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Properties

Label 4000.2.c.i
Level $4000$
Weight $2$
Character orbit 4000.c
Analytic conductor $31.940$
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] = [4000,2,Mod(1249,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4000.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4000.c (of order \(2\), degree \(1\), 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: \(31.9401608085\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 8 x^{14} + 37 x^{12} - 138 x^{11} + 256 x^{10} - 76 x^{9} - 18 x^{8} - 26 x^{7} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{20} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{12} q^{7} + (\beta_{6} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{12} q^{7} + (\beta_{6} - 1) q^{9} - \beta_{4} q^{11} + \beta_{11} q^{13} + (\beta_{13} + \beta_{11} - \beta_{10}) q^{17} + \beta_{2} q^{19} + (\beta_{6} - \beta_{5} - 2) q^{21} + (\beta_{15} + \beta_{12} + \cdots - \beta_1) q^{23}+ \cdots + ( - \beta_{9} + 2 \beta_{7} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} - 24 q^{21} + 32 q^{29} + 16 q^{41} - 32 q^{49} - 56 q^{61} + 56 q^{69} + 40 q^{81} - 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 8 x^{14} + 37 x^{12} - 138 x^{11} + 256 x^{10} - 76 x^{9} - 18 x^{8} - 26 x^{7} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 500152199470 \nu^{15} + 776942372428 \nu^{14} - 1802621238846 \nu^{13} + \cdots - 26582234691229 ) / 79671159324189 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 961097443600 \nu^{15} - 11330194718236 \nu^{14} + 40163968785798 \nu^{13} + \cdots - 205978337694478 ) / 26557053108063 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 999933781768 \nu^{15} - 331786556976 \nu^{14} + 10639506205896 \nu^{13} + \cdots - 62720519870781 ) / 26557053108063 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1153149472876 \nu^{15} - 12285259115044 \nu^{14} + 43042553850492 \nu^{13} + \cdots - 107516795708890 ) / 26557053108063 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 48408826 \nu^{15} - 224980170 \nu^{14} + 552251352 \nu^{13} - 422849619 \nu^{12} + \cdots + 413956605 ) / 724540149 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9769175685334 \nu^{15} - 56624746941510 \nu^{14} + 160343930697744 \nu^{13} + \cdots + 296179030445484 ) / 79671159324189 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10457126440228 \nu^{15} - 47954113385348 \nu^{14} + 114303220327986 \nu^{13} + \cdots - 203437412591926 ) / 79671159324189 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 1943990403076 \nu^{15} - 8217975714212 \nu^{14} + 16866144086297 \nu^{13} + \cdots + 4029135076062 ) / 8852351036021 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 29323675221592 \nu^{15} - 160509829315148 \nu^{14} + 438606035469474 \nu^{13} + \cdots - 402814762522468 ) / 79671159324189 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17787703468290 \nu^{15} + 71441344911916 \nu^{14} - 146044422230592 \nu^{13} + \cdots - 124262490399520 ) / 26557053108063 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 18176491901340 \nu^{15} - 74587406751616 \nu^{14} + 153030325659288 \nu^{13} + \cdots + 91367244471574 ) / 26557053108063 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 58157030369408 \nu^{15} - 240457912925036 \nu^{14} + 492735662581467 \nu^{13} + \cdots + 252733135308872 ) / 79671159324189 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 58903291773492 \nu^{15} + 241498398394376 \nu^{14} - 494482573496748 \nu^{13} + \cdots - 296874601711784 ) / 79671159324189 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 22263043957260 \nu^{15} + 89287469064300 \nu^{14} - 183140782843736 \nu^{13} + \cdots - 154035610750064 ) / 26557053108063 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 90339371262928 \nu^{15} - 373356028591408 \nu^{14} + 764717156699118 \nu^{13} + \cdots + 382093611096439 ) / 79671159324189 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} + \beta_{13} + \beta_{9} - \beta_{7} - 4\beta_{5} - \beta_{4} + 4\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{13} - 2\beta_{10} + 4\beta_{8} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 7 \beta_{14} + 6 \beta_{13} + 2 \beta_{12} - 6 \beta_{11} - 7 \beta_{9} + 6 \beta_{8} + 6 \beta_{7} + \cdots - 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -5\beta_{9} + 2\beta_{7} + 7\beta_{6} + 13\beta_{5} - 5\beta_{4} + 3\beta_{3} + 5\beta_{2} - 28 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 10 \beta_{15} + 55 \beta_{14} - 52 \beta_{13} - 28 \beta_{12} + 60 \beta_{11} + 10 \beta_{10} + \cdots - 38 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 44 \beta_{15} + 48 \beta_{14} - 153 \beta_{13} - 54 \beta_{12} + 82 \beta_{11} + 106 \beta_{10} + \cdots - 76 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 68 \beta_{15} + 227 \beta_{14} - 252 \beta_{13} - 143 \beta_{12} + 273 \beta_{11} + 84 \beta_{10} + \cdots + 289 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 494\beta_{9} - 320\beta_{7} - 431\beta_{6} - 1571\beta_{5} + 346\beta_{4} - 446\beta_{3} - 382\beta_{2} + 1565 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1460 \beta_{15} - 3913 \beta_{14} + 4951 \beta_{13} + 2738 \beta_{12} - 4962 \beta_{11} - 2052 \beta_{10} + \cdots + 7062 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1925 \beta_{15} - 3042 \beta_{14} + 6135 \beta_{13} + 2803 \beta_{12} - 4523 \beta_{11} + \cdots + 5709 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 14652 \beta_{15} - 34858 \beta_{14} + 48267 \beta_{13} + 25876 \beta_{12} - 45628 \beta_{11} + \cdots - 77044 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 46516 \beta_{9} + 32974 \beta_{7} + 32687 \beta_{6} + 156155 \beta_{5} - 25538 \beta_{4} + \cdots - 116258 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 143250 \beta_{15} + 317773 \beta_{14} - 466531 \beta_{13} - 244376 \beta_{12} + 424216 \beta_{11} + \cdots - 792086 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 334296 \beta_{15} + 609651 \beta_{14} - 1068165 \beta_{13} - 519502 \beta_{12} + 863908 \beta_{11} + \cdots - 1203380 \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1383410 \beta_{15} + 2941669 \beta_{14} - 4481457 \beta_{13} - 2311710 \beta_{12} + 3975586 \beta_{11} + \cdots + 7882244 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
−1.70753 + 1.70753i
1.08950 + 1.08950i
2.18004 + 2.18004i
−0.562005 + 0.562005i
0.462175 − 0.462175i
1.15586 + 1.15586i
−0.519164 + 0.519164i
−0.0988705 − 0.0988705i
−0.519164 − 0.519164i
−0.0988705 + 0.0988705i
0.462175 + 0.462175i
1.15586 − 1.15586i
2.18004 − 2.18004i
−0.562005 − 0.562005i
−1.70753 − 1.70753i
1.08950 − 1.08950i
0 − 2.79703i 0 0 0 − 2.30294i 0 −4.82335 0
1249.2 0 − 2.79703i 0 0 0 − 2.30294i 0 −4.82335 0
1249.3 0 − 2.74204i 0 0 0 − 1.42258i 0 −4.51880 0
1249.4 0 − 2.74204i 0 0 0 − 1.42258i 0 −4.51880 0
1249.5 0 − 0.693685i 0 0 0 4.52199i 0 2.51880 0
1249.6 0 − 0.693685i 0 0 0 4.52199i 0 2.51880 0
1249.7 0 − 0.420293i 0 0 0 2.86781i 0 2.82335 0
1249.8 0 − 0.420293i 0 0 0 2.86781i 0 2.82335 0
1249.9 0 0.420293i 0 0 0 − 2.86781i 0 2.82335 0
1249.10 0 0.420293i 0 0 0 − 2.86781i 0 2.82335 0
1249.11 0 0.693685i 0 0 0 − 4.52199i 0 2.51880 0
1249.12 0 0.693685i 0 0 0 − 4.52199i 0 2.51880 0
1249.13 0 2.74204i 0 0 0 1.42258i 0 −4.51880 0
1249.14 0 2.74204i 0 0 0 1.42258i 0 −4.51880 0
1249.15 0 2.79703i 0 0 0 2.30294i 0 −4.82335 0
1249.16 0 2.79703i 0 0 0 2.30294i 0 −4.82335 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4000.2.c.i 16
4.b odd 2 1 inner 4000.2.c.i 16
5.b even 2 1 inner 4000.2.c.i 16
5.c odd 4 1 4000.2.a.o ✓ 8
5.c odd 4 1 4000.2.a.p yes 8
20.d odd 2 1 inner 4000.2.c.i 16
20.e even 4 1 4000.2.a.o ✓ 8
20.e even 4 1 4000.2.a.p yes 8
40.i odd 4 1 8000.2.a.by 8
40.i odd 4 1 8000.2.a.bz 8
40.k even 4 1 8000.2.a.by 8
40.k even 4 1 8000.2.a.bz 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4000.2.a.o ✓ 8 5.c odd 4 1
4000.2.a.o ✓ 8 20.e even 4 1
4000.2.a.p yes 8 5.c odd 4 1
4000.2.a.p yes 8 20.e even 4 1
4000.2.c.i 16 1.a even 1 1 trivial
4000.2.c.i 16 4.b odd 2 1 inner
4000.2.c.i 16 5.b even 2 1 inner
4000.2.c.i 16 20.d odd 2 1 inner
8000.2.a.by 8 40.i odd 4 1
8000.2.a.by 8 40.k even 4 1
8000.2.a.bz 8 40.i odd 4 1
8000.2.a.bz 8 40.k even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4000, [\chi])\):

\( T_{3}^{8} + 16T_{3}^{6} + 69T_{3}^{4} + 40T_{3}^{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{8} + 36T_{7}^{6} + 389T_{7}^{4} + 1540T_{7}^{2} + 1805 \) Copy content Toggle raw display
\( T_{11}^{8} - 64T_{11}^{6} + 1344T_{11}^{4} - 9280T_{11}^{2} + 1280 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 16 T^{6} + 69 T^{4} + \cdots + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 36 T^{6} + \cdots + 1805)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} - 64 T^{6} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 80 T^{6} + \cdots + 20736)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 116 T^{6} + \cdots + 92416)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 140 T^{6} + \cdots + 800000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 116 T^{6} + \cdots + 5)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 19 T^{2} + \cdots + 25)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} - 224 T^{6} + \cdots + 8398080)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 108 T^{6} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 4 T^{3} + \cdots - 695)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + 104 T^{6} + \cdots + 10125)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 176 T^{6} + \cdots + 96605)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 112 T^{2} + 256)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} - 336 T^{6} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 14 T^{3} + \cdots - 355)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + 536 T^{6} + \cdots + 205825280)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} - 364 T^{6} + \cdots + 2151680)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 384 T^{6} + \cdots + 5683456)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 116 T^{6} + \cdots + 1280)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 356 T^{6} + \cdots + 15753125)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} + 5 T - 5)^{8} \) Copy content Toggle raw display
$97$ \( (T^{8} + 416 T^{6} + \cdots + 16128256)^{2} \) Copy content Toggle raw display
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