# Properties

 Label 304.4.a.f Level $304$ Weight $4$ Character orbit 304.a Self dual yes Analytic conductor $17.937$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [304,4,Mod(1,304)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(304, 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("304.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 304.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: $$17.9365806417$$ Analytic rank: $$0$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta + 3) q^{3} - 5 \beta q^{5} + ( - 4 \beta + 17) q^{7} + ( - 5 \beta - 10) q^{9} +O(q^{10})$$ q + (-b + 3) * q^3 - 5*b * q^5 + (-4*b + 17) * q^7 + (-5*b - 10) * q^9 $$q + ( - \beta + 3) q^{3} - 5 \beta q^{5} + ( - 4 \beta + 17) q^{7} + ( - 5 \beta - 10) q^{9} + ( - 5 \beta + 38) q^{11} + (11 \beta - 23) q^{13} + ( - 10 \beta + 40) q^{15} + ( - 6 \beta + 3) q^{17} + 19 q^{19} + ( - 25 \beta + 83) q^{21} + (59 \beta - 27) q^{23} + (25 \beta + 75) q^{25} + (27 \beta - 71) q^{27} + (65 \beta + 45) q^{29} + ( - 80 \beta + 84) q^{31} + ( - 48 \beta + 154) q^{33} + ( - 65 \beta + 160) q^{35} + (8 \beta + 186) q^{37} + (45 \beta - 157) q^{39} + ( - 30 \beta - 56) q^{41} + (117 \beta - 136) q^{43} + (75 \beta + 200) q^{45} + (23 \beta + 216) q^{47} + ( - 120 \beta + 74) q^{49} + ( - 15 \beta + 57) q^{51} + (123 \beta - 199) q^{53} + ( - 165 \beta + 200) q^{55} + ( - 19 \beta + 57) q^{57} + (35 \beta + 419) q^{59} + ( - 175 \beta + 310) q^{61} + ( - 25 \beta - 10) q^{63} + (60 \beta - 440) q^{65} + (61 \beta - 353) q^{67} + (145 \beta - 553) q^{69} + (20 \beta + 846) q^{71} + ( - 64 \beta - 463) q^{73} + ( - 25 \beta + 25) q^{75} + ( - 217 \beta + 806) q^{77} + ( - 10 \beta + 642) q^{79} + (260 \beta - 159) q^{81} + ( - 114 \beta + 102) q^{83} + (15 \beta + 240) q^{85} + (85 \beta - 385) q^{87} + (80 \beta - 484) q^{89} + (235 \beta - 743) q^{91} + ( - 244 \beta + 892) q^{93} - 95 \beta q^{95} + (458 \beta + 126) q^{97} + ( - 115 \beta - 180) q^{99} +O(q^{100})$$ q + (-b + 3) * q^3 - 5*b * q^5 + (-4*b + 17) * q^7 + (-5*b - 10) * q^9 + (-5*b + 38) * q^11 + (11*b - 23) * q^13 + (-10*b + 40) * q^15 + (-6*b + 3) * q^17 + 19 * q^19 + (-25*b + 83) * q^21 + (59*b - 27) * q^23 + (25*b + 75) * q^25 + (27*b - 71) * q^27 + (65*b + 45) * q^29 + (-80*b + 84) * q^31 + (-48*b + 154) * q^33 + (-65*b + 160) * q^35 + (8*b + 186) * q^37 + (45*b - 157) * q^39 + (-30*b - 56) * q^41 + (117*b - 136) * q^43 + (75*b + 200) * q^45 + (23*b + 216) * q^47 + (-120*b + 74) * q^49 + (-15*b + 57) * q^51 + (123*b - 199) * q^53 + (-165*b + 200) * q^55 + (-19*b + 57) * q^57 + (35*b + 419) * q^59 + (-175*b + 310) * q^61 + (-25*b - 10) * q^63 + (60*b - 440) * q^65 + (61*b - 353) * q^67 + (145*b - 553) * q^69 + (20*b + 846) * q^71 + (-64*b - 463) * q^73 + (-25*b + 25) * q^75 + (-217*b + 806) * q^77 + (-10*b + 642) * q^79 + (260*b - 159) * q^81 + (-114*b + 102) * q^83 + (15*b + 240) * q^85 + (85*b - 385) * q^87 + (80*b - 484) * q^89 + (235*b - 743) * q^91 + (-244*b + 892) * q^93 - 95*b * q^95 + (458*b + 126) * q^97 + (-115*b - 180) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 5 q^{3} - 5 q^{5} + 30 q^{7} - 25 q^{9}+O(q^{10})$$ 2 * q + 5 * q^3 - 5 * q^5 + 30 * q^7 - 25 * q^9 $$2 q + 5 q^{3} - 5 q^{5} + 30 q^{7} - 25 q^{9} + 71 q^{11} - 35 q^{13} + 70 q^{15} + 38 q^{19} + 141 q^{21} + 5 q^{23} + 175 q^{25} - 115 q^{27} + 155 q^{29} + 88 q^{31} + 260 q^{33} + 255 q^{35} + 380 q^{37} - 269 q^{39} - 142 q^{41} - 155 q^{43} + 475 q^{45} + 455 q^{47} + 28 q^{49} + 99 q^{51} - 275 q^{53} + 235 q^{55} + 95 q^{57} + 873 q^{59} + 445 q^{61} - 45 q^{63} - 820 q^{65} - 645 q^{67} - 961 q^{69} + 1712 q^{71} - 990 q^{73} + 25 q^{75} + 1395 q^{77} + 1274 q^{79} - 58 q^{81} + 90 q^{83} + 495 q^{85} - 685 q^{87} - 888 q^{89} - 1251 q^{91} + 1540 q^{93} - 95 q^{95} + 710 q^{97} - 475 q^{99}+O(q^{100})$$ 2 * q + 5 * q^3 - 5 * q^5 + 30 * q^7 - 25 * q^9 + 71 * q^11 - 35 * q^13 + 70 * q^15 + 38 * q^19 + 141 * q^21 + 5 * q^23 + 175 * q^25 - 115 * q^27 + 155 * q^29 + 88 * q^31 + 260 * q^33 + 255 * q^35 + 380 * q^37 - 269 * q^39 - 142 * q^41 - 155 * q^43 + 475 * q^45 + 455 * q^47 + 28 * q^49 + 99 * q^51 - 275 * q^53 + 235 * q^55 + 95 * q^57 + 873 * q^59 + 445 * q^61 - 45 * q^63 - 820 * q^65 - 645 * q^67 - 961 * q^69 + 1712 * q^71 - 990 * q^73 + 25 * q^75 + 1395 * q^77 + 1274 * q^79 - 58 * q^81 + 90 * q^83 + 495 * q^85 - 685 * q^87 - 888 * q^89 - 1251 * q^91 + 1540 * q^93 - 95 * q^95 + 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 −16.8614 0 3.51087 0 −26.8614 0
1.2 0 5.37228 0 11.8614 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
$$2$$ $$-1$$
$$19$$ $$-1$$

## 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 304.4.a.f 2
4.b odd 2 1 76.4.a.a 2
8.b even 2 1 1216.4.a.h 2
8.d odd 2 1 1216.4.a.o 2
12.b even 2 1 684.4.a.g 2
20.d odd 2 1 1900.4.a.b 2
20.e even 4 2 1900.4.c.b 4
76.d even 2 1 1444.4.a.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
76.4.a.a 2 4.b odd 2 1
304.4.a.f 2 1.a even 1 1 trivial
684.4.a.g 2 12.b even 2 1
1216.4.a.h 2 8.b even 2 1
1216.4.a.o 2 8.d odd 2 1
1444.4.a.d 2 76.d even 2 1
1900.4.a.b 2 20.d odd 2 1
1900.4.c.b 4 20.e even 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(304))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 5T - 2$$
$5$ $$T^{2} + 5T - 200$$
$7$ $$T^{2} - 30T + 93$$
$11$ $$T^{2} - 71T + 1054$$
$13$ $$T^{2} + 35T - 692$$
$17$ $$T^{2} - 297$$
$19$ $$(T - 19)^{2}$$
$23$ $$T^{2} - 5T - 28712$$
$29$ $$T^{2} - 155T - 28850$$
$31$ $$T^{2} - 88T - 50864$$
$37$ $$T^{2} - 380T + 35572$$
$41$ $$T^{2} + 142T - 2384$$
$43$ $$T^{2} + 155T - 106928$$
$47$ $$T^{2} - 455T + 47392$$
$53$ $$T^{2} + 275T - 105908$$
$59$ $$T^{2} - 873T + 180426$$
$61$ $$T^{2} - 445T - 203150$$
$67$ $$T^{2} + 645T + 73308$$
$71$ $$T^{2} - 1712 T + 729436$$
$73$ $$T^{2} + 990T + 211233$$
$79$ $$T^{2} - 1274 T + 404944$$
$83$ $$T^{2} - 90T - 105192$$
$89$ $$T^{2} + 888T + 144336$$
$97$ $$T^{2} - 710 T - 1604528$$