LMFDB - Embedded newform 1027.6.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1027,6,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, 6); 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, 6, names="a")

Level:	$ N $	$=$	$ 1027 = 13 \cdot 79 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1027.a (trivial)

Newform invariants

comment:select newform

sage:traces = [100] 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:	$164.714182948$
Analytic rank:	$0$
Dimension:	$100$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	\(q-10.8720 q^{2} +0.339250 q^{3} +86.2004 q^{4} +65.1415 q^{5} -3.68833 q^{6} -9.12082 q^{7} -589.267 q^{8} -242.885 q^{9} -708.218 q^{10} +64.6724 q^{11} +29.2435 q^{12} -169.000 q^{13} +99.1616 q^{14} +22.0993 q^{15} +3648.10 q^{16} -780.437 q^{17} +2640.65 q^{18} +633.617 q^{19} +5615.22 q^{20} -3.09424 q^{21} -703.119 q^{22} -1433.91 q^{23} -199.909 q^{24} +1118.41 q^{25} +1837.37 q^{26} -164.837 q^{27} -786.219 q^{28} -7463.13 q^{29} -240.263 q^{30} -4250.83 q^{31} -20805.6 q^{32} +21.9401 q^{33} +8484.91 q^{34} -594.144 q^{35} -20936.8 q^{36} -13420.5 q^{37} -6888.69 q^{38} -57.3333 q^{39} -38385.7 q^{40} +11517.8 q^{41} +33.6406 q^{42} -670.722 q^{43} +5574.79 q^{44} -15821.9 q^{45} +15589.5 q^{46} +1278.19 q^{47} +1237.62 q^{48} -16723.8 q^{49} -12159.3 q^{50} -264.763 q^{51} -14567.9 q^{52} +4249.90 q^{53} +1792.10 q^{54} +4212.85 q^{55} +5374.60 q^{56} +214.955 q^{57} +81139.2 q^{58} +18775.3 q^{59} +1904.97 q^{60} -29203.3 q^{61} +46215.0 q^{62} +2215.31 q^{63} +109459. q^{64} -11008.9 q^{65} -238.533 q^{66} +53910.6 q^{67} -67274.0 q^{68} -486.454 q^{69} +6459.53 q^{70} -11157.0 q^{71} +143124. q^{72} +9511.93 q^{73} +145908. q^{74} +379.421 q^{75} +54618.1 q^{76} -589.866 q^{77} +623.328 q^{78} +6241.00 q^{79} +237642. q^{80} +58965.1 q^{81} -125222. q^{82} +57635.4 q^{83} -266.725 q^{84} -50838.8 q^{85} +7292.10 q^{86} -2531.87 q^{87} -38109.3 q^{88} -50780.3 q^{89} +172015. q^{90} +1541.42 q^{91} -123603. q^{92} -1442.10 q^{93} -13896.4 q^{94} +41274.7 q^{95} -7058.30 q^{96} +83371.2 q^{97} +181821. q^{98} -15708.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 100 q + 32 q^{2} + 64 q^{3} + 1664 q^{4} + 193 q^{5} - 109 q^{6} + 304 q^{7} + 1176 q^{8} + 8698 q^{9} + 967 q^{10} + 558 q^{11} + 3056 q^{12} - 16900 q^{13} + 3555 q^{14} + 3652 q^{15} + 27080 q^{16} + 8143 q^{17}+ \cdots - 202346 q^{99}+O(q^{100}) $ 100 * q + 32 * q^2 + 64 * q^3 + 1664 * q^4 + 193 * q^5 - 109 * q^6 + 304 * q^7 + 1176 * q^8 + 8698 * q^9 + 967 * q^10 + 558 * q^11 + 3056 * q^12 - 16900 * q^13 + 3555 * q^14 + 3652 * q^15 + 27080 * q^16 + 8143 * q^17 + 7692 * q^18 - 15 * q^19 + 3648 * q^20 - 4320 * q^21 + 7608 * q^22 + 15079 * q^23 - 6159 * q^24 + 65899 * q^25 - 5408 * q^26 + 18553 * q^27 + 774 * q^28 + 40682 * q^29 + 5065 * q^30 - 11929 * q^31 + 60269 * q^32 - 18656 * q^33 + 20851 * q^34 + 34786 * q^35 + 167621 * q^36 + 58696 * q^37 + 22925 * q^38 - 10816 * q^39 + 52969 * q^40 + 11359 * q^41 + 169890 * q^42 + 53963 * q^43 + 70269 * q^44 - 13852 * q^45 + 26262 * q^46 + 22935 * q^47 + 169872 * q^48 + 262384 * q^49 + 93839 * q^50 + 90662 * q^51 - 281216 * q^52 + 198075 * q^53 + 182621 * q^54 + 98668 * q^55 + 236320 * q^56 + 130651 * q^57 + 18708 * q^58 - 35663 * q^59 + 339849 * q^60 + 174410 * q^61 + 133206 * q^62 + 179931 * q^63 + 558490 * q^64 - 32617 * q^65 + 40768 * q^66 + 183360 * q^67 + 359028 * q^68 + 160950 * q^69 + 287358 * q^70 + 81142 * q^71 + 454487 * q^72 + 166113 * q^73 + 105143 * q^74 + 155174 * q^75 + 57206 * q^76 + 455275 * q^77 + 18421 * q^78 + 624100 * q^79 + 165857 * q^80 + 930320 * q^81 + 379464 * q^82 + 261187 * q^83 - 204246 * q^84 - 154761 * q^85 + 448672 * q^86 + 61456 * q^87 + 598480 * q^88 - 297246 * q^89 + 668088 * q^90 - 51376 * q^91 + 687016 * q^92 + 261981 * q^93 + 537485 * q^94 + 652107 * q^95 + 204239 * q^96 - 224984 * q^97 + 12148 * q^98 - 202346 * 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 $

$2$	−10.8720	−1.92192	−0.960958	−	0.276693i	$-0.910761\pi$
$2$	−10.8720	−1.92192	−0.960958	+	0.276693i	$0.910761\pi$
$3$	0.339250	0.0217629	0.0108815	−	0.999941i	$-0.496536\pi$
$3$	0.339250	0.0217629	0.0108815	+	0.999941i	$0.496536\pi$
$4$	86.2004	2.69376
$5$	65.1415	1.16529	0.582643	−	0.812728i	$-0.302019\pi$
$5$	65.1415	1.16529	0.582643	+	0.812728i	$0.302019\pi$
$6$	−3.68833	−0.0418265
$7$	−9.12082	−0.0703540	−0.0351770	−	0.999381i	$-0.511200\pi$
$7$	−9.12082	−0.0703540	−0.0351770	+	0.999381i	$0.511200\pi$
$8$	−589.267	−3.25527
$9$	−242.885	−0.999526
$10$	−708.218	−2.23958
$11$	64.6724	0.161153	0.0805763	−	0.996748i	$-0.474324\pi$
$11$	64.6724	0.161153	0.0805763	+	0.996748i	$0.474324\pi$
$12$	29.2435	0.0586242
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	99.1616	0.135215
$15$	22.0993	0.0253600
$16$	3648.10	3.56260
$17$	−780.437	−0.654961	−0.327480	−	0.944858i	$-0.606200\pi$
$17$	−780.437	−0.654961	−0.327480	+	0.944858i	$0.606200\pi$
$18$	2640.65	1.92101
$19$	633.617	0.402664	0.201332	−	0.979523i	$-0.435473\pi$
$19$	633.617	0.402664	0.201332	+	0.979523i	$0.435473\pi$
$20$	5615.22	3.13900
$21$	−3.09424	−0.00153111
$22$	−703.119	−0.309722
$23$	−1433.91	−0.565200	−0.282600	−	0.959238i	$-0.591197\pi$
$23$	−1433.91	−0.565200	−0.282600	+	0.959238i	$0.591197\pi$
$24$	−199.909	−0.0708442
$25$	1118.41	0.357891
$26$	1837.37	0.533044
$27$	−164.837	−0.0435155
$28$	−786.219	−0.189517
$29$	−7463.13	−1.64788	−0.823941	−	0.566676i	$-0.808229\pi$
$29$	−7463.13	−1.64788	−0.823941	+	0.566676i	$0.808229\pi$
$30$	−240.263	−0.0487399
$31$	−4250.83	−0.794455	−0.397228	−	0.917720i	$-0.630028\pi$
$31$	−4250.83	−0.794455	−0.397228	+	0.917720i	$0.630028\pi$
$32$	−20805.6	−3.59174
$33$	21.9401	0.00350715
$34$	8484.91	1.25878
$35$	−594.144	−0.0819825
$36$	−20936.8	−2.69249
$37$	−13420.5	−1.61163	−0.805815	−	0.592168i	$-0.798272\pi$
$37$	−13420.5	−1.61163	−0.805815	+	0.592168i	$0.798272\pi$
$38$	−6888.69	−0.773887
$39$	−57.3333	−0.00603595
$40$	−38385.7	−3.79332
$41$	11517.8	1.07007	0.535033	−	0.844831i	$-0.320299\pi$
$41$	11517.8	1.07007	0.535033	+	0.844831i	$0.320299\pi$
$42$	33.6406	0.00294266
$43$	−670.722	−0.0553187	−0.0276593	−	0.999617i	$-0.508805\pi$
$43$	−670.722	−0.0553187	−0.0276593	+	0.999617i	$0.508805\pi$
$44$	5574.79	0.434107
$45$	−15821.9	−1.16473
$46$	15589.5	1.08627
$47$	1278.19	0.0844014	0.0422007	−	0.999109i	$-0.486563\pi$
$47$	1278.19	0.0844014	0.0422007	+	0.999109i	$0.486563\pi$
$48$	1237.62	0.0775325
$49$	−16723.8	−0.995050
$50$	−12159.3	−0.687836
$51$	−264.763	−0.0142539
$52$	−14567.9	−0.747115
$53$	4249.90	0.207821	0.103910	−	0.994587i	$-0.466865\pi$
$53$	4249.90	0.207821	0.103910	+	0.994587i	$0.466865\pi$
$54$	1792.10	0.0836332
$55$	4212.85	0.187789
$56$	5374.60	0.229021
$57$	214.955	0.00876315
$58$	81139.2	3.16709
$59$	18775.3	0.702193	0.351097	−	0.936339i	$-0.385809\pi$
$59$	18775.3	0.702193	0.351097	+	0.936339i	$0.385809\pi$
$60$	1904.97	0.0683139
$61$	−29203.3	−1.00486	−0.502432	−	0.864617i	$-0.667561\pi$
$61$	−29203.3	−1.00486	−0.502432	+	0.864617i	$0.667561\pi$
$62$	46215.0	1.52688
$63$	2215.31	0.0703207
$64$	109459.	3.34043
$65$	−11008.9	−0.323192
$66$	−238.533	−0.00674045
$67$	53910.6	1.46719	0.733596	−	0.679586i	$-0.237841\pi$
$67$	53910.6	1.46719	0.733596	+	0.679586i	$0.237841\pi$
$68$	−67274.0	−1.76431
$69$	−486.454	−0.0123004
$70$	6459.53	0.157564
$71$	−11157.0	−0.262666	−0.131333	−	0.991338i	$-0.541926\pi$
$71$	−11157.0	−0.262666	−0.131333	+	0.991338i	$0.541926\pi$
$72$	143124.	3.25373
$73$	9511.93	0.208911	0.104456	−	0.994530i	$-0.466690\pi$
$73$	9511.93	0.208911	0.104456	+	0.994530i	$0.466690\pi$
$74$	145908.	3.09742
$75$	379.421	0.00778875
$76$	54618.1	1.08468
$77$	−589.866	−0.0113377
$78$	623.328	0.0116006
$79$	6241.00	0.112509
$79$	6241.00	0.112509
$80$	237642.	4.15144
$81$	58965.1	0.998579
$82$	−125222.	−2.05658
$83$	57635.4	0.918320	0.459160	−	0.888353i	$-0.348150\pi$
$83$	57635.4	0.918320	0.459160	+	0.888353i	$0.348150\pi$
$84$	−266.725	−0.00412445
$85$	−50838.8	−0.763217
$86$	7292.10	0.106318
$87$	−2531.87	−0.0358627
$88$	−38109.3	−0.524595
$89$	−50780.3	−0.679549	−0.339774	−	0.940507i	$-0.610351\pi$
$89$	−50780.3	−0.679549	−0.339774	+	0.940507i	$0.610351\pi$
$90$	172015.	2.23852
$91$	1541.42	0.0195127
$92$	−123603.	−1.52251
$93$	−1442.10	−0.0172897
$94$	−13896.4	−0.162212
$95$	41274.7	0.469219
$96$	−7058.30	−0.0781668
$97$	83371.2	0.899678	0.449839	−	0.893110i	$-0.351481\pi$
$97$	83371.2	0.899678	0.449839	+	0.893110i	$0.351481\pi$
$98$	181821.	1.91240
$99$	−15708.0	−0.161076

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	1027.6.a.d.1.2	✓	100

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1027.6.a.d.1.2	✓	100	1.1	even	1	trivial

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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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1027.a (trivial)

\(f(q)\)	\(=\)	\(q-10.8720 q^{2} +0.339250 q^{3} +86.2004 q^{4} +65.1415 q^{5} -3.68833 q^{6} -9.12082 q^{7} -589.267 q^{8} -242.885 q^{9} -708.218 q^{10} +64.6724 q^{11} +29.2435 q^{12} -169.000 q^{13} +99.1616 q^{14} +22.0993 q^{15} +3648.10 q^{16} -780.437 q^{17} +2640.65 q^{18} +633.617 q^{19} +5615.22 q^{20} -3.09424 q^{21} -703.119 q^{22} -1433.91 q^{23} -199.909 q^{24} +1118.41 q^{25} +1837.37 q^{26} -164.837 q^{27} -786.219 q^{28} -7463.13 q^{29} -240.263 q^{30} -4250.83 q^{31} -20805.6 q^{32} +21.9401 q^{33} +8484.91 q^{34} -594.144 q^{35} -20936.8 q^{36} -13420.5 q^{37} -6888.69 q^{38} -57.3333 q^{39} -38385.7 q^{40} +11517.8 q^{41} +33.6406 q^{42} -670.722 q^{43} +5574.79 q^{44} -15821.9 q^{45} +15589.5 q^{46} +1278.19 q^{47} +1237.62 q^{48} -16723.8 q^{49} -12159.3 q^{50} -264.763 q^{51} -14567.9 q^{52} +4249.90 q^{53} +1792.10 q^{54} +4212.85 q^{55} +5374.60 q^{56} +214.955 q^{57} +81139.2 q^{58} +18775.3 q^{59} +1904.97 q^{60} -29203.3 q^{61} +46215.0 q^{62} +2215.31 q^{63} +109459. q^{64} -11008.9 q^{65} -238.533 q^{66} +53910.6 q^{67} -67274.0 q^{68} -486.454 q^{69} +6459.53 q^{70} -11157.0 q^{71} +143124. q^{72} +9511.93 q^{73} +145908. q^{74} +379.421 q^{75} +54618.1 q^{76} -589.866 q^{77} +623.328 q^{78} +6241.00 q^{79} +237642. q^{80} +58965.1 q^{81} -125222. q^{82} +57635.4 q^{83} -266.725 q^{84} -50838.8 q^{85} +7292.10 q^{86} -2531.87 q^{87} -38109.3 q^{88} -50780.3 q^{89} +172015. q^{90} +1541.42 q^{91} -123603. q^{92} -1442.10 q^{93} -13896.4 q^{94} +41274.7 q^{95} -7058.30 q^{96} +83371.2 q^{97} +181821. q^{98} -15708.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q + 32 q^{2} + 64 q^{3} + 1664 q^{4} + 193 q^{5} - 109 q^{6} + 304 q^{7} + 1176 q^{8} + 8698 q^{9} + 967 q^{10} + 558 q^{11} + 3056 q^{12} - 16900 q^{13} + 3555 q^{14} + 3652 q^{15} + 27080 q^{16} + 8143 q^{17}+ \cdots - 202346 q^{99}+O(q^{100}) \) 100 * q + 32 * q^2 + 64 * q^3 + 1664 * q^4 + 193 * q^5 - 109 * q^6 + 304 * q^7 + 1176 * q^8 + 8698 * q^9 + 967 * q^10 + 558 * q^11 + 3056 * q^12 - 16900 * q^13 + 3555 * q^14 + 3652 * q^15 + 27080 * q^16 + 8143 * q^17 + 7692 * q^18 - 15 * q^19 + 3648 * q^20 - 4320 * q^21 + 7608 * q^22 + 15079 * q^23 - 6159 * q^24 + 65899 * q^25 - 5408 * q^26 + 18553 * q^27 + 774 * q^28 + 40682 * q^29 + 5065 * q^30 - 11929 * q^31 + 60269 * q^32 - 18656 * q^33 + 20851 * q^34 + 34786 * q^35 + 167621 * q^36 + 58696 * q^37 + 22925 * q^38 - 10816 * q^39 + 52969 * q^40 + 11359 * q^41 + 169890 * q^42 + 53963 * q^43 + 70269 * q^44 - 13852 * q^45 + 26262 * q^46 + 22935 * q^47 + 169872 * q^48 + 262384 * q^49 + 93839 * q^50 + 90662 * q^51 - 281216 * q^52 + 198075 * q^53 + 182621 * q^54 + 98668 * q^55 + 236320 * q^56 + 130651 * q^57 + 18708 * q^58 - 35663 * q^59 + 339849 * q^60 + 174410 * q^61 + 133206 * q^62 + 179931 * q^63 + 558490 * q^64 - 32617 * q^65 + 40768 * q^66 + 183360 * q^67 + 359028 * q^68 + 160950 * q^69 + 287358 * q^70 + 81142 * q^71 + 454487 * q^72 + 166113 * q^73 + 105143 * q^74 + 155174 * q^75 + 57206 * q^76 + 455275 * q^77 + 18421 * q^78 + 624100 * q^79 + 165857 * q^80 + 930320 * q^81 + 379464 * q^82 + 261187 * q^83 - 204246 * q^84 - 154761 * q^85 + 448672 * q^86 + 61456 * q^87 + 598480 * q^88 - 297246 * q^89 + 668088 * q^90 - 51376 * q^91 + 687016 * q^92 + 261981 * q^93 + 537485 * q^94 + 652107 * q^95 + 204239 * q^96 - 224984 * q^97 + 12148 * q^98 - 202346 * q^99

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