# Properties

 Label 230.2.b.a Level $230$ Weight $2$ Character orbit 230.b Analytic conductor $1.837$ 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] = [230,2,Mod(139,230)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(230, 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("230.139");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.b (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: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 3x^{2} + 1$$ x^4 + 3*x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q - \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + (2 \beta_{2} + 1) q^{5} + \beta_{2} q^{6} + (3 \beta_{3} + 3 \beta_1) q^{7} + \beta_{3} q^{8} + (\beta_{2} + 2) q^{9}+O(q^{10})$$ q - b3 * q^2 + b1 * q^3 - q^4 + (2*b2 + 1) * q^5 + b2 * q^6 + (3*b3 + 3*b1) * q^7 + b3 * q^8 + (b2 + 2) * q^9 $$q - \beta_{3} q^{2} + \beta_1 q^{3} - q^{4} + (2 \beta_{2} + 1) q^{5} + \beta_{2} q^{6} + (3 \beta_{3} + 3 \beta_1) q^{7} + \beta_{3} q^{8} + (\beta_{2} + 2) q^{9} + ( - \beta_{3} - 2 \beta_1) q^{10} + (\beta_{2} - 4) q^{11} - \beta_1 q^{12} + ( - \beta_{3} + \beta_1) q^{13} + (3 \beta_{2} + 3) q^{14} + (2 \beta_{3} - \beta_1) q^{15} + q^{16} + ( - 4 \beta_{3} - 3 \beta_1) q^{17} + ( - 2 \beta_{3} - \beta_1) q^{18} + (3 \beta_{2} + 5) q^{19} + ( - 2 \beta_{2} - 1) q^{20} - 3 q^{21} + (4 \beta_{3} - \beta_1) q^{22} - \beta_{3} q^{23} - \beta_{2} q^{24} + 5 q^{25} + (\beta_{2} - 1) q^{26} + (\beta_{3} + 4 \beta_1) q^{27} + ( - 3 \beta_{3} - 3 \beta_1) q^{28} - 6 \beta_{2} q^{29} + ( - \beta_{2} + 2) q^{30} + ( - 3 \beta_{2} - 7) q^{31} - \beta_{3} q^{32} + (\beta_{3} - 5 \beta_1) q^{33} + ( - 3 \beta_{2} - 4) q^{34} + (9 \beta_{3} + 3 \beta_1) q^{35} + ( - \beta_{2} - 2) q^{36} + 6 \beta_1 q^{37} + ( - 5 \beta_{3} - 3 \beta_1) q^{38} + (2 \beta_{2} - 1) q^{39} + (\beta_{3} + 2 \beta_1) q^{40} + (\beta_{2} - 4) q^{41} + 3 \beta_{3} q^{42} + ( - 8 \beta_{3} + 2 \beta_1) q^{43} + ( - \beta_{2} + 4) q^{44} + (3 \beta_{2} + 4) q^{45} - q^{46} + ( - 8 \beta_{3} - 6 \beta_1) q^{47} + \beta_1 q^{48} + ( - 9 \beta_{2} - 11) q^{49} - 5 \beta_{3} q^{50} + (\beta_{2} + 3) q^{51} + (\beta_{3} - \beta_1) q^{52} + 2 \beta_{3} q^{53} + (4 \beta_{2} + 1) q^{54} + ( - 9 \beta_{2} - 2) q^{55} + ( - 3 \beta_{2} - 3) q^{56} + (3 \beta_{3} + 2 \beta_1) q^{57} + 6 \beta_1 q^{58} + 6 q^{59} + ( - 2 \beta_{3} + \beta_1) q^{60} + ( - 3 \beta_{2} - 2) q^{61} + (7 \beta_{3} + 3 \beta_1) q^{62} + (9 \beta_{3} + 6 \beta_1) q^{63} - q^{64} + (\beta_{3} - 3 \beta_1) q^{65} + ( - 5 \beta_{2} + 1) q^{66} + (2 \beta_{3} - 2 \beta_1) q^{67} + (4 \beta_{3} + 3 \beta_1) q^{68} + \beta_{2} q^{69} + (3 \beta_{2} + 9) q^{70} + (\beta_{2} + 2) q^{71} + (2 \beta_{3} + \beta_1) q^{72} + (10 \beta_{3} - 4 \beta_1) q^{73} + 6 \beta_{2} q^{74} + 5 \beta_1 q^{75} + ( - 3 \beta_{2} - 5) q^{76} + ( - 9 \beta_{3} - 12 \beta_1) q^{77} + (\beta_{3} - 2 \beta_1) q^{78} + ( - 6 \beta_{2} - 2) q^{79} + (2 \beta_{2} + 1) q^{80} + (6 \beta_{2} + 2) q^{81} + (4 \beta_{3} - \beta_1) q^{82} + (2 \beta_{3} + 6 \beta_1) q^{83} + 3 q^{84} + ( - 10 \beta_{3} - 5 \beta_1) q^{85} + (2 \beta_{2} - 8) q^{86} + ( - 6 \beta_{3} + 6 \beta_1) q^{87} + ( - 4 \beta_{3} + \beta_1) q^{88} + (6 \beta_{2} + 6) q^{89} + ( - 4 \beta_{3} - 3 \beta_1) q^{90} + 3 \beta_{2} q^{91} + \beta_{3} q^{92} + ( - 3 \beta_{3} - 4 \beta_1) q^{93} + ( - 6 \beta_{2} - 8) q^{94} + (7 \beta_{2} + 11) q^{95} + \beta_{2} q^{96} + ( - 8 \beta_{3} - 13 \beta_1) q^{97} + (11 \beta_{3} + 9 \beta_1) q^{98} + ( - 3 \beta_{2} - 7) q^{99}+O(q^{100})$$ q - b3 * q^2 + b1 * q^3 - q^4 + (2*b2 + 1) * q^5 + b2 * q^6 + (3*b3 + 3*b1) * q^7 + b3 * q^8 + (b2 + 2) * q^9 + (-b3 - 2*b1) * q^10 + (b2 - 4) * q^11 - b1 * q^12 + (-b3 + b1) * q^13 + (3*b2 + 3) * q^14 + (2*b3 - b1) * q^15 + q^16 + (-4*b3 - 3*b1) * q^17 + (-2*b3 - b1) * q^18 + (3*b2 + 5) * q^19 + (-2*b2 - 1) * q^20 - 3 * q^21 + (4*b3 - b1) * q^22 - b3 * q^23 - b2 * q^24 + 5 * q^25 + (b2 - 1) * q^26 + (b3 + 4*b1) * q^27 + (-3*b3 - 3*b1) * q^28 - 6*b2 * q^29 + (-b2 + 2) * q^30 + (-3*b2 - 7) * q^31 - b3 * q^32 + (b3 - 5*b1) * q^33 + (-3*b2 - 4) * q^34 + (9*b3 + 3*b1) * q^35 + (-b2 - 2) * q^36 + 6*b1 * q^37 + (-5*b3 - 3*b1) * q^38 + (2*b2 - 1) * q^39 + (b3 + 2*b1) * q^40 + (b2 - 4) * q^41 + 3*b3 * q^42 + (-8*b3 + 2*b1) * q^43 + (-b2 + 4) * q^44 + (3*b2 + 4) * q^45 - q^46 + (-8*b3 - 6*b1) * q^47 + b1 * q^48 + (-9*b2 - 11) * q^49 - 5*b3 * q^50 + (b2 + 3) * q^51 + (b3 - b1) * q^52 + 2*b3 * q^53 + (4*b2 + 1) * q^54 + (-9*b2 - 2) * q^55 + (-3*b2 - 3) * q^56 + (3*b3 + 2*b1) * q^57 + 6*b1 * q^58 + 6 * q^59 + (-2*b3 + b1) * q^60 + (-3*b2 - 2) * q^61 + (7*b3 + 3*b1) * q^62 + (9*b3 + 6*b1) * q^63 - q^64 + (b3 - 3*b1) * q^65 + (-5*b2 + 1) * q^66 + (2*b3 - 2*b1) * q^67 + (4*b3 + 3*b1) * q^68 + b2 * q^69 + (3*b2 + 9) * q^70 + (b2 + 2) * q^71 + (2*b3 + b1) * q^72 + (10*b3 - 4*b1) * q^73 + 6*b2 * q^74 + 5*b1 * q^75 + (-3*b2 - 5) * q^76 + (-9*b3 - 12*b1) * q^77 + (b3 - 2*b1) * q^78 + (-6*b2 - 2) * q^79 + (2*b2 + 1) * q^80 + (6*b2 + 2) * q^81 + (4*b3 - b1) * q^82 + (2*b3 + 6*b1) * q^83 + 3 * q^84 + (-10*b3 - 5*b1) * q^85 + (2*b2 - 8) * q^86 + (-6*b3 + 6*b1) * q^87 + (-4*b3 + b1) * q^88 + (6*b2 + 6) * q^89 + (-4*b3 - 3*b1) * q^90 + 3*b2 * q^91 + b3 * q^92 + (-3*b3 - 4*b1) * q^93 + (-6*b2 - 8) * q^94 + (7*b2 + 11) * q^95 + b2 * q^96 + (-8*b3 - 13*b1) * q^97 + (11*b3 + 9*b1) * q^98 + (-3*b2 - 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 2 q^{6} + 6 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 2 * q^6 + 6 * q^9 $$4 q - 4 q^{4} - 2 q^{6} + 6 q^{9} - 18 q^{11} + 6 q^{14} + 4 q^{16} + 14 q^{19} - 12 q^{21} + 2 q^{24} + 20 q^{25} - 6 q^{26} + 12 q^{29} + 10 q^{30} - 22 q^{31} - 10 q^{34} - 6 q^{36} - 8 q^{39} - 18 q^{41} + 18 q^{44} + 10 q^{45} - 4 q^{46} - 26 q^{49} + 10 q^{51} - 4 q^{54} + 10 q^{55} - 6 q^{56} + 24 q^{59} - 2 q^{61} - 4 q^{64} + 14 q^{66} - 2 q^{69} + 30 q^{70} + 6 q^{71} - 12 q^{74} - 14 q^{76} + 4 q^{79} - 4 q^{81} + 12 q^{84} - 36 q^{86} + 12 q^{89} - 6 q^{91} - 20 q^{94} + 30 q^{95} - 2 q^{96} - 22 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 2 * q^6 + 6 * q^9 - 18 * q^11 + 6 * q^14 + 4 * q^16 + 14 * q^19 - 12 * q^21 + 2 * q^24 + 20 * q^25 - 6 * q^26 + 12 * q^29 + 10 * q^30 - 22 * q^31 - 10 * q^34 - 6 * q^36 - 8 * q^39 - 18 * q^41 + 18 * q^44 + 10 * q^45 - 4 * q^46 - 26 * q^49 + 10 * q^51 - 4 * q^54 + 10 * q^55 - 6 * q^56 + 24 * q^59 - 2 * q^61 - 4 * q^64 + 14 * q^66 - 2 * q^69 + 30 * q^70 + 6 * q^71 - 12 * q^74 - 14 * q^76 + 4 * q^79 - 4 * q^81 + 12 * q^84 - 36 * q^86 + 12 * q^89 - 6 * q^91 - 20 * q^94 + 30 * q^95 - 2 * q^96 - 22 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 3x^{2} + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ v^2 + 1 $$\beta_{3}$$ $$=$$ $$\nu^{3} + 2\nu$$ v^3 + 2*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ b2 - 1 $$\nu^{3}$$ $$=$$ $$\beta_{3} - 2\beta_1$$ b3 - 2*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/230\mathbb{Z}\right)^\times$$.

 $$n$$ $$47$$ $$51$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
1.00000i 1.61803i −1.00000 −2.23607 −1.61803 1.85410i 1.00000i 0.381966 2.23607i
139.2 1.00000i 0.618034i −1.00000 2.23607 0.618034 4.85410i 1.00000i 2.61803 2.23607i
139.3 1.00000i 0.618034i −1.00000 2.23607 0.618034 4.85410i 1.00000i 2.61803 2.23607i
139.4 1.00000i 1.61803i −1.00000 −2.23607 −1.61803 1.85410i 1.00000i 0.381966 2.23607i
 $$n$$: 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 230.2.b.a 4
3.b odd 2 1 2070.2.d.c 4
4.b odd 2 1 1840.2.e.c 4
5.b even 2 1 inner 230.2.b.a 4
5.c odd 4 1 1150.2.a.l 2
5.c odd 4 1 1150.2.a.n 2
15.d odd 2 1 2070.2.d.c 4
20.d odd 2 1 1840.2.e.c 4
20.e even 4 1 9200.2.a.bo 2
20.e even 4 1 9200.2.a.by 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.a 4 1.a even 1 1 trivial
230.2.b.a 4 5.b even 2 1 inner
1150.2.a.l 2 5.c odd 4 1
1150.2.a.n 2 5.c odd 4 1
1840.2.e.c 4 4.b odd 2 1
1840.2.e.c 4 20.d odd 2 1
2070.2.d.c 4 3.b odd 2 1
2070.2.d.c 4 15.d odd 2 1
9200.2.a.bo 2 20.e even 4 1
9200.2.a.by 2 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 3T_{3}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(230, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 3T^{2} + 1$$
$5$ $$(T^{2} - 5)^{2}$$
$7$ $$T^{4} + 27T^{2} + 81$$
$11$ $$(T^{2} + 9 T + 19)^{2}$$
$13$ $$T^{4} + 7T^{2} + 1$$
$17$ $$T^{4} + 35T^{2} + 25$$
$19$ $$(T^{2} - 7 T + 1)^{2}$$
$23$ $$(T^{2} + 1)^{2}$$
$29$ $$(T^{2} - 6 T - 36)^{2}$$
$31$ $$(T^{2} + 11 T + 19)^{2}$$
$37$ $$T^{4} + 108T^{2} + 1296$$
$41$ $$(T^{2} + 9 T + 19)^{2}$$
$43$ $$T^{4} + 172T^{2} + 5776$$
$47$ $$T^{4} + 140T^{2} + 400$$
$53$ $$(T^{2} + 4)^{2}$$
$59$ $$(T - 6)^{4}$$
$61$ $$(T^{2} + T - 11)^{2}$$
$67$ $$T^{4} + 28T^{2} + 16$$
$71$ $$(T^{2} - 3 T + 1)^{2}$$
$73$ $$T^{4} + 328 T^{2} + 15376$$
$79$ $$(T^{2} - 2 T - 44)^{2}$$
$83$ $$T^{4} + 92T^{2} + 1936$$
$89$ $$(T^{2} - 6 T - 36)^{2}$$
$97$ $$T^{4} + 427 T^{2} + 43681$$