Properties

Label 1680.4.a.bd
Level $1680$
Weight $4$
Character orbit 1680.a
Self dual yes
Analytic conductor $99.123$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1680,4,Mod(1,1680)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1680, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1680.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1680.a (trivial)

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: yes
Analytic conductor: \(99.1232088096\)
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: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(\beta = 2\sqrt{5}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 q^{3} - 5 q^{5} + 7 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 3 q^{3} - 5 q^{5} + 7 q^{7} + 9 q^{9} + ( - \beta + 46) q^{11} + (19 \beta + 4) q^{13} + 15 q^{15} + ( - 22 \beta - 22) q^{17} + (13 \beta + 54) q^{19} - 21 q^{21} + ( - 10 \beta + 160) q^{23} + 25 q^{25} - 27 q^{27} + ( - 6 \beta - 118) q^{29} + (51 \beta + 30) q^{31} + (3 \beta - 138) q^{33} - 35 q^{35} + ( - 12 \beta + 102) q^{37} + ( - 57 \beta - 12) q^{39} + (40 \beta + 22) q^{41} + (64 \beta - 68) q^{43} - 45 q^{45} + ( - 84 \beta - 200) q^{47} + 49 q^{49} + (66 \beta + 66) q^{51} + ( - 43 \beta + 8) q^{53} + (5 \beta - 230) q^{55} + ( - 39 \beta - 162) q^{57} + ( - 18 \beta + 232) q^{59} + ( - 42 \beta - 342) q^{61} + 63 q^{63} + ( - 95 \beta - 20) q^{65} + (82 \beta - 368) q^{67} + (30 \beta - 480) q^{69} + ( - 69 \beta + 370) q^{71} + (61 \beta + 212) q^{73} - 75 q^{75} + ( - 7 \beta + 322) q^{77} + ( - 242 \beta + 204) q^{79} + 81 q^{81} + ( - 42 \beta - 304) q^{83} + (110 \beta + 110) q^{85} + (18 \beta + 354) q^{87} + (56 \beta - 666) q^{89} + (133 \beta + 28) q^{91} + ( - 153 \beta - 90) q^{93} + ( - 65 \beta - 270) q^{95} + (43 \beta - 1224) q^{97} + ( - 9 \beta + 414) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 10 q^{5} + 14 q^{7} + 18 q^{9} + 92 q^{11} + 8 q^{13} + 30 q^{15} - 44 q^{17} + 108 q^{19} - 42 q^{21} + 320 q^{23} + 50 q^{25} - 54 q^{27} - 236 q^{29} + 60 q^{31} - 276 q^{33} - 70 q^{35} + 204 q^{37} - 24 q^{39} + 44 q^{41} - 136 q^{43} - 90 q^{45} - 400 q^{47} + 98 q^{49} + 132 q^{51} + 16 q^{53} - 460 q^{55} - 324 q^{57} + 464 q^{59} - 684 q^{61} + 126 q^{63} - 40 q^{65} - 736 q^{67} - 960 q^{69} + 740 q^{71} + 424 q^{73} - 150 q^{75} + 644 q^{77} + 408 q^{79} + 162 q^{81} - 608 q^{83} + 220 q^{85} + 708 q^{87} - 1332 q^{89} + 56 q^{91} - 180 q^{93} - 540 q^{95} - 2448 q^{97} + 828 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
1.61803
−0.618034
0 −3.00000 0 −5.00000 0 7.00000 0 9.00000 0
1.2 0 −3.00000 0 −5.00000 0 7.00000 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(1\)
\(5\) \(1\)
\(7\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.4.a.bd 2
4.b odd 2 1 105.4.a.d 2
12.b even 2 1 315.4.a.l 2
20.d odd 2 1 525.4.a.o 2
20.e even 4 2 525.4.d.k 4
28.d even 2 1 735.4.a.m 2
60.h even 2 1 1575.4.a.n 2
84.h odd 2 1 2205.4.a.be 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.a.d 2 4.b odd 2 1
315.4.a.l 2 12.b even 2 1
525.4.a.o 2 20.d odd 2 1
525.4.d.k 4 20.e even 4 2
735.4.a.m 2 28.d even 2 1
1575.4.a.n 2 60.h even 2 1
1680.4.a.bd 2 1.a even 1 1 trivial
2205.4.a.be 2 84.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1680))\):

\( T_{11}^{2} - 92T_{11} + 2096 \) Copy content Toggle raw display
\( T_{13}^{2} - 8T_{13} - 7204 \) Copy content Toggle raw display
\( T_{17}^{2} + 44T_{17} - 9196 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 3)^{2} \) Copy content Toggle raw display
$5$ \( (T + 5)^{2} \) Copy content Toggle raw display
$7$ \( (T - 7)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 92T + 2096 \) Copy content Toggle raw display
$13$ \( T^{2} - 8T - 7204 \) Copy content Toggle raw display
$17$ \( T^{2} + 44T - 9196 \) Copy content Toggle raw display
$19$ \( T^{2} - 108T - 464 \) Copy content Toggle raw display
$23$ \( T^{2} - 320T + 23600 \) Copy content Toggle raw display
$29$ \( T^{2} + 236T + 13204 \) Copy content Toggle raw display
$31$ \( T^{2} - 60T - 51120 \) Copy content Toggle raw display
$37$ \( T^{2} - 204T + 7524 \) Copy content Toggle raw display
$41$ \( T^{2} - 44T - 31516 \) Copy content Toggle raw display
$43$ \( T^{2} + 136T - 77296 \) Copy content Toggle raw display
$47$ \( T^{2} + 400T - 101120 \) Copy content Toggle raw display
$53$ \( T^{2} - 16T - 36916 \) Copy content Toggle raw display
$59$ \( T^{2} - 464T + 47344 \) Copy content Toggle raw display
$61$ \( T^{2} + 684T + 81684 \) Copy content Toggle raw display
$67$ \( T^{2} + 736T + 944 \) Copy content Toggle raw display
$71$ \( T^{2} - 740T + 41680 \) Copy content Toggle raw display
$73$ \( T^{2} - 424T - 29476 \) Copy content Toggle raw display
$79$ \( T^{2} - 408 T - 1129664 \) Copy content Toggle raw display
$83$ \( T^{2} + 608T + 57136 \) Copy content Toggle raw display
$89$ \( T^{2} + 1332 T + 380836 \) Copy content Toggle raw display
$97$ \( T^{2} + 2448 T + 1461196 \) Copy content Toggle raw display
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