# Properties

 Label 1900.4.a.b Level $1900$ Weight $4$ Character orbit 1900.a Self dual yes Analytic conductor $112.104$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1900,4,Mod(1,1900)]

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, 0, 0]))

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

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

chi := DirichletCharacter("1900.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1900.a (trivial)

## 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: yes Analytic conductor: $$112.103629011$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x - 8$$ x^2 - x - 8 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 76) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{3} + ( - 4 \beta + 17) q^{7} + ( - 5 \beta - 10) q^{9}+O(q^{10})$$ q + (-b + 3) * q^3 + (-4*b + 17) * q^7 + (-5*b - 10) * q^9 $$q + ( - \beta + 3) q^{3} + ( - 4 \beta + 17) q^{7} + ( - 5 \beta - 10) q^{9} + (5 \beta - 38) q^{11} + ( - 11 \beta + 23) q^{13} + (6 \beta - 3) q^{17} - 19 q^{19} + ( - 25 \beta + 83) q^{21} + (59 \beta - 27) q^{23} + (27 \beta - 71) q^{27} + (65 \beta + 45) q^{29} + (80 \beta - 84) q^{31} + (48 \beta - 154) q^{33} + ( - 8 \beta - 186) q^{37} + ( - 45 \beta + 157) q^{39} + ( - 30 \beta - 56) q^{41} + (117 \beta - 136) q^{43} + (23 \beta + 216) q^{47} + ( - 120 \beta + 74) q^{49} + (15 \beta - 57) q^{51} + ( - 123 \beta + 199) q^{53} + (19 \beta - 57) q^{57} + ( - 35 \beta - 419) q^{59} + ( - 175 \beta + 310) q^{61} + ( - 25 \beta - 10) q^{63} + (61 \beta - 353) q^{67} + (145 \beta - 553) q^{69} + ( - 20 \beta - 846) q^{71} + (64 \beta + 463) q^{73} + (217 \beta - 806) q^{77} + (10 \beta - 642) q^{79} + (260 \beta - 159) q^{81} + ( - 114 \beta + 102) q^{83} + (85 \beta - 385) q^{87} + (80 \beta - 484) q^{89} + ( - 235 \beta + 743) q^{91} + (244 \beta - 892) q^{93} + ( - 458 \beta - 126) q^{97} + (115 \beta + 180) q^{99}+O(q^{100})$$ q + (-b + 3) * q^3 + (-4*b + 17) * q^7 + (-5*b - 10) * q^9 + (5*b - 38) * q^11 + (-11*b + 23) * q^13 + (6*b - 3) * q^17 - 19 * q^19 + (-25*b + 83) * q^21 + (59*b - 27) * q^23 + (27*b - 71) * q^27 + (65*b + 45) * q^29 + (80*b - 84) * q^31 + (48*b - 154) * q^33 + (-8*b - 186) * q^37 + (-45*b + 157) * q^39 + (-30*b - 56) * q^41 + (117*b - 136) * q^43 + (23*b + 216) * q^47 + (-120*b + 74) * q^49 + (15*b - 57) * q^51 + (-123*b + 199) * q^53 + (19*b - 57) * q^57 + (-35*b - 419) * q^59 + (-175*b + 310) * q^61 + (-25*b - 10) * q^63 + (61*b - 353) * q^67 + (145*b - 553) * q^69 + (-20*b - 846) * q^71 + (64*b + 463) * q^73 + (217*b - 806) * q^77 + (10*b - 642) * q^79 + (260*b - 159) * q^81 + (-114*b + 102) * q^83 + (85*b - 385) * q^87 + (80*b - 484) * q^89 + (-235*b + 743) * q^91 + (244*b - 892) * q^93 + (-458*b - 126) * q^97 + (115*b + 180) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{3} + 30 q^{7} - 25 q^{9}+O(q^{10})$$ 2 * q + 5 * q^3 + 30 * q^7 - 25 * q^9 $$2 q + 5 q^{3} + 30 q^{7} - 25 q^{9} - 71 q^{11} + 35 q^{13} - 38 q^{19} + 141 q^{21} + 5 q^{23} - 115 q^{27} + 155 q^{29} - 88 q^{31} - 260 q^{33} - 380 q^{37} + 269 q^{39} - 142 q^{41} - 155 q^{43} + 455 q^{47} + 28 q^{49} - 99 q^{51} + 275 q^{53} - 95 q^{57} - 873 q^{59} + 445 q^{61} - 45 q^{63} - 645 q^{67} - 961 q^{69} - 1712 q^{71} + 990 q^{73} - 1395 q^{77} - 1274 q^{79} - 58 q^{81} + 90 q^{83} - 685 q^{87} - 888 q^{89} + 1251 q^{91} - 1540 q^{93} - 710 q^{97} + 475 q^{99}+O(q^{100})$$ 2 * q + 5 * q^3 + 30 * q^7 - 25 * q^9 - 71 * q^11 + 35 * q^13 - 38 * q^19 + 141 * q^21 + 5 * q^23 - 115 * q^27 + 155 * q^29 - 88 * q^31 - 260 * q^33 - 380 * q^37 + 269 * q^39 - 142 * q^41 - 155 * q^43 + 455 * q^47 + 28 * q^49 - 99 * q^51 + 275 * q^53 - 95 * q^57 - 873 * q^59 + 445 * q^61 - 45 * q^63 - 645 * q^67 - 961 * q^69 - 1712 * q^71 + 990 * q^73 - 1395 * q^77 - 1274 * q^79 - 58 * q^81 + 90 * q^83 - 685 * q^87 - 888 * q^89 + 1251 * q^91 - 1540 * q^93 - 710 * q^97 + 475 * q^99

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 3.37228 −2.37228
0 −0.372281 0 0 0 3.51087 0 −26.8614 0
1.2 0 5.37228 0 0 0 26.4891 0 1.86141 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} - 5T_{3} - 2$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1900))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 5T - 2$$
$5$ $$T^{2}$$
$7$ $$T^{2} - 30T + 93$$
$11$ $$T^{2} + 71T + 1054$$
$13$ $$T^{2} - 35T - 692$$
$17$ $$T^{2} - 297$$
$19$ $$(T + 19)^{2}$$
$23$ $$T^{2} - 5T - 28712$$
$29$ $$T^{2} - 155T - 28850$$
$31$ $$T^{2} + 88T - 50864$$
$37$ $$T^{2} + 380T + 35572$$
$41$ $$T^{2} + 142T - 2384$$
$43$ $$T^{2} + 155T - 106928$$
$47$ $$T^{2} - 455T + 47392$$
$53$ $$T^{2} - 275T - 105908$$
$59$ $$T^{2} + 873T + 180426$$
$61$ $$T^{2} - 445T - 203150$$
$67$ $$T^{2} + 645T + 73308$$
$71$ $$T^{2} + 1712 T + 729436$$
$73$ $$T^{2} - 990T + 211233$$
$79$ $$T^{2} + 1274 T + 404944$$
$83$ $$T^{2} - 90T - 105192$$
$89$ $$T^{2} + 888T + 144336$$
$97$ $$T^{2} + 710 T - 1604528$$
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