Properties

Label 208.4.i.e
Level $208$
Weight $4$
Character orbit 208.i
Analytic conductor $12.272$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 208 = 2^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 208.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(12.2723972812\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{17})\)
Defining polynomial: \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + 3 \beta_1) q^{3} + ( - 5 \beta_{3} - 10) q^{5} + ( - \beta_{3} - 7 \beta_{2} - \beta_1 + 7) q^{7} + (15 \beta_{3} + 10 \beta_{2} + 15 \beta_1 - 10) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 3 \beta_1) q^{3} + ( - 5 \beta_{3} - 10) q^{5} + ( - \beta_{3} - 7 \beta_{2} - \beta_1 + 7) q^{7} + (15 \beta_{3} + 10 \beta_{2} + 15 \beta_1 - 10) q^{9} + (\beta_{2} + 15 \beta_1) q^{11} + ( - 7 \beta_{3} + 40 \beta_{2} - 5 \beta_1 + 9) q^{13} + (50 \beta_{2} - 10 \beta_1) q^{15} + ( - 16 \beta_{3} + 43 \beta_{2} - 16 \beta_1 - 43) q^{17} + ( - 5 \beta_{3} + 73 \beta_{2} - 5 \beta_1 - 73) q^{19} + ( - 25 \beta_{3} + 19) q^{21} + (89 \beta_{2} - 33 \beta_1) q^{23} + (75 \beta_{3} + 75) q^{25} + (9 \beta_{3} - 163) q^{27} + (13 \beta_{2} - 60 \beta_1) q^{29} + (100 \beta_{3} + 120) q^{31} + (63 \beta_{3} + 181 \beta_{2} + 63 \beta_1 - 181) q^{33} + ( - 30 \beta_{3} + 50 \beta_{2} - 30 \beta_1 - 50) q^{35} + (73 \beta_{2} + 44 \beta_1) q^{37} + (100 \beta_{3} + 73 \beta_{2} + 155 \beta_1 + 20) q^{39} + ( - 259 \beta_{2} - 20 \beta_1) q^{41} + ( - 97 \beta_{3} - 179 \beta_{2} - 97 \beta_1 + 179) q^{43} + ( - 25 \beta_{3} + 200 \beta_{2} - 25 \beta_1 - 200) q^{45} + ( - 140 \beta_{3} - 100) q^{47} + (290 \beta_{2} - 15 \beta_1) q^{49} + (65 \beta_{3} + 149) q^{51} + ( - 165 \beta_{3} + 190) q^{53} + (290 \beta_{2} - 70 \beta_1) q^{55} + (199 \beta_{3} - 13) q^{57} + (55 \beta_{3} + 377 \beta_{2} + 55 \beta_1 - 377) q^{59} + ( - 200 \beta_{3} + 351 \beta_{2} - 200 \beta_1 - 351) q^{61} + (130 \beta_{2} + 130 \beta_1) q^{63} + ( - 10 \beta_{3} - 500 \beta_{2} + 225 \beta_1 + 50) q^{65} + ( - 283 \beta_{2} + 91 \beta_1) q^{67} + (135 \beta_{3} - 307 \beta_{2} + 135 \beta_1 + 307) q^{69} + ( - 105 \beta_{3} - 11 \beta_{2} - 105 \beta_1 + 11) q^{71} + ( - 85 \beta_{3} + 250) q^{73} + ( - 825 \beta_{2} - 75 \beta_1) q^{75} + ( - 121 \beta_{3} + 67) q^{77} + ( - 40 \beta_{3} - 140) q^{79} + ( - \beta_{2} - 120 \beta_1) q^{81} + ( - 100 \beta_{3} - 180) q^{83} + (295 \beta_{3} - 750 \beta_{2} + 295 \beta_1 + 750) q^{85} + ( - 201 \beta_{3} - 707 \beta_{2} - 201 \beta_1 + 707) q^{87} + ( - 523 \beta_{2} + 125 \beta_1) q^{89} + ( - 65 \beta_{3} - 91 \beta_{2} - 65 \beta_1 + 351) q^{91} + ( - 1080 \beta_{2} - 40 \beta_1) q^{93} + (390 \beta_{3} - 830 \beta_{2} + 390 \beta_1 + 830) q^{95} + ( - 469 \beta_{3} + 27 \beta_{2} - 469 \beta_1 - 27) q^{97} + (390 \beta_{3} - 910) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 5 q^{3} - 30 q^{5} + 15 q^{7} - 35 q^{9} + 17 q^{11} + 125 q^{13} + 90 q^{15} - 70 q^{17} - 141 q^{19} + 126 q^{21} + 145 q^{23} + 150 q^{25} - 670 q^{27} - 34 q^{29} + 280 q^{31} - 425 q^{33} - 70 q^{35} + 190 q^{37} + 181 q^{39} - 538 q^{41} + 455 q^{43} - 375 q^{45} - 120 q^{47} + 565 q^{49} + 466 q^{51} + 1090 q^{53} + 510 q^{55} - 450 q^{57} - 809 q^{59} - 502 q^{61} + 390 q^{63} - 555 q^{65} - 475 q^{67} + 479 q^{69} + 127 q^{71} + 1170 q^{73} - 1725 q^{75} + 510 q^{77} - 480 q^{79} - 122 q^{81} - 520 q^{83} + 1205 q^{85} + 1615 q^{87} - 921 q^{89} + 1287 q^{91} - 2200 q^{93} + 1270 q^{95} + 415 q^{97} - 4420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 5x^{2} + 4x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 5\nu^{2} - 5\nu + 16 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 4 ) / 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 4\beta_{2} + \beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{3} - 4 \) Copy content Toggle raw display

Character values

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

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

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
−0.780776 1.35234i
1.28078 + 2.21837i
−0.780776 + 1.35234i
1.28078 2.21837i
0 −1.84233 3.19101i 0 −17.8078 0 2.71922 4.70983i 0 6.71165 11.6249i 0
81.2 0 4.34233 + 7.52113i 0 2.80776 0 4.78078 8.28055i 0 −24.2116 + 41.9358i 0
113.1 0 −1.84233 + 3.19101i 0 −17.8078 0 2.71922 + 4.70983i 0 6.71165 + 11.6249i 0
113.2 0 4.34233 7.52113i 0 2.80776 0 4.78078 + 8.28055i 0 −24.2116 41.9358i 0
\(n\): e.g. 2-40 or 990-1000
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.e 4
4.b odd 2 1 13.4.c.b 4
12.b even 2 1 117.4.g.d 4
13.c even 3 1 inner 208.4.i.e 4
52.b odd 2 1 169.4.c.f 4
52.f even 4 2 169.4.e.g 8
52.i odd 6 1 169.4.a.j 2
52.i odd 6 1 169.4.c.f 4
52.j odd 6 1 13.4.c.b 4
52.j odd 6 1 169.4.a.f 2
52.l even 12 2 169.4.b.e 4
52.l even 12 2 169.4.e.g 8
156.p even 6 1 117.4.g.d 4
156.p even 6 1 1521.4.a.t 2
156.r even 6 1 1521.4.a.l 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.b 4 4.b odd 2 1
13.4.c.b 4 52.j odd 6 1
117.4.g.d 4 12.b even 2 1
117.4.g.d 4 156.p even 6 1
169.4.a.f 2 52.j odd 6 1
169.4.a.j 2 52.i odd 6 1
169.4.b.e 4 52.l even 12 2
169.4.c.f 4 52.b odd 2 1
169.4.c.f 4 52.i odd 6 1
169.4.e.g 8 52.f even 4 2
169.4.e.g 8 52.l even 12 2
208.4.i.e 4 1.a even 1 1 trivial
208.4.i.e 4 13.c even 3 1 inner
1521.4.a.l 2 156.r even 6 1
1521.4.a.t 2 156.p even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 5T_{3}^{3} + 57T_{3}^{2} + 160T_{3} + 1024 \) acting on \(S_{4}^{\mathrm{new}}(208, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} - 5 T^{3} + 57 T^{2} + \cdots + 1024 \) Copy content Toggle raw display
$5$ \( (T^{2} + 15 T - 50)^{2} \) Copy content Toggle raw display
$7$ \( T^{4} - 15 T^{3} + 173 T^{2} + \cdots + 2704 \) Copy content Toggle raw display
$11$ \( T^{4} - 17 T^{3} + 1173 T^{2} + \cdots + 781456 \) Copy content Toggle raw display
$13$ \( T^{4} - 125 T^{3} + 7956 T^{2} + \cdots + 4826809 \) Copy content Toggle raw display
$17$ \( T^{4} + 70 T^{3} + 4763 T^{2} + \cdots + 18769 \) Copy content Toggle raw display
$19$ \( T^{4} + 141 T^{3} + \cdots + 23658496 \) Copy content Toggle raw display
$23$ \( T^{4} - 145 T^{3} + 20397 T^{2} + \cdots + 394384 \) Copy content Toggle raw display
$29$ \( T^{4} + 34 T^{3} + \cdots + 225330121 \) Copy content Toggle raw display
$31$ \( (T^{2} - 140 T - 37600)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} - 190 T^{3} + 35303 T^{2} + \cdots + 635209 \) Copy content Toggle raw display
$41$ \( T^{4} + 538 T^{3} + \cdots + 4992976921 \) Copy content Toggle raw display
$43$ \( T^{4} - 455 T^{3} + \cdots + 138485824 \) Copy content Toggle raw display
$47$ \( (T^{2} + 60 T - 82400)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 545 T - 41450)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} + 809 T^{3} + \cdots + 22729783696 \) Copy content Toggle raw display
$61$ \( T^{4} + 502 T^{3} + \cdots + 11448786001 \) Copy content Toggle raw display
$67$ \( T^{4} + 475 T^{3} + \cdots + 449948944 \) Copy content Toggle raw display
$71$ \( T^{4} - 127 T^{3} + \cdots + 1833894976 \) Copy content Toggle raw display
$73$ \( (T^{2} - 585 T + 54850)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 240 T + 7600)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 260 T - 25600)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 921 T^{3} + \cdots + 21215087716 \) Copy content Toggle raw display
$97$ \( T^{4} - 415 T^{3} + \cdots + 795268001284 \) Copy content Toggle raw display
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