LMFDB - Newform orbit 75.6.b.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [75,6,Mod(49,75)] 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("75.49"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(75, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 75 = 3 \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	75.b (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,-34] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	no
Analytic conductor:	$12.0287864860$
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:	no (minimal twist has level 15)
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 + 7 i q^{2} - 9 i q^{3} - 17 q^{4} + 63 q^{6} + 12 i q^{7} + 105 i q^{8} - 81 q^{9} + 112 q^{11} + 153 i q^{12} + 974 i q^{13} - 84 q^{14} - 1279 q^{16} + 2182 i q^{17} - 567 i q^{18} - 1420 q^{19} + 108 q^{21} + \cdots - 9072 q^{99} +O(q^{100}) $ q + 7i q^2 - 9i q^3 - 17 * q^4 + 63 * q^6 + 12i q^7 + 105i q^8 - 81 * q^9 + 112 * q^11 + 153i q^12 + 974i q^13 - 84 * q^14 - 1279 * q^16 + 2182i q^17 - 567i q^18 - 1420 * q^19 + 108 * q^21 + 784i q^22 - 3216i q^23 + 945 * q^24 - 6818 * q^26 + 729i q^27 - 204i q^28 + 4150 * q^29 - 5688 * q^31 - 5593i q^32 - 1008i q^33 - 15274 * q^34 + 1377 * q^36 + 6482i q^37 - 9940i q^38 + 8766 * q^39 + 5402 * q^41 + 756i q^42 + 21764i q^43 - 1904 * q^44 + 22512 * q^46 - 368i q^47 + 11511i q^48 + 16663 * q^49 + 19638 * q^51 - 16558i q^52 - 12586i q^53 - 5103 * q^54 - 1260 * q^56 + 12780i q^57 + 29050i q^58 + 25520 * q^59 + 11782 * q^61 - 39816i q^62 - 972i q^63 - 1777 * q^64 + 7056 * q^66 - 13188i q^67 - 37094i q^68 - 28944 * q^69 - 35968 * q^71 - 8505i q^72 - 73186i q^73 - 45374 * q^74 + 24140 * q^76 + 1344i q^77 + 61362i q^78 + 52440 * q^79 + 6561 * q^81 + 37814i q^82 - 69036i q^83 - 1836 * q^84 - 152348 * q^86 - 37350i q^87 + 11760i q^88 + 33870 * q^89 - 11688 * q^91 + 54672i q^92 + 51192i q^93 + 2576 * q^94 - 50337 * q^96 + 143042i q^97 + 116641i q^98 - 9072 * q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 34 q^{4} + 126 q^{6} - 162 q^{9} + 224 q^{11} - 168 q^{14} - 2558 q^{16} - 2840 q^{19} + 216 q^{21} + 1890 q^{24} - 13636 q^{26} + 8300 q^{29} - 11376 q^{31} - 30548 q^{34} + 2754 q^{36} + 17532 q^{39}+ \cdots - 18144 q^{99}+O(q^{100}) $ 2 * q - 34 * q^4 + 126 * q^6 - 162 * q^9 + 224 * q^11 - 168 * q^14 - 2558 * q^16 - 2840 * q^19 + 216 * q^21 + 1890 * q^24 - 13636 * q^26 + 8300 * q^29 - 11376 * q^31 - 30548 * q^34 + 2754 * q^36 + 17532 * q^39 + 10804 * q^41 - 3808 * q^44 + 45024 * q^46 + 33326 * q^49 + 39276 * q^51 - 10206 * q^54 - 2520 * q^56 + 51040 * q^59 + 23564 * q^61 - 3554 * q^64 + 14112 * q^66 - 57888 * q^69 - 71936 * q^71 - 90748 * q^74 + 48280 * q^76 + 104880 * q^79 + 13122 * q^81 - 3672 * q^84 - 304696 * q^86 + 67740 * q^89 - 23376 * q^91 + 5152 * q^94 - 100674 * q^96 - 18144 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/75\mathbb{Z}\right)^\times$.

$n$	$26$	$52$
$\chi(n)$	$1$	$-1$

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} $

49.1

	−	1.00000i
		1.00000i

−

7.00000i

9.00000i

−17.0000

63.0000

−

12.0000i

−

105.000i

−81.0000

49.2

7.00000i

−

9.00000i

−17.0000

63.0000

12.0000i

105.000i

−81.0000

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	75.6.b.a		2
3.b	odd	2	1		225.6.b.a		2
5.b	even	2	1	inner	75.6.b.a		2
5.c	odd	4	1		15.6.a.b	✓	1
5.c	odd	4	1		75.6.a.a		1
15.d	odd	2	1		225.6.b.a		2
15.e	even	4	1		45.6.a.a		1
15.e	even	4	1		225.6.a.h		1
20.e	even	4	1		240.6.a.b		1
35.f	even	4	1		735.6.a.b		1
40.i	odd	4	1		960.6.a.k		1
40.k	even	4	1		960.6.a.x		1
60.l	odd	4	1		720.6.a.q		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
15.6.a.b	✓	1	5.c	odd	4	1
45.6.a.a		1	15.e	even	4	1
75.6.a.a		1	5.c	odd	4	1
75.6.b.a		2	1.a	even	1	1	trivial
75.6.b.a		2	5.b	even	2	1	inner
225.6.a.h		1	15.e	even	4	1
225.6.b.a		2	3.b	odd	2	1
225.6.b.a		2	15.d	odd	2	1
240.6.a.b		1	20.e	even	4	1
720.6.a.q		1	60.l	odd	4	1
735.6.a.b		1	35.f	even	4	1
960.6.a.k		1	40.i	odd	4	1
960.6.a.x		1	40.k	even	4	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2}^{2} + 49 $ T2^2 + 49 acting on $S_{6}^{\mathrm{new}}(75, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 49 $ T^2 + 49
$3$	$ T^{2} + 81 $ T^2 + 81
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 144 $ T^2 + 144
$11$	$ (T - 112)^{2} $ (T - 112)^2
$13$	$ T^{2} + 948676 $ T^2 + 948676
$17$	$ T^{2} + 4761124 $ T^2 + 4761124
$19$	$ (T + 1420)^{2} $ (T + 1420)^2
$23$	$ T^{2} + 10342656 $ T^2 + 10342656
$29$	$ (T - 4150)^{2} $ (T - 4150)^2
$31$	$ (T + 5688)^{2} $ (T + 5688)^2
$37$	$ T^{2} + 42016324 $ T^2 + 42016324
$41$	$ (T - 5402)^{2} $ (T - 5402)^2
$43$	$ T^{2} + 473671696 $ T^2 + 473671696
$47$	$ T^{2} + 135424 $ T^2 + 135424
$53$	$ T^{2} + 158407396 $ T^2 + 158407396
$59$	$ (T - 25520)^{2} $ (T - 25520)^2
$61$	$ (T - 11782)^{2} $ (T - 11782)^2
$67$	$ T^{2} + 173923344 $ T^2 + 173923344
$71$	$ (T + 35968)^{2} $ (T + 35968)^2
$73$	$ T^{2} + 5356190596 $ T^2 + 5356190596
$79$	$ (T - 52440)^{2} $ (T - 52440)^2
$83$	$ T^{2} + 4765969296 $ T^2 + 4765969296
$89$	$ (T - 33870)^{2} $ (T - 33870)^2
$97$	$ T^{2} + 20461013764 $ T^2 + 20461013764
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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:	\( N \)	\(=\)	\( 75 = 3 \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	75.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\( q + 7 i q^{2} - 9 i q^{3} - 17 q^{4} + 63 q^{6} + 12 i q^{7} + 105 i q^{8} - 81 q^{9} + 112 q^{11} + 153 i q^{12} + 974 i q^{13} - 84 q^{14} - 1279 q^{16} + 2182 i q^{17} - 567 i q^{18} - 1420 q^{19} + 108 q^{21} + \cdots - 9072 q^{99} +O(q^{100}) \) q + 7i q^2 - 9i q^3 - 17 * q^4 + 63 * q^6 + 12i q^7 + 105i q^8 - 81 * q^9 + 112 * q^11 + 153i q^12 + 974i q^13 - 84 * q^14 - 1279 * q^16 + 2182i q^17 - 567i q^18 - 1420 * q^19 + 108 * q^21 + 784i q^22 - 3216i q^23 + 945 * q^24 - 6818 * q^26 + 729i q^27 - 204i q^28 + 4150 * q^29 - 5688 * q^31 - 5593i q^32 - 1008i q^33 - 15274 * q^34 + 1377 * q^36 + 6482i q^37 - 9940i q^38 + 8766 * q^39 + 5402 * q^41 + 756i q^42 + 21764i q^43 - 1904 * q^44 + 22512 * q^46 - 368i q^47 + 11511i q^48 + 16663 * q^49 + 19638 * q^51 - 16558i q^52 - 12586i q^53 - 5103 * q^54 - 1260 * q^56 + 12780i q^57 + 29050i q^58 + 25520 * q^59 + 11782 * q^61 - 39816i q^62 - 972i q^63 - 1777 * q^64 + 7056 * q^66 - 13188i q^67 - 37094i q^68 - 28944 * q^69 - 35968 * q^71 - 8505i q^72 - 73186i q^73 - 45374 * q^74 + 24140 * q^76 + 1344i q^77 + 61362i q^78 + 52440 * q^79 + 6561 * q^81 + 37814i q^82 - 69036i q^83 - 1836 * q^84 - 152348 * q^86 - 37350i q^87 + 11760i q^88 + 33870 * q^89 - 11688 * q^91 + 54672i q^92 + 51192i q^93 + 2576 * q^94 - 50337 * q^96 + 143042i q^97 + 116641i q^98 - 9072 * q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 34 q^{4} + 126 q^{6} - 162 q^{9} + 224 q^{11} - 168 q^{14} - 2558 q^{16} - 2840 q^{19} + 216 q^{21} + 1890 q^{24} - 13636 q^{26} + 8300 q^{29} - 11376 q^{31} - 30548 q^{34} + 2754 q^{36} + 17532 q^{39}+ \cdots - 18144 q^{99}+O(q^{100}) \) 2 * q - 34 * q^4 + 126 * q^6 - 162 * q^9 + 224 * q^11 - 168 * q^14 - 2558 * q^16 - 2840 * q^19 + 216 * q^21 + 1890 * q^24 - 13636 * q^26 + 8300 * q^29 - 11376 * q^31 - 30548 * q^34 + 2754 * q^36 + 17532 * q^39 + 10804 * q^41 - 3808 * q^44 + 45024 * q^46 + 33326 * q^49 + 39276 * q^51 - 10206 * q^54 - 2520 * q^56 + 51040 * q^59 + 23564 * q^61 - 3554 * q^64 + 14112 * q^66 - 57888 * q^69 - 71936 * q^71 - 90748 * q^74 + 48280 * q^76 + 104880 * q^79 + 13122 * q^81 - 3672 * q^84 - 304696 * q^86 + 67740 * q^89 - 23376 * q^91 + 5152 * q^94 - 100674 * q^96 - 18144 * q^99

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