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Properties

Label 208.7.t.e
Level $208$
Weight $7$
Character orbit 208.t
Analytic conductor $47.851$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [208,7,Mod(161,208)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(208, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 3]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("208.161");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 208.t (of order \(4\), 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: \(47.8512493929\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 8696 x^{18} + 30747802 x^{16} + 56840077028 x^{14} + 58706009040001 x^{12} + \cdots + 72\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{33}\cdot 3^{4}\cdot 13^{4} \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{6} + \beta_{5} - 6 \beta_{4} + \cdots - 6) q^{5}+ \cdots + (\beta_{7} + \beta_{6} - \beta_{2} + \cdots + 139) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{6} + \beta_{5} - 6 \beta_{4} + \cdots - 6) q^{5}+ \cdots + (33 \beta_{19} - 213 \beta_{18} + \cdots - 78465) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 128 q^{5} + 170 q^{7} + 2792 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 128 q^{5} + 170 q^{7} + 2792 q^{9} - 182 q^{11} - 2394 q^{13} + 13424 q^{15} + 6294 q^{19} - 16724 q^{21} - 56988 q^{27} - 46028 q^{29} + 23110 q^{31} + 79096 q^{33} - 98152 q^{35} - 4064 q^{37} + 142408 q^{39} - 10252 q^{41} - 372280 q^{45} + 145810 q^{47} + 21000 q^{53} + 372636 q^{55} - 440812 q^{57} - 410338 q^{59} + 283072 q^{61} + 253994 q^{63} + 843020 q^{65} + 312226 q^{67} - 921862 q^{71} - 159896 q^{73} - 747668 q^{79} + 553676 q^{81} - 42742 q^{83} - 1762572 q^{85} + 1081080 q^{87} + 1671676 q^{89} - 820942 q^{91} - 1305796 q^{93} + 95436 q^{97} - 1609294 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 8696 x^{18} + 30747802 x^{16} + 56840077028 x^{14} + 58706009040001 x^{12} + \cdots + 72\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 14\!\cdots\!09 \nu^{18} + \cdots + 61\!\cdots\!92 ) / 72\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 33\!\cdots\!91 \nu^{18} + \cdots + 26\!\cdots\!72 ) / 13\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 16\!\cdots\!65 \nu^{18} + \cdots - 67\!\cdots\!28 ) / 39\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 61\!\cdots\!47 \nu^{19} + \cdots + 51\!\cdots\!00 \nu ) / 48\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 61\!\cdots\!47 \nu^{19} + \cdots + 10\!\cdots\!92 \nu ) / 48\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!67 \nu^{19} + \cdots + 52\!\cdots\!80 ) / 88\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 11\!\cdots\!67 \nu^{19} + \cdots + 52\!\cdots\!80 ) / 88\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20\!\cdots\!81 \nu^{19} + \cdots + 30\!\cdots\!08 \nu ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 47\!\cdots\!24 \nu^{19} + \cdots + 15\!\cdots\!52 ) / 88\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 47\!\cdots\!24 \nu^{19} + \cdots - 15\!\cdots\!52 ) / 88\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 53\!\cdots\!61 \nu^{19} + \cdots - 12\!\cdots\!92 ) / 88\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 53\!\cdots\!61 \nu^{19} + \cdots - 12\!\cdots\!04 ) / 88\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 19\!\cdots\!06 \nu^{19} + \cdots - 43\!\cdots\!48 ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 19\!\cdots\!06 \nu^{19} + \cdots + 43\!\cdots\!48 ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 65\!\cdots\!18 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 65\!\cdots\!18 \nu^{19} + \cdots - 12\!\cdots\!36 ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 19\!\cdots\!50 \nu^{19} + \cdots - 17\!\cdots\!72 ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 19\!\cdots\!50 \nu^{19} + \cdots + 17\!\cdots\!72 ) / 29\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 82\!\cdots\!93 \nu^{19} + \cdots + 33\!\cdots\!64 \nu ) / 88\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} - \beta_{6} + \beta_{2} + 6\beta _1 - 869 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + 3 \beta_{16} - 3 \beta_{15} - 2 \beta_{14} - 2 \beta_{13} - 4 \beta_{12} + 4 \beta_{11} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 4 \beta_{18} + 4 \beta_{17} + 34 \beta_{16} + 34 \beta_{15} - 26 \beta_{14} + 26 \beta_{13} + \cdots + 1410348 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 3133 \beta_{19} + 181 \beta_{18} + 181 \beta_{17} - 7882 \beta_{16} + 7882 \beta_{15} + 6079 \beta_{14} + \cdots + 9061 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 26930 \beta_{18} - 26930 \beta_{17} - 131972 \beta_{16} - 131972 \beta_{15} + 141022 \beta_{14} + \cdots - 2591099014 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 7368865 \beta_{19} - 534113 \beta_{18} - 534113 \beta_{17} + 17937032 \beta_{16} - 17937032 \beta_{15} + \cdots - 26511691 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 98524728 \beta_{18} + 98524728 \beta_{17} + 385865406 \beta_{16} + 385865406 \beta_{15} + \cdots + 5082055037528 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 16469002729 \beta_{19} + 1076830749 \beta_{18} + 1076830749 \beta_{17} - 39754288362 \beta_{16} + \cdots + 76623868489 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 297888756214 \beta_{18} - 297888756214 \beta_{17} - 1036858882264 \beta_{16} - 1036858882264 \beta_{15} + \cdots - 10\!\cdots\!30 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 36800953856641 \beta_{19} - 1713714890509 \beta_{18} - 1713714890509 \beta_{17} + 88130477807152 \beta_{16} + \cdots - 208545416578915 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 822572093299124 \beta_{18} + 822572093299124 \beta_{17} + \cdots + 22\!\cdots\!48 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 83\!\cdots\!69 \beta_{19} + \cdots + 54\!\cdots\!21 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 21\!\cdots\!46 \beta_{18} + \cdots - 48\!\cdots\!98 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 18\!\cdots\!65 \beta_{19} + \cdots - 13\!\cdots\!31 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( - 54\!\cdots\!48 \beta_{18} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 43\!\cdots\!41 \beta_{19} + \cdots + 33\!\cdots\!65 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( 13\!\cdots\!54 \beta_{18} + \cdots - 24\!\cdots\!30 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 10\!\cdots\!29 \beta_{19} + \cdots - 81\!\cdots\!71 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
48.5546i
40.5416i
25.3904i
9.54806i
4.43738i
− 3.13944i
− 9.84036i
− 26.1364i
− 39.0354i
− 40.3204i
− 48.5546i
− 40.5416i
− 25.3904i
− 9.54806i
− 4.43738i
3.13944i
9.84036i
26.1364i
39.0354i
40.3204i
0 −47.5546 0 −165.440 + 165.440i 0 162.619 + 162.619i 0 1532.44 0
161.2 0 −39.5416 0 64.9909 − 64.9909i 0 −57.7716 − 57.7716i 0 834.540 0
161.3 0 −24.3904 0 −5.44203 + 5.44203i 0 103.544 + 103.544i 0 −134.107 0
161.4 0 −8.54806 0 −10.2184 + 10.2184i 0 −465.310 − 465.310i 0 −655.931 0
161.5 0 −3.43738 0 122.467 − 122.467i 0 334.376 + 334.376i 0 −717.184 0
161.6 0 4.13944 0 −151.398 + 151.398i 0 −36.3176 − 36.3176i 0 −711.865 0
161.7 0 10.8404 0 −37.3843 + 37.3843i 0 277.706 + 277.706i 0 −611.487 0
161.8 0 27.1364 0 162.773 − 162.773i 0 −243.567 − 243.567i 0 7.38525 0
161.9 0 40.0354 0 −48.7763 + 48.7763i 0 −102.730 − 102.730i 0 873.833 0
161.10 0 41.3204 0 4.42812 − 4.42812i 0 112.451 + 112.451i 0 978.379 0
177.1 0 −47.5546 0 −165.440 − 165.440i 0 162.619 − 162.619i 0 1532.44 0
177.2 0 −39.5416 0 64.9909 + 64.9909i 0 −57.7716 + 57.7716i 0 834.540 0
177.3 0 −24.3904 0 −5.44203 − 5.44203i 0 103.544 − 103.544i 0 −134.107 0
177.4 0 −8.54806 0 −10.2184 − 10.2184i 0 −465.310 + 465.310i 0 −655.931 0
177.5 0 −3.43738 0 122.467 + 122.467i 0 334.376 − 334.376i 0 −717.184 0
177.6 0 4.13944 0 −151.398 − 151.398i 0 −36.3176 + 36.3176i 0 −711.865 0
177.7 0 10.8404 0 −37.3843 − 37.3843i 0 277.706 − 277.706i 0 −611.487 0
177.8 0 27.1364 0 162.773 + 162.773i 0 −243.567 + 243.567i 0 7.38525 0
177.9 0 40.0354 0 −48.7763 − 48.7763i 0 −102.730 + 102.730i 0 873.833 0
177.10 0 41.3204 0 4.42812 + 4.42812i 0 112.451 − 112.451i 0 978.379 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.7.t.e 20
4.b odd 2 1 104.7.l.a ✓ 20
13.d odd 4 1 inner 208.7.t.e 20
52.f even 4 1 104.7.l.a ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
104.7.l.a ✓ 20 4.b odd 2 1
104.7.l.a ✓ 20 52.f even 4 1
208.7.t.e 20 1.a even 1 1 trivial
208.7.t.e 20 13.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{10} - 4343 T_{3}^{8} + 9498 T_{3}^{7} + 5941147 T_{3}^{6} - 19176420 T_{3}^{5} + \cdots - 2714615963136 \) acting on \(S_{7}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \) Copy content Toggle raw display
$3$ \( (T^{10} + \cdots - 2714615963136)^{2} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{20} + \cdots + 18\!\cdots\!44 \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 68\!\cdots\!01 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 33\!\cdots\!84 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 56\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 83\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots - 12\!\cdots\!44)^{2} \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 43\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 32\!\cdots\!64 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 44\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 10\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots + 92\!\cdots\!92)^{2} \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 25\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 38\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 61\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 21\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 46\!\cdots\!88)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 25\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 61\!\cdots\!84 \) Copy content Toggle raw display
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