Properties

Label 117.4.g.b
Level $117$
Weight $4$
Character orbit 117.g
Analytic conductor $6.903$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 117.g (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.90322347067\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + ( - 3 \zeta_{6} + 3) q^{2} - \zeta_{6} q^{4} + 9 q^{5} - 2 \zeta_{6} q^{7} + 21 q^{8} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 3 \zeta_{6} + 3) q^{2} - \zeta_{6} q^{4} + 9 q^{5} - 2 \zeta_{6} q^{7} + 21 q^{8} + ( - 27 \zeta_{6} + 27) q^{10} + ( - 30 \zeta_{6} + 30) q^{11} + (39 \zeta_{6} + 13) q^{13} - 6 q^{14} + ( - 71 \zeta_{6} + 71) q^{16} - 111 \zeta_{6} q^{17} + 46 \zeta_{6} q^{19} - 9 \zeta_{6} q^{20} - 90 \zeta_{6} q^{22} + (6 \zeta_{6} - 6) q^{23} - 44 q^{25} + ( - 39 \zeta_{6} + 156) q^{26} + (2 \zeta_{6} - 2) q^{28} + (105 \zeta_{6} - 105) q^{29} - 100 q^{31} - 45 \zeta_{6} q^{32} - 333 q^{34} - 18 \zeta_{6} q^{35} + (17 \zeta_{6} - 17) q^{37} + 138 q^{38} + 189 q^{40} + (231 \zeta_{6} - 231) q^{41} + 514 \zeta_{6} q^{43} - 30 q^{44} + 18 \zeta_{6} q^{46} + 162 q^{47} + ( - 339 \zeta_{6} + 339) q^{49} + (132 \zeta_{6} - 132) q^{50} + ( - 52 \zeta_{6} + 39) q^{52} - 639 q^{53} + ( - 270 \zeta_{6} + 270) q^{55} - 42 \zeta_{6} q^{56} + 315 \zeta_{6} q^{58} + 600 \zeta_{6} q^{59} - 233 \zeta_{6} q^{61} + (300 \zeta_{6} - 300) q^{62} + 433 q^{64} + (351 \zeta_{6} + 117) q^{65} + (926 \zeta_{6} - 926) q^{67} + (111 \zeta_{6} - 111) q^{68} - 54 q^{70} - 930 \zeta_{6} q^{71} - 253 q^{73} + 51 \zeta_{6} q^{74} + ( - 46 \zeta_{6} + 46) q^{76} - 60 q^{77} - 1324 q^{79} + ( - 639 \zeta_{6} + 639) q^{80} + 693 \zeta_{6} q^{82} - 810 q^{83} - 999 \zeta_{6} q^{85} + 1542 q^{86} + ( - 630 \zeta_{6} + 630) q^{88} + ( - 498 \zeta_{6} + 498) q^{89} + ( - 104 \zeta_{6} + 78) q^{91} + 6 q^{92} + ( - 486 \zeta_{6} + 486) q^{94} + 414 \zeta_{6} q^{95} - 1358 \zeta_{6} q^{97} - 1017 \zeta_{6} q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} - q^{4} + 18 q^{5} - 2 q^{7} + 42 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 3 q^{2} - q^{4} + 18 q^{5} - 2 q^{7} + 42 q^{8} + 27 q^{10} + 30 q^{11} + 65 q^{13} - 12 q^{14} + 71 q^{16} - 111 q^{17} + 46 q^{19} - 9 q^{20} - 90 q^{22} - 6 q^{23} - 88 q^{25} + 273 q^{26} - 2 q^{28} - 105 q^{29} - 200 q^{31} - 45 q^{32} - 666 q^{34} - 18 q^{35} - 17 q^{37} + 276 q^{38} + 378 q^{40} - 231 q^{41} + 514 q^{43} - 60 q^{44} + 18 q^{46} + 324 q^{47} + 339 q^{49} - 132 q^{50} + 26 q^{52} - 1278 q^{53} + 270 q^{55} - 42 q^{56} + 315 q^{58} + 600 q^{59} - 233 q^{61} - 300 q^{62} + 866 q^{64} + 585 q^{65} - 926 q^{67} - 111 q^{68} - 108 q^{70} - 930 q^{71} - 506 q^{73} + 51 q^{74} + 46 q^{76} - 120 q^{77} - 2648 q^{79} + 639 q^{80} + 693 q^{82} - 1620 q^{83} - 999 q^{85} + 3084 q^{86} + 630 q^{88} + 498 q^{89} + 52 q^{91} + 12 q^{92} + 486 q^{94} + 414 q^{95} - 1358 q^{97} - 1017 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
55.1
0.500000 0.866025i
0.500000 + 0.866025i
1.50000 + 2.59808i 0 −0.500000 + 0.866025i 9.00000 0 −1.00000 + 1.73205i 21.0000 0 13.5000 + 23.3827i
100.1 1.50000 2.59808i 0 −0.500000 0.866025i 9.00000 0 −1.00000 1.73205i 21.0000 0 13.5000 23.3827i
\(n\): 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 117.4.g.b 2
3.b odd 2 1 39.4.e.a 2
12.b even 2 1 624.4.q.b 2
13.c even 3 1 inner 117.4.g.b 2
13.c even 3 1 1521.4.a.c 1
13.e even 6 1 1521.4.a.j 1
39.h odd 6 1 507.4.a.a 1
39.i odd 6 1 39.4.e.a 2
39.i odd 6 1 507.4.a.e 1
39.k even 12 2 507.4.b.c 2
156.p even 6 1 624.4.q.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.4.e.a 2 3.b odd 2 1
39.4.e.a 2 39.i odd 6 1
117.4.g.b 2 1.a even 1 1 trivial
117.4.g.b 2 13.c even 3 1 inner
507.4.a.a 1 39.h odd 6 1
507.4.a.e 1 39.i odd 6 1
507.4.b.c 2 39.k even 12 2
624.4.q.b 2 12.b even 2 1
624.4.q.b 2 156.p even 6 1
1521.4.a.c 1 13.c even 3 1
1521.4.a.j 1 13.e even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 3T_{2} + 9 \) acting on \(S_{4}^{\mathrm{new}}(117, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( (T - 9)^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 30T + 900 \) Copy content Toggle raw display
$13$ \( T^{2} - 65T + 2197 \) Copy content Toggle raw display
$17$ \( T^{2} + 111T + 12321 \) Copy content Toggle raw display
$19$ \( T^{2} - 46T + 2116 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 105T + 11025 \) Copy content Toggle raw display
$31$ \( (T + 100)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$41$ \( T^{2} + 231T + 53361 \) Copy content Toggle raw display
$43$ \( T^{2} - 514T + 264196 \) Copy content Toggle raw display
$47$ \( (T - 162)^{2} \) Copy content Toggle raw display
$53$ \( (T + 639)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 600T + 360000 \) Copy content Toggle raw display
$61$ \( T^{2} + 233T + 54289 \) Copy content Toggle raw display
$67$ \( T^{2} + 926T + 857476 \) Copy content Toggle raw display
$71$ \( T^{2} + 930T + 864900 \) Copy content Toggle raw display
$73$ \( (T + 253)^{2} \) Copy content Toggle raw display
$79$ \( (T + 1324)^{2} \) Copy content Toggle raw display
$83$ \( (T + 810)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 498T + 248004 \) Copy content Toggle raw display
$97$ \( T^{2} + 1358 T + 1844164 \) Copy content Toggle raw display
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