Properties

Label 114.4.e.e
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: 6.0.6967728.1
Defining polynomial: \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (2 \beta_{3} + 2) q^{2} + (3 \beta_{3} + 3) q^{3} + 4 \beta_{3} q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{5} + 6 \beta_{3} q^{6} + ( - 2 \beta_{4} + 2 \beta_{2} + \beta_1 - 6) q^{7} - 8 q^{8} + 9 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{3} + 2) q^{2} + (3 \beta_{3} + 3) q^{3} + 4 \beta_{3} q^{4} + ( - \beta_{5} - \beta_{3} + \beta_1 - 1) q^{5} + 6 \beta_{3} q^{6} + ( - 2 \beta_{4} + 2 \beta_{2} + \beta_1 - 6) q^{7} - 8 q^{8} + 9 \beta_{3} q^{9} + ( - 2 \beta_{5} - 2 \beta_{3}) q^{10} + (\beta_{4} - \beta_{2} + 2 \beta_1 - 18) q^{11} - 12 q^{12} + (4 \beta_{4} + 23 \beta_{3} + 2 \beta_{2} + 2) q^{13} + ( - 2 \beta_{5} + 4 \beta_{4} - 16 \beta_{3} + 8 \beta_{2} + 2 \beta_1 - 8) q^{14} + ( - 3 \beta_{5} - 3 \beta_{3}) q^{15} + ( - 16 \beta_{3} - 16) q^{16} + (3 \beta_{5} - \beta_{4} + 18 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 16) q^{17} - 18 q^{18} + ( - 3 \beta_{5} + 3 \beta_{4} + 58 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 48) q^{19} + ( - 4 \beta_1 + 4) q^{20} + ( - 3 \beta_{5} + 6 \beta_{4} - 24 \beta_{3} + 12 \beta_{2} + 3 \beta_1 - 12) q^{21} + ( - 4 \beta_{5} - 2 \beta_{4} - 34 \beta_{3} - 4 \beta_{2} + 4 \beta_1 - 38) q^{22} + ( - 5 \beta_{5} - 20 \beta_{4} - 71 \beta_{3} - 10 \beta_{2} - 10) q^{23} + ( - 24 \beta_{3} - 24) q^{24} + (11 \beta_{5} - 14 \beta_{4} + 87 \beta_{3} - 7 \beta_{2} - 7) q^{25} + (4 \beta_{4} - 4 \beta_{2} - 50) q^{26} - 27 q^{27} + ( - 4 \beta_{5} + 16 \beta_{4} - 32 \beta_{3} + 8 \beta_{2} + 8) q^{28} + ( - \beta_{5} + 26 \beta_{4} - 16 \beta_{3} + 13 \beta_{2} + 13) q^{29} + ( - 6 \beta_1 + 6) q^{30} + ( - 3 \beta_{4} + 3 \beta_{2} + 8 \beta_1 + 33) q^{31} - 32 \beta_{3} q^{32} + ( - 6 \beta_{5} - 3 \beta_{4} - 51 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 57) q^{33} + (6 \beta_{5} - 4 \beta_{4} + 36 \beta_{3} - 2 \beta_{2} - 2) q^{34} + (18 \beta_{5} + 11 \beta_{4} + 93 \beta_{3} + 22 \beta_{2} - 18 \beta_1 + 115) q^{35} + ( - 36 \beta_{3} - 36) q^{36} + ( - 13 \beta_{4} + 13 \beta_{2} + 23 \beta_1 + 94) q^{37} + (4 \beta_{5} - 4 \beta_{4} + 100 \beta_{3} - 10 \beta_{2} - 10 \beta_1 - 26) q^{38} + (6 \beta_{4} - 6 \beta_{2} - 75) q^{39} + (8 \beta_{5} + 8 \beta_{3} - 8 \beta_1 + 8) q^{40} + (4 \beta_{5} - 26 \beta_{4} + 22 \beta_{3} - 52 \beta_{2} - 4 \beta_1 - 30) q^{41} + ( - 6 \beta_{5} + 24 \beta_{4} - 48 \beta_{3} + 12 \beta_{2} + 12) q^{42} + ( - 7 \beta_{5} - 18 \beta_{4} + 126 \beta_{3} - 36 \beta_{2} + 7 \beta_1 + 90) q^{43} + ( - 8 \beta_{5} - 8 \beta_{4} - 68 \beta_{3} - 4 \beta_{2} - 4) q^{44} + ( - 9 \beta_1 + 9) q^{45} + ( - 20 \beta_{4} + 20 \beta_{2} - 10 \beta_1 + 162) q^{46} + ( - 14 \beta_{5} + 36 \beta_{4} - 278 \beta_{3} + 18 \beta_{2} + 18) q^{47} - 48 \beta_{3} q^{48} + ( - 13 \beta_{4} + 13 \beta_{2} + 11 \beta_1 + 377) q^{49} + ( - 14 \beta_{4} + 14 \beta_{2} + 22 \beta_1 - 160) q^{50} + (9 \beta_{5} - 6 \beta_{4} + 54 \beta_{3} - 3 \beta_{2} - 3) q^{51} + ( - 8 \beta_{4} - 92 \beta_{3} - 16 \beta_{2} - 108) q^{52} + (28 \beta_{5} - 26 \beta_{4} - 17 \beta_{3} - 13 \beta_{2} - 13) q^{53} + ( - 54 \beta_{3} - 54) q^{54} + (37 \beta_{5} - 23 \beta_{4} + 502 \beta_{3} - 46 \beta_{2} - 37 \beta_1 + 456) q^{55} + (16 \beta_{4} - 16 \beta_{2} - 8 \beta_1 + 48) q^{56} + (6 \beta_{5} - 6 \beta_{4} + 150 \beta_{3} - 15 \beta_{2} - 15 \beta_1 - 39) q^{57} + (26 \beta_{4} - 26 \beta_{2} - 2 \beta_1 + 6) q^{58} + ( - 9 \beta_{5} + 24 \beta_{4} - 339 \beta_{3} + 48 \beta_{2} + 9 \beta_1 - 291) q^{59} + (12 \beta_{5} + 12 \beta_{3} - 12 \beta_1 + 12) q^{60} + (30 \beta_{5} - 4 \beta_{4} - 121 \beta_{3} - 2 \beta_{2} - 2) q^{61} + ( - 16 \beta_{5} + 6 \beta_{4} + 60 \beta_{3} + 12 \beta_{2} + 16 \beta_1 + 72) q^{62} + ( - 9 \beta_{5} + 36 \beta_{4} - 72 \beta_{3} + 18 \beta_{2} + 18) q^{63} + 64 q^{64} + (18 \beta_{4} - 18 \beta_{2} - 23 \beta_1 + 131) q^{65} + ( - 12 \beta_{5} - 12 \beta_{4} - 102 \beta_{3} - 6 \beta_{2} - 6) q^{66} + (26 \beta_{5} - 10 \beta_{4} + 34 \beta_{3} - 5 \beta_{2} - 5) q^{67} + ( - 4 \beta_{4} + 4 \beta_{2} + 12 \beta_1 - 68) q^{68} + ( - 30 \beta_{4} + 30 \beta_{2} - 15 \beta_1 + 243) q^{69} + (36 \beta_{5} + 44 \beta_{4} + 186 \beta_{3} + 22 \beta_{2} + 22) q^{70} + ( - 31 \beta_{5} - \beta_{4} - 334 \beta_{3} - 2 \beta_{2} + 31 \beta_1 - 336) q^{71} - 72 \beta_{3} q^{72} + ( - 14 \beta_{5} - 40 \beta_{4} + 5 \beta_{3} - 80 \beta_{2} + 14 \beta_1 - 75) q^{73} + ( - 46 \beta_{5} + 26 \beta_{4} + 162 \beta_{3} + 52 \beta_{2} + \cdots + 214) q^{74}+ \cdots + ( - 18 \beta_{5} - 18 \beta_{4} - 153 \beta_{3} - 9 \beta_{2} - 9) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} - 34 q^{7} - 48 q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{2} + 9 q^{3} - 12 q^{4} - 2 q^{5} - 18 q^{6} - 34 q^{7} - 48 q^{8} - 27 q^{9} + 4 q^{10} - 104 q^{11} - 72 q^{12} - 75 q^{13} - 34 q^{14} + 6 q^{15} - 48 q^{16} + 48 q^{17} - 108 q^{18} + 104 q^{19} + 16 q^{20} - 51 q^{21} - 104 q^{22} + 238 q^{23} - 72 q^{24} - 229 q^{25} - 300 q^{26} - 162 q^{27} + 68 q^{28} + 8 q^{29} + 24 q^{30} + 214 q^{31} + 96 q^{32} - 156 q^{33} - 96 q^{34} + 294 q^{35} - 108 q^{36} + 610 q^{37} - 430 q^{38} - 450 q^{39} + 16 q^{40} - 16 q^{41} + 102 q^{42} + 331 q^{43} + 208 q^{44} + 36 q^{45} + 952 q^{46} + 766 q^{47} + 144 q^{48} + 2284 q^{49} - 916 q^{50} - 144 q^{51} - 300 q^{52} + 118 q^{53} - 162 q^{54} + 1400 q^{55} + 272 q^{56} - 645 q^{57} + 32 q^{58} - 936 q^{59} + 24 q^{60} + 399 q^{61} + 214 q^{62} + 153 q^{63} + 384 q^{64} + 740 q^{65} + 312 q^{66} - 61 q^{67} - 384 q^{68} + 1428 q^{69} - 588 q^{70} - 974 q^{71} + 216 q^{72} - 91 q^{73} + 610 q^{74} - 1374 q^{75} - 1276 q^{76} - 72 q^{77} - 450 q^{78} + 321 q^{79} - 32 q^{80} - 243 q^{81} + 32 q^{82} - 4296 q^{83} + 408 q^{84} + 1680 q^{85} - 662 q^{86} + 48 q^{87} + 832 q^{88} - 1116 q^{89} + 36 q^{90} - 1367 q^{91} + 952 q^{92} + 321 q^{93} + 3064 q^{94} - 4198 q^{95} + 576 q^{96} - 1382 q^{97} + 2284 q^{98} + 468 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} + 8x^{4} + 5x^{3} + 50x^{2} - 7x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 10\nu^{5} - 80\nu^{4} + 116\nu^{3} - 500\nu^{2} + 70\nu - 2525 ) / 131 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -52\nu^{5} + 23\nu^{4} - 184\nu^{3} - 544\nu^{2} - 1936\nu + 161 ) / 393 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -56\nu^{5} + 55\nu^{4} - 440\nu^{3} - 344\nu^{2} - 2750\nu - 8 ) / 393 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -58\nu^{5} + 71\nu^{4} - 568\nu^{3} - 244\nu^{2} - 1978\nu - 289 ) / 393 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1036\nu^{5} + 821\nu^{4} - 8140\nu^{3} - 6364\nu^{2} - 52840\nu - 148 ) / 393 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( 2\beta_{4} - 3\beta_{3} + \beta_{2} + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{5} + \beta_{4} - 60\beta_{3} + 2\beta_{2} - 3\beta _1 - 58 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{4} + 5\beta_{2} - \beta _1 - 25 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{5} - 10\beta_{4} + 126\beta_{3} - 5\beta_{2} - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -21\beta_{5} - 62\beta_{4} + 537\beta_{3} - 124\beta_{2} + 21\beta _1 + 413 ) / 6 \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(\beta_{3}\)

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
0.0702177 0.121621i
−1.13654 + 1.96854i
1.56632 2.71294i
0.0702177 + 0.121621i
−1.13654 1.96854i
1.56632 + 2.71294i
1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i −10.1010 17.4954i −3.00000 + 5.19615i −22.8872 −8.00000 −4.50000 + 7.79423i 20.2020 34.9909i
7.2 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 2.60679 + 4.51510i −3.00000 + 5.19615i 31.4905 −8.00000 −4.50000 + 7.79423i −5.21359 + 9.03020i
7.3 1.00000 + 1.73205i 1.50000 + 2.59808i −2.00000 + 3.46410i 6.49420 + 11.2483i −3.00000 + 5.19615i −25.6033 −8.00000 −4.50000 + 7.79423i −12.9884 + 22.4966i
49.1 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i −10.1010 + 17.4954i −3.00000 5.19615i −22.8872 −8.00000 −4.50000 7.79423i 20.2020 + 34.9909i
49.2 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 2.60679 4.51510i −3.00000 5.19615i 31.4905 −8.00000 −4.50000 7.79423i −5.21359 9.03020i
49.3 1.00000 1.73205i 1.50000 2.59808i −2.00000 3.46410i 6.49420 11.2483i −3.00000 5.19615i −25.6033 −8.00000 −4.50000 7.79423i −12.9884 22.4966i
\(n\): 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.e 6
3.b odd 2 1 342.4.g.g 6
19.c even 3 1 inner 114.4.e.e 6
19.c even 3 1 2166.4.a.s 3
19.d odd 6 1 2166.4.a.w 3
57.h odd 6 1 342.4.g.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.e 6 1.a even 1 1 trivial
114.4.e.e 6 19.c even 3 1 inner
342.4.g.g 6 3.b odd 2 1
342.4.g.g 6 57.h odd 6 1
2166.4.a.s 3 19.c even 3 1
2166.4.a.w 3 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 2T_{5}^{5} + 304T_{5}^{4} - 3336T_{5}^{3} + 87264T_{5}^{2} - 410400T_{5} + 1871424 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( (T^{2} - 3 T + 9)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} + 2 T^{5} + 304 T^{4} + \cdots + 1871424 \) Copy content Toggle raw display
$7$ \( (T^{3} + 17 T^{2} - 941 T - 18453)^{2} \) Copy content Toggle raw display
$11$ \( (T^{3} + 52 T^{2} - 888 T - 32688)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 75 T^{5} + \cdots + 174424849 \) Copy content Toggle raw display
$17$ \( T^{6} - 48 T^{5} + \cdots + 107495424 \) Copy content Toggle raw display
$19$ \( T^{6} - 104 T^{5} + \cdots + 322687697779 \) Copy content Toggle raw display
$23$ \( T^{6} - 238 T^{5} + \cdots + 37266922552896 \) Copy content Toggle raw display
$29$ \( T^{6} - 8 T^{5} + \cdots + 93900570624 \) Copy content Toggle raw display
$31$ \( (T^{3} - 107 T^{2} - 14005 T + 1435247)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 305 T^{2} - 125173 T + 28900349)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} + 16 T^{5} + \cdots + 49106682003456 \) Copy content Toggle raw display
$43$ \( T^{6} - 331 T^{5} + \cdots + 9744392803201 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 407898773822016 \) Copy content Toggle raw display
$53$ \( T^{6} - 118 T^{5} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{6} + 936 T^{5} + \cdots + 10\!\cdots\!44 \) Copy content Toggle raw display
$61$ \( T^{6} - 399 T^{5} + \cdots + 61\!\cdots\!69 \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 762088088919249 \) Copy content Toggle raw display
$71$ \( T^{6} + \cdots + 288657109767744 \) Copy content Toggle raw display
$73$ \( T^{6} + 91 T^{5} + \cdots + 27\!\cdots\!21 \) Copy content Toggle raw display
$79$ \( T^{6} - 321 T^{5} + \cdots + 40\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( (T^{3} + 2148 T^{2} + 1271664 T + 211415616)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} + \cdots + 121838150433024 \) Copy content Toggle raw display
$97$ \( T^{6} + 1382 T^{5} + \cdots + 62\!\cdots\!44 \) Copy content Toggle raw display
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