LMFDB - Embedded newform 1980.4.a.q.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1980,4,Mod(1,1980)] 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("1980.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1980.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,25,0,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$116.823781811$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 276x^{3} + 1518x^{2} + 8910x - 55890 $ x^5 - x^4 - 276x^3 + 1518x^2 + 8910*x - 55890
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-17.4852$ of defining polynomial
Character	$\chi$	$=$	1980.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+5.00000 q^{5} -11.1055 q^{7} +11.0000 q^{11} +22.1932 q^{13} -32.7773 q^{17} +19.0060 q^{19} +49.5350 q^{23} +25.0000 q^{25} -189.088 q^{29} -313.571 q^{31} -55.5273 q^{35} +176.028 q^{37} -239.694 q^{41} +340.648 q^{43} +508.846 q^{47} -219.669 q^{49} -45.2446 q^{53} +55.0000 q^{55} -565.688 q^{59} +711.062 q^{61} +110.966 q^{65} -399.799 q^{67} +550.754 q^{71} +32.8102 q^{73} -122.160 q^{77} -277.805 q^{79} -1069.85 q^{83} -163.886 q^{85} +240.529 q^{89} -246.465 q^{91} +95.0298 q^{95} -1278.39 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 25 q^{5} - 16 q^{7} + 55 q^{11} - 4 q^{13} - 102 q^{17} - 10 q^{19} - 178 q^{23} + 125 q^{25} - 318 q^{29} + 72 q^{31} - 80 q^{35} - 294 q^{37} - 270 q^{41} - 110 q^{43} - 666 q^{47} + 349 q^{49}+ \cdots - 1866 q^{97}+O(q^{100}) $ 5 * q + 25 * q^5 - 16 * q^7 + 55 * q^11 - 4 * q^13 - 102 * q^17 - 10 * q^19 - 178 * q^23 + 125 * q^25 - 318 * q^29 + 72 * q^31 - 80 * q^35 - 294 * q^37 - 270 * q^41 - 110 * q^43 - 666 * q^47 + 349 * q^49 - 510 * q^53 + 275 * q^55 + 200 * q^59 - 584 * q^61 - 20 * q^65 - 1588 * q^67 - 146 * q^71 - 178 * q^73 - 176 * q^77 - 844 * q^79 + 996 * q^83 - 510 * q^85 + 208 * q^89 - 1504 * q^91 - 50 * q^95 - 1866 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	−11.1055	−0.599638	−0.299819	−	0.953996i	$-0.596926\pi$
$7$	−11.1055	−0.599638	−0.299819	+	0.953996i	$0.596926\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	22.1932	0.473483	0.236741	−	0.971573i	$-0.423921\pi$
$13$	22.1932	0.473483	0.236741	+	0.971573i	$0.423921\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−32.7773	−0.467627	−0.233813	−	0.972281i	$-0.575121\pi$
$17$	−32.7773	−0.467627	−0.233813	+	0.972281i	$0.575121\pi$
$18$	0	0
$19$	19.0060	0.229488	0.114744	−	0.993395i	$-0.463395\pi$
$19$	19.0060	0.229488	0.114744	+	0.993395i	$0.463395\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	49.5350	0.449076	0.224538	−	0.974465i	$-0.427913\pi$
$23$	49.5350	0.449076	0.224538	+	0.974465i	$0.427913\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−189.088	−1.21078	−0.605392	−	0.795927i	$-0.706984\pi$
$29$	−189.088	−1.21078	−0.605392	+	0.795927i	$0.706984\pi$
$30$	0	0
$31$	−313.571	−1.81674	−0.908372	−	0.418163i	$-0.862674\pi$
$31$	−313.571	−1.81674	−0.908372	+	0.418163i	$0.862674\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−55.5273	−0.268166
$36$	0	0
$37$	176.028	0.782132	0.391066	−	0.920363i	$-0.372106\pi$
$37$	176.028	0.782132	0.391066	+	0.920363i	$0.372106\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−239.694	−0.913021	−0.456511	−	0.889718i	$-0.650901\pi$
$41$	−239.694	−0.913021	−0.456511	+	0.889718i	$0.650901\pi$
$42$	0	0
$43$	340.648	1.20810	0.604050	−	0.796947i	$-0.293553\pi$
$43$	340.648	1.20810	0.604050	+	0.796947i	$0.293553\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	508.846	1.57921	0.789604	−	0.613617i	$-0.210286\pi$
$47$	508.846	1.57921	0.789604	+	0.613617i	$0.210286\pi$
$48$	0	0
$49$	−219.669	−0.640434
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−45.2446	−0.117261	−0.0586304	−	0.998280i	$-0.518673\pi$
$53$	−45.2446	−0.117261	−0.0586304	+	0.998280i	$0.518673\pi$
$54$	0	0
$55$	55.0000	0.134840
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−565.688	−1.24824	−0.624121	−	0.781328i	$-0.714543\pi$
$59$	−565.688	−1.24824	−0.624121	+	0.781328i	$0.714543\pi$
$60$	0	0
$61$	711.062	1.49250	0.746248	−	0.665668i	$-0.231853\pi$
$61$	711.062	1.49250	0.746248	+	0.665668i	$0.231853\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	110.966	0.211748
$66$	0	0
$67$	−399.799	−0.729004	−0.364502	−	0.931203i	$-0.618761\pi$
$67$	−399.799	−0.729004	−0.364502	+	0.931203i	$0.618761\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	550.754	0.920598	0.460299	−	0.887764i	$-0.347742\pi$
$71$	550.754	0.920598	0.460299	+	0.887764i	$0.347742\pi$
$72$	0	0
$73$	32.8102	0.0526047	0.0263024	−	0.999654i	$-0.491627\pi$
$73$	32.8102	0.0526047	0.0263024	+	0.999654i	$0.491627\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−122.160	−0.180798
$78$	0	0
$79$	−277.805	−0.395639	−0.197820	−	0.980238i	$-0.563386\pi$
$79$	−277.805	−0.395639	−0.197820	+	0.980238i	$0.563386\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1069.85	−1.41483	−0.707417	−	0.706796i	$-0.750140\pi$
$83$	−1069.85	−1.41483	−0.707417	+	0.706796i	$0.750140\pi$
$84$	0	0
$85$	−163.886	−0.209129
$86$	0	0
$87$	0	0
$88$	0	0
$89$	240.529	0.286472	0.143236	−	0.989689i	$-0.454249\pi$
$89$	240.529	0.286472	0.143236	+	0.989689i	$0.454249\pi$
$90$	0	0
$91$	−246.465	−0.283918
$92$	0	0
$93$	0	0
$94$	0	0
$95$	95.0298	0.102630
$96$	0	0
$97$	−1278.39	−1.33816	−0.669079	−	0.743191i	$-0.733311\pi$
$97$	−1278.39	−1.33816	−0.669079	+	0.743191i	$0.733311\pi$
$98$	0	0
$99$	0	0

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	1980.4.a.q.1.2	yes	5
3.2	odd	2		1980.4.a.o.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1980.4.a.o.1.2	✓	5	3.2	odd	2
1980.4.a.q.1.2	yes	5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

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:	\( N \)	\(=\)	\( 1980 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1980.a (trivial)

\(f(q)\)	\(=\)	\(q+5.00000 q^{5} -11.1055 q^{7} +11.0000 q^{11} +22.1932 q^{13} -32.7773 q^{17} +19.0060 q^{19} +49.5350 q^{23} +25.0000 q^{25} -189.088 q^{29} -313.571 q^{31} -55.5273 q^{35} +176.028 q^{37} -239.694 q^{41} +340.648 q^{43} +508.846 q^{47} -219.669 q^{49} -45.2446 q^{53} +55.0000 q^{55} -565.688 q^{59} +711.062 q^{61} +110.966 q^{65} -399.799 q^{67} +550.754 q^{71} +32.8102 q^{73} -122.160 q^{77} -277.805 q^{79} -1069.85 q^{83} -163.886 q^{85} +240.529 q^{89} -246.465 q^{91} +95.0298 q^{95} -1278.39 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 25 q^{5} - 16 q^{7} + 55 q^{11} - 4 q^{13} - 102 q^{17} - 10 q^{19} - 178 q^{23} + 125 q^{25} - 318 q^{29} + 72 q^{31} - 80 q^{35} - 294 q^{37} - 270 q^{41} - 110 q^{43} - 666 q^{47} + 349 q^{49}+ \cdots - 1866 q^{97}+O(q^{100}) \) 5 * q + 25 * q^5 - 16 * q^7 + 55 * q^11 - 4 * q^13 - 102 * q^17 - 10 * q^19 - 178 * q^23 + 125 * q^25 - 318 * q^29 + 72 * q^31 - 80 * q^35 - 294 * q^37 - 270 * q^41 - 110 * q^43 - 666 * q^47 + 349 * q^49 - 510 * q^53 + 275 * q^55 + 200 * q^59 - 584 * q^61 - 20 * q^65 - 1588 * q^67 - 146 * q^71 - 178 * q^73 - 176 * q^77 - 844 * q^79 + 996 * q^83 - 510 * q^85 + 208 * q^89 - 1504 * q^91 - 50 * q^95 - 1866 * q^97

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