LMFDB - Embedded newform 513.2.a.e.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(1,513)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$4.09632562369$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.46050$ of defining polynomial
Character	$\chi$	$=$	513.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.46050 q^{2} +4.05408 q^{4} +0.460505 q^{5} -0.593579 q^{7} -5.05408 q^{8} -1.13307 q^{10} -3.59358 q^{11} +0.593579 q^{13} +1.46050 q^{14} +4.32743 q^{16} -4.73385 q^{17} -1.00000 q^{19} +1.86693 q^{20} +8.84202 q^{22} +3.38151 q^{23} -4.78794 q^{25} -1.46050 q^{26} -2.40642 q^{28} +7.83482 q^{29} -1.59358 q^{31} -0.539495 q^{32} +11.6477 q^{34} -0.273346 q^{35} -4.86693 q^{37} +2.46050 q^{38} -2.32743 q^{40} +0.460505 q^{41} -9.48968 q^{43} -14.5687 q^{44} -8.32023 q^{46} -9.00000 q^{47} -6.64766 q^{49} +11.7807 q^{50} +2.40642 q^{52} -7.42840 q^{53} -1.65486 q^{55} +3.00000 q^{56} -19.2776 q^{58} -0.320233 q^{59} +7.51459 q^{61} +3.92101 q^{62} -7.32743 q^{64} +0.273346 q^{65} -13.7089 q^{67} -19.1914 q^{68} +0.672570 q^{70} -9.43560 q^{71} +5.10817 q^{73} +11.9751 q^{74} -4.05408 q^{76} +2.13307 q^{77} +13.8961 q^{79} +1.99280 q^{80} -1.13307 q^{82} -18.1373 q^{83} -2.17996 q^{85} +23.3494 q^{86} +18.1623 q^{88} +12.6228 q^{89} -0.352336 q^{91} +13.7089 q^{92} +22.1445 q^{94} -0.460505 q^{95} -8.32023 q^{97} +16.3566 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 6 q^{8} - 7 q^{10} - 8 q^{11} - q^{13} - 2 q^{14} + 3 q^{16} - 7 q^{17} - 3 q^{19} + 2 q^{20} + q^{22} - 9 q^{23} + 2 q^{25} + 2 q^{26} - 10 q^{28} + 6 q^{29}+ \cdots + 8 q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^4 - 5 * q^5 + q^7 - 6 * q^8 - 7 * q^10 - 8 * q^11 - q^13 - 2 * q^14 + 3 * q^16 - 7 * q^17 - 3 * q^19 + 2 * q^20 + q^22 - 9 * q^23 + 2 * q^25 + 2 * q^26 - 10 * q^28 + 6 * q^29 - 2 * q^31 - 8 * q^32 + 23 * q^34 - 11 * q^37 + q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 19 * q^44 - 23 * q^46 - 27 * q^47 - 8 * q^49 + 27 * q^50 + 10 * q^52 - 2 * q^53 + 15 * q^55 + 9 * q^56 - 4 * q^58 + q^59 + 7 * q^61 - q^62 - 12 * q^64 - 12 * q^67 + q^68 + 12 * q^70 - 3 * q^73 + 14 * q^74 - 3 * q^76 + 10 * q^77 + 7 * q^79 + 14 * q^80 - 7 * q^82 - 5 * q^83 - 9 * q^85 + 37 * q^86 + 27 * q^88 + 4 * q^89 - 13 * q^91 + 12 * q^92 + 9 * q^94 + 5 * q^95 - 23 * q^97 + 8 * q^98

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 $

$2$	−2.46050	−1.73984	−0.869920	−	0.493193i	$-0.835830\pi$
$2$	−2.46050	−1.73984	−0.869920	+	0.493193i	$0.835830\pi$
$3$	0	0
$3$	0	0
$4$	4.05408	2.02704
$5$	0.460505	0.205944	0.102972	−	0.994684i	$-0.467165\pi$
$5$	0.460505	0.205944	0.102972	+	0.994684i	$0.467165\pi$
$6$	0	0
$7$	−0.593579	−0.224352	−0.112176	−	0.993688i	$-0.535782\pi$
$7$	−0.593579	−0.224352	−0.112176	+	0.993688i	$0.535782\pi$
$8$	−5.05408	−1.78689
$9$	0	0
$10$	−1.13307	−0.358310
$11$	−3.59358	−1.08350	−0.541752	−	0.840538i	$-0.682239\pi$
$11$	−3.59358	−1.08350	−0.541752	+	0.840538i	$0.682239\pi$
$12$	0	0
$13$	0.593579	0.164629	0.0823146	−	0.996606i	$-0.473769\pi$
$13$	0.593579	0.164629	0.0823146	+	0.996606i	$0.473769\pi$
$14$	1.46050	0.390336
$15$	0	0
$16$	4.32743	1.08186
$17$	−4.73385	−1.14813	−0.574064	−	0.818811i	$-0.694634\pi$
$17$	−4.73385	−1.14813	−0.574064	+	0.818811i	$0.694634\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.86693	0.417457
$21$	0	0
$22$	8.84202	1.88512
$23$	3.38151	0.705095	0.352547	−	0.935794i	$-0.385316\pi$
$23$	3.38151	0.705095	0.352547	+	0.935794i	$0.385316\pi$
$24$	0	0
$25$	−4.78794	−0.957587
$26$	−1.46050	−0.286429
$27$	0	0
$28$	−2.40642	−0.454771
$29$	7.83482	1.45489	0.727445	−	0.686166i	$-0.240708\pi$
$29$	7.83482	1.45489	0.727445	+	0.686166i	$0.240708\pi$
$30$	0	0
$31$	−1.59358	−0.286215	−0.143108	−	0.989707i	$-0.545710\pi$
$31$	−1.59358	−0.286215	−0.143108	+	0.989707i	$0.545710\pi$
$32$	−0.539495	−0.0953702
$33$	0	0
$34$	11.6477	1.99756
$35$	−0.273346	−0.0462039
$36$	0	0
$37$	−4.86693	−0.800118	−0.400059	−	0.916489i	$-0.631010\pi$
$37$	−4.86693	−0.800118	−0.400059	+	0.916489i	$0.631010\pi$
$38$	2.46050	0.399147
$39$	0	0
$40$	−2.32743	−0.367999
$41$	0.460505	0.0719188	0.0359594	−	0.999353i	$-0.488551\pi$
$41$	0.460505	0.0719188	0.0359594	+	0.999353i	$0.488551\pi$
$42$	0	0
$43$	−9.48968	−1.44716	−0.723582	−	0.690239i	$-0.757505\pi$
$43$	−9.48968	−1.44716	−0.723582	+	0.690239i	$0.757505\pi$
$44$	−14.5687	−2.19631
$45$	0	0
$46$	−8.32023	−1.22675
$47$	−9.00000	−1.31278	−0.656392	−	0.754420i	$-0.727918\pi$
$47$	−9.00000	−1.31278	−0.656392	+	0.754420i	$0.727918\pi$
$48$	0	0
$49$	−6.64766	−0.949666
$50$	11.7807	1.66605
$51$	0	0
$52$	2.40642	0.333711
$53$	−7.42840	−1.02037	−0.510185	−	0.860065i	$-0.670423\pi$
$53$	−7.42840	−1.02037	−0.510185	+	0.860065i	$0.670423\pi$
$54$	0	0
$55$	−1.65486	−0.223141
$56$	3.00000	0.400892
$57$	0	0
$58$	−19.2776	−2.53128
$59$	−0.320233	−0.0416908	−0.0208454	−	0.999783i	$-0.506636\pi$
$59$	−0.320233	−0.0416908	−0.0208454	+	0.999783i	$0.506636\pi$
$60$	0	0
$61$	7.51459	0.962145	0.481072	−	0.876681i	$-0.340247\pi$
$61$	7.51459	0.962145	0.481072	+	0.876681i	$0.340247\pi$
$62$	3.92101	0.497969
$63$	0	0
$64$	−7.32743	−0.915929
$65$	0.273346	0.0339044
$66$	0	0
$67$	−13.7089	−1.67481	−0.837407	−	0.546580i	$-0.815930\pi$
$67$	−13.7089	−1.67481	−0.837407	+	0.546580i	$0.815930\pi$
$68$	−19.1914	−2.32730
$69$	0	0
$70$	0.672570	0.0803874
$71$	−9.43560	−1.11980	−0.559900	−	0.828560i	$-0.689160\pi$
$71$	−9.43560	−1.11980	−0.559900	+	0.828560i	$0.689160\pi$
$72$	0	0
$73$	5.10817	0.597866	0.298933	−	0.954274i	$-0.403369\pi$
$73$	5.10817	0.597866	0.298933	+	0.954274i	$0.403369\pi$
$74$	11.9751	1.39208
$75$	0	0
$76$	−4.05408	−0.465035
$77$	2.13307	0.243086
$78$	0	0
$79$	13.8961	1.56343	0.781717	−	0.623633i	$-0.214344\pi$
$79$	13.8961	1.56343	0.781717	+	0.623633i	$0.214344\pi$
$80$	1.99280	0.222802
$81$	0	0
$82$	−1.13307	−0.125127
$83$	−18.1373	−1.99083	−0.995416	−	0.0956377i	$-0.969511\pi$
$83$	−18.1373	−1.99083	−0.995416	+	0.0956377i	$0.969511\pi$
$84$	0	0
$85$	−2.17996	−0.236450
$86$	23.3494	2.51783
$87$	0	0
$88$	18.1623	1.93610
$89$	12.6228	1.33801	0.669005	−	0.743258i	$-0.266721\pi$
$89$	12.6228	1.33801	0.669005	+	0.743258i	$0.266721\pi$
$90$	0	0
$91$	−0.352336	−0.0369349
$92$	13.7089	1.42926
$93$	0	0
$94$	22.1445	2.28404
$95$	−0.460505	−0.0472468
$96$	0	0
$97$	−8.32023	−0.844792	−0.422396	−	0.906411i	$-0.638811\pi$
$97$	−8.32023	−0.844792	−0.422396	+	0.906411i	$0.638811\pi$
$98$	16.3566	1.65227
$99$	0	0
$100$	−19.4107	−1.94107
$101$	−12.7017	−1.26387	−0.631936	−	0.775021i	$-0.717739\pi$
$101$	−12.7017	−1.26387	−0.631936	+	0.775021i	$0.717739\pi$
$102$	0	0
$103$	2.75583	0.271540	0.135770	−	0.990740i	$-0.456649\pi$
$103$	2.75583	0.271540	0.135770	+	0.990740i	$0.456649\pi$
$104$	−3.00000	−0.294174
$105$	0	0
$106$	18.2776	1.77528
$107$	3.46050	0.334540	0.167270	−	0.985911i	$-0.446505\pi$
$107$	3.46050	0.334540	0.167270	+	0.985911i	$0.446505\pi$
$108$	0	0
$109$	12.3025	1.17837	0.589184	−	0.807999i	$-0.299449\pi$
$109$	12.3025	1.17837	0.589184	+	0.807999i	$0.299449\pi$
$110$	4.07179	0.388230
$111$	0	0
$112$	−2.56867	−0.242717
$113$	3.32023	0.312341	0.156171	−	0.987730i	$-0.450085\pi$
$113$	3.32023	0.312341	0.156171	+	0.987730i	$0.450085\pi$
$114$	0	0
$115$	1.55720	0.145210
$116$	31.7630	2.94912
$117$	0	0
$118$	0.787935	0.0725353
$119$	2.80992	0.257585
$120$	0	0
$121$	1.91381	0.173983
$122$	−18.4897	−1.67398
$123$	0	0
$124$	−6.46050	−0.580170
$125$	−4.50739	−0.403153
$126$	0	0
$127$	1.54669	0.137247	0.0686234	−	0.997643i	$-0.478139\pi$
$127$	1.54669	0.137247	0.0686234	+	0.997643i	$0.478139\pi$
$128$	19.1082	1.68894
$129$	0	0
$130$	−0.672570	−0.0589883
$131$	11.8171	1.03247	0.516233	−	0.856448i	$-0.327334\pi$
$131$	11.8171	1.03247	0.516233	+	0.856448i	$0.327334\pi$
$132$	0	0
$133$	0.593579	0.0514699
$134$	33.7309	2.91391
$135$	0	0
$136$	23.9253	2.05158
$137$	−0.435599	−0.0372157	−0.0186079	−	0.999827i	$-0.505923\pi$
$137$	−0.435599	−0.0372157	−0.0186079	+	0.999827i	$0.505923\pi$
$138$	0	0
$139$	−12.0292	−1.02030	−0.510151	−	0.860085i	$-0.670410\pi$
$139$	−12.0292	−1.02030	−0.510151	+	0.860085i	$0.670410\pi$
$140$	−1.10817	−0.0936573
$141$	0	0
$142$	23.2163	1.94827
$143$	−2.13307	−0.178377
$144$	0	0
$145$	3.60797	0.299626
$146$	−12.5687	−1.04019
$147$	0	0
$148$	−19.7309	−1.62187
$149$	13.7807	1.12896	0.564481	−	0.825446i	$-0.309076\pi$
$149$	13.7807	1.12896	0.564481	+	0.825446i	$0.309076\pi$
$150$	0	0
$151$	−4.69455	−0.382037	−0.191019	−	0.981586i	$-0.561179\pi$
$151$	−4.69455	−0.382037	−0.191019	+	0.981586i	$0.561179\pi$
$152$	5.05408	0.409940
$153$	0	0
$154$	−5.24844	−0.422931
$155$	−0.733851	−0.0589443
$156$	0	0
$157$	23.5907	1.88274	0.941370	−	0.337377i	$-0.109540\pi$
$157$	23.5907	1.88274	0.941370	+	0.337377i	$0.109540\pi$
$158$	−34.1914	−2.72012
$159$	0	0
$160$	−0.248440	−0.0196409
$161$	−2.00720	−0.158189
$162$	0	0
$163$	15.4605	1.21096	0.605480	−	0.795860i	$-0.292981\pi$
$163$	15.4605	1.21096	0.605480	+	0.795860i	$0.292981\pi$
$164$	1.86693	0.145782
$165$	0	0
$166$	44.6270	3.46373
$167$	−12.3202	−0.953368	−0.476684	−	0.879075i	$-0.658161\pi$
$167$	−12.3202	−0.953368	−0.476684	+	0.879075i	$0.658161\pi$
$168$	0	0
$169$	−12.6477	−0.972897
$170$	5.36381	0.411385
$171$	0	0
$172$	−38.4720	−2.93346
$173$	−3.89610	−0.296215	−0.148108	−	0.988971i	$-0.547318\pi$
$173$	−3.89610	−0.296215	−0.148108	+	0.988971i	$0.547318\pi$
$174$	0	0
$175$	2.84202	0.214836
$176$	−15.5510	−1.17220
$177$	0	0
$178$	−31.0584	−2.32792
$179$	−13.1694	−0.984331	−0.492165	−	0.870502i	$-0.663794\pi$
$179$	−13.1694	−0.984331	−0.492165	+	0.870502i	$0.663794\pi$
$180$	0	0
$181$	−6.16945	−0.458572	−0.229286	−	0.973359i	$-0.573639\pi$
$181$	−6.16945	−0.458572	−0.229286	+	0.973359i	$0.573639\pi$
$182$	0.866926	0.0642608
$183$	0	0
$184$	−17.0905	−1.25993
$185$	−2.24124	−0.164779
$186$	0	0
$187$	17.0115	1.24400
$188$	−36.4868	−2.66107
$189$	0	0
$190$	1.13307	0.0822019
$191$	−13.5438	−0.979993	−0.489996	−	0.871725i	$-0.663002\pi$
$191$	−13.5438	−0.979993	−0.489996	+	0.871725i	$0.663002\pi$
$192$	0	0
$193$	−3.78074	−0.272143	−0.136072	−	0.990699i	$-0.543448\pi$
$193$	−3.78074	−0.272143	−0.136072	+	0.990699i	$0.543448\pi$
$194$	20.4720	1.46980
$195$	0	0
$196$	−26.9502	−1.92501
$197$	12.8492	0.915469	0.457734	−	0.889089i	$-0.348661\pi$
$197$	12.8492	0.915469	0.457734	+	0.889089i	$0.348661\pi$
$198$	0	0
$199$	14.8640	1.05368	0.526841	−	0.849964i	$-0.323377\pi$
$199$	14.8640	1.05368	0.526841	+	0.849964i	$0.323377\pi$
$200$	24.1986	1.71110
$201$	0	0
$202$	31.2527	2.19893
$203$	−4.65059	−0.326407
$204$	0	0
$205$	0.212065	0.0148112
$206$	−6.78074	−0.472436
$207$	0	0
$208$	2.56867	0.178105
$209$	3.59358	0.248573
$210$	0	0
$211$	27.1124	1.86650	0.933249	−	0.359231i	$-0.116961\pi$
$211$	27.1124	1.86650	0.933249	+	0.359231i	$0.116961\pi$
$212$	−30.1154	−2.06833
$213$	0	0
$214$	−8.51459	−0.582045
$215$	−4.37005	−0.298035
$216$	0	0
$217$	0.945916	0.0642129
$218$	−30.2704	−2.05017
$219$	0	0
$220$	−6.70895	−0.452317
$221$	−2.80992	−0.189015
$222$	0	0
$223$	−15.3494	−1.02787	−0.513936	−	0.857828i	$-0.671813\pi$
$223$	−15.3494	−1.02787	−0.513936	+	0.857828i	$0.671813\pi$
$224$	0.320233	0.0213965
$225$	0	0
$226$	−8.16945	−0.543424
$227$	7.27042	0.482555	0.241277	−	0.970456i	$-0.422434\pi$
$227$	7.27042	0.482555	0.241277	+	0.970456i	$0.422434\pi$
$228$	0	0
$229$	−19.8640	−1.31265	−0.656325	−	0.754478i	$-0.727890\pi$
$229$	−19.8640	−1.31265	−0.656325	+	0.754478i	$0.727890\pi$
$230$	−3.83151	−0.252642
$231$	0	0
$232$	−39.5979	−2.59973
$233$	4.73093	0.309933	0.154967	−	0.987920i	$-0.450473\pi$
$233$	4.73093	0.309933	0.154967	+	0.987920i	$0.450473\pi$
$234$	0	0
$235$	−4.14454	−0.270360
$236$	−1.29825	−0.0845090
$237$	0	0
$238$	−6.91381	−0.448156
$239$	−3.14747	−0.203593	−0.101796	−	0.994805i	$-0.532459\pi$
$239$	−3.14747	−0.203593	−0.101796	+	0.994805i	$0.532459\pi$
$240$	0	0
$241$	−9.35234	−0.602437	−0.301218	−	0.953555i	$-0.597393\pi$
$241$	−9.35234	−0.602437	−0.301218	+	0.953555i	$0.597393\pi$
$242$	−4.70895	−0.302702
$243$	0	0
$244$	30.4648	1.95031
$245$	−3.06128	−0.195578
$246$	0	0
$247$	−0.593579	−0.0377685
$248$	8.05408	0.511435
$249$	0	0
$250$	11.0905	0.701422
$251$	2.64339	0.166849	0.0834247	−	0.996514i	$-0.473414\pi$
$251$	2.64339	0.166849	0.0834247	+	0.996514i	$0.473414\pi$
$252$	0	0
$253$	−12.1517	−0.763973
$254$	−3.80564	−0.238787
$255$	0	0
$256$	−32.3609	−2.02256
$257$	1.14747	0.0715771	0.0357886	−	0.999359i	$-0.488606\pi$
$257$	1.14747	0.0715771	0.0357886	+	0.999359i	$0.488606\pi$
$258$	0	0
$259$	2.88891	0.179508
$260$	1.10817	0.0687257
$261$	0	0
$262$	−29.0761	−1.79633
$263$	−11.0072	−0.678733	−0.339366	−	0.940654i	$-0.610213\pi$
$263$	−11.0072	−0.678733	−0.339366	+	0.940654i	$0.610213\pi$
$264$	0	0
$265$	−3.42082	−0.210139
$266$	−1.46050	−0.0895493
$267$	0	0
$268$	−55.5772	−3.39492
$269$	19.3858	1.18197	0.590986	−	0.806682i	$-0.298739\pi$
$269$	19.3858	1.18197	0.590986	+	0.806682i	$0.298739\pi$
$270$	0	0
$271$	19.4179	1.17955	0.589776	−	0.807567i	$-0.299216\pi$
$271$	19.4179	1.17955	0.589776	+	0.807567i	$0.299216\pi$
$272$	−20.4854	−1.24211
$273$	0	0
$274$	1.07179	0.0647494
$275$	17.2058	1.03755
$276$	0	0
$277$	28.4868	1.71160	0.855802	−	0.517304i	$-0.173064\pi$
$277$	28.4868	1.71160	0.855802	+	0.517304i	$0.173064\pi$
$278$	29.5979	1.77516
$279$	0	0
$280$	1.38151	0.0825613
$281$	19.5467	1.16606	0.583029	−	0.812451i	$-0.301867\pi$
$281$	19.5467	1.16606	0.583029	+	0.812451i	$0.301867\pi$
$282$	0	0
$283$	−21.5759	−1.28255	−0.641276	−	0.767310i	$-0.721595\pi$
$283$	−21.5759	−1.28255	−0.641276	+	0.767310i	$0.721595\pi$
$284$	−38.2527	−2.26988
$285$	0	0
$286$	5.24844	0.310347
$287$	−0.273346	−0.0161351
$288$	0	0
$289$	5.40935	0.318197
$290$	−8.87744	−0.521301
$291$	0	0
$292$	20.7089	1.21190
$293$	22.6008	1.32035	0.660176	−	0.751111i	$-0.270482\pi$
$293$	22.6008	1.32035	0.660176	+	0.751111i	$0.270482\pi$
$294$	0	0
$295$	−0.147469	−0.00858597
$296$	24.5979	1.42972
$297$	0	0
$298$	−33.9076	−1.96421
$299$	2.00720	0.116079
$300$	0	0
$301$	5.63288	0.324674
$302$	11.5510	0.664683
$303$	0	0
$304$	−4.32743	−0.248195
$305$	3.46050	0.198148
$306$	0	0
$307$	20.2016	1.15296	0.576482	−	0.817110i	$-0.304425\pi$
$307$	20.2016	1.15296	0.576482	+	0.817110i	$0.304425\pi$
$308$	8.64766	0.492746
$309$	0	0
$310$	1.80564	0.102554
$311$	7.02198	0.398180	0.199090	−	0.979981i	$-0.436201\pi$
$311$	7.02198	0.398180	0.199090	+	0.979981i	$0.436201\pi$
$312$	0	0
$313$	26.7922	1.51439	0.757193	−	0.653192i	$-0.226570\pi$
$313$	26.7922	1.51439	0.757193	+	0.653192i	$0.226570\pi$
$314$	−58.0449	−3.27566
$315$	0	0
$316$	56.3360	3.16915
$317$	−15.3887	−0.864316	−0.432158	−	0.901798i	$-0.642248\pi$
$317$	−15.3887	−0.864316	−0.432158	+	0.901798i	$0.642248\pi$
$318$	0	0
$319$	−28.1551	−1.57638
$320$	−3.37432	−0.188630
$321$	0	0
$322$	4.93872	0.275224
$323$	4.73385	0.263399
$324$	0	0
$325$	−2.84202	−0.157647
$326$	−38.0406	−2.10688
$327$	0	0
$328$	−2.32743	−0.128511
$329$	5.34221	0.294526
$330$	0	0
$331$	11.0862	0.609352	0.304676	−	0.952456i	$-0.401452\pi$
$331$	11.0862	0.609352	0.304676	+	0.952456i	$0.401452\pi$
$332$	−73.5303	−4.03550
$333$	0	0
$334$	30.3140	1.65871
$335$	−6.31304	−0.344918
$336$	0	0
$337$	35.5552	1.93682	0.968409	−	0.249369i	$-0.0802232\pi$
$337$	35.5552	1.93682	0.968409	+	0.249369i	$0.0802232\pi$
$338$	31.1196	1.69269
$339$	0	0
$340$	−8.83775	−0.479294
$341$	5.72665	0.310116
$342$	0	0
$343$	8.10097	0.437411
$344$	47.9617	2.58592
$345$	0	0
$346$	9.58638	0.515367
$347$	−35.3317	−1.89671	−0.948353	−	0.317218i	$-0.897251\pi$
$347$	−35.3317	−1.89671	−0.948353	+	0.317218i	$0.897251\pi$
$348$	0	0
$349$	−30.8961	−1.65383	−0.826915	−	0.562327i	$-0.809906\pi$
$349$	−30.8961	−1.65383	−0.826915	+	0.562327i	$0.809906\pi$
$350$	−6.99280	−0.373781
$351$	0	0
$352$	1.93872	0.103334
$353$	−5.82004	−0.309769	−0.154885	−	0.987933i	$-0.549501\pi$
$353$	−5.82004	−0.309769	−0.154885	+	0.987933i	$0.549501\pi$
$354$	0	0
$355$	−4.34514	−0.230616
$356$	51.1737	2.71220
$357$	0	0
$358$	32.4035	1.71258
$359$	−30.8784	−1.62970	−0.814850	−	0.579672i	$-0.803181\pi$
$359$	−30.8784	−1.62970	−0.814850	+	0.579672i	$0.803181\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	15.1800	0.797841
$363$	0	0
$364$	−1.42840	−0.0748686
$365$	2.35234	0.123127
$366$	0	0
$367$	0.464777	0.0242612	0.0121306	−	0.999926i	$-0.496139\pi$
$367$	0.464777	0.0242612	0.0121306	+	0.999926i	$0.496139\pi$
$368$	14.6333	0.762812
$369$	0	0
$370$	5.51459	0.286690
$371$	4.40935	0.228922
$372$	0	0
$373$	−21.4428	−1.11027	−0.555133	−	0.831762i	$-0.687333\pi$
$373$	−21.4428	−1.11027	−0.555133	+	0.831762i	$0.687333\pi$
$374$	−41.8568	−2.16436
$375$	0	0
$376$	45.4868	2.34580
$377$	4.65059	0.239518
$378$	0	0
$379$	−12.7558	−0.655223	−0.327612	−	0.944813i	$-0.606244\pi$
$379$	−12.7558	−0.655223	−0.327612	+	0.944813i	$0.606244\pi$
$380$	−1.86693	−0.0957713
$381$	0	0
$382$	33.3245	1.70503
$383$	−12.8128	−0.654706	−0.327353	−	0.944902i	$-0.606157\pi$
$383$	−12.8128	−0.654706	−0.327353	+	0.944902i	$0.606157\pi$
$384$	0	0
$385$	0.982291	0.0500622
$386$	9.30252	0.473486
$387$	0	0
$388$	−33.7309	−1.71243
$389$	32.1268	1.62890	0.814448	−	0.580237i	$-0.197040\pi$
$389$	32.1268	1.62890	0.814448	+	0.580237i	$0.197040\pi$
$390$	0	0
$391$	−16.0076	−0.809538
$392$	33.5979	1.69695
$393$	0	0
$394$	−31.6156	−1.59277
$395$	6.39922	0.321980
$396$	0	0
$397$	26.7774	1.34392	0.671960	−	0.740587i	$-0.265453\pi$
$397$	26.7774	1.34392	0.671960	+	0.740587i	$0.265453\pi$
$398$	−36.5729	−1.83324
$399$	0	0
$400$	−20.7195	−1.03597
$401$	11.3772	0.568152	0.284076	−	0.958802i	$-0.408313\pi$
$401$	11.3772	0.568152	0.284076	+	0.958802i	$0.408313\pi$
$402$	0	0
$403$	−0.945916	−0.0471194
$404$	−51.4940	−2.56192
$405$	0	0
$406$	11.4428	0.567896
$407$	17.4897	0.866931
$408$	0	0
$409$	−11.3609	−0.561759	−0.280880	−	0.959743i	$-0.590626\pi$
$409$	−11.3609	−0.561759	−0.280880	+	0.959743i	$0.590626\pi$
$410$	−0.521786	−0.0257692
$411$	0	0
$412$	11.1724	0.550423
$413$	0.190084	0.00935341
$414$	0	0
$415$	−8.35234	−0.410000
$416$	−0.320233	−0.0157007
$417$	0	0
$418$	−8.84202	−0.432477
$419$	−1.61556	−0.0789253	−0.0394626	−	0.999221i	$-0.512565\pi$
$419$	−1.61556	−0.0789253	−0.0394626	+	0.999221i	$0.512565\pi$
$420$	0	0
$421$	−10.1111	−0.492785	−0.246392	−	0.969170i	$-0.579245\pi$
$421$	−10.1111	−0.492785	−0.246392	+	0.969170i	$0.579245\pi$
$422$	−66.7103	−3.24741
$423$	0	0
$424$	37.5438	1.82329
$425$	22.6654	1.09943
$426$	0	0
$427$	−4.46050	−0.215859
$428$	14.0292	0.678126
$429$	0	0
$430$	10.7525	0.518532
$431$	−17.4212	−0.839150	−0.419575	−	0.907721i	$-0.637821\pi$
$431$	−17.4212	−0.839150	−0.419575	+	0.907721i	$0.637821\pi$
$432$	0	0
$433$	26.3173	1.26473	0.632365	−	0.774671i	$-0.282084\pi$
$433$	26.3173	1.26473	0.632365	+	0.774671i	$0.282084\pi$
$434$	−2.32743	−0.111720
$435$	0	0
$436$	49.8755	2.38860
$437$	−3.38151	−0.161760
$438$	0	0
$439$	6.29533	0.300460	0.150230	−	0.988651i	$-0.451999\pi$
$439$	6.29533	0.300460	0.150230	+	0.988651i	$0.451999\pi$
$440$	8.36381	0.398729
$441$	0	0
$442$	6.91381	0.328857
$443$	34.1272	1.62143	0.810717	−	0.585439i	$-0.199078\pi$
$443$	34.1272	1.62143	0.810717	+	0.585439i	$0.199078\pi$
$444$	0	0
$445$	5.81284	0.275555
$446$	37.7673	1.78833
$447$	0	0
$448$	4.34941	0.205490
$449$	−21.3360	−1.00691	−0.503453	−	0.864022i	$-0.667937\pi$
$449$	−21.3360	−1.00691	−0.503453	+	0.864022i	$0.667937\pi$
$450$	0	0
$451$	−1.65486	−0.0779244
$452$	13.4605	0.633129
$453$	0	0
$454$	−17.8889	−0.839568
$455$	−0.162253	−0.00760652
$456$	0	0
$457$	−12.0685	−0.564540	−0.282270	−	0.959335i	$-0.591087\pi$
$457$	−12.0685	−0.564540	−0.282270	+	0.959335i	$0.591087\pi$
$458$	48.8755	2.28380
$459$	0	0
$460$	6.31304	0.294347
$461$	−6.67257	−0.310773	−0.155386	−	0.987854i	$-0.549662\pi$
$461$	−6.67257	−0.310773	−0.155386	+	0.987854i	$0.549662\pi$
$462$	0	0
$463$	−27.3638	−1.27170	−0.635852	−	0.771811i	$-0.719351\pi$
$463$	−27.3638	−1.27170	−0.635852	+	0.771811i	$0.719351\pi$
$464$	33.9046	1.57398
$465$	0	0
$466$	−11.6405	−0.539234
$467$	35.4078	1.63848	0.819238	−	0.573454i	$-0.194397\pi$
$467$	35.4078	1.63848	0.819238	+	0.573454i	$0.194397\pi$
$468$	0	0
$469$	8.13735	0.375748
$470$	10.1977	0.470383
$471$	0	0
$472$	1.61849	0.0744968
$473$	34.1019	1.56801
$474$	0	0
$475$	4.78794	0.219686
$476$	11.3916	0.522135
$477$	0	0
$478$	7.74436	0.354219
$479$	6.28054	0.286965	0.143483	−	0.989653i	$-0.454170\pi$
$479$	6.28054	0.286965	0.143483	+	0.989653i	$0.454170\pi$
$480$	0	0
$481$	−2.88891	−0.131723
$482$	23.0115	1.04814
$483$	0	0
$484$	7.75876	0.352671
$485$	−3.83151	−0.173980
$486$	0	0
$487$	−23.8535	−1.08090	−0.540452	−	0.841375i	$-0.681747\pi$
$487$	−23.8535	−1.08090	−0.540452	+	0.841375i	$0.681747\pi$
$488$	−37.9794	−1.71925
$489$	0	0
$490$	7.53230	0.340275
$491$	11.6257	0.524660	0.262330	−	0.964978i	$-0.415509\pi$
$491$	11.6257	0.524660	0.262330	+	0.964978i	$0.415509\pi$
$492$	0	0
$493$	−37.0889	−1.67040
$494$	1.46050	0.0657112
$495$	0	0
$496$	−6.89610	−0.309644
$497$	5.60078	0.251229
$498$	0	0
$499$	4.67684	0.209364	0.104682	−	0.994506i	$-0.466617\pi$
$499$	4.67684	0.209364	0.104682	+	0.994506i	$0.466617\pi$
$500$	−18.2733	−0.817209
$501$	0	0
$502$	−6.50408	−0.290291
$503$	−24.5041	−1.09258	−0.546291	−	0.837595i	$-0.683961\pi$
$503$	−24.5041	−1.09258	−0.546291	+	0.837595i	$0.683961\pi$
$504$	0	0
$505$	−5.84922	−0.260287
$506$	29.8994	1.32919
$507$	0	0
$508$	6.27042	0.278205
$509$	−18.3710	−0.814280	−0.407140	−	0.913366i	$-0.633474\pi$
$509$	−18.3710	−0.814280	−0.407140	+	0.913366i	$0.633474\pi$
$510$	0	0
$511$	−3.03210	−0.134132
$512$	41.4078	1.82998
$513$	0	0
$514$	−2.82335	−0.124533
$515$	1.26907	0.0559221
$516$	0	0
$517$	32.3422	1.42241
$518$	−7.10817	−0.312315
$519$	0	0
$520$	−1.38151	−0.0605834
$521$	18.3743	0.804993	0.402497	−	0.915421i	$-0.368142\pi$
$521$	18.3743	0.804993	0.402497	+	0.915421i	$0.368142\pi$
$522$	0	0
$523$	−40.7601	−1.78231	−0.891157	−	0.453694i	$-0.850106\pi$
$523$	−40.7601	−1.78231	−0.891157	+	0.453694i	$0.850106\pi$
$524$	47.9076	2.09285
$525$	0	0
$526$	27.0833	1.18089
$527$	7.54377	0.328612
$528$	0	0
$529$	−11.5654	−0.502842
$530$	8.41693	0.365608
$531$	0	0
$532$	2.40642	0.104332
$533$	0.273346	0.0118399
$534$	0	0
$535$	1.59358	0.0688964
$536$	69.2862	2.99271
$537$	0	0
$538$	−47.6988	−2.05644
$539$	23.8889	1.02897
$540$	0	0
$541$	−19.2091	−0.825865	−0.412933	−	0.910762i	$-0.635496\pi$
$541$	−19.2091	−0.825865	−0.412933	+	0.910762i	$0.635496\pi$
$542$	−47.7778	−2.05223
$543$	0	0
$544$	2.55389	0.109497
$545$	5.66537	0.242678
$546$	0	0
$547$	1.93113	0.0825692	0.0412846	−	0.999147i	$-0.486855\pi$
$547$	1.93113	0.0825692	0.0412846	+	0.999147i	$0.486855\pi$
$548$	−1.76595	−0.0754378
$549$	0	0
$550$	−42.3350	−1.80517
$551$	−7.83482	−0.333775
$552$	0	0
$553$	−8.24844	−0.350759
$554$	−70.0918	−2.97792
$555$	0	0
$556$	−48.7673	−2.06819
$557$	37.0013	1.56780	0.783899	−	0.620889i	$-0.213228\pi$
$557$	37.0013	1.56780	0.783899	+	0.620889i	$0.213228\pi$
$558$	0	0
$559$	−5.63288	−0.238245
$560$	−1.18289	−0.0499861
$561$	0	0
$562$	−48.0947	−2.02875
$563$	−35.7496	−1.50667	−0.753333	−	0.657639i	$-0.771555\pi$
$563$	−35.7496	−1.50667	−0.753333	+	0.657639i	$0.771555\pi$
$564$	0	0
$565$	1.52898	0.0643248
$566$	53.0875	2.23144
$567$	0	0
$568$	47.6883	2.00096
$569$	−5.88891	−0.246876	−0.123438	−	0.992352i	$-0.539392\pi$
$569$	−5.88891	−0.246876	−0.123438	+	0.992352i	$0.539392\pi$
$570$	0	0
$571$	−7.90330	−0.330743	−0.165371	−	0.986231i	$-0.552882\pi$
$571$	−7.90330	−0.330743	−0.165371	+	0.986231i	$0.552882\pi$
$572$	−8.64766	−0.361577
$573$	0	0
$574$	0.672570	0.0280725
$575$	−16.1905	−0.675189
$576$	0	0
$577$	−20.4897	−0.852997	−0.426498	−	0.904488i	$-0.640253\pi$
$577$	−20.4897	−0.852997	−0.426498	+	0.904488i	$0.640253\pi$
$578$	−13.3097	−0.553611
$579$	0	0
$580$	14.6270	0.607354
$581$	10.7660	0.446647
$582$	0	0
$583$	26.6946	1.10558
$584$	−25.8171	−1.06832
$585$	0	0
$586$	−55.6093	−2.29720
$587$	−32.1623	−1.32748	−0.663739	−	0.747964i	$-0.731031\pi$
$587$	−32.1623	−1.32748	−0.663739	+	0.747964i	$0.731031\pi$
$588$	0	0
$589$	1.59358	0.0656623
$590$	0.362848	0.0149382
$591$	0	0
$592$	−21.0613	−0.865613
$593$	−13.1082	−0.538288	−0.269144	−	0.963100i	$-0.586741\pi$
$593$	−13.1082	−0.538288	−0.269144	+	0.963100i	$0.586741\pi$
$594$	0	0
$595$	1.29398	0.0530480
$596$	55.8683	2.28845
$597$	0	0
$598$	−4.93872	−0.201959
$599$	−4.66964	−0.190797	−0.0953983	−	0.995439i	$-0.530412\pi$
$599$	−4.66964	−0.190797	−0.0953983	+	0.995439i	$0.530412\pi$
$600$	0	0
$601$	−41.8391	−1.70665	−0.853326	−	0.521378i	$-0.825418\pi$
$601$	−41.8391	−1.70665	−0.853326	+	0.521378i	$0.825418\pi$
$602$	−13.8597	−0.564880
$603$	0	0
$604$	−19.0321	−0.774405
$605$	0.881320	0.0358308
$606$	0	0
$607$	8.66245	0.351598	0.175799	−	0.984426i	$-0.443749\pi$
$607$	8.66245	0.351598	0.175799	+	0.984426i	$0.443749\pi$
$608$	0.539495	0.0218794
$609$	0	0
$610$	−8.51459	−0.344746
$611$	−5.34221	−0.216123
$612$	0	0
$613$	−28.1082	−1.13528	−0.567639	−	0.823277i	$-0.692143\pi$
$613$	−28.1082	−1.13528	−0.567639	+	0.823277i	$0.692143\pi$
$614$	−49.7060	−2.00597
$615$	0	0
$616$	−10.7807	−0.434368
$617$	−16.0072	−0.644425	−0.322213	−	0.946667i	$-0.604427\pi$
$617$	−16.0072	−0.644425	−0.322213	+	0.946667i	$0.604427\pi$
$618$	0	0
$619$	−1.00427	−0.0403651	−0.0201826	−	0.999796i	$-0.506425\pi$
$619$	−1.00427	−0.0403651	−0.0201826	+	0.999796i	$0.506425\pi$
$620$	−2.97509	−0.119483
$621$	0	0
$622$	−17.2776	−0.692769
$623$	−7.49261	−0.300185
$624$	0	0
$625$	21.8640	0.874560
$626$	−65.9224	−2.63479
$627$	0	0
$628$	95.6385	3.81639
$629$	23.0393	0.918637
$630$	0	0
$631$	46.0085	1.83157	0.915786	−	0.401667i	$-0.131569\pi$
$631$	46.0085	1.83157	0.915786	+	0.401667i	$0.131569\pi$
$632$	−70.2321	−2.79368
$633$	0	0
$634$	37.8640	1.50377
$635$	0.712259	0.0282652
$636$	0	0
$637$	−3.94592	−0.156343
$638$	69.2757	2.74265
$639$	0	0
$640$	8.79940	0.347827
$641$	10.6300	0.419858	0.209929	−	0.977717i	$-0.432677\pi$
$641$	10.6300	0.419858	0.209929	+	0.977717i	$0.432677\pi$
$642$	0	0
$643$	−30.0541	−1.18522	−0.592609	−	0.805490i	$-0.701902\pi$
$643$	−30.0541	−1.18522	−0.592609	+	0.805490i	$0.701902\pi$
$644$	−8.13735	−0.320656
$645$	0	0
$646$	−11.6477	−0.458271
$647$	−12.7984	−0.503159	−0.251579	−	0.967837i	$-0.580950\pi$
$647$	−12.7984	−0.503159	−0.251579	+	0.967837i	$0.580950\pi$
$648$	0	0
$649$	1.15078	0.0451722
$650$	6.99280	0.274280
$651$	0	0
$652$	62.6782	2.45467
$653$	−12.3494	−0.483270	−0.241635	−	0.970367i	$-0.577684\pi$
$653$	−12.3494	−0.483270	−0.241635	+	0.970367i	$0.577684\pi$
$654$	0	0
$655$	5.44184	0.212630
$656$	1.99280	0.0778059
$657$	0	0
$658$	−13.1445	−0.512428
$659$	16.0115	0.623718	0.311859	−	0.950128i	$-0.399048\pi$
$659$	16.0115	0.623718	0.311859	+	0.950128i	$0.399048\pi$
$660$	0	0
$661$	−10.6549	−0.414426	−0.207213	−	0.978296i	$-0.566439\pi$
$661$	−10.6549	−0.414426	−0.207213	+	0.978296i	$0.566439\pi$
$662$	−27.2776	−1.06017
$663$	0	0
$664$	91.6677	3.55740
$665$	0.273346	0.0105999
$666$	0	0
$667$	26.4936	1.02583
$668$	−49.9473	−1.93252
$669$	0	0
$670$	15.5333	0.600102
$671$	−27.0043	−1.04249
$672$	0	0
$673$	−7.33502	−0.282744	−0.141372	−	0.989957i	$-0.545151\pi$
$673$	−7.33502	−0.282744	−0.141372	+	0.989957i	$0.545151\pi$
$674$	−87.4838	−3.36975
$675$	0	0
$676$	−51.2747	−1.97210
$677$	35.0698	1.34784	0.673921	−	0.738803i	$-0.264609\pi$
$677$	35.0698	1.34784	0.673921	+	0.738803i	$0.264609\pi$
$678$	0	0
$679$	4.93872	0.189531
$680$	11.0177	0.422510
$681$	0	0
$682$	−14.0905	−0.539552
$683$	26.5113	1.01443	0.507213	−	0.861821i	$-0.330676\pi$
$683$	26.5113	1.01443	0.507213	+	0.861821i	$0.330676\pi$
$684$	0	0
$685$	−0.200595	−0.00766436
$686$	−19.9325	−0.761026
$687$	0	0
$688$	−41.0659	−1.56562
$689$	−4.40935	−0.167983
$690$	0	0
$691$	18.6696	0.710227	0.355113	−	0.934823i	$-0.384442\pi$
$691$	18.6696	0.710227	0.355113	+	0.934823i	$0.384442\pi$
$692$	−15.7951	−0.600441
$693$	0	0
$694$	86.9338	3.29996
$695$	−5.53950	−0.210125
$696$	0	0
$697$	−2.17996	−0.0825719
$698$	76.0200	2.87740
$699$	0	0
$700$	11.5218	0.435483
$701$	−13.7191	−0.518162	−0.259081	−	0.965856i	$-0.583420\pi$
$701$	−13.7191	−0.518162	−0.259081	+	0.965856i	$0.583420\pi$
$702$	0	0
$703$	4.86693	0.183560
$704$	26.3317	0.992413
$705$	0	0
$706$	14.3202	0.538949
$707$	7.53950	0.283552
$708$	0	0
$709$	−28.7410	−1.07939	−0.539696	−	0.841860i	$-0.681461\pi$
$709$	−28.7410	−1.07939	−0.539696	+	0.841860i	$0.681461\pi$
$710$	10.6912	0.401235
$711$	0	0
$712$	−63.7965	−2.39087
$713$	−5.38871	−0.201809
$714$	0	0
$715$	−0.982291	−0.0367356
$716$	−53.3901	−1.99528
$717$	0	0
$718$	75.9764	2.83542
$719$	−17.3025	−0.645275	−0.322638	−	0.946523i	$-0.604570\pi$
$719$	−17.3025	−0.645275	−0.322638	+	0.946523i	$0.604570\pi$
$720$	0	0
$721$	−1.63580	−0.0609206
$722$	−2.46050	−0.0915705
$723$	0	0
$724$	−25.0115	−0.929544
$725$	−37.5126	−1.39318
$726$	0	0
$727$	8.57880	0.318170	0.159085	−	0.987265i	$-0.449146\pi$
$727$	8.57880	0.318170	0.159085	+	0.987265i	$0.449146\pi$
$728$	1.78074	0.0659985
$729$	0	0
$730$	−5.78794	−0.214221
$731$	44.9227	1.66153
$732$	0	0
$733$	24.3274	0.898554	0.449277	−	0.893392i	$-0.351682\pi$
$733$	24.3274	0.898554	0.449277	+	0.893392i	$0.351682\pi$
$734$	−1.14359	−0.0422105
$735$	0	0
$736$	−1.82431	−0.0672450
$737$	49.2642	1.81467
$738$	0	0
$739$	−30.7922	−1.13271	−0.566355	−	0.824162i	$-0.691647\pi$
$739$	−30.7922	−1.13271	−0.566355	+	0.824162i	$0.691647\pi$
$740$	−9.08619	−0.334015
$741$	0	0
$742$	−10.8492	−0.398287
$743$	−15.3566	−0.563379	−0.281690	−	0.959506i	$-0.590895\pi$
$743$	−15.3566	−0.563379	−0.281690	+	0.959506i	$0.590895\pi$
$744$	0	0
$745$	6.34610	0.232503
$746$	52.7601	1.93168
$747$	0	0
$748$	68.9659	2.52164
$749$	−2.05408	−0.0750546
$750$	0	0
$751$	−23.5657	−0.859926	−0.429963	−	0.902846i	$-0.641473\pi$
$751$	−23.5657	−0.859926	−0.429963	+	0.902846i	$0.641473\pi$
$752$	−38.9469	−1.42025
$753$	0	0
$754$	−11.4428	−0.416722
$755$	−2.16186	−0.0786783
$756$	0	0
$757$	−6.82043	−0.247893	−0.123946	−	0.992289i	$-0.539555\pi$
$757$	−6.82043	−0.247893	−0.123946	+	0.992289i	$0.539555\pi$
$758$	31.3858	1.13998
$759$	0	0
$760$	2.32743	0.0844248
$761$	−53.8607	−1.95245	−0.976224	−	0.216763i	$-0.930450\pi$
$761$	−53.8607	−1.95245	−0.976224	+	0.216763i	$0.930450\pi$
$762$	0	0
$763$	−7.30252	−0.264369
$764$	−54.9076	−1.98649
$765$	0	0
$766$	31.5261	1.13908
$767$	−0.190084	−0.00686353
$768$	0	0
$769$	14.9722	0.539910	0.269955	−	0.962873i	$-0.412991\pi$
$769$	14.9722	0.539910	0.269955	+	0.962873i	$0.412991\pi$
$770$	−2.41693	−0.0871002
$771$	0	0
$772$	−15.3274	−0.551646
$773$	11.1737	0.401891	0.200945	−	0.979602i	$-0.435599\pi$
$773$	11.1737	0.401891	0.200945	+	0.979602i	$0.435599\pi$
$774$	0	0
$775$	7.62995	0.274076
$776$	42.0512	1.50955
$777$	0	0
$778$	−79.0482	−2.83402
$779$	−0.460505	−0.0164993
$780$	0	0
$781$	33.9076	1.21331
$782$	39.3867	1.40847
$783$	0	0
$784$	−28.7673	−1.02740
$785$	10.8636	0.387739
$786$	0	0
$787$	−5.31304	−0.189389	−0.0946946	−	0.995506i	$-0.530187\pi$
$787$	−5.31304	−0.189389	−0.0946946	+	0.995506i	$0.530187\pi$
$788$	52.0918	1.85569
$789$	0	0
$790$	−15.7453	−0.560193
$791$	−1.97082	−0.0700744
$792$	0	0
$793$	4.46050	0.158397
$794$	−65.8860	−2.33821
$795$	0	0
$796$	60.2599	2.13586
$797$	−7.79806	−0.276221	−0.138111	−	0.990417i	$-0.544103\pi$
$797$	−7.79806	−0.276221	−0.138111	+	0.990417i	$0.544103\pi$
$798$	0	0
$799$	42.6047	1.50724
$800$	2.58307	0.0913252
$801$	0	0
$802$	−27.9938	−0.988494
$803$	−18.3566	−0.647791
$804$	0	0
$805$	−0.924324	−0.0325781
$806$	2.32743	0.0819802
$807$	0	0
$808$	64.1957	2.25840
$809$	−8.91808	−0.313543	−0.156772	−	0.987635i	$-0.550109\pi$
$809$	−8.91808	−0.313543	−0.156772	+	0.987635i	$0.550109\pi$
$810$	0	0
$811$	34.4255	1.20884	0.604421	−	0.796665i	$-0.293405\pi$
$811$	34.4255	1.20884	0.604421	+	0.796665i	$0.293405\pi$
$812$	−18.8539	−0.661641
$813$	0	0
$814$	−43.0335	−1.50832
$815$	7.11964	0.249390
$816$	0	0
$817$	9.48968	0.332002
$818$	27.9535	0.977371
$819$	0	0
$820$	0.859728	0.0300230
$821$	31.8683	1.11221	0.556105	−	0.831112i	$-0.312295\pi$
$821$	31.8683	1.11221	0.556105	+	0.831112i	$0.312295\pi$
$822$	0	0
$823$	−7.29105	−0.254150	−0.127075	−	0.991893i	$-0.540559\pi$
$823$	−7.29105	−0.254150	−0.127075	+	0.991893i	$0.540559\pi$
$824$	−13.9282	−0.485212
$825$	0	0
$826$	−0.467702	−0.0162734
$827$	1.03210	0.0358897	0.0179449	−	0.999839i	$-0.494288\pi$
$827$	1.03210	0.0358897	0.0179449	+	0.999839i	$0.494288\pi$
$828$	0	0
$829$	−37.8459	−1.31444	−0.657221	−	0.753698i	$-0.728268\pi$
$829$	−37.8459	−1.31444	−0.657221	+	0.753698i	$0.728268\pi$
$830$	20.5510	0.713334
$831$	0	0
$832$	−4.34941	−0.150789
$833$	31.4690	1.09034
$834$	0	0
$835$	−5.67353	−0.196341
$836$	14.5687	0.503868
$837$	0	0
$838$	3.97509	0.137317
$839$	32.5772	1.12469	0.562345	−	0.826903i	$-0.309899\pi$
$839$	32.5772	1.12469	0.562345	+	0.826903i	$0.309899\pi$
$840$	0	0
$841$	32.3844	1.11670
$842$	24.8784	0.857366
$843$	0	0
$844$	109.916	3.78347
$845$	−5.82431	−0.200362
$846$	0	0
$847$	−1.13600	−0.0390334
$848$	−32.1459	−1.10389
$849$	0	0
$850$	−55.7683	−1.91284
$851$	−16.4576	−0.564159
$852$	0	0
$853$	28.9971	0.992841	0.496420	−	0.868082i	$-0.334647\pi$
$853$	28.9971	0.992841	0.496420	+	0.868082i	$0.334647\pi$
$854$	10.9751	0.375560
$855$	0	0
$856$	−17.4897	−0.597785
$857$	19.8745	0.678900	0.339450	−	0.940624i	$-0.389759\pi$
$857$	19.8745	0.678900	0.339450	+	0.940624i	$0.389759\pi$
$858$	0	0
$859$	9.65913	0.329565	0.164783	−	0.986330i	$-0.447308\pi$
$859$	9.65913	0.329565	0.164783	+	0.986330i	$0.447308\pi$
$860$	−17.7165	−0.604129
$861$	0	0
$862$	42.8650	1.45999
$863$	37.3973	1.27302	0.636509	−	0.771270i	$-0.280378\pi$
$863$	37.3973	1.27302	0.636509	+	0.771270i	$0.280378\pi$
$864$	0	0
$865$	−1.79417	−0.0610038
$866$	−64.7539	−2.20043
$867$	0	0
$868$	3.83482	0.130162
$869$	−49.9368	−1.69399
$870$	0	0
$871$	−8.13735	−0.275723
$872$	−62.1780	−2.10561
$873$	0	0
$874$	8.32023	0.281436
$875$	2.67549	0.0904482
$876$	0	0
$877$	−11.8128	−0.398891	−0.199446	−	0.979909i	$-0.563914\pi$
$877$	−11.8128	−0.398891	−0.199446	+	0.979909i	$0.563914\pi$
$878$	−15.4897	−0.522751
$879$	0	0
$880$	−7.16129	−0.241407
$881$	−52.6490	−1.77379	−0.886895	−	0.461971i	$-0.847142\pi$
$881$	−52.6490	−1.77379	−0.886895	+	0.461971i	$0.847142\pi$
$882$	0	0
$883$	18.5189	0.623209	0.311605	−	0.950212i	$-0.399134\pi$
$883$	18.5189	0.623209	0.311605	+	0.950212i	$0.399134\pi$
$884$	−11.3916	−0.383142
$885$	0	0
$886$	−83.9702	−2.82103
$887$	−24.4605	−0.821303	−0.410652	−	0.911792i	$-0.634699\pi$
$887$	−24.4605	−0.821303	−0.410652	+	0.911792i	$0.634699\pi$
$888$	0	0
$889$	−0.918085	−0.0307916
$890$	−14.3025	−0.479422
$891$	0	0
$892$	−62.2278	−2.08354
$893$	9.00000	0.301174
$894$	0	0
$895$	−6.06460	−0.202717
$896$	−11.3422	−0.378917
$897$	0	0
$898$	52.4973	1.75186
$899$	−12.4854	−0.416412
$900$	0	0
$901$	35.1649	1.17151
$902$	4.07179	0.135576
$903$	0	0
$904$	−16.7807	−0.558119
$905$	−2.84106	−0.0944401
$906$	0	0
$907$	18.1947	0.604147	0.302073	−	0.953285i	$-0.402321\pi$
$907$	18.1947	0.604147	0.302073	+	0.953285i	$0.402321\pi$
$908$	29.4749	0.978159
$909$	0	0
$910$	0.399223	0.0132341
$911$	−33.9836	−1.12593	−0.562964	−	0.826481i	$-0.690339\pi$
$911$	−33.9836	−1.12593	−0.562964	+	0.826481i	$0.690339\pi$
$912$	0	0
$913$	65.1780	2.15708
$914$	29.6946	0.982208
$915$	0	0
$916$	−80.5303	−2.66080
$917$	−7.01439	−0.231636
$918$	0	0
$919$	3.72958	0.123027	0.0615137	−	0.998106i	$-0.480407\pi$
$919$	3.72958	0.123027	0.0615137	+	0.998106i	$0.480407\pi$
$920$	−7.87024	−0.259474
$921$	0	0
$922$	16.4179	0.540694
$923$	−5.60078	−0.184352
$924$	0	0
$925$	23.3025	0.766182
$926$	67.3288	2.21256
$927$	0	0
$928$	−4.22685	−0.138753
$929$	−52.7280	−1.72995	−0.864975	−	0.501815i	$-0.832666\pi$
$929$	−52.7280	−1.72995	−0.864975	+	0.501815i	$0.832666\pi$
$930$	0	0
$931$	6.64766	0.217868
$932$	19.1796	0.628248
$933$	0	0
$934$	−87.1210	−2.85069
$935$	7.83386	0.256195
$936$	0	0
$937$	27.6080	0.901913	0.450957	−	0.892546i	$-0.351083\pi$
$937$	27.6080	0.901913	0.450957	+	0.892546i	$0.351083\pi$
$938$	−20.0220	−0.653741
$939$	0	0
$940$	−16.8023	−0.548032
$941$	49.3330	1.60821	0.804106	−	0.594486i	$-0.202645\pi$
$941$	49.3330	1.60821	0.804106	+	0.594486i	$0.202645\pi$
$942$	0	0
$943$	1.55720	0.0507095
$944$	−1.38579	−0.0451035
$945$	0	0
$946$	−83.9080	−2.72808
$947$	−44.0482	−1.43138	−0.715688	−	0.698421i	$-0.753887\pi$
$947$	−44.0482	−1.43138	−0.715688	+	0.698421i	$0.753887\pi$
$948$	0	0
$949$	3.03210	0.0984262
$950$	−11.7807	−0.382218
$951$	0	0
$952$	−14.2016	−0.460275
$953$	32.9430	1.06713	0.533564	−	0.845760i	$-0.320852\pi$
$953$	32.9430	1.06713	0.533564	+	0.845760i	$0.320852\pi$
$954$	0	0
$955$	−6.23697	−0.201824
$956$	−12.7601	−0.412691
$957$	0	0
$958$	−15.4533	−0.499274
$959$	0.258562	0.00834942
$960$	0	0
$961$	−28.4605	−0.918081
$962$	7.10817	0.229177
$963$	0	0
$964$	−37.9152	−1.22117
$965$	−1.74105	−0.0560463
$966$	0	0
$967$	−13.3422	−0.429057	−0.214528	−	0.976718i	$-0.568821\pi$
$967$	−13.3422	−0.429057	−0.214528	+	0.976718i	$0.568821\pi$
$968$	−9.67257	−0.310888
$969$	0	0
$970$	9.42744	0.302697
$971$	−2.85253	−0.0915421	−0.0457710	−	0.998952i	$-0.514574\pi$
$971$	−2.85253	−0.0915421	−0.0457710	+	0.998952i	$0.514574\pi$
$972$	0	0
$973$	7.14027	0.228907
$974$	58.6916	1.88060
$975$	0	0
$976$	32.5189	1.04090
$977$	−2.74144	−0.0877064	−0.0438532	−	0.999038i	$-0.513963\pi$
$977$	−2.74144	−0.0877064	−0.0438532	+	0.999038i	$0.513963\pi$
$978$	0	0
$979$	−45.3609	−1.44974
$980$	−12.4107	−0.396445
$981$	0	0
$982$	−28.6050	−0.912824
$983$	−49.7879	−1.58799	−0.793994	−	0.607925i	$-0.792002\pi$
$983$	−49.7879	−1.58799	−0.793994	+	0.607925i	$0.792002\pi$
$984$	0	0
$985$	5.91713	0.188535
$986$	91.2574	2.90623
$987$	0	0
$988$	−2.40642	−0.0765584
$989$	−32.0895	−1.02039
$990$	0	0
$991$	−43.1138	−1.36955	−0.684777	−	0.728752i	$-0.740101\pi$
$991$	−43.1138	−1.36955	−0.684777	+	0.728752i	$0.740101\pi$
$992$	0.859728	0.0272964
$993$	0	0
$994$	−13.7807	−0.437098
$995$	6.84494	0.216999
$996$	0	0
$997$	−4.92860	−0.156090	−0.0780451	−	0.996950i	$-0.524868\pi$
$997$	−4.92860	−0.156090	−0.0780451	+	0.996950i	$0.524868\pi$
$998$	−11.5074	−0.364260
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.e.1.1	✓	3
3.2	odd	2		513.2.a.f.1.3	yes	3
4.3	odd	2		8208.2.a.bf.1.3		3
12.11	even	2		8208.2.a.bp.1.1		3
19.18	odd	2		9747.2.a.ba.1.3		3
57.56	even	2		9747.2.a.x.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.e.1.1	✓	3	1.1	even	1	trivial
513.2.a.f.1.3	yes	3	3.2	odd	2
8208.2.a.bf.1.3		3	4.3	odd	2
8208.2.a.bp.1.1		3	12.11	even	2
9747.2.a.x.1.1		3	57.56	even	2
9747.2.a.ba.1.3		3	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.a (trivial)

\(f(q)\)	\(=\)	\(q-2.46050 q^{2} +4.05408 q^{4} +0.460505 q^{5} -0.593579 q^{7} -5.05408 q^{8} -1.13307 q^{10} -3.59358 q^{11} +0.593579 q^{13} +1.46050 q^{14} +4.32743 q^{16} -4.73385 q^{17} -1.00000 q^{19} +1.86693 q^{20} +8.84202 q^{22} +3.38151 q^{23} -4.78794 q^{25} -1.46050 q^{26} -2.40642 q^{28} +7.83482 q^{29} -1.59358 q^{31} -0.539495 q^{32} +11.6477 q^{34} -0.273346 q^{35} -4.86693 q^{37} +2.46050 q^{38} -2.32743 q^{40} +0.460505 q^{41} -9.48968 q^{43} -14.5687 q^{44} -8.32023 q^{46} -9.00000 q^{47} -6.64766 q^{49} +11.7807 q^{50} +2.40642 q^{52} -7.42840 q^{53} -1.65486 q^{55} +3.00000 q^{56} -19.2776 q^{58} -0.320233 q^{59} +7.51459 q^{61} +3.92101 q^{62} -7.32743 q^{64} +0.273346 q^{65} -13.7089 q^{67} -19.1914 q^{68} +0.672570 q^{70} -9.43560 q^{71} +5.10817 q^{73} +11.9751 q^{74} -4.05408 q^{76} +2.13307 q^{77} +13.8961 q^{79} +1.99280 q^{80} -1.13307 q^{82} -18.1373 q^{83} -2.17996 q^{85} +23.3494 q^{86} +18.1623 q^{88} +12.6228 q^{89} -0.352336 q^{91} +13.7089 q^{92} +22.1445 q^{94} -0.460505 q^{95} -8.32023 q^{97} +16.3566 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{4} - 5 q^{5} + q^{7} - 6 q^{8} - 7 q^{10} - 8 q^{11} - q^{13} - 2 q^{14} + 3 q^{16} - 7 q^{17} - 3 q^{19} + 2 q^{20} + q^{22} - 9 q^{23} + 2 q^{25} + 2 q^{26} - 10 q^{28} + 6 q^{29}+ \cdots + 8 q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^4 - 5 * q^5 + q^7 - 6 * q^8 - 7 * q^10 - 8 * q^11 - q^13 - 2 * q^14 + 3 * q^16 - 7 * q^17 - 3 * q^19 + 2 * q^20 + q^22 - 9 * q^23 + 2 * q^25 + 2 * q^26 - 10 * q^28 + 6 * q^29 - 2 * q^31 - 8 * q^32 + 23 * q^34 - 11 * q^37 + q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 19 * q^44 - 23 * q^46 - 27 * q^47 - 8 * q^49 + 27 * q^50 + 10 * q^52 - 2 * q^53 + 15 * q^55 + 9 * q^56 - 4 * q^58 + q^59 + 7 * q^61 - q^62 - 12 * q^64 - 12 * q^67 + q^68 + 12 * q^70 - 3 * q^73 + 14 * q^74 - 3 * q^76 + 10 * q^77 + 7 * q^79 + 14 * q^80 - 7 * q^82 - 5 * q^83 - 9 * q^85 + 37 * q^86 + 27 * q^88 + 4 * q^89 - 13 * q^91 + 12 * q^92 + 9 * q^94 + 5 * q^95 - 23 * q^97 + 8 * q^98

Introduction

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