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Properties

Label 3960.2.d.h
Level $3960$
Weight $2$
Character orbit 3960.d
Analytic conductor $31.621$
Analytic rank $0$
Dimension $10$
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] = [3960,2,Mod(3169,3960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3960.3169");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3960 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3960.d (of order \(2\), degree \(1\), 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: \(31.6207592004\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 23x^{8} + 187x^{6} + 657x^{4} + 928x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 1320)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{5} q^{5} - \beta_1 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{5} - \beta_1 q^{7} + q^{11} + \beta_{9} q^{13} + (\beta_{7} - \beta_{3} + \beta_1) q^{17} + ( - \beta_{8} + \beta_{6} + \beta_{5}) q^{19} + (\beta_{9} - 2 \beta_{6} + \cdots - \beta_1) q^{23}+ \cdots + ( - \beta_{7} + \beta_{6} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 2 q^{5} + 10 q^{11} - 4 q^{19} - 2 q^{25} + 8 q^{29} + 8 q^{31} - 8 q^{35} - 16 q^{41} - 42 q^{49} + 2 q^{55} + 24 q^{59} + 20 q^{61} - 24 q^{65} - 8 q^{71} - 36 q^{79} + 12 q^{85} + 68 q^{89} - 52 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 23x^{8} + 187x^{6} + 657x^{4} + 928x^{2} + 324 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{9} - 5\nu^{7} + 119\nu^{5} + 945\nu^{3} + 2150\nu ) / 288 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{9} + 97\nu^{7} + 557\nu^{5} + 819\nu^{3} + 576\nu^{2} + 50\nu + 2880 ) / 576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{9} - 5\nu^{7} + 119\nu^{5} + 801\nu^{3} + 1142\nu ) / 144 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} - 15\nu^{8} - 29\nu^{7} - 243\nu^{6} - 265\nu^{5} - 999\nu^{4} - 807\nu^{3} - 585\nu^{2} - 586\nu + 714 ) / 192 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2 \nu^{9} - 27 \nu^{8} + 10 \nu^{7} - 495 \nu^{6} - 238 \nu^{5} - 2835 \nu^{4} - 1602 \nu^{3} + \cdots - 3438 ) / 576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2 \nu^{9} - 27 \nu^{8} - 10 \nu^{7} - 495 \nu^{6} + 238 \nu^{5} - 2835 \nu^{4} + 1602 \nu^{3} + \cdots - 3438 ) / 576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{9} - 97\nu^{7} - 557\nu^{5} - 819\nu^{3} + 526\nu ) / 288 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 5\nu^{8} + 97\nu^{6} + 589\nu^{4} + 1235\nu^{2} + 562 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -\nu^{9} - 19\nu^{7} - 113\nu^{5} - 237\nu^{3} - 126\nu ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - \beta_{3} + 2\beta_{2} - 10 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{6} - 7\beta_{5} - 9\beta_{3} + 4\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{8} - 11\beta_{7} - 15\beta_{6} + 5\beta_{5} + 2\beta_{4} + 11\beta_{3} - 20\beta_{2} + 68 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -2\beta_{9} + 14\beta_{7} - 55\beta_{6} + 55\beta_{5} + 81\beta_{3} - 50\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 36\beta_{8} + 107\beta_{7} + 175\beta_{6} - 15\beta_{5} - 24\beta_{4} - 107\beta_{3} + 190\beta_{2} - 538 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 32\beta_{9} - 232\beta_{7} + 483\beta_{6} - 483\beta_{5} - 745\beta_{3} + 532\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 450 \beta_{8} - 1027 \beta_{7} - 1875 \beta_{6} - 51 \beta_{5} + 230 \beta_{4} + 1027 \beta_{3} + \cdots + 4672 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -398\beta_{9} + 2826\beta_{7} - 4495\beta_{6} + 4495\beta_{5} + 7009\beta_{3} - 5406\beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3960\mathbb{Z}\right)^\times\).

\(n\) \(991\) \(1981\) \(2377\) \(2521\) \(3521\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3169.1
2.35912i
− 2.35912i
1.61972i
− 1.61972i
2.11043i
− 2.11043i
3.13869i
− 3.13869i
− 0.711151i
0.711151i
0 0 0 −2.23248 − 0.126646i 0 − 3.92460i 0 0 0
3169.2 0 0 0 −2.23248 + 0.126646i 0 3.92460i 0 0 0
3169.3 0 0 0 −0.548128 − 2.16785i 0 − 2.99623i 0 0 0
3169.4 0 0 0 −0.548128 + 2.16785i 0 2.99623i 0 0 0
3169.5 0 0 0 0.122287 − 2.23272i 0 − 2.56437i 0 0 0
3169.6 0 0 0 0.122287 + 2.23272i 0 2.56437i 0 0 0
3169.7 0 0 0 1.76211 − 1.37658i 0 2.71268i 0 0 0
3169.8 0 0 0 1.76211 + 1.37658i 0 − 2.71268i 0 0 0
3169.9 0 0 0 1.89621 − 1.18506i 0 4.20542i 0 0 0
3169.10 0 0 0 1.89621 + 1.18506i 0 − 4.20542i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3169.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3960.2.d.h 10
3.b odd 2 1 1320.2.d.c ✓ 10
5.b even 2 1 inner 3960.2.d.h 10
12.b even 2 1 2640.2.d.k 10
15.d odd 2 1 1320.2.d.c ✓ 10
15.e even 4 1 6600.2.a.bx 5
15.e even 4 1 6600.2.a.bz 5
60.h even 2 1 2640.2.d.k 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1320.2.d.c ✓ 10 3.b odd 2 1
1320.2.d.c ✓ 10 15.d odd 2 1
2640.2.d.k 10 12.b even 2 1
2640.2.d.k 10 60.h even 2 1
3960.2.d.h 10 1.a even 1 1 trivial
3960.2.d.h 10 5.b even 2 1 inner
6600.2.a.bx 5 15.e even 4 1
6600.2.a.bz 5 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3960, [\chi])\):

\( T_{7}^{10} + 56T_{7}^{8} + 1204T_{7}^{6} + 12416T_{7}^{4} + 61632T_{7}^{2} + 118336 \) Copy content Toggle raw display
\( T_{29}^{5} - 4T_{29}^{4} - 44T_{29}^{3} + 176T_{29}^{2} + 384T_{29} - 1472 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 2 T^{9} + \cdots + 3125 \) Copy content Toggle raw display
$7$ \( T^{10} + 56 T^{8} + \cdots + 118336 \) Copy content Toggle raw display
$11$ \( (T - 1)^{10} \) Copy content Toggle raw display
$13$ \( T^{10} + 92 T^{8} + \cdots + 419904 \) Copy content Toggle raw display
$17$ \( T^{10} + 68 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$19$ \( (T^{5} + 2 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 188 T^{8} + \cdots + 2359296 \) Copy content Toggle raw display
$29$ \( (T^{5} - 4 T^{4} + \cdots - 1472)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} - 4 T^{4} + \cdots - 1792)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} + 104 T^{8} + \cdots + 16384 \) Copy content Toggle raw display
$41$ \( (T^{5} + 8 T^{4} + \cdots + 3840)^{2} \) Copy content Toggle raw display
$43$ \( T^{10} + 268 T^{8} + \cdots + 9216 \) Copy content Toggle raw display
$47$ \( T^{10} + 332 T^{8} + \cdots + 200704 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 144000000 \) Copy content Toggle raw display
$59$ \( (T^{5} - 12 T^{4} + \cdots + 11392)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} - 10 T^{4} + \cdots - 7184)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 1677721600 \) Copy content Toggle raw display
$71$ \( (T^{5} + 4 T^{4} + \cdots - 1952)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + 368 T^{8} + \cdots + 4804864 \) Copy content Toggle raw display
$79$ \( (T^{5} + 18 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + 440 T^{8} + \cdots + 11505664 \) Copy content Toggle raw display
$89$ \( (T^{5} - 34 T^{4} + \cdots + 6752)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 1266221056 \) Copy content Toggle raw display
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