# Properties

 Label 608.4.a.k Level $608$ Weight $4$ Character orbit 608.a Self dual yes Analytic conductor $35.873$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Learn more

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,4,Mod(1,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, 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("608.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$608 = 2^{5} \cdot 19$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 608.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: $$35.8731612835$$ Analytic rank: $$0$$ 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: $$x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664$$ x^7 - 3*x^6 - 121*x^5 + 402*x^4 + 4234*x^3 - 14542*x^2 - 40996*x + 141664 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{6}$$ 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 $$1,\beta_1,\ldots,\beta_{6}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{5} - \beta_{2} + 6) q^{7} + (\beta_{6} + \beta_{5} - 2 \beta_{2} + 12) q^{9}+O(q^{10})$$ q + b1 * q^3 + (-b3 - 2) * q^5 + (b5 - b2 + 6) * q^7 + (b6 + b5 - 2*b2 + 12) * q^9 $$q + \beta_1 q^{3} + ( - \beta_{3} - 2) q^{5} + (\beta_{5} - \beta_{2} + 6) q^{7} + (\beta_{6} + \beta_{5} - 2 \beta_{2} + 12) q^{9} + (\beta_{4} - 3 \beta_{3} + 2 \beta_{2} - 3 \beta_1 - 2) q^{11} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + 3 \beta_{2} + 2 \beta_1 - 6) q^{13} + (2 \beta_{6} - \beta_{5} - 2 \beta_{3} - 4 \beta_{2} - \beta_1 + 17) q^{15} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + 7 \beta_1 + 7) q^{17} - 19 q^{19} + ( - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} + 17 \beta_1 - 3) q^{21} + (2 \beta_{6} + \beta_{5} - 6 \beta_{4} - 3 \beta_{3} + \beta_{2} + 60) q^{23} + (4 \beta_{5} - 3 \beta_{4} + 9 \beta_{3} - 4 \beta_{2} + 11 \beta_1 - 1) q^{25} + ( - 5 \beta_{6} - 3 \beta_{5} - 2 \beta_{4} - 8 \beta_{3} + 2 \beta_{2} + 8 \beta_1 - 5) q^{27} + (5 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - 7 \beta_{3} + 5 \beta_{2} + 11 \beta_1 + 35) q^{29} + ( - 7 \beta_{6} - 3 \beta_{5} + 6 \beta_{4} + 8 \beta_{3} + 4 \beta_{2} + 9 \beta_1 + 13) q^{31} + (3 \beta_{6} - 5 \beta_{5} + 10 \beta_{4} + 2 \beta_{3} - 7 \beta_1 - 33) q^{33} + ( - 3 \beta_{6} - 2 \beta_{5} - \beta_{4} - 13 \beta_{3} - 2 \beta_{2} + \beta_1 + 22) q^{35} + ( - 2 \beta_{6} - 9 \beta_{5} + 10 \beta_{4} + 8 \beta_{3} - 8 \beta_{2} + \cdots - 89) q^{37}+ \cdots + ( - 26 \beta_{6} - 10 \beta_{5} + 39 \beta_{4} + 23 \beta_{3} + 20 \beta_{2} + \cdots - 266) q^{99}+O(q^{100})$$ q + b1 * q^3 + (-b3 - 2) * q^5 + (b5 - b2 + 6) * q^7 + (b6 + b5 - 2*b2 + 12) * q^9 + (b4 - 3*b3 + 2*b2 - 3*b1 - 2) * q^11 + (-b6 + b5 + 2*b4 + b3 + 3*b2 + 2*b1 - 6) * q^13 + (2*b6 - b5 - 2*b3 - 4*b2 - b1 + 17) * q^15 + (-2*b6 + b5 - b4 - b3 - 2*b2 + 7*b1 + 7) * q^17 - 19 * q^19 + (-2*b5 - 2*b4 + b3 - b2 + 17*b1 - 3) * q^21 + (2*b6 + b5 - 6*b4 - 3*b3 + b2 + 60) * q^23 + (4*b5 - 3*b4 + 9*b3 - 4*b2 + 11*b1 - 1) * q^25 + (-5*b6 - 3*b5 - 2*b4 - 8*b3 + 2*b2 + 8*b1 - 5) * q^27 + (5*b6 - 2*b5 - 6*b4 - 7*b3 + 5*b2 + 11*b1 + 35) * q^29 + (-7*b6 - 3*b5 + 6*b4 + 8*b3 + 4*b2 + 9*b1 + 13) * q^31 + (3*b6 - 5*b5 + 10*b4 + 2*b3 - 7*b1 - 33) * q^33 + (-3*b6 - 2*b5 - b4 - 13*b3 - 2*b2 + b1 + 22) * q^35 + (-2*b6 - 9*b5 + 10*b4 + 8*b3 - 8*b2 + 33*b1 - 89) * q^37 + (5*b6 + 3*b5 + 16*b4 + 23*b3 + 7*b2 - 20*b1 + 110) * q^39 + (-4*b6 + b5 - 2*b4 - 10*b3 + 14*b2 - 7*b1 - 1) * q^41 + (5*b6 + 4*b5 + 3*b4 + 5*b3 + 18*b2 - 21*b1 + 104) * q^43 + (-7*b6 - 6*b5 - 4*b4 - 11*b3 - 2*b2 + 44*b1) * q^45 + (-6*b6 - 16*b4 - 3*b3 + 2*b2 + 14*b1 + 210) * q^47 + (-b5 - 11*b4 - 23*b3 + 14*b2 - 29*b1 + 44) * q^49 + (19*b6 + 4*b5 - 14*b4 + 4*b3 - 32*b2 + 7*b1 + 242) * q^51 + (12*b6 + 2*b5 + 4*b4 + 15*b3 + 21*b2 - 5*b1 + 1) * q^53 + (-7*b6 + 10*b5 + 4*b4 + 33*b3 + 6*b2 + 10*b1 + 266) * q^55 - 19*b1 * q^57 + (-12*b6 - 3*b5 + 22*b4 + 22*b3 - 18*b2 + 20*b1 + 9) * q^59 + (-7*b6 + 30*b5 + 4*b4 + 15*b3 - 8*b2 - 6*b1 + 14) * q^61 + (15*b6 - 6*b5 - 14*b4 - 11*b3 - 12*b2 - 14*b1 + 440) * q^63 + (3*b6 - 10*b5 + 26*b4 - 10*b3 + 6*b2 - 90*b1 - 82) * q^65 + (17*b6 + 2*b5 + 2*b3 - 16*b2 - 15*b1 + 236) * q^67 + (-4*b6 + 3*b5 - 30*b4 + 9*b3 - 27*b2 + 82*b1 + 102) * q^69 + (5*b6 - 18*b5 + 10*b4 - 16*b3 - 22*b2 - 12*b1 + 338) * q^71 + (-b6 + 13*b5 - 9*b4 + 47*b3 - 46*b2 - b1 + 21) * q^73 + (-7*b6 + 15*b5 - 26*b4 + 28*b3 - 2*b2 + 34*b1 + 261) * q^75 + (-19*b6 + 8*b5 + 20*b4 - 15*b3 - 2*b2 - 42*b1 - 84) * q^77 + (-24*b6 - 15*b5 - 4*b4 - 40*b3 + 48*b2 + 3*b1 + 383) * q^79 + (22*b6 - 20*b5 - 18*b4 - 20*b2 - 66*b1 + 45) * q^81 + (7*b6 + 12*b5 + 8*b4 - 50*b3 + 34*b2 - 76*b1 + 108) * q^83 + (20*b6 + 12*b5 - 16*b4 - 39*b3 - 40*b2 - 38*b1 + 368) * q^85 + (17*b5 - 16*b4 - 9*b3 - 49*b2 + 48*b1 + 666) * q^87 + (-32*b6 - b5 + 20*b4 + 18*b3 + 28*b2 - 15*b1 - 65) * q^89 + (-26*b6 - 3*b5 + 16*b4 + 16*b3 + 62*b2 - 12*b1 - 7) * q^91 + (28*b6 + 18*b5 + 30*b4 + 34*b3 + 14*b2 - 88*b1 + 148) * q^93 + (19*b3 + 38) * q^95 + (-7*b6 + 12*b5 - 12*b4 + 14*b3 + 20*b2 - 102*b1 - 24) * q^97 + (-26*b6 - 10*b5 + 39*b4 + 23*b3 + 20*b2 + 35*b1 - 266) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9}+O(q^{10})$$ 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 $$7 q + 3 q^{3} - 17 q^{5} + 42 q^{7} + 86 q^{9} - 33 q^{11} - 35 q^{13} + 120 q^{15} + 66 q^{17} - 133 q^{19} + 33 q^{21} + 389 q^{23} + 44 q^{25} - 39 q^{27} + 233 q^{29} + 158 q^{31} - 206 q^{33} + 123 q^{35} - 436 q^{37} + 807 q^{39} - 94 q^{41} + 645 q^{43} + 103 q^{45} + 1451 q^{47} + 93 q^{49} + 1741 q^{51} + 3 q^{53} + 1971 q^{55} - 57 q^{57} + 297 q^{59} + 93 q^{61} + 2999 q^{63} - 788 q^{65} + 1641 q^{67} + 945 q^{69} + 2392 q^{71} + 324 q^{73} + 1909 q^{75} - 711 q^{77} + 2492 q^{79} + 143 q^{81} + 310 q^{83} + 2353 q^{85} + 4795 q^{87} - 440 q^{89} - 107 q^{91} + 900 q^{93} + 323 q^{95} - 532 q^{97} - 1591 q^{99}+O(q^{100})$$ 7 * q + 3 * q^3 - 17 * q^5 + 42 * q^7 + 86 * q^9 - 33 * q^11 - 35 * q^13 + 120 * q^15 + 66 * q^17 - 133 * q^19 + 33 * q^21 + 389 * q^23 + 44 * q^25 - 39 * q^27 + 233 * q^29 + 158 * q^31 - 206 * q^33 + 123 * q^35 - 436 * q^37 + 807 * q^39 - 94 * q^41 + 645 * q^43 + 103 * q^45 + 1451 * q^47 + 93 * q^49 + 1741 * q^51 + 3 * q^53 + 1971 * q^55 - 57 * q^57 + 297 * q^59 + 93 * q^61 + 2999 * q^63 - 788 * q^65 + 1641 * q^67 + 945 * q^69 + 2392 * q^71 + 324 * q^73 + 1909 * q^75 - 711 * q^77 + 2492 * q^79 + 143 * q^81 + 310 * q^83 + 2353 * q^85 + 4795 * q^87 - 440 * q^89 - 107 * q^91 + 900 * q^93 + 323 * q^95 - 532 * q^97 - 1591 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - 3x^{6} - 121x^{5} + 402x^{4} + 4234x^{3} - 14542x^{2} - 40996x + 141664$$ :

 $$\beta_{1}$$ $$=$$ $$( 31441 \nu^{6} - 53905 \nu^{5} - 3640731 \nu^{4} + 5610384 \nu^{3} + 129105226 \nu^{2} - 93123434 \nu - 1374812288 ) / 46945020$$ (31441*v^6 - 53905*v^5 - 3640731*v^4 + 5610384*v^3 + 129105226*v^2 - 93123434*v - 1374812288) / 46945020 $$\beta_{2}$$ $$=$$ $$( - 39878 \nu^{6} - 170155 \nu^{5} + 4088763 \nu^{4} + 14312913 \nu^{3} - 123786608 \nu^{2} - 269997998 \nu + 1133044114 ) / 23472510$$ (-39878*v^6 - 170155*v^5 + 4088763*v^4 + 14312913*v^3 - 123786608*v^2 - 269997998*v + 1133044114) / 23472510 $$\beta_{3}$$ $$=$$ $$( - 41606 \nu^{6} - 149275 \nu^{5} + 4108191 \nu^{4} + 9529701 \nu^{3} - 122864216 \nu^{2} - 88940246 \nu + 1155407968 ) / 23472510$$ (-41606*v^6 - 149275*v^5 + 4108191*v^4 + 9529701*v^3 - 122864216*v^2 - 88940246*v + 1155407968) / 23472510 $$\beta_{4}$$ $$=$$ $$( 134983 \nu^{6} + 651005 \nu^{5} - 12792033 \nu^{4} - 43729188 \nu^{3} + 362351638 \nu^{2} + 636664498 \nu - 3246819584 ) / 46945020$$ (134983*v^6 + 651005*v^5 - 12792033*v^4 - 43729188*v^3 + 362351638*v^2 + 636664498*v - 3246819584) / 46945020 $$\beta_{5}$$ $$=$$ $$( - 320173 \nu^{6} - 369335 \nu^{5} + 34597563 \nu^{4} + 19633308 \nu^{3} - 1044937738 \nu^{2} - 252259318 \nu + 8164814204 ) / 46945020$$ (-320173*v^6 - 369335*v^5 + 34597563*v^4 + 19633308*v^3 - 1044937738*v^2 - 252259318*v + 8164814204) / 46945020 $$\beta_{6}$$ $$=$$ $$( - 169831 \nu^{6} - 229925 \nu^{5} + 21008001 \nu^{4} + 17685276 \nu^{3} - 766255246 \nu^{2} - 318429586 \nu + 7496458508 ) / 23472510$$ (-169831*v^6 - 229925*v^5 + 21008001*v^4 + 17685276*v^3 - 766255246*v^2 - 318429586*v + 7496458508) / 23472510
 $$\nu$$ $$=$$ $$( \beta_{4} + 2\beta_{3} + \beta_1 ) / 4$$ (b4 + 2*b3 + b1) / 4 $$\nu^{2}$$ $$=$$ $$( \beta_{5} - \beta_{3} - \beta_{2} + 5\beta _1 + 70 ) / 2$$ (b5 - b3 - b2 + 5*b1 + 70) / 2 $$\nu^{3}$$ $$=$$ $$( -3\beta_{6} - 2\beta_{5} + 54\beta_{4} + 76\beta_{3} + 40\beta_{2} + 18\beta _1 - 104 ) / 4$$ (-3*b6 - 2*b5 + 54*b4 + 76*b3 + 40*b2 + 18*b1 - 104) / 4 $$\nu^{4}$$ $$=$$ $$( 39\beta_{6} + 126\beta_{5} - 36\beta_{4} - 182\beta_{3} - 226\beta_{2} + 804\beta _1 + 6554 ) / 4$$ (39*b6 + 126*b5 - 36*b4 - 182*b3 - 226*b2 + 804*b1 + 6554) / 4 $$\nu^{5}$$ $$=$$ $$( -178\beta_{6} - 62\beta_{5} + 1652\beta_{4} + 1821\beta_{3} + 1683\beta_{2} - 558\beta _1 - 5331 ) / 2$$ (-178*b6 - 62*b5 + 1652*b4 + 1821*b3 + 1683*b2 - 558*b1 - 5331) / 2 $$\nu^{6}$$ $$=$$ $$( 4441\beta_{6} + 6522\beta_{5} - 5178\beta_{4} - 14256\beta_{3} - 19324\beta_{2} + 55846\beta _1 + 359232 ) / 4$$ (4441*b6 + 6522*b5 - 5178*b4 - 14256*b3 - 19324*b2 + 55846*b1 + 359232) / 4

## 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
 3.39349 −6.10834 4.17107 −4.00212 6.59304 7.37394 −8.42108
0 −9.45545 0 −11.5945 0 15.9032 0 62.4055 0
1.2 0 −5.20712 0 7.79991 0 26.8592 0 0.114065 0
1.3 0 −2.43763 0 −9.42708 0 −9.58919 0 −21.0580 0
1.4 0 −0.923414 0 −0.670967 0 −28.6780 0 −26.1473 0
1.5 0 5.59727 0 −20.9584 0 13.9085 0 4.32941 0
1.6 0 6.88542 0 4.69330 0 3.62949 0 20.4091 0
1.7 0 8.54092 0 13.1578 0 19.9667 0 45.9472 0
 $$n$$: 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
$$2$$ $$1$$
$$19$$ $$1$$

## 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 608.4.a.k yes 7
4.b odd 2 1 608.4.a.j 7
8.b even 2 1 1216.4.a.bf 7
8.d odd 2 1 1216.4.a.bg 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
608.4.a.j 7 4.b odd 2 1
608.4.a.k yes 7 1.a even 1 1 trivial
1216.4.a.bf 7 8.b even 2 1
1216.4.a.bg 7 8.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(608))$$:

 $$T_{3}^{7} - 3T_{3}^{6} - 133T_{3}^{5} + 367T_{3}^{4} + 4632T_{3}^{3} - 6688T_{3}^{2} - 49248T_{3} - 36480$$ T3^7 - 3*T3^6 - 133*T3^5 + 367*T3^4 + 4632*T3^3 - 6688*T3^2 - 49248*T3 - 36480 $$T_{5}^{7} + 17T_{5}^{6} - 315T_{5}^{5} - 4077T_{5}^{4} + 28950T_{5}^{3} + 216728T_{5}^{2} - 972192T_{5} - 740352$$ T5^7 + 17*T5^6 - 315*T5^5 - 4077*T5^4 + 28950*T5^3 + 216728*T5^2 - 972192*T5 - 740352

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7}$$
$3$ $$T^{7} - 3 T^{6} - 133 T^{5} + \cdots - 36480$$
$5$ $$T^{7} + 17 T^{6} - 315 T^{5} + \cdots - 740352$$
$7$ $$T^{7} - 42 T^{6} + \cdots - 118397740$$
$11$ $$T^{7} + 33 T^{6} + \cdots - 47353183200$$
$13$ $$T^{7} + 35 T^{6} + \cdots - 361563654528$$
$17$ $$T^{7} - 66 T^{6} + \cdots + 821802019554$$
$19$ $$(T + 19)^{7}$$
$23$ $$T^{7} - 389 T^{6} + \cdots - 74757540326400$$
$29$ $$T^{7} + \cdots + 758502349773120$$
$31$ $$T^{7} + \cdots + 542529415987200$$
$37$ $$T^{7} + 436 T^{6} + \cdots - 82\!\cdots\!56$$
$41$ $$T^{7} + \cdots - 883998869667840$$
$43$ $$T^{7} - 645 T^{6} + \cdots - 71\!\cdots\!60$$
$47$ $$T^{7} - 1451 T^{6} + \cdots - 60\!\cdots\!00$$
$53$ $$T^{7} - 3 T^{6} + \cdots + 43\!\cdots\!80$$
$59$ $$T^{7} - 297 T^{6} + \cdots - 52\!\cdots\!20$$
$61$ $$T^{7} - 93 T^{6} + \cdots - 23\!\cdots\!00$$
$67$ $$T^{7} - 1641 T^{6} + \cdots + 28\!\cdots\!20$$
$71$ $$T^{7} - 2392 T^{6} + \cdots + 67\!\cdots\!20$$
$73$ $$T^{7} - 324 T^{6} + \cdots + 22\!\cdots\!06$$
$79$ $$T^{7} - 2492 T^{6} + \cdots + 11\!\cdots\!20$$
$83$ $$T^{7} - 310 T^{6} + \cdots + 13\!\cdots\!00$$
$89$ $$T^{7} + 440 T^{6} + \cdots + 35\!\cdots\!60$$
$97$ $$T^{7} + 532 T^{6} + \cdots + 86\!\cdots\!12$$
show more
show less