# Properties

 Label 1035.4.a.k Level $1035$ Weight $4$ Character orbit 1035.a Self dual yes Analytic conductor $61.067$ Analytic rank $1$ Dimension $5$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1035,4,Mod(1,1035)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1035, 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("1035.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1035 = 3^{2} \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1035.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: $$61.0669768559$$ Analytic rank: $$1$$ Dimension: $$5$$ Coefficient field: $$\mathbb{Q}[x]/(x^{5} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92$$ x^5 - x^4 - 27*x^3 + 7*x^2 + 168*x + 92 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 115) 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 $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{4}+ \cdots + ( - 5 \beta_{4} + 3 \beta_{3} + \cdots - 26) q^{8}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (b4 - b3 - b2 + 3*b1 + 4) * q^4 + 5 * q^5 + (b3 + 3*b2 + b1 - 2) * q^7 + (-5*b4 + 3*b3 + b2 - 5*b1 - 26) * q^8 $$q + ( - \beta_1 - 1) q^{2} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{4}+ \cdots + (11 \beta_{4} - 75 \beta_{3} + \cdots + 509) q^{98}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (b4 - b3 - b2 + 3*b1 + 4) * q^4 + 5 * q^5 + (b3 + 3*b2 + b1 - 2) * q^7 + (-5*b4 + 3*b3 + b2 - 5*b1 - 26) * q^8 + (-5*b1 - 5) * q^10 + (-2*b4 - 4*b3 - 7*b2 - 2*b1 - 1) * q^11 + (b4 - 4*b2 - 6*b1 + 29) * q^13 + (-6*b4 + 8*b3 - b2 + 5*b1 - 18) * q^14 + (13*b4 - 7*b3 + 15*b2 + 19*b1 + 44) * q^16 + (9*b4 + 4*b3 + 2*b2 + 14*b1 - 10) * q^17 + (-5*b4 + 5*b3 - 11*b1 - 29) * q^19 + (5*b4 - 5*b3 - 5*b2 + 15*b1 + 20) * q^20 + (18*b4 - 26*b3 + 5*b2 - 10*b1 + 35) * q^22 + 23 * q^23 + 25 * q^25 + (11*b4 - 11*b3 - 4*b2 - 21*b1 + 55) * q^26 + (23*b4 - 15*b3 + 2*b2 + 15*b1 - 5) * q^28 + (5*b4 - 9*b3 + 21*b2 + 17*b1 - 93) * q^29 + (12*b4 - 8*b3 - 17*b2 + 14*b1 + 8) * q^31 + (-55*b4 + 57*b3 - 37*b2 - 41*b1 - 100) * q^32 + (-41*b4 + 49*b3 - 2*b2 - 8*b1 - 122) * q^34 + (5*b3 + 15*b2 + 5*b1 - 10) * q^35 + (-53*b4 + 24*b3 + 11*b2 + 28*b1 - 7) * q^37 + (31*b4 - 21*b3 + 4*b2 + 61*b1 + 155) * q^38 + (-25*b4 + 15*b3 + 5*b2 - 25*b1 - 130) * q^40 + (19*b4 + 9*b3 + 20*b2 - 67*b1 + 4) * q^41 + (4*b4 - 6*b3 + 60*b2 - 10*b1 - 136) * q^43 + (-64*b4 + 60*b3 - 21*b2 - 46*b1 + 21) * q^44 + (-23*b1 - 23) * q^46 + (8*b4 - 33*b3 - 8*b2 + 29*b1 - 188) * q^47 + (5*b4 - 32*b3 - 10*b2 - 38*b1 - 137) * q^49 + (-25*b1 - 25) * q^50 + (-23*b4 - 7*b3 - 18*b2 + 9*b1 - 51) * q^52 + (-25*b4 + 12*b3 - 81*b2 + 56*b1 - 205) * q^53 + (-10*b4 - 20*b3 - 35*b2 - 10*b1 - 5) * q^55 + (-55*b4 + 9*b3 - 40*b2 - 93*b1 - 23) * q^56 + (-83*b4 + 65*b3 - 23*b2 + 62*b1 - 195) * q^58 + (9*b4 + 88*b3 + 55*b2 - 68*b1 - 275) * q^59 + (-60*b4 + 24*b3 + 101*b2 + 10*b1 - 365) * q^61 + (-24*b4 + 8*b3 - b2 - 69*b1 - 94) * q^62 + (233*b4 - 167*b3 + 43*b2 + 107*b1 + 408) * q^64 + (5*b4 - 20*b2 - 30*b1 + 145) * q^65 + (-7*b4 - 4*b3 - 19*b2 + 28*b1 + 115) * q^67 + (112*b4 - 118*b3 + 109*b2 + 122*b1 + 363) * q^68 + (-30*b4 + 40*b3 - 5*b2 + 25*b1 - 90) * q^70 + (-77*b4 - 35*b3 + 48*b2 + 69*b1 + 4) * q^71 + (84*b4 - 135*b3 + 64*b2 - 49*b1 + 202) * q^73 + (133*b4 - 85*b3 + 147*b2 + 10*b1 - 379) * q^74 + (-143*b4 + 101*b3 - 26*b2 - 227*b1 - 611) * q^76 + (-25*b4 + 73*b3 + 39*b2 + 79*b1 - 548) * q^77 + (-92*b4 + 152*b3 - 4*b2 + 136*b1 + 124) * q^79 + (65*b4 - 35*b3 + 75*b2 + 95*b1 + 220) * q^80 + (-21*b4 + 39*b3 - 116*b2 + 168*b1 + 718) * q^82 + (9*b4 + 74*b3 + 29*b2 + 190*b1 - 35) * q^83 + (45*b4 + 20*b3 + 10*b2 + 70*b1 - 50) * q^85 + (-128*b4 + 116*b3 - 84*b2 + 204*b1 - 4) * q^86 + (196*b4 - 12*b3 + 123*b2 + 250*b1 + 341) * q^88 + (-56*b4 + 134*b3 - 120*b2 + 246*b1 - 104) * q^89 + (-26*b4 + 106*b3 + 81*b2 + 100*b1 - 365) * q^91 + (23*b4 - 23*b3 - 23*b2 + 69*b1 + 92) * q^92 + (-70*b4 + 4*b3 - 12*b2 + 56*b1 - 182) * q^94 + (-25*b4 + 25*b3 - 55*b1 - 145) * q^95 + (26*b4 - 106*b3 - 173*b2 + 128*b1 + 165) * q^97 + (11*b4 - 75*b3 - 70*b2 + 139*b1 + 509) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 6 q^{2} + 22 q^{4} + 25 q^{5} - 3 q^{7} - 138 q^{8}+O(q^{10})$$ 5 * q - 6 * q^2 + 22 * q^4 + 25 * q^5 - 3 * q^7 - 138 * q^8 $$5 q - 6 q^{2} + 22 q^{4} + 25 q^{5} - 3 q^{7} - 138 q^{8} - 30 q^{10} - 23 q^{11} + 132 q^{13} - 93 q^{14} + 282 q^{16} - 23 q^{17} - 161 q^{19} + 110 q^{20} + 193 q^{22} + 115 q^{23} + 125 q^{25} + 257 q^{26} + 17 q^{28} - 401 q^{29} + 32 q^{31} - 670 q^{32} - 663 q^{34} - 15 q^{35} - 38 q^{37} + 875 q^{38} - 690 q^{40} + 12 q^{41} - 566 q^{43} - 47 q^{44} - 138 q^{46} - 919 q^{47} - 738 q^{49} - 150 q^{50} - 305 q^{52} - 1156 q^{53} - 115 q^{55} - 343 q^{56} - 1042 q^{58} - 1324 q^{59} - 1673 q^{61} - 565 q^{62} + 2466 q^{64} + 660 q^{65} + 558 q^{67} + 2267 q^{68} - 465 q^{70} + 108 q^{71} + 1173 q^{73} - 1458 q^{74} - 3477 q^{76} - 2608 q^{77} + 656 q^{79} + 1410 q^{80} + 3505 q^{82} + 82 q^{83} - 115 q^{85} - 112 q^{86} + 2397 q^{88} - 570 q^{89} - 1589 q^{91} + 506 q^{92} - 948 q^{94} - 805 q^{95} + 633 q^{97} + 2555 q^{98}+O(q^{100})$$ 5 * q - 6 * q^2 + 22 * q^4 + 25 * q^5 - 3 * q^7 - 138 * q^8 - 30 * q^10 - 23 * q^11 + 132 * q^13 - 93 * q^14 + 282 * q^16 - 23 * q^17 - 161 * q^19 + 110 * q^20 + 193 * q^22 + 115 * q^23 + 125 * q^25 + 257 * q^26 + 17 * q^28 - 401 * q^29 + 32 * q^31 - 670 * q^32 - 663 * q^34 - 15 * q^35 - 38 * q^37 + 875 * q^38 - 690 * q^40 + 12 * q^41 - 566 * q^43 - 47 * q^44 - 138 * q^46 - 919 * q^47 - 738 * q^49 - 150 * q^50 - 305 * q^52 - 1156 * q^53 - 115 * q^55 - 343 * q^56 - 1042 * q^58 - 1324 * q^59 - 1673 * q^61 - 565 * q^62 + 2466 * q^64 + 660 * q^65 + 558 * q^67 + 2267 * q^68 - 465 * q^70 + 108 * q^71 + 1173 * q^73 - 1458 * q^74 - 3477 * q^76 - 2608 * q^77 + 656 * q^79 + 1410 * q^80 + 3505 * q^82 + 82 * q^83 - 115 * q^85 - 112 * q^86 + 2397 * q^88 - 570 * q^89 - 1589 * q^91 + 506 * q^92 - 948 * q^94 - 805 * q^95 + 633 * q^97 + 2555 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - x^{4} - 27x^{3} + 7x^{2} + 168x + 92$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} + \nu^{3} - 25\nu^{2} - 11\nu + 98 ) / 16$$ (v^4 + v^3 - 25*v^2 - 11*v + 98) / 16 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} + 3\nu^{3} + 17\nu^{2} - 41\nu - 42 ) / 8$$ (-v^4 + 3*v^3 + 17*v^2 - 41*v - 42) / 8 $$\beta_{4}$$ $$=$$ $$( -\nu^{4} + 7\nu^{3} + 25\nu^{2} - 109\nu - 162 ) / 16$$ (-v^4 + 7*v^3 + 25*v^2 - 109*v - 162) / 16
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 11$$ b4 - b3 - b2 + b1 + 11 $$\nu^{3}$$ $$=$$ $$2\beta_{4} + 2\beta_{2} + 15\beta _1 + 8$$ 2*b4 + 2*b2 + 15*b1 + 8 $$\nu^{4}$$ $$=$$ $$23\beta_{4} - 25\beta_{3} - 11\beta_{2} + 21\beta _1 + 169$$ 23*b4 - 25*b3 - 11*b2 + 21*b1 + 169

## 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
 4.60878 3.41740 −0.595043 −2.49214 −3.93900
−5.60878 0 23.4584 5.00000 0 11.4426 −86.7031 0 −28.0439
1.2 −4.41740 0 11.5134 5.00000 0 −8.97260 −15.5200 0 −22.0870
1.3 −0.404957 0 −7.83601 5.00000 0 13.7888 6.41290 0 −2.02479
1.4 1.49214 0 −5.77352 5.00000 0 4.33445 −20.5520 0 7.46070
1.5 2.93900 0 0.637693 5.00000 0 −23.5932 −21.6378 0 14.6950
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$5$$ $$-1$$
$$23$$ $$-1$$

## 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 1035.4.a.k 5
3.b odd 2 1 115.4.a.e 5
12.b even 2 1 1840.4.a.n 5
15.d odd 2 1 575.4.a.j 5
15.e even 4 2 575.4.b.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.e 5 3.b odd 2 1
575.4.a.j 5 15.d odd 2 1
575.4.b.i 10 15.e even 4 2
1035.4.a.k 5 1.a even 1 1 trivial
1840.4.a.n 5 12.b even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1035))$$:

 $$T_{2}^{5} + 6T_{2}^{4} - 13T_{2}^{3} - 72T_{2}^{2} + 82T_{2} + 44$$ T2^5 + 6*T2^4 - 13*T2^3 - 72*T2^2 + 82*T2 + 44 $$T_{7}^{5} + 3T_{7}^{4} - 484T_{7}^{3} + 1757T_{7}^{2} + 34281T_{7} - 144774$$ T7^5 + 3*T7^4 - 484*T7^3 + 1757*T7^2 + 34281*T7 - 144774

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + 6 T^{4} + \cdots + 44$$
$3$ $$T^{5}$$
$5$ $$(T - 5)^{5}$$
$7$ $$T^{5} + 3 T^{4} + \cdots - 144774$$
$11$ $$T^{5} + 23 T^{4} + \cdots + 74136848$$
$13$ $$T^{5} - 132 T^{4} + \cdots + 1550116$$
$17$ $$T^{5} + \cdots + 1039045340$$
$19$ $$T^{5} + 161 T^{4} + \cdots - 801280$$
$23$ $$(T - 23)^{5}$$
$29$ $$T^{5} + \cdots + 6149898500$$
$31$ $$T^{5} - 32 T^{4} + \cdots - 438072447$$
$37$ $$T^{5} + \cdots + 1590700778176$$
$41$ $$T^{5} + \cdots - 114116030755$$
$43$ $$T^{5} + \cdots + 504784881664$$
$47$ $$T^{5} + \cdots - 117787714816$$
$53$ $$T^{5} + \cdots - 5720332226904$$
$59$ $$T^{5} + \cdots - 24279649927232$$
$61$ $$T^{5} + \cdots - 34095834816896$$
$67$ $$T^{5} + \cdots + 5644442112$$
$71$ $$T^{5} + \cdots - 15638892903635$$
$73$ $$T^{5} + \cdots - 100895881632176$$
$79$ $$T^{5} + \cdots - 90481602379776$$
$83$ $$T^{5} + \cdots - 18307318870176$$
$89$ $$T^{5} + \cdots + 115104799418880$$
$97$ $$T^{5} + \cdots - 480989167569272$$