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Properties

Label 3900.2.by.g
Level $3900$
Weight $2$
Character orbit 3900.by
Analytic conductor $31.142$
Analytic rank $0$
Dimension $8$
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] = [3900,2,Mod(1849,3900)]
 
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, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.1849");
 
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.by (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: 8.0.3317760000.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 25x^{4} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{5} + \beta_{3} - \beta_1) q^{7} + \beta_{2} q^{9} + ( - 2 \beta_{2} + 2) q^{11} + (\beta_{5} + \beta_{3} + \beta_1) q^{13} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{17} - 2 \beta_{2} q^{19} + (\beta_{7} + 1) q^{21} + 2 \beta_1 q^{23} - \beta_{3} q^{27} + (\beta_{7} - \beta_{4} + 2 \beta_{2} - 2) q^{29} + q^{31} + (2 \beta_{3} - 2 \beta_1) q^{33} + ( - 2 \beta_{6} + 2 \beta_1) q^{37} + ( - \beta_{7} - 2 \beta_{2} + 1) q^{39} + (\beta_{7} - \beta_{4} - 4 \beta_{2} + 4) q^{41} + ( - 3 \beta_{5} + \beta_{3} - \beta_1) q^{43} + (3 \beta_{6} - 3 \beta_{5}) q^{47} + (2 \beta_{7} - 2 \beta_{4} - 4 \beta_{2} + 4) q^{49} + (\beta_{7} - 2) q^{51} + ( - 2 \beta_{6} + 2 \beta_{5} - 4 \beta_{3}) q^{53} + 2 \beta_{3} q^{57} + (\beta_{4} + 2 \beta_{2}) q^{59} - \beta_{2} q^{61} + ( - \beta_{6} - \beta_1) q^{63} + (\beta_{6} + 7 \beta_1) q^{67} - 2 \beta_{2} q^{69} + ( - \beta_{4} - 2 \beta_{2}) q^{71} + (\beta_{6} - \beta_{5} - 7 \beta_{3}) q^{73} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3}) q^{77} + (2 \beta_{7} - 3) q^{79} + (\beta_{2} - 1) q^{81} + 12 \beta_{3} q^{83} + ( - \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{87} + ( - \beta_{7} + \beta_{4}) q^{89} + ( - 2 \beta_{7} + 9 \beta_{2} - 11) q^{91} - \beta_1 q^{93} + ( - \beta_{5} + 7 \beta_{3} - 7 \beta_1) q^{97} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{9} + 8 q^{11} - 8 q^{19} + 8 q^{21} - 8 q^{29} + 8 q^{31} + 16 q^{41} + 16 q^{49} - 16 q^{51} + 8 q^{59} - 4 q^{61} - 8 q^{69} - 8 q^{71} - 24 q^{79} - 4 q^{81} - 52 q^{91} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 25x^{4} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{2} ) / 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} ) / 25 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} ) / 125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 125\nu ) / 125 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 125\nu ) / 125 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} + 5\nu^{5} + 25\nu^{3} ) / 125 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{5} + 5\nu^{3} + 25\nu ) / 25 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} + \beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 5\beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} + 5\beta_{6} - 5\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 25\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -25\beta_{7} + 25\beta_{6} + 25\beta_{4} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 125\beta_{3} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -125\beta_{5} + 125\beta_{4} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
2.15988 − 0.578737i
−2.15988 + 0.578737i
0.578737 + 2.15988i
−0.578737 − 2.15988i
2.15988 + 0.578737i
−2.15988 − 0.578737i
0.578737 − 2.15988i
−0.578737 + 2.15988i
0 −0.866025 + 0.500000i 0 0 0 −3.60464 − 2.08114i 0 0.500000 − 0.866025i 0
1849.2 0 −0.866025 + 0.500000i 0 0 0 1.87259 + 1.08114i 0 0.500000 − 0.866025i 0
1849.3 0 0.866025 − 0.500000i 0 0 0 −1.87259 − 1.08114i 0 0.500000 − 0.866025i 0
1849.4 0 0.866025 − 0.500000i 0 0 0 3.60464 + 2.08114i 0 0.500000 − 0.866025i 0
3649.1 0 −0.866025 − 0.500000i 0 0 0 −3.60464 + 2.08114i 0 0.500000 + 0.866025i 0
3649.2 0 −0.866025 − 0.500000i 0 0 0 1.87259 − 1.08114i 0 0.500000 + 0.866025i 0
3649.3 0 0.866025 + 0.500000i 0 0 0 −1.87259 + 1.08114i 0 0.500000 + 0.866025i 0
3649.4 0 0.866025 + 0.500000i 0 0 0 3.60464 − 2.08114i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n 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.by.g 8
5.b even 2 1 inner 3900.2.by.g 8
5.c odd 4 1 780.2.q.c ✓ 4
5.c odd 4 1 3900.2.q.j 4
13.c even 3 1 inner 3900.2.by.g 8
15.e even 4 1 2340.2.q.g 4
65.n even 6 1 inner 3900.2.by.g 8
65.q odd 12 1 780.2.q.c ✓ 4
65.q odd 12 1 3900.2.q.j 4
195.bl even 12 1 2340.2.q.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.q.c ✓ 4 5.c odd 4 1
780.2.q.c ✓ 4 65.q odd 12 1
2340.2.q.g 4 15.e even 4 1
2340.2.q.g 4 195.bl even 12 1
3900.2.q.j 4 5.c odd 4 1
3900.2.q.j 4 65.q odd 12 1
3900.2.by.g 8 1.a even 1 1 trivial
3900.2.by.g 8 5.b even 2 1 inner
3900.2.by.g 8 13.c even 3 1 inner
3900.2.by.g 8 65.n even 6 1 inner

Hecke kernels

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

\( T_{7}^{8} - 22T_{7}^{6} + 403T_{7}^{4} - 1782T_{7}^{2} + 6561 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} + 4 \) 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
$7$ \( T^{8} - 22 T^{6} + \cdots + 6561 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} - 14 T^{6} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 28 T^{6} + \cdots + 1296 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 4 T^{2} + 16)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 4 T^{3} + 22 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T - 1)^{8} \) Copy content Toggle raw display
$37$ \( T^{8} - 88 T^{6} + \cdots + 1679616 \) Copy content Toggle raw display
$41$ \( (T^{4} - 8 T^{3} + 58 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} - 182 T^{6} + \cdots + 62742241 \) Copy content Toggle raw display
$47$ \( (T^{2} + 90)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} + 112 T^{2} + 576)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + 22 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + T + 1)^{4} \) Copy content Toggle raw display
$67$ \( T^{8} - 118 T^{6} + \cdots + 2313441 \) Copy content Toggle raw display
$71$ \( (T^{4} + 4 T^{3} + 22 T^{2} + \cdots + 36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 118 T^{2} + 1521)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T - 31)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 144)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 10 T^{2} + 100)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 118 T^{6} + \cdots + 2313441 \) Copy content Toggle raw display
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