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Properties

Label 400.3.b.g
Level $400$
Weight $3$
Character orbit 400.b
Analytic conductor $10.899$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,3,Mod(351,400)]
 
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([1, 0, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.351");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.b (of order \(2\), degree \(1\), 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: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 2x^{2} + x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 80)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{7} + ( - \beta_{3} - 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{7} + ( - \beta_{3} - 9) q^{9} - 2 \beta_1 q^{11} + ( - \beta_{3} - 4) q^{13} - 18 q^{17} - 4 \beta_{2} q^{19} + (\beta_{3} - 6) q^{21} + (3 \beta_{2} + 2 \beta_1) q^{23} + ( - 6 \beta_{2} - 12 \beta_1) q^{27} + (2 \beta_{3} - 18) q^{29} + ( - 2 \beta_{2} + 8 \beta_1) q^{31} + (2 \beta_{3} + 36) q^{33} + ( - \beta_{3} - 20) q^{37} + ( - 6 \beta_{2} - 16 \beta_1) q^{39} + ( - 3 \beta_{3} + 12) q^{41} + (2 \beta_{2} - 5 \beta_1) q^{43} + ( - 9 \beta_{2} + 4 \beta_1) q^{47} + (3 \beta_{3} + 7) q^{49} - 18 \beta_1 q^{51} + (3 \beta_{3} - 12) q^{53} + (4 \beta_{3} - 24) q^{57} - 8 \beta_1 q^{59} + ( - \beta_{3} - 64) q^{61} + ( - 3 \beta_{2} + 6 \beta_1) q^{63} + (2 \beta_{2} - 7 \beta_1) q^{67} + ( - 5 \beta_{3} - 18) q^{69} + (6 \beta_{2} + 16 \beta_1) q^{71} + ( - 4 \beta_{3} + 38) q^{73} + ( - 2 \beta_{3} + 12) q^{77} + 20 \beta_1 q^{79} + (9 \beta_{3} + 99) q^{81} + (6 \beta_{2} + 15 \beta_1) q^{83} + (12 \beta_{2} + 6 \beta_1) q^{87} + ( - 4 \beta_{3} - 6) q^{89} + ( - 8 \beta_{2} + 6 \beta_1) q^{91} + ( - 6 \beta_{3} - 156) q^{93} + ( - 10 \beta_{3} + 26) q^{97} + (12 \beta_{2} + 42 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 36 q^{9} - 16 q^{13} - 72 q^{17} - 24 q^{21} - 72 q^{29} + 144 q^{33} - 80 q^{37} + 48 q^{41} + 28 q^{49} - 48 q^{53} - 96 q^{57} - 256 q^{61} - 72 q^{69} + 152 q^{73} + 48 q^{77} + 396 q^{81} - 24 q^{89} - 624 q^{93} + 104 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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

\(\beta_{1}\)\(=\) \( \nu^{3} + 4\nu + 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -5\nu^{3} + 8\nu^{2} - 12\nu - 1 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -6\nu^{3} - 12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + 6\beta _1 + 6 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} + 3\beta_{2} + 9\beta _1 - 18 ) / 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{3} - 12 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
351.1
0.809017 − 1.40126i
−0.309017 − 0.535233i
−0.309017 + 0.535233i
0.809017 + 1.40126i
0 − 5.60503i 0 0 0 1.32317i 0 −22.4164 0
351.2 0 − 2.14093i 0 0 0 − 9.06914i 0 4.41641 0
351.3 0 2.14093i 0 0 0 9.06914i 0 4.41641 0
351.4 0 5.60503i 0 0 0 − 1.32317i 0 −22.4164 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
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.b.g 4
3.b odd 2 1 3600.3.e.bb 4
4.b odd 2 1 inner 400.3.b.g 4
5.b even 2 1 80.3.b.a ✓ 4
5.c odd 4 2 400.3.h.d 8
8.b even 2 1 1600.3.b.k 4
8.d odd 2 1 1600.3.b.k 4
12.b even 2 1 3600.3.e.bb 4
15.d odd 2 1 720.3.e.c 4
15.e even 4 2 3600.3.j.k 8
20.d odd 2 1 80.3.b.a ✓ 4
20.e even 4 2 400.3.h.d 8
40.e odd 2 1 320.3.b.a 4
40.f even 2 1 320.3.b.a 4
40.i odd 4 2 1600.3.h.p 8
40.k even 4 2 1600.3.h.p 8
60.h even 2 1 720.3.e.c 4
60.l odd 4 2 3600.3.j.k 8
80.k odd 4 2 1280.3.g.f 8
80.q even 4 2 1280.3.g.f 8
120.i odd 2 1 2880.3.e.b 4
120.m even 2 1 2880.3.e.b 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.3.b.a ✓ 4 5.b even 2 1
80.3.b.a ✓ 4 20.d odd 2 1
320.3.b.a 4 40.e odd 2 1
320.3.b.a 4 40.f even 2 1
400.3.b.g 4 1.a even 1 1 trivial
400.3.b.g 4 4.b odd 2 1 inner
400.3.h.d 8 5.c odd 4 2
400.3.h.d 8 20.e even 4 2
720.3.e.c 4 15.d odd 2 1
720.3.e.c 4 60.h even 2 1
1280.3.g.f 8 80.k odd 4 2
1280.3.g.f 8 80.q even 4 2
1600.3.b.k 4 8.b even 2 1
1600.3.b.k 4 8.d odd 2 1
1600.3.h.p 8 40.i odd 4 2
1600.3.h.p 8 40.k even 4 2
2880.3.e.b 4 120.i odd 2 1
2880.3.e.b 4 120.m even 2 1
3600.3.e.bb 4 3.b odd 2 1
3600.3.e.bb 4 12.b even 2 1
3600.3.j.k 8 15.e even 4 2
3600.3.j.k 8 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 \(S_{3}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{4} + 36T_{3}^{2} + 144 \) Copy content Toggle raw display
\( T_{13}^{2} + 8T_{13} - 164 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 36T^{2} + 144 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 84T^{2} + 144 \) Copy content Toggle raw display
$11$ \( T^{4} + 144T^{2} + 2304 \) Copy content Toggle raw display
$13$ \( (T^{2} + 8 T - 164)^{2} \) Copy content Toggle raw display
$17$ \( (T + 18)^{4} \) Copy content Toggle raw display
$19$ \( T^{4} + 1344 T^{2} + 36864 \) Copy content Toggle raw display
$23$ \( T^{4} + 756 T^{2} + 121104 \) Copy content Toggle raw display
$29$ \( (T^{2} + 36 T - 396)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 3024 T^{2} + 2214144 \) Copy content Toggle raw display
$37$ \( (T^{2} + 40 T + 220)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 24 T - 1476)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 1476 T^{2} + 535824 \) Copy content Toggle raw display
$47$ \( T^{4} + 8244 T^{2} + 898704 \) Copy content Toggle raw display
$53$ \( (T^{2} + 24 T - 1476)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 2304 T^{2} + 589824 \) Copy content Toggle raw display
$61$ \( (T^{2} + 128 T + 3916)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 2436 T^{2} + 1468944 \) Copy content Toggle raw display
$71$ \( T^{4} + 9936 T^{2} + 3873024 \) Copy content Toggle raw display
$73$ \( (T^{2} - 76 T - 1436)^{2} \) Copy content Toggle raw display
$79$ \( T^{4} + 14400 T^{2} + 23040000 \) Copy content Toggle raw display
$83$ \( T^{4} + 8964 T^{2} + 4210704 \) Copy content Toggle raw display
$89$ \( (T^{2} + 12 T - 2844)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 52 T - 17324)^{2} \) Copy content Toggle raw display
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