# Properties

 Label 230.2.a.a Level $230$ Weight $2$ Character orbit 230.a Self dual yes Analytic conductor $1.837$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [230,2,Mod(1,230)]

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([0, 0]))

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

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

chi := DirichletCharacter("230.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 230.a (trivial)

## 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: yes Analytic conductor: $$1.83655924649$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{21})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 5$$ x^2 - x - 5 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{21})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 2.79129 −1.79129
−1.00000 −2.79129 1.00000 −1.00000 2.79129 −1.79129 −1.00000 4.79129 1.00000
1.2 −1.00000 1.79129 1.00000 −1.00000 −1.79129 2.79129 −1.00000 0.208712 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$+1$$
$$5$$ $$+1$$
$$23$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.2.a.a 2
3.b odd 2 1 2070.2.a.x 2
4.b odd 2 1 1840.2.a.n 2
5.b even 2 1 1150.2.a.o 2
5.c odd 4 2 1150.2.b.g 4
8.b even 2 1 7360.2.a.bq 2
8.d odd 2 1 7360.2.a.bk 2
20.d odd 2 1 9200.2.a.bs 2
23.b odd 2 1 5290.2.a.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.a 2 1.a even 1 1 trivial
1150.2.a.o 2 5.b even 2 1
1150.2.b.g 4 5.c odd 4 2
1840.2.a.n 2 4.b odd 2 1
2070.2.a.x 2 3.b odd 2 1
5290.2.a.e 2 23.b odd 2 1
7360.2.a.bk 2 8.d odd 2 1
7360.2.a.bq 2 8.b even 2 1
9200.2.a.bs 2 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} - 5$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(230))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} + T - 5$$
$5$ $$(T + 1)^{2}$$
$7$ $$T^{2} - T - 5$$
$11$ $$T^{2} - 3T - 3$$
$13$ $$T^{2} - 7T + 7$$
$17$ $$T^{2} + 3T - 3$$
$19$ $$T^{2} - 7T + 7$$
$23$ $$(T - 1)^{2}$$
$29$ $$T^{2} - 6T - 12$$
$31$ $$T^{2} - 7T - 35$$
$37$ $$(T + 4)^{2}$$
$41$ $$T^{2} + 9T + 15$$
$43$ $$T^{2} - 4T - 80$$
$47$ $$T^{2} + 18T + 60$$
$53$ $$(T - 6)^{2}$$
$59$ $$T^{2} + 18T + 60$$
$61$ $$T^{2} - 7T - 35$$
$67$ $$T^{2} - 4T - 80$$
$71$ $$T^{2} - 3T - 45$$
$73$ $$T^{2} + 2T - 188$$
$79$ $$(T - 8)^{2}$$
$83$ $$(T + 6)^{2}$$
$89$ $$T^{2} - 12T - 48$$
$97$ $$T^{2} - 7T - 119$$