LMFDB - Embedded newform 7.10.a.a.1.1

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			1.1
Root			$7.44622$ 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)

$f(q)$	$=$	\(q-16.8924 q^{2} +109.817 q^{3} -226.645 q^{4} -2438.78 q^{5} -1855.08 q^{6} -2401.00 q^{7} +12477.5 q^{8} -7623.25 q^{9} +41197.0 q^{10} -28548.3 q^{11} -24889.5 q^{12} +138149. q^{13} +40558.8 q^{14} -267819. q^{15} -94733.5 q^{16} -101010. q^{17} +128775. q^{18} -488928. q^{19} +552739. q^{20} -263670. q^{21} +482250. q^{22} -140071. q^{23} +1.37024e6 q^{24} +3.99453e6 q^{25} -2.33367e6 q^{26} -2.99869e6 q^{27} +544175. q^{28} -6.31716e6 q^{29} +4.52413e6 q^{30} -1.00903e6 q^{31} -4.78821e6 q^{32} -3.13508e6 q^{33} +1.70630e6 q^{34} +5.85552e6 q^{35} +1.72777e6 q^{36} +1.19206e7 q^{37} +8.25919e6 q^{38} +1.51711e7 q^{39} -3.04300e7 q^{40} -2.15106e7 q^{41} +4.45404e6 q^{42} +1.65957e7 q^{43} +6.47033e6 q^{44} +1.85915e7 q^{45} +2.36615e6 q^{46} -2.67441e7 q^{47} -1.04033e7 q^{48} +5.76480e6 q^{49} -6.74774e7 q^{50} -1.10926e7 q^{51} -3.13108e7 q^{52} +3.74991e7 q^{53} +5.06552e7 q^{54} +6.96230e7 q^{55} -2.99585e7 q^{56} -5.36926e7 q^{57} +1.06712e8 q^{58} +1.81907e7 q^{59} +6.07000e7 q^{60} -2.50111e7 q^{61} +1.70449e7 q^{62} +1.83034e7 q^{63} +1.29388e8 q^{64} -3.36915e8 q^{65} +5.29592e7 q^{66} -2.18572e8 q^{67} +2.28934e7 q^{68} -1.53822e7 q^{69} -9.89140e7 q^{70} +3.12688e8 q^{71} -9.51193e7 q^{72} -2.89038e8 q^{73} -2.01369e8 q^{74} +4.38667e8 q^{75} +1.10813e8 q^{76} +6.85444e7 q^{77} -2.56276e8 q^{78} +4.68685e8 q^{79} +2.31034e8 q^{80} -1.79258e8 q^{81} +3.63366e8 q^{82} -7.75407e7 q^{83} +5.97597e7 q^{84} +2.46341e8 q^{85} -2.80342e8 q^{86} -6.93730e8 q^{87} -3.56212e8 q^{88} +3.37680e8 q^{89} -3.14055e8 q^{90} -3.31695e8 q^{91} +3.17465e7 q^{92} -1.10808e8 q^{93} +4.51773e8 q^{94} +1.19239e9 q^{95} -5.25827e8 q^{96} -7.36733e8 q^{97} -9.73816e7 q^{98} +2.17631e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−16.8924	−0.746548	−0.373274	−	0.927721i	$-0.621765\pi$
$2$	−16.8924	−0.746548	−0.373274	+	0.927721i	$0.621765\pi$
$3$	109.817	0.782751	0.391375	−	0.920231i	$-0.371999\pi$
$3$	109.817	0.782751	0.391375	+	0.920231i	$0.371999\pi$
$4$	−226.645	−0.442667
$5$	−2438.78	−1.74505	−0.872525	−	0.488569i	$-0.837519\pi$
$5$	−2438.78	−1.74505	−0.872525	+	0.488569i	$0.837519\pi$
$6$	−1855.08	−0.584361
$7$	−2401.00	−0.377964
$7$	−2401.00	−0.377964
$8$	12477.5	1.07702
$9$	−7623.25	−0.387301
$10$	41197.0	1.30276
$11$	−28548.3	−0.587912	−0.293956	−	0.955819i	$-0.594972\pi$
$11$	−28548.3	−0.587912	−0.293956	+	0.955819i	$0.594972\pi$
$12$	−24889.5	−0.346498
$13$	138149.	1.34153	0.670767	−	0.741668i	$-0.265965\pi$
$13$	138149.	1.34153	0.670767	+	0.741668i	$0.265965\pi$
$14$	40558.8	0.282168
$15$	−267819.	−1.36594
$16$	−94733.5	−0.361380
$17$	−101010.	−0.293321	−0.146661	−	0.989187i	$-0.546853\pi$
$17$	−101010.	−0.293321	−0.146661	+	0.989187i	$0.546853\pi$
$18$	128775.	0.289139
$19$	−488928.	−0.860704	−0.430352	−	0.902661i	$-0.641611\pi$
$19$	−488928.	−0.860704	−0.430352	+	0.902661i	$0.641611\pi$
$20$	552739.	0.772476
$21$	−263670.	−0.295852
$22$	482250.	0.438905
$23$	−140071.	−0.104370	−0.0521848	−	0.998637i	$-0.516618\pi$
$23$	−140071.	−0.104370	−0.0521848	+	0.998637i	$0.516618\pi$
$24$	1.37024e6	0.843038
$25$	3.99453e6	2.04520
$26$	−2.33367e6	−1.00152
$27$	−2.99869e6	−1.08591
$28$	544175.	0.167312
$29$	−6.31716e6	−1.65856	−0.829279	−	0.558835i	$-0.811249\pi$
$29$	−6.31716e6	−1.65856	−0.829279	+	0.558835i	$0.811249\pi$
$30$	4.52413e6	1.01974
$31$	−1.00903e6	−0.196234	−0.0981172	−	0.995175i	$-0.531282\pi$
$31$	−1.00903e6	−0.196234	−0.0981172	+	0.995175i	$0.531282\pi$
$32$	−4.78821e6	−0.807232
$33$	−3.13508e6	−0.460189
$34$	1.70630e6	0.218978
$35$	5.85552e6	0.659567
$36$	1.72777e6	0.171445
$37$	1.19206e7	1.04566	0.522832	−	0.852436i	$-0.324876\pi$
$37$	1.19206e7	1.04566	0.522832	+	0.852436i	$0.324876\pi$
$38$	8.25919e6	0.642556
$39$	1.51711e7	1.05009
$40$	−3.04300e7	−1.87945
$41$	−2.15106e7	−1.18884	−0.594422	−	0.804153i	$-0.702619\pi$
$41$	−2.15106e7	−1.18884	−0.594422	+	0.804153i	$0.702619\pi$
$42$	4.45404e6	0.220868
$43$	1.65957e7	0.740265	0.370133	−	0.928979i	$-0.379312\pi$
$43$	1.65957e7	0.740265	0.370133	+	0.928979i	$0.379312\pi$
$44$	6.47033e6	0.260249
$45$	1.85915e7	0.675860
$46$	2.36615e6	0.0779169
$47$	−2.67441e7	−0.799443	−0.399721	−	0.916637i	$-0.630893\pi$
$47$	−2.67441e7	−0.799443	−0.399721	+	0.916637i	$0.630893\pi$
$48$	−1.04033e7	−0.282870
$49$	5.76480e6	0.142857
$50$	−6.74774e7	−1.52684
$51$	−1.10926e7	−0.229597
$52$	−3.13108e7	−0.593853
$53$	3.74991e7	0.652799	0.326399	−	0.945232i	$-0.394165\pi$
$53$	3.74991e7	0.652799	0.326399	+	0.945232i	$0.394165\pi$
$54$	5.06552e7	0.810684
$55$	6.96230e7	1.02594
$56$	−2.99585e7	−0.407075
$57$	−5.36926e7	−0.673717
$58$	1.06712e8	1.23819
$59$	1.81907e7	0.195441	0.0977207	−	0.995214i	$-0.468845\pi$
$59$	1.81907e7	0.195441	0.0977207	+	0.995214i	$0.468845\pi$
$60$	6.07000e7	0.604656
$61$	−2.50111e7	−0.231285	−0.115643	−	0.993291i	$-0.536893\pi$
$61$	−2.50111e7	−0.231285	−0.115643	+	0.993291i	$0.536893\pi$
$62$	1.70449e7	0.146498
$63$	1.83034e7	0.146386
$64$	1.29388e8	0.964017
$65$	−3.36915e8	−2.34105
$66$	5.29592e7	0.343553
$67$	−2.18572e8	−1.32513	−0.662564	−	0.749005i	$-0.730532\pi$
$67$	−2.18572e8	−1.32513	−0.662564	+	0.749005i	$0.730532\pi$
$68$	2.28934e7	0.129844
$69$	−1.53822e7	−0.0816954
$70$	−9.89140e7	−0.492398
$71$	3.12688e8	1.46032	0.730161	−	0.683275i	$-0.239445\pi$
$71$	3.12688e8	1.46032	0.730161	+	0.683275i	$0.239445\pi$
$72$	−9.51193e7	−0.417131
$73$	−2.89038e8	−1.19125	−0.595624	−	0.803264i	$-0.703095\pi$
$73$	−2.89038e8	−1.19125	−0.595624	+	0.803264i	$0.703095\pi$
$74$	−2.01369e8	−0.780638
$75$	4.38667e8	1.60088
$76$	1.10813e8	0.381005
$77$	6.85444e7	0.222210
$78$	−2.56276e8	−0.783940
$79$	4.68685e8	1.35381	0.676907	−	0.736069i	$-0.263320\pi$
$79$	4.68685e8	1.35381	0.676907	+	0.736069i	$0.263320\pi$
$80$	2.31034e8	0.630626
$81$	−1.79258e8	−0.462696
$82$	3.63366e8	0.887529
$83$	−7.75407e7	−0.179341	−0.0896703	−	0.995972i	$-0.528581\pi$
$83$	−7.75407e7	−0.179341	−0.0896703	+	0.995972i	$0.528581\pi$
$84$	5.97597e7	0.130964
$85$	2.46341e8	0.511860
$86$	−2.80342e8	−0.552643
$87$	−6.93730e8	−1.29824
$88$	−3.56212e8	−0.633193
$89$	3.37680e8	0.570493	0.285246	−	0.958454i	$-0.407925\pi$
$89$	3.37680e8	0.570493	0.285246	+	0.958454i	$0.407925\pi$
$90$	−3.14055e8	−0.504562
$91$	−3.31695e8	−0.507052
$92$	3.17465e7	0.0462010
$93$	−1.10808e8	−0.153603
$94$	4.51773e8	0.596822
$95$	1.19239e9	1.50197
$96$	−5.25827e8	−0.631862
$97$	−7.36733e8	−0.844962	−0.422481	−	0.906372i	$-0.638841\pi$
$97$	−7.36733e8	−0.844962	−0.422481	+	0.906372i	$0.638841\pi$
$98$	−9.73816e7	−0.106650
$99$	2.17631e8	0.227699

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.1.1	✓	2
3.2	odd	2		63.10.a.d.1.2		2
4.3	odd	2		112.10.a.e.1.1		2
5.2	odd	4		175.10.b.b.99.1		4
5.3	odd	4		175.10.b.b.99.4		4
5.4	even	2		175.10.a.b.1.2		2
7.2	even	3		49.10.c.c.18.2		4
7.3	odd	6		49.10.c.b.30.2		4
7.4	even	3		49.10.c.c.30.2		4
7.5	odd	6		49.10.c.b.18.2		4
7.6	odd	2		49.10.a.b.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.10.a.a.1.1	✓	2	1.1	even	1	trivial
49.10.a.b.1.1		2	7.6	odd	2
49.10.c.b.18.2		4	7.5	odd	6
49.10.c.b.30.2		4	7.3	odd	6
49.10.c.c.18.2		4	7.2	even	3
49.10.c.c.30.2		4	7.4	even	3
63.10.a.d.1.2		2	3.2	odd	2
112.10.a.e.1.1		2	4.3	odd	2
175.10.a.b.1.2		2	5.4	even	2
175.10.b.b.99.1		4	5.2	odd	4
175.10.b.b.99.4		4	5.3	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)

\(f(q)\)	\(=\)	\(q-16.8924 q^{2} +109.817 q^{3} -226.645 q^{4} -2438.78 q^{5} -1855.08 q^{6} -2401.00 q^{7} +12477.5 q^{8} -7623.25 q^{9} +41197.0 q^{10} -28548.3 q^{11} -24889.5 q^{12} +138149. q^{13} +40558.8 q^{14} -267819. q^{15} -94733.5 q^{16} -101010. q^{17} +128775. q^{18} -488928. q^{19} +552739. q^{20} -263670. q^{21} +482250. q^{22} -140071. q^{23} +1.37024e6 q^{24} +3.99453e6 q^{25} -2.33367e6 q^{26} -2.99869e6 q^{27} +544175. q^{28} -6.31716e6 q^{29} +4.52413e6 q^{30} -1.00903e6 q^{31} -4.78821e6 q^{32} -3.13508e6 q^{33} +1.70630e6 q^{34} +5.85552e6 q^{35} +1.72777e6 q^{36} +1.19206e7 q^{37} +8.25919e6 q^{38} +1.51711e7 q^{39} -3.04300e7 q^{40} -2.15106e7 q^{41} +4.45404e6 q^{42} +1.65957e7 q^{43} +6.47033e6 q^{44} +1.85915e7 q^{45} +2.36615e6 q^{46} -2.67441e7 q^{47} -1.04033e7 q^{48} +5.76480e6 q^{49} -6.74774e7 q^{50} -1.10926e7 q^{51} -3.13108e7 q^{52} +3.74991e7 q^{53} +5.06552e7 q^{54} +6.96230e7 q^{55} -2.99585e7 q^{56} -5.36926e7 q^{57} +1.06712e8 q^{58} +1.81907e7 q^{59} +6.07000e7 q^{60} -2.50111e7 q^{61} +1.70449e7 q^{62} +1.83034e7 q^{63} +1.29388e8 q^{64} -3.36915e8 q^{65} +5.29592e7 q^{66} -2.18572e8 q^{67} +2.28934e7 q^{68} -1.53822e7 q^{69} -9.89140e7 q^{70} +3.12688e8 q^{71} -9.51193e7 q^{72} -2.89038e8 q^{73} -2.01369e8 q^{74} +4.38667e8 q^{75} +1.10813e8 q^{76} +6.85444e7 q^{77} -2.56276e8 q^{78} +4.68685e8 q^{79} +2.31034e8 q^{80} -1.79258e8 q^{81} +3.63366e8 q^{82} -7.75407e7 q^{83} +5.97597e7 q^{84} +2.46341e8 q^{85} -2.80342e8 q^{86} -6.93730e8 q^{87} -3.56212e8 q^{88} +3.37680e8 q^{89} -3.14055e8 q^{90} -3.31695e8 q^{91} +3.17465e7 q^{92} -1.10808e8 q^{93} +4.51773e8 q^{94} +1.19239e9 q^{95} -5.25827e8 q^{96} -7.36733e8 q^{97} -9.73816e7 q^{98} +2.17631e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{2} - 86 q^{3} - 620 q^{4} - 2238 q^{5} - 3988 q^{6} - 4802 q^{7} + 2616 q^{8} + 11038 q^{9} + 43384 q^{10} + 35316 q^{11} + 52136 q^{12} - 26530 q^{13} + 14406 q^{14} - 307136 q^{15} - 752 q^{16}+ \cdots + 1409417860 q^{99}+O(q^{100}) \) 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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