# Properties

 Label 145.2.a.b Level $145$ Weight $2$ Character orbit 145.a Self dual yes Analytic conductor $1.158$ Analytic rank $1$ 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] = [145,2,Mod(1,145)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(145, 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("145.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$145 = 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 145.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: $$1.15783082931$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} - 2 q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - 2 \beta + 2) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9}+O(q^{10})$$ q + (b - 1) * q^2 - 2 * q^3 + (-2*b + 1) * q^4 + q^5 + (-2*b + 2) * q^6 + (-2*b - 2) * q^7 + (b - 3) * q^8 + q^9 $$q + (\beta - 1) q^{2} - 2 q^{3} + ( - 2 \beta + 1) q^{4} + q^{5} + ( - 2 \beta + 2) q^{6} + ( - 2 \beta - 2) q^{7} + (\beta - 3) q^{8} + q^{9} + (\beta - 1) q^{10} + (2 \beta - 2) q^{11} + (4 \beta - 2) q^{12} - 2 q^{13} - 2 q^{14} - 2 q^{15} + 3 q^{16} + 2 \beta q^{17} + (\beta - 1) q^{18} + ( - 2 \beta - 2) q^{19} + ( - 2 \beta + 1) q^{20} + (4 \beta + 4) q^{21} + ( - 4 \beta + 6) q^{22} + (2 \beta - 6) q^{23} + ( - 2 \beta + 6) q^{24} + q^{25} + ( - 2 \beta + 2) q^{26} + 4 q^{27} + (2 \beta + 6) q^{28} + q^{29} + ( - 2 \beta + 2) q^{30} + (6 \beta - 2) q^{31} + (\beta + 3) q^{32} + ( - 4 \beta + 4) q^{33} + ( - 2 \beta + 4) q^{34} + ( - 2 \beta - 2) q^{35} + ( - 2 \beta + 1) q^{36} - 6 \beta q^{37} - 2 q^{38} + 4 q^{39} + (\beta - 3) q^{40} - 6 q^{41} + 4 q^{42} - 6 q^{43} + (6 \beta - 10) q^{44} + q^{45} + ( - 8 \beta + 10) q^{46} + ( - 4 \beta - 6) q^{47} - 6 q^{48} + (8 \beta + 5) q^{49} + (\beta - 1) q^{50} - 4 \beta q^{51} + (4 \beta - 2) q^{52} + ( - 4 \beta + 2) q^{53} + (4 \beta - 4) q^{54} + (2 \beta - 2) q^{55} + (4 \beta + 2) q^{56} + (4 \beta + 4) q^{57} + (\beta - 1) q^{58} + (4 \beta - 2) q^{60} + ( - 4 \beta + 2) q^{61} + ( - 8 \beta + 14) q^{62} + ( - 2 \beta - 2) q^{63} + (2 \beta - 7) q^{64} - 2 q^{65} + (8 \beta - 12) q^{66} + (6 \beta - 2) q^{67} + (2 \beta - 8) q^{68} + ( - 4 \beta + 12) q^{69} - 2 q^{70} + ( - 8 \beta - 4) q^{71} + (\beta - 3) q^{72} + 6 \beta q^{73} + (6 \beta - 12) q^{74} - 2 q^{75} + (2 \beta + 6) q^{76} - 4 q^{77} + (4 \beta - 4) q^{78} + ( - 6 \beta + 6) q^{79} + 3 q^{80} - 11 q^{81} + ( - 6 \beta + 6) q^{82} + ( - 2 \beta + 10) q^{83} + ( - 4 \beta - 12) q^{84} + 2 \beta q^{85} + ( - 6 \beta + 6) q^{86} - 2 q^{87} + ( - 8 \beta + 10) q^{88} + ( - 4 \beta - 2) q^{89} + (\beta - 1) q^{90} + (4 \beta + 4) q^{91} + (14 \beta - 14) q^{92} + ( - 12 \beta + 4) q^{93} + ( - 2 \beta - 2) q^{94} + ( - 2 \beta - 2) q^{95} + ( - 2 \beta - 6) q^{96} + ( - 6 \beta - 4) q^{97} + ( - 3 \beta + 11) q^{98} + (2 \beta - 2) q^{99} +O(q^{100})$$ q + (b - 1) * q^2 - 2 * q^3 + (-2*b + 1) * q^4 + q^5 + (-2*b + 2) * q^6 + (-2*b - 2) * q^7 + (b - 3) * q^8 + q^9 + (b - 1) * q^10 + (2*b - 2) * q^11 + (4*b - 2) * q^12 - 2 * q^13 - 2 * q^14 - 2 * q^15 + 3 * q^16 + 2*b * q^17 + (b - 1) * q^18 + (-2*b - 2) * q^19 + (-2*b + 1) * q^20 + (4*b + 4) * q^21 + (-4*b + 6) * q^22 + (2*b - 6) * q^23 + (-2*b + 6) * q^24 + q^25 + (-2*b + 2) * q^26 + 4 * q^27 + (2*b + 6) * q^28 + q^29 + (-2*b + 2) * q^30 + (6*b - 2) * q^31 + (b + 3) * q^32 + (-4*b + 4) * q^33 + (-2*b + 4) * q^34 + (-2*b - 2) * q^35 + (-2*b + 1) * q^36 - 6*b * q^37 - 2 * q^38 + 4 * q^39 + (b - 3) * q^40 - 6 * q^41 + 4 * q^42 - 6 * q^43 + (6*b - 10) * q^44 + q^45 + (-8*b + 10) * q^46 + (-4*b - 6) * q^47 - 6 * q^48 + (8*b + 5) * q^49 + (b - 1) * q^50 - 4*b * q^51 + (4*b - 2) * q^52 + (-4*b + 2) * q^53 + (4*b - 4) * q^54 + (2*b - 2) * q^55 + (4*b + 2) * q^56 + (4*b + 4) * q^57 + (b - 1) * q^58 + (4*b - 2) * q^60 + (-4*b + 2) * q^61 + (-8*b + 14) * q^62 + (-2*b - 2) * q^63 + (2*b - 7) * q^64 - 2 * q^65 + (8*b - 12) * q^66 + (6*b - 2) * q^67 + (2*b - 8) * q^68 + (-4*b + 12) * q^69 - 2 * q^70 + (-8*b - 4) * q^71 + (b - 3) * q^72 + 6*b * q^73 + (6*b - 12) * q^74 - 2 * q^75 + (2*b + 6) * q^76 - 4 * q^77 + (4*b - 4) * q^78 + (-6*b + 6) * q^79 + 3 * q^80 - 11 * q^81 + (-6*b + 6) * q^82 + (-2*b + 10) * q^83 + (-4*b - 12) * q^84 + 2*b * q^85 + (-6*b + 6) * q^86 - 2 * q^87 + (-8*b + 10) * q^88 + (-4*b - 2) * q^89 + (b - 1) * q^90 + (4*b + 4) * q^91 + (14*b - 14) * q^92 + (-12*b + 4) * q^93 + (-2*b - 2) * q^94 + (-2*b - 2) * q^95 + (-2*b - 6) * q^96 + (-6*b - 4) * q^97 + (-3*b + 11) * q^98 + (2*b - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 $$2 q - 2 q^{2} - 4 q^{3} + 2 q^{4} + 2 q^{5} + 4 q^{6} - 4 q^{7} - 6 q^{8} + 2 q^{9} - 2 q^{10} - 4 q^{11} - 4 q^{12} - 4 q^{13} - 4 q^{14} - 4 q^{15} + 6 q^{16} - 2 q^{18} - 4 q^{19} + 2 q^{20} + 8 q^{21} + 12 q^{22} - 12 q^{23} + 12 q^{24} + 2 q^{25} + 4 q^{26} + 8 q^{27} + 12 q^{28} + 2 q^{29} + 4 q^{30} - 4 q^{31} + 6 q^{32} + 8 q^{33} + 8 q^{34} - 4 q^{35} + 2 q^{36} - 4 q^{38} + 8 q^{39} - 6 q^{40} - 12 q^{41} + 8 q^{42} - 12 q^{43} - 20 q^{44} + 2 q^{45} + 20 q^{46} - 12 q^{47} - 12 q^{48} + 10 q^{49} - 2 q^{50} - 4 q^{52} + 4 q^{53} - 8 q^{54} - 4 q^{55} + 4 q^{56} + 8 q^{57} - 2 q^{58} - 4 q^{60} + 4 q^{61} + 28 q^{62} - 4 q^{63} - 14 q^{64} - 4 q^{65} - 24 q^{66} - 4 q^{67} - 16 q^{68} + 24 q^{69} - 4 q^{70} - 8 q^{71} - 6 q^{72} - 24 q^{74} - 4 q^{75} + 12 q^{76} - 8 q^{77} - 8 q^{78} + 12 q^{79} + 6 q^{80} - 22 q^{81} + 12 q^{82} + 20 q^{83} - 24 q^{84} + 12 q^{86} - 4 q^{87} + 20 q^{88} - 4 q^{89} - 2 q^{90} + 8 q^{91} - 28 q^{92} + 8 q^{93} - 4 q^{94} - 4 q^{95} - 12 q^{96} - 8 q^{97} + 22 q^{98} - 4 q^{99}+O(q^{100})$$ 2 * q - 2 * q^2 - 4 * q^3 + 2 * q^4 + 2 * q^5 + 4 * q^6 - 4 * q^7 - 6 * q^8 + 2 * q^9 - 2 * q^10 - 4 * q^11 - 4 * q^12 - 4 * q^13 - 4 * q^14 - 4 * q^15 + 6 * q^16 - 2 * q^18 - 4 * q^19 + 2 * q^20 + 8 * q^21 + 12 * q^22 - 12 * q^23 + 12 * q^24 + 2 * q^25 + 4 * q^26 + 8 * q^27 + 12 * q^28 + 2 * q^29 + 4 * q^30 - 4 * q^31 + 6 * q^32 + 8 * q^33 + 8 * q^34 - 4 * q^35 + 2 * q^36 - 4 * q^38 + 8 * q^39 - 6 * q^40 - 12 * q^41 + 8 * q^42 - 12 * q^43 - 20 * q^44 + 2 * q^45 + 20 * q^46 - 12 * q^47 - 12 * q^48 + 10 * q^49 - 2 * q^50 - 4 * q^52 + 4 * q^53 - 8 * q^54 - 4 * q^55 + 4 * q^56 + 8 * q^57 - 2 * q^58 - 4 * q^60 + 4 * q^61 + 28 * q^62 - 4 * q^63 - 14 * q^64 - 4 * q^65 - 24 * q^66 - 4 * q^67 - 16 * q^68 + 24 * q^69 - 4 * q^70 - 8 * q^71 - 6 * q^72 - 24 * q^74 - 4 * q^75 + 12 * q^76 - 8 * q^77 - 8 * q^78 + 12 * q^79 + 6 * q^80 - 22 * q^81 + 12 * q^82 + 20 * q^83 - 24 * q^84 + 12 * q^86 - 4 * q^87 + 20 * q^88 - 4 * q^89 - 2 * q^90 + 8 * q^91 - 28 * q^92 + 8 * q^93 - 4 * q^94 - 4 * q^95 - 12 * q^96 - 8 * q^97 + 22 * q^98 - 4 * 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
 −1.41421 1.41421
−2.41421 −2.00000 3.82843 1.00000 4.82843 0.828427 −4.41421 1.00000 −2.41421
1.2 0.414214 −2.00000 −1.82843 1.00000 −0.828427 −4.82843 −1.58579 1.00000 0.414214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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 145.2.a.b 2
3.b odd 2 1 1305.2.a.n 2
4.b odd 2 1 2320.2.a.k 2
5.b even 2 1 725.2.a.c 2
5.c odd 4 2 725.2.b.c 4
7.b odd 2 1 7105.2.a.e 2
8.b even 2 1 9280.2.a.be 2
8.d odd 2 1 9280.2.a.w 2
15.d odd 2 1 6525.2.a.p 2
29.b even 2 1 4205.2.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.a.b 2 1.a even 1 1 trivial
725.2.a.c 2 5.b even 2 1
725.2.b.c 4 5.c odd 4 2
1305.2.a.n 2 3.b odd 2 1
2320.2.a.k 2 4.b odd 2 1
4205.2.a.d 2 29.b even 2 1
6525.2.a.p 2 15.d odd 2 1
7105.2.a.e 2 7.b odd 2 1
9280.2.a.w 2 8.d odd 2 1
9280.2.a.be 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + 2T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(145))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$(T + 2)^{2}$$
$5$ $$(T - 1)^{2}$$
$7$ $$T^{2} + 4T - 4$$
$11$ $$T^{2} + 4T - 4$$
$13$ $$(T + 2)^{2}$$
$17$ $$T^{2} - 8$$
$19$ $$T^{2} + 4T - 4$$
$23$ $$T^{2} + 12T + 28$$
$29$ $$(T - 1)^{2}$$
$31$ $$T^{2} + 4T - 68$$
$37$ $$T^{2} - 72$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T + 6)^{2}$$
$47$ $$T^{2} + 12T + 4$$
$53$ $$T^{2} - 4T - 28$$
$59$ $$T^{2}$$
$61$ $$T^{2} - 4T - 28$$
$67$ $$T^{2} + 4T - 68$$
$71$ $$T^{2} + 8T - 112$$
$73$ $$T^{2} - 72$$
$79$ $$T^{2} - 12T - 36$$
$83$ $$T^{2} - 20T + 92$$
$89$ $$T^{2} + 4T - 28$$
$97$ $$T^{2} + 8T - 56$$