LMFDB - Embedded newform 6024.2.a.k.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6024,2,Mod(1,6024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6024, 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("6024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6024 = 2^{3} \cdot 3 \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6024.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:	$48.1018821776$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 10x^{6} + 25x^{5} + 5x^{4} - 36x^{3} + 11x^{2} + 3x - 1 $ x^8 - 2x^7 - 10x^6 + 25x^5 + 5x^4 - 36x^3 + 11x^2 + 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$-3.14539$ of defining polynomial
Character	$\chi$	$=$	6024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +2.46685 q^{5} -0.744363 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +2.46685 q^{5} -0.744363 q^{7} +1.00000 q^{9} -3.14892 q^{11} -1.92263 q^{13} +2.46685 q^{15} -5.82468 q^{17} -0.218570 q^{19} -0.744363 q^{21} +1.21355 q^{23} +1.08534 q^{25} +1.00000 q^{27} -2.06754 q^{29} +3.35167 q^{31} -3.14892 q^{33} -1.83623 q^{35} -5.53662 q^{37} -1.92263 q^{39} +7.34169 q^{41} -8.25205 q^{43} +2.46685 q^{45} -5.03304 q^{47} -6.44592 q^{49} -5.82468 q^{51} -5.56376 q^{53} -7.76792 q^{55} -0.218570 q^{57} +5.15094 q^{59} +9.53844 q^{61} -0.744363 q^{63} -4.74285 q^{65} +1.46154 q^{67} +1.21355 q^{69} -0.548665 q^{71} -5.81898 q^{73} +1.08534 q^{75} +2.34394 q^{77} -4.72725 q^{79} +1.00000 q^{81} +14.4680 q^{83} -14.3686 q^{85} -2.06754 q^{87} -10.4068 q^{89} +1.43114 q^{91} +3.35167 q^{93} -0.539179 q^{95} -13.9082 q^{97} -3.14892 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9	$ 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} - 6 q^{11} - 4 q^{13} - 5 q^{15} - 12 q^{17} - 10 q^{19} + q^{21} - 6 q^{23} - 7 q^{25} + 8 q^{27} - 2 q^{29} - 3 q^{31} - 6 q^{33} + 4 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} - 5 q^{45} - 15 q^{47} - 21 q^{49} - 12 q^{51} + 13 q^{53} - 33 q^{55} - 10 q^{57} + 8 q^{59} - 38 q^{61} + q^{63} + 16 q^{65} - 31 q^{67} - 6 q^{69} + 5 q^{71} - 26 q^{73} - 7 q^{75} - 24 q^{77} - 25 q^{79} + 8 q^{81} - 7 q^{83} - 18 q^{85} - 2 q^{87} - 14 q^{89} - 6 q^{91} - 3 q^{93} - q^{95} - 51 q^{97} - 6 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9 - 6 * q^11 - 4 * q^13 - 5 * q^15 - 12 * q^17 - 10 * q^19 + q^21 - 6 * q^23 - 7 * q^25 + 8 * q^27 - 2 * q^29 - 3 * q^31 - 6 * q^33 + 4 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 - 5 * q^45 - 15 * q^47 - 21 * q^49 - 12 * q^51 + 13 * q^53 - 33 * q^55 - 10 * q^57 + 8 * q^59 - 38 * q^61 + q^63 + 16 * q^65 - 31 * q^67 - 6 * q^69 + 5 * q^71 - 26 * q^73 - 7 * q^75 - 24 * q^77 - 25 * q^79 + 8 * q^81 - 7 * q^83 - 18 * q^85 - 2 * q^87 - 14 * q^89 - 6 * q^91 - 3 * q^93 - q^95 - 51 * q^97 - 6 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	2.46685	1.10321	0.551604	−	0.834106i	$-0.314016\pi$
$5$	2.46685	1.10321	0.551604	+	0.834106i	$0.314016\pi$
$6$	0	0
$7$	−0.744363	−0.281343	−0.140671	−	0.990056i	$-0.544926\pi$
$7$	−0.744363	−0.281343	−0.140671	+	0.990056i	$0.544926\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.14892	−0.949436	−0.474718	−	0.880138i	$-0.657450\pi$
$11$	−3.14892	−0.949436	−0.474718	+	0.880138i	$0.657450\pi$
$12$	0	0
$13$	−1.92263	−0.533243	−0.266621	−	0.963801i	$-0.585907\pi$
$13$	−1.92263	−0.533243	−0.266621	+	0.963801i	$0.585907\pi$
$14$	0	0
$15$	2.46685	0.636937
$16$	0	0
$17$	−5.82468	−1.41269	−0.706347	−	0.707866i	$-0.749658\pi$
$17$	−5.82468	−1.41269	−0.706347	+	0.707866i	$0.749658\pi$
$18$	0	0
$19$	−0.218570	−0.0501434	−0.0250717	−	0.999686i	$-0.507981\pi$
$19$	−0.218570	−0.0501434	−0.0250717	+	0.999686i	$0.507981\pi$
$20$	0	0
$21$	−0.744363	−0.162433
$22$	0	0
$23$	1.21355	0.253044	0.126522	−	0.991964i	$-0.459619\pi$
$23$	1.21355	0.253044	0.126522	+	0.991964i	$0.459619\pi$
$24$	0	0
$25$	1.08534	0.217068
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.06754	−0.383933	−0.191966	−	0.981402i	$-0.561486\pi$
$29$	−2.06754	−0.383933	−0.191966	+	0.981402i	$0.561486\pi$
$30$	0	0
$31$	3.35167	0.601978	0.300989	−	0.953628i	$-0.402683\pi$
$31$	3.35167	0.601978	0.300989	+	0.953628i	$0.402683\pi$
$32$	0	0
$33$	−3.14892	−0.548157
$34$	0	0
$35$	−1.83623	−0.310380
$36$	0	0
$37$	−5.53662	−0.910214	−0.455107	−	0.890437i	$-0.650399\pi$
$37$	−5.53662	−0.910214	−0.455107	+	0.890437i	$0.650399\pi$
$38$	0	0
$39$	−1.92263	−0.307868
$40$	0	0
$41$	7.34169	1.14658	0.573290	−	0.819352i	$-0.305667\pi$
$41$	7.34169	1.14658	0.573290	+	0.819352i	$0.305667\pi$
$42$	0	0
$43$	−8.25205	−1.25843	−0.629213	−	0.777233i	$-0.716623\pi$
$43$	−8.25205	−1.25843	−0.629213	+	0.777233i	$0.716623\pi$
$44$	0	0
$45$	2.46685	0.367736
$46$	0	0
$47$	−5.03304	−0.734144	−0.367072	−	0.930193i	$-0.619640\pi$
$47$	−5.03304	−0.734144	−0.367072	+	0.930193i	$0.619640\pi$
$48$	0	0
$49$	−6.44592	−0.920846
$50$	0	0
$51$	−5.82468	−0.815619
$52$	0	0
$53$	−5.56376	−0.764241	−0.382120	−	0.924113i	$-0.624806\pi$
$53$	−5.56376	−0.764241	−0.382120	+	0.924113i	$0.624806\pi$
$54$	0	0
$55$	−7.76792	−1.04743
$56$	0	0
$57$	−0.218570	−0.0289503
$58$	0	0
$59$	5.15094	0.670595	0.335297	−	0.942112i	$-0.391163\pi$
$59$	5.15094	0.670595	0.335297	+	0.942112i	$0.391163\pi$
$60$	0	0
$61$	9.53844	1.22127	0.610636	−	0.791911i	$-0.290914\pi$
$61$	9.53844	1.22127	0.610636	+	0.791911i	$0.290914\pi$
$62$	0	0
$63$	−0.744363	−0.0937810
$64$	0	0
$65$	−4.74285	−0.588278
$66$	0	0
$67$	1.46154	0.178555	0.0892775	−	0.996007i	$-0.471544\pi$
$67$	1.46154	0.178555	0.0892775	+	0.996007i	$0.471544\pi$
$68$	0	0
$69$	1.21355	0.146095
$70$	0	0
$71$	−0.548665	−0.0651146	−0.0325573	−	0.999470i	$-0.510365\pi$
$71$	−0.548665	−0.0651146	−0.0325573	+	0.999470i	$0.510365\pi$
$72$	0	0
$73$	−5.81898	−0.681060	−0.340530	−	0.940234i	$-0.610606\pi$
$73$	−5.81898	−0.681060	−0.340530	+	0.940234i	$0.610606\pi$
$74$	0	0
$75$	1.08534	0.125324
$76$	0	0
$77$	2.34394	0.267117
$78$	0	0
$79$	−4.72725	−0.531857	−0.265928	−	0.963993i	$-0.585678\pi$
$79$	−4.72725	−0.531857	−0.265928	+	0.963993i	$0.585678\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	14.4680	1.58807	0.794037	−	0.607870i	$-0.207976\pi$
$83$	14.4680	1.58807	0.794037	+	0.607870i	$0.207976\pi$
$84$	0	0
$85$	−14.3686	−1.55849
$86$	0	0
$87$	−2.06754	−0.221664
$88$	0	0
$89$	−10.4068	−1.10312	−0.551562	−	0.834134i	$-0.685968\pi$
$89$	−10.4068	−1.10312	−0.551562	+	0.834134i	$0.685968\pi$
$90$	0	0
$91$	1.43114	0.150024
$92$	0	0
$93$	3.35167	0.347552
$94$	0	0
$95$	−0.539179	−0.0553186
$96$	0	0
$97$	−13.9082	−1.41217	−0.706084	−	0.708128i	$-0.749540\pi$
$97$	−13.9082	−1.41217	−0.706084	+	0.708128i	$0.749540\pi$
$98$	0	0
$99$	−3.14892	−0.316479
$100$	0	0
$101$	−17.5177	−1.74307	−0.871536	−	0.490332i	$-0.836876\pi$
$101$	−17.5177	−1.74307	−0.871536	+	0.490332i	$0.836876\pi$
$102$	0	0
$103$	5.27753	0.520010	0.260005	−	0.965607i	$-0.416276\pi$
$103$	5.27753	0.520010	0.260005	+	0.965607i	$0.416276\pi$
$104$	0	0
$105$	−1.83623	−0.179198
$106$	0	0
$107$	−10.4150	−1.00686	−0.503430	−	0.864036i	$-0.667929\pi$
$107$	−10.4150	−1.00686	−0.503430	+	0.864036i	$0.667929\pi$
$108$	0	0
$109$	−6.57930	−0.630182	−0.315091	−	0.949061i	$-0.602035\pi$
$109$	−6.57930	−0.630182	−0.315091	+	0.949061i	$0.602035\pi$
$110$	0	0
$111$	−5.53662	−0.525513
$112$	0	0
$113$	0.866743	0.0815363	0.0407682	−	0.999169i	$-0.487019\pi$
$113$	0.866743	0.0815363	0.0407682	+	0.999169i	$0.487019\pi$
$114$	0	0
$115$	2.99365	0.279160
$116$	0	0
$117$	−1.92263	−0.177748
$118$	0	0
$119$	4.33568	0.397451
$120$	0	0
$121$	−1.08428	−0.0985713
$122$	0	0
$123$	7.34169	0.661978
$124$	0	0
$125$	−9.65687	−0.863737
$126$	0	0
$127$	11.0853	0.983659	0.491830	−	0.870691i	$-0.336328\pi$
$127$	11.0853	0.983659	0.491830	+	0.870691i	$0.336328\pi$
$128$	0	0
$129$	−8.25205	−0.726553
$130$	0	0
$131$	−12.1169	−1.05866	−0.529328	−	0.848417i	$-0.677556\pi$
$131$	−12.1169	−1.05866	−0.529328	+	0.848417i	$0.677556\pi$
$132$	0	0
$133$	0.162695	0.0141075
$134$	0	0
$135$	2.46685	0.212312
$136$	0	0
$137$	−21.2051	−1.81168	−0.905838	−	0.423624i	$-0.860758\pi$
$137$	−21.2051	−1.81168	−0.905838	+	0.423624i	$0.860758\pi$
$138$	0	0
$139$	1.75393	0.148766	0.0743830	−	0.997230i	$-0.476301\pi$
$139$	1.75393	0.148766	0.0743830	+	0.997230i	$0.476301\pi$
$140$	0	0
$141$	−5.03304	−0.423858
$142$	0	0
$143$	6.05423	0.506280
$144$	0	0
$145$	−5.10031	−0.423558
$146$	0	0
$147$	−6.44592	−0.531651
$148$	0	0
$149$	−2.53390	−0.207585	−0.103793	−	0.994599i	$-0.533098\pi$
$149$	−2.53390	−0.207585	−0.103793	+	0.994599i	$0.533098\pi$
$150$	0	0
$151$	−15.5191	−1.26293	−0.631463	−	0.775406i	$-0.717545\pi$
$151$	−15.5191	−1.26293	−0.631463	+	0.775406i	$0.717545\pi$
$152$	0	0
$153$	−5.82468	−0.470898
$154$	0	0
$155$	8.26807	0.664107
$156$	0	0
$157$	13.8188	1.10286	0.551431	−	0.834221i	$-0.314082\pi$
$157$	13.8188	1.10286	0.551431	+	0.834221i	$0.314082\pi$
$158$	0	0
$159$	−5.56376	−0.441235
$160$	0	0
$161$	−0.903325	−0.0711920
$162$	0	0
$163$	0.0481229	0.00376927	0.00188464	−	0.999998i	$-0.499400\pi$
$163$	0.0481229	0.00376927	0.00188464	+	0.999998i	$0.499400\pi$
$164$	0	0
$165$	−7.76792	−0.604731
$166$	0	0
$167$	21.7397	1.68227	0.841135	−	0.540825i	$-0.181888\pi$
$167$	21.7397	1.68227	0.841135	+	0.540825i	$0.181888\pi$
$168$	0	0
$169$	−9.30348	−0.715652
$170$	0	0
$171$	−0.218570	−0.0167145
$172$	0	0
$173$	25.9429	1.97241	0.986203	−	0.165542i	$-0.0529374\pi$
$173$	25.9429	1.97241	0.986203	+	0.165542i	$0.0529374\pi$
$174$	0	0
$175$	−0.807887	−0.0610705
$176$	0	0
$177$	5.15094	0.387168
$178$	0	0
$179$	−17.2815	−1.29168	−0.645839	−	0.763474i	$-0.723492\pi$
$179$	−17.2815	−1.29168	−0.645839	+	0.763474i	$0.723492\pi$
$180$	0	0
$181$	3.17744	0.236177	0.118089	−	0.993003i	$-0.462323\pi$
$181$	3.17744	0.236177	0.118089	+	0.993003i	$0.462323\pi$
$182$	0	0
$183$	9.53844	0.705102
$184$	0	0
$185$	−13.6580	−1.00416
$186$	0	0
$187$	18.3415	1.34126
$188$	0	0
$189$	−0.744363	−0.0541445
$190$	0	0
$191$	−4.42875	−0.320453	−0.160227	−	0.987080i	$-0.551223\pi$
$191$	−4.42875	−0.320453	−0.160227	+	0.987080i	$0.551223\pi$
$192$	0	0
$193$	15.7061	1.13055	0.565276	−	0.824902i	$-0.308770\pi$
$193$	15.7061	1.13055	0.565276	+	0.824902i	$0.308770\pi$
$194$	0	0
$195$	−4.74285	−0.339642
$196$	0	0
$197$	−1.59777	−0.113837	−0.0569183	−	0.998379i	$-0.518127\pi$
$197$	−1.59777	−0.113837	−0.0569183	+	0.998379i	$0.518127\pi$
$198$	0	0
$199$	24.8034	1.75827	0.879135	−	0.476573i	$-0.158121\pi$
$199$	24.8034	1.75827	0.879135	+	0.476573i	$0.158121\pi$
$200$	0	0
$201$	1.46154	0.103089
$202$	0	0
$203$	1.53900	0.108017
$204$	0	0
$205$	18.1108	1.26492
$206$	0	0
$207$	1.21355	0.0843479
$208$	0	0
$209$	0.688260	0.0476079
$210$	0	0
$211$	−6.76998	−0.466065	−0.233032	−	0.972469i	$-0.574865\pi$
$211$	−6.76998	−0.466065	−0.233032	+	0.972469i	$0.574865\pi$
$212$	0	0
$213$	−0.548665	−0.0375939
$214$	0	0
$215$	−20.3566	−1.38831
$216$	0	0
$217$	−2.49486	−0.169362
$218$	0	0
$219$	−5.81898	−0.393210
$220$	0	0
$221$	11.1987	0.753309
$222$	0	0
$223$	−16.1145	−1.07911	−0.539553	−	0.841952i	$-0.681407\pi$
$223$	−16.1145	−1.07911	−0.539553	+	0.841952i	$0.681407\pi$
$224$	0	0
$225$	1.08534	0.0723560
$226$	0	0
$227$	26.9090	1.78601	0.893005	−	0.450046i	$-0.148592\pi$
$227$	26.9090	1.78601	0.893005	+	0.450046i	$0.148592\pi$
$228$	0	0
$229$	−3.44540	−0.227679	−0.113839	−	0.993499i	$-0.536315\pi$
$229$	−3.44540	−0.227679	−0.113839	+	0.993499i	$0.536315\pi$
$230$	0	0
$231$	2.34394	0.154220
$232$	0	0
$233$	0.0882963	0.00578448	0.00289224	−	0.999996i	$-0.499079\pi$
$233$	0.0882963	0.00578448	0.00289224	+	0.999996i	$0.499079\pi$
$234$	0	0
$235$	−12.4157	−0.809913
$236$	0	0
$237$	−4.72725	−0.307068
$238$	0	0
$239$	−10.9095	−0.705679	−0.352840	−	0.935684i	$-0.614784\pi$
$239$	−10.9095	−0.705679	−0.352840	+	0.935684i	$0.614784\pi$
$240$	0	0
$241$	−1.18749	−0.0764931	−0.0382465	−	0.999268i	$-0.512177\pi$
$241$	−1.18749	−0.0764931	−0.0382465	+	0.999268i	$0.512177\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−15.9011	−1.01588
$246$	0	0
$247$	0.420230	0.0267386
$248$	0	0
$249$	14.4680	0.916875
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−3.82139	−0.240249
$254$	0	0
$255$	−14.3686	−0.899797
$256$	0	0
$257$	30.1417	1.88018	0.940092	−	0.340920i	$-0.110738\pi$
$257$	30.1417	1.88018	0.940092	+	0.340920i	$0.110738\pi$
$258$	0	0
$259$	4.12126	0.256082
$260$	0	0
$261$	−2.06754	−0.127978
$262$	0	0
$263$	9.53458	0.587928	0.293964	−	0.955817i	$-0.405025\pi$
$263$	9.53458	0.587928	0.293964	+	0.955817i	$0.405025\pi$
$264$	0	0
$265$	−13.7249	−0.843117
$266$	0	0
$267$	−10.4068	−0.636889
$268$	0	0
$269$	−28.6543	−1.74708	−0.873542	−	0.486749i	$-0.838183\pi$
$269$	−28.6543	−1.74708	−0.873542	+	0.486749i	$0.838183\pi$
$270$	0	0
$271$	−7.48943	−0.454950	−0.227475	−	0.973784i	$-0.573047\pi$
$271$	−7.48943	−0.454950	−0.227475	+	0.973784i	$0.573047\pi$
$272$	0	0
$273$	1.43114	0.0866164
$274$	0	0
$275$	−3.41765	−0.206092
$276$	0	0
$277$	−8.69166	−0.522231	−0.261116	−	0.965308i	$-0.584090\pi$
$277$	−8.69166	−0.522231	−0.261116	+	0.965308i	$0.584090\pi$
$278$	0	0
$279$	3.35167	0.200659
$280$	0	0
$281$	3.86372	0.230490	0.115245	−	0.993337i	$-0.463235\pi$
$281$	3.86372	0.230490	0.115245	+	0.993337i	$0.463235\pi$
$282$	0	0
$283$	0.428927	0.0254971	0.0127485	−	0.999919i	$-0.495942\pi$
$283$	0.428927	0.0254971	0.0127485	+	0.999919i	$0.495942\pi$
$284$	0	0
$285$	−0.539179	−0.0319382
$286$	0	0
$287$	−5.46489	−0.322582
$288$	0	0
$289$	16.9269	0.995702
$290$	0	0
$291$	−13.9082	−0.815315
$292$	0	0
$293$	21.4208	1.25142	0.625708	−	0.780058i	$-0.284810\pi$
$293$	21.4208	1.25142	0.625708	+	0.780058i	$0.284810\pi$
$294$	0	0
$295$	12.7066	0.739806
$296$	0	0
$297$	−3.14892	−0.182719
$298$	0	0
$299$	−2.33322	−0.134934
$300$	0	0
$301$	6.14252	0.354049
$302$	0	0
$303$	−17.5177	−1.00636
$304$	0	0
$305$	23.5299	1.34732
$306$	0	0
$307$	28.4410	1.62321	0.811607	−	0.584203i	$-0.198593\pi$
$307$	28.4410	1.62321	0.811607	+	0.584203i	$0.198593\pi$
$308$	0	0
$309$	5.27753	0.300228
$310$	0	0
$311$	9.14902	0.518793	0.259397	−	0.965771i	$-0.416476\pi$
$311$	9.14902	0.518793	0.259397	+	0.965771i	$0.416476\pi$
$312$	0	0
$313$	−1.08746	−0.0614670	−0.0307335	−	0.999528i	$-0.509784\pi$
$313$	−1.08746	−0.0614670	−0.0307335	+	0.999528i	$0.509784\pi$
$314$	0	0
$315$	−1.83623	−0.103460
$316$	0	0
$317$	−29.8287	−1.67535	−0.837675	−	0.546170i	$-0.816085\pi$
$317$	−29.8287	−1.67535	−0.837675	+	0.546170i	$0.816085\pi$
$318$	0	0
$319$	6.51053	0.364519
$320$	0	0
$321$	−10.4150	−0.581311
$322$	0	0
$323$	1.27310	0.0708372
$324$	0	0
$325$	−2.08671	−0.115750
$326$	0	0
$327$	−6.57930	−0.363836
$328$	0	0
$329$	3.74641	0.206546
$330$	0	0
$331$	1.00309	0.0551349	0.0275675	−	0.999620i	$-0.491224\pi$
$331$	1.00309	0.0551349	0.0275675	+	0.999620i	$0.491224\pi$
$332$	0	0
$333$	−5.53662	−0.303405
$334$	0	0
$335$	3.60539	0.196983
$336$	0	0
$337$	−0.267471	−0.0145701	−0.00728504	−	0.999973i	$-0.502319\pi$
$337$	−0.267471	−0.0145701	−0.00728504	+	0.999973i	$0.502319\pi$
$338$	0	0
$339$	0.866743	0.0470750
$340$	0	0
$341$	−10.5542	−0.571540
$342$	0	0
$343$	10.0087	0.540416
$344$	0	0
$345$	2.99365	0.161173
$346$	0	0
$347$	−6.97713	−0.374552	−0.187276	−	0.982307i	$-0.559966\pi$
$347$	−6.97713	−0.374552	−0.187276	+	0.982307i	$0.559966\pi$
$348$	0	0
$349$	1.24983	0.0669019	0.0334509	−	0.999440i	$-0.489350\pi$
$349$	1.24983	0.0669019	0.0334509	+	0.999440i	$0.489350\pi$
$350$	0	0
$351$	−1.92263	−0.102623
$352$	0	0
$353$	0.304431	0.0162032	0.00810161	−	0.999967i	$-0.497421\pi$
$353$	0.304431	0.0162032	0.00810161	+	0.999967i	$0.497421\pi$
$354$	0	0
$355$	−1.35347	−0.0718349
$356$	0	0
$357$	4.33568	0.229469
$358$	0	0
$359$	2.22045	0.117191	0.0585954	−	0.998282i	$-0.481338\pi$
$359$	2.22045	0.117191	0.0585954	+	0.998282i	$0.481338\pi$
$360$	0	0
$361$	−18.9522	−0.997486
$362$	0	0
$363$	−1.08428	−0.0569101
$364$	0	0
$365$	−14.3545	−0.751350
$366$	0	0
$367$	−0.464030	−0.0242222	−0.0121111	−	0.999927i	$-0.503855\pi$
$367$	−0.464030	−0.0242222	−0.0121111	+	0.999927i	$0.503855\pi$
$368$	0	0
$369$	7.34169	0.382193
$370$	0	0
$371$	4.14146	0.215014
$372$	0	0
$373$	1.66576	0.0862499	0.0431250	−	0.999070i	$-0.486269\pi$
$373$	1.66576	0.0862499	0.0431250	+	0.999070i	$0.486269\pi$
$374$	0	0
$375$	−9.65687	−0.498679
$376$	0	0
$377$	3.97512	0.204729
$378$	0	0
$379$	−32.3452	−1.66146	−0.830731	−	0.556675i	$-0.812077\pi$
$379$	−32.3452	−1.66146	−0.830731	+	0.556675i	$0.812077\pi$
$380$	0	0
$381$	11.0853	0.567916
$382$	0	0
$383$	16.6306	0.849783	0.424891	−	0.905244i	$-0.360312\pi$
$383$	16.6306	0.849783	0.424891	+	0.905244i	$0.360312\pi$
$384$	0	0
$385$	5.78215	0.294686
$386$	0	0
$387$	−8.25205	−0.419475
$388$	0	0
$389$	8.51110	0.431530	0.215765	−	0.976445i	$-0.430776\pi$
$389$	8.51110	0.431530	0.215765	+	0.976445i	$0.430776\pi$
$390$	0	0
$391$	−7.06857	−0.357473
$392$	0	0
$393$	−12.1169	−0.611216
$394$	0	0
$395$	−11.6614	−0.586749
$396$	0	0
$397$	−1.45427	−0.0729875	−0.0364938	−	0.999334i	$-0.511619\pi$
$397$	−1.45427	−0.0729875	−0.0364938	+	0.999334i	$0.511619\pi$
$398$	0	0
$399$	0.162695	0.00814496
$400$	0	0
$401$	15.9972	0.798863	0.399432	−	0.916763i	$-0.369207\pi$
$401$	15.9972	0.798863	0.399432	+	0.916763i	$0.369207\pi$
$402$	0	0
$403$	−6.44404	−0.321001
$404$	0	0
$405$	2.46685	0.122579
$406$	0	0
$407$	17.4344	0.864190
$408$	0	0
$409$	0.877963	0.0434125	0.0217062	−	0.999764i	$-0.493090\pi$
$409$	0.877963	0.0434125	0.0217062	+	0.999764i	$0.493090\pi$
$410$	0	0
$411$	−21.2051	−1.04597
$412$	0	0
$413$	−3.83417	−0.188667
$414$	0	0
$415$	35.6905	1.75198
$416$	0	0
$417$	1.75393	0.0858901
$418$	0	0
$419$	8.75196	0.427561	0.213781	−	0.976882i	$-0.431422\pi$
$419$	8.75196	0.427561	0.213781	+	0.976882i	$0.431422\pi$
$420$	0	0
$421$	−18.6430	−0.908604	−0.454302	−	0.890848i	$-0.650111\pi$
$421$	−18.6430	−0.908604	−0.454302	+	0.890848i	$0.650111\pi$
$422$	0	0
$423$	−5.03304	−0.244715
$424$	0	0
$425$	−6.32176	−0.306651
$426$	0	0
$427$	−7.10006	−0.343596
$428$	0	0
$429$	6.05423	0.292301
$430$	0	0
$431$	29.6920	1.43022	0.715108	−	0.699014i	$-0.246378\pi$
$431$	29.6920	1.43022	0.715108	+	0.699014i	$0.246378\pi$
$432$	0	0
$433$	−35.7968	−1.72028	−0.860141	−	0.510056i	$-0.829625\pi$
$433$	−35.7968	−1.72028	−0.860141	+	0.510056i	$0.829625\pi$
$434$	0	0
$435$	−5.10031	−0.244541
$436$	0	0
$437$	−0.265246	−0.0126885
$438$	0	0
$439$	−16.3069	−0.778286	−0.389143	−	0.921177i	$-0.627229\pi$
$439$	−16.3069	−0.778286	−0.389143	+	0.921177i	$0.627229\pi$
$440$	0	0
$441$	−6.44592	−0.306949
$442$	0	0
$443$	29.7342	1.41271	0.706356	−	0.707856i	$-0.250338\pi$
$443$	29.7342	1.41271	0.706356	+	0.707856i	$0.250338\pi$
$444$	0	0
$445$	−25.6721	−1.21697
$446$	0	0
$447$	−2.53390	−0.119850
$448$	0	0
$449$	40.0171	1.88852	0.944262	−	0.329195i	$-0.106777\pi$
$449$	40.0171	1.88852	0.944262	+	0.329195i	$0.106777\pi$
$450$	0	0
$451$	−23.1184	−1.08860
$452$	0	0
$453$	−15.5191	−0.729150
$454$	0	0
$455$	3.53040	0.165508
$456$	0	0
$457$	8.49065	0.397176	0.198588	−	0.980083i	$-0.436364\pi$
$457$	8.49065	0.397176	0.198588	+	0.980083i	$0.436364\pi$
$458$	0	0
$459$	−5.82468	−0.271873
$460$	0	0
$461$	15.3590	0.715340	0.357670	−	0.933848i	$-0.383571\pi$
$461$	15.3590	0.715340	0.357670	+	0.933848i	$0.383571\pi$
$462$	0	0
$463$	−30.2435	−1.40554	−0.702768	−	0.711419i	$-0.748053\pi$
$463$	−30.2435	−1.40554	−0.702768	+	0.711419i	$0.748053\pi$
$464$	0	0
$465$	8.26807	0.383422
$466$	0	0
$467$	4.82158	0.223116	0.111558	−	0.993758i	$-0.464416\pi$
$467$	4.82158	0.223116	0.111558	+	0.993758i	$0.464416\pi$
$468$	0	0
$469$	−1.08791	−0.0502352
$470$	0	0
$471$	13.8188	0.636737
$472$	0	0
$473$	25.9851	1.19480
$474$	0	0
$475$	−0.237223	−0.0108845
$476$	0	0
$477$	−5.56376	−0.254747
$478$	0	0
$479$	−13.0913	−0.598156	−0.299078	−	0.954229i	$-0.596679\pi$
$479$	−13.0913	−0.598156	−0.299078	+	0.954229i	$0.596679\pi$
$480$	0	0
$481$	10.6449	0.485365
$482$	0	0
$483$	−0.903325	−0.0411027
$484$	0	0
$485$	−34.3095	−1.55791
$486$	0	0
$487$	0.431635	0.0195593	0.00977963	−	0.999952i	$-0.496887\pi$
$487$	0.431635	0.0195593	0.00977963	+	0.999952i	$0.496887\pi$
$488$	0	0
$489$	0.0481229	0.00217619
$490$	0	0
$491$	39.2743	1.77243	0.886213	−	0.463277i	$-0.153327\pi$
$491$	39.2743	1.77243	0.886213	+	0.463277i	$0.153327\pi$
$492$	0	0
$493$	12.0428	0.542379
$494$	0	0
$495$	−7.76792	−0.349142
$496$	0	0
$497$	0.408406	0.0183195
$498$	0	0
$499$	−24.4809	−1.09592	−0.547958	−	0.836506i	$-0.684595\pi$
$499$	−24.4809	−1.09592	−0.547958	+	0.836506i	$0.684595\pi$
$500$	0	0
$501$	21.7397	0.971259
$502$	0	0
$503$	−37.9429	−1.69179	−0.845894	−	0.533350i	$-0.820933\pi$
$503$	−37.9429	−1.69179	−0.845894	+	0.533350i	$0.820933\pi$
$504$	0	0
$505$	−43.2134	−1.92297
$506$	0	0
$507$	−9.30348	−0.413182
$508$	0	0
$509$	−21.2545	−0.942089	−0.471045	−	0.882109i	$-0.656123\pi$
$509$	−21.2545	−0.942089	−0.471045	+	0.882109i	$0.656123\pi$
$510$	0	0
$511$	4.33143	0.191611
$512$	0	0
$513$	−0.218570	−0.00965010
$514$	0	0
$515$	13.0189	0.573680
$516$	0	0
$517$	15.8486	0.697022
$518$	0	0
$519$	25.9429	1.13877
$520$	0	0
$521$	15.4325	0.676109	0.338054	−	0.941127i	$-0.390231\pi$
$521$	15.4325	0.676109	0.338054	+	0.941127i	$0.390231\pi$
$522$	0	0
$523$	6.40098	0.279895	0.139948	−	0.990159i	$-0.455307\pi$
$523$	6.40098	0.279895	0.139948	+	0.990159i	$0.455307\pi$
$524$	0	0
$525$	−0.807887	−0.0352591
$526$	0	0
$527$	−19.5224	−0.850410
$528$	0	0
$529$	−21.5273	−0.935969
$530$	0	0
$531$	5.15094	0.223532
$532$	0	0
$533$	−14.1154	−0.611406
$534$	0	0
$535$	−25.6923	−1.11078
$536$	0	0
$537$	−17.2815	−0.745751
$538$	0	0
$539$	20.2977	0.874285
$540$	0	0
$541$	18.4190	0.791894	0.395947	−	0.918273i	$-0.370416\pi$
$541$	18.4190	0.791894	0.395947	+	0.918273i	$0.370416\pi$
$542$	0	0
$543$	3.17744	0.136357
$544$	0	0
$545$	−16.2301	−0.695222
$546$	0	0
$547$	11.8780	0.507866	0.253933	−	0.967222i	$-0.418276\pi$
$547$	11.8780	0.507866	0.253933	+	0.967222i	$0.418276\pi$
$548$	0	0
$549$	9.53844	0.407091
$550$	0	0
$551$	0.451902	0.0192517
$552$	0	0
$553$	3.51879	0.149634
$554$	0	0
$555$	−13.6580	−0.579750
$556$	0	0
$557$	−28.9306	−1.22583	−0.612915	−	0.790149i	$-0.710003\pi$
$557$	−28.9306	−1.22583	−0.612915	+	0.790149i	$0.710003\pi$
$558$	0	0
$559$	15.8657	0.671047
$560$	0	0
$561$	18.3415	0.774378
$562$	0	0
$563$	21.9873	0.926654	0.463327	−	0.886187i	$-0.346656\pi$
$563$	21.9873	0.926654	0.463327	+	0.886187i	$0.346656\pi$
$564$	0	0
$565$	2.13812	0.0899515
$566$	0	0
$567$	−0.744363	−0.0312603
$568$	0	0
$569$	−40.8571	−1.71282	−0.856409	−	0.516298i	$-0.827310\pi$
$569$	−40.8571	−1.71282	−0.856409	+	0.516298i	$0.827310\pi$
$570$	0	0
$571$	9.81972	0.410942	0.205471	−	0.978663i	$-0.434127\pi$
$571$	9.81972	0.410942	0.205471	+	0.978663i	$0.434127\pi$
$572$	0	0
$573$	−4.42875	−0.185014
$574$	0	0
$575$	1.31712	0.0549277
$576$	0	0
$577$	−2.25981	−0.0940770	−0.0470385	−	0.998893i	$-0.514978\pi$
$577$	−2.25981	−0.0940770	−0.0470385	+	0.998893i	$0.514978\pi$
$578$	0	0
$579$	15.7061	0.652724
$580$	0	0
$581$	−10.7695	−0.446793
$582$	0	0
$583$	17.5198	0.725598
$584$	0	0
$585$	−4.74285	−0.196093
$586$	0	0
$587$	22.3307	0.921685	0.460843	−	0.887482i	$-0.347547\pi$
$587$	22.3307	0.921685	0.460843	+	0.887482i	$0.347547\pi$
$588$	0	0
$589$	−0.732575	−0.0301852
$590$	0	0
$591$	−1.59777	−0.0657236
$592$	0	0
$593$	16.8730	0.692891	0.346445	−	0.938070i	$-0.387389\pi$
$593$	16.8730	0.692891	0.346445	+	0.938070i	$0.387389\pi$
$594$	0	0
$595$	10.6955	0.438471
$596$	0	0
$597$	24.8034	1.01514
$598$	0	0
$599$	−30.7846	−1.25782	−0.628912	−	0.777477i	$-0.716499\pi$
$599$	−30.7846	−1.25782	−0.628912	+	0.777477i	$0.716499\pi$
$600$	0	0
$601$	23.1376	0.943802	0.471901	−	0.881652i	$-0.343568\pi$
$601$	23.1376	0.943802	0.471901	+	0.881652i	$0.343568\pi$
$602$	0	0
$603$	1.46154	0.0595183
$604$	0	0
$605$	−2.67476	−0.108745
$606$	0	0
$607$	−0.204666	−0.00830713	−0.00415357	−	0.999991i	$-0.501322\pi$
$607$	−0.204666	−0.00830713	−0.00415357	+	0.999991i	$0.501322\pi$
$608$	0	0
$609$	1.53900	0.0623635
$610$	0	0
$611$	9.67669	0.391477
$612$	0	0
$613$	−3.75629	−0.151715	−0.0758575	−	0.997119i	$-0.524169\pi$
$613$	−3.75629	−0.151715	−0.0758575	+	0.997119i	$0.524169\pi$
$614$	0	0
$615$	18.1108	0.730300
$616$	0	0
$617$	35.9929	1.44902	0.724509	−	0.689265i	$-0.242066\pi$
$617$	35.9929	1.44902	0.724509	+	0.689265i	$0.242066\pi$
$618$	0	0
$619$	2.50131	0.100536	0.0502681	−	0.998736i	$-0.483992\pi$
$619$	2.50131	0.100536	0.0502681	+	0.998736i	$0.483992\pi$
$620$	0	0
$621$	1.21355	0.0486983
$622$	0	0
$623$	7.74647	0.310356
$624$	0	0
$625$	−29.2487	−1.16995
$626$	0	0
$627$	0.688260	0.0274864
$628$	0	0
$629$	32.2490	1.28585
$630$	0	0
$631$	−14.9066	−0.593422	−0.296711	−	0.954967i	$-0.595890\pi$
$631$	−14.9066	−0.593422	−0.296711	+	0.954967i	$0.595890\pi$
$632$	0	0
$633$	−6.76998	−0.269082
$634$	0	0
$635$	27.3457	1.08518
$636$	0	0
$637$	12.3932	0.491035
$638$	0	0
$639$	−0.548665	−0.0217049
$640$	0	0
$641$	9.30722	0.367613	0.183807	−	0.982962i	$-0.441158\pi$
$641$	9.30722	0.367613	0.183807	+	0.982962i	$0.441158\pi$
$642$	0	0
$643$	−29.0540	−1.14578	−0.572890	−	0.819632i	$-0.694178\pi$
$643$	−29.0540	−1.14578	−0.572890	+	0.819632i	$0.694178\pi$
$644$	0	0
$645$	−20.3566	−0.801539
$646$	0	0
$647$	−17.5754	−0.690960	−0.345480	−	0.938426i	$-0.612284\pi$
$647$	−17.5754	−0.690960	−0.345480	+	0.938426i	$0.612284\pi$
$648$	0	0
$649$	−16.2199	−0.636687
$650$	0	0
$651$	−2.49486	−0.0977814
$652$	0	0
$653$	−40.0179	−1.56602	−0.783011	−	0.622007i	$-0.786317\pi$
$653$	−40.0179	−1.56602	−0.783011	+	0.622007i	$0.786317\pi$
$654$	0	0
$655$	−29.8905	−1.16792
$656$	0	0
$657$	−5.81898	−0.227020
$658$	0	0
$659$	−31.4851	−1.22649	−0.613243	−	0.789894i	$-0.710135\pi$
$659$	−31.4851	−1.22649	−0.613243	+	0.789894i	$0.710135\pi$
$660$	0	0
$661$	24.1369	0.938817	0.469409	−	0.882981i	$-0.344467\pi$
$661$	24.1369	0.938817	0.469409	+	0.882981i	$0.344467\pi$
$662$	0	0
$663$	11.1987	0.434923
$664$	0	0
$665$	0.401345	0.0155635
$666$	0	0
$667$	−2.50907	−0.0971517
$668$	0	0
$669$	−16.1145	−0.623022
$670$	0	0
$671$	−30.0358	−1.15952
$672$	0	0
$673$	−17.4344	−0.672047	−0.336023	−	0.941854i	$-0.609082\pi$
$673$	−17.4344	−0.672047	−0.336023	+	0.941854i	$0.609082\pi$
$674$	0	0
$675$	1.08534	0.0417748
$676$	0	0
$677$	5.10467	0.196188	0.0980941	−	0.995177i	$-0.468725\pi$
$677$	5.10467	0.196188	0.0980941	+	0.995177i	$0.468725\pi$
$678$	0	0
$679$	10.3528	0.397303
$680$	0	0
$681$	26.9090	1.03115
$682$	0	0
$683$	−8.45433	−0.323496	−0.161748	−	0.986832i	$-0.551713\pi$
$683$	−8.45433	−0.323496	−0.161748	+	0.986832i	$0.551713\pi$
$684$	0	0
$685$	−52.3098	−1.99866
$686$	0	0
$687$	−3.44540	−0.131450
$688$	0	0
$689$	10.6971	0.407526
$690$	0	0
$691$	16.6033	0.631618	0.315809	−	0.948823i	$-0.397724\pi$
$691$	16.6033	0.631618	0.315809	+	0.948823i	$0.397724\pi$
$692$	0	0
$693$	2.34394	0.0890390
$694$	0	0
$695$	4.32667	0.164120
$696$	0	0
$697$	−42.7630	−1.61977
$698$	0	0
$699$	0.0882963	0.00333967
$700$	0	0
$701$	24.8150	0.937250	0.468625	−	0.883397i	$-0.344750\pi$
$701$	24.8150	0.937250	0.468625	+	0.883397i	$0.344750\pi$
$702$	0	0
$703$	1.21014	0.0456412
$704$	0	0
$705$	−12.4157	−0.467604
$706$	0	0
$707$	13.0395	0.490401
$708$	0	0
$709$	−41.2554	−1.54938	−0.774689	−	0.632342i	$-0.782094\pi$
$709$	−41.2554	−1.54938	−0.774689	+	0.632342i	$0.782094\pi$
$710$	0	0
$711$	−4.72725	−0.177286
$712$	0	0
$713$	4.06744	0.152327
$714$	0	0
$715$	14.9349	0.558532
$716$	0	0
$717$	−10.9095	−0.407424
$718$	0	0
$719$	−1.29971	−0.0484712	−0.0242356	−	0.999706i	$-0.507715\pi$
$719$	−1.29971	−0.0484712	−0.0242356	+	0.999706i	$0.507715\pi$
$720$	0	0
$721$	−3.92840	−0.146301
$722$	0	0
$723$	−1.18749	−0.0441633
$724$	0	0
$725$	−2.24398	−0.0833395
$726$	0	0
$727$	23.6252	0.876212	0.438106	−	0.898923i	$-0.355650\pi$
$727$	23.6252	0.876212	0.438106	+	0.898923i	$0.355650\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	48.0656	1.77777
$732$	0	0
$733$	−14.8868	−0.549858	−0.274929	−	0.961465i	$-0.588654\pi$
$733$	−14.8868	−0.549858	−0.274929	+	0.961465i	$0.588654\pi$
$734$	0	0
$735$	−15.9011	−0.586521
$736$	0	0
$737$	−4.60226	−0.169527
$738$	0	0
$739$	6.22977	0.229166	0.114583	−	0.993414i	$-0.463447\pi$
$739$	6.22977	0.229166	0.114583	+	0.993414i	$0.463447\pi$
$740$	0	0
$741$	0.420230	0.0154375
$742$	0	0
$743$	10.3148	0.378412	0.189206	−	0.981937i	$-0.439409\pi$
$743$	10.3148	0.378412	0.189206	+	0.981937i	$0.439409\pi$
$744$	0	0
$745$	−6.25076	−0.229010
$746$	0	0
$747$	14.4680	0.529358
$748$	0	0
$749$	7.75258	0.283273
$750$	0	0
$751$	6.46696	0.235983	0.117991	−	0.993015i	$-0.462354\pi$
$751$	6.46696	0.235983	0.117991	+	0.993015i	$0.462354\pi$
$752$	0	0
$753$	1.00000	0.0364420
$754$	0	0
$755$	−38.2832	−1.39327
$756$	0	0
$757$	−34.1942	−1.24281	−0.621404	−	0.783490i	$-0.713437\pi$
$757$	−34.1942	−1.24281	−0.621404	+	0.783490i	$0.713437\pi$
$758$	0	0
$759$	−3.82139	−0.138708
$760$	0	0
$761$	−0.0523118	−0.00189630	−0.000948150	−	1.00000i	$-0.500302\pi$
$761$	−0.0523118	−0.00189630	−0.000948150		1.00000i	$0.500302\pi$
$762$	0	0
$763$	4.89739	0.177297
$764$	0	0
$765$	−14.3686	−0.519498
$766$	0	0
$767$	−9.90337	−0.357590
$768$	0	0
$769$	−32.9133	−1.18688	−0.593442	−	0.804877i	$-0.702231\pi$
$769$	−32.9133	−1.18688	−0.593442	+	0.804877i	$0.702231\pi$
$770$	0	0
$771$	30.1417	1.08553
$772$	0	0
$773$	−11.8467	−0.426095	−0.213048	−	0.977042i	$-0.568339\pi$
$773$	−11.8467	−0.426095	−0.213048	+	0.977042i	$0.568339\pi$
$774$	0	0
$775$	3.63770	0.130670
$776$	0	0
$777$	4.12126	0.147849
$778$	0	0
$779$	−1.60467	−0.0574934
$780$	0	0
$781$	1.72770	0.0618221
$782$	0	0
$783$	−2.06754	−0.0738879
$784$	0	0
$785$	34.0889	1.21669
$786$	0	0
$787$	0.923181	0.0329079	0.0164539	−	0.999865i	$-0.494762\pi$
$787$	0.923181	0.0329079	0.0164539	+	0.999865i	$0.494762\pi$
$788$	0	0
$789$	9.53458	0.339440
$790$	0	0
$791$	−0.645172	−0.0229397
$792$	0	0
$793$	−18.3389	−0.651234
$794$	0	0
$795$	−13.7249	−0.486774
$796$	0	0
$797$	23.6049	0.836129	0.418064	−	0.908417i	$-0.362709\pi$
$797$	23.6049	0.836129	0.418064	+	0.908417i	$0.362709\pi$
$798$	0	0
$799$	29.3158	1.03712
$800$	0	0
$801$	−10.4068	−0.367708
$802$	0	0
$803$	18.3235	0.646623
$804$	0	0
$805$	−2.22837	−0.0785396
$806$	0	0
$807$	−28.6543	−1.00868
$808$	0	0
$809$	−43.8750	−1.54256	−0.771282	−	0.636494i	$-0.780384\pi$
$809$	−43.8750	−1.54256	−0.771282	+	0.636494i	$0.780384\pi$
$810$	0	0
$811$	40.4123	1.41907	0.709534	−	0.704672i	$-0.248906\pi$
$811$	40.4123	1.41907	0.709534	+	0.704672i	$0.248906\pi$
$812$	0	0
$813$	−7.48943	−0.262666
$814$	0	0
$815$	0.118712	0.00415829
$816$	0	0
$817$	1.80365	0.0631017
$818$	0	0
$819$	1.43114	0.0500080
$820$	0	0
$821$	30.7948	1.07475	0.537373	−	0.843345i	$-0.319417\pi$
$821$	30.7948	1.07475	0.537373	+	0.843345i	$0.319417\pi$
$822$	0	0
$823$	35.0780	1.22274	0.611371	−	0.791344i	$-0.290618\pi$
$823$	35.0780	1.22274	0.611371	+	0.791344i	$0.290618\pi$
$824$	0	0
$825$	−3.41765	−0.118987
$826$	0	0
$827$	41.9629	1.45919	0.729596	−	0.683878i	$-0.239708\pi$
$827$	41.9629	1.45919	0.729596	+	0.683878i	$0.239708\pi$
$828$	0	0
$829$	−43.0436	−1.49497	−0.747484	−	0.664280i	$-0.768738\pi$
$829$	−43.0436	−1.49497	−0.747484	+	0.664280i	$0.768738\pi$
$830$	0	0
$831$	−8.69166	−0.301510
$832$	0	0
$833$	37.5455	1.30087
$834$	0	0
$835$	53.6286	1.85589
$836$	0	0
$837$	3.35167	0.115851
$838$	0	0
$839$	30.3346	1.04727	0.523633	−	0.851944i	$-0.324576\pi$
$839$	30.3346	1.04727	0.523633	+	0.851944i	$0.324576\pi$
$840$	0	0
$841$	−24.7253	−0.852596
$842$	0	0
$843$	3.86372	0.133074
$844$	0	0
$845$	−22.9503	−0.789513
$846$	0	0
$847$	0.807101	0.0277323
$848$	0	0
$849$	0.428927	0.0147207
$850$	0	0
$851$	−6.71899	−0.230324
$852$	0	0
$853$	−21.9018	−0.749902	−0.374951	−	0.927045i	$-0.622341\pi$
$853$	−21.9018	−0.749902	−0.374951	+	0.927045i	$0.622341\pi$
$854$	0	0
$855$	−0.539179	−0.0184395
$856$	0	0
$857$	15.5345	0.530649	0.265325	−	0.964159i	$-0.414521\pi$
$857$	15.5345	0.530649	0.265325	+	0.964159i	$0.414521\pi$
$858$	0	0
$859$	−53.5975	−1.82872	−0.914362	−	0.404898i	$-0.867307\pi$
$859$	−53.5975	−1.82872	−0.914362	+	0.404898i	$0.867307\pi$
$860$	0	0
$861$	−5.46489	−0.186243
$862$	0	0
$863$	29.0009	0.987203	0.493601	−	0.869688i	$-0.335680\pi$
$863$	29.0009	0.987203	0.493601	+	0.869688i	$0.335680\pi$
$864$	0	0
$865$	63.9973	2.17597
$866$	0	0
$867$	16.9269	0.574869
$868$	0	0
$869$	14.8857	0.504964
$870$	0	0
$871$	−2.81000	−0.0952132
$872$	0	0
$873$	−13.9082	−0.470723
$874$	0	0
$875$	7.18822	0.243006
$876$	0	0
$877$	−4.85804	−0.164044	−0.0820221	−	0.996631i	$-0.526138\pi$
$877$	−4.85804	−0.164044	−0.0820221	+	0.996631i	$0.526138\pi$
$878$	0	0
$879$	21.4208	0.722505
$880$	0	0
$881$	7.58545	0.255560	0.127780	−	0.991803i	$-0.459215\pi$
$881$	7.58545	0.255560	0.127780	+	0.991803i	$0.459215\pi$
$882$	0	0
$883$	45.5707	1.53358	0.766788	−	0.641900i	$-0.221854\pi$
$883$	45.5707	1.53358	0.766788	+	0.641900i	$0.221854\pi$
$884$	0	0
$885$	12.7066	0.427127
$886$	0	0
$887$	21.2305	0.712849	0.356424	−	0.934324i	$-0.383996\pi$
$887$	21.2305	0.712849	0.356424	+	0.934324i	$0.383996\pi$
$888$	0	0
$889$	−8.25147	−0.276746
$890$	0	0
$891$	−3.14892	−0.105493
$892$	0	0
$893$	1.10007	0.0368124
$894$	0	0
$895$	−42.6308	−1.42499
$896$	0	0
$897$	−2.33322	−0.0779040
$898$	0	0
$899$	−6.92972	−0.231119
$900$	0	0
$901$	32.4071	1.07964
$902$	0	0
$903$	6.14252	0.204410
$904$	0	0
$905$	7.83826	0.260553
$906$	0	0
$907$	−26.6699	−0.885560	−0.442780	−	0.896630i	$-0.646008\pi$
$907$	−26.6699	−0.885560	−0.442780	+	0.896630i	$0.646008\pi$
$908$	0	0
$909$	−17.5177	−0.581024
$910$	0	0
$911$	1.24360	0.0412023	0.0206011	−	0.999788i	$-0.493442\pi$
$911$	1.24360	0.0412023	0.0206011	+	0.999788i	$0.493442\pi$
$912$	0	0
$913$	−45.5587	−1.50777
$914$	0	0
$915$	23.5299	0.777874
$916$	0	0
$917$	9.01936	0.297846
$918$	0	0
$919$	22.7386	0.750078	0.375039	−	0.927009i	$-0.377629\pi$
$919$	22.7386	0.750078	0.375039	+	0.927009i	$0.377629\pi$
$920$	0	0
$921$	28.4410	0.937163
$922$	0	0
$923$	1.05488	0.0347219
$924$	0	0
$925$	−6.00911	−0.197578
$926$	0	0
$927$	5.27753	0.173337
$928$	0	0
$929$	48.7961	1.60095	0.800475	−	0.599367i	$-0.204581\pi$
$929$	48.7961	1.60095	0.800475	+	0.599367i	$0.204581\pi$
$930$	0	0
$931$	1.40888	0.0461743
$932$	0	0
$933$	9.14902	0.299525
$934$	0	0
$935$	45.2456	1.47969
$936$	0	0
$937$	−42.1716	−1.37769	−0.688844	−	0.724910i	$-0.741881\pi$
$937$	−42.1716	−1.37769	−0.688844	+	0.724910i	$0.741881\pi$
$938$	0	0
$939$	−1.08746	−0.0354880
$940$	0	0
$941$	−35.6781	−1.16307	−0.581537	−	0.813520i	$-0.697548\pi$
$941$	−35.6781	−1.16307	−0.581537	+	0.813520i	$0.697548\pi$
$942$	0	0
$943$	8.90955	0.290135
$944$	0	0
$945$	−1.83623	−0.0597326
$946$	0	0
$947$	−55.0606	−1.78923	−0.894614	−	0.446840i	$-0.852549\pi$
$947$	−55.0606	−1.78923	−0.894614	+	0.446840i	$0.852549\pi$
$948$	0	0
$949$	11.1878	0.363170
$950$	0	0
$951$	−29.8287	−0.967263
$952$	0	0
$953$	54.9703	1.78066	0.890332	−	0.455312i	$-0.150472\pi$
$953$	54.9703	1.78066	0.890332	+	0.455312i	$0.150472\pi$
$954$	0	0
$955$	−10.9251	−0.353527
$956$	0	0
$957$	6.51053	0.210455
$958$	0	0
$959$	15.7843	0.509702
$960$	0	0
$961$	−19.7663	−0.637622
$962$	0	0
$963$	−10.4150	−0.335620
$964$	0	0
$965$	38.7446	1.24723
$966$	0	0
$967$	−3.91853	−0.126011	−0.0630057	−	0.998013i	$-0.520069\pi$
$967$	−3.91853	−0.126011	−0.0630057	+	0.998013i	$0.520069\pi$
$968$	0	0
$969$	1.27310	0.0408979
$970$	0	0
$971$	−9.09413	−0.291845	−0.145922	−	0.989296i	$-0.546615\pi$
$971$	−9.09413	−0.291845	−0.145922	+	0.989296i	$0.546615\pi$
$972$	0	0
$973$	−1.30556	−0.0418543
$974$	0	0
$975$	−2.08671	−0.0668283
$976$	0	0
$977$	46.7011	1.49410	0.747050	−	0.664768i	$-0.231470\pi$
$977$	46.7011	1.49410	0.747050	+	0.664768i	$0.231470\pi$
$978$	0	0
$979$	32.7704	1.04735
$980$	0	0
$981$	−6.57930	−0.210061
$982$	0	0
$983$	4.82148	0.153781	0.0768907	−	0.997040i	$-0.475501\pi$
$983$	4.82148	0.153781	0.0768907	+	0.997040i	$0.475501\pi$
$984$	0	0
$985$	−3.94146	−0.125586
$986$	0	0
$987$	3.74641	0.119249
$988$	0	0
$989$	−10.0143	−0.318437
$990$	0	0
$991$	−51.4892	−1.63561	−0.817804	−	0.575497i	$-0.804809\pi$
$991$	−51.4892	−1.63561	−0.817804	+	0.575497i	$0.804809\pi$
$992$	0	0
$993$	1.00309	0.0318322
$994$	0	0
$995$	61.1863	1.93974
$996$	0	0
$997$	−36.9815	−1.17122	−0.585608	−	0.810594i	$-0.699144\pi$
$997$	−36.9815	−1.17122	−0.585608	+	0.810594i	$0.699144\pi$
$998$	0	0
$999$	−5.53662	−0.175171

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6024.2.a.k.1.8	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6024.2.a.k.1.8	✓	8	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:	\( N \)	\(=\)	\( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6024.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +2.46685 q^{5} -0.744363 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +2.46685 q^{5} -0.744363 q^{7} +1.00000 q^{9} -3.14892 q^{11} -1.92263 q^{13} +2.46685 q^{15} -5.82468 q^{17} -0.218570 q^{19} -0.744363 q^{21} +1.21355 q^{23} +1.08534 q^{25} +1.00000 q^{27} -2.06754 q^{29} +3.35167 q^{31} -3.14892 q^{33} -1.83623 q^{35} -5.53662 q^{37} -1.92263 q^{39} +7.34169 q^{41} -8.25205 q^{43} +2.46685 q^{45} -5.03304 q^{47} -6.44592 q^{49} -5.82468 q^{51} -5.56376 q^{53} -7.76792 q^{55} -0.218570 q^{57} +5.15094 q^{59} +9.53844 q^{61} -0.744363 q^{63} -4.74285 q^{65} +1.46154 q^{67} +1.21355 q^{69} -0.548665 q^{71} -5.81898 q^{73} +1.08534 q^{75} +2.34394 q^{77} -4.72725 q^{79} +1.00000 q^{81} +14.4680 q^{83} -14.3686 q^{85} -2.06754 q^{87} -10.4068 q^{89} +1.43114 q^{91} +3.35167 q^{93} -0.539179 q^{95} -13.9082 q^{97} -3.14892 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9	\( 8 q + 8 q^{3} - 5 q^{5} + q^{7} + 8 q^{9} - 6 q^{11} - 4 q^{13} - 5 q^{15} - 12 q^{17} - 10 q^{19} + q^{21} - 6 q^{23} - 7 q^{25} + 8 q^{27} - 2 q^{29} - 3 q^{31} - 6 q^{33} + 4 q^{35} - 16 q^{37} - 4 q^{39} + 2 q^{41} - 2 q^{43} - 5 q^{45} - 15 q^{47} - 21 q^{49} - 12 q^{51} + 13 q^{53} - 33 q^{55} - 10 q^{57} + 8 q^{59} - 38 q^{61} + q^{63} + 16 q^{65} - 31 q^{67} - 6 q^{69} + 5 q^{71} - 26 q^{73} - 7 q^{75} - 24 q^{77} - 25 q^{79} + 8 q^{81} - 7 q^{83} - 18 q^{85} - 2 q^{87} - 14 q^{89} - 6 q^{91} - 3 q^{93} - q^{95} - 51 q^{97} - 6 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 - 5 * q^5 + q^7 + 8 * q^9 - 6 * q^11 - 4 * q^13 - 5 * q^15 - 12 * q^17 - 10 * q^19 + q^21 - 6 * q^23 - 7 * q^25 + 8 * q^27 - 2 * q^29 - 3 * q^31 - 6 * q^33 + 4 * q^35 - 16 * q^37 - 4 * q^39 + 2 * q^41 - 2 * q^43 - 5 * q^45 - 15 * q^47 - 21 * q^49 - 12 * q^51 + 13 * q^53 - 33 * q^55 - 10 * q^57 + 8 * q^59 - 38 * q^61 + q^63 + 16 * q^65 - 31 * q^67 - 6 * q^69 + 5 * q^71 - 26 * q^73 - 7 * q^75 - 24 * q^77 - 25 * q^79 + 8 * q^81 - 7 * q^83 - 18 * q^85 - 2 * q^87 - 14 * q^89 - 6 * q^91 - 3 * q^93 - q^95 - 51 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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