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Properties

Label 300.3.k.a
Level $300$
Weight $3$
Character orbit 300.k
Analytic conductor $8.174$
Analytic rank $0$
Dimension $4$
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(157,300)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.157");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 300.k (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: \(8.17440793081\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (2 \beta_{3} + 5 \beta_{2} - 5) q^{7} + 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (2 \beta_{3} + 5 \beta_{2} - 5) q^{7} + 3 \beta_{2} q^{9} + (5 \beta_{3} - 5 \beta_1 - 4) q^{11} - 10 \beta_1 q^{13} + (2 \beta_{3} - 10 \beta_{2} + 10) q^{17} + (10 \beta_{3} - 10 \beta_{2} + 10 \beta_1) q^{19} + (5 \beta_{3} - 5 \beta_1 - 6) q^{21} + (10 \beta_{2} - 6 \beta_1 + 10) q^{23} + 3 \beta_{3} q^{27} + (5 \beta_{3} + 22 \beta_{2} + 5 \beta_1) q^{29} + (10 \beta_{3} - 10 \beta_1 + 4) q^{31} + ( - 15 \beta_{2} - 4 \beta_1 - 15) q^{33} + 6 \beta_{3} q^{37} - 30 \beta_{2} q^{39} + (10 \beta_{3} - 10 \beta_1 + 50) q^{41} + ( - 30 \beta_{2} + 4 \beta_1 - 30) q^{43} + 18 \beta_{3} q^{47} + ( - 20 \beta_{3} - 13 \beta_{2} - 20 \beta_1) q^{49} + ( - 10 \beta_{3} + 10 \beta_1 - 6) q^{51} + (50 \beta_{2} - 12 \beta_1 + 50) q^{53} + ( - 10 \beta_{3} + 30 \beta_{2} - 30) q^{57} + (15 \beta_{3} + 52 \beta_{2} + 15 \beta_1) q^{59} + (10 \beta_{3} - 10 \beta_1 - 78) q^{61} + ( - 15 \beta_{2} - 6 \beta_1 - 15) q^{63} + ( - 12 \beta_{3} - 10 \beta_{2} + 10) q^{67} + (10 \beta_{3} - 18 \beta_{2} + 10 \beta_1) q^{69} + ( - 40 \beta_{3} + 40 \beta_1 - 20) q^{71} + (5 \beta_{2} + 32 \beta_1 + 5) q^{73} + ( - 58 \beta_{3} - 50 \beta_{2} + 50) q^{77} + (10 \beta_{3} + 24 \beta_{2} + 10 \beta_1) q^{79} - 9 q^{81} + (60 \beta_{2} + 34 \beta_1 + 60) q^{83} + (22 \beta_{3} + 15 \beta_{2} - 15) q^{87} + ( - 60 \beta_{3} - 10 \beta_{2} - 60 \beta_1) q^{89} + ( - 50 \beta_{3} + 50 \beta_1 + 60) q^{91} + ( - 30 \beta_{2} + 4 \beta_1 - 30) q^{93} + ( - 16 \beta_{3} + 75 \beta_{2} - 75) q^{97} + ( - 15 \beta_{3} - 12 \beta_{2} - 15 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{7} - 16 q^{11} + 40 q^{17} - 24 q^{21} + 40 q^{23} + 16 q^{31} - 60 q^{33} + 200 q^{41} - 120 q^{43} - 24 q^{51} + 200 q^{53} - 120 q^{57} - 312 q^{61} - 60 q^{63} + 40 q^{67} - 80 q^{71} + 20 q^{73} + 200 q^{77} - 36 q^{81} + 240 q^{83} - 60 q^{87} + 240 q^{91} - 120 q^{93} - 300 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
157.1
−1.22474 + 1.22474i
1.22474 − 1.22474i
−1.22474 − 1.22474i
1.22474 + 1.22474i
0 −1.22474 + 1.22474i 0 0 0 −2.55051 − 2.55051i 0 − 3.00000i 0
157.2 0 1.22474 − 1.22474i 0 0 0 −7.44949 − 7.44949i 0 − 3.00000i 0
193.1 0 −1.22474 − 1.22474i 0 0 0 −2.55051 + 2.55051i 0 3.00000i 0
193.2 0 1.22474 + 1.22474i 0 0 0 −7.44949 + 7.44949i 0 3.00000i 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
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.3.k.a 4
3.b odd 2 1 900.3.l.b 4
4.b odd 2 1 1200.3.bg.o 4
5.b even 2 1 60.3.k.a ✓ 4
5.c odd 4 1 60.3.k.a ✓ 4
5.c odd 4 1 inner 300.3.k.a 4
15.d odd 2 1 180.3.l.b 4
15.e even 4 1 180.3.l.b 4
15.e even 4 1 900.3.l.b 4
20.d odd 2 1 240.3.bg.d 4
20.e even 4 1 240.3.bg.d 4
20.e even 4 1 1200.3.bg.o 4
40.e odd 2 1 960.3.bg.a 4
40.f even 2 1 960.3.bg.b 4
40.i odd 4 1 960.3.bg.b 4
40.k even 4 1 960.3.bg.a 4
60.h even 2 1 720.3.bh.f 4
60.l odd 4 1 720.3.bh.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.3.k.a ✓ 4 5.b even 2 1
60.3.k.a ✓ 4 5.c odd 4 1
180.3.l.b 4 15.d odd 2 1
180.3.l.b 4 15.e even 4 1
240.3.bg.d 4 20.d odd 2 1
240.3.bg.d 4 20.e even 4 1
300.3.k.a 4 1.a even 1 1 trivial
300.3.k.a 4 5.c odd 4 1 inner
720.3.bh.f 4 60.h even 2 1
720.3.bh.f 4 60.l odd 4 1
900.3.l.b 4 3.b odd 2 1
900.3.l.b 4 15.e even 4 1
960.3.bg.a 4 40.e odd 2 1
960.3.bg.a 4 40.k even 4 1
960.3.bg.b 4 40.f even 2 1
960.3.bg.b 4 40.i odd 4 1
1200.3.bg.o 4 4.b odd 2 1
1200.3.bg.o 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 20T_{7}^{3} + 200T_{7}^{2} + 760T_{7} + 1444 \) acting on \(S_{3}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 20 T^{3} + \cdots + 1444 \) Copy content Toggle raw display
$11$ \( (T^{2} + 8 T - 134)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 90000 \) Copy content Toggle raw display
$17$ \( T^{4} - 40 T^{3} + \cdots + 35344 \) Copy content Toggle raw display
$19$ \( T^{4} + 1400 T^{2} + 250000 \) Copy content Toggle raw display
$23$ \( T^{4} - 40 T^{3} + \cdots + 8464 \) Copy content Toggle raw display
$29$ \( T^{4} + 1268 T^{2} + 111556 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 584)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 11664 \) Copy content Toggle raw display
$41$ \( (T^{2} - 100 T + 1900)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 120 T^{3} + \cdots + 3069504 \) Copy content Toggle raw display
$47$ \( T^{4} + 944784 \) Copy content Toggle raw display
$53$ \( T^{4} - 200 T^{3} + \cdots + 20866624 \) Copy content Toggle raw display
$59$ \( T^{4} + 8108 T^{2} + 1833316 \) Copy content Toggle raw display
$61$ \( (T^{2} + 156 T + 5484)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 40 T^{3} + \cdots + 53824 \) Copy content Toggle raw display
$71$ \( (T^{2} + 40 T - 9200)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + \cdots + 9132484 \) Copy content Toggle raw display
$79$ \( T^{4} + 2352T^{2} + 576 \) Copy content Toggle raw display
$83$ \( T^{4} - 240 T^{3} + \cdots + 13927824 \) Copy content Toggle raw display
$89$ \( T^{4} + 43400 T^{2} + 462250000 \) Copy content Toggle raw display
$97$ \( T^{4} + 300 T^{3} + \cdots + 109872324 \) Copy content Toggle raw display
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