LMFDB - Embedded newform 1056.2.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1056,2,Mod(1,1056)] 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("1056.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1056 = 2^{5} \cdot 3 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1056.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,2,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.43220245345$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1056.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.23607 q^{5} +3.23607 q^{7} +1.00000 q^{9} -1.00000 q^{11} +5.23607 q^{13} -1.23607 q^{15} -4.47214 q^{17} +6.47214 q^{19} +3.23607 q^{21} -5.70820 q^{23} -3.47214 q^{25} +1.00000 q^{27} +8.47214 q^{29} -1.00000 q^{33} -4.00000 q^{35} +6.00000 q^{37} +5.23607 q^{39} +10.9443 q^{41} -12.9443 q^{43} -1.23607 q^{45} +5.70820 q^{47} +3.47214 q^{49} -4.47214 q^{51} +6.76393 q^{53} +1.23607 q^{55} +6.47214 q^{57} -5.52786 q^{59} +11.7082 q^{61} +3.23607 q^{63} -6.47214 q^{65} +8.00000 q^{67} -5.70820 q^{69} -5.70820 q^{71} -6.00000 q^{73} -3.47214 q^{75} -3.23607 q^{77} +1.70820 q^{79} +1.00000 q^{81} +4.94427 q^{83} +5.52786 q^{85} +8.47214 q^{87} +2.00000 q^{89} +16.9443 q^{91} -8.00000 q^{95} -8.47214 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + 6 q^{13} + 2 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 8 q^{35} + 12 q^{37} + 6 q^{39} + 4 q^{41} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 + 6 * q^13 + 2 * q^15 + 4 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^33 - 8 * q^35 + 12 * q^37 + 6 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 - 2 * q^47 - 2 * q^49 + 18 * q^53 - 2 * q^55 + 4 * q^57 - 20 * q^59 + 10 * q^61 + 2 * q^63 - 4 * q^65 + 16 * q^67 + 2 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 - 2 * q^77 - 10 * q^79 + 2 * q^81 - 8 * q^83 + 20 * q^85 + 8 * q^87 + 4 * q^89 + 16 * q^91 - 16 * q^95 - 8 * q^97 - 2 * 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	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.23607	−0.552786	−0.276393	−	0.961045i	$-0.589139\pi$
$5$	−1.23607	−0.552786	−0.276393	+	0.961045i	$0.589139\pi$
$6$	0	0
$7$	3.23607	1.22312	0.611559	−	0.791199i	$-0.290543\pi$
$7$	3.23607	1.22312	0.611559	+	0.791199i	$0.290543\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.23607	1.45222	0.726112	−	0.687576i	$-0.241325\pi$
$13$	5.23607	1.45222	0.726112	+	0.687576i	$0.241325\pi$
$14$	0	0
$15$	−1.23607	−0.319151
$16$	0	0
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	0	0
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	0	0
$21$	3.23607	0.706168
$22$	0	0
$23$	−5.70820	−1.19024	−0.595121	−	0.803636i	$-0.702896\pi$
$23$	−5.70820	−1.19024	−0.595121	+	0.803636i	$0.702896\pi$
$24$	0	0
$25$	−3.47214	−0.694427
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−4.00000	−0.676123
$36$	0	0
$37$	6.00000	0.986394	0.493197	−	0.869918i	$-0.335828\pi$
$37$	6.00000	0.986394	0.493197	+	0.869918i	$0.335828\pi$
$38$	0	0
$39$	5.23607	0.838442
$40$	0	0
$41$	10.9443	1.70921	0.854604	−	0.519280i	$-0.173800\pi$
$41$	10.9443	1.70921	0.854604	+	0.519280i	$0.173800\pi$
$42$	0	0
$43$	−12.9443	−1.97398	−0.986991	−	0.160773i	$-0.948601\pi$
$43$	−12.9443	−1.97398	−0.986991	+	0.160773i	$0.948601\pi$
$44$	0	0
$45$	−1.23607	−0.184262
$46$	0	0
$47$	5.70820	0.832627	0.416314	−	0.909221i	$-0.363322\pi$
$47$	5.70820	0.832627	0.416314	+	0.909221i	$0.363322\pi$
$48$	0	0
$49$	3.47214	0.496019
$50$	0	0
$51$	−4.47214	−0.626224
$52$	0	0
$53$	6.76393	0.929098	0.464549	−	0.885548i	$-0.346217\pi$
$53$	6.76393	0.929098	0.464549	+	0.885548i	$0.346217\pi$
$54$	0	0
$55$	1.23607	0.166671
$56$	0	0
$57$	6.47214	0.857255
$58$	0	0
$59$	−5.52786	−0.719667	−0.359833	−	0.933017i	$-0.617166\pi$
$59$	−5.52786	−0.719667	−0.359833	+	0.933017i	$0.617166\pi$
$60$	0	0
$61$	11.7082	1.49908	0.749541	−	0.661958i	$-0.230274\pi$
$61$	11.7082	1.49908	0.749541	+	0.661958i	$0.230274\pi$
$62$	0	0
$63$	3.23607	0.407706
$64$	0	0
$65$	−6.47214	−0.802770
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−5.70820	−0.687187
$70$	0	0
$71$	−5.70820	−0.677439	−0.338720	−	0.940887i	$-0.609994\pi$
$71$	−5.70820	−0.677439	−0.338720	+	0.940887i	$0.609994\pi$
$72$	0	0
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	0	0
$75$	−3.47214	−0.400928
$76$	0	0
$77$	−3.23607	−0.368784
$78$	0	0
$79$	1.70820	0.192188	0.0960940	−	0.995372i	$-0.469365\pi$
$79$	1.70820	0.192188	0.0960940	+	0.995372i	$0.469365\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.94427	0.542704	0.271352	−	0.962480i	$-0.412529\pi$
$83$	4.94427	0.542704	0.271352	+	0.962480i	$0.412529\pi$
$84$	0	0
$85$	5.52786	0.599581
$86$	0	0
$87$	8.47214	0.908308
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	16.9443	1.77624
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	3.52786	0.351036	0.175518	−	0.984476i	$-0.443840\pi$
$101$	3.52786	0.351036	0.175518	+	0.984476i	$0.443840\pi$
$102$	0	0
$103$	−6.47214	−0.637719	−0.318859	−	0.947802i	$-0.603300\pi$
$103$	−6.47214	−0.637719	−0.318859	+	0.947802i	$0.603300\pi$
$104$	0	0
$105$	−4.00000	−0.390360
$106$	0	0
$107$	−16.9443	−1.63806	−0.819032	−	0.573747i	$-0.805489\pi$
$107$	−16.9443	−1.63806	−0.819032	+	0.573747i	$0.805489\pi$
$108$	0	0
$109$	−20.6525	−1.97815	−0.989074	−	0.147418i	$-0.952904\pi$
$109$	−20.6525	−1.97815	−0.989074	+	0.147418i	$0.952904\pi$
$110$	0	0
$111$	6.00000	0.569495
$112$	0	0
$113$	6.94427	0.653262	0.326631	−	0.945152i	$-0.394087\pi$
$113$	6.94427	0.653262	0.326631	+	0.945152i	$0.394087\pi$
$114$	0	0
$115$	7.05573	0.657950
$116$	0	0
$117$	5.23607	0.484075
$118$	0	0
$119$	−14.4721	−1.32666
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	10.9443	0.986812
$124$	0	0
$125$	10.4721	0.936656
$126$	0	0
$127$	−8.18034	−0.725888	−0.362944	−	0.931811i	$-0.618228\pi$
$127$	−8.18034	−0.725888	−0.362944	+	0.931811i	$0.618228\pi$
$128$	0	0
$129$	−12.9443	−1.13968
$130$	0	0
$131$	−16.0000	−1.39793	−0.698963	−	0.715158i	$-0.746355\pi$
$131$	−16.0000	−1.39793	−0.698963	+	0.715158i	$0.746355\pi$
$132$	0	0
$133$	20.9443	1.81610
$134$	0	0
$135$	−1.23607	−0.106384
$136$	0	0
$137$	−7.52786	−0.643149	−0.321574	−	0.946884i	$-0.604212\pi$
$137$	−7.52786	−0.643149	−0.321574	+	0.946884i	$0.604212\pi$
$138$	0	0
$139$	1.52786	0.129592	0.0647959	−	0.997899i	$-0.479360\pi$
$139$	1.52786	0.129592	0.0647959	+	0.997899i	$0.479360\pi$
$140$	0	0
$141$	5.70820	0.480717
$142$	0	0
$143$	−5.23607	−0.437862
$144$	0	0
$145$	−10.4721	−0.869664
$146$	0	0
$147$	3.47214	0.286377
$148$	0	0
$149$	2.00000	0.163846	0.0819232	−	0.996639i	$-0.473894\pi$
$149$	2.00000	0.163846	0.0819232	+	0.996639i	$0.473894\pi$
$150$	0	0
$151$	−11.2361	−0.914378	−0.457189	−	0.889369i	$-0.651144\pi$
$151$	−11.2361	−0.914378	−0.457189	+	0.889369i	$0.651144\pi$
$152$	0	0
$153$	−4.47214	−0.361551
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−16.4721	−1.31462	−0.657310	−	0.753620i	$-0.728306\pi$
$157$	−16.4721	−1.31462	−0.657310	+	0.753620i	$0.728306\pi$
$158$	0	0
$159$	6.76393	0.536415
$160$	0	0
$161$	−18.4721	−1.45581
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	1.23607	0.0962278
$166$	0	0
$167$	24.3607	1.88509	0.942543	−	0.334085i	$-0.108427\pi$
$167$	24.3607	1.88509	0.942543	+	0.334085i	$0.108427\pi$
$168$	0	0
$169$	14.4164	1.10895
$170$	0	0
$171$	6.47214	0.494937
$172$	0	0
$173$	−7.52786	−0.572333	−0.286166	−	0.958180i	$-0.592381\pi$
$173$	−7.52786	−0.572333	−0.286166	+	0.958180i	$0.592381\pi$
$174$	0	0
$175$	−11.2361	−0.849367
$176$	0	0
$177$	−5.52786	−0.415500
$178$	0	0
$179$	−10.4721	−0.782724	−0.391362	−	0.920237i	$-0.627996\pi$
$179$	−10.4721	−0.782724	−0.391362	+	0.920237i	$0.627996\pi$
$180$	0	0
$181$	−8.47214	−0.629729	−0.314864	−	0.949137i	$-0.601959\pi$
$181$	−8.47214	−0.629729	−0.314864	+	0.949137i	$0.601959\pi$
$182$	0	0
$183$	11.7082	0.865495
$184$	0	0
$185$	−7.41641	−0.545265
$186$	0	0
$187$	4.47214	0.327035
$188$	0	0
$189$	3.23607	0.235389
$190$	0	0
$191$	−7.23607	−0.523584	−0.261792	−	0.965124i	$-0.584313\pi$
$191$	−7.23607	−0.523584	−0.261792	+	0.965124i	$0.584313\pi$
$192$	0	0
$193$	−7.52786	−0.541868	−0.270934	−	0.962598i	$-0.587332\pi$
$193$	−7.52786	−0.541868	−0.270934	+	0.962598i	$0.587332\pi$
$194$	0	0
$195$	−6.47214	−0.463479
$196$	0	0
$197$	5.41641	0.385903	0.192952	−	0.981208i	$-0.438194\pi$
$197$	5.41641	0.385903	0.192952	+	0.981208i	$0.438194\pi$
$198$	0	0
$199$	−3.41641	−0.242183	−0.121091	−	0.992641i	$-0.538639\pi$
$199$	−3.41641	−0.242183	−0.121091	+	0.992641i	$0.538639\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	27.4164	1.92425
$204$	0	0
$205$	−13.5279	−0.944827
$206$	0	0
$207$	−5.70820	−0.396748
$208$	0	0
$209$	−6.47214	−0.447687
$210$	0	0
$211$	3.05573	0.210365	0.105182	−	0.994453i	$-0.466457\pi$
$211$	3.05573	0.210365	0.105182	+	0.994453i	$0.466457\pi$
$212$	0	0
$213$	−5.70820	−0.391120
$214$	0	0
$215$	16.0000	1.09119
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.00000	−0.405442
$220$	0	0
$221$	−23.4164	−1.57516
$222$	0	0
$223$	0	0		−	1.00000i	$-0.5\pi$
$223$	0	0			1.00000i	$0.5\pi$
$224$	0	0
$225$	−3.47214	−0.231476
$226$	0	0
$227$	−4.00000	−0.265489	−0.132745	−	0.991150i	$-0.542379\pi$
$227$	−4.00000	−0.265489	−0.132745	+	0.991150i	$0.542379\pi$
$228$	0	0
$229$	9.41641	0.622254	0.311127	−	0.950368i	$-0.399294\pi$
$229$	9.41641	0.622254	0.311127	+	0.950368i	$0.399294\pi$
$230$	0	0
$231$	−3.23607	−0.212918
$232$	0	0
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	−7.05573	−0.460265
$236$	0	0
$237$	1.70820	0.110960
$238$	0	0
$239$	−17.5279	−1.13378	−0.566892	−	0.823792i	$-0.691854\pi$
$239$	−17.5279	−1.13378	−0.566892	+	0.823792i	$0.691854\pi$
$240$	0	0
$241$	18.3607	1.18272	0.591358	−	0.806409i	$-0.298592\pi$
$241$	18.3607	1.18272	0.591358	+	0.806409i	$0.298592\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−4.29180	−0.274193
$246$	0	0
$247$	33.8885	2.15628
$248$	0	0
$249$	4.94427	0.313331
$250$	0	0
$251$	−20.3607	−1.28515	−0.642577	−	0.766221i	$-0.722135\pi$
$251$	−20.3607	−1.28515	−0.642577	+	0.766221i	$0.722135\pi$
$252$	0	0
$253$	5.70820	0.358872
$254$	0	0
$255$	5.52786	0.346168
$256$	0	0
$257$	−14.0000	−0.873296	−0.436648	−	0.899632i	$-0.643834\pi$
$257$	−14.0000	−0.873296	−0.436648	+	0.899632i	$0.643834\pi$
$258$	0	0
$259$	19.4164	1.20648
$260$	0	0
$261$	8.47214	0.524412
$262$	0	0
$263$	−16.0000	−0.986602	−0.493301	−	0.869859i	$-0.664210\pi$
$263$	−16.0000	−0.986602	−0.493301	+	0.869859i	$0.664210\pi$
$264$	0	0
$265$	−8.36068	−0.513592
$266$	0	0
$267$	2.00000	0.122398
$268$	0	0
$269$	−4.29180	−0.261675	−0.130838	−	0.991404i	$-0.541767\pi$
$269$	−4.29180	−0.261675	−0.130838	+	0.991404i	$0.541767\pi$
$270$	0	0
$271$	−12.7639	−0.775354	−0.387677	−	0.921795i	$-0.626722\pi$
$271$	−12.7639	−0.775354	−0.387677	+	0.921795i	$0.626722\pi$
$272$	0	0
$273$	16.9443	1.02551
$274$	0	0
$275$	3.47214	0.209378
$276$	0	0
$277$	−1.23607	−0.0742681	−0.0371341	−	0.999310i	$-0.511823\pi$
$277$	−1.23607	−0.0742681	−0.0371341	+	0.999310i	$0.511823\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.3607	−0.856686	−0.428343	−	0.903616i	$-0.640903\pi$
$281$	−14.3607	−0.856686	−0.428343	+	0.903616i	$0.640903\pi$
$282$	0	0
$283$	20.9443	1.24501	0.622504	−	0.782617i	$-0.286115\pi$
$283$	20.9443	1.24501	0.622504	+	0.782617i	$0.286115\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	35.4164	2.09056
$288$	0	0
$289$	3.00000	0.176471
$290$	0	0
$291$	−8.47214	−0.496645
$292$	0	0
$293$	−7.88854	−0.460854	−0.230427	−	0.973090i	$-0.574012\pi$
$293$	−7.88854	−0.460854	−0.230427	+	0.973090i	$0.574012\pi$
$294$	0	0
$295$	6.83282	0.397822
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−29.8885	−1.72850
$300$	0	0
$301$	−41.8885	−2.41442
$302$	0	0
$303$	3.52786	0.202670
$304$	0	0
$305$	−14.4721	−0.828672
$306$	0	0
$307$	−30.4721	−1.73914	−0.869568	−	0.493813i	$-0.835603\pi$
$307$	−30.4721	−1.73914	−0.869568	+	0.493813i	$0.835603\pi$
$308$	0	0
$309$	−6.47214	−0.368187
$310$	0	0
$311$	2.29180	0.129956	0.0649779	−	0.997887i	$-0.479302\pi$
$311$	2.29180	0.129956	0.0649779	+	0.997887i	$0.479302\pi$
$312$	0	0
$313$	−15.8885	−0.898074	−0.449037	−	0.893513i	$-0.648233\pi$
$313$	−15.8885	−0.898074	−0.449037	+	0.893513i	$0.648233\pi$
$314$	0	0
$315$	−4.00000	−0.225374
$316$	0	0
$317$	−7.70820	−0.432936	−0.216468	−	0.976290i	$-0.569454\pi$
$317$	−7.70820	−0.432936	−0.216468	+	0.976290i	$0.569454\pi$
$318$	0	0
$319$	−8.47214	−0.474349
$320$	0	0
$321$	−16.9443	−0.945737
$322$	0	0
$323$	−28.9443	−1.61050
$324$	0	0
$325$	−18.1803	−1.00846
$326$	0	0
$327$	−20.6525	−1.14208
$328$	0	0
$329$	18.4721	1.01840
$330$	0	0
$331$	−16.9443	−0.931341	−0.465671	−	0.884958i	$-0.654187\pi$
$331$	−16.9443	−0.931341	−0.465671	+	0.884958i	$0.654187\pi$
$332$	0	0
$333$	6.00000	0.328798
$334$	0	0
$335$	−9.88854	−0.540269
$336$	0	0
$337$	−26.9443	−1.46775	−0.733874	−	0.679286i	$-0.762290\pi$
$337$	−26.9443	−1.46775	−0.733874	+	0.679286i	$0.762290\pi$
$338$	0	0
$339$	6.94427	0.377161
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−11.4164	−0.616428
$344$	0	0
$345$	7.05573	0.379868
$346$	0	0
$347$	−17.8885	−0.960307	−0.480154	−	0.877184i	$-0.659419\pi$
$347$	−17.8885	−0.960307	−0.480154	+	0.877184i	$0.659419\pi$
$348$	0	0
$349$	16.2918	0.872080	0.436040	−	0.899927i	$-0.356381\pi$
$349$	16.2918	0.872080	0.436040	+	0.899927i	$0.356381\pi$
$350$	0	0
$351$	5.23607	0.279481
$352$	0	0
$353$	24.8328	1.32172	0.660859	−	0.750510i	$-0.270192\pi$
$353$	24.8328	1.32172	0.660859	+	0.750510i	$0.270192\pi$
$354$	0	0
$355$	7.05573	0.374479
$356$	0	0
$357$	−14.4721	−0.765947
$358$	0	0
$359$	−11.4164	−0.602535	−0.301267	−	0.953540i	$-0.597410\pi$
$359$	−11.4164	−0.602535	−0.301267	+	0.953540i	$0.597410\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	7.41641	0.388193
$366$	0	0
$367$	−36.9443	−1.92848	−0.964238	−	0.265039i	$-0.914615\pi$
$367$	−36.9443	−1.92848	−0.964238	+	0.265039i	$0.914615\pi$
$368$	0	0
$369$	10.9443	0.569736
$370$	0	0
$371$	21.8885	1.13640
$372$	0	0
$373$	18.1803	0.941342	0.470671	−	0.882309i	$-0.344012\pi$
$373$	18.1803	0.941342	0.470671	+	0.882309i	$0.344012\pi$
$374$	0	0
$375$	10.4721	0.540779
$376$	0	0
$377$	44.3607	2.28469
$378$	0	0
$379$	29.8885	1.53527	0.767636	−	0.640886i	$-0.221433\pi$
$379$	29.8885	1.53527	0.767636	+	0.640886i	$0.221433\pi$
$380$	0	0
$381$	−8.18034	−0.419092
$382$	0	0
$383$	30.0689	1.53645	0.768224	−	0.640181i	$-0.221141\pi$
$383$	30.0689	1.53645	0.768224	+	0.640181i	$0.221141\pi$
$384$	0	0
$385$	4.00000	0.203859
$386$	0	0
$387$	−12.9443	−0.657994
$388$	0	0
$389$	27.7082	1.40486	0.702431	−	0.711752i	$-0.252098\pi$
$389$	27.7082	1.40486	0.702431	+	0.711752i	$0.252098\pi$
$390$	0	0
$391$	25.5279	1.29100
$392$	0	0
$393$	−16.0000	−0.807093
$394$	0	0
$395$	−2.11146	−0.106239
$396$	0	0
$397$	14.3607	0.720742	0.360371	−	0.932809i	$-0.382650\pi$
$397$	14.3607	0.720742	0.360371	+	0.932809i	$0.382650\pi$
$398$	0	0
$399$	20.9443	1.04853
$400$	0	0
$401$	13.4164	0.669983	0.334992	−	0.942221i	$-0.391266\pi$
$401$	13.4164	0.669983	0.334992	+	0.942221i	$0.391266\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−1.23607	−0.0614207
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	32.4721	1.60564	0.802822	−	0.596219i	$-0.203331\pi$
$409$	32.4721	1.60564	0.802822	+	0.596219i	$0.203331\pi$
$410$	0	0
$411$	−7.52786	−0.371322
$412$	0	0
$413$	−17.8885	−0.880238
$414$	0	0
$415$	−6.11146	−0.300000
$416$	0	0
$417$	1.52786	0.0748198
$418$	0	0
$419$	7.41641	0.362315	0.181158	−	0.983454i	$-0.442016\pi$
$419$	7.41641	0.362315	0.181158	+	0.983454i	$0.442016\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	0	0
$423$	5.70820	0.277542
$424$	0	0
$425$	15.5279	0.753212
$426$	0	0
$427$	37.8885	1.83356
$428$	0	0
$429$	−5.23607	−0.252800
$430$	0	0
$431$	19.0557	0.917882	0.458941	−	0.888467i	$-0.348229\pi$
$431$	19.0557	0.917882	0.458941	+	0.888467i	$0.348229\pi$
$432$	0	0
$433$	−36.8328	−1.77007	−0.885036	−	0.465522i	$-0.845866\pi$
$433$	−36.8328	−1.77007	−0.885036	+	0.465522i	$0.845866\pi$
$434$	0	0
$435$	−10.4721	−0.502100
$436$	0	0
$437$	−36.9443	−1.76728
$438$	0	0
$439$	−27.2361	−1.29991	−0.649953	−	0.759974i	$-0.725212\pi$
$439$	−27.2361	−1.29991	−0.649953	+	0.759974i	$0.725212\pi$
$440$	0	0
$441$	3.47214	0.165340
$442$	0	0
$443$	29.5279	1.40291	0.701456	−	0.712713i	$-0.252534\pi$
$443$	29.5279	1.40291	0.701456	+	0.712713i	$0.252534\pi$
$444$	0	0
$445$	−2.47214	−0.117190
$446$	0	0
$447$	2.00000	0.0945968
$448$	0	0
$449$	11.5279	0.544034	0.272017	−	0.962293i	$-0.412309\pi$
$449$	11.5279	0.544034	0.272017	+	0.962293i	$0.412309\pi$
$450$	0	0
$451$	−10.9443	−0.515346
$452$	0	0
$453$	−11.2361	−0.527917
$454$	0	0
$455$	−20.9443	−0.981883
$456$	0	0
$457$	31.3050	1.46438	0.732192	−	0.681098i	$-0.238497\pi$
$457$	31.3050	1.46438	0.732192	+	0.681098i	$0.238497\pi$
$458$	0	0
$459$	−4.47214	−0.208741
$460$	0	0
$461$	−18.9443	−0.882323	−0.441161	−	0.897428i	$-0.645433\pi$
$461$	−18.9443	−0.882323	−0.441161	+	0.897428i	$0.645433\pi$
$462$	0	0
$463$	14.4721	0.672577	0.336289	−	0.941759i	$-0.390828\pi$
$463$	14.4721	0.672577	0.336289	+	0.941759i	$0.390828\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.05573	0.326500	0.163250	−	0.986585i	$-0.447802\pi$
$467$	7.05573	0.326500	0.163250	+	0.986585i	$0.447802\pi$
$468$	0	0
$469$	25.8885	1.19542
$470$	0	0
$471$	−16.4721	−0.758996
$472$	0	0
$473$	12.9443	0.595178
$474$	0	0
$475$	−22.4721	−1.03109
$476$	0	0
$477$	6.76393	0.309699
$478$	0	0
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	0	0
$481$	31.4164	1.43246
$482$	0	0
$483$	−18.4721	−0.840511
$484$	0	0
$485$	10.4721	0.475515
$486$	0	0
$487$	1.52786	0.0692341	0.0346171	−	0.999401i	$-0.488979\pi$
$487$	1.52786	0.0692341	0.0346171	+	0.999401i	$0.488979\pi$
$488$	0	0
$489$	20.0000	0.904431
$490$	0	0
$491$	29.8885	1.34885	0.674426	−	0.738343i	$-0.264391\pi$
$491$	29.8885	1.34885	0.674426	+	0.738343i	$0.264391\pi$
$492$	0	0
$493$	−37.8885	−1.70641
$494$	0	0
$495$	1.23607	0.0555571
$496$	0	0
$497$	−18.4721	−0.828589
$498$	0	0
$499$	16.9443	0.758530	0.379265	−	0.925288i	$-0.376177\pi$
$499$	16.9443	0.758530	0.379265	+	0.925288i	$0.376177\pi$
$500$	0	0
$501$	24.3607	1.08835
$502$	0	0
$503$	−19.4164	−0.865735	−0.432867	−	0.901458i	$-0.642498\pi$
$503$	−19.4164	−0.865735	−0.432867	+	0.901458i	$0.642498\pi$
$504$	0	0
$505$	−4.36068	−0.194048
$506$	0	0
$507$	14.4164	0.640255
$508$	0	0
$509$	8.65248	0.383514	0.191757	−	0.981442i	$-0.438581\pi$
$509$	8.65248	0.383514	0.191757	+	0.981442i	$0.438581\pi$
$510$	0	0
$511$	−19.4164	−0.858931
$512$	0	0
$513$	6.47214	0.285752
$514$	0	0
$515$	8.00000	0.352522
$516$	0	0
$517$	−5.70820	−0.251047
$518$	0	0
$519$	−7.52786	−0.330437
$520$	0	0
$521$	16.4721	0.721657	0.360829	−	0.932632i	$-0.382494\pi$
$521$	16.4721	0.721657	0.360829	+	0.932632i	$0.382494\pi$
$522$	0	0
$523$	−9.88854	−0.432396	−0.216198	−	0.976350i	$-0.569366\pi$
$523$	−9.88854	−0.432396	−0.216198	+	0.976350i	$0.569366\pi$
$524$	0	0
$525$	−11.2361	−0.490382
$526$	0	0
$527$	0	0
$528$	0	0
$529$	9.58359	0.416678
$530$	0	0
$531$	−5.52786	−0.239889
$532$	0	0
$533$	57.3050	2.48215
$534$	0	0
$535$	20.9443	0.905500
$536$	0	0
$537$	−10.4721	−0.451906
$538$	0	0
$539$	−3.47214	−0.149555
$540$	0	0
$541$	8.29180	0.356492	0.178246	−	0.983986i	$-0.442958\pi$
$541$	8.29180	0.356492	0.178246	+	0.983986i	$0.442958\pi$
$542$	0	0
$543$	−8.47214	−0.363574
$544$	0	0
$545$	25.5279	1.09349
$546$	0	0
$547$	−3.41641	−0.146075	−0.0730375	−	0.997329i	$-0.523269\pi$
$547$	−3.41641	−0.146075	−0.0730375	+	0.997329i	$0.523269\pi$
$548$	0	0
$549$	11.7082	0.499694
$550$	0	0
$551$	54.8328	2.33596
$552$	0	0
$553$	5.52786	0.235069
$554$	0	0
$555$	−7.41641	−0.314809
$556$	0	0
$557$	−31.8885	−1.35116	−0.675580	−	0.737286i	$-0.736107\pi$
$557$	−31.8885	−1.35116	−0.675580	+	0.737286i	$0.736107\pi$
$558$	0	0
$559$	−67.7771	−2.86667
$560$	0	0
$561$	4.47214	0.188814
$562$	0	0
$563$	7.05573	0.297363	0.148682	−	0.988885i	$-0.452497\pi$
$563$	7.05573	0.297363	0.148682	+	0.988885i	$0.452497\pi$
$564$	0	0
$565$	−8.58359	−0.361114
$566$	0	0
$567$	3.23607	0.135902
$568$	0	0
$569$	19.5279	0.818651	0.409325	−	0.912389i	$-0.365764\pi$
$569$	19.5279	0.818651	0.409325	+	0.912389i	$0.365764\pi$
$570$	0	0
$571$	−12.5836	−0.526607	−0.263303	−	0.964713i	$-0.584812\pi$
$571$	−12.5836	−0.526607	−0.263303	+	0.964713i	$0.584812\pi$
$572$	0	0
$573$	−7.23607	−0.302291
$574$	0	0
$575$	19.8197	0.826537
$576$	0	0
$577$	17.4164	0.725055	0.362527	−	0.931973i	$-0.381914\pi$
$577$	17.4164	0.725055	0.362527	+	0.931973i	$0.381914\pi$
$578$	0	0
$579$	−7.52786	−0.312847
$580$	0	0
$581$	16.0000	0.663792
$582$	0	0
$583$	−6.76393	−0.280133
$584$	0	0
$585$	−6.47214	−0.267590
$586$	0	0
$587$	−0.583592	−0.0240874	−0.0120437	−	0.999927i	$-0.503834\pi$
$587$	−0.583592	−0.0240874	−0.0120437	+	0.999927i	$0.503834\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	5.41641	0.222801
$592$	0	0
$593$	3.52786	0.144872	0.0724360	−	0.997373i	$-0.476923\pi$
$593$	3.52786	0.144872	0.0724360	+	0.997373i	$0.476923\pi$
$594$	0	0
$595$	17.8885	0.733359
$596$	0	0
$597$	−3.41641	−0.139824
$598$	0	0
$599$	38.0689	1.55545	0.777726	−	0.628603i	$-0.216373\pi$
$599$	38.0689	1.55545	0.777726	+	0.628603i	$0.216373\pi$
$600$	0	0
$601$	−7.88854	−0.321780	−0.160890	−	0.986972i	$-0.551437\pi$
$601$	−7.88854	−0.321780	−0.160890	+	0.986972i	$0.551437\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	−1.23607	−0.0502533
$606$	0	0
$607$	2.87539	0.116708	0.0583542	−	0.998296i	$-0.481415\pi$
$607$	2.87539	0.116708	0.0583542	+	0.998296i	$0.481415\pi$
$608$	0	0
$609$	27.4164	1.11097
$610$	0	0
$611$	29.8885	1.20916
$612$	0	0
$613$	25.8197	1.04285	0.521423	−	0.853298i	$-0.325401\pi$
$613$	25.8197	1.04285	0.521423	+	0.853298i	$0.325401\pi$
$614$	0	0
$615$	−13.5279	−0.545496
$616$	0	0
$617$	−23.8885	−0.961717	−0.480858	−	0.876798i	$-0.659675\pi$
$617$	−23.8885	−0.961717	−0.480858	+	0.876798i	$0.659675\pi$
$618$	0	0
$619$	−27.7771	−1.11646	−0.558228	−	0.829688i	$-0.688518\pi$
$619$	−27.7771	−1.11646	−0.558228	+	0.829688i	$0.688518\pi$
$620$	0	0
$621$	−5.70820	−0.229062
$622$	0	0
$623$	6.47214	0.259301
$624$	0	0
$625$	4.41641	0.176656
$626$	0	0
$627$	−6.47214	−0.258472
$628$	0	0
$629$	−26.8328	−1.06989
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	3.05573	0.121454
$634$	0	0
$635$	10.1115	0.401261
$636$	0	0
$637$	18.1803	0.720331
$638$	0	0
$639$	−5.70820	−0.225813
$640$	0	0
$641$	−40.2492	−1.58975	−0.794874	−	0.606774i	$-0.792463\pi$
$641$	−40.2492	−1.58975	−0.794874	+	0.606774i	$0.792463\pi$
$642$	0	0
$643$	27.0557	1.06697	0.533487	−	0.845808i	$-0.320881\pi$
$643$	27.0557	1.06697	0.533487	+	0.845808i	$0.320881\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	−42.6525	−1.67684	−0.838421	−	0.545023i	$-0.816521\pi$
$647$	−42.6525	−1.67684	−0.838421	+	0.545023i	$0.816521\pi$
$648$	0	0
$649$	5.52786	0.216988
$650$	0	0
$651$	0	0
$652$	0	0
$653$	39.1246	1.53106	0.765532	−	0.643398i	$-0.222476\pi$
$653$	39.1246	1.53106	0.765532	+	0.643398i	$0.222476\pi$
$654$	0	0
$655$	19.7771	0.772755
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	−48.9443	−1.90660	−0.953299	−	0.302028i	$-0.902336\pi$
$659$	−48.9443	−1.90660	−0.953299	+	0.302028i	$0.902336\pi$
$660$	0	0
$661$	7.52786	0.292800	0.146400	−	0.989225i	$-0.453231\pi$
$661$	7.52786	0.292800	0.146400	+	0.989225i	$0.453231\pi$
$662$	0	0
$663$	−23.4164	−0.909418
$664$	0	0
$665$	−25.8885	−1.00391
$666$	0	0
$667$	−48.3607	−1.87253
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.7082	−0.451990
$672$	0	0
$673$	16.4721	0.634954	0.317477	−	0.948266i	$-0.397164\pi$
$673$	16.4721	0.634954	0.317477	+	0.948266i	$0.397164\pi$
$674$	0	0
$675$	−3.47214	−0.133643
$676$	0	0
$677$	−7.52786	−0.289319	−0.144660	−	0.989481i	$-0.546209\pi$
$677$	−7.52786	−0.289319	−0.144660	+	0.989481i	$0.546209\pi$
$678$	0	0
$679$	−27.4164	−1.05215
$680$	0	0
$681$	−4.00000	−0.153280
$682$	0	0
$683$	−15.4164	−0.589892	−0.294946	−	0.955514i	$-0.595302\pi$
$683$	−15.4164	−0.589892	−0.294946	+	0.955514i	$0.595302\pi$
$684$	0	0
$685$	9.30495	0.355524
$686$	0	0
$687$	9.41641	0.359258
$688$	0	0
$689$	35.4164	1.34926
$690$	0	0
$691$	0	0		−	1.00000i	$-0.5\pi$
$691$	0	0			1.00000i	$0.5\pi$
$692$	0	0
$693$	−3.23607	−0.122928
$694$	0	0
$695$	−1.88854	−0.0716366
$696$	0	0
$697$	−48.9443	−1.85390
$698$	0	0
$699$	−2.00000	−0.0756469
$700$	0	0
$701$	16.8328	0.635767	0.317883	−	0.948130i	$-0.397028\pi$
$701$	16.8328	0.635767	0.317883	+	0.948130i	$0.397028\pi$
$702$	0	0
$703$	38.8328	1.46461
$704$	0	0
$705$	−7.05573	−0.265734
$706$	0	0
$707$	11.4164	0.429358
$708$	0	0
$709$	−10.3607	−0.389103	−0.194552	−	0.980892i	$-0.562325\pi$
$709$	−10.3607	−0.389103	−0.194552	+	0.980892i	$0.562325\pi$
$710$	0	0
$711$	1.70820	0.0640627
$712$	0	0
$713$	0	0
$714$	0	0
$715$	6.47214	0.242044
$716$	0	0
$717$	−17.5279	−0.654590
$718$	0	0
$719$	38.0689	1.41973	0.709865	−	0.704338i	$-0.248756\pi$
$719$	38.0689	1.41973	0.709865	+	0.704338i	$0.248756\pi$
$720$	0	0
$721$	−20.9443	−0.780005
$722$	0	0
$723$	18.3607	0.682841
$724$	0	0
$725$	−29.4164	−1.09250
$726$	0	0
$727$	27.4164	1.01682	0.508409	−	0.861116i	$-0.330234\pi$
$727$	27.4164	1.01682	0.508409	+	0.861116i	$0.330234\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	57.8885	2.14109
$732$	0	0
$733$	31.1246	1.14961	0.574807	−	0.818289i	$-0.305077\pi$
$733$	31.1246	1.14961	0.574807	+	0.818289i	$0.305077\pi$
$734$	0	0
$735$	−4.29180	−0.158305
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	9.88854	0.363756	0.181878	−	0.983321i	$-0.441782\pi$
$739$	9.88854	0.363756	0.181878	+	0.983321i	$0.441782\pi$
$740$	0	0
$741$	33.8885	1.24493
$742$	0	0
$743$	−37.3050	−1.36859	−0.684293	−	0.729207i	$-0.739889\pi$
$743$	−37.3050	−1.36859	−0.684293	+	0.729207i	$0.739889\pi$
$744$	0	0
$745$	−2.47214	−0.0905721
$746$	0	0
$747$	4.94427	0.180901
$748$	0	0
$749$	−54.8328	−2.00355
$750$	0	0
$751$	33.8885	1.23661	0.618305	−	0.785938i	$-0.287820\pi$
$751$	33.8885	1.23661	0.618305	+	0.785938i	$0.287820\pi$
$752$	0	0
$753$	−20.3607	−0.741984
$754$	0	0
$755$	13.8885	0.505456
$756$	0	0
$757$	−38.9443	−1.41545	−0.707727	−	0.706486i	$-0.750279\pi$
$757$	−38.9443	−1.41545	−0.707727	+	0.706486i	$0.750279\pi$
$758$	0	0
$759$	5.70820	0.207195
$760$	0	0
$761$	−18.5836	−0.673655	−0.336827	−	0.941566i	$-0.609354\pi$
$761$	−18.5836	−0.673655	−0.336827	+	0.941566i	$0.609354\pi$
$762$	0	0
$763$	−66.8328	−2.41951
$764$	0	0
$765$	5.52786	0.199860
$766$	0	0
$767$	−28.9443	−1.04512
$768$	0	0
$769$	24.4721	0.882488	0.441244	−	0.897387i	$-0.354537\pi$
$769$	24.4721	0.882488	0.441244	+	0.897387i	$0.354537\pi$
$770$	0	0
$771$	−14.0000	−0.504198
$772$	0	0
$773$	9.81966	0.353189	0.176594	−	0.984284i	$-0.443492\pi$
$773$	9.81966	0.353189	0.176594	+	0.984284i	$0.443492\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	19.4164	0.696560
$778$	0	0
$779$	70.8328	2.53785
$780$	0	0
$781$	5.70820	0.204256
$782$	0	0
$783$	8.47214	0.302769
$784$	0	0
$785$	20.3607	0.726704
$786$	0	0
$787$	−22.8328	−0.813902	−0.406951	−	0.913450i	$-0.633408\pi$
$787$	−22.8328	−0.813902	−0.406951	+	0.913450i	$0.633408\pi$
$788$	0	0
$789$	−16.0000	−0.569615
$790$	0	0
$791$	22.4721	0.799017
$792$	0	0
$793$	61.3050	2.17700
$794$	0	0
$795$	−8.36068	−0.296523
$796$	0	0
$797$	−3.12461	−0.110679	−0.0553397	−	0.998468i	$-0.517624\pi$
$797$	−3.12461	−0.110679	−0.0553397	+	0.998468i	$0.517624\pi$
$798$	0	0
$799$	−25.5279	−0.903111
$800$	0	0
$801$	2.00000	0.0706665
$802$	0	0
$803$	6.00000	0.211735
$804$	0	0
$805$	22.8328	0.804751
$806$	0	0
$807$	−4.29180	−0.151078
$808$	0	0
$809$	15.8885	0.558611	0.279306	−	0.960202i	$-0.409896\pi$
$809$	15.8885	0.558611	0.279306	+	0.960202i	$0.409896\pi$
$810$	0	0
$811$	−41.8885	−1.47091	−0.735453	−	0.677576i	$-0.763031\pi$
$811$	−41.8885	−1.47091	−0.735453	+	0.677576i	$0.763031\pi$
$812$	0	0
$813$	−12.7639	−0.447651
$814$	0	0
$815$	−24.7214	−0.865951
$816$	0	0
$817$	−83.7771	−2.93099
$818$	0	0
$819$	16.9443	0.592081
$820$	0	0
$821$	50.3607	1.75760	0.878800	−	0.477190i	$-0.158345\pi$
$821$	50.3607	1.75760	0.878800	+	0.477190i	$0.158345\pi$
$822$	0	0
$823$	−25.8885	−0.902418	−0.451209	−	0.892418i	$-0.649007\pi$
$823$	−25.8885	−0.902418	−0.451209	+	0.892418i	$0.649007\pi$
$824$	0	0
$825$	3.47214	0.120884
$826$	0	0
$827$	19.7771	0.687717	0.343858	−	0.939022i	$-0.388266\pi$
$827$	19.7771	0.687717	0.343858	+	0.939022i	$0.388266\pi$
$828$	0	0
$829$	−8.11146	−0.281723	−0.140861	−	0.990029i	$-0.544987\pi$
$829$	−8.11146	−0.281723	−0.140861	+	0.990029i	$0.544987\pi$
$830$	0	0
$831$	−1.23607	−0.0428787
$832$	0	0
$833$	−15.5279	−0.538009
$834$	0	0
$835$	−30.1115	−1.04205
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−23.2361	−0.802198	−0.401099	−	0.916035i	$-0.631372\pi$
$839$	−23.2361	−0.802198	−0.401099	+	0.916035i	$0.631372\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	−14.3607	−0.494608
$844$	0	0
$845$	−17.8197	−0.613015
$846$	0	0
$847$	3.23607	0.111193
$848$	0	0
$849$	20.9443	0.718806
$850$	0	0
$851$	−34.2492	−1.17405
$852$	0	0
$853$	−54.9017	−1.87980	−0.939899	−	0.341452i	$-0.889081\pi$
$853$	−54.9017	−1.87980	−0.939899	+	0.341452i	$0.889081\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	−16.1115	−0.550357	−0.275178	−	0.961393i	$-0.588737\pi$
$857$	−16.1115	−0.550357	−0.275178	+	0.961393i	$0.588737\pi$
$858$	0	0
$859$	29.8885	1.01978	0.509892	−	0.860238i	$-0.329685\pi$
$859$	29.8885	1.01978	0.509892	+	0.860238i	$0.329685\pi$
$860$	0	0
$861$	35.4164	1.20699
$862$	0	0
$863$	16.7639	0.570651	0.285325	−	0.958431i	$-0.407898\pi$
$863$	16.7639	0.570651	0.285325	+	0.958431i	$0.407898\pi$
$864$	0	0
$865$	9.30495	0.316378
$866$	0	0
$867$	3.00000	0.101885
$868$	0	0
$869$	−1.70820	−0.0579468
$870$	0	0
$871$	41.8885	1.41934
$872$	0	0
$873$	−8.47214	−0.286738
$874$	0	0
$875$	33.8885	1.14564
$876$	0	0
$877$	29.2361	0.987232	0.493616	−	0.869680i	$-0.335675\pi$
$877$	29.2361	0.987232	0.493616	+	0.869680i	$0.335675\pi$
$878$	0	0
$879$	−7.88854	−0.266074
$880$	0	0
$881$	−42.9443	−1.44683	−0.723415	−	0.690414i	$-0.757428\pi$
$881$	−42.9443	−1.44683	−0.723415	+	0.690414i	$0.757428\pi$
$882$	0	0
$883$	40.7214	1.37038	0.685191	−	0.728363i	$-0.259719\pi$
$883$	40.7214	1.37038	0.685191	+	0.728363i	$0.259719\pi$
$884$	0	0
$885$	6.83282	0.229683
$886$	0	0
$887$	−24.3607	−0.817952	−0.408976	−	0.912545i	$-0.634114\pi$
$887$	−24.3607	−0.817952	−0.408976	+	0.912545i	$0.634114\pi$
$888$	0	0
$889$	−26.4721	−0.887847
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	36.9443	1.23629
$894$	0	0
$895$	12.9443	0.432679
$896$	0	0
$897$	−29.8885	−0.997949
$898$	0	0
$899$	0	0
$900$	0	0
$901$	−30.2492	−1.00775
$902$	0	0
$903$	−41.8885	−1.39396
$904$	0	0
$905$	10.4721	0.348106
$906$	0	0
$907$	−3.05573	−0.101464	−0.0507319	−	0.998712i	$-0.516155\pi$
$907$	−3.05573	−0.101464	−0.0507319	+	0.998712i	$0.516155\pi$
$908$	0	0
$909$	3.52786	0.117012
$910$	0	0
$911$	20.1803	0.668604	0.334302	−	0.942466i	$-0.391499\pi$
$911$	20.1803	0.668604	0.334302	+	0.942466i	$0.391499\pi$
$912$	0	0
$913$	−4.94427	−0.163632
$914$	0	0
$915$	−14.4721	−0.478434
$916$	0	0
$917$	−51.7771	−1.70983
$918$	0	0
$919$	14.2918	0.471443	0.235721	−	0.971821i	$-0.424255\pi$
$919$	14.2918	0.471443	0.235721	+	0.971821i	$0.424255\pi$
$920$	0	0
$921$	−30.4721	−1.00409
$922$	0	0
$923$	−29.8885	−0.983793
$924$	0	0
$925$	−20.8328	−0.684979
$926$	0	0
$927$	−6.47214	−0.212573
$928$	0	0
$929$	−38.3607	−1.25857	−0.629287	−	0.777173i	$-0.716653\pi$
$929$	−38.3607	−1.25857	−0.629287	+	0.777173i	$0.716653\pi$
$930$	0	0
$931$	22.4721	0.736495
$932$	0	0
$933$	2.29180	0.0750300
$934$	0	0
$935$	−5.52786	−0.180780
$936$	0	0
$937$	27.8885	0.911079	0.455540	−	0.890216i	$-0.349446\pi$
$937$	27.8885	0.911079	0.455540	+	0.890216i	$0.349446\pi$
$938$	0	0
$939$	−15.8885	−0.518503
$940$	0	0
$941$	5.41641	0.176570	0.0882849	−	0.996095i	$-0.471861\pi$
$941$	5.41641	0.176570	0.0882849	+	0.996095i	$0.471861\pi$
$942$	0	0
$943$	−62.4721	−2.03437
$944$	0	0
$945$	−4.00000	−0.130120
$946$	0	0
$947$	−36.7214	−1.19328	−0.596642	−	0.802508i	$-0.703499\pi$
$947$	−36.7214	−1.19328	−0.596642	+	0.802508i	$0.703499\pi$
$948$	0	0
$949$	−31.4164	−1.01982
$950$	0	0
$951$	−7.70820	−0.249956
$952$	0	0
$953$	31.8885	1.03297	0.516486	−	0.856296i	$-0.327240\pi$
$953$	31.8885	1.03297	0.516486	+	0.856296i	$0.327240\pi$
$954$	0	0
$955$	8.94427	0.289430
$956$	0	0
$957$	−8.47214	−0.273865
$958$	0	0
$959$	−24.3607	−0.786647
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−16.9443	−0.546022
$964$	0	0
$965$	9.30495	0.299537
$966$	0	0
$967$	18.8754	0.606992	0.303496	−	0.952833i	$-0.401846\pi$
$967$	18.8754	0.606992	0.303496	+	0.952833i	$0.401846\pi$
$968$	0	0
$969$	−28.9443	−0.929824
$970$	0	0
$971$	44.3607	1.42360	0.711801	−	0.702381i	$-0.247880\pi$
$971$	44.3607	1.42360	0.711801	+	0.702381i	$0.247880\pi$
$972$	0	0
$973$	4.94427	0.158506
$974$	0	0
$975$	−18.1803	−0.582237
$976$	0	0
$977$	−23.5279	−0.752723	−0.376362	−	0.926473i	$-0.622825\pi$
$977$	−23.5279	−0.752723	−0.376362	+	0.926473i	$0.622825\pi$
$978$	0	0
$979$	−2.00000	−0.0639203
$980$	0	0
$981$	−20.6525	−0.659383
$982$	0	0
$983$	47.9574	1.52960	0.764802	−	0.644265i	$-0.222837\pi$
$983$	47.9574	1.52960	0.764802	+	0.644265i	$0.222837\pi$
$984$	0	0
$985$	−6.69505	−0.213322
$986$	0	0
$987$	18.4721	0.587975
$988$	0	0
$989$	73.8885	2.34952
$990$	0	0
$991$	25.8885	0.822377	0.411188	−	0.911550i	$-0.365114\pi$
$991$	25.8885	0.822377	0.411188	+	0.911550i	$0.365114\pi$
$992$	0	0
$993$	−16.9443	−0.537710
$994$	0	0
$995$	4.22291	0.133875
$996$	0	0
$997$	−39.7082	−1.25757	−0.628786	−	0.777579i	$-0.716448\pi$
$997$	−39.7082	−1.25757	−0.628786	+	0.777579i	$0.716448\pi$
$998$	0	0
$999$	6.00000	0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1056.2.a.l.1.1	yes	2
3.2	odd	2		3168.2.a.bf.1.2		2
4.3	odd	2		1056.2.a.k.1.1	✓	2
8.3	odd	2		2112.2.a.bf.1.2		2
8.5	even	2		2112.2.a.be.1.2		2
12.11	even	2		3168.2.a.be.1.2		2
24.5	odd	2		6336.2.a.ct.1.1		2
24.11	even	2		6336.2.a.cs.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1056.2.a.k.1.1	✓	2	4.3	odd	2
1056.2.a.l.1.1	yes	2	1.1	even	1	trivial
2112.2.a.be.1.2		2	8.5	even	2
2112.2.a.bf.1.2		2	8.3	odd	2
3168.2.a.be.1.2		2	12.11	even	2
3168.2.a.bf.1.2		2	3.2	odd	2
6336.2.a.cs.1.1		2	24.11	even	2
6336.2.a.ct.1.1		2	24.5	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:	\( N \)	\(=\)	\( 1056 = 2^{5} \cdot 3 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1056.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.23607 q^{5} +3.23607 q^{7} +1.00000 q^{9} -1.00000 q^{11} +5.23607 q^{13} -1.23607 q^{15} -4.47214 q^{17} +6.47214 q^{19} +3.23607 q^{21} -5.70820 q^{23} -3.47214 q^{25} +1.00000 q^{27} +8.47214 q^{29} -1.00000 q^{33} -4.00000 q^{35} +6.00000 q^{37} +5.23607 q^{39} +10.9443 q^{41} -12.9443 q^{43} -1.23607 q^{45} +5.70820 q^{47} +3.47214 q^{49} -4.47214 q^{51} +6.76393 q^{53} +1.23607 q^{55} +6.47214 q^{57} -5.52786 q^{59} +11.7082 q^{61} +3.23607 q^{63} -6.47214 q^{65} +8.00000 q^{67} -5.70820 q^{69} -5.70820 q^{71} -6.00000 q^{73} -3.47214 q^{75} -3.23607 q^{77} +1.70820 q^{79} +1.00000 q^{81} +4.94427 q^{83} +5.52786 q^{85} +8.47214 q^{87} +2.00000 q^{89} +16.9443 q^{91} -8.00000 q^{95} -8.47214 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 2 q^{11} + 6 q^{13} + 2 q^{15} + 4 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 8 q^{35} + 12 q^{37} + 6 q^{39} + 4 q^{41} - 8 q^{43}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 2 * q^11 + 6 * q^13 + 2 * q^15 + 4 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^33 - 8 * q^35 + 12 * q^37 + 6 * q^39 + 4 * q^41 - 8 * q^43 + 2 * q^45 - 2 * q^47 - 2 * q^49 + 18 * q^53 - 2 * q^55 + 4 * q^57 - 20 * q^59 + 10 * q^61 + 2 * q^63 - 4 * q^65 + 16 * q^67 + 2 * q^69 + 2 * q^71 - 12 * q^73 + 2 * q^75 - 2 * q^77 - 10 * q^79 + 2 * q^81 - 8 * q^83 + 20 * q^85 + 8 * q^87 + 4 * q^89 + 16 * q^91 - 16 * q^95 - 8 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more