Properties

Label 3696.2.d.d
Level $3696$
Weight $2$
Character orbit 3696.d
Analytic conductor $29.513$
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] = [3696,2,Mod(2575,3696)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3696, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("3696.2575");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3696 = 2^{4} \cdot 3 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3696.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: \(29.5127085871\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.836829184.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{2} + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{4} + \beta_1) q^{5} + ( - \beta_{7} + \beta_{5} - \beta_{2} + 1) q^{7} + q^{9} - \beta_{3} q^{11} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{13} + ( - \beta_{4} + \beta_1) q^{15} + (\beta_{4} + 2 \beta_1) q^{17} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 3) q^{19} + ( - \beta_{7} + \beta_{5} - \beta_{2} + 1) q^{21} + ( - \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{23} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{2}) q^{25} + q^{27} + (3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} - 3) q^{29} + ( - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2}) q^{31} - \beta_{3} q^{33} + (\beta_{6} - \beta_{5} - \beta_{4} - 4 \beta_{3} - \beta_{2} + \beta_1) q^{35} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{2} - 1) q^{37} + (\beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - \beta_1) q^{39} + ( - \beta_{6} + \beta_{5} + 2 \beta_{4} - 2 \beta_{3}) q^{41} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{4} + 6 \beta_{3} + 2 \beta_1) q^{43} + ( - \beta_{4} + \beta_1) q^{45} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 5 \beta_{2} + 1) q^{47} + ( - \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 3 \beta_{3} - \beta_1 + 5) q^{49} + (\beta_{4} + 2 \beta_1) q^{51} + (\beta_{6} + \beta_{5} - 5 \beta_{2} - 4) q^{53} + (\beta_{7} + \beta_{2} - 1) q^{55} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 3) q^{57} + ( - \beta_{7} - 5 \beta_{2} + 1) q^{59} + ( - 2 \beta_{6} + 2 \beta_{5} + 4 \beta_{4} + 6 \beta_{3} + 2 \beta_1) q^{61} + ( - \beta_{7} + \beta_{5} - \beta_{2} + 1) q^{63} + (\beta_{7} + 2 \beta_{6} + 2 \beta_{5} + 5 \beta_{2} - 3) q^{65} + ( - \beta_{6} + \beta_{5} - 2 \beta_{4} + 4 \beta_{3} + \beta_1) q^{67} + ( - \beta_{6} + \beta_{5} - 3 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{69} + ( - 4 \beta_{6} + 4 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{71} + (3 \beta_{6} - 3 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + \beta_1) q^{73} + (\beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{2}) q^{75} + ( - \beta_{6} - \beta_{4} + \beta_1) q^{77} + ( - 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{4} - 4 \beta_{3}) q^{79} + q^{81} + (\beta_{6} + \beta_{5} + 3 \beta_{2} + 6) q^{83} + (2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_{2} - 4) q^{85} + (3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} - 3) q^{87} + (2 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} + 2 \beta_{3}) q^{89} + ( - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} - 6 \beta_{4} - 2 \beta_{3} + \beta_{2} + \cdots + 2) q^{91}+ \cdots - \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 8 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 8 q^{7} + 8 q^{9} + 12 q^{19} + 8 q^{21} + 12 q^{25} + 8 q^{27} - 4 q^{29} - 12 q^{37} + 28 q^{47} + 28 q^{49} - 24 q^{53} - 4 q^{55} + 12 q^{57} + 4 q^{59} + 8 q^{63} - 4 q^{65} + 12 q^{75} - 4 q^{77} + 8 q^{81} + 56 q^{83} - 32 q^{85} - 4 q^{87} + 8 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 14x^{6} + 61x^{4} + 84x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 7\nu^{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 12\nu^{5} + 41\nu^{3} + 38\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 11\nu^{3} - 26\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 2\nu^{5} + 10\nu^{4} + 18\nu^{3} + 19\nu^{2} + 32\nu - 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} - 2\nu^{5} + 10\nu^{4} - 18\nu^{3} + 19\nu^{2} - 32\nu - 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 23\nu^{2} + 6 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} - 2\beta_{4} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{7} + 7\beta_{6} + 7\beta_{5} + 4\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -11\beta_{6} + 11\beta_{5} + 18\beta_{4} + 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 51\beta_{7} - 47\beta_{6} - 47\beta_{5} - 40\beta_{2} - 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 91\beta_{6} - 91\beta_{5} - 134\beta_{4} + 8\beta_{3} - 181\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3696\mathbb{Z}\right)^\times\).

\(n\) \(463\) \(673\) \(1585\) \(2465\) \(2773\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2575.1
1.65222i
0.222191i
2.63640i
2.06644i
2.06644i
2.63640i
0.222191i
1.65222i
0 1.00000 0 3.06644i 0 2.60525 + 0.461191i 0 1.00000 0
2575.2 0 1.00000 0 1.63640i 0 −2.50062 0.864220i 0 1.00000 0
2575.3 0 1.00000 0 1.22219i 0 2.37930 1.15711i 0 1.00000 0
2575.4 0 1.00000 0 0.652223i 0 1.51608 + 2.16830i 0 1.00000 0
2575.5 0 1.00000 0 0.652223i 0 1.51608 2.16830i 0 1.00000 0
2575.6 0 1.00000 0 1.22219i 0 2.37930 + 1.15711i 0 1.00000 0
2575.7 0 1.00000 0 1.63640i 0 −2.50062 + 0.864220i 0 1.00000 0
2575.8 0 1.00000 0 3.06644i 0 2.60525 0.461191i 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2575.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3696.2.d.d yes 8
4.b odd 2 1 3696.2.d.c 8
7.b odd 2 1 3696.2.d.c 8
28.d even 2 1 inner 3696.2.d.d yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3696.2.d.c 8 4.b odd 2 1
3696.2.d.c 8 7.b odd 2 1
3696.2.d.d yes 8 1.a even 1 1 trivial
3696.2.d.d yes 8 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3696, [\chi])\):

\( T_{5}^{8} + 14T_{5}^{6} + 49T_{5}^{4} + 56T_{5}^{2} + 16 \) Copy content Toggle raw display
\( T_{19}^{4} - 6T_{19}^{3} - 37T_{19}^{2} + 216T_{19} - 46 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 14 T^{6} + 49 T^{4} + 56 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$7$ \( T^{8} - 8 T^{7} + 18 T^{6} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{8} + 58 T^{6} + 961 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$17$ \( T^{8} + 80 T^{6} + 1720 T^{4} + \cdots + 4624 \) Copy content Toggle raw display
$19$ \( (T^{4} - 6 T^{3} - 37 T^{2} + 216 T - 46)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 100 T^{6} + 2244 T^{4} + \cdots + 16384 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} - 75 T^{2} - 252 T + 98)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 34 T^{2} - 96 T - 64)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 6 T^{3} - 71 T^{2} - 504 T - 784)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 76 T^{6} + 1460 T^{4} + \cdots + 18496 \) Copy content Toggle raw display
$43$ \( T^{8} + 144 T^{6} + 2104 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$47$ \( (T^{4} - 14 T^{3} - 33 T^{2} + 1008 T - 3136)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 12 T^{3} - 78 T^{2} - 728 T + 1784)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 2 T^{3} - 85 T^{2} + 68 T + 1412)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 264 T^{6} + 23824 T^{4} + \cdots + 8667136 \) Copy content Toggle raw display
$67$ \( T^{8} + 82 T^{6} + 385 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{8} + 368 T^{6} + \cdots + 42823936 \) Copy content Toggle raw display
$73$ \( T^{8} + 362 T^{6} + 33841 T^{4} + \cdots + 1263376 \) Copy content Toggle raw display
$79$ \( T^{8} + 396 T^{6} + 40196 T^{4} + \cdots + 262144 \) Copy content Toggle raw display
$83$ \( (T^{4} - 28 T^{3} + 258 T^{2} - 848 T + 544)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + 352 T^{6} + 32224 T^{4} + \cdots + 3748096 \) Copy content Toggle raw display
$97$ \( T^{8} + 332 T^{6} + 21940 T^{4} + \cdots + 107584 \) Copy content Toggle raw display
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