# 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 $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$3.72986904456$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.0.1929141760.2 Defining polynomial: $$x^{4} + 364x^{2} + 3640$$ x^4 + 364*x^2 + 3640 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{10}\cdot 3^{7}$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} + 3 \beta_1 + 549) q^{3} + (\beta_{3} + 8 \beta_{2} - 9824) q^{4} + (3 \beta_{3} - 30 \beta_{2} - 190 \beta_1) q^{5} + (9 \beta_{3} + 24 \beta_{2} + 1845 \beta_1 - 78624) q^{6} + ( - 11 \beta_{3} - 88 \beta_{2} + 206402) q^{7} + ( - 48 \beta_{3} + 480 \beta_{2} - 16768 \beta_1) q^{8} + ( - 189 \beta_{3} - 306 \beta_{2} + 19710 \beta_1 - 406215) q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 + 3*b1 + 549) * q^3 + (b3 + 8*b2 - 9824) * q^4 + (3*b3 - 30*b2 - 190*b1) * q^5 + (9*b3 + 24*b2 + 1845*b1 - 78624) * q^6 + (-11*b3 - 88*b2 + 206402) * q^7 + (-48*b3 + 480*b2 - 16768*b1) * q^8 + (-189*b3 - 306*b2 + 19710*b1 - 406215) * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} + 3 \beta_1 + 549) q^{3} + (\beta_{3} + 8 \beta_{2} - 9824) q^{4} + (3 \beta_{3} - 30 \beta_{2} - 190 \beta_1) q^{5} + (9 \beta_{3} + 24 \beta_{2} + 1845 \beta_1 - 78624) q^{6} + ( - 11 \beta_{3} - 88 \beta_{2} + 206402) q^{7} + ( - 48 \beta_{3} + 480 \beta_{2} - 16768 \beta_1) q^{8} + ( - 189 \beta_{3} - 306 \beta_{2} + 19710 \beta_1 - 406215) q^{9} + ( - 10 \beta_{3} - 80 \beta_{2} + 4979520) q^{10} + (231 \beta_{3} - 2310 \beta_{2} + 64570 \beta_1) q^{11} + (1701 \beta_{3} + 2696 \beta_{2} - 177216 \beta_1 - 39358944) q^{12} + (676 \beta_{3} + 5408 \beta_{2} + 50424218) q^{13} + (528 \beta_{3} - 5280 \beta_{2} + 463010 \beta_1) q^{14} + ( - 8271 \beta_{3} - 16710 \beta_{2} - 117270 \beta_1 - 112432320) q^{15} + ( - 3264 \beta_{3} - 26112 \beta_{2} + 278499328) q^{16} + ( - 10716 \beta_{3} + 107160 \beta_{2} - 2659944 \beta_1) q^{17} + (21546 \beta_{3} + 66960 \beta_{2} + 2439801 \beta_1 - 516559680) q^{18} + ( - 2589 \beta_{3} - 20712 \beta_{2} + 328736810) q^{19} + (49632 \beta_{3} - 496320 \beta_{2} + 2099840 \beta_1) q^{20} + ( - 18711 \beta_{3} - 127994 \beta_{2} + \cdots + 486935946) q^{21}+ \cdots + (1364449779 \beta_{3} + 17632180170 \beta_{2} + \cdots - 21526343879040) q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 + 3*b1 + 549) * q^3 + (b3 + 8*b2 - 9824) * q^4 + (3*b3 - 30*b2 - 190*b1) * q^5 + (9*b3 + 24*b2 + 1845*b1 - 78624) * q^6 + (-11*b3 - 88*b2 + 206402) * q^7 + (-48*b3 + 480*b2 - 16768*b1) * q^8 + (-189*b3 - 306*b2 + 19710*b1 - 406215) * q^9 + (-10*b3 - 80*b2 + 4979520) * q^10 + (231*b3 - 2310*b2 + 64570*b1) * q^11 + (1701*b3 + 2696*b2 - 177216*b1 - 39358944) * q^12 + (676*b3 + 5408*b2 + 50424218) * q^13 + (528*b3 - 5280*b2 + 463010*b1) * q^14 + (-8271*b3 - 16710*b2 - 117270*b1 - 112432320) * q^15 + (-3264*b3 - 26112*b2 + 278499328) * q^16 + (-10716*b3 + 107160*b2 - 2659944*b1) * q^17 + (21546*b3 + 66960*b2 + 2439801*b1 - 516559680) * q^18 + (-2589*b3 - 20712*b2 + 328736810) * q^19 + (49632*b3 - 496320*b2 + 2099840*b1) * q^20 + (-18711*b3 - 127994*b2 + 2244390*b1 + 486935946) * q^21 + (78430*b3 + 627440*b2 - 1692250560) * q^22 + (-94494*b3 + 944940*b2 + 2608556*b1) * q^23 + (-45936*b3 - 208032*b2 - 34669440*b1 + 3356301312) * q^24 + (-294740*b3 - 2357920*b2 - 1720620695) * q^25 + (-32448*b3 + 324480*b2 + 34654490*b1) * q^26 + (116397*b3 + 2020221*b2 + 47961153*b1 + 5215724541) * q^27 + (314466*b3 + 2515728*b2 - 8752875712) * q^28 + (473913*b3 - 4739130*b2 - 123473690*b1) * q^29 + (-17010*b3 - 4908240*b2 + 16416000*b1 + 3073412160) * q^30 + (931225*b3 + 7449800*b2 - 8742677518) * q^31 + (-629760*b3 + 6297600*b2 + 79915008*b1) * q^32 + (75933*b3 + 614130*b2 + 137094210*b1 - 14884309440) * q^33 + (-3302904*b3 - 26423232*b2 + 69711812352) * q^34 + (-250938*b3 + 2509380*b2 - 41782460*b1) * q^35 + (-1058535*b3 + 24846984*b2 - 559647360*b1 - 70597731168) * q^36 + (2336124*b3 + 18688992*b2 + 13894197482) * q^37 + (124272*b3 - 1242720*b2 + 389133002*b1) * q^38 + (1149876*b3 - 55242746*b2 + 51397710*b1 + 4722171714) * q^39 + (4913920*b3 + 39311360*b2 + 26551848960) * q^40 + (2990754*b3 - 29907540*b2 + 405414380*b1) * q^41 + (3012354*b3 + 8973840*b2 + 895310730*b1 - 58820973120) * q^42 + (-7338509*b3 - 58708072*b2 - 80919732262) * q^43 + (20064*b3 - 200640*b2 - 2463950720*b1) * q^44 + (-4441473*b3 + 160116210*b2 + 48720690*b1 + 251755620480) * q^45 + (-3061084*b3 - 24488672*b2 - 68365035648) * q^46 + (-15921708*b3 + 159217080*b2 - 486037064*b1) * q^47 + (-5552064*b3 - 255233536*b2 + 1317734400*b1 + 263759745024) * q^48 + (-4540844*b3 - 36326752*b2 - 561644280141) * q^49 + (14147520*b3 - 141475200*b2 + 5155074025*b1) * q^50 + (-503604*b3 - 20438856*b2 - 5740872840*b1 + 664104220416) * q^51 + (43783194*b3 + 350265552*b2 - 82074486208) * q^52 + (20520639*b3 - 205206390*b2 + 1980312426*b1) * q^53 + (35839827*b3 + 439559784*b2 + 1089013005*b1 - 1256965897824) * q^54 + (-23486980*b3 - 187895840*b2 - 208080512640) * q^55 + (-6443616*b3 + 64436160*b2 - 8502782720*b1) * q^56 + (-4403889*b3 - 310282418*b2 + 1368719646*b1 + 268413364242) * q^57 + (-95038910*b3 - 760311280*b2 + 3235998467520) * q^58 + (-54196629*b3 + 541966290*b2 + 11855514770*b1) * q^59 + (-89646624*b3 - 150613440*b2 + 7733589120*b1 - 2272321658880) * q^60 + (128895340*b3 + 1031162720*b2 - 1042910406598) * q^61 + (-44698800*b3 + 446988000*b2 - 30466294318*b1) * q^62 + (-6941997*b3 - 303408252*b2 + 8094362940*b1 + 736628672178) * q^63 + (-11347968*b3 - 90783744*b2 + 2468520460288) * q^64 + (204746958*b3 - 2047469580*b2 - 9422904140*b1) * q^65 + (133409430*b3 + 1133201520*b2 - 16664313600*b1 - 3592965055680) * q^66 + (-17023241*b3 - 136185928*b2 - 2741059984438) * q^67 + (-17031552*b3 + 170315520*b2 + 103181434368*b1) * q^68 + (230135382*b3 + 445306044*b2 - 2535067620*b1 + 3806832871296) * q^69 + (-56838740*b3 - 454709920*b2 + 1095034711680) * q^70 + (-173344938*b3 + 1733449380*b2 - 57986224860*b1) * q^71 + (-355719600*b3 - 3888203040*b2 - 49107109248*b1 + 6203924213760) * q^72 + (-19955664*b3 - 159645312*b2 - 11161032730702) * q^73 + (-112133952*b3 + 1121339520*b2 - 40602903190*b1) * q^74 + (-501352740*b3 + 3821527415*b2 + 38384204475*b1 + 9066390750765) * q^75 + (354171146*b3 + 2833369168*b2 - 4812373821376) * q^76 + (22495374*b3 - 224953740*b2 + 33453142580*b1) * q^77 + (382854186*b3 + 963122160*b2 + 61414377570*b1 - 1347031183680) * q^78 + (-883814999*b3 - 7070519992*b2 + 10303860644690) * q^79 + (577302528*b3 - 5773025280*b2 - 53676298240*b1) * q^80 + (864875394*b3 - 5080561380*b2 + 73294986780*b1 - 4347204694479) * q^81 + (584859620*b3 + 4678876960*b2 - 10625100071040) * q^82 + (-487252683*b3 + 4872526830*b2 - 39813805394*b1) * q^83 + (534906666*b3 + 6511362064*b2 - 72719091840*b1 - 15486345072576) * q^84 + (1086197520*b3 + 8689580160*b2 + 11323069954560) * q^85 + (352248432*b3 - 3522484320*b2 + 90273005690*b1) * q^86 + (-2147710941*b3 - 4882716210*b2 - 190957483170*b1 - 10412909881920) * q^87 + (-1177749760*b3 - 9421998080*b2 + 36849387294720) * q^88 + (-110846226*b3 + 1108462260*b2 + 136294939380*b1) * q^89 + (-911976570*b3 - 1742141520*b2 + 101806502400*b1 - 1276871843520) * q^90 + (-415138646*b3 - 3321109168*b2 + 5861436097972) * q^91 + (-1401257664*b3 + 14012576640*b2 + 45782513408*b1) * q^92 + (1584013725*b3 + 2104905718*b2 - 163810938954*b1 - 36429316018182) * q^93 + (-1441339544*b3 - 11530716352*b2 + 12738059373312) * q^94 + (781410174*b3 - 7814101740*b2 - 63063955820*b1) * q^95 + (2096520192*b3 + 4468488192*b2 + 98473052160*b1 + 20454457540608) * q^96 + (2850237604*b3 + 22801900832*b2 + 17632495153922) * q^97 + (217960512*b3 - 2179605120*b2 - 455715471309*b1) * q^98 + (1364449779*b3 + 17632180170*b2 + 196983732330*b1 - 21526343879040) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2196 q^{3} - 39296 q^{4} - 314496 q^{6} + 825608 q^{7} - 1624860 q^{9}+O(q^{10})$$ 4 * q + 2196 * q^3 - 39296 * q^4 - 314496 * q^6 + 825608 * q^7 - 1624860 * q^9 $$4 q + 2196 q^{3} - 39296 q^{4} - 314496 q^{6} + 825608 q^{7} - 1624860 q^{9} + 19918080 q^{10} - 157435776 q^{12} + 201696872 q^{13} - 449729280 q^{15} + 1113997312 q^{16} - 2066238720 q^{18} + 1314947240 q^{19} + 1947743784 q^{21} - 6769002240 q^{22} + 13425205248 q^{24} - 6882482780 q^{25} + 20862898164 q^{27} - 35011502848 q^{28} + 12293648640 q^{30} - 34970710072 q^{31} - 59537237760 q^{33} + 278847249408 q^{34} - 282390924672 q^{36} + 55576789928 q^{37} + 18888686856 q^{39} + 106207395840 q^{40} - 235283892480 q^{42} - 323678929048 q^{43} + 1007022481920 q^{45} - 273460142592 q^{46} + 1055038980096 q^{48} - 2246577120564 q^{49} + 2656416881664 q^{51} - 328297944832 q^{52} - 5027863591296 q^{54} - 832322050560 q^{55} + 1073653456968 q^{57} + 12943993870080 q^{58} - 9089286635520 q^{60} - 4171641626392 q^{61} + 2946514688712 q^{63} + 9874081841152 q^{64} - 14371860222720 q^{66} - 10964239937752 q^{67} + 15227331485184 q^{69} + 4380138846720 q^{70} + 24815696855040 q^{72} - 44644130922808 q^{73} + 36265563003060 q^{75} - 19249495285504 q^{76} - 5388124734720 q^{78} + 41215442578760 q^{79} - 17388818777916 q^{81} - 42500400284160 q^{82} - 61945380290304 q^{84} + 45292279818240 q^{85} - 41651639527680 q^{87} + 147397549178880 q^{88} - 5107487374080 q^{90} + 23445744391888 q^{91} - 145717264072728 q^{93} + 50952237493248 q^{94} + 81817830162432 q^{96} + 70529980615688 q^{97} - 86105375516160 q^{99}+O(q^{100})$$ 4 * q + 2196 * q^3 - 39296 * q^4 - 314496 * q^6 + 825608 * q^7 - 1624860 * q^9 + 19918080 * q^10 - 157435776 * q^12 + 201696872 * q^13 - 449729280 * q^15 + 1113997312 * q^16 - 2066238720 * q^18 + 1314947240 * q^19 + 1947743784 * q^21 - 6769002240 * q^22 + 13425205248 * q^24 - 6882482780 * q^25 + 20862898164 * q^27 - 35011502848 * q^28 + 12293648640 * q^30 - 34970710072 * q^31 - 59537237760 * q^33 + 278847249408 * q^34 - 282390924672 * q^36 + 55576789928 * q^37 + 18888686856 * q^39 + 106207395840 * q^40 - 235283892480 * q^42 - 323678929048 * q^43 + 1007022481920 * q^45 - 273460142592 * q^46 + 1055038980096 * q^48 - 2246577120564 * q^49 + 2656416881664 * q^51 - 328297944832 * q^52 - 5027863591296 * q^54 - 832322050560 * q^55 + 1073653456968 * q^57 + 12943993870080 * q^58 - 9089286635520 * q^60 - 4171641626392 * q^61 + 2946514688712 * q^63 + 9874081841152 * q^64 - 14371860222720 * q^66 - 10964239937752 * q^67 + 15227331485184 * q^69 + 4380138846720 * q^70 + 24815696855040 * q^72 - 44644130922808 * q^73 + 36265563003060 * q^75 - 19249495285504 * q^76 - 5388124734720 * q^78 + 41215442578760 * q^79 - 17388818777916 * q^81 - 42500400284160 * q^82 - 61945380290304 * q^84 + 45292279818240 * q^85 - 41651639527680 * q^87 + 147397549178880 * q^88 - 5107487374080 * q^90 + 23445744391888 * q^91 - 145717264072728 * q^93 + 50952237493248 * q^94 + 81817830162432 * q^96 + 70529980615688 * q^97 - 86105375516160 * q^99

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

 $$\beta_{1}$$ $$=$$ $$12\nu$$ 12*v $$\beta_{2}$$ $$=$$ $$2\nu^{3} + 8\nu^{2} + 688\nu + 1456$$ 2*v^3 + 8*v^2 + 688*v + 1456 $$\beta_{3}$$ $$=$$ $$-16\nu^{3} + 80\nu^{2} - 5504\nu + 14560$$ -16*v^3 + 80*v^2 - 5504*v + 14560
 $$\nu$$ $$=$$ $$( \beta_1 ) / 12$$ (b1) / 12 $$\nu^{2}$$ $$=$$ $$( \beta_{3} + 8\beta_{2} - 26208 ) / 144$$ (b3 + 8*b2 - 26208) / 144 $$\nu^{3}$$ $$=$$ $$( -\beta_{3} + 10\beta_{2} - 1032\beta_1 ) / 36$$ (-b3 + 10*b2 - 1032*b1) / 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 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 3.15.b.a 4
3.b odd 2 1 inner 3.15.b.a 4
4.b odd 2 1 48.15.e.b 4
5.b even 2 1 75.15.c.d 4
5.c odd 4 2 75.15.d.b 8
12.b even 2 1 48.15.e.b 4
15.d odd 2 1 75.15.c.d 4
15.e even 4 2 75.15.d.b 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.15.b.a 4 1.a even 1 1 trivial
3.15.b.a 4 3.b odd 2 1 inner
48.15.e.b 4 4.b odd 2 1
48.15.e.b 4 12.b even 2 1
75.15.c.d 4 5.b even 2 1
75.15.c.d 4 15.d odd 2 1
75.15.d.b 8 5.c odd 4 2
75.15.d.b 8 15.e even 4 2

## Hecke kernels

This newform subspace is the entire newspace $$S_{15}^{\mathrm{new}}(3, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 52416 T^{2} + \cdots + 75479040$$
$3$ $$T^{4} - 2196 T^{3} + \cdots + 22876792454961$$
$5$ $$T^{4} + 15648272640 T^{2} + \cdots + 81\!\cdots\!00$$
$7$ $$(T^{2} - 412804 T - 31375221500)^{2}$$
$11$ $$T^{4} + 300096887650560 T^{2} + \cdots + 10\!\cdots\!00$$
$13$ $$(T^{2} - 100848436 T + 22\!\cdots\!00)^{2}$$
$17$ $$T^{4} + \cdots + 43\!\cdots\!60$$
$19$ $$(T^{2} - 657473620 T + 10\!\cdots\!96)^{2}$$
$23$ $$T^{4} + \cdots + 14\!\cdots\!40$$
$29$ $$T^{4} + \cdots + 32\!\cdots\!00$$
$31$ $$(T^{2} + 17485355036 T - 45\!\cdots\!76)^{2}$$
$37$ $$(T^{2} - 27788394964 T - 31\!\cdots\!00)^{2}$$
$41$ $$T^{4} + \cdots + 12\!\cdots\!00$$
$43$ $$(T^{2} + 161839464524 T - 26\!\cdots\!00)^{2}$$
$47$ $$T^{4} + \cdots + 16\!\cdots\!60$$
$53$ $$T^{4} + \cdots + 17\!\cdots\!60$$
$59$ $$T^{4} + \cdots + 34\!\cdots\!00$$
$61$ $$(T^{2} + 2085820813196 T - 90\!\cdots\!96)^{2}$$
$67$ $$(T^{2} + 5482119968876 T + 73\!\cdots\!00)^{2}$$
$71$ $$T^{4} + \cdots + 40\!\cdots\!00$$
$73$ $$(T^{2} + 22322065461404 T + 12\!\cdots\!00)^{2}$$
$79$ $$(T^{2} - 20607721289380 T - 37\!\cdots\!24)^{2}$$
$83$ $$T^{4} + \cdots + 44\!\cdots\!60$$
$89$ $$T^{4} + \cdots + 81\!\cdots\!00$$
$97$ $$(T^{2} - 35264990307844 T - 46\!\cdots\!00)^{2}$$