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Properties

Label 300.4.d.c
Level $300$
Weight $4$
Character orbit 300.d
Analytic conductor $17.701$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [300,4,Mod(49,300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("300.49"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 300.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,0,0,-18,0,12,0,0,0,0,0,0,0,-130] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(17.7005730017\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 i q^{3} + 13 i q^{7} - 9 q^{9} + 6 q^{11} + 5 i q^{13} + 78 i q^{17} - 65 q^{19} + 39 q^{21} + 138 i q^{23} + 27 i q^{27} - 66 q^{29} + 299 q^{31} - 18 i q^{33} + 214 i q^{37} + 15 q^{39} + 360 q^{41} + \cdots - 54 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9} + 12 q^{11} - 130 q^{19} + 78 q^{21} - 132 q^{29} + 598 q^{31} + 30 q^{39} + 720 q^{41} + 348 q^{49} + 468 q^{51} - 1572 q^{59} + 934 q^{61} + 828 q^{69} - 720 q^{71} - 544 q^{79} + 162 q^{81}+ \cdots - 108 q^{99}+O(q^{100}) \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
− 1.00000i
0 − 3.00000i 0 0 0 13.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 − 13.0000i 0 −9.00000 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.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 300.4.d.c 2
3.b odd 2 1 900.4.d.e 2
4.b odd 2 1 1200.4.f.k 2
5.b even 2 1 inner 300.4.d.c 2
5.c odd 4 1 300.4.a.a ✓ 1
5.c odd 4 1 300.4.a.h yes 1
15.d odd 2 1 900.4.d.e 2
15.e even 4 1 900.4.a.f 1
15.e even 4 1 900.4.a.l 1
20.d odd 2 1 1200.4.f.k 2
20.e even 4 1 1200.4.a.d 1
20.e even 4 1 1200.4.a.bi 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
300.4.a.a ✓ 1 5.c odd 4 1
300.4.a.h yes 1 5.c odd 4 1
300.4.d.c 2 1.a even 1 1 trivial
300.4.d.c 2 5.b even 2 1 inner
900.4.a.f 1 15.e even 4 1
900.4.a.l 1 15.e even 4 1
900.4.d.e 2 3.b odd 2 1
900.4.d.e 2 15.d odd 2 1
1200.4.a.d 1 20.e even 4 1
1200.4.a.bi 1 20.e even 4 1
1200.4.f.k 2 4.b odd 2 1
1200.4.f.k 2 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 169 \) acting on \(S_{4}^{\mathrm{new}}(300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 169 \) Copy content Toggle raw display
$11$ \( (T - 6)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 25 \) Copy content Toggle raw display
$17$ \( T^{2} + 6084 \) Copy content Toggle raw display
$19$ \( (T + 65)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 19044 \) Copy content Toggle raw display
$29$ \( (T + 66)^{2} \) Copy content Toggle raw display
$31$ \( (T - 299)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 45796 \) Copy content Toggle raw display
$41$ \( (T - 360)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 41209 \) Copy content Toggle raw display
$47$ \( T^{2} + 6084 \) Copy content Toggle raw display
$53$ \( T^{2} + 404496 \) Copy content Toggle raw display
$59$ \( (T + 786)^{2} \) Copy content Toggle raw display
$61$ \( (T - 467)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 47089 \) Copy content Toggle raw display
$71$ \( (T + 360)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 81796 \) Copy content Toggle raw display
$79$ \( (T + 272)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 248004 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 261121 \) Copy content Toggle raw display
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