# Properties

 Label 6.11.b.a Level 6 Weight 11 Character orbit 6.b Analytic conductor 3.812 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6 = 2 \cdot 3$$ Weight: $$k$$ = $$11$$ Character orbit: $$[\chi]$$ = 6.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$3.81214351604$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{85})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{11}\cdot 3^{4}$$ Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 21 - 3 \beta_{1} + \beta_{2} ) q^{3}$$ $$-512 q^{4}$$ $$+ ( 24 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{5}$$ $$+ ( 1344 + 21 \beta_{1} - 4 \beta_{3} ) q^{6}$$ $$+ ( -11278 - 18 \beta_{1} + 48 \beta_{2} + 3 \beta_{3} ) q^{7}$$ $$-512 \beta_{1} q^{8}$$ $$+ ( 39753 + 1404 \beta_{1} + 42 \beta_{2} + 21 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 21 - 3 \beta_{1} + \beta_{2} ) q^{3}$$ $$-512 q^{4}$$ $$+ ( 24 \beta_{1} + 6 \beta_{2} - 3 \beta_{3} ) q^{5}$$ $$+ ( 1344 + 21 \beta_{1} - 4 \beta_{3} ) q^{6}$$ $$+ ( -11278 - 18 \beta_{1} + 48 \beta_{2} + 3 \beta_{3} ) q^{7}$$ $$-512 \beta_{1} q^{8}$$ $$+ ( 39753 + 1404 \beta_{1} + 42 \beta_{2} + 21 \beta_{3} ) q^{9}$$ $$+ ( -13440 + 144 \beta_{1} - 384 \beta_{2} - 24 \beta_{3} ) q^{10}$$ $$+ ( -3696 \beta_{1} - 210 \beta_{2} + 105 \beta_{3} ) q^{11}$$ $$+ ( -10752 + 1536 \beta_{1} - 512 \beta_{2} ) q^{12}$$ $$+ ( 68810 - 360 \beta_{1} + 960 \beta_{2} + 60 \beta_{3} ) q^{13}$$ $$+ ( -11422 \beta_{1} + 384 \beta_{2} - 192 \beta_{3} ) q^{14}$$ $$+ ( -295200 + 41436 \beta_{1} + 1134 \beta_{2} - 105 \beta_{3} ) q^{15}$$ $$+ 262144 q^{16}$$ $$+ ( -74304 \beta_{1} + 1416 \beta_{2} - 708 \beta_{3} ) q^{17}$$ $$+ ( -726912 + 38745 \beta_{1} + 2688 \beta_{2} - 168 \beta_{3} ) q^{18}$$ $$+ ( -392182 + 1746 \beta_{1} - 4656 \beta_{2} - 291 \beta_{3} ) q^{19}$$ $$+ ( -12288 \beta_{1} - 3072 \beta_{2} + 1536 \beta_{3} ) q^{20}$$ $$+ ( 2407002 + 75144 \beta_{1} - 11278 \beta_{2} + 567 \beta_{3} ) q^{21}$$ $$+ ( 1932672 - 5040 \beta_{1} + 13440 \beta_{2} + 840 \beta_{3} ) q^{22}$$ $$+ ( -90336 \beta_{1} - 4956 \beta_{2} + 2478 \beta_{3} ) q^{23}$$ $$+ ( -688128 - 10752 \beta_{1} + 2048 \beta_{3} ) q^{24}$$ $$+ ( -8433095 + 7560 \beta_{1} - 20160 \beta_{2} - 1260 \beta_{3} ) q^{25}$$ $$+ ( 65930 \beta_{1} + 7680 \beta_{2} - 3840 \beta_{3} ) q^{26}$$ $$+ ( 8654877 - 247779 \beta_{1} + 33579 \beta_{2} - 4797 \beta_{3} ) q^{27}$$ $$+ ( 5774336 + 9216 \beta_{1} - 24576 \beta_{2} - 1536 \beta_{3} ) q^{28}$$ $$+ ( 757128 \beta_{1} + 17682 \beta_{2} - 8841 \beta_{3} ) q^{29}$$ $$+ ( -21432960 - 290160 \beta_{1} - 13440 \beta_{2} - 4536 \beta_{3} ) q^{30}$$ $$+ ( -5446462 - 54810 \beta_{1} + 146160 \beta_{2} + 9135 \beta_{3} ) q^{31}$$ $$+ 262144 \beta_{1} q^{32}$$ $$+ ( 6493536 - 1510236 \beta_{1} - 39690 \beta_{2} + 15099 \beta_{3} ) q^{33}$$ $$+ ( 37771776 + 33984 \beta_{1} - 90624 \beta_{2} - 5664 \beta_{3} ) q^{34}$$ $$+ ( 1956288 \beta_{1} - 57588 \beta_{2} + 28794 \beta_{3} ) q^{35}$$ $$+ ( -20353536 - 718848 \beta_{1} - 21504 \beta_{2} - 10752 \beta_{3} ) q^{36}$$ $$+ ( -17753542 + 102312 \beta_{1} - 272832 \beta_{2} - 17052 \beta_{3} ) q^{37}$$ $$+ ( -378214 \beta_{1} - 37248 \beta_{2} + 18624 \beta_{3} ) q^{38}$$ $$+ ( 54321810 + 619770 \beta_{1} + 68810 \beta_{2} + 11340 \beta_{3} ) q^{39}$$ $$+ ( 6881280 - 73728 \beta_{1} + 196608 \beta_{2} + 12288 \beta_{3} ) q^{40}$$ $$+ ( -4121328 \beta_{1} + 80484 \beta_{2} - 40242 \beta_{3} ) q^{41}$$ $$+ ( -36308352 + 2379786 \beta_{1} + 72576 \beta_{2} + 45112 \beta_{3} ) q^{42}$$ $$+ ( -117672166 - 122094 \beta_{1} + 325584 \beta_{2} + 20349 \beta_{3} ) q^{43}$$ $$+ ( 1892352 \beta_{1} + 107520 \beta_{2} - 53760 \beta_{3} ) q^{44}$$ $$+ ( 78079680 + 4602096 \beta_{1} - 236106 \beta_{2} - 157743 \beta_{3} ) q^{45}$$ $$+ ( 47203584 - 118944 \beta_{1} + 317184 \beta_{2} + 19824 \beta_{3} ) q^{46}$$ $$+ ( -4185600 \beta_{1} + 86760 \beta_{2} - 43380 \beta_{3} ) q^{47}$$ $$+ ( 5505024 - 786432 \beta_{1} + 262144 \beta_{2} ) q^{48}$$ $$+ ( -12514605 + 406008 \beta_{1} - 1082688 \beta_{2} - 67668 \beta_{3} ) q^{49}$$ $$+ ( -8372615 \beta_{1} - 161280 \beta_{2} + 80640 \beta_{3} ) q^{50}$$ $$+ ( -177144192 + 8099568 \beta_{1} + 267624 \beta_{2} + 295092 \beta_{3} ) q^{51}$$ $$+ ( -35230720 + 184320 \beta_{1} - 491520 \beta_{2} - 30720 \beta_{3} ) q^{52}$$ $$+ ( -9081864 \beta_{1} - 356034 \beta_{2} + 178017 \beta_{3} ) q^{53}$$ $$+ ( 120415680 + 8885133 \beta_{1} - 614016 \beta_{2} - 134316 \beta_{3} ) q^{54}$$ $$+ ( 675339840 - 675864 \beta_{1} + 1802304 \beta_{2} + 112644 \beta_{3} ) q^{55}$$ $$+ ( 5848064 \beta_{1} - 196608 \beta_{2} + 98304 \beta_{3} ) q^{56}$$ $$+ ( -264688302 - 2830524 \beta_{1} - 392182 \beta_{2} - 54999 \beta_{3} ) q^{57}$$ $$+ ( -391044480 + 424368 \beta_{1} - 1131648 \beta_{2} - 70728 \beta_{3} ) q^{58}$$ $$+ ( 17198448 \beta_{1} + 139638 \beta_{2} - 69819 \beta_{3} ) q^{59}$$ $$+ ( 151142400 - 21215232 \beta_{1} - 580608 \beta_{2} + 53760 \beta_{3} ) q^{60}$$ $$+ ( -296009686 - 353592 \beta_{1} + 942912 \beta_{2} + 58932 \beta_{3} ) q^{61}$$ $$+ ( -5884942 \beta_{1} + 1169280 \beta_{2} - 584640 \beta_{3} ) q^{62}$$ $$+ ( -226251774 - 28899450 \beta_{1} + 1979652 \beta_{2} - 390171 \beta_{3} ) q^{63}$$ $$-134217728 q^{64}$$ $$+ ( 46190640 \beta_{1} + 614460 \beta_{2} - 307230 \beta_{3} ) q^{65}$$ $$+ ( 780861312 + 5768784 \beta_{1} + 1932672 \beta_{2} + 158760 \beta_{3} ) q^{66}$$ $$+ ( -74341462 + 898506 \beta_{1} - 2396016 \beta_{2} - 149751 \beta_{3} ) q^{67}$$ $$+ ( 38043648 \beta_{1} - 724992 \beta_{2} + 362496 \beta_{3} ) q^{68}$$ $$+ ( 149067072 - 35706888 \beta_{1} - 936684 \beta_{2} + 368778 \beta_{3} ) q^{69}$$ $$+ ( -990562560 - 1382112 \beta_{1} + 3685632 \beta_{2} + 230352 \beta_{3} ) q^{70}$$ $$+ ( -14146272 \beta_{1} - 1281588 \beta_{2} + 640794 \beta_{3} ) q^{71}$$ $$+ ( 372178944 - 19837440 \beta_{1} - 1376256 \beta_{2} + 86016 \beta_{3} ) q^{72}$$ $$+ ( 1633567250 + 832032 \beta_{1} - 2218752 \beta_{2} - 138672 \beta_{3} ) q^{73}$$ $$+ ( -16935046 \beta_{1} - 2182656 \beta_{2} + 1091328 \beta_{3} ) q^{74}$$ $$+ ( -1287507795 + 7949085 \beta_{1} - 8433095 \beta_{2} - 238140 \beta_{3} ) q^{75}$$ $$+ ( 200797184 - 893952 \beta_{1} + 2383872 \beta_{2} + 148992 \beta_{3} ) q^{76}$$ $$+ ( -35848848 \beta_{1} + 918876 \beta_{2} - 459438 \beta_{3} ) q^{77}$$ $$+ ( -330533760 + 53777490 \beta_{1} + 1451520 \beta_{2} - 275240 \beta_{3} ) q^{78}$$ $$+ ( 49820642 + 2886534 \beta_{1} - 7697424 \beta_{2} - 481089 \beta_{3} ) q^{79}$$ $$+ ( 6291456 \beta_{1} + 1572864 \beta_{2} - 786432 \beta_{3} ) q^{80}$$ $$+ ( 364741137 + 70979328 \beta_{1} + 10971828 \beta_{2} + 1192590 \beta_{3} ) q^{81}$$ $$+ ( 2094667008 + 1931616 \beta_{1} - 5150976 \beta_{2} - 321936 \beta_{3} ) q^{82}$$ $$+ ( -24958608 \beta_{1} + 2330154 \beta_{2} - 1165077 \beta_{3} ) q^{83}$$ $$+ ( -1232385024 - 38473728 \beta_{1} + 5774336 \beta_{2} - 290304 \beta_{3} ) q^{84}$$ $$+ ( -3220128000 - 9731232 \beta_{1} + 25949952 \beta_{2} + 1621872 \beta_{3} ) q^{85}$$ $$+ ( -118648918 \beta_{1} + 2604672 \beta_{2} - 1302336 \beta_{3} ) q^{86}$$ $$+ ( 52567200 + 136526292 \beta_{1} + 3341898 \beta_{2} - 3055035 \beta_{3} ) q^{87}$$ $$+ ( -989528064 + 2580480 \beta_{1} - 6881280 \beta_{2} - 430080 \beta_{3} ) q^{88}$$ $$+ ( -118832112 \beta_{1} - 3522372 \beta_{2} + 1761186 \beta_{3} ) q^{89}$$ $$+ ( -2310940800 + 85651344 \beta_{1} - 20191104 \beta_{2} + 944424 \beta_{3} ) q^{90}$$ $$+ ( 2079308020 + 2821500 \beta_{1} - 7524000 \beta_{2} - 470250 \beta_{3} ) q^{91}$$ $$+ ( 46252032 \beta_{1} + 2537472 \beta_{2} - 1268736 \beta_{3} ) q^{92}$$ $$+ ( 7936117098 + 142128336 \beta_{1} - 5446462 \beta_{2} + 1726515 \beta_{3} ) q^{93}$$ $$+ ( 2126369280 + 2082240 \beta_{1} - 5552640 \beta_{2} - 347040 \beta_{3} ) q^{94}$$ $$+ ( -225427488 \beta_{1} - 3330852 \beta_{2} + 1665426 \beta_{3} ) q^{95}$$ $$+ ( 352321536 + 5505024 \beta_{1} - 1048576 \beta_{3} ) q^{96}$$ $$+ ( -9794088766 + 1435608 \beta_{1} - 3828288 \beta_{2} - 239268 \beta_{3} ) q^{97}$$ $$+ ( -9266541 \beta_{1} - 8661504 \beta_{2} + 4330752 \beta_{3} ) q^{98}$$ $$+ ( -656728128 - 271729080 \beta_{1} + 586782 \beta_{2} + 6000813 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 84q^{3}$$ $$\mathstrut -\mathstrut 2048q^{4}$$ $$\mathstrut +\mathstrut 5376q^{6}$$ $$\mathstrut -\mathstrut 45112q^{7}$$ $$\mathstrut +\mathstrut 159012q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 84q^{3}$$ $$\mathstrut -\mathstrut 2048q^{4}$$ $$\mathstrut +\mathstrut 5376q^{6}$$ $$\mathstrut -\mathstrut 45112q^{7}$$ $$\mathstrut +\mathstrut 159012q^{9}$$ $$\mathstrut -\mathstrut 53760q^{10}$$ $$\mathstrut -\mathstrut 43008q^{12}$$ $$\mathstrut +\mathstrut 275240q^{13}$$ $$\mathstrut -\mathstrut 1180800q^{15}$$ $$\mathstrut +\mathstrut 1048576q^{16}$$ $$\mathstrut -\mathstrut 2907648q^{18}$$ $$\mathstrut -\mathstrut 1568728q^{19}$$ $$\mathstrut +\mathstrut 9628008q^{21}$$ $$\mathstrut +\mathstrut 7730688q^{22}$$ $$\mathstrut -\mathstrut 2752512q^{24}$$ $$\mathstrut -\mathstrut 33732380q^{25}$$ $$\mathstrut +\mathstrut 34619508q^{27}$$ $$\mathstrut +\mathstrut 23097344q^{28}$$ $$\mathstrut -\mathstrut 85731840q^{30}$$ $$\mathstrut -\mathstrut 21785848q^{31}$$ $$\mathstrut +\mathstrut 25974144q^{33}$$ $$\mathstrut +\mathstrut 151087104q^{34}$$ $$\mathstrut -\mathstrut 81414144q^{36}$$ $$\mathstrut -\mathstrut 71014168q^{37}$$ $$\mathstrut +\mathstrut 217287240q^{39}$$ $$\mathstrut +\mathstrut 27525120q^{40}$$ $$\mathstrut -\mathstrut 145233408q^{42}$$ $$\mathstrut -\mathstrut 470688664q^{43}$$ $$\mathstrut +\mathstrut 312318720q^{45}$$ $$\mathstrut +\mathstrut 188814336q^{46}$$ $$\mathstrut +\mathstrut 22020096q^{48}$$ $$\mathstrut -\mathstrut 50058420q^{49}$$ $$\mathstrut -\mathstrut 708576768q^{51}$$ $$\mathstrut -\mathstrut 140922880q^{52}$$ $$\mathstrut +\mathstrut 481662720q^{54}$$ $$\mathstrut +\mathstrut 2701359360q^{55}$$ $$\mathstrut -\mathstrut 1058753208q^{57}$$ $$\mathstrut -\mathstrut 1564177920q^{58}$$ $$\mathstrut +\mathstrut 604569600q^{60}$$ $$\mathstrut -\mathstrut 1184038744q^{61}$$ $$\mathstrut -\mathstrut 905007096q^{63}$$ $$\mathstrut -\mathstrut 536870912q^{64}$$ $$\mathstrut +\mathstrut 3123445248q^{66}$$ $$\mathstrut -\mathstrut 297365848q^{67}$$ $$\mathstrut +\mathstrut 596268288q^{69}$$ $$\mathstrut -\mathstrut 3962250240q^{70}$$ $$\mathstrut +\mathstrut 1488715776q^{72}$$ $$\mathstrut +\mathstrut 6534269000q^{73}$$ $$\mathstrut -\mathstrut 5150031180q^{75}$$ $$\mathstrut +\mathstrut 803188736q^{76}$$ $$\mathstrut -\mathstrut 1322135040q^{78}$$ $$\mathstrut +\mathstrut 199282568q^{79}$$ $$\mathstrut +\mathstrut 1458964548q^{81}$$ $$\mathstrut +\mathstrut 8378668032q^{82}$$ $$\mathstrut -\mathstrut 4929540096q^{84}$$ $$\mathstrut -\mathstrut 12880512000q^{85}$$ $$\mathstrut +\mathstrut 210268800q^{87}$$ $$\mathstrut -\mathstrut 3958112256q^{88}$$ $$\mathstrut -\mathstrut 9243763200q^{90}$$ $$\mathstrut +\mathstrut 8317232080q^{91}$$ $$\mathstrut +\mathstrut 31744468392q^{93}$$ $$\mathstrut +\mathstrut 8505477120q^{94}$$ $$\mathstrut +\mathstrut 1409286144q^{96}$$ $$\mathstrut -\mathstrut 39176355064q^{97}$$ $$\mathstrut -\mathstrut 2626912512q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut -\mathstrut$$ $$2$$ $$x^{3}\mathstrut -\mathstrut$$ $$37$$ $$x^{2}\mathstrut +\mathstrut$$ $$38$$ $$x\mathstrut +\mathstrut$$ $$531$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-32 \nu^{3} + 48 \nu^{2} + 464 \nu - 240$$$$)/93$$ $$\beta_{2}$$ $$=$$ $$($$$$-36 \nu^{3} - 132 \nu^{2} + 2196 \nu + 2520$$$$)/31$$ $$\beta_{3}$$ $$=$$ $$($$$$-64 \nu^{3} + 3072 \nu^{2} + 928 \nu - 58512$$$$)/31$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$16$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$60$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$432$$$$)/864$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut -\mathstrut$$ $$6$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$1872$$$$)/96$$ $$\nu^{3}$$ $$=$$ $$($$$$14$$ $$\beta_{3}\mathstrut +\mathstrut$$ $$116$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1731$$ $$\beta_{1}\mathstrut +\mathstrut$$ $$12528$$$$)/432$$

## Character Values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/6\mathbb{Z}\right)^\times$$.

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
5.1
 −4.10977 + 1.41421i 5.10977 + 1.41421i −4.10977 − 1.41421i 5.10977 − 1.41421i
22.6274i −200.269 + 137.627i −512.000 3630.47i 3114.15 + 4531.57i −23226.5 11585.2i 21166.4 55125.0i 82148.2
5.2 22.6274i 242.269 18.8335i −512.000 4818.41i −426.153 5481.92i 670.530 11585.2i 58339.6 9125.53i −109028.
5.3 22.6274i −200.269 137.627i −512.000 3630.47i 3114.15 4531.57i −23226.5 11585.2i 21166.4 + 55125.0i 82148.2
5.4 22.6274i 242.269 + 18.8335i −512.000 4818.41i −426.153 + 5481.92i 670.530 11585.2i 58339.6 + 9125.53i −109028.
 $$n$$: 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
3.b Odd 1 yes

## Hecke kernels

There are no other newforms in $$S_{11}^{\mathrm{new}}(6, [\chi])$$.