LMFDB - Embedded newform 2128.2.a.s.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2128,2,Mod(1,2128)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2128, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2128.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2128 = 2^{4} \cdot 7 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2128.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:	$16.9921655501$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.469.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 4 $ x^3 - x^2 - 5*x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 266)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.16425$ of defining polynomial
Character	$\chi$	$=$	2128.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.683969 q^{3} +4.16425 q^{5} +1.00000 q^{7} -2.53219 q^{9} +O(q^{10})$	$q-0.683969 q^{3} +4.16425 q^{5} +1.00000 q^{7} -2.53219 q^{9} +5.01247 q^{11} -6.32850 q^{13} -2.84822 q^{15} +4.96056 q^{17} +1.00000 q^{19} -0.683969 q^{21} -4.32850 q^{23} +12.3410 q^{25} +3.78384 q^{27} +1.20369 q^{29} +5.69643 q^{31} -3.42837 q^{33} +4.16425 q^{35} +2.79631 q^{37} +4.32850 q^{39} +0.164248 q^{41} +0.796310 q^{43} -10.5447 q^{45} +4.68397 q^{47} +1.00000 q^{49} -3.39287 q^{51} +4.05191 q^{53} +20.8731 q^{55} -0.683969 q^{57} -6.49274 q^{59} -1.64453 q^{61} -2.53219 q^{63} -26.3534 q^{65} -1.03944 q^{67} +2.96056 q^{69} -9.01247 q^{71} +12.0249 q^{73} -8.44084 q^{75} +5.01247 q^{77} -0.467814 q^{79} +5.00853 q^{81} -4.65699 q^{83} +20.6570 q^{85} -0.823287 q^{87} -7.01247 q^{89} -6.32850 q^{91} -3.89619 q^{93} +4.16425 q^{95} -5.64453 q^{97} -12.6925 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + 5 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) $ 3 * q + q^3 + 5 * q^5 + 3 * q^7 + 6 * q^9	$ 3 q + q^{3} + 5 q^{5} + 3 q^{7} + 6 q^{9} - 3 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} + 3 q^{19} + q^{21} + 2 q^{23} + 4 q^{25} + 28 q^{27} + 5 q^{29} - 4 q^{31} - 15 q^{33} + 5 q^{35} + 7 q^{37} - 2 q^{39} - 7 q^{41} + q^{43} + 11 q^{47} + 3 q^{49} + 32 q^{51} + 3 q^{53} + 16 q^{55} + q^{57} + 3 q^{59} + 7 q^{61} + 6 q^{63} - 28 q^{65} - 12 q^{67} - 9 q^{71} - 12 q^{75} - 3 q^{77} - 15 q^{79} + 35 q^{81} + 16 q^{83} + 32 q^{85} - 28 q^{87} - 3 q^{89} - 4 q^{91} - 30 q^{93} + 5 q^{95} - 5 q^{97} - 55 q^{99}+O(q^{100}) $ 3 * q + q^3 + 5 * q^5 + 3 * q^7 + 6 * q^9 - 3 * q^11 - 4 * q^13 + 2 * q^15 + 6 * q^17 + 3 * q^19 + q^21 + 2 * q^23 + 4 * q^25 + 28 * q^27 + 5 * q^29 - 4 * q^31 - 15 * q^33 + 5 * q^35 + 7 * q^37 - 2 * q^39 - 7 * q^41 + q^43 + 11 * q^47 + 3 * q^49 + 32 * q^51 + 3 * q^53 + 16 * q^55 + q^57 + 3 * q^59 + 7 * q^61 + 6 * q^63 - 28 * q^65 - 12 * q^67 - 9 * q^71 - 12 * q^75 - 3 * q^77 - 15 * q^79 + 35 * q^81 + 16 * q^83 + 32 * q^85 - 28 * q^87 - 3 * q^89 - 4 * q^91 - 30 * q^93 + 5 * q^95 - 5 * q^97 - 55 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−0.683969	−0.394890	−0.197445	−	0.980314i	$-0.563264\pi$
$3$	−0.683969	−0.394890	−0.197445	+	0.980314i	$0.563264\pi$
$4$	0	0
$5$	4.16425	1.86231	0.931154	−	0.364626i	$-0.118803\pi$
$5$	4.16425	1.86231	0.931154	+	0.364626i	$0.118803\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.53219	−0.844062
$10$	0	0
$11$	5.01247	1.51132	0.755658	−	0.654967i	$-0.227317\pi$
$11$	5.01247	1.51132	0.755658	+	0.654967i	$0.227317\pi$
$12$	0	0
$13$	−6.32850	−1.75521	−0.877604	−	0.479385i	$-0.840860\pi$
$13$	−6.32850	−1.75521	−0.877604	+	0.479385i	$0.840860\pi$
$14$	0	0
$15$	−2.84822	−0.735406
$16$	0	0
$17$	4.96056	1.20311	0.601556	−	0.798831i	$-0.294548\pi$
$17$	4.96056	1.20311	0.601556	+	0.798831i	$0.294548\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−0.683969	−0.149254
$22$	0	0
$23$	−4.32850	−0.902554	−0.451277	−	0.892384i	$-0.649031\pi$
$23$	−4.32850	−0.902554	−0.451277	+	0.892384i	$0.649031\pi$
$24$	0	0
$25$	12.3410	2.46819
$26$	0	0
$27$	3.78384	0.728201
$28$	0	0
$29$	1.20369	0.223520	0.111760	−	0.993735i	$-0.464351\pi$
$29$	1.20369	0.223520	0.111760	+	0.993735i	$0.464351\pi$
$30$	0	0
$31$	5.69643	1.02311	0.511555	−	0.859251i	$-0.329070\pi$
$31$	5.69643	1.02311	0.511555	+	0.859251i	$0.329070\pi$
$32$	0	0
$33$	−3.42837	−0.596803
$34$	0	0
$35$	4.16425	0.703886
$36$	0	0
$37$	2.79631	0.459710	0.229855	−	0.973225i	$-0.426175\pi$
$37$	2.79631	0.459710	0.229855	+	0.973225i	$0.426175\pi$
$38$	0	0
$39$	4.32850	0.693114
$40$	0	0
$41$	0.164248	0.0256512	0.0128256	−	0.999918i	$-0.495917\pi$
$41$	0.164248	0.0256512	0.0128256	+	0.999918i	$0.495917\pi$
$42$	0	0
$43$	0.796310	0.121436	0.0607180	−	0.998155i	$-0.480661\pi$
$43$	0.796310	0.121436	0.0607180	+	0.998155i	$0.480661\pi$
$44$	0	0
$45$	−10.5447	−1.57190
$46$	0	0
$47$	4.68397	0.683227	0.341614	−	0.939841i	$-0.389027\pi$
$47$	4.68397	0.683227	0.341614	+	0.939841i	$0.389027\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.39287	−0.475097
$52$	0	0
$53$	4.05191	0.556572	0.278286	−	0.960498i	$-0.410234\pi$
$53$	4.05191	0.556572	0.278286	+	0.960498i	$0.410234\pi$
$54$	0	0
$55$	20.8731	2.81453
$56$	0	0
$57$	−0.683969	−0.0905939
$58$	0	0
$59$	−6.49274	−0.845283	−0.422642	−	0.906297i	$-0.638897\pi$
$59$	−6.49274	−0.845283	−0.422642	+	0.906297i	$0.638897\pi$
$60$	0	0
$61$	−1.64453	−0.210560	−0.105280	−	0.994443i	$-0.533574\pi$
$61$	−1.64453	−0.210560	−0.105280	+	0.994443i	$0.533574\pi$
$62$	0	0
$63$	−2.53219	−0.319025
$64$	0	0
$65$	−26.3534	−3.26874
$66$	0	0
$67$	−1.03944	−0.126988	−0.0634941	−	0.997982i	$-0.520224\pi$
$67$	−1.03944	−0.126988	−0.0634941	+	0.997982i	$0.520224\pi$
$68$	0	0
$69$	2.96056	0.356409
$70$	0	0
$71$	−9.01247	−1.06958	−0.534791	−	0.844984i	$-0.679610\pi$
$71$	−9.01247	−1.06958	−0.534791	+	0.844984i	$0.679610\pi$
$72$	0	0
$73$	12.0249	1.40741	0.703706	−	0.710491i	$-0.251527\pi$
$73$	12.0249	1.40741	0.703706	+	0.710491i	$0.251527\pi$
$74$	0	0
$75$	−8.44084	−0.974664
$76$	0	0
$77$	5.01247	0.571223
$78$	0	0
$79$	−0.467814	−0.0526332	−0.0263166	−	0.999654i	$-0.508378\pi$
$79$	−0.467814	−0.0526332	−0.0263166	+	0.999654i	$0.508378\pi$
$80$	0	0
$81$	5.00853	0.556503
$82$	0	0
$83$	−4.65699	−0.511171	−0.255586	−	0.966786i	$-0.582268\pi$
$83$	−4.65699	−0.511171	−0.255586	+	0.966786i	$0.582268\pi$
$84$	0	0
$85$	20.6570	2.24057
$86$	0	0
$87$	−0.823287	−0.0882656
$88$	0	0
$89$	−7.01247	−0.743320	−0.371660	−	0.928369i	$-0.621211\pi$
$89$	−7.01247	−0.743320	−0.371660	+	0.928369i	$0.621211\pi$
$90$	0	0
$91$	−6.32850	−0.663407
$92$	0	0
$93$	−3.89619	−0.404016
$94$	0	0
$95$	4.16425	0.427243
$96$	0	0
$97$	−5.64453	−0.573115	−0.286557	−	0.958063i	$-0.592511\pi$
$97$	−5.64453	−0.573115	−0.286557	+	0.958063i	$0.592511\pi$
$98$	0	0
$99$	−12.6925	−1.27564
$100$	0	0
$101$	−19.3140	−1.92181	−0.960907	−	0.276873i	$-0.910702\pi$
$101$	−19.3140	−1.92181	−0.960907	+	0.276873i	$0.910702\pi$
$102$	0	0
$103$	7.06437	0.696073	0.348037	−	0.937481i	$-0.386848\pi$
$103$	7.06437	0.696073	0.348037	+	0.937481i	$0.386848\pi$
$104$	0	0
$105$	−2.84822	−0.277958
$106$	0	0
$107$	19.0644	1.84302	0.921511	−	0.388352i	$-0.126955\pi$
$107$	19.0644	1.84302	0.921511	+	0.388352i	$0.126955\pi$
$108$	0	0
$109$	12.0519	1.15436	0.577182	−	0.816616i	$-0.304152\pi$
$109$	12.0519	1.15436	0.577182	+	0.816616i	$0.304152\pi$
$110$	0	0
$111$	−1.91259	−0.181535
$112$	0	0
$113$	−9.61755	−0.904743	−0.452371	−	0.891830i	$-0.649422\pi$
$113$	−9.61755	−0.904743	−0.452371	+	0.891830i	$0.649422\pi$
$114$	0	0
$115$	−18.0249	−1.68083
$116$	0	0
$117$	16.0249	1.48151
$118$	0	0
$119$	4.96056	0.454734
$120$	0	0
$121$	14.1248	1.28407
$122$	0	0
$123$	−0.112341	−0.0101294
$124$	0	0
$125$	30.5696	2.73423
$126$	0	0
$127$	7.64453	0.678342	0.339171	−	0.940725i	$-0.389853\pi$
$127$	7.64453	0.678342	0.339171	+	0.940725i	$0.389853\pi$
$128$	0	0
$129$	−0.544651	−0.0479539
$130$	0	0
$131$	13.3679	1.16796	0.583981	−	0.811767i	$-0.301494\pi$
$131$	13.3679	1.16796	0.583981	+	0.811767i	$0.301494\pi$
$132$	0	0
$133$	1.00000	0.0867110
$134$	0	0
$135$	15.7569	1.35614
$136$	0	0
$137$	12.2681	1.04813	0.524066	−	0.851678i	$-0.324415\pi$
$137$	12.2681	1.04813	0.524066	+	0.851678i	$0.324415\pi$
$138$	0	0
$139$	−6.73588	−0.571330	−0.285665	−	0.958330i	$-0.592214\pi$
$139$	−6.73588	−0.571330	−0.285665	+	0.958330i	$0.592214\pi$
$140$	0	0
$141$	−3.20369	−0.269799
$142$	0	0
$143$	−31.7214	−2.65267
$144$	0	0
$145$	5.01247	0.416263
$146$	0	0
$147$	−0.683969	−0.0564128
$148$	0	0
$149$	−12.9606	−1.06177	−0.530885	−	0.847444i	$-0.678140\pi$
$149$	−12.9606	−1.06177	−0.530885	+	0.847444i	$0.678140\pi$
$150$	0	0
$151$	−19.3929	−1.57817	−0.789085	−	0.614285i	$-0.789445\pi$
$151$	−19.3929	−1.57817	−0.789085	+	0.614285i	$0.789445\pi$
$152$	0	0
$153$	−12.5611	−1.01550
$154$	0	0
$155$	23.7214	1.90535
$156$	0	0
$157$	1.09135	0.0870992	0.0435496	−	0.999051i	$-0.486133\pi$
$157$	1.09135	0.0870992	0.0435496	+	0.999051i	$0.486133\pi$
$158$	0	0
$159$	−2.77138	−0.219785
$160$	0	0
$161$	−4.32850	−0.341133
$162$	0	0
$163$	−5.72341	−0.448292	−0.224146	−	0.974556i	$-0.571959\pi$
$163$	−5.72341	−0.448292	−0.224146	+	0.974556i	$0.571959\pi$
$164$	0	0
$165$	−14.2766	−1.11143
$166$	0	0
$167$	16.4323	1.27157	0.635785	−	0.771866i	$-0.280676\pi$
$167$	16.4323	1.27157	0.635785	+	0.771866i	$0.280676\pi$
$168$	0	0
$169$	27.0499	2.08076
$170$	0	0
$171$	−2.53219	−0.193641
$172$	0	0
$173$	8.02493	0.610124	0.305062	−	0.952332i	$-0.401323\pi$
$173$	8.02493	0.610124	0.305062	+	0.952332i	$0.401323\pi$
$174$	0	0
$175$	12.3410	0.932889
$176$	0	0
$177$	4.44084	0.333794
$178$	0	0
$179$	−11.7214	−0.876096	−0.438048	−	0.898952i	$-0.644330\pi$
$179$	−11.7214	−0.876096	−0.438048	+	0.898952i	$0.644330\pi$
$180$	0	0
$181$	10.1038	0.751011	0.375505	−	0.926820i	$-0.377469\pi$
$181$	10.1038	0.751011	0.375505	+	0.926820i	$0.377469\pi$
$182$	0	0
$183$	1.12481	0.0831480
$184$	0	0
$185$	11.6445	0.856123
$186$	0	0
$187$	24.8646	1.81828
$188$	0	0
$189$	3.78384	0.275234
$190$	0	0
$191$	−7.72136	−0.558698	−0.279349	−	0.960190i	$-0.590119\pi$
$191$	−7.72136	−0.558698	−0.279349	+	0.960190i	$0.590119\pi$
$192$	0	0
$193$	−13.2891	−0.956567	−0.478284	−	0.878205i	$-0.658741\pi$
$193$	−13.2891	−0.956567	−0.478284	+	0.878205i	$0.658741\pi$
$194$	0	0
$195$	18.0249	1.29079
$196$	0	0
$197$	11.9211	0.849344	0.424672	−	0.905347i	$-0.360389\pi$
$197$	11.9211	0.849344	0.424672	+	0.905347i	$0.360389\pi$
$198$	0	0
$199$	−20.0854	−1.42381	−0.711907	−	0.702274i	$-0.752168\pi$
$199$	−20.0854	−1.42381	−0.711907	+	0.702274i	$0.752168\pi$
$200$	0	0
$201$	0.710947	0.0501463
$202$	0	0
$203$	1.20369	0.0844825
$204$	0	0
$205$	0.683969	0.0477705
$206$	0	0
$207$	10.9606	0.761811
$208$	0	0
$209$	5.01247	0.346719
$210$	0	0
$211$	23.3929	1.61043	0.805216	−	0.592982i	$-0.202049\pi$
$211$	23.3929	1.61043	0.805216	+	0.592982i	$0.202049\pi$
$212$	0	0
$213$	6.16425	0.422367
$214$	0	0
$215$	3.31603	0.226151
$216$	0	0
$217$	5.69643	0.386699
$218$	0	0
$219$	−8.22468	−0.555772
$220$	0	0
$221$	−31.3929	−2.11171
$222$	0	0
$223$	−3.89619	−0.260908	−0.130454	−	0.991454i	$-0.541643\pi$
$223$	−3.89619	−0.260908	−0.130454	+	0.991454i	$0.541643\pi$
$224$	0	0
$225$	−31.2496	−2.08331
$226$	0	0
$227$	9.92112	0.658488	0.329244	−	0.944245i	$-0.393206\pi$
$227$	9.92112	0.658488	0.329244	+	0.944245i	$0.393206\pi$
$228$	0	0
$229$	20.1642	1.33249	0.666246	−	0.745732i	$-0.267900\pi$
$229$	20.1642	1.33249	0.666246	+	0.745732i	$0.267900\pi$
$230$	0	0
$231$	−3.42837	−0.225570
$232$	0	0
$233$	13.9730	0.915403	0.457702	−	0.889106i	$-0.348673\pi$
$233$	13.9730	0.915403	0.457702	+	0.889106i	$0.348673\pi$
$234$	0	0
$235$	19.5052	1.27238
$236$	0	0
$237$	0.319970	0.0207843
$238$	0	0
$239$	−18.0249	−1.16594	−0.582968	−	0.812495i	$-0.698109\pi$
$239$	−18.0249	−1.16594	−0.582968	+	0.812495i	$0.698109\pi$
$240$	0	0
$241$	−22.9001	−1.47513	−0.737563	−	0.675278i	$-0.764024\pi$
$241$	−22.9001	−1.47513	−0.737563	+	0.675278i	$0.764024\pi$
$242$	0	0
$243$	−14.7772	−0.947959
$244$	0	0
$245$	4.16425	0.266044
$246$	0	0
$247$	−6.32850	−0.402673
$248$	0	0
$249$	3.18524	0.201856
$250$	0	0
$251$	22.6819	1.43167	0.715835	−	0.698269i	$-0.246046\pi$
$251$	22.6819	1.43167	0.715835	+	0.698269i	$0.246046\pi$
$252$	0	0
$253$	−21.6964	−1.36404
$254$	0	0
$255$	−14.1287	−0.884776
$256$	0	0
$257$	−16.4927	−1.02879	−0.514395	−	0.857554i	$-0.671983\pi$
$257$	−16.4927	−1.02879	−0.514395	+	0.857554i	$0.671983\pi$
$258$	0	0
$259$	2.79631	0.173754
$260$	0	0
$261$	−3.04797	−0.188664
$262$	0	0
$263$	−13.2102	−0.814574	−0.407287	−	0.913300i	$-0.633525\pi$
$263$	−13.2102	−0.814574	−0.407287	+	0.913300i	$0.633525\pi$
$264$	0	0
$265$	16.8731	1.03651
$266$	0	0
$267$	4.79631	0.293529
$268$	0	0
$269$	11.9211	0.726843	0.363422	−	0.931625i	$-0.381608\pi$
$269$	11.9211	0.726843	0.363422	+	0.931625i	$0.381608\pi$
$270$	0	0
$271$	−5.22862	−0.317616	−0.158808	−	0.987309i	$-0.550765\pi$
$271$	−5.22862	−0.317616	−0.158808	+	0.987309i	$0.550765\pi$
$272$	0	0
$273$	4.32850	0.261972
$274$	0	0
$275$	61.8586	3.73022
$276$	0	0
$277$	−0.182700	−0.0109774	−0.00548868	−	0.999985i	$-0.501747\pi$
$277$	−0.182700	−0.0109774	−0.00548868	+	0.999985i	$0.501747\pi$
$278$	0	0
$279$	−14.4244	−0.863568
$280$	0	0
$281$	−27.6964	−1.65223	−0.826115	−	0.563501i	$-0.809454\pi$
$281$	−27.6964	−1.65223	−0.826115	+	0.563501i	$0.809454\pi$
$282$	0	0
$283$	23.3929	1.39056	0.695281	−	0.718738i	$-0.255280\pi$
$283$	23.3929	1.39056	0.695281	+	0.718738i	$0.255280\pi$
$284$	0	0
$285$	−2.84822	−0.168714
$286$	0	0
$287$	0.164248	0.00969525
$288$	0	0
$289$	7.60713	0.447478
$290$	0	0
$291$	3.86068	0.226317
$292$	0	0
$293$	−11.5926	−0.677248	−0.338624	−	0.940922i	$-0.609961\pi$
$293$	−11.5926	−0.677248	−0.338624	+	0.940922i	$0.609961\pi$
$294$	0	0
$295$	−27.0374	−1.57418
$296$	0	0
$297$	18.9664	1.10054
$298$	0	0
$299$	27.3929	1.58417
$300$	0	0
$301$	0.796310	0.0458985
$302$	0	0
$303$	13.2102	0.758904
$304$	0	0
$305$	−6.84822	−0.392128
$306$	0	0
$307$	−28.1352	−1.60576	−0.802881	−	0.596139i	$-0.796701\pi$
$307$	−28.1352	−1.60576	−0.802881	+	0.596139i	$0.796701\pi$
$308$	0	0
$309$	−4.83181	−0.274872
$310$	0	0
$311$	5.34096	0.302858	0.151429	−	0.988468i	$-0.451612\pi$
$311$	5.34096	0.302858	0.151429	+	0.988468i	$0.451612\pi$
$312$	0	0
$313$	1.11833	0.0632116	0.0316058	−	0.999500i	$-0.489938\pi$
$313$	1.11833	0.0632116	0.0316058	+	0.999500i	$0.489938\pi$
$314$	0	0
$315$	−10.5447	−0.594124
$316$	0	0
$317$	−27.4448	−1.54145	−0.770726	−	0.637167i	$-0.780106\pi$
$317$	−27.4448	−1.54145	−0.770726	+	0.637167i	$0.780106\pi$
$318$	0	0
$319$	6.03346	0.337809
$320$	0	0
$321$	−13.0394	−0.727791
$322$	0	0
$323$	4.96056	0.276013
$324$	0	0
$325$	−78.0997	−4.33219
$326$	0	0
$327$	−8.24313	−0.455846
$328$	0	0
$329$	4.68397	0.258236
$330$	0	0
$331$	−12.2247	−0.671929	−0.335965	−	0.941875i	$-0.609062\pi$
$331$	−12.2247	−0.671929	−0.335965	+	0.941875i	$0.609062\pi$
$332$	0	0
$333$	−7.08078	−0.388024
$334$	0	0
$335$	−4.32850	−0.236491
$336$	0	0
$337$	−7.14326	−0.389118	−0.194559	−	0.980891i	$-0.562328\pi$
$337$	−7.14326	−0.389118	−0.194559	+	0.980891i	$0.562328\pi$
$338$	0	0
$339$	6.57811	0.357274
$340$	0	0
$341$	28.5532	1.54624
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	12.3285	0.663744
$346$	0	0
$347$	12.6570	0.679463	0.339731	−	0.940522i	$-0.389664\pi$
$347$	12.6570	0.679463	0.339731	+	0.940522i	$0.389664\pi$
$348$	0	0
$349$	−11.3140	−0.605624	−0.302812	−	0.953050i	$-0.597925\pi$
$349$	−11.3140	−0.605624	−0.302812	+	0.953050i	$0.597925\pi$
$350$	0	0
$351$	−23.9460	−1.27815
$352$	0	0
$353$	−6.43231	−0.342357	−0.171179	−	0.985240i	$-0.554758\pi$
$353$	−6.43231	−0.342357	−0.171179	+	0.985240i	$0.554758\pi$
$354$	0	0
$355$	−37.5301	−1.99189
$356$	0	0
$357$	−3.39287	−0.179570
$358$	0	0
$359$	−25.8002	−1.36169	−0.680843	−	0.732430i	$-0.738386\pi$
$359$	−25.8002	−1.36169	−0.680843	+	0.732430i	$0.738386\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.66093	−0.507067
$364$	0	0
$365$	50.0748	2.62103
$366$	0	0
$367$	−3.97302	−0.207390	−0.103695	−	0.994609i	$-0.533067\pi$
$367$	−3.97302	−0.207390	−0.103695	+	0.994609i	$0.533067\pi$
$368$	0	0
$369$	−0.415906	−0.0216512
$370$	0	0
$371$	4.05191	0.210365
$372$	0	0
$373$	21.0459	1.08972	0.544858	−	0.838528i	$-0.316583\pi$
$373$	21.0459	1.08972	0.544858	+	0.838528i	$0.316583\pi$
$374$	0	0
$375$	−20.9087	−1.07972
$376$	0	0
$377$	−7.61755	−0.392324
$378$	0	0
$379$	3.28905	0.168947	0.0844737	−	0.996426i	$-0.473079\pi$
$379$	3.28905	0.168947	0.0844737	+	0.996426i	$0.473079\pi$
$380$	0	0
$381$	−5.22862	−0.267870
$382$	0	0
$383$	2.24961	0.114950	0.0574749	−	0.998347i	$-0.481695\pi$
$383$	2.24961	0.114950	0.0574749	+	0.998347i	$0.481695\pi$
$384$	0	0
$385$	20.8731	1.06379
$386$	0	0
$387$	−2.01640	−0.102500
$388$	0	0
$389$	−4.07888	−0.206808	−0.103404	−	0.994639i	$-0.532973\pi$
$389$	−4.07888	−0.206808	−0.103404	+	0.994639i	$0.532973\pi$
$390$	0	0
$391$	−21.4718	−1.08587
$392$	0	0
$393$	−9.14326	−0.461216
$394$	0	0
$395$	−1.94809	−0.0980192
$396$	0	0
$397$	−32.2226	−1.61721	−0.808604	−	0.588354i	$-0.799776\pi$
$397$	−32.2226	−1.61721	−0.808604	+	0.588354i	$0.799776\pi$
$398$	0	0
$399$	−0.683969	−0.0342413
$400$	0	0
$401$	19.5387	0.975714	0.487857	−	0.872923i	$-0.337779\pi$
$401$	19.5387	0.975714	0.487857	+	0.872923i	$0.337779\pi$
$402$	0	0
$403$	−36.0499	−1.79577
$404$	0	0
$405$	20.8567	1.03638
$406$	0	0
$407$	14.0164	0.694767
$408$	0	0
$409$	5.09135	0.251751	0.125876	−	0.992046i	$-0.459826\pi$
$409$	5.09135	0.251751	0.125876	+	0.992046i	$0.459826\pi$
$410$	0	0
$411$	−8.39098	−0.413896
$412$	0	0
$413$	−6.49274	−0.319487
$414$	0	0
$415$	−19.3929	−0.951958
$416$	0	0
$417$	4.60713	0.225612
$418$	0	0
$419$	10.1458	0.495655	0.247827	−	0.968804i	$-0.420283\pi$
$419$	10.1458	0.495655	0.247827	+	0.968804i	$0.420283\pi$
$420$	0	0
$421$	−34.6570	−1.68908	−0.844539	−	0.535494i	$-0.820126\pi$
$421$	−34.6570	−1.68908	−0.844539	+	0.535494i	$0.820126\pi$
$422$	0	0
$423$	−11.8607	−0.576686
$424$	0	0
$425$	61.2180	2.96951
$426$	0	0
$427$	−1.64453	−0.0795842
$428$	0	0
$429$	21.6964	1.04751
$430$	0	0
$431$	−34.5880	−1.66605	−0.833023	−	0.553238i	$-0.813392\pi$
$431$	−34.5880	−1.66605	−0.833023	+	0.553238i	$0.813392\pi$
$432$	0	0
$433$	−22.9001	−1.10051	−0.550255	−	0.834997i	$-0.685469\pi$
$433$	−22.9001	−1.10051	−0.550255	+	0.834997i	$0.685469\pi$
$434$	0	0
$435$	−3.42837	−0.164378
$436$	0	0
$437$	−4.32850	−0.207060
$438$	0	0
$439$	−6.35343	−0.303232	−0.151616	−	0.988439i	$-0.548448\pi$
$439$	−6.35343	−0.303232	−0.151616	+	0.988439i	$0.548448\pi$
$440$	0	0
$441$	−2.53219	−0.120580
$442$	0	0
$443$	−28.8462	−1.37052	−0.685261	−	0.728297i	$-0.740312\pi$
$443$	−28.8462	−1.37052	−0.685261	+	0.728297i	$0.740312\pi$
$444$	0	0
$445$	−29.2016	−1.38429
$446$	0	0
$447$	8.86462	0.419282
$448$	0	0
$449$	−27.6425	−1.30453	−0.652265	−	0.757991i	$-0.726181\pi$
$449$	−27.6425	−1.30453	−0.652265	+	0.757991i	$0.726181\pi$
$450$	0	0
$451$	0.823287	0.0387671
$452$	0	0
$453$	13.2641	0.623203
$454$	0	0
$455$	−26.3534	−1.23547
$456$	0	0
$457$	−33.9191	−1.58667	−0.793334	−	0.608787i	$-0.791656\pi$
$457$	−33.9191	−1.58667	−0.793334	+	0.608787i	$0.791656\pi$
$458$	0	0
$459$	18.7700	0.876108
$460$	0	0
$461$	3.82723	0.178252	0.0891259	−	0.996020i	$-0.471593\pi$
$461$	3.82723	0.178252	0.0891259	+	0.996020i	$0.471593\pi$
$462$	0	0
$463$	−3.44682	−0.160187	−0.0800937	−	0.996787i	$-0.525522\pi$
$463$	−3.44682	−0.160187	−0.0800937	+	0.996787i	$0.525522\pi$
$464$	0	0
$465$	−16.2247	−0.752402
$466$	0	0
$467$	−11.5137	−0.532792	−0.266396	−	0.963864i	$-0.585833\pi$
$467$	−11.5137	−0.532792	−0.266396	+	0.963864i	$0.585833\pi$
$468$	0	0
$469$	−1.03944	−0.0479970
$470$	0	0
$471$	−0.746450	−0.0343946
$472$	0	0
$473$	3.99147	0.183528
$474$	0	0
$475$	12.3410	0.566242
$476$	0	0
$477$	−10.2602	−0.469782
$478$	0	0
$479$	−13.0210	−0.594944	−0.297472	−	0.954731i	$-0.596144\pi$
$479$	−13.0210	−0.594944	−0.297472	+	0.954731i	$0.596144\pi$
$480$	0	0
$481$	−17.6964	−0.806888
$482$	0	0
$483$	2.96056	0.134710
$484$	0	0
$485$	−23.5052	−1.06732
$486$	0	0
$487$	−37.6240	−1.70491	−0.852454	−	0.522803i	$-0.824886\pi$
$487$	−37.6240	−1.70491	−0.852454	+	0.522803i	$0.824886\pi$
$488$	0	0
$489$	3.91464	0.177026
$490$	0	0
$491$	−23.8422	−1.07598	−0.537992	−	0.842950i	$-0.680817\pi$
$491$	−23.8422	−1.07598	−0.537992	+	0.842950i	$0.680817\pi$
$492$	0	0
$493$	5.97098	0.268919
$494$	0	0
$495$	−52.8547	−2.37564
$496$	0	0
$497$	−9.01247	−0.404264
$498$	0	0
$499$	−5.40344	−0.241891	−0.120946	−	0.992659i	$-0.538593\pi$
$499$	−5.40344	−0.241891	−0.120946	+	0.992659i	$0.538593\pi$
$500$	0	0
$501$	−11.2392	−0.502130
$502$	0	0
$503$	41.5571	1.85294	0.926470	−	0.376368i	$-0.122827\pi$
$503$	41.5571	1.85294	0.926470	+	0.376368i	$0.122827\pi$
$504$	0	0
$505$	−80.4282	−3.57901
$506$	0	0
$507$	−18.5013	−0.821670
$508$	0	0
$509$	16.6819	0.739413	0.369707	−	0.929149i	$-0.379458\pi$
$509$	16.6819	0.739413	0.369707	+	0.929149i	$0.379458\pi$
$510$	0	0
$511$	12.0249	0.531952
$512$	0	0
$513$	3.78384	0.167061
$514$	0	0
$515$	29.4178	1.29630
$516$	0	0
$517$	23.4782	1.03257
$518$	0	0
$519$	−5.48880	−0.240932
$520$	0	0
$521$	13.1852	0.577656	0.288828	−	0.957381i	$-0.406734\pi$
$521$	13.1852	0.577656	0.288828	+	0.957381i	$0.406734\pi$
$522$	0	0
$523$	22.0789	0.965442	0.482721	−	0.875774i	$-0.339648\pi$
$523$	22.0789	0.965442	0.482721	+	0.875774i	$0.339648\pi$
$524$	0	0
$525$	−8.44084	−0.368388
$526$	0	0
$527$	28.2575	1.23092
$528$	0	0
$529$	−4.26412	−0.185397
$530$	0	0
$531$	16.4408	0.713472
$532$	0	0
$533$	−1.03944	−0.0450232
$534$	0	0
$535$	79.3888	3.43228
$536$	0	0
$537$	8.01705	0.345961
$538$	0	0
$539$	5.01247	0.215902
$540$	0	0
$541$	20.6321	0.887042	0.443521	−	0.896264i	$-0.353729\pi$
$541$	20.6321	0.887042	0.443521	+	0.896264i	$0.353729\pi$
$542$	0	0
$543$	−6.91070	−0.296566
$544$	0	0
$545$	50.1871	2.14978
$546$	0	0
$547$	41.1931	1.76129	0.880645	−	0.473776i	$-0.157109\pi$
$547$	41.1931	1.76129	0.880645	+	0.473776i	$0.157109\pi$
$548$	0	0
$549$	4.16425	0.177726
$550$	0	0
$551$	1.20369	0.0512789
$552$	0	0
$553$	−0.467814	−0.0198935
$554$	0	0
$555$	−7.96450	−0.338074
$556$	0	0
$557$	−27.0353	−1.14552	−0.572762	−	0.819722i	$-0.694128\pi$
$557$	−27.0353	−1.14552	−0.572762	+	0.819722i	$0.694128\pi$
$558$	0	0
$559$	−5.03944	−0.213146
$560$	0	0
$561$	−17.0066	−0.718021
$562$	0	0
$563$	40.5761	1.71008	0.855039	−	0.518565i	$-0.173533\pi$
$563$	40.5761	1.71008	0.855039	+	0.518565i	$0.173533\pi$
$564$	0	0
$565$	−40.0499	−1.68491
$566$	0	0
$567$	5.00853	0.210338
$568$	0	0
$569$	20.6819	0.867031	0.433516	−	0.901146i	$-0.357273\pi$
$569$	20.6819	0.867031	0.433516	+	0.901146i	$0.357273\pi$
$570$	0	0
$571$	5.72341	0.239517	0.119759	−	0.992803i	$-0.461788\pi$
$571$	5.72341	0.239517	0.119759	+	0.992803i	$0.461788\pi$
$572$	0	0
$573$	5.28117	0.220624
$574$	0	0
$575$	−53.4178	−2.22768
$576$	0	0
$577$	23.3638	0.972650	0.486325	−	0.873778i	$-0.338337\pi$
$577$	23.3638	0.972650	0.486325	+	0.873778i	$0.338337\pi$
$578$	0	0
$579$	9.08930	0.377739
$580$	0	0
$581$	−4.65699	−0.193205
$582$	0	0
$583$	20.3100	0.841156
$584$	0	0
$585$	66.7318	2.75902
$586$	0	0
$587$	7.01451	0.289520	0.144760	−	0.989467i	$-0.453759\pi$
$587$	7.01451	0.289520	0.144760	+	0.989467i	$0.453759\pi$
$588$	0	0
$589$	5.69643	0.234717
$590$	0	0
$591$	−8.15367	−0.335397
$592$	0	0
$593$	8.63206	0.354476	0.177238	−	0.984168i	$-0.443284\pi$
$593$	8.63206	0.354476	0.177238	+	0.984168i	$0.443284\pi$
$594$	0	0
$595$	20.6570	0.846854
$596$	0	0
$597$	13.7378	0.562249
$598$	0	0
$599$	−17.6695	−0.721954	−0.360977	−	0.932575i	$-0.617557\pi$
$599$	−17.6695	−0.721954	−0.360977	+	0.932575i	$0.617557\pi$
$600$	0	0
$601$	−11.9211	−0.486272	−0.243136	−	0.969992i	$-0.578176\pi$
$601$	−11.9211	−0.486272	−0.243136	+	0.969992i	$0.578176\pi$
$602$	0	0
$603$	2.63206	0.107186
$604$	0	0
$605$	58.8192	2.39134
$606$	0	0
$607$	39.9460	1.62136	0.810680	−	0.585490i	$-0.199098\pi$
$607$	39.9460	1.62136	0.810680	+	0.585490i	$0.199098\pi$
$608$	0	0
$609$	−0.823287	−0.0333613
$610$	0	0
$611$	−29.6425	−1.19921
$612$	0	0
$613$	−25.7753	−1.04106	−0.520528	−	0.853845i	$-0.674265\pi$
$613$	−25.7753	−1.04106	−0.520528	+	0.853845i	$0.674265\pi$
$614$	0	0
$615$	−0.467814	−0.0188641
$616$	0	0
$617$	−5.98155	−0.240808	−0.120404	−	0.992725i	$-0.538419\pi$
$617$	−5.98155	−0.240808	−0.120404	+	0.992725i	$0.538419\pi$
$618$	0	0
$619$	−39.1142	−1.57213	−0.786067	−	0.618141i	$-0.787886\pi$
$619$	−39.1142	−1.57213	−0.786067	+	0.618141i	$0.787886\pi$
$620$	0	0
$621$	−16.3784	−0.657241
$622$	0	0
$623$	−7.01247	−0.280948
$624$	0	0
$625$	65.5945	2.62378
$626$	0	0
$627$	−3.42837	−0.136916
$628$	0	0
$629$	13.8713	0.553083
$630$	0	0
$631$	15.2351	0.606500	0.303250	−	0.952911i	$-0.401928\pi$
$631$	15.2351	0.606500	0.303250	+	0.952911i	$0.401928\pi$
$632$	0	0
$633$	−16.0000	−0.635943
$634$	0	0
$635$	31.8337	1.26328
$636$	0	0
$637$	−6.32850	−0.250744
$638$	0	0
$639$	22.8212	0.902794
$640$	0	0
$641$	10.9186	0.431258	0.215629	−	0.976475i	$-0.430820\pi$
$641$	10.9186	0.431258	0.215629	+	0.976475i	$0.430820\pi$
$642$	0	0
$643$	−0.607132	−0.0239429	−0.0119715	−	0.999928i	$-0.503811\pi$
$643$	−0.607132	−0.0239429	−0.0119715	+	0.999928i	$0.503811\pi$
$644$	0	0
$645$	−2.26806	−0.0893049
$646$	0	0
$647$	16.4139	0.645295	0.322648	−	0.946519i	$-0.395427\pi$
$647$	16.4139	0.645295	0.322648	+	0.946519i	$0.395427\pi$
$648$	0	0
$649$	−32.5447	−1.27749
$650$	0	0
$651$	−3.89619	−0.152704
$652$	0	0
$653$	−35.6923	−1.39675	−0.698375	−	0.715732i	$-0.746093\pi$
$653$	−35.6923	−1.39675	−0.698375	+	0.715732i	$0.746093\pi$
$654$	0	0
$655$	55.6674	2.17511
$656$	0	0
$657$	−30.4494	−1.18794
$658$	0	0
$659$	−3.55064	−0.138313	−0.0691566	−	0.997606i	$-0.522031\pi$
$659$	−3.55064	−0.138313	−0.0691566	+	0.997606i	$0.522031\pi$
$660$	0	0
$661$	45.8921	1.78500	0.892498	−	0.451052i	$-0.148951\pi$
$661$	45.8921	1.78500	0.892498	+	0.451052i	$0.148951\pi$
$662$	0	0
$663$	21.4718	0.833894
$664$	0	0
$665$	4.16425	0.161483
$666$	0	0
$667$	−5.21017	−0.201739
$668$	0	0
$669$	2.66487	0.103030
$670$	0	0
$671$	−8.24313	−0.318223
$672$	0	0
$673$	11.0353	0.425381	0.212691	−	0.977120i	$-0.431777\pi$
$673$	11.0353	0.425381	0.212691	+	0.977120i	$0.431777\pi$
$674$	0	0
$675$	46.6963	1.79734
$676$	0	0
$677$	−0.128745	−0.00494807	−0.00247403	−	0.999997i	$-0.500788\pi$
$677$	−0.128745	−0.00494807	−0.00247403	+	0.999997i	$0.500788\pi$
$678$	0	0
$679$	−5.64453	−0.216617
$680$	0	0
$681$	−6.78574	−0.260030
$682$	0	0
$683$	−20.8817	−0.799015	−0.399508	−	0.916730i	$-0.630819\pi$
$683$	−20.8817	−0.799015	−0.399508	+	0.916730i	$0.630819\pi$
$684$	0	0
$685$	51.0873	1.95194
$686$	0	0
$687$	−13.7917	−0.526187
$688$	0	0
$689$	−25.6425	−0.976901
$690$	0	0
$691$	−23.8791	−0.908405	−0.454203	−	0.890898i	$-0.650076\pi$
$691$	−23.8791	−0.908405	−0.454203	+	0.890898i	$0.650076\pi$
$692$	0	0
$693$	−12.6925	−0.482148
$694$	0	0
$695$	−28.0499	−1.06399
$696$	0	0
$697$	0.814761	0.0308613
$698$	0	0
$699$	−9.55712	−0.361483
$700$	0	0
$701$	−0.839691	−0.0317147	−0.0158574	−	0.999874i	$-0.505048\pi$
$701$	−0.839691	−0.0317147	−0.0158574	+	0.999874i	$0.505048\pi$
$702$	0	0
$703$	2.79631	0.105465
$704$	0	0
$705$	−13.3410	−0.502450
$706$	0	0
$707$	−19.3140	−0.726377
$708$	0	0
$709$	32.4572	1.21896	0.609479	−	0.792802i	$-0.291379\pi$
$709$	32.4572	1.21896	0.609479	+	0.792802i	$0.291379\pi$
$710$	0	0
$711$	1.18459	0.0444257
$712$	0	0
$713$	−24.6570	−0.923412
$714$	0	0
$715$	−132.096	−4.94010
$716$	0	0
$717$	12.3285	0.460416
$718$	0	0
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	7.06437	0.263091
$722$	0	0
$723$	15.6630	0.582512
$724$	0	0
$725$	14.8547	0.551690
$726$	0	0
$727$	−13.2870	−0.492788	−0.246394	−	0.969170i	$-0.579246\pi$
$727$	−13.2870	−0.492788	−0.246394	+	0.969170i	$0.579246\pi$
$728$	0	0
$729$	−4.91842	−0.182164
$730$	0	0
$731$	3.95014	0.146101
$732$	0	0
$733$	19.3993	0.716531	0.358266	−	0.933620i	$-0.383368\pi$
$733$	19.3993	0.716531	0.358266	+	0.933620i	$0.383368\pi$
$734$	0	0
$735$	−2.84822	−0.105058
$736$	0	0
$737$	−5.21017	−0.191919
$738$	0	0
$739$	−27.2536	−1.00254	−0.501269	−	0.865291i	$-0.667133\pi$
$739$	−27.2536	−1.00254	−0.501269	+	0.865291i	$0.667133\pi$
$740$	0	0
$741$	4.32850	0.159011
$742$	0	0
$743$	−39.5322	−1.45030	−0.725148	−	0.688593i	$-0.758229\pi$
$743$	−39.5322	−1.45030	−0.725148	+	0.688593i	$0.758229\pi$
$744$	0	0
$745$	−53.9710	−1.97734
$746$	0	0
$747$	11.7924	0.431460
$748$	0	0
$749$	19.0644	0.696597
$750$	0	0
$751$	−4.13932	−0.151046	−0.0755229	−	0.997144i	$-0.524063\pi$
$751$	−4.13932	−0.151046	−0.0755229	+	0.997144i	$0.524063\pi$
$752$	0	0
$753$	−15.5137	−0.565352
$754$	0	0
$755$	−80.7567	−2.93904
$756$	0	0
$757$	−14.2745	−0.518817	−0.259409	−	0.965768i	$-0.583528\pi$
$757$	−14.2745	−0.518817	−0.259409	+	0.965768i	$0.583528\pi$
$758$	0	0
$759$	14.8397	0.538647
$760$	0	0
$761$	−20.5781	−0.745956	−0.372978	−	0.927840i	$-0.621663\pi$
$761$	−20.5781	−0.745956	−0.372978	+	0.927840i	$0.621663\pi$
$762$	0	0
$763$	12.0519	0.436308
$764$	0	0
$765$	−52.3073	−1.89118
$766$	0	0
$767$	41.0893	1.48365
$768$	0	0
$769$	34.7109	1.25171	0.625854	−	0.779940i	$-0.284750\pi$
$769$	34.7109	1.25171	0.625854	+	0.779940i	$0.284750\pi$
$770$	0	0
$771$	11.2805	0.406258
$772$	0	0
$773$	4.18270	0.150441	0.0752206	−	0.997167i	$-0.476034\pi$
$773$	4.18270	0.150441	0.0752206	+	0.997167i	$0.476034\pi$
$774$	0	0
$775$	70.2995	2.52523
$776$	0	0
$777$	−1.91259	−0.0686138
$778$	0	0
$779$	0.164248	0.00588479
$780$	0	0
$781$	−45.1747	−1.61648
$782$	0	0
$783$	4.55458	0.162767
$784$	0	0
$785$	4.54465	0.162206
$786$	0	0
$787$	−6.77138	−0.241374	−0.120687	−	0.992691i	$-0.538510\pi$
$787$	−6.77138	−0.241374	−0.120687	+	0.992691i	$0.538510\pi$
$788$	0	0
$789$	9.03535	0.321667
$790$	0	0
$791$	−9.61755	−0.341961
$792$	0	0
$793$	10.4074	0.369577
$794$	0	0
$795$	−11.5407	−0.409307
$796$	0	0
$797$	−34.8107	−1.23306	−0.616529	−	0.787333i	$-0.711462\pi$
$797$	−34.8107	−1.23306	−0.616529	+	0.787333i	$0.711462\pi$
$798$	0	0
$799$	23.2351	0.821999
$800$	0	0
$801$	17.7569	0.627408
$802$	0	0
$803$	60.2745	2.12704
$804$	0	0
$805$	−18.0249	−0.635295
$806$	0	0
$807$	−8.15367	−0.287023
$808$	0	0
$809$	−23.3324	−0.820325	−0.410162	−	0.912013i	$-0.634528\pi$
$809$	−23.3324	−0.820325	−0.410162	+	0.912013i	$0.634528\pi$
$810$	0	0
$811$	7.91907	0.278076	0.139038	−	0.990287i	$-0.455599\pi$
$811$	7.91907	0.278076	0.139038	+	0.990287i	$0.455599\pi$
$812$	0	0
$813$	3.57621	0.125423
$814$	0	0
$815$	−23.8337	−0.834858
$816$	0	0
$817$	0.796310	0.0278593
$818$	0	0
$819$	16.0249	0.559956
$820$	0	0
$821$	−29.1852	−1.01857	−0.509286	−	0.860597i	$-0.670090\pi$
$821$	−29.1852	−1.01857	−0.509286	+	0.860597i	$0.670090\pi$
$822$	0	0
$823$	−15.0104	−0.523230	−0.261615	−	0.965172i	$-0.584255\pi$
$823$	−15.0104	−0.523230	−0.261615	+	0.965172i	$0.584255\pi$
$824$	0	0
$825$	−42.3094	−1.47302
$826$	0	0
$827$	−12.3784	−0.430438	−0.215219	−	0.976566i	$-0.569046\pi$
$827$	−12.3784	−0.430438	−0.215219	+	0.976566i	$0.569046\pi$
$828$	0	0
$829$	−32.5611	−1.13089	−0.565446	−	0.824785i	$-0.691296\pi$
$829$	−32.5611	−1.13089	−0.565446	+	0.824785i	$0.691296\pi$
$830$	0	0
$831$	0.124961	0.00433485
$832$	0	0
$833$	4.96056	0.171873
$834$	0	0
$835$	68.4282	2.36806
$836$	0	0
$837$	21.5544	0.745030
$838$	0	0
$839$	−6.35343	−0.219345	−0.109672	−	0.993968i	$-0.534980\pi$
$839$	−6.35343	−0.219345	−0.109672	+	0.993968i	$0.534980\pi$
$840$	0	0
$841$	−27.5511	−0.950039
$842$	0	0
$843$	18.9435	0.652449
$844$	0	0
$845$	112.642	3.87501
$846$	0	0
$847$	14.1248	0.485334
$848$	0	0
$849$	−16.0000	−0.549119
$850$	0	0
$851$	−12.1038	−0.414913
$852$	0	0
$853$	−38.2930	−1.31113	−0.655564	−	0.755140i	$-0.727569\pi$
$853$	−38.2930	−1.31113	−0.655564	+	0.755140i	$0.727569\pi$
$854$	0	0
$855$	−10.5447	−0.360619
$856$	0	0
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	13.8002	0.470858	0.235429	−	0.971891i	$-0.424350\pi$
$859$	13.8002	0.470858	0.235429	+	0.971891i	$0.424350\pi$
$860$	0	0
$861$	−0.112341	−0.00382855
$862$	0	0
$863$	15.0459	0.512169	0.256085	−	0.966654i	$-0.417567\pi$
$863$	15.0459	0.512169	0.256085	+	0.966654i	$0.417567\pi$
$864$	0	0
$865$	33.4178	1.13624
$866$	0	0
$867$	−5.20304	−0.176705
$868$	0	0
$869$	−2.34490	−0.0795453
$870$	0	0
$871$	6.57811	0.222891
$872$	0	0
$873$	14.2930	0.483745
$874$	0	0
$875$	30.5696	1.03344
$876$	0	0
$877$	0.659039	0.0222542	0.0111271	−	0.999938i	$-0.496458\pi$
$877$	0.659039	0.0222542	0.0111271	+	0.999938i	$0.496458\pi$
$878$	0	0
$879$	7.92899	0.267438
$880$	0	0
$881$	39.8501	1.34258	0.671292	−	0.741193i	$-0.265740\pi$
$881$	39.8501	1.34258	0.671292	+	0.741193i	$0.265740\pi$
$882$	0	0
$883$	−13.5158	−0.454842	−0.227421	−	0.973797i	$-0.573029\pi$
$883$	−13.5158	−0.454842	−0.227421	+	0.973797i	$0.573029\pi$
$884$	0	0
$885$	18.4927	0.621627
$886$	0	0
$887$	−52.4822	−1.76218	−0.881089	−	0.472950i	$-0.843189\pi$
$887$	−52.4822	−1.76218	−0.881089	+	0.472950i	$0.843189\pi$
$888$	0	0
$889$	7.64453	0.256389
$890$	0	0
$891$	25.1051	0.841051
$892$	0	0
$893$	4.68397	0.156743
$894$	0	0
$895$	−48.8107	−1.63156
$896$	0	0
$897$	−18.7359	−0.625573
$898$	0	0
$899$	6.85674	0.228685
$900$	0	0
$901$	20.0997	0.669619
$902$	0	0
$903$	−0.544651	−0.0181249
$904$	0	0
$905$	42.0748	1.39861
$906$	0	0
$907$	−27.2891	−0.906118	−0.453059	−	0.891480i	$-0.649667\pi$
$907$	−27.2891	−0.906118	−0.453059	+	0.891480i	$0.649667\pi$
$908$	0	0
$909$	48.9066	1.62213
$910$	0	0
$911$	−32.3968	−1.07335	−0.536677	−	0.843788i	$-0.680321\pi$
$911$	−32.3968	−1.07335	−0.536677	+	0.843788i	$0.680321\pi$
$912$	0	0
$913$	−23.3430	−0.772541
$914$	0	0
$915$	4.68397	0.154847
$916$	0	0
$917$	13.3679	0.441448
$918$	0	0
$919$	−59.0063	−1.94644	−0.973219	−	0.229878i	$-0.926167\pi$
$919$	−59.0063	−1.94644	−0.973219	+	0.229878i	$0.926167\pi$
$920$	0	0
$921$	19.2436	0.634099
$922$	0	0
$923$	57.0353	1.87734
$924$	0	0
$925$	34.5091	1.13465
$926$	0	0
$927$	−17.8883	−0.587529
$928$	0	0
$929$	−55.0353	−1.80565	−0.902826	−	0.430007i	$-0.858511\pi$
$929$	−55.0353	−1.80565	−0.902826	+	0.430007i	$0.858511\pi$
$930$	0	0
$931$	1.00000	0.0327737
$932$	0	0
$933$	−3.65305	−0.119596
$934$	0	0
$935$	103.542	3.38620
$936$	0	0
$937$	17.9829	0.587477	0.293739	−	0.955886i	$-0.405100\pi$
$937$	17.9829	0.587477	0.293739	+	0.955886i	$0.405100\pi$
$938$	0	0
$939$	−0.764901	−0.0249616
$940$	0	0
$941$	15.1433	0.493656	0.246828	−	0.969059i	$-0.420612\pi$
$941$	15.1433	0.493656	0.246828	+	0.969059i	$0.420612\pi$
$942$	0	0
$943$	−0.710947	−0.0231516
$944$	0	0
$945$	15.7569	0.512571
$946$	0	0
$947$	26.9875	0.876977	0.438489	−	0.898737i	$-0.355514\pi$
$947$	26.9875	0.876977	0.438489	+	0.898737i	$0.355514\pi$
$948$	0	0
$949$	−76.0997	−2.47030
$950$	0	0
$951$	18.7714	0.608704
$952$	0	0
$953$	56.5781	1.83274	0.916372	−	0.400327i	$-0.131103\pi$
$953$	56.5781	1.83274	0.916372	+	0.400327i	$0.131103\pi$
$954$	0	0
$955$	−32.1537	−1.04047
$956$	0	0
$957$	−4.12670	−0.133397
$958$	0	0
$959$	12.2681	0.396156
$960$	0	0
$961$	1.44936	0.0467536
$962$	0	0
$963$	−48.2745	−1.55563
$964$	0	0
$965$	−55.3389	−1.78142
$966$	0	0
$967$	26.1786	0.841847	0.420924	−	0.907096i	$-0.361706\pi$
$967$	26.1786	0.841847	0.420924	+	0.907096i	$0.361706\pi$
$968$	0	0
$969$	−3.39287	−0.108995
$970$	0	0
$971$	53.3369	1.71166	0.855831	−	0.517256i	$-0.173046\pi$
$971$	53.3369	1.71166	0.855831	+	0.517256i	$0.173046\pi$
$972$	0	0
$973$	−6.73588	−0.215942
$974$	0	0
$975$	53.4178	1.71074
$976$	0	0
$977$	25.9460	0.830088	0.415044	−	0.909801i	$-0.363766\pi$
$977$	25.9460	0.830088	0.415044	+	0.909801i	$0.363766\pi$
$978$	0	0
$979$	−35.1497	−1.12339
$980$	0	0
$981$	−30.5177	−0.974354
$982$	0	0
$983$	16.6570	0.531276	0.265638	−	0.964073i	$-0.414417\pi$
$983$	16.6570	0.531276	0.265638	+	0.964073i	$0.414417\pi$
$984$	0	0
$985$	49.6425	1.58174
$986$	0	0
$987$	−3.20369	−0.101975
$988$	0	0
$989$	−3.44682	−0.109603
$990$	0	0
$991$	−24.2476	−0.770249	−0.385125	−	0.922865i	$-0.625842\pi$
$991$	−24.2476	−0.770249	−0.385125	+	0.922865i	$0.625842\pi$
$992$	0	0
$993$	8.36130	0.265338
$994$	0	0
$995$	−83.6404	−2.65158
$996$	0	0
$997$	11.9935	0.379839	0.189919	−	0.981800i	$-0.439177\pi$
$997$	11.9935	0.379839	0.189919	+	0.981800i	$0.439177\pi$
$998$	0	0
$999$	10.5808	0.334762

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2128.2.a.s.1.2		3
4.3	odd	2		266.2.a.d.1.2	✓	3
8.3	odd	2		8512.2.a.bm.1.2		3
8.5	even	2		8512.2.a.bj.1.2		3
12.11	even	2		2394.2.a.ba.1.1		3
20.19	odd	2		6650.2.a.cd.1.2		3
28.27	even	2		1862.2.a.r.1.2		3
76.75	even	2		5054.2.a.r.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
266.2.a.d.1.2	✓	3	4.3	odd	2
1862.2.a.r.1.2		3	28.27	even	2
2128.2.a.s.1.2		3	1.1	even	1	trivial
2394.2.a.ba.1.1		3	12.11	even	2
5054.2.a.r.1.2		3	76.75	even	2
6650.2.a.cd.1.2		3	20.19	odd	2
8512.2.a.bj.1.2		3	8.5	even	2
8512.2.a.bm.1.2		3	8.3	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:	\( N \)	\(=\)	\( 2128 = 2^{4} \cdot 7 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2128.a (trivial)

\(f(q)\)	\(=\)	\(q-0.683969 q^{3} +4.16425 q^{5} +1.00000 q^{7} -2.53219 q^{9} +O(q^{10})\)	\(q-0.683969 q^{3} +4.16425 q^{5} +1.00000 q^{7} -2.53219 q^{9} +5.01247 q^{11} -6.32850 q^{13} -2.84822 q^{15} +4.96056 q^{17} +1.00000 q^{19} -0.683969 q^{21} -4.32850 q^{23} +12.3410 q^{25} +3.78384 q^{27} +1.20369 q^{29} +5.69643 q^{31} -3.42837 q^{33} +4.16425 q^{35} +2.79631 q^{37} +4.32850 q^{39} +0.164248 q^{41} +0.796310 q^{43} -10.5447 q^{45} +4.68397 q^{47} +1.00000 q^{49} -3.39287 q^{51} +4.05191 q^{53} +20.8731 q^{55} -0.683969 q^{57} -6.49274 q^{59} -1.64453 q^{61} -2.53219 q^{63} -26.3534 q^{65} -1.03944 q^{67} +2.96056 q^{69} -9.01247 q^{71} +12.0249 q^{73} -8.44084 q^{75} +5.01247 q^{77} -0.467814 q^{79} +5.00853 q^{81} -4.65699 q^{83} +20.6570 q^{85} -0.823287 q^{87} -7.01247 q^{89} -6.32850 q^{91} -3.89619 q^{93} +4.16425 q^{95} -5.64453 q^{97} -12.6925 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + 5 q^{5} + 3 q^{7} + 6 q^{9}+O(q^{10}) \) 3 * q + q^3 + 5 * q^5 + 3 * q^7 + 6 * q^9	\( 3 q + q^{3} + 5 q^{5} + 3 q^{7} + 6 q^{9} - 3 q^{11} - 4 q^{13} + 2 q^{15} + 6 q^{17} + 3 q^{19} + q^{21} + 2 q^{23} + 4 q^{25} + 28 q^{27} + 5 q^{29} - 4 q^{31} - 15 q^{33} + 5 q^{35} + 7 q^{37} - 2 q^{39} - 7 q^{41} + q^{43} + 11 q^{47} + 3 q^{49} + 32 q^{51} + 3 q^{53} + 16 q^{55} + q^{57} + 3 q^{59} + 7 q^{61} + 6 q^{63} - 28 q^{65} - 12 q^{67} - 9 q^{71} - 12 q^{75} - 3 q^{77} - 15 q^{79} + 35 q^{81} + 16 q^{83} + 32 q^{85} - 28 q^{87} - 3 q^{89} - 4 q^{91} - 30 q^{93} + 5 q^{95} - 5 q^{97} - 55 q^{99}+O(q^{100}) \) 3 * q + q^3 + 5 * q^5 + 3 * q^7 + 6 * q^9 - 3 * q^11 - 4 * q^13 + 2 * q^15 + 6 * q^17 + 3 * q^19 + q^21 + 2 * q^23 + 4 * q^25 + 28 * q^27 + 5 * q^29 - 4 * q^31 - 15 * q^33 + 5 * q^35 + 7 * q^37 - 2 * q^39 - 7 * q^41 + q^43 + 11 * q^47 + 3 * q^49 + 32 * q^51 + 3 * q^53 + 16 * q^55 + q^57 + 3 * q^59 + 7 * q^61 + 6 * q^63 - 28 * q^65 - 12 * q^67 - 9 * q^71 - 12 * q^75 - 3 * q^77 - 15 * q^79 + 35 * q^81 + 16 * q^83 + 32 * q^85 - 28 * q^87 - 3 * q^89 - 4 * q^91 - 30 * q^93 + 5 * q^95 - 5 * q^97 - 55 * q^99

Introduction

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