# Properties

 Label 592.8.a.b Level $592$ Weight $8$ Character orbit 592.a Self dual yes Analytic conductor $184.932$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [592,8,Mod(1,592)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(592, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("592.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$592 = 2^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 592.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: $$184.931935087$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\mathbb{Q}[x]/(x^{4} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396$$ x^4 - 2*x^3 - 405*x^2 - 2998*x - 4396 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 74) 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 2 \beta_1 + 14) q^{3} + (4 \beta_{3} - \beta_{2} - 6 \beta_1 + 32) q^{5} + ( - 4 \beta_{2} - 17 \beta_1 + 426) q^{7} + ( - 65 \beta_{3} - 6 \beta_{2} + \cdots + 1108) q^{9}+O(q^{10})$$ q + (-b3 - 2*b1 + 14) * q^3 + (4*b3 - b2 - 6*b1 + 32) * q^5 + (-4*b2 - 17*b1 + 426) * q^7 + (-65*b3 - 6*b2 + 59*b1 + 1108) * q^9 $$q + ( - \beta_{3} - 2 \beta_1 + 14) q^{3} + (4 \beta_{3} - \beta_{2} - 6 \beta_1 + 32) q^{5} + ( - 4 \beta_{2} - 17 \beta_1 + 426) q^{7} + ( - 65 \beta_{3} - 6 \beta_{2} + \cdots + 1108) q^{9}+ \cdots + ( - 173397 \beta_{3} + 13527 \beta_{2} + \cdots + 4397460) q^{99}+O(q^{100})$$ q + (-b3 - 2*b1 + 14) * q^3 + (4*b3 - b2 - 6*b1 + 32) * q^5 + (-4*b2 - 17*b1 + 426) * q^7 + (-65*b3 - 6*b2 + 59*b1 + 1108) * q^9 + (10*b3 - 27*b2 + 167*b1 + 1074) * q^11 + (-119*b3 - 30*b2 - 537*b1 + 2208) * q^13 + (75*b3 + 45*b2 + 225*b1 - 4830) * q^15 + (-218*b3 + 26*b2 - 392*b1 + 5928) * q^17 + (-196*b3 - 196*b2 - 700*b1 - 5574) * q^19 + (-775*b3 - 33*b2 + 487*b1 + 16925) * q^21 + (663*b3 + 268*b2 - 2883*b1 - 4486) * q^23 + (819*b3 - 436*b2 - 1321*b1 - 33563) * q^25 + (-2545*b3 - 1083*b2 + 3172*b1 + 106460) * q^27 + (-191*b3 + 338*b2 + 3445*b1 - 37092) * q^29 + (-716*b3 + 557*b2 + 5304*b1 + 163182) * q^31 + (-2658*b3 - 948*b2 - 2094*b1 - 59493) * q^33 + (3836*b3 - 864*b2 - 3454*b1 + 65038) * q^35 + 50653 * q^37 + (-11066*b3 - 447*b2 + 21296*b1 + 558904) * q^39 + (-3022*b3 - 5007*b2 - 7078*b1 - 173399) * q^41 + (8806*b3 - 136*b2 - 18204*b1 + 43578) * q^43 + (3717*b3 + 3357*b2 + 4797*b1 - 441144) * q^45 + (13798*b3 - 1458*b2 - 26143*b1 + 128346) * q^47 + (1345*b3 - 3695*b2 - 9271*b1 - 394798) * q^49 + (-14844*b3 - 894*b2 + 468*b1 + 717186) * q^51 + (2161*b3 + 9761*b2 - 43421*b1 - 224157) * q^53 + (12215*b3 - 3050*b2 - 24495*b1 + 177100) * q^55 + (-20998*b3 - 4368*b2 + 77536*b1 + 830732) * q^57 + (-4082*b3 - 2372*b2 - 51994*b1 + 478784) * q^59 + (16126*b3 - 8781*b2 - 93394*b1 - 685834) * q^61 + (-57152*b3 - 2706*b2 + 14306*b1 + 819856) * q^63 + (31199*b3 + 499*b2 + 22619*b1 - 3468) * q^65 + (14866*b3 + 23195*b2 - 66346*b1 - 653218) * q^67 + (56334*b3 + 22233*b2 + 27366*b1 - 450576) * q^69 + (3092*b3 - 4470*b2 - 161539*b1 - 1156886) * q^71 + (12119*b3 + 7478*b2 - 122373*b1 - 170551) * q^73 + (36185*b3 + 4635*b2 + 176110*b1 - 1316620) * q^75 + (6097*b3 - 14039*b2 - 52553*b1 + 1119439) * q^77 + (29849*b3 + 2102*b2 - 247611*b1 - 1322648) * q^79 + (-175727*b3 - 49677*b2 - 156811*b1 + 4145617) * q^81 + (25204*b3 - 13742*b2 + 226835*b1 - 633796) * q^83 + (15592*b3 + 9382*b2 + 39892*b1 - 1229744) * q^85 + (59232*b3 - 5529*b2 - 96450*b1 - 1926918) * q^87 + (-20180*b3 + 33884*b2 - 88998*b1 + 2239412) * q^89 + (-75106*b3 - 23600*b2 + 42268*b1 + 4687558) * q^91 + (-146731*b3 - 12807*b2 - 585959*b1 + 1015544) * q^93 + (73296*b3 - 1174*b2 + 32156*b1 + 790488) * q^95 + (51762*b3 + 23962*b2 + 114950*b1 - 5717244) * q^97 + (-173397*b3 + 13527*b2 + 77565*b1 + 4397460) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9}+O(q^{10})$$ 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9 $$4 q + 53 q^{3} + 111 q^{5} + 1666 q^{7} + 4609 q^{9} + 4593 q^{11} + 7847 q^{13} - 18900 q^{15} + 23172 q^{17} - 23696 q^{19} + 69416 q^{21} - 24105 q^{23} - 138149 q^{25} + 433646 q^{27} - 140949 q^{29} + 664609 q^{31} - 240450 q^{33} + 248544 q^{35} + 202612 q^{37} + 2288827 q^{39} - 709737 q^{41} + 128962 q^{43} - 1755342 q^{45} + 445842 q^{47} - 1602774 q^{49} + 2883630 q^{51} - 975870 q^{53} + 644145 q^{55} + 3494630 q^{57} + 1812858 q^{59} - 2955031 q^{61} + 3362482 q^{63} + 666 q^{65} - 2737235 q^{67} - 1781673 q^{69} - 4958184 q^{71} - 931591 q^{73} - 4945810 q^{75} + 4352514 q^{77} - 5813561 q^{79} + 16394896 q^{81} - 2120460 q^{83} - 4845402 q^{85} - 7965333 q^{87} + 8833716 q^{89} + 18886274 q^{91} + 3024182 q^{93} + 3151794 q^{95} - 22666876 q^{97} + 17931894 q^{99}+O(q^{100})$$ 4 * q + 53 * q^3 + 111 * q^5 + 1666 * q^7 + 4609 * q^9 + 4593 * q^11 + 7847 * q^13 - 18900 * q^15 + 23172 * q^17 - 23696 * q^19 + 69416 * q^21 - 24105 * q^23 - 138149 * q^25 + 433646 * q^27 - 140949 * q^29 + 664609 * q^31 - 240450 * q^33 + 248544 * q^35 + 202612 * q^37 + 2288827 * q^39 - 709737 * q^41 + 128962 * q^43 - 1755342 * q^45 + 445842 * q^47 - 1602774 * q^49 + 2883630 * q^51 - 975870 * q^53 + 644145 * q^55 + 3494630 * q^57 + 1812858 * q^59 - 2955031 * q^61 + 3362482 * q^63 + 666 * q^65 - 2737235 * q^67 - 1781673 * q^69 - 4958184 * q^71 - 931591 * q^73 - 4945810 * q^75 + 4352514 * q^77 - 5813561 * q^79 + 16394896 * q^81 - 2120460 * q^83 - 4845402 * q^85 - 7965333 * q^87 + 8833716 * q^89 + 18886274 * q^91 + 3024182 * q^93 + 3151794 * q^95 - 22666876 * q^97 + 17931894 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2x^{3} - 405x^{2} - 2998x - 4396$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{3} + 16\nu^{2} + 241\nu - 516 ) / 10$$ (-v^3 + 16*v^2 + 241*v - 516) / 10 $$\beta_{3}$$ $$=$$ $$( \nu^{3} - 6\nu^{2} - 371\nu - 1454 ) / 10$$ (v^3 - 6*v^2 - 371*v - 1454) / 10
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 13\beta _1 + 197$$ b3 + b2 + 13*b1 + 197 $$\nu^{3}$$ $$=$$ $$16\beta_{3} + 6\beta_{2} + 449\beta _1 + 2636$$ 16*b3 + 6*b2 + 449*b1 + 2636

## 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
 24.1837 −6.83090 −13.3612 −1.99165
0 −55.2289 0 −82.2583 0 −195.531 0 863.231 0
1.2 0 −20.4940 0 375.302 0 980.897 0 −1766.99 0
1.3 0 36.0598 0 −19.7362 0 50.9272 0 −886.690 0
1.4 0 92.6631 0 −162.307 0 829.706 0 6399.45 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
$$2$$ $$-1$$
$$37$$ $$-1$$

## 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 592.8.a.b 4
4.b odd 2 1 74.8.a.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.8.a.a 4 4.b odd 2 1
592.8.a.b 4 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 53T_{3}^{3} - 5274T_{3}^{2} + 107325T_{3} + 3782025$$ acting on $$S_{8}^{\mathrm{new}}(\Gamma_0(592))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 53 T^{3} + \cdots + 3782025$$
$5$ $$T^{4} - 111 T^{3} + \cdots - 98892000$$
$7$ $$T^{4} + \cdots - 8104255332$$
$11$ $$T^{4} + \cdots - 169823359884525$$
$13$ $$T^{4} + \cdots - 32\!\cdots\!32$$
$17$ $$T^{4} + \cdots + 92\!\cdots\!60$$
$19$ $$T^{4} + \cdots + 31\!\cdots\!00$$
$23$ $$T^{4} + \cdots + 46\!\cdots\!92$$
$29$ $$T^{4} + \cdots - 12\!\cdots\!60$$
$31$ $$T^{4} + \cdots + 25\!\cdots\!48$$
$37$ $$(T - 50653)^{4}$$
$41$ $$T^{4} + \cdots + 83\!\cdots\!95$$
$43$ $$T^{4} + \cdots + 23\!\cdots\!44$$
$47$ $$T^{4} + \cdots + 76\!\cdots\!12$$
$53$ $$T^{4} + \cdots - 14\!\cdots\!28$$
$59$ $$T^{4} + \cdots - 78\!\cdots\!72$$
$61$ $$T^{4} + \cdots - 31\!\cdots\!00$$
$67$ $$T^{4} + \cdots - 21\!\cdots\!48$$
$71$ $$T^{4} + \cdots + 68\!\cdots\!00$$
$73$ $$T^{4} + \cdots + 42\!\cdots\!11$$
$79$ $$T^{4} + \cdots + 82\!\cdots\!88$$
$83$ $$T^{4} + \cdots + 23\!\cdots\!20$$
$89$ $$T^{4} + \cdots - 17\!\cdots\!00$$
$97$ $$T^{4} + \cdots + 16\!\cdots\!20$$