LMFDB - Newform orbit 2600.2.k.c

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2600,2,Mod(2001,2600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2600, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2600.2001");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$2600 = 2^{3} \cdot 5^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	2600.k (of order $2$ , degree $1$ , 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:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196$ x^8 - 2x^7 + 2x^6 + 6x^5 + 36x^4 - 52x^3 + 50x^2 + 140*x + 196
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$2^{3}$
Twist minimal:	no (minimal twist has level 520)
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 $1,\beta_1,\ldots,\beta_{7}$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q + \beta_{3} q^{3} + (\beta_{5} - \beta_{4} - \beta_{2}) q^{7} + (\beta_{7} + 2) q^{9} + \beta_{4} q^{11} + ( - \beta_{4} - \beta_{3} + \beta_{2}) q^{13} + (\beta_{7} + \beta_{3} - \beta_1 + 1) q^{17}+ \cdots + (\beta_{6} + \beta_{5} + \cdots - 2 \beta_{2}) q^{99}+O(q^{100})$ q + b3 * q^3 + (b5 - b4 - b2) * q^7 + (b7 + 2) * q^9 + b4 * q^11 + (-b4 - b3 + b2) * q^13 + (b7 + b3 - b1 + 1) * q^17 + (-b6 + b5) * q^19 + (b6 + b5 + b4 - 2b2) q^21 + (b3 + 2b1 + 2) q^23 + (-b7 + 3b3 - b1 + 1) q^27 + (-b7 - 2b3 + 1) q^29 + (b6 + b5 - 2b4 - 4b2) * q^31 + (b5 - b4 - b2) * q^33 + (-3b5 + b4 - 3b2) * q^37 + (-b7 - 2b5 + b4 - 5) q^39 + (-b6 + b5 + b4) * q^41 + (b7 + b1 + 3) * q^43 + (2b6 + b5 - b4 + b2) q^47 + (-2b3 + 4b1 - 5) * q^49 + (4b3 + 4) q^51 + (-b7 + b3 + b1 + 1) * q^53 + (2b6 - 3b5 + b4 - 7b2) q^57 + (b6 + b5 + 2b2) q^59 + (-b7 + 2b3 + 2b1 + 5) * q^61 + (5b5 + b4 + 3b2) * q^63 + (-2b6 - b5 - b4 - b2) q^67 + (b7 + 4b3 - 2b1 + 9) * q^69 + (2b6 + 2b5 - b4 - 2b2) q^71 + (-b5 + 3b4 - b2) q^73 + (-b7 - b3 - 3b1 + 5) q^77 + (-2b3 - 2b1 - 4) * q^79 + (b7 - 4b3 + 2b1 + 6) * q^81 + (3b5 - b4 + 9b2) * q^83 + (-b7 - 3b3 + b1 - 11) q^87 + (-2b5 + 2b4 + 6b2) q^89 + (b7 - b6 - b5 - b4 + 2b3 + 2b2 + 2b1 - 3) q^91 + (7b5 + b4 + 5b2) * q^93 + (2b5 + 2b4 - 4b2) q^97 + (b6 + b5 - 2b4 - 2b2) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$8 q - 4 q^{3} + 16 q^{9} + 4 q^{13} + 4 q^{17} + 12 q^{23} - 4 q^{27} + 16 q^{29} - 40 q^{39} + 24 q^{43} - 32 q^{49} + 16 q^{51} + 4 q^{53} + 32 q^{61} + 56 q^{69} + 44 q^{77} - 24 q^{79} + 64 q^{81} - 76 q^{87}+ \cdots - 32 q^{91}+O(q^{100})$ 8 * q - 4 * q^3 + 16 * q^9 + 4 * q^13 + 4 * q^17 + 12 * q^23 - 4 * q^27 + 16 * q^29 - 40 * q^39 + 24 * q^43 - 32 * q^49 + 16 * q^51 + 4 * q^53 + 32 * q^61 + 56 * q^69 + 44 * q^77 - 24 * q^79 + 64 * q^81 - 76 * q^87 - 32 * q^91

Basis of coefficient ring in terms of a root $\nu$ of $x^{8} - 2x^{7} + 2x^{6} + 6x^{5} + 36x^{4} - 52x^{3} + 50x^{2} + 140x + 196$ x^8 - 2*x^7 + 2*x^6 + 6*x^5 + 36*x^4 - 52*x^3 + 50*x^2 + 140*x + 196 :

$\beta_{1}$	$=$	$( -295\nu^{7} + 581\nu^{6} - 1975\nu^{5} + 12197\nu^{4} - 28834\nu^{3} + 14811\nu^{2} + 52682\nu + 217006 ) / 103451$ (-295v^7 + 581v^6 - 1975v^5 + 12197v^4 - 28834v^3 + 14811v^2 + 52682*v + 217006) / 103451
$\beta_{2}$	$=$	$( - 4733 \nu^{7} + 18790 \nu^{6} - 22920 \nu^{5} - 15070 \nu^{4} - 128766 \nu^{3} + 794160 \nu^{2} + \cdots - 329546 ) / 1448314$ (-4733v^7 + 18790v^6 - 22920v^5 - 15070v^4 - 128766v^3 + 794160v^2 - 577130*v - 329546) / 1448314
$\beta_{3}$	$=$	$( 371\nu^{7} - 380\nu^{6} - 1023\nu^{5} + 15170\nu^{4} + 10312\nu^{3} - 9509\nu^{2} - 26978\nu + 283268 ) / 103451$ (371v^7 - 380v^6 - 1023v^5 + 15170v^4 + 10312v^3 - 9509v^2 - 26978*v + 283268) / 103451
$\beta_{4}$	$=$	$( 4804 \nu^{7} - 30853 \nu^{6} + 39176 \nu^{5} + 9329 \nu^{4} - 16490 \nu^{3} - 693923 \nu^{2} + \cdots + 195258 ) / 724157$ (4804v^7 - 30853v^6 + 39176v^5 + 9329v^4 - 16490v^3 - 693923v^2 + 263566*v + 195258) / 724157
$\beta_{5}$	$=$	$( 23665 \nu^{7} - 93950 \nu^{6} + 114600 \nu^{5} + 75350 \nu^{4} + 643830 \nu^{3} - 2522486 \nu^{2} + \cdots + 1647730 ) / 1448314$ (23665v^7 - 93950v^6 + 114600v^5 + 75350v^4 + 643830v^3 - 2522486v^2 + 2885650*v + 1647730) / 1448314
$\beta_{6}$	$=$	$( 37489 \nu^{7} - 35610 \nu^{6} + 179096 \nu^{5} + 120284 \nu^{4} + 1042570 \nu^{3} + 42688 \nu^{2} + \cdots + 2631678 ) / 1448314$ (37489v^7 - 35610v^6 + 179096v^5 + 120284v^4 + 1042570v^3 + 42688v^2 + 4620886*v + 2631678) / 1448314
$\beta_{7}$	$=$	$( 2968\nu^{7} - 3040\nu^{6} - 8184\nu^{5} + 17909\nu^{4} + 82496\nu^{3} - 76072\nu^{2} - 215824\nu + 93673 ) / 103451$ (2968v^7 - 3040v^6 - 8184v^5 + 17909v^4 + 82496v^3 - 76072v^2 - 215824*v + 93673) / 103451

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

$\nu$	$=$	$( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 ) / 2$ (b5 - b4 - b3 + b2 + b1) / 2
$\nu^{2}$	$=$	$\beta_{5} + 5\beta_{2}$ b5 + 5*b2
$\nu^{3}$	$=$	$( -\beta_{7} - \beta_{6} + 5\beta_{5} - 5\beta_{4} + 5\beta_{3} + 8\beta_{2} - 5\beta _1 - 3 ) / 2$ (-b7 - b6 + 5b5 - 5b4 + 5b3 + 8b2 - 5*b1 - 3) / 2
$\nu^{4}$	$=$	$-\beta_{7} + 8\beta_{3} - 21$ -b7 + 8*b3 - 21
$\nu^{5}$	$=$	$-5\beta_{7} + 5\beta_{6} - 16\beta_{5} + 13\beta_{4} + 16\beta_{3} - 29\beta_{2} - 13\beta _1 - 13$ -5b7 + 5b6 - 16b5 + 13b4 + 16b3 - 29b2 - 13*b1 - 13
$\nu^{6}$	$=$	$13\beta_{6} - 57\beta_{5} + \beta_{4} - 180\beta_{2}$ 13b6 - 57b5 + b4 - 180*b2
$\nu^{7}$	$=$	$41\beta_{7} + 41\beta_{6} - 110\beta_{5} + 70\beta_{4} - 110\beta_{3} - 211\beta_{2} + 70\beta _1 + 101$ 41b7 + 41b6 - 110b5 + 70b4 - 110b3 - 211b2 + 70*b1 + 101

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$ .

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$-1$	$1$	$1$

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}

2001.1

1.90427	−	1.90427i
1.90427	+	1.90427i
−1.61013	−	1.61013i
−1.61013	+	1.61013i
1.47812	−	1.47812i
1.47812	+	1.47812i
−0.772270	−	0.772270i
−0.772270	+	0.772270i

−3.25249

−

1.80854i

7.57872

2001.2

−3.25249

1.80854i

7.57872

2001.3

−1.18501

−

5.22025i

−1.59576

2001.4

−1.18501

5.22025i

−1.59576

2001.5

−0.369700

−

0.956248i

−2.86332

2001.6

−0.369700

0.956248i

−2.86332

2001.7

2.80720

−

3.54454i

4.88037

2001.8

2.80720

3.54454i

4.88037

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
13.b	even	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	2600.2.k.c		8
5.b	even	2	1		520.2.k.b	✓	8
5.c	odd	4	1		2600.2.f.e		8
5.c	odd	4	1		2600.2.f.f		8
13.b	even	2	1	inner	2600.2.k.c		8
15.d	odd	2	1		4680.2.g.k		8
20.d	odd	2	1		1040.2.k.e		8
65.d	even	2	1		520.2.k.b	✓	8
65.g	odd	4	1		6760.2.a.bc		4
65.g	odd	4	1		6760.2.a.bd		4
65.h	odd	4	1		2600.2.f.e		8
65.h	odd	4	1		2600.2.f.f		8
195.e	odd	2	1		4680.2.g.k		8
260.g	odd	2	1		1040.2.k.e		8

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
520.2.k.b	✓	8	5.b	even	2	1
520.2.k.b	✓	8	65.d	even	2	1
1040.2.k.e		8	20.d	odd	2	1
1040.2.k.e		8	260.g	odd	2	1
2600.2.f.e		8	5.c	odd	4	1
2600.2.f.e		8	65.h	odd	4	1
2600.2.f.f		8	5.c	odd	4	1
2600.2.f.f		8	65.h	odd	4	1
2600.2.k.c		8	1.a	even	1	1	trivial
2600.2.k.c		8	13.b	even	2	1	inner
4680.2.g.k		8	15.d	odd	2	1
4680.2.g.k		8	195.e	odd	2	1
6760.2.a.bc		4	65.g	odd	4	1
6760.2.a.bd		4	65.g	odd	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $T_{3}^{4} + 2T_{3}^{3} - 8T_{3}^{2} - 14T_{3} - 4$ T3^4 + 2*T3^3 - 8*T3^2 - 14*T3 - 4 acting on $S_{2}^{\mathrm{new}}(2600, [\chi])$ .

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{8}$ T^8
$3$	$(T^{4} + 2 T^{3} - 8 T^{2} + \cdots - 4)^{2}$ (T^4 + 2T^3 - 8T^2 - 14*T - 4)^2
$5$	$T^{8}$ T^8
$7$	$T^{8} + 44 T^{6} + \cdots + 1024$ T^8 + 44T^6 + 512T^4 + 1552*T^2 + 1024
$11$	$T^{8} + 28 T^{6} + \cdots + 64$ T^8 + 28T^6 + 180T^4 + 260*T^2 + 64
$13$	$T^{8} - 4 T^{7} + \cdots + 28561$ T^8 - 4T^7 + 20T^6 - 44T^5 + 118T^4 - 572T^3 + 3380T^2 - 8788*T + 28561
$17$	$(T^{4} - 2 T^{3} - 40 T^{2} + \cdots - 64)^{2}$ (T^4 - 2T^3 - 40T^2 + 128*T - 64)^2
$19$	$T^{8} + 104 T^{6} + \cdots + 18496$ T^8 + 104T^6 + 3004T^4 + 24036*T^2 + 18496
$23$	$(T^{4} - 6 T^{3} - 64 T^{2} + \cdots + 56)^{2}$ (T^4 - 6T^3 - 64T^2 + 390*T + 56)^2
$29$	$(T^{4} - 8 T^{3} + \cdots - 664)^{2}$ (T^4 - 8T^3 - 44T^2 + 452*T - 664)^2
$31$	$T^{8} + 216 T^{6} + \cdots + 6130576$ T^8 + 216T^6 + 16612T^4 + 535732*T^2 + 6130576
$37$	$T^{8} + 172 T^{6} + \cdots + 53824$ T^8 + 172T^6 + 7728T^4 + 79952*T^2 + 53824
$41$	$T^{8} + 124 T^{6} + \cdots + 16384$ T^8 + 124T^6 + 4144T^4 + 20544*T^2 + 16384
$43$	$(T^{4} - 12 T^{3} + \cdots - 236)^{2}$ (T^4 - 12T^3 - 8T^2 + 338*T - 236)^2
$47$	$T^{8} + 284 T^{6} + \cdots + 7311616$ T^8 + 284T^6 + 24384T^4 + 781456*T^2 + 7311616
$53$	$(T^{4} - 2 T^{3} + \cdots + 128)^{2}$ (T^4 - 2T^3 - 60T^2 + 72*T + 128)^2
$59$	$T^{8} + 96 T^{6} + \cdots + 50176$ T^8 + 96T^6 + 3052T^4 + 33604*T^2 + 50176
$61$	$(T^{4} - 16 T^{3} + \cdots + 2264)^{2}$ (T^4 - 16T^3 - 48T^2 + 876*T + 2264)^2
$67$	$T^{8} + 388 T^{6} + \cdots + 63744256$ T^8 + 388T^6 + 53104T^4 + 3070480*T^2 + 63744256
$71$	$T^{8} + 348 T^{6} + \cdots + 1430416$ T^8 + 348T^6 + 35212T^4 + 1116340*T^2 + 1430416
$73$	$T^{8} + 236 T^{6} + \cdots + 222784$ T^8 + 236T^6 + 14896T^4 + 245712*T^2 + 222784
$79$	$(T^{4} + 12 T^{3} + \cdots + 1024)^{2}$ (T^4 + 12T^3 - 64T^2 - 672*T + 1024)^2
$83$	$T^{8} + 388 T^{6} + \cdots + 891136$ T^8 + 388T^6 + 35088T^4 + 709904*T^2 + 891136
$89$	$T^{8} + 336 T^{6} + \cdots + 16384$ T^8 + 336T^6 + 27136T^4 + 115712*T^2 + 16384
$97$	$T^{8} + 336 T^{6} + \cdots + 215296$ T^8 + 336T^6 + 30176T^4 + 595200*T^2 + 215296
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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
Embeddings:		e.g. 1-3 or 2001.8
Significant digits:
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