# Properties

 Label 400.8.c.b Level $400$ Weight $8$ Character orbit 400.c Analytic conductor $124.954$ 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] = [400,8,Mod(49,400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(400, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("400.49");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 400.c (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: $$124.954010194$$ 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: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 8) 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 $$\beta = 2i$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 42 \beta q^{3} - 228 \beta q^{7} - 4869 q^{9} +O(q^{10})$$ q + 42*b * q^3 - 228*b * q^7 - 4869 * q^9 $$q + 42 \beta q^{3} - 228 \beta q^{7} - 4869 q^{9} + 2524 q^{11} - 5389 \beta q^{13} + 5575 \beta q^{17} + 4124 q^{19} + 38304 q^{21} - 40852 \beta q^{23} - 112644 \beta q^{27} - 99798 q^{29} + 40480 q^{31} + 106008 \beta q^{33} + 209721 \beta q^{37} + 905352 q^{39} + 141402 q^{41} + 345214 \beta q^{43} - 341016 \beta q^{47} + 615607 q^{49} - 936600 q^{51} + 906559 \beta q^{53} + 173208 \beta q^{57} - 966028 q^{59} + 1887670 q^{61} + 1110132 \beta q^{63} + 1482934 \beta q^{67} + 6863136 q^{69} + 2548232 q^{71} - 840163 \beta q^{73} - 575472 \beta q^{77} + 4038064 q^{79} + 8275689 q^{81} + 2692882 \beta q^{83} - 4191516 \beta q^{87} + 6473046 q^{89} - 4914768 q^{91} + 1700160 \beta q^{93} + 3032879 \beta q^{97} - 12289356 q^{99} +O(q^{100})$$ q + 42*b * q^3 - 228*b * q^7 - 4869 * q^9 + 2524 * q^11 - 5389*b * q^13 + 5575*b * q^17 + 4124 * q^19 + 38304 * q^21 - 40852*b * q^23 - 112644*b * q^27 - 99798 * q^29 + 40480 * q^31 + 106008*b * q^33 + 209721*b * q^37 + 905352 * q^39 + 141402 * q^41 + 345214*b * q^43 - 341016*b * q^47 + 615607 * q^49 - 936600 * q^51 + 906559*b * q^53 + 173208*b * q^57 - 966028 * q^59 + 1887670 * q^61 + 1110132*b * q^63 + 1482934*b * q^67 + 6863136 * q^69 + 2548232 * q^71 - 840163*b * q^73 - 575472*b * q^77 + 4038064 * q^79 + 8275689 * q^81 + 2692882*b * q^83 - 4191516*b * q^87 + 6473046 * q^89 - 4914768 * q^91 + 1700160*b * q^93 + 3032879*b * q^97 - 12289356 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 9738 q^{9}+O(q^{10})$$ 2 * q - 9738 * q^9 $$2 q - 9738 q^{9} + 5048 q^{11} + 8248 q^{19} + 76608 q^{21} - 199596 q^{29} + 80960 q^{31} + 1810704 q^{39} + 282804 q^{41} + 1231214 q^{49} - 1873200 q^{51} - 1932056 q^{59} + 3775340 q^{61} + 13726272 q^{69} + 5096464 q^{71} + 8076128 q^{79} + 16551378 q^{81} + 12946092 q^{89} - 9829536 q^{91} - 24578712 q^{99}+O(q^{100})$$ 2 * q - 9738 * q^9 + 5048 * q^11 + 8248 * q^19 + 76608 * q^21 - 199596 * q^29 + 80960 * q^31 + 1810704 * q^39 + 282804 * q^41 + 1231214 * q^49 - 1873200 * q^51 - 1932056 * q^59 + 3775340 * q^61 + 13726272 * q^69 + 5096464 * q^71 + 8076128 * q^79 + 16551378 * q^81 + 12946092 * q^89 - 9829536 * q^91 - 24578712 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/400\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$351$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 − 1.00000i 1.00000i
0 84.0000i 0 0 0 456.000i 0 −4869.00 0
49.2 0 84.0000i 0 0 0 456.000i 0 −4869.00 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 400.8.c.b 2
4.b odd 2 1 200.8.c.a 2
5.b even 2 1 inner 400.8.c.b 2
5.c odd 4 1 16.8.a.c 1
5.c odd 4 1 400.8.a.b 1
15.e even 4 1 144.8.a.g 1
20.d odd 2 1 200.8.c.a 2
20.e even 4 1 8.8.a.a 1
20.e even 4 1 200.8.a.i 1
40.i odd 4 1 64.8.a.a 1
40.k even 4 1 64.8.a.g 1
60.l odd 4 1 72.8.a.d 1
80.i odd 4 1 256.8.b.c 2
80.j even 4 1 256.8.b.e 2
80.s even 4 1 256.8.b.e 2
80.t odd 4 1 256.8.b.c 2
120.q odd 4 1 576.8.a.j 1
120.w even 4 1 576.8.a.k 1
140.j odd 4 1 392.8.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.8.a.a 1 20.e even 4 1
16.8.a.c 1 5.c odd 4 1
64.8.a.a 1 40.i odd 4 1
64.8.a.g 1 40.k even 4 1
72.8.a.d 1 60.l odd 4 1
144.8.a.g 1 15.e even 4 1
200.8.a.i 1 20.e even 4 1
200.8.c.a 2 4.b odd 2 1
200.8.c.a 2 20.d odd 2 1
256.8.b.c 2 80.i odd 4 1
256.8.b.c 2 80.t odd 4 1
256.8.b.e 2 80.j even 4 1
256.8.b.e 2 80.s even 4 1
392.8.a.d 1 140.j odd 4 1
400.8.a.b 1 5.c odd 4 1
400.8.c.b 2 1.a even 1 1 trivial
400.8.c.b 2 5.b even 2 1 inner
576.8.a.j 1 120.q odd 4 1
576.8.a.k 1 120.w even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 7056$$ acting on $$S_{8}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 7056$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 207936$$
$11$ $$(T - 2524)^{2}$$
$13$ $$T^{2} + 116165284$$
$17$ $$T^{2} + 124322500$$
$19$ $$(T - 4124)^{2}$$
$23$ $$T^{2} + 6675543616$$
$29$ $$(T + 99798)^{2}$$
$31$ $$(T - 40480)^{2}$$
$37$ $$T^{2} + 175931591364$$
$41$ $$(T - 141402)^{2}$$
$43$ $$T^{2} + 476690823184$$
$47$ $$T^{2} + 465167649024$$
$53$ $$T^{2} + 3287396881924$$
$59$ $$(T + 966028)^{2}$$
$61$ $$(T - 1887670)^{2}$$
$67$ $$T^{2} + 8796372993424$$
$71$ $$(T - 2548232)^{2}$$
$73$ $$T^{2} + 2823495466276$$
$79$ $$(T - 4038064)^{2}$$
$83$ $$T^{2} + 29006453863696$$
$89$ $$(T - 6473046)^{2}$$
$97$ $$T^{2} + 36793420114564$$