LMFDB - Embedded newform 8281.2.a.bz.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, 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("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,-2,-1,4,-1,9,0,-3,-3,-4,-4,-5,0,0,2,-8,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.6995813.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1 $ x^6 - x^5 - 6x^4 + 4x^3 + 7*x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.04394$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.55469 q^{2} -0.489252 q^{3} +0.417051 q^{4} -1.19151 q^{5} -0.760633 q^{6} -2.46099 q^{8} -2.76063 q^{9} -1.85243 q^{10} +2.11614 q^{11} -0.204043 q^{12} +0.582949 q^{15} -4.66017 q^{16} -0.906303 q^{17} -4.29192 q^{18} +6.69028 q^{19} -0.496921 q^{20} +3.28993 q^{22} +3.59733 q^{23} +1.20404 q^{24} -3.58030 q^{25} +2.81840 q^{27} +8.51545 q^{29} +0.906303 q^{30} -5.28780 q^{31} -2.32313 q^{32} -1.03532 q^{33} -1.40902 q^{34} -1.15133 q^{36} +4.99159 q^{37} +10.4013 q^{38} +2.93230 q^{40} +1.53636 q^{41} +5.43273 q^{43} +0.882538 q^{44} +3.28933 q^{45} +5.59272 q^{46} -3.18673 q^{47} +2.28000 q^{48} -5.56625 q^{50} +0.443410 q^{51} -2.82477 q^{53} +4.38173 q^{54} -2.52140 q^{55} -3.27323 q^{57} +13.2389 q^{58} -10.2460 q^{59} +0.243120 q^{60} -8.26845 q^{61} -8.22088 q^{62} +5.70861 q^{64} -1.60960 q^{66} -3.74363 q^{67} -0.377975 q^{68} -1.76000 q^{69} -2.53020 q^{71} +6.79389 q^{72} -5.73044 q^{73} +7.76035 q^{74} +1.75167 q^{75} +2.79019 q^{76} +6.07240 q^{79} +5.55265 q^{80} +6.90299 q^{81} +2.38856 q^{82} -11.6309 q^{83} +1.07987 q^{85} +8.44619 q^{86} -4.16619 q^{87} -5.20780 q^{88} -17.7511 q^{89} +5.11387 q^{90} +1.50027 q^{92} +2.58707 q^{93} -4.95437 q^{94} -7.97155 q^{95} +1.13659 q^{96} +6.20434 q^{97} -5.84188 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{2} - q^{3} + 4 q^{4} - q^{5} + 9 q^{6} - 3 q^{8} - 3 q^{9} - 4 q^{10} - 4 q^{11} - 5 q^{12} + 2 q^{15} - 8 q^{16} - 5 q^{17} - 3 q^{18} + q^{19} + q^{20} + 5 q^{22} + q^{23} + 11 q^{24}+ \cdots - 10 q^{99}+O(q^{100}) $ 6 * q - 2 * q^2 - q^3 + 4 * q^4 - q^5 + 9 * q^6 - 3 * q^8 - 3 * q^9 - 4 * q^10 - 4 * q^11 - 5 * q^12 + 2 * q^15 - 8 * q^16 - 5 * q^17 - 3 * q^18 + q^19 + q^20 + 5 * q^22 + q^23 + 11 * q^24 - 7 * q^25 - 4 * q^27 - 3 * q^29 + 5 * q^30 - 16 * q^31 - 8 * q^32 - 16 * q^33 + 16 * q^34 + 21 * q^36 + 13 * q^37 + 17 * q^38 + 5 * q^40 + 8 * q^41 + 11 * q^43 - 21 * q^44 + 7 * q^45 - 16 * q^46 + q^47 - 21 * q^48 - 6 * q^50 + 20 * q^51 + 2 * q^53 + 18 * q^54 - 9 * q^55 + 21 * q^57 + 8 * q^58 - 13 * q^59 - 20 * q^60 + 5 * q^61 - 5 * q^62 - 15 * q^64 - 18 * q^66 + 11 * q^67 - 29 * q^68 - 23 * q^69 - 6 * q^71 - 25 * q^72 + 30 * q^73 + 3 * q^74 + 3 * q^75 + 9 * q^76 - 7 * q^79 + 7 * q^80 + 6 * q^81 - q^82 - 27 * q^83 + q^85 + 7 * q^86 - 16 * q^87 - 4 * q^89 - 8 * q^90 + 27 * q^92 + 7 * q^93 - 45 * q^94 + 6 * q^95 - 19 * q^96 + 35 * q^97 - 10 * 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 $

$2$	1.55469	1.09933	0.549665	−	0.835385i	$-0.314755\pi$
$2$	1.55469	1.09933	0.549665	+	0.835385i	$0.314755\pi$
$3$	−0.489252	−0.282470	−0.141235	−	0.989976i	$-0.545107\pi$
$3$	−0.489252	−0.282470	−0.141235	+	0.989976i	$0.545107\pi$
$4$	0.417051	0.208526
$5$	−1.19151	−0.532860	−0.266430	−	0.963854i	$-0.585844\pi$
$5$	−1.19151	−0.532860	−0.266430	+	0.963854i	$0.585844\pi$
$6$	−0.760633	−0.310527
$7$	0	0
$7$	0	0
$8$	−2.46099	−0.870091
$9$	−2.76063	−0.920211
$10$	−1.85243	−0.585789
$11$	2.11614	0.638040	0.319020	−	0.947748i	$-0.396646\pi$
$11$	2.11614	0.638040	0.319020	+	0.947748i	$0.396646\pi$
$12$	−0.204043	−0.0589021
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0.582949	0.150517
$16$	−4.66017	−1.16504
$17$	−0.906303	−0.219811	−0.109905	−	0.993942i	$-0.535055\pi$
$17$	−0.906303	−0.219811	−0.109905	+	0.993942i	$0.535055\pi$
$18$	−4.29192	−1.01162
$19$	6.69028	1.53486	0.767428	−	0.641135i	$-0.221536\pi$
$19$	6.69028	1.53486	0.767428	+	0.641135i	$0.221536\pi$
$20$	−0.496921	−0.111115
$21$	0	0
$22$	3.28993	0.701416
$23$	3.59733	0.750095	0.375048	−	0.927006i	$-0.377626\pi$
$23$	3.59733	0.750095	0.375048	+	0.927006i	$0.377626\pi$
$24$	1.20404	0.245774
$25$	−3.58030	−0.716060
$26$	0	0
$27$	2.81840	0.542401
$28$	0	0
$29$	8.51545	1.58128	0.790639	−	0.612282i	$-0.209748\pi$
$29$	8.51545	1.58128	0.790639	+	0.612282i	$0.209748\pi$
$30$	0.906303	0.165467
$31$	−5.28780	−0.949717	−0.474859	−	0.880062i	$-0.657501\pi$
$31$	−5.28780	−0.949717	−0.474859	+	0.880062i	$0.657501\pi$
$32$	−2.32313	−0.410675
$33$	−1.03532	−0.180227
$34$	−1.40902	−0.241644
$35$	0	0
$36$	−1.15133	−0.191888
$37$	4.99159	0.820612	0.410306	−	0.911948i	$-0.365422\pi$
$37$	4.99159	0.820612	0.410306	+	0.911948i	$0.365422\pi$
$38$	10.4013	1.68731
$39$	0	0
$40$	2.93230	0.463637
$41$	1.53636	0.239939	0.119970	−	0.992778i	$-0.461720\pi$
$41$	1.53636	0.239939	0.119970	+	0.992778i	$0.461720\pi$
$42$	0	0
$43$	5.43273	0.828483	0.414242	−	0.910167i	$-0.364047\pi$
$43$	5.43273	0.828483	0.414242	+	0.910167i	$0.364047\pi$
$44$	0.882538	0.133048
$45$	3.28933	0.490344
$46$	5.59272	0.824602
$47$	−3.18673	−0.464833	−0.232416	−	0.972616i	$-0.574663\pi$
$47$	−3.18673	−0.464833	−0.232416	+	0.972616i	$0.574663\pi$
$48$	2.28000	0.329089
$49$	0	0
$50$	−5.56625	−0.787186
$51$	0.443410	0.0620898
$52$	0	0
$53$	−2.82477	−0.388012	−0.194006	−	0.981000i	$-0.562148\pi$
$53$	−2.82477	−0.388012	−0.194006	+	0.981000i	$0.562148\pi$
$54$	4.38173	0.596278
$55$	−2.52140	−0.339986
$56$	0	0
$57$	−3.27323	−0.433550
$58$	13.2389	1.73835
$59$	−10.2460	−1.33391	−0.666956	−	0.745097i	$-0.732403\pi$
$59$	−10.2460	−1.33391	−0.666956	+	0.745097i	$0.732403\pi$
$60$	0.243120	0.0313866
$61$	−8.26845	−1.05867	−0.529333	−	0.848414i	$-0.677558\pi$
$61$	−8.26845	−1.05867	−0.529333	+	0.848414i	$0.677558\pi$
$62$	−8.22088	−1.04405
$63$	0	0
$64$	5.70861	0.713576
$65$	0	0
$66$	−1.60960	−0.198129
$67$	−3.74363	−0.457358	−0.228679	−	0.973502i	$-0.573441\pi$
$67$	−3.74363	−0.457358	−0.228679	+	0.973502i	$0.573441\pi$
$68$	−0.377975	−0.0458362
$69$	−1.76000	−0.211879
$70$	0	0
$71$	−2.53020	−0.300280	−0.150140	−	0.988665i	$-0.547972\pi$
$71$	−2.53020	−0.300280	−0.150140	+	0.988665i	$0.547972\pi$
$72$	6.79389	0.800668
$73$	−5.73044	−0.670697	−0.335349	−	0.942094i	$-0.608854\pi$
$73$	−5.73044	−0.670697	−0.335349	+	0.942094i	$0.608854\pi$
$74$	7.76035	0.902123
$75$	1.75167	0.202265
$76$	2.79019	0.320057
$77$	0	0
$78$	0	0
$79$	6.07240	0.683198	0.341599	−	0.939846i	$-0.389031\pi$
$79$	6.07240	0.683198	0.341599	+	0.939846i	$0.389031\pi$
$80$	5.55265	0.620805
$81$	6.90299	0.766999
$82$	2.38856	0.263773
$83$	−11.6309	−1.27665	−0.638327	−	0.769766i	$-0.720373\pi$
$83$	−11.6309	−1.27665	−0.638327	+	0.769766i	$0.720373\pi$
$84$	0	0
$85$	1.07987	0.117128
$86$	8.44619	0.910776
$87$	−4.16619	−0.446663
$88$	−5.20780	−0.555153
$89$	−17.7511	−1.88162	−0.940808	−	0.338939i	$-0.889932\pi$
$89$	−17.7511	−1.88162	−0.940808	+	0.338939i	$0.889932\pi$
$90$	5.11387	0.539049
$91$	0	0
$92$	1.50027	0.156414
$93$	2.58707	0.268266
$94$	−4.95437	−0.511004
$95$	−7.97155	−0.817863
$96$	1.13659	0.116003
$97$	6.20434	0.629955	0.314978	−	0.949099i	$-0.398003\pi$
$97$	6.20434	0.629955	0.314978	+	0.949099i	$0.398003\pi$
$98$	0	0
$99$	−5.84188	−0.587131
$100$	−1.49317	−0.149317
$101$	−7.22266	−0.718682	−0.359341	−	0.933206i	$-0.616999\pi$
$101$	−7.22266	−0.718682	−0.359341	+	0.933206i	$0.616999\pi$
$102$	0.689364	0.0682572
$103$	9.92645	0.978082	0.489041	−	0.872261i	$-0.337347\pi$
$103$	9.92645	0.978082	0.489041	+	0.872261i	$0.337347\pi$
$104$	0	0
$105$	0	0
$106$	−4.39164	−0.426553
$107$	−2.20006	−0.212688	−0.106344	−	0.994329i	$-0.533915\pi$
$107$	−2.20006	−0.212688	−0.106344	+	0.994329i	$0.533915\pi$
$108$	1.17542	0.113105
$109$	13.7458	1.31661	0.658305	−	0.752751i	$-0.271274\pi$
$109$	13.7458	1.31661	0.658305	+	0.752751i	$0.271274\pi$
$110$	−3.91999	−0.373757
$111$	−2.44214	−0.231798
$112$	0	0
$113$	−16.0947	−1.51406	−0.757032	−	0.653378i	$-0.773351\pi$
$113$	−16.0947	−1.51406	−0.757032	+	0.653378i	$0.773351\pi$
$114$	−5.08885	−0.476614
$115$	−4.28626	−0.399696
$116$	3.55138	0.329737
$117$	0	0
$118$	−15.9293	−1.46641
$119$	0	0
$120$	−1.43463	−0.130963
$121$	−6.52196	−0.592905
$122$	−12.8549	−1.16382
$123$	−0.751668	−0.0677756
$124$	−2.20528	−0.198040
$125$	10.2235	0.914420
$126$	0	0
$127$	−15.6784	−1.39123	−0.695617	−	0.718413i	$-0.744869\pi$
$127$	−15.6784	−1.39123	−0.695617	+	0.718413i	$0.744869\pi$
$128$	13.5213	1.19513
$129$	−2.65797	−0.234021
$130$	0	0
$131$	−9.53769	−0.833312	−0.416656	−	0.909064i	$-0.636798\pi$
$131$	−9.53769	−0.833312	−0.416656	+	0.909064i	$0.636798\pi$
$132$	−0.431783	−0.0375819
$133$	0	0
$134$	−5.82018	−0.502787
$135$	−3.35815	−0.289024
$136$	2.23040	0.191255
$137$	−2.76461	−0.236197	−0.118098	−	0.993002i	$-0.537680\pi$
$137$	−2.76461	−0.236197	−0.118098	+	0.993002i	$0.537680\pi$
$138$	−2.73625	−0.232925
$139$	−22.7967	−1.93359	−0.966795	−	0.255554i	$-0.917742\pi$
$139$	−22.7967	−1.93359	−0.966795	+	0.255554i	$0.917742\pi$
$140$	0	0
$141$	1.55911	0.131301
$142$	−3.93368	−0.330107
$143$	0	0
$144$	12.8650	1.07209
$145$	−10.1462	−0.842600
$146$	−8.90904	−0.737317
$147$	0	0
$148$	2.08175	0.171119
$149$	−14.4116	−1.18065	−0.590323	−	0.807167i	$-0.701000\pi$
$149$	−14.4116	−1.18065	−0.590323	+	0.807167i	$0.701000\pi$
$150$	2.72329	0.222356
$151$	15.2580	1.24168	0.620840	−	0.783937i	$-0.286792\pi$
$151$	15.2580	1.24168	0.620840	+	0.783937i	$0.286792\pi$
$152$	−16.4647	−1.33546
$153$	2.50197	0.202272
$154$	0	0
$155$	6.30048	0.506067
$156$	0	0
$157$	−11.4149	−0.911008	−0.455504	−	0.890234i	$-0.650541\pi$
$157$	−11.4149	−0.911008	−0.455504	+	0.890234i	$0.650541\pi$
$158$	9.44068	0.751060
$159$	1.38202	0.109602
$160$	2.76803	0.218832
$161$	0	0
$162$	10.7320	0.843185
$163$	−14.4077	−1.12850	−0.564249	−	0.825605i	$-0.690834\pi$
$163$	−14.4077	−1.12850	−0.564249	+	0.825605i	$0.690834\pi$
$164$	0.640742	0.0500335
$165$	1.23360	0.0960357
$166$	−18.0824	−1.40346
$167$	7.77190	0.601407	0.300704	−	0.953718i	$-0.402778\pi$
$167$	7.77190	0.601407	0.300704	+	0.953718i	$0.402778\pi$
$168$	0	0
$169$	0	0
$170$	1.67886	0.128763
$171$	−18.4694	−1.41239
$172$	2.26573	0.172760
$173$	6.09461	0.463365	0.231682	−	0.972791i	$-0.425577\pi$
$173$	6.09461	0.463365	0.231682	+	0.972791i	$0.425577\pi$
$174$	−6.47713	−0.491030
$175$	0	0
$176$	−9.86157	−0.743344
$177$	5.01286	0.376789
$178$	−27.5975	−2.06852
$179$	18.5298	1.38498	0.692490	−	0.721428i	$-0.256514\pi$
$179$	18.5298	1.38498	0.692490	+	0.721428i	$0.256514\pi$
$180$	1.37182	0.102249
$181$	−5.60520	−0.416631	−0.208316	−	0.978062i	$-0.566798\pi$
$181$	−5.60520	−0.416631	−0.208316	+	0.978062i	$0.566798\pi$
$182$	0	0
$183$	4.04535	0.299041
$184$	−8.85299	−0.652651
$185$	−5.94753	−0.437271
$186$	4.02208	0.294913
$187$	−1.91786	−0.140248
$188$	−1.32903	−0.0969295
$189$	0	0
$190$	−12.3933	−0.899102
$191$	0.503703	0.0364466	0.0182233	−	0.999834i	$-0.494199\pi$
$191$	0.503703	0.0364466	0.0182233	+	0.999834i	$0.494199\pi$
$192$	−2.79294	−0.201563
$193$	−3.71244	−0.267227	−0.133614	−	0.991033i	$-0.542658\pi$
$193$	−3.71244	−0.267227	−0.133614	+	0.991033i	$0.542658\pi$
$194$	9.64581	0.692529
$195$	0	0
$196$	0	0
$197$	−7.44451	−0.530399	−0.265200	−	0.964194i	$-0.585438\pi$
$197$	−7.44451	−0.530399	−0.265200	+	0.964194i	$0.585438\pi$
$198$	−9.08230	−0.645451
$199$	7.50556	0.532055	0.266028	−	0.963965i	$-0.414289\pi$
$199$	7.50556	0.532055	0.266028	+	0.963965i	$0.414289\pi$
$200$	8.81108	0.623038
$201$	1.83158	0.129190
$202$	−11.2290	−0.790068
$203$	0	0
$204$	0.184925	0.0129473
$205$	−1.83059	−0.127854
$206$	15.4325	1.07523
$207$	−9.93091	−0.690246
$208$	0	0
$209$	14.1576	0.979299
$210$	0	0
$211$	3.79063	0.260957	0.130479	−	0.991451i	$-0.458349\pi$
$211$	3.79063	0.260957	0.130479	+	0.991451i	$0.458349\pi$
$212$	−1.17807	−0.0809105
$213$	1.23791	0.0848200
$214$	−3.42041	−0.233814
$215$	−6.47316	−0.441466
$216$	−6.93605	−0.471938
$217$	0	0
$218$	21.3704	1.44739
$219$	2.80363	0.189452
$220$	−1.05155	−0.0708958
$221$	0	0
$222$	−3.79676	−0.254822
$223$	4.86879	0.326039	0.163019	−	0.986623i	$-0.447877\pi$
$223$	4.86879	0.326039	0.163019	+	0.986623i	$0.447877\pi$
$224$	0	0
$225$	9.88390	0.658926
$226$	−25.0223	−1.66446
$227$	24.1767	1.60466	0.802332	−	0.596877i	$-0.203592\pi$
$227$	24.1767	1.60466	0.802332	+	0.596877i	$0.203592\pi$
$228$	−1.36510	−0.0904063
$229$	−21.7123	−1.43479	−0.717394	−	0.696668i	$-0.754665\pi$
$229$	−21.7123	−1.43479	−0.717394	+	0.696668i	$0.754665\pi$
$230$	−6.66379	−0.439397
$231$	0	0
$232$	−20.9564	−1.37586
$233$	3.79684	0.248739	0.124370	−	0.992236i	$-0.460309\pi$
$233$	3.79684	0.248739	0.124370	+	0.992236i	$0.460309\pi$
$234$	0	0
$235$	3.79703	0.247691
$236$	−4.27309	−0.278155
$237$	−2.97093	−0.192983
$238$	0	0
$239$	21.9100	1.41724	0.708619	−	0.705592i	$-0.249319\pi$
$239$	21.9100	1.41724	0.708619	+	0.705592i	$0.249319\pi$
$240$	−2.71664	−0.175358
$241$	−20.7488	−1.33655	−0.668273	−	0.743916i	$-0.732966\pi$
$241$	−20.7488	−1.33655	−0.668273	+	0.743916i	$0.732966\pi$
$242$	−10.1396	−0.651798
$243$	−11.8325	−0.759055
$244$	−3.44837	−0.220759
$245$	0	0
$246$	−1.16861	−0.0745077
$247$	0	0
$248$	13.0132	0.826341
$249$	5.69042	0.360616
$250$	15.8944	1.00525
$251$	13.2578	0.836827	0.418413	−	0.908257i	$-0.362586\pi$
$251$	13.2578	0.836827	0.418413	+	0.908257i	$0.362586\pi$
$252$	0	0
$253$	7.61245	0.478591
$254$	−24.3750	−1.52943
$255$	−0.528328	−0.0330852
$256$	9.60425	0.600266
$257$	−13.1711	−0.821590	−0.410795	−	0.911728i	$-0.634749\pi$
$257$	−13.1711	−0.821590	−0.410795	+	0.911728i	$0.634749\pi$
$258$	−4.13231	−0.257266
$259$	0	0
$260$	0	0
$261$	−23.5080	−1.45511
$262$	−14.8281	−0.916084
$263$	−19.1406	−1.18026	−0.590129	−	0.807309i	$-0.700923\pi$
$263$	−19.1406	−1.18026	−0.590129	+	0.807309i	$0.700923\pi$
$264$	2.54792	0.156814
$265$	3.36575	0.206756
$266$	0	0
$267$	8.68477	0.531499
$268$	−1.56129	−0.0953708
$269$	−28.4822	−1.73659	−0.868296	−	0.496047i	$-0.834784\pi$
$269$	−28.4822	−1.73659	−0.868296	+	0.496047i	$0.834784\pi$
$270$	−5.22088	−0.317732
$271$	17.9474	1.09023	0.545114	−	0.838362i	$-0.316486\pi$
$271$	17.9474	1.09023	0.545114	+	0.838362i	$0.316486\pi$
$272$	4.22353	0.256089
$273$	0	0
$274$	−4.29811	−0.259658
$275$	−7.57641	−0.456875
$276$	−0.734010	−0.0441822
$277$	13.4389	0.807463	0.403732	−	0.914877i	$-0.367713\pi$
$277$	13.4389	0.807463	0.403732	+	0.914877i	$0.367713\pi$
$278$	−35.4417	−2.12565
$279$	14.5977	0.873940
$280$	0	0
$281$	−29.9530	−1.78685	−0.893424	−	0.449214i	$-0.851704\pi$
$281$	−29.9530	−1.78685	−0.893424	+	0.449214i	$0.851704\pi$
$282$	2.42393	0.144343
$283$	−9.89122	−0.587972	−0.293986	−	0.955810i	$-0.594982\pi$
$283$	−9.89122	−0.587972	−0.293986	+	0.955810i	$0.594982\pi$
$284$	−1.05523	−0.0626161
$285$	3.90009	0.231021
$286$	0	0
$287$	0	0
$288$	6.41330	0.377907
$289$	−16.1786	−0.951683
$290$	−15.7742	−0.926295
$291$	−3.03548	−0.177943
$292$	−2.38989	−0.139858
$293$	7.91058	0.462141	0.231071	−	0.972937i	$-0.425777\pi$
$293$	7.91058	0.462141	0.231071	+	0.972937i	$0.425777\pi$
$294$	0	0
$295$	12.2082	0.710788
$296$	−12.2842	−0.714007
$297$	5.96412	0.346073
$298$	−22.4056	−1.29792
$299$	0	0
$300$	0.730535	0.0421775
$301$	0	0
$302$	23.7214	1.36502
$303$	3.53370	0.203006
$304$	−31.1779	−1.78817
$305$	9.85196	0.564121
$306$	3.88978	0.222364
$307$	1.27238	0.0726187	0.0363094	−	0.999341i	$-0.488440\pi$
$307$	1.27238	0.0726187	0.0363094	+	0.999341i	$0.488440\pi$
$308$	0	0
$309$	−4.85653	−0.276278
$310$	9.79527	0.556334
$311$	24.7635	1.40421	0.702103	−	0.712075i	$-0.252244\pi$
$311$	24.7635	1.40421	0.702103	+	0.712075i	$0.252244\pi$
$312$	0	0
$313$	2.37651	0.134328	0.0671642	−	0.997742i	$-0.478605\pi$
$313$	2.37651	0.134328	0.0671642	+	0.997742i	$0.478605\pi$
$314$	−17.7466	−1.00150
$315$	0	0
$316$	2.53250	0.142464
$317$	−19.7796	−1.11093	−0.555466	−	0.831539i	$-0.687460\pi$
$317$	−19.7796	−1.11093	−0.555466	+	0.831539i	$0.687460\pi$
$318$	2.14862	0.120488
$319$	18.0199	1.00892
$320$	−6.80187	−0.380236
$321$	1.07638	0.0600779
$322$	0	0
$323$	−6.06342	−0.337378
$324$	2.87890	0.159939
$325$	0	0
$326$	−22.3995	−1.24059
$327$	−6.72516	−0.371902
$328$	−3.78097	−0.208769
$329$	0	0
$330$	1.91786	0.105575
$331$	3.92773	0.215888	0.107944	−	0.994157i	$-0.465573\pi$
$331$	3.92773	0.215888	0.107944	+	0.994157i	$0.465573\pi$
$332$	−4.85067	−0.266215
$333$	−13.7799	−0.755136
$334$	12.0829	0.661145
$335$	4.46058	0.243708
$336$	0	0
$337$	−7.14099	−0.388995	−0.194497	−	0.980903i	$-0.562308\pi$
$337$	−7.14099	−0.388995	−0.194497	+	0.980903i	$0.562308\pi$
$338$	0	0
$339$	7.87437	0.427677
$340$	0.450361	0.0244243
$341$	−11.1897	−0.605958
$342$	−28.7142	−1.55268
$343$	0	0
$344$	−13.3699	−0.720856
$345$	2.09706	0.112902
$346$	9.47522	0.509391
$347$	10.0700	0.540584	0.270292	−	0.962778i	$-0.412880\pi$
$347$	10.0700	0.540584	0.270292	+	0.962778i	$0.412880\pi$
$348$	−1.73752	−0.0931407
$349$	−6.28837	−0.336609	−0.168304	−	0.985735i	$-0.553829\pi$
$349$	−6.28837	−0.336609	−0.168304	+	0.985735i	$0.553829\pi$
$350$	0	0
$351$	0	0
$352$	−4.91606	−0.262027
$353$	34.1672	1.81854	0.909269	−	0.416210i	$-0.136642\pi$
$353$	34.1672	1.81854	0.909269	+	0.416210i	$0.136642\pi$
$354$	7.79342	0.414216
$355$	3.01477	0.160007
$356$	−7.40313	−0.392365
$357$	0	0
$358$	28.8080	1.52255
$359$	18.6865	0.986238	0.493119	−	0.869962i	$-0.335857\pi$
$359$	18.6865	0.986238	0.493119	+	0.869962i	$0.335857\pi$
$360$	−8.09500	−0.426644
$361$	25.7599	1.35578
$362$	−8.71433	−0.458015
$363$	3.19088	0.167478
$364$	0	0
$365$	6.82788	0.357388
$366$	6.28926	0.328745
$367$	−31.0611	−1.62137	−0.810687	−	0.585479i	$-0.800906\pi$
$367$	−31.0611	−1.62137	−0.810687	+	0.585479i	$0.800906\pi$
$368$	−16.7642	−0.873893
$369$	−4.24133	−0.220795
$370$	−9.24655	−0.480705
$371$	0	0
$372$	1.07894	0.0559404
$373$	−2.93704	−0.152074	−0.0760371	−	0.997105i	$-0.524227\pi$
$373$	−2.93704	−0.152074	−0.0760371	+	0.997105i	$0.524227\pi$
$374$	−2.98168	−0.154179
$375$	−5.00188	−0.258296
$376$	7.84252	0.404447
$377$	0	0
$378$	0	0
$379$	10.0851	0.518036	0.259018	−	0.965872i	$-0.416601\pi$
$379$	10.0851	0.518036	0.259018	+	0.965872i	$0.416601\pi$
$380$	−3.32454	−0.170545
$381$	7.67069	0.392981
$382$	0.783100	0.0400669
$383$	3.68931	0.188515	0.0942576	−	0.995548i	$-0.469952\pi$
$383$	3.68931	0.188515	0.0942576	+	0.995548i	$0.469952\pi$
$384$	−6.61534	−0.337588
$385$	0	0
$386$	−5.77168	−0.293771
$387$	−14.9978	−0.762379
$388$	2.58753	0.131362
$389$	22.6667	1.14925	0.574623	−	0.818418i	$-0.305149\pi$
$389$	22.6667	1.14925	0.574623	+	0.818418i	$0.305149\pi$
$390$	0	0
$391$	−3.26027	−0.164879
$392$	0	0
$393$	4.66633	0.235385
$394$	−11.5739	−0.583084
$395$	−7.23533	−0.364049
$396$	−2.43636	−0.122432
$397$	−29.1360	−1.46229	−0.731146	−	0.682221i	$-0.761014\pi$
$397$	−29.1360	−1.46229	−0.731146	+	0.682221i	$0.761014\pi$
$398$	11.6688	0.584904
$399$	0	0
$400$	16.6848	0.834241
$401$	8.12052	0.405519	0.202760	−	0.979229i	$-0.435009\pi$
$401$	8.12052	0.405519	0.202760	+	0.979229i	$0.435009\pi$
$402$	2.84753	0.142022
$403$	0	0
$404$	−3.01222	−0.149864
$405$	−8.22499	−0.408703
$406$	0	0
$407$	10.5629	0.523583
$408$	−1.09123	−0.0540238
$409$	−8.32261	−0.411527	−0.205763	−	0.978602i	$-0.565968\pi$
$409$	−8.32261	−0.411527	−0.205763	+	0.978602i	$0.565968\pi$
$410$	−2.84600	−0.140554
$411$	1.35259	0.0667184
$412$	4.13984	0.203955
$413$	0	0
$414$	−15.4395	−0.758808
$415$	13.8583	0.680278
$416$	0	0
$417$	11.1533	0.546180
$418$	22.0106	1.07657
$419$	−13.0166	−0.635905	−0.317952	−	0.948107i	$-0.602995\pi$
$419$	−13.0166	−0.635905	−0.317952	+	0.948107i	$0.602995\pi$
$420$	0	0
$421$	−8.89681	−0.433604	−0.216802	−	0.976216i	$-0.569563\pi$
$421$	−8.89681	−0.433604	−0.216802	+	0.976216i	$0.569563\pi$
$422$	5.89323	0.286878
$423$	8.79740	0.427744
$424$	6.95174	0.337606
$425$	3.24484	0.157398
$426$	1.92456	0.0932451
$427$	0	0
$428$	−0.917539	−0.0443509
$429$	0	0
$430$	−10.0637	−0.485316
$431$	−8.95743	−0.431464	−0.215732	−	0.976453i	$-0.569214\pi$
$431$	−8.95743	−0.431464	−0.215732	+	0.976453i	$0.569214\pi$
$432$	−13.1342	−0.631920
$433$	−0.172909	−0.00830950	−0.00415475	−	0.999991i	$-0.501323\pi$
$433$	−0.172909	−0.00830950	−0.00415475	+	0.999991i	$0.501323\pi$
$434$	0	0
$435$	4.96407	0.238009
$436$	5.73271	0.274547
$437$	24.0671	1.15129
$438$	4.35876	0.208270
$439$	9.54160	0.455396	0.227698	−	0.973732i	$-0.426880\pi$
$439$	9.54160	0.455396	0.227698	+	0.973732i	$0.426880\pi$
$440$	6.20515	0.295819
$441$	0	0
$442$	0	0
$443$	−13.8735	−0.659151	−0.329576	−	0.944129i	$-0.606906\pi$
$443$	−13.8735	−0.659151	−0.329576	+	0.944129i	$0.606906\pi$
$444$	−1.01850	−0.0483358
$445$	21.1507	1.00264
$446$	7.56945	0.358424
$447$	7.05091	0.333496
$448$	0	0
$449$	−21.2913	−1.00480	−0.502398	−	0.864636i	$-0.667549\pi$
$449$	−21.2913	−1.00480	−0.502398	+	0.864636i	$0.667549\pi$
$450$	15.3664	0.724377
$451$	3.25116	0.153091
$452$	−6.71232	−0.315721
$453$	−7.46501	−0.350737
$454$	37.5872	1.76406
$455$	0	0
$456$	8.05539	0.377228
$457$	9.68564	0.453075	0.226538	−	0.974002i	$-0.427259\pi$
$457$	9.68564	0.453075	0.226538	+	0.974002i	$0.427259\pi$
$458$	−33.7558	−1.57730
$459$	−2.55432	−0.119226
$460$	−1.78759	−0.0833468
$461$	−1.37436	−0.0640101	−0.0320051	−	0.999488i	$-0.510189\pi$
$461$	−1.37436	−0.0640101	−0.0320051	+	0.999488i	$0.510189\pi$
$462$	0	0
$463$	31.7710	1.47653	0.738263	−	0.674513i	$-0.235646\pi$
$463$	31.7710	1.47653	0.738263	+	0.674513i	$0.235646\pi$
$464$	−39.6834	−1.84226
$465$	−3.08252	−0.142948
$466$	5.90290	0.273446
$467$	−29.1209	−1.34756	−0.673778	−	0.738934i	$-0.735330\pi$
$467$	−29.1209	−1.34756	−0.673778	+	0.738934i	$0.735330\pi$
$468$	0	0
$469$	0	0
$470$	5.90319	0.272294
$471$	5.58476	0.257332
$472$	25.2152	1.16062
$473$	11.4964	0.528605
$474$	−4.61887	−0.212152
$475$	−23.9532	−1.09905
$476$	0	0
$477$	7.79816	0.357053
$478$	34.0631	1.55801
$479$	9.72184	0.444202	0.222101	−	0.975024i	$-0.428708\pi$
$479$	9.72184	0.444202	0.222101	+	0.975024i	$0.428708\pi$
$480$	−1.35426	−0.0618134
$481$	0	0
$482$	−32.2578	−1.46930
$483$	0	0
$484$	−2.71999	−0.123636
$485$	−7.39254	−0.335678
$486$	−18.3958	−0.834452
$487$	−17.1133	−0.775478	−0.387739	−	0.921769i	$-0.626744\pi$
$487$	−17.1133	−0.775478	−0.387739	+	0.921769i	$0.626744\pi$
$488$	20.3486	0.921137
$489$	7.04899	0.318766
$490$	0	0
$491$	−25.7213	−1.16079	−0.580394	−	0.814336i	$-0.697101\pi$
$491$	−25.7213	−1.16079	−0.580394	+	0.814336i	$0.697101\pi$
$492$	−0.313484	−0.0141329
$493$	−7.71757	−0.347582
$494$	0	0
$495$	6.96067	0.312859
$496$	24.6421	1.10646
$497$	0	0
$498$	8.84682	0.396435
$499$	5.40396	0.241914	0.120957	−	0.992658i	$-0.461404\pi$
$499$	5.40396	0.241914	0.120957	+	0.992658i	$0.461404\pi$
$500$	4.26373	0.190680
$501$	−3.80241	−0.169879
$502$	20.6118	0.919948
$503$	−12.6169	−0.562562	−0.281281	−	0.959625i	$-0.590759\pi$
$503$	−12.6169	−0.562562	−0.281281	+	0.959625i	$0.590759\pi$
$504$	0	0
$505$	8.60589	0.382957
$506$	11.8350	0.526129
$507$	0	0
$508$	−6.53870	−0.290108
$509$	−1.95876	−0.0868204	−0.0434102	−	0.999057i	$-0.513822\pi$
$509$	−1.95876	−0.0868204	−0.0434102	+	0.999057i	$0.513822\pi$
$510$	−0.821385	−0.0363715
$511$	0	0
$512$	−12.1111	−0.535240
$513$	18.8559	0.832507
$514$	−20.4769	−0.903198
$515$	−11.8275	−0.521181
$516$	−1.10851	−0.0487994
$517$	−6.74357	−0.296582
$518$	0	0
$519$	−2.98180	−0.130886
$520$	0	0
$521$	−39.0954	−1.71280	−0.856401	−	0.516312i	$-0.827305\pi$
$521$	−39.0954	−1.71280	−0.856401	+	0.516312i	$0.827305\pi$
$522$	−36.5476	−1.59965
$523$	−8.71268	−0.380979	−0.190489	−	0.981689i	$-0.561007\pi$
$523$	−8.71268	−0.380979	−0.190489	+	0.981689i	$0.561007\pi$
$524$	−3.97770	−0.173767
$525$	0	0
$526$	−29.7576	−1.29749
$527$	4.79235	0.208758
$528$	4.82479	0.209972
$529$	−10.0592	−0.437357
$530$	5.23269	0.227293
$531$	28.2854	1.22748
$532$	0	0
$533$	0	0
$534$	13.5021	0.584293
$535$	2.62140	0.113333
$536$	9.21304	0.397943
$537$	−9.06571	−0.391214
$538$	−44.2809	−1.90909
$539$	0	0
$540$	−1.40052	−0.0602689
$541$	−21.4994	−0.924330	−0.462165	−	0.886794i	$-0.652927\pi$
$541$	−21.4994	−0.924330	−0.462165	+	0.886794i	$0.652927\pi$
$542$	27.9026	1.19852
$543$	2.74235	0.117686
$544$	2.10546	0.0902707
$545$	−16.3783	−0.701569
$546$	0	0
$547$	−30.2968	−1.29540	−0.647699	−	0.761896i	$-0.724269\pi$
$547$	−30.2968	−1.29540	−0.647699	+	0.761896i	$0.724269\pi$
$548$	−1.15299	−0.0492531
$549$	22.8262	0.974197
$550$	−11.7789	−0.502256
$551$	56.9707	2.42703
$552$	4.33134	0.184354
$553$	0	0
$554$	20.8932	0.887668
$555$	2.90984	0.123516
$556$	−9.50738	−0.403203
$557$	−17.6840	−0.749296	−0.374648	−	0.927167i	$-0.622236\pi$
$557$	−17.6840	−0.749296	−0.374648	+	0.927167i	$0.622236\pi$
$558$	22.6948	0.960749
$559$	0	0
$560$	0	0
$561$	0.938317	0.0396158
$562$	−46.5676	−1.96433
$563$	−41.7390	−1.75909	−0.879545	−	0.475816i	$-0.842153\pi$
$563$	−41.7390	−1.75909	−0.879545	+	0.475816i	$0.842153\pi$
$564$	0.650230	0.0273796
$565$	19.1770	0.806784
$566$	−15.3777	−0.646375
$567$	0	0
$568$	6.22681	0.261271
$569$	5.46775	0.229220	0.114610	−	0.993411i	$-0.463438\pi$
$569$	5.46775	0.229220	0.114610	+	0.993411i	$0.463438\pi$
$570$	6.06342	0.253969
$571$	9.35242	0.391387	0.195693	−	0.980665i	$-0.437304\pi$
$571$	9.35242	0.391387	0.195693	+	0.980665i	$0.437304\pi$
$572$	0	0
$573$	−0.246437	−0.0102951
$574$	0	0
$575$	−12.8795	−0.537113
$576$	−15.7594	−0.656640
$577$	−3.36925	−0.140264	−0.0701318	−	0.997538i	$-0.522342\pi$
$577$	−3.36925	−0.140264	−0.0701318	+	0.997538i	$0.522342\pi$
$578$	−25.1527	−1.04621
$579$	1.81632	0.0754836
$580$	−4.23151	−0.175704
$581$	0	0
$582$	−4.71923	−0.195618
$583$	−5.97761	−0.247567
$584$	14.1026	0.583568
$585$	0	0
$586$	12.2985	0.508045
$587$	−13.1528	−0.542873	−0.271437	−	0.962456i	$-0.587499\pi$
$587$	−13.1528	−0.542873	−0.271437	+	0.962456i	$0.587499\pi$
$588$	0	0
$589$	−35.3769	−1.45768
$590$	18.9799	0.781390
$591$	3.64224	0.149822
$592$	−23.2616	−0.956048
$593$	38.5916	1.58477	0.792384	−	0.610022i	$-0.208839\pi$
$593$	38.5916	1.58477	0.792384	+	0.610022i	$0.208839\pi$
$594$	9.27234	0.380449
$595$	0	0
$596$	−6.01038	−0.246195
$597$	−3.67211	−0.150289
$598$	0	0
$599$	18.4152	0.752426	0.376213	−	0.926533i	$-0.377226\pi$
$599$	18.4152	0.752426	0.376213	+	0.926533i	$0.377226\pi$
$600$	−4.31084	−0.175989
$601$	−41.4037	−1.68889	−0.844445	−	0.535642i	$-0.820070\pi$
$601$	−41.4037	−1.68889	−0.844445	+	0.535642i	$0.820070\pi$
$602$	0	0
$603$	10.3348	0.420865
$604$	6.36338	0.258922
$605$	7.77099	0.315936
$606$	5.49380	0.223170
$607$	12.3051	0.499449	0.249724	−	0.968317i	$-0.419660\pi$
$607$	12.3051	0.499449	0.249724	+	0.968317i	$0.419660\pi$
$608$	−15.5424	−0.630327
$609$	0	0
$610$	15.3167	0.620155
$611$	0	0
$612$	1.04345	0.0421789
$613$	26.2224	1.05911	0.529556	−	0.848275i	$-0.322358\pi$
$613$	26.2224	1.05911	0.529556	+	0.848275i	$0.322358\pi$
$614$	1.97816	0.0798319
$615$	0.895620	0.0361149
$616$	0	0
$617$	−18.8252	−0.757873	−0.378936	−	0.925423i	$-0.623710\pi$
$617$	−18.8252	−0.757873	−0.378936	+	0.925423i	$0.623710\pi$
$618$	−7.55038	−0.303721
$619$	−15.8083	−0.635389	−0.317695	−	0.948193i	$-0.602909\pi$
$619$	−15.8083	−0.635389	−0.317695	+	0.948193i	$0.602909\pi$
$620$	2.62762	0.105528
$621$	10.1387	0.406852
$622$	38.4994	1.54369
$623$	0	0
$624$	0	0
$625$	5.72006	0.228802
$626$	3.69473	0.147671
$627$	−6.92661	−0.276622
$628$	−4.76060	−0.189969
$629$	−4.52389	−0.180379
$630$	0	0
$631$	−16.6763	−0.663875	−0.331937	−	0.943301i	$-0.607702\pi$
$631$	−16.6763	−0.663875	−0.331937	+	0.943301i	$0.607702\pi$
$632$	−14.9441	−0.594445
$633$	−1.85457	−0.0737125
$634$	−30.7511	−1.22128
$635$	18.6810	0.741333
$636$	0.576375	0.0228548
$637$	0	0
$638$	28.0152	1.10913
$639$	6.98497	0.276321
$640$	−16.1108	−0.636837
$641$	49.2464	1.94512	0.972559	−	0.232658i	$-0.0747422\pi$
$641$	49.2464	1.94512	0.972559	+	0.232658i	$0.0747422\pi$
$642$	1.67344	0.0660454
$643$	42.8711	1.69067	0.845335	−	0.534236i	$-0.179401\pi$
$643$	42.8711	1.69067	0.845335	+	0.534236i	$0.179401\pi$
$644$	0	0
$645$	3.16700	0.124701
$646$	−9.42672	−0.370889
$647$	4.25859	0.167422	0.0837112	−	0.996490i	$-0.473323\pi$
$647$	4.25859	0.167422	0.0837112	+	0.996490i	$0.473323\pi$
$648$	−16.9882	−0.667359
$649$	−21.6819	−0.851089
$650$	0	0
$651$	0	0
$652$	−6.00875	−0.235321
$653$	−2.09552	−0.0820040	−0.0410020	−	0.999159i	$-0.513055\pi$
$653$	−2.09552	−0.0820040	−0.0410020	+	0.999159i	$0.513055\pi$
$654$	−10.4555	−0.408843
$655$	11.3643	0.444038
$656$	−7.15971	−0.279540
$657$	15.8196	0.617183
$658$	0	0
$659$	25.4518	0.991463	0.495732	−	0.868476i	$-0.334900\pi$
$659$	25.4518	0.991463	0.495732	+	0.868476i	$0.334900\pi$
$660$	0.514475	0.0200259
$661$	27.8108	1.08171	0.540857	−	0.841115i	$-0.318100\pi$
$661$	27.8108	1.08171	0.540857	+	0.841115i	$0.318100\pi$
$662$	6.10639	0.237332
$663$	0	0
$664$	28.6234	1.11080
$665$	0	0
$666$	−21.4235	−0.830143
$667$	30.6329	1.18611
$668$	3.24128	0.125409
$669$	−2.38207	−0.0920960
$670$	6.93481	0.267915
$671$	−17.4972	−0.675472
$672$	0	0
$673$	15.5207	0.598278	0.299139	−	0.954210i	$-0.403301\pi$
$673$	15.5207	0.598278	0.299139	+	0.954210i	$0.403301\pi$
$674$	−11.1020	−0.427633
$675$	−10.0907	−0.388392
$676$	0	0
$677$	34.5626	1.32835	0.664175	−	0.747577i	$-0.268783\pi$
$677$	34.5626	1.32835	0.664175	+	0.747577i	$0.268783\pi$
$678$	12.2422	0.470158
$679$	0	0
$680$	−2.65755	−0.101912
$681$	−11.8285	−0.453269
$682$	−17.3965	−0.666147
$683$	47.0064	1.79865	0.899325	−	0.437281i	$-0.144058\pi$
$683$	47.0064	1.79865	0.899325	+	0.437281i	$0.144058\pi$
$684$	−7.70269	−0.294520
$685$	3.29407	0.125860
$686$	0	0
$687$	10.6228	0.405284
$688$	−25.3174	−0.965218
$689$	0	0
$690$	3.26027	0.124116
$691$	19.0060	0.723023	0.361512	−	0.932368i	$-0.382261\pi$
$691$	19.0060	0.723023	0.361512	+	0.932368i	$0.382261\pi$
$692$	2.54177	0.0966235
$693$	0	0
$694$	15.6556	0.594280
$695$	27.1625	1.03033
$696$	10.2530	0.388638
$697$	−1.39241	−0.0527413
$698$	−9.77644	−0.370044
$699$	−1.85761	−0.0702612
$700$	0	0
$701$	−45.4648	−1.71718	−0.858591	−	0.512662i	$-0.828659\pi$
$701$	−45.4648	−1.71718	−0.858591	+	0.512662i	$0.828659\pi$
$702$	0	0
$703$	33.3951	1.25952
$704$	12.0802	0.455290
$705$	−1.85770	−0.0699651
$706$	53.1193	1.99917
$707$	0	0
$708$	2.09062	0.0785702
$709$	−9.78779	−0.367588	−0.183794	−	0.982965i	$-0.558838\pi$
$709$	−9.78779	−0.367588	−0.183794	+	0.982965i	$0.558838\pi$
$710$	4.68702	0.175901
$711$	−16.7637	−0.628687
$712$	43.6854	1.63718
$713$	−19.0220	−0.712378
$714$	0	0
$715$	0	0
$716$	7.72786	0.288804
$717$	−10.7195	−0.400326
$718$	29.0517	1.08420
$719$	−27.8403	−1.03827	−0.519133	−	0.854693i	$-0.673745\pi$
$719$	−27.8403	−1.03827	−0.519133	+	0.854693i	$0.673745\pi$
$720$	−15.3288	−0.571271
$721$	0	0
$722$	40.0485	1.49045
$723$	10.1514	0.377533
$724$	−2.33766	−0.0868783
$725$	−30.4879	−1.13229
$726$	4.96082	0.184113
$727$	−14.5650	−0.540186	−0.270093	−	0.962834i	$-0.587055\pi$
$727$	−14.5650	−0.540186	−0.270093	+	0.962834i	$0.587055\pi$
$728$	0	0
$729$	−14.9199	−0.552589
$730$	10.6152	0.392887
$731$	−4.92370	−0.182109
$732$	1.68712	0.0623577
$733$	17.6606	0.652309	0.326155	−	0.945316i	$-0.394247\pi$
$733$	17.6606	0.652309	0.326155	+	0.945316i	$0.394247\pi$
$734$	−48.2902	−1.78243
$735$	0	0
$736$	−8.35705	−0.308045
$737$	−7.92205	−0.291812
$738$	−6.59394	−0.242726
$739$	8.96559	0.329804	0.164902	−	0.986310i	$-0.447269\pi$
$739$	8.96559	0.329804	0.164902	+	0.986310i	$0.447269\pi$
$740$	−2.48042	−0.0911822
$741$	0	0
$742$	0	0
$743$	−26.3679	−0.967343	−0.483671	−	0.875250i	$-0.660697\pi$
$743$	−26.3679	−0.967343	−0.483671	+	0.875250i	$0.660697\pi$
$744$	−6.36674	−0.233416
$745$	17.1716	0.629119
$746$	−4.56618	−0.167180
$747$	32.1086	1.17479
$748$	−0.799847	−0.0292453
$749$	0	0
$750$	−7.77635	−0.283952
$751$	−20.2876	−0.740305	−0.370152	−	0.928971i	$-0.620695\pi$
$751$	−20.2876	−0.740305	−0.370152	+	0.928971i	$0.620695\pi$
$752$	14.8507	0.541550
$753$	−6.48641	−0.236378
$754$	0	0
$755$	−18.1801	−0.661642
$756$	0	0
$757$	24.9984	0.908584	0.454292	−	0.890853i	$-0.349892\pi$
$757$	24.9984	0.908584	0.454292	+	0.890853i	$0.349892\pi$
$758$	15.6791	0.569492
$759$	−3.72440	−0.135187
$760$	19.6179	0.711616
$761$	20.1422	0.730154	0.365077	−	0.930977i	$-0.381043\pi$
$761$	20.1422	0.730154	0.365077	+	0.930977i	$0.381043\pi$
$762$	11.9255	0.432016
$763$	0	0
$764$	0.210070	0.00760006
$765$	−2.98112	−0.107783
$766$	5.73573	0.207240
$767$	0	0
$768$	−4.69889	−0.169557
$769$	8.67220	0.312727	0.156364	−	0.987700i	$-0.450023\pi$
$769$	8.67220	0.312727	0.156364	+	0.987700i	$0.450023\pi$
$770$	0	0
$771$	6.44398	0.232074
$772$	−1.54828	−0.0557237
$773$	2.34567	0.0843679	0.0421839	−	0.999110i	$-0.486568\pi$
$773$	2.34567	0.0843679	0.0421839	+	0.999110i	$0.486568\pi$
$774$	−23.3168	−0.838106
$775$	18.9319	0.680055
$776$	−15.2688	−0.548119
$777$	0	0
$778$	35.2396	1.26340
$779$	10.2787	0.368273
$780$	0	0
$781$	−5.35426	−0.191591
$782$	−5.06870	−0.181256
$783$	23.9999	0.857687
$784$	0	0
$785$	13.6010	0.485440
$786$	7.25468	0.258766
$787$	−34.1166	−1.21613	−0.608063	−	0.793889i	$-0.708053\pi$
$787$	−34.1166	−1.21613	−0.608063	+	0.793889i	$0.708053\pi$
$788$	−3.10474	−0.110602
$789$	9.36455	0.333387
$790$	−11.2487	−0.400210
$791$	0	0
$792$	14.3768	0.510858
$793$	0	0
$794$	−45.2973	−1.60754
$795$	−1.64670	−0.0584024
$796$	3.13020	0.110947
$797$	−34.0844	−1.20733	−0.603666	−	0.797237i	$-0.706294\pi$
$797$	−34.0844	−1.20733	−0.603666	+	0.797237i	$0.706294\pi$
$798$	0	0
$799$	2.88815	0.102175
$800$	8.31749	0.294068
$801$	49.0044	1.73148
$802$	12.6249	0.445800
$803$	−12.1264	−0.427932
$804$	0.763862	0.0269393
$805$	0	0
$806$	0	0
$807$	13.9350	0.490534
$808$	17.7749	0.625319
$809$	−26.5205	−0.932413	−0.466206	−	0.884676i	$-0.654380\pi$
$809$	−26.5205	−0.932413	−0.466206	+	0.884676i	$0.654380\pi$
$810$	−12.7873	−0.449300
$811$	52.5463	1.84515	0.922575	−	0.385818i	$-0.126081\pi$
$811$	52.5463	1.84515	0.922575	+	0.385818i	$0.126081\pi$
$812$	0	0
$813$	−8.78080	−0.307956
$814$	16.4220	0.575590
$815$	17.1669	0.601331
$816$	−2.06637	−0.0723373
$817$	36.3465	1.27160
$818$	−12.9391	−0.452403
$819$	0	0
$820$	−0.763451	−0.0266609
$821$	−30.7546	−1.07334	−0.536671	−	0.843791i	$-0.680318\pi$
$821$	−30.7546	−1.07334	−0.536671	+	0.843791i	$0.680318\pi$
$822$	2.10286	0.0733455
$823$	−29.7038	−1.03541	−0.517705	−	0.855559i	$-0.673213\pi$
$823$	−29.7038	−1.03541	−0.517705	+	0.855559i	$0.673213\pi$
$824$	−24.4289	−0.851021
$825$	3.70677	0.129053
$826$	0	0
$827$	14.8351	0.515866	0.257933	−	0.966163i	$-0.416959\pi$
$827$	14.8351	0.515866	0.257933	+	0.966163i	$0.416959\pi$
$828$	−4.14170	−0.143934
$829$	14.5849	0.506554	0.253277	−	0.967394i	$-0.418492\pi$
$829$	14.5849	0.506554	0.253277	+	0.967394i	$0.418492\pi$
$830$	21.5453	0.747849
$831$	−6.57499	−0.228084
$832$	0	0
$833$	0	0
$834$	17.3399	0.600432
$835$	−9.26030	−0.320466
$836$	5.90443	0.204209
$837$	−14.9031	−0.515128
$838$	−20.2368	−0.699069
$839$	36.8086	1.27077	0.635386	−	0.772194i	$-0.280841\pi$
$839$	36.8086	1.27077	0.635386	+	0.772194i	$0.280841\pi$
$840$	0	0
$841$	43.5128	1.50044
$842$	−13.8318	−0.476674
$843$	14.6546	0.504730
$844$	1.58088	0.0544163
$845$	0	0
$846$	13.6772	0.470232
$847$	0	0
$848$	13.1639	0.452051
$849$	4.83929	0.166084
$850$	5.04470	0.173032
$851$	17.9564	0.615537
$852$	0.516270	0.0176871
$853$	−4.10728	−0.140630	−0.0703152	−	0.997525i	$-0.522401\pi$
$853$	−4.10728	−0.140630	−0.0703152	+	0.997525i	$0.522401\pi$
$854$	0	0
$855$	22.0065	0.752607
$856$	5.41433	0.185058
$857$	−38.3312	−1.30937	−0.654684	−	0.755902i	$-0.727199\pi$
$857$	−38.3312	−1.30937	−0.654684	+	0.755902i	$0.727199\pi$
$858$	0	0
$859$	−39.4369	−1.34557	−0.672785	−	0.739838i	$-0.734902\pi$
$859$	−39.4369	−1.34557	−0.672785	+	0.739838i	$0.734902\pi$
$860$	−2.69964	−0.0920569
$861$	0	0
$862$	−13.9260	−0.474322
$863$	−38.6440	−1.31546	−0.657728	−	0.753255i	$-0.728483\pi$
$863$	−38.6440	−1.31546	−0.657728	+	0.753255i	$0.728483\pi$
$864$	−6.54750	−0.222750
$865$	−7.26180	−0.246909
$866$	−0.268820	−0.00913488
$867$	7.91541	0.268821
$868$	0	0
$869$	12.8500	0.435908
$870$	7.71757	0.261650
$871$	0	0
$872$	−33.8283	−1.14557
$873$	−17.1279	−0.579692
$874$	37.4169	1.26564
$875$	0	0
$876$	1.16926	0.0395055
$877$	−58.0741	−1.96102	−0.980512	−	0.196458i	$-0.937056\pi$
$877$	−58.0741	−1.96102	−0.980512	+	0.196458i	$0.937056\pi$
$878$	14.8342	0.500630
$879$	−3.87026	−0.130541
$880$	11.7502	0.396098
$881$	21.6236	0.728519	0.364259	−	0.931298i	$-0.381322\pi$
$881$	21.6236	0.728519	0.364259	+	0.931298i	$0.381322\pi$
$882$	0	0
$883$	−22.7329	−0.765022	−0.382511	−	0.923951i	$-0.624941\pi$
$883$	−22.7329	−0.765022	−0.382511	+	0.923951i	$0.624941\pi$
$884$	0	0
$885$	−5.97287	−0.200776
$886$	−21.5690	−0.724624
$887$	−16.3317	−0.548365	−0.274182	−	0.961678i	$-0.588407\pi$
$887$	−16.3317	−0.548365	−0.274182	+	0.961678i	$0.588407\pi$
$888$	6.01008	0.201685
$889$	0	0
$890$	32.8827	1.10223
$891$	14.6077	0.489376
$892$	2.03054	0.0679874
$893$	−21.3201	−0.713451
$894$	10.9620	0.366623
$895$	−22.0784	−0.738000
$896$	0	0
$897$	0	0
$898$	−33.1012	−1.10460
$899$	−45.0280	−1.50177
$900$	4.12209	0.137403
$901$	2.56010	0.0852893
$902$	5.05453	0.168297
$903$	0	0
$904$	39.6089	1.31737
$905$	6.67866	0.222006
$906$	−11.6058	−0.385575
$907$	14.4096	0.478463	0.239232	−	0.970963i	$-0.423105\pi$
$907$	14.4096	0.478463	0.239232	+	0.970963i	$0.423105\pi$
$908$	10.0829	0.334614
$909$	19.9391	0.661339
$910$	0	0
$911$	−1.32236	−0.0438118	−0.0219059	−	0.999760i	$-0.506973\pi$
$911$	−1.32236	−0.0438118	−0.0219059	+	0.999760i	$0.506973\pi$
$912$	15.2538	0.505104
$913$	−24.6125	−0.814556
$914$	15.0581	0.498079
$915$	−4.82008	−0.159347
$916$	−9.05513	−0.299190
$917$	0	0
$918$	−3.97117	−0.131068
$919$	−27.4458	−0.905354	−0.452677	−	0.891675i	$-0.649531\pi$
$919$	−27.4458	−0.905354	−0.452677	+	0.891675i	$0.649531\pi$
$920$	10.5484	0.347772
$921$	−0.622515	−0.0205126
$922$	−2.13669	−0.0703682
$923$	0	0
$924$	0	0
$925$	−17.8714	−0.587607
$926$	49.3940	1.62319
$927$	−27.4033	−0.900042
$928$	−19.7825	−0.649391
$929$	−28.6389	−0.939611	−0.469805	−	0.882770i	$-0.655676\pi$
$929$	−28.6389	−0.939611	−0.469805	+	0.882770i	$0.655676\pi$
$930$	−4.79235	−0.157147
$931$	0	0
$932$	1.58348	0.0518685
$933$	−12.1156	−0.396646
$934$	−45.2739	−1.48141
$935$	2.28515	0.0747326
$936$	0	0
$937$	27.9990	0.914688	0.457344	−	0.889290i	$-0.348801\pi$
$937$	27.9990	0.914688	0.457344	+	0.889290i	$0.348801\pi$
$938$	0	0
$939$	−1.16271	−0.0379437
$940$	1.58356	0.0516499
$941$	28.9003	0.942124	0.471062	−	0.882100i	$-0.343871\pi$
$941$	28.9003	0.942124	0.471062	+	0.882100i	$0.343871\pi$
$942$	8.68255	0.282893
$943$	5.52680	0.179977
$944$	47.7480	1.55406
$945$	0	0
$946$	17.8733	0.581111
$947$	30.1235	0.978881	0.489441	−	0.872037i	$-0.337201\pi$
$947$	30.1235	0.978881	0.489441	+	0.872037i	$0.337201\pi$
$948$	−1.23903	−0.0402418
$949$	0	0
$950$	−37.2398	−1.20822
$951$	9.67719	0.313805
$952$	0	0
$953$	4.93022	0.159705	0.0798527	−	0.996807i	$-0.474555\pi$
$953$	4.93022	0.159705	0.0798527	+	0.996807i	$0.474555\pi$
$954$	12.1237	0.392519
$955$	−0.600167	−0.0194210
$956$	9.13757	0.295530
$957$	−8.81625	−0.284989
$958$	15.1144	0.488325
$959$	0	0
$960$	3.32783	0.107405
$961$	−3.03914	−0.0980368
$962$	0	0
$963$	6.07357	0.195718
$964$	−8.65330	−0.278704
$965$	4.42341	0.142395
$966$	0	0
$967$	29.1431	0.937180	0.468590	−	0.883416i	$-0.344762\pi$
$967$	29.1431	0.937180	0.468590	+	0.883416i	$0.344762\pi$
$968$	16.0505	0.515882
$969$	2.96654	0.0952989
$970$	−11.4931	−0.369021
$971$	14.5769	0.467794	0.233897	−	0.972261i	$-0.424852\pi$
$971$	14.5769	0.467794	0.233897	+	0.972261i	$0.424852\pi$
$972$	−4.93476	−0.158282
$973$	0	0
$974$	−26.6058	−0.852506
$975$	0	0
$976$	38.5324	1.23339
$977$	52.5218	1.68032	0.840161	−	0.542337i	$-0.182460\pi$
$977$	52.5218	1.68032	0.840161	+	0.542337i	$0.182460\pi$
$978$	10.9590	0.350429
$979$	−37.5639	−1.20055
$980$	0	0
$981$	−37.9472	−1.21156
$982$	−39.9886	−1.27609
$983$	−6.03769	−0.192572	−0.0962862	−	0.995354i	$-0.530696\pi$
$983$	−6.03769	−0.192572	−0.0962862	+	0.995354i	$0.530696\pi$
$984$	1.84985	0.0589709
$985$	8.87022	0.282629
$986$	−11.9984	−0.382107
$987$	0	0
$988$	0	0
$989$	19.5433	0.621441
$990$	10.8217	0.343935
$991$	31.3484	0.995813	0.497907	−	0.867231i	$-0.334102\pi$
$991$	31.3484	0.995813	0.497907	+	0.867231i	$0.334102\pi$
$992$	12.2842	0.390025
$993$	−1.92165	−0.0609816
$994$	0	0
$995$	−8.94296	−0.283511
$996$	2.37320	0.0751976
$997$	5.48033	0.173564	0.0867819	−	0.996227i	$-0.472342\pi$
$997$	5.48033	0.173564	0.0867819	+	0.996227i	$0.472342\pi$
$998$	8.40146	0.265944
$999$	14.0683	0.445101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.bz.1.5		6
7.2	even	3		1183.2.e.h.508.2		12
7.4	even	3		1183.2.e.h.170.2		12
7.6	odd	2		8281.2.a.ca.1.5		6
13.3	even	3		637.2.f.k.295.2		12
13.9	even	3		637.2.f.k.393.2		12
13.12	even	2		8281.2.a.ce.1.2		6
91.3	odd	6		637.2.g.l.373.2		12
91.9	even	3		91.2.g.b.81.2	yes	12
91.16	even	3		91.2.h.b.74.5	yes	12
91.25	even	6		1183.2.e.g.170.5		12
91.48	odd	6		637.2.f.j.393.2		12
91.51	even	6		1183.2.e.g.508.5		12
91.55	odd	6		637.2.f.j.295.2		12
91.61	odd	6		637.2.g.l.263.2		12
91.68	odd	6		637.2.h.l.165.5		12
91.74	even	3		91.2.h.b.16.5	yes	12
91.81	even	3		91.2.g.b.9.2	✓	12
91.87	odd	6		637.2.h.l.471.5		12
91.90	odd	2		8281.2.a.cf.1.2		6
273.74	odd	6		819.2.s.d.289.2		12
273.107	odd	6		819.2.s.d.802.2		12
273.191	odd	6		819.2.n.d.172.5		12
273.263	odd	6		819.2.n.d.100.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.b.9.2	✓	12	91.81	even	3
91.2.g.b.81.2	yes	12	91.9	even	3
91.2.h.b.16.5	yes	12	91.74	even	3
91.2.h.b.74.5	yes	12	91.16	even	3
637.2.f.j.295.2		12	91.55	odd	6
637.2.f.j.393.2		12	91.48	odd	6
637.2.f.k.295.2		12	13.3	even	3
637.2.f.k.393.2		12	13.9	even	3
637.2.g.l.263.2		12	91.61	odd	6
637.2.g.l.373.2		12	91.3	odd	6
637.2.h.l.165.5		12	91.68	odd	6
637.2.h.l.471.5		12	91.87	odd	6
819.2.n.d.100.5		12	273.263	odd	6
819.2.n.d.172.5		12	273.191	odd	6
819.2.s.d.289.2		12	273.74	odd	6
819.2.s.d.802.2		12	273.107	odd	6
1183.2.e.g.170.5		12	91.25	even	6
1183.2.e.g.508.5		12	91.51	even	6
1183.2.e.h.170.2		12	7.4	even	3
1183.2.e.h.508.2		12	7.2	even	3
8281.2.a.bz.1.5		6	1.1	even	1	trivial
8281.2.a.ca.1.5		6	7.6	odd	2
8281.2.a.ce.1.2		6	13.12	even	2
8281.2.a.cf.1.2		6	91.90	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.55469 q^{2} -0.489252 q^{3} +0.417051 q^{4} -1.19151 q^{5} -0.760633 q^{6} -2.46099 q^{8} -2.76063 q^{9} -1.85243 q^{10} +2.11614 q^{11} -0.204043 q^{12} +0.582949 q^{15} -4.66017 q^{16} -0.906303 q^{17} -4.29192 q^{18} +6.69028 q^{19} -0.496921 q^{20} +3.28993 q^{22} +3.59733 q^{23} +1.20404 q^{24} -3.58030 q^{25} +2.81840 q^{27} +8.51545 q^{29} +0.906303 q^{30} -5.28780 q^{31} -2.32313 q^{32} -1.03532 q^{33} -1.40902 q^{34} -1.15133 q^{36} +4.99159 q^{37} +10.4013 q^{38} +2.93230 q^{40} +1.53636 q^{41} +5.43273 q^{43} +0.882538 q^{44} +3.28933 q^{45} +5.59272 q^{46} -3.18673 q^{47} +2.28000 q^{48} -5.56625 q^{50} +0.443410 q^{51} -2.82477 q^{53} +4.38173 q^{54} -2.52140 q^{55} -3.27323 q^{57} +13.2389 q^{58} -10.2460 q^{59} +0.243120 q^{60} -8.26845 q^{61} -8.22088 q^{62} +5.70861 q^{64} -1.60960 q^{66} -3.74363 q^{67} -0.377975 q^{68} -1.76000 q^{69} -2.53020 q^{71} +6.79389 q^{72} -5.73044 q^{73} +7.76035 q^{74} +1.75167 q^{75} +2.79019 q^{76} +6.07240 q^{79} +5.55265 q^{80} +6.90299 q^{81} +2.38856 q^{82} -11.6309 q^{83} +1.07987 q^{85} +8.44619 q^{86} -4.16619 q^{87} -5.20780 q^{88} -17.7511 q^{89} +5.11387 q^{90} +1.50027 q^{92} +2.58707 q^{93} -4.95437 q^{94} -7.97155 q^{95} +1.13659 q^{96} +6.20434 q^{97} -5.84188 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{2} - q^{3} + 4 q^{4} - q^{5} + 9 q^{6} - 3 q^{8} - 3 q^{9} - 4 q^{10} - 4 q^{11} - 5 q^{12} + 2 q^{15} - 8 q^{16} - 5 q^{17} - 3 q^{18} + q^{19} + q^{20} + 5 q^{22} + q^{23} + 11 q^{24}+ \cdots - 10 q^{99}+O(q^{100}) \) 6 * q - 2 * q^2 - q^3 + 4 * q^4 - q^5 + 9 * q^6 - 3 * q^8 - 3 * q^9 - 4 * q^10 - 4 * q^11 - 5 * q^12 + 2 * q^15 - 8 * q^16 - 5 * q^17 - 3 * q^18 + q^19 + q^20 + 5 * q^22 + q^23 + 11 * q^24 - 7 * q^25 - 4 * q^27 - 3 * q^29 + 5 * q^30 - 16 * q^31 - 8 * q^32 - 16 * q^33 + 16 * q^34 + 21 * q^36 + 13 * q^37 + 17 * q^38 + 5 * q^40 + 8 * q^41 + 11 * q^43 - 21 * q^44 + 7 * q^45 - 16 * q^46 + q^47 - 21 * q^48 - 6 * q^50 + 20 * q^51 + 2 * q^53 + 18 * q^54 - 9 * q^55 + 21 * q^57 + 8 * q^58 - 13 * q^59 - 20 * q^60 + 5 * q^61 - 5 * q^62 - 15 * q^64 - 18 * q^66 + 11 * q^67 - 29 * q^68 - 23 * q^69 - 6 * q^71 - 25 * q^72 + 30 * q^73 + 3 * q^74 + 3 * q^75 + 9 * q^76 - 7 * q^79 + 7 * q^80 + 6 * q^81 - q^82 - 27 * q^83 + q^85 + 7 * q^86 - 16 * q^87 - 4 * q^89 - 8 * q^90 + 27 * q^92 + 7 * q^93 - 45 * q^94 + 6 * q^95 - 19 * q^96 + 35 * q^97 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more