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

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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
0.500000 + 4.44410i
0.500000 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:

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 \(S_{14}^{\mathrm{new}}(8, [\chi])\).