LMFDB - Embedded newform 4024.2.a.g.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4024,2,Mod(1,4024)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4024, 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("4024.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4024 = 2^{3} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4024.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:	$32.1318017734$
Analytic rank:	$0$
Dimension:	$33$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	4024.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.13657 q^{3} -1.27952 q^{5} +1.02417 q^{7} +1.56495 q^{9} +O(q^{10})$	$q-2.13657 q^{3} -1.27952 q^{5} +1.02417 q^{7} +1.56495 q^{9} +1.03872 q^{11} +5.61131 q^{13} +2.73380 q^{15} +3.42406 q^{17} +0.135949 q^{19} -2.18821 q^{21} +8.07345 q^{23} -3.36282 q^{25} +3.06610 q^{27} -3.88941 q^{29} +4.43635 q^{31} -2.21929 q^{33} -1.31045 q^{35} -0.834672 q^{37} -11.9890 q^{39} -7.95342 q^{41} -11.6658 q^{43} -2.00239 q^{45} +1.96362 q^{47} -5.95108 q^{49} -7.31576 q^{51} +6.11245 q^{53} -1.32906 q^{55} -0.290465 q^{57} -10.9540 q^{59} -1.06354 q^{61} +1.60277 q^{63} -7.17981 q^{65} +11.5019 q^{67} -17.2495 q^{69} +6.84947 q^{71} +3.52204 q^{73} +7.18491 q^{75} +1.06382 q^{77} +12.2712 q^{79} -11.2458 q^{81} +9.17881 q^{83} -4.38117 q^{85} +8.31001 q^{87} +12.3139 q^{89} +5.74694 q^{91} -9.47859 q^{93} -0.173950 q^{95} -1.42300 q^{97} +1.62553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	$ 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) $ 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * 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
$3$	−2.13657	−1.23355	−0.616776	−	0.787139i	$-0.711561\pi$
$3$	−2.13657	−1.23355	−0.616776	+	0.787139i	$0.711561\pi$
$4$	0	0
$5$	−1.27952	−0.572221	−0.286110	−	0.958197i	$-0.592362\pi$
$5$	−1.27952	−0.572221	−0.286110	+	0.958197i	$0.592362\pi$
$6$	0	0
$7$	1.02417	0.387100	0.193550	−	0.981090i	$-0.438000\pi$
$7$	1.02417	0.387100	0.193550	+	0.981090i	$0.438000\pi$
$8$	0	0
$9$	1.56495	0.521648
$10$	0	0
$11$	1.03872	0.313185	0.156592	−	0.987663i	$-0.449949\pi$
$11$	1.03872	0.313185	0.156592	+	0.987663i	$0.449949\pi$
$12$	0	0
$13$	5.61131	1.55630	0.778149	−	0.628080i	$-0.216159\pi$
$13$	5.61131	1.55630	0.778149	+	0.628080i	$0.216159\pi$
$14$	0	0
$15$	2.73380	0.705863
$16$	0	0
$17$	3.42406	0.830458	0.415229	−	0.909717i	$-0.363702\pi$
$17$	3.42406	0.830458	0.415229	+	0.909717i	$0.363702\pi$
$18$	0	0
$19$	0.135949	0.0311889	0.0155944	−	0.999878i	$-0.495036\pi$
$19$	0.135949	0.0311889	0.0155944	+	0.999878i	$0.495036\pi$
$20$	0	0
$21$	−2.18821	−0.477508
$22$	0	0
$23$	8.07345	1.68343	0.841715	−	0.539922i	$-0.181546\pi$
$23$	8.07345	1.68343	0.841715	+	0.539922i	$0.181546\pi$
$24$	0	0
$25$	−3.36282	−0.672563
$26$	0	0
$27$	3.06610	0.590071
$28$	0	0
$29$	−3.88941	−0.722245	−0.361123	−	0.932518i	$-0.617606\pi$
$29$	−3.88941	−0.722245	−0.361123	+	0.932518i	$0.617606\pi$
$30$	0	0
$31$	4.43635	0.796792	0.398396	−	0.917213i	$-0.369567\pi$
$31$	4.43635	0.796792	0.398396	+	0.917213i	$0.369567\pi$
$32$	0	0
$33$	−2.21929	−0.386329
$34$	0	0
$35$	−1.31045	−0.221507
$36$	0	0
$37$	−0.834672	−0.137219	−0.0686096	−	0.997644i	$-0.521856\pi$
$37$	−0.834672	−0.137219	−0.0686096	+	0.997644i	$0.521856\pi$
$38$	0	0
$39$	−11.9890	−1.91977
$40$	0	0
$41$	−7.95342	−1.24212	−0.621058	−	0.783764i	$-0.713297\pi$
$41$	−7.95342	−1.24212	−0.621058	+	0.783764i	$0.713297\pi$
$42$	0	0
$43$	−11.6658	−1.77902	−0.889508	−	0.456920i	$-0.848953\pi$
$43$	−11.6658	−1.77902	−0.889508	+	0.456920i	$0.848953\pi$
$44$	0	0
$45$	−2.00239	−0.298498
$46$	0	0
$47$	1.96362	0.286424	0.143212	−	0.989692i	$-0.454257\pi$
$47$	1.96362	0.286424	0.143212	+	0.989692i	$0.454257\pi$
$48$	0	0
$49$	−5.95108	−0.850154
$50$	0	0
$51$	−7.31576	−1.02441
$52$	0	0
$53$	6.11245	0.839609	0.419805	−	0.907615i	$-0.362099\pi$
$53$	6.11245	0.839609	0.419805	+	0.907615i	$0.362099\pi$
$54$	0	0
$55$	−1.32906	−0.179211
$56$	0	0
$57$	−0.290465	−0.0384730
$58$	0	0
$59$	−10.9540	−1.42609	−0.713045	−	0.701118i	$-0.752685\pi$
$59$	−10.9540	−1.42609	−0.713045	+	0.701118i	$0.752685\pi$
$60$	0	0
$61$	−1.06354	−0.136172	−0.0680862	−	0.997679i	$-0.521689\pi$
$61$	−1.06354	−0.136172	−0.0680862	+	0.997679i	$0.521689\pi$
$62$	0	0
$63$	1.60277	0.201930
$64$	0	0
$65$	−7.17981	−0.890546
$66$	0	0
$67$	11.5019	1.40518	0.702588	−	0.711597i	$-0.252028\pi$
$67$	11.5019	1.40518	0.702588	+	0.711597i	$0.252028\pi$
$68$	0	0
$69$	−17.2495	−2.07660
$70$	0	0
$71$	6.84947	0.812883	0.406441	−	0.913677i	$-0.366770\pi$
$71$	6.84947	0.812883	0.406441	+	0.913677i	$0.366770\pi$
$72$	0	0
$73$	3.52204	0.412224	0.206112	−	0.978528i	$-0.433919\pi$
$73$	3.52204	0.412224	0.206112	+	0.978528i	$0.433919\pi$
$74$	0	0
$75$	7.18491	0.829641
$76$	0	0
$77$	1.06382	0.121234
$78$	0	0
$79$	12.2712	1.38062	0.690308	−	0.723515i	$-0.257475\pi$
$79$	12.2712	1.38062	0.690308	+	0.723515i	$0.257475\pi$
$80$	0	0
$81$	−11.2458	−1.24953
$82$	0	0
$83$	9.17881	1.00751	0.503753	−	0.863848i	$-0.331952\pi$
$83$	9.17881	1.00751	0.503753	+	0.863848i	$0.331952\pi$
$84$	0	0
$85$	−4.38117	−0.475205
$86$	0	0
$87$	8.31001	0.890926
$88$	0	0
$89$	12.3139	1.30527	0.652633	−	0.757674i	$-0.273664\pi$
$89$	12.3139	1.30527	0.652633	+	0.757674i	$0.273664\pi$
$90$	0	0
$91$	5.74694	0.602443
$92$	0	0
$93$	−9.47859	−0.982884
$94$	0	0
$95$	−0.173950	−0.0178469
$96$	0	0
$97$	−1.42300	−0.144484	−0.0722418	−	0.997387i	$-0.523015\pi$
$97$	−1.42300	−0.144484	−0.0722418	+	0.997387i	$0.523015\pi$
$98$	0	0
$99$	1.62553	0.163372
$100$	0	0
$101$	12.2195	1.21589	0.607943	−	0.793980i	$-0.291995\pi$
$101$	12.2195	1.21589	0.607943	+	0.793980i	$0.291995\pi$
$102$	0	0
$103$	5.96037	0.587293	0.293647	−	0.955914i	$-0.405131\pi$
$103$	5.96037	0.587293	0.293647	+	0.955914i	$0.405131\pi$
$104$	0	0
$105$	2.79987	0.273240
$106$	0	0
$107$	−6.46582	−0.625074	−0.312537	−	0.949906i	$-0.601179\pi$
$107$	−6.46582	−0.625074	−0.312537	+	0.949906i	$0.601179\pi$
$108$	0	0
$109$	−19.8262	−1.89901	−0.949503	−	0.313758i	$-0.898412\pi$
$109$	−19.8262	−1.89901	−0.949503	+	0.313758i	$0.898412\pi$
$110$	0	0
$111$	1.78334	0.169267
$112$	0	0
$113$	−7.97327	−0.750062	−0.375031	−	0.927012i	$-0.622368\pi$
$113$	−7.97327	−0.750062	−0.375031	+	0.927012i	$0.622368\pi$
$114$	0	0
$115$	−10.3302	−0.963294
$116$	0	0
$117$	8.78140	0.811840
$118$	0	0
$119$	3.50682	0.321470
$120$	0	0
$121$	−9.92107	−0.901915
$122$	0	0
$123$	16.9931	1.53221
$124$	0	0
$125$	10.7004	0.957075
$126$	0	0
$127$	−6.79238	−0.602726	−0.301363	−	0.953509i	$-0.597442\pi$
$127$	−6.79238	−0.602726	−0.301363	+	0.953509i	$0.597442\pi$
$128$	0	0
$129$	24.9248	2.19451
$130$	0	0
$131$	−8.98634	−0.785141	−0.392570	−	0.919722i	$-0.628414\pi$
$131$	−8.98634	−0.785141	−0.392570	+	0.919722i	$0.628414\pi$
$132$	0	0
$133$	0.139235	0.0120732
$134$	0	0
$135$	−3.92315	−0.337651
$136$	0	0
$137$	15.9612	1.36366	0.681830	−	0.731510i	$-0.261184\pi$
$137$	15.9612	1.36366	0.681830	+	0.731510i	$0.261184\pi$
$138$	0	0
$139$	2.18777	0.185564	0.0927822	−	0.995686i	$-0.470424\pi$
$139$	2.18777	0.185564	0.0927822	+	0.995686i	$0.470424\pi$
$140$	0	0
$141$	−4.19542	−0.353318
$142$	0	0
$143$	5.82856	0.487408
$144$	0	0
$145$	4.97659	0.413284
$146$	0	0
$147$	12.7149	1.04871
$148$	0	0
$149$	−3.46161	−0.283586	−0.141793	−	0.989896i	$-0.545287\pi$
$149$	−3.46161	−0.283586	−0.141793	+	0.989896i	$0.545287\pi$
$150$	0	0
$151$	13.9436	1.13472	0.567358	−	0.823471i	$-0.307966\pi$
$151$	13.9436	1.13472	0.567358	+	0.823471i	$0.307966\pi$
$152$	0	0
$153$	5.35847	0.433207
$154$	0	0
$155$	−5.67642	−0.455941
$156$	0	0
$157$	0.827628	0.0660519	0.0330260	−	0.999454i	$-0.489486\pi$
$157$	0.827628	0.0660519	0.0330260	+	0.999454i	$0.489486\pi$
$158$	0	0
$159$	−13.0597	−1.03570
$160$	0	0
$161$	8.26859	0.651656
$162$	0	0
$163$	−6.47269	−0.506980	−0.253490	−	0.967338i	$-0.581579\pi$
$163$	−6.47269	−0.506980	−0.253490	+	0.967338i	$0.581579\pi$
$164$	0	0
$165$	2.83964	0.221066
$166$	0	0
$167$	1.32058	0.102189	0.0510947	−	0.998694i	$-0.483729\pi$
$167$	1.32058	0.102189	0.0510947	+	0.998694i	$0.483729\pi$
$168$	0	0
$169$	18.4868	1.42206
$170$	0	0
$171$	0.212753	0.0162696
$172$	0	0
$173$	−11.1325	−0.846390	−0.423195	−	0.906039i	$-0.639091\pi$
$173$	−11.1325	−0.846390	−0.423195	+	0.906039i	$0.639091\pi$
$174$	0	0
$175$	−3.44410	−0.260349
$176$	0	0
$177$	23.4040	1.75916
$178$	0	0
$179$	−9.68668	−0.724017	−0.362008	−	0.932175i	$-0.617909\pi$
$179$	−9.68668	−0.724017	−0.362008	+	0.932175i	$0.617909\pi$
$180$	0	0
$181$	4.75601	0.353511	0.176756	−	0.984255i	$-0.443440\pi$
$181$	4.75601	0.353511	0.176756	+	0.984255i	$0.443440\pi$
$182$	0	0
$183$	2.27233	0.167976
$184$	0	0
$185$	1.06798	0.0785197
$186$	0	0
$187$	3.55663	0.260086
$188$	0	0
$189$	3.14021	0.228416
$190$	0	0
$191$	−2.05364	−0.148596	−0.0742982	−	0.997236i	$-0.523672\pi$
$191$	−2.05364	−0.148596	−0.0742982	+	0.997236i	$0.523672\pi$
$192$	0	0
$193$	−8.94224	−0.643677	−0.321838	−	0.946795i	$-0.604301\pi$
$193$	−8.94224	−0.643677	−0.321838	+	0.946795i	$0.604301\pi$
$194$	0	0
$195$	15.3402	1.09853
$196$	0	0
$197$	14.9923	1.06816	0.534079	−	0.845435i	$-0.320659\pi$
$197$	14.9923	1.06816	0.534079	+	0.845435i	$0.320659\pi$
$198$	0	0
$199$	19.9435	1.41376	0.706879	−	0.707334i	$-0.250102\pi$
$199$	19.9435	1.41376	0.706879	+	0.707334i	$0.250102\pi$
$200$	0	0
$201$	−24.5746	−1.73336
$202$	0	0
$203$	−3.98342	−0.279581
$204$	0	0
$205$	10.1766	0.710765
$206$	0	0
$207$	12.6345	0.878159
$208$	0	0
$209$	0.141212	0.00976787
$210$	0	0
$211$	27.9334	1.92302	0.961508	−	0.274777i	$-0.0886040\pi$
$211$	27.9334	1.92302	0.961508	+	0.274777i	$0.0886040\pi$
$212$	0	0
$213$	−14.6344	−1.00273
$214$	0	0
$215$	14.9267	1.01799
$216$	0	0
$217$	4.54358	0.308438
$218$	0	0
$219$	−7.52509	−0.508499
$220$	0	0
$221$	19.2135	1.29244
$222$	0	0
$223$	25.6129	1.71517	0.857584	−	0.514344i	$-0.171965\pi$
$223$	25.6129	1.71517	0.857584	+	0.514344i	$0.171965\pi$
$224$	0	0
$225$	−5.26262	−0.350842
$226$	0	0
$227$	16.6687	1.10634	0.553171	−	0.833068i	$-0.313418\pi$
$227$	16.6687	1.10634	0.553171	+	0.833068i	$0.313418\pi$
$228$	0	0
$229$	−9.39519	−0.620852	−0.310426	−	0.950598i	$-0.600472\pi$
$229$	−9.39519	−0.620852	−0.310426	+	0.950598i	$0.600472\pi$
$230$	0	0
$231$	−2.27293	−0.149548
$232$	0	0
$233$	18.6895	1.22439	0.612193	−	0.790708i	$-0.290287\pi$
$233$	18.6895	1.22439	0.612193	+	0.790708i	$0.290287\pi$
$234$	0	0
$235$	−2.51250	−0.163897
$236$	0	0
$237$	−26.2183	−1.70306
$238$	0	0
$239$	18.2393	1.17980	0.589902	−	0.807475i	$-0.299166\pi$
$239$	18.2393	1.17980	0.589902	+	0.807475i	$0.299166\pi$
$240$	0	0
$241$	−14.1962	−0.914456	−0.457228	−	0.889349i	$-0.651158\pi$
$241$	−14.1962	−0.914456	−0.457228	+	0.889349i	$0.651158\pi$
$242$	0	0
$243$	14.8291	0.951290
$244$	0	0
$245$	7.61455	0.486476
$246$	0	0
$247$	0.762852	0.0485391
$248$	0	0
$249$	−19.6112	−1.24281
$250$	0	0
$251$	18.1409	1.14504	0.572521	−	0.819890i	$-0.305965\pi$
$251$	18.1409	1.14504	0.572521	+	0.819890i	$0.305965\pi$
$252$	0	0
$253$	8.38602	0.527224
$254$	0	0
$255$	9.36070	0.586190
$256$	0	0
$257$	10.3998	0.648720	0.324360	−	0.945934i	$-0.394851\pi$
$257$	10.3998	0.648720	0.324360	+	0.945934i	$0.394851\pi$
$258$	0	0
$259$	−0.854846	−0.0531176
$260$	0	0
$261$	−6.08671	−0.376758
$262$	0	0
$263$	27.1545	1.67442	0.837210	−	0.546881i	$-0.184185\pi$
$263$	27.1545	1.67442	0.837210	+	0.546881i	$0.184185\pi$
$264$	0	0
$265$	−7.82102	−0.480442
$266$	0	0
$267$	−26.3095	−1.61011
$268$	0	0
$269$	−16.9134	−1.03123	−0.515614	−	0.856821i	$-0.672436\pi$
$269$	−16.9134	−1.03123	−0.515614	+	0.856821i	$0.672436\pi$
$270$	0	0
$271$	−26.3449	−1.60034	−0.800169	−	0.599775i	$-0.795257\pi$
$271$	−26.3449	−1.60034	−0.800169	+	0.599775i	$0.795257\pi$
$272$	0	0
$273$	−12.2788	−0.743144
$274$	0	0
$275$	−3.49301	−0.210636
$276$	0	0
$277$	−15.6878	−0.942591	−0.471296	−	0.881975i	$-0.656214\pi$
$277$	−15.6878	−0.942591	−0.471296	+	0.881975i	$0.656214\pi$
$278$	0	0
$279$	6.94265	0.415645
$280$	0	0
$281$	15.5461	0.927404	0.463702	−	0.885991i	$-0.346521\pi$
$281$	15.5461	0.927404	0.463702	+	0.885991i	$0.346521\pi$
$282$	0	0
$283$	13.4896	0.801871	0.400935	−	0.916106i	$-0.368685\pi$
$283$	13.4896	0.801871	0.400935	+	0.916106i	$0.368685\pi$
$284$	0	0
$285$	0.371657	0.0220151
$286$	0	0
$287$	−8.14566	−0.480823
$288$	0	0
$289$	−5.27578	−0.310340
$290$	0	0
$291$	3.04034	0.178228
$292$	0	0
$293$	−2.24155	−0.130953	−0.0654765	−	0.997854i	$-0.520857\pi$
$293$	−2.24155	−0.130953	−0.0654765	+	0.997854i	$0.520857\pi$
$294$	0	0
$295$	14.0159	0.816039
$296$	0	0
$297$	3.18481	0.184801
$298$	0	0
$299$	45.3026	2.61992
$300$	0	0
$301$	−11.9477	−0.688657
$302$	0	0
$303$	−26.1079	−1.49986
$304$	0	0
$305$	1.36083	0.0779206
$306$	0	0
$307$	−31.4652	−1.79582	−0.897908	−	0.440184i	$-0.854913\pi$
$307$	−31.4652	−1.79582	−0.897908	+	0.440184i	$0.854913\pi$
$308$	0	0
$309$	−12.7348	−0.724456
$310$	0	0
$311$	22.9866	1.30345	0.651725	−	0.758455i	$-0.274046\pi$
$311$	22.9866	1.30345	0.651725	+	0.758455i	$0.274046\pi$
$312$	0	0
$313$	−10.4702	−0.591812	−0.295906	−	0.955217i	$-0.595621\pi$
$313$	−10.4702	−0.591812	−0.295906	+	0.955217i	$0.595621\pi$
$314$	0	0
$315$	−2.05078	−0.115549
$316$	0	0
$317$	2.71961	0.152748	0.0763742	−	0.997079i	$-0.475666\pi$
$317$	2.71961	0.152748	0.0763742	+	0.997079i	$0.475666\pi$
$318$	0	0
$319$	−4.03999	−0.226196
$320$	0	0
$321$	13.8147	0.771061
$322$	0	0
$323$	0.465498	0.0259010
$324$	0	0
$325$	−18.8698	−1.04671
$326$	0	0
$327$	42.3601	2.34252
$328$	0	0
$329$	2.01108	0.110875
$330$	0	0
$331$	−11.0265	−0.606073	−0.303036	−	0.952979i	$-0.598000\pi$
$331$	−11.0265	−0.606073	−0.303036	+	0.952979i	$0.598000\pi$
$332$	0	0
$333$	−1.30622	−0.0715802
$334$	0	0
$335$	−14.7169	−0.804071
$336$	0	0
$337$	19.3444	1.05376	0.526878	−	0.849941i	$-0.323363\pi$
$337$	19.3444	1.05376	0.526878	+	0.849941i	$0.323363\pi$
$338$	0	0
$339$	17.0355	0.925240
$340$	0	0
$341$	4.60811	0.249543
$342$	0	0
$343$	−13.2641	−0.716194
$344$	0	0
$345$	22.0712	1.18827
$346$	0	0
$347$	−18.4678	−0.991405	−0.495702	−	0.868493i	$-0.665089\pi$
$347$	−18.4678	−0.991405	−0.495702	+	0.868493i	$0.665089\pi$
$348$	0	0
$349$	−15.5342	−0.831524	−0.415762	−	0.909473i	$-0.636485\pi$
$349$	−15.5342	−0.831524	−0.415762	+	0.909473i	$0.636485\pi$
$350$	0	0
$351$	17.2048	0.918327
$352$	0	0
$353$	−0.716588	−0.0381401	−0.0190701	−	0.999818i	$-0.506071\pi$
$353$	−0.716588	−0.0381401	−0.0190701	+	0.999818i	$0.506071\pi$
$354$	0	0
$355$	−8.76407	−0.465148
$356$	0	0
$357$	−7.49259	−0.396550
$358$	0	0
$359$	24.4123	1.28843	0.644216	−	0.764844i	$-0.277184\pi$
$359$	24.4123	1.28843	0.644216	+	0.764844i	$0.277184\pi$
$360$	0	0
$361$	−18.9815	−0.999027
$362$	0	0
$363$	21.1971	1.11256
$364$	0	0
$365$	−4.50653	−0.235883
$366$	0	0
$367$	18.5735	0.969530	0.484765	−	0.874644i	$-0.338905\pi$
$367$	18.5735	0.969530	0.484765	+	0.874644i	$0.338905\pi$
$368$	0	0
$369$	−12.4467	−0.647948
$370$	0	0
$371$	6.26018	0.325013
$372$	0	0
$373$	26.4678	1.37045	0.685224	−	0.728332i	$-0.259704\pi$
$373$	26.4678	1.37045	0.685224	+	0.728332i	$0.259704\pi$
$374$	0	0
$375$	−22.8622	−1.18060
$376$	0	0
$377$	−21.8247	−1.12403
$378$	0	0
$379$	25.6811	1.31915	0.659574	−	0.751639i	$-0.270737\pi$
$379$	25.6811	1.31915	0.659574	+	0.751639i	$0.270737\pi$
$380$	0	0
$381$	14.5124	0.743493
$382$	0	0
$383$	−30.4716	−1.55703	−0.778513	−	0.627629i	$-0.784026\pi$
$383$	−30.4716	−1.55703	−0.778513	+	0.627629i	$0.784026\pi$
$384$	0	0
$385$	−1.36119	−0.0693724
$386$	0	0
$387$	−18.2563	−0.928021
$388$	0	0
$389$	13.9086	0.705196	0.352598	−	0.935775i	$-0.385298\pi$
$389$	13.9086	0.705196	0.352598	+	0.935775i	$0.385298\pi$
$390$	0	0
$391$	27.6440	1.39802
$392$	0	0
$393$	19.2000	0.968511
$394$	0	0
$395$	−15.7013	−0.790017
$396$	0	0
$397$	13.1165	0.658297	0.329148	−	0.944278i	$-0.393238\pi$
$397$	13.1165	0.658297	0.329148	+	0.944278i	$0.393238\pi$
$398$	0	0
$399$	−0.297486	−0.0148929
$400$	0	0
$401$	13.0707	0.652720	0.326360	−	0.945245i	$-0.394178\pi$
$401$	13.0707	0.652720	0.326360	+	0.945245i	$0.394178\pi$
$402$	0	0
$403$	24.8937	1.24005
$404$	0	0
$405$	14.3893	0.715008
$406$	0	0
$407$	−0.866987	−0.0429749
$408$	0	0
$409$	−15.3793	−0.760457	−0.380228	−	0.924893i	$-0.624155\pi$
$409$	−15.3793	−0.760457	−0.380228	+	0.924893i	$0.624155\pi$
$410$	0	0
$411$	−34.1024	−1.68215
$412$	0	0
$413$	−11.2188	−0.552040
$414$	0	0
$415$	−11.7445	−0.576515
$416$	0	0
$417$	−4.67434	−0.228903
$418$	0	0
$419$	9.53979	0.466049	0.233025	−	0.972471i	$-0.425138\pi$
$419$	9.53979	0.466049	0.233025	+	0.972471i	$0.425138\pi$
$420$	0	0
$421$	37.5858	1.83182	0.915909	−	0.401385i	$-0.131471\pi$
$421$	37.5858	1.83182	0.915909	+	0.401385i	$0.131471\pi$
$422$	0	0
$423$	3.07296	0.149412
$424$	0	0
$425$	−11.5145	−0.558535
$426$	0	0
$427$	−1.08925	−0.0527123
$428$	0	0
$429$	−12.4531	−0.601243
$430$	0	0
$431$	29.8765	1.43910	0.719550	−	0.694440i	$-0.244348\pi$
$431$	29.8765	1.43910	0.719550	+	0.694440i	$0.244348\pi$
$432$	0	0
$433$	−2.01490	−0.0968301	−0.0484150	−	0.998827i	$-0.515417\pi$
$433$	−2.01490	−0.0968301	−0.0484150	+	0.998827i	$0.515417\pi$
$434$	0	0
$435$	−10.6329	−0.509806
$436$	0	0
$437$	1.09758	0.0525043
$438$	0	0
$439$	−0.190127	−0.00907428	−0.00453714	−	0.999990i	$-0.501444\pi$
$439$	−0.190127	−0.00907428	−0.00453714	+	0.999990i	$0.501444\pi$
$440$	0	0
$441$	−9.31311	−0.443481
$442$	0	0
$443$	21.7607	1.03388	0.516941	−	0.856021i	$-0.327071\pi$
$443$	21.7607	1.03388	0.516941	+	0.856021i	$0.327071\pi$
$444$	0	0
$445$	−15.7559	−0.746900
$446$	0	0
$447$	7.39598	0.349818
$448$	0	0
$449$	34.4566	1.62611	0.813055	−	0.582187i	$-0.197803\pi$
$449$	34.4566	1.62611	0.813055	+	0.582187i	$0.197803\pi$
$450$	0	0
$451$	−8.26135	−0.389012
$452$	0	0
$453$	−29.7916	−1.39973
$454$	0	0
$455$	−7.35335	−0.344730
$456$	0	0
$457$	−18.1527	−0.849148	−0.424574	−	0.905393i	$-0.639576\pi$
$457$	−18.1527	−0.849148	−0.424574	+	0.905393i	$0.639576\pi$
$458$	0	0
$459$	10.4985	0.490029
$460$	0	0
$461$	−33.1633	−1.54457	−0.772285	−	0.635276i	$-0.780886\pi$
$461$	−33.1633	−1.54457	−0.772285	+	0.635276i	$0.780886\pi$
$462$	0	0
$463$	4.06780	0.189047	0.0945235	−	0.995523i	$-0.469867\pi$
$463$	4.06780	0.189047	0.0945235	+	0.995523i	$0.469867\pi$
$464$	0	0
$465$	12.1281	0.562426
$466$	0	0
$467$	−19.2695	−0.891688	−0.445844	−	0.895111i	$-0.647096\pi$
$467$	−19.2695	−0.891688	−0.445844	+	0.895111i	$0.647096\pi$
$468$	0	0
$469$	11.7799	0.543944
$470$	0	0
$471$	−1.76829	−0.0814784
$472$	0	0
$473$	−12.1174	−0.557160
$474$	0	0
$475$	−0.457172	−0.0209765
$476$	0	0
$477$	9.56564	0.437981
$478$	0	0
$479$	−16.9496	−0.774445	−0.387222	−	0.921986i	$-0.626565\pi$
$479$	−16.9496	−0.774445	−0.387222	+	0.921986i	$0.626565\pi$
$480$	0	0
$481$	−4.68361	−0.213554
$482$	0	0
$483$	−17.6664	−0.803851
$484$	0	0
$485$	1.82076	0.0826765
$486$	0	0
$487$	−40.7726	−1.84758	−0.923791	−	0.382897i	$-0.874926\pi$
$487$	−40.7726	−1.84758	−0.923791	+	0.382897i	$0.874926\pi$
$488$	0	0
$489$	13.8294	0.625386
$490$	0	0
$491$	5.42117	0.244654	0.122327	−	0.992490i	$-0.460964\pi$
$491$	5.42117	0.244654	0.122327	+	0.992490i	$0.460964\pi$
$492$	0	0
$493$	−13.3176	−0.599794
$494$	0	0
$495$	−2.07991	−0.0934850
$496$	0	0
$497$	7.01503	0.314667
$498$	0	0
$499$	−37.7947	−1.69192	−0.845962	−	0.533243i	$-0.820973\pi$
$499$	−37.7947	−1.69192	−0.845962	+	0.533243i	$0.820973\pi$
$500$	0	0
$501$	−2.82151	−0.126056
$502$	0	0
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−15.6352	−0.695756
$506$	0	0
$507$	−39.4984	−1.75419
$508$	0	0
$509$	24.0222	1.06477	0.532383	−	0.846503i	$-0.321297\pi$
$509$	24.0222	1.06477	0.532383	+	0.846503i	$0.321297\pi$
$510$	0	0
$511$	3.60717	0.159572
$512$	0	0
$513$	0.416833	0.0184036
$514$	0	0
$515$	−7.62644	−0.336061
$516$	0	0
$517$	2.03964	0.0897034
$518$	0	0
$519$	23.7854	1.04406
$520$	0	0
$521$	−22.7235	−0.995533	−0.497766	−	0.867311i	$-0.665846\pi$
$521$	−22.7235	−0.995533	−0.497766	+	0.867311i	$0.665846\pi$
$522$	0	0
$523$	−0.797746	−0.0348830	−0.0174415	−	0.999848i	$-0.505552\pi$
$523$	−0.797746	−0.0348830	−0.0174415	+	0.999848i	$0.505552\pi$
$524$	0	0
$525$	7.35857	0.321154
$526$	0	0
$527$	15.1904	0.661702
$528$	0	0
$529$	42.1806	1.83394
$530$	0	0
$531$	−17.1424	−0.743918
$532$	0	0
$533$	−44.6291	−1.93310
$534$	0	0
$535$	8.27317	0.357680
$536$	0	0
$537$	20.6963	0.893112
$538$	0	0
$539$	−6.18148	−0.266255
$540$	0	0
$541$	−2.86985	−0.123384	−0.0616922	−	0.998095i	$-0.519650\pi$
$541$	−2.86985	−0.123384	−0.0616922	+	0.998095i	$0.519650\pi$
$542$	0	0
$543$	−10.1616	−0.436074
$544$	0	0
$545$	25.3681	1.08665
$546$	0	0
$547$	12.3879	0.529667	0.264833	−	0.964294i	$-0.414683\pi$
$547$	12.3879	0.529667	0.264833	+	0.964294i	$0.414683\pi$
$548$	0	0
$549$	−1.66438	−0.0710341
$550$	0	0
$551$	−0.528761	−0.0225260
$552$	0	0
$553$	12.5678	0.534437
$554$	0	0
$555$	−2.28182	−0.0968581
$556$	0	0
$557$	−40.9179	−1.73375	−0.866873	−	0.498529i	$-0.833874\pi$
$557$	−40.9179	−1.73375	−0.866873	+	0.498529i	$0.833874\pi$
$558$	0	0
$559$	−65.4604	−2.76868
$560$	0	0
$561$	−7.59900	−0.320830
$562$	0	0
$563$	−5.71430	−0.240829	−0.120415	−	0.992724i	$-0.538422\pi$
$563$	−5.71430	−0.240829	−0.120415	+	0.992724i	$0.538422\pi$
$564$	0	0
$565$	10.2020	0.429201
$566$	0	0
$567$	−11.5176	−0.483693
$568$	0	0
$569$	34.4607	1.44467	0.722333	−	0.691545i	$-0.243070\pi$
$569$	34.4607	1.44467	0.722333	+	0.691545i	$0.243070\pi$
$570$	0	0
$571$	19.1976	0.803395	0.401698	−	0.915772i	$-0.368420\pi$
$571$	19.1976	0.803395	0.401698	+	0.915772i	$0.368420\pi$
$572$	0	0
$573$	4.38776	0.183301
$574$	0	0
$575$	−27.1495	−1.13221
$576$	0	0
$577$	21.9955	0.915686	0.457843	−	0.889033i	$-0.348622\pi$
$577$	21.9955	0.915686	0.457843	+	0.889033i	$0.348622\pi$
$578$	0	0
$579$	19.1058	0.794008
$580$	0	0
$581$	9.40066	0.390005
$582$	0	0
$583$	6.34909	0.262953
$584$	0	0
$585$	−11.2360	−0.464552
$586$	0	0
$587$	−15.7932	−0.651855	−0.325928	−	0.945395i	$-0.605677\pi$
$587$	−15.7932	−0.651855	−0.325928	+	0.945395i	$0.605677\pi$
$588$	0	0
$589$	0.603118	0.0248510
$590$	0	0
$591$	−32.0322	−1.31763
$592$	0	0
$593$	43.2826	1.77740	0.888702	−	0.458486i	$-0.151608\pi$
$593$	43.2826	1.77740	0.888702	+	0.458486i	$0.151608\pi$
$594$	0	0
$595$	−4.48707	−0.183952
$596$	0	0
$597$	−42.6108	−1.74394
$598$	0	0
$599$	−25.0496	−1.02350	−0.511749	−	0.859135i	$-0.671002\pi$
$599$	−25.0496	−1.02350	−0.511749	+	0.859135i	$0.671002\pi$
$600$	0	0
$601$	−34.9636	−1.42620	−0.713098	−	0.701064i	$-0.752709\pi$
$601$	−34.9636	−1.42620	−0.713098	+	0.701064i	$0.752709\pi$
$602$	0	0
$603$	17.9998	0.733008
$604$	0	0
$605$	12.6943	0.516095
$606$	0	0
$607$	37.1283	1.50699	0.753496	−	0.657453i	$-0.228366\pi$
$607$	37.1283	1.50699	0.753496	+	0.657453i	$0.228366\pi$
$608$	0	0
$609$	8.51086	0.344877
$610$	0	0
$611$	11.0185	0.445760
$612$	0	0
$613$	−11.9914	−0.484329	−0.242165	−	0.970235i	$-0.577857\pi$
$613$	−11.9914	−0.484329	−0.242165	+	0.970235i	$0.577857\pi$
$614$	0	0
$615$	−21.7431	−0.876765
$616$	0	0
$617$	7.55121	0.304000	0.152000	−	0.988380i	$-0.451429\pi$
$617$	7.55121	0.304000	0.152000	+	0.988380i	$0.451429\pi$
$618$	0	0
$619$	−16.8591	−0.677625	−0.338813	−	0.940854i	$-0.610025\pi$
$619$	−16.8591	−0.677625	−0.338813	+	0.940854i	$0.610025\pi$
$620$	0	0
$621$	24.7540	0.993344
$622$	0	0
$623$	12.6115	0.505268
$624$	0	0
$625$	3.12263	0.124905
$626$	0	0
$627$	−0.301711	−0.0120492
$628$	0	0
$629$	−2.85797	−0.113955
$630$	0	0
$631$	39.7580	1.58274	0.791371	−	0.611337i	$-0.209368\pi$
$631$	39.7580	1.58274	0.791371	+	0.611337i	$0.209368\pi$
$632$	0	0
$633$	−59.6818	−2.37214
$634$	0	0
$635$	8.69101	0.344892
$636$	0	0
$637$	−33.3933	−1.32309
$638$	0	0
$639$	10.7191	0.424039
$640$	0	0
$641$	−4.12573	−0.162956	−0.0814782	−	0.996675i	$-0.525964\pi$
$641$	−4.12573	−0.162956	−0.0814782	+	0.996675i	$0.525964\pi$
$642$	0	0
$643$	−33.5320	−1.32237	−0.661187	−	0.750221i	$-0.729947\pi$
$643$	−33.5320	−1.32237	−0.661187	+	0.750221i	$0.729947\pi$
$644$	0	0
$645$	−31.8919	−1.25574
$646$	0	0
$647$	29.0509	1.14211	0.571055	−	0.820912i	$-0.306534\pi$
$647$	29.0509	1.14211	0.571055	+	0.820912i	$0.306534\pi$
$648$	0	0
$649$	−11.3781	−0.446630
$650$	0	0
$651$	−9.70769	−0.380474
$652$	0	0
$653$	−44.4702	−1.74025	−0.870127	−	0.492828i	$-0.835963\pi$
$653$	−44.4702	−1.74025	−0.870127	+	0.492828i	$0.835963\pi$
$654$	0	0
$655$	11.4982	0.449274
$656$	0	0
$657$	5.51180	0.215036
$658$	0	0
$659$	34.3135	1.33666	0.668332	−	0.743863i	$-0.267008\pi$
$659$	34.3135	1.33666	0.668332	+	0.743863i	$0.267008\pi$
$660$	0	0
$661$	−27.5815	−1.07279	−0.536397	−	0.843966i	$-0.680215\pi$
$661$	−27.5815	−1.07279	−0.536397	+	0.843966i	$0.680215\pi$
$662$	0	0
$663$	−41.0510	−1.59429
$664$	0	0
$665$	−0.178154	−0.00690854
$666$	0	0
$667$	−31.4009	−1.21585
$668$	0	0
$669$	−54.7239	−2.11575
$670$	0	0
$671$	−1.10472	−0.0426471
$672$	0	0
$673$	29.3251	1.13040	0.565199	−	0.824954i	$-0.308799\pi$
$673$	29.3251	1.13040	0.565199	+	0.824954i	$0.308799\pi$
$674$	0	0
$675$	−10.3107	−0.396860
$676$	0	0
$677$	15.9656	0.613607	0.306803	−	0.951773i	$-0.400741\pi$
$677$	15.9656	0.613607	0.306803	+	0.951773i	$0.400741\pi$
$678$	0	0
$679$	−1.45739	−0.0559296
$680$	0	0
$681$	−35.6139	−1.36473
$682$	0	0
$683$	−44.2867	−1.69458	−0.847292	−	0.531127i	$-0.821769\pi$
$683$	−44.2867	−1.69458	−0.847292	+	0.531127i	$0.821769\pi$
$684$	0	0
$685$	−20.4228	−0.780315
$686$	0	0
$687$	20.0735	0.765852
$688$	0	0
$689$	34.2988	1.30668
$690$	0	0
$691$	4.66849	0.177598	0.0887989	−	0.996050i	$-0.471697\pi$
$691$	4.66849	0.177598	0.0887989	+	0.996050i	$0.471697\pi$
$692$	0	0
$693$	1.66482	0.0632414
$694$	0	0
$695$	−2.79931	−0.106184
$696$	0	0
$697$	−27.2330	−1.03152
$698$	0	0
$699$	−39.9314	−1.51034
$700$	0	0
$701$	31.3483	1.18401	0.592004	−	0.805935i	$-0.298337\pi$
$701$	31.3483	1.18401	0.592004	+	0.805935i	$0.298337\pi$
$702$	0	0
$703$	−0.113473	−0.00427971
$704$	0	0
$705$	5.36814	0.202176
$706$	0	0
$707$	12.5149	0.470670
$708$	0	0
$709$	18.6245	0.699458	0.349729	−	0.936851i	$-0.386274\pi$
$709$	18.6245	0.699458	0.349729	+	0.936851i	$0.386274\pi$
$710$	0	0
$711$	19.2037	0.720196
$712$	0	0
$713$	35.8167	1.34134
$714$	0	0
$715$	−7.45778	−0.278905
$716$	0	0
$717$	−38.9697	−1.45535
$718$	0	0
$719$	−13.8910	−0.518048	−0.259024	−	0.965871i	$-0.583401\pi$
$719$	−13.8910	−0.518048	−0.259024	+	0.965871i	$0.583401\pi$
$720$	0	0
$721$	6.10444	0.227341
$722$	0	0
$723$	30.3312	1.12803
$724$	0	0
$725$	13.0794	0.485756
$726$	0	0
$727$	−35.5757	−1.31943	−0.659715	−	0.751516i	$-0.729323\pi$
$727$	−35.5757	−1.31943	−0.659715	+	0.751516i	$0.729323\pi$
$728$	0	0
$729$	2.05381	0.0760669
$730$	0	0
$731$	−39.9444	−1.47740
$732$	0	0
$733$	24.4417	0.902775	0.451387	−	0.892328i	$-0.350929\pi$
$733$	24.4417	0.902775	0.451387	+	0.892328i	$0.350929\pi$
$734$	0	0
$735$	−16.2690	−0.600092
$736$	0	0
$737$	11.9472	0.440079
$738$	0	0
$739$	11.7735	0.433096	0.216548	−	0.976272i	$-0.430520\pi$
$739$	11.7735	0.433096	0.216548	+	0.976272i	$0.430520\pi$
$740$	0	0
$741$	−1.62989	−0.0598755
$742$	0	0
$743$	25.7005	0.942861	0.471431	−	0.881903i	$-0.343738\pi$
$743$	25.7005	0.942861	0.471431	+	0.881903i	$0.343738\pi$
$744$	0	0
$745$	4.42921	0.162274
$746$	0	0
$747$	14.3643	0.525564
$748$	0	0
$749$	−6.62210	−0.241966
$750$	0	0
$751$	30.7129	1.12073	0.560366	−	0.828245i	$-0.310661\pi$
$751$	30.7129	1.12073	0.560366	+	0.828245i	$0.310661\pi$
$752$	0	0
$753$	−38.7593	−1.41247
$754$	0	0
$755$	−17.8412	−0.649308
$756$	0	0
$757$	18.2755	0.664234	0.332117	−	0.943238i	$-0.392237\pi$
$757$	18.2755	0.664234	0.332117	+	0.943238i	$0.392237\pi$
$758$	0	0
$759$	−17.9173	−0.650358
$760$	0	0
$761$	39.7157	1.43969	0.719847	−	0.694132i	$-0.244212\pi$
$761$	39.7157	1.43969	0.719847	+	0.694132i	$0.244212\pi$
$762$	0	0
$763$	−20.3054	−0.735105
$764$	0	0
$765$	−6.85630	−0.247890
$766$	0	0
$767$	−61.4664	−2.21942
$768$	0	0
$769$	−30.2312	−1.09016	−0.545082	−	0.838382i	$-0.683502\pi$
$769$	−30.2312	−1.09016	−0.545082	+	0.838382i	$0.683502\pi$
$770$	0	0
$771$	−22.2199	−0.800230
$772$	0	0
$773$	−11.0085	−0.395949	−0.197975	−	0.980207i	$-0.563436\pi$
$773$	−11.0085	−0.395949	−0.197975	+	0.980207i	$0.563436\pi$
$774$	0	0
$775$	−14.9186	−0.535893
$776$	0	0
$777$	1.82644	0.0655232
$778$	0	0
$779$	−1.08126	−0.0387402
$780$	0	0
$781$	7.11465	0.254582
$782$	0	0
$783$	−11.9253	−0.426176
$784$	0	0
$785$	−1.05897	−0.0377963
$786$	0	0
$787$	26.5149	0.945153	0.472576	−	0.881290i	$-0.343324\pi$
$787$	26.5149	0.945153	0.472576	+	0.881290i	$0.343324\pi$
$788$	0	0
$789$	−58.0176	−2.06548
$790$	0	0
$791$	−8.16598	−0.290349
$792$	0	0
$793$	−5.96785	−0.211925
$794$	0	0
$795$	16.7102	0.592649
$796$	0	0
$797$	−12.2392	−0.433534	−0.216767	−	0.976223i	$-0.569551\pi$
$797$	−12.2392	−0.433534	−0.216767	+	0.976223i	$0.569551\pi$
$798$	0	0
$799$	6.72356	0.237863
$800$	0	0
$801$	19.2705	0.680890
$802$	0	0
$803$	3.65840	0.129102
$804$	0	0
$805$	−10.5799	−0.372891
$806$	0	0
$807$	36.1367	1.27207
$808$	0	0
$809$	−15.6458	−0.550075	−0.275038	−	0.961433i	$-0.588690\pi$
$809$	−15.6458	−0.550075	−0.275038	+	0.961433i	$0.588690\pi$
$810$	0	0
$811$	22.3912	0.786261	0.393131	−	0.919483i	$-0.371392\pi$
$811$	22.3912	0.786261	0.393131	+	0.919483i	$0.371392\pi$
$812$	0	0
$813$	56.2878	1.97410
$814$	0	0
$815$	8.28197	0.290105
$816$	0	0
$817$	−1.58595	−0.0554854
$818$	0	0
$819$	8.99364	0.314263
$820$	0	0
$821$	32.3682	1.12966	0.564829	−	0.825208i	$-0.308942\pi$
$821$	32.3682	1.12966	0.564829	+	0.825208i	$0.308942\pi$
$822$	0	0
$823$	−17.1342	−0.597262	−0.298631	−	0.954369i	$-0.596530\pi$
$823$	−17.1342	−0.597262	−0.298631	+	0.954369i	$0.596530\pi$
$824$	0	0
$825$	7.46307	0.259831
$826$	0	0
$827$	−53.5177	−1.86099	−0.930496	−	0.366301i	$-0.880624\pi$
$827$	−53.5177	−1.86099	−0.930496	+	0.366301i	$0.880624\pi$
$828$	0	0
$829$	−22.7575	−0.790400	−0.395200	−	0.918595i	$-0.629325\pi$
$829$	−22.7575	−0.790400	−0.395200	+	0.918595i	$0.629325\pi$
$830$	0	0
$831$	33.5182	1.16273
$832$	0	0
$833$	−20.3769	−0.706017
$834$	0	0
$835$	−1.68971	−0.0584749
$836$	0	0
$837$	13.6023	0.470164
$838$	0	0
$839$	11.3014	0.390167	0.195084	−	0.980787i	$-0.437502\pi$
$839$	11.3014	0.390167	0.195084	+	0.980787i	$0.437502\pi$
$840$	0	0
$841$	−13.8725	−0.478362
$842$	0	0
$843$	−33.2154	−1.14400
$844$	0	0
$845$	−23.6543	−0.813734
$846$	0	0
$847$	−10.1609	−0.349131
$848$	0	0
$849$	−28.8214	−0.989149
$850$	0	0
$851$	−6.73868	−0.230999
$852$	0	0
$853$	42.8197	1.46612	0.733060	−	0.680164i	$-0.238092\pi$
$853$	42.8197	1.46612	0.733060	+	0.680164i	$0.238092\pi$
$854$	0	0
$855$	−0.272222	−0.00930981
$856$	0	0
$857$	44.5283	1.52106	0.760528	−	0.649305i	$-0.224940\pi$
$857$	44.5283	1.52106	0.760528	+	0.649305i	$0.224940\pi$
$858$	0	0
$859$	−33.9900	−1.15972	−0.579861	−	0.814715i	$-0.696893\pi$
$859$	−33.9900	−1.15972	−0.579861	+	0.814715i	$0.696893\pi$
$860$	0	0
$861$	17.4038	0.593120
$862$	0	0
$863$	28.3461	0.964912	0.482456	−	0.875920i	$-0.339745\pi$
$863$	28.3461	0.964912	0.482456	+	0.875920i	$0.339745\pi$
$864$	0	0
$865$	14.2443	0.484322
$866$	0	0
$867$	11.2721	0.382821
$868$	0	0
$869$	12.7463	0.432388
$870$	0	0
$871$	64.5405	2.18687
$872$	0	0
$873$	−2.22692	−0.0753697
$874$	0	0
$875$	10.9591	0.370484
$876$	0	0
$877$	41.9729	1.41733	0.708663	−	0.705547i	$-0.249299\pi$
$877$	41.9729	1.41733	0.708663	+	0.705547i	$0.249299\pi$
$878$	0	0
$879$	4.78924	0.161537
$880$	0	0
$881$	22.6275	0.762339	0.381170	−	0.924505i	$-0.375521\pi$
$881$	22.6275	0.762339	0.381170	+	0.924505i	$0.375521\pi$
$882$	0	0
$883$	−35.2678	−1.18686	−0.593428	−	0.804887i	$-0.702226\pi$
$883$	−35.2678	−1.18686	−0.593428	+	0.804887i	$0.702226\pi$
$884$	0	0
$885$	−29.9460	−1.00663
$886$	0	0
$887$	−41.0378	−1.37791	−0.688957	−	0.724802i	$-0.741931\pi$
$887$	−41.0378	−1.37791	−0.688957	+	0.724802i	$0.741931\pi$
$888$	0	0
$889$	−6.95655	−0.233315
$890$	0	0
$891$	−11.6812	−0.391334
$892$	0	0
$893$	0.266952	0.00893322
$894$	0	0
$895$	12.3943	0.414297
$896$	0	0
$897$	−96.7924	−3.23180
$898$	0	0
$899$	−17.2548	−0.575479
$900$	0	0
$901$	20.9294	0.697260
$902$	0	0
$903$	25.5272	0.849493
$904$	0	0
$905$	−6.08543	−0.202287
$906$	0	0
$907$	−7.47617	−0.248242	−0.124121	−	0.992267i	$-0.539611\pi$
$907$	−7.47617	−0.248242	−0.124121	+	0.992267i	$0.539611\pi$
$908$	0	0
$909$	19.1229	0.634265
$910$	0	0
$911$	9.50199	0.314815	0.157407	−	0.987534i	$-0.449686\pi$
$911$	9.50199	0.314815	0.157407	+	0.987534i	$0.449686\pi$
$912$	0	0
$913$	9.53417	0.315535
$914$	0	0
$915$	−2.90750	−0.0961191
$916$	0	0
$917$	−9.20354	−0.303928
$918$	0	0
$919$	−35.5620	−1.17308	−0.586541	−	0.809920i	$-0.699511\pi$
$919$	−35.5620	−1.17308	−0.586541	+	0.809920i	$0.699511\pi$
$920$	0	0
$921$	67.2278	2.21523
$922$	0	0
$923$	38.4345	1.26509
$924$	0	0
$925$	2.80685	0.0922887
$926$	0	0
$927$	9.32766	0.306360
$928$	0	0
$929$	9.58288	0.314404	0.157202	−	0.987566i	$-0.449753\pi$
$929$	9.58288	0.314404	0.157202	+	0.987566i	$0.449753\pi$
$930$	0	0
$931$	−0.809043	−0.0265153
$932$	0	0
$933$	−49.1126	−1.60787
$934$	0	0
$935$	−4.55079	−0.148827
$936$	0	0
$937$	47.1098	1.53901	0.769505	−	0.638641i	$-0.220503\pi$
$937$	47.1098	1.53901	0.769505	+	0.638641i	$0.220503\pi$
$938$	0	0
$939$	22.3704	0.730030
$940$	0	0
$941$	3.30951	0.107887	0.0539435	−	0.998544i	$-0.482821\pi$
$941$	3.30951	0.107887	0.0539435	+	0.998544i	$0.482821\pi$
$942$	0	0
$943$	−64.2116	−2.09102
$944$	0	0
$945$	−4.01797	−0.130705
$946$	0	0
$947$	−0.981290	−0.0318876	−0.0159438	−	0.999873i	$-0.505075\pi$
$947$	−0.981290	−0.0318876	−0.0159438	+	0.999873i	$0.505075\pi$
$948$	0	0
$949$	19.7633	0.641543
$950$	0	0
$951$	−5.81064	−0.188423
$952$	0	0
$953$	17.8768	0.579085	0.289543	−	0.957165i	$-0.406497\pi$
$953$	17.8768	0.579085	0.289543	+	0.957165i	$0.406497\pi$
$954$	0	0
$955$	2.62769	0.0850299
$956$	0	0
$957$	8.63173	0.279024
$958$	0	0
$959$	16.3470	0.527873
$960$	0	0
$961$	−11.3188	−0.365122
$962$	0	0
$963$	−10.1187	−0.326069
$964$	0	0
$965$	11.4418	0.368325
$966$	0	0
$967$	−54.7970	−1.76215	−0.881076	−	0.472975i	$-0.843180\pi$
$967$	−54.7970	−1.76215	−0.881076	+	0.472975i	$0.843180\pi$
$968$	0	0
$969$	−0.994571	−0.0319502
$970$	0	0
$971$	−18.3961	−0.590360	−0.295180	−	0.955442i	$-0.595380\pi$
$971$	−18.3961	−0.590360	−0.295180	+	0.955442i	$0.595380\pi$
$972$	0	0
$973$	2.24065	0.0718320
$974$	0	0
$975$	40.3167	1.29117
$976$	0	0
$977$	−30.9149	−0.989054	−0.494527	−	0.869162i	$-0.664659\pi$
$977$	−30.9149	−0.989054	−0.494527	+	0.869162i	$0.664659\pi$
$978$	0	0
$979$	12.7906	0.408789
$980$	0	0
$981$	−31.0269	−0.990613
$982$	0	0
$983$	13.0734	0.416978	0.208489	−	0.978025i	$-0.433145\pi$
$983$	13.0734	0.416978	0.208489	+	0.978025i	$0.433145\pi$
$984$	0	0
$985$	−19.1830	−0.611222
$986$	0	0
$987$	−4.29682	−0.136769
$988$	0	0
$989$	−94.1831	−2.99485
$990$	0	0
$991$	−15.9347	−0.506182	−0.253091	−	0.967443i	$-0.581447\pi$
$991$	−15.9347	−0.506182	−0.253091	+	0.967443i	$0.581447\pi$
$992$	0	0
$993$	23.5590	0.747621
$994$	0	0
$995$	−25.5182	−0.808982
$996$	0	0
$997$	−45.9628	−1.45566	−0.727828	−	0.685759i	$-0.759470\pi$
$997$	−45.9628	−1.45566	−0.727828	+	0.685759i	$0.759470\pi$
$998$	0	0
$999$	−2.55919	−0.0809691

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4024.2.a.g.1.8	✓	33
4.3	odd	2		8048.2.a.x.1.26		33

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4024.2.a.g.1.8	✓	33	1.1	even	1	trivial
8048.2.a.x.1.26		33	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4024 = 2^{3} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4024.a (trivial)

\(f(q)\)	\(=\)	\(q-2.13657 q^{3} -1.27952 q^{5} +1.02417 q^{7} +1.56495 q^{9} +O(q^{10})\)	\(q-2.13657 q^{3} -1.27952 q^{5} +1.02417 q^{7} +1.56495 q^{9} +1.03872 q^{11} +5.61131 q^{13} +2.73380 q^{15} +3.42406 q^{17} +0.135949 q^{19} -2.18821 q^{21} +8.07345 q^{23} -3.36282 q^{25} +3.06610 q^{27} -3.88941 q^{29} +4.43635 q^{31} -2.21929 q^{33} -1.31045 q^{35} -0.834672 q^{37} -11.9890 q^{39} -7.95342 q^{41} -11.6658 q^{43} -2.00239 q^{45} +1.96362 q^{47} -5.95108 q^{49} -7.31576 q^{51} +6.11245 q^{53} -1.32906 q^{55} -0.290465 q^{57} -10.9540 q^{59} -1.06354 q^{61} +1.60277 q^{63} -7.17981 q^{65} +11.5019 q^{67} -17.2495 q^{69} +6.84947 q^{71} +3.52204 q^{73} +7.18491 q^{75} +1.06382 q^{77} +12.2712 q^{79} -11.2458 q^{81} +9.17881 q^{83} -4.38117 q^{85} +8.31001 q^{87} +12.3139 q^{89} +5.74694 q^{91} -9.47859 q^{93} -0.173950 q^{95} -1.42300 q^{97} +1.62553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9}+O(q^{10}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9	\( 33 q + 10 q^{3} + 12 q^{7} + 47 q^{9} + 22 q^{11} - 17 q^{13} + 22 q^{15} + 9 q^{17} + 16 q^{19} + 6 q^{21} + 36 q^{23} + 47 q^{25} + 34 q^{27} + 13 q^{29} + 21 q^{31} + 14 q^{33} + 33 q^{35} - 55 q^{37} + 37 q^{39} + 42 q^{41} + 23 q^{43} + 5 q^{45} + 20 q^{47} + 55 q^{49} + 53 q^{51} - 32 q^{53} + 35 q^{55} + 21 q^{57} + 20 q^{59} - 15 q^{61} + 48 q^{63} + 34 q^{65} + 66 q^{67} - 4 q^{69} + 61 q^{71} + 19 q^{73} + 59 q^{75} + 2 q^{77} + 62 q^{79} + 77 q^{81} + 36 q^{83} - 14 q^{85} + 43 q^{87} + 34 q^{89} + 41 q^{91} - 11 q^{93} + 61 q^{95} - 8 q^{97} + 98 q^{99}+O(q^{100}) \) 33 * q + 10 * q^3 + 12 * q^7 + 47 * q^9 + 22 * q^11 - 17 * q^13 + 22 * q^15 + 9 * q^17 + 16 * q^19 + 6 * q^21 + 36 * q^23 + 47 * q^25 + 34 * q^27 + 13 * q^29 + 21 * q^31 + 14 * q^33 + 33 * q^35 - 55 * q^37 + 37 * q^39 + 42 * q^41 + 23 * q^43 + 5 * q^45 + 20 * q^47 + 55 * q^49 + 53 * q^51 - 32 * q^53 + 35 * q^55 + 21 * q^57 + 20 * q^59 - 15 * q^61 + 48 * q^63 + 34 * q^65 + 66 * q^67 - 4 * q^69 + 61 * q^71 + 19 * q^73 + 59 * q^75 + 2 * q^77 + 62 * q^79 + 77 * q^81 + 36 * q^83 - 14 * q^85 + 43 * q^87 + 34 * q^89 + 41 * q^91 - 11 * q^93 + 61 * q^95 - 8 * q^97 + 98 * q^99

Introduction

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