# Properties

 Label 14.6.c.a Level $14$ Weight $6$ Character orbit 14.c Analytic conductor $2.245$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 14.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.24537347738$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{79})$$ Defining polynomial: $$x^{4} + 79 x^{2} + 6241$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 4 \beta_{2} q^{2} + ( -7 - \beta_{1} - 7 \beta_{2} ) q^{3} + ( -16 - 16 \beta_{2} ) q^{4} + ( -4 \beta_{1} + 35 \beta_{2} - 4 \beta_{3} ) q^{5} + ( 28 - 4 \beta_{3} ) q^{6} + ( -21 - 42 \beta_{2} + 7 \beta_{3} ) q^{7} + 64 q^{8} + ( 14 \beta_{1} + 122 \beta_{2} + 14 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + 4 \beta_{2} q^{2} + ( -7 - \beta_{1} - 7 \beta_{2} ) q^{3} + ( -16 - 16 \beta_{2} ) q^{4} + ( -4 \beta_{1} + 35 \beta_{2} - 4 \beta_{3} ) q^{5} + ( 28 - 4 \beta_{3} ) q^{6} + ( -21 - 42 \beta_{2} + 7 \beta_{3} ) q^{7} + 64 q^{8} + ( 14 \beta_{1} + 122 \beta_{2} + 14 \beta_{3} ) q^{9} + ( -140 + 16 \beta_{1} - 140 \beta_{2} ) q^{10} + ( -31 - 7 \beta_{1} - 31 \beta_{2} ) q^{11} + ( 16 \beta_{1} + 112 \beta_{2} + 16 \beta_{3} ) q^{12} + ( 910 - 14 \beta_{3} ) q^{13} + ( 168 - 28 \beta_{1} + 84 \beta_{2} - 28 \beta_{3} ) q^{14} + ( -1019 - 7 \beta_{3} ) q^{15} + 256 \beta_{2} q^{16} + ( -847 - 22 \beta_{1} - 847 \beta_{2} ) q^{17} + ( -488 - 56 \beta_{1} - 488 \beta_{2} ) q^{18} + ( -39 \beta_{1} + 413 \beta_{2} - 39 \beta_{3} ) q^{19} + ( 560 + 64 \beta_{3} ) q^{20} + ( 2065 + 70 \beta_{1} + 2359 \beta_{2} + 42 \beta_{3} ) q^{21} + ( 124 - 28 \beta_{3} ) q^{22} + ( -119 \beta_{1} - 1367 \beta_{2} - 119 \beta_{3} ) q^{23} + ( -448 - 64 \beta_{1} - 448 \beta_{2} ) q^{24} + ( -3156 + 280 \beta_{1} - 3156 \beta_{2} ) q^{25} + ( 56 \beta_{1} + 3640 \beta_{2} + 56 \beta_{3} ) q^{26} + ( 3577 + 23 \beta_{3} ) q^{27} + ( -336 + 112 \beta_{1} + 336 \beta_{2} ) q^{28} + ( -1426 + 238 \beta_{3} ) q^{29} + ( 28 \beta_{1} - 4076 \beta_{2} + 28 \beta_{3} ) q^{30} + ( 1337 + 55 \beta_{1} + 1337 \beta_{2} ) q^{31} + ( -1024 - 1024 \beta_{2} ) q^{32} + ( 80 \beta_{1} + 2429 \beta_{2} + 80 \beta_{3} ) q^{33} + ( 3388 - 88 \beta_{3} ) q^{34} + ( 1470 - 329 \beta_{1} + 9583 \beta_{2} - 161 \beta_{3} ) q^{35} + ( 1952 - 224 \beta_{3} ) q^{36} + ( 126 \beta_{1} - 4573 \beta_{2} + 126 \beta_{3} ) q^{37} + ( -1652 + 156 \beta_{1} - 1652 \beta_{2} ) q^{38} + ( -10794 - 1008 \beta_{1} - 10794 \beta_{2} ) q^{39} + ( -256 \beta_{1} + 2240 \beta_{2} - 256 \beta_{3} ) q^{40} + ( 3066 - 42 \beta_{3} ) q^{41} + ( -9436 - 168 \beta_{1} - 1176 \beta_{2} + 112 \beta_{3} ) q^{42} + ( -8020 - 672 \beta_{3} ) q^{43} + ( 112 \beta_{1} + 496 \beta_{2} + 112 \beta_{3} ) q^{44} + ( 13426 - 2 \beta_{1} + 13426 \beta_{2} ) q^{45} + ( 5468 + 476 \beta_{1} + 5468 \beta_{2} ) q^{46} + ( 87 \beta_{1} - 12663 \beta_{2} + 87 \beta_{3} ) q^{47} + ( 1792 - 256 \beta_{3} ) q^{48} + ( 14161 + 588 \beta_{1} + 294 \beta_{3} ) q^{49} + ( 12624 + 1120 \beta_{3} ) q^{50} + ( 1001 \beta_{1} + 12881 \beta_{2} + 1001 \beta_{3} ) q^{51} + ( -14560 - 224 \beta_{1} - 14560 \beta_{2} ) q^{52} + ( -7479 + 1050 \beta_{1} - 7479 \beta_{2} ) q^{53} + ( -92 \beta_{1} + 14308 \beta_{2} - 92 \beta_{3} ) q^{54} + ( -7763 - 121 \beta_{3} ) q^{55} + ( -1344 - 2688 \beta_{2} + 448 \beta_{3} ) q^{56} + ( -9433 - 140 \beta_{3} ) q^{57} + ( -952 \beta_{1} - 5704 \beta_{2} - 952 \beta_{3} ) q^{58} + ( 553 + 307 \beta_{1} + 553 \beta_{2} ) q^{59} + ( 16304 - 112 \beta_{1} + 16304 \beta_{2} ) q^{60} + ( -46 \beta_{1} - 14021 \beta_{2} - 46 \beta_{3} ) q^{61} + ( -5348 + 220 \beta_{3} ) q^{62} + ( 5124 - 560 \beta_{1} - 28406 \beta_{2} - 1148 \beta_{3} ) q^{63} + 4096 q^{64} + ( -3150 \beta_{1} + 14154 \beta_{2} - 3150 \beta_{3} ) q^{65} + ( -9716 - 320 \beta_{1} - 9716 \beta_{2} ) q^{66} + ( 51321 - 469 \beta_{1} + 51321 \beta_{2} ) q^{67} + ( 352 \beta_{1} + 13552 \beta_{2} + 352 \beta_{3} ) q^{68} + ( -47173 + 2200 \beta_{3} ) q^{69} + ( -38332 + 644 \beta_{1} - 32452 \beta_{2} - 672 \beta_{3} ) q^{70} + ( -5528 + 812 \beta_{3} ) q^{71} + ( 896 \beta_{1} + 7808 \beta_{2} + 896 \beta_{3} ) q^{72} + ( -17535 - 1576 \beta_{1} - 17535 \beta_{2} ) q^{73} + ( 18292 - 504 \beta_{1} + 18292 \beta_{2} ) q^{74} + ( 1196 \beta_{1} - 66388 \beta_{2} + 1196 \beta_{3} ) q^{75} + ( 6608 + 624 \beta_{3} ) q^{76} + ( 14833 + 364 \beta_{1} + 16135 \beta_{2} + 294 \beta_{3} ) q^{77} + ( 43176 - 4032 \beta_{3} ) q^{78} + ( 3269 \beta_{1} + 50881 \beta_{2} + 3269 \beta_{3} ) q^{79} + ( -8960 + 1024 \beta_{1} - 8960 \beta_{2} ) q^{80} + ( 11875 - 14 \beta_{1} + 11875 \beta_{2} ) q^{81} + ( 168 \beta_{1} + 12264 \beta_{2} + 168 \beta_{3} ) q^{82} + ( -22316 - 4396 \beta_{3} ) q^{83} + ( 4704 - 448 \beta_{1} - 33040 \beta_{2} - 1120 \beta_{3} ) q^{84} + ( 1837 + 2618 \beta_{3} ) q^{85} + ( 2688 \beta_{1} - 32080 \beta_{2} + 2688 \beta_{3} ) q^{86} + ( 85190 + 3092 \beta_{1} + 85190 \beta_{2} ) q^{87} + ( -1984 - 448 \beta_{1} - 1984 \beta_{2} ) q^{88} + ( -3252 \beta_{1} - 37737 \beta_{2} - 3252 \beta_{3} ) q^{89} + ( -53704 - 8 \beta_{3} ) q^{90} + ( -50078 - 588 \beta_{1} - 38220 \beta_{2} + 6076 \beta_{3} ) q^{91} + ( -21872 + 1904 \beta_{3} ) q^{92} + ( -1722 \beta_{1} - 26739 \beta_{2} - 1722 \beta_{3} ) q^{93} + ( 50652 - 348 \beta_{1} + 50652 \beta_{2} ) q^{94} + ( -63751 + 3017 \beta_{1} - 63751 \beta_{2} ) q^{95} + ( 1024 \beta_{1} + 7168 \beta_{2} + 1024 \beta_{3} ) q^{96} + ( -4158 - 1078 \beta_{3} ) q^{97} + ( -1176 \beta_{1} + 56644 \beta_{2} + 1176 \beta_{3} ) q^{98} + ( 34750 - 1288 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{2} - 14q^{3} - 32q^{4} - 70q^{5} + 112q^{6} + 256q^{8} - 244q^{9} + O(q^{10})$$ $$4q - 8q^{2} - 14q^{3} - 32q^{4} - 70q^{5} + 112q^{6} + 256q^{8} - 244q^{9} - 280q^{10} - 62q^{11} - 224q^{12} + 3640q^{13} + 504q^{14} - 4076q^{15} - 512q^{16} - 1694q^{17} - 976q^{18} - 826q^{19} + 2240q^{20} + 3542q^{21} + 496q^{22} + 2734q^{23} - 896q^{24} - 6312q^{25} - 7280q^{26} + 14308q^{27} - 2016q^{28} - 5704q^{29} + 8152q^{30} + 2674q^{31} - 2048q^{32} - 4858q^{33} + 13552q^{34} - 13286q^{35} + 7808q^{36} + 9146q^{37} - 3304q^{38} - 21588q^{39} - 4480q^{40} + 12264q^{41} - 35392q^{42} - 32080q^{43} - 992q^{44} + 26852q^{45} + 10936q^{46} + 25326q^{47} + 7168q^{48} + 56644q^{49} + 50496q^{50} - 25762q^{51} - 29120q^{52} - 14958q^{53} - 28616q^{54} - 31052q^{55} - 37732q^{57} + 11408q^{58} + 1106q^{59} + 32608q^{60} + 28042q^{61} - 21392q^{62} + 77308q^{63} + 16384q^{64} - 28308q^{65} - 19432q^{66} + 102642q^{67} - 27104q^{68} - 188692q^{69} - 88424q^{70} - 22112q^{71} - 15616q^{72} - 35070q^{73} + 36584q^{74} + 132776q^{75} + 26432q^{76} + 27062q^{77} + 172704q^{78} - 101762q^{79} - 17920q^{80} + 23750q^{81} - 24528q^{82} - 89264q^{83} + 84896q^{84} + 7348q^{85} + 64160q^{86} + 170380q^{87} - 3968q^{88} + 75474q^{89} - 214816q^{90} - 123872q^{91} - 87488q^{92} + 53478q^{93} + 101304q^{94} - 127502q^{95} - 14336q^{96} - 16632q^{97} - 113288q^{98} + 139000q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 79 x^{2} + 6241$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/79$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3}$$$$/79$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$79 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$79 \beta_{3}$$$$/2$$

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$\beta_{2}$$

## 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
 4.44410 + 7.69740i −4.44410 − 7.69740i 4.44410 − 7.69740i −4.44410 + 7.69740i
−2.00000 + 3.46410i −12.3882 21.4570i −8.00000 13.8564i 18.0528 31.2683i 99.1056 −124.435 36.3731i 64.0000 −185.435 + 321.182i 72.2111 + 125.073i
9.2 −2.00000 + 3.46410i 5.38819 + 9.33263i −8.00000 13.8564i −53.0528 + 91.8901i −43.1056 124.435 36.3731i 64.0000 63.4347 109.872i −212.211 367.560i
11.1 −2.00000 3.46410i −12.3882 + 21.4570i −8.00000 + 13.8564i 18.0528 + 31.2683i 99.1056 −124.435 + 36.3731i 64.0000 −185.435 321.182i 72.2111 125.073i
11.2 −2.00000 3.46410i 5.38819 9.33263i −8.00000 + 13.8564i −53.0528 91.8901i −43.1056 124.435 + 36.3731i 64.0000 63.4347 + 109.872i −212.211 + 367.560i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 14.6.c.a 4
3.b odd 2 1 126.6.g.j 4
4.b odd 2 1 112.6.i.d 4
7.b odd 2 1 98.6.c.e 4
7.c even 3 1 inner 14.6.c.a 4
7.c even 3 1 98.6.a.h 2
7.d odd 6 1 98.6.a.g 2
7.d odd 6 1 98.6.c.e 4
21.g even 6 1 882.6.a.bi 2
21.h odd 6 1 126.6.g.j 4
21.h odd 6 1 882.6.a.ba 2
28.f even 6 1 784.6.a.bb 2
28.g odd 6 1 112.6.i.d 4
28.g odd 6 1 784.6.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.6.c.a 4 1.a even 1 1 trivial
14.6.c.a 4 7.c even 3 1 inner
98.6.a.g 2 7.d odd 6 1
98.6.a.h 2 7.c even 3 1
98.6.c.e 4 7.b odd 2 1
98.6.c.e 4 7.d odd 6 1
112.6.i.d 4 4.b odd 2 1
112.6.i.d 4 28.g odd 6 1
126.6.g.j 4 3.b odd 2 1
126.6.g.j 4 21.h odd 6 1
784.6.a.s 2 28.g odd 6 1
784.6.a.bb 2 28.f even 6 1
882.6.a.ba 2 21.h odd 6 1
882.6.a.bi 2 21.g even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} + 14 T_{3}^{3} + 463 T_{3}^{2} - 3738 T_{3} + 71289$$ acting on $$S_{6}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 16 + 4 T + T^{2} )^{2}$$
$3$ $$71289 - 3738 T + 463 T^{2} + 14 T^{3} + T^{4}$$
$5$ $$14676561 - 268170 T + 8731 T^{2} + 70 T^{3} + T^{4}$$
$7$ $$282475249 - 28322 T^{2} + T^{4}$$
$11$ $$210917529 - 900426 T + 18367 T^{2} + 62 T^{3} + T^{4}$$
$13$ $$( 766164 - 1820 T + T^{2} )^{2}$$
$17$ $$318620736225 + 956203710 T + 2305171 T^{2} + 1694 T^{3} + T^{4}$$
$19$ $$96141544489 - 256115342 T + 992343 T^{2} + 826 T^{3} + T^{4}$$
$23$ $$6792210678969 + 7125315258 T + 10080943 T^{2} - 2734 T^{3} + T^{4}$$
$29$ $$( -15866028 + 2852 T + T^{2} )^{2}$$
$31$ $$691673325561 - 2223882906 T + 6318607 T^{2} - 2674 T^{3} + T^{4}$$
$37$ $$252667333533169 - 145380361898 T + 67753803 T^{2} - 9146 T^{3} + T^{4}$$
$41$ $$( 8842932 - 6132 T + T^{2} )^{2}$$
$43$ $$( -78380144 + 16040 T + T^{2} )^{2}$$
$47$ $$24951287358855225 - 4000489008390 T + 483446511 T^{2} - 25326 T^{3} + T^{4}$$
$53$ $$85529669079884481 - 4374535293522 T + 516196323 T^{2} + 14958 T^{3} + T^{4}$$
$59$ $$868886159765625 + 32601423750 T + 30700111 T^{2} - 1106 T^{3} + T^{4}$$
$61$ $$38384562154446225 - 5493982610970 T + 590433979 T^{2} - 28042 T^{3} + T^{4}$$
$67$ $$6575826121535143225 - 263208715818330 T + 7971042799 T^{2} - 102642 T^{3} + T^{4}$$
$71$ $$( -177793920 + 11056 T + T^{2} )^{2}$$
$73$ $$227907887015854081 - 16742312474370 T + 1707301891 T^{2} + 35070 T^{3} + T^{4}$$
$79$ $$620965930233627225 - 80189872018230 T + 11143518559 T^{2} + 101762 T^{3} + T^{4}$$
$83$ $$( -5608638000 + 44632 T + T^{2} )^{2}$$
$89$ $$3677872821661829025 + 144742383942030 T + 7614102771 T^{2} - 75474 T^{3} + T^{4}$$
$97$ $$( -349929580 + 8316 T + T^{2} )^{2}$$