LMFDB - Embedded newform 3381.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3381,2,Mod(1,3381)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3381, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3381.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3381 = 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3381.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:	$26.9974209234$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.167449$ of defining polynomial
Character	$\chi$	$=$	3381.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} -1.16745 q^{5} +0.167449 q^{6} +0.665102 q^{8} +1.00000 q^{9} +0.195488 q^{10} -1.80451 q^{11} +1.97196 q^{12} -6.97196 q^{13} +1.16745 q^{15} +3.83255 q^{16} -6.13941 q^{17} -0.167449 q^{18} -1.80451 q^{19} +2.30216 q^{20} +0.302164 q^{22} -1.00000 q^{23} -0.665102 q^{24} -3.63706 q^{25} +1.16745 q^{26} -1.00000 q^{27} +5.46961 q^{29} -0.195488 q^{30} +2.19549 q^{31} -1.97196 q^{32} +1.80451 q^{33} +1.02804 q^{34} -1.97196 q^{36} -1.46961 q^{37} +0.302164 q^{38} +6.97196 q^{39} -0.776472 q^{40} -9.74843 q^{41} -6.63706 q^{43} +3.55843 q^{44} -1.16745 q^{45} +0.167449 q^{46} +1.94392 q^{47} -3.83255 q^{48} +0.609023 q^{50} +6.13941 q^{51} +13.7484 q^{52} +1.16745 q^{53} +0.167449 q^{54} +2.10668 q^{55} +1.80451 q^{57} -0.915882 q^{58} -9.11137 q^{59} -2.30216 q^{60} +4.63706 q^{61} -0.367633 q^{62} -7.33490 q^{64} +8.13941 q^{65} -0.302164 q^{66} -15.4463 q^{67} +12.1067 q^{68} +1.00000 q^{69} -9.25078 q^{71} +0.665102 q^{72} +0.474308 q^{73} +0.246086 q^{74} +3.63706 q^{75} +3.55843 q^{76} -1.16745 q^{78} +7.46961 q^{79} -4.47431 q^{80} +1.00000 q^{81} +1.63237 q^{82} -8.13941 q^{83} +7.16745 q^{85} +1.11137 q^{86} -5.46961 q^{87} -1.20018 q^{88} -1.02804 q^{89} +0.195488 q^{90} +1.97196 q^{92} -2.19549 q^{93} -0.325508 q^{94} +2.10668 q^{95} +1.97196 q^{96} +8.47431 q^{97} -1.80451 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} + 3 q^{8} + 3 q^{9} + 12 q^{10} + 6 q^{11} - 6 q^{12} - 9 q^{13} + 3 q^{15} + 12 q^{16} - 6 q^{17} + 6 q^{19} - 3 q^{20} - 9 q^{22} - 3 q^{23} - 3 q^{24} + 3 q^{26} - 3 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^5 + 3 * q^8 + 3 * q^9 + 12 * q^10 + 6 * q^11 - 6 * q^12 - 9 * q^13 + 3 * q^15 + 12 * q^16 - 6 * q^17 + 6 * q^19 - 3 * q^20 - 9 * q^22 - 3 * q^23 - 3 * q^24 + 3 * q^26 - 3 * q^27 + 6 * q^29 - 12 * q^30 + 18 * q^31 + 6 * q^32 - 6 * q^33 + 15 * q^34 + 6 * q^36 + 6 * q^37 - 9 * q^38 + 9 * q^39 + 21 * q^40 + 6 * q^41 - 9 * q^43 + 33 * q^44 - 3 * q^45 - 18 * q^47 - 12 * q^48 - 21 * q^50 + 6 * q^51 + 6 * q^52 + 3 * q^53 - 15 * q^55 - 6 * q^57 + 33 * q^58 - 3 * q^59 + 3 * q^60 + 3 * q^61 - 9 * q^62 - 21 * q^64 + 12 * q^65 + 9 * q^66 - 21 * q^67 + 15 * q^68 + 3 * q^69 + 9 * q^71 + 3 * q^72 - 12 * q^73 - 33 * q^74 + 33 * q^76 - 3 * q^78 + 12 * q^79 + 3 * q^81 - 3 * q^82 - 12 * q^83 + 21 * q^85 - 21 * q^86 - 6 * q^87 - 12 * q^88 - 15 * q^89 + 12 * q^90 - 6 * q^92 - 18 * q^93 - 6 * q^94 - 15 * q^95 - 6 * q^96 + 12 * q^97 + 6 * 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 $

$2$	−0.167449	−0.118404	−0.0592022	−	0.998246i	$-0.518856\pi$
$2$	−0.167449	−0.118404	−0.0592022	+	0.998246i	$0.518856\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.97196	−0.985980
$5$	−1.16745	−0.522099	−0.261050	−	0.965325i	$-0.584069\pi$
$5$	−1.16745	−0.522099	−0.261050	+	0.965325i	$0.584069\pi$
$6$	0.167449	0.0683608
$7$	0	0
$7$	0	0
$8$	0.665102	0.235149
$9$	1.00000	0.333333
$10$	0.195488	0.0618189
$11$	−1.80451	−0.544081	−0.272040	−	0.962286i	$-0.587698\pi$
$11$	−1.80451	−0.544081	−0.272040	+	0.962286i	$0.587698\pi$
$12$	1.97196	0.569256
$13$	−6.97196	−1.93367	−0.966837	−	0.255394i	$-0.917795\pi$
$13$	−6.97196	−1.93367	−0.966837	+	0.255394i	$0.917795\pi$
$14$	0	0
$15$	1.16745	0.301434
$16$	3.83255	0.958138
$17$	−6.13941	−1.48903	−0.744513	−	0.667608i	$-0.767318\pi$
$17$	−6.13941	−1.48903	−0.744513	+	0.667608i	$0.767318\pi$
$18$	−0.167449	−0.0394682
$19$	−1.80451	−0.413983	−0.206992	−	0.978343i	$-0.566367\pi$
$19$	−1.80451	−0.413983	−0.206992	+	0.978343i	$0.566367\pi$
$20$	2.30216	0.514780
$21$	0	0
$22$	0.302164	0.0644216
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−0.665102	−0.135763
$25$	−3.63706	−0.727412
$26$	1.16745	0.228956
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.46961	1.01568	0.507841	−	0.861451i	$-0.330444\pi$
$29$	5.46961	1.01568	0.507841	+	0.861451i	$0.330444\pi$
$30$	−0.195488	−0.0356911
$31$	2.19549	0.394321	0.197161	−	0.980371i	$-0.436828\pi$
$31$	2.19549	0.394321	0.197161	+	0.980371i	$0.436828\pi$
$32$	−1.97196	−0.348597
$33$	1.80451	0.314125
$34$	1.02804	0.176307
$35$	0	0
$36$	−1.97196	−0.328660
$37$	−1.46961	−0.241603	−0.120801	−	0.992677i	$-0.538546\pi$
$37$	−1.46961	−0.241603	−0.120801	+	0.992677i	$0.538546\pi$
$38$	0.302164	0.0490175
$39$	6.97196	1.11641
$40$	−0.776472	−0.122771
$41$	−9.74843	−1.52245	−0.761225	−	0.648488i	$-0.775402\pi$
$41$	−9.74843	−1.52245	−0.761225	+	0.648488i	$0.775402\pi$
$42$	0	0
$43$	−6.63706	−1.01214	−0.506071	−	0.862492i	$-0.668903\pi$
$43$	−6.63706	−1.01214	−0.506071	+	0.862492i	$0.668903\pi$
$44$	3.55843	0.536453
$45$	−1.16745	−0.174033
$46$	0.167449	0.0246890
$47$	1.94392	0.283550	0.141775	−	0.989899i	$-0.454719\pi$
$47$	1.94392	0.283550	0.141775	+	0.989899i	$0.454719\pi$
$48$	−3.83255	−0.553181
$49$	0	0
$50$	0.609023	0.0861289
$51$	6.13941	0.859689
$52$	13.7484	1.90656
$53$	1.16745	0.160361	0.0801807	−	0.996780i	$-0.474450\pi$
$53$	1.16745	0.160361	0.0801807	+	0.996780i	$0.474450\pi$
$54$	0.167449	0.0227869
$55$	2.10668	0.284064
$56$	0	0
$57$	1.80451	0.239013
$58$	−0.915882	−0.120261
$59$	−9.11137	−1.18620	−0.593100	−	0.805129i	$-0.702096\pi$
$59$	−9.11137	−1.18620	−0.593100	+	0.805129i	$0.702096\pi$
$60$	−2.30216	−0.297208
$61$	4.63706	0.593715	0.296857	−	0.954922i	$-0.404061\pi$
$61$	4.63706	0.593715	0.296857	+	0.954922i	$0.404061\pi$
$62$	−0.367633	−0.0466894
$63$	0	0
$64$	−7.33490	−0.916862
$65$	8.13941	1.00957
$66$	−0.302164	−0.0371938
$67$	−15.4463	−1.88706	−0.943531	−	0.331284i	$-0.892518\pi$
$67$	−15.4463	−1.88706	−0.943531	+	0.331284i	$0.892518\pi$
$68$	12.1067	1.46815
$69$	1.00000	0.120386
$70$	0	0
$71$	−9.25078	−1.09787	−0.548933	−	0.835866i	$-0.684966\pi$
$71$	−9.25078	−1.09787	−0.548933	+	0.835866i	$0.684966\pi$
$72$	0.665102	0.0783830
$73$	0.474308	0.0555136	0.0277568	−	0.999615i	$-0.491164\pi$
$73$	0.474308	0.0555136	0.0277568	+	0.999615i	$0.491164\pi$
$74$	0.246086	0.0286069
$75$	3.63706	0.419972
$76$	3.55843	0.408179
$77$	0	0
$78$	−1.16745	−0.132188
$79$	7.46961	0.840397	0.420199	−	0.907432i	$-0.361960\pi$
$79$	7.46961	0.840397	0.420199	+	0.907432i	$0.361960\pi$
$80$	−4.47431	−0.500243
$81$	1.00000	0.111111
$82$	1.63237	0.180265
$83$	−8.13941	−0.893416	−0.446708	−	0.894680i	$-0.647404\pi$
$83$	−8.13941	−0.893416	−0.446708	+	0.894680i	$0.647404\pi$
$84$	0	0
$85$	7.16745	0.777419
$86$	1.11137	0.119842
$87$	−5.46961	−0.586404
$88$	−1.20018	−0.127940
$89$	−1.02804	−0.108972	−0.0544860	−	0.998515i	$-0.517352\pi$
$89$	−1.02804	−0.108972	−0.0544860	+	0.998515i	$0.517352\pi$
$90$	0.195488	0.0206063
$91$	0	0
$92$	1.97196	0.205591
$93$	−2.19549	−0.227662
$94$	−0.325508	−0.0335736
$95$	2.10668	0.216140
$96$	1.97196	0.201262
$97$	8.47431	0.860436	0.430218	−	0.902725i	$-0.358437\pi$
$97$	8.47431	0.860436	0.430218	+	0.902725i	$0.358437\pi$
$98$	0	0
$99$	−1.80451	−0.181360
$100$	7.17214	0.717214
$101$	18.7204	1.86275	0.931374	−	0.364063i	$-0.118611\pi$
$101$	18.7204	1.86275	0.931374	+	0.364063i	$0.118611\pi$
$102$	−1.02804	−0.101791
$103$	−4.66980	−0.460129	−0.230064	−	0.973175i	$-0.573894\pi$
$103$	−4.66980	−0.460129	−0.230064	+	0.973175i	$0.573894\pi$
$104$	−4.63706	−0.454701
$105$	0	0
$106$	−0.195488	−0.0189875
$107$	−1.36294	−0.131760	−0.0658801	−	0.997828i	$-0.520985\pi$
$107$	−1.36294	−0.131760	−0.0658801	+	0.997828i	$0.520985\pi$
$108$	1.97196	0.189752
$109$	−0.497652	−0.0476665	−0.0238332	−	0.999716i	$-0.507587\pi$
$109$	−0.497652	−0.0476665	−0.0238332	+	0.999716i	$0.507587\pi$
$110$	−0.352761	−0.0336345
$111$	1.46961	0.139490
$112$	0	0
$113$	−1.44627	−0.136054	−0.0680268	−	0.997683i	$-0.521670\pi$
$113$	−1.44627	−0.136054	−0.0680268	+	0.997683i	$0.521670\pi$
$114$	−0.302164	−0.0283003
$115$	1.16745	0.108865
$116$	−10.7859	−1.00144
$117$	−6.97196	−0.644558
$118$	1.52569	0.140451
$119$	0	0
$120$	0.776472	0.0708819
$121$	−7.74374	−0.703976
$122$	−0.776472	−0.0702985
$123$	9.74843	0.878987
$124$	−4.32942	−0.388793
$125$	10.0833	0.901881
$126$	0	0
$127$	7.91119	0.702004	0.351002	−	0.936375i	$-0.385841\pi$
$127$	7.91119	0.702004	0.351002	+	0.936375i	$0.385841\pi$
$128$	5.17214	0.457157
$129$	6.63706	0.584361
$130$	−1.36294	−0.119538
$131$	−1.80451	−0.157661	−0.0788305	−	0.996888i	$-0.525119\pi$
$131$	−1.80451	−0.157661	−0.0788305	+	0.996888i	$0.525119\pi$
$132$	−3.55843	−0.309721
$133$	0	0
$134$	2.58647	0.223437
$135$	1.16745	0.100478
$136$	−4.08333	−0.350143
$137$	−13.4696	−1.15079	−0.575393	−	0.817877i	$-0.695151\pi$
$137$	−13.4696	−1.15079	−0.575393	+	0.817877i	$0.695151\pi$
$138$	−0.167449	−0.0142542
$139$	15.1674	1.28649	0.643243	−	0.765662i	$-0.277588\pi$
$139$	15.1674	1.28649	0.643243	+	0.765662i	$0.277588\pi$
$140$	0	0
$141$	−1.94392	−0.163708
$142$	1.54904	0.129992
$143$	12.5810	1.05207
$144$	3.83255	0.319379
$145$	−6.38550	−0.530287
$146$	−0.0794225	−0.00657306
$147$	0	0
$148$	2.89802	0.238216
$149$	19.5529	1.60184	0.800920	−	0.598772i	$-0.204344\pi$
$149$	19.5529	1.60184	0.800920	+	0.598772i	$0.204344\pi$
$150$	−0.609023	−0.0497265
$151$	−13.2741	−1.08023	−0.540116	−	0.841590i	$-0.681620\pi$
$151$	−13.2741	−1.08023	−0.540116	+	0.841590i	$0.681620\pi$
$152$	−1.20018	−0.0973477
$153$	−6.13941	−0.496342
$154$	0	0
$155$	−2.56312	−0.205875
$156$	−13.7484	−1.10076
$157$	13.2835	1.06014	0.530070	−	0.847954i	$-0.322166\pi$
$157$	13.2835	1.06014	0.530070	+	0.847954i	$0.322166\pi$
$158$	−1.25078	−0.0995068
$159$	−1.16745	−0.0925847
$160$	2.30216	0.182002
$161$	0	0
$162$	−0.167449	−0.0131561
$163$	7.44627	0.583237	0.291618	−	0.956535i	$-0.405806\pi$
$163$	7.44627	0.583237	0.291618	+	0.956535i	$0.405806\pi$
$164$	19.2235	1.50111
$165$	−2.10668	−0.164004
$166$	1.36294	0.105784
$167$	−20.7437	−1.60520	−0.802599	−	0.596519i	$-0.796550\pi$
$167$	−20.7437	−1.60520	−0.802599	+	0.596519i	$0.796550\pi$
$168$	0	0
$169$	35.6082	2.73910
$170$	−1.20018	−0.0920499
$171$	−1.80451	−0.137994
$172$	13.0880	0.997953
$173$	17.7484	1.34939	0.674694	−	0.738097i	$-0.264276\pi$
$173$	17.7484	1.34939	0.674694	+	0.738097i	$0.264276\pi$
$174$	0.915882	0.0694329
$175$	0	0
$176$	−6.91588	−0.521304
$177$	9.11137	0.684853
$178$	0.172144	0.0129028
$179$	4.44157	0.331979	0.165989	−	0.986128i	$-0.446918\pi$
$179$	4.44157	0.331979	0.165989	+	0.986128i	$0.446918\pi$
$180$	2.30216	0.171593
$181$	11.6924	0.869086	0.434543	−	0.900651i	$-0.356910\pi$
$181$	11.6924	0.869086	0.434543	+	0.900651i	$0.356910\pi$
$182$	0	0
$183$	−4.63706	−0.342782
$184$	−0.665102	−0.0490319
$185$	1.71570	0.126141
$186$	0.367633	0.0269561
$187$	11.0786	0.810150
$188$	−3.83334	−0.279575
$189$	0	0
$190$	−0.352761	−0.0255920
$191$	21.2741	1.53934	0.769671	−	0.638441i	$-0.220420\pi$
$191$	21.2741	1.53934	0.769671	+	0.638441i	$0.220420\pi$
$192$	7.33490	0.529351
$193$	−3.27412	−0.235677	−0.117838	−	0.993033i	$-0.537596\pi$
$193$	−3.27412	−0.235677	−0.117838	+	0.993033i	$0.537596\pi$
$194$	−1.41902	−0.101879
$195$	−8.13941	−0.582875
$196$	0	0
$197$	0.358242	0.0255237	0.0127619	−	0.999919i	$-0.495938\pi$
$197$	0.358242	0.0255237	0.0127619	+	0.999919i	$0.495938\pi$
$198$	0.302164	0.0214739
$199$	23.3808	1.65742	0.828710	−	0.559678i	$-0.189075\pi$
$199$	23.3808	1.65742	0.828710	+	0.559678i	$0.189075\pi$
$200$	−2.41902	−0.171050
$201$	15.4463	1.08950
$202$	−3.13471	−0.220558
$203$	0	0
$204$	−12.1067	−0.847637
$205$	11.3808	0.794870
$206$	0.781954	0.0544813
$207$	−1.00000	−0.0695048
$208$	−26.7204	−1.85273
$209$	3.25626	0.225240
$210$	0	0
$211$	13.4135	0.923426	0.461713	−	0.887029i	$-0.347235\pi$
$211$	13.4135	0.923426	0.461713	+	0.887029i	$0.347235\pi$
$212$	−2.30216	−0.158113
$213$	9.25078	0.633853
$214$	0.228223	0.0156010
$215$	7.74843	0.528439
$216$	−0.665102	−0.0452544
$217$	0	0
$218$	0.0833315	0.00564392
$219$	−0.474308	−0.0320508
$220$	−4.15428	−0.280082
$221$	42.8037	2.87929
$222$	−0.246086	−0.0165162
$223$	−8.72039	−0.583961	−0.291980	−	0.956424i	$-0.594314\pi$
$223$	−8.72039	−0.583961	−0.291980	+	0.956424i	$0.594314\pi$
$224$	0	0
$225$	−3.63706	−0.242471
$226$	0.242177	0.0161093
$227$	−14.1067	−0.936293	−0.468146	−	0.883651i	$-0.655078\pi$
$227$	−14.1067	−0.936293	−0.468146	+	0.883651i	$0.655078\pi$
$228$	−3.55843	−0.235663
$229$	−5.05529	−0.334063	−0.167032	−	0.985952i	$-0.553418\pi$
$229$	−5.05529	−0.334063	−0.167032	+	0.985952i	$0.553418\pi$
$230$	−0.195488	−0.0128901
$231$	0	0
$232$	3.63785	0.238836
$233$	−27.7157	−1.81572	−0.907858	−	0.419278i	$-0.862283\pi$
$233$	−27.7157	−1.81572	−0.907858	+	0.419278i	$0.862283\pi$
$234$	1.16745	0.0763185
$235$	−2.26943	−0.148041
$236$	17.9673	1.16957
$237$	−7.46961	−0.485204
$238$	0	0
$239$	7.83725	0.506949	0.253475	−	0.967342i	$-0.418427\pi$
$239$	7.83725	0.506949	0.253475	+	0.967342i	$0.418427\pi$
$240$	4.47431	0.288815
$241$	−19.6924	−1.26850	−0.634248	−	0.773130i	$-0.718690\pi$
$241$	−19.6924	−1.26850	−0.634248	+	0.773130i	$0.718690\pi$
$242$	1.29668	0.0833539
$243$	−1.00000	−0.0641500
$244$	−9.14411	−0.585391
$245$	0	0
$246$	−1.63237	−0.104076
$247$	12.5810	0.800509
$248$	1.46022	0.0927242
$249$	8.13941	0.515814
$250$	−1.68845	−0.106787
$251$	30.2227	1.90764	0.953821	−	0.300375i	$-0.0971117\pi$
$251$	30.2227	1.90764	0.953821	+	0.300375i	$0.0971117\pi$
$252$	0	0
$253$	1.80451	0.113449
$254$	−1.32472	−0.0831204
$255$	−7.16745	−0.448843
$256$	13.8037	0.862733
$257$	−12.2227	−0.762434	−0.381217	−	0.924486i	$-0.624495\pi$
$257$	−12.2227	−0.762434	−0.381217	+	0.924486i	$0.624495\pi$
$258$	−1.11137	−0.0691909
$259$	0	0
$260$	−16.0506	−0.995416
$261$	5.46961	0.338561
$262$	0.302164	0.0186678
$263$	5.47900	0.337850	0.168925	−	0.985629i	$-0.445970\pi$
$263$	5.47900	0.337850	0.168925	+	0.985629i	$0.445970\pi$
$264$	1.20018	0.0738662
$265$	−1.36294	−0.0837246
$266$	0	0
$267$	1.02804	0.0629150
$268$	30.4594	1.86061
$269$	−26.8598	−1.63767	−0.818836	−	0.574028i	$-0.805380\pi$
$269$	−26.8598	−1.63767	−0.818836	+	0.574028i	$0.805380\pi$
$270$	−0.195488	−0.0118970
$271$	4.13941	0.251451	0.125726	−	0.992065i	$-0.459874\pi$
$271$	4.13941	0.251451	0.125726	+	0.992065i	$0.459874\pi$
$272$	−23.5296	−1.42669
$273$	0	0
$274$	2.25548	0.136258
$275$	6.56312	0.395771
$276$	−1.97196	−0.118698
$277$	17.2414	1.03593	0.517967	−	0.855400i	$-0.326689\pi$
$277$	17.2414	1.03593	0.517967	+	0.855400i	$0.326689\pi$
$278$	−2.53978	−0.152326
$279$	2.19549	0.131440
$280$	0	0
$281$	−11.8318	−0.705824	−0.352912	−	0.935657i	$-0.614808\pi$
$281$	−11.8318	−0.705824	−0.352912	+	0.935657i	$0.614808\pi$
$282$	0.325508	0.0193837
$283$	13.7812	0.819205	0.409603	−	0.912264i	$-0.365667\pi$
$283$	13.7812	0.819205	0.409603	+	0.912264i	$0.365667\pi$
$284$	18.2422	1.08247
$285$	−2.10668	−0.124789
$286$	−2.10668	−0.124570
$287$	0	0
$288$	−1.97196	−0.116199
$289$	20.6924	1.21720
$290$	1.06925	0.0627883
$291$	−8.47431	−0.496773
$292$	−0.935317	−0.0547353
$293$	−20.9392	−1.22328	−0.611641	−	0.791135i	$-0.709490\pi$
$293$	−20.9392	−1.22328	−0.611641	+	0.791135i	$0.709490\pi$
$294$	0	0
$295$	10.6371	0.619314
$296$	−0.977442	−0.0568127
$297$	1.80451	0.104708
$298$	−3.27412	−0.189665
$299$	6.97196	0.403199
$300$	−7.17214	−0.414084
$301$	0	0
$302$	2.22274	0.127904
$303$	−18.7204	−1.07546
$304$	−6.91588	−0.396653
$305$	−5.41353	−0.309978
$306$	1.02804	0.0587691
$307$	9.80451	0.559573	0.279787	−	0.960062i	$-0.409736\pi$
$307$	9.80451	0.559573	0.279787	+	0.960062i	$0.409736\pi$
$308$	0	0
$309$	4.66980	0.265655
$310$	0.429193	0.0243765
$311$	−12.5810	−0.713402	−0.356701	−	0.934219i	$-0.616099\pi$
$311$	−12.5810	−0.713402	−0.356701	+	0.934219i	$0.616099\pi$
$312$	4.63706	0.262522
$313$	18.1667	1.02684	0.513420	−	0.858137i	$-0.328378\pi$
$313$	18.1667	1.02684	0.513420	+	0.858137i	$0.328378\pi$
$314$	−2.22431	−0.125525
$315$	0	0
$316$	−14.7298	−0.828615
$317$	5.83725	0.327852	0.163926	−	0.986473i	$-0.447584\pi$
$317$	5.83725	0.327852	0.163926	+	0.986473i	$0.447584\pi$
$318$	0.195488	0.0109624
$319$	−9.86998	−0.552613
$320$	8.56312	0.478693
$321$	1.36294	0.0760718
$322$	0	0
$323$	11.0786	0.616432
$324$	−1.97196	−0.109553
$325$	25.3575	1.40658
$326$	−1.24687	−0.0690578
$327$	0.497652	0.0275202
$328$	−6.48370	−0.358002
$329$	0	0
$330$	0.352761	0.0194189
$331$	−28.2788	−1.55434	−0.777172	−	0.629288i	$-0.783347\pi$
$331$	−28.2788	−1.55434	−0.777172	+	0.629288i	$0.783347\pi$
$332$	16.0506	0.880891
$333$	−1.46961	−0.0805343
$334$	3.47352	0.190063
$335$	18.0327	0.985234
$336$	0	0
$337$	10.9720	0.597681	0.298840	−	0.954303i	$-0.403400\pi$
$337$	10.9720	0.597681	0.298840	+	0.954303i	$0.403400\pi$
$338$	−5.96257	−0.324321
$339$	1.44627	0.0785506
$340$	−14.1339	−0.766520
$341$	−3.96178	−0.214543
$342$	0.302164	0.0163392
$343$	0	0
$344$	−4.41432	−0.238004
$345$	−1.16745	−0.0628534
$346$	−2.97196	−0.159774
$347$	−10.4088	−0.558776	−0.279388	−	0.960178i	$-0.590132\pi$
$347$	−10.4088	−0.558776	−0.279388	+	0.960178i	$0.590132\pi$
$348$	10.7859	0.578183
$349$	13.7251	0.734687	0.367344	−	0.930085i	$-0.380267\pi$
$349$	13.7251	0.734687	0.367344	+	0.930085i	$0.380267\pi$
$350$	0	0
$351$	6.97196	0.372136
$352$	3.55843	0.189665
$353$	6.02725	0.320798	0.160399	−	0.987052i	$-0.448722\pi$
$353$	6.02725	0.320798	0.160399	+	0.987052i	$0.448722\pi$
$354$	−1.52569	−0.0810896
$355$	10.7998	0.573195
$356$	2.02725	0.107444
$357$	0	0
$358$	−0.743738	−0.0393078
$359$	−35.1674	−1.85607	−0.928033	−	0.372497i	$-0.878502\pi$
$359$	−35.1674	−1.85607	−0.928033	+	0.372497i	$0.878502\pi$
$360$	−0.776472	−0.0409237
$361$	−15.7437	−0.828618
$362$	−1.95788	−0.102904
$363$	7.74374	0.406441
$364$	0	0
$365$	−0.553731	−0.0289836
$366$	0.776472	0.0405869
$367$	20.5810	1.07432	0.537159	−	0.843481i	$-0.319497\pi$
$367$	20.5810	1.07432	0.537159	+	0.843481i	$0.319497\pi$
$368$	−3.83255	−0.199786
$369$	−9.74843	−0.507483
$370$	−0.287292	−0.0149356
$371$	0	0
$372$	4.32942	0.224470
$373$	5.74843	0.297643	0.148821	−	0.988864i	$-0.452452\pi$
$373$	5.74843	0.297643	0.148821	+	0.988864i	$0.452452\pi$
$374$	−1.85511	−0.0959254
$375$	−10.0833	−0.520701
$376$	1.29291	0.0666765
$377$	−38.1339	−1.96400
$378$	0	0
$379$	21.8972	1.12479	0.562393	−	0.826870i	$-0.309881\pi$
$379$	21.8972	1.12479	0.562393	+	0.826870i	$0.309881\pi$
$380$	−4.15428	−0.213110
$381$	−7.91119	−0.405302
$382$	−3.56233	−0.182265
$383$	−15.7484	−0.804707	−0.402354	−	0.915484i	$-0.631808\pi$
$383$	−15.7484	−0.804707	−0.402354	+	0.915484i	$0.631808\pi$
$384$	−5.17214	−0.263940
$385$	0	0
$386$	0.548250	0.0279052
$387$	−6.63706	−0.337381
$388$	−16.7110	−0.848373
$389$	−32.6410	−1.65496	−0.827481	−	0.561493i	$-0.810227\pi$
$389$	−32.6410	−1.65496	−0.827481	+	0.561493i	$0.810227\pi$
$390$	1.36294	0.0690150
$391$	6.13941	0.310483
$392$	0	0
$393$	1.80451	0.0910256
$394$	−0.0599874	−0.00302212
$395$	−8.72039	−0.438771
$396$	3.55843	0.178818
$397$	−34.2134	−1.71712	−0.858559	−	0.512714i	$-0.828640\pi$
$397$	−34.2134	−1.71712	−0.858559	+	0.512714i	$0.828640\pi$
$398$	−3.91510	−0.196246
$399$	0	0
$400$	−13.9392	−0.696961
$401$	−35.5802	−1.77679	−0.888395	−	0.459080i	$-0.848179\pi$
$401$	−35.5802	−1.77679	−0.888395	+	0.459080i	$0.848179\pi$
$402$	−2.58647	−0.129001
$403$	−15.3069	−0.762489
$404$	−36.9159	−1.83663
$405$	−1.16745	−0.0580110
$406$	0	0
$407$	2.65193	0.131451
$408$	4.08333	0.202155
$409$	−22.4182	−1.10851	−0.554255	−	0.832347i	$-0.686997\pi$
$409$	−22.4182	−1.10851	−0.554255	+	0.832347i	$0.686997\pi$
$410$	−1.90571	−0.0941161
$411$	13.4696	0.664407
$412$	9.20866	0.453678
$413$	0	0
$414$	0.167449	0.00822968
$415$	9.50235	0.466452
$416$	13.7484	0.674072
$417$	−15.1674	−0.742753
$418$	−0.545258	−0.0266695
$419$	24.7859	1.21087	0.605434	−	0.795895i	$-0.292999\pi$
$419$	24.7859	1.21087	0.605434	+	0.795895i	$0.292999\pi$
$420$	0	0
$421$	−35.0265	−1.70709	−0.853543	−	0.521023i	$-0.825551\pi$
$421$	−35.0265	−1.70709	−0.853543	+	0.521023i	$0.825551\pi$
$422$	−2.24609	−0.109338
$423$	1.94392	0.0945167
$424$	0.776472	0.0377088
$425$	22.3294	1.08314
$426$	−1.54904	−0.0750510
$427$	0	0
$428$	2.68766	0.129913
$429$	−12.5810	−0.607416
$430$	−1.29747	−0.0625695
$431$	3.37233	0.162439	0.0812197	−	0.996696i	$-0.474118\pi$
$431$	3.37233	0.162439	0.0812197	+	0.996696i	$0.474118\pi$
$432$	−3.83255	−0.184394
$433$	−7.83176	−0.376371	−0.188185	−	0.982134i	$-0.560261\pi$
$433$	−7.83176	−0.376371	−0.188185	+	0.982134i	$0.560261\pi$
$434$	0	0
$435$	6.38550	0.306161
$436$	0.981351	0.0469982
$437$	1.80451	0.0863215
$438$	0.0794225	0.00379496
$439$	−13.0226	−0.621533	−0.310766	−	0.950486i	$-0.600586\pi$
$439$	−13.0226	−0.621533	−0.310766	+	0.950486i	$0.600586\pi$
$440$	1.40115	0.0667974
$441$	0	0
$442$	−7.16745	−0.340921
$443$	−7.33020	−0.348268	−0.174134	−	0.984722i	$-0.555713\pi$
$443$	−7.33020	−0.348268	−0.174134	+	0.984722i	$0.555713\pi$
$444$	−2.89802	−0.137534
$445$	1.20018	0.0568942
$446$	1.46022	0.0691436
$447$	−19.5529	−0.924823
$448$	0	0
$449$	−9.49296	−0.448000	−0.224000	−	0.974589i	$-0.571912\pi$
$449$	−9.49296	−0.448000	−0.224000	+	0.974589i	$0.571912\pi$
$450$	0.609023	0.0287096
$451$	17.5912	0.828335
$452$	2.85199	0.134146
$453$	13.2741	0.623673
$454$	2.36215	0.110861
$455$	0	0
$456$	1.20018	0.0562037
$457$	−20.1721	−0.943613	−0.471807	−	0.881702i	$-0.656398\pi$
$457$	−20.1721	−0.943613	−0.471807	+	0.881702i	$0.656398\pi$
$458$	0.846505	0.0395546
$459$	6.13941	0.286563
$460$	−2.30216	−0.107339
$461$	−22.3949	−1.04303	−0.521517	−	0.853241i	$-0.674634\pi$
$461$	−22.3949	−1.04303	−0.521517	+	0.853241i	$0.674634\pi$
$462$	0	0
$463$	20.6970	0.961873	0.480937	−	0.876755i	$-0.340297\pi$
$463$	20.6970	0.961873	0.480937	+	0.876755i	$0.340297\pi$
$464$	20.9626	0.973163
$465$	2.56312	0.118862
$466$	4.64097	0.214989
$467$	−3.74843	−0.173457	−0.0867284	−	0.996232i	$-0.527641\pi$
$467$	−3.74843	−0.173457	−0.0867284	+	0.996232i	$0.527641\pi$
$468$	13.7484	0.635522
$469$	0	0
$470$	0.380014	0.0175287
$471$	−13.2835	−0.612072
$472$	−6.05999	−0.278934
$473$	11.9767	0.550687
$474$	1.25078	0.0574503
$475$	6.56312	0.301137
$476$	0	0
$477$	1.16745	0.0534538
$478$	−1.31234	−0.0600251
$479$	29.3014	1.33881	0.669407	−	0.742896i	$-0.266548\pi$
$479$	29.3014	1.33881	0.669407	+	0.742896i	$0.266548\pi$
$480$	−2.30216	−0.105079
$481$	10.2461	0.467181
$482$	3.29747	0.150196
$483$	0	0
$484$	15.2703	0.694107
$485$	−9.89332	−0.449233
$486$	0.167449	0.00759565
$487$	17.0880	0.774332	0.387166	−	0.922010i	$-0.373454\pi$
$487$	17.0880	0.774332	0.387166	+	0.922010i	$0.373454\pi$
$488$	3.08412	0.139611
$489$	−7.44627	−0.336732
$490$	0	0
$491$	−16.3022	−0.735706	−0.367853	−	0.929884i	$-0.619907\pi$
$491$	−16.3022	−0.735706	−0.367853	+	0.929884i	$0.619907\pi$
$492$	−19.2235	−0.866664
$493$	−33.5802	−1.51238
$494$	−2.10668	−0.0947838
$495$	2.10668	0.0946880
$496$	8.41432	0.377814
$497$	0	0
$498$	−1.36294	−0.0610747
$499$	31.7251	1.42021	0.710105	−	0.704096i	$-0.248647\pi$
$499$	31.7251	1.42021	0.710105	+	0.704096i	$0.248647\pi$
$500$	−19.8839	−0.889237
$501$	20.7437	0.926762
$502$	−5.06077	−0.225873
$503$	15.1020	0.673364	0.336682	−	0.941618i	$-0.390695\pi$
$503$	15.1020	0.673364	0.336682	+	0.941618i	$0.390695\pi$
$504$	0	0
$505$	−21.8551	−0.972540
$506$	−0.302164	−0.0134328
$507$	−35.6082	−1.58142
$508$	−15.6006	−0.692163
$509$	28.7804	1.27567	0.637834	−	0.770174i	$-0.279831\pi$
$509$	28.7804	1.27567	0.637834	+	0.770174i	$0.279831\pi$
$510$	1.20018	0.0531450
$511$	0	0
$512$	−12.6557	−0.559309
$513$	1.80451	0.0796711
$514$	2.04669	0.0902755
$515$	5.45175	0.240233
$516$	−13.0880	−0.576168
$517$	−3.50783	−0.154274
$518$	0	0
$519$	−17.7484	−0.779070
$520$	5.41353	0.237399
$521$	−30.4455	−1.33384	−0.666920	−	0.745129i	$-0.732388\pi$
$521$	−30.4455	−1.33384	−0.666920	+	0.745129i	$0.732388\pi$
$522$	−0.915882	−0.0400871
$523$	29.4875	1.28940	0.644699	−	0.764437i	$-0.276983\pi$
$523$	29.4875	1.28940	0.644699	+	0.764437i	$0.276983\pi$
$524$	3.55843	0.155451
$525$	0	0
$526$	−0.917455	−0.0400029
$527$	−13.4790	−0.587155
$528$	6.91588	0.300975
$529$	1.00000	0.0434783
$530$	0.228223	0.00991336
$531$	−9.11137	−0.395400
$532$	0	0
$533$	67.9657	2.94392
$534$	−0.172144	−0.00744941
$535$	1.59116	0.0687919
$536$	−10.2733	−0.443741
$537$	−4.44157	−0.191668
$538$	4.49765	0.193908
$539$	0	0
$540$	−2.30216	−0.0990694
$541$	−9.54355	−0.410309	−0.205155	−	0.978730i	$-0.565770\pi$
$541$	−9.54355	−0.410309	−0.205155	+	0.978730i	$0.565770\pi$
$542$	−0.693141	−0.0297729
$543$	−11.6924	−0.501767
$544$	12.1067	0.519069
$545$	0.580984	0.0248866
$546$	0	0
$547$	−21.2780	−0.909783	−0.454892	−	0.890547i	$-0.650322\pi$
$547$	−21.2780	−0.909783	−0.454892	+	0.890547i	$0.650322\pi$
$548$	26.5615	1.13465
$549$	4.63706	0.197905
$550$	−1.09899	−0.0468611
$551$	−9.86998	−0.420475
$552$	0.665102	0.0283086
$553$	0	0
$554$	−2.88706	−0.122659
$555$	−1.71570	−0.0728274
$556$	−29.9096	−1.26845
$557$	−30.4922	−1.29199	−0.645997	−	0.763340i	$-0.723558\pi$
$557$	−30.4922	−1.29199	−0.645997	+	0.763340i	$0.723558\pi$
$558$	−0.367633	−0.0155631
$559$	46.2733	1.95715
$560$	0	0
$561$	−11.0786	−0.467740
$562$	1.98122	0.0835727
$563$	−20.7476	−0.874409	−0.437205	−	0.899362i	$-0.644031\pi$
$563$	−20.7476	−0.874409	−0.437205	+	0.899362i	$0.644031\pi$
$564$	3.83334	0.161413
$565$	1.68845	0.0710334
$566$	−2.30765	−0.0969976
$567$	0	0
$568$	−6.15271	−0.258162
$569$	−21.4969	−0.901196	−0.450598	−	0.892727i	$-0.648789\pi$
$569$	−21.4969	−0.901196	−0.450598	+	0.892727i	$0.648789\pi$
$570$	0.352761	0.0147755
$571$	−14.1488	−0.592109	−0.296054	−	0.955171i	$-0.595671\pi$
$571$	−14.1488	−0.592109	−0.296054	+	0.955171i	$0.595671\pi$
$572$	−24.8092	−1.03733
$573$	−21.2741	−0.888739
$574$	0	0
$575$	3.63706	0.151676
$576$	−7.33490	−0.305621
$577$	−10.7820	−0.448859	−0.224429	−	0.974490i	$-0.572052\pi$
$577$	−10.7820	−0.448859	−0.224429	+	0.974490i	$0.572052\pi$
$578$	−3.46492	−0.144122
$579$	3.27412	0.136068
$580$	12.5919	0.522852
$581$	0	0
$582$	1.41902	0.0588201
$583$	−2.10668	−0.0872496
$584$	0.315463	0.0130540
$585$	8.13941	0.336523
$586$	3.50626	0.144842
$587$	20.3676	0.840662	0.420331	−	0.907371i	$-0.361914\pi$
$587$	20.3676	0.840662	0.420331	+	0.907371i	$0.361914\pi$
$588$	0	0
$589$	−3.96178	−0.163242
$590$	−1.78117	−0.0733295
$591$	−0.358242	−0.0147361
$592$	−5.63237	−0.231489
$593$	15.9439	0.654738	0.327369	−	0.944897i	$-0.393838\pi$
$593$	15.9439	0.654738	0.327369	+	0.944897i	$0.393838\pi$
$594$	−0.302164	−0.0123979
$595$	0	0
$596$	−38.5576	−1.57938
$597$	−23.3808	−0.956912
$598$	−1.16745	−0.0477405
$599$	18.9159	0.772882	0.386441	−	0.922314i	$-0.373704\pi$
$599$	18.9159	0.772882	0.386441	+	0.922314i	$0.373704\pi$
$600$	2.41902	0.0987559
$601$	48.3388	1.97178	0.985891	−	0.167391i	$-0.0535343\pi$
$601$	48.3388	1.97178	0.985891	+	0.167391i	$0.0535343\pi$
$602$	0	0
$603$	−15.4463	−0.629021
$604$	26.1761	1.06509
$605$	9.04042	0.367545
$606$	3.13471	0.127339
$607$	9.29747	0.377373	0.188686	−	0.982037i	$-0.439577\pi$
$607$	9.29747	0.377373	0.188686	+	0.982037i	$0.439577\pi$
$608$	3.55843	0.144313
$609$	0	0
$610$	0.906492	0.0367028
$611$	−13.5529	−0.548293
$612$	12.1067	0.489383
$613$	41.3108	1.66853	0.834263	−	0.551367i	$-0.185893\pi$
$613$	41.3108	1.66853	0.834263	+	0.551367i	$0.185893\pi$
$614$	−1.64176	−0.0662559
$615$	−11.3808	−0.458918
$616$	0	0
$617$	−36.4782	−1.46856	−0.734279	−	0.678848i	$-0.762480\pi$
$617$	−36.4782	−1.46856	−0.734279	+	0.678848i	$0.762480\pi$
$618$	−0.781954	−0.0314548
$619$	−7.41902	−0.298195	−0.149098	−	0.988822i	$-0.547637\pi$
$619$	−7.41902	−0.298195	−0.149098	+	0.988822i	$0.547637\pi$
$620$	5.05437	0.202989
$621$	1.00000	0.0401286
$622$	2.10668	0.0844700
$623$	0	0
$624$	26.7204	1.06967
$625$	6.41353	0.256541
$626$	−3.04199	−0.121582
$627$	−3.25626	−0.130043
$628$	−26.1946	−1.04528
$629$	9.02256	0.359753
$630$	0	0
$631$	23.0132	0.916140	0.458070	−	0.888916i	$-0.348541\pi$
$631$	23.0132	0.916140	0.458070	+	0.888916i	$0.348541\pi$
$632$	4.96805	0.197618
$633$	−13.4135	−0.533140
$634$	−0.977442	−0.0388192
$635$	−9.23591	−0.366516
$636$	2.30216	0.0912867
$637$	0	0
$638$	1.65272	0.0654318
$639$	−9.25078	−0.365955
$640$	−6.03822	−0.238681
$641$	26.1433	1.03260	0.516300	−	0.856408i	$-0.327309\pi$
$641$	26.1433	1.03260	0.516300	+	0.856408i	$0.327309\pi$
$642$	−0.228223	−0.00900724
$643$	5.47509	0.215917	0.107958	−	0.994155i	$-0.465569\pi$
$643$	5.47509	0.215917	0.107958	+	0.994155i	$0.465569\pi$
$644$	0	0
$645$	−7.74843	−0.305094
$646$	−1.85511	−0.0729883
$647$	−43.8263	−1.72299	−0.861494	−	0.507767i	$-0.830471\pi$
$647$	−43.8263	−1.72299	−0.861494	+	0.507767i	$0.830471\pi$
$648$	0.665102	0.0261277
$649$	16.4416	0.645388
$650$	−4.24609	−0.166545
$651$	0	0
$652$	−14.6838	−0.575060
$653$	28.7298	1.12428	0.562142	−	0.827041i	$-0.309978\pi$
$653$	28.7298	1.12428	0.562142	+	0.827041i	$0.309978\pi$
$654$	−0.0833315	−0.00325852
$655$	2.10668	0.0823146
$656$	−37.3614	−1.45872
$657$	0.474308	0.0185045
$658$	0	0
$659$	−6.47431	−0.252203	−0.126102	−	0.992017i	$-0.540247\pi$
$659$	−6.47431	−0.252203	−0.126102	+	0.992017i	$0.540247\pi$
$660$	4.15428	0.161705
$661$	5.03195	0.195720	0.0978601	−	0.995200i	$-0.468800\pi$
$661$	5.03195	0.195720	0.0978601	+	0.995200i	$0.468800\pi$
$662$	4.73527	0.184041
$663$	−42.8037	−1.66236
$664$	−5.41353	−0.210086
$665$	0	0
$666$	0.246086	0.00953562
$667$	−5.46961	−0.211784
$668$	40.9058	1.58269
$669$	8.72039	0.337150
$670$	−3.01957	−0.116656
$671$	−8.36763	−0.323029
$672$	0	0
$673$	37.9618	1.46332	0.731660	−	0.681670i	$-0.238746\pi$
$673$	37.9618	1.46332	0.731660	+	0.681670i	$0.238746\pi$
$674$	−1.83725	−0.0707681
$675$	3.63706	0.139991
$676$	−70.2180	−2.70069
$677$	13.1947	0.507114	0.253557	−	0.967320i	$-0.418400\pi$
$677$	13.1947	0.507114	0.253557	+	0.967320i	$0.418400\pi$
$678$	−0.242177	−0.00930074
$679$	0	0
$680$	4.76708	0.182809
$681$	14.1067	0.540569
$682$	0.663398	0.0254028
$683$	18.4088	0.704395	0.352197	−	0.935926i	$-0.385435\pi$
$683$	18.4088	0.704395	0.352197	+	0.935926i	$0.385435\pi$
$684$	3.55843	0.136060
$685$	15.7251	0.600825
$686$	0	0
$687$	5.05529	0.192871
$688$	−25.4369	−0.969772
$689$	−8.13941	−0.310087
$690$	0.195488	0.00744212
$691$	−32.5343	−1.23766	−0.618831	−	0.785524i	$-0.712394\pi$
$691$	−32.5343	−1.23766	−0.618831	+	0.785524i	$0.712394\pi$
$692$	−34.9992	−1.33047
$693$	0	0
$694$	1.74295	0.0661615
$695$	−17.7072	−0.671673
$696$	−3.63785	−0.137892
$697$	59.8496	2.26697
$698$	−2.29826	−0.0869902
$699$	27.7157	1.04830
$700$	0	0
$701$	39.9290	1.50810	0.754050	−	0.656817i	$-0.228098\pi$
$701$	39.9290	1.50810	0.754050	+	0.656817i	$0.228098\pi$
$702$	−1.16745	−0.0440625
$703$	2.65193	0.100020
$704$	13.2359	0.498847
$705$	2.26943	0.0854717
$706$	−1.00926	−0.0379840
$707$	0	0
$708$	−17.9673	−0.675251
$709$	−30.2555	−1.13627	−0.568134	−	0.822936i	$-0.692335\pi$
$709$	−30.2555	−1.13627	−0.568134	+	0.822936i	$0.692335\pi$
$710$	−1.80842	−0.0678688
$711$	7.46961	0.280132
$712$	−0.683751	−0.0256246
$713$	−2.19549	−0.0822217
$714$	0	0
$715$	−14.6877	−0.549287
$716$	−8.75861	−0.327325
$717$	−7.83725	−0.292687
$718$	5.88876	0.219767
$719$	−37.9057	−1.41364	−0.706822	−	0.707391i	$-0.749872\pi$
$719$	−37.9057	−1.41364	−0.706822	+	0.707391i	$0.749872\pi$
$720$	−4.47431	−0.166748
$721$	0	0
$722$	2.63628	0.0981120
$723$	19.6924	0.732367
$724$	−23.0569	−0.856902
$725$	−19.8933	−0.738820
$726$	−1.29668	−0.0481244
$727$	42.9010	1.59111	0.795555	−	0.605881i	$-0.207179\pi$
$727$	42.9010	1.59111	0.795555	+	0.605881i	$0.207179\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0.0927218	0.00343179
$731$	40.7476	1.50711
$732$	9.14411	0.337976
$733$	−3.27412	−0.120933	−0.0604663	−	0.998170i	$-0.519259\pi$
$733$	−3.27412	−0.120933	−0.0604663	+	0.998170i	$0.519259\pi$
$734$	−3.44627	−0.127204
$735$	0	0
$736$	1.97196	0.0726874
$737$	27.8730	1.02671
$738$	1.63237	0.0600883
$739$	−17.0226	−0.626185	−0.313092	−	0.949723i	$-0.601365\pi$
$739$	−17.0226	−0.626185	−0.313092	+	0.949723i	$0.601365\pi$
$740$	−3.38329	−0.124372
$741$	−12.5810	−0.462174
$742$	0	0
$743$	20.0506	0.735585	0.367793	−	0.929908i	$-0.380114\pi$
$743$	20.0506	0.735585	0.367793	+	0.929908i	$0.380114\pi$
$744$	−1.46022	−0.0535344
$745$	−22.8271	−0.836319
$746$	−0.962570	−0.0352422
$747$	−8.13941	−0.297805
$748$	−21.8466	−0.798792
$749$	0	0
$750$	1.68845	0.0616533
$751$	−46.7298	−1.70519	−0.852597	−	0.522569i	$-0.824974\pi$
$751$	−46.7298	−1.70519	−0.852597	+	0.522569i	$0.824974\pi$
$752$	7.45018	0.271680
$753$	−30.2227	−1.10138
$754$	6.38550	0.232546
$755$	15.4969	0.563989
$756$	0	0
$757$	22.8092	0.829015	0.414507	−	0.910046i	$-0.363954\pi$
$757$	22.8092	0.829015	0.414507	+	0.910046i	$0.363954\pi$
$758$	−3.66667	−0.133180
$759$	−1.80451	−0.0654996
$760$	1.40115	0.0508252
$761$	−28.1761	−1.02138	−0.510691	−	0.859765i	$-0.670610\pi$
$761$	−28.1761	−1.02138	−0.510691	+	0.859765i	$0.670610\pi$
$762$	1.32472	0.0479896
$763$	0	0
$764$	−41.9517	−1.51776
$765$	7.16745	0.259140
$766$	2.63706	0.0952809
$767$	63.5241	2.29372
$768$	−13.8037	−0.498099
$769$	25.8412	0.931856	0.465928	−	0.884823i	$-0.345721\pi$
$769$	25.8412	0.931856	0.465928	+	0.884823i	$0.345721\pi$
$770$	0	0
$771$	12.2227	0.440191
$772$	6.45645	0.232373
$773$	−44.0833	−1.58557	−0.792784	−	0.609503i	$-0.791369\pi$
$773$	−44.0833	−1.58557	−0.792784	+	0.609503i	$0.791369\pi$
$774$	1.11137	0.0399474
$775$	−7.98513	−0.286834
$776$	5.63628	0.202331
$777$	0	0
$778$	5.46570	0.195955
$779$	17.5912	0.630269
$780$	16.0506	0.574704
$781$	16.6931	0.597328
$782$	−1.02804	−0.0367626
$783$	−5.46961	−0.195468
$784$	0	0
$785$	−15.5078	−0.553498
$786$	−0.302164	−0.0107778
$787$	36.7204	1.30894	0.654470	−	0.756088i	$-0.272892\pi$
$787$	36.7204	1.30894	0.654470	+	0.756088i	$0.272892\pi$
$788$	−0.706440	−0.0251659
$789$	−5.47900	−0.195058
$790$	1.46022	0.0519524
$791$	0	0
$792$	−1.20018	−0.0426467
$793$	−32.3294	−1.14805
$794$	5.72900	0.203315
$795$	1.36294	0.0483384
$796$	−46.1060	−1.63418
$797$	−40.7804	−1.44452	−0.722258	−	0.691624i	$-0.756895\pi$
$797$	−40.7804	−1.44452	−0.722258	+	0.691624i	$0.756895\pi$
$798$	0	0
$799$	−11.9345	−0.422213
$800$	7.17214	0.253574
$801$	−1.02804	−0.0363240
$802$	5.95788	0.210380
$803$	−0.855895	−0.0302039
$804$	−30.4594	−1.07422
$805$	0	0
$806$	2.56312	0.0902821
$807$	26.8598	0.945510
$808$	12.4510	0.438023
$809$	2.46883	0.0867993	0.0433997	−	0.999058i	$-0.486181\pi$
$809$	2.46883	0.0867993	0.0433997	+	0.999058i	$0.486181\pi$
$810$	0.195488	0.00686876
$811$	16.4182	0.576522	0.288261	−	0.957552i	$-0.406923\pi$
$811$	16.4182	0.576522	0.288261	+	0.957552i	$0.406923\pi$
$812$	0	0
$813$	−4.13941	−0.145175
$814$	−0.444064	−0.0155644
$815$	−8.69314	−0.304507
$816$	23.5296	0.823701
$817$	11.9767	0.419010
$818$	3.75391	0.131253
$819$	0	0
$820$	−22.4425	−0.783726
$821$	−34.1012	−1.19014	−0.595070	−	0.803674i	$-0.702876\pi$
$821$	−34.1012	−1.19014	−0.595070	+	0.803674i	$0.702876\pi$
$822$	−2.25548	−0.0786688
$823$	16.1628	0.563398	0.281699	−	0.959503i	$-0.409102\pi$
$823$	16.1628	0.563398	0.281699	+	0.959503i	$0.409102\pi$
$824$	−3.10589	−0.108199
$825$	−6.56312	−0.228499
$826$	0	0
$827$	−18.5522	−0.645122	−0.322561	−	0.946549i	$-0.604544\pi$
$827$	−18.5522	−0.645122	−0.322561	+	0.946549i	$0.604544\pi$
$828$	1.97196	0.0685304
$829$	−45.2453	−1.57143	−0.785717	−	0.618586i	$-0.787706\pi$
$829$	−45.2453	−1.57143	−0.785717	+	0.618586i	$0.787706\pi$
$830$	−1.59116	−0.0552300
$831$	−17.2414	−0.598097
$832$	51.1386	1.77291
$833$	0	0
$834$	2.53978	0.0879453
$835$	24.2173	0.838073
$836$	−6.42122	−0.222083
$837$	−2.19549	−0.0758872
$838$	−4.15037	−0.143372
$839$	1.29747	0.0447936	0.0223968	−	0.999749i	$-0.492870\pi$
$839$	1.29747	0.0447936	0.0223968	+	0.999749i	$0.492870\pi$
$840$	0	0
$841$	0.916669	0.0316093
$842$	5.86515	0.202127
$843$	11.8318	0.407508
$844$	−26.4510	−0.910480
$845$	−41.5708	−1.43008
$846$	−0.325508	−0.0111912
$847$	0	0
$848$	4.47431	0.153648
$849$	−13.7812	−0.472968
$850$	−3.73904	−0.128248
$851$	1.46961	0.0503777
$852$	−18.2422	−0.624967
$853$	−33.2835	−1.13961	−0.569803	−	0.821781i	$-0.692980\pi$
$853$	−33.2835	−1.13961	−0.569803	+	0.821781i	$0.692980\pi$
$854$	0	0
$855$	2.10668	0.0720468
$856$	−0.906492	−0.0309833
$857$	−9.26473	−0.316477	−0.158239	−	0.987401i	$-0.550582\pi$
$857$	−9.26473	−0.316477	−0.158239	+	0.987401i	$0.550582\pi$
$858$	2.10668	0.0719207
$859$	32.7243	1.11654	0.558269	−	0.829660i	$-0.311466\pi$
$859$	32.7243	1.11654	0.558269	+	0.829660i	$0.311466\pi$
$860$	−15.2796	−0.521030
$861$	0	0
$862$	−0.564694	−0.0192335
$863$	−40.1012	−1.36506	−0.682530	−	0.730857i	$-0.739120\pi$
$863$	−40.1012	−1.36506	−0.682530	+	0.730857i	$0.739120\pi$
$864$	1.97196	0.0670875
$865$	−20.7204	−0.704515
$866$	1.31142	0.0445640
$867$	−20.6924	−0.702749
$868$	0	0
$869$	−13.4790	−0.457244
$870$	−1.06925	−0.0362508
$871$	107.691	3.64896
$872$	−0.330989	−0.0112087
$873$	8.47431	0.286812
$874$	−0.302164	−0.0102208
$875$	0	0
$876$	0.935317	0.0316014
$877$	41.6090	1.40504	0.702518	−	0.711666i	$-0.252059\pi$
$877$	41.6090	1.40504	0.702518	+	0.711666i	$0.252059\pi$
$878$	2.18062	0.0735923
$879$	20.9392	0.706263
$880$	8.07394	0.272173
$881$	31.9439	1.07622	0.538109	−	0.842875i	$-0.319139\pi$
$881$	31.9439	1.07622	0.538109	+	0.842875i	$0.319139\pi$
$882$	0	0
$883$	−0.0778502	−0.00261987	−0.00130993	−	0.999999i	$-0.500417\pi$
$883$	−0.0778502	−0.00261987	−0.00130993	+	0.999999i	$0.500417\pi$
$884$	−84.4073	−2.83892
$885$	−10.6371	−0.357561
$886$	1.22744	0.0412365
$887$	−4.80530	−0.161346	−0.0806731	−	0.996741i	$-0.525707\pi$
$887$	−4.80530	−0.161346	−0.0806731	+	0.996741i	$0.525707\pi$
$888$	0.977442	0.0328008
$889$	0	0
$890$	−0.200970	−0.00673652
$891$	−1.80451	−0.0604534
$892$	17.1963	0.575774
$893$	−3.50783	−0.117385
$894$	3.27412	0.109503
$895$	−5.18531	−0.173326
$896$	0	0
$897$	−6.97196	−0.232787
$898$	1.58959	0.0530452
$899$	12.0085	0.400505
$900$	7.17214	0.239071
$901$	−7.16745	−0.238782
$902$	−2.94563	−0.0980786
$903$	0	0
$904$	−0.961916	−0.0319928
$905$	−13.6502	−0.453749
$906$	−2.22274	−0.0738456
$907$	56.9337	1.89045	0.945227	−	0.326414i	$-0.105840\pi$
$907$	56.9337	1.89045	0.945227	+	0.326414i	$0.105840\pi$
$908$	27.8178	0.923166
$909$	18.7204	0.620916
$910$	0	0
$911$	−5.27412	−0.174740	−0.0873698	−	0.996176i	$-0.527846\pi$
$911$	−5.27412	−0.174740	−0.0873698	+	0.996176i	$0.527846\pi$
$912$	6.91588	0.229008
$913$	14.6877	0.486091
$914$	3.37781	0.111728
$915$	5.41353	0.178966
$916$	9.96884	0.329380
$917$	0	0
$918$	−1.02804	−0.0339304
$919$	−26.1955	−0.864109	−0.432055	−	0.901847i	$-0.642211\pi$
$919$	−26.1955	−0.864109	−0.432055	+	0.901847i	$0.642211\pi$
$920$	0.776472	0.0255995
$921$	−9.80451	−0.323070
$922$	3.75001	0.123500
$923$	64.4961	2.12291
$924$	0	0
$925$	5.34507	0.175745
$926$	−3.46570	−0.113890
$927$	−4.66980	−0.153376
$928$	−10.7859	−0.354063
$929$	40.6643	1.33415	0.667076	−	0.744989i	$-0.267545\pi$
$929$	40.6643	1.33415	0.667076	+	0.744989i	$0.267545\pi$
$930$	−0.429193	−0.0140738
$931$	0	0
$932$	54.6543	1.79026
$933$	12.5810	0.411883
$934$	0.627672	0.0205381
$935$	−12.9337	−0.422979
$936$	−4.63706	−0.151567
$937$	−12.4276	−0.405993	−0.202996	−	0.979179i	$-0.565068\pi$
$937$	−12.4276	−0.405993	−0.202996	+	0.979179i	$0.565068\pi$
$938$	0	0
$939$	−18.1667	−0.592847
$940$	4.47523	0.145966
$941$	−36.7149	−1.19687	−0.598436	−	0.801171i	$-0.704211\pi$
$941$	−36.7149	−1.19687	−0.598436	+	0.801171i	$0.704211\pi$
$942$	2.22431	0.0724721
$943$	9.74843	0.317453
$944$	−34.9198	−1.13654
$945$	0	0
$946$	−2.00548	−0.0652038
$947$	2.38159	0.0773912	0.0386956	−	0.999251i	$-0.487680\pi$
$947$	2.38159	0.0773912	0.0386956	+	0.999251i	$0.487680\pi$
$948$	14.7298	0.478401
$949$	−3.30686	−0.107345
$950$	−1.09899	−0.0356559
$951$	−5.83725	−0.189286
$952$	0	0
$953$	−42.5810	−1.37933	−0.689667	−	0.724127i	$-0.742243\pi$
$953$	−42.5810	−1.37933	−0.689667	+	0.724127i	$0.742243\pi$
$954$	−0.195488	−0.00632917
$955$	−24.8365	−0.803689
$956$	−15.4547	−0.499842
$957$	9.86998	0.319051
$958$	−4.90649	−0.158522
$959$	0	0
$960$	−8.56312	−0.276374
$961$	−26.1798	−0.844511
$962$	−1.71570	−0.0553163
$963$	−1.36294	−0.0439201
$964$	38.8326	1.25071
$965$	3.82237	0.123047
$966$	0	0
$967$	−27.3847	−0.880633	−0.440316	−	0.897843i	$-0.645134\pi$
$967$	−27.3847	−0.880633	−0.440316	+	0.897843i	$0.645134\pi$
$968$	−5.15037	−0.165539
$969$	−11.0786	−0.355897
$970$	1.65663	0.0531912
$971$	−16.2555	−0.521663	−0.260832	−	0.965384i	$-0.583997\pi$
$971$	−16.2555	−0.521663	−0.260832	+	0.965384i	$0.583997\pi$
$972$	1.97196	0.0632507
$973$	0	0
$974$	−2.86138	−0.0916844
$975$	−25.3575	−0.812089
$976$	17.7718	0.568861
$977$	−14.1900	−0.453979	−0.226989	−	0.973897i	$-0.572888\pi$
$977$	−14.1900	−0.453979	−0.226989	+	0.973897i	$0.572888\pi$
$978$	1.24687	0.0398706
$979$	1.85511	0.0592895
$980$	0	0
$981$	−0.497652	−0.0158888
$982$	2.72978	0.0871109
$983$	−35.8606	−1.14378	−0.571888	−	0.820332i	$-0.693789\pi$
$983$	−35.8606	−1.14378	−0.571888	+	0.820332i	$0.693789\pi$
$984$	6.48370	0.206693
$985$	−0.418230	−0.0133259
$986$	5.62298	0.179072
$987$	0	0
$988$	−24.8092	−0.789286
$989$	6.63706	0.211046
$990$	−0.352761	−0.0112115
$991$	13.1853	0.418845	0.209423	−	0.977825i	$-0.432842\pi$
$991$	13.1853	0.418845	0.209423	+	0.977825i	$0.432842\pi$
$992$	−4.32942	−0.137459
$993$	28.2788	0.897401
$994$	0	0
$995$	−27.2959	−0.865338
$996$	−16.0506	−0.508583
$997$	−59.2547	−1.87661	−0.938307	−	0.345802i	$-0.887607\pi$
$997$	−59.2547	−1.87661	−0.938307	+	0.345802i	$0.887607\pi$
$998$	−5.31234	−0.168159
$999$	1.46961	0.0464965

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3381.2.a.v.1.2		3
7.6	odd	2		483.2.a.h.1.2	✓	3
21.20	even	2		1449.2.a.l.1.2		3
28.27	even	2		7728.2.a.bt.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.h.1.2	✓	3	7.6	odd	2
1449.2.a.l.1.2		3	21.20	even	2
3381.2.a.v.1.2		3	1.1	even	1	trivial
7728.2.a.bt.1.2		3	28.27	even	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:	\( N \)	\(=\)	\( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3381.a (trivial)

\(f(q)\)	\(=\)	\(q-0.167449 q^{2} -1.00000 q^{3} -1.97196 q^{4} -1.16745 q^{5} +0.167449 q^{6} +0.665102 q^{8} +1.00000 q^{9} +0.195488 q^{10} -1.80451 q^{11} +1.97196 q^{12} -6.97196 q^{13} +1.16745 q^{15} +3.83255 q^{16} -6.13941 q^{17} -0.167449 q^{18} -1.80451 q^{19} +2.30216 q^{20} +0.302164 q^{22} -1.00000 q^{23} -0.665102 q^{24} -3.63706 q^{25} +1.16745 q^{26} -1.00000 q^{27} +5.46961 q^{29} -0.195488 q^{30} +2.19549 q^{31} -1.97196 q^{32} +1.80451 q^{33} +1.02804 q^{34} -1.97196 q^{36} -1.46961 q^{37} +0.302164 q^{38} +6.97196 q^{39} -0.776472 q^{40} -9.74843 q^{41} -6.63706 q^{43} +3.55843 q^{44} -1.16745 q^{45} +0.167449 q^{46} +1.94392 q^{47} -3.83255 q^{48} +0.609023 q^{50} +6.13941 q^{51} +13.7484 q^{52} +1.16745 q^{53} +0.167449 q^{54} +2.10668 q^{55} +1.80451 q^{57} -0.915882 q^{58} -9.11137 q^{59} -2.30216 q^{60} +4.63706 q^{61} -0.367633 q^{62} -7.33490 q^{64} +8.13941 q^{65} -0.302164 q^{66} -15.4463 q^{67} +12.1067 q^{68} +1.00000 q^{69} -9.25078 q^{71} +0.665102 q^{72} +0.474308 q^{73} +0.246086 q^{74} +3.63706 q^{75} +3.55843 q^{76} -1.16745 q^{78} +7.46961 q^{79} -4.47431 q^{80} +1.00000 q^{81} +1.63237 q^{82} -8.13941 q^{83} +7.16745 q^{85} +1.11137 q^{86} -5.46961 q^{87} -1.20018 q^{88} -1.02804 q^{89} +0.195488 q^{90} +1.97196 q^{92} -2.19549 q^{93} -0.325508 q^{94} +2.10668 q^{95} +1.97196 q^{96} +8.47431 q^{97} -1.80451 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} + 3 q^{8} + 3 q^{9} + 12 q^{10} + 6 q^{11} - 6 q^{12} - 9 q^{13} + 3 q^{15} + 12 q^{16} - 6 q^{17} + 6 q^{19} - 3 q^{20} - 9 q^{22} - 3 q^{23} - 3 q^{24} + 3 q^{26} - 3 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 6 * q^4 - 3 * q^5 + 3 * q^8 + 3 * q^9 + 12 * q^10 + 6 * q^11 - 6 * q^12 - 9 * q^13 + 3 * q^15 + 12 * q^16 - 6 * q^17 + 6 * q^19 - 3 * q^20 - 9 * q^22 - 3 * q^23 - 3 * q^24 + 3 * q^26 - 3 * q^27 + 6 * q^29 - 12 * q^30 + 18 * q^31 + 6 * q^32 - 6 * q^33 + 15 * q^34 + 6 * q^36 + 6 * q^37 - 9 * q^38 + 9 * q^39 + 21 * q^40 + 6 * q^41 - 9 * q^43 + 33 * q^44 - 3 * q^45 - 18 * q^47 - 12 * q^48 - 21 * q^50 + 6 * q^51 + 6 * q^52 + 3 * q^53 - 15 * q^55 - 6 * q^57 + 33 * q^58 - 3 * q^59 + 3 * q^60 + 3 * q^61 - 9 * q^62 - 21 * q^64 + 12 * q^65 + 9 * q^66 - 21 * q^67 + 15 * q^68 + 3 * q^69 + 9 * q^71 + 3 * q^72 - 12 * q^73 - 33 * q^74 + 33 * q^76 - 3 * q^78 + 12 * q^79 + 3 * q^81 - 3 * q^82 - 12 * q^83 + 21 * q^85 - 21 * q^86 - 6 * q^87 - 12 * q^88 - 15 * q^89 + 12 * q^90 - 6 * q^92 - 18 * q^93 - 6 * q^94 - 15 * q^95 - 6 * q^96 + 12 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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