Properties

Label 208.4.w.a
Level $208$
Weight $4$
Character orbit 208.w
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.w (of order \(6\), 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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + (7 \zeta_{6} - 7) q^{3} + (16 \zeta_{6} - 8) q^{5} + (13 \zeta_{6} - 26) q^{7} - 22 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (7 \zeta_{6} - 7) q^{3} + (16 \zeta_{6} - 8) q^{5} + (13 \zeta_{6} - 26) q^{7} - 22 \zeta_{6} q^{9} + (13 \zeta_{6} + 13) q^{11} + (52 \zeta_{6} - 39) q^{13} + ( - 56 \zeta_{6} - 56) q^{15} - 27 \zeta_{6} q^{17} + ( - 51 \zeta_{6} + 102) q^{19} + ( - 182 \zeta_{6} + 91) q^{21} + ( - 57 \zeta_{6} + 57) q^{23} - 67 q^{25} - 35 q^{27} + ( - 69 \zeta_{6} + 69) q^{29} + (84 \zeta_{6} - 42) q^{31} + (91 \zeta_{6} - 182) q^{33} - 312 \zeta_{6} q^{35} + ( - 23 \zeta_{6} - 23) q^{37} + ( - 273 \zeta_{6} - 91) q^{39} + ( - 227 \zeta_{6} - 227) q^{41} - 85 \zeta_{6} q^{43} + ( - 176 \zeta_{6} + 352) q^{45} + (396 \zeta_{6} - 198) q^{47} + ( - 164 \zeta_{6} + 164) q^{49} + 189 q^{51} + 426 q^{53} + (312 \zeta_{6} - 312) q^{55} + (714 \zeta_{6} - 357) q^{57} + ( - 11 \zeta_{6} + 22) q^{59} + 17 \zeta_{6} q^{61} + (286 \zeta_{6} + 286) q^{63} + ( - 208 \zeta_{6} - 520) q^{65} + ( - 95 \zeta_{6} - 95) q^{67} + 399 \zeta_{6} q^{69} + (337 \zeta_{6} - 674) q^{71} + (1160 \zeta_{6} - 580) q^{73} + ( - 469 \zeta_{6} + 469) q^{75} - 507 q^{77} + 1244 q^{79} + ( - 839 \zeta_{6} + 839) q^{81} + (492 \zeta_{6} - 246) q^{83} + ( - 216 \zeta_{6} + 432) q^{85} + 483 \zeta_{6} q^{87} + (177 \zeta_{6} + 177) q^{89} + ( - 1183 \zeta_{6} + 338) q^{91} + ( - 294 \zeta_{6} - 294) q^{93} + 1224 \zeta_{6} q^{95} + ( - 713 \zeta_{6} + 1426) q^{97} + ( - 572 \zeta_{6} + 286) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 7 q^{3} - 39 q^{7} - 22 q^{9} + 39 q^{11} - 26 q^{13} - 168 q^{15} - 27 q^{17} + 153 q^{19} + 57 q^{23} - 134 q^{25} - 70 q^{27} + 69 q^{29} - 273 q^{33} - 312 q^{35} - 69 q^{37} - 455 q^{39} - 681 q^{41} - 85 q^{43} + 528 q^{45} + 164 q^{49} + 378 q^{51} + 852 q^{53} - 312 q^{55} + 33 q^{59} + 17 q^{61} + 858 q^{63} - 1248 q^{65} - 285 q^{67} + 399 q^{69} - 1011 q^{71} + 469 q^{75} - 1014 q^{77} + 2488 q^{79} + 839 q^{81} + 648 q^{85} + 483 q^{87} + 531 q^{89} - 507 q^{91} - 882 q^{93} + 1224 q^{95} + 2139 q^{97}+O(q^{100}) \) 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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
0.500000 + 0.866025i
0.500000 0.866025i
0 −3.50000 + 6.06218i 0 13.8564i 0 −19.5000 + 11.2583i 0 −11.0000 19.0526i 0
49.1 0 −3.50000 6.06218i 0 13.8564i 0 −19.5000 11.2583i 0 −11.0000 + 19.0526i 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.e even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 208.4.w.a 2
4.b odd 2 1 13.4.e.a 2
12.b even 2 1 117.4.q.c 2
13.e even 6 1 inner 208.4.w.a 2
52.b odd 2 1 169.4.e.b 2
52.f even 4 2 169.4.c.i 4
52.i odd 6 1 13.4.e.a 2
52.i odd 6 1 169.4.b.b 2
52.j odd 6 1 169.4.b.b 2
52.j odd 6 1 169.4.e.b 2
52.l even 12 2 169.4.a.h 2
52.l even 12 2 169.4.c.i 4
156.r even 6 1 117.4.q.c 2
156.v odd 12 2 1521.4.a.q 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.a 2 4.b odd 2 1
13.4.e.a 2 52.i odd 6 1
117.4.q.c 2 12.b even 2 1
117.4.q.c 2 156.r even 6 1
169.4.a.h 2 52.l even 12 2
169.4.b.b 2 52.i odd 6 1
169.4.b.b 2 52.j odd 6 1
169.4.c.i 4 52.f even 4 2
169.4.c.i 4 52.l even 12 2
169.4.e.b 2 52.b odd 2 1
169.4.e.b 2 52.j odd 6 1
208.4.w.a 2 1.a even 1 1 trivial
208.4.w.a 2 13.e even 6 1 inner
1521.4.a.q 2 156.v odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 7T_{3} + 49 \) 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
$3$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$5$ \( T^{2} + 192 \) Copy content Toggle raw display
$7$ \( T^{2} + 39T + 507 \) Copy content Toggle raw display
$11$ \( T^{2} - 39T + 507 \) Copy content Toggle raw display
$13$ \( T^{2} + 26T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 27T + 729 \) Copy content Toggle raw display
$19$ \( T^{2} - 153T + 7803 \) Copy content Toggle raw display
$23$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$29$ \( T^{2} - 69T + 4761 \) Copy content Toggle raw display
$31$ \( T^{2} + 5292 \) Copy content Toggle raw display
$37$ \( T^{2} + 69T + 1587 \) Copy content Toggle raw display
$41$ \( T^{2} + 681T + 154587 \) Copy content Toggle raw display
$43$ \( T^{2} + 85T + 7225 \) Copy content Toggle raw display
$47$ \( T^{2} + 117612 \) Copy content Toggle raw display
$53$ \( (T - 426)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 33T + 363 \) Copy content Toggle raw display
$61$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$67$ \( T^{2} + 285T + 27075 \) Copy content Toggle raw display
$71$ \( T^{2} + 1011 T + 340707 \) Copy content Toggle raw display
$73$ \( T^{2} + 1009200 \) Copy content Toggle raw display
$79$ \( (T - 1244)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 181548 \) Copy content Toggle raw display
$89$ \( T^{2} - 531T + 93987 \) Copy content Toggle raw display
$97$ \( T^{2} - 2139 T + 1525107 \) Copy content Toggle raw display
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