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Properties

Label 3696.2.t.a
Level $3696$
Weight $2$
Character orbit 3696.t
Analytic conductor $29.513$
Analytic rank $0$
Dimension $12$
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(2815,3696)]
 
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, 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("3696.2815");
 
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.t (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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} - 4 x^{9} + 31 x^{8} - 116 x^{7} + 224 x^{6} - 136 x^{5} + 49 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{6} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{3} - \beta_{2} q^{5} - q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{3} - \beta_{2} q^{5} - q^{7} - q^{9} + ( - \beta_{7} - \beta_{4} - 1) q^{11} + ( - \beta_{11} + \beta_{3}) q^{13} - \beta_1 q^{15} + (\beta_{9} - \beta_{3}) q^{17} + ( - \beta_{5} + \beta_{4}) q^{19} - \beta_{3} q^{21} + (\beta_{9} + \beta_{3}) q^{23} + (\beta_{8} + \beta_{7} + \beta_{6} + \cdots + 2) q^{25}+ \cdots + (\beta_{7} + \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{7} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{7} - 12 q^{9} - 8 q^{11} + 20 q^{25} + 8 q^{37} - 16 q^{39} + 8 q^{43} + 12 q^{49} + 16 q^{51} - 32 q^{53} + 4 q^{55} + 12 q^{63} - 8 q^{69} + 8 q^{77} - 16 q^{79} + 12 q^{81} - 24 q^{83} + 8 q^{89} + 8 q^{93} - 32 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} - 4 x^{9} + 31 x^{8} - 116 x^{7} + 224 x^{6} - 136 x^{5} + 49 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 9600905 \nu^{11} - 1683432 \nu^{10} + 90813788 \nu^{9} - 173211764 \nu^{8} + 741785351 \nu^{7} + \cdots - 2419365472 ) / 4253935576 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 11888499 \nu^{11} + 35291092 \nu^{10} - 57747740 \nu^{9} - 2375116 \nu^{8} - 399044669 \nu^{7} + \cdots + 353249904 ) / 386721416 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70598878 \nu^{11} - 262833251 \nu^{10} + 493401660 \nu^{9} - 138119520 \nu^{8} + \cdots - 2216948288 ) / 2126967788 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 76634307 \nu^{11} + 97893500 \nu^{10} + 118073432 \nu^{9} - 954876208 \nu^{8} + \cdots - 11225854172 ) / 2126967788 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 105368327 \nu^{11} + 269867476 \nu^{10} - 351482036 \nu^{9} - 313990644 \nu^{8} + \cdots - 8382261740 ) / 2126967788 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 226809009 \nu^{11} + 559311356 \nu^{10} - 668887300 \nu^{9} - 979352396 \nu^{8} + \cdots - 13001982472 ) / 4253935576 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 453492903 \nu^{11} + 1841410332 \nu^{10} - 3676831608 \nu^{9} + 1823589372 \nu^{8} + \cdots + 29513156416 ) / 4253935576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 138559268 \nu^{11} - 483638194 \nu^{10} + 845640893 \nu^{9} - 60835412 \nu^{8} + \cdots - 860022936 ) / 1063483894 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 679600883 \nu^{11} - 2574235694 \nu^{10} + 4727331324 \nu^{9} - 1178073644 \nu^{8} + \cdots - 19263185440 ) / 4253935576 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 382518007 \nu^{11} + 1455261551 \nu^{10} - 2664746496 \nu^{9} + 669075044 \nu^{8} + \cdots + 10924540064 ) / 2126967788 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 467590179 \nu^{11} + 1767085315 \nu^{10} - 3279441020 \nu^{9} + 960429316 \nu^{8} + \cdots + 15309157152 ) / 2126967788 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{11} - 2\beta_{8} - \beta_{4} + \beta_{3} + \beta_{2} + \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{11} + \beta_{10} - \beta_{8} + \beta_{7} + 6\beta_{3} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{11} + \beta_{9} + 6\beta_{7} + \beta_{6} + 2\beta_{4} + 3\beta_{3} - 2\beta_{2} + 2\beta _1 - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 7\beta_{8} + 7\beta_{7} + 2\beta_{6} - 7\beta_{5} + 7\beta_{4} + 5\beta_{2} - 32 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23 \beta_{11} - 4 \beta_{10} - 18 \beta_{9} + 74 \beta_{8} + 18 \beta_{6} - 4 \beta_{5} + 23 \beta_{4} + \cdots - 45 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 22\beta_{11} - 23\beta_{10} - 11\beta_{9} + 24\beta_{8} - 24\beta_{7} - 98\beta_{3} + 11\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 147 \beta_{11} - 52 \beta_{10} - 136 \beta_{9} - 478 \beta_{7} - 136 \beta_{6} + 52 \beta_{5} + \cdots + 351 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -339\beta_{8} - 339\beta_{7} - 188\beta_{6} + 303\beta_{5} - 287\beta_{4} - 87\beta_{2} + 1262 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 496 \beta_{11} + 246 \beta_{10} + 491 \beta_{9} - 1598 \beta_{8} - 491 \beta_{6} + 246 \beta_{5} + \cdots + 1361 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -1943\beta_{11} + 2017\beta_{10} + 1474\beta_{9} - 2435\beta_{8} + 2435\beta_{7} + 8362\beta_{3} - 269\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 6897 \beta_{11} + 4132 \beta_{10} + 6982 \beta_{9} + 21862 \beta_{7} + 6982 \beta_{6} - 4132 \beta_{5} + \cdots - 20779 ) / 4 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2815.1
0.728821 − 0.728821i
1.89044 − 1.89044i
0.275806 − 0.275806i
1.33904 − 1.33904i
−0.602099 + 0.602099i
−1.63201 + 1.63201i
0.728821 + 0.728821i
1.89044 + 1.89044i
0.275806 + 0.275806i
1.33904 + 1.33904i
−0.602099 − 0.602099i
−1.63201 − 1.63201i
0 − 1.00000i 0 −4.14524 0 −1.00000 0 −1.00000 0
2815.2 0 − 1.00000i 0 −1.65784 0 −1.00000 0 −1.00000 0
2815.3 0 − 1.00000i 0 −0.932457 0 −1.00000 0 −1.00000 0
2815.4 0 − 1.00000i 0 1.27145 0 −1.00000 0 −1.00000 0
2815.5 0 − 1.00000i 0 1.58002 0 −1.00000 0 −1.00000 0
2815.6 0 − 1.00000i 0 3.88407 0 −1.00000 0 −1.00000 0
2815.7 0 1.00000i 0 −4.14524 0 −1.00000 0 −1.00000 0
2815.8 0 1.00000i 0 −1.65784 0 −1.00000 0 −1.00000 0
2815.9 0 1.00000i 0 −0.932457 0 −1.00000 0 −1.00000 0
2815.10 0 1.00000i 0 1.27145 0 −1.00000 0 −1.00000 0
2815.11 0 1.00000i 0 1.58002 0 −1.00000 0 −1.00000 0
2815.12 0 1.00000i 0 3.88407 0 −1.00000 0 −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2815.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
44.c 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.t.a ✓ 12
4.b odd 2 1 3696.2.t.b yes 12
11.b odd 2 1 3696.2.t.b yes 12
44.c even 2 1 inner 3696.2.t.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3696.2.t.a ✓ 12 1.a even 1 1 trivial
3696.2.t.a ✓ 12 44.c even 2 1 inner
3696.2.t.b yes 12 4.b odd 2 1
3696.2.t.b yes 12 11.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 \(S_{2}^{\mathrm{new}}(3696, [\chi])\):

\( T_{5}^{6} - 20T_{5}^{4} + 4T_{5}^{3} + 65T_{5}^{2} - 12T_{5} - 50 \) Copy content Toggle raw display
\( T_{19}^{6} - 58T_{19}^{4} - 48T_{19}^{3} + 629T_{19}^{2} + 872T_{19} + 80 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{6} \) Copy content Toggle raw display
$5$ \( (T^{6} - 20 T^{4} + \cdots - 50)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + 8 T^{11} + \cdots + 1771561 \) Copy content Toggle raw display
$13$ \( T^{12} + 60 T^{10} + \cdots + 4624 \) Copy content Toggle raw display
$17$ \( T^{12} + 108 T^{10} + \cdots + 107584 \) Copy content Toggle raw display
$19$ \( (T^{6} - 58 T^{4} + \cdots + 80)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 100 T^{10} + \cdots + 2005056 \) Copy content Toggle raw display
$29$ \( T^{12} + 172 T^{10} + \cdots + 3283344 \) Copy content Toggle raw display
$31$ \( T^{12} + 168 T^{10} + \cdots + 82591744 \) Copy content Toggle raw display
$37$ \( (T^{6} - 4 T^{5} + \cdots - 15872)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + 228 T^{10} + \cdots + 262144 \) Copy content Toggle raw display
$43$ \( (T^{6} - 4 T^{5} + \cdots + 2048)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + 228 T^{10} + \cdots + 246016 \) Copy content Toggle raw display
$53$ \( (T^{6} + 16 T^{5} + \cdots - 816)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 32911765056 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 46656000000 \) Copy content Toggle raw display
$67$ \( T^{12} + 340 T^{10} + \cdots + 8856576 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 298598400 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 32251849744 \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots + 89728)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 12 T^{5} + \cdots - 15712)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} - 4 T^{5} + \cdots - 247584)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 16 T^{5} + \cdots - 60176)^{2} \) Copy content Toggle raw display
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