# Properties

 Label 312.4.a.f 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: $$\Q(\sqrt{43})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 43$$ x^2 - 43 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 $$\beta = 2\sqrt{43}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + (\beta + 6) q^{5} + (\beta + 22) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (b + 6) * q^5 + (b + 22) * q^7 + 9 * q^9 $$q + 3 q^{3} + (\beta + 6) q^{5} + (\beta + 22) q^{7} + 9 q^{9} + 26 q^{11} - 13 q^{13} + (3 \beta + 18) q^{15} + ( - 2 \beta - 10) q^{17} + ( - 5 \beta - 30) q^{19} + (3 \beta + 66) q^{21} + ( - 12 \beta - 4) q^{23} + (12 \beta + 83) q^{25} + 27 q^{27} + ( - 12 \beta + 66) q^{29} + ( - 15 \beta - 70) q^{31} + 78 q^{33} + (28 \beta + 304) q^{35} + ( - 14 \beta + 34) q^{37} - 39 q^{39} + ( - 7 \beta + 14) q^{41} + 14 \beta q^{43} + (9 \beta + 54) q^{45} + ( - 6 \beta + 18) q^{47} + (44 \beta + 313) q^{49} + ( - 6 \beta - 30) q^{51} + ( - 4 \beta + 334) q^{53} + (26 \beta + 156) q^{55} + ( - 15 \beta - 90) q^{57} + ( - 22 \beta - 254) q^{59} + ( - 8 \beta + 170) q^{61} + (9 \beta + 198) q^{63} + ( - 13 \beta - 78) q^{65} + ( - 43 \beta - 470) q^{67} + ( - 36 \beta - 12) q^{69} + (68 \beta + 150) q^{71} + ( - 6 \beta + 562) q^{73} + (36 \beta + 249) q^{75} + (26 \beta + 572) q^{77} + ( - 44 \beta - 760) q^{79} + 81 q^{81} + (42 \beta + 262) q^{83} + ( - 22 \beta - 404) q^{85} + ( - 36 \beta + 198) q^{87} + ( - \beta + 950) q^{89} + ( - 13 \beta - 286) q^{91} + ( - 45 \beta - 210) q^{93} + ( - 60 \beta - 1040) q^{95} + ( - 18 \beta - 718) q^{97} + 234 q^{99}+O(q^{100})$$ q + 3 * q^3 + (b + 6) * q^5 + (b + 22) * q^7 + 9 * q^9 + 26 * q^11 - 13 * q^13 + (3*b + 18) * q^15 + (-2*b - 10) * q^17 + (-5*b - 30) * q^19 + (3*b + 66) * q^21 + (-12*b - 4) * q^23 + (12*b + 83) * q^25 + 27 * q^27 + (-12*b + 66) * q^29 + (-15*b - 70) * q^31 + 78 * q^33 + (28*b + 304) * q^35 + (-14*b + 34) * q^37 - 39 * q^39 + (-7*b + 14) * q^41 + 14*b * q^43 + (9*b + 54) * q^45 + (-6*b + 18) * q^47 + (44*b + 313) * q^49 + (-6*b - 30) * q^51 + (-4*b + 334) * q^53 + (26*b + 156) * q^55 + (-15*b - 90) * q^57 + (-22*b - 254) * q^59 + (-8*b + 170) * q^61 + (9*b + 198) * q^63 + (-13*b - 78) * q^65 + (-43*b - 470) * q^67 + (-36*b - 12) * q^69 + (68*b + 150) * q^71 + (-6*b + 562) * q^73 + (36*b + 249) * q^75 + (26*b + 572) * q^77 + (-44*b - 760) * q^79 + 81 * q^81 + (42*b + 262) * q^83 + (-22*b - 404) * q^85 + (-36*b + 198) * q^87 + (-b + 950) * q^89 + (-13*b - 286) * q^91 + (-45*b - 210) * q^93 + (-60*b - 1040) * q^95 + (-18*b - 718) * q^97 + 234 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9}+O(q^{10})$$ 2 * q + 6 * q^3 + 12 * q^5 + 44 * q^7 + 18 * q^9 $$2 q + 6 q^{3} + 12 q^{5} + 44 q^{7} + 18 q^{9} + 52 q^{11} - 26 q^{13} + 36 q^{15} - 20 q^{17} - 60 q^{19} + 132 q^{21} - 8 q^{23} + 166 q^{25} + 54 q^{27} + 132 q^{29} - 140 q^{31} + 156 q^{33} + 608 q^{35} + 68 q^{37} - 78 q^{39} + 28 q^{41} + 108 q^{45} + 36 q^{47} + 626 q^{49} - 60 q^{51} + 668 q^{53} + 312 q^{55} - 180 q^{57} - 508 q^{59} + 340 q^{61} + 396 q^{63} - 156 q^{65} - 940 q^{67} - 24 q^{69} + 300 q^{71} + 1124 q^{73} + 498 q^{75} + 1144 q^{77} - 1520 q^{79} + 162 q^{81} + 524 q^{83} - 808 q^{85} + 396 q^{87} + 1900 q^{89} - 572 q^{91} - 420 q^{93} - 2080 q^{95} - 1436 q^{97} + 468 q^{99}+O(q^{100})$$ 2 * q + 6 * q^3 + 12 * q^5 + 44 * q^7 + 18 * q^9 + 52 * q^11 - 26 * q^13 + 36 * q^15 - 20 * q^17 - 60 * q^19 + 132 * q^21 - 8 * q^23 + 166 * q^25 + 54 * q^27 + 132 * q^29 - 140 * q^31 + 156 * q^33 + 608 * q^35 + 68 * q^37 - 78 * q^39 + 28 * q^41 + 108 * q^45 + 36 * q^47 + 626 * q^49 - 60 * q^51 + 668 * q^53 + 312 * q^55 - 180 * q^57 - 508 * q^59 + 340 * q^61 + 396 * q^63 - 156 * q^65 - 940 * q^67 - 24 * q^69 + 300 * q^71 + 1124 * q^73 + 498 * q^75 + 1144 * q^77 - 1520 * q^79 + 162 * q^81 + 524 * q^83 - 808 * q^85 + 396 * q^87 + 1900 * q^89 - 572 * q^91 - 420 * q^93 - 2080 * q^95 - 1436 * q^97 + 468 * 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
 −6.55744 6.55744
0 3.00000 0 −7.11488 0 8.88512 0 9.00000 0
1.2 0 3.00000 0 19.1149 0 35.1149 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.f 2
3.b odd 2 1 936.4.a.c 2
4.b odd 2 1 624.4.a.l 2
8.b even 2 1 2496.4.a.u 2
8.d odd 2 1 2496.4.a.bd 2
12.b even 2 1 1872.4.a.v 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
312.4.a.f 2 1.a even 1 1 trivial
624.4.a.l 2 4.b odd 2 1
936.4.a.c 2 3.b odd 2 1
1872.4.a.v 2 12.b even 2 1
2496.4.a.u 2 8.b even 2 1
2496.4.a.bd 2 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 12T_{5} - 136$$ acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(312))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$(T - 3)^{2}$$
$5$ $$T^{2} - 12T - 136$$
$7$ $$T^{2} - 44T + 312$$
$11$ $$(T - 26)^{2}$$
$13$ $$(T + 13)^{2}$$
$17$ $$T^{2} + 20T - 588$$
$19$ $$T^{2} + 60T - 3400$$
$23$ $$T^{2} + 8T - 24752$$
$29$ $$T^{2} - 132T - 20412$$
$31$ $$T^{2} + 140T - 33800$$
$37$ $$T^{2} - 68T - 32556$$
$41$ $$T^{2} - 28T - 8232$$
$43$ $$T^{2} - 33712$$
$47$ $$T^{2} - 36T - 5868$$
$53$ $$T^{2} - 668T + 108804$$
$59$ $$T^{2} + 508T - 18732$$
$61$ $$T^{2} - 340T + 17892$$
$67$ $$T^{2} + 940T - 97128$$
$71$ $$T^{2} - 300T - 772828$$
$73$ $$T^{2} - 1124 T + 309652$$
$79$ $$T^{2} + 1520 T + 244608$$
$83$ $$T^{2} - 524T - 234764$$
$89$ $$T^{2} - 1900 T + 902328$$
$97$ $$T^{2} + 1436 T + 459796$$