LMFDB - Embedded newform 3344.2.a.q.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3344,2,Mod(1,3344)] 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("3344.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3344, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,0,0,-3,0,6] 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:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.621.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 3 $ x^3 - 6*x - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 418)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.66908$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.66908 q^{3} -4.12398 q^{5} +4.21417 q^{7} +4.12398 q^{9} +1.00000 q^{11} -2.21417 q^{13} +11.0072 q^{15} -3.45490 q^{17} +1.00000 q^{19} -11.2480 q^{21} +5.45490 q^{23} +12.0072 q^{25} -3.00000 q^{27} +5.57889 q^{29} -7.00724 q^{31} -2.66908 q^{33} -17.3792 q^{35} -2.90981 q^{37} +5.90981 q^{39} -11.9170 q^{41} -1.46214 q^{43} -17.0072 q^{45} -7.58612 q^{47} +10.7593 q^{49} +9.22141 q^{51} -13.2214 q^{53} -4.12398 q^{55} -2.66908 q^{57} -4.79306 q^{59} +8.90981 q^{61} +17.3792 q^{63} +9.13122 q^{65} -1.30437 q^{67} -14.5596 q^{69} +6.80030 q^{71} -1.45490 q^{73} -32.0483 q^{75} +4.21417 q^{77} +9.15777 q^{79} -4.36471 q^{81} +13.2142 q^{83} +14.2480 q^{85} -14.8905 q^{87} +8.24797 q^{89} -9.33092 q^{91} +18.7029 q^{93} -4.12398 q^{95} +2.18038 q^{97} +4.12398 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * 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
$3$	−2.66908	−1.54099	−0.770497	−	0.637444i	$-0.779992\pi$
$3$	−2.66908	−1.54099	−0.770497	+	0.637444i	$0.779992\pi$
$4$	0	0
$5$	−4.12398	−1.84430	−0.922151	−	0.386831i	$-0.873570\pi$
$5$	−4.12398	−1.84430	−0.922151	+	0.386831i	$0.873570\pi$
$6$	0	0
$7$	4.21417	1.59281	0.796404	−	0.604765i	$-0.206733\pi$
$7$	4.21417	1.59281	0.796404	+	0.604765i	$0.206733\pi$
$8$	0	0
$9$	4.12398	1.37466
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−2.21417	−0.614102	−0.307051	−	0.951693i	$-0.599342\pi$
$13$	−2.21417	−0.614102	−0.307051	+	0.951693i	$0.599342\pi$
$14$	0	0
$15$	11.0072	2.84206
$16$	0	0
$17$	−3.45490	−0.837937	−0.418969	−	0.908001i	$-0.637608\pi$
$17$	−3.45490	−0.837937	−0.418969	+	0.908001i	$0.637608\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−11.2480	−2.45451
$22$	0	0
$23$	5.45490	1.13743	0.568713	−	0.822536i	$-0.307441\pi$
$23$	5.45490	1.13743	0.568713	+	0.822536i	$0.307441\pi$
$24$	0	0
$25$	12.0072	2.40145
$26$	0	0
$27$	−3.00000	−0.577350
$28$	0	0
$29$	5.57889	1.03597	0.517987	−	0.855389i	$-0.326682\pi$
$29$	5.57889	1.03597	0.517987	+	0.855389i	$0.326682\pi$
$30$	0	0
$31$	−7.00724	−1.25854	−0.629268	−	0.777188i	$-0.716645\pi$
$31$	−7.00724	−1.25854	−0.629268	+	0.777188i	$0.716645\pi$
$32$	0	0
$33$	−2.66908	−0.464627
$34$	0	0
$35$	−17.3792	−2.93762
$36$	0	0
$37$	−2.90981	−0.478370	−0.239185	−	0.970974i	$-0.576880\pi$
$37$	−2.90981	−0.478370	−0.239185	+	0.970974i	$0.576880\pi$
$38$	0	0
$39$	5.90981	0.946327
$40$	0	0
$41$	−11.9170	−1.86113	−0.930565	−	0.366127i	$-0.880684\pi$
$41$	−11.9170	−1.86113	−0.930565	+	0.366127i	$0.880684\pi$
$42$	0	0
$43$	−1.46214	−0.222974	−0.111487	−	0.993766i	$-0.535561\pi$
$43$	−1.46214	−0.222974	−0.111487	+	0.993766i	$0.535561\pi$
$44$	0	0
$45$	−17.0072	−2.53529
$46$	0	0
$47$	−7.58612	−1.10655	−0.553275	−	0.832999i	$-0.686622\pi$
$47$	−7.58612	−1.10655	−0.553275	+	0.832999i	$0.686622\pi$
$48$	0	0
$49$	10.7593	1.53704
$50$	0	0
$51$	9.22141	1.29126
$52$	0	0
$53$	−13.2214	−1.81610	−0.908050	−	0.418861i	$-0.862429\pi$
$53$	−13.2214	−1.81610	−0.908050	+	0.418861i	$0.862429\pi$
$54$	0	0
$55$	−4.12398	−0.556078
$56$	0	0
$57$	−2.66908	−0.353528
$58$	0	0
$59$	−4.79306	−0.624004	−0.312002	−	0.950082i	$-0.600999\pi$
$59$	−4.79306	−0.624004	−0.312002	+	0.950082i	$0.600999\pi$
$60$	0	0
$61$	8.90981	1.14078	0.570392	−	0.821373i	$-0.306791\pi$
$61$	8.90981	1.14078	0.570392	+	0.821373i	$0.306791\pi$
$62$	0	0
$63$	17.3792	2.18957
$64$	0	0
$65$	9.13122	1.13259
$66$	0	0
$67$	−1.30437	−0.159354	−0.0796769	−	0.996821i	$-0.525389\pi$
$67$	−1.30437	−0.159354	−0.0796769	+	0.996821i	$0.525389\pi$
$68$	0	0
$69$	−14.5596	−1.75277
$70$	0	0
$71$	6.80030	0.807047	0.403524	−	0.914969i	$-0.367785\pi$
$71$	6.80030	0.807047	0.403524	+	0.914969i	$0.367785\pi$
$72$	0	0
$73$	−1.45490	−0.170284	−0.0851418	−	0.996369i	$-0.527134\pi$
$73$	−1.45490	−0.170284	−0.0851418	+	0.996369i	$0.527134\pi$
$74$	0	0
$75$	−32.0483	−3.70061
$76$	0	0
$77$	4.21417	0.480250
$78$	0	0
$79$	9.15777	1.03033	0.515165	−	0.857091i	$-0.327731\pi$
$79$	9.15777	1.03033	0.515165	+	0.857091i	$0.327731\pi$
$80$	0	0
$81$	−4.36471	−0.484968
$82$	0	0
$83$	13.2142	1.45044	0.725222	−	0.688515i	$-0.241737\pi$
$83$	13.2142	1.45044	0.725222	+	0.688515i	$0.241737\pi$
$84$	0	0
$85$	14.2480	1.54541
$86$	0	0
$87$	−14.8905	−1.59643
$88$	0	0
$89$	8.24797	0.874283	0.437141	−	0.899393i	$-0.355991\pi$
$89$	8.24797	0.874283	0.437141	+	0.899393i	$0.355991\pi$
$90$	0	0
$91$	−9.33092	−0.978146
$92$	0	0
$93$	18.7029	1.93940
$94$	0	0
$95$	−4.12398	−0.423112
$96$	0	0
$97$	2.18038	0.221384	0.110692	−	0.993855i	$-0.464693\pi$
$97$	2.18038	0.221384	0.110692	+	0.993855i	$0.464693\pi$
$98$	0	0
$99$	4.12398	0.414476

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	3344.2.a.q.1.1		3
4.3	odd	2		418.2.a.g.1.3	✓	3
12.11	even	2		3762.2.a.bg.1.3		3
44.43	even	2		4598.2.a.bo.1.3		3
76.75	even	2		7942.2.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
418.2.a.g.1.3	✓	3	4.3	odd	2
3344.2.a.q.1.1		3	1.1	even	1	trivial
3762.2.a.bg.1.3		3	12.11	even	2
4598.2.a.bo.1.3		3	44.43	even	2
7942.2.a.bi.1.1		3	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:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-2.66908 q^{3} -4.12398 q^{5} +4.21417 q^{7} +4.12398 q^{9} +1.00000 q^{11} -2.21417 q^{13} +11.0072 q^{15} -3.45490 q^{17} +1.00000 q^{19} -11.2480 q^{21} +5.45490 q^{23} +12.0072 q^{25} -3.00000 q^{27} +5.57889 q^{29} -7.00724 q^{31} -2.66908 q^{33} -17.3792 q^{35} -2.90981 q^{37} +5.90981 q^{39} -11.9170 q^{41} -1.46214 q^{43} -17.0072 q^{45} -7.58612 q^{47} +10.7593 q^{49} +9.22141 q^{51} -13.2214 q^{53} -4.12398 q^{55} -2.66908 q^{57} -4.79306 q^{59} +8.90981 q^{61} +17.3792 q^{63} +9.13122 q^{65} -1.30437 q^{67} -14.5596 q^{69} +6.80030 q^{71} -1.45490 q^{73} -32.0483 q^{75} +4.21417 q^{77} +9.15777 q^{79} -4.36471 q^{81} +13.2142 q^{83} +14.2480 q^{85} -14.8905 q^{87} +8.24797 q^{89} -9.33092 q^{91} +18.7029 q^{93} -4.12398 q^{95} +2.18038 q^{97} +4.12398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 6 q^{7} + 3 q^{9} + 3 q^{11} + 9 q^{15} - 9 q^{17} + 3 q^{19} - 15 q^{21} + 15 q^{23} + 12 q^{25} - 9 q^{27} + 6 q^{29} + 3 q^{31} - 6 q^{37} + 15 q^{39} - 9 q^{41} + 21 q^{43} - 27 q^{45}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^5 + 6 * q^7 + 3 * q^9 + 3 * q^11 + 9 * q^15 - 9 * q^17 + 3 * q^19 - 15 * q^21 + 15 * q^23 + 12 * q^25 - 9 * q^27 + 6 * q^29 + 3 * q^31 - 6 * q^37 + 15 * q^39 - 9 * q^41 + 21 * q^43 - 27 * q^45 + 12 * q^47 + 27 * q^49 - 3 * q^51 - 9 * q^53 - 3 * q^55 + 3 * q^59 + 24 * q^61 - 6 * q^65 + 3 * q^69 - 21 * q^71 - 3 * q^73 - 36 * q^75 + 6 * q^77 + 6 * q^79 - 9 * q^81 + 33 * q^83 + 24 * q^85 - 6 * q^87 + 6 * q^89 - 36 * q^91 + 36 * q^93 - 3 * q^95 + 12 * q^97 + 3 * q^99

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