# Properties

 Label 575.4.a.j Level $575$ Weight $4$ Character orbit 575.a Self dual yes Analytic conductor $33.926$ 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] = [575,4,Mod(1,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, 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("575.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 575.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: $$33.9260982533$$ 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: $$\Z[a_1, a_2, a_3]$$ 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.

 $$f(q)$$ $$=$$ $$q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{4}+ \cdots + ( - 3 \beta_{4} + 6 \beta_{3} + 16) q^{9}+O(q^{10})$$ q + (-b1 - 1) * q^2 + (-b3 + b2 - b1 - 1) * q^3 + (b4 - b3 - b2 + 3*b1 + 4) * q^4 + (-2*b4 - 3*b2 + 2*b1 + 5) * q^6 + (-b3 - 3*b2 - b1 + 2) * q^7 + (-5*b4 + 3*b3 + b2 - 5*b1 - 26) * q^8 + (-3*b4 + 6*b3 + 16) * q^9 $$q + ( - \beta_1 - 1) q^{2} + ( - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{3} + (\beta_{4} - \beta_{3} - \beta_{2} + \cdots + 4) q^{4}+ \cdots + (35 \beta_{4} + 121 \beta_{3} + \cdots + 433) q^{99}+O(q^{100})$$ q + (-b1 - 1) * q^2 + (-b3 + b2 - b1 - 1) * q^3 + (b4 - b3 - b2 + 3*b1 + 4) * q^4 + (-2*b4 - 3*b2 + 2*b1 + 5) * q^6 + (-b3 - 3*b2 - b1 + 2) * q^7 + (-5*b4 + 3*b3 + b2 - 5*b1 - 26) * q^8 + (-3*b4 + 6*b3 + 16) * q^9 + (2*b4 + 4*b3 + 7*b2 + 2*b1 + 1) * q^11 + (10*b4 - 2*b3 + b2 - 4*b1 - 11) * q^12 + (-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 + (15*b4 - 3*b3 + 12*b2 - 4*b1 - 4) * q^18 + (-5*b4 + 5*b3 - 11*b1 - 29) * q^19 + (9*b4 - 5*b3 - 9*b2 - 11*b1 - 8) * q^21 + (-18*b4 + 26*b3 - 5*b2 + 10*b1 - 35) * q^22 + 23 * q^23 + (-14*b4 + 26*b3 - 3*b2 + 25) * q^24 + (-11*b4 + 11*b3 + 4*b2 + 21*b1 - 55) * q^26 + (-9*b4 - 4*b3 + 7*b2 + 8*b1 - 118) * q^27 + (-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 + (-27*b4 - 7*b3 + 14*b2 + 45*b1 - 47) * q^33 + (-41*b4 + 49*b3 - 2*b2 - 8*b1 - 122) * q^34 + (-44*b4 + 14*b3 - 49*b2 + 18*b1 - 107) * q^36 + (53*b4 - 24*b3 - 11*b2 - 28*b1 + 7) * q^37 + (31*b4 - 21*b3 + 4*b2 + 61*b1 + 155) * q^38 + (b4 + 46*b3 + b2 + 22*b1 + 62) * q^39 + (-19*b4 - 9*b3 - 20*b2 + 67*b1 - 4) * q^41 + (-3*b4 - 7*b3 - 25*b2 + 11*b1 + 168) * q^42 + (-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 + (-6*b4 - 6*b3 + 49*b2 + 56*b1 + 125) * q^48 + (5*b4 - 32*b3 - 10*b2 - 38*b1 - 137) * q^49 + (5*b4 - 15*b3 - 30*b2 + 97*b1 - 207) * q^51 + (23*b4 + 7*b3 + 18*b2 - 9*b1 + 51) * q^52 + (-25*b4 + 12*b3 - 81*b2 + 56*b1 - 205) * q^53 + (b4 - 9*b3 + 15*b2 + 101*b1 - 28) * q^54 + (55*b4 - 9*b3 + 40*b2 + 93*b1 + 23) * q^56 + (-27*b4 + 13*b3 - 21*b2 + 15*b1 - 12) * q^57 + (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 + (6*b4 - b3 - 30*b2 + 89*b1 - 61) * q^63 + (233*b4 - 167*b3 + 43*b2 + 107*b1 + 408) * q^64 + (b4 - 15*b3 + 78*b2 - 43*b1 - 579) * q^66 + (7*b4 + 4*b3 + 19*b2 - 28*b1 - 115) * q^67 + (112*b4 - 118*b3 + 109*b2 + 122*b1 + 363) * q^68 + (-23*b3 + 23*b2 - 23*b1 - 23) * q^69 + (77*b4 + 35*b3 - 48*b2 - 69*b1 - 4) * q^71 + (106*b4 - 174*b3 + 73*b2 + 82*b1 + 91) * q^72 + (-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 + (19*b4 + 73*b3 + 65*b2 - 13*b1 - 168) * q^78 + (-92*b4 + 152*b3 - 4*b2 + 136*b1 + 124) * q^79 + (93*b4 + 21*b3 - 27*b2 + 27*b1 - 77) * q^81 + (21*b4 - 39*b3 + 116*b2 - 168*b1 - 718) * q^82 + (9*b4 + 74*b3 + 29*b2 + 190*b1 - 35) * q^83 + (-31*b4 - 15*b3 + 107*b2 - 141*b1 - 152) * q^84 + (128*b4 - 116*b3 + 84*b2 - 204*b1 + 4) * q^86 + (16*b4 - 173*b3 + 26*b2 - 127*b1 - 466) * q^87 + (-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 + (83*b4 - 32*b3 - 87*b2 + 20*b1 - 168) * q^93 + (-70*b4 + 4*b3 - 12*b2 + 56*b1 - 182) * q^94 + (-30*b4 - 78*b3 + 37*b2 - 200*b1 - 1167) * q^96 + (-26*b4 + 106*b3 + 173*b2 - 128*b1 - 165) * q^97 + (11*b4 - 75*b3 - 70*b2 + 139*b1 + 509) * q^98 + (35*b4 + 121*b3 + 58*b2 - 295*b1 + 433) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9}+O(q^{10})$$ 5 * q - 6 * q^2 - 4 * q^3 + 22 * q^4 + 19 * q^6 + 3 * q^7 - 138 * q^8 + 77 * q^9 $$5 q - 6 q^{2} - 4 q^{3} + 22 q^{4} + 19 q^{6} + 3 q^{7} - 138 q^{8} + 77 q^{9} + 23 q^{11} - 47 q^{12} - 132 q^{13} + 93 q^{14} + 282 q^{16} - 23 q^{17} + 15 q^{18} - 161 q^{19} - 60 q^{21} - 193 q^{22} + 115 q^{23} + 105 q^{24} - 257 q^{26} - 577 q^{27} - 17 q^{28} + 401 q^{29} + 32 q^{31} - 670 q^{32} - 189 q^{33} - 663 q^{34} - 659 q^{36} + 38 q^{37} + 875 q^{38} + 335 q^{39} - 12 q^{41} + 798 q^{42} + 566 q^{43} + 47 q^{44} - 138 q^{46} - 919 q^{47} + 773 q^{48} - 738 q^{49} - 993 q^{51} + 305 q^{52} - 1156 q^{53} - 8 q^{54} + 343 q^{56} - 114 q^{57} + 1042 q^{58} + 1324 q^{59} - 1673 q^{61} - 565 q^{62} - 270 q^{63} + 2466 q^{64} - 2781 q^{66} - 558 q^{67} + 2267 q^{68} - 92 q^{69} - 108 q^{71} + 789 q^{72} - 1173 q^{73} + 1458 q^{74} - 3477 q^{76} - 2608 q^{77} - 704 q^{78} + 656 q^{79} - 319 q^{81} - 3505 q^{82} + 82 q^{83} - 718 q^{84} + 112 q^{86} - 2389 q^{87} - 2397 q^{88} + 570 q^{89} - 1589 q^{91} + 506 q^{92} - 911 q^{93} - 948 q^{94} - 5991 q^{96} - 633 q^{97} + 2555 q^{98} + 2021 q^{99}+O(q^{100})$$ 5 * q - 6 * q^2 - 4 * q^3 + 22 * q^4 + 19 * q^6 + 3 * q^7 - 138 * q^8 + 77 * q^9 + 23 * q^11 - 47 * q^12 - 132 * q^13 + 93 * q^14 + 282 * q^16 - 23 * q^17 + 15 * q^18 - 161 * q^19 - 60 * q^21 - 193 * q^22 + 115 * q^23 + 105 * q^24 - 257 * q^26 - 577 * q^27 - 17 * q^28 + 401 * q^29 + 32 * q^31 - 670 * q^32 - 189 * q^33 - 663 * q^34 - 659 * q^36 + 38 * q^37 + 875 * q^38 + 335 * q^39 - 12 * q^41 + 798 * q^42 + 566 * q^43 + 47 * q^44 - 138 * q^46 - 919 * q^47 + 773 * q^48 - 738 * q^49 - 993 * q^51 + 305 * q^52 - 1156 * q^53 - 8 * q^54 + 343 * q^56 - 114 * q^57 + 1042 * q^58 + 1324 * q^59 - 1673 * q^61 - 565 * q^62 - 270 * q^63 + 2466 * q^64 - 2781 * q^66 - 558 * q^67 + 2267 * q^68 - 92 * q^69 - 108 * q^71 + 789 * q^72 - 1173 * q^73 + 1458 * q^74 - 3477 * q^76 - 2608 * q^77 - 704 * q^78 + 656 * q^79 - 319 * q^81 - 3505 * q^82 + 82 * q^83 - 718 * q^84 + 112 * q^86 - 2389 * q^87 - 2397 * q^88 + 570 * q^89 - 1589 * q^91 + 506 * q^92 - 911 * q^93 - 948 * q^94 - 5991 * q^96 - 633 * q^97 + 2555 * q^98 + 2021 * q^99

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 1.89520 23.4584 0 −10.6297 −11.4426 −86.7031 −23.4082 0
1.2 −4.41740 −7.84147 11.5134 0 34.6389 8.97260 −15.5200 34.4886 0
1.3 −0.404957 7.11323 −7.83601 0 −2.88055 −13.7888 6.41290 23.5981 0
1.4 1.49214 −9.02447 −5.77352 0 −13.4658 −4.33445 −20.5520 54.4411 0
1.5 2.93900 3.85751 0.637693 0 11.3372 23.5932 −21.6378 −12.1196 0
 $$n$$: 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
$$5$$ $$+1$$
$$23$$ $$-1$$

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

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.4.a.e 5 5.b even 2 1
575.4.a.j 5 1.a even 1 1 trivial
575.4.b.i 10 5.c odd 4 2
1035.4.a.k 5 15.d odd 2 1
1840.4.a.n 5 20.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 $$S_{4}^{\mathrm{new}}(\Gamma_0(575))$$:

 $$T_{2}^{5} + 6T_{2}^{4} - 13T_{2}^{3} - 72T_{2}^{2} + 82T_{2} + 44$$ T2^5 + 6*T2^4 - 13*T2^3 - 72*T2^2 + 82*T2 + 44 $$T_{3}^{5} + 4T_{3}^{4} - 98T_{3}^{3} - 149T_{3}^{2} + 2536T_{3} - 3680$$ T3^5 + 4*T3^4 - 98*T3^3 - 149*T3^2 + 2536*T3 - 3680

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{5} + 6 T^{4} + \cdots + 44$$
$3$ $$T^{5} + 4 T^{4} + \cdots - 3680$$
$5$ $$T^{5}$$
$7$ $$T^{5} - 3 T^{4} + \cdots + 144774$$
$11$ $$T^{5} - 23 T^{4} + \cdots - 74136848$$
$13$ $$T^{5} + 132 T^{4} + \cdots - 1550116$$
$17$ $$T^{5} + \cdots + 1039045340$$
$19$ $$T^{5} + 161 T^{4} + \cdots - 801280$$
$23$ $$(T - 23)^{5}$$
$29$ $$T^{5} + \cdots - 6149898500$$
$31$ $$T^{5} - 32 T^{4} + \cdots - 438072447$$
$37$ $$T^{5} + \cdots - 1590700778176$$
$41$ $$T^{5} + \cdots + 114116030755$$
$43$ $$T^{5} + \cdots - 504784881664$$
$47$ $$T^{5} + \cdots - 117787714816$$
$53$ $$T^{5} + \cdots - 5720332226904$$
$59$ $$T^{5} + \cdots + 24279649927232$$
$61$ $$T^{5} + \cdots - 34095834816896$$
$67$ $$T^{5} + \cdots - 5644442112$$
$71$ $$T^{5} + \cdots + 15638892903635$$
$73$ $$T^{5} + \cdots + 100895881632176$$
$79$ $$T^{5} + \cdots - 90481602379776$$
$83$ $$T^{5} + \cdots - 18307318870176$$
$89$ $$T^{5} + \cdots - 115104799418880$$
$97$ $$T^{5} + \cdots + 480989167569272$$