LMFDB - Embedded newform 273.4.a.f.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [273,4,Mod(1,273)] 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("273.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(273, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	273.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,-3,12,15,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$16.1075214316$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1038472.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 22x^{2} + 6x + 104 $ x^4 - x^3 - 22x^2 + 6x + 104
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			$-3.33161$ of defining polynomial
Character	$\chi$	$=$	273.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-4.33161 q^{2} +3.00000 q^{3} +10.7629 q^{4} +6.96257 q^{5} -12.9948 q^{6} -7.00000 q^{7} -11.9677 q^{8} +9.00000 q^{9} -30.1592 q^{10} -35.6849 q^{11} +32.2886 q^{12} -13.0000 q^{13} +30.3213 q^{14} +20.8877 q^{15} -34.2635 q^{16} -27.6078 q^{17} -38.9845 q^{18} -123.252 q^{19} +74.9373 q^{20} -21.0000 q^{21} +154.573 q^{22} +204.186 q^{23} -35.9032 q^{24} -76.5226 q^{25} +56.3110 q^{26} +27.0000 q^{27} -75.3402 q^{28} -27.6197 q^{29} -90.4775 q^{30} +174.804 q^{31} +244.158 q^{32} -107.055 q^{33} +119.586 q^{34} -48.7380 q^{35} +96.8659 q^{36} -54.2125 q^{37} +533.878 q^{38} -39.0000 q^{39} -83.3261 q^{40} -271.725 q^{41} +90.9639 q^{42} -513.266 q^{43} -384.072 q^{44} +62.6631 q^{45} -884.457 q^{46} -568.880 q^{47} -102.790 q^{48} +49.0000 q^{49} +331.467 q^{50} -82.8235 q^{51} -139.917 q^{52} +680.814 q^{53} -116.954 q^{54} -248.459 q^{55} +83.7741 q^{56} -369.755 q^{57} +119.638 q^{58} +276.337 q^{59} +224.812 q^{60} -806.359 q^{61} -757.183 q^{62} -63.0000 q^{63} -783.490 q^{64} -90.5134 q^{65} +463.720 q^{66} -691.688 q^{67} -297.140 q^{68} +612.559 q^{69} +211.114 q^{70} -484.700 q^{71} -107.710 q^{72} -576.366 q^{73} +234.828 q^{74} -229.568 q^{75} -1326.54 q^{76} +249.794 q^{77} +168.933 q^{78} -98.4902 q^{79} -238.562 q^{80} +81.0000 q^{81} +1177.01 q^{82} -1065.68 q^{83} -226.020 q^{84} -192.221 q^{85} +2223.27 q^{86} -82.8591 q^{87} +427.067 q^{88} +367.023 q^{89} -271.432 q^{90} +91.0000 q^{91} +2197.63 q^{92} +524.412 q^{93} +2464.17 q^{94} -858.148 q^{95} +732.474 q^{96} +594.655 q^{97} -212.249 q^{98} -321.164 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 24 q^{5} - 9 q^{6} - 28 q^{7} - 39 q^{8} + 36 q^{9} - 26 q^{10} + 8 q^{11} + 45 q^{12} - 52 q^{13} + 21 q^{14} - 72 q^{15} - 181 q^{16} - 6 q^{17} - 27 q^{18} - 332 q^{19}+ \cdots + 72 q^{99}+O(q^{100}) $ 4 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 24 * q^5 - 9 * q^6 - 28 * q^7 - 39 * q^8 + 36 * q^9 - 26 * q^10 + 8 * q^11 + 45 * q^12 - 52 * q^13 + 21 * q^14 - 72 * q^15 - 181 * q^16 - 6 * q^17 - 27 * q^18 - 332 * q^19 - 62 * q^20 - 84 * q^21 - 176 * q^22 + 6 * q^23 - 117 * q^24 - 12 * q^25 + 39 * q^26 + 108 * q^27 - 105 * q^28 + 8 * q^29 - 78 * q^30 - 184 * q^31 + 485 * q^32 + 24 * q^33 - 326 * q^34 + 168 * q^35 + 135 * q^36 - 244 * q^37 + 516 * q^38 - 156 * q^39 + 22 * q^40 - 296 * q^41 + 63 * q^42 - 224 * q^43 - 336 * q^44 - 216 * q^45 - 404 * q^46 - 750 * q^47 - 543 * q^48 + 196 * q^49 + 35 * q^50 - 18 * q^51 - 195 * q^52 + 680 * q^53 - 81 * q^54 - 600 * q^55 + 273 * q^56 - 996 * q^57 - 334 * q^58 - 334 * q^59 - 186 * q^60 - 1212 * q^61 - 928 * q^62 - 252 * q^63 - 697 * q^64 + 312 * q^65 - 528 * q^66 - 388 * q^67 + 1174 * q^68 + 18 * q^69 + 182 * q^70 - 544 * q^71 - 351 * q^72 - 1572 * q^73 + 1482 * q^74 - 36 * q^75 - 1508 * q^76 - 56 * q^77 + 117 * q^78 + 464 * q^79 + 2066 * q^80 + 324 * q^81 + 1806 * q^82 - 3158 * q^83 - 315 * q^84 - 2228 * q^85 + 324 * q^86 + 24 * q^87 + 984 * q^88 + 1156 * q^89 - 234 * q^90 + 364 * q^91 + 1788 * q^92 - 552 * q^93 + 4604 * q^94 + 1128 * q^95 + 1455 * q^96 - 1928 * q^97 - 147 * q^98 + 72 * 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 $

$2$	−4.33161	−1.53146	−0.765728	−	0.643164i	$-0.777621\pi$
$2$	−4.33161	−1.53146	−0.765728	+	0.643164i	$0.777621\pi$
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	10.7629	1.34536
$5$	6.96257	0.622751	0.311376	−	0.950287i	$-0.399210\pi$
$5$	6.96257	0.622751	0.311376	+	0.950287i	$0.399210\pi$
$6$	−12.9948	−0.884187
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−11.9677	−0.528904
$9$	9.00000	0.333333
$10$	−30.1592	−0.953716
$11$	−35.6849	−0.978128	−0.489064	−	0.872248i	$-0.662662\pi$
$11$	−35.6849	−0.978128	−0.489064	+	0.872248i	$0.662662\pi$
$12$	32.2886	0.776744
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	30.3213	0.578836
$15$	20.8877	0.359545
$16$	−34.2635	−0.535367
$17$	−27.6078	−0.393875	−0.196938	−	0.980416i	$-0.563100\pi$
$17$	−27.6078	−0.393875	−0.196938	+	0.980416i	$0.563100\pi$
$18$	−38.9845	−0.510486
$19$	−123.252	−1.48820	−0.744101	−	0.668067i	$-0.767122\pi$
$19$	−123.252	−1.48820	−0.744101	+	0.668067i	$0.767122\pi$
$20$	74.9373	0.837824
$21$	−21.0000	−0.218218
$22$	154.573	1.49796
$23$	204.186	1.85112	0.925561	−	0.378598i	$-0.123594\pi$
$23$	204.186	1.85112	0.925561	+	0.378598i	$0.123594\pi$
$24$	−35.9032	−0.305363
$25$	−76.5226	−0.612181
$26$	56.3110	0.424750
$27$	27.0000	0.192450
$28$	−75.3402	−0.508498
$29$	−27.6197	−0.176857	−0.0884284	−	0.996083i	$-0.528184\pi$
$29$	−27.6197	−0.176857	−0.0884284	+	0.996083i	$0.528184\pi$
$30$	−90.4775	−0.550628
$31$	174.804	1.01277	0.506383	−	0.862309i	$-0.330982\pi$
$31$	174.804	1.01277	0.506383	+	0.862309i	$0.330982\pi$
$32$	244.158	1.34879
$33$	−107.055	−0.564722
$34$	119.586	0.603203
$35$	−48.7380	−0.235378
$36$	96.8659	0.448453
$37$	−54.2125	−0.240878	−0.120439	−	0.992721i	$-0.538430\pi$
$37$	−54.2125	−0.240878	−0.120439	+	0.992721i	$0.538430\pi$
$38$	533.878	2.27912
$39$	−39.0000	−0.160128
$40$	−83.3261	−0.329375
$41$	−271.725	−1.03503	−0.517515	−	0.855674i	$-0.673143\pi$
$41$	−271.725	−1.03503	−0.517515	+	0.855674i	$0.673143\pi$
$42$	90.9639	0.334191
$43$	−513.266	−1.82029	−0.910144	−	0.414293i	$-0.864029\pi$
$43$	−513.266	−1.82029	−0.910144	+	0.414293i	$0.864029\pi$
$44$	−384.072	−1.31593
$45$	62.6631	0.207584
$46$	−884.457	−2.83491
$47$	−568.880	−1.76553	−0.882763	−	0.469818i	$-0.844319\pi$
$47$	−568.880	−1.76553	−0.882763	+	0.469818i	$0.844319\pi$
$48$	−102.790	−0.309094
$49$	49.0000	0.142857
$50$	331.467	0.937529
$51$	−82.8235	−0.227404
$52$	−139.917	−0.373136
$53$	680.814	1.76447	0.882236	−	0.470808i	$-0.156038\pi$
$53$	680.814	1.76447	0.882236	+	0.470808i	$0.156038\pi$
$54$	−116.954	−0.294729
$55$	−248.459	−0.609130
$56$	83.7741	0.199907
$57$	−369.755	−0.859214
$58$	119.638	0.270849
$59$	276.337	0.609764	0.304882	−	0.952390i	$-0.401383\pi$
$59$	276.337	0.609764	0.304882	+	0.952390i	$0.401383\pi$
$60$	224.812	0.483718
$61$	−806.359	−1.69252	−0.846260	−	0.532770i	$-0.821151\pi$
$61$	−806.359	−1.69252	−0.846260	+	0.532770i	$0.821151\pi$
$62$	−757.183	−1.55101
$63$	−63.0000	−0.125988
$64$	−783.490	−1.53025
$65$	−90.5134	−0.172720
$66$	463.720	0.864848
$67$	−691.688	−1.26124	−0.630620	−	0.776092i	$-0.717199\pi$
$67$	−691.688	−1.26124	−0.630620	+	0.776092i	$0.717199\pi$
$68$	−297.140	−0.529904
$69$	612.559	1.06875
$70$	211.114	0.360471
$71$	−484.700	−0.810188	−0.405094	−	0.914275i	$-0.632761\pi$
$71$	−484.700	−0.810188	−0.405094	+	0.914275i	$0.632761\pi$
$72$	−107.710	−0.176301
$73$	−576.366	−0.924090	−0.462045	−	0.886856i	$-0.652884\pi$
$73$	−576.366	−0.924090	−0.462045	+	0.886856i	$0.652884\pi$
$74$	234.828	0.368894
$75$	−229.568	−0.353443
$76$	−1326.54	−2.00217
$77$	249.794	0.369698
$78$	168.933	0.245229
$79$	−98.4902	−0.140266	−0.0701330	−	0.997538i	$-0.522342\pi$
$79$	−98.4902	−0.140266	−0.0701330	+	0.997538i	$0.522342\pi$
$80$	−238.562	−0.333400
$81$	81.0000	0.111111
$82$	1177.01	1.58510
$83$	−1065.68	−1.40932	−0.704658	−	0.709547i	$-0.748900\pi$
$83$	−1065.68	−1.40932	−0.704658	+	0.709547i	$0.748900\pi$
$84$	−226.020	−0.293582
$85$	−192.221	−0.245286
$86$	2223.27	2.78769
$87$	−82.8591	−0.102108
$88$	427.067	0.517336
$89$	367.023	0.437128	0.218564	−	0.975823i	$-0.429863\pi$
$89$	367.023	0.437128	0.218564	+	0.975823i	$0.429863\pi$
$90$	−271.432	−0.317905
$91$	91.0000	0.104828
$92$	2197.63	2.49043
$93$	524.412	0.584720
$94$	2464.17	2.70383
$95$	−858.148	−0.926780
$96$	732.474	0.778727
$97$	594.655	0.622454	0.311227	−	0.950336i	$-0.399260\pi$
$97$	594.655	0.622454	0.311227	+	0.950336i	$0.399260\pi$
$98$	−212.249	−0.218780
$99$	−321.164	−0.326043

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	273.4.a.f.1.1	✓	4
3.2	odd	2		819.4.a.g.1.4		4
7.6	odd	2		1911.4.a.l.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.4.a.f.1.1	✓	4	1.1	even	1	trivial
819.4.a.g.1.4		4	3.2	odd	2
1911.4.a.l.1.1		4	7.6	odd	2

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	273.a (trivial)

\(f(q)\)	\(=\)	\(q-4.33161 q^{2} +3.00000 q^{3} +10.7629 q^{4} +6.96257 q^{5} -12.9948 q^{6} -7.00000 q^{7} -11.9677 q^{8} +9.00000 q^{9} -30.1592 q^{10} -35.6849 q^{11} +32.2886 q^{12} -13.0000 q^{13} +30.3213 q^{14} +20.8877 q^{15} -34.2635 q^{16} -27.6078 q^{17} -38.9845 q^{18} -123.252 q^{19} +74.9373 q^{20} -21.0000 q^{21} +154.573 q^{22} +204.186 q^{23} -35.9032 q^{24} -76.5226 q^{25} +56.3110 q^{26} +27.0000 q^{27} -75.3402 q^{28} -27.6197 q^{29} -90.4775 q^{30} +174.804 q^{31} +244.158 q^{32} -107.055 q^{33} +119.586 q^{34} -48.7380 q^{35} +96.8659 q^{36} -54.2125 q^{37} +533.878 q^{38} -39.0000 q^{39} -83.3261 q^{40} -271.725 q^{41} +90.9639 q^{42} -513.266 q^{43} -384.072 q^{44} +62.6631 q^{45} -884.457 q^{46} -568.880 q^{47} -102.790 q^{48} +49.0000 q^{49} +331.467 q^{50} -82.8235 q^{51} -139.917 q^{52} +680.814 q^{53} -116.954 q^{54} -248.459 q^{55} +83.7741 q^{56} -369.755 q^{57} +119.638 q^{58} +276.337 q^{59} +224.812 q^{60} -806.359 q^{61} -757.183 q^{62} -63.0000 q^{63} -783.490 q^{64} -90.5134 q^{65} +463.720 q^{66} -691.688 q^{67} -297.140 q^{68} +612.559 q^{69} +211.114 q^{70} -484.700 q^{71} -107.710 q^{72} -576.366 q^{73} +234.828 q^{74} -229.568 q^{75} -1326.54 q^{76} +249.794 q^{77} +168.933 q^{78} -98.4902 q^{79} -238.562 q^{80} +81.0000 q^{81} +1177.01 q^{82} -1065.68 q^{83} -226.020 q^{84} -192.221 q^{85} +2223.27 q^{86} -82.8591 q^{87} +427.067 q^{88} +367.023 q^{89} -271.432 q^{90} +91.0000 q^{91} +2197.63 q^{92} +524.412 q^{93} +2464.17 q^{94} -858.148 q^{95} +732.474 q^{96} +594.655 q^{97} -212.249 q^{98} -321.164 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 12 q^{3} + 15 q^{4} - 24 q^{5} - 9 q^{6} - 28 q^{7} - 39 q^{8} + 36 q^{9} - 26 q^{10} + 8 q^{11} + 45 q^{12} - 52 q^{13} + 21 q^{14} - 72 q^{15} - 181 q^{16} - 6 q^{17} - 27 q^{18} - 332 q^{19}+ \cdots + 72 q^{99}+O(q^{100}) \) 4 * q - 3 * q^2 + 12 * q^3 + 15 * q^4 - 24 * q^5 - 9 * q^6 - 28 * q^7 - 39 * q^8 + 36 * q^9 - 26 * q^10 + 8 * q^11 + 45 * q^12 - 52 * q^13 + 21 * q^14 - 72 * q^15 - 181 * q^16 - 6 * q^17 - 27 * q^18 - 332 * q^19 - 62 * q^20 - 84 * q^21 - 176 * q^22 + 6 * q^23 - 117 * q^24 - 12 * q^25 + 39 * q^26 + 108 * q^27 - 105 * q^28 + 8 * q^29 - 78 * q^30 - 184 * q^31 + 485 * q^32 + 24 * q^33 - 326 * q^34 + 168 * q^35 + 135 * q^36 - 244 * q^37 + 516 * q^38 - 156 * q^39 + 22 * q^40 - 296 * q^41 + 63 * q^42 - 224 * q^43 - 336 * q^44 - 216 * q^45 - 404 * q^46 - 750 * q^47 - 543 * q^48 + 196 * q^49 + 35 * q^50 - 18 * q^51 - 195 * q^52 + 680 * q^53 - 81 * q^54 - 600 * q^55 + 273 * q^56 - 996 * q^57 - 334 * q^58 - 334 * q^59 - 186 * q^60 - 1212 * q^61 - 928 * q^62 - 252 * q^63 - 697 * q^64 + 312 * q^65 - 528 * q^66 - 388 * q^67 + 1174 * q^68 + 18 * q^69 + 182 * q^70 - 544 * q^71 - 351 * q^72 - 1572 * q^73 + 1482 * q^74 - 36 * q^75 - 1508 * q^76 - 56 * q^77 + 117 * q^78 + 464 * q^79 + 2066 * q^80 + 324 * q^81 + 1806 * q^82 - 3158 * q^83 - 315 * q^84 - 2228 * q^85 + 324 * q^86 + 24 * q^87 + 984 * q^88 + 1156 * q^89 - 234 * q^90 + 364 * q^91 + 1788 * q^92 - 552 * q^93 + 4604 * q^94 + 1128 * q^95 + 1455 * q^96 - 1928 * q^97 - 147 * q^98 + 72 * q^99

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