Properties

 Label 624.2.cn.d Level $624$ Weight $2$ Character orbit 624.cn Analytic conductor $4.983$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,2,Mod(305,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(12))

chi = DirichletCharacter(H, H._module([0, 0, 6, 5]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("624.305");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 624.cn (of order $$12$$, degree $$4$$, not 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: $$4.98266508613$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$4$$ over $$\Q(\zeta_{12})$$ Coefficient field: 16.0.9349208943630483456.9 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25$$ x^16 - 8*x^15 + 48*x^14 - 196*x^13 + 642*x^12 - 1668*x^11 + 3580*x^10 - 6328*x^9 + 9297*x^8 - 11276*x^7 + 11224*x^6 - 9024*x^5 + 5736*x^4 - 2780*x^3 + 972*x^2 - 220*x + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 3^{2}$$ Twist minimal: no (minimal twist has level 78) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{12} - \beta_{10}) q^{3} + ( - \beta_{13} - \beta_{9} - \beta_{3} + \beta_1) q^{5} + (\beta_{13} + \beta_{11} - \beta_{10} + \beta_{5} + \beta_{2} - 1) q^{7} + (\beta_{15} - \beta_{14} - 2 \beta_{8} - \beta_{7} - \beta_{5}) q^{9}+O(q^{10})$$ q + (-b12 - b10) * q^3 + (-b13 - b9 - b3 + b1) * q^5 + (b13 + b11 - b10 + b5 + b2 - 1) * q^7 + (b15 - b14 - 2*b8 - b7 - b5) * q^9 $$q + ( - \beta_{12} - \beta_{10}) q^{3} + ( - \beta_{13} - \beta_{9} - \beta_{3} + \beta_1) q^{5} + (\beta_{13} + \beta_{11} - \beta_{10} + \beta_{5} + \beta_{2} - 1) q^{7} + (\beta_{15} - \beta_{14} - 2 \beta_{8} - \beta_{7} - \beta_{5}) q^{9} + ( - \beta_{13} + \beta_{11} - \beta_{6} + \beta_{4}) q^{11} + (2 \beta_{13} - \beta_{12} + 2 \beta_{11} - 2 \beta_{10} + \beta_{8} + \beta_{5} - \beta_{2} - \beta_1 - 1) q^{13} + (\beta_{15} - 2 \beta_{13} + 2 \beta_{12} - 3 \beta_{11} + \beta_{10} - 2 \beta_{8} + \beta_{6} - \beta_{5} + \cdots + 1) q^{15}+ \cdots + ( - 3 \beta_{13} + 8 \beta_{12} - 8 \beta_{11} + 8 \beta_{10} - 5 \beta_{9} + 8 \beta_{6} + \cdots + 3 \beta_1) q^{99}+O(q^{100})$$ q + (-b12 - b10) * q^3 + (-b13 - b9 - b3 + b1) * q^5 + (b13 + b11 - b10 + b5 + b2 - 1) * q^7 + (b15 - b14 - 2*b8 - b7 - b5) * q^9 + (-b13 + b11 - b6 + b4) * q^11 + (2*b13 - b12 + 2*b11 - 2*b10 + b8 + b5 - b2 - b1 - 1) * q^13 + (b15 - 2*b13 + 2*b12 - 3*b11 + b10 - 2*b8 + b6 - b5 + 2*b4 - b3 - 2*b2 + 2*b1 + 1) * q^15 + (b14 - 2*b13 + 2*b12 - 4*b11 + 2*b10 - 2*b9 + 2*b6 + 2*b4 + b3 + 2*b1) * q^17 + (b13 + b11 - b10 + 2*b5 - 2*b2 - 2*b1 + 2) * q^19 + (b11 + b8 - b7 + b6 + 2*b5 - b2 - b1 - 1) * q^21 + (b15 + b14 + b12 + b10 - 2*b9 - b6 - b4 - b3 + 2*b1) * q^23 + (-4*b10 + 4*b9 - b8 - 4*b6 - b5 - 2*b2 + 2*b1 + 1) * q^25 + (2*b13 - 3*b12 + 4*b11 - 2*b10 + b9 - 2*b6 - 4*b4 + b1) * q^27 + (b15 + 2*b13 - b12 + 4*b11 - b10 + 2*b9 + b7 - 3*b6 - b4 - 2*b1) * q^29 + (b13 + b12 + b11 - 3*b10 + 2*b9 - 2*b6 - 2*b5 - 2*b2) * q^31 + (b15 + 2*b14 - b8 + 2*b7 + b5 - 2*b3 + b2 - 2) * q^33 + (2*b15 - 2*b14 + b13 - 2*b12 - 2*b10 + 2*b9 - 2*b7 + 2*b6 - b4 - 2*b1) * q^35 + (3*b13 - b12 + 3*b11 - 6*b10 + 3*b9 - 3*b6 + b5 - 2*b1 + 1) * q^37 + (b15 - 2*b13 + 2*b12 - 2*b11 + 4*b10 - b9 + b8 - 2*b7 + b6 + 2*b5 + b4 - 3) * q^39 + (2*b15 - b14 + 4*b13 - 4*b12 + 8*b11 - 4*b10 + 2*b9 - 2*b7 - 4*b6 - 6*b4 - 2*b3 - 2*b1) * q^41 + (-b10 + b9 - b6 - b1) * q^43 + (-b14 - 2*b13 + 2*b12 - 3*b11 + 4*b10 - 3*b9 + b7 + b6 + 3*b5 - b4 - 3*b2 + 5*b1 + 3) * q^45 + (-2*b15 + 3*b12 - 3*b11 + 3*b10 - 3*b9 - b7 + b3 + 3*b1) * q^47 + (-2*b13 - 2*b11 + 4*b10 - 2*b9 - 2*b8 + 2*b6 - b2 - 2*b1 + 2) * q^49 + (2*b15 - 2*b14 + 2*b13 - b12 + 2*b11 - b9 - 4*b2 - 2*b1 + 2) * q^51 + (-b15 - b14 - 4*b13 + 4*b12 - 7*b11 + 4*b10 - 2*b9 + 3*b6 + 6*b4 + 2*b1) * q^53 + (2*b13 - 4*b12 + 2*b11 - 5*b10 + 3*b9 - b8 - 3*b6 - 2*b5 + 3*b2 - 2*b1) * q^55 + (2*b14 - 4*b13 - 2*b11 + 4*b10 - 4*b9 - b8 + b7 + 2*b6 - 2*b5 + 2*b3 - b2 + 2*b1 - 1) * q^57 + (2*b15 + 2*b14 - 2*b13 - b12 - b10 - b9 + 2*b7 + b6 - 4*b3 + b1) * q^59 + (2*b13 - b12 + 2*b11 - 4*b10 + 2*b9 + 8*b8 - 2*b6 + 4*b5 + 3*b2 - 3) * q^61 + (2*b14 - 3*b13 + b12 - b11 + 4*b10 - b9 + b8 + b6 + 2*b5 + 2*b4 - b2 + 2) * q^63 + (2*b15 - b14 + 2*b13 - 2*b12 - 2*b10 + 4*b9 - 3*b7 + 2*b6 + b3 - 4*b1) * q^65 + (b12 + 4*b8 + 4*b5 + 4*b2 + b1 - 4) * q^67 + (-b14 - b13 + 2*b12 - 4*b11 + 3*b10 - b9 + 3*b6 - 3*b5 + 3*b4 - b3 - 2*b2 + b1 - 2) * q^69 + (b15 - 3*b14 - b12 - b11 - b10 + 4*b9 + 3*b7 + 2*b6 + 3*b4 - 4*b1) * q^71 + (2*b13 - 2*b12 + 2*b11 - 2*b10 - b8 - 6*b5 + 5*b2 - 4*b1 + 1) * q^73 + (-4*b15 - 2*b14 - b13 + b12 - b11 + 2*b10 - 2*b9 - 6*b8 + 2*b7 - b6 - b4 + 2*b3 + 3*b1) * q^75 + (2*b15 - 2*b14 + b13 + b12 - 3*b11 + b10 + b7 + 2*b6 + b3) * q^77 + (b13 + b12 + b11 - b10 + 4*b1 + 6) * q^79 + (-2*b15 - 2*b14 - 4*b7 - 2*b3 + 3*b2) * q^81 + (4*b14 - b13 - b9 + 2*b7 + 2*b3 + b1) * q^83 + (3*b13 + 3*b11 - 3*b10 - b8 + b5 + b2 - 2) * q^85 + (-2*b15 + 3*b14 + b13 - b11 + 2*b10 - b9 + 2*b8 + 2*b7 + 2*b6 + b5 - 2*b4 - b3 + 6*b2 - b1 - 6) * q^87 + (-b15 + 3*b14 - 2*b13 + 2*b11 - 6*b9 - 2*b6 - 4*b4 - b3 + 6*b1) * q^89 + (-b13 - b12 - b11 + b10 + 4*b8 - 2*b5 - 2*b2 - 3*b1 + 2) * q^91 + (-3*b15 + b14 + 2*b12 - 2*b11 + 2*b10 - 3*b8 + 2*b7 - 2*b6 + 3*b3 - 3*b2 + 2*b1) * q^93 + (-b15 + b14 + b13 - b12 + 2*b11 - b10 + b9 - b7 - b6 - b4 + 2*b3 - b1) * q^95 + (-4*b13 + 6*b12 - 4*b11 + 10*b10 - 6*b9 - 2*b8 + 6*b6 - 2*b5 + 2*b2 + 2*b1) * q^97 + (-3*b13 + 8*b12 - 8*b11 + 8*b10 - 5*b9 + 8*b6 + 4*b4 + 3*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 8 q^{7}+O(q^{10})$$ 16 * q - 8 * q^7 $$16 q - 8 q^{7} - 24 q^{13} + 16 q^{19} - 24 q^{21} - 16 q^{31} - 24 q^{33} + 16 q^{37} - 48 q^{39} + 24 q^{45} + 24 q^{49} + 24 q^{55} - 24 q^{57} - 24 q^{61} + 24 q^{63} - 32 q^{67} - 48 q^{69} + 56 q^{73} + 96 q^{79} + 24 q^{81} - 24 q^{85} - 48 q^{87} + 16 q^{91} - 24 q^{93} + 16 q^{97}+O(q^{100})$$ 16 * q - 8 * q^7 - 24 * q^13 + 16 * q^19 - 24 * q^21 - 16 * q^31 - 24 * q^33 + 16 * q^37 - 48 * q^39 + 24 * q^45 + 24 * q^49 + 24 * q^55 - 24 * q^57 - 24 * q^61 + 24 * q^63 - 32 * q^67 - 48 * q^69 + 56 * q^73 + 96 * q^79 + 24 * q^81 - 24 * q^85 - 48 * q^87 + 16 * q^91 - 24 * q^93 + 16 * q^97

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 8 x^{15} + 48 x^{14} - 196 x^{13} + 642 x^{12} - 1668 x^{11} + 3580 x^{10} - 6328 x^{9} + 9297 x^{8} - 11276 x^{7} + 11224 x^{6} - 9024 x^{5} + 5736 x^{4} - 2780 x^{3} + \cdots + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( 50 \nu^{14} - 350 \nu^{13} + 2019 \nu^{12} - 7564 \nu^{11} + 23298 \nu^{10} - 55495 \nu^{9} + 109509 \nu^{8} - 173970 \nu^{7} + 227114 \nu^{6} - 237337 \nu^{5} + 197448 \nu^{4} + \cdots + 2945 ) / 65$$ (50*v^14 - 350*v^13 + 2019*v^12 - 7564*v^11 + 23298*v^10 - 55495*v^9 + 109509*v^8 - 173970*v^7 + 227114*v^6 - 237337*v^5 + 197448*v^4 - 125328*v^3 + 58327*v^2 - 17721*v + 2945) / 65 $$\beta_{2}$$ $$=$$ $$( 3456 \nu^{15} - 25920 \nu^{14} + 151876 \nu^{13} - 594074 \nu^{12} + 1879372 \nu^{11} - 4666596 \nu^{10} + 9554736 \nu^{9} - 15945783 \nu^{8} + 21928484 \nu^{7} + \cdots - 125460 ) / 17095$$ (3456*v^15 - 25920*v^14 + 151876*v^13 - 594074*v^12 + 1879372*v^11 - 4666596*v^10 + 9554736*v^9 - 15945783*v^8 + 21928484*v^7 - 24527176*v^6 + 22138664*v^5 - 15739188*v^4 + 8598668*v^3 - 3418428*v^2 + 929924*v - 125460) / 17095 $$\beta_{3}$$ $$=$$ $$( - 4446 \nu^{15} + 26244 \nu^{14} - 145319 \nu^{13} + 473557 \nu^{12} - 1320043 \nu^{11} + 2591696 \nu^{10} - 4078214 \nu^{9} + 4103647 \nu^{8} - 1746621 \nu^{7} + \cdots - 363138 ) / 3419$$ (-4446*v^15 + 26244*v^14 - 145319*v^13 + 473557*v^12 - 1320043*v^11 + 2591696*v^10 - 4078214*v^9 + 4103647*v^8 - 1746621*v^7 - 3595456*v^6 + 9059680*v^5 - 11752436*v^4 + 9921731*v^3 - 5694253*v^2 + 2016505*v - 363138) / 3419 $$\beta_{4}$$ $$=$$ $$( 13659 \nu^{15} - 124929 \nu^{14} + 763986 \nu^{13} - 3301409 \nu^{12} + 11090331 \nu^{11} - 29899787 \nu^{10} + 65696977 \nu^{9} - 119237433 \nu^{8} + \cdots - 2004525 ) / 17095$$ (13659*v^15 - 124929*v^14 + 763986*v^13 - 3301409*v^12 + 11090331*v^11 - 29899787*v^10 + 65696977*v^9 - 119237433*v^8 + 178229161*v^7 - 219256768*v^6 + 218727246*v^5 - 173893084*v^4 + 106330641*v^3 - 47350335*v^2 + 13730184*v - 2004525) / 17095 $$\beta_{5}$$ $$=$$ $$( 26630 \nu^{15} - 182630 \nu^{14} + 1056740 \nu^{13} - 3923413 \nu^{12} + 12097498 \nu^{11} - 28666706 \nu^{10} + 56590650 \nu^{9} - 89528958 \nu^{8} + \cdots + 218885 ) / 17095$$ (26630*v^15 - 182630*v^14 + 1056740*v^13 - 3923413*v^12 + 12097498*v^11 - 28666706*v^10 + 56590650*v^9 - 89528958*v^8 + 116711850*v^7 - 121073088*v^6 + 99711814*v^5 - 61611091*v^4 + 27082796*v^3 - 6949514*v^2 + 510382*v + 218885) / 17095 $$\beta_{6}$$ $$=$$ $$( 31151 \nu^{15} - 209568 \nu^{14} + 1205575 \nu^{13} - 4413594 \nu^{12} + 13470792 \nu^{11} - 31433015 \nu^{10} + 61105562 \nu^{9} - 94697844 \nu^{8} + \cdots + 318955 ) / 17095$$ (31151*v^15 - 209568*v^14 + 1205575*v^13 - 4413594*v^12 + 13470792*v^11 - 31433015*v^10 + 61105562*v^9 - 94697844*v^8 + 120513439*v^7 - 121047543*v^6 + 95524566*v^5 - 55417740*v^4 + 21932579*v^3 - 4315386*v^2 - 222694*v + 318955) / 17095 $$\beta_{7}$$ $$=$$ $$( 4446 \nu^{15} - 40446 \nu^{14} + 244733 \nu^{13} - 1050316 \nu^{12} + 3488215 \nu^{11} - 9316343 \nu^{10} + 20195906 \nu^{9} - 36195170 \nu^{8} + 53194681 \nu^{7} + \cdots - 506866 ) / 3419$$ (4446*v^15 - 40446*v^14 + 244733*v^13 - 1050316*v^12 + 3488215*v^11 - 9316343*v^10 + 20195906*v^9 - 36195170*v^8 + 53194681*v^7 - 64335603*v^6 + 62774562*v^5 - 48766231*v^4 + 28950195*v^3 - 12506136*v^2 + 3501235*v - 506866) / 3419 $$\beta_{8}$$ $$=$$ $$( 26630 \nu^{15} - 216820 \nu^{14} + 1296070 \nu^{13} - 5311527 \nu^{12} + 17314892 \nu^{11} - 44845414 \nu^{10} + 95362110 \nu^{9} - 166733397 \nu^{8} + \cdots - 2071845 ) / 17095$$ (26630*v^15 - 216820*v^14 + 1296070*v^13 - 5311527*v^12 + 17314892*v^11 - 44845414*v^10 + 95362110*v^9 - 166733397*v^8 + 240513840*v^7 - 284726942*v^6 + 273082466*v^5 - 208344314*v^4 + 122001074*v^3 - 52073476*v^2 + 14507768*v - 2071845) / 17095 $$\beta_{9}$$ $$=$$ $$( 34456 \nu^{15} - 251845 \nu^{14} + 1471552 \nu^{13} - 5679382 \nu^{12} + 17866121 \nu^{11} - 43899570 \nu^{10} + 89239997 \nu^{9} - 147407790 \nu^{8} + \cdots - 597160 ) / 17095$$ (34456*v^15 - 251845*v^14 + 1471552*v^13 - 5679382*v^12 + 17866121*v^11 - 43899570*v^10 + 89239997*v^9 - 147407790*v^8 + 200796845*v^7 - 221711386*v^6 + 196859312*v^5 - 136496129*v^4 + 71276310*v^3 - 26093918*v^2 + 5964282*v - 597160) / 17095 $$\beta_{10}$$ $$=$$ $$( 31151 \nu^{15} - 244547 \nu^{14} + 1450428 \nu^{13} - 5830901 \nu^{12} + 18791545 \nu^{11} - 47884454 \nu^{10} + 100424851 \nu^{9} - 172616328 \nu^{8} + \cdots - 1570700 ) / 17095$$ (31151*v^15 - 244547*v^14 + 1450428*v^13 - 5830901*v^12 + 18791545*v^11 - 47884454*v^10 + 100424851*v^9 - 172616328*v^8 + 244795930*v^7 - 283899773*v^6 + 266209462*v^5 - 197414070*v^4 + 111767067*v^3 - 45556942*v^2 + 12002861*v - 1570700) / 17095 $$\beta_{11}$$ $$=$$ $$( 31151 \nu^{15} - 267691 \nu^{14} + 1612436 \nu^{13} - 6769285 \nu^{12} + 22315745 \nu^{11} - 58795272 \nu^{10} + 126534965 \nu^{9} - 224500179 \nu^{8} + \cdots - 2946190 ) / 17095$$ (31151*v^15 - 267691*v^14 + 1612436*v^13 - 6769285*v^12 + 22315745*v^11 - 58795272*v^10 + 126534965*v^9 - 224500179*v^8 + 327817666*v^7 - 393312770*v^6 + 381718799*v^5 - 294631494*v^4 + 174197744*v^3 - 74804909*v^2 + 20879374*v - 2946190) / 17095 $$\beta_{12}$$ $$=$$ $$( - 46358 \nu^{15} + 321385 \nu^{14} - 1862113 \nu^{13} + 6958330 \nu^{12} - 21520004 \nu^{11} + 51288524 \nu^{10} - 101686623 \nu^{9} + 161907582 \nu^{8} + \cdots - 273965 ) / 17095$$ (-46358*v^15 + 321385*v^14 - 1862113*v^13 + 6958330*v^12 - 21520004*v^11 + 51288524*v^10 - 101686623*v^9 + 161907582*v^8 - 212379442*v^7 + 222238161*v^6 - 184926311*v^5 + 116130120*v^4 - 52438780*v^3 + 14435393*v^2 - 1585494*v - 273965) / 17095 $$\beta_{13}$$ $$=$$ $$( 46358 \nu^{15} - 337691 \nu^{14} + 1976255 \nu^{13} - 7626613 \nu^{12} + 24045856 \nu^{11} - 59214555 \nu^{10} + 120883519 \nu^{9} - 200765043 \nu^{8} + \cdots - 1289570 ) / 17095$$ (46358*v^15 - 337691*v^14 + 1976255*v^13 - 7626613*v^12 + 24045856*v^11 - 59214555*v^10 + 120883519*v^9 - 200765043*v^8 + 275748766*v^7 - 307981421*v^6 + 278102740*v^5 - 197668010*v^4 + 107176707*v^3 - 41710597*v^2 + 10489359*v - 1289570) / 17095 $$\beta_{14}$$ $$=$$ $$( 70161 \nu^{15} - 497409 \nu^{14} + 2891923 \nu^{13} - 10958903 \nu^{12} + 34143086 \nu^{11} - 82494331 \nu^{10} + 165437899 \nu^{9} - 268073830 \nu^{8} + \cdots - 701840 ) / 17095$$ (70161*v^15 - 497409*v^14 + 2891923*v^13 - 10958903*v^12 + 34143086*v^11 - 82494331*v^10 + 165437899*v^9 - 268073830*v^8 + 358323284*v^7 - 385802736*v^6 + 333141867*v^5 - 222600632*v^4 + 111186219*v^3 - 38325579*v^2 + 8096306*v - 701840) / 17095 $$\beta_{15}$$ $$=$$ $$( 70161 \nu^{15} - 555006 \nu^{14} + 3295102 \nu^{13} - 13295132 \nu^{12} + 42919133 \nu^{11} - 109669069 \nu^{10} + 230473591 \nu^{9} - 397246280 \nu^{8} + \cdots - 3835485 ) / 17095$$ (70161*v^15 - 555006*v^14 + 3295102*v^13 - 13295132*v^12 + 42919133*v^11 - 109669069*v^10 + 230473591*v^9 - 397246280*v^8 + 564862707*v^7 - 657422824*v^6 + 619023919*v^5 - 461729963*v^4 + 263398784*v^3 - 108502921*v^2 + 28915123*v - 3835485) / 17095
 $$\nu$$ $$=$$ $$( - 2 \beta_{15} - 2 \beta_{14} + 3 \beta_{13} - 3 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 6 \beta_{9} - 3 \beta_{8} - \beta_{7} - 3 \beta_{5} + \beta_{3} - 6 \beta _1 + 3 ) / 6$$ (-2*b15 - 2*b14 + 3*b13 - 3*b12 + 3*b11 - 3*b10 + 6*b9 - 3*b8 - b7 - 3*b5 + b3 - 6*b1 + 3) / 6 $$\nu^{2}$$ $$=$$ $$( -2\beta_{15} - 3\beta_{12} + 3\beta_{11} - 3\beta_{10} + 3\beta_{9} - 3\beta_{5} + \beta_{3} - 6 ) / 3$$ (-2*b15 - 3*b12 + 3*b11 - 3*b10 + 3*b9 - 3*b5 + b3 - 6) / 3 $$\nu^{3}$$ $$=$$ $$( 7 \beta_{15} + 13 \beta_{14} - 21 \beta_{13} + 12 \beta_{12} - 9 \beta_{11} + 12 \beta_{10} - 21 \beta_{9} + 21 \beta_{8} + 2 \beta_{7} - 3 \beta_{6} + 12 \beta_{5} - 6 \beta_{4} + \beta_{3} + 9 \beta_{2} + 39 \beta _1 - 24 ) / 6$$ (7*b15 + 13*b14 - 21*b13 + 12*b12 - 9*b11 + 12*b10 - 21*b9 + 21*b8 + 2*b7 - 3*b6 + 12*b5 - 6*b4 + b3 + 9*b2 + 39*b1 - 24) / 6 $$\nu^{4}$$ $$=$$ $$( 20 \beta_{15} + 2 \beta_{14} - 6 \beta_{13} + 30 \beta_{12} - 18 \beta_{11} + 24 \beta_{10} - 24 \beta_{9} + 9 \beta_{8} - 8 \beta_{7} - 6 \beta_{6} + 27 \beta_{5} - 6 \beta_{4} - 10 \beta_{3} + 9 \beta_{2} + 18 \beta _1 + 18 ) / 3$$ (20*b15 + 2*b14 - 6*b13 + 30*b12 - 18*b11 + 24*b10 - 24*b9 + 9*b8 - 8*b7 - 6*b6 + 27*b5 - 6*b4 - 10*b3 + 9*b2 + 18*b1 + 18) / 3 $$\nu^{5}$$ $$=$$ $$( 7 \beta_{15} - 93 \beta_{14} + 132 \beta_{13} + 3 \beta_{12} + 18 \beta_{11} + 3 \beta_{10} + 57 \beta_{9} - 138 \beta_{8} - 39 \beta_{7} + 39 \beta_{6} - 33 \beta_{5} + 48 \beta_{4} - 56 \beta_{3} - 75 \beta_{2} - 237 \beta _1 + 183 ) / 6$$ (7*b15 - 93*b14 + 132*b13 + 3*b12 + 18*b11 + 3*b10 + 57*b9 - 138*b8 - 39*b7 + 39*b6 - 33*b5 + 48*b4 - 56*b3 - 75*b2 - 237*b1 + 183) / 6 $$\nu^{6}$$ $$=$$ $$( - 135 \beta_{15} - 50 \beta_{14} + 78 \beta_{13} - 207 \beta_{12} + 93 \beta_{11} - 120 \beta_{10} + 147 \beta_{9} - 129 \beta_{8} + 65 \beta_{7} + 117 \beta_{6} - 219 \beta_{5} + 87 \beta_{4} + 45 \beta_{3} - 135 \beta_{2} - 243 \beta _1 - 15 ) / 3$$ (-135*b15 - 50*b14 + 78*b13 - 207*b12 + 93*b11 - 120*b10 + 147*b9 - 129*b8 + 65*b7 + 117*b6 - 219*b5 + 87*b4 + 45*b3 - 135*b2 - 243*b1 - 15) / 3 $$\nu^{7}$$ $$=$$ $$( - 348 \beta_{15} + 604 \beta_{14} - 849 \beta_{13} - 537 \beta_{12} + 3 \beta_{11} - 327 \beta_{10} - 84 \beta_{9} + 801 \beta_{8} + 491 \beta_{7} - 54 \beta_{6} - 207 \beta_{5} - 234 \beta_{4} + 615 \beta_{3} + 378 \beta_{2} + \cdots - 1299 ) / 6$$ (-348*b15 + 604*b14 - 849*b13 - 537*b12 + 3*b11 - 327*b10 - 84*b9 + 801*b8 + 491*b7 - 54*b6 - 207*b5 - 234*b4 + 615*b3 + 378*b2 + 1260*b1 - 1299) / 6 $$\nu^{8}$$ $$=$$ $$264 \beta_{15} + 212 \beta_{14} - 288 \beta_{13} + 392 \beta_{12} - 164 \beta_{11} + 152 \beta_{10} - 304 \beta_{9} + 446 \beta_{8} - 104 \beta_{7} - 380 \beta_{6} + 520 \beta_{5} - 296 \beta_{4} + 8 \beta_{3} + 469 \beta_{2} + \cdots - 202$$ 264*b15 + 212*b14 - 288*b13 + 392*b12 - 164*b11 + 152*b10 - 304*b9 + 446*b8 - 104*b7 - 380*b6 + 520*b5 - 296*b4 + 8*b3 + 469*b2 + 804*b1 - 202 $$\nu^{9}$$ $$=$$ $$( 4153 \beta_{15} - 3395 \beta_{14} + 5103 \beta_{13} + 6606 \beta_{12} - 369 \beta_{11} + 3078 \beta_{10} - 891 \beta_{9} - 3675 \beta_{8} - 4708 \beta_{7} - 2349 \beta_{6} + 4830 \beta_{5} + 36 \beta_{4} - 4925 \beta_{3} + \cdots + 8202 ) / 6$$ (4153*b15 - 3395*b14 + 5103*b13 + 6606*b12 - 369*b11 + 3078*b10 - 891*b9 - 3675*b8 - 4708*b7 - 2349*b6 + 4830*b5 + 36*b4 - 4925*b3 - 27*b2 - 4797*b1 + 8202) / 6 $$\nu^{10}$$ $$=$$ $$( - 3833 \beta_{15} - 6390 \beta_{14} + 8640 \beta_{13} - 5082 \beta_{12} + 2922 \beta_{11} - 861 \beta_{10} + 5766 \beta_{9} - 12003 \beta_{8} + 180 \beta_{7} + 8289 \beta_{6} - 9465 \beta_{5} + 7389 \beta_{4} + \cdots + 8814 ) / 3$$ (-3833*b15 - 6390*b14 + 8640*b13 - 5082*b12 + 2922*b11 - 861*b10 + 5766*b9 - 12003*b8 + 180*b7 + 8289*b6 - 9465*b5 + 7389*b4 - 3011*b3 - 11610*b2 - 20637*b1 + 8814) / 3 $$\nu^{11}$$ $$=$$ $$( - 38777 \beta_{15} + 13891 \beta_{14} - 24690 \beta_{13} - 60285 \beta_{12} + 4404 \beta_{11} - 21675 \beta_{10} + 15753 \beta_{9} + 5778 \beta_{8} + 38147 \beta_{7} + 38061 \beta_{6} - 56559 \beta_{5} + \cdots - 43491 ) / 6$$ (-38777*b15 + 13891*b14 - 24690*b13 - 60285*b12 + 4404*b11 - 21675*b10 + 15753*b9 + 5778*b8 + 38147*b7 + 38061*b6 - 56559*b5 + 15534*b4 + 32242*b3 - 24783*b2 - 3609*b1 - 43491) / 6 $$\nu^{12}$$ $$=$$ $$( 10130 \beta_{15} + 55670 \beta_{14} - 77934 \beta_{13} + 6762 \beta_{12} - 19512 \beta_{11} - 6960 \beta_{10} - 34860 \beta_{9} + 95841 \beta_{8} + 17638 \beta_{7} - 46392 \beta_{6} + 44046 \beta_{5} + \cdots - 89109 ) / 3$$ (10130*b15 + 55670*b14 - 77934*b13 + 6762*b12 - 19512*b11 - 6960*b10 - 34860*b9 + 95841*b8 + 17638*b7 - 46392*b6 + 44046*b5 - 51342*b4 + 40970*b3 + 79695*b2 + 156816*b1 - 89109) / 3 $$\nu^{13}$$ $$=$$ $$( 316318 \beta_{15} + 444 \beta_{14} + 49461 \beta_{13} + 471657 \beta_{12} - 52155 \beta_{11} + 131889 \beta_{10} - 177702 \beta_{9} + 140085 \beta_{8} - 266421 \beta_{7} - 404214 \beta_{6} + \cdots + 151791 ) / 6$$ (316318*b15 + 444*b14 + 49461*b13 + 471657*b12 - 52155*b11 + 131889*b10 - 177702*b9 + 140085*b8 - 266421*b7 - 404214*b6 + 525717*b5 - 231774*b4 - 165647*b3 + 362934*b2 + 336198*b1 + 151791) / 6 $$\nu^{14}$$ $$=$$ $$( 77259 \beta_{15} - 430001 \beta_{14} + 630756 \beta_{13} + 188841 \beta_{12} + 132069 \beta_{11} + 120309 \beta_{10} + 180957 \beta_{9} - 677523 \beta_{8} - 272194 \beta_{7} + 155484 \beta_{6} + \cdots + 760593 ) / 3$$ (77259*b15 - 430001*b14 + 630756*b13 + 188841*b12 + 132069*b11 + 120309*b10 + 180957*b9 - 677523*b8 - 272194*b7 + 155484*b6 - 74946*b5 + 286680*b4 - 405678*b3 - 440622*b2 - 1047270*b1 + 760593) / 3 $$\nu^{15}$$ $$=$$ $$( - 2288517 \beta_{15} - 846113 \beta_{14} + 818229 \beta_{13} - 3234570 \beta_{12} + 590439 \beta_{11} - 688260 \beta_{10} + 1683837 \beta_{9} - 2418363 \beta_{8} + 1547174 \beta_{7} + \cdots + 356820 ) / 6$$ (-2288517*b15 - 846113*b14 + 818229*b13 - 3234570*b12 + 590439*b11 - 688260*b10 + 1683837*b9 - 2418363*b8 + 1547174*b7 + 3514725*b6 - 4219254*b5 + 2420334*b4 + 455763*b3 - 3753099*b2 - 4677825*b1 + 356820) / 6

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$$.

 $$n$$ $$79$$ $$145$$ $$209$$ $$469$$ $$\chi(n)$$ $$1$$ $$-\beta_{5}$$ $$-1$$ $$1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
305.1
 0.5 + 2.74530i 0.5 + 1.33108i 0.5 − 1.74530i 0.5 − 0.331082i 0.5 + 0.410882i 0.5 + 0.589118i 0.5 − 1.00333i 0.5 + 2.00333i 0.5 − 2.74530i 0.5 − 1.33108i 0.5 + 1.74530i 0.5 + 0.331082i 0.5 − 0.410882i 0.5 − 0.589118i 0.5 + 1.00333i 0.5 − 2.00333i
0 −1.73022 0.0795432i 0 −0.313444 0.313444i 0 0.0745867 0.278362i 0 2.98735 + 0.275255i 0
305.2 0 −0.933998 1.45865i 0 −2.76293 2.76293i 0 0.657464 2.45369i 0 −1.25529 + 2.72474i 0
305.3 0 0.933998 + 1.45865i 0 0.313444 + 0.313444i 0 0.0745867 0.278362i 0 −1.25529 + 2.72474i 0
305.4 0 1.73022 + 0.0795432i 0 2.76293 + 2.76293i 0 0.657464 2.45369i 0 2.98735 + 0.275255i 0
353.1 0 −1.45865 0.933998i 0 −0.428520 0.428520i 0 0.735180 0.196991i 0 1.25529 + 2.72474i 0
353.2 0 −0.0795432 1.73022i 0 0.428520 + 0.428520i 0 0.735180 0.196991i 0 −2.98735 + 0.275255i 0
353.3 0 0.0795432 + 1.73022i 0 2.02097 + 2.02097i 0 −3.46723 + 0.929042i 0 −2.98735 + 0.275255i 0
353.4 0 1.45865 + 0.933998i 0 −2.02097 2.02097i 0 −3.46723 + 0.929042i 0 1.25529 + 2.72474i 0
401.1 0 −1.73022 + 0.0795432i 0 −0.313444 + 0.313444i 0 0.0745867 + 0.278362i 0 2.98735 0.275255i 0
401.2 0 −0.933998 + 1.45865i 0 −2.76293 + 2.76293i 0 0.657464 + 2.45369i 0 −1.25529 2.72474i 0
401.3 0 0.933998 1.45865i 0 0.313444 0.313444i 0 0.0745867 + 0.278362i 0 −1.25529 2.72474i 0
401.4 0 1.73022 0.0795432i 0 2.76293 2.76293i 0 0.657464 + 2.45369i 0 2.98735 0.275255i 0
449.1 0 −1.45865 + 0.933998i 0 −0.428520 + 0.428520i 0 0.735180 + 0.196991i 0 1.25529 2.72474i 0
449.2 0 −0.0795432 + 1.73022i 0 0.428520 0.428520i 0 0.735180 + 0.196991i 0 −2.98735 0.275255i 0
449.3 0 0.0795432 1.73022i 0 2.02097 2.02097i 0 −3.46723 0.929042i 0 −2.98735 0.275255i 0
449.4 0 1.45865 0.933998i 0 −2.02097 + 2.02097i 0 −3.46723 0.929042i 0 1.25529 2.72474i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 449.4 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
13.f odd 12 1 inner
39.k even 12 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.cn.d 16
3.b odd 2 1 inner 624.2.cn.d 16
4.b odd 2 1 78.2.k.a 16
12.b even 2 1 78.2.k.a 16
13.f odd 12 1 inner 624.2.cn.d 16
39.k even 12 1 inner 624.2.cn.d 16
52.i odd 6 1 1014.2.g.d 16
52.j odd 6 1 1014.2.g.c 16
52.l even 12 1 78.2.k.a 16
52.l even 12 1 1014.2.g.c 16
52.l even 12 1 1014.2.g.d 16
156.p even 6 1 1014.2.g.c 16
156.r even 6 1 1014.2.g.d 16
156.v odd 12 1 78.2.k.a 16
156.v odd 12 1 1014.2.g.c 16
156.v odd 12 1 1014.2.g.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.k.a 16 4.b odd 2 1
78.2.k.a 16 12.b even 2 1
78.2.k.a 16 52.l even 12 1
78.2.k.a 16 156.v odd 12 1
624.2.cn.d 16 1.a even 1 1 trivial
624.2.cn.d 16 3.b odd 2 1 inner
624.2.cn.d 16 13.f odd 12 1 inner
624.2.cn.d 16 39.k even 12 1 inner
1014.2.g.c 16 52.j odd 6 1
1014.2.g.c 16 52.l even 12 1
1014.2.g.c 16 156.p even 6 1
1014.2.g.c 16 156.v odd 12 1
1014.2.g.d 16 52.i odd 6 1
1014.2.g.d 16 52.l even 12 1
1014.2.g.d 16 156.r even 6 1
1014.2.g.d 16 156.v odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(624, [\chi])$$:

 $$T_{5}^{16} + 300T_{5}^{12} + 15606T_{5}^{8} + 2700T_{5}^{4} + 81$$ T5^16 + 300*T5^12 + 15606*T5^8 + 2700*T5^4 + 81 $$T_{7}^{8} + 4T_{7}^{7} + 2T_{7}^{6} + 16T_{7}^{5} + 46T_{7}^{4} - 112T_{7}^{3} + 68T_{7}^{2} - 16T_{7} + 4$$ T7^8 + 4*T7^7 + 2*T7^6 + 16*T7^5 + 46*T7^4 - 112*T7^3 + 68*T7^2 - 16*T7 + 4

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$T^{16} - 6 T^{12} - 45 T^{8} + \cdots + 6561$$
$5$ $$T^{16} + 300 T^{12} + 15606 T^{8} + \cdots + 81$$
$7$ $$(T^{8} + 4 T^{7} + 2 T^{6} + 16 T^{5} + 46 T^{4} + \cdots + 4)^{2}$$
$11$ $$(T^{8} - 30 T^{6} + 294 T^{4} + 180 T^{2} + \cdots + 36)^{2}$$
$13$ $$(T^{8} + 12 T^{7} + 88 T^{6} + 468 T^{5} + \cdots + 28561)^{2}$$
$17$ $$T^{16} + 72 T^{14} + \cdots + 151807041$$
$19$ $$(T^{8} - 8 T^{7} + 2 T^{6} - 32 T^{5} + \cdots + 5476)^{2}$$
$23$ $$T^{16} + 60 T^{14} + 2832 T^{12} + \cdots + 18974736$$
$29$ $$T^{16} - 132 T^{14} + \cdots + 33871089681$$
$31$ $$(T^{8} + 8 T^{7} + 32 T^{6} - 256 T^{5} + \cdots + 7744)^{2}$$
$37$ $$(T^{8} - 8 T^{7} + 32 T^{6} - 188 T^{5} + \cdots + 375769)^{2}$$
$41$ $$T^{16} + 96 T^{14} + \cdots + 65697655057281$$
$43$ $$(T^{8} - 18 T^{6} + 270 T^{4} - 972 T^{2} + \cdots + 2916)^{2}$$
$47$ $$T^{16} + 8784 T^{12} + \cdots + 41006250000$$
$53$ $$(T^{8} + 156 T^{6} + 7374 T^{4} + \cdots + 522729)^{2}$$
$59$ $$T^{16} + 168 T^{14} + \cdots + 43489065701376$$
$61$ $$(T^{8} + 12 T^{7} + 204 T^{6} + \cdots + 47961)^{2}$$
$67$ $$(T^{8} + 16 T^{7} + 218 T^{6} + 1696 T^{5} + \cdots + 676)^{2}$$
$71$ $$T^{16} + 12 T^{14} + \cdots + 5143987297296$$
$73$ $$(T^{8} - 28 T^{7} + 392 T^{6} + \cdots + 16834609)^{2}$$
$79$ $$(T^{4} - 24 T^{3} + 108 T^{2} + 432 T + 312)^{4}$$
$83$ $$T^{16} + 21360 T^{12} + \cdots + 36804120336$$
$89$ $$T^{16} + 264 T^{14} + \cdots + 59\!\cdots\!56$$
$97$ $$(T^{8} - 8 T^{7} + 296 T^{6} - 608 T^{5} + \cdots + 43264)^{2}$$
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