LMFDB - Newform orbit 575.2.p.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [575,2,Mod(49,575)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(575, base_ring=CyclotomicField(22))

chi = DirichletCharacter(H, H._module([11, 16]))

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

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

chi := DirichletCharacter("575.49");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 575 = 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	575.p (of order $22$, degree $10$, 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:	$4.59139811622$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$2$ over $\Q(\zeta_{22})$
Coefficient field:	$\Q(\zeta_{44})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 $ x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/575\mathbb{Z}\right)^\times$.

$n$	$51$	$277$
$\chi(n)$	$-1 + \zeta_{44}^{2} - \zeta_{44}^{4} + \zeta_{44}^{6} - \zeta_{44}^{8} + \zeta_{44}^{10} - \zeta_{44}^{12} + \zeta_{44}^{14} - \zeta_{44}^{16} + \zeta_{44}^{18}$	$-1$

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} $

49.1

0.755750	−	0.654861i
−0.755750	+	0.654861i
−0.989821	−	0.142315i
0.989821	+	0.142315i
0.909632	−	0.415415i
−0.909632	+	0.415415i
−0.281733	−	0.959493i
0.281733	+	0.959493i
−0.540641	−	0.841254i
0.540641	+	0.841254i
−0.540641	+	0.841254i
0.540641	−	0.841254i
0.755750	+	0.654861i
−0.755750	−	0.654861i
−0.281733	+	0.959493i
0.281733	−	0.959493i
0.909632	+	0.415415i
−0.909632	−	0.415415i
−0.989821	+	0.142315i
0.989821	−	0.142315i

−0.0872586

0.0125459i

−0.909632

−

0.415415i

−1.91153

0.561276i

0.0845850

0.0248364i

0.835939

1.30075i

0.320135

−

0.146201i

−1.30972

−

1.51150i

49.2

0.0872586

−

0.0125459i

0.909632

0.415415i

−1.91153

0.561276i

0.0845850

0.0248364i

−0.835939

−

1.30075i

−0.320135

0.146201i

−1.30972

−

1.51150i

124.1

−0.386758

−

1.31718i

−0.755750

0.654861i

0.0971309

−

0.0624222i

1.15486

0.742184i

−2.02730

−

0.925839i

−2.19475

−

1.90176i

−0.284630

1.97964i

124.2

0.386758

1.31718i

0.755750

−

0.654861i

0.0971309

−

0.0624222i

1.15486

0.742184i

2.02730

0.925839i

2.19475

1.90176i

−0.284630

1.97964i

174.1

−1.81219

1.57028i

−0.540641

0.841254i

0.533654

−

3.71165i

−0.341254

−

2.37347i

−0.386758

1.31718i

2.26844

3.52977i

0.830830

1.81926i

174.2

1.81219

−

1.57028i

0.540641

−

0.841254i

0.533654

−

3.71165i

−0.341254

−

2.37347i

0.386758

−

1.31718i

−2.26844

−

3.52977i

0.830830

1.81926i

324.1

−0.835939

1.30075i

−0.989821

0.142315i

−0.162317

−

0.355426i

0.642315

−

1.40647i

−1.81219

−

1.57028i

−2.46292

−

0.354114i

−1.91899

0.563465i

324.2

0.835939

−

1.30075i

0.989821

−

0.142315i

−0.162317

−

0.355426i

0.642315

−

1.40647i

1.81219

1.57028i

2.46292

0.354114i

−1.91899

0.563465i

349.1

−2.02730

0.925839i

−0.281733

−

0.959493i

1.94306

−

2.24241i

1.45949

1.68434i

0.0872586

−

0.0125459i

−0.607265

2.06815i

1.68251

−

1.08128i

349.2

2.02730

−

0.925839i

0.281733

0.959493i

1.94306

−

2.24241i

1.45949

1.68434i

−0.0872586

0.0125459i

0.607265

−

2.06815i

1.68251

−

1.08128i

374.1

−2.02730

−

0.925839i

−0.281733

0.959493i

1.94306

2.24241i

1.45949

−

1.68434i

0.0872586

0.0125459i

−0.607265

−

2.06815i

1.68251

1.08128i

374.2

2.02730

0.925839i

0.281733

−

0.959493i

1.94306

2.24241i

1.45949

−

1.68434i

−0.0872586

−

0.0125459i

0.607265

2.06815i

1.68251

1.08128i

399.1

−0.0872586

−

0.0125459i

−0.909632

0.415415i

−1.91153

−

0.561276i

0.0845850

−

0.0248364i

0.835939

−

1.30075i

0.320135

0.146201i

−1.30972

1.51150i

399.2

0.0872586

0.0125459i

0.909632

−

0.415415i

−1.91153

−

0.561276i

0.0845850

−

0.0248364i

−0.835939

1.30075i

−0.320135

−

0.146201i

−1.30972

1.51150i

449.1

−0.835939

−

1.30075i

−0.989821

−

0.142315i

−0.162317

0.355426i

0.642315

1.40647i

−1.81219

1.57028i

−2.46292

0.354114i

−1.91899

−

0.563465i

449.2

0.835939

1.30075i

0.989821

0.142315i

−0.162317

0.355426i

0.642315

1.40647i

1.81219

−

1.57028i

2.46292

−

0.354114i

−1.91899

−

0.563465i

499.1

−1.81219

−

1.57028i

−0.540641

−

0.841254i

0.533654

3.71165i

−0.341254

2.37347i

−0.386758

−

1.31718i

2.26844

−

3.52977i

0.830830

−

1.81926i

499.2

1.81219

1.57028i

0.540641

0.841254i

0.533654

3.71165i

−0.341254

2.37347i

0.386758

1.31718i

−2.26844

3.52977i

0.830830

−

1.81926i

524.1

−0.386758

1.31718i

−0.755750

−

0.654861i

0.0971309

0.0624222i

1.15486

−

0.742184i

−2.02730

0.925839i

−2.19475

1.90176i

−0.284630

−

1.97964i

524.2

0.386758

−

1.31718i

0.755750

0.654861i

0.0971309

0.0624222i

1.15486

−

0.742184i

2.02730

−

0.925839i

2.19475

−

1.90176i

−0.284630

−

1.97964i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
5.b	even	2	1	inner
23.c	even	11	1	inner
115.j	even	22	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	575.2.p.a		20
5.b	even	2	1	inner	575.2.p.a		20
5.c	odd	4	1		115.2.g.a	✓	10
5.c	odd	4	1		575.2.k.a		10
23.c	even	11	1	inner	575.2.p.a		20
115.j	even	22	1	inner	575.2.p.a		20
115.k	odd	44	1		115.2.g.a	✓	10
115.k	odd	44	1		575.2.k.a		10
115.k	odd	44	1		2645.2.a.n		5
115.l	even	44	1		2645.2.a.o		5

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
115.2.g.a	✓	10	5.c	odd	4	1
115.2.g.a	✓	10	115.k	odd	44	1
575.2.k.a		10	5.c	odd	4	1
575.2.k.a		10	115.k	odd	44	1
575.2.p.a		20	1.a	even	1	1	trivial
575.2.p.a		20	5.b	even	2	1	inner
575.2.p.a		20	23.c	even	11	1	inner
575.2.p.a		20	115.j	even	22	1	inner
2645.2.a.n		5	115.k	odd	44	1
2645.2.a.o		5	115.l	even	44	1

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{20} - 3 T_{2}^{18} + 42 T_{2}^{16} - 5 T_{2}^{14} + 378 T_{2}^{12} + 1781 T_{2}^{10} + 7626 T_{2}^{8} + \cdots + 1 $ T2^20 - 3*T2^18 + 42*T2^16 - 5*T2^14 + 378*T2^12 + 1781*T2^10 + 7626*T2^8 + 15237*T2^6 + 16329*T2^4 - 246*T2^2 + 1 acting on $S_{2}^{\mathrm{new}}(575, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{20} - 3 T^{18} + \cdots + 1 $ T^20 - 3T^18 + 42T^16 - 5T^14 + 378T^12 + 1781T^10 + 7626T^8 + 15237T^6 + 16329T^4 - 246*T^2 + 1
$3$	$ T^{20} - T^{18} + \cdots + 1 $ T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	$ T^{20} $ T^20
$7$	$ T^{20} - 3 T^{18} + \cdots + 1 $ T^20 - 3T^18 + 42T^16 - 5T^14 + 378T^12 + 1781T^10 + 7626T^8 + 15237T^6 + 16329T^4 - 246*T^2 + 1
$11$	$ (T^{10} - 8 T^{9} + \cdots + 7921)^{2} $ (T^10 - 8T^9 + 31T^8 - 50T^7 - 7T^6 + 353T^5 - 602T^4 + 482T^3 + 2447T^2 - 4539*T + 7921)^2
$13$	$ T^{20} - 22 T^{18} + \cdots + 14641 $ T^20 - 22T^18 + 121T^16 - 121T^14 + 4114T^12 - 2420T^10 + 25289T^8 - 1331T^6 + 14641T^2 + 14641
$17$	$ T^{20} + \cdots + 217611987121 $ T^20 - T^18 - 87T^16 - 4324T^14 + 34112T^12 - 396518T^10 + 20088102T^8 - 214330359T^6 + 4440100391T^4 - 34380705789T^2 + 217611987121
$19$	$ (T^{10} - 13 T^{9} + \cdots + 11881)^{2} $ (T^10 - 13T^9 + 48T^8 - 85T^7 + 665T^6 - 890T^5 + 4068T^4 - 8037T^3 + 17867T^2 - 20928*T + 11881)^2
$23$	$ T^{20} + \cdots + 41426511213649 $ T^20 - 45T^18 + 2245T^16 - 59797T^14 + 1905289T^12 - 39268725T^10 + 1007897881T^8 - 16733652277T^6 + 332340570805T^4 - 3523994337645*T^2 + 41426511213649
$29$	$ (T^{10} + 2 T^{9} + \cdots + 127893481)^{2} $ (T^10 + 2T^9 + 70T^8 + 338T^7 + 4185T^6 - 4599T^5 + 41578T^4 - 348308T^3 - 1971142T^2 - 13491637*T + 127893481)^2
$31$	$ (T^{10} + 20 T^{9} + \cdots + 7921)^{2} $ (T^10 + 20T^9 + 158T^8 + 344T^7 + 27T^6 - 3112T^5 + 3815T^4 - 6068T^3 + 8616T^2 + 445*T + 7921)^2
$37$	$ T^{20} + \cdots + 131912598473041 $ T^20 - 169T^18 + 28561T^16 - 1973629T^14 + 105917617T^12 + 4918104839T^10 - 133367878607T^8 - 16856673051685T^6 + 518624629976737T^4 + 214561956126647*T^2 + 131912598473041
$41$	$ (T^{10} + 5 T^{9} + \cdots + 193293409)^{2} $ (T^10 + 5T^9 + 91T^8 + 1082T^7 + 4684T^6 + 37676T^5 + 382255T^4 + 1304768T^3 + 4364782T^2 + 46825304*T + 193293409)^2
$43$	$ T^{20} + \cdots + 18324914739121 $ T^20 - 269T^18 + 19913T^16 - 296058T^14 + 51852458T^12 - 1361845882T^10 + 30324162599T^8 + 321911617860T^6 + 1445205162130T^4 + 3136376600348*T^2 + 18324914739121
$47$	$ (T^{10} + 298 T^{8} + \cdots + 96255721)^{2} $ (T^10 + 298T^8 + 30477T^6 + 1293564T^4 + 20991512T^2 + 96255721)^2
$53$	$ T^{20} + \cdots + 1985193642961 $ T^20 - 322T^18 + 44735T^16 - 2204273T^14 - 69364511T^12 + 4290759274T^10 + 402078735226T^8 - 619721438950T^6 + 350775932336076T^4 + 21921979594720*T^2 + 1985193642961
$59$	$ (T^{10} + 5 T^{9} + \cdots + 437688241)^{2} $ (T^10 + 5T^9 + 201T^8 + 587T^7 + 14672T^6 + 2080T^5 + 295674T^4 + 1827862T^3 + 40431593T^2 - 275424965*T + 437688241)^2
$61$	$ (T^{10} + 3 T^{9} + \cdots + 436921)^{2} $ (T^10 + 3T^9 + 75T^8 + 192T^7 + 1126T^6 + 15951T^5 + 71481T^4 + 161258T^3 + 157327T^2 - 564494*T + 436921)^2
$67$	$ T^{20} + \cdots + 655026269697361 $ T^20 - 180T^18 + 28990T^16 - 1166251T^14 + 150817989T^12 + 6646473184T^10 + 237665561412T^8 + 3193640193167T^6 + 9453082583272T^4 - 247445145230528*T^2 + 655026269697361
$71$	$ (T^{10} + 22 T^{9} + \cdots + 1437601)^{2} $ (T^10 + 22T^9 + 275T^8 + 3333T^7 + 27962T^6 + 113036T^5 + 275759T^4 + 486783T^3 + 931216T^2 + 1411223*T + 1437601)^2
$73$	$ T^{20} + \cdots + 61\!\cdots\!21 $ T^20 + 109T^18 + 2872T^16 - 716343T^14 - 29900903T^12 - 1352297552T^10 + 321144829270T^8 - 5917766441489T^6 + 347713644075645T^4 - 1915821071152330*T^2 + 6181684471968721
$79$	$ (T^{10} + 51 T^{9} + \cdots + 246772681)^{2} $ (T^10 + 51T^9 + 1193T^8 + 17008T^7 + 167896T^6 + 1264064T^5 + 7864278T^4 + 35175355T^3 + 113149199T^2 + 230812337*T + 246772681)^2
$83$	$ T^{20} + \cdots + 100598673352321 $ T^20 + 217T^18 + 7280T^16 - 2508115T^14 + 87969721T^12 - 964772656T^10 + 51699658574T^8 - 140284443121T^6 + 10935349571041T^4 - 42450361785154*T^2 + 100598673352321
$89$	$ (T^{10} + 57 T^{9} + \cdots + 43256929)^{2} $ (T^10 + 57T^9 + 1544T^8 + 26969T^7 + 336506T^6 + 3006321T^5 + 18714199T^4 + 71841315T^3 + 144725243T^2 + 81350913*T + 43256929)^2
$97$	$ T^{20} + \cdots + 38\!\cdots\!41 $ T^20 + 68T^18 + 32608T^16 - 2597389T^14 + 435881000T^12 + 25719103024T^10 + 851170881405T^8 + 19346189264673T^6 + 753525078139606T^4 - 3353601655763056*T^2 + 3849846217142641
show more
show less

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 575 = 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	575.p (of order \(22\), degree \(10\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \zeta_{44}^{15} + \cdots - \zeta_{44}^{3}) q^{2}+ \cdots + 2 \zeta_{44}^{12} q^{9}+O(q^{10}) \) q + (-z^15 + z^13 + z^9 + z^5 - z^3) * q^2 - z^17 * q^3 + (z^18 - z^10 - z^8 + z^6 + z^4) * q^4 + (z^18 - z^16 + z^14 - z^12 + z^6 + z^2) * q^6 + (-z^19 + z^9 + z^5 - z^3 + z) * q^7 + (-2z^19 + z^17 - z^15 + 2z^13 + 2z^9 - 2z^7 + 2z^5 - 2z^3 + 2z) q^8 + 2z^12 q^9	\( q + ( - \zeta_{44}^{15} + \cdots - \zeta_{44}^{3}) q^{2}+ \cdots + ( - 4 \zeta_{44}^{18} + 4 \zeta_{44}^{16} + \cdots + 2) q^{99}+O(q^{100}) \) q + (-z^15 + z^13 + z^9 + z^5 - z^3) * q^2 - z^17 * q^3 + (z^18 - z^10 - z^8 + z^6 + z^4) * q^4 + (z^18 - z^16 + z^14 - z^12 + z^6 + z^2) * q^6 + (-z^19 + z^9 + z^5 - z^3 + z) * q^7 + (-2z^19 + z^17 - z^15 + 2z^13 + 2z^9 - 2z^7 + 2z^5 - 2z^3 + 2z) q^8 + 2z^12 q^9 + (2z^18 - z^16 + z^14 - z^12 + z^10 - 2z^8) * q^11 + (-z^19 + z^17 - z^15 + 2z^13 - z^11 + z^9 - z^7 - 2z^3 + 2z) q^12 + (z^13 - z^9 + z^7 - z^3) * q^13 + (2z^18 - z^16 + 2z^14 - 2z^12 + 2z^10 - z^8 + 2z^6 + z^2 - 1) q^14 + (3z^18 - 2z^16 + 2z^14 - 2z^12 + 3z^10 + z^6 - z^4 + z^2 - 1) q^16 + (-3z^19 + z^17 + z^13 - 3z^11 - z^7 + z) * q^17 + (2z^19 - 2z^13 + 2z^11 - 2z^9 + 2z^7 - 2z) * q^18 + (-z^14 - z^12 + 2z^10 + z^8 - z^6 - 2z^4 + z^2 + 1) * q^19 + (-z^16 - z^12 + z^10 - z^8 + z^6 + z^2) * q^21 + (-3z^19 + 2z^17 - 4z^15 + 3z^13 - 2z^11 + 3z^9 - 4z^7 + 2z^5 - 3z^3) q^22 + (-3z^19 + 2z^17 - 2z^15 + 2z^9 - 4z^7 + 2z^5 - 2z^3 + 4z) * q^23 + (-2z^16 - z^12 + z^10 + 2z^6) * q^24 + (2z^18 - 2z^16 - 2z^8 + 3z^6 - 2z^4) q^26 + 5z^7 q^27 + (-z^19 + 2z^13 - 2z^7 + z) * q^28 + (-z^18 + 2z^16 - 2z^14 + z^12 + 2z^8 - 6z^6 + 2z^4 - 6z^2 + 2) * q^29 + (z^18 - 3z^16 + z^14 + 3z^10 + z^8 + 2z^6 - 2z^4 - z^2 - 3) * q^31 + (z^19 - 2z^17 + z^13 - z^11 + 2z^7 - z^5) * q^32 + (2z^13 - z^11 + z^9 - z^7 + z^5 - 2z^3) * q^33 + (-4z^18 - z^16 - z^14 + z^12 + z^10 + 4z^8 - z^4 + z^2 - 1) * q^34 + (4z^16 - 2z^14 + 2z^12 - 2z^10 - 2z^6 + 2z^4 - 2z^2 + 4) q^36 + (-6z^15 + 6z^9 + z) * q^37 + (3z^19 - 3z^17 + 3z^15 - 3z^13 - z^9 + 3z^7 - z^5 + 3z^3 - z) * q^38 + (z^18 - z^16 + z^14 - z^12 + z^10 + z^6 - 2z^4 + 2z^2 - 1) * q^39 + (3z^18 + 3z^16 - z^14 + 2z^12 + z^10 + 2z^8 - z^6 + 3z^4 + 3z^2) * q^41 + (-z^19 + z^17 + 2z^13 - z^11 + 2z^9 - 2z^7 + 2z^5 - z^3 + 2z) q^42 + (4z^17 - z^11 - 2z^9 + 3z^7 - 3z^5 + 2z^3 + z) q^43 + (z^18 + 2z^16 - 3z^14 - z^12 - z^10 + z^8 + z^6 + 3z^4 - 2z^2 - 1) * q^44 + (-2z^18 - 2z^14 - z^12 + 5z^10 + 2z^8 + z^6 + 4z^4 - 3z^2 - 1) * q^46 + (3z^19 - 4z^17 + 4z^15 - 5z^13 - 2z^11 - 5z^9 + 4z^7 - 4z^5 + 3z^3) q^47 + (z^15 + 2z^13 - z^11 + z^9 - z^7 + 2z^5 + z^3) * q^48 + (-z^18 + z^16 + z^10 - 4z^6 + z^2) q^49 + (-z^18 - 3z^14 + z^12 + z^8 - 3z^6 - z^2) * q^51 + (3z^17 - 4z^15 + 3z^13 - 3z^7 + 3z^5 - 3z^3 + 3z) q^52 + (z^19 - 2z^17 + 3z^15 - 3z^13 + 2z^11 - z^9 - 7z^5 + 2z^3 - 7z) q^53 + (5z^18 + 5z^14 - 5z^8 + 5z^6 - 5z^4 + 5z^2) * q^54 + (-z^16 + z^14 - z^12 + 4z^10 + 4z^6 - z^4 + z^2 - 1) * q^56 + (z^19 - 3z^17 + 2z^15 - 2z^13 + 2z^11 - 3z^9 + z^7 + 3z^3 - 3z) q^57 + (z^17 - 3z^15 - 6z^13 - 3z^11 + z^9 + 7z^3 - 7z) q^58 + (-z^16 - 2z^14 - 8z^8 - 2z^2 - 1) q^59 + (-3z^18 + z^16 - 3z^14 - 2z^8 + 2z^2) * q^61 + (6z^19 + 4z^17 + 2z^15 - 2z^13 - 4z^11 - 6z^9 + z^5 + 4z^3 + z) q^62 + (2z^19 + 2z^11 + 2z^7 - 2z^5 + 2z^3 - 2z) * q^63 + (4z^18 - 2z^16 + 8z^14 - 3z^12 - 3z^8 + 8z^6 - 2z^4 + 4z^2) * q^64 + (3z^18 - 3z^16 - z^12 - z^10 + z^6 - z^2 - 1) * q^66 + (-4z^17 + 4z^15 - 5z^13 + 5z^11 + z^9 + 5z^7 - 5z^5 + 4z^3 - 4z) * q^67 + (8z^19 - 4z^17 + 4z^15 - 5z^13 + z^11 - 5z^9 + 4z^7 - 4z^5 + 8z^3) * q^68 + (-2z^18 - 2z^16 - z^14 - 2z^8 + 2z^6 - 2z^2) q^69 + (2z^18 + 3z^16 + 2z^14 - z^12 - 2z^10 + 2z^8 + z^6 - 2z^4 - 3z^2 - 2) q^71 + (4z^11 + 2z^7 - 2z^5 - 4z) * q^72 + (z^19 - z^17 - 3z^13 - 3z^11 - 2z^9 + 4z^7 - 2z^5 - 3z^3 - 3z) q^73 + (6z^18 + 5z^16 + z^14 - 5z^10 + z^6 + 5z^4 + 6z^2) q^74 + (-5z^18 + 2z^16 + 5z^14 + 2z^12 - 5z^10 - z^6 + z^4 - z^2 + 1) q^76 + (z^19 - z^17 + z^15 - z^13 + z^9 - 3z^7 - 3z^3 + z) * q^77 + (2z^19 - 2z^17 + 2z^15 - 2z^9 + 2z^7 - 2z^5 + z) * q^78 + (-3z^16 - 5z^14 + 3z^12 - 2z^10 + 4z^8 - 2z^6 + 3z^4 - 5z^2 - 3) * q^79 - z^2 * q^81 + (-2z^19 + 6z^15 - z^13 + 8z^11 - z^9 + z^7 - 8z^5 + z^3 - 6z) q^82 + (-z^19 - 4z^17 + 3z^15 - 3z^13 + 3z^11 - 3z^9 + 4z^7 + z^5 + 2z) q^83 + (-z^18 - z^14 + 2z^8 - 2z^2) * q^84 + (-6z^18 + 2z^16 - 2z^14 + 6z^12 - 7z^4) q^86 + (4z^19 - 2z^15 + z^13 - z^7 + 2z^5 - 4z) * q^87 + (-2z^19 + 2z^17 + 2z^13 - 3z^11 + 2z^9 - 8z^7 + 2z^5 - 3z^3 + 2z) q^88 + (-5z^12 - z^10 + 2z^8 - 6z^6 + 2z^4 - z^2 - 5) * q^89 + (z^12 - z^10 - 2) * q^91 + (-2z^19 + 4z^17 + 5z^15 - 2z^13 - 4z^11 - 2z^9 - 3z^7 + 3z^5 + 5z^3 - z) q^92 + (3z^19 + z^17 + 2z^15 - z^13 - z^11 - z^9 + 2z^7 + z^5 + 3z^3) * q^93 + (-7z^18 - 4z^16 + 7z^14 + z^12 - z^10 + z^8 - z^6 - 7z^4 + 4z^2 + 7) q^94 + (z^14 - 2z^12 + z^8 - z^6 + 2z^2 - 1) * q^96 + (-4z^19 + 4z^17 - 4z^15 + 5z^13 + 2z^11 - 2z^9 - 5z^7 + 4z^5 - 4z^3 + 4z) * q^97 + (2z^19 - 7z^17 + 4z^15 - 7z^13 + 2z^11 + 6z^7 - 6z^5 + 6z^3 - 6z) q^98 + (-4z^18 + 4z^16 - 4z^14 + 4z^12 - 4z^10 - 2z^6 + 2z^4 - 2z^2 + 2) * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 2 q^{4} + 12 q^{6} - 4 q^{9}+O(q^{10}) \) 20 * q + 2 * q^4 + 12 * q^6 - 4 * q^9	\( 20 q + 2 q^{4} + 12 q^{6} - 4 q^{9} + 16 q^{11} + 6 q^{14} + 10 q^{16} + 26 q^{19} + 12 q^{21} + 12 q^{24} + 22 q^{26} - 4 q^{29} - 40 q^{31} - 32 q^{34} + 48 q^{36} - 10 q^{41} - 38 q^{44} - 32 q^{46} - 8 q^{49} - 20 q^{51} + 60 q^{54} + 6 q^{56} - 10 q^{59} - 6 q^{61} + 68 q^{64} - 8 q^{66} + 2 q^{69} - 44 q^{71} - 2 q^{74} - 4 q^{76} - 102 q^{79} - 2 q^{81} - 12 q^{84} - 18 q^{86} - 114 q^{89} - 44 q^{91} + 162 q^{94} - 14 q^{96} - 12 q^{99}+O(q^{100}) \) 20 * q + 2 * q^4 + 12 * q^6 - 4 * q^9 + 16 * q^11 + 6 * q^14 + 10 * q^16 + 26 * q^19 + 12 * q^21 + 12 * q^24 + 22 * q^26 - 4 * q^29 - 40 * q^31 - 32 * q^34 + 48 * q^36 - 10 * q^41 - 38 * q^44 - 32 * q^46 - 8 * q^49 - 20 * q^51 + 60 * q^54 + 6 * q^56 - 10 * q^59 - 6 * q^61 + 68 * q^64 - 8 * q^66 + 2 * q^69 - 44 * q^71 - 2 * q^74 - 4 * q^76 - 102 * q^79 - 2 * q^81 - 12 * q^84 - 18 * q^86 - 114 * q^89 - 44 * q^91 + 162 * q^94 - 14 * q^96 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

$q$-expansion

Character values

Embeddings

Inner twists

Twists

Hecke kernels

Hecke characteristic polynomials

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	575.2.p.a

Level	$575$
Weight	$2$
Character orbit	575.p
Analytic conductor	$4.591$
Analytic rank	$0$
Dimension	$20$
CM	no
Inner twists	$4$

Self dual:	no
Analytic conductor:	\(4.59139811622\)
Analytic rank:	\(0\)
Dimension:	\(20\)
Relative dimension:	\(2\) over \(\Q(\zeta_{22})\)
Coefficient field:	\(\Q(\zeta_{44})\)
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	\( x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) x^20 - x^18 + x^16 - x^14 + x^12 - x^10 + x^8 - x^6 + x^4 - x^2 + 1
Coefficient ring:	\(\Z[a_1, a_2, a_3]\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 115)
Sato-Tate group:	$\mathrm{SU}(2)[C_{22}]$

\(n\)	\(51\)	\(277\)
\(\chi(n)\)	\(-1 + \zeta_{44}^{2} - \zeta_{44}^{4} + \zeta_{44}^{6} - \zeta_{44}^{8} + \zeta_{44}^{10} - \zeta_{44}^{12} + \zeta_{44}^{14} - \zeta_{44}^{16} + \zeta_{44}^{18}\)	\(-1\)

\(n\):		e.g. 2-40 or 990-1000
Embeddings:		e.g. 1-3 or 49.2
Significant digits:
Format:

$p$	$F_p(T)$
$2$	\( T^{20} - 3 T^{18} + \cdots + 1 \) T^20 - 3T^18 + 42T^16 - 5T^14 + 378T^12 + 1781T^10 + 7626T^8 + 15237T^6 + 16329T^4 - 246*T^2 + 1
$3$	\( T^{20} - T^{18} + \cdots + 1 \) T^20 - T^18 + T^16 - T^14 + T^12 - T^10 + T^8 - T^6 + T^4 - T^2 + 1
$5$	\( T^{20} \) T^20
$7$	\( T^{20} - 3 T^{18} + \cdots + 1 \) T^20 - 3T^18 + 42T^16 - 5T^14 + 378T^12 + 1781T^10 + 7626T^8 + 15237T^6 + 16329T^4 - 246*T^2 + 1
$11$	\( (T^{10} - 8 T^{9} + \cdots + 7921)^{2} \) (T^10 - 8T^9 + 31T^8 - 50T^7 - 7T^6 + 353T^5 - 602T^4 + 482T^3 + 2447T^2 - 4539*T + 7921)^2
$13$	\( T^{20} - 22 T^{18} + \cdots + 14641 \) T^20 - 22T^18 + 121T^16 - 121T^14 + 4114T^12 - 2420T^10 + 25289T^8 - 1331T^6 + 14641T^2 + 14641
$17$	\( T^{20} + \cdots + 217611987121 \) T^20 - T^18 - 87T^16 - 4324T^14 + 34112T^12 - 396518T^10 + 20088102T^8 - 214330359T^6 + 4440100391T^4 - 34380705789T^2 + 217611987121
$19$	\( (T^{10} - 13 T^{9} + \cdots + 11881)^{2} \) (T^10 - 13T^9 + 48T^8 - 85T^7 + 665T^6 - 890T^5 + 4068T^4 - 8037T^3 + 17867T^2 - 20928*T + 11881)^2
$23$	\( T^{20} + \cdots + 41426511213649 \) T^20 - 45T^18 + 2245T^16 - 59797T^14 + 1905289T^12 - 39268725T^10 + 1007897881T^8 - 16733652277T^6 + 332340570805T^4 - 3523994337645*T^2 + 41426511213649
$29$	\( (T^{10} + 2 T^{9} + \cdots + 127893481)^{2} \) (T^10 + 2T^9 + 70T^8 + 338T^7 + 4185T^6 - 4599T^5 + 41578T^4 - 348308T^3 - 1971142T^2 - 13491637*T + 127893481)^2
$31$	\( (T^{10} + 20 T^{9} + \cdots + 7921)^{2} \) (T^10 + 20T^9 + 158T^8 + 344T^7 + 27T^6 - 3112T^5 + 3815T^4 - 6068T^3 + 8616T^2 + 445*T + 7921)^2
$37$	\( T^{20} + \cdots + 131912598473041 \) T^20 - 169T^18 + 28561T^16 - 1973629T^14 + 105917617T^12 + 4918104839T^10 - 133367878607T^8 - 16856673051685T^6 + 518624629976737T^4 + 214561956126647*T^2 + 131912598473041
$41$	\( (T^{10} + 5 T^{9} + \cdots + 193293409)^{2} \) (T^10 + 5T^9 + 91T^8 + 1082T^7 + 4684T^6 + 37676T^5 + 382255T^4 + 1304768T^3 + 4364782T^2 + 46825304*T + 193293409)^2
$43$	\( T^{20} + \cdots + 18324914739121 \) T^20 - 269T^18 + 19913T^16 - 296058T^14 + 51852458T^12 - 1361845882T^10 + 30324162599T^8 + 321911617860T^6 + 1445205162130T^4 + 3136376600348*T^2 + 18324914739121
$47$	\( (T^{10} + 298 T^{8} + \cdots + 96255721)^{2} \) (T^10 + 298T^8 + 30477T^6 + 1293564T^4 + 20991512T^2 + 96255721)^2
$53$	\( T^{20} + \cdots + 1985193642961 \) T^20 - 322T^18 + 44735T^16 - 2204273T^14 - 69364511T^12 + 4290759274T^10 + 402078735226T^8 - 619721438950T^6 + 350775932336076T^4 + 21921979594720*T^2 + 1985193642961
$59$	\( (T^{10} + 5 T^{9} + \cdots + 437688241)^{2} \) (T^10 + 5T^9 + 201T^8 + 587T^7 + 14672T^6 + 2080T^5 + 295674T^4 + 1827862T^3 + 40431593T^2 - 275424965*T + 437688241)^2
$61$	\( (T^{10} + 3 T^{9} + \cdots + 436921)^{2} \) (T^10 + 3T^9 + 75T^8 + 192T^7 + 1126T^6 + 15951T^5 + 71481T^4 + 161258T^3 + 157327T^2 - 564494*T + 436921)^2
$67$	\( T^{20} + \cdots + 655026269697361 \) T^20 - 180T^18 + 28990T^16 - 1166251T^14 + 150817989T^12 + 6646473184T^10 + 237665561412T^8 + 3193640193167T^6 + 9453082583272T^4 - 247445145230528*T^2 + 655026269697361
$71$	\( (T^{10} + 22 T^{9} + \cdots + 1437601)^{2} \) (T^10 + 22T^9 + 275T^8 + 3333T^7 + 27962T^6 + 113036T^5 + 275759T^4 + 486783T^3 + 931216T^2 + 1411223*T + 1437601)^2
$73$	\( T^{20} + \cdots + 61\!\cdots\!21 \) T^20 + 109T^18 + 2872T^16 - 716343T^14 - 29900903T^12 - 1352297552T^10 + 321144829270T^8 - 5917766441489T^6 + 347713644075645T^4 - 1915821071152330*T^2 + 6181684471968721
$79$	\( (T^{10} + 51 T^{9} + \cdots + 246772681)^{2} \) (T^10 + 51T^9 + 1193T^8 + 17008T^7 + 167896T^6 + 1264064T^5 + 7864278T^4 + 35175355T^3 + 113149199T^2 + 230812337*T + 246772681)^2
$83$	\( T^{20} + \cdots + 100598673352321 \) T^20 + 217T^18 + 7280T^16 - 2508115T^14 + 87969721T^12 - 964772656T^10 + 51699658574T^8 - 140284443121T^6 + 10935349571041T^4 - 42450361785154*T^2 + 100598673352321
$89$	\( (T^{10} + 57 T^{9} + \cdots + 43256929)^{2} \) (T^10 + 57T^9 + 1544T^8 + 26969T^7 + 336506T^6 + 3006321T^5 + 18714199T^4 + 71841315T^3 + 144725243T^2 + 81350913*T + 43256929)^2
$97$	\( T^{20} + \cdots + 38\!\cdots\!41 \) T^20 + 68T^18 + 32608T^16 - 2597389T^14 + 435881000T^12 + 25719103024T^10 + 851170881405T^8 + 19346189264673T^6 + 753525078139606T^4 - 3353601655763056*T^2 + 3849846217142641
show more
show less