LMFDB - Embedded newform 8047.2.a.b.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8047,2,Mod(1,8047)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8047, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8047.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8047 = 13 \cdot 619 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8047.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$64.2556185065$
Analytic rank:	$1$
Dimension:	$142$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8047.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.45771 q^{2} +2.98686 q^{3} +4.04034 q^{4} -1.52748 q^{5} -7.34083 q^{6} +5.18217 q^{7} -5.01456 q^{8} +5.92132 q^{9} +O(q^{10})$	\(q-2.45771 q^{2} +2.98686 q^{3} +4.04034 q^{4} -1.52748 q^{5} -7.34083 q^{6} +5.18217 q^{7} -5.01456 q^{8} +5.92132 q^{9} +3.75410 q^{10} -3.38966 q^{11} +12.0679 q^{12} +1.00000 q^{13} -12.7363 q^{14} -4.56236 q^{15} +4.24365 q^{16} -6.38035 q^{17} -14.5529 q^{18} -0.342245 q^{19} -6.17152 q^{20} +15.4784 q^{21} +8.33080 q^{22} -2.04112 q^{23} -14.9778 q^{24} -2.66681 q^{25} -2.45771 q^{26} +8.72558 q^{27} +20.9377 q^{28} -10.3843 q^{29} +11.2130 q^{30} +0.408753 q^{31} -0.400547 q^{32} -10.1244 q^{33} +15.6810 q^{34} -7.91565 q^{35} +23.9241 q^{36} -6.11794 q^{37} +0.841139 q^{38} +2.98686 q^{39} +7.65962 q^{40} +7.78956 q^{41} -38.0415 q^{42} -5.88168 q^{43} -13.6954 q^{44} -9.04469 q^{45} +5.01647 q^{46} -7.49849 q^{47} +12.6752 q^{48} +19.8549 q^{49} +6.55425 q^{50} -19.0572 q^{51} +4.04034 q^{52} +5.42105 q^{53} -21.4449 q^{54} +5.17763 q^{55} -25.9863 q^{56} -1.02224 q^{57} +25.5216 q^{58} -11.6026 q^{59} -18.4335 q^{60} -3.06923 q^{61} -1.00460 q^{62} +30.6853 q^{63} -7.50287 q^{64} -1.52748 q^{65} +24.8829 q^{66} +12.7491 q^{67} -25.7788 q^{68} -6.09652 q^{69} +19.4544 q^{70} +4.94261 q^{71} -29.6928 q^{72} -5.49144 q^{73} +15.0361 q^{74} -7.96539 q^{75} -1.38279 q^{76} -17.5658 q^{77} -7.34083 q^{78} +3.51488 q^{79} -6.48208 q^{80} +8.29811 q^{81} -19.1445 q^{82} +10.9590 q^{83} +62.5380 q^{84} +9.74584 q^{85} +14.4555 q^{86} -31.0165 q^{87} +16.9976 q^{88} -7.93955 q^{89} +22.2292 q^{90} +5.18217 q^{91} -8.24679 q^{92} +1.22089 q^{93} +18.4291 q^{94} +0.522772 q^{95} -1.19638 q^{96} -14.5818 q^{97} -48.7976 q^{98} -20.0713 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) $ 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See 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 $

$2$	−2.45771	−1.73786	−0.868932	−	0.494932i	$-0.835193\pi$
$2$	−2.45771	−1.73786	−0.868932	+	0.494932i	$0.835193\pi$
$3$	2.98686	1.72446	0.862232	−	0.506514i	$-0.169066\pi$
$3$	2.98686	1.72446	0.862232	+	0.506514i	$0.169066\pi$
$4$	4.04034	2.02017
$5$	−1.52748	−0.683109	−0.341554	−	0.939862i	$-0.610953\pi$
$5$	−1.52748	−0.683109	−0.341554	+	0.939862i	$0.610953\pi$
$6$	−7.34083	−2.99688
$7$	5.18217	1.95868	0.979339	−	0.202227i	$-0.0648178\pi$
$7$	5.18217	1.95868	0.979339	+	0.202227i	$0.0648178\pi$
$8$	−5.01456	−1.77291
$9$	5.92132	1.97377
$10$	3.75410	1.18715
$11$	−3.38966	−1.02202	−0.511010	−	0.859575i	$-0.670729\pi$
$11$	−3.38966	−1.02202	−0.511010	+	0.859575i	$0.670729\pi$
$12$	12.0679	3.48371
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−12.7363	−3.40391
$15$	−4.56236	−1.17800
$16$	4.24365	1.06091
$17$	−6.38035	−1.54746	−0.773731	−	0.633514i	$-0.781612\pi$
$17$	−6.38035	−1.54746	−0.773731	+	0.633514i	$0.781612\pi$
$18$	−14.5529	−3.43015
$19$	−0.342245	−0.0785164	−0.0392582	−	0.999229i	$-0.512499\pi$
$19$	−0.342245	−0.0785164	−0.0392582	+	0.999229i	$0.512499\pi$
$20$	−6.17152	−1.37999
$21$	15.4784	3.37767
$22$	8.33080	1.77613
$23$	−2.04112	−0.425602	−0.212801	−	0.977096i	$-0.568259\pi$
$23$	−2.04112	−0.425602	−0.212801	+	0.977096i	$0.568259\pi$
$24$	−14.9778	−3.05732
$25$	−2.66681	−0.533362
$26$	−2.45771	−0.481997
$27$	8.72558	1.67924
$28$	20.9377	3.95686
$29$	−10.3843	−1.92832	−0.964159	−	0.265325i	$-0.914521\pi$
$29$	−10.3843	−1.92832	−0.964159	+	0.265325i	$0.914521\pi$
$30$	11.2130	2.04720
$31$	0.408753	0.0734142	0.0367071	−	0.999326i	$-0.488313\pi$
$31$	0.408753	0.0734142	0.0367071	+	0.999326i	$0.488313\pi$
$32$	−0.400547	−0.0708073
$33$	−10.1244	−1.76244
$34$	15.6810	2.68928
$35$	−7.91565	−1.33799
$36$	23.9241	3.98736
$37$	−6.11794	−1.00578	−0.502891	−	0.864350i	$-0.667730\pi$
$37$	−6.11794	−1.00578	−0.502891	+	0.864350i	$0.667730\pi$
$38$	0.841139	0.136451
$39$	2.98686	0.478280
$40$	7.65962	1.21109
$41$	7.78956	1.21652	0.608262	−	0.793736i	$-0.291867\pi$
$41$	7.78956	1.21652	0.608262	+	0.793736i	$0.291867\pi$
$42$	−38.0415	−5.86992
$43$	−5.88168	−0.896947	−0.448474	−	0.893796i	$-0.648032\pi$
$43$	−5.88168	−0.896947	−0.448474	+	0.893796i	$0.648032\pi$
$44$	−13.6954	−2.06465
$45$	−9.04469	−1.34830
$46$	5.01647	0.739638
$47$	−7.49849	−1.09377	−0.546883	−	0.837209i	$-0.684186\pi$
$47$	−7.49849	−1.09377	−0.546883	+	0.837209i	$0.684186\pi$
$48$	12.6752	1.82950
$49$	19.8549	2.83642
$50$	6.55425	0.926911
$51$	−19.0572	−2.66854
$52$	4.04034	0.560294
$53$	5.42105	0.744638	0.372319	−	0.928105i	$-0.378563\pi$
$53$	5.42105	0.744638	0.372319	+	0.928105i	$0.378563\pi$
$54$	−21.4449	−2.91829
$55$	5.17763	0.698151
$56$	−25.9863	−3.47257
$57$	−1.02224	−0.135399
$58$	25.5216	3.35115
$59$	−11.6026	−1.51053	−0.755267	−	0.655417i	$-0.772493\pi$
$59$	−11.6026	−1.51053	−0.755267	+	0.655417i	$0.772493\pi$
$60$	−18.4335	−2.37975
$61$	−3.06923	−0.392974	−0.196487	−	0.980506i	$-0.562953\pi$
$61$	−3.06923	−0.392974	−0.196487	+	0.980506i	$0.562953\pi$
$62$	−1.00460	−0.127584
$63$	30.6853	3.86599
$64$	−7.50287	−0.937859
$65$	−1.52748	−0.189460
$66$	24.8829	3.06287
$67$	12.7491	1.55755	0.778777	−	0.627301i	$-0.215840\pi$
$67$	12.7491	1.55755	0.778777	+	0.627301i	$0.215840\pi$
$68$	−25.7788	−3.12613
$69$	−6.09652	−0.733935
$70$	19.4544	2.32524
$71$	4.94261	0.586580	0.293290	−	0.956023i	$-0.405250\pi$
$71$	4.94261	0.586580	0.293290	+	0.956023i	$0.405250\pi$
$72$	−29.6928	−3.49933
$73$	−5.49144	−0.642724	−0.321362	−	0.946956i	$-0.604141\pi$
$73$	−5.49144	−0.642724	−0.321362	+	0.946956i	$0.604141\pi$
$74$	15.0361	1.74791
$75$	−7.96539	−0.919764
$76$	−1.38279	−0.158616
$77$	−17.5658	−2.00181
$78$	−7.34083	−0.831185
$79$	3.51488	0.395455	0.197728	−	0.980257i	$-0.436644\pi$
$79$	3.51488	0.395455	0.197728	+	0.980257i	$0.436644\pi$
$80$	−6.48208	−0.724719
$81$	8.29811	0.922012
$82$	−19.1445	−2.11415
$83$	10.9590	1.20290	0.601452	−	0.798909i	$-0.294589\pi$
$83$	10.9590	1.20290	0.601452	+	0.798909i	$0.294589\pi$
$84$	62.5380	6.82346
$85$	9.74584	1.05708
$86$	14.4555	1.55877
$87$	−31.0165	−3.32531
$88$	16.9976	1.81195
$89$	−7.93955	−0.841591	−0.420795	−	0.907156i	$-0.638249\pi$
$89$	−7.93955	−0.841591	−0.420795	+	0.907156i	$0.638249\pi$
$90$	22.2292	2.34317
$91$	5.18217	0.543239
$92$	−8.24679	−0.859788
$93$	1.22089	0.126600
$94$	18.4291	1.90082
$95$	0.522772	0.0536352
$96$	−1.19638	−0.122105
$97$	−14.5818	−1.48056	−0.740278	−	0.672301i	$-0.765306\pi$
$97$	−14.5818	−1.48056	−0.740278	+	0.672301i	$0.765306\pi$
$98$	−48.7976	−4.92931
$99$	−20.0713	−2.01724
$100$	−10.7748	−1.07748
$101$	−4.97015	−0.494548	−0.247274	−	0.968946i	$-0.579535\pi$
$101$	−4.97015	−0.494548	−0.247274	+	0.968946i	$0.579535\pi$
$102$	46.8371	4.63756
$103$	−3.95903	−0.390095	−0.195047	−	0.980794i	$-0.562486\pi$
$103$	−3.95903	−0.390095	−0.195047	+	0.980794i	$0.562486\pi$
$104$	−5.01456	−0.491718
$105$	−23.6429	−2.30731
$106$	−13.3234	−1.29408
$107$	−8.86480	−0.856993	−0.428496	−	0.903544i	$-0.640957\pi$
$107$	−8.86480	−0.856993	−0.428496	+	0.903544i	$0.640957\pi$
$108$	35.2543	3.39235
$109$	−2.94574	−0.282151	−0.141075	−	0.989999i	$-0.545056\pi$
$109$	−2.94574	−0.282151	−0.141075	+	0.989999i	$0.545056\pi$
$110$	−12.7251	−1.21329
$111$	−18.2734	−1.73444
$112$	21.9913	2.07799
$113$	−0.152845	−0.0143784	−0.00718922	−	0.999974i	$-0.502288\pi$
$113$	−0.152845	−0.0143784	−0.00718922	+	0.999974i	$0.502288\pi$
$114$	2.51236	0.235304
$115$	3.11776	0.290732
$116$	−41.9561	−3.89553
$117$	5.92132	0.547427
$118$	28.5159	2.62510
$119$	−33.0641	−3.03098
$120$	22.8782	2.08849
$121$	0.489785	0.0445259
$122$	7.54327	0.682936
$123$	23.2663	2.09785
$124$	1.65150	0.148309
$125$	11.7109	1.04745
$126$	−75.4156	−6.71856
$127$	−10.9358	−0.970395	−0.485198	−	0.874405i	$-0.661252\pi$
$127$	−10.9358	−0.970395	−0.485198	+	0.874405i	$0.661252\pi$
$128$	19.2410	1.70068
$129$	−17.5677	−1.54675
$130$	3.75410	0.329256
$131$	−2.20261	−0.192443	−0.0962217	−	0.995360i	$-0.530676\pi$
$131$	−2.20261	−0.192443	−0.0962217	+	0.995360i	$0.530676\pi$
$132$	−40.9061	−3.56042
$133$	−1.77357	−0.153788
$134$	−31.3337	−2.70682
$135$	−13.3281	−1.14710
$136$	31.9946	2.74352
$137$	12.0345	1.02817	0.514087	−	0.857738i	$-0.328131\pi$
$137$	12.0345	1.02817	0.514087	+	0.857738i	$0.328131\pi$
$138$	14.9835	1.27548
$139$	−17.2365	−1.46198	−0.730991	−	0.682387i	$-0.760942\pi$
$139$	−17.2365	−1.46198	−0.730991	+	0.682387i	$0.760942\pi$
$140$	−31.9819	−2.70296
$141$	−22.3969	−1.88616
$142$	−12.1475	−1.01940
$143$	−3.38966	−0.283457
$144$	25.1280	2.09400
$145$	15.8618	1.31725
$146$	13.4964	1.11697
$147$	59.3038	4.89130
$148$	−24.7185	−2.03185
$149$	−0.979981	−0.0802832	−0.0401416	−	0.999194i	$-0.512781\pi$
$149$	−0.979981	−0.0802832	−0.0401416	+	0.999194i	$0.512781\pi$
$150$	19.5766	1.59842
$151$	17.6666	1.43769	0.718844	−	0.695171i	$-0.244672\pi$
$151$	17.6666	1.43769	0.718844	+	0.695171i	$0.244672\pi$
$152$	1.71621	0.139203
$153$	−37.7801	−3.05434
$154$	43.1716	3.47887
$155$	−0.624361	−0.0501499
$156$	12.0679	0.966207
$157$	−21.5511	−1.71996	−0.859982	−	0.510324i	$-0.829525\pi$
$157$	−21.5511	−1.71996	−0.859982	+	0.510324i	$0.829525\pi$
$158$	−8.63856	−0.687247
$159$	16.1919	1.28410
$160$	0.611826	0.0483691
$161$	−10.5774	−0.833617
$162$	−20.3943	−1.60233
$163$	−17.7714	−1.39196	−0.695980	−	0.718061i	$-0.745030\pi$
$163$	−17.7714	−1.39196	−0.695980	+	0.718061i	$0.745030\pi$
$164$	31.4724	2.45759
$165$	15.4648	1.20394
$166$	−26.9340	−2.09048
$167$	−9.21042	−0.712724	−0.356362	−	0.934348i	$-0.615983\pi$
$167$	−9.21042	−0.712724	−0.356362	+	0.934348i	$0.615983\pi$
$168$	−77.6174	−5.98831
$169$	1.00000	0.0769231
$170$	−23.9525	−1.83707
$171$	−2.02654	−0.154974
$172$	−23.7640	−1.81198
$173$	13.5281	1.02852	0.514262	−	0.857633i	$-0.328066\pi$
$173$	13.5281	1.02852	0.514262	+	0.857633i	$0.328066\pi$
$174$	76.2295	5.77894
$175$	−13.8199	−1.04468
$176$	−14.3845	−1.08427
$177$	−34.6554	−2.60486
$178$	19.5131	1.46257
$179$	−3.82914	−0.286204	−0.143102	−	0.989708i	$-0.545708\pi$
$179$	−3.82914	−0.286204	−0.143102	+	0.989708i	$0.545708\pi$
$180$	−36.5436	−2.72380
$181$	−17.0122	−1.26451	−0.632253	−	0.774762i	$-0.717870\pi$
$181$	−17.0122	−1.26451	−0.632253	+	0.774762i	$0.717870\pi$
$182$	−12.7363	−0.944076
$183$	−9.16735	−0.677670
$184$	10.2353	0.754555
$185$	9.34501	0.687059
$186$	−3.00059	−0.220014
$187$	21.6272	1.58154
$188$	−30.2964	−2.20959
$189$	45.2175	3.28909
$190$	−1.28482	−0.0932107
$191$	20.3595	1.47316	0.736582	−	0.676348i	$-0.236438\pi$
$191$	20.3595	1.47316	0.736582	+	0.676348i	$0.236438\pi$
$192$	−22.4100	−1.61730
$193$	−14.7942	−1.06491	−0.532456	−	0.846458i	$-0.678731\pi$
$193$	−14.7942	−1.06491	−0.532456	+	0.846458i	$0.678731\pi$
$194$	35.8378	2.57300
$195$	−4.56236	−0.326717
$196$	80.2206	5.73004
$197$	15.9375	1.13550	0.567749	−	0.823202i	$-0.307814\pi$
$197$	15.9375	1.13550	0.567749	+	0.823202i	$0.307814\pi$
$198$	49.3293	3.50568
$199$	12.9588	0.918628	0.459314	−	0.888274i	$-0.348095\pi$
$199$	12.9588	0.918628	0.459314	+	0.888274i	$0.348095\pi$
$200$	13.3729	0.945605
$201$	38.0798	2.68595
$202$	12.2152	0.859457
$203$	−53.8133	−3.77695
$204$	−76.9975	−5.39091
$205$	−11.8984	−0.831019
$206$	9.73014	0.677931
$207$	−12.0861	−0.840042
$208$	4.24365	0.294244
$209$	1.16009	0.0802454
$210$	58.1075	4.00980
$211$	9.28866	0.639458	0.319729	−	0.947509i	$-0.396408\pi$
$211$	9.28866	0.639458	0.319729	+	0.947509i	$0.396408\pi$
$212$	21.9029	1.50429
$213$	14.7629	1.01154
$214$	21.7871	1.48934
$215$	8.98413	0.612713
$216$	−43.7549	−2.97715
$217$	2.11823	0.143795
$218$	7.23977	0.490339
$219$	−16.4021	−1.10835
$220$	20.9194	1.41038
$221$	−6.38035	−0.429189
$222$	44.9108	3.01421
$223$	16.0288	1.07337	0.536685	−	0.843783i	$-0.319676\pi$
$223$	16.0288	1.07337	0.536685	+	0.843783i	$0.319676\pi$
$224$	−2.07570	−0.138689
$225$	−15.7911	−1.05274
$226$	0.375648	0.0249878
$227$	16.1569	1.07237	0.536185	−	0.844101i	$-0.319865\pi$
$227$	16.1569	1.07237	0.536185	+	0.844101i	$0.319865\pi$
$228$	−4.13018	−0.273528
$229$	16.6606	1.10096	0.550482	−	0.834847i	$-0.314444\pi$
$229$	16.6606	1.10096	0.550482	+	0.834847i	$0.314444\pi$
$230$	−7.66254	−0.505253
$231$	−52.4666	−3.45205
$232$	52.0727	3.41874
$233$	3.65358	0.239354	0.119677	−	0.992813i	$-0.461814\pi$
$233$	3.65358	0.239354	0.119677	+	0.992813i	$0.461814\pi$
$234$	−14.5529	−0.951353
$235$	11.4538	0.747162
$236$	−46.8786	−3.05153
$237$	10.4985	0.681948
$238$	81.2619	5.26743
$239$	−14.6949	−0.950532	−0.475266	−	0.879842i	$-0.657648\pi$
$239$	−14.6949	−0.950532	−0.475266	+	0.879842i	$0.657648\pi$
$240$	−19.3611	−1.24975
$241$	−9.73367	−0.627000	−0.313500	−	0.949588i	$-0.601502\pi$
$241$	−9.73367	−0.627000	−0.313500	+	0.949588i	$0.601502\pi$
$242$	−1.20375	−0.0773800
$243$	−1.39147	−0.0892629
$244$	−12.4007	−0.793875
$245$	−30.3279	−1.93758
$246$	−57.1818	−3.64578
$247$	−0.342245	−0.0217765
$248$	−2.04972	−0.130157
$249$	32.7329	2.07436
$250$	−28.7820	−1.82033
$251$	11.9580	0.754782	0.377391	−	0.926054i	$-0.376821\pi$
$251$	11.9580	0.754782	0.377391	+	0.926054i	$0.376821\pi$
$252$	123.979	7.80995
$253$	6.91868	0.434974
$254$	26.8770	1.68641
$255$	29.1095	1.82290
$256$	−32.2830	−2.01769
$257$	−8.02412	−0.500531	−0.250265	−	0.968177i	$-0.580518\pi$
$257$	−8.02412	−0.500531	−0.250265	+	0.968177i	$0.580518\pi$
$258$	43.1764	2.68805
$259$	−31.7042	−1.97000
$260$	−6.17152	−0.382742
$261$	−61.4889	−3.80607
$262$	5.41339	0.334440
$263$	−24.6066	−1.51731	−0.758655	−	0.651493i	$-0.774143\pi$
$263$	−24.6066	−1.51731	−0.758655	+	0.651493i	$0.774143\pi$
$264$	50.7695	3.12465
$265$	−8.28053	−0.508669
$266$	4.35893	0.267263
$267$	−23.7143	−1.45129
$268$	51.5108	3.14652
$269$	4.39893	0.268207	0.134104	−	0.990967i	$-0.457184\pi$
$269$	4.39893	0.268207	0.134104	+	0.990967i	$0.457184\pi$
$270$	32.7567	1.99351
$271$	−21.8040	−1.32450	−0.662249	−	0.749284i	$-0.730398\pi$
$271$	−21.8040	−1.32450	−0.662249	+	0.749284i	$0.730398\pi$
$272$	−27.0760	−1.64172
$273$	15.4784	0.936797
$274$	−29.5772	−1.78682
$275$	9.03958	0.545107
$276$	−24.6320	−1.48267
$277$	−15.6534	−0.940522	−0.470261	−	0.882527i	$-0.655840\pi$
$277$	−15.6534	−0.940522	−0.470261	+	0.882527i	$0.655840\pi$
$278$	42.3623	2.54072
$279$	2.42036	0.144903
$280$	39.6935	2.37214
$281$	23.6332	1.40984	0.704920	−	0.709287i	$-0.250983\pi$
$281$	23.6332	1.40984	0.704920	+	0.709287i	$0.250983\pi$
$282$	55.0451	3.27789
$283$	−6.48779	−0.385659	−0.192830	−	0.981232i	$-0.561766\pi$
$283$	−6.48779	−0.385659	−0.192830	+	0.981232i	$0.561766\pi$
$284$	19.9698	1.18499
$285$	1.56144	0.0924920
$286$	8.33080	0.492610
$287$	40.3669	2.38278
$288$	−2.37177	−0.139758
$289$	23.7089	1.39464
$290$	−38.9837	−2.28920
$291$	−43.5537	−2.55317
$292$	−22.1873	−1.29841
$293$	−0.736764	−0.0430422	−0.0215211	−	0.999768i	$-0.506851\pi$
$293$	−0.736764	−0.0430422	−0.0215211	+	0.999768i	$0.506851\pi$
$294$	−145.752	−8.50041
$295$	17.7228	1.03186
$296$	30.6788	1.78317
$297$	−29.5767	−1.71622
$298$	2.40851	0.139521
$299$	−2.04112	−0.118041
$300$	−32.1829	−1.85808
$301$	−30.4799	−1.75683
$302$	−43.4194	−2.49851
$303$	−14.8451	−0.852830
$304$	−1.45237	−0.0832990
$305$	4.68818	0.268444
$306$	92.8526	5.30803
$307$	28.2791	1.61397	0.806986	−	0.590571i	$-0.201097\pi$
$307$	28.2791	1.61397	0.806986	+	0.590571i	$0.201097\pi$
$308$	−70.9717	−4.04399
$309$	−11.8251	−0.672704
$310$	1.53450	0.0871537
$311$	−32.5481	−1.84563	−0.922817	−	0.385239i	$-0.874119\pi$
$311$	−32.5481	−1.84563	−0.922817	+	0.385239i	$0.874119\pi$
$312$	−14.9778	−0.847949
$313$	4.91397	0.277754	0.138877	−	0.990310i	$-0.455651\pi$
$313$	4.91397	0.277754	0.138877	+	0.990310i	$0.455651\pi$
$314$	52.9663	2.98906
$315$	−46.8712	−2.64089
$316$	14.2013	0.798886
$317$	−16.3378	−0.917622	−0.458811	−	0.888534i	$-0.651725\pi$
$317$	−16.3378	−0.917622	−0.458811	+	0.888534i	$0.651725\pi$
$318$	−39.7950	−2.23159
$319$	35.1993	1.97078
$320$	11.4605	0.640660
$321$	−26.4779	−1.47785
$322$	25.9962	1.44871
$323$	2.18364	0.121501
$324$	33.5272	1.86262
$325$	−2.66681	−0.147928
$326$	43.6768	2.41904
$327$	−8.79851	−0.486559
$328$	−39.0612	−2.15679
$329$	−38.8585	−2.14234
$330$	−38.0081	−2.09228
$331$	1.98972	0.109365	0.0546825	−	0.998504i	$-0.482585\pi$
$331$	1.98972	0.109365	0.0546825	+	0.998504i	$0.482585\pi$
$332$	44.2779	2.43007
$333$	−36.2263	−1.98519
$334$	22.6365	1.23862
$335$	−19.4740	−1.06398
$336$	65.6850	3.58341
$337$	−21.7932	−1.18715	−0.593577	−	0.804777i	$-0.702285\pi$
$337$	−21.7932	−1.18715	−0.593577	+	0.804777i	$0.702285\pi$
$338$	−2.45771	−0.133682
$339$	−0.456526	−0.0247951
$340$	39.3765	2.13549
$341$	−1.38553	−0.0750308
$342$	4.98066	0.269323
$343$	66.6164	3.59695
$344$	29.4940	1.59021
$345$	9.31230	0.501357
$346$	−33.2482	−1.78743
$347$	30.1582	1.61898	0.809488	−	0.587136i	$-0.199745\pi$
$347$	30.1582	1.61898	0.809488	+	0.587136i	$0.199745\pi$
$348$	−125.317	−6.71770
$349$	15.6396	0.837166	0.418583	−	0.908178i	$-0.362527\pi$
$349$	15.6396	0.837166	0.418583	+	0.908178i	$0.362527\pi$
$350$	33.9653	1.81552
$351$	8.72558	0.465737
$352$	1.35772	0.0723666
$353$	−14.3140	−0.761855	−0.380928	−	0.924605i	$-0.624395\pi$
$353$	−14.3140	−0.761855	−0.380928	+	0.924605i	$0.624395\pi$
$354$	85.1730	4.52689
$355$	−7.54973	−0.400698
$356$	−32.0785	−1.70016
$357$	−98.7577	−5.22681
$358$	9.41092	0.497383
$359$	−25.7071	−1.35677	−0.678383	−	0.734708i	$-0.737319\pi$
$359$	−25.7071	−1.35677	−0.678383	+	0.734708i	$0.737319\pi$
$360$	45.3551	2.39042
$361$	−18.8829	−0.993835
$362$	41.8110	2.19754
$363$	1.46292	0.0767833
$364$	20.9377	1.09744
$365$	8.38805	0.439051
$366$	22.5307	1.17770
$367$	−4.35062	−0.227100	−0.113550	−	0.993532i	$-0.536222\pi$
$367$	−4.35062	−0.227100	−0.113550	+	0.993532i	$0.536222\pi$
$368$	−8.66178	−0.451526
$369$	46.1245	2.40115
$370$	−22.9673	−1.19401
$371$	28.0928	1.45851
$372$	4.93280	0.255754
$373$	14.6009	0.756003	0.378002	−	0.925805i	$-0.376611\pi$
$373$	14.6009	0.756003	0.378002	+	0.925805i	$0.376611\pi$
$374$	−53.1534	−2.74850
$375$	34.9788	1.80630
$376$	37.6016	1.93915
$377$	−10.3843	−0.534819
$378$	−111.131	−5.71598
$379$	5.91846	0.304011	0.152005	−	0.988380i	$-0.451427\pi$
$379$	5.91846	0.304011	0.152005	+	0.988380i	$0.451427\pi$
$380$	2.11217	0.108352
$381$	−32.6637	−1.67341
$382$	−50.0378	−2.56016
$383$	−35.6472	−1.82148	−0.910742	−	0.412975i	$-0.864490\pi$
$383$	−35.6472	−1.82148	−0.910742	+	0.412975i	$0.864490\pi$
$384$	57.4701	2.93276
$385$	26.8314	1.36745
$386$	36.3599	1.85067
$387$	−34.8273	−1.77037
$388$	−58.9153	−2.99097
$389$	10.4391	0.529286	0.264643	−	0.964347i	$-0.414746\pi$
$389$	10.4391	0.529286	0.264643	+	0.964347i	$0.414746\pi$
$390$	11.2130	0.567790
$391$	13.0230	0.658603
$392$	−99.5636	−5.02872
$393$	−6.57890	−0.331862
$394$	−39.1697	−1.97334
$395$	−5.36890	−0.270139
$396$	−81.0947	−4.07516
$397$	−3.04317	−0.152732	−0.0763661	−	0.997080i	$-0.524332\pi$
$397$	−3.04317	−0.152732	−0.0763661	+	0.997080i	$0.524332\pi$
$398$	−31.8491	−1.59645
$399$	−5.29741	−0.265202
$400$	−11.3170	−0.565851
$401$	14.2306	0.710643	0.355321	−	0.934744i	$-0.384371\pi$
$401$	14.2306	0.710643	0.355321	+	0.934744i	$0.384371\pi$
$402$	−93.5892	−4.66781
$403$	0.408753	0.0203614
$404$	−20.0811	−0.999071
$405$	−12.6752	−0.629834
$406$	132.257	6.56383
$407$	20.7377	1.02793
$408$	95.5634	4.73109
$409$	−20.9710	−1.03695	−0.518475	−	0.855093i	$-0.673500\pi$
$409$	−20.9710	−1.03695	−0.518475	+	0.855093i	$0.673500\pi$
$410$	29.2428	1.44420
$411$	35.9452	1.77305
$412$	−15.9958	−0.788057
$413$	−60.1269	−2.95865
$414$	29.7041	1.45988
$415$	−16.7396	−0.821714
$416$	−0.400547	−0.0196384
$417$	−51.4830	−2.52113
$418$	−2.85117	−0.139455
$419$	−19.5540	−0.955273	−0.477637	−	0.878558i	$-0.658506\pi$
$419$	−19.5540	−0.955273	−0.477637	+	0.878558i	$0.658506\pi$
$420$	−95.5254	−4.66116
$421$	3.20576	0.156239	0.0781196	−	0.996944i	$-0.475108\pi$
$421$	3.20576	0.156239	0.0781196	+	0.996944i	$0.475108\pi$
$422$	−22.8288	−1.11129
$423$	−44.4010	−2.15885
$424$	−27.1841	−1.32018
$425$	17.0152	0.825358
$426$	−36.2829	−1.75791
$427$	−15.9053	−0.769710
$428$	−35.8168	−1.73127
$429$	−10.1244	−0.488812
$430$	−22.0804	−1.06481
$431$	−8.66242	−0.417254	−0.208627	−	0.977995i	$-0.566899\pi$
$431$	−8.66242	−0.417254	−0.208627	+	0.977995i	$0.566899\pi$
$432$	37.0283	1.78153
$433$	−25.1018	−1.20632	−0.603158	−	0.797622i	$-0.706091\pi$
$433$	−25.1018	−1.20632	−0.603158	+	0.797622i	$0.706091\pi$
$434$	−5.20599	−0.249896
$435$	47.3770	2.27155
$436$	−11.9018	−0.569992
$437$	0.698561	0.0334167
$438$	40.3117	1.92617
$439$	−18.3193	−0.874333	−0.437167	−	0.899381i	$-0.644018\pi$
$439$	−18.3193	−0.874333	−0.437167	+	0.899381i	$0.644018\pi$
$440$	−25.9635	−1.23776
$441$	117.567	5.59845
$442$	15.6810	0.745871
$443$	38.1649	1.81327	0.906636	−	0.421914i	$-0.138642\pi$
$443$	38.1649	1.81327	0.906636	+	0.421914i	$0.138642\pi$
$444$	−73.8308	−3.50385
$445$	12.1275	0.574898
$446$	−39.3942	−1.86537
$447$	−2.92707	−0.138445
$448$	−38.8812	−1.83696
$449$	5.16433	0.243720	0.121860	−	0.992547i	$-0.461114\pi$
$449$	5.16433	0.243720	0.121860	+	0.992547i	$0.461114\pi$
$450$	38.8098	1.82951
$451$	−26.4039	−1.24331
$452$	−0.617545	−0.0290469
$453$	52.7677	2.47924
$454$	−39.7089	−1.86363
$455$	−7.91565	−0.371092
$456$	5.12607	0.240050
$457$	27.3878	1.28115	0.640574	−	0.767896i	$-0.278696\pi$
$457$	27.3878	1.28115	0.640574	+	0.767896i	$0.278696\pi$
$458$	−40.9470	−1.91333
$459$	−55.6723	−2.59856
$460$	12.5968	0.587329
$461$	25.5838	1.19155	0.595777	−	0.803150i	$-0.296844\pi$
$461$	25.5838	1.19155	0.595777	+	0.803150i	$0.296844\pi$
$462$	128.948	5.99918
$463$	32.2289	1.49781	0.748903	−	0.662680i	$-0.230581\pi$
$463$	32.2289	1.49781	0.748903	+	0.662680i	$0.230581\pi$
$464$	−44.0674	−2.04578
$465$	−1.86488	−0.0864817
$466$	−8.97943	−0.415964
$467$	27.7214	1.28279	0.641396	−	0.767210i	$-0.278356\pi$
$467$	27.7214	1.28279	0.641396	+	0.767210i	$0.278356\pi$
$468$	23.9241	1.10589
$469$	66.0682	3.05075
$470$	−28.1500	−1.29846
$471$	−64.3700	−2.96602
$472$	58.1821	2.67805
$473$	19.9369	0.916699
$474$	−25.8022	−1.18513
$475$	0.912703	0.0418777
$476$	−133.590	−6.12309
$477$	32.0998	1.46975
$478$	36.1157	1.65189
$479$	4.46868	0.204179	0.102090	−	0.994775i	$-0.467447\pi$
$479$	4.46868	0.204179	0.102090	+	0.994775i	$0.467447\pi$
$480$	1.82744	0.0834108
$481$	−6.11794	−0.278954
$482$	23.9225	1.08964
$483$	−31.5932	−1.43754
$484$	1.97890	0.0899499
$485$	22.2734	1.01138
$486$	3.41983	0.155127
$487$	39.7406	1.80082	0.900408	−	0.435046i	$-0.143268\pi$
$487$	39.7406	1.80082	0.900408	+	0.435046i	$0.143268\pi$
$488$	15.3908	0.696710
$489$	−53.0805	−2.40038
$490$	74.5373	3.36725
$491$	30.1692	1.36152	0.680758	−	0.732509i	$-0.261651\pi$
$491$	30.1692	1.36152	0.680758	+	0.732509i	$0.261651\pi$
$492$	94.0038	4.23802
$493$	66.2555	2.98400
$494$	0.841139	0.0378446
$495$	30.6584	1.37799
$496$	1.73461	0.0778861
$497$	25.6135	1.14892
$498$	−80.4480	−3.60496
$499$	22.9322	1.02658	0.513292	−	0.858214i	$-0.328426\pi$
$499$	22.9322	1.02658	0.513292	+	0.858214i	$0.328426\pi$
$500$	47.3159	2.11603
$501$	−27.5102	−1.22907
$502$	−29.3893	−1.31171
$503$	−11.0690	−0.493542	−0.246771	−	0.969074i	$-0.579370\pi$
$503$	−11.0690	−0.493542	−0.246771	+	0.969074i	$0.579370\pi$
$504$	−153.873	−6.85406
$505$	7.59179	0.337830
$506$	−17.0041	−0.755925
$507$	2.98686	0.132651
$508$	−44.1843	−1.96036
$509$	−22.0427	−0.977026	−0.488513	−	0.872557i	$-0.662461\pi$
$509$	−22.0427	−0.977026	−0.488513	+	0.872557i	$0.662461\pi$
$510$	−71.5426	−3.16796
$511$	−28.4576	−1.25889
$512$	40.8603	1.80579
$513$	−2.98629	−0.131848
$514$	19.7209	0.869854
$515$	6.04733	0.266477
$516$	−70.9796	−3.12470
$517$	25.4173	1.11785
$518$	77.9198	3.42360
$519$	40.4066	1.77365
$520$	7.65962	0.335897
$521$	−7.27689	−0.318806	−0.159403	−	0.987214i	$-0.550957\pi$
$521$	−7.27689	−0.318806	−0.159403	+	0.987214i	$0.550957\pi$
$522$	151.122	6.61442
$523$	13.0643	0.571261	0.285631	−	0.958340i	$-0.407797\pi$
$523$	13.0643	0.571261	0.285631	+	0.958340i	$0.407797\pi$
$524$	−8.89931	−0.388768
$525$	−41.2780	−1.80152
$526$	60.4759	2.63688
$527$	−2.60799	−0.113606
$528$	−42.9645	−1.86979
$529$	−18.8338	−0.818863
$530$	20.3511	0.883997
$531$	−68.7030	−2.98145
$532$	−7.16583	−0.310678
$533$	7.78956	0.337403
$534$	58.2829	2.52215
$535$	13.5408	0.585419
$536$	−63.9312	−2.76141
$537$	−11.4371	−0.493548
$538$	−10.8113	−0.466108
$539$	−67.3014	−2.89888
$540$	−53.8501	−2.31734
$541$	34.6307	1.48889	0.744446	−	0.667683i	$-0.232714\pi$
$541$	34.6307	1.48889	0.744446	+	0.667683i	$0.232714\pi$
$542$	53.5878	2.30179
$543$	−50.8130	−2.18059
$544$	2.55563	0.109572
$545$	4.49955	0.192740
$546$	−38.0415	−1.62802
$547$	31.6921	1.35506	0.677529	−	0.735496i	$-0.263051\pi$
$547$	31.6921	1.35506	0.677529	+	0.735496i	$0.263051\pi$
$548$	48.6233	2.07708
$549$	−18.1739	−0.775643
$550$	−22.2167	−0.947322
$551$	3.55398	0.151405
$552$	30.5714	1.30120
$553$	18.2147	0.774569
$554$	38.4715	1.63450
$555$	27.9122	1.18481
$556$	−69.6413	−2.95345
$557$	−17.6442	−0.747609	−0.373805	−	0.927507i	$-0.621947\pi$
$557$	−17.6442	−0.747609	−0.373805	+	0.927507i	$0.621947\pi$
$558$	−5.94854	−0.251822
$559$	−5.88168	−0.248768
$560$	−33.5913	−1.41949
$561$	64.5974	2.72731
$562$	−58.0836	−2.45011
$563$	42.5493	1.79324	0.896619	−	0.442804i	$-0.146016\pi$
$563$	42.5493	1.79324	0.896619	+	0.442804i	$0.146016\pi$
$564$	−90.4911	−3.81036
$565$	0.233467	0.00982204
$566$	15.9451	0.670223
$567$	43.0022	1.80592
$568$	−24.7850	−1.03996
$569$	−0.473941	−0.0198686	−0.00993432	−	0.999951i	$-0.503162\pi$
$569$	−0.473941	−0.0198686	−0.00993432	+	0.999951i	$0.503162\pi$
$570$	−3.83758	−0.160738
$571$	0.385004	0.0161119	0.00805596	−	0.999968i	$-0.497436\pi$
$571$	0.385004	0.0161119	0.00805596	+	0.999968i	$0.497436\pi$
$572$	−13.6954	−0.572632
$573$	60.8111	2.54042
$574$	−99.2100	−4.14095
$575$	5.44327	0.227000
$576$	−44.4269	−1.85112
$577$	13.8601	0.577004	0.288502	−	0.957479i	$-0.406843\pi$
$577$	13.8601	0.577004	0.288502	+	0.957479i	$0.406843\pi$
$578$	−58.2695	−2.42369
$579$	−44.1883	−1.83640
$580$	64.0870	2.66107
$581$	56.7913	2.35610
$582$	107.042	4.43705
$583$	−18.3755	−0.761035
$584$	27.5371	1.13949
$585$	−9.04469	−0.373952
$586$	1.81075	0.0748015
$587$	43.1254	1.77998	0.889989	−	0.455983i	$-0.150712\pi$
$587$	43.1254	1.77998	0.889989	+	0.455983i	$0.150712\pi$
$588$	239.608	9.88125
$589$	−0.139894	−0.00576422
$590$	−43.5574	−1.79323
$591$	47.6030	1.95812
$592$	−25.9624	−1.06705
$593$	−6.34374	−0.260506	−0.130253	−	0.991481i	$-0.541579\pi$
$593$	−6.34374	−0.260506	−0.130253	+	0.991481i	$0.541579\pi$
$594$	72.6911	2.98255
$595$	50.5046	2.07049
$596$	−3.95945	−0.162186
$597$	38.7062	1.58414
$598$	5.01647	0.205139
$599$	−45.1339	−1.84412	−0.922060	−	0.387048i	$-0.873495\pi$
$599$	−45.1339	−1.84412	−0.922060	+	0.387048i	$0.873495\pi$
$600$	39.9429	1.63066
$601$	−16.1640	−0.659342	−0.329671	−	0.944096i	$-0.606938\pi$
$601$	−16.1640	−0.659342	−0.329671	+	0.944096i	$0.606938\pi$
$602$	74.9107	3.05313
$603$	75.4917	3.07426
$604$	71.3791	2.90437
$605$	−0.748136	−0.0304161
$606$	36.4850	1.48210
$607$	−8.48631	−0.344449	−0.172224	−	0.985058i	$-0.555095\pi$
$607$	−8.48631	−0.344449	−0.172224	+	0.985058i	$0.555095\pi$
$608$	0.137085	0.00555954
$609$	−160.733	−6.51322
$610$	−11.5222	−0.466519
$611$	−7.49849	−0.303356
$612$	−152.644	−6.17029
$613$	−28.3373	−1.14453	−0.572267	−	0.820068i	$-0.693936\pi$
$613$	−28.3373	−1.14453	−0.572267	+	0.820068i	$0.693936\pi$
$614$	−69.5018	−2.80486
$615$	−35.5388	−1.43306
$616$	88.0847	3.54903
$617$	11.1325	0.448177	0.224088	−	0.974569i	$-0.428060\pi$
$617$	11.1325	0.448177	0.224088	+	0.974569i	$0.428060\pi$
$618$	29.0626	1.16907
$619$	1.00000	0.0401934
$619$	1.00000	0.0401934
$620$	−2.52263	−0.101311
$621$	−17.8099	−0.714687
$622$	79.9938	3.20746
$623$	−41.1441	−1.64840
$624$	12.6752	0.507413
$625$	−4.55405	−0.182162
$626$	−12.0771	−0.482699
$627$	3.46504	0.138380
$628$	−87.0737	−3.47462
$629$	39.0346	1.55641
$630$	115.196	4.58951
$631$	−23.8515	−0.949512	−0.474756	−	0.880117i	$-0.657464\pi$
$631$	−23.8515	−0.949512	−0.474756	+	0.880117i	$0.657464\pi$
$632$	−17.6256	−0.701108
$633$	27.7439	1.10272
$634$	40.1536	1.59470
$635$	16.7042	0.662885
$636$	65.4207	2.59410
$637$	19.8549	0.786681
$638$	−86.5096	−3.42495
$639$	29.2668	1.15778
$640$	−29.3902	−1.16175
$641$	24.7696	0.978341	0.489171	−	0.872188i	$-0.337300\pi$
$641$	24.7696	0.978341	0.489171	+	0.872188i	$0.337300\pi$
$642$	65.0750	2.56831
$643$	−42.9256	−1.69282	−0.846411	−	0.532531i	$-0.821241\pi$
$643$	−42.9256	−1.69282	−0.846411	+	0.532531i	$0.821241\pi$
$644$	−42.7363	−1.68405
$645$	26.8343	1.05660
$646$	−5.36676	−0.211152
$647$	15.8630	0.623638	0.311819	−	0.950142i	$-0.399062\pi$
$647$	15.8630	0.623638	0.311819	+	0.950142i	$0.399062\pi$
$648$	−41.6113	−1.63465
$649$	39.3290	1.54380
$650$	6.55425	0.257079
$651$	6.32685	0.247969
$652$	−71.8023	−2.81199
$653$	43.4405	1.69996	0.849978	−	0.526818i	$-0.176615\pi$
$653$	43.4405	1.69996	0.849978	+	0.526818i	$0.176615\pi$
$654$	21.6242	0.845572
$655$	3.36444	0.131460
$656$	33.0562	1.29063
$657$	−32.5166	−1.26859
$658$	95.5028	3.72309
$659$	0.240191	0.00935649	0.00467825	−	0.999989i	$-0.498511\pi$
$659$	0.240191	0.00935649	0.00467825	+	0.999989i	$0.498511\pi$
$660$	62.4832	2.43215
$661$	−34.8936	−1.35720	−0.678601	−	0.734507i	$-0.737413\pi$
$661$	−34.8936	−1.35720	−0.678601	+	0.734507i	$0.737413\pi$
$662$	−4.89016	−0.190061
$663$	−19.0572	−0.740120
$664$	−54.9544	−2.13264
$665$	2.70909	0.105054
$666$	89.0337	3.44999
$667$	21.1956	0.820696
$668$	−37.2132	−1.43982
$669$	47.8759	1.85099
$670$	47.8615	1.84905
$671$	10.4036	0.401628
$672$	−6.19983	−0.239164
$673$	−29.2862	−1.12890	−0.564449	−	0.825468i	$-0.690911\pi$
$673$	−29.2862	−1.12890	−0.564449	+	0.825468i	$0.690911\pi$
$674$	53.5615	2.06311
$675$	−23.2695	−0.895643
$676$	4.04034	0.155398
$677$	48.2299	1.85363	0.926813	−	0.375523i	$-0.122537\pi$
$677$	48.2299	1.85363	0.926813	+	0.375523i	$0.122537\pi$
$678$	1.12201	0.0430905
$679$	−75.5653	−2.89993
$680$	−48.8711	−1.87412
$681$	48.2583	1.84926
$682$	3.40524	0.130393
$683$	−30.3203	−1.16017	−0.580087	−	0.814554i	$-0.696982\pi$
$683$	−30.3203	−1.16017	−0.580087	+	0.814554i	$0.696982\pi$
$684$	−8.18792	−0.313073
$685$	−18.3824	−0.702354
$686$	−163.724	−6.25100
$687$	49.7629	1.89857
$688$	−24.9598	−0.951583
$689$	5.42105	0.206525
$690$	−22.8869	−0.871291
$691$	5.73060	0.218002	0.109001	−	0.994042i	$-0.465235\pi$
$691$	5.73060	0.218002	0.109001	+	0.994042i	$0.465235\pi$
$692$	54.6581	2.07779
$693$	−104.013	−3.95112
$694$	−74.1201	−2.81356
$695$	26.3284	0.998693
$696$	155.534	5.89550
$697$	−49.7001	−1.88253
$698$	−38.4375	−1.45488
$699$	10.9127	0.412757
$700$	−55.8370	−2.11044
$701$	10.1621	0.383817	0.191909	−	0.981413i	$-0.438532\pi$
$701$	10.1621	0.383817	0.191909	+	0.981413i	$0.438532\pi$
$702$	−21.4449	−0.809387
$703$	2.09383	0.0789704
$704$	25.4322	0.958511
$705$	34.2108	1.28845
$706$	35.1796	1.32400
$707$	−25.7562	−0.968660
$708$	−140.020	−5.26226
$709$	−0.120200	−0.00451421	−0.00225710	−	0.999997i	$-0.500718\pi$
$709$	−0.120200	−0.00451421	−0.00225710	+	0.999997i	$0.500718\pi$
$710$	18.5551	0.696359
$711$	20.8128	0.780539
$712$	39.8133	1.49207
$713$	−0.834312	−0.0312452
$714$	242.718	9.08349
$715$	5.17763	0.193632
$716$	−15.4710	−0.578180
$717$	−43.8915	−1.63916
$718$	63.1805	2.35787
$719$	−16.6608	−0.621345	−0.310672	−	0.950517i	$-0.600554\pi$
$719$	−16.6608	−0.621345	−0.310672	+	0.950517i	$0.600554\pi$
$720$	−38.3825	−1.43043
$721$	−20.5164	−0.764070
$722$	46.4086	1.72715
$723$	−29.0731	−1.08124
$724$	−68.7350	−2.55452
$725$	27.6930	1.02849
$726$	−3.59543	−0.133439
$727$	−28.7802	−1.06740	−0.533699	−	0.845674i	$-0.679198\pi$
$727$	−28.7802	−1.06740	−0.533699	+	0.845674i	$0.679198\pi$
$728$	−25.9863	−0.963116
$729$	−29.0505	−1.07594
$730$	−20.6154	−0.763010
$731$	37.5272	1.38799
$732$	−37.0392	−1.36901
$733$	−18.6476	−0.688764	−0.344382	−	0.938830i	$-0.611912\pi$
$733$	−18.6476	−0.688764	−0.344382	+	0.938830i	$0.611912\pi$
$734$	10.6926	0.394669
$735$	−90.5853	−3.34129
$736$	0.817562	0.0301357
$737$	−43.2152	−1.59185
$738$	−113.361	−4.17286
$739$	−13.7404	−0.505448	−0.252724	−	0.967538i	$-0.581326\pi$
$739$	−13.7404	−0.505448	−0.252724	+	0.967538i	$0.581326\pi$
$740$	37.7570	1.38798
$741$	−1.02224	−0.0375528
$742$	−69.0440	−2.53468
$743$	−4.89938	−0.179741	−0.0898704	−	0.995953i	$-0.528645\pi$
$743$	−4.89938	−0.179741	−0.0898704	+	0.995953i	$0.528645\pi$
$744$	−6.12221	−0.224451
$745$	1.49690	0.0548422
$746$	−35.8847	−1.31383
$747$	64.8916	2.37426
$748$	87.3812	3.19497
$749$	−45.9389	−1.67857
$750$	−85.9676	−3.13909
$751$	−28.2207	−1.02979	−0.514893	−	0.857254i	$-0.672168\pi$
$751$	−28.2207	−1.02979	−0.514893	+	0.857254i	$0.672168\pi$
$752$	−31.8210	−1.16039
$753$	35.7169	1.30159
$754$	25.5216	0.929443
$755$	−26.9853	−0.982097
$756$	182.694	6.64451
$757$	36.4334	1.32419	0.662097	−	0.749418i	$-0.269667\pi$
$757$	36.4334	1.32419	0.662097	+	0.749418i	$0.269667\pi$
$758$	−14.5459	−0.528329
$759$	20.6651	0.750097
$760$	−2.62147	−0.0950906
$761$	−0.411195	−0.0149058	−0.00745291	−	0.999972i	$-0.502372\pi$
$761$	−0.411195	−0.0149058	−0.00745291	+	0.999972i	$0.502372\pi$
$762$	80.2778	2.90816
$763$	−15.2653	−0.552642
$764$	82.2594	2.97604
$765$	57.7083	2.08645
$766$	87.6104	3.16549
$767$	−11.6026	−0.418947
$768$	−96.4247	−3.47943
$769$	−4.19905	−0.151422	−0.0757109	−	0.997130i	$-0.524123\pi$
$769$	−4.19905	−0.151422	−0.0757109	+	0.997130i	$0.524123\pi$
$770$	−65.9437	−2.37645
$771$	−23.9669	−0.863147
$772$	−59.7737	−2.15130
$773$	26.5578	0.955217	0.477608	−	0.878573i	$-0.341504\pi$
$773$	26.5578	0.955217	0.477608	+	0.878573i	$0.341504\pi$
$774$	85.5954	3.07666
$775$	−1.09007	−0.0391564
$776$	73.1212	2.62490
$777$	−94.6960	−3.39720
$778$	−25.6564	−0.919826
$779$	−2.66594	−0.0955171
$780$	−18.4335	−0.660024
$781$	−16.7538	−0.599497
$782$	−32.0068	−1.14456
$783$	−90.6092	−3.23811
$784$	84.2573	3.00919
$785$	32.9188	1.17492
$786$	16.1690	0.576730
$787$	26.9943	0.962241	0.481121	−	0.876654i	$-0.340230\pi$
$787$	26.9943	0.962241	0.481121	+	0.876654i	$0.340230\pi$
$788$	64.3928	2.29390
$789$	−73.4965	−2.61655
$790$	13.1952	0.469464
$791$	−0.792069	−0.0281627
$792$	100.649	3.57639
$793$	−3.06923	−0.108991
$794$	7.47922	0.265428
$795$	−24.7328	−0.877181
$796$	52.3581	1.85578
$797$	6.12022	0.216789	0.108395	−	0.994108i	$-0.465429\pi$
$797$	6.12022	0.216789	0.108395	+	0.994108i	$0.465429\pi$
$798$	13.0195	0.460885
$799$	47.8430	1.69256
$800$	1.06818	0.0377660
$801$	−47.0127	−1.66111
$802$	−34.9747	−1.23500
$803$	18.6141	0.656877
$804$	153.855	5.42606
$805$	16.1568	0.569451
$806$	−1.00460	−0.0353854
$807$	13.1390	0.462514
$808$	24.9231	0.876791
$809$	−33.7005	−1.18485	−0.592423	−	0.805627i	$-0.701828\pi$
$809$	−33.7005	−1.18485	−0.592423	+	0.805627i	$0.701828\pi$
$810$	31.1519	1.09457
$811$	34.5015	1.21151	0.605756	−	0.795651i	$-0.292871\pi$
$811$	34.5015	1.21151	0.605756	+	0.795651i	$0.292871\pi$
$812$	−217.424	−7.63008
$813$	−65.1254	−2.28405
$814$	−50.9673	−1.78640
$815$	27.1453	0.950860
$816$	−80.8721	−2.83109
$817$	2.01297	0.0704251
$818$	51.5407	1.80208
$819$	30.6853	1.07223
$820$	−48.0735	−1.67880
$821$	−34.6963	−1.21091	−0.605454	−	0.795880i	$-0.707008\pi$
$821$	−34.6963	−1.21091	−0.605454	+	0.795880i	$0.707008\pi$
$822$	−88.3429	−3.08131
$823$	6.51841	0.227217	0.113609	−	0.993526i	$-0.463759\pi$
$823$	6.51841	0.227217	0.113609	+	0.993526i	$0.463759\pi$
$824$	19.8528	0.691604
$825$	27.0000	0.940018
$826$	147.774	5.14173
$827$	12.7726	0.444147	0.222074	−	0.975030i	$-0.428717\pi$
$827$	12.7726	0.444147	0.222074	+	0.975030i	$0.428717\pi$
$828$	−48.8319	−1.69703
$829$	−7.22018	−0.250767	−0.125384	−	0.992108i	$-0.540016\pi$
$829$	−7.22018	−0.250767	−0.125384	+	0.992108i	$0.540016\pi$
$830$	41.1410	1.42803
$831$	−46.7545	−1.62190
$832$	−7.50287	−0.260115
$833$	−126.681	−4.38925
$834$	126.530	4.38139
$835$	14.0687	0.486868
$836$	4.68717	0.162109
$837$	3.56661	0.123280
$838$	48.0579	1.66013
$839$	−48.0940	−1.66039	−0.830195	−	0.557473i	$-0.811771\pi$
$839$	−48.0940	−1.66039	−0.830195	+	0.557473i	$0.811771\pi$
$840$	118.559	4.09067
$841$	78.8339	2.71841
$842$	−7.87883	−0.271522
$843$	70.5891	2.43122
$844$	37.5293	1.29181
$845$	−1.52748	−0.0525468
$846$	109.125	3.75178
$847$	2.53815	0.0872119
$848$	23.0050	0.789996
$849$	−19.3781	−0.665055
$850$	−41.8184	−1.43436
$851$	12.4874	0.428063
$852$	59.6471	2.04347
$853$	9.47360	0.324370	0.162185	−	0.986760i	$-0.448146\pi$
$853$	9.47360	0.324370	0.162185	+	0.986760i	$0.448146\pi$
$854$	39.0905	1.33765
$855$	3.09550	0.105864
$856$	44.4530	1.51937
$857$	34.7861	1.18827	0.594136	−	0.804365i	$-0.297494\pi$
$857$	34.7861	1.18827	0.594136	+	0.804365i	$0.297494\pi$
$858$	24.8829	0.849489
$859$	−0.201976	−0.00689133	−0.00344567	−	0.999994i	$-0.501097\pi$
$859$	−0.201976	−0.00689133	−0.00344567	+	0.999994i	$0.501097\pi$
$860$	36.2989	1.23778
$861$	120.570	4.10902
$862$	21.2897	0.725131
$863$	34.9891	1.19104	0.595522	−	0.803339i	$-0.296945\pi$
$863$	34.9891	1.19104	0.595522	+	0.803339i	$0.296945\pi$
$864$	−3.49500	−0.118902
$865$	−20.6639	−0.702593
$866$	61.6929	2.09641
$867$	70.8150	2.40500
$868$	8.55836	0.290490
$869$	−11.9143	−0.404163
$870$	−116.439	−3.94765
$871$	12.7491	0.431988
$872$	14.7716	0.500229
$873$	−86.3435	−2.92228
$874$	−1.71686	−0.0580737
$875$	60.6878	2.05162
$876$	−66.2702	−2.23906
$877$	−19.0414	−0.642981	−0.321491	−	0.946913i	$-0.604184\pi$
$877$	−19.0414	−0.642981	−0.321491	+	0.946913i	$0.604184\pi$
$878$	45.0236	1.51947
$879$	−2.20061	−0.0742248
$880$	21.9720	0.740677
$881$	−37.6093	−1.26709	−0.633544	−	0.773707i	$-0.718400\pi$
$881$	−37.6093	−1.26709	−0.633544	+	0.773707i	$0.718400\pi$
$882$	−288.947	−9.72934
$883$	−3.78623	−0.127417	−0.0637084	−	0.997969i	$-0.520293\pi$
$883$	−3.78623	−0.127417	−0.0637084	+	0.997969i	$0.520293\pi$
$884$	−25.7788	−0.867034
$885$	52.9354	1.77940
$886$	−93.7984	−3.15122
$887$	−34.9355	−1.17302	−0.586510	−	0.809942i	$-0.699498\pi$
$887$	−34.9355	−1.17302	−0.586510	+	0.809942i	$0.699498\pi$
$888$	91.6331	3.07501
$889$	−56.6712	−1.90069
$890$	−29.8058	−0.999094
$891$	−28.1278	−0.942315
$892$	64.7619	2.16839
$893$	2.56632	0.0858786
$894$	7.19388	0.240599
$895$	5.84893	0.195508
$896$	99.7101	3.33108
$897$	−6.09652	−0.203557
$898$	−12.6924	−0.423552
$899$	−4.24462	−0.141566
$900$	−63.8012	−2.12671
$901$	−34.5882	−1.15230
$902$	64.8932	2.16071
$903$	−91.0391	−3.02959
$904$	0.766449	0.0254917
$905$	25.9857	0.863795
$906$	−129.688	−4.30858
$907$	−16.9678	−0.563405	−0.281703	−	0.959502i	$-0.590899\pi$
$907$	−16.9678	−0.563405	−0.281703	+	0.959502i	$0.590899\pi$
$908$	65.2793	2.16637
$909$	−29.4299	−0.976127
$910$	19.4544	0.644906
$911$	−4.74641	−0.157255	−0.0786277	−	0.996904i	$-0.525054\pi$
$911$	−4.74641	−0.157255	−0.0786277	+	0.996904i	$0.525054\pi$
$912$	−4.33802	−0.143646
$913$	−37.1472	−1.22939
$914$	−67.3113	−2.22646
$915$	14.0029	0.462922
$916$	67.3145	2.22413
$917$	−11.4143	−0.376934
$918$	136.826	4.51594
$919$	23.7505	0.783455	0.391728	−	0.920081i	$-0.371878\pi$
$919$	23.7505	0.783455	0.391728	+	0.920081i	$0.371878\pi$
$920$	−15.6342	−0.515443
$921$	84.4656	2.78324
$922$	−62.8775	−2.07076
$923$	4.94261	0.162688
$924$	−211.983	−6.97371
$925$	16.3154	0.536447
$926$	−79.2094	−2.60298
$927$	−23.4427	−0.769959
$928$	4.15940	0.136539
$929$	−39.6588	−1.30116	−0.650581	−	0.759437i	$-0.725475\pi$
$929$	−39.6588	−1.30116	−0.650581	+	0.759437i	$0.725475\pi$
$930$	4.58333	0.150293
$931$	−6.79525	−0.222705
$932$	14.7617	0.483535
$933$	−97.2166	−3.18273
$934$	−68.1311	−2.22932
$935$	−33.0351	−1.08036
$936$	−29.6928	−0.970540
$937$	−3.41241	−0.111479	−0.0557393	−	0.998445i	$-0.517752\pi$
$937$	−3.41241	−0.111479	−0.0557393	+	0.998445i	$0.517752\pi$
$938$	−162.376	−5.30178
$939$	14.6773	0.478977
$940$	46.2771	1.50939
$941$	4.98237	0.162421	0.0812103	−	0.996697i	$-0.474121\pi$
$941$	4.98237	0.162421	0.0812103	+	0.996697i	$0.474121\pi$
$942$	158.203	5.15453
$943$	−15.8994	−0.517755
$944$	−49.2375	−1.60254
$945$	−69.0687	−2.24680
$946$	−48.9991	−1.59310
$947$	−29.4914	−0.958340	−0.479170	−	0.877722i	$-0.659062\pi$
$947$	−29.4914	−0.958340	−0.479170	+	0.877722i	$0.659062\pi$
$948$	42.4173	1.37765
$949$	−5.49144	−0.178260
$950$	−2.24316	−0.0727777
$951$	−48.7987	−1.58241
$952$	165.802	5.37366
$953$	45.8479	1.48516	0.742579	−	0.669758i	$-0.233602\pi$
$953$	45.8479	1.48516	0.742579	+	0.669758i	$0.233602\pi$
$954$	−78.8919	−2.55422
$955$	−31.0987	−1.00633
$956$	−59.3722	−1.92023
$957$	105.135	3.39854
$958$	−10.9827	−0.354836
$959$	62.3647	2.01386
$960$	34.2308	1.10479
$961$	−30.8329	−0.994610
$962$	15.0361	0.484784
$963$	−52.4914	−1.69151
$964$	−39.3273	−1.26665
$965$	22.5978	0.727450
$966$	77.6470	2.49825
$967$	12.7421	0.409760	0.204880	−	0.978787i	$-0.434320\pi$
$967$	12.7421	0.409760	0.204880	+	0.978787i	$0.434320\pi$
$968$	−2.45606	−0.0789406
$969$	6.52223	0.209524
$970$	−54.7414	−1.75764
$971$	−7.61524	−0.244385	−0.122192	−	0.992506i	$-0.538992\pi$
$971$	−7.61524	−0.244385	−0.122192	+	0.992506i	$0.538992\pi$
$972$	−5.62201	−0.180326
$973$	−89.3226	−2.86355
$974$	−97.6707	−3.12957
$975$	−7.96539	−0.255097
$976$	−13.0247	−0.416911
$977$	−24.6855	−0.789759	−0.394880	−	0.918733i	$-0.629214\pi$
$977$	−24.6855	−0.789759	−0.394880	+	0.918733i	$0.629214\pi$
$978$	130.457	4.17154
$979$	26.9124	0.860123
$980$	−122.535	−3.91424
$981$	−17.4427	−0.556902
$982$	−74.1471	−2.36613
$983$	−25.3886	−0.809771	−0.404886	−	0.914367i	$-0.632689\pi$
$983$	−25.3886	−0.809771	−0.404886	+	0.914367i	$0.632689\pi$
$984$	−116.670	−3.71931
$985$	−24.3441	−0.775668
$986$	−162.837	−5.18578
$987$	−116.065	−3.69438
$988$	−1.38279	−0.0439923
$989$	12.0052	0.381743
$990$	−75.3495	−2.39476
$991$	56.7807	1.80370	0.901849	−	0.432051i	$-0.142210\pi$
$991$	56.7807	1.80370	0.901849	+	0.432051i	$0.142210\pi$
$992$	−0.163725	−0.00519827
$993$	5.94302	0.188596
$994$	−62.9505	−1.99667
$995$	−19.7943	−0.627523
$996$	132.252	4.19056
$997$	2.70596	0.0856986	0.0428493	−	0.999082i	$-0.486356\pi$
$997$	2.70596	0.0856986	0.0428493	+	0.999082i	$0.486356\pi$
$998$	−56.3606	−1.78406
$999$	−53.3826	−1.68895

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8047.2.a.b.1.14	✓	142

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8047.2.a.b.1.14	✓	142	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8047 = 13 \cdot 619 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8047.a (trivial)

\(f(q)\)	\(=\)	\(q-2.45771 q^{2} +2.98686 q^{3} +4.04034 q^{4} -1.52748 q^{5} -7.34083 q^{6} +5.18217 q^{7} -5.01456 q^{8} +5.92132 q^{9} +O(q^{10})\)	\(q-2.45771 q^{2} +2.98686 q^{3} +4.04034 q^{4} -1.52748 q^{5} -7.34083 q^{6} +5.18217 q^{7} -5.01456 q^{8} +5.92132 q^{9} +3.75410 q^{10} -3.38966 q^{11} +12.0679 q^{12} +1.00000 q^{13} -12.7363 q^{14} -4.56236 q^{15} +4.24365 q^{16} -6.38035 q^{17} -14.5529 q^{18} -0.342245 q^{19} -6.17152 q^{20} +15.4784 q^{21} +8.33080 q^{22} -2.04112 q^{23} -14.9778 q^{24} -2.66681 q^{25} -2.45771 q^{26} +8.72558 q^{27} +20.9377 q^{28} -10.3843 q^{29} +11.2130 q^{30} +0.408753 q^{31} -0.400547 q^{32} -10.1244 q^{33} +15.6810 q^{34} -7.91565 q^{35} +23.9241 q^{36} -6.11794 q^{37} +0.841139 q^{38} +2.98686 q^{39} +7.65962 q^{40} +7.78956 q^{41} -38.0415 q^{42} -5.88168 q^{43} -13.6954 q^{44} -9.04469 q^{45} +5.01647 q^{46} -7.49849 q^{47} +12.6752 q^{48} +19.8549 q^{49} +6.55425 q^{50} -19.0572 q^{51} +4.04034 q^{52} +5.42105 q^{53} -21.4449 q^{54} +5.17763 q^{55} -25.9863 q^{56} -1.02224 q^{57} +25.5216 q^{58} -11.6026 q^{59} -18.4335 q^{60} -3.06923 q^{61} -1.00460 q^{62} +30.6853 q^{63} -7.50287 q^{64} -1.52748 q^{65} +24.8829 q^{66} +12.7491 q^{67} -25.7788 q^{68} -6.09652 q^{69} +19.4544 q^{70} +4.94261 q^{71} -29.6928 q^{72} -5.49144 q^{73} +15.0361 q^{74} -7.96539 q^{75} -1.38279 q^{76} -17.5658 q^{77} -7.34083 q^{78} +3.51488 q^{79} -6.48208 q^{80} +8.29811 q^{81} -19.1445 q^{82} +10.9590 q^{83} +62.5380 q^{84} +9.74584 q^{85} +14.4555 q^{86} -31.0165 q^{87} +16.9976 q^{88} -7.93955 q^{89} +22.2292 q^{90} +5.18217 q^{91} -8.24679 q^{92} +1.22089 q^{93} +18.4291 q^{94} +0.522772 q^{95} -1.19638 q^{96} -14.5818 q^{97} -48.7976 q^{98} -20.0713 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9}+O(q^{10}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9	\( 142 q - 13 q^{2} - 26 q^{3} + 129 q^{4} - 37 q^{5} - 15 q^{6} - 14 q^{7} - 39 q^{8} + 98 q^{9} - 25 q^{10} - 25 q^{11} - 62 q^{12} + 142 q^{13} - 57 q^{14} - 14 q^{15} + 111 q^{16} - 141 q^{17} - 29 q^{18} - 3 q^{19} - 87 q^{20} - 19 q^{21} - 24 q^{22} - 69 q^{23} - 40 q^{24} + 87 q^{25} - 13 q^{26} - 95 q^{27} - 34 q^{28} - 147 q^{29} - 2 q^{30} - 21 q^{31} - 66 q^{32} - 62 q^{33} - 6 q^{34} - 59 q^{35} + 74 q^{36} - 37 q^{37} - 76 q^{38} - 26 q^{39} - 61 q^{40} - 97 q^{41} - 29 q^{42} - 33 q^{43} - 57 q^{44} - 86 q^{45} - q^{46} - 102 q^{47} - 141 q^{48} + 70 q^{49} - 28 q^{50} - 13 q^{51} + 129 q^{52} - 137 q^{53} - 29 q^{54} - 24 q^{55} - 130 q^{56} - 65 q^{57} - 15 q^{58} - 56 q^{59} + 11 q^{60} - 77 q^{61} - 150 q^{62} - 32 q^{63} + 73 q^{64} - 37 q^{65} - 32 q^{66} - 9 q^{67} - 226 q^{68} - 113 q^{69} + 6 q^{70} - 18 q^{71} - 82 q^{72} - 117 q^{73} - 70 q^{74} - 83 q^{75} + 40 q^{76} - 214 q^{77} - 15 q^{78} - 52 q^{79} - 161 q^{80} - 10 q^{81} - 36 q^{82} - 74 q^{83} + 53 q^{84} + 2 q^{85} + 17 q^{86} - 49 q^{87} - 29 q^{88} - 171 q^{89} - 57 q^{90} - 14 q^{91} - 187 q^{92} - 39 q^{93} + 13 q^{94} - 150 q^{95} - 47 q^{96} - 126 q^{97} - 85 q^{98}+O(q^{100}) \) 142 * q - 13 * q^2 - 26 * q^3 + 129 * q^4 - 37 * q^5 - 15 * q^6 - 14 * q^7 - 39 * q^8 + 98 * q^9 - 25 * q^10 - 25 * q^11 - 62 * q^12 + 142 * q^13 - 57 * q^14 - 14 * q^15 + 111 * q^16 - 141 * q^17 - 29 * q^18 - 3 * q^19 - 87 * q^20 - 19 * q^21 - 24 * q^22 - 69 * q^23 - 40 * q^24 + 87 * q^25 - 13 * q^26 - 95 * q^27 - 34 * q^28 - 147 * q^29 - 2 * q^30 - 21 * q^31 - 66 * q^32 - 62 * q^33 - 6 * q^34 - 59 * q^35 + 74 * q^36 - 37 * q^37 - 76 * q^38 - 26 * q^39 - 61 * q^40 - 97 * q^41 - 29 * q^42 - 33 * q^43 - 57 * q^44 - 86 * q^45 - q^46 - 102 * q^47 - 141 * q^48 + 70 * q^49 - 28 * q^50 - 13 * q^51 + 129 * q^52 - 137 * q^53 - 29 * q^54 - 24 * q^55 - 130 * q^56 - 65 * q^57 - 15 * q^58 - 56 * q^59 + 11 * q^60 - 77 * q^61 - 150 * q^62 - 32 * q^63 + 73 * q^64 - 37 * q^65 - 32 * q^66 - 9 * q^67 - 226 * q^68 - 113 * q^69 + 6 * q^70 - 18 * q^71 - 82 * q^72 - 117 * q^73 - 70 * q^74 - 83 * q^75 + 40 * q^76 - 214 * q^77 - 15 * q^78 - 52 * q^79 - 161 * q^80 - 10 * q^81 - 36 * q^82 - 74 * q^83 + 53 * q^84 + 2 * q^85 + 17 * q^86 - 49 * q^87 - 29 * q^88 - 171 * q^89 - 57 * q^90 - 14 * q^91 - 187 * q^92 - 39 * q^93 + 13 * q^94 - 150 * q^95 - 47 * q^96 - 126 * q^97 - 85 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more