3} - 2$$ T3 - 2 $$T_{7}$$ T7

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 2)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 4$$
$13$ $$T^{2} + 4T + 13$$
$17$ $$T^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T - 6)^{2}$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} + 36$$
$37$ $$T^{2} + 36$$
$41$ $$T^{2} + 64$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T + 12)^{2}$$
$59$ $$T^{2} + 4$$
$61$ $$(T - 6)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2} + 4$$
$73$ $$T^{2} + 36$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 16$$
$89$ $$T^{2} + 64$$
$97$ $$T^{2} + 36$$