Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,4,Mod(49,300)]
 
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, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//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);
 
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

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: \(17.7005730017\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
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: 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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 i q^{3} + 32 i q^{7} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 i q^{3} + 32 i q^{7} - 9 q^{9} + 36 q^{11} + 10 i q^{13} - 78 i q^{17} - 140 q^{19} - 96 q^{21} + 192 i q^{23} - 27 i q^{27} - 6 q^{29} - 16 q^{31} + 108 i q^{33} - 34 i q^{37} - 30 q^{39} - 390 q^{41} + 52 i q^{43} + 408 i q^{47} - 681 q^{49} + 234 q^{51} + 114 i q^{53} - 420 i q^{57} - 516 q^{59} - 58 q^{61} - 288 i q^{63} - 892 i q^{67} - 576 q^{69} - 120 q^{71} + 646 i q^{73} + 1152 i q^{77} + 1168 q^{79} + 81 q^{81} + 732 i q^{83} - 18 i q^{87} + 1590 q^{89} - 320 q^{91} - 48 i q^{93} + 194 i q^{97} - 324 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 18 q^{9} + 72 q^{11} - 280 q^{19} - 192 q^{21} - 12 q^{29} - 32 q^{31} - 60 q^{39} - 780 q^{41} - 1362 q^{49} + 468 q^{51} - 1032 q^{59} - 116 q^{61} - 1152 q^{69} - 240 q^{71} + 2336 q^{79} + 162 q^{81} + 3180 q^{89} - 640 q^{91} - 648 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.00000i
1.00000i
0 3.00000i 0 0 0 32.0000i 0 −9.00000 0
49.2 0 3.00000i 0 0 0 32.0000i 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.d 2
3.b odd 2 1 900.4.d.b 2
4.b odd 2 1 1200.4.f.e 2
5.b even 2 1 inner 300.4.d.d 2
5.c odd 4 1 60.4.a.b 1
5.c odd 4 1 300.4.a.e 1
15.d odd 2 1 900.4.d.b 2
15.e even 4 1 180.4.a.c 1
15.e even 4 1 900.4.a.b 1
20.d odd 2 1 1200.4.f.e 2
20.e even 4 1 240.4.a.j 1
20.e even 4 1 1200.4.a.s 1
40.i odd 4 1 960.4.a.bb 1
40.k even 4 1 960.4.a.a 1
45.k odd 12 2 1620.4.i.a 2
45.l even 12 2 1620.4.i.g 2
60.l odd 4 1 720.4.a.c 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.4.a.b 1 5.c odd 4 1
180.4.a.c 1 15.e even 4 1
240.4.a.j 1 20.e even 4 1
300.4.a.e 1 5.c odd 4 1
300.4.d.d 2 1.a even 1 1 trivial
300.4.d.d 2 5.b even 2 1 inner
720.4.a.c 1 60.l odd 4 1
900.4.a.b 1 15.e even 4 1
900.4.d.b 2 3.b odd 2 1
900.4.d.b 2 15.d odd 2 1
960.4.a.a 1 40.k even 4 1
960.4.a.bb 1 40.i odd 4 1
1200.4.a.s 1 20.e even 4 1
1200.4.f.e 2 4.b odd 2 1
1200.4.f.e 2 20.d odd 2 1
1620.4.i.a 2 45.k odd 12 2
1620.4.i.g 2 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{2} + 1024 \) 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
$7$ \( T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( (T - 36)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 100 \) Copy content Toggle raw display
$17$ \( T^{2} + 6084 \) Copy content Toggle raw display
$19$ \( (T + 140)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36864 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T + 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 1156 \) Copy content Toggle raw display
$41$ \( (T + 390)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 2704 \) Copy content Toggle raw display
$47$ \( T^{2} + 166464 \) Copy content Toggle raw display
$53$ \( T^{2} + 12996 \) Copy content Toggle raw display
$59$ \( (T + 516)^{2} \) Copy content Toggle raw display
$61$ \( (T + 58)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 795664 \) Copy content Toggle raw display
$71$ \( (T + 120)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 417316 \) Copy content Toggle raw display
$79$ \( (T - 1168)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 535824 \) Copy content Toggle raw display
$89$ \( (T - 1590)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 37636 \) Copy content Toggle raw display
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