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Properties

Label 400.2.u.e
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
Twist minimal: no (minimal twist has level 200)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{3} + (\beta_{6} - \beta_{4}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{6} + 2 \beta_{3} + \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{3} + (\beta_{6} - \beta_{4}) q^{5} + ( - \beta_{5} - \beta_{4} + \beta_1 - 1) q^{7} + (\beta_{6} + 2 \beta_{3} + \beta_{2}) q^{9} + ( - \beta_{7} + \beta_{5} + \cdots + 3 \beta_{2}) q^{11}+ \cdots + (2 \beta_{5} + 2 \beta_{4} + 3 \beta_{3} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{3} + 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{3} + 5 q^{5} - 4 q^{7} + 8 q^{9} + 2 q^{11} + 8 q^{13} + 5 q^{15} + 10 q^{17} - 3 q^{19} - 3 q^{21} - 6 q^{23} - 5 q^{25} + 18 q^{27} + 6 q^{29} + 10 q^{31} - 16 q^{33} + 15 q^{35} - 2 q^{37} + q^{39} - 4 q^{41} + 12 q^{43} + 12 q^{47} - 4 q^{49} + 20 q^{51} - 13 q^{53} - 20 q^{55} - 6 q^{57} - 29 q^{59} + 15 q^{61} - 4 q^{63} - 50 q^{65} - 28 q^{67} - 12 q^{69} + 16 q^{71} + q^{73} - 15 q^{75} - 11 q^{77} - 23 q^{79} + 32 q^{81} - 14 q^{83} - 15 q^{85} - 3 q^{87} - 7 q^{89} - 29 q^{91} + 20 q^{93} - 45 q^{95} - 3 q^{97} + 22 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/400\mathbb{Z}\right)^\times\).

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

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
−0.357358 + 1.86824i
1.66637 − 0.917186i
1.17421 − 0.0566033i
−0.983224 + 0.644389i
1.17421 + 0.0566033i
−0.983224 − 0.644389i
−0.357358 − 1.86824i
1.66637 + 0.917186i
0 1.30902 − 0.951057i 0 0.279141 − 2.21858i 0 −3.77447 0 −0.118034 + 0.363271i 0
81.2 0 1.30902 − 0.951057i 0 1.52988 + 1.63079i 0 2.77447 0 −0.118034 + 0.363271i 0
161.1 0 0.190983 − 0.587785i 0 −1.39991 + 1.74363i 0 −1.83337 0 2.11803 + 1.53884i 0
161.2 0 0.190983 − 0.587785i 0 2.09089 − 0.792578i 0 0.833366 0 2.11803 + 1.53884i 0
241.1 0 0.190983 + 0.587785i 0 −1.39991 − 1.74363i 0 −1.83337 0 2.11803 − 1.53884i 0
241.2 0 0.190983 + 0.587785i 0 2.09089 + 0.792578i 0 0.833366 0 2.11803 − 1.53884i 0
321.1 0 1.30902 + 0.951057i 0 0.279141 + 2.21858i 0 −3.77447 0 −0.118034 − 0.363271i 0
321.2 0 1.30902 + 0.951057i 0 1.52988 − 1.63079i 0 2.77447 0 −0.118034 − 0.363271i 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.e 8
4.b odd 2 1 200.2.m.b ✓ 8
20.d odd 2 1 1000.2.m.b 8
20.e even 4 2 1000.2.q.b 16
25.d even 5 1 inner 400.2.u.e 8
25.d even 5 1 10000.2.a.q 4
25.e even 10 1 10000.2.a.z 4
100.h odd 10 1 1000.2.m.b 8
100.h odd 10 1 5000.2.a.f 4
100.j odd 10 1 200.2.m.b ✓ 8
100.j odd 10 1 5000.2.a.i 4
100.l even 20 2 1000.2.q.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.b ✓ 8 4.b odd 2 1
200.2.m.b ✓ 8 100.j odd 10 1
400.2.u.e 8 1.a even 1 1 trivial
400.2.u.e 8 25.d even 5 1 inner
1000.2.m.b 8 20.d odd 2 1
1000.2.m.b 8 100.h odd 10 1
1000.2.q.b 16 20.e even 4 2
1000.2.q.b 16 100.l even 20 2
5000.2.a.f 4 100.h odd 10 1
5000.2.a.i 4 100.j odd 10 1
10000.2.a.q 4 25.d even 5 1
10000.2.a.z 4 25.e even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 3T_{3}^{3} + 4T_{3}^{2} - 2T_{3} + 1 \) 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
$3$ \( (T^{4} - 3 T^{3} + 4 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 5 T^{7} + \cdots + 625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 2 T^{3} - 11 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 2 T^{7} + \cdots + 3481 \) Copy content Toggle raw display
$13$ \( T^{8} - 8 T^{7} + \cdots + 6241 \) Copy content Toggle raw display
$17$ \( T^{8} - 10 T^{7} + \cdots + 24025 \) Copy content Toggle raw display
$19$ \( T^{8} + 3 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$23$ \( T^{8} + 6 T^{7} + \cdots + 2401 \) Copy content Toggle raw display
$29$ \( T^{8} - 6 T^{7} + \cdots + 116281 \) Copy content Toggle raw display
$31$ \( T^{8} - 10 T^{7} + \cdots + 15625 \) Copy content Toggle raw display
$37$ \( T^{8} + 2 T^{7} + \cdots + 4532641 \) Copy content Toggle raw display
$41$ \( T^{8} + 4 T^{7} + \cdots + 7921 \) Copy content Toggle raw display
$43$ \( (T^{4} - 6 T^{3} + \cdots + 5056)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 12 T^{7} + \cdots + 96721 \) Copy content Toggle raw display
$53$ \( T^{8} + 13 T^{7} + \cdots + 495616 \) Copy content Toggle raw display
$59$ \( T^{8} + 29 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$61$ \( T^{8} - 15 T^{7} + \cdots + 21025 \) Copy content Toggle raw display
$67$ \( T^{8} + 28 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$71$ \( T^{8} - 16 T^{7} + \cdots + 15768841 \) Copy content Toggle raw display
$73$ \( T^{8} - T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$79$ \( T^{8} + 23 T^{7} + \cdots + 9759376 \) Copy content Toggle raw display
$83$ \( T^{8} + 14 T^{7} + \cdots + 8288641 \) Copy content Toggle raw display
$89$ \( T^{8} + 7 T^{7} + \cdots + 80656 \) Copy content Toggle raw display
$97$ \( T^{8} + 3 T^{7} + \cdots + 4330561 \) Copy content Toggle raw display
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