# Properties

 Label 145.4.a.a Level $145$ Weight $4$ Character orbit 145.a Self dual yes Analytic conductor $8.555$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [145,4,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, 4, names="a")

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

chi := DirichletCharacter("145.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$145 = 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ 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: $$8.55527695083$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 8 q^{3} - 7 q^{4} - 5 q^{5} - 8 q^{6} - 14 q^{7} - 15 q^{8} + 37 q^{9}+O(q^{10})$$ q + q^2 - 8 * q^3 - 7 * q^4 - 5 * q^5 - 8 * q^6 - 14 * q^7 - 15 * q^8 + 37 * q^9 $$q + q^{2} - 8 q^{3} - 7 q^{4} - 5 q^{5} - 8 q^{6} - 14 q^{7} - 15 q^{8} + 37 q^{9} - 5 q^{10} + 62 q^{11} + 56 q^{12} + 42 q^{13} - 14 q^{14} + 40 q^{15} + 41 q^{16} - 114 q^{17} + 37 q^{18} - 70 q^{19} + 35 q^{20} + 112 q^{21} + 62 q^{22} + 62 q^{23} + 120 q^{24} + 25 q^{25} + 42 q^{26} - 80 q^{27} + 98 q^{28} - 29 q^{29} + 40 q^{30} + 142 q^{31} + 161 q^{32} - 496 q^{33} - 114 q^{34} + 70 q^{35} - 259 q^{36} + 146 q^{37} - 70 q^{38} - 336 q^{39} + 75 q^{40} + 162 q^{41} + 112 q^{42} + 352 q^{43} - 434 q^{44} - 185 q^{45} + 62 q^{46} - 444 q^{47} - 328 q^{48} - 147 q^{49} + 25 q^{50} + 912 q^{51} - 294 q^{52} - 238 q^{53} - 80 q^{54} - 310 q^{55} + 210 q^{56} + 560 q^{57} - 29 q^{58} + 840 q^{59} - 280 q^{60} + 2 q^{61} + 142 q^{62} - 518 q^{63} - 167 q^{64} - 210 q^{65} - 496 q^{66} - 154 q^{67} + 798 q^{68} - 496 q^{69} + 70 q^{70} + 892 q^{71} - 555 q^{72} - 38 q^{73} + 146 q^{74} - 200 q^{75} + 490 q^{76} - 868 q^{77} - 336 q^{78} + 1050 q^{79} - 205 q^{80} - 359 q^{81} + 162 q^{82} - 778 q^{83} - 784 q^{84} + 570 q^{85} + 352 q^{86} + 232 q^{87} - 930 q^{88} + 1410 q^{89} - 185 q^{90} - 588 q^{91} - 434 q^{92} - 1136 q^{93} - 444 q^{94} + 350 q^{95} - 1288 q^{96} + 466 q^{97} - 147 q^{98} + 2294 q^{99}+O(q^{100})$$ q + q^2 - 8 * q^3 - 7 * q^4 - 5 * q^5 - 8 * q^6 - 14 * q^7 - 15 * q^8 + 37 * q^9 - 5 * q^10 + 62 * q^11 + 56 * q^12 + 42 * q^13 - 14 * q^14 + 40 * q^15 + 41 * q^16 - 114 * q^17 + 37 * q^18 - 70 * q^19 + 35 * q^20 + 112 * q^21 + 62 * q^22 + 62 * q^23 + 120 * q^24 + 25 * q^25 + 42 * q^26 - 80 * q^27 + 98 * q^28 - 29 * q^29 + 40 * q^30 + 142 * q^31 + 161 * q^32 - 496 * q^33 - 114 * q^34 + 70 * q^35 - 259 * q^36 + 146 * q^37 - 70 * q^38 - 336 * q^39 + 75 * q^40 + 162 * q^41 + 112 * q^42 + 352 * q^43 - 434 * q^44 - 185 * q^45 + 62 * q^46 - 444 * q^47 - 328 * q^48 - 147 * q^49 + 25 * q^50 + 912 * q^51 - 294 * q^52 - 238 * q^53 - 80 * q^54 - 310 * q^55 + 210 * q^56 + 560 * q^57 - 29 * q^58 + 840 * q^59 - 280 * q^60 + 2 * q^61 + 142 * q^62 - 518 * q^63 - 167 * q^64 - 210 * q^65 - 496 * q^66 - 154 * q^67 + 798 * q^68 - 496 * q^69 + 70 * q^70 + 892 * q^71 - 555 * q^72 - 38 * q^73 + 146 * q^74 - 200 * q^75 + 490 * q^76 - 868 * q^77 - 336 * q^78 + 1050 * q^79 - 205 * q^80 - 359 * q^81 + 162 * q^82 - 778 * q^83 - 784 * q^84 + 570 * q^85 + 352 * q^86 + 232 * q^87 - 930 * q^88 + 1410 * q^89 - 185 * q^90 - 588 * q^91 - 434 * q^92 - 1136 * q^93 - 444 * q^94 + 350 * q^95 - 1288 * q^96 + 466 * q^97 - 147 * q^98 + 2294 * 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
 0
1.00000 −8.00000 −7.00000 −5.00000 −8.00000 −14.0000 −15.0000 37.0000 −5.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$+1$$
$$29$$ $$+1$$

## 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 145.4.a.a 1
3.b odd 2 1 1305.4.a.b 1
4.b odd 2 1 2320.4.a.f 1
5.b even 2 1 725.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.a.a 1 1.a even 1 1 trivial
725.4.a.a 1 5.b even 2 1
1305.4.a.b 1 3.b odd 2 1
2320.4.a.f 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(145))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 8$$
$5$ $$T + 5$$
$7$ $$T + 14$$
$11$ $$T - 62$$
$13$ $$T - 42$$
$17$ $$T + 114$$
$19$ $$T + 70$$
$23$ $$T - 62$$
$29$ $$T + 29$$
$31$ $$T - 142$$
$37$ $$T - 146$$
$41$ $$T - 162$$
$43$ $$T - 352$$
$47$ $$T + 444$$
$53$ $$T + 238$$
$59$ $$T - 840$$
$61$ $$T - 2$$
$67$ $$T + 154$$
$71$ $$T - 892$$
$73$ $$T + 38$$
$79$ $$T - 1050$$
$83$ $$T + 778$$
$89$ $$T - 1410$$
$97$ $$T - 466$$