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Properties

Label 450.2.l.d
Level $450$
Weight $2$
Character orbit 450.l
Analytic conductor $3.593$
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] = [450,2,Mod(19,450)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(450, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 9]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("450.19");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 450.l (of order \(10\), degree \(4\), 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: \(3.59326809096\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{11} q^{2} + (\beta_{5} + \beta_{4} + \beta_{3} + 1) q^{4} + (\beta_{14} - \beta_{13} + \cdots - \beta_{8}) q^{5}+ \cdots + \beta_{12} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{11} q^{2} + (\beta_{5} + \beta_{4} + \beta_{3} + 1) q^{4} + (\beta_{14} - \beta_{13} + \cdots - \beta_{8}) q^{5}+ \cdots + (\beta_{15} - \beta_{14} + \cdots + 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 4 q^{16} - 16 q^{19} - 20 q^{22} + 20 q^{25} - 10 q^{28} + 6 q^{31} - 26 q^{34} + 10 q^{37} + 20 q^{46} + 28 q^{49} - 20 q^{55} + 32 q^{61} + 4 q^{64} - 40 q^{67} - 30 q^{70} - 24 q^{76} - 36 q^{79} - 70 q^{85} + 10 q^{88} + 52 q^{91} - 70 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1807 \nu^{14} - 3112 \nu^{12} + 10523 \nu^{10} + 91863 \nu^{8} + 793 \nu^{6} - 961270 \nu^{4} + \cdots - 1433125 ) / 633875 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + \cdots + 563878625 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} + \cdots - 167096475 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} + \cdots - 215410900 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 454006 \nu^{14} + 189729 \nu^{12} + 1810849 \nu^{10} + 19437794 \nu^{8} - 33302741 \nu^{6} + \cdots - 184475875 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2758534 \nu^{14} - 1453576 \nu^{12} - 9476246 \nu^{10} - 120179351 \nu^{8} + 220846139 \nu^{6} + \cdots + 1026496375 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2293 \nu^{15} - 486 \nu^{13} + 12284 \nu^{11} + 101834 \nu^{9} - 117086 \nu^{7} + \cdots - 955700 \nu ) / 633875 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 3183877 \nu^{15} + 4006568 \nu^{13} + 6543803 \nu^{11} + 134015793 \nu^{9} + \cdots - 1245786250 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 4538397 \nu^{14} + 2417483 \nu^{12} + 16184468 \nu^{10} + 196094683 \nu^{8} + \cdots - 1833539875 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8189122 \nu^{15} - 3158678 \nu^{13} - 32153713 \nu^{11} - 353467203 \nu^{9} + \cdots + 3331016750 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 8211971 \nu^{15} - 5580699 \nu^{13} - 25580004 \nu^{11} - 355005349 \nu^{9} + \cdots + 3295922875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 9262311 \nu^{15} - 650299 \nu^{13} - 34740779 \nu^{11} - 420559724 \nu^{9} + \cdots + 5129227500 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 441862 \nu^{15} + 108648 \nu^{13} + 1544473 \nu^{11} + 20140738 \nu^{9} - 29602957 \nu^{7} + \cdots - 215410900 \nu ) / 34356025 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 723142 \nu^{15} - 177867 \nu^{13} - 2483482 \nu^{11} - 32522627 \nu^{9} + 47878273 \nu^{7} + \cdots + 382507375 \nu ) / 34356025 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} - \beta_{6} + 2\beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} - \beta_{13} - 4\beta_{12} + \beta_{11} - 4\beta_{9} - \beta_{8} - \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{10} + 5\beta_{7} - 4\beta_{6} - 2\beta_{5} - \beta_{4} - 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{15} - 6\beta_{14} + 13\beta_{11} + 13\beta_{9} + 19\beta_{8} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{7} + 7\beta_{6} + 15\beta_{5} + 8\beta_{4} + 29\beta_{3} - 2\beta_{2} + 29 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 23\beta_{15} - 38\beta_{13} - 15\beta_{12} - 4\beta_{9} - 15\beta_{8} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 27\beta_{10} + 15\beta_{7} - 27\beta_{6} + 57\beta_{5} - 53\beta_{4} + 57\beta_{3} + 15\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 95\beta_{15} + 95\beta_{14} - 57\beta_{13} - 96\beta_{12} + 57\beta_{11} - 57\beta_{9} - 57\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 152\beta_{10} + 248\beta_{7} - 209\beta_{4} - 209\beta_{3} - 190 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -39\beta_{14} - 39\beta_{13} + 305\beta_{12} + 344\beta_{11} + 857\beta_{9} + 857\beta_{8} + 19\beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -494\beta_{10} + 799\beta_{6} + 935\beta_{5} + 1012\beta_{3} - 494\beta_{2} + 441 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 1506 \beta_{15} + 935 \beta_{14} - 1506 \beta_{13} + 281 \beta_{12} - 1104 \beta_{11} + 281 \beta_{9} + \cdots - 571 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( -450\beta_{10} - 450\beta_{7} + 281\beta_{6} + 2358\beta_{5} - 3376\beta_{4} + 281\beta_{3} - 169\beta_{2} - 3545 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 3826\beta_{15} + 6184\beta_{14} - 1012\beta_{11} - 1012\beta_{9} - 1631\beta_{8} - 3826\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/450\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(1 + \beta_{3} + \beta_{4} + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
1.86824 − 0.357358i
−0.917186 + 1.66637i
0.917186 − 1.66637i
−1.86824 + 0.357358i
−0.0566033 − 1.17421i
0.644389 + 0.983224i
−0.644389 − 0.983224i
0.0566033 + 1.17421i
−0.0566033 + 1.17421i
0.644389 − 0.983224i
−0.644389 + 0.983224i
0.0566033 − 1.17421i
1.86824 + 0.357358i
−0.917186 − 1.66637i
0.917186 + 1.66637i
−1.86824 − 0.357358i
−0.951057 + 0.309017i 0 0.809017 − 0.587785i −1.95894 − 1.07822i 0 3.03582i −0.587785 + 0.809017i 0 2.19625 + 0.420099i
19.2 −0.951057 + 0.309017i 0 0.809017 − 0.587785i 0.420099 + 2.19625i 0 − 0.407162i −0.587785 + 0.809017i 0 −1.07822 − 1.95894i
19.3 0.951057 − 0.309017i 0 0.809017 − 0.587785i −0.420099 − 2.19625i 0 − 0.407162i 0.587785 − 0.809017i 0 −1.07822 − 1.95894i
19.4 0.951057 − 0.309017i 0 0.809017 − 0.587785i 1.95894 + 1.07822i 0 3.03582i 0.587785 − 0.809017i 0 2.19625 + 0.420099i
109.1 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i −1.87020 + 1.22570i 0 − 3.26086i 0.951057 + 0.309017i 0 0.107666 − 2.23347i
109.2 −0.587785 + 0.809017i 0 −0.309017 − 0.951057i 2.23347 − 0.107666i 0 − 0.992398i 0.951057 + 0.309017i 0 −1.22570 + 1.87020i
109.3 0.587785 − 0.809017i 0 −0.309017 − 0.951057i −2.23347 + 0.107666i 0 − 0.992398i −0.951057 − 0.309017i 0 −1.22570 + 1.87020i
109.4 0.587785 − 0.809017i 0 −0.309017 − 0.951057i 1.87020 − 1.22570i 0 − 3.26086i −0.951057 − 0.309017i 0 0.107666 − 2.23347i
289.1 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i −1.87020 − 1.22570i 0 3.26086i 0.951057 − 0.309017i 0 0.107666 + 2.23347i
289.2 −0.587785 − 0.809017i 0 −0.309017 + 0.951057i 2.23347 + 0.107666i 0 0.992398i 0.951057 − 0.309017i 0 −1.22570 − 1.87020i
289.3 0.587785 + 0.809017i 0 −0.309017 + 0.951057i −2.23347 − 0.107666i 0 0.992398i −0.951057 + 0.309017i 0 −1.22570 − 1.87020i
289.4 0.587785 + 0.809017i 0 −0.309017 + 0.951057i 1.87020 + 1.22570i 0 3.26086i −0.951057 + 0.309017i 0 0.107666 + 2.23347i
379.1 −0.951057 − 0.309017i 0 0.809017 + 0.587785i −1.95894 + 1.07822i 0 − 3.03582i −0.587785 − 0.809017i 0 2.19625 − 0.420099i
379.2 −0.951057 − 0.309017i 0 0.809017 + 0.587785i 0.420099 − 2.19625i 0 0.407162i −0.587785 − 0.809017i 0 −1.07822 + 1.95894i
379.3 0.951057 + 0.309017i 0 0.809017 + 0.587785i −0.420099 + 2.19625i 0 0.407162i 0.587785 + 0.809017i 0 −1.07822 + 1.95894i
379.4 0.951057 + 0.309017i 0 0.809017 + 0.587785i 1.95894 − 1.07822i 0 − 3.03582i 0.587785 + 0.809017i 0 2.19625 − 0.420099i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.e even 10 1 inner
75.h odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.2.l.d ✓ 16
3.b odd 2 1 inner 450.2.l.d ✓ 16
25.e even 10 1 inner 450.2.l.d ✓ 16
75.h odd 10 1 inner 450.2.l.d ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.l.d ✓ 16 1.a even 1 1 trivial
450.2.l.d ✓ 16 3.b odd 2 1 inner
450.2.l.d ✓ 16 25.e even 10 1 inner
450.2.l.d ✓ 16 75.h odd 10 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 21T_{7}^{6} + 121T_{7}^{4} + 116T_{7}^{2} + 16 \) acting on \(S_{2}^{\mathrm{new}}(450, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} - 10 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( (T^{8} + 21 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + 17 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$13$ \( (T^{8} + 14 T^{6} + \cdots + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} - 72 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$19$ \( (T^{8} + 8 T^{7} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 10485760000 \) Copy content Toggle raw display
$29$ \( T^{16} + 32 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$31$ \( (T^{8} - 3 T^{7} + \cdots + 5180176)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} - 5 T^{7} + \cdots + 633616)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + 100 T^{14} + \cdots + 160000 \) Copy content Toggle raw display
$43$ \( (T^{8} + 176 T^{6} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 80 T^{6} + \cdots + 160000)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 104060401 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 9250941239296 \) Copy content Toggle raw display
$61$ \( (T^{8} - 16 T^{7} + \cdots + 495616)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 20 T^{7} + \cdots + 25563136)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 60523872256 \) Copy content Toggle raw display
$73$ \( (T^{8} - 180 T^{6} + \cdots + 19360000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 18 T^{7} + \cdots + 20214016)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 70\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 26639462656 \) Copy content Toggle raw display
$97$ \( (T^{8} + 35 T^{7} + \cdots + 12313081)^{2} \) Copy content Toggle raw display
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