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Properties

Label 3900.2.cd.k
Level $3900$
Weight $2$
Character orbit 3900.cd
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(901,3900)]
 
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, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3900.901");
 
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.cd (of order \(6\), 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: \(8\)
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: \( 3^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + (\beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{7} + \beta_{6} + \beta_{5} + \cdots + 1) q^{7}+ \cdots + (\beta_{6} - 4 \beta_{3} + \beta_{2} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} + 6 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} + 6 q^{7} - 4 q^{9} - 24 q^{11} - 2 q^{13} + 2 q^{17} - 12 q^{19} + 8 q^{27} - 14 q^{29} + 24 q^{33} - 12 q^{37} + 4 q^{39} + 18 q^{41} + 10 q^{43} + 32 q^{49} - 4 q^{51} - 16 q^{53} + 6 q^{59} - 24 q^{61} - 6 q^{63} + 18 q^{67} - 6 q^{71} + 48 q^{77} + 8 q^{79} - 4 q^{81} - 14 q^{87} - 6 q^{89} + 24 q^{91} - 30 q^{97}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( - 576 \nu^{7} + 7405 \nu^{6} + 23585 \nu^{5} - 109530 \nu^{4} + 61525 \nu^{3} + 275115 \nu^{2} + \cdots + 1014949 ) / 284739 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2445 \nu^{7} + 1688 \nu^{6} - 29423 \nu^{5} + 82314 \nu^{4} - 26350 \nu^{3} - 232023 \nu^{2} + \cdots - 727753 ) / 284739 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 62\nu^{7} + 84\nu^{6} - 716\nu^{5} + 773\nu^{4} + 2168\nu^{3} - 3968\nu^{2} + 6246\nu + 1235 ) / 5811 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4113 \nu^{7} - 21733 \nu^{6} - 12695 \nu^{5} + 150348 \nu^{4} - 159037 \nu^{3} - 251610 \nu^{2} + \cdots - 489268 ) / 284739 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30\nu^{7} - 34\nu^{6} - 107\nu^{5} + 330\nu^{4} + 173\nu^{3} + 342\nu^{2} + 2098\nu + 1157 ) / 1911 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4789 \nu^{7} - 27128 \nu^{6} + 7741 \nu^{5} + 124660 \nu^{4} - 307042 \nu^{3} + 123518 \nu^{2} + \cdots - 902005 ) / 284739 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 8347 \nu^{7} + 12560 \nu^{6} + 26225 \nu^{5} - 142777 \nu^{4} + 81028 \nu^{3} - 64325 \nu^{2} + \cdots + 443482 ) / 284739 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 2\beta_{5} - \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + \beta_{6} + 2\beta_{5} - 4\beta_{4} - \beta_{3} - 2\beta_{2} - 4\beta _1 + 6 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{7} - 6\beta_{6} + 7\beta_{5} + 3\beta_{4} - 5\beta_{3} - 4\beta_{2} - 3\beta _1 - 15 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -22\beta_{7} - 14\beta_{6} - 16\beta_{5} + 2\beta_{4} - 13\beta_{3} - 8\beta_{2} - 10\beta _1 + 21 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -29\beta_{7} - 48\beta_{6} - 16\beta_{5} + 51\beta_{4} - 55\beta_{3} + 16\beta_{2} + 39\beta _1 - 96 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -37\beta_{7} - 15\beta_{6} - 55\beta_{5} + 2\beta_{4} - 5\beta_{3} + 10\beta_{2} + 15\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -40\beta_{7} - 45\beta_{6} - 80\beta_{5} + 195\beta_{4} + 40\beta_{3} + 155\beta_{2} + 363\beta _1 - 603 ) / 3 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(301\) \(1301\) \(1951\) \(3277\)
\(\chi(n)\) \(1 - \beta_{3}\) \(1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
0.338876 + 1.46735i
−2.39244 + 0.0909984i
1.02715 − 1.10132i
2.02641 + 1.27503i
0.338876 − 1.46735i
−2.39244 − 0.0909984i
1.02715 + 1.10132i
2.02641 − 1.27503i
0 −0.500000 + 0.866025i 0 0 0 −4.40205 + 2.54152i 0 −0.500000 − 0.866025i 0
901.2 0 −0.500000 + 0.866025i 0 0 0 0.272995 − 0.157614i 0 −0.500000 − 0.866025i 0
901.3 0 −0.500000 + 0.866025i 0 0 0 3.30397 − 1.90755i 0 −0.500000 − 0.866025i 0
901.4 0 −0.500000 + 0.866025i 0 0 0 3.82508 − 2.20841i 0 −0.500000 − 0.866025i 0
2701.1 0 −0.500000 − 0.866025i 0 0 0 −4.40205 − 2.54152i 0 −0.500000 + 0.866025i 0
2701.2 0 −0.500000 − 0.866025i 0 0 0 0.272995 + 0.157614i 0 −0.500000 + 0.866025i 0
2701.3 0 −0.500000 − 0.866025i 0 0 0 3.30397 + 1.90755i 0 −0.500000 + 0.866025i 0
2701.4 0 −0.500000 − 0.866025i 0 0 0 3.82508 + 2.20841i 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.e 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.cd.k 8
5.b even 2 1 780.2.cc.c ✓ 8
5.c odd 4 1 3900.2.bw.h 8
5.c odd 4 1 3900.2.bw.k 8
13.e even 6 1 inner 3900.2.cd.k 8
15.d odd 2 1 2340.2.dj.b 8
65.l even 6 1 780.2.cc.c ✓ 8
65.r odd 12 1 3900.2.bw.h 8
65.r odd 12 1 3900.2.bw.k 8
195.y odd 6 1 2340.2.dj.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
780.2.cc.c ✓ 8 5.b even 2 1
780.2.cc.c ✓ 8 65.l even 6 1
2340.2.dj.b 8 15.d odd 2 1
2340.2.dj.b 8 195.y odd 6 1
3900.2.bw.h 8 5.c odd 4 1
3900.2.bw.h 8 65.r odd 12 1
3900.2.bw.k 8 5.c odd 4 1
3900.2.bw.k 8 65.r odd 12 1
3900.2.cd.k 8 1.a even 1 1 trivial
3900.2.cd.k 8 13.e 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 \(S_{2}^{\mathrm{new}}(3900, [\chi])\):

\( T_{7}^{8} - 6T_{7}^{7} - 12T_{7}^{6} + 144T_{7}^{5} + 279T_{7}^{4} - 3888T_{7}^{3} + 9396T_{7}^{2} - 4374T_{7} + 729 \) 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
\( T_{17}^{8} - 2T_{17}^{7} + 24T_{17}^{6} - 56T_{17}^{5} + 514T_{17}^{4} - 1032T_{17}^{3} + 1944T_{17}^{2} - 864T_{17} + 324 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{2} + T + 1)^{4} \) 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
$13$ \( T^{8} + 2 T^{7} + \cdots + 28561 \) Copy content Toggle raw display
$17$ \( T^{8} - 2 T^{7} + \cdots + 324 \) Copy content Toggle raw display
$19$ \( T^{8} + 12 T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$23$ \( T^{8} + 48 T^{6} + \cdots + 46656 \) 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
$37$ \( T^{8} + 12 T^{7} + \cdots + 186624 \) Copy content Toggle raw display
$41$ \( T^{8} - 18 T^{7} + \cdots + 11819844 \) Copy content Toggle raw display
$43$ \( T^{8} - 10 T^{7} + \cdots + 729 \) Copy content Toggle raw display
$47$ \( T^{8} + 204 T^{6} + \cdots + 1542564 \) Copy content Toggle raw display
$53$ \( (T^{4} + 8 T^{3} + \cdots - 1152)^{2} \) 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
$67$ \( T^{8} - 18 T^{7} + \cdots + 21609 \) Copy content Toggle raw display
$71$ \( T^{8} + 6 T^{7} + \cdots + 91355364 \) Copy content Toggle raw display
$73$ \( T^{8} + 444 T^{6} + \cdots + 32867289 \) Copy content Toggle raw display
$79$ \( (T^{4} - 4 T^{3} + \cdots - 767)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 240 T^{6} + \cdots + 46656 \) Copy content Toggle raw display
$89$ \( T^{8} + 6 T^{7} + \cdots + 54756 \) Copy content Toggle raw display
$97$ \( T^{8} + 30 T^{7} + \cdots + 257049 \) Copy content Toggle raw display
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