# Properties

 Label 10.3.c.a Level $10$ Weight $3$ Character orbit 10.c Analytic conductor $0.272$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [10,3,Mod(3,10)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(10, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([3]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("10.3");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$10 = 2 \cdot 5$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 10.c (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: $$0.272480264360$$ 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: $$1$$ 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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - i - 1) q^{2} + (2 i - 2) q^{3} + 2 i q^{4} - 5 i q^{5} + 4 q^{6} + (2 i + 2) q^{7} + ( - 2 i + 2) q^{8} + i q^{9} +O(q^{10})$$ q + (-i - 1) * q^2 + (2*i - 2) * q^3 + 2*i * q^4 - 5*i * q^5 + 4 * q^6 + (2*i + 2) * q^7 + (-2*i + 2) * q^8 + i * q^9 $$q + ( - i - 1) q^{2} + (2 i - 2) q^{3} + 2 i q^{4} - 5 i q^{5} + 4 q^{6} + (2 i + 2) q^{7} + ( - 2 i + 2) q^{8} + i q^{9} + (5 i - 5) q^{10} - 8 q^{11} + ( - 4 i - 4) q^{12} + ( - 3 i + 3) q^{13} - 4 i q^{14} + (10 i + 10) q^{15} - 4 q^{16} + (7 i + 7) q^{17} + ( - i + 1) q^{18} - 20 i q^{19} + 10 q^{20} - 8 q^{21} + (8 i + 8) q^{22} + (2 i - 2) q^{23} + 8 i q^{24} - 25 q^{25} - 6 q^{26} + ( - 20 i - 20) q^{27} + (4 i - 4) q^{28} + 40 i q^{29} - 20 i q^{30} + 52 q^{31} + (4 i + 4) q^{32} + ( - 16 i + 16) q^{33} - 14 i q^{34} + ( - 10 i + 10) q^{35} - 2 q^{36} + ( - 3 i - 3) q^{37} + (20 i - 20) q^{38} + 12 i q^{39} + ( - 10 i - 10) q^{40} - 8 q^{41} + (8 i + 8) q^{42} + (42 i - 42) q^{43} - 16 i q^{44} + 5 q^{45} + 4 q^{46} + ( - 18 i - 18) q^{47} + ( - 8 i + 8) q^{48} - 41 i q^{49} + (25 i + 25) q^{50} - 28 q^{51} + (6 i + 6) q^{52} + ( - 53 i + 53) q^{53} + 40 i q^{54} + 40 i q^{55} + 8 q^{56} + (40 i + 40) q^{57} + ( - 40 i + 40) q^{58} - 20 i q^{59} + (20 i - 20) q^{60} - 48 q^{61} + ( - 52 i - 52) q^{62} + (2 i - 2) q^{63} - 8 i q^{64} + ( - 15 i - 15) q^{65} - 32 q^{66} + (62 i + 62) q^{67} + (14 i - 14) q^{68} - 8 i q^{69} - 20 q^{70} - 28 q^{71} + (2 i + 2) q^{72} + (47 i - 47) q^{73} + 6 i q^{74} + ( - 50 i + 50) q^{75} + 40 q^{76} + ( - 16 i - 16) q^{77} + ( - 12 i + 12) q^{78} + 20 i q^{80} + 71 q^{81} + (8 i + 8) q^{82} + ( - 18 i + 18) q^{83} - 16 i q^{84} + ( - 35 i + 35) q^{85} + 84 q^{86} + ( - 80 i - 80) q^{87} + (16 i - 16) q^{88} + 80 i q^{89} + ( - 5 i - 5) q^{90} + 12 q^{91} + ( - 4 i - 4) q^{92} + (104 i - 104) q^{93} + 36 i q^{94} - 100 q^{95} - 16 q^{96} + ( - 63 i - 63) q^{97} + (41 i - 41) q^{98} - 8 i q^{99} +O(q^{100})$$ q + (-i - 1) * q^2 + (2*i - 2) * q^3 + 2*i * q^4 - 5*i * q^5 + 4 * q^6 + (2*i + 2) * q^7 + (-2*i + 2) * q^8 + i * q^9 + (5*i - 5) * q^10 - 8 * q^11 + (-4*i - 4) * q^12 + (-3*i + 3) * q^13 - 4*i * q^14 + (10*i + 10) * q^15 - 4 * q^16 + (7*i + 7) * q^17 + (-i + 1) * q^18 - 20*i * q^19 + 10 * q^20 - 8 * q^21 + (8*i + 8) * q^22 + (2*i - 2) * q^23 + 8*i * q^24 - 25 * q^25 - 6 * q^26 + (-20*i - 20) * q^27 + (4*i - 4) * q^28 + 40*i * q^29 - 20*i * q^30 + 52 * q^31 + (4*i + 4) * q^32 + (-16*i + 16) * q^33 - 14*i * q^34 + (-10*i + 10) * q^35 - 2 * q^36 + (-3*i - 3) * q^37 + (20*i - 20) * q^38 + 12*i * q^39 + (-10*i - 10) * q^40 - 8 * q^41 + (8*i + 8) * q^42 + (42*i - 42) * q^43 - 16*i * q^44 + 5 * q^45 + 4 * q^46 + (-18*i - 18) * q^47 + (-8*i + 8) * q^48 - 41*i * q^49 + (25*i + 25) * q^50 - 28 * q^51 + (6*i + 6) * q^52 + (-53*i + 53) * q^53 + 40*i * q^54 + 40*i * q^55 + 8 * q^56 + (40*i + 40) * q^57 + (-40*i + 40) * q^58 - 20*i * q^59 + (20*i - 20) * q^60 - 48 * q^61 + (-52*i - 52) * q^62 + (2*i - 2) * q^63 - 8*i * q^64 + (-15*i - 15) * q^65 - 32 * q^66 + (62*i + 62) * q^67 + (14*i - 14) * q^68 - 8*i * q^69 - 20 * q^70 - 28 * q^71 + (2*i + 2) * q^72 + (47*i - 47) * q^73 + 6*i * q^74 + (-50*i + 50) * q^75 + 40 * q^76 + (-16*i - 16) * q^77 + (-12*i + 12) * q^78 + 20*i * q^80 + 71 * q^81 + (8*i + 8) * q^82 + (-18*i + 18) * q^83 - 16*i * q^84 + (-35*i + 35) * q^85 + 84 * q^86 + (-80*i - 80) * q^87 + (16*i - 16) * q^88 + 80*i * q^89 + (-5*i - 5) * q^90 + 12 * q^91 + (-4*i - 4) * q^92 + (104*i - 104) * q^93 + 36*i * q^94 - 100 * q^95 - 16 * q^96 + (-63*i - 63) * q^97 + (41*i - 41) * q^98 - 8*i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 8 q^{6} + 4 q^{7} + 4 q^{8}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 8 * q^6 + 4 * q^7 + 4 * q^8 $$2 q - 2 q^{2} - 4 q^{3} + 8 q^{6} + 4 q^{7} + 4 q^{8} - 10 q^{10} - 16 q^{11} - 8 q^{12} + 6 q^{13} + 20 q^{15} - 8 q^{16} + 14 q^{17} + 2 q^{18} + 20 q^{20} - 16 q^{21} + 16 q^{22} - 4 q^{23} - 50 q^{25} - 12 q^{26} - 40 q^{27} - 8 q^{28} + 104 q^{31} + 8 q^{32} + 32 q^{33} + 20 q^{35} - 4 q^{36} - 6 q^{37} - 40 q^{38} - 20 q^{40} - 16 q^{41} + 16 q^{42} - 84 q^{43} + 10 q^{45} + 8 q^{46} - 36 q^{47} + 16 q^{48} + 50 q^{50} - 56 q^{51} + 12 q^{52} + 106 q^{53} + 16 q^{56} + 80 q^{57} + 80 q^{58} - 40 q^{60} - 96 q^{61} - 104 q^{62} - 4 q^{63} - 30 q^{65} - 64 q^{66} + 124 q^{67} - 28 q^{68} - 40 q^{70} - 56 q^{71} + 4 q^{72} - 94 q^{73} + 100 q^{75} + 80 q^{76} - 32 q^{77} + 24 q^{78} + 142 q^{81} + 16 q^{82} + 36 q^{83} + 70 q^{85} + 168 q^{86} - 160 q^{87} - 32 q^{88} - 10 q^{90} + 24 q^{91} - 8 q^{92} - 208 q^{93} - 200 q^{95} - 32 q^{96} - 126 q^{97} - 82 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 8 * q^6 + 4 * q^7 + 4 * q^8 - 10 * q^10 - 16 * q^11 - 8 * q^12 + 6 * q^13 + 20 * q^15 - 8 * q^16 + 14 * q^17 + 2 * q^18 + 20 * q^20 - 16 * q^21 + 16 * q^22 - 4 * q^23 - 50 * q^25 - 12 * q^26 - 40 * q^27 - 8 * q^28 + 104 * q^31 + 8 * q^32 + 32 * q^33 + 20 * q^35 - 4 * q^36 - 6 * q^37 - 40 * q^38 - 20 * q^40 - 16 * q^41 + 16 * q^42 - 84 * q^43 + 10 * q^45 + 8 * q^46 - 36 * q^47 + 16 * q^48 + 50 * q^50 - 56 * q^51 + 12 * q^52 + 106 * q^53 + 16 * q^56 + 80 * q^57 + 80 * q^58 - 40 * q^60 - 96 * q^61 - 104 * q^62 - 4 * q^63 - 30 * q^65 - 64 * q^66 + 124 * q^67 - 28 * q^68 - 40 * q^70 - 56 * q^71 + 4 * q^72 - 94 * q^73 + 100 * q^75 + 80 * q^76 - 32 * q^77 + 24 * q^78 + 142 * q^81 + 16 * q^82 + 36 * q^83 + 70 * q^85 + 168 * q^86 - 160 * q^87 - 32 * q^88 - 10 * q^90 + 24 * q^91 - 8 * q^92 - 208 * q^93 - 200 * q^95 - 32 * q^96 - 126 * q^97 - 82 * q^98

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/10\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$\chi(n)$$ $$i$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 − 1.00000i 1.00000i
−1.00000 + 1.00000i −2.00000 2.00000i 2.00000i 5.00000i 4.00000 2.00000 2.00000i 2.00000 + 2.00000i 1.00000i −5.00000 5.00000i
7.1 −1.00000 1.00000i −2.00000 + 2.00000i 2.00000i 5.00000i 4.00000 2.00000 + 2.00000i 2.00000 2.00000i 1.00000i −5.00000 + 5.00000i
 $$n$$: 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.3.c.a 2
3.b odd 2 1 90.3.g.b 2
4.b odd 2 1 80.3.p.c 2
5.b even 2 1 50.3.c.c 2
5.c odd 4 1 inner 10.3.c.a 2
5.c odd 4 1 50.3.c.c 2
7.b odd 2 1 490.3.f.b 2
8.b even 2 1 320.3.p.h 2
8.d odd 2 1 320.3.p.a 2
12.b even 2 1 720.3.bh.c 2
15.d odd 2 1 450.3.g.b 2
15.e even 4 1 90.3.g.b 2
15.e even 4 1 450.3.g.b 2
20.d odd 2 1 400.3.p.b 2
20.e even 4 1 80.3.p.c 2
20.e even 4 1 400.3.p.b 2
35.f even 4 1 490.3.f.b 2
40.i odd 4 1 320.3.p.h 2
40.k even 4 1 320.3.p.a 2
60.l odd 4 1 720.3.bh.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.3.c.a 2 1.a even 1 1 trivial
10.3.c.a 2 5.c odd 4 1 inner
50.3.c.c 2 5.b even 2 1
50.3.c.c 2 5.c odd 4 1
80.3.p.c 2 4.b odd 2 1
80.3.p.c 2 20.e even 4 1
90.3.g.b 2 3.b odd 2 1
90.3.g.b 2 15.e even 4 1
320.3.p.a 2 8.d odd 2 1
320.3.p.a 2 40.k even 4 1
320.3.p.h 2 8.b even 2 1
320.3.p.h 2 40.i odd 4 1
400.3.p.b 2 20.d odd 2 1
400.3.p.b 2 20.e even 4 1
450.3.g.b 2 15.d odd 2 1
450.3.g.b 2 15.e even 4 1
490.3.f.b 2 7.b odd 2 1
490.3.f.b 2 35.f even 4 1
720.3.bh.c 2 12.b even 2 1
720.3.bh.c 2 60.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(10, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T + 2$$
$3$ $$T^{2} + 4T + 8$$
$5$ $$T^{2} + 25$$
$7$ $$T^{2} - 4T + 8$$
$11$ $$(T + 8)^{2}$$
$13$ $$T^{2} - 6T + 18$$
$17$ $$T^{2} - 14T + 98$$
$19$ $$T^{2} + 400$$
$23$ $$T^{2} + 4T + 8$$
$29$ $$T^{2} + 1600$$
$31$ $$(T - 52)^{2}$$
$37$ $$T^{2} + 6T + 18$$
$41$ $$(T + 8)^{2}$$
$43$ $$T^{2} + 84T + 3528$$
$47$ $$T^{2} + 36T + 648$$
$53$ $$T^{2} - 106T + 5618$$
$59$ $$T^{2} + 400$$
$61$ $$(T + 48)^{2}$$
$67$ $$T^{2} - 124T + 7688$$
$71$ $$(T + 28)^{2}$$
$73$ $$T^{2} + 94T + 4418$$
$79$ $$T^{2}$$
$83$ $$T^{2} - 36T + 648$$
$89$ $$T^{2} + 6400$$
$97$ $$T^{2} + 126T + 7938$$