Properties

Label 14.4.c.a
Level $14$
Weight $4$
Character orbit 14.c
Analytic conductor $0.826$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 14 = 2 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 14.c (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.826026740080\)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 2) q^{2} + 5 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 9 \zeta_{6} + 9) q^{5} - 10 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} + 8 q^{8} + ( - 2 \zeta_{6} + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 2) q^{2} + 5 \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + ( - 9 \zeta_{6} + 9) q^{5} - 10 q^{6} + ( - 14 \zeta_{6} - 7) q^{7} + 8 q^{8} + ( - 2 \zeta_{6} + 2) q^{9} + 18 \zeta_{6} q^{10} + 57 \zeta_{6} q^{11} + ( - 20 \zeta_{6} + 20) q^{12} - 70 q^{13} + ( - 14 \zeta_{6} + 42) q^{14} + 45 q^{15} + (16 \zeta_{6} - 16) q^{16} - 51 \zeta_{6} q^{17} + 4 \zeta_{6} q^{18} + (5 \zeta_{6} - 5) q^{19} - 36 q^{20} + ( - 105 \zeta_{6} + 70) q^{21} - 114 q^{22} + (69 \zeta_{6} - 69) q^{23} + 40 \zeta_{6} q^{24} + 44 \zeta_{6} q^{25} + ( - 140 \zeta_{6} + 140) q^{26} + 145 q^{27} + (84 \zeta_{6} - 56) q^{28} + 114 q^{29} + (90 \zeta_{6} - 90) q^{30} - 23 \zeta_{6} q^{31} - 32 \zeta_{6} q^{32} + (285 \zeta_{6} - 285) q^{33} + 102 q^{34} + (63 \zeta_{6} - 189) q^{35} - 8 q^{36} + ( - 253 \zeta_{6} + 253) q^{37} - 10 \zeta_{6} q^{38} - 350 \zeta_{6} q^{39} + ( - 72 \zeta_{6} + 72) q^{40} - 42 q^{41} + (140 \zeta_{6} + 70) q^{42} - 124 q^{43} + ( - 228 \zeta_{6} + 228) q^{44} - 18 \zeta_{6} q^{45} - 138 \zeta_{6} q^{46} + (201 \zeta_{6} - 201) q^{47} - 80 q^{48} + (392 \zeta_{6} - 147) q^{49} - 88 q^{50} + ( - 255 \zeta_{6} + 255) q^{51} + 280 \zeta_{6} q^{52} + 393 \zeta_{6} q^{53} + (290 \zeta_{6} - 290) q^{54} + 513 q^{55} + ( - 112 \zeta_{6} - 56) q^{56} - 25 q^{57} + (228 \zeta_{6} - 228) q^{58} - 219 \zeta_{6} q^{59} - 180 \zeta_{6} q^{60} + ( - 709 \zeta_{6} + 709) q^{61} + 46 q^{62} + (14 \zeta_{6} - 42) q^{63} + 64 q^{64} + (630 \zeta_{6} - 630) q^{65} - 570 \zeta_{6} q^{66} - 419 \zeta_{6} q^{67} + (204 \zeta_{6} - 204) q^{68} - 345 q^{69} + ( - 378 \zeta_{6} + 252) q^{70} - 96 q^{71} + ( - 16 \zeta_{6} + 16) q^{72} + 313 \zeta_{6} q^{73} + 506 \zeta_{6} q^{74} + (220 \zeta_{6} - 220) q^{75} + 20 q^{76} + ( - 1197 \zeta_{6} + 798) q^{77} + 700 q^{78} + (461 \zeta_{6} - 461) q^{79} + 144 \zeta_{6} q^{80} + 671 \zeta_{6} q^{81} + ( - 84 \zeta_{6} + 84) q^{82} - 588 q^{83} + (140 \zeta_{6} - 420) q^{84} - 459 q^{85} + ( - 248 \zeta_{6} + 248) q^{86} + 570 \zeta_{6} q^{87} + 456 \zeta_{6} q^{88} + ( - 1017 \zeta_{6} + 1017) q^{89} + 36 q^{90} + (980 \zeta_{6} + 490) q^{91} + 276 q^{92} + ( - 115 \zeta_{6} + 115) q^{93} - 402 \zeta_{6} q^{94} + 45 \zeta_{6} q^{95} + ( - 160 \zeta_{6} + 160) q^{96} - 1834 q^{97} + ( - 294 \zeta_{6} - 490) q^{98} + 114 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 5 q^{3} - 4 q^{4} + 9 q^{5} - 20 q^{6} - 28 q^{7} + 16 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 5 q^{3} - 4 q^{4} + 9 q^{5} - 20 q^{6} - 28 q^{7} + 16 q^{8} + 2 q^{9} + 18 q^{10} + 57 q^{11} + 20 q^{12} - 140 q^{13} + 70 q^{14} + 90 q^{15} - 16 q^{16} - 51 q^{17} + 4 q^{18} - 5 q^{19} - 72 q^{20} + 35 q^{21} - 228 q^{22} - 69 q^{23} + 40 q^{24} + 44 q^{25} + 140 q^{26} + 290 q^{27} - 28 q^{28} + 228 q^{29} - 90 q^{30} - 23 q^{31} - 32 q^{32} - 285 q^{33} + 204 q^{34} - 315 q^{35} - 16 q^{36} + 253 q^{37} - 10 q^{38} - 350 q^{39} + 72 q^{40} - 84 q^{41} + 280 q^{42} - 248 q^{43} + 228 q^{44} - 18 q^{45} - 138 q^{46} - 201 q^{47} - 160 q^{48} + 98 q^{49} - 176 q^{50} + 255 q^{51} + 280 q^{52} + 393 q^{53} - 290 q^{54} + 1026 q^{55} - 224 q^{56} - 50 q^{57} - 228 q^{58} - 219 q^{59} - 180 q^{60} + 709 q^{61} + 92 q^{62} - 70 q^{63} + 128 q^{64} - 630 q^{65} - 570 q^{66} - 419 q^{67} - 204 q^{68} - 690 q^{69} + 126 q^{70} - 192 q^{71} + 16 q^{72} + 313 q^{73} + 506 q^{74} - 220 q^{75} + 40 q^{76} + 399 q^{77} + 1400 q^{78} - 461 q^{79} + 144 q^{80} + 671 q^{81} + 84 q^{82} - 1176 q^{83} - 700 q^{84} - 918 q^{85} + 248 q^{86} + 570 q^{87} + 456 q^{88} + 1017 q^{89} + 72 q^{90} + 1960 q^{91} + 552 q^{92} + 115 q^{93} - 402 q^{94} + 45 q^{95} + 160 q^{96} - 3668 q^{97} - 1274 q^{98} + 228 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/14\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 + \zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
0.500000 + 0.866025i
0.500000 0.866025i
−1.00000 + 1.73205i 2.50000 + 4.33013i −2.00000 3.46410i 4.50000 7.79423i −10.0000 −14.0000 12.1244i 8.00000 1.00000 1.73205i 9.00000 + 15.5885i
11.1 −1.00000 1.73205i 2.50000 4.33013i −2.00000 + 3.46410i 4.50000 + 7.79423i −10.0000 −14.0000 + 12.1244i 8.00000 1.00000 + 1.73205i 9.00000 15.5885i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.4.c.a 2
3.b odd 2 1 126.4.g.d 2
4.b odd 2 1 112.4.i.a 2
5.b even 2 1 350.4.e.e 2
5.c odd 4 2 350.4.j.b 4
7.b odd 2 1 98.4.c.a 2
7.c even 3 1 inner 14.4.c.a 2
7.c even 3 1 98.4.a.d 1
7.d odd 6 1 98.4.a.f 1
7.d odd 6 1 98.4.c.a 2
8.b even 2 1 448.4.i.b 2
8.d odd 2 1 448.4.i.e 2
21.c even 2 1 882.4.g.u 2
21.g even 6 1 882.4.a.c 1
21.g even 6 1 882.4.g.u 2
21.h odd 6 1 126.4.g.d 2
21.h odd 6 1 882.4.a.f 1
28.f even 6 1 784.4.a.c 1
28.g odd 6 1 112.4.i.a 2
28.g odd 6 1 784.4.a.p 1
35.i odd 6 1 2450.4.a.d 1
35.j even 6 1 350.4.e.e 2
35.j even 6 1 2450.4.a.q 1
35.l odd 12 2 350.4.j.b 4
56.k odd 6 1 448.4.i.e 2
56.p even 6 1 448.4.i.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.a 2 1.a even 1 1 trivial
14.4.c.a 2 7.c even 3 1 inner
98.4.a.d 1 7.c even 3 1
98.4.a.f 1 7.d odd 6 1
98.4.c.a 2 7.b odd 2 1
98.4.c.a 2 7.d odd 6 1
112.4.i.a 2 4.b odd 2 1
112.4.i.a 2 28.g odd 6 1
126.4.g.d 2 3.b odd 2 1
126.4.g.d 2 21.h odd 6 1
350.4.e.e 2 5.b even 2 1
350.4.e.e 2 35.j even 6 1
350.4.j.b 4 5.c odd 4 2
350.4.j.b 4 35.l odd 12 2
448.4.i.b 2 8.b even 2 1
448.4.i.b 2 56.p even 6 1
448.4.i.e 2 8.d odd 2 1
448.4.i.e 2 56.k odd 6 1
784.4.a.c 1 28.f even 6 1
784.4.a.p 1 28.g odd 6 1
882.4.a.c 1 21.g even 6 1
882.4.a.f 1 21.h odd 6 1
882.4.g.u 2 21.c even 2 1
882.4.g.u 2 21.g even 6 1
2450.4.a.d 1 35.i odd 6 1
2450.4.a.q 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - 5T_{3} + 25 \) acting on \(S_{4}^{\mathrm{new}}(14, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$3$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$5$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$7$ \( T^{2} + 28T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} - 57T + 3249 \) Copy content Toggle raw display
$13$ \( (T + 70)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 69T + 4761 \) Copy content Toggle raw display
$29$ \( (T - 114)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 23T + 529 \) Copy content Toggle raw display
$37$ \( T^{2} - 253T + 64009 \) Copy content Toggle raw display
$41$ \( (T + 42)^{2} \) Copy content Toggle raw display
$43$ \( (T + 124)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 201T + 40401 \) Copy content Toggle raw display
$53$ \( T^{2} - 393T + 154449 \) Copy content Toggle raw display
$59$ \( T^{2} + 219T + 47961 \) Copy content Toggle raw display
$61$ \( T^{2} - 709T + 502681 \) Copy content Toggle raw display
$67$ \( T^{2} + 419T + 175561 \) Copy content Toggle raw display
$71$ \( (T + 96)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} - 313T + 97969 \) Copy content Toggle raw display
$79$ \( T^{2} + 461T + 212521 \) Copy content Toggle raw display
$83$ \( (T + 588)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 1017 T + 1034289 \) Copy content Toggle raw display
$97$ \( (T + 1834)^{2} \) Copy content Toggle raw display
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