# Properties

 Label 14.5.d.a Level 14 Weight 5 Character orbit 14.d 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.d (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.44717948317$$ 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: $$2^{2}$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -6 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{3} + 8 \beta_{2} q^{4} + ( 9 + 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{5} + ( 16 - 6 \beta_{1} + 32 \beta_{2} - 3 \beta_{3} ) q^{6} + ( -35 - 56 \beta_{2} ) q^{7} + 8 \beta_{3} q^{8} + ( 42 - 36 \beta_{1} + 42 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -6 + 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} ) q^{3} + 8 \beta_{2} q^{4} + ( 9 + 2 \beta_{1} - 9 \beta_{2} + 4 \beta_{3} ) q^{5} + ( 16 - 6 \beta_{1} + 32 \beta_{2} - 3 \beta_{3} ) q^{6} + ( -35 - 56 \beta_{2} ) q^{7} + 8 \beta_{3} q^{8} + ( 42 - 36 \beta_{1} + 42 \beta_{2} ) q^{9} + ( -32 + 9 \beta_{1} - 16 \beta_{2} - 9 \beta_{3} ) q^{10} + ( -6 \beta_{1} + 27 \beta_{2} - 6 \beta_{3} ) q^{11} + ( 24 + 16 \beta_{1} - 24 \beta_{2} + 32 \beta_{3} ) q^{12} + ( 8 + 72 \beta_{1} + 16 \beta_{2} + 36 \beta_{3} ) q^{13} + ( -35 \beta_{1} - 56 \beta_{3} ) q^{14} + ( -177 - 72 \beta_{3} ) q^{15} + ( -64 - 64 \beta_{2} ) q^{16} + ( 306 + 32 \beta_{1} + 153 \beta_{2} - 32 \beta_{3} ) q^{17} + ( 42 \beta_{1} - 288 \beta_{2} + 42 \beta_{3} ) q^{18} + ( 5 + 90 \beta_{1} - 5 \beta_{2} + 180 \beta_{3} ) q^{19} + ( 72 - 32 \beta_{1} + 144 \beta_{2} - 16 \beta_{3} ) q^{20} + ( 42 - 182 \beta_{1} + 273 \beta_{2} - 154 \beta_{3} ) q^{21} + ( 48 + 27 \beta_{3} ) q^{22} + ( -243 - 240 \beta_{1} - 243 \beta_{2} ) q^{23} + ( -256 + 24 \beta_{1} - 128 \beta_{2} - 24 \beta_{3} ) q^{24} + ( 108 \beta_{1} + 286 \beta_{2} + 108 \beta_{3} ) q^{25} + ( -288 + 8 \beta_{1} + 288 \beta_{2} + 16 \beta_{3} ) q^{26} + ( -459 + 60 \beta_{1} - 918 \beta_{2} + 30 \beta_{3} ) q^{27} + ( 448 + 168 \beta_{2} ) q^{28} + ( 810 - 24 \beta_{3} ) q^{29} + ( 576 - 177 \beta_{1} + 576 \beta_{2} ) q^{30} + ( -182 + 180 \beta_{1} - 91 \beta_{2} - 180 \beta_{3} ) q^{31} + ( -64 \beta_{1} - 64 \beta_{3} ) q^{32} + ( 177 + 72 \beta_{1} - 177 \beta_{2} + 144 \beta_{3} ) q^{33} + ( 256 + 306 \beta_{1} + 512 \beta_{2} + 153 \beta_{3} ) q^{34} + ( -819 + 154 \beta_{1} - 693 \beta_{2} - 28 \beta_{3} ) q^{35} + ( -336 - 288 \beta_{3} ) q^{36} + ( -223 + 270 \beta_{1} - 223 \beta_{2} ) q^{37} + ( -1440 + 5 \beta_{1} - 720 \beta_{2} - 5 \beta_{3} ) q^{38} + ( -276 \beta_{1} + 1656 \beta_{2} - 276 \beta_{3} ) q^{39} + ( 128 + 72 \beta_{1} - 128 \beta_{2} + 144 \beta_{3} ) q^{40} + ( 72 - 416 \beta_{1} + 144 \beta_{2} - 208 \beta_{3} ) q^{41} + ( 1232 + 42 \beta_{1} - 224 \beta_{2} + 273 \beta_{3} ) q^{42} + ( 586 + 648 \beta_{3} ) q^{43} + ( -216 + 48 \beta_{1} - 216 \beta_{2} ) q^{44} + ( 1908 - 408 \beta_{1} + 954 \beta_{2} + 408 \beta_{3} ) q^{45} + ( -243 \beta_{1} - 1920 \beta_{2} - 243 \beta_{3} ) q^{46} + ( 117 - 316 \beta_{1} - 117 \beta_{2} - 632 \beta_{3} ) q^{47} + ( 192 - 256 \beta_{1} + 384 \beta_{2} - 128 \beta_{3} ) q^{48} + ( -1911 + 784 \beta_{2} ) q^{49} + ( -864 + 286 \beta_{3} ) q^{50} + ( 159 + 630 \beta_{1} + 159 \beta_{2} ) q^{51} + ( -128 - 288 \beta_{1} - 64 \beta_{2} + 288 \beta_{3} ) q^{52} + ( 774 \beta_{1} - 1377 \beta_{2} + 774 \beta_{3} ) q^{53} + ( -240 - 459 \beta_{1} + 240 \beta_{2} - 918 \beta_{3} ) q^{54} + ( 339 - 216 \beta_{1} + 678 \beta_{2} - 108 \beta_{3} ) q^{55} + ( 448 \beta_{1} + 168 \beta_{3} ) q^{56} + ( -4365 - 840 \beta_{3} ) q^{57} + ( 192 + 810 \beta_{1} + 192 \beta_{2} ) q^{58} + ( 4122 - 146 \beta_{1} + 2061 \beta_{2} + 146 \beta_{3} ) q^{59} + ( 576 \beta_{1} - 1416 \beta_{2} + 576 \beta_{3} ) q^{60} + ( 1281 - 738 \beta_{1} - 1281 \beta_{2} - 1476 \beta_{3} ) q^{61} + ( 1440 - 182 \beta_{1} + 2880 \beta_{2} - 91 \beta_{3} ) q^{62} + ( 882 + 1260 \beta_{1} - 1470 \beta_{2} + 2016 \beta_{3} ) q^{63} + 512 q^{64} + ( -1512 + 924 \beta_{1} - 1512 \beta_{2} ) q^{65} + ( -1152 + 177 \beta_{1} - 576 \beta_{2} - 177 \beta_{3} ) q^{66} + ( -1674 \beta_{1} + 2531 \beta_{2} - 1674 \beta_{3} ) q^{67} + ( -1224 + 256 \beta_{1} + 1224 \beta_{2} + 512 \beta_{3} ) q^{68} + ( -3111 + 468 \beta_{1} - 6222 \beta_{2} + 234 \beta_{3} ) q^{69} + ( 224 - 819 \beta_{1} + 1456 \beta_{2} - 693 \beta_{3} ) q^{70} + ( 4698 + 456 \beta_{3} ) q^{71} + ( 2304 - 336 \beta_{1} + 2304 \beta_{2} ) q^{72} + ( -5758 - 900 \beta_{1} - 2879 \beta_{2} + 900 \beta_{3} ) q^{73} + ( -223 \beta_{1} + 2160 \beta_{2} - 223 \beta_{3} ) q^{74} + ( -870 + 248 \beta_{1} + 870 \beta_{2} + 496 \beta_{3} ) q^{75} + ( 40 - 1440 \beta_{1} + 80 \beta_{2} - 720 \beta_{3} ) q^{76} + ( 1512 - 126 \beta_{1} + 567 \beta_{2} + 210 \beta_{3} ) q^{77} + ( 2208 + 1656 \beta_{3} ) q^{78} + ( 397 - 972 \beta_{1} + 397 \beta_{2} ) q^{79} + ( -1152 + 128 \beta_{1} - 576 \beta_{2} - 128 \beta_{3} ) q^{80} + ( -108 \beta_{1} + 2169 \beta_{2} - 108 \beta_{3} ) q^{81} + ( 1664 + 72 \beta_{1} - 1664 \beta_{2} + 144 \beta_{3} ) q^{82} + ( 2448 - 200 \beta_{1} + 4896 \beta_{2} - 100 \beta_{3} ) q^{83} + ( -2184 + 1232 \beta_{1} - 1848 \beta_{2} - 224 \beta_{3} ) q^{84} + ( 2595 + 54 \beta_{3} ) q^{85} + ( -5184 + 586 \beta_{1} - 5184 \beta_{2} ) q^{86} + ( -4092 + 1548 \beta_{1} - 2046 \beta_{2} - 1548 \beta_{3} ) q^{87} + ( -216 \beta_{1} + 384 \beta_{2} - 216 \beta_{3} ) q^{88} + ( -2079 + 564 \beta_{1} + 2079 \beta_{2} + 1128 \beta_{3} ) q^{89} + ( -3264 + 1908 \beta_{1} - 6528 \beta_{2} + 954 \beta_{3} ) q^{90} + ( 616 - 504 \beta_{1} - 112 \beta_{2} - 3276 \beta_{3} ) q^{91} + ( 1944 - 1920 \beta_{3} ) q^{92} + ( 9459 - 2166 \beta_{1} + 9459 \beta_{2} ) q^{93} + ( 5056 + 117 \beta_{1} + 2528 \beta_{2} - 117 \beta_{3} ) q^{94} + ( 2460 \beta_{1} - 4455 \beta_{2} + 2460 \beta_{3} ) q^{95} + ( 1024 + 192 \beta_{1} - 1024 \beta_{2} + 384 \beta_{3} ) q^{96} + ( 216 + 2448 \beta_{1} + 432 \beta_{2} + 1224 \beta_{3} ) q^{97} + ( -1911 \beta_{1} + 784 \beta_{3} ) q^{98} + ( -2862 - 1224 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 18q^{3} - 16q^{4} + 54q^{5} - 28q^{7} + 84q^{9} + O(q^{10})$$ $$4q - 18q^{3} - 16q^{4} + 54q^{5} - 28q^{7} + 84q^{9} - 96q^{10} - 54q^{11} + 144q^{12} - 708q^{15} - 128q^{16} + 918q^{17} + 576q^{18} + 30q^{19} - 378q^{21} + 192q^{22} - 486q^{23} - 768q^{24} - 572q^{25} - 1728q^{26} + 1456q^{28} + 3240q^{29} + 1152q^{30} - 546q^{31} + 1062q^{33} - 1890q^{35} - 1344q^{36} - 446q^{37} - 4320q^{38} - 3312q^{39} + 768q^{40} + 5376q^{42} + 2344q^{43} - 432q^{44} + 5724q^{45} + 3840q^{46} + 702q^{47} - 9212q^{49} - 3456q^{50} + 318q^{51} - 384q^{52} + 2754q^{53} - 1440q^{54} - 17460q^{57} + 384q^{58} + 12366q^{59} + 2832q^{60} + 7686q^{61} + 6468q^{63} + 2048q^{64} - 3024q^{65} - 3456q^{66} - 5062q^{67} - 7344q^{68} - 2016q^{70} + 18792q^{71} + 4608q^{72} - 17274q^{73} - 4320q^{74} - 5220q^{75} + 4914q^{77} + 8832q^{78} + 794q^{79} - 3456q^{80} - 4338q^{81} + 9984q^{82} - 5040q^{84} + 10380q^{85} - 10368q^{86} - 12276q^{87} - 768q^{88} - 12474q^{89} + 2688q^{91} + 7776q^{92} + 18918q^{93} + 15168q^{94} + 8910q^{95} + 6144q^{96} - 11448q^{99} + O(q^{100})$$

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

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

## 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
−1.41421 2.44949i −12.9853 7.49706i −4.00000 + 6.92820i 21.9853 12.6932i 42.4098i −7.00000 48.4974i 22.6274 71.9117 + 124.555i −62.1838 35.9018i
3.2 1.41421 + 2.44949i 3.98528 + 2.30090i −4.00000 + 6.92820i 5.01472 2.89525i 13.0159i −7.00000 48.4974i −22.6274 −29.9117 51.8086i 14.1838 + 8.18900i
5.1 −1.41421 + 2.44949i −12.9853 + 7.49706i −4.00000 6.92820i 21.9853 + 12.6932i 42.4098i −7.00000 + 48.4974i 22.6274 71.9117 124.555i −62.1838 + 35.9018i
5.2 1.41421 2.44949i 3.98528 2.30090i −4.00000 6.92820i 5.01472 + 2.89525i 13.0159i −7.00000 + 48.4974i −22.6274 −29.9117 + 51.8086i 14.1838 8.18900i
 $$n$$: 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 14.5.d.a 4
3.b odd 2 1 126.5.n.a 4
4.b odd 2 1 112.5.s.b 4
5.b even 2 1 350.5.k.a 4
5.c odd 4 2 350.5.i.a 8
7.b odd 2 1 98.5.d.a 4
7.c even 3 1 98.5.b.b 4
7.c even 3 1 98.5.d.a 4
7.d odd 6 1 inner 14.5.d.a 4
7.d odd 6 1 98.5.b.b 4
21.g even 6 1 126.5.n.a 4
21.g even 6 1 882.5.c.b 4
21.h odd 6 1 882.5.c.b 4
28.f even 6 1 112.5.s.b 4
28.f even 6 1 784.5.c.b 4
28.g odd 6 1 784.5.c.b 4
35.i odd 6 1 350.5.k.a 4
35.k even 12 2 350.5.i.a 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.5.d.a 4 1.a even 1 1 trivial
14.5.d.a 4 7.d odd 6 1 inner
98.5.b.b 4 7.c even 3 1
98.5.b.b 4 7.d odd 6 1
98.5.d.a 4 7.b odd 2 1
98.5.d.a 4 7.c even 3 1
112.5.s.b 4 4.b odd 2 1
112.5.s.b 4 28.f even 6 1
126.5.n.a 4 3.b odd 2 1
126.5.n.a 4 21.g even 6 1
350.5.i.a 8 5.c odd 4 2
350.5.i.a 8 35.k even 12 2
350.5.k.a 4 5.b even 2 1
350.5.k.a 4 35.i odd 6 1
784.5.c.b 4 28.f even 6 1
784.5.c.b 4 28.g odd 6 1
882.5.c.b 4 21.g even 6 1
882.5.c.b 4 21.h odd 6 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 8 T^{2} + 64 T^{4}$$
$3$ $$1 + 18 T + 201 T^{2} + 1674 T^{3} + 10836 T^{4} + 135594 T^{5} + 1318761 T^{6} + 9565938 T^{7} + 43046721 T^{8}$$
$5$ $$1 - 54 T + 2369 T^{2} - 75438 T^{3} + 2168484 T^{4} - 47148750 T^{5} + 925390625 T^{6} - 13183593750 T^{7} + 152587890625 T^{8}$$
$7$ $$( 1 + 14 T + 2401 T^{2} )^{2}$$
$11$ $$1 + 54 T - 26807 T^{2} + 23814 T^{3} + 626404692 T^{4} + 348660774 T^{5} - 5746318522967 T^{6} + 169475132342934 T^{7} + 45949729863572161 T^{8}$$
$13$ $$1 - 51652 T^{2} + 2274555846 T^{4} - 42134123201092 T^{6} + 665416609183179841 T^{8}$$
$17$ $$1 - 918 T + 493601 T^{2} - 195252174 T^{3} + 61724271876 T^{4} - 16307656824654 T^{5} + 3443240848635041 T^{6} - 534847213776920598 T^{7} + 48661191875666868481 T^{8}$$
$19$ $$1 - 30 T + 66617 T^{2} - 1989510 T^{3} - 12546522252 T^{4} - 259274932710 T^{5} + 1131394019102297 T^{6} - 66399447571984830 T^{7} +$$$$28\!\cdots\!81$$$$T^{8}$$
$23$ $$1 + 486 T + 78265 T^{2} - 195250986 T^{3} - 119466109356 T^{4} - 54639231173226 T^{5} + 6129009263017465 T^{6} + 10650507473961876006 T^{7} +$$$$61\!\cdots\!61$$$$T^{8}$$
$29$ $$( 1 - 1620 T + 2066054 T^{2} - 1145795220 T^{3} + 500246412961 T^{4} )^{2}$$
$31$ $$1 + 546 T + 1193657 T^{2} + 597479610 T^{3} + 436340752596 T^{4} + 551784966906810 T^{5} + 1018059357078711737 T^{6} +$$$$43\!\cdots\!06$$$$T^{7} +$$$$72\!\cdots\!81$$$$T^{8}$$
$37$ $$1 + 446 T - 3015935 T^{2} - 237928066 T^{3} + 6449986888804 T^{4} - 445915502102626 T^{5} - 10593409721861231135 T^{6} +$$$$29\!\cdots\!26$$$$T^{7} +$$$$12\!\cdots\!41$$$$T^{8}$$
$41$ $$1 - 9195268 T^{2} + 37043496050310 T^{4} - 73423527441728999428 T^{6} +$$$$63\!\cdots\!41$$$$T^{8}$$
$43$ $$( 1 - 1172 T + 3821766 T^{2} - 4006834772 T^{3} + 11688200277601 T^{4} )^{2}$$
$47$ $$1 - 702 T + 7568153 T^{2} - 5197527270 T^{3} + 31807801869972 T^{4} - 25362275066400870 T^{5} +$$$$18\!\cdots\!33$$$$T^{6} -$$$$81\!\cdots\!82$$$$T^{7} +$$$$56\!\cdots\!21$$$$T^{8}$$
$53$ $$1 - 2754 T - 5299967 T^{2} + 7976903166 T^{3} + 43904732373732 T^{4} + 62941602870162846 T^{5} -$$$$32\!\cdots\!87$$$$T^{6} -$$$$13\!\cdots\!14$$$$T^{7} +$$$$38\!\cdots\!21$$$$T^{8}$$
$59$ $$1 - 12366 T + 87438953 T^{2} - 450942278166 T^{3} + 1800614696429652 T^{4} - 5464230374699839926 T^{5} +$$$$12\!\cdots\!13$$$$T^{6} -$$$$22\!\cdots\!46$$$$T^{7} +$$$$21\!\cdots\!41$$$$T^{8}$$
$61$ $$1 - 7686 T + 39234641 T^{2} - 150208335774 T^{3} + 462871617507012 T^{4} - 2079760734001415934 T^{5} +$$$$75\!\cdots\!21$$$$T^{6} -$$$$20\!\cdots\!06$$$$T^{7} +$$$$36\!\cdots\!61$$$$T^{8}$$
$67$ $$1 + 5062 T + 1333849 T^{2} - 81053994314 T^{3} - 332413001385740 T^{4} - 1633328846954725994 T^{5} +$$$$54\!\cdots\!09$$$$T^{6} +$$$$41\!\cdots\!82$$$$T^{7} +$$$$16\!\cdots\!81$$$$T^{8}$$
$71$ $$( 1 - 9396 T + 71231078 T^{2} - 238768154676 T^{3} + 645753531245761 T^{4} )^{2}$$
$73$ $$1 + 17274 T + 161686097 T^{2} + 1074829823970 T^{3} + 5889761488255716 T^{4} + 30523276375087636770 T^{5} +$$$$13\!\cdots\!57$$$$T^{6} +$$$$39\!\cdots\!54$$$$T^{7} +$$$$65\!\cdots\!61$$$$T^{8}$$
$79$ $$1 - 794 T - 69869063 T^{2} + 5876126422 T^{3} + 3428515016079124 T^{4} + 228875600103140182 T^{5} -$$$$10\!\cdots\!43$$$$T^{6} -$$$$46\!\cdots\!54$$$$T^{7} +$$$$23\!\cdots\!21$$$$T^{8}$$
$83$ $$1 - 153397060 T^{2} + 10369989980918982 T^{4} -$$$$34\!\cdots\!60$$$$T^{6} +$$$$50\!\cdots\!81$$$$T^{8}$$
$89$ $$1 + 12474 T + 182683793 T^{2} + 1631810023074 T^{3} + 16430717819326692 T^{4} +$$$$10\!\cdots\!34$$$$T^{5} +$$$$71\!\cdots\!33$$$$T^{6} +$$$$30\!\cdots\!54$$$$T^{7} +$$$$15\!\cdots\!61$$$$T^{8}$$
$97$ $$1 - 281924740 T^{2} + 35525126061727494 T^{4} -$$$$22\!\cdots\!40$$$$T^{6} +$$$$61\!\cdots\!21$$$$T^{8}$$