LMFDB - Newform orbit 600.3.l.e

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [600,3,Mod(401,600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("600.401"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 600 = 2^{3} \cdot 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	600.l (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,4,0,0,0,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.3488158616$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.574198272.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} - 34x^{3} + 81x^{2} + 156x + 198 $ x^6 - 2x^5 - 7x^4 - 34x^3 + 81x^2 + 156*x + 198
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4}\cdot 3 $
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 $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + (\beta_{2} + 1) q^{3} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 2) q^{7} + ( - \beta_{5} + \beta_{2} - \beta_1 + 3) q^{9} + (2 \beta_{2} - \beta_1 + 1) q^{11} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 4) q^{13}+ \cdots + (\beta_{5} + 3 \beta_{3} - 7 \beta_{2} + \cdots - 42) q^{99}+O(q^{100}) $ q + (b2 + 1) * q^3 + (-b5 - b4 + b3 + b2 - b1 - 2) * q^7 + (-b5 + b2 - b1 + 3) * q^9 + (2b2 - b1 + 1) q^11 + (b5 + b4 - b3 - b2 + b1 - 4) * q^13 + (b5 + 2b4 + b3 - b1 + 1) q^17 + (-b5 - 2b4 + b3 - 2b2 - 2b1 + 7) q^19 + (-b5 + 3b3 - b1 - 4) q^21 + (2b5 - b4 + 2b3 - 3b2 + 2b1 - 2) * q^23 + (-b5 - 3b4 + 3b3 + 7b2 - 4b1 - 3) * q^27 + (-2b5 + b4 - 2b3 - 5b2 + 2b1 - 2) * q^29 + (-3b5 - 3b4 + 3b3 + 3b2 - 3b1 - 20) q^31 + (-b5 + b4 + 2b2 - 3b1 - 12) * q^33 + (-4b5 - 2b4 + 4b3 + 10b2 - 2b1 - 12) q^37 + (b5 - 3b3 - 6b2 + b1 - 2) * q^39 + (3b5 - 4b4 + 3b3 + 6b2 - b1 + 1) * q^41 + (-5b5 - 3b4 + 5b3 + 11b2 - 3b1) q^43 + (6b5 + 2b4 + 6b3 + 2b2 - 2b1 + 2) q^47 + (7b5 + 9b4 - 7b3 - b2 + 9b1 + 17) * q^49 + (5b5 + 11b4 - 3b3 - 6b2 + b1 + 3) * q^51 + (-3b4 + 15b2 - 6b1 + 6) q^53 + (2b5 + 3b3 + 11b2 + 2b1 - 26) * q^57 + (-2b5 - 12b4 - 2b3 + 12b2) * q^59 + (-b5 + 3b4 + b3 + 13b2 + 3b1 + 14) q^61 + (3b5 + 9b4 + 3b3 - 9b2 + 3b1 + 24) q^63 + (b5 - b3 - 4b2 - 71) q^67 + (4b5 + 13b4 - 6b3 - 15b2 + 14b1 + 24) q^69 + (4b5 - 13b4 + 4b3 + 17b2 - 2b1 + 2) q^71 + (-3b5 - 7b4 + 3b3 - 9b2 - 7b1 - 17) q^73 + (4b5 + 9b4 + 4b3 - 9b2) * q^77 + (8b5 + 2b4 - 8b3 - 26b2 + 2b1 - 16) q^79 + (-4b5 + 9b4 + 3b3 - 2b2 - 4b1 - 36) q^81 + (-2b5 - 3b4 - 2b3 + 9b2 - 3b1 + 3) q^83 + (-17b4 + 6b3 + 7b2 - 2b1 + 24) * q^87 + (-b5 - 3b4 - b3 - 11b2 + 7b1 - 7) q^89 + (-b5 - 3b4 + b3 - 5b2 - 3b1 - 54) q^91 + (-3b5 + 9b3 - 14b2 - 3b1 - 26) * q^93 + (-b5 + 5b4 + b3 + 19b2 + 5b1 + 84) q^97 + (b5 + 3b3 - 7b2 - 8b1 - 42) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{3} - 10 q^{7} + 16 q^{9} - 26 q^{13} + 50 q^{19} - 18 q^{21} - 26 q^{27} - 114 q^{31} - 82 q^{33} - 76 q^{37} - 6 q^{39} - 2 q^{43} + 76 q^{49} - 6 q^{51} - 172 q^{57} + 62 q^{61} + 150 q^{63}+ \cdots - 250 q^{99}+O(q^{100}) $ 6 * q + 4 * q^3 - 10 * q^7 + 16 * q^9 - 26 * q^13 + 50 * q^19 - 18 * q^21 - 26 * q^27 - 114 * q^31 - 82 * q^33 - 76 * q^37 - 6 * q^39 - 2 * q^43 + 76 * q^49 - 6 * q^51 - 172 * q^57 + 62 * q^61 + 150 * q^63 - 422 * q^67 + 156 * q^69 - 72 * q^73 - 76 * q^79 - 224 * q^81 + 172 * q^87 - 310 * q^91 - 110 * q^93 + 470 * q^97 - 250 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} - 7x^{4} - 34x^{3} + 81x^{2} + 156x + 198 $ x^6 - 2*x^5 - 7*x^4 - 34*x^3 + 81*x^2 + 156*x + 198 :

$\beta_{1}$	$=$	$ ( -8\nu^{5} + 5\nu^{4} + 26\nu^{3} + 529\nu^{2} - 2760\nu + 267 ) / 885 $ (-8v^5 + 5v^4 + 26v^3 + 529v^2 - 2760*v + 267) / 885
$\beta_{2}$	$=$	$ ( 32\nu^{5} - 20\nu^{4} - 104\nu^{3} - 1231\nu^{2} + 420\nu + 702 ) / 885 $ (32v^5 - 20v^4 - 104v^3 - 1231v^2 + 420*v + 702) / 885
$\beta_{3}$	$=$	$ ( -46\nu^{5} + 250\nu^{4} + 592\nu^{3} + 608\nu^{2} - 10560\nu - 11961 ) / 885 $ (-46v^5 + 250v^4 + 592v^3 + 608v^2 - 10560*v - 11961) / 885
$\beta_{4}$	$=$	$ ( -64\nu^{5} + 40\nu^{4} + 208\nu^{3} + 1577\nu^{2} - 840\nu + 366 ) / 885 $ (-64v^5 + 40v^4 + 208v^3 + 1577v^2 - 840*v + 366) / 885
$\beta_{5}$	$=$	$ ( 72\nu^{5} + 250\nu^{4} - 234\nu^{3} - 3286\nu^{2} - 7020\nu - 2403 ) / 885 $ (72v^5 + 250v^4 - 234v^3 - 3286v^2 - 7020*v - 2403) / 885

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

$\nu$	$=$	$ ( -\beta_{4} - 3\beta_{2} - 4\beta _1 + 4 ) / 12 $ (-b4 - 3b2 - 4b1 + 4) / 12
$\nu^{2}$	$=$	$ -\beta_{4} - 2\beta_{2} + 2 $ -b4 - 2*b2 + 2
$\nu^{3}$	$=$	$ ( -3\beta_{5} - 5\beta_{4} + 4\beta_{3} + \beta_{2} - 6\beta _1 + 49 ) / 2 $ (-3b5 - 5b4 + 4b3 + b2 - 6b1 + 49) / 2
$\nu^{4}$	$=$	$ 3\beta_{5} - 4\beta_{4} - 17\beta_{2} - 9\beta _1 + 26 $ 3b5 - 4b4 - 17b2 - 9b1 + 26
$\nu^{5}$	$=$	$ ( -6\beta_{5} - 96\beta_{4} + 13\beta_{3} - 110\beta_{2} - 22\beta _1 + 293 ) / 2 $ (-6b5 - 96b4 + 13b3 - 110b2 - 22*b1 + 293) / 2

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/600\mathbb{Z}\right)^\times$.

$n$	$151$	$301$	$401$	$577$
$\chi(n)$	$1$	$1$	$-1$	$1$

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} $

401.1

3.56627	−	0.139571i
3.56627	+	0.139571i
−0.788754	−	0.994180i
−0.788754	+	0.994180i
−1.77752	−	2.54797i
−1.77752	+	2.54797i

−2.56627

−

1.55378i

1.34301

4.17150

7.97487i

401.2

−2.56627

1.55378i

1.34301

4.17150

−

7.97487i

401.3

1.78875

−

2.40839i

−12.2014

−2.60072

−

8.61605i

401.4

1.78875

2.40839i

−12.2014

−2.60072

8.61605i

401.5

2.77752

−

1.13375i

5.85843

6.42922

−

6.29803i

401.6

2.77752

1.13375i

5.85843

6.42922

6.29803i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	600.3.l.e	yes	6
3.b	odd	2	1	inner	600.3.l.e	yes	6
4.b	odd	2	1		1200.3.l.v		6
5.b	even	2	1		600.3.l.d	✓	6
5.c	odd	4	2		600.3.c.c		12
12.b	even	2	1		1200.3.l.v		6
15.d	odd	2	1		600.3.l.d	✓	6
15.e	even	4	2		600.3.c.c		12
20.d	odd	2	1		1200.3.l.w		6
20.e	even	4	2		1200.3.c.l		12
60.h	even	2	1		1200.3.l.w		6
60.l	odd	4	2		1200.3.c.l		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
600.3.c.c		12	5.c	odd	4	2
600.3.c.c		12	15.e	even	4	2
600.3.l.d	✓	6	5.b	even	2	1
600.3.l.d	✓	6	15.d	odd	2	1
600.3.l.e	yes	6	1.a	even	1	1	trivial
600.3.l.e	yes	6	3.b	odd	2	1	inner
1200.3.c.l		12	20.e	even	4	2
1200.3.c.l		12	60.l	odd	4	2
1200.3.l.v		6	4.b	odd	2	1
1200.3.l.v		6	12.b	even	2	1
1200.3.l.w		6	20.d	odd	2	1
1200.3.l.w		6	60.h	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{3} + 5T_{7}^{2} - 80T_{7} + 96 $ T7^3 + 5*T7^2 - 80*T7 + 96 acting on $S_{3}^{\mathrm{new}}(600, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ T^{6} - 4 T^{5} + \cdots + 729 $ T^6 - 4T^5 + 30T^3 - 324*T + 729
$5$	$ T^{6} $ T^6
$7$	$ (T^{3} + 5 T^{2} - 80 T + 96)^{2} $ (T^3 + 5T^2 - 80T + 96)^2
$11$	$ T^{6} + 246 T^{4} + \cdots + 161312 $ T^6 + 246T^4 + 15129T^2 + 161312
$13$	$ (T^{3} + 13 T^{2} + \cdots - 540)^{2} $ (T^3 + 13T^2 - 32T - 540)^2
$17$	$ T^{6} + 1174 T^{4} + \cdots + 9435168 $ T^6 + 1174T^4 + 282073T^2 + 9435168
$19$	$ (T^{3} - 25 T^{2} + \cdots + 877)^{2} $ (T^3 - 25T^2 - 205T + 877)^2
$23$	$ T^{6} + 2680 T^{4} + \cdots + 423055872 $ T^6 + 2680T^4 + 1985680T^2 + 423055872
$29$	$ T^{6} + 2616 T^{4} + \cdots + 37601792 $ T^6 + 2616T^4 + 1599888T^2 + 37601792
$31$	$ (T^{3} + 57 T^{2} + \cdots - 1804)^{2} $ (T^3 + 57T^2 + 288T - 1804)^2
$37$	$ (T^{3} + 38 T^{2} + \cdots - 69696)^{2} $ (T^3 + 38T^2 - 1824T - 69696)^2
$41$	$ T^{6} + 4806 T^{4} + \cdots + 22418208 $ T^6 + 4806T^4 + 743337T^2 + 22418208
$43$	$ (T^{3} + T^{2} + \cdots - 59628)^{2} $ (T^3 + T^2 - 3032*T - 59628)^2
$47$	$ T^{6} + \cdots + 187644280832 $ T^6 + 17304T^4 + 99107472T^2 + 187644280832
$53$	$ T^{6} + \cdots + 24652657152 $ T^6 + 9720T^4 + 28689552T^2 + 24652657152
$59$	$ T^{6} + \cdots + 91824979968 $ T^6 + 17056T^4 + 82182400T^2 + 91824979968
$61$	$ (T^{3} - 31 T^{2} + \cdots + 117664)^{2} $ (T^3 - 31T^2 - 4288T + 117664)^2
$67$	$ (T^{3} + 211 T^{2} + \cdots + 324893)^{2} $ (T^3 + 211T^2 + 14495T + 324893)^2
$71$	$ T^{6} + \cdots + 15890452992 $ T^6 + 20600T^4 + 36573840T^2 + 15890452992
$73$	$ (T^{3} + 36 T^{2} + \cdots - 210978)^{2} $ (T^3 + 36T^2 - 5427T - 210978)^2
$79$	$ (T^{3} + 38 T^{2} + \cdots + 370080)^{2} $ (T^3 + 38T^2 - 14016T + 370080)^2
$83$	$ T^{6} + 5206 T^{4} + \cdots + 1152 $ T^6 + 5206T^4 + 449113T^2 + 1152
$89$	$ T^{6} + \cdots + 5739061248 $ T^6 + 13478T^4 + 17838441T^2 + 5739061248
$97$	$ (T^{3} - 235 T^{2} + \cdots + 613636)^{2} $ (T^3 - 235T^2 + 8048T + 613636)^2
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Level:	\( N \)	\(=\)	\( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	600.l (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + (\beta_{2} + 1) q^{3} + ( - \beta_{5} - \beta_{4} + \beta_{3} + \cdots - 2) q^{7} + ( - \beta_{5} + \beta_{2} - \beta_1 + 3) q^{9} + (2 \beta_{2} - \beta_1 + 1) q^{11} + (\beta_{5} + \beta_{4} - \beta_{3} + \cdots - 4) q^{13}+ \cdots + (\beta_{5} + 3 \beta_{3} - 7 \beta_{2} + \cdots - 42) q^{99}+O(q^{100}) \) q + (b2 + 1) * q^3 + (-b5 - b4 + b3 + b2 - b1 - 2) * q^7 + (-b5 + b2 - b1 + 3) * q^9 + (2b2 - b1 + 1) q^11 + (b5 + b4 - b3 - b2 + b1 - 4) * q^13 + (b5 + 2b4 + b3 - b1 + 1) q^17 + (-b5 - 2b4 + b3 - 2b2 - 2b1 + 7) q^19 + (-b5 + 3b3 - b1 - 4) q^21 + (2b5 - b4 + 2b3 - 3b2 + 2b1 - 2) * q^23 + (-b5 - 3b4 + 3b3 + 7b2 - 4b1 - 3) * q^27 + (-2b5 + b4 - 2b3 - 5b2 + 2b1 - 2) * q^29 + (-3b5 - 3b4 + 3b3 + 3b2 - 3b1 - 20) q^31 + (-b5 + b4 + 2b2 - 3b1 - 12) * q^33 + (-4b5 - 2b4 + 4b3 + 10b2 - 2b1 - 12) q^37 + (b5 - 3b3 - 6b2 + b1 - 2) * q^39 + (3b5 - 4b4 + 3b3 + 6b2 - b1 + 1) * q^41 + (-5b5 - 3b4 + 5b3 + 11b2 - 3b1) q^43 + (6b5 + 2b4 + 6b3 + 2b2 - 2b1 + 2) q^47 + (7b5 + 9b4 - 7b3 - b2 + 9b1 + 17) * q^49 + (5b5 + 11b4 - 3b3 - 6b2 + b1 + 3) * q^51 + (-3b4 + 15b2 - 6b1 + 6) q^53 + (2b5 + 3b3 + 11b2 + 2b1 - 26) * q^57 + (-2b5 - 12b4 - 2b3 + 12b2) * q^59 + (-b5 + 3b4 + b3 + 13b2 + 3b1 + 14) q^61 + (3b5 + 9b4 + 3b3 - 9b2 + 3b1 + 24) q^63 + (b5 - b3 - 4b2 - 71) q^67 + (4b5 + 13b4 - 6b3 - 15b2 + 14b1 + 24) q^69 + (4b5 - 13b4 + 4b3 + 17b2 - 2b1 + 2) q^71 + (-3b5 - 7b4 + 3b3 - 9b2 - 7b1 - 17) q^73 + (4b5 + 9b4 + 4b3 - 9b2) * q^77 + (8b5 + 2b4 - 8b3 - 26b2 + 2b1 - 16) q^79 + (-4b5 + 9b4 + 3b3 - 2b2 - 4b1 - 36) q^81 + (-2b5 - 3b4 - 2b3 + 9b2 - 3b1 + 3) q^83 + (-17b4 + 6b3 + 7b2 - 2b1 + 24) * q^87 + (-b5 - 3b4 - b3 - 11b2 + 7b1 - 7) q^89 + (-b5 - 3b4 + b3 - 5b2 - 3b1 - 54) q^91 + (-3b5 + 9b3 - 14b2 - 3b1 - 26) * q^93 + (-b5 + 5b4 + b3 + 19b2 + 5b1 + 84) q^97 + (b5 + 3b3 - 7b2 - 8b1 - 42) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{3} - 10 q^{7} + 16 q^{9} - 26 q^{13} + 50 q^{19} - 18 q^{21} - 26 q^{27} - 114 q^{31} - 82 q^{33} - 76 q^{37} - 6 q^{39} - 2 q^{43} + 76 q^{49} - 6 q^{51} - 172 q^{57} + 62 q^{61} + 150 q^{63}+ \cdots - 250 q^{99}+O(q^{100}) \) 6 * q + 4 * q^3 - 10 * q^7 + 16 * q^9 - 26 * q^13 + 50 * q^19 - 18 * q^21 - 26 * q^27 - 114 * q^31 - 82 * q^33 - 76 * q^37 - 6 * q^39 - 2 * q^43 + 76 * q^49 - 6 * q^51 - 172 * q^57 + 62 * q^61 + 150 * q^63 - 422 * q^67 + 156 * q^69 - 72 * q^73 - 76 * q^79 - 224 * q^81 + 172 * q^87 - 310 * q^91 - 110 * q^93 + 470 * q^97 - 250 * q^99

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