LMFDB - Embedded newform 4056.2.a.bg.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4056,2,Mod(1,4056)] 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("4056.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4056, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4056 = 2^{3} \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4056.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,-6,0,1,0,7,0,6,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.3873230598$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.27700337.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 19x^{4} + 17x^{3} + 103x^{2} - 71x - 127 $ x^6 - x^5 - 19x^4 + 17x^3 + 103x^2 - 71x - 127
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.72245$ of defining polynomial
Character	$\chi$	$=$	4056.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.65870 q^{5} +3.72245 q^{7} +1.00000 q^{9} +3.03794 q^{11} -2.65870 q^{15} +3.27997 q^{17} +7.35072 q^{19} -3.72245 q^{21} -4.11266 q^{23} +2.06870 q^{25} -1.00000 q^{27} -0.411722 q^{29} +5.12551 q^{31} -3.03794 q^{33} +9.89688 q^{35} +9.87882 q^{37} +1.73202 q^{41} -10.0415 q^{43} +2.65870 q^{45} +1.09391 q^{47} +6.85662 q^{49} -3.27997 q^{51} -13.6404 q^{53} +8.07698 q^{55} -7.35072 q^{57} +12.5581 q^{59} +1.36307 q^{61} +3.72245 q^{63} -9.46192 q^{67} +4.11266 q^{69} +6.26504 q^{71} -13.9720 q^{73} -2.06870 q^{75} +11.3086 q^{77} +5.45858 q^{79} +1.00000 q^{81} -14.3060 q^{83} +8.72046 q^{85} +0.411722 q^{87} +2.40802 q^{89} -5.12551 q^{93} +19.5434 q^{95} -11.1646 q^{97} +3.03794 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} - 7 q^{21} - 6 q^{23} + 17 q^{25} - 6 q^{27} + 3 q^{29} + 9 q^{31} - 8 q^{33} + 4 q^{35} + 6 q^{37} - 15 q^{41} + 11 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 - 7 * q^21 - 6 * q^23 + 17 * q^25 - 6 * q^27 + 3 * q^29 + 9 * q^31 - 8 * q^33 + 4 * q^35 + 6 * q^37 - 15 * q^41 + 11 * q^43 + q^45 + 5 * q^47 + 5 * q^49 + 5 * q^51 + 12 * q^53 + 7 * q^55 - 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 - 49 * q^73 - 17 * q^75 - 25 * q^77 + 14 * q^79 + 6 * q^81 - 3 * q^83 - 19 * q^85 - 3 * q^87 + 13 * q^89 - 9 * q^93 + 43 * q^95 - 14 * q^97 + 8 * 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
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.65870	1.18901	0.594504	−	0.804093i	$-0.297349\pi$
$5$	2.65870	1.18901	0.594504	+	0.804093i	$0.297349\pi$
$6$	0	0
$7$	3.72245	1.40695	0.703477	−	0.710718i	$-0.251630\pi$
$7$	3.72245	1.40695	0.703477	+	0.710718i	$0.251630\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.03794	0.915974	0.457987	−	0.888959i	$-0.348571\pi$
$11$	3.03794	0.915974	0.457987	+	0.888959i	$0.348571\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−2.65870	−0.686474
$16$	0	0
$17$	3.27997	0.795510	0.397755	−	0.917492i	$-0.369789\pi$
$17$	3.27997	0.795510	0.397755	+	0.917492i	$0.369789\pi$
$18$	0	0
$19$	7.35072	1.68637	0.843186	−	0.537622i	$-0.180677\pi$
$19$	7.35072	1.68637	0.843186	+	0.537622i	$0.180677\pi$
$20$	0	0
$21$	−3.72245	−0.812305
$22$	0	0
$23$	−4.11266	−0.857550	−0.428775	−	0.903411i	$-0.641055\pi$
$23$	−4.11266	−0.857550	−0.428775	+	0.903411i	$0.641055\pi$
$24$	0	0
$25$	2.06870	0.413739
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−0.411722	−0.0764549	−0.0382275	−	0.999269i	$-0.512171\pi$
$29$	−0.411722	−0.0764549	−0.0382275	+	0.999269i	$0.512171\pi$
$30$	0	0
$31$	5.12551	0.920569	0.460285	−	0.887771i	$-0.347747\pi$
$31$	5.12551	0.920569	0.460285	+	0.887771i	$0.347747\pi$
$32$	0	0
$33$	−3.03794	−0.528838
$34$	0	0
$35$	9.89688	1.67288
$36$	0	0
$37$	9.87882	1.62407	0.812034	−	0.583611i	$-0.198361\pi$
$37$	9.87882	1.62407	0.812034	+	0.583611i	$0.198361\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1.73202	0.270496	0.135248	−	0.990812i	$-0.456817\pi$
$41$	1.73202	0.270496	0.135248	+	0.990812i	$0.456817\pi$
$42$	0	0
$43$	−10.0415	−1.53131	−0.765654	−	0.643253i	$-0.777584\pi$
$43$	−10.0415	−1.53131	−0.765654	+	0.643253i	$0.777584\pi$
$44$	0	0
$45$	2.65870	0.396336
$46$	0	0
$47$	1.09391	0.159563	0.0797815	−	0.996812i	$-0.474578\pi$
$47$	1.09391	0.159563	0.0797815	+	0.996812i	$0.474578\pi$
$48$	0	0
$49$	6.85662	0.979517
$50$	0	0
$51$	−3.27997	−0.459288
$52$	0	0
$53$	−13.6404	−1.87365	−0.936825	−	0.349797i	$-0.886250\pi$
$53$	−13.6404	−1.87365	−0.936825	+	0.349797i	$0.886250\pi$
$54$	0	0
$55$	8.07698	1.08910
$56$	0	0
$57$	−7.35072	−0.973627
$58$	0	0
$59$	12.5581	1.63493	0.817466	−	0.575977i	$-0.195378\pi$
$59$	12.5581	1.63493	0.817466	+	0.575977i	$0.195378\pi$
$60$	0	0
$61$	1.36307	0.174523	0.0872615	−	0.996185i	$-0.472188\pi$
$61$	1.36307	0.174523	0.0872615	+	0.996185i	$0.472188\pi$
$62$	0	0
$63$	3.72245	0.468984
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.46192	−1.15596	−0.577979	−	0.816052i	$-0.696158\pi$
$67$	−9.46192	−1.15596	−0.577979	+	0.816052i	$0.696158\pi$
$68$	0	0
$69$	4.11266	0.495106
$70$	0	0
$71$	6.26504	0.743524	0.371762	−	0.928328i	$-0.378754\pi$
$71$	6.26504	0.743524	0.371762	+	0.928328i	$0.378754\pi$
$72$	0	0
$73$	−13.9720	−1.63530	−0.817649	−	0.575717i	$-0.804723\pi$
$73$	−13.9720	−1.63530	−0.817649	+	0.575717i	$0.804723\pi$
$74$	0	0
$75$	−2.06870	−0.238872
$76$	0	0
$77$	11.3086	1.28873
$78$	0	0
$79$	5.45858	0.614139	0.307069	−	0.951687i	$-0.400652\pi$
$79$	5.45858	0.614139	0.307069	+	0.951687i	$0.400652\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−14.3060	−1.57029	−0.785144	−	0.619313i	$-0.787411\pi$
$83$	−14.3060	−1.57029	−0.785144	+	0.619313i	$0.787411\pi$
$84$	0	0
$85$	8.72046	0.945867
$86$	0	0
$87$	0.411722	0.0441413
$88$	0	0
$89$	2.40802	0.255249	0.127625	−	0.991823i	$-0.459265\pi$
$89$	2.40802	0.255249	0.127625	+	0.991823i	$0.459265\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−5.12551	−0.531491
$94$	0	0
$95$	19.5434	2.00511
$96$	0	0
$97$	−11.1646	−1.13359	−0.566794	−	0.823859i	$-0.691817\pi$
$97$	−11.1646	−1.13359	−0.566794	+	0.823859i	$0.691817\pi$
$98$	0	0
$99$	3.03794	0.305325
$100$	0	0
$101$	−16.1645	−1.60843	−0.804214	−	0.594340i	$-0.797414\pi$
$101$	−16.1645	−1.60843	−0.804214	+	0.594340i	$0.797414\pi$
$102$	0	0
$103$	4.90839	0.483638	0.241819	−	0.970321i	$-0.422256\pi$
$103$	4.90839	0.483638	0.241819	+	0.970321i	$0.422256\pi$
$104$	0	0
$105$	−9.89688	−0.965837
$106$	0	0
$107$	−1.03391	−0.0999515	−0.0499758	−	0.998750i	$-0.515914\pi$
$107$	−1.03391	−0.0999515	−0.0499758	+	0.998750i	$0.515914\pi$
$108$	0	0
$109$	−8.64406	−0.827950	−0.413975	−	0.910288i	$-0.635860\pi$
$109$	−8.64406	−0.827950	−0.413975	+	0.910288i	$0.635860\pi$
$110$	0	0
$111$	−9.87882	−0.937656
$112$	0	0
$113$	−17.7568	−1.67042	−0.835210	−	0.549930i	$-0.814654\pi$
$113$	−17.7568	−1.67042	−0.835210	+	0.549930i	$0.814654\pi$
$114$	0	0
$115$	−10.9343	−1.01963
$116$	0	0
$117$	0	0
$118$	0	0
$119$	12.2095	1.11924
$120$	0	0
$121$	−1.77091	−0.160992
$122$	0	0
$123$	−1.73202	−0.156171
$124$	0	0
$125$	−7.79346	−0.697069
$126$	0	0
$127$	−9.38988	−0.833217	−0.416609	−	0.909086i	$-0.636781\pi$
$127$	−9.38988	−0.833217	−0.416609	+	0.909086i	$0.636781\pi$
$128$	0	0
$129$	10.0415	0.884101
$130$	0	0
$131$	5.82373	0.508821	0.254411	−	0.967096i	$-0.418119\pi$
$131$	5.82373	0.508821	0.254411	+	0.967096i	$0.418119\pi$
$132$	0	0
$133$	27.3627	2.37265
$134$	0	0
$135$	−2.65870	−0.228825
$136$	0	0
$137$	13.5842	1.16058	0.580288	−	0.814412i	$-0.302940\pi$
$137$	13.5842	1.16058	0.580288	+	0.814412i	$0.302940\pi$
$138$	0	0
$139$	20.3632	1.72719	0.863594	−	0.504188i	$-0.168208\pi$
$139$	20.3632	1.72719	0.863594	+	0.504188i	$0.168208\pi$
$140$	0	0
$141$	−1.09391	−0.0921238
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−1.09465	−0.0909055
$146$	0	0
$147$	−6.85662	−0.565524
$148$	0	0
$149$	−20.4792	−1.67772	−0.838859	−	0.544349i	$-0.816777\pi$
$149$	−20.4792	−1.67772	−0.838859	+	0.544349i	$0.816777\pi$
$150$	0	0
$151$	−15.6614	−1.27451	−0.637253	−	0.770655i	$-0.719929\pi$
$151$	−15.6614	−1.27451	−0.637253	+	0.770655i	$0.719929\pi$
$152$	0	0
$153$	3.27997	0.265170
$154$	0	0
$155$	13.6272	1.09456
$156$	0	0
$157$	11.4398	0.912993	0.456496	−	0.889725i	$-0.349104\pi$
$157$	11.4398	0.912993	0.456496	+	0.889725i	$0.349104\pi$
$158$	0	0
$159$	13.6404	1.08175
$160$	0	0
$161$	−15.3092	−1.20653
$162$	0	0
$163$	16.7092	1.30877	0.654384	−	0.756163i	$-0.272928\pi$
$163$	16.7092	1.30877	0.654384	+	0.756163i	$0.272928\pi$
$164$	0	0
$165$	−8.07698	−0.628792
$166$	0	0
$167$	10.4991	0.812444	0.406222	−	0.913774i	$-0.366846\pi$
$167$	10.4991	0.812444	0.406222	+	0.913774i	$0.366846\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	7.35072	0.562124
$172$	0	0
$173$	−4.45612	−0.338792	−0.169396	−	0.985548i	$-0.554182\pi$
$173$	−4.45612	−0.338792	−0.169396	+	0.985548i	$0.554182\pi$
$174$	0	0
$175$	7.70061	0.582112
$176$	0	0
$177$	−12.5581	−0.943928
$178$	0	0
$179$	−4.21904	−0.315346	−0.157673	−	0.987491i	$-0.550399\pi$
$179$	−4.21904	−0.315346	−0.157673	+	0.987491i	$0.550399\pi$
$180$	0	0
$181$	−6.39554	−0.475377	−0.237688	−	0.971341i	$-0.576390\pi$
$181$	−6.39554	−0.475377	−0.237688	+	0.971341i	$0.576390\pi$
$182$	0	0
$183$	−1.36307	−0.100761
$184$	0	0
$185$	26.2648	1.93103
$186$	0	0
$187$	9.96436	0.728666
$188$	0	0
$189$	−3.72245	−0.270768
$190$	0	0
$191$	9.58457	0.693515	0.346758	−	0.937955i	$-0.387283\pi$
$191$	9.58457	0.693515	0.346758	+	0.937955i	$0.387283\pi$
$192$	0	0
$193$	−17.1851	−1.23701	−0.618507	−	0.785779i	$-0.712262\pi$
$193$	−17.1851	−1.23701	−0.618507	+	0.785779i	$0.712262\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−12.2123	−0.870090	−0.435045	−	0.900409i	$-0.643267\pi$
$197$	−12.2123	−0.870090	−0.435045	+	0.900409i	$0.643267\pi$
$198$	0	0
$199$	−12.2265	−0.866710	−0.433355	−	0.901223i	$-0.642670\pi$
$199$	−12.2265	−0.866710	−0.433355	+	0.901223i	$0.642670\pi$
$200$	0	0
$201$	9.46192	0.667392
$202$	0	0
$203$	−1.53261	−0.107568
$204$	0	0
$205$	4.60492	0.321622
$206$	0	0
$207$	−4.11266	−0.285850
$208$	0	0
$209$	22.3311	1.54467
$210$	0	0
$211$	10.8819	0.749138	0.374569	−	0.927199i	$-0.377791\pi$
$211$	10.8819	0.749138	0.374569	+	0.927199i	$0.377791\pi$
$212$	0	0
$213$	−6.26504	−0.429274
$214$	0	0
$215$	−26.6972	−1.82074
$216$	0	0
$217$	19.0795	1.29520
$218$	0	0
$219$	13.9720	0.944140
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.88623	0.394171	0.197086	−	0.980386i	$-0.436852\pi$
$223$	5.88623	0.394171	0.197086	+	0.980386i	$0.436852\pi$
$224$	0	0
$225$	2.06870	0.137913
$226$	0	0
$227$	13.1454	0.872491	0.436246	−	0.899828i	$-0.356308\pi$
$227$	13.1454	0.872491	0.436246	+	0.899828i	$0.356308\pi$
$228$	0	0
$229$	23.8710	1.57744	0.788721	−	0.614752i	$-0.210744\pi$
$229$	23.8710	1.57744	0.788721	+	0.614752i	$0.210744\pi$
$230$	0	0
$231$	−11.3086	−0.744050
$232$	0	0
$233$	−2.65905	−0.174200	−0.0871000	−	0.996200i	$-0.527760\pi$
$233$	−2.65905	−0.174200	−0.0871000	+	0.996200i	$0.527760\pi$
$234$	0	0
$235$	2.90838	0.189722
$236$	0	0
$237$	−5.45858	−0.354573
$238$	0	0
$239$	−8.99863	−0.582073	−0.291037	−	0.956712i	$-0.594000\pi$
$239$	−8.99863	−0.582073	−0.291037	+	0.956712i	$0.594000\pi$
$240$	0	0
$241$	−1.45787	−0.0939098	−0.0469549	−	0.998897i	$-0.514952\pi$
$241$	−1.45787	−0.0939098	−0.0469549	+	0.998897i	$0.514952\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	18.2297	1.16465
$246$	0	0
$247$	0	0
$248$	0	0
$249$	14.3060	0.906607
$250$	0	0
$251$	−3.98236	−0.251364	−0.125682	−	0.992071i	$-0.540112\pi$
$251$	−3.98236	−0.251364	−0.125682	+	0.992071i	$0.540112\pi$
$252$	0	0
$253$	−12.4940	−0.785493
$254$	0	0
$255$	−8.72046	−0.546097
$256$	0	0
$257$	−15.2959	−0.954131	−0.477066	−	0.878868i	$-0.658300\pi$
$257$	−15.2959	−0.954131	−0.477066	+	0.878868i	$0.658300\pi$
$258$	0	0
$259$	36.7734	2.28499
$260$	0	0
$261$	−0.411722	−0.0254850
$262$	0	0
$263$	−14.8408	−0.915125	−0.457562	−	0.889178i	$-0.651277\pi$
$263$	−14.8408	−0.915125	−0.457562	+	0.889178i	$0.651277\pi$
$264$	0	0
$265$	−36.2657	−2.22778
$266$	0	0
$267$	−2.40802	−0.147368
$268$	0	0
$269$	−12.7970	−0.780246	−0.390123	−	0.920763i	$-0.627567\pi$
$269$	−12.7970	−0.780246	−0.390123	+	0.920763i	$0.627567\pi$
$270$	0	0
$271$	27.0802	1.64500	0.822502	−	0.568762i	$-0.192578\pi$
$271$	27.0802	1.64500	0.822502	+	0.568762i	$0.192578\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.28458	0.378974
$276$	0	0
$277$	7.02829	0.422289	0.211145	−	0.977455i	$-0.432281\pi$
$277$	7.02829	0.422289	0.211145	+	0.977455i	$0.432281\pi$
$278$	0	0
$279$	5.12551	0.306856
$280$	0	0
$281$	−18.8159	−1.12246	−0.561232	−	0.827658i	$-0.689673\pi$
$281$	−18.8159	−1.12246	−0.561232	+	0.827658i	$0.689673\pi$
$282$	0	0
$283$	−4.79334	−0.284934	−0.142467	−	0.989800i	$-0.545504\pi$
$283$	−4.79334	−0.284934	−0.142467	+	0.989800i	$0.545504\pi$
$284$	0	0
$285$	−19.5434	−1.15765
$286$	0	0
$287$	6.44735	0.380575
$288$	0	0
$289$	−6.24179	−0.367164
$290$	0	0
$291$	11.1646	0.654478
$292$	0	0
$293$	29.4230	1.71891	0.859456	−	0.511209i	$-0.170802\pi$
$293$	29.4230	1.71891	0.859456	+	0.511209i	$0.170802\pi$
$294$	0	0
$295$	33.3884	1.94395
$296$	0	0
$297$	−3.03794	−0.176279
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−37.3788	−2.15448
$302$	0	0
$303$	16.1645	0.928627
$304$	0	0
$305$	3.62399	0.207509
$306$	0	0
$307$	30.5800	1.74529	0.872646	−	0.488354i	$-0.162402\pi$
$307$	30.5800	1.74529	0.872646	+	0.488354i	$0.162402\pi$
$308$	0	0
$309$	−4.90839	−0.279229
$310$	0	0
$311$	3.05106	0.173009	0.0865047	−	0.996251i	$-0.472430\pi$
$311$	3.05106	0.173009	0.0865047	+	0.996251i	$0.472430\pi$
$312$	0	0
$313$	26.7958	1.51459	0.757294	−	0.653074i	$-0.226521\pi$
$313$	26.7958	1.51459	0.757294	+	0.653074i	$0.226521\pi$
$314$	0	0
$315$	9.89688	0.557626
$316$	0	0
$317$	33.1691	1.86296	0.931481	−	0.363791i	$-0.118518\pi$
$317$	33.1691	1.86296	0.931481	+	0.363791i	$0.118518\pi$
$318$	0	0
$319$	−1.25079	−0.0700307
$320$	0	0
$321$	1.03391	0.0577070
$322$	0	0
$323$	24.1102	1.34153
$324$	0	0
$325$	0	0
$326$	0	0
$327$	8.64406	0.478017
$328$	0	0
$329$	4.07202	0.224498
$330$	0	0
$331$	25.8226	1.41934	0.709670	−	0.704534i	$-0.248844\pi$
$331$	25.8226	1.41934	0.709670	+	0.704534i	$0.248844\pi$
$332$	0	0
$333$	9.87882	0.541356
$334$	0	0
$335$	−25.1564	−1.37444
$336$	0	0
$337$	12.9469	0.705265	0.352632	−	0.935762i	$-0.385287\pi$
$337$	12.9469	0.705265	0.352632	+	0.935762i	$0.385287\pi$
$338$	0	0
$339$	17.7568	0.964418
$340$	0	0
$341$	15.5710	0.843217
$342$	0	0
$343$	−0.533732	−0.0288188
$344$	0	0
$345$	10.9343	0.588685
$346$	0	0
$347$	4.05605	0.217740	0.108870	−	0.994056i	$-0.465277\pi$
$347$	4.05605	0.217740	0.108870	+	0.994056i	$0.465277\pi$
$348$	0	0
$349$	−5.87479	−0.314470	−0.157235	−	0.987561i	$-0.550258\pi$
$349$	−5.87479	−0.314470	−0.157235	+	0.987561i	$0.550258\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9.15183	0.487103	0.243551	−	0.969888i	$-0.421688\pi$
$353$	9.15183	0.487103	0.243551	+	0.969888i	$0.421688\pi$
$354$	0	0
$355$	16.6569	0.884056
$356$	0	0
$357$	−12.2095	−0.646196
$358$	0	0
$359$	−9.33002	−0.492420	−0.246210	−	0.969217i	$-0.579185\pi$
$359$	−9.33002	−0.492420	−0.246210	+	0.969217i	$0.579185\pi$
$360$	0	0
$361$	35.0331	1.84385
$362$	0	0
$363$	1.77091	0.0929489
$364$	0	0
$365$	−37.1474	−1.94438
$366$	0	0
$367$	−16.8167	−0.877826	−0.438913	−	0.898530i	$-0.644636\pi$
$367$	−16.8167	−0.877826	−0.438913	+	0.898530i	$0.644636\pi$
$368$	0	0
$369$	1.73202	0.0901653
$370$	0	0
$371$	−50.7756	−2.63614
$372$	0	0
$373$	26.8491	1.39020	0.695098	−	0.718915i	$-0.255361\pi$
$373$	26.8491	1.39020	0.695098	+	0.718915i	$0.255361\pi$
$374$	0	0
$375$	7.79346	0.402453
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−24.6170	−1.26449	−0.632244	−	0.774769i	$-0.717866\pi$
$379$	−24.6170	−1.26449	−0.632244	+	0.774769i	$0.717866\pi$
$380$	0	0
$381$	9.38988	0.481058
$382$	0	0
$383$	−6.17173	−0.315360	−0.157680	−	0.987490i	$-0.550402\pi$
$383$	−6.17173	−0.315360	−0.157680	+	0.987490i	$0.550402\pi$
$384$	0	0
$385$	30.0661	1.53231
$386$	0	0
$387$	−10.0415	−0.510436
$388$	0	0
$389$	−10.2737	−0.520896	−0.260448	−	0.965488i	$-0.583870\pi$
$389$	−10.2737	−0.520896	−0.260448	+	0.965488i	$0.583870\pi$
$390$	0	0
$391$	−13.4894	−0.682189
$392$	0	0
$393$	−5.82373	−0.293768
$394$	0	0
$395$	14.5127	0.730215
$396$	0	0
$397$	28.8265	1.44676	0.723381	−	0.690449i	$-0.242587\pi$
$397$	28.8265	1.44676	0.723381	+	0.690449i	$0.242587\pi$
$398$	0	0
$399$	−27.3627	−1.36985
$400$	0	0
$401$	−2.97045	−0.148337	−0.0741685	−	0.997246i	$-0.523630\pi$
$401$	−2.97045	−0.148337	−0.0741685	+	0.997246i	$0.523630\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	2.65870	0.132112
$406$	0	0
$407$	30.0113	1.48760
$408$	0	0
$409$	−8.43242	−0.416956	−0.208478	−	0.978027i	$-0.566851\pi$
$409$	−8.43242	−0.416956	−0.208478	+	0.978027i	$0.566851\pi$
$410$	0	0
$411$	−13.5842	−0.670059
$412$	0	0
$413$	46.7470	2.30027
$414$	0	0
$415$	−38.0354	−1.86709
$416$	0	0
$417$	−20.3632	−0.997192
$418$	0	0
$419$	9.07273	0.443232	0.221616	−	0.975134i	$-0.428867\pi$
$419$	9.07273	0.443232	0.221616	+	0.975134i	$0.428867\pi$
$420$	0	0
$421$	−13.4921	−0.657565	−0.328782	−	0.944406i	$-0.606638\pi$
$421$	−13.4921	−0.657565	−0.328782	+	0.944406i	$0.606638\pi$
$422$	0	0
$423$	1.09391	0.0531877
$424$	0	0
$425$	6.78526	0.329134
$426$	0	0
$427$	5.07395	0.245546
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−13.3245	−0.641820	−0.320910	−	0.947110i	$-0.603989\pi$
$431$	−13.3245	−0.641820	−0.320910	+	0.947110i	$0.603989\pi$
$432$	0	0
$433$	7.47760	0.359351	0.179675	−	0.983726i	$-0.442495\pi$
$433$	7.47760	0.359351	0.179675	+	0.983726i	$0.442495\pi$
$434$	0	0
$435$	1.09465	0.0524843
$436$	0	0
$437$	−30.2311	−1.44615
$438$	0	0
$439$	−6.83485	−0.326209	−0.163105	−	0.986609i	$-0.552151\pi$
$439$	−6.83485	−0.326209	−0.163105	+	0.986609i	$0.552151\pi$
$440$	0	0
$441$	6.85662	0.326506
$442$	0	0
$443$	−10.2666	−0.487779	−0.243889	−	0.969803i	$-0.578423\pi$
$443$	−10.2666	−0.487779	−0.243889	+	0.969803i	$0.578423\pi$
$444$	0	0
$445$	6.40220	0.303493
$446$	0	0
$447$	20.4792	0.968631
$448$	0	0
$449$	−17.9414	−0.846706	−0.423353	−	0.905965i	$-0.639147\pi$
$449$	−17.9414	−0.846706	−0.423353	+	0.905965i	$0.639147\pi$
$450$	0	0
$451$	5.26177	0.247767
$452$	0	0
$453$	15.6614	0.735836
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0.298024	0.0139410	0.00697050	−	0.999976i	$-0.497781\pi$
$457$	0.298024	0.0139410	0.00697050	+	0.999976i	$0.497781\pi$
$458$	0	0
$459$	−3.27997	−0.153096
$460$	0	0
$461$	22.7589	1.05999	0.529993	−	0.848002i	$-0.322194\pi$
$461$	22.7589	1.05999	0.529993	+	0.848002i	$0.322194\pi$
$462$	0	0
$463$	−5.34050	−0.248194	−0.124097	−	0.992270i	$-0.539603\pi$
$463$	−5.34050	−0.248194	−0.124097	+	0.992270i	$0.539603\pi$
$464$	0	0
$465$	−13.6272	−0.631947
$466$	0	0
$467$	−34.4899	−1.59600	−0.798002	−	0.602655i	$-0.794110\pi$
$467$	−34.4899	−1.59600	−0.798002	+	0.602655i	$0.794110\pi$
$468$	0	0
$469$	−35.2215	−1.62638
$470$	0	0
$471$	−11.4398	−0.527116
$472$	0	0
$473$	−30.5054	−1.40264
$474$	0	0
$475$	15.2064	0.697718
$476$	0	0
$477$	−13.6404	−0.624550
$478$	0	0
$479$	17.4714	0.798287	0.399143	−	0.916888i	$-0.369308\pi$
$479$	17.4714	0.798287	0.399143	+	0.916888i	$0.369308\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	15.3092	0.696592
$484$	0	0
$485$	−29.6832	−1.34785
$486$	0	0
$487$	−20.5339	−0.930481	−0.465240	−	0.885184i	$-0.654032\pi$
$487$	−20.5339	−0.930481	−0.465240	+	0.885184i	$0.654032\pi$
$488$	0	0
$489$	−16.7092	−0.755617
$490$	0	0
$491$	23.1011	1.04254	0.521268	−	0.853393i	$-0.325459\pi$
$491$	23.1011	1.04254	0.521268	+	0.853393i	$0.325459\pi$
$492$	0	0
$493$	−1.35044	−0.0608206
$494$	0	0
$495$	8.07698	0.363033
$496$	0	0
$497$	23.3213	1.04610
$498$	0	0
$499$	8.19788	0.366988	0.183494	−	0.983021i	$-0.441259\pi$
$499$	8.19788	0.366988	0.183494	+	0.983021i	$0.441259\pi$
$500$	0	0
$501$	−10.4991	−0.469065
$502$	0	0
$503$	−21.1314	−0.942203	−0.471102	−	0.882079i	$-0.656143\pi$
$503$	−21.1314	−0.942203	−0.471102	+	0.882079i	$0.656143\pi$
$504$	0	0
$505$	−42.9766	−1.91243
$506$	0	0
$507$	0	0
$508$	0	0
$509$	24.3644	1.07993	0.539967	−	0.841686i	$-0.318437\pi$
$509$	24.3644	1.07993	0.539967	+	0.841686i	$0.318437\pi$
$510$	0	0
$511$	−52.0100	−2.30079
$512$	0	0
$513$	−7.35072	−0.324542
$514$	0	0
$515$	13.0500	0.575050
$516$	0	0
$517$	3.32323	0.146156
$518$	0	0
$519$	4.45612	0.195602
$520$	0	0
$521$	29.3990	1.28799	0.643996	−	0.765029i	$-0.277275\pi$
$521$	29.3990	1.28799	0.643996	+	0.765029i	$0.277275\pi$
$522$	0	0
$523$	−27.0198	−1.18149	−0.590746	−	0.806858i	$-0.701166\pi$
$523$	−27.0198	−1.18149	−0.590746	+	0.806858i	$0.701166\pi$
$524$	0	0
$525$	−7.70061	−0.336082
$526$	0	0
$527$	16.8115	0.732322
$528$	0	0
$529$	−6.08600	−0.264609
$530$	0	0
$531$	12.5581	0.544977
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−2.74885	−0.118843
$536$	0	0
$537$	4.21904	0.182065
$538$	0	0
$539$	20.8300	0.897212
$540$	0	0
$541$	12.4490	0.535225	0.267613	−	0.963527i	$-0.413765\pi$
$541$	12.4490	0.535225	0.267613	+	0.963527i	$0.413765\pi$
$542$	0	0
$543$	6.39554	0.274459
$544$	0	0
$545$	−22.9820	−0.984439
$546$	0	0
$547$	42.5858	1.82084	0.910418	−	0.413689i	$-0.135760\pi$
$547$	42.5858	1.82084	0.910418	+	0.413689i	$0.135760\pi$
$548$	0	0
$549$	1.36307	0.0581743
$550$	0	0
$551$	−3.02646	−0.128931
$552$	0	0
$553$	20.3193	0.864064
$554$	0	0
$555$	−26.2648	−1.11488
$556$	0	0
$557$	−20.8025	−0.881430	−0.440715	−	0.897647i	$-0.645275\pi$
$557$	−20.8025	−0.881430	−0.440715	+	0.897647i	$0.645275\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−9.96436	−0.420696
$562$	0	0
$563$	−15.8229	−0.666854	−0.333427	−	0.942776i	$-0.608205\pi$
$563$	−15.8229	−0.666854	−0.333427	+	0.942776i	$0.608205\pi$
$564$	0	0
$565$	−47.2101	−1.98614
$566$	0	0
$567$	3.72245	0.156328
$568$	0	0
$569$	32.0491	1.34357	0.671784	−	0.740747i	$-0.265528\pi$
$569$	32.0491	1.34357	0.671784	+	0.740747i	$0.265528\pi$
$570$	0	0
$571$	15.5134	0.649214	0.324607	−	0.945849i	$-0.394768\pi$
$571$	15.5134	0.649214	0.324607	+	0.945849i	$0.394768\pi$
$572$	0	0
$573$	−9.58457	−0.400401
$574$	0	0
$575$	−8.50785	−0.354802
$576$	0	0
$577$	42.4506	1.76724	0.883620	−	0.468205i	$-0.155099\pi$
$577$	42.4506	1.76724	0.883620	+	0.468205i	$0.155099\pi$
$578$	0	0
$579$	17.1851	0.714190
$580$	0	0
$581$	−53.2534	−2.20932
$582$	0	0
$583$	−41.4387	−1.71621
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.2335	−0.835126	−0.417563	−	0.908648i	$-0.637116\pi$
$587$	−20.2335	−0.835126	−0.417563	+	0.908648i	$0.637116\pi$
$588$	0	0
$589$	37.6762	1.55242
$590$	0	0
$591$	12.2123	0.502347
$592$	0	0
$593$	5.48873	0.225395	0.112698	−	0.993629i	$-0.464051\pi$
$593$	5.48873	0.225395	0.112698	+	0.993629i	$0.464051\pi$
$594$	0	0
$595$	32.4615	1.33079
$596$	0	0
$597$	12.2265	0.500396
$598$	0	0
$599$	−33.7217	−1.37783	−0.688915	−	0.724842i	$-0.741913\pi$
$599$	−33.7217	−1.37783	−0.688915	+	0.724842i	$0.741913\pi$
$600$	0	0
$601$	2.21511	0.0903562	0.0451781	−	0.998979i	$-0.485614\pi$
$601$	2.21511	0.0903562	0.0451781	+	0.998979i	$0.485614\pi$
$602$	0	0
$603$	−9.46192	−0.385319
$604$	0	0
$605$	−4.70833	−0.191421
$606$	0	0
$607$	11.4943	0.466539	0.233270	−	0.972412i	$-0.425058\pi$
$607$	11.4943	0.466539	0.233270	+	0.972412i	$0.425058\pi$
$608$	0	0
$609$	1.53261	0.0621047
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−25.0756	−1.01279	−0.506397	−	0.862300i	$-0.669023\pi$
$613$	−25.0756	−1.01279	−0.506397	+	0.862300i	$0.669023\pi$
$614$	0	0
$615$	−4.60492	−0.185688
$616$	0	0
$617$	−41.2483	−1.66059	−0.830296	−	0.557322i	$-0.811829\pi$
$617$	−41.2483	−1.66059	−0.830296	+	0.557322i	$0.811829\pi$
$618$	0	0
$619$	−5.88692	−0.236615	−0.118308	−	0.992977i	$-0.537747\pi$
$619$	−5.88692	−0.236615	−0.118308	+	0.992977i	$0.537747\pi$
$620$	0	0
$621$	4.11266	0.165035
$622$	0	0
$623$	8.96371	0.359124
$624$	0	0
$625$	−31.0640	−1.24256
$626$	0	0
$627$	−22.3311	−0.891817
$628$	0	0
$629$	32.4022	1.29196
$630$	0	0
$631$	33.2398	1.32326	0.661628	−	0.749832i	$-0.269866\pi$
$631$	33.2398	1.32326	0.661628	+	0.749832i	$0.269866\pi$
$632$	0	0
$633$	−10.8819	−0.432515
$634$	0	0
$635$	−24.9649	−0.990701
$636$	0	0
$637$	0	0
$638$	0	0
$639$	6.26504	0.247841
$640$	0	0
$641$	−17.8240	−0.704005	−0.352002	−	0.935999i	$-0.614499\pi$
$641$	−17.8240	−0.704005	−0.352002	+	0.935999i	$0.614499\pi$
$642$	0	0
$643$	−13.8564	−0.546442	−0.273221	−	0.961951i	$-0.588089\pi$
$643$	−13.8564	−0.546442	−0.273221	+	0.961951i	$0.588089\pi$
$644$	0	0
$645$	26.6972	1.05120
$646$	0	0
$647$	−44.4689	−1.74825	−0.874127	−	0.485697i	$-0.838566\pi$
$647$	−44.4689	−1.74825	−0.874127	+	0.485697i	$0.838566\pi$
$648$	0	0
$649$	38.1509	1.49755
$650$	0	0
$651$	−19.0795	−0.747783
$652$	0	0
$653$	−1.11583	−0.0436657	−0.0218328	−	0.999762i	$-0.506950\pi$
$653$	−1.11583	−0.0436657	−0.0218328	+	0.999762i	$0.506950\pi$
$654$	0	0
$655$	15.4835	0.604992
$656$	0	0
$657$	−13.9720	−0.545099
$658$	0	0
$659$	1.56758	0.0610644	0.0305322	−	0.999534i	$-0.490280\pi$
$659$	1.56758	0.0610644	0.0305322	+	0.999534i	$0.490280\pi$
$660$	0	0
$661$	−2.63693	−0.102565	−0.0512824	−	0.998684i	$-0.516331\pi$
$661$	−2.63693	−0.102565	−0.0512824	+	0.998684i	$0.516331\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	72.7492	2.82109
$666$	0	0
$667$	1.69328	0.0655639
$668$	0	0
$669$	−5.88623	−0.227575
$670$	0	0
$671$	4.14092	0.159858
$672$	0	0
$673$	−31.8531	−1.22785	−0.613923	−	0.789366i	$-0.710410\pi$
$673$	−31.8531	−1.22785	−0.613923	+	0.789366i	$0.710410\pi$
$674$	0	0
$675$	−2.06870	−0.0796241
$676$	0	0
$677$	−25.1260	−0.965671	−0.482835	−	0.875711i	$-0.660393\pi$
$677$	−25.1260	−0.965671	−0.482835	+	0.875711i	$0.660393\pi$
$678$	0	0
$679$	−41.5595	−1.59491
$680$	0	0
$681$	−13.1454	−0.503733
$682$	0	0
$683$	−35.1090	−1.34341	−0.671704	−	0.740820i	$-0.734437\pi$
$683$	−35.1090	−1.34341	−0.671704	+	0.740820i	$0.734437\pi$
$684$	0	0
$685$	36.1163	1.37993
$686$	0	0
$687$	−23.8710	−0.910736
$688$	0	0
$689$	0	0
$690$	0	0
$691$	14.7925	0.562732	0.281366	−	0.959600i	$-0.409212\pi$
$691$	14.7925	0.562732	0.281366	+	0.959600i	$0.409212\pi$
$692$	0	0
$693$	11.3086	0.429577
$694$	0	0
$695$	54.1398	2.05364
$696$	0	0
$697$	5.68097	0.215182
$698$	0	0
$699$	2.65905	0.100574
$700$	0	0
$701$	−28.9889	−1.09490	−0.547448	−	0.836839i	$-0.684401\pi$
$701$	−28.9889	−1.09490	−0.547448	+	0.836839i	$0.684401\pi$
$702$	0	0
$703$	72.6164	2.73878
$704$	0	0
$705$	−2.90838	−0.109536
$706$	0	0
$707$	−60.1715	−2.26298
$708$	0	0
$709$	−8.48462	−0.318647	−0.159323	−	0.987226i	$-0.550931\pi$
$709$	−8.48462	−0.318647	−0.159323	+	0.987226i	$0.550931\pi$
$710$	0	0
$711$	5.45858	0.204713
$712$	0	0
$713$	−21.0795	−0.789434
$714$	0	0
$715$	0	0
$716$	0	0
$717$	8.99863	0.336060
$718$	0	0
$719$	−5.39831	−0.201323	−0.100661	−	0.994921i	$-0.532096\pi$
$719$	−5.39831	−0.201323	−0.100661	+	0.994921i	$0.532096\pi$
$720$	0	0
$721$	18.2712	0.680456
$722$	0	0
$723$	1.45787	0.0542188
$724$	0	0
$725$	−0.851728	−0.0316324
$726$	0	0
$727$	9.28457	0.344346	0.172173	−	0.985067i	$-0.444921\pi$
$727$	9.28457	0.344346	0.172173	+	0.985067i	$0.444921\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−32.9357	−1.21817
$732$	0	0
$733$	24.8811	0.919005	0.459502	−	0.888177i	$-0.348028\pi$
$733$	24.8811	0.919005	0.459502	+	0.888177i	$0.348028\pi$
$734$	0	0
$735$	−18.2297	−0.672413
$736$	0	0
$737$	−28.7447	−1.05883
$738$	0	0
$739$	−18.1422	−0.667373	−0.333686	−	0.942684i	$-0.608293\pi$
$739$	−18.1422	−0.667373	−0.333686	+	0.942684i	$0.608293\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−49.5850	−1.81910	−0.909549	−	0.415596i	$-0.863573\pi$
$743$	−49.5850	−1.81910	−0.909549	+	0.415596i	$0.863573\pi$
$744$	0	0
$745$	−54.4480	−1.99482
$746$	0	0
$747$	−14.3060	−0.523430
$748$	0	0
$749$	−3.84866	−0.140627
$750$	0	0
$751$	41.6714	1.52061	0.760305	−	0.649567i	$-0.225050\pi$
$751$	41.6714	1.52061	0.760305	+	0.649567i	$0.225050\pi$
$752$	0	0
$753$	3.98236	0.145125
$754$	0	0
$755$	−41.6389	−1.51540
$756$	0	0
$757$	−14.3032	−0.519857	−0.259929	−	0.965628i	$-0.583699\pi$
$757$	−14.3032	−0.519857	−0.259929	+	0.965628i	$0.583699\pi$
$758$	0	0
$759$	12.4940	0.453505
$760$	0	0
$761$	−52.4720	−1.90211	−0.951054	−	0.309023i	$-0.899998\pi$
$761$	−52.4720	−1.90211	−0.951054	+	0.309023i	$0.899998\pi$
$762$	0	0
$763$	−32.1770	−1.16489
$764$	0	0
$765$	8.72046	0.315289
$766$	0	0
$767$	0	0
$768$	0	0
$769$	6.79004	0.244855	0.122428	−	0.992477i	$-0.460932\pi$
$769$	6.79004	0.244855	0.122428	+	0.992477i	$0.460932\pi$
$770$	0	0
$771$	15.2959	0.550868
$772$	0	0
$773$	39.0260	1.40367	0.701834	−	0.712340i	$-0.252365\pi$
$773$	39.0260	1.40367	0.701834	+	0.712340i	$0.252365\pi$
$774$	0	0
$775$	10.6031	0.380876
$776$	0	0
$777$	−36.7734	−1.31924
$778$	0	0
$779$	12.7316	0.456157
$780$	0	0
$781$	19.0328	0.681048
$782$	0	0
$783$	0.411722	0.0147138
$784$	0	0
$785$	30.4149	1.08556
$786$	0	0
$787$	−24.5427	−0.874853	−0.437426	−	0.899254i	$-0.644110\pi$
$787$	−24.5427	−0.874853	−0.437426	+	0.899254i	$0.644110\pi$
$788$	0	0
$789$	14.8408	0.528348
$790$	0	0
$791$	−66.0988	−2.35020
$792$	0	0
$793$	0	0
$794$	0	0
$795$	36.2657	1.28621
$796$	0	0
$797$	15.7291	0.557152	0.278576	−	0.960414i	$-0.410138\pi$
$797$	15.7291	0.557152	0.278576	+	0.960414i	$0.410138\pi$
$798$	0	0
$799$	3.58799	0.126934
$800$	0	0
$801$	2.40802	0.0850831
$802$	0	0
$803$	−42.4461	−1.49789
$804$	0	0
$805$	−40.7025	−1.43458
$806$	0	0
$807$	12.7970	0.450475
$808$	0	0
$809$	−54.2376	−1.90689	−0.953446	−	0.301563i	$-0.902492\pi$
$809$	−54.2376	−1.90689	−0.953446	+	0.301563i	$0.902492\pi$
$810$	0	0
$811$	29.9691	1.05236	0.526178	−	0.850374i	$-0.323624\pi$
$811$	29.9691	1.05236	0.526178	+	0.850374i	$0.323624\pi$
$812$	0	0
$813$	−27.0802	−0.949744
$814$	0	0
$815$	44.4248	1.55613
$816$	0	0
$817$	−73.8120	−2.58235
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−13.4089	−0.467973	−0.233986	−	0.972240i	$-0.575177\pi$
$821$	−13.4089	−0.467973	−0.233986	+	0.972240i	$0.575177\pi$
$822$	0	0
$823$	−38.7542	−1.35089	−0.675443	−	0.737412i	$-0.736048\pi$
$823$	−38.7542	−1.35089	−0.675443	+	0.737412i	$0.736048\pi$
$824$	0	0
$825$	−6.28458	−0.218801
$826$	0	0
$827$	31.1229	1.08225	0.541124	−	0.840943i	$-0.317999\pi$
$827$	31.1229	1.08225	0.541124	+	0.840943i	$0.317999\pi$
$828$	0	0
$829$	41.0222	1.42476	0.712380	−	0.701794i	$-0.247617\pi$
$829$	41.0222	1.42476	0.712380	+	0.701794i	$0.247617\pi$
$830$	0	0
$831$	−7.02829	−0.243809
$832$	0	0
$833$	22.4895	0.779215
$834$	0	0
$835$	27.9140	0.966002
$836$	0	0
$837$	−5.12551	−0.177164
$838$	0	0
$839$	−14.1452	−0.488347	−0.244173	−	0.969732i	$-0.578517\pi$
$839$	−14.1452	−0.488347	−0.244173	+	0.969732i	$0.578517\pi$
$840$	0	0
$841$	−28.8305	−0.994155
$842$	0	0
$843$	18.8159	0.648055
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−6.59214	−0.226508
$848$	0	0
$849$	4.79334	0.164507
$850$	0	0
$851$	−40.6282	−1.39272
$852$	0	0
$853$	−7.63902	−0.261555	−0.130778	−	0.991412i	$-0.541747\pi$
$853$	−7.63902	−0.261555	−0.130778	+	0.991412i	$0.541747\pi$
$854$	0	0
$855$	19.5434	0.668370
$856$	0	0
$857$	6.11628	0.208928	0.104464	−	0.994529i	$-0.466687\pi$
$857$	6.11628	0.208928	0.104464	+	0.994529i	$0.466687\pi$
$858$	0	0
$859$	31.6762	1.08078	0.540388	−	0.841416i	$-0.318277\pi$
$859$	31.6762	1.08078	0.540388	+	0.841416i	$0.318277\pi$
$860$	0	0
$861$	−6.44735	−0.219725
$862$	0	0
$863$	28.0865	0.956076	0.478038	−	0.878339i	$-0.341348\pi$
$863$	28.0865	0.956076	0.478038	+	0.878339i	$0.341348\pi$
$864$	0	0
$865$	−11.8475	−0.402827
$866$	0	0
$867$	6.24179	0.211982
$868$	0	0
$869$	16.5829	0.562535
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−11.1646	−0.377863
$874$	0	0
$875$	−29.0108	−0.980743
$876$	0	0
$877$	46.6646	1.57575	0.787877	−	0.615833i	$-0.211181\pi$
$877$	46.6646	1.57575	0.787877	+	0.615833i	$0.211181\pi$
$878$	0	0
$879$	−29.4230	−0.992415
$880$	0	0
$881$	−18.8403	−0.634746	−0.317373	−	0.948301i	$-0.602801\pi$
$881$	−18.8403	−0.634746	−0.317373	+	0.948301i	$0.602801\pi$
$882$	0	0
$883$	−26.5979	−0.895092	−0.447546	−	0.894261i	$-0.647702\pi$
$883$	−26.5979	−0.895092	−0.447546	+	0.894261i	$0.647702\pi$
$884$	0	0
$885$	−33.3884	−1.12234
$886$	0	0
$887$	−14.0186	−0.470697	−0.235349	−	0.971911i	$-0.575623\pi$
$887$	−14.0186	−0.470697	−0.235349	+	0.971911i	$0.575623\pi$
$888$	0	0
$889$	−34.9533	−1.17230
$890$	0	0
$891$	3.03794	0.101775
$892$	0	0
$893$	8.04102	0.269083
$894$	0	0
$895$	−11.2172	−0.374948
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.11029	−0.0703821
$900$	0	0
$901$	−44.7401	−1.49051
$902$	0	0
$903$	37.3788	1.24389
$904$	0	0
$905$	−17.0038	−0.565227
$906$	0	0
$907$	−18.8130	−0.624675	−0.312338	−	0.949971i	$-0.601112\pi$
$907$	−18.8130	−0.624675	−0.312338	+	0.949971i	$0.601112\pi$
$908$	0	0
$909$	−16.1645	−0.536143
$910$	0	0
$911$	7.29460	0.241681	0.120840	−	0.992672i	$-0.461441\pi$
$911$	7.29460	0.241681	0.120840	+	0.992672i	$0.461441\pi$
$912$	0	0
$913$	−43.4608	−1.43834
$914$	0	0
$915$	−3.62399	−0.119805
$916$	0	0
$917$	21.6785	0.715888
$918$	0	0
$919$	33.9015	1.11831	0.559154	−	0.829064i	$-0.311126\pi$
$919$	33.9015	1.11831	0.559154	+	0.829064i	$0.311126\pi$
$920$	0	0
$921$	−30.5800	−1.00764
$922$	0	0
$923$	0	0
$924$	0	0
$925$	20.4363	0.671940
$926$	0	0
$927$	4.90839	0.161213
$928$	0	0
$929$	−13.1814	−0.432467	−0.216234	−	0.976342i	$-0.569377\pi$
$929$	−13.1814	−0.432467	−0.216234	+	0.976342i	$0.569377\pi$
$930$	0	0
$931$	50.4011	1.65183
$932$	0	0
$933$	−3.05106	−0.0998871
$934$	0	0
$935$	26.4923	0.866389
$936$	0	0
$937$	−33.2186	−1.08520	−0.542602	−	0.839990i	$-0.682561\pi$
$937$	−33.2186	−1.08520	−0.542602	+	0.839990i	$0.682561\pi$
$938$	0	0
$939$	−26.7958	−0.874448
$940$	0	0
$941$	−7.07347	−0.230588	−0.115294	−	0.993331i	$-0.536781\pi$
$941$	−7.07347	−0.230588	−0.115294	+	0.993331i	$0.536781\pi$
$942$	0	0
$943$	−7.12321	−0.231964
$944$	0	0
$945$	−9.89688	−0.321946
$946$	0	0
$947$	56.8558	1.84756	0.923782	−	0.382919i	$-0.125081\pi$
$947$	56.8558	1.84756	0.923782	+	0.382919i	$0.125081\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−33.1691	−1.07558
$952$	0	0
$953$	22.6448	0.733539	0.366769	−	0.930312i	$-0.380464\pi$
$953$	22.6448	0.733539	0.366769	+	0.930312i	$0.380464\pi$
$954$	0	0
$955$	25.4825	0.824595
$956$	0	0
$957$	1.25079	0.0404322
$958$	0	0
$959$	50.5664	1.63288
$960$	0	0
$961$	−4.72911	−0.152552
$962$	0	0
$963$	−1.03391	−0.0333172
$964$	0	0
$965$	−45.6902	−1.47082
$966$	0	0
$967$	32.2338	1.03657	0.518284	−	0.855208i	$-0.326571\pi$
$967$	32.2338	1.03657	0.518284	+	0.855208i	$0.326571\pi$
$968$	0	0
$969$	−24.1102	−0.774530
$970$	0	0
$971$	−14.4937	−0.465125	−0.232563	−	0.972581i	$-0.574711\pi$
$971$	−14.4937	−0.465125	−0.232563	+	0.972581i	$0.574711\pi$
$972$	0	0
$973$	75.8011	2.43007
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.0769	0.418366	0.209183	−	0.977876i	$-0.432920\pi$
$977$	13.0769	0.418366	0.209183	+	0.977876i	$0.432920\pi$
$978$	0	0
$979$	7.31541	0.233802
$980$	0	0
$981$	−8.64406	−0.275983
$982$	0	0
$983$	36.0315	1.14923	0.574613	−	0.818425i	$-0.305152\pi$
$983$	36.0315	1.14923	0.574613	+	0.818425i	$0.305152\pi$
$984$	0	0
$985$	−32.4688	−1.03454
$986$	0	0
$987$	−4.07202	−0.129614
$988$	0	0
$989$	41.2971	1.31317
$990$	0	0
$991$	36.5518	1.16111	0.580554	−	0.814222i	$-0.302836\pi$
$991$	36.5518	1.16111	0.580554	+	0.814222i	$0.302836\pi$
$992$	0	0
$993$	−25.8226	−0.819457
$994$	0	0
$995$	−32.5065	−1.03053
$996$	0	0
$997$	−40.0370	−1.26798	−0.633992	−	0.773340i	$-0.718585\pi$
$997$	−40.0370	−1.26798	−0.633992	+	0.773340i	$0.718585\pi$
$998$	0	0
$999$	−9.87882	−0.312552

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4056.2.a.bg.1.5	yes	6
4.3	odd	2		8112.2.a.cw.1.5		6
13.5	odd	4		4056.2.c.q.337.4		12
13.8	odd	4		4056.2.c.q.337.9		12
13.12	even	2		4056.2.a.bf.1.2	✓	6
52.51	odd	2		8112.2.a.cv.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4056.2.a.bf.1.2	✓	6	13.12	even	2
4056.2.a.bg.1.5	yes	6	1.1	even	1	trivial
4056.2.c.q.337.4		12	13.5	odd	4
4056.2.c.q.337.9		12	13.8	odd	4
8112.2.a.cv.1.2		6	52.51	odd	2
8112.2.a.cw.1.5		6	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4056 = 2^{3} \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4056.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.65870 q^{5} +3.72245 q^{7} +1.00000 q^{9} +3.03794 q^{11} -2.65870 q^{15} +3.27997 q^{17} +7.35072 q^{19} -3.72245 q^{21} -4.11266 q^{23} +2.06870 q^{25} -1.00000 q^{27} -0.411722 q^{29} +5.12551 q^{31} -3.03794 q^{33} +9.89688 q^{35} +9.87882 q^{37} +1.73202 q^{41} -10.0415 q^{43} +2.65870 q^{45} +1.09391 q^{47} +6.85662 q^{49} -3.27997 q^{51} -13.6404 q^{53} +8.07698 q^{55} -7.35072 q^{57} +12.5581 q^{59} +1.36307 q^{61} +3.72245 q^{63} -9.46192 q^{67} +4.11266 q^{69} +6.26504 q^{71} -13.9720 q^{73} -2.06870 q^{75} +11.3086 q^{77} +5.45858 q^{79} +1.00000 q^{81} -14.3060 q^{83} +8.72046 q^{85} +0.411722 q^{87} +2.40802 q^{89} -5.12551 q^{93} +19.5434 q^{95} -11.1646 q^{97} +3.03794 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + q^{5} + 7 q^{7} + 6 q^{9} + 8 q^{11} - q^{15} - 5 q^{17} + 19 q^{19} - 7 q^{21} - 6 q^{23} + 17 q^{25} - 6 q^{27} + 3 q^{29} + 9 q^{31} - 8 q^{33} + 4 q^{35} + 6 q^{37} - 15 q^{41} + 11 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + q^5 + 7 * q^7 + 6 * q^9 + 8 * q^11 - q^15 - 5 * q^17 + 19 * q^19 - 7 * q^21 - 6 * q^23 + 17 * q^25 - 6 * q^27 + 3 * q^29 + 9 * q^31 - 8 * q^33 + 4 * q^35 + 6 * q^37 - 15 * q^41 + 11 * q^43 + q^45 + 5 * q^47 + 5 * q^49 + 5 * q^51 + 12 * q^53 + 7 * q^55 - 19 * q^57 - 5 * q^59 + 17 * q^61 + 7 * q^63 + 13 * q^67 + 6 * q^69 + 26 * q^71 - 49 * q^73 - 17 * q^75 - 25 * q^77 + 14 * q^79 + 6 * q^81 - 3 * q^83 - 19 * q^85 - 3 * q^87 + 13 * q^89 - 9 * q^93 + 43 * q^95 - 14 * q^97 + 8 * q^99

Introduction

L-functions

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Database

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