LMFDB - Embedded newform 4012.2.a.j.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.461334 q^{3} +1.69592 q^{5} +4.42760 q^{7} -2.78717 q^{9} +O(q^{10})$	$q+0.461334 q^{3} +1.69592 q^{5} +4.42760 q^{7} -2.78717 q^{9} -0.360678 q^{11} -2.93094 q^{13} +0.782386 q^{15} +1.00000 q^{17} -1.66217 q^{19} +2.04260 q^{21} +5.11562 q^{23} -2.12385 q^{25} -2.66982 q^{27} +3.08082 q^{29} -5.67476 q^{31} -0.166393 q^{33} +7.50886 q^{35} +9.56664 q^{37} -1.35214 q^{39} +6.23775 q^{41} +9.43684 q^{43} -4.72682 q^{45} -1.63599 q^{47} +12.6036 q^{49} +0.461334 q^{51} +7.92237 q^{53} -0.611681 q^{55} -0.766816 q^{57} +1.00000 q^{59} +9.66627 q^{61} -12.3405 q^{63} -4.97064 q^{65} -0.237376 q^{67} +2.36001 q^{69} +0.460364 q^{71} +14.3876 q^{73} -0.979806 q^{75} -1.59694 q^{77} +7.06666 q^{79} +7.12983 q^{81} -3.90940 q^{83} +1.69592 q^{85} +1.42129 q^{87} -10.0885 q^{89} -12.9770 q^{91} -2.61796 q^{93} -2.81891 q^{95} -4.59257 q^{97} +1.00527 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * 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
$3$	0.461334	0.266351	0.133176	−	0.991092i	$-0.457483\pi$
$3$	0.461334	0.266351	0.133176	+	0.991092i	$0.457483\pi$
$4$	0	0
$5$	1.69592	0.758439	0.379219	−	0.925307i	$-0.376193\pi$
$5$	1.69592	0.758439	0.379219	+	0.925307i	$0.376193\pi$
$6$	0	0
$7$	4.42760	1.67348	0.836738	−	0.547604i	$-0.184460\pi$
$7$	4.42760	1.67348	0.836738	+	0.547604i	$0.184460\pi$
$8$	0	0
$9$	−2.78717	−0.929057
$10$	0	0
$11$	−0.360678	−0.108748	−0.0543742	−	0.998521i	$-0.517316\pi$
$11$	−0.360678	−0.108748	−0.0543742	+	0.998521i	$0.517316\pi$
$12$	0	0
$13$	−2.93094	−0.812896	−0.406448	−	0.913674i	$-0.633233\pi$
$13$	−2.93094	−0.812896	−0.406448	+	0.913674i	$0.633233\pi$
$14$	0	0
$15$	0.782386	0.202011
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−1.66217	−0.381328	−0.190664	−	0.981655i	$-0.561064\pi$
$19$	−1.66217	−0.381328	−0.190664	+	0.981655i	$0.561064\pi$
$20$	0	0
$21$	2.04260	0.445732
$22$	0	0
$23$	5.11562	1.06668	0.533340	−	0.845901i	$-0.320937\pi$
$23$	5.11562	1.06668	0.533340	+	0.845901i	$0.320937\pi$
$24$	0	0
$25$	−2.12385	−0.424771
$26$	0	0
$27$	−2.66982	−0.513807
$28$	0	0
$29$	3.08082	0.572094	0.286047	−	0.958216i	$-0.407659\pi$
$29$	3.08082	0.572094	0.286047	+	0.958216i	$0.407659\pi$
$30$	0	0
$31$	−5.67476	−1.01922	−0.509608	−	0.860407i	$-0.670210\pi$
$31$	−5.67476	−1.01922	−0.509608	+	0.860407i	$0.670210\pi$
$32$	0	0
$33$	−0.166393	−0.0289653
$34$	0	0
$35$	7.50886	1.26923
$36$	0	0
$37$	9.56664	1.57275	0.786373	−	0.617752i	$-0.211957\pi$
$37$	9.56664	1.57275	0.786373	+	0.617752i	$0.211957\pi$
$38$	0	0
$39$	−1.35214	−0.216516
$40$	0	0
$41$	6.23775	0.974173	0.487087	−	0.873354i	$-0.338060\pi$
$41$	6.23775	0.974173	0.487087	+	0.873354i	$0.338060\pi$
$42$	0	0
$43$	9.43684	1.43910	0.719552	−	0.694438i	$-0.244347\pi$
$43$	9.43684	1.43910	0.719552	+	0.694438i	$0.244347\pi$
$44$	0	0
$45$	−4.72682	−0.704633
$46$	0	0
$47$	−1.63599	−0.238634	−0.119317	−	0.992856i	$-0.538070\pi$
$47$	−1.63599	−0.238634	−0.119317	+	0.992856i	$0.538070\pi$
$48$	0	0
$49$	12.6036	1.80052
$50$	0	0
$51$	0.461334	0.0645997
$52$	0	0
$53$	7.92237	1.08822	0.544111	−	0.839013i	$-0.316867\pi$
$53$	7.92237	1.08822	0.544111	+	0.839013i	$0.316867\pi$
$54$	0	0
$55$	−0.611681	−0.0824791
$56$	0	0
$57$	−0.766816	−0.101567
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	9.66627	1.23764	0.618820	−	0.785533i	$-0.287611\pi$
$61$	9.66627	1.23764	0.618820	+	0.785533i	$0.287611\pi$
$62$	0	0
$63$	−12.3405	−1.55475
$64$	0	0
$65$	−4.97064	−0.616532
$66$	0	0
$67$	−0.237376	−0.0290001	−0.0145001	−	0.999895i	$-0.504616\pi$
$67$	−0.237376	−0.0290001	−0.0145001	+	0.999895i	$0.504616\pi$
$68$	0	0
$69$	2.36001	0.284112
$70$	0	0
$71$	0.460364	0.0546352	0.0273176	−	0.999627i	$-0.491303\pi$
$71$	0.460364	0.0546352	0.0273176	+	0.999627i	$0.491303\pi$
$72$	0	0
$73$	14.3876	1.68394	0.841968	−	0.539528i	$-0.181397\pi$
$73$	14.3876	1.68394	0.841968	+	0.539528i	$0.181397\pi$
$74$	0	0
$75$	−0.979806	−0.113138
$76$	0	0
$77$	−1.59694	−0.181988
$78$	0	0
$79$	7.06666	0.795062	0.397531	−	0.917589i	$-0.369867\pi$
$79$	7.06666	0.795062	0.397531	+	0.917589i	$0.369867\pi$
$80$	0	0
$81$	7.12983	0.792204
$82$	0	0
$83$	−3.90940	−0.429112	−0.214556	−	0.976712i	$-0.568831\pi$
$83$	−3.90940	−0.429112	−0.214556	+	0.976712i	$0.568831\pi$
$84$	0	0
$85$	1.69592	0.183948
$86$	0	0
$87$	1.42129	0.152378
$88$	0	0
$89$	−10.0885	−1.06938	−0.534690	−	0.845048i	$-0.679572\pi$
$89$	−10.0885	−1.06938	−0.534690	+	0.845048i	$0.679572\pi$
$90$	0	0
$91$	−12.9770	−1.36036
$92$	0	0
$93$	−2.61796	−0.271470
$94$	0	0
$95$	−2.81891	−0.289214
$96$	0	0
$97$	−4.59257	−0.466305	−0.233152	−	0.972440i	$-0.574904\pi$
$97$	−4.59257	−0.466305	−0.233152	+	0.972440i	$0.574904\pi$
$98$	0	0
$99$	1.00527	0.101034
$100$	0	0
$101$	6.76014	0.672659	0.336330	−	0.941744i	$-0.390814\pi$
$101$	6.76014	0.672659	0.336330	+	0.941744i	$0.390814\pi$
$102$	0	0
$103$	2.84508	0.280334	0.140167	−	0.990128i	$-0.455236\pi$
$103$	2.84508	0.280334	0.140167	+	0.990128i	$0.455236\pi$
$104$	0	0
$105$	3.46409	0.338061
$106$	0	0
$107$	0.879143	0.0849900	0.0424950	−	0.999097i	$-0.486469\pi$
$107$	0.879143	0.0849900	0.0424950	+	0.999097i	$0.486469\pi$
$108$	0	0
$109$	11.3788	1.08989	0.544945	−	0.838472i	$-0.316551\pi$
$109$	11.3788	1.08989	0.544945	+	0.838472i	$0.316551\pi$
$110$	0	0
$111$	4.41342	0.418903
$112$	0	0
$113$	−7.82243	−0.735872	−0.367936	−	0.929851i	$-0.619936\pi$
$113$	−7.82243	−0.735872	−0.367936	+	0.929851i	$0.619936\pi$
$114$	0	0
$115$	8.67569	0.809012
$116$	0	0
$117$	8.16902	0.755227
$118$	0	0
$119$	4.42760	0.405877
$120$	0	0
$121$	−10.8699	−0.988174
$122$	0	0
$123$	2.87769	0.259472
$124$	0	0
$125$	−12.0815	−1.08060
$126$	0	0
$127$	1.62529	0.144222	0.0721108	−	0.997397i	$-0.477026\pi$
$127$	1.62529	0.144222	0.0721108	+	0.997397i	$0.477026\pi$
$128$	0	0
$129$	4.35354	0.383307
$130$	0	0
$131$	0.664430	0.0580515	0.0290258	−	0.999579i	$-0.490760\pi$
$131$	0.664430	0.0580515	0.0290258	+	0.999579i	$0.490760\pi$
$132$	0	0
$133$	−7.35943	−0.638143
$134$	0	0
$135$	−4.52780	−0.389691
$136$	0	0
$137$	−13.3135	−1.13745	−0.568726	−	0.822527i	$-0.692564\pi$
$137$	−13.3135	−1.13745	−0.568726	+	0.822527i	$0.692564\pi$
$138$	0	0
$139$	10.2938	0.873105	0.436553	−	0.899679i	$-0.356199\pi$
$139$	10.2938	0.873105	0.436553	+	0.899679i	$0.356199\pi$
$140$	0	0
$141$	−0.754739	−0.0635605
$142$	0	0
$143$	1.05712	0.0884012
$144$	0	0
$145$	5.22483	0.433899
$146$	0	0
$147$	5.81449	0.479571
$148$	0	0
$149$	−5.82667	−0.477339	−0.238670	−	0.971101i	$-0.576711\pi$
$149$	−5.82667	−0.477339	−0.238670	+	0.971101i	$0.576711\pi$
$150$	0	0
$151$	18.3463	1.49300	0.746500	−	0.665385i	$-0.231733\pi$
$151$	18.3463	1.49300	0.746500	+	0.665385i	$0.231733\pi$
$152$	0	0
$153$	−2.78717	−0.225329
$154$	0	0
$155$	−9.62394	−0.773013
$156$	0	0
$157$	−12.6470	−1.00934	−0.504669	−	0.863313i	$-0.668385\pi$
$157$	−12.6470	−1.00934	−0.504669	+	0.863313i	$0.668385\pi$
$158$	0	0
$159$	3.65486	0.289849
$160$	0	0
$161$	22.6499	1.78506
$162$	0	0
$163$	−24.3672	−1.90859	−0.954294	−	0.298869i	$-0.903390\pi$
$163$	−24.3672	−1.90859	−0.954294	+	0.298869i	$0.903390\pi$
$164$	0	0
$165$	−0.282189	−0.0219684
$166$	0	0
$167$	13.4539	1.04109	0.520547	−	0.853833i	$-0.325728\pi$
$167$	13.4539	1.04109	0.520547	+	0.853833i	$0.325728\pi$
$168$	0	0
$169$	−4.40960	−0.339200
$170$	0	0
$171$	4.63275	0.354275
$172$	0	0
$173$	13.3972	1.01857	0.509285	−	0.860598i	$-0.329910\pi$
$173$	13.3972	1.01857	0.509285	+	0.860598i	$0.329910\pi$
$174$	0	0
$175$	−9.40357	−0.710843
$176$	0	0
$177$	0.461334	0.0346760
$178$	0	0
$179$	−16.7228	−1.24992	−0.624960	−	0.780656i	$-0.714885\pi$
$179$	−16.7228	−1.24992	−0.624960	+	0.780656i	$0.714885\pi$
$180$	0	0
$181$	−4.33746	−0.322401	−0.161200	−	0.986922i	$-0.551537\pi$
$181$	−4.33746	−0.322401	−0.161200	+	0.986922i	$0.551537\pi$
$182$	0	0
$183$	4.45938	0.329647
$184$	0	0
$185$	16.2243	1.19283
$186$	0	0
$187$	−0.360678	−0.0263754
$188$	0	0
$189$	−11.8209	−0.859843
$190$	0	0
$191$	−5.87382	−0.425015	−0.212507	−	0.977159i	$-0.568163\pi$
$191$	−5.87382	−0.425015	−0.212507	+	0.977159i	$0.568163\pi$
$192$	0	0
$193$	−8.22925	−0.592355	−0.296177	−	0.955133i	$-0.595712\pi$
$193$	−8.22925	−0.592355	−0.296177	+	0.955133i	$0.595712\pi$
$194$	0	0
$195$	−2.29312	−0.164214
$196$	0	0
$197$	−3.34058	−0.238006	−0.119003	−	0.992894i	$-0.537970\pi$
$197$	−3.34058	−0.238006	−0.119003	+	0.992894i	$0.537970\pi$
$198$	0	0
$199$	−11.2558	−0.797901	−0.398951	−	0.916972i	$-0.630625\pi$
$199$	−11.2558	−0.797901	−0.398951	+	0.916972i	$0.630625\pi$
$200$	0	0
$201$	−0.109510	−0.00772422
$202$	0	0
$203$	13.6406	0.957386
$204$	0	0
$205$	10.5787	0.738851
$206$	0	0
$207$	−14.2581	−0.991007
$208$	0	0
$209$	0.599508	0.0414688
$210$	0	0
$211$	−26.3351	−1.81299	−0.906493	−	0.422221i	$-0.861250\pi$
$211$	−26.3351	−1.81299	−0.906493	+	0.422221i	$0.861250\pi$
$212$	0	0
$213$	0.212382	0.0145521
$214$	0	0
$215$	16.0041	1.09147
$216$	0	0
$217$	−25.1256	−1.70563
$218$	0	0
$219$	6.63747	0.448519
$220$	0	0
$221$	−2.93094	−0.197156
$222$	0	0
$223$	8.24892	0.552389	0.276194	−	0.961102i	$-0.410927\pi$
$223$	8.24892	0.552389	0.276194	+	0.961102i	$0.410927\pi$
$224$	0	0
$225$	5.91954	0.394636
$226$	0	0
$227$	−12.4534	−0.826559	−0.413280	−	0.910604i	$-0.635617\pi$
$227$	−12.4534	−0.826559	−0.413280	+	0.910604i	$0.635617\pi$
$228$	0	0
$229$	−0.0800369	−0.00528899	−0.00264449	−	0.999997i	$-0.500842\pi$
$229$	−0.0800369	−0.00528899	−0.00264449	+	0.999997i	$0.500842\pi$
$230$	0	0
$231$	−0.736722	−0.0484727
$232$	0	0
$233$	−15.8462	−1.03812	−0.519058	−	0.854739i	$-0.673717\pi$
$233$	−15.8462	−1.03812	−0.519058	+	0.854739i	$0.673717\pi$
$234$	0	0
$235$	−2.77451	−0.180989
$236$	0	0
$237$	3.26009	0.211766
$238$	0	0
$239$	9.61530	0.621962	0.310981	−	0.950416i	$-0.399342\pi$
$239$	9.61530	0.621962	0.310981	+	0.950416i	$0.399342\pi$
$240$	0	0
$241$	−8.43556	−0.543382	−0.271691	−	0.962384i	$-0.587583\pi$
$241$	−8.43556	−0.543382	−0.271691	+	0.962384i	$0.587583\pi$
$242$	0	0
$243$	11.2987	0.724811
$244$	0	0
$245$	21.3748	1.36558
$246$	0	0
$247$	4.87172	0.309980
$248$	0	0
$249$	−1.80354	−0.114295
$250$	0	0
$251$	23.6359	1.49189	0.745943	−	0.666010i	$-0.231999\pi$
$251$	23.6359	1.49189	0.745943	+	0.666010i	$0.231999\pi$
$252$	0	0
$253$	−1.84509	−0.116000
$254$	0	0
$255$	0.782386	0.0489949
$256$	0	0
$257$	−2.33869	−0.145883	−0.0729417	−	0.997336i	$-0.523239\pi$
$257$	−2.33869	−0.145883	−0.0729417	+	0.997336i	$0.523239\pi$
$258$	0	0
$259$	42.3572	2.63195
$260$	0	0
$261$	−8.58678	−0.531508
$262$	0	0
$263$	20.8909	1.28819	0.644095	−	0.764945i	$-0.277234\pi$
$263$	20.8909	1.28819	0.644095	+	0.764945i	$0.277234\pi$
$264$	0	0
$265$	13.4357	0.825349
$266$	0	0
$267$	−4.65417	−0.284831
$268$	0	0
$269$	−16.4826	−1.00496	−0.502481	−	0.864588i	$-0.667579\pi$
$269$	−16.4826	−1.00496	−0.502481	+	0.864588i	$0.667579\pi$
$270$	0	0
$271$	26.8623	1.63177	0.815883	−	0.578217i	$-0.196251\pi$
$271$	26.8623	1.63177	0.815883	+	0.578217i	$0.196251\pi$
$272$	0	0
$273$	−5.98674	−0.362334
$274$	0	0
$275$	0.766027	0.0461932
$276$	0	0
$277$	−14.9887	−0.900582	−0.450291	−	0.892882i	$-0.648680\pi$
$277$	−14.9887	−0.900582	−0.450291	+	0.892882i	$0.648680\pi$
$278$	0	0
$279$	15.8165	0.946910
$280$	0	0
$281$	10.4137	0.621230	0.310615	−	0.950536i	$-0.399465\pi$
$281$	10.4137	0.621230	0.310615	+	0.950536i	$0.399465\pi$
$282$	0	0
$283$	−12.7292	−0.756674	−0.378337	−	0.925668i	$-0.623504\pi$
$283$	−12.7292	−0.756674	−0.378337	+	0.925668i	$0.623504\pi$
$284$	0	0
$285$	−1.30046	−0.0770325
$286$	0	0
$287$	27.6183	1.63025
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−2.11871	−0.124201
$292$	0	0
$293$	20.1760	1.17869	0.589347	−	0.807880i	$-0.299385\pi$
$293$	20.1760	1.17869	0.589347	+	0.807880i	$0.299385\pi$
$294$	0	0
$295$	1.69592	0.0987403
$296$	0	0
$297$	0.962945	0.0558757
$298$	0	0
$299$	−14.9936	−0.867100
$300$	0	0
$301$	41.7826	2.40831
$302$	0	0
$303$	3.11868	0.179164
$304$	0	0
$305$	16.3932	0.938674
$306$	0	0
$307$	−15.1821	−0.866487	−0.433243	−	0.901277i	$-0.642631\pi$
$307$	−15.1821	−0.866487	−0.433243	+	0.901277i	$0.642631\pi$
$308$	0	0
$309$	1.31253	0.0746673
$310$	0	0
$311$	22.2298	1.26053	0.630267	−	0.776378i	$-0.282945\pi$
$311$	22.2298	1.26053	0.630267	+	0.776378i	$0.282945\pi$
$312$	0	0
$313$	−10.0577	−0.568492	−0.284246	−	0.958751i	$-0.591743\pi$
$313$	−10.0577	−0.568492	−0.284246	+	0.958751i	$0.591743\pi$
$314$	0	0
$315$	−20.9285	−1.17919
$316$	0	0
$317$	−19.3239	−1.08534	−0.542669	−	0.839947i	$-0.682586\pi$
$317$	−19.3239	−1.08534	−0.542669	+	0.839947i	$0.682586\pi$
$318$	0	0
$319$	−1.11118	−0.0622144
$320$	0	0
$321$	0.405579	0.0226372
$322$	0	0
$323$	−1.66217	−0.0924856
$324$	0	0
$325$	6.22488	0.345294
$326$	0	0
$327$	5.24942	0.290294
$328$	0	0
$329$	−7.24352	−0.399348
$330$	0	0
$331$	−24.1533	−1.32759	−0.663793	−	0.747916i	$-0.731055\pi$
$331$	−24.1533	−1.32759	−0.663793	+	0.747916i	$0.731055\pi$
$332$	0	0
$333$	−26.6639	−1.46117
$334$	0	0
$335$	−0.402571	−0.0219948
$336$	0	0
$337$	−4.59825	−0.250483	−0.125241	−	0.992126i	$-0.539971\pi$
$337$	−4.59825	−0.250483	−0.125241	+	0.992126i	$0.539971\pi$
$338$	0	0
$339$	−3.60875	−0.196001
$340$	0	0
$341$	2.04676	0.110838
$342$	0	0
$343$	24.8107	1.33965
$344$	0	0
$345$	4.00239	0.215481
$346$	0	0
$347$	19.0214	1.02112	0.510562	−	0.859841i	$-0.329437\pi$
$347$	19.0214	1.02112	0.510562	+	0.859841i	$0.329437\pi$
$348$	0	0
$349$	9.49879	0.508459	0.254229	−	0.967144i	$-0.418178\pi$
$349$	9.49879	0.508459	0.254229	+	0.967144i	$0.418178\pi$
$350$	0	0
$351$	7.82507	0.417672
$352$	0	0
$353$	23.9142	1.27282	0.636412	−	0.771349i	$-0.280418\pi$
$353$	23.9142	1.27282	0.636412	+	0.771349i	$0.280418\pi$
$354$	0	0
$355$	0.780741	0.0414374
$356$	0	0
$357$	2.04260	0.108106
$358$	0	0
$359$	31.8803	1.68258	0.841288	−	0.540586i	$-0.181798\pi$
$359$	31.8803	1.68258	0.841288	+	0.540586i	$0.181798\pi$
$360$	0	0
$361$	−16.2372	−0.854589
$362$	0	0
$363$	−5.01466	−0.263201
$364$	0	0
$365$	24.4002	1.27716
$366$	0	0
$367$	−3.40884	−0.177940	−0.0889699	−	0.996034i	$-0.528358\pi$
$367$	−3.40884	−0.177940	−0.0889699	+	0.996034i	$0.528358\pi$
$368$	0	0
$369$	−17.3857	−0.905062
$370$	0	0
$371$	35.0771	1.82111
$372$	0	0
$373$	−9.85859	−0.510459	−0.255229	−	0.966881i	$-0.582151\pi$
$373$	−9.85859	−0.510459	−0.255229	+	0.966881i	$0.582151\pi$
$374$	0	0
$375$	−5.57360	−0.287820
$376$	0	0
$377$	−9.02970	−0.465053
$378$	0	0
$379$	17.4002	0.893788	0.446894	−	0.894587i	$-0.352530\pi$
$379$	17.4002	0.893788	0.446894	+	0.894587i	$0.352530\pi$
$380$	0	0
$381$	0.749803	0.0384136
$382$	0	0
$383$	22.5299	1.15123	0.575613	−	0.817723i	$-0.304764\pi$
$383$	22.5299	1.15123	0.575613	+	0.817723i	$0.304764\pi$
$384$	0	0
$385$	−2.70828	−0.138027
$386$	0	0
$387$	−26.3021	−1.33701
$388$	0	0
$389$	14.2181	0.720886	0.360443	−	0.932781i	$-0.382626\pi$
$389$	14.2181	0.720886	0.360443	+	0.932781i	$0.382626\pi$
$390$	0	0
$391$	5.11562	0.258708
$392$	0	0
$393$	0.306524	0.0154621
$394$	0	0
$395$	11.9845	0.603006
$396$	0	0
$397$	−35.8064	−1.79707	−0.898536	−	0.438900i	$-0.855368\pi$
$397$	−35.8064	−1.79707	−0.898536	+	0.438900i	$0.855368\pi$
$398$	0	0
$399$	−3.39515	−0.169970
$400$	0	0
$401$	5.88641	0.293953	0.146977	−	0.989140i	$-0.453046\pi$
$401$	5.88641	0.293953	0.146977	+	0.989140i	$0.453046\pi$
$402$	0	0
$403$	16.6324	0.828517
$404$	0	0
$405$	12.0916	0.600838
$406$	0	0
$407$	−3.45048	−0.171034
$408$	0	0
$409$	−8.00208	−0.395677	−0.197839	−	0.980235i	$-0.563392\pi$
$409$	−8.00208	−0.395677	−0.197839	+	0.980235i	$0.563392\pi$
$410$	0	0
$411$	−6.14199	−0.302962
$412$	0	0
$413$	4.42760	0.217868
$414$	0	0
$415$	−6.63003	−0.325455
$416$	0	0
$417$	4.74886	0.232553
$418$	0	0
$419$	−13.7184	−0.670189	−0.335095	−	0.942184i	$-0.608768\pi$
$419$	−13.7184	−0.670189	−0.335095	+	0.942184i	$0.608768\pi$
$420$	0	0
$421$	−33.3517	−1.62546	−0.812730	−	0.582640i	$-0.802020\pi$
$421$	−33.3517	−1.62546	−0.812730	+	0.582640i	$0.802020\pi$
$422$	0	0
$423$	4.55979	0.221705
$424$	0	0
$425$	−2.12385	−0.103022
$426$	0	0
$427$	42.7984	2.07116
$428$	0	0
$429$	0.487688	0.0235458
$430$	0	0
$431$	−38.7568	−1.86685	−0.933425	−	0.358773i	$-0.883195\pi$
$431$	−38.7568	−1.86685	−0.933425	+	0.358773i	$0.883195\pi$
$432$	0	0
$433$	−15.7278	−0.755830	−0.377915	−	0.925840i	$-0.623359\pi$
$433$	−15.7278	−0.755830	−0.377915	+	0.925840i	$0.623359\pi$
$434$	0	0
$435$	2.41039	0.115569
$436$	0	0
$437$	−8.50303	−0.406755
$438$	0	0
$439$	−0.780691	−0.0372604	−0.0186302	−	0.999826i	$-0.505931\pi$
$439$	−0.780691	−0.0372604	−0.0186302	+	0.999826i	$0.505931\pi$
$440$	0	0
$441$	−35.1285	−1.67279
$442$	0	0
$443$	11.9634	0.568397	0.284198	−	0.958765i	$-0.408273\pi$
$443$	11.9634	0.568397	0.284198	+	0.958765i	$0.408273\pi$
$444$	0	0
$445$	−17.1093	−0.811059
$446$	0	0
$447$	−2.68804	−0.127140
$448$	0	0
$449$	30.6727	1.44754	0.723768	−	0.690044i	$-0.242409\pi$
$449$	30.6727	1.44754	0.723768	+	0.690044i	$0.242409\pi$
$450$	0	0
$451$	−2.24982	−0.105940
$452$	0	0
$453$	8.46377	0.397663
$454$	0	0
$455$	−22.0080	−1.03175
$456$	0	0
$457$	−33.0145	−1.54435	−0.772176	−	0.635408i	$-0.780832\pi$
$457$	−33.0145	−1.54435	−0.772176	+	0.635408i	$0.780832\pi$
$458$	0	0
$459$	−2.66982	−0.124616
$460$	0	0
$461$	27.4100	1.27661	0.638306	−	0.769782i	$-0.279635\pi$
$461$	27.4100	1.27661	0.638306	+	0.769782i	$0.279635\pi$
$462$	0	0
$463$	22.8263	1.06083	0.530414	−	0.847739i	$-0.322036\pi$
$463$	22.8263	1.06083	0.530414	+	0.847739i	$0.322036\pi$
$464$	0	0
$465$	−4.43985	−0.205893
$466$	0	0
$467$	−29.0899	−1.34612	−0.673061	−	0.739587i	$-0.735021\pi$
$467$	−29.0899	−1.34612	−0.673061	+	0.739587i	$0.735021\pi$
$468$	0	0
$469$	−1.05101	−0.0485310
$470$	0	0
$471$	−5.83447	−0.268838
$472$	0	0
$473$	−3.40366	−0.156500
$474$	0	0
$475$	3.53021	0.161977
$476$	0	0
$477$	−22.0810	−1.01102
$478$	0	0
$479$	−35.4242	−1.61857	−0.809286	−	0.587415i	$-0.800146\pi$
$479$	−35.4242	−1.61857	−0.809286	+	0.587415i	$0.800146\pi$
$480$	0	0
$481$	−28.0392	−1.27848
$482$	0	0
$483$	10.4492	0.475454
$484$	0	0
$485$	−7.78863	−0.353663
$486$	0	0
$487$	−2.33964	−0.106019	−0.0530096	−	0.998594i	$-0.516881\pi$
$487$	−2.33964	−0.106019	−0.0530096	+	0.998594i	$0.516881\pi$
$488$	0	0
$489$	−11.2414	−0.508355
$490$	0	0
$491$	2.16343	0.0976341	0.0488170	−	0.998808i	$-0.484455\pi$
$491$	2.16343	0.0976341	0.0488170	+	0.998808i	$0.484455\pi$
$492$	0	0
$493$	3.08082	0.138753
$494$	0	0
$495$	1.70486	0.0766278
$496$	0	0
$497$	2.03831	0.0914306
$498$	0	0
$499$	−17.9461	−0.803379	−0.401689	−	0.915776i	$-0.631577\pi$
$499$	−17.9461	−0.803379	−0.401689	+	0.915776i	$0.631577\pi$
$500$	0	0
$501$	6.20674	0.277296
$502$	0	0
$503$	24.0074	1.07044	0.535219	−	0.844713i	$-0.320229\pi$
$503$	24.0074	1.07044	0.535219	+	0.844713i	$0.320229\pi$
$504$	0	0
$505$	11.4647	0.510171
$506$	0	0
$507$	−2.03430	−0.0903464
$508$	0	0
$509$	17.1190	0.758785	0.379393	−	0.925236i	$-0.376133\pi$
$509$	17.1190	0.758785	0.379393	+	0.925236i	$0.376133\pi$
$510$	0	0
$511$	63.7023	2.81803
$512$	0	0
$513$	4.43769	0.195929
$514$	0	0
$515$	4.82503	0.212616
$516$	0	0
$517$	0.590067	0.0259511
$518$	0	0
$519$	6.18058	0.271297
$520$	0	0
$521$	34.6192	1.51670	0.758348	−	0.651850i	$-0.226007\pi$
$521$	34.6192	1.51670	0.758348	+	0.651850i	$0.226007\pi$
$522$	0	0
$523$	3.39108	0.148281	0.0741407	−	0.997248i	$-0.476379\pi$
$523$	3.39108	0.148281	0.0741407	+	0.997248i	$0.476379\pi$
$524$	0	0
$525$	−4.33819	−0.189334
$526$	0	0
$527$	−5.67476	−0.247196
$528$	0	0
$529$	3.16957	0.137807
$530$	0	0
$531$	−2.78717	−0.120953
$532$	0	0
$533$	−18.2825	−0.791901
$534$	0	0
$535$	1.49096	0.0644597
$536$	0	0
$537$	−7.71480	−0.332918
$538$	0	0
$539$	−4.54586	−0.195804
$540$	0	0
$541$	−15.7098	−0.675417	−0.337708	−	0.941251i	$-0.609652\pi$
$541$	−15.7098	−0.675417	−0.337708	+	0.941251i	$0.609652\pi$
$542$	0	0
$543$	−2.00102	−0.0858718
$544$	0	0
$545$	19.2975	0.826615
$546$	0	0
$547$	39.4320	1.68599	0.842996	−	0.537920i	$-0.180790\pi$
$547$	39.4320	1.68599	0.842996	+	0.537920i	$0.180790\pi$
$548$	0	0
$549$	−26.9416	−1.14984
$550$	0	0
$551$	−5.12085	−0.218156
$552$	0	0
$553$	31.2884	1.33052
$554$	0	0
$555$	7.48480	0.317712
$556$	0	0
$557$	−46.2115	−1.95804	−0.979022	−	0.203753i	$-0.934686\pi$
$557$	−46.2115	−1.95804	−0.979022	+	0.203753i	$0.934686\pi$
$558$	0	0
$559$	−27.6588	−1.16984
$560$	0	0
$561$	−0.166393	−0.00702512
$562$	0	0
$563$	−42.3083	−1.78308	−0.891541	−	0.452940i	$-0.850375\pi$
$563$	−42.3083	−1.78308	−0.891541	+	0.452940i	$0.850375\pi$
$564$	0	0
$565$	−13.2662	−0.558114
$566$	0	0
$567$	31.5681	1.32573
$568$	0	0
$569$	−16.7948	−0.704075	−0.352037	−	0.935986i	$-0.614511\pi$
$569$	−16.7948	−0.704075	−0.352037	+	0.935986i	$0.614511\pi$
$570$	0	0
$571$	−25.1890	−1.05413	−0.527063	−	0.849826i	$-0.676707\pi$
$571$	−25.1890	−1.05413	−0.527063	+	0.849826i	$0.676707\pi$
$572$	0	0
$573$	−2.70979	−0.113203
$574$	0	0
$575$	−10.8648	−0.453094
$576$	0	0
$577$	21.7537	0.905617	0.452809	−	0.891608i	$-0.350422\pi$
$577$	21.7537	0.905617	0.452809	+	0.891608i	$0.350422\pi$
$578$	0	0
$579$	−3.79644	−0.157774
$580$	0	0
$581$	−17.3093	−0.718109
$582$	0	0
$583$	−2.85742	−0.118342
$584$	0	0
$585$	13.8540	0.572793
$586$	0	0
$587$	−13.2716	−0.547778	−0.273889	−	0.961761i	$-0.588310\pi$
$587$	−13.2716	−0.547778	−0.273889	+	0.961761i	$0.588310\pi$
$588$	0	0
$589$	9.43241	0.388656
$590$	0	0
$591$	−1.54112	−0.0633933
$592$	0	0
$593$	15.2146	0.624789	0.312395	−	0.949952i	$-0.398869\pi$
$593$	15.2146	0.624789	0.312395	+	0.949952i	$0.398869\pi$
$594$	0	0
$595$	7.50886	0.307833
$596$	0	0
$597$	−5.19267	−0.212522
$598$	0	0
$599$	31.6039	1.29130	0.645649	−	0.763634i	$-0.276587\pi$
$599$	31.6039	1.29130	0.645649	+	0.763634i	$0.276587\pi$
$600$	0	0
$601$	−14.4363	−0.588869	−0.294435	−	0.955672i	$-0.595131\pi$
$601$	−14.4363	−0.588869	−0.294435	+	0.955672i	$0.595131\pi$
$602$	0	0
$603$	0.661608	0.0269428
$604$	0	0
$605$	−18.4345	−0.749469
$606$	0	0
$607$	−35.3647	−1.43541	−0.717705	−	0.696347i	$-0.754807\pi$
$607$	−35.3647	−1.43541	−0.717705	+	0.696347i	$0.754807\pi$
$608$	0	0
$609$	6.29290	0.255001
$610$	0	0
$611$	4.79499	0.193985
$612$	0	0
$613$	8.17000	0.329983	0.164992	−	0.986295i	$-0.447240\pi$
$613$	8.17000	0.329983	0.164992	+	0.986295i	$0.447240\pi$
$614$	0	0
$615$	4.88033	0.196794
$616$	0	0
$617$	7.04428	0.283592	0.141796	−	0.989896i	$-0.454712\pi$
$617$	7.04428	0.283592	0.141796	+	0.989896i	$0.454712\pi$
$618$	0	0
$619$	−14.8377	−0.596377	−0.298188	−	0.954507i	$-0.596382\pi$
$619$	−14.8377	−0.596377	−0.298188	+	0.954507i	$0.596382\pi$
$620$	0	0
$621$	−13.6578	−0.548068
$622$	0	0
$623$	−44.6679	−1.78958
$624$	0	0
$625$	−9.86998	−0.394799
$626$	0	0
$627$	0.276574	0.0110453
$628$	0	0
$629$	9.56664	0.381447
$630$	0	0
$631$	−9.17644	−0.365308	−0.182654	−	0.983177i	$-0.558469\pi$
$631$	−9.17644	−0.365308	−0.182654	+	0.983177i	$0.558469\pi$
$632$	0	0
$633$	−12.1493	−0.482891
$634$	0	0
$635$	2.75637	0.109383
$636$	0	0
$637$	−36.9405	−1.46364
$638$	0	0
$639$	−1.28311	−0.0507592
$640$	0	0
$641$	−5.00622	−0.197734	−0.0988669	−	0.995101i	$-0.531522\pi$
$641$	−5.00622	−0.197734	−0.0988669	+	0.995101i	$0.531522\pi$
$642$	0	0
$643$	−37.1960	−1.46687	−0.733434	−	0.679760i	$-0.762084\pi$
$643$	−37.1960	−1.46687	−0.733434	+	0.679760i	$0.762084\pi$
$644$	0	0
$645$	7.38325	0.290715
$646$	0	0
$647$	−25.9526	−1.02030	−0.510151	−	0.860085i	$-0.670411\pi$
$647$	−25.9526	−1.02030	−0.510151	+	0.860085i	$0.670411\pi$
$648$	0	0
$649$	−0.360678	−0.0141578
$650$	0	0
$651$	−11.5913	−0.454298
$652$	0	0
$653$	2.70153	0.105719	0.0528594	−	0.998602i	$-0.483166\pi$
$653$	2.70153	0.105719	0.0528594	+	0.998602i	$0.483166\pi$
$654$	0	0
$655$	1.12682	0.0440285
$656$	0	0
$657$	−40.1006	−1.56447
$658$	0	0
$659$	−6.70948	−0.261364	−0.130682	−	0.991424i	$-0.541717\pi$
$659$	−6.70948	−0.261364	−0.130682	+	0.991424i	$0.541717\pi$
$660$	0	0
$661$	38.5778	1.50050	0.750251	−	0.661153i	$-0.229933\pi$
$661$	38.5778	1.50050	0.750251	+	0.661153i	$0.229933\pi$
$662$	0	0
$663$	−1.35214	−0.0525128
$664$	0	0
$665$	−12.4810	−0.483993
$666$	0	0
$667$	15.7603	0.610242
$668$	0	0
$669$	3.80551	0.147129
$670$	0	0
$671$	−3.48641	−0.134591
$672$	0	0
$673$	8.77856	0.338389	0.169194	−	0.985583i	$-0.445883\pi$
$673$	8.77856	0.338389	0.169194	+	0.985583i	$0.445883\pi$
$674$	0	0
$675$	5.67030	0.218250
$676$	0	0
$677$	−21.2819	−0.817930	−0.408965	−	0.912550i	$-0.634110\pi$
$677$	−21.2819	−0.817930	−0.408965	+	0.912550i	$0.634110\pi$
$678$	0	0
$679$	−20.3341	−0.780349
$680$	0	0
$681$	−5.74516	−0.220155
$682$	0	0
$683$	−9.44420	−0.361372	−0.180686	−	0.983541i	$-0.557832\pi$
$683$	−9.44420	−0.361372	−0.180686	+	0.983541i	$0.557832\pi$
$684$	0	0
$685$	−22.5787	−0.862688
$686$	0	0
$687$	−0.0369237	−0.00140873
$688$	0	0
$689$	−23.2200	−0.884611
$690$	0	0
$691$	40.1712	1.52819	0.764093	−	0.645106i	$-0.223187\pi$
$691$	40.1712	1.52819	0.764093	+	0.645106i	$0.223187\pi$
$692$	0	0
$693$	4.45094	0.169077
$694$	0	0
$695$	17.4574	0.662197
$696$	0	0
$697$	6.23775	0.236272
$698$	0	0
$699$	−7.31038	−0.276504
$700$	0	0
$701$	−35.6428	−1.34621	−0.673106	−	0.739546i	$-0.735040\pi$
$701$	−35.6428	−1.34621	−0.673106	+	0.739546i	$0.735040\pi$
$702$	0	0
$703$	−15.9014	−0.599732
$704$	0	0
$705$	−1.27998	−0.0482068
$706$	0	0
$707$	29.9312	1.12568
$708$	0	0
$709$	14.9968	0.563216	0.281608	−	0.959529i	$-0.409132\pi$
$709$	14.9968	0.563216	0.281608	+	0.959529i	$0.409132\pi$
$710$	0	0
$711$	−19.6960	−0.738658
$712$	0	0
$713$	−29.0299	−1.08718
$714$	0	0
$715$	1.79280	0.0670469
$716$	0	0
$717$	4.43587	0.165660
$718$	0	0
$719$	43.8518	1.63540	0.817699	−	0.575646i	$-0.195249\pi$
$719$	43.8518	1.63540	0.817699	+	0.575646i	$0.195249\pi$
$720$	0	0
$721$	12.5969	0.469132
$722$	0	0
$723$	−3.89161	−0.144731
$724$	0	0
$725$	−6.54321	−0.243009
$726$	0	0
$727$	−19.1803	−0.711357	−0.355678	−	0.934608i	$-0.615750\pi$
$727$	−19.1803	−0.711357	−0.355678	+	0.934608i	$0.615750\pi$
$728$	0	0
$729$	−16.1770	−0.599149
$730$	0	0
$731$	9.43684	0.349034
$732$	0	0
$733$	−17.6250	−0.650993	−0.325497	−	0.945543i	$-0.605531\pi$
$733$	−17.6250	−0.650993	−0.325497	+	0.945543i	$0.605531\pi$
$734$	0	0
$735$	9.86091	0.363725
$736$	0	0
$737$	0.0856163	0.00315372
$738$	0	0
$739$	−1.05440	−0.0387869	−0.0193934	−	0.999812i	$-0.506174\pi$
$739$	−1.05440	−0.0387869	−0.0193934	+	0.999812i	$0.506174\pi$
$740$	0	0
$741$	2.24749	0.0825636
$742$	0	0
$743$	−7.16638	−0.262909	−0.131454	−	0.991322i	$-0.541965\pi$
$743$	−7.16638	−0.262909	−0.131454	+	0.991322i	$0.541965\pi$
$744$	0	0
$745$	−9.88157	−0.362033
$746$	0	0
$747$	10.8962	0.398670
$748$	0	0
$749$	3.89250	0.142229
$750$	0	0
$751$	−31.4838	−1.14886	−0.574430	−	0.818553i	$-0.694776\pi$
$751$	−31.4838	−1.14886	−0.574430	+	0.818553i	$0.694776\pi$
$752$	0	0
$753$	10.9041	0.397366
$754$	0	0
$755$	31.1139	1.13235
$756$	0	0
$757$	−11.9148	−0.433052	−0.216526	−	0.976277i	$-0.569473\pi$
$757$	−11.9148	−0.433052	−0.216526	+	0.976277i	$0.569473\pi$
$758$	0	0
$759$	−0.851203	−0.0308967
$760$	0	0
$761$	46.9522	1.70202	0.851008	−	0.525153i	$-0.175992\pi$
$761$	46.9522	1.70202	0.851008	+	0.525153i	$0.175992\pi$
$762$	0	0
$763$	50.3807	1.82390
$764$	0	0
$765$	−4.72682	−0.170899
$766$	0	0
$767$	−2.93094	−0.105830
$768$	0	0
$769$	−47.3906	−1.70895	−0.854475	−	0.519493i	$-0.826121\pi$
$769$	−47.3906	−1.70895	−0.854475	+	0.519493i	$0.826121\pi$
$770$	0	0
$771$	−1.07892	−0.0388562
$772$	0	0
$773$	11.6695	0.419723	0.209862	−	0.977731i	$-0.432699\pi$
$773$	11.6695	0.419723	0.209862	+	0.977731i	$0.432699\pi$
$774$	0	0
$775$	12.0523	0.432933
$776$	0	0
$777$	19.5408	0.701024
$778$	0	0
$779$	−10.3682	−0.371479
$780$	0	0
$781$	−0.166043	−0.00594149
$782$	0	0
$783$	−8.22524	−0.293946
$784$	0	0
$785$	−21.4482	−0.765521
$786$	0	0
$787$	41.7719	1.48901	0.744503	−	0.667619i	$-0.232686\pi$
$787$	41.7719	1.48901	0.744503	+	0.667619i	$0.232686\pi$
$788$	0	0
$789$	9.63770	0.343111
$790$	0	0
$791$	−34.6346	−1.23146
$792$	0	0
$793$	−28.3312	−1.00607
$794$	0	0
$795$	6.19835	0.219833
$796$	0	0
$797$	−20.7311	−0.734335	−0.367167	−	0.930155i	$-0.619672\pi$
$797$	−20.7311	−0.734335	−0.367167	+	0.930155i	$0.619672\pi$
$798$	0	0
$799$	−1.63599	−0.0578773
$800$	0	0
$801$	28.1184	0.993515
$802$	0	0
$803$	−5.18927	−0.183125
$804$	0	0
$805$	38.4125	1.35386
$806$	0	0
$807$	−7.60398	−0.267673
$808$	0	0
$809$	12.7854	0.449510	0.224755	−	0.974415i	$-0.427842\pi$
$809$	12.7854	0.449510	0.224755	+	0.974415i	$0.427842\pi$
$810$	0	0
$811$	−36.4760	−1.28084	−0.640422	−	0.768023i	$-0.721241\pi$
$811$	−36.4760	−1.28084	−0.640422	+	0.768023i	$0.721241\pi$
$812$	0	0
$813$	12.3925	0.434623
$814$	0	0
$815$	−41.3249	−1.44755
$816$	0	0
$817$	−15.6856	−0.548771
$818$	0	0
$819$	36.1692	1.26385
$820$	0	0
$821$	32.1871	1.12334	0.561668	−	0.827362i	$-0.310160\pi$
$821$	32.1871	1.12334	0.561668	+	0.827362i	$0.310160\pi$
$822$	0	0
$823$	11.3577	0.395906	0.197953	−	0.980212i	$-0.436571\pi$
$823$	11.3577	0.395906	0.197953	+	0.980212i	$0.436571\pi$
$824$	0	0
$825$	0.353394	0.0123036
$826$	0	0
$827$	29.7249	1.03364	0.516819	−	0.856095i	$-0.327116\pi$
$827$	29.7249	1.03364	0.516819	+	0.856095i	$0.327116\pi$
$828$	0	0
$829$	−17.4682	−0.606697	−0.303349	−	0.952880i	$-0.598105\pi$
$829$	−17.4682	−0.606697	−0.303349	+	0.952880i	$0.598105\pi$
$830$	0	0
$831$	−6.91479	−0.239871
$832$	0	0
$833$	12.6036	0.436690
$834$	0	0
$835$	22.8167	0.789605
$836$	0	0
$837$	15.1506	0.523680
$838$	0	0
$839$	−41.2429	−1.42386	−0.711932	−	0.702249i	$-0.752179\pi$
$839$	−41.2429	−1.42386	−0.711932	+	0.702249i	$0.752179\pi$
$840$	0	0
$841$	−19.5085	−0.672708
$842$	0	0
$843$	4.80420	0.165465
$844$	0	0
$845$	−7.47834	−0.257263
$846$	0	0
$847$	−48.1276	−1.65368
$848$	0	0
$849$	−5.87243	−0.201541
$850$	0	0
$851$	48.9393	1.67762
$852$	0	0
$853$	37.8948	1.29749	0.648747	−	0.761004i	$-0.275293\pi$
$853$	37.8948	1.29749	0.648747	+	0.761004i	$0.275293\pi$
$854$	0	0
$855$	7.85678	0.268696
$856$	0	0
$857$	−15.8334	−0.540858	−0.270429	−	0.962740i	$-0.587165\pi$
$857$	−15.8334	−0.540858	−0.270429	+	0.962740i	$0.587165\pi$
$858$	0	0
$859$	−19.3066	−0.658731	−0.329366	−	0.944202i	$-0.606835\pi$
$859$	−19.3066	−0.658731	−0.329366	+	0.944202i	$0.606835\pi$
$860$	0	0
$861$	12.7412	0.434221
$862$	0	0
$863$	−21.5504	−0.733585	−0.366793	−	0.930303i	$-0.619544\pi$
$863$	−21.5504	−0.733585	−0.366793	+	0.930303i	$0.619544\pi$
$864$	0	0
$865$	22.7206	0.772523
$866$	0	0
$867$	0.461334	0.0156677
$868$	0	0
$869$	−2.54879	−0.0864618
$870$	0	0
$871$	0.695735	0.0235741
$872$	0	0
$873$	12.8003	0.433223
$874$	0	0
$875$	−53.4920	−1.80836
$876$	0	0
$877$	−32.4507	−1.09578	−0.547891	−	0.836550i	$-0.684569\pi$
$877$	−32.4507	−1.09578	−0.547891	+	0.836550i	$0.684569\pi$
$878$	0	0
$879$	9.30787	0.313947
$880$	0	0
$881$	19.7585	0.665681	0.332840	−	0.942983i	$-0.391993\pi$
$881$	19.7585	0.665681	0.332840	+	0.942983i	$0.391993\pi$
$882$	0	0
$883$	17.9247	0.603213	0.301606	−	0.953433i	$-0.402477\pi$
$883$	17.9247	0.603213	0.301606	+	0.953433i	$0.402477\pi$
$884$	0	0
$885$	0.782386	0.0262996
$886$	0	0
$887$	32.9987	1.10799	0.553994	−	0.832520i	$-0.313103\pi$
$887$	32.9987	1.10799	0.553994	+	0.832520i	$0.313103\pi$
$888$	0	0
$889$	7.19615	0.241351
$890$	0	0
$891$	−2.57157	−0.0861510
$892$	0	0
$893$	2.71930	0.0909979
$894$	0	0
$895$	−28.3605	−0.947988
$896$	0	0
$897$	−6.91704	−0.230953
$898$	0	0
$899$	−17.4829	−0.583088
$900$	0	0
$901$	7.92237	0.263932
$902$	0	0
$903$	19.2757	0.641456
$904$	0	0
$905$	−7.35598	−0.244521
$906$	0	0
$907$	4.58462	0.152230	0.0761149	−	0.997099i	$-0.475748\pi$
$907$	4.58462	0.152230	0.0761149	+	0.997099i	$0.475748\pi$
$908$	0	0
$909$	−18.8417	−0.624939
$910$	0	0
$911$	38.3889	1.27188	0.635941	−	0.771738i	$-0.280612\pi$
$911$	38.3889	1.27188	0.635941	+	0.771738i	$0.280612\pi$
$912$	0	0
$913$	1.41003	0.0466653
$914$	0	0
$915$	7.56276	0.250017
$916$	0	0
$917$	2.94183	0.0971478
$918$	0	0
$919$	38.1050	1.25697	0.628485	−	0.777822i	$-0.283675\pi$
$919$	38.1050	1.25697	0.628485	+	0.777822i	$0.283675\pi$
$920$	0	0
$921$	−7.00401	−0.230790
$922$	0	0
$923$	−1.34930	−0.0444127
$924$	0	0
$925$	−20.3181	−0.668056
$926$	0	0
$927$	−7.92972	−0.260446
$928$	0	0
$929$	11.7771	0.386395	0.193198	−	0.981160i	$-0.438114\pi$
$929$	11.7771	0.386395	0.193198	+	0.981160i	$0.438114\pi$
$930$	0	0
$931$	−20.9494	−0.686589
$932$	0	0
$933$	10.2553	0.335745
$934$	0	0
$935$	−0.611681	−0.0200041
$936$	0	0
$937$	−16.7672	−0.547762	−0.273881	−	0.961764i	$-0.588307\pi$
$937$	−16.7672	−0.547762	−0.273881	+	0.961764i	$0.588307\pi$
$938$	0	0
$939$	−4.63994	−0.151419
$940$	0	0
$941$	46.3865	1.51216	0.756078	−	0.654481i	$-0.227113\pi$
$941$	46.3865	1.51216	0.756078	+	0.654481i	$0.227113\pi$
$942$	0	0
$943$	31.9100	1.03913
$944$	0	0
$945$	−20.0473	−0.652139
$946$	0	0
$947$	−41.1386	−1.33683	−0.668413	−	0.743791i	$-0.733026\pi$
$947$	−41.1386	−1.33683	−0.668413	+	0.743791i	$0.733026\pi$
$948$	0	0
$949$	−42.1690	−1.36886
$950$	0	0
$951$	−8.91477	−0.289081
$952$	0	0
$953$	−2.19758	−0.0711865	−0.0355933	−	0.999366i	$-0.511332\pi$
$953$	−2.19758	−0.0711865	−0.0355933	+	0.999366i	$0.511332\pi$
$954$	0	0
$955$	−9.96153	−0.322348
$956$	0	0
$957$	−0.512627	−0.0165709
$958$	0	0
$959$	−58.9470	−1.90350
$960$	0	0
$961$	1.20286	0.0388020
$962$	0	0
$963$	−2.45032	−0.0789606
$964$	0	0
$965$	−13.9562	−0.449265
$966$	0	0
$967$	−10.8200	−0.347949	−0.173974	−	0.984750i	$-0.555661\pi$
$967$	−10.8200	−0.347949	−0.173974	+	0.984750i	$0.555661\pi$
$968$	0	0
$969$	−0.766816	−0.0246337
$970$	0	0
$971$	−5.11287	−0.164080	−0.0820398	−	0.996629i	$-0.526143\pi$
$971$	−5.11287	−0.164080	−0.0820398	+	0.996629i	$0.526143\pi$
$972$	0	0
$973$	45.5766	1.46112
$974$	0	0
$975$	2.87175	0.0919696
$976$	0	0
$977$	−40.8060	−1.30550	−0.652750	−	0.757574i	$-0.726385\pi$
$977$	−40.8060	−1.30550	−0.652750	+	0.757574i	$0.726385\pi$
$978$	0	0
$979$	3.63870	0.116293
$980$	0	0
$981$	−31.7146	−1.01257
$982$	0	0
$983$	57.3044	1.82773	0.913863	−	0.406023i	$-0.133085\pi$
$983$	57.3044	1.82773	0.913863	+	0.406023i	$0.133085\pi$
$984$	0	0
$985$	−5.66536	−0.180513
$986$	0	0
$987$	−3.34168	−0.106367
$988$	0	0
$989$	48.2753	1.53506
$990$	0	0
$991$	35.1617	1.11695	0.558473	−	0.829522i	$-0.311387\pi$
$991$	35.1617	1.11695	0.558473	+	0.829522i	$0.311387\pi$
$992$	0	0
$993$	−11.1428	−0.353605
$994$	0	0
$995$	−19.0889	−0.605159
$996$	0	0
$997$	−22.5775	−0.715037	−0.357518	−	0.933906i	$-0.616377\pi$
$997$	−22.5775	−0.715037	−0.357518	+	0.933906i	$0.616377\pi$
$998$	0	0
$999$	−25.5412	−0.808087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.11	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.11	✓	21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+0.461334 q^{3} +1.69592 q^{5} +4.42760 q^{7} -2.78717 q^{9} +O(q^{10})\)	\(q+0.461334 q^{3} +1.69592 q^{5} +4.42760 q^{7} -2.78717 q^{9} -0.360678 q^{11} -2.93094 q^{13} +0.782386 q^{15} +1.00000 q^{17} -1.66217 q^{19} +2.04260 q^{21} +5.11562 q^{23} -2.12385 q^{25} -2.66982 q^{27} +3.08082 q^{29} -5.67476 q^{31} -0.166393 q^{33} +7.50886 q^{35} +9.56664 q^{37} -1.35214 q^{39} +6.23775 q^{41} +9.43684 q^{43} -4.72682 q^{45} -1.63599 q^{47} +12.6036 q^{49} +0.461334 q^{51} +7.92237 q^{53} -0.611681 q^{55} -0.766816 q^{57} +1.00000 q^{59} +9.66627 q^{61} -12.3405 q^{63} -4.97064 q^{65} -0.237376 q^{67} +2.36001 q^{69} +0.460364 q^{71} +14.3876 q^{73} -0.979806 q^{75} -1.59694 q^{77} +7.06666 q^{79} +7.12983 q^{81} -3.90940 q^{83} +1.69592 q^{85} +1.42129 q^{87} -10.0885 q^{89} -12.9770 q^{91} -2.61796 q^{93} -2.81891 q^{95} -4.59257 q^{97} +1.00527 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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