LMFDB - Embedded newform 8001.2.a.t.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8001,2,Mod(1,8001)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8001, 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("8001.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8001 = 3^{2} \cdot 7 \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8001.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:	$63.8883066572$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} + \cdots + 1 $ x^16 - 2x^15 - 20x^14 + 38x^13 + 155x^12 - 275x^11 - 593x^10 + 957x^9 + 1177x^8 - 1655x^7 - 1150x^6 + 1279x^5 + 474x^4 - 280x^3 - 83x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 889)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.79993$ of defining polynomial
Character	$\chi$	$=$	8001.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.79993 q^{2} +1.23975 q^{4} -3.74624 q^{5} -1.00000 q^{7} -1.36840 q^{8} +O(q^{10})$	\(q+1.79993 q^{2} +1.23975 q^{4} -3.74624 q^{5} -1.00000 q^{7} -1.36840 q^{8} -6.74297 q^{10} -2.34824 q^{11} -4.15115 q^{13} -1.79993 q^{14} -4.94252 q^{16} +2.99424 q^{17} -8.02172 q^{19} -4.64440 q^{20} -4.22667 q^{22} -2.42457 q^{23} +9.03429 q^{25} -7.47179 q^{26} -1.23975 q^{28} +0.671992 q^{29} -0.209987 q^{31} -6.15940 q^{32} +5.38942 q^{34} +3.74624 q^{35} +2.33234 q^{37} -14.4385 q^{38} +5.12634 q^{40} +6.97272 q^{41} -11.8100 q^{43} -2.91123 q^{44} -4.36406 q^{46} +7.41991 q^{47} +1.00000 q^{49} +16.2611 q^{50} -5.14639 q^{52} -14.1093 q^{53} +8.79706 q^{55} +1.36840 q^{56} +1.20954 q^{58} -2.29601 q^{59} +4.09924 q^{61} -0.377963 q^{62} -1.20145 q^{64} +15.5512 q^{65} -7.27412 q^{67} +3.71211 q^{68} +6.74297 q^{70} +6.82849 q^{71} -1.77452 q^{73} +4.19805 q^{74} -9.94493 q^{76} +2.34824 q^{77} -13.8383 q^{79} +18.5159 q^{80} +12.5504 q^{82} +1.29121 q^{83} -11.2171 q^{85} -21.2571 q^{86} +3.21332 q^{88} +14.0263 q^{89} +4.15115 q^{91} -3.00586 q^{92} +13.3553 q^{94} +30.0513 q^{95} -6.65752 q^{97} +1.79993 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	$ 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

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 $

$2$	1.79993	1.27274	0.636372	−	0.771383i	$-0.280435\pi$
$2$	1.79993	1.27274	0.636372	+	0.771383i	$0.280435\pi$
$3$	0	0
$3$	0	0
$4$	1.23975	0.619875
$5$	−3.74624	−1.67537	−0.837684	−	0.546155i	$-0.816091\pi$
$5$	−3.74624	−1.67537	−0.837684	+	0.546155i	$0.816091\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.36840	−0.483801
$9$	0	0
$10$	−6.74297	−2.13231
$11$	−2.34824	−0.708021	−0.354010	−	0.935242i	$-0.615182\pi$
$11$	−2.34824	−0.708021	−0.354010	+	0.935242i	$0.615182\pi$
$12$	0	0
$13$	−4.15115	−1.15132	−0.575661	−	0.817688i	$-0.695255\pi$
$13$	−4.15115	−1.15132	−0.575661	+	0.817688i	$0.695255\pi$
$14$	−1.79993	−0.481052
$15$	0	0
$16$	−4.94252	−1.23563
$17$	2.99424	0.726209	0.363105	−	0.931748i	$-0.381717\pi$
$17$	2.99424	0.726209	0.363105	+	0.931748i	$0.381717\pi$
$18$	0	0
$19$	−8.02172	−1.84031	−0.920155	−	0.391555i	$-0.871937\pi$
$19$	−8.02172	−1.84031	−0.920155	+	0.391555i	$0.871937\pi$
$20$	−4.64440	−1.03852
$21$	0	0
$22$	−4.22667	−0.901128
$23$	−2.42457	−0.505558	−0.252779	−	0.967524i	$-0.581345\pi$
$23$	−2.42457	−0.505558	−0.252779	+	0.967524i	$0.581345\pi$
$24$	0	0
$25$	9.03429	1.80686
$26$	−7.47179	−1.46534
$27$	0	0
$28$	−1.23975	−0.234291
$29$	0.671992	0.124786	0.0623929	−	0.998052i	$-0.480127\pi$
$29$	0.671992	0.124786	0.0623929	+	0.998052i	$0.480127\pi$
$30$	0	0
$31$	−0.209987	−0.0377148	−0.0188574	−	0.999822i	$-0.506003\pi$
$31$	−0.209987	−0.0377148	−0.0188574	+	0.999822i	$0.506003\pi$
$32$	−6.15940	−1.08884
$33$	0	0
$34$	5.38942	0.924278
$35$	3.74624	0.633230
$36$	0	0
$37$	2.33234	0.383435	0.191717	−	0.981450i	$-0.438594\pi$
$37$	2.33234	0.383435	0.191717	+	0.981450i	$0.438594\pi$
$38$	−14.4385	−2.34224
$39$	0	0
$40$	5.12634	0.810546
$41$	6.97272	1.08896	0.544478	−	0.838775i	$-0.316728\pi$
$41$	6.97272	1.08896	0.544478	+	0.838775i	$0.316728\pi$
$42$	0	0
$43$	−11.8100	−1.80100	−0.900502	−	0.434852i	$-0.856801\pi$
$43$	−11.8100	−1.80100	−0.900502	+	0.434852i	$0.856801\pi$
$44$	−2.91123	−0.438884
$45$	0	0
$46$	−4.36406	−0.643446
$47$	7.41991	1.08231	0.541153	−	0.840924i	$-0.317988\pi$
$47$	7.41991	1.08231	0.541153	+	0.840924i	$0.317988\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	16.2611	2.29967
$51$	0	0
$52$	−5.14639	−0.713676
$53$	−14.1093	−1.93807	−0.969034	−	0.246928i	$-0.920579\pi$
$53$	−14.1093	−1.93807	−0.969034	+	0.246928i	$0.920579\pi$
$54$	0	0
$55$	8.79706	1.18620
$56$	1.36840	0.182860
$57$	0	0
$58$	1.20954	0.158820
$59$	−2.29601	−0.298915	−0.149458	−	0.988768i	$-0.547753\pi$
$59$	−2.29601	−0.298915	−0.149458	+	0.988768i	$0.547753\pi$
$60$	0	0
$61$	4.09924	0.524854	0.262427	−	0.964952i	$-0.415477\pi$
$61$	4.09924	0.524854	0.262427	+	0.964952i	$0.415477\pi$
$62$	−0.377963	−0.0480013
$63$	0	0
$64$	−1.20145	−0.150181
$65$	15.5512	1.92889
$66$	0	0
$67$	−7.27412	−0.888675	−0.444337	−	0.895860i	$-0.646561\pi$
$67$	−7.27412	−0.888675	−0.444337	+	0.895860i	$0.646561\pi$
$68$	3.71211	0.450159
$69$	0	0
$70$	6.74297	0.805939
$71$	6.82849	0.810393	0.405196	−	0.914230i	$-0.367203\pi$
$71$	6.82849	0.810393	0.405196	+	0.914230i	$0.367203\pi$
$72$	0	0
$73$	−1.77452	−0.207692	−0.103846	−	0.994593i	$-0.533115\pi$
$73$	−1.77452	−0.207692	−0.103846	+	0.994593i	$0.533115\pi$
$74$	4.19805	0.488014
$75$	0	0
$76$	−9.94493	−1.14076
$77$	2.34824	0.267607
$78$	0	0
$79$	−13.8383	−1.55693	−0.778463	−	0.627690i	$-0.784000\pi$
$79$	−13.8383	−1.55693	−0.778463	+	0.627690i	$0.784000\pi$
$80$	18.5159	2.07014
$81$	0	0
$82$	12.5504	1.38596
$83$	1.29121	0.141729	0.0708646	−	0.997486i	$-0.477424\pi$
$83$	1.29121	0.141729	0.0708646	+	0.997486i	$0.477424\pi$
$84$	0	0
$85$	−11.2171	−1.21667
$86$	−21.2571	−2.29222
$87$	0	0
$88$	3.21332	0.342541
$89$	14.0263	1.48679	0.743393	−	0.668854i	$-0.233215\pi$
$89$	14.0263	1.48679	0.743393	+	0.668854i	$0.233215\pi$
$90$	0	0
$91$	4.15115	0.435159
$92$	−3.00586	−0.313383
$93$	0	0
$94$	13.3553	1.37750
$95$	30.0513	3.08320
$96$	0	0
$97$	−6.65752	−0.675969	−0.337985	−	0.941152i	$-0.609745\pi$
$97$	−6.65752	−0.675969	−0.337985	+	0.941152i	$0.609745\pi$
$98$	1.79993	0.181820
$99$	0	0
$100$	11.2003	1.12003
$101$	9.60285	0.955519	0.477760	−	0.878491i	$-0.341449\pi$
$101$	9.60285	0.955519	0.477760	+	0.878491i	$0.341449\pi$
$102$	0	0
$103$	12.1832	1.20045	0.600224	−	0.799832i	$-0.295078\pi$
$103$	12.1832	1.20045	0.600224	+	0.799832i	$0.295078\pi$
$104$	5.68043	0.557012
$105$	0	0
$106$	−25.3958	−2.46666
$107$	12.8230	1.23965	0.619823	−	0.784742i	$-0.287205\pi$
$107$	12.8230	1.23965	0.619823	+	0.784742i	$0.287205\pi$
$108$	0	0
$109$	−2.55762	−0.244975	−0.122488	−	0.992470i	$-0.539087\pi$
$109$	−2.55762	−0.244975	−0.122488	+	0.992470i	$0.539087\pi$
$110$	15.8341	1.50972
$111$	0	0
$112$	4.94252	0.467024
$113$	−15.7491	−1.48155	−0.740775	−	0.671753i	$-0.765542\pi$
$113$	−15.7491	−1.48155	−0.740775	+	0.671753i	$0.765542\pi$
$114$	0	0
$115$	9.08303	0.846997
$116$	0.833102	0.0773516
$117$	0	0
$118$	−4.13266	−0.380442
$119$	−2.99424	−0.274481
$120$	0	0
$121$	−5.48578	−0.498707
$122$	7.37835	0.668005
$123$	0	0
$124$	−0.260332	−0.0233785
$125$	−15.1134	−1.35179
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	10.1563	0.897696
$129$	0	0
$130$	27.9911	2.45498
$131$	13.6229	1.19023	0.595117	−	0.803639i	$-0.297106\pi$
$131$	13.6229	1.19023	0.595117	+	0.803639i	$0.297106\pi$
$132$	0	0
$133$	8.02172	0.695571
$134$	−13.0929	−1.13105
$135$	0	0
$136$	−4.09731	−0.351341
$137$	−4.65045	−0.397314	−0.198657	−	0.980069i	$-0.563658\pi$
$137$	−4.65045	−0.397314	−0.198657	+	0.980069i	$0.563658\pi$
$138$	0	0
$139$	14.7158	1.24817	0.624087	−	0.781355i	$-0.285471\pi$
$139$	14.7158	1.24817	0.624087	+	0.781355i	$0.285471\pi$
$140$	4.64440	0.392523
$141$	0	0
$142$	12.2908	1.03142
$143$	9.74790	0.815160
$144$	0	0
$145$	−2.51744	−0.209062
$146$	−3.19401	−0.264338
$147$	0	0
$148$	2.89152	0.237681
$149$	13.5765	1.11223	0.556113	−	0.831106i	$-0.312292\pi$
$149$	13.5765	1.11223	0.556113	+	0.831106i	$0.312292\pi$
$150$	0	0
$151$	21.5583	1.75439	0.877195	−	0.480134i	$-0.159412\pi$
$151$	21.5583	1.75439	0.877195	+	0.480134i	$0.159412\pi$
$152$	10.9769	0.890344
$153$	0	0
$154$	4.22667	0.340594
$155$	0.786662	0.0631862
$156$	0	0
$157$	20.2197	1.61371	0.806856	−	0.590749i	$-0.201167\pi$
$157$	20.2197	1.61371	0.806856	+	0.590749i	$0.201167\pi$
$158$	−24.9079	−1.98157
$159$	0	0
$160$	23.0746	1.82420
$161$	2.42457	0.191083
$162$	0	0
$163$	−23.3416	−1.82826	−0.914128	−	0.405425i	$-0.867124\pi$
$163$	−23.3416	−1.82826	−0.914128	+	0.405425i	$0.867124\pi$
$164$	8.64443	0.675017
$165$	0	0
$166$	2.32410	0.180385
$167$	3.29629	0.255074	0.127537	−	0.991834i	$-0.459293\pi$
$167$	3.29629	0.255074	0.127537	+	0.991834i	$0.459293\pi$
$168$	0	0
$169$	4.23207	0.325544
$170$	−20.1900	−1.54851
$171$	0	0
$172$	−14.6414	−1.11640
$173$	−18.3410	−1.39444	−0.697220	−	0.716857i	$-0.745580\pi$
$173$	−18.3410	−1.39444	−0.697220	+	0.716857i	$0.745580\pi$
$174$	0	0
$175$	−9.03429	−0.682928
$176$	11.6062	0.874851
$177$	0	0
$178$	25.2464	1.89230
$179$	23.8274	1.78095	0.890473	−	0.455035i	$-0.150373\pi$
$179$	23.8274	1.78095	0.890473	+	0.455035i	$0.150373\pi$
$180$	0	0
$181$	−18.2618	−1.35739	−0.678695	−	0.734420i	$-0.737454\pi$
$181$	−18.2618	−1.35739	−0.678695	+	0.734420i	$0.737454\pi$
$182$	7.47179	0.553846
$183$	0	0
$184$	3.31778	0.244590
$185$	−8.73750	−0.642394
$186$	0	0
$187$	−7.03118	−0.514171
$188$	9.19884	0.670894
$189$	0	0
$190$	54.0902	3.92412
$191$	−9.99052	−0.722888	−0.361444	−	0.932394i	$-0.617716\pi$
$191$	−9.99052	−0.722888	−0.361444	+	0.932394i	$0.617716\pi$
$192$	0	0
$193$	14.9070	1.07303	0.536515	−	0.843891i	$-0.319740\pi$
$193$	14.9070	1.07303	0.536515	+	0.843891i	$0.319740\pi$
$194$	−11.9831	−0.860335
$195$	0	0
$196$	1.23975	0.0885536
$197$	−16.6543	−1.18657	−0.593283	−	0.804994i	$-0.702169\pi$
$197$	−16.6543	−1.18657	−0.593283	+	0.804994i	$0.702169\pi$
$198$	0	0
$199$	−10.1573	−0.720029	−0.360014	−	0.932947i	$-0.617228\pi$
$199$	−10.1573	−0.720029	−0.360014	+	0.932947i	$0.617228\pi$
$200$	−12.3625	−0.874161
$201$	0	0
$202$	17.2845	1.21613
$203$	−0.671992	−0.0471646
$204$	0	0
$205$	−26.1215	−1.82440
$206$	21.9290	1.52786
$207$	0	0
$208$	20.5172	1.42261
$209$	18.8369	1.30298
$210$	0	0
$211$	14.5011	0.998297	0.499149	−	0.866516i	$-0.333646\pi$
$211$	14.5011	0.998297	0.499149	+	0.866516i	$0.333646\pi$
$212$	−17.4921	−1.20136
$213$	0	0
$214$	23.0805	1.57775
$215$	44.2430	3.01734
$216$	0	0
$217$	0.209987	0.0142549
$218$	−4.60353	−0.311791
$219$	0	0
$220$	10.9062	0.735293
$221$	−12.4295	−0.836101
$222$	0	0
$223$	6.05256	0.405309	0.202655	−	0.979250i	$-0.435043\pi$
$223$	6.05256	0.405309	0.202655	+	0.979250i	$0.435043\pi$
$224$	6.15940	0.411542
$225$	0	0
$226$	−28.3473	−1.88563
$227$	−6.53235	−0.433567	−0.216784	−	0.976220i	$-0.569557\pi$
$227$	−6.53235	−0.433567	−0.216784	+	0.976220i	$0.569557\pi$
$228$	0	0
$229$	−10.9498	−0.723584	−0.361792	−	0.932259i	$-0.617835\pi$
$229$	−10.9498	−0.723584	−0.361792	+	0.932259i	$0.617835\pi$
$230$	16.3488	1.07801
$231$	0	0
$232$	−0.919552	−0.0603715
$233$	−23.4858	−1.53861	−0.769304	−	0.638883i	$-0.779397\pi$
$233$	−23.4858	−1.53861	−0.769304	+	0.638883i	$0.779397\pi$
$234$	0	0
$235$	−27.7968	−1.81326
$236$	−2.84648	−0.185290
$237$	0	0
$238$	−5.38942	−0.349344
$239$	4.96656	0.321260	0.160630	−	0.987015i	$-0.448647\pi$
$239$	4.96656	0.321260	0.160630	+	0.987015i	$0.448647\pi$
$240$	0	0
$241$	−0.500641	−0.0322491	−0.0161246	−	0.999870i	$-0.505133\pi$
$241$	−0.500641	−0.0322491	−0.0161246	+	0.999870i	$0.505133\pi$
$242$	−9.87402	−0.634726
$243$	0	0
$244$	5.08204	0.325344
$245$	−3.74624	−0.239338
$246$	0	0
$247$	33.2994	2.11879
$248$	0.287346	0.0182465
$249$	0	0
$250$	−27.2031	−1.72048
$251$	1.46017	0.0921648	0.0460824	−	0.998938i	$-0.485326\pi$
$251$	1.46017	0.0921648	0.0460824	+	0.998938i	$0.485326\pi$
$252$	0	0
$253$	5.69348	0.357946
$254$	−1.79993	−0.112938
$255$	0	0
$256$	20.6835	1.29272
$257$	4.41040	0.275113	0.137557	−	0.990494i	$-0.456075\pi$
$257$	4.41040	0.275113	0.137557	+	0.990494i	$0.456075\pi$
$258$	0	0
$259$	−2.33234	−0.144925
$260$	19.2796	1.19567
$261$	0	0
$262$	24.5202	1.51486
$263$	7.62333	0.470075	0.235037	−	0.971986i	$-0.424479\pi$
$263$	7.62333	0.470075	0.235037	+	0.971986i	$0.424479\pi$
$264$	0	0
$265$	52.8570	3.24698
$266$	14.4385	0.885284
$267$	0	0
$268$	−9.01808	−0.550867
$269$	−8.62676	−0.525983	−0.262991	−	0.964798i	$-0.584709\pi$
$269$	−8.62676	−0.525983	−0.262991	+	0.964798i	$0.584709\pi$
$270$	0	0
$271$	−8.57975	−0.521183	−0.260591	−	0.965449i	$-0.583918\pi$
$271$	−8.57975	−0.521183	−0.260591	+	0.965449i	$0.583918\pi$
$272$	−14.7991	−0.897326
$273$	0	0
$274$	−8.37048	−0.505679
$275$	−21.2147	−1.27929
$276$	0	0
$277$	−1.27651	−0.0766982	−0.0383491	−	0.999264i	$-0.512210\pi$
$277$	−1.27651	−0.0766982	−0.0383491	+	0.999264i	$0.512210\pi$
$278$	26.4874	1.58861
$279$	0	0
$280$	−5.12634	−0.306357
$281$	−24.6384	−1.46980	−0.734901	−	0.678175i	$-0.762771\pi$
$281$	−24.6384	−1.46980	−0.734901	+	0.678175i	$0.762771\pi$
$282$	0	0
$283$	−23.3964	−1.39077	−0.695386	−	0.718636i	$-0.744767\pi$
$283$	−23.3964	−1.39077	−0.695386	+	0.718636i	$0.744767\pi$
$284$	8.46562	0.502342
$285$	0	0
$286$	17.5455	1.03749
$287$	−6.97272	−0.411587
$288$	0	0
$289$	−8.03454	−0.472620
$290$	−4.53122	−0.266082
$291$	0	0
$292$	−2.19996	−0.128743
$293$	2.37228	0.138590	0.0692951	−	0.997596i	$-0.477925\pi$
$293$	2.37228	0.138590	0.0692951	+	0.997596i	$0.477925\pi$
$294$	0	0
$295$	8.60140	0.500793
$296$	−3.19157	−0.185506
$297$	0	0
$298$	24.4367	1.41558
$299$	10.0648	0.582061
$300$	0	0
$301$	11.8100	0.680715
$302$	38.8034	2.23289
$303$	0	0
$304$	39.6475	2.27394
$305$	−15.3567	−0.879324
$306$	0	0
$307$	−24.8651	−1.41913	−0.709564	−	0.704641i	$-0.751108\pi$
$307$	−24.8651	−1.41913	−0.709564	+	0.704641i	$0.751108\pi$
$308$	2.91123	0.165883
$309$	0	0
$310$	1.41594	0.0804199
$311$	8.44319	0.478770	0.239385	−	0.970925i	$-0.423054\pi$
$311$	8.44319	0.478770	0.239385	+	0.970925i	$0.423054\pi$
$312$	0	0
$313$	−25.5138	−1.44212	−0.721061	−	0.692871i	$-0.756345\pi$
$313$	−25.5138	−1.44212	−0.721061	+	0.692871i	$0.756345\pi$
$314$	36.3941	2.05384
$315$	0	0
$316$	−17.1560	−0.965100
$317$	−5.25226	−0.294996	−0.147498	−	0.989062i	$-0.547122\pi$
$317$	−5.25226	−0.294996	−0.147498	+	0.989062i	$0.547122\pi$
$318$	0	0
$319$	−1.57800	−0.0883509
$320$	4.50091	0.251609
$321$	0	0
$322$	4.36406	0.243200
$323$	−24.0189	−1.33645
$324$	0	0
$325$	−37.5027	−2.08028
$326$	−42.0133	−2.32690
$327$	0	0
$328$	−9.54146	−0.526839
$329$	−7.41991	−0.409073
$330$	0	0
$331$	0.0258723	0.00142207	0.000711036	−	1.00000i	$-0.499774\pi$
$331$	0.0258723	0.00142207	0.000711036		1.00000i	$0.499774\pi$
$332$	1.60078	0.0878544
$333$	0	0
$334$	5.93309	0.324644
$335$	27.2506	1.48886
$336$	0	0
$337$	−17.4019	−0.947942	−0.473971	−	0.880540i	$-0.657180\pi$
$337$	−17.4019	−0.947942	−0.473971	+	0.880540i	$0.657180\pi$
$338$	7.61744	0.414334
$339$	0	0
$340$	−13.9064	−0.754182
$341$	0.493100	0.0267029
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	16.1607	0.871328
$345$	0	0
$346$	−33.0125	−1.77476
$347$	16.7652	0.900001	0.450001	−	0.893028i	$-0.351424\pi$
$347$	16.7652	0.900001	0.450001	+	0.893028i	$0.351424\pi$
$348$	0	0
$349$	−26.3395	−1.40992	−0.704960	−	0.709247i	$-0.749035\pi$
$349$	−26.3395	−1.40992	−0.704960	+	0.709247i	$0.749035\pi$
$350$	−16.2611	−0.869193
$351$	0	0
$352$	14.4637	0.770920
$353$	−6.19555	−0.329756	−0.164878	−	0.986314i	$-0.552723\pi$
$353$	−6.19555	−0.329756	−0.164878	+	0.986314i	$0.552723\pi$
$354$	0	0
$355$	−25.5811	−1.35771
$356$	17.3891	0.921622
$357$	0	0
$358$	42.8877	2.26669
$359$	6.67732	0.352415	0.176208	−	0.984353i	$-0.443617\pi$
$359$	6.67732	0.352415	0.176208	+	0.984353i	$0.443617\pi$
$360$	0	0
$361$	45.3480	2.38674
$362$	−32.8700	−1.72761
$363$	0	0
$364$	5.14639	0.269744
$365$	6.64777	0.347960
$366$	0	0
$367$	26.0481	1.35970	0.679849	−	0.733352i	$-0.262045\pi$
$367$	26.0481	1.35970	0.679849	+	0.733352i	$0.262045\pi$
$368$	11.9835	0.624683
$369$	0	0
$370$	−15.7269	−0.817603
$371$	14.1093	0.732521
$372$	0	0
$373$	23.9872	1.24201	0.621004	−	0.783807i	$-0.286725\pi$
$373$	23.9872	1.24201	0.621004	+	0.783807i	$0.286725\pi$
$374$	−12.6556	−0.654408
$375$	0	0
$376$	−10.1534	−0.523621
$377$	−2.78954	−0.143669
$378$	0	0
$379$	15.5530	0.798906	0.399453	−	0.916754i	$-0.369200\pi$
$379$	15.5530	0.798906	0.399453	+	0.916754i	$0.369200\pi$
$380$	37.2561	1.91120
$381$	0	0
$382$	−17.9822	−0.920051
$383$	−12.0746	−0.616984	−0.308492	−	0.951227i	$-0.599824\pi$
$383$	−12.0746	−0.616984	−0.308492	+	0.951227i	$0.599824\pi$
$384$	0	0
$385$	−8.79706	−0.448340
$386$	26.8316	1.36569
$387$	0	0
$388$	−8.25367	−0.419016
$389$	−10.2542	−0.519909	−0.259954	−	0.965621i	$-0.583708\pi$
$389$	−10.2542	−0.519909	−0.259954	+	0.965621i	$0.583708\pi$
$390$	0	0
$391$	−7.25975	−0.367141
$392$	−1.36840	−0.0691145
$393$	0	0
$394$	−29.9765	−1.51019
$395$	51.8414	2.60842
$396$	0	0
$397$	31.3596	1.57389	0.786947	−	0.617021i	$-0.211661\pi$
$397$	31.3596	1.57389	0.786947	+	0.617021i	$0.211661\pi$
$398$	−18.2824	−0.916412
$399$	0	0
$400$	−44.6522	−2.23261
$401$	14.6531	0.731740	0.365870	−	0.930666i	$-0.380771\pi$
$401$	14.6531	0.731740	0.365870	+	0.930666i	$0.380771\pi$
$402$	0	0
$403$	0.871690	0.0434219
$404$	11.9051	0.592303
$405$	0	0
$406$	−1.20954	−0.0600284
$407$	−5.47689	−0.271480
$408$	0	0
$409$	25.3477	1.25336	0.626681	−	0.779276i	$-0.284413\pi$
$409$	25.3477	1.25336	0.626681	+	0.779276i	$0.284413\pi$
$410$	−47.0168	−2.32200
$411$	0	0
$412$	15.1042	0.744128
$413$	2.29601	0.112979
$414$	0	0
$415$	−4.83720	−0.237449
$416$	25.5686	1.25360
$417$	0	0
$418$	33.9051	1.65835
$419$	8.98733	0.439060	0.219530	−	0.975606i	$-0.429548\pi$
$419$	8.98733	0.439060	0.219530	+	0.975606i	$0.429548\pi$
$420$	0	0
$421$	8.28734	0.403900	0.201950	−	0.979396i	$-0.435272\pi$
$421$	8.28734	0.403900	0.201950	+	0.979396i	$0.435272\pi$
$422$	26.1010	1.27058
$423$	0	0
$424$	19.3072	0.937640
$425$	27.0508	1.31216
$426$	0	0
$427$	−4.09924	−0.198376
$428$	15.8973	0.768425
$429$	0	0
$430$	79.6342	3.84030
$431$	5.02472	0.242032	0.121016	−	0.992651i	$-0.461385\pi$
$431$	5.02472	0.242032	0.121016	+	0.992651i	$0.461385\pi$
$432$	0	0
$433$	−14.1923	−0.682038	−0.341019	−	0.940056i	$-0.610772\pi$
$433$	−14.1923	−0.682038	−0.341019	+	0.940056i	$0.610772\pi$
$434$	0.377963	0.0181428
$435$	0	0
$436$	−3.17081	−0.151854
$437$	19.4493	0.930384
$438$	0	0
$439$	24.3851	1.16384	0.581918	−	0.813247i	$-0.302302\pi$
$439$	24.3851	1.16384	0.581918	+	0.813247i	$0.302302\pi$
$440$	−12.0379	−0.573883
$441$	0	0
$442$	−22.3723	−1.06414
$443$	−20.8938	−0.992697	−0.496348	−	0.868123i	$-0.665326\pi$
$443$	−20.8938	−0.992697	−0.496348	+	0.868123i	$0.665326\pi$
$444$	0	0
$445$	−52.5459	−2.49092
$446$	10.8942	0.515855
$447$	0	0
$448$	1.20145	0.0567631
$449$	−25.3562	−1.19663	−0.598315	−	0.801261i	$-0.704163\pi$
$449$	−25.3562	−1.19663	−0.598315	+	0.801261i	$0.704163\pi$
$450$	0	0
$451$	−16.3736	−0.771004
$452$	−19.5249	−0.918376
$453$	0	0
$454$	−11.7578	−0.551820
$455$	−15.5512	−0.729052
$456$	0	0
$457$	21.3577	0.999072	0.499536	−	0.866293i	$-0.333504\pi$
$457$	21.3577	0.999072	0.499536	+	0.866293i	$0.333504\pi$
$458$	−19.7089	−0.920936
$459$	0	0
$460$	11.2607	0.525032
$461$	−8.09807	−0.377165	−0.188582	−	0.982057i	$-0.560389\pi$
$461$	−8.09807	−0.377165	−0.188582	+	0.982057i	$0.560389\pi$
$462$	0	0
$463$	−19.1276	−0.888937	−0.444469	−	0.895794i	$-0.646607\pi$
$463$	−19.1276	−0.888937	−0.444469	+	0.895794i	$0.646607\pi$
$464$	−3.32133	−0.154189
$465$	0	0
$466$	−42.2729	−1.95825
$467$	3.22927	0.149433	0.0747165	−	0.997205i	$-0.476195\pi$
$467$	3.22927	0.149433	0.0747165	+	0.997205i	$0.476195\pi$
$468$	0	0
$469$	7.27412	0.335887
$470$	−50.0322	−2.30781
$471$	0	0
$472$	3.14185	0.144616
$473$	27.7326	1.27515
$474$	0	0
$475$	−72.4706	−3.32518
$476$	−3.71211	−0.170144
$477$	0	0
$478$	8.93947	0.408882
$479$	−30.2222	−1.38089	−0.690443	−	0.723387i	$-0.742584\pi$
$479$	−30.2222	−1.38089	−0.690443	+	0.723387i	$0.742584\pi$
$480$	0	0
$481$	−9.68191	−0.441457
$482$	−0.901119	−0.0410448
$483$	0	0
$484$	−6.80099	−0.309136
$485$	24.9407	1.13250
$486$	0	0
$487$	5.13201	0.232554	0.116277	−	0.993217i	$-0.462904\pi$
$487$	5.13201	0.232554	0.116277	+	0.993217i	$0.462904\pi$
$488$	−5.60939	−0.253925
$489$	0	0
$490$	−6.74297	−0.304616
$491$	11.0295	0.497755	0.248878	−	0.968535i	$-0.419938\pi$
$491$	11.0295	0.497755	0.248878	+	0.968535i	$0.419938\pi$
$492$	0	0
$493$	2.01210	0.0906206
$494$	59.9366	2.69667
$495$	0	0
$496$	1.03787	0.0466016
$497$	−6.82849	−0.306300
$498$	0	0
$499$	−2.61282	−0.116966	−0.0584829	−	0.998288i	$-0.518626\pi$
$499$	−2.61282	−0.116966	−0.0584829	+	0.998288i	$0.518626\pi$
$500$	−18.7369	−0.837938
$501$	0	0
$502$	2.62820	0.117302
$503$	−18.5752	−0.828229	−0.414114	−	0.910225i	$-0.635909\pi$
$503$	−18.5752	−0.828229	−0.414114	+	0.910225i	$0.635909\pi$
$504$	0	0
$505$	−35.9746	−1.60085
$506$	10.2479	0.455573
$507$	0	0
$508$	−1.23975	−0.0550050
$509$	15.0258	0.666005	0.333002	−	0.942926i	$-0.391938\pi$
$509$	15.0258	0.666005	0.333002	+	0.942926i	$0.391938\pi$
$510$	0	0
$511$	1.77452	0.0785001
$512$	16.9163	0.747601
$513$	0	0
$514$	7.93842	0.350148
$515$	−45.6412	−2.01119
$516$	0	0
$517$	−17.4237	−0.766295
$518$	−4.19805	−0.184452
$519$	0	0
$520$	−21.2802	−0.933200
$521$	−13.1803	−0.577439	−0.288720	−	0.957414i	$-0.593230\pi$
$521$	−13.1803	−0.577439	−0.288720	+	0.957414i	$0.593230\pi$
$522$	0	0
$523$	42.0462	1.83855	0.919277	−	0.393611i	$-0.128774\pi$
$523$	42.0462	1.83855	0.919277	+	0.393611i	$0.128774\pi$
$524$	16.8889	0.737797
$525$	0	0
$526$	13.7215	0.598284
$527$	−0.628752	−0.0273889
$528$	0	0
$529$	−17.1214	−0.744411
$530$	95.1389	4.13257
$531$	0	0
$532$	9.94493	0.431167
$533$	−28.9448	−1.25374
$534$	0	0
$535$	−48.0380	−2.07686
$536$	9.95388	0.429942
$537$	0	0
$538$	−15.5276	−0.669441
$539$	−2.34824	−0.101146
$540$	0	0
$541$	22.9431	0.986400	0.493200	−	0.869916i	$-0.335827\pi$
$541$	22.9431	0.986400	0.493200	+	0.869916i	$0.335827\pi$
$542$	−15.4430	−0.663332
$543$	0	0
$544$	−18.4427	−0.790724
$545$	9.58144	0.410424
$546$	0	0
$547$	15.5106	0.663185	0.331593	−	0.943423i	$-0.392414\pi$
$547$	15.5106	0.663185	0.331593	+	0.943423i	$0.392414\pi$
$548$	−5.76539	−0.246285
$549$	0	0
$550$	−38.1849	−1.62821
$551$	−5.39053	−0.229644
$552$	0	0
$553$	13.8383	0.588463
$554$	−2.29764	−0.0976172
$555$	0	0
$556$	18.2439	0.773712
$557$	−26.5955	−1.12689	−0.563444	−	0.826154i	$-0.690524\pi$
$557$	−26.5955	−1.12689	−0.563444	+	0.826154i	$0.690524\pi$
$558$	0	0
$559$	49.0250	2.07354
$560$	−18.5159	−0.782438
$561$	0	0
$562$	−44.3473	−1.87068
$563$	−5.96057	−0.251208	−0.125604	−	0.992080i	$-0.540087\pi$
$563$	−5.96057	−0.251208	−0.125604	+	0.992080i	$0.540087\pi$
$564$	0	0
$565$	58.9999	2.48214
$566$	−42.1119	−1.77010
$567$	0	0
$568$	−9.34409	−0.392069
$569$	3.66744	0.153747	0.0768735	−	0.997041i	$-0.475506\pi$
$569$	3.66744	0.153747	0.0768735	+	0.997041i	$0.475506\pi$
$570$	0	0
$571$	11.4664	0.479855	0.239927	−	0.970791i	$-0.422876\pi$
$571$	11.4664	0.479855	0.239927	+	0.970791i	$0.422876\pi$
$572$	12.0850	0.505297
$573$	0	0
$574$	−12.5504	−0.523844
$575$	−21.9043	−0.913473
$576$	0	0
$577$	−23.6760	−0.985646	−0.492823	−	0.870130i	$-0.664035\pi$
$577$	−23.6760	−0.985646	−0.492823	+	0.870130i	$0.664035\pi$
$578$	−14.4616	−0.601524
$579$	0	0
$580$	−3.12100	−0.129592
$581$	−1.29121	−0.0535686
$582$	0	0
$583$	33.1321	1.37219
$584$	2.42825	0.100482
$585$	0	0
$586$	4.26994	0.176390
$587$	21.6054	0.891751	0.445875	−	0.895095i	$-0.352892\pi$
$587$	21.6054	0.891751	0.445875	+	0.895095i	$0.352892\pi$
$588$	0	0
$589$	1.68446	0.0694070
$590$	15.4819	0.637381
$591$	0	0
$592$	−11.5276	−0.473783
$593$	20.8560	0.856452	0.428226	−	0.903672i	$-0.359139\pi$
$593$	20.8560	0.856452	0.428226	+	0.903672i	$0.359139\pi$
$594$	0	0
$595$	11.2171	0.459857
$596$	16.8314	0.689442
$597$	0	0
$598$	18.1159	0.740814
$599$	19.5817	0.800088	0.400044	−	0.916496i	$-0.368995\pi$
$599$	19.5817	0.800088	0.400044	+	0.916496i	$0.368995\pi$
$600$	0	0
$601$	−5.51087	−0.224793	−0.112397	−	0.993663i	$-0.535853\pi$
$601$	−5.51087	−0.224793	−0.112397	+	0.993663i	$0.535853\pi$
$602$	21.2571	0.866376
$603$	0	0
$604$	26.7269	1.08750
$605$	20.5510	0.835518
$606$	0	0
$607$	42.5940	1.72884	0.864418	−	0.502773i	$-0.167687\pi$
$607$	42.5940	1.72884	0.864418	+	0.502773i	$0.167687\pi$
$608$	49.4090	2.00380
$609$	0	0
$610$	−27.6411	−1.11915
$611$	−30.8012	−1.24608
$612$	0	0
$613$	−26.4808	−1.06955	−0.534774	−	0.844995i	$-0.679603\pi$
$613$	−26.4808	−1.06955	−0.534774	+	0.844995i	$0.679603\pi$
$614$	−44.7555	−1.80619
$615$	0	0
$616$	−3.21332	−0.129468
$617$	24.1799	0.973447	0.486724	−	0.873556i	$-0.338192\pi$
$617$	24.1799	0.973447	0.486724	+	0.873556i	$0.338192\pi$
$618$	0	0
$619$	−45.7593	−1.83922	−0.919611	−	0.392830i	$-0.871496\pi$
$619$	−45.7593	−1.83922	−0.919611	+	0.392830i	$0.871496\pi$
$620$	0.975265	0.0391676
$621$	0	0
$622$	15.1972	0.609351
$623$	−14.0263	−0.561953
$624$	0	0
$625$	11.4470	0.457880
$626$	−45.9230	−1.83545
$627$	0	0
$628$	25.0674	1.00030
$629$	6.98358	0.278454
$630$	0	0
$631$	43.6793	1.73885	0.869423	−	0.494069i	$-0.164491\pi$
$631$	43.6793	1.73885	0.869423	+	0.494069i	$0.164491\pi$
$632$	18.9362	0.753243
$633$	0	0
$634$	−9.45370	−0.375455
$635$	3.74624	0.148665
$636$	0	0
$637$	−4.15115	−0.164475
$638$	−2.84028	−0.112448
$639$	0	0
$640$	−38.0478	−1.50397
$641$	49.0383	1.93690	0.968448	−	0.249215i	$-0.0801727\pi$
$641$	49.0383	1.93690	0.968448	+	0.249215i	$0.0801727\pi$
$642$	0	0
$643$	−21.5863	−0.851281	−0.425641	−	0.904892i	$-0.639951\pi$
$643$	−21.5863	−0.851281	−0.425641	+	0.904892i	$0.639951\pi$
$644$	3.00586	0.118448
$645$	0	0
$646$	−43.2324	−1.70096
$647$	−17.8322	−0.701058	−0.350529	−	0.936552i	$-0.613998\pi$
$647$	−17.8322	−0.701058	−0.350529	+	0.936552i	$0.613998\pi$
$648$	0	0
$649$	5.39158	0.211638
$650$	−67.5023	−2.64766
$651$	0	0
$652$	−28.9378	−1.13329
$653$	−38.4815	−1.50590	−0.752948	−	0.658080i	$-0.771369\pi$
$653$	−38.4815	−1.50590	−0.752948	+	0.658080i	$0.771369\pi$
$654$	0	0
$655$	−51.0345	−1.99408
$656$	−34.4628	−1.34555
$657$	0	0
$658$	−13.3553	−0.520645
$659$	−7.96570	−0.310300	−0.155150	−	0.987891i	$-0.549586\pi$
$659$	−7.96570	−0.310300	−0.155150	+	0.987891i	$0.549586\pi$
$660$	0	0
$661$	11.3105	0.439929	0.219965	−	0.975508i	$-0.429406\pi$
$661$	11.3105	0.439929	0.219965	+	0.975508i	$0.429406\pi$
$662$	0.0465684	0.00180993
$663$	0	0
$664$	−1.76689	−0.0685688
$665$	−30.0513	−1.16534
$666$	0	0
$667$	−1.62929	−0.0630865
$668$	4.08657	0.158114
$669$	0	0
$670$	49.0491	1.89493
$671$	−9.62600	−0.371608
$672$	0	0
$673$	16.2398	0.625997	0.312999	−	0.949754i	$-0.398666\pi$
$673$	16.2398	0.625997	0.312999	+	0.949754i	$0.398666\pi$
$674$	−31.3222	−1.20649
$675$	0	0
$676$	5.24671	0.201797
$677$	21.5318	0.827533	0.413766	−	0.910383i	$-0.364213\pi$
$677$	21.5318	0.827533	0.413766	+	0.910383i	$0.364213\pi$
$678$	0	0
$679$	6.65752	0.255492
$680$	15.3495	0.588626
$681$	0	0
$682$	0.887546	0.0339859
$683$	−3.19273	−0.122167	−0.0610833	−	0.998133i	$-0.519456\pi$
$683$	−3.19273	−0.122167	−0.0610833	+	0.998133i	$0.519456\pi$
$684$	0	0
$685$	17.4217	0.665648
$686$	−1.79993	−0.0687217
$687$	0	0
$688$	58.3710	2.22537
$689$	58.5701	2.23134
$690$	0	0
$691$	45.6605	1.73701	0.868504	−	0.495683i	$-0.165082\pi$
$691$	45.6605	1.73701	0.868504	+	0.495683i	$0.165082\pi$
$692$	−22.7382	−0.864378
$693$	0	0
$694$	30.1761	1.14547
$695$	−55.1287	−2.09115
$696$	0	0
$697$	20.8780	0.790810
$698$	−47.4092	−1.79447
$699$	0	0
$700$	−11.2003	−0.423330
$701$	−29.9301	−1.13044	−0.565221	−	0.824939i	$-0.691209\pi$
$701$	−29.9301	−1.13044	−0.565221	+	0.824939i	$0.691209\pi$
$702$	0	0
$703$	−18.7094	−0.705638
$704$	2.82129	0.106331
$705$	0	0
$706$	−11.1516	−0.419695
$707$	−9.60285	−0.361152
$708$	0	0
$709$	6.61221	0.248327	0.124163	−	0.992262i	$-0.460375\pi$
$709$	6.61221	0.248327	0.124163	+	0.992262i	$0.460375\pi$
$710$	−46.0443	−1.72801
$711$	0	0
$712$	−19.1936	−0.719310
$713$	0.509130	0.0190671
$714$	0	0
$715$	−36.5179	−1.36569
$716$	29.5401	1.10396
$717$	0	0
$718$	12.0187	0.448534
$719$	22.2182	0.828599	0.414300	−	0.910141i	$-0.364027\pi$
$719$	22.2182	0.828599	0.414300	+	0.910141i	$0.364027\pi$
$720$	0	0
$721$	−12.1832	−0.453727
$722$	81.6233	3.03770
$723$	0	0
$724$	−22.6401	−0.841412
$725$	6.07097	0.225470
$726$	0	0
$727$	32.0322	1.18801	0.594005	−	0.804461i	$-0.297546\pi$
$727$	32.0322	1.18801	0.594005	+	0.804461i	$0.297546\pi$
$728$	−5.68043	−0.210531
$729$	0	0
$730$	11.9655	0.442864
$731$	−35.3619	−1.30791
$732$	0	0
$733$	29.1028	1.07494	0.537468	−	0.843284i	$-0.319381\pi$
$733$	29.1028	1.07494	0.537468	+	0.843284i	$0.319381\pi$
$734$	46.8847	1.73055
$735$	0	0
$736$	14.9339	0.550471
$737$	17.0814	0.629200
$738$	0	0
$739$	−7.26489	−0.267243	−0.133622	−	0.991032i	$-0.542661\pi$
$739$	−7.26489	−0.267243	−0.133622	+	0.991032i	$0.542661\pi$
$740$	−10.8323	−0.398204
$741$	0	0
$742$	25.3958	0.932311
$743$	−8.90194	−0.326581	−0.163290	−	0.986578i	$-0.552211\pi$
$743$	−8.90194	−0.326581	−0.163290	+	0.986578i	$0.552211\pi$
$744$	0	0
$745$	−50.8606	−1.86339
$746$	43.1752	1.58076
$747$	0	0
$748$	−8.71691	−0.318722
$749$	−12.8230	−0.468542
$750$	0	0
$751$	14.6447	0.534391	0.267196	−	0.963642i	$-0.413903\pi$
$751$	14.6447	0.534391	0.267196	+	0.963642i	$0.413903\pi$
$752$	−36.6731	−1.33733
$753$	0	0
$754$	−5.02098	−0.182853
$755$	−80.7625	−2.93925
$756$	0	0
$757$	−45.0380	−1.63694	−0.818468	−	0.574552i	$-0.805176\pi$
$757$	−45.0380	−1.63694	−0.818468	+	0.574552i	$0.805176\pi$
$758$	27.9944	1.01680
$759$	0	0
$760$	−41.1221	−1.49165
$761$	38.0393	1.37892	0.689462	−	0.724322i	$-0.257847\pi$
$761$	38.0393	1.37892	0.689462	+	0.724322i	$0.257847\pi$
$762$	0	0
$763$	2.55762	0.0925920
$764$	−12.3857	−0.448100
$765$	0	0
$766$	−21.7335	−0.785262
$767$	9.53109	0.344148
$768$	0	0
$769$	−25.1058	−0.905337	−0.452668	−	0.891679i	$-0.649528\pi$
$769$	−25.1058	−0.905337	−0.452668	+	0.891679i	$0.649528\pi$
$770$	−15.8341	−0.570621
$771$	0	0
$772$	18.4810	0.665145
$773$	18.0142	0.647927	0.323963	−	0.946070i	$-0.394985\pi$
$773$	18.0142	0.647927	0.323963	+	0.946070i	$0.394985\pi$
$774$	0	0
$775$	−1.89709	−0.0681454
$776$	9.11014	0.327035
$777$	0	0
$778$	−18.4569	−0.661710
$779$	−55.9332	−2.00402
$780$	0	0
$781$	−16.0349	−0.573775
$782$	−13.0670	−0.467277
$783$	0	0
$784$	−4.94252	−0.176519
$785$	−75.7480	−2.70356
$786$	0	0
$787$	−32.2498	−1.14958	−0.574790	−	0.818301i	$-0.694916\pi$
$787$	−32.2498	−1.14958	−0.574790	+	0.818301i	$0.694916\pi$
$788$	−20.6471	−0.735523
$789$	0	0
$790$	93.3109	3.31985
$791$	15.7491	0.559973
$792$	0	0
$793$	−17.0166	−0.604277
$794$	56.4451	2.00316
$795$	0	0
$796$	−12.5925	−0.446328
$797$	20.9431	0.741843	0.370921	−	0.928664i	$-0.379042\pi$
$797$	20.9431	0.741843	0.370921	+	0.928664i	$0.379042\pi$
$798$	0	0
$799$	22.2170	0.785980
$800$	−55.6458	−1.96738
$801$	0	0
$802$	26.3745	0.931317
$803$	4.16699	0.147050
$804$	0	0
$805$	−9.08303	−0.320135
$806$	1.56898	0.0552650
$807$	0	0
$808$	−13.1405	−0.462282
$809$	−44.6972	−1.57147	−0.785735	−	0.618563i	$-0.787715\pi$
$809$	−44.6972	−1.57147	−0.785735	+	0.618563i	$0.787715\pi$
$810$	0	0
$811$	−11.5507	−0.405600	−0.202800	−	0.979220i	$-0.565004\pi$
$811$	−11.5507	−0.405600	−0.202800	+	0.979220i	$0.565004\pi$
$812$	−0.833102	−0.0292361
$813$	0	0
$814$	−9.85803	−0.345524
$815$	87.4433	3.06300
$816$	0	0
$817$	94.7363	3.31440
$818$	45.6240	1.59521
$819$	0	0
$820$	−32.3841	−1.13090
$821$	−0.564615	−0.0197052	−0.00985259	−	0.999951i	$-0.503136\pi$
$821$	−0.564615	−0.0197052	−0.00985259	+	0.999951i	$0.503136\pi$
$822$	0	0
$823$	38.8293	1.35350	0.676752	−	0.736211i	$-0.263387\pi$
$823$	38.8293	1.35350	0.676752	+	0.736211i	$0.263387\pi$
$824$	−16.6715	−0.580779
$825$	0	0
$826$	4.13266	0.143794
$827$	−34.1533	−1.18763	−0.593814	−	0.804603i	$-0.702378\pi$
$827$	−34.1533	−1.18763	−0.593814	+	0.804603i	$0.702378\pi$
$828$	0	0
$829$	−50.3105	−1.74736	−0.873678	−	0.486504i	$-0.838272\pi$
$829$	−50.3105	−1.74736	−0.873678	+	0.486504i	$0.838272\pi$
$830$	−8.70662	−0.302211
$831$	0	0
$832$	4.98740	0.172907
$833$	2.99424	0.103744
$834$	0	0
$835$	−12.3487	−0.427343
$836$	23.3531	0.807683
$837$	0	0
$838$	16.1766	0.558810
$839$	24.4893	0.845465	0.422733	−	0.906254i	$-0.361071\pi$
$839$	24.4893	0.845465	0.422733	+	0.906254i	$0.361071\pi$
$840$	0	0
$841$	−28.5484	−0.984429
$842$	14.9166	0.514062
$843$	0	0
$844$	17.9777	0.618819
$845$	−15.8543	−0.545406
$846$	0	0
$847$	5.48578	0.188493
$848$	69.7357	2.39473
$849$	0	0
$850$	48.6896	1.67004
$851$	−5.65493	−0.193849
$852$	0	0
$853$	20.8723	0.714655	0.357327	−	0.933979i	$-0.383688\pi$
$853$	20.8723	0.714655	0.357327	+	0.933979i	$0.383688\pi$
$854$	−7.37835	−0.252482
$855$	0	0
$856$	−17.5469	−0.599742
$857$	−14.2491	−0.486741	−0.243370	−	0.969933i	$-0.578253\pi$
$857$	−14.2491	−0.486741	−0.243370	+	0.969933i	$0.578253\pi$
$858$	0	0
$859$	−32.1064	−1.09545	−0.547727	−	0.836657i	$-0.684507\pi$
$859$	−32.1064	−1.09545	−0.547727	+	0.836657i	$0.684507\pi$
$860$	54.8502	1.87038
$861$	0	0
$862$	9.04414	0.308045
$863$	12.5269	0.426421	0.213211	−	0.977006i	$-0.431608\pi$
$863$	12.5269	0.426421	0.213211	+	0.977006i	$0.431608\pi$
$864$	0	0
$865$	68.7097	2.33620
$866$	−25.5451	−0.868059
$867$	0	0
$868$	0.260332	0.00883624
$869$	32.4955	1.10234
$870$	0	0
$871$	30.1960	1.02315
$872$	3.49984	0.118519
$873$	0	0
$874$	35.0073	1.18414
$875$	15.1134	0.510927
$876$	0	0
$877$	6.04916	0.204266	0.102133	−	0.994771i	$-0.467433\pi$
$877$	6.04916	0.204266	0.102133	+	0.994771i	$0.467433\pi$
$878$	43.8914	1.48126
$879$	0	0
$880$	−43.4796	−1.46570
$881$	21.7051	0.731263	0.365631	−	0.930760i	$-0.380853\pi$
$881$	21.7051	0.731263	0.365631	+	0.930760i	$0.380853\pi$
$882$	0	0
$883$	17.6690	0.594608	0.297304	−	0.954783i	$-0.403913\pi$
$883$	17.6690	0.594608	0.297304	+	0.954783i	$0.403913\pi$
$884$	−15.4095	−0.518278
$885$	0	0
$886$	−37.6075	−1.26345
$887$	18.8379	0.632514	0.316257	−	0.948674i	$-0.397574\pi$
$887$	18.8379	0.632514	0.316257	+	0.948674i	$0.397574\pi$
$888$	0	0
$889$	1.00000	0.0335389
$890$	−94.5790	−3.17030
$891$	0	0
$892$	7.50366	0.251241
$893$	−59.5205	−1.99178
$894$	0	0
$895$	−89.2633	−2.98374
$896$	−10.1563	−0.339297
$897$	0	0
$898$	−45.6393	−1.52300
$899$	−0.141110	−0.00470627
$900$	0	0
$901$	−42.2467	−1.40744
$902$	−29.4714	−0.981289
$903$	0	0
$904$	21.5510	0.716776
$905$	68.4131	2.27413
$906$	0	0
$907$	44.4847	1.47709	0.738545	−	0.674204i	$-0.235513\pi$
$907$	44.4847	1.47709	0.738545	+	0.674204i	$0.235513\pi$
$908$	−8.09848	−0.268757
$909$	0	0
$910$	−27.9911	−0.927896
$911$	13.6987	0.453859	0.226930	−	0.973911i	$-0.427131\pi$
$911$	13.6987	0.453859	0.226930	+	0.973911i	$0.427131\pi$
$912$	0	0
$913$	−3.03208	−0.100347
$914$	38.4424	1.27156
$915$	0	0
$916$	−13.5750	−0.448532
$917$	−13.6229	−0.449867
$918$	0	0
$919$	−42.1448	−1.39023	−0.695114	−	0.718899i	$-0.744646\pi$
$919$	−42.1448	−1.39023	−0.695114	+	0.718899i	$0.744646\pi$
$920$	−12.4292	−0.409778
$921$	0	0
$922$	−14.5760	−0.480034
$923$	−28.3461	−0.933024
$924$	0	0
$925$	21.0711	0.692812
$926$	−34.4284	−1.13139
$927$	0	0
$928$	−4.13906	−0.135871
$929$	−5.12478	−0.168139	−0.0840693	−	0.996460i	$-0.526792\pi$
$929$	−5.12478	−0.168139	−0.0840693	+	0.996460i	$0.526792\pi$
$930$	0	0
$931$	−8.02172	−0.262901
$932$	−29.1166	−0.953744
$933$	0	0
$934$	5.81247	0.190190
$935$	26.3405	0.861426
$936$	0	0
$937$	−50.8922	−1.66258	−0.831288	−	0.555843i	$-0.812396\pi$
$937$	−50.8922	−1.66258	−0.831288	+	0.555843i	$0.812396\pi$
$938$	13.0929	0.427498
$939$	0	0
$940$	−34.4610	−1.12399
$941$	−30.3249	−0.988565	−0.494282	−	0.869301i	$-0.664569\pi$
$941$	−30.3249	−0.988565	−0.494282	+	0.869301i	$0.664569\pi$
$942$	0	0
$943$	−16.9059	−0.550531
$944$	11.3481	0.369348
$945$	0	0
$946$	49.9168	1.62294
$947$	−4.01496	−0.130469	−0.0652343	−	0.997870i	$-0.520779\pi$
$947$	−4.01496	−0.130469	−0.0652343	+	0.997870i	$0.520779\pi$
$948$	0	0
$949$	7.36630	0.239120
$950$	−130.442	−4.23210
$951$	0	0
$952$	4.09731	0.132794
$953$	9.25641	0.299844	0.149922	−	0.988698i	$-0.452098\pi$
$953$	9.25641	0.299844	0.149922	+	0.988698i	$0.452098\pi$
$954$	0	0
$955$	37.4268	1.21110
$956$	6.15729	0.199141
$957$	0	0
$958$	−54.3978	−1.75751
$959$	4.65045	0.150171
$960$	0	0
$961$	−30.9559	−0.998578
$962$	−17.4268	−0.561861
$963$	0	0
$964$	−0.620669	−0.0199904
$965$	−55.8452	−1.79772
$966$	0	0
$967$	−38.7039	−1.24463	−0.622317	−	0.782765i	$-0.713809\pi$
$967$	−38.7039	−1.24463	−0.622317	+	0.782765i	$0.713809\pi$
$968$	7.50672	0.241275
$969$	0	0
$970$	44.8915	1.44138
$971$	29.7333	0.954188	0.477094	−	0.878852i	$-0.341690\pi$
$971$	29.7333	0.954188	0.477094	+	0.878852i	$0.341690\pi$
$972$	0	0
$973$	−14.7158	−0.471766
$974$	9.23726	0.295981
$975$	0	0
$976$	−20.2606	−0.648526
$977$	31.2156	0.998674	0.499337	−	0.866408i	$-0.333577\pi$
$977$	31.2156	0.998674	0.499337	+	0.866408i	$0.333577\pi$
$978$	0	0
$979$	−32.9371	−1.05268
$980$	−4.64440	−0.148360
$981$	0	0
$982$	19.8524	0.633515
$983$	−19.5414	−0.623273	−0.311637	−	0.950201i	$-0.600877\pi$
$983$	−19.5414	−0.623273	−0.311637	+	0.950201i	$0.600877\pi$
$984$	0	0
$985$	62.3908	1.98794
$986$	3.62165	0.115337
$987$	0	0
$988$	41.2829	1.31338
$989$	28.6341	0.910513
$990$	0	0
$991$	−3.11750	−0.0990306	−0.0495153	−	0.998773i	$-0.515768\pi$
$991$	−3.11750	−0.0990306	−0.0495153	+	0.998773i	$0.515768\pi$
$992$	1.29340	0.0410653
$993$	0	0
$994$	−12.2908	−0.389841
$995$	38.0515	1.20631
$996$	0	0
$997$	−1.31113	−0.0415240	−0.0207620	−	0.999784i	$-0.506609\pi$
$997$	−1.31113	−0.0415240	−0.0207620	+	0.999784i	$0.506609\pi$
$998$	−4.70289	−0.148867
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8001.2.a.t.1.13		16
3.2	odd	2		889.2.a.c.1.4	✓	16
21.20	even	2		6223.2.a.k.1.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
889.2.a.c.1.4	✓	16	3.2	odd	2
6223.2.a.k.1.4		16	21.20	even	2
8001.2.a.t.1.13		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8001.a (trivial)

\(f(q)\)	\(=\)	\(q+1.79993 q^{2} +1.23975 q^{4} -3.74624 q^{5} -1.00000 q^{7} -1.36840 q^{8} +O(q^{10})\)	\(q+1.79993 q^{2} +1.23975 q^{4} -3.74624 q^{5} -1.00000 q^{7} -1.36840 q^{8} -6.74297 q^{10} -2.34824 q^{11} -4.15115 q^{13} -1.79993 q^{14} -4.94252 q^{16} +2.99424 q^{17} -8.02172 q^{19} -4.64440 q^{20} -4.22667 q^{22} -2.42457 q^{23} +9.03429 q^{25} -7.47179 q^{26} -1.23975 q^{28} +0.671992 q^{29} -0.209987 q^{31} -6.15940 q^{32} +5.38942 q^{34} +3.74624 q^{35} +2.33234 q^{37} -14.4385 q^{38} +5.12634 q^{40} +6.97272 q^{41} -11.8100 q^{43} -2.91123 q^{44} -4.36406 q^{46} +7.41991 q^{47} +1.00000 q^{49} +16.2611 q^{50} -5.14639 q^{52} -14.1093 q^{53} +8.79706 q^{55} +1.36840 q^{56} +1.20954 q^{58} -2.29601 q^{59} +4.09924 q^{61} -0.377963 q^{62} -1.20145 q^{64} +15.5512 q^{65} -7.27412 q^{67} +3.71211 q^{68} +6.74297 q^{70} +6.82849 q^{71} -1.77452 q^{73} +4.19805 q^{74} -9.94493 q^{76} +2.34824 q^{77} -13.8383 q^{79} +18.5159 q^{80} +12.5504 q^{82} +1.29121 q^{83} -11.2171 q^{85} -21.2571 q^{86} +3.21332 q^{88} +14.0263 q^{89} +4.15115 q^{91} -3.00586 q^{92} +13.3553 q^{94} +30.0513 q^{95} -6.65752 q^{97} +1.79993 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8	\( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 16 * q + 2 * q^2 + 12 * q^4 + 9 * q^5 - 16 * q^7 + 6 * q^8 - 2 * q^10 + 22 * q^11 - 4 * q^13 - 2 * q^14 + 12 * q^16 + 18 * q^17 - 15 * q^19 + 40 * q^20 - 11 * q^22 + 5 * q^23 + 15 * q^25 + 24 * q^26 - 12 * q^28 + 12 * q^29 - 32 * q^31 + 9 * q^32 - 14 * q^34 - 9 * q^35 - 2 * q^37 - 3 * q^38 - 14 * q^40 + 45 * q^41 - 3 * q^43 + 54 * q^44 + 49 * q^47 + 16 * q^49 + 6 * q^50 + 38 * q^52 - 16 * q^53 + 7 * q^55 - 6 * q^56 + 16 * q^58 + 35 * q^59 - 11 * q^61 - 17 * q^62 - 2 * q^64 - 14 * q^65 + 17 * q^67 + 71 * q^68 + 2 * q^70 + 81 * q^71 - 15 * q^73 - 13 * q^74 + 14 * q^76 - 22 * q^77 - 34 * q^79 + 33 * q^80 - 14 * q^82 + 39 * q^83 - 17 * q^85 - 36 * q^86 + 61 * q^88 + 32 * q^89 + 4 * q^91 - 37 * q^92 + 13 * q^94 + 33 * q^95 - 4 * q^97 + 2 * q^98

Introduction

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