# Properties

 Label 700.6.e.d Level $700$ Weight $6$ Character orbit 700.e Analytic conductor $112.269$ Analytic rank $0$ Dimension $2$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,6,Mod(449,700)]

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, 1, 0]))

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

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

chi := DirichletCharacter("700.449");

S:= CuspForms(chi, 6);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 700.e (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: $$112.268673869$$ Analytic rank: $$0$$ 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 28) 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.

 $$f(q)$$ $$=$$ $$q + 2 i q^{3} + 49 i q^{7} + 239 q^{9}+O(q^{10})$$ q + 2*i * q^3 + 49*i * q^7 + 239 * q^9 $$q + 2 i q^{3} + 49 i q^{7} + 239 q^{9} - 720 q^{11} - 572 i q^{13} + 1254 i q^{17} + 94 q^{19} - 98 q^{21} - 96 i q^{23} + 964 i q^{27} + 4374 q^{29} - 6244 q^{31} - 1440 i q^{33} - 10798 i q^{37} + 1144 q^{39} + 12006 q^{41} + 9160 i q^{43} - 25836 i q^{47} - 2401 q^{49} - 2508 q^{51} - 1014 i q^{53} + 188 i q^{57} - 1242 q^{59} + 7592 q^{61} + 11711 i q^{63} + 41132 i q^{67} + 192 q^{69} - 37632 q^{71} + 13438 i q^{73} - 35280 i q^{77} - 6248 q^{79} + 56149 q^{81} + 25254 i q^{83} + 8748 i q^{87} + 45126 q^{89} + 28028 q^{91} - 12488 i q^{93} + 107222 i q^{97} - 172080 q^{99} +O(q^{100})$$ q + 2*i * q^3 + 49*i * q^7 + 239 * q^9 - 720 * q^11 - 572*i * q^13 + 1254*i * q^17 + 94 * q^19 - 98 * q^21 - 96*i * q^23 + 964*i * q^27 + 4374 * q^29 - 6244 * q^31 - 1440*i * q^33 - 10798*i * q^37 + 1144 * q^39 + 12006 * q^41 + 9160*i * q^43 - 25836*i * q^47 - 2401 * q^49 - 2508 * q^51 - 1014*i * q^53 + 188*i * q^57 - 1242 * q^59 + 7592 * q^61 + 11711*i * q^63 + 41132*i * q^67 + 192 * q^69 - 37632 * q^71 + 13438*i * q^73 - 35280*i * q^77 - 6248 * q^79 + 56149 * q^81 + 25254*i * q^83 + 8748*i * q^87 + 45126 * q^89 + 28028 * q^91 - 12488*i * q^93 + 107222*i * q^97 - 172080 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 478 q^{9}+O(q^{10})$$ 2 * q + 478 * q^9 $$2 q + 478 q^{9} - 1440 q^{11} + 188 q^{19} - 196 q^{21} + 8748 q^{29} - 12488 q^{31} + 2288 q^{39} + 24012 q^{41} - 4802 q^{49} - 5016 q^{51} - 2484 q^{59} + 15184 q^{61} + 384 q^{69} - 75264 q^{71} - 12496 q^{79} + 112298 q^{81} + 90252 q^{89} + 56056 q^{91} - 344160 q^{99}+O(q^{100})$$ 2 * q + 478 * q^9 - 1440 * q^11 + 188 * q^19 - 196 * q^21 + 8748 * q^29 - 12488 * q^31 + 2288 * q^39 + 24012 * q^41 - 4802 * q^49 - 5016 * q^51 - 2484 * q^59 + 15184 * q^61 + 384 * q^69 - 75264 * q^71 - 12496 * q^79 + 112298 * q^81 + 90252 * q^89 + 56056 * q^91 - 344160 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 2.00000i 0 0 0 49.0000i 0 239.000 0
449.2 0 2.00000i 0 0 0 49.0000i 0 239.000 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 700.6.e.d 2
5.b even 2 1 inner 700.6.e.d 2
5.c odd 4 1 28.6.a.a 1
5.c odd 4 1 700.6.a.d 1
15.e even 4 1 252.6.a.d 1
20.e even 4 1 112.6.a.e 1
35.f even 4 1 196.6.a.d 1
35.k even 12 2 196.6.e.e 2
35.l odd 12 2 196.6.e.f 2
40.i odd 4 1 448.6.a.i 1
40.k even 4 1 448.6.a.h 1
60.l odd 4 1 1008.6.a.bb 1
140.j odd 4 1 784.6.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.6.a.a 1 5.c odd 4 1
112.6.a.e 1 20.e even 4 1
196.6.a.d 1 35.f even 4 1
196.6.e.e 2 35.k even 12 2
196.6.e.f 2 35.l odd 12 2
252.6.a.d 1 15.e even 4 1
448.6.a.h 1 40.k even 4 1
448.6.a.i 1 40.i odd 4 1
700.6.a.d 1 5.c odd 4 1
700.6.e.d 2 1.a even 1 1 trivial
700.6.e.d 2 5.b even 2 1 inner
784.6.a.f 1 140.j odd 4 1
1008.6.a.bb 1 60.l odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 4$$ acting on $$S_{6}^{\mathrm{new}}(700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 4$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T + 720)^{2}$$
$13$ $$T^{2} + 327184$$
$17$ $$T^{2} + 1572516$$
$19$ $$(T - 94)^{2}$$
$23$ $$T^{2} + 9216$$
$29$ $$(T - 4374)^{2}$$
$31$ $$(T + 6244)^{2}$$
$37$ $$T^{2} + 116596804$$
$41$ $$(T - 12006)^{2}$$
$43$ $$T^{2} + 83905600$$
$47$ $$T^{2} + 667498896$$
$53$ $$T^{2} + 1028196$$
$59$ $$(T + 1242)^{2}$$
$61$ $$(T - 7592)^{2}$$
$67$ $$T^{2} + 1691841424$$
$71$ $$(T + 37632)^{2}$$
$73$ $$T^{2} + 180579844$$
$79$ $$(T + 6248)^{2}$$
$83$ $$T^{2} + 637764516$$
$89$ $$(T - 45126)^{2}$$
$97$ $$T^{2} + 11496557284$$