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Properties

Label 600.3.g.b
Level $600$
Weight $3$
Character orbit 600.g
Analytic conductor $16.349$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [600,3,Mod(451,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.451"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 4 x^{14} - 12 x^{13} + 22 x^{12} - 64 x^{11} + 144 x^{10} - 272 x^{9} + 656 x^{8} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{4} q^{3} + \beta_{2} q^{4} + \beta_{6} q^{6} - \beta_{5} q^{7} + ( - \beta_{3} - 1) q^{8} + 3 q^{9} + ( - \beta_{7} + \beta_{2} - \beta_1 - 2) q^{11} + \beta_{9} q^{12} + (\beta_{15} - \beta_{14} + \cdots + \beta_1) q^{13}+ \cdots + ( - 3 \beta_{7} + 3 \beta_{2} + \cdots - 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 4 q^{4} + 6 q^{6} - 20 q^{8} + 48 q^{9} - 32 q^{11} + 22 q^{14} - 8 q^{16} - 6 q^{18} - 32 q^{19} + 44 q^{22} + 66 q^{26} + 76 q^{28} - 132 q^{32} - 108 q^{34} - 12 q^{36} + 30 q^{38} - 54 q^{42}+ \cdots - 96 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 4 x^{14} - 12 x^{13} + 22 x^{12} - 64 x^{11} + 144 x^{10} - 272 x^{9} + 656 x^{8} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17 \nu^{15} + 29 \nu^{14} + 42 \nu^{13} - 40 \nu^{12} + 82 \nu^{11} - 710 \nu^{10} + 2600 \nu^{9} + \cdots - 806912 ) / 897024 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9 \nu^{15} + 9 \nu^{14} - 298 \nu^{13} + 408 \nu^{12} - 1018 \nu^{11} + 2010 \nu^{10} + \cdots + 544768 ) / 299008 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 63 \nu^{15} + 26 \nu^{14} - 164 \nu^{13} + 292 \nu^{12} - 378 \nu^{11} - 152 \nu^{10} + \cdots + 1114112 ) / 897024 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 20 \nu^{15} + 47 \nu^{14} + 204 \nu^{13} + 56 \nu^{12} - 1604 \nu^{11} + 994 \nu^{10} + \cdots + 950272 ) / 448512 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 41 \nu^{15} - 100 \nu^{14} - 304 \nu^{13} + 924 \nu^{12} - 898 \nu^{11} + 3668 \nu^{10} + \cdots + 1302528 ) / 598016 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 25 \nu^{15} - 22 \nu^{14} + 116 \nu^{13} - 252 \nu^{12} + 1046 \nu^{11} - 2936 \nu^{10} + \cdots - 1032192 ) / 224256 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 150 \nu^{15} + 113 \nu^{14} + 70 \nu^{13} - 1604 \nu^{12} - 96 \nu^{11} - 4082 \nu^{10} + \cdots - 1294336 ) / 897024 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13 \nu^{15} + 317 \nu^{14} - 390 \nu^{13} + 416 \nu^{12} + 1006 \nu^{11} + 1618 \nu^{10} + \cdots + 1847296 ) / 897024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( \nu^{15} - 4 \nu^{12} - 2 \nu^{11} - 20 \nu^{10} + 16 \nu^{9} + 16 \nu^{8} + 112 \nu^{7} + \cdots - 4096 ) / 4096 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 67 \nu^{15} - 834 \nu^{14} - 1132 \nu^{13} - 284 \nu^{12} + 478 \nu^{11} + 7632 \nu^{10} + \cdots + 11386880 ) / 1794048 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 26 \nu^{15} - 35 \nu^{14} + 133 \nu^{13} - 270 \nu^{12} + 532 \nu^{11} - 1582 \nu^{10} + \cdots - 496128 ) / 112128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( \nu^{15} - 2 \nu^{14} + 4 \nu^{13} - 12 \nu^{12} + 22 \nu^{11} - 64 \nu^{10} + 144 \nu^{9} + \cdots - 28672 ) / 4096 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{15} + \beta_{12} - \beta_{11} - \beta_{7} - \beta_{5} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} - 2 \beta_{14} - \beta_{13} - \beta_{11} - 2 \beta_{10} - \beta_{9} - 3 \beta_{8} + \cdots + 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4 \beta_{15} - 6 \beta_{14} + 2 \beta_{13} + 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8 \beta_{15} - 12 \beta_{14} - 4 \beta_{13} + 4 \beta_{12} + 4 \beta_{8} + 4 \beta_{6} - 8 \beta_{5} + \cdots + 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{15} + 8 \beta_{14} - 8 \beta_{13} + 6 \beta_{12} + 2 \beta_{11} - 8 \beta_{10} - 12 \beta_{9} + \cdots - 36 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 4 \beta_{14} + 14 \beta_{13} + 12 \beta_{12} + 2 \beta_{11} - 12 \beta_{10} - 26 \beta_{9} - 6 \beta_{8} + \cdots - 108 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 80 \beta_{15} - 68 \beta_{14} - 32 \beta_{12} + 32 \beta_{11} - 44 \beta_{10} - 112 \beta_{9} + \cdots + 12 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 24 \beta_{15} - 80 \beta_{14} + 80 \beta_{13} - 48 \beta_{12} + 144 \beta_{11} - 104 \beta_{10} + \cdots - 476 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 180 \beta_{15} - 208 \beta_{14} - 184 \beta_{13} - 228 \beta_{12} - 20 \beta_{11} - 256 \beta_{10} + \cdots + 544 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 88 \beta_{15} + 408 \beta_{14} - 244 \beta_{13} - 528 \beta_{12} - 36 \beta_{11} - 104 \beta_{10} + \cdots - 24 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 672 \beta_{15} - 216 \beta_{14} - 232 \beta_{13} + 720 \beta_{12} + 776 \beta_{11} + 56 \beta_{10} + \cdots + 952 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 800 \beta_{15} + 400 \beta_{14} - 544 \beta_{13} + 1200 \beta_{12} + 976 \beta_{11} - 1088 \beta_{10} + \cdots + 10712 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
1.97080 + 0.340537i
1.97080 − 0.340537i
1.83242 + 0.801397i
1.83242 − 0.801397i
1.10374 + 1.66786i
1.10374 − 1.66786i
0.479713 + 1.94162i
0.479713 − 1.94162i
−0.169347 + 1.99282i
−0.169347 − 1.99282i
−0.935441 + 1.76775i
−0.935441 − 1.76775i
−1.53917 + 1.27709i
−1.53917 − 1.27709i
−1.74272 + 0.981294i
−1.74272 − 0.981294i
−1.97080 − 0.340537i −1.73205 3.76807 + 1.34226i 0 3.41352 + 0.589828i 11.8458i −6.96900 − 3.92848i 3.00000 0
451.2 −1.97080 + 0.340537i −1.73205 3.76807 − 1.34226i 0 3.41352 − 0.589828i − 11.8458i −6.96900 + 3.92848i 3.00000 0
451.3 −1.83242 − 0.801397i 1.73205 2.71552 + 2.93699i 0 −3.17384 − 1.38806i 0.610378i −2.62228 − 7.55802i 3.00000 0
451.4 −1.83242 + 0.801397i 1.73205 2.71552 − 2.93699i 0 −3.17384 + 1.38806i − 0.610378i −2.62228 + 7.55802i 3.00000 0
451.5 −1.10374 − 1.66786i −1.73205 −1.56350 + 3.68178i 0 1.91174 + 2.88881i 2.13685i 7.86638 − 1.45605i 3.00000 0
451.6 −1.10374 + 1.66786i −1.73205 −1.56350 − 3.68178i 0 1.91174 − 2.88881i − 2.13685i 7.86638 + 1.45605i 3.00000 0
451.7 −0.479713 − 1.94162i 1.73205 −3.53975 + 1.86284i 0 −0.830888 − 3.36298i − 8.37149i 5.31498 + 5.97921i 3.00000 0
451.8 −0.479713 + 1.94162i 1.73205 −3.53975 − 1.86284i 0 −0.830888 + 3.36298i 8.37149i 5.31498 − 5.97921i 3.00000 0
451.9 0.169347 − 1.99282i −1.73205 −3.94264 − 0.674955i 0 −0.293318 + 3.45166i − 2.71814i −2.01274 + 7.74267i 3.00000 0
451.10 0.169347 + 1.99282i −1.73205 −3.94264 + 0.674955i 0 −0.293318 − 3.45166i 2.71814i −2.01274 − 7.74267i 3.00000 0
451.11 0.935441 − 1.76775i 1.73205 −2.24990 − 3.30726i 0 1.62023 − 3.06184i 10.9881i −7.95106 + 0.883521i 3.00000 0
451.12 0.935441 + 1.76775i 1.73205 −2.24990 + 3.30726i 0 1.62023 + 3.06184i − 10.9881i −7.95106 − 0.883521i 3.00000 0
451.13 1.53917 − 1.27709i −1.73205 0.738072 − 3.93132i 0 −2.66592 + 2.21199i 8.70188i −3.88464 − 6.99354i 3.00000 0
451.14 1.53917 + 1.27709i −1.73205 0.738072 + 3.93132i 0 −2.66592 − 2.21199i − 8.70188i −3.88464 + 6.99354i 3.00000 0
451.15 1.74272 − 0.981294i 1.73205 2.07413 − 3.42023i 0 3.01847 − 1.69965i − 6.06690i 0.258360 − 7.99583i 3.00000 0
451.16 1.74272 + 0.981294i 1.73205 2.07413 + 3.42023i 0 3.01847 + 1.69965i 6.06690i 0.258360 + 7.99583i 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.3.g.b ✓ 16
4.b odd 2 1 2400.3.g.d 16
5.b even 2 1 600.3.g.c yes 16
5.c odd 4 2 600.3.p.c 32
8.b even 2 1 2400.3.g.d 16
8.d odd 2 1 inner 600.3.g.b ✓ 16
20.d odd 2 1 2400.3.g.c 16
20.e even 4 2 2400.3.p.c 32
40.e odd 2 1 600.3.g.c yes 16
40.f even 2 1 2400.3.g.c 16
40.i odd 4 2 2400.3.p.c 32
40.k even 4 2 600.3.p.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.3.g.b ✓ 16 1.a even 1 1 trivial
600.3.g.b ✓ 16 8.d odd 2 1 inner
600.3.g.c yes 16 5.b even 2 1
600.3.g.c yes 16 40.e odd 2 1
600.3.p.c 32 5.c odd 4 2
600.3.p.c 32 40.k even 4 2
2400.3.g.c 16 20.d odd 2 1
2400.3.g.c 16 40.f even 2 1
2400.3.g.d 16 4.b odd 2 1
2400.3.g.d 16 8.b even 2 1
2400.3.p.c 32 20.e even 4 2
2400.3.p.c 32 40.i odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{16} + 456 T_{7}^{14} + 80796 T_{7}^{12} + 7020664 T_{7}^{10} + 309600966 T_{7}^{8} + \cdots + 41593807233 \) Copy content Toggle raw display
\( T_{17}^{8} - 1040 T_{17}^{6} + 3248 T_{17}^{5} + 147888 T_{17}^{4} + 183424 T_{17}^{3} - 3135808 T_{17}^{2} + \cdots - 3681728 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 2 T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 41593807233 \) Copy content Toggle raw display
$11$ \( (T^{8} + 16 T^{7} + \cdots + 4975168)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 14328667961793 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1040 T^{6} + \cdots - 3681728)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 16 T^{7} + \cdots + 67183681)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 96\!\cdots\!88 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 31\!\cdots\!32 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 36\!\cdots\!93 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 91\!\cdots\!72 \) Copy content Toggle raw display
$41$ \( (T^{8} - 6080 T^{6} + \cdots + 82365625408)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 48 T^{7} + \cdots - 256825457759)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 20\!\cdots\!12 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 51\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 4317848463104)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 37\!\cdots\!93 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots - 8301889426559)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 33\!\cdots\!28 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 5531610691328)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 26\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 298413965580032)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 37741159803743)^{2} \) Copy content Toggle raw display
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