# Properties

 Label 300.3.c.c Level $300$ Weight $3$ Character orbit 300.c Analytic conductor $8.174$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [300,3,Mod(151,300)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(300, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("300.151");

S:= CuspForms(chi, 3);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$300 = 2^{2} \cdot 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 300.c (of order $$2$$, degree $$1$$, 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: $$8.17440793081$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - x + 1$$ x^2 - x + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 60) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$\beta = \sqrt{-3}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta + 1) q^{2} - \beta q^{3} + (2 \beta - 2) q^{4} + ( - \beta + 3) q^{6} - 6 \beta q^{7} - 8 q^{8} - 3 q^{9} +O(q^{10})$$ q + (b + 1) * q^2 - b * q^3 + (2*b - 2) * q^4 + (-b + 3) * q^6 - 6*b * q^7 - 8 * q^8 - 3 * q^9 $$q + (\beta + 1) q^{2} - \beta q^{3} + (2 \beta - 2) q^{4} + ( - \beta + 3) q^{6} - 6 \beta q^{7} - 8 q^{8} - 3 q^{9} - 6 \beta q^{11} + (2 \beta + 6) q^{12} + 18 q^{13} + ( - 6 \beta + 18) q^{14} + ( - 8 \beta - 8) q^{16} + 10 q^{17} + ( - 3 \beta - 3) q^{18} - 8 \beta q^{19} - 18 q^{21} + ( - 6 \beta + 18) q^{22} - 4 \beta q^{23} + 8 \beta q^{24} + (18 \beta + 18) q^{26} + 3 \beta q^{27} + (12 \beta + 36) q^{28} - 36 q^{29} + 4 \beta q^{31} + ( - 16 \beta + 16) q^{32} - 18 q^{33} + (10 \beta + 10) q^{34} + ( - 6 \beta + 6) q^{36} + 54 q^{37} + ( - 8 \beta + 24) q^{38} - 18 \beta q^{39} + 18 q^{41} + ( - 18 \beta - 18) q^{42} - 12 \beta q^{43} + (12 \beta + 36) q^{44} + ( - 4 \beta + 12) q^{46} + (8 \beta - 24) q^{48} - 59 q^{49} - 10 \beta q^{51} + (36 \beta - 36) q^{52} - 26 q^{53} + (3 \beta - 9) q^{54} + 48 \beta q^{56} - 24 q^{57} + ( - 36 \beta - 36) q^{58} + 18 \beta q^{59} - 74 q^{61} + (4 \beta - 12) q^{62} + 18 \beta q^{63} + 64 q^{64} + ( - 18 \beta - 18) q^{66} - 24 \beta q^{67} + (20 \beta - 20) q^{68} - 12 q^{69} + 60 \beta q^{71} + 24 q^{72} + 36 q^{73} + (54 \beta + 54) q^{74} + (16 \beta + 48) q^{76} - 108 q^{77} + ( - 18 \beta + 54) q^{78} - 52 \beta q^{79} + 9 q^{81} + (18 \beta + 18) q^{82} + 52 \beta q^{83} + ( - 36 \beta + 36) q^{84} + ( - 12 \beta + 36) q^{86} + 36 \beta q^{87} + 48 \beta q^{88} - 18 q^{89} - 108 \beta q^{91} + (8 \beta + 24) q^{92} + 12 q^{93} + ( - 16 \beta - 48) q^{96} - 72 q^{97} + ( - 59 \beta - 59) q^{98} + 18 \beta q^{99} +O(q^{100})$$ q + (b + 1) * q^2 - b * q^3 + (2*b - 2) * q^4 + (-b + 3) * q^6 - 6*b * q^7 - 8 * q^8 - 3 * q^9 - 6*b * q^11 + (2*b + 6) * q^12 + 18 * q^13 + (-6*b + 18) * q^14 + (-8*b - 8) * q^16 + 10 * q^17 + (-3*b - 3) * q^18 - 8*b * q^19 - 18 * q^21 + (-6*b + 18) * q^22 - 4*b * q^23 + 8*b * q^24 + (18*b + 18) * q^26 + 3*b * q^27 + (12*b + 36) * q^28 - 36 * q^29 + 4*b * q^31 + (-16*b + 16) * q^32 - 18 * q^33 + (10*b + 10) * q^34 + (-6*b + 6) * q^36 + 54 * q^37 + (-8*b + 24) * q^38 - 18*b * q^39 + 18 * q^41 + (-18*b - 18) * q^42 - 12*b * q^43 + (12*b + 36) * q^44 + (-4*b + 12) * q^46 + (8*b - 24) * q^48 - 59 * q^49 - 10*b * q^51 + (36*b - 36) * q^52 - 26 * q^53 + (3*b - 9) * q^54 + 48*b * q^56 - 24 * q^57 + (-36*b - 36) * q^58 + 18*b * q^59 - 74 * q^61 + (4*b - 12) * q^62 + 18*b * q^63 + 64 * q^64 + (-18*b - 18) * q^66 - 24*b * q^67 + (20*b - 20) * q^68 - 12 * q^69 + 60*b * q^71 + 24 * q^72 + 36 * q^73 + (54*b + 54) * q^74 + (16*b + 48) * q^76 - 108 * q^77 + (-18*b + 54) * q^78 - 52*b * q^79 + 9 * q^81 + (18*b + 18) * q^82 + 52*b * q^83 + (-36*b + 36) * q^84 + (-12*b + 36) * q^86 + 36*b * q^87 + 48*b * q^88 - 18 * q^89 - 108*b * q^91 + (8*b + 24) * q^92 + 12 * q^93 + (-16*b - 48) * q^96 - 72 * q^97 + (-59*b - 59) * q^98 + 18*b * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{2} - 4 q^{4} + 6 q^{6} - 16 q^{8} - 6 q^{9}+O(q^{10})$$ 2 * q + 2 * q^2 - 4 * q^4 + 6 * q^6 - 16 * q^8 - 6 * q^9 $$2 q + 2 q^{2} - 4 q^{4} + 6 q^{6} - 16 q^{8} - 6 q^{9} + 12 q^{12} + 36 q^{13} + 36 q^{14} - 16 q^{16} + 20 q^{17} - 6 q^{18} - 36 q^{21} + 36 q^{22} + 36 q^{26} + 72 q^{28} - 72 q^{29} + 32 q^{32} - 36 q^{33} + 20 q^{34} + 12 q^{36} + 108 q^{37} + 48 q^{38} + 36 q^{41} - 36 q^{42} + 72 q^{44} + 24 q^{46} - 48 q^{48} - 118 q^{49} - 72 q^{52} - 52 q^{53} - 18 q^{54} - 48 q^{57} - 72 q^{58} - 148 q^{61} - 24 q^{62} + 128 q^{64} - 36 q^{66} - 40 q^{68} - 24 q^{69} + 48 q^{72} + 72 q^{73} + 108 q^{74} + 96 q^{76} - 216 q^{77} + 108 q^{78} + 18 q^{81} + 36 q^{82} + 72 q^{84} + 72 q^{86} - 36 q^{89} + 48 q^{92} + 24 q^{93} - 96 q^{96} - 144 q^{97} - 118 q^{98}+O(q^{100})$$ 2 * q + 2 * q^2 - 4 * q^4 + 6 * q^6 - 16 * q^8 - 6 * q^9 + 12 * q^12 + 36 * q^13 + 36 * q^14 - 16 * q^16 + 20 * q^17 - 6 * q^18 - 36 * q^21 + 36 * q^22 + 36 * q^26 + 72 * q^28 - 72 * q^29 + 32 * q^32 - 36 * q^33 + 20 * q^34 + 12 * q^36 + 108 * q^37 + 48 * q^38 + 36 * q^41 - 36 * q^42 + 72 * q^44 + 24 * q^46 - 48 * q^48 - 118 * q^49 - 72 * q^52 - 52 * q^53 - 18 * q^54 - 48 * q^57 - 72 * q^58 - 148 * q^61 - 24 * q^62 + 128 * q^64 - 36 * q^66 - 40 * q^68 - 24 * q^69 + 48 * q^72 + 72 * q^73 + 108 * q^74 + 96 * q^76 - 216 * q^77 + 108 * q^78 + 18 * q^81 + 36 * q^82 + 72 * q^84 + 72 * q^86 - 36 * q^89 + 48 * q^92 + 24 * q^93 - 96 * q^96 - 144 * q^97 - 118 * q^98

## Character values

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

 $$n$$ $$101$$ $$151$$ $$277$$ $$\chi(n)$$ $$1$$ $$-1$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
151.1
 0.5 − 0.866025i 0.5 + 0.866025i
1.00000 1.73205i 1.73205i −2.00000 3.46410i 0 3.00000 + 1.73205i 10.3923i −8.00000 −3.00000 0
151.2 1.00000 + 1.73205i 1.73205i −2.00000 + 3.46410i 0 3.00000 1.73205i 10.3923i −8.00000 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.c.c 2
3.b odd 2 1 900.3.c.f 2
4.b odd 2 1 inner 300.3.c.c 2
5.b even 2 1 300.3.c.a 2
5.c odd 4 2 60.3.f.a 4
12.b even 2 1 900.3.c.f 2
15.d odd 2 1 900.3.c.j 2
15.e even 4 2 180.3.f.e 4
20.d odd 2 1 300.3.c.a 2
20.e even 4 2 60.3.f.a 4
40.i odd 4 2 960.3.j.b 4
40.k even 4 2 960.3.j.b 4
60.h even 2 1 900.3.c.j 2
60.l odd 4 2 180.3.f.e 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.f.a 4 5.c odd 4 2
60.3.f.a 4 20.e even 4 2
180.3.f.e 4 15.e even 4 2
180.3.f.e 4 60.l odd 4 2
300.3.c.a 2 5.b even 2 1
300.3.c.a 2 20.d odd 2 1
300.3.c.c 2 1.a even 1 1 trivial
300.3.c.c 2 4.b odd 2 1 inner
900.3.c.f 2 3.b odd 2 1
900.3.c.f 2 12.b even 2 1
900.3.c.j 2 15.d odd 2 1
900.3.c.j 2 60.h even 2 1
960.3.j.b 4 40.i odd 4 2
960.3.j.b 4 40.k even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(300, [\chi])$$:

 $$T_{7}^{2} + 108$$ T7^2 + 108 $$T_{13} - 18$$ T13 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} - 2T + 4$$
$3$ $$T^{2} + 3$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 108$$
$11$ $$T^{2} + 108$$
$13$ $$(T - 18)^{2}$$
$17$ $$(T - 10)^{2}$$
$19$ $$T^{2} + 192$$
$23$ $$T^{2} + 48$$
$29$ $$(T + 36)^{2}$$
$31$ $$T^{2} + 48$$
$37$ $$(T - 54)^{2}$$
$41$ $$(T - 18)^{2}$$
$43$ $$T^{2} + 432$$
$47$ $$T^{2}$$
$53$ $$(T + 26)^{2}$$
$59$ $$T^{2} + 972$$
$61$ $$(T + 74)^{2}$$
$67$ $$T^{2} + 1728$$
$71$ $$T^{2} + 10800$$
$73$ $$(T - 36)^{2}$$
$79$ $$T^{2} + 8112$$
$83$ $$T^{2} + 8112$$
$89$ $$(T + 18)^{2}$$
$97$ $$(T + 72)^{2}$$