# Properties

 Label 700.6.e.b 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.

 $$f(q)$$ $$=$$ $$q + 26 i q^{3} + 49 i q^{7} - 433 q^{9}+O(q^{10})$$ q + 26*i * q^3 + 49*i * q^7 - 433 * q^9 $$q + 26 i q^{3} + 49 i q^{7} - 433 q^{9} + 8 q^{11} + 684 i q^{13} + 2218 i q^{17} + 2698 q^{19} - 1274 q^{21} + 3344 i q^{23} - 4940 i q^{27} + 3254 q^{29} + 4788 q^{31} + 208 i q^{33} + 11470 i q^{37} - 17784 q^{39} + 13350 q^{41} - 928 i q^{43} - 1212 i q^{47} - 2401 q^{49} - 57668 q^{51} + 13110 i q^{53} + 70148 i q^{57} - 34702 q^{59} - 1032 q^{61} - 21217 i q^{63} - 10108 i q^{67} - 86944 q^{69} + 62720 q^{71} - 18926 i q^{73} + 392 i q^{77} - 11400 q^{79} + 23221 q^{81} + 88958 i q^{83} + 84604 i q^{87} - 19722 q^{89} - 33516 q^{91} + 124488 i q^{93} - 17062 i q^{97} - 3464 q^{99} +O(q^{100})$$ q + 26*i * q^3 + 49*i * q^7 - 433 * q^9 + 8 * q^11 + 684*i * q^13 + 2218*i * q^17 + 2698 * q^19 - 1274 * q^21 + 3344*i * q^23 - 4940*i * q^27 + 3254 * q^29 + 4788 * q^31 + 208*i * q^33 + 11470*i * q^37 - 17784 * q^39 + 13350 * q^41 - 928*i * q^43 - 1212*i * q^47 - 2401 * q^49 - 57668 * q^51 + 13110*i * q^53 + 70148*i * q^57 - 34702 * q^59 - 1032 * q^61 - 21217*i * q^63 - 10108*i * q^67 - 86944 * q^69 + 62720 * q^71 - 18926*i * q^73 + 392*i * q^77 - 11400 * q^79 + 23221 * q^81 + 88958*i * q^83 + 84604*i * q^87 - 19722 * q^89 - 33516 * q^91 + 124488*i * q^93 - 17062*i * q^97 - 3464 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 866 q^{9}+O(q^{10})$$ 2 * q - 866 * q^9 $$2 q - 866 q^{9} + 16 q^{11} + 5396 q^{19} - 2548 q^{21} + 6508 q^{29} + 9576 q^{31} - 35568 q^{39} + 26700 q^{41} - 4802 q^{49} - 115336 q^{51} - 69404 q^{59} - 2064 q^{61} - 173888 q^{69} + 125440 q^{71} - 22800 q^{79} + 46442 q^{81} - 39444 q^{89} - 67032 q^{91} - 6928 q^{99}+O(q^{100})$$ 2 * q - 866 * q^9 + 16 * q^11 + 5396 * q^19 - 2548 * q^21 + 6508 * q^29 + 9576 * q^31 - 35568 * q^39 + 26700 * q^41 - 4802 * q^49 - 115336 * q^51 - 69404 * q^59 - 2064 * q^61 - 173888 * q^69 + 125440 * q^71 - 22800 * q^79 + 46442 * q^81 - 39444 * q^89 - 67032 * q^91 - 6928 * 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 26.0000i 0 0 0 49.0000i 0 −433.000 0
449.2 0 26.0000i 0 0 0 49.0000i 0 −433.000 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.b 2
5.b even 2 1 inner 700.6.e.b 2
5.c odd 4 1 28.6.a.b 1
5.c odd 4 1 700.6.a.b 1
15.e even 4 1 252.6.a.a 1
20.e even 4 1 112.6.a.b 1
35.f even 4 1 196.6.a.a 1
35.k even 12 2 196.6.e.i 2
35.l odd 12 2 196.6.e.a 2
40.i odd 4 1 448.6.a.b 1
40.k even 4 1 448.6.a.o 1
60.l odd 4 1 1008.6.a.l 1
140.j odd 4 1 784.6.a.m 1

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 676$$ acting on $$S_{6}^{\mathrm{new}}(700, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 676$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 2401$$
$11$ $$(T - 8)^{2}$$
$13$ $$T^{2} + 467856$$
$17$ $$T^{2} + 4919524$$
$19$ $$(T - 2698)^{2}$$
$23$ $$T^{2} + 11182336$$
$29$ $$(T - 3254)^{2}$$
$31$ $$(T - 4788)^{2}$$
$37$ $$T^{2} + 131560900$$
$41$ $$(T - 13350)^{2}$$
$43$ $$T^{2} + 861184$$
$47$ $$T^{2} + 1468944$$
$53$ $$T^{2} + 171872100$$
$59$ $$(T + 34702)^{2}$$
$61$ $$(T + 1032)^{2}$$
$67$ $$T^{2} + 102171664$$
$71$ $$(T - 62720)^{2}$$
$73$ $$T^{2} + 358193476$$
$79$ $$(T + 11400)^{2}$$
$83$ $$T^{2} + 7913525764$$
$89$ $$(T + 19722)^{2}$$
$97$ $$T^{2} + 291111844$$