Properties

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

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(274,325)]

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, 0]))

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.b (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: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{2}$$ v^2 $$\beta_{2}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}$$ v^3 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}$$ -v^3 + v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} + \beta_{2} ) / 2$$ (b3 + b2) / 2 $$\zeta_{8}^{2}$$ $$=$$ $$\beta_1$$ b1 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} + \beta_{2} ) / 2$$ (-b3 + b2) / 2

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
274.1
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 1.41421i −3.82843 0 3.41421 4.82843i 4.41421i 1.00000 0
274.2 0.414214i 1.41421i 1.82843 0 0.585786 0.828427i 1.58579i 1.00000 0
274.3 0.414214i 1.41421i 1.82843 0 0.585786 0.828427i 1.58579i 1.00000 0
274.4 2.41421i 1.41421i −3.82843 0 3.41421 4.82843i 4.41421i 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
5.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.b.f 4
3.b odd 2 1 2925.2.c.r 4
5.b even 2 1 inner 325.2.b.f 4
5.c odd 4 1 65.2.a.b 2
5.c odd 4 1 325.2.a.i 2
15.d odd 2 1 2925.2.c.r 4
15.e even 4 1 585.2.a.m 2
15.e even 4 1 2925.2.a.u 2
20.e even 4 1 1040.2.a.j 2
20.e even 4 1 5200.2.a.bu 2
35.f even 4 1 3185.2.a.j 2
40.i odd 4 1 4160.2.a.bf 2
40.k even 4 1 4160.2.a.z 2
55.e even 4 1 7865.2.a.j 2
60.l odd 4 1 9360.2.a.cd 2
65.f even 4 1 845.2.c.b 4
65.h odd 4 1 845.2.a.g 2
65.h odd 4 1 4225.2.a.r 2
65.k even 4 1 845.2.c.b 4
65.o even 12 2 845.2.m.f 8
65.q odd 12 2 845.2.e.h 4
65.r odd 12 2 845.2.e.c 4
65.t even 12 2 845.2.m.f 8
195.s even 4 1 7605.2.a.x 2

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

Hecke kernels

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

 $$T_{2}^{4} + 6T_{2}^{2} + 1$$ T2^4 + 6*T2^2 + 1 $$T_{3}^{2} + 2$$ T3^2 + 2

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$(T^{2} + 2)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24T^{2} + 16$$
$11$ $$(T^{2} - 4 T + 2)^{2}$$
$13$ $$(T^{2} + 1)^{2}$$
$17$ $$T^{4} + 24T^{2} + 16$$
$19$ $$(T^{2} + 4 T + 2)^{2}$$
$23$ $$(T^{2} + 2)^{2}$$
$29$ $$(T^{2} - 32)^{2}$$
$31$ $$(T^{2} - 12 T + 18)^{2}$$
$37$ $$(T^{2} + 72)^{2}$$
$41$ $$(T^{2} + 12 T + 28)^{2}$$
$43$ $$T^{4} + 132T^{2} + 1156$$
$47$ $$T^{4} + 24T^{2} + 16$$
$53$ $$T^{4} + 216T^{2} + 1296$$
$59$ $$(T^{2} + 12 T + 18)^{2}$$
$61$ $$(T + 8)^{4}$$
$67$ $$(T^{2} + 4)^{2}$$
$71$ $$(T^{2} - 4 T - 94)^{2}$$
$73$ $$(T^{2} + 72)^{2}$$
$79$ $$(T^{2} - 72)^{2}$$
$83$ $$T^{4} + 88T^{2} + 784$$
$89$ $$(T + 6)^{4}$$
$97$ $$T^{4} + 72T^{2} + 784$$