LMFDB - Newform orbit 2664.2.r.n

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2664,2,Mod(433,2664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2664, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))

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

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

chi := DirichletCharacter("2664.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2664 = 2^{3} \cdot 3^{2} \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2664.r (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:	$21.2721470985$
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:	$x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4$ x^10 - x^9 + 9x^8 + 4x^7 + 54x^6 + 6x^5 + 98x^4 - 8x^3 + 148x^2 - 24x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$2^{4}$
Twist minimal:	no (minimal twist has level 296)
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.

$f(q)$	$=$	$q + ( - \beta_{9} - \beta_{6} - \beta_{2}) q^{5} + ( - \beta_{8} - \beta_{7} + 1) q^{7} + (\beta_{3} - 2) q^{11} + ( - \beta_{7} - \beta_{6} - \beta_{2} + \cdots + 1) q^{13} - \beta_{7} q^{17} + (\beta_{7} + \beta_{6} + \beta_{2} - 1) q^{19}+ \cdots + ( - \beta_{5} - \beta_{4} + 3 \beta_{2} - 5) q^{97}+O(q^{100})$ q + (-b9 - b6 - b2) * q^5 + (-b8 - b7 + 1) * q^7 + (b3 - 2) * q^11 + (-b7 - b6 - b2 - b1 + 1) * q^13 - b7 * q^17 + (b7 + b6 + b2 - 1) * q^19 + (b4 - b2 + 2) * q^23 + (b9 - b8 - 3b7 - b5 - b4 + b3 - b1) q^25 + (-2b5 - b4) q^29 + (b3 - 2) * q^31 + (-3b9 + 2b8 - 2b7 - 2b6 + 3b5 + 2b4 + b3 - b1) * q^35 + (-b9 + 3b7 + b5 - b4 - b2 + b1 - 3) q^37 + (-b9 - b8 - 2b6 - 2b2 - b1) * q^41 + (-b5 - b2 + 1) * q^43 + (2b5 - 2b4 - b3 - 4) * q^47 + (3b9 + b8 - 3b7 + b6 - 3b5 + b4 + 2b3 - 2b1) q^49 + (b9 - b8 + 3b6 - b5 - b4) q^53 + (4b9 + 2b8 + 2b7 + 2b6 + 2b2 + b1 - 2) q^55 + (b7 - b6 - b3 + b1) * q^59 + (2b9 + b8 - 3b7 - b6 - b2 - b1 + 3) * q^61 + (-4b9 - 2b8 - 7b7 - b6 + 4b5 - 2b4 + b3 - b1) q^65 + (-2b9 - b7 - b6 - b2 + 1) q^67 + (-b8 - 3b7 + 2b6 + 2b2 + 3) q^71 + (3b5 - b4 + b3 - 5) q^73 + (2b9 + 4b8 + 2b7 + 2b6 + 2b2 - 2) q^77 + (4b9 + 2b8 - 3b7 - b6 - b2 + 3) q^79 + (5b9 + 4b7 - 5b5) q^83 + (b5 + b2) * q^85 + (b9 + b8 - 4b7 + 2b6 - b5 + b4 - b3 + b1) * q^89 + (-2b9 - 2b8 - 3b7 - 3b6 + 2b5 - 2b4 + b3 - b1) * q^91 + (2b9 + 5b7 + b6 - 2b5) q^95 + (-b5 - b4 + 3b2 - 5) q^97
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 3 q^{5} + 3 q^{7} - 16 q^{11} + q^{13} - 5 q^{17} - 3 q^{19} + 12 q^{23} - 12 q^{25} - 16 q^{31} - 5 q^{35} - 12 q^{37} - 9 q^{41} + 4 q^{43} - 32 q^{47} - 18 q^{49} - 5 q^{53} + 4 q^{55} + 5 q^{59}+ \cdots - 36 q^{97}+O(q^{100})$ 10 * q - 3 * q^5 + 3 * q^7 - 16 * q^11 + q^13 - 5 * q^17 - 3 * q^19 + 12 * q^23 - 12 * q^25 - 16 * q^31 - 5 * q^35 - 12 * q^37 - 9 * q^41 + 4 * q^43 - 32 * q^47 - 18 * q^49 - 5 * q^53 + 4 * q^55 + 5 * q^59 + 15 * q^61 - 23 * q^65 + q^67 + 17 * q^71 - 36 * q^73 + 4 * q^77 + 21 * q^79 + 15 * q^83 + 6 * q^85 - 29 * q^89 - q^91 + 21 * q^95 - 36 * q^97

Basis of coefficient ring in terms of a root $\nu$ of $x^{10} - x^{9} + 9x^{8} + 4x^{7} + 54x^{6} + 6x^{5} + 98x^{4} - 8x^{3} + 148x^{2} - 24x + 4$ x^10 - x^9 + 9*x^8 + 4*x^7 + 54*x^6 + 6*x^5 + 98*x^4 - 8*x^3 + 148*x^2 - 24*x + 4 :

$\beta_{1}$	$=$	$2\nu$ 2*v
$\beta_{2}$	$=$	$( - 13 \nu^{9} - 605 \nu^{8} + 1210 \nu^{7} - 7028 \nu^{6} + 1815 \nu^{5} - 27830 \nu^{4} + \cdots + 5505 ) / 15127$ (-13v^9 - 605v^8 + 1210v^7 - 7028v^6 + 1815v^5 - 27830v^4 + 3582v^3 - 52030v^2 + 8470*v + 5505) / 15127
$\beta_{3}$	$=$	$( 40 \nu^{9} + 33 \nu^{8} - 66 \nu^{7} + 1012 \nu^{6} - 99 \nu^{5} + 1518 \nu^{4} - 15676 \nu^{3} + \cdots + 5004 ) / 15127$ (40v^9 + 33v^8 - 66v^7 + 1012v^6 - 99v^5 + 1518v^4 - 15676v^3 + 2838v^2 - 462*v + 5004) / 15127
$\beta_{4}$	$=$	$( 146 \nu^{9} + 1309 \nu^{8} - 2618 \nu^{7} + 17092 \nu^{6} - 3927 \nu^{5} + 60214 \nu^{4} + \cdots + 49383 ) / 15127$ (146v^9 + 1309v^8 - 2618v^7 + 17092v^6 - 3927v^5 + 60214v^4 - 39065v^3 + 112574v^2 - 18326*v + 49383) / 15127
$\beta_{5}$	$=$	$( - 153 \nu^{9} + 360 \nu^{8} - 720 \nu^{7} + 235 \nu^{6} - 1080 \nu^{5} + 16560 \nu^{4} + \cdots + 50660 ) / 15127$ (-153v^9 + 360v^8 - 720v^7 + 235v^6 - 1080v^5 + 16560v^4 + 13067v^3 + 30960v^2 - 5040*v + 50660) / 15127
$\beta_{6}$	$=$	$( 863 \nu^{9} - 1395 \nu^{8} + 7112 \nu^{7} + 4762 \nu^{6} + 27956 \nu^{5} + 7143 \nu^{4} + \cdots - 11542 ) / 30254$ (863v^9 - 1395v^8 + 7112v^7 + 4762v^6 + 27956v^5 + 7143v^4 + 54444v^3 + 74520v^2 + 71394*v - 11542) / 30254
$\beta_{7}$	$=$	$( - 1251 \nu^{9} + 1291 \nu^{8} - 11226 \nu^{7} - 5070 \nu^{6} - 66542 \nu^{5} - 7605 \nu^{4} + \cdots + 29562 ) / 30254$ (-1251v^9 + 1291v^8 - 11226v^7 - 5070v^6 - 66542v^5 - 7605v^4 - 121080v^3 - 5668v^2 - 182310*v + 29562) / 30254
$\beta_{8}$	$=$	$( - 2432 \nu^{9} - 1142 \nu^{8} - 17165 \nu^{7} - 36462 \nu^{6} - 156488 \nu^{5} - 123845 \nu^{4} + \cdots - 3432 ) / 30254$ (-2432v^9 - 1142v^8 - 17165v^7 - 36462v^6 - 156488v^5 - 123845v^4 - 276076v^3 - 80924v^2 - 550194*v - 3432) / 30254
$\beta_{9}$	$=$	$( 4505 \nu^{9} - 4117 \nu^{8} + 40649 \nu^{7} + 22134 \nu^{6} + 243578 \nu^{5} + 65616 \nu^{4} + \cdots + 2796 ) / 30254$ (4505v^9 - 4117v^8 + 40649v^7 + 22134v^6 + 243578v^5 + 65616v^4 + 441952v^3 + 30596v^2 + 597888*v + 2796) / 30254

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

$\nu$	$=$	$( \beta_1 ) / 2$ (b1) / 2
$\nu^{2}$	$=$	$( -\beta_{9} + \beta_{8} - 7\beta_{7} - 2\beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_1 ) / 2$ (-b9 + b8 - 7b7 - 2b6 + b5 + b4 - b3 + b1) / 2
$\nu^{3}$	$=$	$\beta_{4} - 3\beta_{3} + 2\beta_{2} - 3$ b4 - 3b3 + 2b2 - 3
$\nu^{4}$	$=$	$2\beta_{9} - 4\beta_{8} + 21\beta_{7} + 9\beta_{6} + 9\beta_{2} - 6\beta _1 - 21$ 2b9 - 4b8 + 21b7 + 9b6 + 9b2 - 6b1 - 21
$\nu^{5}$	$=$	$\beta_{9} - 11\beta_{8} + 40\beta_{7} + 23\beta_{6} - \beta_{5} - 11\beta_{4} + 23\beta_{3} - 23\beta_1$ b9 - 11b8 + 40b7 + 23b6 - b5 - 11b4 + 23b3 - 23b1
$\nu^{6}$	$=$	$-11\beta_{5} - 35\beta_{4} + 60\beta_{3} - 79\beta_{2} + 160$ -11b5 - 35b4 + 60b3 - 79b2 + 160
$\nu^{7}$	$=$	$-16\beta_{9} + 104\beta_{8} - 409\beta_{7} - 223\beta_{6} - 223\beta_{2} + 197\beta _1 + 409$ -16b9 + 104b8 - 409b7 - 223b6 - 223b2 + 197b1 + 409
$\nu^{8}$	$=$	$-78\beta_{9} + 316\beta_{8} - 1363\beta_{7} - 705\beta_{6} + 78\beta_{5} + 316\beta_{4} - 565\beta_{3} + 565\beta_1$ -78b9 + 316b8 - 1363b7 - 705b6 + 78b5 + 316b4 - 565b3 + 565b1
$\nu^{9}$	$=$	$176\beta_{5} + 954\beta_{4} - 1757\beta_{3} + 2073\beta_{2} - 3877$ 176b5 + 954b4 - 1757b3 + 2073b2 - 3877

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2664\mathbb{Z}\right)^\times$ .

$n$	$1297$	$1333$	$1999$	$2369$
$\chi(n)$	$-1 + \beta_{7}$	$1$	$1$	$1$

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}

433.1

0.0825878	−	0.143046i
−0.831426	+	1.44007i
0.657199	−	1.13830i
1.50928	−	2.61415i
−0.917643	+	1.58940i
0.0825878	+	0.143046i
−0.831426	−	1.44007i
0.657199	+	1.13830i
1.50928	+	2.61415i
−0.917643	−	1.58940i

−1.87297

3.24408i

2.03658

−

3.52745i

433.2

−1.19680

2.07292i

−2.55985

4.43380i

433.3

−0.883889

1.53094i

0.597948

−

1.03568i

433.4

0.717824

−

1.24331i

−0.0542045

0.0938850i

433.5

1.73584

−

3.00655i

1.47954

−

2.56263i

1009.1

−1.87297

−

3.24408i

2.03658

3.52745i

1009.2

−1.19680

−

2.07292i

−2.55985

−

4.43380i

1009.3

−0.883889

−

1.53094i

0.597948

1.03568i

1009.4

0.717824

1.24331i

−0.0542045

−

0.0938850i

1009.5

1.73584

3.00655i

1.47954

2.56263i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2664.2.r.n		10
3.b	odd	2	1		296.2.i.c	✓	10
12.b	even	2	1		592.2.i.h		10
37.c	even	3	1	inner	2664.2.r.n		10
111.i	odd	6	1		296.2.i.c	✓	10
444.t	even	6	1		592.2.i.h		10

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
296.2.i.c	✓	10	3.b	odd	2	1
296.2.i.c	✓	10	111.i	odd	6	1
592.2.i.h		10	12.b	even	2	1
592.2.i.h		10	444.t	even	6	1
2664.2.r.n		10	1.a	even	1	1	trivial
2664.2.r.n		10	37.c	even	3	1	inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(2664, [\chi])$ :

T_{5}^{10} + 3 T_{5}^{9} + 23 T_{5}^{8} + 42 T_{5}^{7} + 301 T_{5}^{6} + 541 T_{5}^{5} + 1821 T_{5}^{4} + \cdots + 6241

T_{11}^{5} + 8T_{11}^{4} - 8T_{11}^{3} - 144T_{11}^{2} - 112T_{11} + 128

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10}$ T^10
$3$	$T^{10}$ T^10
$5$	$T^{10} + 3 T^{9} + \cdots + 6241$ T^10 + 3T^9 + 23T^8 + 42T^7 + 301T^6 + 541T^5 + 1821T^4 + 1330T^3 + 3759T^2 + 1659*T + 6241
$7$	$T^{10} - 3 T^{9} + \cdots + 64$ T^10 - 3T^9 + 31T^8 - 110T^7 + 812T^6 - 2328T^5 + 6312T^4 - 5984T^3 + 4800T^2 + 512*T + 64
$11$	$(T^{5} + 8 T^{4} + \cdots + 128)^{2}$ (T^5 + 8T^4 - 8T^3 - 144T^2 - 112T + 128)^2
$13$	$T^{10} - T^{9} + \cdots + 2304$ T^10 - T^9 + 45T^8 + 12T^7 + 1728T^6 - 304T^5 + 10064T^4 - 640T^3 + 50944T^2 - 10752T + 2304
$17$	$(T^{2} + T + 1)^{5}$ (T^2 + T + 1)^5
$19$	$T^{10} + 3 T^{9} + \cdots + 256$ T^10 + 3T^9 + 21T^8 + 28T^7 + 224T^6 + 304T^5 + 1168T^4 - 128T^3 + 768T^2 + 256*T + 256
$23$	$(T^{5} - 6 T^{4} + \cdots - 1152)^{2}$ (T^5 - 6T^4 - 36T^3 + 248T^2 + 8T - 1152)^2
$29$	$(T^{5} - 72 T^{3} + \cdots - 358)^{2}$ (T^5 - 72T^3 + 222T^2 + 135*T - 358)^2
$31$	$(T^{5} + 8 T^{4} + \cdots + 128)^{2}$ (T^5 + 8T^4 - 8T^3 - 144T^2 - 112T + 128)^2
$37$	$T^{10} + 12 T^{9} + \cdots + 69343957$ T^10 + 12T^9 + 61T^8 - 26T^7 - 1967T^6 - 14918T^5 - 72779T^4 - 35594T^3 + 3089833T^2 + 22489932*T + 69343957
$41$	$T^{10} + 9 T^{9} + \cdots + 8543929$ T^10 + 9T^9 + 167T^8 + 1494T^7 + 21053T^6 + 156719T^5 + 1015477T^4 + 3410678T^3 + 8594719T^2 + 10087273*T + 8543929
$43$	$(T^{5} - 2 T^{4} - 16 T^{3} + \cdots + 32)^{2}$ (T^5 - 2T^4 - 16T^3 + 8T^2 + 56T + 32)^2
$47$	$(T^{5} + 16 T^{4} + \cdots - 23424)^{2}$ (T^5 + 16T^4 - 72T^3 - 2416T^2 - 13616T - 23424)^2
$53$	$T^{10} + 5 T^{9} + \cdots + 2304$ T^10 + 5T^9 + 141T^8 + 28T^7 + 11648T^6 + 2032T^5 + 478224T^4 - 1000576T^3 + 11090176T^2 + 159744*T + 2304
$59$	$T^{10} - 5 T^{9} + \cdots + 147456$ T^10 - 5T^9 + 69T^8 - 420T^7 + 4176T^6 - 20096T^5 + 76160T^4 - 171008T^3 + 286720T^2 - 245760*T + 147456
$61$	$T^{10} - 15 T^{9} + \cdots + 40921609$ T^10 - 15T^9 + 247T^8 - 1834T^7 + 21565T^6 - 162937T^5 + 1159957T^4 - 4967314T^3 + 16610647T^2 - 31031847*T + 40921609
$67$	$T^{10} - T^{9} + \cdots + 20736$ T^10 - T^9 + 41T^8 - 40T^7 + 1512T^6 - 1488T^5 + 6576T^4 - 6400T^3 + 22144T^2 - 18432T + 20736
$71$	$T^{10} - 17 T^{9} + \cdots + 40000$ T^10 - 17T^9 + 243T^8 - 1246T^7 + 6540T^6 - 5448T^5 + 79304T^4 - 129760T^3 + 184000T^2 - 96000*T + 40000
$73$	$(T^{5} + 18 T^{4} + \cdots + 1184)^{2}$ (T^5 + 18T^4 - 24T^3 - 912T^2 + 1296T + 1184)^2
$79$	$T^{10} + \cdots + 12908595456$ T^10 - 21T^9 + 545T^8 - 5032T^7 + 89688T^6 - 619216T^5 + 10308912T^4 - 34831360T^3 + 419561344T^2 + 352664064*T + 12908595456
$83$	$T^{10} - 15 T^{9} + \cdots + 45481536$ T^10 - 15T^9 + 395T^8 - 2130T^7 + 62000T^6 - 344544T^5 + 5714440T^4 + 2387040T^3 + 19780960T^2 - 13488000*T + 45481536
$89$	$T^{10} + \cdots + 1372480209$ T^10 + 29T^9 + 623T^8 + 7646T^7 + 79789T^6 + 576523T^5 + 4361213T^4 + 24802846T^3 + 146221375T^2 + 484093149*T + 1372480209
$97$	$(T^{5} + 18 T^{4} + \cdots + 88282)^{2}$ (T^5 + 18T^4 - 102T^3 - 2660T^2 + 429T + 88282)^2
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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
Embeddings:		e.g. 1-3 or 433.5
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