LMFDB - Newform orbit 294.4.e.l

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [294,4,Mod(67,294)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(294, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4]))

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("294.67");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$294 = 2 \cdot 3 \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	294.e (of order $3$ , degree $2$ , 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:	$17.3465615417$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{1345})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{4} - x^{3} + 337x^{2} + 336x + 112896$ x^4 - x^3 + 337x^2 + 336x + 112896
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
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 $1,\beta_1,\beta_2,\beta_3$ for the coefficient ring described below. We also show the integral $q$ -expansion of the trace form.

$f(q)$	$=$	$q - 2 \beta_{2} q^{2} + ( - 3 \beta_{2} + 3) q^{3} + (4 \beta_{2} - 4) q^{4} + (2 \beta_{2} + \beta_1) q^{5} - 6 q^{6} + 8 q^{8} - 9 \beta_{2} q^{9} + ( - 2 \beta_{3} - 4 \beta_{2} + \cdots + 6) q^{10}+ \cdots + (9 \beta_{3} + 297) q^{99}+O(q^{100})$ q - 2b2 q^2 + (-3b2 + 3) q^3 + (4b2 - 4) q^4 + (2b2 + b1) q^5 - 6 * q^6 + 8 * q^8 - 9b2 q^9 + (-2b3 - 4b2 - 2b1 + 6) q^10 + (-b3 + 34b2 - b1 - 33) q^11 + 12b2 q^12 + (-b3 - 20) * q^13 + (-3b3 + 9) q^15 - 16b2 q^16 + (4b3 + 44b2 + 4b1 - 48) q^17 + (18b2 - 18) q^18 + (23b2 - 3b1) * q^19 + (4b3 - 12) q^20 + (2b3 + 66) q^22 + (-76b2 + 4b1) * q^23 + (-24b2 + 24) q^24 + (5b3 + 215b2 + 5b1 - 220) q^25 + (42b2 - 2b1) * q^26 - 27 * q^27 + (11b3 + 33) q^29 + (-12b2 - 6b1) * q^30 + (-2b3 + 261b2 - 2b1 - 259) q^31 + (32b2 - 32) q^32 + (102b2 - 3b1) * q^33 + (-8b3 + 96) q^34 + 36 * q^36 + (b2 - 9b1) q^37 + (6b3 - 46b2 + 6b1 + 40) q^38 + (-3b3 + 63b2 - 3b1 - 60) q^39 + (16b2 + 8b1) * q^40 + (-6b3 + 216) q^41 + (-15b3 - 46) q^43 + (-136b2 + 4b1) * q^44 + (-9b3 - 18b2 - 9b1 + 27) q^45 + (-8b3 + 152b2 - 8b1 - 144) q^46 + (282b2 + 12b1) * q^47 - 48 * q^48 + (-10b3 + 440) q^50 + (132b2 + 12b1) * q^51 + (4b3 - 84b2 + 4b1 + 80) q^52 + (-3b3 - 120b2 - 3b1 + 123) q^53 + 54b2 q^54 + (31b3 + 237) q^55 + (9b3 + 60) q^57 + (-88b2 + 22b1) * q^58 + (-25b3 + 16b2 - 25b1 + 9) q^59 + (12b3 + 24b2 + 12b1 - 36) q^60 + (114b2 - 4b1) * q^61 + (4b3 + 518) q^62 + 64 * q^64 + (294b2 - 18b1) * q^65 + (6b3 - 204b2 + 6b1 + 198) q^66 + (-11b3 + 349b2 - 11b1 - 338) q^67 + (-176b2 - 16b1) * q^68 + (-12b3 - 216) q^69 + (-20b3 + 246) q^71 - 72b2 q^72 + (35b3 + 443b2 + 35b1 - 478) q^73 + (18b3 - 2b2 + 18b1 - 16) q^74 + (645b2 + 15b1) * q^75 + (-12b3 - 80) q^76 + (6b3 + 120) q^78 + (267b2 - 8b1) * q^79 + (-16b3 - 32b2 - 16b1 + 48) q^80 + (81b2 - 81) q^81 + (-420b2 - 12b1) * q^82 + (-25b3 + 123) q^83 + (56b3 - 1488) q^85 + (122b2 - 30b1) * q^86 + (33b3 - 132b2 + 33b1 + 99) q^87 + (-8b3 + 272b2 - 8b1 - 264) q^88 + (408b2 - 42b1) * q^89 + (18b3 - 54) q^90 + (16b3 + 288) q^92 + (783b2 - 6b1) * q^93 + (-24b3 - 564b2 - 24b1 + 588) q^94 + (14b3 - 962b2 + 14b1 + 948) q^95 + 96b2 q^96 + (-35b3 - 959) q^97 + (9b3 + 297) q^99
$\operatorname{Tr}(f)(q)$	$=$	$4 q - 4 q^{2} + 6 q^{3} - 8 q^{4} + 5 q^{5} - 24 q^{6} + 32 q^{8} - 18 q^{9} + 10 q^{10} - 67 q^{11} + 24 q^{12} - 82 q^{13} + 30 q^{15} - 32 q^{16} - 92 q^{17} - 36 q^{18} + 43 q^{19} - 40 q^{20} + 268 q^{22}+ \cdots + 1206 q^{99}+O(q^{100})$ 4 * q - 4 * q^2 + 6 * q^3 - 8 * q^4 + 5 * q^5 - 24 * q^6 + 32 * q^8 - 18 * q^9 + 10 * q^10 - 67 * q^11 + 24 * q^12 - 82 * q^13 + 30 * q^15 - 32 * q^16 - 92 * q^17 - 36 * q^18 + 43 * q^19 - 40 * q^20 + 268 * q^22 - 148 * q^23 + 48 * q^24 - 435 * q^25 + 82 * q^26 - 108 * q^27 + 154 * q^29 - 30 * q^30 - 520 * q^31 - 64 * q^32 + 201 * q^33 + 368 * q^34 + 144 * q^36 - 7 * q^37 + 86 * q^38 - 123 * q^39 + 40 * q^40 + 852 * q^41 - 214 * q^43 - 268 * q^44 + 45 * q^45 - 296 * q^46 + 576 * q^47 - 192 * q^48 + 1740 * q^50 + 276 * q^51 + 164 * q^52 + 243 * q^53 + 108 * q^54 + 1010 * q^55 + 258 * q^57 - 154 * q^58 - 7 * q^59 - 60 * q^60 + 224 * q^61 + 2080 * q^62 + 256 * q^64 + 570 * q^65 + 402 * q^66 - 687 * q^67 - 368 * q^68 - 888 * q^69 + 944 * q^71 - 144 * q^72 - 921 * q^73 - 14 * q^74 + 1305 * q^75 - 344 * q^76 + 492 * q^78 + 526 * q^79 + 80 * q^80 - 162 * q^81 - 852 * q^82 + 442 * q^83 - 5840 * q^85 + 214 * q^86 + 231 * q^87 - 536 * q^88 + 774 * q^89 - 180 * q^90 + 1184 * q^92 + 1560 * q^93 + 1152 * q^94 + 1910 * q^95 + 192 * q^96 - 3906 * q^97 + 1206 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $x^{4} - x^{3} + 337x^{2} + 336x + 112896$ x^4 - x^3 + 337*x^2 + 336*x + 112896 :

$\beta_{1}$	$=$	$\nu$ v
$\beta_{2}$	$=$	$( -\nu^{3} + 337\nu^{2} - 337\nu + 112896 ) / 113232$ (-v^3 + 337v^2 - 337v + 112896) / 113232
$\beta_{3}$	$=$	$( \nu^{3} + 673 ) / 337$ (v^3 + 673) / 337

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

$\nu$	$=$	$\beta_1$ b1
$\nu^{2}$	$=$	$\beta_{3} + 336\beta_{2} + \beta _1 - 337$ b3 + 336*b2 + b1 - 337
$\nu^{3}$	$=$	$337\beta_{3} - 673$ 337*b3 - 673

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/294\mathbb{Z}\right)^\times$ .

$n$	$197$	$199$
$\chi(n)$	$1$	$-\beta_{2}$

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}

67.1

−8.91856	−	15.4474i
9.41856	+	16.3134i
−8.91856	+	15.4474i
9.41856	−	16.3134i

−1.00000

−

1.73205i

1.50000

−

2.59808i

−2.00000

3.46410i

−7.91856

−

13.7153i

−6.00000

8.00000

−4.50000

−

7.79423i

−15.8371

27.4307i

67.2

−1.00000

−

1.73205i

1.50000

−

2.59808i

−2.00000

3.46410i

10.4186

18.0455i

−6.00000

8.00000

−4.50000

−

7.79423i

20.8371

−

36.0910i

79.1

−1.00000

1.73205i

1.50000

2.59808i

−2.00000

−

3.46410i

−7.91856

13.7153i

−6.00000

8.00000

−4.50000

7.79423i

−15.8371

−

27.4307i

79.2

−1.00000

1.73205i

1.50000

2.59808i

−2.00000

−

3.46410i

10.4186

−

18.0455i

−6.00000

8.00000

−4.50000

7.79423i

20.8371

36.0910i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	294.4.e.l		4
3.b	odd	2	1		882.4.g.bf		4
7.b	odd	2	1		42.4.e.c	✓	4
7.c	even	3	1		294.4.a.m		2
7.c	even	3	1	inner	294.4.e.l		4
7.d	odd	6	1		42.4.e.c	✓	4
7.d	odd	6	1		294.4.a.n		2
21.c	even	2	1		126.4.g.g		4
21.g	even	6	1		126.4.g.g		4
21.g	even	6	1		882.4.a.v		2
21.h	odd	6	1		882.4.a.z		2
21.h	odd	6	1		882.4.g.bf		4
28.d	even	2	1		336.4.q.j		4
28.f	even	6	1		336.4.q.j		4
28.f	even	6	1		2352.4.a.bq		2
28.g	odd	6	1		2352.4.a.ca		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.4.e.c	✓	4	7.b	odd	2	1
42.4.e.c	✓	4	7.d	odd	6	1
126.4.g.g		4	21.c	even	2	1
126.4.g.g		4	21.g	even	6	1
294.4.a.m		2	7.c	even	3	1
294.4.a.n		2	7.d	odd	6	1
294.4.e.l		4	1.a	even	1	1	trivial
294.4.e.l		4	7.c	even	3	1	inner
336.4.q.j		4	28.d	even	2	1
336.4.q.j		4	28.f	even	6	1
882.4.a.v		2	21.g	even	6	1
882.4.a.z		2	21.h	odd	6	1
882.4.g.bf		4	3.b	odd	2	1
882.4.g.bf		4	21.h	odd	6	1
2352.4.a.bq		2	28.f	even	6	1
2352.4.a.ca		2	28.g	odd	6	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(294, [\chi])$ :

T_{5}^{4} - 5T_{5}^{3} + 355T_{5}^{2} + 1650T_{5} + 108900

T_{11}^{4} + 67T_{11}^{3} + 3703T_{11}^{2} + 52662T_{11} + 617796

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$(T^{2} + 2 T + 4)^{2}$ (T^2 + 2*T + 4)^2
$3$	$(T^{2} - 3 T + 9)^{2}$ (T^2 - 3*T + 9)^2
$5$	$T^{4} - 5 T^{3} + \cdots + 108900$ T^4 - 5T^3 + 355T^2 + 1650*T + 108900
$7$	$T^{4}$ T^4
$11$	$T^{4} + 67 T^{3} + \cdots + 617796$ T^4 + 67T^3 + 3703T^2 + 52662*T + 617796
$13$	$(T^{2} + 41 T + 84)^{2}$ (T^2 + 41*T + 84)^2
$17$	$T^{4} + 92 T^{3} + \cdots + 10653696$ T^4 + 92T^3 + 11728T^2 - 300288*T + 10653696
$19$	$T^{4} - 43 T^{3} + \cdots + 6574096$ T^4 - 43T^3 + 4413T^2 + 110252*T + 6574096
$23$	$T^{4} + 148 T^{3} + \cdots + 9216$ T^4 + 148T^3 + 21808T^2 + 14208*T + 9216
$29$	$(T^{2} - 77 T - 39204)^{2}$ (T^2 - 77*T - 39204)^2
$31$	$T^{4} + \cdots + 4389725025$ T^4 + 520T^3 + 204145T^2 + 34452600*T + 4389725025
$37$	$T^{4} + 7 T^{3} + \cdots + 741146176$ T^4 + 7T^3 + 27273T^2 - 190568*T + 741146176
$41$	$(T^{2} - 426 T + 33264)^{2}$ (T^2 - 426*T + 33264)^2
$43$	$(T^{2} + 107 T - 72794)^{2}$ (T^2 + 107*T - 72794)^2
$47$	$T^{4} + \cdots + 1191906576$ T^4 - 576T^3 + 297252T^2 - 19885824*T + 1191906576
$53$	$T^{4} - 243 T^{3} + \cdots + 137733696$ T^4 - 243T^3 + 47313T^2 - 2851848*T + 137733696
$59$	$T^{4} + \cdots + 44160500736$ T^4 + 7T^3 + 210193T^2 - 1471008*T + 44160500736
$61$	$T^{4} - 224 T^{3} + \cdots + 51322896$ T^4 - 224T^3 + 43012T^2 - 1604736*T + 51322896
$67$	$T^{4} + \cdots + 5976217636$ T^4 + 687T^3 + 394663T^2 + 53109222*T + 5976217636
$71$	$(T^{2} - 472 T - 78804)^{2}$ (T^2 - 472*T - 78804)^2
$73$	$T^{4} + \cdots + 39938423716$ T^4 + 921T^3 + 1048087T^2 - 184058166*T + 39938423716
$79$	$T^{4} + \cdots + 2270427201$ T^4 - 526T^3 + 229027T^2 - 25063374*T + 2270427201
$83$	$(T^{2} - 221 T - 197946)^{2}$ (T^2 - 221*T - 197946)^2
$89$	$T^{4} + \cdots + 196582277376$ T^4 - 774T^3 + 1042452T^2 + 343173024*T + 196582277376
$97$	$(T^{2} + 1953 T + 541646)^{2}$ (T^2 + 1953*T + 541646)^2
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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 $\F_{q}$

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Introduction

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Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	294.4.e.l

Level	$294$
Weight	$4$
Character orbit	294.e
Analytic conductor	$17.347$
Analytic rank	$0$
Dimension	$4$
Inner twists	$2$