# Properties

 Label 1900.2.i.e Level $1900$ Weight $2$ Character orbit 1900.i Analytic conductor $15.172$ 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] = [1900,2,Mod(201,1900)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1900, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

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

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

chi := DirichletCharacter("1900.201");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.i (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: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$12$$ Relative dimension: $$6$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + 14 x^{3} + 145 x^{2} + 33 x + 9$$ x^12 - 3*x^11 + 15*x^10 - 16*x^9 + 79*x^8 - 65*x^7 + 298*x^6 + 13*x^5 + 233*x^4 + 14*x^3 + 145*x^2 + 33*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$3$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{6} + \beta_1 - 1) q^{3} + \beta_{4} q^{7} + (\beta_{10} + \beta_{6} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b6 + b1 - 1) * q^3 + b4 * q^7 + (b10 + b6 - b3 + b2 - b1 + 1) * q^9 $$q + ( - \beta_{6} + \beta_1 - 1) q^{3} + \beta_{4} q^{7} + (\beta_{10} + \beta_{6} - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{9} - \beta_{5} q^{11} + (\beta_{11} + \beta_{8} + 2 \beta_{6} + \beta_{2} - \beta_1 + 1) q^{13} + ( - \beta_{10} + \beta_{9} + \beta_{7}) q^{17} + ( - \beta_{11} + \beta_{10} - \beta_{9} + \beta_{6} - \beta_{5} + \beta_{2} - \beta_1 + 1) q^{19} + (\beta_{10} + \beta_{9} + \beta_{7}) q^{21} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{4} + \beta_{3}) q^{23} + ( - \beta_{4} + 2 \beta_{3} + 2) q^{27} + ( - \beta_{11} - \beta_{8} - \beta_{6} - \beta_{2} + \beta_1 - 1) q^{29} + ( - \beta_{5} + \beta_{4} + 2 \beta_{3}) q^{31} + (\beta_{11} - \beta_{10} + \beta_{9} - \beta_{7} - 2 \beta_{6} + \beta_1 - 2) q^{33} + ( - 2 \beta_{5} - 2) q^{37} + ( - \beta_{8} + \beta_{5} + \beta_{3} - 3 \beta_{2} + 2) q^{39} + ( - 3 \beta_{10} + 2 \beta_{9} - 2 \beta_{6} + \beta_1 - 2) q^{41} + ( - 2 \beta_{10} + 2 \beta_{9} + \beta_{7} - 5 \beta_{6} + 3 \beta_1 - 5) q^{43} + (2 \beta_{10} + \beta_{9} - 3 \beta_{6} + \beta_{5} - 2 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{47} + ( - \beta_{8} - \beta_{3} - 2 \beta_{2}) q^{49} + (\beta_{11} - \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 1) q^{51} + (2 \beta_{11} - 2 \beta_{10} + \beta_{9} + 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + \beta_{5} + \beta_{4} + \cdots + 1) q^{53}+ \cdots + (2 \beta_{10} + \beta_{7} + 4 \beta_{6} + \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 2) q^{99}+O(q^{100})$$ q + (-b6 + b1 - 1) * q^3 + b4 * q^7 + (b10 + b6 - b3 + b2 - b1 + 1) * q^9 - b5 * q^11 + (b11 + b8 + 2*b6 + b2 - b1 + 1) * q^13 + (-b10 + b9 + b7) * q^17 + (-b11 + b10 - b9 + b6 - b5 + b2 - b1 + 1) * q^19 + (b10 + b9 + b7) * q^21 + (b11 - b10 + b8 - b7 - b4 + b3) * q^23 + (-b4 + 2*b3 + 2) * q^27 + (-b11 - b8 - b6 - b2 + b1 - 1) * q^29 + (-b5 + b4 + 2*b3) * q^31 + (b11 - b10 + b9 - b7 - 2*b6 + b1 - 2) * q^33 + (-2*b5 - 2) * q^37 + (-b8 + b5 + b3 - 3*b2 + 2) * q^39 + (-3*b10 + 2*b9 - 2*b6 + b1 - 2) * q^41 + (-2*b10 + 2*b9 + b7 - 5*b6 + 3*b1 - 5) * q^43 + (2*b10 + b9 - 3*b6 + b5 - 2*b3 - b2 + b1 - 1) * q^47 + (-b8 - b3 - 2*b2) * q^49 + (b11 - b10 + b8 - b7 - b6 - b4 + b3 + b2 - b1 + 1) * q^51 + (2*b11 - 2*b10 + b9 + 2*b8 + b7 + 2*b6 + b5 + b4 + 2*b3 + b2 - b1 + 1) * q^53 + (b11 - b10 - b9 - b7 - b6 - 2*b5 + b3 - b2 + b1 + 1) * q^57 + (-2*b11 - b9 + 2*b7 - b6 + b1 - 1) * q^59 + (b11 + b9 + b8 + 3*b7 + b6 + b5 + 3*b4) * q^61 + (b11 - 3*b10 + b8 - 3*b6 + 3*b3 - 3*b2 + 3*b1 - 3) * q^63 + (b11 - b10 + b9 + b8 - b7 + 3*b6 + b5 - b4 + b3 + b2 - b1 + 1) * q^67 + (-b8 + 2*b5 + 2*b4 - 2*b3 + b2) * q^69 + (b11 - b10 + b9 + 2*b7 + b6 + b1 + 1) * q^71 + (2*b11 + b10 + b9 + b7 + 2*b6 + 2) * q^73 + (b8 + 3*b2 + 1) * q^77 + (-b11 + b10 - b9 + 2*b6 - 4*b1 + 2) * q^79 + (-b9 - 3*b7 - b6 + 3*b1 - 1) * q^81 + (b8 - 2*b5 - b4 + b3 + b2 + 2) * q^83 + (b8 - b5 - b3 + 2*b2 - 2) * q^87 + (-b11 - b10 - b8 - 2*b7 + b6 - 2*b4 + b3) * q^89 + (2*b11 - 2*b9 + 2*b8 - 3*b7 + b6 - 2*b5 - 3*b4 + b2 - b1 + 1) * q^91 + (b11 - 2*b10 + 2*b9 - 2*b7 - 4*b6 + 5*b1 - 4) * q^93 + (2*b11 - b10 + 2*b7 + 3*b6 - b1 + 3) * q^97 + (2*b10 + b7 + 4*b6 + b4 - 2*b3 + 2*b2 - 2*b1 + 2) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q - 3 q^{3} - 3 q^{9}+O(q^{10})$$ 12 * q - 3 * q^3 - 3 * q^9 $$12 q - 3 q^{3} - 3 q^{9} + 2 q^{11} - 7 q^{13} + q^{17} + q^{21} + 2 q^{23} + 24 q^{27} + q^{29} + 2 q^{31} - 10 q^{33} - 20 q^{37} + 36 q^{39} - 7 q^{41} - 19 q^{43} + 14 q^{47} + 8 q^{49} + 11 q^{51} - 6 q^{53} + 28 q^{57} - 5 q^{61} + 11 q^{63} - 14 q^{67} - 14 q^{69} + 8 q^{71} + 9 q^{73} - 2 q^{77} + q^{79} + 2 q^{81} + 26 q^{83} - 30 q^{87} - 8 q^{89} + 3 q^{91} - 9 q^{93} + 11 q^{97} - 18 q^{99}+O(q^{100})$$ 12 * q - 3 * q^3 - 3 * q^9 + 2 * q^11 - 7 * q^13 + q^17 + q^21 + 2 * q^23 + 24 * q^27 + q^29 + 2 * q^31 - 10 * q^33 - 20 * q^37 + 36 * q^39 - 7 * q^41 - 19 * q^43 + 14 * q^47 + 8 * q^49 + 11 * q^51 - 6 * q^53 + 28 * q^57 - 5 * q^61 + 11 * q^63 - 14 * q^67 - 14 * q^69 + 8 * q^71 + 9 * q^73 - 2 * q^77 + q^79 + 2 * q^81 + 26 * q^83 - 30 * q^87 - 8 * q^89 + 3 * q^91 - 9 * q^93 + 11 * q^97 - 18 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} - 3 x^{11} + 15 x^{10} - 16 x^{9} + 79 x^{8} - 65 x^{7} + 298 x^{6} + 13 x^{5} + 233 x^{4} + 14 x^{3} + 145 x^{2} + 33 x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( - 638495 \nu^{11} - 5059670 \nu^{10} + 2185864 \nu^{9} - 61632615 \nu^{8} - 86473310 \nu^{7} - 313073128 \nu^{6} - 445497855 \nu^{5} + \cdots - 2681192682 ) / 2135068398$$ (-638495*v^11 - 5059670*v^10 + 2185864*v^9 - 61632615*v^8 - 86473310*v^7 - 313073128*v^6 - 445497855*v^5 - 1233567830*v^4 - 3200881200*v^3 - 892972135*v^2 - 207837960*v - 2681192682) / 2135068398 $$\beta_{3}$$ $$=$$ $$( - 2112220 \nu^{11} + 5607653 \nu^{10} - 24678133 \nu^{9} + 8533470 \nu^{8} - 89367331 \nu^{7} + 19282261 \nu^{6} - 259923180 \nu^{5} + \cdots - 1951111485 ) / 711689466$$ (-2112220*v^11 + 5607653*v^10 - 24678133*v^9 + 8533470*v^8 - 89367331*v^7 + 19282261*v^6 - 259923180*v^5 - 606181345*v^4 + 772282665*v^3 - 452450816*v^2 - 105738663*v - 1951111485) / 711689466 $$\beta_{4}$$ $$=$$ $$( 3389210 \nu^{11} + 4511687 \nu^{10} + 20306405 \nu^{9} + 114731760 \nu^{8} + 262313951 \nu^{7} + 606863995 \nu^{6} + 1150918890 \nu^{5} + \cdots + 1619981121 ) / 711689466$$ (3389210*v^11 + 4511687*v^10 + 20306405*v^9 + 114731760*v^8 + 262313951*v^7 + 606863995*v^6 + 1150918890*v^5 + 3073317005*v^4 + 4917790269*v^3 + 2238395086*v^2 + 521414583*v + 1619981121) / 711689466 $$\beta_{5}$$ $$=$$ $$( - 35281678 \nu^{11} + 70497029 \nu^{10} - 385595812 \nu^{9} - 77727516 \nu^{8} - 1696717843 \nu^{7} - 1136318216 \nu^{6} + \cdots - 5196018312 ) / 2135068398$$ (-35281678*v^11 + 70497029*v^10 - 385595812*v^9 - 77727516*v^8 - 1696717843*v^7 - 1136318216*v^6 - 5600322462*v^5 - 13728352537*v^4 - 422891238*v^3 - 10151561138*v^2 - 2369519571*v - 5196018312) / 2135068398 $$\beta_{6}$$ $$=$$ $$( - 60680476 \nu^{11} + 182679923 \nu^{10} - 905147470 \nu^{9} + 968701752 \nu^{8} - 4732124989 \nu^{7} + 4030704250 \nu^{6} + \cdots - 1794617748 ) / 2135068398$$ (-60680476*v^11 + 182679923*v^10 - 905147470*v^9 + 968701752*v^8 - 4732124989*v^7 + 4030704250*v^6 - 17769708720*v^5 - 343348333*v^4 - 12904983078*v^3 + 2351354536*v^2 - 7905696885*v - 1794617748) / 2135068398 $$\beta_{7}$$ $$=$$ $$( 62867497 \nu^{11} - 203695424 \nu^{10} + 980933620 \nu^{9} - 1201195899 \nu^{8} + 5086161106 \nu^{7} - 5139259390 \nu^{6} + \cdots - 397480590 ) / 2135068398$$ (62867497*v^11 - 203695424*v^10 + 980933620*v^9 - 1201195899*v^8 + 5086161106*v^7 - 5139259390*v^6 + 19505876175*v^5 - 3449941442*v^4 + 14173622286*v^3 - 2952954247*v^2 + 17820058356*v - 397480590) / 2135068398 $$\beta_{8}$$ $$=$$ $$( 102677425 \nu^{11} - 177003953 \nu^{10} + 1095480130 \nu^{9} + 493877715 \nu^{8} + 5185680031 \nu^{7} + 4686279104 \nu^{6} + \cdots + 17785843362 ) / 2135068398$$ (102677425*v^11 - 177003953*v^10 + 1095480130*v^9 + 493877715*v^8 + 5185680031*v^7 + 4686279104*v^6 + 17827730625*v^5 + 44330893465*v^4 + 9517460142*v^3 + 32695561541*v^2 + 7628969433*v + 17785843362) / 2135068398 $$\beta_{9}$$ $$=$$ $$( - 131028322 \nu^{11} + 444681245 \nu^{10} - 2136716515 \nu^{9} + 2925836544 \nu^{8} - 11360311729 \nu^{7} + 12770142025 \nu^{6} + \cdots + 924096825 ) / 2135068398$$ (-131028322*v^11 + 444681245*v^10 - 2136716515*v^9 + 2925836544*v^8 - 11360311729*v^7 + 12770142025*v^6 - 42996086940*v^5 + 14406379883*v^4 - 31258077591*v^3 + 7259257192*v^2 - 16532556735*v + 924096825) / 2135068398 $$\beta_{10}$$ $$=$$ $$( 175066273 \nu^{11} - 536276480 \nu^{10} + 2643593875 \nu^{9} - 2942137461 \nu^{8} + 13841799664 \nu^{7} - 12347339095 \nu^{6} + \cdots - 1015605495 ) / 2135068398$$ (175066273*v^11 - 536276480*v^10 + 2643593875*v^9 - 2942137461*v^8 + 13841799664*v^7 - 12347339095*v^6 + 52083858765*v^5 - 2022066866*v^4 + 37830916029*v^3 - 7169319793*v^2 + 21056968308*v - 1015605495) / 2135068398 $$\beta_{11}$$ $$=$$ $$( 137992489 \nu^{11} - 469201302 \nu^{10} + 2243299983 \nu^{9} - 3049379221 \nu^{8} + 11871320512 \nu^{7} - 13293030539 \nu^{6} + \cdots - 965971179 ) / 711689466$$ (137992489*v^11 - 469201302*v^10 + 2243299983*v^9 - 3049379221*v^8 + 11871320512*v^7 - 13293030539*v^6 + 45104060701*v^5 - 14256401492*v^4 + 32789448329*v^3 - 7561467697*v^2 + 19342466968*v - 965971179) / 711689466
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{10} + 3\beta_{6} - \beta_{3} - \beta_{2} + \beta _1 - 1$$ b10 + 3*b6 - b3 - b2 + b1 - 1 $$\nu^{3}$$ $$=$$ $$-\beta_{4} - \beta_{3} - 6\beta_{2} - 8$$ -b4 - b3 - 6*b2 - 8 $$\nu^{4}$$ $$=$$ $$-7\beta_{10} - \beta_{9} + \beta_{7} - 17\beta_{6} - 10\beta _1 - 17$$ -7*b10 - b9 + b7 - 17*b6 - 10*b1 - 17 $$\nu^{5}$$ $$=$$ $$- \beta_{11} - 10 \beta_{10} - 3 \beta_{9} - \beta_{8} + 8 \beta_{7} - 21 \beta_{6} - 3 \beta_{5} + 8 \beta_{4} + 10 \beta_{3} + 42 \beta_{2} - 42 \beta _1 + 42$$ -b11 - 10*b10 - 3*b9 - b8 + 8*b7 - 21*b6 - 3*b5 + 8*b4 + 10*b3 + 42*b2 - 42*b1 + 42 $$\nu^{6}$$ $$=$$ $$-3\beta_{8} - 15\beta_{5} + 13\beta_{4} + 47\beta_{3} + 87\beta_{2} + 197$$ -3*b8 - 15*b5 + 13*b4 + 47*b3 + 87*b2 + 197 $$\nu^{7}$$ $$=$$ $$15\beta_{11} + 85\beta_{10} + 46\beta_{9} - 62\beta_{7} + 184\beta_{6} + 309\beta _1 + 184$$ 15*b11 + 85*b10 + 46*b9 - 62*b7 + 184*b6 + 309*b1 + 184 $$\nu^{8}$$ $$=$$ $$46 \beta_{11} + 325 \beta_{10} + 169 \beta_{9} + 46 \beta_{8} - 131 \beta_{7} + 750 \beta_{6} + 169 \beta_{5} - 131 \beta_{4} - 325 \beta_{3} - 724 \beta_{2} + 724 \beta _1 - 724$$ 46*b11 + 325*b10 + 169*b9 + 46*b8 - 131*b7 + 750*b6 + 169*b5 - 131*b4 - 325*b3 - 724*b2 + 724*b1 - 724 $$\nu^{9}$$ $$=$$ $$169\beta_{8} + 515\beta_{5} - 494\beta_{4} - 686\beta_{3} - 2339\beta_{2} - 3848$$ 169*b8 + 515*b5 - 494*b4 - 686*b3 - 2339*b2 - 3848 $$\nu^{10}$$ $$=$$ $$-515\beta_{11} - 2318\beta_{10} - 1693\beta_{9} + 1201\beta_{7} - 5301\beta_{6} - 5904\beta _1 - 5301$$ -515*b11 - 2318*b10 - 1693*b9 + 1201*b7 - 5301*b6 - 5904*b1 - 5301 $$\nu^{11}$$ $$=$$ $$- 1693 \beta_{11} - 5412 \beta_{10} - 5102 \beta_{9} - 1693 \beta_{8} + 4011 \beta_{7} - 12008 \beta_{6} - 5102 \beta_{5} + 4011 \beta_{4} + 5412 \beta_{3} + 18049 \beta_{2} - 18049 \beta _1 + 18049$$ -1693*b11 - 5412*b10 - 5102*b9 - 1693*b8 + 4011*b7 - 12008*b6 - 5102*b5 + 4011*b4 + 5412*b3 + 18049*b2 - 18049*b1 + 18049

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$1$$ $$\beta_{6}$$ $$1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
201.1
 −1.08347 + 1.87662i −0.432807 + 0.749643i −0.126563 + 0.219213i 0.430479 − 0.745611i 1.28913 − 2.23284i 1.42323 − 2.46510i −1.08347 − 1.87662i −0.432807 − 0.749643i −0.126563 − 0.219213i 0.430479 + 0.745611i 1.28913 + 2.23284i 1.42323 + 2.46510i
0 −1.58347 + 2.74265i 0 0 0 −3.03607 0 −3.51475 6.08773i 0
201.2 0 −0.932807 + 1.61567i 0 0 0 3.93019 0 −0.240257 0.416137i 0
201.3 0 −0.626563 + 1.08524i 0 0 0 2.18534 0 0.714838 + 1.23814i 0
201.4 0 −0.0695210 + 0.120414i 0 0 0 −3.40785 0 1.49033 + 2.58133i 0
201.5 0 0.789132 1.36682i 0 0 0 −1.39989 0 0.254541 + 0.440878i 0
201.6 0 0.923228 1.59908i 0 0 0 1.72830 0 −0.204702 0.354554i 0
501.1 0 −1.58347 2.74265i 0 0 0 −3.03607 0 −3.51475 + 6.08773i 0
501.2 0 −0.932807 1.61567i 0 0 0 3.93019 0 −0.240257 + 0.416137i 0
501.3 0 −0.626563 1.08524i 0 0 0 2.18534 0 0.714838 1.23814i 0
501.4 0 −0.0695210 0.120414i 0 0 0 −3.40785 0 1.49033 2.58133i 0
501.5 0 0.789132 + 1.36682i 0 0 0 −1.39989 0 0.254541 0.440878i 0
501.6 0 0.923228 + 1.59908i 0 0 0 1.72830 0 −0.204702 + 0.354554i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 201.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.i.e 12
5.b even 2 1 1900.2.i.f yes 12
5.c odd 4 2 1900.2.s.e 24
19.c even 3 1 inner 1900.2.i.e 12
95.i even 6 1 1900.2.i.f yes 12
95.m odd 12 2 1900.2.s.e 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1900.2.i.e 12 1.a even 1 1 trivial
1900.2.i.e 12 19.c even 3 1 inner
1900.2.i.f yes 12 5.b even 2 1
1900.2.i.f yes 12 95.i even 6 1
1900.2.s.e 24 5.c odd 4 2
1900.2.s.e 24 95.m odd 12 2

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} + 3 T_{3}^{11} + 15 T_{3}^{10} + 16 T_{3}^{9} + 79 T_{3}^{8} + 77 T_{3}^{7} + 274 T_{3}^{6} + 149 T_{3}^{5} + 473 T_{3}^{4} + 286 T_{3}^{3} + 505 T_{3}^{2} + 69 T_{3} + 9$$ acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{12}$$
$3$ $$T^{12} + 3 T^{11} + 15 T^{10} + 16 T^{9} + \cdots + 9$$
$5$ $$T^{12}$$
$7$ $$(T^{6} - 23 T^{4} - 2 T^{3} + 141 T^{2} + \cdots - 215)^{2}$$
$11$ $$(T^{6} - T^{5} - 22 T^{4} + 43 T^{3} + 66 T^{2} + \cdots + 27)^{2}$$
$13$ $$T^{12} + 7 T^{11} + 69 T^{10} + \cdots + 199809$$
$17$ $$T^{12} - T^{11} + 49 T^{10} + \cdots + 762129$$
$19$ $$T^{12} + 33 T^{10} + 126 T^{9} + \cdots + 47045881$$
$23$ $$T^{12} - 2 T^{11} + 75 T^{10} + \cdots + 455625$$
$29$ $$T^{12} - T^{11} + 41 T^{10} - 94 T^{9} + \cdots + 729$$
$31$ $$(T^{6} - T^{5} - 120 T^{4} + 53 T^{3} + \cdots + 2213)^{2}$$
$37$ $$(T^{6} + 10 T^{5} - 48 T^{4} - 280 T^{3} + \cdots - 2880)^{2}$$
$41$ $$T^{12} + 7 T^{11} + 241 T^{10} + \cdots + 303282225$$
$43$ $$T^{12} + 19 T^{11} + \cdots + 1154300625$$
$47$ $$T^{12} - 14 T^{11} + \cdots + 2546514369$$
$53$ $$T^{12} + 6 T^{11} + 185 T^{10} + \cdots + 19210689$$
$59$ $$T^{12} + 227 T^{10} + \cdots + 624650049$$
$61$ $$T^{12} + 5 T^{11} + 275 T^{10} + \cdots + 3932289$$
$67$ $$T^{12} + 14 T^{11} + 200 T^{10} + \cdots + 5089536$$
$71$ $$T^{12} - 8 T^{11} + 161 T^{10} + \cdots + 531441$$
$73$ $$T^{12} - 9 T^{11} + 279 T^{10} + \cdots + 352125225$$
$79$ $$T^{12} - T^{11} + 181 T^{10} + \cdots + 526105969$$
$83$ $$(T^{6} - 13 T^{5} - 122 T^{4} + \cdots + 27345)^{2}$$
$89$ $$T^{12} + 8 T^{11} + 223 T^{10} + \cdots + 3908529$$
$97$ $$T^{12} - 11 T^{11} + 287 T^{10} + \cdots + 22325625$$