Properties

Label 600.2.bp.a
Level $600$
Weight $2$
Character orbit 600.bp
Analytic conductor $4.791$
Analytic rank $0$
Dimension $16$
CM discriminant -24
Inner twists $4$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,2,Mod(53,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(20))
 
chi = DirichletCharacter(H, H._module([0, 10, 10, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.53");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.bp (of order \(20\), degree \(8\), 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: \(4.79102412128\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{20})\)
Coefficient field: 16.0.6879707136000000000000.9
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 9x^{12} + 81x^{8} - 729x^{4} + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{20}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{12} - \beta_{8} + \beta_{6} + \beta_{4} - 1) q^{2} + ( - \beta_{13} + \beta_{9} - \beta_{5} + \beta_1) q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{6} - \beta_{4} + 1) q^{5} + (\beta_{13} + \beta_{3}) q^{6} + (\beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{14} + 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{2}) q^{8} - 3 \beta_{14} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{12} - \beta_{8} + \beta_{6} + \beta_{4} - 1) q^{2} + ( - \beta_{13} + \beta_{9} - \beta_{5} + \beta_1) q^{3} - 2 \beta_{2} q^{4} + ( - \beta_{12} + \beta_{11} + \beta_{8} + \beta_{6} - \beta_{4} + 1) q^{5} + (\beta_{13} + \beta_{3}) q^{6} + (\beta_{14} - 2 \beta_{13} + \beta_{12} - \beta_{10} + \beta_{9} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} - \beta_{2} + \beta_1) q^{7} + ( - 2 \beta_{14} + 2 \beta_{10} - 2 \beta_{8} - 2 \beta_{6} + 2 \beta_{2}) q^{8} - 3 \beta_{14} q^{9} + (\beta_{13} + 2 \beta_{12} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_1) q^{10} + ( - 2 \beta_{12} - 2 \beta_{10} + \beta_{7} - \beta_{5} + 2 \beta_{2} + 2) q^{11} + (2 \beta_{15} - 2 \beta_{11} + 2 \beta_{7} - 2 \beta_{3}) q^{12} + ( - 2 \beta_{15} + 2 \beta_{13} + 2 \beta_{11} - \beta_{10} - \beta_{8} - 2 \beta_{7} + \beta_{6} + \cdots - \beta_{2}) q^{14}+ \cdots + (3 \beta_{15} - 6 \beta_{14} - 6 \beta_{12} - 3 \beta_{11} + 6 \beta_{8} + 3 \beta_{7} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 4 q^{5} + 4 q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 4 q^{5} + 4 q^{7} + 8 q^{8} + 8 q^{10} + 24 q^{11} - 12 q^{15} + 16 q^{16} + 48 q^{18} + 8 q^{20} - 36 q^{21} - 16 q^{22} + 32 q^{28} - 12 q^{30} - 64 q^{32} + 12 q^{33} - 8 q^{35} - 24 q^{36} + 24 q^{42} + 48 q^{45} + 4 q^{50} + 28 q^{55} + 24 q^{56} - 8 q^{58} - 80 q^{59} + 12 q^{63} - 36 q^{66} - 32 q^{70} - 24 q^{72} - 28 q^{73} + 24 q^{75} + 12 q^{77} + 64 q^{80} + 36 q^{81} - 24 q^{83} - 36 q^{87} + 32 q^{88} - 24 q^{93} - 16 q^{97} - 72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 9x^{12} + 81x^{8} - 729x^{4} + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} ) / 81 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} ) / 81 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( \nu^{10} ) / 243 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} ) / 243 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{12} ) / 729 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( \nu^{13} ) / 729 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( \nu^{14} ) / 2187 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} ) / 2187 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 81\beta_{8} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 81\beta_{9} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 243\beta_{10} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 243\beta_{11} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 729\beta_{12} \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 729\beta_{13} \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 2187\beta_{14} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 2187\beta_{15} \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−1.54327 + 0.786335i
1.54327 0.786335i
0.270952 1.71073i
−0.270952 + 1.71073i
−1.71073 0.270952i
1.71073 + 0.270952i
0.786335 + 1.54327i
−0.786335 1.54327i
−1.54327 0.786335i
1.54327 + 0.786335i
0.270952 + 1.71073i
−0.270952 1.71073i
−1.71073 + 0.270952i
1.71073 0.270952i
0.786335 1.54327i
−0.786335 + 1.54327i
−0.642040 1.26007i −0.270952 1.71073i −1.17557 + 1.61803i −2.04641 0.901229i −1.98168 + 1.43977i 3.64268 3.64268i 2.79360 + 0.442463i −2.85317 + 0.927051i 0.178260 + 3.15725i
53.2 −0.642040 1.26007i 0.270952 + 1.71073i −1.17557 + 1.61803i −0.473739 + 2.18531i 1.98168 1.43977i −3.20022 + 3.20022i 2.79360 + 0.442463i −2.85317 + 0.927051i 3.05781 0.806108i
77.1 −1.39680 0.221232i −0.786335 1.54327i 1.90211 + 0.618034i −1.48949 1.66775i 0.756934 + 2.32960i −2.44450 2.44450i −2.52015 1.28408i −1.76336 + 2.42705i 1.71157 + 2.65905i
77.2 −1.39680 0.221232i 0.786335 + 1.54327i 1.90211 + 0.618034i 1.93196 1.12585i −0.756934 2.32960i 3.72858 + 3.72858i −2.52015 1.28408i −1.76336 + 2.42705i −2.94764 + 1.14518i
173.1 −0.221232 + 1.39680i −1.54327 + 0.786335i −1.90211 0.618034i 1.66775 1.48949i −0.756934 2.32960i −2.83274 + 2.83274i 1.28408 2.52015i 1.76336 2.42705i 1.71157 + 2.65905i
173.2 −0.221232 + 1.39680i 1.54327 0.786335i −1.90211 0.618034i 1.12585 + 1.93196i 0.756934 + 2.32960i 0.312596 0.312596i 1.28408 2.52015i 1.76336 2.42705i −2.94764 + 1.14518i
197.1 1.26007 0.642040i −1.71073 + 0.270952i 1.17557 1.61803i 2.18531 + 0.473739i −1.98168 + 1.43977i 1.93871 + 1.93871i 0.442463 2.79360i 2.85317 0.927051i 3.05781 0.806108i
197.2 1.26007 0.642040i 1.71073 0.270952i 1.17557 1.61803i −0.901229 + 2.04641i 1.98168 1.43977i 0.854897 + 0.854897i 0.442463 2.79360i 2.85317 0.927051i 0.178260 + 3.15725i
317.1 −0.642040 + 1.26007i −0.270952 + 1.71073i −1.17557 1.61803i −2.04641 + 0.901229i −1.98168 1.43977i 3.64268 + 3.64268i 2.79360 0.442463i −2.85317 0.927051i 0.178260 3.15725i
317.2 −0.642040 + 1.26007i 0.270952 1.71073i −1.17557 1.61803i −0.473739 2.18531i 1.98168 + 1.43977i −3.20022 3.20022i 2.79360 0.442463i −2.85317 0.927051i 3.05781 + 0.806108i
413.1 −1.39680 + 0.221232i −0.786335 + 1.54327i 1.90211 0.618034i −1.48949 + 1.66775i 0.756934 2.32960i −2.44450 + 2.44450i −2.52015 + 1.28408i −1.76336 2.42705i 1.71157 2.65905i
413.2 −1.39680 + 0.221232i 0.786335 1.54327i 1.90211 0.618034i 1.93196 + 1.12585i −0.756934 + 2.32960i 3.72858 3.72858i −2.52015 + 1.28408i −1.76336 2.42705i −2.94764 1.14518i
437.1 −0.221232 1.39680i −1.54327 0.786335i −1.90211 + 0.618034i 1.66775 + 1.48949i −0.756934 + 2.32960i −2.83274 2.83274i 1.28408 + 2.52015i 1.76336 + 2.42705i 1.71157 2.65905i
437.2 −0.221232 1.39680i 1.54327 + 0.786335i −1.90211 + 0.618034i 1.12585 1.93196i 0.756934 2.32960i 0.312596 + 0.312596i 1.28408 + 2.52015i 1.76336 + 2.42705i −2.94764 1.14518i
533.1 1.26007 + 0.642040i −1.71073 0.270952i 1.17557 + 1.61803i 2.18531 0.473739i −1.98168 1.43977i 1.93871 1.93871i 0.442463 + 2.79360i 2.85317 + 0.927051i 3.05781 + 0.806108i
533.2 1.26007 + 0.642040i 1.71073 + 0.270952i 1.17557 + 1.61803i −0.901229 2.04641i 1.98168 + 1.43977i 0.854897 0.854897i 0.442463 + 2.79360i 2.85317 + 0.927051i 0.178260 3.15725i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 53.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
24.h odd 2 1 CM by \(\Q(\sqrt{-6}) \)
25.f odd 20 1 inner
600.bp even 20 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.bp.a 16
3.b odd 2 1 600.2.bp.b yes 16
8.b even 2 1 600.2.bp.b yes 16
24.h odd 2 1 CM 600.2.bp.a 16
25.f odd 20 1 inner 600.2.bp.a 16
75.l even 20 1 600.2.bp.b yes 16
200.x odd 20 1 600.2.bp.b yes 16
600.bp even 20 1 inner 600.2.bp.a 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.bp.a 16 1.a even 1 1 trivial
600.2.bp.a 16 24.h odd 2 1 CM
600.2.bp.a 16 25.f odd 20 1 inner
600.2.bp.a 16 600.bp even 20 1 inner
600.2.bp.b yes 16 3.b odd 2 1
600.2.bp.b yes 16 8.b even 2 1
600.2.bp.b yes 16 75.l even 20 1
600.2.bp.b yes 16 200.x odd 20 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{16} - 4 T_{7}^{15} + 8 T_{7}^{14} + 40 T_{7}^{13} + 986 T_{7}^{12} - 3060 T_{7}^{11} + 5152 T_{7}^{10} + 22372 T_{7}^{9} + 308571 T_{7}^{8} - 664020 T_{7}^{7} + 684320 T_{7}^{6} + 229620 T_{7}^{5} + \cdots + 6225025 \) Copy content Toggle raw display
\( T_{11}^{16} - 24 T_{11}^{15} + 332 T_{11}^{14} - 3192 T_{11}^{13} + 23945 T_{11}^{12} - 145128 T_{11}^{11} + 738072 T_{11}^{10} - 3051736 T_{11}^{9} + 9764486 T_{11}^{8} - 21796120 T_{11}^{7} + 35405100 T_{11}^{6} + \cdots + 515517025 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 2 T^{7} + 2 T^{6} - 4 T^{4} + 8 T^{2} + \cdots + 16)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 9 T^{12} + 81 T^{8} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{16} - 4 T^{15} + 8 T^{14} + \cdots + 390625 \) Copy content Toggle raw display
$7$ \( T^{16} - 4 T^{15} + 8 T^{14} + \cdots + 6225025 \) Copy content Toggle raw display
$11$ \( T^{16} - 24 T^{15} + \cdots + 515517025 \) Copy content Toggle raw display
$13$ \( T^{16} \) Copy content Toggle raw display
$17$ \( T^{16} \) Copy content Toggle raw display
$19$ \( T^{16} \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 116 T^{14} + \cdots + 39062500000000 \) Copy content Toggle raw display
$31$ \( T^{16} + 238 T^{14} + \cdots + 7611645084241 \) Copy content Toggle raw display
$37$ \( T^{16} \) Copy content Toggle raw display
$41$ \( T^{16} \) Copy content Toggle raw display
$43$ \( T^{16} \) Copy content Toggle raw display
$47$ \( T^{16} \) Copy content Toggle raw display
$53$ \( T^{16} + 5300 T^{13} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{16} + 80 T^{15} + \cdots + 34\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} \) Copy content Toggle raw display
$67$ \( T^{16} \) Copy content Toggle raw display
$71$ \( T^{16} \) Copy content Toggle raw display
$73$ \( T^{16} + 28 T^{15} + \cdots + 39062500000000 \) Copy content Toggle raw display
$79$ \( (T^{8} - 216 T^{6} - 3950 T^{5} + \cdots + 55636681)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + 24 T^{15} + \cdots + 24\!\cdots\!41 \) Copy content Toggle raw display
$89$ \( T^{16} \) Copy content Toggle raw display
$97$ \( T^{16} + 16 T^{15} + \cdots + 22\!\cdots\!25 \) Copy content Toggle raw display
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