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Properties

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

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [400,2,Mod(101,400)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.101");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.l (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: 16.0.534694406811304329216.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 80)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + ( - \beta_{14} + \beta_{12} + \cdots - \beta_{2}) q^{3}+ \cdots + ( - \beta_{11} - \beta_{6} - 2 \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + ( - \beta_{14} + \beta_{12} + \cdots - \beta_{2}) q^{3}+ \cdots + ( - 2 \beta_{11} + 6 \beta_{6} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 4 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 4 q^{6} + 8 q^{11} + 4 q^{14} + 16 q^{16} + 8 q^{19} - 16 q^{21} + 32 q^{24} + 32 q^{26} + 16 q^{29} + 16 q^{31} - 48 q^{34} + 60 q^{36} + 8 q^{44} - 28 q^{46} - 16 q^{49} - 16 q^{51} - 40 q^{54} - 56 q^{56} + 24 q^{59} + 16 q^{64} - 72 q^{66} - 32 q^{69} - 88 q^{76} - 16 q^{79} - 16 q^{81} + 80 q^{84} - 28 q^{86} - 16 q^{91} - 12 q^{94} + 56 q^{96} - 88 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} - 2x^{12} + 4x^{10} + 4x^{8} + 16x^{6} - 32x^{4} - 128x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{14} + 2\nu^{12} + 6\nu^{10} + 28\nu^{8} + 20\nu^{6} - 96\nu^{4} - 416\nu^{2} + 320 ) / 576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{11} + 2\nu^{9} + 2\nu^{7} + 4\nu^{5} - 20\nu^{3} - 32\nu ) / 48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} - 2\nu^{12} + 6\nu^{10} - 4\nu^{8} + 52\nu^{6} + 48\nu^{4} - 112\nu^{2} - 32 ) / 288 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{14} - 2\nu^{12} + 18\nu^{10} + 20\nu^{8} - 20\nu^{6} - 64\nu^{2} + 256 ) / 288 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{14} + 4\nu^{12} - 4\nu^{8} - 8\nu^{6} + 104\nu^{2} + 112 ) / 144 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{14} + 2\nu^{12} + 18\nu^{10} - 20\nu^{8} - 4\nu^{6} - 144\nu^{4} - 128\nu^{2} + 704 ) / 576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{15} + 2\nu^{13} + 6\nu^{11} + 4\nu^{9} - 12\nu^{7} + 48\nu^{3} + 64\nu ) / 192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{14} - 2\nu^{10} + 4\nu^{6} + 8\nu^{4} - 32\nu^{2} + 32 ) / 96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 7\nu^{15} + 2\nu^{13} - 54\nu^{11} - 20\nu^{9} + 44\nu^{7} + 144\nu^{5} + 256\nu^{3} - 1792\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{15} + 2\nu^{13} + 2\nu^{11} - 4\nu^{9} - 4\nu^{7} - 16\nu^{5} + 32\nu^{3} + 128\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3\nu^{14} - 2\nu^{12} - 6\nu^{10} - 12\nu^{8} - 20\nu^{6} + 96\nu^{4} - 320 ) / 192 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( -11\nu^{15} + 2\nu^{13} + 30\nu^{11} + 28\nu^{9} - 124\nu^{7} - 192\nu^{5} - 224\nu^{3} + 896\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -11\nu^{15} - 10\nu^{13} + 30\nu^{11} + 4\nu^{9} + 68\nu^{7} - 336\nu^{5} - 128\nu^{3} + 896\nu ) / 1152 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -\nu^{15} + \nu^{13} + 2\nu^{11} + 2\nu^{9} - 4\nu^{7} - 44\nu^{5} + 8\nu^{3} + 96\nu ) / 96 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} + \nu^{13} - 2\nu^{11} - 6\nu^{9} + 4\nu^{7} + 20\nu^{5} + 8\nu^{3} - 128\nu ) / 96 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} - \beta_{13} - \beta_{12} + \beta_{10} - \beta_{9} - \beta_{7} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} - \beta_{8} - \beta_{6} - 2\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{15} - \beta_{12} + \beta_{9} + \beta_{7} - \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{8} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} - 3\beta_{14} + 2\beta_{10} + \beta_{7} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -2\beta_{11} - 2\beta_{8} - 2\beta_{6} - 2\beta_{4} + 4\beta_{3} + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{13} - 8\beta_{12} + 2\beta_{10} - 2\beta_{9} + 2\beta_{7} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -4\beta_{11} + 2\beta_{8} - 10\beta_{6} + 4\beta_{4} - 2\beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -8\beta_{15} + 6\beta_{14} - 6\beta_{13} - 10\beta_{12} - 6\beta_{10} - 2\beta_{9} + 8\beta_{7} + 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2\beta_{11} - 10\beta_{8} + 6\beta_{6} + 4\beta_{5} + 8\beta_{4} + 8\beta_{3} - 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 16\beta_{15} + 12\beta_{13} - 16\beta_{10} - 12\beta_{9} + 20\beta_{7} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 8\beta_{11} - 4\beta_{8} - 4\beta_{6} + 36\beta_{5} - 4\beta_{4} + 8\beta_{3} + 36\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 20\beta_{15} + 24\beta_{14} - 20\beta_{13} - 20\beta_{12} + 20\beta_{10} - 20\beta_{9} + 12\beta_{7} + 32\beta_{2} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12\beta_{11} - 60\beta_{8} - 12\beta_{6} - 16\beta_{4} + 8\beta_{3} + 32\beta _1 + 52 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 24\beta_{15} + 72\beta_{14} - 72\beta_{13} - 64\beta_{12} - 72\beta_{10} - 80\beta_{9} - 24\beta_{7} - 32\beta_{2} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
−1.40501 + 0.161069i
−1.32661 + 0.490008i
−0.841995 + 1.13624i
−0.238945 − 1.39388i
0.238945 + 1.39388i
0.841995 − 1.13624i
1.32661 − 0.490008i
1.40501 − 0.161069i
−1.40501 − 0.161069i
−1.32661 − 0.490008i
−0.841995 − 1.13624i
−0.238945 + 1.39388i
0.238945 − 1.39388i
0.841995 + 1.13624i
1.32661 + 0.490008i
1.40501 + 0.161069i
−1.40501 − 0.161069i 0.734294 − 0.734294i 1.94811 + 0.452606i 0 −1.14996 + 0.913419i 1.71452i −2.66422 − 0.949697i 1.92163i 0
101.2 −1.32661 − 0.490008i −1.99154 + 1.99154i 1.51978 + 1.30010i 0 3.61786 − 1.66612i − 1.09033i −1.37910 − 2.46943i − 4.93244i 0
101.3 −0.841995 − 1.13624i 1.86033 − 1.86033i −0.582088 + 1.91342i 0 −3.68016 − 0.547394i − 3.61392i 2.66422 − 0.949697i − 3.92163i 0
101.4 −0.238945 + 1.39388i 0.183790 − 0.183790i −1.88581 − 0.666123i 0 0.212266 + 0.300098i − 3.84853i 1.37910 − 2.46943i 2.93244i 0
101.5 0.238945 − 1.39388i −0.183790 + 0.183790i −1.88581 − 0.666123i 0 0.212266 + 0.300098i 3.84853i −1.37910 + 2.46943i 2.93244i 0
101.6 0.841995 + 1.13624i −1.86033 + 1.86033i −0.582088 + 1.91342i 0 −3.68016 − 0.547394i 3.61392i −2.66422 + 0.949697i − 3.92163i 0
101.7 1.32661 + 0.490008i 1.99154 − 1.99154i 1.51978 + 1.30010i 0 3.61786 − 1.66612i 1.09033i 1.37910 + 2.46943i − 4.93244i 0
101.8 1.40501 + 0.161069i −0.734294 + 0.734294i 1.94811 + 0.452606i 0 −1.14996 + 0.913419i − 1.71452i 2.66422 + 0.949697i 1.92163i 0
301.1 −1.40501 + 0.161069i 0.734294 + 0.734294i 1.94811 − 0.452606i 0 −1.14996 − 0.913419i − 1.71452i −2.66422 + 0.949697i − 1.92163i 0
301.2 −1.32661 + 0.490008i −1.99154 − 1.99154i 1.51978 − 1.30010i 0 3.61786 + 1.66612i 1.09033i −1.37910 + 2.46943i 4.93244i 0
301.3 −0.841995 + 1.13624i 1.86033 + 1.86033i −0.582088 − 1.91342i 0 −3.68016 + 0.547394i 3.61392i 2.66422 + 0.949697i 3.92163i 0
301.4 −0.238945 − 1.39388i 0.183790 + 0.183790i −1.88581 + 0.666123i 0 0.212266 − 0.300098i 3.84853i 1.37910 + 2.46943i − 2.93244i 0
301.5 0.238945 + 1.39388i −0.183790 − 0.183790i −1.88581 + 0.666123i 0 0.212266 − 0.300098i − 3.84853i −1.37910 − 2.46943i − 2.93244i 0
301.6 0.841995 − 1.13624i −1.86033 − 1.86033i −0.582088 − 1.91342i 0 −3.68016 + 0.547394i − 3.61392i −2.66422 − 0.949697i 3.92163i 0
301.7 1.32661 − 0.490008i 1.99154 + 1.99154i 1.51978 − 1.30010i 0 3.61786 + 1.66612i − 1.09033i 1.37910 − 2.46943i 4.93244i 0
301.8 1.40501 − 0.161069i −0.734294 − 0.734294i 1.94811 − 0.452606i 0 −1.14996 − 0.913419i 1.71452i 2.66422 − 0.949697i − 1.92163i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 101.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
16.e even 4 1 inner
80.q 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.l.i 16
4.b odd 2 1 1600.2.l.h 16
5.b even 2 1 inner 400.2.l.i 16
5.c odd 4 2 80.2.q.c ✓ 16
15.e even 4 2 720.2.bm.f 16
16.e even 4 1 inner 400.2.l.i 16
16.f odd 4 1 1600.2.l.h 16
20.d odd 2 1 1600.2.l.h 16
20.e even 4 2 320.2.q.c 16
40.i odd 4 2 640.2.q.e 16
40.k even 4 2 640.2.q.f 16
80.i odd 4 1 80.2.q.c ✓ 16
80.i odd 4 1 640.2.q.e 16
80.j even 4 1 320.2.q.c 16
80.j even 4 1 640.2.q.f 16
80.k odd 4 1 1600.2.l.h 16
80.q even 4 1 inner 400.2.l.i 16
80.s even 4 1 320.2.q.c 16
80.s even 4 1 640.2.q.f 16
80.t odd 4 1 80.2.q.c ✓ 16
80.t odd 4 1 640.2.q.e 16
240.bb even 4 1 720.2.bm.f 16
240.bf even 4 1 720.2.bm.f 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
80.2.q.c ✓ 16 5.c odd 4 2
80.2.q.c ✓ 16 80.i odd 4 1
80.2.q.c ✓ 16 80.t odd 4 1
320.2.q.c 16 20.e even 4 2
320.2.q.c 16 80.j even 4 1
320.2.q.c 16 80.s even 4 1
400.2.l.i 16 1.a even 1 1 trivial
400.2.l.i 16 5.b even 2 1 inner
400.2.l.i 16 16.e even 4 1 inner
400.2.l.i 16 80.q even 4 1 inner
640.2.q.e 16 40.i odd 4 2
640.2.q.e 16 80.i odd 4 1
640.2.q.e 16 80.t odd 4 1
640.2.q.f 16 40.k even 4 2
640.2.q.f 16 80.j even 4 1
640.2.q.f 16 80.s even 4 1
720.2.bm.f 16 15.e even 4 2
720.2.bm.f 16 240.bb even 4 1
720.2.bm.f 16 240.bf even 4 1
1600.2.l.h 16 4.b odd 2 1
1600.2.l.h 16 16.f odd 4 1
1600.2.l.h 16 20.d odd 2 1
1600.2.l.h 16 80.k odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\):

\( T_{3}^{16} + 112T_{3}^{12} + 3144T_{3}^{8} + 3520T_{3}^{4} + 16 \) Copy content Toggle raw display
\( T_{7}^{8} + 32T_{7}^{6} + 312T_{7}^{4} + 896T_{7}^{2} + 676 \) 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
$3$ \( T^{16} + 112 T^{12} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 32 T^{6} + \cdots + 676)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 4 T^{7} + 8 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 224 T^{4} + 7744)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 28 T^{2} + 88)^{4} \) Copy content Toggle raw display
$19$ \( (T^{8} - 4 T^{7} + \cdots + 59536)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 32 T^{6} + 168 T^{4} + \cdots + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} - 8 T^{7} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} + \cdots + 208)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + 1536 T^{4} + 147456)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 192 T^{6} + \cdots + 219024)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 9721171216 \) Copy content Toggle raw display
$47$ \( (T^{8} - 80 T^{6} + \cdots + 27556)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + 4672 T^{12} + \cdots + 4096 \) Copy content Toggle raw display
$59$ \( (T^{8} - 12 T^{7} + \cdots + 144)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 576)^{4} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 36804120336 \) Copy content Toggle raw display
$71$ \( (T^{8} + 256 T^{6} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 456 T^{6} + \cdots + 48776256)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots - 368)^{4} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( (T^{8} + 208 T^{6} + \cdots + 135424)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 200 T^{6} + \cdots + 2383936)^{2} \) Copy content Toggle raw display
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