LMFDB - Embedded newform 2960.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2960,2,Mod(1,2960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2960, 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("2960.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2960 = 2^{4} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2960.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:	$23.6357189983$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	2960.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +1.00000 q^{5} +4.73205 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} +1.00000 q^{5} +4.73205 q^{7} -2.46410 q^{9} +5.46410 q^{11} -5.46410 q^{13} -0.732051 q^{15} +5.46410 q^{17} -6.19615 q^{19} -3.46410 q^{21} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} +4.92820 q^{29} -0.732051 q^{31} -4.00000 q^{33} +4.73205 q^{35} +1.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} -6.92820 q^{43} -2.46410 q^{45} +4.73205 q^{47} +15.3923 q^{49} -4.00000 q^{51} -6.00000 q^{53} +5.46410 q^{55} +4.53590 q^{57} +10.1962 q^{59} -4.92820 q^{61} -11.6603 q^{63} -5.46410 q^{65} +3.66025 q^{67} -5.85641 q^{69} -2.92820 q^{71} -0.928203 q^{73} -0.732051 q^{75} +25.8564 q^{77} -8.73205 q^{79} +4.46410 q^{81} +8.73205 q^{83} +5.46410 q^{85} -3.60770 q^{87} -2.00000 q^{89} -25.8564 q^{91} +0.535898 q^{93} -6.19615 q^{95} -2.00000 q^{97} -13.4641 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{17} - 2 q^{19} + 16 q^{23} + 2 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{31} - 8 q^{33} + 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{47} + 10 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} + 16 q^{57} + 10 q^{59} + 4 q^{61} - 6 q^{63} - 4 q^{65} - 10 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 14 q^{79} + 2 q^{81} + 14 q^{83} + 4 q^{85} - 28 q^{87} - 4 q^{89} - 24 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 20 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 4 * q^11 - 4 * q^13 + 2 * q^15 + 4 * q^17 - 2 * q^19 + 16 * q^23 + 2 * q^25 + 8 * q^27 - 4 * q^29 + 2 * q^31 - 8 * q^33 + 6 * q^35 + 2 * q^37 + 8 * q^39 - 4 * q^41 + 2 * q^45 + 6 * q^47 + 10 * q^49 - 8 * q^51 - 12 * q^53 + 4 * q^55 + 16 * q^57 + 10 * q^59 + 4 * q^61 - 6 * q^63 - 4 * q^65 - 10 * q^67 + 16 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 + 24 * q^77 - 14 * q^79 + 2 * q^81 + 14 * q^83 + 4 * q^85 - 28 * q^87 - 4 * q^89 - 24 * q^91 + 8 * q^93 - 2 * q^95 - 4 * q^97 - 20 * 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	5.46410	1.64749	0.823744	−	0.566961i	$-0.191881\pi$
$11$	5.46410	1.64749	0.823744	+	0.566961i	$0.191881\pi$
$12$	0	0
$13$	−5.46410	−1.51547	−0.757735	−	0.652563i	$-0.773694\pi$
$13$	−5.46410	−1.51547	−0.757735	+	0.652563i	$0.773694\pi$
$14$	0	0
$15$	−0.732051	−0.189015
$16$	0	0
$17$	5.46410	1.32524	0.662620	−	0.748956i	$-0.269445\pi$
$17$	5.46410	1.32524	0.662620	+	0.748956i	$0.269445\pi$
$18$	0	0
$19$	−6.19615	−1.42149	−0.710747	−	0.703447i	$-0.751643\pi$
$19$	−6.19615	−1.42149	−0.710747	+	0.703447i	$0.751643\pi$
$20$	0	0
$21$	−3.46410	−0.755929
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	4.92820	0.915144	0.457572	−	0.889172i	$-0.348719\pi$
$29$	4.92820	0.915144	0.457572	+	0.889172i	$0.348719\pi$
$30$	0	0
$31$	−0.732051	−0.131480	−0.0657401	−	0.997837i	$-0.520941\pi$
$31$	−0.732051	−0.131480	−0.0657401	+	0.997837i	$0.520941\pi$
$32$	0	0
$33$	−4.00000	−0.696311
$34$	0	0
$35$	4.73205	0.799863
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	0	0
$39$	4.00000	0.640513
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−6.92820	−1.05654	−0.528271	−	0.849076i	$-0.677159\pi$
$43$	−6.92820	−1.05654	−0.528271	+	0.849076i	$0.677159\pi$
$44$	0	0
$45$	−2.46410	−0.367327
$46$	0	0
$47$	4.73205	0.690241	0.345120	−	0.938558i	$-0.387838\pi$
$47$	4.73205	0.690241	0.345120	+	0.938558i	$0.387838\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	−4.00000	−0.560112
$52$	0	0
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	5.46410	0.736779
$56$	0	0
$57$	4.53590	0.600794
$58$	0	0
$59$	10.1962	1.32743	0.663713	−	0.747987i	$-0.268980\pi$
$59$	10.1962	1.32743	0.663713	+	0.747987i	$0.268980\pi$
$60$	0	0
$61$	−4.92820	−0.630992	−0.315496	−	0.948927i	$-0.602171\pi$
$61$	−4.92820	−0.630992	−0.315496	+	0.948927i	$0.602171\pi$
$62$	0	0
$63$	−11.6603	−1.46905
$64$	0	0
$65$	−5.46410	−0.677738
$66$	0	0
$67$	3.66025	0.447171	0.223586	−	0.974684i	$-0.428224\pi$
$67$	3.66025	0.447171	0.223586	+	0.974684i	$0.428224\pi$
$68$	0	0
$69$	−5.85641	−0.705028
$70$	0	0
$71$	−2.92820	−0.347514	−0.173757	−	0.984789i	$-0.555591\pi$
$71$	−2.92820	−0.347514	−0.173757	+	0.984789i	$0.555591\pi$
$72$	0	0
$73$	−0.928203	−0.108638	−0.0543190	−	0.998524i	$-0.517299\pi$
$73$	−0.928203	−0.108638	−0.0543190	+	0.998524i	$0.517299\pi$
$74$	0	0
$75$	−0.732051	−0.0845299
$76$	0	0
$77$	25.8564	2.94661
$78$	0	0
$79$	−8.73205	−0.982432	−0.491216	−	0.871038i	$-0.663448\pi$
$79$	−8.73205	−0.982432	−0.491216	+	0.871038i	$0.663448\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	8.73205	0.958467	0.479234	−	0.877687i	$-0.340915\pi$
$83$	8.73205	0.958467	0.479234	+	0.877687i	$0.340915\pi$
$84$	0	0
$85$	5.46410	0.592665
$86$	0	0
$87$	−3.60770	−0.386786
$88$	0	0
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−25.8564	−2.71049
$92$	0	0
$93$	0.535898	0.0555701
$94$	0	0
$95$	−6.19615	−0.635712
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−13.4641	−1.35319
$100$	0	0
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	0	0
$103$	6.53590	0.644001	0.322001	−	0.946739i	$-0.395645\pi$
$103$	6.53590	0.644001	0.322001	+	0.946739i	$0.395645\pi$
$104$	0	0
$105$	−3.46410	−0.338062
$106$	0	0
$107$	−3.26795	−0.315925	−0.157962	−	0.987445i	$-0.550492\pi$
$107$	−3.26795	−0.315925	−0.157962	+	0.987445i	$0.550492\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−0.732051	−0.0694832
$112$	0	0
$113$	−10.5359	−0.991134	−0.495567	−	0.868570i	$-0.665040\pi$
$113$	−10.5359	−0.991134	−0.495567	+	0.868570i	$0.665040\pi$
$114$	0	0
$115$	8.00000	0.746004
$116$	0	0
$117$	13.4641	1.24476
$118$	0	0
$119$	25.8564	2.37025
$120$	0	0
$121$	18.8564	1.71422
$122$	0	0
$123$	1.46410	0.132014
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.66025	−0.324795	−0.162398	−	0.986725i	$-0.551923\pi$
$127$	−3.66025	−0.324795	−0.162398	+	0.986725i	$0.551923\pi$
$128$	0	0
$129$	5.07180	0.446547
$130$	0	0
$131$	18.5885	1.62408	0.812041	−	0.583601i	$-0.198357\pi$
$131$	18.5885	1.62408	0.812041	+	0.583601i	$0.198357\pi$
$132$	0	0
$133$	−29.3205	−2.54241
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	0	0
$139$	−6.92820	−0.587643	−0.293821	−	0.955860i	$-0.594927\pi$
$139$	−6.92820	−0.587643	−0.293821	+	0.955860i	$0.594927\pi$
$140$	0	0
$141$	−3.46410	−0.291730
$142$	0	0
$143$	−29.8564	−2.49672
$144$	0	0
$145$	4.92820	0.409265
$146$	0	0
$147$	−11.2679	−0.929365
$148$	0	0
$149$	4.39230	0.359832	0.179916	−	0.983682i	$-0.442417\pi$
$149$	4.39230	0.359832	0.179916	+	0.983682i	$0.442417\pi$
$150$	0	0
$151$	12.3923	1.00847	0.504236	−	0.863566i	$-0.331774\pi$
$151$	12.3923	1.00847	0.504236	+	0.863566i	$0.331774\pi$
$152$	0	0
$153$	−13.4641	−1.08851
$154$	0	0
$155$	−0.732051	−0.0587997
$156$	0	0
$157$	3.07180	0.245156	0.122578	−	0.992459i	$-0.460884\pi$
$157$	3.07180	0.245156	0.122578	+	0.992459i	$0.460884\pi$
$158$	0	0
$159$	4.39230	0.348332
$160$	0	0
$161$	37.8564	2.98350
$162$	0	0
$163$	11.3205	0.886691	0.443345	−	0.896351i	$-0.353792\pi$
$163$	11.3205	0.886691	0.443345	+	0.896351i	$0.353792\pi$
$164$	0	0
$165$	−4.00000	−0.311400
$166$	0	0
$167$	1.46410	0.113296	0.0566478	−	0.998394i	$-0.481959\pi$
$167$	1.46410	0.113296	0.0566478	+	0.998394i	$0.481959\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	0	0
$171$	15.2679	1.16757
$172$	0	0
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	4.73205	0.357709
$176$	0	0
$177$	−7.46410	−0.561036
$178$	0	0
$179$	0.339746	0.0253938	0.0126969	−	0.999919i	$-0.495958\pi$
$179$	0.339746	0.0253938	0.0126969	+	0.999919i	$0.495958\pi$
$180$	0	0
$181$	5.46410	0.406143	0.203072	−	0.979164i	$-0.434908\pi$
$181$	5.46410	0.406143	0.203072	+	0.979164i	$0.434908\pi$
$182$	0	0
$183$	3.60770	0.266688
$184$	0	0
$185$	1.00000	0.0735215
$186$	0	0
$187$	29.8564	2.18332
$188$	0	0
$189$	18.9282	1.37682
$190$	0	0
$191$	8.73205	0.631829	0.315915	−	0.948788i	$-0.397689\pi$
$191$	8.73205	0.631829	0.315915	+	0.948788i	$0.397689\pi$
$192$	0	0
$193$	15.8564	1.14137	0.570685	−	0.821169i	$-0.306678\pi$
$193$	15.8564	1.14137	0.570685	+	0.821169i	$0.306678\pi$
$194$	0	0
$195$	4.00000	0.286446
$196$	0	0
$197$	−22.7846	−1.62334	−0.811668	−	0.584119i	$-0.801440\pi$
$197$	−22.7846	−1.62334	−0.811668	+	0.584119i	$0.801440\pi$
$198$	0	0
$199$	12.0526	0.854383	0.427192	−	0.904161i	$-0.359503\pi$
$199$	12.0526	0.854383	0.427192	+	0.904161i	$0.359503\pi$
$200$	0	0
$201$	−2.67949	−0.188997
$202$	0	0
$203$	23.3205	1.63678
$204$	0	0
$205$	−2.00000	−0.139686
$206$	0	0
$207$	−19.7128	−1.37014
$208$	0	0
$209$	−33.8564	−2.34190
$210$	0	0
$211$	17.8564	1.22929	0.614643	−	0.788806i	$-0.289300\pi$
$211$	17.8564	1.22929	0.614643	+	0.788806i	$0.289300\pi$
$212$	0	0
$213$	2.14359	0.146877
$214$	0	0
$215$	−6.92820	−0.472500
$216$	0	0
$217$	−3.46410	−0.235159
$218$	0	0
$219$	0.679492	0.0459158
$220$	0	0
$221$	−29.8564	−2.00836
$222$	0	0
$223$	−16.0526	−1.07496	−0.537479	−	0.843277i	$-0.680623\pi$
$223$	−16.0526	−1.07496	−0.537479	+	0.843277i	$0.680623\pi$
$224$	0	0
$225$	−2.46410	−0.164273
$226$	0	0
$227$	−24.3923	−1.61897	−0.809487	−	0.587138i	$-0.800255\pi$
$227$	−24.3923	−1.61897	−0.809487	+	0.587138i	$0.800255\pi$
$228$	0	0
$229$	−11.8564	−0.783493	−0.391747	−	0.920073i	$-0.628129\pi$
$229$	−11.8564	−0.783493	−0.391747	+	0.920073i	$0.628129\pi$
$230$	0	0
$231$	−18.9282	−1.24538
$232$	0	0
$233$	28.9282	1.89515	0.947575	−	0.319534i	$-0.103526\pi$
$233$	28.9282	1.89515	0.947575	+	0.319534i	$0.103526\pi$
$234$	0	0
$235$	4.73205	0.308685
$236$	0	0
$237$	6.39230	0.415225
$238$	0	0
$239$	20.7321	1.34104	0.670522	−	0.741889i	$-0.266070\pi$
$239$	20.7321	1.34104	0.670522	+	0.741889i	$0.266070\pi$
$240$	0	0
$241$	4.92820	0.317453	0.158727	−	0.987323i	$-0.449261\pi$
$241$	4.92820	0.317453	0.158727	+	0.987323i	$0.449261\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	15.3923	0.983378
$246$	0	0
$247$	33.8564	2.15423
$248$	0	0
$249$	−6.39230	−0.405096
$250$	0	0
$251$	−19.2679	−1.21618	−0.608091	−	0.793867i	$-0.708064\pi$
$251$	−19.2679	−1.21618	−0.608091	+	0.793867i	$0.708064\pi$
$252$	0	0
$253$	43.7128	2.74820
$254$	0	0
$255$	−4.00000	−0.250490
$256$	0	0
$257$	−18.5359	−1.15624	−0.578119	−	0.815953i	$-0.696213\pi$
$257$	−18.5359	−1.15624	−0.578119	+	0.815953i	$0.696213\pi$
$258$	0	0
$259$	4.73205	0.294035
$260$	0	0
$261$	−12.1436	−0.751670
$262$	0	0
$263$	8.05256	0.496542	0.248271	−	0.968691i	$-0.420138\pi$
$263$	8.05256	0.496542	0.248271	+	0.968691i	$0.420138\pi$
$264$	0	0
$265$	−6.00000	−0.368577
$266$	0	0
$267$	1.46410	0.0896016
$268$	0	0
$269$	20.3923	1.24334	0.621670	−	0.783279i	$-0.286454\pi$
$269$	20.3923	1.24334	0.621670	+	0.783279i	$0.286454\pi$
$270$	0	0
$271$	24.7846	1.50556	0.752779	−	0.658273i	$-0.228713\pi$
$271$	24.7846	1.50556	0.752779	+	0.658273i	$0.228713\pi$
$272$	0	0
$273$	18.9282	1.14559
$274$	0	0
$275$	5.46410	0.329498
$276$	0	0
$277$	22.2487	1.33680	0.668398	−	0.743804i	$-0.266980\pi$
$277$	22.2487	1.33680	0.668398	+	0.743804i	$0.266980\pi$
$278$	0	0
$279$	1.80385	0.107994
$280$	0	0
$281$	−8.92820	−0.532612	−0.266306	−	0.963889i	$-0.585803\pi$
$281$	−8.92820	−0.532612	−0.266306	+	0.963889i	$0.585803\pi$
$282$	0	0
$283$	−16.3923	−0.974421	−0.487211	−	0.873284i	$-0.661986\pi$
$283$	−16.3923	−0.974421	−0.487211	+	0.873284i	$0.661986\pi$
$284$	0	0
$285$	4.53590	0.268683
$286$	0	0
$287$	−9.46410	−0.558648
$288$	0	0
$289$	12.8564	0.756259
$290$	0	0
$291$	1.46410	0.0858272
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	10.1962	0.593643
$296$	0	0
$297$	21.8564	1.26824
$298$	0	0
$299$	−43.7128	−2.52798
$300$	0	0
$301$	−32.7846	−1.88967
$302$	0	0
$303$	6.92820	0.398015
$304$	0	0
$305$	−4.92820	−0.282188
$306$	0	0
$307$	−18.5885	−1.06090	−0.530450	−	0.847716i	$-0.677977\pi$
$307$	−18.5885	−1.06090	−0.530450	+	0.847716i	$0.677977\pi$
$308$	0	0
$309$	−4.78461	−0.272187
$310$	0	0
$311$	−2.87564	−0.163063	−0.0815314	−	0.996671i	$-0.525981\pi$
$311$	−2.87564	−0.163063	−0.0815314	+	0.996671i	$0.525981\pi$
$312$	0	0
$313$	−23.8564	−1.34844	−0.674222	−	0.738529i	$-0.735521\pi$
$313$	−23.8564	−1.34844	−0.674222	+	0.738529i	$0.735521\pi$
$314$	0	0
$315$	−11.6603	−0.656981
$316$	0	0
$317$	−4.14359	−0.232727	−0.116364	−	0.993207i	$-0.537124\pi$
$317$	−4.14359	−0.232727	−0.116364	+	0.993207i	$0.537124\pi$
$318$	0	0
$319$	26.9282	1.50769
$320$	0	0
$321$	2.39230	0.133525
$322$	0	0
$323$	−33.8564	−1.88382
$324$	0	0
$325$	−5.46410	−0.303094
$326$	0	0
$327$	−1.46410	−0.0809650
$328$	0	0
$329$	22.3923	1.23453
$330$	0	0
$331$	33.1244	1.82068	0.910340	−	0.413862i	$-0.135820\pi$
$331$	33.1244	1.82068	0.910340	+	0.413862i	$0.135820\pi$
$332$	0	0
$333$	−2.46410	−0.135032
$334$	0	0
$335$	3.66025	0.199981
$336$	0	0
$337$	26.0000	1.41631	0.708155	−	0.706057i	$-0.249528\pi$
$337$	26.0000	1.41631	0.708155	+	0.706057i	$0.249528\pi$
$338$	0	0
$339$	7.71281	0.418902
$340$	0	0
$341$	−4.00000	−0.216612
$342$	0	0
$343$	39.7128	2.14429
$344$	0	0
$345$	−5.85641	−0.315298
$346$	0	0
$347$	−30.9282	−1.66031	−0.830156	−	0.557530i	$-0.811749\pi$
$347$	−30.9282	−1.66031	−0.830156	+	0.557530i	$0.811749\pi$
$348$	0	0
$349$	−15.3205	−0.820088	−0.410044	−	0.912066i	$-0.634487\pi$
$349$	−15.3205	−0.820088	−0.410044	+	0.912066i	$0.634487\pi$
$350$	0	0
$351$	−21.8564	−1.16661
$352$	0	0
$353$	−11.8564	−0.631053	−0.315526	−	0.948917i	$-0.602181\pi$
$353$	−11.8564	−0.631053	−0.315526	+	0.948917i	$0.602181\pi$
$354$	0	0
$355$	−2.92820	−0.155413
$356$	0	0
$357$	−18.9282	−1.00179
$358$	0	0
$359$	12.3923	0.654041	0.327020	−	0.945017i	$-0.393955\pi$
$359$	12.3923	0.654041	0.327020	+	0.945017i	$0.393955\pi$
$360$	0	0
$361$	19.3923	1.02065
$362$	0	0
$363$	−13.8038	−0.724514
$364$	0	0
$365$	−0.928203	−0.0485844
$366$	0	0
$367$	4.33975	0.226533	0.113266	−	0.993565i	$-0.463869\pi$
$367$	4.33975	0.226533	0.113266	+	0.993565i	$0.463869\pi$
$368$	0	0
$369$	4.92820	0.256552
$370$	0	0
$371$	−28.3923	−1.47406
$372$	0	0
$373$	−10.7846	−0.558406	−0.279203	−	0.960232i	$-0.590070\pi$
$373$	−10.7846	−0.558406	−0.279203	+	0.960232i	$0.590070\pi$
$374$	0	0
$375$	−0.732051	−0.0378029
$376$	0	0
$377$	−26.9282	−1.38687
$378$	0	0
$379$	−32.3923	−1.66388	−0.831940	−	0.554865i	$-0.812770\pi$
$379$	−32.3923	−1.66388	−0.831940	+	0.554865i	$0.812770\pi$
$380$	0	0
$381$	2.67949	0.137275
$382$	0	0
$383$	8.00000	0.408781	0.204390	−	0.978889i	$-0.434479\pi$
$383$	8.00000	0.408781	0.204390	+	0.978889i	$0.434479\pi$
$384$	0	0
$385$	25.8564	1.31776
$386$	0	0
$387$	17.0718	0.867808
$388$	0	0
$389$	−11.8564	−0.601144	−0.300572	−	0.953759i	$-0.597178\pi$
$389$	−11.8564	−0.601144	−0.300572	+	0.953759i	$0.597178\pi$
$390$	0	0
$391$	43.7128	2.21065
$392$	0	0
$393$	−13.6077	−0.686417
$394$	0	0
$395$	−8.73205	−0.439357
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	0	0
$399$	21.4641	1.07455
$400$	0	0
$401$	−32.9282	−1.64436	−0.822178	−	0.569230i	$-0.807241\pi$
$401$	−32.9282	−1.64436	−0.822178	+	0.569230i	$0.807241\pi$
$402$	0	0
$403$	4.00000	0.199254
$404$	0	0
$405$	4.46410	0.221823
$406$	0	0
$407$	5.46410	0.270845
$408$	0	0
$409$	3.07180	0.151891	0.0759453	−	0.997112i	$-0.475803\pi$
$409$	3.07180	0.151891	0.0759453	+	0.997112i	$0.475803\pi$
$410$	0	0
$411$	−1.46410	−0.0722188
$412$	0	0
$413$	48.2487	2.37416
$414$	0	0
$415$	8.73205	0.428640
$416$	0	0
$417$	5.07180	0.248367
$418$	0	0
$419$	−14.2487	−0.696095	−0.348048	−	0.937477i	$-0.613155\pi$
$419$	−14.2487	−0.696095	−0.348048	+	0.937477i	$0.613155\pi$
$420$	0	0
$421$	−0.143594	−0.00699832	−0.00349916	−	0.999994i	$-0.501114\pi$
$421$	−0.143594	−0.00699832	−0.00349916	+	0.999994i	$0.501114\pi$
$422$	0	0
$423$	−11.6603	−0.566941
$424$	0	0
$425$	5.46410	0.265048
$426$	0	0
$427$	−23.3205	−1.12856
$428$	0	0
$429$	21.8564	1.05524
$430$	0	0
$431$	2.19615	0.105785	0.0528925	−	0.998600i	$-0.483156\pi$
$431$	2.19615	0.105785	0.0528925	+	0.998600i	$0.483156\pi$
$432$	0	0
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	0	0
$435$	−3.60770	−0.172976
$436$	0	0
$437$	−49.5692	−2.37122
$438$	0	0
$439$	−13.5167	−0.645115	−0.322558	−	0.946550i	$-0.604543\pi$
$439$	−13.5167	−0.645115	−0.322558	+	0.946550i	$0.604543\pi$
$440$	0	0
$441$	−37.9282	−1.80610
$442$	0	0
$443$	14.8756	0.706763	0.353382	−	0.935479i	$-0.385032\pi$
$443$	14.8756	0.706763	0.353382	+	0.935479i	$0.385032\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	0	0
$447$	−3.21539	−0.152083
$448$	0	0
$449$	−21.7128	−1.02469	−0.512345	−	0.858779i	$-0.671223\pi$
$449$	−21.7128	−1.02469	−0.512345	+	0.858779i	$0.671223\pi$
$450$	0	0
$451$	−10.9282	−0.514589
$452$	0	0
$453$	−9.07180	−0.426230
$454$	0	0
$455$	−25.8564	−1.21217
$456$	0	0
$457$	−31.8564	−1.49018	−0.745090	−	0.666964i	$-0.767593\pi$
$457$	−31.8564	−1.49018	−0.745090	+	0.666964i	$0.767593\pi$
$458$	0	0
$459$	21.8564	1.02017
$460$	0	0
$461$	14.7846	0.688588	0.344294	−	0.938862i	$-0.388118\pi$
$461$	14.7846	0.688588	0.344294	+	0.938862i	$0.388118\pi$
$462$	0	0
$463$	−18.9282	−0.879668	−0.439834	−	0.898079i	$-0.644963\pi$
$463$	−18.9282	−0.879668	−0.439834	+	0.898079i	$0.644963\pi$
$464$	0	0
$465$	0.535898	0.0248517
$466$	0	0
$467$	25.1769	1.16505	0.582524	−	0.812813i	$-0.302065\pi$
$467$	25.1769	1.16505	0.582524	+	0.812813i	$0.302065\pi$
$468$	0	0
$469$	17.3205	0.799787
$470$	0	0
$471$	−2.24871	−0.103615
$472$	0	0
$473$	−37.8564	−1.74064
$474$	0	0
$475$	−6.19615	−0.284299
$476$	0	0
$477$	14.7846	0.676941
$478$	0	0
$479$	−4.05256	−0.185166	−0.0925831	−	0.995705i	$-0.529512\pi$
$479$	−4.05256	−0.185166	−0.0925831	+	0.995705i	$0.529512\pi$
$480$	0	0
$481$	−5.46410	−0.249142
$482$	0	0
$483$	−27.7128	−1.26098
$484$	0	0
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−4.39230	−0.199034	−0.0995172	−	0.995036i	$-0.531730\pi$
$487$	−4.39230	−0.199034	−0.0995172	+	0.995036i	$0.531730\pi$
$488$	0	0
$489$	−8.28719	−0.374760
$490$	0	0
$491$	14.9282	0.673700	0.336850	−	0.941558i	$-0.390638\pi$
$491$	14.9282	0.673700	0.336850	+	0.941558i	$0.390638\pi$
$492$	0	0
$493$	26.9282	1.21279
$494$	0	0
$495$	−13.4641	−0.605166
$496$	0	0
$497$	−13.8564	−0.621545
$498$	0	0
$499$	−9.41154	−0.421319	−0.210659	−	0.977560i	$-0.567561\pi$
$499$	−9.41154	−0.421319	−0.210659	+	0.977560i	$0.567561\pi$
$500$	0	0
$501$	−1.07180	−0.0478843
$502$	0	0
$503$	18.9282	0.843967	0.421983	−	0.906604i	$-0.361334\pi$
$503$	18.9282	0.843967	0.421983	+	0.906604i	$0.361334\pi$
$504$	0	0
$505$	−9.46410	−0.421147
$506$	0	0
$507$	−12.3397	−0.548027
$508$	0	0
$509$	−2.00000	−0.0886484	−0.0443242	−	0.999017i	$-0.514113\pi$
$509$	−2.00000	−0.0886484	−0.0443242	+	0.999017i	$0.514113\pi$
$510$	0	0
$511$	−4.39230	−0.194304
$512$	0	0
$513$	−24.7846	−1.09427
$514$	0	0
$515$	6.53590	0.288006
$516$	0	0
$517$	25.8564	1.13716
$518$	0	0
$519$	−7.32051	−0.321335
$520$	0	0
$521$	26.5359	1.16256	0.581279	−	0.813704i	$-0.302552\pi$
$521$	26.5359	1.16256	0.581279	+	0.813704i	$0.302552\pi$
$522$	0	0
$523$	−36.7846	−1.60848	−0.804239	−	0.594306i	$-0.797427\pi$
$523$	−36.7846	−1.60848	−0.804239	+	0.594306i	$0.797427\pi$
$524$	0	0
$525$	−3.46410	−0.151186
$526$	0	0
$527$	−4.00000	−0.174243
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−25.1244	−1.09030
$532$	0	0
$533$	10.9282	0.473353
$534$	0	0
$535$	−3.26795	−0.141286
$536$	0	0
$537$	−0.248711	−0.0107327
$538$	0	0
$539$	84.1051	3.62266
$540$	0	0
$541$	−30.0000	−1.28980	−0.644900	−	0.764267i	$-0.723101\pi$
$541$	−30.0000	−1.28980	−0.644900	+	0.764267i	$0.723101\pi$
$542$	0	0
$543$	−4.00000	−0.171656
$544$	0	0
$545$	2.00000	0.0856706
$546$	0	0
$547$	4.00000	0.171028	0.0855138	−	0.996337i	$-0.472747\pi$
$547$	4.00000	0.171028	0.0855138	+	0.996337i	$0.472747\pi$
$548$	0	0
$549$	12.1436	0.518276
$550$	0	0
$551$	−30.5359	−1.30087
$552$	0	0
$553$	−41.3205	−1.75713
$554$	0	0
$555$	−0.732051	−0.0310738
$556$	0	0
$557$	44.1051	1.86879	0.934397	−	0.356234i	$-0.115939\pi$
$557$	44.1051	1.86879	0.934397	+	0.356234i	$0.115939\pi$
$558$	0	0
$559$	37.8564	1.60116
$560$	0	0
$561$	−21.8564	−0.922778
$562$	0	0
$563$	−30.9282	−1.30347	−0.651734	−	0.758447i	$-0.725958\pi$
$563$	−30.9282	−1.30347	−0.651734	+	0.758447i	$0.725958\pi$
$564$	0	0
$565$	−10.5359	−0.443249
$566$	0	0
$567$	21.1244	0.887140
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	−6.39230	−0.267042
$574$	0	0
$575$	8.00000	0.333623
$576$	0	0
$577$	−24.3923	−1.01546	−0.507732	−	0.861515i	$-0.669516\pi$
$577$	−24.3923	−1.01546	−0.507732	+	0.861515i	$0.669516\pi$
$578$	0	0
$579$	−11.6077	−0.482399
$580$	0	0
$581$	41.3205	1.71426
$582$	0	0
$583$	−32.7846	−1.35780
$584$	0	0
$585$	13.4641	0.556672
$586$	0	0
$587$	−44.7846	−1.84846	−0.924229	−	0.381838i	$-0.875291\pi$
$587$	−44.7846	−1.84846	−0.924229	+	0.381838i	$0.875291\pi$
$588$	0	0
$589$	4.53590	0.186898
$590$	0	0
$591$	16.6795	0.686103
$592$	0	0
$593$	−34.7846	−1.42843	−0.714216	−	0.699925i	$-0.753217\pi$
$593$	−34.7846	−1.42843	−0.714216	+	0.699925i	$0.753217\pi$
$594$	0	0
$595$	25.8564	1.06001
$596$	0	0
$597$	−8.82309	−0.361105
$598$	0	0
$599$	−9.46410	−0.386693	−0.193346	−	0.981131i	$-0.561934\pi$
$599$	−9.46410	−0.386693	−0.193346	+	0.981131i	$0.561934\pi$
$600$	0	0
$601$	27.6077	1.12614	0.563071	−	0.826409i	$-0.309620\pi$
$601$	27.6077	1.12614	0.563071	+	0.826409i	$0.309620\pi$
$602$	0	0
$603$	−9.01924	−0.367292
$604$	0	0
$605$	18.8564	0.766622
$606$	0	0
$607$	−0.784610	−0.0318463	−0.0159232	−	0.999873i	$-0.505069\pi$
$607$	−0.784610	−0.0318463	−0.0159232	+	0.999873i	$0.505069\pi$
$608$	0	0
$609$	−17.0718	−0.691784
$610$	0	0
$611$	−25.8564	−1.04604
$612$	0	0
$613$	16.9282	0.683724	0.341862	−	0.939750i	$-0.388943\pi$
$613$	16.9282	0.683724	0.341862	+	0.939750i	$0.388943\pi$
$614$	0	0
$615$	1.46410	0.0590383
$616$	0	0
$617$	−0.928203	−0.0373681	−0.0186840	−	0.999825i	$-0.505948\pi$
$617$	−0.928203	−0.0373681	−0.0186840	+	0.999825i	$0.505948\pi$
$618$	0	0
$619$	17.8564	0.717710	0.358855	−	0.933393i	$-0.383167\pi$
$619$	17.8564	0.717710	0.358855	+	0.933393i	$0.383167\pi$
$620$	0	0
$621$	32.0000	1.28412
$622$	0	0
$623$	−9.46410	−0.379171
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	24.7846	0.989802
$628$	0	0
$629$	5.46410	0.217868
$630$	0	0
$631$	−30.9808	−1.23332	−0.616662	−	0.787228i	$-0.711516\pi$
$631$	−30.9808	−1.23332	−0.616662	+	0.787228i	$0.711516\pi$
$632$	0	0
$633$	−13.0718	−0.519557
$634$	0	0
$635$	−3.66025	−0.145253
$636$	0	0
$637$	−84.1051	−3.33237
$638$	0	0
$639$	7.21539	0.285436
$640$	0	0
$641$	−6.53590	−0.258152	−0.129076	−	0.991635i	$-0.541201\pi$
$641$	−6.53590	−0.258152	−0.129076	+	0.991635i	$0.541201\pi$
$642$	0	0
$643$	−34.5359	−1.36196	−0.680981	−	0.732301i	$-0.738447\pi$
$643$	−34.5359	−1.36196	−0.680981	+	0.732301i	$0.738447\pi$
$644$	0	0
$645$	5.07180	0.199702
$646$	0	0
$647$	35.7128	1.40402	0.702008	−	0.712169i	$-0.252287\pi$
$647$	35.7128	1.40402	0.702008	+	0.712169i	$0.252287\pi$
$648$	0	0
$649$	55.7128	2.18692
$650$	0	0
$651$	2.53590	0.0993897
$652$	0	0
$653$	−18.0000	−0.704394	−0.352197	−	0.935926i	$-0.614565\pi$
$653$	−18.0000	−0.704394	−0.352197	+	0.935926i	$0.614565\pi$
$654$	0	0
$655$	18.5885	0.726311
$656$	0	0
$657$	2.28719	0.0892317
$658$	0	0
$659$	6.92820	0.269884	0.134942	−	0.990853i	$-0.456915\pi$
$659$	6.92820	0.269884	0.134942	+	0.990853i	$0.456915\pi$
$660$	0	0
$661$	−31.0718	−1.20855	−0.604276	−	0.796775i	$-0.706538\pi$
$661$	−31.0718	−1.20855	−0.604276	+	0.796775i	$0.706538\pi$
$662$	0	0
$663$	21.8564	0.848832
$664$	0	0
$665$	−29.3205	−1.13700
$666$	0	0
$667$	39.4256	1.52657
$668$	0	0
$669$	11.7513	0.454331
$670$	0	0
$671$	−26.9282	−1.03955
$672$	0	0
$673$	−32.9282	−1.26929	−0.634644	−	0.772804i	$-0.718853\pi$
$673$	−32.9282	−1.26929	−0.634644	+	0.772804i	$0.718853\pi$
$674$	0	0
$675$	4.00000	0.153960
$676$	0	0
$677$	−4.14359	−0.159251	−0.0796256	−	0.996825i	$-0.525372\pi$
$677$	−4.14359	−0.159251	−0.0796256	+	0.996825i	$0.525372\pi$
$678$	0	0
$679$	−9.46410	−0.363199
$680$	0	0
$681$	17.8564	0.684259
$682$	0	0
$683$	4.78461	0.183078	0.0915390	−	0.995801i	$-0.470821\pi$
$683$	4.78461	0.183078	0.0915390	+	0.995801i	$0.470821\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	0	0
$687$	8.67949	0.331143
$688$	0	0
$689$	32.7846	1.24899
$690$	0	0
$691$	0.392305	0.0149240	0.00746199	−	0.999972i	$-0.497625\pi$
$691$	0.392305	0.0149240	0.00746199	+	0.999972i	$0.497625\pi$
$692$	0	0
$693$	−63.7128	−2.42025
$694$	0	0
$695$	−6.92820	−0.262802
$696$	0	0
$697$	−10.9282	−0.413935
$698$	0	0
$699$	−21.1769	−0.800984
$700$	0	0
$701$	−11.8564	−0.447810	−0.223905	−	0.974611i	$-0.571881\pi$
$701$	−11.8564	−0.447810	−0.223905	+	0.974611i	$0.571881\pi$
$702$	0	0
$703$	−6.19615	−0.233692
$704$	0	0
$705$	−3.46410	−0.130466
$706$	0	0
$707$	−44.7846	−1.68430
$708$	0	0
$709$	43.8564	1.64706	0.823531	−	0.567271i	$-0.192001\pi$
$709$	43.8564	1.64706	0.823531	+	0.567271i	$0.192001\pi$
$710$	0	0
$711$	21.5167	0.806938
$712$	0	0
$713$	−5.85641	−0.219324
$714$	0	0
$715$	−29.8564	−1.11657
$716$	0	0
$717$	−15.1769	−0.566792
$718$	0	0
$719$	12.3923	0.462155	0.231077	−	0.972935i	$-0.425775\pi$
$719$	12.3923	0.462155	0.231077	+	0.972935i	$0.425775\pi$
$720$	0	0
$721$	30.9282	1.15183
$722$	0	0
$723$	−3.60770	−0.134172
$724$	0	0
$725$	4.92820	0.183029
$726$	0	0
$727$	−8.78461	−0.325803	−0.162902	−	0.986642i	$-0.552085\pi$
$727$	−8.78461	−0.325803	−0.162902	+	0.986642i	$0.552085\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−37.8564	−1.40017
$732$	0	0
$733$	27.8564	1.02890	0.514450	−	0.857520i	$-0.327996\pi$
$733$	27.8564	1.02890	0.514450	+	0.857520i	$0.327996\pi$
$734$	0	0
$735$	−11.2679	−0.415625
$736$	0	0
$737$	20.0000	0.736709
$738$	0	0
$739$	−6.92820	−0.254858	−0.127429	−	0.991848i	$-0.540673\pi$
$739$	−6.92820	−0.254858	−0.127429	+	0.991848i	$0.540673\pi$
$740$	0	0
$741$	−24.7846	−0.910485
$742$	0	0
$743$	−17.1244	−0.628232	−0.314116	−	0.949385i	$-0.601708\pi$
$743$	−17.1244	−0.628232	−0.314116	+	0.949385i	$0.601708\pi$
$744$	0	0
$745$	4.39230	0.160922
$746$	0	0
$747$	−21.5167	−0.787253
$748$	0	0
$749$	−15.4641	−0.565046
$750$	0	0
$751$	20.3923	0.744126	0.372063	−	0.928208i	$-0.378651\pi$
$751$	20.3923	0.744126	0.372063	+	0.928208i	$0.378651\pi$
$752$	0	0
$753$	14.1051	0.514019
$754$	0	0
$755$	12.3923	0.451002
$756$	0	0
$757$	1.71281	0.0622532	0.0311266	−	0.999515i	$-0.490090\pi$
$757$	1.71281	0.0622532	0.0311266	+	0.999515i	$0.490090\pi$
$758$	0	0
$759$	−32.0000	−1.16153
$760$	0	0
$761$	10.0000	0.362500	0.181250	−	0.983437i	$-0.441986\pi$
$761$	10.0000	0.362500	0.181250	+	0.983437i	$0.441986\pi$
$762$	0	0
$763$	9.46410	0.342623
$764$	0	0
$765$	−13.4641	−0.486796
$766$	0	0
$767$	−55.7128	−2.01167
$768$	0	0
$769$	7.07180	0.255016	0.127508	−	0.991838i	$-0.459302\pi$
$769$	7.07180	0.255016	0.127508	+	0.991838i	$0.459302\pi$
$770$	0	0
$771$	13.5692	0.488683
$772$	0	0
$773$	2.78461	0.100155	0.0500777	−	0.998745i	$-0.484053\pi$
$773$	2.78461	0.100155	0.0500777	+	0.998745i	$0.484053\pi$
$774$	0	0
$775$	−0.732051	−0.0262960
$776$	0	0
$777$	−3.46410	−0.124274
$778$	0	0
$779$	12.3923	0.444000
$780$	0	0
$781$	−16.0000	−0.572525
$782$	0	0
$783$	19.7128	0.704478
$784$	0	0
$785$	3.07180	0.109637
$786$	0	0
$787$	−22.1962	−0.791207	−0.395604	−	0.918421i	$-0.629465\pi$
$787$	−22.1962	−0.791207	−0.395604	+	0.918421i	$0.629465\pi$
$788$	0	0
$789$	−5.89488	−0.209863
$790$	0	0
$791$	−49.8564	−1.77269
$792$	0	0
$793$	26.9282	0.956249
$794$	0	0
$795$	4.39230	0.155779
$796$	0	0
$797$	10.5359	0.373201	0.186600	−	0.982436i	$-0.440253\pi$
$797$	10.5359	0.373201	0.186600	+	0.982436i	$0.440253\pi$
$798$	0	0
$799$	25.8564	0.914734
$800$	0	0
$801$	4.92820	0.174129
$802$	0	0
$803$	−5.07180	−0.178980
$804$	0	0
$805$	37.8564	1.33426
$806$	0	0
$807$	−14.9282	−0.525498
$808$	0	0
$809$	43.5692	1.53181	0.765906	−	0.642952i	$-0.222291\pi$
$809$	43.5692	1.53181	0.765906	+	0.642952i	$0.222291\pi$
$810$	0	0
$811$	41.8564	1.46978	0.734889	−	0.678188i	$-0.237234\pi$
$811$	41.8564	1.46978	0.734889	+	0.678188i	$0.237234\pi$
$812$	0	0
$813$	−18.1436	−0.636324
$814$	0	0
$815$	11.3205	0.396540
$816$	0	0
$817$	42.9282	1.50187
$818$	0	0
$819$	63.7128	2.22631
$820$	0	0
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	0	0
$823$	−48.4449	−1.68868	−0.844341	−	0.535806i	$-0.820008\pi$
$823$	−48.4449	−1.68868	−0.844341	+	0.535806i	$0.820008\pi$
$824$	0	0
$825$	−4.00000	−0.139262
$826$	0	0
$827$	16.3923	0.570016	0.285008	−	0.958525i	$-0.408004\pi$
$827$	16.3923	0.570016	0.285008	+	0.958525i	$0.408004\pi$
$828$	0	0
$829$	−51.5692	−1.79107	−0.895537	−	0.444988i	$-0.853208\pi$
$829$	−51.5692	−1.79107	−0.895537	+	0.444988i	$0.853208\pi$
$830$	0	0
$831$	−16.2872	−0.564996
$832$	0	0
$833$	84.1051	2.91407
$834$	0	0
$835$	1.46410	0.0506673
$836$	0	0
$837$	−2.92820	−0.101214
$838$	0	0
$839$	8.78461	0.303278	0.151639	−	0.988436i	$-0.451545\pi$
$839$	8.78461	0.303278	0.151639	+	0.988436i	$0.451545\pi$
$840$	0	0
$841$	−4.71281	−0.162511
$842$	0	0
$843$	6.53590	0.225108
$844$	0	0
$845$	16.8564	0.579878
$846$	0	0
$847$	89.2295	3.06596
$848$	0	0
$849$	12.0000	0.411839
$850$	0	0
$851$	8.00000	0.274236
$852$	0	0
$853$	−30.0000	−1.02718	−0.513590	−	0.858036i	$-0.671685\pi$
$853$	−30.0000	−1.02718	−0.513590	+	0.858036i	$0.671685\pi$
$854$	0	0
$855$	15.2679	0.522153
$856$	0	0
$857$	−31.8564	−1.08819	−0.544097	−	0.839022i	$-0.683128\pi$
$857$	−31.8564	−1.08819	−0.544097	+	0.839022i	$0.683128\pi$
$858$	0	0
$859$	−44.4449	−1.51644	−0.758220	−	0.651999i	$-0.773931\pi$
$859$	−44.4449	−1.51644	−0.758220	+	0.651999i	$0.773931\pi$
$860$	0	0
$861$	6.92820	0.236113
$862$	0	0
$863$	20.4449	0.695951	0.347976	−	0.937504i	$-0.386869\pi$
$863$	20.4449	0.695951	0.347976	+	0.937504i	$0.386869\pi$
$864$	0	0
$865$	10.0000	0.340010
$866$	0	0
$867$	−9.41154	−0.319633
$868$	0	0
$869$	−47.7128	−1.61855
$870$	0	0
$871$	−20.0000	−0.677674
$872$	0	0
$873$	4.92820	0.166794
$874$	0	0
$875$	4.73205	0.159973
$876$	0	0
$877$	−9.21539	−0.311182	−0.155591	−	0.987822i	$-0.549728\pi$
$877$	−9.21539	−0.311182	−0.155591	+	0.987822i	$0.549728\pi$
$878$	0	0
$879$	−4.39230	−0.148149
$880$	0	0
$881$	12.6795	0.427183	0.213591	−	0.976923i	$-0.431484\pi$
$881$	12.6795	0.427183	0.213591	+	0.976923i	$0.431484\pi$
$882$	0	0
$883$	−14.2487	−0.479507	−0.239754	−	0.970834i	$-0.577067\pi$
$883$	−14.2487	−0.479507	−0.239754	+	0.970834i	$0.577067\pi$
$884$	0	0
$885$	−7.46410	−0.250903
$886$	0	0
$887$	−9.80385	−0.329181	−0.164590	−	0.986362i	$-0.552630\pi$
$887$	−9.80385	−0.329181	−0.164590	+	0.986362i	$0.552630\pi$
$888$	0	0
$889$	−17.3205	−0.580911
$890$	0	0
$891$	24.3923	0.817173
$892$	0	0
$893$	−29.3205	−0.981173
$894$	0	0
$895$	0.339746	0.0113565
$896$	0	0
$897$	32.0000	1.06845
$898$	0	0
$899$	−3.60770	−0.120323
$900$	0	0
$901$	−32.7846	−1.09221
$902$	0	0
$903$	24.0000	0.798670
$904$	0	0
$905$	5.46410	0.181633
$906$	0	0
$907$	−54.2487	−1.80130	−0.900649	−	0.434546i	$-0.856909\pi$
$907$	−54.2487	−1.80130	−0.900649	+	0.434546i	$0.856909\pi$
$908$	0	0
$909$	23.3205	0.773492
$910$	0	0
$911$	−37.9090	−1.25598	−0.627990	−	0.778221i	$-0.716122\pi$
$911$	−37.9090	−1.25598	−0.627990	+	0.778221i	$0.716122\pi$
$912$	0	0
$913$	47.7128	1.57906
$914$	0	0
$915$	3.60770	0.119267
$916$	0	0
$917$	87.9615	2.90475
$918$	0	0
$919$	16.7321	0.551939	0.275970	−	0.961166i	$-0.411001\pi$
$919$	16.7321	0.551939	0.275970	+	0.961166i	$0.411001\pi$
$920$	0	0
$921$	13.6077	0.448389
$922$	0	0
$923$	16.0000	0.526646
$924$	0	0
$925$	1.00000	0.0328798
$926$	0	0
$927$	−16.1051	−0.528961
$928$	0	0
$929$	11.8564	0.388996	0.194498	−	0.980903i	$-0.437692\pi$
$929$	11.8564	0.388996	0.194498	+	0.980903i	$0.437692\pi$
$930$	0	0
$931$	−95.3731	−3.12573
$932$	0	0
$933$	2.10512	0.0689185
$934$	0	0
$935$	29.8564	0.976409
$936$	0	0
$937$	−23.8564	−0.779355	−0.389677	−	0.920951i	$-0.627413\pi$
$937$	−23.8564	−0.779355	−0.389677	+	0.920951i	$0.627413\pi$
$938$	0	0
$939$	17.4641	0.569919
$940$	0	0
$941$	7.60770	0.248004	0.124002	−	0.992282i	$-0.460427\pi$
$941$	7.60770	0.248004	0.124002	+	0.992282i	$0.460427\pi$
$942$	0	0
$943$	−16.0000	−0.521032
$944$	0	0
$945$	18.9282	0.615734
$946$	0	0
$947$	−17.1769	−0.558175	−0.279087	−	0.960266i	$-0.590032\pi$
$947$	−17.1769	−0.558175	−0.279087	+	0.960266i	$0.590032\pi$
$948$	0	0
$949$	5.07180	0.164637
$950$	0	0
$951$	3.03332	0.0983622
$952$	0	0
$953$	15.8564	0.513639	0.256820	−	0.966459i	$-0.417325\pi$
$953$	15.8564	0.513639	0.256820	+	0.966459i	$0.417325\pi$
$954$	0	0
$955$	8.73205	0.282563
$956$	0	0
$957$	−19.7128	−0.637225
$958$	0	0
$959$	9.46410	0.305612
$960$	0	0
$961$	−30.4641	−0.982713
$962$	0	0
$963$	8.05256	0.259490
$964$	0	0
$965$	15.8564	0.510436
$966$	0	0
$967$	44.3923	1.42756	0.713780	−	0.700370i	$-0.246982\pi$
$967$	44.3923	1.42756	0.713780	+	0.700370i	$0.246982\pi$
$968$	0	0
$969$	24.7846	0.796196
$970$	0	0
$971$	−1.07180	−0.0343956	−0.0171978	−	0.999852i	$-0.505474\pi$
$971$	−1.07180	−0.0343956	−0.0171978	+	0.999852i	$0.505474\pi$
$972$	0	0
$973$	−32.7846	−1.05103
$974$	0	0
$975$	4.00000	0.128103
$976$	0	0
$977$	−34.0000	−1.08776	−0.543878	−	0.839164i	$-0.683045\pi$
$977$	−34.0000	−1.08776	−0.543878	+	0.839164i	$0.683045\pi$
$978$	0	0
$979$	−10.9282	−0.349267
$980$	0	0
$981$	−4.92820	−0.157345
$982$	0	0
$983$	−20.4449	−0.652090	−0.326045	−	0.945354i	$-0.605716\pi$
$983$	−20.4449	−0.652090	−0.326045	+	0.945354i	$0.605716\pi$
$984$	0	0
$985$	−22.7846	−0.725978
$986$	0	0
$987$	−16.3923	−0.521773
$988$	0	0
$989$	−55.4256	−1.76243
$990$	0	0
$991$	−44.0526	−1.39938	−0.699688	−	0.714449i	$-0.746678\pi$
$991$	−44.0526	−1.39938	−0.699688	+	0.714449i	$0.746678\pi$
$992$	0	0
$993$	−24.2487	−0.769510
$994$	0	0
$995$	12.0526	0.382092
$996$	0	0
$997$	−31.8564	−1.00890	−0.504451	−	0.863440i	$-0.668305\pi$
$997$	−31.8564	−1.00890	−0.504451	+	0.863440i	$0.668305\pi$
$998$	0	0
$999$	4.00000	0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2960.2.a.q.1.1		2
4.3	odd	2		370.2.a.e.1.2	✓	2
12.11	even	2		3330.2.a.bd.1.1		2
20.3	even	4		1850.2.b.l.149.4		4
20.7	even	4		1850.2.b.l.149.1		4
20.19	odd	2		1850.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.e.1.2	✓	2	4.3	odd	2
1850.2.a.x.1.1		2	20.19	odd	2
1850.2.b.l.149.1		4	20.7	even	4
1850.2.b.l.149.4		4	20.3	even	4
2960.2.a.q.1.1		2	1.1	even	1	trivial
3330.2.a.bd.1.1		2	12.11	even	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:	\( N \)	\(=\)	\( 2960 = 2^{4} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2960.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +1.00000 q^{5} +4.73205 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} +1.00000 q^{5} +4.73205 q^{7} -2.46410 q^{9} +5.46410 q^{11} -5.46410 q^{13} -0.732051 q^{15} +5.46410 q^{17} -6.19615 q^{19} -3.46410 q^{21} +8.00000 q^{23} +1.00000 q^{25} +4.00000 q^{27} +4.92820 q^{29} -0.732051 q^{31} -4.00000 q^{33} +4.73205 q^{35} +1.00000 q^{37} +4.00000 q^{39} -2.00000 q^{41} -6.92820 q^{43} -2.46410 q^{45} +4.73205 q^{47} +15.3923 q^{49} -4.00000 q^{51} -6.00000 q^{53} +5.46410 q^{55} +4.53590 q^{57} +10.1962 q^{59} -4.92820 q^{61} -11.6603 q^{63} -5.46410 q^{65} +3.66025 q^{67} -5.85641 q^{69} -2.92820 q^{71} -0.928203 q^{73} -0.732051 q^{75} +25.8564 q^{77} -8.73205 q^{79} +4.46410 q^{81} +8.73205 q^{83} +5.46410 q^{85} -3.60770 q^{87} -2.00000 q^{89} -25.8564 q^{91} +0.535898 q^{93} -6.19615 q^{95} -2.00000 q^{97} -13.4641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 6 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 2 q^{15} + 4 q^{17} - 2 q^{19} + 16 q^{23} + 2 q^{25} + 8 q^{27} - 4 q^{29} + 2 q^{31} - 8 q^{33} + 6 q^{35} + 2 q^{37} + 8 q^{39} - 4 q^{41} + 2 q^{45} + 6 q^{47} + 10 q^{49} - 8 q^{51} - 12 q^{53} + 4 q^{55} + 16 q^{57} + 10 q^{59} + 4 q^{61} - 6 q^{63} - 4 q^{65} - 10 q^{67} + 16 q^{69} + 8 q^{71} + 12 q^{73} + 2 q^{75} + 24 q^{77} - 14 q^{79} + 2 q^{81} + 14 q^{83} + 4 q^{85} - 28 q^{87} - 4 q^{89} - 24 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 20 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 6 * q^7 + 2 * q^9 + 4 * q^11 - 4 * q^13 + 2 * q^15 + 4 * q^17 - 2 * q^19 + 16 * q^23 + 2 * q^25 + 8 * q^27 - 4 * q^29 + 2 * q^31 - 8 * q^33 + 6 * q^35 + 2 * q^37 + 8 * q^39 - 4 * q^41 + 2 * q^45 + 6 * q^47 + 10 * q^49 - 8 * q^51 - 12 * q^53 + 4 * q^55 + 16 * q^57 + 10 * q^59 + 4 * q^61 - 6 * q^63 - 4 * q^65 - 10 * q^67 + 16 * q^69 + 8 * q^71 + 12 * q^73 + 2 * q^75 + 24 * q^77 - 14 * q^79 + 2 * q^81 + 14 * q^83 + 4 * q^85 - 28 * q^87 - 4 * q^89 - 24 * q^91 + 8 * q^93 - 2 * q^95 - 4 * q^97 - 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more