# Properties

 Label 7.10.c.a Level $7$ Weight $10$ Character orbit 7.c Analytic conductor $3.605$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7,10,Mod(2,7)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("7.2");

S:= CuspForms(chi, 10);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$7$$ Weight: $$k$$ $$=$$ $$10$$ Character orbit: $$[\chi]$$ $$=$$ 7.c (of order $$3$$, degree $$2$$, 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: $$3.60525085315$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000$$ x^10 - x^9 + 430*x^8 + 61*x^7 + 146753*x^6 + 23608*x^5 + 16136944*x^4 + 30575648*x^3 + 1399072384*x^2 + 1034227200*x + 761760000 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$2^{12}\cdot 3^{3}\cdot 7^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( - 4 \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} - 33 \beta_{3} - \beta_1 + 33) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 191 \beta_{3} + 7 \beta_1 - 191) q^{4} + ( - \beta_{9} + 307 \beta_{3}) q^{5} + ( - 2 \beta_{6} + 11 \beta_{5} + 7 \beta_{4} + 60 \beta_{2} - 897) q^{6} + ( - \beta_{9} - 10 \beta_{8} - 10 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + \cdots - 797) q^{7}+ \cdots + (7 \beta_{9} + 16 \beta_{8} - 86 \beta_{7} - 7284 \beta_{3} + 258 \beta_{2} - 258 \beta_1) q^{9}+O(q^{10})$$ q + (-4*b3 + b2 - b1) * q^2 + (-b7 - b5 - 33*b3 - b1 + 33) * q^3 + (-b8 + b7 + b5 + b4 + 191*b3 + 7*b1 - 191) * q^4 + (-b9 + 307*b3) * q^5 + (-2*b6 + 11*b5 + 7*b4 + 60*b2 - 897) * q^6 + (-b9 - 10*b8 - 10*b7 + 3*b6 - 9*b5 + 2*b4 + 1430*b3 - 67*b2 + 26*b1 - 797) * q^7 + (-66*b5 - 10*b4 - 26*b2 + 3462) * q^8 + (7*b9 + 16*b8 - 86*b7 - 7284*b3 + 258*b2 - 258*b1) * q^9 $$q + ( - 4 \beta_{3} + \beta_{2} - \beta_1) q^{2} + ( - \beta_{7} - \beta_{5} - 33 \beta_{3} - \beta_1 + 33) q^{3} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} + 191 \beta_{3} + 7 \beta_1 - 191) q^{4} + ( - \beta_{9} + 307 \beta_{3}) q^{5} + ( - 2 \beta_{6} + 11 \beta_{5} + 7 \beta_{4} + 60 \beta_{2} - 897) q^{6} + ( - \beta_{9} - 10 \beta_{8} - 10 \beta_{7} + 3 \beta_{6} - 9 \beta_{5} + \cdots - 797) q^{7}+ \cdots + ( - 309295 \beta_{6} + 1130938 \beta_{5} + \cdots - 200734674) q^{99}+O(q^{100})$$ q + (-4*b3 + b2 - b1) * q^2 + (-b7 - b5 - 33*b3 - b1 + 33) * q^3 + (-b8 + b7 + b5 + b4 + 191*b3 + 7*b1 - 191) * q^4 + (-b9 + 307*b3) * q^5 + (-2*b6 + 11*b5 + 7*b4 + 60*b2 - 897) * q^6 + (-b9 - 10*b8 - 10*b7 + 3*b6 - 9*b5 + 2*b4 + 1430*b3 - 67*b2 + 26*b1 - 797) * q^7 + (-66*b5 - 10*b4 - 26*b2 + 3462) * q^8 + (7*b9 + 16*b8 - 86*b7 - 7284*b3 + 258*b2 - 258*b1) * q^9 + (4*b9 + 14*b8 + 170*b7 - 4*b6 + 170*b5 - 14*b4 - 910*b3 - 299*b1 + 910) * q^10 + (-7*b9 + 86*b8 - 9*b7 + 7*b6 - 9*b5 - 86*b4 - 8425*b3 + 3*b1 + 8425) * q^11 + (16*b9 - 77*b8 + 341*b7 + 28419*b3 - 2959*b2 + 2959*b1) * q^12 + (27*b6 + 286*b5 + 28*b4 + 1042*b2 - 32504) * q^13 + (10*b9 + 9*b8 - 789*b7 - 44*b6 - 246*b5 + 50*b4 - 47991*b3 - 1038*b2 - 3242*b1 + 21074) * q^14 + (-7*b6 - 1055*b5 + 86*b4 + 6201*b2 + 13101) * q^15 + (-112*b9 - 120*b8 - 664*b7 + 62344*b3 + 4560*b2 - 4560*b1) * q^16 + (-65*b9 - 168*b8 + 750*b7 + 65*b6 + 750*b5 + 168*b4 - 68163*b3 - 8798*b1 + 68163) * q^17 + (112*b9 - 824*b8 + 712*b7 - 112*b6 + 712*b5 + 824*b4 + 208608*b3 + 14898*b1 - 208608) * q^18 + (-109*b9 + 826*b8 - 1397*b7 + 4211*b3 - 19463*b2 + 19463*b1) * q^19 + (-128*b6 - 1395*b5 - 707*b4 + 1923*b2 - 35363) * q^20 + (588*b8 + 2450*b7 + 245*b6 - 784*b4 - 238875*b3 - 16758*b2 - 18130*b1 - 25725) * q^21 + (126*b6 + 5491*b5 - 305*b4 + 34660*b2 - 22441) * q^22 + (728*b9 + 148*b8 + 4941*b7 + 526371*b3 + 7791*b2 - 7791*b1) * q^23 + (432*b9 + 966*b8 - 7230*b7 - 432*b6 - 7230*b5 - 966*b4 - 1708074*b3 - 38142*b1 + 1708074) * q^24 + (-714*b9 + 2840*b8 - 2840*b7 + 714*b6 - 2840*b5 - 2840*b4 + 259928*b3 - 1520*b1 - 259928) * q^25 + (408*b9 - 3052*b8 + 684*b7 + 855644*b3 - 50734*b2 + 50734*b1) * q^26 + (131*b6 - 521*b5 + 2282*b4 + 6339*b2 - 1835079) * q^27 + (-432*b9 - 2591*b8 + 8455*b7 - 496*b6 + 10385*b5 + 3993*b4 - 152527*b3 + 11054*b2 + 33933*b1 - 2152215) * q^28 + (-959*b6 + 5538*b5 + 1268*b4 - 5930*b2 + 1530044) * q^29 + (-2254*b9 + 485*b8 + 1055*b7 + 4115493*b3 - 6210*b2 + 6210*b1) * q^30 + (-1397*b9 - 4158*b8 + 73*b7 + 1397*b6 + 73*b5 + 4158*b4 - 3799953*b3 + 90345*b1 + 3799953) * q^31 + (2016*b9 - 2216*b8 - 9432*b7 - 2016*b6 - 9432*b5 + 2216*b4 + 1215128*b3 - 81384*b1 - 1215128) * q^32 + (-908*b9 + 2296*b8 - 8312*b7 + 237315*b3 + 245136*b2 - 245136*b1) * q^33 + (904*b6 + 4116*b5 + 3892*b4 + 26341*b2 - 6302736) * q^34 + (2499*b9 - 1078*b8 - 18375*b7 - 833*b6 - 33810*b5 - 8526*b4 - 4262559*b3 + 54341*b2 + 80213*b1 - 2273306) * q^35 + (3808*b6 - 37402*b5 - 6938*b4 - 388038*b2 + 7314918) * q^36 + (2030*b9 + 2868*b8 - 41494*b7 + 7949757*b3 - 132990*b2 + 132990*b1) * q^37 + (1578*b9 + 15085*b8 + 53511*b7 - 1578*b6 + 53511*b5 - 15085*b4 - 13360763*b3 + 221252*b1 + 13360763) * q^38 + (-371*b9 - 8078*b8 + 74690*b7 + 371*b6 + 74690*b5 + 8078*b4 + 8898414*b3 + 25270*b1 - 8898414) * q^39 + (1184*b9 + 13286*b8 + 62130*b7 + 1028214*b3 + 357926*b2 - 357926*b1) * q^40 + (-2635*b6 + 26982*b5 - 32816*b4 - 435718*b2 - 5185700) * q^41 + (-5488*b9 + 27440*b8 - 92512*b7 + 6076*b6 + 22442*b5 + 1666*b4 - 10695720*b3 + 317667*b2 + 94374*b1 - 13136802) * q^42 + (-7028*b6 - 66624*b5 + 18832*b4 - 132344*b2 + 10269764) * q^43 + (7504*b9 - 26253*b8 + 48597*b7 + 20018819*b3 - 166623*b2 + 166623*b1) * q^44 + (3209*b9 - 36148*b8 - 18730*b7 - 3209*b6 - 18730*b5 + 36148*b4 - 16879164*b3 + 391218*b1 + 16879164) * q^45 + (-13090*b9 + 12107*b8 - 158127*b7 + 13090*b6 - 158127*b5 - 12107*b4 + 2609163*b3 - 582710*b1 - 2609163) * q^46 + (-129*b9 - 31990*b8 - 113391*b7 + 6414639*b3 + 88975*b2 - 88975*b1) * q^47 + (-2608*b6 - 90256*b5 + 45248*b4 + 769288*b2 - 18775680) * q^48 + (-826*b9 - 47264*b8 + 202244*b7 - 9037*b6 - 2534*b5 + 27328*b4 + 3142748*b3 - 139622*b2 - 301532*b1 - 6311697) * q^49 + (-2856*b6 + 290460*b5 + 44*b4 + 676184*b2 - 52660) * q^50 + (-22953*b9 + 57234*b8 - 37389*b7 + 8644815*b3 + 486345*b2 - 486345*b1) * q^51 + (-10720*b9 + 25634*b8 - 130050*b7 + 10720*b6 - 130050*b5 - 25634*b4 - 21387550*b3 - 1415726*b1 + 21387550) * q^52 + (28672*b9 + 11844*b8 + 153342*b7 - 28672*b6 + 153342*b5 - 11844*b4 + 5847531*b3 + 1630882*b1 - 5847531) * q^53 + (-6130*b9 + 1799*b8 + 90677*b7 + 11291151*b3 - 2529708*b2 + 2529708*b1) * q^54 + (19856*b6 + 91565*b5 + 41692*b4 - 495623*b2 - 14307503) * q^55 + (27296*b9 - 31036*b8 + 130860*b7 - 7408*b6 + 106014*b5 - 52730*b4 + 6271140*b3 - 2086366*b2 + 623692*b1 + 9077142) * q^56 + (41650*b6 - 112292*b5 - 98600*b4 + 1674308*b2 - 12832221) * q^57 + (12376*b9 - 10068*b8 + 274612*b7 - 9635948*b3 + 1024314*b2 - 1024314*b1) * q^58 + (2652*b9 + 107072*b8 - 244461*b7 - 2652*b6 - 244461*b5 - 107072*b4 - 10188249*b3 - 1888789*b1 + 10188249) * q^59 + (9520*b9 + 1519*b8 - 149975*b7 - 9520*b6 - 149975*b5 - 1519*b4 - 26869857*b3 - 820029*b1 + 26869857) * q^60 + (23626*b9 + 75684*b8 - 296018*b7 - 22604563*b3 - 307914*b2 + 307914*b1) * q^61 + (-13758*b6 - 57327*b5 - 97867*b4 + 2282626*b2 + 45842373) * q^62 + (-40749*b9 + 122718*b8 - 98034*b7 + 35020*b6 + 10244*b5 + 32680*b4 - 6105438*b3 + 1330818*b2 - 3933990*b1 + 35103588) * q^63 + (-72576*b6 + 64368*b5 + 111408*b4 + 904752*b2 - 19537232) * q^64 + (27853*b9 - 110432*b8 - 56490*b7 - 67114040*b3 - 1411242*b2 + 1411242*b1) * q^65 + (15664*b9 - 275408*b8 + 595120*b7 - 15664*b6 + 595120*b5 + 275408*b4 + 168072516*b3 + 1398237*b1 - 168072516) * q^66 + (-106498*b9 - 45732*b8 + 133127*b7 + 106498*b6 + 133127*b5 + 45732*b4 + 26836615*b3 + 3520839*b1 - 26836615) * q^67 + (-36448*b9 - 138033*b8 + 488529*b7 + 7892559*b3 - 3144119*b2 + 3144119*b1) * q^68 + (-56583*b6 - 146436*b5 - 49728*b4 - 5336292*b2 + 134567397) * q^69 + (-18326*b9 - 13867*b8 - 853825*b7 - 28910*b6 - 374115*b5 + 68257*b4 + 63096173*b3 + 702660*b2 + 3414712*b1 + 37265529) * q^70 + (2478*b6 - 577176*b5 + 247788*b4 + 618504*b2 - 118841568) * q^71 + (-18816*b9 + 116436*b8 - 1384836*b7 - 192184524*b3 + 3874644*b2 - 3874644*b1) * q^72 + (22698*b9 + 81536*b8 + 681424*b7 - 22698*b6 + 681424*b5 - 81536*b4 + 173198403*b3 + 2667688*b1 - 173198403) * q^73 + (69132*b9 - 135790*b8 + 77382*b7 - 69132*b6 + 77382*b5 + 135790*b4 - 120976554*b3 - 6502975*b1 + 120976554) * q^74 + (-13718*b9 + 151564*b8 + 496520*b7 - 38065836*b3 + 11219744*b2 - 11219744*b1) * q^75 + (87696*b6 + 429807*b5 - 142569*b4 + 5990733*b2 + 107515751) * q^76 + (70363*b9 + 81568*b8 - 510240*b7 + 9180*b6 - 606342*b5 - 261140*b4 - 983785*b3 + 3572614*b2 - 2401520*b1 + 190811013) * q^77 + (131740*b6 - 1104922*b5 - 454370*b4 - 13203372*b2 + 57522990) * q^78 + (-7728*b9 - 190092*b8 + 1186521*b7 - 100369637*b3 - 11509413*b2 + 11509413*b1) * q^79 + (-90032*b9 + 400120*b8 - 527880*b7 + 90032*b6 - 527880*b5 - 400120*b4 + 252792184*b3 + 3541920*b1 - 252792184) * q^80 + (179333*b9 + 243152*b8 + 386246*b7 - 179333*b6 + 386246*b5 - 243152*b4 - 109718097*b3 + 1207626*b1 + 109718097) * q^81 + (130136*b9 + 212268*b8 - 1325932*b7 - 271025884*b3 + 5363238*b2 - 5363238*b1) * q^82 + (48406*b6 + 977208*b5 + 779044*b4 - 6939344*b2 - 11965020) * q^83 + (43904*b9 - 526897*b8 + 2793441*b7 - 26656*b6 + 1853866*b5 + 201194*b4 + 331413999*b3 - 6083497*b2 + 18467071*b1 - 118515222) * q^84 + (-110432*b6 + 1989570*b5 - 501212*b4 - 6540962*b2 - 117971535) * q^85 + (-142800*b9 + 686976*b8 + 1318944*b7 - 137616464*b3 + 6522284*b2 - 6522284*b1) * q^86 + (-41041*b9 - 114898*b8 - 1964398*b7 + 41041*b6 - 1964398*b5 + 114898*b4 + 70609326*b3 + 8113446*b1 - 70609326) * q^87 + (-139216*b9 + 430550*b8 - 403278*b7 + 139216*b6 - 403278*b5 - 430550*b4 - 185530650*b3 - 11927534*b1 + 185530650) * q^88 + (-144296*b9 - 985432*b8 + 654420*b7 + 330210049*b3 + 5558276*b2 - 5558276*b1) * q^89 + (-96920*b6 - 2520700*b5 - 125468*b4 + 5212662*b2 + 195712764) * q^90 + (-77812*b9 - 89768*b8 + 162092*b7 - 76881*b6 + 412678*b5 + 502250*b4 - 81478180*b3 + 14868854*b2 - 10990700*b1 + 109658962) * q^91 + (28336*b6 + 2452731*b5 + 1387667*b4 + 10684041*b2 - 135172605) * q^92 + (215992*b9 - 591932*b8 - 1890534*b7 - 85029801*b3 - 1356734*b2 + 1356734*b1) * q^93 + (291278*b9 - 863485*b8 - 322695*b7 - 291278*b6 - 322695*b5 + 863485*b4 + 48863379*b3 - 13086278*b1 - 48863379) * q^94 + (-326914*b9 - 684952*b8 - 4148415*b7 + 326914*b6 - 4148415*b5 + 684952*b4 + 241561379*b3 + 9505677*b1 - 241561379) * q^95 + (-39392*b9 + 439096*b8 - 1174520*b7 - 283520904*b3 - 13074456*b2 + 13074456*b1) * q^96 + (-123697*b6 - 1405558*b5 - 1237712*b4 - 6851474*b2 + 90081768) * q^97 + (-330232*b9 + 1446564*b8 - 1487556*b7 + 306656*b6 - 3896424*b5 - 781704*b4 - 93269792*b3 - 13252771*b2 - 8106357*b1 - 185733352) * q^98 + (-309295*b6 + 1130938*b5 + 219734*b4 + 25930614*b2 - 200734674) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9}+O(q^{10})$$ 10 * q - 18 * q^2 + 161 * q^3 - 940 * q^4 + 1533 * q^5 - 8708 * q^6 - 1036 * q^7 + 34272 * q^8 - 35734 * q^9 $$10 q - 18 q^{2} + 161 q^{3} - 940 q^{4} + 1533 q^{5} - 8708 q^{6} - 1036 q^{7} + 34272 q^{8} - 35734 q^{9} + 4298 q^{10} + 42213 q^{11} + 135604 q^{12} - 319676 q^{13} - 39522 q^{14} + 151394 q^{15} + 322064 q^{16} + 324681 q^{17} - 1012868 q^{18} - 16121 q^{19} - 350616 q^{20} - 1557857 q^{21} - 62692 q^{22} + 2638863 q^{23} + 8449728 q^{24} - 1304092 q^{25} + 4179252 q^{26} - 18331558 q^{27} - 22156316 q^{28} + 15292500 q^{29} + 20557942 q^{30} + 19179237 q^{31} - 6263520 q^{32} + 1689359 q^{33} - 62909700 q^{34} - 43746759 q^{35} + 71476528 q^{36} + 39566985 q^{37} + 67365270 q^{38} - 44299486 q^{39} + 5721744 q^{40} - 53436852 q^{41} - 183129856 q^{42} + 101835992 q^{43} + 99704916 q^{44} + 85098230 q^{45} - 14489202 q^{46} + 32509659 q^{47} - 185141600 q^{48} - 49024598 q^{49} + 3328464 q^{50} + 44168403 q^{51} + 103893272 q^{52} - 25714707 q^{53} + 51200926 q^{54} - 144695222 q^{55} + 115352832 q^{56} - 121710346 q^{57} - 46645516 q^{58} + 46776513 q^{59} + 132391756 q^{60} - 113075039 q^{61} + 467465628 q^{62} + 318071530 q^{63} - 192008960 q^{64} - 338113566 q^{65} - 836682602 q^{66} - 126707879 q^{67} + 32262636 q^{68} + 1323616182 q^{69} + 697712470 q^{70} - 1188736032 q^{71} - 950557728 q^{72} - 859257651 q^{73} + 591757530 q^{74} - 169061732 q^{75} + 1101475592 q^{76} + 1911891891 q^{77} + 519432424 q^{78} - 527065417 q^{79} - 1257352656 q^{80} + 551662715 q^{81} - 1341703076 q^{82} - 144863208 q^{83} + 486452204 q^{84} - 1197360222 q^{85} - 678648216 q^{86} - 340781350 q^{87} + 903700608 q^{88} + 1661554797 q^{89} + 1967758744 q^{90} + 726641384 q^{91} - 1301840952 q^{92} - 423057489 q^{93} - 272580882 q^{94} - 1197123495 q^{95} - 1441922272 q^{96} + 869770188 q^{97} - 2404833858 q^{98} - 1900777180 q^{99}+O(q^{100})$$ 10 * q - 18 * q^2 + 161 * q^3 - 940 * q^4 + 1533 * q^5 - 8708 * q^6 - 1036 * q^7 + 34272 * q^8 - 35734 * q^9 + 4298 * q^10 + 42213 * q^11 + 135604 * q^12 - 319676 * q^13 - 39522 * q^14 + 151394 * q^15 + 322064 * q^16 + 324681 * q^17 - 1012868 * q^18 - 16121 * q^19 - 350616 * q^20 - 1557857 * q^21 - 62692 * q^22 + 2638863 * q^23 + 8449728 * q^24 - 1304092 * q^25 + 4179252 * q^26 - 18331558 * q^27 - 22156316 * q^28 + 15292500 * q^29 + 20557942 * q^30 + 19179237 * q^31 - 6263520 * q^32 + 1689359 * q^33 - 62909700 * q^34 - 43746759 * q^35 + 71476528 * q^36 + 39566985 * q^37 + 67365270 * q^38 - 44299486 * q^39 + 5721744 * q^40 - 53436852 * q^41 - 183129856 * q^42 + 101835992 * q^43 + 99704916 * q^44 + 85098230 * q^45 - 14489202 * q^46 + 32509659 * q^47 - 185141600 * q^48 - 49024598 * q^49 + 3328464 * q^50 + 44168403 * q^51 + 103893272 * q^52 - 25714707 * q^53 + 51200926 * q^54 - 144695222 * q^55 + 115352832 * q^56 - 121710346 * q^57 - 46645516 * q^58 + 46776513 * q^59 + 132391756 * q^60 - 113075039 * q^61 + 467465628 * q^62 + 318071530 * q^63 - 192008960 * q^64 - 338113566 * q^65 - 836682602 * q^66 - 126707879 * q^67 + 32262636 * q^68 + 1323616182 * q^69 + 697712470 * q^70 - 1188736032 * q^71 - 950557728 * q^72 - 859257651 * q^73 + 591757530 * q^74 - 169061732 * q^75 + 1101475592 * q^76 + 1911891891 * q^77 + 519432424 * q^78 - 527065417 * q^79 - 1257352656 * q^80 + 551662715 * q^81 - 1341703076 * q^82 - 144863208 * q^83 + 486452204 * q^84 - 1197360222 * q^85 - 678648216 * q^86 - 340781350 * q^87 + 903700608 * q^88 + 1661554797 * q^89 + 1967758744 * q^90 + 726641384 * q^91 - 1301840952 * q^92 - 423057489 * q^93 - 272580882 * q^94 - 1197123495 * q^95 - 1441922272 * q^96 + 869770188 * q^97 - 2404833858 * q^98 - 1900777180 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - x^{9} + 430 x^{8} + 61 x^{7} + 146753 x^{6} + 23608 x^{5} + 16136944 x^{4} + 30575648 x^{3} + 1399072384 x^{2} + 1034227200 x + 761760000$$ :

 $$\beta_{1}$$ $$=$$ $$2\nu$$ 2*v $$\beta_{2}$$ $$=$$ $$( 174976544647 \nu^{9} + 543280294119 \nu^{8} - 2080323777608 \nu^{7} + 306747495137243 \nu^{6} - 527825025841115 \nu^{5} + \cdots - 72\!\cdots\!00 ) / 49\!\cdots\!80$$ (174976544647*v^9 + 543280294119*v^8 - 2080323777608*v^7 + 306747495137243*v^6 - 527825025841115*v^5 + 79251473775272906*v^4 - 8455568604050729964*v^3 + 8633477355188858864*v^2 + 6390233594342311200*v - 728499442105846008000) / 495155599943444664480 $$\beta_{3}$$ $$=$$ $$( 439915122044593 \nu^{9} - 359425911506973 \nu^{8} + \cdots + 45\!\cdots\!00 ) / 45\!\cdots\!00$$ (439915122044593*v^9 - 359425911506973*v^8 + 189413411414469730*v^7 + 25877873507020493*v^6 + 64699967753173288309*v^5 + 10142716689341838644*v^4 + 7135341367123388280552*v^3 + 9561128363649180087824*v^2 + 619444498139966557897152*v + 457911692363235156681600) / 455543151947969091321600 $$\beta_{4}$$ $$=$$ $$( 9108600924689 \nu^{9} + 744234087958953 \nu^{8} + \cdots + 14\!\cdots\!20 ) / 23\!\cdots\!40$$ (9108600924689*v^9 + 744234087958953*v^8 - 2849814149431096*v^7 + 263155948009015381*v^6 - 723062074883078005*v^5 + 108565778922995107222*v^4 - 451992977525138313948*v^3 + 11826911844414978154768*v^2 + 8753915285372555954400*v + 1498083923068056865537920) / 2310726133069408434240 $$\beta_{5}$$ $$=$$ $$( - 44987322634457 \nu^{9} - 60129676436289 \nu^{8} + 230247989821048 \nu^{7} + \cdots + 26\!\cdots\!80 ) / 69\!\cdots\!20$$ (-44987322634457*v^9 - 60129676436289*v^8 + 230247989821048*v^7 - 51400958387300173*v^6 + 58419104028555565*v^5 - 8771467558811176486*v^4 + 1145664200820139449324*v^3 - 955543953106873546384*v^2 - 707264155426816927200*v + 261687922576606706402880) / 6932178399208225302720 $$\beta_{6}$$ $$=$$ $$( - 10228081391315 \nu^{9} + \cdots + 13\!\cdots\!84 ) / 69\!\cdots\!72$$ (-10228081391315*v^9 + 1952450458321161*v^8 - 7476304878007352*v^7 + 897435986095250741*v^6 - 1896907038176377685*v^5 + 284815366892851129814*v^4 - 297921422945101274472*v^3 + 31027145658538503980816*v^2 + 22965336024722535832800*v + 1379362333342473784891584) / 693217839920822530272 $$\beta_{7}$$ $$=$$ $$( - 24\!\cdots\!57 \nu^{9} + \cdots - 60\!\cdots\!00 ) / 15\!\cdots\!00$$ (-24593260288252157*v^9 + 42692492469767637*v^8 - 13545670542188346350*v^7 + 9874684454720909303*v^6 - 5152118620797417711701*v^5 + 2305464026693842938944*v^4 - 830194281200809808698368*v^3 - 538575932743903852034896*v^2 - 81614206561156272103139328*v - 60375780584089979592902400) / 1594401031817891819625600 $$\beta_{8}$$ $$=$$ $$( 20\!\cdots\!03 \nu^{9} + \cdots + 20\!\cdots\!00 ) / 31\!\cdots\!00$$ (2067492150283470103*v^9 - 1639595498403371523*v^8 + 883784357122680443350*v^7 + 146171516473387214363*v^6 + 300834508490249091273979*v^5 + 53897632103545345431524*v^4 + 32599014210284667922809672*v^3 + 57713122284011650146364784*v^2 + 2809343727409862629926750912*v + 2076654230999456261004489600) / 3188802063635783639251200 $$\beta_{9}$$ $$=$$ $$( 10\!\cdots\!87 \nu^{9} + \cdots + 10\!\cdots\!00 ) / 53\!\cdots\!00$$ (1042602787806930287*v^9 - 1349683788498539907*v^8 + 446731469300652376670*v^7 + 69678488713722111187*v^6 + 152247759899824475413931*v^5 + 26156328254249575048396*v^4 + 16592811419754117068838168*v^3 + 40035573671756716045423216*v^2 + 1433445544800945125082504768*v + 1059612788993006122246454400) / 531467010605963939875200
 $$\nu$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\nu^{2}$$ $$=$$ $$( \beta_{8} - \beta_{7} - 687\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b8 - b7 - 687*b3 - b2 + b1) / 4 $$\nu^{3}$$ $$=$$ $$( -27\beta_{5} + \beta_{4} - 507\beta_{2} - 375 ) / 4$$ (-27*b5 + b4 - 507*b2 - 375) / 4 $$\nu^{4}$$ $$=$$ $$( 14 \beta_{9} - 169 \beta_{8} + 155 \beta_{7} - 14 \beta_{6} + 155 \beta_{5} + 169 \beta_{4} + 87639 \beta_{3} - 134 \beta _1 - 87639 ) / 2$$ (14*b9 - 169*b8 + 155*b7 - 14*b6 + 155*b5 + 169*b4 + 87639*b3 - 134*b1 - 87639) / 2 $$\nu^{5}$$ $$=$$ $$( 28\beta_{9} - 583\beta_{8} - 11457\beta_{7} + 99345\beta_{3} + 142643\beta_{2} - 142643\beta_1 ) / 4$$ (28*b9 - 583*b8 - 11457*b7 + 99345*b3 + 142643*b2 - 142643*b1) / 4 $$\nu^{6}$$ $$=$$ $$( 12040\beta_{6} - 89029\beta_{5} - 107929\beta_{4} + 71655\beta_{2} + 49481343 ) / 4$$ (12040*b6 - 89029*b5 - 107929*b4 + 71655*b2 + 49481343) / 4 $$\nu^{7}$$ $$=$$ $$( - 9450 \beta_{9} + 115386 \beta_{8} + 1949460 \beta_{7} + 9450 \beta_{6} + 1949460 \beta_{5} - 115386 \beta_{4} - 13417998 \beta_{3} + 21095143 \beta _1 + 13417998 ) / 2$$ (-9450*b9 + 115386*b8 + 1949460*b7 + 9450*b6 + 1949460*b5 - 115386*b4 - 13417998*b3 + 21095143*b1 + 13417998) / 2 $$\nu^{8}$$ $$=$$ $$( - 4129692 \beta_{9} + 33731905 \beta_{8} - 25531489 \beta_{7} - 14657685447 \beta_{3} - 22648273 \beta_{2} + 22648273 \beta_1 ) / 4$$ (-4129692*b9 + 33731905*b8 - 25531489*b7 - 14657685447*b3 - 22648273*b2 + 22648273*b1) / 4 $$\nu^{9}$$ $$=$$ $$( -8200416\beta_{6} - 1242725823\beta_{5} + 81699181\beta_{4} - 12756581151\beta_{2} - 8505431979 ) / 4$$ (-8200416*b6 - 1242725823*b5 + 81699181*b4 - 12756581151*b2 - 8505431979) / 4

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/7\mathbb{Z}\right)^\times$$.

 $$n$$ $$3$$ $$\chi(n)$$ $$-\beta_{3}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2.1
 −8.71912 − 15.1020i −5.11725 − 8.86334i −0.371984 − 0.644295i 5.89912 + 10.2176i 8.80924 + 15.2580i −8.71912 + 15.1020i −5.11725 + 8.86334i −0.371984 + 0.644295i 5.89912 − 10.2176i 8.80924 − 15.2580i
−19.4382 + 33.6680i 113.728 + 196.982i −499.691 865.489i −162.760 + 281.909i −8842.67 5234.95 3598.46i 18947.7 −16026.5 + 27758.7i −6327.54 10959.6i
2.2 −12.2345 + 21.1908i −79.7348 138.105i −43.3662 75.1124i 1014.15 1756.56i 3902.06 −4235.51 4734.35i −10405.9 −2873.78 + 4977.54i 24815.2 + 42981.2i
2.3 −2.74397 + 4.75269i −1.70307 2.94981i 240.941 + 417.323i −828.924 + 1435.74i 18.6927 2822.68 + 5690.88i −5454.36 9835.70 17035.9i −4549.08 7879.24i
2.4 9.79824 16.9710i 104.977 + 181.826i 63.9892 + 110.832i 983.791 1703.98i 4114.37 −5768.52 + 2660.41i 12541.3 −12198.9 + 21129.2i −19278.8 33391.9i
2.5 15.6185 27.0520i −56.7670 98.3234i −231.874 401.617i −239.755 + 415.269i −3546.46 1428.40 6189.77i 1507.26 3396.51 5882.92i 7489.23 + 12971.7i
4.1 −19.4382 33.6680i 113.728 196.982i −499.691 + 865.489i −162.760 281.909i −8842.67 5234.95 + 3598.46i 18947.7 −16026.5 27758.7i −6327.54 + 10959.6i
4.2 −12.2345 21.1908i −79.7348 + 138.105i −43.3662 + 75.1124i 1014.15 + 1756.56i 3902.06 −4235.51 + 4734.35i −10405.9 −2873.78 4977.54i 24815.2 42981.2i
4.3 −2.74397 4.75269i −1.70307 + 2.94981i 240.941 417.323i −828.924 1435.74i 18.6927 2822.68 5690.88i −5454.36 9835.70 + 17035.9i −4549.08 + 7879.24i
4.4 9.79824 + 16.9710i 104.977 181.826i 63.9892 110.832i 983.791 + 1703.98i 4114.37 −5768.52 2660.41i 12541.3 −12198.9 21129.2i −19278.8 + 33391.9i
4.5 15.6185 + 27.0520i −56.7670 + 98.3234i −231.874 + 401.617i −239.755 415.269i −3546.46 1428.40 + 6189.77i 1507.26 3396.51 + 5882.92i 7489.23 12971.7i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7.10.c.a 10
3.b odd 2 1 63.10.e.b 10
4.b odd 2 1 112.10.i.c 10
7.b odd 2 1 49.10.c.g 10
7.c even 3 1 inner 7.10.c.a 10
7.c even 3 1 49.10.a.e 5
7.d odd 6 1 49.10.a.f 5
7.d odd 6 1 49.10.c.g 10
21.h odd 6 1 63.10.e.b 10
28.g odd 6 1 112.10.i.c 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.10.c.a 10 1.a even 1 1 trivial
7.10.c.a 10 7.c even 3 1 inner
49.10.a.e 5 7.c even 3 1
49.10.a.f 5 7.d odd 6 1
49.10.c.g 10 7.b odd 2 1
49.10.c.g 10 7.d odd 6 1
63.10.e.b 10 3.b odd 2 1
63.10.e.b 10 21.h odd 6 1
112.10.i.c 10 4.b odd 2 1
112.10.i.c 10 28.g odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{10}^{\mathrm{new}}(7, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 18 T^{9} + \cdots + 10212166139904$$
$3$ $$T^{10} - 161 T^{9} + \cdots + 86\!\cdots\!89$$
$5$ $$T^{10} - 1533 T^{9} + \cdots + 10\!\cdots\!25$$
$7$ $$T^{10} + 1036 T^{9} + \cdots + 10\!\cdots\!07$$
$11$ $$T^{10} - 42213 T^{9} + \cdots + 91\!\cdots\!25$$
$13$ $$(T^{5} + 159838 T^{4} + \cdots + 79\!\cdots\!92)^{2}$$
$17$ $$T^{10} - 324681 T^{9} + \cdots + 29\!\cdots\!81$$
$19$ $$T^{10} + 16121 T^{9} + \cdots + 39\!\cdots\!01$$
$23$ $$T^{10} - 2638863 T^{9} + \cdots + 42\!\cdots\!01$$
$29$ $$(T^{5} - 7646250 T^{4} + \cdots + 73\!\cdots\!00)^{2}$$
$31$ $$T^{10} - 19179237 T^{9} + \cdots + 14\!\cdots\!69$$
$37$ $$T^{10} - 39566985 T^{9} + \cdots + 47\!\cdots\!25$$
$41$ $$(T^{5} + 26718426 T^{4} + \cdots - 20\!\cdots\!40)^{2}$$
$43$ $$(T^{5} - 50917996 T^{4} + \cdots + 72\!\cdots\!36)^{2}$$
$47$ $$T^{10} - 32509659 T^{9} + \cdots + 34\!\cdots\!25$$
$53$ $$T^{10} + 25714707 T^{9} + \cdots + 47\!\cdots\!21$$
$59$ $$T^{10} - 46776513 T^{9} + \cdots + 79\!\cdots\!25$$
$61$ $$T^{10} + 113075039 T^{9} + \cdots + 27\!\cdots\!41$$
$67$ $$T^{10} + 126707879 T^{9} + \cdots + 20\!\cdots\!25$$
$71$ $$(T^{5} + 594368016 T^{4} + \cdots + 25\!\cdots\!92)^{2}$$
$73$ $$T^{10} + 859257651 T^{9} + \cdots + 87\!\cdots\!29$$
$79$ $$T^{10} + 527065417 T^{9} + \cdots + 26\!\cdots\!21$$
$83$ $$(T^{5} + 72431604 T^{4} + \cdots + 39\!\cdots\!36)^{2}$$
$89$ $$T^{10} - 1661554797 T^{9} + \cdots + 21\!\cdots\!41$$
$97$ $$(T^{5} - 434885094 T^{4} + \cdots - 84\!\cdots\!56)^{2}$$