# Properties

 Label 14.3.d.a Level 14 Weight 3 Character orbit 14.d Analytic conductor 0.381 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.381472370104$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{2}, \sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + 6 \beta_{1} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{3} + 2 \beta_{2} q^{4} + ( -1 - 2 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{5} + ( -2 - 2 \beta_{1} - 4 \beta_{2} - \beta_{3} ) q^{6} + ( 3 + 4 \beta_{1} + 2 \beta_{2} + 5 \beta_{3} ) q^{7} + 2 \beta_{3} q^{8} + 6 \beta_{1} q^{9} + ( 8 - \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{10} + ( -3 \beta_{1} - 9 \beta_{2} - 3 \beta_{3} ) q^{11} + ( 2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{12} + ( 6 - 4 \beta_{1} + 12 \beta_{2} - 2 \beta_{3} ) q^{13} + ( -10 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{14} + ( -9 + 3 \beta_{3} ) q^{15} + ( -4 - 4 \beta_{2} ) q^{16} + ( -10 - 2 \beta_{1} - 5 \beta_{2} + 2 \beta_{3} ) q^{17} + 12 \beta_{2} q^{18} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{19} + ( -2 + 8 \beta_{1} - 4 \beta_{2} + 4 \beta_{3} ) q^{20} + ( 8 - 8 \beta_{1} - 11 \beta_{2} - 10 \beta_{3} ) q^{21} + ( 6 - 9 \beta_{3} ) q^{22} + ( 15 - 9 \beta_{1} + 15 \beta_{2} ) q^{23} + ( 8 + 2 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{24} + ( 12 \beta_{1} - 2 \beta_{2} + 12 \beta_{3} ) q^{25} + ( 4 + 6 \beta_{1} - 4 \beta_{2} + 12 \beta_{3} ) q^{26} + ( -3 + 6 \beta_{1} - 6 \beta_{2} + 3 \beta_{3} ) q^{27} + ( -4 - 10 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{28} + ( 12 - 6 \beta_{3} ) q^{29} + ( -6 - 9 \beta_{1} - 6 \beta_{2} ) q^{30} + ( -14 + 15 \beta_{1} - 7 \beta_{2} - 15 \beta_{3} ) q^{31} + ( -4 \beta_{1} - 4 \beta_{3} ) q^{32} + ( -15 + 12 \beta_{1} + 15 \beta_{2} + 24 \beta_{3} ) q^{33} + ( -4 - 10 \beta_{1} - 8 \beta_{2} - 5 \beta_{3} ) q^{34} + ( 7 - 7 \beta_{1} + 35 \beta_{2} - 14 \beta_{3} ) q^{35} + 12 \beta_{3} q^{36} + ( -31 - 24 \beta_{1} - 31 \beta_{2} ) q^{37} + ( -4 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{38} + ( -12 \beta_{1} - 6 \beta_{2} - 12 \beta_{3} ) q^{39} + ( -8 - 2 \beta_{1} + 8 \beta_{2} - 4 \beta_{3} ) q^{40} + ( -2 + 20 \beta_{1} - 4 \beta_{2} + 10 \beta_{3} ) q^{41} + ( 20 + 8 \beta_{1} + 4 \beta_{2} - 11 \beta_{3} ) q^{42} + ( -2 - 6 \beta_{3} ) q^{43} + ( 18 + 6 \beta_{1} + 18 \beta_{2} ) q^{44} + ( 48 - 6 \beta_{1} + 24 \beta_{2} + 6 \beta_{3} ) q^{45} + ( 15 \beta_{1} - 18 \beta_{2} + 15 \beta_{3} ) q^{46} + ( 29 + \beta_{1} - 29 \beta_{2} + 2 \beta_{3} ) q^{47} + ( 4 + 8 \beta_{1} + 8 \beta_{2} + 4 \beta_{3} ) q^{48} + ( -25 + 4 \beta_{1} - 40 \beta_{2} + 26 \beta_{3} ) q^{49} + ( -24 - 2 \beta_{3} ) q^{50} + ( 27 + 21 \beta_{1} + 27 \beta_{2} ) q^{51} + ( -24 + 4 \beta_{1} - 12 \beta_{2} - 4 \beta_{3} ) q^{52} + ( -12 \beta_{1} + 39 \beta_{2} - 12 \beta_{3} ) q^{53} + ( -6 - 3 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{54} + ( -3 - 30 \beta_{1} - 6 \beta_{2} - 15 \beta_{3} ) q^{55} + ( 4 - 4 \beta_{1} - 16 \beta_{2} + 2 \beta_{3} ) q^{56} + 3 q^{57} + ( 12 + 12 \beta_{1} + 12 \beta_{2} ) q^{58} + ( -26 - 25 \beta_{1} - 13 \beta_{2} + 25 \beta_{3} ) q^{59} + ( -6 \beta_{1} - 18 \beta_{2} - 6 \beta_{3} ) q^{60} + ( -7 - 32 \beta_{1} + 7 \beta_{2} - 64 \beta_{3} ) q^{61} + ( 30 - 14 \beta_{1} + 60 \beta_{2} - 7 \beta_{3} ) q^{62} + ( -60 + 18 \beta_{1} - 12 \beta_{2} + 12 \beta_{3} ) q^{63} + 8 q^{64} + ( -42 + 42 \beta_{1} - 42 \beta_{2} ) q^{65} + ( -48 - 15 \beta_{1} - 24 \beta_{2} + 15 \beta_{3} ) q^{66} + ( 45 \beta_{1} + 29 \beta_{2} + 45 \beta_{3} ) q^{67} + ( 10 - 4 \beta_{1} - 10 \beta_{2} - 8 \beta_{3} ) q^{68} + ( 3 - 12 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{69} + ( 28 + 7 \beta_{1} + 14 \beta_{2} + 35 \beta_{3} ) q^{70} + ( -6 + 30 \beta_{3} ) q^{71} + ( -24 - 24 \beta_{2} ) q^{72} + ( 106 + 16 \beta_{1} + 53 \beta_{2} - 16 \beta_{3} ) q^{73} + ( -31 \beta_{1} - 48 \beta_{2} - 31 \beta_{3} ) q^{74} + ( 22 - 10 \beta_{1} - 22 \beta_{2} - 20 \beta_{3} ) q^{75} + ( 2 - 4 \beta_{1} + 4 \beta_{2} - 2 \beta_{3} ) q^{76} + ( 42 + 42 \beta_{1} + 21 \beta_{2} ) q^{77} + ( 24 - 6 \beta_{3} ) q^{78} + ( 55 + 15 \beta_{1} + 55 \beta_{2} ) q^{79} + ( 8 - 8 \beta_{1} + 4 \beta_{2} + 8 \beta_{3} ) q^{80} + ( -54 \beta_{1} - 9 \beta_{2} - 54 \beta_{3} ) q^{81} + ( -20 - 2 \beta_{1} + 20 \beta_{2} - 4 \beta_{3} ) q^{82} + ( -68 + 8 \beta_{1} - 136 \beta_{2} + 4 \beta_{3} ) q^{83} + ( 22 + 20 \beta_{1} + 38 \beta_{2} + 4 \beta_{3} ) q^{84} + ( -9 + 24 \beta_{3} ) q^{85} + ( 12 - 2 \beta_{1} + 12 \beta_{2} ) q^{86} + ( -48 - 18 \beta_{1} - 24 \beta_{2} + 18 \beta_{3} ) q^{87} + ( 18 \beta_{1} + 12 \beta_{2} + 18 \beta_{3} ) q^{88} + ( -63 + 24 \beta_{1} + 63 \beta_{2} + 48 \beta_{3} ) q^{89} + ( -12 + 48 \beta_{1} - 24 \beta_{2} + 24 \beta_{3} ) q^{90} + ( 30 - 44 \beta_{1} + 48 \beta_{2} + 8 \beta_{3} ) q^{91} + ( -30 - 18 \beta_{3} ) q^{92} + ( -69 - 24 \beta_{1} - 69 \beta_{2} ) q^{93} + ( -4 + 29 \beta_{1} - 2 \beta_{2} - 29 \beta_{3} ) q^{94} + ( -9 \beta_{1} + 15 \beta_{2} - 9 \beta_{3} ) q^{95} + ( -8 + 4 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} ) q^{96} + ( 22 - 52 \beta_{1} + 44 \beta_{2} - 26 \beta_{3} ) q^{97} + ( -52 - 25 \beta_{1} - 44 \beta_{2} - 40 \beta_{3} ) q^{98} + ( 36 - 54 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{3} - 4q^{4} - 6q^{5} + 8q^{7} + O(q^{10})$$ $$4q - 6q^{3} - 4q^{4} - 6q^{5} + 8q^{7} + 24q^{10} + 18q^{11} + 12q^{12} - 36q^{14} - 36q^{15} - 8q^{16} - 30q^{17} - 24q^{18} + 6q^{19} + 54q^{21} + 24q^{22} + 30q^{23} + 24q^{24} + 4q^{25} + 24q^{26} - 20q^{28} + 48q^{29} - 12q^{30} - 42q^{31} - 90q^{33} - 42q^{35} - 62q^{37} - 12q^{38} + 12q^{39} - 48q^{40} + 72q^{42} - 8q^{43} + 36q^{44} + 144q^{45} + 36q^{46} + 174q^{47} - 20q^{49} - 96q^{50} + 54q^{51} - 72q^{52} - 78q^{53} - 36q^{54} + 48q^{56} + 12q^{57} + 24q^{58} - 78q^{59} + 36q^{60} - 42q^{61} - 216q^{63} + 32q^{64} - 84q^{65} - 144q^{66} - 58q^{67} + 60q^{68} + 84q^{70} - 24q^{71} - 48q^{72} + 318q^{73} + 96q^{74} + 132q^{75} + 126q^{77} + 96q^{78} + 110q^{79} + 24q^{80} + 18q^{81} - 120q^{82} + 12q^{84} - 36q^{85} + 24q^{86} - 144q^{87} - 24q^{88} - 378q^{89} + 24q^{91} - 120q^{92} - 138q^{93} - 12q^{94} - 30q^{95} - 48q^{96} - 120q^{98} + 144q^{99} + O(q^{100})$$

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

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

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$1 + \beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
3.1
 −0.707107 − 1.22474i 0.707107 + 1.22474i −0.707107 + 1.22474i 0.707107 − 1.22474i
−0.707107 1.22474i 0.621320 + 0.358719i −1.00000 + 1.73205i −5.74264 + 3.31552i 1.01461i 6.24264 3.16693i 2.82843 −4.24264 7.34847i 8.12132 + 4.68885i
3.2 0.707107 + 1.22474i −3.62132 2.09077i −1.00000 + 1.73205i 2.74264 1.58346i 5.91359i −2.24264 + 6.63103i −2.82843 4.24264 + 7.34847i 3.87868 + 2.23936i
5.1 −0.707107 + 1.22474i 0.621320 0.358719i −1.00000 1.73205i −5.74264 3.31552i 1.01461i 6.24264 + 3.16693i 2.82843 −4.24264 + 7.34847i 8.12132 4.68885i
5.2 0.707107 1.22474i −3.62132 + 2.09077i −1.00000 1.73205i 2.74264 + 1.58346i 5.91359i −2.24264 6.63103i −2.82843 4.24264 7.34847i 3.87868 2.23936i
 $$n$$: 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.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.3.d.a 4
3.b odd 2 1 126.3.n.c 4
4.b odd 2 1 112.3.s.b 4
5.b even 2 1 350.3.k.a 4
5.c odd 4 2 350.3.i.a 8
7.b odd 2 1 98.3.d.a 4
7.c even 3 1 98.3.b.b 4
7.c even 3 1 98.3.d.a 4
7.d odd 6 1 inner 14.3.d.a 4
7.d odd 6 1 98.3.b.b 4
8.b even 2 1 448.3.s.d 4
8.d odd 2 1 448.3.s.c 4
12.b even 2 1 1008.3.cg.l 4
21.c even 2 1 882.3.n.b 4
21.g even 6 1 126.3.n.c 4
21.g even 6 1 882.3.c.f 4
21.h odd 6 1 882.3.c.f 4
21.h odd 6 1 882.3.n.b 4
28.d even 2 1 784.3.s.c 4
28.f even 6 1 112.3.s.b 4
28.f even 6 1 784.3.c.e 4
28.g odd 6 1 784.3.c.e 4
28.g odd 6 1 784.3.s.c 4
35.i odd 6 1 350.3.k.a 4
35.k even 12 2 350.3.i.a 8
56.j odd 6 1 448.3.s.d 4
56.m even 6 1 448.3.s.c 4
84.j odd 6 1 1008.3.cg.l 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.3.d.a 4 1.a even 1 1 trivial
14.3.d.a 4 7.d odd 6 1 inner
98.3.b.b 4 7.c even 3 1
98.3.b.b 4 7.d odd 6 1
98.3.d.a 4 7.b odd 2 1
98.3.d.a 4 7.c even 3 1
112.3.s.b 4 4.b odd 2 1
112.3.s.b 4 28.f even 6 1
126.3.n.c 4 3.b odd 2 1
126.3.n.c 4 21.g even 6 1
350.3.i.a 8 5.c odd 4 2
350.3.i.a 8 35.k even 12 2
350.3.k.a 4 5.b even 2 1
350.3.k.a 4 35.i odd 6 1
448.3.s.c 4 8.d odd 2 1
448.3.s.c 4 56.m even 6 1
448.3.s.d 4 8.b even 2 1
448.3.s.d 4 56.j odd 6 1
784.3.c.e 4 28.f even 6 1
784.3.c.e 4 28.g odd 6 1
784.3.s.c 4 28.d even 2 1
784.3.s.c 4 28.g odd 6 1
882.3.c.f 4 21.g even 6 1
882.3.c.f 4 21.h odd 6 1
882.3.n.b 4 21.c even 2 1
882.3.n.b 4 21.h odd 6 1
1008.3.cg.l 4 12.b even 2 1
1008.3.cg.l 4 84.j odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} + 4 T^{4}$$
$3$ $$1 + 6 T + 27 T^{2} + 90 T^{3} + 252 T^{4} + 810 T^{5} + 2187 T^{6} + 4374 T^{7} + 6561 T^{8}$$
$5$ $$( 1 + 2 T + 25 T^{2} )^{2}( 1 + 2 T - 21 T^{2} + 50 T^{3} + 625 T^{4} )$$
$7$ $$1 - 8 T + 42 T^{2} - 392 T^{3} + 2401 T^{4}$$
$11$ $$1 - 18 T + 19 T^{2} - 1134 T^{3} + 39180 T^{4} - 137214 T^{5} + 278179 T^{6} - 31888098 T^{7} + 214358881 T^{8}$$
$13$ $$1 - 412 T^{2} + 89190 T^{4} - 11767132 T^{6} + 815730721 T^{8}$$
$17$ $$1 + 30 T + 929 T^{2} + 18870 T^{3} + 398820 T^{4} + 5453430 T^{5} + 77591009 T^{6} + 724127070 T^{7} + 6975757441 T^{8}$$
$19$ $$1 - 6 T + 731 T^{2} - 4314 T^{3} + 390972 T^{4} - 1557354 T^{5} + 95264651 T^{6} - 282275286 T^{7} + 16983563041 T^{8}$$
$23$ $$1 - 30 T - 221 T^{2} - 1890 T^{3} + 500700 T^{4} - 999810 T^{5} - 61844861 T^{6} - 4441076670 T^{7} + 78310985281 T^{8}$$
$29$ $$( 1 - 24 T + 1754 T^{2} - 20184 T^{3} + 707281 T^{4} )^{2}$$
$31$ $$1 + 42 T + 1307 T^{2} + 30198 T^{3} + 158508 T^{4} + 29020278 T^{5} + 1207041947 T^{6} + 37275154602 T^{7} + 852891037441 T^{8}$$
$37$ $$1 + 62 T + 1297 T^{2} - 11842 T^{3} - 649388 T^{4} - 16211698 T^{5} + 2430786817 T^{6} + 159075037358 T^{7} + 3512479453921 T^{8}$$
$41$ $$1 - 5500 T^{2} + 13185222 T^{4} - 15541685500 T^{6} + 7984925229121 T^{8}$$
$43$ $$( 1 + 4 T + 3630 T^{2} + 7396 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 - 174 T + 17027 T^{2} - 1206690 T^{3} + 65507772 T^{4} - 2665578210 T^{5} + 83086328387 T^{6} - 1875583467246 T^{7} + 23811286661761 T^{8}$$
$53$ $$1 + 78 T - 767 T^{2} + 96174 T^{3} + 21955764 T^{4} + 270152766 T^{5} - 6051998927 T^{6} + 1728820168062 T^{7} + 62259690411361 T^{8}$$
$59$ $$1 + 78 T + 5747 T^{2} + 290082 T^{3} + 8773068 T^{4} + 1009775442 T^{5} + 69638473667 T^{6} + 3290081623998 T^{7} + 146830437604321 T^{8}$$
$61$ $$1 + 42 T + 2033 T^{2} + 60690 T^{3} - 9569868 T^{4} + 225827490 T^{5} + 28148594753 T^{6} + 2163855723162 T^{7} + 191707312997281 T^{8}$$
$67$ $$1 + 58 T - 2405 T^{2} - 186122 T^{3} - 1970756 T^{4} - 835501658 T^{5} - 48463446005 T^{6} + 5246586165802 T^{7} + 406067677556641 T^{8}$$
$71$ $$( 1 + 12 T + 8318 T^{2} + 60492 T^{3} + 25411681 T^{4} )^{2}$$
$73$ $$1 - 318 T + 51257 T^{2} - 5580582 T^{3} + 459199092 T^{4} - 29738921478 T^{5} + 1455608638937 T^{6} - 48124283959902 T^{7} + 806460091894081 T^{8}$$
$79$ $$1 - 110 T - 2957 T^{2} - 283250 T^{3} + 112247068 T^{4} - 1767763250 T^{5} - 115175389517 T^{6} - 26739620107310 T^{7} + 1517108809906561 T^{8}$$
$83$ $$1 + 380 T^{2} + 89625894 T^{4} + 18034161980 T^{6} + 2252292232139041 T^{8}$$
$89$ $$1 + 378 T + 71921 T^{2} + 9182754 T^{3} + 904668996 T^{4} + 72736594434 T^{5} + 4512484714961 T^{6} + 187858927983258 T^{7} + 3936588805702081 T^{8}$$
$97$ $$1 - 26620 T^{2} + 330657414 T^{4} - 2356649460220 T^{6} + 7837433594376961 T^{8}$$