# Properties

 Label 1323.4.a.bq Level $1323$ Weight $4$ Character orbit 1323.a Self dual yes Analytic conductor $78.060$ Analytic rank $0$ Dimension $12$ 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: $$12$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 72x^{10} + 1809x^{8} - 19062x^{6} + 78324x^{4} - 73368x^{2} + 19600$$ x^12 - 72*x^10 + 1809*x^8 - 19062*x^6 + 78324*x^4 - 73368*x^2 + 19600 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{6}\cdot 3^{4}\cdot 7^{2}$$ 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 a basis $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + 4) q^{4} + (\beta_{5} + 3) q^{5} + (\beta_{9} - \beta_{7} + \cdots + 5 \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b2 + 4) * q^4 + (b5 + 3) * q^5 + (b9 - b7 + b3 + 5*b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{2} + 4) q^{4} + (\beta_{5} + 3) q^{5} + (\beta_{9} - \beta_{7} + \cdots + 5 \beta_1) q^{8}+ \cdots + ( - 11 \beta_{11} + 63 \beta_{9} + \cdots - 25 \beta_1) q^{97}+O(q^{100})$$ q + b1 * q^2 + (b2 + 4) * q^4 + (b5 + 3) * q^5 + (b9 - b7 + b3 + 5*b1) * q^8 + (b9 + b8 - b7 + b3 + 5*b1) * q^10 + (-b11 - b9 - 4*b1) * q^11 + (-b11 - b9 - b8 - b7 + b3 + b1) * q^13 + (b10 + b6 + 4*b5 + 2*b4 + 4*b2 + 35) * q^16 + (b6 + b4 + 2*b2 + 24) * q^17 + (-b11 - b9 - 2*b8 + 6*b1) * q^19 + (b10 - 2*b5 + b4 + 11*b2 + 44) * q^20 + (-b10 - 3*b6 - 3*b4 - 6*b2 - 49) * q^22 + (-6*b9 - 3*b7 + 3*b3 + 2*b1) * q^23 + (-b10 + 3*b6 + 6*b5 - 6*b2 + 25) * q^25 + (-2*b6 + 2*b5 - 2*b4 + 5*b2 + 12) * q^26 + (-b11 + 7*b9 + 3*b8 + 4*b7 - b3 + 5*b1) * q^29 + (-b11 + 3*b9 + 10*b8 - b7 + 3*b3 + 22*b1) * q^31 + (4*b11 + 16*b9 + b8 - 8*b7 + 3*b3 + 39*b1) * q^32 + (2*b11 + 12*b9 - 3*b8 - 4*b7 + 2*b3 + 42*b1) * q^34 + (-2*b10 - b6 + 17*b5 + b4 - 2*b2 + 12) * q^37 + (-b10 - b6 - 4*b5 - b4 + 6*b2 + 69) * q^38 + (2*b11 + 7*b9 - 10*b8 - 7*b7 + 4*b3 + 99*b1) * q^40 + (-4*b10 + 3*b6 - b5 - b4 + 16*b2 + 148) * q^41 + (3*b10 - b6 + 11*b5 + 4*b4 - 2*b2 + 10) * q^43 + (-28*b9 + 9*b8 + 10*b7 - 10*b3 - 75*b1) * q^44 + (3*b10 - 6*b6 + 12*b5 - 12*b4 + 5*b2) * q^46 + (-b10 + 2*b6 - 14*b5 - b4 + 18*b2 + 87) * q^47 + (4*b11 + 12*b9 - 3*b8 + 16*b7 - b3 - 9*b1) * q^50 + (4*b11 - 5*b9 + 16*b8 + 5*b7 - b3 + 53*b1) * q^52 + (-2*b11 - 4*b9 - 7*b8 + b7 - 11*b3 - 47*b1) * q^53 + (b11 - 25*b9 - 13*b8 + 10*b7 - 6*b3 - 29*b1) * q^55 + (-2*b10 + 2*b6 + 2*b5 + 7*b4 + 8*b2 + 94) * q^58 + (4*b10 - 10*b6 + 8*b5 - 10*b4 - 16*b2 + 111) * q^59 + (3*b11 - 15*b9 + 5*b8 + 12*b7 - 10*b3 + 77*b1) * q^61 + (2*b10 - 9*b6 + 32*b5 - 7*b4 + 31*b2 + 295) * q^62 + (-b10 + 15*b6 - 18*b5 + 24*b4 + 58*b2 + 269) * q^64 + (-b11 - 33*b9 - 33*b8 - 19*b7 + 4*b3 + 41*b1) * q^65 + (4*b6 + b5 - 16*b4 - 10*b2 + 86) * q^67 + (4*b10 + 11*b6 + 2*b5 + 23*b4 + 65*b2 + 369) * q^68 + (-2*b11 + 14*b9 - 21*b8 + 44*b7 - 11*b3 + b1) * q^71 + (-4*b11 - 8*b9 - 14*b8 + 9*b7 - 24*b3 + 14*b1) * q^73 + (-6*b11 + 17*b9 + 20*b8 - 33*b7 + 5*b3 + 12*b1) * q^74 + (4*b11 + 15*b8 - 2*b7 - 2*b3 + 63*b1) * q^76 + (-b10 - 10*b6 - 4*b5 + 7*b4 - 12*b2 - 147) * q^79 + (-2*b10 + 21*b6 + 12*b5 + 21*b4 + 58*b2 + 867) * q^80 + (-2*b11 + 21*b9 - 10*b8 + 3*b7 + 3*b3 + 290*b1) * q^82 + (6*b10 + 3*b6 - 57*b5 + 13*b4 + 22*b2 + 254) * q^83 + (2*b10 + 4*b6 + 31*b5 + 14*b4 + 22*b2 + 207) * q^85 + (4*b11 + 29*b9 + 14*b8 - 41*b7 + 16*b3 + 6*b1) * q^86 + (-2*b10 - 13*b6 - 22*b5 - 41*b4 - 142*b2 - 659) * q^88 + (-2*b10 - 8*b6 + 39*b5 - 14*b4 - 42*b2 + 78) * q^89 + (-6*b11 - 31*b9 + 30*b8 + 73*b7 + 11*b3 + 89*b1) * q^92 + (2*b11 + 6*b9 - 20*b8 + 14*b7 + 3*b3 + 229*b1) * q^94 + (-3*b11 - 25*b9 - 17*b8 - 46*b7 + 14*b3 - 13*b1) * q^95 + (-11*b11 + 63*b9 + 11*b8 - 12*b3 - 25*b1) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 48 q^{4} + 40 q^{5}+O(q^{10})$$ 12 * q + 48 * q^4 + 40 * q^5 $$12 q + 48 q^{4} + 40 q^{5} + 444 q^{16} + 292 q^{17} + 524 q^{20} - 600 q^{22} + 324 q^{25} + 144 q^{26} + 216 q^{37} + 808 q^{38} + 1768 q^{41} + 180 q^{43} + 984 q^{47} + 1164 q^{58} + 1324 q^{59} + 3640 q^{62} + 3252 q^{64} + 972 q^{67} + 4528 q^{68} - 1752 q^{79} + 10536 q^{80} + 2872 q^{83} + 2664 q^{85} - 8160 q^{88} + 1036 q^{89}+O(q^{100})$$ 12 * q + 48 * q^4 + 40 * q^5 + 444 * q^16 + 292 * q^17 + 524 * q^20 - 600 * q^22 + 324 * q^25 + 144 * q^26 + 216 * q^37 + 808 * q^38 + 1768 * q^41 + 180 * q^43 + 984 * q^47 + 1164 * q^58 + 1324 * q^59 + 3640 * q^62 + 3252 * q^64 + 972 * q^67 + 4528 * q^68 - 1752 * q^79 + 10536 * q^80 + 2872 * q^83 + 2664 * q^85 - 8160 * q^88 + 1036 * q^89

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 72x^{10} + 1809x^{8} - 19062x^{6} + 78324x^{4} - 73368x^{2} + 19600$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 12$$ v^2 - 12 $$\beta_{3}$$ $$=$$ $$( -3\nu^{11} + 211\nu^{9} - 5112\nu^{7} + 50696\nu^{5} - 182112\nu^{3} + 41584\nu ) / 2520$$ (-3*v^11 + 211*v^9 - 5112*v^7 + 50696*v^5 - 182112*v^3 + 41584*v) / 2520 $$\beta_{4}$$ $$=$$ $$( -\nu^{10} + 77\nu^{8} - 2096\nu^{6} + 23956\nu^{4} - 101392\nu^{2} + 56000 ) / 504$$ (-v^10 + 77*v^8 - 2096*v^6 + 23956*v^4 - 101392*v^2 + 56000) / 504 $$\beta_{5}$$ $$=$$ $$( 19\nu^{10} - 1351\nu^{8} + 33188\nu^{6} - 334232\nu^{4} + 1229080\nu^{2} - 591136 ) / 3528$$ (19*v^10 - 1351*v^8 + 33188*v^6 - 334232*v^4 + 1229080*v^2 - 591136) / 3528 $$\beta_{6}$$ $$=$$ $$( 2\nu^{10} - 147\nu^{8} + 3793\nu^{6} - 40968\nu^{4} + 163892\nu^{2} - 82572 ) / 252$$ (2*v^10 - 147*v^8 + 3793*v^6 - 40968*v^4 + 163892*v^2 - 82572) / 252 $$\beta_{7}$$ $$=$$ $$( 13\nu^{11} - 931\nu^{9} + 23132\nu^{7} - 237326\nu^{5} + 898432\nu^{3} - 449344\nu ) / 2520$$ (13*v^11 - 931*v^9 + 23132*v^7 - 237326*v^5 + 898432*v^3 - 449344*v) / 2520 $$\beta_{8}$$ $$=$$ $$( 19\nu^{11} - 1351\nu^{9} + 33188\nu^{7} - 334232\nu^{5} + 1225552\nu^{3} - 524104\nu ) / 3528$$ (19*v^11 - 1351*v^9 + 33188*v^7 - 334232*v^5 + 1225552*v^3 - 524104*v) / 3528 $$\beta_{9}$$ $$=$$ $$( 8\nu^{11} - 571\nu^{9} + 14122\nu^{7} - 144011\nu^{5} + 541532\nu^{3} - 271924\nu ) / 1260$$ (8*v^11 - 571*v^9 + 14122*v^7 - 144011*v^5 + 541532*v^3 - 271924*v) / 1260 $$\beta_{10}$$ $$=$$ $$( -15\nu^{10} + 1064\nu^{8} - 26085\nu^{6} + 263104\nu^{4} - 981684\nu^{2} + 501368 ) / 588$$ (-15*v^10 + 1064*v^8 - 26085*v^6 + 263104*v^4 - 981684*v^2 + 501368) / 588 $$\beta_{11}$$ $$=$$ $$( -274\nu^{11} + 19523\nu^{9} - 481631\nu^{7} + 4898098\nu^{5} - 18464716\nu^{3} + 10048472\nu ) / 17640$$ (-274*v^11 + 19523*v^9 - 481631*v^7 + 4898098*v^5 - 18464716*v^3 + 10048472*v) / 17640
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 12$$ b2 + 12 $$\nu^{3}$$ $$=$$ $$\beta_{9} - \beta_{7} + \beta_{3} + 21\beta_1$$ b9 - b7 + b3 + 21*b1 $$\nu^{4}$$ $$=$$ $$\beta_{10} + \beta_{6} + 4\beta_{5} + 2\beta_{4} + 28\beta_{2} + 259$$ b10 + b6 + 4*b5 + 2*b4 + 28*b2 + 259 $$\nu^{5}$$ $$=$$ $$4\beta_{11} + 48\beta_{9} + \beta_{8} - 40\beta_{7} + 35\beta_{3} + 519\beta_1$$ 4*b11 + 48*b9 + b8 - 40*b7 + 35*b3 + 519*b1 $$\nu^{6}$$ $$=$$ $$39\beta_{10} + 55\beta_{6} + 142\beta_{5} + 104\beta_{4} + 794\beta_{2} + 6533$$ 39*b10 + 55*b6 + 142*b5 + 104*b4 + 794*b2 + 6533 $$\nu^{7}$$ $$=$$ $$188\beta_{11} + 1780\beta_{9} - 23\beta_{8} - 1360\beta_{7} + 1043\beta_{3} + 13923\beta_1$$ 188*b11 + 1780*b9 - 23*b8 - 1360*b7 + 1043*b3 + 13923*b1 $$\nu^{8}$$ $$=$$ $$1231\beta_{10} + 2179\beta_{6} + 4126\beta_{5} + 4088\beta_{4} + 23038\beta_{2} + 177921$$ 1231*b10 + 2179*b6 + 4126*b5 + 4088*b4 + 23038*b2 + 177921 $$\nu^{9}$$ $$=$$ $$6820\beta_{11} + 60408\beta_{9} - 2411\beta_{8} - 44332\beta_{7} + 30179\beta_{3} + 390803\beta_1$$ 6820*b11 + 60408*b9 - 2411*b8 - 44332*b7 + 30179*b3 + 390803*b1 $$\nu^{10}$$ $$=$$ $$36999\beta_{10} + 76459\beta_{6} + 115894\beta_{5} + 144200\beta_{4} + 679078\beta_{2} + 5050649$$ 36999*b10 + 76459*b6 + 115894*b5 + 144200*b4 + 679078*b2 + 5050649 $$\nu^{11}$$ $$=$$ $$226916\beta_{11} + 1966008\beta_{9} - 113483\beta_{8} - 1415820\beta_{7} + 875227\beta_{3} + 11271171\beta_1$$ 226916*b11 + 1966008*b9 - 113483*b8 - 1415820*b7 + 875227*b3 + 11271171*b1

## 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
 −5.52482 −4.58553 −3.36380 −2.82234 −0.796935 −0.730397 0.730397 0.796935 2.82234 3.36380 4.58553 5.52482
−5.52482 0 22.5236 12.5636 0 0 −80.2405 0 −69.4118
1.2 −4.58553 0 13.0270 8.26880 0 0 −23.0516 0 −37.9168
1.3 −3.36380 0 3.31513 −2.29822 0 0 15.7589 0 7.73076
1.4 −2.82234 0 −0.0343991 −15.1035 0 0 22.6758 0 42.6273
1.5 −0.796935 0 −7.36489 20.8368 0 0 12.2448 0 −16.6056
1.6 −0.730397 0 −7.46652 −4.26748 0 0 11.2967 0 3.11695
1.7 0.730397 0 −7.46652 −4.26748 0 0 −11.2967 0 −3.11695
1.8 0.796935 0 −7.36489 20.8368 0 0 −12.2448 0 16.6056
1.9 2.82234 0 −0.0343991 −15.1035 0 0 −22.6758 0 −42.6273
1.10 3.36380 0 3.31513 −2.29822 0 0 −15.7589 0 −7.73076
1.11 4.58553 0 13.0270 8.26880 0 0 23.0516 0 37.9168
1.12 5.52482 0 22.5236 12.5636 0 0 80.2405 0 69.4118
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.12 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
21.c even 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.bq yes 12
3.b odd 2 1 1323.4.a.bp 12
7.b odd 2 1 1323.4.a.bp 12
21.c even 2 1 inner 1323.4.a.bq yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1323.4.a.bp 12 3.b odd 2 1
1323.4.a.bp 12 7.b odd 2 1
1323.4.a.bq yes 12 1.a even 1 1 trivial
1323.4.a.bq yes 12 21.c even 2 1 inner

## Hecke kernels

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

 $$T_{2}^{12} - 72T_{2}^{10} + 1809T_{2}^{8} - 19062T_{2}^{6} + 78324T_{2}^{4} - 73368T_{2}^{2} + 19600$$ T2^12 - 72*T2^10 + 1809*T2^8 - 19062*T2^6 + 78324*T2^4 - 73368*T2^2 + 19600 $$T_{5}^{6} - 20T_{5}^{5} - 256T_{5}^{4} + 5100T_{5}^{3} + 5545T_{5}^{2} - 156200T_{5} - 320650$$ T5^6 - 20*T5^5 - 256*T5^4 + 5100*T5^3 + 5545*T5^2 - 156200*T5 - 320650 $$T_{13}^{12} - 12558 T_{13}^{10} + 54627765 T_{13}^{8} - 100319568900 T_{13}^{6} + 76101602881500 T_{13}^{4} + \cdots + 465502200250000$$ T13^12 - 12558*T13^10 + 54627765*T13^8 - 100319568900*T13^6 + 76101602881500*T13^4 - 18116300510130000*T13^2 + 465502200250000

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} - 72 T^{10} + \cdots + 19600$$
$3$ $$T^{12}$$
$5$ $$(T^{6} - 20 T^{5} + \cdots - 320650)^{2}$$
$7$ $$T^{12}$$
$11$ $$T^{12} + \cdots + 23\!\cdots\!00$$
$13$ $$T^{12} + \cdots + 465502200250000$$
$17$ $$(T^{6} - 146 T^{5} + \cdots - 289548173)^{2}$$
$19$ $$T^{12} + \cdots + 10\!\cdots\!76$$
$23$ $$T^{12} + \cdots + 95\!\cdots\!16$$
$29$ $$T^{12} + \cdots + 61\!\cdots\!00$$
$31$ $$T^{12} + \cdots + 49\!\cdots\!64$$
$37$ $$(T^{6} + \cdots - 15605287117700)^{2}$$
$41$ $$(T^{6} + \cdots - 473293981068200)^{2}$$
$43$ $$(T^{6} + \cdots - 20524022438525)^{2}$$
$47$ $$(T^{6} + \cdots + 67715557786894)^{2}$$
$53$ $$T^{12} + \cdots + 12\!\cdots\!16$$
$59$ $$(T^{6} + \cdots + 250989605837225)^{2}$$
$61$ $$T^{12} + \cdots + 29\!\cdots\!00$$
$67$ $$(T^{6} + \cdots - 11\!\cdots\!00)^{2}$$
$71$ $$T^{12} + \cdots + 14\!\cdots\!00$$
$73$ $$T^{12} + \cdots + 16\!\cdots\!00$$
$79$ $$(T^{6} + \cdots + 14\!\cdots\!70)^{2}$$
$83$ $$(T^{6} + \cdots + 87\!\cdots\!72)^{2}$$
$89$ $$(T^{6} + \cdots + 51\!\cdots\!00)^{2}$$
$97$ $$T^{12} + \cdots + 51\!\cdots\!00$$