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Properties

Label 3900.2.by.j
Level $3900$
Weight $2$
Character orbit 3900.by
Analytic conductor $31.142$
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] = [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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} + 192x^{8} - 952x^{6} + 3520x^{4} - 2304x^{2} + 1296 \) 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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} + \beta_{11} q^{7} + (\beta_{6} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} + \beta_{11} q^{7} + (\beta_{6} + 1) q^{9} + 2 \beta_{5} q^{11} + (\beta_{11} + \beta_{8} - \beta_{3} + \beta_1) q^{13} + (\beta_{11} - \beta_{10} + \cdots + 2 \beta_1) q^{17}+ \cdots - 2 \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{9} + 16 q^{19} - 4 q^{21} + 52 q^{31} - 8 q^{39} - 28 q^{41} + 4 q^{49} + 8 q^{51} - 8 q^{59} + 38 q^{61} - 36 q^{71} + 36 q^{79} - 6 q^{81} + 8 q^{89} - 34 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 192x^{8} - 952x^{6} + 3520x^{4} - 2304x^{2} + 1296 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{10} + 12\nu^{8} - 105\nu^{6} + 220\nu^{4} - 144\nu^{2} + 3132 ) / 3744 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{11} - 12\nu^{9} + 144\nu^{7} - 220\nu^{5} + 144\nu^{3} + 18240\nu ) / 14976 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{10} + 60\nu^{8} - 642\nu^{6} + 1100\nu^{4} - 720\nu^{2} - 25992 ) / 7488 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{10} - 12\nu^{8} + 144\nu^{6} - 532\nu^{4} + 2640\nu^{2} - 1728 ) / 1872 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{10} - 61\nu^{8} + 732\nu^{6} - 3376\nu^{4} + 13420\nu^{2} - 8784 ) / 7488 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{11} - 61\nu^{9} + 732\nu^{7} - 3376\nu^{5} + 13420\nu^{3} - 1296\nu ) / 7488 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{11} + 36\nu^{9} - 380\nu^{7} + 660\nu^{5} - 432\nu^{3} - 21232\nu ) / 4992 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{10} + 17\nu^{8} - 204\nu^{6} + 1084\nu^{4} - 3740\nu^{2} + 2448 ) / 576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4\nu^{11} - 61\nu^{9} + 732\nu^{7} - 3376\nu^{5} + 12484\nu^{3} - 1296\nu ) / 5616 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -97\nu^{11} + 1606\nu^{9} - 19272\nu^{7} + 98716\nu^{5} - 353320\nu^{3} + 231264\nu ) / 44928 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} + 5\beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{10} + 8\beta_{7} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{9} + 40\beta_{6} - 14\beta_{5} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -6\beta_{11} - 78\beta_{10} + 70\beta_{7} + 78\beta_{3} - 70\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -64\beta_{4} + 160\beta_{2} - 356 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 96\beta_{8} + 864\beta_{3} - 644\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -548\beta_{9} - 3316\beta_{6} + 1700\beta_{5} - 548\beta_{4} + 1700\beta_{2} - 3316 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 1152\beta_{11} + 9048\beta_{10} + 1152\beta_{8} - 6112\beta_{7} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -4960\beta_{9} - 31712\beta_{6} + 17464\beta_{5} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 12504\beta_{11} + 92280\beta_{10} - 59096\beta_{7} - 92280\beta_{3} + 59096\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
−0.708892 + 0.409279i
2.72575 − 1.57372i
−2.01686 + 1.16444i
2.01686 − 1.16444i
−2.72575 + 1.57372i
0.708892 − 0.409279i
−0.708892 − 0.409279i
2.72575 + 1.57372i
−2.01686 − 1.16444i
2.01686 + 1.16444i
−2.72575 − 1.57372i
0.708892 + 0.409279i
0 −0.866025 + 0.500000i 0 0 0 −3.04097 − 1.75570i 0 0.500000 − 0.866025i 0
1849.2 0 −0.866025 + 0.500000i 0 0 0 1.52324 + 0.879443i 0 0.500000 − 0.866025i 0
1849.3 0 −0.866025 + 0.500000i 0 0 0 2.38375 + 1.37626i 0 0.500000 − 0.866025i 0
1849.4 0 0.866025 − 0.500000i 0 0 0 −2.38375 − 1.37626i 0 0.500000 − 0.866025i 0
1849.5 0 0.866025 − 0.500000i 0 0 0 −1.52324 − 0.879443i 0 0.500000 − 0.866025i 0
1849.6 0 0.866025 − 0.500000i 0 0 0 3.04097 + 1.75570i 0 0.500000 − 0.866025i 0
3649.1 0 −0.866025 − 0.500000i 0 0 0 −3.04097 + 1.75570i 0 0.500000 + 0.866025i 0
3649.2 0 −0.866025 − 0.500000i 0 0 0 1.52324 − 0.879443i 0 0.500000 + 0.866025i 0
3649.3 0 −0.866025 − 0.500000i 0 0 0 2.38375 − 1.37626i 0 0.500000 + 0.866025i 0
3649.4 0 0.866025 + 0.500000i 0 0 0 −2.38375 + 1.37626i 0 0.500000 + 0.866025i 0
3649.5 0 0.866025 + 0.500000i 0 0 0 −1.52324 + 0.879443i 0 0.500000 + 0.866025i 0
3649.6 0 0.866025 + 0.500000i 0 0 0 3.04097 − 1.75570i 0 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.6
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.j 12
5.b even 2 1 inner 3900.2.by.j 12
5.c odd 4 1 780.2.q.e ✓ 6
5.c odd 4 1 3900.2.q.n 6
13.c even 3 1 inner 3900.2.by.j 12
15.e even 4 1 2340.2.q.h 6
65.n even 6 1 inner 3900.2.by.j 12
65.q odd 12 1 780.2.q.e ✓ 6
65.q odd 12 1 3900.2.q.n 6
195.bl even 12 1 2340.2.q.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.q.e ✓ 6 5.c odd 4 1
780.2.q.e ✓ 6 65.q odd 12 1
2340.2.q.h 6 15.e even 4 1
2340.2.q.h 6 195.bl even 12 1
3900.2.q.n 6 5.c odd 4 1
3900.2.q.n 6 65.q odd 12 1
3900.2.by.j 12 1.a even 1 1 trivial
3900.2.by.j 12 5.b even 2 1 inner
3900.2.by.j 12 13.c even 3 1 inner
3900.2.by.j 12 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 \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{12} - 23T_{7}^{10} + 374T_{7}^{8} - 2987T_{7}^{6} + 17378T_{7}^{4} - 44795T_{7}^{2} + 83521 \) Copy content Toggle raw display
\( T_{11}^{6} + 32T_{11}^{4} + 96T_{11}^{3} + 1024T_{11}^{2} + 1536T_{11} + 2304 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{2} + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 23 T^{10} + \cdots + 83521 \) Copy content Toggle raw display
$11$ \( (T^{6} + 32 T^{4} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 21 T^{10} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 252047376 \) Copy content Toggle raw display
$19$ \( (T^{6} - 8 T^{5} + \cdots + 2704)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( (T^{6} + 8 T^{4} + 12 T^{3} + \cdots + 36)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 13 T^{2} + \cdots + 13)^{4} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 533794816 \) Copy content Toggle raw display
$41$ \( (T^{6} + 14 T^{5} + \cdots + 86436)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 141158161 \) Copy content Toggle raw display
$47$ \( (T^{6} + 80 T^{4} + \cdots + 10404)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 96 T^{4} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + 4 T^{5} + \cdots + 191844)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 19 T^{5} + \cdots + 13689)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 168823196161 \) Copy content Toggle raw display
$71$ \( (T^{6} + 18 T^{5} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 79 T^{4} + \cdots + 3969)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 9 T^{2} + \cdots + 169)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + 364 T^{4} + \cdots + 876096)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 4 T^{5} + \cdots + 298116)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 492884401 \) Copy content Toggle raw display
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