LMFDB - Embedded newform 8281.2.a.br.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8281,2,Mod(1,8281)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8281.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8281, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,4,0,0,0,0,-12,0,0,0,0,0,0,-20,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{3}, \sqrt{13})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 11x^{2} + 12x - 3 $ x^4 - 2x^3 - 11x^2 + 12*x - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 637)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.03483$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205 q^{2} +1.00000 q^{4} -3.60555 q^{5} -1.73205 q^{8} -3.00000 q^{9} -6.24500 q^{10} +3.46410 q^{11} -5.00000 q^{16} +6.24500 q^{17} -5.19615 q^{18} +7.21110 q^{19} -3.60555 q^{20} +6.00000 q^{22} -4.00000 q^{23} +8.00000 q^{25} +1.00000 q^{29} -5.19615 q^{32} +10.8167 q^{34} -3.00000 q^{36} -1.73205 q^{37} +12.4900 q^{38} +6.24500 q^{40} +3.60555 q^{41} -6.00000 q^{43} +3.46410 q^{44} +10.8167 q^{45} -6.92820 q^{46} -7.21110 q^{47} +13.8564 q^{50} +5.00000 q^{53} -12.4900 q^{55} +1.73205 q^{58} -7.21110 q^{59} +6.24500 q^{61} +1.00000 q^{64} -13.8564 q^{67} +6.24500 q^{68} -10.3923 q^{71} +5.19615 q^{72} +10.8167 q^{73} -3.00000 q^{74} +7.21110 q^{76} -6.00000 q^{79} +18.0278 q^{80} +9.00000 q^{81} +6.24500 q^{82} +7.21110 q^{83} -22.5167 q^{85} -10.3923 q^{86} -6.00000 q^{88} -7.21110 q^{89} +18.7350 q^{90} -4.00000 q^{92} -12.4900 q^{94} -26.0000 q^{95} +7.21110 q^{97} -10.3923 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{4} - 12 q^{9} - 20 q^{16} + 24 q^{22} - 16 q^{23} + 32 q^{25} + 4 q^{29} - 12 q^{36} - 24 q^{43} + 20 q^{53} + 4 q^{64} - 12 q^{74} - 24 q^{79} + 36 q^{81} - 24 q^{88} - 16 q^{92} - 104 q^{95}+O(q^{100}) $ 4 * q + 4 * q^4 - 12 * q^9 - 20 * q^16 + 24 * q^22 - 16 * q^23 + 32 * q^25 + 4 * q^29 - 12 * q^36 - 24 * q^43 + 20 * q^53 + 4 * q^64 - 12 * q^74 - 24 * q^79 + 36 * q^81 - 24 * q^88 - 16 * q^92 - 104 * q^95

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 $

$2$	1.73205	1.22474	0.612372	−	0.790569i	$-0.290215\pi$
$2$	1.73205	1.22474	0.612372	+	0.790569i	$0.290215\pi$
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	1.00000	0.500000
$5$	−3.60555	−1.61245	−0.806226	−	0.591608i	$-0.798493\pi$
$5$	−3.60555	−1.61245	−0.806226	+	0.591608i	$0.798493\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.73205	−0.612372
$9$	−3.00000	−1.00000
$10$	−6.24500	−1.97484
$11$	3.46410	1.04447	0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410	1.04447	0.522233	+	0.852803i	$0.325099\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	0	0
$16$	−5.00000	−1.25000
$17$	6.24500	1.51463	0.757317	−	0.653047i	$-0.226510\pi$
$17$	6.24500	1.51463	0.757317	+	0.653047i	$0.226510\pi$
$18$	−5.19615	−1.22474
$19$	7.21110	1.65434	0.827170	−	0.561951i	$-0.189949\pi$
$19$	7.21110	1.65434	0.827170	+	0.561951i	$0.189949\pi$
$20$	−3.60555	−0.806226
$21$	0	0
$22$	6.00000	1.27920
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	8.00000	1.60000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−5.19615	−0.918559
$33$	0	0
$34$	10.8167	1.85504
$35$	0	0
$36$	−3.00000	−0.500000
$37$	−1.73205	−0.284747	−0.142374	−	0.989813i	$-0.545473\pi$
$37$	−1.73205	−0.284747	−0.142374	+	0.989813i	$0.545473\pi$
$38$	12.4900	2.02614
$39$	0	0
$40$	6.24500	0.987421
$41$	3.60555	0.563093	0.281546	−	0.959548i	$-0.409153\pi$
$41$	3.60555	0.563093	0.281546	+	0.959548i	$0.409153\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	3.46410	0.522233
$45$	10.8167	1.61245
$46$	−6.92820	−1.02151
$47$	−7.21110	−1.05185	−0.525924	−	0.850532i	$-0.676280\pi$
$47$	−7.21110	−1.05185	−0.525924	+	0.850532i	$0.676280\pi$
$48$	0	0
$49$	0	0
$50$	13.8564	1.95959
$51$	0	0
$52$	0	0
$53$	5.00000	0.686803	0.343401	−	0.939189i	$-0.388421\pi$
$53$	5.00000	0.686803	0.343401	+	0.939189i	$0.388421\pi$
$54$	0	0
$55$	−12.4900	−1.68415
$56$	0	0
$57$	0	0
$58$	1.73205	0.227429
$59$	−7.21110	−0.938806	−0.469403	−	0.882984i	$-0.655531\pi$
$59$	−7.21110	−0.938806	−0.469403	+	0.882984i	$0.655531\pi$
$60$	0	0
$61$	6.24500	0.799590	0.399795	−	0.916605i	$-0.369081\pi$
$61$	6.24500	0.799590	0.399795	+	0.916605i	$0.369081\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−13.8564	−1.69283	−0.846415	−	0.532524i	$-0.821244\pi$
$67$	−13.8564	−1.69283	−0.846415	+	0.532524i	$0.821244\pi$
$68$	6.24500	0.757317
$69$	0	0
$70$	0	0
$71$	−10.3923	−1.23334	−0.616670	−	0.787222i	$-0.711519\pi$
$71$	−10.3923	−1.23334	−0.616670	+	0.787222i	$0.711519\pi$
$72$	5.19615	0.612372
$73$	10.8167	1.26599	0.632997	−	0.774154i	$-0.281825\pi$
$73$	10.8167	1.26599	0.632997	+	0.774154i	$0.281825\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	7.21110	0.827170
$77$	0	0
$78$	0	0
$79$	−6.00000	−0.675053	−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000	−0.675053	−0.337526	+	0.941316i	$0.609590\pi$
$80$	18.0278	2.01556
$81$	9.00000	1.00000
$82$	6.24500	0.689645
$83$	7.21110	0.791521	0.395761	−	0.918354i	$-0.370481\pi$
$83$	7.21110	0.791521	0.395761	+	0.918354i	$0.370481\pi$
$84$	0	0
$85$	−22.5167	−2.44227
$86$	−10.3923	−1.12063
$87$	0	0
$88$	−6.00000	−0.639602
$89$	−7.21110	−0.764375	−0.382188	−	0.924085i	$-0.624829\pi$
$89$	−7.21110	−0.764375	−0.382188	+	0.924085i	$0.624829\pi$
$90$	18.7350	1.97484
$91$	0	0
$92$	−4.00000	−0.417029
$93$	0	0
$94$	−12.4900	−1.28824
$95$	−26.0000	−2.66754
$96$	0	0
$97$	7.21110	0.732177	0.366088	−	0.930580i	$-0.380697\pi$
$97$	7.21110	0.732177	0.366088	+	0.930580i	$0.380697\pi$
$98$	0	0
$99$	−10.3923	−1.04447
$100$	8.00000	0.800000
$101$	6.24500	0.621401	0.310700	−	0.950508i	$-0.399436\pi$
$101$	6.24500	0.621401	0.310700	+	0.950508i	$0.399436\pi$
$102$	0	0
$103$	−12.4900	−1.23068	−0.615338	−	0.788263i	$-0.710980\pi$
$103$	−12.4900	−1.23068	−0.615338	+	0.788263i	$0.710980\pi$
$104$	0	0
$105$	0	0
$106$	8.66025	0.841158
$107$	2.00000	0.193347	0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000	0.193347	0.0966736	+	0.995316i	$0.469180\pi$
$108$	0	0
$109$	−6.92820	−0.663602	−0.331801	−	0.943349i	$-0.607656\pi$
$109$	−6.92820	−0.663602	−0.331801	+	0.943349i	$0.607656\pi$
$110$	−21.6333	−2.06265
$111$	0	0
$112$	0	0
$113$	5.00000	0.470360	0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000	0.470360	0.235180	+	0.971952i	$0.424432\pi$
$114$	0	0
$115$	14.4222	1.34488
$116$	1.00000	0.0928477
$117$	0	0
$118$	−12.4900	−1.14980
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	10.8167	0.979294
$123$	0	0
$124$	0	0
$125$	−10.8167	−0.967471
$126$	0	0
$127$	−18.0000	−1.59724	−0.798621	−	0.601834i	$-0.794437\pi$
$127$	−18.0000	−1.59724	−0.798621	+	0.601834i	$0.794437\pi$
$128$	12.1244	1.07165
$129$	0	0
$130$	0	0
$131$	−12.4900	−1.09126	−0.545628	−	0.838027i	$-0.683709\pi$
$131$	−12.4900	−1.09126	−0.545628	+	0.838027i	$0.683709\pi$
$132$	0	0
$133$	0	0
$134$	−24.0000	−2.07328
$135$	0	0
$136$	−10.8167	−0.927520
$137$	−8.66025	−0.739895	−0.369948	−	0.929053i	$-0.620624\pi$
$137$	−8.66025	−0.739895	−0.369948	+	0.929053i	$0.620624\pi$
$138$	0	0
$139$	−12.4900	−1.05939	−0.529694	−	0.848189i	$-0.677693\pi$
$139$	−12.4900	−1.05939	−0.529694	+	0.848189i	$0.677693\pi$
$140$	0	0
$141$	0	0
$142$	−18.0000	−1.51053
$143$	0	0
$144$	15.0000	1.25000
$145$	−3.60555	−0.299425
$146$	18.7350	1.55052
$147$	0	0
$148$	−1.73205	−0.142374
$149$	15.5885	1.27706	0.638528	−	0.769599i	$-0.279544\pi$
$149$	15.5885	1.27706	0.638528	+	0.769599i	$0.279544\pi$
$150$	0	0
$151$	3.46410	0.281905	0.140952	−	0.990016i	$-0.454984\pi$
$151$	3.46410	0.281905	0.140952	+	0.990016i	$0.454984\pi$
$152$	−12.4900	−1.01307
$153$	−18.7350	−1.51463
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−6.24500	−0.498405	−0.249203	−	0.968451i	$-0.580168\pi$
$157$	−6.24500	−0.498405	−0.249203	+	0.968451i	$0.580168\pi$
$158$	−10.3923	−0.826767
$159$	0	0
$160$	18.7350	1.48113
$161$	0	0
$162$	15.5885	1.22474
$163$	10.3923	0.813988	0.406994	−	0.913431i	$-0.366577\pi$
$163$	10.3923	0.813988	0.406994	+	0.913431i	$0.366577\pi$
$164$	3.60555	0.281546
$165$	0	0
$166$	12.4900	0.969412
$167$	−7.21110	−0.558012	−0.279006	−	0.960289i	$-0.590005\pi$
$167$	−7.21110	−0.558012	−0.279006	+	0.960289i	$0.590005\pi$
$168$	0	0
$169$	0	0
$170$	−39.0000	−2.99116
$171$	−21.6333	−1.65434
$172$	−6.00000	−0.457496
$173$	−24.9800	−1.89919	−0.949597	−	0.313474i	$-0.898507\pi$
$173$	−24.9800	−1.89919	−0.949597	+	0.313474i	$0.898507\pi$
$174$	0	0
$175$	0	0
$176$	−17.3205	−1.30558
$177$	0	0
$178$	−12.4900	−0.936165
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	10.8167	0.806226
$181$	−6.24500	−0.464187	−0.232094	−	0.972693i	$-0.574558\pi$
$181$	−6.24500	−0.464187	−0.232094	+	0.972693i	$0.574558\pi$
$182$	0	0
$183$	0	0
$184$	6.92820	0.510754
$185$	6.24500	0.459141
$186$	0	0
$187$	21.6333	1.58198
$188$	−7.21110	−0.525924
$189$	0	0
$190$	−45.0333	−3.26706
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	0	0
$193$	8.66025	0.623379	0.311689	−	0.950184i	$-0.399105\pi$
$193$	8.66025	0.623379	0.311689	+	0.950184i	$0.399105\pi$
$194$	12.4900	0.896729
$195$	0	0
$196$	0	0
$197$	0	0		−	1.00000i	$-0.5\pi$
$197$	0	0			1.00000i	$0.5\pi$
$198$	−18.0000	−1.27920
$199$	−12.4900	−0.885392	−0.442696	−	0.896672i	$-0.645978\pi$
$199$	−12.4900	−0.885392	−0.442696	+	0.896672i	$0.645978\pi$
$200$	−13.8564	−0.979796
$201$	0	0
$202$	10.8167	0.761057
$203$	0	0
$204$	0	0
$205$	−13.0000	−0.907959
$206$	−21.6333	−1.50726
$207$	12.0000	0.834058
$208$	0	0
$209$	24.9800	1.72790
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	5.00000	0.343401
$213$	0	0
$214$	3.46410	0.236801
$215$	21.6333	1.47538
$216$	0	0
$217$	0	0
$218$	−12.0000	−0.812743
$219$	0	0
$220$	−12.4900	−0.842075
$221$	0	0
$222$	0	0
$223$	7.21110	0.482891	0.241446	−	0.970414i	$-0.422378\pi$
$223$	7.21110	0.482891	0.241446	+	0.970414i	$0.422378\pi$
$224$	0	0
$225$	−24.0000	−1.60000
$226$	8.66025	0.576072
$227$	−14.4222	−0.957235	−0.478618	−	0.878023i	$-0.658862\pi$
$227$	−14.4222	−0.957235	−0.478618	+	0.878023i	$0.658862\pi$
$228$	0	0
$229$	21.6333	1.42957	0.714785	−	0.699345i	$-0.246525\pi$
$229$	21.6333	1.42957	0.714785	+	0.699345i	$0.246525\pi$
$230$	24.9800	1.64713
$231$	0	0
$232$	−1.73205	−0.113715
$233$	−2.00000	−0.131024	−0.0655122	−	0.997852i	$-0.520868\pi$
$233$	−2.00000	−0.131024	−0.0655122	+	0.997852i	$0.520868\pi$
$234$	0	0
$235$	26.0000	1.69605
$236$	−7.21110	−0.469403
$237$	0	0
$238$	0	0
$239$	−17.3205	−1.12037	−0.560185	−	0.828367i	$-0.689270\pi$
$239$	−17.3205	−1.12037	−0.560185	+	0.828367i	$0.689270\pi$
$240$	0	0
$241$	10.8167	0.696762	0.348381	−	0.937353i	$-0.386732\pi$
$241$	10.8167	0.696762	0.348381	+	0.937353i	$0.386732\pi$
$242$	1.73205	0.111340
$243$	0	0
$244$	6.24500	0.399795
$245$	0	0
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0	0
$250$	−18.7350	−1.18491
$251$	−12.4900	−0.788362	−0.394181	−	0.919033i	$-0.628972\pi$
$251$	−12.4900	−0.788362	−0.394181	+	0.919033i	$0.628972\pi$
$252$	0	0
$253$	−13.8564	−0.871145
$254$	−31.1769	−1.95621
$255$	0	0
$256$	19.0000	1.18750
$257$	−18.7350	−1.16866	−0.584328	−	0.811517i	$-0.698642\pi$
$257$	−18.7350	−1.16866	−0.584328	+	0.811517i	$0.698642\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.00000	−0.185695
$262$	−21.6333	−1.33651
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	−18.0278	−1.10744
$266$	0	0
$267$	0	0
$268$	−13.8564	−0.846415
$269$	24.9800	1.52306	0.761528	−	0.648131i	$-0.224449\pi$
$269$	24.9800	1.52306	0.761528	+	0.648131i	$0.224449\pi$
$270$	0	0
$271$	7.21110	0.438043	0.219022	−	0.975720i	$-0.429713\pi$
$271$	7.21110	0.438043	0.219022	+	0.975720i	$0.429713\pi$
$272$	−31.2250	−1.89329
$273$	0	0
$274$	−15.0000	−0.906183
$275$	27.7128	1.67115
$276$	0	0
$277$	−25.0000	−1.50210	−0.751052	−	0.660243i	$-0.770453\pi$
$277$	−25.0000	−1.50210	−0.751052	+	0.660243i	$0.770453\pi$
$278$	−21.6333	−1.29748
$279$	0	0
$280$	0	0
$281$	−22.5167	−1.34323	−0.671616	−	0.740900i	$-0.734399\pi$
$281$	−22.5167	−1.34323	−0.671616	+	0.740900i	$0.734399\pi$
$282$	0	0
$283$	−12.4900	−0.742453	−0.371227	−	0.928542i	$-0.621063\pi$
$283$	−12.4900	−0.742453	−0.371227	+	0.928542i	$0.621063\pi$
$284$	−10.3923	−0.616670
$285$	0	0
$286$	0	0
$287$	0	0
$288$	15.5885	0.918559
$289$	22.0000	1.29412
$290$	−6.24500	−0.366719
$291$	0	0
$292$	10.8167	0.632997
$293$	−18.0278	−1.05319	−0.526596	−	0.850115i	$-0.676532\pi$
$293$	−18.0278	−1.05319	−0.526596	+	0.850115i	$0.676532\pi$
$294$	0	0
$295$	26.0000	1.51378
$296$	3.00000	0.174371
$297$	0	0
$298$	27.0000	1.56407
$299$	0	0
$300$	0	0
$301$	0	0
$302$	6.00000	0.345261
$303$	0	0
$304$	−36.0555	−2.06793
$305$	−22.5167	−1.28930
$306$	−32.4500	−1.85504
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.9800	−1.41649	−0.708243	−	0.705969i	$-0.750512\pi$
$311$	−24.9800	−1.41649	−0.708243	+	0.705969i	$0.750512\pi$
$312$	0	0
$313$	0	0		−	1.00000i	$-0.5\pi$
$313$	0	0			1.00000i	$0.5\pi$
$314$	−10.8167	−0.610419
$315$	0	0
$316$	−6.00000	−0.337526
$317$	15.5885	0.875535	0.437767	−	0.899088i	$-0.355769\pi$
$317$	15.5885	0.875535	0.437767	+	0.899088i	$0.355769\pi$
$318$	0	0
$319$	3.46410	0.193952
$320$	−3.60555	−0.201556
$321$	0	0
$322$	0	0
$323$	45.0333	2.50572
$324$	9.00000	0.500000
$325$	0	0
$326$	18.0000	0.996928
$327$	0	0
$328$	−6.24500	−0.344822
$329$	0	0
$330$	0	0
$331$	10.3923	0.571213	0.285606	−	0.958347i	$-0.407805\pi$
$331$	10.3923	0.571213	0.285606	+	0.958347i	$0.407805\pi$
$332$	7.21110	0.395761
$333$	5.19615	0.284747
$334$	−12.4900	−0.683422
$335$	49.9600	2.72961
$336$	0	0
$337$	−11.0000	−0.599208	−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000	−0.599208	−0.299604	+	0.954064i	$0.596855\pi$
$338$	0	0
$339$	0	0
$340$	−22.5167	−1.22114
$341$	0	0
$342$	−37.4700	−2.02614
$343$	0	0
$344$	10.3923	0.560316
$345$	0	0
$346$	−43.2666	−2.32603
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	−21.6333	−1.15800	−0.579002	−	0.815326i	$-0.696558\pi$
$349$	−21.6333	−1.15800	−0.579002	+	0.815326i	$0.696558\pi$
$350$	0	0
$351$	0	0
$352$	−18.0000	−0.959403
$353$	25.2389	1.34333	0.671664	−	0.740855i	$-0.265580\pi$
$353$	25.2389	1.34333	0.671664	+	0.740855i	$0.265580\pi$
$354$	0	0
$355$	37.4700	1.98870
$356$	−7.21110	−0.382188
$357$	0	0
$358$	−27.7128	−1.46467
$359$	−27.7128	−1.46263	−0.731313	−	0.682042i	$-0.761092\pi$
$359$	−27.7128	−1.46263	−0.731313	+	0.682042i	$0.761092\pi$
$360$	−18.7350	−0.987421
$361$	33.0000	1.73684
$362$	−10.8167	−0.568511
$363$	0	0
$364$	0	0
$365$	−39.0000	−2.04135
$366$	0	0
$367$	24.9800	1.30394	0.651972	−	0.758243i	$-0.273942\pi$
$367$	24.9800	1.30394	0.651972	+	0.758243i	$0.273942\pi$
$368$	20.0000	1.04257
$369$	−10.8167	−0.563093
$370$	10.8167	0.562331
$371$	0	0
$372$	0	0
$373$	15.0000	0.776671	0.388335	−	0.921518i	$-0.373050\pi$
$373$	15.0000	0.776671	0.388335	+	0.921518i	$0.373050\pi$
$374$	37.4700	1.93753
$375$	0	0
$376$	12.4900	0.644122
$377$	0	0
$378$	0	0
$379$	−20.7846	−1.06763	−0.533817	−	0.845600i	$-0.679243\pi$
$379$	−20.7846	−1.06763	−0.533817	+	0.845600i	$0.679243\pi$
$380$	−26.0000	−1.33377
$381$	0	0
$382$	6.92820	0.354478
$383$	36.0555	1.84235	0.921175	−	0.389148i	$-0.127230\pi$
$383$	36.0555	1.84235	0.921175	+	0.389148i	$0.127230\pi$
$384$	0	0
$385$	0	0
$386$	15.0000	0.763480
$387$	18.0000	0.914991
$388$	7.21110	0.366088
$389$	−17.0000	−0.861934	−0.430967	−	0.902368i	$-0.641828\pi$
$389$	−17.0000	−0.861934	−0.430967	+	0.902368i	$0.641828\pi$
$390$	0	0
$391$	−24.9800	−1.26329
$392$	0	0
$393$	0	0
$394$	0	0
$395$	21.6333	1.08849
$396$	−10.3923	−0.522233
$397$	−7.21110	−0.361915	−0.180957	−	0.983491i	$-0.557920\pi$
$397$	−7.21110	−0.361915	−0.180957	+	0.983491i	$0.557920\pi$
$398$	−21.6333	−1.08438
$399$	0	0
$400$	−40.0000	−2.00000
$401$	8.66025	0.432472	0.216236	−	0.976341i	$-0.430622\pi$
$401$	8.66025	0.432472	0.216236	+	0.976341i	$0.430622\pi$
$402$	0	0
$403$	0	0
$404$	6.24500	0.310700
$405$	−32.4500	−1.61245
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	25.2389	1.24798	0.623991	−	0.781432i	$-0.285510\pi$
$409$	25.2389	1.24798	0.623991	+	0.781432i	$0.285510\pi$
$410$	−22.5167	−1.11202
$411$	0	0
$412$	−12.4900	−0.615338
$413$	0	0
$414$	20.7846	1.02151
$415$	−26.0000	−1.27629
$416$	0	0
$417$	0	0
$418$	43.2666	2.11624
$419$	0	0		−	1.00000i	$-0.5\pi$
$419$	0	0			1.00000i	$0.5\pi$
$420$	0	0
$421$	22.5167	1.09739	0.548697	−	0.836021i	$-0.315124\pi$
$421$	22.5167	1.09739	0.548697	+	0.836021i	$0.315124\pi$
$422$	−34.6410	−1.68630
$423$	21.6333	1.05185
$424$	−8.66025	−0.420579
$425$	49.9600	2.42342
$426$	0	0
$427$	0	0
$428$	2.00000	0.0966736
$429$	0	0
$430$	37.4700	1.80696
$431$	27.7128	1.33488	0.667440	−	0.744664i	$-0.267390\pi$
$431$	27.7128	1.33488	0.667440	+	0.744664i	$0.267390\pi$
$432$	0	0
$433$	18.7350	0.900346	0.450173	−	0.892941i	$-0.351362\pi$
$433$	18.7350	0.900346	0.450173	+	0.892941i	$0.351362\pi$
$434$	0	0
$435$	0	0
$436$	−6.92820	−0.331801
$437$	−28.8444	−1.37982
$438$	0	0
$439$	12.4900	0.596115	0.298057	−	0.954548i	$-0.403661\pi$
$439$	12.4900	0.596115	0.298057	+	0.954548i	$0.403661\pi$
$440$	21.6333	1.03133
$441$	0	0
$442$	0	0
$443$	16.0000	0.760183	0.380091	−	0.924949i	$-0.375893\pi$
$443$	16.0000	0.760183	0.380091	+	0.924949i	$0.375893\pi$
$444$	0	0
$445$	26.0000	1.23252
$446$	12.4900	0.591418
$447$	0	0
$448$	0	0
$449$	−6.92820	−0.326962	−0.163481	−	0.986546i	$-0.552272\pi$
$449$	−6.92820	−0.326962	−0.163481	+	0.986546i	$0.552272\pi$
$450$	−41.5692	−1.95959
$451$	12.4900	0.588131
$452$	5.00000	0.235180
$453$	0	0
$454$	−24.9800	−1.17237
$455$	0	0
$456$	0	0
$457$	39.8372	1.86350	0.931752	−	0.363095i	$-0.118280\pi$
$457$	39.8372	1.86350	0.931752	+	0.363095i	$0.118280\pi$
$458$	37.4700	1.75086
$459$	0	0
$460$	14.4222	0.672439
$461$	−39.6611	−1.84720	−0.923600	−	0.383358i	$-0.874768\pi$
$461$	−39.6611	−1.84720	−0.923600	+	0.383358i	$0.874768\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	−5.00000	−0.232119
$465$	0	0
$466$	−3.46410	−0.160471
$467$	12.4900	0.577968	0.288984	−	0.957334i	$-0.406683\pi$
$467$	12.4900	0.577968	0.288984	+	0.957334i	$0.406683\pi$
$468$	0	0
$469$	0	0
$470$	45.0333	2.07723
$471$	0	0
$472$	12.4900	0.574899
$473$	−20.7846	−0.955677
$474$	0	0
$475$	57.6888	2.64694
$476$	0	0
$477$	−15.0000	−0.686803
$478$	−30.0000	−1.37217
$479$	−14.4222	−0.658967	−0.329484	−	0.944161i	$-0.606875\pi$
$479$	−14.4222	−0.658967	−0.329484	+	0.944161i	$0.606875\pi$
$480$	0	0
$481$	0	0
$482$	18.7350	0.853356
$483$	0	0
$484$	1.00000	0.0454545
$485$	−26.0000	−1.18060
$486$	0	0
$487$	−20.7846	−0.941841	−0.470920	−	0.882176i	$-0.656078\pi$
$487$	−20.7846	−0.941841	−0.470920	+	0.882176i	$0.656078\pi$
$488$	−10.8167	−0.489647
$489$	0	0
$490$	0	0
$491$	4.00000	0.180517	0.0902587	−	0.995918i	$-0.471231\pi$
$491$	4.00000	0.180517	0.0902587	+	0.995918i	$0.471231\pi$
$492$	0	0
$493$	6.24500	0.281261
$494$	0	0
$495$	37.4700	1.68415
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−27.7128	−1.24060	−0.620298	−	0.784366i	$-0.712988\pi$
$499$	−27.7128	−1.24060	−0.620298	+	0.784366i	$0.712988\pi$
$500$	−10.8167	−0.483735
$501$	0	0
$502$	−21.6333	−0.965542
$503$	−12.4900	−0.556901	−0.278451	−	0.960451i	$-0.589821\pi$
$503$	−12.4900	−0.556901	−0.278451	+	0.960451i	$0.589821\pi$
$504$	0	0
$505$	−22.5167	−1.00198
$506$	−24.0000	−1.06693
$507$	0	0
$508$	−18.0000	−0.798621
$509$	−3.60555	−0.159813	−0.0799066	−	0.996802i	$-0.525462\pi$
$509$	−3.60555	−0.159813	−0.0799066	+	0.996802i	$0.525462\pi$
$510$	0	0
$511$	0	0
$512$	8.66025	0.382733
$513$	0	0
$514$	−32.4500	−1.43131
$515$	45.0333	1.98441
$516$	0	0
$517$	−24.9800	−1.09862
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−6.24500	−0.273598	−0.136799	−	0.990599i	$-0.543681\pi$
$521$	−6.24500	−0.273598	−0.136799	+	0.990599i	$0.543681\pi$
$522$	−5.19615	−0.227429
$523$	24.9800	1.09230	0.546149	−	0.837688i	$-0.316093\pi$
$523$	24.9800	1.09230	0.546149	+	0.837688i	$0.316093\pi$
$524$	−12.4900	−0.545628
$525$	0	0
$526$	3.46410	0.151042
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−31.2250	−1.35633
$531$	21.6333	0.938806
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−7.21110	−0.311763
$536$	24.0000	1.03664
$537$	0	0
$538$	43.2666	1.86536
$539$	0	0
$540$	0	0
$541$	5.19615	0.223400	0.111700	−	0.993742i	$-0.464370\pi$
$541$	5.19615	0.223400	0.111700	+	0.993742i	$0.464370\pi$
$542$	12.4900	0.536491
$543$	0	0
$544$	−32.4500	−1.39128
$545$	24.9800	1.07003
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−8.66025	−0.369948
$549$	−18.7350	−0.799590
$550$	48.0000	2.04673
$551$	7.21110	0.307203
$552$	0	0
$553$	0	0
$554$	−43.3013	−1.83969
$555$	0	0
$556$	−12.4900	−0.529694
$557$	−1.73205	−0.0733893	−0.0366947	−	0.999327i	$-0.511683\pi$
$557$	−1.73205	−0.0733893	−0.0366947	+	0.999327i	$0.511683\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	−39.0000	−1.64512
$563$	12.4900	0.526391	0.263195	−	0.964743i	$-0.415224\pi$
$563$	12.4900	0.526391	0.263195	+	0.964743i	$0.415224\pi$
$564$	0	0
$565$	−18.0278	−0.758433
$566$	−21.6333	−0.909316
$567$	0	0
$568$	18.0000	0.755263
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−24.0000	−1.00437	−0.502184	−	0.864761i	$-0.667470\pi$
$571$	−24.0000	−1.00437	−0.502184	+	0.864761i	$0.667470\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−32.0000	−1.33449
$576$	−3.00000	−0.125000
$577$	−18.0278	−0.750505	−0.375253	−	0.926923i	$-0.622444\pi$
$577$	−18.0278	−0.750505	−0.375253	+	0.926923i	$0.622444\pi$
$578$	38.1051	1.58496
$579$	0	0
$580$	−3.60555	−0.149712
$581$	0	0
$582$	0	0
$583$	17.3205	0.717342
$584$	−18.7350	−0.775260
$585$	0	0
$586$	−31.2250	−1.28989
$587$	28.8444	1.19054	0.595268	−	0.803527i	$-0.297046\pi$
$587$	28.8444	1.19054	0.595268	+	0.803527i	$0.297046\pi$
$588$	0	0
$589$	0	0
$590$	45.0333	1.85399
$591$	0	0
$592$	8.66025	0.355934
$593$	18.0278	0.740311	0.370156	−	0.928970i	$-0.379304\pi$
$593$	18.0278	0.740311	0.370156	+	0.928970i	$0.379304\pi$
$594$	0	0
$595$	0	0
$596$	15.5885	0.638528
$597$	0	0
$598$	0	0
$599$	−10.0000	−0.408589	−0.204294	−	0.978909i	$-0.565490\pi$
$599$	−10.0000	−0.408589	−0.204294	+	0.978909i	$0.565490\pi$
$600$	0	0
$601$	43.7150	1.78317	0.891586	−	0.452852i	$-0.149593\pi$
$601$	43.7150	1.78317	0.891586	+	0.452852i	$0.149593\pi$
$602$	0	0
$603$	41.5692	1.69283
$604$	3.46410	0.140952
$605$	−3.60555	−0.146587
$606$	0	0
$607$	−37.4700	−1.52086	−0.760430	−	0.649420i	$-0.775012\pi$
$607$	−37.4700	−1.52086	−0.760430	+	0.649420i	$0.775012\pi$
$608$	−37.4700	−1.51961
$609$	0	0
$610$	−39.0000	−1.57906
$611$	0	0
$612$	−18.7350	−0.757317
$613$	1.73205	0.0699569	0.0349784	−	0.999388i	$-0.488864\pi$
$613$	1.73205	0.0699569	0.0349784	+	0.999388i	$0.488864\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.5167	0.906487	0.453243	−	0.891387i	$-0.350267\pi$
$617$	22.5167	0.906487	0.453243	+	0.891387i	$0.350267\pi$
$618$	0	0
$619$	36.0555	1.44919	0.724597	−	0.689173i	$-0.242026\pi$
$619$	36.0555	1.44919	0.724597	+	0.689173i	$0.242026\pi$
$620$	0	0
$621$	0	0
$622$	−43.2666	−1.73483
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	0	0
$627$	0	0
$628$	−6.24500	−0.249203
$629$	−10.8167	−0.431288
$630$	0	0
$631$	13.8564	0.551615	0.275807	−	0.961213i	$-0.411055\pi$
$631$	13.8564	0.551615	0.275807	+	0.961213i	$0.411055\pi$
$632$	10.3923	0.413384
$633$	0	0
$634$	27.0000	1.07231
$635$	64.8999	2.57547
$636$	0	0
$637$	0	0
$638$	6.00000	0.237542
$639$	31.1769	1.23334
$640$	−43.7150	−1.72799
$641$	29.0000	1.14543	0.572716	−	0.819754i	$-0.305890\pi$
$641$	29.0000	1.14543	0.572716	+	0.819754i	$0.305890\pi$
$642$	0	0
$643$	0	0		−	1.00000i	$-0.5\pi$
$643$	0	0			1.00000i	$0.5\pi$
$644$	0	0
$645$	0	0
$646$	78.0000	3.06887
$647$	−12.4900	−0.491032	−0.245516	−	0.969392i	$-0.578957\pi$
$647$	−12.4900	−0.491032	−0.245516	+	0.969392i	$0.578957\pi$
$648$	−15.5885	−0.612372
$649$	−24.9800	−0.980550
$650$	0	0
$651$	0	0
$652$	10.3923	0.406994
$653$	38.0000	1.48705	0.743527	−	0.668705i	$-0.233151\pi$
$653$	38.0000	1.48705	0.743527	+	0.668705i	$0.233151\pi$
$654$	0	0
$655$	45.0333	1.75960
$656$	−18.0278	−0.703866
$657$	−32.4500	−1.26599
$658$	0	0
$659$	−4.00000	−0.155818	−0.0779089	−	0.996960i	$-0.524824\pi$
$659$	−4.00000	−0.155818	−0.0779089	+	0.996960i	$0.524824\pi$
$660$	0	0
$661$	−10.8167	−0.420719	−0.210360	−	0.977624i	$-0.567463\pi$
$661$	−10.8167	−0.420719	−0.210360	+	0.977624i	$0.567463\pi$
$662$	18.0000	0.699590
$663$	0	0
$664$	−12.4900	−0.484706
$665$	0	0
$666$	9.00000	0.348743
$667$	−4.00000	−0.154881
$668$	−7.21110	−0.279006
$669$	0	0
$670$	86.5332	3.34307
$671$	21.6333	0.835145
$672$	0	0
$673$	−3.00000	−0.115642	−0.0578208	−	0.998327i	$-0.518415\pi$
$673$	−3.00000	−0.115642	−0.0578208	+	0.998327i	$0.518415\pi$
$674$	−19.0526	−0.733877
$675$	0	0
$676$	0	0
$677$	−24.9800	−0.960059	−0.480030	−	0.877252i	$-0.659374\pi$
$677$	−24.9800	−0.960059	−0.480030	+	0.877252i	$0.659374\pi$
$678$	0	0
$679$	0	0
$680$	39.0000	1.49558
$681$	0	0
$682$	0	0
$683$	45.0333	1.72315	0.861576	−	0.507628i	$-0.169478\pi$
$683$	45.0333	1.72315	0.861576	+	0.507628i	$0.169478\pi$
$684$	−21.6333	−0.827170
$685$	31.2250	1.19305
$686$	0	0
$687$	0	0
$688$	30.0000	1.14374
$689$	0	0
$690$	0	0
$691$	−28.8444	−1.09729	−0.548647	−	0.836054i	$-0.684857\pi$
$691$	−28.8444	−1.09729	−0.548647	+	0.836054i	$0.684857\pi$
$692$	−24.9800	−0.949597
$693$	0	0
$694$	−48.4974	−1.84094
$695$	45.0333	1.70821
$696$	0	0
$697$	22.5167	0.852879
$698$	−37.4700	−1.41826
$699$	0	0
$700$	0	0
$701$	−14.0000	−0.528773	−0.264386	−	0.964417i	$-0.585169\pi$
$701$	−14.0000	−0.528773	−0.264386	+	0.964417i	$0.585169\pi$
$702$	0	0
$703$	−12.4900	−0.471069
$704$	3.46410	0.130558
$705$	0	0
$706$	43.7150	1.64524
$707$	0	0
$708$	0	0
$709$	−32.9090	−1.23592	−0.617961	−	0.786209i	$-0.712041\pi$
$709$	−32.9090	−1.23592	−0.617961	+	0.786209i	$0.712041\pi$
$710$	64.8999	2.43565
$711$	18.0000	0.675053
$712$	12.4900	0.468082
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−16.0000	−0.597948
$717$	0	0
$718$	−48.0000	−1.79134
$719$	−37.4700	−1.39739	−0.698697	−	0.715417i	$-0.746237\pi$
$719$	−37.4700	−1.39739	−0.698697	+	0.715417i	$0.746237\pi$
$720$	−54.0833	−2.01556
$721$	0	0
$722$	57.1577	2.12719
$723$	0	0
$724$	−6.24500	−0.232094
$725$	8.00000	0.297113
$726$	0	0
$727$	24.9800	0.926457	0.463228	−	0.886239i	$-0.346691\pi$
$727$	24.9800	0.926457	0.463228	+	0.886239i	$0.346691\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	−67.5500	−2.50014
$731$	−37.4700	−1.38588
$732$	0	0
$733$	−10.8167	−0.399522	−0.199761	−	0.979845i	$-0.564017\pi$
$733$	−10.8167	−0.399522	−0.199761	+	0.979845i	$0.564017\pi$
$734$	43.2666	1.59700
$735$	0	0
$736$	20.7846	0.766131
$737$	−48.0000	−1.76810
$738$	−18.7350	−0.689645
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	6.24500	0.229571
$741$	0	0
$742$	0	0
$743$	20.7846	0.762513	0.381257	−	0.924469i	$-0.375491\pi$
$743$	20.7846	0.762513	0.381257	+	0.924469i	$0.375491\pi$
$744$	0	0
$745$	−56.2050	−2.05919
$746$	25.9808	0.951223
$747$	−21.6333	−0.791521
$748$	21.6333	0.790992
$749$	0	0
$750$	0	0
$751$	−42.0000	−1.53260	−0.766301	−	0.642482i	$-0.777905\pi$
$751$	−42.0000	−1.53260	−0.766301	+	0.642482i	$0.777905\pi$
$752$	36.0555	1.31481
$753$	0	0
$754$	0	0
$755$	−12.4900	−0.454557
$756$	0	0
$757$	−14.0000	−0.508839	−0.254419	−	0.967094i	$-0.581884\pi$
$757$	−14.0000	−0.508839	−0.254419	+	0.967094i	$0.581884\pi$
$758$	−36.0000	−1.30758
$759$	0	0
$760$	45.0333	1.63353
$761$	−36.0555	−1.30701	−0.653506	−	0.756922i	$-0.726702\pi$
$761$	−36.0555	−1.30701	−0.653506	+	0.756922i	$0.726702\pi$
$762$	0	0
$763$	0	0
$764$	4.00000	0.144715
$765$	67.5500	2.44227
$766$	62.4500	2.25641
$767$	0	0
$768$	0	0
$769$	21.6333	0.780117	0.390059	−	0.920790i	$-0.372455\pi$
$769$	21.6333	0.780117	0.390059	+	0.920790i	$0.372455\pi$
$770$	0	0
$771$	0	0
$772$	8.66025	0.311689
$773$	−7.21110	−0.259365	−0.129683	−	0.991556i	$-0.541396\pi$
$773$	−7.21110	−0.259365	−0.129683	+	0.991556i	$0.541396\pi$
$774$	31.1769	1.12063
$775$	0	0
$776$	−12.4900	−0.448365
$777$	0	0
$778$	−29.4449	−1.05565
$779$	26.0000	0.931547
$780$	0	0
$781$	−36.0000	−1.28818
$782$	−43.2666	−1.54721
$783$	0	0
$784$	0	0
$785$	22.5167	0.803654
$786$	0	0
$787$	−14.4222	−0.514096	−0.257048	−	0.966399i	$-0.582750\pi$
$787$	−14.4222	−0.514096	−0.257048	+	0.966399i	$0.582750\pi$
$788$	0	0
$789$	0	0
$790$	37.4700	1.33312
$791$	0	0
$792$	18.0000	0.639602
$793$	0	0
$794$	−12.4900	−0.443253
$795$	0	0
$796$	−12.4900	−0.442696
$797$	24.9800	0.884837	0.442418	−	0.896809i	$-0.354121\pi$
$797$	24.9800	0.884837	0.442418	+	0.896809i	$0.354121\pi$
$798$	0	0
$799$	−45.0333	−1.59316
$800$	−41.5692	−1.46969
$801$	21.6333	0.764375
$802$	15.0000	0.529668
$803$	37.4700	1.32229
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	−10.8167	−0.380529
$809$	−7.00000	−0.246107	−0.123053	−	0.992400i	$-0.539269\pi$
$809$	−7.00000	−0.246107	−0.123053	+	0.992400i	$0.539269\pi$
$810$	−56.2050	−1.97484
$811$	43.2666	1.51930	0.759648	−	0.650334i	$-0.225371\pi$
$811$	43.2666	1.51930	0.759648	+	0.650334i	$0.225371\pi$
$812$	0	0
$813$	0	0
$814$	−10.3923	−0.364250
$815$	−37.4700	−1.31252
$816$	0	0
$817$	−43.2666	−1.51371
$818$	43.7150	1.52846
$819$	0	0
$820$	−13.0000	−0.453980
$821$	13.8564	0.483592	0.241796	−	0.970327i	$-0.422264\pi$
$821$	13.8564	0.483592	0.241796	+	0.970327i	$0.422264\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	21.6333	0.753632
$825$	0	0
$826$	0	0
$827$	−17.3205	−0.602293	−0.301147	−	0.953578i	$-0.597369\pi$
$827$	−17.3205	−0.602293	−0.301147	+	0.953578i	$0.597369\pi$
$828$	12.0000	0.417029
$829$	18.7350	0.650693	0.325347	−	0.945595i	$-0.394519\pi$
$829$	18.7350	0.650693	0.325347	+	0.945595i	$0.394519\pi$
$830$	−45.0333	−1.56313
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	26.0000	0.899767
$836$	24.9800	0.863951
$837$	0	0
$838$	0	0
$839$	−50.4777	−1.74268	−0.871342	−	0.490676i	$-0.836750\pi$
$839$	−50.4777	−1.74268	−0.871342	+	0.490676i	$0.836750\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	39.0000	1.34403
$843$	0	0
$844$	−20.0000	−0.688428
$845$	0	0
$846$	37.4700	1.28824
$847$	0	0
$848$	−25.0000	−0.858504
$849$	0	0
$850$	86.5332	2.96807
$851$	6.92820	0.237496
$852$	0	0
$853$	25.2389	0.864162	0.432081	−	0.901835i	$-0.357779\pi$
$853$	25.2389	0.864162	0.432081	+	0.901835i	$0.357779\pi$
$854$	0	0
$855$	78.0000	2.66754
$856$	−3.46410	−0.118401
$857$	31.2250	1.06663	0.533313	−	0.845918i	$-0.320947\pi$
$857$	31.2250	1.06663	0.533313	+	0.845918i	$0.320947\pi$
$858$	0	0
$859$	−49.9600	−1.70461	−0.852306	−	0.523043i	$-0.824797\pi$
$859$	−49.9600	−1.70461	−0.852306	+	0.523043i	$0.824797\pi$
$860$	21.6333	0.737690
$861$	0	0
$862$	48.0000	1.63489
$863$	−38.1051	−1.29711	−0.648557	−	0.761166i	$-0.724627\pi$
$863$	−38.1051	−1.29711	−0.648557	+	0.761166i	$0.724627\pi$
$864$	0	0
$865$	90.0666	3.06236
$866$	32.4500	1.10269
$867$	0	0
$868$	0	0
$869$	−20.7846	−0.705070
$870$	0	0
$871$	0	0
$872$	12.0000	0.406371
$873$	−21.6333	−0.732177
$874$	−49.9600	−1.68992
$875$	0	0
$876$	0	0
$877$	−46.7654	−1.57915	−0.789577	−	0.613651i	$-0.789700\pi$
$877$	−46.7654	−1.57915	−0.789577	+	0.613651i	$0.789700\pi$
$878$	21.6333	0.730089
$879$	0	0
$880$	62.4500	2.10519
$881$	−43.7150	−1.47280	−0.736398	−	0.676549i	$-0.763475\pi$
$881$	−43.7150	−1.47280	−0.736398	+	0.676549i	$0.763475\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	0	0
$886$	27.7128	0.931030
$887$	−24.9800	−0.838746	−0.419373	−	0.907814i	$-0.637750\pi$
$887$	−24.9800	−0.838746	−0.419373	+	0.907814i	$0.637750\pi$
$888$	0	0
$889$	0	0
$890$	45.0333	1.50952
$891$	31.1769	1.04447
$892$	7.21110	0.241446
$893$	−52.0000	−1.74011
$894$	0	0
$895$	57.6888	1.92832
$896$	0	0
$897$	0	0
$898$	−12.0000	−0.400445
$899$	0	0
$900$	−24.0000	−0.800000
$901$	31.2250	1.04026
$902$	21.6333	0.720310
$903$	0	0
$904$	−8.66025	−0.288036
$905$	22.5167	0.748479
$906$	0	0
$907$	22.0000	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$907$	22.0000	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$908$	−14.4222	−0.478618
$909$	−18.7350	−0.621401
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	24.9800	0.826717
$914$	69.0000	2.28232
$915$	0	0
$916$	21.6333	0.714785
$917$	0	0
$918$	0	0
$919$	−2.00000	−0.0659739	−0.0329870	−	0.999456i	$-0.510502\pi$
$919$	−2.00000	−0.0659739	−0.0329870	+	0.999456i	$0.510502\pi$
$920$	−24.9800	−0.823566
$921$	0	0
$922$	−68.6950	−2.26235
$923$	0	0
$924$	0	0
$925$	−13.8564	−0.455596
$926$	18.0000	0.591517
$927$	37.4700	1.23068
$928$	−5.19615	−0.170572
$929$	−25.2389	−0.828060	−0.414030	−	0.910263i	$-0.635879\pi$
$929$	−25.2389	−0.828060	−0.414030	+	0.910263i	$0.635879\pi$
$930$	0	0
$931$	0	0
$932$	−2.00000	−0.0655122
$933$	0	0
$934$	21.6333	0.707863
$935$	−78.0000	−2.55087
$936$	0	0
$937$	−6.24500	−0.204015	−0.102008	−	0.994784i	$-0.532527\pi$
$937$	−6.24500	−0.204015	−0.102008	+	0.994784i	$0.532527\pi$
$938$	0	0
$939$	0	0
$940$	26.0000	0.848026
$941$	7.21110	0.235075	0.117538	−	0.993068i	$-0.462500\pi$
$941$	7.21110	0.235075	0.117538	+	0.993068i	$0.462500\pi$
$942$	0	0
$943$	−14.4222	−0.469652
$944$	36.0555	1.17351
$945$	0	0
$946$	−36.0000	−1.17046
$947$	−20.7846	−0.675409	−0.337705	−	0.941252i	$-0.609650\pi$
$947$	−20.7846	−0.675409	−0.337705	+	0.941252i	$0.609650\pi$
$948$	0	0
$949$	0	0
$950$	99.9200	3.24183
$951$	0	0
$952$	0	0
$953$	10.0000	0.323932	0.161966	−	0.986796i	$-0.448217\pi$
$953$	10.0000	0.323932	0.161966	+	0.986796i	$0.448217\pi$
$954$	−25.9808	−0.841158
$955$	−14.4222	−0.466692
$956$	−17.3205	−0.560185
$957$	0	0
$958$	−24.9800	−0.807067
$959$	0	0
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−6.00000	−0.193347
$964$	10.8167	0.348381
$965$	−31.2250	−1.00517
$966$	0	0
$967$	48.4974	1.55957	0.779786	−	0.626046i	$-0.215328\pi$
$967$	48.4974	1.55957	0.779786	+	0.626046i	$0.215328\pi$
$968$	−1.73205	−0.0556702
$969$	0	0
$970$	−45.0333	−1.44593
$971$	−12.4900	−0.400823	−0.200412	−	0.979712i	$-0.564228\pi$
$971$	−12.4900	−0.400823	−0.200412	+	0.979712i	$0.564228\pi$
$972$	0	0
$973$	0	0
$974$	−36.0000	−1.15351
$975$	0	0
$976$	−31.2250	−0.999488
$977$	−5.19615	−0.166240	−0.0831198	−	0.996540i	$-0.526488\pi$
$977$	−5.19615	−0.166240	−0.0831198	+	0.996540i	$0.526488\pi$
$978$	0	0
$979$	−24.9800	−0.798364
$980$	0	0
$981$	20.7846	0.663602
$982$	6.92820	0.221088
$983$	−50.4777	−1.60999	−0.804995	−	0.593282i	$-0.797832\pi$
$983$	−50.4777	−1.60999	−0.804995	+	0.593282i	$0.797832\pi$
$984$	0	0
$985$	0	0
$986$	10.8167	0.344472
$987$	0	0
$988$	0	0
$989$	24.0000	0.763156
$990$	64.8999	2.06265
$991$	−12.0000	−0.381193	−0.190596	−	0.981669i	$-0.561042\pi$
$991$	−12.0000	−0.381193	−0.190596	+	0.981669i	$0.561042\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	45.0333	1.42765
$996$	0	0
$997$	−31.2250	−0.988905	−0.494453	−	0.869205i	$-0.664631\pi$
$997$	−31.2250	−0.988905	−0.494453	+	0.869205i	$0.664631\pi$
$998$	−48.0000	−1.51941
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.br.1.3		4
7.6	odd	2	inner	8281.2.a.br.1.4		4
13.6	odd	12		637.2.q.d.491.2	yes	4
13.11	odd	12		637.2.q.d.589.1	yes	4
13.12	even	2	inner	8281.2.a.br.1.2		4
91.6	even	12		637.2.q.d.491.1	✓	4
91.11	odd	12		637.2.u.f.30.2		4
91.19	even	12		637.2.u.f.361.1		4
91.24	even	12		637.2.u.f.30.1		4
91.32	odd	12		637.2.k.e.569.1		4
91.37	odd	12		637.2.k.e.459.1		4
91.45	even	12		637.2.k.e.569.2		4
91.58	odd	12		637.2.u.f.361.2		4
91.76	even	12		637.2.q.d.589.2	yes	4
91.89	even	12		637.2.k.e.459.2		4
91.90	odd	2	inner	8281.2.a.br.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
637.2.k.e.459.1		4	91.37	odd	12
637.2.k.e.459.2		4	91.89	even	12
637.2.k.e.569.1		4	91.32	odd	12
637.2.k.e.569.2		4	91.45	even	12
637.2.q.d.491.1	✓	4	91.6	even	12
637.2.q.d.491.2	yes	4	13.6	odd	12
637.2.q.d.589.1	yes	4	13.11	odd	12
637.2.q.d.589.2	yes	4	91.76	even	12
637.2.u.f.30.1		4	91.24	even	12
637.2.u.f.30.2		4	91.11	odd	12
637.2.u.f.361.1		4	91.19	even	12
637.2.u.f.361.2		4	91.58	odd	12
8281.2.a.br.1.1		4	91.90	odd	2	inner
8281.2.a.br.1.2		4	13.12	even	2	inner
8281.2.a.br.1.3		4	1.1	even	1	trivial
8281.2.a.br.1.4		4	7.6	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+1.73205 q^{2} +1.00000 q^{4} -3.60555 q^{5} -1.73205 q^{8} -3.00000 q^{9} -6.24500 q^{10} +3.46410 q^{11} -5.00000 q^{16} +6.24500 q^{17} -5.19615 q^{18} +7.21110 q^{19} -3.60555 q^{20} +6.00000 q^{22} -4.00000 q^{23} +8.00000 q^{25} +1.00000 q^{29} -5.19615 q^{32} +10.8167 q^{34} -3.00000 q^{36} -1.73205 q^{37} +12.4900 q^{38} +6.24500 q^{40} +3.60555 q^{41} -6.00000 q^{43} +3.46410 q^{44} +10.8167 q^{45} -6.92820 q^{46} -7.21110 q^{47} +13.8564 q^{50} +5.00000 q^{53} -12.4900 q^{55} +1.73205 q^{58} -7.21110 q^{59} +6.24500 q^{61} +1.00000 q^{64} -13.8564 q^{67} +6.24500 q^{68} -10.3923 q^{71} +5.19615 q^{72} +10.8167 q^{73} -3.00000 q^{74} +7.21110 q^{76} -6.00000 q^{79} +18.0278 q^{80} +9.00000 q^{81} +6.24500 q^{82} +7.21110 q^{83} -22.5167 q^{85} -10.3923 q^{86} -6.00000 q^{88} -7.21110 q^{89} +18.7350 q^{90} -4.00000 q^{92} -12.4900 q^{94} -26.0000 q^{95} +7.21110 q^{97} -10.3923 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{4} - 12 q^{9} - 20 q^{16} + 24 q^{22} - 16 q^{23} + 32 q^{25} + 4 q^{29} - 12 q^{36} - 24 q^{43} + 20 q^{53} + 4 q^{64} - 12 q^{74} - 24 q^{79} + 36 q^{81} - 24 q^{88} - 16 q^{92} - 104 q^{95}+O(q^{100}) \) 4 * q + 4 * q^4 - 12 * q^9 - 20 * q^16 + 24 * q^22 - 16 * q^23 + 32 * q^25 + 4 * q^29 - 12 * q^36 - 24 * q^43 + 20 * q^53 + 4 * q^64 - 12 * q^74 - 24 * q^79 + 36 * q^81 - 24 * q^88 - 16 * q^92 - 104 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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