↑

Properties

Label 900.6.j.b
Level $900$
Weight $6$
Character orbit 900.j
Analytic conductor $144.345$
Analytic rank $0$
Dimension $20$
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] = [900,6,Mod(557,900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(900, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("900.557");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 900 = 2^{2} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 900.j (of order \(4\), degree \(2\), 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: \(144.345437832\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 222025 x^{16} + 11247583920 x^{12} + 151104106237945 x^{8} + \cdots + 67\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{24}\cdot 5^{4} \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - 8 \beta_{3} + 8) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - 8 \beta_{3} + 8) q^{7} + \beta_{18} q^{11} + ( - \beta_{10} + \beta_{8} - 13 \beta_{3} - 13) q^{13} + (\beta_{18} - \beta_{17} + \cdots + \beta_{4}) q^{17}+ \cdots + ( - 8 \beta_{11} - 101 \beta_{9} + \cdots - 10527) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 152 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 152 q^{7} - 252 q^{13} + 9440 q^{31} - 29988 q^{37} - 6720 q^{43} + 67200 q^{61} - 139552 q^{67} - 134828 q^{73} - 220560 q^{91} - 207132 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 222025 x^{16} + 11247583920 x^{12} + 151104106237945 x^{8} + \cdots + 67\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 92\!\cdots\!13 \nu^{16} + \cdots - 69\!\cdots\!98 ) / 18\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 34\!\cdots\!41 \nu^{17} + \cdots + 32\!\cdots\!66 \nu ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35886151085753 \nu^{18} + \cdots + 11\!\cdots\!93 \nu^{2} ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 11\!\cdots\!49 \nu^{17} + \cdots + 18\!\cdots\!49 \nu ) / 26\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 61\!\cdots\!31 \nu^{18} + \cdots - 94\!\cdots\!40 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61\!\cdots\!31 \nu^{18} + \cdots + 94\!\cdots\!40 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 30\!\cdots\!21 \nu^{18} + \cdots - 29\!\cdots\!96 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 30\!\cdots\!21 \nu^{18} + \cdots - 29\!\cdots\!96 ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 31\!\cdots\!05 \nu^{18} + \cdots - 24\!\cdots\!08 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!05 \nu^{18} + \cdots - 24\!\cdots\!08 ) / 21\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 43\!\cdots\!91 \nu^{18} + \cdots - 34\!\cdots\!51 \nu^{2} ) / 85\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 38\!\cdots\!11 \nu^{19} + \cdots - 53\!\cdots\!92 \nu ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 38\!\cdots\!11 \nu^{19} + \cdots - 53\!\cdots\!92 \nu ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 89\!\cdots\!81 \nu^{19} + \cdots + 10\!\cdots\!52 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 89\!\cdots\!81 \nu^{19} + \cdots + 10\!\cdots\!52 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 45\!\cdots\!79 \nu^{19} + \cdots - 10\!\cdots\!52 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 78\!\cdots\!49 \nu^{19} + \cdots + 60\!\cdots\!88 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 78\!\cdots\!49 \nu^{19} + \cdots - 60\!\cdots\!88 \nu ) / 24\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 16\!\cdots\!53 \nu^{19} + \cdots + 24\!\cdots\!33 \nu^{3} ) / 24\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 117\beta_{15} + 117\beta_{14} + 19\beta_{13} + 19\beta_{12} - 15\beta_{4} + 21\beta_{2} ) / 2160 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{11} - 54 \beta_{10} + 54 \beta_{9} + 258 \beta_{8} - 258 \beta_{7} - 5 \beta_{6} + \cdots + 66236 \beta_{3} ) / 432 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1995 \beta_{19} - 540 \beta_{18} - 540 \beta_{17} - 2091 \beta_{16} + 5991 \beta_{15} + \cdots + 5471 \beta_{12} ) / 540 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 23157 \beta_{10} + 23157 \beta_{9} - 102246 \beta_{8} - 102246 \beta_{7} - 4483 \beta_{6} + \cdots - 19101842 ) / 432 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1163565 \beta_{18} - 1163565 \beta_{17} - 10373247 \beta_{15} - 10373247 \beta_{14} + \cdots - 2573346 \beta_{2} ) / 2160 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 390829 \beta_{11} + 4658787 \beta_{10} - 4658787 \beta_{9} - 20195328 \beta_{8} + \cdots - 3187097938 \beta_{3} ) / 216 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1282918605 \beta_{19} + 535538250 \beta_{18} + 535538250 \beta_{17} + 760557639 \beta_{16} + \cdots - 8318799119 \beta_{12} ) / 2160 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3700432944 \beta_{10} - 3700432944 \beta_{9} + 16003907034 \beta_{8} + 16003907034 \beta_{7} + \cdots + 2302838678480 ) / 432 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 114707303415 \beta_{18} + 114707303415 \beta_{17} + 666858981456 \beta_{15} + \cdots + 116021266863 \beta_{2} ) / 1080 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 126247710515 \beta_{11} - 1463955757839 \beta_{10} + 1463955757839 \beta_{9} + \cdots + 869085809766278 \beta_{3} ) / 432 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 199511398737780 \beta_{19} - 94674634837515 \beta_{18} - 94674634837515 \beta_{17} + \cdots + 14\!\cdots\!24 \beta_{12} ) / 2160 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 144640405871514 \beta_{10} + 144640405871514 \beta_{9} - 627571154757876 \beta_{8} + \cdots - 83\!\cdots\!54 ) / 108 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 38\!\cdots\!20 \beta_{18} + \cdots - 25\!\cdots\!61 \beta_{2} ) / 2160 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 19\!\cdots\!98 \beta_{11} + \cdots - 13\!\cdots\!72 \beta_{3} ) / 432 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 77\!\cdots\!45 \beta_{19} + \cdots - 59\!\cdots\!81 \beta_{12} ) / 540 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 90\!\cdots\!13 \beta_{10} + \cdots + 51\!\cdots\!50 ) / 432 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 60\!\cdots\!85 \beta_{18} + \cdots + 35\!\cdots\!26 \beta_{2} ) / 2160 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 15\!\cdots\!21 \beta_{11} + \cdots + 10\!\cdots\!54 \beta_{3} ) / 216 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( - 48\!\cdots\!05 \beta_{19} + \cdots + 37\!\cdots\!19 \beta_{12} ) / 2160 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/900\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(-\beta_{3}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
557.1
−14.0576 + 14.0576i
14.0576 − 14.0576i
−8.59490 + 8.59490i
8.59490 − 8.59490i
2.20900 − 2.20900i
−2.20900 + 2.20900i
10.2322 − 10.2322i
−10.2322 + 10.2322i
−1.85337 + 1.85337i
1.85337 − 1.85337i
14.0576 + 14.0576i
−14.0576 − 14.0576i
8.59490 + 8.59490i
−8.59490 − 8.59490i
−2.20900 − 2.20900i
2.20900 + 2.20900i
−10.2322 − 10.2322i
10.2322 + 10.2322i
1.85337 + 1.85337i
−1.85337 − 1.85337i
0 0 0 0 0 −141.299 − 141.299i 0 0 0
557.2 0 0 0 0 0 −141.299 − 141.299i 0 0 0
557.3 0 0 0 0 0 −35.3941 − 35.3941i 0 0 0
557.4 0 0 0 0 0 −35.3941 − 35.3941i 0 0 0
557.5 0 0 0 0 0 −3.20270 − 3.20270i 0 0 0
557.6 0 0 0 0 0 −3.20270 − 3.20270i 0 0 0
557.7 0 0 0 0 0 41.6720 + 41.6720i 0 0 0
557.8 0 0 0 0 0 41.6720 + 41.6720i 0 0 0
557.9 0 0 0 0 0 176.224 + 176.224i 0 0 0
557.10 0 0 0 0 0 176.224 + 176.224i 0 0 0
593.1 0 0 0 0 0 −141.299 + 141.299i 0 0 0
593.2 0 0 0 0 0 −141.299 + 141.299i 0 0 0
593.3 0 0 0 0 0 −35.3941 + 35.3941i 0 0 0
593.4 0 0 0 0 0 −35.3941 + 35.3941i 0 0 0
593.5 0 0 0 0 0 −3.20270 + 3.20270i 0 0 0
593.6 0 0 0 0 0 −3.20270 + 3.20270i 0 0 0
593.7 0 0 0 0 0 41.6720 − 41.6720i 0 0 0
593.8 0 0 0 0 0 41.6720 − 41.6720i 0 0 0
593.9 0 0 0 0 0 176.224 − 176.224i 0 0 0
593.10 0 0 0 0 0 176.224 − 176.224i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.6.j.b 20
3.b odd 2 1 inner 900.6.j.b 20
5.b even 2 1 180.6.j.a ✓ 20
5.c odd 4 1 180.6.j.a ✓ 20
5.c odd 4 1 inner 900.6.j.b 20
15.d odd 2 1 180.6.j.a ✓ 20
15.e even 4 1 180.6.j.a ✓ 20
15.e even 4 1 inner 900.6.j.b 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
180.6.j.a ✓ 20 5.b even 2 1
180.6.j.a ✓ 20 5.c odd 4 1
180.6.j.a ✓ 20 15.d odd 2 1
180.6.j.a ✓ 20 15.e even 4 1
900.6.j.b 20 1.a even 1 1 trivial
900.6.j.b 20 3.b odd 2 1 inner
900.6.j.b 20 5.c odd 4 1 inner
900.6.j.b 20 15.e even 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{10} - 76 T_{7}^{9} + 2888 T_{7}^{8} + 3499520 T_{7}^{7} + 2465098368 T_{7}^{6} + \cdots + 44\!\cdots\!68 \) acting on \(S_{6}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 44\!\cdots\!68)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 27\!\cdots\!68)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 15\!\cdots\!68)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 36\!\cdots\!76)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 60\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 44\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots - 37\!\cdots\!32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 91\!\cdots\!68)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 20\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 19\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots - 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 69\!\cdots\!00)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 82\!\cdots\!68)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 12\!\cdots\!32)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 63\!\cdots\!68)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 96\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots - 21\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 52\!\cdots\!68)^{2} \) Copy content Toggle raw display
show more
show less