# Properties

 Label 33.4.e.a Level $33$ Weight $4$ Character orbit 33.e Analytic conductor $1.947$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$33 = 3 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 33.e (of order $$5$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.94706303019$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{10})$$ Defining polynomial: $$x^{4} - x^{3} + x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{10}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 4 - 2 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{2} -3 \zeta_{10}^{3} q^{3} + 12 \zeta_{10}^{2} q^{4} + ( 1 - 12 \zeta_{10} + 12 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( 6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{6} + ( 6 \zeta_{10} - 25 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{7} + ( -16 + 16 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{8} -9 \zeta_{10} q^{9} +O(q^{10})$$ $$q + ( 4 - 2 \zeta_{10} + 4 \zeta_{10}^{2} ) q^{2} -3 \zeta_{10}^{3} q^{3} + 12 \zeta_{10}^{2} q^{4} + ( 1 - 12 \zeta_{10} + 12 \zeta_{10}^{2} - \zeta_{10}^{3} ) q^{5} + ( 6 + 6 \zeta_{10} - 6 \zeta_{10}^{2} - 6 \zeta_{10}^{3} ) q^{6} + ( 6 \zeta_{10} - 25 \zeta_{10}^{2} + 6 \zeta_{10}^{3} ) q^{7} + ( -16 + 16 \zeta_{10} + 8 \zeta_{10}^{3} ) q^{8} -9 \zeta_{10} q^{9} + ( -42 + 26 \zeta_{10}^{2} - 26 \zeta_{10}^{3} ) q^{10} + ( -15 + 35 \zeta_{10} - 25 \zeta_{10}^{2} + 41 \zeta_{10}^{3} ) q^{11} + 36 q^{12} + ( -22 - 11 \zeta_{10} - 22 \zeta_{10}^{2} ) q^{13} + ( 88 - 88 \zeta_{10} - 14 \zeta_{10}^{3} ) q^{14} + ( 33 \zeta_{10} - 36 \zeta_{10}^{2} + 33 \zeta_{10}^{3} ) q^{15} + ( 16 - 16 \zeta_{10} + 16 \zeta_{10}^{2} - 16 \zeta_{10}^{3} ) q^{16} + ( 78 - 21 \zeta_{10} + 21 \zeta_{10}^{2} - 78 \zeta_{10}^{3} ) q^{17} + ( -36 \zeta_{10} + 18 \zeta_{10}^{2} - 36 \zeta_{10}^{3} ) q^{18} + ( -45 + 45 \zeta_{10} + 83 \zeta_{10}^{3} ) q^{19} + ( -132 + 144 \zeta_{10} - 132 \zeta_{10}^{2} ) q^{20} + ( -57 + 18 \zeta_{10}^{2} - 18 \zeta_{10}^{3} ) q^{21} + ( -42 - 12 \zeta_{10} - 48 \zeta_{10}^{2} + 172 \zeta_{10}^{3} ) q^{22} + ( -15 + 44 \zeta_{10}^{2} - 44 \zeta_{10}^{3} ) q^{23} + ( 48 - 24 \zeta_{10} + 48 \zeta_{10}^{2} ) q^{24} + ( -143 + 143 \zeta_{10} + 3 \zeta_{10}^{3} ) q^{25} + ( -88 \zeta_{10} - 66 \zeta_{10}^{2} - 88 \zeta_{10}^{3} ) q^{26} + ( -27 + 27 \zeta_{10} - 27 \zeta_{10}^{2} + 27 \zeta_{10}^{3} ) q^{27} + ( 228 - 300 \zeta_{10} + 300 \zeta_{10}^{2} - 228 \zeta_{10}^{3} ) q^{28} + ( 192 \zeta_{10} - 30 \zeta_{10}^{2} + 192 \zeta_{10}^{3} ) q^{29} + ( 78 - 78 \zeta_{10} + 126 \zeta_{10}^{3} ) q^{30} + ( -77 + 33 \zeta_{10} - 77 \zeta_{10}^{2} ) q^{31} + ( 96 + 192 \zeta_{10}^{2} - 192 \zeta_{10}^{3} ) q^{32} + ( 30 + 18 \zeta_{10} + 105 \zeta_{10}^{2} - 60 \zeta_{10}^{3} ) q^{33} + ( 384 + 198 \zeta_{10}^{2} - 198 \zeta_{10}^{3} ) q^{34} + ( 281 - 366 \zeta_{10} + 281 \zeta_{10}^{2} ) q^{35} -108 \zeta_{10}^{3} q^{36} + ( -22 \zeta_{10} - 245 \zeta_{10}^{2} - 22 \zeta_{10}^{3} ) q^{37} + ( -346 + 104 \zeta_{10} - 104 \zeta_{10}^{2} + 346 \zeta_{10}^{3} ) q^{38} + ( -99 + 33 \zeta_{10} - 33 \zeta_{10}^{2} + 99 \zeta_{10}^{3} ) q^{39} + ( 104 \zeta_{10} - 272 \zeta_{10}^{2} + 104 \zeta_{10}^{3} ) q^{40} + ( -34 + 34 \zeta_{10} - 27 \zeta_{10}^{3} ) q^{41} + ( -264 + 222 \zeta_{10} - 264 \zeta_{10}^{2} ) q^{42} + ( -75 - 84 \zeta_{10}^{2} + 84 \zeta_{10}^{3} ) q^{43} + ( -192 - 300 \zeta_{10} + 120 \zeta_{10}^{2} + 120 \zeta_{10}^{3} ) q^{44} + ( -9 + 99 \zeta_{10}^{2} - 99 \zeta_{10}^{3} ) q^{45} + ( -148 + 294 \zeta_{10} - 148 \zeta_{10}^{2} ) q^{46} + ( 55 - 55 \zeta_{10} - 200 \zeta_{10}^{3} ) q^{47} -48 \zeta_{10}^{2} q^{48} + ( -54 + 318 \zeta_{10} - 318 \zeta_{10}^{2} + 54 \zeta_{10}^{3} ) q^{49} + ( -578 + 852 \zeta_{10} - 852 \zeta_{10}^{2} + 578 \zeta_{10}^{3} ) q^{50} + ( -171 \zeta_{10} - 63 \zeta_{10}^{2} - 171 \zeta_{10}^{3} ) q^{51} + ( 264 - 264 \zeta_{10} - 396 \zeta_{10}^{3} ) q^{52} + ( 143 - 241 \zeta_{10} + 143 \zeta_{10}^{2} ) q^{53} + ( -54 - 108 \zeta_{10}^{2} + 108 \zeta_{10}^{3} ) q^{54} + ( 295 - 571 \zeta_{10} + 202 \zeta_{10}^{2} - 51 \zeta_{10}^{3} ) q^{55} + ( 56 + 352 \zeta_{10}^{2} - 352 \zeta_{10}^{3} ) q^{56} + ( 135 + 114 \zeta_{10} + 135 \zeta_{10}^{2} ) q^{57} + ( -264 + 264 \zeta_{10} + 1092 \zeta_{10}^{3} ) q^{58} + ( -451 \zeta_{10} + 418 \zeta_{10}^{2} - 451 \zeta_{10}^{3} ) q^{59} + ( 36 - 432 \zeta_{10} + 432 \zeta_{10}^{2} - 36 \zeta_{10}^{3} ) q^{60} + ( 717 - 438 \zeta_{10} + 438 \zeta_{10}^{2} - 717 \zeta_{10}^{3} ) q^{61} + ( -22 \zeta_{10} - 374 \zeta_{10}^{2} - 22 \zeta_{10}^{3} ) q^{62} + ( 54 - 54 \zeta_{10} + 171 \zeta_{10}^{3} ) q^{63} + 832 \zeta_{10} q^{64} + ( 209 + 99 \zeta_{10}^{2} - 99 \zeta_{10}^{3} ) q^{65} + ( -180 + 552 \zeta_{10} - 36 \zeta_{10}^{2} + 162 \zeta_{10}^{3} ) q^{66} + ( -243 - 561 \zeta_{10}^{2} + 561 \zeta_{10}^{3} ) q^{67} + ( 684 + 252 \zeta_{10} + 684 \zeta_{10}^{2} ) q^{68} + ( 132 - 132 \zeta_{10} + 45 \zeta_{10}^{3} ) q^{69} + ( -902 \zeta_{10} + 1856 \zeta_{10}^{2} - 902 \zeta_{10}^{3} ) q^{70} + ( -433 + 708 \zeta_{10} - 708 \zeta_{10}^{2} + 433 \zeta_{10}^{3} ) q^{71} + ( 72 + 72 \zeta_{10} - 72 \zeta_{10}^{2} - 72 \zeta_{10}^{3} ) q^{72} + ( -318 \zeta_{10} + 346 \zeta_{10}^{2} - 318 \zeta_{10}^{3} ) q^{73} + ( 1024 - 1024 \zeta_{10} - 622 \zeta_{10}^{3} ) q^{74} + ( 429 - 420 \zeta_{10} + 429 \zeta_{10}^{2} ) q^{75} + ( -996 - 540 \zeta_{10}^{2} + 540 \zeta_{10}^{3} ) q^{76} + ( 94 + 745 \zeta_{10} - 496 \zeta_{10}^{2} - 34 \zeta_{10}^{3} ) q^{77} + ( -462 - 264 \zeta_{10}^{2} + 264 \zeta_{10}^{3} ) q^{78} + ( -702 + 637 \zeta_{10} - 702 \zeta_{10}^{2} ) q^{79} + ( -176 + 176 \zeta_{10} - 16 \zeta_{10}^{3} ) q^{80} + 81 \zeta_{10}^{2} q^{81} + ( -82 + 258 \zeta_{10} - 258 \zeta_{10}^{2} + 82 \zeta_{10}^{3} ) q^{82} + ( -35 - 431 \zeta_{10} + 431 \zeta_{10}^{2} + 35 \zeta_{10}^{3} ) q^{83} + ( 216 \zeta_{10} - 900 \zeta_{10}^{2} + 216 \zeta_{10}^{3} ) q^{84} + ( -174 + 174 \zeta_{10} + 549 \zeta_{10}^{3} ) q^{85} + ( -132 - 354 \zeta_{10} - 132 \zeta_{10}^{2} ) q^{86} + ( 486 + 576 \zeta_{10}^{2} - 576 \zeta_{10}^{3} ) q^{87} + ( -496 - 192 \zeta_{10} + 24 \zeta_{10}^{2} - 240 \zeta_{10}^{3} ) q^{88} + ( -1279 + 154 \zeta_{10}^{2} - 154 \zeta_{10}^{3} ) q^{89} + ( -234 + 612 \zeta_{10} - 234 \zeta_{10}^{2} ) q^{90} + ( -352 + 352 \zeta_{10} + 495 \zeta_{10}^{3} ) q^{91} + ( 528 \zeta_{10} - 708 \zeta_{10}^{2} + 528 \zeta_{10}^{3} ) q^{92} + ( -132 - 99 \zeta_{10} + 99 \zeta_{10}^{2} + 132 \zeta_{10}^{3} ) q^{93} + ( 620 + 70 \zeta_{10} - 70 \zeta_{10}^{2} - 620 \zeta_{10}^{3} ) q^{94} + ( -373 \zeta_{10} - 39 \zeta_{10}^{2} - 373 \zeta_{10}^{3} ) q^{95} + ( 576 - 576 \zeta_{10} - 288 \zeta_{10}^{3} ) q^{96} + ( 891 - 282 \zeta_{10} + 891 \zeta_{10}^{2} ) q^{97} + ( 948 - 744 \zeta_{10}^{2} + 744 \zeta_{10}^{3} ) q^{98} + ( 369 - 234 \zeta_{10} + 54 \zeta_{10}^{2} - 144 \zeta_{10}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 10q^{2} - 3q^{3} - 12q^{4} - 21q^{5} + 30q^{6} + 37q^{7} - 40q^{8} - 9q^{9} + O(q^{10})$$ $$4q + 10q^{2} - 3q^{3} - 12q^{4} - 21q^{5} + 30q^{6} + 37q^{7} - 40q^{8} - 9q^{9} - 220q^{10} + 41q^{11} + 144q^{12} - 77q^{13} + 250q^{14} + 102q^{15} + 16q^{16} + 192q^{17} - 90q^{18} - 52q^{19} - 252q^{20} - 264q^{21} + 40q^{22} - 148q^{23} + 120q^{24} - 426q^{25} - 110q^{26} - 27q^{27} + 84q^{28} + 414q^{29} + 360q^{30} - 198q^{31} - 27q^{33} + 1140q^{34} + 477q^{35} - 108q^{36} + 201q^{37} - 830q^{38} - 231q^{39} + 480q^{40} - 129q^{41} - 570q^{42} - 132q^{43} - 1068q^{44} - 234q^{45} - 150q^{46} - 35q^{47} + 48q^{48} + 474q^{49} - 30q^{50} - 279q^{51} + 396q^{52} + 188q^{53} + 356q^{55} - 480q^{56} + 519q^{57} + 300q^{58} - 1320q^{59} - 756q^{60} + 1275q^{61} + 330q^{62} + 333q^{63} + 832q^{64} + 638q^{65} + 30q^{66} + 150q^{67} + 2304q^{68} + 441q^{69} - 3660q^{70} + 117q^{71} + 360q^{72} - 982q^{73} + 2450q^{74} + 867q^{75} - 2904q^{76} + 1583q^{77} - 1320q^{78} - 1469q^{79} - 544q^{80} - 81q^{81} + 270q^{82} - 967q^{83} + 1332q^{84} + 27q^{85} - 750q^{86} + 792q^{87} - 2440q^{88} - 5424q^{89} - 90q^{90} - 561q^{91} + 1764q^{92} - 594q^{93} + 2000q^{94} - 707q^{95} + 1440q^{96} + 2391q^{97} + 5280q^{98} + 1044q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/33\mathbb{Z}\right)^\times$$.

 $$n$$ $$13$$ $$23$$ $$\chi(n)$$ $$\zeta_{10}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 0.809017 + 0.587785i −0.309017 − 0.951057i 0.809017 − 0.587785i −0.309017 + 0.951057i
3.61803 + 2.62866i 0.927051 2.85317i 3.70820 + 11.4127i −4.69098 + 3.40820i 10.8541 7.88597i −4.72542 14.5434i −5.52786 + 17.0130i −7.28115 5.29007i −25.9311
16.1 1.38197 + 4.25325i −2.42705 1.76336i −9.70820 + 7.05342i −5.80902 + 17.8783i 4.14590 12.7598i 23.2254 16.8743i −14.4721 10.5146i 2.78115 + 8.55951i −84.0689
25.1 3.61803 2.62866i 0.927051 + 2.85317i 3.70820 11.4127i −4.69098 3.40820i 10.8541 + 7.88597i −4.72542 + 14.5434i −5.52786 17.0130i −7.28115 + 5.29007i −25.9311
31.1 1.38197 4.25325i −2.42705 + 1.76336i −9.70820 7.05342i −5.80902 17.8783i 4.14590 + 12.7598i 23.2254 + 16.8743i −14.4721 + 10.5146i 2.78115 8.55951i −84.0689
 $$n$$: 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
11.c even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 33.4.e.a 4
3.b odd 2 1 99.4.f.a 4
11.c even 5 1 inner 33.4.e.a 4
11.c even 5 1 363.4.a.n 2
11.d odd 10 1 363.4.a.o 2
33.f even 10 1 1089.4.a.q 2
33.h odd 10 1 99.4.f.a 4
33.h odd 10 1 1089.4.a.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.e.a 4 1.a even 1 1 trivial
33.4.e.a 4 11.c even 5 1 inner
99.4.f.a 4 3.b odd 2 1
99.4.f.a 4 33.h odd 10 1
363.4.a.n 2 11.c even 5 1
363.4.a.o 2 11.d odd 10 1
1089.4.a.p 2 33.h odd 10 1
1089.4.a.q 2 33.f even 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 10 T_{2}^{3} + 60 T_{2}^{2} - 200 T_{2} + 400$$ acting on $$S_{4}^{\mathrm{new}}(33, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 10 T + 52 T^{2} - 200 T^{3} + 624 T^{4} - 1600 T^{5} + 3328 T^{6} - 5120 T^{7} + 4096 T^{8}$$
$3$ $$1 + 3 T + 9 T^{2} + 27 T^{3} + 81 T^{4}$$
$5$ $$1 + 21 T + 371 T^{2} + 5331 T^{3} + 75256 T^{4} + 666375 T^{5} + 5796875 T^{6} + 41015625 T^{7} + 244140625 T^{8}$$
$7$ $$1 - 37 T + 276 T^{2} + 12019 T^{3} - 398611 T^{4} + 4122517 T^{5} + 32471124 T^{6} - 1493083459 T^{7} + 13841287201 T^{8}$$
$11$ $$1 - 41 T + 1881 T^{2} - 54571 T^{3} + 1771561 T^{4}$$
$13$ $$1 + 77 T + 102 T^{2} - 92675 T^{3} - 2336149 T^{4} - 203606975 T^{5} + 492334518 T^{6} + 816546451721 T^{7} + 23298085122481 T^{8}$$
$17$ $$1 - 192 T + 9721 T^{2} + 336564 T^{3} - 58738571 T^{4} + 1653538932 T^{5} + 234641308249 T^{6} - 22768872287424 T^{7} + 582622237229761 T^{8}$$
$19$ $$1 + 52 T + 4395 T^{2} + 634622 T^{3} + 83733539 T^{4} + 4352872298 T^{5} + 206766646995 T^{6} + 16779760284508 T^{7} + 2213314919066161 T^{8}$$
$23$ $$( 1 + 74 T + 23283 T^{2} + 900358 T^{3} + 148035889 T^{4} )^{2}$$
$29$ $$1 - 414 T + 118207 T^{2} - 26000652 T^{3} + 4686132205 T^{4} - 634129901628 T^{5} + 70312280305447 T^{6} - 6005958434009766 T^{7} + 353814783205469041 T^{8}$$
$31$ $$1 + 198 T - 7527 T^{2} + 1052546 T^{3} + 1096985415 T^{4} + 31356397886 T^{5} - 6680240206887 T^{6} + 5235045187812858 T^{7} + 787662783788549761 T^{8}$$
$37$ $$1 - 201 T + 16698 T^{2} + 5740795 T^{3} - 1719990309 T^{4} + 290788489135 T^{5} + 42842499577482 T^{6} - 26122309698810477 T^{7} + 6582952005840035281 T^{8}$$
$41$ $$1 + 129 T - 62650 T^{2} - 6085179 T^{3} + 3934113919 T^{4} - 419396621859 T^{5} - 297594030698650 T^{6} + 42232269536820969 T^{7} + 22563490300366186081 T^{8}$$
$43$ $$( 1 + 66 T + 151283 T^{2} + 5247462 T^{3} + 6321363049 T^{4} )^{2}$$
$47$ $$1 + 35 T - 62723 T^{2} - 27766685 T^{3} + 7503450804 T^{4} - 2882820536755 T^{5} - 676104723080867 T^{6} + 39169566558596845 T^{7} +$$$$11\!\cdots\!41$$$$T^{8}$$
$53$ $$1 - 188 T - 43463 T^{2} - 48799240 T^{3} + 31190841441 T^{4} - 7265084453480 T^{5} - 963329627749727 T^{6} - 620355555258801004 T^{7} +$$$$49\!\cdots\!41$$$$T^{8}$$
$59$ $$1 + 1320 T + 594431 T^{2} + 124006410 T^{3} + 29587885771 T^{4} + 25468312479390 T^{5} + 25073416792753271 T^{6} + 11435154480624519480 T^{7} +$$$$17\!\cdots\!81$$$$T^{8}$$
$61$ $$1 - 1275 T + 398429 T^{2} + 290507655 T^{3} - 280558765844 T^{4} + 65939718039555 T^{5} + 20527211236278869 T^{6} - 14910036268363529775 T^{7} +$$$$26\!\cdots\!21$$$$T^{8}$$
$67$ $$( 1 - 75 T + 209531 T^{2} - 22557225 T^{3} + 90458382169 T^{4} )^{2}$$
$71$ $$1 - 117 T + 251153 T^{2} - 117855189 T^{3} + 146557541980 T^{4} - 42181668550179 T^{5} + 32172770607610913 T^{6} - 5364274584058536627 T^{7} +$$$$16\!\cdots\!41$$$$T^{8}$$
$73$ $$1 + 982 T + 25167 T^{2} - 89899270 T^{3} + 52391431241 T^{4} - 34972344317590 T^{5} + 3808628473015263 T^{6} + 57811898147519090566 T^{7} +$$$$22\!\cdots\!21$$$$T^{8}$$
$79$ $$1 + 1469 T + 1436772 T^{2} + 1372338877 T^{3} + 1206137476805 T^{4} + 676616587577203 T^{5} + 349261249643818212 T^{6} +$$$$17\!\cdots\!11$$$$T^{7} +$$$$59\!\cdots\!41$$$$T^{8}$$
$83$ $$1 + 967 T + 281752 T^{2} + 542972135 T^{3} + 732624179601 T^{4} + 310464408155245 T^{5} + 92116104077462488 T^{6} +$$$$18\!\cdots\!01$$$$T^{7} +$$$$10\!\cdots\!61$$$$T^{8}$$
$89$ $$( 1 + 2712 T + 3219029 T^{2} + 1911875928 T^{3} + 496981290961 T^{4} )^{2}$$
$97$ $$1 - 2391 T + 2091113 T^{2} - 1694593545 T^{3} + 1933106673916 T^{4} - 1546609774495785 T^{5} + 1741838588143095977 T^{6} -$$$$18\!\cdots\!47$$$$T^{7} +$$$$69\!\cdots\!41$$$$T^{8}$$