Properties

Label 114.4.e.d
Level $114$
Weight $4$
Character orbit 114.e
Analytic conductor $6.726$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 114 = 2 \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 114.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.72621774065\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.627014547.1
Defining polynomial: \(x^{6} + 26 x^{4} + 169 x^{2} + 147\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -2 \beta_{1} q^{2} + 3 \beta_{1} q^{3} + ( -4 + 4 \beta_{1} ) q^{4} + ( -3 \beta_{1} + \beta_{2} + \beta_{5} ) q^{5} + ( 6 - 6 \beta_{1} ) q^{6} + ( 2 - \beta_{2} ) q^{7} + 8 q^{8} + ( -9 + 9 \beta_{1} ) q^{9} +O(q^{10})\) \( q -2 \beta_{1} q^{2} + 3 \beta_{1} q^{3} + ( -4 + 4 \beta_{1} ) q^{4} + ( -3 \beta_{1} + \beta_{2} + \beta_{5} ) q^{5} + ( 6 - 6 \beta_{1} ) q^{6} + ( 2 - \beta_{2} ) q^{7} + 8 q^{8} + ( -9 + 9 \beta_{1} ) q^{9} + ( -6 + 6 \beta_{1} - 2 \beta_{5} ) q^{10} + ( -15 + \beta_{3} ) q^{11} -12 q^{12} + ( 1 - \beta_{1} + 6 \beta_{5} ) q^{13} + ( -4 \beta_{1} + 2 \beta_{2} + 2 \beta_{5} ) q^{14} + ( 9 - 9 \beta_{1} + 3 \beta_{5} ) q^{15} -16 \beta_{1} q^{16} + ( 30 \beta_{1} + 7 \beta_{2} - \beta_{4} + 7 \beta_{5} ) q^{17} + 18 q^{18} + ( 10 - 6 \beta_{1} + 4 \beta_{2} - \beta_{4} - 3 \beta_{5} ) q^{19} + ( 12 - 4 \beta_{2} ) q^{20} + ( 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{5} ) q^{21} + ( 30 \beta_{1} + 2 \beta_{4} ) q^{22} + ( -3 + 3 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} + 9 \beta_{5} ) q^{23} + 24 \beta_{1} q^{24} + ( 29 - 29 \beta_{1} + \beta_{3} + \beta_{4} - 5 \beta_{5} ) q^{25} + ( -2 + 12 \beta_{2} ) q^{26} -27 q^{27} + ( -8 + 8 \beta_{1} - 4 \beta_{5} ) q^{28} + ( -36 + 36 \beta_{1} + \beta_{3} + \beta_{4} + 15 \beta_{5} ) q^{29} + ( -18 + 6 \beta_{2} ) q^{30} + ( -28 + 24 \beta_{2} - \beta_{3} ) q^{31} + ( -32 + 32 \beta_{1} ) q^{32} + ( -45 \beta_{1} - 3 \beta_{4} ) q^{33} + ( 60 - 60 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 14 \beta_{5} ) q^{34} + ( -93 \beta_{1} + 4 \beta_{2} - \beta_{4} + 4 \beta_{5} ) q^{35} -36 \beta_{1} q^{36} + ( 77 - 17 \beta_{2} - 3 \beta_{3} ) q^{37} + ( -12 - 8 \beta_{1} - 14 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} ) q^{38} + ( 3 - 18 \beta_{2} ) q^{39} + ( -24 \beta_{1} + 8 \beta_{2} + 8 \beta_{5} ) q^{40} + ( 222 \beta_{1} - 20 \beta_{2} - 2 \beta_{4} - 20 \beta_{5} ) q^{41} + ( 12 - 12 \beta_{1} + 6 \beta_{5} ) q^{42} + ( 28 \beta_{1} - 21 \beta_{2} + 2 \beta_{4} - 21 \beta_{5} ) q^{43} + ( 60 - 60 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} ) q^{44} + ( 27 - 9 \beta_{2} ) q^{45} + ( 6 + 18 \beta_{2} - 4 \beta_{3} ) q^{46} + ( -102 + 102 \beta_{1} - 6 \beta_{3} - 6 \beta_{4} - 10 \beta_{5} ) q^{47} + ( 48 - 48 \beta_{1} ) q^{48} + ( -252 - 3 \beta_{2} - \beta_{3} ) q^{49} + ( -58 - 10 \beta_{2} - 2 \beta_{3} ) q^{50} + ( -90 + 90 \beta_{1} - 3 \beta_{3} - 3 \beta_{4} + 21 \beta_{5} ) q^{51} + ( 4 \beta_{1} - 24 \beta_{2} - 24 \beta_{5} ) q^{52} + ( 435 - 435 \beta_{1} - \beta_{3} - \beta_{4} + 18 \beta_{5} ) q^{53} + 54 \beta_{1} q^{54} + ( 66 \beta_{1} - 55 \beta_{2} + 5 \beta_{4} - 55 \beta_{5} ) q^{55} + ( 16 - 8 \beta_{2} ) q^{56} + ( 18 + 12 \beta_{1} + 21 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 12 \beta_{5} ) q^{57} + ( 72 + 30 \beta_{2} - 2 \beta_{3} ) q^{58} + ( -81 \beta_{1} + 9 \beta_{2} + 9 \beta_{5} ) q^{59} + ( 36 \beta_{1} - 12 \beta_{2} - 12 \beta_{5} ) q^{60} + ( 157 - 157 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} - 32 \beta_{5} ) q^{61} + ( 56 \beta_{1} - 48 \beta_{2} - 2 \beta_{4} - 48 \beta_{5} ) q^{62} + ( -18 + 18 \beta_{1} - 9 \beta_{5} ) q^{63} + 64 q^{64} + ( -525 + 13 \beta_{2} + 6 \beta_{3} ) q^{65} + ( -90 + 90 \beta_{1} + 6 \beta_{3} + 6 \beta_{4} ) q^{66} + ( 244 - 244 \beta_{1} - \beta_{3} - \beta_{4} - 12 \beta_{5} ) q^{67} + ( -120 - 28 \beta_{2} - 4 \beta_{3} ) q^{68} + ( -9 - 27 \beta_{2} + 6 \beta_{3} ) q^{69} + ( -186 + 186 \beta_{1} + 2 \beta_{3} + 2 \beta_{4} - 8 \beta_{5} ) q^{70} + ( 18 \beta_{1} - 19 \beta_{2} + 11 \beta_{4} - 19 \beta_{5} ) q^{71} + ( -72 + 72 \beta_{1} ) q^{72} + ( 217 \beta_{1} + 66 \beta_{2} + 4 \beta_{4} + 66 \beta_{5} ) q^{73} + ( -154 \beta_{1} + 34 \beta_{2} - 6 \beta_{4} + 34 \beta_{5} ) q^{74} + ( 87 + 15 \beta_{2} + 3 \beta_{3} ) q^{75} + ( -16 + 40 \beta_{1} + 12 \beta_{2} - 4 \beta_{3} + 28 \beta_{5} ) q^{76} + ( -51 + 55 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -6 \beta_{1} + 36 \beta_{2} + 36 \beta_{5} ) q^{78} + ( 172 \beta_{1} + 30 \beta_{2} - 3 \beta_{4} + 30 \beta_{5} ) q^{79} + ( -48 + 48 \beta_{1} - 16 \beta_{5} ) q^{80} -81 \beta_{1} q^{81} + ( 444 - 444 \beta_{1} + 4 \beta_{3} + 4 \beta_{4} + 40 \beta_{5} ) q^{82} + ( 846 + 37 \beta_{2} - 5 \beta_{3} ) q^{83} + ( -24 + 12 \beta_{2} ) q^{84} + ( -540 + 540 \beta_{1} + 12 \beta_{3} + 12 \beta_{4} - 24 \beta_{5} ) q^{85} + ( 56 - 56 \beta_{1} - 4 \beta_{3} - 4 \beta_{4} + 42 \beta_{5} ) q^{86} + ( -108 - 45 \beta_{2} + 3 \beta_{3} ) q^{87} + ( -120 + 8 \beta_{3} ) q^{88} + ( -543 + 543 \beta_{1} + 11 \beta_{3} + 11 \beta_{4} + 34 \beta_{5} ) q^{89} + ( -54 \beta_{1} + 18 \beta_{2} + 18 \beta_{5} ) q^{90} + ( 524 - 524 \beta_{1} - 6 \beta_{3} - 6 \beta_{4} + 7 \beta_{5} ) q^{91} + ( -12 \beta_{1} - 36 \beta_{2} - 8 \beta_{4} - 36 \beta_{5} ) q^{92} + ( -84 \beta_{1} + 72 \beta_{2} + 3 \beta_{4} + 72 \beta_{5} ) q^{93} + ( 204 - 20 \beta_{2} + 12 \beta_{3} ) q^{94} + ( 222 + 357 \beta_{1} - 4 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 44 \beta_{5} ) q^{95} -96 q^{96} + ( -320 \beta_{1} + 13 \beta_{2} + \beta_{4} + 13 \beta_{5} ) q^{97} + ( 504 \beta_{1} + 6 \beta_{2} - 2 \beta_{4} + 6 \beta_{5} ) q^{98} + ( 135 - 135 \beta_{1} - 9 \beta_{3} - 9 \beta_{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 6q^{2} + 9q^{3} - 12q^{4} - 10q^{5} + 18q^{6} + 14q^{7} + 48q^{8} - 27q^{9} + O(q^{10}) \) \( 6q - 6q^{2} + 9q^{3} - 12q^{4} - 10q^{5} + 18q^{6} + 14q^{7} + 48q^{8} - 27q^{9} - 20q^{10} - 88q^{11} - 72q^{12} + 9q^{13} - 14q^{14} + 30q^{15} - 48q^{16} + 84q^{17} + 108q^{18} + 32q^{19} + 80q^{20} + 21q^{21} + 88q^{22} + 2q^{23} + 72q^{24} + 83q^{25} - 36q^{26} - 162q^{27} - 28q^{28} - 92q^{29} - 120q^{30} - 218q^{31} - 96q^{32} - 132q^{33} + 168q^{34} - 282q^{35} - 108q^{36} + 490q^{37} - 74q^{38} + 54q^{39} - 80q^{40} + 688q^{41} + 42q^{42} + 103q^{43} + 176q^{44} + 180q^{45} - 8q^{46} - 322q^{47} + 144q^{48} - 1508q^{49} - 332q^{50} - 252q^{51} + 36q^{52} + 1322q^{53} + 162q^{54} + 248q^{55} + 112q^{56} + 111q^{57} + 368q^{58} - 252q^{59} + 120q^{60} + 435q^{61} + 218q^{62} - 63q^{63} + 384q^{64} - 3164q^{65} - 264q^{66} + 719q^{67} - 672q^{68} + 12q^{69} - 564q^{70} + 62q^{71} - 216q^{72} + 581q^{73} - 490q^{74} + 498q^{75} + 20q^{76} - 408q^{77} - 54q^{78} + 489q^{79} - 160q^{80} - 243q^{81} + 1376q^{82} + 4992q^{83} - 168q^{84} - 1632q^{85} + 206q^{86} - 552q^{87} - 704q^{88} - 1584q^{89} - 180q^{90} + 1573q^{91} + 8q^{92} - 327q^{93} + 1288q^{94} + 2362q^{95} - 576q^{96} - 974q^{97} + 1508q^{98} + 396q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} + 26 x^{4} + 169 x^{2} + 147\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\((\)\( \nu^{3} + 13 \nu + 7 \)\()/14\)
\(\beta_{2}\)\(=\)\((\)\( 2 \nu^{4} + 40 \nu^{2} + 119 \)\()/7\)
\(\beta_{3}\)\(=\)\((\)\( 10 \nu^{4} + 116 \nu^{2} - 119 \)\()/7\)
\(\beta_{4}\)\(=\)\((\)\( 2 \nu^{5} - 10 \nu^{4} + 43 \nu^{3} - 116 \nu^{2} + 431 \nu + 119 \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( 2 \nu^{5} - 2 \nu^{4} + 43 \nu^{3} - 40 \nu^{2} + 179 \nu - 119 \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(-2 \beta_{5} + 2 \beta_{4} + \beta_{3} - \beta_{2}\)\()/36\)
\(\nu^{2}\)\(=\)\((\)\(-\beta_{3} + 5 \beta_{2} - 102\)\()/12\)
\(\nu^{3}\)\(=\)\((\)\(26 \beta_{5} - 26 \beta_{4} - 13 \beta_{3} + 13 \beta_{2} + 504 \beta_{1} - 252\)\()/36\)
\(\nu^{4}\)\(=\)\((\)\(10 \beta_{3} - 29 \beta_{2} + 663\)\()/6\)
\(\nu^{5}\)\(=\)\((\)\(-64 \beta_{5} + 190 \beta_{4} + 95 \beta_{3} - 32 \beta_{2} - 5418 \beta_{1} + 2709\)\()/18\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/114\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
2.99107i
4.00355i
1.01248i
2.99107i
4.00355i
1.01248i
−1.00000 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −7.12716 12.3446i 3.00000 5.19615i 13.2543 8.00000 −4.50000 + 7.79423i −14.2543 + 24.6892i
7.2 −1.00000 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −2.09405 3.62701i 3.00000 5.19615i 3.18810 8.00000 −4.50000 + 7.79423i −4.18810 + 7.25401i
7.3 −1.00000 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 4.22121 + 7.31135i 3.00000 5.19615i −9.44242 8.00000 −4.50000 + 7.79423i 8.44242 14.6227i
49.1 −1.00000 + 1.73205i 1.50000 2.59808i −2.00000 3.46410i −7.12716 + 12.3446i 3.00000 + 5.19615i 13.2543 8.00000 −4.50000 7.79423i −14.2543 24.6892i
49.2 −1.00000 + 1.73205i 1.50000 2.59808i −2.00000 3.46410i −2.09405 + 3.62701i 3.00000 + 5.19615i 3.18810 8.00000 −4.50000 7.79423i −4.18810 7.25401i
49.3 −1.00000 + 1.73205i 1.50000 2.59808i −2.00000 3.46410i 4.22121 7.31135i 3.00000 + 5.19615i −9.44242 8.00000 −4.50000 7.79423i 8.44242 + 14.6227i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 114.4.e.d 6
3.b odd 2 1 342.4.g.h 6
19.c even 3 1 inner 114.4.e.d 6
19.c even 3 1 2166.4.a.u 3
19.d odd 6 1 2166.4.a.t 3
57.h odd 6 1 342.4.g.h 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.d 6 1.a even 1 1 trivial
114.4.e.d 6 19.c even 3 1 inner
342.4.g.h 6 3.b odd 2 1
342.4.g.h 6 57.h odd 6 1
2166.4.a.t 3 19.d odd 6 1
2166.4.a.u 3 19.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 10 T_{5}^{5} + 196 T_{5}^{4} + 48 T_{5}^{3} + 14256 T_{5}^{2} + 48384 T_{5} + 254016 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 4 + 2 T + T^{2} )^{3} \)
$3$ \( ( 9 - 3 T + T^{2} )^{3} \)
$5$ \( 254016 + 48384 T + 14256 T^{2} + 48 T^{3} + 196 T^{4} + 10 T^{5} + T^{6} \)
$7$ \( ( 399 - 113 T - 7 T^{2} + T^{3} )^{2} \)
$11$ \( ( -217224 - 4740 T + 44 T^{2} + T^{3} )^{2} \)
$13$ \( 1420159225 - 174443865 T + 21766806 T^{2} - 33709 T^{3} + 4710 T^{4} - 9 T^{5} + T^{6} \)
$17$ \( 673712640000 - 8273664000 T + 170553600 T^{2} - 794880 T^{3} + 17136 T^{4} - 84 T^{5} + T^{6} \)
$19$ \( 322687697779 - 1505468192 T - 8443429 T^{2} + 1033600 T^{3} - 1231 T^{4} - 32 T^{5} + T^{6} \)
$23$ \( 11401114176 + 3223780992 T + 911343312 T^{2} + 273936 T^{3} + 30196 T^{4} - 2 T^{5} + T^{6} \)
$29$ \( 1696079056896 - 39257616384 T + 788845824 T^{2} - 5377920 T^{3} + 38608 T^{4} + 92 T^{5} + T^{6} \)
$31$ \( ( -9721957 - 73489 T + 109 T^{2} + T^{3} )^{2} \)
$37$ \( ( -1317799 - 71005 T - 245 T^{2} + T^{3} )^{2} \)
$41$ \( 105675768336384 + 910385464320 T + 14915425536 T^{2} - 81489024 T^{3} + 384784 T^{4} - 688 T^{5} + T^{6} \)
$43$ \( 39054112950241 + 495553041713 T + 5644333322 T^{2} + 20666249 T^{3} + 89906 T^{4} - 103 T^{5} + T^{6} \)
$47$ \( 136793983242816 - 1943436862944 T + 23844396384 T^{2} - 76896600 T^{3} + 269848 T^{4} + 322 T^{5} + T^{6} \)
$53$ \( 4486100708289600 - 35729472185280 T + 196021376784 T^{2} - 571261536 T^{3} + 1214236 T^{4} - 1322 T^{5} + T^{6} \)
$59$ \( 134994517056 - 3928411872 T + 206907696 T^{2} + 3429216 T^{3} + 52812 T^{4} + 252 T^{5} + T^{6} \)
$61$ \( 5456924641 - 10530828147 T + 20354632134 T^{2} + 61864553 T^{3} + 331782 T^{4} - 435 T^{5} + T^{6} \)
$67$ \( 92437321911489 - 1437617323191 T + 15445546402 T^{2} - 88281047 T^{3} + 367434 T^{4} - 719 T^{5} + T^{6} \)
$71$ \( 12330917402872896 - 79753610220768 T + 522713246112 T^{2} - 177560184 T^{3} + 722056 T^{4} - 62 T^{5} + T^{6} \)
$73$ \( 4992458989302225 + 36054247673115 T + 219322540726 T^{2} + 437780959 T^{3} + 847830 T^{4} - 581 T^{5} + T^{6} \)
$79$ \( 2685185649775729 - 4885526114337 T + 34228288914 T^{2} - 57534145 T^{3} + 333402 T^{4} - 489 T^{5} + T^{6} \)
$83$ \( ( -372278592 + 1783728 T - 2496 T^{2} + T^{3} )^{2} \)
$89$ \( 156435026228942400 - 28918743806880 T + 631847538576 T^{2} + 906853104 T^{3} + 2435940 T^{4} + 1584 T^{5} + T^{6} \)
$97$ \( 585649680040000 + 7025318060000 T + 60703095200 T^{2} + 234351800 T^{3} + 658376 T^{4} + 974 T^{5} + T^{6} \)
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