LMFDB - Embedded newform 961.6.a.c.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [961,6,Mod(1,961)] 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("961.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(961, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 961 = 31^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	961.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,7,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$154.128850840$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 199x^{6} + 256x^{5} + 12633x^{4} - 18583x^{3} - 260319x^{2} + 410640x + 275908 $ x^8 - x^7 - 199x^6 + 256x^5 + 12633x^4 - 18583x^3 - 260319x^2 + 410640x + 275908
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2}\cdot 5\cdot 13 $
Twist minimal:	no (minimal twist has level 31)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-7.89102$ of defining polynomial
Character	$\chi$	$=$	961.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+8.89102 q^{2} -18.9313 q^{3} +47.0502 q^{4} -33.8630 q^{5} -168.319 q^{6} -134.850 q^{7} +133.811 q^{8} +115.395 q^{9} -301.077 q^{10} +191.415 q^{11} -890.722 q^{12} -766.219 q^{13} -1198.95 q^{14} +641.071 q^{15} -315.888 q^{16} -1357.39 q^{17} +1025.98 q^{18} -164.832 q^{19} -1593.26 q^{20} +2552.89 q^{21} +1701.88 q^{22} -3313.68 q^{23} -2533.22 q^{24} -1978.30 q^{25} -6812.46 q^{26} +2415.73 q^{27} -6344.71 q^{28} +6675.50 q^{29} +5699.78 q^{30} -7090.52 q^{32} -3623.74 q^{33} -12068.6 q^{34} +4566.42 q^{35} +5429.35 q^{36} -10334.6 q^{37} -1465.53 q^{38} +14505.5 q^{39} -4531.25 q^{40} +13434.9 q^{41} +22697.7 q^{42} -16907.4 q^{43} +9006.12 q^{44} -3907.62 q^{45} -29462.0 q^{46} -18714.2 q^{47} +5980.17 q^{48} +1377.48 q^{49} -17589.1 q^{50} +25697.2 q^{51} -36050.7 q^{52} -11410.8 q^{53} +21478.3 q^{54} -6481.90 q^{55} -18044.4 q^{56} +3120.50 q^{57} +59351.9 q^{58} +7475.61 q^{59} +30162.5 q^{60} +55634.9 q^{61} -15561.0 q^{63} -52933.5 q^{64} +25946.5 q^{65} -32218.8 q^{66} +26388.2 q^{67} -63865.5 q^{68} +62732.4 q^{69} +40600.1 q^{70} -14258.2 q^{71} +15441.1 q^{72} -6721.72 q^{73} -91885.2 q^{74} +37451.8 q^{75} -7755.39 q^{76} -25812.3 q^{77} +128969. q^{78} +29059.9 q^{79} +10696.9 q^{80} -73774.0 q^{81} +119450. q^{82} +37568.9 q^{83} +120114. q^{84} +45965.3 q^{85} -150324. q^{86} -126376. q^{87} +25613.5 q^{88} -91193.1 q^{89} -34742.7 q^{90} +103325. q^{91} -155909. q^{92} -166388. q^{94} +5581.72 q^{95} +134233. q^{96} +95596.4 q^{97} +12247.2 q^{98} +22088.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 7 q^{2} + 2 q^{3} + 149 q^{4} + 128 q^{5} - 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} - 574 q^{11} + 46 q^{12} + 122 q^{13} - 309 q^{14} + 524 q^{15} + 833 q^{16} - 1932 q^{17} - 6845 q^{18}+ \cdots + 180150 q^{99}+O(q^{100}) $ 8 * q + 7 * q^2 + 2 * q^3 + 149 * q^4 + 128 * q^5 - 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 - 574 * q^11 + 46 * q^12 + 122 * q^13 - 309 * q^14 + 524 * q^15 + 833 * q^16 - 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 - 996 * q^21 + 9000 * q^22 + 4136 * q^23 + 17468 * q^24 + 3308 * q^25 - 1524 * q^26 + 2624 * q^27 - 21245 * q^28 - 5050 * q^29 + 30450 * q^30 + 21045 * q^32 + 27920 * q^33 + 24738 * q^34 + 25700 * q^35 + 21637 * q^36 - 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 + 9994 * q^42 - 21058 * q^43 - 13168 * q^44 - 1344 * q^45 - 6198 * q^46 + 5348 * q^47 + 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 + 60386 * q^52 - 4926 * q^53 + 24476 * q^54 + 56892 * q^55 - 87652 * q^56 - 22072 * q^57 + 25002 * q^58 + 16944 * q^59 + 138556 * q^60 + 73682 * q^61 - 138784 * q^63 - 1300 * q^64 - 63316 * q^65 - 128796 * q^66 - 134768 * q^67 + 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 - 70792 * q^73 - 307468 * q^74 + 302902 * q^75 - 65405 * q^76 - 107932 * q^77 + 36820 * q^78 + 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 - 21882 * q^83 - 299144 * q^84 + 130464 * q^85 - 480478 * q^86 - 440244 * q^87 + 71982 * q^88 - 164392 * q^89 + 397969 * q^90 + 60028 * q^91 - 31012 * q^92 - 170388 * q^94 - 7404 * q^95 + 331010 * q^96 - 131948 * q^97 + 288768 * q^98 + 180150 * 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 $

$2$	8.89102	1.57172	0.785862	−	0.618402i	$-0.212220\pi$
$2$	8.89102	1.57172	0.785862	+	0.618402i	$0.212220\pi$
$3$	−18.9313	−1.21444	−0.607222	−	0.794532i	$-0.707716\pi$
$3$	−18.9313	−1.21444	−0.607222	+	0.794532i	$0.707716\pi$
$4$	47.0502	1.47032
$5$	−33.8630	−0.605760	−0.302880	−	0.953029i	$-0.597948\pi$
$5$	−33.8630	−0.605760	−0.302880	+	0.953029i	$0.597948\pi$
$6$	−168.319	−1.90877
$7$	−134.850	−1.04017	−0.520086	−	0.854114i	$-0.674100\pi$
$7$	−134.850	−1.04017	−0.520086	+	0.854114i	$0.674100\pi$
$8$	133.811	0.739209
$9$	115.395	0.474876
$10$	−301.077	−0.952088
$11$	191.415	0.476974	0.238487	−	0.971146i	$-0.423349\pi$
$11$	191.415	0.476974	0.238487	+	0.971146i	$0.423349\pi$
$12$	−890.722	−1.78562
$13$	−766.219	−1.25746	−0.628730	−	0.777623i	$-0.716425\pi$
$13$	−766.219	−1.25746	−0.628730	+	0.777623i	$0.716425\pi$
$14$	−1198.95	−1.63486
$15$	641.071	0.735662
$16$	−315.888	−0.308484
$17$	−1357.39	−1.13915	−0.569577	−	0.821938i	$-0.692893\pi$
$17$	−1357.39	−1.13915	−0.569577	+	0.821938i	$0.692893\pi$
$18$	1025.98	0.746375
$19$	−164.832	−0.104751	−0.0523755	−	0.998627i	$-0.516679\pi$
$19$	−164.832	−0.104751	−0.0523755	+	0.998627i	$0.516679\pi$
$20$	−1593.26	−0.890659
$21$	2552.89	1.26323
$22$	1701.88	0.749672
$23$	−3313.68	−1.30615	−0.653073	−	0.757295i	$-0.726520\pi$
$23$	−3313.68	−1.30615	−0.653073	+	0.757295i	$0.726520\pi$
$24$	−2533.22	−0.897729
$25$	−1978.30	−0.633055
$26$	−6812.46	−1.97638
$27$	2415.73	0.637734
$28$	−6344.71	−1.52938
$29$	6675.50	1.47397	0.736985	−	0.675909i	$-0.236249\pi$
$29$	6675.50	1.47397	0.736985	+	0.675909i	$0.236249\pi$
$30$	5699.78	1.15626
$31$	0	0
$31$	0	0
$32$	−7090.52	−1.22406
$33$	−3623.74	−0.579259
$34$	−12068.6	−1.79044
$35$	4566.42	0.630095
$36$	5429.35	0.698219
$37$	−10334.6	−1.24105	−0.620526	−	0.784186i	$-0.713081\pi$
$37$	−10334.6	−1.24105	−0.620526	+	0.784186i	$0.713081\pi$
$38$	−1465.53	−0.164640
$39$	14505.5	1.52712
$40$	−4531.25	−0.447783
$41$	13434.9	1.24817	0.624085	−	0.781356i	$-0.285472\pi$
$41$	13434.9	1.24817	0.624085	+	0.781356i	$0.285472\pi$
$42$	22697.7	1.98545
$43$	−16907.4	−1.39446	−0.697228	−	0.716850i	$-0.745583\pi$
$43$	−16907.4	−1.39446	−0.697228	+	0.716850i	$0.745583\pi$
$44$	9006.12	0.701303
$45$	−3907.62	−0.287661
$46$	−29462.0	−2.05290
$47$	−18714.2	−1.23574	−0.617869	−	0.786281i	$-0.712004\pi$
$47$	−18714.2	−1.23574	−0.617869	+	0.786281i	$0.712004\pi$
$48$	5980.17	0.374637
$49$	1377.48	0.0819589
$50$	−17589.1	−0.994988
$51$	25697.2	1.38344
$52$	−36050.7	−1.84887
$53$	−11410.8	−0.557990	−0.278995	−	0.960293i	$-0.590001\pi$
$53$	−11410.8	−0.557990	−0.278995	+	0.960293i	$0.590001\pi$
$54$	21478.3	1.00234
$55$	−6481.90	−0.288932
$56$	−18044.4	−0.768905
$57$	3120.50	0.127214
$58$	59351.9	2.31667
$59$	7475.61	0.279587	0.139793	−	0.990181i	$-0.455356\pi$
$59$	7475.61	0.279587	0.139793	+	0.990181i	$0.455356\pi$
$60$	30162.5	1.08166
$61$	55634.9	1.91436	0.957179	−	0.289498i	$-0.0934884\pi$
$61$	55634.9	1.91436	0.957179	+	0.289498i	$0.0934884\pi$
$62$	0	0
$63$	−15561.0	−0.493953
$64$	−52933.5	−1.61540
$65$	25946.5	0.761719
$66$	−32218.8	−0.910435
$67$	26388.2	0.718161	0.359081	−	0.933307i	$-0.383090\pi$
$67$	26388.2	0.718161	0.359081	+	0.933307i	$0.383090\pi$
$68$	−63865.5	−1.67492
$69$	62732.4	1.58624
$70$	40600.1	0.990335
$71$	−14258.2	−0.335674	−0.167837	−	0.985815i	$-0.553678\pi$
$71$	−14258.2	−0.335674	−0.167837	+	0.985815i	$0.553678\pi$
$72$	15441.1	0.351033
$73$	−6721.72	−0.147630	−0.0738148	−	0.997272i	$-0.523517\pi$
$73$	−6721.72	−0.147630	−0.0738148	+	0.997272i	$0.523517\pi$
$74$	−91885.2	−1.95059
$75$	37451.8	0.768810
$76$	−7755.39	−0.154017
$77$	−25812.3	−0.496135
$78$	128969.	2.40021
$79$	29059.9	0.523874	0.261937	−	0.965085i	$-0.415639\pi$
$79$	29059.9	0.523874	0.261937	+	0.965085i	$0.415639\pi$
$80$	10696.9	0.186867
$81$	−73774.0	−1.24937
$82$	119450.	1.96178
$83$	37568.9	0.598595	0.299298	−	0.954160i	$-0.403248\pi$
$83$	37568.9	0.598595	0.299298	+	0.954160i	$0.403248\pi$
$84$	120114.	1.85735
$85$	45965.3	0.690054
$86$	−150324.	−2.19170
$87$	−126376.	−1.79005
$88$	25613.5	0.352584
$89$	−91193.1	−1.22036	−0.610179	−	0.792264i	$-0.708902\pi$
$89$	−91193.1	−1.22036	−0.610179	+	0.792264i	$0.708902\pi$
$90$	−34742.7	−0.452124
$91$	103325.	1.30798
$92$	−155909.	−1.92045
$93$	0	0
$94$	−166388.	−1.94224
$95$	5581.72	0.0634540
$96$	134233.	1.48655
$97$	95596.4	1.03160	0.515801	−	0.856708i	$-0.327494\pi$
$97$	95596.4	1.03160	0.515801	+	0.856708i	$0.327494\pi$
$98$	12247.2	0.128817
$99$	22088.4	0.226504

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	961.6.a.c.1.7		8
31.30	odd	2		31.6.a.b.1.7	✓	8
93.92	even	2		279.6.a.f.1.2		8
124.123	even	2		496.6.a.h.1.3		8
155.154	odd	2		775.6.a.b.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
31.6.a.b.1.7	✓	8	31.30	odd	2
279.6.a.f.1.2		8	93.92	even	2
496.6.a.h.1.3		8	124.123	even	2
775.6.a.b.1.2		8	155.154	odd	2
961.6.a.c.1.7		8	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:	\( N \)	\(=\)	\( 961 = 31^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	961.a (trivial)

\(f(q)\)	\(=\)	\(q+8.89102 q^{2} -18.9313 q^{3} +47.0502 q^{4} -33.8630 q^{5} -168.319 q^{6} -134.850 q^{7} +133.811 q^{8} +115.395 q^{9} -301.077 q^{10} +191.415 q^{11} -890.722 q^{12} -766.219 q^{13} -1198.95 q^{14} +641.071 q^{15} -315.888 q^{16} -1357.39 q^{17} +1025.98 q^{18} -164.832 q^{19} -1593.26 q^{20} +2552.89 q^{21} +1701.88 q^{22} -3313.68 q^{23} -2533.22 q^{24} -1978.30 q^{25} -6812.46 q^{26} +2415.73 q^{27} -6344.71 q^{28} +6675.50 q^{29} +5699.78 q^{30} -7090.52 q^{32} -3623.74 q^{33} -12068.6 q^{34} +4566.42 q^{35} +5429.35 q^{36} -10334.6 q^{37} -1465.53 q^{38} +14505.5 q^{39} -4531.25 q^{40} +13434.9 q^{41} +22697.7 q^{42} -16907.4 q^{43} +9006.12 q^{44} -3907.62 q^{45} -29462.0 q^{46} -18714.2 q^{47} +5980.17 q^{48} +1377.48 q^{49} -17589.1 q^{50} +25697.2 q^{51} -36050.7 q^{52} -11410.8 q^{53} +21478.3 q^{54} -6481.90 q^{55} -18044.4 q^{56} +3120.50 q^{57} +59351.9 q^{58} +7475.61 q^{59} +30162.5 q^{60} +55634.9 q^{61} -15561.0 q^{63} -52933.5 q^{64} +25946.5 q^{65} -32218.8 q^{66} +26388.2 q^{67} -63865.5 q^{68} +62732.4 q^{69} +40600.1 q^{70} -14258.2 q^{71} +15441.1 q^{72} -6721.72 q^{73} -91885.2 q^{74} +37451.8 q^{75} -7755.39 q^{76} -25812.3 q^{77} +128969. q^{78} +29059.9 q^{79} +10696.9 q^{80} -73774.0 q^{81} +119450. q^{82} +37568.9 q^{83} +120114. q^{84} +45965.3 q^{85} -150324. q^{86} -126376. q^{87} +25613.5 q^{88} -91193.1 q^{89} -34742.7 q^{90} +103325. q^{91} -155909. q^{92} -166388. q^{94} +5581.72 q^{95} +134233. q^{96} +95596.4 q^{97} +12247.2 q^{98} +22088.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 7 q^{2} + 2 q^{3} + 149 q^{4} + 128 q^{5} - 72 q^{6} + 88 q^{7} + 924 q^{8} + 1512 q^{9} + 1581 q^{10} - 574 q^{11} + 46 q^{12} + 122 q^{13} - 309 q^{14} + 524 q^{15} + 833 q^{16} - 1932 q^{17} - 6845 q^{18}+ \cdots + 180150 q^{99}+O(q^{100}) \) 8 * q + 7 * q^2 + 2 * q^3 + 149 * q^4 + 128 * q^5 - 72 * q^6 + 88 * q^7 + 924 * q^8 + 1512 * q^9 + 1581 * q^10 - 574 * q^11 + 46 * q^12 + 122 * q^13 - 309 * q^14 + 524 * q^15 + 833 * q^16 - 1932 * q^17 - 6845 * q^18 - 1796 * q^19 - 37 * q^20 - 996 * q^21 + 9000 * q^22 + 4136 * q^23 + 17468 * q^24 + 3308 * q^25 - 1524 * q^26 + 2624 * q^27 - 21245 * q^28 - 5050 * q^29 + 30450 * q^30 + 21045 * q^32 + 27920 * q^33 + 24738 * q^34 + 25700 * q^35 + 21637 * q^36 - 23674 * q^37 - 4289 * q^38 + 38652 * q^39 + 36606 * q^40 + 44828 * q^41 + 9994 * q^42 - 21058 * q^43 - 13168 * q^44 - 1344 * q^45 - 6198 * q^46 + 5348 * q^47 + 69588 * q^48 + 19356 * q^49 + 33242 * q^50 + 22300 * q^51 + 60386 * q^52 - 4926 * q^53 + 24476 * q^54 + 56892 * q^55 - 87652 * q^56 - 22072 * q^57 + 25002 * q^58 + 16944 * q^59 + 138556 * q^60 + 73682 * q^61 - 138784 * q^63 - 1300 * q^64 - 63316 * q^65 - 128796 * q^66 - 134768 * q^67 + 9524 * q^68 + 3992 * q^69 - 142737 * q^70 + 123724 * q^71 + 93232 * q^72 - 70792 * q^73 - 307468 * q^74 + 302902 * q^75 - 65405 * q^76 - 107932 * q^77 + 36820 * q^78 + 26036 * q^79 + 576783 * q^80 + 82528 * q^81 + 142335 * q^82 - 21882 * q^83 - 299144 * q^84 + 130464 * q^85 - 480478 * q^86 - 440244 * q^87 + 71982 * q^88 - 164392 * q^89 + 397969 * q^90 + 60028 * q^91 - 31012 * q^92 - 170388 * q^94 - 7404 * q^95 + 331010 * q^96 - 131948 * q^97 + 288768 * q^98 + 180150 * q^99

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