# Properties

 Label 325.2.k.c Level $325$ Weight $2$ Character orbit 325.k Analytic conductor $2.595$ Analytic rank $0$ Dimension $12$ Inner twists $4$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(57,325)]

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([1, 3]))

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

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

chi := DirichletCharacter("325.57");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$325 = 5^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 325.k (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: $$12$$ Relative dimension: $$6$$ over $$\Q(i)$$ Coefficient field: $$\mathbb{Q}[x]/(x^{12} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{12} + 28x^{10} + 260x^{8} + 972x^{6} + 1524x^{4} + 840x^{2} + 100$$ x^12 + 28*x^10 + 260*x^8 + 972*x^6 + 1524*x^4 + 840*x^2 + 100 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{11}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( - \beta_{5} + 2) q^{4} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{8} - 2 \beta_{6}) q^{9}+O(q^{10})$$ q + b2 * q^2 + b4 * q^3 + (-b5 + 2) * q^4 + (-b10 + b8 + b6 + b5 - 1) * q^6 + b9 * q^7 + (-b4 + b3 + b2 - b1) * q^8 + (-b8 - 2*b6) * q^9 $$q + \beta_{2} q^{2} + \beta_{4} q^{3} + ( - \beta_{5} + 2) q^{4} + ( - \beta_{10} + \beta_{8} + \beta_{6} + \cdots - 1) q^{6}+ \cdots + ( - \beta_{10} + 2 \beta_{8} + \cdots + 1) q^{99}+O(q^{100})$$ q + b2 * q^2 + b4 * q^3 + (-b5 + 2) * q^4 + (-b10 + b8 + b6 + b5 - 1) * q^6 + b9 * q^7 + (-b4 + b3 + b2 - b1) * q^8 + (-b8 - 2*b6) * q^9 + b11 * q^11 + (b9 - b7 + 3*b4 - b2 + b1) * q^12 + (b9 + b3 - b2) * q^13 + (b8 + b6) * q^14 + (-b11 + b10 - 2*b5 + 3) * q^16 + (b7 + b2) * q^17 + (-b9 + 3*b7 - b4 - b3) * q^18 + (-b11 - b6 - 1) * q^19 + (-b8 + b5) * q^21 + (-b7 - 3*b3 + b2) * q^22 + (b9 + b7 - b3 - b2 - b1) * q^23 + (-b10 + 3*b8 + 6*b6 + 3*b5 - 6) * q^24 + (-b11 + 2*b8 + 2*b6 - 3) * q^26 + (b9 - b7 + b2 - b1) * q^27 + (-b9 - 2*b7 + b4 + b3) * q^28 + (b11 + b10 - b8 - b6) * q^29 + (b10 - b8 + 3*b6 - b5 - 3) * q^31 + (-3*b4 + 3*b3 + b2) * q^32 + (b4 - b3 + 4*b2 + b1) * q^33 + (-b8 - 4*b6 - b5 + 4) * q^34 + (b11 + b10 - 4*b8 - 11*b6) * q^36 + (-b9 + 2*b7) * q^37 + (2*b7 + 3*b3 - 2*b2) * q^38 + (b10 - 2*b8 - b6 + b5 - 4) * q^39 + (-b8 - b6 - b5 + 1) * q^41 + (-b9 + b7 - 2*b3 - b2 + b1) * q^42 + (-3*b7 - b3 + 3*b2) * q^43 + (b11 - 2*b8 + b6 + 2*b5 + 1) * q^44 + (b11 - b8 - 4*b6 + b5 - 4) * q^46 + (-b9 + 2*b7 + 2*b4 + 2*b3) * q^47 + (b9 - 6*b7 + 3*b4 - 6*b2 + b1) * q^48 + (b11 - b10 - 2*b5 + 1) * q^49 + (-b11 - b10 + 2*b8 + 2*b6) * q^51 + (-3*b7 + 2*b4 + 3*b3 - 2*b2) * q^52 + (2*b8 + 5*b6 - 2*b5 + 5) * q^54 + (-b11 - b10 + b8 + 7*b6) * q^56 + (-2*b4 + 2*b3 - 4*b2 - b1) * q^57 + (-b9 - 4*b4 - 4*b3) * q^58 + (-b10 + b8 + 4*b6 + b5 - 4) * q^59 + (3*b5 - 3) * q^61 + (-b9 - 3*b7 - 5*b4 - 3*b2 - b1) * q^62 + (-b4 + b3 + 2*b2 + b1) * q^63 + (-b11 + b10 - 3*b5 + 4) * q^64 + (b11 - b10 - b5 + 13) * q^66 + (2*b4 - 2*b3 + b1) * q^67 + (-b9 + 3*b7 - 2*b4 + 3*b2 - b1) * q^68 + (-b11 + b10 - b5 + 7) * q^69 + (b10 + b8 - 2*b6 + b5 + 2) * q^71 + (-2*b9 + 7*b7 - 5*b4 - 5*b3) * q^72 + (2*b4 - 2*b3 + 3*b1) * q^73 + (-3*b8 - 9*b6) * q^74 + (-b11 + b8 - 3*b6 - b5 - 3) * q^76 + (-3*b9 - b7 - b2 - 3*b1) * q^77 + (-2*b9 + 2*b7 - 4*b4 - 3*b3 - 6*b2 + b1) * q^78 + (-b11 - b10 + 4*b8 - 2*b6) * q^79 + (b11 - b10 + b5 + 4) * q^81 + (-b9 + 2*b7 - 2*b4 + 2*b2 - b1) * q^82 + (3*b9 + 4*b7 + 3*b4 + 3*b3) * q^83 + (2*b11 - 2*b8 - 7*b6 + 2*b5 - 7) * q^84 + (b11 + 2*b8 + 11*b6 - 2*b5 + 11) * q^86 + (-5*b7 + 5*b2) * q^87 + (-2*b9 + 2*b7 - b3 - 2*b2 + 2*b1) * q^88 + (2*b10 - 2*b8 + b6 - 2*b5 - 1) * q^89 + (b11 - b10 - b6 - b5 - 6) * q^91 + (-3*b9 + 2*b7 - 3*b3 - 2*b2 + 3*b1) * q^92 + (b9 - 6*b7 - 3*b4 - 3*b3) * q^93 + (-2*b11 - 2*b10 + b8 - 5*b6) * q^94 + (-b10 + 4*b8 + 16*b6 + 4*b5 - 16) * q^96 + (-2*b4 + 2*b3 - 4*b2) * q^97 + (b4 - b3 + 5*b2 - 2*b1) * q^98 + (-b10 + 2*b8 - b6 + 2*b5 + 1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$12 q + 20 q^{4} - 12 q^{6}+O(q^{10})$$ 12 * q + 20 * q^4 - 12 * q^6 $$12 q + 20 q^{4} - 12 q^{6} - 4 q^{11} + 36 q^{16} - 8 q^{19} + 4 q^{21} - 64 q^{24} - 32 q^{26} - 36 q^{31} + 44 q^{34} - 40 q^{39} + 8 q^{41} + 16 q^{44} - 48 q^{46} - 4 q^{49} + 52 q^{54} - 48 q^{59} - 24 q^{61} + 44 q^{64} + 144 q^{66} + 88 q^{69} + 32 q^{71} - 36 q^{76} + 44 q^{81} - 84 q^{84} + 120 q^{86} - 12 q^{89} - 84 q^{91} - 180 q^{96} + 16 q^{99}+O(q^{100})$$ 12 * q + 20 * q^4 - 12 * q^6 - 4 * q^11 + 36 * q^16 - 8 * q^19 + 4 * q^21 - 64 * q^24 - 32 * q^26 - 36 * q^31 + 44 * q^34 - 40 * q^39 + 8 * q^41 + 16 * q^44 - 48 * q^46 - 4 * q^49 + 52 * q^54 - 48 * q^59 - 24 * q^61 + 44 * q^64 + 144 * q^66 + 88 * q^69 + 32 * q^71 - 36 * q^76 + 44 * q^81 - 84 * q^84 + 120 * q^86 - 12 * q^89 - 84 * q^91 - 180 * q^96 + 16 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{12} + 28x^{10} + 260x^{8} + 972x^{6} + 1524x^{4} + 840x^{2} + 100$$ :

 $$\beta_{1}$$ $$=$$ $$( -13\nu^{10} - 324\nu^{8} - 2360\nu^{6} - 4786\nu^{4} - 632\nu^{2} + 1340 ) / 300$$ (-13*v^10 - 324*v^8 - 2360*v^6 - 4786*v^4 - 632*v^2 + 1340) / 300 $$\beta_{2}$$ $$=$$ $$( -6\nu^{10} - 153\nu^{8} - 1180\nu^{6} - 2942\nu^{4} - 2174\nu^{2} - 220 ) / 100$$ (-6*v^10 - 153*v^8 - 1180*v^6 - 2942*v^4 - 2174*v^2 - 220) / 100 $$\beta_{3}$$ $$=$$ $$( 19 \nu^{11} - 110 \nu^{10} + 462 \nu^{9} - 2730 \nu^{8} + 3230 \nu^{7} - 19750 \nu^{6} + 6418 \nu^{5} + \cdots + 1300 ) / 3000$$ (19*v^11 - 110*v^10 + 462*v^9 - 2730*v^8 + 3230*v^7 - 19750*v^6 + 6418*v^5 - 40220*v^4 + 5516*v^3 - 13240*v^2 + 6880*v + 1300) / 3000 $$\beta_{4}$$ $$=$$ $$( 19 \nu^{11} + 110 \nu^{10} + 462 \nu^{9} + 2730 \nu^{8} + 3230 \nu^{7} + 19750 \nu^{6} + 6418 \nu^{5} + \cdots - 1300 ) / 3000$$ (19*v^11 + 110*v^10 + 462*v^9 + 2730*v^8 + 3230*v^7 + 19750*v^6 + 6418*v^5 + 40220*v^4 + 5516*v^3 + 13240*v^2 + 6880*v - 1300) / 3000 $$\beta_{5}$$ $$=$$ $$( 6\nu^{10} + 153\nu^{8} + 1180\nu^{6} + 2942\nu^{4} + 2224\nu^{2} + 470 ) / 50$$ (6*v^10 + 153*v^8 + 1180*v^6 + 2942*v^4 + 2224*v^2 + 470) / 50 $$\beta_{6}$$ $$=$$ $$( 16\nu^{11} + 418\nu^{9} + 3395\nu^{7} + 9652\nu^{5} + 9674\nu^{3} + 2570\nu ) / 500$$ (16*v^11 + 418*v^9 + 3395*v^7 + 9652*v^5 + 9674*v^3 + 2570*v) / 500 $$\beta_{7}$$ $$=$$ $$( -16\nu^{11} - 418\nu^{9} - 3395\nu^{7} - 9652\nu^{5} - 9674\nu^{3} - 2070\nu ) / 500$$ (-16*v^11 - 418*v^9 - 3395*v^7 - 9652*v^5 - 9674*v^3 - 2070*v) / 500 $$\beta_{8}$$ $$=$$ $$( -18\nu^{11} - 489\nu^{9} - 4285\nu^{7} - 14246\nu^{5} - 18152\nu^{3} - 7110\nu ) / 500$$ (-18*v^11 - 489*v^9 - 4285*v^7 - 14246*v^5 - 18152*v^3 - 7110*v) / 500 $$\beta_{9}$$ $$=$$ $$( 59\nu^{11} + 1407\nu^{9} + 9130\nu^{7} + 10148\nu^{5} - 22274\nu^{3} - 20320\nu ) / 1500$$ (59*v^11 + 1407*v^9 + 9130*v^7 + 10148*v^5 - 22274*v^3 - 20320*v) / 1500 $$\beta_{10}$$ $$=$$ $$( - 47 \nu^{11} + 60 \nu^{10} - 1206 \nu^{9} + 1580 \nu^{8} - 9440 \nu^{7} + 13000 \nu^{6} + \cdots + 8200 ) / 1000$$ (-47*v^11 + 60*v^10 - 1206*v^9 + 1580*v^8 - 9440*v^7 + 13000*v^6 - 24734*v^5 + 37620*v^4 - 23208*v^3 + 37040*v^2 - 10940*v + 8200) / 1000 $$\beta_{11}$$ $$=$$ $$( - 47 \nu^{11} - 60 \nu^{10} - 1206 \nu^{9} - 1580 \nu^{8} - 9440 \nu^{7} - 13000 \nu^{6} + \cdots - 8200 ) / 1000$$ (-47*v^11 - 60*v^10 - 1206*v^9 - 1580*v^8 - 9440*v^7 - 13000*v^6 - 24734*v^5 - 37620*v^4 - 23208*v^3 - 37040*v^2 - 10940*v - 8200) / 1000
 $$\nu$$ $$=$$ $$\beta_{7} + \beta_{6}$$ b7 + b6 $$\nu^{2}$$ $$=$$ $$\beta_{5} + 2\beta_{2} - 5$$ b5 + 2*b2 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{9} - 3\beta_{8} - 8\beta_{7} - 13\beta_{6} + \beta_{4} + \beta_{3}$$ b9 - 3*b8 - 8*b7 - 13*b6 + b4 + b3 $$\nu^{4}$$ $$=$$ $$-\beta_{11} + \beta_{10} - 14\beta_{5} + 4\beta_{4} - 4\beta_{3} - 24\beta_{2} + 4\beta _1 + 48$$ -b11 + b10 - 14*b5 + 4*b4 - 4*b3 - 24*b2 + 4*b1 + 48 $$\nu^{5}$$ $$=$$ $$-5\beta_{11} - 5\beta_{10} - 18\beta_{9} + 50\beta_{8} + 84\beta_{7} + 156\beta_{6} - 21\beta_{4} - 21\beta_{3}$$ -5*b11 - 5*b10 - 18*b9 + 50*b8 + 84*b7 + 156*b6 - 21*b4 - 21*b3 $$\nu^{6}$$ $$=$$ $$26\beta_{11} - 26\beta_{10} + 194\beta_{5} - 86\beta_{4} + 86\beta_{3} + 280\beta_{2} - 68\beta _1 - 552$$ 26*b11 - 26*b10 + 194*b5 - 86*b4 + 86*b3 + 280*b2 - 68*b1 - 552 $$\nu^{7}$$ $$=$$ $$112 \beta_{11} + 112 \beta_{10} + 262 \beta_{9} - 714 \beta_{8} - 974 \beta_{7} - 1912 \beta_{6} + \cdots + 358 \beta_{3}$$ 112*b11 + 112*b10 + 262*b9 - 714*b8 - 974*b7 - 1912*b6 + 358*b4 + 358*b3 $$\nu^{8}$$ $$=$$ $$-470\beta_{11} + 470\beta_{10} - 2666\beta_{5} + 1408\beta_{4} - 1408\beta_{3} - 3376\beta_{2} + 976\beta _1 + 6786$$ -470*b11 + 470*b10 - 2666*b5 + 1408*b4 - 1408*b3 - 3376*b2 + 976*b1 + 6786 $$\nu^{9}$$ $$=$$ $$- 1878 \beta_{11} - 1878 \beta_{10} - 3642 \beta_{9} + 9834 \beta_{8} + 11888 \beta_{7} + \cdots - 5484 \beta_{3}$$ -1878*b11 - 1878*b10 - 3642*b9 + 9834*b8 + 11888*b7 + 24082*b6 - 5484*b4 - 5484*b3 $$\nu^{10}$$ $$=$$ $$7362 \beta_{11} - 7362 \beta_{10} + 36332 \beta_{5} - 20952 \beta_{4} + 20952 \beta_{3} + 42048 \beta_{2} + \cdots - 86244$$ 7362*b11 - 7362*b10 + 36332*b5 - 20952*b4 + 20952*b3 + 42048*b2 - 13476*b1 - 86244 $$\nu^{11}$$ $$=$$ $$28314 \beta_{11} + 28314 \beta_{10} + 49808 \beta_{9} - 133760 \beta_{8} - 149900 \beta_{7} + \cdots + 79370 \beta_{3}$$ 28314*b11 + 28314*b10 + 49808*b9 - 133760*b8 - 149900*b7 - 309816*b6 + 79370*b4 + 79370*b3

## Character values

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

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

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
57.1
 3.66222i 2.88559i 1.59763i 0.402372i − 0.885586i − 1.66222i − 3.66222i − 2.88559i − 1.59763i − 0.402372i 0.885586i 1.66222i
−2.66222 2.01090 2.01090i 5.08741 0 −5.35344 + 5.35344i 1.53534i −8.21935 5.08741i 0
57.2 −1.88559 −1.50921 + 1.50921i 1.55543 0 2.84575 2.84575i 0.294568i 0.838267 1.55543i 0
57.3 −0.597628 0.823760 0.823760i −1.64284 0 −0.492302 + 0.492302i 4.42221i 2.17707 1.64284i 0
57.4 0.597628 −0.823760 + 0.823760i −1.64284 0 −0.492302 + 0.492302i 4.42221i −2.17707 1.64284i 0
57.5 1.88559 1.50921 1.50921i 1.55543 0 2.84575 2.84575i 0.294568i −0.838267 1.55543i 0
57.6 2.66222 −2.01090 + 2.01090i 5.08741 0 −5.35344 + 5.35344i 1.53534i 8.21935 5.08741i 0
268.1 −2.66222 2.01090 + 2.01090i 5.08741 0 −5.35344 5.35344i 1.53534i −8.21935 5.08741i 0
268.2 −1.88559 −1.50921 1.50921i 1.55543 0 2.84575 + 2.84575i 0.294568i 0.838267 1.55543i 0
268.3 −0.597628 0.823760 + 0.823760i −1.64284 0 −0.492302 0.492302i 4.42221i 2.17707 1.64284i 0
268.4 0.597628 −0.823760 0.823760i −1.64284 0 −0.492302 0.492302i 4.42221i −2.17707 1.64284i 0
268.5 1.88559 1.50921 + 1.50921i 1.55543 0 2.84575 + 2.84575i 0.294568i −0.838267 1.55543i 0
268.6 2.66222 −2.01090 2.01090i 5.08741 0 −5.35344 5.35344i 1.53534i 8.21935 5.08741i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 57.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
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.k.c yes 12
5.b even 2 1 inner 325.2.k.c yes 12
5.c odd 4 2 325.2.f.c 12
13.d odd 4 1 325.2.f.c 12
65.f even 4 1 inner 325.2.k.c yes 12
65.g odd 4 1 325.2.f.c 12
65.k even 4 1 inner 325.2.k.c yes 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.2.f.c 12 5.c odd 4 2
325.2.f.c 12 13.d odd 4 1
325.2.f.c 12 65.g odd 4 1
325.2.k.c yes 12 1.a even 1 1 trivial
325.2.k.c yes 12 5.b even 2 1 inner
325.2.k.c yes 12 65.f even 4 1 inner
325.2.k.c yes 12 65.k even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - 11T_{2}^{4} + 29T_{2}^{2} - 9$$ acting on $$S_{2}^{\mathrm{new}}(325, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{6} - 11 T^{4} + 29 T^{2} - 9)^{2}$$
$3$ $$T^{12} + 88 T^{8} + \cdots + 2500$$
$5$ $$T^{12}$$
$7$ $$(T^{6} + 22 T^{4} + 48 T^{2} + 4)^{2}$$
$11$ $$(T^{6} + 2 T^{5} + \cdots + 1458)^{2}$$
$13$ $$T^{12} + 30 T^{10} + \cdots + 4826809$$
$17$ $$T^{12} + 252 T^{8} + \cdots + 5184$$
$19$ $$(T^{6} + 4 T^{5} + 8 T^{4} + \cdots + 50)^{2}$$
$23$ $$T^{12} + 5052 T^{8} + \cdots + 324$$
$29$ $$(T^{6} + 96 T^{4} + \cdots + 22500)^{2}$$
$31$ $$(T^{6} + 18 T^{5} + \cdots + 8450)^{2}$$
$37$ $$(T^{6} + 74 T^{4} + \cdots + 8100)^{2}$$
$41$ $$(T^{6} - 4 T^{5} + \cdots + 288)^{2}$$
$43$ $$T^{12} + \cdots + 189833284$$
$47$ $$(T^{6} + 146 T^{4} + \cdots + 6084)^{2}$$
$53$ $$T^{12}$$
$59$ $$(T^{6} + 24 T^{5} + \cdots + 162)^{2}$$
$61$ $$(T^{3} + 6 T^{2} + \cdots - 162)^{4}$$
$67$ $$(T^{6} - 126 T^{4} + \cdots - 3844)^{2}$$
$71$ $$(T^{6} - 16 T^{5} + \cdots + 18)^{2}$$
$73$ $$(T^{6} - 286 T^{4} + \cdots - 422500)^{2}$$
$79$ $$(T^{6} + 384 T^{4} + \cdots + 1971216)^{2}$$
$83$ $$(T^{6} + 398 T^{4} + \cdots + 1028196)^{2}$$
$89$ $$(T^{6} + 6 T^{5} + \cdots + 145800)^{2}$$
$97$ $$(T^{6} - 192 T^{4} + \cdots - 1024)^{2}$$