LMFDB - Embedded newform 889.2.a.d.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [889,2,Mod(1,889)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(889, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("889.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 889 = 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	889.a (trivial)

Newform invariants

comment:select newform

sage:traces = [20] 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:	$7.09870073969$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 152 x^{17} - 274 x^{16} - 1061 x^{15} + 3125 x^{14} + 2977 x^{13} - 15474 x^{12} + \cdots + 31 $ x^20 - 8x^19 + 152x^17 - 274x^16 - 1061x^15 + 3125x^14 + 2977x^13 - 15474x^12 - 56x^11 + 39579x^10 - 17664x^9 - 52271x^8 + 35701x^7 + 32493x^6 - 25504x^5 - 8607x^4 + 6812x^3 + 609x^2 - 425x + 31
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.60447$ of defining polynomial
Character	$\chi$	$=$	889.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.60447 q^{2} +2.96301 q^{3} +0.574339 q^{4} +0.248535 q^{5} -4.75407 q^{6} +1.00000 q^{7} +2.28744 q^{8} +5.77943 q^{9} -0.398768 q^{10} +2.06439 q^{11} +1.70177 q^{12} +4.61355 q^{13} -1.60447 q^{14} +0.736411 q^{15} -4.81881 q^{16} -1.38495 q^{17} -9.27295 q^{18} -2.86500 q^{19} +0.142743 q^{20} +2.96301 q^{21} -3.31226 q^{22} +5.17882 q^{23} +6.77770 q^{24} -4.93823 q^{25} -7.40233 q^{26} +8.23548 q^{27} +0.574339 q^{28} -9.35535 q^{29} -1.18155 q^{30} -7.06776 q^{31} +3.15679 q^{32} +6.11681 q^{33} +2.22211 q^{34} +0.248535 q^{35} +3.31935 q^{36} +9.60134 q^{37} +4.59682 q^{38} +13.6700 q^{39} +0.568508 q^{40} +7.82024 q^{41} -4.75407 q^{42} +10.4189 q^{43} +1.18566 q^{44} +1.43639 q^{45} -8.30929 q^{46} -7.14182 q^{47} -14.2782 q^{48} +1.00000 q^{49} +7.92327 q^{50} -4.10361 q^{51} +2.64974 q^{52} +9.73442 q^{53} -13.2136 q^{54} +0.513073 q^{55} +2.28744 q^{56} -8.48903 q^{57} +15.0104 q^{58} -1.47644 q^{59} +0.422949 q^{60} -11.9646 q^{61} +11.3400 q^{62} +5.77943 q^{63} +4.57264 q^{64} +1.14663 q^{65} -9.81427 q^{66} -14.5589 q^{67} -0.795429 q^{68} +15.3449 q^{69} -0.398768 q^{70} +9.83898 q^{71} +13.2201 q^{72} -4.36287 q^{73} -15.4051 q^{74} -14.6320 q^{75} -1.64548 q^{76} +2.06439 q^{77} -21.9332 q^{78} -2.06143 q^{79} -1.19764 q^{80} +7.06353 q^{81} -12.5474 q^{82} +15.1003 q^{83} +1.70177 q^{84} -0.344207 q^{85} -16.7169 q^{86} -27.7200 q^{87} +4.72216 q^{88} -12.9515 q^{89} -2.30465 q^{90} +4.61355 q^{91} +2.97440 q^{92} -20.9418 q^{93} +11.4589 q^{94} -0.712053 q^{95} +9.35359 q^{96} -5.88747 q^{97} -1.60447 q^{98} +11.9310 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 8 q^{2} + 24 q^{4} + 3 q^{5} + 6 q^{6} + 20 q^{7} + 24 q^{8} + 30 q^{9} - 8 q^{10} + 26 q^{11} - 4 q^{12} - 4 q^{13} + 8 q^{14} + 10 q^{15} + 24 q^{16} + 4 q^{17} + 5 q^{18} + q^{19} - 2 q^{20}+ \cdots + 15 q^{99}+O(q^{100}) $ 20 * q + 8 * q^2 + 24 * q^4 + 3 * q^5 + 6 * q^6 + 20 * q^7 + 24 * q^8 + 30 * q^9 - 8 * q^10 + 26 * q^11 - 4 * q^12 - 4 * q^13 + 8 * q^14 + 10 * q^15 + 24 * q^16 + 4 * q^17 + 5 * q^18 + q^19 - 2 * q^20 + q^22 + 31 * q^23 - 6 * q^24 + 27 * q^25 + 4 * q^26 - 18 * q^27 + 24 * q^28 + 16 * q^29 - 5 * q^30 + 6 * q^31 + 41 * q^32 - 18 * q^33 - 10 * q^34 + 3 * q^35 + 18 * q^36 + 2 * q^37 + 3 * q^38 + 43 * q^39 - 38 * q^40 + 25 * q^41 + 6 * q^42 + 13 * q^43 + 66 * q^44 - 2 * q^45 + 20 * q^46 + 19 * q^47 - 16 * q^48 + 20 * q^49 - 4 * q^50 + 4 * q^51 + 20 * q^52 + 24 * q^53 + 5 * q^54 - 3 * q^55 + 24 * q^56 - 4 * q^57 + 12 * q^58 + 23 * q^59 + 24 * q^60 - 27 * q^61 + 7 * q^62 + 30 * q^63 + 2 * q^64 + 26 * q^65 + 26 * q^66 + 9 * q^67 - 25 * q^68 - 3 * q^69 - 8 * q^70 + 63 * q^71 + 27 * q^72 - 21 * q^73 + 21 * q^74 - 52 * q^75 - 10 * q^76 + 26 * q^77 - 70 * q^78 + 18 * q^79 - 23 * q^80 + 40 * q^81 - 42 * q^82 - q^83 - 4 * q^84 - 41 * q^85 - 12 * q^86 - 9 * q^87 + 57 * q^88 - 16 * q^89 + q^90 - 4 * q^91 + 17 * q^92 - 41 * q^93 + 7 * q^94 + 75 * q^95 - 81 * q^96 - 32 * q^97 + 8 * q^98 + 15 * 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 $\theta_p = \textrm{Arg}(\alpha_p)$.

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 $

$2$	−1.60447	−1.13453	−0.567267	−	0.823534i	$-0.691999\pi$
$2$	−1.60447	−1.13453	−0.567267	+	0.823534i	$0.691999\pi$
$3$	2.96301	1.71069	0.855347	−	0.518055i	$-0.173344\pi$
$3$	2.96301	1.71069	0.855347	+	0.518055i	$0.173344\pi$
$4$	0.574339	0.287169
$5$	0.248535	0.111148	0.0555740	−	0.998455i	$-0.482301\pi$
$5$	0.248535	0.111148	0.0555740	+	0.998455i	$0.482301\pi$
$6$	−4.75407	−1.94084
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.28744	0.808731
$9$	5.77943	1.92648
$10$	−0.398768	−0.126101
$11$	2.06439	0.622437	0.311219	−	0.950338i	$-0.399263\pi$
$11$	2.06439	0.622437	0.311219	+	0.950338i	$0.399263\pi$
$12$	1.70177	0.491259
$13$	4.61355	1.27957	0.639785	−	0.768554i	$-0.279023\pi$
$13$	4.61355	1.27957	0.639785	+	0.768554i	$0.279023\pi$
$14$	−1.60447	−0.428814
$15$	0.736411	0.190140
$16$	−4.81881	−1.20470
$17$	−1.38495	−0.335899	−0.167950	−	0.985796i	$-0.553715\pi$
$17$	−1.38495	−0.335899	−0.167950	+	0.985796i	$0.553715\pi$
$18$	−9.27295	−2.18566
$19$	−2.86500	−0.657277	−0.328638	−	0.944456i	$-0.606590\pi$
$19$	−2.86500	−0.657277	−0.328638	+	0.944456i	$0.606590\pi$
$20$	0.142743	0.0319183
$21$	2.96301	0.646582
$22$	−3.31226	−0.706177
$23$	5.17882	1.07986	0.539930	−	0.841710i	$-0.318451\pi$
$23$	5.17882	1.07986	0.539930	+	0.841710i	$0.318451\pi$
$24$	6.77770	1.38349
$25$	−4.93823	−0.987646
$26$	−7.40233	−1.45172
$27$	8.23548	1.58492
$28$	0.574339	0.108540
$29$	−9.35535	−1.73724	−0.868622	−	0.495475i	$-0.834994\pi$
$29$	−9.35535	−1.73724	−0.868622	+	0.495475i	$0.834994\pi$
$30$	−1.18155	−0.215721
$31$	−7.06776	−1.26941	−0.634703	−	0.772756i	$-0.718878\pi$
$31$	−7.06776	−1.26941	−0.634703	+	0.772756i	$0.718878\pi$
$32$	3.15679	0.558046
$33$	6.11681	1.06480
$34$	2.22211	0.381089
$35$	0.248535	0.0420100
$36$	3.31935	0.553225
$37$	9.60134	1.57845	0.789226	−	0.614103i	$-0.210482\pi$
$37$	9.60134	1.57845	0.789226	+	0.614103i	$0.210482\pi$
$38$	4.59682	0.745703
$39$	13.6700	2.18895
$40$	0.568508	0.0898889
$41$	7.82024	1.22132	0.610658	−	0.791894i	$-0.290905\pi$
$41$	7.82024	1.22132	0.610658	+	0.791894i	$0.290905\pi$
$42$	−4.75407	−0.733570
$43$	10.4189	1.58887	0.794437	−	0.607346i	$-0.207766\pi$
$43$	10.4189	1.58887	0.794437	+	0.607346i	$0.207766\pi$
$44$	1.18566	0.178745
$45$	1.43639	0.214124
$46$	−8.30929	−1.22514
$47$	−7.14182	−1.04174	−0.520871	−	0.853636i	$-0.674393\pi$
$47$	−7.14182	−1.04174	−0.520871	+	0.853636i	$0.674393\pi$
$48$	−14.2782	−2.06088
$49$	1.00000	0.142857
$50$	7.92327	1.12052
$51$	−4.10361	−0.574621
$52$	2.64974	0.367453
$53$	9.73442	1.33713	0.668563	−	0.743656i	$-0.266910\pi$
$53$	9.73442	1.33713	0.668563	+	0.743656i	$0.266910\pi$
$54$	−13.2136	−1.79815
$55$	0.513073	0.0691827
$56$	2.28744	0.305672
$57$	−8.48903	−1.12440
$58$	15.0104	1.97096
$59$	−1.47644	−0.192217	−0.0961083	−	0.995371i	$-0.530640\pi$
$59$	−1.47644	−0.192217	−0.0961083	+	0.995371i	$0.530640\pi$
$60$	0.422949	0.0546025
$61$	−11.9646	−1.53191	−0.765957	−	0.642892i	$-0.777734\pi$
$61$	−11.9646	−1.53191	−0.765957	+	0.642892i	$0.777734\pi$
$62$	11.3400	1.44019
$63$	5.77943	0.728140
$64$	4.57264	0.571580
$65$	1.14663	0.142222
$66$	−9.81427	−1.20805
$67$	−14.5589	−1.77865	−0.889325	−	0.457276i	$-0.848825\pi$
$67$	−14.5589	−1.77865	−0.889325	+	0.457276i	$0.848825\pi$
$68$	−0.795429	−0.0964599
$69$	15.3449	1.84731
$70$	−0.398768	−0.0476618
$71$	9.83898	1.16767	0.583836	−	0.811872i	$-0.301551\pi$
$71$	9.83898	1.16767	0.583836	+	0.811872i	$0.301551\pi$
$72$	13.2201	1.55800
$73$	−4.36287	−0.510636	−0.255318	−	0.966857i	$-0.582180\pi$
$73$	−4.36287	−0.510636	−0.255318	+	0.966857i	$0.582180\pi$
$74$	−15.4051	−1.79081
$75$	−14.6320	−1.68956
$76$	−1.64548	−0.188750
$77$	2.06439	0.235259
$78$	−21.9332	−2.48344
$79$	−2.06143	−0.231929	−0.115964	−	0.993253i	$-0.536996\pi$
$79$	−2.06143	−0.231929	−0.115964	+	0.993253i	$0.536996\pi$
$80$	−1.19764	−0.133900
$81$	7.06353	0.784836
$82$	−12.5474	−1.38563
$83$	15.1003	1.65747	0.828735	−	0.559642i	$-0.189061\pi$
$83$	15.1003	1.65747	0.828735	+	0.559642i	$0.189061\pi$
$84$	1.70177	0.185678
$85$	−0.344207	−0.0373345
$86$	−16.7169	−1.80263
$87$	−27.7200	−2.97190
$88$	4.72216	0.503384
$89$	−12.9515	−1.37286	−0.686429	−	0.727197i	$-0.740823\pi$
$89$	−12.9515	−1.37286	−0.686429	+	0.727197i	$0.740823\pi$
$90$	−2.30465	−0.242931
$91$	4.61355	0.483632
$92$	2.97440	0.310102
$93$	−20.9418	−2.17157
$94$	11.4589	1.18189
$95$	−0.712053	−0.0730551
$96$	9.35359	0.954647
$97$	−5.88747	−0.597782	−0.298891	−	0.954287i	$-0.596617\pi$
$97$	−5.88747	−0.597782	−0.298891	+	0.954287i	$0.596617\pi$
$98$	−1.60447	−0.162076
$99$	11.9310	1.19911
$100$	−2.83622	−0.283622
$101$	12.2737	1.22128	0.610642	−	0.791907i	$-0.290912\pi$
$101$	12.2737	1.22128	0.610642	+	0.791907i	$0.290912\pi$
$102$	6.58414	0.651927
$103$	10.0831	0.993514	0.496757	−	0.867890i	$-0.334524\pi$
$103$	10.0831	0.993514	0.496757	+	0.867890i	$0.334524\pi$
$104$	10.5532	1.03483
$105$	0.736411	0.0718663
$106$	−15.6186	−1.51702
$107$	11.0209	1.06543	0.532713	−	0.846296i	$-0.321172\pi$
$107$	11.0209	1.06543	0.532713	+	0.846296i	$0.321172\pi$
$108$	4.72995	0.455140
$109$	−3.92874	−0.376305	−0.188152	−	0.982140i	$-0.560250\pi$
$109$	−3.92874	−0.376305	−0.188152	+	0.982140i	$0.560250\pi$
$110$	−0.823212	−0.0784902
$111$	28.4489	2.70025
$112$	−4.81881	−0.455335
$113$	5.84341	0.549702	0.274851	−	0.961487i	$-0.411372\pi$
$113$	5.84341	0.549702	0.274851	+	0.961487i	$0.411372\pi$
$114$	13.6204	1.27567
$115$	1.28712	0.120024
$116$	−5.37314	−0.498883
$117$	26.6637	2.46506
$118$	2.36892	0.218076
$119$	−1.38495	−0.126958
$120$	1.68449	0.153773
$121$	−6.73829	−0.612572
$122$	19.1969	1.73801
$123$	23.1714	2.08930
$124$	−4.05929	−0.364535
$125$	−2.47000	−0.220923
$126$	−9.27295	−0.826100
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−13.6503	−1.20652
$129$	30.8714	2.71808
$130$	−1.83974	−0.161355
$131$	5.23380	0.457279	0.228640	−	0.973511i	$-0.426572\pi$
$131$	5.23380	0.457279	0.228640	+	0.973511i	$0.426572\pi$
$132$	3.51312	0.305778
$133$	−2.86500	−0.248427
$134$	23.3593	2.01794
$135$	2.04680	0.176161
$136$	−3.16798	−0.271652
$137$	−2.25200	−0.192402	−0.0962008	−	0.995362i	$-0.530669\pi$
$137$	−2.25200	−0.192402	−0.0962008	+	0.995362i	$0.530669\pi$
$138$	−24.6205	−2.09584
$139$	−21.4076	−1.81577	−0.907885	−	0.419220i	$-0.862304\pi$
$139$	−21.4076	−1.81577	−0.907885	+	0.419220i	$0.862304\pi$
$140$	0.142743	0.0120640
$141$	−21.1613	−1.78210
$142$	−15.7864	−1.32476
$143$	9.52418	0.796452
$144$	−27.8500	−2.32083
$145$	−2.32513	−0.193091
$146$	7.00012	0.579334
$147$	2.96301	0.244385
$148$	5.51442	0.453283
$149$	−5.55086	−0.454744	−0.227372	−	0.973808i	$-0.573013\pi$
$149$	−5.55086	−0.454744	−0.227372	+	0.973808i	$0.573013\pi$
$150$	23.4767	1.91687
$151$	−1.74871	−0.142308	−0.0711540	−	0.997465i	$-0.522668\pi$
$151$	−1.74871	−0.142308	−0.0711540	+	0.997465i	$0.522668\pi$
$152$	−6.55351	−0.531560
$153$	−8.00421	−0.647102
$154$	−3.31226	−0.266910
$155$	−1.75658	−0.141092
$156$	7.85121	0.628600
$157$	8.65739	0.690935	0.345467	−	0.938431i	$-0.387720\pi$
$157$	8.65739	0.690935	0.345467	+	0.938431i	$0.387720\pi$
$158$	3.30751	0.263132
$159$	28.8432	2.28741
$160$	0.784571	0.0620258
$161$	5.17882	0.408149
$162$	−11.3332	−0.890424
$163$	−8.20158	−0.642397	−0.321199	−	0.947012i	$-0.604086\pi$
$163$	−8.20158	−0.642397	−0.321199	+	0.947012i	$0.604086\pi$
$164$	4.49146	0.350724
$165$	1.52024	0.118351
$166$	−24.2280	−1.88046
$167$	−5.68096	−0.439606	−0.219803	−	0.975544i	$-0.570541\pi$
$167$	−5.68096	−0.439606	−0.219803	+	0.975544i	$0.570541\pi$
$168$	6.77770	0.522911
$169$	8.28488	0.637298
$170$	0.552272	0.0423573
$171$	−16.5581	−1.26623
$172$	5.98400	0.456276
$173$	−10.8690	−0.826356	−0.413178	−	0.910650i	$-0.635581\pi$
$173$	−10.8690	−0.826356	−0.413178	+	0.910650i	$0.635581\pi$
$174$	44.4760	3.37172
$175$	−4.93823	−0.373295
$176$	−9.94791	−0.749852
$177$	−4.37472	−0.328824
$178$	20.7804	1.55756
$179$	−20.4990	−1.53217	−0.766083	−	0.642742i	$-0.777797\pi$
$179$	−20.4990	−1.53217	−0.766083	+	0.642742i	$0.777797\pi$
$180$	0.824974	0.0614899
$181$	8.86824	0.659171	0.329586	−	0.944126i	$-0.393091\pi$
$181$	8.86824	0.659171	0.329586	+	0.944126i	$0.393091\pi$
$182$	−7.40233	−0.548697
$183$	−35.4513	−2.62064
$184$	11.8462	0.873316
$185$	2.38627	0.175442
$186$	33.6007	2.46372
$187$	−2.85907	−0.209076
$188$	−4.10182	−0.299156
$189$	8.23548	0.599043
$190$	1.14247	0.0828835
$191$	−18.2214	−1.31846	−0.659228	−	0.751943i	$-0.729117\pi$
$191$	−18.2214	−1.31846	−0.659228	+	0.751943i	$0.729117\pi$
$192$	13.5488	0.977799
$193$	13.8127	0.994263	0.497132	−	0.867675i	$-0.334387\pi$
$193$	13.8127	0.994263	0.497132	+	0.867675i	$0.334387\pi$
$194$	9.44629	0.678204
$195$	3.39747	0.243298
$196$	0.574339	0.0410242
$197$	8.13105	0.579314	0.289657	−	0.957131i	$-0.406459\pi$
$197$	8.13105	0.579314	0.289657	+	0.957131i	$0.406459\pi$
$198$	−19.1430	−1.36043
$199$	−21.0663	−1.49335	−0.746676	−	0.665188i	$-0.768351\pi$
$199$	−21.0663	−1.49335	−0.746676	+	0.665188i	$0.768351\pi$
$200$	−11.2959	−0.798740
$201$	−43.1381	−3.04273
$202$	−19.6929	−1.38559
$203$	−9.35535	−0.656617
$204$	−2.35686	−0.165013
$205$	1.94360	0.135747
$206$	−16.1780	−1.12718
$207$	29.9307	2.08032
$208$	−22.2318	−1.54150
$209$	−5.91449	−0.409113
$210$	−1.18155	−0.0815349
$211$	−22.9120	−1.57733	−0.788663	−	0.614826i	$-0.789226\pi$
$211$	−22.9120	−1.57733	−0.788663	+	0.614826i	$0.789226\pi$
$212$	5.59085	0.383981
$213$	29.1530	1.99753
$214$	−17.6827	−1.20876
$215$	2.58947	0.176600
$216$	18.8382	1.28177
$217$	−7.06776	−0.479791
$218$	6.30356	0.426931
$219$	−12.9272	−0.873542
$220$	0.294677	0.0198671
$221$	−6.38953	−0.429806
$222$	−45.6455	−3.06353
$223$	−0.0417173	−0.00279359	−0.00139680	−	0.999999i	$-0.500445\pi$
$223$	−0.0417173	−0.00279359	−0.00139680	+	0.999999i	$0.500445\pi$
$224$	3.15679	0.210922
$225$	−28.5402	−1.90268
$226$	−9.37560	−0.623656
$227$	26.8446	1.78174	0.890870	−	0.454259i	$-0.150096\pi$
$227$	26.8446	1.78174	0.890870	+	0.454259i	$0.150096\pi$
$228$	−4.87558	−0.322893
$229$	10.6928	0.706602	0.353301	−	0.935510i	$-0.385059\pi$
$229$	10.6928	0.706602	0.353301	+	0.935510i	$0.385059\pi$
$230$	−2.06515	−0.136172
$231$	6.11681	0.402457
$232$	−21.3998	−1.40496
$233$	7.67199	0.502609	0.251304	−	0.967908i	$-0.419140\pi$
$233$	7.67199	0.502609	0.251304	+	0.967908i	$0.419140\pi$
$234$	−42.7813	−2.79670
$235$	−1.77499	−0.115788
$236$	−0.847979	−0.0551987
$237$	−6.10804	−0.396760
$238$	2.22211	0.144038
$239$	4.88761	0.316153	0.158077	−	0.987427i	$-0.449471\pi$
$239$	4.88761	0.316153	0.158077	+	0.987427i	$0.449471\pi$
$240$	−3.54863	−0.229063
$241$	−21.2974	−1.37189	−0.685943	−	0.727656i	$-0.740610\pi$
$241$	−21.2974	−1.37189	−0.685943	+	0.727656i	$0.740610\pi$
$242$	10.8114	0.694984
$243$	−3.77714	−0.242304
$244$	−6.87175	−0.439918
$245$	0.248535	0.0158783
$246$	−37.1780	−2.37038
$247$	−13.2178	−0.841031
$248$	−16.1671	−1.02661
$249$	44.7422	2.83542
$250$	3.96304	0.250645
$251$	−11.3807	−0.718341	−0.359170	−	0.933272i	$-0.616940\pi$
$251$	−11.3807	−0.718341	−0.359170	+	0.933272i	$0.616940\pi$
$252$	3.31935	0.209099
$253$	10.6911	0.672145
$254$	1.60447	0.100674
$255$	−1.01989	−0.0638680
$256$	12.7562	0.797263
$257$	4.17847	0.260646	0.130323	−	0.991472i	$-0.458399\pi$
$257$	4.17847	0.260646	0.130323	+	0.991472i	$0.458399\pi$
$258$	−49.5324	−3.08376
$259$	9.60134	0.596598
$260$	0.658553	0.0408417
$261$	−54.0686	−3.34676
$262$	−8.39749	−0.518799
$263$	−19.8837	−1.22608	−0.613040	−	0.790052i	$-0.710053\pi$
$263$	−19.8837	−1.22608	−0.613040	+	0.790052i	$0.710053\pi$
$264$	13.9918	0.861137
$265$	2.41934	0.148619
$266$	4.59682	0.281849
$267$	−38.3755	−2.34854
$268$	−8.36172	−0.510773
$269$	−1.42998	−0.0871876	−0.0435938	−	0.999049i	$-0.513881\pi$
$269$	−1.42998	−0.0871876	−0.0435938	+	0.999049i	$0.513881\pi$
$270$	−3.28404	−0.199861
$271$	28.2370	1.71527	0.857637	−	0.514255i	$-0.171932\pi$
$271$	28.2370	1.71527	0.857637	+	0.514255i	$0.171932\pi$
$272$	6.67380	0.404659
$273$	13.6700	0.827347
$274$	3.61328	0.218286
$275$	−10.1944	−0.614748
$276$	8.81317	0.530491
$277$	11.5810	0.695834	0.347917	−	0.937525i	$-0.386889\pi$
$277$	11.5810	0.695834	0.347917	+	0.937525i	$0.386889\pi$
$278$	34.3480	2.06005
$279$	−40.8476	−2.44548
$280$	0.568508	0.0339748
$281$	−15.9817	−0.953389	−0.476694	−	0.879069i	$-0.658165\pi$
$281$	−15.9817	−0.953389	−0.476694	+	0.879069i	$0.658165\pi$
$282$	33.9527	2.02186
$283$	−8.61636	−0.512190	−0.256095	−	0.966652i	$-0.582436\pi$
$283$	−8.61636	−0.512190	−0.256095	+	0.966652i	$0.582436\pi$
$284$	5.65091	0.335320
$285$	−2.10982	−0.124975
$286$	−15.2813	−0.903602
$287$	7.82024	0.461614
$288$	18.2444	1.07506
$289$	−15.0819	−0.887172
$290$	3.73061	0.219069
$291$	−17.4446	−1.02262
$292$	−2.50577	−0.146639
$293$	5.27199	0.307993	0.153997	−	0.988071i	$-0.450786\pi$
$293$	5.27199	0.307993	0.153997	+	0.988071i	$0.450786\pi$
$294$	−4.75407	−0.277263
$295$	−0.366948	−0.0213645
$296$	21.9625	1.27654
$297$	17.0013	0.986513
$298$	8.90621	0.515923
$299$	23.8928	1.38176
$300$	−8.40374	−0.485190
$301$	10.4189	0.600538
$302$	2.80576	0.161453
$303$	36.3672	2.08924
$304$	13.8059	0.791823
$305$	−2.97362	−0.170269
$306$	12.8425	0.734159
$307$	−4.86994	−0.277942	−0.138971	−	0.990296i	$-0.544379\pi$
$307$	−4.86994	−0.277942	−0.138971	+	0.990296i	$0.544379\pi$
$308$	1.18566	0.0675592
$309$	29.8762	1.69960
$310$	2.81839	0.160074
$311$	8.28578	0.469844	0.234922	−	0.972014i	$-0.424517\pi$
$311$	8.28578	0.469844	0.234922	+	0.972014i	$0.424517\pi$
$312$	31.2693	1.77027
$313$	−29.8516	−1.68731	−0.843657	−	0.536883i	$-0.819602\pi$
$313$	−29.8516	−1.68731	−0.843657	+	0.536883i	$0.819602\pi$
$314$	−13.8906	−0.783890
$315$	1.43639	0.0809314
$316$	−1.18396	−0.0666029
$317$	15.3754	0.863570	0.431785	−	0.901977i	$-0.357884\pi$
$317$	15.3754	0.863570	0.431785	+	0.901977i	$0.357884\pi$
$318$	−46.2782	−2.59515
$319$	−19.3131	−1.08133
$320$	1.13646	0.0635300
$321$	32.6549	1.82262
$322$	−8.30929	−0.463059
$323$	3.96788	0.220779
$324$	4.05686	0.225381
$325$	−22.7828	−1.26376
$326$	13.1592	0.728822
$327$	−11.6409	−0.643743
$328$	17.8883	0.987716
$329$	−7.14182	−0.393741
$330$	−2.43919	−0.134273
$331$	−31.5246	−1.73275	−0.866374	−	0.499396i	$-0.833555\pi$
$331$	−31.5246	−1.73275	−0.866374	+	0.499396i	$0.833555\pi$
$332$	8.67267	0.475974
$333$	55.4903	3.04085
$334$	9.11496	0.498748
$335$	−3.61839	−0.197694
$336$	−14.2782	−0.778939
$337$	−9.40882	−0.512531	−0.256265	−	0.966606i	$-0.582492\pi$
$337$	−9.40882	−0.512531	−0.256265	+	0.966606i	$0.582492\pi$
$338$	−13.2929	−0.723037
$339$	17.3141	0.940372
$340$	−0.197692	−0.0107213
$341$	−14.5906	−0.790126
$342$	26.5670	1.43658
$343$	1.00000	0.0539949
$344$	23.8327	1.28497
$345$	3.81374	0.205325
$346$	17.4391	0.937530
$347$	18.4781	0.991955	0.495978	−	0.868335i	$-0.334810\pi$
$347$	18.4781	0.991955	0.495978	+	0.868335i	$0.334810\pi$
$348$	−15.9207	−0.853437
$349$	5.28963	0.283147	0.141574	−	0.989928i	$-0.454784\pi$
$349$	5.28963	0.283147	0.141574	+	0.989928i	$0.454784\pi$
$350$	7.92327	0.423516
$351$	37.9948	2.02801
$352$	6.51684	0.347349
$353$	−14.4032	−0.766605	−0.383303	−	0.923623i	$-0.625213\pi$
$353$	−14.4032	−0.766605	−0.383303	+	0.923623i	$0.625213\pi$
$354$	7.01912	0.373062
$355$	2.44533	0.129785
$356$	−7.43855	−0.394243
$357$	−4.10361	−0.217186
$358$	32.8901	1.73830
$359$	6.83211	0.360585	0.180293	−	0.983613i	$-0.442296\pi$
$359$	6.83211	0.360585	0.180293	+	0.983613i	$0.442296\pi$
$360$	3.28565	0.173169
$361$	−10.7918	−0.567987
$362$	−14.2289	−0.747853
$363$	−19.9656	−1.04792
$364$	2.64974	0.138884
$365$	−1.08433	−0.0567562
$366$	56.8807	2.97320
$367$	−36.1081	−1.88483	−0.942413	−	0.334451i	$-0.891449\pi$
$367$	−36.1081	−1.88483	−0.942413	+	0.334451i	$0.891449\pi$
$368$	−24.9558	−1.30091
$369$	45.1965	2.35284
$370$	−3.82870	−0.199045
$371$	9.73442	0.505386
$372$	−12.0277	−0.623608
$373$	−13.9607	−0.722858	−0.361429	−	0.932400i	$-0.617711\pi$
$373$	−13.9607	−0.722858	−0.361429	+	0.932400i	$0.617711\pi$
$374$	4.58731	0.237204
$375$	−7.31862	−0.377932
$376$	−16.3365	−0.842489
$377$	−43.1614	−2.22293
$378$	−13.2136	−0.679635
$379$	17.2040	0.883712	0.441856	−	0.897086i	$-0.354320\pi$
$379$	17.2040	0.883712	0.441856	+	0.897086i	$0.354320\pi$
$380$	−0.408959	−0.0209792
$381$	−2.96301	−0.151800
$382$	29.2358	1.49583
$383$	−3.16174	−0.161558	−0.0807788	−	0.996732i	$-0.525741\pi$
$383$	−3.16174	−0.161558	−0.0807788	+	0.996732i	$0.525741\pi$
$384$	−40.4459	−2.06399
$385$	0.513073	0.0261486
$386$	−22.1622	−1.12803
$387$	60.2156	3.06093
$388$	−3.38140	−0.171665
$389$	3.60627	0.182845	0.0914226	−	0.995812i	$-0.470859\pi$
$389$	3.60627	0.182845	0.0914226	+	0.995812i	$0.470859\pi$
$390$	−5.45116	−0.276030
$391$	−7.17240	−0.362724
$392$	2.28744	0.115533
$393$	15.5078	0.782265
$394$	−13.0461	−0.657251
$395$	−0.512337	−0.0257785
$396$	6.85243	0.344348
$397$	−1.35977	−0.0682447	−0.0341224	−	0.999418i	$-0.510864\pi$
$397$	−1.35977	−0.0682447	−0.0341224	+	0.999418i	$0.510864\pi$
$398$	33.8004	1.69426
$399$	−8.48903	−0.424983
$400$	23.7964	1.18982
$401$	18.2667	0.912195	0.456097	−	0.889930i	$-0.349247\pi$
$401$	18.2667	0.912195	0.456097	+	0.889930i	$0.349247\pi$
$402$	69.2140	3.45208
$403$	−32.6075	−1.62429
$404$	7.04928	0.350715
$405$	1.75553	0.0872331
$406$	15.0104	0.744955
$407$	19.8209	0.982487
$408$	−9.38676	−0.464714
$409$	24.3580	1.20443	0.602213	−	0.798335i	$-0.294286\pi$
$409$	24.3580	1.20443	0.602213	+	0.798335i	$0.294286\pi$
$410$	−3.11846	−0.154010
$411$	−6.67271	−0.329140
$412$	5.79109	0.285307
$413$	−1.47644	−0.0726511
$414$	−48.0230	−2.36020
$415$	3.75294	0.184225
$416$	14.5640	0.714059
$417$	−63.4310	−3.10623
$418$	9.48964	0.464154
$419$	4.53561	0.221579	0.110789	−	0.993844i	$-0.464662\pi$
$419$	4.53561	0.221579	0.110789	+	0.993844i	$0.464662\pi$
$420$	0.422949	0.0206378
$421$	−1.42660	−0.0695283	−0.0347642	−	0.999396i	$-0.511068\pi$
$421$	−1.42660	−0.0695283	−0.0347642	+	0.999396i	$0.511068\pi$
$422$	36.7617	1.78953
$423$	−41.2757	−2.00689
$424$	22.2669	1.08137
$425$	6.83919	0.331749
$426$	−46.7752	−2.26627
$427$	−11.9646	−0.579009
$428$	6.32970	0.305958
$429$	28.2202	1.36249
$430$	−4.15474	−0.200359
$431$	26.6902	1.28562	0.642812	−	0.766024i	$-0.277768\pi$
$431$	26.6902	1.28562	0.642812	+	0.766024i	$0.277768\pi$
$432$	−39.6852	−1.90936
$433$	−4.65866	−0.223881	−0.111940	−	0.993715i	$-0.535707\pi$
$433$	−4.65866	−0.223881	−0.111940	+	0.993715i	$0.535707\pi$
$434$	11.3400	0.544339
$435$	−6.88938	−0.330320
$436$	−2.25643	−0.108063
$437$	−14.8373	−0.709767
$438$	20.7414	0.991064
$439$	18.5623	0.885929	0.442964	−	0.896539i	$-0.353927\pi$
$439$	18.5623	0.885929	0.442964	+	0.896539i	$0.353927\pi$
$440$	1.17362	0.0559502
$441$	5.77943	0.275211
$442$	10.2518	0.487630
$443$	26.2295	1.24620	0.623101	−	0.782142i	$-0.285873\pi$
$443$	26.2295	1.24620	0.623101	+	0.782142i	$0.285873\pi$
$444$	16.3393	0.775428
$445$	−3.21890	−0.152591
$446$	0.0669343	0.00316943
$447$	−16.4473	−0.777928
$448$	4.57264	0.216037
$449$	−11.5318	−0.544220	−0.272110	−	0.962266i	$-0.587721\pi$
$449$	−11.5318	−0.544220	−0.272110	+	0.962266i	$0.587721\pi$
$450$	45.7920	2.15865
$451$	16.1440	0.760192
$452$	3.35609	0.157857
$453$	−5.18144	−0.243445
$454$	−43.0715	−2.02145
$455$	1.14663	0.0537548
$456$	−19.4181	−0.909337
$457$	25.5261	1.19406	0.597031	−	0.802218i	$-0.296347\pi$
$457$	25.5261	1.19406	0.597031	+	0.802218i	$0.296347\pi$
$458$	−17.1564	−0.801664
$459$	−11.4057	−0.532373
$460$	0.739241	0.0344673
$461$	−7.68056	−0.357719	−0.178860	−	0.983875i	$-0.557241\pi$
$461$	−7.68056	−0.357719	−0.178860	+	0.983875i	$0.557241\pi$
$462$	−9.81427	−0.456601
$463$	−0.801721	−0.0372591	−0.0186296	−	0.999826i	$-0.505930\pi$
$463$	−0.801721	−0.0372591	−0.0186296	+	0.999826i	$0.505930\pi$
$464$	45.0817	2.09286
$465$	−5.20477	−0.241366
$466$	−12.3095	−0.570227
$467$	−31.8617	−1.47439	−0.737193	−	0.675682i	$-0.763849\pi$
$467$	−31.8617	−1.47439	−0.737193	+	0.675682i	$0.763849\pi$
$468$	15.3140	0.707890
$469$	−14.5589	−0.672266
$470$	2.84793	0.131365
$471$	25.6519	1.18198
$472$	−3.37727	−0.155452
$473$	21.5088	0.988974
$474$	9.80019	0.450138
$475$	14.1480	0.649157
$476$	−0.795429	−0.0364584
$477$	56.2594	2.57594
$478$	−7.84205	−0.358687
$479$	26.2469	1.19925	0.599627	−	0.800280i	$-0.295316\pi$
$479$	26.2469	1.19925	0.599627	+	0.800280i	$0.295316\pi$
$480$	2.32469	0.106107
$481$	44.2963	2.01974
$482$	34.1711	1.55645
$483$	15.3449	0.698218
$484$	−3.87006	−0.175912
$485$	−1.46324	−0.0664423
$486$	6.06033	0.274902
$487$	9.60895	0.435423	0.217712	−	0.976013i	$-0.430141\pi$
$487$	9.60895	0.435423	0.217712	+	0.976013i	$0.430141\pi$
$488$	−27.3683	−1.23891
$489$	−24.3014	−1.09895
$490$	−0.398768	−0.0180145
$491$	5.73767	0.258938	0.129469	−	0.991584i	$-0.458673\pi$
$491$	5.73767	0.258938	0.129469	+	0.991584i	$0.458673\pi$
$492$	13.3083	0.599982
$493$	12.9567	0.583539
$494$	21.2077	0.954179
$495$	2.96527	0.133279
$496$	34.0582	1.52926
$497$	9.83898	0.441339
$498$	−71.7878	−3.21689
$499$	−17.7188	−0.793201	−0.396601	−	0.917991i	$-0.629810\pi$
$499$	−17.7188	−0.793201	−0.396601	+	0.917991i	$0.629810\pi$
$500$	−1.41861	−0.0634423
$501$	−16.8328	−0.752032
$502$	18.2600	0.814983
$503$	5.84637	0.260677	0.130338	−	0.991470i	$-0.458394\pi$
$503$	5.84637	0.260677	0.130338	+	0.991470i	$0.458394\pi$
$504$	13.2201	0.588869
$505$	3.05045	0.135743
$506$	−17.1536	−0.762572
$507$	24.5482	1.09022
$508$	−0.574339	−0.0254822
$509$	−42.4028	−1.87947	−0.939736	−	0.341902i	$-0.888929\pi$
$509$	−42.4028	−1.87947	−0.939736	+	0.341902i	$0.888929\pi$
$510$	1.63639	0.0724605
$511$	−4.36287	−0.193002
$512$	6.83350	0.302001
$513$	−23.5947	−1.04173
$514$	−6.70425	−0.295712
$515$	2.50599	0.110427
$516$	17.7307	0.780549
$517$	−14.7435	−0.648419
$518$	−15.4051	−0.676862
$519$	−32.2050	−1.41364
$520$	2.62284	0.115019
$521$	−43.4989	−1.90572	−0.952861	−	0.303408i	$-0.901875\pi$
$521$	−43.4989	−1.90572	−0.952861	+	0.303408i	$0.901875\pi$
$522$	86.7517	3.79702
$523$	14.8939	0.651264	0.325632	−	0.945497i	$-0.394423\pi$
$523$	14.8939	0.651264	0.325632	+	0.945497i	$0.394423\pi$
$524$	3.00597	0.131316
$525$	−14.6320	−0.638594
$526$	31.9028	1.39103
$527$	9.78847	0.426393
$528$	−29.4758	−1.28277
$529$	3.82022	0.166097
$530$	−3.88177	−0.168613
$531$	−8.53301	−0.370301
$532$	−1.64548	−0.0713407
$533$	36.0791	1.56276
$534$	61.5725	2.66450
$535$	2.73907	0.118420
$536$	−33.3025	−1.43845
$537$	−60.7387	−2.62107
$538$	2.29437	0.0989173
$539$	2.06439	0.0889196
$540$	1.17556	0.0505880
$541$	−14.6373	−0.629305	−0.314652	−	0.949207i	$-0.601888\pi$
$541$	−14.6373	−0.629305	−0.314652	+	0.949207i	$0.601888\pi$
$542$	−45.3055	−1.94604
$543$	26.2767	1.12764
$544$	−4.37198	−0.187447
$545$	−0.976428	−0.0418256
$546$	−21.9332	−0.938653
$547$	10.2687	0.439059	0.219529	−	0.975606i	$-0.429548\pi$
$547$	10.2687	0.439059	0.219529	+	0.975606i	$0.429548\pi$
$548$	−1.29341	−0.0552518
$549$	−69.1487	−2.95120
$550$	16.3567	0.697453
$551$	26.8031	1.14185
$552$	35.1005	1.49398
$553$	−2.06143	−0.0876609
$554$	−18.5814	−0.789448
$555$	7.07053	0.300127
$556$	−12.2952	−0.521433
$557$	16.1907	0.686023	0.343012	−	0.939331i	$-0.388553\pi$
$557$	16.1907	0.686023	0.343012	+	0.939331i	$0.388553\pi$
$558$	65.5390	2.77449
$559$	48.0684	2.03308
$560$	−1.19764	−0.0506096
$561$	−8.47146	−0.357665
$562$	25.6422	1.08165
$563$	16.6969	0.703692	0.351846	−	0.936058i	$-0.385554\pi$
$563$	16.6969	0.703692	0.351846	+	0.936058i	$0.385554\pi$
$564$	−12.1537	−0.511765
$565$	1.45229	0.0610983
$566$	13.8247	0.581097
$567$	7.06353	0.296640
$568$	22.5061	0.944333
$569$	−13.1417	−0.550928	−0.275464	−	0.961311i	$-0.588832\pi$
$569$	−13.1417	−0.550928	−0.275464	+	0.961311i	$0.588832\pi$
$570$	3.38515	0.141788
$571$	5.86566	0.245470	0.122735	−	0.992439i	$-0.460833\pi$
$571$	5.86566	0.245470	0.122735	+	0.992439i	$0.460833\pi$
$572$	5.47010	0.228716
$573$	−53.9903	−2.25548
$574$	−12.5474	−0.523717
$575$	−25.5742	−1.06652
$576$	26.4273	1.10114
$577$	32.8604	1.36800	0.683998	−	0.729484i	$-0.260240\pi$
$577$	32.8604	1.36800	0.683998	+	0.729484i	$0.260240\pi$
$578$	24.1986	1.00653
$579$	40.9273	1.70088
$580$	−1.33541	−0.0554499
$581$	15.1003	0.626465
$582$	27.9895	1.16020
$583$	20.0956	0.832276
$584$	−9.97980	−0.412967
$585$	6.62686	0.273987
$586$	−8.45878	−0.349429
$587$	−1.60407	−0.0662069	−0.0331035	−	0.999452i	$-0.510539\pi$
$587$	−1.60407	−0.0662069	−0.0331035	+	0.999452i	$0.510539\pi$
$588$	1.70177	0.0701799
$589$	20.2491	0.834352
$590$	0.588758	0.0242388
$591$	24.0924	0.991029
$592$	−46.2671	−1.90156
$593$	−16.5682	−0.680376	−0.340188	−	0.940357i	$-0.610491\pi$
$593$	−16.5682	−0.680376	−0.340188	+	0.940357i	$0.610491\pi$
$594$	−27.2781	−1.11923
$595$	−0.344207	−0.0141111
$596$	−3.18807	−0.130589
$597$	−62.4197	−2.55467
$598$	−38.3354	−1.56765
$599$	21.7895	0.890295	0.445147	−	0.895457i	$-0.353151\pi$
$599$	21.7895	0.890295	0.445147	+	0.895457i	$0.353151\pi$
$600$	−33.4699	−1.36640
$601$	19.1184	0.779856	0.389928	−	0.920845i	$-0.372500\pi$
$601$	19.1184	0.779856	0.389928	+	0.920845i	$0.372500\pi$
$602$	−16.7169	−0.681331
$603$	−84.1420	−3.42653
$604$	−1.00435	−0.0408665
$605$	−1.67470	−0.0680862
$606$	−58.3503	−2.37032
$607$	9.16116	0.371840	0.185920	−	0.982565i	$-0.440473\pi$
$607$	9.16116	0.371840	0.185920	+	0.982565i	$0.440473\pi$
$608$	−9.04420	−0.366791
$609$	−27.7200	−1.12327
$610$	4.77110	0.193176
$611$	−32.9492	−1.33298
$612$	−4.59712	−0.185828
$613$	−19.8971	−0.803636	−0.401818	−	0.915720i	$-0.631621\pi$
$613$	−19.8971	−0.803636	−0.401818	+	0.915720i	$0.631621\pi$
$614$	7.81369	0.315335
$615$	5.75891	0.232222
$616$	4.72216	0.190261
$617$	10.7473	0.432672	0.216336	−	0.976319i	$-0.430589\pi$
$617$	10.7473	0.432672	0.216336	+	0.976319i	$0.430589\pi$
$618$	−47.9356	−1.92825
$619$	30.7703	1.23676	0.618381	−	0.785878i	$-0.287789\pi$
$619$	30.7703	1.23676	0.618381	+	0.785878i	$0.287789\pi$
$620$	−1.00887	−0.0405173
$621$	42.6501	1.71149
$622$	−13.2943	−0.533054
$623$	−12.9515	−0.518891
$624$	−65.8732	−2.63704
$625$	24.0773	0.963091
$626$	47.8962	1.91432
$627$	−17.5247	−0.699868
$628$	4.97227	0.198415
$629$	−13.2974	−0.530200
$630$	−2.30465	−0.0918194
$631$	30.5498	1.21617	0.608085	−	0.793872i	$-0.291938\pi$
$631$	30.5498	1.21617	0.608085	+	0.793872i	$0.291938\pi$
$632$	−4.71539	−0.187568
$633$	−67.8884	−2.69832
$634$	−24.6695	−0.979750
$635$	−0.248535	−0.00986280
$636$	16.5658	0.656875
$637$	4.61355	0.182796
$638$	30.9874	1.22680
$639$	56.8637	2.24949
$640$	−3.39256	−0.134103
$641$	−14.6914	−0.580276	−0.290138	−	0.956985i	$-0.593701\pi$
$641$	−14.6914	−0.580276	−0.290138	+	0.956985i	$0.593701\pi$
$642$	−52.3940	−2.06783
$643$	−10.9887	−0.433353	−0.216677	−	0.976243i	$-0.569522\pi$
$643$	−10.9887	−0.433353	−0.216677	+	0.976243i	$0.569522\pi$
$644$	2.97440	0.117208
$645$	7.67262	0.302109
$646$	−6.36636	−0.250481
$647$	1.94049	0.0762884	0.0381442	−	0.999272i	$-0.487855\pi$
$647$	1.94049	0.0762884	0.0381442	+	0.999272i	$0.487855\pi$
$648$	16.1574	0.634722
$649$	−3.04796	−0.119643
$650$	36.5544	1.43378
$651$	−20.9418	−0.820775
$652$	−4.71048	−0.184477
$653$	−30.5882	−1.19701	−0.598504	−	0.801119i	$-0.704238\pi$
$653$	−30.5882	−1.19701	−0.598504	+	0.801119i	$0.704238\pi$
$654$	18.6775	0.730349
$655$	1.30078	0.0508257
$656$	−37.6843	−1.47132
$657$	−25.2149	−0.983728
$658$	11.4589	0.446713
$659$	6.09333	0.237363	0.118681	−	0.992932i	$-0.462133\pi$
$659$	6.09333	0.237363	0.118681	+	0.992932i	$0.462133\pi$
$660$	0.873132	0.0339866
$661$	31.6109	1.22952	0.614761	−	0.788714i	$-0.289253\pi$
$661$	31.6109	1.22952	0.614761	+	0.788714i	$0.289253\pi$
$662$	50.5804	1.96586
$663$	−18.9322	−0.735267
$664$	34.5409	1.34045
$665$	−0.712053	−0.0276122
$666$	−89.0328	−3.44995
$667$	−48.4497	−1.87598
$668$	−3.26280	−0.126241
$669$	−0.123609	−0.00477899
$670$	5.80561	0.224290
$671$	−24.6997	−0.953520
$672$	9.35359	0.360823
$673$	−45.7773	−1.76458	−0.882292	−	0.470702i	$-0.844001\pi$
$673$	−45.7773	−1.76458	−0.882292	+	0.470702i	$0.844001\pi$
$674$	15.0962	0.581484
$675$	−40.6687	−1.56534
$676$	4.75833	0.183013
$677$	−39.1821	−1.50589	−0.752944	−	0.658084i	$-0.771367\pi$
$677$	−39.1821	−1.50589	−0.752944	+	0.658084i	$0.771367\pi$
$678$	−27.7800	−1.06688
$679$	−5.88747	−0.225940
$680$	−0.787353	−0.0301936
$681$	79.5409	3.04801
$682$	23.4103	0.896426
$683$	−31.2759	−1.19674	−0.598369	−	0.801221i	$-0.704184\pi$
$683$	−31.2759	−1.19674	−0.598369	+	0.801221i	$0.704184\pi$
$684$	−9.50995	−0.363622
$685$	−0.559701	−0.0213851
$686$	−1.60447	−0.0612591
$687$	31.6830	1.20878
$688$	−50.2069	−1.91412
$689$	44.9103	1.71094
$690$	−6.11905	−0.232948
$691$	−0.988316	−0.0375973	−0.0187986	−	0.999823i	$-0.505984\pi$
$691$	−0.988316	−0.0375973	−0.0187986	+	0.999823i	$0.505984\pi$
$692$	−6.24250	−0.237304
$693$	11.9310	0.453221
$694$	−29.6476	−1.12541
$695$	−5.32053	−0.201819
$696$	−63.4078	−2.40346
$697$	−10.8306	−0.410239
$698$	−8.48707	−0.321240
$699$	22.7322	0.859810
$700$	−2.83622	−0.107199
$701$	10.9677	0.414244	0.207122	−	0.978315i	$-0.433590\pi$
$701$	10.9677	0.414244	0.207122	+	0.978315i	$0.433590\pi$
$702$	−60.9618	−2.30085
$703$	−27.5079	−1.03748
$704$	9.43972	0.355773
$705$	−5.25931	−0.198077
$706$	23.1096	0.869740
$707$	12.2737	0.461602
$708$	−2.51257	−0.0944282
$709$	46.4954	1.74617	0.873086	−	0.487567i	$-0.162116\pi$
$709$	46.4954	1.74617	0.873086	+	0.487567i	$0.162116\pi$
$710$	−3.92347	−0.147245
$711$	−11.9139	−0.446806
$712$	−29.6258	−1.11027
$713$	−36.6027	−1.37078
$714$	6.58414	0.246405
$715$	2.36709	0.0885241
$716$	−11.7734	−0.439991
$717$	14.4820	0.540842
$718$	−10.9620	−0.409096
$719$	−23.9536	−0.893319	−0.446660	−	0.894704i	$-0.647386\pi$
$719$	−23.9536	−0.893319	−0.446660	+	0.894704i	$0.647386\pi$
$720$	−6.92169	−0.257956
$721$	10.0831	0.375513
$722$	17.3151	0.644401
$723$	−63.1044	−2.34688
$724$	5.09337	0.189294
$725$	46.1989	1.71578
$726$	32.0343	1.18891
$727$	11.2434	0.416995	0.208498	−	0.978023i	$-0.433143\pi$
$727$	11.2434	0.416995	0.208498	+	0.978023i	$0.433143\pi$
$728$	10.5532	0.391128
$729$	−32.3823	−1.19934
$730$	1.73977	0.0643919
$731$	−14.4297	−0.533701
$732$	−20.3611	−0.752566
$733$	3.07023	0.113402	0.0567008	−	0.998391i	$-0.481942\pi$
$733$	3.07023	0.113402	0.0567008	+	0.998391i	$0.481942\pi$
$734$	57.9345	2.13840
$735$	0.736411	0.0271629
$736$	16.3484	0.602612
$737$	−30.0552	−1.10710
$738$	−72.5167	−2.66938
$739$	0.267916	0.00985545	0.00492772	−	0.999988i	$-0.498431\pi$
$739$	0.267916	0.00985545	0.00492772	+	0.999988i	$0.498431\pi$
$740$	1.37053	0.0503815
$741$	−39.1646	−1.43875
$742$	−15.6186	−0.573378
$743$	14.7553	0.541320	0.270660	−	0.962675i	$-0.412758\pi$
$743$	14.7553	0.541320	0.270660	+	0.962675i	$0.412758\pi$
$744$	−47.9032	−1.75621
$745$	−1.37958	−0.0505439
$746$	22.3996	0.820107
$747$	87.2709	3.19308
$748$	−1.64208	−0.0600402
$749$	11.0209	0.402693
$750$	11.7425	0.428777
$751$	31.8578	1.16251	0.581253	−	0.813723i	$-0.302563\pi$
$751$	31.8578	1.16251	0.581253	+	0.813723i	$0.302563\pi$
$752$	34.4151	1.25499
$753$	−33.7210	−1.22886
$754$	69.2514	2.52199
$755$	−0.434615	−0.0158173
$756$	4.72995	0.172027
$757$	44.9482	1.63367	0.816835	−	0.576872i	$-0.195727\pi$
$757$	44.9482	1.63367	0.816835	+	0.576872i	$0.195727\pi$
$758$	−27.6034	−1.00260
$759$	31.6779	1.14983
$760$	−1.62878	−0.0590819
$761$	2.11329	0.0766067	0.0383033	−	0.999266i	$-0.487805\pi$
$761$	2.11329	0.0766067	0.0383033	+	0.999266i	$0.487805\pi$
$762$	4.75407	0.172222
$763$	−3.92874	−0.142230
$764$	−10.4653	−0.378620
$765$	−1.98932	−0.0719241
$766$	5.07294	0.183293
$767$	−6.81165	−0.245955
$768$	37.7968	1.36387
$769$	−45.5871	−1.64391	−0.821956	−	0.569550i	$-0.807117\pi$
$769$	−45.5871	−1.64391	−0.821956	+	0.569550i	$0.807117\pi$
$770$	−0.823212	−0.0296665
$771$	12.3808	0.445885
$772$	7.93319	0.285522
$773$	26.2249	0.943244	0.471622	−	0.881801i	$-0.343669\pi$
$773$	26.2249	0.943244	0.471622	+	0.881801i	$0.343669\pi$
$774$	−96.6144	−3.47273
$775$	34.9022	1.25372
$776$	−13.4672	−0.483445
$777$	28.4489	1.02060
$778$	−5.78617	−0.207444
$779$	−22.4050	−0.802742
$780$	1.95130	0.0698677
$781$	20.3115	0.726803
$782$	11.5079	0.411523
$783$	−77.0458	−2.75339
$784$	−4.81881	−0.172100
$785$	2.15166	0.0767961
$786$	−24.8819	−0.887507
$787$	26.9352	0.960137	0.480069	−	0.877231i	$-0.340612\pi$
$787$	26.9352	0.960137	0.480069	+	0.877231i	$0.340612\pi$
$788$	4.66998	0.166361
$789$	−58.9155	−2.09745
$790$	0.822031	0.0292466
$791$	5.84341	0.207768
$792$	27.2914	0.969758
$793$	−55.1994	−1.96019
$794$	2.18171	0.0774260
$795$	7.16853	0.254242
$796$	−12.0992	−0.428845
$797$	−9.96337	−0.352921	−0.176460	−	0.984308i	$-0.556465\pi$
$797$	−9.96337	−0.352921	−0.176460	+	0.984308i	$0.556465\pi$
$798$	13.6204	0.482158
$799$	9.89104	0.349920
$800$	−15.5889	−0.551152
$801$	−74.8524	−2.64478
$802$	−29.3084	−1.03492
$803$	−9.00668	−0.317839
$804$	−24.7759	−0.873778
$805$	1.28712	0.0453649
$806$	52.3179	1.84282
$807$	−4.23705	−0.149151
$808$	28.0754	0.987690
$809$	−12.3868	−0.435495	−0.217747	−	0.976005i	$-0.569871\pi$
$809$	−12.3868	−0.435495	−0.217747	+	0.976005i	$0.569871\pi$
$810$	−2.81671	−0.0989689
$811$	45.3995	1.59419	0.797096	−	0.603853i	$-0.206368\pi$
$811$	45.3995	1.59419	0.797096	+	0.603853i	$0.206368\pi$
$812$	−5.37314	−0.188560
$813$	83.6665	2.93431
$814$	−31.8022	−1.11467
$815$	−2.03838	−0.0714013
$816$	19.7745	0.692247
$817$	−29.8503	−1.04433
$818$	−39.0818	−1.36646
$819$	26.6637	0.931706
$820$	1.11628	0.0389823
$821$	36.9896	1.29095	0.645474	−	0.763782i	$-0.276660\pi$
$821$	36.9896	1.29095	0.645474	+	0.763782i	$0.276660\pi$
$822$	10.7062	0.373421
$823$	−4.32363	−0.150712	−0.0753561	−	0.997157i	$-0.524009\pi$
$823$	−4.32363	−0.150712	−0.0753561	+	0.997157i	$0.524009\pi$
$824$	23.0644	0.803486
$825$	−30.2062	−1.05165
$826$	2.36892	0.0824252
$827$	1.50024	0.0521684	0.0260842	−	0.999660i	$-0.491696\pi$
$827$	1.50024	0.0521684	0.0260842	+	0.999660i	$0.491696\pi$
$828$	17.1903	0.597405
$829$	−39.3301	−1.36599	−0.682995	−	0.730423i	$-0.739323\pi$
$829$	−39.3301	−1.36599	−0.682995	+	0.730423i	$0.739323\pi$
$830$	−6.02150	−0.209009
$831$	34.3146	1.19036
$832$	21.0961	0.731377
$833$	−1.38495	−0.0479856
$834$	101.773	3.52412
$835$	−1.41192	−0.0488614
$836$	−3.39692	−0.117485
$837$	−58.2064	−2.01191
$838$	−7.27727	−0.251389
$839$	−3.56845	−0.123197	−0.0615983	−	0.998101i	$-0.519620\pi$
$839$	−3.56845	−0.123197	−0.0615983	+	0.998101i	$0.519620\pi$
$840$	1.68449	0.0581206
$841$	58.5225	2.01802
$842$	2.28895	0.0788823
$843$	−47.3540	−1.63096
$844$	−13.1592	−0.452959
$845$	2.05908	0.0708345
$846$	66.2257	2.27689
$847$	−6.73829	−0.231530
$848$	−46.9083	−1.61084
$849$	−25.5304	−0.876200
$850$	−10.9733	−0.376381
$851$	49.7237	1.70451
$852$	16.7437	0.573629
$853$	23.6630	0.810205	0.405102	−	0.914271i	$-0.367236\pi$
$853$	23.6630	0.810205	0.405102	+	0.914271i	$0.367236\pi$
$854$	19.1969	0.656906
$855$	−4.11526	−0.140739
$856$	25.2095	0.861644
$857$	−31.5762	−1.07862	−0.539311	−	0.842107i	$-0.681315\pi$
$857$	−31.5762	−1.07862	−0.539311	+	0.842107i	$0.681315\pi$
$858$	−45.2787	−1.54579
$859$	27.0910	0.924333	0.462167	−	0.886793i	$-0.347072\pi$
$859$	27.0910	0.924333	0.462167	+	0.886793i	$0.347072\pi$
$860$	1.48723	0.0507142
$861$	23.1714	0.789681
$862$	−42.8238	−1.45858
$863$	25.8856	0.881157	0.440579	−	0.897714i	$-0.354773\pi$
$863$	25.8856	0.881157	0.440579	+	0.897714i	$0.354773\pi$
$864$	25.9977	0.884458
$865$	−2.70133	−0.0918479
$866$	7.47470	0.254001
$867$	−44.6879	−1.51768
$868$	−4.05929	−0.137781
$869$	−4.25560	−0.144361
$870$	11.0538	0.374760
$871$	−67.1681	−2.27591
$872$	−8.98675	−0.304330
$873$	−34.0262	−1.15161
$874$	23.8061	0.805255
$875$	−2.47000	−0.0835011
$876$	−7.42461	−0.250854
$877$	32.1756	1.08649	0.543246	−	0.839574i	$-0.317195\pi$
$877$	32.1756	1.08649	0.543246	+	0.839574i	$0.317195\pi$
$878$	−29.7827	−1.00512
$879$	15.6210	0.526882
$880$	−2.47240	−0.0833446
$881$	−14.9202	−0.502673	−0.251337	−	0.967900i	$-0.580870\pi$
$881$	−14.9202	−0.502673	−0.251337	+	0.967900i	$0.580870\pi$
$882$	−9.27295	−0.312236
$883$	−37.8634	−1.27420	−0.637102	−	0.770780i	$-0.719867\pi$
$883$	−37.8634	−1.27420	−0.637102	+	0.770780i	$0.719867\pi$
$884$	−3.66975	−0.123427
$885$	−1.08727	−0.0365482
$886$	−42.0846	−1.41386
$887$	−3.91895	−0.131585	−0.0657927	−	0.997833i	$-0.520958\pi$
$887$	−3.91895	−0.131585	−0.0657927	+	0.997833i	$0.520958\pi$
$888$	65.0750	2.18378
$889$	−1.00000	−0.0335389
$890$	5.16464	0.173119
$891$	14.5819	0.488511
$892$	−0.0239598	−0.000802234 0
$893$	20.4613	0.684713
$894$	26.3892	0.882587
$895$	−5.09471	−0.170297
$896$	−13.6503	−0.456023
$897$	70.7946	2.36376
$898$	18.5025	0.617437
$899$	66.1213	2.20527
$900$	−16.3917	−0.546390
$901$	−13.4817	−0.449139
$902$	−25.9027	−0.862465
$903$	30.8714	1.02734
$904$	13.3664	0.444561
$905$	2.20407	0.0732656
$906$	8.31349	0.276197
$907$	−2.40898	−0.0799890	−0.0399945	−	0.999200i	$-0.512734\pi$
$907$	−2.40898	−0.0799890	−0.0399945	+	0.999200i	$0.512734\pi$
$908$	15.4179	0.511661
$909$	70.9353	2.35277
$910$	−1.83974	−0.0609866
$911$	−2.93744	−0.0973218	−0.0486609	−	0.998815i	$-0.515495\pi$
$911$	−2.93744	−0.0973218	−0.0486609	+	0.998815i	$0.515495\pi$
$912$	40.9071	1.35457
$913$	31.1728	1.03167
$914$	−40.9560	−1.35470
$915$	−8.81088	−0.291279
$916$	6.14130	0.202914
$917$	5.23380	0.172835
$918$	18.3002	0.603996
$919$	26.8776	0.886609	0.443304	−	0.896371i	$-0.353806\pi$
$919$	26.8776	0.886609	0.443304	+	0.896371i	$0.353806\pi$
$920$	2.94420	0.0970674
$921$	−14.4297	−0.475474
$922$	12.3233	0.405845
$923$	45.3927	1.49412
$924$	3.51312	0.115573
$925$	−47.4136	−1.55895
$926$	1.28634	0.0422718
$927$	58.2744	1.91398
$928$	−29.5328	−0.969463
$929$	−51.6928	−1.69599	−0.847993	−	0.530007i	$-0.822189\pi$
$929$	−51.6928	−1.69599	−0.847993	+	0.530007i	$0.822189\pi$
$930$	8.35093	0.273838
$931$	−2.86500	−0.0938967
$932$	4.40632	0.144334
$933$	24.5509	0.803759
$934$	51.1214	1.67274
$935$	−0.710579	−0.0232384
$936$	60.9916	1.99357
$937$	−45.0224	−1.47082	−0.735410	−	0.677623i	$-0.763010\pi$
$937$	−45.0224	−1.47082	−0.735410	+	0.677623i	$0.763010\pi$
$938$	23.3593	0.762710
$939$	−88.4507	−2.88648
$940$	−1.01945	−0.0332506
$941$	7.32770	0.238876	0.119438	−	0.992842i	$-0.461891\pi$
$941$	7.32770	0.238876	0.119438	+	0.992842i	$0.461891\pi$
$942$	−41.1579	−1.34100
$943$	40.4996	1.31885
$944$	7.11471	0.231564
$945$	2.04680	0.0665825
$946$	−34.5103	−1.12203
$947$	42.2672	1.37350	0.686749	−	0.726894i	$-0.259037\pi$
$947$	42.2672	1.37350	0.686749	+	0.726894i	$0.259037\pi$
$948$	−3.50808	−0.113937
$949$	−20.1284	−0.653394
$950$	−22.7002	−0.736491
$951$	45.5575	1.47730
$952$	−3.16798	−0.102675
$953$	52.4838	1.70012	0.850059	−	0.526688i	$-0.176566\pi$
$953$	52.4838	1.70012	0.850059	+	0.526688i	$0.176566\pi$
$954$	−90.2668	−2.92249
$955$	−4.52865	−0.146544
$956$	2.80714	0.0907895
$957$	−57.2249	−1.84982
$958$	−42.1126	−1.36060
$959$	−2.25200	−0.0727210
$960$	3.36734	0.108681
$961$	18.9532	0.611394
$962$	−71.0723	−2.29146
$963$	63.6943	2.05252
$964$	−12.2319	−0.393963
$965$	3.43295	0.110510
$966$	−24.6205	−0.792152
$967$	27.7940	0.893796	0.446898	−	0.894585i	$-0.352529\pi$
$967$	27.7940	0.893796	0.446898	+	0.894585i	$0.352529\pi$
$968$	−15.4134	−0.495406
$969$	11.7569	0.377685
$970$	2.34773	0.0753811
$971$	51.6768	1.65839	0.829194	−	0.558961i	$-0.188800\pi$
$971$	51.6768	1.65839	0.829194	+	0.558961i	$0.188800\pi$
$972$	−2.16936	−0.0695822
$973$	−21.4076	−0.686296
$974$	−15.4173	−0.494003
$975$	−67.5056	−2.16191
$976$	57.6553	1.84550
$977$	51.0146	1.63210	0.816050	−	0.577981i	$-0.196159\pi$
$977$	51.0146	1.63210	0.816050	+	0.577981i	$0.196159\pi$
$978$	38.9909	1.24679
$979$	−26.7370	−0.854518
$980$	0.142743	0.00455976
$981$	−22.7059	−0.724943
$982$	−9.20595	−0.293774
$983$	−49.2089	−1.56952	−0.784761	−	0.619799i	$-0.787214\pi$
$983$	−49.2089	−1.56952	−0.784761	+	0.619799i	$0.787214\pi$
$984$	53.0032	1.68968
$985$	2.02085	0.0643896
$986$	−20.7886	−0.662045
$987$	−21.1613	−0.673571
$988$	−7.59152	−0.241518
$989$	53.9579	1.71576
$990$	−4.75770	−0.151210
$991$	20.6408	0.655677	0.327839	−	0.944734i	$-0.393680\pi$
$991$	20.6408	0.655677	0.327839	+	0.944734i	$0.393680\pi$
$992$	−22.3114	−0.708388
$993$	−93.4076	−2.96420
$994$	−15.7864	−0.500714
$995$	−5.23571	−0.165983
$996$	25.6972	0.814247
$997$	48.8298	1.54645	0.773227	−	0.634129i	$-0.218641\pi$
$997$	48.8298	1.54645	0.773227	+	0.634129i	$0.218641\pi$
$998$	28.4293	0.899914
$999$	79.0717	2.50172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	889.2.a.d.1.5	✓	20
3.2	odd	2		8001.2.a.w.1.16		20
7.6	odd	2		6223.2.a.l.1.5		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.d.1.5	✓	20	1.1	even	1	trivial
6223.2.a.l.1.5		20	7.6	odd	2
8001.2.a.w.1.16		20	3.2	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 889 = 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	889.a (trivial)

\(f(q)\)	\(=\)	\(q-1.60447 q^{2} +2.96301 q^{3} +0.574339 q^{4} +0.248535 q^{5} -4.75407 q^{6} +1.00000 q^{7} +2.28744 q^{8} +5.77943 q^{9} -0.398768 q^{10} +2.06439 q^{11} +1.70177 q^{12} +4.61355 q^{13} -1.60447 q^{14} +0.736411 q^{15} -4.81881 q^{16} -1.38495 q^{17} -9.27295 q^{18} -2.86500 q^{19} +0.142743 q^{20} +2.96301 q^{21} -3.31226 q^{22} +5.17882 q^{23} +6.77770 q^{24} -4.93823 q^{25} -7.40233 q^{26} +8.23548 q^{27} +0.574339 q^{28} -9.35535 q^{29} -1.18155 q^{30} -7.06776 q^{31} +3.15679 q^{32} +6.11681 q^{33} +2.22211 q^{34} +0.248535 q^{35} +3.31935 q^{36} +9.60134 q^{37} +4.59682 q^{38} +13.6700 q^{39} +0.568508 q^{40} +7.82024 q^{41} -4.75407 q^{42} +10.4189 q^{43} +1.18566 q^{44} +1.43639 q^{45} -8.30929 q^{46} -7.14182 q^{47} -14.2782 q^{48} +1.00000 q^{49} +7.92327 q^{50} -4.10361 q^{51} +2.64974 q^{52} +9.73442 q^{53} -13.2136 q^{54} +0.513073 q^{55} +2.28744 q^{56} -8.48903 q^{57} +15.0104 q^{58} -1.47644 q^{59} +0.422949 q^{60} -11.9646 q^{61} +11.3400 q^{62} +5.77943 q^{63} +4.57264 q^{64} +1.14663 q^{65} -9.81427 q^{66} -14.5589 q^{67} -0.795429 q^{68} +15.3449 q^{69} -0.398768 q^{70} +9.83898 q^{71} +13.2201 q^{72} -4.36287 q^{73} -15.4051 q^{74} -14.6320 q^{75} -1.64548 q^{76} +2.06439 q^{77} -21.9332 q^{78} -2.06143 q^{79} -1.19764 q^{80} +7.06353 q^{81} -12.5474 q^{82} +15.1003 q^{83} +1.70177 q^{84} -0.344207 q^{85} -16.7169 q^{86} -27.7200 q^{87} +4.72216 q^{88} -12.9515 q^{89} -2.30465 q^{90} +4.61355 q^{91} +2.97440 q^{92} -20.9418 q^{93} +11.4589 q^{94} -0.712053 q^{95} +9.35359 q^{96} -5.88747 q^{97} -1.60447 q^{98} +11.9310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 8 q^{2} + 24 q^{4} + 3 q^{5} + 6 q^{6} + 20 q^{7} + 24 q^{8} + 30 q^{9} - 8 q^{10} + 26 q^{11} - 4 q^{12} - 4 q^{13} + 8 q^{14} + 10 q^{15} + 24 q^{16} + 4 q^{17} + 5 q^{18} + q^{19} - 2 q^{20}+ \cdots + 15 q^{99}+O(q^{100}) \) 20 * q + 8 * q^2 + 24 * q^4 + 3 * q^5 + 6 * q^6 + 20 * q^7 + 24 * q^8 + 30 * q^9 - 8 * q^10 + 26 * q^11 - 4 * q^12 - 4 * q^13 + 8 * q^14 + 10 * q^15 + 24 * q^16 + 4 * q^17 + 5 * q^18 + q^19 - 2 * q^20 + q^22 + 31 * q^23 - 6 * q^24 + 27 * q^25 + 4 * q^26 - 18 * q^27 + 24 * q^28 + 16 * q^29 - 5 * q^30 + 6 * q^31 + 41 * q^32 - 18 * q^33 - 10 * q^34 + 3 * q^35 + 18 * q^36 + 2 * q^37 + 3 * q^38 + 43 * q^39 - 38 * q^40 + 25 * q^41 + 6 * q^42 + 13 * q^43 + 66 * q^44 - 2 * q^45 + 20 * q^46 + 19 * q^47 - 16 * q^48 + 20 * q^49 - 4 * q^50 + 4 * q^51 + 20 * q^52 + 24 * q^53 + 5 * q^54 - 3 * q^55 + 24 * q^56 - 4 * q^57 + 12 * q^58 + 23 * q^59 + 24 * q^60 - 27 * q^61 + 7 * q^62 + 30 * q^63 + 2 * q^64 + 26 * q^65 + 26 * q^66 + 9 * q^67 - 25 * q^68 - 3 * q^69 - 8 * q^70 + 63 * q^71 + 27 * q^72 - 21 * q^73 + 21 * q^74 - 52 * q^75 - 10 * q^76 + 26 * q^77 - 70 * q^78 + 18 * q^79 - 23 * q^80 + 40 * q^81 - 42 * q^82 - q^83 - 4 * q^84 - 41 * q^85 - 12 * q^86 - 9 * q^87 + 57 * q^88 - 16 * q^89 + q^90 - 4 * q^91 + 17 * q^92 - 41 * q^93 + 7 * q^94 + 75 * q^95 - 81 * q^96 - 32 * q^97 + 8 * q^98 + 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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