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Properties

Label 600.3.g.d
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 = [16,4] 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: \( x^{16} - 4 x^{15} + x^{14} + 24 x^{13} - 44 x^{12} - 32 x^{11} + 180 x^{10} - 64 x^{9} - 352 x^{8} + \cdots + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{19}\cdot 3 \)
Twist minimal: no (minimal twist has level 120)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_{5} q^{3} + ( - \beta_{7} + 1) q^{4} - \beta_{6} q^{6} + (\beta_{6} + \beta_{3}) q^{7} + ( - \beta_{13} + \beta_{5} - \beta_{2} - 2) q^{8} + 3 q^{9} + (\beta_{15} - \beta_{14} + \beta_{11} + \cdots + 4) q^{11}+ \cdots + (3 \beta_{15} - 3 \beta_{14} + \cdots + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{2} + 14 q^{4} + 6 q^{6} - 20 q^{8} + 48 q^{9} + 64 q^{11} - 20 q^{14} - 14 q^{16} + 12 q^{18} - 32 q^{19} - 28 q^{22} - 54 q^{24} + 36 q^{26} + 28 q^{28} - 36 q^{32} - 72 q^{34} + 42 q^{36}+ \cdots + 192 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + x^{14} + 24 x^{13} - 44 x^{12} - 32 x^{11} + 180 x^{10} - 64 x^{9} - 352 x^{8} + \cdots + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + \nu^{13} + 24 \nu^{12} - 44 \nu^{11} - 32 \nu^{10} + 180 \nu^{9} + \cdots - 81920 ) / 16384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 4 \nu^{14} + \nu^{13} + 24 \nu^{12} - 44 \nu^{11} - 32 \nu^{10} + 180 \nu^{9} + \cdots - 65536 ) / 16384 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 5 \nu^{15} + 23 \nu^{14} - 9 \nu^{13} - 61 \nu^{12} + 188 \nu^{11} + 340 \nu^{10} - 596 \nu^{9} + \cdots + 299008 ) / 36864 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{15} + 84 \nu^{14} + 95 \nu^{13} - 616 \nu^{12} + 760 \nu^{11} + 1248 \nu^{10} + \cdots + 1343488 ) / 147456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11 \nu^{15} + 20 \nu^{14} + 27 \nu^{13} - 136 \nu^{12} + 50 \nu^{11} + 352 \nu^{10} + \cdots + 163840 ) / 73728 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 5 \nu^{15} - 2 \nu^{14} + 35 \nu^{13} - 66 \nu^{12} - 52 \nu^{11} + 260 \nu^{10} - 196 \nu^{9} + \cdots + 36864 ) / 36864 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{15} - 3 \nu^{14} - 3 \nu^{13} + 25 \nu^{12} - 20 \nu^{11} - 76 \nu^{10} + 148 \nu^{9} + \cdots - 57344 ) / 4096 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43 \nu^{15} + 116 \nu^{14} + 85 \nu^{13} - 816 \nu^{12} + 388 \nu^{11} + 2128 \nu^{10} + \cdots + 1032192 ) / 147456 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 25 \nu^{15} + 62 \nu^{14} + 85 \nu^{13} - 438 \nu^{12} + 154 \nu^{11} + 1336 \nu^{10} + \cdots + 663552 ) / 73728 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 61 \nu^{15} - 240 \nu^{14} - 167 \nu^{13} + 1468 \nu^{12} - 736 \nu^{11} - 4224 \nu^{10} + \cdots - 2768896 ) / 147456 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 73 \nu^{15} - 260 \nu^{14} - 91 \nu^{13} + 1656 \nu^{12} - 1696 \nu^{11} - 3856 \nu^{10} + \cdots - 2703360 ) / 147456 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 85 \nu^{15} - 196 \nu^{14} - 363 \nu^{13} + 1688 \nu^{12} - 604 \nu^{11} - 5264 \nu^{10} + \cdots - 3031040 ) / 147456 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 26 \nu^{15} - 80 \nu^{14} - 63 \nu^{13} + 580 \nu^{12} - 461 \nu^{11} - 1552 \nu^{10} + \cdots - 1024000 ) / 36864 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 37 \nu^{15} + 87 \nu^{14} + 155 \nu^{13} - 697 \nu^{12} + 352 \nu^{11} + 1848 \nu^{10} + \cdots + 974848 ) / 36864 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 85 \nu^{15} + 214 \nu^{14} + 309 \nu^{13} - 1670 \nu^{12} + 982 \nu^{11} + 4976 \nu^{10} + \cdots + 2883584 ) / 73728 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{12} - \beta_{11} - \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{4} + \cdots + 5 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{12} - 4\beta_{9} - 2\beta_{6} + 4\beta_{5} + \beta_{2} + \beta _1 - 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7 \beta_{15} + 4 \beta_{14} - 4 \beta_{13} - 7 \beta_{12} - 3 \beta_{11} - 3 \beta_{9} - 5 \beta_{8} + \cdots + 3 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 8 \beta_{13} + 6 \beta_{12} - 8 \beta_{11} - 8 \beta_{10} - 4 \beta_{9} - 8 \beta_{8} - 8 \beta_{7} + \cdots - 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 17 \beta_{15} - 4 \beta_{14} - 28 \beta_{13} - 9 \beta_{12} + 3 \beta_{11} + 3 \beta_{9} - 19 \beta_{8} + \cdots + 17 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 4 \beta_{15} - 16 \beta_{14} + 24 \beta_{13} + 6 \beta_{12} - 20 \beta_{11} + 8 \beta_{10} + 24 \beta_{9} + \cdots - 35 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 81 \beta_{15} - 76 \beta_{14} - 20 \beta_{13} - 63 \beta_{12} + 45 \beta_{11} + 32 \beta_{10} + \cdots - 341 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 184 \beta_{15} - 64 \beta_{14} - 40 \beta_{13} - 26 \beta_{12} - 64 \beta_{11} + 152 \beta_{10} + \cdots + 519 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 335 \beta_{15} - 180 \beta_{14} - 76 \beta_{13} + 7 \beta_{12} + 115 \beta_{11} + 128 \beta_{10} + \cdots - 1655 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 68 \beta_{15} + 80 \beta_{14} + 216 \beta_{13} + 334 \beta_{12} - 604 \beta_{11} + 360 \beta_{10} + \cdots + 581 ) / 4 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 97 \beta_{15} + 308 \beta_{14} + 812 \beta_{13} + 257 \beta_{12} + 1117 \beta_{11} - 160 \beta_{10} + \cdots - 2925 ) / 4 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2544 \beta_{15} + 3872 \beta_{14} + 2232 \beta_{13} - 258 \beta_{12} - 904 \beta_{11} - 616 \beta_{10} + \cdots + 3583 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 575 \beta_{15} + 3020 \beta_{14} + 7156 \beta_{13} + 5063 \beta_{12} - 221 \beta_{11} - 7744 \beta_{10} + \cdots - 3839 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 10396 \beta_{15} + 6768 \beta_{14} - 10056 \beta_{13} + 150 \beta_{12} - 84 \beta_{11} - 6040 \beta_{10} + \cdots + 11053 ) / 4 \) 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
−1.95484 + 0.422617i
−1.95484 − 0.422617i
−1.86802 + 0.714481i
−1.86802 − 0.714481i
−1.08168 + 1.68225i
−1.08168 − 1.68225i
0.444906 + 1.94989i
0.444906 − 1.94989i
1.13200 + 1.64881i
1.13200 − 1.64881i
1.56126 + 1.24999i
1.56126 − 1.24999i
1.81470 + 0.840753i
1.81470 − 0.840753i
1.95167 + 0.436996i
1.95167 − 0.436996i
−1.95484 − 0.422617i 1.73205 3.64279 + 1.65230i 0 −3.38588 − 0.731994i − 8.99346i −6.42278 − 4.76948i 3.00000 0
451.2 −1.95484 + 0.422617i 1.73205 3.64279 − 1.65230i 0 −3.38588 + 0.731994i 8.99346i −6.42278 + 4.76948i 3.00000 0
451.3 −1.86802 − 0.714481i −1.73205 2.97903 + 2.66934i 0 3.23551 + 1.23752i 0.274624i −3.65772 − 7.11485i 3.00000 0
451.4 −1.86802 + 0.714481i −1.73205 2.97903 − 2.66934i 0 3.23551 − 1.23752i − 0.274624i −3.65772 + 7.11485i 3.00000 0
451.5 −1.08168 − 1.68225i −1.73205 −1.65995 + 3.63931i 0 1.87352 + 2.91375i − 3.06445i 7.91777 − 1.14409i 3.00000 0
451.6 −1.08168 + 1.68225i −1.73205 −1.65995 − 3.63931i 0 1.87352 − 2.91375i 3.06445i 7.91777 + 1.14409i 3.00000 0
451.7 0.444906 − 1.94989i 1.73205 −3.60412 − 1.73503i 0 0.770599 − 3.37730i 5.48395i −4.98661 + 6.25570i 3.00000 0
451.8 0.444906 + 1.94989i 1.73205 −3.60412 + 1.73503i 0 0.770599 + 3.37730i − 5.48395i −4.98661 − 6.25570i 3.00000 0
451.9 1.13200 − 1.64881i −1.73205 −1.43715 − 3.73291i 0 −1.96068 + 2.85582i − 11.6935i −7.78171 − 1.85607i 3.00000 0
451.10 1.13200 + 1.64881i −1.73205 −1.43715 + 3.73291i 0 −1.96068 − 2.85582i 11.6935i −7.78171 + 1.85607i 3.00000 0
451.11 1.56126 − 1.24999i 1.73205 0.875058 − 3.90311i 0 2.70418 − 2.16504i 7.58970i −3.51265 − 7.18758i 3.00000 0
451.12 1.56126 + 1.24999i 1.73205 0.875058 + 3.90311i 0 2.70418 + 2.16504i − 7.58970i −3.51265 + 7.18758i 3.00000 0
451.13 1.81470 − 0.840753i 1.73205 2.58627 − 3.05143i 0 3.14315 − 1.45623i − 6.88771i 2.12781 − 7.71184i 3.00000 0
451.14 1.81470 + 0.840753i 1.73205 2.58627 + 3.05143i 0 3.14315 + 1.45623i 6.88771i 2.12781 + 7.71184i 3.00000 0
451.15 1.95167 − 0.436996i −1.73205 3.61807 − 1.70575i 0 −3.38040 + 0.756900i 8.35441i 6.31589 − 4.91015i 3.00000 0
451.16 1.95167 + 0.436996i −1.73205 3.61807 + 1.70575i 0 −3.38040 − 0.756900i − 8.35441i 6.31589 + 4.91015i 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.d 16
4.b odd 2 1 2400.3.g.b 16
5.b even 2 1 120.3.g.a ✓ 16
5.c odd 4 2 600.3.p.b 32
8.b even 2 1 2400.3.g.b 16
8.d odd 2 1 inner 600.3.g.d 16
15.d odd 2 1 360.3.g.c 16
20.d odd 2 1 480.3.g.a 16
20.e even 4 2 2400.3.p.b 32
40.e odd 2 1 120.3.g.a ✓ 16
40.f even 2 1 480.3.g.a 16
40.i odd 4 2 2400.3.p.b 32
40.k even 4 2 600.3.p.b 32
60.h even 2 1 1440.3.g.c 16
120.i odd 2 1 1440.3.g.c 16
120.m even 2 1 360.3.g.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.g.a ✓ 16 5.b even 2 1
120.3.g.a ✓ 16 40.e odd 2 1
360.3.g.c 16 15.d odd 2 1
360.3.g.c 16 120.m even 2 1
480.3.g.a 16 20.d odd 2 1
480.3.g.a 16 40.f even 2 1
600.3.g.d 16 1.a even 1 1 trivial
600.3.g.d 16 8.d odd 2 1 inner
600.3.p.b 32 5.c odd 4 2
600.3.p.b 32 40.k even 4 2
1440.3.g.c 16 60.h even 2 1
1440.3.g.c 16 120.i odd 2 1
2400.3.g.b 16 4.b odd 2 1
2400.3.g.b 16 8.b even 2 1
2400.3.p.b 32 20.e even 4 2
2400.3.p.b 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 \(S_{3}^{\mathrm{new}}(600, [\chi])\):

\( T_{7}^{16} + 432 T_{7}^{14} + 74976 T_{7}^{12} + 6766336 T_{7}^{10} + 340312320 T_{7}^{8} + \cdots + 44930433024 \) Copy content Toggle raw display
\( T_{17}^{8} - 1256 T_{17}^{6} - 2176 T_{17}^{5} + 470160 T_{17}^{4} + 1704448 T_{17}^{3} + \cdots - 648544256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 4 T^{15} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 44930433024 \) Copy content Toggle raw display
$11$ \( (T^{8} - 32 T^{7} + \cdots - 16941056)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( (T^{8} - 1256 T^{6} + \cdots - 648544256)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 16 T^{7} + \cdots + 2351435776)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 75\!\cdots\!24 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 596041052258304 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 11\!\cdots\!04 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 41\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{8} - 9488 T^{6} + \cdots - 166457351936)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 96 T^{7} + \cdots - 686683848704)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 62\!\cdots\!64 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots - 29039705882624)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 50\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 81320576352256)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 92\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 37300922708224)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 73\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 4495382675456)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 3381143764736)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 525348544432384)^{2} \) Copy content Toggle raw display
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