LMFDB - Embedded newform 1027.6.a.d.1.13

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			1.13
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)

$f(q)$	$=$	\(q-8.84537 q^{2} -18.8653 q^{3} +46.2406 q^{4} +54.8723 q^{5} +166.871 q^{6} -196.721 q^{7} -125.963 q^{8} +112.901 q^{9} -485.366 q^{10} +331.415 q^{11} -872.344 q^{12} -169.000 q^{13} +1740.07 q^{14} -1035.18 q^{15} -365.506 q^{16} -612.597 q^{17} -998.650 q^{18} +407.448 q^{19} +2537.33 q^{20} +3711.20 q^{21} -2931.49 q^{22} -1240.84 q^{23} +2376.34 q^{24} -114.029 q^{25} +1494.87 q^{26} +2454.36 q^{27} -9096.48 q^{28} -3782.13 q^{29} +9156.59 q^{30} -10150.4 q^{31} +7263.87 q^{32} -6252.26 q^{33} +5418.65 q^{34} -10794.5 q^{35} +5220.60 q^{36} +1901.13 q^{37} -3604.03 q^{38} +3188.24 q^{39} -6911.90 q^{40} -17094.3 q^{41} -32826.9 q^{42} +11300.9 q^{43} +15324.8 q^{44} +6195.13 q^{45} +10975.7 q^{46} -126.597 q^{47} +6895.39 q^{48} +21892.0 q^{49} +1008.63 q^{50} +11556.8 q^{51} -7814.66 q^{52} +13918.8 q^{53} -21709.8 q^{54} +18185.5 q^{55} +24779.6 q^{56} -7686.65 q^{57} +33454.3 q^{58} +27031.2 q^{59} -47867.6 q^{60} +8798.37 q^{61} +89783.9 q^{62} -22209.9 q^{63} -52555.4 q^{64} -9273.42 q^{65} +55303.6 q^{66} +7111.24 q^{67} -28326.8 q^{68} +23408.8 q^{69} +95481.5 q^{70} -58204.8 q^{71} -14221.4 q^{72} -15968.3 q^{73} -16816.2 q^{74} +2151.19 q^{75} +18840.7 q^{76} -65196.2 q^{77} -28201.2 q^{78} +6241.00 q^{79} -20056.2 q^{80} -73737.3 q^{81} +151206. q^{82} +62227.4 q^{83} +171608. q^{84} -33614.6 q^{85} -99961.0 q^{86} +71351.1 q^{87} -41746.2 q^{88} -25318.5 q^{89} -54798.3 q^{90} +33245.8 q^{91} -57377.0 q^{92} +191490. q^{93} +1119.80 q^{94} +22357.6 q^{95} -137035. q^{96} +132647. q^{97} -193643. q^{98} +37417.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−8.84537	−1.56366	−0.781828	−	0.623494i	$-0.785712\pi$
$2$	−8.84537	−1.56366	−0.781828	+	0.623494i	$0.785712\pi$
$3$	−18.8653	−1.21021	−0.605106	−	0.796145i	$-0.706869\pi$
$3$	−18.8653	−1.21021	−0.605106	+	0.796145i	$0.706869\pi$
$4$	46.2406	1.44502
$5$	54.8723	0.981586	0.490793	−	0.871276i	$-0.336707\pi$
$5$	54.8723	0.981586	0.490793	+	0.871276i	$0.336707\pi$
$6$	166.871	1.89235
$7$	−196.721	−1.51742	−0.758708	−	0.651431i	$-0.774169\pi$
$7$	−196.721	−1.51742	−0.758708	+	0.651431i	$0.774169\pi$
$8$	−125.963	−0.695856
$9$	112.901	0.464613
$10$	−485.366	−1.53486
$11$	331.415	0.825830	0.412915	−	0.910770i	$-0.364511\pi$
$11$	331.415	0.825830	0.412915	+	0.910770i	$0.364511\pi$
$12$	−872.344	−1.74878
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	1740.07	2.37272
$15$	−1035.18	−1.18793
$16$	−365.506	−0.356939
$17$	−612.597	−0.514106	−0.257053	−	0.966397i	$-0.582751\pi$
$17$	−612.597	−0.514106	−0.257053	+	0.966397i	$0.582751\pi$
$18$	−998.650	−0.726494
$19$	407.448	0.258934	0.129467	−	0.991584i	$-0.458673\pi$
$19$	407.448	0.258934	0.129467	+	0.991584i	$0.458673\pi$
$20$	2537.33	1.41841
$21$	3711.20	1.83639
$22$	−2931.49	−1.29131
$23$	−1240.84	−0.489097	−0.244548	−	0.969637i	$-0.578640\pi$
$23$	−1240.84	−0.489097	−0.244548	+	0.969637i	$0.578640\pi$
$24$	2376.34	0.842133
$25$	−114.029	−0.0364892
$26$	1494.87	0.433680
$27$	2454.36	0.647932
$28$	−9096.48	−2.19269
$29$	−3782.13	−0.835105	−0.417553	−	0.908653i	$-0.637112\pi$
$29$	−3782.13	−0.835105	−0.417553	+	0.908653i	$0.637112\pi$
$30$	9156.59	1.85751
$31$	−10150.4	−1.89705	−0.948524	−	0.316706i	$-0.897423\pi$
$31$	−10150.4	−1.89705	−0.948524	+	0.316706i	$0.897423\pi$
$32$	7263.87	1.25399
$33$	−6252.26	−0.999430
$34$	5418.65	0.803884
$35$	−10794.5	−1.48947
$36$	5220.60	0.671374
$37$	1901.13	0.228301	0.114151	−	0.993463i	$-0.463585\pi$
$37$	1901.13	0.228301	0.114151	+	0.993463i	$0.463585\pi$
$38$	−3604.03	−0.404883
$39$	3188.24	0.335652
$40$	−6911.90	−0.683043
$41$	−17094.3	−1.58815	−0.794077	−	0.607817i	$-0.792046\pi$
$41$	−17094.3	−1.58815	−0.794077	+	0.607817i	$0.792046\pi$
$42$	−32826.9	−2.87149
$43$	11300.9	0.932059	0.466030	−	0.884769i	$-0.345684\pi$
$43$	11300.9	0.932059	0.466030	+	0.884769i	$0.345684\pi$
$44$	15324.8	1.19334
$45$	6195.13	0.456057
$46$	10975.7	0.764779
$47$	−126.597	−0.00835946	−0.00417973	−	0.999991i	$-0.501330\pi$
$47$	−126.597	−0.00835946	−0.00417973	+	0.999991i	$0.501330\pi$
$48$	6895.39	0.431972
$49$	21892.0	1.30255
$50$	1008.63	0.0570566
$51$	11556.8	0.622177
$52$	−7814.66	−0.400776
$53$	13918.8	0.680629	0.340315	−	0.940312i	$-0.389466\pi$
$53$	13918.8	0.680629	0.340315	+	0.940312i	$0.389466\pi$
$54$	−21709.8	−1.01314
$55$	18185.5	0.810623
$56$	24779.6	1.05590
$57$	−7686.65	−0.313365
$58$	33454.3	1.30582
$59$	27031.2	1.01096	0.505482	−	0.862837i	$-0.331315\pi$
$59$	27031.2	1.01096	0.505482	+	0.862837i	$0.331315\pi$
$60$	−47867.6	−1.71658
$61$	8798.37	0.302746	0.151373	−	0.988477i	$-0.451631\pi$
$61$	8798.37	0.302746	0.151373	+	0.988477i	$0.451631\pi$
$62$	89783.9	2.96633
$63$	−22209.9	−0.705011
$64$	−52555.4	−1.60386
$65$	−9273.42	−0.272243
$66$	55303.6	1.56276
$67$	7111.24	0.193535	0.0967673	−	0.995307i	$-0.469150\pi$
$67$	7111.24	0.193535	0.0967673	+	0.995307i	$0.469150\pi$
$68$	−28326.8	−0.742893
$69$	23408.8	0.591911
$70$	95481.5	2.32902
$71$	−58204.8	−1.37029	−0.685145	−	0.728406i	$-0.740261\pi$
$71$	−58204.8	−1.37029	−0.685145	+	0.728406i	$0.740261\pi$
$72$	−14221.4	−0.323304
$73$	−15968.3	−0.350712	−0.175356	−	0.984505i	$-0.556108\pi$
$73$	−15968.3	−0.350712	−0.175356	+	0.984505i	$0.556108\pi$
$74$	−16816.2	−0.356984
$75$	2151.19	0.0441597
$76$	18840.7	0.374164
$77$	−65196.2	−1.25313
$78$	−28201.2	−0.524845
$79$	6241.00	0.112509
$79$	6241.00	0.112509
$80$	−20056.2	−0.350367
$81$	−73737.3	−1.24875
$82$	151206.	2.48333
$83$	62227.4	0.991485	0.495743	−	0.868469i	$-0.334896\pi$
$83$	62227.4	0.991485	0.495743	+	0.868469i	$0.334896\pi$
$84$	171608.	2.65362
$85$	−33614.6	−0.504639
$86$	−99961.0	−1.45742
$87$	71351.1	1.01065
$88$	−41746.2	−0.574659
$89$	−25318.5	−0.338815	−0.169407	−	0.985546i	$-0.554185\pi$
$89$	−25318.5	−0.338815	−0.169407	+	0.985546i	$0.554185\pi$
$90$	−54798.3	−0.713116
$91$	33245.8	0.420855
$92$	−57377.0	−0.706754
$93$	191490.	2.29583
$94$	1119.80	0.0130713
$95$	22357.6	0.254166
$96$	−137035.	−1.51759
$97$	132647.	1.43143	0.715713	−	0.698395i	$-0.246102\pi$
$97$	132647.	1.43143	0.715713	+	0.698395i	$0.246102\pi$
$98$	−193643.	−2.03674
$99$	37417.1	0.383691

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.1.13	✓	100

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1027.6.a.d.1.13	✓	100	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q-8.84537 q^{2} -18.8653 q^{3} +46.2406 q^{4} +54.8723 q^{5} +166.871 q^{6} -196.721 q^{7} -125.963 q^{8} +112.901 q^{9} -485.366 q^{10} +331.415 q^{11} -872.344 q^{12} -169.000 q^{13} +1740.07 q^{14} -1035.18 q^{15} -365.506 q^{16} -612.597 q^{17} -998.650 q^{18} +407.448 q^{19} +2537.33 q^{20} +3711.20 q^{21} -2931.49 q^{22} -1240.84 q^{23} +2376.34 q^{24} -114.029 q^{25} +1494.87 q^{26} +2454.36 q^{27} -9096.48 q^{28} -3782.13 q^{29} +9156.59 q^{30} -10150.4 q^{31} +7263.87 q^{32} -6252.26 q^{33} +5418.65 q^{34} -10794.5 q^{35} +5220.60 q^{36} +1901.13 q^{37} -3604.03 q^{38} +3188.24 q^{39} -6911.90 q^{40} -17094.3 q^{41} -32826.9 q^{42} +11300.9 q^{43} +15324.8 q^{44} +6195.13 q^{45} +10975.7 q^{46} -126.597 q^{47} +6895.39 q^{48} +21892.0 q^{49} +1008.63 q^{50} +11556.8 q^{51} -7814.66 q^{52} +13918.8 q^{53} -21709.8 q^{54} +18185.5 q^{55} +24779.6 q^{56} -7686.65 q^{57} +33454.3 q^{58} +27031.2 q^{59} -47867.6 q^{60} +8798.37 q^{61} +89783.9 q^{62} -22209.9 q^{63} -52555.4 q^{64} -9273.42 q^{65} +55303.6 q^{66} +7111.24 q^{67} -28326.8 q^{68} +23408.8 q^{69} +95481.5 q^{70} -58204.8 q^{71} -14221.4 q^{72} -15968.3 q^{73} -16816.2 q^{74} +2151.19 q^{75} +18840.7 q^{76} -65196.2 q^{77} -28201.2 q^{78} +6241.00 q^{79} -20056.2 q^{80} -73737.3 q^{81} +151206. q^{82} +62227.4 q^{83} +171608. q^{84} -33614.6 q^{85} -99961.0 q^{86} +71351.1 q^{87} -41746.2 q^{88} -25318.5 q^{89} -54798.3 q^{90} +33245.8 q^{91} -57377.0 q^{92} +191490. q^{93} +1119.80 q^{94} +22357.6 q^{95} -137035. q^{96} +132647. q^{97} -193643. q^{98} +37417.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 100 q + 32 q^{2} + 64 q^{3} + 1664 q^{4} + 193 q^{5} - 109 q^{6} + 304 q^{7} + 1176 q^{8} + 8698 q^{9} + 967 q^{10} + 558 q^{11} + 3056 q^{12} - 16900 q^{13} + 3555 q^{14} + 3652 q^{15} + 27080 q^{16} + 8143 q^{17}+ \cdots - 202346 q^{99}+O(q^{100}) \) 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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