# Properties

 Label 14.8.c.a Level $14$ Weight $8$ Character orbit 14.c Analytic conductor $4.373$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$14 = 2 \cdot 7$$ Weight: $$k$$ $$=$$ $$8$$ Character orbit: $$[\chi]$$ $$=$$ 14.c (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.37339035678$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{2389})$$ Defining polynomial: $$x^{4} - x^{3} + 598 x^{2} + 597 x + 356409$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -8 + 8 \beta_{1} ) q^{2} + ( 28 \beta_{1} - \beta_{2} ) q^{3} -64 \beta_{1} q^{4} + ( 119 - 119 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( -224 - 8 \beta_{3} ) q^{6} + ( 14 + 56 \beta_{1} - 14 \beta_{2} - 21 \beta_{3} ) q^{7} + 512 q^{8} + ( -986 + 986 \beta_{1} - 56 \beta_{2} - 56 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -8 + 8 \beta_{1} ) q^{2} + ( 28 \beta_{1} - \beta_{2} ) q^{3} -64 \beta_{1} q^{4} + ( 119 - 119 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} ) q^{5} + ( -224 - 8 \beta_{3} ) q^{6} + ( 14 + 56 \beta_{1} - 14 \beta_{2} - 21 \beta_{3} ) q^{7} + 512 q^{8} + ( -986 + 986 \beta_{1} - 56 \beta_{2} - 56 \beta_{3} ) q^{9} + ( 952 \beta_{1} + 16 \beta_{2} ) q^{10} + ( -2924 \beta_{1} + 7 \beta_{2} ) q^{11} + ( 1792 - 1792 \beta_{1} + 64 \beta_{2} + 64 \beta_{3} ) q^{12} + ( 658 + 196 \beta_{3} ) q^{13} + ( -560 + 112 \beta_{1} + 168 \beta_{2} + 56 \beta_{3} ) q^{14} + ( -1446 + 63 \beta_{3} ) q^{15} + ( -4096 + 4096 \beta_{1} ) q^{16} + ( 23821 \beta_{1} - 260 \beta_{2} ) q^{17} + ( -7888 \beta_{1} + 448 \beta_{2} ) q^{18} + ( 20524 - 20524 \beta_{1} + 243 \beta_{2} + 243 \beta_{3} ) q^{19} + ( -7616 + 128 \beta_{3} ) q^{20} + ( -35014 - 14763 \beta_{1} + 126 \beta_{2} - 448 \beta_{3} ) q^{21} + ( 23392 + 56 \beta_{3} ) q^{22} + ( 24658 - 24658 \beta_{1} - 847 \beta_{2} - 847 \beta_{3} ) q^{23} + ( 14336 \beta_{1} - 512 \beta_{2} ) q^{24} + ( 54408 \beta_{1} - 476 \beta_{2} ) q^{25} + ( -5264 + 5264 \beta_{1} - 1568 \beta_{2} - 1568 \beta_{3} ) q^{26} + ( -100156 - 367 \beta_{3} ) q^{27} + ( 3584 - 4480 \beta_{1} - 448 \beta_{2} + 896 \beta_{3} ) q^{28} + ( 86410 - 868 \beta_{3} ) q^{29} + ( 11568 - 11568 \beta_{1} - 504 \beta_{2} - 504 \beta_{3} ) q^{30} + ( 35126 \beta_{1} + 3457 \beta_{2} ) q^{31} -32768 \beta_{1} q^{32} + ( 98595 - 98595 \beta_{1} + 3120 \beta_{2} + 3120 \beta_{3} ) q^{33} + ( -190568 - 2080 \beta_{3} ) q^{34} + ( 41776 - 102004 \beta_{1} - 2527 \beta_{2} - 973 \beta_{3} ) q^{35} + ( 63104 + 3584 \beta_{3} ) q^{36} + ( 44083 - 44083 \beta_{1} + 10962 \beta_{2} + 10962 \beta_{3} ) q^{37} + ( 164192 \beta_{1} - 1944 \beta_{2} ) q^{38} + ( 486668 \beta_{1} - 6146 \beta_{2} ) q^{39} + ( 60928 - 60928 \beta_{1} - 1024 \beta_{2} - 1024 \beta_{3} ) q^{40} + ( -578130 + 924 \beta_{3} ) q^{41} + ( 398216 - 280112 \beta_{1} + 3584 \beta_{2} + 4592 \beta_{3} ) q^{42} + ( 28772 - 8064 \beta_{3} ) q^{43} + ( -187136 + 187136 \beta_{1} - 448 \beta_{2} - 448 \beta_{3} ) q^{44} + ( -150234 \beta_{1} - 4692 \beta_{2} ) q^{45} + ( 197264 \beta_{1} + 6776 \beta_{2} ) q^{46} + ( 706146 - 706146 \beta_{1} - 10215 \beta_{2} - 10215 \beta_{3} ) q^{47} + ( -114688 - 4096 \beta_{3} ) q^{48} + ( 582365 - 931784 \beta_{1} + 392 \beta_{2} - 2156 \beta_{3} ) q^{49} + ( -435264 - 3808 \beta_{3} ) q^{50} + ( -1288128 + 1288128 \beta_{1} - 31101 \beta_{2} - 31101 \beta_{3} ) q^{51} + ( -42112 \beta_{1} + 12544 \beta_{2} ) q^{52} + ( 1180587 \beta_{1} + 1890 \beta_{2} ) q^{53} + ( 801248 - 801248 \beta_{1} + 2936 \beta_{2} + 2936 \beta_{3} ) q^{54} + ( -314510 + 5015 \beta_{3} ) q^{55} + ( 7168 + 28672 \beta_{1} - 7168 \beta_{2} - 10752 \beta_{3} ) q^{56} + ( 1155199 + 27328 \beta_{3} ) q^{57} + ( -691280 + 691280 \beta_{1} + 6944 \beta_{2} + 6944 \beta_{3} ) q^{58} + ( -921256 \beta_{1} - 15541 \beta_{2} ) q^{59} + ( 92544 \beta_{1} + 4032 \beta_{2} ) q^{60} + ( -639121 + 639121 \beta_{1} + 30350 \beta_{2} + 30350 \beta_{3} ) q^{61} + ( -281008 + 27656 \beta_{3} ) q^{62} + ( 867468 - 2795660 \beta_{1} + 19922 \beta_{2} + 2982 \beta_{3} ) q^{63} + 262144 q^{64} + ( -858186 + 858186 \beta_{1} + 22008 \beta_{2} + 22008 \beta_{3} ) q^{65} + ( 788760 \beta_{1} - 24960 \beta_{2} ) q^{66} + ( 1060740 \beta_{1} + 46823 \beta_{2} ) q^{67} + ( 1524544 - 1524544 \beta_{1} + 16640 \beta_{2} + 16640 \beta_{3} ) q^{68} + ( -1333059 + 942 \beta_{3} ) q^{69} + ( 481824 + 334208 \beta_{1} + 7784 \beta_{2} - 12432 \beta_{3} ) q^{70} + ( 1800080 - 14168 \beta_{3} ) q^{71} + ( -504832 + 504832 \beta_{1} - 28672 \beta_{2} - 28672 \beta_{3} ) q^{72} + ( 817341 \beta_{1} + 50252 \beta_{2} ) q^{73} + ( 352664 \beta_{1} - 87696 \beta_{2} ) q^{74} + ( -2660588 + 2660588 \beta_{1} - 67736 \beta_{2} - 67736 \beta_{3} ) q^{75} + ( -1313536 - 15552 \beta_{3} ) q^{76} + ( 397866 - 87619 \beta_{1} - 19978 \beta_{2} + 41328 \beta_{3} ) q^{77} + ( -3893344 - 49168 \beta_{3} ) q^{78} + ( 4596302 - 4596302 \beta_{1} + 8141 \beta_{2} + 8141 \beta_{3} ) q^{79} + ( 487424 \beta_{1} + 8192 \beta_{2} ) q^{80} + ( -1524749 \beta_{1} - 12040 \beta_{2} ) q^{81} + ( 4625040 - 4625040 \beta_{1} - 7392 \beta_{2} - 7392 \beta_{3} ) q^{82} + ( -14140 - 68936 \beta_{3} ) q^{83} + ( -944832 + 3185728 \beta_{1} - 36736 \beta_{2} - 8064 \beta_{3} ) q^{84} + ( 1592419 - 16702 \beta_{3} ) q^{85} + ( -230176 + 230176 \beta_{1} + 64512 \beta_{2} + 64512 \beta_{3} ) q^{86} + ( 345828 \beta_{1} - 62106 \beta_{2} ) q^{87} + ( -1497088 \beta_{1} + 3584 \beta_{2} ) q^{88} + ( -4468275 + 4468275 \beta_{1} - 42324 \beta_{2} - 42324 \beta_{3} ) q^{89} + ( 1201872 - 37536 \beta_{3} ) q^{90} + ( -9823912 + 6592264 \beta_{1} - 20188 \beta_{2} - 11074 \beta_{3} ) q^{91} + ( -1578112 + 54208 \beta_{3} ) q^{92} + ( 7275245 - 7275245 \beta_{1} + 61670 \beta_{2} + 61670 \beta_{3} ) q^{93} + ( 5649168 \beta_{1} + 81720 \beta_{2} ) q^{94} + ( -1281302 \beta_{1} - 12131 \beta_{2} ) q^{95} + ( 917504 - 917504 \beta_{1} + 32768 \beta_{2} + 32768 \beta_{3} ) q^{96} + ( 2937102 - 32284 \beta_{3} ) q^{97} + ( 2795352 + 4658920 \beta_{1} + 17248 \beta_{2} + 20384 \beta_{3} ) q^{98} + ( 3819552 + 170646 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 16q^{2} + 56q^{3} - 128q^{4} + 238q^{5} - 896q^{6} + 168q^{7} + 2048q^{8} - 1972q^{9} + O(q^{10})$$ $$4q - 16q^{2} + 56q^{3} - 128q^{4} + 238q^{5} - 896q^{6} + 168q^{7} + 2048q^{8} - 1972q^{9} + 1904q^{10} - 5848q^{11} + 3584q^{12} + 2632q^{13} - 2016q^{14} - 5784q^{15} - 8192q^{16} + 47642q^{17} - 15776q^{18} + 41048q^{19} - 30464q^{20} - 169582q^{21} + 93568q^{22} + 49316q^{23} + 28672q^{24} + 108816q^{25} - 10528q^{26} - 400624q^{27} + 5376q^{28} + 345640q^{29} + 23136q^{30} + 70252q^{31} - 65536q^{32} + 197190q^{33} - 762272q^{34} - 36904q^{35} + 252416q^{36} + 88166q^{37} + 328384q^{38} + 973336q^{39} + 121856q^{40} - 2312520q^{41} + 1032640q^{42} + 115088q^{43} - 374272q^{44} - 300468q^{45} + 394528q^{46} + 1412292q^{47} - 458752q^{48} + 465892q^{49} - 1741056q^{50} - 2576256q^{51} - 84224q^{52} + 2361174q^{53} + 1602496q^{54} - 1258040q^{55} + 86016q^{56} + 4620796q^{57} - 1382560q^{58} - 1842512q^{59} + 185088q^{60} - 1278242q^{61} - 1124032q^{62} - 2121448q^{63} + 1048576q^{64} - 1716372q^{65} + 1577520q^{66} + 2121480q^{67} + 3049088q^{68} - 5332236q^{69} + 2595712q^{70} + 7200320q^{71} - 1009664q^{72} + 1634682q^{73} + 705328q^{74} - 5321176q^{75} - 5254144q^{76} + 1416226q^{77} - 15573376q^{78} + 9192604q^{79} + 974848q^{80} - 3049498q^{81} + 9250080q^{82} - 56560q^{83} + 2592128q^{84} + 6369676q^{85} - 460352q^{86} + 691656q^{87} - 2994176q^{88} - 8936550q^{89} + 4807488q^{90} - 26111120q^{91} - 6312448q^{92} + 14550490q^{93} + 11298336q^{94} - 2562604q^{95} + 1835008q^{96} + 11748408q^{97} + 20499248q^{98} + 15278208q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} + 598 x^{2} + 597 x + 356409$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{3} + 598 \nu^{2} - 598 \nu + 356409$$$$)/357006$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 598 \nu^{2} + 714610 \nu - 356409$$$$)/357006$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} + 896$$$$)/299$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} + 1195 \beta_{1} - 1195$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$299 \beta_{3} - 896$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1
 12.4693 + 21.5975i −11.9693 − 20.7315i 12.4693 − 21.5975i −11.9693 + 20.7315i
−4.00000 + 6.92820i −10.4387 18.0804i −32.0000 55.4256i 108.377 187.715i 167.019 726.284 544.110i 512.000 875.567 1516.53i 867.019 + 1501.72i
9.2 −4.00000 + 6.92820i 38.4387 + 66.5778i −32.0000 55.4256i 10.6226 18.3989i −615.019 −642.284 + 641.104i 512.000 −1861.57 + 3224.33i 84.9808 + 147.191i
11.1 −4.00000 6.92820i −10.4387 + 18.0804i −32.0000 + 55.4256i 108.377 + 187.715i 167.019 726.284 + 544.110i 512.000 875.567 + 1516.53i 867.019 1501.72i
11.2 −4.00000 6.92820i 38.4387 66.5778i −32.0000 + 55.4256i 10.6226 + 18.3989i −615.019 −642.284 641.104i 512.000 −1861.57 3224.33i 84.9808 147.191i
 $$n$$: 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 14.8.c.a 4
3.b odd 2 1 126.8.g.e 4
4.b odd 2 1 112.8.i.a 4
7.b odd 2 1 98.8.c.h 4
7.c even 3 1 inner 14.8.c.a 4
7.c even 3 1 98.8.a.h 2
7.d odd 6 1 98.8.a.k 2
7.d odd 6 1 98.8.c.h 4
21.h odd 6 1 126.8.g.e 4
28.g odd 6 1 112.8.i.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.8.c.a 4 1.a even 1 1 trivial
14.8.c.a 4 7.c even 3 1 inner
98.8.a.h 2 7.c even 3 1
98.8.a.k 2 7.d odd 6 1
98.8.c.h 4 7.b odd 2 1
98.8.c.h 4 7.d odd 6 1
112.8.i.a 4 4.b odd 2 1
112.8.i.a 4 28.g odd 6 1
126.8.g.e 4 3.b odd 2 1
126.8.g.e 4 21.h odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 56 T_{3}^{3} + 4741 T_{3}^{2} + 89880 T_{3} + 2576025$$ acting on $$S_{8}^{\mathrm{new}}(14, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 64 + 8 T + T^{2} )^{2}$$
$3$ $$2576025 + 89880 T + 4741 T^{2} - 56 T^{3} + T^{4}$$
$5$ $$21206025 - 1095990 T + 52039 T^{2} - 238 T^{3} + T^{4}$$
$7$ $$678223072849 - 138355224 T - 218834 T^{2} - 168 T^{3} + T^{4}$$
$11$ $$71110682271225 + 49314517320 T + 25766389 T^{2} + 5848 T^{3} + T^{4}$$
$13$ $$( -91342860 - 1316 T + T^{2} )^{2}$$
$17$ $$164790239668336881 - 19339966944522 T + 1863816523 T^{2} - 47642 T^{3} + T^{4}$$
$19$ $$78493276127245225 - 11500275107720 T + 1404771789 T^{2} - 41048 T^{3} + T^{4}$$
$23$ $$1222955395138220769 + 54537239624292 T + 3537940993 T^{2} - 49316 T^{3} + T^{4}$$
$29$ $$( 5666758164 - 172820 T + T^{2} )^{2}$$
$31$ $$74\!\cdots\!25$$$$+ 1919055786031020 T + 32252085889 T^{2} - 70252 T^{3} + T^{4}$$
$37$ $$81\!\cdots\!29$$$$+ 25138942165957282 T + 292905178383 T^{2} - 88166 T^{3} + T^{4}$$
$41$ $$( 332194626036 + 1156260 T + T^{2} )^{2}$$
$43$ $$( -154524293360 - 57544 T + T^{2} )^{2}$$
$47$ $$62\!\cdots\!81$$$$- 352167779849094972 T + 1745209651473 T^{2} - 1412292 T^{3} + T^{4}$$
$53$ $$19\!\cdots\!61$$$$- 3270820811450183406 T + 4189890740607 T^{2} - 2361174 T^{3} + T^{4}$$
$59$ $$73\!\cdots\!29$$$$+ 500638023944439024 T + 3123135537517 T^{2} + 1842512 T^{3} + T^{4}$$
$61$ $$32\!\cdots\!81$$$$- 2290719592631767878 T + 3425988610423 T^{2} + 1278242 T^{3} + T^{4}$$
$67$ $$16\!\cdots\!61$$$$+ 8724498066914483880 T + 8613135705781 T^{2} - 2121480 T^{3} + T^{4}$$
$71$ $$( 2760738723264 - 3600160 T + T^{2} )^{2}$$
$73$ $$28\!\cdots\!25$$$$+ 8769755399259278550 T + 8036993441899 T^{2} - 1634682 T^{3} + T^{4}$$
$79$ $$43\!\cdots\!25$$$$-$$$$19\!\cdots\!80$$$$T + 63536309305321 T^{2} - 9192604 T^{3} + T^{4}$$
$83$ $$( -11352739197744 + 28280 T + T^{2} )^{2}$$
$89$ $$24\!\cdots\!21$$$$+$$$$14\!\cdots\!50$$$$T + 64175910238539 T^{2} + 8936550 T^{3} + T^{4}$$
$97$ $$( 6136617007220 - 5874204 T + T^{2} )^{2}$$