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Properties

Label 900.6.j.a
Level $900$
Weight $6$
Character orbit 900.j
Analytic conductor $144.345$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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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: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 16112194x^{12} + 72373894590801x^{8} + 60780662400876311824x^{4} + 14178241191207403341807616 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{49}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{24} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{7} + (\beta_{14} - \beta_{11}) q^{11} + (\beta_{8} + \beta_{6}) q^{13} - \beta_{13} q^{17} + ( - \beta_{5} + 224 \beta_{4}) q^{19} + \beta_{2} q^{23} + (8 \beta_{15} + 10 \beta_{12}) q^{29} + (2 \beta_1 + 352) q^{31} + ( - 3 \beta_{9} + 89 \beta_{7}) q^{37} + ( - 4 \beta_{14} - 3 \beta_{11}) q^{41} + ( - 8 \beta_{8} + 80 \beta_{6}) q^{43} + ( - 2 \beta_{13} - 6 \beta_{10}) q^{47} + (4 \beta_{5} - 2989 \beta_{4}) q^{49} + ( - 11 \beta_{3} - 4 \beta_{2}) q^{53} + ( - 59 \beta_{15} + \beta_{12}) q^{59} + ( - 16 \beta_1 - 7450) q^{61} + (32 \beta_{9} - 174 \beta_{7}) q^{67} + (2 \beta_{14} - 82 \beta_{11}) q^{71} + (14 \beta_{8} - 306 \beta_{6}) q^{73} + ( - 5 \beta_{13} + 16 \beta_{10}) q^{77} + (12 \beta_{5} - 20720 \beta_{4}) q^{79} + (32 \beta_{3} + 14 \beta_{2}) q^{83} + (124 \beta_{15} + 477 \beta_{12}) q^{89} + (43 \beta_1 - 8628) q^{91} + ( - 124 \beta_{9} - 244 \beta_{7}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 5632 q^{31} - 119200 q^{61} - 138048 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 16112194x^{12} + 72373894590801x^{8} + 60780662400876311824x^{4} + 14178241191207403341807616 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 864 \nu^{12} - 10445698697 \nu^{8} + \cdots + 73\!\cdots\!76 ) / 37\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25\!\cdots\!49 \nu^{13} + \cdots + 10\!\cdots\!68 \nu ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 59\!\cdots\!71 \nu^{13} + \cdots + 13\!\cdots\!48 \nu ) / 12\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 9588928073 \nu^{14} + \cdots + 92\!\cdots\!32 \nu^{2} ) / 41\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1111307 \nu^{14} - 19788294245206 \nu^{10} + \cdots - 13\!\cdots\!52 \nu^{2} ) / 30\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 93\!\cdots\!29 \nu^{14} + \cdots - 17\!\cdots\!28 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 93\!\cdots\!29 \nu^{14} + \cdots - 17\!\cdots\!28 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35\!\cdots\!23 \nu^{14} + \cdots + 72\!\cdots\!96 ) / 60\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 35\!\cdots\!23 \nu^{14} + \cdots + 72\!\cdots\!96 ) / 60\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 24\!\cdots\!11 \nu^{15} + \cdots - 20\!\cdots\!40 \nu^{3} ) / 11\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 49\!\cdots\!83 \nu^{15} + \cdots + 98\!\cdots\!16 \nu ) / 21\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 49\!\cdots\!83 \nu^{15} + \cdots + 98\!\cdots\!16 \nu ) / 21\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 88\!\cdots\!79 \nu^{15} + \cdots + 65\!\cdots\!28 \nu^{3} ) / 28\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 38\!\cdots\!31 \nu^{15} + \cdots - 79\!\cdots\!88 \nu ) / 45\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 38\!\cdots\!31 \nu^{15} + \cdots + 79\!\cdots\!88 \nu ) / 45\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{12} - \beta_{11} + 6\beta_{3} - 3\beta_{2} ) / 324 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{6} + 2\beta_{5} + 6910\beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -729\beta_{15} - 729\beta_{14} + 9909\beta_{13} + 4831\beta_{12} - 4831\beta_{11} + 9450\beta_{10} ) / 324 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -222\beta_{9} - 222\beta_{8} - 4745\beta_{7} - 4745\beta_{6} + 6912\beta _1 - 16112194 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4197825 \beta_{15} + 4197825 \beta_{14} + 18110431 \beta_{12} + 18110431 \beta_{11} + \cdots + 26836890 \beta_{2} ) / 324 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 1150626 \beta_{9} + 1150626 \beta_{8} - 18766761 \beta_{7} + 18766761 \beta_{6} + \cdots - 42271528810 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 17004150261 \beta_{15} + 17004150261 \beta_{14} - 58848770409 \beta_{13} + \cdots - 74426052990 \beta_{10} ) / 324 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4439240094 \beta_{9} + 4439240094 \beta_{8} + 68034906025 \beta_{7} + 68034906025 \beta_{6} + \cdots + 114855006312034 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 60875592361785 \beta_{15} - 60875592361785 \beta_{14} - 228669950022991 \beta_{12} + \cdots - 205233878893842 \beta_{2} ) / 324 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 15452236999962 \beta_{9} - 15452236999962 \beta_{8} + 233147671639721 \beta_{7} + \cdots + 31\!\cdots\!50 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 20\!\cdots\!33 \beta_{15} + \cdots + 56\!\cdots\!50 \beta_{10} ) / 324 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 51\!\cdots\!26 \beta_{9} + \cdots - 86\!\cdots\!74 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 66\!\cdots\!25 \beta_{15} + \cdots + 15\!\cdots\!34 \beta_{2} ) / 324 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 16\!\cdots\!78 \beta_{9} + \cdots - 23\!\cdots\!30 \beta_{4} ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 21\!\cdots\!45 \beta_{15} + \cdots - 42\!\cdots\!50 \beta_{10} ) / 324 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(451\) \(577\)
\(\chi(n)\) \(-1\) \(1\) \(\beta_{4}\)

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
37.4215 + 37.4215i
−37.4215 − 37.4215i
−19.1476 − 19.1476i
19.1476 + 19.1476i
18.4405 + 18.4405i
−18.4405 − 18.4405i
−36.7144 − 36.7144i
36.7144 + 36.7144i
−37.4215 + 37.4215i
37.4215 − 37.4215i
19.1476 − 19.1476i
−19.1476 + 19.1476i
−18.4405 + 18.4405i
18.4405 − 18.4405i
36.7144 − 36.7144i
−36.7144 + 36.7144i
0 0 0 0 0 −104.844 − 104.844i 0 0 0
557.2 0 0 0 0 0 −104.844 − 104.844i 0 0 0
557.3 0 0 0 0 0 −53.1577 − 53.1577i 0 0 0
557.4 0 0 0 0 0 −53.1577 − 53.1577i 0 0 0
557.5 0 0 0 0 0 53.1577 + 53.1577i 0 0 0
557.6 0 0 0 0 0 53.1577 + 53.1577i 0 0 0
557.7 0 0 0 0 0 104.844 + 104.844i 0 0 0
557.8 0 0 0 0 0 104.844 + 104.844i 0 0 0
593.1 0 0 0 0 0 −104.844 + 104.844i 0 0 0
593.2 0 0 0 0 0 −104.844 + 104.844i 0 0 0
593.3 0 0 0 0 0 −53.1577 + 53.1577i 0 0 0
593.4 0 0 0 0 0 −53.1577 + 53.1577i 0 0 0
593.5 0 0 0 0 0 53.1577 − 53.1577i 0 0 0
593.6 0 0 0 0 0 53.1577 − 53.1577i 0 0 0
593.7 0 0 0 0 0 104.844 − 104.844i 0 0 0
593.8 0 0 0 0 0 104.844 − 104.844i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 557.8
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
5.c odd 4 2 inner
15.d odd 2 1 inner
15.e even 4 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.6.j.a ✓ 16
3.b odd 2 1 inner 900.6.j.a ✓ 16
5.b even 2 1 inner 900.6.j.a ✓ 16
5.c odd 4 2 inner 900.6.j.a ✓ 16
15.d odd 2 1 inner 900.6.j.a ✓ 16
15.e even 4 2 inner 900.6.j.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.6.j.a ✓ 16 1.a even 1 1 trivial
900.6.j.a ✓ 16 3.b odd 2 1 inner
900.6.j.a ✓ 16 5.b even 2 1 inner
900.6.j.a ✓ 16 5.c odd 4 2 inner
900.6.j.a ✓ 16 15.d odd 2 1 inner
900.6.j.a ✓ 16 15.e even 4 2 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 515258568T_{7}^{4} + 15436811079361296 \) acting on \(S_{6}^{\mathrm{new}}(900, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 15\!\cdots\!96)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 470196 T^{2} + 52003153764)^{4} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 98\!\cdots\!76)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 16958615831056)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 31\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 205190498078784)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 704 T - 16549136)^{8} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 19\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 11469858811524)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 70\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 63\!\cdots\!84)^{4} \) Copy content Toggle raw display
$61$ \( (T^{2} + 14900 T - 1011572060)^{8} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 52\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 72\!\cdots\!44)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 48\!\cdots\!16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots + 29\!\cdots\!00)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 16\!\cdots\!24)^{4} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 20\!\cdots\!16)^{2} \) Copy content Toggle raw display
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