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Properties

Label 1200.3.j.e
Level $1200$
Weight $3$
Character orbit 1200.j
Analytic conductor $32.698$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,3,Mod(799,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("1200.799");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1200.j (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: \(32.6976317232\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{14}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 240)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{6} - 2 \beta_1) q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{6} - 2 \beta_1) q^{7} + 3 q^{9} + ( - \beta_{5} + \beta_{2}) q^{11} + (\beta_{7} - 2 \beta_{3}) q^{13} + ( - \beta_{7} - 6 \beta_{3}) q^{17} + (2 \beta_{5} + 4 \beta_{2}) q^{19} + (\beta_{4} + 6) q^{21} + ( - 2 \beta_{6} - 4 \beta_1) q^{23} - 3 \beta_1 q^{27} + ( - 3 \beta_{4} - 12) q^{29} + ( - 2 \beta_{5} + 4 \beta_{2}) q^{31} + (\beta_{7} - 3 \beta_{3}) q^{33} + (\beta_{7} - 20 \beta_{3}) q^{37} + ( - 3 \beta_{5} + 2 \beta_{2}) q^{39} + 42 q^{41} + (8 \beta_{6} - 4 \beta_1) q^{43} + (4 \beta_{6} + 8 \beta_1) q^{47} + (4 \beta_{4} + 23) q^{49} + (3 \beta_{5} + 6 \beta_{2}) q^{51} + ( - 3 \beta_{7} + 24 \beta_{3}) q^{53} + ( - 2 \beta_{7} - 12 \beta_{3}) q^{57} + (\beta_{5} + 11 \beta_{2}) q^{59} + ( - 2 \beta_{4} + 86) q^{61} + ( - 3 \beta_{6} - 6 \beta_1) q^{63} + ( - 6 \beta_{6} + 16 \beta_1) q^{67} + (2 \beta_{4} + 12) q^{69} + (14 \beta_{5} - 2 \beta_{2}) q^{71} + ( - 6 \beta_{7} - 11 \beta_{3}) q^{73} + (4 \beta_{7} - 36 \beta_{3}) q^{77} - 2 \beta_{5} q^{79} + 9 q^{81} + ( - 6 \beta_{6} + 48 \beta_1) q^{83} + (9 \beta_{6} + 12 \beta_1) q^{87} + (2 \beta_{4} + 66) q^{89} + ( - 10 \beta_{5} + 34 \beta_{2}) q^{91} + (2 \beta_{7} - 12 \beta_{3}) q^{93} + (4 \beta_{7} - 43 \beta_{3}) q^{97} + ( - 3 \beta_{5} + 3 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 24 q^{9} + 48 q^{21} - 96 q^{29} + 336 q^{41} + 184 q^{49} + 688 q^{61} + 96 q^{69} + 72 q^{81} + 528 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{7} - 4\nu^{5} + 10\nu^{3} - 7\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -3\nu^{6} + 8\nu^{4} - 24\nu^{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 3\nu^{7} - 8\nu^{5} + 20\nu^{3} - \nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{6} - 27 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -4\nu^{6} + 12\nu^{4} - 28\nu^{2} + 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 16\nu^{5} - 44\nu^{3} + 31\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 9\nu^{7} - 24\nu^{5} + 66\nu^{3} - 3\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 3\beta_{6} - 3\beta_{3} + 6\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + \beta_{4} - 9\beta_{2} + 18 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 6\beta_{3} ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{5} - \beta_{4} - 7\beta_{2} - 14 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 5\beta_{7} - 15\beta_{6} - 33\beta_{3} - 66\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -2\beta_{4} - 27 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -13\beta_{7} - 39\beta_{6} + 87\beta_{3} - 174\beta_1 ) / 24 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1200\mathbb{Z}\right)^\times\).

\(n\) \(401\) \(577\) \(751\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
799.1
1.40126 + 0.809017i
1.40126 − 0.809017i
−0.535233 + 0.309017i
−0.535233 − 0.309017i
0.535233 − 0.309017i
0.535233 + 0.309017i
−1.40126 − 0.809017i
−1.40126 + 0.809017i
0 −1.73205 0 0 0 −11.2101 0 3.00000 0
799.2 0 −1.73205 0 0 0 −11.2101 0 3.00000 0
799.3 0 −1.73205 0 0 0 4.28187 0 3.00000 0
799.4 0 −1.73205 0 0 0 4.28187 0 3.00000 0
799.5 0 1.73205 0 0 0 −4.28187 0 3.00000 0
799.6 0 1.73205 0 0 0 −4.28187 0 3.00000 0
799.7 0 1.73205 0 0 0 11.2101 0 3.00000 0
799.8 0 1.73205 0 0 0 11.2101 0 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 799.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1200.3.j.e 8
3.b odd 2 1 3600.3.j.p 8
4.b odd 2 1 inner 1200.3.j.e 8
5.b even 2 1 inner 1200.3.j.e 8
5.c odd 4 1 240.3.e.b ✓ 4
5.c odd 4 1 1200.3.e.j 4
12.b even 2 1 3600.3.j.p 8
15.d odd 2 1 3600.3.j.p 8
15.e even 4 1 720.3.e.e 4
15.e even 4 1 3600.3.e.z 4
20.d odd 2 1 inner 1200.3.j.e 8
20.e even 4 1 240.3.e.b ✓ 4
20.e even 4 1 1200.3.e.j 4
40.i odd 4 1 960.3.e.a 4
40.k even 4 1 960.3.e.a 4
60.h even 2 1 3600.3.j.p 8
60.l odd 4 1 720.3.e.e 4
60.l odd 4 1 3600.3.e.z 4
120.q odd 4 1 2880.3.e.d 4
120.w even 4 1 2880.3.e.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
240.3.e.b ✓ 4 5.c odd 4 1
240.3.e.b ✓ 4 20.e even 4 1
720.3.e.e 4 15.e even 4 1
720.3.e.e 4 60.l odd 4 1
960.3.e.a 4 40.i odd 4 1
960.3.e.a 4 40.k even 4 1
1200.3.e.j 4 5.c odd 4 1
1200.3.e.j 4 20.e even 4 1
1200.3.j.e 8 1.a even 1 1 trivial
1200.3.j.e 8 4.b odd 2 1 inner
1200.3.j.e 8 5.b even 2 1 inner
1200.3.j.e 8 20.d odd 2 1 inner
2880.3.e.d 4 120.q odd 4 1
2880.3.e.d 4 120.w even 4 1
3600.3.e.z 4 15.e even 4 1
3600.3.e.z 4 60.l odd 4 1
3600.3.j.p 8 3.b odd 2 1
3600.3.j.p 8 12.b even 2 1
3600.3.j.p 8 15.d odd 2 1
3600.3.j.p 8 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1200, [\chi])\):

\( T_{7}^{4} - 144T_{7}^{2} + 2304 \) Copy content Toggle raw display
\( T_{13}^{4} + 392T_{13}^{2} + 26896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} - 144 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 144 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 392 T^{2} + 26896)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 648 T^{2} + 1296)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 864 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 576 T^{2} + 36864)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 24 T - 1476)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 864 T^{2} + 2304)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 3560 T^{2} + 2016400)^{2} \) Copy content Toggle raw display
$41$ \( (T - 42)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} - 7776 T^{2} + 14379264)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 2304 T^{2} + 589824)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 7848 T^{2} + 467856)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 3024 T^{2} + 1937664)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 172 T + 6676)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 5856 T^{2} + 1937664)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 23616 T^{2} + 137170944)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 13928 T^{2} + 35952016)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 240)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 18144 T^{2} + 22581504)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 132 T + 3636)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 20552 T^{2} + 20394256)^{2} \) Copy content Toggle raw display
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