LMFDB - Embedded newform 3249.2.a.t.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3249, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,5,-2,0,-1,-3,0,4,0,0,1,-3,0,5,0,0,0,-22,0,-18,-14,0,5, 21,0,17,4,0,-15,-17,0,0,-18,0,3,0,0,24,-4,0,-3,-12,0,-20,18,0,-2,-23,0, -5,-6,0,12,-21,0,-8,0,0,13,23,0,-15,6,0,-9,0,0,24,-18,0,19,-1,0,0,-12, 0,-11,-10,0,8,4,0,0,-17,0,-6,-16,0,-7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.571993 q^{2} -1.67282 q^{4} +2.67282 q^{5} -3.67282 q^{7} +2.10083 q^{8} -1.52884 q^{10} +3.81681 q^{11} -0.143987 q^{13} +2.10083 q^{14} +2.14399 q^{16} -4.47116 q^{20} -2.18319 q^{22} -7.52884 q^{23} +2.14399 q^{25} +0.0823593 q^{26} +6.14399 q^{28} -5.34565 q^{29} -8.81681 q^{31} -5.42801 q^{32} -9.81681 q^{35} +1.00000 q^{37} +5.61515 q^{40} +5.34565 q^{41} +2.81681 q^{43} -6.38485 q^{44} +4.30644 q^{46} +6.00000 q^{47} +6.48963 q^{49} -1.22635 q^{50} +0.240864 q^{52} +8.01847 q^{53} +10.2017 q^{55} -7.71598 q^{56} +3.05767 q^{58} +3.81681 q^{59} +11.4896 q^{61} +5.04316 q^{62} -1.18319 q^{64} -0.384851 q^{65} -5.38485 q^{67} +5.61515 q^{70} -13.6336 q^{71} -0.345647 q^{73} -0.571993 q^{74} -14.0185 q^{77} -6.52884 q^{79} +5.73050 q^{80} -3.05767 q^{82} +2.28797 q^{83} -1.61120 q^{86} +8.01847 q^{88} -8.67282 q^{89} +0.528837 q^{91} +12.5944 q^{92} -3.43196 q^{94} +5.91369 q^{97} -3.71203 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} + 5 q^{16} - 22 q^{20} - 18 q^{22} - 14 q^{23} + 5 q^{25} + 21 q^{26} + 17 q^{28} + 4 q^{29} - 15 q^{31} - 17 q^{32} - 18 q^{35}+ \cdots - 14 q^{98}+O(q^{100}) $ 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8 + 4 * q^10 + q^13 - 3 * q^14 + 5 * q^16 - 22 * q^20 - 18 * q^22 - 14 * q^23 + 5 * q^25 + 21 * q^26 + 17 * q^28 + 4 * q^29 - 15 * q^31 - 17 * q^32 - 18 * q^35 + 3 * q^37 + 24 * q^40 - 4 * q^41 - 3 * q^43 - 12 * q^44 - 20 * q^46 + 18 * q^47 - 2 * q^49 - 23 * q^50 - 5 * q^52 - 6 * q^53 + 12 * q^55 - 21 * q^56 - 8 * q^58 + 13 * q^61 + 23 * q^62 - 15 * q^64 + 6 * q^65 - 9 * q^67 + 24 * q^70 - 18 * q^71 + 19 * q^73 - q^74 - 12 * q^77 - 11 * q^79 - 10 * q^80 + 8 * q^82 + 4 * q^83 - 17 * q^86 - 6 * q^88 - 16 * q^89 - 7 * q^91 + 2 * q^92 - 6 * q^94 + 2 * q^97 - 14 * 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 $

$2$	−0.571993	−0.404460	−0.202230	−	0.979338i	$-0.564819\pi$
$2$	−0.571993	−0.404460	−0.202230	+	0.979338i	$0.564819\pi$
$3$	0	0
$3$	0	0
$4$	−1.67282	−0.836412
$5$	2.67282	1.19532	0.597662	−	0.801749i	$-0.296097\pi$
$5$	2.67282	1.19532	0.597662	+	0.801749i	$0.296097\pi$
$6$	0	0
$7$	−3.67282	−1.38820	−0.694098	−	0.719880i	$-0.744197\pi$
$7$	−3.67282	−1.38820	−0.694098	+	0.719880i	$0.744197\pi$
$8$	2.10083	0.742756
$9$	0	0
$10$	−1.52884	−0.483461
$11$	3.81681	1.15081	0.575406	−	0.817868i	$-0.304844\pi$
$11$	3.81681	1.15081	0.575406	+	0.817868i	$0.304844\pi$
$12$	0	0
$13$	−0.143987	−0.0399347	−0.0199673	−	0.999801i	$-0.506356\pi$
$13$	−0.143987	−0.0399347	−0.0199673	+	0.999801i	$0.506356\pi$
$14$	2.10083	0.561471
$15$	0	0
$16$	2.14399	0.535997
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−4.47116	−0.999782
$21$	0	0
$22$	−2.18319	−0.465458
$23$	−7.52884	−1.56987	−0.784936	−	0.619577i	$-0.787304\pi$
$23$	−7.52884	−1.56987	−0.784936	+	0.619577i	$0.787304\pi$
$24$	0	0
$25$	2.14399	0.428797
$26$	0.0823593	0.0161520
$27$	0	0
$28$	6.14399	1.16110
$29$	−5.34565	−0.992662	−0.496331	−	0.868133i	$-0.665320\pi$
$29$	−5.34565	−0.992662	−0.496331	+	0.868133i	$0.665320\pi$
$30$	0	0
$31$	−8.81681	−1.58355	−0.791773	−	0.610816i	$-0.790842\pi$
$31$	−8.81681	−1.58355	−0.791773	+	0.610816i	$0.790842\pi$
$32$	−5.42801	−0.959545
$33$	0	0
$34$	0	0
$35$	−9.81681	−1.65934
$36$	0	0
$37$	1.00000	0.164399	0.0821995	−	0.996616i	$-0.473806\pi$
$37$	1.00000	0.164399	0.0821995	+	0.996616i	$0.473806\pi$
$38$	0	0
$39$	0	0
$40$	5.61515	0.887833
$41$	5.34565	0.834850	0.417425	−	0.908711i	$-0.362933\pi$
$41$	5.34565	0.834850	0.417425	+	0.908711i	$0.362933\pi$
$42$	0	0
$43$	2.81681	0.429560	0.214780	−	0.976663i	$-0.431097\pi$
$43$	2.81681	0.429560	0.214780	+	0.976663i	$0.431097\pi$
$44$	−6.38485	−0.962552
$45$	0	0
$46$	4.30644	0.634951
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	6.48963	0.927091
$50$	−1.22635	−0.173431
$51$	0	0
$52$	0.240864	0.0334018
$53$	8.01847	1.10142	0.550711	−	0.834696i	$-0.314357\pi$
$53$	8.01847	1.10142	0.550711	+	0.834696i	$0.314357\pi$
$54$	0	0
$55$	10.2017	1.37559
$56$	−7.71598	−1.03109
$57$	0	0
$58$	3.05767	0.401492
$59$	3.81681	0.496906	0.248453	−	0.968644i	$-0.420078\pi$
$59$	3.81681	0.496906	0.248453	+	0.968644i	$0.420078\pi$
$60$	0	0
$61$	11.4896	1.47110	0.735548	−	0.677472i	$-0.236925\pi$
$61$	11.4896	1.47110	0.735548	+	0.677472i	$0.236925\pi$
$62$	5.04316	0.640481
$63$	0	0
$64$	−1.18319	−0.147899
$65$	−0.384851	−0.0477348
$66$	0	0
$67$	−5.38485	−0.657864	−0.328932	−	0.944354i	$-0.606689\pi$
$67$	−5.38485	−0.657864	−0.328932	+	0.944354i	$0.606689\pi$
$68$	0	0
$69$	0	0
$70$	5.61515	0.671139
$71$	−13.6336	−1.61801	−0.809007	−	0.587800i	$-0.799994\pi$
$71$	−13.6336	−1.61801	−0.809007	+	0.587800i	$0.799994\pi$
$72$	0	0
$73$	−0.345647	−0.0404550	−0.0202275	−	0.999795i	$-0.506439\pi$
$73$	−0.345647	−0.0404550	−0.0202275	+	0.999795i	$0.506439\pi$
$74$	−0.571993	−0.0664929
$75$	0	0
$76$	0	0
$77$	−14.0185	−1.59755
$78$	0	0
$79$	−6.52884	−0.734552	−0.367276	−	0.930112i	$-0.619709\pi$
$79$	−6.52884	−0.734552	−0.367276	+	0.930112i	$0.619709\pi$
$80$	5.73050	0.640689
$81$	0	0
$82$	−3.05767	−0.337664
$83$	2.28797	0.251138	0.125569	−	0.992085i	$-0.459924\pi$
$83$	2.28797	0.251138	0.125569	+	0.992085i	$0.459924\pi$
$84$	0	0
$85$	0	0
$86$	−1.61120	−0.173740
$87$	0	0
$88$	8.01847	0.854772
$89$	−8.67282	−0.919317	−0.459659	−	0.888096i	$-0.652028\pi$
$89$	−8.67282	−0.919317	−0.459659	+	0.888096i	$0.652028\pi$
$90$	0	0
$91$	0.528837	0.0554372
$92$	12.5944	1.31306
$93$	0	0
$94$	−3.43196	−0.353980
$95$	0	0
$96$	0	0
$97$	5.91369	0.600444	0.300222	−	0.953869i	$-0.402939\pi$
$97$	5.91369	0.600444	0.300222	+	0.953869i	$0.402939\pi$
$98$	−3.71203	−0.374971
$99$	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	3249.2.a.t.1.2		3
3.2	odd	2		1083.2.a.o.1.2		3
19.8	odd	6		171.2.f.b.64.2		6
19.12	odd	6		171.2.f.b.163.2		6
19.18	odd	2		3249.2.a.y.1.2		3
57.8	even	6		57.2.e.b.7.2	✓	6
57.50	even	6		57.2.e.b.49.2	yes	6
57.56	even	2		1083.2.a.l.1.2		3
76.27	even	6		2736.2.s.z.577.1		6
76.31	even	6		2736.2.s.z.1873.1		6
228.107	odd	6		912.2.q.l.49.3		6
228.179	odd	6		912.2.q.l.577.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.b.7.2	✓	6	57.8	even	6
57.2.e.b.49.2	yes	6	57.50	even	6
171.2.f.b.64.2		6	19.8	odd	6
171.2.f.b.163.2		6	19.12	odd	6
912.2.q.l.49.3		6	228.107	odd	6
912.2.q.l.577.3		6	228.179	odd	6
1083.2.a.l.1.2		3	57.56	even	2
1083.2.a.o.1.2		3	3.2	odd	2
2736.2.s.z.577.1		6	76.27	even	6
2736.2.s.z.1873.1		6	76.31	even	6
3249.2.a.t.1.2		3	1.1	even	1	trivial
3249.2.a.y.1.2		3	19.18	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$ over number fields
Genus 2 curves
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-0.571993 q^{2} -1.67282 q^{4} +2.67282 q^{5} -3.67282 q^{7} +2.10083 q^{8} -1.52884 q^{10} +3.81681 q^{11} -0.143987 q^{13} +2.10083 q^{14} +2.14399 q^{16} -4.47116 q^{20} -2.18319 q^{22} -7.52884 q^{23} +2.14399 q^{25} +0.0823593 q^{26} +6.14399 q^{28} -5.34565 q^{29} -8.81681 q^{31} -5.42801 q^{32} -9.81681 q^{35} +1.00000 q^{37} +5.61515 q^{40} +5.34565 q^{41} +2.81681 q^{43} -6.38485 q^{44} +4.30644 q^{46} +6.00000 q^{47} +6.48963 q^{49} -1.22635 q^{50} +0.240864 q^{52} +8.01847 q^{53} +10.2017 q^{55} -7.71598 q^{56} +3.05767 q^{58} +3.81681 q^{59} +11.4896 q^{61} +5.04316 q^{62} -1.18319 q^{64} -0.384851 q^{65} -5.38485 q^{67} +5.61515 q^{70} -13.6336 q^{71} -0.345647 q^{73} -0.571993 q^{74} -14.0185 q^{77} -6.52884 q^{79} +5.73050 q^{80} -3.05767 q^{82} +2.28797 q^{83} -1.61120 q^{86} +8.01847 q^{88} -8.67282 q^{89} +0.528837 q^{91} +12.5944 q^{92} -3.43196 q^{94} +5.91369 q^{97} -3.71203 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 5 q^{4} - 2 q^{5} - q^{7} - 3 q^{8} + 4 q^{10} + q^{13} - 3 q^{14} + 5 q^{16} - 22 q^{20} - 18 q^{22} - 14 q^{23} + 5 q^{25} + 21 q^{26} + 17 q^{28} + 4 q^{29} - 15 q^{31} - 17 q^{32} - 18 q^{35}+ \cdots - 14 q^{98}+O(q^{100}) \) 3 * q - q^2 + 5 * q^4 - 2 * q^5 - q^7 - 3 * q^8 + 4 * q^10 + q^13 - 3 * q^14 + 5 * q^16 - 22 * q^20 - 18 * q^22 - 14 * q^23 + 5 * q^25 + 21 * q^26 + 17 * q^28 + 4 * q^29 - 15 * q^31 - 17 * q^32 - 18 * q^35 + 3 * q^37 + 24 * q^40 - 4 * q^41 - 3 * q^43 - 12 * q^44 - 20 * q^46 + 18 * q^47 - 2 * q^49 - 23 * q^50 - 5 * q^52 - 6 * q^53 + 12 * q^55 - 21 * q^56 - 8 * q^58 + 13 * q^61 + 23 * q^62 - 15 * q^64 + 6 * q^65 - 9 * q^67 + 24 * q^70 - 18 * q^71 + 19 * q^73 - q^74 - 12 * q^77 - 11 * q^79 - 10 * q^80 + 8 * q^82 + 4 * q^83 - 17 * q^86 - 6 * q^88 - 16 * q^89 - 7 * q^91 + 2 * q^92 - 6 * q^94 + 2 * q^97 - 14 * q^98

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