Properties

Label 114.4.e.b
Level $114$
Weight $4$
Character orbit 114.e
Analytic conductor $6.726$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 6) q^{5} + 6 \zeta_{6} q^{6} + 19 q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} + (3 \zeta_{6} - 3) q^{3} - 4 \zeta_{6} q^{4} + ( - 6 \zeta_{6} + 6) q^{5} + 6 \zeta_{6} q^{6} + 19 q^{7} - 8 q^{8} - 9 \zeta_{6} q^{9} - 12 \zeta_{6} q^{10} + 32 q^{11} + 12 q^{12} - 81 \zeta_{6} q^{13} + ( - 38 \zeta_{6} + 38) q^{14} + 18 \zeta_{6} q^{15} + (16 \zeta_{6} - 16) q^{16} + ( - 124 \zeta_{6} + 124) q^{17} - 18 q^{18} + (38 \zeta_{6} + 57) q^{19} - 24 q^{20} + (57 \zeta_{6} - 57) q^{21} + ( - 64 \zeta_{6} + 64) q^{22} - 98 \zeta_{6} q^{23} + ( - 24 \zeta_{6} + 24) q^{24} + 89 \zeta_{6} q^{25} - 162 q^{26} + 27 q^{27} - 76 \zeta_{6} q^{28} + 300 \zeta_{6} q^{29} + 36 q^{30} - 225 q^{31} + 32 \zeta_{6} q^{32} + (96 \zeta_{6} - 96) q^{33} - 248 \zeta_{6} q^{34} + ( - 114 \zeta_{6} + 114) q^{35} + (36 \zeta_{6} - 36) q^{36} - 293 q^{37} + ( - 114 \zeta_{6} + 190) q^{38} + 243 q^{39} + (48 \zeta_{6} - 48) q^{40} + (176 \zeta_{6} - 176) q^{41} + 114 \zeta_{6} q^{42} + ( - 111 \zeta_{6} + 111) q^{43} - 128 \zeta_{6} q^{44} - 54 q^{45} - 196 q^{46} + 550 \zeta_{6} q^{47} - 48 \zeta_{6} q^{48} + 18 q^{49} + 178 q^{50} + 372 \zeta_{6} q^{51} + (324 \zeta_{6} - 324) q^{52} + 482 \zeta_{6} q^{53} + ( - 54 \zeta_{6} + 54) q^{54} + ( - 192 \zeta_{6} + 192) q^{55} - 152 q^{56} + (171 \zeta_{6} - 285) q^{57} + 600 q^{58} + ( - 496 \zeta_{6} + 496) q^{59} + ( - 72 \zeta_{6} + 72) q^{60} - 155 \zeta_{6} q^{61} + (450 \zeta_{6} - 450) q^{62} - 171 \zeta_{6} q^{63} + 64 q^{64} - 486 q^{65} + 192 \zeta_{6} q^{66} - 465 \zeta_{6} q^{67} - 496 q^{68} + 294 q^{69} - 228 \zeta_{6} q^{70} + ( - 110 \zeta_{6} + 110) q^{71} + 72 \zeta_{6} q^{72} + (817 \zeta_{6} - 817) q^{73} + (586 \zeta_{6} - 586) q^{74} - 267 q^{75} + ( - 380 \zeta_{6} + 152) q^{76} + 608 q^{77} + ( - 486 \zeta_{6} + 486) q^{78} + (259 \zeta_{6} - 259) q^{79} + 96 \zeta_{6} q^{80} + (81 \zeta_{6} - 81) q^{81} + 352 \zeta_{6} q^{82} - 56 q^{83} + 228 q^{84} - 744 \zeta_{6} q^{85} - 222 \zeta_{6} q^{86} - 900 q^{87} - 256 q^{88} - 308 \zeta_{6} q^{89} + (108 \zeta_{6} - 108) q^{90} - 1539 \zeta_{6} q^{91} + (392 \zeta_{6} - 392) q^{92} + ( - 675 \zeta_{6} + 675) q^{93} + 1100 q^{94} + ( - 342 \zeta_{6} + 570) q^{95} - 96 q^{96} + ( - 1150 \zeta_{6} + 1150) q^{97} + ( - 36 \zeta_{6} + 36) q^{98} - 288 \zeta_{6} q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} + 6 q^{6} + 38 q^{7} - 16 q^{8} - 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 3 q^{3} - 4 q^{4} + 6 q^{5} + 6 q^{6} + 38 q^{7} - 16 q^{8} - 9 q^{9} - 12 q^{10} + 64 q^{11} + 24 q^{12} - 81 q^{13} + 38 q^{14} + 18 q^{15} - 16 q^{16} + 124 q^{17} - 36 q^{18} + 152 q^{19} - 48 q^{20} - 57 q^{21} + 64 q^{22} - 98 q^{23} + 24 q^{24} + 89 q^{25} - 324 q^{26} + 54 q^{27} - 76 q^{28} + 300 q^{29} + 72 q^{30} - 450 q^{31} + 32 q^{32} - 96 q^{33} - 248 q^{34} + 114 q^{35} - 36 q^{36} - 586 q^{37} + 266 q^{38} + 486 q^{39} - 48 q^{40} - 176 q^{41} + 114 q^{42} + 111 q^{43} - 128 q^{44} - 108 q^{45} - 392 q^{46} + 550 q^{47} - 48 q^{48} + 36 q^{49} + 356 q^{50} + 372 q^{51} - 324 q^{52} + 482 q^{53} + 54 q^{54} + 192 q^{55} - 304 q^{56} - 399 q^{57} + 1200 q^{58} + 496 q^{59} + 72 q^{60} - 155 q^{61} - 450 q^{62} - 171 q^{63} + 128 q^{64} - 972 q^{65} + 192 q^{66} - 465 q^{67} - 992 q^{68} + 588 q^{69} - 228 q^{70} + 110 q^{71} + 72 q^{72} - 817 q^{73} - 586 q^{74} - 534 q^{75} - 76 q^{76} + 1216 q^{77} + 486 q^{78} - 259 q^{79} + 96 q^{80} - 81 q^{81} + 352 q^{82} - 112 q^{83} + 456 q^{84} - 744 q^{85} - 222 q^{86} - 1800 q^{87} - 512 q^{88} - 308 q^{89} - 108 q^{90} - 1539 q^{91} - 392 q^{92} + 675 q^{93} + 2200 q^{94} + 798 q^{95} - 192 q^{96} + 1150 q^{97} + 36 q^{98} - 288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(77\) \(97\)
\(\chi(n)\) \(1\) \(-\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
0.500000 0.866025i
0.500000 + 0.866025i
1.00000 + 1.73205i −1.50000 2.59808i −2.00000 + 3.46410i 3.00000 + 5.19615i 3.00000 5.19615i 19.0000 −8.00000 −4.50000 + 7.79423i −6.00000 + 10.3923i
49.1 1.00000 1.73205i −1.50000 + 2.59808i −2.00000 3.46410i 3.00000 5.19615i 3.00000 + 5.19615i 19.0000 −8.00000 −4.50000 7.79423i −6.00000 10.3923i
\(n\): e.g. 2-40 or 990-1000
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.b 2
3.b odd 2 1 342.4.g.a 2
19.c even 3 1 inner 114.4.e.b 2
19.c even 3 1 2166.4.a.b 1
19.d odd 6 1 2166.4.a.e 1
57.h odd 6 1 342.4.g.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.4.e.b 2 1.a even 1 1 trivial
114.4.e.b 2 19.c even 3 1 inner
342.4.g.a 2 3.b odd 2 1
342.4.g.a 2 57.h odd 6 1
2166.4.a.b 1 19.c even 3 1
2166.4.a.e 1 19.d odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 6T_{5} + 36 \) acting on \(S_{4}^{\mathrm{new}}(114, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$5$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$7$ \( (T - 19)^{2} \) Copy content Toggle raw display
$11$ \( (T - 32)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 81T + 6561 \) Copy content Toggle raw display
$17$ \( T^{2} - 124T + 15376 \) Copy content Toggle raw display
$19$ \( T^{2} - 152T + 6859 \) Copy content Toggle raw display
$23$ \( T^{2} + 98T + 9604 \) Copy content Toggle raw display
$29$ \( T^{2} - 300T + 90000 \) Copy content Toggle raw display
$31$ \( (T + 225)^{2} \) Copy content Toggle raw display
$37$ \( (T + 293)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 176T + 30976 \) Copy content Toggle raw display
$43$ \( T^{2} - 111T + 12321 \) Copy content Toggle raw display
$47$ \( T^{2} - 550T + 302500 \) Copy content Toggle raw display
$53$ \( T^{2} - 482T + 232324 \) Copy content Toggle raw display
$59$ \( T^{2} - 496T + 246016 \) Copy content Toggle raw display
$61$ \( T^{2} + 155T + 24025 \) Copy content Toggle raw display
$67$ \( T^{2} + 465T + 216225 \) Copy content Toggle raw display
$71$ \( T^{2} - 110T + 12100 \) Copy content Toggle raw display
$73$ \( T^{2} + 817T + 667489 \) Copy content Toggle raw display
$79$ \( T^{2} + 259T + 67081 \) Copy content Toggle raw display
$83$ \( (T + 56)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 308T + 94864 \) Copy content Toggle raw display
$97$ \( T^{2} - 1150 T + 1322500 \) Copy content Toggle raw display
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