# Properties

 Label 400.9.b.c Level $400$ Weight $9$ Character orbit 400.b Analytic conductor $162.951$ Analytic rank $0$ 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] = [400,9,Mod(351,400)]

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([1, 0, 0]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("400.351");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 400.b (of order $$2$$, degree $$1$$, 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: $$162.951444024$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 9$$ x^2 - x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}\cdot 3$$ Twist minimal: no (minimal twist has level 16) 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 $$\beta = 24\sqrt{-35}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta q^{3} + 18 \beta q^{7} - 13599 q^{9} +O(q^{10})$$ q - b * q^3 + 18*b * q^7 - 13599 * q^9 $$q - \beta q^{3} + 18 \beta q^{7} - 13599 q^{9} + 135 \beta q^{11} + 27710 q^{13} - 50370 q^{17} + 765 \beta q^{19} + 362880 q^{21} - 1242 \beta q^{23} + 7038 \beta q^{27} + 54978 q^{29} + 8280 \beta q^{31} + 2721600 q^{33} - 793730 q^{37} - 27710 \beta q^{39} - 75582 q^{41} - 3519 \beta q^{43} - 20196 \beta q^{47} - 767039 q^{49} + 50370 \beta q^{51} - 11166210 q^{53} + 15422400 q^{57} - 153765 \beta q^{59} - 23826622 q^{61} - 244782 \beta q^{63} - 52785 \beta q^{67} - 25038720 q^{69} - 71010 \beta q^{71} - 6516610 q^{73} - 48988800 q^{77} - 343620 \beta q^{79} + 52663041 q^{81} - 517293 \beta q^{83} - 54978 \beta q^{87} + 86795778 q^{89} + 498780 \beta q^{91} + 166924800 q^{93} + 46670270 q^{97} - 1835865 \beta q^{99} +O(q^{100})$$ q - b * q^3 + 18*b * q^7 - 13599 * q^9 + 135*b * q^11 + 27710 * q^13 - 50370 * q^17 + 765*b * q^19 + 362880 * q^21 - 1242*b * q^23 + 7038*b * q^27 + 54978 * q^29 + 8280*b * q^31 + 2721600 * q^33 - 793730 * q^37 - 27710*b * q^39 - 75582 * q^41 - 3519*b * q^43 - 20196*b * q^47 - 767039 * q^49 + 50370*b * q^51 - 11166210 * q^53 + 15422400 * q^57 - 153765*b * q^59 - 23826622 * q^61 - 244782*b * q^63 - 52785*b * q^67 - 25038720 * q^69 - 71010*b * q^71 - 6516610 * q^73 - 48988800 * q^77 - 343620*b * q^79 + 52663041 * q^81 - 517293*b * q^83 - 54978*b * q^87 + 86795778 * q^89 + 498780*b * q^91 + 166924800 * q^93 + 46670270 * q^97 - 1835865*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 27198 q^{9}+O(q^{10})$$ 2 * q - 27198 * q^9 $$2 q - 27198 q^{9} + 55420 q^{13} - 100740 q^{17} + 725760 q^{21} + 109956 q^{29} + 5443200 q^{33} - 1587460 q^{37} - 151164 q^{41} - 1534078 q^{49} - 22332420 q^{53} + 30844800 q^{57} - 47653244 q^{61} - 50077440 q^{69} - 13033220 q^{73} - 97977600 q^{77} + 105326082 q^{81} + 173591556 q^{89} + 333849600 q^{93} + 93340540 q^{97}+O(q^{100})$$ 2 * q - 27198 * q^9 + 55420 * q^13 - 100740 * q^17 + 725760 * q^21 + 109956 * q^29 + 5443200 * q^33 - 1587460 * q^37 - 151164 * q^41 - 1534078 * q^49 - 22332420 * q^53 + 30844800 * q^57 - 47653244 * q^61 - 50077440 * q^69 - 13033220 * q^73 - 97977600 * q^77 + 105326082 * q^81 + 173591556 * q^89 + 333849600 * q^93 + 93340540 * q^97

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
351.1
 0.5 + 2.95804i 0.5 − 2.95804i
0 141.986i 0 0 0 2555.75i 0 −13599.0 0
351.2 0 141.986i 0 0 0 2555.75i 0 −13599.0 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
4.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.9.b.c 2
4.b odd 2 1 inner 400.9.b.c 2
5.b even 2 1 16.9.c.a 2
5.c odd 4 2 400.9.h.b 4
15.d odd 2 1 144.9.g.g 2
20.d odd 2 1 16.9.c.a 2
20.e even 4 2 400.9.h.b 4
40.e odd 2 1 64.9.c.d 2
40.f even 2 1 64.9.c.d 2
60.h even 2 1 144.9.g.g 2
80.k odd 4 2 256.9.d.f 4
80.q even 4 2 256.9.d.f 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.9.c.a 2 5.b even 2 1
16.9.c.a 2 20.d odd 2 1
64.9.c.d 2 40.e odd 2 1
64.9.c.d 2 40.f even 2 1
144.9.g.g 2 15.d odd 2 1
144.9.g.g 2 60.h even 2 1
256.9.d.f 4 80.k odd 4 2
256.9.d.f 4 80.q even 4 2
400.9.b.c 2 1.a even 1 1 trivial
400.9.b.c 2 4.b odd 2 1 inner
400.9.h.b 4 5.c odd 4 2
400.9.h.b 4 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{9}^{\mathrm{new}}(400, [\chi])$$:

 $$T_{3}^{2} + 20160$$ T3^2 + 20160 $$T_{13} - 27710$$ T13 - 27710

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} + 20160$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 6531840$$
$11$ $$T^{2} + 367416000$$
$13$ $$(T - 27710)^{2}$$
$17$ $$(T + 50370)^{2}$$
$19$ $$T^{2} + 11798136000$$
$23$ $$T^{2} + 31098090240$$
$29$ $$(T - 54978)^{2}$$
$31$ $$T^{2} + 1382137344000$$
$37$ $$(T + 793730)^{2}$$
$41$ $$(T + 75582)^{2}$$
$43$ $$T^{2} + 249648557760$$
$47$ $$T^{2} + 8222828866560$$
$53$ $$(T + 11166210)^{2}$$
$59$ $$T^{2} + \cdots + 476656492536000$$
$61$ $$(T + 23826622)^{2}$$
$67$ $$T^{2} + 56170925496000$$
$71$ $$T^{2} + \cdots + 101655189216000$$
$73$ $$(T + 6516610)^{2}$$
$79$ $$T^{2} + 23\!\cdots\!00$$
$83$ $$T^{2} + 53\!\cdots\!40$$
$89$ $$(T - 86795778)^{2}$$
$97$ $$(T - 46670270)^{2}$$