# Properties

 Label 1452.4.a.t Level $1452$ Weight $4$ Character orbit 1452.a Self dual yes Analytic conductor $85.671$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1452,4,Mod(1,1452)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1452, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1452.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1452 = 2^{2} \cdot 3 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1452.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: $$85.6707733283$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: $$\mathbb{Q}[x]/(x^{6} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611$$ x^6 - x^5 - 174*x^4 + 63*x^3 + 7614*x^2 + 1579*x - 12611 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 11$$ Twist minimal: no (minimal twist has level 132) 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 3 q^{3} + ( - \beta_{2} + 2) q^{5} + ( - \beta_{3} + \beta_{2} - 4) q^{7} + 9 q^{9}+O(q^{10})$$ q + 3 * q^3 + (-b2 + 2) * q^5 + (-b3 + b2 - 4) * q^7 + 9 * q^9 $$q + 3 q^{3} + ( - \beta_{2} + 2) q^{5} + ( - \beta_{3} + \beta_{2} - 4) q^{7} + 9 q^{9} + ( - \beta_{4} + 3 \beta_{2} + \cdots - 10) q^{13}+ \cdots + ( - 26 \beta_{5} + 12 \beta_{4} + \cdots - 669) q^{97}+O(q^{100})$$ q + 3 * q^3 + (-b2 + 2) * q^5 + (-b3 + b2 - 4) * q^7 + 9 * q^9 + (-b4 + 3*b2 + 3*b1 - 10) * q^13 + (-3*b2 + 6) * q^15 + (3*b5 + b4 + 4*b3 + 2*b2 - b1 - 5) * q^17 + (-5*b5 + 2*b3 - 3*b2 - 4*b1 - 47) * q^19 + (-3*b3 + 3*b2 - 12) * q^21 + (-7*b5 - b4 + 3*b3 - 2*b2 - b1 - 22) * q^23 + (12*b5 + 2*b4 - 2*b3 - b2 - 3*b1 + 23) * q^25 + 27 * q^27 + (10*b5 + 4*b4 - 3*b3 + 3*b2 - 5*b1 - 22) * q^29 + (-9*b5 + 6*b4 + 7*b3 - 16*b2 - 6*b1 - 43) * q^31 + (-15*b5 + 3*b4 - 2*b3 - 149) * q^35 + (4*b5 - 3*b4 + 14*b3 + 15*b2 - b1 + 54) * q^37 + (-3*b4 + 9*b2 + 9*b1 - 30) * q^39 + (9*b5 - 11*b4 - 9*b3 + 16*b2 + 8*b1 - 73) * q^41 + (-4*b5 - 3*b4 + 8*b3 + 5*b2 + 4*b1 - 165) * q^43 + (-9*b2 + 18) * q^45 + (-8*b5 - 15*b4 + 5*b3 + 5*b2 + 2*b1 - 118) * q^47 + (10*b5 - 3*b4 + 13*b3 - 2*b2 + 14*b1 + 39) * q^49 + (9*b5 + 3*b4 + 12*b3 + 6*b2 - 3*b1 - 15) * q^51 + (-3*b5 + 2*b4 - 5*b3 + 22*b2 + 31*b1 - 116) * q^53 + (-15*b5 + 6*b3 - 9*b2 - 12*b1 - 141) * q^57 + (7*b5 + b4 - 25*b3 + 13*b2 + b1 - 7) * q^59 + (18*b5 + 14*b4 - 13*b3 + 4*b2 - 18*b1 - 169) * q^61 + (-9*b3 + 9*b2 - 36) * q^63 + (-16*b5 - 2*b4 + 9*b3 + 30*b2 + 3*b1 - 198) * q^65 + (-17*b5 + 4*b4 - 10*b3 - 21*b2 - 4*b1 + 297) * q^67 + (-21*b5 - 3*b4 + 9*b3 - 6*b2 - 3*b1 - 66) * q^69 + (15*b5 + 20*b4 - 30*b3 + 11*b2 - 23*b1 - 38) * q^71 + (-26*b5 + 26*b4 - b3 - 39*b2 + 17*b1 - 246) * q^73 + (36*b5 + 6*b4 - 6*b3 - 3*b2 - 9*b1 + 69) * q^75 + (44*b5 - 15*b4 - 17*b3 + 12*b2 - 5*b1 - 243) * q^79 + 81 * q^81 + (-3*b5 + b4 + b2 + 8*b1 - 508) * q^83 + (-2*b5 - 51*b4 + 28*b3 - 23*b2 + 17*b1 - 232) * q^85 + (30*b5 + 12*b4 - 9*b3 + 9*b2 - 15*b1 - 66) * q^87 + (19*b5 - 9*b4 - 27*b3 + 60*b2 + 20*b1 + 247) * q^89 + (34*b5 - 47*b4 - 24*b3 + 5*b2 + 8*b1 + 199) * q^91 + (-27*b5 + 18*b4 + 21*b3 - 48*b2 - 18*b1 - 129) * q^93 + (-10*b5 + 33*b4 - 17*b3 + 77*b2 + 10*b1 - 268) * q^95 + (-26*b5 + 12*b4 - 14*b3 - 23*b2 - 43*b1 - 669) * q^97 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9}+O(q^{10})$$ 6 * q + 18 * q^3 + 13 * q^5 - 23 * q^7 + 54 * q^9 $$6 q + 18 q^{3} + 13 q^{5} - 23 q^{7} + 54 q^{9} - 66 q^{13} + 39 q^{15} - 44 q^{17} - 270 q^{19} - 69 q^{21} - 124 q^{23} + 131 q^{25} + 162 q^{27} - 141 q^{29} - 253 q^{31} - 884 q^{35} + 288 q^{37} - 198 q^{39} - 428 q^{41} - 1006 q^{43} + 117 q^{45} - 674 q^{47} + 181 q^{49} - 132 q^{51} - 773 q^{53} - 810 q^{57} - 17 q^{59} - 1016 q^{61} - 207 q^{63} - 1220 q^{65} + 1836 q^{67} - 372 q^{69} - 208 q^{71} - 1521 q^{73} + 393 q^{75} - 1425 q^{79} + 486 q^{81} - 3065 q^{83} - 1304 q^{85} - 423 q^{87} + 1444 q^{89} + 1328 q^{91} - 759 q^{93} - 1760 q^{95} - 3887 q^{97}+O(q^{100})$$ 6 * q + 18 * q^3 + 13 * q^5 - 23 * q^7 + 54 * q^9 - 66 * q^13 + 39 * q^15 - 44 * q^17 - 270 * q^19 - 69 * q^21 - 124 * q^23 + 131 * q^25 + 162 * q^27 - 141 * q^29 - 253 * q^31 - 884 * q^35 + 288 * q^37 - 198 * q^39 - 428 * q^41 - 1006 * q^43 + 117 * q^45 - 674 * q^47 + 181 * q^49 - 132 * q^51 - 773 * q^53 - 810 * q^57 - 17 * q^59 - 1016 * q^61 - 207 * q^63 - 1220 * q^65 + 1836 * q^67 - 372 * q^69 - 208 * q^71 - 1521 * q^73 + 393 * q^75 - 1425 * q^79 + 486 * q^81 - 3065 * q^83 - 1304 * q^85 - 423 * q^87 + 1444 * q^89 + 1328 * q^91 - 759 * q^93 - 1760 * q^95 - 3887 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} - 174x^{4} + 63x^{3} + 7614x^{2} + 1579x - 12611$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{5} - 9\nu^{4} - 151\nu^{3} + 781\nu^{2} - 545\nu - 725 ) / 3234$$ (v^5 - 9*v^4 - 151*v^3 + 781*v^2 - 545*v - 725) / 3234 $$\beta_{2}$$ $$=$$ $$( \nu^{5} - 58\nu^{4} - 641\nu^{3} + 8572\nu^{2} + 57471\nu - 187954 ) / 12936$$ (v^5 - 58*v^4 - 641*v^3 + 8572*v^2 + 57471*v - 187954) / 12936 $$\beta_{3}$$ $$=$$ $$( 2\nu^{5} - 67\nu^{4} + 286\nu^{3} + 6119\nu^{2} - 36860\nu - 43149 ) / 4312$$ (2*v^5 - 67*v^4 + 286*v^3 + 6119*v^2 - 36860*v - 43149) / 4312 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 9\nu^{4} - 151\nu^{3} + 781\nu^{2} + 5923\nu - 1019 ) / 588$$ (v^5 - 9*v^4 - 151*v^3 + 781*v^2 + 5923*v - 1019) / 588 $$\beta_{5}$$ $$=$$ $$( -31\nu^{5} - 15\nu^{4} + 4975\nu^{3} - 103\nu^{2} - 188023\nu + 98915 ) / 12936$$ (-31*v^5 - 15*v^4 + 4975*v^3 - 103*v^2 - 188023*v + 98915) / 12936
 $$\nu$$ $$=$$ $$( 2\beta_{4} - 11\beta _1 + 1 ) / 22$$ (2*b4 - 11*b1 + 1) / 22 $$\nu^{2}$$ $$=$$ $$( -22\beta_{5} - 42\beta_{4} + 22\beta_{3} + 66\beta_{2} + 11\beta _1 + 1277 ) / 22$$ (-22*b5 - 42*b4 + 22*b3 + 66*b2 + 11*b1 + 1277) / 22 $$\nu^{3}$$ $$=$$ $$( -33\beta_{5} + 24\beta_{4} + 77\beta_{3} - 33\beta_{2} - 495\beta _1 + 474 ) / 11$$ (-33*b5 + 24*b4 + 77*b3 - 33*b2 - 495*b1 + 474) / 11 $$\nu^{4}$$ $$=$$ $$( -2838\beta_{5} - 4790\beta_{4} + 1958\beta_{3} + 5346\beta_{2} + 77\beta _1 + 110685 ) / 22$$ (-2838*b5 - 4790*b4 + 1958*b3 + 5346*b2 + 77*b1 + 110685) / 22 $$\nu^{5}$$ $$=$$ $$( -18326\beta_{5} - 1970\beta_{4} + 23694\beta_{3} - 13398\beta_{2} - 92235\beta _1 + 158471 ) / 22$$ (-18326*b5 - 1970*b4 + 23694*b3 - 13398*b2 - 92235*b1 + 158471) / 22

## 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
 8.85455 10.2999 −9.47700 −8.43670 1.20018 −1.44098
0 3.00000 0 −17.0090 0 13.6696 0 9.00000 0
1.2 0 3.00000 0 −3.88318 0 −1.97618 0 9.00000 0
1.3 0 3.00000 0 −0.981524 0 17.7745 0 9.00000 0
1.4 0 3.00000 0 3.10899 0 −29.8524 0 9.00000 0
1.5 0 3.00000 0 10.3377 0 5.80080 0 9.00000 0
1.6 0 3.00000 0 21.4270 0 −28.4163 0 9.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$-1$$

## 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 1452.4.a.t 6
11.b odd 2 1 1452.4.a.u 6
11.d odd 10 2 132.4.i.c 12
33.f even 10 2 396.4.j.c 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
132.4.i.c 12 11.d odd 10 2
396.4.j.c 12 33.f even 10 2
1452.4.a.t 6 1.a even 1 1 trivial
1452.4.a.u 6 11.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1452))$$:

 $$T_{5}^{6} - 13T_{5}^{5} - 356T_{5}^{4} + 3363T_{5}^{3} + 10396T_{5}^{2} - 38845T_{5} - 44645$$ T5^6 - 13*T5^5 - 356*T5^4 + 3363*T5^3 + 10396*T5^2 - 38845*T5 - 44645 $$T_{7}^{6} + 23T_{7}^{5} - 855T_{7}^{4} - 9990T_{7}^{3} + 262475T_{7}^{2} - 644817T_{7} - 2362741$$ T7^6 + 23*T7^5 - 855*T7^4 - 9990*T7^3 + 262475*T7^2 - 644817*T7 - 2362741

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$(T - 3)^{6}$$
$5$ $$T^{6} - 13 T^{5} + \cdots - 44645$$
$7$ $$T^{6} + 23 T^{5} + \cdots - 2362741$$
$11$ $$T^{6}$$
$13$ $$T^{6} + \cdots + 2715277100$$
$17$ $$T^{6} + \cdots - 33421191664$$
$19$ $$T^{6} + \cdots - 61426596100$$
$23$ $$T^{6} + \cdots + 18587649476$$
$29$ $$T^{6} + \cdots + 3236332781296$$
$31$ $$T^{6} + \cdots - 2389629513941$$
$37$ $$T^{6} + \cdots - 33044892890544$$
$41$ $$T^{6} + \cdots + 6483755263220$$
$43$ $$T^{6} + \cdots - 27670010268564$$
$47$ $$T^{6} + \cdots + 40658968741616$$
$53$ $$T^{6} + \cdots + 28\!\cdots\!79$$
$59$ $$T^{6} + \cdots - 672659313542861$$
$61$ $$T^{6} + \cdots + 12926954748556$$
$67$ $$T^{6} + \cdots - 362260302137744$$
$71$ $$T^{6} + \cdots + 583597535659220$$
$73$ $$T^{6} + \cdots + 28\!\cdots\!96$$
$79$ $$T^{6} + \cdots + 10\!\cdots\!81$$
$83$ $$T^{6} + \cdots + 14\!\cdots\!01$$
$89$ $$T^{6} + \cdots + 769184681261380$$
$97$ $$T^{6} + \cdots + 56\!\cdots\!59$$