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Properties

Label 1000.2.m.c
Level $1000$
Weight $2$
Character orbit 1000.m
Analytic conductor $7.985$
Analytic rank $0$
Dimension $16$
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] = [1000,2,Mod(201,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.m (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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - x^{15} + 12 x^{14} - 18 x^{13} + 100 x^{12} + 23 x^{11} + 567 x^{10} + 556 x^{9} + 3841 x^{8} + \cdots + 6400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{10} q^{7} + (\beta_{11} + \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{10} q^{7} + (\beta_{11} + \beta_{9} + \beta_{8} + \cdots + \beta_1) q^{9}+ \cdots + ( - 2 \beta_{15} + 2 \beta_{14} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - q^{3} + 6 q^{7} - 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - q^{3} + 6 q^{7} - 11 q^{9} - 10 q^{11} - q^{13} + 4 q^{17} - 3 q^{21} - 11 q^{23} - 13 q^{27} + 5 q^{29} - 9 q^{31} - 16 q^{33} - 30 q^{37} + 14 q^{39} - 2 q^{41} + 42 q^{43} + 16 q^{47} + 18 q^{49} + 100 q^{51} - 11 q^{53} + 64 q^{57} - 53 q^{59} + 4 q^{61} + 38 q^{63} + 14 q^{67} - 7 q^{69} - 6 q^{71} + 24 q^{73} - 23 q^{77} - 22 q^{79} - 6 q^{81} - 33 q^{83} - 37 q^{87} + 20 q^{89} - 27 q^{91} - 40 q^{93} - 11 q^{97} + 122 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - x^{15} + 12 x^{14} - 18 x^{13} + 100 x^{12} + 23 x^{11} + 567 x^{10} + 556 x^{9} + 3841 x^{8} + \cdots + 6400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 13\!\cdots\!95 \nu^{15} + \cdots + 19\!\cdots\!40 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!43 \nu^{15} + \cdots + 16\!\cdots\!80 ) / 98\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 19\!\cdots\!45 \nu^{15} + \cdots + 85\!\cdots\!00 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 49\!\cdots\!88 \nu^{15} + \cdots - 11\!\cdots\!60 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 85\!\cdots\!65 \nu^{15} + \cdots - 11\!\cdots\!20 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 62\!\cdots\!01 \nu^{15} + \cdots + 33\!\cdots\!00 ) / 98\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 32\!\cdots\!57 \nu^{15} + \cdots + 37\!\cdots\!60 ) / 31\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 83\!\cdots\!53 \nu^{15} + \cdots + 53\!\cdots\!20 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29\!\cdots\!95 \nu^{15} + \cdots + 35\!\cdots\!20 ) / 19\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13\!\cdots\!29 \nu^{15} + \cdots - 13\!\cdots\!00 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17\!\cdots\!22 \nu^{15} + \cdots + 10\!\cdots\!00 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 28\!\cdots\!43 \nu^{15} + \cdots - 11\!\cdots\!40 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 30\!\cdots\!73 \nu^{15} + \cdots - 96\!\cdots\!00 ) / 79\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14\!\cdots\!85 \nu^{15} + \cdots - 76\!\cdots\!20 ) / 31\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + 5\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( - \beta_{15} + \beta_{13} - 2 \beta_{12} + 2 \beta_{10} + \beta_{9} + 2 \beta_{8} - 3 \beta_{6} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} - 10 \beta_{12} - 13 \beta_{11} + 10 \beta_{10} + 10 \beta_{9} - 28 \beta_{6} - 38 \beta_{5} + \cdots - 28 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{15} - 25 \beta_{14} + 8 \beta_{13} - 8 \beta_{12} + 2 \beta_{11} - 12 \beta_{10} + 61 \beta_{7} + \cdots - 44 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} - 94 \beta_{14} + 91 \beta_{13} + 4 \beta_{11} - 94 \beta_{9} - \beta_{8} + 334 \beta_{5} + \cdots + 99 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 56 \beta_{15} + 128 \beta_{14} + 56 \beta_{12} - 573 \beta_{11} + 128 \beta_{10} - 136 \beta_{9} + \cdots - 533 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 841 \beta_{15} - 841 \beta_{13} + 853 \beta_{12} - 897 \beta_{10} - 521 \beta_{9} - 853 \beta_{8} + \cdots - 1001 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 1361 \beta_{14} - 365 \beta_{13} + 2350 \beta_{12} + 5559 \beta_{11} - 2670 \beta_{10} - 2670 \beta_{9} + \cdots + 5399 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 7970 \beta_{15} + 8690 \beta_{14} - 44 \beta_{13} + 44 \beta_{12} - 1656 \beta_{11} + 4246 \beta_{10} + \cdots + 19872 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2068 \beta_{15} + 26617 \beta_{14} - 21023 \beta_{13} - 7502 \beta_{11} + 26617 \beta_{9} + \cdots - 27647 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1027 \beta_{15} - 46599 \beta_{14} + 1027 \beta_{12} + 164489 \beta_{11} - 46599 \beta_{10} + \cdots + 141409 \beta_1 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 217988 \beta_{15} + 217988 \beta_{13} - 225384 \beta_{12} + 264056 \beta_{10} + 108348 \beta_{9} + \cdots + 310308 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 504128 \beta_{14} - 29480 \beta_{13} - 726745 \beta_{12} - 1679657 \beta_{11} + 836385 \beta_{10} + \cdots - 1850652 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2234465 \beta_{15} - 2616810 \beta_{14} - 41623 \beta_{13} + 41623 \beta_{12} + 1084728 \beta_{11} + \cdots - 5585371 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(\beta_{5}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
0.772523 + 2.37758i
0.372462 + 1.14632i
−0.462625 − 1.42381i
−0.991378 − 3.05115i
2.48200 − 1.80328i
0.777729 − 0.565053i
−0.701201 + 0.509453i
−1.74951 + 1.27109i
2.48200 + 1.80328i
0.777729 + 0.565053i
−0.701201 − 0.509453i
−1.74951 − 1.27109i
0.772523 − 2.37758i
0.372462 − 1.14632i
−0.462625 + 1.42381i
−0.991378 + 3.05115i
0 −0.772523 − 2.37758i 0 0 0 1.96923 0 −2.62905 + 1.91012i 0
201.2 0 −0.372462 − 1.14632i 0 0 0 −1.59935 0 1.25173 − 0.909432i 0
201.3 0 0.462625 + 1.42381i 0 0 0 4.94031 0 0.613832 − 0.445975i 0
201.4 0 0.991378 + 3.05115i 0 0 0 −2.69216 0 −5.89962 + 4.28633i 0
401.1 0 −2.48200 + 1.80328i 0 0 0 1.36851 0 1.98145 − 6.09828i 0
401.2 0 −0.777729 + 0.565053i 0 0 0 −0.364298 0 −0.641474 + 1.97425i 0
401.3 0 0.701201 − 0.509453i 0 0 0 −3.82614 0 −0.694910 + 2.13871i 0
401.4 0 1.74951 − 1.27109i 0 0 0 3.20389 0 0.518051 − 1.59440i 0
601.1 0 −2.48200 − 1.80328i 0 0 0 1.36851 0 1.98145 + 6.09828i 0
601.2 0 −0.777729 − 0.565053i 0 0 0 −0.364298 0 −0.641474 − 1.97425i 0
601.3 0 0.701201 + 0.509453i 0 0 0 −3.82614 0 −0.694910 − 2.13871i 0
601.4 0 1.74951 + 1.27109i 0 0 0 3.20389 0 0.518051 + 1.59440i 0
801.1 0 −0.772523 + 2.37758i 0 0 0 1.96923 0 −2.62905 − 1.91012i 0
801.2 0 −0.372462 + 1.14632i 0 0 0 −1.59935 0 1.25173 + 0.909432i 0
801.3 0 0.462625 − 1.42381i 0 0 0 4.94031 0 0.613832 + 0.445975i 0
801.4 0 0.991378 − 3.05115i 0 0 0 −2.69216 0 −5.89962 − 4.28633i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.4
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 1000.2.m.c 16
5.b even 2 1 200.2.m.c ✓ 16
5.c odd 4 2 1000.2.q.d 32
20.d odd 2 1 400.2.u.g 16
25.d even 5 1 inner 1000.2.m.c 16
25.d even 5 1 5000.2.a.m 8
25.e even 10 1 200.2.m.c ✓ 16
25.e even 10 1 5000.2.a.l 8
25.f odd 20 2 1000.2.q.d 32
100.h odd 10 1 400.2.u.g 16
100.h odd 10 1 10000.2.a.bk 8
100.j odd 10 1 10000.2.a.bh 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.c ✓ 16 5.b even 2 1
200.2.m.c ✓ 16 25.e even 10 1
400.2.u.g 16 20.d odd 2 1
400.2.u.g 16 100.h odd 10 1
1000.2.m.c 16 1.a even 1 1 trivial
1000.2.m.c 16 25.d even 5 1 inner
1000.2.q.d 32 5.c odd 4 2
1000.2.q.d 32 25.f odd 20 2
5000.2.a.l 8 25.e even 10 1
5000.2.a.m 8 25.d even 5 1
10000.2.a.bh 8 100.j odd 10 1
10000.2.a.bk 8 100.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + T_{3}^{15} + 12 T_{3}^{14} + 18 T_{3}^{13} + 100 T_{3}^{12} - 23 T_{3}^{11} + 567 T_{3}^{10} + \cdots + 6400 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + T^{15} + \cdots + 6400 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 3 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 10 T^{15} + \cdots + 59969536 \) Copy content Toggle raw display
$13$ \( T^{16} + T^{15} + \cdots + 17161 \) Copy content Toggle raw display
$17$ \( T^{16} - 4 T^{15} + \cdots + 1488400 \) Copy content Toggle raw display
$19$ \( T^{16} + 27 T^{14} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15499254016 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 173189977921 \) Copy content Toggle raw display
$31$ \( T^{16} + 9 T^{15} + \cdots + 102400 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 7351519081 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 300264545296 \) Copy content Toggle raw display
$43$ \( (T^{8} - 21 T^{7} + \cdots - 1321984)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 88881451868416 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 280104942704896 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 1029137207296 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 1369239321025 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 31860822016 \) Copy content Toggle raw display
$71$ \( T^{16} + 6 T^{15} + \cdots + 39942400 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 32453120071696 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 36986982400 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 631045939456 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 2234872512601 \) Copy content Toggle raw display
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