LMFDB - Embedded newform 3520.2.a.bj.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3520,2,Mod(1,3520)] 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("3520.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-2,0,-1,0,11,0,-2,0,-4] 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:	$28.1073415115$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 110)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.37228$ of defining polynomial
Character	$\chi$	$=$	3520.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-3.37228 q^{3} -1.00000 q^{5} -3.37228 q^{7} +8.37228 q^{9} -1.00000 q^{11} -2.00000 q^{13} +3.37228 q^{15} +1.37228 q^{17} +0.627719 q^{19} +11.3723 q^{21} -2.74456 q^{23} +1.00000 q^{25} -18.1168 q^{27} -1.37228 q^{29} -3.37228 q^{31} +3.37228 q^{33} +3.37228 q^{35} -9.37228 q^{37} +6.74456 q^{39} -11.4891 q^{41} -4.00000 q^{43} -8.37228 q^{45} -2.74456 q^{47} +4.37228 q^{49} -4.62772 q^{51} +4.11684 q^{53} +1.00000 q^{55} -2.11684 q^{57} -2.74456 q^{59} +5.37228 q^{61} -28.2337 q^{63} +2.00000 q^{65} +8.00000 q^{67} +9.25544 q^{69} -10.1168 q^{71} -15.4891 q^{73} -3.37228 q^{75} +3.37228 q^{77} +1.25544 q^{79} +35.9783 q^{81} -2.74456 q^{83} -1.37228 q^{85} +4.62772 q^{87} -1.37228 q^{89} +6.74456 q^{91} +11.3723 q^{93} -0.627719 q^{95} -12.7446 q^{97} -8.37228 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 2 q^{5} - q^{7} + 11 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} - 19 q^{27} + 3 q^{29} - q^{31} + q^{33} + q^{35} - 13 q^{37} + 2 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) $ 2 * q - q^3 - 2 * q^5 - q^7 + 11 * q^9 - 2 * q^11 - 4 * q^13 + q^15 - 3 * q^17 + 7 * q^19 + 17 * q^21 + 6 * q^23 + 2 * q^25 - 19 * q^27 + 3 * q^29 - q^31 + q^33 + q^35 - 13 * q^37 + 2 * q^39 - 8 * q^43 - 11 * q^45 + 6 * q^47 + 3 * q^49 - 15 * q^51 - 9 * q^53 + 2 * q^55 + 13 * q^57 + 6 * q^59 + 5 * q^61 - 22 * q^63 + 4 * q^65 + 16 * q^67 + 30 * q^69 - 3 * q^71 - 8 * q^73 - q^75 + q^77 + 14 * q^79 + 26 * q^81 + 6 * q^83 + 3 * q^85 + 15 * q^87 + 3 * q^89 + 2 * q^91 + 17 * q^93 - 7 * q^95 - 14 * q^97 - 11 * 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
$3$	−3.37228	−1.94699	−0.973494	−	0.228714i	$-0.926548\pi$
$3$	−3.37228	−1.94699	−0.973494	+	0.228714i	$0.926548\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.37228	−1.27460	−0.637301	−	0.770615i	$-0.719949\pi$
$7$	−3.37228	−1.27460	−0.637301	+	0.770615i	$0.719949\pi$
$8$	0	0
$9$	8.37228	2.79076
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	3.37228	0.870719
$16$	0	0
$17$	1.37228	0.332827	0.166414	−	0.986056i	$-0.446781\pi$
$17$	1.37228	0.332827	0.166414	+	0.986056i	$0.446781\pi$
$18$	0	0
$19$	0.627719	0.144009	0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719	0.144009	0.0720043	+	0.997404i	$0.477060\pi$
$20$	0	0
$21$	11.3723	2.48164
$22$	0	0
$23$	−2.74456	−0.572281	−0.286140	−	0.958188i	$-0.592372\pi$
$23$	−2.74456	−0.572281	−0.286140	+	0.958188i	$0.592372\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−18.1168	−3.48659
$28$	0	0
$29$	−1.37228	−0.254826	−0.127413	−	0.991850i	$-0.540667\pi$
$29$	−1.37228	−0.254826	−0.127413	+	0.991850i	$0.540667\pi$
$30$	0	0
$31$	−3.37228	−0.605680	−0.302840	−	0.953041i	$-0.597935\pi$
$31$	−3.37228	−0.605680	−0.302840	+	0.953041i	$0.597935\pi$
$32$	0	0
$33$	3.37228	0.587039
$34$	0	0
$35$	3.37228	0.570020
$36$	0	0
$37$	−9.37228	−1.54079	−0.770397	−	0.637565i	$-0.779942\pi$
$37$	−9.37228	−1.54079	−0.770397	+	0.637565i	$0.779942\pi$
$38$	0	0
$39$	6.74456	1.07999
$40$	0	0
$41$	−11.4891	−1.79430	−0.897150	−	0.441726i	$-0.854366\pi$
$41$	−11.4891	−1.79430	−0.897150	+	0.441726i	$0.854366\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	−8.37228	−1.24807
$46$	0	0
$47$	−2.74456	−0.400336	−0.200168	−	0.979762i	$-0.564149\pi$
$47$	−2.74456	−0.400336	−0.200168	+	0.979762i	$0.564149\pi$
$48$	0	0
$49$	4.37228	0.624612
$50$	0	0
$51$	−4.62772	−0.648010
$52$	0	0
$53$	4.11684	0.565492	0.282746	−	0.959195i	$-0.408755\pi$
$53$	4.11684	0.565492	0.282746	+	0.959195i	$0.408755\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	−2.11684	−0.280383
$58$	0	0
$59$	−2.74456	−0.357312	−0.178656	−	0.983912i	$-0.557175\pi$
$59$	−2.74456	−0.357312	−0.178656	+	0.983912i	$0.557175\pi$
$60$	0	0
$61$	5.37228	0.687850	0.343925	−	0.938997i	$-0.388243\pi$
$61$	5.37228	0.687850	0.343925	+	0.938997i	$0.388243\pi$
$62$	0	0
$63$	−28.2337	−3.55711
$64$	0	0
$65$	2.00000	0.248069
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	9.25544	1.11422
$70$	0	0
$71$	−10.1168	−1.20065	−0.600324	−	0.799757i	$-0.704962\pi$
$71$	−10.1168	−1.20065	−0.600324	+	0.799757i	$0.704962\pi$
$72$	0	0
$73$	−15.4891	−1.81286	−0.906432	−	0.422351i	$-0.861205\pi$
$73$	−15.4891	−1.81286	−0.906432	+	0.422351i	$0.861205\pi$
$74$	0	0
$75$	−3.37228	−0.389398
$76$	0	0
$77$	3.37228	0.384307
$78$	0	0
$79$	1.25544	0.141248	0.0706239	−	0.997503i	$-0.477501\pi$
$79$	1.25544	0.141248	0.0706239	+	0.997503i	$0.477501\pi$
$80$	0	0
$81$	35.9783	3.99758
$82$	0	0
$83$	−2.74456	−0.301255	−0.150627	−	0.988591i	$-0.548129\pi$
$83$	−2.74456	−0.301255	−0.150627	+	0.988591i	$0.548129\pi$
$84$	0	0
$85$	−1.37228	−0.148845
$86$	0	0
$87$	4.62772	0.496144
$88$	0	0
$89$	−1.37228	−0.145462	−0.0727308	−	0.997352i	$-0.523171\pi$
$89$	−1.37228	−0.145462	−0.0727308	+	0.997352i	$0.523171\pi$
$90$	0	0
$91$	6.74456	0.707022
$92$	0	0
$93$	11.3723	1.17925
$94$	0	0
$95$	−0.627719	−0.0644026
$96$	0	0
$97$	−12.7446	−1.29401	−0.647007	−	0.762484i	$-0.723980\pi$
$97$	−12.7446	−1.29401	−0.647007	+	0.762484i	$0.723980\pi$
$98$	0	0
$99$	−8.37228	−0.841446

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	3520.2.a.bj.1.1		2
4.3	odd	2		3520.2.a.bq.1.2		2
8.3	odd	2		110.2.a.d.1.1	✓	2
8.5	even	2		880.2.a.n.1.2		2
24.5	odd	2		7920.2.a.bq.1.1		2
24.11	even	2		990.2.a.m.1.2		2
40.3	even	4		550.2.b.f.199.3		4
40.13	odd	4		4400.2.b.p.4049.4		4
40.19	odd	2		550.2.a.n.1.2		2
40.27	even	4		550.2.b.f.199.2		4
40.29	even	2		4400.2.a.bl.1.1		2
40.37	odd	4		4400.2.b.p.4049.1		4
56.27	even	2		5390.2.a.bp.1.2		2
88.21	odd	2		9680.2.a.bt.1.2		2
88.43	even	2		1210.2.a.r.1.1		2
120.59	even	2		4950.2.a.bw.1.1		2
120.83	odd	4		4950.2.c.bc.199.1		4
120.107	odd	4		4950.2.c.bc.199.4		4
440.219	even	2		6050.2.a.cb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
110.2.a.d.1.1	✓	2	8.3	odd	2
550.2.a.n.1.2		2	40.19	odd	2
550.2.b.f.199.2		4	40.27	even	4
550.2.b.f.199.3		4	40.3	even	4
880.2.a.n.1.2		2	8.5	even	2
990.2.a.m.1.2		2	24.11	even	2
1210.2.a.r.1.1		2	88.43	even	2
3520.2.a.bj.1.1		2	1.1	even	1	trivial
3520.2.a.bq.1.2		2	4.3	odd	2
4400.2.a.bl.1.1		2	40.29	even	2
4400.2.b.p.4049.1		4	40.37	odd	4
4400.2.b.p.4049.4		4	40.13	odd	4
4950.2.a.bw.1.1		2	120.59	even	2
4950.2.c.bc.199.1		4	120.83	odd	4
4950.2.c.bc.199.4		4	120.107	odd	4
5390.2.a.bp.1.2		2	56.27	even	2
6050.2.a.cb.1.2		2	440.219	even	2
7920.2.a.bq.1.1		2	24.5	odd	2
9680.2.a.bt.1.2		2	88.21	odd	2

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Level:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.a (trivial)

\(f(q)\)	\(=\)	\(q-3.37228 q^{3} -1.00000 q^{5} -3.37228 q^{7} +8.37228 q^{9} -1.00000 q^{11} -2.00000 q^{13} +3.37228 q^{15} +1.37228 q^{17} +0.627719 q^{19} +11.3723 q^{21} -2.74456 q^{23} +1.00000 q^{25} -18.1168 q^{27} -1.37228 q^{29} -3.37228 q^{31} +3.37228 q^{33} +3.37228 q^{35} -9.37228 q^{37} +6.74456 q^{39} -11.4891 q^{41} -4.00000 q^{43} -8.37228 q^{45} -2.74456 q^{47} +4.37228 q^{49} -4.62772 q^{51} +4.11684 q^{53} +1.00000 q^{55} -2.11684 q^{57} -2.74456 q^{59} +5.37228 q^{61} -28.2337 q^{63} +2.00000 q^{65} +8.00000 q^{67} +9.25544 q^{69} -10.1168 q^{71} -15.4891 q^{73} -3.37228 q^{75} +3.37228 q^{77} +1.25544 q^{79} +35.9783 q^{81} -2.74456 q^{83} -1.37228 q^{85} +4.62772 q^{87} -1.37228 q^{89} +6.74456 q^{91} +11.3723 q^{93} -0.627719 q^{95} -12.7446 q^{97} -8.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 2 q^{5} - q^{7} + 11 q^{9} - 2 q^{11} - 4 q^{13} + q^{15} - 3 q^{17} + 7 q^{19} + 17 q^{21} + 6 q^{23} + 2 q^{25} - 19 q^{27} + 3 q^{29} - q^{31} + q^{33} + q^{35} - 13 q^{37} + 2 q^{39}+ \cdots - 11 q^{99}+O(q^{100}) \) 2 * q - q^3 - 2 * q^5 - q^7 + 11 * q^9 - 2 * q^11 - 4 * q^13 + q^15 - 3 * q^17 + 7 * q^19 + 17 * q^21 + 6 * q^23 + 2 * q^25 - 19 * q^27 + 3 * q^29 - q^31 + q^33 + q^35 - 13 * q^37 + 2 * q^39 - 8 * q^43 - 11 * q^45 + 6 * q^47 + 3 * q^49 - 15 * q^51 - 9 * q^53 + 2 * q^55 + 13 * q^57 + 6 * q^59 + 5 * q^61 - 22 * q^63 + 4 * q^65 + 16 * q^67 + 30 * q^69 - 3 * q^71 - 8 * q^73 - q^75 + q^77 + 14 * q^79 + 26 * q^81 + 6 * q^83 + 3 * q^85 + 15 * q^87 + 3 * q^89 + 2 * q^91 + 17 * q^93 - 7 * q^95 - 14 * q^97 - 11 * q^99

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