# Properties

 Label 1900.4.c.b Level $1900$ Weight $4$ Character orbit 1900.c Analytic conductor $112.104$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,4,Mod(1749,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("1900.1749");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1900.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: $$112.103629011$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 17x^{2} + 64$$ x^4 + 17*x^2 + 64 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$5^{2}$$ Twist minimal: no (minimal twist has level 76) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{2} - 5 \beta_1) q^{7} + (\beta_{3} + 12) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b2 - 5*b1) * q^7 + (b3 + 12) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{2} - 5 \beta_1) q^{7} + (\beta_{3} + 12) q^{9} + (\beta_{3} - 36) q^{11} + ( - 2 \beta_{2} + 9 \beta_1) q^{13} + ( - 3 \beta_{2} + 3 \beta_1) q^{17} + 19 q^{19} + ( - 5 \beta_{3} + 73) q^{21} + (30 \beta_{2} - 29 \beta_1) q^{23} + ( - 2 \beta_{2} + 25 \beta_1) q^{27} + ( - 13 \beta_{3} - 71) q^{29} + (16 \beta_{3} - 52) q^{31} + ( - 2 \beta_{2} - 50 \beta_1) q^{33} + (42 \beta_{2} + 34 \beta_1) q^{37} + (9 \beta_{3} - 139) q^{39} + ( - 6 \beta_{3} - 68) q^{41} + (43 \beta_{2} - 74 \beta_1) q^{43} + ( - 57 \beta_{2} - 34 \beta_1) q^{47} + (24 \beta_{3} - 26) q^{49} + (3 \beta_{3} - 51) q^{51} + ( - 34 \beta_{2} + 89 \beta_1) q^{53} + 19 \beta_1 q^{57} + (7 \beta_{3} + 433) q^{59} + ( - 35 \beta_{3} + 240) q^{61} + ( - 17 \beta_{2} + 8 \beta_1) q^{63} + (34 \beta_{2} + 95 \beta_1) q^{67} + ( - 29 \beta_{3} + 495) q^{69} + ( - 4 \beta_{3} - 854) q^{71} + (131 \beta_{2} + 67 \beta_1) q^{73} + (31 \beta_{2} + 248 \beta_1) q^{77} + ( - 2 \beta_{3} + 638) q^{79} + (52 \beta_{3} - 55) q^{81} + ( - 48 \beta_{2} + 66 \beta_1) q^{83} + (26 \beta_{2} + 111 \beta_1) q^{87} + ( - 16 \beta_{3} + 452) q^{89} + ( - 47 \beta_{3} + 649) q^{91} + ( - 32 \beta_{2} - 276 \beta_1) q^{93} + (300 \beta_{2} - 158 \beta_1) q^{97} + ( - 23 \beta_{3} - 226) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b2 - 5*b1) * q^7 + (b3 + 12) * q^9 + (b3 - 36) * q^11 + (-2*b2 + 9*b1) * q^13 + (-3*b2 + 3*b1) * q^17 + 19 * q^19 + (-5*b3 + 73) * q^21 + (30*b2 - 29*b1) * q^23 + (-2*b2 + 25*b1) * q^27 + (-13*b3 - 71) * q^29 + (16*b3 - 52) * q^31 + (-2*b2 - 50*b1) * q^33 + (42*b2 + 34*b1) * q^37 + (9*b3 - 139) * q^39 + (-6*b3 - 68) * q^41 + (43*b2 - 74*b1) * q^43 + (-57*b2 - 34*b1) * q^47 + (24*b3 - 26) * q^49 + (3*b3 - 51) * q^51 + (-34*b2 + 89*b1) * q^53 + 19*b1 * q^57 + (7*b3 + 433) * q^59 + (-35*b3 + 240) * q^61 + (-17*b2 + 8*b1) * q^63 + (34*b2 + 95*b1) * q^67 + (-29*b3 + 495) * q^69 + (-4*b3 - 854) * q^71 + (131*b2 + 67*b1) * q^73 + (31*b2 + 248*b1) * q^77 + (-2*b3 + 638) * q^79 + (52*b3 - 55) * q^81 + (-48*b2 + 66*b1) * q^83 + (26*b2 + 111*b1) * q^87 + (-16*b3 + 452) * q^89 + (-47*b3 + 649) * q^91 + (-32*b2 - 276*b1) * q^93 + (300*b2 - 158*b1) * q^97 + (-23*b3 - 226) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 50 q^{9}+O(q^{10})$$ 4 * q + 50 * q^9 $$4 q + 50 q^{9} - 142 q^{11} + 76 q^{19} + 282 q^{21} - 310 q^{29} - 176 q^{31} - 538 q^{39} - 284 q^{41} - 56 q^{49} - 198 q^{51} + 1746 q^{59} + 890 q^{61} + 1922 q^{69} - 3424 q^{71} + 2548 q^{79} - 116 q^{81} + 1776 q^{89} + 2502 q^{91} - 950 q^{99}+O(q^{100})$$ 4 * q + 50 * q^9 - 142 * q^11 + 76 * q^19 + 282 * q^21 - 310 * q^29 - 176 * q^31 - 538 * q^39 - 284 * q^41 - 56 * q^49 - 198 * q^51 + 1746 * q^59 + 890 * q^61 + 1922 * q^69 - 3424 * q^71 + 2548 * q^79 - 116 * q^81 + 1776 * q^89 + 2502 * q^91 - 950 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 17x^{2} + 64$$ :

 $$\beta_{1}$$ $$=$$ $$( -\nu^{3} - 5\nu ) / 4$$ (-v^3 - 5*v) / 4 $$\beta_{2}$$ $$=$$ $$( -3\nu^{3} - 35\nu ) / 8$$ (-3*v^3 - 35*v) / 8 $$\beta_{3}$$ $$=$$ $$5\nu^{2} + 43$$ 5*v^2 + 43
 $$\nu$$ $$=$$ $$( -2\beta_{2} + 3\beta_1 ) / 5$$ (-2*b2 + 3*b1) / 5 $$\nu^{2}$$ $$=$$ $$( \beta_{3} - 43 ) / 5$$ (b3 - 43) / 5 $$\nu^{3}$$ $$=$$ $$2\beta_{2} - 7\beta_1$$ 2*b2 - 7*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1749.1
 − 3.37228i − 2.37228i 2.37228i 3.37228i
0 5.37228i 0 0 0 26.4891i 0 −1.86141 0
1749.2 0 0.372281i 0 0 0 3.51087i 0 26.8614 0
1749.3 0 0.372281i 0 0 0 3.51087i 0 26.8614 0
1749.4 0 5.37228i 0 0 0 26.4891i 0 −1.86141 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 1900.4.c.b 4
5.b even 2 1 inner 1900.4.c.b 4
5.c odd 4 1 76.4.a.a 2
5.c odd 4 1 1900.4.a.b 2
15.e even 4 1 684.4.a.g 2
20.e even 4 1 304.4.a.f 2
40.i odd 4 1 1216.4.a.o 2
40.k even 4 1 1216.4.a.h 2
95.g even 4 1 1444.4.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.a 2 5.c odd 4 1
304.4.a.f 2 20.e even 4 1
684.4.a.g 2 15.e even 4 1
1216.4.a.h 2 40.k even 4 1
1216.4.a.o 2 40.i odd 4 1
1444.4.a.d 2 95.g even 4 1
1900.4.a.b 2 5.c odd 4 1
1900.4.c.b 4 1.a even 1 1 trivial
1900.4.c.b 4 5.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 29T_{3}^{2} + 4$$ acting on $$S_{4}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 29T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 714T^{2} + 8649$$
$11$ $$(T^{2} + 71 T + 1054)^{2}$$
$13$ $$T^{4} + 2609 T^{2} + 478864$$
$17$ $$(T^{2} + 297)^{2}$$
$19$ $$(T - 19)^{4}$$
$23$ $$T^{4} + 57449 T^{2} + \cdots + 824378944$$
$29$ $$(T^{2} + 155 T - 28850)^{2}$$
$31$ $$(T^{2} + 88 T - 50864)^{2}$$
$37$ $$T^{4} + 73256 T^{2} + \cdots + 1265367184$$
$41$ $$(T^{2} + 142 T - 2384)^{2}$$
$43$ $$T^{4} + 237881 T^{2} + \cdots + 11433597184$$
$47$ $$T^{4} + 112241 T^{2} + \cdots + 2246001664$$
$53$ $$T^{4} + 287441 T^{2} + \cdots + 11216504464$$
$59$ $$(T^{2} - 873 T + 180426)^{2}$$
$61$ $$(T^{2} - 445 T - 203150)^{2}$$
$67$ $$T^{4} + 269409 T^{2} + \cdots + 5374062864$$
$71$ $$(T^{2} + 1712 T + 729436)^{2}$$
$73$ $$T^{4} + 557634 T^{2} + \cdots + 44619380289$$
$79$ $$(T^{2} - 1274 T + 404944)^{2}$$
$83$ $$T^{4} + 218484 T^{2} + \cdots + 11065356864$$
$89$ $$(T^{2} - 888 T + 144336)^{2}$$
$97$ $$T^{4} + 3713156 T^{2} + \cdots + 2574510102784$$