LMFDB - Embedded newform 1027.4.a.c.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1027,4,Mod(1,1027)] 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("1027.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1027, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1027 = 13 \cdot 79 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1027.a (trivial)

Newform invariants

comment:select newform

sage:traces = [56,-16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$60.5949615759$
Analytic rank:	$1$
Dimension:	$56$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	1027.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-5.12043 q^{2} -9.71533 q^{3} +18.2188 q^{4} -12.2172 q^{5} +49.7467 q^{6} -17.9634 q^{7} -52.3248 q^{8} +67.3877 q^{9} +62.5571 q^{10} -54.8069 q^{11} -177.002 q^{12} -13.0000 q^{13} +91.9803 q^{14} +118.694 q^{15} +122.175 q^{16} -104.614 q^{17} -345.054 q^{18} +142.317 q^{19} -222.582 q^{20} +174.520 q^{21} +280.635 q^{22} -195.988 q^{23} +508.353 q^{24} +24.2590 q^{25} +66.5656 q^{26} -392.380 q^{27} -327.272 q^{28} -70.8312 q^{29} -607.763 q^{30} -226.300 q^{31} -206.990 q^{32} +532.467 q^{33} +535.668 q^{34} +219.462 q^{35} +1227.72 q^{36} -212.055 q^{37} -728.722 q^{38} +126.299 q^{39} +639.260 q^{40} -97.0472 q^{41} -893.619 q^{42} +295.759 q^{43} -998.517 q^{44} -823.286 q^{45} +1003.55 q^{46} -103.728 q^{47} -1186.97 q^{48} -20.3169 q^{49} -124.216 q^{50} +1016.36 q^{51} -236.845 q^{52} -56.8790 q^{53} +2009.16 q^{54} +669.584 q^{55} +939.930 q^{56} -1382.65 q^{57} +362.686 q^{58} +464.587 q^{59} +2162.46 q^{60} -155.737 q^{61} +1158.75 q^{62} -1210.51 q^{63} +82.4777 q^{64} +158.823 q^{65} -2726.46 q^{66} +185.564 q^{67} -1905.94 q^{68} +1904.09 q^{69} -1123.74 q^{70} +1019.03 q^{71} -3526.05 q^{72} +876.721 q^{73} +1085.81 q^{74} -235.684 q^{75} +2592.84 q^{76} +984.517 q^{77} -646.707 q^{78} +79.0000 q^{79} -1492.63 q^{80} +1992.64 q^{81} +496.924 q^{82} +662.048 q^{83} +3179.55 q^{84} +1278.08 q^{85} -1514.41 q^{86} +688.149 q^{87} +2867.76 q^{88} +110.089 q^{89} +4215.58 q^{90} +233.524 q^{91} -3570.68 q^{92} +2198.58 q^{93} +531.133 q^{94} -1738.70 q^{95} +2010.97 q^{96} +334.855 q^{97} +104.031 q^{98} -3693.31 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 16 q^{2} - 17 q^{3} + 208 q^{4} - 35 q^{5} + 29 q^{6} - 37 q^{7} - 156 q^{8} + 415 q^{9} - 67 q^{10} - 114 q^{11} - 196 q^{12} - 728 q^{13} - 183 q^{14} - 17 q^{15} + 680 q^{16} - 383 q^{17} - 342 q^{18}+ \cdots - 11354 q^{99}+O(q^{100}) $ 56 * q - 16 * q^2 - 17 * q^3 + 208 * q^4 - 35 * q^5 + 29 * q^6 - 37 * q^7 - 156 * q^8 + 415 * q^9 - 67 * q^10 - 114 * q^11 - 196 * q^12 - 728 * q^13 - 183 * q^14 - 17 * q^15 + 680 * q^16 - 383 * q^17 - 342 * q^18 + 36 * q^19 - 384 * q^20 - 213 * q^21 - 148 * q^22 - 734 * q^23 + 21 * q^24 + 813 * q^25 + 208 * q^26 - 851 * q^27 - 1334 * q^28 - 1480 * q^29 - 1859 * q^30 - 244 * q^31 - 1039 * q^32 - 176 * q^33 - 615 * q^34 - 341 * q^35 + 269 * q^36 - 1231 * q^37 - 1603 * q^38 + 221 * q^39 - 671 * q^40 - 374 * q^41 - 2676 * q^42 - 1059 * q^43 - 2019 * q^44 - 994 * q^45 - 230 * q^46 - 951 * q^47 - 2664 * q^48 + 1843 * q^49 - 1135 * q^50 - 1855 * q^51 - 2704 * q^52 - 2892 * q^53 - 955 * q^54 - 2156 * q^55 - 2264 * q^56 - 2120 * q^57 - 616 * q^58 + 28 * q^59 - 171 * q^60 - 1290 * q^61 - 1158 * q^62 - 1092 * q^63 + 1498 * q^64 + 455 * q^65 + 304 * q^66 - 2820 * q^67 - 3642 * q^68 - 3252 * q^69 + 1144 * q^70 + 349 * q^71 - 5713 * q^72 - 890 * q^73 - 1519 * q^74 - 1954 * q^75 + 1678 * q^76 - 5768 * q^77 - 377 * q^78 + 4424 * q^79 - 2875 * q^80 + 1004 * q^81 - 384 * q^82 - 3482 * q^83 - 2502 * q^84 - 6769 * q^85 - 3806 * q^86 - 4820 * q^87 - 2304 * q^88 + 1890 * q^89 - 7524 * q^90 + 481 * q^91 - 9584 * q^92 - 4380 * q^93 - 1157 * q^94 - 6756 * q^95 + 647 * q^96 - 3052 * q^97 - 4070 * q^98 - 11354 * 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 $

$2$	−5.12043	−1.81035	−0.905173	−	0.425043i	$-0.860259\pi$
$2$	−5.12043	−1.81035	−0.905173	+	0.425043i	$0.860259\pi$
$3$	−9.71533	−1.86972	−0.934858	−	0.355021i	$-0.884474\pi$
$3$	−9.71533	−1.86972	−0.934858	+	0.355021i	$0.884474\pi$
$4$	18.2188	2.27735
$5$	−12.2172	−1.09274	−0.546368	−	0.837545i	$-0.683990\pi$
$5$	−12.2172	−1.09274	−0.546368	+	0.837545i	$0.683990\pi$
$6$	49.7467	3.38483
$7$	−17.9634	−0.969932	−0.484966	−	0.874533i	$-0.661168\pi$
$7$	−17.9634	−0.969932	−0.484966	+	0.874533i	$0.661168\pi$
$8$	−52.3248	−2.31245
$9$	67.3877	2.49584
$10$	62.5571	1.97823
$11$	−54.8069	−1.50226	−0.751132	−	0.660152i	$-0.770492\pi$
$11$	−54.8069	−1.50226	−0.751132	+	0.660152i	$0.770492\pi$
$12$	−177.002	−4.25800
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	91.9803	1.75591
$15$	118.694	2.04311
$16$	122.175	1.90898
$17$	−104.614	−1.49250	−0.746252	−	0.665663i	$-0.768149\pi$
$17$	−104.614	−1.49250	−0.746252	+	0.665663i	$0.768149\pi$
$18$	−345.054	−4.51834
$19$	142.317	1.71840	0.859201	−	0.511638i	$-0.170961\pi$
$19$	142.317	1.71840	0.859201	+	0.511638i	$0.170961\pi$
$20$	−222.582	−2.48854
$21$	174.520	1.81350
$22$	280.635	2.71962
$23$	−195.988	−1.77680	−0.888400	−	0.459070i	$-0.848183\pi$
$23$	−195.988	−1.77680	−0.888400	+	0.459070i	$0.848183\pi$
$24$	508.353	4.32363
$25$	24.2590	0.194072
$26$	66.5656	0.502100
$27$	−392.380	−2.79680
$28$	−327.272	−2.20888
$29$	−70.8312	−0.453553	−0.226776	−	0.973947i	$-0.572819\pi$
$29$	−70.8312	−0.453553	−0.226776	+	0.973947i	$0.572819\pi$
$30$	−607.763	−3.69873
$31$	−226.300	−1.31112	−0.655558	−	0.755145i	$-0.727567\pi$
$31$	−226.300	−1.31112	−0.655558	+	0.755145i	$0.727567\pi$
$32$	−206.990	−1.14347
$33$	532.467	2.80881
$34$	535.668	2.70195
$35$	219.462	1.05988
$36$	1227.72	5.68391
$37$	−212.055	−0.942205	−0.471103	−	0.882078i	$-0.656144\pi$
$37$	−212.055	−0.942205	−0.471103	+	0.882078i	$0.656144\pi$
$38$	−728.722	−3.11090
$39$	126.299	0.518566
$40$	639.260	2.52690
$41$	−97.0472	−0.369664	−0.184832	−	0.982770i	$-0.559174\pi$
$41$	−97.0472	−0.369664	−0.184832	+	0.982770i	$0.559174\pi$
$42$	−893.619	−3.28306
$43$	295.759	1.04890	0.524451	−	0.851441i	$-0.324271\pi$
$43$	295.759	1.04890	0.524451	+	0.851441i	$0.324271\pi$
$44$	−998.517	−3.42118
$45$	−823.286	−2.72730
$46$	1003.55	3.21662
$47$	−103.728	−0.321921	−0.160961	−	0.986961i	$-0.551459\pi$
$47$	−103.728	−0.321921	−0.160961	+	0.986961i	$0.551459\pi$
$48$	−1186.97	−3.56925
$49$	−20.3169	−0.0592328
$50$	−124.216	−0.351337
$51$	1016.36	2.79056
$52$	−236.845	−0.631624
$53$	−56.8790	−0.147414	−0.0737068	−	0.997280i	$-0.523483\pi$
$53$	−56.8790	−0.147414	−0.0737068	+	0.997280i	$0.523483\pi$
$54$	2009.16	5.06317
$55$	669.584	1.64158
$56$	939.930	2.24292
$57$	−1382.65	−3.21293
$58$	362.686	0.821087
$59$	464.587	1.02515	0.512577	−	0.858641i	$-0.328691\pi$
$59$	464.587	1.02515	0.512577	+	0.858641i	$0.328691\pi$
$60$	2162.46	4.65287
$61$	−155.737	−0.326887	−0.163443	−	0.986553i	$-0.552260\pi$
$61$	−155.737	−0.326887	−0.163443	+	0.986553i	$0.552260\pi$
$62$	1158.75	2.37357
$63$	−1210.51	−2.42080
$64$	82.4777	0.161089
$65$	158.823	0.303070
$66$	−2726.46	−5.08491
$67$	185.564	0.338362	0.169181	−	0.985585i	$-0.445888\pi$
$67$	185.564	0.338362	0.169181	+	0.985585i	$0.445888\pi$
$68$	−1905.94	−3.39896
$69$	1904.09	3.32211
$70$	−1123.74	−1.91875
$71$	1019.03	1.70333	0.851663	−	0.524090i	$-0.175594\pi$
$71$	1019.03	1.70333	0.851663	+	0.524090i	$0.175594\pi$
$72$	−3526.05	−5.77151
$73$	876.721	1.40565	0.702825	−	0.711363i	$-0.251922\pi$
$73$	876.721	1.40565	0.702825	+	0.711363i	$0.251922\pi$
$74$	1085.81	1.70572
$75$	−235.684	−0.362859
$76$	2592.84	3.91341
$77$	984.517	1.45709
$78$	−646.707	−0.938784
$79$	79.0000	0.112509
$79$	79.0000	0.112509
$80$	−1492.63	−2.08601
$81$	1992.64	2.73338
$82$	496.924	0.669220
$83$	662.048	0.875533	0.437766	−	0.899089i	$-0.355770\pi$
$83$	662.048	0.875533	0.437766	+	0.899089i	$0.355770\pi$
$84$	3179.55	4.12997
$85$	1278.08	1.63091
$86$	−1514.41	−1.89888
$87$	688.149	0.848015
$88$	2867.76	3.47391
$89$	110.089	0.131117	0.0655584	−	0.997849i	$-0.479117\pi$
$89$	110.089	0.131117	0.0655584	+	0.997849i	$0.479117\pi$
$90$	4215.58	4.93735
$91$	233.524	0.269011
$92$	−3570.68	−4.04640
$93$	2198.58	2.45142
$94$	531.133	0.582789
$95$	−1738.70	−1.87776
$96$	2010.97	2.13796
$97$	334.855	0.350509	0.175255	−	0.984523i	$-0.443925\pi$
$97$	334.855	0.350509	0.175255	+	0.984523i	$0.443925\pi$
$98$	104.031	0.107232
$99$	−3693.31	−3.74941

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	1027.4.a.c.1.5	✓	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1027.4.a.c.1.5	✓	56	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 1027 = 13 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1027.a (trivial)

\(f(q)\)	\(=\)	\(q-5.12043 q^{2} -9.71533 q^{3} +18.2188 q^{4} -12.2172 q^{5} +49.7467 q^{6} -17.9634 q^{7} -52.3248 q^{8} +67.3877 q^{9} +62.5571 q^{10} -54.8069 q^{11} -177.002 q^{12} -13.0000 q^{13} +91.9803 q^{14} +118.694 q^{15} +122.175 q^{16} -104.614 q^{17} -345.054 q^{18} +142.317 q^{19} -222.582 q^{20} +174.520 q^{21} +280.635 q^{22} -195.988 q^{23} +508.353 q^{24} +24.2590 q^{25} +66.5656 q^{26} -392.380 q^{27} -327.272 q^{28} -70.8312 q^{29} -607.763 q^{30} -226.300 q^{31} -206.990 q^{32} +532.467 q^{33} +535.668 q^{34} +219.462 q^{35} +1227.72 q^{36} -212.055 q^{37} -728.722 q^{38} +126.299 q^{39} +639.260 q^{40} -97.0472 q^{41} -893.619 q^{42} +295.759 q^{43} -998.517 q^{44} -823.286 q^{45} +1003.55 q^{46} -103.728 q^{47} -1186.97 q^{48} -20.3169 q^{49} -124.216 q^{50} +1016.36 q^{51} -236.845 q^{52} -56.8790 q^{53} +2009.16 q^{54} +669.584 q^{55} +939.930 q^{56} -1382.65 q^{57} +362.686 q^{58} +464.587 q^{59} +2162.46 q^{60} -155.737 q^{61} +1158.75 q^{62} -1210.51 q^{63} +82.4777 q^{64} +158.823 q^{65} -2726.46 q^{66} +185.564 q^{67} -1905.94 q^{68} +1904.09 q^{69} -1123.74 q^{70} +1019.03 q^{71} -3526.05 q^{72} +876.721 q^{73} +1085.81 q^{74} -235.684 q^{75} +2592.84 q^{76} +984.517 q^{77} -646.707 q^{78} +79.0000 q^{79} -1492.63 q^{80} +1992.64 q^{81} +496.924 q^{82} +662.048 q^{83} +3179.55 q^{84} +1278.08 q^{85} -1514.41 q^{86} +688.149 q^{87} +2867.76 q^{88} +110.089 q^{89} +4215.58 q^{90} +233.524 q^{91} -3570.68 q^{92} +2198.58 q^{93} +531.133 q^{94} -1738.70 q^{95} +2010.97 q^{96} +334.855 q^{97} +104.031 q^{98} -3693.31 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 16 q^{2} - 17 q^{3} + 208 q^{4} - 35 q^{5} + 29 q^{6} - 37 q^{7} - 156 q^{8} + 415 q^{9} - 67 q^{10} - 114 q^{11} - 196 q^{12} - 728 q^{13} - 183 q^{14} - 17 q^{15} + 680 q^{16} - 383 q^{17} - 342 q^{18}+ \cdots - 11354 q^{99}+O(q^{100}) \) 56 * q - 16 * q^2 - 17 * q^3 + 208 * q^4 - 35 * q^5 + 29 * q^6 - 37 * q^7 - 156 * q^8 + 415 * q^9 - 67 * q^10 - 114 * q^11 - 196 * q^12 - 728 * q^13 - 183 * q^14 - 17 * q^15 + 680 * q^16 - 383 * q^17 - 342 * q^18 + 36 * q^19 - 384 * q^20 - 213 * q^21 - 148 * q^22 - 734 * q^23 + 21 * q^24 + 813 * q^25 + 208 * q^26 - 851 * q^27 - 1334 * q^28 - 1480 * q^29 - 1859 * q^30 - 244 * q^31 - 1039 * q^32 - 176 * q^33 - 615 * q^34 - 341 * q^35 + 269 * q^36 - 1231 * q^37 - 1603 * q^38 + 221 * q^39 - 671 * q^40 - 374 * q^41 - 2676 * q^42 - 1059 * q^43 - 2019 * q^44 - 994 * q^45 - 230 * q^46 - 951 * q^47 - 2664 * q^48 + 1843 * q^49 - 1135 * q^50 - 1855 * q^51 - 2704 * q^52 - 2892 * q^53 - 955 * q^54 - 2156 * q^55 - 2264 * q^56 - 2120 * q^57 - 616 * q^58 + 28 * q^59 - 171 * q^60 - 1290 * q^61 - 1158 * q^62 - 1092 * q^63 + 1498 * q^64 + 455 * q^65 + 304 * q^66 - 2820 * q^67 - 3642 * q^68 - 3252 * q^69 + 1144 * q^70 + 349 * q^71 - 5713 * q^72 - 890 * q^73 - 1519 * q^74 - 1954 * q^75 + 1678 * q^76 - 5768 * q^77 - 377 * q^78 + 4424 * q^79 - 2875 * q^80 + 1004 * q^81 - 384 * q^82 - 3482 * q^83 - 2502 * q^84 - 6769 * q^85 - 3806 * q^86 - 4820 * q^87 - 2304 * q^88 + 1890 * q^89 - 7524 * q^90 + 481 * q^91 - 9584 * q^92 - 4380 * q^93 - 1157 * q^94 - 6756 * q^95 + 647 * q^96 - 3052 * q^97 - 4070 * q^98 - 11354 * q^99

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