# 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 $$2$$ and degree $$1$$)

## 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: $$2^{2}$$ 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.

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

## 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. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
8.b Even 1 yes

## Hecke kernels

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