LMFDB - Newform orbit 608.2.bh.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [608,2,Mod(15,608)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(608, base_ring=CyclotomicField(18))

chi = DirichletCharacter(H, H._module([9, 9, 11]))

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

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

chi := DirichletCharacter("608.15");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 608 = 2^{5} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	608.bh (of order $18$, degree $6$, 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:	$4.85490444289$
Analytic rank:	$0$
Dimension:	$12$
Relative dimension:	$2$ over $\Q(\zeta_{18})$
Coefficient field:	12.0.101559956668416.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{U}(1)[D_{18}]$

$q$-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 $1,\beta_1,\ldots,\beta_{11}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + (\beta_{10} + 2 \beta_{9} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) $ q + (-b8 - b6 + b5 - b3 + b2 + 1) * q^3 + (b10 + 2b9 - 3b8 + b6 - 2b3 + 2b1) * q^9	$ q + ( - \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + ( - 7 \beta_{11} + 9 \beta_{10} + \cdots + 7) q^{99}+O(q^{100}) $ q + (-b8 - b6 + b5 - b3 + b2 + 1) * q^3 + (b10 + 2b9 - 3b8 + b6 - 2b3 + 2b1) * q^9 + (-b11 - 3b10 - 3b8 - b7 + b5 + 3b4) q^11 + (-3b10 + 4b7 - 2b1) q^17 + (-3b11 - b2) q^19 + (-5b8 + 5b2) * q^25 + (3b11 - 3b7 + 5b6 - 3b4 + b3 - 3b2 + 3b1 + 5) * q^27 + (2b11 - b10 - 4b7 - 5b6 - b4 + 2b3 - 5b2 + 5) q^33 + (-4b11 - 4b9 + 3b6 + 4b3 - 3b2) q^41 + (5b10 + 3b7 - 5b4 - 6b1) * q^43 + (7b6 - 7) q^49 + (-8b10 - b9 + 5b7 + b6 + b4 + 5b3 - b1 - 8) q^51 + (-5b10 + 7b8 - 4b7 - 2b5 + 5b4 - 7b2) * q^57 + (5b9 - 3b8 + 3b6 - 5b5 - 5b3 + 3b2) * q^59 + (7b10 - 3b9 - 7b6 + 3b3 + 3b1) q^67 + (-6b11 + 6b9 - b8 + 6b5 - 1) q^73 + (5b11 - 5b10 - 5b8 - 5b5 + 5b1) q^75 + (4b11 + 9b10 - 6b9 - 3b8 - 7b6 + 6b5 - 9b4 - 4b3 + 10b2 + 10) q^81 + (b11 + 9b8 - b7 - b5 + 9b4 + b1) * q^83 + (9b8 - 2b5 + 9b2) q^89 + (6b11 - 5b8 + 5b6 - 6b5 - 6b3 - 5) q^97 + (-7b11 + 9b10 - 2b9 + 2b7 - 9b6 + 14b5 - 2b4 - 3b3 - b2 - 5b1 + 7) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 6 q^{3} + 6 q^{9}+O(q^{10}) $ 12 * q + 6 * q^3 + 6 * q^9	$ 12 q + 6 q^{3} + 6 q^{9} + 90 q^{27} + 30 q^{33} + 18 q^{41} - 42 q^{49} - 90 q^{51} + 18 q^{59} - 42 q^{67} - 12 q^{73} + 78 q^{81} - 30 q^{97} + 30 q^{99}+O(q^{100}) $ 12 * q + 6 * q^3 + 6 * q^9 + 90 * q^27 + 30 * q^33 + 18 * q^41 - 42 * q^49 - 90 * q^51 + 18 * q^59 - 42 * q^67 - 12 * q^73 + 78 * q^81 - 30 * q^97 + 30 * q^99

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root $\nu$ of $ x^{12} - 8x^{6} + 64 $ x^12 - 8*x^6 + 64 :

$\beta_{1}$	$=$	$ \nu $ v
$\beta_{2}$	$=$	$ ( \nu^{2} ) / 2 $ (v^2) / 2
$\beta_{3}$	$=$	$ ( \nu^{3} ) / 2 $ (v^3) / 2
$\beta_{4}$	$=$	$ ( \nu^{4} ) / 4 $ (v^4) / 4
$\beta_{5}$	$=$	$ ( \nu^{5} ) / 4 $ (v^5) / 4
$\beta_{6}$	$=$	$ ( \nu^{6} ) / 8 $ (v^6) / 8
$\beta_{7}$	$=$	$ ( \nu^{7} ) / 8 $ (v^7) / 8
$\beta_{8}$	$=$	$ ( \nu^{8} ) / 16 $ (v^8) / 16
$\beta_{9}$	$=$	$ ( \nu^{9} ) / 16 $ (v^9) / 16
$\beta_{10}$	$=$	$ ( \nu^{10} ) / 32 $ (v^10) / 32
$\beta_{11}$	$=$	$ ( \nu^{11} ) / 32 $ (v^11) / 32

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

$\nu$	$=$	$ \beta_1 $ b1
$\nu^{2}$	$=$	$ 2\beta_{2} $ 2*b2
$\nu^{3}$	$=$	$ 2\beta_{3} $ 2*b3
$\nu^{4}$	$=$	$ 4\beta_{4} $ 4*b4
$\nu^{5}$	$=$	$ 4\beta_{5} $ 4*b5
$\nu^{6}$	$=$	$ 8\beta_{6} $ 8*b6
$\nu^{7}$	$=$	$ 8\beta_{7} $ 8*b7
$\nu^{8}$	$=$	$ 16\beta_{8} $ 16*b8
$\nu^{9}$	$=$	$ 16\beta_{9} $ 16*b9
$\nu^{10}$	$=$	$ 32\beta_{10} $ 32*b10
$\nu^{11}$	$=$	$ 32\beta_{11} $ 32*b11

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

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/608\mathbb{Z}\right)^\times$.

$n$	$97$	$191$	$229$
$\chi(n)$	$\beta_{2}$	$-1$	$-1$

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} $

15.1

−0.483690	+	1.32893i
0.483690	−	1.32893i
−0.909039	+	1.08335i
0.909039	−	1.08335i
1.39273	−	0.245576i
−1.39273	+	0.245576i
−0.909039	−	1.08335i
0.909039	+	1.08335i
−0.483690	−	1.32893i
0.483690	+	1.32893i
1.39273	+	0.245576i
−1.39273	−	0.245576i

−1.94383

0.342749i

0.841902

−

0.306427i

15.2

3.29112

−

0.580314i

7.67564

−

2.79370i

79.1

−1.18075

−

3.24408i

−6.83175

5.73252i

79.2

0.301363

0.827987i

1.70339

−

1.42931i

143.1

0.950339

1.13257i

0.141375

−

0.801775i

143.2

1.58175

1.88506i

−0.530560

3.00895i

431.1

−1.18075

3.24408i

−6.83175

−

5.73252i

431.2

0.301363

−

0.827987i

1.70339

1.42931i

527.1

−1.94383

−

0.342749i

0.841902

0.306427i

527.2

3.29112

0.580314i

7.67564

2.79370i

591.1

0.950339

−

1.13257i

0.141375

0.801775i

591.2

1.58175

−

1.88506i

−0.530560

−

3.00895i

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
8.d	odd	2	1	CM by $\Q(\sqrt{-2}) $
19.f	odd	18	1	inner
152.v	even	18	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	608.2.bh.a		12
4.b	odd	2	1		152.2.v.a	✓	12
8.b	even	2	1		152.2.v.a	✓	12
8.d	odd	2	1	CM	608.2.bh.a		12
19.f	odd	18	1	inner	608.2.bh.a		12
76.k	even	18	1		152.2.v.a	✓	12
152.s	odd	18	1		152.2.v.a	✓	12
152.v	even	18	1	inner	608.2.bh.a		12

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
152.2.v.a	✓	12	4.b	odd	2	1
152.2.v.a	✓	12	8.b	even	2	1
152.2.v.a	✓	12	76.k	even	18	1
152.2.v.a	✓	12	152.s	odd	18	1
608.2.bh.a		12	1.a	even	1	1	trivial
608.2.bh.a		12	8.d	odd	2	1	CM
608.2.bh.a		12	19.f	odd	18	1	inner
608.2.bh.a		12	152.v	even	18	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{3}^{12} - 6 T_{3}^{11} + 15 T_{3}^{10} - 48 T_{3}^{9} + 138 T_{3}^{8} - 48 T_{3}^{7} - 932 T_{3}^{6} + \cdots + 5329 $ T3^12 - 6*T3^11 + 15*T3^10 - 48*T3^9 + 138*T3^8 - 48*T3^7 - 932*T3^6 + 2706*T3^5 - 951*T3^4 - 4422*T3^3 + 10941*T3^2 - 8322*T3 + 5329 acting on $S_{2}^{\mathrm{new}}(608, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{12} $ T^12
$3$	$ T^{12} - 6 T^{11} + \cdots + 5329 $ T^12 - 6T^11 + 15T^10 - 48T^9 + 138T^8 - 48T^7 - 932T^6 + 2706T^5 - 951T^4 - 4422T^3 + 10941T^2 - 8322*T + 5329
$5$	$ T^{12} $ T^12
$7$	$ T^{12} $ T^12
$11$	$ T^{12} + 66 T^{10} + \cdots + 13461561 $ T^12 + 66T^10 - 36T^9 + 3267T^8 - 1782T^7 + 64860T^6 - 58806T^5 + 954459T^4 - 514782T^3 + 4348377T^2 + 2179386T + 13461561
$13$	$ T^{12} $ T^12
$17$	$ T^{12} + 486 T^{9} + \cdots + 11390625 $ T^12 + 486T^9 + 239571T^6 - 1640250*T^3 + 11390625
$19$	$ T^{12} - 106 T^{9} + \cdots + 47045881 $ T^12 - 106T^9 + 4377T^6 - 727054*T^3 + 47045881
$23$	$ T^{12} $ T^12
$29$	$ T^{12} $ T^12
$31$	$ T^{12} $ T^12
$37$	$ T^{12} $ T^12
$41$	$ T^{12} + \cdots + 41437894969 $ T^12 - 18T^11 + 93T^10 - 1296T^9 + 28488T^8 - 351576T^7 + 2202202T^6 + 1888506T^5 - 71266908T^4 + 623389716T^3 + 830094720T^2 - 2345045760*T + 41437894969
$43$	$ T^{12} + \cdots + 594823321 $ T^12 + 1870T^9 + 3521289T^6 - 45607430*T^3 + 594823321
$47$	$ T^{12} $ T^12
$53$	$ T^{12} $ T^12
$59$	$ T^{12} + \cdots + 260443853569 $ T^12 - 18T^11 + 39T^10 - 1296T^9 + 52032T^8 - 669096T^7 + 1827118T^6 + 45627534T^5 - 340620204T^4 + 1916313012T^3 + 17399909400T^2 - 42255903600*T + 260443853569
$61$	$ T^{12} $ T^12
$67$	$ T^{12} + \cdots + 87534914769 $ T^12 + 42T^11 + 975T^10 + 16464T^9 + 245898T^8 + 3091536T^7 + 24364572T^6 + 63399042T^5 - 569203191T^4 - 2899434006T^3 + 17973774909T^2 - 51929282034*T + 87534914769
$71$	$ T^{12} $ T^12
$73$	$ T^{12} + \cdots + 964620586801 $ T^12 + 12T^11 + 498T^10 + 4970T^9 + 71757T^8 + 451728T^7 + 3008127T^6 - 26934210T^5 + 650190528T^4 + 7953790574T^3 - 20405021655T^2 - 265610952138*T + 964620586801
$79$	$ T^{12} $ T^12
$83$	$ T^{12} + \cdots + 11478765321 $ T^12 + 498T^10 + 2700T^9 + 186003T^8 + 1008450T^7 + 32913276T^6 + 251104050T^5 + 4351281723T^4 + 21130911450T^3 + 106354097361T^2 + 36014774850T + 11478765321
$89$	$ T^{12} + \cdots + 168425239515625 $ T^12 + 14418T^9 + 82270783T^6 + 187115001750*T^3 + 168425239515625
$97$	$ T^{12} + \cdots + 7482560872329 $ T^12 + 30T^11 + 309T^10 + 6000T^9 + 89946T^8 - 742320T^7 - 35557716T^6 - 418010670T^5 + 624284289T^4 + 38143164510T^3 + 456480510471T^2 + 2245484670030*T + 7482560872329
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Classical	Maass
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 608 = 2^{5} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	608.bh (of order \(18\), degree \(6\), not minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + (\beta_{10} + 2 \beta_{9} + \cdots + 2 \beta_1) q^{9}+O(q^{10}) \) q + (-b8 - b6 + b5 - b3 + b2 + 1) * q^3 + (b10 + 2b9 - 3b8 + b6 - 2b3 + 2b1) * q^9	\( q + ( - \beta_{8} - \beta_{6} + \beta_{5} + \cdots + 1) q^{3}+ \cdots + ( - 7 \beta_{11} + 9 \beta_{10} + \cdots + 7) q^{99}+O(q^{100}) \) q + (-b8 - b6 + b5 - b3 + b2 + 1) * q^3 + (b10 + 2b9 - 3b8 + b6 - 2b3 + 2b1) * q^9 + (-b11 - 3b10 - 3b8 - b7 + b5 + 3b4) q^11 + (-3b10 + 4b7 - 2b1) q^17 + (-3b11 - b2) q^19 + (-5b8 + 5b2) * q^25 + (3b11 - 3b7 + 5b6 - 3b4 + b3 - 3b2 + 3b1 + 5) * q^27 + (2b11 - b10 - 4b7 - 5b6 - b4 + 2b3 - 5b2 + 5) q^33 + (-4b11 - 4b9 + 3b6 + 4b3 - 3b2) q^41 + (5b10 + 3b7 - 5b4 - 6b1) * q^43 + (7b6 - 7) q^49 + (-8b10 - b9 + 5b7 + b6 + b4 + 5b3 - b1 - 8) q^51 + (-5b10 + 7b8 - 4b7 - 2b5 + 5b4 - 7b2) * q^57 + (5b9 - 3b8 + 3b6 - 5b5 - 5b3 + 3b2) * q^59 + (7b10 - 3b9 - 7b6 + 3b3 + 3b1) q^67 + (-6b11 + 6b9 - b8 + 6b5 - 1) q^73 + (5b11 - 5b10 - 5b8 - 5b5 + 5b1) q^75 + (4b11 + 9b10 - 6b9 - 3b8 - 7b6 + 6b5 - 9b4 - 4b3 + 10b2 + 10) q^81 + (b11 + 9b8 - b7 - b5 + 9b4 + b1) * q^83 + (9b8 - 2b5 + 9b2) q^89 + (6b11 - 5b8 + 5b6 - 6b5 - 6b3 - 5) q^97 + (-7b11 + 9b10 - 2b9 + 2b7 - 9b6 + 14b5 - 2b4 - 3b3 - b2 - 5b1 + 7) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 6 q^{3} + 6 q^{9}+O(q^{10}) \) 12 * q + 6 * q^3 + 6 * q^9	\( 12 q + 6 q^{3} + 6 q^{9} + 90 q^{27} + 30 q^{33} + 18 q^{41} - 42 q^{49} - 90 q^{51} + 18 q^{59} - 42 q^{67} - 12 q^{73} + 78 q^{81} - 30 q^{97} + 30 q^{99}+O(q^{100}) \) 12 * q + 6 * q^3 + 6 * q^9 + 90 * q^27 + 30 * q^33 + 18 * q^41 - 42 * q^49 - 90 * q^51 + 18 * q^59 - 42 * q^67 - 12 * q^73 + 78 * q^81 - 30 * q^97 + 30 * q^99

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