Properties

Label 1200.4.a.bs
Level $1200$
Weight $4$
Character orbit 1200.a
Self dual yes
Analytic conductor $70.802$
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] = [1200,4,Mod(1,1200)]
 
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([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("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.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: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{181}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 600)
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 \(\beta = 2\sqrt{181}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 3 q^{3} + (\beta + 3) q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 3 q^{3} + (\beta + 3) q^{7} + 9 q^{9} + ( - \beta - 4) q^{11} + (2 \beta - 7) q^{13} + \beta q^{17} + ( - 3 \beta - 43) q^{19} + (3 \beta + 9) q^{21} + (3 \beta + 64) q^{23} + 27 q^{27} + ( - 5 \beta + 172) q^{29} + (7 \beta + 79) q^{31} + ( - 3 \beta - 12) q^{33} + (6 \beta - 18) q^{37} + (6 \beta - 21) q^{39} + ( - 7 \beta - 122) q^{41} + ( - 7 \beta + 195) q^{43} + (4 \beta + 378) q^{47} + (6 \beta + 390) q^{49} + 3 \beta q^{51} + ( - 7 \beta - 134) q^{53} + ( - 9 \beta - 129) q^{57} + ( - 8 \beta - 2) q^{59} + (8 \beta - 517) q^{61} + (9 \beta + 27) q^{63} + ( - 9 \beta + 811) q^{67} + (9 \beta + 192) q^{69} + ( - 33 \beta + 138) q^{71} + (10 \beta - 322) q^{73} + ( - 7 \beta - 736) q^{77} + (20 \beta + 472) q^{79} + 81 q^{81} + ( - 32 \beta - 242) q^{83} + ( - 15 \beta + 516) q^{87} + (4 \beta + 1112) q^{89} + ( - \beta + 1427) q^{91} + (21 \beta + 237) q^{93} + ( - 32 \beta - 255) q^{97} + ( - 9 \beta - 36) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 6 q^{3} + 6 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 6 q^{3} + 6 q^{7} + 18 q^{9} - 8 q^{11} - 14 q^{13} - 86 q^{19} + 18 q^{21} + 128 q^{23} + 54 q^{27} + 344 q^{29} + 158 q^{31} - 24 q^{33} - 36 q^{37} - 42 q^{39} - 244 q^{41} + 390 q^{43} + 756 q^{47} + 780 q^{49} - 268 q^{53} - 258 q^{57} - 4 q^{59} - 1034 q^{61} + 54 q^{63} + 1622 q^{67} + 384 q^{69} + 276 q^{71} - 644 q^{73} - 1472 q^{77} + 944 q^{79} + 162 q^{81} - 484 q^{83} + 1032 q^{87} + 2224 q^{89} + 2854 q^{91} + 474 q^{93} - 510 q^{97} - 72 q^{99}+O(q^{100}) \) 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
−6.22681
7.22681
0 3.00000 0 0 0 −23.9072 0 9.00000 0
1.2 0 3.00000 0 0 0 29.9072 0 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 1200.4.a.bs 2
4.b odd 2 1 600.4.a.r 2
5.b even 2 1 1200.4.a.bm 2
5.c odd 4 2 1200.4.f.w 4
12.b even 2 1 1800.4.a.bj 2
20.d odd 2 1 600.4.a.w yes 2
20.e even 4 2 600.4.f.k 4
60.h even 2 1 1800.4.a.bq 2
60.l odd 4 2 1800.4.f.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.4.a.r 2 4.b odd 2 1
600.4.a.w yes 2 20.d odd 2 1
600.4.f.k 4 20.e even 4 2
1200.4.a.bm 2 5.b even 2 1
1200.4.a.bs 2 1.a even 1 1 trivial
1200.4.f.w 4 5.c odd 4 2
1800.4.a.bj 2 12.b even 2 1
1800.4.a.bq 2 60.h even 2 1
1800.4.f.y 4 60.l odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1200))\):

\( T_{7}^{2} - 6T_{7} - 715 \) Copy content Toggle raw display
\( T_{11}^{2} + 8T_{11} - 708 \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 6T - 715 \) Copy content Toggle raw display
$11$ \( T^{2} + 8T - 708 \) Copy content Toggle raw display
$13$ \( T^{2} + 14T - 2847 \) Copy content Toggle raw display
$17$ \( T^{2} - 724 \) Copy content Toggle raw display
$19$ \( T^{2} + 86T - 4667 \) Copy content Toggle raw display
$23$ \( T^{2} - 128T - 2420 \) Copy content Toggle raw display
$29$ \( T^{2} - 344T + 11484 \) Copy content Toggle raw display
$31$ \( T^{2} - 158T - 29235 \) Copy content Toggle raw display
$37$ \( T^{2} + 36T - 25740 \) Copy content Toggle raw display
$41$ \( T^{2} + 244T - 20592 \) Copy content Toggle raw display
$43$ \( T^{2} - 390T + 2549 \) Copy content Toggle raw display
$47$ \( T^{2} - 756T + 131300 \) Copy content Toggle raw display
$53$ \( T^{2} + 268T - 17520 \) Copy content Toggle raw display
$59$ \( T^{2} + 4T - 46332 \) Copy content Toggle raw display
$61$ \( T^{2} + 1034 T + 220953 \) Copy content Toggle raw display
$67$ \( T^{2} - 1622 T + 599077 \) Copy content Toggle raw display
$71$ \( T^{2} - 276T - 769392 \) Copy content Toggle raw display
$73$ \( T^{2} + 644T + 31284 \) Copy content Toggle raw display
$79$ \( T^{2} - 944T - 66816 \) Copy content Toggle raw display
$83$ \( T^{2} + 484T - 682812 \) Copy content Toggle raw display
$89$ \( T^{2} - 2224 T + 1224960 \) Copy content Toggle raw display
$97$ \( T^{2} + 510T - 676351 \) Copy content Toggle raw display
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