# Properties

 Label 325.2.m.a Level $325$ Weight $2$ Character orbit 325.m Analytic conductor $2.595$ Analytic rank $0$ Dimension $4$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(49,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

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

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

chi := DirichletCharacter("325.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.m (of order $$6$$, degree $$2$$, not 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 13) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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 primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

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

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$-1$$ $$1 - \zeta_{12}^{2}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
−0.866025 1.50000i −1.73205 + 1.00000i −0.500000 + 0.866025i 0 3.00000 + 1.73205i 0 −1.73205 0.500000 0.866025i 0
49.2 0.866025 + 1.50000i 1.73205 1.00000i −0.500000 + 0.866025i 0 3.00000 + 1.73205i 0 1.73205 0.500000 0.866025i 0
199.1 −0.866025 + 1.50000i −1.73205 1.00000i −0.500000 0.866025i 0 3.00000 1.73205i 0 −1.73205 0.500000 + 0.866025i 0
199.2 0.866025 1.50000i 1.73205 + 1.00000i −0.500000 0.866025i 0 3.00000 1.73205i 0 1.73205 0.500000 + 0.866025i 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
5.b even 2 1 inner
13.e even 6 1 inner
65.l even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.m.a 4
5.b even 2 1 inner 325.2.m.a 4
5.c odd 4 1 13.2.e.a 2
5.c odd 4 1 325.2.n.a 2
13.e even 6 1 inner 325.2.m.a 4
15.e even 4 1 117.2.q.c 2
20.e even 4 1 208.2.w.b 2
35.f even 4 1 637.2.q.a 2
35.k even 12 1 637.2.k.c 2
35.k even 12 1 637.2.u.b 2
35.l odd 12 1 637.2.k.a 2
35.l odd 12 1 637.2.u.c 2
40.i odd 4 1 832.2.w.d 2
40.k even 4 1 832.2.w.a 2
60.l odd 4 1 1872.2.by.d 2
65.f even 4 1 169.2.c.a 4
65.h odd 4 1 169.2.e.a 2
65.k even 4 1 169.2.c.a 4
65.l even 6 1 inner 325.2.m.a 4
65.o even 12 1 169.2.a.a 2
65.o even 12 1 169.2.c.a 4
65.o even 12 1 4225.2.a.v 2
65.q odd 12 1 169.2.b.a 2
65.q odd 12 1 169.2.e.a 2
65.r odd 12 1 13.2.e.a 2
65.r odd 12 1 169.2.b.a 2
65.r odd 12 1 325.2.n.a 2
65.t even 12 1 169.2.a.a 2
65.t even 12 1 169.2.c.a 4
65.t even 12 1 4225.2.a.v 2
195.bc odd 12 1 1521.2.a.k 2
195.bf even 12 1 117.2.q.c 2
195.bf even 12 1 1521.2.b.a 2
195.bl even 12 1 1521.2.b.a 2
195.bn odd 12 1 1521.2.a.k 2
260.be odd 12 1 2704.2.a.o 2
260.bg even 12 1 208.2.w.b 2
260.bg even 12 1 2704.2.f.b 2
260.bj even 12 1 2704.2.f.b 2
260.bl odd 12 1 2704.2.a.o 2
455.cf odd 12 1 8281.2.a.q 2
455.cr even 12 1 637.2.u.b 2
455.ct odd 12 1 637.2.u.c 2
455.cw even 12 1 637.2.k.c 2
455.cz even 12 1 637.2.q.a 2
455.da odd 12 1 637.2.k.a 2
455.ds odd 12 1 8281.2.a.q 2
520.co odd 12 1 832.2.w.d 2
520.cs even 12 1 832.2.w.a 2
780.cw odd 12 1 1872.2.by.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.2.e.a 2 5.c odd 4 1
13.2.e.a 2 65.r odd 12 1
117.2.q.c 2 15.e even 4 1
117.2.q.c 2 195.bf even 12 1
169.2.a.a 2 65.o even 12 1
169.2.a.a 2 65.t even 12 1
169.2.b.a 2 65.q odd 12 1
169.2.b.a 2 65.r odd 12 1
169.2.c.a 4 65.f even 4 1
169.2.c.a 4 65.k even 4 1
169.2.c.a 4 65.o even 12 1
169.2.c.a 4 65.t even 12 1
169.2.e.a 2 65.h odd 4 1
169.2.e.a 2 65.q odd 12 1
208.2.w.b 2 20.e even 4 1
208.2.w.b 2 260.bg even 12 1
325.2.m.a 4 1.a even 1 1 trivial
325.2.m.a 4 5.b even 2 1 inner
325.2.m.a 4 13.e even 6 1 inner
325.2.m.a 4 65.l even 6 1 inner
325.2.n.a 2 5.c odd 4 1
325.2.n.a 2 65.r odd 12 1
637.2.k.a 2 35.l odd 12 1
637.2.k.a 2 455.da odd 12 1
637.2.k.c 2 35.k even 12 1
637.2.k.c 2 455.cw even 12 1
637.2.q.a 2 35.f even 4 1
637.2.q.a 2 455.cz even 12 1
637.2.u.b 2 35.k even 12 1
637.2.u.b 2 455.cr even 12 1
637.2.u.c 2 35.l odd 12 1
637.2.u.c 2 455.ct odd 12 1
832.2.w.a 2 40.k even 4 1
832.2.w.a 2 520.cs even 12 1
832.2.w.d 2 40.i odd 4 1
832.2.w.d 2 520.co odd 12 1
1521.2.a.k 2 195.bc odd 12 1
1521.2.a.k 2 195.bn odd 12 1
1521.2.b.a 2 195.bf even 12 1
1521.2.b.a 2 195.bl even 12 1
1872.2.by.d 2 60.l odd 4 1
1872.2.by.d 2 780.cw odd 12 1
2704.2.a.o 2 260.be odd 12 1
2704.2.a.o 2 260.bl odd 12 1
2704.2.f.b 2 260.bg even 12 1
2704.2.f.b 2 260.bj even 12 1
4225.2.a.v 2 65.o even 12 1
4225.2.a.v 2 65.t even 12 1
8281.2.a.q 2 455.cf odd 12 1
8281.2.a.q 2 455.ds odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 3T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 3T^{2} + 9$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$T^{4}$$
$13$ $$T^{4} - T^{2} + 169$$
$17$ $$T^{4} - 9T^{2} + 81$$
$19$ $$(T^{2} - 6 T + 12)^{2}$$
$23$ $$T^{4} - 36T^{2} + 1296$$
$29$ $$(T^{2} - 3 T + 9)^{2}$$
$31$ $$(T^{2} + 12)^{2}$$
$37$ $$T^{4} + 75T^{2} + 5625$$
$41$ $$(T^{2} + 9 T + 27)^{2}$$
$43$ $$T^{4} - 64T^{2} + 4096$$
$47$ $$(T^{2} - 12)^{2}$$
$53$ $$(T^{2} + 9)^{2}$$
$59$ $$(T^{2} + 12 T + 48)^{2}$$
$61$ $$(T^{2} + T + 1)^{2}$$
$67$ $$T^{4} + 12T^{2} + 144$$
$71$ $$(T^{2} - 6 T + 12)^{2}$$
$73$ $$(T^{2} - 3)^{2}$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} - 192)^{2}$$
$89$ $$(T^{2} - 12 T + 48)^{2}$$
$97$ $$T^{4} + 48T^{2} + 2304$$