LMFDB - Newform orbit 325.4.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [325,4,Mod(51,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("325.51"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	325.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$19.1756207519$
Analytic rank:	$1$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
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 + i q^{2} - 2 q^{3} + 7 q^{4} - 2 i q^{6} - 24 i q^{7} + 15 i q^{8} - 23 q^{9} - 10 i q^{11} - 14 q^{12} + (9 i - 46) q^{13} + 24 q^{14} + 41 q^{16} - 96 q^{17} - 23 i q^{18} + 114 i q^{19} + 48 i q^{21} + \cdots + 230 i q^{99} +O(q^{100}) $ q + i * q^2 - 2 * q^3 + 7 * q^4 - 2i q^6 - 24i q^7 + 15i q^8 - 23 * q^9 - 10i q^11 - 14 * q^12 + (9i - 46) q^13 + 24 * q^14 + 41 * q^16 - 96 * q^17 - 23i q^18 + 114i q^19 + 48i q^21 + 10 * q^22 - 102 * q^23 - 30i q^24 + (-46i - 9) q^26 + 100 * q^27 - 168i q^28 - 30 * q^29 + 210i q^31 + 161i q^32 + 20i q^33 - 96i q^34 - 161 * q^36 + 126i q^37 - 114 * q^38 + (-18i + 92) q^39 - 200i q^41 - 48 * q^42 - 282 * q^43 - 70i q^44 - 102i q^46 - 544i q^47 - 82 * q^48 - 233 * q^49 + 192 * q^51 + (63i - 322) q^52 - 372 * q^53 + 100i q^54 + 360 * q^56 - 228i q^57 - 30i q^58 + 154i q^59 - 738 * q^61 - 210 * q^62 + 552i q^63 + 167 * q^64 - 20 * q^66 + 396i q^67 - 672 * q^68 + 204 * q^69 - 710i q^71 - 345i q^72 - 762i q^73 - 126 * q^74 + 798i q^76 - 240 * q^77 + (92i + 18) q^78 - 240 * q^79 + 421 * q^81 + 200 * q^82 + 68i q^83 + 336i q^84 - 282i q^86 + 60 * q^87 + 150 * q^88 - 1336i q^89 + (1104i + 216) q^91 - 714 * q^92 - 420i q^93 + 544 * q^94 - 322i q^96 + 1626i q^97 - 233i q^98 + 230i q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{3} + 14 q^{4} - 46 q^{9} - 28 q^{12} - 92 q^{13} + 48 q^{14} + 82 q^{16} - 192 q^{17} + 20 q^{22} - 204 q^{23} - 18 q^{26} + 200 q^{27} - 60 q^{29} - 322 q^{36} - 228 q^{38} + 184 q^{39} - 96 q^{42}+ \cdots + 1088 q^{94}+O(q^{100}) $ 2 * q - 4 * q^3 + 14 * q^4 - 46 * q^9 - 28 * q^12 - 92 * q^13 + 48 * q^14 + 82 * q^16 - 192 * q^17 + 20 * q^22 - 204 * q^23 - 18 * q^26 + 200 * q^27 - 60 * q^29 - 322 * q^36 - 228 * q^38 + 184 * q^39 - 96 * q^42 - 564 * q^43 - 164 * q^48 - 466 * q^49 + 384 * q^51 - 644 * q^52 - 744 * q^53 + 720 * q^56 - 1476 * q^61 - 420 * q^62 + 334 * q^64 - 40 * q^66 - 1344 * q^68 + 408 * q^69 - 252 * q^74 - 480 * q^77 + 36 * q^78 - 480 * q^79 + 842 * q^81 + 400 * q^82 + 120 * q^87 + 300 * q^88 + 432 * q^91 - 1428 * q^92 + 1088 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$1$	$-1$

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} $

51.1

	−	1.00000i
		1.00000i

−

1.00000i

−2.00000

7.00000

2.00000i

24.0000i

−

15.0000i

−23.0000

51.2

1.00000i

−2.00000

7.00000

−

2.00000i

−

24.0000i

15.0000i

−23.0000

Inner twists

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

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	325.4.c.a		2
5.b	even	2	1		325.4.c.c		2
5.c	odd	4	1		65.4.d.a	✓	2
5.c	odd	4	1		65.4.d.b	yes	2
13.b	even	2	1	inner	325.4.c.a		2
65.d	even	2	1		325.4.c.c		2
65.h	odd	4	1		65.4.d.a	✓	2
65.h	odd	4	1		65.4.d.b	yes	2

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
65.4.d.a	✓	2	5.c	odd	4	1
65.4.d.a	✓	2	65.h	odd	4	1
65.4.d.b	yes	2	5.c	odd	4	1
65.4.d.b	yes	2	65.h	odd	4	1
325.4.c.a		2	1.a	even	1	1	trivial
325.4.c.a		2	13.b	even	2	1	inner
325.4.c.c		2	5.b	even	2	1
325.4.c.c		2	65.d	even	2	1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $S_{4}^{\mathrm{new}}(325, [\chi])$:

$ T_{2}^{2} + 1 $

T2^2 + 1

$ T_{3} + 2 $

T3 + 2

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{2} + 1 $ T^2 + 1
$3$	$ (T + 2)^{2} $ (T + 2)^2
$5$	$ T^{2} $ T^2
$7$	$ T^{2} + 576 $ T^2 + 576
$11$	$ T^{2} + 100 $ T^2 + 100
$13$	$ T^{2} + 92T + 2197 $ T^2 + 92*T + 2197
$17$	$ (T + 96)^{2} $ (T + 96)^2
$19$	$ T^{2} + 12996 $ T^2 + 12996
$23$	$ (T + 102)^{2} $ (T + 102)^2
$29$	$ (T + 30)^{2} $ (T + 30)^2
$31$	$ T^{2} + 44100 $ T^2 + 44100
$37$	$ T^{2} + 15876 $ T^2 + 15876
$41$	$ T^{2} + 40000 $ T^2 + 40000
$43$	$ (T + 282)^{2} $ (T + 282)^2
$47$	$ T^{2} + 295936 $ T^2 + 295936
$53$	$ (T + 372)^{2} $ (T + 372)^2
$59$	$ T^{2} + 23716 $ T^2 + 23716
$61$	$ (T + 738)^{2} $ (T + 738)^2
$67$	$ T^{2} + 156816 $ T^2 + 156816
$71$	$ T^{2} + 504100 $ T^2 + 504100
$73$	$ T^{2} + 580644 $ T^2 + 580644
$79$	$ (T + 240)^{2} $ (T + 240)^2
$83$	$ T^{2} + 4624 $ T^2 + 4624
$89$	$ T^{2} + 1784896 $ T^2 + 1784896
$97$	$ T^{2} + 2643876 $ T^2 + 2643876
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Level:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	325.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + i q^{2} - 2 q^{3} + 7 q^{4} - 2 i q^{6} - 24 i q^{7} + 15 i q^{8} - 23 q^{9} - 10 i q^{11} - 14 q^{12} + (9 i - 46) q^{13} + 24 q^{14} + 41 q^{16} - 96 q^{17} - 23 i q^{18} + 114 i q^{19} + 48 i q^{21} + \cdots + 230 i q^{99} +O(q^{100}) \) q + i * q^2 - 2 * q^3 + 7 * q^4 - 2i q^6 - 24i q^7 + 15i q^8 - 23 * q^9 - 10i q^11 - 14 * q^12 + (9i - 46) q^13 + 24 * q^14 + 41 * q^16 - 96 * q^17 - 23i q^18 + 114i q^19 + 48i q^21 + 10 * q^22 - 102 * q^23 - 30i q^24 + (-46i - 9) q^26 + 100 * q^27 - 168i q^28 - 30 * q^29 + 210i q^31 + 161i q^32 + 20i q^33 - 96i q^34 - 161 * q^36 + 126i q^37 - 114 * q^38 + (-18i + 92) q^39 - 200i q^41 - 48 * q^42 - 282 * q^43 - 70i q^44 - 102i q^46 - 544i q^47 - 82 * q^48 - 233 * q^49 + 192 * q^51 + (63i - 322) q^52 - 372 * q^53 + 100i q^54 + 360 * q^56 - 228i q^57 - 30i q^58 + 154i q^59 - 738 * q^61 - 210 * q^62 + 552i q^63 + 167 * q^64 - 20 * q^66 + 396i q^67 - 672 * q^68 + 204 * q^69 - 710i q^71 - 345i q^72 - 762i q^73 - 126 * q^74 + 798i q^76 - 240 * q^77 + (92i + 18) q^78 - 240 * q^79 + 421 * q^81 + 200 * q^82 + 68i q^83 + 336i q^84 - 282i q^86 + 60 * q^87 + 150 * q^88 - 1336i q^89 + (1104i + 216) q^91 - 714 * q^92 - 420i q^93 + 544 * q^94 - 322i q^96 + 1626i q^97 - 233i q^98 + 230i q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{3} + 14 q^{4} - 46 q^{9} - 28 q^{12} - 92 q^{13} + 48 q^{14} + 82 q^{16} - 192 q^{17} + 20 q^{22} - 204 q^{23} - 18 q^{26} + 200 q^{27} - 60 q^{29} - 322 q^{36} - 228 q^{38} + 184 q^{39} - 96 q^{42}+ \cdots + 1088 q^{94}+O(q^{100}) \) 2 * q - 4 * q^3 + 14 * q^4 - 46 * q^9 - 28 * q^12 - 92 * q^13 + 48 * q^14 + 82 * q^16 - 192 * q^17 + 20 * q^22 - 204 * q^23 - 18 * q^26 + 200 * q^27 - 60 * q^29 - 322 * q^36 - 228 * q^38 + 184 * q^39 - 96 * q^42 - 564 * q^43 - 164 * q^48 - 466 * q^49 + 384 * q^51 - 644 * q^52 - 744 * q^53 + 720 * q^56 - 1476 * q^61 - 420 * q^62 + 334 * q^64 - 40 * q^66 - 1344 * q^68 + 408 * q^69 - 252 * q^74 - 480 * q^77 + 36 * q^78 - 480 * q^79 + 842 * q^81 + 400 * q^82 + 120 * q^87 + 300 * q^88 + 432 * q^91 - 1428 * q^92 + 1088 * q^94

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