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Properties

Label 600.3.l.f
Level $600$
Weight $3$
Character orbit 600.l
Analytic conductor $16.349$
Analytic rank $0$
Dimension $8$
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] = [600,3,Mod(401,600)]
 
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, 0, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.l (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: \(16.3488158616\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.681615360000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} - 1) q^{7} + (\beta_{7} + \beta_{6} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + (\beta_{5} + \beta_{4} + \beta_{3} - 1) q^{7} + (\beta_{7} + \beta_{6} + 2) q^{9} + (\beta_{7} - \beta_{5} + 2 \beta_{4} + \cdots + \beta_1) q^{11}+ \cdots + ( - 15 \beta_{7} + 4 \beta_{6} + \cdots - 22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} + 8 q^{13} - 8 q^{19} + 28 q^{21} - 20 q^{27} + 120 q^{31} + 112 q^{33} - 8 q^{37} - 72 q^{39} + 328 q^{43} + 64 q^{49} + 64 q^{51} - 72 q^{57} + 8 q^{61} - 88 q^{63} - 152 q^{67} + 100 q^{69} - 32 q^{73} + 88 q^{79} + 224 q^{81} + 152 q^{87} + 560 q^{91} + 368 q^{93} - 144 q^{97} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 7\nu^{7} + 53\nu^{6} - 530\nu^{5} - 280\nu^{4} + 3613\nu^{3} + 12557\nu^{2} - 20988\nu - 26976 ) / 9300 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 69\nu^{7} - 319\nu^{6} - 375\nu^{5} + 2045\nu^{4} + 6186\nu^{3} - 16366\nu^{2} + 1704\nu - 3912 ) / 18600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -83\nu^{7} + 213\nu^{6} - 115\nu^{5} + 65\nu^{4} - 2562\nu^{3} - 998\nu^{2} - 52728\nu + 2064 ) / 18600 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 21\nu^{7} - 89\nu^{6} - 71\nu^{5} + 927\nu^{4} + 950\nu^{3} - 5698\nu^{2} + 2136\nu + 10584 ) / 3720 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 27\nu^{7} - 17\nu^{6} - 295\nu^{5} + 5\nu^{4} + 2488\nu^{3} + 672\nu^{2} + 1152\nu + 5424 ) / 3720 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -77\nu^{7} + 192\nu^{6} + 405\nu^{5} - 20\nu^{4} - 6418\nu^{3} - 5602\nu^{2} + 16968\nu + 27036 ) / 9300 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 8\nu^{7} - 28\nu^{6} - 30\nu^{5} + 145\nu^{4} + 232\nu^{3} - 507\nu^{2} + 1788\nu - 804 ) / 930 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{6} - \beta_{4} - 4\beta_{3} + 3\beta_{2} + 2\beta _1 + 1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{5} + 2\beta_{4} - 4\beta_{3} - 3\beta_{2} + 5\beta _1 + 13 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -6\beta_{7} - \beta_{5} + 2\beta_{4} - 6\beta_{3} + 5\beta_{2} + 3\beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -24\beta_{7} + 6\beta_{6} - 15\beta_{5} + 64\beta_{4} - 32\beta_{3} - 45\beta_{2} + 37\beta _1 - 97 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -150\beta_{7} - 84\beta_{6} - 51\beta_{5} + 146\beta_{4} - 130\beta_{3} - 117\beta_{2} - 7\beta _1 - 257 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -124\beta_{7} - 26\beta_{6} + 75\beta_{5} + 180\beta_{4} - 44\beta_{3} - 239\beta_{2} - 25\beta _1 - 771 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 210 \beta_{7} - 1032 \beta_{6} + 765 \beta_{5} + 1342 \beta_{4} + 346 \beta_{3} - 3093 \beta_{2} + \cdots - 6073 ) / 6 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
1.54294 − 1.41421i
1.54294 + 1.41421i
−2.22255 − 1.41421i
−2.22255 + 1.41421i
−0.542939 − 1.41421i
−0.542939 + 1.41421i
3.22255 + 1.41421i
3.22255 − 1.41421i
0 −2.98254 − 0.323191i 0 0 0 −4.72640 0 8.79110 + 1.92786i 0
401.2 0 −2.98254 + 0.323191i 0 0 0 −4.72640 0 8.79110 − 1.92786i 0
401.3 0 −0.291610 − 2.98579i 0 0 0 −4.46268 0 −8.82993 + 1.74137i 0
401.4 0 −0.291610 + 2.98579i 0 0 0 −4.46268 0 −8.82993 − 1.74137i 0
401.5 0 2.40140 − 1.79813i 0 0 0 10.2132 0 2.53346 − 8.63606i 0
401.6 0 2.40140 + 1.79813i 0 0 0 10.2132 0 2.53346 + 8.63606i 0
401.7 0 2.87275 − 0.864473i 0 0 0 −9.02416 0 7.50537 − 4.96683i 0
401.8 0 2.87275 + 0.864473i 0 0 0 −9.02416 0 7.50537 + 4.96683i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b 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.l.f 8
3.b odd 2 1 inner 600.3.l.f 8
4.b odd 2 1 1200.3.l.x 8
5.b even 2 1 120.3.l.a ✓ 8
5.c odd 4 2 600.3.c.d 16
12.b even 2 1 1200.3.l.x 8
15.d odd 2 1 120.3.l.a ✓ 8
15.e even 4 2 600.3.c.d 16
20.d odd 2 1 240.3.l.d 8
20.e even 4 2 1200.3.c.m 16
40.e odd 2 1 960.3.l.g 8
40.f even 2 1 960.3.l.h 8
60.h even 2 1 240.3.l.d 8
60.l odd 4 2 1200.3.c.m 16
120.i odd 2 1 960.3.l.h 8
120.m even 2 1 960.3.l.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
120.3.l.a ✓ 8 5.b even 2 1
120.3.l.a ✓ 8 15.d odd 2 1
240.3.l.d 8 20.d odd 2 1
240.3.l.d 8 60.h even 2 1
600.3.c.d 16 5.c odd 4 2
600.3.c.d 16 15.e even 4 2
600.3.l.f 8 1.a even 1 1 trivial
600.3.l.f 8 3.b odd 2 1 inner
960.3.l.g 8 40.e odd 2 1
960.3.l.g 8 120.m even 2 1
960.3.l.h 8 40.f even 2 1
960.3.l.h 8 120.i odd 2 1
1200.3.c.m 16 20.e even 4 2
1200.3.c.m 16 60.l odd 4 2
1200.3.l.x 8 4.b odd 2 1
1200.3.l.x 8 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 8T_{7}^{3} - 82T_{7}^{2} - 872T_{7} - 1944 \) acting on \(S_{3}^{\mathrm{new}}(600, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 4 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 8 T^{3} + \cdots - 1944)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 888 T^{6} + \cdots + 232989696 \) Copy content Toggle raw display
$13$ \( (T^{4} - 4 T^{3} + \cdots - 3456)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 1104 T^{6} + \cdots + 15872256 \) Copy content Toggle raw display
$19$ \( (T^{4} + 4 T^{3} + \cdots + 16736)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 93650688576 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 3474395136 \) Copy content Toggle raw display
$31$ \( (T^{4} - 60 T^{3} + \cdots - 151296)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 4 T^{3} + \cdots - 31104)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 43961355472896 \) Copy content Toggle raw display
$43$ \( (T^{4} - 164 T^{3} + \cdots + 1582656)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 13517317696 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 6801580544256 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 15563214360576 \) Copy content Toggle raw display
$61$ \( (T^{4} - 4 T^{3} + \cdots + 30631296)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 76 T^{3} + \cdots - 668224)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 35499479924736 \) Copy content Toggle raw display
$73$ \( (T^{4} + 16 T^{3} + \cdots + 2938896)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 44 T^{3} + \cdots - 12384)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 336130569170496 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 13\!\cdots\!76 \) Copy content Toggle raw display
$97$ \( (T^{4} + 72 T^{3} + \cdots + 10270096)^{2} \) Copy content Toggle raw display
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