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Properties

Label 208.4.i.h
Level $208$
Weight $4$
Character orbit 208.i
Analytic conductor $12.272$
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] = [208,4,Mod(81,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.i (of order \(3\), degree \(2\), 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: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} + 132 x^{10} - 149 x^{9} + 12792 x^{8} - 16413 x^{7} + 432175 x^{6} + 21798 x^{5} + \cdots + 16842816 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3 \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{5} - \beta_{4} + \beta_{3} + 2) q^{5} + (\beta_{8} - \beta_{7}) q^{7} + (\beta_{11} + \beta_{10} + \cdots + 15 \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{5} - \beta_{4} + \beta_{3} + 2) q^{5} + (\beta_{8} - \beta_{7}) q^{7} + (\beta_{11} + \beta_{10} + \cdots + 15 \beta_{2}) q^{9}+ \cdots + ( - 19 \beta_{10} - 15 \beta_{9} + \cdots - 87) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 3 q^{3} + 18 q^{5} + q^{7} - 93 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 3 q^{3} + 18 q^{5} + q^{7} - 93 q^{9} + 11 q^{11} + 49 q^{13} + 168 q^{15} - 50 q^{17} + 211 q^{19} - 166 q^{21} - 103 q^{23} + 786 q^{25} + 486 q^{27} + 48 q^{29} - 380 q^{31} + 133 q^{33} - 226 q^{35} - 476 q^{37} - 655 q^{39} - 10 q^{41} + 13 q^{43} + 57 q^{45} - 244 q^{47} + 31 q^{49} + 1490 q^{51} - 966 q^{53} + 1510 q^{55} + 2354 q^{57} - 731 q^{59} + 704 q^{61} - 518 q^{63} + 251 q^{65} + 901 q^{67} - 3479 q^{69} - 673 q^{71} + 3102 q^{73} - 2739 q^{75} + 266 q^{77} + 1680 q^{79} - 3018 q^{81} - 4032 q^{83} + 1373 q^{85} - 1485 q^{87} - 295 q^{89} + 2009 q^{91} - 3220 q^{93} + 3128 q^{95} - 2575 q^{97} - 916 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3 x^{11} + 132 x^{10} - 149 x^{9} + 12792 x^{8} - 16413 x^{7} + 432175 x^{6} + 21798 x^{5} + \cdots + 16842816 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 367279577462643 \nu^{11} + \cdots - 41\!\cdots\!52 ) / 97\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 21391958672173 \nu^{11} + 36131866392471 \nu^{10} + \cdots - 83\!\cdots\!28 ) / 51\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 14\!\cdots\!46 \nu^{11} + \cdots + 22\!\cdots\!52 ) / 22\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 88\!\cdots\!68 \nu^{11} + \cdots + 15\!\cdots\!20 ) / 67\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 47\!\cdots\!83 \nu^{11} + \cdots - 42\!\cdots\!88 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 10\!\cdots\!91 \nu^{11} + \cdots + 62\!\cdots\!12 ) / 35\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 11\!\cdots\!27 \nu^{11} + \cdots + 18\!\cdots\!20 ) / 22\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\!\cdots\!44 \nu^{11} + \cdots - 10\!\cdots\!72 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\!\cdots\!05 \nu^{11} + \cdots + 10\!\cdots\!32 ) / 20\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!54 \nu^{11} + \cdots - 40\!\cdots\!00 ) / 20\!\cdots\!08 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} - \beta_{3} + 41\beta_{2} + 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} - \beta_{9} - 4\beta_{7} - 12\beta_{5} + 12\beta_{4} - 78\beta_{3} - \beta_{2} + \beta _1 - 100 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 96 \beta_{11} + 3 \beta_{9} - 51 \beta_{8} - 93 \beta_{7} - 93 \beta_{6} - 6 \beta_{4} - 3147 \beta_{2} + \cdots - 3147 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 215 \beta_{11} - 215 \beta_{10} + 161 \beta_{9} - 423 \beta_{8} + 477 \beta_{7} - 107 \beta_{6} + \cdots + 6955 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 9080 \beta_{10} + 8018 \beta_{9} + 531 \beta_{8} + 15476 \beta_{7} + 531 \beta_{6} + 474 \beta_{5} + \cdots + 290429 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 28947 \beta_{11} - 7458 \beta_{9} + 56901 \beta_{8} + 21489 \beta_{7} + 21489 \beta_{6} - 142728 \beta_{4} + \cdots + 515091 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 856396 \beta_{11} + 856396 \beta_{10} - 792037 \beta_{9} + 685728 \beta_{8} - 750087 \beta_{7} + \cdots - 2673074 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3423583 \beta_{10} - 1794691 \beta_{9} - 814446 \beta_{8} - 8756302 \beta_{7} - 814446 \beta_{6} + \cdots - 123676390 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 80925648 \beta_{11} + 6961611 \beta_{9} - 76148943 \beta_{8} - 73964037 \beta_{7} - 73964037 \beta_{6} + \cdots - 2177824551 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 382231691 \beta_{11} - 382231691 \beta_{10} + 299121137 \beta_{9} - 554376411 \beta_{8} + \cdots + 5607136087 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−4.55155 − 7.88352i
−2.64171 − 4.57558i
−0.379776 − 0.657792i
0.876590 + 1.51830i
3.21754 + 5.57295i
4.97891 + 8.62372i
−4.55155 + 7.88352i
−2.64171 + 4.57558i
−0.379776 + 0.657792i
0.876590 − 1.51830i
3.21754 − 5.57295i
4.97891 − 8.62372i
0 −5.05155 − 8.74955i 0 −18.1244 0 −0.830837 + 1.43905i 0 −37.5364 + 65.0149i 0
81.2 0 −3.14171 − 5.44160i 0 20.6420 0 7.77998 − 13.4753i 0 −6.24068 + 10.8092i 0
81.3 0 −0.879776 − 1.52382i 0 2.15935 0 −9.24060 + 16.0052i 0 11.9520 − 20.7014i 0
81.4 0 0.376590 + 0.652272i 0 −7.26675 0 −1.42656 + 2.47088i 0 13.2164 − 22.8914i 0
81.5 0 2.71754 + 4.70692i 0 −5.68956 0 15.3229 − 26.5400i 0 −1.27007 + 2.19983i 0
81.6 0 4.47891 + 7.75769i 0 17.2794 0 −11.1048 + 19.2342i 0 −26.6212 + 46.1093i 0
113.1 0 −5.05155 + 8.74955i 0 −18.1244 0 −0.830837 − 1.43905i 0 −37.5364 − 65.0149i 0
113.2 0 −3.14171 + 5.44160i 0 20.6420 0 7.77998 + 13.4753i 0 −6.24068 − 10.8092i 0
113.3 0 −0.879776 + 1.52382i 0 2.15935 0 −9.24060 − 16.0052i 0 11.9520 + 20.7014i 0
113.4 0 0.376590 − 0.652272i 0 −7.26675 0 −1.42656 − 2.47088i 0 13.2164 + 22.8914i 0
113.5 0 2.71754 − 4.70692i 0 −5.68956 0 15.3229 + 26.5400i 0 −1.27007 − 2.19983i 0
113.6 0 4.47891 − 7.75769i 0 17.2794 0 −11.1048 − 19.2342i 0 −26.6212 − 46.1093i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.i.h 12
4.b odd 2 1 104.4.i.c ✓ 12
13.c even 3 1 inner 208.4.i.h 12
52.i odd 6 1 1352.4.a.m 6
52.j odd 6 1 104.4.i.c ✓ 12
52.j odd 6 1 1352.4.a.n 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.4.i.c ✓ 12 4.b odd 2 1
104.4.i.c ✓ 12 52.j odd 6 1
208.4.i.h 12 1.a even 1 1 trivial
208.4.i.h 12 13.c even 3 1 inner
1352.4.a.m 6 52.i odd 6 1
1352.4.a.n 6 52.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} + 3 T_{3}^{11} + 132 T_{3}^{10} + 117 T_{3}^{9} + 12720 T_{3}^{8} + 14325 T_{3}^{7} + \cdots + 16777216 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 3 T^{11} + \cdots + 16777216 \) Copy content Toggle raw display
$5$ \( (T^{6} - 9 T^{5} + \cdots - 577144)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 861065220096 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 81\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 55\!\cdots\!01 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 82\!\cdots\!64 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 95\!\cdots\!81 \) Copy content Toggle raw display
$31$ \( (T^{6} + 190 T^{5} + \cdots - 75539398656)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 76\!\cdots\!01 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 38\!\cdots\!61 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 26204469977088)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 76899859894728)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 36\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 46\!\cdots\!29 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 27\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 12\!\cdots\!72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 18\!\cdots\!68)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 26\!\cdots\!08)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 15\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 43\!\cdots\!16 \) Copy content Toggle raw display
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