Properties

Label 208.4.i.b
Level $208$
Weight $4$
Character orbit 208.i
Analytic conductor $12.272$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + 17 q^{5} + 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + 17 q^{5} + 20 \zeta_{6} q^{7} + 23 \zeta_{6} q^{9} + (32 \zeta_{6} - 32) q^{11} + ( - 13 \zeta_{6} - 39) q^{13} + ( - 34 \zeta_{6} + 34) q^{15} + 13 \zeta_{6} q^{17} + 30 \zeta_{6} q^{19} + 40 q^{21} + ( - 78 \zeta_{6} + 78) q^{23} + 164 q^{25} + 100 q^{27} + (197 \zeta_{6} - 197) q^{29} + 74 q^{31} + 64 \zeta_{6} q^{33} + 340 \zeta_{6} q^{35} + ( - 227 \zeta_{6} + 227) q^{37} + (78 \zeta_{6} - 104) q^{39} + ( - 165 \zeta_{6} + 165) q^{41} - 156 \zeta_{6} q^{43} + 391 \zeta_{6} q^{45} + 162 q^{47} + (57 \zeta_{6} - 57) q^{49} + 26 q^{51} + 93 q^{53} + (544 \zeta_{6} - 544) q^{55} + 60 q^{57} - 864 \zeta_{6} q^{59} - 145 \zeta_{6} q^{61} + (460 \zeta_{6} - 460) q^{63} + ( - 221 \zeta_{6} - 663) q^{65} + ( - 862 \zeta_{6} + 862) q^{67} - 156 \zeta_{6} q^{69} + 654 \zeta_{6} q^{71} + 215 q^{73} + ( - 328 \zeta_{6} + 328) q^{75} - 640 q^{77} + 76 q^{79} + (421 \zeta_{6} - 421) q^{81} - 628 q^{83} + 221 \zeta_{6} q^{85} + 394 \zeta_{6} q^{87} + ( - 266 \zeta_{6} + 266) q^{89} + ( - 1040 \zeta_{6} + 260) q^{91} + ( - 148 \zeta_{6} + 148) q^{93} + 510 \zeta_{6} q^{95} - 238 \zeta_{6} q^{97} - 736 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 34 q^{5} + 20 q^{7} + 23 q^{9} - 32 q^{11} - 91 q^{13} + 34 q^{15} + 13 q^{17} + 30 q^{19} + 80 q^{21} + 78 q^{23} + 328 q^{25} + 200 q^{27} - 197 q^{29} + 148 q^{31} + 64 q^{33} + 340 q^{35} + 227 q^{37} - 130 q^{39} + 165 q^{41} - 156 q^{43} + 391 q^{45} + 324 q^{47} - 57 q^{49} + 52 q^{51} + 186 q^{53} - 544 q^{55} + 120 q^{57} - 864 q^{59} - 145 q^{61} - 460 q^{63} - 1547 q^{65} + 862 q^{67} - 156 q^{69} + 654 q^{71} + 430 q^{73} + 328 q^{75} - 1280 q^{77} + 152 q^{79} - 421 q^{81} - 1256 q^{83} + 221 q^{85} + 394 q^{87} + 266 q^{89} - 520 q^{91} + 148 q^{93} + 510 q^{95} - 238 q^{97} - 1472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(53\) \(79\) \(145\)
\(\chi(n)\) \(1\) \(1\) \(-\zeta_{6}\)

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
0.500000 0.866025i
0.500000 + 0.866025i
0 1.00000 + 1.73205i 0 17.0000 0 10.0000 17.3205i 0 11.5000 19.9186i 0
113.1 0 1.00000 1.73205i 0 17.0000 0 10.0000 + 17.3205i 0 11.5000 + 19.9186i 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.b 2
4.b odd 2 1 13.4.c.a 2
12.b even 2 1 117.4.g.c 2
13.c even 3 1 inner 208.4.i.b 2
52.b odd 2 1 169.4.c.d 2
52.f even 4 2 169.4.e.c 4
52.i odd 6 1 169.4.a.a 1
52.i odd 6 1 169.4.c.d 2
52.j odd 6 1 13.4.c.a 2
52.j odd 6 1 169.4.a.d 1
52.l even 12 2 169.4.b.c 2
52.l even 12 2 169.4.e.c 4
156.p even 6 1 117.4.g.c 2
156.p even 6 1 1521.4.a.b 1
156.r even 6 1 1521.4.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.c.a 2 4.b odd 2 1
13.4.c.a 2 52.j odd 6 1
117.4.g.c 2 12.b even 2 1
117.4.g.c 2 156.p even 6 1
169.4.a.a 1 52.i odd 6 1
169.4.a.d 1 52.j odd 6 1
169.4.b.c 2 52.l even 12 2
169.4.c.d 2 52.b odd 2 1
169.4.c.d 2 52.i odd 6 1
169.4.e.c 4 52.f even 4 2
169.4.e.c 4 52.l even 12 2
208.4.i.b 2 1.a even 1 1 trivial
208.4.i.b 2 13.c even 3 1 inner
1521.4.a.b 1 156.p even 6 1
1521.4.a.k 1 156.r even 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$5$ \( (T - 17)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 20T + 400 \) Copy content Toggle raw display
$11$ \( T^{2} + 32T + 1024 \) Copy content Toggle raw display
$13$ \( T^{2} + 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$19$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$23$ \( T^{2} - 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} + 197T + 38809 \) Copy content Toggle raw display
$31$ \( (T - 74)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 227T + 51529 \) Copy content Toggle raw display
$41$ \( T^{2} - 165T + 27225 \) Copy content Toggle raw display
$43$ \( T^{2} + 156T + 24336 \) Copy content Toggle raw display
$47$ \( (T - 162)^{2} \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 864T + 746496 \) Copy content Toggle raw display
$61$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$67$ \( T^{2} - 862T + 743044 \) Copy content Toggle raw display
$71$ \( T^{2} - 654T + 427716 \) Copy content Toggle raw display
$73$ \( (T - 215)^{2} \) Copy content Toggle raw display
$79$ \( (T - 76)^{2} \) Copy content Toggle raw display
$83$ \( (T + 628)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 266T + 70756 \) Copy content Toggle raw display
$97$ \( T^{2} + 238T + 56644 \) Copy content Toggle raw display
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