# Properties

 Label 325.2.d.f Level $325$ Weight $2$ Character orbit 325.d Analytic conductor $2.595$ Analytic rank $0$ Dimension $6$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(324,325)]

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([1, 1]))

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

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

chi := DirichletCharacter("325.324");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.d (of order $$2$$, degree $$1$$, 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: $$6$$ Coefficient field: 6.0.5089536.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18$$ x^6 - 2*x^5 + 2*x^4 + 2*x^3 + 16*x^2 - 24*x + 18 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{3} + 2) q^{4} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{6} + (\beta_{3} - 1) q^{7} + ( - \beta_{3} + \beta_1 + 1) q^{8} + (\beta_{3} + 2 \beta_1 - 2) q^{9}+O(q^{10})$$ q + b1 * q^2 - b5 * q^3 + (-b3 + 2) * q^4 + (b5 - b4 - b2) * q^6 + (b3 - 1) * q^7 + (-b3 + b1 + 1) * q^8 + (b3 + 2*b1 - 2) * q^9 $$q + \beta_1 q^{2} - \beta_{5} q^{3} + ( - \beta_{3} + 2) q^{4} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{6} + (\beta_{3} - 1) q^{7} + ( - \beta_{3} + \beta_1 + 1) q^{8} + (\beta_{3} + 2 \beta_1 - 2) q^{9} + ( - \beta_{5} + \beta_{4} - \beta_{2}) q^{11} + ( - 3 \beta_{5} - \beta_{4}) q^{12} + (\beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{13}+ \cdots + (7 \beta_{5} + 2 \beta_{4} - 5 \beta_{2}) q^{99}+O(q^{100})$$ q + b1 * q^2 - b5 * q^3 + (-b3 + 2) * q^4 + (b5 - b4 - b2) * q^6 + (b3 - 1) * q^7 + (-b3 + b1 + 1) * q^8 + (b3 + 2*b1 - 2) * q^9 + (-b5 + b4 - b2) * q^11 + (-3*b5 - b4) * q^12 + (b5 + b4 + b3 + b1 - 1) * q^13 + (b3 - 2*b1 - 1) * q^14 + (2*b1 + 1) * q^16 + 2*b2 * q^17 + (-b3 - 3*b1 + 7) * q^18 + (b5 + 2*b4 - b2) * q^19 + (2*b5 + b4) * q^21 + (b5 - b4 - 2*b2) * q^22 + (b5 - b4) * q^23 + (-b5 - 2*b4 - b2) * q^24 + (b5 + 2*b4 + b2 - 2*b1 + 3) * q^26 + (2*b5 - b4 - 2*b2) * q^27 + (b3 - 2*b1 - 7) * q^28 + (-b3 + 1) * q^29 + (b5 - b4 - b2) * q^31 + (-b1 + 6) * q^32 + (b3 + 2*b1 - 7) * q^33 + (4*b5 + 2*b4 + 2*b2) * q^34 + (4*b1 - 7) * q^36 + (b3 - 7) * q^37 + (b5 + 2*b4) * q^38 + (3*b5 - b3 - b2 + 3) * q^39 + (-2*b5 - b4 + 2*b2) * q^41 + (3*b4 + 2*b2) * q^42 + (-b5 - 3*b4 + 2*b2) * q^43 + (-5*b5 - 4*b4 + b2) * q^44 + (-3*b5 + b2) * q^46 + (-b3 + 1) * q^47 + (b5 - 2*b4 - 2*b2) * q^48 + (2*b1 - 1) * q^49 + 4*b1 * q^51 + (3*b5 + 2*b4 + 2*b2 + b1 - 6) * q^52 + (-2*b5 - b4 - 2*b2) * q^53 + (-8*b5 - b4) * q^54 + (b3 - 4*b1 - 7) * q^56 + (-b3 + 1) * q^57 + (-b3 + 2*b1 + 1) * q^58 + (b5 + 2*b4 + b2) * q^59 + (-b3 - 2*b1 - 1) * q^61 + (-5*b5 - b4) * q^62 + (b3 - 2*b1 + 5) * q^63 + (b3 + 2*b1 - 6) * q^64 + (-b3 - 8*b1 + 7) * q^66 + (-b3 - 2*b1 + 7) * q^67 + (4*b5 + 8*b4 + 2*b2) * q^68 + (-b3 - 4*b1 + 7) * q^69 + (-b5 - 2*b4 + b2) * q^71 + (-2*b3 - b1 + 2) * q^72 + (b3 + 4*b1 + 5) * q^73 + (b3 - 8*b1 - 1) * q^74 + (b5 - b4 + 3*b2) * q^76 + (4*b5 + 5*b4 - 2*b2) * q^77 + (-5*b5 + 2*b4 - b3 + 2*b2 + 4*b1 + 1) * q^78 + (2*b3 + 4*b1 - 6) * q^79 + (b3 - 4*b1 + 6) * q^81 + (4*b5 - b4) * q^82 + (-b3 - 2*b1 - 5) * q^83 + (6*b5 + 3*b4 + 2*b2) * q^84 + (-b5 - 2*b4 + b2) * q^86 + (-2*b5 - b4) * q^87 + (-3*b5 - 6*b4) * q^88 + (-4*b5 - 2*b4 + 2*b2) * q^89 + (-2*b5 - b4 + b3 - 2*b2 + 5) * q^91 + (3*b5 - 2*b2) * q^92 + (-b3 - 6*b1 + 7) * q^93 + (-b3 + 2*b1 + 1) * q^94 + (-7*b5 + b4 + b2) * q^96 + (-2*b3 - 4) * q^97 + (-2*b3 - b1 + 8) * q^98 + (7*b5 + 2*b4 - 5*b2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 2 q^{2} + 10 q^{4} - 4 q^{7} + 6 q^{8} - 6 q^{9}+O(q^{10})$$ 6 * q + 2 * q^2 + 10 * q^4 - 4 * q^7 + 6 * q^8 - 6 * q^9 $$6 q + 2 q^{2} + 10 q^{4} - 4 q^{7} + 6 q^{8} - 6 q^{9} - 2 q^{13} - 8 q^{14} + 10 q^{16} + 34 q^{18} + 14 q^{26} - 44 q^{28} + 4 q^{29} + 34 q^{32} - 36 q^{33} - 34 q^{36} - 40 q^{37} + 16 q^{39} + 4 q^{47} - 2 q^{49} + 8 q^{51} - 34 q^{52} - 48 q^{56} + 4 q^{57} + 8 q^{58} - 12 q^{61} + 28 q^{63} - 30 q^{64} + 24 q^{66} + 36 q^{67} + 32 q^{69} + 6 q^{72} + 40 q^{73} - 20 q^{74} + 12 q^{78} - 24 q^{79} + 30 q^{81} - 36 q^{83} + 32 q^{91} + 28 q^{93} + 8 q^{94} - 28 q^{97} + 42 q^{98}+O(q^{100})$$ 6 * q + 2 * q^2 + 10 * q^4 - 4 * q^7 + 6 * q^8 - 6 * q^9 - 2 * q^13 - 8 * q^14 + 10 * q^16 + 34 * q^18 + 14 * q^26 - 44 * q^28 + 4 * q^29 + 34 * q^32 - 36 * q^33 - 34 * q^36 - 40 * q^37 + 16 * q^39 + 4 * q^47 - 2 * q^49 + 8 * q^51 - 34 * q^52 - 48 * q^56 + 4 * q^57 + 8 * q^58 - 12 * q^61 + 28 * q^63 - 30 * q^64 + 24 * q^66 + 36 * q^67 + 32 * q^69 + 6 * q^72 + 40 * q^73 - 20 * q^74 + 12 * q^78 - 24 * q^79 + 30 * q^81 - 36 * q^83 + 32 * q^91 + 28 * q^93 + 8 * q^94 - 28 * q^97 + 42 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 16x^{2} - 24x + 18$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} - 24\nu^{4} + 6\nu^{3} + \nu^{2} - 6\nu - 285 ) / 131$$ (v^5 - 24*v^4 + 6*v^3 + v^2 - 6*v - 285) / 131 $$\beta_{2}$$ $$=$$ $$( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} + 226\nu - 138 ) / 131$$ (6*v^5 - 13*v^4 + 36*v^3 + 6*v^2 + 226*v - 138) / 131 $$\beta_{3}$$ $$=$$ $$( 6\nu^{5} - 13\nu^{4} + 36\nu^{3} + 6\nu^{2} - 36\nu - 7 ) / 131$$ (6*v^5 - 13*v^4 + 36*v^3 + 6*v^2 - 36*v - 7) / 131 $$\beta_{4}$$ $$=$$ $$( 46\nu^{5} - 56\nu^{4} + 14\nu^{3} + 308\nu^{2} + 772\nu - 534 ) / 393$$ (46*v^5 - 56*v^4 + 14*v^3 + 308*v^2 + 772*v - 534) / 393 $$\beta_{5}$$ $$=$$ $$( -92\nu^{5} + 112\nu^{4} - 28\nu^{3} - 223\nu^{2} - 1544\nu + 1068 ) / 393$$ (-92*v^5 + 112*v^4 - 28*v^3 - 223*v^2 - 1544*v + 1068) / 393
 $$\nu$$ $$=$$ $$( -\beta_{3} + \beta_{2} + 1 ) / 2$$ (-b3 + b2 + 1) / 2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{4}$$ b5 + 2*b4 $$\nu^{3}$$ $$=$$ $$( 2\beta_{5} + \beta_{4} + 4\beta_{3} + 4\beta_{2} - 2\beta _1 - 4 ) / 2$$ (2*b5 + b4 + 4*b3 + 4*b2 - 2*b1 - 4) / 2 $$\nu^{4}$$ $$=$$ $$\beta_{3} - 6\beta _1 - 13$$ b3 - 6*b1 - 13 $$\nu^{5}$$ $$=$$ $$-7\beta_{5} - 5\beta_{4} + 9\beta_{3} - 9\beta_{2} - 7\beta _1 - 12$$ -7*b5 - 5*b4 + 9*b3 - 9*b2 - 7*b1 - 12

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
324.1
 0.675970 − 0.675970i 0.675970 + 0.675970i −1.33641 + 1.33641i −1.33641 − 1.33641i 1.66044 + 1.66044i 1.66044 − 1.66044i
−2.08613 3.08613i 2.35194 0 6.43807i −1.35194 −0.734191 −6.52420 0
324.2 −2.08613 3.08613i 2.35194 0 6.43807i −1.35194 −0.734191 −6.52420 0
324.3 0.571993 0.428007i −1.67282 0 0.244817i 2.67282 −2.10083 2.81681 0
324.4 0.571993 0.428007i −1.67282 0 0.244817i 2.67282 −2.10083 2.81681 0
324.5 2.51414 1.51414i 4.32088 0 3.80675i −3.32088 5.83502 0.707389 0
324.6 2.51414 1.51414i 4.32088 0 3.80675i −3.32088 5.83502 0.707389 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 324.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.d 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.d.f 6
5.b even 2 1 325.2.d.e 6
5.c odd 4 1 65.2.c.a 6
5.c odd 4 1 325.2.c.g 6
13.b even 2 1 325.2.d.e 6
15.e even 4 1 585.2.b.g 6
20.e even 4 1 1040.2.k.d 6
65.d even 2 1 inner 325.2.d.f 6
65.f even 4 1 845.2.a.i 3
65.f even 4 1 4225.2.a.bc 3
65.h odd 4 1 65.2.c.a 6
65.h odd 4 1 325.2.c.g 6
65.k even 4 1 845.2.a.k 3
65.k even 4 1 4225.2.a.be 3
65.o even 12 2 845.2.e.i 6
65.q odd 12 2 845.2.m.h 12
65.r odd 12 2 845.2.m.h 12
65.t even 12 2 845.2.e.k 6
195.j odd 4 1 7605.2.a.bs 3
195.s even 4 1 585.2.b.g 6
195.u odd 4 1 7605.2.a.cc 3
260.p even 4 1 1040.2.k.d 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.c.a 6 5.c odd 4 1
65.2.c.a 6 65.h odd 4 1
325.2.c.g 6 5.c odd 4 1
325.2.c.g 6 65.h odd 4 1
325.2.d.e 6 5.b even 2 1
325.2.d.e 6 13.b even 2 1
325.2.d.f 6 1.a even 1 1 trivial
325.2.d.f 6 65.d even 2 1 inner
585.2.b.g 6 15.e even 4 1
585.2.b.g 6 195.s even 4 1
845.2.a.i 3 65.f even 4 1
845.2.a.k 3 65.k even 4 1
845.2.e.i 6 65.o even 12 2
845.2.e.k 6 65.t even 12 2
845.2.m.h 12 65.q odd 12 2
845.2.m.h 12 65.r odd 12 2
1040.2.k.d 6 20.e even 4 1
1040.2.k.d 6 260.p even 4 1
4225.2.a.bc 3 65.f even 4 1
4225.2.a.be 3 65.k even 4 1
7605.2.a.bs 3 195.j odd 4 1
7605.2.a.cc 3 195.u odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - T_{2}^{2} - 5T_{2} + 3$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{3} - T^{2} - 5 T + 3)^{2}$$
$3$ $$T^{6} + 12 T^{4} + \cdots + 4$$
$5$ $$T^{6}$$
$7$ $$(T^{3} + 2 T^{2} - 8 T - 12)^{2}$$
$11$ $$T^{6} + 48 T^{4} + \cdots + 2916$$
$13$ $$T^{6} + 2 T^{5} + \cdots + 2197$$
$17$ $$T^{6} + 80 T^{4} + \cdots + 9216$$
$19$ $$T^{6} + 44 T^{4} + \cdots + 36$$
$23$ $$T^{6} + 32 T^{4} + \cdots + 36$$
$29$ $$(T^{3} - 2 T^{2} - 8 T + 12)^{2}$$
$31$ $$T^{6} + 56 T^{4} + \cdots + 36$$
$37$ $$(T^{3} + 20 T^{2} + \cdots + 228)^{2}$$
$41$ $$T^{6} + 92 T^{4} + \cdots + 5184$$
$43$ $$T^{6} + 120 T^{4} + \cdots + 4$$
$47$ $$(T^{3} - 2 T^{2} - 8 T + 12)^{2}$$
$53$ $$T^{6} + 156 T^{4} + \cdots + 5184$$
$59$ $$T^{6} + 84 T^{4} + \cdots + 324$$
$61$ $$(T^{3} + 6 T^{2} - 12 T - 76)^{2}$$
$67$ $$(T^{3} - 18 T^{2} + \cdots - 108)^{2}$$
$71$ $$T^{6} + 44 T^{4} + \cdots + 36$$
$73$ $$(T^{3} - 20 T^{2} + \cdots + 516)^{2}$$
$79$ $$(T^{3} + 12 T^{2} + \cdots - 32)^{2}$$
$83$ $$(T^{3} + 18 T^{2} + \cdots + 36)^{2}$$
$89$ $$T^{6} + 192 T^{4} + \cdots + 82944$$
$97$ $$(T^{3} + 14 T^{2} + \cdots - 24)^{2}$$