LMFDB - Embedded newform 55.6.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [55,6,Mod(1,55)] 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("55.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(55, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 55 = 5 \cdot 11 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	55.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.82111008971$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.21865.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 30x + 40 $ x^3 - x^2 - 30*x + 40
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			$5.25849$ of defining polynomial
Character	$\chi$	$=$	55.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-8.45512 q^{2} -26.7755 q^{3} +39.4891 q^{4} +25.0000 q^{5} +226.390 q^{6} +22.2927 q^{7} -63.3212 q^{8} +473.926 q^{9} -211.378 q^{10} +121.000 q^{11} -1057.34 q^{12} -225.547 q^{13} -188.488 q^{14} -669.387 q^{15} -728.262 q^{16} -1059.59 q^{17} -4007.11 q^{18} +2525.94 q^{19} +987.227 q^{20} -596.899 q^{21} -1023.07 q^{22} -337.423 q^{23} +1695.46 q^{24} +625.000 q^{25} +1907.03 q^{26} -6183.16 q^{27} +880.320 q^{28} -7644.60 q^{29} +5659.75 q^{30} +6754.80 q^{31} +8183.83 q^{32} -3239.83 q^{33} +8958.95 q^{34} +557.318 q^{35} +18714.9 q^{36} -5663.43 q^{37} -21357.2 q^{38} +6039.13 q^{39} -1583.03 q^{40} -13317.4 q^{41} +5046.85 q^{42} +9007.03 q^{43} +4778.18 q^{44} +11848.2 q^{45} +2852.96 q^{46} -16644.2 q^{47} +19499.6 q^{48} -16310.0 q^{49} -5284.45 q^{50} +28371.0 q^{51} -8906.65 q^{52} +21261.8 q^{53} +52279.4 q^{54} +3025.00 q^{55} -1411.60 q^{56} -67633.4 q^{57} +64636.1 q^{58} -44329.0 q^{59} -26433.5 q^{60} -36591.1 q^{61} -57112.7 q^{62} +10565.1 q^{63} -45890.9 q^{64} -5638.68 q^{65} +27393.2 q^{66} -45840.6 q^{67} -41842.2 q^{68} +9034.68 q^{69} -4712.19 q^{70} -31877.6 q^{71} -30009.6 q^{72} +49936.7 q^{73} +47885.0 q^{74} -16734.7 q^{75} +99747.3 q^{76} +2697.42 q^{77} -51061.6 q^{78} +48257.1 q^{79} -18206.6 q^{80} +50393.1 q^{81} +112600. q^{82} +66052.0 q^{83} -23571.0 q^{84} -26489.7 q^{85} -76155.6 q^{86} +204688. q^{87} -7661.87 q^{88} -124047. q^{89} -100178. q^{90} -5028.06 q^{91} -13324.5 q^{92} -180863. q^{93} +140729. q^{94} +63148.6 q^{95} -219126. q^{96} +73854.3 q^{97} +137903. q^{98} +57345.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 7 q^{2} - 36 q^{3} + 41 q^{4} + 75 q^{5} + 101 q^{6} - 102 q^{7} - 15 q^{8} + 249 q^{9} - 175 q^{10} + 363 q^{11} - 1237 q^{12} - 1646 q^{13} - 963 q^{14} - 900 q^{15} - 2687 q^{16} - 1742 q^{17}+ \cdots + 30129 q^{99}+O(q^{100}) $ 3 * q - 7 * q^2 - 36 * q^3 + 41 * q^4 + 75 * q^5 + 101 * q^6 - 102 * q^7 - 15 * q^8 + 249 * q^9 - 175 * q^10 + 363 * q^11 - 1237 * q^12 - 1646 * q^13 - 963 * q^14 - 900 * q^15 - 2687 * q^16 - 1742 * q^17 - 3076 * q^18 - 10 * q^19 + 1025 * q^20 + 1236 * q^21 - 847 * q^22 - 3876 * q^23 + 5055 * q^24 + 1875 * q^25 - 2894 * q^26 - 4920 * q^27 - 209 * q^28 + 1970 * q^29 + 2525 * q^30 + 6596 * q^31 + 2353 * q^32 - 4356 * q^33 + 16887 * q^34 - 2550 * q^35 + 20138 * q^36 - 22032 * q^37 - 2205 * q^38 + 19528 * q^39 - 375 * q^40 - 7214 * q^41 + 16911 * q^42 + 29644 * q^43 + 4961 * q^44 + 6225 * q^45 + 21126 * q^46 - 8432 * q^47 + 33799 * q^48 - 34939 * q^49 - 4375 * q^50 + 16006 * q^51 - 15462 * q^52 - 31656 * q^53 + 90335 * q^54 + 9075 * q^55 + 16235 * q^56 - 94600 * q^57 + 94035 * q^58 - 31440 * q^59 - 30925 * q^60 - 49374 * q^61 - 100169 * q^62 + 12924 * q^63 + 9601 * q^64 - 41150 * q^65 + 12221 * q^66 - 111892 * q^67 - 30099 * q^68 - 13032 * q^69 - 24075 * q^70 - 58444 * q^71 - 68490 * q^72 + 5554 * q^73 - 35743 * q^74 - 22500 * q^75 + 128385 * q^76 - 12342 * q^77 + 60418 * q^78 + 76080 * q^79 - 67175 * q^80 + 30843 * q^81 + 18026 * q^82 + 44844 * q^83 - 6863 * q^84 - 43550 * q^85 - 157244 * q^86 + 119090 * q^87 - 1815 * q^88 + 25460 * q^89 - 76900 * q^90 + 123236 * q^91 + 14338 * q^92 - 101062 * q^93 + 411002 * q^94 - 250 * q^95 - 186809 * q^96 - 150482 * q^97 + 209376 * q^98 + 30129 * 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 $

$2$	−8.45512	−1.49467	−0.747334	−	0.664448i	$-0.768667\pi$
$2$	−8.45512	−1.49467	−0.747334	+	0.664448i	$0.768667\pi$
$3$	−26.7755	−1.71765	−0.858824	−	0.512271i	$-0.828804\pi$
$3$	−26.7755	−1.71765	−0.858824	+	0.512271i	$0.828804\pi$
$4$	39.4891	1.23403
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	226.390	2.56731
$7$	22.2927	0.171956	0.0859782	−	0.996297i	$-0.472598\pi$
$7$	22.2927	0.171956	0.0859782	+	0.996297i	$0.472598\pi$
$8$	−63.3212	−0.349804
$9$	473.926	1.95031
$10$	−211.378	−0.668436
$11$	121.000	0.301511
$11$	121.000	0.301511
$12$	−1057.34	−2.11964
$13$	−225.547	−0.370151	−0.185075	−	0.982724i	$-0.559253\pi$
$13$	−225.547	−0.370151	−0.185075	+	0.982724i	$0.559253\pi$
$14$	−188.488	−0.257018
$15$	−669.387	−0.768155
$16$	−728.262	−0.711194
$17$	−1059.59	−0.889232	−0.444616	−	0.895721i	$-0.646660\pi$
$17$	−1059.59	−0.889232	−0.444616	+	0.895721i	$0.646660\pi$
$18$	−4007.11	−2.91507
$19$	2525.94	1.60524	0.802620	−	0.596491i	$-0.203439\pi$
$19$	2525.94	1.60524	0.802620	+	0.596491i	$0.203439\pi$
$20$	987.227	0.551877
$21$	−596.899	−0.295360
$22$	−1023.07	−0.450660
$23$	−337.423	−0.133001	−0.0665006	−	0.997786i	$-0.521183\pi$
$23$	−337.423	−0.133001	−0.0665006	+	0.997786i	$0.521183\pi$
$24$	1695.46	0.600839
$25$	625.000	0.200000
$26$	1907.03	0.553253
$27$	−6183.16	−1.63230
$28$	880.320	0.212200
$29$	−7644.60	−1.68795	−0.843976	−	0.536381i	$-0.819791\pi$
$29$	−7644.60	−1.68795	−0.843976	+	0.536381i	$0.819791\pi$
$30$	5659.75	1.14814
$31$	6754.80	1.26243	0.631217	−	0.775607i	$-0.282556\pi$
$31$	6754.80	1.26243	0.631217	+	0.775607i	$0.282556\pi$
$32$	8183.83	1.41280
$33$	−3239.83	−0.517890
$34$	8958.95	1.32911
$35$	557.318	0.0769012
$36$	18714.9	2.40675
$37$	−5663.43	−0.680104	−0.340052	−	0.940407i	$-0.610445\pi$
$37$	−5663.43	−0.680104	−0.340052	+	0.940407i	$0.610445\pi$
$38$	−21357.2	−2.39930
$39$	6039.13	0.635789
$40$	−1583.03	−0.156437
$41$	−13317.4	−1.23725	−0.618627	−	0.785685i	$-0.712311\pi$
$41$	−13317.4	−1.23725	−0.618627	+	0.785685i	$0.712311\pi$
$42$	5046.85	0.441466
$43$	9007.03	0.742866	0.371433	−	0.928460i	$-0.378867\pi$
$43$	9007.03	0.742866	0.371433	+	0.928460i	$0.378867\pi$
$44$	4778.18	0.372075
$45$	11848.2	0.872207
$46$	2852.96	0.198793
$47$	−16644.2	−1.09905	−0.549526	−	0.835477i	$-0.685192\pi$
$47$	−16644.2	−1.09905	−0.549526	+	0.835477i	$0.685192\pi$
$48$	19499.6	1.22158
$49$	−16310.0	−0.970431
$50$	−5284.45	−0.298934
$51$	28371.0	1.52739
$52$	−8906.65	−0.456779
$53$	21261.8	1.03971	0.519853	−	0.854256i	$-0.325987\pi$
$53$	21261.8	1.03971	0.519853	+	0.854256i	$0.325987\pi$
$54$	52279.4	2.43975
$55$	3025.00	0.134840
$56$	−1411.60	−0.0601509
$57$	−67633.4	−2.75724
$58$	64636.1	2.52293
$59$	−44329.0	−1.65790	−0.828948	−	0.559325i	$-0.811060\pi$
$59$	−44329.0	−1.65790	−0.828948	+	0.559325i	$0.811060\pi$
$60$	−26433.5	−0.947930
$61$	−36591.1	−1.25907	−0.629537	−	0.776970i	$-0.716755\pi$
$61$	−36591.1	−1.25907	−0.629537	+	0.776970i	$0.716755\pi$
$62$	−57112.7	−1.88692
$63$	10565.1	0.335369
$64$	−45890.9	−1.40048
$65$	−5638.68	−0.165537
$66$	27393.2	0.774074
$67$	−45840.6	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$67$	−45840.6	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$68$	−41842.2	−1.09734
$69$	9034.68	0.228449
$70$	−4712.19	−0.114942
$71$	−31877.6	−0.750481	−0.375240	−	0.926928i	$-0.622440\pi$
$71$	−31877.6	−0.750481	−0.375240	+	0.926928i	$0.622440\pi$
$72$	−30009.6	−0.682227
$73$	49936.7	1.09676	0.548382	−	0.836228i	$-0.315244\pi$
$73$	49936.7	1.09676	0.548382	+	0.836228i	$0.315244\pi$
$74$	47885.0	1.01653
$75$	−16734.7	−0.343530
$76$	99747.3	1.98092
$77$	2697.42	0.0518468
$78$	−51061.6	−0.950294
$79$	48257.1	0.869949	0.434974	−	0.900443i	$-0.356757\pi$
$79$	48257.1	0.869949	0.434974	+	0.900443i	$0.356757\pi$
$80$	−18206.6	−0.318056
$81$	50393.1	0.853411
$82$	112600.	1.84928
$83$	66052.0	1.05242	0.526212	−	0.850353i	$-0.323612\pi$
$83$	66052.0	1.05242	0.526212	+	0.850353i	$0.323612\pi$
$84$	−23571.0	−0.364485
$85$	−26489.7	−0.397677
$86$	−76155.6	−1.11034
$87$	204688.	2.89931
$88$	−7661.87	−0.105470
$89$	−124047.	−1.66002	−0.830009	−	0.557750i	$-0.811665\pi$
$89$	−124047.	−1.66002	−0.830009	+	0.557750i	$0.811665\pi$
$90$	−100178.	−1.30366
$91$	−5028.06	−0.0636498
$92$	−13324.5	−0.164128
$93$	−180863.	−2.16842
$94$	140729.	1.64272
$95$	63148.6	0.717885
$96$	−219126.	−2.42670
$97$	73854.3	0.796978	0.398489	−	0.917173i	$-0.369535\pi$
$97$	73854.3	0.796978	0.398489	+	0.917173i	$0.369535\pi$
$98$	137903.	1.45047
$99$	57345.1	0.588042

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	55.6.a.a.1.1	✓	3
3.2	odd	2		495.6.a.f.1.3		3
4.3	odd	2		880.6.a.l.1.3		3
5.2	odd	4		275.6.b.c.199.1		6
5.3	odd	4		275.6.b.c.199.6		6
5.4	even	2		275.6.a.c.1.3		3
11.10	odd	2		605.6.a.b.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
55.6.a.a.1.1	✓	3	1.1	even	1	trivial
275.6.a.c.1.3		3	5.4	even	2
275.6.b.c.199.1		6	5.2	odd	4
275.6.b.c.199.6		6	5.3	odd	4
495.6.a.f.1.3		3	3.2	odd	2
605.6.a.b.1.3		3	11.10	odd	2
880.6.a.l.1.3		3	4.3	odd	2

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

Level:	\( N \)	\(=\)	\( 55 = 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	55.a (trivial)

\(f(q)\)	\(=\)	\(q-8.45512 q^{2} -26.7755 q^{3} +39.4891 q^{4} +25.0000 q^{5} +226.390 q^{6} +22.2927 q^{7} -63.3212 q^{8} +473.926 q^{9} -211.378 q^{10} +121.000 q^{11} -1057.34 q^{12} -225.547 q^{13} -188.488 q^{14} -669.387 q^{15} -728.262 q^{16} -1059.59 q^{17} -4007.11 q^{18} +2525.94 q^{19} +987.227 q^{20} -596.899 q^{21} -1023.07 q^{22} -337.423 q^{23} +1695.46 q^{24} +625.000 q^{25} +1907.03 q^{26} -6183.16 q^{27} +880.320 q^{28} -7644.60 q^{29} +5659.75 q^{30} +6754.80 q^{31} +8183.83 q^{32} -3239.83 q^{33} +8958.95 q^{34} +557.318 q^{35} +18714.9 q^{36} -5663.43 q^{37} -21357.2 q^{38} +6039.13 q^{39} -1583.03 q^{40} -13317.4 q^{41} +5046.85 q^{42} +9007.03 q^{43} +4778.18 q^{44} +11848.2 q^{45} +2852.96 q^{46} -16644.2 q^{47} +19499.6 q^{48} -16310.0 q^{49} -5284.45 q^{50} +28371.0 q^{51} -8906.65 q^{52} +21261.8 q^{53} +52279.4 q^{54} +3025.00 q^{55} -1411.60 q^{56} -67633.4 q^{57} +64636.1 q^{58} -44329.0 q^{59} -26433.5 q^{60} -36591.1 q^{61} -57112.7 q^{62} +10565.1 q^{63} -45890.9 q^{64} -5638.68 q^{65} +27393.2 q^{66} -45840.6 q^{67} -41842.2 q^{68} +9034.68 q^{69} -4712.19 q^{70} -31877.6 q^{71} -30009.6 q^{72} +49936.7 q^{73} +47885.0 q^{74} -16734.7 q^{75} +99747.3 q^{76} +2697.42 q^{77} -51061.6 q^{78} +48257.1 q^{79} -18206.6 q^{80} +50393.1 q^{81} +112600. q^{82} +66052.0 q^{83} -23571.0 q^{84} -26489.7 q^{85} -76155.6 q^{86} +204688. q^{87} -7661.87 q^{88} -124047. q^{89} -100178. q^{90} -5028.06 q^{91} -13324.5 q^{92} -180863. q^{93} +140729. q^{94} +63148.6 q^{95} -219126. q^{96} +73854.3 q^{97} +137903. q^{98} +57345.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 7 q^{2} - 36 q^{3} + 41 q^{4} + 75 q^{5} + 101 q^{6} - 102 q^{7} - 15 q^{8} + 249 q^{9} - 175 q^{10} + 363 q^{11} - 1237 q^{12} - 1646 q^{13} - 963 q^{14} - 900 q^{15} - 2687 q^{16} - 1742 q^{17}+ \cdots + 30129 q^{99}+O(q^{100}) \) 3 * q - 7 * q^2 - 36 * q^3 + 41 * q^4 + 75 * q^5 + 101 * q^6 - 102 * q^7 - 15 * q^8 + 249 * q^9 - 175 * q^10 + 363 * q^11 - 1237 * q^12 - 1646 * q^13 - 963 * q^14 - 900 * q^15 - 2687 * q^16 - 1742 * q^17 - 3076 * q^18 - 10 * q^19 + 1025 * q^20 + 1236 * q^21 - 847 * q^22 - 3876 * q^23 + 5055 * q^24 + 1875 * q^25 - 2894 * q^26 - 4920 * q^27 - 209 * q^28 + 1970 * q^29 + 2525 * q^30 + 6596 * q^31 + 2353 * q^32 - 4356 * q^33 + 16887 * q^34 - 2550 * q^35 + 20138 * q^36 - 22032 * q^37 - 2205 * q^38 + 19528 * q^39 - 375 * q^40 - 7214 * q^41 + 16911 * q^42 + 29644 * q^43 + 4961 * q^44 + 6225 * q^45 + 21126 * q^46 - 8432 * q^47 + 33799 * q^48 - 34939 * q^49 - 4375 * q^50 + 16006 * q^51 - 15462 * q^52 - 31656 * q^53 + 90335 * q^54 + 9075 * q^55 + 16235 * q^56 - 94600 * q^57 + 94035 * q^58 - 31440 * q^59 - 30925 * q^60 - 49374 * q^61 - 100169 * q^62 + 12924 * q^63 + 9601 * q^64 - 41150 * q^65 + 12221 * q^66 - 111892 * q^67 - 30099 * q^68 - 13032 * q^69 - 24075 * q^70 - 58444 * q^71 - 68490 * q^72 + 5554 * q^73 - 35743 * q^74 - 22500 * q^75 + 128385 * q^76 - 12342 * q^77 + 60418 * q^78 + 76080 * q^79 - 67175 * q^80 + 30843 * q^81 + 18026 * q^82 + 44844 * q^83 - 6863 * q^84 - 43550 * q^85 - 157244 * q^86 + 119090 * q^87 - 1815 * q^88 + 25460 * q^89 - 76900 * q^90 + 123236 * q^91 + 14338 * q^92 - 101062 * q^93 + 411002 * q^94 - 250 * q^95 - 186809 * q^96 - 150482 * q^97 + 209376 * q^98 + 30129 * q^99

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