LMFDB - Newform orbit 7225.2.a.bs

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.a (trivial)

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:	yes
Analytic conductor:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17$ x^12 - 4x^11 - 10x^10 + 52x^9 + 21x^8 - 232x^7 + 44x^6 + 424x^5 - 137x^4 - 316x^3 + 46x^2 + 92*x + 17
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$ -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.

$f(q)$	$=$	$q + \beta_1 q^{2} + (\beta_{5} + 1) q^{3} + (\beta_{7} + \beta_{6} + 1) q^{4} + (\beta_{11} + \beta_{9} - \beta_{7} + \cdots - 1) q^{6} + (\beta_{11} + \beta_{10} + \beta_1 + 1) q^{7} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \cdots + 1) q^{8}+ \cdots + ( - \beta_{11} - 5 \beta_{10} + \cdots - 3) q^{99}+O(q^{100})$ q + b1 * q^2 + (b5 + 1) * q^3 + (b7 + b6 + 1) * q^4 + (b11 + b9 - b7 - b6 + b5 + 2b1 - 1) q^6 + (b11 + b10 + b1 + 1) * q^7 + (-b10 - b9 + b7 + 2b6 - b5 - b4 + b3 + 1) q^8 + (b11 + b10 - b8 - b7 - b6 + b5 + b4 + b2 + b1 + 1) * q^9 + (-b11 - b10 - b9 + b7 + b6 - b4 - b2 - 1) * q^11 + (2b11 + b10 + 2b9 - 2b8 - b7 - b6 + b5 + b4 - 2b3 + b2 + b1 + 1) * q^12 + (-b11 - b10 + b4 - b3 + 1) * q^13 + (b11 + b10 - b7 + b6 - b3 + b2 + 3b1) q^14 + (b11 - b10 - b9 - b8 + b7 + 2b6 - 2b5 + b1 + 1) * q^16 + (3b11 + b10 + 2b9 - 2b7 - 2b6 + b4 + b3 + 3b1 - 1) q^18 + (-2b11 - b9 + b8 + b7 + 2b6 + b5 - b4 - b2) * q^19 + (2b11 + b8 - b7 - b6 + 2b5 + 2b1 + 2) q^21 + (-b11 - b10 + 2b7 + 2b6 - b4 - 3b1 + 2) q^22 + (b9 + b5 - b2 - b1 + 2) * q^23 + (3b11 + b10 + b9 - 4b8 - b7 - 2b6 - b4 - b3 + b2 + 2b1 - 1) * q^24 + (-2b11 - 2b10 - b9 + b8 + 2b7 + b6 - b5 - b4 + 4b3 - b2 - 2b1 + 2) q^26 + (-b11 - 2b9 + 2b8 - b7 + b6 + b5 + b2 + 3) * q^27 + (2b11 + b9 + b7 + b6 - 2b3 + 2b1 + 3) q^28 + (b11 + b10 + b8 + b5 + b4 - b3 + b1 - 1) * q^29 + (-b11 + 2b8 - b7 - b6 + b5 + b4 - b3 + b1 - 2) q^31 + (b11 - 2b8 + 2b7 + 3b6 - 2b5 + b4 + b3 + b2 + 2) * q^32 + (2b11 - 3b8 - 3b6 - 3b5 - b4 + b3 + b1 - 1) * q^33 + (b9 - b8 - b7 + 2b6 + b5 + b4 - 2b3 + b2 + 1) * q^36 + (-2b11 - b10 - 2b9 + 2b7 + 2b6 - b5 - b4 + 2b3 - 2b2 - 2b1 + 3) q^37 + (-b11 + 3b8 + b7 + b5 - b4 - b3 - b2 + 2) q^38 + (-b11 + b10 + 2b9 - b8 + b5 + b4 - b3 - b1 - 1) q^39 + (b10 + b6 + b5 + b4 + 2b3 - b2) q^41 + (3b11 + b10 + 2b9 - b8 - 3b7 + b6 + 3b5 + b4 - b3 + 2b2 + 5b1 - 1) * q^42 + (-2b11 - 2b9 + b8 + 3b6 - b5 - b4 - b3 - b1 + 1) q^43 + (2b11 + b9 - b8 - b7 - 2b6 - b5 + b2 + 3b1 - 2) q^44 + (b9 - b8 - 3b7 - 3b6 + 3b5 - b3 + b2 + 3b1 - 4) * q^46 + (b10 - b9 + b8 - b6 + b4 - b3 + 3) * q^47 + (4b11 + b10 + 2b9 - 4b8 - 2b6 - 2b4 - 2b3 + 2b1 - 3) q^48 + (b11 + b10 + b9 - 3b6 + 2b5 - 2b3 + 3b1 + 1) * q^49 + (-3b11 - b10 - 3b9 + b8 + 4b7 + 4b6 - 2b5 + 4b3 - b2 - 3b1 + 3) q^52 + (-4b11 - 2b9 + b8 + b7 + b6 + b5 - 2b3 - b1) q^53 + (b11 + b10 + 5b8 - b5 + b4 + b3 - 2b2 + 3b1 - 1) q^54 + (-3b10 - b9 - 3b8 + 3b7 + 3b6 - 3b4 + 2b3 + b1 + 2) * q^56 + (2b11 + b10 + b9 - 2b8 - 3b7 - 3b6 + 2b4 + 2b2 + 2b1 + 2) q^57 + (-b9 + 2b8 - 2b7 + 2b6 - b4 + 2b3 + b2 + b1 - 2) * q^58 + (2b11 + 2b9 - 2b8 - 2b7 - 4b6 + 2b5 + 2b4 - 2b3 + 2b2 + 4b1) * q^59 + (b11 - b10 + b8 - 3b6 + 3b5 - b4 + b3 + b1 - 1) * q^61 + (-3b11 - b9 + 4b8 + b5 + 2b3 - b2 - 4b1 + 2) * q^62 + (4b11 + b10 + b9 - 2b8 - 4b7 - 6b6 + 2b5 + 2b4 + 2b3 + b2 + 5b1 + 4) * q^63 + (-b10 - b9 + 2b7 + 2b6 - 2b5 - b4 + 5b3 + 2b1 + 2) q^64 + (b10 + b9 - 6b8 + 2b7 + 2b6 - 2b5 + b4 - 3b3 + 2b2 - 4b1 + 3) q^66 + (-b11 - b10 + 4b7 + 2b6 - b4 + b3 - 2b1 + 1) q^67 + (b11 + b10 + b9 - b6 + 2b4 + b1 + 4) q^69 + (-2b11 - b10 - b9 + 3b8 + b7 + b6 + 2b5 - b4 + b3 - b1 + 2) q^71 + (-b11 - 2b10 - 2b9 - b8 + 3b7 + b5 - 3b4 - b3 - b2 - b1) * q^72 + (-2b11 + b9 + 2b8 + 2b5 + 2b3 - b2 + b1 + 1) * q^73 + (-4b11 - 2b10 - 2b9 + 2b8 + 2b7 + 3b6 - 2b5 + 2b3 - 2b1) q^74 + (b8 - 2b7 - 2b6 - 2b3 + 2b2 + 2b1) q^76 + (-2b11 - 3b10 - 3b9 + b8 + 2b7 + 3b6 - 3b4 - b3 - 2b1 - 1) q^77 + (b9 - 2b7 - 6b6 + 2b5 - b4 - b2 + b1 - 2) q^78 + (b10 - 3b8 + b7 + 2b6 - 2b5 + 2b4 - 2b2 - 3) q^79 + (-3b11 - 2b9 + 2b8 - b7 + 2b6 + 2b5 - b4 + b3 - 3b1 + 4) * q^81 + (2b8 - 3b7 - b6 + b5 + 2b4 + 2b3 + b2 + b1 - 1) * q^82 + (-b11 + b10 + b9 - 4b8 - b7 + b6 + b5 + 2b4 + 2b3 - b2 + 2b1 + 3) * q^83 + (8b11 + 3b10 + 6b9 - 6b8 - b7 - 2b6 + b5 + 2b4 - 3b3 + b2 + 6b1 + 1) * q^84 + (b10 - b9 + 4b8 + 2b7 - 2b5 - b4 - b3 - 2b2 - 1) * q^86 + (4b11 + 3b10 + 2b9 - 3b7 - 3b6 + 3b4 - b3 + b2 + 3b1 + 2) q^87 + (4b11 + 3b10 + b9 - 4b8 - 2b7 - 2b6 + 3b4 - 2b3 + b2 + 5b1 + 3) * q^88 + (-2b11 + b8 + b7 + 3b6 - 2b3) q^89 + (2b11 + 2b10 + 4b6 - 2b5 + 2b4 + b3 - 6) q^91 + (5b11 + 3b10 + 5b9 - 2b8 - 2b7 - 8b6 + 4b5 + 2b4 - 4b3 + b2) q^92 + (2b10 + 2b9 + 2b8 - 3b7 - 2b6 + 3b4 + b3 + 2b1 - 3) q^93 + (-3b11 - 2b9 + 4b8 - b7 + 2b6 - 2b5 - b4 + 3b3 + b1 - 3) * q^94 + (2b11 + b10 + 4b9 - 3b8 - b7 + b6 + 4b5 - 7b3 + 2b2 - 1) * q^96 + (-5b11 - 2b10 - 3b9 + 2b8 + 5b7 + 3b6 + b5 - 4b4 - 4b2 - 3b1 + 4) q^97 + (b10 + 2b9 - b8 - b7 + 3b5 - 2b4 - 2b3 + b2 + 3b1 + 3) q^98 + (-b11 - 5b10 - 3b9 + 4b8 + 5b7 - 2b5 - 7b4 + 4b3 - 3b2 - 4b1 - 3) q^99
$\operatorname{Tr}(f)(q)$	$=$	$12 q + 4 q^{2} + 8 q^{3} + 12 q^{4} - 8 q^{6} + 16 q^{7} + 12 q^{8} + 12 q^{9} - 16 q^{11} + 16 q^{12} + 8 q^{13} + 16 q^{14} + 12 q^{16} - 4 q^{18} + 16 q^{21} + 16 q^{22} + 16 q^{23} + 16 q^{26} + 32 q^{27}+ \cdots - 56 q^{99}+O(q^{100})$ 12 * q + 4 * q^2 + 8 * q^3 + 12 * q^4 - 8 * q^6 + 16 * q^7 + 12 * q^8 + 12 * q^9 - 16 * q^11 + 16 * q^12 + 8 * q^13 + 16 * q^14 + 12 * q^16 - 4 * q^18 + 16 * q^21 + 16 * q^22 + 16 * q^23 + 16 * q^26 + 32 * q^27 + 40 * q^28 - 16 * q^29 - 24 * q^31 + 28 * q^32 + 12 * q^36 + 24 * q^37 + 24 * q^38 - 8 * q^39 - 8 * q^41 + 16 * q^43 - 8 * q^44 - 40 * q^46 + 32 * q^47 - 24 * q^48 + 20 * q^49 + 24 * q^52 - 8 * q^54 + 24 * q^56 + 32 * q^57 - 16 * q^58 + 8 * q^59 - 24 * q^61 + 8 * q^62 + 48 * q^63 + 36 * q^64 + 40 * q^66 + 8 * q^67 + 48 * q^69 + 16 * q^71 - 12 * q^72 + 16 * q^73 + 16 * q^76 - 24 * q^77 - 24 * q^78 - 40 * q^79 + 36 * q^81 - 16 * q^82 + 40 * q^83 + 32 * q^84 - 8 * q^86 + 32 * q^87 + 48 * q^88 + 8 * q^89 - 72 * q^91 - 8 * q^92 - 24 * q^93 - 16 * q^94 - 8 * q^96 + 32 * q^97 + 60 * q^98 - 56 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{12} - 4 x^{11} - 10 x^{10} + 52 x^{9} + 21 x^{8} - 232 x^{7} + 44 x^{6} + 424 x^{5} - 137 x^{4} + \cdots + 17$ x^12 - 4*x^11 - 10*x^10 + 52*x^9 + 21*x^8 - 232*x^7 + 44*x^6 + 424*x^5 - 137*x^4 - 316*x^3 + 46*x^2 + 92*x + 17 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( 54 \nu^{11} - 195 \nu^{10} - 568 \nu^{9} + 2455 \nu^{8} + 1476 \nu^{7} - 10314 \nu^{6} + 1071 \nu^{5} + \cdots + 1470 ) / 41$ (54v^11 - 195v^10 - 568v^9 + 2455v^8 + 1476v^7 - 10314v^6 + 1071v^5 + 16568v^4 - 5884v^3 - 8961v^2 + 2901*v + 1470) / 41
$\beta_{3}$	$=$	$( - 141 \nu^{11} + 352 \nu^{10} + 1925 \nu^{9} - 4399 \nu^{8} - 9389 \nu^{7} + 18116 \nu^{6} + \cdots + 1191 ) / 41$ (-141v^11 + 352v^10 + 1925v^9 - 4399v^8 - 9389v^7 + 18116v^6 + 20184v^5 - 27485v^4 - 20329v^3 + 11023v^2 + 8504*v + 1191) / 41
$\beta_{4}$	$=$	$( - 212 \nu^{11} + 515 \nu^{10} + 2933 \nu^{9} - 6428 \nu^{8} - 14596 \nu^{7} + 26388 \nu^{6} + \cdots + 2356 ) / 41$ (-212v^11 + 515v^10 + 2933v^9 - 6428v^8 - 14596v^7 + 26388v^6 + 32299v^5 - 39646v^4 - 33533v^3 + 14990v^2 + 14163*v + 2356) / 41
$\beta_{5}$	$=$	$( - 232 \nu^{11} + 569 \nu^{10} + 3195 \nu^{9} - 7111 \nu^{8} - 15785 \nu^{7} + 29265 \nu^{6} + \cdots + 2217 ) / 41$ (-232v^11 + 569v^10 + 3195v^9 - 7111v^8 - 15785v^7 + 29265v^6 + 34540v^5 - 44241v^4 - 35311v^3 + 17293v^2 + 14642*v + 2217) / 41
$\beta_{6}$	$=$	$( - 332 \nu^{11} + 839 \nu^{10} + 4505 \nu^{9} - 10485 \nu^{8} - 21771 \nu^{7} + 43158 \nu^{6} + \cdots + 2793 ) / 41$ (-332v^11 + 839v^10 + 4505v^9 - 10485v^8 - 21771v^7 + 43158v^6 + 46196v^5 - 65289v^4 - 45964v^3 + 25856v^2 + 19292*v + 2793) / 41
$\beta_{7}$	$=$	$( 332 \nu^{11} - 839 \nu^{10} - 4505 \nu^{9} + 10485 \nu^{8} + 21771 \nu^{7} - 43158 \nu^{6} + \cdots - 2916 ) / 41$ (332v^11 - 839v^10 - 4505v^9 + 10485v^8 + 21771v^7 - 43158v^6 - 46196v^5 + 65289v^4 + 45964v^3 - 25815v^2 - 19292*v - 2916) / 41
$\beta_{8}$	$=$	$- 8 \nu^{11} + 19 \nu^{10} + 112 \nu^{9} - 237 \nu^{8} - 568 \nu^{7} + 971 \nu^{6} + 1294 \nu^{5} + \cdots + 108$ -8v^11 + 19v^10 + 112v^9 - 237v^8 - 568v^7 + 971v^6 + 1294v^5 - 1449v^4 - 1388v^3 + 524v^2 + 590*v + 108
$\beta_{9}$	$=$	$( 362 \nu^{11} - 879 \nu^{10} - 5021 \nu^{9} + 10997 \nu^{8} + 25092 \nu^{7} - 45321 \nu^{6} + \cdots - 3999 ) / 41$ (362v^11 - 879v^10 - 5021v^9 + 10997v^8 + 25092v^7 - 45321v^6 - 55892v^5 + 68594v^4 + 58348v^3 - 26502v^2 - 24418*v - 3999) / 41
$\beta_{10}$	$=$	$( - 391 \nu^{11} + 986 \nu^{10} + 5323 \nu^{9} - 12342 \nu^{8} - 25871 \nu^{7} + 50942 \nu^{6} + \cdots + 3328 ) / 41$ (-391v^11 + 986v^10 + 5323v^9 - 12342v^8 - 25871v^7 + 50942v^6 + 55433v^5 - 77481v^4 - 55838v^3 + 31139v^2 + 23573*v + 3328) / 41
$\beta_{11}$	$=$	$( - 489 \nu^{11} + 1185 \nu^{10} + 6779 \nu^{9} - 14799 \nu^{8} - 33866 \nu^{7} + 60804 \nu^{6} + \cdots + 5644 ) / 41$ (-489v^11 + 1185v^10 + 6779v^9 - 14799v^8 - 33866v^7 + 60804v^6 + 75479v^5 - 91448v^4 - 79056v^3 + 34564v^2 + 33296*v + 5644) / 41

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{7} + \beta_{6} + 3$ b7 + b6 + 3
$\nu^{3}$	$=$	$-\beta_{10} - \beta_{9} + \beta_{7} + 2\beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + 4\beta _1 + 1$ -b10 - b9 + b7 + 2b6 - b5 - b4 + b3 + 4b1 + 1
$\nu^{4}$	$=$	$\beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + 7\beta_{7} + 8\beta_{6} - 2\beta_{5} + \beta _1 + 15$ b11 - b10 - b9 - b8 + 7b7 + 8b6 - 2*b5 + b1 + 15
$\nu^{5}$	$=$	$\beta_{11} - 8 \beta_{10} - 8 \beta_{9} - 2 \beta_{8} + 10 \beta_{7} + 19 \beta_{6} - 10 \beta_{5} + \cdots + 10$ b11 - 8b10 - 8b9 - 2b8 + 10b7 + 19b6 - 10b5 - 7b4 + 9b3 + b2 + 20*b1 + 10
$\nu^{6}$	$=$	$10 \beta_{11} - 11 \beta_{10} - 11 \beta_{9} - 10 \beta_{8} + 48 \beta_{7} + 58 \beta_{6} - 22 \beta_{5} + \cdots + 88$ 10b11 - 11b10 - 11b9 - 10b8 + 48b7 + 58b6 - 22b5 - b4 + 5b3 + 12*b1 + 88
$\nu^{7}$	$=$	$9 \beta_{11} - 58 \beta_{10} - 59 \beta_{9} - 21 \beta_{8} + 83 \beta_{7} + 151 \beta_{6} - 80 \beta_{5} + \cdots + 91$ 9b11 - 58b10 - 59b9 - 21b8 + 83b7 + 151b6 - 80b5 - 43b4 + 67b3 + 10b2 + 113*b1 + 91
$\nu^{8}$	$=$	$71 \beta_{11} - 98 \beta_{10} - 99 \beta_{9} - 72 \beta_{8} + 335 \beta_{7} + 421 \beta_{6} - 189 \beta_{5} + \cdots + 560$ 71b11 - 98b10 - 99b9 - 72b8 + 335b7 + 421b6 - 189b5 - 16b4 + 71b3 - b2 + 107b1 + 560
$\nu^{9}$	$=$	$55 \beta_{11} - 417 \beta_{10} - 436 \beta_{9} - 158 \beta_{8} + 657 \beta_{7} + 1146 \beta_{6} + \cdots + 772$ 55b11 - 417b10 - 436b9 - 158b8 + 657b7 + 1146b6 - 606b5 - 264b4 + 484b3 + 72b2 + 681*b1 + 772
$\nu^{10}$	$=$	$432 \beta_{11} - 810 \beta_{10} - 841 \beta_{9} - 458 \beta_{8} + 2378 \beta_{7} + 3087 \beta_{6} + \cdots + 3733$ 432b11 - 810b10 - 841b9 - 458b8 + 2378b7 + 3087b6 - 1507b5 - 173b4 + 718b3 - 17b2 + 850*b1 + 3733
$\nu^{11}$	$=$	$248 \beta_{11} - 3015 \beta_{10} - 3254 \beta_{9} - 1032 \beta_{8} + 5096 \beta_{7} + 8565 \beta_{6} + \cdots + 6241$ 248b11 - 3015b10 - 3254b9 - 1032b8 + 5096b7 + 8565b6 - 4536b5 - 1660b4 + 3510b3 + 449b2 + 4250*b1 + 6241

Display $\beta_i$ in terms of $\nu^j$

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}

1.1

−2.35190
−2.04505
−1.43840
−0.747914
−0.360254
−0.301687
0.962871
1.55041
1.55555
1.80583
2.63994
2.73061

−2.35190

1.56935

3.53144

−3.69096

3.58212

−3.60181

−0.537139

1.2

−2.04505

3.19566

2.18224

−6.53528

−1.17743

−0.372688

7.21221

1.3

−1.43840

−0.109907

0.0689897

0.158090

0.695085

2.77756

−2.98792

1.4

−0.747914

3.07503

−1.44062

−2.29986

3.23262

2.57329

6.45581

1.5

−0.360254

−0.0542373

−1.87022

0.0195392

−0.298718

1.39426

−2.99706

1.6

−0.301687

−1.06101

−1.90899

0.320094

−2.50984

1.17929

−1.87425

1.7

0.962871

−2.64897

−1.07288

−2.55062

3.09463

−2.95879

4.01706

1.8

1.55041

1.14040

0.403772

1.76809

−3.74146

−2.47481

−1.69949

1.9

1.55555

3.00797

0.419729

4.67904

3.45467

−2.45819

6.04787

1.10

1.80583

−0.687917

1.26102

−1.24226

4.34193

−1.33447

−2.52677

1.11

2.63994

2.12055

4.96928

5.59814

4.06194

7.83873

1.49675

1.12

2.73061

−1.54691

5.45623

−4.22400

1.26445

9.43761

−0.607075

Atkin-Lehner signs

$p$	Sign
$5$	$+1$
$17$	$-1$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	7225.2.a.bs		12
5.b	even	2	1		1445.2.a.p		12
17.b	even	2	1		7225.2.a.bq		12
17.e	odd	16	2		425.2.m.b		24
85.c	even	2	1		1445.2.a.q		12
85.j	even	4	2		1445.2.d.j		24
85.o	even	16	2		425.2.n.f		24
85.p	odd	16	2		85.2.l.a	✓	24
85.r	even	16	2		425.2.n.c		24
255.be	even	16	2		765.2.be.b		24

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
85.2.l.a	✓	24	85.p	odd	16	2
425.2.m.b		24	17.e	odd	16	2
425.2.n.c		24	85.r	even	16	2
425.2.n.f		24	85.o	even	16	2
765.2.be.b		24	255.be	even	16	2
1445.2.a.p		12	5.b	even	2	1
1445.2.a.q		12	85.c	even	2	1
1445.2.d.j		24	85.j	even	4	2
7225.2.a.bq		12	17.b	even	2	1
7225.2.a.bs		12	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(\Gamma_0(7225))$ :

T_{2}^{12} - 4 T_{2}^{11} - 10 T_{2}^{10} + 52 T_{2}^{9} + 21 T_{2}^{8} - 232 T_{2}^{7} + 44 T_{2}^{6} + \cdots + 17

T_{3}^{12} - 8 T_{3}^{11} + 8 T_{3}^{10} + 80 T_{3}^{9} - 186 T_{3}^{8} - 176 T_{3}^{7} + 680 T_{3}^{6} + \cdots + 2

T_{7}^{12} - 16 T_{7}^{11} + 76 T_{7}^{10} + 72 T_{7}^{9} - 1680 T_{7}^{8} + 4048 T_{7}^{7} + \cdots + 6338

T_{11}^{12} + 16 T_{11}^{11} + 60 T_{11}^{10} - 232 T_{11}^{9} - 1658 T_{11}^{8} + 480 T_{11}^{7} + \cdots - 1598

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{12} - 4 T^{11} + \cdots + 17$ T^12 - 4T^11 - 10T^10 + 52T^9 + 21T^8 - 232T^7 + 44T^6 + 424T^5 - 137T^4 - 316T^3 + 46T^2 + 92*T + 17
$3$	$T^{12} - 8 T^{11} + \cdots + 2$ T^12 - 8T^11 + 8T^10 + 80T^9 - 186T^8 - 176T^7 + 680T^6 + 56T^5 - 854T^4 + 16T^3 + 356T^2 + 56*T + 2
$5$	$T^{12}$ T^12
$7$	$T^{12} - 16 T^{11} + \cdots + 6338$ T^12 - 16T^11 + 76T^10 + 72T^9 - 1680T^8 + 4048T^7 + 4204T^6 - 28888T^5 + 26714T^4 + 27608T^3 - 45684T^2 + 6056*T + 6338
$11$	$T^{12} + 16 T^{11} + \cdots - 1598$ T^12 + 16T^11 + 60T^10 - 232T^9 - 1658T^8 + 480T^7 + 14580T^6 + 4176T^5 - 55616T^4 - 7808T^3 + 83664T^2 - 26032*T - 1598
$13$	$T^{12} - 8 T^{11} + \cdots - 23932$ T^12 - 8T^11 - 60T^10 + 520T^9 + 1114T^8 - 10856T^7 - 7068T^6 + 83680T^5 + 4272T^4 - 229968T^3 + 26960T^2 + 188128*T - 23932
$17$	$T^{12}$ T^12
$19$	$T^{12} - 92 T^{10} + \cdots - 3008$ T^12 - 92T^10 + 16T^9 + 2348T^8 - 2624T^7 - 21184T^6 + 42624T^5 + 40048T^4 - 155136T^3 + 105664T^2 - 5376T - 3008
$23$	$T^{12} - 16 T^{11} + \cdots - 2206$ T^12 - 16T^11 + 12T^10 + 888T^9 - 3650T^8 - 6704T^7 + 45872T^6 + 1344T^5 - 125762T^4 - 56016T^3 + 39476T^2 + 14840*T - 2206
$29$	$T^{12} + 16 T^{11} + \cdots - 6008$ T^12 + 16T^11 - 4T^10 - 1360T^9 - 7570T^8 + 3872T^7 + 147712T^6 + 460736T^5 + 546220T^4 + 241024T^3 - 6256T^2 - 31424*T - 6008
$31$	$T^{12} + 24 T^{11} + \cdots + 352546$ T^12 + 24T^11 + 168T^10 - 152T^9 - 6448T^8 - 21536T^7 + 18564T^6 + 190864T^5 + 134176T^4 - 502896T^3 - 503840T^2 + 467920*T + 352546
$37$	$T^{12} - 24 T^{11} + \cdots + 26248$ T^12 - 24T^11 + 100T^10 + 1304T^9 - 8202T^8 - 27200T^7 + 166392T^6 + 334912T^5 - 970300T^4 - 2334912T^3 - 1073760T^2 - 59488*T + 26248
$41$	$T^{12} + 8 T^{11} + \cdots - 427904$ T^12 + 8T^11 - 152T^10 - 992T^9 + 5200T^8 + 26432T^7 - 68128T^6 - 221056T^5 + 260096T^4 + 779264T^3 - 139520T^2 - 958976*T - 427904
$43$	$T^{12} - 16 T^{11} + \cdots + 63172$ T^12 - 16T^11 - 72T^10 + 1896T^9 - 446T^8 - 68256T^7 + 95300T^6 + 712176T^5 - 1402148T^4 - 1843152T^3 + 5300088T^2 - 2495344*T + 63172
$47$	$T^{12} - 32 T^{11} + \cdots + 45712388$ T^12 - 32T^11 + 312T^10 + 160T^9 - 20586T^8 + 100056T^7 + 176452T^6 - 2518016T^5 + 4426592T^4 + 10892048T^3 - 35225424T^2 + 408896*T + 45712388
$53$	$T^{12} - 340 T^{10} + \cdots - 8342512$ T^12 - 340T^10 - 552T^9 + 34572T^8 + 103632T^7 - 929528T^6 - 3557472T^5 + 2929672T^4 + 18540064T^3 + 1552704T^2 - 25680960T - 8342512
$59$	$T^{12} + \cdots + 296882176$ T^12 - 8T^11 - 264T^10 + 1824T^9 + 25168T^8 - 139008T^7 - 1074176T^6 + 4081664T^5 + 21752576T^4 - 42518528T^3 - 177895424T^2 + 84303872*T + 296882176
$61$	$T^{12} + 24 T^{11} + \cdots - 12501472$ T^12 + 24T^11 - 32T^10 - 4528T^9 - 24776T^8 + 152768T^7 + 1198704T^6 - 1845504T^5 - 18182512T^4 + 8825984T^3 + 73022400T^2 + 33484032*T - 12501472
$67$	$T^{12} + \cdots - 961394684$ T^12 - 8T^11 - 300T^10 + 2376T^9 + 29802T^8 - 229096T^7 - 1294284T^6 + 9713504T^5 + 23898224T^4 - 185423152T^3 - 104248400T^2 + 1276344416*T - 961394684
$71$	$T^{12} - 16 T^{11} + \cdots + 1907842$ T^12 - 16T^11 - 44T^10 + 2200T^9 - 11870T^8 - 26704T^7 + 438988T^6 - 1225504T^5 - 447508T^4 + 6242304T^3 - 4106280T^2 - 7287232*T + 1907842
$73$	$T^{12} - 16 T^{11} + \cdots + 15430176$ T^12 - 16T^11 - 180T^10 + 2432T^9 + 17076T^8 - 106048T^7 - 709424T^6 + 1465504T^5 + 12013184T^4 + 699648T^3 - 62005824T^2 - 57939840*T + 15430176
$79$	$T^{12} + \cdots + 23846133218$ T^12 + 40T^11 + 312T^10 - 6632T^9 - 109828T^8 + 19136T^7 + 8681748T^6 + 36349952T^5 - 194513172T^4 - 1447213792T^3 - 2787128T^2 + 15022243824*T + 23846133218
$83$	$T^{12} + \cdots - 3068795644$ T^12 - 40T^11 + 164T^10 + 11312T^9 - 124422T^8 - 741112T^7 + 13028748T^6 + 6486784T^5 - 478529200T^4 + 619550304T^3 + 5797401776T^2 - 12208069952*T - 3068795644
$89$	$T^{12} - 8 T^{11} + \cdots + 186436$ T^12 - 8T^11 - 176T^10 + 1776T^9 + 1416T^8 - 40112T^7 + 9884T^6 + 312336T^5 - 80816T^4 - 867552T^3 + 131184T^2 + 646752*T + 186436
$97$	$T^{12} + \cdots - 3976256888$ T^12 - 32T^11 - 132T^10 + 11448T^9 - 28570T^8 - 1426608T^7 + 4901992T^6 + 78430016T^5 - 190414124T^4 - 1778107072T^3 + 672188992T^2 + 6584458336*T - 3976256888
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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

$n$ :		e.g. 2-40 or 990-1000
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