LMFDB - Embedded newform 2070.2.a.x.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2070,2,Mod(1,2070)] 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("2070.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2070, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,2,0,1,2,0,2,-3,0,7] 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:	$16.5290332184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 230)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.79129$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} +0.791288 q^{11} +5.79129 q^{13} -1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +5.79129 q^{19} +1.00000 q^{20} +0.791288 q^{22} -1.00000 q^{23} +1.00000 q^{25} +5.79129 q^{26} -1.79129 q^{28} -7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -0.791288 q^{34} -1.79129 q^{35} -4.00000 q^{37} +5.79129 q^{38} +1.00000 q^{40} +6.79129 q^{41} +11.1652 q^{43} +0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} -3.79129 q^{49} +1.00000 q^{50} +5.79129 q^{52} -6.00000 q^{53} +0.791288 q^{55} -1.79129 q^{56} -7.58258 q^{58} +13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +1.00000 q^{64} +5.79129 q^{65} +11.1652 q^{67} -0.791288 q^{68} -1.79129 q^{70} -8.37386 q^{71} +12.7477 q^{73} -4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} +8.00000 q^{79} +1.00000 q^{80} +6.79129 q^{82} +6.00000 q^{83} -0.791288 q^{85} +11.1652 q^{86} +0.791288 q^{88} -15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} +4.41742 q^{94} +5.79129 q^{95} -7.95644 q^{97} -3.79129 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 2 q^{20} - 3 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + q^{28} - 6 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 2 * q^20 - 3 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + q^28 - 6 * q^29 + 7 * q^31 + 2 * q^32 + 3 * q^34 + q^35 - 8 * q^37 + 7 * q^38 + 2 * q^40 + 9 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 7 * q^52 - 12 * q^53 - 3 * q^55 + q^56 - 6 * q^58 + 18 * q^59 + 7 * q^61 + 7 * q^62 + 2 * q^64 + 7 * q^65 + 4 * q^67 + 3 * q^68 + q^70 - 3 * q^71 - 2 * q^73 - 8 * q^74 + 7 * q^76 - 12 * q^77 + 16 * q^79 + 2 * q^80 + 9 * q^82 + 12 * q^83 + 3 * q^85 + 4 * q^86 - 3 * q^88 - 12 * q^89 - 7 * q^91 - 2 * q^92 + 18 * q^94 + 7 * q^95 + 7 * q^97 - 3 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.79129	−0.677043	−0.338522	−	0.940959i	$-0.609927\pi$
$7$	−1.79129	−0.677043	−0.338522	+	0.940959i	$0.609927\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	0.791288	0.238582	0.119291	−	0.992859i	$-0.461938\pi$
$11$	0.791288	0.238582	0.119291	+	0.992859i	$0.461938\pi$
$12$	0	0
$13$	5.79129	1.60621	0.803107	−	0.595835i	$-0.203179\pi$
$13$	5.79129	1.60621	0.803107	+	0.595835i	$0.203179\pi$
$14$	−1.79129	−0.478742
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.791288	−0.191915	−0.0959577	−	0.995385i	$-0.530591\pi$
$17$	−0.791288	−0.191915	−0.0959577	+	0.995385i	$0.530591\pi$
$18$	0	0
$19$	5.79129	1.32861	0.664306	−	0.747460i	$-0.268727\pi$
$19$	5.79129	1.32861	0.664306	+	0.747460i	$0.268727\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	0.791288	0.168703
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	5.79129	1.13576
$27$	0	0
$28$	−1.79129	−0.338522
$29$	−7.58258	−1.40805	−0.704024	−	0.710176i	$-0.748615\pi$
$29$	−7.58258	−1.40805	−0.704024	+	0.710176i	$0.748615\pi$
$30$	0	0
$31$	−3.37386	−0.605964	−0.302982	−	0.952996i	$-0.597982\pi$
$31$	−3.37386	−0.605964	−0.302982	+	0.952996i	$0.597982\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.791288	−0.135705
$35$	−1.79129	−0.302783
$36$	0	0
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	5.79129	0.939471
$39$	0	0
$40$	1.00000	0.158114
$41$	6.79129	1.06062	0.530310	−	0.847804i	$-0.322075\pi$
$41$	6.79129	1.06062	0.530310	+	0.847804i	$0.322075\pi$
$42$	0	0
$43$	11.1652	1.70267	0.851335	−	0.524623i	$-0.175794\pi$
$43$	11.1652	1.70267	0.851335	+	0.524623i	$0.175794\pi$
$44$	0.791288	0.119291
$45$	0	0
$46$	−1.00000	−0.147442
$47$	4.41742	0.644348	0.322174	−	0.946681i	$-0.395586\pi$
$47$	4.41742	0.644348	0.322174	+	0.946681i	$0.395586\pi$
$48$	0	0
$49$	−3.79129	−0.541613
$50$	1.00000	0.141421
$51$	0	0
$52$	5.79129	0.803107
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	0.791288	0.106697
$56$	−1.79129	−0.239371
$57$	0	0
$58$	−7.58258	−0.995641
$59$	13.5826	1.76830	0.884150	−	0.467202i	$-0.154738\pi$
$59$	13.5826	1.76830	0.884150	+	0.467202i	$0.154738\pi$
$60$	0	0
$61$	10.3739	1.32824	0.664119	−	0.747627i	$-0.268807\pi$
$61$	10.3739	1.32824	0.664119	+	0.747627i	$0.268807\pi$
$62$	−3.37386	−0.428481
$63$	0	0
$64$	1.00000	0.125000
$65$	5.79129	0.718321
$66$	0	0
$67$	11.1652	1.36404	0.682020	−	0.731333i	$-0.261102\pi$
$67$	11.1652	1.36404	0.682020	+	0.731333i	$0.261102\pi$
$68$	−0.791288	−0.0959577
$69$	0	0
$70$	−1.79129	−0.214100
$71$	−8.37386	−0.993795	−0.496897	−	0.867809i	$-0.665527\pi$
$71$	−8.37386	−0.993795	−0.496897	+	0.867809i	$0.665527\pi$
$72$	0	0
$73$	12.7477	1.49201	0.746004	−	0.665941i	$-0.231970\pi$
$73$	12.7477	1.49201	0.746004	+	0.665941i	$0.231970\pi$
$74$	−4.00000	−0.464991
$75$	0	0
$76$	5.79129	0.664306
$77$	−1.41742	−0.161530
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	6.79129	0.749972
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−0.791288	−0.0858272
$86$	11.1652	1.20397
$87$	0	0
$88$	0.791288	0.0843516
$89$	−15.1652	−1.60750	−0.803751	−	0.594965i	$-0.797166\pi$
$89$	−15.1652	−1.60750	−0.803751	+	0.594965i	$0.797166\pi$
$90$	0	0
$91$	−10.3739	−1.08748
$92$	−1.00000	−0.104257
$93$	0	0
$94$	4.41742	0.455623
$95$	5.79129	0.594174
$96$	0	0
$97$	−7.95644	−0.807854	−0.403927	−	0.914791i	$-0.632355\pi$
$97$	−7.95644	−0.807854	−0.403927	+	0.914791i	$0.632355\pi$
$98$	−3.79129	−0.382978
$99$	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	2070.2.a.x.1.1		2
3.2	odd	2		230.2.a.a.1.1	✓	2
12.11	even	2		1840.2.a.n.1.2		2
15.2	even	4		1150.2.b.g.599.2		4
15.8	even	4		1150.2.b.g.599.3		4
15.14	odd	2		1150.2.a.o.1.2		2
24.5	odd	2		7360.2.a.bq.1.2		2
24.11	even	2		7360.2.a.bk.1.1		2
60.59	even	2		9200.2.a.bs.1.1		2
69.68	even	2		5290.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
230.2.a.a.1.1	✓	2	3.2	odd	2
1150.2.a.o.1.2		2	15.14	odd	2
1150.2.b.g.599.2		4	15.2	even	4
1150.2.b.g.599.3		4	15.8	even	4
1840.2.a.n.1.2		2	12.11	even	2
2070.2.a.x.1.1		2	1.1	even	1	trivial
5290.2.a.e.1.1		2	69.68	even	2
7360.2.a.bk.1.1		2	24.11	even	2
7360.2.a.bq.1.2		2	24.5	odd	2
9200.2.a.bs.1.1		2	60.59	even	2

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Level:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.79129 q^{7} +1.00000 q^{8} +1.00000 q^{10} +0.791288 q^{11} +5.79129 q^{13} -1.79129 q^{14} +1.00000 q^{16} -0.791288 q^{17} +5.79129 q^{19} +1.00000 q^{20} +0.791288 q^{22} -1.00000 q^{23} +1.00000 q^{25} +5.79129 q^{26} -1.79129 q^{28} -7.58258 q^{29} -3.37386 q^{31} +1.00000 q^{32} -0.791288 q^{34} -1.79129 q^{35} -4.00000 q^{37} +5.79129 q^{38} +1.00000 q^{40} +6.79129 q^{41} +11.1652 q^{43} +0.791288 q^{44} -1.00000 q^{46} +4.41742 q^{47} -3.79129 q^{49} +1.00000 q^{50} +5.79129 q^{52} -6.00000 q^{53} +0.791288 q^{55} -1.79129 q^{56} -7.58258 q^{58} +13.5826 q^{59} +10.3739 q^{61} -3.37386 q^{62} +1.00000 q^{64} +5.79129 q^{65} +11.1652 q^{67} -0.791288 q^{68} -1.79129 q^{70} -8.37386 q^{71} +12.7477 q^{73} -4.00000 q^{74} +5.79129 q^{76} -1.41742 q^{77} +8.00000 q^{79} +1.00000 q^{80} +6.79129 q^{82} +6.00000 q^{83} -0.791288 q^{85} +11.1652 q^{86} +0.791288 q^{88} -15.1652 q^{89} -10.3739 q^{91} -1.00000 q^{92} +4.41742 q^{94} +5.79129 q^{95} -7.95644 q^{97} -3.79129 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + q^{7} + 2 q^{8} + 2 q^{10} - 3 q^{11} + 7 q^{13} + q^{14} + 2 q^{16} + 3 q^{17} + 7 q^{19} + 2 q^{20} - 3 q^{22} - 2 q^{23} + 2 q^{25} + 7 q^{26} + q^{28} - 6 q^{29}+ \cdots - 3 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + q^7 + 2 * q^8 + 2 * q^10 - 3 * q^11 + 7 * q^13 + q^14 + 2 * q^16 + 3 * q^17 + 7 * q^19 + 2 * q^20 - 3 * q^22 - 2 * q^23 + 2 * q^25 + 7 * q^26 + q^28 - 6 * q^29 + 7 * q^31 + 2 * q^32 + 3 * q^34 + q^35 - 8 * q^37 + 7 * q^38 + 2 * q^40 + 9 * q^41 + 4 * q^43 - 3 * q^44 - 2 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 7 * q^52 - 12 * q^53 - 3 * q^55 + q^56 - 6 * q^58 + 18 * q^59 + 7 * q^61 + 7 * q^62 + 2 * q^64 + 7 * q^65 + 4 * q^67 + 3 * q^68 + q^70 - 3 * q^71 - 2 * q^73 - 8 * q^74 + 7 * q^76 - 12 * q^77 + 16 * q^79 + 2 * q^80 + 9 * q^82 + 12 * q^83 + 3 * q^85 + 4 * q^86 - 3 * q^88 - 12 * q^89 - 7 * q^91 - 2 * q^92 + 18 * q^94 + 7 * q^95 + 7 * q^97 - 3 * q^98

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