LMFDB - Embedded newform 1160.2.a.g.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1160,2,Mod(1,1160)] 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("1160.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1160 = 2^{3} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1160.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,-3,0,-3] 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:	$9.26264663447$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	1160.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.47283 q^{3} -1.00000 q^{5} +1.11491 q^{7} -0.830760 q^{9} -2.22982 q^{11} +1.47283 q^{13} +1.47283 q^{15} -4.06058 q^{17} +5.51396 q^{19} -1.64207 q^{21} +1.24302 q^{23} +1.00000 q^{25} +5.64207 q^{27} -1.00000 q^{29} +1.83076 q^{31} +3.28415 q^{33} -1.11491 q^{35} +1.05433 q^{37} -2.16924 q^{39} -4.22982 q^{41} +7.83076 q^{43} +0.830760 q^{45} +2.71585 q^{47} -5.75698 q^{49} +5.98055 q^{51} +9.34472 q^{53} +2.22982 q^{55} -8.12115 q^{57} +0.904539 q^{59} +13.2166 q^{61} -0.926221 q^{63} -1.47283 q^{65} +1.43171 q^{67} -1.83076 q^{69} +6.56829 q^{71} +11.7221 q^{73} -1.47283 q^{75} -2.48604 q^{77} +8.98680 q^{79} -5.81756 q^{81} +6.94567 q^{83} +4.06058 q^{85} +1.47283 q^{87} -3.05433 q^{89} +1.64207 q^{91} -2.69641 q^{93} -5.51396 q^{95} -12.6483 q^{97} +1.85244 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - q^{13} - q^{15} + 5 q^{17} + 2 q^{19} - 4 q^{21} + 11 q^{23} + 3 q^{25} + 16 q^{27} - 3 q^{29} + q^{31} + 8 q^{33} + 3 q^{35} + 14 q^{37} - 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) $ 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 2 * q^9 + 6 * q^11 - q^13 - q^15 + 5 * q^17 + 2 * q^19 - 4 * q^21 + 11 * q^23 + 3 * q^25 + 16 * q^27 - 3 * q^29 + q^31 + 8 * q^33 + 3 * q^35 + 14 * q^37 - 11 * q^39 + 19 * q^43 - 2 * q^45 + 10 * q^47 - 10 * q^49 + 26 * q^51 + 9 * q^53 - 6 * q^55 + 10 * q^57 + q^59 + 7 * q^61 + q^65 + 8 * q^67 - q^69 + 16 * q^71 + 9 * q^73 + q^75 - 22 * q^77 + 7 * q^79 + 7 * q^81 + 10 * q^83 - 5 * q^85 - q^87 - 20 * q^89 + 4 * q^91 - 18 * q^93 - 2 * q^95 - 9 * 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
$3$	−1.47283	−0.850341	−0.425171	−	0.905113i	$-0.639786\pi$
$3$	−1.47283	−0.850341	−0.425171	+	0.905113i	$0.639786\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.11491	0.421395	0.210698	−	0.977551i	$-0.432426\pi$
$7$	1.11491	0.421395	0.210698	+	0.977551i	$0.432426\pi$
$8$	0	0
$9$	−0.830760	−0.276920
$10$	0	0
$11$	−2.22982	−0.672315	−0.336157	−	0.941806i	$-0.609127\pi$
$11$	−2.22982	−0.672315	−0.336157	+	0.941806i	$0.609127\pi$
$12$	0	0
$13$	1.47283	0.408491	0.204245	−	0.978920i	$-0.434526\pi$
$13$	1.47283	0.408491	0.204245	+	0.978920i	$0.434526\pi$
$14$	0	0
$15$	1.47283	0.380284
$16$	0	0
$17$	−4.06058	−0.984834	−0.492417	−	0.870359i	$-0.663887\pi$
$17$	−4.06058	−0.984834	−0.492417	+	0.870359i	$0.663887\pi$
$18$	0	0
$19$	5.51396	1.26499	0.632495	−	0.774565i	$-0.282031\pi$
$19$	5.51396	1.26499	0.632495	+	0.774565i	$0.282031\pi$
$20$	0	0
$21$	−1.64207	−0.358330
$22$	0	0
$23$	1.24302	0.259187	0.129594	−	0.991567i	$-0.458633\pi$
$23$	1.24302	0.259187	0.129594	+	0.991567i	$0.458633\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.64207	1.08582
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	1.83076	0.328814	0.164407	−	0.986393i	$-0.447429\pi$
$31$	1.83076	0.328814	0.164407	+	0.986393i	$0.447429\pi$
$32$	0	0
$33$	3.28415	0.571697
$34$	0	0
$35$	−1.11491	−0.188454
$36$	0	0
$37$	1.05433	0.173331	0.0866656	−	0.996237i	$-0.472379\pi$
$37$	1.05433	0.173331	0.0866656	+	0.996237i	$0.472379\pi$
$38$	0	0
$39$	−2.16924	−0.347356
$40$	0	0
$41$	−4.22982	−0.660586	−0.330293	−	0.943878i	$-0.607148\pi$
$41$	−4.22982	−0.660586	−0.330293	+	0.943878i	$0.607148\pi$
$42$	0	0
$43$	7.83076	1.19418	0.597090	−	0.802174i	$-0.296324\pi$
$43$	7.83076	1.19418	0.597090	+	0.802174i	$0.296324\pi$
$44$	0	0
$45$	0.830760	0.123842
$46$	0	0
$47$	2.71585	0.396148	0.198074	−	0.980187i	$-0.436531\pi$
$47$	2.71585	0.396148	0.198074	+	0.980187i	$0.436531\pi$
$48$	0	0
$49$	−5.75698	−0.822426
$50$	0	0
$51$	5.98055	0.837445
$52$	0	0
$53$	9.34472	1.28360	0.641798	−	0.766874i	$-0.278189\pi$
$53$	9.34472	1.28360	0.641798	+	0.766874i	$0.278189\pi$
$54$	0	0
$55$	2.22982	0.300668
$56$	0	0
$57$	−8.12115	−1.07567
$58$	0	0
$59$	0.904539	0.117761	0.0588805	−	0.998265i	$-0.481247\pi$
$59$	0.904539	0.117761	0.0588805	+	0.998265i	$0.481247\pi$
$60$	0	0
$61$	13.2166	1.69221	0.846107	−	0.533013i	$-0.178940\pi$
$61$	13.2166	1.69221	0.846107	+	0.533013i	$0.178940\pi$
$62$	0	0
$63$	−0.926221	−0.116693
$64$	0	0
$65$	−1.47283	−0.182683
$66$	0	0
$67$	1.43171	0.174911	0.0874553	−	0.996168i	$-0.472126\pi$
$67$	1.43171	0.174911	0.0874553	+	0.996168i	$0.472126\pi$
$68$	0	0
$69$	−1.83076	−0.220398
$70$	0	0
$71$	6.56829	0.779513	0.389757	−	0.920918i	$-0.372559\pi$
$71$	6.56829	0.779513	0.389757	+	0.920918i	$0.372559\pi$
$72$	0	0
$73$	11.7221	1.37197	0.685984	−	0.727617i	$-0.259372\pi$
$73$	11.7221	1.37197	0.685984	+	0.727617i	$0.259372\pi$
$74$	0	0
$75$	−1.47283	−0.170068
$76$	0	0
$77$	−2.48604	−0.283310
$78$	0	0
$79$	8.98680	1.01109	0.505547	−	0.862799i	$-0.331291\pi$
$79$	8.98680	1.01109	0.505547	+	0.862799i	$0.331291\pi$
$80$	0	0
$81$	−5.81756	−0.646395
$82$	0	0
$83$	6.94567	0.762386	0.381193	−	0.924495i	$-0.375513\pi$
$83$	6.94567	0.762386	0.381193	+	0.924495i	$0.375513\pi$
$84$	0	0
$85$	4.06058	0.440431
$86$	0	0
$87$	1.47283	0.157904
$88$	0	0
$89$	−3.05433	−0.323759	−0.161879	−	0.986811i	$-0.551756\pi$
$89$	−3.05433	−0.323759	−0.161879	+	0.986811i	$0.551756\pi$
$90$	0	0
$91$	1.64207	0.172136
$92$	0	0
$93$	−2.69641	−0.279604
$94$	0	0
$95$	−5.51396	−0.565721
$96$	0	0
$97$	−12.6483	−1.28424	−0.642121	−	0.766603i	$-0.721945\pi$
$97$	−12.6483	−1.28424	−0.642121	+	0.766603i	$0.721945\pi$
$98$	0	0
$99$	1.85244	0.186177

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	1160.2.a.g.1.1	✓	3
4.3	odd	2		2320.2.a.p.1.3		3
5.4	even	2		5800.2.a.q.1.3		3
8.3	odd	2		9280.2.a.bq.1.1		3
8.5	even	2		9280.2.a.bo.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1160.2.a.g.1.1	✓	3	1.1	even	1	trivial
2320.2.a.p.1.3		3	4.3	odd	2
5800.2.a.q.1.3		3	5.4	even	2
9280.2.a.bo.1.3		3	8.5	even	2
9280.2.a.bq.1.1		3	8.3	odd	2

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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:	\( N \)	\(=\)	\( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.47283 q^{3} -1.00000 q^{5} +1.11491 q^{7} -0.830760 q^{9} -2.22982 q^{11} +1.47283 q^{13} +1.47283 q^{15} -4.06058 q^{17} +5.51396 q^{19} -1.64207 q^{21} +1.24302 q^{23} +1.00000 q^{25} +5.64207 q^{27} -1.00000 q^{29} +1.83076 q^{31} +3.28415 q^{33} -1.11491 q^{35} +1.05433 q^{37} -2.16924 q^{39} -4.22982 q^{41} +7.83076 q^{43} +0.830760 q^{45} +2.71585 q^{47} -5.75698 q^{49} +5.98055 q^{51} +9.34472 q^{53} +2.22982 q^{55} -8.12115 q^{57} +0.904539 q^{59} +13.2166 q^{61} -0.926221 q^{63} -1.47283 q^{65} +1.43171 q^{67} -1.83076 q^{69} +6.56829 q^{71} +11.7221 q^{73} -1.47283 q^{75} -2.48604 q^{77} +8.98680 q^{79} -5.81756 q^{81} +6.94567 q^{83} +4.06058 q^{85} +1.47283 q^{87} -3.05433 q^{89} +1.64207 q^{91} -2.69641 q^{93} -5.51396 q^{95} -12.6483 q^{97} +1.85244 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - q^{13} - q^{15} + 5 q^{17} + 2 q^{19} - 4 q^{21} + 11 q^{23} + 3 q^{25} + 16 q^{27} - 3 q^{29} + q^{31} + 8 q^{33} + 3 q^{35} + 14 q^{37} - 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) \) 3 * q + q^3 - 3 * q^5 - 3 * q^7 + 2 * q^9 + 6 * q^11 - q^13 - q^15 + 5 * q^17 + 2 * q^19 - 4 * q^21 + 11 * q^23 + 3 * q^25 + 16 * q^27 - 3 * q^29 + q^31 + 8 * q^33 + 3 * q^35 + 14 * q^37 - 11 * q^39 + 19 * q^43 - 2 * q^45 + 10 * q^47 - 10 * q^49 + 26 * q^51 + 9 * q^53 - 6 * q^55 + 10 * q^57 + q^59 + 7 * q^61 + q^65 + 8 * q^67 - q^69 + 16 * q^71 + 9 * q^73 + q^75 - 22 * q^77 + 7 * q^79 + 7 * q^81 + 10 * q^83 - 5 * q^85 - q^87 - 20 * q^89 + 4 * q^91 - 18 * q^93 - 2 * q^95 - 9 * q^97

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