↑

Properties

Label 400.2.bi.b
Level $400$
Weight $2$
Character orbit 400.bi
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(47,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([10, 0, 17]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("400.47");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 400.bi (of order \(20\), degree \(8\), 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: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 806 x^{12} - 2288 x^{11} + 5530 x^{10} - 11062 x^{9} + \cdots + 521 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{15} + \beta_{11}) q^{3} + ( - 2 \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 1) q^{5}+ \cdots + ( - 2 \beta_{8} + \beta_{7} + \cdots - 2 \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{15} + \beta_{11}) q^{3} + ( - 2 \beta_{9} - \beta_{8} + \beta_{7} + \cdots + 1) q^{5}+ \cdots + (2 \beta_{15} + 2 \beta_{14} + \cdots - 4 \beta_{5}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 20 q^{9} + 20 q^{17} - 20 q^{21} - 20 q^{25} + 20 q^{29} + 60 q^{33} - 20 q^{37} + 28 q^{41} - 20 q^{45} + 60 q^{53} - 20 q^{57} - 12 q^{61} - 20 q^{65} - 80 q^{69} - 40 q^{73} - 60 q^{77} + 56 q^{81} - 60 q^{85} + 40 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} + 52 x^{14} - 224 x^{13} + 806 x^{12} - 2288 x^{11} + 5530 x^{10} - 11062 x^{9} + \cdots + 521 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 72 \nu^{15} + 110382 \nu^{14} - 635648 \nu^{13} + 3762239 \nu^{12} - 12727642 \nu^{11} + \cdots + 13296359 ) / 472739 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1627 \nu^{15} + 65707 \nu^{14} - 528267 \nu^{13} + 3297265 \nu^{12} - 13402321 \nu^{11} + \cdots + 15325911 ) / 472739 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 72 \nu^{15} + 111462 \nu^{14} - 917260 \nu^{13} + 5576337 \nu^{12} - 23116010 \nu^{11} + \cdots + 36219750 ) / 472739 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1627 \nu^{15} - 90112 \nu^{14} + 562466 \nu^{13} - 3149416 \nu^{12} + 11098236 \nu^{11} + \cdots - 9227466 ) / 472739 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 21035 \nu^{15} + 61366 \nu^{14} - 371195 \nu^{13} + 325732 \nu^{12} - 234915 \nu^{11} + \cdots - 20012687 ) / 472739 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 21035 \nu^{15} - 254159 \nu^{14} + 1720746 \nu^{13} - 8486422 \nu^{12} + 31654892 \nu^{11} + \cdots - 31624145 ) / 472739 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22328 \nu^{15} - 56538 \nu^{14} + 237488 \nu^{13} + 618475 \nu^{12} - 4585262 \nu^{11} + \cdots + 20838924 ) / 472739 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 39575 \nu^{15} + 242672 \nu^{14} - 1469107 \nu^{13} + 5186852 \nu^{12} - 16505738 \nu^{11} + \cdots - 16756569 ) / 472739 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 39575 \nu^{15} + 350953 \nu^{14} - 2227074 \nu^{13} + 9835012 \nu^{12} - 34541127 \nu^{11} + \cdots + 30649381 ) / 472739 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 122649 \nu^{15} + 804984 \nu^{14} - 4850190 \nu^{13} + 17864761 \nu^{12} - 57312966 \nu^{11} + \cdots - 18228555 ) / 472739 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 122649 \nu^{15} - 1034751 \nu^{14} + 6458559 \nu^{13} - 27739460 \nu^{12} + 95652363 \nu^{11} + \cdots - 65219122 ) / 472739 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 163851 \nu^{15} - 1098153 \nu^{14} + 6604458 \nu^{13} - 24938631 \nu^{12} + 80947517 \nu^{11} + \cdots - 30081005 ) / 472739 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 184886 \nu^{15} - 1613771 \nu^{14} + 10155417 \nu^{13} - 44026027 \nu^{12} + 152415484 \nu^{11} + \cdots - 56800813 ) / 472739 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 199182 \nu^{15} - 1385584 \nu^{14} + 8293993 \nu^{13} - 31768997 \nu^{12} + 102573179 \nu^{11} + \cdots - 15723891 ) / 472739 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 199182 \nu^{15} + 1602146 \nu^{14} - 9809927 \nu^{13} + 40592578 \nu^{12} - 135807523 \nu^{11} + \cdots + 27806466 ) / 472739 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} + \beta_{12} - \beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{6} - \beta_{4} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} - \beta_{11} + \beta_{9} - \beta_{3} - \beta_{2} - \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} - 2 \beta_{14} - 4 \beta_{13} - \beta_{12} + 2 \beta_{11} - 5 \beta_{10} - 7 \beta_{9} + \cdots - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} - 2 \beta_{14} - 6 \beta_{12} + 6 \beta_{11} - 2 \beta_{10} - 12 \beta_{9} - 6 \beta_{8} + \cdots + 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 13 \beta_{15} + 13 \beta_{14} + 25 \beta_{13} - 10 \beta_{12} - \beta_{11} + 26 \beta_{10} + \cdots + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 23 \beta_{15} + 26 \beta_{14} + 6 \beta_{13} + 32 \beta_{12} - 37 \beta_{11} + 24 \beta_{10} + \cdots - 45 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 64 \beta_{15} - 43 \beta_{14} - 170 \beta_{13} + 138 \beta_{12} - 52 \beta_{11} - 130 \beta_{10} + \cdots - 153 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 216 \beta_{15} - 236 \beta_{14} - 104 \beta_{13} - 152 \beta_{12} + 236 \beta_{11} - 224 \beta_{10} + \cdots + 336 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 116 \beta_{15} - 190 \beta_{14} + 1123 \beta_{13} - 1310 \beta_{12} + 773 \beta_{11} + 538 \beta_{10} + \cdots + 1648 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1796 \beta_{15} + 1772 \beta_{14} + 1272 \beta_{13} + 478 \beta_{12} - 1423 \beta_{11} + 1908 \beta_{10} + \cdots - 2160 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2519 \beta_{15} + 5291 \beta_{14} - 6489 \beta_{13} + 10869 \beta_{12} - 8433 \beta_{11} + \cdots - 17214 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 13176 \beta_{15} - 11244 \beta_{14} - 12948 \beta_{13} + 1674 \beta_{12} + 7104 \beta_{11} - 14936 \beta_{10} + \cdots + 9754 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 46021 \beta_{15} - 65092 \beta_{14} + 26733 \beta_{13} - 80972 \beta_{12} + 79063 \beta_{11} + \cdots + 157912 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 82485 \beta_{15} + 55568 \beta_{14} + 115070 \beta_{13} - 52940 \beta_{12} - 17975 \beta_{11} + \cdots - 1879 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 530656 \beta_{15} + 627739 \beta_{14} + 11791 \beta_{13} + 530574 \beta_{12} - 657455 \beta_{11} + \cdots - 1269893 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(351\)
\(\chi(n)\) \(1\) \(\beta_{8}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
47.1
0.500000 + 1.77088i
0.500000 − 2.77088i
0.500000 + 1.82107i
0.500000 − 0.821074i
0.500000 − 1.82107i
0.500000 + 0.821074i
0.500000 − 0.509304i
0.500000 + 1.50930i
0.500000 + 0.509304i
0.500000 − 1.50930i
0.500000 − 0.941940i
0.500000 + 1.94194i
0.500000 + 0.941940i
0.500000 − 1.94194i
0.500000 − 1.77088i
0.500000 + 2.77088i
0 −0.502390 − 3.17196i 0 −0.530249 + 2.17229i 0 −0.398699 + 0.398699i 0 −6.95579 + 2.26007i 0
47.2 0 0.502390 + 3.17196i 0 −0.530249 + 2.17229i 0 0.398699 − 0.398699i 0 −6.95579 + 2.26007i 0
63.1 0 −1.66465 − 0.848182i 0 2.06909 + 0.847859i 0 1.19176 + 1.19176i 0 0.288294 + 0.396802i 0
63.2 0 1.66465 + 0.848182i 0 2.06909 + 0.847859i 0 −1.19176 − 1.19176i 0 0.288294 + 0.396802i 0
127.1 0 −1.66465 + 0.848182i 0 2.06909 − 0.847859i 0 1.19176 − 1.19176i 0 0.288294 − 0.396802i 0
127.2 0 1.66465 − 0.848182i 0 2.06909 − 0.847859i 0 −1.19176 + 1.19176i 0 0.288294 − 0.396802i 0
223.1 0 −0.648013 − 1.27180i 0 0.166977 − 2.22982i 0 2.92911 + 2.92911i 0 0.565808 − 0.778768i 0
223.2 0 0.648013 + 1.27180i 0 0.166977 − 2.22982i 0 −2.92911 − 2.92911i 0 0.565808 − 0.778768i 0
287.1 0 −0.648013 + 1.27180i 0 0.166977 + 2.22982i 0 2.92911 − 2.92911i 0 0.565808 + 0.778768i 0
287.2 0 0.648013 − 1.27180i 0 0.166977 + 2.22982i 0 −2.92911 + 2.92911i 0 0.565808 + 0.778768i 0
303.1 0 −2.01411 + 0.319003i 0 −1.70582 − 1.44575i 0 3.13704 + 3.13704i 0 1.10169 − 0.357960i 0
303.2 0 2.01411 − 0.319003i 0 −1.70582 − 1.44575i 0 −3.13704 − 3.13704i 0 1.10169 − 0.357960i 0
367.1 0 −2.01411 − 0.319003i 0 −1.70582 + 1.44575i 0 3.13704 − 3.13704i 0 1.10169 + 0.357960i 0
367.2 0 2.01411 + 0.319003i 0 −1.70582 + 1.44575i 0 −3.13704 + 3.13704i 0 1.10169 + 0.357960i 0
383.1 0 −0.502390 + 3.17196i 0 −0.530249 − 2.17229i 0 −0.398699 − 0.398699i 0 −6.95579 − 2.26007i 0
383.2 0 0.502390 − 3.17196i 0 −0.530249 − 2.17229i 0 0.398699 + 0.398699i 0 −6.95579 − 2.26007i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 47.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
25.f odd 20 1 inner
100.l even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 400.2.bi.b ✓ 16
4.b odd 2 1 inner 400.2.bi.b ✓ 16
25.f odd 20 1 inner 400.2.bi.b ✓ 16
100.l even 20 1 inner 400.2.bi.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
400.2.bi.b ✓ 16 1.a even 1 1 trivial
400.2.bi.b ✓ 16 4.b odd 2 1 inner
400.2.bi.b ✓ 16 25.f odd 20 1 inner
400.2.bi.b ✓ 16 100.l even 20 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 10T_{3}^{14} - 45T_{3}^{12} - 360T_{3}^{10} + 2685T_{3}^{8} - 6200T_{3}^{6} + 4275T_{3}^{4} - 3050T_{3}^{2} + 93025 \) acting on \(S_{2}^{\mathrm{new}}(400, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 10 T^{14} + \cdots + 93025 \) Copy content Toggle raw display
$5$ \( (T^{8} + 5 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 690 T^{12} + \cdots + 93025 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 14884000000 \) Copy content Toggle raw display
$13$ \( (T^{8} + 5 T^{6} + \cdots + 42025)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 10 T^{7} + \cdots + 400)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + 130 T^{14} + \cdots + 58140625 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 12123111025 \) Copy content Toggle raw display
$29$ \( (T^{8} - 10 T^{7} + \cdots + 19321)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + 10 T^{14} + \cdots + 58140625 \) Copy content Toggle raw display
$37$ \( (T^{8} + 10 T^{7} + \cdots + 9025)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} - 14 T^{7} + \cdots + 55696)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 262866417025 \) Copy content Toggle raw display
$47$ \( T^{16} + 20 T^{12} + \cdots + 93025 \) Copy content Toggle raw display
$53$ \( (T^{8} - 30 T^{7} + \cdots + 9025)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 7576944390625 \) Copy content Toggle raw display
$61$ \( (T^{8} + 6 T^{7} + \cdots + 546121)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 9302500000000 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 7576944390625 \) Copy content Toggle raw display
$73$ \( (T^{8} + 20 T^{7} + \cdots + 1452025)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 805005849390625 \) Copy content Toggle raw display
$83$ \( T^{16} + 90 T^{14} + \cdots + 93025 \) Copy content Toggle raw display
$89$ \( (T^{8} + 44 T^{6} + \cdots + 56490256)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 20 T^{7} + \cdots + 255025)^{2} \) Copy content Toggle raw display
show more
show less