# Properties

 Label 65.4.a.a Level $65$ Weight $4$ Character orbit 65.a Self dual yes Analytic conductor $3.835$ 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] = [65,4,Mod(1,65)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(65, 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("65.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$65 = 5 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 65.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: $$3.83512415037$$ 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)

 $$f(q)$$ $$=$$ $$q + 5 q^{2} + 2 q^{3} + 17 q^{4} - 5 q^{5} + 10 q^{6} - 12 q^{7} + 45 q^{8} - 23 q^{9}+O(q^{10})$$ q + 5 * q^2 + 2 * q^3 + 17 * q^4 - 5 * q^5 + 10 * q^6 - 12 * q^7 + 45 * q^8 - 23 * q^9 $$q + 5 q^{2} + 2 q^{3} + 17 q^{4} - 5 q^{5} + 10 q^{6} - 12 q^{7} + 45 q^{8} - 23 q^{9} - 25 q^{10} + 14 q^{11} + 34 q^{12} - 13 q^{13} - 60 q^{14} - 10 q^{15} + 89 q^{16} + 98 q^{17} - 115 q^{18} - 26 q^{19} - 85 q^{20} - 24 q^{21} + 70 q^{22} - 114 q^{23} + 90 q^{24} + 25 q^{25} - 65 q^{26} - 100 q^{27} - 204 q^{28} + 58 q^{29} - 50 q^{30} + 306 q^{31} + 85 q^{32} + 28 q^{33} + 490 q^{34} + 60 q^{35} - 391 q^{36} + 86 q^{37} - 130 q^{38} - 26 q^{39} - 225 q^{40} - 374 q^{41} - 120 q^{42} - 314 q^{43} + 238 q^{44} + 115 q^{45} - 570 q^{46} + 620 q^{47} + 178 q^{48} - 199 q^{49} + 125 q^{50} + 196 q^{51} - 221 q^{52} + 362 q^{53} - 500 q^{54} - 70 q^{55} - 540 q^{56} - 52 q^{57} + 290 q^{58} + 266 q^{59} - 170 q^{60} + 634 q^{61} + 1530 q^{62} + 276 q^{63} - 287 q^{64} + 65 q^{65} + 140 q^{66} + 612 q^{67} + 1666 q^{68} - 228 q^{69} + 300 q^{70} - 686 q^{71} - 1035 q^{72} + 202 q^{73} + 430 q^{74} + 50 q^{75} - 442 q^{76} - 168 q^{77} - 130 q^{78} - 516 q^{79} - 445 q^{80} + 421 q^{81} - 1870 q^{82} + 48 q^{83} - 408 q^{84} - 490 q^{85} - 1570 q^{86} + 116 q^{87} + 630 q^{88} - 1230 q^{89} + 575 q^{90} + 156 q^{91} - 1938 q^{92} + 612 q^{93} + 3100 q^{94} + 130 q^{95} + 170 q^{96} + 350 q^{97} - 995 q^{98} - 322 q^{99}+O(q^{100})$$ q + 5 * q^2 + 2 * q^3 + 17 * q^4 - 5 * q^5 + 10 * q^6 - 12 * q^7 + 45 * q^8 - 23 * q^9 - 25 * q^10 + 14 * q^11 + 34 * q^12 - 13 * q^13 - 60 * q^14 - 10 * q^15 + 89 * q^16 + 98 * q^17 - 115 * q^18 - 26 * q^19 - 85 * q^20 - 24 * q^21 + 70 * q^22 - 114 * q^23 + 90 * q^24 + 25 * q^25 - 65 * q^26 - 100 * q^27 - 204 * q^28 + 58 * q^29 - 50 * q^30 + 306 * q^31 + 85 * q^32 + 28 * q^33 + 490 * q^34 + 60 * q^35 - 391 * q^36 + 86 * q^37 - 130 * q^38 - 26 * q^39 - 225 * q^40 - 374 * q^41 - 120 * q^42 - 314 * q^43 + 238 * q^44 + 115 * q^45 - 570 * q^46 + 620 * q^47 + 178 * q^48 - 199 * q^49 + 125 * q^50 + 196 * q^51 - 221 * q^52 + 362 * q^53 - 500 * q^54 - 70 * q^55 - 540 * q^56 - 52 * q^57 + 290 * q^58 + 266 * q^59 - 170 * q^60 + 634 * q^61 + 1530 * q^62 + 276 * q^63 - 287 * q^64 + 65 * q^65 + 140 * q^66 + 612 * q^67 + 1666 * q^68 - 228 * q^69 + 300 * q^70 - 686 * q^71 - 1035 * q^72 + 202 * q^73 + 430 * q^74 + 50 * q^75 - 442 * q^76 - 168 * q^77 - 130 * q^78 - 516 * q^79 - 445 * q^80 + 421 * q^81 - 1870 * q^82 + 48 * q^83 - 408 * q^84 - 490 * q^85 - 1570 * q^86 + 116 * q^87 + 630 * q^88 - 1230 * q^89 + 575 * q^90 + 156 * q^91 - 1938 * q^92 + 612 * q^93 + 3100 * q^94 + 130 * q^95 + 170 * q^96 + 350 * q^97 - 995 * q^98 - 322 * 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
5.00000 2.00000 17.0000 −5.00000 10.0000 −12.0000 45.0000 −23.0000 −25.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$13$$ $$1$$

## 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 65.4.a.a 1
3.b odd 2 1 585.4.a.a 1
4.b odd 2 1 1040.4.a.a 1
5.b even 2 1 325.4.a.a 1
5.c odd 4 2 325.4.b.a 2
13.b even 2 1 845.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.4.a.a 1 1.a even 1 1 trivial
325.4.a.a 1 5.b even 2 1
325.4.b.a 2 5.c odd 4 2
585.4.a.a 1 3.b odd 2 1
845.4.a.a 1 13.b even 2 1
1040.4.a.a 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 5$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(65))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 5$$
$3$ $$T - 2$$
$5$ $$T + 5$$
$7$ $$T + 12$$
$11$ $$T - 14$$
$13$ $$T + 13$$
$17$ $$T - 98$$
$19$ $$T + 26$$
$23$ $$T + 114$$
$29$ $$T - 58$$
$31$ $$T - 306$$
$37$ $$T - 86$$
$41$ $$T + 374$$
$43$ $$T + 314$$
$47$ $$T - 620$$
$53$ $$T - 362$$
$59$ $$T - 266$$
$61$ $$T - 634$$
$67$ $$T - 612$$
$71$ $$T + 686$$
$73$ $$T - 202$$
$79$ $$T + 516$$
$83$ $$T - 48$$
$89$ $$T + 1230$$
$97$ $$T - 350$$