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Properties

Label 3900.2.q.q
Level $3900$
Weight $2$
Character orbit 3900.q
Analytic conductor $31.142$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3900,2,Mod(601,3900)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.601");
 
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.q (of order \(3\), degree \(2\), 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: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 39 x^{14} + 6 x^{13} + 874 x^{12} + 524 x^{11} + 12451 x^{10} + 15026 x^{9} + \cdots + 2250000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 780)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{3} + \beta_1 q^{7} + ( - \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{3} + \beta_1 q^{7} + ( - \beta_{2} - 1) q^{9} - \beta_{10} q^{11} - \beta_{3} q^{13} + (\beta_{14} - \beta_{13} - \beta_{11} + \cdots - 1) q^{17}+ \cdots + ( - \beta_{11} + \beta_{10} + \cdots + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{3} + 2 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{3} + 2 q^{7} - 8 q^{9} - 2 q^{11} - 6 q^{17} + 4 q^{19} - 4 q^{21} - 12 q^{23} + 16 q^{27} + 12 q^{31} - 2 q^{33} + 6 q^{37} - 6 q^{41} - 12 q^{43} + 28 q^{47} - 18 q^{49} + 12 q^{51} - 8 q^{57} - 6 q^{59} - 20 q^{61} + 2 q^{63} + 2 q^{67} - 12 q^{69} + 18 q^{71} + 24 q^{73} + 16 q^{77} + 4 q^{79} - 8 q^{81} - 44 q^{83} + 12 q^{89} - 30 q^{91} - 6 q^{93} - 4 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} + 39 x^{14} + 6 x^{13} + 874 x^{12} + 524 x^{11} + 12451 x^{10} + 15026 x^{9} + \cdots + 2250000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 80\!\cdots\!14 \nu^{15} + \cdots + 18\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 57\!\cdots\!69 \nu^{15} + \cdots - 33\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\!\cdots\!21 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 99\!\cdots\!43 \nu^{15} + \cdots + 11\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 71\!\cdots\!71 \nu^{15} + \cdots + 36\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40\!\cdots\!03 \nu^{15} + \cdots + 16\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 12\!\cdots\!31 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 13\!\cdots\!83 \nu^{15} + \cdots + 38\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 51\!\cdots\!34 \nu^{15} + \cdots - 12\!\cdots\!00 ) / 23\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 26\!\cdots\!63 \nu^{15} + \cdots + 50\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!07 \nu^{15} + \cdots + 26\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 33\!\cdots\!57 \nu^{15} + \cdots + 31\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 21\!\cdots\!81 \nu^{15} + \cdots + 14\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 46\!\cdots\!97 \nu^{15} + \cdots - 43\!\cdots\!00 ) / 95\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{13} - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} + 9\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 5 \beta_{14} - 2 \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} - 14 \beta_{6} + \beta_{5} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{15} + 25 \beta_{14} + 18 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} + 18 \beta_{9} - 25 \beta_{7} + \cdots - 129 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 54 \beta_{13} + 22 \beta_{12} + 58 \beta_{11} - 40 \beta_{10} + 33 \beta_{9} - 118 \beta_{7} + \cdots - 346 \beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 61 \beta_{15} - 540 \beta_{14} + 405 \beta_{11} - 320 \beta_{10} - 85 \beta_{9} + 16 \beta_{8} + \cdots + 2276 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 423 \beta_{15} - 2501 \beta_{14} - 1252 \beta_{13} - 423 \beta_{12} + 207 \beta_{11} - 1342 \beta_{9} + \cdots + 8303 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6631 \beta_{13} - 1463 \beta_{12} - 6683 \beta_{11} + 6069 \beta_{10} - 4709 \beta_{9} + \cdots + 44132 \beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 8214 \beta_{15} + 52278 \beta_{14} - 33968 \beta_{11} + 23544 \beta_{10} + 10424 \beta_{9} + \cdots - 183174 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 32400 \beta_{15} + 237877 \beta_{14} + 134887 \beta_{13} + 32400 \beta_{12} - 26852 \beta_{11} + \cdots - 892725 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 588350 \beta_{13} + 163531 \beta_{12} + 615642 \beta_{11} - 509881 \beta_{10} + 401667 \beta_{9} + \cdots - 3914705 \beta_{2} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 692750 \beta_{15} - 4972585 \beta_{14} + 3387625 \beta_{11} - 2453934 \beta_{10} - 933691 \beta_{9} + \cdots + 18396093 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 3314542 \beta_{15} - 22768018 \beta_{14} - 12457032 \beta_{13} - 3314542 \beta_{12} + 2319191 \beta_{11} + \cdots + 82593262 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 57957186 \beta_{13} - 14563645 \beta_{12} - 59387402 \beta_{11} + 50592740 \beta_{10} + \cdots + 382130084 \beta_{2} \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 67871457 \beta_{15} + 475270403 \beta_{14} - 319997967 \beta_{11} + 227237031 \beta_{10} + \cdots - 1733208029 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
601.1
−1.55573 + 2.69460i
−1.44931 + 2.51027i
−1.35434 + 2.34579i
−0.706955 + 1.22448i
0.549724 − 0.952150i
0.944936 − 1.63668i
2.28429 − 3.95650i
2.28738 − 3.96186i
−1.55573 − 2.69460i
−1.44931 − 2.51027i
−1.35434 − 2.34579i
−0.706955 − 1.22448i
0.549724 + 0.952150i
0.944936 + 1.63668i
2.28429 + 3.95650i
2.28738 + 3.96186i
0 −0.500000 − 0.866025i 0 0 0 −1.55573 + 2.69460i 0 −0.500000 + 0.866025i 0
601.2 0 −0.500000 − 0.866025i 0 0 0 −1.44931 + 2.51027i 0 −0.500000 + 0.866025i 0
601.3 0 −0.500000 − 0.866025i 0 0 0 −1.35434 + 2.34579i 0 −0.500000 + 0.866025i 0
601.4 0 −0.500000 − 0.866025i 0 0 0 −0.706955 + 1.22448i 0 −0.500000 + 0.866025i 0
601.5 0 −0.500000 − 0.866025i 0 0 0 0.549724 − 0.952150i 0 −0.500000 + 0.866025i 0
601.6 0 −0.500000 − 0.866025i 0 0 0 0.944936 − 1.63668i 0 −0.500000 + 0.866025i 0
601.7 0 −0.500000 − 0.866025i 0 0 0 2.28429 − 3.95650i 0 −0.500000 + 0.866025i 0
601.8 0 −0.500000 − 0.866025i 0 0 0 2.28738 − 3.96186i 0 −0.500000 + 0.866025i 0
2401.1 0 −0.500000 + 0.866025i 0 0 0 −1.55573 − 2.69460i 0 −0.500000 − 0.866025i 0
2401.2 0 −0.500000 + 0.866025i 0 0 0 −1.44931 − 2.51027i 0 −0.500000 − 0.866025i 0
2401.3 0 −0.500000 + 0.866025i 0 0 0 −1.35434 − 2.34579i 0 −0.500000 − 0.866025i 0
2401.4 0 −0.500000 + 0.866025i 0 0 0 −0.706955 − 1.22448i 0 −0.500000 − 0.866025i 0
2401.5 0 −0.500000 + 0.866025i 0 0 0 0.549724 + 0.952150i 0 −0.500000 − 0.866025i 0
2401.6 0 −0.500000 + 0.866025i 0 0 0 0.944936 + 1.63668i 0 −0.500000 − 0.866025i 0
2401.7 0 −0.500000 + 0.866025i 0 0 0 2.28429 + 3.95650i 0 −0.500000 − 0.866025i 0
2401.8 0 −0.500000 + 0.866025i 0 0 0 2.28738 + 3.96186i 0 −0.500000 − 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 601.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3900.2.q.q 16
5.b even 2 1 3900.2.q.r 16
5.c odd 4 2 780.2.bx.a ✓ 32
13.c even 3 1 inner 3900.2.q.q 16
15.e even 4 2 2340.2.de.c 32
65.n even 6 1 3900.2.q.r 16
65.q odd 12 2 780.2.bx.a ✓ 32
195.bl even 12 2 2340.2.de.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.bx.a ✓ 32 5.c odd 4 2
780.2.bx.a ✓ 32 65.q odd 12 2
2340.2.de.c 32 15.e even 4 2
2340.2.de.c 32 195.bl even 12 2
3900.2.q.q 16 1.a even 1 1 trivial
3900.2.q.q 16 13.c even 3 1 inner
3900.2.q.r 16 5.b even 2 1
3900.2.q.r 16 65.n even 6 1

Hecke kernels

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

\( T_{7}^{16} - 2 T_{7}^{15} + 39 T_{7}^{14} + 6 T_{7}^{13} + 874 T_{7}^{12} + 524 T_{7}^{11} + \cdots + 2250000 \) Copy content Toggle raw display
\( T_{11}^{16} + 2 T_{11}^{15} + 66 T_{11}^{14} + 132 T_{11}^{13} + 3064 T_{11}^{12} + 5792 T_{11}^{11} + \cdots + 10036224 \) Copy content Toggle raw display
\( T_{23}^{16} + 12 T_{23}^{15} + 190 T_{23}^{14} + 1256 T_{23}^{13} + 14504 T_{23}^{12} + 85624 T_{23}^{11} + \cdots + 90326016 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{15} + \cdots + 2250000 \) Copy content Toggle raw display
$11$ \( T^{16} + 2 T^{15} + \cdots + 10036224 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + 6 T^{15} + \cdots + 22500 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14457657600 \) Copy content Toggle raw display
$23$ \( T^{16} + 12 T^{15} + \cdots + 90326016 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 12994632036 \) Copy content Toggle raw display
$31$ \( (T^{8} - 6 T^{7} + \cdots + 266320)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 6 T^{15} + \cdots + 20736 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 196917612516 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 4533598224 \) Copy content Toggle raw display
$47$ \( (T^{8} - 14 T^{7} + \cdots - 5000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} - 223 T^{6} + \cdots + 559872)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 689085932544 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 230535379881 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 5890736743056 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} - 12 T^{7} + \cdots + 198327)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} - 2 T^{7} + \cdots - 2315952)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 22 T^{7} + \cdots - 4658592)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 131543466177600 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 232089380870400 \) Copy content Toggle raw display
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