# Properties

 Label 475.2.e.e Level $475$ Weight $2$ Character orbit 475.e Analytic conductor $3.793$ Analytic rank $0$ Dimension $8$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [475,2,Mod(26,475)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(475, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("475.26");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$475 = 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 475.e (of order $$3$$, degree $$2$$, minimal)

## 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: no Analytic conductor: $$3.79289409601$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(\zeta_{3})$$ Coefficient field: 8.0.4601315889.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9$$ x^8 - x^7 + 6*x^6 - 3*x^5 + 26*x^4 - 14*x^3 + 31*x^2 + 12*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 95) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{7} q^{2} + (\beta_{5} - \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + (\beta_{7} + \beta_{5} - \beta_{4} + \cdots - 1) q^{9}+O(q^{10})$$ q - b7 * q^2 + (b5 - b1) * q^3 + (-b7 - b6 + b5 + b4 - 1) * q^4 + (-b7 - b6 + b4 + b3 + b1) * q^6 + (b6 - b2 + 1) * q^7 + (-2*b6 + b4 - b3 + 2*b2 - 3) * q^8 + (b7 + b5 - b4 - 2*b3 - 2*b1 - 1) * q^9 $$q - \beta_{7} q^{2} + (\beta_{5} - \beta_1) q^{3} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + (5 \beta_{7} + 12 \beta_{5} + \cdots - 12) q^{99}+O(q^{100})$$ q - b7 * q^2 + (b5 - b1) * q^3 + (-b7 - b6 + b5 + b4 - 1) * q^4 + (-b7 - b6 + b4 + b3 + b1) * q^6 + (b6 - b2 + 1) * q^7 + (-2*b6 + b4 - b3 + 2*b2 - 3) * q^8 + (b7 + b5 - b4 - 2*b3 - 2*b1 - 1) * q^9 + (-b6 + 2*b3 + b2) * q^11 + (-b6 + 2*b4 + b2 - 1) * q^12 + (-b6 - 2*b5 + b3 + b1 + 2) * q^13 + (-2*b7 - b2 - b1) * q^14 + (4*b7 - b5 + 2*b2 + b1) * q^16 + (-b7 - 2*b1) * q^17 + (-b6 + 2*b3 + b2 + 3) * q^18 + (b4 + 2*b2 - b1 + 1) * q^19 + (-b7 + b5 - b1) * q^21 + (b7 - b2 + 3*b1) * q^22 + (b7 - b6 - b4 - b3 - b1) * q^23 + (2*b7 - 6*b5 + b2 + 3*b1) * q^24 + (-b4 - 2*b3) * q^26 + (b6 - 3*b4 - b3 - b2 - 4) * q^27 + (-3*b7 - 2*b6 + 4*b5 + 3*b4 - 4) * q^28 + (2*b7 + 2*b6 - 2*b4 + b3 + b1) * q^29 + (-b6 - 3*b4 - b3 + b2 - 1) * q^31 + (5*b7 + 3*b6 - 6*b5 - 5*b4 - b3 - b1 + 6) * q^32 + (3*b7 + 6*b5 - 2*b1) * q^33 + (-b7 - 3*b6 + 3*b5 + b4 + 2*b3 + 2*b1 - 3) * q^34 + (2*b5 - b2 - b1) * q^36 + (-b6 + 2*b4 + b2 + 1) * q^37 + (2*b7 + b6 - 3*b5 - 2*b4 + 3*b3 + b2 + 3*b1) * q^38 + (2*b4 + 3*b3 + 5) * q^39 + (2*b7 + 3*b5 - 3*b2 - 2*b1) * q^41 + (-2*b7 - 2*b6 + 3*b5 + 2*b4 + b3 + b1 - 3) * q^42 + (-b7 - b5 + b2 + 4*b1) * q^43 + (b6 - 3*b5 + 3) * q^44 + (-b6 + b2 + 3) * q^46 + (-2*b7 + 3*b6 + 3*b5 + 2*b4 - 2*b3 - 2*b1 - 3) * q^47 + (5*b7 + 4*b6 - 4*b5 - 5*b4 - 2*b3 - 2*b1 + 4) * q^48 + (b6 - b4 - b3 - b2 - 3) * q^49 + (b7 - b6 + 6*b5 - b4 - b3 - b1 - 6) * q^51 + (-b7 - b5 - b2) * q^52 + (-3*b7 - 2*b6 + 3*b4 + 2*b3 + 2*b1) * q^53 + (9*b5 - 3*b2 - 2*b1) * q^54 + (-3*b6 + 5*b4 + 3*b2 - 9) * q^56 + (4*b7 + 4*b5 - 3*b4 - b3 + b2 - 3*b1 - 3) * q^57 + (5*b6 - 4*b4 + b3 - 5*b2 + 6) * q^58 + 5*b1 * q^59 + (-4*b7 + 2*b6 - b5 + 4*b4 + 1) * q^61 + (-b7 + 9*b5 - b2) * q^62 + (2*b6 + b5 - b3 - b1 - 1) * q^63 + (3*b6 - 6*b4 + 2*b3 - 3*b2 + 13) * q^64 + (-3*b7 + b6 - 9*b5 + 3*b4 + 2*b3 + 2*b1 + 9) * q^66 + (-2*b7 - 2*b5 + 2*b4 + 2*b3 + 2*b1 + 2) * q^67 + (-2*b6 + 5*b4 - b3 + 2*b2 - 3) * q^68 + (b6 - b4 - 2*b3 - b2 - 3) * q^69 + (-3*b7 - 6*b5 + b1) * q^71 + (-3*b7 - 4*b6 - 6*b5 + 3*b4 + 4*b3 + 4*b1 + 6) * q^72 + (b7 - 4*b5 + b2 - 3*b1) * q^73 + (2*b7 - 6*b5 + 3*b2 + b1) * q^74 + (2*b7 + 3*b6 - 2*b5 - 5*b4 + b3 - 2*b2 + b1 + 8) * q^76 + (-2*b6 - b4 + 3*b3 + 2*b2 - 3) * q^77 + (-3*b7 - 6*b5 - b2 + 3*b1) * q^78 + (-2*b7 - 5*b5 + b2 + b1) * q^79 + (-2*b7 - 4*b5 - 3*b2 + 2*b1) * q^81 + (-4*b7 - 3*b6 - 6*b5 + 4*b4 - b3 - b1 + 6) * q^82 + (2*b6 - b4 + 2*b3 - 2*b2) * q^83 + (-3*b6 + 5*b4 - b3 + 3*b2 - 4) * q^84 + (b7 + 4*b6 + 3*b5 - b4 - 3*b3 - 3*b1 - 3) * q^86 + (2*b6 - 3*b4 - b3 - 2*b2 + 3) * q^87 + (3*b6 - 2*b4 - 5*b3 - 3*b2) * q^88 + (-2*b7 + 3*b6 + 3*b5 + 2*b4 - 3*b3 - 3*b1 - 3) * q^89 + (b6 + b5 + 2*b3 + 2*b1 - 1) * q^91 + (-b2 - b1) * q^92 + (-3*b7 - 4*b5 - 3*b2 + 5*b1) * q^93 + (-b6 + 2*b4 + 5*b3 + b2 - 6) * q^94 + (5*b6 - 9*b4 - 5*b2 + 3) * q^96 + (-b5 + 4*b2 + 5*b1) * q^97 + (b7 + 3*b5 - b2 - 2*b1) * q^98 + (5*b7 + 12*b5 - 5*b4 - 5*b3 - 5*b1 - 12) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + q^{2} + 3 q^{3} - 5 q^{4} - 2 q^{6} + 8 q^{7} - 24 q^{8} - q^{9}+O(q^{10})$$ 8 * q + q^2 + 3 * q^3 - 5 * q^4 - 2 * q^6 + 8 * q^7 - 24 * q^8 - q^9 $$8 q + q^{2} + 3 q^{3} - 5 q^{4} - 2 q^{6} + 8 q^{7} - 24 q^{8} - q^{9} - 4 q^{11} - 12 q^{12} + 7 q^{13} + q^{14} - 7 q^{16} - q^{17} + 20 q^{18} + 5 q^{19} + 4 q^{21} + 2 q^{22} + 2 q^{23} - 23 q^{24} + 6 q^{26} - 24 q^{27} - 19 q^{28} + q^{29} + 30 q^{32} + 19 q^{33} - 15 q^{34} + 7 q^{36} + 4 q^{37} - 13 q^{38} + 30 q^{39} + 8 q^{41} - 15 q^{42} + q^{43} + 12 q^{44} + 24 q^{46} - 12 q^{47} + 23 q^{48} - 20 q^{49} - 22 q^{51} - 3 q^{52} - 5 q^{53} + 34 q^{54} - 82 q^{56} - 7 q^{57} + 54 q^{58} + 5 q^{59} + 37 q^{62} - 3 q^{63} + 112 q^{64} + 31 q^{66} + 4 q^{67} - 32 q^{68} - 18 q^{69} - 20 q^{71} + 17 q^{72} - 20 q^{73} - 25 q^{74} + 63 q^{76} - 28 q^{77} - 18 q^{78} - 17 q^{79} - 12 q^{81} + 21 q^{82} - 2 q^{83} - 40 q^{84} - 8 q^{86} + 32 q^{87} + 14 q^{88} - 11 q^{89} - 6 q^{91} - q^{92} - 8 q^{93} - 62 q^{94} + 42 q^{96} + q^{97} + 9 q^{98} - 38 q^{99}+O(q^{100})$$ 8 * q + q^2 + 3 * q^3 - 5 * q^4 - 2 * q^6 + 8 * q^7 - 24 * q^8 - q^9 - 4 * q^11 - 12 * q^12 + 7 * q^13 + q^14 - 7 * q^16 - q^17 + 20 * q^18 + 5 * q^19 + 4 * q^21 + 2 * q^22 + 2 * q^23 - 23 * q^24 + 6 * q^26 - 24 * q^27 - 19 * q^28 + q^29 + 30 * q^32 + 19 * q^33 - 15 * q^34 + 7 * q^36 + 4 * q^37 - 13 * q^38 + 30 * q^39 + 8 * q^41 - 15 * q^42 + q^43 + 12 * q^44 + 24 * q^46 - 12 * q^47 + 23 * q^48 - 20 * q^49 - 22 * q^51 - 3 * q^52 - 5 * q^53 + 34 * q^54 - 82 * q^56 - 7 * q^57 + 54 * q^58 + 5 * q^59 + 37 * q^62 - 3 * q^63 + 112 * q^64 + 31 * q^66 + 4 * q^67 - 32 * q^68 - 18 * q^69 - 20 * q^71 + 17 * q^72 - 20 * q^73 - 25 * q^74 + 63 * q^76 - 28 * q^77 - 18 * q^78 - 17 * q^79 - 12 * q^81 + 21 * q^82 - 2 * q^83 - 40 * q^84 - 8 * q^86 + 32 * q^87 + 14 * q^88 - 11 * q^89 - 6 * q^91 - q^92 - 8 * q^93 - 62 * q^94 + 42 * q^96 + q^97 + 9 * q^98 - 38 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243$$ (-26*v^7 - 189*v^6 + 729*v^5 - 911*v^4 + 3051*v^3 - 3618*v^2 + 14317*v - 1215) / 4243 $$\beta_{3}$$ $$=$$ $$( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243$$ (115*v^7 + 20*v^6 + 529*v^5 + 276*v^4 + 3314*v^3 + 989*v^2 + 483*v + 1947) / 4243 $$\beta_{4}$$ $$=$$ $$( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243$$ (-135*v^7 + 161*v^6 - 621*v^5 - 324*v^4 - 2599*v^3 - 1161*v^2 - 567*v - 11694) / 4243 $$\beta_{5}$$ $$=$$ $$( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729$$ (-649*v^7 + 994*v^6 - 3834*v^5 + 3534*v^4 - 16046*v^3 + 19028*v^2 - 17152*v + 6390) / 12729 $$\beta_{6}$$ $$=$$ $$( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243$$ (434*v^7 - 109*v^6 + 2845*v^5 + 193*v^4 + 12064*v^3 + 338*v^2 + 16249*v + 6573) / 4243 $$\beta_{7}$$ $$=$$ $$( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243$$ (514*v^7 - 833*v^6 + 3213*v^5 - 3858*v^4 + 13447*v^3 - 15946*v^2 + 16585*v - 5355) / 4243
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3\beta_{5} - \beta_{4} - 3$$ b7 + 3*b5 - b4 - 3 $$\nu^{3}$$ $$=$$ $$-\beta_{6} + 4\beta_{3} + \beta_{2}$$ -b6 + 4*b3 + b2 $$\nu^{4}$$ $$=$$ $$-5\beta_{7} - 12\beta_{5} + \beta_{2}$$ -5*b7 - 12*b5 + b2 $$\nu^{5}$$ $$=$$ $$-\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1$$ -b7 + 6*b6 + b4 - 17*b3 - 17*b1 $$\nu^{6}$$ $$=$$ $$7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51$$ 7*b6 + 23*b4 - b3 - 7*b2 + 51 $$\nu^{7}$$ $$=$$ $$8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1$$ 8*b7 + 3*b5 - 30*b2 + 74*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/475\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$\chi(n)$$ $$1$$ $$-1 + \beta_{5}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
26.1
 1.07988 + 1.87040i −1.02359 − 1.77290i 0.689667 + 1.19454i −0.245959 − 0.426014i 1.07988 − 1.87040i −1.02359 + 1.77290i 0.689667 − 1.19454i −0.245959 + 0.426014i
−0.832272 1.44154i −0.579878 1.00438i −0.385355 + 0.667454i 0 −0.965233 + 1.67183i 2.43525 −2.04621 0.827483 1.43324i 0
26.2 −0.595455 1.03136i 1.52359 + 2.63893i 0.290867 0.503797i 0 1.81445 3.14272i 0.609175 −3.07461 −3.14263 + 5.44319i 0
26.3 0.548719 + 0.950409i −0.189667 0.328513i 0.397815 0.689035i 0 0.208148 0.360522i −1.89307 3.06803 1.42805 2.47346i 0
26.4 1.37901 + 2.38851i 0.745959 + 1.29204i −2.80333 + 4.85550i 0 −2.05737 + 3.56347i 2.84864 −9.94721 0.387090 0.670459i 0
201.1 −0.832272 + 1.44154i −0.579878 + 1.00438i −0.385355 0.667454i 0 −0.965233 1.67183i 2.43525 −2.04621 0.827483 + 1.43324i 0
201.2 −0.595455 + 1.03136i 1.52359 2.63893i 0.290867 + 0.503797i 0 1.81445 + 3.14272i 0.609175 −3.07461 −3.14263 5.44319i 0
201.3 0.548719 0.950409i −0.189667 + 0.328513i 0.397815 + 0.689035i 0 0.208148 + 0.360522i −1.89307 3.06803 1.42805 + 2.47346i 0
201.4 1.37901 2.38851i 0.745959 1.29204i −2.80333 4.85550i 0 −2.05737 3.56347i 2.84864 −9.94721 0.387090 + 0.670459i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 26.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.e.e 8
5.b even 2 1 95.2.e.c 8
5.c odd 4 2 475.2.j.c 16
15.d odd 2 1 855.2.k.h 8
19.c even 3 1 inner 475.2.e.e 8
19.c even 3 1 9025.2.a.bg 4
19.d odd 6 1 9025.2.a.bp 4
20.d odd 2 1 1520.2.q.o 8
95.h odd 6 1 1805.2.a.i 4
95.i even 6 1 95.2.e.c 8
95.i even 6 1 1805.2.a.o 4
95.m odd 12 2 475.2.j.c 16
285.n odd 6 1 855.2.k.h 8
380.p odd 6 1 1520.2.q.o 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 5.b even 2 1
95.2.e.c 8 95.i even 6 1
475.2.e.e 8 1.a even 1 1 trivial
475.2.e.e 8 19.c even 3 1 inner
475.2.j.c 16 5.c odd 4 2
475.2.j.c 16 95.m odd 12 2
855.2.k.h 8 15.d odd 2 1
855.2.k.h 8 285.n odd 6 1
1520.2.q.o 8 20.d odd 2 1
1520.2.q.o 8 380.p odd 6 1
1805.2.a.i 4 95.h odd 6 1
1805.2.a.o 4 95.i even 6 1
9025.2.a.bg 4 19.c even 3 1
9025.2.a.bp 4 19.d odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} + 4T_{2}^{5} + 31T_{2}^{4} + 6T_{2}^{3} + 37T_{2}^{2} + 6T_{2} + 36$$ acting on $$S_{2}^{\mathrm{new}}(475, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{7} + \cdots + 36$$
$3$ $$T^{8} - 3 T^{7} + \cdots + 4$$
$5$ $$T^{8}$$
$7$ $$(T^{4} - 4 T^{3} - T^{2} + \cdots - 8)^{2}$$
$11$ $$(T^{4} + 2 T^{3} - 25 T^{2} + \cdots + 3)^{2}$$
$13$ $$T^{8} - 7 T^{7} + \cdots + 256$$
$17$ $$T^{8} + T^{7} + \cdots + 11664$$
$19$ $$T^{8} - 5 T^{7} + \cdots + 130321$$
$23$ $$T^{8} - 2 T^{7} + \cdots + 36$$
$29$ $$T^{8} - T^{7} + \cdots + 19881$$
$31$ $$(T^{4} - 67 T^{2} + \cdots + 1063)^{2}$$
$37$ $$(T^{4} - 2 T^{3} + \cdots - 118)^{2}$$
$41$ $$T^{8} - 8 T^{7} + \cdots + 5008644$$
$43$ $$T^{8} - T^{7} + \cdots + 630436$$
$47$ $$T^{8} + 12 T^{7} + \cdots + 5363856$$
$53$ $$T^{8} + 5 T^{7} + \cdots + 2916$$
$59$ $$T^{8} - 5 T^{7} + \cdots + 3515625$$
$61$ $$T^{8} + 130 T^{6} + \cdots + 9296401$$
$67$ $$T^{8} - 4 T^{7} + \cdots + 4096$$
$71$ $$T^{8} + 20 T^{7} + \cdots + 59049$$
$73$ $$T^{8} + 20 T^{7} + \cdots + 2979076$$
$79$ $$T^{8} + 17 T^{7} + \cdots + 33856$$
$83$ $$(T^{4} + T^{3} - 62 T^{2} + \cdots + 366)^{2}$$
$89$ $$T^{8} + 11 T^{7} + \cdots + 14561856$$
$97$ $$T^{8} - T^{7} + \cdots + 55383364$$