LMFDB - Embedded newform 11.10.a.a.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [11,10,Mod(1,11)] 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("11.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 11 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	11.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:	$5.66539419780$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.2659452.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 306x - 836 $ x^3 - 306*x - 836
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2\cdot 3 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-15.9214$ of defining polynomial
Character	$\chi$	$=$	11.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+31.8429 q^{2} -261.882 q^{3} +501.969 q^{4} -1908.19 q^{5} -8339.07 q^{6} +3060.06 q^{7} -319.425 q^{8} +48899.1 q^{9} -60762.2 q^{10} -14641.0 q^{11} -131457. q^{12} -100860. q^{13} +97441.1 q^{14} +499720. q^{15} -267179. q^{16} +141383. q^{17} +1.55709e6 q^{18} -491244. q^{19} -957851. q^{20} -801374. q^{21} -466212. q^{22} +1.02113e6 q^{23} +83651.6 q^{24} +1.68806e6 q^{25} -3.21167e6 q^{26} -7.65118e6 q^{27} +1.53605e6 q^{28} -1.58245e6 q^{29} +1.59125e7 q^{30} -2.39728e6 q^{31} -8.34421e6 q^{32} +3.83421e6 q^{33} +4.50205e6 q^{34} -5.83917e6 q^{35} +2.45458e7 q^{36} -1.40297e7 q^{37} -1.56426e7 q^{38} +2.64134e7 q^{39} +609523. q^{40} -8.90588e6 q^{41} -2.55181e7 q^{42} +3.79365e7 q^{43} -7.34932e6 q^{44} -9.33088e7 q^{45} +3.25158e7 q^{46} +3.02797e7 q^{47} +6.99694e7 q^{48} -3.09896e7 q^{49} +5.37527e7 q^{50} -3.70257e7 q^{51} -5.06286e7 q^{52} -1.24582e6 q^{53} -2.43635e8 q^{54} +2.79378e7 q^{55} -977459. q^{56} +1.28648e8 q^{57} -5.03897e7 q^{58} +1.27784e8 q^{59} +2.50844e8 q^{60} -6.35208e7 q^{61} -7.63362e7 q^{62} +1.49634e8 q^{63} -1.28908e8 q^{64} +1.92460e8 q^{65} +1.22092e8 q^{66} +7.51979e7 q^{67} +7.09700e7 q^{68} -2.67416e8 q^{69} -1.85936e8 q^{70} -9.00530e7 q^{71} -1.56196e7 q^{72} -1.51420e8 q^{73} -4.46747e8 q^{74} -4.42072e8 q^{75} -2.46589e8 q^{76} -4.48023e7 q^{77} +8.41079e8 q^{78} -5.42941e8 q^{79} +5.09829e8 q^{80} +1.04122e9 q^{81} -2.83589e8 q^{82} +4.30849e8 q^{83} -4.02265e8 q^{84} -2.69786e8 q^{85} +1.20801e9 q^{86} +4.14414e8 q^{87} +4.67670e6 q^{88} -5.75029e8 q^{89} -2.97122e9 q^{90} -3.08638e8 q^{91} +5.12577e8 q^{92} +6.27804e8 q^{93} +9.64194e8 q^{94} +9.37386e8 q^{95} +2.18520e9 q^{96} -2.55198e8 q^{97} -9.86799e8 q^{98} -7.15932e8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9} - 86724 q^{10} - 43923 q^{11} - 71040 q^{12} - 93258 q^{13} + 324264 q^{14} + 540330 q^{15} + 22656 q^{16}+ \cdots - 213070473 q^{99}+O(q^{100}) $ 3 * q - 186 * q^3 + 912 * q^4 - 1824 * q^5 - 10956 * q^6 - 7260 * q^7 - 20064 * q^8 + 14553 * q^9 - 86724 * q^10 - 43923 * q^11 - 71040 * q^12 - 93258 * q^13 + 324264 * q^14 + 540330 * q^15 + 22656 * q^16 + 18678 * q^17 + 1997028 * q^18 - 1027356 * q^19 - 156576 * q^20 - 1287924 * q^21 + 1674690 * q^23 - 947232 * q^24 - 1560729 * q^25 - 2162088 * q^26 - 10285434 * q^27 - 2570304 * q^28 - 2693658 * q^29 + 14272500 * q^30 - 4525302 * q^31 + 160512 * q^32 + 2723226 * q^33 + 19899408 * q^34 - 7933860 * q^35 + 20908368 * q^36 - 8820204 * q^37 + 23297712 * q^38 + 24922920 * q^39 - 5145888 * q^40 - 9771102 * q^41 - 8130360 * q^42 + 18795744 * q^43 - 13352592 * q^44 - 91915398 * q^45 - 68671020 * q^46 - 31155816 * q^47 + 76391808 * q^48 - 54229677 * q^49 + 103405236 * q^50 - 62110884 * q^51 - 86345952 * q^52 + 47500122 * q^53 - 153828180 * q^54 + 26705184 * q^55 + 113407680 * q^56 + 62103096 * q^57 - 120644760 * q^58 + 332138370 * q^59 + 290804160 * q^60 - 49031730 * q^61 - 86992620 * q^62 + 319652784 * q^63 - 421220352 * q^64 + 161689572 * q^65 + 160406796 * q^66 + 330560082 * q^67 - 382273056 * q^68 - 104664822 * q^69 - 22907160 * q^70 - 57835050 * q^71 + 302075136 * q^72 - 458816886 * q^73 - 528939708 * q^74 - 562184580 * q^75 - 1324671744 * q^76 + 106293660 * q^77 + 903869520 * q^78 - 798908748 * q^79 + 442084224 * q^80 + 1544572395 * q^81 - 376730664 * q^82 + 1239784920 * q^83 - 809016384 * q^84 - 632001744 * q^85 + 627638088 * q^86 + 505901880 * q^87 + 293757024 * q^88 - 699523368 * q^89 - 2575080288 * q^90 - 268926960 * q^91 + 3537302976 * q^92 + 614745786 * q^93 + 3327514080 * q^94 + 107303856 * q^95 + 2520807168 * q^96 - 2207436012 * q^97 - 1261919472 * q^98 - 213070473 * 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 $

$2$	31.8429	1.40727	0.703635	−	0.710562i	$-0.251559\pi$
$2$	31.8429	1.40727	0.703635	+	0.710562i	$0.251559\pi$
$3$	−261.882	−1.86664	−0.933318	−	0.359050i	$-0.883101\pi$
$3$	−261.882	−1.86664	−0.933318	+	0.359050i	$0.883101\pi$
$4$	501.969	0.980408
$5$	−1908.19	−1.36539	−0.682694	−	0.730704i	$-0.739192\pi$
$5$	−1908.19	−1.36539	−0.682694	+	0.730704i	$0.739192\pi$
$6$	−8339.07	−2.62686
$7$	3060.06	0.481713	0.240857	−	0.970561i	$-0.422572\pi$
$7$	3060.06	0.481713	0.240857	+	0.970561i	$0.422572\pi$
$8$	−319.425	−0.0275717
$9$	48899.1	2.48433
$10$	−60762.2	−1.92147
$11$	−14641.0	−0.301511
$11$	−14641.0	−0.301511
$12$	−131457.	−1.83007
$13$	−100860.	−0.979431	−0.489716	−	0.871882i	$-0.662899\pi$
$13$	−100860.	−0.979431	−0.489716	+	0.871882i	$0.662899\pi$
$14$	97441.1	0.677901
$15$	499720.	2.54869
$16$	−267179.	−1.01921
$17$	141383.	0.410561	0.205281	−	0.978703i	$-0.434189\pi$
$17$	141383.	0.410561	0.205281	+	0.978703i	$0.434189\pi$
$18$	1.55709e6	3.49613
$19$	−491244.	−0.864780	−0.432390	−	0.901687i	$-0.642330\pi$
$19$	−491244.	−0.864780	−0.432390	+	0.901687i	$0.642330\pi$
$20$	−957851.	−1.33864
$21$	−801374.	−0.899184
$22$	−466212.	−0.424308
$23$	1.02113e6	0.760864	0.380432	−	0.924809i	$-0.375775\pi$
$23$	1.02113e6	0.760864	0.380432	+	0.924809i	$0.375775\pi$
$24$	83651.6	0.0514664
$25$	1.68806e6	0.864287
$26$	−3.21167e6	−1.37832
$27$	−7.65118e6	−2.77071
$28$	1.53605e6	0.472275
$29$	−1.58245e6	−0.415469	−0.207734	−	0.978185i	$-0.566609\pi$
$29$	−1.58245e6	−0.415469	−0.207734	+	0.978185i	$0.566609\pi$
$30$	1.59125e7	3.58669
$31$	−2.39728e6	−0.466220	−0.233110	−	0.972450i	$-0.574890\pi$
$31$	−2.39728e6	−0.466220	−0.233110	+	0.972450i	$0.574890\pi$
$32$	−8.34421e6	−1.40673
$33$	3.83421e6	0.562812
$34$	4.50205e6	0.577770
$35$	−5.83917e6	−0.657726
$36$	2.45458e7	2.43566
$37$	−1.40297e7	−1.23067	−0.615335	−	0.788266i	$-0.710979\pi$
$37$	−1.40297e7	−1.23067	−0.615335	+	0.788266i	$0.710979\pi$
$38$	−1.56426e7	−1.21698
$39$	2.64134e7	1.82824
$40$	609523.	0.0376461
$41$	−8.90588e6	−0.492209	−0.246104	−	0.969243i	$-0.579151\pi$
$41$	−8.90588e6	−0.492209	−0.246104	+	0.969243i	$0.579151\pi$
$42$	−2.55181e7	−1.26539
$43$	3.79365e7	1.69219	0.846095	−	0.533032i	$-0.178947\pi$
$43$	3.79365e7	1.69219	0.846095	+	0.533032i	$0.178947\pi$
$44$	−7.34932e6	−0.295604
$45$	−9.33088e7	−3.39208
$46$	3.25158e7	1.07074
$47$	3.02797e7	0.905131	0.452566	−	0.891731i	$-0.350509\pi$
$47$	3.02797e7	0.905131	0.452566	+	0.891731i	$0.350509\pi$
$48$	6.99694e7	1.90249
$49$	−3.09896e7	−0.767952
$50$	5.37527e7	1.21628
$51$	−3.70257e7	−0.766369
$52$	−5.06286e7	−0.960242
$53$	−1.24582e6	−0.0216877	−0.0108439	−	0.999941i	$-0.503452\pi$
$53$	−1.24582e6	−0.0216877	−0.0108439	+	0.999941i	$0.503452\pi$
$54$	−2.43635e8	−3.89914
$55$	2.79378e7	0.411680
$56$	−977459.	−0.0132817
$57$	1.28648e8	1.61423
$58$	−5.03897e7	−0.584676
$59$	1.27784e8	1.37291	0.686457	−	0.727170i	$-0.259165\pi$
$59$	1.27784e8	1.37291	0.686457	+	0.727170i	$0.259165\pi$
$60$	2.50844e8	2.49875
$61$	−6.35208e7	−0.587397	−0.293699	−	0.955898i	$-0.594886\pi$
$61$	−6.35208e7	−0.587397	−0.293699	+	0.955898i	$0.594886\pi$
$62$	−7.63362e7	−0.656097
$63$	1.49634e8	1.19674
$64$	−1.28908e8	−0.960439
$65$	1.92460e8	1.33730
$66$	1.22092e8	0.792028
$67$	7.51979e7	0.455900	0.227950	−	0.973673i	$-0.426798\pi$
$67$	7.51979e7	0.455900	0.227950	+	0.973673i	$0.426798\pi$
$68$	7.09700e7	0.402517
$69$	−2.67416e8	−1.42026
$70$	−1.85936e8	−0.925598
$71$	−9.00530e7	−0.420567	−0.210284	−	0.977640i	$-0.567439\pi$
$71$	−9.00530e7	−0.420567	−0.210284	+	0.977640i	$0.567439\pi$
$72$	−1.56196e7	−0.0684973
$73$	−1.51420e8	−0.624064	−0.312032	−	0.950072i	$-0.601010\pi$
$73$	−1.51420e8	−0.624064	−0.312032	+	0.950072i	$0.601010\pi$
$74$	−4.46747e8	−1.73188
$75$	−4.42072e8	−1.61331
$76$	−2.46589e8	−0.847837
$77$	−4.48023e7	−0.145242
$78$	8.41079e8	2.57283
$79$	−5.42941e8	−1.56831	−0.784153	−	0.620568i	$-0.786902\pi$
$79$	−5.42941e8	−1.56831	−0.784153	+	0.620568i	$0.786902\pi$
$80$	5.09829e8	1.39162
$81$	1.04122e9	2.68758
$82$	−2.83589e8	−0.692671
$83$	4.30849e8	0.996493	0.498246	−	0.867036i	$-0.333978\pi$
$83$	4.30849e8	0.996493	0.498246	+	0.867036i	$0.333978\pi$
$84$	−4.02265e8	−0.881567
$85$	−2.69786e8	−0.560576
$86$	1.20801e9	2.38137
$87$	4.14414e8	0.775529
$88$	4.67670e6	0.00831319
$89$	−5.75029e8	−0.971481	−0.485741	−	0.874103i	$-0.661450\pi$
$89$	−5.75029e8	−0.971481	−0.485741	+	0.874103i	$0.661450\pi$
$90$	−2.97122e9	−4.77357
$91$	−3.08638e8	−0.471805
$92$	5.12577e8	0.745957
$93$	6.27804e8	0.870263
$94$	9.64194e8	1.27376
$95$	9.37386e8	1.18076
$96$	2.18520e9	2.62585
$97$	−2.55198e8	−0.292688	−0.146344	−	0.989234i	$-0.546751\pi$
$97$	−2.55198e8	−0.292688	−0.146344	+	0.989234i	$0.546751\pi$
$98$	−9.86799e8	−1.08072
$99$	−7.15932e8	−0.749055

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	11.10.a.a.1.3	✓	3
3.2	odd	2		99.10.a.b.1.1		3
4.3	odd	2		176.10.a.g.1.3		3
5.4	even	2		275.10.a.a.1.1		3
11.10	odd	2		121.10.a.b.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
11.10.a.a.1.3	✓	3	1.1	even	1	trivial
99.10.a.b.1.1		3	3.2	odd	2
121.10.a.b.1.1		3	11.10	odd	2
176.10.a.g.1.3		3	4.3	odd	2
275.10.a.a.1.1		3	5.4	even	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:	\( N \)	\(=\)	\( 11 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	11.a (trivial)

\(f(q)\)	\(=\)	\(q+31.8429 q^{2} -261.882 q^{3} +501.969 q^{4} -1908.19 q^{5} -8339.07 q^{6} +3060.06 q^{7} -319.425 q^{8} +48899.1 q^{9} -60762.2 q^{10} -14641.0 q^{11} -131457. q^{12} -100860. q^{13} +97441.1 q^{14} +499720. q^{15} -267179. q^{16} +141383. q^{17} +1.55709e6 q^{18} -491244. q^{19} -957851. q^{20} -801374. q^{21} -466212. q^{22} +1.02113e6 q^{23} +83651.6 q^{24} +1.68806e6 q^{25} -3.21167e6 q^{26} -7.65118e6 q^{27} +1.53605e6 q^{28} -1.58245e6 q^{29} +1.59125e7 q^{30} -2.39728e6 q^{31} -8.34421e6 q^{32} +3.83421e6 q^{33} +4.50205e6 q^{34} -5.83917e6 q^{35} +2.45458e7 q^{36} -1.40297e7 q^{37} -1.56426e7 q^{38} +2.64134e7 q^{39} +609523. q^{40} -8.90588e6 q^{41} -2.55181e7 q^{42} +3.79365e7 q^{43} -7.34932e6 q^{44} -9.33088e7 q^{45} +3.25158e7 q^{46} +3.02797e7 q^{47} +6.99694e7 q^{48} -3.09896e7 q^{49} +5.37527e7 q^{50} -3.70257e7 q^{51} -5.06286e7 q^{52} -1.24582e6 q^{53} -2.43635e8 q^{54} +2.79378e7 q^{55} -977459. q^{56} +1.28648e8 q^{57} -5.03897e7 q^{58} +1.27784e8 q^{59} +2.50844e8 q^{60} -6.35208e7 q^{61} -7.63362e7 q^{62} +1.49634e8 q^{63} -1.28908e8 q^{64} +1.92460e8 q^{65} +1.22092e8 q^{66} +7.51979e7 q^{67} +7.09700e7 q^{68} -2.67416e8 q^{69} -1.85936e8 q^{70} -9.00530e7 q^{71} -1.56196e7 q^{72} -1.51420e8 q^{73} -4.46747e8 q^{74} -4.42072e8 q^{75} -2.46589e8 q^{76} -4.48023e7 q^{77} +8.41079e8 q^{78} -5.42941e8 q^{79} +5.09829e8 q^{80} +1.04122e9 q^{81} -2.83589e8 q^{82} +4.30849e8 q^{83} -4.02265e8 q^{84} -2.69786e8 q^{85} +1.20801e9 q^{86} +4.14414e8 q^{87} +4.67670e6 q^{88} -5.75029e8 q^{89} -2.97122e9 q^{90} -3.08638e8 q^{91} +5.12577e8 q^{92} +6.27804e8 q^{93} +9.64194e8 q^{94} +9.37386e8 q^{95} +2.18520e9 q^{96} -2.55198e8 q^{97} -9.86799e8 q^{98} -7.15932e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 186 q^{3} + 912 q^{4} - 1824 q^{5} - 10956 q^{6} - 7260 q^{7} - 20064 q^{8} + 14553 q^{9} - 86724 q^{10} - 43923 q^{11} - 71040 q^{12} - 93258 q^{13} + 324264 q^{14} + 540330 q^{15} + 22656 q^{16}+ \cdots - 213070473 q^{99}+O(q^{100}) \) 3 * q - 186 * q^3 + 912 * q^4 - 1824 * q^5 - 10956 * q^6 - 7260 * q^7 - 20064 * q^8 + 14553 * q^9 - 86724 * q^10 - 43923 * q^11 - 71040 * q^12 - 93258 * q^13 + 324264 * q^14 + 540330 * q^15 + 22656 * q^16 + 18678 * q^17 + 1997028 * q^18 - 1027356 * q^19 - 156576 * q^20 - 1287924 * q^21 + 1674690 * q^23 - 947232 * q^24 - 1560729 * q^25 - 2162088 * q^26 - 10285434 * q^27 - 2570304 * q^28 - 2693658 * q^29 + 14272500 * q^30 - 4525302 * q^31 + 160512 * q^32 + 2723226 * q^33 + 19899408 * q^34 - 7933860 * q^35 + 20908368 * q^36 - 8820204 * q^37 + 23297712 * q^38 + 24922920 * q^39 - 5145888 * q^40 - 9771102 * q^41 - 8130360 * q^42 + 18795744 * q^43 - 13352592 * q^44 - 91915398 * q^45 - 68671020 * q^46 - 31155816 * q^47 + 76391808 * q^48 - 54229677 * q^49 + 103405236 * q^50 - 62110884 * q^51 - 86345952 * q^52 + 47500122 * q^53 - 153828180 * q^54 + 26705184 * q^55 + 113407680 * q^56 + 62103096 * q^57 - 120644760 * q^58 + 332138370 * q^59 + 290804160 * q^60 - 49031730 * q^61 - 86992620 * q^62 + 319652784 * q^63 - 421220352 * q^64 + 161689572 * q^65 + 160406796 * q^66 + 330560082 * q^67 - 382273056 * q^68 - 104664822 * q^69 - 22907160 * q^70 - 57835050 * q^71 + 302075136 * q^72 - 458816886 * q^73 - 528939708 * q^74 - 562184580 * q^75 - 1324671744 * q^76 + 106293660 * q^77 + 903869520 * q^78 - 798908748 * q^79 + 442084224 * q^80 + 1544572395 * q^81 - 376730664 * q^82 + 1239784920 * q^83 - 809016384 * q^84 - 632001744 * q^85 + 627638088 * q^86 + 505901880 * q^87 + 293757024 * q^88 - 699523368 * q^89 - 2575080288 * q^90 - 268926960 * q^91 + 3537302976 * q^92 + 614745786 * q^93 + 3327514080 * q^94 + 107303856 * q^95 + 2520807168 * q^96 - 2207436012 * q^97 - 1261919472 * q^98 - 213070473 * q^99

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