↑

Properties

Label 1800.2.d.u
Level $1800$
Weight $2$
Character orbit 1800.d
Analytic conductor $14.373$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1800,2,Mod(1549,1800)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1800, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("1800.1549");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1800.d (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: \(14.3730723638\)
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: \( x^{16} - 4x^{14} - 4x^{12} + 12x^{10} + 389x^{8} + 816x^{6} + 2924x^{4} + 1040x^{2} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{10}\cdot 5^{2} \)
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.

\(f(q)\) \(=\) \( q + \beta_{10} q^{2} + \beta_{8} q^{4} - \beta_{5} q^{7} + \beta_{9} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{10} q^{2} + \beta_{8} q^{4} - \beta_{5} q^{7} + \beta_{9} q^{8} + \beta_{2} q^{11} + \beta_1 q^{13} + \beta_{6} q^{14} + ( - \beta_{13} - 1) q^{16} + ( - \beta_{14} - 2 \beta_{10} + \cdots + \beta_{3}) q^{17}+ \cdots + ( - 2 \beta_{14} + 2 \beta_{9}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{16} + 16 q^{31} + 48 q^{34} - 16 q^{46} + 80 q^{76} + 32 q^{79} - 128 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4x^{14} - 4x^{12} + 12x^{10} + 389x^{8} + 816x^{6} + 2924x^{4} + 1040x^{2} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 191 \nu^{14} + 1662 \nu^{12} - 3442 \nu^{10} - 4576 \nu^{8} - 60311 \nu^{6} + 209482 \nu^{4} + \cdots + 1156700 ) / 296400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 116047 \nu^{15} - 484014 \nu^{13} - 379486 \nu^{11} + 1806272 \nu^{9} + 42548887 \nu^{7} + \cdots + 184956500 \nu ) / 47221200 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8495195 \nu^{15} + 37639014 \nu^{13} + 20427782 \nu^{11} - 123075472 \nu^{9} + \cdots + 1752197660 \nu ) / 897202800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 255958 \nu^{15} + 949761 \nu^{13} + 1397704 \nu^{11} - 3157778 \nu^{9} - 100444018 \nu^{7} + \cdots - 413218250 \nu ) / 23610600 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 3015005 \nu^{14} + 12778626 \nu^{12} + 8623370 \nu^{10} - 37296688 \nu^{8} + \cdots - 1548946300 ) / 179440560 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 192367 \nu^{15} - 820212 \nu^{13} - 568102 \nu^{11} + 2509676 \nu^{9} + 74492815 \nu^{7} + \cdots + 53425520 \nu ) / 9444240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 434957 \nu^{14} - 1796049 \nu^{12} - 1501046 \nu^{10} + 5208502 \nu^{8} + 169576937 \nu^{6} + \cdots + 251739250 ) / 23610600 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 221941 \nu^{14} + 931797 \nu^{12} + 731698 \nu^{10} - 2918006 \nu^{8} - 86040781 \nu^{6} + \cdots - 96791450 ) / 11805300 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 4756699 \nu^{15} - 19443027 \nu^{13} - 17540254 \nu^{11} + 59115446 \nu^{9} + \cdots + 3839813390 \nu ) / 224300700 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 31361273 \nu^{15} - 130722258 \nu^{13} - 103199090 \nu^{11} + 390046384 \nu^{9} + \cdots + 17059383820 \nu ) / 897202800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 17084159 \nu^{14} - 71489958 \nu^{12} - 56074742 \nu^{10} + 223812784 \nu^{8} + \cdots + 9339039700 ) / 448601400 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17089541 \nu^{14} + 71942982 \nu^{12} + 52933058 \nu^{10} - 213850936 \nu^{8} + \cdots - 7917583300 ) / 448601400 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 44168 \nu^{14} - 185416 \nu^{12} - 137304 \nu^{10} + 545818 \nu^{8} + 17083488 \nu^{6} + \cdots + 22699425 ) / 983775 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28996501 \nu^{15} + 122199045 \nu^{13} + 89763982 \nu^{11} - 368587310 \nu^{9} + \cdots - 13206993530 \nu ) / 448601400 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 1782543 \nu^{15} - 7439456 \nu^{13} - 5835014 \nu^{11} + 22467588 \nu^{9} + 688678943 \nu^{7} + \cdots + 1008190600 \nu ) / 15740400 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{15} - 5\beta_{10} + 5\beta_{9} - \beta_{6} + 2\beta_{4} - \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{13} - \beta_{12} - 4\beta_{11} - 5\beta_{8} - 10\beta_{7} + \beta_{5} + 5\beta _1 + 5 ) / 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{15} - 15\beta_{14} - 15\beta_{10} + 10\beta_{9} - 17\beta_{6} - \beta_{4} - 5\beta_{3} - 12\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 10\beta_{13} - 37\beta_{12} - 3\beta_{11} + 20\beta_{8} - 40\beta_{7} + 37\beta_{5} + 15\beta _1 + 30 ) / 10 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 29\beta_{15} - 40\beta_{14} - 165\beta_{10} + 30\beta_{9} - 19\beta_{6} - 2\beta_{4} + 15\beta_{3} - 94\beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 135\beta_{13} - 59\beta_{12} + 24\beta_{11} + 125\beta_{8} - 110\beta_{7} + 289\beta_{5} - 15\beta _1 + 95 ) / 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 72 \beta_{15} - 440 \beta_{14} - 695 \beta_{10} + 185 \beta_{9} + 117 \beta_{6} + \cdots - 313 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 655 \beta_{13} + 232 \beta_{12} + 848 \beta_{11} + 1200 \beta_{8} - 480 \beta_{7} + 1288 \beta_{5} + \cdots - 1505 ) / 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 474 \beta_{15} - 2310 \beta_{14} - 3135 \beta_{10} - 805 \beta_{9} + 1879 \beta_{6} + \cdots - 341 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 2275 \beta_{13} + 2779 \beta_{12} + 5196 \beta_{11} + 9205 \beta_{8} + 2930 \beta_{7} + 4611 \beta_{5} + \cdots - 13045 ) / 10 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 2753 \beta_{15} - 6780 \beta_{14} - 4545 \beta_{10} - 10280 \beta_{9} + 15683 \beta_{6} + \cdots + 4788 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2580 \beta_{13} + 30373 \beta_{12} + 30307 \beta_{11} + 39780 \beta_{8} + 37800 \beta_{7} + \cdots - 97020 ) / 10 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 24131 \beta_{15} - 3740 \beta_{14} + 68685 \beta_{10} - 79970 \beta_{9} + 92911 \beta_{6} + \cdots + 71526 \beta_{2} ) / 10 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 63635 \beta_{13} + 186251 \beta_{12} + 137084 \beta_{11} + 131715 \beta_{8} + 273950 \beta_{7} + \cdots - 507355 ) / 10 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 89772 \beta_{15} + 233250 \beta_{14} + 748335 \beta_{10} - 482025 \beta_{9} + \cdots + 544037 \beta_{2} ) / 10 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1549.1
0.0382181 − 0.438926i
0.660816 − 1.70872i
0.660816 + 1.70872i
0.0382181 + 0.438926i
2.33132 + 0.608977i
1.06152 + 1.23157i
1.06152 − 1.23157i
2.33132 − 0.608977i
−1.06152 + 1.23157i
−2.33132 + 0.608977i
−2.33132 − 0.608977i
−1.06152 − 1.23157i
−0.660816 − 1.70872i
−0.0382181 − 0.438926i
−0.0382181 + 0.438926i
−0.660816 + 1.70872i
−1.26979 − 0.622597i 0 1.22474 + 1.58114i 0 0 − 1.44949i −0.570759 − 2.77024i 0 0
1549.2 −1.26979 − 0.622597i 0 1.22474 + 1.58114i 0 0 1.44949i −0.570759 − 2.77024i 0 0
1549.3 −1.26979 + 0.622597i 0 1.22474 − 1.58114i 0 0 − 1.44949i −0.570759 + 2.77024i 0 0
1549.4 −1.26979 + 0.622597i 0 1.22474 − 1.58114i 0 0 1.44949i −0.570759 + 2.77024i 0 0
1549.5 −0.622597 − 1.26979i 0 −1.22474 + 1.58114i 0 0 − 3.44949i 2.77024 + 0.570759i 0 0
1549.6 −0.622597 − 1.26979i 0 −1.22474 + 1.58114i 0 0 3.44949i 2.77024 + 0.570759i 0 0
1549.7 −0.622597 + 1.26979i 0 −1.22474 − 1.58114i 0 0 − 3.44949i 2.77024 − 0.570759i 0 0
1549.8 −0.622597 + 1.26979i 0 −1.22474 − 1.58114i 0 0 3.44949i 2.77024 − 0.570759i 0 0
1549.9 0.622597 − 1.26979i 0 −1.22474 − 1.58114i 0 0 − 3.44949i −2.77024 + 0.570759i 0 0
1549.10 0.622597 − 1.26979i 0 −1.22474 − 1.58114i 0 0 3.44949i −2.77024 + 0.570759i 0 0
1549.11 0.622597 + 1.26979i 0 −1.22474 + 1.58114i 0 0 − 3.44949i −2.77024 − 0.570759i 0 0
1549.12 0.622597 + 1.26979i 0 −1.22474 + 1.58114i 0 0 3.44949i −2.77024 − 0.570759i 0 0
1549.13 1.26979 − 0.622597i 0 1.22474 − 1.58114i 0 0 − 1.44949i 0.570759 − 2.77024i 0 0
1549.14 1.26979 − 0.622597i 0 1.22474 − 1.58114i 0 0 1.44949i 0.570759 − 2.77024i 0 0
1549.15 1.26979 + 0.622597i 0 1.22474 + 1.58114i 0 0 − 1.44949i 0.570759 + 2.77024i 0 0
1549.16 1.26979 + 0.622597i 0 1.22474 + 1.58114i 0 0 1.44949i 0.570759 + 2.77024i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1549.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
8.b even 2 1 inner
15.d odd 2 1 inner
24.h odd 2 1 inner
40.f even 2 1 inner
120.i odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.2.d.u 16
3.b odd 2 1 inner 1800.2.d.u 16
4.b odd 2 1 7200.2.d.u 16
5.b even 2 1 inner 1800.2.d.u 16
5.c odd 4 1 1800.2.k.r ✓ 8
5.c odd 4 1 1800.2.k.s yes 8
8.b even 2 1 inner 1800.2.d.u 16
8.d odd 2 1 7200.2.d.u 16
12.b even 2 1 7200.2.d.u 16
15.d odd 2 1 inner 1800.2.d.u 16
15.e even 4 1 1800.2.k.r ✓ 8
15.e even 4 1 1800.2.k.s yes 8
20.d odd 2 1 7200.2.d.u 16
20.e even 4 1 7200.2.k.q 8
20.e even 4 1 7200.2.k.t 8
24.f even 2 1 7200.2.d.u 16
24.h odd 2 1 inner 1800.2.d.u 16
40.e odd 2 1 7200.2.d.u 16
40.f even 2 1 inner 1800.2.d.u 16
40.i odd 4 1 1800.2.k.r ✓ 8
40.i odd 4 1 1800.2.k.s yes 8
40.k even 4 1 7200.2.k.q 8
40.k even 4 1 7200.2.k.t 8
60.h even 2 1 7200.2.d.u 16
60.l odd 4 1 7200.2.k.q 8
60.l odd 4 1 7200.2.k.t 8
120.i odd 2 1 inner 1800.2.d.u 16
120.m even 2 1 7200.2.d.u 16
120.q odd 4 1 7200.2.k.q 8
120.q odd 4 1 7200.2.k.t 8
120.w even 4 1 1800.2.k.r ✓ 8
120.w even 4 1 1800.2.k.s yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1800.2.d.u 16 1.a even 1 1 trivial
1800.2.d.u 16 3.b odd 2 1 inner
1800.2.d.u 16 5.b even 2 1 inner
1800.2.d.u 16 8.b even 2 1 inner
1800.2.d.u 16 15.d odd 2 1 inner
1800.2.d.u 16 24.h odd 2 1 inner
1800.2.d.u 16 40.f even 2 1 inner
1800.2.d.u 16 120.i odd 2 1 inner
1800.2.k.r ✓ 8 5.c odd 4 1
1800.2.k.r ✓ 8 15.e even 4 1
1800.2.k.r ✓ 8 40.i odd 4 1
1800.2.k.r ✓ 8 120.w even 4 1
1800.2.k.s yes 8 5.c odd 4 1
1800.2.k.s yes 8 15.e even 4 1
1800.2.k.s yes 8 40.i odd 4 1
1800.2.k.s yes 8 120.w even 4 1
7200.2.d.u 16 4.b odd 2 1
7200.2.d.u 16 8.d odd 2 1
7200.2.d.u 16 12.b even 2 1
7200.2.d.u 16 20.d odd 2 1
7200.2.d.u 16 24.f even 2 1
7200.2.d.u 16 40.e odd 2 1
7200.2.d.u 16 60.h even 2 1
7200.2.d.u 16 120.m even 2 1
7200.2.k.q 8 20.e even 4 1
7200.2.k.q 8 40.k even 4 1
7200.2.k.q 8 60.l odd 4 1
7200.2.k.q 8 120.q odd 4 1
7200.2.k.t 8 20.e even 4 1
7200.2.k.t 8 40.k even 4 1
7200.2.k.t 8 60.l odd 4 1
7200.2.k.t 8 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1800, [\chi])\):

\( T_{7}^{4} + 14T_{7}^{2} + 25 \) Copy content Toggle raw display
\( T_{11}^{4} + 32T_{11}^{2} + 40 \) Copy content Toggle raw display
\( T_{13}^{2} - 15 \) Copy content Toggle raw display
\( T_{37}^{2} - 40 \) Copy content Toggle raw display
\( T_{41}^{4} - 80T_{41}^{2} + 1000 \) Copy content Toggle raw display
\( T_{53}^{4} - 80T_{53}^{2} + 1000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 2 T^{4} + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{4} + 14 T^{2} + 25)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 32 T^{2} + 40)^{4} \) Copy content Toggle raw display
$13$ \( (T^{2} - 15)^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} + 48 T^{2} + 360)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} + 50 T^{2} + 25)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + 32 T^{2} + 40)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 48 T^{2} + 360)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 2 T - 5)^{8} \) Copy content Toggle raw display
$37$ \( (T^{2} - 40)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 80 T^{2} + 1000)^{4} \) Copy content Toggle raw display
$43$ \( (T^{4} - 50 T^{2} + 25)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 128 T^{2} + 2560)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 80 T^{2} + 1000)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} + 128 T^{2} + 2560)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 110 T^{2} + 625)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} - 210 T^{2} + 5625)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} - 320 T^{2} + 25000)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 56 T^{2} + 400)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 20)^{8} \) Copy content Toggle raw display
$83$ \( (T^{4} - 192 T^{2} + 5760)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} - 320 T^{2} + 16000)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 434 T^{2} + 46225)^{4} \) Copy content Toggle raw display
show more
show less