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: \(x^{2} - x + 1\)
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.

\(f(q)\) \(=\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( -7 + 7 \zeta_{6} ) q^{5} + 2 q^{6} + ( -19 + 18 \zeta_{6} ) q^{7} -8 q^{8} + ( 26 - 26 \zeta_{6} ) q^{9} +O(q^{10})\) \( q + ( 2 - 2 \zeta_{6} ) q^{2} + \zeta_{6} q^{3} -4 \zeta_{6} q^{4} + ( -7 + 7 \zeta_{6} ) q^{5} + 2 q^{6} + ( -19 + 18 \zeta_{6} ) q^{7} -8 q^{8} + ( 26 - 26 \zeta_{6} ) q^{9} + 14 \zeta_{6} q^{10} -35 \zeta_{6} q^{11} + ( 4 - 4 \zeta_{6} ) q^{12} + 66 q^{13} + ( -2 + 38 \zeta_{6} ) q^{14} -7 q^{15} + ( -16 + 16 \zeta_{6} ) q^{16} -59 \zeta_{6} q^{17} -52 \zeta_{6} q^{18} + ( -137 + 137 \zeta_{6} ) q^{19} + 28 q^{20} + ( -18 - \zeta_{6} ) q^{21} -70 q^{22} + ( 7 - 7 \zeta_{6} ) q^{23} -8 \zeta_{6} q^{24} + 76 \zeta_{6} q^{25} + ( 132 - 132 \zeta_{6} ) q^{26} + 53 q^{27} + ( 72 + 4 \zeta_{6} ) q^{28} + 106 q^{29} + ( -14 + 14 \zeta_{6} ) q^{30} -75 \zeta_{6} q^{31} + 32 \zeta_{6} q^{32} + ( 35 - 35 \zeta_{6} ) q^{33} -118 q^{34} + ( 7 - 133 \zeta_{6} ) q^{35} -104 q^{36} + ( -11 + 11 \zeta_{6} ) q^{37} + 274 \zeta_{6} q^{38} + 66 \zeta_{6} q^{39} + ( 56 - 56 \zeta_{6} ) q^{40} -498 q^{41} + ( -38 + 36 \zeta_{6} ) q^{42} + 260 q^{43} + ( -140 + 140 \zeta_{6} ) q^{44} + 182 \zeta_{6} q^{45} -14 \zeta_{6} q^{46} + ( 171 - 171 \zeta_{6} ) q^{47} -16 q^{48} + ( 37 - 360 \zeta_{6} ) q^{49} + 152 q^{50} + ( 59 - 59 \zeta_{6} ) q^{51} -264 \zeta_{6} q^{52} + 417 \zeta_{6} q^{53} + ( 106 - 106 \zeta_{6} ) q^{54} + 245 q^{55} + ( 152 - 144 \zeta_{6} ) q^{56} -137 q^{57} + ( 212 - 212 \zeta_{6} ) q^{58} + 17 \zeta_{6} q^{59} + 28 \zeta_{6} q^{60} + ( -51 + 51 \zeta_{6} ) q^{61} -150 q^{62} + ( -26 + 494 \zeta_{6} ) q^{63} + 64 q^{64} + ( -462 + 462 \zeta_{6} ) q^{65} -70 \zeta_{6} q^{66} -439 \zeta_{6} q^{67} + ( -236 + 236 \zeta_{6} ) q^{68} + 7 q^{69} + ( -252 - 14 \zeta_{6} ) q^{70} -784 q^{71} + ( -208 + 208 \zeta_{6} ) q^{72} -295 \zeta_{6} q^{73} + 22 \zeta_{6} q^{74} + ( -76 + 76 \zeta_{6} ) q^{75} + 548 q^{76} + ( 630 + 35 \zeta_{6} ) q^{77} + 132 q^{78} + ( 495 - 495 \zeta_{6} ) q^{79} -112 \zeta_{6} q^{80} -649 \zeta_{6} q^{81} + ( -996 + 996 \zeta_{6} ) q^{82} + 932 q^{83} + ( -4 + 76 \zeta_{6} ) q^{84} + 413 q^{85} + ( 520 - 520 \zeta_{6} ) q^{86} + 106 \zeta_{6} q^{87} + 280 \zeta_{6} q^{88} + ( 873 - 873 \zeta_{6} ) q^{89} + 364 q^{90} + ( -1254 + 1188 \zeta_{6} ) q^{91} -28 q^{92} + ( 75 - 75 \zeta_{6} ) q^{93} -342 \zeta_{6} q^{94} -959 \zeta_{6} q^{95} + ( -32 + 32 \zeta_{6} ) q^{96} -290 q^{97} + ( -646 - 74 \zeta_{6} ) q^{98} -910 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + q^{3} - 4q^{4} - 7q^{5} + 4q^{6} - 20q^{7} - 16q^{8} + 26q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + q^{3} - 4q^{4} - 7q^{5} + 4q^{6} - 20q^{7} - 16q^{8} + 26q^{9} + 14q^{10} - 35q^{11} + 4q^{12} + 132q^{13} + 34q^{14} - 14q^{15} - 16q^{16} - 59q^{17} - 52q^{18} - 137q^{19} + 56q^{20} - 37q^{21} - 140q^{22} + 7q^{23} - 8q^{24} + 76q^{25} + 132q^{26} + 106q^{27} + 148q^{28} + 212q^{29} - 14q^{30} - 75q^{31} + 32q^{32} + 35q^{33} - 236q^{34} - 119q^{35} - 208q^{36} - 11q^{37} + 274q^{38} + 66q^{39} + 56q^{40} - 996q^{41} - 40q^{42} + 520q^{43} - 140q^{44} + 182q^{45} - 14q^{46} + 171q^{47} - 32q^{48} - 286q^{49} + 304q^{50} + 59q^{51} - 264q^{52} + 417q^{53} + 106q^{54} + 490q^{55} + 160q^{56} - 274q^{57} + 212q^{58} + 17q^{59} + 28q^{60} - 51q^{61} - 300q^{62} + 442q^{63} + 128q^{64} - 462q^{65} - 70q^{66} - 439q^{67} - 236q^{68} + 14q^{69} - 518q^{70} - 1568q^{71} - 208q^{72} - 295q^{73} + 22q^{74} - 76q^{75} + 1096q^{76} + 1295q^{77} + 264q^{78} + 495q^{79} - 112q^{80} - 649q^{81} - 996q^{82} + 1864q^{83} + 68q^{84} + 826q^{85} + 520q^{86} + 106q^{87} + 280q^{88} + 873q^{89} + 728q^{90} - 1320q^{91} - 56q^{92} + 75q^{93} - 342q^{94} - 959q^{95} - 32q^{96} - 580q^{97} - 1366q^{98} - 1820q^{99} + O(q^{100}) \)

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 \(S_{4}^{\mathrm{new}}(14, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 2 T + 4 T^{2} \)
$3$ \( 1 - T - 26 T^{2} - 27 T^{3} + 729 T^{4} \)
$5$ \( 1 + 7 T - 76 T^{2} + 875 T^{3} + 15625 T^{4} \)
$7$ \( 1 + 20 T + 343 T^{2} \)
$11$ \( 1 + 35 T - 106 T^{2} + 46585 T^{3} + 1771561 T^{4} \)
$13$ \( ( 1 - 66 T + 2197 T^{2} )^{2} \)
$17$ \( 1 + 59 T - 1432 T^{2} + 289867 T^{3} + 24137569 T^{4} \)
$19$ \( 1 + 137 T + 11910 T^{2} + 939683 T^{3} + 47045881 T^{4} \)
$23$ \( 1 - 7 T - 12118 T^{2} - 85169 T^{3} + 148035889 T^{4} \)
$29$ \( ( 1 - 106 T + 24389 T^{2} )^{2} \)
$31$ \( 1 + 75 T - 24166 T^{2} + 2234325 T^{3} + 887503681 T^{4} \)
$37$ \( 1 + 11 T - 50532 T^{2} + 557183 T^{3} + 2565726409 T^{4} \)
$41$ \( ( 1 + 498 T + 68921 T^{2} )^{2} \)
$43$ \( ( 1 - 260 T + 79507 T^{2} )^{2} \)
$47$ \( 1 - 171 T - 74582 T^{2} - 17753733 T^{3} + 10779215329 T^{4} \)
$53$ \( 1 - 417 T + 25012 T^{2} - 62081709 T^{3} + 22164361129 T^{4} \)
$59$ \( 1 - 17 T - 205090 T^{2} - 3491443 T^{3} + 42180533641 T^{4} \)
$61$ \( 1 + 51 T - 224380 T^{2} + 11576031 T^{3} + 51520374361 T^{4} \)
$67$ \( 1 + 439 T - 108042 T^{2} + 132034957 T^{3} + 90458382169 T^{4} \)
$71$ \( ( 1 + 784 T + 357911 T^{2} )^{2} \)
$73$ \( 1 + 295 T - 301992 T^{2} + 114760015 T^{3} + 151334226289 T^{4} \)
$79$ \( 1 - 495 T - 248014 T^{2} - 244054305 T^{3} + 243087455521 T^{4} \)
$83$ \( ( 1 - 932 T + 571787 T^{2} )^{2} \)
$89$ \( 1 - 873 T + 57160 T^{2} - 615437937 T^{3} + 496981290961 T^{4} \)
$97$ \( ( 1 + 290 T + 912673 T^{2} )^{2} \)
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