Properties

 Label 325.4.c.b Level $325$ Weight $4$ Character orbit 325.c Analytic conductor $19.176$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,4,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, 4, names="a")

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

chi := DirichletCharacter("325.51");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$19.1756207519$$ 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: $$3$$ Twist minimal: no (minimal twist has level 13) 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.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{3} - q^{4} + i q^{6} - 5 i q^{7} + 7 i q^{8} - 26 q^{9} +O(q^{10})$$ q + i * q^2 + q^3 - q^4 + i * q^6 - 5*i * q^7 + 7*i * q^8 - 26 * q^9 $$q + i q^{2} + q^{3} - q^{4} + i q^{6} - 5 i q^{7} + 7 i q^{8} - 26 q^{9} - 16 i q^{11} - q^{12} + ( - 13 i - 26) q^{13} + 45 q^{14} - 71 q^{16} + 45 q^{17} - 26 i q^{18} - 2 i q^{19} - 5 i q^{21} + 144 q^{22} - 162 q^{23} + 7 i q^{24} + ( - 26 i + 117) q^{26} - 53 q^{27} + 5 i q^{28} - 144 q^{29} - 88 i q^{31} - 15 i q^{32} - 16 i q^{33} + 45 i q^{34} + 26 q^{36} - 101 i q^{37} + 18 q^{38} + ( - 13 i - 26) q^{39} + 64 i q^{41} + 45 q^{42} + 97 q^{43} + 16 i q^{44} - 162 i q^{46} - 37 i q^{47} - 71 q^{48} + 118 q^{49} + 45 q^{51} + (13 i + 26) q^{52} + 414 q^{53} - 53 i q^{54} + 315 q^{56} - 2 i q^{57} - 144 i q^{58} + 174 i q^{59} + 376 q^{61} + 792 q^{62} + 130 i q^{63} - 433 q^{64} + 144 q^{66} - 12 i q^{67} - 45 q^{68} - 162 q^{69} - 119 i q^{71} - 182 i q^{72} + 366 i q^{73} + 909 q^{74} + 2 i q^{76} - 720 q^{77} + ( - 26 i + 117) q^{78} - 830 q^{79} + 649 q^{81} - 576 q^{82} - 146 i q^{83} + 5 i q^{84} + 97 i q^{86} - 144 q^{87} + 1008 q^{88} - 146 i q^{89} + (130 i - 585) q^{91} + 162 q^{92} - 88 i q^{93} + 333 q^{94} - 15 i q^{96} - 284 i q^{97} + 118 i q^{98} + 416 i q^{99} +O(q^{100})$$ q + i * q^2 + q^3 - q^4 + i * q^6 - 5*i * q^7 + 7*i * q^8 - 26 * q^9 - 16*i * q^11 - q^12 + (-13*i - 26) * q^13 + 45 * q^14 - 71 * q^16 + 45 * q^17 - 26*i * q^18 - 2*i * q^19 - 5*i * q^21 + 144 * q^22 - 162 * q^23 + 7*i * q^24 + (-26*i + 117) * q^26 - 53 * q^27 + 5*i * q^28 - 144 * q^29 - 88*i * q^31 - 15*i * q^32 - 16*i * q^33 + 45*i * q^34 + 26 * q^36 - 101*i * q^37 + 18 * q^38 + (-13*i - 26) * q^39 + 64*i * q^41 + 45 * q^42 + 97 * q^43 + 16*i * q^44 - 162*i * q^46 - 37*i * q^47 - 71 * q^48 + 118 * q^49 + 45 * q^51 + (13*i + 26) * q^52 + 414 * q^53 - 53*i * q^54 + 315 * q^56 - 2*i * q^57 - 144*i * q^58 + 174*i * q^59 + 376 * q^61 + 792 * q^62 + 130*i * q^63 - 433 * q^64 + 144 * q^66 - 12*i * q^67 - 45 * q^68 - 162 * q^69 - 119*i * q^71 - 182*i * q^72 + 366*i * q^73 + 909 * q^74 + 2*i * q^76 - 720 * q^77 + (-26*i + 117) * q^78 - 830 * q^79 + 649 * q^81 - 576 * q^82 - 146*i * q^83 + 5*i * q^84 + 97*i * q^86 - 144 * q^87 + 1008 * q^88 - 146*i * q^89 + (130*i - 585) * q^91 + 162 * q^92 - 88*i * q^93 + 333 * q^94 - 15*i * q^96 - 284*i * q^97 + 118*i * q^98 + 416*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{4} - 52 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^4 - 52 * q^9 $$2 q + 2 q^{3} - 2 q^{4} - 52 q^{9} - 2 q^{12} - 52 q^{13} + 90 q^{14} - 142 q^{16} + 90 q^{17} + 288 q^{22} - 324 q^{23} + 234 q^{26} - 106 q^{27} - 288 q^{29} + 52 q^{36} + 36 q^{38} - 52 q^{39} + 90 q^{42} + 194 q^{43} - 142 q^{48} + 236 q^{49} + 90 q^{51} + 52 q^{52} + 828 q^{53} + 630 q^{56} + 752 q^{61} + 1584 q^{62} - 866 q^{64} + 288 q^{66} - 90 q^{68} - 324 q^{69} + 1818 q^{74} - 1440 q^{77} + 234 q^{78} - 1660 q^{79} + 1298 q^{81} - 1152 q^{82} - 288 q^{87} + 2016 q^{88} - 1170 q^{91} + 324 q^{92} + 666 q^{94}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^4 - 52 * q^9 - 2 * q^12 - 52 * q^13 + 90 * q^14 - 142 * q^16 + 90 * q^17 + 288 * q^22 - 324 * q^23 + 234 * q^26 - 106 * q^27 - 288 * q^29 + 52 * q^36 + 36 * q^38 - 52 * q^39 + 90 * q^42 + 194 * q^43 - 142 * q^48 + 236 * q^49 + 90 * q^51 + 52 * q^52 + 828 * q^53 + 630 * q^56 + 752 * q^61 + 1584 * q^62 - 866 * q^64 + 288 * q^66 - 90 * q^68 - 324 * q^69 + 1818 * q^74 - 1440 * q^77 + 234 * q^78 - 1660 * q^79 + 1298 * q^81 - 1152 * q^82 - 288 * q^87 + 2016 * q^88 - 1170 * q^91 + 324 * q^92 + 666 * 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
3.00000i 1.00000 −1.00000 0 3.00000i 15.0000i 21.0000i −26.0000 0
51.2 3.00000i 1.00000 −1.00000 0 3.00000i 15.0000i 21.0000i −26.0000 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 325.4.c.b 2
5.b even 2 1 13.4.b.a 2
5.c odd 4 1 325.4.d.a 2
5.c odd 4 1 325.4.d.b 2
13.b even 2 1 inner 325.4.c.b 2
15.d odd 2 1 117.4.b.a 2
20.d odd 2 1 208.4.f.b 2
40.e odd 2 1 832.4.f.c 2
40.f even 2 1 832.4.f.e 2
65.d even 2 1 13.4.b.a 2
65.g odd 4 1 169.4.a.b 1
65.g odd 4 1 169.4.a.c 1
65.h odd 4 1 325.4.d.a 2
65.h odd 4 1 325.4.d.b 2
65.l even 6 2 169.4.e.d 4
65.n even 6 2 169.4.e.d 4
65.s odd 12 2 169.4.c.b 2
65.s odd 12 2 169.4.c.c 2
195.e odd 2 1 117.4.b.a 2
195.n even 4 1 1521.4.a.d 1
195.n even 4 1 1521.4.a.i 1
260.g odd 2 1 208.4.f.b 2
520.b odd 2 1 832.4.f.c 2
520.p even 2 1 832.4.f.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.b.a 2 5.b even 2 1
13.4.b.a 2 65.d even 2 1
117.4.b.a 2 15.d odd 2 1
117.4.b.a 2 195.e odd 2 1
169.4.a.b 1 65.g odd 4 1
169.4.a.c 1 65.g odd 4 1
169.4.c.b 2 65.s odd 12 2
169.4.c.c 2 65.s odd 12 2
169.4.e.d 4 65.l even 6 2
169.4.e.d 4 65.n even 6 2
208.4.f.b 2 20.d odd 2 1
208.4.f.b 2 260.g odd 2 1
325.4.c.b 2 1.a even 1 1 trivial
325.4.c.b 2 13.b even 2 1 inner
325.4.d.a 2 5.c odd 4 1
325.4.d.a 2 65.h odd 4 1
325.4.d.b 2 5.c odd 4 1
325.4.d.b 2 65.h odd 4 1
832.4.f.c 2 40.e odd 2 1
832.4.f.c 2 520.b odd 2 1
832.4.f.e 2 40.f even 2 1
832.4.f.e 2 520.p even 2 1
1521.4.a.d 1 195.n even 4 1
1521.4.a.i 1 195.n even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(325, [\chi])$$:

 $$T_{2}^{2} + 9$$ T2^2 + 9 $$T_{3} - 1$$ T3 - 1

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 9$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 225$$
$11$ $$T^{2} + 2304$$
$13$ $$T^{2} + 52T + 2197$$
$17$ $$(T - 45)^{2}$$
$19$ $$T^{2} + 36$$
$23$ $$(T + 162)^{2}$$
$29$ $$(T + 144)^{2}$$
$31$ $$T^{2} + 69696$$
$37$ $$T^{2} + 91809$$
$41$ $$T^{2} + 36864$$
$43$ $$(T - 97)^{2}$$
$47$ $$T^{2} + 12321$$
$53$ $$(T - 414)^{2}$$
$59$ $$T^{2} + 272484$$
$61$ $$(T - 376)^{2}$$
$67$ $$T^{2} + 1296$$
$71$ $$T^{2} + 127449$$
$73$ $$T^{2} + 1205604$$
$79$ $$(T + 830)^{2}$$
$83$ $$T^{2} + 191844$$
$89$ $$T^{2} + 191844$$
$97$ $$T^{2} + 725904$$