Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,3,Mod(101,300)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.101");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.g (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: \(\Q(\sqrt{-5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(\beta = \sqrt{-5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta - 2) q^{3} + 8 q^{7} + ( - 4 \beta - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta - 2) q^{3} + 8 q^{7} + ( - 4 \beta - 1) q^{9} - 4 \beta q^{11} + 12 q^{13} + 14 \beta q^{17} + 6 q^{19} + (8 \beta - 16) q^{21} + 2 \beta q^{23} + (7 \beta + 22) q^{27} + 12 \beta q^{29} + 34 q^{31} + (8 \beta + 20) q^{33} + 44 q^{37} + (12 \beta - 24) q^{39} + 8 \beta q^{41} - 28 q^{43} - 2 \beta q^{47} + 15 q^{49} + ( - 28 \beta - 70) q^{51} + 18 \beta q^{53} + (6 \beta - 12) q^{57} - 44 \beta q^{59} + 74 q^{61} + ( - 32 \beta - 8) q^{63} - 92 q^{67} + ( - 4 \beta - 10) q^{69} + 24 \beta q^{71} + 56 q^{73} - 32 \beta q^{77} + 78 q^{79} + (8 \beta - 79) q^{81} - 46 \beta q^{83} + ( - 24 \beta - 60) q^{87} + 8 \beta q^{89} + 96 q^{91} + (34 \beta - 68) q^{93} - 32 q^{97} + (4 \beta - 80) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{3} + 16 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{3} + 16 q^{7} - 2 q^{9} + 24 q^{13} + 12 q^{19} - 32 q^{21} + 44 q^{27} + 68 q^{31} + 40 q^{33} + 88 q^{37} - 48 q^{39} - 56 q^{43} + 30 q^{49} - 140 q^{51} - 24 q^{57} + 148 q^{61} - 16 q^{63} - 184 q^{67} - 20 q^{69} + 112 q^{73} + 156 q^{79} - 158 q^{81} - 120 q^{87} + 192 q^{91} - 136 q^{93} - 64 q^{97} - 160 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
2.23607i
2.23607i
0 −2.00000 2.23607i 0 0 0 8.00000 0 −1.00000 + 8.94427i 0
101.2 0 −2.00000 + 2.23607i 0 0 0 8.00000 0 −1.00000 8.94427i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.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.g.e 2
3.b odd 2 1 inner 300.3.g.e 2
4.b odd 2 1 1200.3.l.q 2
5.b even 2 1 300.3.g.h 2
5.c odd 4 2 60.3.b.a 4
12.b even 2 1 1200.3.l.q 2
15.d odd 2 1 300.3.g.h 2
15.e even 4 2 60.3.b.a 4
20.d odd 2 1 1200.3.l.h 2
20.e even 4 2 240.3.c.d 4
40.i odd 4 2 960.3.c.h 4
40.k even 4 2 960.3.c.g 4
45.k odd 12 4 1620.3.t.b 8
45.l even 12 4 1620.3.t.b 8
60.h even 2 1 1200.3.l.h 2
60.l odd 4 2 240.3.c.d 4
120.q odd 4 2 960.3.c.g 4
120.w even 4 2 960.3.c.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.b.a 4 5.c odd 4 2
60.3.b.a 4 15.e even 4 2
240.3.c.d 4 20.e even 4 2
240.3.c.d 4 60.l odd 4 2
300.3.g.e 2 1.a even 1 1 trivial
300.3.g.e 2 3.b odd 2 1 inner
300.3.g.h 2 5.b even 2 1
300.3.g.h 2 15.d odd 2 1
960.3.c.g 4 40.k even 4 2
960.3.c.g 4 120.q odd 4 2
960.3.c.h 4 40.i odd 4 2
960.3.c.h 4 120.w even 4 2
1200.3.l.h 2 20.d odd 2 1
1200.3.l.h 2 60.h even 2 1
1200.3.l.q 2 4.b odd 2 1
1200.3.l.q 2 12.b even 2 1
1620.3.t.b 8 45.k odd 12 4
1620.3.t.b 8 45.l even 12 4

Hecke kernels

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

\( T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{2} + 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 4T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T - 8)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 80 \) Copy content Toggle raw display
$13$ \( (T - 12)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 980 \) Copy content Toggle raw display
$19$ \( (T - 6)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 20 \) Copy content Toggle raw display
$29$ \( T^{2} + 720 \) Copy content Toggle raw display
$31$ \( (T - 34)^{2} \) Copy content Toggle raw display
$37$ \( (T - 44)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 320 \) Copy content Toggle raw display
$43$ \( (T + 28)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 20 \) Copy content Toggle raw display
$53$ \( T^{2} + 1620 \) Copy content Toggle raw display
$59$ \( T^{2} + 9680 \) Copy content Toggle raw display
$61$ \( (T - 74)^{2} \) Copy content Toggle raw display
$67$ \( (T + 92)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 2880 \) Copy content Toggle raw display
$73$ \( (T - 56)^{2} \) Copy content Toggle raw display
$79$ \( (T - 78)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 10580 \) Copy content Toggle raw display
$89$ \( T^{2} + 320 \) Copy content Toggle raw display
$97$ \( (T + 32)^{2} \) Copy content Toggle raw display
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