# Properties

 Label 600.4.f.a Level $600$ Weight $4$ Character orbit 600.f Analytic conductor $35.401$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [600,4,Mod(49,600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("600.49");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$600 = 2^{3} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 600.f (of order $$2$$, degree $$1$$, not minimal)

## 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: no Analytic conductor: $$35.4011460034$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 120) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 i q^{3} + 20 i q^{7} - 9 q^{9}+O(q^{10})$$ q + 3*i * q^3 + 20*i * q^7 - 9 * q^9 $$q + 3 i q^{3} + 20 i q^{7} - 9 q^{9} - 56 q^{11} + 86 i q^{13} - 106 i q^{17} - 4 q^{19} - 60 q^{21} - 136 i q^{23} - 27 i q^{27} + 206 q^{29} - 152 q^{31} - 168 i q^{33} + 282 i q^{37} - 258 q^{39} - 246 q^{41} - 412 i q^{43} + 40 i q^{47} - 57 q^{49} + 318 q^{51} + 126 i q^{53} - 12 i q^{57} - 56 q^{59} - 2 q^{61} - 180 i q^{63} - 388 i q^{67} + 408 q^{69} - 672 q^{71} - 1170 i q^{73} - 1120 i q^{77} - 408 q^{79} + 81 q^{81} - 668 i q^{83} + 618 i q^{87} - 66 q^{89} - 1720 q^{91} - 456 i q^{93} - 926 i q^{97} + 504 q^{99} +O(q^{100})$$ q + 3*i * q^3 + 20*i * q^7 - 9 * q^9 - 56 * q^11 + 86*i * q^13 - 106*i * q^17 - 4 * q^19 - 60 * q^21 - 136*i * q^23 - 27*i * q^27 + 206 * q^29 - 152 * q^31 - 168*i * q^33 + 282*i * q^37 - 258 * q^39 - 246 * q^41 - 412*i * q^43 + 40*i * q^47 - 57 * q^49 + 318 * q^51 + 126*i * q^53 - 12*i * q^57 - 56 * q^59 - 2 * q^61 - 180*i * q^63 - 388*i * q^67 + 408 * q^69 - 672 * q^71 - 1170*i * q^73 - 1120*i * q^77 - 408 * q^79 + 81 * q^81 - 668*i * q^83 + 618*i * q^87 - 66 * q^89 - 1720 * q^91 - 456*i * q^93 - 926*i * q^97 + 504 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 18 q^{9}+O(q^{10})$$ 2 * q - 18 * q^9 $$2 q - 18 q^{9} - 112 q^{11} - 8 q^{19} - 120 q^{21} + 412 q^{29} - 304 q^{31} - 516 q^{39} - 492 q^{41} - 114 q^{49} + 636 q^{51} - 112 q^{59} - 4 q^{61} + 816 q^{69} - 1344 q^{71} - 816 q^{79} + 162 q^{81} - 132 q^{89} - 3440 q^{91} + 1008 q^{99}+O(q^{100})$$ 2 * q - 18 * q^9 - 112 * q^11 - 8 * q^19 - 120 * q^21 + 412 * q^29 - 304 * q^31 - 516 * q^39 - 492 * q^41 - 114 * q^49 + 636 * q^51 - 112 * q^59 - 4 * q^61 + 816 * q^69 - 1344 * q^71 - 816 * q^79 + 162 * q^81 - 132 * q^89 - 3440 * q^91 + 1008 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$$.

 $$n$$ $$151$$ $$301$$ $$401$$ $$577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 3.00000i 0 0 0 20.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 20.0000i 0 −9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.4.f.a 2
3.b odd 2 1 1800.4.f.v 2
4.b odd 2 1 1200.4.f.t 2
5.b even 2 1 inner 600.4.f.a 2
5.c odd 4 1 120.4.a.b 1
5.c odd 4 1 600.4.a.i 1
15.d odd 2 1 1800.4.f.v 2
15.e even 4 1 360.4.a.n 1
15.e even 4 1 1800.4.a.f 1
20.d odd 2 1 1200.4.f.t 2
20.e even 4 1 240.4.a.g 1
20.e even 4 1 1200.4.a.p 1
40.i odd 4 1 960.4.a.bj 1
40.k even 4 1 960.4.a.k 1
60.l odd 4 1 720.4.a.q 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.4.a.b 1 5.c odd 4 1
240.4.a.g 1 20.e even 4 1
360.4.a.n 1 15.e even 4 1
600.4.a.i 1 5.c odd 4 1
600.4.f.a 2 1.a even 1 1 trivial
600.4.f.a 2 5.b even 2 1 inner
720.4.a.q 1 60.l odd 4 1
960.4.a.k 1 40.k even 4 1
960.4.a.bj 1 40.i odd 4 1
1200.4.a.p 1 20.e even 4 1
1200.4.f.t 2 4.b odd 2 1
1200.4.f.t 2 20.d odd 2 1
1800.4.a.f 1 15.e even 4 1
1800.4.f.v 2 3.b odd 2 1
1800.4.f.v 2 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(600, [\chi])$$:

 $$T_{7}^{2} + 400$$ T7^2 + 400 $$T_{11} + 56$$ T11 + 56

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 9$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 400$$
$11$ $$(T + 56)^{2}$$
$13$ $$T^{2} + 7396$$
$17$ $$T^{2} + 11236$$
$19$ $$(T + 4)^{2}$$
$23$ $$T^{2} + 18496$$
$29$ $$(T - 206)^{2}$$
$31$ $$(T + 152)^{2}$$
$37$ $$T^{2} + 79524$$
$41$ $$(T + 246)^{2}$$
$43$ $$T^{2} + 169744$$
$47$ $$T^{2} + 1600$$
$53$ $$T^{2} + 15876$$
$59$ $$(T + 56)^{2}$$
$61$ $$(T + 2)^{2}$$
$67$ $$T^{2} + 150544$$
$71$ $$(T + 672)^{2}$$
$73$ $$T^{2} + 1368900$$
$79$ $$(T + 408)^{2}$$
$83$ $$T^{2} + 446224$$
$89$ $$(T + 66)^{2}$$
$97$ $$T^{2} + 857476$$