Properties

Label 208.4.w.b
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: \(\Z[a_1, \ldots, a_{9}]\)
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 1) q^{5} + ( - 8 \zeta_{6} + 16) q^{7} + 23 \zeta_{6} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{3} + ( - 2 \zeta_{6} + 1) q^{5} + ( - 8 \zeta_{6} + 16) q^{7} + 23 \zeta_{6} q^{9} + ( - 8 \zeta_{6} - 8) q^{11} + (13 \zeta_{6} + 39) q^{13} + ( - 2 \zeta_{6} - 2) q^{15} - 117 \zeta_{6} q^{17} + ( - 66 \zeta_{6} + 132) q^{19} + ( - 32 \zeta_{6} + 16) q^{21} + (78 \zeta_{6} - 78) q^{23} + 122 q^{25} + 100 q^{27} + ( - 141 \zeta_{6} + 141) q^{29} + ( - 180 \zeta_{6} + 90) q^{31} + (16 \zeta_{6} - 32) q^{33} - 24 \zeta_{6} q^{35} + ( - 83 \zeta_{6} - 83) q^{37} + ( - 78 \zeta_{6} + 104) q^{39} + (157 \zeta_{6} + 157) q^{41} + 104 \zeta_{6} q^{43} + ( - 23 \zeta_{6} + 46) q^{45} + ( - 348 \zeta_{6} + 174) q^{47} + (151 \zeta_{6} - 151) q^{49} - 234 q^{51} + 93 q^{53} + (24 \zeta_{6} - 24) q^{55} + ( - 264 \zeta_{6} + 132) q^{57} + ( - 164 \zeta_{6} + 328) q^{59} - 145 \zeta_{6} q^{61} + (184 \zeta_{6} + 184) q^{63} + ( - 91 \zeta_{6} + 65) q^{65} + (454 \zeta_{6} + 454) q^{67} + 156 \zeta_{6} q^{69} + (610 \zeta_{6} - 1220) q^{71} + (530 \zeta_{6} - 265) q^{73} + ( - 244 \zeta_{6} + 244) q^{75} - 192 q^{77} - 1276 q^{79} + (421 \zeta_{6} - 421) q^{81} + (912 \zeta_{6} - 456) q^{83} + (117 \zeta_{6} - 234) q^{85} - 282 \zeta_{6} q^{87} + ( - 564 \zeta_{6} - 564) q^{89} + ( - 208 \zeta_{6} + 728) q^{91} + ( - 180 \zeta_{6} - 180) q^{93} - 198 \zeta_{6} q^{95} + ( - 116 \zeta_{6} + 232) q^{97} + ( - 368 \zeta_{6} + 184) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} + 24 q^{7} + 23 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{3} + 24 q^{7} + 23 q^{9} - 24 q^{11} + 91 q^{13} - 6 q^{15} - 117 q^{17} + 198 q^{19} - 78 q^{23} + 244 q^{25} + 200 q^{27} + 141 q^{29} - 48 q^{33} - 24 q^{35} - 249 q^{37} + 130 q^{39} + 471 q^{41} + 104 q^{43} + 69 q^{45} - 151 q^{49} - 468 q^{51} + 186 q^{53} - 24 q^{55} + 492 q^{59} - 145 q^{61} + 552 q^{63} + 39 q^{65} + 1362 q^{67} + 156 q^{69} - 1830 q^{71} + 244 q^{75} - 384 q^{77} - 2552 q^{79} - 421 q^{81} - 351 q^{85} - 282 q^{87} - 1692 q^{89} + 1248 q^{91} - 540 q^{93} - 198 q^{95} + 348 q^{97}+O(q^{100}) \) 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 1.00000 1.73205i 0 1.73205i 0 12.0000 6.92820i 0 11.5000 + 19.9186i 0
49.1 0 1.00000 + 1.73205i 0 1.73205i 0 12.0000 + 6.92820i 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.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.b 2
4.b odd 2 1 13.4.e.b 2
12.b even 2 1 117.4.q.a 2
13.e even 6 1 inner 208.4.w.b 2
52.b odd 2 1 169.4.e.a 2
52.f even 4 2 169.4.c.h 4
52.i odd 6 1 13.4.e.b 2
52.i odd 6 1 169.4.b.d 2
52.j odd 6 1 169.4.b.d 2
52.j odd 6 1 169.4.e.a 2
52.l even 12 2 169.4.a.i 2
52.l even 12 2 169.4.c.h 4
156.r even 6 1 117.4.q.a 2
156.v odd 12 2 1521.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
13.4.e.b 2 4.b odd 2 1
13.4.e.b 2 52.i odd 6 1
117.4.q.a 2 12.b even 2 1
117.4.q.a 2 156.r even 6 1
169.4.a.i 2 52.l even 12 2
169.4.b.d 2 52.i odd 6 1
169.4.b.d 2 52.j odd 6 1
169.4.c.h 4 52.f even 4 2
169.4.c.h 4 52.l even 12 2
169.4.e.a 2 52.b odd 2 1
169.4.e.a 2 52.j odd 6 1
208.4.w.b 2 1.a even 1 1 trivial
208.4.w.b 2 13.e even 6 1 inner
1521.4.a.o 2 156.v odd 12 2

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
$5$ \( T^{2} + 3 \) Copy content Toggle raw display
$7$ \( T^{2} - 24T + 192 \) Copy content Toggle raw display
$11$ \( T^{2} + 24T + 192 \) Copy content Toggle raw display
$13$ \( T^{2} - 91T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 117T + 13689 \) Copy content Toggle raw display
$19$ \( T^{2} - 198T + 13068 \) Copy content Toggle raw display
$23$ \( T^{2} + 78T + 6084 \) Copy content Toggle raw display
$29$ \( T^{2} - 141T + 19881 \) Copy content Toggle raw display
$31$ \( T^{2} + 24300 \) Copy content Toggle raw display
$37$ \( T^{2} + 249T + 20667 \) Copy content Toggle raw display
$41$ \( T^{2} - 471T + 73947 \) Copy content Toggle raw display
$43$ \( T^{2} - 104T + 10816 \) Copy content Toggle raw display
$47$ \( T^{2} + 90828 \) Copy content Toggle raw display
$53$ \( (T - 93)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 492T + 80688 \) Copy content Toggle raw display
$61$ \( T^{2} + 145T + 21025 \) Copy content Toggle raw display
$67$ \( T^{2} - 1362 T + 618348 \) Copy content Toggle raw display
$71$ \( T^{2} + 1830 T + 1116300 \) Copy content Toggle raw display
$73$ \( T^{2} + 210675 \) Copy content Toggle raw display
$79$ \( (T + 1276)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 623808 \) Copy content Toggle raw display
$89$ \( T^{2} + 1692 T + 954288 \) Copy content Toggle raw display
$97$ \( T^{2} - 348T + 40368 \) Copy content Toggle raw display
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