# Properties

 Label 4600.2.e.u Level $4600$ Weight $2$ Character orbit 4600.e Analytic conductor $36.731$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(4049,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4600.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$4600 = 2^{3} \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4600.e (of order $$2$$, degree $$1$$, not 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: $$36.7311849298$$ Analytic rank: $$0$$ Dimension: $$10$$ Coefficient field: $$\mathbb{Q}[x]/(x^{10} + \cdots)$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256$$ x^10 + 28*x^8 + 260*x^6 + 897*x^4 + 1056*x^2 + 256 Coefficient ring: $$\Z[a_1, \ldots, a_{23}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 920) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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 $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + \beta_{3} - 2) q^{9}+O(q^{10})$$ q + b1 * q^3 + (b8 - b6) * q^7 + (-b5 + b3 - 2) * q^9 $$q + \beta_1 q^{3} + (\beta_{8} - \beta_{6}) q^{7} + ( - \beta_{5} + \beta_{3} - 2) q^{9} + (\beta_{4} + \beta_{3}) q^{11} + ( - \beta_{9} - \beta_{8} + \beta_{2}) q^{13} + ( - \beta_{9} + \beta_{6} - \beta_{2} - \beta_1) q^{17} + ( - \beta_{5} - 2 \beta_{4} - 1) q^{19} + (\beta_{5} - \beta_{4} + \beta_{3} + 1) q^{21} - \beta_{2} q^{23} + (\beta_{8} + 2 \beta_{6} - \beta_{2} - 3 \beta_1) q^{27} + (\beta_{7} - \beta_{3} - 1) q^{29} + (\beta_{7} - \beta_{5} - 2 \beta_{4} - \beta_{3} + 4) q^{31} + (\beta_{8} + 2 \beta_{6} + 3 \beta_{2} - 2 \beta_1) q^{33} + ( - \beta_{6} - 3 \beta_{2}) q^{37} + (2 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{3} - 3) q^{39} + ( - \beta_{7} - \beta_{4} + 5) q^{41} + ( - \beta_{9} - 2 \beta_{6} + 2 \beta_{2}) q^{47} + (\beta_{7} + 2 \beta_{5} + \beta_{4} - \beta_{3} - 6) q^{49} + (2 \beta_{7} + \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + 3) q^{51} + ( - 2 \beta_{9} - 2 \beta_{8} + \beta_{6} + \beta_{2} - 2 \beta_1) q^{53} + (\beta_{9} - \beta_{8} - 9 \beta_{2} - 3 \beta_1) q^{57} + ( - \beta_{7} - 2 \beta_{5} + 1) q^{59} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3}) q^{61} + (3 \beta_{9} + \beta_{8} - \beta_{6} - 8 \beta_{2} + \beta_1) q^{63} + ( - 2 \beta_{8} + \beta_{6} - \beta_{2} + 2 \beta_1) q^{67} + \beta_{4} q^{69} + ( - \beta_{7} - 2 \beta_{5} - 3 \beta_{4} + 1) q^{71} + (\beta_{9} + 2 \beta_{6}) q^{73} + (\beta_{9} - \beta_{8} - 2 \beta_{6} - 3 \beta_{2} - \beta_1) q^{77} - 2 \beta_{3} q^{79} + (\beta_{5} - 3 \beta_{4} - 5 \beta_{3} + 10) q^{81} + ( - \beta_{6} - 9 \beta_{2}) q^{83} + (\beta_{9} - 2 \beta_{6} + 2 \beta_{2} + 2 \beta_1) q^{87} + ( - 2 \beta_{7} + 2 \beta_{5} + 2 \beta_{4} - 2) q^{89} + ( - 4 \beta_{5} - 5 \beta_{4} + \beta_{3} + 4) q^{91} + (2 \beta_{9} - \beta_{8} - 2 \beta_{6} - 7 \beta_{2} + 5 \beta_1) q^{93} + ( - \beta_{9} + 2 \beta_{8} + 2 \beta_{6} - 6 \beta_{2} - \beta_1) q^{97} + (3 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 11) q^{99}+O(q^{100})$$ q + b1 * q^3 + (b8 - b6) * q^7 + (-b5 + b3 - 2) * q^9 + (b4 + b3) * q^11 + (-b9 - b8 + b2) * q^13 + (-b9 + b6 - b2 - b1) * q^17 + (-b5 - 2*b4 - 1) * q^19 + (b5 - b4 + b3 + 1) * q^21 - b2 * q^23 + (b8 + 2*b6 - b2 - 3*b1) * q^27 + (b7 - b3 - 1) * q^29 + (b7 - b5 - 2*b4 - b3 + 4) * q^31 + (b8 + 2*b6 + 3*b2 - 2*b1) * q^33 + (-b6 - 3*b2) * q^37 + (2*b7 - b5 - b4 + 2*b3 - 3) * q^39 + (-b7 - b4 + 5) * q^41 + (-b9 - 2*b6 + 2*b2) * q^47 + (b7 + 2*b5 + b4 - b3 - 6) * q^49 + (2*b7 + b5 - 2*b4 - 2*b3 + 3) * q^51 + (-2*b9 - 2*b8 + b6 + b2 - 2*b1) * q^53 + (b9 - b8 - 9*b2 - 3*b1) * q^57 + (-b7 - 2*b5 + 1) * q^59 + (-2*b5 - b4 + b3) * q^61 + (3*b9 + b8 - b6 - 8*b2 + b1) * q^63 + (-2*b8 + b6 - b2 + 2*b1) * q^67 + b4 * q^69 + (-b7 - 2*b5 - 3*b4 + 1) * q^71 + (b9 + 2*b6) * q^73 + (b9 - b8 - 2*b6 - 3*b2 - b1) * q^77 - 2*b3 * q^79 + (b5 - 3*b4 - 5*b3 + 10) * q^81 + (-b6 - 9*b2) * q^83 + (b9 - 2*b6 + 2*b2 + 2*b1) * q^87 + (-2*b7 + 2*b5 + 2*b4 - 2) * q^89 + (-4*b5 - 5*b4 + b3 + 4) * q^91 + (2*b9 - b8 - 2*b6 - 7*b2 + 5*b1) * q^93 + (-b9 + 2*b8 + 2*b6 - 6*b2 - b1) * q^97 + (3*b5 - 4*b4 - 4*b3 + 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 26 q^{9}+O(q^{10})$$ 10 * q - 26 * q^9 $$10 q - 26 q^{9} - 2 q^{11} - 14 q^{19} + 12 q^{21} - 8 q^{29} + 38 q^{31} - 38 q^{39} + 50 q^{41} - 50 q^{49} + 38 q^{51} + 2 q^{59} - 10 q^{61} + 2 q^{71} + 4 q^{79} + 114 q^{81} - 12 q^{89} + 22 q^{91} + 130 q^{99}+O(q^{100})$$ 10 * q - 26 * q^9 - 2 * q^11 - 14 * q^19 + 12 * q^21 - 8 * q^29 + 38 * q^31 - 38 * q^39 + 50 * q^41 - 50 * q^49 + 38 * q^51 + 2 * q^59 - 10 * q^61 + 2 * q^71 + 4 * q^79 + 114 * q^81 - 12 * q^89 + 22 * q^91 + 130 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} + 28x^{8} + 260x^{6} + 897x^{4} + 1056x^{2} + 256$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{9} - 12\nu^{7} - 68\nu^{5} - 833\nu^{3} - 2064\nu ) / 1024$$ (-v^9 - 12*v^7 - 68*v^5 - 833*v^3 - 2064*v) / 1024 $$\beta_{3}$$ $$=$$ $$( 3\nu^{8} + 36\nu^{6} - 52\nu^{4} - 829\nu^{2} - 208 ) / 256$$ (3*v^8 + 36*v^6 - 52*v^4 - 829*v^2 - 208) / 256 $$\beta_{4}$$ $$=$$ $$( -\nu^{8} - 12\nu^{6} - 4\nu^{4} + 63\nu^{2} - 16 ) / 64$$ (-v^8 - 12*v^6 - 4*v^4 + 63*v^2 - 16) / 64 $$\beta_{5}$$ $$=$$ $$( 3\nu^{8} + 36\nu^{6} - 52\nu^{4} - 1085\nu^{2} - 1488 ) / 256$$ (3*v^8 + 36*v^6 - 52*v^4 - 1085*v^2 - 1488) / 256 $$\beta_{6}$$ $$=$$ $$( -15\nu^{9} - 436\nu^{7} - 4092\nu^{5} - 12495\nu^{3} - 4592\nu ) / 2048$$ (-15*v^9 - 436*v^7 - 4092*v^5 - 12495*v^3 - 4592*v) / 2048 $$\beta_{7}$$ $$=$$ $$( 25\nu^{8} + 556\nu^{6} + 3748\nu^{4} + 7513\nu^{2} + 2192 ) / 512$$ (25*v^8 + 556*v^6 + 3748*v^4 + 7513*v^2 + 2192) / 512 $$\beta_{8}$$ $$=$$ $$( 7\nu^{9} + 212\nu^{7} + 2012\nu^{5} + 6343\nu^{3} + 5872\nu ) / 512$$ (7*v^9 + 212*v^7 + 2012*v^5 + 6343*v^3 + 5872*v) / 512 $$\beta_{9}$$ $$=$$ $$( 25\nu^{9} + 556\nu^{7} + 3748\nu^{5} + 7513\nu^{3} + 1680\nu ) / 1024$$ (25*v^9 + 556*v^7 + 3748*v^5 + 7513*v^3 + 1680*v) / 1024
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{3} - 5$$ -b5 + b3 - 5 $$\nu^{3}$$ $$=$$ $$\beta_{8} + 2\beta_{6} - \beta_{2} - 9\beta_1$$ b8 + 2*b6 - b2 - 9*b1 $$\nu^{4}$$ $$=$$ $$10\beta_{5} - 3\beta_{4} - 14\beta_{3} + 46$$ 10*b5 - 3*b4 - 14*b3 + 46 $$\nu^{5}$$ $$=$$ $$-\beta_{9} - 13\beta_{8} - 28\beta_{6} + 3\beta_{2} + 94\beta_1$$ -b9 - 13*b8 - 28*b6 + 3*b2 + 94*b1 $$\nu^{6}$$ $$=$$ $$2\beta_{7} - 107\beta_{5} + 49\beta_{4} + 164\beta_{3} - 485$$ 2*b7 - 107*b5 + 49*b4 + 164*b3 - 485 $$\nu^{7}$$ $$=$$ $$12\beta_{9} + 156\beta_{8} + 328\beta_{6} + 24\beta_{2} - 1025\beta_1$$ 12*b9 + 156*b8 + 328*b6 + 24*b2 - 1025*b1 $$\nu^{8}$$ $$=$$ $$-24\beta_{7} + 1181\beta_{5} - 640\beta_{4} - 1849\beta_{3} + 5305$$ -24*b7 + 1181*b5 - 640*b4 - 1849*b3 + 5305 $$\nu^{9}$$ $$=$$ $$-76\beta_{9} - 1821\beta_{8} - 3698\beta_{6} - 683\beta_{2} + 11341\beta_1$$ -76*b9 - 1821*b8 - 3698*b6 - 683*b2 + 11341*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times$$.

 $$n$$ $$1151$$ $$1201$$ $$2301$$ $$2577$$ $$\chi(n)$$ $$1$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4049.1
 − 3.36002i − 3.30649i − 1.93283i − 1.31091i − 0.568386i 0.568386i 1.31091i 1.93283i 3.30649i 3.36002i
0 3.36002i 0 0 0 1.90754i 0 −8.28974 0
4049.2 0 3.30649i 0 0 0 2.55040i 0 −7.93288 0
4049.3 0 1.93283i 0 0 0 2.38236i 0 −0.735829 0
4049.4 0 1.31091i 0 0 0 4.66212i 0 1.28151 0
4049.5 0 0.568386i 0 0 0 4.73770i 0 2.67694 0
4049.6 0 0.568386i 0 0 0 4.73770i 0 2.67694 0
4049.7 0 1.31091i 0 0 0 4.66212i 0 1.28151 0
4049.8 0 1.93283i 0 0 0 2.38236i 0 −0.735829 0
4049.9 0 3.30649i 0 0 0 2.55040i 0 −7.93288 0
4049.10 0 3.36002i 0 0 0 1.90754i 0 −8.28974 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 4049.10 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4600.2.e.u 10
5.b even 2 1 inner 4600.2.e.u 10
5.c odd 4 1 920.2.a.j 5
5.c odd 4 1 4600.2.a.be 5
15.e even 4 1 8280.2.a.bs 5
20.e even 4 1 1840.2.a.v 5
20.e even 4 1 9200.2.a.cu 5
40.i odd 4 1 7360.2.a.co 5
40.k even 4 1 7360.2.a.cp 5

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
920.2.a.j 5 5.c odd 4 1
1840.2.a.v 5 20.e even 4 1
4600.2.a.be 5 5.c odd 4 1
4600.2.e.u 10 1.a even 1 1 trivial
4600.2.e.u 10 5.b even 2 1 inner
7360.2.a.co 5 40.i odd 4 1
7360.2.a.cp 5 40.k even 4 1
8280.2.a.bs 5 15.e even 4 1
9200.2.a.cu 5 20.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(4600, [\chi])$$:

 $$T_{3}^{10} + 28T_{3}^{8} + 260T_{3}^{6} + 897T_{3}^{4} + 1056T_{3}^{2} + 256$$ T3^10 + 28*T3^8 + 260*T3^6 + 897*T3^4 + 1056*T3^2 + 256 $$T_{7}^{10} + 60T_{7}^{8} + 1268T_{7}^{6} + 11441T_{7}^{4} + 45568T_{7}^{2} + 65536$$ T7^10 + 60*T7^8 + 1268*T7^6 + 11441*T7^4 + 45568*T7^2 + 65536 $$T_{11}^{5} + T_{11}^{4} - 35T_{11}^{3} - 28T_{11}^{2} + 172T_{11} + 64$$ T11^5 + T11^4 - 35*T11^3 - 28*T11^2 + 172*T11 + 64 $$T_{13}^{10} + 108T_{13}^{8} + 3916T_{13}^{6} + 56825T_{13}^{4} + 285000T_{13}^{2} + 250000$$ T13^10 + 108*T13^8 + 3916*T13^6 + 56825*T13^4 + 285000*T13^2 + 250000

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10}$$
$3$ $$T^{10} + 28 T^{8} + 260 T^{6} + \cdots + 256$$
$5$ $$T^{10}$$
$7$ $$T^{10} + 60 T^{8} + 1268 T^{6} + \cdots + 65536$$
$11$ $$(T^{5} + T^{4} - 35 T^{3} - 28 T^{2} + 172 T + 64)^{2}$$
$13$ $$T^{10} + 108 T^{8} + 3916 T^{6} + \cdots + 250000$$
$17$ $$T^{10} + 108 T^{8} + 3096 T^{6} + \cdots + 1024$$
$19$ $$(T^{5} + 7 T^{4} - 41 T^{3} - 180 T^{2} + \cdots - 512)^{2}$$
$23$ $$(T^{2} + 1)^{5}$$
$29$ $$(T^{5} + 4 T^{4} - 41 T^{3} + 36 T^{2} + \cdots - 8)^{2}$$
$31$ $$(T^{5} - 19 T^{4} + 72 T^{3} + 183 T^{2} + \cdots - 128)^{2}$$
$37$ $$T^{10} + 97 T^{8} + 2664 T^{6} + \cdots + 4096$$
$41$ $$(T^{5} - 25 T^{4} + 212 T^{3} - 653 T^{2} + \cdots + 2182)^{2}$$
$43$ $$T^{10}$$
$47$ $$T^{10} + 341 T^{8} + 35052 T^{6} + \cdots + 262144$$
$53$ $$T^{10} + 329 T^{8} + \cdots + 410953984$$
$59$ $$(T^{5} - T^{4} - 142 T^{3} + 236 T^{2} + \cdots - 13568)^{2}$$
$61$ $$(T^{5} + 5 T^{4} - 115 T^{3} - 568 T^{2} + \cdots + 7664)^{2}$$
$67$ $$T^{10} + 333 T^{8} + \cdots + 67108864$$
$71$ $$(T^{5} - T^{4} - 214 T^{3} + 407 T^{2} + \cdots - 3968)^{2}$$
$73$ $$T^{10} + 317 T^{8} + 29652 T^{6} + \cdots + 1763584$$
$79$ $$(T^{5} - 2 T^{4} - 128 T^{3} - 128 T^{2} + \cdots + 1024)^{2}$$
$83$ $$T^{10} + 457 T^{8} + \cdots + 1698758656$$
$89$ $$(T^{5} + 6 T^{4} - 136 T^{3} - 512 T^{2} + \cdots + 8192)^{2}$$
$97$ $$T^{10} + 703 T^{8} + \cdots + 2461747456$$