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Properties

Label 400.2.u.d
Level $400$
Weight $2$
Character orbit 400.u
Analytic conductor $3.194$
Analytic rank $0$
Dimension $8$
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] = [400,2,Mod(81,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.81");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.u (of order \(5\), degree \(4\), 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: \(3.19401608085\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.58140625.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 50)
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{6} + \beta_{5} + \cdots + \beta_{2}) q^{3}+ \cdots + (\beta_{7} + 3 \beta_{6} - \beta_{4} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + \beta_{5} + \cdots + \beta_{2}) q^{3}+ \cdots + (2 \beta_{5} + 2 \beta_{4} + \beta_{3} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 3 q^{3} - 4 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 3 q^{3} - 4 q^{7} - q^{9} - q^{11} - 13 q^{13} + 10 q^{15} - 11 q^{17} - 20 q^{19} - 19 q^{21} + 3 q^{23} + 30 q^{25} - 15 q^{27} - 15 q^{29} + 9 q^{31} + 19 q^{33} + 15 q^{35} - 6 q^{37} + 12 q^{39} - 9 q^{41} - 12 q^{43} + 15 q^{45} + q^{47} - 4 q^{49} - 26 q^{51} + 7 q^{53} + 25 q^{55} - 10 q^{59} + 6 q^{61} + 8 q^{63} - 10 q^{65} + 11 q^{67} + 43 q^{69} + 9 q^{71} - 8 q^{73} - 30 q^{75} + 33 q^{77} + 10 q^{79} - 17 q^{81} - 27 q^{83} + 5 q^{85} - 15 q^{89} - q^{91} - 46 q^{93} + 30 q^{95} - 36 q^{97} + 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 4x^{6} - 7x^{5} + 11x^{4} + 5x^{3} - 10x^{2} - 25x + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -64\nu^{7} + 206\nu^{6} - 318\nu^{5} + 399\nu^{4} - 732\nu^{3} - 126\nu^{2} + 2175\nu + 235 ) / 1355 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 406\nu^{7} - 714\nu^{6} + 747\nu^{5} - 1896\nu^{4} + 2103\nu^{3} + 4949\nu^{2} + 1065\nu - 7800 ) / 1355 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -420\nu^{7} + 776\nu^{6} - 698\nu^{5} + 1924\nu^{4} - 2297\nu^{3} - 5129\nu^{2} - 1055\nu + 10265 ) / 1355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 504\nu^{7} - 877\nu^{6} + 946\nu^{5} - 2363\nu^{4} + 2919\nu^{3} + 5125\nu^{2} + 3705\nu - 11505 ) / 1355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -613\nu^{7} + 1321\nu^{6} - 1513\nu^{5} + 3394\nu^{4} - 4352\nu^{3} - 6178\nu^{2} - 530\nu + 12985 ) / 1355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -714\nu^{7} + 1265\nu^{6} - 1295\nu^{5} + 3325\nu^{4} - 3390\nu^{3} - 8367\nu^{2} - 2200\nu + 14605 ) / 1355 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -1020\nu^{7} + 1962\nu^{6} - 2121\nu^{5} + 5563\nu^{4} - 6314\nu^{3} - 10443\nu^{2} - 3530\nu + 21445 ) / 1355 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{6} - 2\beta_{5} + \beta_{4} + \beta_{3} - 2\beta_{2} + 3\beta _1 + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} + 3\beta_{6} - 4\beta_{5} - 3\beta_{4} + 2\beta_{3} + 11\beta_{2} + 6\beta _1 - 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} + 16\beta_{6} + 2\beta_{5} + 4\beta_{4} - 6\beta_{3} + 17\beta_{2} - 3\beta _1 + 2 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{7} - 3\beta_{6} - 6\beta_{5} + 13\beta_{4} - 7\beta_{3} - 6\beta_{2} - 6\beta _1 + 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 16\beta_{7} - 6\beta_{6} - 22\beta_{5} + 6\beta_{4} + 51\beta_{3} + 43\beta_{2} + 8\beta _1 - 67 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -23\beta_{7} + 63\beta_{6} + 46\beta_{5} + 12\beta_{4} + 52\beta_{3} + 156\beta_{2} - 34\beta _1 - 144 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -31\beta_{7} - 9\beta_{6} + 137\beta_{5} + 84\beta_{4} - 31\beta_{3} - 33\beta_{2} - 168\beta _1 + 62 ) / 5 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−0.983224 − 0.644389i
1.17421 + 0.0566033i
1.66637 − 0.917186i
−0.357358 + 1.86824i
1.66637 + 0.917186i
−0.357358 − 1.86824i
−0.983224 + 0.644389i
1.17421 − 0.0566033i
0 −1.09089 + 0.792578i 0 −2.15743 − 0.587785i 0 0.833366 0 −0.365190 + 1.12394i 0
81.2 0 2.39991 − 1.74363i 0 2.15743 − 0.587785i 0 −1.83337 0 1.79224 − 5.51595i 0
161.1 0 −0.529876 + 1.63079i 0 2.02373 + 0.951057i 0 2.77447 0 0.0483405 + 0.0351215i 0
161.2 0 0.720859 − 2.21858i 0 −2.02373 + 0.951057i 0 −3.77447 0 −1.97539 − 1.43521i 0
241.1 0 −0.529876 − 1.63079i 0 2.02373 − 0.951057i 0 2.77447 0 0.0483405 − 0.0351215i 0
241.2 0 0.720859 + 2.21858i 0 −2.02373 − 0.951057i 0 −3.77447 0 −1.97539 + 1.43521i 0
321.1 0 −1.09089 − 0.792578i 0 −2.15743 + 0.587785i 0 0.833366 0 −0.365190 − 1.12394i 0
321.2 0 2.39991 + 1.74363i 0 2.15743 + 0.587785i 0 −1.83337 0 1.79224 + 5.51595i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.u.d 8
4.b odd 2 1 50.2.d.b ✓ 8
12.b even 2 1 450.2.h.e 8
20.d odd 2 1 250.2.d.d 8
20.e even 4 2 250.2.e.c 16
25.d even 5 1 inner 400.2.u.d 8
25.d even 5 1 10000.2.a.t 4
25.e even 10 1 10000.2.a.x 4
100.h odd 10 1 250.2.d.d 8
100.h odd 10 1 1250.2.a.f 4
100.j odd 10 1 50.2.d.b ✓ 8
100.j odd 10 1 1250.2.a.l 4
100.l even 20 2 250.2.e.c 16
100.l even 20 2 1250.2.b.e 8
300.n even 10 1 450.2.h.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b ✓ 8 4.b odd 2 1
50.2.d.b ✓ 8 100.j odd 10 1
250.2.d.d 8 20.d odd 2 1
250.2.d.d 8 100.h odd 10 1
250.2.e.c 16 20.e even 4 2
250.2.e.c 16 100.l even 20 2
400.2.u.d 8 1.a even 1 1 trivial
400.2.u.d 8 25.d even 5 1 inner
450.2.h.e 8 12.b even 2 1
450.2.h.e 8 300.n even 10 1
1250.2.a.f 4 100.h odd 10 1
1250.2.a.l 4 100.j odd 10 1
1250.2.b.e 8 100.l even 20 2
10000.2.a.t 4 25.d even 5 1
10000.2.a.x 4 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 3T_{3}^{7} + 8T_{3}^{6} - 6T_{3}^{5} + 25T_{3}^{4} + 24T_{3}^{3} + 128T_{3}^{2} + 192T_{3} + 256 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 3 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} - 15 T^{6} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 11 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{8} + 13 T^{7} + \cdots + 39601 \) Copy content Toggle raw display
$17$ \( T^{8} + 11 T^{7} + \cdots + 11881 \) Copy content Toggle raw display
$19$ \( T^{8} + 20 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$23$ \( T^{8} - 3 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{8} + 15 T^{7} + \cdots + 25 \) Copy content Toggle raw display
$31$ \( T^{8} - 9 T^{7} + \cdots + 1597696 \) Copy content Toggle raw display
$37$ \( T^{8} + 6 T^{7} + \cdots + 5041 \) Copy content Toggle raw display
$41$ \( T^{8} + 9 T^{7} + \cdots + 7921 \) Copy content Toggle raw display
$43$ \( (T^{4} + 6 T^{3} + \cdots + 176)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - T^{7} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} - 7 T^{7} + \cdots + 65536 \) Copy content Toggle raw display
$59$ \( T^{8} + 10 T^{7} + \cdots + 102400 \) Copy content Toggle raw display
$61$ \( T^{8} - 6 T^{7} + \cdots + 2920681 \) Copy content Toggle raw display
$67$ \( T^{8} - 11 T^{7} + \cdots + 891136 \) Copy content Toggle raw display
$71$ \( T^{8} - 9 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots + 1175056 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} + \cdots + 102400 \) Copy content Toggle raw display
$83$ \( T^{8} + 27 T^{7} + \cdots + 13424896 \) Copy content Toggle raw display
$89$ \( T^{8} + 15 T^{7} + \cdots + 9610000 \) Copy content Toggle raw display
$97$ \( (T^{4} + 18 T^{3} + 124 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
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