# Properties

 Label 3.15.b.a Level 3 Weight 15 Character orbit 3.b Analytic conductor 3.730 Analytic rank 0 Dimension 4 CM No Inner twists 2

# Related objects

## Newspace parameters

 Level: $$N$$ = $$3$$ Weight: $$k$$ = $$15$$ Character orbit: $$[\chi]$$ = 3.b (of order $$2$$ and degree $$1$$)

## Newform invariants

 Self dual: No Analytic conductor: $$3.72986904456$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.1929141760.2 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{10}\cdot 3^{7}$$ 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.

 $$f(q)$$ $$=$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 549 + 3 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( -9824 + 8 \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( -190 \beta_{1} - 30 \beta_{2} + 3 \beta_{3} ) q^{5}$$ $$+ ( -78624 + 1845 \beta_{1} + 24 \beta_{2} + 9 \beta_{3} ) q^{6}$$ $$+ ( 206402 - 88 \beta_{2} - 11 \beta_{3} ) q^{7}$$ $$+ ( -16768 \beta_{1} + 480 \beta_{2} - 48 \beta_{3} ) q^{8}$$ $$+ ( -406215 + 19710 \beta_{1} - 306 \beta_{2} - 189 \beta_{3} ) q^{9}$$ $$+O(q^{10})$$ $$q$$ $$+ \beta_{1} q^{2}$$ $$+ ( 549 + 3 \beta_{1} - \beta_{2} ) q^{3}$$ $$+ ( -9824 + 8 \beta_{2} + \beta_{3} ) q^{4}$$ $$+ ( -190 \beta_{1} - 30 \beta_{2} + 3 \beta_{3} ) q^{5}$$ $$+ ( -78624 + 1845 \beta_{1} + 24 \beta_{2} + 9 \beta_{3} ) q^{6}$$ $$+ ( 206402 - 88 \beta_{2} - 11 \beta_{3} ) q^{7}$$ $$+ ( -16768 \beta_{1} + 480 \beta_{2} - 48 \beta_{3} ) q^{8}$$ $$+ ( -406215 + 19710 \beta_{1} - 306 \beta_{2} - 189 \beta_{3} ) q^{9}$$ $$+ ( 4979520 - 80 \beta_{2} - 10 \beta_{3} ) q^{10}$$ $$+ ( 64570 \beta_{1} - 2310 \beta_{2} + 231 \beta_{3} ) q^{11}$$ $$+ ( -39358944 - 177216 \beta_{1} + 2696 \beta_{2} + 1701 \beta_{3} ) q^{12}$$ $$+ ( 50424218 + 5408 \beta_{2} + 676 \beta_{3} ) q^{13}$$ $$+ ( 463010 \beta_{1} - 5280 \beta_{2} + 528 \beta_{3} ) q^{14}$$ $$+ ( -112432320 - 117270 \beta_{1} - 16710 \beta_{2} - 8271 \beta_{3} ) q^{15}$$ $$+ ( 278499328 - 26112 \beta_{2} - 3264 \beta_{3} ) q^{16}$$ $$+ ( -2659944 \beta_{1} + 107160 \beta_{2} - 10716 \beta_{3} ) q^{17}$$ $$+ ( -516559680 + 2439801 \beta_{1} + 66960 \beta_{2} + 21546 \beta_{3} ) q^{18}$$ $$+ ( 328736810 - 20712 \beta_{2} - 2589 \beta_{3} ) q^{19}$$ $$+ ( 2099840 \beta_{1} - 496320 \beta_{2} + 49632 \beta_{3} ) q^{20}$$ $$+ ( 486935946 + 2244390 \beta_{1} - 127994 \beta_{2} - 18711 \beta_{3} ) q^{21}$$ $$+ ( -1692250560 + 627440 \beta_{2} + 78430 \beta_{3} ) q^{22}$$ $$+ ( 2608556 \beta_{1} + 944940 \beta_{2} - 94494 \beta_{3} ) q^{23}$$ $$+ ( 3356301312 - 34669440 \beta_{1} - 208032 \beta_{2} - 45936 \beta_{3} ) q^{24}$$ $$+ ( -1720620695 - 2357920 \beta_{2} - 294740 \beta_{3} ) q^{25}$$ $$+ ( 34654490 \beta_{1} + 324480 \beta_{2} - 32448 \beta_{3} ) q^{26}$$ $$+ ( 5215724541 + 47961153 \beta_{1} + 2020221 \beta_{2} + 116397 \beta_{3} ) q^{27}$$ $$+ ( -8752875712 + 2515728 \beta_{2} + 314466 \beta_{3} ) q^{28}$$ $$+ ( -123473690 \beta_{1} - 4739130 \beta_{2} + 473913 \beta_{3} ) q^{29}$$ $$+ ( 3073412160 + 16416000 \beta_{1} - 4908240 \beta_{2} - 17010 \beta_{3} ) q^{30}$$ $$+ ( -8742677518 + 7449800 \beta_{2} + 931225 \beta_{3} ) q^{31}$$ $$+ ( 79915008 \beta_{1} + 6297600 \beta_{2} - 629760 \beta_{3} ) q^{32}$$ $$+ ( -14884309440 + 137094210 \beta_{1} + 614130 \beta_{2} + 75933 \beta_{3} ) q^{33}$$ $$+ ( 69711812352 - 26423232 \beta_{2} - 3302904 \beta_{3} ) q^{34}$$ $$+ ( -41782460 \beta_{1} + 2509380 \beta_{2} - 250938 \beta_{3} ) q^{35}$$ $$+ ( -70597731168 - 559647360 \beta_{1} + 24846984 \beta_{2} - 1058535 \beta_{3} ) q^{36}$$ $$+ ( 13894197482 + 18688992 \beta_{2} + 2336124 \beta_{3} ) q^{37}$$ $$+ ( 389133002 \beta_{1} - 1242720 \beta_{2} + 124272 \beta_{3} ) q^{38}$$ $$+ ( 4722171714 + 51397710 \beta_{1} - 55242746 \beta_{2} + 1149876 \beta_{3} ) q^{39}$$ $$+ ( 26551848960 + 39311360 \beta_{2} + 4913920 \beta_{3} ) q^{40}$$ $$+ ( 405414380 \beta_{1} - 29907540 \beta_{2} + 2990754 \beta_{3} ) q^{41}$$ $$+ ( -58820973120 + 895310730 \beta_{1} + 8973840 \beta_{2} + 3012354 \beta_{3} ) q^{42}$$ $$+ ( -80919732262 - 58708072 \beta_{2} - 7338509 \beta_{3} ) q^{43}$$ $$+ ( -2463950720 \beta_{1} - 200640 \beta_{2} + 20064 \beta_{3} ) q^{44}$$ $$+ ( 251755620480 + 48720690 \beta_{1} + 160116210 \beta_{2} - 4441473 \beta_{3} ) q^{45}$$ $$+ ( -68365035648 - 24488672 \beta_{2} - 3061084 \beta_{3} ) q^{46}$$ $$+ ( -486037064 \beta_{1} + 159217080 \beta_{2} - 15921708 \beta_{3} ) q^{47}$$ $$+ ( 263759745024 + 1317734400 \beta_{1} - 255233536 \beta_{2} - 5552064 \beta_{3} ) q^{48}$$ $$+ ( -561644280141 - 36326752 \beta_{2} - 4540844 \beta_{3} ) q^{49}$$ $$+ ( 5155074025 \beta_{1} - 141475200 \beta_{2} + 14147520 \beta_{3} ) q^{50}$$ $$+ ( 664104220416 - 5740872840 \beta_{1} - 20438856 \beta_{2} - 503604 \beta_{3} ) q^{51}$$ $$+ ( -82074486208 + 350265552 \beta_{2} + 43783194 \beta_{3} ) q^{52}$$ $$+ ( 1980312426 \beta_{1} - 205206390 \beta_{2} + 20520639 \beta_{3} ) q^{53}$$ $$+ ( -1256965897824 + 1089013005 \beta_{1} + 439559784 \beta_{2} + 35839827 \beta_{3} ) q^{54}$$ $$+ ( -208080512640 - 187895840 \beta_{2} - 23486980 \beta_{3} ) q^{55}$$ $$+ ( -8502782720 \beta_{1} + 64436160 \beta_{2} - 6443616 \beta_{3} ) q^{56}$$ $$+ ( 268413364242 + 1368719646 \beta_{1} - 310282418 \beta_{2} - 4403889 \beta_{3} ) q^{57}$$ $$+ ( 3235998467520 - 760311280 \beta_{2} - 95038910 \beta_{3} ) q^{58}$$ $$+ ( 11855514770 \beta_{1} + 541966290 \beta_{2} - 54196629 \beta_{3} ) q^{59}$$ $$+ ( -2272321658880 + 7733589120 \beta_{1} - 150613440 \beta_{2} - 89646624 \beta_{3} ) q^{60}$$ $$+ ( -1042910406598 + 1031162720 \beta_{2} + 128895340 \beta_{3} ) q^{61}$$ $$+ ( -30466294318 \beta_{1} + 446988000 \beta_{2} - 44698800 \beta_{3} ) q^{62}$$ $$+ ( 736628672178 + 8094362940 \beta_{1} - 303408252 \beta_{2} - 6941997 \beta_{3} ) q^{63}$$ $$+ ( 2468520460288 - 90783744 \beta_{2} - 11347968 \beta_{3} ) q^{64}$$ $$+ ( -9422904140 \beta_{1} - 2047469580 \beta_{2} + 204746958 \beta_{3} ) q^{65}$$ $$+ ( -3592965055680 - 16664313600 \beta_{1} + 1133201520 \beta_{2} + 133409430 \beta_{3} ) q^{66}$$ $$+ ( -2741059984438 - 136185928 \beta_{2} - 17023241 \beta_{3} ) q^{67}$$ $$+ ( 103181434368 \beta_{1} + 170315520 \beta_{2} - 17031552 \beta_{3} ) q^{68}$$ $$+ ( 3806832871296 - 2535067620 \beta_{1} + 445306044 \beta_{2} + 230135382 \beta_{3} ) q^{69}$$ $$+ ( 1095034711680 - 454709920 \beta_{2} - 56838740 \beta_{3} ) q^{70}$$ $$+ ( -57986224860 \beta_{1} + 1733449380 \beta_{2} - 173344938 \beta_{3} ) q^{71}$$ $$+ ( 6203924213760 - 49107109248 \beta_{1} - 3888203040 \beta_{2} - 355719600 \beta_{3} ) q^{72}$$ $$+ ( -11161032730702 - 159645312 \beta_{2} - 19955664 \beta_{3} ) q^{73}$$ $$+ ( -40602903190 \beta_{1} + 1121339520 \beta_{2} - 112133952 \beta_{3} ) q^{74}$$ $$+ ( 9066390750765 + 38384204475 \beta_{1} + 3821527415 \beta_{2} - 501352740 \beta_{3} ) q^{75}$$ $$+ ( -4812373821376 + 2833369168 \beta_{2} + 354171146 \beta_{3} ) q^{76}$$ $$+ ( 33453142580 \beta_{1} - 224953740 \beta_{2} + 22495374 \beta_{3} ) q^{77}$$ $$+ ( -1347031183680 + 61414377570 \beta_{1} + 963122160 \beta_{2} + 382854186 \beta_{3} ) q^{78}$$ $$+ ( 10303860644690 - 7070519992 \beta_{2} - 883814999 \beta_{3} ) q^{79}$$ $$+ ( -53676298240 \beta_{1} - 5773025280 \beta_{2} + 577302528 \beta_{3} ) q^{80}$$ $$+ ( -4347204694479 + 73294986780 \beta_{1} - 5080561380 \beta_{2} + 864875394 \beta_{3} ) q^{81}$$ $$+ ( -10625100071040 + 4678876960 \beta_{2} + 584859620 \beta_{3} ) q^{82}$$ $$+ ( -39813805394 \beta_{1} + 4872526830 \beta_{2} - 487252683 \beta_{3} ) q^{83}$$ $$+ ( -15486345072576 - 72719091840 \beta_{1} + 6511362064 \beta_{2} + 534906666 \beta_{3} ) q^{84}$$ $$+ ( 11323069954560 + 8689580160 \beta_{2} + 1086197520 \beta_{3} ) q^{85}$$ $$+ ( 90273005690 \beta_{1} - 3522484320 \beta_{2} + 352248432 \beta_{3} ) q^{86}$$ $$+ ( -10412909881920 - 190957483170 \beta_{1} - 4882716210 \beta_{2} - 2147710941 \beta_{3} ) q^{87}$$ $$+ ( 36849387294720 - 9421998080 \beta_{2} - 1177749760 \beta_{3} ) q^{88}$$ $$+ ( 136294939380 \beta_{1} + 1108462260 \beta_{2} - 110846226 \beta_{3} ) q^{89}$$ $$+ ( -1276871843520 + 101806502400 \beta_{1} - 1742141520 \beta_{2} - 911976570 \beta_{3} ) q^{90}$$ $$+ ( 5861436097972 - 3321109168 \beta_{2} - 415138646 \beta_{3} ) q^{91}$$ $$+ ( 45782513408 \beta_{1} + 14012576640 \beta_{2} - 1401257664 \beta_{3} ) q^{92}$$ $$+ ( -36429316018182 - 163810938954 \beta_{1} + 2104905718 \beta_{2} + 1584013725 \beta_{3} ) q^{93}$$ $$+ ( 12738059373312 - 11530716352 \beta_{2} - 1441339544 \beta_{3} ) q^{94}$$ $$+ ( -63063955820 \beta_{1} - 7814101740 \beta_{2} + 781410174 \beta_{3} ) q^{95}$$ $$+ ( 20454457540608 + 98473052160 \beta_{1} + 4468488192 \beta_{2} + 2096520192 \beta_{3} ) q^{96}$$ $$+ ( 17632495153922 + 22801900832 \beta_{2} + 2850237604 \beta_{3} ) q^{97}$$ $$+ ( -455715471309 \beta_{1} - 2179605120 \beta_{2} + 217960512 \beta_{3} ) q^{98}$$ $$+ ( -21526343879040 + 196983732330 \beta_{1} + 17632180170 \beta_{2} + 1364449779 \beta_{3} ) q^{99}$$ $$+O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q$$ $$\mathstrut +\mathstrut 2196q^{3}$$ $$\mathstrut -\mathstrut 39296q^{4}$$ $$\mathstrut -\mathstrut 314496q^{6}$$ $$\mathstrut +\mathstrut 825608q^{7}$$ $$\mathstrut -\mathstrut 1624860q^{9}$$ $$\mathstrut +\mathstrut O(q^{10})$$ $$4q$$ $$\mathstrut +\mathstrut 2196q^{3}$$ $$\mathstrut -\mathstrut 39296q^{4}$$ $$\mathstrut -\mathstrut 314496q^{6}$$ $$\mathstrut +\mathstrut 825608q^{7}$$ $$\mathstrut -\mathstrut 1624860q^{9}$$ $$\mathstrut +\mathstrut 19918080q^{10}$$ $$\mathstrut -\mathstrut 157435776q^{12}$$ $$\mathstrut +\mathstrut 201696872q^{13}$$ $$\mathstrut -\mathstrut 449729280q^{15}$$ $$\mathstrut +\mathstrut 1113997312q^{16}$$ $$\mathstrut -\mathstrut 2066238720q^{18}$$ $$\mathstrut +\mathstrut 1314947240q^{19}$$ $$\mathstrut +\mathstrut 1947743784q^{21}$$ $$\mathstrut -\mathstrut 6769002240q^{22}$$ $$\mathstrut +\mathstrut 13425205248q^{24}$$ $$\mathstrut -\mathstrut 6882482780q^{25}$$ $$\mathstrut +\mathstrut 20862898164q^{27}$$ $$\mathstrut -\mathstrut 35011502848q^{28}$$ $$\mathstrut +\mathstrut 12293648640q^{30}$$ $$\mathstrut -\mathstrut 34970710072q^{31}$$ $$\mathstrut -\mathstrut 59537237760q^{33}$$ $$\mathstrut +\mathstrut 278847249408q^{34}$$ $$\mathstrut -\mathstrut 282390924672q^{36}$$ $$\mathstrut +\mathstrut 55576789928q^{37}$$ $$\mathstrut +\mathstrut 18888686856q^{39}$$ $$\mathstrut +\mathstrut 106207395840q^{40}$$ $$\mathstrut -\mathstrut 235283892480q^{42}$$ $$\mathstrut -\mathstrut 323678929048q^{43}$$ $$\mathstrut +\mathstrut 1007022481920q^{45}$$ $$\mathstrut -\mathstrut 273460142592q^{46}$$ $$\mathstrut +\mathstrut 1055038980096q^{48}$$ $$\mathstrut -\mathstrut 2246577120564q^{49}$$ $$\mathstrut +\mathstrut 2656416881664q^{51}$$ $$\mathstrut -\mathstrut 328297944832q^{52}$$ $$\mathstrut -\mathstrut 5027863591296q^{54}$$ $$\mathstrut -\mathstrut 832322050560q^{55}$$ $$\mathstrut +\mathstrut 1073653456968q^{57}$$ $$\mathstrut +\mathstrut 12943993870080q^{58}$$ $$\mathstrut -\mathstrut 9089286635520q^{60}$$ $$\mathstrut -\mathstrut 4171641626392q^{61}$$ $$\mathstrut +\mathstrut 2946514688712q^{63}$$ $$\mathstrut +\mathstrut 9874081841152q^{64}$$ $$\mathstrut -\mathstrut 14371860222720q^{66}$$ $$\mathstrut -\mathstrut 10964239937752q^{67}$$ $$\mathstrut +\mathstrut 15227331485184q^{69}$$ $$\mathstrut +\mathstrut 4380138846720q^{70}$$ $$\mathstrut +\mathstrut 24815696855040q^{72}$$ $$\mathstrut -\mathstrut 44644130922808q^{73}$$ $$\mathstrut +\mathstrut 36265563003060q^{75}$$ $$\mathstrut -\mathstrut 19249495285504q^{76}$$ $$\mathstrut -\mathstrut 5388124734720q^{78}$$ $$\mathstrut +\mathstrut 41215442578760q^{79}$$ $$\mathstrut -\mathstrut 17388818777916q^{81}$$ $$\mathstrut -\mathstrut 42500400284160q^{82}$$ $$\mathstrut -\mathstrut 61945380290304q^{84}$$ $$\mathstrut +\mathstrut 45292279818240q^{85}$$ $$\mathstrut -\mathstrut 41651639527680q^{87}$$ $$\mathstrut +\mathstrut 147397549178880q^{88}$$ $$\mathstrut -\mathstrut 5107487374080q^{90}$$ $$\mathstrut +\mathstrut 23445744391888q^{91}$$ $$\mathstrut -\mathstrut 145717264072728q^{93}$$ $$\mathstrut +\mathstrut 50952237493248q^{94}$$ $$\mathstrut +\mathstrut 81817830162432q^{96}$$ $$\mathstrut +\mathstrut 70529980615688q^{97}$$ $$\mathstrut -\mathstrut 86105375516160q^{99}$$ $$\mathstrut +\mathstrut O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4}\mathstrut +\mathstrut$$ $$364$$ $$x^{2}\mathstrut +\mathstrut$$ $$3640$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$12 \nu$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{3} + 8 \nu^{2} + 688 \nu + 1456$$ $$\beta_{3}$$ $$=$$ $$-16 \nu^{3} + 80 \nu^{2} - 5504 \nu + 14560$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/12$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3}\mathstrut +\mathstrut$$ $$8$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$26208$$$$)/144$$ $$\nu^{3}$$ $$=$$ $$($$$$-$$$$\beta_{3}\mathstrut +\mathstrut$$ $$10$$ $$\beta_{2}\mathstrut -\mathstrut$$ $$1032$$ $$\beta_{1}$$$$)/36$$

## 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
 − 18.8072i − 3.20795i 3.20795i 18.8072i
225.686i 1922.67 1042.26i −34550.1 23159.5i −235223. 433920.i 478389. 4.09983e6i 2.61037e6 4.00784e6i 5.22678e6
2.2 38.4954i −824.672 + 2025.56i 14902.1 122931.i 77974.7 + 31746.1i −65585.1 1.20437e6i −3.42280e6 3.34084e6i 4.73226e6
2.3 38.4954i −824.672 2025.56i 14902.1 122931.i 77974.7 31746.1i −65585.1 1.20437e6i −3.42280e6 + 3.34084e6i 4.73226e6
2.4 225.686i 1922.67 + 1042.26i −34550.1 23159.5i −235223. + 433920.i 478389. 4.09983e6i 2.61037e6 + 4.00784e6i 5.22678e6
 $$n$$: 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 $$S_{15}^{\mathrm{new}}(3, [\chi])$$.