# Properties

 Label 300.2.m.b Level $300$ Weight $2$ Character orbit 300.m Analytic conductor $2.396$ Analytic rank $0$ Dimension $8$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(61,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 0, 8]))

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

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

chi := DirichletCharacter("300.61");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.m (of order $$5$$, degree $$4$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{5})$$ Coefficient field: 8.0.26265625.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16$$ x^8 - 3*x^7 + 2*x^6 + x^4 + 8*x^2 - 24*x + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{3} q^{3} + ( - \beta_{7} + \beta_{6} - 1) q^{5} + (\beta_{7} + \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{7} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{9}+O(q^{10})$$ q + b3 * q^3 + (-b7 + b6 - 1) * q^5 + (b7 + b6 + 2*b3 - b2) * q^7 + (b6 + b3 + b1 - 1) * q^9 $$q + \beta_{3} q^{3} + ( - \beta_{7} + \beta_{6} - 1) q^{5} + (\beta_{7} + \beta_{6} + 2 \beta_{3} - \beta_{2}) q^{7} + (\beta_{6} + \beta_{3} + \beta_1 - 1) q^{9} + ( - \beta_{5} - \beta_{3} - \beta_{2} + 1) q^{11} + (\beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + 3 \beta_1 - 2) q^{13} + (\beta_{6} + \beta_{4} + \beta_1) q^{15} + ( - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{2} - 2 \beta_1 + 1) q^{17} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} - \beta_1) q^{19} + (\beta_{7} + \beta_{6} + 2 \beta_{3} + \beta_1) q^{21} + ( - 2 \beta_{5} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{23} + (2 \beta_{6} + \beta_{4} + 3 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 1) q^{25} + \beta_1 q^{27} + (\beta_{7} - \beta_{3}) q^{29} + ( - 3 \beta_{7} - 4 \beta_{6} - \beta_{5} - 2 \beta_{4} - 3 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{31} + (\beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_{2} + 1) q^{33} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - 6 \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{35} + (2 \beta_{6} + 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 2) q^{37} + (\beta_{5} + \beta_{2} + \beta_1) q^{39} + ( - 3 \beta_{6} - 4 \beta_{4} - \beta_{3} - \beta_1 + 3) q^{41} + ( - 2 \beta_{7} - \beta_{6} - 3 \beta_{3} + 2 \beta_{2}) q^{43} + ( - \beta_{6} + \beta_{5} - \beta_1 + 1) q^{45} + ( - \beta_{7} - 6 \beta_{6} - 2 \beta_{4} - 7 \beta_{3} - 2 \beta_{2} - 6 \beta_1) q^{47} + (3 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 5 \beta_{3} - 3 \beta_{2} + \cdots - 3) q^{49}+ \cdots + ( - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{2} - \beta_1) q^{99}+O(q^{100})$$ q + b3 * q^3 + (-b7 + b6 - 1) * q^5 + (b7 + b6 + 2*b3 - b2) * q^7 + (b6 + b3 + b1 - 1) * q^9 + (-b5 - b3 - b2 + 1) * q^11 + (b7 + 2*b6 - b5 + 2*b4 + 3*b3 + 3*b1 - 2) * q^13 + (b6 + b4 + b1) * q^15 + (-2*b6 + b5 - b4 - b2 - 2*b1 + 1) * q^17 + (b7 + b6 + b5 + b3 - b2 - b1) * q^19 + (b7 + b6 + 2*b3 + b1) * q^21 + (-2*b5 + b3 - b2 - b1 - 1) * q^23 + (2*b6 + b4 + 3*b3 + 2*b2 + 4*b1 - 1) * q^25 + b1 * q^27 + (b7 - b3) * q^29 + (-3*b7 - 4*b6 - b5 - 2*b4 - 3*b3 + b2 - b1 + 2) * q^31 + (b7 - b5 + 2*b4 + b3 + b2 + 1) * q^33 + (-2*b7 - b6 + b5 - 6*b3 + b2 - 2*b1 + 1) * q^35 + (2*b6 + 3*b4 + 2*b3 + 2*b1 - 2) * q^37 + (b5 + b2 + b1) * q^39 + (-3*b6 - 4*b4 - b3 - b1 + 3) * q^41 + (-2*b7 - b6 - 3*b3 + 2*b2) * q^43 + (-b6 + b5 - b1 + 1) * q^45 + (-b7 - 6*b6 - 2*b4 - 7*b3 - 2*b2 - 6*b1) * q^47 + (3*b7 + 2*b6 + 2*b5 - 2*b4 + 5*b3 - 3*b2 - 2*b1 - 3) * q^49 + (b7 + b6 + 2*b3 - b2 - 2) * q^51 + (-b7 - 3*b6 + 2*b4 - 2*b3 + 2*b2 - 3*b1) * q^53 + (-2*b7 + 5*b6 + b5 - 2*b4 - 2*b3 + 2*b2 - 2) * q^55 + (b7 + b5 - b4 + b3 - b2 - b1 + 1) * q^57 + (-b7 + 3*b6 + b5 - 2*b4 - 5*b3 - 5*b1 - 3) * q^59 + (-2*b5 - 8*b3 + 2*b2 + 3*b1 + 8) * q^61 + (-b4 + b3 + b1) * q^63 + (-2*b7 - 2*b6 + 3*b5 - 3*b4 - 4*b3 - b2 + 4) * q^65 + (2*b7 - 10*b6 + 2*b5 + 2*b3 - 2*b2 - 3*b1 + 1) * q^67 + (b7 + 4*b6 - 2*b5 + 3*b4 + b3 + 2*b2 + 3*b1 - 1) * q^69 + (-b7 - 3*b6 - 7*b3 - 3*b1) * q^71 + (2*b5 + 3*b3 + 2*b2 - 2*b1 - 3) * q^73 + (-2*b7 - b6 + b5 - 2*b4 + 2*b1 - 1) * q^75 + (-3*b3 - b2 + b1 + 3) * q^77 + (4*b7 - 4*b6 - b4 + b3 - b2 - 4*b1) * q^79 - b6 * q^81 + (-2*b7 - 3*b6 - 6*b5 + 4*b4 - 2*b3 + 6*b2 + 2*b1 + 4) * q^83 + (b7 + 2*b6 - b5 + 2*b4 + 3*b1 - 5) * q^85 + (-2*b6 - b4 - 2*b3 - 2*b1 + 2) * q^87 + (b5 + 6*b3 + 3*b2 + 7*b1 - 6) * q^89 + (-5*b6 - b4 - b3 - b1 + 5) * q^91 + (-b7 + 2*b6 - 3*b5 + 3*b4 + b3 + b2 + 3*b1 - 4) * q^93 + (-2*b7 - 5*b6 - 6*b3 + b2 - b1) * q^95 + (-4*b7 - 2*b6 + 4*b4 - b3 + 4*b2 - 2*b1) * q^97 + (-b7 + b6 + b5 - b4 + b2 - b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{3} - 5 q^{5} + 8 q^{7} - 2 q^{9}+O(q^{10})$$ 8 * q + 2 * q^3 - 5 * q^5 + 8 * q^7 - 2 * q^9 $$8 q + 2 q^{3} - 5 q^{5} + 8 q^{7} - 2 q^{9} + 8 q^{11} + 5 q^{15} + 3 q^{17} + 5 q^{19} + 7 q^{21} - 7 q^{23} + 5 q^{25} + 2 q^{27} - 3 q^{29} - 3 q^{31} + 7 q^{33} - 10 q^{35} - q^{37} + 10 q^{41} - 12 q^{43} + 5 q^{45} - 33 q^{47} - 8 q^{49} - 8 q^{51} - 19 q^{53} - 15 q^{55} + 10 q^{57} - 38 q^{59} + 46 q^{61} + 3 q^{63} + 25 q^{65} - 8 q^{67} + 2 q^{69} - 25 q^{71} - 26 q^{73} - 5 q^{75} + 23 q^{77} - 16 q^{79} - 2 q^{81} + 8 q^{83} - 30 q^{85} + 3 q^{87} - 30 q^{89} + 25 q^{91} - 22 q^{93} - 25 q^{95} - 14 q^{97} - 2 q^{99}+O(q^{100})$$ 8 * q + 2 * q^3 - 5 * q^5 + 8 * q^7 - 2 * q^9 + 8 * q^11 + 5 * q^15 + 3 * q^17 + 5 * q^19 + 7 * q^21 - 7 * q^23 + 5 * q^25 + 2 * q^27 - 3 * q^29 - 3 * q^31 + 7 * q^33 - 10 * q^35 - q^37 + 10 * q^41 - 12 * q^43 + 5 * q^45 - 33 * q^47 - 8 * q^49 - 8 * q^51 - 19 * q^53 - 15 * q^55 + 10 * q^57 - 38 * q^59 + 46 * q^61 + 3 * q^63 + 25 * q^65 - 8 * q^67 + 2 * q^69 - 25 * q^71 - 26 * q^73 - 5 * q^75 + 23 * q^77 - 16 * q^79 - 2 * q^81 + 8 * q^83 - 30 * q^85 + 3 * q^87 - 30 * q^89 + 25 * q^91 - 22 * q^93 - 25 * q^95 - 14 * q^97 - 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8$$ (v^7 - v^6 + v^3 + 2*v^2 + 4*v - 8) / 8 $$\beta_{2}$$ $$=$$ $$( \nu^{7} - \nu^{6} + \nu^{3} - 2\nu^{2} + 12\nu - 12 ) / 4$$ (v^7 - v^6 + v^3 - 2*v^2 + 12*v - 12) / 4 $$\beta_{3}$$ $$=$$ $$( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8$$ (-7*v^7 + 9*v^6 + 2*v^5 + 4*v^4 + v^3 - 4*v^2 - 60*v + 64) / 8 $$\beta_{4}$$ $$=$$ $$( 11\nu^{7} - 15\nu^{6} - 4\nu^{5} - 8\nu^{4} + 3\nu^{3} + 2\nu^{2} + 92\nu - 96 ) / 8$$ (11*v^7 - 15*v^6 - 4*v^5 - 8*v^4 + 3*v^3 + 2*v^2 + 92*v - 96) / 8 $$\beta_{5}$$ $$=$$ $$( 11\nu^{7} - 13\nu^{6} - 6\nu^{5} - 8\nu^{4} + 3\nu^{3} + 4\nu^{2} + 96\nu - 88 ) / 8$$ (11*v^7 - 13*v^6 - 6*v^5 - 8*v^4 + 3*v^3 + 4*v^2 + 96*v - 88) / 8 $$\beta_{6}$$ $$=$$ $$( 13\nu^{7} - 21\nu^{6} - 4\nu^{5} - 4\nu^{4} + 5\nu^{3} + 10\nu^{2} + 120\nu - 144 ) / 8$$ (13*v^7 - 21*v^6 - 4*v^5 - 4*v^4 + 5*v^3 + 10*v^2 + 120*v - 144) / 8 $$\beta_{7}$$ $$=$$ $$( 19\nu^{7} - 31\nu^{6} - 4\nu^{5} - 8\nu^{4} + 11\nu^{3} + 18\nu^{2} + 180\nu - 224 ) / 8$$ (19*v^7 - 31*v^6 - 4*v^5 - 8*v^4 + 11*v^3 + 18*v^2 + 180*v - 224) / 8
 $$\nu$$ $$=$$ $$( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{4} - \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 5$$ (b7 - b6 + b5 - 2*b4 - b3 + b2 - 4*b1 + 4) / 5 $$\nu^{2}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} - 4\beta_{4} - 2\beta_{3} - 3\beta_{2} + 2\beta _1 + 3 ) / 5$$ (2*b7 - 2*b6 + 2*b5 - 4*b4 - 2*b3 - 3*b2 + 2*b1 + 3) / 5 $$\nu^{3}$$ $$=$$ $$( 2\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 8\beta_{3} + 2\beta_{2} + 7\beta _1 + 3 ) / 5$$ (2*b7 - 2*b6 + 2*b5 + b4 + 8*b3 + 2*b2 + 7*b1 + 3) / 5 $$\nu^{4}$$ $$=$$ $$-\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} + 2\beta _1 + 1$$ -b7 + 2*b6 - b4 + b2 + 2*b1 + 1 $$\nu^{5}$$ $$=$$ $$( 6\beta_{7} - 11\beta_{6} - 9\beta_{5} + 3\beta_{4} - 11\beta_{3} + 6\beta_{2} + 6\beta _1 + 19 ) / 5$$ (6*b7 - 11*b6 - 9*b5 + 3*b4 - 11*b3 + 6*b2 + 6*b1 + 19) / 5 $$\nu^{6}$$ $$=$$ $$( 2\beta_{7} - 7\beta_{6} + 7\beta_{5} - 9\beta_{4} - 7\beta_{3} + 7\beta_{2} + 12\beta _1 - 12 ) / 5$$ (2*b7 - 7*b6 + 7*b5 - 9*b4 - 7*b3 + 7*b2 + 12*b1 - 12) / 5 $$\nu^{7}$$ $$=$$ $$( -8\beta_{7} + 3\beta_{6} - 3\beta_{5} + 6\beta_{4} - 7\beta_{3} + 7\beta_{2} + 57\beta _1 + 3 ) / 5$$ (-8*b7 + 3*b6 - 3*b5 + 6*b4 - 7*b3 + 7*b2 + 57*b1 + 3) / 5

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/300\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
61.1
 1.40799 − 0.132563i −1.21700 + 0.720348i 1.33631 + 0.462894i −0.0272949 − 1.41395i 1.33631 − 0.462894i −0.0272949 + 1.41395i 1.40799 + 0.132563i −1.21700 − 0.720348i
0 −0.309017 + 0.951057i 0 −1.96485 + 1.06740i 0 −1.74037 0 −0.809017 0.587785i 0
61.2 0 −0.309017 + 0.951057i 0 −0.962197 2.01846i 0 1.50430 0 −0.809017 0.587785i 0
121.1 0 0.809017 + 0.587785i 0 −1.57146 1.59076i 0 4.32440 0 0.309017 + 0.951057i 0
121.2 0 0.809017 + 0.587785i 0 1.99851 + 1.00297i 0 −0.0883282 0 0.309017 + 0.951057i 0
181.1 0 0.809017 0.587785i 0 −1.57146 + 1.59076i 0 4.32440 0 0.309017 0.951057i 0
181.2 0 0.809017 0.587785i 0 1.99851 1.00297i 0 −0.0883282 0 0.309017 0.951057i 0
241.1 0 −0.309017 0.951057i 0 −1.96485 1.06740i 0 −1.74037 0 −0.809017 + 0.587785i 0
241.2 0 −0.309017 0.951057i 0 −0.962197 + 2.01846i 0 1.50430 0 −0.809017 + 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.2 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.m.b 8
3.b odd 2 1 900.2.n.b 8
5.b even 2 1 1500.2.m.a 8
5.c odd 4 2 1500.2.o.b 16
25.d even 5 1 inner 300.2.m.b 8
25.d even 5 1 7500.2.a.e 4
25.e even 10 1 1500.2.m.a 8
25.e even 10 1 7500.2.a.f 4
25.f odd 20 2 1500.2.o.b 16
25.f odd 20 2 7500.2.d.c 8
75.j odd 10 1 900.2.n.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.m.b 8 1.a even 1 1 trivial
300.2.m.b 8 25.d even 5 1 inner
900.2.n.b 8 3.b odd 2 1
900.2.n.b 8 75.j odd 10 1
1500.2.m.a 8 5.b even 2 1
1500.2.m.a 8 25.e even 10 1
1500.2.o.b 16 5.c odd 4 2
1500.2.o.b 16 25.f odd 20 2
7500.2.a.e 4 25.d even 5 1
7500.2.a.f 4 25.e even 10 1
7500.2.d.c 8 25.f odd 20 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{4} - 4T_{7}^{3} - 4T_{7}^{2} + 11T_{7} + 1$$ acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8}$$
$3$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$5$ $$T^{8} + 5 T^{7} + 10 T^{6} - 5 T^{5} + \cdots + 625$$
$7$ $$(T^{4} - 4 T^{3} - 4 T^{2} + 11 T + 1)^{2}$$
$11$ $$T^{8} - 8 T^{7} + 23 T^{6} + 39 T^{5} + \cdots + 361$$
$13$ $$T^{8} - 5 T^{6} - 5 T^{5} + 310 T^{4} + \cdots + 2025$$
$17$ $$T^{8} - 3 T^{7} + 13 T^{6} - T^{5} + \cdots + 81$$
$19$ $$T^{8} - 5 T^{7} + 5 T^{6} + 45 T^{5} + \cdots + 25$$
$23$ $$T^{8} + 7 T^{7} + 18 T^{6} + \cdots + 29241$$
$29$ $$T^{8} + 3 T^{7} + 8 T^{6} + 21 T^{5} + \cdots + 1$$
$31$ $$T^{8} + 3 T^{7} + 58 T^{6} + \cdots + 962361$$
$37$ $$T^{8} + T^{7} + 57 T^{6} + \cdots + 1042441$$
$41$ $$T^{8} - 10 T^{7} + 155 T^{6} + \cdots + 7317025$$
$43$ $$(T^{4} + 6 T^{3} - 19 T^{2} - 64 T + 131)^{2}$$
$47$ $$T^{8} + 33 T^{7} + 568 T^{6} + \cdots + 16801801$$
$53$ $$T^{8} + 19 T^{7} + 217 T^{6} + \cdots + 9801$$
$59$ $$T^{8} + 38 T^{7} + 773 T^{6} + \cdots + 13697401$$
$61$ $$T^{8} - 46 T^{7} + 1037 T^{6} + \cdots + 62552281$$
$67$ $$T^{8} + 8 T^{7} + 183 T^{6} + \cdots + 408321$$
$71$ $$T^{8} + 25 T^{7} + 275 T^{6} + \cdots + 25$$
$73$ $$T^{8} + 26 T^{7} + 267 T^{6} + \cdots + 121$$
$79$ $$T^{8} + 16 T^{7} + 117 T^{6} + \cdots + 408321$$
$83$ $$T^{8} - 8 T^{7} - 17 T^{6} + \cdots + 46908801$$
$89$ $$T^{8} + 30 T^{7} + 625 T^{6} + \cdots + 97515625$$
$97$ $$T^{8} + 14 T^{7} + 87 T^{6} + \cdots + 64304361$$