# Properties

 Label 14.5.b.a Level 14 Weight 5 Character orbit 14.b Analytic conductor 1.447 Analytic rank 0 Dimension 4 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 14.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.44717948317$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.1308672.3 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + 8 q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{6} + ( -19 + \beta_{1} - 9 \beta_{2} - \beta_{3} ) q^{7} + 8 \beta_{2} q^{8} + ( -63 + 6 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{2} q^{2} -\beta_{1} q^{3} + 8 q^{4} + ( \beta_{1} + \beta_{3} ) q^{5} -\beta_{3} q^{6} + ( -19 + \beta_{1} - 9 \beta_{2} - \beta_{3} ) q^{7} + 8 \beta_{2} q^{8} + ( -63 + 6 \beta_{2} ) q^{9} + ( 8 \beta_{1} + \beta_{3} ) q^{10} + ( 90 - 36 \beta_{2} ) q^{11} -8 \beta_{1} q^{12} + ( -\beta_{1} + \beta_{3} ) q^{13} + ( -72 - 8 \beta_{1} - 19 \beta_{2} + \beta_{3} ) q^{14} + ( 96 + 138 \beta_{2} ) q^{15} + 64 q^{16} + ( -20 \beta_{1} - 2 \beta_{3} ) q^{17} + ( 48 - 63 \beta_{2} ) q^{18} + ( 19 \beta_{1} - 10 \beta_{3} ) q^{19} + ( 8 \beta_{1} + 8 \beta_{3} ) q^{20} + ( 192 + 19 \beta_{1} - 150 \beta_{2} + 9 \beta_{3} ) q^{21} + ( -288 + 90 \beta_{2} ) q^{22} + ( -198 + 96 \beta_{2} ) q^{23} -8 \beta_{3} q^{24} + ( -575 - 234 \beta_{2} ) q^{25} + ( 8 \beta_{1} - \beta_{3} ) q^{26} + ( -18 \beta_{1} - 6 \beta_{3} ) q^{27} + ( -152 + 8 \beta_{1} - 72 \beta_{2} - 8 \beta_{3} ) q^{28} + ( 306 + 228 \beta_{2} ) q^{29} + ( 1104 + 96 \beta_{2} ) q^{30} + ( 72 \beta_{1} + 18 \beta_{3} ) q^{31} + 64 \beta_{2} q^{32} + ( -90 \beta_{1} + 36 \beta_{3} ) q^{33} + ( -16 \beta_{1} - 20 \beta_{3} ) q^{34} + ( 1008 - 91 \beta_{1} - 42 \beta_{2} - 28 \beta_{3} ) q^{35} + ( -504 + 48 \beta_{2} ) q^{36} + ( -974 - 108 \beta_{2} ) q^{37} + ( -80 \beta_{1} + 19 \beta_{3} ) q^{38} + ( -192 + 150 \beta_{2} ) q^{39} + ( 64 \beta_{1} + 8 \beta_{3} ) q^{40} + ( 110 \beta_{1} + 2 \beta_{3} ) q^{41} + ( -1200 + 72 \beta_{1} + 192 \beta_{2} + 19 \beta_{3} ) q^{42} + ( 922 - 360 \beta_{2} ) q^{43} + ( 720 - 288 \beta_{2} ) q^{44} + ( -15 \beta_{1} - 57 \beta_{3} ) q^{45} + ( 768 - 198 \beta_{2} ) q^{46} + ( 124 \beta_{1} - 38 \beta_{3} ) q^{47} -64 \beta_{1} q^{48} + ( -383 + 106 \beta_{1} + 684 \beta_{2} + 20 \beta_{3} ) q^{49} + ( -1872 - 575 \beta_{2} ) q^{50} + ( -2784 - 168 \beta_{2} ) q^{51} + ( -8 \beta_{1} + 8 \beta_{3} ) q^{52} + ( 1458 + 1032 \beta_{2} ) q^{53} + ( -48 \beta_{1} - 18 \beta_{3} ) q^{54} + ( -198 \beta_{1} + 54 \beta_{3} ) q^{55} + ( -576 - 64 \beta_{1} - 152 \beta_{2} + 8 \beta_{3} ) q^{56} + ( 3216 - 1554 \beta_{2} ) q^{57} + ( 1824 + 306 \beta_{2} ) q^{58} + ( 5 \beta_{1} + 50 \beta_{3} ) q^{59} + ( 768 + 1104 \beta_{2} ) q^{60} + ( -221 \beta_{1} - 85 \beta_{3} ) q^{61} + ( 144 \beta_{1} + 72 \beta_{3} ) q^{62} + ( 765 - 111 \beta_{1} + 453 \beta_{2} + 69 \beta_{3} ) q^{63} + 512 q^{64} + ( -1008 + 42 \beta_{2} ) q^{65} + ( 288 \beta_{1} - 90 \beta_{3} ) q^{66} + ( -262 + 288 \beta_{2} ) q^{67} + ( -160 \beta_{1} - 16 \beta_{3} ) q^{68} + ( 198 \beta_{1} - 96 \beta_{3} ) q^{69} + ( -336 - 224 \beta_{1} + 1008 \beta_{2} - 91 \beta_{3} ) q^{70} + ( -5382 - 1098 \beta_{2} ) q^{71} + ( 384 - 504 \beta_{2} ) q^{72} + ( -266 \beta_{1} + 104 \beta_{3} ) q^{73} + ( -864 - 974 \beta_{2} ) q^{74} + ( 575 \beta_{1} + 234 \beta_{3} ) q^{75} + ( 152 \beta_{1} - 80 \beta_{3} ) q^{76} + ( 882 + 378 \beta_{1} - 126 \beta_{2} - 126 \beta_{3} ) q^{77} + ( 1200 - 192 \beta_{2} ) q^{78} + ( 3194 - 1350 \beta_{2} ) q^{79} + ( 64 \beta_{1} + 64 \beta_{3} ) q^{80} + ( -7407 - 270 \beta_{2} ) q^{81} + ( 16 \beta_{1} + 110 \beta_{3} ) q^{82} + ( -397 \beta_{1} - 100 \beta_{3} ) q^{83} + ( 1536 + 152 \beta_{1} - 1200 \beta_{2} + 72 \beta_{3} ) q^{84} + ( 4128 + 2952 \beta_{2} ) q^{85} + ( -2880 + 922 \beta_{2} ) q^{86} + ( -306 \beta_{1} - 228 \beta_{3} ) q^{87} + ( -2304 + 720 \beta_{2} ) q^{88} + ( -270 \beta_{1} - 36 \beta_{3} ) q^{89} + ( -456 \beta_{1} - 15 \beta_{3} ) q^{90} + ( 1392 - 53 \beta_{1} - 342 \beta_{2} - 10 \beta_{3} ) q^{91} + ( -1584 + 768 \beta_{2} ) q^{92} + ( 9504 + 2160 \beta_{2} ) q^{93} + ( -304 \beta_{1} + 124 \beta_{3} ) q^{94} + ( 9216 - 1662 \beta_{2} ) q^{95} -64 \beta_{3} q^{96} + ( 344 \beta_{1} + 286 \beta_{3} ) q^{97} + ( 5472 + 160 \beta_{1} - 383 \beta_{2} + 106 \beta_{3} ) q^{98} + ( -7398 + 2808 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 32q^{4} - 76q^{7} - 252q^{9} + O(q^{10})$$ $$4q + 32q^{4} - 76q^{7} - 252q^{9} + 360q^{11} - 288q^{14} + 384q^{15} + 256q^{16} + 192q^{18} + 768q^{21} - 1152q^{22} - 792q^{23} - 2300q^{25} - 608q^{28} + 1224q^{29} + 4416q^{30} + 4032q^{35} - 2016q^{36} - 3896q^{37} - 768q^{39} - 4800q^{42} + 3688q^{43} + 2880q^{44} + 3072q^{46} - 1532q^{49} - 7488q^{50} - 11136q^{51} + 5832q^{53} - 2304q^{56} + 12864q^{57} + 7296q^{58} + 3072q^{60} + 3060q^{63} + 2048q^{64} - 4032q^{65} - 1048q^{67} - 1344q^{70} - 21528q^{71} + 1536q^{72} - 3456q^{74} + 3528q^{77} + 4800q^{78} + 12776q^{79} - 29628q^{81} + 6144q^{84} + 16512q^{85} - 11520q^{86} - 9216q^{88} + 5568q^{91} - 6336q^{92} + 38016q^{93} + 36864q^{95} + 21888q^{98} - 29592q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 72 x^{2} + 1278$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{2} + 72$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$4 \nu^{3} + 144 \nu$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$3 \beta_{2} - 72$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$3 \beta_{3} - 72 \beta_{1}$$$$)/4$$

## Character values

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

 $$n$$ $$3$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1
 6.34371i − 6.34371i 5.63537i − 5.63537i
−2.82843 12.6874i 8.00000 23.1980i 35.8854i 6.45584 + 48.5729i −22.6274 −79.9706 65.6139i
13.2 −2.82843 12.6874i 8.00000 23.1980i 35.8854i 6.45584 48.5729i −22.6274 −79.9706 65.6139i
13.3 2.82843 11.2707i 8.00000 43.1492i 31.8784i −44.4558 20.6077i 22.6274 −46.0294 122.044i
13.4 2.82843 11.2707i 8.00000 43.1492i 31.8784i −44.4558 + 20.6077i 22.6274 −46.0294 122.044i
 $$n$$: 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.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 14.5.b.a 4
3.b odd 2 1 126.5.c.a 4
4.b odd 2 1 112.5.c.c 4
5.b even 2 1 350.5.b.a 4
5.c odd 4 2 350.5.d.a 8
7.b odd 2 1 inner 14.5.b.a 4
7.c even 3 2 98.5.d.d 8
7.d odd 6 2 98.5.d.d 8
8.b even 2 1 448.5.c.e 4
8.d odd 2 1 448.5.c.f 4
12.b even 2 1 1008.5.f.h 4
21.c even 2 1 126.5.c.a 4
28.d even 2 1 112.5.c.c 4
35.c odd 2 1 350.5.b.a 4
35.f even 4 2 350.5.d.a 8
56.e even 2 1 448.5.c.f 4
56.h odd 2 1 448.5.c.e 4
84.h odd 2 1 1008.5.f.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.5.b.a 4 1.a even 1 1 trivial
14.5.b.a 4 7.b odd 2 1 inner
98.5.d.d 8 7.c even 3 2
98.5.d.d 8 7.d odd 6 2
112.5.c.c 4 4.b odd 2 1
112.5.c.c 4 28.d even 2 1
126.5.c.a 4 3.b odd 2 1
126.5.c.a 4 21.c even 2 1
350.5.b.a 4 5.b even 2 1
350.5.b.a 4 35.c odd 2 1
350.5.d.a 8 5.c odd 4 2
350.5.d.a 8 35.f even 4 2
448.5.c.e 4 8.b even 2 1
448.5.c.e 4 56.h odd 2 1
448.5.c.f 4 8.d odd 2 1
448.5.c.f 4 56.e even 2 1
1008.5.f.h 4 12.b even 2 1
1008.5.f.h 4 84.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{5}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - 8 T^{2} )^{2}$$
$3$ $$1 - 36 T^{2} + 13158 T^{4} - 236196 T^{6} + 43046721 T^{8}$$
$5$ $$1 - 100 T^{2} + 345702 T^{4} - 39062500 T^{6} + 152587890625 T^{8}$$
$7$ $$1 + 76 T + 3654 T^{2} + 182476 T^{3} + 5764801 T^{4}$$
$11$ $$( 1 - 180 T + 27014 T^{2} - 2635380 T^{3} + 214358881 T^{4} )^{2}$$
$13$ $$1 - 111460 T^{2} + 4736358630 T^{4} - 90921346162660 T^{6} + 665416609183179841 T^{8}$$
$17$ $$1 - 217348 T^{2} + 25123879686 T^{4} - 1516166928286468 T^{6} + 48661191875666868481 T^{8}$$
$19$ $$1 - 150436 T^{2} + 9183572838 T^{4} - 2554939289635876 T^{6} +$$$$28\!\cdots\!81$$$$T^{8}$$
$23$ $$( 1 + 396 T + 525158 T^{2} + 110817036 T^{3} + 78310985281 T^{4} )^{2}$$
$29$ $$( 1 - 612 T + 1092326 T^{2} - 432855972 T^{3} + 500246412961 T^{4} )^{2}$$
$31$ $$1 - 1703428 T^{2} + 1577900136966 T^{4} - 1452838474126047748 T^{6} +$$$$72\!\cdots\!81$$$$T^{8}$$
$37$ $$( 1 + 1948 T + 4603686 T^{2} + 3650865628 T^{3} + 3512479453921 T^{4} )^{2}$$
$41$ $$1 - 7851268 T^{2} + 31379741059206 T^{4} - 62691787933790375428 T^{6} +$$$$63\!\cdots\!41$$$$T^{8}$$
$43$ $$( 1 - 1844 T + 6650886 T^{2} - 6304269044 T^{3} + 11688200277601 T^{4} )^{2}$$
$47$ $$1 - 10858756 T^{2} + 58650967963398 T^{4} -$$$$25\!\cdots\!16$$$$T^{6} +$$$$56\!\cdots\!21$$$$T^{8}$$
$53$ $$( 1 - 2916 T + 9386534 T^{2} - 23008642596 T^{3} + 62259690411361 T^{4} )^{2}$$
$59$ $$1 - 42750244 T^{2} + 750538168343526 T^{4} -$$$$62\!\cdots\!24$$$$T^{6} +$$$$21\!\cdots\!41$$$$T^{8}$$
$61$ $$1 - 28277476 T^{2} + 401277478909158 T^{4} -$$$$54\!\cdots\!56$$$$T^{6} +$$$$36\!\cdots\!61$$$$T^{8}$$
$67$ $$( 1 + 524 T + 39707334 T^{2} + 10559187404 T^{3} + 406067677556641 T^{4} )^{2}$$
$71$ $$( 1 + 10764 T + 70144454 T^{2} + 273531334284 T^{3} + 645753531245761 T^{4} )^{2}$$
$73$ $$1 - 62983684 T^{2} + 1969418053172358 T^{4} -$$$$50\!\cdots\!04$$$$T^{6} +$$$$65\!\cdots\!61$$$$T^{8}$$
$79$ $$( 1 - 6388 T + 73521798 T^{2} - 248813117428 T^{3} + 1517108809906561 T^{4} )^{2}$$
$83$ $$1 - 129024292 T^{2} + 7865129580692070 T^{4} -$$$$29\!\cdots\!72$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8}$$
$89$ $$1 - 228854020 T^{2} + 20924407354852230 T^{4} -$$$$90\!\cdots\!20$$$$T^{6} +$$$$15\!\cdots\!61$$$$T^{8}$$
$97$ $$1 - 150468100 T^{2} + 16842128301249030 T^{4} -$$$$11\!\cdots\!00$$$$T^{6} +$$$$61\!\cdots\!21$$$$T^{8}$$