# Properties

 Label 675.2.k.c Level $675$ Weight $2$ Character orbit 675.k Analytic conductor $5.390$ Analytic rank $0$ Dimension $16$ CM no Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [675,2,Mod(199,675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(675, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([2, 3]))

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

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

chi := DirichletCharacter("675.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 675.k (of order $$6$$, degree $$2$$, 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: $$5.38990213644$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81$$ x^16 - 12*x^14 + 102*x^12 - 406*x^10 + 1167*x^8 - 1842*x^6 + 2023*x^4 - 441*x^2 + 81 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 225) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + \beta_1) q^{8}+O(q^{10})$$ q + b1 * q^2 + (b11 + b7 + b3 + 1) * q^4 - b2 * q^7 + (-b15 + b14 - b13 - b9 - b8 + b4 - b2 + b1) * q^8 $$q + \beta_1 q^{2} + (\beta_{11} + \beta_{7} + \beta_{3} + 1) q^{4} - \beta_{2} q^{7} + ( - \beta_{15} + \beta_{14} - \beta_{13} - \beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + \beta_1) q^{8} + (\beta_{10} - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} - 1) q^{11} + (2 \beta_{14} + \beta_{9}) q^{13} + (\beta_{12} + \beta_{11} + \beta_{10} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{3} + 1) q^{14} + (2 \beta_{10} + \beta_{6} + 3 \beta_{3} + 1) q^{16} + ( - \beta_{15} - 2 \beta_{14} + 2 \beta_{13} + 2 \beta_{8} - 2 \beta_{4} - \beta_{2}) q^{17} + ( - 2 \beta_{11} + \beta_{5} - 2) q^{19} + (\beta_{15} + \beta_{14} + 2 \beta_{9}) q^{22} + ( - \beta_{15} + \beta_{14} + 5 \beta_{13} - \beta_{9}) q^{23} + ( - 2 \beta_{12} - 3 \beta_{11} - \beta_{5} + 1) q^{26} + (2 \beta_{14} + \beta_{13} - 2 \beta_{9} + \beta_{8} + 2 \beta_{4} + 2 \beta_1) q^{28} + ( - \beta_{10} - \beta_{7} - 2 \beta_{6} + \beta_{5} - 2 \beta_{3} - 2) q^{29} + (2 \beta_{12} + \beta_{11} + 2 \beta_{10} - 2 \beta_{7} - \beta_{6} + 2 \beta_{5} + \beta_{3} + 1) q^{31} + (3 \beta_{14} - 2 \beta_{9}) q^{32} + ( - \beta_{10} + \beta_{7} - \beta_{5} - 5 \beta_{3}) q^{34} + ( - \beta_{15} + \beta_{14} + 3 \beta_{13} - 4 \beta_{9} + 3 \beta_{8} + \beta_{4} - \beta_{2} + 4 \beta_1) q^{37} + (5 \beta_{8} - 2 \beta_{4} - \beta_{2}) q^{38} + ( - \beta_{12} + \beta_{11} - \beta_{10} - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} + 1) q^{41} + (\beta_{8} - 2 \beta_{4} - 2 \beta_{2} - 2 \beta_1) q^{43} + (2 \beta_{12} + \beta_{11} + \beta_{5} + 6) q^{44} + ( - 2 \beta_{12} + \beta_{11} - 4 \beta_{5} + 2) q^{46} + ( - 5 \beta_{8} - \beta_{4} - 2 \beta_{2}) q^{47} + ( - \beta_{12} + \beta_{11} - \beta_{10} - 2 \beta_{7} - \beta_{5} + \beta_{3} + 1) q^{49} + (3 \beta_{8} - \beta_{4} + \beta_{2} - \beta_1) q^{52} + ( - 3 \beta_{14} - 6 \beta_{13} - \beta_{9} - 6 \beta_{8} - 3 \beta_{4} + \beta_1) q^{53} + ( - 6 \beta_{6} + 3 \beta_{3} - 6) q^{56} + (2 \beta_{15} - 3 \beta_{14} + \beta_{13} + 3 \beta_{9}) q^{58} + ( - 3 \beta_{12} - \beta_{11} - 3 \beta_{10} + 3 \beta_{6} - 3 \beta_{5} - \beta_{3} - 1) q^{59} + ( - \beta_{10} - 2 \beta_{7} + \beta_{6} + 2 \beta_{5} - 2 \beta_{3} + 1) q^{61} + (3 \beta_{14} + 3 \beta_{4}) q^{62} + (\beta_{12} - 2 \beta_{11} - \beta_{5} - 5) q^{64} + ( - \beta_{15} + 3 \beta_{13} + 3 \beta_{9}) q^{67} + (2 \beta_{15} - 2 \beta_{14} + 5 \beta_{13} + 4 \beta_{9}) q^{68} + (\beta_{11} - 3 \beta_{5} + 4) q^{71} + ( - 2 \beta_{15} - 3 \beta_{14} + \beta_{13} - 3 \beta_{9} + \beta_{8} - 3 \beta_{4} + \cdots + 3 \beta_1) q^{73}+ \cdots + (3 \beta_{15} - 6 \beta_{13} + 4 \beta_{9} - 6 \beta_{8} + 3 \beta_{2} - 4 \beta_1) q^{98}+O(q^{100})$$ q + b1 * q^2 + (b11 + b7 + b3 + 1) * q^4 - b2 * q^7 + (-b15 + b14 - b13 - b9 - b8 + b4 - b2 + b1) * q^8 + (b10 - 2*b7 - b6 + 2*b5 - 1) * q^11 + (2*b14 + b9) * q^13 + (b12 + b11 + b10 + b7 + 2*b6 + b5 + b3 + 1) * q^14 + (2*b10 + b6 + 3*b3 + 1) * q^16 + (-b15 - 2*b14 + 2*b13 + 2*b8 - 2*b4 - b2) * q^17 + (-2*b11 + b5 - 2) * q^19 + (b15 + b14 + 2*b9) * q^22 + (-b15 + b14 + 5*b13 - b9) * q^23 + (-2*b12 - 3*b11 - b5 + 1) * q^26 + (2*b14 + b13 - 2*b9 + b8 + 2*b4 + 2*b1) * q^28 + (-b10 - b7 - 2*b6 + b5 - 2*b3 - 2) * q^29 + (2*b12 + b11 + 2*b10 - 2*b7 - b6 + 2*b5 + b3 + 1) * q^31 + (3*b14 - 2*b9) * q^32 + (-b10 + b7 - b5 - 5*b3) * q^34 + (-b15 + b14 + 3*b13 - 4*b9 + 3*b8 + b4 - b2 + 4*b1) * q^37 + (5*b8 - 2*b4 - b2) * q^38 + (-b12 + b11 - b10 - b7 + b6 - b5 + b3 + 1) * q^41 + (b8 - 2*b4 - 2*b2 - 2*b1) * q^43 + (2*b12 + b11 + b5 + 6) * q^44 + (-2*b12 + b11 - 4*b5 + 2) * q^46 + (-5*b8 - b4 - 2*b2) * q^47 + (-b12 + b11 - b10 - 2*b7 - b5 + b3 + 1) * q^49 + (3*b8 - b4 + b2 - b1) * q^52 + (-3*b14 - 6*b13 - b9 - 6*b8 - 3*b4 + b1) * q^53 + (-6*b6 + 3*b3 - 6) * q^56 + (2*b15 - 3*b14 + b13 + 3*b9) * q^58 + (-3*b12 - b11 - 3*b10 + 3*b6 - 3*b5 - b3 - 1) * q^59 + (-b10 - 2*b7 + b6 + 2*b5 - 2*b3 + 1) * q^61 + (3*b14 + 3*b4) * q^62 + (b12 - 2*b11 - b5 - 5) * q^64 + (-b15 + 3*b13 + 3*b9) * q^67 + (2*b15 - 2*b14 + 5*b13 + 4*b9) * q^68 + (b11 - 3*b5 + 4) * q^71 + (-2*b15 - 3*b14 + b13 - 3*b9 + b8 - 3*b4 - 2*b2 + 3*b1) * q^73 + (2*b10 + 5*b7 - 5*b6 - 5*b5 + 4*b3 - 5) * q^74 + (-b12 - 4*b11 - b10 - b7 - b5 - 4*b3 - 4) * q^76 + (-2*b15 - 4*b14 - 5*b13 + b9) * q^77 + (b10 - 3*b7 - 3*b6 + 3*b5 - 3) * q^79 + (2*b15 - 5*b13 + 3*b9 - 5*b8 + 2*b2 - 3*b1) * q^82 + (9*b8 + 3*b1) * q^83 + (-5*b11 + 6*b6 - 5*b3 - 5) * q^86 + (b8 + 5*b4 + b2 + 2*b1) * q^88 + (-3*b12 - 3*b11 - 3*b5 - 3) * q^89 + (3*b12 - b11 + 2*b5 - 5) * q^91 + (2*b8 + b4 + 2*b2 - 2*b1) * q^92 + (b12 + 5*b11 + b10 + 2*b7 + 3*b6 + b5 + 5*b3 + 5) * q^94 + (-5*b8 + b4 - b2 - 2*b1) * q^97 + (3*b15 - 6*b13 + 4*b9 - 6*b8 + 3*b2 - 4*b1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q + 8 q^{4}+O(q^{10})$$ 16 * q + 8 * q^4 $$16 q + 8 q^{4} - 2 q^{11} - 6 q^{14} - 8 q^{16} - 8 q^{19} + 40 q^{26} - 2 q^{29} + 8 q^{31} + 18 q^{34} - 10 q^{41} + 88 q^{44} - 6 q^{49} - 60 q^{56} - 34 q^{59} + 26 q^{61} - 76 q^{64} + 32 q^{71} - 80 q^{74} - 22 q^{76} - 14 q^{79} - 68 q^{86} - 36 q^{89} - 68 q^{91} + 6 q^{94}+O(q^{100})$$ 16 * q + 8 * q^4 - 2 * q^11 - 6 * q^14 - 8 * q^16 - 8 * q^19 + 40 * q^26 - 2 * q^29 + 8 * q^31 + 18 * q^34 - 10 * q^41 + 88 * q^44 - 6 * q^49 - 60 * q^56 - 34 * q^59 + 26 * q^61 - 76 * q^64 + 32 * q^71 - 80 * q^74 - 22 * q^76 - 14 * q^79 - 68 * q^86 - 36 * q^89 - 68 * q^91 + 6 * q^94

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 12x^{14} + 102x^{12} - 406x^{10} + 1167x^{8} - 1842x^{6} + 2023x^{4} - 441x^{2} + 81$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 28 \nu^{15} + 943 \nu^{13} - 2470 \nu^{11} + 8108 \nu^{9} + 208132 \nu^{7} + 15890 \nu^{5} - 3474 \nu^{3} + 4310244 \nu ) / 795285$$ (-28*v^15 + 943*v^13 - 2470*v^11 + 8108*v^9 + 208132*v^7 + 15890*v^5 - 3474*v^3 + 4310244*v) / 795285 $$\beta_{3}$$ $$=$$ $$( 26\nu^{14} - 74\nu^{12} - 10\nu^{10} + 10439\nu^{8} - 38576\nu^{6} + 94895\nu^{4} - 82467\nu^{2} + 18063 ) / 29025$$ (26*v^14 - 74*v^12 - 10*v^10 + 10439*v^8 - 38576*v^6 + 94895*v^4 - 82467*v^2 + 18063) / 29025 $$\beta_{4}$$ $$=$$ $$( 1540 \nu^{15} - 16519 \nu^{13} + 135850 \nu^{11} - 445940 \nu^{9} + 1153589 \nu^{7} - 873950 \nu^{5} + 191070 \nu^{3} + 2158308 \nu ) / 795285$$ (1540*v^15 - 16519*v^13 + 135850*v^11 - 445940*v^9 + 1153589*v^7 - 873950*v^5 + 191070*v^3 + 2158308*v) / 795285 $$\beta_{5}$$ $$=$$ $$( 266 \nu^{14} - 2999 \nu^{12} + 23465 \nu^{10} - 77026 \nu^{8} + 148849 \nu^{6} - 150955 \nu^{4} + 33003 \nu^{2} - 126387 ) / 92475$$ (266*v^14 - 2999*v^12 + 23465*v^10 - 77026*v^8 + 148849*v^6 - 150955*v^4 + 33003*v^2 - 126387) / 92475 $$\beta_{6}$$ $$=$$ $$( - 11836 \nu^{14} + 138574 \nu^{12} - 1170340 \nu^{10} + 4500371 \nu^{8} - 12811274 \nu^{6} + 19088030 \nu^{4} - 21981813 \nu^{2} + 814212 ) / 3976425$$ (-11836*v^14 + 138574*v^12 - 1170340*v^10 + 4500371*v^8 - 12811274*v^6 + 19088030*v^4 - 21981813*v^2 + 814212) / 3976425 $$\beta_{7}$$ $$=$$ $$( - 12338 \nu^{14} + 121397 \nu^{12} - 1025270 \nu^{10} + 3257143 \nu^{8} - 11223247 \nu^{6} + 16721965 \nu^{4} - 26841054 \nu^{2} + 713286 ) / 3976425$$ (-12338*v^14 + 121397*v^12 - 1025270*v^10 + 3257143*v^8 - 11223247*v^6 + 16721965*v^4 - 26841054*v^2 + 713286) / 3976425 $$\beta_{8}$$ $$=$$ $$( 3766 \nu^{15} - 41414 \nu^{13} + 332215 \nu^{11} - 1090526 \nu^{9} + 2480389 \nu^{7} - 2137205 \nu^{5} + 467253 \nu^{3} + 2721243 \nu ) / 1325475$$ (3766*v^15 - 41414*v^13 + 332215*v^11 - 1090526*v^9 + 2480389*v^7 - 2137205*v^5 + 467253*v^3 + 2721243*v) / 1325475 $$\beta_{9}$$ $$=$$ $$( - 11836 \nu^{15} + 138574 \nu^{13} - 1170340 \nu^{11} + 4500371 \nu^{9} - 12811274 \nu^{7} + 19088030 \nu^{5} - 21981813 \nu^{3} + 4790637 \nu ) / 3976425$$ (-11836*v^15 + 138574*v^13 - 1170340*v^11 + 4500371*v^9 - 12811274*v^7 + 19088030*v^5 - 21981813*v^3 + 4790637*v) / 3976425 $$\beta_{10}$$ $$=$$ $$( 13873 \nu^{14} - 220432 \nu^{12} + 2012770 \nu^{10} - 10392128 \nu^{8} + 31813907 \nu^{6} - 59345540 \nu^{4} + 58685109 \nu^{2} - 12812391 ) / 3976425$$ (13873*v^14 - 220432*v^12 + 2012770*v^10 - 10392128*v^8 + 31813907*v^6 - 59345540*v^4 + 58685109*v^2 - 12812391) / 3976425 $$\beta_{11}$$ $$=$$ $$( - 14896 \nu^{14} + 165889 \nu^{12} - 1314040 \nu^{10} + 4313456 \nu^{8} - 9114389 \nu^{6} + 8453480 \nu^{4} - 1848168 \nu^{2} - 5535918 ) / 3976425$$ (-14896*v^14 + 165889*v^12 - 1314040*v^10 + 4313456*v^8 - 9114389*v^6 + 8453480*v^4 - 1848168*v^2 - 5535918) / 3976425 $$\beta_{12}$$ $$=$$ $$( 18998 \nu^{14} - 206837 \nu^{12} + 1675895 \nu^{10} - 5501278 \nu^{8} + 13209112 \nu^{6} - 10781365 \nu^{4} + 2357109 \nu^{2} + 14978844 ) / 3976425$$ (18998*v^14 - 206837*v^12 + 1675895*v^10 - 5501278*v^8 + 13209112*v^6 - 10781365*v^4 + 2357109*v^2 + 14978844) / 3976425 $$\beta_{13}$$ $$=$$ $$( 19253 \nu^{15} - 245932 \nu^{13} + 2129695 \nu^{11} - 9130758 \nu^{9} + 26781707 \nu^{7} - 44578415 \nu^{5} + 47402299 \nu^{3} - 10338741 \nu ) / 3976425$$ (19253*v^15 - 245932*v^13 + 2129695*v^11 - 9130758*v^9 + 26781707*v^7 - 44578415*v^5 + 47402299*v^3 - 10338741*v) / 3976425 $$\beta_{14}$$ $$=$$ $$( 30232 \nu^{15} - 363428 \nu^{13} + 3087680 \nu^{11} - 12331002 \nu^{9} + 35467228 \nu^{7} - 57068185 \nu^{5} + 61524806 \nu^{3} - 13412214 \nu ) / 3976425$$ (30232*v^15 - 363428*v^13 + 3087680*v^11 - 12331002*v^9 + 35467228*v^7 - 57068185*v^5 + 61524806*v^3 - 13412214*v) / 3976425 $$\beta_{15}$$ $$=$$ $$( 66701 \nu^{15} - 773434 \nu^{13} + 6504640 \nu^{11} - 24700761 \nu^{9} + 70028009 \nu^{7} - 105967505 \nu^{5} + 119626108 \nu^{3} - 26067942 \nu ) / 3976425$$ (66701*v^15 - 773434*v^13 + 6504640*v^11 - 24700761*v^9 + 70028009*v^7 - 105967505*v^5 + 119626108*v^3 - 26067942*v) / 3976425
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{11} + \beta_{7} - 2\beta_{6} + \beta_{3} + 1$$ b11 + b7 - 2*b6 + b3 + 1 $$\nu^{3}$$ $$=$$ $$-\beta_{15} + \beta_{14} - \beta_{13} - 5\beta_{9} - \beta_{8} + \beta_{4} - \beta_{2} + 5\beta_1$$ -b15 + b14 - b13 - 5*b9 - b8 + b4 - b2 + 5*b1 $$\nu^{4}$$ $$=$$ $$2\beta_{10} + 6\beta_{7} - 7\beta_{6} - 6\beta_{5} + 9\beta_{3} - 7$$ 2*b10 + 6*b7 - 7*b6 - 6*b5 + 9*b3 - 7 $$\nu^{5}$$ $$=$$ $$-8\beta_{15} + 11\beta_{14} - 8\beta_{13} - 30\beta_{9}$$ -8*b15 + 11*b14 - 8*b13 - 30*b9 $$\nu^{6}$$ $$=$$ $$-19\beta_{12} - 68\beta_{11} - 57\beta_{5} - 101$$ -19*b12 - 68*b11 - 57*b5 - 101 $$\nu^{7}$$ $$=$$ $$60\beta_{8} - 87\beta_{4} + 57\beta_{2} - 196\beta_1$$ 60*b8 - 87*b4 + 57*b2 - 196*b1 $$\nu^{8}$$ $$=$$ $$-144\beta_{12} - 487\beta_{11} - 144\beta_{10} - 253\beta_{7} + 191\beta_{6} - 144\beta_{5} - 487\beta_{3} - 487$$ -144*b12 - 487*b11 - 144*b10 - 253*b7 + 191*b6 - 144*b5 - 487*b3 - 487 $$\nu^{9}$$ $$=$$ $$397 \beta_{15} - 631 \beta_{14} + 433 \beta_{13} + 1328 \beta_{9} + 433 \beta_{8} - 631 \beta_{4} + 397 \beta_{2} - 1328 \beta_1$$ 397*b15 - 631*b14 + 433*b13 + 1328*b9 + 433*b8 - 631*b4 + 397*b2 - 1328*b1 $$\nu^{10}$$ $$=$$ $$-1028\beta_{10} - 1725\beta_{7} + 1231\beta_{6} + 1725\beta_{5} - 3420\beta_{3} + 1231$$ -1028*b10 - 1725*b7 + 1231*b6 + 1725*b5 - 3420*b3 + 1231 $$\nu^{11}$$ $$=$$ $$2753\beta_{15} - 4448\beta_{14} + 3059\beta_{13} + 9129\beta_{9}$$ 2753*b15 - 4448*b14 + 3059*b13 + 9129*b9 $$\nu^{12}$$ $$=$$ $$7201\beta_{12} + 23837\beta_{11} + 19083\beta_{5} + 32141$$ 7201*b12 + 23837*b11 + 19083*b5 + 32141 $$\nu^{13}$$ $$=$$ $$-21390\beta_{8} + 31038\beta_{4} - 19083\beta_{2} + 63106\beta_1$$ -21390*b8 + 31038*b4 - 19083*b2 + 63106*b1 $$\nu^{14}$$ $$=$$ $$50121 \beta_{12} + 165655 \beta_{11} + 50121 \beta_{10} + 82189 \beta_{7} - 57008 \beta_{6} + 50121 \beta_{5} + 165655 \beta_{3} + 165655$$ 50121*b12 + 165655*b11 + 50121*b10 + 82189*b7 - 57008*b6 + 50121*b5 + 165655*b3 + 165655 $$\nu^{15}$$ $$=$$ $$- 132310 \beta_{15} + 215776 \beta_{14} - 148879 \beta_{13} - 437162 \beta_{9} - 148879 \beta_{8} + 215776 \beta_{4} - 132310 \beta_{2} + 437162 \beta_1$$ -132310*b15 + 215776*b14 - 148879*b13 - 437162*b9 - 148879*b8 + 215776*b4 - 132310*b2 + 437162*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/675\mathbb{Z}\right)^\times$$.

 $$n$$ $$326$$ $$352$$ $$\chi(n)$$ $$\beta_{6}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −2.28087 + 1.31686i −1.41485 + 0.816862i −1.27588 + 0.736627i −0.409850 + 0.236627i 0.409850 − 0.236627i 1.27588 − 0.736627i 1.41485 − 0.816862i 2.28087 − 1.31686i −2.28087 − 1.31686i −1.41485 − 0.816862i −1.27588 − 0.736627i −0.409850 − 0.236627i 0.409850 + 0.236627i 1.27588 + 0.736627i 1.41485 + 0.816862i 2.28087 + 1.31686i
−2.28087 + 1.31686i 0 2.46825 4.27513i 0 0 −1.55662 + 0.898714i 7.73393i 0 0
199.2 −1.41485 + 0.816862i 0 0.334526 0.579416i 0 0 0.437645 0.252674i 2.17440i 0 0
199.3 −1.27588 + 0.736627i 0 0.0852394 0.147639i 0 0 3.34791 1.93291i 2.69535i 0 0
199.4 −0.409850 + 0.236627i 0 −0.888015 + 1.53809i 0 0 2.21967 1.28153i 1.78702i 0 0
199.5 0.409850 0.236627i 0 −0.888015 + 1.53809i 0 0 −2.21967 + 1.28153i 1.78702i 0 0
199.6 1.27588 0.736627i 0 0.0852394 0.147639i 0 0 −3.34791 + 1.93291i 2.69535i 0 0
199.7 1.41485 0.816862i 0 0.334526 0.579416i 0 0 −0.437645 + 0.252674i 2.17440i 0 0
199.8 2.28087 1.31686i 0 2.46825 4.27513i 0 0 1.55662 0.898714i 7.73393i 0 0
424.1 −2.28087 1.31686i 0 2.46825 + 4.27513i 0 0 −1.55662 0.898714i 7.73393i 0 0
424.2 −1.41485 0.816862i 0 0.334526 + 0.579416i 0 0 0.437645 + 0.252674i 2.17440i 0 0
424.3 −1.27588 0.736627i 0 0.0852394 + 0.147639i 0 0 3.34791 + 1.93291i 2.69535i 0 0
424.4 −0.409850 0.236627i 0 −0.888015 1.53809i 0 0 2.21967 + 1.28153i 1.78702i 0 0
424.5 0.409850 + 0.236627i 0 −0.888015 1.53809i 0 0 −2.21967 1.28153i 1.78702i 0 0
424.6 1.27588 + 0.736627i 0 0.0852394 + 0.147639i 0 0 −3.34791 1.93291i 2.69535i 0 0
424.7 1.41485 + 0.816862i 0 0.334526 + 0.579416i 0 0 −0.437645 0.252674i 2.17440i 0 0
424.8 2.28087 + 1.31686i 0 2.46825 + 4.27513i 0 0 1.55662 + 0.898714i 7.73393i 0 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 424.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
9.c even 3 1 inner
45.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.k.c 16
3.b odd 2 1 225.2.k.c 16
5.b even 2 1 inner 675.2.k.c 16
5.c odd 4 1 675.2.e.c 8
5.c odd 4 1 675.2.e.e 8
9.c even 3 1 inner 675.2.k.c 16
9.c even 3 1 2025.2.b.o 8
9.d odd 6 1 225.2.k.c 16
9.d odd 6 1 2025.2.b.n 8
15.d odd 2 1 225.2.k.c 16
15.e even 4 1 225.2.e.c 8
15.e even 4 1 225.2.e.e yes 8
45.h odd 6 1 225.2.k.c 16
45.h odd 6 1 2025.2.b.n 8
45.j even 6 1 inner 675.2.k.c 16
45.j even 6 1 2025.2.b.o 8
45.k odd 12 1 675.2.e.c 8
45.k odd 12 1 675.2.e.e 8
45.k odd 12 1 2025.2.a.p 4
45.k odd 12 1 2025.2.a.z 4
45.l even 12 1 225.2.e.c 8
45.l even 12 1 225.2.e.e yes 8
45.l even 12 1 2025.2.a.q 4
45.l even 12 1 2025.2.a.y 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c 8 15.e even 4 1
225.2.e.c 8 45.l even 12 1
225.2.e.e yes 8 15.e even 4 1
225.2.e.e yes 8 45.l even 12 1
225.2.k.c 16 3.b odd 2 1
225.2.k.c 16 9.d odd 6 1
225.2.k.c 16 15.d odd 2 1
225.2.k.c 16 45.h odd 6 1
675.2.e.c 8 5.c odd 4 1
675.2.e.c 8 45.k odd 12 1
675.2.e.e 8 5.c odd 4 1
675.2.e.e 8 45.k odd 12 1
675.2.k.c 16 1.a even 1 1 trivial
675.2.k.c 16 5.b even 2 1 inner
675.2.k.c 16 9.c even 3 1 inner
675.2.k.c 16 45.j even 6 1 inner
2025.2.a.p 4 45.k odd 12 1
2025.2.a.q 4 45.l even 12 1
2025.2.a.y 4 45.l even 12 1
2025.2.a.z 4 45.k odd 12 1
2025.2.b.n 8 9.d odd 6 1
2025.2.b.n 8 45.h odd 6 1
2025.2.b.o 8 9.c even 3 1
2025.2.b.o 8 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{16} - 12T_{2}^{14} + 102T_{2}^{12} - 406T_{2}^{10} + 1167T_{2}^{8} - 1842T_{2}^{6} + 2023T_{2}^{4} - 441T_{2}^{2} + 81$$ acting on $$S_{2}^{\mathrm{new}}(675, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16} - 12 T^{14} + 102 T^{12} + \cdots + 81$$
$3$ $$T^{16}$$
$5$ $$T^{16}$$
$7$ $$T^{16} - 25 T^{14} + 451 T^{12} + \cdots + 6561$$
$11$ $$(T^{8} + T^{7} + 26 T^{6} - 107 T^{5} + \cdots + 81)^{2}$$
$13$ $$T^{16} - 64 T^{14} + \cdots + 131079601$$
$17$ $$(T^{8} + 81 T^{6} + 2214 T^{4} + \cdots + 91809)^{2}$$
$19$ $$(T^{4} + 2 T^{3} - 27 T^{2} - 80 T - 25)^{4}$$
$23$ $$T^{16} - 111 T^{14} + \cdots + 3486784401$$
$29$ $$(T^{8} + T^{7} + 41 T^{6} + 244 T^{5} + \cdots + 16641)^{2}$$
$31$ $$(T^{8} - 4 T^{7} + 58 T^{6} + 114 T^{5} + \cdots + 59049)^{2}$$
$37$ $$(T^{8} + 199 T^{6} + 9513 T^{4} + \cdots + 418609)^{2}$$
$41$ $$(T^{8} + 5 T^{7} + 50 T^{6} + 197 T^{5} + \cdots + 42849)^{2}$$
$43$ $$T^{16} - 196 T^{14} + \cdots + 205144679041$$
$47$ $$T^{16} - 186 T^{14} + \cdots + 21071715921$$
$53$ $$(T^{8} + 228 T^{6} + 13614 T^{4} + \cdots + 221841)^{2}$$
$59$ $$(T^{8} + 17 T^{7} + 287 T^{6} + \cdots + 5349969)^{2}$$
$61$ $$(T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1)^{2}$$
$67$ $$T^{16} - 217 T^{14} + \cdots + 3486784401$$
$71$ $$(T^{4} - 8 T^{3} - 40 T^{2} + 263 T + 381)^{4}$$
$73$ $$(T^{8} + 196 T^{6} + 8478 T^{4} + \cdots + 12769)^{2}$$
$79$ $$(T^{8} + 7 T^{7} + 82 T^{6} - 93 T^{5} + \cdots + 42849)^{2}$$
$83$ $$T^{16} - 324 T^{14} + \cdots + 282429536481$$
$89$ $$(T^{4} + 9 T^{3} - 99 T^{2} - 405 T + 2025)^{4}$$
$97$ $$T^{16} - 199 T^{14} + \cdots + 824843587681$$