Properties

Label 600.2.b.g
Level $600$
Weight $2$
Character orbit 600.b
Analytic conductor $4.791$
Analytic rank $0$
Dimension $12$
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] = [600,2,Mod(251,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.251");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.b (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: 12.0.537291533250985984.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5x^{10} + 14x^{8} - 30x^{6} + 56x^{4} - 80x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{8} q^{2} - \beta_{5} q^{3} + (\beta_{4} + 1) q^{4} + (\beta_{10} - \beta_{4}) q^{6} + ( - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{5} - \beta_{3} + 1) q^{7} + ( - \beta_{11} + \beta_{8} - \beta_{2}) q^{8} + (\beta_{11} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{8} q^{2} - \beta_{5} q^{3} + (\beta_{4} + 1) q^{4} + (\beta_{10} - \beta_{4}) q^{6} + ( - \beta_{10} + \beta_{9} + \beta_{7} - \beta_{5} - \beta_{3} + 1) q^{7} + ( - \beta_{11} + \beta_{8} - \beta_{2}) q^{8} + (\beta_{11} - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{2} + \beta_1) q^{9} + ( - \beta_{11} - \beta_{6} - \beta_{2}) q^{11} + (\beta_{11} + \beta_{9} + \beta_{8} + \beta_{7} - \beta_{5} + 1) q^{12} + (\beta_{10} - \beta_{9} - \beta_{4} - \beta_{3} + \beta_1 - 1) q^{13} + ( - \beta_{11} - \beta_{8} + \beta_{7} - 2 \beta_{6} + \beta_{5}) q^{14} + (\beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{16} + (\beta_{11} + \beta_{8} - \beta_{7} + \beta_{6} - \beta_{5}) q^{17} + ( - \beta_{11} - \beta_{9} - 2 \beta_{7} + \beta_{3} + \beta_1) q^{18} + ( - \beta_{7} + \beta_{5} - \beta_{4} - \beta_1 - 1) q^{19} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_1 + 1) q^{21} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{22} + (\beta_{11} + 2 \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} - \beta_{2}) q^{23} + (\beta_{10} + \beta_{3} + \beta_{2} - \beta_1) q^{24} + (\beta_{10} + \beta_{9} + \beta_{8} - 2 \beta_{6} - 2 \beta_{2}) q^{26} + (\beta_{11} + 2 \beta_{8} + \beta_{6} - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{27} + ( - 2 \beta_{10} + 2 \beta_{9} + \beta_{7} - \beta_{5} - \beta_{3} - 3 \beta_1 + 3) q^{28} + ( - \beta_{11} - 2 \beta_{10} - 2 \beta_{9} - 3 \beta_{8} + \beta_{6}) q^{29} + (\beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{31} + ( - 2 \beta_{11} + \beta_{10} + \beta_{9} - 2 \beta_{6} - 2 \beta_{2}) q^{32} + ( - \beta_{8} + \beta_{7} + \beta_{4} + \beta_{2} + \beta_1 - 1) q^{33} + (2 \beta_{10} - 2 \beta_{9} - \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_1 - 2) q^{34} + (\beta_{11} - \beta_{9} + \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 1) q^{36}+ \cdots + (2 \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{4} - 2 \beta_{2} + \beta_1 + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 2 q^{3} + 10 q^{4} + 7 q^{6} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 2 q^{3} + 10 q^{4} + 7 q^{6} - 2 q^{9} + 3 q^{12} - 6 q^{16} + 5 q^{18} - 4 q^{19} + 2 q^{22} + 5 q^{24} - 8 q^{27} + 20 q^{28} - 18 q^{33} - 2 q^{34} + 19 q^{36} + 14 q^{42} + 40 q^{43} - 16 q^{46} + 27 q^{48} - 36 q^{49} - 30 q^{51} + 4 q^{52} - 30 q^{54} - 42 q^{57} - 52 q^{58} + 10 q^{64} + 7 q^{66} + 60 q^{67} + 39 q^{72} - 12 q^{73} - 38 q^{76} - 54 q^{78} - 10 q^{81} - 58 q^{82} - 34 q^{84} - 34 q^{88} - 24 q^{91} + 28 q^{94} - 31 q^{96} + 32 q^{97} + 58 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5x^{10} + 14x^{8} - 30x^{6} + 56x^{4} - 80x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} + \nu^{7} - 2\nu^{5} + 6\nu^{3} ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{10} - 5\nu^{8} + 14\nu^{6} - 14\nu^{4} + 40\nu^{2} - 48 ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{10} + 5\nu^{8} - 14\nu^{6} + 30\nu^{4} - 56\nu^{2} + 64 ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 6 \nu^{10} + \nu^{9} - 22 \nu^{8} - 10 \nu^{7} + 44 \nu^{6} + 22 \nu^{5} - 100 \nu^{4} - 32 \nu^{3} + 160 \nu^{2} + 16 \nu - 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{11} + 5\nu^{9} - 14\nu^{7} + 30\nu^{5} - 24\nu^{3} + 16\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - \nu^{11} - 6 \nu^{10} + \nu^{9} + 22 \nu^{8} - 10 \nu^{7} - 44 \nu^{6} + 22 \nu^{5} + 100 \nu^{4} - 32 \nu^{3} - 160 \nu^{2} + 16 \nu + 160 ) / 64 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -\nu^{11} + 5\nu^{9} - 14\nu^{7} + 30\nu^{5} - 56\nu^{3} + 80\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 5 \nu^{11} + 2 \nu^{10} + 13 \nu^{9} - 18 \nu^{8} - 26 \nu^{7} + 36 \nu^{6} + 62 \nu^{5} - 76 \nu^{4} - 80 \nu^{3} + 160 \nu^{2} + 80 \nu - 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5 \nu^{11} - 2 \nu^{10} + 13 \nu^{9} + 18 \nu^{8} - 26 \nu^{7} - 36 \nu^{6} + 62 \nu^{5} + 76 \nu^{4} - 80 \nu^{3} - 160 \nu^{2} + 80 \nu + 224 ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 3\nu^{9} + 8\nu^{7} - 14\nu^{5} + 20\nu^{3} - 24\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} - \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} - \beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{11} + \beta_{10} + \beta_{9} + \beta_{8} + 2\beta_{6} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{10} - \beta_{9} + \beta_{7} - \beta_{5} - 2\beta_{4} + 2\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 4\beta_{11} + 3\beta_{10} + 3\beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{5} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 5\beta_{10} - 5\beta_{9} + \beta_{7} - \beta_{5} - 8\beta_{4} + 2\beta _1 - 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( \beta_{10} + \beta_{9} - 7\beta_{7} + 2\beta_{6} - 7\beta_{5} - 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 11\beta_{10} - 11\beta_{9} - 9\beta_{7} + 9\beta_{5} + 2\beta_{4} + 2\beta_{3} + 12\beta _1 - 22 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4\beta_{11} - 7\beta_{10} - 7\beta_{9} + 10\beta_{8} - 5\beta_{7} + 14\beta_{6} - 5\beta_{5} - 6\beta_{2} \) Copy content Toggle raw display

Character values

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

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

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
251.1
−1.39298 + 0.244153i
−1.39298 0.244153i
−1.26128 + 0.639662i
−1.26128 0.639662i
−0.847808 + 1.13191i
−0.847808 1.13191i
0.847808 + 1.13191i
0.847808 1.13191i
1.26128 + 0.639662i
1.26128 0.639662i
1.39298 + 0.244153i
1.39298 0.244153i
−1.39298 0.244153i 1.31310 + 1.12950i 1.88078 + 0.680200i 0 −1.55335 1.89397i 4.34495i −2.45381 1.40670i 0.448458 + 2.96629i 0
251.2 −1.39298 + 0.244153i 1.31310 1.12950i 1.88078 0.680200i 0 −1.55335 + 1.89397i 4.34495i −2.45381 + 1.40670i 0.448458 2.96629i 0
251.3 −1.26128 0.639662i −1.57067 0.730070i 1.18166 + 1.61359i 0 1.51406 + 1.92552i 1.25539i −0.458259 2.79106i 1.93400 + 2.29339i 0
251.4 −1.26128 + 0.639662i −1.57067 + 0.730070i 1.18166 1.61359i 0 1.51406 1.92552i 1.25539i −0.458259 + 2.79106i 1.93400 2.29339i 0
251.5 −0.847808 1.13191i −0.242431 + 1.71500i −0.562443 + 1.91929i 0 2.14676 1.17958i 3.08957i 2.64930 0.990551i −2.88245 0.831539i 0
251.6 −0.847808 + 1.13191i −0.242431 1.71500i −0.562443 1.91929i 0 2.14676 + 1.17958i 3.08957i 2.64930 + 0.990551i −2.88245 + 0.831539i 0
251.7 0.847808 1.13191i −0.242431 + 1.71500i −0.562443 1.91929i 0 1.73569 + 1.72840i 3.08957i −2.64930 0.990551i −2.88245 0.831539i 0
251.8 0.847808 + 1.13191i −0.242431 1.71500i −0.562443 + 1.91929i 0 1.73569 1.72840i 3.08957i −2.64930 + 0.990551i −2.88245 + 0.831539i 0
251.9 1.26128 0.639662i −1.57067 0.730070i 1.18166 1.61359i 0 −2.44805 + 0.0838735i 1.25539i 0.458259 2.79106i 1.93400 + 2.29339i 0
251.10 1.26128 + 0.639662i −1.57067 + 0.730070i 1.18166 + 1.61359i 0 −2.44805 0.0838735i 1.25539i 0.458259 + 2.79106i 1.93400 2.29339i 0
251.11 1.39298 0.244153i 1.31310 + 1.12950i 1.88078 0.680200i 0 2.10489 + 1.25277i 4.34495i 2.45381 1.40670i 0.448458 + 2.96629i 0
251.12 1.39298 + 0.244153i 1.31310 1.12950i 1.88078 + 0.680200i 0 2.10489 1.25277i 4.34495i 2.45381 + 1.40670i 0.448458 2.96629i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 251.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
8.d odd 2 1 inner
24.f even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 600.2.b.g 12
3.b odd 2 1 inner 600.2.b.g 12
4.b odd 2 1 2400.2.b.h 12
5.b even 2 1 600.2.b.h yes 12
5.c odd 4 2 600.2.m.e 24
8.b even 2 1 2400.2.b.h 12
8.d odd 2 1 inner 600.2.b.g 12
12.b even 2 1 2400.2.b.h 12
15.d odd 2 1 600.2.b.h yes 12
15.e even 4 2 600.2.m.e 24
20.d odd 2 1 2400.2.b.g 12
20.e even 4 2 2400.2.m.e 24
24.f even 2 1 inner 600.2.b.g 12
24.h odd 2 1 2400.2.b.h 12
40.e odd 2 1 600.2.b.h yes 12
40.f even 2 1 2400.2.b.g 12
40.i odd 4 2 2400.2.m.e 24
40.k even 4 2 600.2.m.e 24
60.h even 2 1 2400.2.b.g 12
60.l odd 4 2 2400.2.m.e 24
120.i odd 2 1 2400.2.b.g 12
120.m even 2 1 600.2.b.h yes 12
120.q odd 4 2 600.2.m.e 24
120.w even 4 2 2400.2.m.e 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
600.2.b.g 12 1.a even 1 1 trivial
600.2.b.g 12 3.b odd 2 1 inner
600.2.b.g 12 8.d odd 2 1 inner
600.2.b.g 12 24.f even 2 1 inner
600.2.b.h yes 12 5.b even 2 1
600.2.b.h yes 12 15.d odd 2 1
600.2.b.h yes 12 40.e odd 2 1
600.2.b.h yes 12 120.m even 2 1
600.2.m.e 24 5.c odd 4 2
600.2.m.e 24 15.e even 4 2
600.2.m.e 24 40.k even 4 2
600.2.m.e 24 120.q odd 4 2
2400.2.b.g 12 20.d odd 2 1
2400.2.b.g 12 40.f even 2 1
2400.2.b.g 12 60.h even 2 1
2400.2.b.g 12 120.i odd 2 1
2400.2.b.h 12 4.b odd 2 1
2400.2.b.h 12 8.b even 2 1
2400.2.b.h 12 12.b even 2 1
2400.2.b.h 12 24.h odd 2 1
2400.2.m.e 24 20.e even 4 2
2400.2.m.e 24 40.i odd 4 2
2400.2.m.e 24 60.l odd 4 2
2400.2.m.e 24 120.w even 4 2

Hecke kernels

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

\( T_{7}^{6} + 30T_{7}^{4} + 225T_{7}^{2} + 284 \) Copy content Toggle raw display
\( T_{11}^{6} + 19T_{11}^{4} + 112T_{11}^{2} + 200 \) Copy content Toggle raw display
\( T_{23}^{6} - 104T_{23}^{4} + 2136T_{23}^{2} - 9088 \) Copy content Toggle raw display
\( T_{43}^{3} - 10T_{43}^{2} - 29T_{43} + 148 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 5 T^{10} + 14 T^{8} - 30 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T^{6} + T^{5} + T^{4} + 2 T^{3} + 3 T^{2} + \cdots + 27)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 30 T^{4} + 225 T^{2} + 284)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 19 T^{4} + 112 T^{2} + 200)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 58 T^{4} + 841 T^{2} + 284)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 35 T^{4} + 176 T^{2} + 8)^{2} \) Copy content Toggle raw display
$19$ \( (T^{3} + T^{2} - 19 T + 25)^{4} \) Copy content Toggle raw display
$23$ \( (T^{6} - 104 T^{4} + 2136 T^{2} + \cdots - 9088)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 140 T^{4} + 5752 T^{2} + \cdots - 56800)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 98 T^{4} + 2969 T^{2} + \cdots + 28400)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 108 T^{4} + 3200 T^{2} + \cdots + 18176)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 155 T^{4} + 6608 T^{2} + \cdots + 57800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 10 T^{2} - 29 T + 148)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} - 212 T^{4} + 11520 T^{2} + \cdots - 36352)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} - 280 T^{4} + 23576 T^{2} + \cdots - 581632)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 140 T^{4} + 2816 T^{2} + \cdots + 512)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 154 T^{4} + 5929 T^{2} + \cdots + 7100)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} - 15 T^{2} + 65 T - 83)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} - 236 T^{4} + 14776 T^{2} + \cdots - 274912)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 3 T^{2} - 88 T + 244)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 348 T^{4} + 37376 T^{2} + \cdots + 1163264)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 387 T^{4} + 35168 T^{2} + \cdots + 175232)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 187 T^{4} + 5088 T^{2} + \cdots + 36992)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 8 T^{2} - 99 T + 242)^{4} \) Copy content Toggle raw display
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