LMFDB - Newform orbit 18.23.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [18,23,Mod(17,18)] 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("18.17"); S:= CuspForms(chi, 23); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 23, names="a")

Level:	$ N $	$=$	$ 18 = 2 \cdot 3^{2} $
Weight:	$ k $	$=$	$ 23 $
Character orbit:	$[\chi]$	$=$	18.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$55.2073382715$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
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 $\beta = \sqrt{-2}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 1024 \beta q^{2} - 2097152 q^{4} + 19826925 \beta q^{5} - 1666948276 q^{7} - 2147483648 \beta q^{8} - 40605542400 q^{10} - 130862199588 \beta q^{11} + 399915433112 q^{13} - 1706955034624 \beta q^{14} + \cdots - 11\!\cdots\!52 \beta q^{98} +O(q^{100}) $ q + 1024b q^2 - 2097152 * q^4 + 19826925b q^5 - 1666948276 * q^7 - 2147483648b q^8 - 40605542400 * q^10 - 130862199588b q^11 + 399915433112 * q^13 - 1706955034624b q^14 + 4398046511104 * q^16 - 3621515694201b q^17 - 142603811032336 * q^19 - 41580075417600b q^20 + 268005784756224 * q^22 + 102452165990052b q^23 + 1597971881104375 * q^25 + 409513403506688b q^26 + 3495843910909952 * q^28 - 13395230660362317b q^29 + 5818906190320124 * q^31 + 4503599627370496b q^32 + 7416864141723648 * q^34 - 33050458447131300b q^35 + 99489731621460014 * q^37 - 146026302497112064b q^38 + 85155994455244800 * q^40 - 35342963445366687b q^41 + 846050138097114680 * q^43 + 274437923590373376b q^44 - 209822035947626496 * q^46 + 1667654247291032820b q^47 - 1131104493723615873 * q^49 + 1636323206250880000b q^50 - 838683450381697024 * q^52 + 2862388151293990203b q^53 + 5189190033132613800 * q^55 + 3579744164771790848b q^56 + 27433432392422025216 * q^58 + 31055427451064918904b q^59 + 64862304214149354050 * q^61 + 5958559938887806976b q^62 - 9223372036854775808 * q^64 + 7929093298654140600b q^65 + 203569144366409204072 * q^67 + 7594868881125015552b q^68 + 67687338899724902400 * q^70 + 51271853098342781484b q^71 + 555984492003263138096 * q^73 + 101877485180375054336b q^74 + 299061867514085507072 * q^76 + 218140517996784510288b q^77 - 76399175592178414876 * q^79 + 87199738322170675200b q^80 + 72382389136110974976 * q^82 + 1537459924599341476356b q^83 + 143607040110492323850 * q^85 + 866355341411445432320b q^86 - 562048867513084674048 * q^88 - 2536415336230224554151b q^89 - 666638341771841714912 * q^91 - 214857764810369531904b q^92 - 3415355898452035215360 * q^94 - 2827395066052298446800b q^95 - 5299032201716862372016 * q^97 - 1158251001572982653952b q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4194304 q^{4} - 3333896552 q^{7} - 81211084800 q^{10} + 799830866224 q^{13} + 8796093022208 q^{16} - 285207622064672 q^{19} + 536011569512448 q^{22} + 31\!\cdots\!50 q^{25} + 69\!\cdots\!04 q^{28}+ \cdots - 10\!\cdots\!32 q^{97}+O(q^{100}) $ 2 * q - 4194304 * q^4 - 3333896552 * q^7 - 81211084800 * q^10 + 799830866224 * q^13 + 8796093022208 * q^16 - 285207622064672 * q^19 + 536011569512448 * q^22 + 3195943762208750 * q^25 + 6991687821819904 * q^28 + 11637812380640248 * q^31 + 14833728283447296 * q^34 + 198979463242920028 * q^37 + 170311988910489600 * q^40 + 1692100276194229360 * q^43 - 419644071895252992 * q^46 - 2262208987447231746 * q^49 - 1677366900763394048 * q^52 + 10378380066265227600 * q^55 + 54866864784844050432 * q^58 + 129724608428298708100 * q^61 - 18446744073709551616 * q^64 + 407138288732818408144 * q^67 + 135374677799449804800 * q^70 + 1111968984006526276192 * q^73 + 598123735028171014144 * q^76 - 152798351184356829752 * q^79 + 144764778272221949952 * q^82 + 287214080220984647700 * q^85 - 1124097735026169348096 * q^88 - 1333276683543683429824 * q^91 - 6830711796904070430720 * q^94 - 10598064403433724744032 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/18\mathbb{Z}\right)^\times$.

$n$	$11$
$\chi(n)$	$-1$

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} $

17.1

	−	1.41421i
		1.41421i

−

1448.15i

−2.09715e6

−

2.80395e7i

−1.66695e9

3.03700e9i

−4.06055e10

17.2

1448.15i

−2.09715e6

2.80395e7i

−1.66695e9

−

3.03700e9i

−4.06055e10

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	18.23.b.a	✓	2
3.b	odd	2	1	inner	18.23.b.a	✓	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
18.23.b.a	✓	2	1.a	even	1	1	trivial
18.23.b.a	✓	2	3.b	odd	2	1	inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5}^{2} + 786213909911250 $ T5^2 + 786213909911250 acting on $S_{23}^{\mathrm{new}}(18, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 2097152 $ T^2 + 2097152
$3$	$ T^{2} $ T^2
$5$	$ T^{2} + 786213909911250 $ T^2 + 786213909911250
$7$	$ (T + 1666948276)^{2} $ (T + 1666948276)^2
$11$	$ T^{2} + 34\!\cdots\!88 $ T^2 + 34249830562019094739488
$13$	$ (T - 399915433112)^{2} $ (T - 399915433112)^2
$17$	$ T^{2} + 26\!\cdots\!02 $ T^2 + 26230751846688301890056802
$19$	$ (T + 142603811032336)^{2} $ (T + 142603811032336)^2
$23$	$ T^{2} + 20\!\cdots\!08 $ T^2 + 20992892632106335410725925408
$29$	$ T^{2} + 35\!\cdots\!78 $ T^2 + 358864408888621350348419427216978
$31$	$ (T - 58\!\cdots\!24)^{2} $ (T - 5818906190320124)^2
$37$	$ (T - 99\!\cdots\!14)^{2} $ (T - 99489731621460014)^2
$41$	$ T^{2} + 24\!\cdots\!38 $ T^2 + 2498250130201051756997295778711938
$43$	$ (T - 84\!\cdots\!80)^{2} $ (T - 846050138097114680)^2
$47$	$ T^{2} + 55\!\cdots\!00 $ T^2 + 5562141377015642491326946364634304800
$53$	$ T^{2} + 16\!\cdots\!18 $ T^2 + 16386531857336453896481999090919962418
$59$	$ T^{2} + 19\!\cdots\!32 $ T^2 + 1928879148336713052055490163744193122432
$61$	$ (T - 64\!\cdots\!50)^{2} $ (T - 64862304214149354050)^2
$67$	$ (T - 20\!\cdots\!72)^{2} $ (T - 203569144366409204072)^2
$71$	$ T^{2} + 52\!\cdots\!12 $ T^2 + 5257605840276084562777684748019546484512
$73$	$ (T - 55\!\cdots\!96)^{2} $ (T - 555984492003263138096)^2
$79$	$ (T + 76\!\cdots\!76)^{2} $ (T + 76399175592178414876)^2
$83$	$ T^{2} + 47\!\cdots\!72 $ T^2 + 4727566039498025555471794638337179414077472
$89$	$ T^{2} + 12\!\cdots\!02 $ T^2 + 12866805515727766151595442129612053462661602
$97$	$ (T + 52\!\cdots\!16)^{2} $ (T + 5299032201716862372016)^2
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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 18 = 2 \cdot 3^{2} \)
Weight:	\( k \)	\(=\)	\( 23 \)
Character orbit:	\([\chi]\)	\(=\)	18.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + 1024 \beta q^{2} - 2097152 q^{4} + 19826925 \beta q^{5} - 1666948276 q^{7} - 2147483648 \beta q^{8} - 40605542400 q^{10} - 130862199588 \beta q^{11} + 399915433112 q^{13} - 1706955034624 \beta q^{14} + \cdots - 11\!\cdots\!52 \beta q^{98} +O(q^{100}) \) q + 1024b q^2 - 2097152 * q^4 + 19826925b q^5 - 1666948276 * q^7 - 2147483648b q^8 - 40605542400 * q^10 - 130862199588b q^11 + 399915433112 * q^13 - 1706955034624b q^14 + 4398046511104 * q^16 - 3621515694201b q^17 - 142603811032336 * q^19 - 41580075417600b q^20 + 268005784756224 * q^22 + 102452165990052b q^23 + 1597971881104375 * q^25 + 409513403506688b q^26 + 3495843910909952 * q^28 - 13395230660362317b q^29 + 5818906190320124 * q^31 + 4503599627370496b q^32 + 7416864141723648 * q^34 - 33050458447131300b q^35 + 99489731621460014 * q^37 - 146026302497112064b q^38 + 85155994455244800 * q^40 - 35342963445366687b q^41 + 846050138097114680 * q^43 + 274437923590373376b q^44 - 209822035947626496 * q^46 + 1667654247291032820b q^47 - 1131104493723615873 * q^49 + 1636323206250880000b q^50 - 838683450381697024 * q^52 + 2862388151293990203b q^53 + 5189190033132613800 * q^55 + 3579744164771790848b q^56 + 27433432392422025216 * q^58 + 31055427451064918904b q^59 + 64862304214149354050 * q^61 + 5958559938887806976b q^62 - 9223372036854775808 * q^64 + 7929093298654140600b q^65 + 203569144366409204072 * q^67 + 7594868881125015552b q^68 + 67687338899724902400 * q^70 + 51271853098342781484b q^71 + 555984492003263138096 * q^73 + 101877485180375054336b q^74 + 299061867514085507072 * q^76 + 218140517996784510288b q^77 - 76399175592178414876 * q^79 + 87199738322170675200b q^80 + 72382389136110974976 * q^82 + 1537459924599341476356b q^83 + 143607040110492323850 * q^85 + 866355341411445432320b q^86 - 562048867513084674048 * q^88 - 2536415336230224554151b q^89 - 666638341771841714912 * q^91 - 214857764810369531904b q^92 - 3415355898452035215360 * q^94 - 2827395066052298446800b q^95 - 5299032201716862372016 * q^97 - 1158251001572982653952b q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4194304 q^{4} - 3333896552 q^{7} - 81211084800 q^{10} + 799830866224 q^{13} + 8796093022208 q^{16} - 285207622064672 q^{19} + 536011569512448 q^{22} + 31\!\cdots\!50 q^{25} + 69\!\cdots\!04 q^{28}+ \cdots - 10\!\cdots\!32 q^{97}+O(q^{100}) \) 2 * q - 4194304 * q^4 - 3333896552 * q^7 - 81211084800 * q^10 + 799830866224 * q^13 + 8796093022208 * q^16 - 285207622064672 * q^19 + 536011569512448 * q^22 + 3195943762208750 * q^25 + 6991687821819904 * q^28 + 11637812380640248 * q^31 + 14833728283447296 * q^34 + 198979463242920028 * q^37 + 170311988910489600 * q^40 + 1692100276194229360 * q^43 - 419644071895252992 * q^46 - 2262208987447231746 * q^49 - 1677366900763394048 * q^52 + 10378380066265227600 * q^55 + 54866864784844050432 * q^58 + 129724608428298708100 * q^61 - 18446744073709551616 * q^64 + 407138288732818408144 * q^67 + 135374677799449804800 * q^70 + 1111968984006526276192 * q^73 + 598123735028171014144 * q^76 - 152798351184356829752 * q^79 + 144764778272221949952 * q^82 + 287214080220984647700 * q^85 - 1124097735026169348096 * q^88 - 1333276683543683429824 * q^91 - 6830711796904070430720 * q^94 - 10598064403433724744032 * q^97

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