LMFDB - Newform orbit 30.8.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [30,8,Mod(19,30)] 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("30.19"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 30 = 2 \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	30.c (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:	$9.37155076452$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-1}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
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 $i = \sqrt{-1}$. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + 8 i q^{2} - 27 i q^{3} - 64 q^{4} + (275 i - 50) q^{5} + 216 q^{6} - 1126 i q^{7} - 512 i q^{8} - 729 q^{9} + ( - 400 i - 2200) q^{10} - 5518 q^{11} + 1728 i q^{12} - 12798 i q^{13} + 9008 q^{14} + (1350 i + 7425) q^{15} + \cdots + 4022622 q^{99} +O(q^{100}) $ q + 8i q^2 - 27i q^3 - 64 * q^4 + (275i - 50) q^5 + 216 * q^6 - 1126i q^7 - 512i q^8 - 729 * q^9 + (-400i - 2200) q^10 - 5518 * q^11 + 1728i q^12 - 12798i q^13 + 9008 * q^14 + (1350i + 7425) q^15 + 4096 * q^16 - 32206i q^17 - 5832i q^18 + 4440 * q^19 + (-17600i + 3200) q^20 - 30402 * q^21 - 44144i q^22 + 95452i q^23 - 13824 * q^24 + (-27500i - 73125) q^25 + 102384 * q^26 + 19683i q^27 + 72064i q^28 - 19440 * q^29 + (59400i - 10800) q^30 - 240248 * q^31 + 32768i q^32 + 148986i q^33 + 257648 * q^34 + (56300i + 309650) q^35 + 46656 * q^36 + 77834i q^37 + 35520i q^38 - 345546 * q^39 + (25600i + 140800) q^40 + 299522 * q^41 - 243216i q^42 + 416212i q^43 + 353152 * q^44 + (-200475i + 36450) q^45 - 763616 * q^46 - 322976i q^47 - 110592i q^48 - 444333 * q^49 + (-585000i + 220000) q^50 - 869562 * q^51 + 819072i q^52 - 880878i q^53 - 157464 * q^54 + (-1517450i + 275900) q^55 - 576512 * q^56 - 119880i q^57 - 155520i q^58 + 1845110 * q^59 + (-86400i - 475200) q^60 - 861718 * q^61 - 1921984i q^62 + 820854i q^63 - 262144 * q^64 + (639900i + 3519450) q^65 - 1191888 * q^66 + 673864i q^67 + 2061184i q^68 + 2577204 * q^69 + (2477200i - 450400) q^70 - 3426948 * q^71 + 373248i q^72 - 4678748i q^73 - 622672 * q^74 + (1974375i - 742500) q^75 - 284160 * q^76 + 6213268i q^77 - 2764368i q^78 + 3137760 * q^79 + (1126400i - 204800) q^80 + 531441 * q^81 + 2396176i q^82 + 484132i q^83 + 1945728 * q^84 + (1610300i + 8856650) q^85 - 3329696 * q^86 + 524880i q^87 + 2825216i q^88 - 6258710 * q^89 + (291600i + 1603800) q^90 - 14410548 * q^91 - 6108928i q^92 + 6486696i q^93 + 2583808 * q^94 + (1221000i - 222000) q^95 + 884736 * q^96 - 8657576i q^97 - 3554664i q^98 + 4022622 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 128 q^{4} - 100 q^{5} + 432 q^{6} - 1458 q^{9} - 4400 q^{10} - 11036 q^{11} + 18016 q^{14} + 14850 q^{15} + 8192 q^{16} + 8880 q^{19} + 6400 q^{20} - 60804 q^{21} - 27648 q^{24} - 146250 q^{25} + 204768 q^{26}+ \cdots + 8045244 q^{99}+O(q^{100}) $ 2 * q - 128 * q^4 - 100 * q^5 + 432 * q^6 - 1458 * q^9 - 4400 * q^10 - 11036 * q^11 + 18016 * q^14 + 14850 * q^15 + 8192 * q^16 + 8880 * q^19 + 6400 * q^20 - 60804 * q^21 - 27648 * q^24 - 146250 * q^25 + 204768 * q^26 - 38880 * q^29 - 21600 * q^30 - 480496 * q^31 + 515296 * q^34 + 619300 * q^35 + 93312 * q^36 - 691092 * q^39 + 281600 * q^40 + 599044 * q^41 + 706304 * q^44 + 72900 * q^45 - 1527232 * q^46 - 888666 * q^49 + 440000 * q^50 - 1739124 * q^51 - 314928 * q^54 + 551800 * q^55 - 1153024 * q^56 + 3690220 * q^59 - 950400 * q^60 - 1723436 * q^61 - 524288 * q^64 + 7038900 * q^65 - 2383776 * q^66 + 5154408 * q^69 - 900800 * q^70 - 6853896 * q^71 - 1245344 * q^74 - 1485000 * q^75 - 568320 * q^76 + 6275520 * q^79 - 409600 * q^80 + 1062882 * q^81 + 3891456 * q^84 + 17713300 * q^85 - 6659392 * q^86 - 12517420 * q^89 + 3207600 * q^90 - 28821096 * q^91 + 5167616 * q^94 - 444000 * q^95 + 1769472 * q^96 + 8045244 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/30\mathbb{Z}\right)^\times$.

$n$	$7$	$11$
$\chi(n)$	$-1$	$1$

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} $

19.1

	−	1.00000i
		1.00000i

−

8.00000i

27.0000i

−64.0000

−50.0000

−

275.000i

216.000

1126.00i

512.000i

−729.000

−2200.00

400.000i

19.2

8.00000i

−

27.0000i

−64.0000

−50.0000

275.000i

216.000

−

1126.00i

−

512.000i

−729.000

−2200.00

−

400.000i

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	30.8.c.a	✓	2
3.b	odd	2	1		90.8.c.a		2
4.b	odd	2	1		240.8.f.a		2
5.b	even	2	1	inner	30.8.c.a	✓	2
5.c	odd	4	1		150.8.a.d		1
5.c	odd	4	1		150.8.a.m		1
15.d	odd	2	1		90.8.c.a		2
15.e	even	4	1		450.8.a.b		1
15.e	even	4	1		450.8.a.y		1
20.d	odd	2	1		240.8.f.a		2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
30.8.c.a	✓	2	1.a	even	1	1	trivial
30.8.c.a	✓	2	5.b	even	2	1	inner
90.8.c.a		2	3.b	odd	2	1
90.8.c.a		2	15.d	odd	2	1
150.8.a.d		1	5.c	odd	4	1
150.8.a.m		1	5.c	odd	4	1
240.8.f.a		2	4.b	odd	2	1
240.8.f.a		2	20.d	odd	2	1
450.8.a.b		1	15.e	even	4	1
450.8.a.y		1	15.e	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{7}^{2} + 1267876 $ T7^2 + 1267876 acting on $S_{8}^{\mathrm{new}}(30, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 64 $ T^2 + 64
$3$	$ T^{2} + 729 $ T^2 + 729
$5$	$ T^{2} + 100T + 78125 $ T^2 + 100*T + 78125
$7$	$ T^{2} + 1267876 $ T^2 + 1267876
$11$	$ (T + 5518)^{2} $ (T + 5518)^2
$13$	$ T^{2} + 163788804 $ T^2 + 163788804
$17$	$ T^{2} + 1037226436 $ T^2 + 1037226436
$19$	$ (T - 4440)^{2} $ (T - 4440)^2
$23$	$ T^{2} + 9111084304 $ T^2 + 9111084304
$29$	$ (T + 19440)^{2} $ (T + 19440)^2
$31$	$ (T + 240248)^{2} $ (T + 240248)^2
$37$	$ T^{2} + 6058131556 $ T^2 + 6058131556
$41$	$ (T - 299522)^{2} $ (T - 299522)^2
$43$	$ T^{2} + 173232428944 $ T^2 + 173232428944
$47$	$ T^{2} + 104313496576 $ T^2 + 104313496576
$53$	$ T^{2} + 775946050884 $ T^2 + 775946050884
$59$	$ (T - 1845110)^{2} $ (T - 1845110)^2
$61$	$ (T + 861718)^{2} $ (T + 861718)^2
$67$	$ T^{2} + 454092690496 $ T^2 + 454092690496
$71$	$ (T + 3426948)^{2} $ (T + 3426948)^2
$73$	$ T^{2} + 21890682847504 $ T^2 + 21890682847504
$79$	$ (T - 3137760)^{2} $ (T - 3137760)^2
$83$	$ T^{2} + 234383793424 $ T^2 + 234383793424
$89$	$ (T + 6258710)^{2} $ (T + 6258710)^2
$97$	$ T^{2} + 74953622195776 $ T^2 + 74953622195776
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Elliptic curves over $\Q$
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Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 30 = 2 \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	30.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + 8 i q^{2} - 27 i q^{3} - 64 q^{4} + (275 i - 50) q^{5} + 216 q^{6} - 1126 i q^{7} - 512 i q^{8} - 729 q^{9} + ( - 400 i - 2200) q^{10} - 5518 q^{11} + 1728 i q^{12} - 12798 i q^{13} + 9008 q^{14} + (1350 i + 7425) q^{15} + \cdots + 4022622 q^{99} +O(q^{100}) \) q + 8i q^2 - 27i q^3 - 64 * q^4 + (275i - 50) q^5 + 216 * q^6 - 1126i q^7 - 512i q^8 - 729 * q^9 + (-400i - 2200) q^10 - 5518 * q^11 + 1728i q^12 - 12798i q^13 + 9008 * q^14 + (1350i + 7425) q^15 + 4096 * q^16 - 32206i q^17 - 5832i q^18 + 4440 * q^19 + (-17600i + 3200) q^20 - 30402 * q^21 - 44144i q^22 + 95452i q^23 - 13824 * q^24 + (-27500i - 73125) q^25 + 102384 * q^26 + 19683i q^27 + 72064i q^28 - 19440 * q^29 + (59400i - 10800) q^30 - 240248 * q^31 + 32768i q^32 + 148986i q^33 + 257648 * q^34 + (56300i + 309650) q^35 + 46656 * q^36 + 77834i q^37 + 35520i q^38 - 345546 * q^39 + (25600i + 140800) q^40 + 299522 * q^41 - 243216i q^42 + 416212i q^43 + 353152 * q^44 + (-200475i + 36450) q^45 - 763616 * q^46 - 322976i q^47 - 110592i q^48 - 444333 * q^49 + (-585000i + 220000) q^50 - 869562 * q^51 + 819072i q^52 - 880878i q^53 - 157464 * q^54 + (-1517450i + 275900) q^55 - 576512 * q^56 - 119880i q^57 - 155520i q^58 + 1845110 * q^59 + (-86400i - 475200) q^60 - 861718 * q^61 - 1921984i q^62 + 820854i q^63 - 262144 * q^64 + (639900i + 3519450) q^65 - 1191888 * q^66 + 673864i q^67 + 2061184i q^68 + 2577204 * q^69 + (2477200i - 450400) q^70 - 3426948 * q^71 + 373248i q^72 - 4678748i q^73 - 622672 * q^74 + (1974375i - 742500) q^75 - 284160 * q^76 + 6213268i q^77 - 2764368i q^78 + 3137760 * q^79 + (1126400i - 204800) q^80 + 531441 * q^81 + 2396176i q^82 + 484132i q^83 + 1945728 * q^84 + (1610300i + 8856650) q^85 - 3329696 * q^86 + 524880i q^87 + 2825216i q^88 - 6258710 * q^89 + (291600i + 1603800) q^90 - 14410548 * q^91 - 6108928i q^92 + 6486696i q^93 + 2583808 * q^94 + (1221000i - 222000) q^95 + 884736 * q^96 - 8657576i q^97 - 3554664i q^98 + 4022622 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 128 q^{4} - 100 q^{5} + 432 q^{6} - 1458 q^{9} - 4400 q^{10} - 11036 q^{11} + 18016 q^{14} + 14850 q^{15} + 8192 q^{16} + 8880 q^{19} + 6400 q^{20} - 60804 q^{21} - 27648 q^{24} - 146250 q^{25} + 204768 q^{26}+ \cdots + 8045244 q^{99}+O(q^{100}) \) 2 * q - 128 * q^4 - 100 * q^5 + 432 * q^6 - 1458 * q^9 - 4400 * q^10 - 11036 * q^11 + 18016 * q^14 + 14850 * q^15 + 8192 * q^16 + 8880 * q^19 + 6400 * q^20 - 60804 * q^21 - 27648 * q^24 - 146250 * q^25 + 204768 * q^26 - 38880 * q^29 - 21600 * q^30 - 480496 * q^31 + 515296 * q^34 + 619300 * q^35 + 93312 * q^36 - 691092 * q^39 + 281600 * q^40 + 599044 * q^41 + 706304 * q^44 + 72900 * q^45 - 1527232 * q^46 - 888666 * q^49 + 440000 * q^50 - 1739124 * q^51 - 314928 * q^54 + 551800 * q^55 - 1153024 * q^56 + 3690220 * q^59 - 950400 * q^60 - 1723436 * q^61 - 524288 * q^64 + 7038900 * q^65 - 2383776 * q^66 + 5154408 * q^69 - 900800 * q^70 - 6853896 * q^71 - 1245344 * q^74 - 1485000 * q^75 - 568320 * q^76 + 6275520 * q^79 - 409600 * q^80 + 1062882 * q^81 + 3891456 * q^84 + 17713300 * q^85 - 6659392 * q^86 - 12517420 * q^89 + 3207600 * q^90 - 28821096 * q^91 + 5167616 * q^94 - 444000 * q^95 + 1769472 * q^96 + 8045244 * q^99

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