Properties

Label 600.2.d.f
Level $600$
Weight $2$
Character orbit 600.d
Analytic conductor $4.791$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,2,Mod(349,600)]
 
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([0, 1, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 600.d (of order \(2\), degree \(1\), not 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: \(6\)
Coefficient field: 6.0.399424.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 120)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + q^{3} - \beta_1 q^{4} - \beta_{2} q^{6} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + q^{3} - \beta_1 q^{4} - \beta_{2} q^{6} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{7} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{8} + q^{9} + ( - 2 \beta_{5} + \beta_{3}) q^{11} - \beta_1 q^{12} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1) q^{13} + (\beta_{5} - \beta_{4} + 2 \beta_1 + 3) q^{14} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{16} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{17} - \beta_{2} q^{18} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{19} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{21} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 3) q^{22} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{23} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{24} + ( - 2 \beta_1 + 4) q^{26} + q^{27} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3) q^{28} - \beta_{3} q^{29} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{31} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{32} + ( - 2 \beta_{5} + \beta_{3}) q^{33} + (2 \beta_{5} + 2 \beta_{3} - 2) q^{34} - \beta_1 q^{36} + ( - \beta_{4} - \beta_{3} + \beta_{2} - 3 \beta_1) q^{37} + (2 \beta_1 + 4) q^{38} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1) q^{39} + (2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 - 2) q^{41} + (\beta_{5} - \beta_{4} + 2 \beta_1 + 3) q^{42} + (2 \beta_{4} + 2 \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{5} + 3 \beta_{4} - 4 \beta_{3} - 2 \beta_1 + 1) q^{44} + ( - 2 \beta_1 - 4) q^{46} + (4 \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1) q^{47} + (\beta_{4} + \beta_{3} + \beta_{2} + 2) q^{48} + (2 \beta_{4} - 2 \beta_{3} + 6 \beta_{2} - 2 \beta_1 - 5) q^{49} + ( - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{51} + ( - 2 \beta_{5} + 2 \beta_{3} - 6 \beta_{2} - 2) q^{52} + 2 q^{53} - \beta_{2} q^{54} + (\beta_{5} - 3 \beta_{4} - 4 \beta_{2} - 2 \beta_1 - 1) q^{56} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{57} + (\beta_{5} - \beta_{4} - 1) q^{58} + ( - 2 \beta_{5} - \beta_{3}) q^{59} + ( - 4 \beta_{5} + 2 \beta_{3}) q^{61} + (2 \beta_{2} + 2 \beta_1 - 4) q^{62} + ( - \beta_{4} + \beta_{2} + \beta_1) q^{63} + (2 \beta_{5} + \beta_{4} - \beta_{3} + 3 \beta_{2} - 2 \beta_1 - 4) q^{64} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 2 \beta_1 + 3) q^{66} + 4 q^{67} + ( - 4 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{68} + (\beta_{4} - \beta_{3} - \beta_{2} - \beta_1) q^{69} + (2 \beta_{3} - 4 \beta_{2} + 4 \beta_1) q^{71} + ( - \beta_{5} + \beta_{3} - \beta_{2} - 1) q^{72} - 3 \beta_{3} q^{73} + ( - 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{74} + (2 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 2) q^{76} + ( - 4 \beta_{4} - 4 \beta_{2} - 4 \beta_1 - 8) q^{77} + ( - 2 \beta_1 + 4) q^{78} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} + \beta_1 - 6) q^{79} + q^{81} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 6 \beta_{2} - 2) q^{82} + ( - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1 + 4) q^{83} + (\beta_{5} - \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 3) q^{84} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} - 2) q^{86} - \beta_{3} q^{87} + (3 \beta_{5} - \beta_{4} - 2 \beta_1 - 11) q^{88} + (2 \beta_{4} + 2 \beta_{2} + 2 \beta_1 + 6) q^{89} + (4 \beta_{5} - 4 \beta_{4} - 2 \beta_{3} + 4 \beta_{2} + 4 \beta_1) q^{91} + ( - 2 \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{92} + (\beta_{4} - \beta_{3} + 3 \beta_{2} - \beta_1 - 2) q^{93} + ( - 2 \beta_{5} - 2 \beta_{4} - 4 \beta_{3} - 2 \beta_1 - 2) q^{94} + ( - \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{96} + ( - 4 \beta_{5} + 2 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{97} + (5 \beta_{2} + 4 \beta_1 - 8) q^{98} + ( - 2 \beta_{5} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 2 q^{4} - 6 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 2 q^{4} - 6 q^{8} + 6 q^{9} + 2 q^{12} + 16 q^{14} + 10 q^{16} + 12 q^{22} - 6 q^{24} + 28 q^{26} + 6 q^{27} + 20 q^{28} - 12 q^{31} - 10 q^{32} - 12 q^{34} + 2 q^{36} + 8 q^{37} + 20 q^{38} - 20 q^{41} + 16 q^{42} - 8 q^{43} + 4 q^{44} - 20 q^{46} + 10 q^{48} - 30 q^{49} - 12 q^{52} + 12 q^{53} + 4 q^{56} - 4 q^{58} - 28 q^{62} - 22 q^{64} + 12 q^{66} + 24 q^{67} - 8 q^{71} - 6 q^{72} - 12 q^{74} + 12 q^{76} - 32 q^{77} + 28 q^{78} - 36 q^{79} + 6 q^{81} - 16 q^{82} + 32 q^{83} + 20 q^{84} - 16 q^{86} - 60 q^{88} + 28 q^{89} - 12 q^{92} - 12 q^{93} - 4 q^{94} - 10 q^{96} - 56 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} + 3x^{4} - 6x^{3} + 6x^{2} - 8x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + \nu^{3} - 2\nu^{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} + 3\nu^{3} - 4\nu^{2} + 2\nu - 8 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -\nu^{5} + \nu^{4} - 2\nu^{3} + 3\nu^{2} - \nu + 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + \nu^{4} - 2\nu^{3} + 3\nu^{2} - 3\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{5} - \beta_{4} + 2\beta_{3} - 2\beta_{2} + 2\beta _1 + 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + \beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{5} + \beta_{4} - 2\beta_{3} + 6\beta_{2} + 2\beta _1 + 5 ) / 2 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.264658 1.38923i
0.264658 + 1.38923i
1.40680 + 0.144584i
1.40680 0.144584i
−0.671462 1.24464i
−0.671462 + 1.24464i
−1.38923 0.264658i 1.00000 1.85991 + 0.735342i 0 −1.38923 0.264658i 0.941367i −2.38923 1.51380i 1.00000 0
349.2 −1.38923 + 0.264658i 1.00000 1.85991 0.735342i 0 −1.38923 + 0.264658i 0.941367i −2.38923 + 1.51380i 1.00000 0
349.3 0.144584 1.40680i 1.00000 −1.95819 0.406803i 0 0.144584 1.40680i 3.62721i −0.855416 + 2.69597i 1.00000 0
349.4 0.144584 + 1.40680i 1.00000 −1.95819 + 0.406803i 0 0.144584 + 1.40680i 3.62721i −0.855416 2.69597i 1.00000 0
349.5 1.24464 0.671462i 1.00000 1.09828 1.67146i 0 1.24464 0.671462i 4.68585i 0.244644 2.81783i 1.00000 0
349.6 1.24464 + 0.671462i 1.00000 1.09828 + 1.67146i 0 1.24464 + 0.671462i 4.68585i 0.244644 + 2.81783i 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 349.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
40.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.d.f 6
3.b odd 2 1 1800.2.d.r 6
4.b odd 2 1 2400.2.d.e 6
5.b even 2 1 600.2.d.e 6
5.c odd 4 1 120.2.k.b 6
5.c odd 4 1 600.2.k.c 6
8.b even 2 1 600.2.d.e 6
8.d odd 2 1 2400.2.d.f 6
12.b even 2 1 7200.2.d.r 6
15.d odd 2 1 1800.2.d.q 6
15.e even 4 1 360.2.k.f 6
15.e even 4 1 1800.2.k.p 6
20.d odd 2 1 2400.2.d.f 6
20.e even 4 1 480.2.k.b 6
20.e even 4 1 2400.2.k.c 6
24.f even 2 1 7200.2.d.q 6
24.h odd 2 1 1800.2.d.q 6
40.e odd 2 1 2400.2.d.e 6
40.f even 2 1 inner 600.2.d.f 6
40.i odd 4 1 120.2.k.b 6
40.i odd 4 1 600.2.k.c 6
40.k even 4 1 480.2.k.b 6
40.k even 4 1 2400.2.k.c 6
60.h even 2 1 7200.2.d.q 6
60.l odd 4 1 1440.2.k.f 6
60.l odd 4 1 7200.2.k.p 6
80.i odd 4 1 3840.2.a.bp 3
80.j even 4 1 3840.2.a.bo 3
80.s even 4 1 3840.2.a.br 3
80.t odd 4 1 3840.2.a.bq 3
120.i odd 2 1 1800.2.d.r 6
120.m even 2 1 7200.2.d.r 6
120.q odd 4 1 1440.2.k.f 6
120.q odd 4 1 7200.2.k.p 6
120.w even 4 1 360.2.k.f 6
120.w even 4 1 1800.2.k.p 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.2.k.b 6 5.c odd 4 1
120.2.k.b 6 40.i odd 4 1
360.2.k.f 6 15.e even 4 1
360.2.k.f 6 120.w even 4 1
480.2.k.b 6 20.e even 4 1
480.2.k.b 6 40.k even 4 1
600.2.d.e 6 5.b even 2 1
600.2.d.e 6 8.b even 2 1
600.2.d.f 6 1.a even 1 1 trivial
600.2.d.f 6 40.f even 2 1 inner
600.2.k.c 6 5.c odd 4 1
600.2.k.c 6 40.i odd 4 1
1440.2.k.f 6 60.l odd 4 1
1440.2.k.f 6 120.q odd 4 1
1800.2.d.q 6 15.d odd 2 1
1800.2.d.q 6 24.h odd 2 1
1800.2.d.r 6 3.b odd 2 1
1800.2.d.r 6 120.i odd 2 1
1800.2.k.p 6 15.e even 4 1
1800.2.k.p 6 120.w even 4 1
2400.2.d.e 6 4.b odd 2 1
2400.2.d.e 6 40.e odd 2 1
2400.2.d.f 6 8.d odd 2 1
2400.2.d.f 6 20.d odd 2 1
2400.2.k.c 6 20.e even 4 1
2400.2.k.c 6 40.k even 4 1
3840.2.a.bo 3 80.j even 4 1
3840.2.a.bp 3 80.i odd 4 1
3840.2.a.bq 3 80.t odd 4 1
3840.2.a.br 3 80.s even 4 1
7200.2.d.q 6 24.f even 2 1
7200.2.d.q 6 60.h even 2 1
7200.2.d.r 6 12.b even 2 1
7200.2.d.r 6 120.m even 2 1
7200.2.k.p 6 60.l odd 4 1
7200.2.k.p 6 120.q odd 4 1

Hecke kernels

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

\( T_{7}^{6} + 36T_{7}^{4} + 320T_{7}^{2} + 256 \) Copy content Toggle raw display
\( T_{11}^{6} + 64T_{11}^{4} + 1088T_{11}^{2} + 4096 \) Copy content Toggle raw display
\( T_{13}^{3} - 28T_{13} + 16 \) Copy content Toggle raw display
\( T_{37}^{3} - 4T_{37}^{2} - 60T_{37} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} - T^{4} + 2 T^{3} - 2 T^{2} + 8 \) Copy content Toggle raw display
$3$ \( (T - 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 36 T^{4} + 320 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{6} + 64 T^{4} + 1088 T^{2} + \cdots + 4096 \) Copy content Toggle raw display
$13$ \( (T^{3} - 28 T + 16)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} + 68 T^{4} + 640 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$19$ \( T^{6} + 40 T^{4} + 272 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{6} + 40 T^{4} + 272 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} + 4)^{3} \) Copy content Toggle raw display
$31$ \( (T^{3} + 6 T^{2} - 16 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 4 T^{2} - 60 T + 256)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} + 10 T^{2} - 36 T - 232)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 4 T^{2} - 64 T - 128)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 200 T^{4} + 12432 T^{2} + \cdots + 246016 \) Copy content Toggle raw display
$53$ \( (T - 2)^{6} \) Copy content Toggle raw display
$59$ \( T^{6} + 80 T^{4} + 576 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$61$ \( T^{6} + 256 T^{4} + 17408 T^{2} + \cdots + 262144 \) Copy content Toggle raw display
$67$ \( (T - 4)^{6} \) Copy content Toggle raw display
$71$ \( (T^{3} + 4 T^{2} - 112 T + 64)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 36)^{3} \) Copy content Toggle raw display
$79$ \( (T^{3} + 18 T^{2} + 80 T + 64)^{2} \) Copy content Toggle raw display
$83$ \( (T^{3} - 16 T^{2} + 16 T + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{3} - 14 T^{2} - 4 T + 184)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 332 T^{4} + 11824 T^{2} + \cdots + 107584 \) Copy content Toggle raw display
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