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Properties

Label 3762.2.g.g
Level $3762$
Weight $2$
Character orbit 3762.g
Analytic conductor $30.040$
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] = [3762,2,Mod(2089,3762)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3762, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3762.2089");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3762 = 2 \cdot 3^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3762.g (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: \(30.0397212404\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.14584320320.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 4x^{6} + 11x^{5} - 11x^{4} + 32x^{3} + 44x^{2} - 18x + 46 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 418)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} - \beta_{2} q^{5} + (\beta_{5} + \beta_{3}) q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} - \beta_{2} q^{5} + (\beta_{5} + \beta_{3}) q^{7} - q^{8} + \beta_{2} q^{10} + ( - \beta_{5} - \beta_{4} + \beta_1 + 1) q^{11} + (\beta_{2} + 2 \beta_1 - 1) q^{13} + ( - \beta_{5} - \beta_{3}) q^{14} + q^{16} + ( - \beta_{7} + \beta_{5} + \beta_{3}) q^{17} + ( - 2 \beta_{7} + \beta_{5} + \cdots + \beta_1) q^{19}+ \cdots + (\beta_{6} - \beta_{4} + \beta_1 - 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{2} + 8 q^{4} - 2 q^{5} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{2} + 8 q^{4} - 2 q^{5} - 8 q^{8} + 2 q^{10} + 6 q^{11} - 10 q^{13} + 8 q^{16} - 2 q^{20} - 6 q^{22} - 12 q^{23} - 2 q^{25} + 10 q^{26} - 14 q^{29} - 8 q^{32} + 2 q^{40} + 22 q^{41} + 6 q^{44} + 12 q^{46} - 24 q^{47} + 10 q^{49} + 2 q^{50} - 10 q^{52} + 14 q^{58} + 8 q^{64} - 16 q^{65} + 16 q^{77} + 12 q^{79} - 2 q^{80} - 22 q^{82} - 6 q^{88} - 12 q^{92} + 24 q^{94} - 12 q^{95} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 4x^{6} + 11x^{5} - 11x^{4} + 32x^{3} + 44x^{2} - 18x + 46 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 53\nu^{7} + 1084\nu^{6} - 556\nu^{5} + 2891\nu^{4} + 14726\nu^{3} - 4474\nu^{2} + 10896\nu + 54404 ) / 23578 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -355\nu^{7} + 302\nu^{6} - 2504\nu^{5} - 3349\nu^{4} + 1014\nu^{3} - 26086\nu^{2} + 12432\nu - 4506 ) / 23578 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -955\nu^{7} - 848\nu^{6} + 1566\nu^{5} - 23621\nu^{4} + 11362\nu^{3} + 4544\nu^{2} - 107360\nu + 42008 ) / 23578 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1247\nu^{7} - 2522\nu^{6} + 11386\nu^{5} + 1735\nu^{4} + 2150\nu^{3} + 67788\nu^{2} + 5014\nu + 53088 ) / 23578 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 686\nu^{7} - 650\nu^{6} + 2813\nu^{5} + 4944\nu^{4} - 4915\nu^{3} + 8599\nu^{2} + 7126\nu - 1388 ) / 11789 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1501\nu^{7} - 4000\nu^{6} + 10056\nu^{5} - 1315\nu^{4} - 19364\nu^{3} + 58358\nu^{2} - 1490\nu + 5968 ) / 23578 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1727\nu^{7} - 1602\nu^{6} + 8130\nu^{5} + 13237\nu^{4} - 10844\nu^{3} + 43284\nu^{2} + 48976\nu + 1730 ) / 23578 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} - 2\beta_{5} - \beta_{4} - 2\beta_{2} + \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -15\beta_{7} - 2\beta_{6} + 9\beta_{5} + 8\beta_{4} - 6\beta_{3} - 3\beta_{2} - 4\beta _1 + 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11\beta_{7} - 21\beta_{6} + 6\beta_{5} + 13\beta_{4} + 6\beta_{3} + 14\beta_{2} - 23\beta _1 + 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 58\beta_{7} - 16\beta_{6} - 22\beta_{5} - 20\beta_{4} + 22\beta_{3} - \beta_{2} - 7\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 107\beta_{7} + 72\beta_{6} - 47\beta_{5} - 122\beta_{4} + 46\beta_{3} - 23\beta_{2} + 126\beta _1 - 134 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3762\mathbb{Z}\right)^\times\).

\(n\) \(343\) \(2377\) \(2927\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2089.1
1.20394 + 1.50360i
1.20394 − 1.50360i
0.682410 − 2.29682i
0.682410 + 2.29682i
0.274776 + 0.839339i
0.274776 − 0.839339i
−1.66113 − 0.0964267i
−1.66113 + 0.0964267i
−1.00000 0 1.00000 −2.40788 0 − 3.19266i −1.00000 0 2.40788
2089.2 −1.00000 0 1.00000 −2.40788 0 3.19266i −1.00000 0 2.40788
2089.3 −1.00000 0 1.00000 −1.36482 0 − 0.331974i −1.00000 0 1.36482
2089.4 −1.00000 0 1.00000 −1.36482 0 0.331974i −1.00000 0 1.36482
2089.5 −1.00000 0 1.00000 −0.549551 0 − 2.61555i −1.00000 0 0.549551
2089.6 −1.00000 0 1.00000 −0.549551 0 2.61555i −1.00000 0 0.549551
2089.7 −1.00000 0 1.00000 3.32225 0 − 2.41983i −1.00000 0 −3.32225
2089.8 −1.00000 0 1.00000 3.32225 0 2.41983i −1.00000 0 −3.32225
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2089.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
209.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3762.2.g.g 8
3.b odd 2 1 418.2.b.d yes 8
11.b odd 2 1 3762.2.g.h 8
19.b odd 2 1 3762.2.g.h 8
33.d even 2 1 418.2.b.c ✓ 8
57.d even 2 1 418.2.b.c ✓ 8
209.d even 2 1 inner 3762.2.g.g 8
627.b odd 2 1 418.2.b.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
418.2.b.c ✓ 8 33.d even 2 1
418.2.b.c ✓ 8 57.d even 2 1
418.2.b.d yes 8 3.b odd 2 1
418.2.b.d yes 8 627.b odd 2 1
3762.2.g.g 8 1.a even 1 1 trivial
3762.2.g.g 8 209.d even 2 1 inner
3762.2.g.h 8 11.b odd 2 1
3762.2.g.h 8 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3762, [\chi])\):

\( T_{5}^{4} + T_{5}^{3} - 9T_{5}^{2} - 16T_{5} - 6 \) Copy content Toggle raw display
\( T_{13}^{4} + 5T_{13}^{3} - 16T_{13}^{2} - 79T_{13} - 1 \) Copy content Toggle raw display
\( T_{23}^{4} + 6T_{23}^{3} - 7T_{23}^{2} - 44T_{23} + 48 \) Copy content Toggle raw display
\( T_{29}^{4} + 7T_{29}^{3} + 12T_{29}^{2} - T_{29} - 9 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} - 9 T^{2} + \cdots - 6)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 23 T^{6} + \cdots + 45 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( (T^{4} + 5 T^{3} - 16 T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 38 T^{6} + \cdots + 1280 \) Copy content Toggle raw display
$19$ \( T^{8} - 64 T^{5} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( (T^{4} + 6 T^{3} - 7 T^{2} + \cdots + 48)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 7 T^{3} + 12 T^{2} + \cdots - 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + 121 T^{6} + \cdots + 95220 \) Copy content Toggle raw display
$37$ \( T^{8} + 152 T^{6} + \cdots + 184320 \) Copy content Toggle raw display
$41$ \( (T^{4} - 11 T^{3} + \cdots + 450)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 145 T^{6} + \cdots + 162000 \) Copy content Toggle raw display
$47$ \( (T^{4} + 12 T^{3} + \cdots + 768)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 266 T^{6} + \cdots + 2366720 \) Copy content Toggle raw display
$59$ \( T^{8} + 290 T^{6} + \cdots + 1458000 \) Copy content Toggle raw display
$61$ \( T^{8} + 288 T^{6} + \cdots + 3732480 \) Copy content Toggle raw display
$67$ \( T^{8} + 259 T^{6} + \cdots + 595125 \) Copy content Toggle raw display
$71$ \( T^{8} + 313 T^{6} + \cdots + 4762880 \) Copy content Toggle raw display
$73$ \( T^{8} + 182 T^{6} + \cdots + 2592000 \) Copy content Toggle raw display
$79$ \( (T^{4} - 6 T^{3} + \cdots - 3200)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 265 T^{6} + \cdots + 36980 \) Copy content Toggle raw display
$89$ \( T^{8} + 568 T^{6} + \cdots + 184832000 \) Copy content Toggle raw display
$97$ \( T^{8} + 64 T^{6} + \cdots + 46080 \) Copy content Toggle raw display
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