LMFDB - Embedded newform 3520.2.a.bz.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3520,2,Mod(1,3520)] 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("3520.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3520 = 2^{6} \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3520.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,0,0,-5,0,2,0,11,0,-5,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$28.1073415115$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.792644.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 4x - 2 $ x^5 - x^4 - 7x^3 + 4x^2 + 4*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1760)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.466307$ of defining polynomial
Character	$\chi$	$=$	3520.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.94897 q^{3} -1.00000 q^{5} +3.88159 q^{7} +5.69645 q^{9} -1.00000 q^{11} +3.81764 q^{13} +2.94897 q^{15} -3.62906 q^{17} -4.69645 q^{19} -11.4467 q^{21} -8.49773 q^{23} +1.00000 q^{25} -7.95176 q^{27} -4.05381 q^{29} +7.44670 q^{31} +2.94897 q^{33} -3.88159 q^{35} +8.59440 q^{37} -11.2581 q^{39} +0.134771 q^{41} -6.71281 q^{43} -5.69645 q^{45} -5.00278 q^{47} +8.06673 q^{49} +10.7020 q^{51} +8.59440 q^{53} +1.00000 q^{55} +13.8497 q^{57} +3.76318 q^{59} +0.0538074 q^{61} +22.1113 q^{63} -3.81764 q^{65} +12.5304 q^{67} +25.0596 q^{69} -15.6073 q^{71} -5.68287 q^{73} -2.94897 q^{75} -3.88159 q^{77} -5.89795 q^{79} +6.36018 q^{81} -10.8204 q^{83} +3.62906 q^{85} +11.9546 q^{87} +13.0994 q^{89} +14.8185 q^{91} -21.9601 q^{93} +4.69645 q^{95} +13.2581 q^{97} -5.69645 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{5} + 2 q^{7} + 11 q^{9} - 5 q^{11} - 6 q^{13} + 2 q^{17} - 6 q^{19} - 12 q^{21} - 12 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{31} - 2 q^{35} - 4 q^{37} - 16 q^{39} + 6 q^{41} - 4 q^{43} - 11 q^{45}+ \cdots - 11 q^{99}+O(q^{100}) $ 5 * q - 5 * q^5 + 2 * q^7 + 11 * q^9 - 5 * q^11 - 6 * q^13 + 2 * q^17 - 6 * q^19 - 12 * q^21 - 12 * q^23 + 5 * q^25 - 10 * q^29 - 8 * q^31 - 2 * q^35 - 4 * q^37 - 16 * q^39 + 6 * q^41 - 4 * q^43 - 11 * q^45 + 23 * q^49 - 14 * q^51 - 4 * q^53 + 5 * q^55 - 16 * q^59 - 10 * q^61 - 4 * q^63 + 6 * q^65 + 8 * q^67 - 24 * q^69 - 24 * q^71 + 2 * q^73 - 2 * q^77 + 21 * q^81 - 4 * q^83 - 2 * q^85 + 2 * q^87 + 24 * q^89 - 20 * q^91 - 2 * q^93 + 6 * q^95 + 26 * q^97 - 11 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−2.94897	−1.70259	−0.851295	−	0.524687i	$-0.824183\pi$
$3$	−2.94897	−1.70259	−0.851295	+	0.524687i	$0.824183\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.88159	1.46710	0.733551	−	0.679634i	$-0.237861\pi$
$7$	3.88159	1.46710	0.733551	+	0.679634i	$0.237861\pi$
$8$	0	0
$9$	5.69645	1.89882
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	3.81764	1.05882	0.529412	−	0.848365i	$-0.322413\pi$
$13$	3.81764	1.05882	0.529412	+	0.848365i	$0.322413\pi$
$14$	0	0
$15$	2.94897	0.761422
$16$	0	0
$17$	−3.62906	−0.880177	−0.440088	−	0.897954i	$-0.645053\pi$
$17$	−3.62906	−0.880177	−0.440088	+	0.897954i	$0.645053\pi$
$18$	0	0
$19$	−4.69645	−1.07744	−0.538720	−	0.842485i	$-0.681092\pi$
$19$	−4.69645	−1.07744	−0.538720	+	0.842485i	$0.681092\pi$
$20$	0	0
$21$	−11.4467	−2.49788
$22$	0	0
$23$	−8.49773	−1.77190	−0.885950	−	0.463782i	$-0.846492\pi$
$23$	−8.49773	−1.77190	−0.885950	+	0.463782i	$0.846492\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−7.95176	−1.53032
$28$	0	0
$29$	−4.05381	−0.752773	−0.376387	−	0.926463i	$-0.622834\pi$
$29$	−4.05381	−0.752773	−0.376387	+	0.926463i	$0.622834\pi$
$30$	0	0
$31$	7.44670	1.33747	0.668734	−	0.743502i	$-0.266837\pi$
$31$	7.44670	1.33747	0.668734	+	0.743502i	$0.266837\pi$
$32$	0	0
$33$	2.94897	0.513350
$34$	0	0
$35$	−3.88159	−0.656108
$36$	0	0
$37$	8.59440	1.41291	0.706455	−	0.707758i	$-0.250293\pi$
$37$	8.59440	1.41291	0.706455	+	0.707758i	$0.250293\pi$
$38$	0	0
$39$	−11.2581	−1.80274
$40$	0	0
$41$	0.134771	0.0210477	0.0105239	−	0.999945i	$-0.496650\pi$
$41$	0.134771	0.0210477	0.0105239	+	0.999945i	$0.496650\pi$
$42$	0	0
$43$	−6.71281	−1.02369	−0.511847	−	0.859077i	$-0.671038\pi$
$43$	−6.71281	−1.02369	−0.511847	+	0.859077i	$0.671038\pi$
$44$	0	0
$45$	−5.69645	−0.849176
$46$	0	0
$47$	−5.00278	−0.729731	−0.364865	−	0.931060i	$-0.618885\pi$
$47$	−5.00278	−0.729731	−0.364865	+	0.931060i	$0.618885\pi$
$48$	0	0
$49$	8.06673	1.15239
$50$	0	0
$51$	10.7020	1.49858
$52$	0	0
$53$	8.59440	1.18053	0.590266	−	0.807209i	$-0.299023\pi$
$53$	8.59440	1.18053	0.590266	+	0.807209i	$0.299023\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0	0
$57$	13.8497	1.83444
$58$	0	0
$59$	3.76318	0.489924	0.244962	−	0.969533i	$-0.421225\pi$
$59$	3.76318	0.489924	0.244962	+	0.969533i	$0.421225\pi$
$60$	0	0
$61$	0.0538074	0.00688934	0.00344467	−	0.999994i	$-0.498904\pi$
$61$	0.0538074	0.00688934	0.00344467	+	0.999994i	$0.498904\pi$
$62$	0	0
$63$	22.1113	2.78576
$64$	0	0
$65$	−3.81764	−0.473520
$66$	0	0
$67$	12.5304	1.53084	0.765419	−	0.643532i	$-0.222532\pi$
$67$	12.5304	1.53084	0.765419	+	0.643532i	$0.222532\pi$
$68$	0	0
$69$	25.0596	3.01682
$70$	0	0
$71$	−15.6073	−1.85225	−0.926124	−	0.377219i	$-0.876880\pi$
$71$	−15.6073	−1.85225	−0.926124	+	0.377219i	$0.876880\pi$
$72$	0	0
$73$	−5.68287	−0.665130	−0.332565	−	0.943080i	$-0.607914\pi$
$73$	−5.68287	−0.665130	−0.332565	+	0.943080i	$0.607914\pi$
$74$	0	0
$75$	−2.94897	−0.340518
$76$	0	0
$77$	−3.88159	−0.442348
$78$	0	0
$79$	−5.89795	−0.663571	−0.331786	−	0.943355i	$-0.607651\pi$
$79$	−5.89795	−0.663571	−0.331786	+	0.943355i	$0.607651\pi$
$80$	0	0
$81$	6.36018	0.706686
$82$	0	0
$83$	−10.8204	−1.18770	−0.593848	−	0.804577i	$-0.702392\pi$
$83$	−10.8204	−1.18770	−0.593848	+	0.804577i	$0.702392\pi$
$84$	0	0
$85$	3.62906	0.393627
$86$	0	0
$87$	11.9546	1.28166
$88$	0	0
$89$	13.0994	1.38854	0.694269	−	0.719715i	$-0.255728\pi$
$89$	13.0994	1.38854	0.694269	+	0.719715i	$0.255728\pi$
$90$	0	0
$91$	14.8185	1.55340
$92$	0	0
$93$	−21.9601	−2.27716
$94$	0	0
$95$	4.69645	0.481845
$96$	0	0
$97$	13.2581	1.34616	0.673079	−	0.739570i	$-0.264971\pi$
$97$	13.2581	1.34616	0.673079	+	0.739570i	$0.264971\pi$
$98$	0	0
$99$	−5.69645	−0.572515
$100$	0	0
$101$	−15.6611	−1.55834	−0.779170	−	0.626813i	$-0.784359\pi$
$101$	−15.6611	−1.55834	−0.779170	+	0.626813i	$0.784359\pi$
$102$	0	0
$103$	2.76040	0.271990	0.135995	−	0.990710i	$-0.456577\pi$
$103$	2.76040	0.271990	0.135995	+	0.990710i	$0.456577\pi$
$104$	0	0
$105$	11.4467	1.11708
$106$	0	0
$107$	12.3739	1.19623	0.598117	−	0.801409i	$-0.295916\pi$
$107$	12.3739	1.19623	0.598117	+	0.801409i	$0.295916\pi$
$108$	0	0
$109$	−8.13477	−0.779170	−0.389585	−	0.920991i	$-0.627382\pi$
$109$	−8.13477	−0.779170	−0.389585	+	0.920991i	$0.627382\pi$
$110$	0	0
$111$	−25.3447	−2.40561
$112$	0	0
$113$	3.23228	0.304068	0.152034	−	0.988375i	$-0.451418\pi$
$113$	3.23228	0.304068	0.152034	+	0.988375i	$0.451418\pi$
$114$	0	0
$115$	8.49773	0.792417
$116$	0	0
$117$	21.7470	2.01051
$118$	0	0
$119$	−14.0865	−1.29131
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−0.397437	−0.0358357
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.7128	−0.950608	−0.475304	−	0.879822i	$-0.657662\pi$
$127$	−10.7128	−0.950608	−0.475304	+	0.879822i	$0.657662\pi$
$128$	0	0
$129$	19.7959	1.74293
$130$	0	0
$131$	−4.53583	−0.396298	−0.198149	−	0.980172i	$-0.563493\pi$
$131$	−4.53583	−0.396298	−0.198149	+	0.980172i	$0.563493\pi$
$132$	0	0
$133$	−18.2297	−1.58071
$134$	0	0
$135$	7.95176	0.684378
$136$	0	0
$137$	−0.242386	−0.0207084	−0.0103542	−	0.999946i	$-0.503296\pi$
$137$	−0.242386	−0.0207084	−0.0103542	+	0.999946i	$0.503296\pi$
$138$	0	0
$139$	−19.3657	−1.64258	−0.821290	−	0.570511i	$-0.806745\pi$
$139$	−19.3657	−1.64258	−0.821290	+	0.570511i	$0.806745\pi$
$140$	0	0
$141$	14.7531	1.24243
$142$	0	0
$143$	−3.81764	−0.319247
$144$	0	0
$145$	4.05381	0.336650
$146$	0	0
$147$	−23.7886	−1.96205
$148$	0	0
$149$	0.214421	0.0175661	0.00878304	−	0.999961i	$-0.497204\pi$
$149$	0.214421	0.0175661	0.00878304	+	0.999961i	$0.497204\pi$
$150$	0	0
$151$	−17.5332	−1.42683	−0.713417	−	0.700740i	$-0.752853\pi$
$151$	−17.5332	−1.42683	−0.713417	+	0.700740i	$0.752853\pi$
$152$	0	0
$153$	−20.6728	−1.67129
$154$	0	0
$155$	−7.44670	−0.598134
$156$	0	0
$157$	−18.5672	−1.48183	−0.740914	−	0.671600i	$-0.765607\pi$
$157$	−18.5672	−1.48183	−0.740914	+	0.671600i	$0.765607\pi$
$158$	0	0
$159$	−25.3447	−2.00996
$160$	0	0
$161$	−32.9847	−2.59956
$162$	0	0
$163$	5.05103	0.395627	0.197813	−	0.980240i	$-0.436616\pi$
$163$	5.05103	0.395627	0.197813	+	0.980240i	$0.436616\pi$
$164$	0	0
$165$	−2.94897	−0.229577
$166$	0	0
$167$	3.72097	0.287938	0.143969	−	0.989582i	$-0.454013\pi$
$167$	3.72097	0.287938	0.143969	+	0.989582i	$0.454013\pi$
$168$	0	0
$169$	1.57438	0.121106
$170$	0	0
$171$	−26.7531	−2.04586
$172$	0	0
$173$	1.88697	0.143464	0.0717320	−	0.997424i	$-0.477147\pi$
$173$	1.88697	0.143464	0.0717320	+	0.997424i	$0.477147\pi$
$174$	0	0
$175$	3.88159	0.293421
$176$	0	0
$177$	−11.0975	−0.834140
$178$	0	0
$179$	−7.49495	−0.560199	−0.280099	−	0.959971i	$-0.590367\pi$
$179$	−7.49495	−0.560199	−0.280099	+	0.959971i	$0.590367\pi$
$180$	0	0
$181$	17.1302	1.27328	0.636640	−	0.771161i	$-0.280324\pi$
$181$	17.1302	1.27328	0.636640	+	0.771161i	$0.280324\pi$
$182$	0	0
$183$	−0.158677	−0.0117297
$184$	0	0
$185$	−8.59440	−0.631873
$186$	0	0
$187$	3.62906	0.265383
$188$	0	0
$189$	−30.8654	−2.24513
$190$	0	0
$191$	−23.7959	−1.72181	−0.860905	−	0.508765i	$-0.830102\pi$
$191$	−23.7959	−1.72181	−0.860905	+	0.508765i	$0.830102\pi$
$192$	0	0
$193$	−15.4250	−1.11031	−0.555156	−	0.831746i	$-0.687342\pi$
$193$	−15.4250	−1.11031	−0.555156	+	0.831746i	$0.687342\pi$
$194$	0	0
$195$	11.2581	0.806211
$196$	0	0
$197$	−7.44048	−0.530113	−0.265056	−	0.964233i	$-0.585391\pi$
$197$	−7.44048	−0.530113	−0.265056	+	0.964233i	$0.585391\pi$
$198$	0	0
$199$	16.9799	1.20368	0.601838	−	0.798618i	$-0.294435\pi$
$199$	16.9799	1.20368	0.601838	+	0.798618i	$0.294435\pi$
$200$	0	0
$201$	−36.9520	−2.60639
$202$	0	0
$203$	−15.7352	−1.10440
$204$	0	0
$205$	−0.134771	−0.00941284
$206$	0	0
$207$	−48.4069	−3.36451
$208$	0	0
$209$	4.69645	0.324860
$210$	0	0
$211$	−24.3317	−1.67507	−0.837533	−	0.546387i	$-0.816003\pi$
$211$	−24.3317	−1.67507	−0.837533	+	0.546387i	$0.816003\pi$
$212$	0	0
$213$	46.0256	3.15362
$214$	0	0
$215$	6.71281	0.457810
$216$	0	0
$217$	28.9050	1.96220
$218$	0	0
$219$	16.7586	1.13244
$220$	0	0
$221$	−13.8545	−0.931952
$222$	0	0
$223$	9.27232	0.620921	0.310460	−	0.950586i	$-0.399517\pi$
$223$	9.27232	0.620921	0.310460	+	0.950586i	$0.399517\pi$
$224$	0	0
$225$	5.69645	0.379763
$226$	0	0
$227$	23.6321	1.56851	0.784257	−	0.620436i	$-0.213044\pi$
$227$	23.6321	1.56851	0.784257	+	0.620436i	$0.213044\pi$
$228$	0	0
$229$	−7.70461	−0.509135	−0.254568	−	0.967055i	$-0.581933\pi$
$229$	−7.70461	−0.509135	−0.254568	+	0.967055i	$0.581933\pi$
$230$	0	0
$231$	11.4467	0.753138
$232$	0	0
$233$	14.5225	0.951399	0.475699	−	0.879608i	$-0.342195\pi$
$233$	14.5225	0.951399	0.475699	+	0.879608i	$0.342195\pi$
$234$	0	0
$235$	5.00278	0.326345
$236$	0	0
$237$	17.3929	1.12979
$238$	0	0
$239$	7.92379	0.512547	0.256274	−	0.966604i	$-0.417505\pi$
$239$	7.92379	0.512547	0.256274	+	0.966604i	$0.417505\pi$
$240$	0	0
$241$	−15.9307	−1.02618	−0.513092	−	0.858333i	$-0.671500\pi$
$241$	−15.9307	−1.02618	−0.513092	+	0.858333i	$0.671500\pi$
$242$	0	0
$243$	5.09927	0.327118
$244$	0	0
$245$	−8.06673	−0.515364
$246$	0	0
$247$	−17.9294	−1.14082
$248$	0	0
$249$	31.9091	2.02216
$250$	0	0
$251$	13.7687	0.869075	0.434538	−	0.900654i	$-0.356912\pi$
$251$	13.7687	0.869075	0.434538	+	0.900654i	$0.356912\pi$
$252$	0	0
$253$	8.49773	0.534248
$254$	0	0
$255$	−10.7020	−0.670186
$256$	0	0
$257$	−8.02584	−0.500638	−0.250319	−	0.968163i	$-0.580536\pi$
$257$	−8.02584	−0.500638	−0.250319	+	0.968163i	$0.580536\pi$
$258$	0	0
$259$	33.3599	2.07288
$260$	0	0
$261$	−23.0923	−1.42938
$262$	0	0
$263$	15.2800	0.942208	0.471104	−	0.882078i	$-0.343856\pi$
$263$	15.2800	0.942208	0.471104	+	0.882078i	$0.343856\pi$
$264$	0	0
$265$	−8.59440	−0.527950
$266$	0	0
$267$	−38.6299	−2.36411
$268$	0	0
$269$	−10.0327	−0.611706	−0.305853	−	0.952079i	$-0.598942\pi$
$269$	−10.0327	−0.611706	−0.305853	+	0.952079i	$0.598942\pi$
$270$	0	0
$271$	9.15607	0.556192	0.278096	−	0.960553i	$-0.410297\pi$
$271$	9.15607	0.556192	0.278096	+	0.960553i	$0.410297\pi$
$272$	0	0
$273$	−43.6994	−2.64481
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−15.3453	−0.922010	−0.461005	−	0.887397i	$-0.652511\pi$
$277$	−15.3453	−0.922010	−0.461005	+	0.887397i	$0.652511\pi$
$278$	0	0
$279$	42.4198	2.53960
$280$	0	0
$281$	−19.6611	−1.17288	−0.586442	−	0.809991i	$-0.699472\pi$
$281$	−19.6611	−1.17288	−0.586442	+	0.809991i	$0.699472\pi$
$282$	0	0
$283$	−9.81032	−0.583163	−0.291581	−	0.956546i	$-0.594181\pi$
$283$	−9.81032	−0.583163	−0.291581	+	0.956546i	$0.594181\pi$
$284$	0	0
$285$	−13.8497	−0.820386
$286$	0	0
$287$	0.523127	0.0308792
$288$	0	0
$289$	−3.82991	−0.225289
$290$	0	0
$291$	−39.0979	−2.29196
$292$	0	0
$293$	−21.0051	−1.22713	−0.613566	−	0.789643i	$-0.710266\pi$
$293$	−21.0051	−1.22713	−0.613566	+	0.789643i	$0.710266\pi$
$294$	0	0
$295$	−3.76318	−0.219101
$296$	0	0
$297$	7.95176	0.461408
$298$	0	0
$299$	−32.4413	−1.87613
$300$	0	0
$301$	−26.0564	−1.50186
$302$	0	0
$303$	46.1843	2.65322
$304$	0	0
$305$	−0.0538074	−0.00308100
$306$	0	0
$307$	20.6857	1.18059	0.590296	−	0.807187i	$-0.299011\pi$
$307$	20.6857	1.18059	0.590296	+	0.807187i	$0.299011\pi$
$308$	0	0
$309$	−8.14033	−0.463087
$310$	0	0
$311$	−6.90893	−0.391770	−0.195885	−	0.980627i	$-0.562758\pi$
$311$	−6.90893	−0.391770	−0.195885	+	0.980627i	$0.562758\pi$
$312$	0	0
$313$	17.5918	0.994347	0.497173	−	0.867651i	$-0.334371\pi$
$313$	17.5918	0.994347	0.497173	+	0.867651i	$0.334371\pi$
$314$	0	0
$315$	−22.1113	−1.24583
$316$	0	0
$317$	18.8368	1.05798	0.528990	−	0.848628i	$-0.322571\pi$
$317$	18.8368	1.05798	0.528990	+	0.848628i	$0.322571\pi$
$318$	0	0
$319$	4.05381	0.226970
$320$	0	0
$321$	−36.4904	−2.03670
$322$	0	0
$323$	17.0437	0.948337
$324$	0	0
$325$	3.81764	0.211765
$326$	0	0
$327$	23.9892	1.32661
$328$	0	0
$329$	−19.4187	−1.07059
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	0	0
$333$	48.9575	2.68286
$334$	0	0
$335$	−12.5304	−0.684612
$336$	0	0
$337$	−34.8437	−1.89806	−0.949029	−	0.315190i	$-0.897932\pi$
$337$	−34.8437	−1.89806	−0.949029	+	0.315190i	$0.897932\pi$
$338$	0	0
$339$	−9.53192	−0.517703
$340$	0	0
$341$	−7.44670	−0.403262
$342$	0	0
$343$	4.14060	0.223571
$344$	0	0
$345$	−25.0596	−1.34916
$346$	0	0
$347$	19.7410	1.05975	0.529876	−	0.848075i	$-0.322239\pi$
$347$	19.7410	1.05975	0.529876	+	0.848075i	$0.322239\pi$
$348$	0	0
$349$	−23.8217	−1.27515	−0.637574	−	0.770389i	$-0.720062\pi$
$349$	−23.8217	−1.27515	−0.637574	+	0.770389i	$0.720062\pi$
$350$	0	0
$351$	−30.3569	−1.62033
$352$	0	0
$353$	−15.7701	−0.839355	−0.419678	−	0.907673i	$-0.637857\pi$
$353$	−15.7701	−0.839355	−0.419678	+	0.907673i	$0.637857\pi$
$354$	0	0
$355$	15.6073	0.828350
$356$	0	0
$357$	41.5408	2.19857
$358$	0	0
$359$	−0.235509	−0.0124297	−0.00621485	−	0.999981i	$-0.501978\pi$
$359$	−0.235509	−0.0124297	−0.00621485	+	0.999981i	$0.501978\pi$
$360$	0	0
$361$	3.05662	0.160875
$362$	0	0
$363$	−2.94897	−0.154781
$364$	0	0
$365$	5.68287	0.297455
$366$	0	0
$367$	−8.89517	−0.464324	−0.232162	−	0.972677i	$-0.574580\pi$
$367$	−8.89517	−0.464324	−0.232162	+	0.972677i	$0.574580\pi$
$368$	0	0
$369$	0.767718	0.0399658
$370$	0	0
$371$	33.3599	1.73196
$372$	0	0
$373$	−16.8988	−0.874988	−0.437494	−	0.899221i	$-0.644134\pi$
$373$	−16.8988	−0.874988	−0.437494	+	0.899221i	$0.644134\pi$
$374$	0	0
$375$	2.94897	0.152284
$376$	0	0
$377$	−15.4760	−0.797054
$378$	0	0
$379$	−26.3449	−1.35324	−0.676622	−	0.736330i	$-0.736557\pi$
$379$	−26.3449	−1.35324	−0.676622	+	0.736330i	$0.736557\pi$
$380$	0	0
$381$	31.5918	1.61850
$382$	0	0
$383$	19.6796	1.00558	0.502791	−	0.864408i	$-0.332306\pi$
$383$	19.6796	1.00558	0.502791	+	0.864408i	$0.332306\pi$
$384$	0	0
$385$	3.88159	0.197824
$386$	0	0
$387$	−38.2392	−1.94381
$388$	0	0
$389$	14.4099	0.730610	0.365305	−	0.930888i	$-0.380965\pi$
$389$	14.4099	0.730610	0.365305	+	0.930888i	$0.380965\pi$
$390$	0	0
$391$	30.8388	1.55958
$392$	0	0
$393$	13.3761	0.674733
$394$	0	0
$395$	5.89795	0.296758
$396$	0	0
$397$	−28.5817	−1.43447	−0.717237	−	0.696829i	$-0.754593\pi$
$397$	−28.5817	−1.43447	−0.717237	+	0.696829i	$0.754593\pi$
$398$	0	0
$399$	53.7588	2.69131
$400$	0	0
$401$	−5.62478	−0.280888	−0.140444	−	0.990089i	$-0.544853\pi$
$401$	−5.62478	−0.280888	−0.140444	+	0.990089i	$0.544853\pi$
$402$	0	0
$403$	28.4288	1.41614
$404$	0	0
$405$	−6.36018	−0.316040
$406$	0	0
$407$	−8.59440	−0.426008
$408$	0	0
$409$	−1.09751	−0.0542684	−0.0271342	−	0.999632i	$-0.508638\pi$
$409$	−1.09751	−0.0542684	−0.0271342	+	0.999632i	$0.508638\pi$
$410$	0	0
$411$	0.714791	0.0352580
$412$	0	0
$413$	14.6071	0.718769
$414$	0	0
$415$	10.8204	0.531154
$416$	0	0
$417$	57.1091	2.79664
$418$	0	0
$419$	16.5051	0.806325	0.403162	−	0.915129i	$-0.367911\pi$
$419$	16.5051	0.806325	0.403162	+	0.915129i	$0.367911\pi$
$420$	0	0
$421$	−1.89239	−0.0922292	−0.0461146	−	0.998936i	$-0.514684\pi$
$421$	−1.89239	−0.0922292	−0.0461146	+	0.998936i	$0.514684\pi$
$422$	0	0
$423$	−28.4981	−1.38562
$424$	0	0
$425$	−3.62906	−0.176035
$426$	0	0
$427$	0.208858	0.0101074
$428$	0	0
$429$	11.2581	0.543547
$430$	0	0
$431$	−15.6870	−0.755615	−0.377807	−	0.925884i	$-0.623322\pi$
$431$	−15.6870	−0.755615	−0.377807	+	0.925884i	$0.623322\pi$
$432$	0	0
$433$	−29.1195	−1.39939	−0.699696	−	0.714441i	$-0.746681\pi$
$433$	−29.1195	−1.39939	−0.699696	+	0.714441i	$0.746681\pi$
$434$	0	0
$435$	−11.9546	−0.573178
$436$	0	0
$437$	39.9091	1.90911
$438$	0	0
$439$	−24.2863	−1.15912	−0.579561	−	0.814929i	$-0.696776\pi$
$439$	−24.2863	−1.15912	−0.579561	+	0.814929i	$0.696776\pi$
$440$	0	0
$441$	45.9517	2.18818
$442$	0	0
$443$	−27.8307	−1.32228	−0.661139	−	0.750263i	$-0.729927\pi$
$443$	−27.8307	−1.32228	−0.661139	+	0.750263i	$0.729927\pi$
$444$	0	0
$445$	−13.0994	−0.620973
$446$	0	0
$447$	−0.632323	−0.0299079
$448$	0	0
$449$	−36.0426	−1.70096	−0.850478	−	0.526011i	$-0.823687\pi$
$449$	−36.0426	−1.70096	−0.850478	+	0.526011i	$0.823687\pi$
$450$	0	0
$451$	−0.134771	−0.00634613
$452$	0	0
$453$	51.7050	2.42931
$454$	0	0
$455$	−14.8185	−0.694703
$456$	0	0
$457$	19.3595	0.905600	0.452800	−	0.891612i	$-0.350425\pi$
$457$	19.3595	0.905600	0.452800	+	0.891612i	$0.350425\pi$
$458$	0	0
$459$	28.8574	1.34695
$460$	0	0
$461$	22.7307	1.05867	0.529336	−	0.848412i	$-0.322441\pi$
$461$	22.7307	1.05867	0.529336	+	0.848412i	$0.322441\pi$
$462$	0	0
$463$	−27.8200	−1.29290	−0.646452	−	0.762955i	$-0.723748\pi$
$463$	−27.8200	−1.29290	−0.646452	+	0.762955i	$0.723748\pi$
$464$	0	0
$465$	21.9601	1.01838
$466$	0	0
$467$	−35.1604	−1.62703	−0.813514	−	0.581545i	$-0.802448\pi$
$467$	−35.1604	−1.62703	−0.813514	+	0.581545i	$0.802448\pi$
$468$	0	0
$469$	48.6380	2.24590
$470$	0	0
$471$	54.7543	2.52295
$472$	0	0
$473$	6.71281	0.308655
$474$	0	0
$475$	−4.69645	−0.215488
$476$	0	0
$477$	48.9575	2.24161
$478$	0	0
$479$	−19.0213	−0.869105	−0.434553	−	0.900646i	$-0.643094\pi$
$479$	−19.0213	−0.869105	−0.434553	+	0.900646i	$0.643094\pi$
$480$	0	0
$481$	32.8103	1.49602
$482$	0	0
$483$	97.2710	4.42598
$484$	0	0
$485$	−13.2581	−0.602020
$486$	0	0
$487$	23.1048	1.04698	0.523490	−	0.852032i	$-0.324630\pi$
$487$	23.1048	1.04698	0.523490	+	0.852032i	$0.324630\pi$
$488$	0	0
$489$	−14.8953	−0.673591
$490$	0	0
$491$	−30.8626	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$491$	−30.8626	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$492$	0	0
$493$	14.7115	0.662574
$494$	0	0
$495$	5.69645	0.256036
$496$	0	0
$497$	−60.5812	−2.71744
$498$	0	0
$499$	26.6566	1.19331	0.596656	−	0.802497i	$-0.296496\pi$
$499$	26.6566	1.19331	0.596656	+	0.802497i	$0.296496\pi$
$500$	0	0
$501$	−10.9731	−0.490240
$502$	0	0
$503$	−8.23360	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$503$	−8.23360	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$504$	0	0
$505$	15.6611	0.696911
$506$	0	0
$507$	−4.64282	−0.206195
$508$	0	0
$509$	−17.3782	−0.770274	−0.385137	−	0.922859i	$-0.625846\pi$
$509$	−17.3782	−0.770274	−0.385137	+	0.922859i	$0.625846\pi$
$510$	0	0
$511$	−22.0586	−0.975813
$512$	0	0
$513$	37.3450	1.64882
$514$	0	0
$515$	−2.76040	−0.121638
$516$	0	0
$517$	5.00278	0.220022
$518$	0	0
$519$	−5.56464	−0.244260
$520$	0	0
$521$	−10.6307	−0.465741	−0.232871	−	0.972508i	$-0.574812\pi$
$521$	−10.6307	−0.465741	−0.232871	+	0.972508i	$0.574812\pi$
$522$	0	0
$523$	33.2605	1.45438	0.727189	−	0.686437i	$-0.240826\pi$
$523$	33.2605	1.45438	0.727189	+	0.686437i	$0.240826\pi$
$524$	0	0
$525$	−11.4467	−0.499575
$526$	0	0
$527$	−27.0246	−1.17721
$528$	0	0
$529$	49.2114	2.13963
$530$	0	0
$531$	21.4367	0.930275
$532$	0	0
$533$	0.514508	0.0222858
$534$	0	0
$535$	−12.3739	−0.534972
$536$	0	0
$537$	22.1024	0.953789
$538$	0	0
$539$	−8.06673	−0.347459
$540$	0	0
$541$	−31.7545	−1.36523	−0.682617	−	0.730776i	$-0.739158\pi$
$541$	−31.7545	−1.36523	−0.682617	+	0.730776i	$0.739158\pi$
$542$	0	0
$543$	−50.5166	−2.16788
$544$	0	0
$545$	8.13477	0.348455
$546$	0	0
$547$	−25.5735	−1.09344	−0.546722	−	0.837314i	$-0.684124\pi$
$547$	−25.5735	−1.09344	−0.546722	+	0.837314i	$0.684124\pi$
$548$	0	0
$549$	0.306511	0.0130816
$550$	0	0
$551$	19.0385	0.811067
$552$	0	0
$553$	−22.8934	−0.973527
$554$	0	0
$555$	25.3447	1.07582
$556$	0	0
$557$	−6.80754	−0.288445	−0.144222	−	0.989545i	$-0.546068\pi$
$557$	−6.80754	−0.288445	−0.144222	+	0.989545i	$0.546068\pi$
$558$	0	0
$559$	−25.6271	−1.08391
$560$	0	0
$561$	−10.7020	−0.451839
$562$	0	0
$563$	3.59502	0.151512	0.0757560	−	0.997126i	$-0.475863\pi$
$563$	3.59502	0.151512	0.0757560	+	0.997126i	$0.475863\pi$
$564$	0	0
$565$	−3.23228	−0.135983
$566$	0	0
$567$	24.6876	1.03678
$568$	0	0
$569$	25.4299	1.06608	0.533038	−	0.846091i	$-0.321050\pi$
$569$	25.4299	1.06608	0.533038	+	0.846091i	$0.321050\pi$
$570$	0	0
$571$	15.3363	0.641803	0.320901	−	0.947113i	$-0.396014\pi$
$571$	15.3363	0.641803	0.320901	+	0.947113i	$0.396014\pi$
$572$	0	0
$573$	70.1735	2.93154
$574$	0	0
$575$	−8.49773	−0.354380
$576$	0	0
$577$	−7.39158	−0.307716	−0.153858	−	0.988093i	$-0.549170\pi$
$577$	−7.39158	−0.307716	−0.153858	+	0.988093i	$0.549170\pi$
$578$	0	0
$579$	45.4878	1.89041
$580$	0	0
$581$	−42.0004	−1.74247
$582$	0	0
$583$	−8.59440	−0.355944
$584$	0	0
$585$	−21.7470	−0.899128
$586$	0	0
$587$	22.1106	0.912603	0.456301	−	0.889825i	$-0.349174\pi$
$587$	22.1106	0.912603	0.456301	+	0.889825i	$0.349174\pi$
$588$	0	0
$589$	−34.9731	−1.44104
$590$	0	0
$591$	21.9418	0.902565
$592$	0	0
$593$	−10.4234	−0.428039	−0.214020	−	0.976829i	$-0.568656\pi$
$593$	−10.4234	−0.428039	−0.214020	+	0.976829i	$0.568656\pi$
$594$	0	0
$595$	14.0865	0.577491
$596$	0	0
$597$	−50.0734	−2.04937
$598$	0	0
$599$	8.44216	0.344937	0.172469	−	0.985015i	$-0.444826\pi$
$599$	8.44216	0.344937	0.172469	+	0.985015i	$0.444826\pi$
$600$	0	0
$601$	2.85512	0.116463	0.0582315	−	0.998303i	$-0.481454\pi$
$601$	2.85512	0.116463	0.0582315	+	0.998303i	$0.481454\pi$
$602$	0	0
$603$	71.3791	2.90678
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	17.6392	0.715953	0.357977	−	0.933731i	$-0.383467\pi$
$607$	17.6392	0.715953	0.357977	+	0.933731i	$0.383467\pi$
$608$	0	0
$609$	46.4027	1.88033
$610$	0	0
$611$	−19.0988	−0.772656
$612$	0	0
$613$	−31.3440	−1.26597	−0.632986	−	0.774163i	$-0.718171\pi$
$613$	−31.3440	−1.26597	−0.632986	+	0.774163i	$0.718171\pi$
$614$	0	0
$615$	0.397437	0.0160262
$616$	0	0
$617$	40.7956	1.64237	0.821185	−	0.570663i	$-0.193314\pi$
$617$	40.7956	1.64237	0.821185	+	0.570663i	$0.193314\pi$
$618$	0	0
$619$	10.7518	0.432150	0.216075	−	0.976377i	$-0.430674\pi$
$619$	10.7518	0.432150	0.216075	+	0.976377i	$0.430674\pi$
$620$	0	0
$621$	67.5719	2.71157
$622$	0	0
$623$	50.8467	2.03713
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−13.8497	−0.553104
$628$	0	0
$629$	−31.1896	−1.24361
$630$	0	0
$631$	−22.5972	−0.899581	−0.449790	−	0.893134i	$-0.648501\pi$
$631$	−22.5972	−0.899581	−0.449790	+	0.893134i	$0.648501\pi$
$632$	0	0
$633$	71.7536	2.85195
$634$	0	0
$635$	10.7128	0.425125
$636$	0	0
$637$	30.7959	1.22018
$638$	0	0
$639$	−88.9063	−3.51708
$640$	0	0
$641$	19.8525	0.784127	0.392064	−	0.919938i	$-0.371761\pi$
$641$	19.8525	0.784127	0.392064	+	0.919938i	$0.371761\pi$
$642$	0	0
$643$	1.25513	0.0494975	0.0247487	−	0.999694i	$-0.492121\pi$
$643$	1.25513	0.0494975	0.0247487	+	0.999694i	$0.492121\pi$
$644$	0	0
$645$	−19.7959	−0.779463
$646$	0	0
$647$	−37.9888	−1.49349	−0.746747	−	0.665109i	$-0.768385\pi$
$647$	−37.9888	−1.49349	−0.746747	+	0.665109i	$0.768385\pi$
$648$	0	0
$649$	−3.76318	−0.147718
$650$	0	0
$651$	−85.2402	−3.34083
$652$	0	0
$653$	22.5126	0.880987	0.440494	−	0.897756i	$-0.354803\pi$
$653$	22.5126	0.880987	0.440494	+	0.897756i	$0.354803\pi$
$654$	0	0
$655$	4.53583	0.177230
$656$	0	0
$657$	−32.3722	−1.26296
$658$	0	0
$659$	−39.3655	−1.53346	−0.766731	−	0.641969i	$-0.778118\pi$
$659$	−39.3655	−1.53346	−0.766731	+	0.641969i	$0.778118\pi$
$660$	0	0
$661$	48.9429	1.90366	0.951829	−	0.306630i	$-0.0992014\pi$
$661$	48.9429	1.90366	0.951829	+	0.306630i	$0.0992014\pi$
$662$	0	0
$663$	40.8564	1.58673
$664$	0	0
$665$	18.2297	0.706917
$666$	0	0
$667$	34.4482	1.33384
$668$	0	0
$669$	−27.3438	−1.05717
$670$	0	0
$671$	−0.0538074	−0.00207721
$672$	0	0
$673$	10.8217	0.417148	0.208574	−	0.978007i	$-0.433118\pi$
$673$	10.8217	0.417148	0.208574	+	0.978007i	$0.433118\pi$
$674$	0	0
$675$	−7.95176	−0.306063
$676$	0	0
$677$	39.2364	1.50798	0.753988	−	0.656888i	$-0.228128\pi$
$677$	39.2364	1.50798	0.753988	+	0.656888i	$0.228128\pi$
$678$	0	0
$679$	51.4626	1.97495
$680$	0	0
$681$	−69.6903	−2.67054
$682$	0	0
$683$	21.1152	0.807949	0.403974	−	0.914770i	$-0.367628\pi$
$683$	21.1152	0.807949	0.403974	+	0.914770i	$0.367628\pi$
$684$	0	0
$685$	0.242386	0.00926110
$686$	0	0
$687$	22.7207	0.866849
$688$	0	0
$689$	32.8103	1.24997
$690$	0	0
$691$	−34.6458	−1.31799	−0.658994	−	0.752148i	$-0.729018\pi$
$691$	−34.6458	−1.31799	−0.658994	+	0.752148i	$0.729018\pi$
$692$	0	0
$693$	−22.1113	−0.839938
$694$	0	0
$695$	19.3657	0.734584
$696$	0	0
$697$	−0.489093	−0.0185257
$698$	0	0
$699$	−42.8264	−1.61984
$700$	0	0
$701$	0.0538074	0.00203228	0.00101614	−	0.999999i	$-0.499677\pi$
$701$	0.0538074	0.00203228	0.00101614	+	0.999999i	$0.499677\pi$
$702$	0	0
$703$	−40.3631	−1.52232
$704$	0	0
$705$	−14.7531	−0.555633
$706$	0	0
$707$	−60.7900	−2.28624
$708$	0	0
$709$	4.16618	0.156464	0.0782320	−	0.996935i	$-0.475073\pi$
$709$	4.16618	0.156464	0.0782320	+	0.996935i	$0.475073\pi$
$710$	0	0
$711$	−33.5974	−1.26000
$712$	0	0
$713$	−63.2801	−2.36986
$714$	0	0
$715$	3.81764	0.142772
$716$	0	0
$717$	−23.3671	−0.872659
$718$	0	0
$719$	26.2471	0.978853	0.489427	−	0.872045i	$-0.337206\pi$
$719$	26.2471	0.978853	0.489427	+	0.872045i	$0.337206\pi$
$720$	0	0
$721$	10.7147	0.399037
$722$	0	0
$723$	46.9791	1.74717
$724$	0	0
$725$	−4.05381	−0.150555
$726$	0	0
$727$	1.38778	0.0514699	0.0257349	−	0.999669i	$-0.491807\pi$
$727$	1.38778	0.0514699	0.0257349	+	0.999669i	$0.491807\pi$
$728$	0	0
$729$	−34.1181	−1.26364
$730$	0	0
$731$	24.3612	0.901031
$732$	0	0
$733$	−44.3176	−1.63691	−0.818453	−	0.574573i	$-0.805168\pi$
$733$	−44.3176	−1.63691	−0.818453	+	0.574573i	$0.805168\pi$
$734$	0	0
$735$	23.7886	0.877455
$736$	0	0
$737$	−12.5304	−0.461565
$738$	0	0
$739$	−11.1505	−0.410178	−0.205089	−	0.978743i	$-0.565748\pi$
$739$	−11.1505	−0.410178	−0.205089	+	0.978743i	$0.565748\pi$
$740$	0	0
$741$	52.8732	1.94235
$742$	0	0
$743$	30.8973	1.13351	0.566755	−	0.823886i	$-0.308199\pi$
$743$	30.8973	1.13351	0.566755	+	0.823886i	$0.308199\pi$
$744$	0	0
$745$	−0.214421	−0.00785579
$746$	0	0
$747$	−61.6380	−2.25522
$748$	0	0
$749$	48.0305	1.75500
$750$	0	0
$751$	20.4133	0.744893	0.372446	−	0.928054i	$-0.378519\pi$
$751$	20.4133	0.744893	0.372446	+	0.928054i	$0.378519\pi$
$752$	0	0
$753$	−40.6037	−1.47968
$754$	0	0
$755$	17.5332	0.638100
$756$	0	0
$757$	−38.6019	−1.40301	−0.701505	−	0.712665i	$-0.747488\pi$
$757$	−38.6019	−1.40301	−0.701505	+	0.712665i	$0.747488\pi$
$758$	0	0
$759$	−25.0596	−0.909605
$760$	0	0
$761$	49.8781	1.80808	0.904040	−	0.427448i	$-0.140587\pi$
$761$	49.8781	1.80808	0.904040	+	0.427448i	$0.140587\pi$
$762$	0	0
$763$	−31.5758	−1.14312
$764$	0	0
$765$	20.6728	0.747425
$766$	0	0
$767$	14.3665	0.518743
$768$	0	0
$769$	15.1901	0.547769	0.273885	−	0.961763i	$-0.411691\pi$
$769$	15.1901	0.547769	0.273885	+	0.961763i	$0.411691\pi$
$770$	0	0
$771$	23.6680	0.852382
$772$	0	0
$773$	−24.0075	−0.863490	−0.431745	−	0.901996i	$-0.642102\pi$
$773$	−24.0075	−0.863490	−0.431745	+	0.901996i	$0.642102\pi$
$774$	0	0
$775$	7.44670	0.267493
$776$	0	0
$777$	−98.3775	−3.52927
$778$	0	0
$779$	−0.632946	−0.0226777
$780$	0	0
$781$	15.6073	0.558474
$782$	0	0
$783$	32.2349	1.15198
$784$	0	0
$785$	18.5672	0.662693
$786$	0	0
$787$	23.8892	0.851557	0.425778	−	0.904828i	$-0.360000\pi$
$787$	23.8892	0.851557	0.425778	+	0.904828i	$0.360000\pi$
$788$	0	0
$789$	−45.0605	−1.60420
$790$	0	0
$791$	12.5464	0.446098
$792$	0	0
$793$	0.205417	0.00729459
$794$	0	0
$795$	25.3447	0.898882
$796$	0	0
$797$	−5.74421	−0.203470	−0.101735	−	0.994812i	$-0.532439\pi$
$797$	−5.74421	−0.203470	−0.101735	+	0.994812i	$0.532439\pi$
$798$	0	0
$799$	18.1554	0.642292
$800$	0	0
$801$	74.6203	2.63658
$802$	0	0
$803$	5.68287	0.200544
$804$	0	0
$805$	32.9847	1.16256
$806$	0	0
$807$	29.5862	1.04148
$808$	0	0
$809$	−51.5392	−1.81202	−0.906011	−	0.423254i	$-0.860888\pi$
$809$	−51.5392	−1.81202	−0.906011	+	0.423254i	$0.860888\pi$
$810$	0	0
$811$	12.8424	0.450958	0.225479	−	0.974248i	$-0.427605\pi$
$811$	12.8424	0.450958	0.225479	+	0.974248i	$0.427605\pi$
$812$	0	0
$813$	−27.0010	−0.946967
$814$	0	0
$815$	−5.05103	−0.176930
$816$	0	0
$817$	31.5264	1.10297
$818$	0	0
$819$	84.4129	2.94962
$820$	0	0
$821$	−3.63918	−0.127008	−0.0635041	−	0.997982i	$-0.520228\pi$
$821$	−3.63918	−0.127008	−0.0635041	+	0.997982i	$0.520228\pi$
$822$	0	0
$823$	−3.60606	−0.125699	−0.0628497	−	0.998023i	$-0.520019\pi$
$823$	−3.60606	−0.125699	−0.0628497	+	0.998023i	$0.520019\pi$
$824$	0	0
$825$	2.94897	0.102670
$826$	0	0
$827$	18.8247	0.654600	0.327300	−	0.944920i	$-0.393861\pi$
$827$	18.8247	0.654600	0.327300	+	0.944920i	$0.393861\pi$
$828$	0	0
$829$	−30.9192	−1.07387	−0.536935	−	0.843624i	$-0.680418\pi$
$829$	−30.9192	−1.07387	−0.536935	+	0.843624i	$0.680418\pi$
$830$	0	0
$831$	45.2529	1.56981
$832$	0	0
$833$	−29.2747	−1.01431
$834$	0	0
$835$	−3.72097	−0.128770
$836$	0	0
$837$	−59.2144	−2.04675
$838$	0	0
$839$	−9.10439	−0.314318	−0.157159	−	0.987573i	$-0.550234\pi$
$839$	−9.10439	−0.314318	−0.157159	+	0.987573i	$0.550234\pi$
$840$	0	0
$841$	−12.5666	−0.433333
$842$	0	0
$843$	57.9801	1.99694
$844$	0	0
$845$	−1.57438	−0.0541605
$846$	0	0
$847$	3.88159	0.133373
$848$	0	0
$849$	28.9304	0.992888
$850$	0	0
$851$	−73.0329	−2.50353
$852$	0	0
$853$	−11.6135	−0.397640	−0.198820	−	0.980036i	$-0.563711\pi$
$853$	−11.6135	−0.397640	−0.198820	+	0.980036i	$0.563711\pi$
$854$	0	0
$855$	26.7531	0.914936
$856$	0	0
$857$	18.1578	0.620257	0.310128	−	0.950695i	$-0.399628\pi$
$857$	18.1578	0.620257	0.310128	+	0.950695i	$0.399628\pi$
$858$	0	0
$859$	15.8067	0.539317	0.269658	−	0.962956i	$-0.413089\pi$
$859$	15.8067	0.539317	0.269658	+	0.962956i	$0.413089\pi$
$860$	0	0
$861$	−1.54269	−0.0525746
$862$	0	0
$863$	−24.9446	−0.849126	−0.424563	−	0.905398i	$-0.639572\pi$
$863$	−24.9446	−0.849126	−0.424563	+	0.905398i	$0.639572\pi$
$864$	0	0
$865$	−1.88697	−0.0641590
$866$	0	0
$867$	11.2943	0.383574
$868$	0	0
$869$	5.89795	0.200074
$870$	0	0
$871$	47.8368	1.62089
$872$	0	0
$873$	75.5242	2.55611
$874$	0	0
$875$	−3.88159	−0.131222
$876$	0	0
$877$	−16.8334	−0.568423	−0.284211	−	0.958762i	$-0.591732\pi$
$877$	−16.8334	−0.568423	−0.284211	+	0.958762i	$0.591732\pi$
$878$	0	0
$879$	61.9436	2.08930
$880$	0	0
$881$	54.1571	1.82460	0.912299	−	0.409524i	$-0.134305\pi$
$881$	54.1571	1.82460	0.912299	+	0.409524i	$0.134305\pi$
$882$	0	0
$883$	−3.64736	−0.122743	−0.0613717	−	0.998115i	$-0.519548\pi$
$883$	−3.64736	−0.122743	−0.0613717	+	0.998115i	$0.519548\pi$
$884$	0	0
$885$	11.0975	0.373039
$886$	0	0
$887$	13.4181	0.450535	0.225267	−	0.974297i	$-0.427674\pi$
$887$	13.4181	0.450535	0.225267	+	0.974297i	$0.427674\pi$
$888$	0	0
$889$	−41.5827	−1.39464
$890$	0	0
$891$	−6.36018	−0.213074
$892$	0	0
$893$	23.4953	0.786240
$894$	0	0
$895$	7.49495	0.250528
$896$	0	0
$897$	95.6685	3.19428
$898$	0	0
$899$	−30.1875	−1.00681
$900$	0	0
$901$	−31.1896	−1.03908
$902$	0	0
$903$	76.8395	2.55706
$904$	0	0
$905$	−17.1302	−0.569428
$906$	0	0
$907$	−24.1162	−0.800764	−0.400382	−	0.916348i	$-0.631123\pi$
$907$	−24.1162	−0.800764	−0.400382	+	0.916348i	$0.631123\pi$
$908$	0	0
$909$	−89.2128	−2.95900
$910$	0	0
$911$	−30.5864	−1.01337	−0.506686	−	0.862131i	$-0.669130\pi$
$911$	−30.5864	−1.01337	−0.506686	+	0.862131i	$0.669130\pi$
$912$	0	0
$913$	10.8204	0.358104
$914$	0	0
$915$	0.158677	0.00524569
$916$	0	0
$917$	−17.6062	−0.581409
$918$	0	0
$919$	37.8523	1.24863	0.624315	−	0.781172i	$-0.285378\pi$
$919$	37.8523	1.24863	0.624315	+	0.781172i	$0.285378\pi$
$920$	0	0
$921$	−61.0014	−2.01007
$922$	0	0
$923$	−59.5831	−1.96120
$924$	0	0
$925$	8.59440	0.282582
$926$	0	0
$927$	15.7244	0.516459
$928$	0	0
$929$	−16.7782	−0.550476	−0.275238	−	0.961376i	$-0.588757\pi$
$929$	−16.7782	−0.550476	−0.275238	+	0.961376i	$0.588757\pi$
$930$	0	0
$931$	−37.8850	−1.24163
$932$	0	0
$933$	20.3743	0.667023
$934$	0	0
$935$	−3.62906	−0.118683
$936$	0	0
$937$	−10.6495	−0.347903	−0.173952	−	0.984754i	$-0.555654\pi$
$937$	−10.6495	−0.347903	−0.173952	+	0.984754i	$0.555654\pi$
$938$	0	0
$939$	−51.8777	−1.69297
$940$	0	0
$941$	38.6996	1.26157	0.630786	−	0.775957i	$-0.282733\pi$
$941$	38.6996	1.26157	0.630786	+	0.775957i	$0.282733\pi$
$942$	0	0
$943$	−1.14525	−0.0372945
$944$	0	0
$945$	30.8654	1.00405
$946$	0	0
$947$	33.6258	1.09269	0.546346	−	0.837559i	$-0.316018\pi$
$947$	33.6258	1.09269	0.546346	+	0.837559i	$0.316018\pi$
$948$	0	0
$949$	−21.6952	−0.704255
$950$	0	0
$951$	−55.5492	−1.80131
$952$	0	0
$953$	55.0684	1.78384	0.891921	−	0.452192i	$-0.149358\pi$
$953$	55.0684	1.78384	0.891921	+	0.452192i	$0.149358\pi$
$954$	0	0
$955$	23.7959	0.770017
$956$	0	0
$957$	−11.9546	−0.386436
$958$	0	0
$959$	−0.940843	−0.0303814
$960$	0	0
$961$	24.4534	0.788819
$962$	0	0
$963$	70.4875	2.27143
$964$	0	0
$965$	15.4250	0.496547
$966$	0	0
$967$	−16.9287	−0.544391	−0.272196	−	0.962242i	$-0.587750\pi$
$967$	−16.9287	−0.544391	−0.272196	+	0.962242i	$0.587750\pi$
$968$	0	0
$969$	−50.2614	−1.61463
$970$	0	0
$971$	0.877074	0.0281466	0.0140733	−	0.999901i	$-0.495520\pi$
$971$	0.877074	0.0281466	0.0140733	+	0.999901i	$0.495520\pi$
$972$	0	0
$973$	−75.1698	−2.40983
$974$	0	0
$975$	−11.2581	−0.360549
$976$	0	0
$977$	24.2591	0.776119	0.388059	−	0.921634i	$-0.373146\pi$
$977$	24.2591	0.776119	0.388059	+	0.921634i	$0.373146\pi$
$978$	0	0
$979$	−13.0994	−0.418660
$980$	0	0
$981$	−46.3393	−1.47950
$982$	0	0
$983$	10.1929	0.325103	0.162551	−	0.986700i	$-0.448028\pi$
$983$	10.1929	0.325103	0.162551	+	0.986700i	$0.448028\pi$
$984$	0	0
$985$	7.44048	0.237074
$986$	0	0
$987$	57.2654	1.82278
$988$	0	0
$989$	57.0436	1.81388
$990$	0	0
$991$	18.5817	0.590267	0.295133	−	0.955456i	$-0.404636\pi$
$991$	18.5817	0.590267	0.295133	+	0.955456i	$0.404636\pi$
$992$	0	0
$993$	11.7959	0.374331
$994$	0	0
$995$	−16.9799	−0.538300
$996$	0	0
$997$	−41.6842	−1.32015	−0.660076	−	0.751199i	$-0.729476\pi$
$997$	−41.6842	−1.32015	−0.660076	+	0.751199i	$0.729476\pi$
$998$	0	0
$999$	−68.3405	−2.16220

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3520.2.a.bz.1.1		5
4.3	odd	2		3520.2.a.by.1.5		5
8.3	odd	2		1760.2.a.u.1.1	✓	5
8.5	even	2		1760.2.a.v.1.5	yes	5
40.19	odd	2		8800.2.a.bv.1.5		5
40.29	even	2		8800.2.a.bu.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1760.2.a.u.1.1	✓	5	8.3	odd	2
1760.2.a.v.1.5	yes	5	8.5	even	2
3520.2.a.by.1.5		5	4.3	odd	2
3520.2.a.bz.1.1		5	1.1	even	1	trivial
8800.2.a.bu.1.1		5	40.29	even	2
8800.2.a.bv.1.5		5	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3520 = 2^{6} \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3520.a (trivial)

\(f(q)\)	\(=\)	\(q-2.94897 q^{3} -1.00000 q^{5} +3.88159 q^{7} +5.69645 q^{9} -1.00000 q^{11} +3.81764 q^{13} +2.94897 q^{15} -3.62906 q^{17} -4.69645 q^{19} -11.4467 q^{21} -8.49773 q^{23} +1.00000 q^{25} -7.95176 q^{27} -4.05381 q^{29} +7.44670 q^{31} +2.94897 q^{33} -3.88159 q^{35} +8.59440 q^{37} -11.2581 q^{39} +0.134771 q^{41} -6.71281 q^{43} -5.69645 q^{45} -5.00278 q^{47} +8.06673 q^{49} +10.7020 q^{51} +8.59440 q^{53} +1.00000 q^{55} +13.8497 q^{57} +3.76318 q^{59} +0.0538074 q^{61} +22.1113 q^{63} -3.81764 q^{65} +12.5304 q^{67} +25.0596 q^{69} -15.6073 q^{71} -5.68287 q^{73} -2.94897 q^{75} -3.88159 q^{77} -5.89795 q^{79} +6.36018 q^{81} -10.8204 q^{83} +3.62906 q^{85} +11.9546 q^{87} +13.0994 q^{89} +14.8185 q^{91} -21.9601 q^{93} +4.69645 q^{95} +13.2581 q^{97} -5.69645 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{5} + 2 q^{7} + 11 q^{9} - 5 q^{11} - 6 q^{13} + 2 q^{17} - 6 q^{19} - 12 q^{21} - 12 q^{23} + 5 q^{25} - 10 q^{29} - 8 q^{31} - 2 q^{35} - 4 q^{37} - 16 q^{39} + 6 q^{41} - 4 q^{43} - 11 q^{45}+ \cdots - 11 q^{99}+O(q^{100}) \) 5 * q - 5 * q^5 + 2 * q^7 + 11 * q^9 - 5 * q^11 - 6 * q^13 + 2 * q^17 - 6 * q^19 - 12 * q^21 - 12 * q^23 + 5 * q^25 - 10 * q^29 - 8 * q^31 - 2 * q^35 - 4 * q^37 - 16 * q^39 + 6 * q^41 - 4 * q^43 - 11 * q^45 + 23 * q^49 - 14 * q^51 - 4 * q^53 + 5 * q^55 - 16 * q^59 - 10 * q^61 - 4 * q^63 + 6 * q^65 + 8 * q^67 - 24 * q^69 - 24 * q^71 + 2 * q^73 - 2 * q^77 + 21 * q^81 - 4 * q^83 - 2 * q^85 + 2 * q^87 + 24 * q^89 - 20 * q^91 - 2 * q^93 + 6 * q^95 + 26 * q^97 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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