LMFDB - Embedded newform 4024.2.a.f.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.19201 q^{3} -0.00829431 q^{5} +3.59968 q^{7} -1.57911 q^{9} +O(q^{10})$	$q-1.19201 q^{3} -0.00829431 q^{5} +3.59968 q^{7} -1.57911 q^{9} -0.236191 q^{11} +1.29395 q^{13} +0.00988691 q^{15} -1.05359 q^{17} +2.02775 q^{19} -4.29086 q^{21} +0.342412 q^{23} -4.99993 q^{25} +5.45835 q^{27} +4.97523 q^{29} +4.91753 q^{31} +0.281542 q^{33} -0.0298569 q^{35} -10.6369 q^{37} -1.54240 q^{39} +8.34963 q^{41} -1.40443 q^{43} +0.0130976 q^{45} -0.719341 q^{47} +5.95770 q^{49} +1.25589 q^{51} +12.5435 q^{53} +0.00195904 q^{55} -2.41710 q^{57} -1.26898 q^{59} +9.37657 q^{61} -5.68429 q^{63} -0.0107324 q^{65} -0.639380 q^{67} -0.408159 q^{69} -16.7878 q^{71} -0.333178 q^{73} +5.95997 q^{75} -0.850211 q^{77} +0.976942 q^{79} -1.76909 q^{81} -8.02026 q^{83} +0.00873880 q^{85} -5.93053 q^{87} +14.7851 q^{89} +4.65780 q^{91} -5.86175 q^{93} -0.0168188 q^{95} +4.97360 q^{97} +0.372971 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	$ 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) $ 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−1.19201	−0.688208	−0.344104	−	0.938932i	$-0.611817\pi$
$3$	−1.19201	−0.688208	−0.344104	+	0.938932i	$0.611817\pi$
$4$	0	0
$5$	−0.00829431	−0.00370933	−0.00185466	−	0.999998i	$-0.500590\pi$
$5$	−0.00829431	−0.00370933	−0.00185466	+	0.999998i	$0.500590\pi$
$6$	0	0
$7$	3.59968	1.36055	0.680276	−	0.732956i	$-0.261860\pi$
$7$	3.59968	1.36055	0.680276	+	0.732956i	$0.261860\pi$
$8$	0	0
$9$	−1.57911	−0.526370
$10$	0	0
$11$	−0.236191	−0.0712142	−0.0356071	−	0.999366i	$-0.511336\pi$
$11$	−0.236191	−0.0712142	−0.0356071	+	0.999366i	$0.511336\pi$
$12$	0	0
$13$	1.29395	0.358877	0.179438	−	0.983769i	$-0.442572\pi$
$13$	1.29395	0.358877	0.179438	+	0.983769i	$0.442572\pi$
$14$	0	0
$15$	0.00988691	0.00255279
$16$	0	0
$17$	−1.05359	−0.255533	−0.127766	−	0.991804i	$-0.540781\pi$
$17$	−1.05359	−0.255533	−0.127766	+	0.991804i	$0.540781\pi$
$18$	0	0
$19$	2.02775	0.465198	0.232599	−	0.972573i	$-0.425277\pi$
$19$	2.02775	0.465198	0.232599	+	0.972573i	$0.425277\pi$
$20$	0	0
$21$	−4.29086	−0.936342
$22$	0	0
$23$	0.342412	0.0713978	0.0356989	−	0.999363i	$-0.488634\pi$
$23$	0.342412	0.0713978	0.0356989	+	0.999363i	$0.488634\pi$
$24$	0	0
$25$	−4.99993	−0.999986
$26$	0	0
$27$	5.45835	1.05046
$28$	0	0
$29$	4.97523	0.923878	0.461939	−	0.886912i	$-0.347154\pi$
$29$	4.97523	0.923878	0.461939	+	0.886912i	$0.347154\pi$
$30$	0	0
$31$	4.91753	0.883214	0.441607	−	0.897209i	$-0.354409\pi$
$31$	4.91753	0.883214	0.441607	+	0.897209i	$0.354409\pi$
$32$	0	0
$33$	0.281542	0.0490102
$34$	0	0
$35$	−0.0298569	−0.00504673
$36$	0	0
$37$	−10.6369	−1.74870	−0.874349	−	0.485298i	$-0.838711\pi$
$37$	−10.6369	−1.74870	−0.874349	+	0.485298i	$0.838711\pi$
$38$	0	0
$39$	−1.54240	−0.246982
$40$	0	0
$41$	8.34963	1.30399	0.651997	−	0.758222i	$-0.273931\pi$
$41$	8.34963	1.30399	0.651997	+	0.758222i	$0.273931\pi$
$42$	0	0
$43$	−1.40443	−0.214174	−0.107087	−	0.994250i	$-0.534152\pi$
$43$	−1.40443	−0.214174	−0.107087	+	0.994250i	$0.534152\pi$
$44$	0	0
$45$	0.0130976	0.00195248
$46$	0	0
$47$	−0.719341	−0.104927	−0.0524633	−	0.998623i	$-0.516707\pi$
$47$	−0.719341	−0.104927	−0.0524633	+	0.998623i	$0.516707\pi$
$48$	0	0
$49$	5.95770	0.851100
$50$	0	0
$51$	1.25589	0.175860
$52$	0	0
$53$	12.5435	1.72298	0.861489	−	0.507776i	$-0.169532\pi$
$53$	12.5435	1.72298	0.861489	+	0.507776i	$0.169532\pi$
$54$	0	0
$55$	0.00195904	0.000264157 0
$56$	0	0
$57$	−2.41710	−0.320153
$58$	0	0
$59$	−1.26898	−0.165207	−0.0826036	−	0.996582i	$-0.526324\pi$
$59$	−1.26898	−0.165207	−0.0826036	+	0.996582i	$0.526324\pi$
$60$	0	0
$61$	9.37657	1.20055	0.600274	−	0.799795i	$-0.295058\pi$
$61$	9.37657	1.20055	0.600274	+	0.799795i	$0.295058\pi$
$62$	0	0
$63$	−5.68429	−0.716153
$64$	0	0
$65$	−0.0107324	−0.00133119
$66$	0	0
$67$	−0.639380	−0.0781127	−0.0390563	−	0.999237i	$-0.512435\pi$
$67$	−0.639380	−0.0781127	−0.0390563	+	0.999237i	$0.512435\pi$
$68$	0	0
$69$	−0.408159	−0.0491366
$70$	0	0
$71$	−16.7878	−1.99234	−0.996172	−	0.0874194i	$-0.972138\pi$
$71$	−16.7878	−1.99234	−0.996172	+	0.0874194i	$0.972138\pi$
$72$	0	0
$73$	−0.333178	−0.0389955	−0.0194978	−	0.999810i	$-0.506207\pi$
$73$	−0.333178	−0.0389955	−0.0194978	+	0.999810i	$0.506207\pi$
$74$	0	0
$75$	5.95997	0.688199
$76$	0	0
$77$	−0.850211	−0.0968906
$78$	0	0
$79$	0.976942	0.109915	0.0549573	−	0.998489i	$-0.482498\pi$
$79$	0.976942	0.109915	0.0549573	+	0.998489i	$0.482498\pi$
$80$	0	0
$81$	−1.76909	−0.196565
$82$	0	0
$83$	−8.02026	−0.880338	−0.440169	−	0.897915i	$-0.645081\pi$
$83$	−8.02026	−0.880338	−0.440169	+	0.897915i	$0.645081\pi$
$84$	0	0
$85$	0.00873880	0.000947856 0
$86$	0	0
$87$	−5.93053	−0.635820
$88$	0	0
$89$	14.7851	1.56722	0.783611	−	0.621251i	$-0.213375\pi$
$89$	14.7851	1.56722	0.783611	+	0.621251i	$0.213375\pi$
$90$	0	0
$91$	4.65780	0.488270
$92$	0	0
$93$	−5.86175	−0.607835
$94$	0	0
$95$	−0.0168188	−0.00172557
$96$	0	0
$97$	4.97360	0.504993	0.252496	−	0.967598i	$-0.418748\pi$
$97$	4.97360	0.504993	0.252496	+	0.967598i	$0.418748\pi$
$98$	0	0
$99$	0.372971	0.0374850
$100$	0	0
$101$	6.51261	0.648029	0.324014	−	0.946052i	$-0.394967\pi$
$101$	6.51261	0.648029	0.324014	+	0.946052i	$0.394967\pi$
$102$	0	0
$103$	−0.895767	−0.0882625	−0.0441313	−	0.999026i	$-0.514052\pi$
$103$	−0.895767	−0.0882625	−0.0441313	+	0.999026i	$0.514052\pi$
$104$	0	0
$105$	0.0355897	0.00347320
$106$	0	0
$107$	−0.749199	−0.0724278	−0.0362139	−	0.999344i	$-0.511530\pi$
$107$	−0.749199	−0.0724278	−0.0362139	+	0.999344i	$0.511530\pi$
$108$	0	0
$109$	−0.920480	−0.0881660	−0.0440830	−	0.999028i	$-0.514037\pi$
$109$	−0.920480	−0.0881660	−0.0440830	+	0.999028i	$0.514037\pi$
$110$	0	0
$111$	12.6793	1.20347
$112$	0	0
$113$	−1.99729	−0.187889	−0.0939446	−	0.995577i	$-0.529948\pi$
$113$	−1.99729	−0.187889	−0.0939446	+	0.995577i	$0.529948\pi$
$114$	0	0
$115$	−0.00284007	−0.000264838 0
$116$	0	0
$117$	−2.04328	−0.188902
$118$	0	0
$119$	−3.79259	−0.347666
$120$	0	0
$121$	−10.9442	−0.994929
$122$	0	0
$123$	−9.95286	−0.897419
$124$	0	0
$125$	0.0829426	0.00741861
$126$	0	0
$127$	−0.709657	−0.0629719	−0.0314859	−	0.999504i	$-0.510024\pi$
$127$	−0.709657	−0.0629719	−0.0314859	+	0.999504i	$0.510024\pi$
$128$	0	0
$129$	1.67410	0.147396
$130$	0	0
$131$	11.5110	1.00572	0.502862	−	0.864367i	$-0.332280\pi$
$131$	11.5110	1.00572	0.502862	+	0.864367i	$0.332280\pi$
$132$	0	0
$133$	7.29925	0.632925
$134$	0	0
$135$	−0.0452733	−0.00389650
$136$	0	0
$137$	16.6759	1.42471	0.712357	−	0.701817i	$-0.247628\pi$
$137$	16.6759	1.42471	0.712357	+	0.701817i	$0.247628\pi$
$138$	0	0
$139$	16.8999	1.43343	0.716717	−	0.697365i	$-0.245644\pi$
$139$	16.8999	1.43343	0.716717	+	0.697365i	$0.245644\pi$
$140$	0	0
$141$	0.857463	0.0722114
$142$	0	0
$143$	−0.305619	−0.0255571
$144$	0	0
$145$	−0.0412661	−0.00342697
$146$	0	0
$147$	−7.10165	−0.585734
$148$	0	0
$149$	−6.82365	−0.559015	−0.279508	−	0.960143i	$-0.590171\pi$
$149$	−6.82365	−0.559015	−0.279508	+	0.960143i	$0.590171\pi$
$150$	0	0
$151$	−14.7773	−1.20256	−0.601279	−	0.799039i	$-0.705342\pi$
$151$	−14.7773	−1.20256	−0.601279	+	0.799039i	$0.705342\pi$
$152$	0	0
$153$	1.66373	0.134505
$154$	0	0
$155$	−0.0407875	−0.00327613
$156$	0	0
$157$	−3.61485	−0.288497	−0.144248	−	0.989542i	$-0.546076\pi$
$157$	−3.61485	−0.288497	−0.144248	+	0.989542i	$0.546076\pi$
$158$	0	0
$159$	−14.9520	−1.18577
$160$	0	0
$161$	1.23257	0.0971405
$162$	0	0
$163$	4.24645	0.332607	0.166304	−	0.986075i	$-0.446817\pi$
$163$	4.24645	0.332607	0.166304	+	0.986075i	$0.446817\pi$
$164$	0	0
$165$	−0.00233520	−0.000181795 0
$166$	0	0
$167$	13.4947	1.04425	0.522126	−	0.852868i	$-0.325139\pi$
$167$	13.4947	1.04425	0.522126	+	0.852868i	$0.325139\pi$
$168$	0	0
$169$	−11.3257	−0.871208
$170$	0	0
$171$	−3.20204	−0.244866
$172$	0	0
$173$	17.6053	1.33850	0.669252	−	0.743036i	$-0.266615\pi$
$173$	17.6053	1.33850	0.669252	+	0.743036i	$0.266615\pi$
$174$	0	0
$175$	−17.9982	−1.36053
$176$	0	0
$177$	1.51264	0.113697
$178$	0	0
$179$	10.9093	0.815403	0.407701	−	0.913115i	$-0.366330\pi$
$179$	10.9093	0.815403	0.407701	+	0.913115i	$0.366330\pi$
$180$	0	0
$181$	−6.96914	−0.518012	−0.259006	−	0.965876i	$-0.583395\pi$
$181$	−6.96914	−0.518012	−0.259006	+	0.965876i	$0.583395\pi$
$182$	0	0
$183$	−11.1770	−0.826226
$184$	0	0
$185$	0.0882259	0.00648649
$186$	0	0
$187$	0.248848	0.0181976
$188$	0	0
$189$	19.6483	1.42920
$190$	0	0
$191$	20.9001	1.51228	0.756140	−	0.654410i	$-0.227083\pi$
$191$	20.9001	1.51228	0.756140	+	0.654410i	$0.227083\pi$
$192$	0	0
$193$	9.90789	0.713186	0.356593	−	0.934260i	$-0.383938\pi$
$193$	9.90789	0.713186	0.356593	+	0.934260i	$0.383938\pi$
$194$	0	0
$195$	0.0127932	0.000916137 0
$196$	0	0
$197$	6.85518	0.488411	0.244206	−	0.969723i	$-0.421473\pi$
$197$	6.85518	0.488411	0.244206	+	0.969723i	$0.421473\pi$
$198$	0	0
$199$	−11.3990	−0.808050	−0.404025	−	0.914748i	$-0.632389\pi$
$199$	−11.3990	−0.808050	−0.404025	+	0.914748i	$0.632389\pi$
$200$	0	0
$201$	0.762148	0.0537578
$202$	0	0
$203$	17.9093	1.25698
$204$	0	0
$205$	−0.0692545	−0.00483694
$206$	0	0
$207$	−0.540706	−0.0375817
$208$	0	0
$209$	−0.478936	−0.0331287
$210$	0	0
$211$	8.29917	0.571338	0.285669	−	0.958328i	$-0.407784\pi$
$211$	8.29917	0.571338	0.285669	+	0.958328i	$0.407784\pi$
$212$	0	0
$213$	20.0112	1.37115
$214$	0	0
$215$	0.0116488	0.000794443 0
$216$	0	0
$217$	17.7015	1.20166
$218$	0	0
$219$	0.397152	0.0268370
$220$	0	0
$221$	−1.36329	−0.0917048
$222$	0	0
$223$	−6.17678	−0.413628	−0.206814	−	0.978380i	$-0.566309\pi$
$223$	−6.17678	−0.413628	−0.206814	+	0.978380i	$0.566309\pi$
$224$	0	0
$225$	7.89544	0.526363
$226$	0	0
$227$	4.81764	0.319758	0.159879	−	0.987137i	$-0.448890\pi$
$227$	4.81764	0.319758	0.159879	+	0.987137i	$0.448890\pi$
$228$	0	0
$229$	14.1332	0.933945	0.466973	−	0.884272i	$-0.345345\pi$
$229$	14.1332	0.933945	0.466973	+	0.884272i	$0.345345\pi$
$230$	0	0
$231$	1.01346	0.0666809
$232$	0	0
$233$	20.4582	1.34026	0.670131	−	0.742242i	$-0.266238\pi$
$233$	20.4582	1.34026	0.670131	+	0.742242i	$0.266238\pi$
$234$	0	0
$235$	0.00596644	0.000389208 0
$236$	0	0
$237$	−1.16453	−0.0756441
$238$	0	0
$239$	14.0320	0.907653	0.453826	−	0.891090i	$-0.350059\pi$
$239$	14.0320	0.907653	0.453826	+	0.891090i	$0.350059\pi$
$240$	0	0
$241$	5.55506	0.357833	0.178917	−	0.983864i	$-0.442741\pi$
$241$	5.55506	0.357833	0.178917	+	0.983864i	$0.442741\pi$
$242$	0	0
$243$	−14.2663	−0.915182
$244$	0	0
$245$	−0.0494151	−0.00315701
$246$	0	0
$247$	2.62380	0.166949
$248$	0	0
$249$	9.56024	0.605855
$250$	0	0
$251$	11.7296	0.740369	0.370184	−	0.928958i	$-0.379295\pi$
$251$	11.7296	0.740369	0.370184	+	0.928958i	$0.379295\pi$
$252$	0	0
$253$	−0.0808746	−0.00508454
$254$	0	0
$255$	−0.0104167	−0.000652322 0
$256$	0	0
$257$	12.7180	0.793326	0.396663	−	0.917964i	$-0.370168\pi$
$257$	12.7180	0.793326	0.396663	+	0.917964i	$0.370168\pi$
$258$	0	0
$259$	−38.2895	−2.37919
$260$	0	0
$261$	−7.85644	−0.486301
$262$	0	0
$263$	−24.3705	−1.50275	−0.751374	−	0.659877i	$-0.770608\pi$
$263$	−24.3705	−1.50275	−0.751374	+	0.659877i	$0.770608\pi$
$264$	0	0
$265$	−0.104039	−0.00639109
$266$	0	0
$267$	−17.6241	−1.07858
$268$	0	0
$269$	4.98980	0.304234	0.152117	−	0.988363i	$-0.451391\pi$
$269$	4.98980	0.304234	0.152117	+	0.988363i	$0.451391\pi$
$270$	0	0
$271$	0.392505	0.0238430	0.0119215	−	0.999929i	$-0.496205\pi$
$271$	0.392505	0.0238430	0.0119215	+	0.999929i	$0.496205\pi$
$272$	0	0
$273$	−5.55215	−0.336031
$274$	0	0
$275$	1.18094	0.0712132
$276$	0	0
$277$	−3.81870	−0.229444	−0.114722	−	0.993398i	$-0.536598\pi$
$277$	−3.81870	−0.229444	−0.114722	+	0.993398i	$0.536598\pi$
$278$	0	0
$279$	−7.76531	−0.464897
$280$	0	0
$281$	17.0967	1.01990	0.509950	−	0.860204i	$-0.329664\pi$
$281$	17.0967	1.01990	0.509950	+	0.860204i	$0.329664\pi$
$282$	0	0
$283$	5.29789	0.314927	0.157463	−	0.987525i	$-0.449668\pi$
$283$	5.29789	0.314927	0.157463	+	0.987525i	$0.449668\pi$
$284$	0	0
$285$	0.0200482	0.00118755
$286$	0	0
$287$	30.0560	1.77415
$288$	0	0
$289$	−15.8899	−0.934703
$290$	0	0
$291$	−5.92859	−0.347540
$292$	0	0
$293$	30.5744	1.78618	0.893088	−	0.449882i	$-0.148534\pi$
$293$	30.5744	1.78618	0.893088	+	0.449882i	$0.148534\pi$
$294$	0	0
$295$	0.0105253	0.000612808 0
$296$	0	0
$297$	−1.28921	−0.0748077
$298$	0	0
$299$	0.443063	0.0256230
$300$	0	0
$301$	−5.05552	−0.291395
$302$	0	0
$303$	−7.76310	−0.445978
$304$	0	0
$305$	−0.0777722	−0.00445322
$306$	0	0
$307$	21.4467	1.22403	0.612015	−	0.790846i	$-0.290359\pi$
$307$	21.4467	1.22403	0.612015	+	0.790846i	$0.290359\pi$
$308$	0	0
$309$	1.06776	0.0607430
$310$	0	0
$311$	−23.7786	−1.34836	−0.674181	−	0.738567i	$-0.735503\pi$
$311$	−23.7786	−1.34836	−0.674181	+	0.738567i	$0.735503\pi$
$312$	0	0
$313$	16.5848	0.937428	0.468714	−	0.883350i	$-0.344718\pi$
$313$	16.5848	0.937428	0.468714	+	0.883350i	$0.344718\pi$
$314$	0	0
$315$	0.0471473	0.00265645
$316$	0	0
$317$	4.29146	0.241032	0.120516	−	0.992711i	$-0.461545\pi$
$317$	4.29146	0.241032	0.120516	+	0.992711i	$0.461545\pi$
$318$	0	0
$319$	−1.17510	−0.0657932
$320$	0	0
$321$	0.893054	0.0498454
$322$	0	0
$323$	−2.13642	−0.118873
$324$	0	0
$325$	−6.46965	−0.358872
$326$	0	0
$327$	1.09722	0.0606765
$328$	0	0
$329$	−2.58940	−0.142758
$330$	0	0
$331$	12.5642	0.690589	0.345294	−	0.938494i	$-0.387779\pi$
$331$	12.5642	0.690589	0.345294	+	0.938494i	$0.387779\pi$
$332$	0	0
$333$	16.7968	0.920461
$334$	0	0
$335$	0.00530321	0.000289746 0
$336$	0	0
$337$	2.69627	0.146875	0.0734377	−	0.997300i	$-0.476603\pi$
$337$	2.69627	0.146875	0.0734377	+	0.997300i	$0.476603\pi$
$338$	0	0
$339$	2.38079	0.129307
$340$	0	0
$341$	−1.16147	−0.0628974
$342$	0	0
$343$	−3.75194	−0.202586
$344$	0	0
$345$	0.00338540	0.000182264 0
$346$	0	0
$347$	−17.1120	−0.918622	−0.459311	−	0.888276i	$-0.651904\pi$
$347$	−17.1120	−0.918622	−0.459311	+	0.888276i	$0.651904\pi$
$348$	0	0
$349$	5.58668	0.299048	0.149524	−	0.988758i	$-0.452226\pi$
$349$	5.58668	0.299048	0.149524	+	0.988758i	$0.452226\pi$
$350$	0	0
$351$	7.06282	0.376985
$352$	0	0
$353$	−5.32114	−0.283216	−0.141608	−	0.989923i	$-0.545227\pi$
$353$	−5.32114	−0.283216	−0.141608	+	0.989923i	$0.545227\pi$
$354$	0	0
$355$	0.139243	0.00739026
$356$	0	0
$357$	4.52081	0.239266
$358$	0	0
$359$	−23.6161	−1.24641	−0.623205	−	0.782059i	$-0.714170\pi$
$359$	−23.6161	−1.24641	−0.623205	+	0.782059i	$0.714170\pi$
$360$	0	0
$361$	−14.8882	−0.783591
$362$	0	0
$363$	13.0456	0.684718
$364$	0	0
$365$	0.00276348	0.000144647 0
$366$	0	0
$367$	−26.4302	−1.37964	−0.689822	−	0.723979i	$-0.742311\pi$
$367$	−26.4302	−1.37964	−0.689822	+	0.723979i	$0.742311\pi$
$368$	0	0
$369$	−13.1850	−0.686383
$370$	0	0
$371$	45.1525	2.34420
$372$	0	0
$373$	35.5135	1.83882	0.919411	−	0.393299i	$-0.128666\pi$
$373$	35.5135	1.83882	0.919411	+	0.393299i	$0.128666\pi$
$374$	0	0
$375$	−0.0988685	−0.00510555
$376$	0	0
$377$	6.43769	0.331558
$378$	0	0
$379$	30.5639	1.56996	0.784982	−	0.619519i	$-0.212672\pi$
$379$	30.5639	1.56996	0.784982	+	0.619519i	$0.212672\pi$
$380$	0	0
$381$	0.845919	0.0433378
$382$	0	0
$383$	24.6088	1.25745	0.628727	−	0.777626i	$-0.283576\pi$
$383$	24.6088	1.25745	0.628727	+	0.777626i	$0.283576\pi$
$384$	0	0
$385$	0.00705192	0.000359399 0
$386$	0	0
$387$	2.21776	0.112735
$388$	0	0
$389$	13.5310	0.686049	0.343024	−	0.939326i	$-0.388549\pi$
$389$	13.5310	0.686049	0.343024	+	0.939326i	$0.388549\pi$
$390$	0	0
$391$	−0.360762	−0.0182445
$392$	0	0
$393$	−13.7213	−0.692148
$394$	0	0
$395$	−0.00810306	−0.000407709 0
$396$	0	0
$397$	8.79828	0.441573	0.220786	−	0.975322i	$-0.429138\pi$
$397$	8.79828	0.441573	0.220786	+	0.975322i	$0.429138\pi$
$398$	0	0
$399$	−8.70079	−0.435584
$400$	0	0
$401$	39.0670	1.95091	0.975456	−	0.220196i	$-0.0706697\pi$
$401$	39.0670	1.95091	0.975456	+	0.220196i	$0.0706697\pi$
$402$	0	0
$403$	6.36302	0.316965
$404$	0	0
$405$	0.0146734	0.000729125 0
$406$	0	0
$407$	2.51234	0.124532
$408$	0	0
$409$	4.62095	0.228491	0.114246	−	0.993453i	$-0.463555\pi$
$409$	4.62095	0.228491	0.114246	+	0.993453i	$0.463555\pi$
$410$	0	0
$411$	−19.8778	−0.980500
$412$	0	0
$413$	−4.56793	−0.224773
$414$	0	0
$415$	0.0665225	0.00326546
$416$	0	0
$417$	−20.1449	−0.986500
$418$	0	0
$419$	−1.61521	−0.0789081	−0.0394540	−	0.999221i	$-0.512562\pi$
$419$	−1.61521	−0.0789081	−0.0394540	+	0.999221i	$0.512562\pi$
$420$	0	0
$421$	36.5018	1.77899	0.889494	−	0.456947i	$-0.151057\pi$
$421$	36.5018	1.77899	0.889494	+	0.456947i	$0.151057\pi$
$422$	0	0
$423$	1.13592	0.0552302
$424$	0	0
$425$	5.26787	0.255529
$426$	0	0
$427$	33.7527	1.63341
$428$	0	0
$429$	0.364301	0.0175886
$430$	0	0
$431$	−4.63846	−0.223427	−0.111713	−	0.993740i	$-0.535634\pi$
$431$	−4.63846	−0.223427	−0.111713	+	0.993740i	$0.535634\pi$
$432$	0	0
$433$	−10.8951	−0.523586	−0.261793	−	0.965124i	$-0.584314\pi$
$433$	−10.8951	−0.523586	−0.261793	+	0.965124i	$0.584314\pi$
$434$	0	0
$435$	0.0491897	0.00235847
$436$	0	0
$437$	0.694326	0.0332141
$438$	0	0
$439$	−15.4147	−0.735701	−0.367851	−	0.929885i	$-0.619906\pi$
$439$	−15.4147	−0.735701	−0.367851	+	0.929885i	$0.619906\pi$
$440$	0	0
$441$	−9.40786	−0.447994
$442$	0	0
$443$	−16.2641	−0.772729	−0.386364	−	0.922346i	$-0.626269\pi$
$443$	−16.2641	−0.772729	−0.386364	+	0.922346i	$0.626269\pi$
$444$	0	0
$445$	−0.122633	−0.00581335
$446$	0	0
$447$	8.13387	0.384719
$448$	0	0
$449$	−32.6158	−1.53923	−0.769617	−	0.638506i	$-0.779553\pi$
$449$	−32.6158	−1.53923	−0.769617	+	0.638506i	$0.779553\pi$
$450$	0	0
$451$	−1.97211	−0.0928629
$452$	0	0
$453$	17.6147	0.827610
$454$	0	0
$455$	−0.0386332	−0.00181115
$456$	0	0
$457$	9.38180	0.438862	0.219431	−	0.975628i	$-0.429580\pi$
$457$	9.38180	0.438862	0.219431	+	0.975628i	$0.429580\pi$
$458$	0	0
$459$	−5.75086	−0.268427
$460$	0	0
$461$	−12.3689	−0.576078	−0.288039	−	0.957619i	$-0.593003\pi$
$461$	−12.3689	−0.576078	−0.288039	+	0.957619i	$0.593003\pi$
$462$	0	0
$463$	−20.5079	−0.953083	−0.476541	−	0.879152i	$-0.658110\pi$
$463$	−20.5079	−0.953083	−0.476541	+	0.879152i	$0.658110\pi$
$464$	0	0
$465$	0.0486192	0.00225466
$466$	0	0
$467$	0.212142	0.00981674	0.00490837	−	0.999988i	$-0.498438\pi$
$467$	0.212142	0.00981674	0.00490837	+	0.999988i	$0.498438\pi$
$468$	0	0
$469$	−2.30156	−0.106276
$470$	0	0
$471$	4.30895	0.198546
$472$	0	0
$473$	0.331714	0.0152522
$474$	0	0
$475$	−10.1386	−0.465191
$476$	0	0
$477$	−19.8075	−0.906924
$478$	0	0
$479$	8.00300	0.365666	0.182833	−	0.983144i	$-0.441473\pi$
$479$	8.00300	0.365666	0.182833	+	0.983144i	$0.441473\pi$
$480$	0	0
$481$	−13.7636	−0.627566
$482$	0	0
$483$	−1.46924	−0.0668528
$484$	0	0
$485$	−0.0412526	−0.00187319
$486$	0	0
$487$	−4.88745	−0.221471	−0.110736	−	0.993850i	$-0.535321\pi$
$487$	−4.88745	−0.221471	−0.110736	+	0.993850i	$0.535321\pi$
$488$	0	0
$489$	−5.06181	−0.228903
$490$	0	0
$491$	16.6330	0.750637	0.375318	−	0.926896i	$-0.377533\pi$
$491$	16.6330	0.750637	0.375318	+	0.926896i	$0.377533\pi$
$492$	0	0
$493$	−5.24185	−0.236081
$494$	0	0
$495$	−0.00309354	−0.000139044 0
$496$	0	0
$497$	−60.4307	−2.71069
$498$	0	0
$499$	−14.8792	−0.666084	−0.333042	−	0.942912i	$-0.608075\pi$
$499$	−14.8792	−0.666084	−0.333042	+	0.942912i	$0.608075\pi$
$500$	0	0
$501$	−16.0858	−0.718663
$502$	0	0
$503$	−1.00000	−0.0445878
$503$	−1.00000	−0.0445878
$504$	0	0
$505$	−0.0540176	−0.00240375
$506$	0	0
$507$	13.5004	0.599572
$508$	0	0
$509$	−14.3413	−0.635669	−0.317834	−	0.948146i	$-0.602956\pi$
$509$	−14.3413	−0.635669	−0.317834	+	0.948146i	$0.602956\pi$
$510$	0	0
$511$	−1.19933	−0.0530554
$512$	0	0
$513$	11.0682	0.488672
$514$	0	0
$515$	0.00742977	0.000327395 0
$516$	0	0
$517$	0.169902	0.00747227
$518$	0	0
$519$	−20.9857	−0.921169
$520$	0	0
$521$	−4.98720	−0.218493	−0.109247	−	0.994015i	$-0.534844\pi$
$521$	−4.98720	−0.218493	−0.109247	+	0.994015i	$0.534844\pi$
$522$	0	0
$523$	−29.1762	−1.27579	−0.637893	−	0.770125i	$-0.720194\pi$
$523$	−29.1762	−1.27579	−0.637893	+	0.770125i	$0.720194\pi$
$524$	0	0
$525$	21.4540	0.936330
$526$	0	0
$527$	−5.18105	−0.225690
$528$	0	0
$529$	−22.8828	−0.994902
$530$	0	0
$531$	2.00386	0.0869601
$532$	0	0
$533$	10.8040	0.467973
$534$	0	0
$535$	0.00621409	0.000268659 0
$536$	0	0
$537$	−13.0041	−0.561167
$538$	0	0
$539$	−1.40715	−0.0606104
$540$	0	0
$541$	8.79189	0.377993	0.188996	−	0.981978i	$-0.439477\pi$
$541$	8.79189	0.377993	0.188996	+	0.981978i	$0.439477\pi$
$542$	0	0
$543$	8.30729	0.356500
$544$	0	0
$545$	0.00763475	0.000327037 0
$546$	0	0
$547$	29.8955	1.27824	0.639121	−	0.769107i	$-0.279298\pi$
$547$	29.8955	1.27824	0.639121	+	0.769107i	$0.279298\pi$
$548$	0	0
$549$	−14.8066	−0.631932
$550$	0	0
$551$	10.0885	0.429786
$552$	0	0
$553$	3.51668	0.149544
$554$	0	0
$555$	−0.105166	−0.00446406
$556$	0	0
$557$	−15.3637	−0.650980	−0.325490	−	0.945545i	$-0.605529\pi$
$557$	−15.3637	−0.650980	−0.325490	+	0.945545i	$0.605529\pi$
$558$	0	0
$559$	−1.81726	−0.0768621
$560$	0	0
$561$	−0.296630	−0.0125237
$562$	0	0
$563$	−6.86317	−0.289248	−0.144624	−	0.989487i	$-0.546197\pi$
$563$	−6.86317	−0.289248	−0.144624	+	0.989487i	$0.546197\pi$
$564$	0	0
$565$	0.0165661	0.000696943 0
$566$	0	0
$567$	−6.36815	−0.267437
$568$	0	0
$569$	−33.6082	−1.40893	−0.704464	−	0.709740i	$-0.748812\pi$
$569$	−33.6082	−1.40893	−0.704464	+	0.709740i	$0.748812\pi$
$570$	0	0
$571$	−0.850711	−0.0356011	−0.0178006	−	0.999842i	$-0.505666\pi$
$571$	−0.850711	−0.0356011	−0.0178006	+	0.999842i	$0.505666\pi$
$572$	0	0
$573$	−24.9132	−1.04076
$574$	0	0
$575$	−1.71204	−0.0713969
$576$	0	0
$577$	−25.5379	−1.06316	−0.531579	−	0.847009i	$-0.678401\pi$
$577$	−25.5379	−1.06316	−0.531579	+	0.847009i	$0.678401\pi$
$578$	0	0
$579$	−11.8103	−0.490820
$580$	0	0
$581$	−28.8704	−1.19774
$582$	0	0
$583$	−2.96265	−0.122700
$584$	0	0
$585$	0.0169476	0.000700699 0
$586$	0	0
$587$	−31.0960	−1.28347	−0.641735	−	0.766927i	$-0.721785\pi$
$587$	−31.0960	−1.28347	−0.641735	+	0.766927i	$0.721785\pi$
$588$	0	0
$589$	9.97151	0.410869
$590$	0	0
$591$	−8.17145	−0.336128
$592$	0	0
$593$	2.09175	0.0858977	0.0429489	−	0.999077i	$-0.486325\pi$
$593$	2.09175	0.0858977	0.0429489	+	0.999077i	$0.486325\pi$
$594$	0	0
$595$	0.0314569	0.00128961
$596$	0	0
$597$	13.5877	0.556107
$598$	0	0
$599$	−30.8627	−1.26101	−0.630507	−	0.776183i	$-0.717153\pi$
$599$	−30.8627	−1.26101	−0.630507	+	0.776183i	$0.717153\pi$
$600$	0	0
$601$	−20.9919	−0.856278	−0.428139	−	0.903713i	$-0.640831\pi$
$601$	−20.9919	−0.856278	−0.428139	+	0.903713i	$0.640831\pi$
$602$	0	0
$603$	1.00965	0.0411161
$604$	0	0
$605$	0.0907747	0.00369052
$606$	0	0
$607$	−30.3208	−1.23068	−0.615341	−	0.788261i	$-0.710982\pi$
$607$	−30.3208	−1.23068	−0.615341	+	0.788261i	$0.710982\pi$
$608$	0	0
$609$	−21.3480	−0.865066
$610$	0	0
$611$	−0.930790	−0.0376557
$612$	0	0
$613$	−14.4127	−0.582125	−0.291062	−	0.956704i	$-0.594009\pi$
$613$	−14.4127	−0.582125	−0.291062	+	0.956704i	$0.594009\pi$
$614$	0	0
$615$	0.0825521	0.00332882
$616$	0	0
$617$	−28.9846	−1.16688	−0.583438	−	0.812158i	$-0.698293\pi$
$617$	−28.9846	−1.16688	−0.583438	+	0.812158i	$0.698293\pi$
$618$	0	0
$619$	−21.8662	−0.878875	−0.439437	−	0.898273i	$-0.644822\pi$
$619$	−21.8662	−0.878875	−0.439437	+	0.898273i	$0.644822\pi$
$620$	0	0
$621$	1.86900	0.0750006
$622$	0	0
$623$	53.2218	2.13229
$624$	0	0
$625$	24.9990	0.999959
$626$	0	0
$627$	0.570897	0.0227994
$628$	0	0
$629$	11.2069	0.446850
$630$	0	0
$631$	31.8171	1.26662	0.633310	−	0.773898i	$-0.281696\pi$
$631$	31.8171	1.26662	0.633310	+	0.773898i	$0.281696\pi$
$632$	0	0
$633$	−9.89270	−0.393200
$634$	0	0
$635$	0.00588612	0.000233584 0
$636$	0	0
$637$	7.70896	0.305440
$638$	0	0
$639$	26.5097	1.04871
$640$	0	0
$641$	14.7049	0.580808	0.290404	−	0.956904i	$-0.406210\pi$
$641$	14.7049	0.580808	0.290404	+	0.956904i	$0.406210\pi$
$642$	0	0
$643$	8.85646	0.349265	0.174632	−	0.984634i	$-0.444126\pi$
$643$	8.85646	0.349265	0.174632	+	0.984634i	$0.444126\pi$
$644$	0	0
$645$	−0.0138855	−0.000546742 0
$646$	0	0
$647$	8.64142	0.339729	0.169865	−	0.985467i	$-0.445667\pi$
$647$	8.64142	0.339729	0.169865	+	0.985467i	$0.445667\pi$
$648$	0	0
$649$	0.299722	0.0117651
$650$	0	0
$651$	−21.1004	−0.826990
$652$	0	0
$653$	−10.1436	−0.396950	−0.198475	−	0.980106i	$-0.563599\pi$
$653$	−10.1436	−0.396950	−0.198475	+	0.980106i	$0.563599\pi$
$654$	0	0
$655$	−0.0954762	−0.00373056
$656$	0	0
$657$	0.526125	0.0205261
$658$	0	0
$659$	−3.71237	−0.144614	−0.0723068	−	0.997382i	$-0.523036\pi$
$659$	−3.71237	−0.144614	−0.0723068	+	0.997382i	$0.523036\pi$
$660$	0	0
$661$	37.1173	1.44370	0.721848	−	0.692052i	$-0.243293\pi$
$661$	37.1173	1.44370	0.721848	+	0.692052i	$0.243293\pi$
$662$	0	0
$663$	1.62506	0.0631120
$664$	0	0
$665$	−0.0605423	−0.00234773
$666$	0	0
$667$	1.70358	0.0659629
$668$	0	0
$669$	7.36279	0.284662
$670$	0	0
$671$	−2.21466	−0.0854960
$672$	0	0
$673$	−23.3631	−0.900582	−0.450291	−	0.892882i	$-0.648680\pi$
$673$	−23.3631	−0.900582	−0.450291	+	0.892882i	$0.648680\pi$
$674$	0	0
$675$	−27.2914	−1.05045
$676$	0	0
$677$	−13.8968	−0.534099	−0.267049	−	0.963683i	$-0.586049\pi$
$677$	−13.8968	−0.534099	−0.267049	+	0.963683i	$0.586049\pi$
$678$	0	0
$679$	17.9034	0.687069
$680$	0	0
$681$	−5.74268	−0.220060
$682$	0	0
$683$	−14.4073	−0.551282	−0.275641	−	0.961261i	$-0.588890\pi$
$683$	−14.4073	−0.551282	−0.275641	+	0.961261i	$0.588890\pi$
$684$	0	0
$685$	−0.138315	−0.00528473
$686$	0	0
$687$	−16.8469	−0.642748
$688$	0	0
$689$	16.2306	0.618336
$690$	0	0
$691$	41.4400	1.57645	0.788225	−	0.615387i	$-0.211000\pi$
$691$	41.4400	1.57645	0.788225	+	0.615387i	$0.211000\pi$
$692$	0	0
$693$	1.34258	0.0510003
$694$	0	0
$695$	−0.140173	−0.00531708
$696$	0	0
$697$	−8.79709	−0.333213
$698$	0	0
$699$	−24.3864	−0.922380
$700$	0	0
$701$	26.7145	1.00899	0.504497	−	0.863413i	$-0.331678\pi$
$701$	26.7145	1.00899	0.504497	+	0.863413i	$0.331678\pi$
$702$	0	0
$703$	−21.5690	−0.813490
$704$	0	0
$705$	−0.00711206	−0.000267856 0
$706$	0	0
$707$	23.4433	0.881676
$708$	0	0
$709$	−40.0011	−1.50227	−0.751136	−	0.660147i	$-0.770494\pi$
$709$	−40.0011	−1.50227	−0.751136	+	0.660147i	$0.770494\pi$
$710$	0	0
$711$	−1.54270	−0.0578557
$712$	0	0
$713$	1.68382	0.0630596
$714$	0	0
$715$	0.00253490	9.47997e−5 0
$716$	0	0
$717$	−16.7263	−0.624654
$718$	0	0
$719$	−9.58794	−0.357570	−0.178785	−	0.983888i	$-0.557217\pi$
$719$	−9.58794	−0.357570	−0.178785	+	0.983888i	$0.557217\pi$
$720$	0	0
$721$	−3.22447	−0.120086
$722$	0	0
$723$	−6.62170	−0.246264
$724$	0	0
$725$	−24.8758	−0.923865
$726$	0	0
$727$	48.2983	1.79128	0.895642	−	0.444775i	$-0.146716\pi$
$727$	48.2983	1.79128	0.895642	+	0.444775i	$0.146716\pi$
$728$	0	0
$729$	22.3128	0.826401
$730$	0	0
$731$	1.47970	0.0547286
$732$	0	0
$733$	−1.79102	−0.0661529	−0.0330764	−	0.999453i	$-0.510530\pi$
$733$	−1.79102	−0.0661529	−0.0330764	+	0.999453i	$0.510530\pi$
$734$	0	0
$735$	0.0589033	0.00217268
$736$	0	0
$737$	0.151016	0.00556273
$738$	0	0
$739$	−44.7292	−1.64539	−0.822696	−	0.568482i	$-0.807531\pi$
$739$	−44.7292	−1.64539	−0.822696	+	0.568482i	$0.807531\pi$
$740$	0	0
$741$	−3.12760	−0.114895
$742$	0	0
$743$	19.9301	0.731165	0.365583	−	0.930779i	$-0.380870\pi$
$743$	19.9301	0.731165	0.365583	+	0.930779i	$0.380870\pi$
$744$	0	0
$745$	0.0565975	0.00207357
$746$	0	0
$747$	12.6649	0.463383
$748$	0	0
$749$	−2.69688	−0.0985418
$750$	0	0
$751$	−31.9906	−1.16735	−0.583676	−	0.811987i	$-0.698386\pi$
$751$	−31.9906	−1.16735	−0.583676	+	0.811987i	$0.698386\pi$
$752$	0	0
$753$	−13.9819	−0.509528
$754$	0	0
$755$	0.122567	0.00446068
$756$	0	0
$757$	−26.1197	−0.949337	−0.474669	−	0.880165i	$-0.657432\pi$
$757$	−26.1197	−0.949337	−0.474669	+	0.880165i	$0.657432\pi$
$758$	0	0
$759$	0.0964034	0.00349922
$760$	0	0
$761$	0.758468	0.0274944	0.0137472	−	0.999906i	$-0.495624\pi$
$761$	0.758468	0.0274944	0.0137472	+	0.999906i	$0.495624\pi$
$762$	0	0
$763$	−3.31343	−0.119954
$764$	0	0
$765$	−0.0137995	−0.000498923 0
$766$	0	0
$767$	−1.64200	−0.0592890
$768$	0	0
$769$	−39.3451	−1.41882	−0.709410	−	0.704796i	$-0.751039\pi$
$769$	−39.3451	−1.41882	−0.709410	+	0.704796i	$0.751039\pi$
$770$	0	0
$771$	−15.1600	−0.545973
$772$	0	0
$773$	−42.3512	−1.52327	−0.761634	−	0.648008i	$-0.775602\pi$
$773$	−42.3512	−1.52327	−0.761634	+	0.648008i	$0.775602\pi$
$774$	0	0
$775$	−24.5873	−0.883202
$776$	0	0
$777$	45.6415	1.63738
$778$	0	0
$779$	16.9310	0.606615
$780$	0	0
$781$	3.96512	0.141883
$782$	0	0
$783$	27.1566	0.970497
$784$	0	0
$785$	0.0299827	0.00107013
$786$	0	0
$787$	−30.8443	−1.09948	−0.549741	−	0.835335i	$-0.685274\pi$
$787$	−30.8443	−1.09948	−0.549741	+	0.835335i	$0.685274\pi$
$788$	0	0
$789$	29.0499	1.03420
$790$	0	0
$791$	−7.18961	−0.255633
$792$	0	0
$793$	12.1328	0.430848
$794$	0	0
$795$	0.124016	0.00439840
$796$	0	0
$797$	36.7910	1.30320	0.651602	−	0.758561i	$-0.274097\pi$
$797$	36.7910	1.30320	0.651602	+	0.758561i	$0.274097\pi$
$798$	0	0
$799$	0.757890	0.0268122
$800$	0	0
$801$	−23.3474	−0.824939
$802$	0	0
$803$	0.0786936	0.00277704
$804$	0	0
$805$	−0.0102234	−0.000360326 0
$806$	0	0
$807$	−5.94790	−0.209376
$808$	0	0
$809$	49.6272	1.74480	0.872399	−	0.488794i	$-0.162563\pi$
$809$	49.6272	1.74480	0.872399	+	0.488794i	$0.162563\pi$
$810$	0	0
$811$	−24.8106	−0.871219	−0.435609	−	0.900136i	$-0.643467\pi$
$811$	−24.8106	−0.871219	−0.435609	+	0.900136i	$0.643467\pi$
$812$	0	0
$813$	−0.467870	−0.0164089
$814$	0	0
$815$	−0.0352213	−0.00123375
$816$	0	0
$817$	−2.84784	−0.0996334
$818$	0	0
$819$	−7.35517	−0.257011
$820$	0	0
$821$	−9.55505	−0.333474	−0.166737	−	0.986001i	$-0.553323\pi$
$821$	−9.55505	−0.333474	−0.166737	+	0.986001i	$0.553323\pi$
$822$	0	0
$823$	17.5403	0.611416	0.305708	−	0.952125i	$-0.401107\pi$
$823$	17.5403	0.611416	0.305708	+	0.952125i	$0.401107\pi$
$824$	0	0
$825$	−1.40769	−0.0490095
$826$	0	0
$827$	−8.65118	−0.300831	−0.150415	−	0.988623i	$-0.548061\pi$
$827$	−8.65118	−0.300831	−0.150415	+	0.988623i	$0.548061\pi$
$828$	0	0
$829$	−19.0755	−0.662519	−0.331260	−	0.943540i	$-0.607474\pi$
$829$	−19.0755	−0.662519	−0.331260	+	0.943540i	$0.607474\pi$
$830$	0	0
$831$	4.55194	0.157905
$832$	0	0
$833$	−6.27697	−0.217484
$834$	0	0
$835$	−0.111929	−0.00387347
$836$	0	0
$837$	26.8416	0.927781
$838$	0	0
$839$	39.8734	1.37658	0.688292	−	0.725434i	$-0.258361\pi$
$839$	39.8734	1.37658	0.688292	+	0.725434i	$0.258361\pi$
$840$	0	0
$841$	−4.24705	−0.146450
$842$	0	0
$843$	−20.3794	−0.701904
$844$	0	0
$845$	0.0939389	0.00323160
$846$	0	0
$847$	−39.3957	−1.35365
$848$	0	0
$849$	−6.31515	−0.216735
$850$	0	0
$851$	−3.64221	−0.124853
$852$	0	0
$853$	−22.6151	−0.774327	−0.387163	−	0.922011i	$-0.626545\pi$
$853$	−22.6151	−0.774327	−0.387163	+	0.922011i	$0.626545\pi$
$854$	0	0
$855$	0.0265587	0.000908289 0
$856$	0	0
$857$	12.5924	0.430147	0.215074	−	0.976598i	$-0.431001\pi$
$857$	12.5924	0.430147	0.215074	+	0.976598i	$0.431001\pi$
$858$	0	0
$859$	−23.0993	−0.788139	−0.394070	−	0.919081i	$-0.628933\pi$
$859$	−23.0993	−0.788139	−0.394070	+	0.919081i	$0.628933\pi$
$860$	0	0
$861$	−35.8271	−1.22098
$862$	0	0
$863$	−36.2456	−1.23381	−0.616907	−	0.787036i	$-0.711614\pi$
$863$	−36.2456	−1.23381	−0.616907	+	0.787036i	$0.711614\pi$
$864$	0	0
$865$	−0.146024	−0.00496495
$866$	0	0
$867$	18.9410	0.643270
$868$	0	0
$869$	−0.230745	−0.00782748
$870$	0	0
$871$	−0.827324	−0.0280328
$872$	0	0
$873$	−7.85386	−0.265813
$874$	0	0
$875$	0.298567	0.0100934
$876$	0	0
$877$	−31.7167	−1.07100	−0.535498	−	0.844536i	$-0.679876\pi$
$877$	−31.7167	−1.07100	−0.535498	+	0.844536i	$0.679876\pi$
$878$	0	0
$879$	−36.4450	−1.22926
$880$	0	0
$881$	42.6588	1.43721	0.718605	−	0.695418i	$-0.244781\pi$
$881$	42.6588	1.43721	0.718605	+	0.695418i	$0.244781\pi$
$882$	0	0
$883$	17.2625	0.580928	0.290464	−	0.956886i	$-0.406190\pi$
$883$	17.2625	0.580928	0.290464	+	0.956886i	$0.406190\pi$
$884$	0	0
$885$	−0.0125463	−0.000421739 0
$886$	0	0
$887$	−42.2551	−1.41879	−0.709393	−	0.704813i	$-0.751031\pi$
$887$	−42.2551	−1.41879	−0.709393	+	0.704813i	$0.751031\pi$
$888$	0	0
$889$	−2.55454	−0.0856765
$890$	0	0
$891$	0.417842	0.0139982
$892$	0	0
$893$	−1.45864	−0.0488116
$894$	0	0
$895$	−0.0904855	−0.00302460
$896$	0	0
$897$	−0.528136	−0.0176340
$898$	0	0
$899$	24.4658	0.815982
$900$	0	0
$901$	−13.2157	−0.440278
$902$	0	0
$903$	6.02623	0.200540
$904$	0	0
$905$	0.0578042	0.00192148
$906$	0	0
$907$	−20.2593	−0.672700	−0.336350	−	0.941737i	$-0.609192\pi$
$907$	−20.2593	−0.672700	−0.336350	+	0.941737i	$0.609192\pi$
$908$	0	0
$909$	−10.2841	−0.341103
$910$	0	0
$911$	−23.3060	−0.772163	−0.386082	−	0.922465i	$-0.626172\pi$
$911$	−23.3060	−0.772163	−0.386082	+	0.922465i	$0.626172\pi$
$912$	0	0
$913$	1.89431	0.0626925
$914$	0	0
$915$	0.0927054	0.00306474
$916$	0	0
$917$	41.4361	1.36834
$918$	0	0
$919$	−2.60692	−0.0859944	−0.0429972	−	0.999075i	$-0.513691\pi$
$919$	−2.60692	−0.0859944	−0.0429972	+	0.999075i	$0.513691\pi$
$920$	0	0
$921$	−25.5647	−0.842387
$922$	0	0
$923$	−21.7225	−0.715005
$924$	0	0
$925$	53.1838	1.74867
$926$	0	0
$927$	1.41451	0.0464587
$928$	0	0
$929$	−7.70044	−0.252643	−0.126322	−	0.991989i	$-0.540317\pi$
$929$	−7.70044	−0.252643	−0.126322	+	0.991989i	$0.540317\pi$
$930$	0	0
$931$	12.0807	0.395930
$932$	0	0
$933$	28.3444	0.927953
$934$	0	0
$935$	−0.00206402	−6.75008e−5 0
$936$	0	0
$937$	48.1224	1.57209	0.786044	−	0.618170i	$-0.212126\pi$
$937$	48.1224	1.57209	0.786044	+	0.618170i	$0.212126\pi$
$938$	0	0
$939$	−19.7693	−0.645145
$940$	0	0
$941$	−0.920597	−0.0300106	−0.0150053	−	0.999887i	$-0.504777\pi$
$941$	−0.920597	−0.0300106	−0.0150053	+	0.999887i	$0.504777\pi$
$942$	0	0
$943$	2.85902	0.0931024
$944$	0	0
$945$	−0.162969	−0.00530139
$946$	0	0
$947$	32.4149	1.05334	0.526672	−	0.850068i	$-0.323440\pi$
$947$	32.4149	1.05334	0.526672	+	0.850068i	$0.323440\pi$
$948$	0	0
$949$	−0.431115	−0.0139946
$950$	0	0
$951$	−5.11547	−0.165880
$952$	0	0
$953$	−38.2734	−1.23980	−0.619898	−	0.784682i	$-0.712826\pi$
$953$	−38.2734	−1.23980	−0.619898	+	0.784682i	$0.712826\pi$
$954$	0	0
$955$	−0.173352	−0.00560955
$956$	0	0
$957$	1.40074	0.0452794
$958$	0	0
$959$	60.0277	1.93840
$960$	0	0
$961$	−6.81794	−0.219934
$962$	0	0
$963$	1.18307	0.0381238
$964$	0	0
$965$	−0.0821792	−0.00264544
$966$	0	0
$967$	42.9606	1.38152	0.690759	−	0.723085i	$-0.257277\pi$
$967$	42.9606	1.38152	0.690759	+	0.723085i	$0.257277\pi$
$968$	0	0
$969$	2.54663	0.0818096
$970$	0	0
$971$	25.0577	0.804141	0.402070	−	0.915609i	$-0.368291\pi$
$971$	25.0577	0.804141	0.402070	+	0.915609i	$0.368291\pi$
$972$	0	0
$973$	60.8344	1.95026
$974$	0	0
$975$	7.71190	0.246978
$976$	0	0
$977$	−18.2373	−0.583463	−0.291732	−	0.956500i	$-0.594231\pi$
$977$	−18.2373	−0.583463	−0.291732	+	0.956500i	$0.594231\pi$
$978$	0	0
$979$	−3.49212	−0.111609
$980$	0	0
$981$	1.45354	0.0464079
$982$	0	0
$983$	−23.3237	−0.743910	−0.371955	−	0.928251i	$-0.621313\pi$
$983$	−23.3237	−0.743910	−0.371955	+	0.928251i	$0.621313\pi$
$984$	0	0
$985$	−0.0568590	−0.00181168
$986$	0	0
$987$	3.08659	0.0982473
$988$	0	0
$989$	−0.480895	−0.0152916
$990$	0	0
$991$	30.8997	0.981561	0.490780	−	0.871283i	$-0.336712\pi$
$991$	30.8997	0.981561	0.490780	+	0.871283i	$0.336712\pi$
$992$	0	0
$993$	−14.9766	−0.475269
$994$	0	0
$995$	0.0945465	0.00299732
$996$	0	0
$997$	29.1902	0.924463	0.462232	−	0.886759i	$-0.347049\pi$
$997$	29.1902	0.924463	0.462232	+	0.886759i	$0.347049\pi$
$998$	0	0
$999$	−58.0600	−1.83694

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.f.1.12	✓	33
4.3	odd	2		8048.2.a.y.1.22		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.f.1.12	✓	33	1.1	even	1	trivial
8048.2.a.y.1.22		33	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-1.19201 q^{3} -0.00829431 q^{5} +3.59968 q^{7} -1.57911 q^{9} +O(q^{10})\)	\(q-1.19201 q^{3} -0.00829431 q^{5} +3.59968 q^{7} -1.57911 q^{9} -0.236191 q^{11} +1.29395 q^{13} +0.00988691 q^{15} -1.05359 q^{17} +2.02775 q^{19} -4.29086 q^{21} +0.342412 q^{23} -4.99993 q^{25} +5.45835 q^{27} +4.97523 q^{29} +4.91753 q^{31} +0.281542 q^{33} -0.0298569 q^{35} -10.6369 q^{37} -1.54240 q^{39} +8.34963 q^{41} -1.40443 q^{43} +0.0130976 q^{45} -0.719341 q^{47} +5.95770 q^{49} +1.25589 q^{51} +12.5435 q^{53} +0.00195904 q^{55} -2.41710 q^{57} -1.26898 q^{59} +9.37657 q^{61} -5.68429 q^{63} -0.0107324 q^{65} -0.639380 q^{67} -0.408159 q^{69} -16.7878 q^{71} -0.333178 q^{73} +5.95997 q^{75} -0.850211 q^{77} +0.976942 q^{79} -1.76909 q^{81} -8.02026 q^{83} +0.00873880 q^{85} -5.93053 q^{87} +14.7851 q^{89} +4.65780 q^{91} -5.86175 q^{93} -0.0168188 q^{95} +4.97360 q^{97} +0.372971 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9}+O(q^{10}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9	\( 33 q - 2 q^{3} + 12 q^{5} + 4 q^{7} + 43 q^{9} + 22 q^{11} + 25 q^{13} - 4 q^{15} + 17 q^{17} + 6 q^{19} + 18 q^{21} + 16 q^{23} + 47 q^{25} - 20 q^{27} + 47 q^{29} - 7 q^{31} - 6 q^{33} + 19 q^{35} + 75 q^{37} + 21 q^{39} + 22 q^{41} - 5 q^{43} + 33 q^{45} + 10 q^{47} + 31 q^{49} + 9 q^{51} + 64 q^{53} - 3 q^{55} + 5 q^{57} + 28 q^{59} + 49 q^{61} - 10 q^{63} + 46 q^{65} - 14 q^{67} + 30 q^{69} + 35 q^{71} + 19 q^{73} - 33 q^{75} + 32 q^{77} - 12 q^{79} + 57 q^{81} + 82 q^{85} - 5 q^{87} + 42 q^{89} - 15 q^{91} + 55 q^{93} + 33 q^{95} + 4 q^{97} + 22 q^{99}+O(q^{100}) \) 33 * q - 2 * q^3 + 12 * q^5 + 4 * q^7 + 43 * q^9 + 22 * q^11 + 25 * q^13 - 4 * q^15 + 17 * q^17 + 6 * q^19 + 18 * q^21 + 16 * q^23 + 47 * q^25 - 20 * q^27 + 47 * q^29 - 7 * q^31 - 6 * q^33 + 19 * q^35 + 75 * q^37 + 21 * q^39 + 22 * q^41 - 5 * q^43 + 33 * q^45 + 10 * q^47 + 31 * q^49 + 9 * q^51 + 64 * q^53 - 3 * q^55 + 5 * q^57 + 28 * q^59 + 49 * q^61 - 10 * q^63 + 46 * q^65 - 14 * q^67 + 30 * q^69 + 35 * q^71 + 19 * q^73 - 33 * q^75 + 32 * q^77 - 12 * q^79 + 57 * q^81 + 82 * q^85 - 5 * q^87 + 42 * q^89 - 15 * q^91 + 55 * q^93 + 33 * q^95 + 4 * q^97 + 22 * q^99

Introduction

L-functions

Modular forms

Varieties

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