# Properties

 Label 325.2.f.d Level $325$ Weight $2$ Character orbit 325.f Analytic conductor $2.595$ Analytic rank $0$ Dimension $16$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(18,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(4))

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

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

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

chi := DirichletCharacter("325.18");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.f (of order $$4$$, 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: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{16} + 111x^{12} + 329x^{8} + 168x^{4} + 16$$ x^16 + 111*x^12 + 329*x^8 + 168*x^4 + 16 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{4}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q - \beta_{13} q^{2} + \beta_{3} q^{3} + \beta_1 q^{4} + ( - \beta_{8} + \beta_{4} + 1) q^{6} + \beta_{12} q^{7} + (\beta_{14} + \beta_{11} + \beta_{5}) q^{8} + (\beta_{15} + \beta_{7} + \beta_{6} + 1) q^{9}+O(q^{10})$$ q - b13 * q^2 + b3 * q^3 + b1 * q^4 + (-b8 + b4 + 1) * q^6 + b12 * q^7 + (b14 + b11 + b5) * q^8 + (b15 + b7 + b6 + 1) * q^9 $$q - \beta_{13} q^{2} + \beta_{3} q^{3} + \beta_1 q^{4} + ( - \beta_{8} + \beta_{4} + 1) q^{6} + \beta_{12} q^{7} + (\beta_{14} + \beta_{11} + \beta_{5}) q^{8} + (\beta_{15} + \beta_{7} + \beta_{6} + 1) q^{9} + (\beta_{15} - \beta_{9} - \beta_{7} + \cdots - 1) q^{11}+ \cdots + ( - 4 \beta_{15} - \beta_{8} + \cdots + 5) q^{99}+O(q^{100})$$ q - b13 * q^2 + b3 * q^3 + b1 * q^4 + (-b8 + b4 + 1) * q^6 + b12 * q^7 + (b14 + b11 + b5) * q^8 + (b15 + b7 + b6 + 1) * q^9 + (b15 - b9 - b7 + b1 - 1) * q^11 + (-b13 - b12) * q^12 + (-b13 + b10 - b5 + b3 + b2) * q^13 + (b15 + 2*b4) * q^14 + (-b7 + b6 - b1 - 1) * q^16 + (-b14 + b13 + b12 - b11 - b10 + b3) * q^17 + (-2*b14 - b10 + 2*b5 + b3 - b2) * q^18 + (-b15 + 2*b9 + 2*b4 - b1 - 2) * q^19 + (-b9 - b4 + 1) * q^21 + (3*b13 - 3*b12 + b11 - b10 + b5 - b2) * q^22 + (b13 - b12 + b11 - b10 - 3*b2) * q^23 + (-b15 - 2*b8 + b1) * q^24 + (2*b15 - b9 - b8 + b7 + 2*b1 + 1) * q^26 + (-b11 + b10 - 2*b5 + b2) * q^27 + (-b14 - 2*b12 - b10 + b5) * q^28 + (-2*b15 - b9 + b8 - b7 - b6 - b4 - 1) * q^29 + (2*b15 + b6 - 2*b1) * q^31 + (b11 + b3 + b2) * q^32 + (-b14 - b5 - 2*b3 - 2*b2) * q^33 + (-b15 - b8 - b6 + 3*b4 + b1 + 3) * q^34 + (-3*b15 + b9 - b8 + 2*b4) * q^36 + (b14 + 3*b12 + 3*b10 - b5 - 3*b3 + 3*b2) * q^37 + (-b11 + b10 - 2*b5 + 4*b2) * q^38 + (b15 + b9 + 2*b6 - 3*b4 - 2*b1 - 2) * q^39 + (-2*b15 + b8 - 2*b6 + 2*b1) * q^41 + (-b13 + b12 - 2*b2) * q^42 + (2*b5 + b2) * q^43 + (-3*b15 - b9 - 3*b7 - 5*b4 - 3*b1 + 2) * q^44 + (-2*b15 + 3*b9 + b4 - 2*b1 - 1) * q^46 + (b14 + 2*b12 - b10 - b5) * q^47 + (2*b14 + b11 + b10 - 4*b3) * q^48 + (-b1 - 5) * q^49 + (2*b15 - 2*b9 + 2*b8 + b7 + b6 - 4*b4 + 1) * q^51 + (b13 - 3*b12 + 2*b11 + 3*b5 - b2) * q^52 + (3*b14 - b13 - b12 - b11 - b10 - 2*b3) * q^53 + (3*b15 - b9 + 2*b7 - b4 + 3*b1 + 3) * q^54 + (-3*b15 - b7 - b6 - 1) * q^56 + (4*b14 - 4*b13 + 2*b11 + 4*b5 - 2*b3 - 2*b2) * q^57 + (3*b14 + 3*b12 + 2*b10 - 3*b5 + b3 - b2) * q^58 + (-b15 + 2*b8 - 3*b6 - 3*b4 + b1 - 3) * q^59 + (3*b9 + 3*b8 + 2*b7 - 2*b6 - 2*b1 - 1) * q^61 + (-5*b14 - 3*b13 - 3*b12 - 2*b11 - 2*b10 + b3) * q^62 + (-2*b14 - b11 - 2*b5 + b3 + b2) * q^63 + (-b9 - b8 - 2*b7 + 2*b6 - 3*b1) * q^64 + (2*b9 + 2*b8 + b7 - b6 + 2*b1 - 3) * q^66 + (-b14 + 4*b13 - 3*b11 - b5 + b3 + b2) * q^67 + (b14 - b11 - b10 - b3) * q^68 + (b9 + b8 + 3*b7 - 3*b6 + 3*b1 + 8) * q^69 + (-b15 - 3*b8 + b1) * q^71 + (-b14 + 4*b12 + b10 + b5) * q^72 + (b14 + 3*b13 - 3*b11 + b5 + 2*b3 + 2*b2) * q^73 + (8*b15 - 3*b9 + 3*b8 + b7 + b6 + 1) * q^74 + (b15 + 2*b7 + b1 + 2) * q^76 + (-b14 - 3*b13 - 3*b12 - b11 - b10 + b3) * q^77 + (-5*b14 + 3*b12 - 2*b11 - b10 - b5 + 2*b3 + 2*b2) * q^78 + (b15 - b9 + b8 + b7 + b6 + 5*b4 + 1) * q^79 + (2*b9 + 2*b8 + 2*b7 - 2*b6 - 1) * q^81 + (6*b14 + 2*b13 + 2*b12 + 2*b11 + 2*b10) * q^82 + (-b14 + 5*b12 + 4*b10 + b5 - 2*b3 + 2*b2) * q^83 + (b15 + 2*b4 + b1 - 2) * q^84 + (-2*b15 - b9 - 2*b7 - b4 - 2*b1 - 1) * q^86 + (4*b13 - 4*b12 - 2*b5 - 3*b2) * q^87 + (-2*b13 + 2*b12 - b11 + b10 - 7*b5 - b2) * q^88 + (2*b15 + 3*b8 - 2*b6 - 5*b4 - 2*b1 - 5) * q^89 + (2*b15 - b9 + b8 + b6 - 2*b1) * q^91 + (-b13 + b12 - 4*b5) * q^92 + (b14 + 3*b12 + b10 - b5 - 2*b3 + 2*b2) * q^93 + (3*b15 + b7 + b6 + 4*b4 + 1) * q^94 + (b15 + 2*b6 - 4*b4 - b1 - 4) * q^96 + (-2*b14 + 3*b13 - b11 - 2*b5 - 2*b3 - 2*b2) * q^97 + (-b14 + 3*b13 - b11 - b5) * q^98 + (-4*b15 - b8 - b6 + 5*b4 + 4*b1 + 5) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16 q - 8 q^{4} + 12 q^{6}+O(q^{10})$$ 16 * q - 8 * q^4 + 12 * q^6 $$16 q - 8 q^{4} + 12 q^{6} - 20 q^{11} - 8 q^{16} - 16 q^{19} + 12 q^{21} - 16 q^{24} - 16 q^{26} + 8 q^{31} + 44 q^{34} - 28 q^{39} + 4 q^{41} + 76 q^{44} + 12 q^{46} - 72 q^{49} + 4 q^{54} - 24 q^{59} + 24 q^{61} + 16 q^{64} - 48 q^{66} + 112 q^{69} - 20 q^{71} + 8 q^{76} - 40 q^{84} + 12 q^{86} - 36 q^{89} + 8 q^{91} - 72 q^{96} + 52 q^{99}+O(q^{100})$$ 16 * q - 8 * q^4 + 12 * q^6 - 20 * q^11 - 8 * q^16 - 16 * q^19 + 12 * q^21 - 16 * q^24 - 16 * q^26 + 8 * q^31 + 44 * q^34 - 28 * q^39 + 4 * q^41 + 76 * q^44 + 12 * q^46 - 72 * q^49 + 4 * q^54 - 24 * q^59 + 24 * q^61 + 16 * q^64 - 48 * q^66 + 112 * q^69 - 20 * q^71 + 8 * q^76 - 40 * q^84 + 12 * q^86 - 36 * q^89 + 8 * q^91 - 72 * q^96 + 52 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} + 111x^{12} + 329x^{8} + 168x^{4} + 16$$ :

 $$\beta_{1}$$ $$=$$ $$( -357\nu^{12} - 39679\nu^{8} - 123305\nu^{4} - 62220 ) / 12928$$ (-357*v^12 - 39679*v^8 - 123305*v^4 - 62220) / 12928 $$\beta_{2}$$ $$=$$ $$( 201\nu^{13} + 22123\nu^{9} + 45469\nu^{5} - 4676\nu ) / 6464$$ (201*v^13 + 22123*v^9 + 45469*v^5 - 4676*v) / 6464 $$\beta_{3}$$ $$=$$ $$( 447\nu^{15} + 48813\nu^{11} + 58571\nu^{7} - 106780\nu^{3} ) / 25856$$ (447*v^15 + 48813*v^11 + 58571*v^7 - 106780*v^3) / 25856 $$\beta_{4}$$ $$=$$ $$( 357\nu^{14} + 39679\nu^{10} + 123305\nu^{6} + 88076\nu^{2} ) / 25856$$ (357*v^14 + 39679*v^10 + 123305*v^6 + 88076*v^2) / 25856 $$\beta_{5}$$ $$=$$ $$( -1447\nu^{13} - 160357\nu^{9} - 446803\nu^{5} - 128452\nu ) / 25856$$ (-1447*v^13 - 160357*v^9 - 446803*v^5 - 128452*v) / 25856 $$\beta_{6}$$ $$=$$ $$( 1537 \nu^{14} + 1466 \nu^{12} + 169491 \nu^{10} + 161998 \nu^{8} + 382069 \nu^{6} + 400386 \nu^{4} + \cdots + 8024 ) / 51712$$ (1537*v^14 + 1466*v^12 + 169491*v^10 + 161998*v^8 + 382069*v^6 + 400386*v^4 - 66404*v^2 + 8024) / 51712 $$\beta_{7}$$ $$=$$ $$( 1537 \nu^{14} - 1466 \nu^{12} + 169491 \nu^{10} - 161998 \nu^{8} + 382069 \nu^{6} - 400386 \nu^{4} + \cdots - 59736 ) / 51712$$ (1537*v^14 - 1466*v^12 + 169491*v^10 - 161998*v^8 + 382069*v^6 - 400386*v^4 - 66404*v^2 - 59736) / 51712 $$\beta_{8}$$ $$=$$ $$( 201\nu^{14} + 201\nu^{12} + 22123\nu^{10} + 22123\nu^{8} + 45469\nu^{6} + 45469\nu^{4} - 4676\nu^{2} + 1788 ) / 6464$$ (201*v^14 + 201*v^12 + 22123*v^10 + 22123*v^8 + 45469*v^6 + 45469*v^4 - 4676*v^2 + 1788) / 6464 $$\beta_{9}$$ $$=$$ $$( -201\nu^{14} + 201\nu^{12} - 22123\nu^{10} + 22123\nu^{8} - 45469\nu^{6} + 45469\nu^{4} + 4676\nu^{2} + 1788 ) / 6464$$ (-201*v^14 + 201*v^12 - 22123*v^10 + 22123*v^8 - 45469*v^6 + 45469*v^4 + 4676*v^2 + 1788) / 6464 $$\beta_{10}$$ $$=$$ $$( - 1259 \nu^{15} + 357 \nu^{13} - 139697 \nu^{11} + 39679 \nu^{9} - 408359 \nu^{7} + \cdots + 62220 \nu ) / 25856$$ (-1259*v^15 + 357*v^13 - 139697*v^11 + 39679*v^9 - 408359*v^7 + 123305*v^5 - 183412*v^3 + 62220*v) / 25856 $$\beta_{11}$$ $$=$$ $$( - 1259 \nu^{15} - 357 \nu^{13} - 139697 \nu^{11} - 39679 \nu^{9} - 408359 \nu^{7} + \cdots - 62220 \nu ) / 25856$$ (-1259*v^15 - 357*v^13 - 139697*v^11 - 39679*v^9 - 408359*v^7 - 123305*v^5 - 183412*v^3 - 62220*v) / 25856 $$\beta_{12}$$ $$=$$ $$( - 1616 \nu^{15} + 357 \nu^{13} - 179376 \nu^{11} + 39679 \nu^{9} - 531664 \nu^{7} + \cdots + 88076 \nu ) / 25856$$ (-1616*v^15 + 357*v^13 - 179376*v^11 + 39679*v^9 - 531664*v^7 + 123305*v^5 - 271488*v^3 + 88076*v) / 25856 $$\beta_{13}$$ $$=$$ $$( - 1616 \nu^{15} - 357 \nu^{13} - 179376 \nu^{11} - 39679 \nu^{9} - 531664 \nu^{7} + \cdots - 88076 \nu ) / 25856$$ (-1616*v^15 - 357*v^13 - 179376*v^11 - 39679*v^9 - 531664*v^7 - 123305*v^5 - 271488*v^3 - 88076*v) / 25856 $$\beta_{14}$$ $$=$$ $$( -6573\nu^{15} - 728279\nu^{11} - 2015505\nu^{7} - 667244\nu^{3} ) / 51712$$ (-6573*v^15 - 728279*v^11 - 2015505*v^7 - 667244*v^3) / 51712 $$\beta_{15}$$ $$=$$ $$( 1259\nu^{14} + 139697\nu^{10} + 408359\nu^{6} + 183412\nu^{2} ) / 12928$$ (1259*v^14 + 139697*v^10 + 408359*v^6 + 183412*v^2) / 12928
 $$\nu$$ $$=$$ $$( -\beta_{13} + \beta_{12} + \beta_{11} - \beta_{10} ) / 2$$ (-b13 + b12 + b11 - b10) / 2 $$\nu^{2}$$ $$=$$ $$( -2\beta_{15} + \beta_{9} - \beta_{8} + 2\beta_{7} + 2\beta_{6} + 10\beta_{4} + 2 ) / 2$$ (-2*b15 + b9 - b8 + 2*b7 + 2*b6 + 10*b4 + 2) / 2 $$\nu^{3}$$ $$=$$ $$-2\beta_{14} - 2\beta_{13} - 2\beta_{12} + 5\beta_{11} + 5\beta_{10} - \beta_{3}$$ -2*b14 - 2*b13 - 2*b12 + 5*b11 + 5*b10 - b3 $$\nu^{4}$$ $$=$$ $$( 13\beta_{9} + 13\beta_{8} + 24\beta_{7} - 24\beta_{6} - 20\beta _1 - 72 ) / 2$$ (13*b9 + 13*b8 + 24*b7 - 24*b6 - 20*b1 - 72) / 2 $$\nu^{5}$$ $$=$$ $$( 35\beta_{13} - 35\beta_{12} - 103\beta_{11} + 103\beta_{10} + 48\beta_{5} + 26\beta_{2} ) / 2$$ (35*b13 - 35*b12 - 103*b11 + 103*b10 + 48*b5 + 26*b2) / 2 $$\nu^{6}$$ $$=$$ $$103\beta_{15} - 70\beta_{9} + 70\beta_{8} - 127\beta_{7} - 127\beta_{6} - 495\beta_{4} - 127$$ 103*b15 - 70*b9 + 70*b8 - 127*b7 - 127*b6 - 495*b4 - 127 $$\nu^{7}$$ $$=$$ $$( 508\beta_{14} + 355\beta_{13} + 355\beta_{12} - 1069\beta_{11} - 1069\beta_{10} + 280\beta_{3} ) / 2$$ (508*b14 + 355*b13 + 355*b12 - 1069*b11 - 1069*b10 + 280*b3) / 2 $$\nu^{8}$$ $$=$$ $$( -1463\beta_{9} - 1463\beta_{8} - 2646\beta_{7} + 2646\beta_{6} + 2138\beta _1 + 7632 ) / 2$$ (-1463*b9 - 1463*b8 - 2646*b7 + 2646*b6 + 2138*b1 + 7632) / 2 $$\nu^{9}$$ $$=$$ $$-1838\beta_{13} + 1838\beta_{12} + 5553\beta_{11} - 5553\beta_{10} - 2646\beta_{5} - 1463\beta_{2}$$ -1838*b13 + 1838*b12 + 5553*b11 - 5553*b10 - 2646*b5 - 1463*b2 $$\nu^{10}$$ $$=$$ $$( - 22212 \beta_{15} + 15215 \beta_{9} - 15215 \beta_{8} + 27504 \beta_{7} + 27504 \beta_{6} + \cdots + 27504 ) / 2$$ (-22212*b15 + 15215*b9 - 15215*b8 + 27504*b7 + 27504*b6 + 106784*b4 + 27504) / 2 $$\nu^{11}$$ $$=$$ $$( -55008\beta_{14} - 38177\beta_{13} - 38177\beta_{12} + 115397\beta_{11} + 115397\beta_{10} - 30430\beta_{3} ) / 2$$ (-55008*b14 - 38177*b13 - 38177*b12 + 115397*b11 + 115397*b10 - 30430*b3) / 2 $$\nu^{12}$$ $$=$$ $$79058\beta_{9} + 79058\beta_{8} + 142901\beta_{7} - 142901\beta_{6} - 115397\beta _1 - 411872$$ 79058*b9 + 79058*b8 + 142901*b7 - 142901*b6 - 115397*b1 - 411872 $$\nu^{13}$$ $$=$$ $$( 396657 \beta_{13} - 396657 \beta_{12} - 1199055 \beta_{11} + 1199055 \beta_{10} + \cdots + 316232 \beta_{2} ) / 2$$ (396657*b13 - 396657*b12 - 1199055*b11 + 1199055*b10 + 571604*b5 + 316232*b2) / 2 $$\nu^{14}$$ $$=$$ $$( 2398110 \beta_{15} - 1642973 \beta_{9} + 1642973 \beta_{8} - 2969714 \beta_{7} - 2969714 \beta_{6} + \cdots - 2969714 ) / 2$$ (2398110*b15 - 1642973*b9 + 1642973*b8 - 2969714*b7 - 2969714*b6 - 11528962*b4 - 2969714) / 2 $$\nu^{15}$$ $$=$$ $$2969714 \beta_{14} + 2060754 \beta_{13} + 2060754 \beta_{12} - 6229523 \beta_{11} + \cdots + 1642973 \beta_{3}$$ 2969714*b14 + 2060754*b13 + 2060754*b12 - 6229523*b11 - 6229523*b10 + 1642973*b3

## Character values

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

 $$n$$ $$27$$ $$301$$ $$\chi(n)$$ $$\beta_{4}$$ $$\beta_{4}$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
18.1
 0.419746 − 0.419746i 0.592914 − 0.592914i 0.881421 − 0.881421i 2.27933 − 2.27933i −2.27933 + 2.27933i −0.881421 + 0.881421i −0.592914 + 0.592914i −0.419746 + 0.419746i −0.419746 − 0.419746i −0.592914 − 0.592914i −0.881421 − 0.881421i −2.27933 − 2.27933i 2.27933 + 2.27933i 0.881421 + 0.881421i 0.592914 + 0.592914i 0.419746 + 0.419746i
2.38239i 0.648130 + 0.648130i −3.67579 0 1.54410 1.54410i 2.38239 3.99239i 2.15985i 0
18.2 1.68658i 1.99698 + 1.99698i −0.844569 0 3.36807 3.36807i 1.68658 1.94873i 4.97584i 0
18.3 1.13453i −1.59157 1.59157i 0.712838 0 −1.80569 + 1.80569i 1.13453 3.07780i 2.06618i 0
18.4 0.438725i −0.242722 0.242722i 1.80752 0 −0.106488 + 0.106488i 0.438725 1.67045i 2.88217i 0
18.5 0.438725i 0.242722 + 0.242722i 1.80752 0 −0.106488 + 0.106488i −0.438725 1.67045i 2.88217i 0
18.6 1.13453i 1.59157 + 1.59157i 0.712838 0 −1.80569 + 1.80569i −1.13453 3.07780i 2.06618i 0
18.7 1.68658i −1.99698 1.99698i −0.844569 0 3.36807 3.36807i −1.68658 1.94873i 4.97584i 0
18.8 2.38239i −0.648130 0.648130i −3.67579 0 1.54410 1.54410i −2.38239 3.99239i 2.15985i 0
307.1 2.38239i −0.648130 + 0.648130i −3.67579 0 1.54410 + 1.54410i −2.38239 3.99239i 2.15985i 0
307.2 1.68658i −1.99698 + 1.99698i −0.844569 0 3.36807 + 3.36807i −1.68658 1.94873i 4.97584i 0
307.3 1.13453i 1.59157 1.59157i 0.712838 0 −1.80569 1.80569i −1.13453 3.07780i 2.06618i 0
307.4 0.438725i 0.242722 0.242722i 1.80752 0 −0.106488 0.106488i −0.438725 1.67045i 2.88217i 0
307.5 0.438725i −0.242722 + 0.242722i 1.80752 0 −0.106488 0.106488i 0.438725 1.67045i 2.88217i 0
307.6 1.13453i −1.59157 + 1.59157i 0.712838 0 −1.80569 1.80569i 1.13453 3.07780i 2.06618i 0
307.7 1.68658i 1.99698 1.99698i −0.844569 0 3.36807 + 3.36807i 1.68658 1.94873i 4.97584i 0
307.8 2.38239i 0.648130 0.648130i −3.67579 0 1.54410 + 1.54410i 2.38239 3.99239i 2.15985i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 18.8 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
65.f even 4 1 inner
65.k even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.f.d 16
5.b even 2 1 inner 325.2.f.d 16
5.c odd 4 2 325.2.k.d yes 16
13.d odd 4 1 325.2.k.d yes 16
65.f even 4 1 inner 325.2.f.d 16
65.g odd 4 1 325.2.k.d yes 16
65.k even 4 1 inner 325.2.f.d 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.f.d 16 1.a even 1 1 trivial
325.2.f.d 16 5.b even 2 1 inner
325.2.f.d 16 65.f even 4 1 inner
325.2.f.d 16 65.k even 4 1 inner
325.2.k.d yes 16 5.c odd 4 2
325.2.k.d yes 16 13.d odd 4 1
325.2.k.d yes 16 65.g odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{8} + 10T_{2}^{6} + 29T_{2}^{4} + 26T_{2}^{2} + 4$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{8} + 10 T^{6} + 29 T^{4} + \cdots + 4)^{2}$$
$3$ $$T^{16} + 90 T^{12} + \cdots + 16$$
$5$ $$T^{16}$$
$7$ $$(T^{8} - 10 T^{6} + 29 T^{4} + \cdots + 4)^{2}$$
$11$ $$(T^{8} + 10 T^{7} + \cdots + 33124)^{2}$$
$13$ $$T^{16} + \cdots + 815730721$$
$17$ $$T^{16} + 1890 T^{12} + \cdots + 160000$$
$19$ $$(T^{8} + 8 T^{7} + \cdots + 4096)^{2}$$
$23$ $$T^{16} + \cdots + 48796810000$$
$29$ $$(T^{8} + 120 T^{6} + \cdots + 169)^{2}$$
$31$ $$(T^{8} - 4 T^{7} + \cdots + 7396)^{2}$$
$37$ $$(T^{8} - 294 T^{6} + \cdots + 25040016)^{2}$$
$41$ $$(T^{8} - 2 T^{7} + 2 T^{6} + \cdots + 64)^{2}$$
$43$ $$T^{16} + 1530 T^{12} + \cdots + 6250000$$
$47$ $$(T^{8} - 88 T^{6} + \cdots + 64)^{2}$$
$53$ $$T^{16} + \cdots + 51769445841$$
$59$ $$(T^{8} + 12 T^{7} + \cdots + 11236)^{2}$$
$61$ $$(T^{4} - 6 T^{3} + \cdots + 2259)^{4}$$
$67$ $$(T^{8} + 308 T^{6} + \cdots + 7761796)^{2}$$
$71$ $$(T^{8} + 10 T^{7} + \cdots + 355216)^{2}$$
$73$ $$(T^{8} + 382 T^{6} + \cdots + 13512976)^{2}$$
$79$ $$(T^{8} + 234 T^{6} + \cdots + 6091024)^{2}$$
$83$ $$(T^{8} - 586 T^{6} + \cdots + 145395364)^{2}$$
$89$ $$(T^{8} + 18 T^{7} + \cdots + 1882384)^{2}$$
$97$ $$(T^{8} + 242 T^{6} + \cdots + 4096)^{2}$$