# Properties

 Label 312.4.a.b Level $312$ Weight $4$ Character orbit 312.a Self dual yes Analytic conductor $18.409$ 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] = [312,4,Mod(1,312)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(312, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("312.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$312 = 2^{3} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 312.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: $$18.4085959218$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{55})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 55$$ x^2 - 55 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: yes 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 $$\beta = 2\sqrt{55}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - 3 q^{3} + (\beta - 2) q^{5} + (\beta + 10) q^{7} + 9 q^{9}+O(q^{10})$$ q - 3 * q^3 + (b - 2) * q^5 + (b + 10) * q^7 + 9 * q^9 $$q - 3 q^{3} + (\beta - 2) q^{5} + (\beta + 10) q^{7} + 9 q^{9} + (2 \beta - 10) q^{11} - 13 q^{13} + ( - 3 \beta + 6) q^{15} + ( - 2 \beta - 34) q^{17} + ( - \beta + 14) q^{19} + ( - 3 \beta - 30) q^{21} + (4 \beta + 60) q^{23} + ( - 4 \beta + 99) q^{25} - 27 q^{27} - 6 q^{29} + ( - 3 \beta + 230) q^{31} + ( - 6 \beta + 30) q^{33} + (8 \beta + 200) q^{35} + (2 \beta + 114) q^{37} + 39 q^{39} + ( - 11 \beta + 46) q^{41} + ( - 10 \beta + 216) q^{43} + (9 \beta - 18) q^{45} + (8 \beta + 358) q^{47} + (20 \beta - 23) q^{49} + (6 \beta + 102) q^{51} + (16 \beta - 178) q^{53} + ( - 14 \beta + 460) q^{55} + (3 \beta - 42) q^{57} + 78 q^{59} + (48 \beta - 102) q^{61} + (9 \beta + 90) q^{63} + ( - 13 \beta + 26) q^{65} + ( - 51 \beta + 102) q^{67} + ( - 12 \beta - 180) q^{69} + ( - 6 \beta - 302) q^{71} + ( - 22 \beta + 242) q^{73} + (12 \beta - 297) q^{75} + (10 \beta + 340) q^{77} + (20 \beta + 880) q^{79} + 81 q^{81} + ( - 20 \beta - 134) q^{83} + ( - 30 \beta - 372) q^{85} + 18 q^{87} + ( - 93 \beta + 38) q^{89} + ( - 13 \beta - 130) q^{91} + (9 \beta - 690) q^{93} + (16 \beta - 248) q^{95} + (54 \beta + 2) q^{97} + (18 \beta - 90) q^{99}+O(q^{100})$$ q - 3 * q^3 + (b - 2) * q^5 + (b + 10) * q^7 + 9 * q^9 + (2*b - 10) * q^11 - 13 * q^13 + (-3*b + 6) * q^15 + (-2*b - 34) * q^17 + (-b + 14) * q^19 + (-3*b - 30) * q^21 + (4*b + 60) * q^23 + (-4*b + 99) * q^25 - 27 * q^27 - 6 * q^29 + (-3*b + 230) * q^31 + (-6*b + 30) * q^33 + (8*b + 200) * q^35 + (2*b + 114) * q^37 + 39 * q^39 + (-11*b + 46) * q^41 + (-10*b + 216) * q^43 + (9*b - 18) * q^45 + (8*b + 358) * q^47 + (20*b - 23) * q^49 + (6*b + 102) * q^51 + (16*b - 178) * q^53 + (-14*b + 460) * q^55 + (3*b - 42) * q^57 + 78 * q^59 + (48*b - 102) * q^61 + (9*b + 90) * q^63 + (-13*b + 26) * q^65 + (-51*b + 102) * q^67 + (-12*b - 180) * q^69 + (-6*b - 302) * q^71 + (-22*b + 242) * q^73 + (12*b - 297) * q^75 + (10*b + 340) * q^77 + (20*b + 880) * q^79 + 81 * q^81 + (-20*b - 134) * q^83 + (-30*b - 372) * q^85 + 18 * q^87 + (-93*b + 38) * q^89 + (-13*b - 130) * q^91 + (9*b - 690) * q^93 + (16*b - 248) * q^95 + (54*b + 2) * q^97 + (18*b - 90) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9 $$2 q - 6 q^{3} - 4 q^{5} + 20 q^{7} + 18 q^{9} - 20 q^{11} - 26 q^{13} + 12 q^{15} - 68 q^{17} + 28 q^{19} - 60 q^{21} + 120 q^{23} + 198 q^{25} - 54 q^{27} - 12 q^{29} + 460 q^{31} + 60 q^{33} + 400 q^{35} + 228 q^{37} + 78 q^{39} + 92 q^{41} + 432 q^{43} - 36 q^{45} + 716 q^{47} - 46 q^{49} + 204 q^{51} - 356 q^{53} + 920 q^{55} - 84 q^{57} + 156 q^{59} - 204 q^{61} + 180 q^{63} + 52 q^{65} + 204 q^{67} - 360 q^{69} - 604 q^{71} + 484 q^{73} - 594 q^{75} + 680 q^{77} + 1760 q^{79} + 162 q^{81} - 268 q^{83} - 744 q^{85} + 36 q^{87} + 76 q^{89} - 260 q^{91} - 1380 q^{93} - 496 q^{95} + 4 q^{97} - 180 q^{99}+O(q^{100})$$ 2 * q - 6 * q^3 - 4 * q^5 + 20 * q^7 + 18 * q^9 - 20 * q^11 - 26 * q^13 + 12 * q^15 - 68 * q^17 + 28 * q^19 - 60 * q^21 + 120 * q^23 + 198 * q^25 - 54 * q^27 - 12 * q^29 + 460 * q^31 + 60 * q^33 + 400 * q^35 + 228 * q^37 + 78 * q^39 + 92 * q^41 + 432 * q^43 - 36 * q^45 + 716 * q^47 - 46 * q^49 + 204 * q^51 - 356 * q^53 + 920 * q^55 - 84 * q^57 + 156 * q^59 - 204 * q^61 + 180 * q^63 + 52 * q^65 + 204 * q^67 - 360 * q^69 - 604 * q^71 + 484 * q^73 - 594 * q^75 + 680 * q^77 + 1760 * q^79 + 162 * q^81 - 268 * q^83 - 744 * q^85 + 36 * q^87 + 76 * q^89 - 260 * q^91 - 1380 * q^93 - 496 * q^95 + 4 * q^97 - 180 * 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
 −7.41620 7.41620
0 −3.00000 0 −16.8324 0 −4.83240 0 9.00000 0
1.2 0 −3.00000 0 12.8324 0 24.8324 0 9.00000 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
$$2$$ $$+1$$
$$3$$ $$+1$$
$$13$$ $$+1$$

## 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 312.4.a.b 2
3.b odd 2 1 936.4.a.h 2
4.b odd 2 1 624.4.a.n 2
8.b even 2 1 2496.4.a.bi 2
8.d odd 2 1 2496.4.a.x 2
12.b even 2 1 1872.4.a.bd 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.b 2 1.a even 1 1 trivial
624.4.a.n 2 4.b odd 2 1
936.4.a.h 2 3.b odd 2 1
1872.4.a.bd 2 12.b even 2 1
2496.4.a.x 2 8.d odd 2 1
2496.4.a.bi 2 8.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} + 4T_{5} - 216$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(312))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T + 3)^{2}$$
$5$ $$T^{2} + 4T - 216$$
$7$ $$T^{2} - 20T - 120$$
$11$ $$T^{2} + 20T - 780$$
$13$ $$(T + 13)^{2}$$
$17$ $$T^{2} + 68T + 276$$
$19$ $$T^{2} - 28T - 24$$
$23$ $$T^{2} - 120T + 80$$
$29$ $$(T + 6)^{2}$$
$31$ $$T^{2} - 460T + 50920$$
$37$ $$T^{2} - 228T + 12116$$
$41$ $$T^{2} - 92T - 24504$$
$43$ $$T^{2} - 432T + 24656$$
$47$ $$T^{2} - 716T + 114084$$
$53$ $$T^{2} + 356T - 24636$$
$59$ $$(T - 78)^{2}$$
$61$ $$T^{2} + 204T - 496476$$
$67$ $$T^{2} - 204T - 561816$$
$71$ $$T^{2} + 604T + 83284$$
$73$ $$T^{2} - 484T - 47916$$
$79$ $$T^{2} - 1760 T + 686400$$
$83$ $$T^{2} + 268T - 70044$$
$89$ $$T^{2} - 76T - 1901336$$
$97$ $$T^{2} - 4T - 641516$$