↑

Properties

Label 400.2.j.c
Level $400$
Weight $2$
Character orbit 400.j
Analytic conductor $3.194$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,2,Mod(43,400)]
 
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, 1, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.43");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.j (of order \(4\), degree \(2\), 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} + 6x^{12} - 12x^{10} + 36x^{8} - 48x^{6} + 96x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} - \beta_{8} q^{3} + \beta_{15} q^{4} + ( - \beta_{11} - 1) q^{6} + (\beta_{14} + \beta_{6} - \beta_{3}) q^{7} + (\beta_{10} - \beta_{6} + \beta_{5}) q^{8} + ( - \beta_{11} + \beta_{2} + \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{12} q^{2} - \beta_{8} q^{3} + \beta_{15} q^{4} + ( - \beta_{11} - 1) q^{6} + (\beta_{14} + \beta_{6} - \beta_{3}) q^{7} + (\beta_{10} - \beta_{6} + \beta_{5}) q^{8} + ( - \beta_{11} + \beta_{2} + \beta_1 - 1) q^{9} + ( - \beta_{15} - \beta_{9} + \beta_{7} + \cdots + 1) q^{11}+ \cdots + (\beta_{15} - \beta_{9} - \beta_{7} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 12 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 12 q^{6} - 16 q^{9} + 8 q^{11} - 20 q^{14} - 16 q^{16} - 8 q^{19} - 24 q^{24} + 16 q^{26} + 16 q^{29} + 48 q^{34} - 4 q^{36} - 40 q^{44} + 36 q^{46} - 48 q^{51} + 32 q^{54} + 64 q^{56} - 8 q^{59} - 16 q^{61} + 16 q^{64} + 32 q^{66} - 16 q^{69} - 32 q^{71} - 72 q^{74} - 32 q^{76} + 80 q^{79} + 16 q^{81} + 136 q^{84} - 60 q^{86} - 64 q^{91} - 28 q^{94} - 56 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} + 6x^{12} - 12x^{10} + 36x^{8} - 48x^{6} + 96x^{4} - 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} - 18\nu^{12} - 42\nu^{10} - 12\nu^{8} + 228\nu^{6} + 336\nu^{4} - 1248\nu^{2} - 320 ) / 1344 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{14} - 3\nu^{12} - 30\nu^{8} + 24\nu^{6} - 84\nu^{4} + 240\nu^{2} - 352 ) / 168 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{15} - 2\nu^{13} - 42\nu^{11} + 36\nu^{9} - 124\nu^{7} + 336\nu^{5} - 64\nu^{3} + 512\nu ) / 896 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{14} + 2\nu^{12} - 14\nu^{10} + 76\nu^{8} + 12\nu^{6} + 336\nu^{4} - 160\nu^{2} + 832 ) / 448 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{15} - 4\nu^{13} - 12\nu^{9} + 32\nu^{7} - 72\nu^{3} - 96\nu ) / 224 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{15} - 18\nu^{13} + 42\nu^{11} - 12\nu^{9} + 60\nu^{7} + 96\nu^{3} + 128\nu ) / 2688 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -11\nu^{14} + 30\nu^{12} - 42\nu^{10} + 132\nu^{8} - 156\nu^{6} + 336\nu^{4} - 384\nu^{2} + 832 ) / 1344 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{15} - 6\nu^{11} + 24\nu^{9} - 36\nu^{7} + 24\nu^{5} - 144\nu^{3} + 128\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -13\nu^{14} - 18\nu^{12} + 42\nu^{10} - 12\nu^{8} + 60\nu^{6} + 96\nu^{2} + 128 ) / 1344 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\nu^{15} + 30\nu^{13} + 126\nu^{11} + 132\nu^{9} + 180\nu^{7} + 1008\nu^{5} + 960\nu^{3} + 3968\nu ) / 2688 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{14} + 6\nu^{10} - 24\nu^{8} + 36\nu^{6} - 24\nu^{4} + 144\nu^{2} - 224 ) / 96 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} - 2\nu^{13} + 6\nu^{11} - 12\nu^{9} + 36\nu^{7} - 48\nu^{5} + 96\nu^{3} - 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 11\nu^{15} - 30\nu^{13} + 42\nu^{11} - 132\nu^{9} + 156\nu^{7} - 336\nu^{5} + 384\nu^{3} - 2176\nu ) / 1344 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\nu^{15} - 30\nu^{13} + 42\nu^{11} - 132\nu^{9} + 156\nu^{7} - 336\nu^{5} + 384\nu^{3} + 512\nu ) / 1344 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -\nu^{14} + 2\nu^{12} - 6\nu^{10} + 12\nu^{8} - 36\nu^{6} + 48\nu^{4} - 96\nu^{2} + 128 ) / 64 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{13} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{12} - \beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} + \beta_{11} + \beta_{9} - \beta_{7} + \beta_{4} - 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{14} + \beta_{13} - \beta_{12} + \beta_{10} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{15} + 4\beta_{7} + 2\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{14} - 4\beta_{13} + 4\beta_{12} + 4\beta_{5} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{15} - 4\beta_{11} + 4\beta_{7} + 2\beta_{4} - 2\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -4\beta_{14} + 10\beta_{12} + 2\beta_{10} + 10\beta_{8} - 2\beta_{6} - 2\beta_{5} + 2\beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4\beta_{11} + 12\beta_{9} - 4\beta_{7} - 2\beta_{4} - 10\beta_{2} - 6\beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12\beta_{13} - 12\beta_{12} + 8\beta_{10} + 16\beta_{6} - 4\beta_{5} - 12\beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -4\beta_{15} + 12\beta_{11} - 12\beta_{9} + 36\beta_{7} - 12\beta_{4} - 12\beta_{2} + 12 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( -12\beta_{14} - 12\beta_{13} - 16\beta_{12} - 24\beta_{8} - 24\beta_{6} - 24\beta_{3} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -48\beta_{9} - 48\beta_{7} + 12\beta_{4} - 12\beta_{2} - 12\beta _1 - 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 16\beta_{14} + 32\beta_{13} + 24\beta_{10} + 24\beta_{8} - 120\beta_{6} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(-\beta_{7}\) \(\beta_{7}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
43.1
1.35949 − 0.389597i
1.24570 + 0.669507i
0.859408 + 1.12313i
0.601202 − 1.28006i
−0.601202 + 1.28006i
−0.859408 − 1.12313i
−1.24570 − 0.669507i
−1.35949 + 0.389597i
1.35949 + 0.389597i
1.24570 − 0.669507i
0.859408 − 1.12313i
0.601202 + 1.28006i
−0.601202 − 1.28006i
−0.859408 + 1.12313i
−1.24570 + 0.669507i
−1.35949 − 0.389597i
−1.35949 − 0.389597i − 0.207468i 1.69643 + 1.05931i 0 −0.0808289 + 0.282051i −1.32185 + 1.32185i −1.89357 − 2.10104i 2.95696 0
43.2 −1.24570 + 0.669507i 3.07455i 1.10352 − 1.66801i 0 −2.05844 − 3.82996i 1.47763 − 1.47763i −0.257910 + 2.81664i −6.45288 0
43.3 −0.859408 + 1.12313i − 1.54564i −0.522835 − 1.93045i 0 1.73595 + 1.32833i −1.17442 + 1.17442i 2.61747 + 1.07183i 0.611000 0
43.4 −0.601202 − 1.28006i − 2.02856i −1.27711 + 1.53915i 0 −2.59669 + 1.21958i 3.26957 − 3.26957i 2.73801 + 0.709444i −1.11507 0
43.5 0.601202 + 1.28006i 2.02856i −1.27711 + 1.53915i 0 −2.59669 + 1.21958i −3.26957 + 3.26957i −2.73801 − 0.709444i −1.11507 0
43.6 0.859408 − 1.12313i 1.54564i −0.522835 − 1.93045i 0 1.73595 + 1.32833i 1.17442 − 1.17442i −2.61747 − 1.07183i 0.611000 0
43.7 1.24570 − 0.669507i − 3.07455i 1.10352 − 1.66801i 0 −2.05844 − 3.82996i −1.47763 + 1.47763i 0.257910 − 2.81664i −6.45288 0
43.8 1.35949 + 0.389597i 0.207468i 1.69643 + 1.05931i 0 −0.0808289 + 0.282051i 1.32185 − 1.32185i 1.89357 + 2.10104i 2.95696 0
307.1 −1.35949 + 0.389597i 0.207468i 1.69643 − 1.05931i 0 −0.0808289 − 0.282051i −1.32185 − 1.32185i −1.89357 + 2.10104i 2.95696 0
307.2 −1.24570 − 0.669507i − 3.07455i 1.10352 + 1.66801i 0 −2.05844 + 3.82996i 1.47763 + 1.47763i −0.257910 − 2.81664i −6.45288 0
307.3 −0.859408 − 1.12313i 1.54564i −0.522835 + 1.93045i 0 1.73595 − 1.32833i −1.17442 − 1.17442i 2.61747 − 1.07183i 0.611000 0
307.4 −0.601202 + 1.28006i 2.02856i −1.27711 − 1.53915i 0 −2.59669 − 1.21958i 3.26957 + 3.26957i 2.73801 − 0.709444i −1.11507 0
307.5 0.601202 − 1.28006i − 2.02856i −1.27711 − 1.53915i 0 −2.59669 − 1.21958i −3.26957 − 3.26957i −2.73801 + 0.709444i −1.11507 0
307.6 0.859408 + 1.12313i − 1.54564i −0.522835 + 1.93045i 0 1.73595 − 1.32833i 1.17442 + 1.17442i −2.61747 + 1.07183i 0.611000 0
307.7 1.24570 + 0.669507i 3.07455i 1.10352 + 1.66801i 0 −2.05844 + 3.82996i −1.47763 − 1.47763i 0.257910 + 2.81664i −6.45288 0
307.8 1.35949 − 0.389597i − 0.207468i 1.69643 − 1.05931i 0 −0.0808289 − 0.282051i 1.32185 + 1.32185i 1.89357 − 2.10104i 2.95696 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 43.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
80.j even 4 1 inner
80.s even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.j.c ✓ 16
4.b odd 2 1 1600.2.j.c 16
5.b even 2 1 inner 400.2.j.c ✓ 16
5.c odd 4 2 400.2.s.c yes 16
16.e even 4 1 1600.2.s.c 16
16.f odd 4 1 400.2.s.c yes 16
20.d odd 2 1 1600.2.j.c 16
20.e even 4 2 1600.2.s.c 16
80.i odd 4 1 1600.2.j.c 16
80.j even 4 1 inner 400.2.j.c ✓ 16
80.k odd 4 1 400.2.s.c yes 16
80.q even 4 1 1600.2.s.c 16
80.s even 4 1 inner 400.2.j.c ✓ 16
80.t odd 4 1 1600.2.j.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.j.c ✓ 16 1.a even 1 1 trivial
400.2.j.c ✓ 16 5.b even 2 1 inner
400.2.j.c ✓ 16 80.j even 4 1 inner
400.2.j.c ✓ 16 80.s even 4 1 inner
400.2.s.c yes 16 5.c odd 4 2
400.2.s.c yes 16 16.f odd 4 1
400.2.s.c yes 16 80.k odd 4 1
1600.2.j.c 16 4.b odd 2 1
1600.2.j.c 16 20.d odd 2 1
1600.2.j.c 16 80.i odd 4 1
1600.2.j.c 16 80.t odd 4 1
1600.2.s.c 16 16.e even 4 1
1600.2.s.c 16 20.e even 4 2
1600.2.s.c 16 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 16T_{3}^{6} + 72T_{3}^{4} + 96T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 2 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{8} + 16 T^{6} + 72 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 496 T^{12} + \cdots + 810000 \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + \cdots + 7056)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} - 56 T^{6} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 1600000000 \) Copy content Toggle raw display
$19$ \( (T^{8} + 4 T^{7} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + 2448 T^{12} + \cdots + 10000 \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + \cdots + 242064)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 160 T^{6} + \cdots + 1849600)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 144 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 192 T^{6} + \cdots + 90000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 96 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + 20464 T^{12} + \cdots + 38416 \) Copy content Toggle raw display
$53$ \( (T^{8} + 200 T^{6} + \cdots + 1915456)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 4 T^{7} + \cdots + 1192464)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + 8 T^{7} + \cdots + 3625216)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} - 336 T^{6} + \cdots + 9771876)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} + \cdots - 192)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 14281868906496 \) Copy content Toggle raw display
$79$ \( (T^{4} - 20 T^{3} + \cdots - 6000)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + 256 T^{6} + \cdots + 4235364)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 136 T^{2} + \cdots - 240)^{4} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 77\!\cdots\!00 \) Copy content Toggle raw display
show more
show less