# Properties

 Label 1305.4.a.e Level $1305$ Weight $4$ Character orbit 1305.a Self dual yes Analytic conductor $76.997$ Analytic rank $0$ Dimension $2$ 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: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{34})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 34$$ x^2 - 34 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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)

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 4\sqrt{34}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 8 q^{4} + 5 q^{5} + \beta q^{7}+O(q^{10})$$ q - 8 * q^4 + 5 * q^5 + b * q^7 $$q - 8 q^{4} + 5 q^{5} + \beta q^{7} + (2 \beta + 26) q^{11} + (2 \beta - 2) q^{13} + 64 q^{16} + (2 \beta - 28) q^{17} + (4 \beta - 18) q^{19} - 40 q^{20} + (3 \beta - 24) q^{23} + 25 q^{25} - 8 \beta q^{28} - 29 q^{29} + 74 q^{31} + 5 \beta q^{35} + ( - 4 \beta + 280) q^{37} + ( - 4 \beta - 430) q^{41} + (6 \beta - 40) q^{43} + ( - 16 \beta - 208) q^{44} + (6 \beta + 24) q^{47} + 201 q^{49} + ( - 16 \beta + 16) q^{52} + ( - 4 \beta - 394) q^{53} + (10 \beta + 130) q^{55} + ( - 6 \beta - 252) q^{59} + (20 \beta + 382) q^{61} - 512 q^{64} + (10 \beta - 10) q^{65} + ( - 21 \beta + 188) q^{67} + ( - 16 \beta + 224) q^{68} - 444 q^{71} + (26 \beta - 440) q^{73} + ( - 32 \beta + 144) q^{76} + (26 \beta + 1088) q^{77} + ( - 4 \beta - 86) q^{79} + 320 q^{80} + ( - 21 \beta + 156) q^{83} + (10 \beta - 140) q^{85} + (16 \beta + 658) q^{89} + ( - 2 \beta + 1088) q^{91} + ( - 24 \beta + 192) q^{92} + (20 \beta - 90) q^{95} + (6 \beta + 1244) q^{97} +O(q^{100})$$ q - 8 * q^4 + 5 * q^5 + b * q^7 + (2*b + 26) * q^11 + (2*b - 2) * q^13 + 64 * q^16 + (2*b - 28) * q^17 + (4*b - 18) * q^19 - 40 * q^20 + (3*b - 24) * q^23 + 25 * q^25 - 8*b * q^28 - 29 * q^29 + 74 * q^31 + 5*b * q^35 + (-4*b + 280) * q^37 + (-4*b - 430) * q^41 + (6*b - 40) * q^43 + (-16*b - 208) * q^44 + (6*b + 24) * q^47 + 201 * q^49 + (-16*b + 16) * q^52 + (-4*b - 394) * q^53 + (10*b + 130) * q^55 + (-6*b - 252) * q^59 + (20*b + 382) * q^61 - 512 * q^64 + (10*b - 10) * q^65 + (-21*b + 188) * q^67 + (-16*b + 224) * q^68 - 444 * q^71 + (26*b - 440) * q^73 + (-32*b + 144) * q^76 + (26*b + 1088) * q^77 + (-4*b - 86) * q^79 + 320 * q^80 + (-21*b + 156) * q^83 + (10*b - 140) * q^85 + (16*b + 658) * q^89 + (-2*b + 1088) * q^91 + (-24*b + 192) * q^92 + (20*b - 90) * q^95 + (6*b + 1244) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 16 q^{4} + 10 q^{5}+O(q^{10})$$ 2 * q - 16 * q^4 + 10 * q^5 $$2 q - 16 q^{4} + 10 q^{5} + 52 q^{11} - 4 q^{13} + 128 q^{16} - 56 q^{17} - 36 q^{19} - 80 q^{20} - 48 q^{23} + 50 q^{25} - 58 q^{29} + 148 q^{31} + 560 q^{37} - 860 q^{41} - 80 q^{43} - 416 q^{44} + 48 q^{47} + 402 q^{49} + 32 q^{52} - 788 q^{53} + 260 q^{55} - 504 q^{59} + 764 q^{61} - 1024 q^{64} - 20 q^{65} + 376 q^{67} + 448 q^{68} - 888 q^{71} - 880 q^{73} + 288 q^{76} + 2176 q^{77} - 172 q^{79} + 640 q^{80} + 312 q^{83} - 280 q^{85} + 1316 q^{89} + 2176 q^{91} + 384 q^{92} - 180 q^{95} + 2488 q^{97}+O(q^{100})$$ 2 * q - 16 * q^4 + 10 * q^5 + 52 * q^11 - 4 * q^13 + 128 * q^16 - 56 * q^17 - 36 * q^19 - 80 * q^20 - 48 * q^23 + 50 * q^25 - 58 * q^29 + 148 * q^31 + 560 * q^37 - 860 * q^41 - 80 * q^43 - 416 * q^44 + 48 * q^47 + 402 * q^49 + 32 * q^52 - 788 * q^53 + 260 * q^55 - 504 * q^59 + 764 * q^61 - 1024 * q^64 - 20 * q^65 + 376 * q^67 + 448 * q^68 - 888 * q^71 - 880 * q^73 + 288 * q^76 + 2176 * q^77 - 172 * q^79 + 640 * q^80 + 312 * q^83 - 280 * q^85 + 1316 * q^89 + 2176 * q^91 + 384 * q^92 - 180 * q^95 + 2488 * q^97

## 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
 −5.83095 5.83095
0 0 −8.00000 5.00000 0 −23.3238 0 0 0
1.2 0 0 −8.00000 5.00000 0 23.3238 0 0 0
 $$n$$: 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.e 2
3.b odd 2 1 435.4.a.e 2
15.d odd 2 1 2175.4.a.d 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2}$$
$5$ $$(T - 5)^{2}$$
$7$ $$T^{2} - 544$$
$11$ $$T^{2} - 52T - 1500$$
$13$ $$T^{2} + 4T - 2172$$
$17$ $$T^{2} + 56T - 1392$$
$19$ $$T^{2} + 36T - 8380$$
$23$ $$T^{2} + 48T - 4320$$
$29$ $$(T + 29)^{2}$$
$31$ $$(T - 74)^{2}$$
$37$ $$T^{2} - 560T + 69696$$
$41$ $$T^{2} + 860T + 176196$$
$43$ $$T^{2} + 80T - 17984$$
$47$ $$T^{2} - 48T - 19008$$
$53$ $$T^{2} + 788T + 146532$$
$59$ $$T^{2} + 504T + 43920$$
$61$ $$T^{2} - 764T - 71676$$
$67$ $$T^{2} - 376T - 204560$$
$71$ $$(T + 444)^{2}$$
$73$ $$T^{2} + 880T - 174144$$
$79$ $$T^{2} + 172T - 1308$$
$83$ $$T^{2} - 312T - 215568$$
$89$ $$T^{2} - 1316 T + 293700$$
$97$ $$T^{2} - 2488 T + 1527952$$