LMFDB - Newform orbit 725.2.c.h

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [725,2,Mod(376,725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(725, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("725.376");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$725 = 5^{2} \cdot 29$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	725.c (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:	$5.78915414654$
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:	$x^{10} + 16x^{8} + 89x^{6} + 209x^{4} + 185x^{2} + 25$ x^10 + 16x^8 + 89x^6 + 209x^4 + 185x^2 + 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$5$
Twist minimal:	yes
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.

$f(q)$	$=$	$q + \beta_1 q^{2} - \beta_{4} q^{3} + (\beta_{2} - 1) q^{4} + ( - \beta_{9} + \beta_{3} + 1) q^{6} + \beta_{9} q^{7} + (\beta_{8} - \beta_{7} - \beta_{5} - \beta_1) q^{8} + (\beta_{6} + \beta_{3} - 1) q^{9}+ \cdots + ( - 2 \beta_{8} + 4 \beta_{7} + \cdots - 4 \beta_1) q^{99}+O(q^{100})$ q + b1 * q^2 - b4 * q^3 + (b2 - 1) * q^4 + (-b9 + b3 + 1) * q^6 + b9 * q^7 + (b8 - b7 - b5 - b1) * q^8 + (b6 + b3 - 1) * q^9 + (b5 - b4 + b1) * q^11 + (b7 + b4) * q^12 + (b6 - b2 - 1) * q^13 + (-b8 - b5 + b1) * q^14 + (-b9 + 2b6 + b3 - 2b2) * q^16 + b8 * q^17 + (-2b8 + 2b7 - b5 + 2b4 - b1) q^18 + (2b7 + b5) q^19 + (b8 - b7 + b5 + b1) * q^21 + (b3 + 2b2 - 2) q^22 + (b9 - b6 - b3 + b2) * q^23 + (-b6 + b2) * q^24 + (-2b8 + 2b7 + b5 - b4 + b1) * q^26 + (b8 + b7 + b4 - 2b1) q^27 + (-b6 + b2 - 1) * q^28 + (-b9 + b7 - b5 + b4 - b2 + 2) * q^29 + (b8 - b7 + 2b5 - b4 + b1) q^31 + (-2b8 + 3b7 + b4 + b1) * q^32 + (-2b9 + b6 + 2b3 - 1) * q^33 + (b9 + b6 - b2 - 2) * q^34 + (b9 - 2b6 - 2b3 + 2b2 + 1) q^36 + (-2b8 - 2b5 + 2b4 + b1) q^37 + (3b9 - 2b6 - 2b3 + 3b2 - 2) * q^38 + (2b8 - b7 + b4 + b1) q^39 + (-b8 - b5 + 3b4 - b1) q^41 + (b9 + 2b6 + b3 - 4) q^42 + (-b7 + b5 + 3b1) q^43 + (b8 - b7 - b5 + b4 - 4b1) q^44 + (2b8 - 3b7 - b5 - 2b4 - b1) q^46 + (b8 - 2b7 - 2b5 - 2b4 + b1) q^47 + (2b8 - b5 + 3b4 - 2b1) q^48 + (b9 - b6 + b3 + 2b2 + 1) q^49 + (-b9 - 2b6 + 3) q^51 + (-2b6 - b3 + 4b2 - 2) * q^52 + (b9 + 2b6 - b3 - 2b2) * q^53 + (3b9 - 2b3 - 2b2 + 2) q^54 + (-2b7 - 3b5 + b4 - b1) * q^56 + (b9 - 2b6 - 2b3 - 2b2 + 2) q^57 + (b9 + b7 - b6 + 2b5 - 2b3 + 3b1 - 2) q^58 + (3b9 - b6 - b3 - 1) q^59 + (-3b8 + 2b7 + b5) * q^61 + (b9 + 2b6 + 2b3 + b2 - 3) * q^62 + (-b9 - b6 + 2b3 + b2 + 6) q^63 + (-b6 - 2b3 + 2b2 - 3) * q^64 + (-b8 + 3b7 + 5b4 - 3b1) q^66 + (b6 - b3 + 2b2 - 5) q^67 + (-b8 + 2b7 - b4 + b1) q^68 + (-b7 + b5 - 3b4 + 3b1) * q^69 + (b6 - 2b3 - 3b2 - 2) * q^71 + (b8 - 2b7 - 3b5 - 4b1) q^72 + (3b5 - 3b4) * q^73 + (-2b9 - 2b6 - 2b3 + b2 - 1) q^74 + (4b8 - 3b7 - 2b5 - 4b4 - 5b1) q^76 + (b7 + 3b5 - 3b4) * q^77 + (2b9 + 3b6 - 2b2 - 7) q^78 + (b7 - b5 + 3b4 + 3b1) * q^79 + (2b9 - b6 - b3 - b2 + 2) q^81 + (b9 - b6 - 3b3 - b2 + 2) q^82 + (-2b9 + 3b3 - 2b2 + 1) q^83 + (-2b8 + b7 + b4 - b1) q^84 + (b6 + b3 + 3b2 - 8) q^86 + (2b9 - b8 - 2b6 - b5 - 2b4 - 2b3 - b2 - b1 + 2) * q^87 + (2b6 + 2b3 - 3b2 + 6) q^88 + (-2b8 - b7 - b5 - b4 + 3b1) * q^89 + (-3b9 - b6 - 2b2 + 3) * q^91 + (-2b9 + 3b6 + 3b3 - 5b2 + 4) * q^92 + (-5b9 + 3b3 + b2 + 4) * q^93 + (-5b9 + 3b6 + 4b3 - 4b2 - 1) * q^94 + (4b9 - 3b3 - 3b2 - 1) q^96 + (b8 - 2b7 + 2b4 - 5b1) q^97 + (b8 - 2b7 - 4b5 + 4b4 - 2b1) * q^98 + (-2b8 + 4b7 + b5 + 4b4 - 4b1) * q^99
$\operatorname{Tr}(f)(q)$	$=$	$10 q - 12 q^{4} + 2 q^{6} + 4 q^{7} - 10 q^{9} - 4 q^{13} + 4 q^{16} - 28 q^{22} + 2 q^{23} - 6 q^{24} - 16 q^{28} + 18 q^{29} - 22 q^{33} - 10 q^{34} + 10 q^{36} - 14 q^{38} - 32 q^{42} + 2 q^{49} + 18 q^{51}+ \cdots + 24 q^{96}+O(q^{100})$ 10 * q - 12 * q^4 + 2 * q^6 + 4 * q^7 - 10 * q^9 - 4 * q^13 + 4 * q^16 - 28 * q^22 + 2 * q^23 - 6 * q^24 - 16 * q^28 + 18 * q^29 - 22 * q^33 - 10 * q^34 + 10 * q^36 - 14 * q^38 - 32 * q^42 + 2 * q^49 + 18 * q^51 - 32 * q^52 + 20 * q^53 + 44 * q^54 + 28 * q^57 - 12 * q^58 + 2 * q^59 - 28 * q^62 + 42 * q^63 - 30 * q^64 - 46 * q^67 - 2 * q^71 - 20 * q^74 - 46 * q^78 + 30 * q^81 + 34 * q^82 - 6 * q^83 - 86 * q^86 + 30 * q^87 + 66 * q^88 + 18 * q^91 + 42 * q^92 + 6 * q^93 - 26 * q^94 + 24 * q^96

Basis of coefficient ring in terms of a root $\nu$ of $x^{10} + 16x^{8} + 89x^{6} + 209x^{4} + 185x^{2} + 25$ x^10 + 16*x^8 + 89*x^6 + 209*x^4 + 185*x^2 + 25 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$\nu^{2} + 3$ v^2 + 3
$\beta_{3}$	$=$	$( \nu^{8} + 10\nu^{6} + 19\nu^{4} - 15\nu^{2} - 20 ) / 5$ (v^8 + 10v^6 + 19v^4 - 15*v^2 - 20) / 5
$\beta_{4}$	$=$	$( \nu^{9} + 12\nu^{7} + 39\nu^{5} + 28\nu^{3} - 15\nu ) / 5$ (v^9 + 12v^7 + 39v^5 + 28v^3 - 15v) / 5
$\beta_{5}$	$=$	$( \nu^{9} + 13\nu^{7} + 49\nu^{5} + 47\nu^{3} - 25\nu ) / 5$ (v^9 + 13v^7 + 49v^5 + 47v^3 - 25v) / 5
$\beta_{6}$	$=$	$( -2\nu^{8} - 25\nu^{6} - 88\nu^{4} - 80\nu^{2} + 15 ) / 5$ (-2v^8 - 25v^6 - 88v^4 - 80v^2 + 15) / 5
$\beta_{7}$	$=$	$( \nu^{9} + 14\nu^{7} + 64\nu^{5} + 116\nu^{3} + 70\nu ) / 5$ (v^9 + 14v^7 + 64v^5 + 116v^3 + 70v) / 5
$\beta_{8}$	$=$	$( 2\nu^{9} + 27\nu^{7} + 113\nu^{5} + 168\nu^{3} + 70\nu ) / 5$ (2v^9 + 27v^7 + 113v^5 + 168v^3 + 70*v) / 5
$\beta_{9}$	$=$	$( -3\nu^{8} - 40\nu^{6} - 162\nu^{4} - 215\nu^{2} - 40 ) / 5$ (-3v^8 - 40v^6 - 162v^4 - 215v^2 - 40) / 5

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{2} - 3$ b2 - 3
$\nu^{3}$	$=$	$\beta_{8} - \beta_{7} - \beta_{5} - 5\beta_1$ b8 - b7 - b5 - 5*b1
$\nu^{4}$	$=$	$-\beta_{9} + 2\beta_{6} + \beta_{3} - 8\beta_{2} + 14$ -b9 + 2b6 + b3 - 8b2 + 14
$\nu^{5}$	$=$	$-10\beta_{8} + 11\beta_{7} + 8\beta_{5} + \beta_{4} + 29\beta_1$ -10b8 + 11b7 + 8b5 + b4 + 29b1
$\nu^{6}$	$=$	$10\beta_{9} - 21\beta_{6} - 12\beta_{3} + 58\beta_{2} - 79$ 10b9 - 21b6 - 12b3 + 58b2 - 79
$\nu^{7}$	$=$	$81\beta_{8} - 91\beta_{7} - 56\beta_{5} - 15\beta_{4} - 185\beta_1$ 81b8 - 91b7 - 56b5 - 15b4 - 185*b1
$\nu^{8}$	$=$	$-81\beta_{9} + 172\beta_{6} + 106\beta_{3} - 413\beta_{2} + 499$ -81b9 + 172b6 + 106b3 - 413b2 + 499
$\nu^{9}$	$=$	$-610\beta_{8} + 691\beta_{7} + 388\beta_{5} + 146\beta_{4} + 1244\beta_1$ -610b8 + 691b7 + 388b5 + 146b4 + 1244*b1

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/725\mathbb{Z}\right)^\times$ .

$n$	$176$	$552$
$\chi(n)$	$-1$	$1$

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}

376.1

	−	2.66826i
	−	1.94602i
	−	1.75883i
	−	1.35526i
	−	0.403970i
		0.403970i
		1.35526i
		1.75883i
		1.94602i
		2.66826i

−

2.66826i

0.671523i

−5.11962

1.79180

1.28715

8.32395i

2.54906

376.2

−

1.94602i

−

3.02664i

−1.78699

−5.88990

1.26169

−

0.414527i

−6.16057

376.3

−

1.75883i

2.79103i

−1.09347

4.90893

−3.15518

−

1.59443i

−4.78982

376.4

−

1.35526i

0.587096i

0.163265

0.795669

4.41763

−

2.93179i

2.65532

376.5

−

0.403970i

−

1.50133i

1.83681

−0.606490

−1.81129

−

1.54995i

0.746019

376.6

0.403970i

1.50133i

1.83681

−0.606490

−1.81129

1.54995i

0.746019

376.7

1.35526i

−

0.587096i

0.163265

0.795669

4.41763

2.93179i

2.65532

376.8

1.75883i

−

2.79103i

−1.09347

4.90893

−3.15518

1.59443i

−4.78982

376.9

1.94602i

3.02664i

−1.78699

−5.88990

1.26169

0.414527i

−6.16057

376.10

2.66826i

−

0.671523i

−5.11962

1.79180

1.28715

−

8.32395i

2.54906

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	725.2.c.h	yes	10
5.b	even	2	1		725.2.c.g	✓	10
5.c	odd	4	2		725.2.d.d		20
29.b	even	2	1	inner	725.2.c.h	yes	10
145.d	even	2	1		725.2.c.g	✓	10
145.h	odd	4	2		725.2.d.d		20

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
725.2.c.g	✓	10	5.b	even	2	1
725.2.c.g	✓	10	145.d	even	2	1
725.2.c.h	yes	10	1.a	even	1	1	trivial
725.2.c.h	yes	10	29.b	even	2	1	inner
725.2.d.d		20	5.c	odd	4	2
725.2.d.d		20	145.h	odd	4	2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{2}^{\mathrm{new}}(725, [\chi])$ :

T_{2}^{10} + 16T_{2}^{8} + 89T_{2}^{6} + 209T_{2}^{4} + 185T_{2}^{2} + 25

T_{3}^{10} + 20T_{3}^{8} + 125T_{3}^{6} + 251T_{3}^{4} + 145T_{3}^{2} + 25

T_{7}^{5} - 2T_{7}^{4} - 16T_{7}^{3} + 17T_{7}^{2} + 38T_{7} - 41

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$T^{10} + 16 T^{8} + \cdots + 25$ T^10 + 16T^8 + 89T^6 + 209T^4 + 185T^2 + 25
$3$	$T^{10} + 20 T^{8} + \cdots + 25$ T^10 + 20T^8 + 125T^6 + 251T^4 + 145T^2 + 25
$5$	$T^{10}$ T^10
$7$	$(T^{5} - 2 T^{4} - 16 T^{3} + \cdots - 41)^{2}$ (T^5 - 2T^4 - 16T^3 + 17T^2 + 38T - 41)^2
$11$	$T^{10} + 44 T^{8} + \cdots + 625$ T^10 + 44T^8 + 611T^6 + 2960T^4 + 2650T^2 + 625
$13$	$(T^{5} + 2 T^{4} - 28 T^{3} + \cdots - 23)^{2}$ (T^5 + 2T^4 - 28T^3 - 31T^2 + 100T - 23)^2
$17$	$T^{10} + 47 T^{8} + \cdots + 25$ T^10 + 47T^8 + 563T^6 + 1529T^4 + 460T^2 + 25
$19$	$T^{10} + 140 T^{8} + \cdots + 6579225$ T^10 + 140T^8 + 7433T^6 + 182979T^4 + 1988955T^2 + 6579225
$23$	$(T^{5} - T^{4} - 38 T^{3} + \cdots - 27)^{2}$ (T^5 - T^4 - 38T^3 + 63T^2 + 27*T - 27)^2
$29$	$T^{10} - 18 T^{9} + \cdots + 20511149$ T^10 - 18T^9 + 173T^8 - 1224T^7 + 6958T^6 - 36396T^5 + 201782T^4 - 1029384T^3 + 4219297T^2 - 12731058*T + 20511149
$31$	$T^{10} + 173 T^{8} + \cdots + 455625$ T^10 + 173T^8 + 10586T^6 + 263115T^4 + 2083725T^2 + 455625
$37$	$T^{10} + 256 T^{8} + \cdots + 51194025$ T^10 + 256T^8 + 20897T^6 + 717741T^4 + 10525545T^2 + 51194025
$41$	$T^{10} + 173 T^{8} + \cdots + 625$ T^10 + 173T^8 + 7859T^6 + 106370T^4 + 50725T^2 + 625
$43$	$T^{10} + 182 T^{8} + \cdots + 1476225$ T^10 + 182T^8 + 12155T^6 + 354294T^4 + 3849120T^2 + 1476225
$47$	$T^{10} + 341 T^{8} + \cdots + 84916225$ T^10 + 341T^8 + 37907T^6 + 1617926T^4 + 21953935T^2 + 84916225
$53$	$(T^{5} - 10 T^{4} + \cdots - 25353)^{2}$ (T^5 - 10T^4 - 155T^3 + 1863T^2 - 621T - 25353)^2
$59$	$(T^{5} - T^{4} - 124 T^{3} + \cdots + 2565)^{2}$ (T^5 - T^4 - 124T^3 + 27T^2 + 3807*T + 2565)^2
$61$	$T^{10} + \cdots + 200930625$ T^10 + 401T^8 + 49871T^6 + 2073465T^4 + 34554600T^2 + 200930625
$67$	$(T^{5} + 23 T^{4} + \cdots - 4783)^{2}$ (T^5 + 23T^4 + 43T^3 - 1924T^2 - 10931T - 4783)^2
$71$	$(T^{5} + T^{4} + \cdots - 15525)^{2}$ (T^5 + T^4 - 238T^3 + 189T^2 + 10665*T - 15525)^2
$73$	$T^{10} + 288 T^{8} + \cdots + 1476225$ T^10 + 288T^8 + 19116T^6 + 292329T^4 + 1443420T^2 + 1476225
$79$	$T^{10} + \cdots + 610337025$ T^10 + 410T^8 + 59840T^6 + 3672891T^4 + 86503140T^2 + 610337025
$83$	$(T^{5} + 3 T^{4} + \cdots - 59373)^{2}$ (T^5 + 3T^4 - 335T^3 + 27T^2 + 25866T - 59373)^2
$89$	$T^{10} + \cdots + 2883153025$ T^10 + 434T^8 + 71672T^6 + 5604599T^4 + 206764960T^2 + 2883153025
$97$	$T^{10} + \cdots + 312759225$ T^10 + 625T^8 + 111983T^6 + 7153866T^4 + 125185095T^2 + 312759225
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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 376.10
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