# Properties

 Label 1323.4.a.bo Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1323 = 3^{3} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1323.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: $$78.0595269376$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9$$ x^8 - 54*x^6 - 6*x^5 + 555*x^4 + 642*x^3 - 218*x^2 - 54*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{5}\cdot 3^{4}\cdot 7^{2}$$ Twist minimal: no (minimal twist has level 189) 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 a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_1 + 6) q^{4} + \beta_{3} q^{5} + ( - \beta_{4} - 5 \beta_{2}) q^{8}+O(q^{10})$$ q - b2 * q^2 + (b1 + 6) * q^4 + b3 * q^5 + (-b4 - 5*b2) * q^8 $$q - \beta_{2} q^{2} + (\beta_1 + 6) q^{4} + \beta_{3} q^{5} + ( - \beta_{4} - 5 \beta_{2}) q^{8} + (\beta_{6} + 5) q^{10} + ( - \beta_{5} - \beta_{4} + \cdots - 3 \beta_{2}) q^{11}+ \cdots + (2 \beta_{7} + 8 \beta_{6} + 46 \beta_1 + 69) q^{97}+O(q^{100})$$ q - b2 * q^2 + (b1 + 6) * q^4 + b3 * q^5 + (-b4 - 5*b2) * q^8 + (b6 + 5) * q^10 + (-b5 - b4 - b3 - 3*b2) * q^11 + (b6 + 3*b1 - 11) * q^13 + (b7 - b6 + 11*b1 + 20) * q^16 + (-2*b5 + b3 + 8*b2) * q^17 + (b7 + b6 - b1 + 1) * q^19 + (b5 + b4 + b3 - 3*b2) * q^20 + (-b7 - 2*b6 + 10*b1 + 29) * q^22 + (-b5 - b4 - 2*b3 - 15*b2) * q^23 + (4*b6 - 10*b1 + 49) * q^25 + (b5 - 2*b4 + 9*b3 - 8*b2) * q^26 + (-2*b5 - 4*b4 + 7*b3 - 20*b2) * q^29 + (-b7 + 14*b1 + 100) * q^31 + (4*b5 - 5*b4 - 6*b3 - 57*b2) * q^32 + (-4*b7 + b6 - 22*b1 - 119) * q^34 + (-b6 - 15*b1 + 87) * q^37 + (6*b5 + b4 + 12*b3 + 10*b2) * q^38 + (b7 - 6*b6 - 4*b1 + 15) * q^40 + (-2*b5 + 4*b4 + 5*b3 - 28*b2) * q^41 + (-2*b7 + 6*b6 + 20*b1 + 179) * q^43 + (b5 - 3*b4 - 13*b3 - 81*b2) * q^44 + (-b7 - 3*b6 + 22*b1 + 192) * q^46 + (-2*b5 - 8*b4 + 15*b3 + 8*b2) * q^47 + (4*b5 + 14*b4 + 36*b3 + 29*b2) * q^50 + (4*b7 - b6 + 19*b1 + 247) * q^52 + (-3*b5 + 9*b4 - 2*b3 - 37*b2) * q^53 + (6*b7 - 12*b6 - 8*b1 - 154) * q^55 + (3*b6 + 62*b1 + 295) * q^58 + (b5 + 5*b4 + 27*b3 + 91*b2) * q^59 + (-b7 + 2*b6 - 22*b1 + 26) * q^61 + (-5*b5 - 13*b4 - 3*b3 - 200*b2) * q^62 + (5*b7 - 3*b6 + 67*b1 + 622) * q^64 + (7*b5 + 17*b4 + 5*b3 - 105*b2) * q^65 + (-b7 - 13*b6 - b1 + 92) * q^67 + (-3*b5 + 27*b4 - 11*b3 + 203*b2) * q^68 + (5*b5 + 3*b4 - 45*b3 + 21*b2) * q^71 + (-7*b7 - 7*b6 + b1 - 567) * q^73 + (-b5 + 14*b4 - 9*b3 + 16*b2) * q^74 + (3*b7 + 5*b6 + 26*b1 - 50) * q^76 + (-4*b7 + b6 + 29*b1 + 69) * q^79 + (-9*b5 - 11*b4 - 59*b3 + 27*b2) * q^80 + (-8*b7 + 9*b6 - 42*b1 + 413) * q^82 + (8*b5 - 2*b4 + 10*b3 - 98*b2) * q^83 + (10*b7 - 4*b6 - 38*b1 + 164) * q^85 + (-4*b5 - 12*b4 + 48*b3 - 311*b2) * q^86 + (13*b7 + 50*b1 + 837) * q^88 + (4*b4 + 17*b3 + 4*b2) * q^89 + (-16*b4 - 14*b3 - 234*b2) * q^92 + (4*b7 + 7*b6 + 90*b1 - 65) * q^94 + (-13*b5 + 5*b4 + 44*b3 - 129*b2) * q^95 + (2*b7 + 8*b6 + 46*b1 + 69) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 48 q^{4}+O(q^{10})$$ 8 * q + 48 * q^4 $$8 q + 48 q^{4} + 44 q^{10} - 84 q^{13} + 156 q^{16} + 12 q^{19} + 224 q^{22} + 408 q^{25} + 800 q^{31} - 948 q^{34} + 692 q^{37} + 96 q^{40} + 1456 q^{43} + 1524 q^{46} + 1972 q^{52} - 1280 q^{55} + 2372 q^{58} + 216 q^{61} + 4964 q^{64} + 684 q^{67} - 4564 q^{73} - 380 q^{76} + 556 q^{79} + 3340 q^{82} + 1296 q^{85} + 6696 q^{88} - 492 q^{94} + 584 q^{97}+O(q^{100})$$ 8 * q + 48 * q^4 + 44 * q^10 - 84 * q^13 + 156 * q^16 + 12 * q^19 + 224 * q^22 + 408 * q^25 + 800 * q^31 - 948 * q^34 + 692 * q^37 + 96 * q^40 + 1456 * q^43 + 1524 * q^46 + 1972 * q^52 - 1280 * q^55 + 2372 * q^58 + 216 * q^61 + 4964 * q^64 + 684 * q^67 - 4564 * q^73 - 380 * q^76 + 556 * q^79 + 3340 * q^82 + 1296 * q^85 + 6696 * q^88 - 492 * q^94 + 584 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 54x^{6} - 6x^{5} + 555x^{4} + 642x^{3} - 218x^{2} - 54x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$( 43\nu^{7} - 264\nu^{6} - 1786\nu^{5} + 12816\nu^{4} - 1106\nu^{3} - 63870\nu^{2} + 6267\nu - 2010 ) / 714$$ (43*v^7 - 264*v^6 - 1786*v^5 + 12816*v^4 - 1106*v^3 - 63870*v^2 + 6267*v - 2010) / 714 $$\beta_{2}$$ $$=$$ $$( -482\nu^{7} + 618\nu^{6} + 24611\nu^{5} - 27567\nu^{4} - 201047\nu^{3} - 103161\nu^{2} + 10425\nu + 12543 ) / 3570$$ (-482*v^7 + 618*v^6 + 24611*v^5 - 27567*v^4 - 201047*v^3 - 103161*v^2 + 10425*v + 12543) / 3570 $$\beta_{3}$$ $$=$$ $$( 739\nu^{7} + 519\nu^{6} - 41122\nu^{5} - 29901\nu^{4} + 467824\nu^{3} + 644097\nu^{2} - 262695\nu - 55416 ) / 3570$$ (739*v^7 + 519*v^6 - 41122*v^5 - 29901*v^4 + 467824*v^3 + 644097*v^2 - 262695*v - 55416) / 3570 $$\beta_{4}$$ $$=$$ $$( 1159 \nu^{7} + 1884 \nu^{6} - 65167 \nu^{5} - 102981 \nu^{4} + 757939 \nu^{3} + 1525047 \nu^{2} + \cdots - 164091 ) / 3570$$ (1159*v^7 + 1884*v^6 - 65167*v^5 - 102981*v^4 + 757939*v^3 + 1525047*v^2 + 141030*v - 164091) / 3570 $$\beta_{5}$$ $$=$$ $$( -12\nu^{7} + 29\nu^{6} + 602\nu^{5} - 1397\nu^{4} - 4532\nu^{3} + 3815\nu^{2} + 4496\nu - 516 ) / 34$$ (-12*v^7 + 29*v^6 + 602*v^5 - 1397*v^4 - 4532*v^3 + 3815*v^2 + 4496*v - 516) / 34 $$\beta_{6}$$ $$=$$ $$( -359\nu^{7} - 270\nu^{6} + 19826\nu^{5} + 15801\nu^{4} - 219296\nu^{3} - 336171\nu^{2} + 55359\nu + 22626 ) / 357$$ (-359*v^7 - 270*v^6 + 19826*v^5 + 15801*v^4 - 219296*v^3 - 336171*v^2 + 55359*v + 22626) / 357 $$\beta_{7}$$ $$=$$ $$( 128\nu^{7} - 162\nu^{6} - 6563\nu^{5} + 7257\nu^{4} + 54671\nu^{3} + 25941\nu^{2} - 8217\nu + 642 ) / 51$$ (128*v^7 - 162*v^6 - 6563*v^5 + 7257*v^4 + 54671*v^3 + 25941*v^2 - 8217*v + 642) / 51
 $$\nu$$ $$=$$ $$( \beta_{7} + 2\beta_{6} + \beta_{5} + 2\beta_{4} + 5\beta_{3} + 14\beta_{2} + \beta _1 - 1 ) / 42$$ (b7 + 2*b6 + b5 + 2*b4 + 5*b3 + 14*b2 + b1 - 1) / 42 $$\nu^{2}$$ $$=$$ $$( -5\beta_{7} + 4\beta_{6} - 6\beta_{5} + 9\beta_{4} + 12\beta_{3} - 63\beta_{2} + 9\beta _1 + 565 ) / 42$$ (-5*b7 + 4*b6 - 6*b5 + 9*b4 + 12*b3 - 63*b2 + 9*b1 + 565) / 42 $$\nu^{3}$$ $$=$$ $$( 65\beta_{7} + 144\beta_{6} + 115\beta_{5} + 83\beta_{4} + 449\beta_{3} + 749\beta_{2} + 58\beta _1 + 117 ) / 84$$ (65*b7 + 144*b6 + 115*b5 + 83*b4 + 449*b3 + 749*b2 + 58*b1 + 117) / 84 $$\nu^{4}$$ $$=$$ $$( -218\beta_{7} + 166\beta_{6} - 285\beta_{5} + 312\beta_{4} + 339\beta_{3} - 3318\beta_{2} - 99\beta _1 + 18880 ) / 42$$ (-218*b7 + 166*b6 - 285*b5 + 312*b4 + 339*b3 - 3318*b2 - 99*b1 + 18880) / 42 $$\nu^{5}$$ $$=$$ $$( 2577 \beta_{7} + 5672 \beta_{6} + 5063 \beta_{5} + 2377 \beta_{4} + 17713 \beta_{3} + 25375 \beta_{2} + \cdots - 19531 ) / 84$$ (2577*b7 + 5672*b6 + 5063*b5 + 2377*b4 + 17713*b3 + 25375*b2 + 176*b1 - 19531) / 84 $$\nu^{6}$$ $$=$$ $$( - 1577 \beta_{7} + 983 \beta_{6} - 2046 \beta_{5} + 1823 \beta_{4} + 1614 \beta_{3} - 25221 \beta_{2} + \cdots + 118967 ) / 7$$ (-1577*b7 + 983*b6 - 2046*b5 + 1823*b4 + 1614*b3 - 25221*b2 - 1773*b1 + 118967) / 7 $$\nu^{7}$$ $$=$$ $$( 53663 \beta_{7} + 112030 \beta_{6} + 107141 \beta_{5} + 37672 \beta_{4} + 349141 \beta_{3} + \cdots - 807479 ) / 42$$ (53663*b7 + 112030*b6 + 107141*b5 + 37672*b4 + 349141*b3 + 500836*b2 - 17485*b1 - 807479) / 42

## 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
 4.35333 −2.07765 0.128152 −0.256959 0.356871 6.13087 −6.51461 −2.12000
−5.46178 0 21.8311 −0.199136 0 0 −75.5424 0 1.08764
1.2 −3.49236 0 4.19658 7.87531 0 0 13.2829 0 −27.5034
1.3 −3.29273 0 2.84210 −21.7165 0 0 16.9836 0 71.5066
1.4 −1.76924 0 −4.86977 13.0512 0 0 22.7698 0 −23.0908
1.5 1.76924 0 −4.86977 −13.0512 0 0 −22.7698 0 −23.0908
1.6 3.29273 0 2.84210 21.7165 0 0 −16.9836 0 71.5066
1.7 3.49236 0 4.19658 −7.87531 0 0 −13.2829 0 −27.5034
1.8 5.46178 0 21.8311 0.199136 0 0 75.5424 0 1.08764
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$+1$$
$$7$$ $$+1$$

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1323.4.a.bo 8
3.b odd 2 1 inner 1323.4.a.bo 8
7.b odd 2 1 1323.4.a.bn 8
7.c even 3 2 189.4.e.h 16
21.c even 2 1 1323.4.a.bn 8
21.h odd 6 2 189.4.e.h 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.h 16 7.c even 3 2
189.4.e.h 16 21.h odd 6 2
1323.4.a.bn 8 7.b odd 2 1
1323.4.a.bn 8 21.c even 2 1
1323.4.a.bo 8 1.a even 1 1 trivial
1323.4.a.bo 8 3.b odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1323))$$:

 $$T_{2}^{8} - 56T_{2}^{6} + 985T_{2}^{4} - 6510T_{2}^{2} + 12348$$ T2^8 - 56*T2^6 + 985*T2^4 - 6510*T2^2 + 12348 $$T_{5}^{8} - 704T_{5}^{6} + 120172T_{5}^{4} - 4986912T_{5}^{2} + 197568$$ T5^8 - 704*T5^6 + 120172*T5^4 - 4986912*T5^2 + 197568 $$T_{13}^{4} + 42T_{13}^{3} - 4475T_{13}^{2} - 94812T_{13} + 5259952$$ T13^4 + 42*T13^3 - 4475*T13^2 - 94812*T13 + 5259952

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - 56 T^{6} + \cdots + 12348$$
$3$ $$T^{8}$$
$5$ $$T^{8} - 704 T^{6} + \cdots + 197568$$
$7$ $$T^{8}$$
$11$ $$T^{8} + \cdots + 11628770291712$$
$13$ $$(T^{4} + 42 T^{3} + \cdots + 5259952)^{2}$$
$17$ $$T^{8} + \cdots + 5799554111232$$
$19$ $$(T^{4} - 6 T^{3} + \cdots - 11890928)^{2}$$
$23$ $$T^{8} + \cdots + 19258163854272$$
$29$ $$T^{8} + \cdots + 18\!\cdots\!32$$
$31$ $$(T^{4} - 400 T^{3} + \cdots - 343595259)^{2}$$
$37$ $$(T^{4} - 346 T^{3} + \cdots - 405399024)^{2}$$
$41$ $$T^{8} + \cdots + 13\!\cdots\!88$$
$43$ $$(T^{4} - 728 T^{3} + \cdots - 12025459739)^{2}$$
$47$ $$T^{8} + \cdots + 21\!\cdots\!00$$
$53$ $$T^{8} + \cdots + 44\!\cdots\!48$$
$59$ $$T^{8} + \cdots + 19\!\cdots\!92$$
$61$ $$(T^{4} - 108 T^{3} + \cdots - 1314437615)^{2}$$
$67$ $$(T^{4} - 342 T^{3} + \cdots - 16835689556)^{2}$$
$71$ $$T^{8} + \cdots + 87\!\cdots\!00$$
$73$ $$(T^{4} + 2282 T^{3} + \cdots - 330036608412)^{2}$$
$79$ $$(T^{4} - 278 T^{3} + \cdots - 2954434736)^{2}$$
$83$ $$T^{8} + \cdots + 16\!\cdots\!72$$
$89$ $$T^{8} + \cdots + 69\!\cdots\!12$$
$97$ $$(T^{4} - 292 T^{3} + \cdots + 100316149821)^{2}$$