# Properties

 Label 5225.2.a.n Level $5225$ Weight $2$ Character orbit 5225.a Self dual yes Analytic conductor $41.722$ Analytic rank $0$ Dimension $7$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5225,2,Mod(1,5225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$5225 = 5^{2} \cdot 11 \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5225.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: $$41.7218350561$$ 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: $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ x^7 - x^6 - 14*x^5 + 10*x^4 + 59*x^3 - 27*x^2 - 66*x + 30 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 209) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + 2) q^{4} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{6} + 2) q^{9}+O(q^{10})$$ q + b1 * q^2 + b2 * q^3 + (b3 + b2 + 2) * q^4 + (b6 + b5 + b4 - b2 - 1) * q^6 + (b5 - b2 - 2) * q^7 + (b6 + b5 + b3 + b1) * q^8 + (-b6 + 2) * q^9 $$q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} + \beta_{2} + 2) q^{4} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{6}+ \cdots + (\beta_{6} - 2) q^{99}+O(q^{100})$$ q + b1 * q^2 + b2 * q^3 + (b3 + b2 + 2) * q^4 + (b6 + b5 + b4 - b2 - 1) * q^6 + (b5 - b2 - 2) * q^7 + (b6 + b5 + b3 + b1) * q^8 + (-b6 + 2) * q^9 - q^11 + (-b6 + b5 + 2*b2 + 3) * q^12 + (b5 - b4 + b3 + b2 - b1) * q^13 + (-b6 + b4 + b2 - b1 + 2) * q^14 + (b3 + 3*b2 + 2*b1 + 4) * q^16 + (-b6 - b5 + b3) * q^17 + (b5 + b4 - b3 - 3*b2 + 2*b1 - 2) * q^18 + q^19 + (b6 + b4 + b3 - b2 + b1 - 3) * q^21 - b1 * q^22 + (-2*b5 + b3 + b2 - 1) * q^23 + (2*b5 + 3*b4 - b3 - 3*b2 + 4*b1 - 1) * q^24 + (b6 + 2*b4 + b3 - b2 + 2*b1 - 2) * q^26 + (b6 - b4 + 2*b3 + 2*b2 - 3*b1) * q^27 + (b6 + 2*b5 + 2*b4 - 3*b3 - 3*b2 + 2*b1 - 4) * q^28 + (b3 - 3) * q^29 + (b5 - b4 - b1 + 3) * q^31 + (b6 + b5 + 2*b4 + b3 + 3*b1 + 6) * q^32 - b2 * q^33 + (-2*b4 - 2*b2 - 2) * q^34 + (-b6 + 4*b2 - 2*b1 + 6) * q^36 + (b6 + b5 - 2*b4 + b2 - 1) * q^37 + b1 * q^38 + (-2*b6 - b5 + b4 + b3 + 2*b2 + 5) * q^39 + (b6 + b4 - b2 - b1 - 2) * q^41 + (-b6 - 3*b4 + 2*b3 + 6*b2 - 2*b1 + 7) * q^42 + (2*b5 - b4 - b3 - b2 - 1) * q^43 + (-b3 - b2 - 2) * q^44 + (b6 - b5 - 4*b4 + b3 - 2*b1 - 2) * q^46 + (-2*b6 - 2*b5 + 2*b1) * q^47 + (-b6 + 3*b5 + 2*b4 + 2*b2 + 11) * q^48 + (-2*b5 - b4 - b3 + 3*b2 + 4) * q^49 + (b6 + b5 - 2*b4 + b3 + 2*b2 - 4*b1 - 4) * q^51 + (-b6 - b4 + 5*b2 + b1 + 10) * q^52 + (-2*b4 - 2*b3) * q^53 + (2*b6 - b5 - b4 + b3 + 2*b1 - 9) * q^54 + (-b6 + 2*b5 + b4 - 2*b3 + 3*b2 - 3*b1 + 6) * q^56 + b2 * q^57 + (-b4 + b3 + b2 - 2*b1 + 1) * q^58 + (b6 + b5 - 4*b4 + 2*b3 + 3*b2 - 2*b1 - 3) * q^59 + (b6 + b5 - b3 - 2*b1 + 2) * q^61 + (-b5 + 2*b4 - b2 + 4*b1 - 2) * q^62 + (b6 + b4 - 2*b3 - 4*b2 + 4*b1 - 1) * q^63 + (4*b5 + b3 + b2 + 4*b1 + 6) * q^64 + (-b6 - b5 - b4 + b2 + 1) * q^66 + (-b5 - b4 + 2*b3 + 2*b2 - 3*b1 - 1) * q^67 + (-4*b5 - 2*b4 + 2*b2 - 2*b1 + 4) * q^68 + (-b6 + b5 - 2*b4 - 2*b3 - 3*b2 - 2*b1 - 1) * q^69 + (-2*b5 + 2*b4 + 2*b3 + b2 + 2) * q^71 + (4*b6 + 3*b5 + 3*b4 - b3 - 3*b2 + 2*b1 - 10) * q^72 + (2*b6 + 2*b4 - 2*b2) * q^73 + (b6 - 3*b5 + 2*b4 + 3*b3 + 2*b2 + 4) * q^74 + (b3 + b2 + 2) * q^76 + (-b5 + b2 + 2) * q^77 + (2*b6 + 5*b5 + b4 - 2*b3 - 7*b2 + 5*b1 - 7) * q^78 + (-b6 - b5 - b3 + 4*b1 + 8) * q^79 + (-3*b6 - 2*b5 - b4 - 2*b3 + 2*b1 + 2) * q^81 + (-b6 - 2*b4 - b3 + 3*b2 - 2*b1 - 2) * q^82 + (-b6 + b5 + b4 + 2*b3 - b2 - 2*b1 + 1) * q^83 + (4*b6 + b5 + 3*b4 - 7*b2 + 7*b1 - 5) * q^84 + (-b6 - b5 + 4*b4 + 3) * q^86 + (b5 - 3*b2 - 2) * q^87 + (-b6 - b5 - b3 - b1) * q^88 + (b6 - b5 + 2*b4 + 4*b3 + b2 - 1) * q^89 + (3*b6 + b5 + 2*b4 - 4*b3 - 4*b2 + 2*b1 + 1) * q^91 + (-6*b5 - 4*b4 + 2*b3 - 2*b1) * q^92 + (-b6 - 2*b5 + b4 + b3 + 5*b2 + 2) * q^93 + (-2*b4 - 4*b2 - 2*b1 + 2) * q^94 + (2*b6 + 6*b5 + 3*b4 - b3 + b2 + 6*b1 - 1) * q^96 + (-b6 - b5 + 2*b4 + 2*b3 - b2 + 4*b1 + 3) * q^97 + (3*b6 - b5 - 4*b2 + b1 - 5) * q^98 + (b6 - 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9}+O(q^{10})$$ 7 * q + q^2 - 2 * q^3 + 15 * q^4 - 2 * q^6 - 10 * q^7 + 9 * q^8 + 11 * q^9 $$7 q + q^{2} - 2 q^{3} + 15 q^{4} - 2 q^{6} - 10 q^{7} + 9 q^{8} + 11 q^{9} - 7 q^{11} + 16 q^{12} + 4 q^{13} + 6 q^{14} + 27 q^{16} - 2 q^{17} - 9 q^{18} + 7 q^{19} - 14 q^{21} - q^{22} - 10 q^{23} - 2 q^{24} - 8 q^{26} + 4 q^{27} - 26 q^{28} - 18 q^{29} + 24 q^{31} + 49 q^{32} + 2 q^{33} - 6 q^{34} + 29 q^{36} + q^{38} + 24 q^{39} - 12 q^{41} + 44 q^{42} - 2 q^{43} - 15 q^{44} - 4 q^{46} - 8 q^{47} + 72 q^{48} + 17 q^{49} - 24 q^{51} + 60 q^{52} - 2 q^{53} - 52 q^{54} + 26 q^{56} - 2 q^{57} + 8 q^{58} - 10 q^{59} + 14 q^{61} - 14 q^{62} + 55 q^{64} + 2 q^{66} - 8 q^{67} + 18 q^{68} - 6 q^{69} + 10 q^{71} - 53 q^{72} + 6 q^{73} + 26 q^{74} + 15 q^{76} + 10 q^{77} - 22 q^{78} + 52 q^{79} - q^{81} - 24 q^{82} + 10 q^{83} - 6 q^{84} + 8 q^{86} - 6 q^{87} - 9 q^{88} + 12 q^{91} - 2 q^{93} + 24 q^{94} + 6 q^{96} + 24 q^{97} - 19 q^{98} - 11 q^{99}+O(q^{100})$$ 7 * q + q^2 - 2 * q^3 + 15 * q^4 - 2 * q^6 - 10 * q^7 + 9 * q^8 + 11 * q^9 - 7 * q^11 + 16 * q^12 + 4 * q^13 + 6 * q^14 + 27 * q^16 - 2 * q^17 - 9 * q^18 + 7 * q^19 - 14 * q^21 - q^22 - 10 * q^23 - 2 * q^24 - 8 * q^26 + 4 * q^27 - 26 * q^28 - 18 * q^29 + 24 * q^31 + 49 * q^32 + 2 * q^33 - 6 * q^34 + 29 * q^36 + q^38 + 24 * q^39 - 12 * q^41 + 44 * q^42 - 2 * q^43 - 15 * q^44 - 4 * q^46 - 8 * q^47 + 72 * q^48 + 17 * q^49 - 24 * q^51 + 60 * q^52 - 2 * q^53 - 52 * q^54 + 26 * q^56 - 2 * q^57 + 8 * q^58 - 10 * q^59 + 14 * q^61 - 14 * q^62 + 55 * q^64 + 2 * q^66 - 8 * q^67 + 18 * q^68 - 6 * q^69 + 10 * q^71 - 53 * q^72 + 6 * q^73 + 26 * q^74 + 15 * q^76 + 10 * q^77 - 22 * q^78 + 52 * q^79 - q^81 - 24 * q^82 + 10 * q^83 - 6 * q^84 + 8 * q^86 - 6 * q^87 - 9 * q^88 + 12 * q^91 - 2 * q^93 + 24 * q^94 + 6 * q^96 + 24 * q^97 - 19 * q^98 - 11 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{7} - x^{6} - 14x^{5} + 10x^{4} + 59x^{3} - 27x^{2} - 66x + 30$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{4} - 7\nu^{2} - 2\nu + 4 ) / 2$$ (v^4 - 7*v^2 - 2*v + 4) / 2 $$\beta_{3}$$ $$=$$ $$( -\nu^{4} + 9\nu^{2} + 2\nu - 12 ) / 2$$ (-v^4 + 9*v^2 + 2*v - 12) / 2 $$\beta_{4}$$ $$=$$ $$( \nu^{5} - 9\nu^{3} + 14\nu - 6 ) / 2$$ (v^5 - 9*v^3 + 14*v - 6) / 2 $$\beta_{5}$$ $$=$$ $$( \nu^{6} - 10\nu^{4} + 23\nu^{2} - 4\nu - 10 ) / 4$$ (v^6 - 10*v^4 + 23*v^2 - 4*v - 10) / 4 $$\beta_{6}$$ $$=$$ $$( -\nu^{6} + 12\nu^{4} + 4\nu^{3} - 41\nu^{2} - 20\nu + 34 ) / 4$$ (-v^6 + 12*v^4 + 4*v^3 - 41*v^2 - 20*v + 34) / 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4$$ b3 + b2 + 4 $$\nu^{3}$$ $$=$$ $$\beta_{6} + \beta_{5} + \beta_{3} + 5\beta_1$$ b6 + b5 + b3 + 5*b1 $$\nu^{4}$$ $$=$$ $$7\beta_{3} + 9\beta_{2} + 2\beta _1 + 24$$ 7*b3 + 9*b2 + 2*b1 + 24 $$\nu^{5}$$ $$=$$ $$9\beta_{6} + 9\beta_{5} + 2\beta_{4} + 9\beta_{3} + 31\beta _1 + 6$$ 9*b6 + 9*b5 + 2*b4 + 9*b3 + 31*b1 + 6 $$\nu^{6}$$ $$=$$ $$4\beta_{5} + 47\beta_{3} + 67\beta_{2} + 24\beta _1 + 158$$ 4*b5 + 47*b3 + 67*b2 + 24*b1 + 158

## 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
 −2.55401 −2.03821 −1.45416 0.456669 1.19313 2.61330 2.78328
−2.55401 2.99825 4.52299 0 −7.65758 −4.42321 −6.44376 5.98952 0
1.2 −2.03821 −1.87275 2.15429 0 3.81704 −1.92338 −0.314472 0.507178 0
1.3 −1.45416 −1.71116 0.114592 0 2.48830 2.00933 2.74169 −0.0719365 0
1.4 0.456669 0.835165 −1.79145 0 0.381394 −4.69915 −1.73144 −2.30250 0
1.5 1.19313 −3.16232 −0.576442 0 −3.77306 1.30958 −3.07403 7.00029 0
1.6 2.61330 −1.19599 4.82936 0 −3.12549 −3.61829 7.39397 −1.56960 0
1.7 2.78328 2.10880 5.74667 0 5.86939 1.34513 10.4280 1.44705 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
$$5$$ $$1$$
$$11$$ $$1$$
$$19$$ $$-1$$

## 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 5225.2.a.n 7
5.b even 2 1 209.2.a.d 7
15.d odd 2 1 1881.2.a.p 7
20.d odd 2 1 3344.2.a.ba 7
55.d odd 2 1 2299.2.a.q 7
95.d odd 2 1 3971.2.a.i 7

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
209.2.a.d 7 5.b even 2 1
1881.2.a.p 7 15.d odd 2 1
2299.2.a.q 7 55.d odd 2 1
3344.2.a.ba 7 20.d odd 2 1
3971.2.a.i 7 95.d odd 2 1
5225.2.a.n 7 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5225))$$:

 $$T_{2}^{7} - T_{2}^{6} - 14T_{2}^{5} + 10T_{2}^{4} + 59T_{2}^{3} - 27T_{2}^{2} - 66T_{2} + 30$$ T2^7 - T2^6 - 14*T2^5 + 10*T2^4 + 59*T2^3 - 27*T2^2 - 66*T2 + 30 $$T_{7}^{7} + 10T_{7}^{6} + 17T_{7}^{5} - 86T_{7}^{4} - 185T_{7}^{3} + 316T_{7}^{2} + 394T_{7} - 512$$ T7^7 + 10*T7^6 + 17*T7^5 - 86*T7^4 - 185*T7^3 + 316*T7^2 + 394*T7 - 512

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{7} - T^{6} + \cdots + 30$$
$3$ $$T^{7} + 2 T^{6} + \cdots - 64$$
$5$ $$T^{7}$$
$7$ $$T^{7} + 10 T^{6} + \cdots - 512$$
$11$ $$(T + 1)^{7}$$
$13$ $$T^{7} - 4 T^{6} + \cdots + 5716$$
$17$ $$T^{7} + 2 T^{6} + \cdots + 17088$$
$19$ $$(T - 1)^{7}$$
$23$ $$T^{7} + 10 T^{6} + \cdots - 1920$$
$29$ $$T^{7} + 18 T^{6} + \cdots - 276$$
$31$ $$T^{7} - 24 T^{6} + \cdots + 4$$
$37$ $$T^{7} - 121 T^{5} + \cdots + 8992$$
$41$ $$T^{7} + 12 T^{6} + \cdots - 1824$$
$43$ $$T^{7} + 2 T^{6} + \cdots - 4976$$
$47$ $$T^{7} + 8 T^{6} + \cdots - 79872$$
$53$ $$T^{7} + 2 T^{6} + \cdots - 768$$
$59$ $$T^{7} + 10 T^{6} + \cdots - 6552192$$
$61$ $$T^{7} - 14 T^{6} + \cdots - 36544$$
$67$ $$T^{7} + 8 T^{6} + \cdots - 13544$$
$71$ $$T^{7} - 10 T^{6} + \cdots + 39756$$
$73$ $$T^{7} - 6 T^{6} + \cdots - 67328$$
$79$ $$T^{7} - 52 T^{6} + \cdots - 203264$$
$83$ $$T^{7} - 10 T^{6} + \cdots - 576936$$
$89$ $$T^{7} - 401 T^{5} + \cdots - 8199552$$
$97$ $$T^{7} - 24 T^{6} + \cdots + 17393056$$