Properties

 Label 102.2.f.a Level $102$ Weight $2$ Character orbit 102.f Analytic conductor $0.814$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,2,Mod(13,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("102.13");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$102 = 2 \cdot 3 \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 102.f (of order $$4$$, degree $$2$$, 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: $$0.814474100617$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ 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: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \zeta_{8}^{2} q^{2} + \zeta_{8} q^{3} - q^{4} + ( - \zeta_{8}^{2} + 2 \zeta_{8} - 1) q^{5} - \zeta_{8}^{3} q^{6} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} +O(q^{10})$$ q - z^2 * q^2 + z * q^3 - q^4 + (-z^2 + 2*z - 1) * q^5 - z^3 * q^6 + (-2*z^3 - 2*z^2 + 2) * q^7 + z^2 * q^8 + z^2 * q^9 $$q - \zeta_{8}^{2} q^{2} + \zeta_{8} q^{3} - q^{4} + ( - \zeta_{8}^{2} + 2 \zeta_{8} - 1) q^{5} - \zeta_{8}^{3} q^{6} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{7} + \zeta_{8}^{2} q^{8} + \zeta_{8}^{2} q^{9} + ( - 2 \zeta_{8}^{3} + \zeta_{8}^{2} - 1) q^{10} + 4 \zeta_{8}^{3} q^{11} - \zeta_{8} q^{12} + (2 \zeta_{8}^{3} - 2 \zeta_{8}) q^{13} + ( - 2 \zeta_{8}^{2} - 2 \zeta_{8} - 2) q^{14} + ( - \zeta_{8}^{3} + 2 \zeta_{8}^{2} - \zeta_{8}) q^{15} + q^{16} + ( - 2 \zeta_{8}^{3} - 2 \zeta_{8} - 3) q^{17} + q^{18} + (2 \zeta_{8}^{3} + 4 \zeta_{8}^{2} + 2 \zeta_{8}) q^{19} + (\zeta_{8}^{2} - 2 \zeta_{8} + 1) q^{20} + ( - 2 \zeta_{8}^{3} + 2 \zeta_{8} + 2) q^{21} + 4 \zeta_{8} q^{22} + (6 \zeta_{8}^{3} - 2 \zeta_{8}^{2} + 2) q^{23} + \zeta_{8}^{3} q^{24} + ( - 4 \zeta_{8}^{3} + \cdots - 4 \zeta_{8}) q^{25} + \cdots - 4 \zeta_{8} q^{99} +O(q^{100})$$ q - z^2 * q^2 + z * q^3 - q^4 + (-z^2 + 2*z - 1) * q^5 - z^3 * q^6 + (-2*z^3 - 2*z^2 + 2) * q^7 + z^2 * q^8 + z^2 * q^9 + (-2*z^3 + z^2 - 1) * q^10 + 4*z^3 * q^11 - z * q^12 + (2*z^3 - 2*z) * q^13 + (-2*z^2 - 2*z - 2) * q^14 + (-z^3 + 2*z^2 - z) * q^15 + q^16 + (-2*z^3 - 2*z - 3) * q^17 + q^18 + (2*z^3 + 4*z^2 + 2*z) * q^19 + (z^2 - 2*z + 1) * q^20 + (-2*z^3 + 2*z + 2) * q^21 + 4*z * q^22 + (6*z^3 - 2*z^2 + 2) * q^23 + z^3 * q^24 + (-4*z^3 + z^2 - 4*z) * q^25 + (2*z^3 + 2*z) * q^26 + z^3 * q^27 + (2*z^3 + 2*z^2 - 2) * q^28 + (z^2 - 2*z + 1) * q^29 + (z^3 - z + 2) * q^30 + (2*z^2 - 2*z + 2) * q^31 - z^2 * q^32 - 4 * q^33 + (2*z^3 + 3*z^2 - 2*z) * q^34 + (-2*z^3 + 2*z) * q^35 - z^2 * q^36 + (-3*z^2 + 2*z - 3) * q^37 + (-2*z^3 + 2*z + 4) * q^38 + (-2*z^2 - 2) * q^39 + (2*z^3 - z^2 + 1) * q^40 + (-4*z^3 + 5*z^2 - 5) * q^41 + (-2*z^3 - 2*z^2 - 2*z) * q^42 + (4*z^3 - 4*z^2 + 4*z) * q^43 - 4*z^3 * q^44 + (2*z^3 - z^2 + 1) * q^45 + (-2*z^2 + 6*z - 2) * q^46 + (-6*z^3 + 6*z + 4) * q^47 + z * q^48 + (-8*z^3 - 5*z^2 - 8*z) * q^49 + (4*z^3 - 4*z + 1) * q^50 + (-2*z^2 - 3*z + 2) * q^51 + (-2*z^3 + 2*z) * q^52 + (-2*z^3 - 2*z) * q^53 + z * q^54 + (-4*z^3 + 4*z - 8) * q^55 + (2*z^2 + 2*z + 2) * q^56 + (4*z^3 + 2*z^2 - 2) * q^57 + (2*z^3 - z^2 + 1) * q^58 + (-6*z^3 - 4*z^2 - 6*z) * q^59 + (z^3 - 2*z^2 + z) * q^60 + (6*z^3 - 5*z^2 + 5) * q^61 + (2*z^3 - 2*z^2 + 2) * q^62 + (2*z^2 + 2*z + 2) * q^63 - q^64 + (-4*z^2 + 4*z - 4) * q^65 + 4*z^2 * q^66 + (2*z^3 - 2*z + 4) * q^67 + (2*z^3 + 2*z + 3) * q^68 + (-2*z^3 + 2*z - 6) * q^69 + (-2*z^3 - 2*z) * q^70 + (6*z^2 - 2*z + 6) * q^71 - q^72 + (z^2 + 8*z + 1) * q^73 + (-2*z^3 + 3*z^2 - 3) * q^74 + (z^3 - 4*z^2 + 4) * q^75 + (-2*z^3 - 4*z^2 - 2*z) * q^76 + (8*z^3 + 8*z^2 + 8*z) * q^77 + (2*z^2 - 2) * q^78 + (6*z^3 + 6*z^2 - 6) * q^79 + (-z^2 + 2*z - 1) * q^80 - q^81 + (5*z^2 - 4*z + 5) * q^82 + (-6*z^3 + 4*z^2 - 6*z) * q^83 + (2*z^3 - 2*z - 2) * q^84 + (4*z^3 - z^2 - 6*z + 7) * q^85 + (-4*z^3 + 4*z - 4) * q^86 + (z^3 - 2*z^2 + z) * q^87 - 4*z * q^88 + (-z^2 + 2*z - 1) * q^90 + (8*z^3 + 4*z^2 - 4) * q^91 + (-6*z^3 + 2*z^2 - 2) * q^92 + (2*z^3 - 2*z^2 + 2*z) * q^93 + (-6*z^3 - 4*z^2 - 6*z) * q^94 + 4*z^3 * q^95 - z^3 * q^96 + (-z^2 - 12*z - 1) * q^97 + (8*z^3 - 8*z - 5) * q^98 - 4*z * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 4 q^{5} + 8 q^{7}+O(q^{10})$$ 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^7 $$4 q - 4 q^{4} - 4 q^{5} + 8 q^{7} - 4 q^{10} - 8 q^{14} + 4 q^{16} - 12 q^{17} + 4 q^{18} + 4 q^{20} + 8 q^{21} + 8 q^{23} - 8 q^{28} + 4 q^{29} + 8 q^{30} + 8 q^{31} - 16 q^{33} - 12 q^{37} + 16 q^{38} - 8 q^{39} + 4 q^{40} - 20 q^{41} + 4 q^{45} - 8 q^{46} + 16 q^{47} + 4 q^{50} + 8 q^{51} - 32 q^{55} + 8 q^{56} - 8 q^{57} + 4 q^{58} + 20 q^{61} + 8 q^{62} + 8 q^{63} - 4 q^{64} - 16 q^{65} + 16 q^{67} + 12 q^{68} - 24 q^{69} + 24 q^{71} - 4 q^{72} + 4 q^{73} - 12 q^{74} + 16 q^{75} - 8 q^{78} - 24 q^{79} - 4 q^{80} - 4 q^{81} + 20 q^{82} - 8 q^{84} + 28 q^{85} - 16 q^{86} - 4 q^{90} - 16 q^{91} - 8 q^{92} - 4 q^{97} - 20 q^{98}+O(q^{100})$$ 4 * q - 4 * q^4 - 4 * q^5 + 8 * q^7 - 4 * q^10 - 8 * q^14 + 4 * q^16 - 12 * q^17 + 4 * q^18 + 4 * q^20 + 8 * q^21 + 8 * q^23 - 8 * q^28 + 4 * q^29 + 8 * q^30 + 8 * q^31 - 16 * q^33 - 12 * q^37 + 16 * q^38 - 8 * q^39 + 4 * q^40 - 20 * q^41 + 4 * q^45 - 8 * q^46 + 16 * q^47 + 4 * q^50 + 8 * q^51 - 32 * q^55 + 8 * q^56 - 8 * q^57 + 4 * q^58 + 20 * q^61 + 8 * q^62 + 8 * q^63 - 4 * q^64 - 16 * q^65 + 16 * q^67 + 12 * q^68 - 24 * q^69 + 24 * q^71 - 4 * q^72 + 4 * q^73 - 12 * q^74 + 16 * q^75 - 8 * q^78 - 24 * q^79 - 4 * q^80 - 4 * q^81 + 20 * q^82 - 8 * q^84 + 28 * q^85 - 16 * q^86 - 4 * q^90 - 16 * q^91 - 8 * q^92 - 4 * q^97 - 20 * q^98

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/102\mathbb{Z}\right)^\times$$.

 $$n$$ $$35$$ $$37$$ $$\chi(n)$$ $$1$$ $$\zeta_{8}^{2}$$

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i 0.707107 − 0.707107i
1.00000i −0.707107 0.707107i −1.00000 −2.41421 2.41421i −0.707107 + 0.707107i 0.585786 0.585786i 1.00000i 1.00000i −2.41421 + 2.41421i
13.2 1.00000i 0.707107 + 0.707107i −1.00000 0.414214 + 0.414214i 0.707107 0.707107i 3.41421 3.41421i 1.00000i 1.00000i 0.414214 0.414214i
55.1 1.00000i −0.707107 + 0.707107i −1.00000 −2.41421 + 2.41421i −0.707107 0.707107i 0.585786 + 0.585786i 1.00000i 1.00000i −2.41421 2.41421i
55.2 1.00000i 0.707107 0.707107i −1.00000 0.414214 0.414214i 0.707107 + 0.707107i 3.41421 + 3.41421i 1.00000i 1.00000i 0.414214 + 0.414214i
 $$n$$: 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
17.c even 4 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 102.2.f.a 4
3.b odd 2 1 306.2.g.g 4
4.b odd 2 1 816.2.bd.a 4
12.b even 2 1 2448.2.be.t 4
17.b even 2 1 1734.2.f.i 4
17.c even 4 1 inner 102.2.f.a 4
17.c even 4 1 1734.2.f.i 4
17.d even 8 1 1734.2.a.n 2
17.d even 8 1 1734.2.a.o 2
17.d even 8 2 1734.2.b.h 4
51.f odd 4 1 306.2.g.g 4
51.g odd 8 1 5202.2.a.o 2
51.g odd 8 1 5202.2.a.x 2
68.f odd 4 1 816.2.bd.a 4
204.l even 4 1 2448.2.be.t 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
102.2.f.a 4 1.a even 1 1 trivial
102.2.f.a 4 17.c even 4 1 inner
306.2.g.g 4 3.b odd 2 1
306.2.g.g 4 51.f odd 4 1
816.2.bd.a 4 4.b odd 2 1
816.2.bd.a 4 68.f odd 4 1
1734.2.a.n 2 17.d even 8 1
1734.2.a.o 2 17.d even 8 1
1734.2.b.h 4 17.d even 8 2
1734.2.f.i 4 17.b even 2 1
1734.2.f.i 4 17.c even 4 1
2448.2.be.t 4 12.b even 2 1
2448.2.be.t 4 204.l even 4 1
5202.2.a.o 2 51.g odd 8 1
5202.2.a.x 2 51.g odd 8 1

Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(102, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 1$$
$5$ $$T^{4} + 4 T^{3} + \cdots + 4$$
$7$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$11$ $$T^{4} + 256$$
$13$ $$(T^{2} - 8)^{2}$$
$17$ $$(T^{2} + 6 T + 17)^{2}$$
$19$ $$T^{4} + 48T^{2} + 64$$
$23$ $$T^{4} - 8 T^{3} + \cdots + 784$$
$29$ $$T^{4} - 4 T^{3} + \cdots + 4$$
$31$ $$T^{4} - 8 T^{3} + \cdots + 16$$
$37$ $$T^{4} + 12 T^{3} + \cdots + 196$$
$41$ $$T^{4} + 20 T^{3} + \cdots + 1156$$
$43$ $$T^{4} + 96T^{2} + 256$$
$47$ $$(T^{2} - 8 T - 56)^{2}$$
$53$ $$(T^{2} + 8)^{2}$$
$59$ $$T^{4} + 176T^{2} + 3136$$
$61$ $$T^{4} - 20 T^{3} + \cdots + 196$$
$67$ $$(T^{2} - 8 T + 8)^{2}$$
$71$ $$T^{4} - 24 T^{3} + \cdots + 4624$$
$73$ $$T^{4} - 4 T^{3} + \cdots + 3844$$
$79$ $$T^{4} + 24 T^{3} + \cdots + 1296$$
$83$ $$T^{4} + 176T^{2} + 3136$$
$89$ $$T^{4}$$
$97$ $$T^{4} + 4 T^{3} + \cdots + 20164$$