# Properties

 Label 325.2.e.d Level $325$ Weight $2$ Character orbit 325.e Analytic conductor $2.595$ Analytic rank $0$ Dimension $10$ Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(126,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([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("325.126");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.e (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: $$2.59513806569$$ 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: $$x^{10} + 8x^{8} - 2x^{7} + 52x^{6} - 5x^{5} + 97x^{4} + 60x^{3} + 141x^{2} + 36x + 9$$ x^10 + 8*x^8 - 2*x^7 + 52*x^6 - 5*x^5 + 97*x^4 + 60*x^3 + 141*x^2 + 36*x + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + ( - \beta_{8} - \beta_{6} + \beta_{4} + 1) q^{3} + (\beta_{9} - \beta_{6}) q^{4} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{6}+ \cdots + ( - 2 \beta_{8} - \beta_{7} + \cdots + \beta_1) q^{9}+O(q^{10})$$ q - b1 * q^2 + (-b8 - b6 + b4 + 1) * q^3 + (b9 - b6) * q^4 + (-b7 - b6 - b5 + b2) * q^6 + (b7 + b5 - b1) * q^7 + (b4 - b3 + b2) * q^8 + (-2*b8 - b7 - 2*b6 - b5 + b1) * q^9 $$q - \beta_1 q^{2} + ( - \beta_{8} - \beta_{6} + \beta_{4} + 1) q^{3} + (\beta_{9} - \beta_{6}) q^{4} + ( - \beta_{7} - \beta_{6} + \cdots + \beta_{2}) q^{6}+ \cdots + (3 \beta_{5} + 5 \beta_{4} - \beta_{3} + \cdots + 9) q^{99}+O(q^{100})$$ q - b1 * q^2 + (-b8 - b6 + b4 + 1) * q^3 + (b9 - b6) * q^4 + (-b7 - b6 - b5 + b2) * q^6 + (b7 + b5 - b1) * q^7 + (b4 - b3 + b2) * q^8 + (-2*b8 - b7 - 2*b6 - b5 + b1) * q^9 + (b7 + b6 - 1) * q^11 + (b5 - b2) * q^12 + (b9 - b7 - b6 - b5 - b4 - 1) * q^13 + (-b5 + 2*b4 - b3 + 2*b2 - 1) * q^14 + (-b8 - b7 + b4 + 3*b1) * q^16 + (-b9 + b7 + b6 + b5 - 2*b2 + b1) * q^17 + (-b5 - 2*b4 + b3 - 1) * q^18 + (b9 + b6) * q^19 + (b4 - b3 - 2*b2 - 1) * q^21 + (2*b8 - b7 + 2*b6 - b5 + b2) * q^22 + (2*b9 - b8 - 3*b6 + b4 + 2*b3 + b1 + 3) * q^23 + (-2*b8 - b7 - 4*b6 + 2*b4 + b1 + 4) * q^24 + (b9 + b7 - 2*b6 + b5 - b4 + b2) * q^26 + (-2*b5 - 2*b4 + b3 + 2*b2 - 6) * q^27 + (-2*b9 + b8 - b7 + 3*b6 - b4 - 2*b3 + 2*b1 - 3) * q^28 + (-b9 + b8 - b4 - b3 + 2*b1) * q^29 + (-2*b4 + b3 - b2 - 1) * q^31 + (-b9 + 6*b6) * q^32 + (b9 + 2*b8 + b7 + b6 + b5 - 3*b2 + 2*b1) * q^33 + (-b5 + b4 + 2*b3 - b2 + 5) * q^34 + (2*b9 - b8 - b7 - 3*b6 + b4 + 2*b3 - 2*b1 + 3) * q^36 + (2*b8 - b7 - 3*b6 - 2*b4 + 3) * q^37 + (b4 - b3 + b2) * q^38 + (2*b8 + 4*b6 - 3*b4 + b3 + 2*b2 - b1 - 4) * q^39 + (-2*b9 + b8 + 2*b7 + 2*b6 - b4 - 2*b3 - 3*b1 - 2) * q^41 + (b9 - b8 - b7 - 7*b6 + b4 + b3 + 4*b1 + 7) * q^42 + (-2*b9 + 2*b6 + 3*b2 - 3*b1) * q^43 + (b5 - 2*b4 - 2*b2 - 2) * q^44 + (b9 + 2*b8 - b7 + 2*b6 - b5 + 7*b2 - 6*b1) * q^46 + (-b5 - b4 - 2*b3 - 2*b2 + 2) * q^47 + (-b9 - 2*b8 + b7 - b6 + b5 - b1) * q^48 + (-2*b9 - b8 + b7 + b6 + b4 - 2*b3 - 2*b1 - 1) * q^49 + (b5 + b4 - b3 - 3*b2 + 1) * q^51 + (-2*b8 - b7 + 3*b6 - b5 + 3*b4 - b3 + 4*b2 + b1 - 3) * q^52 + (-b5 + 3*b4 + b3 - 1) * q^53 + (b9 + 5*b8 + 6*b6 - 5*b4 + b3 - 6) * q^54 + (-2*b9 + 3*b6 - 5*b2 + 5*b1) * q^56 + (b5 + 2*b4 - b2 + 2) * q^57 + (-3*b9 - b8 + b7 + 7*b6 + b5 - 2*b2 + b1) * q^58 + (-2*b9 + 4*b8 + b7 + 4*b6 + b5 + b2 - 2*b1) * q^59 + (-b9 - 2*b8 - 2*b6 - b2 + b1) * q^61 + (2*b9 + b8 + 2*b7 - b6 - b4 + 2*b3 - 3*b1 + 1) * q^62 + (-b8 - b7 - 6*b6 + b4 - 2*b1 + 6) * q^63 + (2*b5 + b4 + b3 - 4*b2) * q^64 + (b5 + 3*b4 + b3 + 3*b2 + 10) * q^66 + (-b9 - 5*b8 + b6 + 5*b4 - b3 + b1 - 1) * q^67 + (2*b9 + 4*b8 - 3*b6 - 4*b4 + 2*b3 - 7*b1 + 3) * q^68 + (-2*b8 + 2*b7 - 4*b6 + 2*b5 - 3*b2 + b1) * q^69 + (-3*b9 - b8 + 2*b7 - 4*b6 + 2*b5 - 2*b1) * q^71 + (2*b9 - 4*b8 - 2*b7 - 11*b6 - 2*b5 + 5*b2 - 3*b1) * q^72 + (b5 - b4 - 2*b3 + b2 + 2) * q^73 + (-2*b8 + 3*b7 + 3*b5 + b2 - 4*b1) * q^74 + (-b8 - b7 + 4*b6 + b4 + 3*b1 - 4) * q^76 + (-b5 - b4 - b3 - 3*b2 - 7) * q^77 + (b8 + 3*b7 + 6*b6 + 2*b5 - b4 - b3 - 4*b2 - b1 - 7) * q^78 + (2*b5 + b4 + 2*b3 - 4*b2 - 2) * q^79 + (2*b9 + 5*b8 + 2*b7 + 9*b6 - 5*b4 + 2*b3 + 5*b1 - 9) * q^81 + (b9 + 2*b8 - b7 - 4*b6 - b5 - 2*b2 + 3*b1) * q^82 + (4*b5 + 3*b4 + b3 + 4) * q^83 + (-b9 + b8 + 7*b6 + 3*b2 - 3*b1) * q^84 + (-2*b4 - b3 - 6*b2 - 9) * q^86 + (2*b7 + 5*b6 + 2*b5 - b2 - b1) * q^87 + (b9 + 2*b8 + b7 + b6 - 2*b4 + b3 + b1 - 1) * q^88 + (-b9 + b8 + 2*b7 - b6 - b4 - b3 - 3*b1 + 1) * q^89 + (-b9 + b8 - 2*b7 - 4*b6 - b5 - b3 + 4*b2 + 2*b1 + 4) * q^91 + (3*b5 + b4 - 3*b3 - 12) * q^92 + (4*b8 + 2*b7 + 9*b6 - 4*b4 - 2*b1 - 9) * q^93 + (b9 - 6*b6 + b3 + 6) * q^94 + (-b5 + 5*b4 + b2 + 5) * q^96 + (2*b9 - b8 - b7 + 7*b6 - b5 + 4*b2 - 3*b1) * q^97 + (-2*b7 - 5*b6 - 2*b5 - 3*b2 + 5*b1) * q^98 + (3*b5 + 5*b4 - b3 - 3*b2 + 9) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q + 3 q^{3} - 6 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9}+O(q^{10})$$ 10 * q + 3 * q^3 - 6 * q^4 - 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 $$10 q + 3 q^{3} - 6 q^{4} - 3 q^{6} - 2 q^{7} - 6 q^{8} - 4 q^{9} - 3 q^{11} - 4 q^{12} - 10 q^{13} - 16 q^{14} - 4 q^{16} + 4 q^{17} + 4 q^{18} + 4 q^{19} - 16 q^{21} + 8 q^{22} + 15 q^{23} + 14 q^{24} - 9 q^{26} - 42 q^{27} - 17 q^{28} + q^{29} + 31 q^{32} - 2 q^{33} + 54 q^{34} + 13 q^{36} + 17 q^{37} - 6 q^{38} - 10 q^{39} - 6 q^{41} + 32 q^{42} + 12 q^{43} - 16 q^{44} + 7 q^{46} + 24 q^{47} - 2 q^{48} - 7 q^{49} - 23 q^{52} - 16 q^{53} - 19 q^{54} + 17 q^{56} + 8 q^{57} + 38 q^{58} + 12 q^{59} - 5 q^{61} + 13 q^{62} + 26 q^{63} - 10 q^{64} + 86 q^{66} - 16 q^{67} + 25 q^{68} - 20 q^{69} - 19 q^{71} - 45 q^{72} + 16 q^{73} - 2 q^{74} - 24 q^{76} - 64 q^{77} - 42 q^{78} - 28 q^{79} - 29 q^{81} - 23 q^{82} + 14 q^{83} + 34 q^{84} - 84 q^{86} + 21 q^{87} + 2 q^{88} + 10 q^{89} + 17 q^{91} - 142 q^{92} - 33 q^{93} + 31 q^{94} + 34 q^{96} + 37 q^{97} - 21 q^{98} + 56 q^{99}+O(q^{100})$$ 10 * q + 3 * q^3 - 6 * q^4 - 3 * q^6 - 2 * q^7 - 6 * q^8 - 4 * q^9 - 3 * q^11 - 4 * q^12 - 10 * q^13 - 16 * q^14 - 4 * q^16 + 4 * q^17 + 4 * q^18 + 4 * q^19 - 16 * q^21 + 8 * q^22 + 15 * q^23 + 14 * q^24 - 9 * q^26 - 42 * q^27 - 17 * q^28 + q^29 + 31 * q^32 - 2 * q^33 + 54 * q^34 + 13 * q^36 + 17 * q^37 - 6 * q^38 - 10 * q^39 - 6 * q^41 + 32 * q^42 + 12 * q^43 - 16 * q^44 + 7 * q^46 + 24 * q^47 - 2 * q^48 - 7 * q^49 - 23 * q^52 - 16 * q^53 - 19 * q^54 + 17 * q^56 + 8 * q^57 + 38 * q^58 + 12 * q^59 - 5 * q^61 + 13 * q^62 + 26 * q^63 - 10 * q^64 + 86 * q^66 - 16 * q^67 + 25 * q^68 - 20 * q^69 - 19 * q^71 - 45 * q^72 + 16 * q^73 - 2 * q^74 - 24 * q^76 - 64 * q^77 - 42 * q^78 - 28 * q^79 - 29 * q^81 - 23 * q^82 + 14 * q^83 + 34 * q^84 - 84 * q^86 + 21 * q^87 + 2 * q^88 + 10 * q^89 + 17 * q^91 - 142 * q^92 - 33 * q^93 + 31 * q^94 + 34 * q^96 + 37 * q^97 - 21 * q^98 + 56 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 8x^{8} - 2x^{7} + 52x^{6} - 5x^{5} + 97x^{4} + 60x^{3} + 141x^{2} + 36x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( 1304 \nu^{9} + 1776 \nu^{8} - 1628 \nu^{7} + 8447 \nu^{6} - 14800 \nu^{5} + 93980 \nu^{4} + \cdots - 242496 ) / 983355$$ (1304*v^9 + 1776*v^8 - 1628*v^7 + 8447*v^6 - 14800*v^5 + 93980*v^4 - 511687*v^3 + 165612*v^2 + 42624*v - 242496) / 983355 $$\beta_{3}$$ $$=$$ $$( 592 \nu^{9} - 4020 \nu^{8} + 3685 \nu^{7} - 27536 \nu^{6} + 33500 \nu^{5} - 212725 \nu^{4} + \cdots - 987267 ) / 327785$$ (592*v^9 - 4020*v^8 + 3685*v^7 - 27536*v^6 + 33500*v^5 - 212725*v^4 + 29124*v^3 - 374865*v^2 - 96480*v - 987267) / 327785 $$\beta_{4}$$ $$=$$ $$( - 4744 \nu^{9} - 20940 \nu^{8} + 19195 \nu^{7} - 124843 \nu^{6} + 174500 \nu^{5} - 1108075 \nu^{4} + \cdots - 1749321 ) / 983355$$ (-4744*v^9 - 20940*v^8 + 19195*v^7 - 124843*v^6 + 174500*v^5 - 1108075*v^4 + 1662452*v^3 - 1952655*v^2 - 502560*v - 1749321) / 983355 $$\beta_{5}$$ $$=$$ $$( 3213 \nu^{9} - 12516 \nu^{8} + 11473 \nu^{7} - 129958 \nu^{6} + 104300 \nu^{5} - 662305 \nu^{4} + \cdots - 553661 ) / 327785$$ (3213*v^9 - 12516*v^8 + 11473*v^7 - 129958*v^6 + 104300*v^5 - 662305*v^4 + 22966*v^3 - 1167117*v^2 - 300384*v - 553661) / 327785 $$\beta_{6}$$ $$=$$ $$( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 983355$$ (26944*v^9 + 1304*v^8 + 217328*v^7 - 55516*v^6 + 1409535*v^5 - 149520*v^4 + 2707548*v^3 + 1104953*v^2 + 3964716*v + 1012608) / 983355 $$\beta_{7}$$ $$=$$ $$( - 47884 \nu^{9} + 55843 \nu^{8} - 351659 \nu^{7} + 476704 \nu^{6} - 2541330 \nu^{5} + 2272140 \nu^{4} + \cdots + 13443 ) / 983355$$ (-47884*v^9 + 55843*v^8 - 351659*v^7 + 476704*v^6 - 2541330*v^5 + 2272140*v^4 - 4375563*v^3 - 938609*v^2 - 4559898*v + 13443) / 983355 $$\beta_{8}$$ $$=$$ $$( - 75292 \nu^{9} - 9639 \nu^{8} - 564788 \nu^{7} + 116165 \nu^{6} - 3525310 \nu^{5} + \cdots - 1809360 ) / 983355$$ (-75292*v^9 - 9639*v^8 - 564788*v^7 + 116165*v^6 - 3525310*v^5 + 63560*v^4 - 5316409*v^3 - 4586418*v^2 - 7114821*v - 1809360) / 983355 $$\beta_{9}$$ $$=$$ $$( 26944 \nu^{9} + 1304 \nu^{8} + 217328 \nu^{7} - 55516 \nu^{6} + 1409535 \nu^{5} - 149520 \nu^{4} + \cdots + 1012608 ) / 327785$$ (26944*v^9 + 1304*v^8 + 217328*v^7 - 55516*v^6 + 1409535*v^5 - 149520*v^4 + 2707548*v^3 + 1432738*v^2 + 3964716*v + 1012608) / 327785
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{9} - 3\beta_{6}$$ b9 - 3*b6 $$\nu^{3}$$ $$=$$ $$-\beta_{4} + \beta_{3} - 5\beta_{2}$$ -b4 + b3 - 5*b2 $$\nu^{4}$$ $$=$$ $$-6\beta_{9} - \beta_{8} - \beta_{7} + 14\beta_{6} + \beta_{4} - 6\beta_{3} + 3\beta _1 - 14$$ -6*b9 - b8 - b7 + 14*b6 + b4 - 6*b3 + 3*b1 - 14 $$\nu^{5}$$ $$=$$ $$9\beta_{9} + 8\beta_{8} - 6\beta_{6} + 28\beta_{2} - 28\beta_1$$ 9*b9 + 8*b8 - 6*b6 + 28*b2 - 28*b1 $$\nu^{6}$$ $$=$$ $$-8\beta_{5} - 9\beta_{4} + 37\beta_{3} - 24\beta_{2} + 76$$ -8*b5 - 9*b4 + 37*b3 - 24*b2 + 76 $$\nu^{7}$$ $$=$$ $$-69\beta_{9} - 53\beta_{8} - \beta_{7} + 71\beta_{6} + 53\beta_{4} - 69\beta_{3} + 167\beta _1 - 71$$ -69*b9 - 53*b8 - b7 + 71*b6 + 53*b4 - 69*b3 + 167*b1 - 71 $$\nu^{8}$$ $$=$$ $$236\beta_{9} + 71\beta_{8} + 52\beta_{7} - 446\beta_{6} + 52\beta_{5} + 211\beta_{2} - 263\beta_1$$ 236*b9 + 71*b8 + 52*b7 - 446*b6 + 52*b5 + 211*b2 - 263*b1 $$\nu^{9}$$ $$=$$ $$-19\beta_{5} - 340\beta_{4} + 499\beta_{3} - 1022\beta_{2} + 614$$ -19*b5 - 340*b4 + 499*b3 - 1022*b2 + 614

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$$.

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$1$$ $$-\beta_{6}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
126.1
 1.08066 − 1.87176i 0.904178 − 1.56608i −0.134432 + 0.232843i −0.547998 + 0.949161i −1.30241 + 2.25583i 1.08066 + 1.87176i 0.904178 + 1.56608i −0.134432 − 0.232843i −0.547998 − 0.949161i −1.30241 − 2.25583i
−1.08066 + 1.87176i 0.980433 1.69816i −1.33565 2.31341i 0 2.11903 + 3.67026i −1.53837 2.66453i 1.45089 −0.422497 0.731787i 0
126.2 −0.904178 + 1.56608i −0.929015 + 1.60910i −0.635076 1.09998i 0 −1.67999 2.90983i 2.08417 + 3.60988i −1.31983 −0.226138 0.391682i 0
126.3 0.134432 0.232843i −0.301414 + 0.522064i 0.963856 + 1.66945i 0 0.0810394 + 0.140364i −0.715471 1.23923i 1.05602 1.31830 + 2.28336i 0
126.4 0.547998 0.949161i 1.68234 2.91389i 0.399395 + 0.691773i 0 −1.84383 3.19362i 0.795836 + 1.37843i 3.06747 −4.16051 7.20621i 0
126.5 1.30241 2.25583i 0.0676602 0.117191i −2.39253 4.14398i 0 −0.176242 0.305261i −1.62616 2.81660i −7.25455 1.49084 + 2.58222i 0
276.1 −1.08066 1.87176i 0.980433 + 1.69816i −1.33565 + 2.31341i 0 2.11903 3.67026i −1.53837 + 2.66453i 1.45089 −0.422497 + 0.731787i 0
276.2 −0.904178 1.56608i −0.929015 1.60910i −0.635076 + 1.09998i 0 −1.67999 + 2.90983i 2.08417 3.60988i −1.31983 −0.226138 + 0.391682i 0
276.3 0.134432 + 0.232843i −0.301414 0.522064i 0.963856 1.66945i 0 0.0810394 0.140364i −0.715471 + 1.23923i 1.05602 1.31830 2.28336i 0
276.4 0.547998 + 0.949161i 1.68234 + 2.91389i 0.399395 0.691773i 0 −1.84383 + 3.19362i 0.795836 1.37843i 3.06747 −4.16051 + 7.20621i 0
276.5 1.30241 + 2.25583i 0.0676602 + 0.117191i −2.39253 + 4.14398i 0 −0.176242 + 0.305261i −1.62616 + 2.81660i −7.25455 1.49084 2.58222i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 126.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.e.d yes 10
5.b even 2 1 325.2.e.c 10
5.c odd 4 2 325.2.o.c 20
13.c even 3 1 inner 325.2.e.d yes 10
13.c even 3 1 4225.2.a.bn 5
13.e even 6 1 4225.2.a.bm 5
65.l even 6 1 4225.2.a.bo 5
65.n even 6 1 325.2.e.c 10
65.n even 6 1 4225.2.a.bp 5
65.q odd 12 2 325.2.o.c 20

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.e.c 10 5.b even 2 1
325.2.e.c 10 65.n even 6 1
325.2.e.d yes 10 1.a even 1 1 trivial
325.2.e.d yes 10 13.c even 3 1 inner
325.2.o.c 20 5.c odd 4 2
325.2.o.c 20 65.q odd 12 2
4225.2.a.bm 5 13.e even 6 1
4225.2.a.bn 5 13.c even 3 1
4225.2.a.bo 5 65.l even 6 1
4225.2.a.bp 5 65.n even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 8T_{2}^{8} + 2T_{2}^{7} + 52T_{2}^{6} + 5T_{2}^{5} + 97T_{2}^{4} - 60T_{2}^{3} + 141T_{2}^{2} - 36T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 8 T^{8} + \cdots + 9$$
$3$ $$T^{10} - 3 T^{9} + \cdots + 1$$
$5$ $$T^{10}$$
$7$ $$T^{10} + 2 T^{9} + \cdots + 9025$$
$11$ $$T^{10} + 3 T^{9} + \cdots + 9$$
$13$ $$T^{10} + 10 T^{9} + \cdots + 371293$$
$17$ $$T^{10} - 4 T^{9} + \cdots + 239121$$
$19$ $$T^{10} - 4 T^{9} + \cdots + 225$$
$23$ $$T^{10} - 15 T^{9} + \cdots + 281961$$
$29$ $$T^{10} - T^{9} + \cdots + 881721$$
$31$ $$(T^{5} - 57 T^{3} + \cdots + 225)^{2}$$
$37$ $$T^{10} - 17 T^{9} + \cdots + 826281$$
$41$ $$T^{10} + 6 T^{9} + \cdots + 50625$$
$43$ $$T^{10} - 12 T^{9} + \cdots + 14190289$$
$47$ $$(T^{5} - 12 T^{4} + \cdots - 2025)^{2}$$
$53$ $$(T^{5} + 8 T^{4} + \cdots - 6075)^{2}$$
$59$ $$T^{10} - 12 T^{9} + \cdots + 4782969$$
$61$ $$T^{10} + 5 T^{9} + \cdots + 403225$$
$67$ $$T^{10} + \cdots + 2567550241$$
$71$ $$T^{10} + \cdots + 101787921$$
$73$ $$(T^{5} - 8 T^{4} + \cdots - 10125)^{2}$$
$79$ $$(T^{5} + 14 T^{4} + \cdots + 16875)^{2}$$
$83$ $$(T^{5} - 7 T^{4} + \cdots + 53775)^{2}$$
$89$ $$T^{10} - 10 T^{9} + \cdots + 2537649$$
$97$ $$T^{10} + \cdots + 1683378841$$