LMFDB - Embedded newform 7.10.a.a.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7,10,Mod(1,7)] 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("7.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	7.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2] 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:	$3.60525085315$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{193}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 48 $ x^2 - x - 48
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-6.44622$ of defining polynomial
Character	$\chi$	$=$	7.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+10.8924 q^{2} -195.817 q^{3} -393.355 q^{4} +200.782 q^{5} -2132.92 q^{6} -2401.00 q^{7} -9861.52 q^{8} +18661.3 q^{9} +2187.01 q^{10} +63864.3 q^{11} +77025.5 q^{12} -164679. q^{13} -26152.8 q^{14} -39316.5 q^{15} +93981.5 q^{16} -362910. q^{17} +203267. q^{18} -436498. q^{19} -78978.6 q^{20} +470156. q^{21} +695638. q^{22} +918199. q^{23} +1.93105e6 q^{24} -1.91281e6 q^{25} -1.79375e6 q^{26} +200076. q^{27} +944445. q^{28} -3.68643e6 q^{29} -428253. q^{30} +3.47629e6 q^{31} +6.07279e6 q^{32} -1.25057e7 q^{33} -3.95298e6 q^{34} -482078. q^{35} -7.34049e6 q^{36} +1.88149e7 q^{37} -4.75453e6 q^{38} +3.22469e7 q^{39} -1.98002e6 q^{40} +2.40714e6 q^{41} +5.12115e6 q^{42} -1.25306e7 q^{43} -2.51213e7 q^{44} +3.74685e6 q^{45} +1.00014e7 q^{46} -5.54509e7 q^{47} -1.84032e7 q^{48} +5.76480e6 q^{49} -2.08352e7 q^{50} +7.10639e7 q^{51} +6.47772e7 q^{52} -9.26889e7 q^{53} +2.17931e6 q^{54} +1.28228e7 q^{55} +2.36775e7 q^{56} +8.54737e7 q^{57} -4.01542e7 q^{58} -2.52600e7 q^{59} +1.54653e7 q^{60} +6.93275e7 q^{61} +3.78653e7 q^{62} -4.48057e7 q^{63} +1.80290e7 q^{64} -3.30646e7 q^{65} -1.36218e8 q^{66} -2.33494e7 q^{67} +1.42752e8 q^{68} -1.79799e8 q^{69} -5.25101e6 q^{70} -1.06194e8 q^{71} -1.84028e8 q^{72} -2.10115e8 q^{73} +2.04940e8 q^{74} +3.74561e8 q^{75} +1.71699e8 q^{76} -1.53338e8 q^{77} +3.51247e8 q^{78} -149606. q^{79} +1.88698e7 q^{80} -4.06488e8 q^{81} +2.62197e7 q^{82} +5.21565e8 q^{83} -1.84938e8 q^{84} -7.28659e7 q^{85} -1.36489e8 q^{86} +7.21865e8 q^{87} -6.29799e8 q^{88} +2.98587e8 q^{89} +4.08123e7 q^{90} +3.95394e8 q^{91} -3.61178e8 q^{92} -6.80716e8 q^{93} -6.03996e8 q^{94} -8.76410e7 q^{95} -1.18915e9 q^{96} -8.95983e8 q^{97} +6.27928e7 q^{98} +1.19179e9 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{2} - 86 q^{3} - 620 q^{4} - 2238 q^{5} - 3988 q^{6} - 4802 q^{7} + 2616 q^{8} + 11038 q^{9} + 43384 q^{10} + 35316 q^{11} + 52136 q^{12} - 26530 q^{13} + 14406 q^{14} - 307136 q^{15} - 752 q^{16}+ \cdots + 1409417860 q^{99}+O(q^{100}) $ 2 * q - 6 * q^2 - 86 * q^3 - 620 * q^4 - 2238 * q^5 - 3988 * q^6 - 4802 * q^7 + 2616 * q^8 + 11038 * q^9 + 43384 * q^10 + 35316 * q^11 + 52136 * q^12 - 26530 * q^13 + 14406 * q^14 - 307136 * q^15 - 752 * q^16 - 463920 * q^17 + 332042 * q^18 - 925426 * q^19 + 473760 * q^20 + 206486 * q^21 + 1177888 * q^22 + 778128 * q^23 + 3301296 * q^24 + 2081722 * q^25 - 4127424 * q^26 - 2798612 * q^27 + 1488620 * q^28 - 10003584 * q^29 + 4095872 * q^30 + 2467260 * q^31 + 1284576 * q^32 - 15640784 * q^33 - 2246676 * q^34 + 5373438 * q^35 - 5612716 * q^36 + 30735552 * q^37 + 3504660 * q^38 + 47417944 * q^39 - 32409984 * q^40 - 19103448 * q^41 + 9575188 * q^42 + 4065100 * q^43 - 18650976 * q^44 + 22338298 * q^45 + 12367584 * q^46 - 82195020 * q^47 - 28806496 * q^48 + 11529602 * q^49 - 88312626 * q^50 + 59971356 * q^51 + 33466384 * q^52 - 55189812 * q^53 + 52834472 * q^54 + 82445816 * q^55 - 6281016 * q^56 + 31781116 * q^57 + 66558004 * q^58 - 7069218 * q^59 + 76165376 * q^60 + 44316386 * q^61 + 54910200 * q^62 - 26502238 * q^63 + 147417152 * q^64 - 369979260 * q^65 - 83258464 * q^66 - 241921336 * q^67 + 165645816 * q^68 - 195181152 * q^69 - 104164984 * q^70 + 206493816 * q^71 - 279147720 * q^72 - 499153188 * q^73 + 3571524 * q^74 + 813228014 * q^75 + 282511768 * q^76 - 84793716 * q^77 + 94970960 * q^78 + 468535096 * q^79 + 249904128 * q^80 - 585745634 * q^81 + 389586092 * q^82 + 444023958 * q^83 - 125178536 * q^84 + 173475060 * q^85 - 416830608 * q^86 + 28134340 * q^87 - 986010816 * q^88 + 636267396 * q^89 - 273242728 * q^90 + 63698530 * q^91 - 329431488 * q^92 - 791523960 * q^93 - 152223192 * q^94 + 1104747984 * q^95 - 1714981184 * q^96 - 1632716064 * q^97 - 34588806 * q^98 + 1409417860 * 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 $

$2$	10.8924	0.481383	0.240691	−	0.970602i	$-0.422626\pi$
$2$	10.8924	0.481383	0.240691	+	0.970602i	$0.422626\pi$
$3$	−195.817	−1.39574	−0.697870	−	0.716225i	$-0.745869\pi$
$3$	−195.817	−1.39574	−0.697870	+	0.716225i	$0.745869\pi$
$4$	−393.355	−0.768271
$5$	200.782	0.143668	0.0718340	−	0.997417i	$-0.477115\pi$
$5$	200.782	0.143668	0.0718340	+	0.997417i	$0.477115\pi$
$6$	−2132.92	−0.671885
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	−9861.52	−0.851215
$9$	18661.3	0.948090
$10$	2187.01	0.0691593
$11$	63864.3	1.31520	0.657599	−	0.753369i	$-0.271572\pi$
$11$	63864.3	1.31520	0.657599	+	0.753369i	$0.271572\pi$
$12$	77025.5	1.07231
$13$	−164679.	−1.59916	−0.799581	−	0.600558i	$-0.794945\pi$
$13$	−164679.	−1.59916	−0.799581	+	0.600558i	$0.794945\pi$
$14$	−26152.8	−0.181946
$15$	−39316.5	−0.200523
$16$	93981.5	0.358511
$17$	−362910.	−1.05385	−0.526925	−	0.849912i	$-0.676655\pi$
$17$	−362910.	−1.05385	−0.526925	+	0.849912i	$0.676655\pi$
$18$	203267.	0.456394
$19$	−436498.	−0.768406	−0.384203	−	0.923249i	$-0.625524\pi$
$19$	−436498.	−0.768406	−0.384203	+	0.923249i	$0.625524\pi$
$20$	−78978.6	−0.110376
$21$	470156.	0.527540
$22$	695638.	0.633113
$23$	918199.	0.684166	0.342083	−	0.939670i	$-0.388868\pi$
$23$	918199.	0.684166	0.342083	+	0.939670i	$0.388868\pi$
$24$	1.93105e6	1.18807
$25$	−1.91281e6	−0.979359
$26$	−1.79375e6	−0.769809
$27$	200076.	0.0724531
$28$	944445.	0.290379
$29$	−3.68643e6	−0.967865	−0.483932	−	0.875105i	$-0.660792\pi$
$29$	−3.68643e6	−0.967865	−0.483932	+	0.875105i	$0.660792\pi$
$30$	−428253.	−0.0965284
$31$	3.47629e6	0.676064	0.338032	−	0.941135i	$-0.390239\pi$
$31$	3.47629e6	0.676064	0.338032	+	0.941135i	$0.390239\pi$
$32$	6.07279e6	1.02380
$33$	−1.25057e7	−1.83567
$34$	−3.95298e6	−0.507305
$35$	−482078.	−0.0543014
$36$	−7.34049e6	−0.728390
$37$	1.88149e7	1.65042	0.825210	−	0.564826i	$-0.191057\pi$
$37$	1.88149e7	1.65042	0.825210	+	0.564826i	$0.191057\pi$
$38$	−4.75453e6	−0.369897
$39$	3.22469e7	2.23201
$40$	−1.98002e6	−0.122292
$41$	2.40714e6	0.133038	0.0665188	−	0.997785i	$-0.478811\pi$
$41$	2.40714e6	0.133038	0.0665188	+	0.997785i	$0.478811\pi$
$42$	5.12115e6	0.253949
$43$	−1.25306e7	−0.558938	−0.279469	−	0.960155i	$-0.590158\pi$
$43$	−1.25306e7	−0.558938	−0.279469	+	0.960155i	$0.590158\pi$
$44$	−2.51213e7	−1.01043
$45$	3.74685e6	0.136210
$46$	1.00014e7	0.329346
$47$	−5.54509e7	−1.65756	−0.828779	−	0.559577i	$-0.810964\pi$
$47$	−5.54509e7	−1.65756	−0.828779	+	0.559577i	$0.810964\pi$
$48$	−1.84032e7	−0.500388
$49$	5.76480e6	0.142857
$50$	−2.08352e7	−0.471447
$51$	7.10639e7	1.47090
$52$	6.47772e7	1.22859
$53$	−9.26889e7	−1.61356	−0.806782	−	0.590849i	$-0.798793\pi$
$53$	−9.26889e7	−1.61356	−0.806782	+	0.590849i	$0.798793\pi$
$54$	2.17931e6	0.0348777
$55$	1.28228e7	0.188952
$56$	2.36775e7	0.321729
$57$	8.54737e7	1.07250
$58$	−4.01542e7	−0.465913
$59$	−2.52600e7	−0.271393	−0.135696	−	0.990750i	$-0.543327\pi$
$59$	−2.52600e7	−0.271393	−0.135696	+	0.990750i	$0.543327\pi$
$60$	1.54653e7	0.154056
$61$	6.93275e7	0.641093	0.320547	−	0.947233i	$-0.396133\pi$
$61$	6.93275e7	0.641093	0.320547	+	0.947233i	$0.396133\pi$
$62$	3.78653e7	0.325446
$63$	−4.48057e7	−0.358344
$64$	1.80290e7	0.134326
$65$	−3.30646e7	−0.229748
$66$	−1.36218e8	−0.883661
$67$	−2.33494e7	−0.141559	−0.0707796	−	0.997492i	$-0.522549\pi$
$67$	−2.33494e7	−0.141559	−0.0707796	+	0.997492i	$0.522549\pi$
$68$	1.42752e8	0.809643
$69$	−1.79799e8	−0.954918
$70$	−5.25101e6	−0.0261398
$71$	−1.06194e8	−0.495950	−0.247975	−	0.968766i	$-0.579765\pi$
$71$	−1.06194e8	−0.495950	−0.247975	+	0.968766i	$0.579765\pi$
$72$	−1.84028e8	−0.807028
$73$	−2.10115e8	−0.865974	−0.432987	−	0.901400i	$-0.642540\pi$
$73$	−2.10115e8	−0.865974	−0.432987	+	0.901400i	$0.642540\pi$
$74$	2.04940e8	0.794483
$75$	3.74561e8	1.36693
$76$	1.71699e8	0.590344
$77$	−1.53338e8	−0.497098
$78$	3.51247e8	1.07445
$79$	−149606.	−0.000432144 0	−0.000216072	−	1.00000i	$-0.500069\pi$
$79$	−149606.	−0.000432144 0	−0.000216072		1.00000i	$0.500069\pi$
$80$	1.88698e7	0.0515066
$81$	−4.06488e8	−1.04922
$82$	2.62197e7	0.0640420
$83$	5.21565e8	1.20630	0.603152	−	0.797626i	$-0.293911\pi$
$83$	5.21565e8	1.20630	0.603152	+	0.797626i	$0.293911\pi$
$84$	−1.84938e8	−0.405294
$85$	−7.28659e7	−0.151405
$86$	−1.36489e8	−0.269063
$87$	7.21865e8	1.35089
$88$	−6.29799e8	−1.11952
$89$	2.98587e8	0.504448	0.252224	−	0.967669i	$-0.418838\pi$
$89$	2.98587e8	0.504448	0.252224	+	0.967669i	$0.418838\pi$
$90$	4.08123e7	0.0655692
$91$	3.95394e8	0.604426
$92$	−3.61178e8	−0.525625
$93$	−6.80716e8	−0.943610
$94$	−6.03996e8	−0.797919
$95$	−8.76410e7	−0.110395
$96$	−1.18915e9	−1.42895
$97$	−8.95983e8	−1.02761	−0.513803	−	0.857908i	$-0.671764\pi$
$97$	−8.95983e8	−1.02761	−0.513803	+	0.857908i	$0.671764\pi$
$98$	6.27928e7	0.0687689
$99$	1.19179e9	1.24693

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	7.10.a.a.1.2	✓	2
3.2	odd	2		63.10.a.d.1.1		2
4.3	odd	2		112.10.a.e.1.2		2
5.2	odd	4		175.10.b.b.99.3		4
5.3	odd	4		175.10.b.b.99.2		4
5.4	even	2		175.10.a.b.1.1		2
7.2	even	3		49.10.c.c.18.1		4
7.3	odd	6		49.10.c.b.30.1		4
7.4	even	3		49.10.c.c.30.1		4
7.5	odd	6		49.10.c.b.18.1		4
7.6	odd	2		49.10.a.b.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.a.1.2	✓	2	1.1	even	1	trivial
49.10.a.b.1.2		2	7.6	odd	2
49.10.c.b.18.1		4	7.5	odd	6
49.10.c.b.30.1		4	7.3	odd	6
49.10.c.c.18.1		4	7.2	even	3
49.10.c.c.30.1		4	7.4	even	3
63.10.a.d.1.1		2	3.2	odd	2
112.10.a.e.1.2		2	4.3	odd	2
175.10.a.b.1.1		2	5.4	even	2
175.10.b.b.99.2		4	5.3	odd	4
175.10.b.b.99.3		4	5.2	odd	4

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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:	\( N \)	\(=\)	\( 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	7.a (trivial)

\(f(q)\)	\(=\)	\(q+10.8924 q^{2} -195.817 q^{3} -393.355 q^{4} +200.782 q^{5} -2132.92 q^{6} -2401.00 q^{7} -9861.52 q^{8} +18661.3 q^{9} +2187.01 q^{10} +63864.3 q^{11} +77025.5 q^{12} -164679. q^{13} -26152.8 q^{14} -39316.5 q^{15} +93981.5 q^{16} -362910. q^{17} +203267. q^{18} -436498. q^{19} -78978.6 q^{20} +470156. q^{21} +695638. q^{22} +918199. q^{23} +1.93105e6 q^{24} -1.91281e6 q^{25} -1.79375e6 q^{26} +200076. q^{27} +944445. q^{28} -3.68643e6 q^{29} -428253. q^{30} +3.47629e6 q^{31} +6.07279e6 q^{32} -1.25057e7 q^{33} -3.95298e6 q^{34} -482078. q^{35} -7.34049e6 q^{36} +1.88149e7 q^{37} -4.75453e6 q^{38} +3.22469e7 q^{39} -1.98002e6 q^{40} +2.40714e6 q^{41} +5.12115e6 q^{42} -1.25306e7 q^{43} -2.51213e7 q^{44} +3.74685e6 q^{45} +1.00014e7 q^{46} -5.54509e7 q^{47} -1.84032e7 q^{48} +5.76480e6 q^{49} -2.08352e7 q^{50} +7.10639e7 q^{51} +6.47772e7 q^{52} -9.26889e7 q^{53} +2.17931e6 q^{54} +1.28228e7 q^{55} +2.36775e7 q^{56} +8.54737e7 q^{57} -4.01542e7 q^{58} -2.52600e7 q^{59} +1.54653e7 q^{60} +6.93275e7 q^{61} +3.78653e7 q^{62} -4.48057e7 q^{63} +1.80290e7 q^{64} -3.30646e7 q^{65} -1.36218e8 q^{66} -2.33494e7 q^{67} +1.42752e8 q^{68} -1.79799e8 q^{69} -5.25101e6 q^{70} -1.06194e8 q^{71} -1.84028e8 q^{72} -2.10115e8 q^{73} +2.04940e8 q^{74} +3.74561e8 q^{75} +1.71699e8 q^{76} -1.53338e8 q^{77} +3.51247e8 q^{78} -149606. q^{79} +1.88698e7 q^{80} -4.06488e8 q^{81} +2.62197e7 q^{82} +5.21565e8 q^{83} -1.84938e8 q^{84} -7.28659e7 q^{85} -1.36489e8 q^{86} +7.21865e8 q^{87} -6.29799e8 q^{88} +2.98587e8 q^{89} +4.08123e7 q^{90} +3.95394e8 q^{91} -3.61178e8 q^{92} -6.80716e8 q^{93} -6.03996e8 q^{94} -8.76410e7 q^{95} -1.18915e9 q^{96} -8.95983e8 q^{97} +6.27928e7 q^{98} +1.19179e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} - 86 q^{3} - 620 q^{4} - 2238 q^{5} - 3988 q^{6} - 4802 q^{7} + 2616 q^{8} + 11038 q^{9} + 43384 q^{10} + 35316 q^{11} + 52136 q^{12} - 26530 q^{13} + 14406 q^{14} - 307136 q^{15} - 752 q^{16}+ \cdots + 1409417860 q^{99}+O(q^{100}) \) 2 * q - 6 * q^2 - 86 * q^3 - 620 * q^4 - 2238 * q^5 - 3988 * q^6 - 4802 * q^7 + 2616 * q^8 + 11038 * q^9 + 43384 * q^10 + 35316 * q^11 + 52136 * q^12 - 26530 * q^13 + 14406 * q^14 - 307136 * q^15 - 752 * q^16 - 463920 * q^17 + 332042 * q^18 - 925426 * q^19 + 473760 * q^20 + 206486 * q^21 + 1177888 * q^22 + 778128 * q^23 + 3301296 * q^24 + 2081722 * q^25 - 4127424 * q^26 - 2798612 * q^27 + 1488620 * q^28 - 10003584 * q^29 + 4095872 * q^30 + 2467260 * q^31 + 1284576 * q^32 - 15640784 * q^33 - 2246676 * q^34 + 5373438 * q^35 - 5612716 * q^36 + 30735552 * q^37 + 3504660 * q^38 + 47417944 * q^39 - 32409984 * q^40 - 19103448 * q^41 + 9575188 * q^42 + 4065100 * q^43 - 18650976 * q^44 + 22338298 * q^45 + 12367584 * q^46 - 82195020 * q^47 - 28806496 * q^48 + 11529602 * q^49 - 88312626 * q^50 + 59971356 * q^51 + 33466384 * q^52 - 55189812 * q^53 + 52834472 * q^54 + 82445816 * q^55 - 6281016 * q^56 + 31781116 * q^57 + 66558004 * q^58 - 7069218 * q^59 + 76165376 * q^60 + 44316386 * q^61 + 54910200 * q^62 - 26502238 * q^63 + 147417152 * q^64 - 369979260 * q^65 - 83258464 * q^66 - 241921336 * q^67 + 165645816 * q^68 - 195181152 * q^69 - 104164984 * q^70 + 206493816 * q^71 - 279147720 * q^72 - 499153188 * q^73 + 3571524 * q^74 + 813228014 * q^75 + 282511768 * q^76 - 84793716 * q^77 + 94970960 * q^78 + 468535096 * q^79 + 249904128 * q^80 - 585745634 * q^81 + 389586092 * q^82 + 444023958 * q^83 - 125178536 * q^84 + 173475060 * q^85 - 416830608 * q^86 + 28134340 * q^87 - 986010816 * q^88 + 636267396 * q^89 - 273242728 * q^90 + 63698530 * q^91 - 329431488 * q^92 - 791523960 * q^93 - 152223192 * q^94 + 1104747984 * q^95 - 1714981184 * q^96 - 1632716064 * q^97 - 34588806 * q^98 + 1409417860 * q^99

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