LMFDB - Embedded newform 513.2.a.h.1.2

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 = [4,0,0,4] 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:	$0$
Dimension:	$4$
Coefficient field:	4.4.27648.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 3 $ x^4 - 6*x^2 + 3
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			1.2
Root			$-0.741964$ 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)

$f(q)$	$=$	\(q-0.741964 q^{2} -1.44949 q^{4} +3.30136 q^{5} +4.44949 q^{7} +2.55940 q^{8} -2.44949 q^{10} -4.04332 q^{11} +2.00000 q^{13} -3.30136 q^{14} +1.00000 q^{16} -6.60272 q^{17} +1.00000 q^{19} -4.78529 q^{20} +3.00000 q^{22} +4.78529 q^{23} +5.89898 q^{25} -1.48393 q^{26} -6.44949 q^{28} +2.55940 q^{29} +7.44949 q^{31} -5.86076 q^{32} +4.89898 q^{34} +14.6894 q^{35} +4.44949 q^{37} -0.741964 q^{38} +8.44949 q^{40} +3.70982 q^{41} -5.34847 q^{43} +5.86076 q^{44} -3.55051 q^{46} +0.741964 q^{47} +12.7980 q^{49} -4.37683 q^{50} -2.89898 q^{52} -10.9795 q^{53} -13.3485 q^{55} +11.3880 q^{56} -1.89898 q^{58} +11.3880 q^{59} -8.34847 q^{61} -5.52725 q^{62} +2.34847 q^{64} +6.60272 q^{65} -2.34847 q^{67} +9.57058 q^{68} -10.8990 q^{70} -10.2376 q^{71} -8.34847 q^{73} -3.30136 q^{74} -1.44949 q^{76} -17.9907 q^{77} +5.00000 q^{79} +3.30136 q^{80} -2.75255 q^{82} +5.86076 q^{83} -21.7980 q^{85} +3.96837 q^{86} -10.3485 q^{88} -5.52725 q^{89} +8.89898 q^{91} -6.93623 q^{92} -0.550510 q^{94} +3.30136 q^{95} +5.55051 q^{97} -9.49562 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} + 8 q^{7} + 8 q^{13} + 4 q^{16} + 4 q^{19} + 12 q^{22} + 4 q^{25} - 16 q^{28} + 20 q^{31} + 8 q^{37} + 24 q^{40} + 8 q^{43} - 24 q^{46} + 12 q^{49} + 8 q^{52} - 24 q^{55} + 12 q^{58} - 4 q^{61}+ \cdots + 32 q^{97}+O(q^{100}) $ 4 * q + 4 * q^4 + 8 * q^7 + 8 * q^13 + 4 * q^16 + 4 * q^19 + 12 * q^22 + 4 * q^25 - 16 * q^28 + 20 * q^31 + 8 * q^37 + 24 * q^40 + 8 * q^43 - 24 * q^46 + 12 * q^49 + 8 * q^52 - 24 * q^55 + 12 * q^58 - 4 * q^61 - 20 * q^64 + 20 * q^67 - 24 * q^70 - 4 * q^73 + 4 * q^76 + 20 * q^79 - 60 * q^82 - 48 * q^85 - 12 * q^88 + 16 * q^91 - 12 * q^94 + 32 * q^97

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 $

$2$	−0.741964	−0.524648	−0.262324	−	0.964980i	$-0.584489\pi$
$2$	−0.741964	−0.524648	−0.262324	+	0.964980i	$0.584489\pi$
$3$	0	0
$3$	0	0
$4$	−1.44949	−0.724745
$5$	3.30136	1.47641	0.738207	−	0.674575i	$-0.235673\pi$
$5$	3.30136	1.47641	0.738207	+	0.674575i	$0.235673\pi$
$6$	0	0
$7$	4.44949	1.68175	0.840875	−	0.541230i	$-0.182041\pi$
$7$	4.44949	1.68175	0.840875	+	0.541230i	$0.182041\pi$
$8$	2.55940	0.904883
$9$	0	0
$10$	−2.44949	−0.774597
$11$	−4.04332	−1.21911	−0.609554	−	0.792745i	$-0.708651\pi$
$11$	−4.04332	−1.21911	−0.609554	+	0.792745i	$0.708651\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−3.30136	−0.882326
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.60272	−1.60139	−0.800697	−	0.599069i	$-0.795538\pi$
$17$	−6.60272	−1.60139	−0.800697	+	0.599069i	$0.795538\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−4.78529	−1.07002
$21$	0	0
$22$	3.00000	0.639602
$23$	4.78529	0.997801	0.498901	−	0.866659i	$-0.333737\pi$
$23$	4.78529	0.997801	0.498901	+	0.866659i	$0.333737\pi$
$24$	0	0
$25$	5.89898	1.17980
$26$	−1.48393	−0.291022
$27$	0	0
$28$	−6.44949	−1.21884
$29$	2.55940	0.475268	0.237634	−	0.971355i	$-0.423628\pi$
$29$	2.55940	0.475268	0.237634	+	0.971355i	$0.423628\pi$
$30$	0	0
$31$	7.44949	1.33797	0.668984	−	0.743277i	$-0.266729\pi$
$31$	7.44949	1.33797	0.668984	+	0.743277i	$0.266729\pi$
$32$	−5.86076	−1.03605
$33$	0	0
$34$	4.89898	0.840168
$35$	14.6894	2.48296
$36$	0	0
$37$	4.44949	0.731492	0.365746	−	0.930715i	$-0.380814\pi$
$37$	4.44949	0.731492	0.365746	+	0.930715i	$0.380814\pi$
$38$	−0.741964	−0.120362
$39$	0	0
$40$	8.44949	1.33598
$41$	3.70982	0.579376	0.289688	−	0.957121i	$-0.406448\pi$
$41$	3.70982	0.579376	0.289688	+	0.957121i	$0.406448\pi$
$42$	0	0
$43$	−5.34847	−0.815634	−0.407817	−	0.913064i	$-0.633710\pi$
$43$	−5.34847	−0.815634	−0.407817	+	0.913064i	$0.633710\pi$
$44$	5.86076	0.883542
$45$	0	0
$46$	−3.55051	−0.523494
$47$	0.741964	0.108227	0.0541133	−	0.998535i	$-0.482767\pi$
$47$	0.741964	0.108227	0.0541133	+	0.998535i	$0.482767\pi$
$48$	0	0
$49$	12.7980	1.82828
$50$	−4.37683	−0.618977
$51$	0	0
$52$	−2.89898	−0.402016
$53$	−10.9795	−1.50816	−0.754079	−	0.656784i	$-0.771916\pi$
$53$	−10.9795	−1.50816	−0.754079	+	0.656784i	$0.771916\pi$
$54$	0	0
$55$	−13.3485	−1.79991
$56$	11.3880	1.52179
$57$	0	0
$58$	−1.89898	−0.249348
$59$	11.3880	1.48259	0.741296	−	0.671178i	$-0.234211\pi$
$59$	11.3880	1.48259	0.741296	+	0.671178i	$0.234211\pi$
$60$	0	0
$61$	−8.34847	−1.06891	−0.534456	−	0.845196i	$-0.679483\pi$
$61$	−8.34847	−1.06891	−0.534456	+	0.845196i	$0.679483\pi$
$62$	−5.52725	−0.701962
$63$	0	0
$64$	2.34847	0.293559
$65$	6.60272	0.818967
$66$	0	0
$67$	−2.34847	−0.286911	−0.143456	−	0.989657i	$-0.545821\pi$
$67$	−2.34847	−0.286911	−0.143456	+	0.989657i	$0.545821\pi$
$68$	9.57058	1.16060
$69$	0	0
$70$	−10.8990	−1.30268
$71$	−10.2376	−1.21498	−0.607489	−	0.794328i	$-0.707823\pi$
$71$	−10.2376	−1.21498	−0.607489	+	0.794328i	$0.707823\pi$
$72$	0	0
$73$	−8.34847	−0.977114	−0.488557	−	0.872532i	$-0.662477\pi$
$73$	−8.34847	−0.977114	−0.488557	+	0.872532i	$0.662477\pi$
$74$	−3.30136	−0.383775
$75$	0	0
$76$	−1.44949	−0.166268
$77$	−17.9907	−2.05023
$78$	0	0
$79$	5.00000	0.562544	0.281272	−	0.959628i	$-0.409244\pi$
$79$	5.00000	0.562544	0.281272	+	0.959628i	$0.409244\pi$
$80$	3.30136	0.369103
$81$	0	0
$82$	−2.75255	−0.303968
$83$	5.86076	0.643302	0.321651	−	0.946858i	$-0.395762\pi$
$83$	5.86076	0.643302	0.321651	+	0.946858i	$0.395762\pi$
$84$	0	0
$85$	−21.7980	−2.36432
$86$	3.96837	0.427920
$87$	0	0
$88$	−10.3485	−1.10315
$89$	−5.52725	−0.585887	−0.292944	−	0.956130i	$-0.594635\pi$
$89$	−5.52725	−0.585887	−0.292944	+	0.956130i	$0.594635\pi$
$90$	0	0
$91$	8.89898	0.932867
$92$	−6.93623	−0.723152
$93$	0	0
$94$	−0.550510	−0.0567808
$95$	3.30136	0.338712
$96$	0	0
$97$	5.55051	0.563569	0.281784	−	0.959478i	$-0.409074\pi$
$97$	5.55051	0.563569	0.281784	+	0.959478i	$0.409074\pi$
$98$	−9.49562	−0.959203
$99$	0	0
$100$	−8.55051	−0.855051
$101$	−7.75314	−0.771467	−0.385733	−	0.922610i	$-0.626052\pi$
$101$	−7.75314	−0.771467	−0.385733	+	0.922610i	$0.626052\pi$
$102$	0	0
$103$	11.7980	1.16249	0.581244	−	0.813730i	$-0.302566\pi$
$103$	11.7980	1.16249	0.581244	+	0.813730i	$0.302566\pi$
$104$	5.11879	0.501939
$105$	0	0
$106$	8.14643	0.791251
$107$	17.9907	1.73923	0.869615	−	0.493731i	$-0.164367\pi$
$107$	17.9907	1.73923	0.869615	+	0.493731i	$0.164367\pi$
$108$	0	0
$109$	8.24745	0.789962	0.394981	−	0.918689i	$-0.370751\pi$
$109$	8.24745	0.789962	0.394981	+	0.918689i	$0.370751\pi$
$110$	9.90408	0.944317
$111$	0	0
$112$	4.44949	0.420437
$113$	−4.04332	−0.380364	−0.190182	−	0.981749i	$-0.560908\pi$
$113$	−4.04332	−0.380364	−0.190182	+	0.981749i	$0.560908\pi$
$114$	0	0
$115$	15.7980	1.47317
$116$	−3.70982	−0.344448
$117$	0	0
$118$	−8.44949	−0.777839
$119$	−29.3787	−2.69314
$120$	0	0
$121$	5.34847	0.486224
$122$	6.19426	0.560802
$123$	0	0
$124$	−10.7980	−0.969685
$125$	2.96786	0.265453
$126$	0	0
$127$	−13.7980	−1.22437	−0.612185	−	0.790714i	$-0.709709\pi$
$127$	−13.7980	−1.22437	−0.612185	+	0.790714i	$0.709709\pi$
$128$	9.97903	0.882030
$129$	0	0
$130$	−4.89898	−0.429669
$131$	−17.3237	−1.51358	−0.756790	−	0.653658i	$-0.773234\pi$
$131$	−17.3237	−1.51358	−0.756790	+	0.653658i	$0.773234\pi$
$132$	0	0
$133$	4.44949	0.385820
$134$	1.74248	0.150527
$135$	0	0
$136$	−16.8990	−1.44908
$137$	2.96786	0.253561	0.126780	−	0.991931i	$-0.459536\pi$
$137$	2.96786	0.253561	0.126780	+	0.991931i	$0.459536\pi$
$138$	0	0
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−21.2921	−1.79951
$141$	0	0
$142$	7.59592	0.637435
$143$	−8.08665	−0.676239
$144$	0	0
$145$	8.44949	0.701692
$146$	6.19426	0.512641
$147$	0	0
$148$	−6.44949	−0.530145
$149$	1.48393	0.121568	0.0607840	−	0.998151i	$-0.480640\pi$
$149$	1.48393	0.121568	0.0607840	+	0.998151i	$0.480640\pi$
$150$	0	0
$151$	−9.69694	−0.789126	−0.394563	−	0.918869i	$-0.629104\pi$
$151$	−9.69694	−0.789126	−0.394563	+	0.918869i	$0.629104\pi$
$152$	2.55940	0.207594
$153$	0	0
$154$	13.3485	1.07565
$155$	24.5934	1.97539
$156$	0	0
$157$	−14.3485	−1.14513	−0.572566	−	0.819858i	$-0.694052\pi$
$157$	−14.3485	−1.14513	−0.572566	+	0.819858i	$0.694052\pi$
$158$	−3.70982	−0.295137
$159$	0	0
$160$	−19.3485	−1.52963
$161$	21.2921	1.67805
$162$	0	0
$163$	−16.2474	−1.27260	−0.636299	−	0.771442i	$-0.719536\pi$
$163$	−16.2474	−1.27260	−0.636299	+	0.771442i	$0.719536\pi$
$164$	−5.37734	−0.419900
$165$	0	0
$166$	−4.34847	−0.337507
$167$	2.15094	0.166445	0.0832223	−	0.996531i	$-0.473479\pi$
$167$	2.15094	0.166445	0.0832223	+	0.996531i	$0.473479\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	16.1733	1.24044
$171$	0	0
$172$	7.75255	0.591126
$173$	−16.9153	−1.28604	−0.643022	−	0.765848i	$-0.722320\pi$
$173$	−16.9153	−1.28604	−0.643022	+	0.765848i	$0.722320\pi$
$174$	0	0
$175$	26.2474	1.98412
$176$	−4.04332	−0.304777
$177$	0	0
$178$	4.10102	0.307384
$179$	6.26922	0.468583	0.234292	−	0.972166i	$-0.424723\pi$
$179$	6.26922	0.468583	0.234292	+	0.972166i	$0.424723\pi$
$180$	0	0
$181$	−15.1464	−1.12583	−0.562913	−	0.826516i	$-0.690319\pi$
$181$	−15.1464	−1.12583	−0.562913	+	0.826516i	$0.690319\pi$
$182$	−6.60272	−0.489426
$183$	0	0
$184$	12.2474	0.902894
$185$	14.6894	1.07998
$186$	0	0
$187$	26.6969	1.95227
$188$	−1.07547	−0.0784366
$189$	0	0
$190$	−2.44949	−0.177705
$191$	−8.49511	−0.614684	−0.307342	−	0.951599i	$-0.599440\pi$
$191$	−8.49511	−0.614684	−0.307342	+	0.951599i	$0.599440\pi$
$192$	0	0
$193$	−24.6969	−1.77772	−0.888862	−	0.458175i	$-0.848503\pi$
$193$	−24.6969	−1.77772	−0.888862	+	0.458175i	$0.848503\pi$
$194$	−4.11828	−0.295675
$195$	0	0
$196$	−18.5505	−1.32504
$197$	7.26973	0.517947	0.258973	−	0.965884i	$-0.416616\pi$
$197$	7.26973	0.517947	0.258973	+	0.965884i	$0.416616\pi$
$198$	0	0
$199$	14.0000	0.992434	0.496217	−	0.868199i	$-0.334722\pi$
$199$	14.0000	0.992434	0.496217	+	0.868199i	$0.334722\pi$
$200$	15.0978	1.06758
$201$	0	0
$202$	5.75255	0.404748
$203$	11.3880	0.799281
$204$	0	0
$205$	12.2474	0.855399
$206$	−8.75366	−0.609896
$207$	0	0
$208$	2.00000	0.138675
$209$	−4.04332	−0.279683
$210$	0	0
$211$	12.3485	0.850104	0.425052	−	0.905169i	$-0.360256\pi$
$211$	12.3485	0.850104	0.425052	+	0.905169i	$0.360256\pi$
$212$	15.9147	1.09303
$213$	0	0
$214$	−13.3485	−0.912483
$215$	−17.6572	−1.20421
$216$	0	0
$217$	33.1464	2.25013
$218$	−6.11931	−0.414452
$219$	0	0
$220$	19.3485	1.30447
$221$	−13.2054	−0.888294
$222$	0	0
$223$	23.0000	1.54019	0.770097	−	0.637927i	$-0.220208\pi$
$223$	23.0000	1.54019	0.770097	+	0.637927i	$0.220208\pi$
$224$	−26.0774	−1.74237
$225$	0	0
$226$	3.00000	0.199557
$227$	−24.2599	−1.61019	−0.805095	−	0.593147i	$-0.797885\pi$
$227$	−24.2599	−1.61019	−0.805095	+	0.593147i	$0.797885\pi$
$228$	0	0
$229$	−11.8990	−0.786307	−0.393153	−	0.919473i	$-0.628616\pi$
$229$	−11.8990	−0.786307	−0.393153	+	0.919473i	$0.628616\pi$
$230$	−11.7215	−0.772894
$231$	0	0
$232$	6.55051	0.430062
$233$	23.9264	1.56747	0.783737	−	0.621093i	$-0.213311\pi$
$233$	23.9264	1.56747	0.783737	+	0.621093i	$0.213311\pi$
$234$	0	0
$235$	2.44949	0.159787
$236$	−16.5068	−1.07450
$237$	0	0
$238$	21.7980	1.41295
$239$	6.67767	0.431943	0.215971	−	0.976400i	$-0.430708\pi$
$239$	6.67767	0.431943	0.215971	+	0.976400i	$0.430708\pi$
$240$	0	0
$241$	−23.3485	−1.50401	−0.752004	−	0.659159i	$-0.770912\pi$
$241$	−23.3485	−1.50401	−0.752004	+	0.659159i	$0.770912\pi$
$242$	−3.96837	−0.255097
$243$	0	0
$244$	12.1010	0.774688
$245$	42.2507	2.69930
$246$	0	0
$247$	2.00000	0.127257
$248$	19.0662	1.21070
$249$	0	0
$250$	−2.20204	−0.139269
$251$	−15.0229	−0.948235	−0.474118	−	0.880461i	$-0.657233\pi$
$251$	−15.0229	−0.948235	−0.474118	+	0.880461i	$0.657233\pi$
$252$	0	0
$253$	−19.3485	−1.21643
$254$	10.2376	0.642363
$255$	0	0
$256$	−12.1010	−0.756314
$257$	−20.2916	−1.26575	−0.632877	−	0.774253i	$-0.718126\pi$
$257$	−20.2916	−1.26575	−0.632877	+	0.774253i	$0.718126\pi$
$258$	0	0
$259$	19.7980	1.23019
$260$	−9.57058	−0.593542
$261$	0	0
$262$	12.8536	0.794096
$263$	9.97903	0.615334	0.307667	−	0.951494i	$-0.400452\pi$
$263$	9.97903	0.615334	0.307667	+	0.951494i	$0.400452\pi$
$264$	0	0
$265$	−36.2474	−2.22666
$266$	−3.30136	−0.202419
$267$	0	0
$268$	3.40408	0.207937
$269$	11.3880	0.694339	0.347170	−	0.937802i	$-0.387143\pi$
$269$	11.3880	0.694339	0.347170	+	0.937802i	$0.387143\pi$
$270$	0	0
$271$	2.24745	0.136523	0.0682614	−	0.997667i	$-0.478255\pi$
$271$	2.24745	0.136523	0.0682614	+	0.997667i	$0.478255\pi$
$272$	−6.60272	−0.400349
$273$	0	0
$274$	−2.20204	−0.133030
$275$	−23.8515	−1.43830
$276$	0	0
$277$	−13.0000	−0.781094	−0.390547	−	0.920583i	$-0.627714\pi$
$277$	−13.0000	−0.781094	−0.390547	+	0.920583i	$0.627714\pi$
$278$	7.41964	0.445000
$279$	0	0
$280$	37.5959	2.24679
$281$	23.7765	1.41839	0.709194	−	0.705013i	$-0.249059\pi$
$281$	23.7765	1.41839	0.709194	+	0.705013i	$0.249059\pi$
$282$	0	0
$283$	6.89898	0.410102	0.205051	−	0.978751i	$-0.434264\pi$
$283$	6.89898	0.410102	0.205051	+	0.978751i	$0.434264\pi$
$284$	14.8393	0.880549
$285$	0	0
$286$	6.00000	0.354787
$287$	16.5068	0.974366
$288$	0	0
$289$	26.5959	1.56447
$290$	−6.26922	−0.368141
$291$	0	0
$292$	12.1010	0.708159
$293$	−4.04332	−0.236214	−0.118107	−	0.993001i	$-0.537683\pi$
$293$	−4.04332	−0.236214	−0.118107	+	0.993001i	$0.537683\pi$
$294$	0	0
$295$	37.5959	2.18892
$296$	11.3880	0.661915
$297$	0	0
$298$	−1.10102	−0.0637804
$299$	9.57058	0.553481
$300$	0	0
$301$	−23.7980	−1.37169
$302$	7.19478	0.414013
$303$	0	0
$304$	1.00000	0.0573539
$305$	−27.5613	−1.57816
$306$	0	0
$307$	−10.7980	−0.616272	−0.308136	−	0.951342i	$-0.599705\pi$
$307$	−10.7980	−0.616272	−0.308136	+	0.951342i	$0.599705\pi$
$308$	26.0774	1.48590
$309$	0	0
$310$	−18.2474	−1.03639
$311$	7.67819	0.435390	0.217695	−	0.976017i	$-0.430146\pi$
$311$	7.67819	0.435390	0.217695	+	0.976017i	$0.430146\pi$
$312$	0	0
$313$	11.2474	0.635743	0.317872	−	0.948134i	$-0.397032\pi$
$313$	11.2474	0.635743	0.317872	+	0.948134i	$0.397032\pi$
$314$	10.6460	0.600791
$315$	0	0
$316$	−7.24745	−0.407701
$317$	20.1417	1.13127	0.565634	−	0.824656i	$-0.308631\pi$
$317$	20.1417	1.13127	0.565634	+	0.824656i	$0.308631\pi$
$318$	0	0
$319$	−10.3485	−0.579403
$320$	7.75314	0.433414
$321$	0	0
$322$	−15.7980	−0.880386
$323$	−6.60272	−0.367385
$324$	0	0
$325$	11.7980	0.654433
$326$	12.0550	0.667666
$327$	0	0
$328$	9.49490	0.524268
$329$	3.30136	0.182010
$330$	0	0
$331$	−17.5959	−0.967159	−0.483580	−	0.875300i	$-0.660664\pi$
$331$	−17.5959	−0.967159	−0.483580	+	0.875300i	$0.660664\pi$
$332$	−8.49511	−0.466230
$333$	0	0
$334$	−1.59592	−0.0873247
$335$	−7.75314	−0.423599
$336$	0	0
$337$	3.34847	0.182403	0.0912014	−	0.995832i	$-0.470929\pi$
$337$	3.34847	0.182403	0.0912014	+	0.995832i	$0.470929\pi$
$338$	6.67767	0.363218
$339$	0	0
$340$	31.5959	1.71353
$341$	−30.1207	−1.63113
$342$	0	0
$343$	25.7980	1.39296
$344$	−13.6889	−0.738054
$345$	0	0
$346$	12.5505	0.674720
$347$	27.5613	1.47957	0.739784	−	0.672844i	$-0.234928\pi$
$347$	27.5613	1.47957	0.739784	+	0.672844i	$0.234928\pi$
$348$	0	0
$349$	−14.5959	−0.781302	−0.390651	−	0.920539i	$-0.627750\pi$
$349$	−14.5959	−0.781302	−0.390651	+	0.920539i	$0.627750\pi$
$350$	−19.4747	−1.04096
$351$	0	0
$352$	23.6969	1.26305
$353$	−23.4430	−1.24775	−0.623873	−	0.781526i	$-0.714442\pi$
$353$	−23.4430	−1.24775	−0.623873	+	0.781526i	$0.714442\pi$
$354$	0	0
$355$	−33.7980	−1.79381
$356$	8.01169	0.424619
$357$	0	0
$358$	−4.65153	−0.245841
$359$	−29.0452	−1.53295	−0.766474	−	0.642275i	$-0.777991\pi$
$359$	−29.0452	−1.53295	−0.766474	+	0.642275i	$0.777991\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	11.2381	0.590661
$363$	0	0
$364$	−12.8990	−0.676090
$365$	−27.5613	−1.44262
$366$	0	0
$367$	−6.69694	−0.349577	−0.174789	−	0.984606i	$-0.555924\pi$
$367$	−6.69694	−0.349577	−0.174789	+	0.984606i	$0.555924\pi$
$368$	4.78529	0.249450
$369$	0	0
$370$	−10.8990	−0.566611
$371$	−48.8534	−2.53634
$372$	0	0
$373$	24.0454	1.24502	0.622512	−	0.782610i	$-0.286112\pi$
$373$	24.0454	1.24502	0.622512	+	0.782610i	$0.286112\pi$
$374$	−19.8082	−1.02426
$375$	0	0
$376$	1.89898	0.0979324
$377$	5.11879	0.263631
$378$	0	0
$379$	35.4949	1.82325	0.911625	−	0.411022i	$-0.134828\pi$
$379$	35.4949	1.82325	0.911625	+	0.411022i	$0.134828\pi$
$380$	−4.78529	−0.245480
$381$	0	0
$382$	6.30306	0.322493
$383$	−15.6899	−0.801716	−0.400858	−	0.916140i	$-0.631288\pi$
$383$	−15.6899	−0.801716	−0.400858	+	0.916140i	$0.631288\pi$
$384$	0	0
$385$	−59.3939	−3.02699
$386$	18.3242	0.932679
$387$	0	0
$388$	−8.04541	−0.408444
$389$	−2.48444	−0.125966	−0.0629831	−	0.998015i	$-0.520061\pi$
$389$	−2.48444	−0.125966	−0.0629831	+	0.998015i	$0.520061\pi$
$390$	0	0
$391$	−31.5959	−1.59787
$392$	32.7551	1.65438
$393$	0	0
$394$	−5.39388	−0.271740
$395$	16.5068	0.830547
$396$	0	0
$397$	−13.0000	−0.652451	−0.326226	−	0.945292i	$-0.605777\pi$
$397$	−13.0000	−0.652451	−0.326226	+	0.945292i	$0.605777\pi$
$398$	−10.3875	−0.520678
$399$	0	0
$400$	5.89898	0.294949
$401$	−7.60324	−0.379687	−0.189844	−	0.981814i	$-0.560798\pi$
$401$	−7.60324	−0.379687	−0.189844	+	0.981814i	$0.560798\pi$
$402$	0	0
$403$	14.8990	0.742171
$404$	11.2381	0.559116
$405$	0	0
$406$	−8.44949	−0.419341
$407$	−17.9907	−0.891767
$408$	0	0
$409$	2.24745	0.111129	0.0555646	−	0.998455i	$-0.482304\pi$
$409$	2.24745	0.111129	0.0555646	+	0.998455i	$0.482304\pi$
$410$	−9.08716	−0.448783
$411$	0	0
$412$	−17.1010	−0.842507
$413$	50.6708	2.49335
$414$	0	0
$415$	19.3485	0.949779
$416$	−11.7215	−0.574694
$417$	0	0
$418$	3.00000	0.146735
$419$	18.3992	0.898859	0.449430	−	0.893316i	$-0.351627\pi$
$419$	18.3992	0.898859	0.449430	+	0.893316i	$0.351627\pi$
$420$	0	0
$421$	−23.1010	−1.12587	−0.562937	−	0.826500i	$-0.690329\pi$
$421$	−23.1010	−1.12587	−0.562937	+	0.826500i	$0.690329\pi$
$422$	−9.16212	−0.446005
$423$	0	0
$424$	−28.1010	−1.36471
$425$	−38.9493	−1.88932
$426$	0	0
$427$	−37.1464	−1.79764
$428$	−26.0774	−1.26050
$429$	0	0
$430$	13.1010	0.631787
$431$	−11.3880	−0.548541	−0.274271	−	0.961653i	$-0.588436\pi$
$431$	−11.3880	−0.548541	−0.274271	+	0.961653i	$0.588436\pi$
$432$	0	0
$433$	6.65153	0.319652	0.159826	−	0.987145i	$-0.448907\pi$
$433$	6.65153	0.319652	0.159826	+	0.987145i	$0.448907\pi$
$434$	−24.5934	−1.18052
$435$	0	0
$436$	−11.9546	−0.572521
$437$	4.78529	0.228911
$438$	0	0
$439$	2.30306	0.109919	0.0549596	−	0.998489i	$-0.482497\pi$
$439$	2.30306	0.109919	0.0549596	+	0.998489i	$0.482497\pi$
$440$	−34.1640	−1.62871
$441$	0	0
$442$	9.79796	0.466041
$443$	8.01169	0.380647	0.190324	−	0.981721i	$-0.439046\pi$
$443$	8.01169	0.380647	0.190324	+	0.981721i	$0.439046\pi$
$444$	0	0
$445$	−18.2474	−0.865012
$446$	−17.0652	−0.808059
$447$	0	0
$448$	10.4495	0.493692
$449$	17.5823	0.829759	0.414879	−	0.909876i	$-0.363824\pi$
$449$	17.5823	0.829759	0.414879	+	0.909876i	$0.363824\pi$
$450$	0	0
$451$	−15.0000	−0.706322
$452$	5.86076	0.275667
$453$	0	0
$454$	18.0000	0.844782
$455$	29.3787	1.37730
$456$	0	0
$457$	4.14643	0.193962	0.0969809	−	0.995286i	$-0.469081\pi$
$457$	4.14643	0.193962	0.0969809	+	0.995286i	$0.469081\pi$
$458$	8.82861	0.412534
$459$	0	0
$460$	−22.8990	−1.06767
$461$	9.57058	0.445746	0.222873	−	0.974847i	$-0.428456\pi$
$461$	9.57058	0.445746	0.222873	+	0.974847i	$0.428456\pi$
$462$	0	0
$463$	−2.89898	−0.134727	−0.0673635	−	0.997728i	$-0.521459\pi$
$463$	−2.89898	−0.134727	−0.0673635	+	0.997728i	$0.521459\pi$
$464$	2.55940	0.118817
$465$	0	0
$466$	−17.7526	−0.822371
$467$	9.49562	0.439405	0.219702	−	0.975567i	$-0.429491\pi$
$467$	9.49562	0.439405	0.219702	+	0.975567i	$0.429491\pi$
$468$	0	0
$469$	−10.4495	−0.482513
$470$	−1.81743	−0.0838319
$471$	0	0
$472$	29.1464	1.34157
$473$	21.6256	0.994346
$474$	0	0
$475$	5.89898	0.270664
$476$	42.5842	1.95184
$477$	0	0
$478$	−4.95459	−0.226618
$479$	37.9488	1.73392	0.866962	−	0.498374i	$-0.166069\pi$
$479$	37.9488	1.73392	0.866962	+	0.498374i	$0.166069\pi$
$480$	0	0
$481$	8.89898	0.405759
$482$	17.3237	0.789074
$483$	0	0
$484$	−7.75255	−0.352389
$485$	18.3242	0.832061
$486$	0	0
$487$	6.34847	0.287677	0.143838	−	0.989601i	$-0.454055\pi$
$487$	6.34847	0.287677	0.143838	+	0.989601i	$0.454055\pi$
$488$	−21.3670	−0.967241
$489$	0	0
$490$	−31.3485	−1.41618
$491$	2.89290	0.130555	0.0652774	−	0.997867i	$-0.479207\pi$
$491$	2.89290	0.130555	0.0652774	+	0.997867i	$0.479207\pi$
$492$	0	0
$493$	−16.8990	−0.761092
$494$	−1.48393	−0.0667651
$495$	0	0
$496$	7.44949	0.334492
$497$	−45.5520	−2.04329
$498$	0	0
$499$	−34.0000	−1.52205	−0.761025	−	0.648723i	$-0.775303\pi$
$499$	−34.0000	−1.52205	−0.761025	+	0.648723i	$0.775303\pi$
$500$	−4.30188	−0.192386
$501$	0	0
$502$	11.1464	0.497489
$503$	15.9147	0.709603	0.354802	−	0.934942i	$-0.384548\pi$
$503$	15.9147	0.709603	0.354802	+	0.934942i	$0.384548\pi$
$504$	0	0
$505$	−25.5959	−1.13900
$506$	14.3559	0.638196
$507$	0	0
$508$	20.0000	0.887357
$509$	9.64553	0.427531	0.213765	−	0.976885i	$-0.431427\pi$
$509$	9.64553	0.427531	0.213765	+	0.976885i	$0.431427\pi$
$510$	0	0
$511$	−37.1464	−1.64326
$512$	−10.9795	−0.485232
$513$	0	0
$514$	15.0556	0.664075
$515$	38.9493	1.71631
$516$	0	0
$517$	−3.00000	−0.131940
$518$	−14.6894	−0.645414
$519$	0	0
$520$	16.8990	0.741069
$521$	−9.90408	−0.433906	−0.216953	−	0.976182i	$-0.569612\pi$
$521$	−9.90408	−0.433906	−0.216953	+	0.976182i	$0.569612\pi$
$522$	0	0
$523$	21.0454	0.920251	0.460126	−	0.887854i	$-0.347804\pi$
$523$	21.0454	0.920251	0.460126	+	0.887854i	$0.347804\pi$
$524$	25.1106	1.09696
$525$	0	0
$526$	−7.40408	−0.322833
$527$	−49.1869	−2.14261
$528$	0	0
$529$	−0.101021	−0.00439220
$530$	26.8943	1.16821
$531$	0	0
$532$	−6.44949	−0.279621
$533$	7.41964	0.321380
$534$	0	0
$535$	59.3939	2.56782
$536$	−6.01066	−0.259621
$537$	0	0
$538$	−8.44949	−0.364283
$539$	−51.7463	−2.22887
$540$	0	0
$541$	−23.0454	−0.990799	−0.495400	−	0.868665i	$-0.664978\pi$
$541$	−23.0454	−0.990799	−0.495400	+	0.868665i	$0.664978\pi$
$542$	−1.66753	−0.0716264
$543$	0	0
$544$	38.6969	1.65912
$545$	27.2278	1.16631
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−4.30188	−0.183767
$549$	0	0
$550$	17.6969	0.754600
$551$	2.55940	0.109034
$552$	0	0
$553$	22.2474	0.946058
$554$	9.64553	0.409799
$555$	0	0
$556$	14.4949	0.614721
$557$	−26.7444	−1.13320	−0.566598	−	0.823994i	$-0.691741\pi$
$557$	−26.7444	−1.13320	−0.566598	+	0.823994i	$0.691741\pi$
$558$	0	0
$559$	−10.6969	−0.452432
$560$	14.6894	0.620739
$561$	0	0
$562$	−17.6413	−0.744154
$563$	25.2605	1.06460	0.532301	−	0.846555i	$-0.321328\pi$
$563$	25.2605	1.06460	0.532301	+	0.846555i	$0.321328\pi$
$564$	0	0
$565$	−13.3485	−0.561574
$566$	−5.11879	−0.215159
$567$	0	0
$568$	−26.2020	−1.09941
$569$	9.90408	0.415201	0.207600	−	0.978214i	$-0.433435\pi$
$569$	9.90408	0.415201	0.207600	+	0.978214i	$0.433435\pi$
$570$	0	0
$571$	−13.5505	−0.567071	−0.283536	−	0.958962i	$-0.591507\pi$
$571$	−13.5505	−0.567071	−0.283536	+	0.958962i	$0.591507\pi$
$572$	11.7215	0.490101
$573$	0	0
$574$	−12.2474	−0.511199
$575$	28.2283	1.17720
$576$	0	0
$577$	40.6969	1.69424	0.847118	−	0.531405i	$-0.178336\pi$
$577$	40.6969	1.69424	0.847118	+	0.531405i	$0.178336\pi$
$578$	−19.7332	−0.820793
$579$	0	0
$580$	−12.2474	−0.508548
$581$	26.0774	1.08187
$582$	0	0
$583$	44.3939	1.83861
$584$	−21.3670	−0.884175
$585$	0	0
$586$	3.00000	0.123929
$587$	24.7434	1.02127	0.510634	−	0.859798i	$-0.329411\pi$
$587$	24.7434	1.02127	0.510634	+	0.859798i	$0.329411\pi$
$588$	0	0
$589$	7.44949	0.306951
$590$	−27.8948	−1.14841
$591$	0	0
$592$	4.44949	0.182873
$593$	−25.9275	−1.06471	−0.532357	−	0.846520i	$-0.678694\pi$
$593$	−25.9275	−1.06471	−0.532357	+	0.846520i	$0.678694\pi$
$594$	0	0
$595$	−96.9898	−3.97619
$596$	−2.15094	−0.0881058
$597$	0	0
$598$	−7.10102	−0.290382
$599$	12.0550	0.492555	0.246277	−	0.969199i	$-0.420793\pi$
$599$	12.0550	0.492555	0.246277	+	0.969199i	$0.420793\pi$
$600$	0	0
$601$	1.14643	0.0467638	0.0233819	−	0.999727i	$-0.492557\pi$
$601$	1.14643	0.0467638	0.0233819	+	0.999727i	$0.492557\pi$
$602$	17.6572	0.719655
$603$	0	0
$604$	14.0556	0.571915
$605$	17.6572	0.717868
$606$	0	0
$607$	−1.24745	−0.0506324	−0.0253162	−	0.999679i	$-0.508059\pi$
$607$	−1.24745	−0.0506324	−0.0253162	+	0.999679i	$0.508059\pi$
$608$	−5.86076	−0.237685
$609$	0	0
$610$	20.4495	0.827976
$611$	1.48393	0.0600333
$612$	0	0
$613$	−23.1010	−0.933041	−0.466521	−	0.884510i	$-0.654493\pi$
$613$	−23.1010	−0.933041	−0.466521	+	0.884510i	$0.654493\pi$
$614$	8.01169	0.323326
$615$	0	0
$616$	−46.0454	−1.85522
$617$	3.15145	0.126873	0.0634364	−	0.997986i	$-0.479794\pi$
$617$	3.15145	0.126873	0.0634364	+	0.997986i	$0.479794\pi$
$618$	0	0
$619$	35.7980	1.43884	0.719421	−	0.694575i	$-0.244407\pi$
$619$	35.7980	1.43884	0.719421	+	0.694575i	$0.244407\pi$
$620$	−35.6480	−1.43166
$621$	0	0
$622$	−5.69694	−0.228426
$623$	−24.5934	−0.985316
$624$	0	0
$625$	−19.6969	−0.787878
$626$	−8.34520	−0.333541
$627$	0	0
$628$	20.7980	0.829929
$629$	−29.3787	−1.17141
$630$	0	0
$631$	−27.3939	−1.09053	−0.545267	−	0.838263i	$-0.683572\pi$
$631$	−27.3939	−1.09053	−0.545267	+	0.838263i	$0.683572\pi$
$632$	12.7970	0.509037
$633$	0	0
$634$	−14.9444	−0.593517
$635$	−45.5520	−1.80768
$636$	0	0
$637$	25.5959	1.01415
$638$	7.67819	0.303982
$639$	0	0
$640$	32.9444	1.30224
$641$	22.5175	0.889386	0.444693	−	0.895683i	$-0.353313\pi$
$641$	22.5175	0.889386	0.444693	+	0.895683i	$0.353313\pi$
$642$	0	0
$643$	−11.3485	−0.447540	−0.223770	−	0.974642i	$-0.571836\pi$
$643$	−11.3485	−0.447540	−0.223770	+	0.974642i	$0.571836\pi$
$644$	−30.8627	−1.21616
$645$	0	0
$646$	4.89898	0.192748
$647$	−21.2171	−0.834132	−0.417066	−	0.908876i	$-0.636942\pi$
$647$	−21.2171	−0.834132	−0.417066	+	0.908876i	$0.636942\pi$
$648$	0	0
$649$	−46.0454	−1.80744
$650$	−8.75366	−0.343347
$651$	0	0
$652$	23.5505	0.922309
$653$	−0.816917	−0.0319684	−0.0159842	−	0.999872i	$-0.505088\pi$
$653$	−0.816917	−0.0319684	−0.0159842	+	0.999872i	$0.505088\pi$
$654$	0	0
$655$	−57.1918	−2.23467
$656$	3.70982	0.144844
$657$	0	0
$658$	−2.44949	−0.0954911
$659$	13.6889	0.533242	0.266621	−	0.963801i	$-0.414093\pi$
$659$	13.6889	0.533242	0.266621	+	0.963801i	$0.414093\pi$
$660$	0	0
$661$	39.5959	1.54010	0.770051	−	0.637982i	$-0.220231\pi$
$661$	39.5959	1.54010	0.770051	+	0.637982i	$0.220231\pi$
$662$	13.0555	0.507418
$663$	0	0
$664$	15.0000	0.582113
$665$	14.6894	0.569629
$666$	0	0
$667$	12.2474	0.474223
$668$	−3.11776	−0.120630
$669$	0	0
$670$	5.75255	0.222240
$671$	33.7556	1.30312
$672$	0	0
$673$	15.1010	0.582102	0.291051	−	0.956708i	$-0.405995\pi$
$673$	15.1010	0.582102	0.291051	+	0.956708i	$0.405995\pi$
$674$	−2.48444	−0.0956972
$675$	0	0
$676$	13.0454	0.501746
$677$	−11.5379	−0.443438	−0.221719	−	0.975111i	$-0.571167\pi$
$677$	−11.5379	−0.443438	−0.221719	+	0.975111i	$0.571167\pi$
$678$	0	0
$679$	24.6969	0.947782
$680$	−55.7896	−2.13943
$681$	0	0
$682$	22.3485	0.855767
$683$	12.7220	0.486795	0.243397	−	0.969927i	$-0.421738\pi$
$683$	12.7220	0.486795	0.243397	+	0.969927i	$0.421738\pi$
$684$	0	0
$685$	9.79796	0.374361
$686$	−19.1412	−0.730813
$687$	0	0
$688$	−5.34847	−0.203908
$689$	−21.9591	−0.836575
$690$	0	0
$691$	−26.0454	−0.990814	−0.495407	−	0.868661i	$-0.664981\pi$
$691$	−26.0454	−0.990814	−0.495407	+	0.868661i	$0.664981\pi$
$692$	24.5185	0.932053
$693$	0	0
$694$	−20.4495	−0.776252
$695$	−33.0136	−1.25228
$696$	0	0
$697$	−24.4949	−0.927810
$698$	10.8296	0.409908
$699$	0	0
$700$	−38.0454	−1.43798
$701$	25.9275	0.979267	0.489634	−	0.871928i	$-0.337131\pi$
$701$	25.9275	0.979267	0.489634	+	0.871928i	$0.337131\pi$
$702$	0	0
$703$	4.44949	0.167816
$704$	−9.49562	−0.357880
$705$	0	0
$706$	17.3939	0.654627
$707$	−34.4975	−1.29741
$708$	0	0
$709$	2.00000	0.0751116	0.0375558	−	0.999295i	$-0.488043\pi$
$709$	2.00000	0.0751116	0.0375558	+	0.999295i	$0.488043\pi$
$710$	25.0769	0.941118
$711$	0	0
$712$	−14.1464	−0.530160
$713$	35.6480	1.33503
$714$	0	0
$715$	−26.6969	−0.998409
$716$	−9.08716	−0.339603
$717$	0	0
$718$	21.5505	0.804258
$719$	−2.37580	−0.0886023	−0.0443012	−	0.999018i	$-0.514106\pi$
$719$	−2.37580	−0.0886023	−0.0443012	+	0.999018i	$0.514106\pi$
$720$	0	0
$721$	52.4949	1.95501
$722$	−0.741964	−0.0276130
$723$	0	0
$724$	21.9546	0.815936
$725$	15.0978	0.560719
$726$	0	0
$727$	49.3939	1.83192	0.915959	−	0.401272i	$-0.131432\pi$
$727$	49.3939	1.83192	0.915959	+	0.401272i	$0.131432\pi$
$728$	22.7760	0.844135
$729$	0	0
$730$	20.4495	0.756870
$731$	35.3144	1.30615
$732$	0	0
$733$	33.0454	1.22056	0.610280	−	0.792186i	$-0.291057\pi$
$733$	33.0454	1.22056	0.610280	+	0.792186i	$0.291057\pi$
$734$	4.96889	0.183405
$735$	0	0
$736$	−28.0454	−1.03377
$737$	9.49562	0.349776
$738$	0	0
$739$	−14.6515	−0.538965	−0.269483	−	0.963005i	$-0.586853\pi$
$739$	−14.6515	−0.538965	−0.269483	+	0.963005i	$0.586853\pi$
$740$	−21.2921	−0.782713
$741$	0	0
$742$	36.2474	1.33069
$743$	5.11879	0.187790	0.0938951	−	0.995582i	$-0.470068\pi$
$743$	5.11879	0.187790	0.0938951	+	0.995582i	$0.470068\pi$
$744$	0	0
$745$	4.89898	0.179485
$746$	−17.8408	−0.653199
$747$	0	0
$748$	−38.6969	−1.41490
$749$	80.0496	2.92495
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	0.741964	0.0270566
$753$	0	0
$754$	−3.79796	−0.138314
$755$	−32.0131	−1.16508
$756$	0	0
$757$	31.6969	1.15204	0.576022	−	0.817434i	$-0.304604\pi$
$757$	31.6969	1.15204	0.576022	+	0.817434i	$0.304604\pi$
$758$	−26.3359	−0.956564
$759$	0	0
$760$	8.44949	0.306495
$761$	47.7030	1.72923	0.864616	−	0.502434i	$-0.167562\pi$
$761$	47.7030	1.72923	0.864616	+	0.502434i	$0.167562\pi$
$762$	0	0
$763$	36.6969	1.32852
$764$	12.3136	0.445489
$765$	0	0
$766$	11.6413	0.420618
$767$	22.7760	0.822394
$768$	0	0
$769$	31.3939	1.13209	0.566046	−	0.824374i	$-0.308472\pi$
$769$	31.3939	1.13209	0.566046	+	0.824374i	$0.308472\pi$
$770$	44.0681	1.58810
$771$	0	0
$772$	35.7980	1.28840
$773$	−34.4226	−1.23809	−0.619047	−	0.785354i	$-0.712481\pi$
$773$	−34.4226	−1.23809	−0.619047	+	0.785354i	$0.712481\pi$
$774$	0	0
$775$	43.9444	1.57853
$776$	14.2060	0.509964
$777$	0	0
$778$	1.84337	0.0660879
$779$	3.70982	0.132918
$780$	0	0
$781$	41.3939	1.48119
$782$	23.4430	0.838321
$783$	0	0
$784$	12.7980	0.457070
$785$	−47.3695	−1.69069
$786$	0	0
$787$	−19.2474	−0.686097	−0.343049	−	0.939318i	$-0.611460\pi$
$787$	−19.2474	−0.686097	−0.343049	+	0.939318i	$0.611460\pi$
$788$	−10.5374	−0.375379
$789$	0	0
$790$	−12.2474	−0.435745
$791$	−17.9907	−0.639677
$792$	0	0
$793$	−16.6969	−0.592926
$794$	9.64553	0.342307
$795$	0	0
$796$	−20.2929	−0.719261
$797$	−34.5725	−1.22462	−0.612310	−	0.790618i	$-0.709760\pi$
$797$	−34.5725	−1.22462	−0.612310	+	0.790618i	$0.709760\pi$
$798$	0	0
$799$	−4.89898	−0.173313
$800$	−34.5725	−1.22232
$801$	0	0
$802$	5.64133	0.199202
$803$	33.7556	1.19121
$804$	0	0
$805$	70.2929	2.47750
$806$	−11.0545	−0.389378
$807$	0	0
$808$	−19.8434	−0.698087
$809$	1.00052	0.0351762	0.0175881	−	0.999845i	$-0.494401\pi$
$809$	1.00052	0.0351762	0.0175881	+	0.999845i	$0.494401\pi$
$810$	0	0
$811$	−14.1010	−0.495154	−0.247577	−	0.968868i	$-0.579634\pi$
$811$	−14.1010	−0.495154	−0.247577	+	0.968868i	$0.579634\pi$
$812$	−16.5068	−0.579275
$813$	0	0
$814$	13.3485	0.467864
$815$	−53.6387	−1.87888
$816$	0	0
$817$	−5.34847	−0.187119
$818$	−1.66753	−0.0583037
$819$	0	0
$820$	−17.7526	−0.619946
$821$	−18.6577	−0.651160	−0.325580	−	0.945515i	$-0.605559\pi$
$821$	−18.6577	−0.651160	−0.325580	+	0.945515i	$0.605559\pi$
$822$	0	0
$823$	44.2474	1.54237	0.771185	−	0.636612i	$-0.219665\pi$
$823$	44.2474	1.54237	0.771185	+	0.636612i	$0.219665\pi$
$824$	30.1957	1.05192
$825$	0	0
$826$	−37.5959	−1.30813
$827$	−43.7346	−1.52080	−0.760401	−	0.649454i	$-0.774997\pi$
$827$	−43.7346	−1.52080	−0.760401	+	0.649454i	$0.774997\pi$
$828$	0	0
$829$	−44.2929	−1.53835	−0.769177	−	0.639035i	$-0.779334\pi$
$829$	−44.2929	−1.53835	−0.769177	+	0.639035i	$0.779334\pi$
$830$	−14.3559	−0.498299
$831$	0	0
$832$	4.69694	0.162837
$833$	−84.5013	−2.92780
$834$	0	0
$835$	7.10102	0.245741
$836$	5.86076	0.202699
$837$	0	0
$838$	−13.6515	−0.471584
$839$	−36.1314	−1.24739	−0.623697	−	0.781667i	$-0.714370\pi$
$839$	−36.1314	−1.24739	−0.623697	+	0.781667i	$0.714370\pi$
$840$	0	0
$841$	−22.4495	−0.774120
$842$	17.1401	0.590688
$843$	0	0
$844$	−17.8990	−0.616108
$845$	−29.7122	−1.02213
$846$	0	0
$847$	23.7980	0.817708
$848$	−10.9795	−0.377039
$849$	0	0
$850$	28.8990	0.991227
$851$	21.2921	0.729883
$852$	0	0
$853$	−27.6969	−0.948325	−0.474163	−	0.880437i	$-0.657249\pi$
$853$	−27.6969	−0.948325	−0.474163	+	0.880437i	$0.657249\pi$
$854$	27.5613	0.943128
$855$	0	0
$856$	46.0454	1.57380
$857$	−5.37734	−0.183687	−0.0918433	−	0.995773i	$-0.529276\pi$
$857$	−5.37734	−0.183687	−0.0918433	+	0.995773i	$0.529276\pi$
$858$	0	0
$859$	44.0000	1.50126	0.750630	−	0.660722i	$-0.229750\pi$
$859$	44.0000	1.50126	0.750630	+	0.660722i	$0.229750\pi$
$860$	25.5940	0.872747
$861$	0	0
$862$	8.44949	0.287791
$863$	56.6065	1.92691	0.963454	−	0.267872i	$-0.0863205\pi$
$863$	56.6065	1.92691	0.963454	+	0.267872i	$0.0863205\pi$
$864$	0	0
$865$	−55.8434	−1.89873
$866$	−4.93519	−0.167705
$867$	0	0
$868$	−48.0454	−1.63077
$869$	−20.2166	−0.685802
$870$	0	0
$871$	−4.69694	−0.159150
$872$	21.1085	0.714824
$873$	0	0
$874$	−3.55051	−0.120098
$875$	13.2054	0.446425
$876$	0	0
$877$	57.3485	1.93652	0.968260	−	0.249945i	$-0.0804125\pi$
$877$	57.3485	1.93652	0.968260	+	0.249945i	$0.0804125\pi$
$878$	−1.70879	−0.0576688
$879$	0	0
$880$	−13.3485	−0.449977
$881$	16.1733	0.544892	0.272446	−	0.962171i	$-0.412167\pi$
$881$	16.1733	0.544892	0.272446	+	0.962171i	$0.412167\pi$
$882$	0	0
$883$	12.0454	0.405360	0.202680	−	0.979245i	$-0.435035\pi$
$883$	12.0454	0.405360	0.202680	+	0.979245i	$0.435035\pi$
$884$	19.1412	0.643787
$885$	0	0
$886$	−5.94439	−0.199706
$887$	23.7765	0.798338	0.399169	−	0.916877i	$-0.369299\pi$
$887$	23.7765	0.798338	0.399169	+	0.916877i	$0.369299\pi$
$888$	0	0
$889$	−61.3939	−2.05908
$890$	13.5389	0.453827
$891$	0	0
$892$	−33.3383	−1.11625
$893$	0.741964	0.0248289
$894$	0	0
$895$	20.6969	0.691822
$896$	44.4016	1.48335
$897$	0	0
$898$	−13.0454	−0.435331
$899$	19.0662	0.635893
$900$	0	0
$901$	72.4949	2.41516
$902$	11.1295	0.370570
$903$	0	0
$904$	−10.3485	−0.344185
$905$	−50.0038	−1.66218
$906$	0	0
$907$	−2.40408	−0.0798262	−0.0399131	−	0.999203i	$-0.512708\pi$
$907$	−2.40408	−0.0798262	−0.0399131	+	0.999203i	$0.512708\pi$
$908$	35.1645	1.16698
$909$	0	0
$910$	−21.7980	−0.722595
$911$	−2.63435	−0.0872799	−0.0436400	−	0.999047i	$-0.513895\pi$
$911$	−2.63435	−0.0872799	−0.0436400	+	0.999047i	$0.513895\pi$
$912$	0	0
$913$	−23.6969	−0.784254
$914$	−3.07650	−0.101762
$915$	0	0
$916$	17.2474	0.569872
$917$	−77.0817	−2.54546
$918$	0	0
$919$	−46.2474	−1.52556	−0.762781	−	0.646657i	$-0.776167\pi$
$919$	−46.2474	−1.52556	−0.762781	+	0.646657i	$0.776167\pi$
$920$	40.4332	1.33304
$921$	0	0
$922$	−7.10102	−0.233860
$923$	−20.4752	−0.673948
$924$	0	0
$925$	26.2474	0.863011
$926$	2.15094	0.0706842
$927$	0	0
$928$	−15.0000	−0.492399
$929$	−48.0365	−1.57603	−0.788013	−	0.615659i	$-0.788890\pi$
$929$	−48.0365	−1.57603	−0.788013	+	0.615659i	$0.788890\pi$
$930$	0	0
$931$	12.7980	0.419436
$932$	−34.6811	−1.13602
$933$	0	0
$934$	−7.04541	−0.230533
$935$	88.1362	2.88236
$936$	0	0
$937$	−48.3939	−1.58096	−0.790480	−	0.612488i	$-0.790169\pi$
$937$	−48.3939	−1.58096	−0.790480	+	0.612488i	$0.790169\pi$
$938$	7.75314	0.253149
$939$	0	0
$940$	−3.55051	−0.115805
$941$	8.42015	0.274489	0.137245	−	0.990537i	$-0.456175\pi$
$941$	8.42015	0.274489	0.137245	+	0.990537i	$0.456175\pi$
$942$	0	0
$943$	17.7526	0.578103
$944$	11.3880	0.370648
$945$	0	0
$946$	−16.0454	−0.521681
$947$	1.07547	0.0349480	0.0174740	−	0.999847i	$-0.494438\pi$
$947$	1.07547	0.0349480	0.0174740	+	0.999847i	$0.494438\pi$
$948$	0	0
$949$	−16.6969	−0.542006
$950$	−4.37683	−0.142003
$951$	0	0
$952$	−75.1918	−2.43698
$953$	2.07598	0.0672477	0.0336239	−	0.999435i	$-0.489295\pi$
$953$	2.07598	0.0672477	0.0336239	+	0.999435i	$0.489295\pi$
$954$	0	0
$955$	−28.0454	−0.907528
$956$	−9.67922	−0.313048
$957$	0	0
$958$	−28.1566	−0.909700
$959$	13.2054	0.426426
$960$	0	0
$961$	24.4949	0.790158
$962$	−6.60272	−0.212880
$963$	0	0
$964$	33.8434	1.09002
$965$	−81.5335	−2.62466
$966$	0	0
$967$	30.0454	0.966195	0.483098	−	0.875567i	$-0.339512\pi$
$967$	30.0454	0.966195	0.483098	+	0.875567i	$0.339512\pi$
$968$	13.6889	0.439976
$969$	0	0
$970$	−13.5959	−0.436539
$971$	26.5608	0.852376	0.426188	−	0.904635i	$-0.359856\pi$
$971$	26.5608	0.852376	0.426188	+	0.904635i	$0.359856\pi$
$972$	0	0
$973$	−44.4949	−1.42644
$974$	−4.71033	−0.150929
$975$	0	0
$976$	−8.34847	−0.267228
$977$	62.4673	1.99851	0.999253	−	0.0386481i	$-0.0123051\pi$
$977$	62.4673	1.99851	0.999253	+	0.0386481i	$0.0123051\pi$
$978$	0	0
$979$	22.3485	0.714260
$980$	−61.2419	−1.95630
$981$	0	0
$982$	−2.14643	−0.0684953
$983$	−5.60221	−0.178683	−0.0893413	−	0.996001i	$-0.528476\pi$
$983$	−5.60221	−0.178683	−0.0893413	+	0.996001i	$0.528476\pi$
$984$	0	0
$985$	24.0000	0.764704
$986$	12.5384	0.399305
$987$	0	0
$988$	−2.89898	−0.0922288
$989$	−25.5940	−0.813841
$990$	0	0
$991$	10.6969	0.339799	0.169900	−	0.985461i	$-0.445656\pi$
$991$	10.6969	0.339799	0.169900	+	0.985461i	$0.445656\pi$
$992$	−43.6596	−1.38620
$993$	0	0
$994$	33.7980	1.07201
$995$	46.2190	1.46524
$996$	0	0
$997$	2.00000	0.0633406	0.0316703	−	0.999498i	$-0.489917\pi$
$997$	2.00000	0.0633406	0.0316703	+	0.999498i	$0.489917\pi$
$998$	25.2268	0.798539
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.h.1.2	✓	4
3.2	odd	2	inner	513.2.a.h.1.3	yes	4
4.3	odd	2		8208.2.a.bu.1.4		4
12.11	even	2		8208.2.a.bu.1.1		4
19.18	odd	2		9747.2.a.be.1.3		4
57.56	even	2		9747.2.a.be.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.h.1.2	✓	4	1.1	even	1	trivial
513.2.a.h.1.3	yes	4	3.2	odd	2	inner
8208.2.a.bu.1.1		4	12.11	even	2
8208.2.a.bu.1.4		4	4.3	odd	2
9747.2.a.be.1.2		4	57.56	even	2
9747.2.a.be.1.3		4	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)

\(f(q)\)	\(=\)	\(q-0.741964 q^{2} -1.44949 q^{4} +3.30136 q^{5} +4.44949 q^{7} +2.55940 q^{8} -2.44949 q^{10} -4.04332 q^{11} +2.00000 q^{13} -3.30136 q^{14} +1.00000 q^{16} -6.60272 q^{17} +1.00000 q^{19} -4.78529 q^{20} +3.00000 q^{22} +4.78529 q^{23} +5.89898 q^{25} -1.48393 q^{26} -6.44949 q^{28} +2.55940 q^{29} +7.44949 q^{31} -5.86076 q^{32} +4.89898 q^{34} +14.6894 q^{35} +4.44949 q^{37} -0.741964 q^{38} +8.44949 q^{40} +3.70982 q^{41} -5.34847 q^{43} +5.86076 q^{44} -3.55051 q^{46} +0.741964 q^{47} +12.7980 q^{49} -4.37683 q^{50} -2.89898 q^{52} -10.9795 q^{53} -13.3485 q^{55} +11.3880 q^{56} -1.89898 q^{58} +11.3880 q^{59} -8.34847 q^{61} -5.52725 q^{62} +2.34847 q^{64} +6.60272 q^{65} -2.34847 q^{67} +9.57058 q^{68} -10.8990 q^{70} -10.2376 q^{71} -8.34847 q^{73} -3.30136 q^{74} -1.44949 q^{76} -17.9907 q^{77} +5.00000 q^{79} +3.30136 q^{80} -2.75255 q^{82} +5.86076 q^{83} -21.7980 q^{85} +3.96837 q^{86} -10.3485 q^{88} -5.52725 q^{89} +8.89898 q^{91} -6.93623 q^{92} -0.550510 q^{94} +3.30136 q^{95} +5.55051 q^{97} -9.49562 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} + 8 q^{7} + 8 q^{13} + 4 q^{16} + 4 q^{19} + 12 q^{22} + 4 q^{25} - 16 q^{28} + 20 q^{31} + 8 q^{37} + 24 q^{40} + 8 q^{43} - 24 q^{46} + 12 q^{49} + 8 q^{52} - 24 q^{55} + 12 q^{58} - 4 q^{61}+ \cdots + 32 q^{97}+O(q^{100}) \) 4 * q + 4 * q^4 + 8 * q^7 + 8 * q^13 + 4 * q^16 + 4 * q^19 + 12 * q^22 + 4 * q^25 - 16 * q^28 + 20 * q^31 + 8 * q^37 + 24 * q^40 + 8 * q^43 - 24 * q^46 + 12 * q^49 + 8 * q^52 - 24 * q^55 + 12 * q^58 - 4 * q^61 - 20 * q^64 + 20 * q^67 - 24 * q^70 - 4 * q^73 + 4 * q^76 + 20 * q^79 - 60 * q^82 - 48 * q^85 - 12 * q^88 + 16 * q^91 - 12 * q^94 + 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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