# Properties

 Label 400.9.h.b Level $400$ Weight $9$ Character orbit 400.h Analytic conductor $162.951$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [400,9,Mod(399,400)]

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, 1]))

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

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

chi := DirichletCharacter("400.399");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$400 = 2^{4} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 400.h (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: $$162.951444024$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{35})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 17x^{2} + 81$$ x^4 - 17*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{9}\cdot 3^{2}\cdot 5^{2}$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{3} - 18 \beta_{2} q^{7} + 13599 q^{9}+O(q^{10})$$ q - b2 * q^3 - 18*b2 * q^7 + 13599 * q^9 $$q - \beta_{2} q^{3} - 18 \beta_{2} q^{7} + 13599 q^{9} - 27 \beta_{3} q^{11} + 2771 \beta_1 q^{13} + 5037 \beta_1 q^{17} + 153 \beta_{3} q^{19} + 362880 q^{21} - 1242 \beta_{2} q^{23} - 7038 \beta_{2} q^{27} - 54978 q^{29} - 1656 \beta_{3} q^{31} + 272160 \beta_1 q^{33} + 79373 \beta_1 q^{37} - 5542 \beta_{3} q^{39} - 75582 q^{41} - 3519 \beta_{2} q^{43} + 20196 \beta_{2} q^{47} + 767039 q^{49} - 10074 \beta_{3} q^{51} - 1116621 \beta_1 q^{53} - 1542240 \beta_1 q^{57} - 30753 \beta_{3} q^{59} - 23826622 q^{61} - 244782 \beta_{2} q^{63} + 52785 \beta_{2} q^{67} + 25038720 q^{69} + 14202 \beta_{3} q^{71} - 651661 \beta_1 q^{73} + 4898880 \beta_1 q^{77} - 68724 \beta_{3} q^{79} + 52663041 q^{81} - 517293 \beta_{2} q^{83} + 54978 \beta_{2} q^{87} - 86795778 q^{89} - 99756 \beta_{3} q^{91} + 16692480 \beta_1 q^{93} - 4667027 \beta_1 q^{97} - 367173 \beta_{3} q^{99}+O(q^{100})$$ q - b2 * q^3 - 18*b2 * q^7 + 13599 * q^9 - 27*b3 * q^11 + 2771*b1 * q^13 + 5037*b1 * q^17 + 153*b3 * q^19 + 362880 * q^21 - 1242*b2 * q^23 - 7038*b2 * q^27 - 54978 * q^29 - 1656*b3 * q^31 + 272160*b1 * q^33 + 79373*b1 * q^37 - 5542*b3 * q^39 - 75582 * q^41 - 3519*b2 * q^43 + 20196*b2 * q^47 + 767039 * q^49 - 10074*b3 * q^51 - 1116621*b1 * q^53 - 1542240*b1 * q^57 - 30753*b3 * q^59 - 23826622 * q^61 - 244782*b2 * q^63 + 52785*b2 * q^67 + 25038720 * q^69 + 14202*b3 * q^71 - 651661*b1 * q^73 + 4898880*b1 * q^77 - 68724*b3 * q^79 + 52663041 * q^81 - 517293*b2 * q^83 + 54978*b2 * q^87 - 86795778 * q^89 - 99756*b3 * q^91 + 16692480*b1 * q^93 - 4667027*b1 * q^97 - 367173*b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 54396 q^{9}+O(q^{10})$$ 4 * q + 54396 * q^9 $$4 q + 54396 q^{9} + 1451520 q^{21} - 219912 q^{29} - 302328 q^{41} + 3068156 q^{49} - 95306488 q^{61} + 100154880 q^{69} + 210652164 q^{81} - 347183112 q^{89}+O(q^{100})$$ 4 * q + 54396 * q^9 + 1451520 * q^21 - 219912 * q^29 - 302328 * q^41 + 3068156 * q^49 - 95306488 * q^61 + 100154880 * q^69 + 210652164 * q^81 - 347183112 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 17x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$( 10\nu^{3} - 80\nu ) / 9$$ (10*v^3 - 80*v) / 9 $$\beta_{2}$$ $$=$$ $$( -8\nu^{3} + 208\nu ) / 3$$ (-8*v^3 + 208*v) / 3 $$\beta_{3}$$ $$=$$ $$240\nu^{2} - 2040$$ 240*v^2 - 2040
 $$\nu$$ $$=$$ $$( 5\beta_{2} + 12\beta_1 ) / 240$$ (5*b2 + 12*b1) / 240 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 2040 ) / 240$$ (b3 + 2040) / 240 $$\nu^{3}$$ $$=$$ $$( 5\beta_{2} + 39\beta_1 ) / 30$$ (5*b2 + 39*b1) / 30

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
399.1
 2.95804 + 0.500000i 2.95804 − 0.500000i −2.95804 − 0.500000i −2.95804 + 0.500000i
0 −141.986 0 0 0 −2555.75 0 13599.0 0
399.2 0 −141.986 0 0 0 −2555.75 0 13599.0 0
399.3 0 141.986 0 0 0 2555.75 0 13599.0 0
399.4 0 141.986 0 0 0 2555.75 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
5.b even 2 1 inner
20.d 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.h.b 4
4.b odd 2 1 inner 400.9.h.b 4
5.b even 2 1 inner 400.9.h.b 4
5.c odd 4 1 16.9.c.a 2
5.c odd 4 1 400.9.b.c 2
15.e even 4 1 144.9.g.g 2
20.d odd 2 1 inner 400.9.h.b 4
20.e even 4 1 16.9.c.a 2
20.e even 4 1 400.9.b.c 2
40.i odd 4 1 64.9.c.d 2
40.k even 4 1 64.9.c.d 2
60.l odd 4 1 144.9.g.g 2
80.i odd 4 1 256.9.d.f 4
80.j even 4 1 256.9.d.f 4
80.s even 4 1 256.9.d.f 4
80.t odd 4 1 256.9.d.f 4

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 20160$$ acting on $$S_{9}^{\mathrm{new}}(400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} - 20160)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 6531840)^{2}$$
$11$ $$(T^{2} + 367416000)^{2}$$
$13$ $$(T^{2} + 767844100)^{2}$$
$17$ $$(T^{2} + 2537136900)^{2}$$
$19$ $$(T^{2} + 11798136000)^{2}$$
$23$ $$(T^{2} - 31098090240)^{2}$$
$29$ $$(T + 54978)^{4}$$
$31$ $$(T^{2} + 1382137344000)^{2}$$
$37$ $$(T^{2} + 630007312900)^{2}$$
$41$ $$(T + 75582)^{4}$$
$43$ $$(T^{2} - 249648557760)^{2}$$
$47$ $$(T^{2} - 8222828866560)^{2}$$
$53$ $$(T^{2} + 124684245764100)^{2}$$
$59$ $$(T^{2} + 476656492536000)^{2}$$
$61$ $$(T + 23826622)^{4}$$
$67$ $$(T^{2} - 56170925496000)^{2}$$
$71$ $$(T^{2} + 101655189216000)^{2}$$
$73$ $$(T^{2} + 42466205892100)^{2}$$
$79$ $$(T^{2} + 23\!\cdots\!00)^{2}$$
$83$ $$(T^{2} - 53\!\cdots\!40)^{2}$$
$89$ $$(T + 86795778)^{4}$$
$97$ $$(T^{2} + 21\!\cdots\!00)^{2}$$