LMFDB - Embedded newform 158.6.a.b.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [158,6,Mod(1,158)] 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("158.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(158, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 158 = 2 \cdot 79 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	158.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,24] 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:	$25.3406435305$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} - 517x^{4} + 294x^{3} + 75132x^{2} + 2488x - 3309088 $ x^6 - 3x^5 - 517x^4 + 294x^3 + 75132x^2 + 2488*x - 3309088
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$10.5956$ of defining polynomial
Character	$\chi$	$=$	158.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+4.00000 q^{2} -22.7897 q^{3} +16.0000 q^{4} +26.1459 q^{5} -91.1588 q^{6} -22.0965 q^{7} +64.0000 q^{8} +276.371 q^{9} +104.584 q^{10} -249.053 q^{11} -364.635 q^{12} +776.896 q^{13} -88.3861 q^{14} -595.857 q^{15} +256.000 q^{16} +31.5001 q^{17} +1105.48 q^{18} -1028.56 q^{19} +418.334 q^{20} +503.574 q^{21} -996.213 q^{22} -3705.75 q^{23} -1458.54 q^{24} -2441.39 q^{25} +3107.59 q^{26} -760.509 q^{27} -353.545 q^{28} +622.712 q^{29} -2383.43 q^{30} -6440.79 q^{31} +1024.00 q^{32} +5675.85 q^{33} +126.001 q^{34} -577.734 q^{35} +4421.93 q^{36} -7990.38 q^{37} -4114.25 q^{38} -17705.2 q^{39} +1673.34 q^{40} -1130.28 q^{41} +2014.29 q^{42} -13395.5 q^{43} -3984.85 q^{44} +7225.96 q^{45} -14823.0 q^{46} -9771.71 q^{47} -5834.16 q^{48} -16318.7 q^{49} -9765.57 q^{50} -717.879 q^{51} +12430.3 q^{52} -13779.8 q^{53} -3042.04 q^{54} -6511.72 q^{55} -1414.18 q^{56} +23440.6 q^{57} +2490.85 q^{58} -651.135 q^{59} -9533.72 q^{60} +8903.08 q^{61} -25763.2 q^{62} -6106.84 q^{63} +4096.00 q^{64} +20312.7 q^{65} +22703.4 q^{66} +42401.8 q^{67} +504.002 q^{68} +84453.0 q^{69} -2310.94 q^{70} +55578.6 q^{71} +17687.7 q^{72} +25226.4 q^{73} -31961.5 q^{74} +55638.6 q^{75} -16457.0 q^{76} +5503.22 q^{77} -70821.0 q^{78} +6241.00 q^{79} +6693.35 q^{80} -49826.3 q^{81} -4521.11 q^{82} -62278.3 q^{83} +8057.18 q^{84} +823.600 q^{85} -53582.0 q^{86} -14191.4 q^{87} -15939.4 q^{88} -26647.4 q^{89} +28903.8 q^{90} -17166.7 q^{91} -59292.0 q^{92} +146784. q^{93} -39086.8 q^{94} -26892.7 q^{95} -23336.7 q^{96} +116661. q^{97} -65275.0 q^{98} -68831.1 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 24 q^{2} - 36 q^{3} + 96 q^{4} - 94 q^{5} - 144 q^{6} - 194 q^{7} + 384 q^{8} + 516 q^{9} - 376 q^{10} - 258 q^{11} - 576 q^{12} - 2370 q^{13} - 776 q^{14} - 516 q^{15} + 1536 q^{16} + 862 q^{17}+ \cdots + 255952 q^{99}+O(q^{100}) $ 6 * q + 24 * q^2 - 36 * q^3 + 96 * q^4 - 94 * q^5 - 144 * q^6 - 194 * q^7 + 384 * q^8 + 516 * q^9 - 376 * q^10 - 258 * q^11 - 576 * q^12 - 2370 * q^13 - 776 * q^14 - 516 * q^15 + 1536 * q^16 + 862 * q^17 + 2064 * q^18 - 3222 * q^19 - 1504 * q^20 - 10176 * q^21 - 1032 * q^22 - 12906 * q^23 - 2304 * q^24 - 4518 * q^25 - 9480 * q^26 - 11562 * q^27 - 3104 * q^28 - 3400 * q^29 - 2064 * q^30 - 6372 * q^31 + 6144 * q^32 - 11168 * q^33 + 3448 * q^34 - 4224 * q^35 + 8256 * q^36 + 4468 * q^37 - 12888 * q^38 + 3002 * q^39 - 6016 * q^40 - 45328 * q^41 - 40704 * q^42 - 10068 * q^43 - 4128 * q^44 - 54726 * q^45 - 51624 * q^46 - 58924 * q^47 - 9216 * q^48 + 35244 * q^49 - 18072 * q^50 + 6264 * q^51 - 37920 * q^52 + 7126 * q^53 - 46248 * q^54 - 43708 * q^55 - 12416 * q^56 + 56796 * q^57 - 13600 * q^58 - 32414 * q^59 - 8256 * q^60 + 36390 * q^61 - 25488 * q^62 + 68214 * q^63 + 24576 * q^64 + 70502 * q^65 - 44672 * q^66 - 37880 * q^67 + 13792 * q^68 - 4224 * q^69 - 16896 * q^70 - 34396 * q^71 + 33024 * q^72 + 70966 * q^73 + 17872 * q^74 + 161060 * q^75 - 51552 * q^76 + 116222 * q^77 + 12008 * q^78 + 37446 * q^79 - 24064 * q^80 + 291474 * q^81 - 181312 * q^82 - 9330 * q^83 - 162816 * q^84 - 72620 * q^85 - 40272 * q^86 + 369600 * q^87 - 16512 * q^88 + 194380 * q^89 - 218904 * q^90 - 51630 * q^91 - 206496 * q^92 - 190574 * q^93 - 235696 * q^94 + 164272 * q^95 - 36864 * q^96 - 93248 * q^97 + 140976 * q^98 + 255952 * 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 $

$2$	4.00000	0.707107
$2$	4.00000	0.707107
$3$	−22.7897	−1.46196	−0.730980	−	0.682399i	$-0.760937\pi$
$3$	−22.7897	−1.46196	−0.730980	+	0.682399i	$0.760937\pi$
$4$	16.0000	0.500000
$5$	26.1459	0.467712	0.233856	−	0.972271i	$-0.424866\pi$
$5$	26.1459	0.467712	0.233856	+	0.972271i	$0.424866\pi$
$6$	−91.1588	−1.03376
$7$	−22.0965	−0.170443	−0.0852215	−	0.996362i	$-0.527160\pi$
$7$	−22.0965	−0.170443	−0.0852215	+	0.996362i	$0.527160\pi$
$8$	64.0000	0.353553
$9$	276.371	1.13733
$10$	104.584	0.330722
$11$	−249.053	−0.620598	−0.310299	−	0.950639i	$-0.600429\pi$
$11$	−249.053	−0.620598	−0.310299	+	0.950639i	$0.600429\pi$
$12$	−364.635	−0.730980
$13$	776.896	1.27498	0.637492	−	0.770457i	$-0.279972\pi$
$13$	776.896	1.27498	0.637492	+	0.770457i	$0.279972\pi$
$14$	−88.3861	−0.120521
$15$	−595.857	−0.683777
$16$	256.000	0.250000
$17$	31.5001	0.0264357	0.0132178	−	0.999913i	$-0.495793\pi$
$17$	31.5001	0.0264357	0.0132178	+	0.999913i	$0.495793\pi$
$18$	1105.48	0.804212
$19$	−1028.56	−0.653652	−0.326826	−	0.945085i	$-0.605979\pi$
$19$	−1028.56	−0.653652	−0.326826	+	0.945085i	$0.605979\pi$
$20$	418.334	0.233856
$21$	503.574	0.249181
$22$	−996.213	−0.438829
$23$	−3705.75	−1.46069	−0.730343	−	0.683081i	$-0.760640\pi$
$23$	−3705.75	−1.46069	−0.730343	+	0.683081i	$0.760640\pi$
$24$	−1458.54	−0.516881
$25$	−2441.39	−0.781245
$26$	3107.59	0.901550
$27$	−760.509	−0.200768
$28$	−353.545	−0.0852215
$29$	622.712	0.137497	0.0687483	−	0.997634i	$-0.478099\pi$
$29$	622.712	0.137497	0.0687483	+	0.997634i	$0.478099\pi$
$30$	−2383.43	−0.483503
$31$	−6440.79	−1.20375	−0.601873	−	0.798592i	$-0.705579\pi$
$31$	−6440.79	−1.20375	−0.601873	+	0.798592i	$0.705579\pi$
$32$	1024.00	0.176777
$33$	5675.85	0.907290
$34$	126.001	0.0186928
$35$	−577.734	−0.0797182
$36$	4421.93	0.568664
$37$	−7990.38	−0.959540	−0.479770	−	0.877394i	$-0.659280\pi$
$37$	−7990.38	−0.959540	−0.479770	+	0.877394i	$0.659280\pi$
$38$	−4114.25	−0.462202
$39$	−17705.2	−1.86398
$40$	1673.34	0.165361
$41$	−1130.28	−0.105009	−0.0525044	−	0.998621i	$-0.516720\pi$
$41$	−1130.28	−0.105009	−0.0525044	+	0.998621i	$0.516720\pi$
$42$	2014.29	0.176197
$43$	−13395.5	−1.10481	−0.552405	−	0.833576i	$-0.686290\pi$
$43$	−13395.5	−1.10481	−0.552405	+	0.833576i	$0.686290\pi$
$44$	−3984.85	−0.310299
$45$	7225.96	0.531942
$46$	−14823.0	−1.03286
$47$	−9771.71	−0.645247	−0.322624	−	0.946527i	$-0.604565\pi$
$47$	−9771.71	−0.645247	−0.322624	+	0.946527i	$0.604565\pi$
$48$	−5834.16	−0.365490
$49$	−16318.7	−0.970949
$50$	−9765.57	−0.552424
$51$	−717.879	−0.0386479
$52$	12430.3	0.637492
$53$	−13779.8	−0.673835	−0.336918	−	0.941534i	$-0.609384\pi$
$53$	−13779.8	−0.673835	−0.336918	+	0.941534i	$0.609384\pi$
$54$	−3042.04	−0.141965
$55$	−6511.72	−0.290261
$56$	−1414.18	−0.0602607
$57$	23440.6	0.955613
$58$	2490.85	0.0972248
$59$	−651.135	−0.0243523	−0.0121762	−	0.999926i	$-0.503876\pi$
$59$	−651.135	−0.0243523	−0.0121762	+	0.999926i	$0.503876\pi$
$60$	−9533.72	−0.341888
$61$	8903.08	0.306348	0.153174	−	0.988199i	$-0.451050\pi$
$61$	8903.08	0.306348	0.153174	+	0.988199i	$0.451050\pi$
$62$	−25763.2	−0.851177
$63$	−6106.84	−0.193850
$64$	4096.00	0.125000
$65$	20312.7	0.596325
$66$	22703.4	0.641551
$67$	42401.8	1.15398	0.576988	−	0.816752i	$-0.304228\pi$
$67$	42401.8	1.15398	0.576988	+	0.816752i	$0.304228\pi$
$68$	504.002	0.0132178
$69$	84453.0	2.13546
$70$	−2310.94	−0.0563693
$71$	55578.6	1.30846	0.654232	−	0.756294i	$-0.272992\pi$
$71$	55578.6	1.30846	0.654232	+	0.756294i	$0.272992\pi$
$72$	17687.7	0.402106
$73$	25226.4	0.554050	0.277025	−	0.960863i	$-0.410652\pi$
$73$	25226.4	0.554050	0.277025	+	0.960863i	$0.410652\pi$
$74$	−31961.5	−0.678497
$75$	55638.6	1.14215
$76$	−16457.0	−0.326826
$77$	5503.22	0.105777
$78$	−70821.0	−1.31803
$79$	6241.00	0.112509
$79$	6241.00	0.112509
$80$	6693.35	0.116928
$81$	−49826.3	−0.843813
$82$	−4521.11	−0.0742524
$83$	−62278.3	−0.992297	−0.496148	−	0.868238i	$-0.665253\pi$
$83$	−62278.3	−0.992297	−0.496148	+	0.868238i	$0.665253\pi$
$84$	8057.18	0.124590
$85$	823.600	0.0123643
$86$	−53582.0	−0.781219
$87$	−14191.4	−0.201015
$88$	−15939.4	−0.219415
$89$	−26647.4	−0.356598	−0.178299	−	0.983976i	$-0.557059\pi$
$89$	−26647.4	−0.356598	−0.178299	+	0.983976i	$0.557059\pi$
$90$	28903.8	0.376140
$91$	−17166.7	−0.217312
$92$	−59292.0	−0.730343
$93$	146784.	1.75983
$94$	−39086.8	−0.456259
$95$	−26892.7	−0.305721
$96$	−23336.7	−0.258441
$97$	116661.	1.25891	0.629457	−	0.777035i	$-0.283277\pi$
$97$	116661.	1.25891	0.629457	+	0.777035i	$0.283277\pi$
$98$	−65275.0	−0.686565
$99$	−68831.1	−0.705824

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	158.6.a.b.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
158.6.a.b.1.2	✓	6	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 158 = 2 \cdot 79 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	158.a (trivial)

\(f(q)\)	\(=\)	\(q+4.00000 q^{2} -22.7897 q^{3} +16.0000 q^{4} +26.1459 q^{5} -91.1588 q^{6} -22.0965 q^{7} +64.0000 q^{8} +276.371 q^{9} +104.584 q^{10} -249.053 q^{11} -364.635 q^{12} +776.896 q^{13} -88.3861 q^{14} -595.857 q^{15} +256.000 q^{16} +31.5001 q^{17} +1105.48 q^{18} -1028.56 q^{19} +418.334 q^{20} +503.574 q^{21} -996.213 q^{22} -3705.75 q^{23} -1458.54 q^{24} -2441.39 q^{25} +3107.59 q^{26} -760.509 q^{27} -353.545 q^{28} +622.712 q^{29} -2383.43 q^{30} -6440.79 q^{31} +1024.00 q^{32} +5675.85 q^{33} +126.001 q^{34} -577.734 q^{35} +4421.93 q^{36} -7990.38 q^{37} -4114.25 q^{38} -17705.2 q^{39} +1673.34 q^{40} -1130.28 q^{41} +2014.29 q^{42} -13395.5 q^{43} -3984.85 q^{44} +7225.96 q^{45} -14823.0 q^{46} -9771.71 q^{47} -5834.16 q^{48} -16318.7 q^{49} -9765.57 q^{50} -717.879 q^{51} +12430.3 q^{52} -13779.8 q^{53} -3042.04 q^{54} -6511.72 q^{55} -1414.18 q^{56} +23440.6 q^{57} +2490.85 q^{58} -651.135 q^{59} -9533.72 q^{60} +8903.08 q^{61} -25763.2 q^{62} -6106.84 q^{63} +4096.00 q^{64} +20312.7 q^{65} +22703.4 q^{66} +42401.8 q^{67} +504.002 q^{68} +84453.0 q^{69} -2310.94 q^{70} +55578.6 q^{71} +17687.7 q^{72} +25226.4 q^{73} -31961.5 q^{74} +55638.6 q^{75} -16457.0 q^{76} +5503.22 q^{77} -70821.0 q^{78} +6241.00 q^{79} +6693.35 q^{80} -49826.3 q^{81} -4521.11 q^{82} -62278.3 q^{83} +8057.18 q^{84} +823.600 q^{85} -53582.0 q^{86} -14191.4 q^{87} -15939.4 q^{88} -26647.4 q^{89} +28903.8 q^{90} -17166.7 q^{91} -59292.0 q^{92} +146784. q^{93} -39086.8 q^{94} -26892.7 q^{95} -23336.7 q^{96} +116661. q^{97} -65275.0 q^{98} -68831.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 24 q^{2} - 36 q^{3} + 96 q^{4} - 94 q^{5} - 144 q^{6} - 194 q^{7} + 384 q^{8} + 516 q^{9} - 376 q^{10} - 258 q^{11} - 576 q^{12} - 2370 q^{13} - 776 q^{14} - 516 q^{15} + 1536 q^{16} + 862 q^{17}+ \cdots + 255952 q^{99}+O(q^{100}) \) 6 * q + 24 * q^2 - 36 * q^3 + 96 * q^4 - 94 * q^5 - 144 * q^6 - 194 * q^7 + 384 * q^8 + 516 * q^9 - 376 * q^10 - 258 * q^11 - 576 * q^12 - 2370 * q^13 - 776 * q^14 - 516 * q^15 + 1536 * q^16 + 862 * q^17 + 2064 * q^18 - 3222 * q^19 - 1504 * q^20 - 10176 * q^21 - 1032 * q^22 - 12906 * q^23 - 2304 * q^24 - 4518 * q^25 - 9480 * q^26 - 11562 * q^27 - 3104 * q^28 - 3400 * q^29 - 2064 * q^30 - 6372 * q^31 + 6144 * q^32 - 11168 * q^33 + 3448 * q^34 - 4224 * q^35 + 8256 * q^36 + 4468 * q^37 - 12888 * q^38 + 3002 * q^39 - 6016 * q^40 - 45328 * q^41 - 40704 * q^42 - 10068 * q^43 - 4128 * q^44 - 54726 * q^45 - 51624 * q^46 - 58924 * q^47 - 9216 * q^48 + 35244 * q^49 - 18072 * q^50 + 6264 * q^51 - 37920 * q^52 + 7126 * q^53 - 46248 * q^54 - 43708 * q^55 - 12416 * q^56 + 56796 * q^57 - 13600 * q^58 - 32414 * q^59 - 8256 * q^60 + 36390 * q^61 - 25488 * q^62 + 68214 * q^63 + 24576 * q^64 + 70502 * q^65 - 44672 * q^66 - 37880 * q^67 + 13792 * q^68 - 4224 * q^69 - 16896 * q^70 - 34396 * q^71 + 33024 * q^72 + 70966 * q^73 + 17872 * q^74 + 161060 * q^75 - 51552 * q^76 + 116222 * q^77 + 12008 * q^78 + 37446 * q^79 - 24064 * q^80 + 291474 * q^81 - 181312 * q^82 - 9330 * q^83 - 162816 * q^84 - 72620 * q^85 - 40272 * q^86 + 369600 * q^87 - 16512 * q^88 + 194380 * q^89 - 218904 * q^90 - 51630 * q^91 - 206496 * q^92 - 190574 * q^93 - 235696 * q^94 + 164272 * q^95 - 36864 * q^96 - 93248 * q^97 + 140976 * q^98 + 255952 * q^99

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