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Properties

Label 400.3.k.c
Level $400$
Weight $3$
Character orbit 400.k
Analytic conductor $10.899$
Analytic rank $0$
Dimension $6$
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(99,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 3, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 400.k (of order \(4\), degree \(2\), 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: \(10.8992105744\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(i)\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} + (\beta_{5} + \beta_{3} - \beta_1 + 1) q^{4} + (\beta_{4} + 4 \beta_{3} - \beta_{2} + \cdots - 2) q^{6}+ \cdots + (2 \beta_{4} + \beta_{3} - 4 \beta_{2} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{4} - \beta_{3} + \beta_1) q^{3} + (\beta_{5} + \beta_{3} - \beta_1 + 1) q^{4} + (\beta_{4} + 4 \beta_{3} - \beta_{2} + \cdots - 2) q^{6}+ \cdots + ( - 13 \beta_{5} - 4 \beta_{4} + \cdots + 45) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 8 q^{6} + 28 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} - 2 q^{3} + 8 q^{4} - 8 q^{6} + 28 q^{8} - 18 q^{11} + 28 q^{12} + 2 q^{13} - 12 q^{14} - 40 q^{16} + 38 q^{18} - 30 q^{19} - 20 q^{21} + 72 q^{22} - 48 q^{24} + 96 q^{26} + 64 q^{27} - 32 q^{28} + 18 q^{29} + 88 q^{32} + 76 q^{34} - 52 q^{36} - 46 q^{37} - 52 q^{38} + 196 q^{39} + 80 q^{42} + 114 q^{43} - 20 q^{44} + 28 q^{46} + 192 q^{47} + 56 q^{48} + 46 q^{49} + 156 q^{51} - 100 q^{52} + 78 q^{53} - 32 q^{54} - 168 q^{56} + 36 q^{57} + 28 q^{58} - 206 q^{59} + 30 q^{61} + 80 q^{62} + 284 q^{63} - 64 q^{64} + 196 q^{66} + 226 q^{67} - 264 q^{68} + 116 q^{69} - 260 q^{71} + 300 q^{72} + 48 q^{73} + 92 q^{74} - 188 q^{76} - 212 q^{77} - 292 q^{78} + 86 q^{81} - 112 q^{82} + 318 q^{83} - 232 q^{84} + 268 q^{86} - 176 q^{88} + 188 q^{91} - 224 q^{92} + 32 q^{93} - 48 q^{94} - 80 q^{96} - 362 q^{98} + 226 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 3\nu^{3} + 6\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} - 3\nu^{3} + 2\nu + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{5} + 4\nu^{4} - 9\nu^{3} + 12\nu^{2} - 10\nu + 20 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{5} + 2\nu^{4} - 5\nu^{3} + 8\nu^{2} - 4\nu + 14 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{3} - \beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} + \beta_{3} - 2\beta_{2} + 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} + 6\beta_{3} + \beta_{2} + \beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3\beta_{5} + 3\beta_{4} - 3\beta_{3} + 2\beta _1 + 4 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(-\beta_{3}\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
0.264658 − 1.38923i
1.40680 + 0.144584i
−0.671462 + 1.24464i
0.264658 + 1.38923i
1.40680 − 0.144584i
−0.671462 − 1.24464i
−1.65389 − 1.12457i −3.24914 − 3.24914i 1.47068 + 3.71982i 0 1.71982 + 9.02760i − 4.61555i 1.75086 − 7.80605i 12.1138i 0
99.2 −1.26222 + 1.55139i 2.10278 + 2.10278i −0.813607 − 3.91638i 0 −5.91638 + 0.608056i − 3.04888i 7.10278 + 3.68111i − 0.156674i 0
99.3 1.91611 + 0.573183i 0.146365 + 0.146365i 3.34292 + 2.19656i 0 0.196558 + 0.364346i 9.66442i 5.14637 + 6.12494i − 8.95715i 0
299.1 −1.65389 + 1.12457i −3.24914 + 3.24914i 1.47068 − 3.71982i 0 1.71982 − 9.02760i 4.61555i 1.75086 + 7.80605i − 12.1138i 0
299.2 −1.26222 − 1.55139i 2.10278 − 2.10278i −0.813607 + 3.91638i 0 −5.91638 − 0.608056i 3.04888i 7.10278 − 3.68111i 0.156674i 0
299.3 1.91611 − 0.573183i 0.146365 − 0.146365i 3.34292 − 2.19656i 0 0.196558 − 0.364346i − 9.66442i 5.14637 − 6.12494i 8.95715i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.k odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.3.k.c 6
5.b even 2 1 400.3.k.d 6
5.c odd 4 1 16.3.f.a ✓ 6
5.c odd 4 1 400.3.r.c 6
15.e even 4 1 144.3.m.a 6
16.f odd 4 1 400.3.k.d 6
20.e even 4 1 64.3.f.a 6
40.i odd 4 1 128.3.f.b 6
40.k even 4 1 128.3.f.a 6
60.l odd 4 1 576.3.m.a 6
80.i odd 4 1 64.3.f.a 6
80.j even 4 1 128.3.f.b 6
80.j even 4 1 400.3.r.c 6
80.k odd 4 1 inner 400.3.k.c 6
80.s even 4 1 16.3.f.a ✓ 6
80.t odd 4 1 128.3.f.a 6
120.q odd 4 1 1152.3.m.b 6
120.w even 4 1 1152.3.m.a 6
160.u even 8 2 1024.3.d.k 12
160.v odd 8 2 1024.3.c.j 12
160.ba even 8 2 1024.3.c.j 12
160.bb odd 8 2 1024.3.d.k 12
240.z odd 4 1 144.3.m.a 6
240.bb even 4 1 576.3.m.a 6
240.bd odd 4 1 1152.3.m.a 6
240.bf even 4 1 1152.3.m.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
16.3.f.a ✓ 6 5.c odd 4 1
16.3.f.a ✓ 6 80.s even 4 1
64.3.f.a 6 20.e even 4 1
64.3.f.a 6 80.i odd 4 1
128.3.f.a 6 40.k even 4 1
128.3.f.a 6 80.t odd 4 1
128.3.f.b 6 40.i odd 4 1
128.3.f.b 6 80.j even 4 1
144.3.m.a 6 15.e even 4 1
144.3.m.a 6 240.z odd 4 1
400.3.k.c 6 1.a even 1 1 trivial
400.3.k.c 6 80.k odd 4 1 inner
400.3.k.d 6 5.b even 2 1
400.3.k.d 6 16.f odd 4 1
400.3.r.c 6 5.c odd 4 1
400.3.r.c 6 80.j even 4 1
576.3.m.a 6 60.l odd 4 1
576.3.m.a 6 240.bb even 4 1
1024.3.c.j 12 160.v odd 8 2
1024.3.c.j 12 160.ba even 8 2
1024.3.d.k 12 160.u even 8 2
1024.3.d.k 12 160.bb odd 8 2
1152.3.m.a 6 120.w even 4 1
1152.3.m.a 6 240.bd odd 4 1
1152.3.m.b 6 120.q odd 4 1
1152.3.m.b 6 240.bf even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 2T_{3}^{5} + 2T_{3}^{4} - 32T_{3}^{3} + 196T_{3}^{2} - 56T_{3} + 8 \) acting on \(S_{3}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 2 T^{5} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{6} + 2 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 124 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$11$ \( T^{6} + 18 T^{5} + \cdots + 587528 \) Copy content Toggle raw display
$13$ \( T^{6} - 2 T^{5} + \cdots + 1286408 \) Copy content Toggle raw display
$17$ \( T^{6} + 524 T^{4} + \cdots + 2383936 \) Copy content Toggle raw display
$19$ \( T^{6} + 30 T^{5} + \cdots + 13448 \) Copy content Toggle raw display
$23$ \( T^{6} + 572 T^{4} + \cdots + 937024 \) Copy content Toggle raw display
$29$ \( T^{6} - 18 T^{5} + \cdots + 19046792 \) Copy content Toggle raw display
$31$ \( T^{6} + 1920 T^{4} + \cdots + 16777216 \) Copy content Toggle raw display
$37$ \( T^{6} + 46 T^{5} + \cdots + 42632 \) Copy content Toggle raw display
$41$ \( T^{6} + 4992 T^{4} + \cdots + 67108864 \) Copy content Toggle raw display
$43$ \( T^{6} - 114 T^{5} + \cdots + 42632 \) Copy content Toggle raw display
$47$ \( (T^{3} - 96 T^{2} + \cdots + 77824)^{2} \) Copy content Toggle raw display
$53$ \( T^{6} - 78 T^{5} + \cdots + 783752 \) Copy content Toggle raw display
$59$ \( T^{6} + \cdots + 8410007432 \) Copy content Toggle raw display
$61$ \( T^{6} - 30 T^{5} + \cdots + 151449608 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 87233303432 \) Copy content Toggle raw display
$71$ \( (T^{3} + 130 T^{2} + \cdots - 391864)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} - 24 T^{2} + \cdots - 85504)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} + \cdots + 1550483193856 \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 105636303368 \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 25681985536 \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 270325125184 \) Copy content Toggle raw display
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