# Properties

 Label 1305.4.a.c 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)

 $$f(q)$$ $$=$$ $$q + q^{2} - 7 q^{4} - 5 q^{5} + 4 q^{7} - 15 q^{8}+O(q^{10})$$ q + q^2 - 7 * q^4 - 5 * q^5 + 4 * q^7 - 15 * q^8 $$q + q^{2} - 7 q^{4} - 5 q^{5} + 4 q^{7} - 15 q^{8} - 5 q^{10} + 36 q^{11} - 22 q^{13} + 4 q^{14} + 41 q^{16} + 2 q^{17} - 56 q^{19} + 35 q^{20} + 36 q^{22} + 40 q^{23} + 25 q^{25} - 22 q^{26} - 28 q^{28} - 29 q^{29} + 152 q^{31} + 161 q^{32} + 2 q^{34} - 20 q^{35} + 34 q^{37} - 56 q^{38} + 75 q^{40} + 250 q^{41} - 412 q^{43} - 252 q^{44} + 40 q^{46} + 120 q^{47} - 327 q^{49} + 25 q^{50} + 154 q^{52} + 762 q^{53} - 180 q^{55} - 60 q^{56} - 29 q^{58} + 188 q^{59} - 54 q^{61} + 152 q^{62} - 167 q^{64} + 110 q^{65} - 244 q^{67} - 14 q^{68} - 20 q^{70} - 600 q^{71} + 6 q^{73} + 34 q^{74} + 392 q^{76} + 144 q^{77} - 640 q^{79} - 205 q^{80} + 250 q^{82} - 664 q^{83} - 10 q^{85} - 412 q^{86} - 540 q^{88} - 150 q^{89} - 88 q^{91} - 280 q^{92} + 120 q^{94} + 280 q^{95} - 1690 q^{97} - 327 q^{98}+O(q^{100})$$ q + q^2 - 7 * q^4 - 5 * q^5 + 4 * q^7 - 15 * q^8 - 5 * q^10 + 36 * q^11 - 22 * q^13 + 4 * q^14 + 41 * q^16 + 2 * q^17 - 56 * q^19 + 35 * q^20 + 36 * q^22 + 40 * q^23 + 25 * q^25 - 22 * q^26 - 28 * q^28 - 29 * q^29 + 152 * q^31 + 161 * q^32 + 2 * q^34 - 20 * q^35 + 34 * q^37 - 56 * q^38 + 75 * q^40 + 250 * q^41 - 412 * q^43 - 252 * q^44 + 40 * q^46 + 120 * q^47 - 327 * q^49 + 25 * q^50 + 154 * q^52 + 762 * q^53 - 180 * q^55 - 60 * q^56 - 29 * q^58 + 188 * q^59 - 54 * q^61 + 152 * q^62 - 167 * q^64 + 110 * q^65 - 244 * q^67 - 14 * q^68 - 20 * q^70 - 600 * q^71 + 6 * q^73 + 34 * q^74 + 392 * q^76 + 144 * q^77 - 640 * q^79 - 205 * q^80 + 250 * q^82 - 664 * q^83 - 10 * q^85 - 412 * q^86 - 540 * q^88 - 150 * q^89 - 88 * q^91 - 280 * q^92 + 120 * q^94 + 280 * q^95 - 1690 * q^97 - 327 * 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 4.00000 −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.c 1
3.b odd 2 1 435.4.a.b 1
15.d odd 2 1 2175.4.a.b 1

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T$$
$5$ $$T + 5$$
$7$ $$T - 4$$
$11$ $$T - 36$$
$13$ $$T + 22$$
$17$ $$T - 2$$
$19$ $$T + 56$$
$23$ $$T - 40$$
$29$ $$T + 29$$
$31$ $$T - 152$$
$37$ $$T - 34$$
$41$ $$T - 250$$
$43$ $$T + 412$$
$47$ $$T - 120$$
$53$ $$T - 762$$
$59$ $$T - 188$$
$61$ $$T + 54$$
$67$ $$T + 244$$
$71$ $$T + 600$$
$73$ $$T - 6$$
$79$ $$T + 640$$
$83$ $$T + 664$$
$89$ $$T + 150$$
$97$ $$T + 1690$$