# Properties

 Label 375.2.g.c Level $375$ Weight $2$ Character orbit 375.g Analytic conductor $2.994$ Analytic rank $0$ Dimension $12$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [375,2,Mod(76,375)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(375, base_ring=CyclotomicField(10))

chi = DirichletCharacter(H, H._module([0, 2]))

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

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

chi := DirichletCharacter("375.76");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 375.g (of order $$5$$, 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: $$2.99439007580$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$3$$ over $$\Q(\zeta_{5})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1$$ x^12 + 3*x^10 - 2*x^9 + 34*x^8 - 22*x^7 + 236*x^6 - 179*x^5 + 877*x^4 - 409*x^3 + 96*x^2 - 11*x + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$5$$ Twist minimal: no (minimal twist has level 75) Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + \beta_{8} q^{3} + ( - \beta_{10} + \beta_{8} - \beta_{4} - 1) q^{4} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{6} + ( - \beta_{5} + 1) q^{7} + (\beta_{8} - \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{8} + \beta_{4} q^{9}+O(q^{10})$$ q - b2 * q^2 + b8 * q^3 + (-b10 + b8 - b4 - 1) * q^4 + (-b5 - b3 + b2 - b1) * q^6 + (-b5 + 1) * q^7 + (b8 - b7 - 2*b5 + b4 - b3 + 2*b2 - b1) * q^8 + b4 * q^9 $$q - \beta_{2} q^{2} + \beta_{8} q^{3} + ( - \beta_{10} + \beta_{8} - \beta_{4} - 1) q^{4} + ( - \beta_{5} - \beta_{3} + \beta_{2} - \beta_1) q^{6} + ( - \beta_{5} + 1) q^{7} + (\beta_{8} - \beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + 2 \beta_{2} - \beta_1) q^{8} + \beta_{4} q^{9} + ( - \beta_{9} - \beta_{8} - \beta_{2} - 1) q^{11} + ( - \beta_{7} + \beta_{6} + 2 \beta_{4} + 1) q^{12} + ( - \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + 1) q^{13} + (\beta_{11} + 3 \beta_{8} - 4 \beta_{7} + 4 \beta_{4} - \beta_{2} + 3) q^{14} + (\beta_{11} + \beta_{10} - 3 \beta_{7} - \beta_{6} + 5 \beta_{4} + \beta_{2} - \beta_1 + 3) q^{16} + (\beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} - 2 \beta_{7} - \beta_{4} - \beta_{3} - \beta_1 - 1) q^{17} + \beta_{5} q^{18} + ( - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{4} + \beta_{3} + \beta_1 + 1) q^{19} + (\beta_{8} - \beta_1) q^{21} + ( - \beta_{10} + 2 \beta_{8} + 2 \beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_1 - 1) q^{22} + ( - \beta_{11} - \beta_{9} - 3 \beta_{8} + \beta_{7} - \beta_{4} - \beta_{2} - 3) q^{23} + ( - \beta_{8} + \beta_{7} + 2 \beta_{5} + \beta_{3} - \beta_{2}) q^{24} + ( - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} - 5) q^{26} + ( - \beta_{8} + \beta_{7} - \beta_{4} - 1) q^{27} + ( - \beta_{10} + \beta_{8} + \beta_{5} + \beta_{3} - 3 \beta_1) q^{28} + ( - \beta_{9} - \beta_{8} - \beta_{6} - \beta_{5} - \beta_{3} - 1) q^{29} + (\beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{5} - \beta_{4} - 2 \beta_{2} + \cdots - 1) q^{31}+ \cdots + (\beta_{11} - \beta_{6} + \beta_{5}) q^{99}+O(q^{100})$$ q - b2 * q^2 + b8 * q^3 + (-b10 + b8 - b4 - 1) * q^4 + (-b5 - b3 + b2 - b1) * q^6 + (-b5 + 1) * q^7 + (b8 - b7 - 2*b5 + b4 - b3 + 2*b2 - b1) * q^8 + b4 * q^9 + (-b9 - b8 - b2 - 1) * q^11 + (-b7 + b6 + 2*b4 + 1) * q^12 + (-b11 - b10 - b7 + b6 + b4 - b3 + 1) * q^13 + (b11 + 3*b8 - 4*b7 + 4*b4 - b2 + 3) * q^14 + (b11 + b10 - 3*b7 - b6 + 5*b4 + b2 - b1 + 3) * q^16 + (b11 - b10 - b9 - 2*b8 - 2*b7 - b4 - b3 - b1 - 1) * q^17 + b5 * q^18 + (-b11 + b9 - b8 - b7 + b6 - 2*b4 + b3 + b1 + 1) * q^19 + (b8 - b1) * q^21 + (-b10 + 2*b8 + 2*b5 - b4 + 2*b3 + 2*b1 - 1) * q^22 + (-b11 - b9 - 3*b8 + b7 - b4 - b2 - 3) * q^23 + (-b8 + b7 + 2*b5 + b3 - b2) * q^24 + (-b10 - b9 - b7 + b5 - b3 + b2 - 5) * q^26 + (-b8 + b7 - b4 - 1) * q^27 + (-b10 + b8 + b5 + b3 - 3*b1) * q^28 + (-b9 - b8 - b6 - b5 - b3 - 1) * q^29 + (b11 - b10 - b9 - 2*b8 + 2*b7 + 2*b5 - b4 - 2*b2 - 1) * q^31 + (-b11 + b10 + b9 - b8 + 2*b7 + b6 + b5 + 3*b3 - 3*b2 + 3) * q^32 + (b10 - b6 - b5 - b3 + b2 - b1) * q^33 + (-b11 - b10 + 2*b7 + b6 + b4 + 3*b3 + b2 - b1 - 2) * q^34 + (-b11 - b8 + 2*b7 - 2*b4 - 1) * q^36 + (b11 + b10 - b6 + 4*b4 - 2*b3 + b2 - b1) * q^37 + (b11 + b10 - 2*b7 - b6 + b3 - 2*b2 + 2*b1 + 2) * q^38 + (-b9 - b8 + b7 - b4 + b2 - 1) * q^39 + (-b11 - b10 - b7 + 2*b3 - 3*b2 + 3*b1 + 1) * q^41 + (-b11 + b9 + 4*b7 + b6 - b5 - b4 - b3 + b2 - b1 + 1) * q^42 + (-b10 - b9 - b7 - b5 + 2*b3 - 2*b2) * q^43 + (-5*b8 + b7 - 3*b5 - 5*b4 - 2*b3 + 3*b2 - 2*b1) * q^44 + (-b10 + 2*b8 + 3*b5 - 2*b4 + 3*b3 + 6*b1 - 2) * q^46 + (b10 + 4*b8 + 3*b4 + 2*b1 + 3) * q^47 + (b9 - b8 + 5*b7 + b5 - 5*b4 - b2 + b1 - 1) * q^48 + (b10 + b9 + b7 - 2*b5 - 2) * q^49 + (-b11 + b10 + b9 + 2*b8 - b7 + b6 - b3 + b2 + 4) * q^51 + (-b11 + b9 - 4*b8 - b5 + 5*b2 - b1 - 4) * q^52 + (b10 + 2*b8 + 2*b5 - 4*b4 + 2*b3 + 2*b1 - 4) * q^53 + b1 * q^54 + (-2*b11 + b10 + 2*b9 - 2*b8 + 8*b7 + b6 - b5 - 4*b4 - b3 + b2 - b1 + 2) * q^56 + (-b10 - b9 + b8 - 2*b7 + b3 - b2 + 2) * q^57 + (b11 - b10 - b9 + 3*b8 - 6*b7 + 4*b4 + 2*b3 + 2*b1 - 1) * q^58 + (2*b11 + 2*b10 + 3*b7 - 2*b6 + b4 - 2*b3 + b2 - b1 - 3) * q^59 + (b11 + b9 - 2*b8 + 2*b7 - 3*b5 - 2*b4 + b2 - 3*b1 - 2) * q^61 + (-2*b11 - 2*b10 + 7*b7 - 8*b4 - b3 + b2 - b1 - 7) * q^62 + (b4 - b3) * q^63 + (-b11 + b9 + 7*b8 - b7 + 2*b5 + b4 - 4*b2 + 2*b1 + 7) * q^64 + (-b7 + b6 + 3*b4 + 2*b3 - 2*b2 + 2*b1 + 1) * q^66 + (-b11 + 2*b10 + b9 - b6 + b5 - b4 - b3 - b2 - b1 + 1) * q^67 + (b11 + 2*b10 + 2*b9 - 7*b8 + 9*b7 - b6 + b5 - 2*b3 + 2*b2 + 5) * q^68 + (b11 + b10 - b9 - 2*b8 - b7 - 2*b6 - b5 - b4 - b3 + b2 - b1 - 1) * q^69 + (2*b10 + b9 - 2*b8 + b6 - b5 - 3*b4 - b3 - 2*b1 - 2) * q^71 + (-b5 - b4 - b3 + b1 - 1) * q^72 + (2*b11 + b9 - b8 + 4*b7 + b5 - 4*b4 + b1 - 1) * q^73 + (-b11 - b10 - b9 - 3*b8 + 2*b7 + b6 + 4*b5 + 2*b3 - 2*b2 - 5) * q^74 + (2*b11 + b10 + b9 + 4*b8 - 3*b7 - 2*b6 + 2*b3 - 2*b2 - 3) * q^76 + (b11 - b9 + b8 - 3*b7 + 2*b5 + 3*b4 - b2 + 2*b1 + 1) * q^77 + (b10 - 4*b8 + b5 + b4 + b3 + 2*b1 + 1) * q^78 + (2*b10 + 2*b9 + b8 + 2*b6 - b5 + b4 - b3 + b1 + 3) * q^79 - b7 * q^81 + (3*b11 - b10 - b9 + 10*b8 - 11*b7 - 3*b6 - 2*b5 - b3 + b2 + 1) * q^82 + (-2*b11 - b10 + 2*b9 + b8 + 3*b7 + 3*b6 - b4 - b3 - b1 + 2) * q^83 + (b6 + b4 - 3*b3 - b2 + b1) * q^84 + (b11 + 2*b9 + 9*b8 - 3*b7 - b5 + 3*b4 + 2*b2 - b1 + 9) * q^86 + (b11 + b10 - b6 + b2 - b1) * q^87 + (b11 + b10 - 4*b7 + 7*b4 - b2 + b1 + 4) * q^88 + (-2*b11 + 5*b8 + 3*b7 + 3*b5 - 3*b4 + b2 + 3*b1 + 5) * q^89 + (-b11 - b10 - 2*b7 + 2*b6 + 6*b4 - 3*b3 + b2 - b1 + 2) * q^91 + (b11 + b10 - b9 - 4*b8 - 8*b7 - 2*b6 - 4*b5 - 3*b4 - 2*b3 + 4*b2 - 2*b1 - 1) * q^92 + (-b11 + b10 + b9 + 2*b8 - b7 + b6 - 2*b5 - 2*b3 + 2*b2) * q^93 + (2*b11 - 2*b9 - b8 - 7*b7 - 2*b6 + b5 + b4 - 2*b3 - b2 - 2*b1 - 2) * q^94 + (-b10 - b9 + b8 - b6 - 3*b5 - 2*b4 - 3*b3 - 2*b1 - 3) * q^96 + (b10 - 2*b9 - 2*b8 - 2*b6 + 2*b5 + 2*b3 - b1 - 2) * q^97 + (2*b11 + 6*b8 - 7*b7 + b5 + 7*b4 + b1 + 6) * q^98 + (b11 - b6 + b5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{3} - 10 q^{4} + 12 q^{7} - 9 q^{8} - 3 q^{9}+O(q^{10})$$ 12 * q - 3 * q^3 - 10 * q^4 + 12 * q^7 - 9 * q^8 - 3 * q^9 $$12 q - 3 q^{3} - 10 q^{4} + 12 q^{7} - 9 q^{8} - 3 q^{9} - 4 q^{11} + 2 q^{13} + 6 q^{14} + 16 q^{16} + q^{17} + 7 q^{19} - 3 q^{21} - 13 q^{22} - 19 q^{23} + 6 q^{24} - 56 q^{26} - 3 q^{27} - q^{28} - q^{29} + 13 q^{31} + 32 q^{32} + q^{33} - 25 q^{34} - 8 q^{37} + 22 q^{38} + 2 q^{39} + 8 q^{41} + 16 q^{42} + 4 q^{43} + 33 q^{44} - 22 q^{46} + 13 q^{47} + 16 q^{48} - 28 q^{49} + 26 q^{51} - 44 q^{52} - 44 q^{53} + 45 q^{56} + 22 q^{57} - 41 q^{58} - 22 q^{59} - 8 q^{61} - 41 q^{62} - 3 q^{63} + 49 q^{64} - 3 q^{66} + 6 q^{67} + 100 q^{68} + 6 q^{69} - 21 q^{71} - 9 q^{72} + 16 q^{73} - 44 q^{74} - 52 q^{76} - q^{77} + 19 q^{78} + 10 q^{79} - 3 q^{81} - 26 q^{82} + 10 q^{83} - 6 q^{84} + 56 q^{86} + 4 q^{87} + 16 q^{88} + 57 q^{89} - 7 q^{91} - 3 q^{92} - 22 q^{93} - 23 q^{94} - 23 q^{96} - 4 q^{97} + 18 q^{98} + 6 q^{99}+O(q^{100})$$ 12 * q - 3 * q^3 - 10 * q^4 + 12 * q^7 - 9 * q^8 - 3 * q^9 - 4 * q^11 + 2 * q^13 + 6 * q^14 + 16 * q^16 + q^17 + 7 * q^19 - 3 * q^21 - 13 * q^22 - 19 * q^23 + 6 * q^24 - 56 * q^26 - 3 * q^27 - q^28 - q^29 + 13 * q^31 + 32 * q^32 + q^33 - 25 * q^34 - 8 * q^37 + 22 * q^38 + 2 * q^39 + 8 * q^41 + 16 * q^42 + 4 * q^43 + 33 * q^44 - 22 * q^46 + 13 * q^47 + 16 * q^48 - 28 * q^49 + 26 * q^51 - 44 * q^52 - 44 * q^53 + 45 * q^56 + 22 * q^57 - 41 * q^58 - 22 * q^59 - 8 * q^61 - 41 * q^62 - 3 * q^63 + 49 * q^64 - 3 * q^66 + 6 * q^67 + 100 * q^68 + 6 * q^69 - 21 * q^71 - 9 * q^72 + 16 * q^73 - 44 * q^74 - 52 * q^76 - q^77 + 19 * q^78 + 10 * q^79 - 3 * q^81 - 26 * q^82 + 10 * q^83 - 6 * q^84 + 56 * q^86 + 4 * q^87 + 16 * q^88 + 57 * q^89 - 7 * q^91 - 3 * q^92 - 22 * q^93 - 23 * q^94 - 23 * q^96 - 4 * q^97 + 18 * q^98 + 6 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 3x^{10} - 2x^{9} + 34x^{8} - 22x^{7} + 236x^{6} - 179x^{5} + 877x^{4} - 409x^{3} + 96x^{2} - 11x + 1$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 78219134 \nu^{11} - 275969449 \nu^{10} - 322545241 \nu^{9} - 688323383 \nu^{8} - 2382459144 \nu^{7} - 7518729011 \nu^{6} + \cdots - 1397761481 ) / 30308822750$$ (-78219134*v^11 - 275969449*v^10 - 322545241*v^9 - 688323383*v^8 - 2382459144*v^7 - 7518729011*v^6 - 15194306845*v^5 - 49826077334*v^4 - 45143977692*v^3 - 198535658181*v^2 + 35416016020*v - 1397761481) / 30308822750 $$\beta_{3}$$ $$=$$ $$( - 444010163 \nu^{11} - 328887363 \nu^{10} - 1432734827 \nu^{9} - 101573726 \nu^{8} - 14512620193 \nu^{7} - 1187789257 \nu^{6} + \cdots - 6770591262 ) / 30308822750$$ (-444010163*v^11 - 328887363*v^10 - 1432734827*v^9 - 101573726*v^8 - 14512620193*v^7 - 1187789257*v^6 - 101054177500*v^5 + 3452612852*v^4 - 350578208249*v^3 - 92883364032*v^2 + 22525707095*v - 6770591262) / 30308822750 $$\beta_{4}$$ $$=$$ $$( - 1397761481 \nu^{11} + 78219134 \nu^{10} - 3917314994 \nu^{9} + 3118068203 \nu^{8} - 46835566971 \nu^{7} + 33133211726 \nu^{6} + \cdots - 20040639729 ) / 30308822750$$ (-1397761481*v^11 + 78219134*v^10 - 3917314994*v^9 + 3118068203*v^8 - 46835566971*v^7 + 33133211726*v^6 - 322352980505*v^5 + 265393611944*v^4 - 1176010741503*v^3 + 616828423421*v^2 + 64350556005*v - 20040639729) / 30308822750 $$\beta_{5}$$ $$=$$ $$( 1818269283 \nu^{11} + 240671723 \nu^{10} + 5444121987 \nu^{9} - 2938441244 \nu^{8} + 61232986233 \nu^{7} - 31823778603 \nu^{6} + \cdots - 3685345568 ) / 30308822750$$ (1818269283*v^11 + 240671723*v^10 + 5444121987*v^9 - 2938441244*v^8 + 61232986233*v^7 - 31823778603*v^6 + 424398953870*v^5 - 268840754862*v^4 + 1546641523369*v^3 - 535050339608*v^2 + 65280664295*v - 3685345568) / 30308822750 $$\beta_{6}$$ $$=$$ $$( 1905700356 \nu^{11} - 2131145819 \nu^{10} + 4372551549 \nu^{9} - 10545703663 \nu^{8} + 64978959816 \nu^{7} - 112688230641 \nu^{6} + \cdots + 25111873119 ) / 30308822750$$ (1905700356*v^11 - 2131145819*v^10 + 4372551549*v^9 - 10545703663*v^8 + 64978959816*v^7 - 112688230641*v^6 + 451494146575*v^5 - 826296544974*v^4 + 1740835657738*v^3 - 2476340056991*v^2 - 76145058940*v + 25111873119) / 30308822750 $$\beta_{7}$$ $$=$$ $$( - 3685345568 \nu^{11} - 1818269283 \nu^{10} - 11296708427 \nu^{9} + 1926569149 \nu^{8} - 122363308068 \nu^{7} + 19844616263 \nu^{6} + \cdots - 24741863047 ) / 30308822750$$ (-3685345568*v^11 - 1818269283*v^10 - 11296708427*v^9 + 1926569149*v^8 - 122363308068*v^7 + 19844616263*v^6 - 837917775445*v^5 + 235277902802*v^4 - 2963207308274*v^3 - 39335186057*v^2 + 181257165080*v - 24741863047) / 30308822750 $$\beta_{8}$$ $$=$$ $$( 6770591262 \nu^{11} - 444010163 \nu^{10} + 19982886423 \nu^{9} - 14973917351 \nu^{8} + 230098529182 \nu^{7} - 163465627957 \nu^{6} + \cdots - 51950796787 ) / 30308822750$$ (6770591262*v^11 - 444010163*v^10 + 19982886423*v^9 - 14973917351*v^8 + 230098529182*v^7 - 163465627957*v^6 + 1596671748575*v^5 - 1312990013398*v^4 + 5941261149626*v^3 - 3119750034407*v^2 + 557093397120*v - 51950796787) / 30308822750 $$\beta_{9}$$ $$=$$ $$( - 3594254357 \nu^{11} + 648267081 \nu^{10} - 10484886687 \nu^{9} + 9161303434 \nu^{8} - 122610276713 \nu^{7} + 100643733429 \nu^{6} + \cdots + 13852270880 ) / 6061764550$$ (-3594254357*v^11 + 648267081*v^10 - 10484886687*v^9 + 9161303434*v^8 - 122610276713*v^7 + 100643733429*v^6 - 852450509356*v^5 + 790744249492*v^4 - 3198526129079*v^3 + 1992575852018*v^2 - 355758053867*v + 13852270880) / 6061764550 $$\beta_{10}$$ $$=$$ $$( 21468863544 \nu^{11} - 1399563761 \nu^{10} + 63167876941 \nu^{9} - 47451650867 \nu^{8} + 728953008894 \nu^{7} - 518817498679 \nu^{6} + \cdots - 164302304099 ) / 30308822750$$ (21468863544*v^11 - 1399563761*v^10 + 63167876941*v^9 - 47451650867*v^8 + 728953008894*v^7 - 518817498679*v^6 + 5055738779435*v^5 - 4156383014316*v^4 + 18791172393242*v^3 - 9866805339769*v^2 + 1590614018810*v - 164302304099) / 30308822750 $$\beta_{11}$$ $$=$$ $$( - 29133222771 \nu^{11} - 6153218841 \nu^{10} - 88476638949 \nu^{9} + 39572030488 \nu^{8} - 981450539091 \nu^{7} + \cdots + 45677018676 ) / 30308822750$$ (-29133222771*v^11 - 6153218841*v^10 - 88476638949*v^9 + 39572030488*v^8 - 981450539091*v^7 + 433510281051*v^6 - 6776249219160*v^5 + 3779688498924*v^4 - 24698086195263*v^3 + 6669444982566*v^2 - 1191999052005*v + 45677018676) / 30308822750
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{7} + \beta_{6} + 4\beta_{4} + 1$$ -b7 + b6 + 4*b4 + 1 $$\nu^{3}$$ $$=$$ $$\beta_{7} + \beta_{5} - \beta_{4} - 6\beta_{2} + \beta_1$$ b7 + b5 - b4 - 6*b2 + b1 $$\nu^{4}$$ $$=$$ $$6\beta_{11} - \beta_{10} - 6\beta_{9} + 3\beta_{8} - 25\beta_{7} - 5\beta_{6} - \beta_{5} + 9\beta_{4} + \beta_{2} - 6$$ 6*b11 - b10 - 6*b9 + 3*b8 - 25*b7 - 5*b6 - b5 + 9*b4 + b2 - 6 $$\nu^{5}$$ $$=$$ $$-\beta_{11} + \beta_{10} + \beta_{9} - 9\beta_{8} + 10\beta_{7} + \beta_{6} + 37\beta_{5} + 11\beta_{3} - 11\beta_{2} + 3$$ -b11 + b10 + b9 - 9*b8 + 10*b7 + b6 + 37*b5 + 11*b3 - 11*b2 + 3 $$\nu^{6}$$ $$=$$ $$-37\beta_{10} - 11\beta_{9} + 79\beta_{8} - 11\beta_{6} - 12\beta_{5} - 71\beta_{4} - 12\beta_{3} + 2\beta _1 - 82$$ -37*b10 - 11*b9 + 79*b8 - 11*b6 - 12*b5 - 71*b4 - 12*b3 + 2*b1 - 82 $$\nu^{7}$$ $$=$$ $$12\beta_{11} + 12\beta_{10} - 87\beta_{7} + 2\beta_{6} + 116\beta_{4} + 142\beta_{3} + 93\beta_{2} - 93\beta _1 + 87$$ 12*b11 + 12*b10 - 87*b7 + 2*b6 + 116*b4 + 142*b3 + 93*b2 - 93*b1 + 87 $$\nu^{8}$$ $$=$$ $$- 235 \beta_{11} + 93 \beta_{9} - 438 \beta_{8} + 1059 \beta_{7} + 111 \beta_{5} - 1059 \beta_{4} - 144 \beta_{2} + 111 \beta _1 - 438$$ -235*b11 + 93*b9 - 438*b8 + 1059*b7 + 111*b5 - 1059*b4 - 144*b2 + 111*b1 - 438 $$\nu^{9}$$ $$=$$ $$144 \beta_{11} - 111 \beta_{10} - 144 \beta_{9} + 568 \beta_{8} - 1015 \beta_{7} - 33 \beta_{6} - 1529 \beta_{5} + 712 \beta_{4} - 815 \beta_{3} + 1529 \beta_{2} - 815 \beta _1 - 144$$ 144*b11 - 111*b10 - 144*b9 + 568*b8 - 1015*b7 - 33*b6 - 1529*b5 + 712*b4 - 815*b3 + 1529*b2 - 815*b1 - 144 $$\nu^{10}$$ $$=$$ $$- 714 \beta_{11} + 1529 \beta_{10} + 1529 \beta_{9} - 2286 \beta_{8} + 3815 \beta_{7} + 714 \beta_{6} + 1303 \beta_{5} + 934 \beta_{3} - 934 \beta_{2} + 4799$$ -714*b11 + 1529*b10 + 1529*b9 - 2286*b8 + 3815*b7 + 714*b6 + 1303*b5 + 934*b3 - 934*b2 + 4799 $$\nu^{11}$$ $$=$$ $$- 1303 \beta_{10} - 934 \beta_{9} + 1821 \beta_{8} - 934 \beta_{6} - 5243 \beta_{5} - 5634 \beta_{4} - 5243 \beta_{3} + 4900 \beta _1 - 6568$$ -1303*b10 - 934*b9 + 1821*b8 - 934*b6 - 5243*b5 - 5634*b4 - 5243*b3 + 4900*b1 - 6568

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/375\mathbb{Z}\right)^\times$$.

 $$n$$ $$127$$ $$251$$ $$\chi(n)$$ $$\beta_{8}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
76.1
 0.623865 + 1.92006i 0.0437845 + 0.134755i −0.667650 − 2.05481i 1.97423 − 1.43436i 0.199632 − 0.145041i −2.17386 + 1.57940i 1.97423 + 1.43436i 0.199632 + 0.145041i −2.17386 − 1.57940i 0.623865 − 1.92006i 0.0437845 − 0.134755i −0.667650 + 2.05481i
−1.63330 1.18666i 0.309017 + 0.951057i 0.641469 + 1.97424i 0 0.623865 1.92006i −1.01887 0.0473123 0.145612i −0.809017 + 0.587785i 0
76.2 −0.114629 0.0832830i 0.309017 + 0.951057i −0.611830 1.88302i 0 0.0437845 0.134755i 0.858311 −0.174259 + 0.536314i −0.809017 + 0.587785i 0
76.3 1.74793 + 1.26995i 0.309017 + 0.951057i 0.824463 + 2.53744i 0 −0.667650 + 2.05481i 3.16056 −0.446002 + 1.37265i −0.809017 + 0.587785i 0
151.1 −0.754089 2.32085i −0.809017 + 0.587785i −3.19965 + 2.32468i 0 1.97423 + 1.43436i 3.44028 3.85959 + 2.80415i 0.309017 0.951057i 0
151.2 −0.0762527 0.234682i −0.809017 + 0.587785i 1.56877 1.13978i 0 0.199632 + 0.145041i 1.24676 −0.786373 0.571334i 0.309017 0.951057i 0
151.3 0.830342 + 2.55553i −0.809017 + 0.587785i −4.22323 + 3.06835i 0 −2.17386 1.57940i −1.68704 −7.00026 5.08599i 0.309017 0.951057i 0
226.1 −0.754089 + 2.32085i −0.809017 0.587785i −3.19965 2.32468i 0 1.97423 1.43436i 3.44028 3.85959 2.80415i 0.309017 + 0.951057i 0
226.2 −0.0762527 + 0.234682i −0.809017 0.587785i 1.56877 + 1.13978i 0 0.199632 0.145041i 1.24676 −0.786373 + 0.571334i 0.309017 + 0.951057i 0
226.3 0.830342 2.55553i −0.809017 0.587785i −4.22323 3.06835i 0 −2.17386 + 1.57940i −1.68704 −7.00026 + 5.08599i 0.309017 + 0.951057i 0
301.1 −1.63330 + 1.18666i 0.309017 0.951057i 0.641469 1.97424i 0 0.623865 + 1.92006i −1.01887 0.0473123 + 0.145612i −0.809017 0.587785i 0
301.2 −0.114629 + 0.0832830i 0.309017 0.951057i −0.611830 + 1.88302i 0 0.0437845 + 0.134755i 0.858311 −0.174259 0.536314i −0.809017 0.587785i 0
301.3 1.74793 1.26995i 0.309017 0.951057i 0.824463 2.53744i 0 −0.667650 2.05481i 3.16056 −0.446002 1.37265i −0.809017 0.587785i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 301.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 375.2.g.c 12
5.b even 2 1 75.2.g.c 12
5.c odd 4 2 375.2.i.d 24
15.d odd 2 1 225.2.h.d 12
25.d even 5 1 inner 375.2.g.c 12
25.d even 5 1 1875.2.a.k 6
25.e even 10 1 75.2.g.c 12
25.e even 10 1 1875.2.a.j 6
25.f odd 20 2 375.2.i.d 24
25.f odd 20 2 1875.2.b.f 12
75.h odd 10 1 225.2.h.d 12
75.h odd 10 1 5625.2.a.p 6
75.j odd 10 1 5625.2.a.q 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.c 12 5.b even 2 1
75.2.g.c 12 25.e even 10 1
225.2.h.d 12 15.d odd 2 1
225.2.h.d 12 75.h odd 10 1
375.2.g.c 12 1.a even 1 1 trivial
375.2.g.c 12 25.d even 5 1 inner
375.2.i.d 24 5.c odd 4 2
375.2.i.d 24 25.f odd 20 2
1875.2.a.j 6 25.e even 10 1
1875.2.a.k 6 25.d even 5 1
1875.2.b.f 12 25.f odd 20 2
5625.2.a.p 6 75.h odd 10 1
5625.2.a.q 6 75.j odd 10 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{12} + 8 T_{2}^{10} + 3 T_{2}^{9} + 34 T_{2}^{8} + 8 T_{2}^{7} + 91 T_{2}^{6} + 96 T_{2}^{5} + 852 T_{2}^{4} + 321 T_{2}^{3} + 96 T_{2}^{2} + 14 T_{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(375, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12} + 8 T^{10} + 3 T^{9} + 34 T^{8} + \cdots + 1$$
$3$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{3}$$
$5$ $$T^{12}$$
$7$ $$(T^{6} - 6 T^{5} + 4 T^{4} + 25 T^{3} + \cdots + 20)^{2}$$
$11$ $$T^{12} + 4 T^{11} + 11 T^{10} + \cdots + 59536$$
$13$ $$T^{12} - 2 T^{11} + 19 T^{10} + \cdots + 10201$$
$17$ $$T^{12} - T^{11} + 29 T^{10} + \cdots + 21520321$$
$19$ $$T^{12} - 7 T^{11} + 73 T^{10} + \cdots + 144400$$
$23$ $$T^{12} + 19 T^{11} + 156 T^{10} + \cdots + 4080400$$
$29$ $$T^{12} + T^{11} + 12 T^{10} + \cdots + 4431025$$
$31$ $$T^{12} - 13 T^{11} + 44 T^{10} + \cdots + 8410000$$
$37$ $$T^{12} + 8 T^{11} + 44 T^{10} + \cdots + 36300625$$
$41$ $$T^{12} - 8 T^{11} + \cdots + 6831849025$$
$43$ $$(T^{6} - 2 T^{5} - 91 T^{4} + 174 T^{3} + \cdots - 6284)^{2}$$
$47$ $$T^{12} - 13 T^{11} + 178 T^{10} + \cdots + 5216656$$
$53$ $$T^{12} + 44 T^{11} + \cdots + 1189905025$$
$59$ $$T^{12} + 22 T^{11} + 213 T^{10} + \cdots + 15366400$$
$61$ $$T^{12} + 8 T^{11} + \cdots + 28314456361$$
$67$ $$T^{12} - 6 T^{11} + 29 T^{10} + \cdots + 215619856$$
$71$ $$T^{12} + 21 T^{11} + 259 T^{10} + \cdots + 38416$$
$73$ $$T^{12} - 16 T^{11} + \cdots + 493062025$$
$79$ $$T^{12} - 10 T^{11} - 35 T^{10} + \cdots + 64000000$$
$83$ $$T^{12} - 10 T^{11} + \cdots + 1683953296$$
$89$ $$T^{12} - 57 T^{11} + \cdots + 142170473025$$
$97$ $$T^{12} + 4 T^{11} + 139 T^{10} + \cdots + 5755201$$