LMFDB - Embedded newform 7600.2.a.cf.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7600,2,Mod(1,7600)] 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("7600.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 7600 = 2^{4} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7600.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,2,0,0,0,4,0,8,0,-4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.6863055362$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.51658$ of defining polynomial
Character	$\chi$	$=$	7600.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53844 q^{3} +5.03316 q^{7} -0.633188 q^{9} +3.03316 q^{11} +4.57160 q^{13} +1.07689 q^{17} -1.00000 q^{19} +7.74324 q^{21} +4.11005 q^{23} -5.58946 q^{27} -1.07689 q^{29} -5.58946 q^{31} +4.66635 q^{33} -0.0947438 q^{37} +7.03316 q^{39} +10.6663 q^{41} +5.03316 q^{43} -12.2995 q^{47} +18.3327 q^{49} +1.65673 q^{51} +4.09474 q^{53} -1.53844 q^{57} +1.39997 q^{59} -5.69951 q^{61} -3.18694 q^{63} +5.28168 q^{67} +6.32308 q^{69} +5.67692 q^{71} -9.07689 q^{73} +15.2664 q^{77} +5.39997 q^{79} -6.69951 q^{81} +1.95627 q^{83} -1.65673 q^{87} -2.18949 q^{89} +23.0096 q^{91} -8.59907 q^{93} +2.16106 q^{97} -1.92056 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 4 q^{21} - 8 q^{23} - 4 q^{27} + 4 q^{29} - 4 q^{31} - 8 q^{33} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} - 12 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 + 4 * q^7 + 8 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^17 - 4 * q^19 - 4 * q^21 - 8 * q^23 - 4 * q^27 + 4 * q^29 - 4 * q^31 - 8 * q^33 + 6 * q^37 + 12 * q^39 + 16 * q^41 + 4 * q^43 - 12 * q^47 + 20 * q^49 + 36 * q^51 + 10 * q^53 - 2 * q^57 + 20 * q^61 + 20 * q^63 - 18 * q^67 + 28 * q^69 + 20 * q^71 - 28 * q^73 + 40 * q^77 + 16 * q^79 + 16 * q^81 - 36 * q^87 + 4 * q^89 + 36 * q^91 + 40 * q^93 - 30 * q^97 + 4 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$. You can download additional coefficients here.

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.53844	0.888221	0.444111	−	0.895972i	$-0.353520\pi$
$3$	1.53844	0.888221	0.444111	+	0.895972i	$0.353520\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	5.03316	1.90236	0.951178	−	0.308644i	$-0.0998750\pi$
$7$	5.03316	1.90236	0.951178	+	0.308644i	$0.0998750\pi$
$8$	0	0
$9$	−0.633188	−0.211063
$10$	0	0
$11$	3.03316	0.914532	0.457266	−	0.889330i	$-0.348829\pi$
$11$	3.03316	0.914532	0.457266	+	0.889330i	$0.348829\pi$
$12$	0	0
$13$	4.57160	1.26793	0.633967	−	0.773360i	$-0.281425\pi$
$13$	4.57160	1.26793	0.633967	+	0.773360i	$0.281425\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.07689	0.261184	0.130592	−	0.991436i	$-0.458312\pi$
$17$	1.07689	0.261184	0.130592	+	0.991436i	$0.458312\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	7.74324	1.68971
$22$	0	0
$23$	4.11005	0.857004	0.428502	−	0.903541i	$-0.359041\pi$
$23$	4.11005	0.857004	0.428502	+	0.903541i	$0.359041\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−5.58946	−1.07569
$28$	0	0
$29$	−1.07689	−0.199973	−0.0999866	−	0.994989i	$-0.531880\pi$
$29$	−1.07689	−0.199973	−0.0999866	+	0.994989i	$0.531880\pi$
$30$	0	0
$31$	−5.58946	−1.00390	−0.501948	−	0.864898i	$-0.667383\pi$
$31$	−5.58946	−1.00390	−0.501948	+	0.864898i	$0.667383\pi$
$32$	0	0
$33$	4.66635	0.812307
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.0947438	−0.0155758	−0.00778789	−	0.999970i	$-0.502479\pi$
$37$	−0.0947438	−0.0155758	−0.00778789	+	0.999970i	$0.502479\pi$
$38$	0	0
$39$	7.03316	1.12621
$40$	0	0
$41$	10.6663	1.66580	0.832902	−	0.553421i	$-0.186678\pi$
$41$	10.6663	1.66580	0.832902	+	0.553421i	$0.186678\pi$
$42$	0	0
$43$	5.03316	0.767550	0.383775	−	0.923427i	$-0.374624\pi$
$43$	5.03316	0.767550	0.383775	+	0.923427i	$0.374624\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.2995	−1.79407	−0.897036	−	0.441958i	$-0.854284\pi$
$47$	−12.2995	−1.79407	−0.897036	+	0.441958i	$0.854284\pi$
$48$	0	0
$49$	18.3327	2.61896
$50$	0	0
$51$	1.65673	0.231989
$52$	0	0
$53$	4.09474	0.562456	0.281228	−	0.959641i	$-0.409258\pi$
$53$	4.09474	0.562456	0.281228	+	0.959641i	$0.409258\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.53844	−0.203772
$58$	0	0
$59$	1.39997	0.182261	0.0911304	−	0.995839i	$-0.470952\pi$
$59$	1.39997	0.182261	0.0911304	+	0.995839i	$0.470952\pi$
$60$	0	0
$61$	−5.69951	−0.729747	−0.364874	−	0.931057i	$-0.618888\pi$
$61$	−5.69951	−0.729747	−0.364874	+	0.931057i	$0.618888\pi$
$62$	0	0
$63$	−3.18694	−0.401516
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.28168	0.645260	0.322630	−	0.946525i	$-0.395433\pi$
$67$	5.28168	0.645260	0.322630	+	0.946525i	$0.395433\pi$
$68$	0	0
$69$	6.32308	0.761210
$70$	0	0
$71$	5.67692	0.673726	0.336863	−	0.941554i	$-0.390634\pi$
$71$	5.67692	0.673726	0.336863	+	0.941554i	$0.390634\pi$
$72$	0	0
$73$	−9.07689	−1.06237	−0.531185	−	0.847256i	$-0.678253\pi$
$73$	−9.07689	−1.06237	−0.531185	+	0.847256i	$0.678253\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	15.2664	1.73977
$78$	0	0
$79$	5.39997	0.607544	0.303772	−	0.952745i	$-0.401754\pi$
$79$	5.39997	0.607544	0.303772	+	0.952745i	$0.401754\pi$
$80$	0	0
$81$	−6.69951	−0.744390
$82$	0	0
$83$	1.95627	0.214729	0.107364	−	0.994220i	$-0.465759\pi$
$83$	1.95627	0.214729	0.107364	+	0.994220i	$0.465759\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−1.65673	−0.177621
$88$	0	0
$89$	−2.18949	−0.232085	−0.116043	−	0.993244i	$-0.537021\pi$
$89$	−2.18949	−0.232085	−0.116043	+	0.993244i	$0.537021\pi$
$90$	0	0
$91$	23.0096	2.41206
$92$	0	0
$93$	−8.59907	−0.891682
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.16106	0.219423	0.109711	−	0.993963i	$-0.465007\pi$
$97$	2.16106	0.219423	0.109711	+	0.993963i	$0.465007\pi$
$98$	0	0
$99$	−1.92056	−0.193024

Currently showing only $a_p$; display all $a_n$ Currently showing all $a_n$; display only $a_p$

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7600.2.a.cf.1.3		4
4.3	odd	2		475.2.a.i.1.2		4
5.4	even	2		1520.2.a.t.1.2		4
12.11	even	2		4275.2.a.bo.1.3		4
20.3	even	4		475.2.b.e.324.5		8
20.7	even	4		475.2.b.e.324.4		8
20.19	odd	2		95.2.a.b.1.3	✓	4
40.19	odd	2		6080.2.a.cc.1.2		4
40.29	even	2		6080.2.a.ch.1.3		4
60.59	even	2		855.2.a.m.1.2		4
76.75	even	2		9025.2.a.bf.1.3		4
140.139	even	2		4655.2.a.y.1.3		4
380.379	even	2		1805.2.a.p.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.a.b.1.3	✓	4	20.19	odd	2
475.2.a.i.1.2		4	4.3	odd	2
475.2.b.e.324.4		8	20.7	even	4
475.2.b.e.324.5		8	20.3	even	4
855.2.a.m.1.2		4	60.59	even	2
1520.2.a.t.1.2		4	5.4	even	2
1805.2.a.p.1.2		4	380.379	even	2
4275.2.a.bo.1.3		4	12.11	even	2
4655.2.a.y.1.3		4	140.139	even	2
6080.2.a.cc.1.2		4	40.19	odd	2
6080.2.a.ch.1.3		4	40.29	even	2
7600.2.a.cf.1.3		4	1.1	even	1	trivial
9025.2.a.bf.1.3		4	76.75	even	2

Overview	Random
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Classical	Maass
Hilbert	Bianchi

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:	\( N \)	\(=\)	\( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7600.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53844 q^{3} +5.03316 q^{7} -0.633188 q^{9} +3.03316 q^{11} +4.57160 q^{13} +1.07689 q^{17} -1.00000 q^{19} +7.74324 q^{21} +4.11005 q^{23} -5.58946 q^{27} -1.07689 q^{29} -5.58946 q^{31} +4.66635 q^{33} -0.0947438 q^{37} +7.03316 q^{39} +10.6663 q^{41} +5.03316 q^{43} -12.2995 q^{47} +18.3327 q^{49} +1.65673 q^{51} +4.09474 q^{53} -1.53844 q^{57} +1.39997 q^{59} -5.69951 q^{61} -3.18694 q^{63} +5.28168 q^{67} +6.32308 q^{69} +5.67692 q^{71} -9.07689 q^{73} +15.2664 q^{77} +5.39997 q^{79} -6.69951 q^{81} +1.95627 q^{83} -1.65673 q^{87} -2.18949 q^{89} +23.0096 q^{91} -8.59907 q^{93} +2.16106 q^{97} -1.92056 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} + 4 q^{7} + 8 q^{9} - 4 q^{11} - 2 q^{13} - 4 q^{17} - 4 q^{19} - 4 q^{21} - 8 q^{23} - 4 q^{27} + 4 q^{29} - 4 q^{31} - 8 q^{33} + 6 q^{37} + 12 q^{39} + 16 q^{41} + 4 q^{43} - 12 q^{47}+ \cdots + 4 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 + 4 * q^7 + 8 * q^9 - 4 * q^11 - 2 * q^13 - 4 * q^17 - 4 * q^19 - 4 * q^21 - 8 * q^23 - 4 * q^27 + 4 * q^29 - 4 * q^31 - 8 * q^33 + 6 * q^37 + 12 * q^39 + 16 * q^41 + 4 * q^43 - 12 * q^47 + 20 * q^49 + 36 * q^51 + 10 * q^53 - 2 * q^57 + 20 * q^61 + 20 * q^63 - 18 * q^67 + 28 * q^69 + 20 * q^71 - 28 * q^73 + 40 * q^77 + 16 * q^79 + 16 * q^81 - 36 * q^87 + 4 * q^89 + 36 * q^91 + 40 * q^93 - 30 * q^97 + 4 * q^99

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