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Properties

Label 3900.2.bw.k
Level $3900$
Weight $2$
Character orbit 3900.bw
Analytic conductor $31.142$
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] = [3900,2,Mod(49,3900)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3900, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3900 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3900.bw (of order \(6\), degree \(2\), 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: \(31.1416567883\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.454201344.7
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 780)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{7}+ \cdots - \beta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} + 3 \beta_{6} + \beta_{5} + \cdots + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 6 q^{7} + 4 q^{9} - 24 q^{11} + 10 q^{13} - 18 q^{17} + 12 q^{19} - 12 q^{23} + 14 q^{29} + 4 q^{33} - 4 q^{37} - 4 q^{39} + 18 q^{41} - 18 q^{43} - 36 q^{47} - 32 q^{49} - 4 q^{51} - 16 q^{57} - 6 q^{59} - 24 q^{61} - 6 q^{63} + 14 q^{67} - 6 q^{71} - 12 q^{73} - 8 q^{79} - 4 q^{81} - 48 q^{83} - 6 q^{87} + 6 q^{89} + 24 q^{91} - 8 q^{93} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 4x^{6} + 24x^{5} - 25x^{4} - 12x^{3} + 128x^{2} - 182x + 169 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -72\nu^{7} + 13\nu^{6} - 1615\nu^{5} + 2736\nu^{4} + 6778\nu^{3} - 23106\nu^{2} + 17495\nu - 5462 ) / 21903 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1772 \nu^{7} + 2348 \nu^{6} + 2031 \nu^{5} - 64082 \nu^{4} + 89748 \nu^{3} + 49487 \nu^{2} + \cdots + 460200 ) / 284739 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2105 \nu^{7} + 5146 \nu^{6} + 8251 \nu^{5} - 29525 \nu^{4} + 17057 \nu^{3} - 62854 \nu^{2} + \cdots + 155675 ) / 284739 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2365\nu^{7} + 80\nu^{6} - 7692\nu^{5} + 19645\nu^{4} + 42834\nu^{3} - 11896\nu^{2} + 58801\nu + 328068 ) / 284739 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 62\nu^{7} + 84\nu^{6} - 716\nu^{5} + 773\nu^{4} + 2168\nu^{3} - 3968\nu^{2} + 6246\nu - 4576 ) / 5811 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 5049 \nu^{7} - 21902 \nu^{6} + 8300 \nu^{5} + 114780 \nu^{4} - 247151 \nu^{3} + 48768 \nu^{2} + \cdots - 418262 ) / 284739 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} - 3\beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - 3\beta_{6} + 5\beta_{5} - 2\beta_{4} + 2\beta_{2} + \beta _1 - 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{7} - 7\beta_{6} + 6\beta_{5} - 8\beta_{4} - 8\beta_{3} + 4\beta_{2} - 8\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} - 13\beta_{6} + 28\beta_{5} + 15\beta_{4} - 2\beta_{3} + 4\beta_{2} - 45 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{7} + 5\beta_{6} + 2\beta_{5} + 11\beta_{4} - 32\beta_{3} - 13\beta_{2} - 45\beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 50\beta_{7} + 65\beta_{6} + 71\beta_{5} + 136\beta_{4} + 56\beta_{3} - 56\beta_{2} + 25\beta _1 - 136 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3900\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
1.02715 + 1.10132i
0.338876 − 1.46735i
−2.39244 − 0.0909984i
2.02641 − 1.27503i
1.02715 − 1.10132i
0.338876 + 1.46735i
−2.39244 + 0.0909984i
2.02641 + 1.27503i
0 −0.866025 + 0.500000i 0 0 0 −1.90755 + 3.30397i 0 0.500000 − 0.866025i 0
49.2 0 −0.866025 + 0.500000i 0 0 0 2.54152 − 4.40205i 0 0.500000 − 0.866025i 0
49.3 0 0.866025 − 0.500000i 0 0 0 0.157614 − 0.272995i 0 0.500000 − 0.866025i 0
49.4 0 0.866025 − 0.500000i 0 0 0 2.20841 − 3.82508i 0 0.500000 − 0.866025i 0
2149.1 0 −0.866025 − 0.500000i 0 0 0 −1.90755 − 3.30397i 0 0.500000 + 0.866025i 0
2149.2 0 −0.866025 − 0.500000i 0 0 0 2.54152 + 4.40205i 0 0.500000 + 0.866025i 0
2149.3 0 0.866025 + 0.500000i 0 0 0 0.157614 + 0.272995i 0 0.500000 + 0.866025i 0
2149.4 0 0.866025 + 0.500000i 0 0 0 2.20841 + 3.82508i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.l even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.2.bw.k 8
5.b even 2 1 3900.2.bw.h 8
5.c odd 4 1 780.2.cc.c ✓ 8
5.c odd 4 1 3900.2.cd.k 8
13.e even 6 1 3900.2.bw.h 8
15.e even 4 1 2340.2.dj.b 8
65.l even 6 1 inner 3900.2.bw.k 8
65.r odd 12 1 780.2.cc.c ✓ 8
65.r odd 12 1 3900.2.cd.k 8
195.bf even 12 1 2340.2.dj.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.cc.c ✓ 8 5.c odd 4 1
780.2.cc.c ✓ 8 65.r odd 12 1
2340.2.dj.b 8 15.e even 4 1
2340.2.dj.b 8 195.bf even 12 1
3900.2.bw.h 8 5.b even 2 1
3900.2.bw.h 8 13.e even 6 1
3900.2.bw.k 8 1.a even 1 1 trivial
3900.2.bw.k 8 65.l even 6 1 inner
3900.2.cd.k 8 5.c odd 4 1
3900.2.cd.k 8 65.r odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{8} - 6T_{7}^{7} + 48T_{7}^{6} - 108T_{7}^{5} + 711T_{7}^{4} - 1404T_{7}^{3} + 7776T_{7}^{2} - 2430T_{7} + 729 \) Copy content Toggle raw display
\( T_{11}^{8} + 24T_{11}^{7} + 248T_{11}^{6} + 1344T_{11}^{5} + 3784T_{11}^{4} + 4032T_{11}^{3} - 2304T_{11}^{2} - 5184T_{11} + 5184 \) Copy content Toggle raw display
\( T_{17}^{8} + 18T_{17}^{7} + 140T_{17}^{6} + 576T_{17}^{5} + 1258T_{17}^{4} + 1152T_{17}^{3} - 144T_{17}^{2} - 648T_{17} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 6 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$11$ \( T^{8} + 24 T^{7} + \cdots + 5184 \) Copy content Toggle raw display
$13$ \( T^{8} - 10 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} + 18 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$19$ \( T^{8} - 12 T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$23$ \( T^{8} + 12 T^{7} + \cdots + 46656 \) Copy content Toggle raw display
$29$ \( T^{8} - 14 T^{7} + \cdots + 54756 \) Copy content Toggle raw display
$31$ \( T^{8} + 284 T^{6} + \cdots + 15657849 \) Copy content Toggle raw display
$37$ \( T^{8} + 4 T^{7} + \cdots + 186624 \) Copy content Toggle raw display
$41$ \( T^{8} - 18 T^{7} + \cdots + 11819844 \) Copy content Toggle raw display
$43$ \( T^{8} + 18 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$47$ \( (T^{4} + 18 T^{3} + \cdots - 1242)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 320 T^{6} + \cdots + 1327104 \) Copy content Toggle raw display
$59$ \( T^{8} + 6 T^{7} + \cdots + 171396 \) Copy content Toggle raw display
$61$ \( T^{8} + 24 T^{7} + \cdots + 64818601 \) Copy content Toggle raw display
$67$ \( T^{8} - 14 T^{7} + \cdots + 21609 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + \cdots + 91355364 \) Copy content Toggle raw display
$73$ \( (T^{4} + 6 T^{3} + \cdots + 5733)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots - 767)^{2} \) Copy content Toggle raw display
$83$ \( (T^{4} + 24 T^{3} + \cdots + 216)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 6 T^{7} + \cdots + 54756 \) Copy content Toggle raw display
$97$ \( T^{8} + 14 T^{7} + \cdots + 257049 \) Copy content Toggle raw display
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