Properties

 Label 1323.4.a.bh Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $1$ Dimension $7$ CM no Inner twists $1$

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: $$1$$ Dimension: $$7$$ Coefficient field: $$\mathbb{Q}[x]/(x^{7} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504$$ x^7 - x^6 - 43*x^5 + 10*x^4 + 513*x^3 + 258*x^2 - 936*x - 504 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3^{3}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 4) q^{4} - \beta_{4} q^{5} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 11) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b2 + b1 + 4) * q^4 - b4 * q^5 + (-b3 - b2 - 4*b1 - 11) * q^8 $$q - \beta_1 q^{2} + (\beta_{2} + \beta_1 + 4) q^{4} - \beta_{4} q^{5} + ( - \beta_{3} - \beta_{2} - 4 \beta_1 - 11) q^{8} + (2 \beta_{6} + 2 \beta_{4} + \beta_{3} + \cdots - 3) q^{10}+ \cdots + ( - 18 \beta_{6} + 18 \beta_{5} + \cdots + 526) q^{97}+O(q^{100})$$ q - b1 * q^2 + (b2 + b1 + 4) * q^4 - b4 * q^5 + (-b3 - b2 - 4*b1 - 11) * q^8 + (2*b6 + 2*b4 + b3 + 2*b1 - 3) * q^10 + (b6 + b3 - 3*b1 - 14) * q^11 + (-2*b5 + b4 + 2*b1 + 17) * q^13 + (-b5 + 2*b4 - b2 + 12*b1 + 18) * q^16 + (b6 + 3*b5 - b3 + 4*b2 + 3*b1 + 3) * q^17 + (-3*b6 + 3*b4 - 3*b3 - 2*b2 - 11*b1 - 23) * q^19 + (-8*b6 + 3*b5 - 6*b4 + b3 - 3*b2 - 3*b1 - 11) * q^20 + (-2*b6 + 2*b5 - 6*b4 + 2*b3 + b2 + 18*b1 + 38) * q^22 + (-5*b6 + 3*b5 - b4 + b3 - 6*b2 + 3*b1 + 4) * q^23 + (-3*b6 - b5 + 2*b4 + 3*b3 + 2*b2 + 27*b1 + 51) * q^25 + (-2*b6 - 6*b5 - 2*b4 - 5*b3 - 6*b2 - 25*b1 - 29) * q^26 + (5*b6 - 3*b5 - 3*b3 - 14*b2 + 3*b1 - 44) * q^29 + (-8*b6 + 2*b5 + 3*b4 + 2*b3 - 2*b2 + 12*b1 - 3) * q^31 + (-4*b6 - 3*b5 - 4*b4 + 5*b3 - 6*b2 + 3*b1 - 55) * q^32 + (-2*b6 + 9*b5 - 2*b4 + 2*b3 + 7*b2 - 31*b1 - 12) * q^34 + (b6 + 5*b5 + 6*b4 - b3 + 6*b2 + 35*b1 - 23) * q^37 + (-6*b5 + 12*b4 - 7*b3 + 17*b2 + 39*b1 + 133) * q^38 + (12*b6 + 2*b5 + 26*b4 - 2*b2 + 47*b1 + 8) * q^40 + (7*b6 - 3*b5 + 5*b4 + 7*b3 - 6*b2 - 23*b1 - 74) * q^41 + (3*b6 - 8*b5 - 14*b4 - 3*b3 + 8*b2 - 5*b1) * q^43 + (8*b6 + 6*b5 + 16*b4 + b3 - 22*b2 - 17*b1 - 129) * q^44 + (12*b6 + 5*b5 + 20*b4 + 9*b3 - 5*b2 + 50*b1 - 61) * q^46 + (10*b6 + 3*b5 + b4 + 4*b3 - 18*b2 + 22*b1 + 40) * q^47 + (2*b6 - 3*b5 + 2*b4 - 6*b3 - 41*b2 - 89*b1 - 344) * q^50 + (8*b6 - 9*b5 + 14*b4 - 11*b3 + 26*b2 + 64*b1 + 133) * q^52 + (-3*b5 + b4 - 12*b3 - 22*b1 - 189) * q^53 + (7*b6 - 5*b5 + 20*b4 + 5*b3 - 16*b2 - 91*b1 - 11) * q^55 + (-10*b6 - 7*b5 - 14*b4 + 10*b3 + 5*b2 + 122*b1 - 28) * q^58 + (7*b6 + 3*b5 - 5*b4 + 21*b3 + 38*b2 + 27*b1 + 5) * q^59 + (-6*b6 - 9*b5 + 6*b4 + 12*b3 + 10*b2 + 28*b1 - 17) * q^61 + (10*b6 + 22*b4 - 3*b3 - 22*b2 + 15*b1 - 175) * q^62 + (16*b6 - 2*b4 + 5*b3 - 20*b2 + 14*b1 - 245) * q^64 + (-b6 - 9*b5 - 35*b4 + 3*b3 - 14*b2 - 31*b1 - 230) * q^65 + (-3*b6 - 9*b5 - 15*b4 - 39*b3 - 10*b2 + 35*b1 - 136) * q^67 + (3*b5 + 8*b4 + 21*b3 + 9*b2 - 2*b1 + 369) * q^68 + (-11*b6 + 12*b5 + 31*b4 + 29*b3 - 8*b2 + 59*b1 - 123) * q^71 + (5*b6 + 5*b5 + 10*b4 - 5*b3 - 58*b2 + 47*b1 - 204) * q^73 + (-14*b6 + 15*b5 - 14*b4 - 2*b3 - 21*b2 - 59*b1 - 368) * q^74 + (-25*b5 - 34*b4 - 24*b3 - 14*b2 - 253*b1 - 234) * q^76 + (19*b6 + 2*b5 + 33*b4 - 19*b3 - 64*b2 + 163*b1 - 194) * q^79 + (-12*b6 - 6*b5 - 52*b4 - 16*b3 - 7*b2 - 77*b1 - 332) * q^80 + (-24*b6 + 5*b5 - 52*b4 + 9*b3 + 3*b2 + 130*b1 + 287) * q^82 + (32*b6 - 15*b5 - 5*b4 - 18*b3 + 10*b2 + 112*b1 - 194) * q^83 + (-33*b6 + 32*b5 + 23*b4 - 9*b3 - 20*b2 - 31*b1 + 53) * q^85 + (22*b6 - 24*b5 + 22*b4 - 10*b3 + b2 - 48*b1 + 18) * q^86 + (-32*b6 + 11*b5 - 18*b4 + 11*b3 + 26*b2 + 130*b1 - 13) * q^88 + (-6*b6 + 33*b5 + 40*b4 - 22*b3 - 40*b2 - 60*b1 + 175) * q^89 + (-24*b6 + 12*b5 - 98*b4 + 8*b3 - 7*b2 - 2*b1 - 524) * q^92 + (-22*b6 + 23*b5 - 50*b4 + 37*b3 - 18*b2 + 66*b1 - 229) * q^94 + (40*b6 + 12*b5 + 39*b4 - 8*b3 + 48*b2 + 234*b1 - 527) * q^95 + (-18*b6 + 18*b5 + 30*b4 + 6*b3 + 68*b2 - 70*b1 + 526) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q - q^{2} + 31 q^{4} - q^{5} - 84 q^{8}+O(q^{10})$$ 7 * q - q^2 + 31 * q^4 - q^5 - 84 * q^8 $$7 q - q^{2} + 31 q^{4} - q^{5} - 84 q^{8} - 12 q^{10} - 98 q^{11} + 124 q^{13} + 139 q^{16} + 30 q^{17} - 182 q^{19} - 110 q^{20} + 276 q^{22} + 6 q^{23} + 388 q^{25} - 245 q^{26} - 323 q^{29} - 26 q^{31} - 398 q^{32} - 114 q^{34} - 112 q^{37} + 1015 q^{38} + 147 q^{40} - 524 q^{41} + 8 q^{43} - 937 q^{44} - 339 q^{46} + 288 q^{47} - 2576 q^{50} + 1075 q^{52} - 1353 q^{53} - 156 q^{55} - 81 q^{58} + 165 q^{59} - 56 q^{61} - 1215 q^{62} - 1706 q^{64} - 1694 q^{65} - 988 q^{67} + 2625 q^{68} - 792 q^{71} - 1487 q^{73} - 2736 q^{74} - 1952 q^{76} - 1273 q^{79} - 2501 q^{80} + 2049 q^{82} - 1170 q^{83} + 216 q^{85} + 160 q^{86} + 9 q^{88} + 1058 q^{89} - 3834 q^{92} - 1653 q^{94} - 3260 q^{95} + 3730 q^{97}+O(q^{100})$$ 7 * q - q^2 + 31 * q^4 - q^5 - 84 * q^8 - 12 * q^10 - 98 * q^11 + 124 * q^13 + 139 * q^16 + 30 * q^17 - 182 * q^19 - 110 * q^20 + 276 * q^22 + 6 * q^23 + 388 * q^25 - 245 * q^26 - 323 * q^29 - 26 * q^31 - 398 * q^32 - 114 * q^34 - 112 * q^37 + 1015 * q^38 + 147 * q^40 - 524 * q^41 + 8 * q^43 - 937 * q^44 - 339 * q^46 + 288 * q^47 - 2576 * q^50 + 1075 * q^52 - 1353 * q^53 - 156 * q^55 - 81 * q^58 + 165 * q^59 - 56 * q^61 - 1215 * q^62 - 1706 * q^64 - 1694 * q^65 - 988 * q^67 + 2625 * q^68 - 792 * q^71 - 1487 * q^73 - 2736 * q^74 - 1952 * q^76 - 1273 * q^79 - 2501 * q^80 + 2049 * q^82 - 1170 * q^83 + 216 * q^85 + 160 * q^86 + 9 * q^88 + 1058 * q^89 - 3834 * q^92 - 1653 * q^94 - 3260 * q^95 + 3730 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 43x^{5} + 10x^{4} + 513x^{3} + 258x^{2} - 936x - 504$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 12$$ v^2 - v - 12 $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 19\nu + 1$$ v^3 - v^2 - 19*v + 1 $$\beta_{4}$$ $$=$$ $$( -\nu^{6} + 4\nu^{5} + 52\nu^{4} - 103\nu^{3} - 729\nu^{2} + 207\nu + 1008 ) / 42$$ (-v^6 + 4*v^5 + 52*v^4 - 103*v^3 - 729*v^2 + 207*v + 1008) / 42 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 4\nu^{5} + 31\nu^{4} - 103\nu^{3} - 246\nu^{2} + 480\nu + 294 ) / 21$$ (-v^6 + 4*v^5 + 31*v^4 - 103*v^3 - 246*v^2 + 480*v + 294) / 21 $$\beta_{6}$$ $$=$$ $$( 5\nu^{6} + \nu^{5} - 197\nu^{4} - 52\nu^{3} + 1965\nu^{2} + 372\nu - 2436 ) / 84$$ (5*v^6 + v^5 - 197*v^4 - 52*v^3 + 1965*v^2 + 372*v - 2436) / 84
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta _1 + 12$$ b2 + b1 + 12 $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 20\beta _1 + 11$$ b3 + b2 + 20*b1 + 11 $$\nu^{4}$$ $$=$$ $$-\beta_{5} + 2\beta_{4} + 23\beta_{2} + 36\beta _1 + 242$$ -b5 + 2*b4 + 23*b2 + 36*b1 + 242 $$\nu^{5}$$ $$=$$ $$4\beta_{6} + 3\beta_{5} + 4\beta_{4} + 27\beta_{3} + 38\beta_{2} + 445\beta _1 + 407$$ 4*b6 + 3*b5 + 4*b4 + 27*b3 + 38*b2 + 445*b1 + 407 $$\nu^{6}$$ $$=$$ $$16\beta_{6} - 40\beta_{5} + 78\beta_{4} + 5\beta_{3} + 516\beta_{2} + 1070\beta _1 + 5339$$ 16*b6 - 40*b5 + 78*b4 + 5*b3 + 516*b2 + 1070*b1 + 5339

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
 5.09184 4.76215 1.44197 −0.541672 −1.63185 −3.62964 −4.49279
−5.09184 0 17.9269 −18.5985 0 0 −50.5460 0 94.7009
1.2 −4.76215 0 14.6781 18.7013 0 0 −31.8020 0 −89.0586
1.3 −1.44197 0 −5.92074 6.60364 0 0 20.0732 0 −9.52221
1.4 0.541672 0 −7.70659 −16.7289 0 0 −8.50782 0 −9.06158
1.5 1.63185 0 −5.33705 12.3790 0 0 −21.7641 0 20.2007
1.6 3.62964 0 5.17430 4.84163 0 0 −10.2563 0 17.5734
1.7 4.49279 0 12.1851 −8.19814 0 0 18.8030 0 −36.8325
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

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

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 1323.4.a.bh 7
3.b odd 2 1 1323.4.a.bk 7
7.b odd 2 1 1323.4.a.bi 7
7.d odd 6 2 189.4.e.g yes 14
21.c even 2 1 1323.4.a.bj 7
21.g even 6 2 189.4.e.f 14

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
189.4.e.f 14 21.g even 6 2
189.4.e.g yes 14 7.d odd 6 2
1323.4.a.bh 7 1.a even 1 1 trivial
1323.4.a.bi 7 7.b odd 2 1
1323.4.a.bj 7 21.c even 2 1
1323.4.a.bk 7 3.b odd 2 1

Hecke kernels

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

 $$T_{2}^{7} + T_{2}^{6} - 43T_{2}^{5} - 10T_{2}^{4} + 513T_{2}^{3} - 258T_{2}^{2} - 936T_{2} + 504$$ T2^7 + T2^6 - 43*T2^5 - 10*T2^4 + 513*T2^3 - 258*T2^2 - 936*T2 + 504 $$T_{5}^{7} + T_{5}^{6} - 631T_{5}^{5} + 311T_{5}^{4} + 112338T_{5}^{3} - 287178T_{5}^{2} - 4846491T_{5} + 18879633$$ T5^7 + T5^6 - 631*T5^5 + 311*T5^4 + 112338*T5^3 - 287178*T5^2 - 4846491*T5 + 18879633 $$T_{13}^{7} - 124 T_{13}^{6} - 354 T_{13}^{5} + 620994 T_{13}^{4} - 26760399 T_{13}^{3} + \cdots - 86896171316$$ T13^7 - 124*T13^6 - 354*T13^5 + 620994*T13^4 - 26760399*T13^3 + 167423118*T13^2 + 7969485833*T13 - 86896171316

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} + T^{6} + \cdots + 504$$
$3$ $$T^{7}$$
$5$ $$T^{7} + T^{6} + \cdots + 18879633$$
$7$ $$T^{7}$$
$11$ $$T^{7} + 98 T^{6} + \cdots - 934983936$$
$13$ $$T^{7} + \cdots - 86896171316$$
$17$ $$T^{7} + \cdots + 37518015618$$
$19$ $$T^{7} + \cdots + 2945814380716$$
$23$ $$T^{7} + \cdots + 20028475364274$$
$29$ $$T^{7} + \cdots + 495493458583509$$
$31$ $$T^{7} + \cdots + 379691567407224$$
$37$ $$T^{7} + \cdots + 36979101568584$$
$41$ $$T^{7} + \cdots - 6874920774882$$
$43$ $$T^{7} + \cdots - 765727707363728$$
$47$ $$T^{7} + \cdots - 22\!\cdots\!06$$
$53$ $$T^{7} + \cdots + 33\!\cdots\!73$$
$59$ $$T^{7} + \cdots + 18\!\cdots\!05$$
$61$ $$T^{7} + \cdots + 46\!\cdots\!86$$
$67$ $$T^{7} + \cdots - 95\!\cdots\!50$$
$71$ $$T^{7} + \cdots - 23\!\cdots\!58$$
$73$ $$T^{7} + \cdots + 39\!\cdots\!45$$
$79$ $$T^{7} + \cdots - 25\!\cdots\!31$$
$83$ $$T^{7} + \cdots + 86\!\cdots\!12$$
$89$ $$T^{7} + \cdots + 12\!\cdots\!14$$
$97$ $$T^{7} + \cdots - 43\!\cdots\!00$$