Properties

 Label 700.6.a.b Level $700$ Weight $6$ Character orbit 700.a Self dual yes Analytic conductor $112.269$ 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] = [700,6,Mod(1,700)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("700.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 700.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: $$112.268673869$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) 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)

 $$f(q)$$ $$=$$ $$q - 26 q^{3} + 49 q^{7} + 433 q^{9}+O(q^{10})$$ q - 26 * q^3 + 49 * q^7 + 433 * q^9 $$q - 26 q^{3} + 49 q^{7} + 433 q^{9} + 8 q^{11} - 684 q^{13} + 2218 q^{17} - 2698 q^{19} - 1274 q^{21} - 3344 q^{23} - 4940 q^{27} - 3254 q^{29} + 4788 q^{31} - 208 q^{33} + 11470 q^{37} + 17784 q^{39} + 13350 q^{41} + 928 q^{43} - 1212 q^{47} + 2401 q^{49} - 57668 q^{51} - 13110 q^{53} + 70148 q^{57} + 34702 q^{59} - 1032 q^{61} + 21217 q^{63} - 10108 q^{67} + 86944 q^{69} + 62720 q^{71} + 18926 q^{73} + 392 q^{77} + 11400 q^{79} + 23221 q^{81} - 88958 q^{83} + 84604 q^{87} + 19722 q^{89} - 33516 q^{91} - 124488 q^{93} - 17062 q^{97} + 3464 q^{99}+O(q^{100})$$ q - 26 * q^3 + 49 * q^7 + 433 * q^9 + 8 * q^11 - 684 * q^13 + 2218 * q^17 - 2698 * q^19 - 1274 * q^21 - 3344 * q^23 - 4940 * q^27 - 3254 * q^29 + 4788 * q^31 - 208 * q^33 + 11470 * q^37 + 17784 * q^39 + 13350 * q^41 + 928 * q^43 - 1212 * q^47 + 2401 * q^49 - 57668 * q^51 - 13110 * q^53 + 70148 * q^57 + 34702 * q^59 - 1032 * q^61 + 21217 * q^63 - 10108 * q^67 + 86944 * q^69 + 62720 * q^71 + 18926 * q^73 + 392 * q^77 + 11400 * q^79 + 23221 * q^81 - 88958 * q^83 + 84604 * q^87 + 19722 * q^89 - 33516 * q^91 - 124488 * q^93 - 17062 * q^97 + 3464 * q^99

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
0 −26.0000 0 0 0 49.0000 0 433.000 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
$$2$$ $$-1$$
$$5$$ $$+1$$
$$7$$ $$-1$$

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 700.6.a.b 1
5.b even 2 1 28.6.a.b 1
5.c odd 4 2 700.6.e.b 2
15.d odd 2 1 252.6.a.a 1
20.d odd 2 1 112.6.a.b 1
35.c odd 2 1 196.6.a.a 1
35.i odd 6 2 196.6.e.i 2
35.j even 6 2 196.6.e.a 2
40.e odd 2 1 448.6.a.o 1
40.f even 2 1 448.6.a.b 1
60.h even 2 1 1008.6.a.l 1
140.c even 2 1 784.6.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.b 1 5.b even 2 1
112.6.a.b 1 20.d odd 2 1
196.6.a.a 1 35.c odd 2 1
196.6.e.a 2 35.j even 6 2
196.6.e.i 2 35.i odd 6 2
252.6.a.a 1 15.d odd 2 1
448.6.a.b 1 40.f even 2 1
448.6.a.o 1 40.e odd 2 1
700.6.a.b 1 1.a even 1 1 trivial
700.6.e.b 2 5.c odd 4 2
784.6.a.m 1 140.c even 2 1
1008.6.a.l 1 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3} + 26$$ acting on $$S_{6}^{\mathrm{new}}(\Gamma_0(700))$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T + 26$$
$5$ $$T$$
$7$ $$T - 49$$
$11$ $$T - 8$$
$13$ $$T + 684$$
$17$ $$T - 2218$$
$19$ $$T + 2698$$
$23$ $$T + 3344$$
$29$ $$T + 3254$$
$31$ $$T - 4788$$
$37$ $$T - 11470$$
$41$ $$T - 13350$$
$43$ $$T - 928$$
$47$ $$T + 1212$$
$53$ $$T + 13110$$
$59$ $$T - 34702$$
$61$ $$T + 1032$$
$67$ $$T + 10108$$
$71$ $$T - 62720$$
$73$ $$T - 18926$$
$79$ $$T - 11400$$
$83$ $$T + 88958$$
$89$ $$T - 19722$$
$97$ $$T + 17062$$