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Properties

Label 208.12.f.b
Level $208$
Weight $12$
Character orbit 208.f
Analytic conductor $159.815$
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] = [208,12,Mod(129,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.129");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 208.f (of order \(2\), degree \(1\), not 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: \(159.815381556\)
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: \( x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{31}\cdot 3^{4}\cdot 13^{4} \)
Twist minimal: no (minimal twist has level 13)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 41) q^{3} - \beta_{5} q^{5} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{7} + (\beta_{3} - 41 \beta_{2} + 54567) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 41) q^{3} - \beta_{5} q^{5} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{7} + (\beta_{3} - 41 \beta_{2} + 54567) q^{9} + ( - \beta_{11} + \beta_{7} + \cdots + 12 \beta_1) q^{11}+ \cdots + ( - 84701 \beta_{11} + \cdots + 5021579 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 488 q^{3} + 654644 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 488 q^{3} + 654644 q^{9} + 3208868 q^{13} + 12198768 q^{17} - 5810592 q^{23} + 6102388 q^{25} + 52613336 q^{27} - 244463112 q^{29} + 562027560 q^{35} + 2199109744 q^{39} - 2294519976 q^{43} - 3573617796 q^{49} - 7713246552 q^{51} - 4602062760 q^{53} + 6178744976 q^{55} - 13775649944 q^{61} - 7598401512 q^{65} - 25419983328 q^{69} - 68016370832 q^{75} - 80478036048 q^{77} - 18046097296 q^{79} - 132677486692 q^{81} - 94507900752 q^{87} - 104793638664 q^{91} - 145093149648 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 256\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1291753 \nu^{10} + 21684945577 \nu^{8} + 126271453345720 \nu^{6} + \cdots + 35\!\cdots\!32 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 131671877 \nu^{10} - 3200315071493 \nu^{8} + \cdots + 15\!\cdots\!12 ) / 61\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 16792789 \nu^{10} - 281904292501 \nu^{8} + \cdots + 25\!\cdots\!84 ) / 38\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 30769879 \nu^{11} + 554264573911 \nu^{9} + \cdots + 26\!\cdots\!76 \nu ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1464330623 \nu^{11} + 27874125571007 \nu^{9} + \cdots + 18\!\cdots\!12 \nu ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 514611463 \nu^{11} - 8978411151367 \nu^{9} + \cdots - 30\!\cdots\!72 \nu ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 211150913 \nu^{11} + 111834642526 \nu^{10} + 3657808714817 \nu^{9} + \cdots - 35\!\cdots\!56 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 2009607071 \nu^{10} - 33436695519839 \nu^{8} + \cdots - 31\!\cdots\!24 ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 92261903 \nu^{11} + 1625373026255 \nu^{9} + \cdots + 11\!\cdots\!44 \nu ) / 22\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 7070370157 \nu^{11} - 132121808647213 \nu^{9} + \cdots - 14\!\cdots\!08 \nu ) / 56\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 208\beta_{2} - 786544 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 64\beta_{10} - 64\beta_{7} + 128\beta_{6} - 1664\beta_{5} - 5249\beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 896 \beta_{9} - 768 \beta_{8} - 384 \beta_{7} - 384 \beta_{5} - 7001 \beta_{4} + 2688 \beta_{3} + \cdots + 4165080176 ) / 256 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 20480 \beta_{11} - 506688 \beta_{10} + 1086272 \beta_{7} - 1595008 \beta_{6} + \cdots + 31419289 \beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 10675584 \beta_{9} + 7796480 \beta_{8} + 3898240 \beta_{7} + 3898240 \beta_{5} + 47611313 \beta_{4} + \cdots - 25133095064816 ) / 256 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 187633664 \beta_{11} + 3611237696 \beta_{10} - 11165496640 \beta_{7} + 14298880640 \beta_{6} + \cdots - 200672930801 \beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 97276273024 \beta_{9} - 66287849216 \beta_{8} - 33143924608 \beta_{7} - 33143924608 \beta_{5} + \cdots + 16\!\cdots\!40 ) / 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1385340702720 \beta_{11} - 25675095255872 \beta_{10} + 97529799394112 \beta_{7} + \cdots + 13\!\cdots\!25 \beta_1 ) / 256 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 800164560446848 \beta_{9} + 531289489831680 \beta_{8} + 265644744915840 \beta_{7} + \cdots - 10\!\cdots\!60 ) / 256 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 97\!\cdots\!80 \beta_{11} + \cdots - 92\!\cdots\!61 \beta_1 ) / 256 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/208\mathbb{Z}\right)^\times\).

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
69.4876i
− 69.4876i
3.38741i
− 3.38741i
− 65.0925i
65.0925i
37.0312i
− 37.0312i
85.7436i
− 85.7436i
− 25.1535i
25.1535i
0 −612.411 0 − 159.313i 0 48715.3i 0 197900. 0
129.2 0 −612.411 0 159.313i 0 − 48715.3i 0 197900. 0
129.3 0 −494.534 0 − 9091.88i 0 − 45027.4i 0 67417.3 0
129.4 0 −494.534 0 9091.88i 0 45027.4i 0 67417.3 0
129.5 0 −15.7424 0 − 6676.77i 0 81596.2i 0 −176899. 0
129.6 0 −15.7424 0 6676.77i 0 − 81596.2i 0 −176899. 0
129.7 0 143.287 0 − 5266.14i 0 16188.3i 0 −156616. 0
129.8 0 143.287 0 5266.14i 0 − 16188.3i 0 −156616. 0
129.9 0 584.699 0 − 6975.15i 0 41147.2i 0 164726. 0
129.10 0 584.699 0 6975.15i 0 − 41147.2i 0 164726. 0
129.11 0 638.702 0 − 9287.93i 0 − 25239.3i 0 230793. 0
129.12 0 638.702 0 9287.93i 0 25239.3i 0 230793. 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.12.f.b 12
4.b odd 2 1 13.12.b.a ✓ 12
12.b even 2 1 117.12.b.b 12
13.b even 2 1 inner 208.12.f.b 12
52.b odd 2 1 13.12.b.a ✓ 12
52.f even 4 2 169.12.a.e 12
156.h even 2 1 117.12.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.12.b.a ✓ 12 4.b odd 2 1
13.12.b.a ✓ 12 52.b odd 2 1
117.12.b.b 12 12.b even 2 1
117.12.b.b 12 156.h even 2 1
169.12.a.e 12 52.f even 4 2
208.12.f.b 12 1.a even 1 1 trivial
208.12.f.b 12 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 244 T_{3}^{5} - 665334 T_{3}^{4} + 129598956 T_{3}^{3} + 109163403621 T_{3}^{2} + \cdots - 255121008509808 \) acting on \(S_{12}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{6} + \cdots - 255121008509808)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 33\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 38\!\cdots\!96 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 14\!\cdots\!68)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 44\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots - 65\!\cdots\!28)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 13\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 68\!\cdots\!08)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 37\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 34\!\cdots\!84 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
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