# Properties

 Label 1305.4.a.b 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 145) 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)

 $$f(q)$$ $$=$$ $$q - q^{2} - 7 q^{4} + 5 q^{5} - 14 q^{7} + 15 q^{8}+O(q^{10})$$ q - q^2 - 7 * q^4 + 5 * q^5 - 14 * q^7 + 15 * q^8 $$q - q^{2} - 7 q^{4} + 5 q^{5} - 14 q^{7} + 15 q^{8} - 5 q^{10} - 62 q^{11} + 42 q^{13} + 14 q^{14} + 41 q^{16} + 114 q^{17} - 70 q^{19} - 35 q^{20} + 62 q^{22} - 62 q^{23} + 25 q^{25} - 42 q^{26} + 98 q^{28} + 29 q^{29} + 142 q^{31} - 161 q^{32} - 114 q^{34} - 70 q^{35} + 146 q^{37} + 70 q^{38} + 75 q^{40} - 162 q^{41} + 352 q^{43} + 434 q^{44} + 62 q^{46} + 444 q^{47} - 147 q^{49} - 25 q^{50} - 294 q^{52} + 238 q^{53} - 310 q^{55} - 210 q^{56} - 29 q^{58} - 840 q^{59} + 2 q^{61} - 142 q^{62} - 167 q^{64} + 210 q^{65} - 154 q^{67} - 798 q^{68} + 70 q^{70} - 892 q^{71} - 38 q^{73} - 146 q^{74} + 490 q^{76} + 868 q^{77} + 1050 q^{79} + 205 q^{80} + 162 q^{82} + 778 q^{83} + 570 q^{85} - 352 q^{86} - 930 q^{88} - 1410 q^{89} - 588 q^{91} + 434 q^{92} - 444 q^{94} - 350 q^{95} + 466 q^{97} + 147 q^{98}+O(q^{100})$$ q - q^2 - 7 * q^4 + 5 * q^5 - 14 * q^7 + 15 * q^8 - 5 * q^10 - 62 * q^11 + 42 * q^13 + 14 * q^14 + 41 * q^16 + 114 * q^17 - 70 * q^19 - 35 * q^20 + 62 * q^22 - 62 * q^23 + 25 * q^25 - 42 * q^26 + 98 * q^28 + 29 * q^29 + 142 * q^31 - 161 * q^32 - 114 * q^34 - 70 * q^35 + 146 * q^37 + 70 * q^38 + 75 * q^40 - 162 * q^41 + 352 * q^43 + 434 * q^44 + 62 * q^46 + 444 * q^47 - 147 * q^49 - 25 * q^50 - 294 * q^52 + 238 * q^53 - 310 * q^55 - 210 * q^56 - 29 * q^58 - 840 * q^59 + 2 * q^61 - 142 * q^62 - 167 * q^64 + 210 * q^65 - 154 * q^67 - 798 * q^68 + 70 * q^70 - 892 * q^71 - 38 * q^73 - 146 * q^74 + 490 * q^76 + 868 * q^77 + 1050 * q^79 + 205 * q^80 + 162 * q^82 + 778 * q^83 + 570 * q^85 - 352 * q^86 - 930 * q^88 - 1410 * q^89 - 588 * q^91 + 434 * q^92 - 444 * q^94 - 350 * q^95 + 466 * q^97 + 147 * 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
−1.00000 0 −7.00000 5.00000 0 −14.0000 15.0000 0 −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
$$3$$ $$-1$$
$$5$$ $$-1$$
$$29$$ $$-1$$

## 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.b 1
3.b odd 2 1 145.4.a.a 1
12.b even 2 1 2320.4.a.f 1
15.d odd 2 1 725.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.4.a.a 1 3.b odd 2 1
725.4.a.a 1 15.d odd 2 1
1305.4.a.b 1 1.a even 1 1 trivial
2320.4.a.f 1 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1305))$$.

## Hecke characteristic polynomials

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