# Properties

 Label 1305.4.a.d Level $1305$ Weight $4$ Character orbit 1305.a Self dual yes Analytic conductor $76.997$ Analytic rank $1$ 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] = [1305,4,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1305.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1305 = 3^{2} \cdot 5 \cdot 29$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1305.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: $$76.9974925575$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 435) 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)

 $$f(q)$$ $$=$$ $$q + 2 q^{2} - 4 q^{4} - 5 q^{5} + 29 q^{7} - 24 q^{8}+O(q^{10})$$ q + 2 * q^2 - 4 * q^4 - 5 * q^5 + 29 * q^7 - 24 * q^8 $$q + 2 q^{2} - 4 q^{4} - 5 q^{5} + 29 q^{7} - 24 q^{8} - 10 q^{10} + 15 q^{11} + 3 q^{13} + 58 q^{14} - 16 q^{16} - 121 q^{17} - 40 q^{19} + 20 q^{20} + 30 q^{22} + 116 q^{23} + 25 q^{25} + 6 q^{26} - 116 q^{28} - 29 q^{29} - 116 q^{31} + 160 q^{32} - 242 q^{34} - 145 q^{35} + 36 q^{37} - 80 q^{38} + 120 q^{40} + 170 q^{41} + 230 q^{43} - 60 q^{44} + 232 q^{46} - 231 q^{47} + 498 q^{49} + 50 q^{50} - 12 q^{52} - 456 q^{53} - 75 q^{55} - 696 q^{56} - 58 q^{58} - 576 q^{59} + 342 q^{61} - 232 q^{62} + 448 q^{64} - 15 q^{65} - 269 q^{67} + 484 q^{68} - 290 q^{70} - 302 q^{71} - 372 q^{73} + 72 q^{74} + 160 q^{76} + 435 q^{77} - 348 q^{79} + 80 q^{80} + 340 q^{82} + 512 q^{83} + 605 q^{85} + 460 q^{86} - 360 q^{88} - 1525 q^{89} + 87 q^{91} - 464 q^{92} - 462 q^{94} + 200 q^{95} - 560 q^{97} + 996 q^{98}+O(q^{100})$$ q + 2 * q^2 - 4 * q^4 - 5 * q^5 + 29 * q^7 - 24 * q^8 - 10 * q^10 + 15 * q^11 + 3 * q^13 + 58 * q^14 - 16 * q^16 - 121 * q^17 - 40 * q^19 + 20 * q^20 + 30 * q^22 + 116 * q^23 + 25 * q^25 + 6 * q^26 - 116 * q^28 - 29 * q^29 - 116 * q^31 + 160 * q^32 - 242 * q^34 - 145 * q^35 + 36 * q^37 - 80 * q^38 + 120 * q^40 + 170 * q^41 + 230 * q^43 - 60 * q^44 + 232 * q^46 - 231 * q^47 + 498 * q^49 + 50 * q^50 - 12 * q^52 - 456 * q^53 - 75 * q^55 - 696 * q^56 - 58 * q^58 - 576 * q^59 + 342 * q^61 - 232 * q^62 + 448 * q^64 - 15 * q^65 - 269 * q^67 + 484 * q^68 - 290 * q^70 - 302 * q^71 - 372 * q^73 + 72 * q^74 + 160 * q^76 + 435 * q^77 - 348 * q^79 + 80 * q^80 + 340 * q^82 + 512 * q^83 + 605 * q^85 + 460 * q^86 - 360 * q^88 - 1525 * q^89 + 87 * q^91 - 464 * q^92 - 462 * q^94 + 200 * q^95 - 560 * q^97 + 996 * q^98

## 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
2.00000 0 −4.00000 −5.00000 0 29.0000 −24.0000 0 −10.0000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$+1$$
$$29$$ $$+1$$

## 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 1305.4.a.d 1
3.b odd 2 1 435.4.a.a 1
15.d odd 2 1 2175.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
435.4.a.a 1 3.b odd 2 1
1305.4.a.d 1 1.a even 1 1 trivial
2175.4.a.c 1 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} - 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1305))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 2$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 29$$
$11$ $$T - 15$$
$13$ $$T - 3$$
$17$ $$T + 121$$
$19$ $$T + 40$$
$23$ $$T - 116$$
$29$ $$T + 29$$
$31$ $$T + 116$$
$37$ $$T - 36$$
$41$ $$T - 170$$
$43$ $$T - 230$$
$47$ $$T + 231$$
$53$ $$T + 456$$
$59$ $$T + 576$$
$61$ $$T - 342$$
$67$ $$T + 269$$
$71$ $$T + 302$$
$73$ $$T + 372$$
$79$ $$T + 348$$
$83$ $$T - 512$$
$89$ $$T + 1525$$
$97$ $$T + 560$$