Properties

 Label 325.2.a.i 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: 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} + \beta q^{3} + (2 \beta + 1) q^{4} + (\beta + 2) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta + 3) q^{8} - q^{9}+O(q^{10})$$ q + (b + 1) * q^2 + b * q^3 + (2*b + 1) * q^4 + (b + 2) * q^6 + (-2*b - 2) * q^7 + (b + 3) * q^8 - q^9 $$q + (\beta + 1) q^{2} + \beta q^{3} + (2 \beta + 1) q^{4} + (\beta + 2) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta + 3) q^{8} - q^{9} + (\beta + 2) q^{11} + (\beta + 4) q^{12} + q^{13} + ( - 4 \beta - 6) q^{14} + 3 q^{16} + ( - 2 \beta + 2) q^{17} + ( - \beta - 1) q^{18} + ( - \beta + 2) q^{19} + ( - 2 \beta - 4) q^{21} + (3 \beta + 4) q^{22} - \beta q^{23} + (3 \beta + 2) q^{24} + (\beta + 1) q^{26} - 4 \beta q^{27} + ( - 6 \beta - 10) q^{28} - 4 \beta q^{29} + ( - 3 \beta + 6) q^{31} + (\beta - 3) q^{32} + (2 \beta + 2) q^{33} - 2 q^{34} + ( - 2 \beta - 1) q^{36} + 6 \beta q^{37} + \beta q^{38} + \beta q^{39} + (2 \beta - 6) q^{41} + ( - 6 \beta - 8) q^{42} + (5 \beta + 4) q^{43} + (5 \beta + 6) q^{44} + ( - \beta - 2) q^{46} + (2 \beta + 2) q^{47} + 3 \beta q^{48} + (8 \beta + 5) q^{49} + (2 \beta - 4) q^{51} + (2 \beta + 1) q^{52} + ( - 6 \beta + 6) q^{53} + ( - 4 \beta - 8) q^{54} + ( - 8 \beta - 10) q^{56} + (2 \beta - 2) q^{57} + ( - 4 \beta - 8) q^{58} + ( - 3 \beta + 6) q^{59} - 8 q^{61} + 3 \beta q^{62} + (2 \beta + 2) q^{63} + ( - 2 \beta - 7) q^{64} + (4 \beta + 6) q^{66} + 2 q^{67} + (2 \beta - 6) q^{68} - 2 q^{69} + (7 \beta + 2) q^{71} + ( - \beta - 3) q^{72} - 6 \beta q^{73} + (6 \beta + 12) q^{74} + (3 \beta - 2) q^{76} + ( - 6 \beta - 8) q^{77} + (\beta + 2) q^{78} - 6 \beta q^{79} - 5 q^{81} + ( - 4 \beta - 2) q^{82} + ( - 2 \beta + 6) q^{83} + ( - 10 \beta - 12) q^{84} + (9 \beta + 14) q^{86} - 8 q^{87} + (5 \beta + 8) q^{88} + 6 q^{89} + ( - 2 \beta - 2) q^{91} + ( - \beta - 4) q^{92} + (6 \beta - 6) q^{93} + (4 \beta + 6) q^{94} + ( - 3 \beta + 2) q^{96} + (4 \beta + 2) q^{97} + (13 \beta + 21) q^{98} + ( - \beta - 2) q^{99} +O(q^{100})$$ q + (b + 1) * q^2 + b * q^3 + (2*b + 1) * q^4 + (b + 2) * q^6 + (-2*b - 2) * q^7 + (b + 3) * q^8 - q^9 + (b + 2) * q^11 + (b + 4) * q^12 + q^13 + (-4*b - 6) * q^14 + 3 * q^16 + (-2*b + 2) * q^17 + (-b - 1) * q^18 + (-b + 2) * q^19 + (-2*b - 4) * q^21 + (3*b + 4) * q^22 - b * q^23 + (3*b + 2) * q^24 + (b + 1) * q^26 - 4*b * q^27 + (-6*b - 10) * q^28 - 4*b * q^29 + (-3*b + 6) * q^31 + (b - 3) * q^32 + (2*b + 2) * q^33 - 2 * q^34 + (-2*b - 1) * q^36 + 6*b * q^37 + b * q^38 + b * q^39 + (2*b - 6) * q^41 + (-6*b - 8) * q^42 + (5*b + 4) * q^43 + (5*b + 6) * q^44 + (-b - 2) * q^46 + (2*b + 2) * q^47 + 3*b * q^48 + (8*b + 5) * q^49 + (2*b - 4) * q^51 + (2*b + 1) * q^52 + (-6*b + 6) * q^53 + (-4*b - 8) * q^54 + (-8*b - 10) * q^56 + (2*b - 2) * q^57 + (-4*b - 8) * q^58 + (-3*b + 6) * q^59 - 8 * q^61 + 3*b * q^62 + (2*b + 2) * q^63 + (-2*b - 7) * q^64 + (4*b + 6) * q^66 + 2 * q^67 + (2*b - 6) * q^68 - 2 * q^69 + (7*b + 2) * q^71 + (-b - 3) * q^72 - 6*b * q^73 + (6*b + 12) * q^74 + (3*b - 2) * q^76 + (-6*b - 8) * q^77 + (b + 2) * q^78 - 6*b * q^79 - 5 * q^81 + (-4*b - 2) * q^82 + (-2*b + 6) * q^83 + (-10*b - 12) * q^84 + (9*b + 14) * q^86 - 8 * q^87 + (5*b + 8) * q^88 + 6 * q^89 + (-2*b - 2) * q^91 + (-b - 4) * q^92 + (6*b - 6) * q^93 + (4*b + 6) * q^94 + (-3*b + 2) * q^96 + (4*b + 2) * q^97 + (13*b + 21) * q^98 + (-b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9 $$2 q + 2 q^{2} + 2 q^{4} + 4 q^{6} - 4 q^{7} + 6 q^{8} - 2 q^{9} + 4 q^{11} + 8 q^{12} + 2 q^{13} - 12 q^{14} + 6 q^{16} + 4 q^{17} - 2 q^{18} + 4 q^{19} - 8 q^{21} + 8 q^{22} + 4 q^{24} + 2 q^{26} - 20 q^{28} + 12 q^{31} - 6 q^{32} + 4 q^{33} - 4 q^{34} - 2 q^{36} - 12 q^{41} - 16 q^{42} + 8 q^{43} + 12 q^{44} - 4 q^{46} + 4 q^{47} + 10 q^{49} - 8 q^{51} + 2 q^{52} + 12 q^{53} - 16 q^{54} - 20 q^{56} - 4 q^{57} - 16 q^{58} + 12 q^{59} - 16 q^{61} + 4 q^{63} - 14 q^{64} + 12 q^{66} + 4 q^{67} - 12 q^{68} - 4 q^{69} + 4 q^{71} - 6 q^{72} + 24 q^{74} - 4 q^{76} - 16 q^{77} + 4 q^{78} - 10 q^{81} - 4 q^{82} + 12 q^{83} - 24 q^{84} + 28 q^{86} - 16 q^{87} + 16 q^{88} + 12 q^{89} - 4 q^{91} - 8 q^{92} - 12 q^{93} + 12 q^{94} + 4 q^{96} + 4 q^{97} + 42 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^6 - 4 * q^7 + 6 * q^8 - 2 * q^9 + 4 * q^11 + 8 * q^12 + 2 * q^13 - 12 * q^14 + 6 * q^16 + 4 * q^17 - 2 * q^18 + 4 * q^19 - 8 * q^21 + 8 * q^22 + 4 * q^24 + 2 * q^26 - 20 * q^28 + 12 * q^31 - 6 * q^32 + 4 * q^33 - 4 * q^34 - 2 * q^36 - 12 * q^41 - 16 * q^42 + 8 * q^43 + 12 * q^44 - 4 * q^46 + 4 * q^47 + 10 * q^49 - 8 * q^51 + 2 * q^52 + 12 * q^53 - 16 * q^54 - 20 * q^56 - 4 * q^57 - 16 * q^58 + 12 * q^59 - 16 * q^61 + 4 * q^63 - 14 * q^64 + 12 * q^66 + 4 * q^67 - 12 * q^68 - 4 * q^69 + 4 * q^71 - 6 * q^72 + 24 * q^74 - 4 * q^76 - 16 * q^77 + 4 * q^78 - 10 * q^81 - 4 * q^82 + 12 * q^83 - 24 * q^84 + 28 * q^86 - 16 * q^87 + 16 * q^88 + 12 * q^89 - 4 * q^91 - 8 * q^92 - 12 * q^93 + 12 * q^94 + 4 * q^96 + 4 * q^97 + 42 * q^98 - 4 * 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
−0.414214 −1.41421 −1.82843 0 0.585786 0.828427 1.58579 −1.00000 0
1.2 2.41421 1.41421 3.82843 0 3.41421 −4.82843 4.41421 −1.00000 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.i 2
3.b odd 2 1 2925.2.a.u 2
4.b odd 2 1 5200.2.a.bu 2
5.b even 2 1 65.2.a.b 2
5.c odd 4 2 325.2.b.f 4
13.b even 2 1 4225.2.a.r 2
15.d odd 2 1 585.2.a.m 2
15.e even 4 2 2925.2.c.r 4
20.d odd 2 1 1040.2.a.j 2
35.c odd 2 1 3185.2.a.j 2
40.e odd 2 1 4160.2.a.z 2
40.f even 2 1 4160.2.a.bf 2
55.d odd 2 1 7865.2.a.j 2
60.h even 2 1 9360.2.a.cd 2
65.d even 2 1 845.2.a.g 2
65.g odd 4 2 845.2.c.b 4
65.l even 6 2 845.2.e.c 4
65.n even 6 2 845.2.e.h 4
65.s odd 12 4 845.2.m.f 8
195.e odd 2 1 7605.2.a.x 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.a.b 2 5.b even 2 1
325.2.a.i 2 1.a even 1 1 trivial
325.2.b.f 4 5.c odd 4 2
585.2.a.m 2 15.d odd 2 1
845.2.a.g 2 65.d even 2 1
845.2.c.b 4 65.g odd 4 2
845.2.e.c 4 65.l even 6 2
845.2.e.h 4 65.n even 6 2
845.2.m.f 8 65.s odd 12 4
1040.2.a.j 2 20.d odd 2 1
2925.2.a.u 2 3.b odd 2 1
2925.2.c.r 4 15.e even 4 2
3185.2.a.j 2 35.c odd 2 1
4160.2.a.z 2 40.e odd 2 1
4160.2.a.bf 2 40.f even 2 1
4225.2.a.r 2 13.b even 2 1
5200.2.a.bu 2 4.b odd 2 1
7605.2.a.x 2 195.e odd 2 1
7865.2.a.j 2 55.d odd 2 1
9360.2.a.cd 2 60.h even 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 $$T_{3}^{2} - 2$$ T3^2 - 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T - 1$$
$3$ $$T^{2} - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 4T - 4$$
$11$ $$T^{2} - 4T + 2$$
$13$ $$(T - 1)^{2}$$
$17$ $$T^{2} - 4T - 4$$
$19$ $$T^{2} - 4T + 2$$
$23$ $$T^{2} - 2$$
$29$ $$T^{2} - 32$$
$31$ $$T^{2} - 12T + 18$$
$37$ $$T^{2} - 72$$
$41$ $$T^{2} + 12T + 28$$
$43$ $$T^{2} - 8T - 34$$
$47$ $$T^{2} - 4T - 4$$
$53$ $$T^{2} - 12T - 36$$
$59$ $$T^{2} - 12T + 18$$
$61$ $$(T + 8)^{2}$$
$67$ $$(T - 2)^{2}$$
$71$ $$T^{2} - 4T - 94$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 72$$
$83$ $$T^{2} - 12T + 28$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} - 4T - 28$$