LMFDB - Embedded newform 8281.2.a.cu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

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")

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

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 1183)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
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)

$f(q)$	$=$	\(q-2.37543 q^{2} -0.696489 q^{3} +3.64268 q^{4} -2.62567 q^{5} +1.65446 q^{6} -3.90209 q^{8} -2.51490 q^{9} +6.23710 q^{10} +4.46749 q^{11} -2.53709 q^{12} +1.82875 q^{15} +1.98378 q^{16} -2.35221 q^{17} +5.97398 q^{18} +6.82253 q^{19} -9.56448 q^{20} -10.6122 q^{22} +1.91154 q^{23} +2.71776 q^{24} +1.89414 q^{25} +3.84107 q^{27} +7.24773 q^{29} -4.34408 q^{30} +7.55820 q^{31} +3.09183 q^{32} -3.11156 q^{33} +5.58752 q^{34} -9.16100 q^{36} -5.94991 q^{37} -16.2065 q^{38} +10.2456 q^{40} +0.271667 q^{41} -3.28422 q^{43} +16.2737 q^{44} +6.60330 q^{45} -4.54073 q^{46} +8.10744 q^{47} -1.38168 q^{48} -4.49940 q^{50} +1.63829 q^{51} +7.19152 q^{53} -9.12421 q^{54} -11.7302 q^{55} -4.75182 q^{57} -17.2165 q^{58} +9.04443 q^{59} +6.66156 q^{60} +7.88752 q^{61} -17.9540 q^{62} -11.3120 q^{64} +7.39130 q^{66} -7.00361 q^{67} -8.56836 q^{68} -1.33136 q^{69} +6.84206 q^{71} +9.81337 q^{72} -0.348014 q^{73} +14.1336 q^{74} -1.31925 q^{75} +24.8523 q^{76} -12.5248 q^{79} -5.20875 q^{80} +4.86944 q^{81} -0.645327 q^{82} +13.4536 q^{83} +6.17612 q^{85} +7.80146 q^{86} -5.04797 q^{87} -17.4325 q^{88} -6.00512 q^{89} -15.6857 q^{90} +6.96312 q^{92} -5.26420 q^{93} -19.2587 q^{94} -17.9137 q^{95} -2.15343 q^{96} +12.6274 q^{97} -11.2353 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} + 5 q^{10} - q^{11} + 5 q^{12} + 5 q^{15} + 17 q^{16} - 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} - 21 q^{27}+ \cdots - 39 q^{99}+O(q^{100}) $ 24 * q - q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 + 5 * q^10 - q^11 + 5 * q^12 + 5 * q^15 + 17 * q^16 - 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 - 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 - 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 - 4 * q^37 - 29 * q^38 - 4 * q^40 + 49 * q^41 + 13 * q^43 + 10 * q^44 + 58 * q^45 - 10 * q^46 + 62 * q^47 + 89 * q^48 - 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 - 14 * q^55 - 13 * q^57 + 56 * q^58 + 79 * q^59 + 22 * q^60 + 13 * q^61 + 12 * q^62 + 18 * q^64 - 38 * q^66 - 2 * q^67 - 12 * q^68 - 28 * q^69 - 19 * q^71 + 81 * q^72 + 17 * q^73 + 17 * q^74 + 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 - 22 * q^82 + 81 * q^83 - 34 * q^85 + 22 * q^86 + 70 * q^87 - 33 * q^88 + 72 * q^89 + q^90 - 4 * q^92 + 19 * q^93 - 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 - 39 * 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 $

$2$	−2.37543	−1.67969	−0.839843	−	0.542830i	$-0.817353\pi$
$2$	−2.37543	−1.67969	−0.839843	+	0.542830i	$0.817353\pi$
$3$	−0.696489	−0.402118	−0.201059	−	0.979579i	$-0.564438\pi$
$3$	−0.696489	−0.402118	−0.201059	+	0.979579i	$0.564438\pi$
$4$	3.64268	1.82134
$5$	−2.62567	−1.17423	−0.587117	−	0.809502i	$-0.699737\pi$
$5$	−2.62567	−1.17423	−0.587117	+	0.809502i	$0.699737\pi$
$6$	1.65446	0.675432
$7$	0	0
$7$	0	0
$8$	−3.90209	−1.37960
$9$	−2.51490	−0.838301
$10$	6.23710	1.97234
$11$	4.46749	1.34700	0.673500	−	0.739188i	$-0.264790\pi$
$11$	4.46749	1.34700	0.673500	+	0.739188i	$0.264790\pi$
$12$	−2.53709	−0.732395
$13$	0	0
$13$	0	0
$14$	0	0
$15$	1.82875	0.472181
$16$	1.98378	0.495945
$17$	−2.35221	−0.570495	−0.285247	−	0.958454i	$-0.592076\pi$
$17$	−2.35221	−0.570495	−0.285247	+	0.958454i	$0.592076\pi$
$18$	5.97398	1.40808
$19$	6.82253	1.56520	0.782598	−	0.622527i	$-0.213894\pi$
$19$	6.82253	1.56520	0.782598	+	0.622527i	$0.213894\pi$
$20$	−9.56448	−2.13868
$21$	0	0
$22$	−10.6122	−2.26253
$23$	1.91154	0.398583	0.199291	−	0.979940i	$-0.436136\pi$
$23$	1.91154	0.398583	0.199291	+	0.979940i	$0.436136\pi$
$24$	2.71776	0.554761
$25$	1.89414	0.378828
$26$	0	0
$27$	3.84107	0.739214
$28$	0	0
$29$	7.24773	1.34587	0.672935	−	0.739702i	$-0.265033\pi$
$29$	7.24773	1.34587	0.672935	+	0.739702i	$0.265033\pi$
$30$	−4.34408	−0.793116
$31$	7.55820	1.35749	0.678746	−	0.734373i	$-0.262524\pi$
$31$	7.55820	1.35749	0.678746	+	0.734373i	$0.262524\pi$
$32$	3.09183	0.546564
$33$	−3.11156	−0.541653
$34$	5.58752	0.958251
$35$	0	0
$36$	−9.16100	−1.52683
$37$	−5.94991	−0.978159	−0.489079	−	0.872239i	$-0.662667\pi$
$37$	−5.94991	−0.978159	−0.489079	+	0.872239i	$0.662667\pi$
$38$	−16.2065	−2.62904
$39$	0	0
$40$	10.2456	1.61997
$41$	0.271667	0.0424273	0.0212136	−	0.999775i	$-0.493247\pi$
$41$	0.271667	0.0424273	0.0212136	+	0.999775i	$0.493247\pi$
$42$	0	0
$43$	−3.28422	−0.500840	−0.250420	−	0.968137i	$-0.580569\pi$
$43$	−3.28422	−0.500840	−0.250420	+	0.968137i	$0.580569\pi$
$44$	16.2737	2.45335
$45$	6.60330	0.984362
$46$	−4.54073	−0.669494
$47$	8.10744	1.18259	0.591296	−	0.806455i	$-0.298617\pi$
$47$	8.10744	1.18259	0.591296	+	0.806455i	$0.298617\pi$
$48$	−1.38168	−0.199429
$49$	0	0
$50$	−4.49940	−0.636311
$51$	1.63829	0.229406
$52$	0	0
$53$	7.19152	0.987831	0.493916	−	0.869510i	$-0.335565\pi$
$53$	7.19152	0.987831	0.493916	+	0.869510i	$0.335565\pi$
$54$	−9.12421	−1.24165
$55$	−11.7302	−1.58169
$56$	0	0
$57$	−4.75182	−0.629394
$58$	−17.2165	−2.26064
$59$	9.04443	1.17748	0.588742	−	0.808321i	$-0.299623\pi$
$59$	9.04443	1.17748	0.588742	+	0.808321i	$0.299623\pi$
$60$	6.66156	0.860004
$61$	7.88752	1.00989	0.504947	−	0.863150i	$-0.331512\pi$
$61$	7.88752	1.00989	0.504947	+	0.863150i	$0.331512\pi$
$62$	−17.9540	−2.28016
$63$	0	0
$64$	−11.3120	−1.41400
$65$	0	0
$66$	7.39130	0.909807
$67$	−7.00361	−0.855628	−0.427814	−	0.903867i	$-0.640716\pi$
$67$	−7.00361	−0.855628	−0.427814	+	0.903867i	$0.640716\pi$
$68$	−8.56836	−1.03907
$69$	−1.33136	−0.160277
$70$	0	0
$71$	6.84206	0.812004	0.406002	−	0.913872i	$-0.366923\pi$
$71$	6.84206	0.812004	0.406002	+	0.913872i	$0.366923\pi$
$72$	9.81337	1.15652
$73$	−0.348014	−0.0407319	−0.0203660	−	0.999793i	$-0.506483\pi$
$73$	−0.348014	−0.0407319	−0.0203660	+	0.999793i	$0.506483\pi$
$74$	14.1336	1.64300
$75$	−1.31925	−0.152334
$76$	24.8523	2.85076
$77$	0	0
$78$	0	0
$79$	−12.5248	−1.40915	−0.704577	−	0.709628i	$-0.748863\pi$
$79$	−12.5248	−1.40915	−0.704577	+	0.709628i	$0.748863\pi$
$80$	−5.20875	−0.582356
$81$	4.86944	0.541049
$82$	−0.645327	−0.0712645
$83$	13.4536	1.47673	0.738363	−	0.674404i	$-0.235599\pi$
$83$	13.4536	1.47673	0.738363	+	0.674404i	$0.235599\pi$
$84$	0	0
$85$	6.17612	0.669895
$86$	7.80146	0.841253
$87$	−5.04797	−0.541199
$88$	−17.4325	−1.85832
$89$	−6.00512	−0.636541	−0.318271	−	0.948000i	$-0.603102\pi$
$89$	−6.00512	−0.636541	−0.318271	+	0.948000i	$0.603102\pi$
$90$	−15.6857	−1.65342
$91$	0	0
$92$	6.96312	0.725956
$93$	−5.26420	−0.545873
$94$	−19.2587	−1.98638
$95$	−17.9137	−1.83791
$96$	−2.15343	−0.219784
$97$	12.6274	1.28212	0.641060	−	0.767491i	$-0.278495\pi$
$97$	12.6274	1.28212	0.641060	+	0.767491i	$0.278495\pi$
$98$	0	0
$99$	−11.2353	−1.12919
$100$	6.89975	0.689975
$101$	−13.5168	−1.34497	−0.672486	−	0.740110i	$-0.734773\pi$
$101$	−13.5168	−1.34497	−0.672486	+	0.740110i	$0.734773\pi$
$102$	−3.89165	−0.385330
$103$	−13.4747	−1.32770	−0.663852	−	0.747864i	$-0.731080\pi$
$103$	−13.4747	−1.32770	−0.663852	+	0.747864i	$0.731080\pi$
$104$	0	0
$105$	0	0
$106$	−17.0830	−1.65925
$107$	−2.26178	−0.218654	−0.109327	−	0.994006i	$-0.534870\pi$
$107$	−2.26178	−0.218654	−0.109327	+	0.994006i	$0.534870\pi$
$108$	13.9918	1.34636
$109$	14.5403	1.39270	0.696352	−	0.717700i	$-0.254805\pi$
$109$	14.5403	1.39270	0.696352	+	0.717700i	$0.254805\pi$
$110$	27.8642	2.65675
$111$	4.14405	0.393336
$112$	0	0
$113$	−2.50654	−0.235795	−0.117898	−	0.993026i	$-0.537615\pi$
$113$	−2.50654	−0.235795	−0.117898	+	0.993026i	$0.537615\pi$
$114$	11.2876	1.05718
$115$	−5.01906	−0.468030
$116$	26.4012	2.45129
$117$	0	0
$118$	−21.4844	−1.97780
$119$	0	0
$120$	−7.13595	−0.651420
$121$	8.95847	0.814406
$122$	−18.7363	−1.69630
$123$	−0.189213	−0.0170608
$124$	27.5321	2.47246
$125$	8.15497	0.729402
$126$	0	0
$127$	1.37935	0.122397	0.0611987	−	0.998126i	$-0.480508\pi$
$127$	1.37935	0.122397	0.0611987	+	0.998126i	$0.480508\pi$
$128$	20.6873	1.82851
$129$	2.28743	0.201397
$130$	0	0
$131$	13.2587	1.15842	0.579209	−	0.815179i	$-0.303362\pi$
$131$	13.2587	1.15842	0.579209	+	0.815179i	$0.303362\pi$
$132$	−11.3344	−0.986536
$133$	0	0
$134$	16.6366	1.43719
$135$	−10.0854	−0.868011
$136$	9.17853	0.787052
$137$	−19.2541	−1.64499	−0.822494	−	0.568774i	$-0.807418\pi$
$137$	−19.2541	−1.64499	−0.822494	+	0.568774i	$0.807418\pi$
$138$	3.16257	0.269216
$139$	9.81673	0.832644	0.416322	−	0.909217i	$-0.363319\pi$
$139$	9.81673	0.832644	0.416322	+	0.909217i	$0.363319\pi$
$140$	0	0
$141$	−5.64675	−0.475542
$142$	−16.2529	−1.36391
$143$	0	0
$144$	−4.98902	−0.415752
$145$	−19.0301	−1.58037
$146$	0.826684	0.0684168
$147$	0	0
$148$	−21.6736	−1.78156
$149$	6.34667	0.519939	0.259970	−	0.965617i	$-0.416287\pi$
$149$	6.34667	0.519939	0.259970	+	0.965617i	$0.416287\pi$
$150$	3.13378	0.255872
$151$	−3.17436	−0.258326	−0.129163	−	0.991623i	$-0.541229\pi$
$151$	−3.17436	−0.258326	−0.129163	+	0.991623i	$0.541229\pi$
$152$	−26.6221	−2.15934
$153$	5.91558	0.478246
$154$	0	0
$155$	−19.8453	−1.59401
$156$	0	0
$157$	−0.402481	−0.0321215	−0.0160607	−	0.999871i	$-0.505113\pi$
$157$	−0.402481	−0.0321215	−0.0160607	+	0.999871i	$0.505113\pi$
$158$	29.7519	2.36693
$159$	−5.00882	−0.397225
$160$	−8.11813	−0.641795
$161$	0	0
$162$	−11.5670	−0.908792
$163$	−11.8864	−0.931012	−0.465506	−	0.885045i	$-0.654128\pi$
$163$	−11.8864	−0.931012	−0.465506	+	0.885045i	$0.654128\pi$
$164$	0.989597	0.0772746
$165$	8.16993	0.636028
$166$	−31.9582	−2.48043
$167$	−0.438220	−0.0339105	−0.0169552	−	0.999856i	$-0.505397\pi$
$167$	−0.438220	−0.0339105	−0.0169552	+	0.999856i	$0.505397\pi$
$168$	0	0
$169$	0	0
$170$	−14.6710	−1.12521
$171$	−17.1580	−1.31211
$172$	−11.9634	−0.912200
$173$	−5.47503	−0.416258	−0.208129	−	0.978101i	$-0.566737\pi$
$173$	−5.47503	−0.416258	−0.208129	+	0.978101i	$0.566737\pi$
$174$	11.9911	0.909044
$175$	0	0
$176$	8.86253	0.668038
$177$	−6.29935	−0.473488
$178$	14.2648	1.06919
$179$	0.541906	0.0405040	0.0202520	−	0.999795i	$-0.493553\pi$
$179$	0.541906	0.0405040	0.0202520	+	0.999795i	$0.493553\pi$
$180$	24.0537	1.79286
$181$	10.9671	0.815178	0.407589	−	0.913165i	$-0.366370\pi$
$181$	10.9671	0.815178	0.407589	+	0.913165i	$0.366370\pi$
$182$	0	0
$183$	−5.49358	−0.406097
$184$	−7.45898	−0.549884
$185$	15.6225	1.14859
$186$	12.5048	0.916894
$187$	−10.5085	−0.768456
$188$	29.5329	2.15390
$189$	0	0
$190$	42.5528	3.08711
$191$	4.90268	0.354745	0.177373	−	0.984144i	$-0.443240\pi$
$191$	4.90268	0.354745	0.177373	+	0.984144i	$0.443240\pi$
$192$	7.87870	0.568596
$193$	−17.8596	−1.28556	−0.642782	−	0.766049i	$-0.722220\pi$
$193$	−17.8596	−1.28556	−0.642782	+	0.766049i	$0.722220\pi$
$194$	−29.9956	−2.15356
$195$	0	0
$196$	0	0
$197$	2.43488	0.173478	0.0867390	−	0.996231i	$-0.472355\pi$
$197$	2.43488	0.173478	0.0867390	+	0.996231i	$0.472355\pi$
$198$	26.6887	1.89668
$199$	−5.14098	−0.364435	−0.182217	−	0.983258i	$-0.558327\pi$
$199$	−5.14098	−0.364435	−0.182217	+	0.983258i	$0.558327\pi$
$200$	−7.39109	−0.522629
$201$	4.87794	0.344064
$202$	32.1082	2.25913
$203$	0	0
$204$	5.96777	0.417827
$205$	−0.713308	−0.0498196
$206$	32.0083	2.23012
$207$	−4.80733	−0.334132
$208$	0	0
$209$	30.4796	2.10832
$210$	0	0
$211$	−18.5097	−1.27426	−0.637129	−	0.770758i	$-0.719878\pi$
$211$	−18.5097	−1.27426	−0.637129	+	0.770758i	$0.719878\pi$
$212$	26.1964	1.79918
$213$	−4.76543	−0.326522
$214$	5.37270	0.367270
$215$	8.62329	0.588103
$216$	−14.9882	−1.01982
$217$	0	0
$218$	−34.5394	−2.33931
$219$	0.242388	0.0163791
$220$	−42.7292	−2.88080
$221$	0	0
$222$	−9.84391	−0.660680
$223$	−8.79953	−0.589260	−0.294630	−	0.955611i	$-0.595196\pi$
$223$	−8.79953	−0.589260	−0.294630	+	0.955611i	$0.595196\pi$
$224$	0	0
$225$	−4.76357	−0.317571
$226$	5.95411	0.396062
$227$	8.49873	0.564081	0.282040	−	0.959403i	$-0.408989\pi$
$227$	8.49873	0.564081	0.282040	+	0.959403i	$0.408989\pi$
$228$	−17.3094	−1.14634
$229$	−23.0879	−1.52569	−0.762846	−	0.646580i	$-0.776199\pi$
$229$	−23.0879	−1.52569	−0.762846	+	0.646580i	$0.776199\pi$
$230$	11.9224	0.786143
$231$	0	0
$232$	−28.2813	−1.85676
$233$	14.6156	0.957498	0.478749	−	0.877952i	$-0.341090\pi$
$233$	14.6156	0.957498	0.478749	+	0.877952i	$0.341090\pi$
$234$	0	0
$235$	−21.2875	−1.38864
$236$	32.9460	2.14460
$237$	8.72341	0.566646
$238$	0	0
$239$	−11.9223	−0.771192	−0.385596	−	0.922668i	$-0.626004\pi$
$239$	−11.9223	−0.771192	−0.385596	+	0.922668i	$0.626004\pi$
$240$	3.62784	0.234176
$241$	22.2083	1.43057	0.715283	−	0.698835i	$-0.246298\pi$
$241$	22.2083	1.43057	0.715283	+	0.698835i	$0.246298\pi$
$242$	−21.2803	−1.36795
$243$	−14.9147	−0.956780
$244$	28.7318	1.83936
$245$	0	0
$246$	0.449463	0.0286567
$247$	0	0
$248$	−29.4928	−1.87279
$249$	−9.37030	−0.593818
$250$	−19.3716	−1.22517
$251$	11.4119	0.720314	0.360157	−	0.932892i	$-0.382723\pi$
$251$	11.4119	0.720314	0.360157	+	0.932892i	$0.382723\pi$
$252$	0	0
$253$	8.53977	0.536891
$254$	−3.27655	−0.205589
$255$	−4.30160	−0.269377
$256$	−26.5172	−1.65732
$257$	11.2453	0.701464	0.350732	−	0.936476i	$-0.385933\pi$
$257$	11.2453	0.701464	0.350732	+	0.936476i	$0.385933\pi$
$258$	−5.43363	−0.338283
$259$	0	0
$260$	0	0
$261$	−18.2273	−1.12824
$262$	−31.4951	−1.94578
$263$	−5.24132	−0.323194	−0.161597	−	0.986857i	$-0.551664\pi$
$263$	−5.24132	−0.323194	−0.161597	+	0.986857i	$0.551664\pi$
$264$	12.1416	0.747263
$265$	−18.8826	−1.15995
$266$	0	0
$267$	4.18250	0.255965
$268$	−25.5120	−1.55839
$269$	−6.80855	−0.415125	−0.207562	−	0.978222i	$-0.566553\pi$
$269$	−6.80855	−0.415125	−0.207562	+	0.978222i	$0.566553\pi$
$270$	23.9572	1.45799
$271$	4.04340	0.245619	0.122809	−	0.992430i	$-0.460810\pi$
$271$	4.04340	0.245619	0.122809	+	0.992430i	$0.460810\pi$
$272$	−4.66627	−0.282934
$273$	0	0
$274$	45.7368	2.76306
$275$	8.46204	0.510280
$276$	−4.84974	−0.291920
$277$	−3.18579	−0.191416	−0.0957079	−	0.995409i	$-0.530511\pi$
$277$	−3.18579	−0.191416	−0.0957079	+	0.995409i	$0.530511\pi$
$278$	−23.3190	−1.39858
$279$	−19.0081	−1.13799
$280$	0	0
$281$	−13.8238	−0.824660	−0.412330	−	0.911035i	$-0.635285\pi$
$281$	−13.8238	−0.824660	−0.412330	+	0.911035i	$0.635285\pi$
$282$	13.4135	0.798761
$283$	−4.71357	−0.280193	−0.140096	−	0.990138i	$-0.544741\pi$
$283$	−4.71357	−0.280193	−0.140096	+	0.990138i	$0.544741\pi$
$284$	24.9235	1.47894
$285$	12.4767	0.739056
$286$	0	0
$287$	0	0
$288$	−7.77566	−0.458185
$289$	−11.4671	−0.674536
$290$	45.2049	2.65452
$291$	−8.79486	−0.515564
$292$	−1.26770	−0.0741868
$293$	13.5197	0.789831	0.394916	−	0.918717i	$-0.370774\pi$
$293$	13.5197	0.789831	0.394916	+	0.918717i	$0.370774\pi$
$294$	0	0
$295$	−23.7477	−1.38264
$296$	23.2171	1.34946
$297$	17.1599	0.995721
$298$	−15.0761	−0.873334
$299$	0	0
$300$	−4.80560	−0.277451
$301$	0	0
$302$	7.54049	0.433906
$303$	9.41430	0.540838
$304$	13.5344	0.776252
$305$	−20.7100	−1.18585
$306$	−14.0521	−0.803303
$307$	0.157347	0.00898024	0.00449012	−	0.999990i	$-0.498571\pi$
$307$	0.157347	0.00898024	0.00449012	+	0.999990i	$0.498571\pi$
$308$	0	0
$309$	9.38500	0.533894
$310$	47.1413	2.67744
$311$	2.01577	0.114304	0.0571519	−	0.998365i	$-0.481798\pi$
$311$	2.01577	0.114304	0.0571519	+	0.998365i	$0.481798\pi$
$312$	0	0
$313$	15.9070	0.899117	0.449559	−	0.893251i	$-0.351581\pi$
$313$	15.9070	0.899117	0.449559	+	0.893251i	$0.351581\pi$
$314$	0.956066	0.0539539
$315$	0	0
$316$	−45.6240	−2.56655
$317$	22.2854	1.25167	0.625835	−	0.779955i	$-0.284758\pi$
$317$	22.2854	1.25167	0.625835	+	0.779955i	$0.284758\pi$
$318$	11.8981	0.667213
$319$	32.3792	1.81289
$320$	29.7016	1.66037
$321$	1.57530	0.0879249
$322$	0	0
$323$	−16.0480	−0.892936
$324$	17.7378	0.985436
$325$	0	0
$326$	28.2353	1.56381
$327$	−10.1271	−0.560032
$328$	−1.06007	−0.0585325
$329$	0	0
$330$	−19.4071	−1.06833
$331$	−5.42392	−0.298126	−0.149063	−	0.988828i	$-0.547626\pi$
$331$	−5.42392	−0.298126	−0.149063	+	0.988828i	$0.547626\pi$
$332$	49.0073	2.68962
$333$	14.9634	0.819991
$334$	1.04096	0.0569589
$335$	18.3892	1.00471
$336$	0	0
$337$	15.1236	0.823833	0.411917	−	0.911222i	$-0.364860\pi$
$337$	15.1236	0.823833	0.411917	+	0.911222i	$0.364860\pi$
$338$	0	0
$339$	1.74578	0.0948176
$340$	22.4977	1.22011
$341$	33.7662	1.82854
$342$	40.7577	2.20392
$343$	0	0
$344$	12.8153	0.690956
$345$	3.49572	0.188203
$346$	13.0056	0.699183
$347$	19.5336	1.04862	0.524309	−	0.851528i	$-0.324324\pi$
$347$	19.5336	1.04862	0.524309	+	0.851528i	$0.324324\pi$
$348$	−18.3882	−0.985709
$349$	−19.2006	−1.02779	−0.513893	−	0.857854i	$-0.671797\pi$
$349$	−19.2006	−1.02779	−0.513893	+	0.857854i	$0.671797\pi$
$350$	0	0
$351$	0	0
$352$	13.8127	0.736221
$353$	−0.870180	−0.0463150	−0.0231575	−	0.999732i	$-0.507372\pi$
$353$	−0.870180	−0.0463150	−0.0231575	+	0.999732i	$0.507372\pi$
$354$	14.9637	0.795311
$355$	−17.9650	−0.953483
$356$	−21.8747	−1.15936
$357$	0	0
$358$	−1.28726	−0.0680339
$359$	−2.85875	−0.150879	−0.0754397	−	0.997150i	$-0.524036\pi$
$359$	−2.85875	−0.150879	−0.0754397	+	0.997150i	$0.524036\pi$
$360$	−25.7667	−1.35802
$361$	27.5469	1.44984
$362$	−26.0516	−1.36924
$363$	−6.23948	−0.327488
$364$	0	0
$365$	0.913769	0.0478289
$366$	13.0496	0.682115
$367$	−16.1982	−0.845537	−0.422769	−	0.906238i	$-0.638942\pi$
$367$	−16.1982	−0.845537	−0.422769	+	0.906238i	$0.638942\pi$
$368$	3.79207	0.197675
$369$	−0.683216	−0.0355668
$370$	−37.1102	−1.92927
$371$	0	0
$372$	−19.1758	−0.994221
$373$	−13.3611	−0.691813	−0.345907	−	0.938269i	$-0.612429\pi$
$373$	−13.3611	−0.691813	−0.345907	+	0.938269i	$0.612429\pi$
$374$	24.9622	1.29076
$375$	−5.67985	−0.293306
$376$	−31.6359	−1.63150
$377$	0	0
$378$	0	0
$379$	1.03567	0.0531987	0.0265994	−	0.999646i	$-0.491532\pi$
$379$	1.03567	0.0531987	0.0265994	+	0.999646i	$0.491532\pi$
$380$	−65.2540	−3.34746
$381$	−0.960702	−0.0492183
$382$	−11.6460	−0.595860
$383$	−34.8972	−1.78316	−0.891581	−	0.452861i	$-0.850403\pi$
$383$	−34.8972	−1.78316	−0.891581	+	0.452861i	$0.850403\pi$
$384$	−14.4085	−0.735279
$385$	0	0
$386$	42.4243	2.15934
$387$	8.25950	0.419854
$388$	45.9977	2.33518
$389$	15.8129	0.801746	0.400873	−	0.916134i	$-0.368707\pi$
$389$	15.8129	0.801746	0.400873	+	0.916134i	$0.368707\pi$
$390$	0	0
$391$	−4.49633	−0.227389
$392$	0	0
$393$	−9.23454	−0.465821
$394$	−5.78389	−0.291388
$395$	32.8861	1.65468
$396$	−40.9267	−2.05664
$397$	35.1469	1.76397	0.881986	−	0.471276i	$-0.156206\pi$
$397$	35.1469	1.76397	0.881986	+	0.471276i	$0.156206\pi$
$398$	12.2121	0.612135
$399$	0	0
$400$	3.75756	0.187878
$401$	−2.60906	−0.130290	−0.0651452	−	0.997876i	$-0.520751\pi$
$401$	−2.60906	−0.130290	−0.0651452	+	0.997876i	$0.520751\pi$
$402$	−11.5872	−0.577919
$403$	0	0
$404$	−49.2374	−2.44965
$405$	−12.7855	−0.635319
$406$	0	0
$407$	−26.5812	−1.31758
$408$	−6.39275	−0.316488
$409$	−24.8983	−1.23114	−0.615571	−	0.788082i	$-0.711074\pi$
$409$	−24.8983	−1.23114	−0.615571	+	0.788082i	$0.711074\pi$
$410$	1.69442	0.0836812
$411$	13.4103	0.661480
$412$	−49.0842	−2.41820
$413$	0	0
$414$	11.4195	0.561237
$415$	−35.3247	−1.73402
$416$	0	0
$417$	−6.83725	−0.334822
$418$	−72.4022	−3.54131
$419$	4.08971	0.199796	0.0998978	−	0.994998i	$-0.468148\pi$
$419$	4.08971	0.199796	0.0998978	+	0.994998i	$0.468148\pi$
$420$	0	0
$421$	27.7416	1.35204	0.676021	−	0.736882i	$-0.263703\pi$
$421$	27.7416	1.35204	0.676021	+	0.736882i	$0.263703\pi$
$422$	43.9685	2.14035
$423$	−20.3894	−0.991368
$424$	−28.0619	−1.36281
$425$	−4.45541	−0.216119
$426$	11.3200	0.548453
$427$	0	0
$428$	−8.23894	−0.398244
$429$	0	0
$430$	−20.4840	−0.987828
$431$	−23.7707	−1.14499	−0.572496	−	0.819907i	$-0.694025\pi$
$431$	−23.7707	−1.14499	−0.572496	+	0.819907i	$0.694025\pi$
$432$	7.61985	0.366610
$433$	−28.1511	−1.35285	−0.676427	−	0.736510i	$-0.736473\pi$
$433$	−28.1511	−1.35285	−0.676427	+	0.736510i	$0.736473\pi$
$434$	0	0
$435$	13.2543	0.635495
$436$	52.9656	2.53659
$437$	13.0415	0.623860
$438$	−0.575776	−0.0275117
$439$	22.4172	1.06991	0.534956	−	0.844880i	$-0.320328\pi$
$439$	22.4172	1.06991	0.534956	+	0.844880i	$0.320328\pi$
$440$	45.7721	2.18210
$441$	0	0
$442$	0	0
$443$	32.6314	1.55037	0.775183	−	0.631737i	$-0.217658\pi$
$443$	32.6314	1.55037	0.775183	+	0.631737i	$0.217658\pi$
$444$	15.0955	0.716399
$445$	15.7675	0.747449
$446$	20.9027	0.989772
$447$	−4.42039	−0.209077
$448$	0	0
$449$	6.70185	0.316280	0.158140	−	0.987417i	$-0.449450\pi$
$449$	6.70185	0.316280	0.158140	+	0.987417i	$0.449450\pi$
$450$	11.3155	0.533420
$451$	1.21367	0.0571495
$452$	−9.13052	−0.429464
$453$	2.21091	0.103878
$454$	−20.1882	−0.947478
$455$	0	0
$456$	18.5420	0.868310
$457$	33.1389	1.55017	0.775085	−	0.631857i	$-0.217707\pi$
$457$	33.1389	1.55017	0.775085	+	0.631857i	$0.217707\pi$
$458$	54.8438	2.56268
$459$	−9.03500	−0.421718
$460$	−18.2829	−0.852443
$461$	−21.1892	−0.986877	−0.493439	−	0.869781i	$-0.664260\pi$
$461$	−21.1892	−0.986877	−0.493439	+	0.869781i	$0.664260\pi$
$462$	0	0
$463$	−8.78862	−0.408442	−0.204221	−	0.978925i	$-0.565466\pi$
$463$	−8.78862	−0.408442	−0.204221	+	0.978925i	$0.565466\pi$
$464$	14.3779	0.667478
$465$	13.8221	0.640983
$466$	−34.7183	−1.60829
$467$	−31.7094	−1.46733	−0.733667	−	0.679509i	$-0.762193\pi$
$467$	−31.7094	−1.46733	−0.733667	+	0.679509i	$0.762193\pi$
$468$	0	0
$469$	0	0
$470$	50.5669	2.33248
$471$	0.280323	0.0129166
$472$	−35.2922	−1.62445
$473$	−14.6722	−0.674630
$474$	−20.7219	−0.951788
$475$	12.9228	0.592939
$476$	0	0
$477$	−18.0860	−0.828100
$478$	28.3207	1.29536
$479$	−37.3844	−1.70814	−0.854068	−	0.520162i	$-0.825872\pi$
$479$	−37.3844	−1.70814	−0.854068	+	0.520162i	$0.825872\pi$
$480$	5.65419	0.258077
$481$	0	0
$482$	−52.7544	−2.40290
$483$	0	0
$484$	32.6329	1.48331
$485$	−33.1554	−1.50551
$486$	35.4289	1.60709
$487$	−16.6299	−0.753573	−0.376787	−	0.926300i	$-0.622971\pi$
$487$	−16.6299	−0.753573	−0.376787	+	0.926300i	$0.622971\pi$
$488$	−30.7778	−1.39325
$489$	8.27872	0.374377
$490$	0	0
$491$	−13.4173	−0.605515	−0.302758	−	0.953068i	$-0.597907\pi$
$491$	−13.4173	−0.605515	−0.302758	+	0.953068i	$0.597907\pi$
$492$	−0.689244	−0.0310735
$493$	−17.0482	−0.767812
$494$	0	0
$495$	29.5002	1.32593
$496$	14.9938	0.673242
$497$	0	0
$498$	22.2585	0.997428
$499$	6.01428	0.269236	0.134618	−	0.990898i	$-0.457019\pi$
$499$	6.01428	0.269236	0.134618	+	0.990898i	$0.457019\pi$
$500$	29.7060	1.32849
$501$	0.305216	0.0136360
$502$	−27.1083	−1.20990
$503$	12.0169	0.535807	0.267904	−	0.963446i	$-0.413669\pi$
$503$	12.0169	0.535807	0.267904	+	0.963446i	$0.413669\pi$
$504$	0	0
$505$	35.4906	1.57931
$506$	−20.2857	−0.901807
$507$	0	0
$508$	5.02453	0.222928
$509$	35.8748	1.59012	0.795062	−	0.606528i	$-0.207438\pi$
$509$	35.8748	1.59012	0.795062	+	0.606528i	$0.207438\pi$
$510$	10.2182	0.452468
$511$	0	0
$512$	21.6153	0.955271
$513$	26.2058	1.15702
$514$	−26.7125	−1.17824
$515$	35.3802	1.55904
$516$	8.33238	0.366812
$517$	36.2199	1.59295
$518$	0	0
$519$	3.81330	0.167385
$520$	0	0
$521$	9.89671	0.433583	0.216791	−	0.976218i	$-0.430441\pi$
$521$	9.89671	0.433583	0.216791	+	0.976218i	$0.430441\pi$
$522$	43.2978	1.89509
$523$	2.36683	0.103494	0.0517472	−	0.998660i	$-0.483521\pi$
$523$	2.36683	0.103494	0.0517472	+	0.998660i	$0.483521\pi$
$524$	48.2972	2.10987
$525$	0	0
$526$	12.4504	0.542864
$527$	−17.7785	−0.774442
$528$	−6.17266	−0.268630
$529$	−19.3460	−0.841132
$530$	44.8542	1.94834
$531$	−22.7459	−0.987086
$532$	0	0
$533$	0	0
$534$	−9.93525	−0.429940
$535$	5.93868	0.256751
$536$	27.3287	1.18042
$537$	−0.377432	−0.0162874
$538$	16.1733	0.697279
$539$	0	0
$540$	−36.7379	−1.58095
$541$	12.5733	0.540569	0.270284	−	0.962781i	$-0.412882\pi$
$541$	12.5733	0.540569	0.270284	+	0.962781i	$0.412882\pi$
$542$	−9.60482	−0.412562
$543$	−7.63847	−0.327798
$544$	−7.27264	−0.311812
$545$	−38.1779	−1.63536
$546$	0	0
$547$	−1.03013	−0.0440453	−0.0220226	−	0.999757i	$-0.507011\pi$
$547$	−1.03013	−0.0440453	−0.0220226	+	0.999757i	$0.507011\pi$
$548$	−70.1366	−2.99609
$549$	−19.8364	−0.846595
$550$	−20.1010	−0.857110
$551$	49.4479	2.10655
$552$	5.19510	0.221118
$553$	0	0
$554$	7.56764	0.321518
$555$	−10.8809	−0.461868
$556$	35.7593	1.51653
$557$	−18.6475	−0.790122	−0.395061	−	0.918655i	$-0.629276\pi$
$557$	−18.6475	−0.790122	−0.395061	+	0.918655i	$0.629276\pi$
$558$	45.1526	1.91146
$559$	0	0
$560$	0	0
$561$	7.31904	0.309010
$562$	32.8376	1.38517
$563$	−27.7721	−1.17045	−0.585226	−	0.810870i	$-0.698994\pi$
$563$	−27.7721	−1.17045	−0.585226	+	0.810870i	$0.698994\pi$
$564$	−20.5693	−0.866125
$565$	6.58134	0.276879
$566$	11.1968	0.470636
$567$	0	0
$568$	−26.6983	−1.12024
$569$	−7.56843	−0.317285	−0.158642	−	0.987336i	$-0.550712\pi$
$569$	−7.56843	−0.317285	−0.158642	+	0.987336i	$0.550712\pi$
$570$	−29.6376	−1.24138
$571$	2.60591	0.109054	0.0545270	−	0.998512i	$-0.482635\pi$
$571$	2.60591	0.109054	0.0545270	+	0.998512i	$0.482635\pi$
$572$	0	0
$573$	−3.41466	−0.142650
$574$	0	0
$575$	3.62071	0.150994
$576$	28.4486	1.18536
$577$	−29.4257	−1.22501	−0.612505	−	0.790467i	$-0.709838\pi$
$577$	−29.4257	−1.22501	−0.612505	+	0.790467i	$0.709838\pi$
$578$	27.2394	1.13301
$579$	12.4390	0.516949
$580$	−69.3208	−2.87839
$581$	0	0
$582$	20.8916	0.865985
$583$	32.1280	1.33061
$584$	1.35798	0.0561936
$585$	0	0
$586$	−32.1152	−1.32667
$587$	−15.1833	−0.626682	−0.313341	−	0.949641i	$-0.601448\pi$
$587$	−15.1833	−0.626682	−0.313341	+	0.949641i	$0.601448\pi$
$588$	0	0
$589$	51.5660	2.12474
$590$	56.4110	2.32241
$591$	−1.69587	−0.0697586
$592$	−11.8033	−0.485113
$593$	6.15116	0.252598	0.126299	−	0.991992i	$-0.459690\pi$
$593$	6.15116	0.252598	0.126299	+	0.991992i	$0.459690\pi$
$594$	−40.7623	−1.67250
$595$	0	0
$596$	23.1189	0.946987
$597$	3.58064	0.146546
$598$	0	0
$599$	5.99925	0.245123	0.122561	−	0.992461i	$-0.460889\pi$
$599$	5.99925	0.245123	0.122561	+	0.992461i	$0.460889\pi$
$600$	5.14782	0.210159
$601$	−9.62476	−0.392602	−0.196301	−	0.980544i	$-0.562893\pi$
$601$	−9.62476	−0.392602	−0.196301	+	0.980544i	$0.562893\pi$
$602$	0	0
$603$	17.6134	0.717273
$604$	−11.5632	−0.470500
$605$	−23.5220	−0.956304
$606$	−22.3631	−0.908437
$607$	8.92852	0.362397	0.181199	−	0.983447i	$-0.442002\pi$
$607$	8.92852	0.362397	0.181199	+	0.983447i	$0.442002\pi$
$608$	21.0941	0.855480
$609$	0	0
$610$	49.1953	1.99186
$611$	0	0
$612$	21.5486	0.871050
$613$	−7.73833	−0.312548	−0.156274	−	0.987714i	$-0.549948\pi$
$613$	−7.73833	−0.312548	−0.156274	+	0.987714i	$0.549948\pi$
$614$	−0.373766	−0.0150840
$615$	0.496811	0.0200334
$616$	0	0
$617$	−5.97203	−0.240425	−0.120212	−	0.992748i	$-0.538358\pi$
$617$	−5.97203	−0.240425	−0.120212	+	0.992748i	$0.538358\pi$
$618$	−22.2935	−0.896774
$619$	−21.4983	−0.864091	−0.432046	−	0.901852i	$-0.642208\pi$
$619$	−21.4983	−0.864091	−0.432046	+	0.901852i	$0.642208\pi$
$620$	−72.2903	−2.90325
$621$	7.34235	0.294638
$622$	−4.78833	−0.191994
$623$	0	0
$624$	0	0
$625$	−30.8829	−1.23532
$626$	−37.7860	−1.51023
$627$	−21.2287	−0.847793
$628$	−1.46611	−0.0585042
$629$	13.9954	0.558034
$630$	0	0
$631$	−9.61734	−0.382860	−0.191430	−	0.981506i	$-0.561313\pi$
$631$	−9.61734	−0.382860	−0.191430	+	0.981506i	$0.561313\pi$
$632$	48.8730	1.94406
$633$	12.8918	0.512402
$634$	−52.9374	−2.10241
$635$	−3.62171	−0.143723
$636$	−18.2455	−0.723483
$637$	0	0
$638$	−76.9146	−3.04508
$639$	−17.2071	−0.680703
$640$	−54.3179	−2.14710
$641$	47.7161	1.88467	0.942335	−	0.334670i	$-0.108625\pi$
$641$	47.7161	1.88467	0.942335	+	0.334670i	$0.108625\pi$
$642$	−3.74203	−0.147686
$643$	8.58409	0.338524	0.169262	−	0.985571i	$-0.445862\pi$
$643$	8.58409	0.338524	0.169262	+	0.985571i	$0.445862\pi$
$644$	0	0
$645$	−6.00603	−0.236487
$646$	38.1210	1.49985
$647$	3.29974	0.129726	0.0648631	−	0.997894i	$-0.479339\pi$
$647$	3.29974	0.129726	0.0648631	+	0.997894i	$0.479339\pi$
$648$	−19.0010	−0.746429
$649$	40.4059	1.58607
$650$	0	0
$651$	0	0
$652$	−43.2983	−1.69569
$653$	−17.2337	−0.674406	−0.337203	−	0.941432i	$-0.609481\pi$
$653$	−17.2337	−0.674406	−0.337203	+	0.941432i	$0.609481\pi$
$654$	24.0564	0.940678
$655$	−34.8129	−1.36025
$656$	0.538928	0.0210416
$657$	0.875221	0.0341456
$658$	0	0
$659$	−2.82156	−0.109912	−0.0549561	−	0.998489i	$-0.517502\pi$
$659$	−2.82156	−0.109912	−0.0549561	+	0.998489i	$0.517502\pi$
$660$	29.7605	1.15842
$661$	32.2767	1.25542	0.627709	−	0.778448i	$-0.283993\pi$
$661$	32.2767	1.25542	0.627709	+	0.778448i	$0.283993\pi$
$662$	12.8842	0.500757
$663$	0	0
$664$	−52.4972	−2.03729
$665$	0	0
$666$	−35.5447	−1.37733
$667$	13.8543	0.536441
$668$	−1.59630	−0.0617626
$669$	6.12878	0.236952
$670$	−43.6823	−1.68759
$671$	35.2374	1.36033
$672$	0	0
$673$	−39.9538	−1.54011	−0.770053	−	0.637980i	$-0.779770\pi$
$673$	−39.9538	−1.54011	−0.770053	+	0.637980i	$0.779770\pi$
$674$	−35.9250	−1.38378
$675$	7.27552	0.280035
$676$	0	0
$677$	−34.2338	−1.31571	−0.657855	−	0.753144i	$-0.728536\pi$
$677$	−34.2338	−1.31571	−0.657855	+	0.753144i	$0.728536\pi$
$678$	−4.14698	−0.159264
$679$	0	0
$680$	−24.0998	−0.924184
$681$	−5.91928	−0.226827
$682$	−80.2093	−3.07137
$683$	47.7479	1.82702	0.913512	−	0.406812i	$-0.133359\pi$
$683$	47.7479	1.82702	0.913512	+	0.406812i	$0.133359\pi$
$684$	−62.5012	−2.38979
$685$	50.5549	1.93160
$686$	0	0
$687$	16.0805	0.613509
$688$	−6.51519	−0.248389
$689$	0	0
$690$	−8.30386	−0.316122
$691$	37.8096	1.43835	0.719173	−	0.694831i	$-0.244521\pi$
$691$	37.8096	1.43835	0.719173	+	0.694831i	$0.244521\pi$
$692$	−19.9438	−0.758149
$693$	0	0
$694$	−46.4008	−1.76135
$695$	−25.7755	−0.977720
$696$	19.6976	0.746636
$697$	−0.639018	−0.0242045
$698$	45.6098	1.72636
$699$	−10.1796	−0.385027
$700$	0	0
$701$	8.43524	0.318595	0.159297	−	0.987231i	$-0.449077\pi$
$701$	8.43524	0.318595	0.159297	+	0.987231i	$0.449077\pi$
$702$	0	0
$703$	−40.5934	−1.53101
$704$	−50.5363	−1.90466
$705$	14.8265	0.558398
$706$	2.06706	0.0777947
$707$	0	0
$708$	−22.9465	−0.862384
$709$	19.1245	0.718237	0.359119	−	0.933292i	$-0.383077\pi$
$709$	19.1245	0.718237	0.359119	+	0.933292i	$0.383077\pi$
$710$	42.6747	1.60155
$711$	31.4987	1.18129
$712$	23.4325	0.878170
$713$	14.4478	0.541073
$714$	0	0
$715$	0	0
$716$	1.97399	0.0737716
$717$	8.30378	0.310110
$718$	6.79078	0.253430
$719$	−4.57280	−0.170537	−0.0852684	−	0.996358i	$-0.527175\pi$
$719$	−4.57280	−0.170537	−0.0852684	+	0.996358i	$0.527175\pi$
$720$	13.0995	0.488190
$721$	0	0
$722$	−65.4359	−2.43527
$723$	−15.4679	−0.575257
$724$	39.9497	1.48472
$725$	13.7282	0.509853
$726$	14.8215	0.550076
$727$	30.5472	1.13293	0.566466	−	0.824085i	$-0.308310\pi$
$727$	30.5472	1.13293	0.566466	+	0.824085i	$0.308310\pi$
$728$	0	0
$729$	−4.22038	−0.156310
$730$	−2.17060	−0.0803374
$731$	7.72518	0.285726
$732$	−20.0114	−0.739641
$733$	41.2339	1.52301	0.761504	−	0.648160i	$-0.224461\pi$
$733$	41.2339	1.52301	0.761504	+	0.648160i	$0.224461\pi$
$734$	38.4777	1.42024
$735$	0	0
$736$	5.91015	0.217851
$737$	−31.2886	−1.15253
$738$	1.62293	0.0597411
$739$	−42.4268	−1.56070	−0.780348	−	0.625346i	$-0.784958\pi$
$739$	−42.4268	−1.56070	−0.780348	+	0.625346i	$0.784958\pi$
$740$	56.9078	2.09197
$741$	0	0
$742$	0	0
$743$	49.0471	1.79936	0.899681	−	0.436548i	$-0.143799\pi$
$743$	49.0471	1.79936	0.899681	+	0.436548i	$0.143799\pi$
$744$	20.5414	0.753084
$745$	−16.6642	−0.610531
$746$	31.7385	1.16203
$747$	−33.8345	−1.23794
$748$	−38.2791	−1.39962
$749$	0	0
$750$	13.4921	0.492662
$751$	−42.3269	−1.54453	−0.772266	−	0.635300i	$-0.780877\pi$
$751$	−42.3269	−1.54453	−0.772266	+	0.635300i	$0.780877\pi$
$752$	16.0834	0.586501
$753$	−7.94829	−0.289652
$754$	0	0
$755$	8.33482	0.303335
$756$	0	0
$757$	47.5394	1.72785	0.863925	−	0.503620i	$-0.167999\pi$
$757$	47.5394	1.72785	0.863925	+	0.503620i	$0.167999\pi$
$758$	−2.46016	−0.0893571
$759$	−5.94786	−0.215894
$760$	69.9009	2.53557
$761$	37.2674	1.35094	0.675472	−	0.737386i	$-0.263940\pi$
$761$	37.2674	1.35094	0.675472	+	0.737386i	$0.263940\pi$
$762$	2.28208	0.0826712
$763$	0	0
$764$	17.8589	0.646113
$765$	−15.5323	−0.561573
$766$	82.8959	2.99515
$767$	0	0
$768$	18.4689	0.666441
$769$	28.9830	1.04516	0.522578	−	0.852592i	$-0.324970\pi$
$769$	28.9830	1.04516	0.522578	+	0.852592i	$0.324970\pi$
$770$	0	0
$771$	−7.83225	−0.282071
$772$	−65.0569	−2.34145
$773$	39.0260	1.40367	0.701833	−	0.712341i	$-0.252365\pi$
$773$	39.0260	1.40367	0.701833	+	0.712341i	$0.252365\pi$
$774$	−19.6199	−0.705223
$775$	14.3163	0.514256
$776$	−49.2733	−1.76881
$777$	0	0
$778$	−37.5625	−1.34668
$779$	1.85346	0.0664070
$780$	0	0
$781$	30.5669	1.09377
$782$	10.6807	0.381943
$783$	27.8391	0.994887
$784$	0	0
$785$	1.05678	0.0377181
$786$	21.9360	0.782432
$787$	47.3720	1.68863	0.844315	−	0.535848i	$-0.180008\pi$
$787$	47.3720	1.68863	0.844315	+	0.535848i	$0.180008\pi$
$788$	8.86949	0.315963
$789$	3.65053	0.129962
$790$	−78.1186	−2.77934
$791$	0	0
$792$	43.8411	1.55783
$793$	0	0
$794$	−83.4891	−2.96292
$795$	13.1515	0.466436
$796$	−18.7270	−0.663760
$797$	−45.2740	−1.60369	−0.801844	−	0.597533i	$-0.796148\pi$
$797$	−45.2740	−1.60369	−0.801844	+	0.597533i	$0.796148\pi$
$798$	0	0
$799$	−19.0704	−0.674662
$800$	5.85636	0.207054
$801$	15.1023	0.533613
$802$	6.19766	0.218847
$803$	−1.55475	−0.0548659
$804$	17.7688	0.626658
$805$	0	0
$806$	0	0
$807$	4.74209	0.166929
$808$	52.7437	1.85552
$809$	−1.71154	−0.0601746	−0.0300873	−	0.999547i	$-0.509579\pi$
$809$	−1.71154	−0.0601746	−0.0300873	+	0.999547i	$0.509579\pi$
$810$	30.3712	1.06714
$811$	33.5397	1.17774	0.588868	−	0.808229i	$-0.299574\pi$
$811$	33.5397	1.17774	0.588868	+	0.808229i	$0.299574\pi$
$812$	0	0
$813$	−2.81618	−0.0987678
$814$	63.1418	2.21312
$815$	31.2096	1.09323
$816$	3.25001	0.113773
$817$	−22.4067	−0.783912
$818$	59.1443	2.06793
$819$	0	0
$820$	−2.59836	−0.0907385
$821$	−10.6944	−0.373237	−0.186619	−	0.982432i	$-0.559753\pi$
$821$	−10.6944	−0.373237	−0.186619	+	0.982432i	$0.559753\pi$
$822$	−31.8552	−1.11108
$823$	24.3322	0.848166	0.424083	−	0.905623i	$-0.360597\pi$
$823$	24.3322	0.848166	0.424083	+	0.905623i	$0.360597\pi$
$824$	52.5796	1.83170
$825$	−5.89372	−0.205193
$826$	0	0
$827$	38.9990	1.35613	0.678065	−	0.735002i	$-0.262819\pi$
$827$	38.9990	1.35613	0.678065	+	0.735002i	$0.262819\pi$
$828$	−17.5116	−0.608569
$829$	42.2385	1.46700	0.733502	−	0.679688i	$-0.237885\pi$
$829$	42.2385	1.46700	0.733502	+	0.679688i	$0.237885\pi$
$830$	83.9115	2.91261
$831$	2.21887	0.0769718
$832$	0	0
$833$	0	0
$834$	16.2414	0.562395
$835$	1.15062	0.0398189
$836$	111.028	3.83997
$837$	29.0316	1.00348
$838$	−9.71484	−0.335594
$839$	38.5208	1.32989	0.664943	−	0.746894i	$-0.268456\pi$
$839$	38.5208	1.32989	0.664943	+	0.746894i	$0.268456\pi$
$840$	0	0
$841$	23.5296	0.811367
$842$	−65.8983	−2.27100
$843$	9.62815	0.331611
$844$	−67.4248	−2.32086
$845$	0	0
$846$	48.4337	1.66519
$847$	0	0
$848$	14.2664	0.489910
$849$	3.28295	0.112671
$850$	10.5835	0.363012
$851$	−11.3735	−0.389877
$852$	−17.3589	−0.594708
$853$	−16.6794	−0.571090	−0.285545	−	0.958365i	$-0.592175\pi$
$853$	−16.6794	−0.571090	−0.285545	+	0.958365i	$0.592175\pi$
$854$	0	0
$855$	45.0512	1.54072
$856$	8.82565	0.301655
$857$	13.7073	0.468232	0.234116	−	0.972209i	$-0.424780\pi$
$857$	13.7073	0.468232	0.234116	+	0.972209i	$0.424780\pi$
$858$	0	0
$859$	52.5804	1.79402	0.897011	−	0.442009i	$-0.145734\pi$
$859$	52.5804	1.79402	0.897011	+	0.442009i	$0.145734\pi$
$860$	31.4119	1.07114
$861$	0	0
$862$	56.4656	1.92323
$863$	−35.2625	−1.20035	−0.600175	−	0.799868i	$-0.704903\pi$
$863$	−35.2625	−1.20035	−0.600175	+	0.799868i	$0.704903\pi$
$864$	11.8760	0.404028
$865$	14.3756	0.488785
$866$	66.8710	2.27237
$867$	7.98672	0.271243
$868$	0	0
$869$	−55.9546	−1.89813
$870$	−31.4847	−1.06743
$871$	0	0
$872$	−56.7374	−1.92137
$873$	−31.7567	−1.07480
$874$	−30.9793	−1.04789
$875$	0	0
$876$	0.882943	0.0298319
$877$	−16.1911	−0.546735	−0.273368	−	0.961910i	$-0.588138\pi$
$877$	−16.1911	−0.546735	−0.273368	+	0.961910i	$0.588138\pi$
$878$	−53.2505	−1.79712
$879$	−9.41635	−0.317606
$880$	−23.2701	−0.784434
$881$	−14.9633	−0.504126	−0.252063	−	0.967711i	$-0.581109\pi$
$881$	−14.9633	−0.504126	−0.252063	+	0.967711i	$0.581109\pi$
$882$	0	0
$883$	17.2375	0.580087	0.290044	−	0.957013i	$-0.406330\pi$
$883$	17.2375	0.580087	0.290044	+	0.957013i	$0.406330\pi$
$884$	0	0
$885$	16.5400	0.555986
$886$	−77.5138	−2.60413
$887$	3.54129	0.118905	0.0594524	−	0.998231i	$-0.481065\pi$
$887$	3.54129	0.118905	0.0594524	+	0.998231i	$0.481065\pi$
$888$	−16.1704	−0.542644
$889$	0	0
$890$	−37.4545	−1.25548
$891$	21.7542	0.728793
$892$	−32.0539	−1.07324
$893$	55.3133	1.85099
$894$	10.5003	0.351184
$895$	−1.42287	−0.0475612
$896$	0	0
$897$	0	0
$898$	−15.9198	−0.531251
$899$	54.7798	1.82701
$900$	−17.3522	−0.578406
$901$	−16.9160	−0.563552
$902$	−2.88299	−0.0959932
$903$	0	0
$904$	9.78073	0.325302
$905$	−28.7960	−0.957211
$906$	−5.25187	−0.174482
$907$	−5.56457	−0.184769	−0.0923843	−	0.995723i	$-0.529449\pi$
$907$	−5.56457	−0.184769	−0.0923843	+	0.995723i	$0.529449\pi$
$908$	30.9582	1.02738
$909$	33.9934	1.12749
$910$	0	0
$911$	−19.4554	−0.644587	−0.322293	−	0.946640i	$-0.604454\pi$
$911$	−19.4554	−0.644587	−0.322293	+	0.946640i	$0.604454\pi$
$912$	−9.42658	−0.312145
$913$	60.1039	1.98915
$914$	−78.7191	−2.60380
$915$	14.4243	0.476853
$916$	−84.1020	−2.77881
$917$	0	0
$918$	21.4621	0.708353
$919$	30.1088	0.993198	0.496599	−	0.867980i	$-0.334582\pi$
$919$	30.1088	0.993198	0.496599	+	0.867980i	$0.334582\pi$
$920$	19.5848	0.645692
$921$	−0.109590	−0.00361112
$922$	50.3334	1.65764
$923$	0	0
$924$	0	0
$925$	−11.2699	−0.370554
$926$	20.8768	0.686054
$927$	33.8876	1.11302
$928$	22.4088	0.735605
$929$	58.2420	1.91086	0.955428	−	0.295224i	$-0.0953941\pi$
$929$	58.2420	1.91086	0.955428	+	0.295224i	$0.0953941\pi$
$930$	−32.8334	−1.07665
$931$	0	0
$932$	53.2399	1.74393
$933$	−1.40396	−0.0459636
$934$	75.3235	2.46466
$935$	27.5918	0.902348
$936$	0	0
$937$	−3.88483	−0.126912	−0.0634560	−	0.997985i	$-0.520212\pi$
$937$	−3.88483	−0.126912	−0.0634560	+	0.997985i	$0.520212\pi$
$938$	0	0
$939$	−11.0791	−0.361551
$940$	−77.5435	−2.52919
$941$	34.2935	1.11794	0.558968	−	0.829189i	$-0.311197\pi$
$941$	34.2935	1.11794	0.558968	+	0.829189i	$0.311197\pi$
$942$	−0.665890	−0.0216959
$943$	0.519301	0.0169108
$944$	17.9422	0.583968
$945$	0	0
$946$	34.8529	1.13317
$947$	27.1474	0.882171	0.441086	−	0.897465i	$-0.354594\pi$
$947$	27.1474	0.882171	0.441086	+	0.897465i	$0.354594\pi$
$948$	31.7766	1.03206
$949$	0	0
$950$	−30.6973	−0.995952
$951$	−15.5215	−0.503320
$952$	0	0
$953$	−2.32970	−0.0754663	−0.0377332	−	0.999288i	$-0.512014\pi$
$953$	−2.32970	−0.0754663	−0.0377332	+	0.999288i	$0.512014\pi$
$954$	42.9620	1.39095
$955$	−12.8728	−0.416554
$956$	−43.4293	−1.40460
$957$	−22.5518	−0.728995
$958$	88.8041	2.86913
$959$	0	0
$960$	−20.6868	−0.667665
$961$	26.1264	0.842786
$962$	0	0
$963$	5.68815	0.183298
$964$	80.8980	2.60555
$965$	46.8934	1.50955
$966$	0	0
$967$	38.2143	1.22889	0.614445	−	0.788960i	$-0.289380\pi$
$967$	38.2143	1.22889	0.614445	+	0.788960i	$0.289380\pi$
$968$	−34.9567	−1.12355
$969$	11.1773	0.359066
$970$	78.7584	2.52878
$971$	50.5565	1.62243	0.811217	−	0.584745i	$-0.198806\pi$
$971$	50.5565	1.62243	0.811217	+	0.584745i	$0.198806\pi$
$972$	−54.3297	−1.74262
$973$	0	0
$974$	39.5033	1.26577
$975$	0	0
$976$	15.6471	0.500852
$977$	−26.2341	−0.839304	−0.419652	−	0.907685i	$-0.637848\pi$
$977$	−26.2341	−0.839304	−0.419652	+	0.907685i	$0.637848\pi$
$978$	−19.6656	−0.628835
$979$	−26.8278	−0.857420
$980$	0	0
$981$	−36.5674	−1.16751
$982$	31.8720	1.01708
$983$	−36.9281	−1.17782	−0.588912	−	0.808197i	$-0.700444\pi$
$983$	−36.9281	−1.17782	−0.588912	+	0.808197i	$0.700444\pi$
$984$	0.738327	0.0235370
$985$	−6.39318	−0.203704
$986$	40.4968	1.28968
$987$	0	0
$988$	0	0
$989$	−6.27791	−0.199626
$990$	−70.0757	−2.22715
$991$	−24.5679	−0.780425	−0.390213	−	0.920725i	$-0.627598\pi$
$991$	−24.5679	−0.780425	−0.390213	+	0.920725i	$0.627598\pi$
$992$	23.3687	0.741957
$993$	3.77770	0.119882
$994$	0	0
$995$	13.4985	0.427932
$996$	−34.1330	−1.08155
$997$	24.2418	0.767747	0.383873	−	0.923386i	$-0.374590\pi$
$997$	24.2418	0.767747	0.383873	+	0.923386i	$0.374590\pi$
$998$	−14.2865	−0.452232
$999$	−22.8540	−0.723069

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.cu.1.3		24
7.2	even	3		1183.2.e.l.508.22	yes	48
7.4	even	3		1183.2.e.l.170.22	yes	48
7.6	odd	2		8281.2.a.ct.1.3		24
13.12	even	2		8281.2.a.cv.1.22		24
91.25	even	6		1183.2.e.k.170.3	✓	48
91.51	even	6		1183.2.e.k.508.3	yes	48
91.90	odd	2		8281.2.a.cw.1.22		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1183.2.e.k.170.3	✓	48	91.25	even	6
1183.2.e.k.508.3	yes	48	91.51	even	6
1183.2.e.l.170.22	yes	48	7.4	even	3
1183.2.e.l.508.22	yes	48	7.2	even	3
8281.2.a.ct.1.3		24	7.6	odd	2
8281.2.a.cu.1.3		24	1.1	even	1	trivial
8281.2.a.cv.1.22		24	13.12	even	2
8281.2.a.cw.1.22		24	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.37543 q^{2} -0.696489 q^{3} +3.64268 q^{4} -2.62567 q^{5} +1.65446 q^{6} -3.90209 q^{8} -2.51490 q^{9} +6.23710 q^{10} +4.46749 q^{11} -2.53709 q^{12} +1.82875 q^{15} +1.98378 q^{16} -2.35221 q^{17} +5.97398 q^{18} +6.82253 q^{19} -9.56448 q^{20} -10.6122 q^{22} +1.91154 q^{23} +2.71776 q^{24} +1.89414 q^{25} +3.84107 q^{27} +7.24773 q^{29} -4.34408 q^{30} +7.55820 q^{31} +3.09183 q^{32} -3.11156 q^{33} +5.58752 q^{34} -9.16100 q^{36} -5.94991 q^{37} -16.2065 q^{38} +10.2456 q^{40} +0.271667 q^{41} -3.28422 q^{43} +16.2737 q^{44} +6.60330 q^{45} -4.54073 q^{46} +8.10744 q^{47} -1.38168 q^{48} -4.49940 q^{50} +1.63829 q^{51} +7.19152 q^{53} -9.12421 q^{54} -11.7302 q^{55} -4.75182 q^{57} -17.2165 q^{58} +9.04443 q^{59} +6.66156 q^{60} +7.88752 q^{61} -17.9540 q^{62} -11.3120 q^{64} +7.39130 q^{66} -7.00361 q^{67} -8.56836 q^{68} -1.33136 q^{69} +6.84206 q^{71} +9.81337 q^{72} -0.348014 q^{73} +14.1336 q^{74} -1.31925 q^{75} +24.8523 q^{76} -12.5248 q^{79} -5.20875 q^{80} +4.86944 q^{81} -0.645327 q^{82} +13.4536 q^{83} +6.17612 q^{85} +7.80146 q^{86} -5.04797 q^{87} -17.4325 q^{88} -6.00512 q^{89} -15.6857 q^{90} +6.96312 q^{92} -5.26420 q^{93} -19.2587 q^{94} -17.9137 q^{95} -2.15343 q^{96} +12.6274 q^{97} -11.2353 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - q^{2} + 23 q^{4} + 13 q^{5} + 14 q^{6} + 26 q^{9} + 5 q^{10} - q^{11} + 5 q^{12} + 5 q^{15} + 17 q^{16} - 5 q^{17} + 24 q^{19} + 34 q^{20} - 14 q^{22} + 11 q^{23} + 32 q^{24} + 33 q^{25} - 21 q^{27}+ \cdots - 39 q^{99}+O(q^{100}) \) 24 * q - q^2 + 23 * q^4 + 13 * q^5 + 14 * q^6 + 26 * q^9 + 5 * q^10 - q^11 + 5 * q^12 + 5 * q^15 + 17 * q^16 - 5 * q^17 + 24 * q^19 + 34 * q^20 - 14 * q^22 + 11 * q^23 + 32 * q^24 + 33 * q^25 - 21 * q^27 + 4 * q^29 - 22 * q^30 + 40 * q^31 - 6 * q^32 + 24 * q^33 + 36 * q^34 - 15 * q^36 - 4 * q^37 - 29 * q^38 - 4 * q^40 + 49 * q^41 + 13 * q^43 + 10 * q^44 + 58 * q^45 - 10 * q^46 + 62 * q^47 + 89 * q^48 - 23 * q^50 - 21 * q^51 - 18 * q^53 + 12 * q^54 - 14 * q^55 - 13 * q^57 + 56 * q^58 + 79 * q^59 + 22 * q^60 + 13 * q^61 + 12 * q^62 + 18 * q^64 - 38 * q^66 - 2 * q^67 - 12 * q^68 - 28 * q^69 - 19 * q^71 + 81 * q^72 + 17 * q^73 + 17 * q^74 + 24 * q^75 + 58 * q^76 - 9 * q^79 + 63 * q^80 + 16 * q^81 - 22 * q^82 + 81 * q^83 - 34 * q^85 + 22 * q^86 + 70 * q^87 - 33 * q^88 + 72 * q^89 + q^90 - 4 * q^92 + 19 * q^93 - 30 * q^94 + 13 * q^95 + 11 * q^96 + 45 * q^97 - 39 * q^99

Introduction

L-functions

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