LMFDB - Embedded newform 9200.2.a.bz.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9200,2,Mod(1,9200)] 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("9200.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9200, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9200 = 2^{4} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9200.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,0,0,-4,0,6,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$73.4623698596$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 4600)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9200.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+3.23607 q^{3} -4.23607 q^{7} +7.47214 q^{9} -1.00000 q^{11} +2.23607 q^{13} -6.47214 q^{17} -1.00000 q^{19} -13.7082 q^{21} -1.00000 q^{23} +14.4721 q^{27} -1.76393 q^{29} -0.472136 q^{31} -3.23607 q^{33} -11.2361 q^{37} +7.23607 q^{39} -5.94427 q^{41} +2.52786 q^{43} +11.7082 q^{47} +10.9443 q^{49} -20.9443 q^{51} -1.23607 q^{53} -3.23607 q^{57} -3.23607 q^{59} -7.23607 q^{61} -31.6525 q^{63} +12.9443 q^{67} -3.23607 q^{69} -10.0000 q^{71} +9.47214 q^{73} +4.23607 q^{77} -15.1803 q^{79} +24.4164 q^{81} +9.00000 q^{83} -5.70820 q^{87} +2.00000 q^{89} -9.47214 q^{91} -1.52786 q^{93} -6.18034 q^{97} -7.47214 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{7} + 6 q^{9} - 2 q^{11} - 4 q^{17} - 2 q^{19} - 14 q^{21} - 2 q^{23} + 20 q^{27} - 8 q^{29} + 8 q^{31} - 2 q^{33} - 18 q^{37} + 10 q^{39} + 6 q^{41} + 14 q^{43} + 10 q^{47} + 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^7 + 6 * q^9 - 2 * q^11 - 4 * q^17 - 2 * q^19 - 14 * q^21 - 2 * q^23 + 20 * q^27 - 8 * q^29 + 8 * q^31 - 2 * q^33 - 18 * q^37 + 10 * q^39 + 6 * q^41 + 14 * q^43 + 10 * q^47 + 4 * q^49 - 24 * q^51 + 2 * q^53 - 2 * q^57 - 2 * q^59 - 10 * q^61 - 32 * q^63 + 8 * q^67 - 2 * q^69 - 20 * q^71 + 10 * q^73 + 4 * q^77 - 8 * q^79 + 22 * q^81 + 18 * q^83 + 2 * q^87 + 4 * q^89 - 10 * q^91 - 12 * q^93 + 10 * q^97 - 6 * 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
$3$	3.23607	1.86834	0.934172	−	0.356822i	$-0.116140\pi$
$3$	3.23607	1.86834	0.934172	+	0.356822i	$0.116140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−4.23607	−1.60108	−0.800542	−	0.599277i	$-0.795455\pi$
$7$	−4.23607	−1.60108	−0.800542	+	0.599277i	$0.795455\pi$
$8$	0	0
$9$	7.47214	2.49071
$10$	0	0
$11$	−1.00000	−0.301511	−0.150756	−	0.988571i	$-0.548171\pi$
$11$	−1.00000	−0.301511	−0.150756	+	0.988571i	$0.548171\pi$
$12$	0	0
$13$	2.23607	0.620174	0.310087	−	0.950708i	$-0.399642\pi$
$13$	2.23607	0.620174	0.310087	+	0.950708i	$0.399642\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	−13.7082	−2.99138
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	14.4721	2.78516
$28$	0	0
$29$	−1.76393	−0.327554	−0.163777	−	0.986497i	$-0.552368\pi$
$29$	−1.76393	−0.327554	−0.163777	+	0.986497i	$0.552368\pi$
$30$	0	0
$31$	−0.472136	−0.0847981	−0.0423991	−	0.999101i	$-0.513500\pi$
$31$	−0.472136	−0.0847981	−0.0423991	+	0.999101i	$0.513500\pi$
$32$	0	0
$33$	−3.23607	−0.563327
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−11.2361	−1.84720	−0.923599	−	0.383360i	$-0.874767\pi$
$37$	−11.2361	−1.84720	−0.923599	+	0.383360i	$0.874767\pi$
$38$	0	0
$39$	7.23607	1.15870
$40$	0	0
$41$	−5.94427	−0.928339	−0.464170	−	0.885746i	$-0.653647\pi$
$41$	−5.94427	−0.928339	−0.464170	+	0.885746i	$0.653647\pi$
$42$	0	0
$43$	2.52786	0.385496	0.192748	−	0.981248i	$-0.438260\pi$
$43$	2.52786	0.385496	0.192748	+	0.981248i	$0.438260\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	11.7082	1.70782	0.853909	−	0.520423i	$-0.174226\pi$
$47$	11.7082	1.70782	0.853909	+	0.520423i	$0.174226\pi$
$48$	0	0
$49$	10.9443	1.56347
$50$	0	0
$51$	−20.9443	−2.93278
$52$	0	0
$53$	−1.23607	−0.169787	−0.0848935	−	0.996390i	$-0.527055\pi$
$53$	−1.23607	−0.169787	−0.0848935	+	0.996390i	$0.527055\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−3.23607	−0.428628
$58$	0	0
$59$	−3.23607	−0.421300	−0.210650	−	0.977562i	$-0.567558\pi$
$59$	−3.23607	−0.421300	−0.210650	+	0.977562i	$0.567558\pi$
$60$	0	0
$61$	−7.23607	−0.926484	−0.463242	−	0.886232i	$-0.653314\pi$
$61$	−7.23607	−0.926484	−0.463242	+	0.886232i	$0.653314\pi$
$62$	0	0
$63$	−31.6525	−3.98784
$64$	0	0
$65$	0	0
$66$	0	0
$67$	12.9443	1.58139	0.790697	−	0.612207i	$-0.209718\pi$
$67$	12.9443	1.58139	0.790697	+	0.612207i	$0.209718\pi$
$68$	0	0
$69$	−3.23607	−0.389577
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	0	0
$73$	9.47214	1.10863	0.554315	−	0.832307i	$-0.312980\pi$
$73$	9.47214	1.10863	0.554315	+	0.832307i	$0.312980\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	4.23607	0.482745
$78$	0	0
$79$	−15.1803	−1.70792	−0.853961	−	0.520337i	$-0.825806\pi$
$79$	−15.1803	−1.70792	−0.853961	+	0.520337i	$0.825806\pi$
$80$	0	0
$81$	24.4164	2.71293
$82$	0	0
$83$	9.00000	0.987878	0.493939	−	0.869496i	$-0.335557\pi$
$83$	9.00000	0.987878	0.493939	+	0.869496i	$0.335557\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5.70820	−0.611984
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	−9.47214	−0.992950
$92$	0	0
$93$	−1.52786	−0.158432
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−6.18034	−0.627518	−0.313759	−	0.949503i	$-0.601588\pi$
$97$	−6.18034	−0.627518	−0.313759	+	0.949503i	$0.601588\pi$
$98$	0	0
$99$	−7.47214	−0.750978

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	9200.2.a.bz.1.2		2
4.3	odd	2		4600.2.a.q.1.1	✓	2
5.4	even	2		9200.2.a.bn.1.1		2
20.3	even	4		4600.2.e.l.4049.1		4
20.7	even	4		4600.2.e.l.4049.4		4
20.19	odd	2		4600.2.a.u.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.q.1.1	✓	2	4.3	odd	2
4600.2.a.u.1.2	yes	2	20.19	odd	2
4600.2.e.l.4049.1		4	20.3	even	4
4600.2.e.l.4049.4		4	20.7	even	4
9200.2.a.bn.1.1		2	5.4	even	2
9200.2.a.bz.1.2		2	1.1	even	1	trivial

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Classical	Maass
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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:	\( N \)	\(=\)	\( 9200 = 2^{4} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9200.a (trivial)

\(f(q)\)	\(=\)	\(q+3.23607 q^{3} -4.23607 q^{7} +7.47214 q^{9} -1.00000 q^{11} +2.23607 q^{13} -6.47214 q^{17} -1.00000 q^{19} -13.7082 q^{21} -1.00000 q^{23} +14.4721 q^{27} -1.76393 q^{29} -0.472136 q^{31} -3.23607 q^{33} -11.2361 q^{37} +7.23607 q^{39} -5.94427 q^{41} +2.52786 q^{43} +11.7082 q^{47} +10.9443 q^{49} -20.9443 q^{51} -1.23607 q^{53} -3.23607 q^{57} -3.23607 q^{59} -7.23607 q^{61} -31.6525 q^{63} +12.9443 q^{67} -3.23607 q^{69} -10.0000 q^{71} +9.47214 q^{73} +4.23607 q^{77} -15.1803 q^{79} +24.4164 q^{81} +9.00000 q^{83} -5.70820 q^{87} +2.00000 q^{89} -9.47214 q^{91} -1.52786 q^{93} -6.18034 q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{7} + 6 q^{9} - 2 q^{11} - 4 q^{17} - 2 q^{19} - 14 q^{21} - 2 q^{23} + 20 q^{27} - 8 q^{29} + 8 q^{31} - 2 q^{33} - 18 q^{37} + 10 q^{39} + 6 q^{41} + 14 q^{43} + 10 q^{47} + 4 q^{49}+ \cdots - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^7 + 6 * q^9 - 2 * q^11 - 4 * q^17 - 2 * q^19 - 14 * q^21 - 2 * q^23 + 20 * q^27 - 8 * q^29 + 8 * q^31 - 2 * q^33 - 18 * q^37 + 10 * q^39 + 6 * q^41 + 14 * q^43 + 10 * q^47 + 4 * q^49 - 24 * q^51 + 2 * q^53 - 2 * q^57 - 2 * q^59 - 10 * q^61 - 32 * q^63 + 8 * q^67 - 2 * q^69 - 20 * q^71 + 10 * q^73 + 4 * q^77 - 8 * q^79 + 22 * q^81 + 18 * q^83 + 2 * q^87 + 4 * q^89 - 10 * q^91 - 12 * q^93 + 10 * q^97 - 6 * q^99

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