# Properties

 Label 8.14.b.a Level 8 Weight 14 Character orbit 8.b Analytic conductor 8.578 Analytic rank 1 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8 = 2^{3}$$ Weight: $$k$$ $$=$$ $$14$$ Character orbit: $$[\chi]$$ $$=$$ 8.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.57847431615$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-79})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 2\sqrt{-79}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( -56 - 4 \beta ) q^{2} + 129 \beta q^{3} + ( -1920 + 448 \beta ) q^{4} -1270 \beta q^{5} + ( 163056 - 7224 \beta ) q^{6} -175832 q^{7} + ( 673792 - 17408 \beta ) q^{8} -3664233 q^{9} +O(q^{10})$$ $$q + ( -56 - 4 \beta ) q^{2} + 129 \beta q^{3} + ( -1920 + 448 \beta ) q^{4} -1270 \beta q^{5} + ( 163056 - 7224 \beta ) q^{6} -175832 q^{7} + ( 673792 - 17408 \beta ) q^{8} -3664233 q^{9} + ( -1605280 + 71120 \beta ) q^{10} + 148245 \beta q^{11} + ( -18262272 - 247680 \beta ) q^{12} -1748546 \beta q^{13} + ( 9846592 + 703328 \beta ) q^{14} + 51770280 q^{15} + ( -59736064 - 1720320 \beta ) q^{16} -133520302 q^{17} + ( 205197048 + 14656932 \beta ) q^{18} -1956439 \beta q^{19} + ( 179791360 + 2438400 \beta ) q^{20} -22682328 \beta q^{21} + ( 187381680 - 8301720 \beta ) q^{22} -35585416 q^{23} + ( 709619712 + 86919168 \beta ) q^{24} + 711026725 q^{25} + ( -2210162144 + 97918576 \beta ) q^{26} -267018390 \beta q^{27} + ( 337597440 - 78772736 \beta ) q^{28} + 89285286 \beta q^{29} + ( -2899135680 - 207081120 \beta ) q^{30} -5765001568 q^{31} + ( 1170735104 + 335282176 \beta ) q^{32} -6043059180 q^{33} + ( 7477136912 + 534081208 \beta ) q^{34} + 223306640 \beta q^{35} + ( 7035327360 - 1641576384 \beta ) q^{36} + 740167642 \beta q^{37} + ( -2472938896 + 109560584 \beta ) q^{38} + 71277729144 q^{39} + ( -6986178560 - 855715840 \beta ) q^{40} -23546348918 q^{41} + ( -28670462592 + 1270210368 \beta ) q^{42} + 821222629 \beta q^{43} + ( -20986748160 - 284630400 \beta ) q^{44} + 4653575910 \beta q^{45} + ( 1992783296 + 142341664 \beta ) q^{46} -68107736592 q^{47} + ( 70127124480 - 7705952256 \beta ) q^{48} -65972118183 q^{49} + ( -39817496600 - 2844106900 \beta ) q^{50} -17224118958 \beta q^{51} + ( 247538160128 + 3357208320 \beta ) q^{52} + 9353966274 \beta q^{53} + ( -337511244960 + 14953029840 \beta ) q^{54} + 59493683400 q^{55} + ( -118474194944 + 3060883456 \beta ) q^{56} + 79752279396 q^{57} + ( 112856601504 - 4999976016 \beta ) q^{58} -7179956339 \beta q^{59} + ( -99398937600 + 23193085440 \beta ) q^{60} -23861087370 \beta q^{61} + ( 322840087808 + 23060006272 \beta ) q^{62} + 644289416856 q^{63} + ( 358235504640 - 23458742272 \beta ) q^{64} -701726480720 q^{65} + ( 338411314080 + 24172236720 \beta ) q^{66} + 21163131297 \beta q^{67} + ( 256358979840 - 59817095296 \beta ) q^{68} -4590518664 \beta q^{69} + ( 282259592960 - 12505171840 \beta ) q^{70} -1309471657368 q^{71} + ( -2468930881536 + 63786968064 \beta ) q^{72} + 478647871914 q^{73} + ( 935571899488 - 41449387952 \beta ) q^{74} + 91722447525 \beta q^{75} + ( 276969156352 + 3756362880 \beta ) q^{76} -26066214840 \beta q^{77} + ( -3991552832064 - 285110916576 \beta ) q^{78} -364547231600 q^{79} + ( -690398822400 + 75864801280 \beta ) q^{80} + 5042766700701 q^{81} + ( 1318595539408 + 94185395672 \beta ) q^{82} + 49098397129 \beta q^{83} + ( 3211091810304 + 43550069760 \beta ) q^{84} + 169570783540 \beta q^{85} + ( 1038025403056 - 45988467224 \beta ) q^{86} -3639625398504 q^{87} + ( 815485071360 + 99886295040 \beta ) q^{88} -102457641350 q^{89} + ( 5882119950240 - 260600250960 \beta ) q^{90} + 307450340272 \beta q^{91} + ( 68323998720 - 15942266368 \beta ) q^{92} -743685202272 \beta q^{93} + ( 3814033249152 + 272430946368 \beta ) q^{94} -785158099480 q^{95} + ( -13667442622464 + 151024828416 \beta ) q^{96} -6157717373342 q^{97} + ( 3694438618248 + 263888472732 \beta ) q^{98} -543204221085 \beta q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 112q^{2} - 3840q^{4} + 326112q^{6} - 351664q^{7} + 1347584q^{8} - 7328466q^{9} + O(q^{10})$$ $$2q - 112q^{2} - 3840q^{4} + 326112q^{6} - 351664q^{7} + 1347584q^{8} - 7328466q^{9} - 3210560q^{10} - 36524544q^{12} + 19693184q^{14} + 103540560q^{15} - 119472128q^{16} - 267040604q^{17} + 410394096q^{18} + 359582720q^{20} + 374763360q^{22} - 71170832q^{23} + 1419239424q^{24} + 1422053450q^{25} - 4420324288q^{26} + 675194880q^{28} - 5798271360q^{30} - 11530003136q^{31} + 2341470208q^{32} - 12086118360q^{33} + 14954273824q^{34} + 14070654720q^{36} - 4945877792q^{38} + 142555458288q^{39} - 13972357120q^{40} - 47092697836q^{41} - 57340925184q^{42} - 41973496320q^{44} + 3985566592q^{46} - 136215473184q^{47} + 140254248960q^{48} - 131944236366q^{49} - 79634993200q^{50} + 495076320256q^{52} - 675022489920q^{54} + 118987366800q^{55} - 236948389888q^{56} + 159504558792q^{57} + 225713203008q^{58} - 198797875200q^{60} + 645680175616q^{62} + 1288578833712q^{63} + 716471009280q^{64} - 1403452961440q^{65} + 676822628160q^{66} + 512717959680q^{68} + 564519185920q^{70} - 2618943314736q^{71} - 4937861763072q^{72} + 957295743828q^{73} + 1871143798976q^{74} + 553938312704q^{76} - 7983105664128q^{78} - 729094463200q^{79} - 1380797644800q^{80} + 10085533401402q^{81} + 2637191078816q^{82} + 6422183620608q^{84} + 2076050806112q^{86} - 7279250797008q^{87} + 1630970142720q^{88} - 204915282700q^{89} + 11764239900480q^{90} + 136647997440q^{92} + 7628066498304q^{94} - 1570316198960q^{95} - 27334885244928q^{96} - 12315434746684q^{97} + 7388877236496q^{98} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/8\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$7$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 0.5 + 4.44410i 0.5 − 4.44410i
−56.0000 71.1056i 2293.15i −1920.00 + 7963.82i 22576.0i 163056. 128417.i −175832. 673792. 309451.i −3.66423e6 −1.60528e6 + 1.26426e6i
5.2 −56.0000 + 71.1056i 2293.15i −1920.00 7963.82i 22576.0i 163056. + 128417.i −175832. 673792. + 309451.i −3.66423e6 −1.60528e6 1.26426e6i
 $$n$$: 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
8.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.14.b.a 2
3.b odd 2 1 72.14.d.b 2
4.b odd 2 1 32.14.b.a 2
8.b even 2 1 inner 8.14.b.a 2
8.d odd 2 1 32.14.b.a 2
24.h odd 2 1 72.14.d.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.14.b.a 2 1.a even 1 1 trivial
8.14.b.a 2 8.b even 2 1 inner
32.14.b.a 2 4.b odd 2 1
32.14.b.a 2 8.d odd 2 1
72.14.d.b 2 3.b odd 2 1
72.14.d.b 2 24.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + 5258556$$ acting on $$S_{14}^{\mathrm{new}}(8, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 112 T + 8192 T^{2}$$
$3$ $$1 + 2069910 T^{2} + 2541865828329 T^{4}$$
$5$ $$1 - 1931729850 T^{2} + 1490116119384765625 T^{4}$$
$7$ $$( 1 + 175832 T + 96889010407 T^{2} )^{2}$$
$11$ $$1 - 62100824999962 T^{2} +$$$$11\!\cdots\!61$$$$T^{4}$$
$13$ $$1 + 360392330876150 T^{2} +$$$$91\!\cdots\!09$$$$T^{4}$$
$17$ $$( 1 + 133520302 T + 9904578032905937 T^{2} )^{2}$$
$19$ $$1 - 82896428399326282 T^{2} +$$$$17\!\cdots\!81$$$$T^{4}$$
$23$ $$( 1 + 35585416 T + 504036361936467383 T^{2} )^{2}$$
$29$ $$1 - 18002148940349036842 T^{2} +$$$$10\!\cdots\!21$$$$T^{4}$$
$31$ $$( 1 + 5765001568 T + 24417546297445042591 T^{2} )^{2}$$
$37$ $$1 -$$$$31\!\cdots\!70$$$$T^{2} +$$$$59\!\cdots\!09$$$$T^{4}$$
$41$ $$( 1 + 23546348918 T +$$$$92\!\cdots\!21$$$$T^{2} )^{2}$$
$43$ $$1 -$$$$32\!\cdots\!30$$$$T^{2} +$$$$29\!\cdots\!49$$$$T^{4}$$
$47$ $$( 1 + 68107736592 T +$$$$54\!\cdots\!27$$$$T^{2} )^{2}$$
$53$ $$1 -$$$$24\!\cdots\!30$$$$T^{2} +$$$$67\!\cdots\!29$$$$T^{4}$$
$59$ $$1 -$$$$19\!\cdots\!22$$$$T^{2} +$$$$11\!\cdots\!41$$$$T^{4}$$
$61$ $$1 -$$$$14\!\cdots\!62$$$$T^{2} +$$$$26\!\cdots\!61$$$$T^{4}$$
$67$ $$1 -$$$$95\!\cdots\!30$$$$T^{2} +$$$$30\!\cdots\!69$$$$T^{4}$$
$71$ $$( 1 + 1309471657368 T +$$$$11\!\cdots\!11$$$$T^{2} )^{2}$$
$73$ $$( 1 - 478647871914 T +$$$$16\!\cdots\!33$$$$T^{2} )^{2}$$
$79$ $$( 1 + 364547231600 T +$$$$46\!\cdots\!39$$$$T^{2} )^{2}$$
$83$ $$1 -$$$$16\!\cdots\!70$$$$T^{2} +$$$$78\!\cdots\!69$$$$T^{4}$$
$89$ $$( 1 + 102457641350 T +$$$$21\!\cdots\!69$$$$T^{2} )^{2}$$
$97$ $$( 1 + 6157717373342 T +$$$$67\!\cdots\!77$$$$T^{2} )^{2}$$