Properties

 Label 230.2.a.b 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: $$\Q(\sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 3$$ x^2 - x - 3 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + (\beta + 1) q^{3} + q^{4} + q^{5} + ( - \beta - 1) q^{6} + ( - \beta + 2) q^{7} - q^{8} + (3 \beta + 1) q^{9}+O(q^{10})$$ q - q^2 + (b + 1) * q^3 + q^4 + q^5 + (-b - 1) * q^6 + (-b + 2) * q^7 - q^8 + (3*b + 1) * q^9 $$q - q^{2} + (\beta + 1) q^{3} + q^{4} + q^{5} + ( - \beta - 1) q^{6} + ( - \beta + 2) q^{7} - q^{8} + (3 \beta + 1) q^{9} - q^{10} + ( - \beta - 3) q^{11} + (\beta + 1) q^{12} + ( - \beta + 2) q^{13} + (\beta - 2) q^{14} + (\beta + 1) q^{15} + q^{16} + ( - 3 \beta + 3) q^{17} + ( - 3 \beta - 1) q^{18} + ( - 3 \beta + 2) q^{19} + q^{20} - q^{21} + (\beta + 3) q^{22} - q^{23} + ( - \beta - 1) q^{24} + q^{25} + (\beta - 2) q^{26} + (4 \beta + 7) q^{27} + ( - \beta + 2) q^{28} + 2 \beta q^{29} + ( - \beta - 1) q^{30} + (3 \beta - 4) q^{31} - q^{32} + ( - 5 \beta - 6) q^{33} + (3 \beta - 3) q^{34} + ( - \beta + 2) q^{35} + (3 \beta + 1) q^{36} + 8 q^{37} + (3 \beta - 2) q^{38} - q^{39} - q^{40} + ( - 3 \beta - 3) q^{41} + q^{42} + (4 \beta - 4) q^{43} + ( - \beta - 3) q^{44} + (3 \beta + 1) q^{45} + q^{46} + 2 \beta q^{47} + (\beta + 1) q^{48} - 3 \beta q^{49} - q^{50} + ( - 3 \beta - 6) q^{51} + ( - \beta + 2) q^{52} + (4 \beta - 6) q^{53} + ( - 4 \beta - 7) q^{54} + ( - \beta - 3) q^{55} + (\beta - 2) q^{56} + ( - 4 \beta - 7) q^{57} - 2 \beta q^{58} + ( - 2 \beta - 6) q^{59} + (\beta + 1) q^{60} + ( - 5 \beta + 5) q^{61} + ( - 3 \beta + 4) q^{62} + (2 \beta - 7) q^{63} + q^{64} + ( - \beta + 2) q^{65} + (5 \beta + 6) q^{66} - 4 q^{67} + ( - 3 \beta + 3) q^{68} + ( - \beta - 1) q^{69} + (\beta - 2) q^{70} + (\beta - 15) q^{71} + ( - 3 \beta - 1) q^{72} + (6 \beta + 2) q^{73} - 8 q^{74} + (\beta + 1) q^{75} + ( - 3 \beta + 2) q^{76} + (2 \beta - 3) q^{77} + q^{78} + (8 \beta - 4) q^{79} + q^{80} + (6 \beta + 16) q^{81} + (3 \beta + 3) q^{82} + ( - 4 \beta + 6) q^{83} - q^{84} + ( - 3 \beta + 3) q^{85} + ( - 4 \beta + 4) q^{86} + (4 \beta + 6) q^{87} + (\beta + 3) q^{88} + ( - 3 \beta - 1) q^{90} + ( - 3 \beta + 7) q^{91} - q^{92} + (2 \beta + 5) q^{93} - 2 \beta q^{94} + ( - 3 \beta + 2) q^{95} + ( - \beta - 1) q^{96} + ( - \beta + 5) q^{97} + 3 \beta q^{98} + ( - 13 \beta - 12) q^{99} +O(q^{100})$$ q - q^2 + (b + 1) * q^3 + q^4 + q^5 + (-b - 1) * q^6 + (-b + 2) * q^7 - q^8 + (3*b + 1) * q^9 - q^10 + (-b - 3) * q^11 + (b + 1) * q^12 + (-b + 2) * q^13 + (b - 2) * q^14 + (b + 1) * q^15 + q^16 + (-3*b + 3) * q^17 + (-3*b - 1) * q^18 + (-3*b + 2) * q^19 + q^20 - q^21 + (b + 3) * q^22 - q^23 + (-b - 1) * q^24 + q^25 + (b - 2) * q^26 + (4*b + 7) * q^27 + (-b + 2) * q^28 + 2*b * q^29 + (-b - 1) * q^30 + (3*b - 4) * q^31 - q^32 + (-5*b - 6) * q^33 + (3*b - 3) * q^34 + (-b + 2) * q^35 + (3*b + 1) * q^36 + 8 * q^37 + (3*b - 2) * q^38 - q^39 - q^40 + (-3*b - 3) * q^41 + q^42 + (4*b - 4) * q^43 + (-b - 3) * q^44 + (3*b + 1) * q^45 + q^46 + 2*b * q^47 + (b + 1) * q^48 - 3*b * q^49 - q^50 + (-3*b - 6) * q^51 + (-b + 2) * q^52 + (4*b - 6) * q^53 + (-4*b - 7) * q^54 + (-b - 3) * q^55 + (b - 2) * q^56 + (-4*b - 7) * q^57 - 2*b * q^58 + (-2*b - 6) * q^59 + (b + 1) * q^60 + (-5*b + 5) * q^61 + (-3*b + 4) * q^62 + (2*b - 7) * q^63 + q^64 + (-b + 2) * q^65 + (5*b + 6) * q^66 - 4 * q^67 + (-3*b + 3) * q^68 + (-b - 1) * q^69 + (b - 2) * q^70 + (b - 15) * q^71 + (-3*b - 1) * q^72 + (6*b + 2) * q^73 - 8 * q^74 + (b + 1) * q^75 + (-3*b + 2) * q^76 + (2*b - 3) * q^77 + q^78 + (8*b - 4) * q^79 + q^80 + (6*b + 16) * q^81 + (3*b + 3) * q^82 + (-4*b + 6) * q^83 - q^84 + (-3*b + 3) * q^85 + (-4*b + 4) * q^86 + (4*b + 6) * q^87 + (b + 3) * q^88 + (-3*b - 1) * q^90 + (-3*b + 7) * q^91 - q^92 + (2*b + 5) * q^93 - 2*b * q^94 + (-3*b + 2) * q^95 + (-b - 1) * q^96 + (-b + 5) * q^97 + 3*b * q^98 + (-13*b - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 2 q^{8} + 5 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 2 * q^8 + 5 * q^9 $$2 q - 2 q^{2} + 3 q^{3} + 2 q^{4} + 2 q^{5} - 3 q^{6} + 3 q^{7} - 2 q^{8} + 5 q^{9} - 2 q^{10} - 7 q^{11} + 3 q^{12} + 3 q^{13} - 3 q^{14} + 3 q^{15} + 2 q^{16} + 3 q^{17} - 5 q^{18} + q^{19} + 2 q^{20} - 2 q^{21} + 7 q^{22} - 2 q^{23} - 3 q^{24} + 2 q^{25} - 3 q^{26} + 18 q^{27} + 3 q^{28} + 2 q^{29} - 3 q^{30} - 5 q^{31} - 2 q^{32} - 17 q^{33} - 3 q^{34} + 3 q^{35} + 5 q^{36} + 16 q^{37} - q^{38} - 2 q^{39} - 2 q^{40} - 9 q^{41} + 2 q^{42} - 4 q^{43} - 7 q^{44} + 5 q^{45} + 2 q^{46} + 2 q^{47} + 3 q^{48} - 3 q^{49} - 2 q^{50} - 15 q^{51} + 3 q^{52} - 8 q^{53} - 18 q^{54} - 7 q^{55} - 3 q^{56} - 18 q^{57} - 2 q^{58} - 14 q^{59} + 3 q^{60} + 5 q^{61} + 5 q^{62} - 12 q^{63} + 2 q^{64} + 3 q^{65} + 17 q^{66} - 8 q^{67} + 3 q^{68} - 3 q^{69} - 3 q^{70} - 29 q^{71} - 5 q^{72} + 10 q^{73} - 16 q^{74} + 3 q^{75} + q^{76} - 4 q^{77} + 2 q^{78} + 2 q^{80} + 38 q^{81} + 9 q^{82} + 8 q^{83} - 2 q^{84} + 3 q^{85} + 4 q^{86} + 16 q^{87} + 7 q^{88} - 5 q^{90} + 11 q^{91} - 2 q^{92} + 12 q^{93} - 2 q^{94} + q^{95} - 3 q^{96} + 9 q^{97} + 3 q^{98} - 37 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 + 3 * q^3 + 2 * q^4 + 2 * q^5 - 3 * q^6 + 3 * q^7 - 2 * q^8 + 5 * q^9 - 2 * q^10 - 7 * q^11 + 3 * q^12 + 3 * q^13 - 3 * q^14 + 3 * q^15 + 2 * q^16 + 3 * q^17 - 5 * q^18 + q^19 + 2 * q^20 - 2 * q^21 + 7 * q^22 - 2 * q^23 - 3 * q^24 + 2 * q^25 - 3 * q^26 + 18 * q^27 + 3 * q^28 + 2 * q^29 - 3 * q^30 - 5 * q^31 - 2 * q^32 - 17 * q^33 - 3 * q^34 + 3 * q^35 + 5 * q^36 + 16 * q^37 - q^38 - 2 * q^39 - 2 * q^40 - 9 * q^41 + 2 * q^42 - 4 * q^43 - 7 * q^44 + 5 * q^45 + 2 * q^46 + 2 * q^47 + 3 * q^48 - 3 * q^49 - 2 * q^50 - 15 * q^51 + 3 * q^52 - 8 * q^53 - 18 * q^54 - 7 * q^55 - 3 * q^56 - 18 * q^57 - 2 * q^58 - 14 * q^59 + 3 * q^60 + 5 * q^61 + 5 * q^62 - 12 * q^63 + 2 * q^64 + 3 * q^65 + 17 * q^66 - 8 * q^67 + 3 * q^68 - 3 * q^69 - 3 * q^70 - 29 * q^71 - 5 * q^72 + 10 * q^73 - 16 * q^74 + 3 * q^75 + q^76 - 4 * q^77 + 2 * q^78 + 2 * q^80 + 38 * q^81 + 9 * q^82 + 8 * q^83 - 2 * q^84 + 3 * q^85 + 4 * q^86 + 16 * q^87 + 7 * q^88 - 5 * q^90 + 11 * q^91 - 2 * q^92 + 12 * q^93 - 2 * q^94 + q^95 - 3 * q^96 + 9 * q^97 + 3 * q^98 - 37 * 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
 −1.30278 2.30278
−1.00000 −0.302776 1.00000 1.00000 0.302776 3.30278 −1.00000 −2.90833 −1.00000
1.2 −1.00000 3.30278 1.00000 1.00000 −3.30278 −0.302776 −1.00000 7.90833 −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.b 2
3.b odd 2 1 2070.2.a.w 2
4.b odd 2 1 1840.2.a.j 2
5.b even 2 1 1150.2.a.m 2
5.c odd 4 2 1150.2.b.f 4
8.b even 2 1 7360.2.a.bc 2
8.d odd 2 1 7360.2.a.bu 2
20.d odd 2 1 9200.2.a.ca 2
23.b odd 2 1 5290.2.a.j 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.b 2 1.a even 1 1 trivial
1150.2.a.m 2 5.b even 2 1
1150.2.b.f 4 5.c odd 4 2
1840.2.a.j 2 4.b odd 2 1
2070.2.a.w 2 3.b odd 2 1
5290.2.a.j 2 23.b odd 2 1
7360.2.a.bc 2 8.b even 2 1
7360.2.a.bu 2 8.d odd 2 1
9200.2.a.ca 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 3T_{3} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(230))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T + 1)^{2}$$
$3$ $$T^{2} - 3T - 1$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} - 3T - 1$$
$11$ $$T^{2} + 7T + 9$$
$13$ $$T^{2} - 3T - 1$$
$17$ $$T^{2} - 3T - 27$$
$19$ $$T^{2} - T - 29$$
$23$ $$(T + 1)^{2}$$
$29$ $$T^{2} - 2T - 12$$
$31$ $$T^{2} + 5T - 23$$
$37$ $$(T - 8)^{2}$$
$41$ $$T^{2} + 9T - 9$$
$43$ $$T^{2} + 4T - 48$$
$47$ $$T^{2} - 2T - 12$$
$53$ $$T^{2} + 8T - 36$$
$59$ $$T^{2} + 14T + 36$$
$61$ $$T^{2} - 5T - 75$$
$67$ $$(T + 4)^{2}$$
$71$ $$T^{2} + 29T + 207$$
$73$ $$T^{2} - 10T - 92$$
$79$ $$T^{2} - 208$$
$83$ $$T^{2} - 8T - 36$$
$89$ $$T^{2}$$
$97$ $$T^{2} - 9T + 17$$