Properties

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

Related objects

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,2,Mod(7,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 0, 1]))

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

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

chi := DirichletCharacter("300.7");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 300.j (of order $$4$$, 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: $$2.39551206064$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$4$$ over $$\Q(i)$$ Coefficient field: 8.0.157351936.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{8} + x^{4} + 16$$ x^8 + x^4 + 16 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + x^{4} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{4} + 2 ) / 3$$ (v^4 + 2) / 3 $$\beta_{2}$$ $$=$$ $$( \nu^{5} + 5\nu ) / 6$$ (v^5 + 5*v) / 6 $$\beta_{3}$$ $$=$$ $$( \nu^{6} + 5\nu^{2} ) / 12$$ (v^6 + 5*v^2) / 12 $$\beta_{4}$$ $$=$$ $$( -\nu^{5} + \nu ) / 6$$ (-v^5 + v) / 6 $$\beta_{5}$$ $$=$$ $$( -\nu^{6} + 7\nu^{2} ) / 12$$ (-v^6 + 7*v^2) / 12 $$\beta_{6}$$ $$=$$ $$( -\nu^{7} + 7\nu^{3} ) / 24$$ (-v^7 + 7*v^3) / 24 $$\beta_{7}$$ $$=$$ $$( \nu^{7} + 5\nu^{3} ) / 12$$ (v^7 + 5*v^3) / 12
 $$\nu$$ $$=$$ $$\beta_{4} + \beta_{2}$$ b4 + b2 $$\nu^{2}$$ $$=$$ $$\beta_{5} + \beta_{3}$$ b5 + b3 $$\nu^{3}$$ $$=$$ $$\beta_{7} + 2\beta_{6}$$ b7 + 2*b6 $$\nu^{4}$$ $$=$$ $$3\beta _1 - 2$$ 3*b1 - 2 $$\nu^{5}$$ $$=$$ $$-5\beta_{4} + \beta_{2}$$ -5*b4 + b2 $$\nu^{6}$$ $$=$$ $$-5\beta_{5} + 7\beta_{3}$$ -5*b5 + 7*b3 $$\nu^{7}$$ $$=$$ $$7\beta_{7} - 10\beta_{6}$$ 7*b7 - 10*b6

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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.28897 − 0.581861i −0.581861 − 1.28897i 0.581861 + 1.28897i 1.28897 + 0.581861i −1.28897 + 0.581861i −0.581861 + 1.28897i 0.581861 − 1.28897i 1.28897 − 0.581861i
−1.28897 0.581861i −0.707107 0.707107i 1.32288 + 1.50000i 0 0.500000 + 1.32288i −1.41421 + 1.41421i −0.832353 2.70318i 1.00000i 0
7.2 −0.581861 1.28897i 0.707107 + 0.707107i −1.32288 + 1.50000i 0 0.500000 1.32288i 1.41421 1.41421i 2.70318 + 0.832353i 1.00000i 0
7.3 0.581861 + 1.28897i −0.707107 0.707107i −1.32288 + 1.50000i 0 0.500000 1.32288i −1.41421 + 1.41421i −2.70318 0.832353i 1.00000i 0
7.4 1.28897 + 0.581861i 0.707107 + 0.707107i 1.32288 + 1.50000i 0 0.500000 + 1.32288i 1.41421 1.41421i 0.832353 + 2.70318i 1.00000i 0
43.1 −1.28897 + 0.581861i −0.707107 + 0.707107i 1.32288 1.50000i 0 0.500000 1.32288i −1.41421 1.41421i −0.832353 + 2.70318i 1.00000i 0
43.2 −0.581861 + 1.28897i 0.707107 0.707107i −1.32288 1.50000i 0 0.500000 + 1.32288i 1.41421 + 1.41421i 2.70318 0.832353i 1.00000i 0
43.3 0.581861 1.28897i −0.707107 + 0.707107i −1.32288 1.50000i 0 0.500000 + 1.32288i −1.41421 1.41421i −2.70318 + 0.832353i 1.00000i 0
43.4 1.28897 0.581861i 0.707107 0.707107i 1.32288 1.50000i 0 0.500000 1.32288i 1.41421 + 1.41421i 0.832353 2.70318i 1.00000i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 7.4 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
5.c odd 4 2 inner
20.d odd 2 1 inner
20.e even 4 2 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.2.j.b 8
3.b odd 2 1 900.2.k.h 8
4.b odd 2 1 inner 300.2.j.b 8
5.b even 2 1 inner 300.2.j.b 8
5.c odd 4 2 inner 300.2.j.b 8
12.b even 2 1 900.2.k.h 8
15.d odd 2 1 900.2.k.h 8
15.e even 4 2 900.2.k.h 8
20.d odd 2 1 inner 300.2.j.b 8
20.e even 4 2 inner 300.2.j.b 8
60.h even 2 1 900.2.k.h 8
60.l odd 4 2 900.2.k.h 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.2.j.b 8 1.a even 1 1 trivial
300.2.j.b 8 4.b odd 2 1 inner
300.2.j.b 8 5.b even 2 1 inner
300.2.j.b 8 5.c odd 4 2 inner
300.2.j.b 8 20.d odd 2 1 inner
300.2.j.b 8 20.e even 4 2 inner
900.2.k.h 8 3.b odd 2 1
900.2.k.h 8 12.b even 2 1
900.2.k.h 8 15.d odd 2 1
900.2.k.h 8 15.e even 4 2
900.2.k.h 8 60.h even 2 1
900.2.k.h 8 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{4} + 16$$ T7^4 + 16 $$T_{19}^{2} - 28$$ T19^2 - 28

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} + T^{4} + 16$$
$3$ $$(T^{4} + 1)^{2}$$
$5$ $$T^{8}$$
$7$ $$(T^{4} + 16)^{2}$$
$11$ $$(T^{2} + 28)^{4}$$
$13$ $$(T^{4} + 784)^{2}$$
$17$ $$T^{8}$$
$19$ $$(T^{2} - 28)^{4}$$
$23$ $$(T^{4} + 256)^{2}$$
$29$ $$(T^{2} + 64)^{4}$$
$31$ $$(T^{2} + 28)^{4}$$
$37$ $$(T^{4} + 784)^{2}$$
$41$ $$(T + 2)^{8}$$
$43$ $$(T^{4} + 4096)^{2}$$
$47$ $$T^{8}$$
$53$ $$(T^{4} + 12544)^{2}$$
$59$ $$(T^{2} - 28)^{4}$$
$61$ $$(T - 6)^{8}$$
$67$ $$(T^{4} + 20736)^{2}$$
$71$ $$T^{8}$$
$73$ $$(T^{4} + 12544)^{2}$$
$79$ $$(T^{2} - 28)^{4}$$
$83$ $$(T^{4} + 20736)^{2}$$
$89$ $$(T^{2} + 36)^{4}$$
$97$ $$(T^{4} + 12544)^{2}$$