LMFDB - Newform orbit 121.6.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [121,6,Mod(1,121)] 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("121.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(121, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 121 = 11^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	121.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.4064421974$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{38}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 38 $ x^2 - 38
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-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 $\beta = \sqrt{38}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + \beta q^{2} + 7 q^{3} + 6 q^{4} - 19 q^{5} + 7 \beta q^{6} - 8 \beta q^{7} - 26 \beta q^{8} - 194 q^{9} - 19 \beta q^{10} + 42 q^{12} - 120 \beta q^{13} - 304 q^{14} - 133 q^{15} - 1180 q^{16} + 296 \beta q^{17} + \cdots - 14375 \beta q^{98} +O(q^{100}) $ q + b * q^2 + 7 * q^3 + 6 * q^4 - 19 * q^5 + 7b q^6 - 8b q^7 - 26b q^8 - 194 * q^9 - 19b q^10 + 42 * q^12 - 120b q^13 - 304 * q^14 - 133 * q^15 - 1180 * q^16 + 296b q^17 - 194b q^18 - 344b q^19 - 114 * q^20 - 56b q^21 + 3071 * q^23 - 182b q^24 - 2764 * q^25 - 4560 * q^26 - 3059 * q^27 - 48b q^28 + 80b q^29 - 133b q^30 - 1581 * q^31 - 348b q^32 + 11248 * q^34 + 152b q^35 - 1164 * q^36 - 9145 * q^37 - 13072 * q^38 - 840b q^39 + 494b q^40 + 2824b q^41 - 2128 * q^42 + 2288b q^43 + 3686 * q^45 + 3071b q^46 - 16636 * q^47 - 8260 * q^48 - 14375 * q^49 - 2764b q^50 + 2072b q^51 - 720b q^52 - 16266 * q^53 - 3059b q^54 + 7904 * q^56 - 2408b q^57 + 3040 * q^58 + 14505 * q^59 - 798 * q^60 + 1264b q^61 - 1581b q^62 + 1552b q^63 + 24536 * q^64 + 2280b q^65 - 10635 * q^67 + 1776b q^68 + 21497 * q^69 + 5776 * q^70 + 31045 * q^71 + 5044b q^72 - 5464b q^73 - 9145b q^74 - 19348 * q^75 - 2064b q^76 - 31920 * q^78 - 13584b q^79 + 22420 * q^80 + 25729 * q^81 + 107312 * q^82 + 13760b q^83 - 336b q^84 - 5624b q^85 + 86944 * q^86 + 560b q^87 - 109481 * q^89 + 3686b q^90 + 36480 * q^91 + 18426 * q^92 - 11067 * q^93 - 16636b q^94 + 6536b q^95 - 2436b q^96 - 13615 * q^97 - 14375b q^98
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 14 q^{3} + 12 q^{4} - 38 q^{5} - 388 q^{9} + 84 q^{12} - 608 q^{14} - 266 q^{15} - 2360 q^{16} - 228 q^{20} + 6142 q^{23} - 5528 q^{25} - 9120 q^{26} - 6118 q^{27} - 3162 q^{31} + 22496 q^{34} - 2328 q^{36}+ \cdots - 27230 q^{97}+O(q^{100}) $ 2 * q + 14 * q^3 + 12 * q^4 - 38 * q^5 - 388 * q^9 + 84 * q^12 - 608 * q^14 - 266 * q^15 - 2360 * q^16 - 228 * q^20 + 6142 * q^23 - 5528 * q^25 - 9120 * q^26 - 6118 * q^27 - 3162 * q^31 + 22496 * q^34 - 2328 * q^36 - 18290 * q^37 - 26144 * q^38 - 4256 * q^42 + 7372 * q^45 - 33272 * q^47 - 16520 * q^48 - 28750 * q^49 - 32532 * q^53 + 15808 * q^56 + 6080 * q^58 + 29010 * q^59 - 1596 * q^60 + 49072 * q^64 - 21270 * q^67 + 42994 * q^69 + 11552 * q^70 + 62090 * q^71 - 38696 * q^75 - 63840 * q^78 + 44840 * q^80 + 51458 * q^81 + 214624 * q^82 + 173888 * q^86 - 218962 * q^89 + 72960 * q^91 + 36852 * q^92 - 22134 * q^93 - 27230 * q^97

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} $

1.1

−6.16441
6.16441

−6.16441

7.00000

6.00000

−19.0000

−43.1509

49.3153

160.275

−194.000

117.124

1.2

6.16441

7.00000

6.00000

−19.0000

43.1509

−49.3153

−160.275

−194.000

−117.124

Atkin-Lehner signs

$ p $	Sign
$11$	$ +1 $

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	121.6.a.c	✓	2
3.b	odd	2	1		1089.6.a.m		2
11.b	odd	2	1	inner	121.6.a.c	✓	2
33.d	even	2	1		1089.6.a.m		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
121.6.a.c	✓	2	1.a	even	1	1	trivial
121.6.a.c	✓	2	11.b	odd	2	1	inner
1089.6.a.m		2	3.b	odd	2	1
1089.6.a.m		2	33.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} - 38 $ T2^2 - 38 acting on $S_{6}^{\mathrm{new}}(\Gamma_0(121))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} - 38 $ T^2 - 38
$3$	$ (T - 7)^{2} $ (T - 7)^2
$5$	$ (T + 19)^{2} $ (T + 19)^2
$7$	$ T^{2} - 2432 $ T^2 - 2432
$11$	$ T^{2} $ T^2
$13$	$ T^{2} - 547200 $ T^2 - 547200
$17$	$ T^{2} - 3329408 $ T^2 - 3329408
$19$	$ T^{2} - 4496768 $ T^2 - 4496768
$23$	$ (T - 3071)^{2} $ (T - 3071)^2
$29$	$ T^{2} - 243200 $ T^2 - 243200
$31$	$ (T + 1581)^{2} $ (T + 1581)^2
$37$	$ (T + 9145)^{2} $ (T + 9145)^2
$41$	$ T^{2} - 303049088 $ T^2 - 303049088
$43$	$ T^{2} - 198927872 $ T^2 - 198927872
$47$	$ (T + 16636)^{2} $ (T + 16636)^2
$53$	$ (T + 16266)^{2} $ (T + 16266)^2
$59$	$ (T - 14505)^{2} $ (T - 14505)^2
$61$	$ T^{2} - 60712448 $ T^2 - 60712448
$67$	$ (T + 10635)^{2} $ (T + 10635)^2
$71$	$ (T - 31045)^{2} $ (T - 31045)^2
$73$	$ T^{2} - 1134501248 $ T^2 - 1134501248
$79$	$ T^{2} - 7011952128 $ T^2 - 7011952128
$83$	$ T^{2} - 7194828800 $ T^2 - 7194828800
$89$	$ (T + 109481)^{2} $ (T + 109481)^2
$97$	$ (T + 13615)^{2} $ (T + 13615)^2
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Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 121 = 11^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	121.a (trivial)

\(f(q)\)	\(=\)	\( q + \beta q^{2} + 7 q^{3} + 6 q^{4} - 19 q^{5} + 7 \beta q^{6} - 8 \beta q^{7} - 26 \beta q^{8} - 194 q^{9} - 19 \beta q^{10} + 42 q^{12} - 120 \beta q^{13} - 304 q^{14} - 133 q^{15} - 1180 q^{16} + 296 \beta q^{17} + \cdots - 14375 \beta q^{98} +O(q^{100}) \) q + b * q^2 + 7 * q^3 + 6 * q^4 - 19 * q^5 + 7b q^6 - 8b q^7 - 26b q^8 - 194 * q^9 - 19b q^10 + 42 * q^12 - 120b q^13 - 304 * q^14 - 133 * q^15 - 1180 * q^16 + 296b q^17 - 194b q^18 - 344b q^19 - 114 * q^20 - 56b q^21 + 3071 * q^23 - 182b q^24 - 2764 * q^25 - 4560 * q^26 - 3059 * q^27 - 48b q^28 + 80b q^29 - 133b q^30 - 1581 * q^31 - 348b q^32 + 11248 * q^34 + 152b q^35 - 1164 * q^36 - 9145 * q^37 - 13072 * q^38 - 840b q^39 + 494b q^40 + 2824b q^41 - 2128 * q^42 + 2288b q^43 + 3686 * q^45 + 3071b q^46 - 16636 * q^47 - 8260 * q^48 - 14375 * q^49 - 2764b q^50 + 2072b q^51 - 720b q^52 - 16266 * q^53 - 3059b q^54 + 7904 * q^56 - 2408b q^57 + 3040 * q^58 + 14505 * q^59 - 798 * q^60 + 1264b q^61 - 1581b q^62 + 1552b q^63 + 24536 * q^64 + 2280b q^65 - 10635 * q^67 + 1776b q^68 + 21497 * q^69 + 5776 * q^70 + 31045 * q^71 + 5044b q^72 - 5464b q^73 - 9145b q^74 - 19348 * q^75 - 2064b q^76 - 31920 * q^78 - 13584b q^79 + 22420 * q^80 + 25729 * q^81 + 107312 * q^82 + 13760b q^83 - 336b q^84 - 5624b q^85 + 86944 * q^86 + 560b q^87 - 109481 * q^89 + 3686b q^90 + 36480 * q^91 + 18426 * q^92 - 11067 * q^93 - 16636b q^94 + 6536b q^95 - 2436b q^96 - 13615 * q^97 - 14375b q^98
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 14 q^{3} + 12 q^{4} - 38 q^{5} - 388 q^{9} + 84 q^{12} - 608 q^{14} - 266 q^{15} - 2360 q^{16} - 228 q^{20} + 6142 q^{23} - 5528 q^{25} - 9120 q^{26} - 6118 q^{27} - 3162 q^{31} + 22496 q^{34} - 2328 q^{36}+ \cdots - 27230 q^{97}+O(q^{100}) \) 2 * q + 14 * q^3 + 12 * q^4 - 38 * q^5 - 388 * q^9 + 84 * q^12 - 608 * q^14 - 266 * q^15 - 2360 * q^16 - 228 * q^20 + 6142 * q^23 - 5528 * q^25 - 9120 * q^26 - 6118 * q^27 - 3162 * q^31 + 22496 * q^34 - 2328 * q^36 - 18290 * q^37 - 26144 * q^38 - 4256 * q^42 + 7372 * q^45 - 33272 * q^47 - 16520 * q^48 - 28750 * q^49 - 32532 * q^53 + 15808 * q^56 + 6080 * q^58 + 29010 * q^59 - 1596 * q^60 + 49072 * q^64 - 21270 * q^67 + 42994 * q^69 + 11552 * q^70 + 62090 * q^71 - 38696 * q^75 - 63840 * q^78 + 44840 * q^80 + 51458 * q^81 + 214624 * q^82 + 173888 * q^86 - 218962 * q^89 + 72960 * q^91 + 36852 * q^92 - 22134 * q^93 - 27230 * q^97

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