# Properties

 Label 15.9.c.a Level $15$ Weight $9$ Character orbit 15.c Analytic conductor $6.111$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [15,9,Mod(11,15)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(15, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 9, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("15.11");

S:= CuspForms(chi, 9);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$15 = 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 15.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$6.11067915092$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 4 x^{9} - 433 x^{8} - 2220 x^{7} + 49747 x^{6} + 744964 x^{5} + 4580249 x^{4} + 16418988 x^{3} + \cdots + 53656344$$ x^10 - 4*x^9 - 433*x^8 - 2220*x^7 + 49747*x^6 + 744964*x^5 + 4580249*x^4 + 16418988*x^3 + 38943804*x^2 + 57910464*x + 53656344 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{4}\cdot 3^{10}\cdot 5^{10}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{2} - 2 \beta_1 - 11) q^{3} + ( - \beta_{3} - 79) q^{4} + ( - \beta_{4} - 3 \beta_1) q^{5} + (\beta_{8} - \beta_{4} - 2 \beta_{3} + \cdots - 529) q^{6}+ \cdots + ( - \beta_{9} + 2 \beta_{8} + \cdots + 385) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b2 - 2*b1 - 11) * q^3 + (-b3 - 79) * q^4 + (-b4 - 3*b1) * q^5 + (b8 - b4 - 2*b3 + 7*b1 - 529) * q^6 + (b9 + b8 - 2*b6 - 3*b5 + b4 + 2*b3 - 6*b2 - 4*b1 + 717) * q^7 + (-2*b9 - 2*b8 - b7 - 6*b5 + 3*b4 + 7*b2 + 148*b1 - 3) * q^8 + (-b9 + 2*b8 + 2*b7 + 3*b6 - 5*b5 - b3 + 26*b2 - 115*b1 + 385) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{2} - 2 \beta_1 - 11) q^{3} + ( - \beta_{3} - 79) q^{4} + ( - \beta_{4} - 3 \beta_1) q^{5} + (\beta_{8} - \beta_{4} - 2 \beta_{3} + \cdots - 529) q^{6}+ \cdots + ( - 25421 \beta_{9} + 15604 \beta_{8} + \cdots - 3323878) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-b2 - 2*b1 - 11) * q^3 + (-b3 - 79) * q^4 + (-b4 - 3*b1) * q^5 + (b8 - b4 - 2*b3 + 7*b1 - 529) * q^6 + (b9 + b8 - 2*b6 - 3*b5 + b4 + 2*b3 - 6*b2 - 4*b1 + 717) * q^7 + (-2*b9 - 2*b8 - b7 - 6*b5 + 3*b4 + 7*b2 + 148*b1 - 3) * q^8 + (-b9 + 2*b8 + 2*b7 + 3*b6 - 5*b5 - b3 + 26*b2 - 115*b1 + 385) * q^9 + (5*b6 - 5*b3 + 20*b2 + 10*b1 - 880) * q^10 + (-5*b9 - b7 - 10*b5 - 7*b4 - 8*b2 + 332*b1 - 2) * q^11 + (-3*b9 - 6*b8 - 6*b7 + 18*b6 - 12*b5 + 24*b4 - 18*b3 + 25*b2 + 770*b1 - 391) * q^12 + (7*b9 + 10*b8 - 3*b7 - 36*b6 - 18*b5 + 7*b4 - 14*b3 - 34*b2 - 30*b1 - 5552) * q^13 + (-3*b9 + 16*b8 + 15*b7 + 10*b5 - 65*b4 + 54*b2 - 1550*b1 - 16) * q^14 + (15*b9 + 5*b8 + 16*b4 - 25*b3 + 30*b2 - 277*b1 - 2135) * q^15 + (-3*b9 - 12*b8 + 9*b7 - 38*b6 - 3*b4 + 119*b3 - 446*b2 - 202*b1 + 28163) * q^16 + (38*b9 + 26*b8 - 14*b7 + 102*b5 - 150*b4 - 454*b2 - 572*b1 + 132) * q^17 + (-11*b9 - 44*b8 - 11*b7 + 60*b6 + 32*b5 + 207*b4 - 110*b3 + 712*b2 - 83*b1 - 42148) * q^18 + (-49*b9 - 46*b8 - 3*b7 + 6*b6 + 150*b5 - 49*b4 - 122*b3 + 22*b2 + 54*b1 - 23192) * q^19 + (5*b9 - 60*b8 - 20*b7 - 50*b5 + 54*b4 + 275*b2 + 1672*b1 - 55) * q^20 + (-6*b9 + 24*b8 + 69*b7 + 81*b6 + 255*b5 + 147*b4 + 171*b3 - 756*b2 - 1809*b1 + 29214) * q^21 + (-8*b9 - 2*b8 - 6*b7 + 62*b6 + 30*b5 - 8*b4 + 638*b3 + 424*b2 + 208*b1 + 113182) * q^22 + (76*b9 - 165*b8 - b7 - 13*b5 - 286*b4 + 1396*b2 - 7152*b1 - 251) * q^23 + (39*b9 + 60*b8 - 45*b7 - 252*b6 - 270*b5 + 855*b4 + 831*b3 - 840*b2 + 6816*b1 + 114063) * q^24 - 78125 * q^25 + (-116*b9 + 162*b8 + 122*b7 - 70*b5 - 1480*b4 + 244*b2 + 10064*b1 - 158) * q^26 + (-59*b9 + 121*b8 - 38*b7 - 174*b6 + 131*b5 + 1017*b4 - 1076*b3 - 683*b2 - 2270*b1 - 33862) * q^27 + (174*b9 + 258*b8 - 84*b7 + 240*b6 - 438*b5 + 174*b4 - 1990*b3 + 3996*b2 + 1656*b1 - 335604) * q^28 + (9*b9 - 8*b8 - 213*b7 + 10*b5 - 1343*b4 - 2706*b2 - 2890*b1 + 632) * q^29 + (-30*b9 - 105*b8 + 30*b7 - 135*b6 - 300*b5 + 465*b4 - 765*b3 + 190*b2 + 8660*b1 - 96955) * q^30 + (-39*b9 - 336*b8 + 297*b7 + 350*b6 - 180*b5 - 39*b4 + 290*b3 - 8182*b2 - 3458*b1 + 89814) * q^31 + (-466*b9 + 472*b8 + 19*b7 - 460*b5 - 2575*b4 - 3535*b2 - 30700*b1 + 421) * q^32 + (141*b9 + 4*b8 - 135*b7 - 342*b6 - 414*b5 + 2429*b4 + 1192*b3 + 480*b2 - 7286*b1 + 126986) * q^33 + (264*b9 + 396*b8 - 132*b7 + 602*b6 - 660*b5 + 264*b4 - 3778*b3 + 7160*b2 + 3052*b1 - 113924) * q^34 + (10*b9 + 5*b8 + 335*b7 + 25*b5 - 628*b4 + 4300*b2 - 17364*b1 - 985) * q^35 + (-242*b9 - 704*b8 + 268*b7 - 894*b6 - 2290*b5 + 3348*b4 + 1027*b3 - 4076*b2 + 47122*b1 - 66481) * q^36 + (-131*b9 + 418*b8 - 549*b7 - 652*b6 + 942*b5 - 131*b4 + 4902*b3 + 14698*b2 + 6382*b1 + 466508) * q^37 + (-150*b9 - 26*b8 - 504*b7 - 326*b5 - 1982*b4 - 6168*b2 + 73962*b1 + 1310) * q^38 + (21*b9 + 168*b8 - 111*b7 + 1818*b6 + 2910*b5 + 183*b4 + 3744*b3 + 5306*b2 + 21628*b1 + 184114) * q^39 + (35*b9 - 310*b8 + 345*b7 - 70*b6 - 450*b5 + 35*b4 + 3045*b3 - 11250*b2 - 4970*b1 + 294615) * q^40 + (1115*b9 - 1420*b8 + 961*b7 + 810*b5 + 537*b4 + 24158*b2 + 70428*b1 - 4608) * q^41 + (375*b9 + 1170*b8 - 639*b7 + 522*b6 + 2142*b5 + 345*b4 - 7872*b3 + 6708*b2 - 77532*b1 - 542838) * q^42 + (-1238*b9 - 1445*b8 + 207*b7 - 788*b6 + 3507*b5 - 1238*b4 + 3988*b3 - 12252*b2 - 4474*b1 + 777145) * q^43 + (244*b9 + 492*b8 + 98*b7 + 980*b5 - 142*b4 - 3398*b2 - 232760*b1 + 934) * q^44 + (-340*b9 - 550*b8 + 605*b7 + 1155*b6 + 1075*b5 - 1476*b4 - 2455*b3 - 2170*b2 + 50492*b1 - 214730) * q^45 + (-95*b9 - 1160*b8 + 1065*b7 - 1336*b6 - 780*b5 - 95*b4 - 5748*b3 - 39614*b2 - 17582*b1 - 2537724) * q^46 + (1726*b9 - 1161*b8 - 2815*b7 + 2291*b5 + 9860*b4 - 27872*b2 - 34780*b1 + 7849) * q^47 + (282*b9 + 1824*b8 + 501*b7 + 216*b6 + 1632*b5 - 6477*b4 + 7434*b3 - 16673*b2 - 204220*b1 + 2247929) * q^48 + (-27*b9 + 792*b8 - 819*b7 - 5462*b6 + 900*b5 - 27*b4 + 1066*b3 + 4306*b2 + 542*b1 + 934391) * q^49 + 78125*b1 * q^50 + (1248*b9 - 2624*b8 + 2232*b7 + 4626*b6 + 270*b5 - 10486*b4 - 5738*b3 + 19776*b2 - 117662*b1 - 2778934) * q^51 + (914*b9 + 2648*b8 - 1734*b7 + 2804*b6 - 1008*b5 + 914*b4 + 996*b3 + 68532*b2 + 29884*b1 + 2094512) * q^52 + (-1302*b9 + 6432*b8 + 2970*b7 + 3828*b5 + 2880*b4 - 17976*b2 + 95534*b1 + 2652) * q^53 + (-2215*b9 - 667*b8 - 1633*b7 - 3534*b6 - 5930*b5 - 6066*b4 - 2224*b3 + 12170*b2 + 403763*b1 - 803429) * q^54 + (135*b9 - 660*b8 + 795*b7 - 960*b6 - 1200*b5 + 135*b4 + 1810*b3 - 29010*b2 - 13050*b1 - 431920) * q^55 + (-3446*b9 - 6868*b8 + 1760*b7 - 13760*b5 + 19048*b4 + 88138*b2 + 572864*b1 - 22462) * q^56 + (579*b9 - 102*b8 - 4791*b7 + 360*b6 - 7962*b5 - 12975*b4 + 3690*b3 + 31226*b2 + 21700*b1 - 289772) * q^57 + (1502*b9 + 1658*b8 - 156*b7 + 3770*b6 - 4350*b5 + 1502*b4 - 470*b3 + 23076*b2 + 9724*b1 - 439366) * q^58 + (-2807*b9 + 1574*b8 - 2101*b7 - 4040*b5 + 26095*b4 - 38672*b2 - 377372*b1 + 6644) * q^59 + (1260*b9 + 460*b8 + 540*b7 - 3060*b6 - 3150*b5 - 1774*b4 + 10735*b3 - 27000*b2 + 251188*b1 + 2310365) * q^60 + (2695*b9 - 200*b8 + 2895*b7 + 2134*b6 - 10980*b5 + 2695*b4 - 2594*b3 - 78714*b2 - 36262*b1 + 2254558) * q^61 + (-1004*b9 + 8678*b8 + 1046*b7 + 6670*b5 - 1232*b4 - 63500*b2 - 275206*b1 + 13214) * q^62 + (-5844*b9 - 2919*b8 - 3*b7 + 1332*b6 - 1329*b5 - 30564*b4 - 19182*b3 + 16350*b2 - 572538*b1 + 869781) * q^63 + (-1355*b9 - 380*b8 - 975*b7 + 13258*b6 + 5040*b5 - 1355*b4 + 3225*b3 + 81522*b2 + 40166*b1 - 2305007) * q^64 + (-1165*b9 - 2770*b8 + 1285*b7 - 5100*b5 + 8947*b4 + 42800*b2 - 299114*b1 - 10560) * q^65 + (2034*b9 + 2718*b8 + 3600*b7 - 14976*b6 + 4410*b5 - 13698*b4 + 4356*b3 - 82036*b2 - 562466*b1 - 2726570) * q^66 + (-738*b9 + 1335*b8 - 2073*b7 - 2792*b6 + 4287*b5 - 738*b4 - 54508*b3 + 53692*b2 + 23438*b1 - 4696311) * q^67 + (4902*b9 - 14048*b8 - 8244*b7 - 4244*b5 + 32344*b4 + 14358*b2 + 1174448*b1 + 1538) * q^68 + (9285*b9 + 9792*b8 + 1296*b7 - 4905*b6 + 2475*b5 - 35484*b4 - 19023*b3 - 73302*b2 + 670035*b1 + 3366024) * q^69 + (-2305*b9 - 2620*b8 + 315*b7 + 7480*b6 + 6600*b5 - 2305*b4 - 27780*b3 + 15230*b2 + 10550*b1 - 6311640) * q^70 + (3550*b9 + 9590*b8 - 172*b7 + 16690*b5 + 6070*b4 - 92096*b2 + 632136*b1 + 23246) * q^71 + (-8556*b9 + 930*b8 + 7347*b7 - 5112*b6 + 25710*b5 - 9693*b4 + 80112*b3 - 94839*b2 - 482676*b1 + 5466327) * q^72 + (3658*b9 + 4924*b8 - 1266*b7 - 15196*b6 - 9708*b5 + 3658*b4 + 25396*b3 - 12956*b2 - 12668*b1 - 12986750) * q^73 + (12920*b9 - 3490*b8 + 2098*b7 + 22350*b5 - 26316*b4 + 45764*b2 - 1848672*b1 - 354) * q^74 + (78125*b2 + 156250*b1 + 859375) * q^75 + (-9518*b9 - 8252*b8 - 1266*b7 + 5756*b6 + 29820*b5 - 9518*b4 + 61230*b3 + 44500*b2 + 29236*b1 + 19911650) * q^76 + (-4768*b9 - 10252*b8 + 3820*b7 - 19788*b5 + 34500*b4 + 146696*b2 + 555092*b1 - 36732) * q^77 + (16818*b9 - 6728*b8 - 8208*b7 - 342*b6 + 12762*b5 + 79316*b4 - 29858*b3 + 99876*b2 - 1256630*b1 + 6019412) * q^78 + (-3373*b9 - 6802*b8 + 3429*b7 + 16150*b6 + 6690*b5 - 3373*b4 + 81854*b3 - 51874*b2 - 15706*b1 + 16276056) * q^79 + (4020*b9 + 4260*b8 + 795*b7 + 12300*b5 - 19111*b4 - 32025*b2 - 972668*b1 + 10155) * q^80 + (-20347*b9 - 7108*b8 - 12223*b7 + 28182*b6 - 4700*b5 + 2493*b4 + 22358*b3 + 17870*b2 + 553742*b1 - 9350723) * q^81 + (-6222*b9 - 16818*b8 + 10596*b7 - 17572*b6 + 8070*b5 - 6222*b4 + 37072*b3 - 421804*b2 - 183488*b1 + 20378318) * q^82 + (-2976*b9 - 10299*b8 + 4755*b7 - 16251*b5 - 2208*b4 + 157482*b2 + 1454346*b1 - 37839) * q^83 + (-7554*b9 - 23208*b8 + 498*b7 + 22914*b6 - 570*b5 + 122262*b4 - 89442*b3 + 144522*b2 + 2849850*b1 - 19798110) * q^84 + (6130*b9 + 12670*b8 - 6540*b7 - 16650*b6 - 11850*b5 + 6130*b4 - 24800*b3 + 154940*b2 + 58260*b1 - 11109790) * q^85 + (4987*b9 + 33626*b8 - 865*b7 + 43600*b5 - 144539*b4 - 318866*b2 - 1870292*b1 + 74834) * q^86 + (21633*b9 + 5762*b8 + 9837*b7 + 3978*b6 - 30258*b5 + 32599*b4 + 12122*b3 + 80976*b2 - 294796*b1 - 16844138) * q^87 + (-1018*b9 + 3128*b8 - 4146*b7 + 22300*b6 + 7200*b5 - 1018*b4 - 105750*b3 + 219836*b2 + 102644*b1 - 48556686) * q^88 + (-25158*b9 + 8016*b8 + 726*b7 - 42300*b5 - 53466*b4 - 37548*b2 - 479856*b1 - 11304) * q^89 + (-5495*b9 - 4130*b8 - 6965*b7 + 10815*b6 - 3850*b5 + 33075*b4 + 34735*b3 - 8600*b2 + 1013050*b1 + 17205470) * q^90 + (12924*b9 + 28086*b8 - 15162*b7 - 47688*b6 - 23610*b5 + 12924*b4 - 22088*b3 + 320280*b2 + 116892*b1 + 44352930) * q^91 + (-5538*b9 + 854*b8 + 4794*b7 - 10222*b5 - 207496*b4 + 60174*b2 + 2272400*b1 - 18212) * q^92 + (-7497*b9 + 13230*b8 + 18819*b7 + 2034*b6 - 58680*b5 + 4203*b4 - 44784*b3 - 60678*b2 - 1459314*b1 + 46313604) * q^93 + (22561*b9 + 16684*b8 + 5877*b7 - 114796*b6 - 73560*b5 + 22561*b4 - 1240*b3 - 602126*b2 - 311870*b1 - 9116072) * q^94 + (-1395*b9 - 7010*b8 - 18795*b7 - 9800*b5 + 32273*b4 - 179850*b2 - 356346*b1 + 40970) * q^95 + (25731*b9 + 45812*b8 - 13383*b7 - 4158*b6 - 18180*b5 + 208249*b4 - 93073*b3 + 311076*b2 - 2727490*b1 - 36169007) * q^96 + (13428*b9 + 2124*b8 + 11304*b7 + 64120*b6 - 51588*b5 + 13428*b4 - 153720*b3 - 78392*b2 - 30016*b1 - 25925026) * q^97 + (-6856*b9 + 25846*b8 + 11530*b7 + 12134*b5 - 275452*b4 - 75868*b2 - 942931*b1 + 10246) * q^98 + (-25421*b9 + 15604*b8 + 13675*b7 - 12972*b6 + 39230*b5 - 72639*b4 + 106132*b3 - 202880*b2 + 621724*b1 - 3323878) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 112 q^{3} - 786 q^{4} - 5282 q^{6} + 7156 q^{7} + 3922 q^{9}+O(q^{10})$$ 10 * q - 112 * q^3 - 786 * q^4 - 5282 * q^6 + 7156 * q^7 + 3922 * q^9 $$10 q - 112 q^{3} - 786 q^{4} - 5282 q^{6} + 7156 q^{7} + 3922 q^{9} - 8750 q^{10} - 3812 q^{12} - 55464 q^{13} - 21250 q^{15} + 280386 q^{16} - 419800 q^{18} - 231516 q^{19} + 289572 q^{21} + 1129940 q^{22} + 1136334 q^{24} - 781250 q^{25} - 335512 q^{27} - 3340724 q^{28} - 965000 q^{30} + 881620 q^{31} + 1266460 q^{33} - 1111276 q^{34} - 668662 q^{36} + 4672616 q^{37} + 1826792 q^{39} + 2913750 q^{40} - 5392860 q^{42} + 7731336 q^{43} - 2142500 q^{45} - 25424604 q^{46} + 22413388 q^{48} + 9354214 q^{49} - 27732692 q^{51} + 21064016 q^{52} - 7979798 q^{54} - 4377500 q^{55} - 2856304 q^{57} - 4351100 q^{58} + 23016250 q^{60} + 22417020 q^{61} + 8830596 q^{63} - 22935002 q^{64} - 27419800 q^{66} - 46646024 q^{67} + 33562632 q^{69} - 62992500 q^{70} + 54175560 q^{72} - 129964884 q^{73} + 8750000 q^{75} + 198922436 q^{76} + 60388360 q^{78} + 162310924 q^{79} - 93575390 q^{81} + 202877560 q^{82} - 197346768 q^{84} - 110682500 q^{85} - 168322540 q^{87} - 484775700 q^{88} + 171878750 q^{90} + 444288464 q^{91} + 463412376 q^{93} - 92050036 q^{94} - 360807406 q^{96} - 258825724 q^{97} - 33965200 q^{99}+O(q^{100})$$ 10 * q - 112 * q^3 - 786 * q^4 - 5282 * q^6 + 7156 * q^7 + 3922 * q^9 - 8750 * q^10 - 3812 * q^12 - 55464 * q^13 - 21250 * q^15 + 280386 * q^16 - 419800 * q^18 - 231516 * q^19 + 289572 * q^21 + 1129940 * q^22 + 1136334 * q^24 - 781250 * q^25 - 335512 * q^27 - 3340724 * q^28 - 965000 * q^30 + 881620 * q^31 + 1266460 * q^33 - 1111276 * q^34 - 668662 * q^36 + 4672616 * q^37 + 1826792 * q^39 + 2913750 * q^40 - 5392860 * q^42 + 7731336 * q^43 - 2142500 * q^45 - 25424604 * q^46 + 22413388 * q^48 + 9354214 * q^49 - 27732692 * q^51 + 21064016 * q^52 - 7979798 * q^54 - 4377500 * q^55 - 2856304 * q^57 - 4351100 * q^58 + 23016250 * q^60 + 22417020 * q^61 + 8830596 * q^63 - 22935002 * q^64 - 27419800 * q^66 - 46646024 * q^67 + 33562632 * q^69 - 62992500 * q^70 + 54175560 * q^72 - 129964884 * q^73 + 8750000 * q^75 + 198922436 * q^76 + 60388360 * q^78 + 162310924 * q^79 - 93575390 * q^81 + 202877560 * q^82 - 197346768 * q^84 - 110682500 * q^85 - 168322540 * q^87 - 484775700 * q^88 + 171878750 * q^90 + 444288464 * q^91 + 463412376 * q^93 - 92050036 * q^94 - 360807406 * q^96 - 258825724 * q^97 - 33965200 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 4 x^{9} - 433 x^{8} - 2220 x^{7} + 49747 x^{6} + 744964 x^{5} + 4580249 x^{4} + 16418988 x^{3} + \cdots + 53656344$$ :

 $$\beta_{1}$$ $$=$$ $$( - 422181448349 \nu^{9} + 6737848279449 \nu^{8} + 139312194537578 \nu^{7} + \cdots + 41\!\cdots\!04 ) / 13\!\cdots\!40$$ (-422181448349*v^9 + 6737848279449*v^8 + 139312194537578*v^7 - 990054798966676*v^6 - 24034596572099589*v^5 - 70247377969529165*v^4 + 729928185416215536*v^3 + 7040333543796566448*v^2 + 26368627529241277344*v + 41556905602917131304) / 1314779316311147940 $$\beta_{2}$$ $$=$$ $$( - 2454900232379 \nu^{9} + 8124069454611 \nu^{8} + \cdots - 14\!\cdots\!12 ) / 52\!\cdots\!76$$ (-2454900232379*v^9 + 8124069454611*v^8 + 1018980904094492*v^7 + 6837493686202016*v^6 - 98994081243372909*v^5 - 2012334306130205963*v^4 - 15257574842705758860*v^3 - 58096265667022204896*v^2 - 115329530767248237816*v - 144722949260663611512) / 525911726524459176 $$\beta_{3}$$ $$=$$ $$( - 6879197511740 \nu^{9} + 17194261561041 \nu^{8} + \cdots - 24\!\cdots\!12 ) / 26\!\cdots\!88$$ (-6879197511740*v^9 + 17194261561041*v^8 + 3086200896610772*v^7 + 19289070838345274*v^6 - 346055941008215076*v^5 - 5720291048548337495*v^4 - 35870255949722934420*v^3 - 118607149615439990316*v^2 - 216018284274101641824*v - 242344663543802056212) / 262955863262229588 $$\beta_{4}$$ $$=$$ $$( - 5233252857042 \nu^{9} + 39209403295667 \nu^{8} + \cdots - 12\!\cdots\!68 ) / 14\!\cdots\!60$$ (-5233252857042*v^9 + 39209403295667*v^8 + 2097358502904124*v^7 + 4634362891972442*v^6 - 265297550568774812*v^5 - 2976113610472692595*v^4 - 15067978055253639362*v^3 - 46657419841784316366*v^2 - 102719484330363856848*v - 120274107094158683868) / 146086590701238660 $$\beta_{5}$$ $$=$$ $$( 136514483776663 \nu^{9} - 765773291245803 \nu^{8} + \cdots + 15\!\cdots\!32 ) / 26\!\cdots\!80$$ (136514483776663*v^9 - 765773291245803*v^8 - 61967928024908296*v^7 - 141989555456732008*v^6 + 8227762673774026893*v^5 + 82592668460512852195*v^4 + 320989486618021556628*v^3 + 714069530509762112064*v^2 + 1075459840329855982392*v + 155785885869354786432) / 2629558632622295880 $$\beta_{6}$$ $$=$$ $$( - 13995282752553 \nu^{9} + 72894475678928 \nu^{8} + \cdots - 11\!\cdots\!92 ) / 14\!\cdots\!60$$ (-13995282752553*v^9 + 72894475678928*v^8 + 6004988054004996*v^7 + 23591943424282108*v^6 - 739668907135884083*v^5 - 9533425812763576680*v^4 - 50851973329178405488*v^3 - 153315418716108067164*v^2 - 263730388041193353792*v - 118989054392825258892) / 146086590701238660 $$\beta_{7}$$ $$=$$ $$( - 157357802456293 \nu^{9} + \cdots + 20\!\cdots\!48 ) / 87\!\cdots\!60$$ (-157357802456293*v^9 + 2471413966226433*v^8 + 49586159199625936*v^7 - 345218347874953892*v^6 - 7570175981484697803*v^5 - 24243396055331133145*v^4 + 79723035452777208912*v^3 + 595528623514187813676*v^2 + 1534710704358321076248*v + 2015158102835327035848) / 876519544207431960 $$\beta_{8}$$ $$=$$ $$( - 21633109424708 \nu^{9} + 115213115150653 \nu^{8} + \cdots - 16\!\cdots\!44 ) / 87\!\cdots\!96$$ (-21633109424708*v^9 + 115213115150653*v^8 + 9406667353071860*v^7 + 33472643851545358*v^6 - 1189227811906060870*v^5 - 14497894251150512177*v^4 - 70773983640517845866*v^3 - 179305270644491554866*v^2 - 264621836004730077216*v - 169818536166651896544) / 87651954420743196 $$\beta_{9}$$ $$=$$ $$( 143077927645465 \nu^{9} - 910044828994923 \nu^{8} + \cdots + 11\!\cdots\!96 ) / 52\!\cdots\!76$$ (143077927645465*v^9 - 910044828994923*v^8 - 59671279512767176*v^7 - 186552193363709416*v^6 + 7631915356160788491*v^5 + 91248706097458495843*v^4 + 432162112144171606476*v^3 + 1015244254252052021256*v^2 + 1566812423520347603400*v + 1199573014933664274096) / 525911726524459176
 $$\nu$$ $$=$$ $$( - \beta_{9} - 9 \beta_{8} + 8 \beta_{7} - 9 \beta_{6} - 5 \beta_{5} - 55 \beta_{4} + 174 \beta_{3} + \cdots + 2822 ) / 6750$$ (-b9 - 9*b8 + 8*b7 - 9*b6 - 5*b5 - 55*b4 + 174*b3 - 294*b2 - 292*b1 + 2822) / 6750 $$\nu^{2}$$ $$=$$ $$( 61 \beta_{9} + 124 \beta_{8} + 87 \beta_{7} - 576 \beta_{6} + 180 \beta_{5} + 55 \beta_{4} + \cdots + 596958 ) / 6750$$ (61*b9 + 124*b8 + 87*b7 - 576*b6 + 180*b5 + 55*b4 + 2136*b3 - 4216*b2 - 10188*b1 + 596958) / 6750 $$\nu^{3}$$ $$=$$ $$( 21 \beta_{9} + 334 \beta_{8} + 2477 \beta_{7} - 6966 \beta_{6} + 1780 \beta_{5} - 11841 \beta_{4} + \cdots + 8080028 ) / 6750$$ (21*b9 + 334*b8 + 2477*b7 - 6966*b6 + 1780*b5 - 11841*b4 + 45576*b3 - 89086*b2 - 354826*b1 + 8080028) / 6750 $$\nu^{4}$$ $$=$$ $$( 20657 \beta_{9} + 50548 \beta_{8} + 47629 \beta_{7} - 148302 \beta_{6} + 90560 \beta_{5} + \cdots + 162283666 ) / 6750$$ (20657*b9 + 50548*b8 + 47629*b7 - 148302*b6 + 90560*b5 - 187903*b4 + 790872*b3 - 1710022*b2 - 7704530*b1 + 162283666) / 6750 $$\nu^{5}$$ $$=$$ $$( 313201 \beta_{9} + 1080634 \beta_{8} + 1007967 \beta_{7} - 2524536 \beta_{6} + 1886880 \beta_{5} + \cdots + 2815161348 ) / 6750$$ (313201*b9 + 1080634*b8 + 1007967*b7 - 2524536*b6 + 1886880*b5 - 5270459*b4 + 14199996*b3 - 31357936*b2 - 183589140*b1 + 2815161348) / 6750 $$\nu^{6}$$ $$=$$ $$( 8412225 \beta_{9} + 27927400 \beta_{8} + 20048825 \beta_{7} - 44262990 \beta_{6} + 46838500 \beta_{5} + \cdots + 49105415690 ) / 6750$$ (8412225*b9 + 27927400*b8 + 20048825*b7 - 44262990*b6 + 46838500*b5 - 110137143*b4 + 247192740*b3 - 594762910*b2 - 3974883634*b1 + 49105415690) / 6750 $$\nu^{7}$$ $$=$$ $$( 173861933 \beta_{9} + 628042342 \beta_{8} + 398846731 \beta_{7} - 755515368 \beta_{6} + \cdots + 838451682124 ) / 6750$$ (173861933*b9 + 628042342*b8 + 398846731*b7 - 755515368*b6 + 1011745340*b5 - 2421792763*b4 + 4196669748*b3 - 10733826748*b2 - 85121005388*b1 + 838451682124) / 6750 $$\nu^{8}$$ $$=$$ $$( 3726354937 \beta_{9} + 13831361068 \beta_{8} + 7774933389 \beta_{7} - 12398581422 \beta_{6} + \cdots + 13782771805146 ) / 6750$$ (3726354937*b9 + 13831361068*b8 + 7774933389*b7 - 12398581422*b6 + 21952533960*b5 - 50434932071*b4 + 69336318792*b3 - 194440320262*b2 - 1768199744754*b1 + 13782771805146) / 6750 $$\nu^{9}$$ $$=$$ $$( 76838521617 \beta_{9} + 294645646858 \beta_{8} + 149605572119 \beta_{7} - 197196436872 \beta_{6} + \cdots + 218981655222716 ) / 6750$$ (76838521617*b9 + 294645646858*b8 + 149605572119*b7 - 197196436872*b6 + 457649934160*b5 - 1037467341315*b4 + 1097007348492*b3 - 3423351196312*b2 - 36075692056876*b1 + 218981655222716) / 6750

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/15\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 18.9110 − 2.23607i −0.616448 + 2.23607i −7.95862 − 2.23607i −1.94191 + 2.23607i −6.39402 − 2.23607i −6.39402 + 2.23607i −1.94191 − 2.23607i −7.95862 + 2.23607i −0.616448 − 2.23607i 18.9110 + 2.23607i
29.5009i 41.5815 69.5124i −614.301 279.508i −2050.68 1226.69i 3174.27 10570.2i −3102.96 5780.86i −8245.74
11.2 23.7888i −80.3707 10.0775i −309.906 279.508i −239.732 + 1911.92i −692.753 1282.36i 6357.89 + 1619.88i 6649.17
11.3 10.2357i −57.8099 + 56.7364i 151.230 279.508i 580.739 + 591.726i 3448.05 4168.30i 122.959 6559.85i −2860.98
11.4 8.27106i 70.4414 39.9876i 187.590 279.508i −330.740 582.625i 860.291 3668.96i 3362.98 5633.57i 2311.83
11.5 7.97572i −29.8424 75.3023i 192.388 279.508i −600.590 + 238.014i −3211.86 3576.22i −4779.87 + 4494.40i −2229.28
11.6 7.97572i −29.8424 + 75.3023i 192.388 279.508i −600.590 238.014i −3211.86 3576.22i −4779.87 4494.40i −2229.28
11.7 8.27106i 70.4414 + 39.9876i 187.590 279.508i −330.740 + 582.625i 860.291 3668.96i 3362.98 + 5633.57i 2311.83
11.8 10.2357i −57.8099 56.7364i 151.230 279.508i 580.739 591.726i 3448.05 4168.30i 122.959 + 6559.85i −2860.98
11.9 23.7888i −80.3707 + 10.0775i −309.906 279.508i −239.732 1911.92i −692.753 1282.36i 6357.89 1619.88i 6649.17
11.10 29.5009i 41.5815 + 69.5124i −614.301 279.508i −2050.68 + 1226.69i 3174.27 10570.2i −3102.96 + 5780.86i −8245.74
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 11.10 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 15.9.c.a 10
3.b odd 2 1 inner 15.9.c.a 10
4.b odd 2 1 240.9.l.b 10
5.b even 2 1 75.9.c.g 10
5.c odd 4 2 75.9.d.c 20
12.b even 2 1 240.9.l.b 10
15.d odd 2 1 75.9.c.g 10
15.e even 4 2 75.9.d.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.9.c.a 10 1.a even 1 1 trivial
15.9.c.a 10 3.b odd 2 1 inner
75.9.c.g 10 5.b even 2 1
75.9.c.g 10 15.d odd 2 1
75.9.d.c 20 5.c odd 4 2
75.9.d.c 20 15.e even 4 2
240.9.l.b 10 4.b odd 2 1
240.9.l.b 10 12.b even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(15, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + \cdots + 224550432000$$
$3$ $$T^{10} + \cdots + 12\!\cdots\!01$$
$5$ $$(T^{2} + 78125)^{5}$$
$7$ $$(T^{5} + \cdots - 20\!\cdots\!00)^{2}$$
$11$ $$T^{10} + \cdots + 68\!\cdots\!00$$
$13$ $$(T^{5} + \cdots + 34\!\cdots\!00)^{2}$$
$17$ $$T^{10} + \cdots + 11\!\cdots\!00$$
$19$ $$(T^{5} + \cdots - 11\!\cdots\!32)^{2}$$
$23$ $$T^{10} + \cdots + 63\!\cdots\!00$$
$29$ $$T^{10} + \cdots + 20\!\cdots\!00$$
$31$ $$(T^{5} + \cdots + 13\!\cdots\!68)^{2}$$
$37$ $$(T^{5} + \cdots + 10\!\cdots\!00)^{2}$$
$41$ $$T^{10} + \cdots + 56\!\cdots\!00$$
$43$ $$(T^{5} + \cdots + 23\!\cdots\!00)^{2}$$
$47$ $$T^{10} + \cdots + 18\!\cdots\!00$$
$53$ $$T^{10} + \cdots + 48\!\cdots\!00$$
$59$ $$T^{10} + \cdots + 86\!\cdots\!00$$
$61$ $$(T^{5} + \cdots + 10\!\cdots\!68)^{2}$$
$67$ $$(T^{5} + \cdots + 17\!\cdots\!00)^{2}$$
$71$ $$T^{10} + \cdots + 15\!\cdots\!00$$
$73$ $$(T^{5} + \cdots + 14\!\cdots\!00)^{2}$$
$79$ $$(T^{5} + \cdots - 23\!\cdots\!32)^{2}$$
$83$ $$T^{10} + \cdots + 11\!\cdots\!00$$
$89$ $$T^{10} + \cdots + 10\!\cdots\!00$$
$97$ $$(T^{5} + \cdots + 13\!\cdots\!00)^{2}$$