Properties

Label 14.4.c.b
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.

\(f(q)\) \(=\) \( q + ( - 2 \zeta_{6} + 2) q^{2} + \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + (7 \zeta_{6} - 7) q^{5} + 2 q^{6} + (18 \zeta_{6} - 19) q^{7} - 8 q^{8} + ( - 26 \zeta_{6} + 26) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \zeta_{6} + 2) q^{2} + \zeta_{6} q^{3} - 4 \zeta_{6} q^{4} + (7 \zeta_{6} - 7) q^{5} + 2 q^{6} + (18 \zeta_{6} - 19) q^{7} - 8 q^{8} + ( - 26 \zeta_{6} + 26) q^{9} + 14 \zeta_{6} q^{10} - 35 \zeta_{6} q^{11} + ( - 4 \zeta_{6} + 4) q^{12} + 66 q^{13} + (38 \zeta_{6} - 2) q^{14} - 7 q^{15} + (16 \zeta_{6} - 16) q^{16} - 59 \zeta_{6} q^{17} - 52 \zeta_{6} q^{18} + (137 \zeta_{6} - 137) q^{19} + 28 q^{20} + ( - \zeta_{6} - 18) q^{21} - 70 q^{22} + ( - 7 \zeta_{6} + 7) q^{23} - 8 \zeta_{6} q^{24} + 76 \zeta_{6} q^{25} + ( - 132 \zeta_{6} + 132) q^{26} + 53 q^{27} + (4 \zeta_{6} + 72) q^{28} + 106 q^{29} + (14 \zeta_{6} - 14) q^{30} - 75 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( - 35 \zeta_{6} + 35) q^{33} - 118 q^{34} + ( - 133 \zeta_{6} + 7) q^{35} - 104 q^{36} + (11 \zeta_{6} - 11) q^{37} + 274 \zeta_{6} q^{38} + 66 \zeta_{6} q^{39} + ( - 56 \zeta_{6} + 56) q^{40} - 498 q^{41} + (36 \zeta_{6} - 38) q^{42} + 260 q^{43} + (140 \zeta_{6} - 140) q^{44} + 182 \zeta_{6} q^{45} - 14 \zeta_{6} q^{46} + ( - 171 \zeta_{6} + 171) q^{47} - 16 q^{48} + ( - 360 \zeta_{6} + 37) q^{49} + 152 q^{50} + ( - 59 \zeta_{6} + 59) q^{51} - 264 \zeta_{6} q^{52} + 417 \zeta_{6} q^{53} + ( - 106 \zeta_{6} + 106) q^{54} + 245 q^{55} + ( - 144 \zeta_{6} + 152) q^{56} - 137 q^{57} + ( - 212 \zeta_{6} + 212) q^{58} + 17 \zeta_{6} q^{59} + 28 \zeta_{6} q^{60} + (51 \zeta_{6} - 51) q^{61} - 150 q^{62} + (494 \zeta_{6} - 26) q^{63} + 64 q^{64} + (462 \zeta_{6} - 462) q^{65} - 70 \zeta_{6} q^{66} - 439 \zeta_{6} q^{67} + (236 \zeta_{6} - 236) q^{68} + 7 q^{69} + ( - 14 \zeta_{6} - 252) q^{70} - 784 q^{71} + (208 \zeta_{6} - 208) q^{72} - 295 \zeta_{6} q^{73} + 22 \zeta_{6} q^{74} + (76 \zeta_{6} - 76) q^{75} + 548 q^{76} + (35 \zeta_{6} + 630) q^{77} + 132 q^{78} + ( - 495 \zeta_{6} + 495) q^{79} - 112 \zeta_{6} q^{80} - 649 \zeta_{6} q^{81} + (996 \zeta_{6} - 996) q^{82} + 932 q^{83} + (76 \zeta_{6} - 4) q^{84} + 413 q^{85} + ( - 520 \zeta_{6} + 520) q^{86} + 106 \zeta_{6} q^{87} + 280 \zeta_{6} q^{88} + ( - 873 \zeta_{6} + 873) q^{89} + 364 q^{90} + (1188 \zeta_{6} - 1254) q^{91} - 28 q^{92} + ( - 75 \zeta_{6} + 75) q^{93} - 342 \zeta_{6} q^{94} - 959 \zeta_{6} q^{95} + (32 \zeta_{6} - 32) q^{96} - 290 q^{97} + ( - 74 \zeta_{6} - 646) q^{98} - 910 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + q^{3} - 4 q^{4} - 7 q^{5} + 4 q^{6} - 20 q^{7} - 16 q^{8} + 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + q^{3} - 4 q^{4} - 7 q^{5} + 4 q^{6} - 20 q^{7} - 16 q^{8} + 26 q^{9} + 14 q^{10} - 35 q^{11} + 4 q^{12} + 132 q^{13} + 34 q^{14} - 14 q^{15} - 16 q^{16} - 59 q^{17} - 52 q^{18} - 137 q^{19} + 56 q^{20} - 37 q^{21} - 140 q^{22} + 7 q^{23} - 8 q^{24} + 76 q^{25} + 132 q^{26} + 106 q^{27} + 148 q^{28} + 212 q^{29} - 14 q^{30} - 75 q^{31} + 32 q^{32} + 35 q^{33} - 236 q^{34} - 119 q^{35} - 208 q^{36} - 11 q^{37} + 274 q^{38} + 66 q^{39} + 56 q^{40} - 996 q^{41} - 40 q^{42} + 520 q^{43} - 140 q^{44} + 182 q^{45} - 14 q^{46} + 171 q^{47} - 32 q^{48} - 286 q^{49} + 304 q^{50} + 59 q^{51} - 264 q^{52} + 417 q^{53} + 106 q^{54} + 490 q^{55} + 160 q^{56} - 274 q^{57} + 212 q^{58} + 17 q^{59} + 28 q^{60} - 51 q^{61} - 300 q^{62} + 442 q^{63} + 128 q^{64} - 462 q^{65} - 70 q^{66} - 439 q^{67} - 236 q^{68} + 14 q^{69} - 518 q^{70} - 1568 q^{71} - 208 q^{72} - 295 q^{73} + 22 q^{74} - 76 q^{75} + 1096 q^{76} + 1295 q^{77} + 264 q^{78} + 495 q^{79} - 112 q^{80} - 649 q^{81} - 996 q^{82} + 1864 q^{83} + 68 q^{84} + 826 q^{85} + 520 q^{86} + 106 q^{87} + 280 q^{88} + 873 q^{89} + 728 q^{90} - 1320 q^{91} - 56 q^{92} + 75 q^{93} - 342 q^{94} - 959 q^{95} - 32 q^{96} - 580 q^{97} - 1366 q^{98} - 1820 q^{99}+O(q^{100}) \) 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 0.500000 + 0.866025i −2.00000 3.46410i −3.50000 + 6.06218i 2.00000 −10.0000 + 15.5885i −8.00000 13.0000 22.5167i 7.00000 + 12.1244i
11.1 1.00000 + 1.73205i 0.500000 0.866025i −2.00000 + 3.46410i −3.50000 6.06218i 2.00000 −10.0000 15.5885i −8.00000 13.0000 + 22.5167i 7.00000 12.1244i
\(n\): 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.b 2
3.b odd 2 1 126.4.g.c 2
4.b odd 2 1 112.4.i.b 2
5.b even 2 1 350.4.e.b 2
5.c odd 4 2 350.4.j.d 4
7.b odd 2 1 98.4.c.e 2
7.c even 3 1 inner 14.4.c.b 2
7.c even 3 1 98.4.a.b 1
7.d odd 6 1 98.4.a.c 1
7.d odd 6 1 98.4.c.e 2
8.b even 2 1 448.4.i.c 2
8.d odd 2 1 448.4.i.d 2
21.c even 2 1 882.4.g.d 2
21.g even 6 1 882.4.a.p 1
21.g even 6 1 882.4.g.d 2
21.h odd 6 1 126.4.g.c 2
21.h odd 6 1 882.4.a.k 1
28.f even 6 1 784.4.a.j 1
28.g odd 6 1 112.4.i.b 2
28.g odd 6 1 784.4.a.l 1
35.i odd 6 1 2450.4.a.bf 1
35.j even 6 1 350.4.e.b 2
35.j even 6 1 2450.4.a.bh 1
35.l odd 12 2 350.4.j.d 4
56.k odd 6 1 448.4.i.d 2
56.p even 6 1 448.4.i.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.4.c.b 2 1.a even 1 1 trivial
14.4.c.b 2 7.c even 3 1 inner
98.4.a.b 1 7.c even 3 1
98.4.a.c 1 7.d odd 6 1
98.4.c.e 2 7.b odd 2 1
98.4.c.e 2 7.d odd 6 1
112.4.i.b 2 4.b odd 2 1
112.4.i.b 2 28.g odd 6 1
126.4.g.c 2 3.b odd 2 1
126.4.g.c 2 21.h odd 6 1
350.4.e.b 2 5.b even 2 1
350.4.e.b 2 35.j even 6 1
350.4.j.d 4 5.c odd 4 2
350.4.j.d 4 35.l odd 12 2
448.4.i.c 2 8.b even 2 1
448.4.i.c 2 56.p even 6 1
448.4.i.d 2 8.d odd 2 1
448.4.i.d 2 56.k odd 6 1
784.4.a.j 1 28.f even 6 1
784.4.a.l 1 28.g odd 6 1
882.4.a.k 1 21.h odd 6 1
882.4.a.p 1 21.g even 6 1
882.4.g.d 2 21.c even 2 1
882.4.g.d 2 21.g even 6 1
2450.4.a.bf 1 35.i odd 6 1
2450.4.a.bh 1 35.j even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} - T_{3} + 1 \) 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
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$7$ \( T^{2} + 20T + 343 \) Copy content Toggle raw display
$11$ \( T^{2} + 35T + 1225 \) Copy content Toggle raw display
$13$ \( (T - 66)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 59T + 3481 \) Copy content Toggle raw display
$19$ \( T^{2} + 137T + 18769 \) Copy content Toggle raw display
$23$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$29$ \( (T - 106)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 75T + 5625 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 498)^{2} \) Copy content Toggle raw display
$43$ \( (T - 260)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} - 171T + 29241 \) Copy content Toggle raw display
$53$ \( T^{2} - 417T + 173889 \) Copy content Toggle raw display
$59$ \( T^{2} - 17T + 289 \) Copy content Toggle raw display
$61$ \( T^{2} + 51T + 2601 \) Copy content Toggle raw display
$67$ \( T^{2} + 439T + 192721 \) Copy content Toggle raw display
$71$ \( (T + 784)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 295T + 87025 \) Copy content Toggle raw display
$79$ \( T^{2} - 495T + 245025 \) Copy content Toggle raw display
$83$ \( (T - 932)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} - 873T + 762129 \) Copy content Toggle raw display
$97$ \( (T + 290)^{2} \) Copy content Toggle raw display
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