Properties

Label 400.6.a.w
Level $400$
Weight $6$
Character orbit 400.a
Self dual yes
Analytic conductor $64.154$
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] = [400,6,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{241}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 60 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 25)
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 \(\beta = \sqrt{241}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 10) q^{3} + ( - 2 \beta + 100) q^{7} + ( - 20 \beta + 98) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 10) q^{3} + ( - 2 \beta + 100) q^{7} + ( - 20 \beta + 98) q^{9} + (25 \beta + 98) q^{11} + (16 \beta + 180) q^{13} + ( - 68 \beta - 745) q^{17} + ( - 35 \beta + 1590) q^{19} + ( - 120 \beta + 1482) q^{21} + ( - 6 \beta + 780) q^{23} + ( - 55 \beta + 3370) q^{27} + (40 \beta - 1960) q^{29} + ( - 550 \beta + 548) q^{31} + (152 \beta - 5045) q^{33} + (192 \beta - 1010) q^{37} + ( - 20 \beta - 2056) q^{39} + ( - 200 \beta + 13877) q^{41} + (1064 \beta - 1500) q^{43} + ( - 772 \beta + 12880) q^{47} + ( - 400 \beta - 5843) q^{49} + (65 \beta + 8938) q^{51} + (376 \beta + 13490) q^{53} + ( - 1940 \beta + 24335) q^{57} + ( - 980 \beta - 5980) q^{59} + ( - 1000 \beta - 12198) q^{61} + ( - 2196 \beta + 19440) q^{63} + (793 \beta - 20030) q^{67} + ( - 840 \beta + 9246) q^{69} + (500 \beta + 43648) q^{71} + (556 \beta + 35145) q^{73} + (2304 \beta - 2250) q^{77} + (2510 \beta - 32740) q^{79} + (940 \beta + 23141) q^{81} + (429 \beta + 46290) q^{83} + (2360 \beta - 29240) q^{87} + (5220 \beta - 36405) q^{89} + (1240 \beta + 10288) q^{91} + ( - 6048 \beta + 138030) q^{93} + (5472 \beta + 63070) q^{97} + (490 \beta - 110896) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 20 q^{3} + 200 q^{7} + 196 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 20 q^{3} + 200 q^{7} + 196 q^{9} + 196 q^{11} + 360 q^{13} - 1490 q^{17} + 3180 q^{19} + 2964 q^{21} + 1560 q^{23} + 6740 q^{27} - 3920 q^{29} + 1096 q^{31} - 10090 q^{33} - 2020 q^{37} - 4112 q^{39} + 27754 q^{41} - 3000 q^{43} + 25760 q^{47} - 11686 q^{49} + 17876 q^{51} + 26980 q^{53} + 48670 q^{57} - 11960 q^{59} - 24396 q^{61} + 38880 q^{63} - 40060 q^{67} + 18492 q^{69} + 87296 q^{71} + 70290 q^{73} - 4500 q^{77} - 65480 q^{79} + 46282 q^{81} + 92580 q^{83} - 58480 q^{87} - 72810 q^{89} + 20576 q^{91} + 276060 q^{93} + 126140 q^{97} - 221792 q^{99}+O(q^{100}) \) 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
8.26209
−7.26209
0 −5.52417 0 0 0 68.9517 0 −212.483 0
1.2 0 25.5242 0 0 0 131.048 0 408.483 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 400.6.a.w 2
4.b odd 2 1 25.6.a.b 2
5.b even 2 1 400.6.a.o 2
5.c odd 4 2 400.6.c.n 4
12.b even 2 1 225.6.a.s 2
20.d odd 2 1 25.6.a.d yes 2
20.e even 4 2 25.6.b.b 4
60.h even 2 1 225.6.a.l 2
60.l odd 4 2 225.6.b.i 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
25.6.a.b 2 4.b odd 2 1
25.6.a.d yes 2 20.d odd 2 1
25.6.b.b 4 20.e even 4 2
225.6.a.l 2 60.h even 2 1
225.6.a.s 2 12.b even 2 1
225.6.b.i 4 60.l odd 4 2
400.6.a.o 2 5.b even 2 1
400.6.a.w 2 1.a even 1 1 trivial
400.6.c.n 4 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 20T_{3} - 141 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(400))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 20T - 141 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 200T + 9036 \) Copy content Toggle raw display
$11$ \( T^{2} - 196T - 141021 \) Copy content Toggle raw display
$13$ \( T^{2} - 360T - 29296 \) Copy content Toggle raw display
$17$ \( T^{2} + 1490 T - 559359 \) Copy content Toggle raw display
$19$ \( T^{2} - 3180 T + 2232875 \) Copy content Toggle raw display
$23$ \( T^{2} - 1560 T + 599724 \) Copy content Toggle raw display
$29$ \( T^{2} + 3920 T + 3456000 \) Copy content Toggle raw display
$31$ \( T^{2} - 1096 T - 72602196 \) Copy content Toggle raw display
$37$ \( T^{2} + 2020 T - 7864124 \) Copy content Toggle raw display
$41$ \( T^{2} - 27754 T + 182931129 \) Copy content Toggle raw display
$43$ \( T^{2} + 3000 T - 270585136 \) Copy content Toggle raw display
$47$ \( T^{2} - 25760 T + 22262256 \) Copy content Toggle raw display
$53$ \( T^{2} - 26980 T + 147908484 \) Copy content Toggle raw display
$59$ \( T^{2} + 11960 T - 195696000 \) Copy content Toggle raw display
$61$ \( T^{2} + 24396 T - 92208796 \) Copy content Toggle raw display
$67$ \( T^{2} + 40060 T + 249648291 \) Copy content Toggle raw display
$71$ \( T^{2} - 87296 T + 1844897904 \) Copy content Toggle raw display
$73$ \( T^{2} - 70290 T + 1160669249 \) Copy content Toggle raw display
$79$ \( T^{2} + 65480 T - 446416500 \) Copy content Toggle raw display
$83$ \( T^{2} - 92580 T + 2098410219 \) Copy content Toggle raw display
$89$ \( T^{2} + 72810 T - 5241540375 \) Copy content Toggle raw display
$97$ \( T^{2} - 126140 T - 3238386044 \) Copy content Toggle raw display
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