# Properties

 Label 3.11.b.a Level 3 Weight 11 Character orbit 3.b Analytic conductor 1.906 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3$$ Weight: $$k$$ $$=$$ $$11$$ Character orbit: $$[\chi]$$ $$=$$ 3.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.90607175802$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-5})$$ Defining polynomial: $$x^{2} + 5$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}\cdot 3$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = 12\sqrt{-5}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta q^{2} + ( -27 + 9 \beta ) q^{3} + 304 q^{4} -106 \beta q^{5} + ( -6480 - 27 \beta ) q^{6} + 17234 q^{7} + 1328 \beta q^{8} + ( -57591 - 486 \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{2} + ( -27 + 9 \beta ) q^{3} + 304 q^{4} -106 \beta q^{5} + ( -6480 - 27 \beta ) q^{6} + 17234 q^{7} + 1328 \beta q^{8} + ( -57591 - 486 \beta ) q^{9} + 76320 q^{10} -6962 \beta q^{11} + ( -8208 + 2736 \beta ) q^{12} -169654 q^{13} + 17234 \beta q^{14} + ( 686880 + 2862 \beta ) q^{15} -644864 q^{16} -12792 \beta q^{17} + ( 349920 - 57591 \beta ) q^{18} -949462 q^{19} -32224 \beta q^{20} + ( -465318 + 155106 \beta ) q^{21} + 5012640 q^{22} + 99044 \beta q^{23} + ( -8605440 - 35856 \beta ) q^{24} + 1675705 q^{25} -169654 \beta q^{26} + ( 4704237 - 505197 \beta ) q^{27} + 5239136 q^{28} + 118594 \beta q^{29} + ( -2060640 + 686880 \beta ) q^{30} -29793118 q^{31} + 715008 \beta q^{32} + ( 45113760 + 187974 \beta ) q^{33} + 9210240 q^{34} -1826804 \beta q^{35} + ( -17507664 - 147744 \beta ) q^{36} -60811846 q^{37} -949462 \beta q^{38} + ( 4580658 - 1526886 \beta ) q^{39} + 101352960 q^{40} + 6770372 \beta q^{41} + ( -111676320 - 465318 \beta ) q^{42} + 107419706 q^{43} -2116448 \beta q^{44} + ( -37091520 + 6104646 \beta ) q^{45} -71311680 q^{46} -9987608 \beta q^{47} + ( 17411328 - 5803776 \beta ) q^{48} + 14535507 q^{49} + 1675705 \beta q^{50} + ( 82892160 + 345384 \beta ) q^{51} -51574816 q^{52} + 7158798 \beta q^{53} + ( 363741840 + 4704237 \beta ) q^{54} -531339840 q^{55} + 22886752 \beta q^{56} + ( 25635474 - 8545158 \beta ) q^{57} -85387680 q^{58} -24192682 \beta q^{59} + ( 208811520 + 870048 \beta ) q^{60} + 1030793642 q^{61} -29793118 \beta q^{62} + ( -992523294 - 8375724 \beta ) q^{63} -1175146496 q^{64} + 17983324 \beta q^{65} + ( -135341280 + 45113760 \beta ) q^{66} + 1876742474 q^{67} -3888768 \beta q^{68} + ( -641805120 - 2674188 \beta ) q^{69} + 1315298880 q^{70} + 100003596 \beta q^{71} + ( 464693760 - 76480848 \beta ) q^{72} -2846528494 q^{73} -60811846 \beta q^{74} + ( -45244035 + 15081345 \beta ) q^{75} -288636448 q^{76} -119983108 \beta q^{77} + ( 1099357920 + 4580658 \beta ) q^{78} + 1488647618 q^{79} + 68355584 \beta q^{80} + ( 3146662161 + 55978452 \beta ) q^{81} -4874667840 q^{82} + 47175562 \beta q^{83} + ( -141456672 + 47152224 \beta ) q^{84} -976285440 q^{85} + 107419706 \beta q^{86} + ( -768489120 - 3202038 \beta ) q^{87} + 6656785920 q^{88} -224371428 \beta q^{89} + ( -4395345120 - 37091520 \beta ) q^{90} -2923817036 q^{91} + 30109376 \beta q^{92} + ( 804414186 - 268138062 \beta ) q^{93} + 7191077760 q^{94} + 100642972 \beta q^{95} + ( -4633251840 - 19305216 \beta ) q^{96} -1592948926 q^{97} + 14535507 \beta q^{98} + ( -2436143040 + 400948542 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 54q^{3} + 608q^{4} - 12960q^{6} + 34468q^{7} - 115182q^{9} + O(q^{10})$$ $$2q - 54q^{3} + 608q^{4} - 12960q^{6} + 34468q^{7} - 115182q^{9} + 152640q^{10} - 16416q^{12} - 339308q^{13} + 1373760q^{15} - 1289728q^{16} + 699840q^{18} - 1898924q^{19} - 930636q^{21} + 10025280q^{22} - 17210880q^{24} + 3351410q^{25} + 9408474q^{27} + 10478272q^{28} - 4121280q^{30} - 59586236q^{31} + 90227520q^{33} + 18420480q^{34} - 35015328q^{36} - 121623692q^{37} + 9161316q^{39} + 202705920q^{40} - 223352640q^{42} + 214839412q^{43} - 74183040q^{45} - 142623360q^{46} + 34822656q^{48} + 29071014q^{49} + 165784320q^{51} - 103149632q^{52} + 727483680q^{54} - 1062679680q^{55} + 51270948q^{57} - 170775360q^{58} + 417623040q^{60} + 2061587284q^{61} - 1985046588q^{63} - 2350292992q^{64} - 270682560q^{66} + 3753484948q^{67} - 1283610240q^{69} + 2630597760q^{70} + 929387520q^{72} - 5693056988q^{73} - 90488070q^{75} - 577272896q^{76} + 2198715840q^{78} + 2977295236q^{79} + 6293324322q^{81} - 9749335680q^{82} - 282913344q^{84} - 1952570880q^{85} - 1536978240q^{87} + 13313571840q^{88} - 8790690240q^{90} - 5847634072q^{91} + 1608828372q^{93} + 14382155520q^{94} - 9266503680q^{96} - 3185897852q^{97} - 4872286080q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 − 2.23607i 2.23607i
26.8328i −27.0000 241.495i 304.000 2844.28i −6480.00 + 724.486i 17234.0 35634.0i −57591.0 + 13040.7i 76320.0
2.2 26.8328i −27.0000 + 241.495i 304.000 2844.28i −6480.00 724.486i 17234.0 35634.0i −57591.0 13040.7i 76320.0
 $$n$$: 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3.11.b.a 2
3.b odd 2 1 inner 3.11.b.a 2
4.b odd 2 1 48.11.e.c 2
5.b even 2 1 75.11.c.d 2
5.c odd 4 2 75.11.d.b 4
8.b even 2 1 192.11.e.e 2
8.d odd 2 1 192.11.e.d 2
9.c even 3 2 81.11.d.d 4
9.d odd 6 2 81.11.d.d 4
12.b even 2 1 48.11.e.c 2
15.d odd 2 1 75.11.c.d 2
15.e even 4 2 75.11.d.b 4
24.f even 2 1 192.11.e.d 2
24.h odd 2 1 192.11.e.e 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.11.b.a 2 1.a even 1 1 trivial
3.11.b.a 2 3.b odd 2 1 inner
48.11.e.c 2 4.b odd 2 1
48.11.e.c 2 12.b even 2 1
75.11.c.d 2 5.b even 2 1
75.11.c.d 2 15.d odd 2 1
75.11.d.b 4 5.c odd 4 2
75.11.d.b 4 15.e even 4 2
81.11.d.d 4 9.c even 3 2
81.11.d.d 4 9.d odd 6 2
192.11.e.d 2 8.d odd 2 1
192.11.e.d 2 24.f even 2 1
192.11.e.e 2 8.b even 2 1
192.11.e.e 2 24.h odd 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{11}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - 1328 T^{2} + 1048576 T^{4}$$
$3$ $$1 + 54 T + 59049 T^{2}$$
$5$ $$1 - 11441330 T^{2} + 95367431640625 T^{4}$$
$7$ $$( 1 - 17234 T + 282475249 T^{2} )^{2}$$
$11$ $$1 - 16976849522 T^{2} +$$$$67\!\cdots\!01$$$$T^{4}$$
$13$ $$( 1 + 169654 T + 137858491849 T^{2} )^{2}$$
$17$ $$1 - 3914170410818 T^{2} +$$$$40\!\cdots\!01$$$$T^{4}$$
$19$ $$( 1 + 949462 T + 6131066257801 T^{2} )^{2}$$
$23$ $$1 - 75790028393378 T^{2} +$$$$17\!\cdots\!01$$$$T^{4}$$
$29$ $$1 - 831288000078482 T^{2} +$$$$17\!\cdots\!01$$$$T^{4}$$
$31$ $$( 1 + 29793118 T + 819628286980801 T^{2} )^{2}$$
$37$ $$( 1 + 60811846 T + 4808584372417849 T^{2} )^{2}$$
$41$ $$1 + 6157996032931678 T^{2} +$$$$18\!\cdots\!01$$$$T^{4}$$
$43$ $$( 1 - 107419706 T + 21611482313284249 T^{2} )^{2}$$
$47$ $$1 - 33376598707262018 T^{2} +$$$$27\!\cdots\!01$$$$T^{4}$$
$53$ $$1 - 312876100791567218 T^{2} +$$$$30\!\cdots\!01$$$$T^{4}$$
$59$ $$1 - 600827685707033522 T^{2} +$$$$26\!\cdots\!01$$$$T^{4}$$
$61$ $$( 1 - 1030793642 T + 713342911662882601 T^{2} )^{2}$$
$67$ $$( 1 - 1876742474 T + 1822837804551761449 T^{2} )^{2}$$
$71$ $$1 + 690030731290713118 T^{2} +$$$$10\!\cdots\!01$$$$T^{4}$$
$73$ $$( 1 + 2846528494 T + 4297625829703557649 T^{2} )^{2}$$
$79$ $$( 1 - 1488647618 T + 9468276082626847201 T^{2} )^{2}$$
$83$ $$1 - 29429698146400299218 T^{2} +$$$$24\!\cdots\!01$$$$T^{4}$$
$89$ $$1 - 26116812713945754722 T^{2} +$$$$97\!\cdots\!01$$$$T^{4}$$
$97$ $$( 1 + 1592948926 T + 73742412689492826049 T^{2} )^{2}$$