LMFDB - Embedded newform 8024.2.a.y.1.17

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8024,2,Mod(1,8024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8024, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8024 = 2^{3} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8024.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:	$64.0719625819$
Analytic rank:	$1$
Dimension:	$23$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.17
Character	$\chi$	$=$	8024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.25653 q^{3} +3.93457 q^{5} -3.56420 q^{7} -1.42113 q^{9} +O(q^{10})$	$q+1.25653 q^{3} +3.93457 q^{5} -3.56420 q^{7} -1.42113 q^{9} -3.07833 q^{11} +2.74994 q^{13} +4.94392 q^{15} -1.00000 q^{17} +2.16402 q^{19} -4.47854 q^{21} -5.55312 q^{23} +10.4809 q^{25} -5.55529 q^{27} -2.25685 q^{29} -0.00422060 q^{31} -3.86802 q^{33} -14.0236 q^{35} +5.75808 q^{37} +3.45539 q^{39} -2.70729 q^{41} -11.1211 q^{43} -5.59152 q^{45} +3.90158 q^{47} +5.70355 q^{49} -1.25653 q^{51} -6.71321 q^{53} -12.1119 q^{55} +2.71917 q^{57} -1.00000 q^{59} -11.2654 q^{61} +5.06518 q^{63} +10.8198 q^{65} -5.44196 q^{67} -6.97767 q^{69} -3.00896 q^{71} -0.207216 q^{73} +13.1695 q^{75} +10.9718 q^{77} -3.57390 q^{79} -2.71703 q^{81} -4.16115 q^{83} -3.93457 q^{85} -2.83581 q^{87} +5.88005 q^{89} -9.80134 q^{91} -0.00530333 q^{93} +8.51450 q^{95} -9.10944 q^{97} +4.37469 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9	$ 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) $ 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * 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
$3$	1.25653	0.725460	0.362730	−	0.931894i	$-0.381845\pi$
$3$	1.25653	0.725460	0.362730	+	0.931894i	$0.381845\pi$
$4$	0	0
$5$	3.93457	1.75959	0.879797	−	0.475350i	$-0.157678\pi$
$5$	3.93457	1.75959	0.879797	+	0.475350i	$0.157678\pi$
$6$	0	0
$7$	−3.56420	−1.34714	−0.673571	−	0.739122i	$-0.735241\pi$
$7$	−3.56420	−1.34714	−0.673571	+	0.739122i	$0.735241\pi$
$8$	0	0
$9$	−1.42113	−0.473708
$10$	0	0
$11$	−3.07833	−0.928151	−0.464075	−	0.885796i	$-0.653613\pi$
$11$	−3.07833	−0.928151	−0.464075	+	0.885796i	$0.653613\pi$
$12$	0	0
$13$	2.74994	0.762695	0.381348	−	0.924432i	$-0.375460\pi$
$13$	2.74994	0.762695	0.381348	+	0.924432i	$0.375460\pi$
$14$	0	0
$15$	4.94392	1.27651
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	2.16402	0.496461	0.248230	−	0.968701i	$-0.420151\pi$
$19$	2.16402	0.496461	0.248230	+	0.968701i	$0.420151\pi$
$20$	0	0
$21$	−4.47854	−0.977297
$22$	0	0
$23$	−5.55312	−1.15790	−0.578952	−	0.815361i	$-0.696538\pi$
$23$	−5.55312	−1.15790	−0.578952	+	0.815361i	$0.696538\pi$
$24$	0	0
$25$	10.4809	2.09617
$26$	0	0
$27$	−5.55529	−1.06912
$28$	0	0
$29$	−2.25685	−0.419086	−0.209543	−	0.977799i	$-0.567198\pi$
$29$	−2.25685	−0.419086	−0.209543	+	0.977799i	$0.567198\pi$
$30$	0	0
$31$	−0.00422060	−0.000758043 0	−0.000379021	−	1.00000i	$-0.500121\pi$
$31$	−0.00422060	−0.000758043 0	−0.000379021		1.00000i	$0.500121\pi$
$32$	0	0
$33$	−3.86802	−0.673336
$34$	0	0
$35$	−14.0236	−2.37042
$36$	0	0
$37$	5.75808	0.946622	0.473311	−	0.880895i	$-0.343059\pi$
$37$	5.75808	0.946622	0.473311	+	0.880895i	$0.343059\pi$
$38$	0	0
$39$	3.45539	0.553305
$40$	0	0
$41$	−2.70729	−0.422807	−0.211404	−	0.977399i	$-0.567803\pi$
$41$	−2.70729	−0.422807	−0.211404	+	0.977399i	$0.567803\pi$
$42$	0	0
$43$	−11.1211	−1.69595	−0.847976	−	0.530034i	$-0.822179\pi$
$43$	−11.1211	−1.69595	−0.847976	+	0.530034i	$0.822179\pi$
$44$	0	0
$45$	−5.59152	−0.833534
$46$	0	0
$47$	3.90158	0.569104	0.284552	−	0.958661i	$-0.408155\pi$
$47$	3.90158	0.569104	0.284552	+	0.958661i	$0.408155\pi$
$48$	0	0
$49$	5.70355	0.814792
$50$	0	0
$51$	−1.25653	−0.175950
$52$	0	0
$53$	−6.71321	−0.922130	−0.461065	−	0.887366i	$-0.652533\pi$
$53$	−6.71321	−0.922130	−0.461065	+	0.887366i	$0.652533\pi$
$54$	0	0
$55$	−12.1119	−1.63317
$56$	0	0
$57$	2.71917	0.360162
$58$	0	0
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−11.2654	−1.44239	−0.721193	−	0.692734i	$-0.756406\pi$
$61$	−11.2654	−1.44239	−0.721193	+	0.692734i	$0.756406\pi$
$62$	0	0
$63$	5.06518	0.638153
$64$	0	0
$65$	10.8198	1.34203
$66$	0	0
$67$	−5.44196	−0.664841	−0.332421	−	0.943131i	$-0.607865\pi$
$67$	−5.44196	−0.664841	−0.332421	+	0.943131i	$0.607865\pi$
$68$	0	0
$69$	−6.97767	−0.840013
$70$	0	0
$71$	−3.00896	−0.357098	−0.178549	−	0.983931i	$-0.557140\pi$
$71$	−3.00896	−0.357098	−0.178549	+	0.983931i	$0.557140\pi$
$72$	0	0
$73$	−0.207216	−0.0242527	−0.0121264	−	0.999926i	$-0.503860\pi$
$73$	−0.207216	−0.0242527	−0.0121264	+	0.999926i	$0.503860\pi$
$74$	0	0
$75$	13.1695	1.52069
$76$	0	0
$77$	10.9718	1.25035
$78$	0	0
$79$	−3.57390	−0.402096	−0.201048	−	0.979581i	$-0.564435\pi$
$79$	−3.57390	−0.402096	−0.201048	+	0.979581i	$0.564435\pi$
$80$	0	0
$81$	−2.71703	−0.301892
$82$	0	0
$83$	−4.16115	−0.456746	−0.228373	−	0.973574i	$-0.573341\pi$
$83$	−4.16115	−0.456746	−0.228373	+	0.973574i	$0.573341\pi$
$84$	0	0
$85$	−3.93457	−0.426764
$86$	0	0
$87$	−2.83581	−0.304030
$88$	0	0
$89$	5.88005	0.623284	0.311642	−	0.950200i	$-0.399121\pi$
$89$	5.88005	0.623284	0.311642	+	0.950200i	$0.399121\pi$
$90$	0	0
$91$	−9.80134	−1.02746
$92$	0	0
$93$	−0.00530333	−0.000549929 0
$94$	0	0
$95$	8.51450	0.873569
$96$	0	0
$97$	−9.10944	−0.924923	−0.462462	−	0.886639i	$-0.653034\pi$
$97$	−9.10944	−0.924923	−0.462462	+	0.886639i	$0.653034\pi$
$98$	0	0
$99$	4.37469	0.439673
$100$	0	0
$101$	3.36167	0.334499	0.167249	−	0.985915i	$-0.446512\pi$
$101$	3.36167	0.334499	0.167249	+	0.985915i	$0.446512\pi$
$102$	0	0
$103$	8.82433	0.869487	0.434744	−	0.900554i	$-0.356839\pi$
$103$	8.82433	0.869487	0.434744	+	0.900554i	$0.356839\pi$
$104$	0	0
$105$	−17.6211	−1.71965
$106$	0	0
$107$	−2.28460	−0.220861	−0.110430	−	0.993884i	$-0.535223\pi$
$107$	−2.28460	−0.220861	−0.110430	+	0.993884i	$0.535223\pi$
$108$	0	0
$109$	5.60607	0.536964	0.268482	−	0.963285i	$-0.413478\pi$
$109$	5.60607	0.536964	0.268482	+	0.963285i	$0.413478\pi$
$110$	0	0
$111$	7.23522	0.686736
$112$	0	0
$113$	−4.76980	−0.448705	−0.224353	−	0.974508i	$-0.572027\pi$
$113$	−4.76980	−0.448705	−0.224353	+	0.974508i	$0.572027\pi$
$114$	0	0
$115$	−21.8491	−2.03744
$116$	0	0
$117$	−3.90800	−0.361295
$118$	0	0
$119$	3.56420	0.326730
$120$	0	0
$121$	−1.52390	−0.138536
$122$	0	0
$123$	−3.40179	−0.306730
$124$	0	0
$125$	21.5648	1.92881
$126$	0	0
$127$	7.44183	0.660355	0.330178	−	0.943919i	$-0.392891\pi$
$127$	7.44183	0.660355	0.330178	+	0.943919i	$0.392891\pi$
$128$	0	0
$129$	−13.9740	−1.23035
$130$	0	0
$131$	9.99213	0.873017	0.436508	−	0.899700i	$-0.356215\pi$
$131$	9.99213	0.873017	0.436508	+	0.899700i	$0.356215\pi$
$132$	0	0
$133$	−7.71302	−0.668803
$134$	0	0
$135$	−21.8577	−1.88121
$136$	0	0
$137$	20.7866	1.77592	0.887960	−	0.459920i	$-0.152122\pi$
$137$	20.7866	1.77592	0.887960	+	0.459920i	$0.152122\pi$
$138$	0	0
$139$	−17.9028	−1.51850	−0.759248	−	0.650802i	$-0.774433\pi$
$139$	−17.9028	−1.51850	−0.759248	+	0.650802i	$0.774433\pi$
$140$	0	0
$141$	4.90247	0.412862
$142$	0	0
$143$	−8.46521	−0.707896
$144$	0	0
$145$	−8.87974	−0.737422
$146$	0	0
$147$	7.16669	0.591099
$148$	0	0
$149$	−5.86005	−0.480074	−0.240037	−	0.970764i	$-0.577160\pi$
$149$	−5.86005	−0.480074	−0.240037	+	0.970764i	$0.577160\pi$
$150$	0	0
$151$	−5.31582	−0.432595	−0.216297	−	0.976328i	$-0.569398\pi$
$151$	−5.31582	−0.432595	−0.216297	+	0.976328i	$0.569398\pi$
$152$	0	0
$153$	1.42113	0.114891
$154$	0	0
$155$	−0.0166063	−0.00133385
$156$	0	0
$157$	−7.46330	−0.595636	−0.297818	−	0.954623i	$-0.596259\pi$
$157$	−7.46330	−0.595636	−0.297818	+	0.954623i	$0.596259\pi$
$158$	0	0
$159$	−8.43537	−0.668968
$160$	0	0
$161$	19.7924	1.55986
$162$	0	0
$163$	−3.30975	−0.259240	−0.129620	−	0.991564i	$-0.541376\pi$
$163$	−3.30975	−0.259240	−0.129620	+	0.991564i	$0.541376\pi$
$164$	0	0
$165$	−15.2190	−1.18480
$166$	0	0
$167$	−0.270878	−0.0209612	−0.0104806	−	0.999945i	$-0.503336\pi$
$167$	−0.270878	−0.0209612	−0.0104806	+	0.999945i	$0.503336\pi$
$168$	0	0
$169$	−5.43785	−0.418296
$170$	0	0
$171$	−3.07535	−0.235178
$172$	0	0
$173$	4.72125	0.358950	0.179475	−	0.983763i	$-0.442560\pi$
$173$	4.72125	0.358950	0.179475	+	0.983763i	$0.442560\pi$
$174$	0	0
$175$	−37.3559	−2.82384
$176$	0	0
$177$	−1.25653	−0.0944468
$178$	0	0
$179$	−13.6902	−1.02325	−0.511625	−	0.859209i	$-0.670956\pi$
$179$	−13.6902	−1.02325	−0.511625	+	0.859209i	$0.670956\pi$
$180$	0	0
$181$	0.731964	0.0544065	0.0272032	−	0.999630i	$-0.491340\pi$
$181$	0.731964	0.0544065	0.0272032	+	0.999630i	$0.491340\pi$
$182$	0	0
$183$	−14.1553	−1.04639
$184$	0	0
$185$	22.6556	1.66567
$186$	0	0
$187$	3.07833	0.225110
$188$	0	0
$189$	19.8002	1.44025
$190$	0	0
$191$	−21.2217	−1.53555	−0.767775	−	0.640720i	$-0.778636\pi$
$191$	−21.2217	−1.53555	−0.767775	+	0.640720i	$0.778636\pi$
$192$	0	0
$193$	−20.5205	−1.47710	−0.738550	−	0.674199i	$-0.764489\pi$
$193$	−20.5205	−1.47710	−0.738550	+	0.674199i	$0.764489\pi$
$194$	0	0
$195$	13.5955	0.973591
$196$	0	0
$197$	−11.6131	−0.827398	−0.413699	−	0.910414i	$-0.635763\pi$
$197$	−11.6131	−0.827398	−0.413699	+	0.910414i	$0.635763\pi$
$198$	0	0
$199$	16.4856	1.16863	0.584315	−	0.811527i	$-0.301363\pi$
$199$	16.4856	1.16863	0.584315	+	0.811527i	$0.301363\pi$
$200$	0	0
$201$	−6.83800	−0.482316
$202$	0	0
$203$	8.04387	0.564569
$204$	0	0
$205$	−10.6520	−0.743969
$206$	0	0
$207$	7.89167	0.548509
$208$	0	0
$209$	−6.66157	−0.460790
$210$	0	0
$211$	14.3344	0.986818	0.493409	−	0.869797i	$-0.335751\pi$
$211$	14.3344	0.986818	0.493409	+	0.869797i	$0.335751\pi$
$212$	0	0
$213$	−3.78086	−0.259060
$214$	0	0
$215$	−43.7568	−2.98419
$216$	0	0
$217$	0.0150431	0.00102119
$218$	0	0
$219$	−0.260373	−0.0175944
$220$	0	0
$221$	−2.74994	−0.184981
$222$	0	0
$223$	−6.09525	−0.408168	−0.204084	−	0.978953i	$-0.565422\pi$
$223$	−6.09525	−0.408168	−0.204084	+	0.978953i	$0.565422\pi$
$224$	0	0
$225$	−14.8946	−0.992973
$226$	0	0
$227$	−9.70183	−0.643933	−0.321967	−	0.946751i	$-0.604344\pi$
$227$	−9.70183	−0.643933	−0.321967	+	0.946751i	$0.604344\pi$
$228$	0	0
$229$	−3.16516	−0.209159	−0.104580	−	0.994517i	$-0.533350\pi$
$229$	−3.16516	−0.209159	−0.104580	+	0.994517i	$0.533350\pi$
$230$	0	0
$231$	13.7864	0.907079
$232$	0	0
$233$	−0.535750	−0.0350982	−0.0175491	−	0.999846i	$-0.505586\pi$
$233$	−0.535750	−0.0350982	−0.0175491	+	0.999846i	$0.505586\pi$
$234$	0	0
$235$	15.3511	1.00139
$236$	0	0
$237$	−4.49073	−0.291704
$238$	0	0
$239$	−3.70234	−0.239485	−0.119742	−	0.992805i	$-0.538207\pi$
$239$	−3.70234	−0.239485	−0.119742	+	0.992805i	$0.538207\pi$
$240$	0	0
$241$	8.15946	0.525597	0.262798	−	0.964851i	$-0.415355\pi$
$241$	8.15946	0.525597	0.262798	+	0.964851i	$0.415355\pi$
$242$	0	0
$243$	13.2518	0.850105
$244$	0	0
$245$	22.4410	1.43370
$246$	0	0
$247$	5.95093	0.378648
$248$	0	0
$249$	−5.22862	−0.331351
$250$	0	0
$251$	−25.5064	−1.60995	−0.804974	−	0.593310i	$-0.797821\pi$
$251$	−25.5064	−1.60995	−0.804974	+	0.593310i	$0.797821\pi$
$252$	0	0
$253$	17.0943	1.07471
$254$	0	0
$255$	−4.94392	−0.309600
$256$	0	0
$257$	−10.2586	−0.639914	−0.319957	−	0.947432i	$-0.603669\pi$
$257$	−10.2586	−0.639914	−0.319957	+	0.947432i	$0.603669\pi$
$258$	0	0
$259$	−20.5230	−1.27523
$260$	0	0
$261$	3.20727	0.198525
$262$	0	0
$263$	13.6021	0.838742	0.419371	−	0.907815i	$-0.362251\pi$
$263$	13.6021	0.838742	0.419371	+	0.907815i	$0.362251\pi$
$264$	0	0
$265$	−26.4136	−1.62257
$266$	0	0
$267$	7.38848	0.452168
$268$	0	0
$269$	29.5611	1.80237	0.901187	−	0.433431i	$-0.142697\pi$
$269$	29.5611	1.80237	0.901187	+	0.433431i	$0.142697\pi$
$270$	0	0
$271$	−30.3524	−1.84378	−0.921890	−	0.387452i	$-0.873355\pi$
$271$	−30.3524	−1.84378	−0.921890	+	0.387452i	$0.873355\pi$
$272$	0	0
$273$	−12.3157	−0.745380
$274$	0	0
$275$	−32.2635	−1.94556
$276$	0	0
$277$	−3.89540	−0.234052	−0.117026	−	0.993129i	$-0.537336\pi$
$277$	−3.89540	−0.234052	−0.117026	+	0.993129i	$0.537336\pi$
$278$	0	0
$279$	0.00599801	0.000359091 0
$280$	0	0
$281$	6.74069	0.402116	0.201058	−	0.979579i	$-0.435562\pi$
$281$	6.74069	0.402116	0.201058	+	0.979579i	$0.435562\pi$
$282$	0	0
$283$	−10.1403	−0.602778	−0.301389	−	0.953501i	$-0.597450\pi$
$283$	−10.1403	−0.602778	−0.301389	+	0.953501i	$0.597450\pi$
$284$	0	0
$285$	10.6987	0.633739
$286$	0	0
$287$	9.64932	0.569581
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−11.4463	−0.670995
$292$	0	0
$293$	−26.3820	−1.54125	−0.770627	−	0.637287i	$-0.780057\pi$
$293$	−26.3820	−1.54125	−0.770627	+	0.637287i	$0.780057\pi$
$294$	0	0
$295$	−3.93457	−0.229080
$296$	0	0
$297$	17.1010	0.992301
$298$	0	0
$299$	−15.2707	−0.883129
$300$	0	0
$301$	39.6379	2.28469
$302$	0	0
$303$	4.22405	0.242665
$304$	0	0
$305$	−44.3245	−2.53802
$306$	0	0
$307$	−11.4704	−0.654652	−0.327326	−	0.944911i	$-0.606148\pi$
$307$	−11.4704	−0.654652	−0.327326	+	0.944911i	$0.606148\pi$
$308$	0	0
$309$	11.0881	0.630778
$310$	0	0
$311$	0.376698	0.0213606	0.0106803	−	0.999943i	$-0.496600\pi$
$311$	0.376698	0.0213606	0.0106803	+	0.999943i	$0.496600\pi$
$312$	0	0
$313$	13.2217	0.747337	0.373669	−	0.927562i	$-0.378100\pi$
$313$	13.2217	0.747337	0.373669	+	0.927562i	$0.378100\pi$
$314$	0	0
$315$	19.9293	1.12289
$316$	0	0
$317$	19.2248	1.07977	0.539887	−	0.841737i	$-0.318467\pi$
$317$	19.2248	1.07977	0.539887	+	0.841737i	$0.318467\pi$
$318$	0	0
$319$	6.94732	0.388975
$320$	0	0
$321$	−2.87068	−0.160226
$322$	0	0
$323$	−2.16402	−0.120409
$324$	0	0
$325$	28.8217	1.59874
$326$	0	0
$327$	7.04422	0.389546
$328$	0	0
$329$	−13.9060	−0.766665
$330$	0	0
$331$	17.4854	0.961083	0.480541	−	0.876972i	$-0.340440\pi$
$331$	17.4854	0.961083	0.480541	+	0.876972i	$0.340440\pi$
$332$	0	0
$333$	−8.18295	−0.448423
$334$	0	0
$335$	−21.4118	−1.16985
$336$	0	0
$337$	12.9599	0.705969	0.352984	−	0.935629i	$-0.385167\pi$
$337$	12.9599	0.705969	0.352984	+	0.935629i	$0.385167\pi$
$338$	0	0
$339$	−5.99341	−0.325517
$340$	0	0
$341$	0.0129924	0.000703578 0
$342$	0	0
$343$	4.62082	0.249501
$344$	0	0
$345$	−27.4542	−1.47808
$346$	0	0
$347$	20.0786	1.07788	0.538938	−	0.842345i	$-0.318826\pi$
$347$	20.0786	1.07788	0.538938	+	0.842345i	$0.318826\pi$
$348$	0	0
$349$	−29.7123	−1.59046	−0.795230	−	0.606308i	$-0.792650\pi$
$349$	−29.7123	−1.59046	−0.795230	+	0.606308i	$0.792650\pi$
$350$	0	0
$351$	−15.2767	−0.815410
$352$	0	0
$353$	−31.3807	−1.67023	−0.835114	−	0.550076i	$-0.814599\pi$
$353$	−31.3807	−1.67023	−0.835114	+	0.550076i	$0.814599\pi$
$354$	0	0
$355$	−11.8390	−0.628348
$356$	0	0
$357$	4.47854	0.237029
$358$	0	0
$359$	7.60943	0.401610	0.200805	−	0.979631i	$-0.435644\pi$
$359$	7.60943	0.401610	0.200805	+	0.979631i	$0.435644\pi$
$360$	0	0
$361$	−14.3170	−0.753527
$362$	0	0
$363$	−1.91483	−0.100503
$364$	0	0
$365$	−0.815305	−0.0426750
$366$	0	0
$367$	4.28204	0.223521	0.111760	−	0.993735i	$-0.464351\pi$
$367$	4.28204	0.223521	0.111760	+	0.993735i	$0.464351\pi$
$368$	0	0
$369$	3.84739	0.200287
$370$	0	0
$371$	23.9272	1.24224
$372$	0	0
$373$	−5.76119	−0.298303	−0.149152	−	0.988814i	$-0.547654\pi$
$373$	−5.76119	−0.298303	−0.149152	+	0.988814i	$0.547654\pi$
$374$	0	0
$375$	27.0969	1.39928
$376$	0	0
$377$	−6.20619	−0.319635
$378$	0	0
$379$	−4.74598	−0.243785	−0.121892	−	0.992543i	$-0.538896\pi$
$379$	−4.74598	−0.243785	−0.121892	+	0.992543i	$0.538896\pi$
$380$	0	0
$381$	9.35090	0.479061
$382$	0	0
$383$	22.8239	1.16625	0.583125	−	0.812383i	$-0.301830\pi$
$383$	22.8239	1.16625	0.583125	+	0.812383i	$0.301830\pi$
$384$	0	0
$385$	43.1693	2.20011
$386$	0	0
$387$	15.8045	0.803387
$388$	0	0
$389$	7.05806	0.357858	0.178929	−	0.983862i	$-0.442737\pi$
$389$	7.05806	0.357858	0.178929	+	0.983862i	$0.442737\pi$
$390$	0	0
$391$	5.55312	0.280833
$392$	0	0
$393$	12.5554	0.633338
$394$	0	0
$395$	−14.0618	−0.707525
$396$	0	0
$397$	35.6750	1.79048	0.895239	−	0.445587i	$-0.147005\pi$
$397$	35.6750	1.79048	0.895239	+	0.445587i	$0.147005\pi$
$398$	0	0
$399$	−9.69166	−0.485190
$400$	0	0
$401$	−0.811108	−0.0405048	−0.0202524	−	0.999795i	$-0.506447\pi$
$401$	−0.811108	−0.0405048	−0.0202524	+	0.999795i	$0.506447\pi$
$402$	0	0
$403$	−0.0116064	−0.000578156 0
$404$	0	0
$405$	−10.6903	−0.531207
$406$	0	0
$407$	−17.7253	−0.878608
$408$	0	0
$409$	25.8513	1.27826	0.639131	−	0.769098i	$-0.279294\pi$
$409$	25.8513	1.27826	0.639131	+	0.769098i	$0.279294\pi$
$410$	0	0
$411$	26.1191	1.28836
$412$	0	0
$413$	3.56420	0.175383
$414$	0	0
$415$	−16.3723	−0.803687
$416$	0	0
$417$	−22.4955	−1.10161
$418$	0	0
$419$	36.4443	1.78042	0.890210	−	0.455550i	$-0.150557\pi$
$419$	36.4443	1.78042	0.890210	+	0.455550i	$0.150557\pi$
$420$	0	0
$421$	12.2534	0.597196	0.298598	−	0.954379i	$-0.403481\pi$
$421$	12.2534	0.597196	0.298598	+	0.954379i	$0.403481\pi$
$422$	0	0
$423$	−5.54464	−0.269590
$424$	0	0
$425$	−10.4809	−0.508396
$426$	0	0
$427$	40.1522	1.94310
$428$	0	0
$429$	−10.6368	−0.513550
$430$	0	0
$431$	18.8301	0.907014	0.453507	−	0.891253i	$-0.350173\pi$
$431$	18.8301	0.907014	0.453507	+	0.891253i	$0.350173\pi$
$432$	0	0
$433$	−24.7361	−1.18874	−0.594371	−	0.804191i	$-0.702599\pi$
$433$	−24.7361	−1.18874	−0.594371	+	0.804191i	$0.702599\pi$
$434$	0	0
$435$	−11.1577	−0.534970
$436$	0	0
$437$	−12.0171	−0.574854
$438$	0	0
$439$	−11.7338	−0.560023	−0.280011	−	0.959997i	$-0.590338\pi$
$439$	−11.7338	−0.560023	−0.280011	+	0.959997i	$0.590338\pi$
$440$	0	0
$441$	−8.10545	−0.385974
$442$	0	0
$443$	28.7954	1.36811	0.684056	−	0.729429i	$-0.260214\pi$
$443$	28.7954	1.36811	0.684056	+	0.729429i	$0.260214\pi$
$444$	0	0
$445$	23.1355	1.09673
$446$	0	0
$447$	−7.36335	−0.348275
$448$	0	0
$449$	−10.5553	−0.498136	−0.249068	−	0.968486i	$-0.580124\pi$
$449$	−10.5553	−0.498136	−0.249068	+	0.968486i	$0.580124\pi$
$450$	0	0
$451$	8.33392	0.392429
$452$	0	0
$453$	−6.67950	−0.313830
$454$	0	0
$455$	−38.5641	−1.80791
$456$	0	0
$457$	−21.3776	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$457$	−21.3776	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$458$	0	0
$459$	5.55529	0.259299
$460$	0	0
$461$	−16.4658	−0.766890	−0.383445	−	0.923564i	$-0.625262\pi$
$461$	−16.4658	−0.766890	−0.383445	+	0.923564i	$0.625262\pi$
$462$	0	0
$463$	3.46666	0.161109	0.0805547	−	0.996750i	$-0.474331\pi$
$463$	3.46666	0.161109	0.0805547	+	0.996750i	$0.474331\pi$
$464$	0	0
$465$	−0.0208663	−0.000967652 0
$466$	0	0
$467$	−7.16923	−0.331753	−0.165876	−	0.986147i	$-0.553045\pi$
$467$	−7.16923	−0.331753	−0.165876	+	0.986147i	$0.553045\pi$
$468$	0	0
$469$	19.3963	0.895636
$470$	0	0
$471$	−9.37788	−0.432110
$472$	0	0
$473$	34.2344	1.57410
$474$	0	0
$475$	22.6808	1.04067
$476$	0	0
$477$	9.54031	0.436821
$478$	0	0
$479$	−10.5300	−0.481128	−0.240564	−	0.970633i	$-0.577332\pi$
$479$	−10.5300	−0.481128	−0.240564	+	0.970633i	$0.577332\pi$
$480$	0	0
$481$	15.8344	0.721984
$482$	0	0
$483$	24.8698	1.13162
$484$	0	0
$485$	−35.8417	−1.62749
$486$	0	0
$487$	15.2222	0.689783	0.344892	−	0.938643i	$-0.387916\pi$
$487$	15.2222	0.689783	0.344892	+	0.938643i	$0.387916\pi$
$488$	0	0
$489$	−4.15882	−0.188068
$490$	0	0
$491$	−6.06208	−0.273578	−0.136789	−	0.990600i	$-0.543678\pi$
$491$	−6.06208	−0.273578	−0.136789	+	0.990600i	$0.543678\pi$
$492$	0	0
$493$	2.25685	0.101643
$494$	0	0
$495$	17.2125	0.773645
$496$	0	0
$497$	10.7246	0.481062
$498$	0	0
$499$	29.3216	1.31261	0.656307	−	0.754494i	$-0.272118\pi$
$499$	29.3216	1.31261	0.656307	+	0.754494i	$0.272118\pi$
$500$	0	0
$501$	−0.340368	−0.0152065
$502$	0	0
$503$	5.46335	0.243599	0.121799	−	0.992555i	$-0.461134\pi$
$503$	5.46335	0.243599	0.121799	+	0.992555i	$0.461134\pi$
$504$	0	0
$505$	13.2267	0.588582
$506$	0	0
$507$	−6.83283	−0.303457
$508$	0	0
$509$	−15.8506	−0.702564	−0.351282	−	0.936270i	$-0.614254\pi$
$509$	−15.8506	−0.702564	−0.351282	+	0.936270i	$0.614254\pi$
$510$	0	0
$511$	0.738559	0.0326719
$512$	0	0
$513$	−12.0218	−0.530774
$514$	0	0
$515$	34.7200	1.52994
$516$	0	0
$517$	−12.0104	−0.528215
$518$	0	0
$519$	5.93240	0.260404
$520$	0	0
$521$	36.7668	1.61078	0.805392	−	0.592742i	$-0.201955\pi$
$521$	36.7668	1.61078	0.805392	+	0.592742i	$0.201955\pi$
$522$	0	0
$523$	39.5785	1.73065	0.865324	−	0.501213i	$-0.167113\pi$
$523$	39.5785	1.73065	0.865324	+	0.501213i	$0.167113\pi$
$524$	0	0
$525$	−46.9389	−2.04858
$526$	0	0
$527$	0.00422060	0.000183852 0
$528$	0	0
$529$	7.83710	0.340743
$530$	0	0
$531$	1.42113	0.0616716
$532$	0	0
$533$	−7.44487	−0.322473
$534$	0	0
$535$	−8.98893	−0.388625
$536$	0	0
$537$	−17.2021	−0.742327
$538$	0	0
$539$	−17.5574	−0.756250
$540$	0	0
$541$	5.85716	0.251819	0.125909	−	0.992042i	$-0.459815\pi$
$541$	5.85716	0.251819	0.125909	+	0.992042i	$0.459815\pi$
$542$	0	0
$543$	0.919737	0.0394697
$544$	0	0
$545$	22.0575	0.944839
$546$	0	0
$547$	13.8724	0.593140	0.296570	−	0.955011i	$-0.404157\pi$
$547$	13.8724	0.593140	0.296570	+	0.955011i	$0.404157\pi$
$548$	0	0
$549$	16.0095	0.683271
$550$	0	0
$551$	−4.88387	−0.208060
$552$	0	0
$553$	12.7381	0.541680
$554$	0	0
$555$	28.4675	1.20838
$556$	0	0
$557$	−9.00157	−0.381409	−0.190704	−	0.981648i	$-0.561077\pi$
$557$	−9.00157	−0.381409	−0.190704	+	0.981648i	$0.561077\pi$
$558$	0	0
$559$	−30.5823	−1.29350
$560$	0	0
$561$	3.86802	0.163308
$562$	0	0
$563$	5.20078	0.219187	0.109593	−	0.993977i	$-0.465045\pi$
$563$	5.20078	0.219187	0.109593	+	0.993977i	$0.465045\pi$
$564$	0	0
$565$	−18.7671	−0.789539
$566$	0	0
$567$	9.68404	0.406692
$568$	0	0
$569$	−28.1274	−1.17916	−0.589580	−	0.807710i	$-0.700707\pi$
$569$	−28.1274	−1.17916	−0.589580	+	0.807710i	$0.700707\pi$
$570$	0	0
$571$	−8.55424	−0.357984	−0.178992	−	0.983851i	$-0.557284\pi$
$571$	−8.55424	−0.357984	−0.178992	+	0.983851i	$0.557284\pi$
$572$	0	0
$573$	−26.6658	−1.11398
$574$	0	0
$575$	−58.2014	−2.42717
$576$	0	0
$577$	−31.2295	−1.30010	−0.650052	−	0.759890i	$-0.725253\pi$
$577$	−31.2295	−1.30010	−0.650052	+	0.759890i	$0.725253\pi$
$578$	0	0
$579$	−25.7847	−1.07158
$580$	0	0
$581$	14.8312	0.615301
$582$	0	0
$583$	20.6655	0.855876
$584$	0	0
$585$	−15.3763	−0.635733
$586$	0	0
$587$	−16.9886	−0.701195	−0.350598	−	0.936526i	$-0.614021\pi$
$587$	−16.9886	−0.701195	−0.350598	+	0.936526i	$0.614021\pi$
$588$	0	0
$589$	−0.00913348	−0.000376339 0
$590$	0	0
$591$	−14.5922	−0.600244
$592$	0	0
$593$	21.8405	0.896883	0.448441	−	0.893812i	$-0.351979\pi$
$593$	21.8405	0.896883	0.448441	+	0.893812i	$0.351979\pi$
$594$	0	0
$595$	14.0236	0.574912
$596$	0	0
$597$	20.7146	0.847794
$598$	0	0
$599$	13.4138	0.548073	0.274037	−	0.961719i	$-0.411641\pi$
$599$	13.4138	0.548073	0.274037	+	0.961719i	$0.411641\pi$
$600$	0	0
$601$	18.3687	0.749273	0.374637	−	0.927172i	$-0.377768\pi$
$601$	18.3687	0.749273	0.374637	+	0.927172i	$0.377768\pi$
$602$	0	0
$603$	7.73371	0.314941
$604$	0	0
$605$	−5.99589	−0.243768
$606$	0	0
$607$	33.6872	1.36732	0.683661	−	0.729799i	$-0.260386\pi$
$607$	33.6872	1.36732	0.683661	+	0.729799i	$0.260386\pi$
$608$	0	0
$609$	10.1074	0.409572
$610$	0	0
$611$	10.7291	0.434053
$612$	0	0
$613$	16.2898	0.657938	0.328969	−	0.944341i	$-0.393299\pi$
$613$	16.2898	0.657938	0.328969	+	0.944341i	$0.393299\pi$
$614$	0	0
$615$	−13.3846	−0.539719
$616$	0	0
$617$	−18.7025	−0.752934	−0.376467	−	0.926430i	$-0.622861\pi$
$617$	−18.7025	−0.752934	−0.376467	+	0.926430i	$0.622861\pi$
$618$	0	0
$619$	−12.5456	−0.504251	−0.252125	−	0.967695i	$-0.581130\pi$
$619$	−12.5456	−0.504251	−0.252125	+	0.967695i	$0.581130\pi$
$620$	0	0
$621$	30.8492	1.23793
$622$	0	0
$623$	−20.9577	−0.839653
$624$	0	0
$625$	32.4440	1.29776
$626$	0	0
$627$	−8.37048	−0.334285
$628$	0	0
$629$	−5.75808	−0.229590
$630$	0	0
$631$	24.8839	0.990614	0.495307	−	0.868718i	$-0.335056\pi$
$631$	24.8839	0.990614	0.495307	+	0.868718i	$0.335056\pi$
$632$	0	0
$633$	18.0116	0.715897
$634$	0	0
$635$	29.2804	1.16196
$636$	0	0
$637$	15.6844	0.621438
$638$	0	0
$639$	4.27611	0.169160
$640$	0	0
$641$	−1.43843	−0.0568145	−0.0284073	−	0.999596i	$-0.509044\pi$
$641$	−1.43843	−0.0568145	−0.0284073	+	0.999596i	$0.509044\pi$
$642$	0	0
$643$	24.2853	0.957720	0.478860	−	0.877891i	$-0.341050\pi$
$643$	24.2853	0.957720	0.478860	+	0.877891i	$0.341050\pi$
$644$	0	0
$645$	−54.9818	−2.16491
$646$	0	0
$647$	−37.6283	−1.47932	−0.739660	−	0.672981i	$-0.765014\pi$
$647$	−37.6283	−1.47932	−0.739660	+	0.672981i	$0.765014\pi$
$648$	0	0
$649$	3.07833	0.120835
$650$	0	0
$651$	0.0189021	0.000740833 0
$652$	0	0
$653$	35.0033	1.36979	0.684893	−	0.728644i	$-0.259849\pi$
$653$	35.0033	1.36979	0.684893	+	0.728644i	$0.259849\pi$
$654$	0	0
$655$	39.3148	1.53615
$656$	0	0
$657$	0.294479	0.0114887
$658$	0	0
$659$	−39.9805	−1.55742	−0.778710	−	0.627384i	$-0.784126\pi$
$659$	−39.9805	−1.55742	−0.778710	+	0.627384i	$0.784126\pi$
$660$	0	0
$661$	−21.0305	−0.817991	−0.408995	−	0.912536i	$-0.634121\pi$
$661$	−21.0305	−0.817991	−0.408995	+	0.912536i	$0.634121\pi$
$662$	0	0
$663$	−3.45539	−0.134196
$664$	0	0
$665$	−30.3474	−1.17682
$666$	0	0
$667$	12.5325	0.485262
$668$	0	0
$669$	−7.65888	−0.296109
$670$	0	0
$671$	34.6786	1.33875
$672$	0	0
$673$	24.4543	0.942643	0.471322	−	0.881961i	$-0.343777\pi$
$673$	24.4543	0.942643	0.471322	+	0.881961i	$0.343777\pi$
$674$	0	0
$675$	−58.2242	−2.24105
$676$	0	0
$677$	−13.5133	−0.519357	−0.259679	−	0.965695i	$-0.583617\pi$
$677$	−13.5133	−0.519357	−0.259679	+	0.965695i	$0.583617\pi$
$678$	0	0
$679$	32.4679	1.24600
$680$	0	0
$681$	−12.1907	−0.467148
$682$	0	0
$683$	35.1926	1.34661	0.673304	−	0.739366i	$-0.264874\pi$
$683$	35.1926	1.34661	0.673304	+	0.739366i	$0.264874\pi$
$684$	0	0
$685$	81.7864	3.12490
$686$	0	0
$687$	−3.97712	−0.151737
$688$	0	0
$689$	−18.4609	−0.703304
$690$	0	0
$691$	−32.3054	−1.22896	−0.614478	−	0.788934i	$-0.710633\pi$
$691$	−32.3054	−1.22896	−0.614478	+	0.788934i	$0.710633\pi$
$692$	0	0
$693$	−15.5923	−0.592302
$694$	0	0
$695$	−70.4398	−2.67193
$696$	0	0
$697$	2.70729	0.102546
$698$	0	0
$699$	−0.673188	−0.0254623
$700$	0	0
$701$	4.10932	0.155207	0.0776034	−	0.996984i	$-0.475273\pi$
$701$	4.10932	0.155207	0.0776034	+	0.996984i	$0.475273\pi$
$702$	0	0
$703$	12.4606	0.469961
$704$	0	0
$705$	19.2891	0.726470
$706$	0	0
$707$	−11.9817	−0.450617
$708$	0	0
$709$	−33.2063	−1.24709	−0.623545	−	0.781787i	$-0.714308\pi$
$709$	−33.2063	−1.24709	−0.623545	+	0.781787i	$0.714308\pi$
$710$	0	0
$711$	5.07897	0.190476
$712$	0	0
$713$	0.0234375	0.000877741 0
$714$	0	0
$715$	−33.3070	−1.24561
$716$	0	0
$717$	−4.65212	−0.173736
$718$	0	0
$719$	−18.9088	−0.705181	−0.352591	−	0.935778i	$-0.614699\pi$
$719$	−18.9088	−0.705181	−0.352591	+	0.935778i	$0.614699\pi$
$720$	0	0
$721$	−31.4517	−1.17132
$722$	0	0
$723$	10.2526	0.381299
$724$	0	0
$725$	−23.6537	−0.878477
$726$	0	0
$727$	−38.0421	−1.41090	−0.705452	−	0.708758i	$-0.749256\pi$
$727$	−38.0421	−1.41090	−0.705452	+	0.708758i	$0.749256\pi$
$728$	0	0
$729$	24.8024	0.918609
$730$	0	0
$731$	11.1211	0.411329
$732$	0	0
$733$	5.15052	0.190239	0.0951194	−	0.995466i	$-0.469677\pi$
$733$	5.15052	0.190239	0.0951194	+	0.995466i	$0.469677\pi$
$734$	0	0
$735$	28.1979	1.04009
$736$	0	0
$737$	16.7521	0.617073
$738$	0	0
$739$	20.6112	0.758196	0.379098	−	0.925356i	$-0.376234\pi$
$739$	20.6112	0.758196	0.379098	+	0.925356i	$0.376234\pi$
$740$	0	0
$741$	7.47753	0.274694
$742$	0	0
$743$	−5.34340	−0.196030	−0.0980152	−	0.995185i	$-0.531249\pi$
$743$	−5.34340	−0.196030	−0.0980152	+	0.995185i	$0.531249\pi$
$744$	0	0
$745$	−23.0568	−0.844736
$746$	0	0
$747$	5.91352	0.216364
$748$	0	0
$749$	8.14279	0.297531
$750$	0	0
$751$	−5.59110	−0.204022	−0.102011	−	0.994783i	$-0.532528\pi$
$751$	−5.59110	−0.204022	−0.102011	+	0.994783i	$0.532528\pi$
$752$	0	0
$753$	−32.0496	−1.16795
$754$	0	0
$755$	−20.9155	−0.761191
$756$	0	0
$757$	33.6951	1.22467	0.612335	−	0.790598i	$-0.290230\pi$
$757$	33.6951	1.22467	0.612335	+	0.790598i	$0.290230\pi$
$758$	0	0
$759$	21.4796	0.779659
$760$	0	0
$761$	−5.38882	−0.195344	−0.0976722	−	0.995219i	$-0.531140\pi$
$761$	−5.38882	−0.195344	−0.0976722	+	0.995219i	$0.531140\pi$
$762$	0	0
$763$	−19.9812	−0.723368
$764$	0	0
$765$	5.59152	0.202162
$766$	0	0
$767$	−2.74994	−0.0992945
$768$	0	0
$769$	28.2792	1.01977	0.509886	−	0.860242i	$-0.329688\pi$
$769$	28.2792	1.01977	0.509886	+	0.860242i	$0.329688\pi$
$770$	0	0
$771$	−12.8903	−0.464232
$772$	0	0
$773$	−44.2573	−1.59182	−0.795911	−	0.605413i	$-0.793008\pi$
$773$	−44.2573	−1.59182	−0.795911	+	0.605413i	$0.793008\pi$
$774$	0	0
$775$	−0.0442355	−0.00158899
$776$	0	0
$777$	−25.7878	−0.925131
$778$	0	0
$779$	−5.85863	−0.209907
$780$	0	0
$781$	9.26257	0.331441
$782$	0	0
$783$	12.5375	0.448052
$784$	0	0
$785$	−29.3649	−1.04808
$786$	0	0
$787$	−39.5510	−1.40984	−0.704920	−	0.709287i	$-0.749017\pi$
$787$	−39.5510	−1.40984	−0.704920	+	0.709287i	$0.749017\pi$
$788$	0	0
$789$	17.0915	0.608474
$790$	0	0
$791$	17.0005	0.604470
$792$	0	0
$793$	−30.9791	−1.10010
$794$	0	0
$795$	−33.1896	−1.17711
$796$	0	0
$797$	16.6094	0.588336	0.294168	−	0.955754i	$-0.404957\pi$
$797$	16.6094	0.588336	0.294168	+	0.955754i	$0.404957\pi$
$798$	0	0
$799$	−3.90158	−0.138028
$800$	0	0
$801$	−8.35629	−0.295255
$802$	0	0
$803$	0.637877	0.0225102
$804$	0	0
$805$	77.8748	2.74472
$806$	0	0
$807$	37.1445	1.30755
$808$	0	0
$809$	−26.7324	−0.939862	−0.469931	−	0.882703i	$-0.655721\pi$
$809$	−26.7324	−0.939862	−0.469931	+	0.882703i	$0.655721\pi$
$810$	0	0
$811$	−24.8178	−0.871470	−0.435735	−	0.900075i	$-0.643511\pi$
$811$	−24.8178	−0.871470	−0.435735	+	0.900075i	$0.643511\pi$
$812$	0	0
$813$	−38.1389	−1.33759
$814$	0	0
$815$	−13.0225	−0.456157
$816$	0	0
$817$	−24.0663	−0.841974
$818$	0	0
$819$	13.9289	0.486716
$820$	0	0
$821$	−42.0803	−1.46861	−0.734306	−	0.678819i	$-0.762492\pi$
$821$	−42.0803	−1.46861	−0.734306	+	0.678819i	$0.762492\pi$
$822$	0	0
$823$	−20.8093	−0.725366	−0.362683	−	0.931912i	$-0.618139\pi$
$823$	−20.8093	−0.725366	−0.362683	+	0.931912i	$0.618139\pi$
$824$	0	0
$825$	−40.5401	−1.41143
$826$	0	0
$827$	6.28377	0.218508	0.109254	−	0.994014i	$-0.465154\pi$
$827$	6.28377	0.218508	0.109254	+	0.994014i	$0.465154\pi$
$828$	0	0
$829$	32.7063	1.13594	0.567969	−	0.823050i	$-0.307730\pi$
$829$	32.7063	1.13594	0.567969	+	0.823050i	$0.307730\pi$
$830$	0	0
$831$	−4.89470	−0.169795
$832$	0	0
$833$	−5.70355	−0.197616
$834$	0	0
$835$	−1.06579	−0.0368832
$836$	0	0
$837$	0.0234467	0.000810436 0
$838$	0	0
$839$	0.383506	0.0132401	0.00662005	−	0.999978i	$-0.497893\pi$
$839$	0.383506	0.0132401	0.00662005	+	0.999978i	$0.497893\pi$
$840$	0	0
$841$	−23.9066	−0.824367
$842$	0	0
$843$	8.46990	0.291719
$844$	0	0
$845$	−21.3956	−0.736031
$846$	0	0
$847$	5.43149	0.186628
$848$	0	0
$849$	−12.7416	−0.437291
$850$	0	0
$851$	−31.9753	−1.09610
$852$	0	0
$853$	22.1179	0.757301	0.378651	−	0.925540i	$-0.376388\pi$
$853$	22.1179	0.757301	0.378651	+	0.925540i	$0.376388\pi$
$854$	0	0
$855$	−12.1002	−0.413817
$856$	0	0
$857$	43.2225	1.47645	0.738227	−	0.674552i	$-0.235663\pi$
$857$	43.2225	1.47645	0.738227	+	0.674552i	$0.235663\pi$
$858$	0	0
$859$	−0.633535	−0.0216159	−0.0108080	−	0.999942i	$-0.503440\pi$
$859$	−0.633535	−0.0216159	−0.0108080	+	0.999942i	$0.503440\pi$
$860$	0	0
$861$	12.1247	0.413208
$862$	0	0
$863$	16.2376	0.552736	0.276368	−	0.961052i	$-0.410869\pi$
$863$	16.2376	0.552736	0.276368	+	0.961052i	$0.410869\pi$
$864$	0	0
$865$	18.5761	0.631606
$866$	0	0
$867$	1.25653	0.0426741
$868$	0	0
$869$	11.0016	0.373205
$870$	0	0
$871$	−14.9651	−0.507071
$872$	0	0
$873$	12.9457	0.438144
$874$	0	0
$875$	−76.8613	−2.59839
$876$	0	0
$877$	−43.7849	−1.47851	−0.739256	−	0.673424i	$-0.764823\pi$
$877$	−43.7849	−1.47851	−0.739256	+	0.673424i	$0.764823\pi$
$878$	0	0
$879$	−33.1499	−1.11812
$880$	0	0
$881$	48.8665	1.64635	0.823176	−	0.567786i	$-0.192200\pi$
$881$	48.8665	1.64635	0.823176	+	0.567786i	$0.192200\pi$
$882$	0	0
$883$	−33.4586	−1.12597	−0.562985	−	0.826467i	$-0.690347\pi$
$883$	−33.4586	−1.12597	−0.562985	+	0.826467i	$0.690347\pi$
$884$	0	0
$885$	−4.94392	−0.166188
$886$	0	0
$887$	−55.1309	−1.85111	−0.925557	−	0.378609i	$-0.876402\pi$
$887$	−55.1309	−1.85111	−0.925557	+	0.378609i	$0.876402\pi$
$888$	0	0
$889$	−26.5242	−0.889593
$890$	0	0
$891$	8.36390	0.280201
$892$	0	0
$893$	8.44311	0.282538
$894$	0	0
$895$	−53.8649	−1.80051
$896$	0	0
$897$	−19.1882	−0.640674
$898$	0	0
$899$	0.00952527	0.000317685 0
$900$	0	0
$901$	6.71321	0.223649
$902$	0	0
$903$	49.8063	1.65745
$904$	0	0
$905$	2.87996	0.0957333
$906$	0	0
$907$	−30.8505	−1.02438	−0.512188	−	0.858874i	$-0.671165\pi$
$907$	−30.8505	−1.02438	−0.512188	+	0.858874i	$0.671165\pi$
$908$	0	0
$909$	−4.77735	−0.158455
$910$	0	0
$911$	−20.6821	−0.685229	−0.342614	−	0.939476i	$-0.611312\pi$
$911$	−20.6821	−0.685229	−0.342614	+	0.939476i	$0.611312\pi$
$912$	0	0
$913$	12.8094	0.423929
$914$	0	0
$915$	−55.6952	−1.84123
$916$	0	0
$917$	−35.6140	−1.17608
$918$	0	0
$919$	−59.6795	−1.96864	−0.984322	−	0.176378i	$-0.943562\pi$
$919$	−59.6795	−1.96864	−0.984322	+	0.176378i	$0.943562\pi$
$920$	0	0
$921$	−14.4130	−0.474924
$922$	0	0
$923$	−8.27446	−0.272357
$924$	0	0
$925$	60.3496	1.98428
$926$	0	0
$927$	−12.5405	−0.411883
$928$	0	0
$929$	16.2162	0.532035	0.266017	−	0.963968i	$-0.414292\pi$
$929$	16.2162	0.532035	0.266017	+	0.963968i	$0.414292\pi$
$930$	0	0
$931$	12.3426	0.404512
$932$	0	0
$933$	0.473333	0.0154962
$934$	0	0
$935$	12.1119	0.396101
$936$	0	0
$937$	19.3249	0.631315	0.315658	−	0.948873i	$-0.397775\pi$
$937$	19.3249	0.631315	0.315658	+	0.948873i	$0.397775\pi$
$938$	0	0
$939$	16.6136	0.542163
$940$	0	0
$941$	38.3566	1.25039	0.625195	−	0.780468i	$-0.285019\pi$
$941$	38.3566	1.25039	0.625195	+	0.780468i	$0.285019\pi$
$942$	0	0
$943$	15.0339	0.489570
$944$	0	0
$945$	77.9052	2.53426
$946$	0	0
$947$	3.83161	0.124511	0.0622553	−	0.998060i	$-0.480171\pi$
$947$	3.83161	0.124511	0.0622553	+	0.998060i	$0.480171\pi$
$948$	0	0
$949$	−0.569830	−0.0184975
$950$	0	0
$951$	24.1566	0.783333
$952$	0	0
$953$	33.9178	1.09871	0.549353	−	0.835590i	$-0.314874\pi$
$953$	33.9178	1.09871	0.549353	+	0.835590i	$0.314874\pi$
$954$	0	0
$955$	−83.4984	−2.70194
$956$	0	0
$957$	8.72954	0.282186
$958$	0	0
$959$	−74.0877	−2.39242
$960$	0	0
$961$	−31.0000	−0.999999
$962$	0	0
$963$	3.24671	0.104624
$964$	0	0
$965$	−80.7395	−2.59910
$966$	0	0
$967$	−7.04620	−0.226591	−0.113295	−	0.993561i	$-0.536141\pi$
$967$	−7.04620	−0.226591	−0.113295	+	0.993561i	$0.536141\pi$
$968$	0	0
$969$	−2.71917	−0.0873522
$970$	0	0
$971$	40.7548	1.30788	0.653942	−	0.756545i	$-0.273114\pi$
$971$	40.7548	1.30788	0.653942	+	0.756545i	$0.273114\pi$
$972$	0	0
$973$	63.8092	2.04563
$974$	0	0
$975$	36.2154	1.15982
$976$	0	0
$977$	−47.2211	−1.51074	−0.755368	−	0.655301i	$-0.772542\pi$
$977$	−47.2211	−1.51074	−0.755368	+	0.655301i	$0.772542\pi$
$978$	0	0
$979$	−18.1007	−0.578502
$980$	0	0
$981$	−7.96693	−0.254365
$982$	0	0
$983$	−30.9654	−0.987642	−0.493821	−	0.869564i	$-0.664400\pi$
$983$	−30.9654	−0.987642	−0.493821	+	0.869564i	$0.664400\pi$
$984$	0	0
$985$	−45.6925	−1.45588
$986$	0	0
$987$	−17.4734	−0.556184
$988$	0	0
$989$	61.7568	1.96375
$990$	0	0
$991$	19.7724	0.628092	0.314046	−	0.949408i	$-0.398315\pi$
$991$	19.7724	0.628092	0.314046	+	0.949408i	$0.398315\pi$
$992$	0	0
$993$	21.9709	0.697227
$994$	0	0
$995$	64.8636	2.05631
$996$	0	0
$997$	−37.6434	−1.19218	−0.596090	−	0.802918i	$-0.703280\pi$
$997$	−37.6434	−1.19218	−0.596090	+	0.802918i	$0.703280\pi$
$998$	0	0
$999$	−31.9878	−1.01205

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8024.2.a.y.1.17	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8024.2.a.y.1.17	✓	23	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8024 = 2^{3} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.25653 q^{3} +3.93457 q^{5} -3.56420 q^{7} -1.42113 q^{9} +O(q^{10})\)	\(q+1.25653 q^{3} +3.93457 q^{5} -3.56420 q^{7} -1.42113 q^{9} -3.07833 q^{11} +2.74994 q^{13} +4.94392 q^{15} -1.00000 q^{17} +2.16402 q^{19} -4.47854 q^{21} -5.55312 q^{23} +10.4809 q^{25} -5.55529 q^{27} -2.25685 q^{29} -0.00422060 q^{31} -3.86802 q^{33} -14.0236 q^{35} +5.75808 q^{37} +3.45539 q^{39} -2.70729 q^{41} -11.1211 q^{43} -5.59152 q^{45} +3.90158 q^{47} +5.70355 q^{49} -1.25653 q^{51} -6.71321 q^{53} -12.1119 q^{55} +2.71917 q^{57} -1.00000 q^{59} -11.2654 q^{61} +5.06518 q^{63} +10.8198 q^{65} -5.44196 q^{67} -6.97767 q^{69} -3.00896 q^{71} -0.207216 q^{73} +13.1695 q^{75} +10.9718 q^{77} -3.57390 q^{79} -2.71703 q^{81} -4.16115 q^{83} -3.93457 q^{85} -2.83581 q^{87} +5.88005 q^{89} -9.80134 q^{91} -0.00530333 q^{93} +8.51450 q^{95} -9.10944 q^{97} +4.37469 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9}+O(q^{10}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9	\( 23 q - 6 q^{3} - q^{7} + 23 q^{9} - 3 q^{11} - 7 q^{13} - 2 q^{15} - 23 q^{17} - 16 q^{19} - 11 q^{21} - 29 q^{23} + 31 q^{25} - 3 q^{27} - 5 q^{29} - 41 q^{31} + 8 q^{33} - 22 q^{35} + 5 q^{37} + 16 q^{39} + 11 q^{41} + 13 q^{43} - 26 q^{45} - 39 q^{47} + 16 q^{49} + 6 q^{51} - 2 q^{53} - 35 q^{55} + 13 q^{57} - 23 q^{59} - 37 q^{61} + 33 q^{65} - 34 q^{67} - 66 q^{69} - 13 q^{71} - 14 q^{73} - 81 q^{75} - 4 q^{77} - 61 q^{79} - q^{81} - 9 q^{83} - 16 q^{87} + 28 q^{89} - 18 q^{91} - 62 q^{93} - 33 q^{95} - 5 q^{99}+O(q^{100}) \) 23 * q - 6 * q^3 - q^7 + 23 * q^9 - 3 * q^11 - 7 * q^13 - 2 * q^15 - 23 * q^17 - 16 * q^19 - 11 * q^21 - 29 * q^23 + 31 * q^25 - 3 * q^27 - 5 * q^29 - 41 * q^31 + 8 * q^33 - 22 * q^35 + 5 * q^37 + 16 * q^39 + 11 * q^41 + 13 * q^43 - 26 * q^45 - 39 * q^47 + 16 * q^49 + 6 * q^51 - 2 * q^53 - 35 * q^55 + 13 * q^57 - 23 * q^59 - 37 * q^61 + 33 * q^65 - 34 * q^67 - 66 * q^69 - 13 * q^71 - 14 * q^73 - 81 * q^75 - 4 * q^77 - 61 * q^79 - q^81 - 9 * q^83 - 16 * q^87 + 28 * q^89 - 18 * q^91 - 62 * q^93 - 33 * q^95 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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