LMFDB - Embedded newform 2421.2.a.i.1.8

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2421,2,Mod(1,2421)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2421.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2421, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2421 = 3^{2} \cdot 269 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2421.a (trivial)

Newform invariants

comment:select newform

sage:traces = [16,-1,0,25,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$19.3317823294$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 314 x^{12} - 283 x^{11} - 1803 x^{10} + 1435 x^{9} + \cdots + 172 $ x^16 - x^15 - 28x^14 + 27x^13 + 314x^12 - 283x^11 - 1803x^10 + 1435x^9 + 5637x^8 - 3547x^7 - 9470x^6 + 3701x^5 + 7860x^4 - 1001x^3 - 2363x^2 - 43x + 172
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 269)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.282877$ of defining polynomial
Character	$\chi$	$=$	2421.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.282877 q^{2} -1.91998 q^{4} -1.97313 q^{5} +2.44119 q^{7} +1.10887 q^{8} +0.558153 q^{10} -3.43591 q^{11} +4.69311 q^{13} -0.690556 q^{14} +3.52629 q^{16} -1.90195 q^{17} -1.23155 q^{19} +3.78838 q^{20} +0.971939 q^{22} -6.50000 q^{23} -1.10675 q^{25} -1.32757 q^{26} -4.68704 q^{28} +5.54253 q^{29} +4.88221 q^{31} -3.21525 q^{32} +0.538017 q^{34} -4.81680 q^{35} -10.2236 q^{37} +0.348377 q^{38} -2.18795 q^{40} +5.60633 q^{41} -2.85084 q^{43} +6.59688 q^{44} +1.83870 q^{46} +1.80999 q^{47} -1.04058 q^{49} +0.313073 q^{50} -9.01068 q^{52} -2.04207 q^{53} +6.77951 q^{55} +2.70697 q^{56} -1.56785 q^{58} +1.02290 q^{59} +12.6403 q^{61} -1.38106 q^{62} -6.14306 q^{64} -9.26012 q^{65} +1.65225 q^{67} +3.65171 q^{68} +1.36256 q^{70} +4.59265 q^{71} +5.17859 q^{73} +2.89202 q^{74} +2.36455 q^{76} -8.38772 q^{77} +13.5494 q^{79} -6.95784 q^{80} -1.58590 q^{82} +13.9606 q^{83} +3.75280 q^{85} +0.806435 q^{86} -3.80998 q^{88} +9.64059 q^{89} +11.4568 q^{91} +12.4799 q^{92} -0.512003 q^{94} +2.43001 q^{95} -16.2488 q^{97} +0.294355 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - q^{2} + 25 q^{4} + q^{5} + 11 q^{7} + 7 q^{10} - 16 q^{11} - q^{13} + 9 q^{14} + 39 q^{16} + 2 q^{17} + 35 q^{19} + 16 q^{20} - 3 q^{22} - q^{23} + 13 q^{25} + 14 q^{26} + 11 q^{28} - 2 q^{29}+ \cdots - 24 q^{98}+O(q^{100}) $ 16 * q - q^2 + 25 * q^4 + q^5 + 11 * q^7 + 7 * q^10 - 16 * q^11 - q^13 + 9 * q^14 + 39 * q^16 + 2 * q^17 + 35 * q^19 + 16 * q^20 - 3 * q^22 - q^23 + 13 * q^25 + 14 * q^26 + 11 * q^28 - 2 * q^29 + 13 * q^31 + 29 * q^32 - 10 * q^34 - 9 * q^35 + 4 * q^37 + 32 * q^38 + 6 * q^40 + 2 * q^41 + 51 * q^43 - 8 * q^44 + 20 * q^46 + 9 * q^47 + 27 * q^49 + 58 * q^50 - 38 * q^52 + 29 * q^53 - 8 * q^55 + 65 * q^56 - 26 * q^58 + 13 * q^59 + 14 * q^61 + 47 * q^62 + 32 * q^64 + 29 * q^65 + 25 * q^67 + 45 * q^68 - 44 * q^70 + 9 * q^71 + 4 * q^73 - 8 * q^74 + 58 * q^76 + 10 * q^77 + 51 * q^79 + 7 * q^80 - 21 * q^82 + 2 * q^83 + 4 * q^85 - 10 * q^86 - 18 * q^88 + 35 * q^89 + 65 * q^91 - 69 * q^92 + 24 * q^94 - 18 * q^95 - 14 * q^97 - 24 * q^98

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 $

$2$	−0.282877	−0.200024	−0.100012	−	0.994986i	$-0.531888\pi$
$2$	−0.282877	−0.200024	−0.100012	+	0.994986i	$0.531888\pi$
$3$	0	0
$3$	0	0
$4$	−1.91998	−0.959990
$5$	−1.97313	−0.882412	−0.441206	−	0.897406i	$-0.645449\pi$
$5$	−1.97313	−0.882412	−0.441206	+	0.897406i	$0.645449\pi$
$6$	0	0
$7$	2.44119	0.922684	0.461342	−	0.887222i	$-0.347368\pi$
$7$	2.44119	0.922684	0.461342	+	0.887222i	$0.347368\pi$
$8$	1.10887	0.392045
$9$	0	0
$10$	0.558153	0.176503
$11$	−3.43591	−1.03597	−0.517983	−	0.855391i	$-0.673317\pi$
$11$	−3.43591	−1.03597	−0.517983	+	0.855391i	$0.673317\pi$
$12$	0	0
$13$	4.69311	1.30163	0.650817	−	0.759235i	$-0.274427\pi$
$13$	4.69311	1.30163	0.650817	+	0.759235i	$0.274427\pi$
$14$	−0.690556	−0.184559
$15$	0	0
$16$	3.52629	0.881572
$17$	−1.90195	−0.461291	−0.230645	−	0.973038i	$-0.574084\pi$
$17$	−1.90195	−0.461291	−0.230645	+	0.973038i	$0.574084\pi$
$18$	0	0
$19$	−1.23155	−0.282537	−0.141269	−	0.989971i	$-0.545118\pi$
$19$	−1.23155	−0.282537	−0.141269	+	0.989971i	$0.545118\pi$
$20$	3.78838	0.847107
$21$	0	0
$22$	0.971939	0.207218
$23$	−6.50000	−1.35534	−0.677672	−	0.735365i	$-0.737011\pi$
$23$	−6.50000	−1.35534	−0.677672	+	0.735365i	$0.737011\pi$
$24$	0	0
$25$	−1.10675	−0.221349
$26$	−1.32757	−0.260358
$27$	0	0
$28$	−4.68704	−0.885768
$29$	5.54253	1.02922	0.514611	−	0.857424i	$-0.327936\pi$
$29$	5.54253	1.02922	0.514611	+	0.857424i	$0.327936\pi$
$30$	0	0
$31$	4.88221	0.876871	0.438436	−	0.898763i	$-0.355533\pi$
$31$	4.88221	0.876871	0.438436	+	0.898763i	$0.355533\pi$
$32$	−3.21525	−0.568380
$33$	0	0
$34$	0.538017	0.0922692
$35$	−4.81680	−0.814187
$36$	0	0
$37$	−10.2236	−1.68075	−0.840377	−	0.542003i	$-0.817666\pi$
$37$	−10.2236	−1.68075	−0.840377	+	0.542003i	$0.817666\pi$
$38$	0.348377	0.0565142
$39$	0	0
$40$	−2.18795	−0.345945
$41$	5.60633	0.875561	0.437780	−	0.899082i	$-0.355765\pi$
$41$	5.60633	0.875561	0.437780	+	0.899082i	$0.355765\pi$
$42$	0	0
$43$	−2.85084	−0.434749	−0.217374	−	0.976088i	$-0.569749\pi$
$43$	−2.85084	−0.434749	−0.217374	+	0.976088i	$0.569749\pi$
$44$	6.59688	0.994518
$45$	0	0
$46$	1.83870	0.271101
$47$	1.80999	0.264014	0.132007	−	0.991249i	$-0.457858\pi$
$47$	1.80999	0.264014	0.132007	+	0.991249i	$0.457858\pi$
$48$	0	0
$49$	−1.04058	−0.148654
$50$	0.313073	0.0442752
$51$	0	0
$52$	−9.01068	−1.24956
$53$	−2.04207	−0.280500	−0.140250	−	0.990116i	$-0.544791\pi$
$53$	−2.04207	−0.280500	−0.140250	+	0.990116i	$0.544791\pi$
$54$	0	0
$55$	6.77951	0.914149
$56$	2.70697	0.361734
$57$	0	0
$58$	−1.56785	−0.205869
$59$	1.02290	0.133170	0.0665850	−	0.997781i	$-0.478790\pi$
$59$	1.02290	0.133170	0.0665850	+	0.997781i	$0.478790\pi$
$60$	0	0
$61$	12.6403	1.61843	0.809213	−	0.587515i	$-0.199894\pi$
$61$	12.6403	1.61843	0.809213	+	0.587515i	$0.199894\pi$
$62$	−1.38106	−0.175395
$63$	0	0
$64$	−6.14306	−0.767882
$65$	−9.26012	−1.14858
$66$	0	0
$67$	1.65225	0.201854	0.100927	−	0.994894i	$-0.467819\pi$
$67$	1.65225	0.201854	0.100927	+	0.994894i	$0.467819\pi$
$68$	3.65171	0.442835
$69$	0	0
$70$	1.36256	0.162857
$71$	4.59265	0.545047	0.272523	−	0.962149i	$-0.412142\pi$
$71$	4.59265	0.545047	0.272523	+	0.962149i	$0.412142\pi$
$72$	0	0
$73$	5.17859	0.606108	0.303054	−	0.952973i	$-0.401994\pi$
$73$	5.17859	0.606108	0.303054	+	0.952973i	$0.401994\pi$
$74$	2.89202	0.336191
$75$	0	0
$76$	2.36455	0.271233
$77$	−8.38772	−0.955870
$78$	0	0
$79$	13.5494	1.52443	0.762213	−	0.647326i	$-0.224113\pi$
$79$	13.5494	1.52443	0.762213	+	0.647326i	$0.224113\pi$
$80$	−6.95784	−0.777910
$81$	0	0
$82$	−1.58590	−0.175133
$83$	13.9606	1.53237	0.766187	−	0.642618i	$-0.222152\pi$
$83$	13.9606	1.53237	0.766187	+	0.642618i	$0.222152\pi$
$84$	0	0
$85$	3.75280	0.407049
$86$	0.806435	0.0869601
$87$	0	0
$88$	−3.80998	−0.406145
$89$	9.64059	1.02190	0.510950	−	0.859610i	$-0.329294\pi$
$89$	9.64059	1.02190	0.510950	+	0.859610i	$0.329294\pi$
$90$	0	0
$91$	11.4568	1.20100
$92$	12.4799	1.30112
$93$	0	0
$94$	−0.512003	−0.0528090
$95$	2.43001	0.249314
$96$	0	0
$97$	−16.2488	−1.64982	−0.824908	−	0.565267i	$-0.808773\pi$
$97$	−16.2488	−1.64982	−0.824908	+	0.565267i	$0.808773\pi$
$98$	0.294355	0.0297344
$99$	0	0
$100$	2.12493	0.212493
$101$	−15.4748	−1.53980	−0.769900	−	0.638165i	$-0.779694\pi$
$101$	−15.4748	−1.53980	−0.769900	+	0.638165i	$0.779694\pi$
$102$	0	0
$103$	11.6936	1.15220	0.576101	−	0.817379i	$-0.304574\pi$
$103$	11.6936	1.15220	0.576101	+	0.817379i	$0.304574\pi$
$104$	5.20405	0.510299
$105$	0	0
$106$	0.577654	0.0561067
$107$	−0.0867365	−0.00838514	−0.00419257	−	0.999991i	$-0.501335\pi$
$107$	−0.0867365	−0.00838514	−0.00419257	+	0.999991i	$0.501335\pi$
$108$	0	0
$109$	3.18524	0.305091	0.152545	−	0.988296i	$-0.451253\pi$
$109$	3.18524	0.305091	0.152545	+	0.988296i	$0.451253\pi$
$110$	−1.91776	−0.182852
$111$	0	0
$112$	8.60835	0.813413
$113$	2.15866	0.203070	0.101535	−	0.994832i	$-0.467625\pi$
$113$	2.15866	0.203070	0.101535	+	0.994832i	$0.467625\pi$
$114$	0	0
$115$	12.8254	1.19597
$116$	−10.6416	−0.988043
$117$	0	0
$118$	−0.289354	−0.0266372
$119$	−4.64303	−0.425626
$120$	0	0
$121$	0.805487	0.0732261
$122$	−3.57565	−0.323724
$123$	0	0
$124$	−9.37375	−0.841788
$125$	12.0494	1.07773
$126$	0	0
$127$	3.13665	0.278333	0.139166	−	0.990269i	$-0.455558\pi$
$127$	3.13665	0.278333	0.139166	+	0.990269i	$0.455558\pi$
$128$	8.16822	0.721975
$129$	0	0
$130$	2.61947	0.229743
$131$	19.1069	1.66938	0.834689	−	0.550721i	$-0.185647\pi$
$131$	19.1069	1.66938	0.834689	+	0.550721i	$0.185647\pi$
$132$	0	0
$133$	−3.00645	−0.260693
$134$	−0.467383	−0.0403757
$135$	0	0
$136$	−2.10902	−0.180847
$137$	7.77105	0.663925	0.331963	−	0.943293i	$-0.392289\pi$
$137$	7.77105	0.663925	0.331963	+	0.943293i	$0.392289\pi$
$138$	0	0
$139$	7.50884	0.636891	0.318446	−	0.947941i	$-0.396839\pi$
$139$	7.50884	0.636891	0.318446	+	0.947941i	$0.396839\pi$
$140$	9.24816	0.781612
$141$	0	0
$142$	−1.29915	−0.109022
$143$	−16.1251	−1.34845
$144$	0	0
$145$	−10.9361	−0.908198
$146$	−1.46490	−0.121236
$147$	0	0
$148$	19.6292	1.61351
$149$	21.5548	1.76584	0.882921	−	0.469522i	$-0.155574\pi$
$149$	21.5548	1.76584	0.882921	+	0.469522i	$0.155574\pi$
$150$	0	0
$151$	−8.16873	−0.664762	−0.332381	−	0.943145i	$-0.607852\pi$
$151$	−8.16873	−0.664762	−0.332381	+	0.943145i	$0.607852\pi$
$152$	−1.36563	−0.110767
$153$	0	0
$154$	2.37269	0.191197
$155$	−9.63325	−0.773761
$156$	0	0
$157$	19.5185	1.55774	0.778872	−	0.627183i	$-0.215792\pi$
$157$	19.5185	1.55774	0.778872	+	0.627183i	$0.215792\pi$
$158$	−3.83281	−0.304922
$159$	0	0
$160$	6.34411	0.501546
$161$	−15.8677	−1.25055
$162$	0	0
$163$	5.85613	0.458687	0.229344	−	0.973345i	$-0.426342\pi$
$163$	5.85613	0.458687	0.229344	+	0.973345i	$0.426342\pi$
$164$	−10.7640	−0.840530
$165$	0	0
$166$	−3.94912	−0.306511
$167$	16.1245	1.24775	0.623876	−	0.781523i	$-0.285557\pi$
$167$	16.1245	1.24775	0.623876	+	0.781523i	$0.285557\pi$
$168$	0	0
$169$	9.02526	0.694251
$170$	−1.06158	−0.0814194
$171$	0	0
$172$	5.47355	0.417354
$173$	13.5232	1.02815	0.514074	−	0.857746i	$-0.328136\pi$
$173$	13.5232	1.02815	0.514074	+	0.857746i	$0.328136\pi$
$174$	0	0
$175$	−2.70178	−0.204236
$176$	−12.1160	−0.913279
$177$	0	0
$178$	−2.72710	−0.204405
$179$	−22.1712	−1.65715	−0.828577	−	0.559875i	$-0.810849\pi$
$179$	−22.1712	−1.65715	−0.828577	+	0.559875i	$0.810849\pi$
$180$	0	0
$181$	3.61117	0.268416	0.134208	−	0.990953i	$-0.457151\pi$
$181$	3.61117	0.268416	0.134208	+	0.990953i	$0.457151\pi$
$182$	−3.24085	−0.240228
$183$	0	0
$184$	−7.20766	−0.531356
$185$	20.1726	1.48312
$186$	0	0
$187$	6.53494	0.477882
$188$	−3.47514	−0.253451
$189$	0	0
$190$	−0.687394	−0.0498688
$191$	7.41632	0.536626	0.268313	−	0.963332i	$-0.413534\pi$
$191$	7.41632	0.536626	0.268313	+	0.963332i	$0.413534\pi$
$192$	0	0
$193$	−23.0099	−1.65629	−0.828145	−	0.560514i	$-0.810604\pi$
$193$	−23.0099	−1.65629	−0.828145	+	0.560514i	$0.810604\pi$
$194$	4.59640	0.330003
$195$	0	0
$196$	1.99789	0.142707
$197$	−0.810204	−0.0577247	−0.0288623	−	0.999583i	$-0.509188\pi$
$197$	−0.810204	−0.0577247	−0.0288623	+	0.999583i	$0.509188\pi$
$198$	0	0
$199$	16.0676	1.13900	0.569500	−	0.821991i	$-0.307137\pi$
$199$	16.0676	1.13900	0.569500	+	0.821991i	$0.307137\pi$
$200$	−1.22724	−0.0867789
$201$	0	0
$202$	4.37746	0.307997
$203$	13.5304	0.949647
$204$	0	0
$205$	−11.0620	−0.772605
$206$	−3.30784	−0.230468
$207$	0	0
$208$	16.5493	1.14748
$209$	4.23150	0.292699
$210$	0	0
$211$	−9.98098	−0.687119	−0.343559	−	0.939131i	$-0.611633\pi$
$211$	−9.98098	−0.687119	−0.343559	+	0.939131i	$0.611633\pi$
$212$	3.92074	0.269278
$213$	0	0
$214$	0.0245357	0.00167723
$215$	5.62508	0.383627
$216$	0	0
$217$	11.9184	0.809075
$218$	−0.901030	−0.0610255
$219$	0	0
$220$	−13.0165	−0.877574
$221$	−8.92606	−0.600432
$222$	0	0
$223$	−19.6088	−1.31310	−0.656550	−	0.754283i	$-0.727985\pi$
$223$	−19.6088	−1.31310	−0.656550	+	0.754283i	$0.727985\pi$
$224$	−7.84903	−0.524436
$225$	0	0
$226$	−0.610634	−0.0406188
$227$	11.4505	0.759998	0.379999	−	0.924987i	$-0.375924\pi$
$227$	11.4505	0.759998	0.379999	+	0.924987i	$0.375924\pi$
$228$	0	0
$229$	22.6433	1.49631	0.748156	−	0.663523i	$-0.230939\pi$
$229$	22.6433	1.49631	0.748156	+	0.663523i	$0.230939\pi$
$230$	−3.62799	−0.239223
$231$	0	0
$232$	6.14595	0.403501
$233$	15.8879	1.04085	0.520427	−	0.853906i	$-0.325773\pi$
$233$	15.8879	1.04085	0.520427	+	0.853906i	$0.325773\pi$
$234$	0	0
$235$	−3.57134	−0.232969
$236$	−1.96395	−0.127842
$237$	0	0
$238$	1.31340	0.0851353
$239$	−7.35894	−0.476010	−0.238005	−	0.971264i	$-0.576493\pi$
$239$	−7.35894	−0.476010	−0.238005	+	0.971264i	$0.576493\pi$
$240$	0	0
$241$	−13.1161	−0.844882	−0.422441	−	0.906390i	$-0.638827\pi$
$241$	−13.1161	−0.844882	−0.422441	+	0.906390i	$0.638827\pi$
$242$	−0.227853	−0.0146470
$243$	0	0
$244$	−24.2692	−1.55367
$245$	2.05320	0.131174
$246$	0	0
$247$	−5.77980	−0.367760
$248$	5.41374	0.343773
$249$	0	0
$250$	−3.40850	−0.215572
$251$	0.274032	0.0172967	0.00864837	−	0.999963i	$-0.497247\pi$
$251$	0.274032	0.0172967	0.00864837	+	0.999963i	$0.497247\pi$
$252$	0	0
$253$	22.3334	1.40409
$254$	−0.887284	−0.0556732
$255$	0	0
$256$	9.97552	0.623470
$257$	5.65777	0.352922	0.176461	−	0.984308i	$-0.443535\pi$
$257$	5.65777	0.352922	0.176461	+	0.984308i	$0.443535\pi$
$258$	0	0
$259$	−24.9578	−1.55080
$260$	17.7793	1.10262
$261$	0	0
$262$	−5.40490	−0.333916
$263$	−18.5754	−1.14541	−0.572703	−	0.819763i	$-0.694105\pi$
$263$	−18.5754	−1.14541	−0.572703	+	0.819763i	$0.694105\pi$
$264$	0	0
$265$	4.02928	0.247517
$266$	0.850455	0.0521447
$267$	0	0
$268$	−3.17229	−0.193778
$269$	−1.00000	−0.0609711
$269$	−1.00000	−0.0609711
$270$	0	0
$271$	−5.19643	−0.315661	−0.157830	−	0.987466i	$-0.550450\pi$
$271$	−5.19643	−0.315661	−0.157830	+	0.987466i	$0.550450\pi$
$272$	−6.70683	−0.406661
$273$	0	0
$274$	−2.19825	−0.132801
$275$	3.80269	0.229311
$276$	0	0
$277$	6.13328	0.368513	0.184257	−	0.982878i	$-0.441012\pi$
$277$	6.13328	0.368513	0.184257	+	0.982878i	$0.441012\pi$
$278$	−2.12407	−0.127393
$279$	0	0
$280$	−5.34120	−0.319198
$281$	−19.4792	−1.16203	−0.581017	−	0.813891i	$-0.697345\pi$
$281$	−19.4792	−1.16203	−0.581017	+	0.813891i	$0.697345\pi$
$282$	0	0
$283$	1.43510	0.0853076	0.0426538	−	0.999090i	$-0.486419\pi$
$283$	1.43510	0.0853076	0.0426538	+	0.999090i	$0.486419\pi$
$284$	−8.81780	−0.523240
$285$	0	0
$286$	4.56141	0.269722
$287$	13.6861	0.807866
$288$	0	0
$289$	−13.3826	−0.787211
$290$	3.09358	0.181661
$291$	0	0
$292$	−9.94279	−0.581858
$293$	29.8473	1.74370	0.871848	−	0.489776i	$-0.162922\pi$
$293$	29.8473	1.74370	0.871848	+	0.489776i	$0.162922\pi$
$294$	0	0
$295$	−2.01832	−0.117511
$296$	−11.3367	−0.658931
$297$	0	0
$298$	−6.09736	−0.353210
$299$	−30.5052	−1.76416
$300$	0	0
$301$	−6.95944	−0.401136
$302$	2.31074	0.132968
$303$	0	0
$304$	−4.34280	−0.249077
$305$	−24.9410	−1.42812
$306$	0	0
$307$	−14.6428	−0.835710	−0.417855	−	0.908514i	$-0.637218\pi$
$307$	−14.6428	−0.835710	−0.417855	+	0.908514i	$0.637218\pi$
$308$	16.1043	0.917626
$309$	0	0
$310$	2.72502	0.154771
$311$	14.9926	0.850155	0.425077	−	0.905157i	$-0.360247\pi$
$311$	14.9926	0.850155	0.425077	+	0.905157i	$0.360247\pi$
$312$	0	0
$313$	−14.3883	−0.813272	−0.406636	−	0.913590i	$-0.633298\pi$
$313$	−14.3883	−0.813272	−0.406636	+	0.913590i	$0.633298\pi$
$314$	−5.52132	−0.311586
$315$	0	0
$316$	−26.0146	−1.46344
$317$	−18.4565	−1.03662	−0.518309	−	0.855193i	$-0.673438\pi$
$317$	−18.4565	−1.03662	−0.518309	+	0.855193i	$0.673438\pi$
$318$	0	0
$319$	−19.0436	−1.06624
$320$	12.1211	0.677589
$321$	0	0
$322$	4.48861	0.250141
$323$	2.34235	0.130332
$324$	0	0
$325$	−5.19408	−0.288116
$326$	−1.65656	−0.0917484
$327$	0	0
$328$	6.21669	0.343259
$329$	4.41853	0.243601
$330$	0	0
$331$	29.1435	1.60187	0.800935	−	0.598752i	$-0.204336\pi$
$331$	29.1435	1.60187	0.800935	+	0.598752i	$0.204336\pi$
$332$	−26.8041	−1.47106
$333$	0	0
$334$	−4.56125	−0.249580
$335$	−3.26011	−0.178119
$336$	0	0
$337$	−13.3401	−0.726682	−0.363341	−	0.931656i	$-0.618364\pi$
$337$	−13.3401	−0.726682	−0.363341	+	0.931656i	$0.618364\pi$
$338$	−2.55303	−0.138867
$339$	0	0
$340$	−7.20531	−0.390763
$341$	−16.7748	−0.908409
$342$	0	0
$343$	−19.6286	−1.05984
$344$	−3.16121	−0.170441
$345$	0	0
$346$	−3.82539	−0.205654
$347$	−15.4841	−0.831232	−0.415616	−	0.909540i	$-0.636434\pi$
$347$	−15.4841	−0.831232	−0.415616	+	0.909540i	$0.636434\pi$
$348$	0	0
$349$	4.95774	0.265382	0.132691	−	0.991157i	$-0.457638\pi$
$349$	4.95774	0.265382	0.132691	+	0.991157i	$0.457638\pi$
$350$	0.764271	0.0408520
$351$	0	0
$352$	11.0473	0.588823
$353$	19.7221	1.04970	0.524851	−	0.851194i	$-0.324121\pi$
$353$	19.7221	1.04970	0.524851	+	0.851194i	$0.324121\pi$
$354$	0	0
$355$	−9.06190	−0.480956
$356$	−18.5098	−0.981015
$357$	0	0
$358$	6.27172	0.331471
$359$	−10.0142	−0.528527	−0.264263	−	0.964451i	$-0.585129\pi$
$359$	−10.0142	−0.528527	−0.264263	+	0.964451i	$0.585129\pi$
$360$	0	0
$361$	−17.4833	−0.920173
$362$	−1.02152	−0.0536897
$363$	0	0
$364$	−21.9968	−1.15295
$365$	−10.2180	−0.534837
$366$	0	0
$367$	−1.28284	−0.0669637	−0.0334819	−	0.999439i	$-0.510660\pi$
$367$	−1.28284	−0.0669637	−0.0334819	+	0.999439i	$0.510660\pi$
$368$	−22.9209	−1.19483
$369$	0	0
$370$	−5.70634	−0.296659
$371$	−4.98509	−0.258813
$372$	0	0
$373$	33.4573	1.73235	0.866176	−	0.499739i	$-0.166571\pi$
$373$	33.4573	1.73235	0.866176	+	0.499739i	$0.166571\pi$
$374$	−1.84858	−0.0955878
$375$	0	0
$376$	2.00704	0.103505
$377$	26.0117	1.33967
$378$	0	0
$379$	32.2690	1.65755	0.828775	−	0.559583i	$-0.189039\pi$
$379$	32.2690	1.65755	0.828775	+	0.559583i	$0.189039\pi$
$380$	−4.66558	−0.239339
$381$	0	0
$382$	−2.09790	−0.107338
$383$	13.8572	0.708071	0.354036	−	0.935232i	$-0.384809\pi$
$383$	13.8572	0.708071	0.354036	+	0.935232i	$0.384809\pi$
$384$	0	0
$385$	16.5501	0.843471
$386$	6.50897	0.331298
$387$	0	0
$388$	31.1974	1.58381
$389$	−29.3155	−1.48636	−0.743178	−	0.669094i	$-0.766683\pi$
$389$	−29.3155	−1.48636	−0.743178	+	0.669094i	$0.766683\pi$
$390$	0	0
$391$	12.3627	0.625208
$392$	−1.15387	−0.0582791
$393$	0	0
$394$	0.229188	0.0115463
$395$	−26.7348	−1.34517
$396$	0	0
$397$	−4.33222	−0.217428	−0.108714	−	0.994073i	$-0.534673\pi$
$397$	−4.33222	−0.217428	−0.108714	+	0.994073i	$0.534673\pi$
$398$	−4.54514	−0.227827
$399$	0	0
$400$	−3.90271	−0.195135
$401$	−11.9708	−0.597794	−0.298897	−	0.954285i	$-0.596619\pi$
$401$	−11.9708	−0.597794	−0.298897	+	0.954285i	$0.596619\pi$
$402$	0	0
$403$	22.9127	1.14137
$404$	29.7113	1.47819
$405$	0	0
$406$	−3.82743	−0.189952
$407$	35.1275	1.74120
$408$	0	0
$409$	10.5969	0.523985	0.261992	−	0.965070i	$-0.415620\pi$
$409$	10.5969	0.523985	0.261992	+	0.965070i	$0.415620\pi$
$410$	3.12919	0.154540
$411$	0	0
$412$	−22.4514	−1.10610
$413$	2.49709	0.122874
$414$	0	0
$415$	−27.5461	−1.35218
$416$	−15.0895	−0.739823
$417$	0	0
$418$	−1.19699	−0.0585468
$419$	−0.320387	−0.0156519	−0.00782595	−	0.999969i	$-0.502491\pi$
$419$	−0.320387	−0.0156519	−0.00782595	+	0.999969i	$0.502491\pi$
$420$	0	0
$421$	24.0501	1.17213	0.586064	−	0.810265i	$-0.300677\pi$
$421$	24.0501	1.17213	0.586064	+	0.810265i	$0.300677\pi$
$422$	2.82338	0.137440
$423$	0	0
$424$	−2.26439	−0.109969
$425$	2.10498	0.102106
$426$	0	0
$427$	30.8574	1.49330
$428$	0.166532	0.00804965
$429$	0	0
$430$	−1.59120	−0.0767346
$431$	22.7364	1.09518	0.547588	−	0.836748i	$-0.315546\pi$
$431$	22.7364	1.09518	0.547588	+	0.836748i	$0.315546\pi$
$432$	0	0
$433$	−26.8292	−1.28933	−0.644663	−	0.764467i	$-0.723002\pi$
$433$	−26.8292	−1.28933	−0.644663	+	0.764467i	$0.723002\pi$
$434$	−3.37144	−0.161834
$435$	0	0
$436$	−6.11561	−0.292884
$437$	8.00508	0.382935
$438$	0	0
$439$	26.3083	1.25562	0.627812	−	0.778365i	$-0.283951\pi$
$439$	26.3083	1.25562	0.627812	+	0.778365i	$0.283951\pi$
$440$	7.51760	0.358387
$441$	0	0
$442$	2.52497	0.120101
$443$	−6.31703	−0.300131	−0.150066	−	0.988676i	$-0.547949\pi$
$443$	−6.31703	−0.300131	−0.150066	+	0.988676i	$0.547949\pi$
$444$	0	0
$445$	−19.0222	−0.901737
$446$	5.54686	0.262651
$447$	0	0
$448$	−14.9964	−0.708513
$449$	−23.1873	−1.09428	−0.547138	−	0.837042i	$-0.684283\pi$
$449$	−23.1873	−1.09428	−0.547138	+	0.837042i	$0.684283\pi$
$450$	0	0
$451$	−19.2628	−0.907052
$452$	−4.14459	−0.194945
$453$	0	0
$454$	−3.23908	−0.152018
$455$	−22.6057	−1.05977
$456$	0	0
$457$	28.8658	1.35029	0.675144	−	0.737686i	$-0.264082\pi$
$457$	28.8658	1.35029	0.675144	+	0.737686i	$0.264082\pi$
$458$	−6.40526	−0.299298
$459$	0	0
$460$	−24.6244	−1.14812
$461$	9.40857	0.438201	0.219100	−	0.975702i	$-0.429688\pi$
$461$	9.40857	0.438201	0.219100	+	0.975702i	$0.429688\pi$
$462$	0	0
$463$	38.4998	1.78924	0.894618	−	0.446831i	$-0.147447\pi$
$463$	38.4998	1.78924	0.894618	+	0.446831i	$0.147447\pi$
$464$	19.5446	0.907333
$465$	0	0
$466$	−4.49432	−0.208196
$467$	−29.2898	−1.35537	−0.677686	−	0.735352i	$-0.737017\pi$
$467$	−29.2898	−1.35537	−0.677686	+	0.735352i	$0.737017\pi$
$468$	0	0
$469$	4.03346	0.186248
$470$	1.01025	0.0465993
$471$	0	0
$472$	1.13426	0.0522087
$473$	9.79522	0.450385
$474$	0	0
$475$	1.36302	0.0625394
$476$	8.91453	0.408597
$477$	0	0
$478$	2.08167	0.0952134
$479$	14.2259	0.650000	0.325000	−	0.945714i	$-0.394636\pi$
$479$	14.2259	0.650000	0.325000	+	0.945714i	$0.394636\pi$
$480$	0	0
$481$	−47.9806	−2.18773
$482$	3.71023	0.168997
$483$	0	0
$484$	−1.54652	−0.0702964
$485$	32.0610	1.45582
$486$	0	0
$487$	−34.8431	−1.57889	−0.789446	−	0.613820i	$-0.789632\pi$
$487$	−34.8431	−1.57889	−0.789446	+	0.613820i	$0.789632\pi$
$488$	14.0165	0.634496
$489$	0	0
$490$	−0.580802	−0.0262380
$491$	29.5777	1.33482	0.667411	−	0.744689i	$-0.267402\pi$
$491$	29.5777	1.33482	0.667411	+	0.744689i	$0.267402\pi$
$492$	0	0
$493$	−10.5416	−0.474771
$494$	1.63497	0.0735608
$495$	0	0
$496$	17.2161	0.773025
$497$	11.2115	0.502906
$498$	0	0
$499$	−22.3770	−1.00173	−0.500866	−	0.865525i	$-0.666985\pi$
$499$	−22.3770	−1.00173	−0.500866	+	0.865525i	$0.666985\pi$
$500$	−23.1347	−1.03461
$501$	0	0
$502$	−0.0775172	−0.00345976
$503$	−1.82300	−0.0812836	−0.0406418	−	0.999174i	$-0.512940\pi$
$503$	−1.82300	−0.0812836	−0.0406418	+	0.999174i	$0.512940\pi$
$504$	0	0
$505$	30.5338	1.35874
$506$	−6.31760	−0.280852
$507$	0	0
$508$	−6.02231	−0.267197
$509$	2.76277	0.122458	0.0612288	−	0.998124i	$-0.480498\pi$
$509$	2.76277	0.122458	0.0612288	+	0.998124i	$0.480498\pi$
$510$	0	0
$511$	12.6419	0.559246
$512$	−19.1583	−0.846684
$513$	0	0
$514$	−1.60045	−0.0705928
$515$	−23.0730	−1.01672
$516$	0	0
$517$	−6.21895	−0.273509
$518$	7.05998	0.310198
$519$	0	0
$520$	−10.2683	−0.450294
$521$	−14.7532	−0.646352	−0.323176	−	0.946339i	$-0.604751\pi$
$521$	−14.7532	−0.646352	−0.323176	+	0.946339i	$0.604751\pi$
$522$	0	0
$523$	3.00571	0.131431	0.0657154	−	0.997838i	$-0.479067\pi$
$523$	3.00571	0.131431	0.0657154	+	0.997838i	$0.479067\pi$
$524$	−36.6849	−1.60259
$525$	0	0
$526$	5.25454	0.229109
$527$	−9.28573	−0.404493
$528$	0	0
$529$	19.2500	0.836956
$530$	−1.13979	−0.0495093
$531$	0	0
$532$	5.77233	0.250262
$533$	26.3111	1.13966
$534$	0	0
$535$	0.171143	0.00739914
$536$	1.83213	0.0791360
$537$	0	0
$538$	0.282877	0.0121957
$539$	3.57534	0.154001
$540$	0	0
$541$	22.5866	0.971076	0.485538	−	0.874216i	$-0.338624\pi$
$541$	22.5866	0.971076	0.485538	+	0.874216i	$0.338624\pi$
$542$	1.46995	0.0631397
$543$	0	0
$544$	6.11524	0.262189
$545$	−6.28491	−0.269216
$546$	0	0
$547$	6.83246	0.292135	0.146067	−	0.989275i	$-0.453338\pi$
$547$	6.83246	0.292135	0.146067	+	0.989275i	$0.453338\pi$
$548$	−14.9203	−0.637362
$549$	0	0
$550$	−1.07569	−0.0458676
$551$	−6.82591	−0.290793
$552$	0	0
$553$	33.0767	1.40656
$554$	−1.73496	−0.0737114
$555$	0	0
$556$	−14.4168	−0.611409
$557$	−22.0266	−0.933298	−0.466649	−	0.884443i	$-0.654539\pi$
$557$	−22.0266	−0.933298	−0.466649	+	0.884443i	$0.654539\pi$
$558$	0	0
$559$	−13.3793	−0.565883
$560$	−16.9854	−0.717765
$561$	0	0
$562$	5.51022	0.232435
$563$	36.1440	1.52329	0.761645	−	0.647995i	$-0.224392\pi$
$563$	36.1440	1.52329	0.761645	+	0.647995i	$0.224392\pi$
$564$	0	0
$565$	−4.25932	−0.179191
$566$	−0.405955	−0.0170636
$567$	0	0
$568$	5.09265	0.213683
$569$	4.48775	0.188136	0.0940682	−	0.995566i	$-0.470013\pi$
$569$	4.48775	0.188136	0.0940682	+	0.995566i	$0.470013\pi$
$570$	0	0
$571$	−15.3385	−0.641896	−0.320948	−	0.947097i	$-0.604001\pi$
$571$	−15.3385	−0.641896	−0.320948	+	0.947097i	$0.604001\pi$
$572$	30.9599	1.29450
$573$	0	0
$574$	−3.87148	−0.161593
$575$	7.19386	0.300005
$576$	0	0
$577$	25.5418	1.06332	0.531659	−	0.846958i	$-0.321569\pi$
$577$	25.5418	1.06332	0.531659	+	0.846958i	$0.321569\pi$
$578$	3.78562	0.157461
$579$	0	0
$580$	20.9972	0.871861
$581$	34.0805	1.41390
$582$	0	0
$583$	7.01638	0.290589
$584$	5.74239	0.237622
$585$	0	0
$586$	−8.44309	−0.348781
$587$	−6.30137	−0.260085	−0.130043	−	0.991508i	$-0.541511\pi$
$587$	−6.30137	−0.260085	−0.130043	+	0.991508i	$0.541511\pi$
$588$	0	0
$589$	−6.01269	−0.247749
$590$	0.570934	0.0235050
$591$	0	0
$592$	−36.0514	−1.48171
$593$	11.9031	0.488801	0.244400	−	0.969674i	$-0.421409\pi$
$593$	11.9031	0.488801	0.244400	+	0.969674i	$0.421409\pi$
$594$	0	0
$595$	9.16131	0.375577
$596$	−41.3849	−1.69519
$597$	0	0
$598$	8.62920	0.352874
$599$	44.5084	1.81856	0.909282	−	0.416180i	$-0.136632\pi$
$599$	44.5084	1.81856	0.909282	+	0.416180i	$0.136632\pi$
$600$	0	0
$601$	18.3848	0.749930	0.374965	−	0.927039i	$-0.377655\pi$
$601$	18.3848	0.749930	0.374965	+	0.927039i	$0.377655\pi$
$602$	1.96866	0.0802367
$603$	0	0
$604$	15.6838	0.638165
$605$	−1.58933	−0.0646156
$606$	0	0
$607$	−21.1238	−0.857389	−0.428694	−	0.903450i	$-0.641026\pi$
$607$	−21.1238	−0.857389	−0.428694	+	0.903450i	$0.641026\pi$
$608$	3.95974	0.160589
$609$	0	0
$610$	7.05523	0.285658
$611$	8.49446	0.343649
$612$	0	0
$613$	2.99053	0.120786	0.0603931	−	0.998175i	$-0.480765\pi$
$613$	2.99053	0.120786	0.0603931	+	0.998175i	$0.480765\pi$
$614$	4.14211	0.167162
$615$	0	0
$616$	−9.30090	−0.374744
$617$	10.4767	0.421775	0.210888	−	0.977510i	$-0.432365\pi$
$617$	10.4767	0.421775	0.210888	+	0.977510i	$0.432365\pi$
$618$	0	0
$619$	−9.34147	−0.375465	−0.187733	−	0.982220i	$-0.560114\pi$
$619$	−9.34147	−0.375465	−0.187733	+	0.982220i	$0.560114\pi$
$620$	18.4957	0.742804
$621$	0	0
$622$	−4.24107	−0.170051
$623$	23.5345	0.942892
$624$	0	0
$625$	−18.2414	−0.729655
$626$	4.07010	0.162674
$627$	0	0
$628$	−37.4751	−1.49542
$629$	19.4448	0.775316
$630$	0	0
$631$	−21.8463	−0.869688	−0.434844	−	0.900506i	$-0.643196\pi$
$631$	−21.8463	−0.869688	−0.434844	+	0.900506i	$0.643196\pi$
$632$	15.0245	0.597644
$633$	0	0
$634$	5.22090	0.207348
$635$	−6.18902	−0.245604
$636$	0	0
$637$	−4.88355	−0.193493
$638$	5.38700	0.213273
$639$	0	0
$640$	−16.1170	−0.637079
$641$	−39.9747	−1.57891	−0.789453	−	0.613811i	$-0.789636\pi$
$641$	−39.9747	−1.57891	−0.789453	+	0.613811i	$0.789636\pi$
$642$	0	0
$643$	−15.1256	−0.596495	−0.298247	−	0.954489i	$-0.596402\pi$
$643$	−15.1256	−0.596495	−0.298247	+	0.954489i	$0.596402\pi$
$644$	30.4658	1.20052
$645$	0	0
$646$	−0.662596	−0.0260695
$647$	−43.0209	−1.69133	−0.845664	−	0.533716i	$-0.820795\pi$
$647$	−43.0209	−1.69133	−0.845664	+	0.533716i	$0.820795\pi$
$648$	0	0
$649$	−3.51459	−0.137960
$650$	1.46928	0.0576301
$651$	0	0
$652$	−11.2437	−0.440336
$653$	−33.2486	−1.30112	−0.650559	−	0.759455i	$-0.725466\pi$
$653$	−33.2486	−1.30112	−0.650559	+	0.759455i	$0.725466\pi$
$654$	0	0
$655$	−37.7005	−1.47308
$656$	19.7695	0.771870
$657$	0	0
$658$	−1.24990	−0.0487261
$659$	−42.2827	−1.64710	−0.823549	−	0.567245i	$-0.808009\pi$
$659$	−42.2827	−1.64710	−0.823549	+	0.567245i	$0.808009\pi$
$660$	0	0
$661$	−16.5817	−0.644955	−0.322477	−	0.946577i	$-0.604516\pi$
$661$	−16.5817	−0.644955	−0.322477	+	0.946577i	$0.604516\pi$
$662$	−8.24400	−0.320412
$663$	0	0
$664$	15.4805	0.600759
$665$	5.93213	0.230038
$666$	0	0
$667$	−36.0264	−1.39495
$668$	−30.9588	−1.19783
$669$	0	0
$670$	0.922208	0.0356280
$671$	−43.4310	−1.67663
$672$	0	0
$673$	12.5066	0.482096	0.241048	−	0.970513i	$-0.422509\pi$
$673$	12.5066	0.482096	0.241048	+	0.970513i	$0.422509\pi$
$674$	3.77361	0.145354
$675$	0	0
$676$	−17.3283	−0.666474
$677$	−41.3779	−1.59028	−0.795141	−	0.606424i	$-0.792603\pi$
$677$	−41.3779	−1.59028	−0.795141	+	0.606424i	$0.792603\pi$
$678$	0	0
$679$	−39.6664	−1.52226
$680$	4.16137	0.159581
$681$	0	0
$682$	4.74521	0.181703
$683$	12.2490	0.468694	0.234347	−	0.972153i	$-0.424705\pi$
$683$	12.2490	0.468694	0.234347	+	0.972153i	$0.424705\pi$
$684$	0	0
$685$	−15.3333	−0.585856
$686$	5.55247	0.211994
$687$	0	0
$688$	−10.0529	−0.383262
$689$	−9.58367	−0.365109
$690$	0	0
$691$	−29.4843	−1.12164	−0.560819	−	0.827939i	$-0.689514\pi$
$691$	−29.4843	−1.12164	−0.560819	+	0.827939i	$0.689514\pi$
$692$	−25.9643	−0.987013
$693$	0	0
$694$	4.38010	0.166266
$695$	−14.8159	−0.562000
$696$	0	0
$697$	−10.6630	−0.403888
$698$	−1.40243	−0.0530827
$699$	0	0
$700$	5.18737	0.196064
$701$	16.4542	0.621465	0.310733	−	0.950497i	$-0.399426\pi$
$701$	16.4542	0.621465	0.310733	+	0.950497i	$0.399426\pi$
$702$	0	0
$703$	12.5909	0.474875
$704$	21.1070	0.795500
$705$	0	0
$706$	−5.57892	−0.209966
$707$	−37.7770	−1.42075
$708$	0	0
$709$	−46.2347	−1.73638	−0.868190	−	0.496232i	$-0.834717\pi$
$709$	−46.2347	−1.73638	−0.868190	+	0.496232i	$0.834717\pi$
$710$	2.56340	0.0962027
$711$	0	0
$712$	10.6902	0.400631
$713$	−31.7344	−1.18846
$714$	0	0
$715$	31.8170	1.18989
$716$	42.5683	1.59085
$717$	0	0
$718$	2.83277	0.105718
$719$	23.8436	0.889218	0.444609	−	0.895725i	$-0.353343\pi$
$719$	23.8436	0.889218	0.444609	+	0.895725i	$0.353343\pi$
$720$	0	0
$721$	28.5462	1.06312
$722$	4.94561	0.184057
$723$	0	0
$724$	−6.93338	−0.257677
$725$	−6.13418	−0.227818
$726$	0	0
$727$	38.1659	1.41549	0.707747	−	0.706466i	$-0.249712\pi$
$727$	38.1659	1.41549	0.707747	+	0.706466i	$0.249712\pi$
$728$	12.7041	0.470845
$729$	0	0
$730$	2.89045	0.106980
$731$	5.42215	0.200546
$732$	0	0
$733$	−50.9077	−1.88032	−0.940159	−	0.340737i	$-0.889323\pi$
$733$	−50.9077	−1.88032	−0.940159	+	0.340737i	$0.889323\pi$
$734$	0.362885	0.0133943
$735$	0	0
$736$	20.8991	0.770351
$737$	−5.67698	−0.209114
$738$	0	0
$739$	48.7104	1.79184	0.895920	−	0.444216i	$-0.146518\pi$
$739$	48.7104	1.79184	0.895920	+	0.444216i	$0.146518\pi$
$740$	−38.7309	−1.42378
$741$	0	0
$742$	1.41017	0.0517688
$743$	−0.675629	−0.0247864	−0.0123932	−	0.999923i	$-0.503945\pi$
$743$	−0.675629	−0.0247864	−0.0123932	+	0.999923i	$0.503945\pi$
$744$	0	0
$745$	−42.5306	−1.55820
$746$	−9.46428	−0.346512
$747$	0	0
$748$	−12.5470	−0.458762
$749$	−0.211741	−0.00773683
$750$	0	0
$751$	20.4428	0.745967	0.372983	−	0.927838i	$-0.378335\pi$
$751$	20.4428	0.745967	0.372983	+	0.927838i	$0.378335\pi$
$752$	6.38253	0.232747
$753$	0	0
$754$	−7.35810	−0.267966
$755$	16.1180	0.586594
$756$	0	0
$757$	−21.5003	−0.781441	−0.390720	−	0.920509i	$-0.627774\pi$
$757$	−21.5003	−0.781441	−0.390720	+	0.920509i	$0.627774\pi$
$758$	−9.12815	−0.331549
$759$	0	0
$760$	2.69457	0.0977423
$761$	−29.7165	−1.07722	−0.538612	−	0.842554i	$-0.681051\pi$
$761$	−29.7165	−1.07722	−0.538612	+	0.842554i	$0.681051\pi$
$762$	0	0
$763$	7.77579	0.281503
$764$	−14.2392	−0.515156
$765$	0	0
$766$	−3.91988	−0.141631
$767$	4.80057	0.173339
$768$	0	0
$769$	1.14904	0.0414355	0.0207178	−	0.999785i	$-0.493405\pi$
$769$	1.14904	0.0414355	0.0207178	+	0.999785i	$0.493405\pi$
$770$	−4.68163	−0.168714
$771$	0	0
$772$	44.1786	1.59002
$773$	−3.89004	−0.139915	−0.0699576	−	0.997550i	$-0.522286\pi$
$773$	−3.89004	−0.139915	−0.0699576	+	0.997550i	$0.522286\pi$
$774$	0	0
$775$	−5.40337	−0.194095
$776$	−18.0178	−0.646802
$777$	0	0
$778$	8.29268	0.297307
$779$	−6.90448	−0.247379
$780$	0	0
$781$	−15.7799	−0.564650
$782$	−3.49711	−0.125056
$783$	0	0
$784$	−3.66938	−0.131049
$785$	−38.5126	−1.37457
$786$	0	0
$787$	19.9603	0.711508	0.355754	−	0.934580i	$-0.384224\pi$
$787$	19.9603	0.711508	0.355754	+	0.934580i	$0.384224\pi$
$788$	1.55558	0.0554151
$789$	0	0
$790$	7.56264	0.269067
$791$	5.26970	0.187369
$792$	0	0
$793$	59.3223	2.10660
$794$	1.22548	0.0434907
$795$	0	0
$796$	−30.8494	−1.09343
$797$	42.9686	1.52203	0.761014	−	0.648736i	$-0.224702\pi$
$797$	42.9686	1.52203	0.761014	+	0.648736i	$0.224702\pi$
$798$	0	0
$799$	−3.44251	−0.121787
$800$	3.55846	0.125811
$801$	0	0
$802$	3.38626	0.119573
$803$	−17.7932	−0.627908
$804$	0	0
$805$	31.3092	1.10350
$806$	−6.48148	−0.228300
$807$	0	0
$808$	−17.1595	−0.603671
$809$	−28.0766	−0.987119	−0.493560	−	0.869712i	$-0.664305\pi$
$809$	−28.0766	−0.987119	−0.493560	+	0.869712i	$0.664305\pi$
$810$	0	0
$811$	−1.23855	−0.0434915	−0.0217458	−	0.999764i	$-0.506922\pi$
$811$	−1.23855	−0.0434915	−0.0217458	+	0.999764i	$0.506922\pi$
$812$	−25.9781	−0.911652
$813$	0	0
$814$	−9.93673	−0.348282
$815$	−11.5549	−0.404751
$816$	0	0
$817$	3.51095	0.122833
$818$	−2.99762	−0.104809
$819$	0	0
$820$	21.2389	0.741694
$821$	−9.91227	−0.345941	−0.172970	−	0.984927i	$-0.555336\pi$
$821$	−9.91227	−0.345941	−0.172970	+	0.984927i	$0.555336\pi$
$822$	0	0
$823$	4.82339	0.168133	0.0840663	−	0.996460i	$-0.473209\pi$
$823$	4.82339	0.168133	0.0840663	+	0.996460i	$0.473209\pi$
$824$	12.9667	0.451715
$825$	0	0
$826$	−0.706369	−0.0245777
$827$	−24.6328	−0.856566	−0.428283	−	0.903645i	$-0.640881\pi$
$827$	−24.6328	−0.856566	−0.428283	+	0.903645i	$0.640881\pi$
$828$	0	0
$829$	14.6300	0.508122	0.254061	−	0.967188i	$-0.418234\pi$
$829$	14.6300	0.508122	0.254061	+	0.967188i	$0.418234\pi$
$830$	7.79215	0.270469
$831$	0	0
$832$	−28.8300	−0.999502
$833$	1.97913	0.0685728
$834$	0	0
$835$	−31.8158	−1.10103
$836$	−8.12440	−0.280988
$837$	0	0
$838$	0.0906298	0.00313076
$839$	42.5008	1.46729	0.733645	−	0.679533i	$-0.237818\pi$
$839$	42.5008	1.46729	0.733645	+	0.679533i	$0.237818\pi$
$840$	0	0
$841$	1.71964	0.0592979
$842$	−6.80320	−0.234454
$843$	0	0
$844$	19.1633	0.659627
$845$	−17.8080	−0.612615
$846$	0	0
$847$	1.96635	0.0675646
$848$	−7.20094	−0.247281
$849$	0	0
$850$	−0.595449	−0.0204237
$851$	66.4535	2.27800
$852$	0	0
$853$	27.8087	0.952153	0.476077	−	0.879404i	$-0.342058\pi$
$853$	27.8087	0.952153	0.476077	+	0.879404i	$0.342058\pi$
$854$	−8.72884	−0.298695
$855$	0	0
$856$	−0.0961796	−0.00328735
$857$	25.9123	0.885149	0.442574	−	0.896732i	$-0.354065\pi$
$857$	25.9123	0.885149	0.442574	+	0.896732i	$0.354065\pi$
$858$	0	0
$859$	53.1905	1.81484	0.907418	−	0.420229i	$-0.138050\pi$
$859$	53.1905	1.81484	0.907418	+	0.420229i	$0.138050\pi$
$860$	−10.8000	−0.368278
$861$	0	0
$862$	−6.43160	−0.219061
$863$	27.3604	0.931360	0.465680	−	0.884953i	$-0.345810\pi$
$863$	27.3604	0.931360	0.465680	+	0.884953i	$0.345810\pi$
$864$	0	0
$865$	−26.6830	−0.907251
$866$	7.58934	0.257896
$867$	0	0
$868$	−22.8831	−0.776704
$869$	−46.5545	−1.57925
$870$	0	0
$871$	7.75419	0.262741
$872$	3.53202	0.119609
$873$	0	0
$874$	−2.26445	−0.0765961
$875$	29.4150	0.994407
$876$	0	0
$877$	−58.1170	−1.96247	−0.981235	−	0.192813i	$-0.938239\pi$
$877$	−58.1170	−1.96247	−0.981235	+	0.192813i	$0.938239\pi$
$878$	−7.44199	−0.251155
$879$	0	0
$880$	23.9065	0.805888
$881$	46.7912	1.57644	0.788218	−	0.615396i	$-0.211004\pi$
$881$	46.7912	1.57644	0.788218	+	0.615396i	$0.211004\pi$
$882$	0	0
$883$	25.8997	0.871593	0.435797	−	0.900045i	$-0.356467\pi$
$883$	25.8997	0.871593	0.435797	+	0.900045i	$0.356467\pi$
$884$	17.1379	0.576409
$885$	0	0
$886$	1.78694	0.0600334
$887$	−47.8720	−1.60738	−0.803692	−	0.595046i	$-0.797134\pi$
$887$	−47.8720	−1.60738	−0.803692	+	0.595046i	$0.797134\pi$
$888$	0	0
$889$	7.65716	0.256813
$890$	5.38092	0.180369
$891$	0	0
$892$	37.6484	1.26056
$893$	−2.22909	−0.0745937
$894$	0	0
$895$	43.7468	1.46229
$896$	19.9402	0.666155
$897$	0	0
$898$	6.55915	0.218882
$899$	27.0598	0.902495
$900$	0	0
$901$	3.88392	0.129392
$902$	5.44900	0.181432
$903$	0	0
$904$	2.39367	0.0796124
$905$	−7.12532	−0.236854
$906$	0	0
$907$	−55.0189	−1.82687	−0.913437	−	0.406980i	$-0.866582\pi$
$907$	−55.0189	−1.82687	−0.913437	+	0.406980i	$0.866582\pi$
$908$	−21.9848	−0.729591
$909$	0	0
$910$	6.39463	0.211980
$911$	46.9311	1.55490	0.777448	−	0.628947i	$-0.216514\pi$
$911$	46.9311	1.55490	0.777448	+	0.628947i	$0.216514\pi$
$912$	0	0
$913$	−47.9674	−1.58749
$914$	−8.16547	−0.270090
$915$	0	0
$916$	−43.4747	−1.43645
$917$	46.6436	1.54031
$918$	0	0
$919$	−10.0327	−0.330948	−0.165474	−	0.986214i	$-0.552915\pi$
$919$	−10.0327	−0.330948	−0.165474	+	0.986214i	$0.552915\pi$
$920$	14.2217	0.468874
$921$	0	0
$922$	−2.66146	−0.0876506
$923$	21.5538	0.709452
$924$	0	0
$925$	11.3150	0.372034
$926$	−10.8907	−0.357890
$927$	0	0
$928$	−17.8206	−0.584990
$929$	24.5291	0.804773	0.402387	−	0.915470i	$-0.368181\pi$
$929$	24.5291	0.804773	0.402387	+	0.915470i	$0.368181\pi$
$930$	0	0
$931$	1.28153	0.0420003
$932$	−30.5045	−0.999209
$933$	0	0
$934$	8.28540	0.271107
$935$	−12.8943	−0.421689
$936$	0	0
$937$	8.89765	0.290673	0.145337	−	0.989382i	$-0.453573\pi$
$937$	8.89765	0.290673	0.145337	+	0.989382i	$0.453573\pi$
$938$	−1.14097	−0.0372540
$939$	0	0
$940$	6.85691	0.223648
$941$	36.1992	1.18006	0.590030	−	0.807381i	$-0.299116\pi$
$941$	36.1992	1.18006	0.590030	+	0.807381i	$0.299116\pi$
$942$	0	0
$943$	−36.4411	−1.18669
$944$	3.60704	0.117399
$945$	0	0
$946$	−2.77084	−0.0900877
$947$	−15.3584	−0.499081	−0.249541	−	0.968364i	$-0.580280\pi$
$947$	−15.3584	−0.499081	−0.249541	+	0.968364i	$0.580280\pi$
$948$	0	0
$949$	24.3037	0.788931
$950$	−0.385565	−0.0125094
$951$	0	0
$952$	−5.14852	−0.166864
$953$	5.78646	0.187442	0.0937209	−	0.995599i	$-0.470124\pi$
$953$	5.78646	0.187442	0.0937209	+	0.995599i	$0.470124\pi$
$954$	0	0
$955$	−14.6334	−0.473525
$956$	14.1290	0.456965
$957$	0	0
$958$	−4.02418	−0.130015
$959$	18.9706	0.612593
$960$	0	0
$961$	−7.16401	−0.231097
$962$	13.5726	0.437597
$963$	0	0
$964$	25.1826	0.811079
$965$	45.4016	1.46153
$966$	0	0
$967$	−27.8199	−0.894629	−0.447315	−	0.894377i	$-0.647620\pi$
$967$	−27.8199	−0.894629	−0.447315	+	0.894377i	$0.647620\pi$
$968$	0.893181	0.0287079
$969$	0	0
$970$	−9.06931	−0.291198
$971$	26.2776	0.843290	0.421645	−	0.906761i	$-0.361453\pi$
$971$	26.2776	0.843290	0.421645	+	0.906761i	$0.361453\pi$
$972$	0	0
$973$	18.3305	0.587649
$974$	9.85630	0.315816
$975$	0	0
$976$	44.5734	1.42676
$977$	38.8447	1.24275	0.621377	−	0.783512i	$-0.286574\pi$
$977$	38.8447	1.24275	0.621377	+	0.783512i	$0.286574\pi$
$978$	0	0
$979$	−33.1242	−1.05865
$980$	−3.94210	−0.125926
$981$	0	0
$982$	−8.36684	−0.266996
$983$	−15.9632	−0.509147	−0.254574	−	0.967053i	$-0.581935\pi$
$983$	−15.9632	−0.509147	−0.254574	+	0.967053i	$0.581935\pi$
$984$	0	0
$985$	1.59864	0.0509369
$986$	2.98198	0.0949655
$987$	0	0
$988$	11.0971	0.353046
$989$	18.5304	0.589234
$990$	0	0
$991$	14.6111	0.464138	0.232069	−	0.972699i	$-0.425450\pi$
$991$	14.6111	0.464138	0.232069	+	0.972699i	$0.425450\pi$
$992$	−15.6975	−0.498396
$993$	0	0
$994$	−3.17148	−0.100593
$995$	−31.7035	−1.00507
$996$	0	0
$997$	44.1438	1.39805	0.699024	−	0.715098i	$-0.253618\pi$
$997$	44.1438	1.39805	0.699024	+	0.715098i	$0.253618\pi$
$998$	6.32992	0.200370
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2421.2.a.i.1.8		16
3.2	odd	2		269.2.a.c.1.9	✓	16
12.11	even	2		4304.2.a.l.1.12		16
15.14	odd	2		6725.2.a.i.1.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
269.2.a.c.1.9	✓	16	3.2	odd	2
2421.2.a.i.1.8		16	1.1	even	1	trivial
4304.2.a.l.1.12		16	12.11	even	2
6725.2.a.i.1.8		16	15.14	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:	\( N \)	\(=\)	\( 2421 = 3^{2} \cdot 269 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2421.a (trivial)

\(f(q)\)	\(=\)	\(q-0.282877 q^{2} -1.91998 q^{4} -1.97313 q^{5} +2.44119 q^{7} +1.10887 q^{8} +0.558153 q^{10} -3.43591 q^{11} +4.69311 q^{13} -0.690556 q^{14} +3.52629 q^{16} -1.90195 q^{17} -1.23155 q^{19} +3.78838 q^{20} +0.971939 q^{22} -6.50000 q^{23} -1.10675 q^{25} -1.32757 q^{26} -4.68704 q^{28} +5.54253 q^{29} +4.88221 q^{31} -3.21525 q^{32} +0.538017 q^{34} -4.81680 q^{35} -10.2236 q^{37} +0.348377 q^{38} -2.18795 q^{40} +5.60633 q^{41} -2.85084 q^{43} +6.59688 q^{44} +1.83870 q^{46} +1.80999 q^{47} -1.04058 q^{49} +0.313073 q^{50} -9.01068 q^{52} -2.04207 q^{53} +6.77951 q^{55} +2.70697 q^{56} -1.56785 q^{58} +1.02290 q^{59} +12.6403 q^{61} -1.38106 q^{62} -6.14306 q^{64} -9.26012 q^{65} +1.65225 q^{67} +3.65171 q^{68} +1.36256 q^{70} +4.59265 q^{71} +5.17859 q^{73} +2.89202 q^{74} +2.36455 q^{76} -8.38772 q^{77} +13.5494 q^{79} -6.95784 q^{80} -1.58590 q^{82} +13.9606 q^{83} +3.75280 q^{85} +0.806435 q^{86} -3.80998 q^{88} +9.64059 q^{89} +11.4568 q^{91} +12.4799 q^{92} -0.512003 q^{94} +2.43001 q^{95} -16.2488 q^{97} +0.294355 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - q^{2} + 25 q^{4} + q^{5} + 11 q^{7} + 7 q^{10} - 16 q^{11} - q^{13} + 9 q^{14} + 39 q^{16} + 2 q^{17} + 35 q^{19} + 16 q^{20} - 3 q^{22} - q^{23} + 13 q^{25} + 14 q^{26} + 11 q^{28} - 2 q^{29}+ \cdots - 24 q^{98}+O(q^{100}) \) 16 * q - q^2 + 25 * q^4 + q^5 + 11 * q^7 + 7 * q^10 - 16 * q^11 - q^13 + 9 * q^14 + 39 * q^16 + 2 * q^17 + 35 * q^19 + 16 * q^20 - 3 * q^22 - q^23 + 13 * q^25 + 14 * q^26 + 11 * q^28 - 2 * q^29 + 13 * q^31 + 29 * q^32 - 10 * q^34 - 9 * q^35 + 4 * q^37 + 32 * q^38 + 6 * q^40 + 2 * q^41 + 51 * q^43 - 8 * q^44 + 20 * q^46 + 9 * q^47 + 27 * q^49 + 58 * q^50 - 38 * q^52 + 29 * q^53 - 8 * q^55 + 65 * q^56 - 26 * q^58 + 13 * q^59 + 14 * q^61 + 47 * q^62 + 32 * q^64 + 29 * q^65 + 25 * q^67 + 45 * q^68 - 44 * q^70 + 9 * q^71 + 4 * q^73 - 8 * q^74 + 58 * q^76 + 10 * q^77 + 51 * q^79 + 7 * q^80 - 21 * q^82 + 2 * q^83 + 4 * q^85 - 10 * q^86 - 18 * q^88 + 35 * q^89 + 65 * q^91 - 69 * q^92 + 24 * q^94 - 18 * q^95 - 14 * q^97 - 24 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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