LMFDB - Embedded newform 6008.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6008,2,Mod(1,6008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6008, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6008 = 2^{3} \cdot 751 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$47.9741215344$
Analytic rank:	$1$
Dimension:	$44$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	6008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.30487 q^{3} +0.148423 q^{5} -4.48376 q^{7} +7.92214 q^{9} +O(q^{10})$	$q-3.30487 q^{3} +0.148423 q^{5} -4.48376 q^{7} +7.92214 q^{9} -5.76809 q^{11} +2.52864 q^{13} -0.490519 q^{15} -1.39645 q^{17} +6.53547 q^{19} +14.8182 q^{21} -1.51253 q^{23} -4.97797 q^{25} -16.2670 q^{27} -7.16181 q^{29} +4.58421 q^{31} +19.0628 q^{33} -0.665494 q^{35} -3.08289 q^{37} -8.35682 q^{39} +1.44541 q^{41} -0.693289 q^{43} +1.17583 q^{45} +0.309604 q^{47} +13.1041 q^{49} +4.61509 q^{51} +9.64897 q^{53} -0.856119 q^{55} -21.5989 q^{57} +14.8274 q^{59} +14.2407 q^{61} -35.5209 q^{63} +0.375309 q^{65} -6.92913 q^{67} +4.99872 q^{69} +7.49241 q^{71} +1.54792 q^{73} +16.4515 q^{75} +25.8627 q^{77} -9.02035 q^{79} +29.9938 q^{81} -0.305722 q^{83} -0.207266 q^{85} +23.6688 q^{87} -8.31641 q^{89} -11.3378 q^{91} -15.1502 q^{93} +0.970016 q^{95} -14.7898 q^{97} -45.6956 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	$ 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) $ 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−3.30487	−1.90806	−0.954032	−	0.299703i	$-0.903112\pi$
$3$	−3.30487	−1.90806	−0.954032	+	0.299703i	$0.903112\pi$
$4$	0	0
$5$	0.148423	0.0663769	0.0331885	−	0.999449i	$-0.489434\pi$
$5$	0.148423	0.0663769	0.0331885	+	0.999449i	$0.489434\pi$
$6$	0	0
$7$	−4.48376	−1.69470	−0.847351	−	0.531034i	$-0.821804\pi$
$7$	−4.48376	−1.69470	−0.847351	+	0.531034i	$0.821804\pi$
$8$	0	0
$9$	7.92214	2.64071
$10$	0	0
$11$	−5.76809	−1.73914	−0.869572	−	0.493806i	$-0.835605\pi$
$11$	−5.76809	−1.73914	−0.869572	+	0.493806i	$0.835605\pi$
$12$	0	0
$13$	2.52864	0.701319	0.350659	−	0.936503i	$-0.385957\pi$
$13$	2.52864	0.701319	0.350659	+	0.936503i	$0.385957\pi$
$14$	0	0
$15$	−0.490519	−0.126651
$16$	0	0
$17$	−1.39645	−0.338690	−0.169345	−	0.985557i	$-0.554165\pi$
$17$	−1.39645	−0.338690	−0.169345	+	0.985557i	$0.554165\pi$
$18$	0	0
$19$	6.53547	1.49934	0.749670	−	0.661812i	$-0.230212\pi$
$19$	6.53547	1.49934	0.749670	+	0.661812i	$0.230212\pi$
$20$	0	0
$21$	14.8182	3.23360
$22$	0	0
$23$	−1.51253	−0.315385	−0.157692	−	0.987488i	$-0.550405\pi$
$23$	−1.51253	−0.315385	−0.157692	+	0.987488i	$0.550405\pi$
$24$	0	0
$25$	−4.97797	−0.995594
$26$	0	0
$27$	−16.2670	−3.13058
$28$	0	0
$29$	−7.16181	−1.32991	−0.664957	−	0.746881i	$-0.731550\pi$
$29$	−7.16181	−1.32991	−0.664957	+	0.746881i	$0.731550\pi$
$30$	0	0
$31$	4.58421	0.823348	0.411674	−	0.911331i	$-0.364944\pi$
$31$	4.58421	0.823348	0.411674	+	0.911331i	$0.364944\pi$
$32$	0	0
$33$	19.0628	3.31840
$34$	0	0
$35$	−0.665494	−0.112489
$36$	0	0
$37$	−3.08289	−0.506824	−0.253412	−	0.967358i	$-0.581553\pi$
$37$	−3.08289	−0.506824	−0.253412	+	0.967358i	$0.581553\pi$
$38$	0	0
$39$	−8.35682	−1.33816
$40$	0	0
$41$	1.44541	0.225735	0.112868	−	0.993610i	$-0.463996\pi$
$41$	1.44541	0.225735	0.112868	+	0.993610i	$0.463996\pi$
$42$	0	0
$43$	−0.693289	−0.105726	−0.0528628	−	0.998602i	$-0.516835\pi$
$43$	−0.693289	−0.105726	−0.0528628	+	0.998602i	$0.516835\pi$
$44$	0	0
$45$	1.17583	0.175282
$46$	0	0
$47$	0.309604	0.0451604	0.0225802	−	0.999745i	$-0.492812\pi$
$47$	0.309604	0.0451604	0.0225802	+	0.999745i	$0.492812\pi$
$48$	0	0
$49$	13.1041	1.87201
$50$	0	0
$51$	4.61509	0.646242
$52$	0	0
$53$	9.64897	1.32539	0.662694	−	0.748890i	$-0.269413\pi$
$53$	9.64897	1.32539	0.662694	+	0.748890i	$0.269413\pi$
$54$	0	0
$55$	−0.856119	−0.115439
$56$	0	0
$57$	−21.5989	−2.86084
$58$	0	0
$59$	14.8274	1.93036	0.965179	−	0.261589i	$-0.0842466\pi$
$59$	14.8274	1.93036	0.965179	+	0.261589i	$0.0842466\pi$
$60$	0	0
$61$	14.2407	1.82333	0.911666	−	0.410931i	$-0.134796\pi$
$61$	14.2407	1.82333	0.911666	+	0.410931i	$0.134796\pi$
$62$	0	0
$63$	−35.5209	−4.47522
$64$	0	0
$65$	0.375309	0.0465514
$66$	0	0
$67$	−6.92913	−0.846527	−0.423264	−	0.906007i	$-0.639116\pi$
$67$	−6.92913	−0.846527	−0.423264	+	0.906007i	$0.639116\pi$
$68$	0	0
$69$	4.99872	0.601775
$70$	0	0
$71$	7.49241	0.889186	0.444593	−	0.895733i	$-0.353348\pi$
$71$	7.49241	0.889186	0.444593	+	0.895733i	$0.353348\pi$
$72$	0	0
$73$	1.54792	0.181171	0.0905853	−	0.995889i	$-0.471126\pi$
$73$	1.54792	0.181171	0.0905853	+	0.995889i	$0.471126\pi$
$74$	0	0
$75$	16.4515	1.89966
$76$	0	0
$77$	25.8627	2.94733
$78$	0	0
$79$	−9.02035	−1.01487	−0.507434	−	0.861690i	$-0.669406\pi$
$79$	−9.02035	−1.01487	−0.507434	+	0.861690i	$0.669406\pi$
$80$	0	0
$81$	29.9938	3.33265
$82$	0	0
$83$	−0.305722	−0.0335574	−0.0167787	−	0.999859i	$-0.505341\pi$
$83$	−0.305722	−0.0335574	−0.0167787	+	0.999859i	$0.505341\pi$
$84$	0	0
$85$	−0.207266	−0.0224812
$86$	0	0
$87$	23.6688	2.53756
$88$	0	0
$89$	−8.31641	−0.881537	−0.440769	−	0.897621i	$-0.645294\pi$
$89$	−8.31641	−0.881537	−0.440769	+	0.897621i	$0.645294\pi$
$90$	0	0
$91$	−11.3378	−1.18853
$92$	0	0
$93$	−15.1502	−1.57100
$94$	0	0
$95$	0.970016	0.0995216
$96$	0	0
$97$	−14.7898	−1.50168	−0.750841	−	0.660483i	$-0.770351\pi$
$97$	−14.7898	−1.50168	−0.750841	+	0.660483i	$0.770351\pi$
$98$	0	0
$99$	−45.6956	−4.59258
$100$	0	0
$101$	12.3697	1.23083	0.615413	−	0.788205i	$-0.288989\pi$
$101$	12.3697	1.23083	0.615413	+	0.788205i	$0.288989\pi$
$102$	0	0
$103$	3.83180	0.377558	0.188779	−	0.982020i	$-0.439547\pi$
$103$	3.83180	0.377558	0.188779	+	0.982020i	$0.439547\pi$
$104$	0	0
$105$	2.19937	0.214636
$106$	0	0
$107$	9.02224	0.872213	0.436107	−	0.899895i	$-0.356357\pi$
$107$	9.02224	0.872213	0.436107	+	0.899895i	$0.356357\pi$
$108$	0	0
$109$	−13.7229	−1.31441	−0.657207	−	0.753710i	$-0.728262\pi$
$109$	−13.7229	−1.31441	−0.657207	+	0.753710i	$0.728262\pi$
$110$	0	0
$111$	10.1885	0.967054
$112$	0	0
$113$	15.5587	1.46364	0.731819	−	0.681499i	$-0.238671\pi$
$113$	15.5587	1.46364	0.731819	+	0.681499i	$0.238671\pi$
$114$	0	0
$115$	−0.224495	−0.0209343
$116$	0	0
$117$	20.0322	1.85198
$118$	0	0
$119$	6.26136	0.573978
$120$	0	0
$121$	22.2708	2.02462
$122$	0	0
$123$	−4.77688	−0.430717
$124$	0	0
$125$	−1.48096	−0.132461
$126$	0	0
$127$	−0.872690	−0.0774387	−0.0387193	−	0.999250i	$-0.512328\pi$
$127$	−0.872690	−0.0774387	−0.0387193	+	0.999250i	$0.512328\pi$
$128$	0	0
$129$	2.29123	0.201731
$130$	0	0
$131$	5.63571	0.492394	0.246197	−	0.969220i	$-0.420819\pi$
$131$	5.63571	0.492394	0.246197	+	0.969220i	$0.420819\pi$
$132$	0	0
$133$	−29.3035	−2.54093
$134$	0	0
$135$	−2.41440	−0.207799
$136$	0	0
$137$	17.8345	1.52370	0.761852	−	0.647751i	$-0.224290\pi$
$137$	17.8345	1.52370	0.761852	+	0.647751i	$0.224290\pi$
$138$	0	0
$139$	17.6861	1.50011	0.750057	−	0.661373i	$-0.230026\pi$
$139$	17.6861	1.50011	0.750057	+	0.661373i	$0.230026\pi$
$140$	0	0
$141$	−1.02320	−0.0861690
$142$	0	0
$143$	−14.5854	−1.21969
$144$	0	0
$145$	−1.06298	−0.0882757
$146$	0	0
$147$	−43.3073	−3.57192
$148$	0	0
$149$	−5.56767	−0.456121	−0.228061	−	0.973647i	$-0.573238\pi$
$149$	−5.56767	−0.456121	−0.228061	+	0.973647i	$0.573238\pi$
$150$	0	0
$151$	−9.71377	−0.790496	−0.395248	−	0.918575i	$-0.629341\pi$
$151$	−9.71377	−0.790496	−0.395248	+	0.918575i	$0.629341\pi$
$152$	0	0
$153$	−11.0629	−0.894382
$154$	0	0
$155$	0.680404	0.0546513
$156$	0	0
$157$	−9.10929	−0.727000	−0.363500	−	0.931594i	$-0.618418\pi$
$157$	−9.10929	−0.727000	−0.363500	+	0.931594i	$0.618418\pi$
$158$	0	0
$159$	−31.8886	−2.52893
$160$	0	0
$161$	6.78183	0.534483
$162$	0	0
$163$	−1.64943	−0.129194	−0.0645968	−	0.997911i	$-0.520576\pi$
$163$	−1.64943	−0.129194	−0.0645968	+	0.997911i	$0.520576\pi$
$164$	0	0
$165$	2.82936	0.220265
$166$	0	0
$167$	−5.97735	−0.462541	−0.231271	−	0.972889i	$-0.574288\pi$
$167$	−5.97735	−0.462541	−0.231271	+	0.972889i	$0.574288\pi$
$168$	0	0
$169$	−6.60597	−0.508152
$170$	0	0
$171$	51.7749	3.95932
$172$	0	0
$173$	7.18636	0.546369	0.273184	−	0.961962i	$-0.411923\pi$
$173$	7.18636	0.546369	0.273184	+	0.961962i	$0.411923\pi$
$174$	0	0
$175$	22.3200	1.68723
$176$	0	0
$177$	−49.0024	−3.68325
$178$	0	0
$179$	2.15011	0.160707	0.0803535	−	0.996766i	$-0.474395\pi$
$179$	2.15011	0.160707	0.0803535	+	0.996766i	$0.474395\pi$
$180$	0	0
$181$	8.78730	0.653155	0.326578	−	0.945170i	$-0.394105\pi$
$181$	8.78730	0.653155	0.326578	+	0.945170i	$0.394105\pi$
$182$	0	0
$183$	−47.0635	−3.47904
$184$	0	0
$185$	−0.457573	−0.0336414
$186$	0	0
$187$	8.05486	0.589030
$188$	0	0
$189$	72.9373	5.30541
$190$	0	0
$191$	2.48682	0.179940	0.0899700	−	0.995944i	$-0.471323\pi$
$191$	2.48682	0.179940	0.0899700	+	0.995944i	$0.471323\pi$
$192$	0	0
$193$	−12.6742	−0.912307	−0.456153	−	0.889901i	$-0.650773\pi$
$193$	−12.6742	−0.912307	−0.456153	+	0.889901i	$0.650773\pi$
$194$	0	0
$195$	−1.24035	−0.0888231
$196$	0	0
$197$	−3.79838	−0.270623	−0.135312	−	0.990803i	$-0.543204\pi$
$197$	−3.79838	−0.270623	−0.135312	+	0.990803i	$0.543204\pi$
$198$	0	0
$199$	1.88765	0.133812	0.0669061	−	0.997759i	$-0.478687\pi$
$199$	1.88765	0.133812	0.0669061	+	0.997759i	$0.478687\pi$
$200$	0	0
$201$	22.8998	1.61523
$202$	0	0
$203$	32.1118	2.25381
$204$	0	0
$205$	0.214533	0.0149836
$206$	0	0
$207$	−11.9825	−0.832841
$208$	0	0
$209$	−37.6972	−2.60757
$210$	0	0
$211$	−20.0694	−1.38163	−0.690816	−	0.723030i	$-0.742749\pi$
$211$	−20.0694	−1.38163	−0.690816	+	0.723030i	$0.742749\pi$
$212$	0	0
$213$	−24.7614	−1.69662
$214$	0	0
$215$	−0.102900	−0.00701774
$216$	0	0
$217$	−20.5545	−1.39533
$218$	0	0
$219$	−5.11568	−0.345685
$220$	0	0
$221$	−3.53113	−0.237529
$222$	0	0
$223$	−10.0681	−0.674212	−0.337106	−	0.941467i	$-0.609448\pi$
$223$	−10.0681	−0.674212	−0.337106	+	0.941467i	$0.609448\pi$
$224$	0	0
$225$	−39.4362	−2.62908
$226$	0	0
$227$	5.46529	0.362744	0.181372	−	0.983415i	$-0.441946\pi$
$227$	5.46529	0.362744	0.181372	+	0.983415i	$0.441946\pi$
$228$	0	0
$229$	6.35786	0.420139	0.210070	−	0.977686i	$-0.432631\pi$
$229$	6.35786	0.420139	0.210070	+	0.977686i	$0.432631\pi$
$230$	0	0
$231$	−85.4728	−5.62370
$232$	0	0
$233$	−17.2664	−1.13116	−0.565580	−	0.824694i	$-0.691348\pi$
$233$	−17.2664	−1.13116	−0.565580	+	0.824694i	$0.691348\pi$
$234$	0	0
$235$	0.0459525	0.00299761
$236$	0	0
$237$	29.8110	1.93644
$238$	0	0
$239$	11.7402	0.759412	0.379706	−	0.925107i	$-0.376025\pi$
$239$	11.7402	0.759412	0.379706	+	0.925107i	$0.376025\pi$
$240$	0	0
$241$	−9.45760	−0.609217	−0.304609	−	0.952478i	$-0.598526\pi$
$241$	−9.45760	−0.609217	−0.304609	+	0.952478i	$0.598526\pi$
$242$	0	0
$243$	−50.3246	−3.22832
$244$	0	0
$245$	1.94495	0.124258
$246$	0	0
$247$	16.5259	1.05152
$248$	0	0
$249$	1.01037	0.0640297
$250$	0	0
$251$	−23.0926	−1.45759	−0.728795	−	0.684732i	$-0.759919\pi$
$251$	−23.0926	−1.45759	−0.728795	+	0.684732i	$0.759919\pi$
$252$	0	0
$253$	8.72442	0.548500
$254$	0	0
$255$	0.684987	0.0428956
$256$	0	0
$257$	7.24364	0.451846	0.225923	−	0.974145i	$-0.427460\pi$
$257$	7.24364	0.451846	0.225923	+	0.974145i	$0.427460\pi$
$258$	0	0
$259$	13.8229	0.858916
$260$	0	0
$261$	−56.7368	−3.51192
$262$	0	0
$263$	−0.940250	−0.0579783	−0.0289891	−	0.999580i	$-0.509229\pi$
$263$	−0.940250	−0.0579783	−0.0289891	+	0.999580i	$0.509229\pi$
$264$	0	0
$265$	1.43213	0.0879752
$266$	0	0
$267$	27.4846	1.68203
$268$	0	0
$269$	26.2956	1.60327	0.801635	−	0.597814i	$-0.203964\pi$
$269$	26.2956	1.60327	0.801635	+	0.597814i	$0.203964\pi$
$270$	0	0
$271$	−10.0257	−0.609020	−0.304510	−	0.952509i	$-0.598493\pi$
$271$	−10.0257	−0.609020	−0.304510	+	0.952509i	$0.598493\pi$
$272$	0	0
$273$	37.4700	2.26779
$274$	0	0
$275$	28.7134	1.73148
$276$	0	0
$277$	15.6662	0.941290	0.470645	−	0.882323i	$-0.344021\pi$
$277$	15.6662	0.941290	0.470645	+	0.882323i	$0.344021\pi$
$278$	0	0
$279$	36.3167	2.17423
$280$	0	0
$281$	−27.6006	−1.64651	−0.823256	−	0.567670i	$-0.807845\pi$
$281$	−27.6006	−1.64651	−0.823256	+	0.567670i	$0.807845\pi$
$282$	0	0
$283$	−17.3803	−1.03315	−0.516577	−	0.856241i	$-0.672794\pi$
$283$	−17.3803	−1.03315	−0.516577	+	0.856241i	$0.672794\pi$
$284$	0	0
$285$	−3.20577	−0.189894
$286$	0	0
$287$	−6.48087	−0.382554
$288$	0	0
$289$	−15.0499	−0.885289
$290$	0	0
$291$	48.8784	2.86531
$292$	0	0
$293$	−10.9514	−0.639785	−0.319893	−	0.947454i	$-0.603647\pi$
$293$	−10.9514	−0.639785	−0.319893	+	0.947454i	$0.603647\pi$
$294$	0	0
$295$	2.20073	0.128131
$296$	0	0
$297$	93.8294	5.44454
$298$	0	0
$299$	−3.82465	−0.221185
$300$	0	0
$301$	3.10854	0.179173
$302$	0	0
$303$	−40.8800	−2.34850
$304$	0	0
$305$	2.11365	0.121027
$306$	0	0
$307$	20.6860	1.18061	0.590306	−	0.807180i	$-0.299007\pi$
$307$	20.6860	1.18061	0.590306	+	0.807180i	$0.299007\pi$
$308$	0	0
$309$	−12.6636	−0.720406
$310$	0	0
$311$	26.8238	1.52104	0.760519	−	0.649315i	$-0.224944\pi$
$311$	26.8238	1.52104	0.760519	+	0.649315i	$0.224944\pi$
$312$	0	0
$313$	26.2321	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$313$	26.2321	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$314$	0	0
$315$	−5.27214	−0.297051
$316$	0	0
$317$	−26.9213	−1.51205	−0.756025	−	0.654542i	$-0.772861\pi$
$317$	−26.9213	−1.51205	−0.756025	+	0.654542i	$0.772861\pi$
$318$	0	0
$319$	41.3099	2.31291
$320$	0	0
$321$	−29.8173	−1.66424
$322$	0	0
$323$	−9.12648	−0.507811
$324$	0	0
$325$	−12.5875	−0.698229
$326$	0	0
$327$	45.3523	2.50799
$328$	0	0
$329$	−1.38819	−0.0765334
$330$	0	0
$331$	34.1535	1.87725	0.938624	−	0.344943i	$-0.112102\pi$
$331$	34.1535	1.87725	0.938624	+	0.344943i	$0.112102\pi$
$332$	0	0
$333$	−24.4231	−1.33838
$334$	0	0
$335$	−1.02844	−0.0561899
$336$	0	0
$337$	−3.46160	−0.188565	−0.0942826	−	0.995545i	$-0.530056\pi$
$337$	−3.46160	−0.188565	−0.0942826	+	0.995545i	$0.530056\pi$
$338$	0	0
$339$	−51.4194	−2.79272
$340$	0	0
$341$	−26.4421	−1.43192
$342$	0	0
$343$	−27.3693	−1.47780
$344$	0	0
$345$	0.741926	0.0399440
$346$	0	0
$347$	19.8963	1.06809	0.534045	−	0.845456i	$-0.320671\pi$
$347$	19.8963	1.06809	0.534045	+	0.845456i	$0.320671\pi$
$348$	0	0
$349$	−31.7447	−1.69925	−0.849627	−	0.527385i	$-0.823173\pi$
$349$	−31.7447	−1.69925	−0.849627	+	0.527385i	$0.823173\pi$
$350$	0	0
$351$	−41.1334	−2.19554
$352$	0	0
$353$	−25.3293	−1.34814	−0.674072	−	0.738665i	$-0.735456\pi$
$353$	−25.3293	−1.34814	−0.674072	+	0.738665i	$0.735456\pi$
$354$	0	0
$355$	1.11205	0.0590214
$356$	0	0
$357$	−20.6930	−1.09519
$358$	0	0
$359$	−12.7839	−0.674706	−0.337353	−	0.941378i	$-0.609532\pi$
$359$	−12.7839	−0.674706	−0.337353	+	0.941378i	$0.609532\pi$
$360$	0	0
$361$	23.7124	1.24802
$362$	0	0
$363$	−73.6021	−3.86311
$364$	0	0
$365$	0.229748	0.0120256
$366$	0	0
$367$	−18.0515	−0.942279	−0.471140	−	0.882059i	$-0.656157\pi$
$367$	−18.0515	−0.942279	−0.471140	+	0.882059i	$0.656157\pi$
$368$	0	0
$369$	11.4507	0.596101
$370$	0	0
$371$	−43.2637	−2.24614
$372$	0	0
$373$	6.21241	0.321666	0.160833	−	0.986982i	$-0.448582\pi$
$373$	6.21241	0.321666	0.160833	+	0.986982i	$0.448582\pi$
$374$	0	0
$375$	4.89439	0.252745
$376$	0	0
$377$	−18.1096	−0.932694
$378$	0	0
$379$	32.9698	1.69355	0.846773	−	0.531955i	$-0.178542\pi$
$379$	32.9698	1.69355	0.846773	+	0.531955i	$0.178542\pi$
$380$	0	0
$381$	2.88412	0.147758
$382$	0	0
$383$	−20.5101	−1.04802	−0.524008	−	0.851713i	$-0.675564\pi$
$383$	−20.5101	−1.04802	−0.524008	+	0.851713i	$0.675564\pi$
$384$	0	0
$385$	3.83863	0.195635
$386$	0	0
$387$	−5.49233	−0.279191
$388$	0	0
$389$	−2.55097	−0.129339	−0.0646697	−	0.997907i	$-0.520599\pi$
$389$	−2.55097	−0.129339	−0.0646697	+	0.997907i	$0.520599\pi$
$390$	0	0
$391$	2.11218	0.106818
$392$	0	0
$393$	−18.6253	−0.939520
$394$	0	0
$395$	−1.33883	−0.0673639
$396$	0	0
$397$	−39.1921	−1.96699	−0.983497	−	0.180923i	$-0.942091\pi$
$397$	−39.1921	−1.96699	−0.983497	+	0.180923i	$0.942091\pi$
$398$	0	0
$399$	96.8440	4.84827
$400$	0	0
$401$	−3.38811	−0.169194	−0.0845971	−	0.996415i	$-0.526960\pi$
$401$	−3.38811	−0.169194	−0.0845971	+	0.996415i	$0.526960\pi$
$402$	0	0
$403$	11.5918	0.577430
$404$	0	0
$405$	4.45178	0.221211
$406$	0	0
$407$	17.7824	0.881440
$408$	0	0
$409$	−30.7071	−1.51837	−0.759185	−	0.650875i	$-0.774402\pi$
$409$	−30.7071	−1.51837	−0.759185	+	0.650875i	$0.774402\pi$
$410$	0	0
$411$	−58.9406	−2.90733
$412$	0	0
$413$	−66.4823	−3.27138
$414$	0	0
$415$	−0.0453763	−0.00222744
$416$	0	0
$417$	−58.4501	−2.86231
$418$	0	0
$419$	−27.3851	−1.33785	−0.668924	−	0.743331i	$-0.733245\pi$
$419$	−27.3851	−1.33785	−0.668924	+	0.743331i	$0.733245\pi$
$420$	0	0
$421$	8.24185	0.401683	0.200842	−	0.979624i	$-0.435632\pi$
$421$	8.24185	0.401683	0.200842	+	0.979624i	$0.435632\pi$
$422$	0	0
$423$	2.45272	0.119256
$424$	0	0
$425$	6.95150	0.337197
$426$	0	0
$427$	−63.8518	−3.09000
$428$	0	0
$429$	48.2029	2.32726
$430$	0	0
$431$	40.8702	1.96865	0.984325	−	0.176365i	$-0.0564338\pi$
$431$	40.8702	1.96865	0.984325	+	0.176365i	$0.0564338\pi$
$432$	0	0
$433$	−21.7992	−1.04760	−0.523800	−	0.851841i	$-0.675486\pi$
$433$	−21.7992	−1.04760	−0.523800	+	0.851841i	$0.675486\pi$
$434$	0	0
$435$	3.51300	0.168436
$436$	0	0
$437$	−9.88511	−0.472869
$438$	0	0
$439$	−14.8708	−0.709744	−0.354872	−	0.934915i	$-0.615476\pi$
$439$	−14.8708	−0.709744	−0.354872	+	0.934915i	$0.615476\pi$
$440$	0	0
$441$	103.812	4.94345
$442$	0	0
$443$	1.70472	0.0809936	0.0404968	−	0.999180i	$-0.487106\pi$
$443$	1.70472	0.0809936	0.0404968	+	0.999180i	$0.487106\pi$
$444$	0	0
$445$	−1.23435	−0.0585138
$446$	0	0
$447$	18.4004	0.870309
$448$	0	0
$449$	−38.6971	−1.82623	−0.913114	−	0.407704i	$-0.866330\pi$
$449$	−38.6971	−1.82623	−0.913114	+	0.407704i	$0.866330\pi$
$450$	0	0
$451$	−8.33725	−0.392586
$452$	0	0
$453$	32.1027	1.50832
$454$	0	0
$455$	−1.68280	−0.0788907
$456$	0	0
$457$	0.532922	0.0249291	0.0124645	−	0.999922i	$-0.496032\pi$
$457$	0.532922	0.0249291	0.0124645	+	0.999922i	$0.496032\pi$
$458$	0	0
$459$	22.7161	1.06030
$460$	0	0
$461$	34.6301	1.61288	0.806442	−	0.591313i	$-0.201390\pi$
$461$	34.6301	1.61288	0.806442	+	0.591313i	$0.201390\pi$
$462$	0	0
$463$	−25.4364	−1.18213	−0.591065	−	0.806624i	$-0.701292\pi$
$463$	−25.4364	−1.18213	−0.591065	+	0.806624i	$0.701292\pi$
$464$	0	0
$465$	−2.24864	−0.104278
$466$	0	0
$467$	0.501459	0.0232048	0.0116024	−	0.999933i	$-0.496307\pi$
$467$	0.501459	0.0232048	0.0116024	+	0.999933i	$0.496307\pi$
$468$	0	0
$469$	31.0685	1.43461
$470$	0	0
$471$	30.1050	1.38716
$472$	0	0
$473$	3.99895	0.183872
$474$	0	0
$475$	−32.5334	−1.49273
$476$	0	0
$477$	76.4405	3.49997
$478$	0	0
$479$	8.29956	0.379217	0.189608	−	0.981860i	$-0.439278\pi$
$479$	8.29956	0.379217	0.189608	+	0.981860i	$0.439278\pi$
$480$	0	0
$481$	−7.79553	−0.355446
$482$	0	0
$483$	−22.4130	−1.01983
$484$	0	0
$485$	−2.19516	−0.0996770
$486$	0	0
$487$	−31.4129	−1.42346	−0.711728	−	0.702455i	$-0.752087\pi$
$487$	−31.4129	−1.42346	−0.711728	+	0.702455i	$0.752087\pi$
$488$	0	0
$489$	5.45116	0.246510
$490$	0	0
$491$	−13.5894	−0.613282	−0.306641	−	0.951825i	$-0.599205\pi$
$491$	−13.5894	−0.613282	−0.306641	+	0.951825i	$0.599205\pi$
$492$	0	0
$493$	10.0011	0.450428
$494$	0	0
$495$	−6.78229	−0.304841
$496$	0	0
$497$	−33.5942	−1.50690
$498$	0	0
$499$	−44.0750	−1.97307	−0.986535	−	0.163551i	$-0.947705\pi$
$499$	−44.0750	−1.97307	−0.986535	+	0.163551i	$0.947705\pi$
$500$	0	0
$501$	19.7543	0.882558
$502$	0	0
$503$	9.64921	0.430237	0.215119	−	0.976588i	$-0.430986\pi$
$503$	9.64921	0.430237	0.215119	+	0.976588i	$0.430986\pi$
$504$	0	0
$505$	1.83594	0.0816985
$506$	0	0
$507$	21.8318	0.969586
$508$	0	0
$509$	−14.2454	−0.631415	−0.315708	−	0.948857i	$-0.602242\pi$
$509$	−14.2454	−0.631415	−0.315708	+	0.948857i	$0.602242\pi$
$510$	0	0
$511$	−6.94051	−0.307030
$512$	0	0
$513$	−106.312	−4.69381
$514$	0	0
$515$	0.568729	0.0250612
$516$	0	0
$517$	−1.78582	−0.0785404
$518$	0	0
$519$	−23.7500	−1.04251
$520$	0	0
$521$	−29.8294	−1.30685	−0.653425	−	0.756991i	$-0.726669\pi$
$521$	−29.8294	−1.30685	−0.653425	+	0.756991i	$0.726669\pi$
$522$	0	0
$523$	−22.7909	−0.996575	−0.498287	−	0.867012i	$-0.666038\pi$
$523$	−22.7909	−0.996575	−0.498287	+	0.867012i	$0.666038\pi$
$524$	0	0
$525$	−73.7647	−3.21935
$526$	0	0
$527$	−6.40164	−0.278860
$528$	0	0
$529$	−20.7122	−0.900532
$530$	0	0
$531$	117.464	5.09752
$532$	0	0
$533$	3.65492	0.158312
$534$	0	0
$535$	1.33911	0.0578948
$536$	0	0
$537$	−7.10583	−0.306639
$538$	0	0
$539$	−75.5855	−3.25570
$540$	0	0
$541$	−26.6036	−1.14378	−0.571888	−	0.820332i	$-0.693789\pi$
$541$	−26.6036	−1.14378	−0.571888	+	0.820332i	$0.693789\pi$
$542$	0	0
$543$	−29.0408	−1.24626
$544$	0	0
$545$	−2.03680	−0.0872468
$546$	0	0
$547$	6.86528	0.293538	0.146769	−	0.989171i	$-0.453113\pi$
$547$	6.86528	0.293538	0.146769	+	0.989171i	$0.453113\pi$
$548$	0	0
$549$	112.817	4.81490
$550$	0	0
$551$	−46.8058	−1.99399
$552$	0	0
$553$	40.4451	1.71990
$554$	0	0
$555$	1.51222	0.0641901
$556$	0	0
$557$	34.0133	1.44119	0.720596	−	0.693356i	$-0.243868\pi$
$557$	34.0133	1.44119	0.720596	+	0.693356i	$0.243868\pi$
$558$	0	0
$559$	−1.75308	−0.0741473
$560$	0	0
$561$	−26.6202	−1.12391
$562$	0	0
$563$	−28.3031	−1.19284	−0.596418	−	0.802674i	$-0.703410\pi$
$563$	−28.3031	−1.19284	−0.596418	+	0.802674i	$0.703410\pi$
$564$	0	0
$565$	2.30927	0.0971519
$566$	0	0
$567$	−134.485	−5.64784
$568$	0	0
$569$	6.45557	0.270632	0.135316	−	0.990803i	$-0.456795\pi$
$569$	6.45557	0.270632	0.135316	+	0.990803i	$0.456795\pi$
$570$	0	0
$571$	27.4735	1.14973	0.574865	−	0.818248i	$-0.305055\pi$
$571$	27.4735	1.14973	0.574865	+	0.818248i	$0.305055\pi$
$572$	0	0
$573$	−8.21860	−0.343337
$574$	0	0
$575$	7.52934	0.313995
$576$	0	0
$577$	−12.6039	−0.524706	−0.262353	−	0.964972i	$-0.584499\pi$
$577$	−12.6039	−0.524706	−0.262353	+	0.964972i	$0.584499\pi$
$578$	0	0
$579$	41.8864	1.74074
$580$	0	0
$581$	1.37079	0.0568698
$582$	0	0
$583$	−55.6561	−2.30504
$584$	0	0
$585$	2.97325	0.122929
$586$	0	0
$587$	−22.3671	−0.923190	−0.461595	−	0.887091i	$-0.652723\pi$
$587$	−22.3671	−0.923190	−0.461595	+	0.887091i	$0.652723\pi$
$588$	0	0
$589$	29.9600	1.23448
$590$	0	0
$591$	12.5531	0.516367
$592$	0	0
$593$	42.8316	1.75888	0.879442	−	0.476007i	$-0.157916\pi$
$593$	42.8316	1.75888	0.879442	+	0.476007i	$0.157916\pi$
$594$	0	0
$595$	0.929332	0.0380989
$596$	0	0
$597$	−6.23844	−0.255322
$598$	0	0
$599$	8.45315	0.345386	0.172693	−	0.984976i	$-0.444753\pi$
$599$	8.45315	0.345386	0.172693	+	0.984976i	$0.444753\pi$
$600$	0	0
$601$	−3.33543	−0.136055	−0.0680275	−	0.997683i	$-0.521671\pi$
$601$	−3.33543	−0.136055	−0.0680275	+	0.997683i	$0.521671\pi$
$602$	0	0
$603$	−54.8935	−2.23543
$604$	0	0
$605$	3.30551	0.134388
$606$	0	0
$607$	40.6823	1.65124	0.825622	−	0.564223i	$-0.190824\pi$
$607$	40.6823	1.65124	0.825622	+	0.564223i	$0.190824\pi$
$608$	0	0
$609$	−106.125	−4.30041
$610$	0	0
$611$	0.782878	0.0316718
$612$	0	0
$613$	24.7532	0.999774	0.499887	−	0.866091i	$-0.333375\pi$
$613$	24.7532	0.999774	0.499887	+	0.866091i	$0.333375\pi$
$614$	0	0
$615$	−0.709001	−0.0285897
$616$	0	0
$617$	21.6587	0.871945	0.435972	−	0.899960i	$-0.356405\pi$
$617$	21.6587	0.871945	0.435972	+	0.899960i	$0.356405\pi$
$618$	0	0
$619$	12.7606	0.512891	0.256446	−	0.966559i	$-0.417449\pi$
$619$	12.7606	0.512891	0.256446	+	0.966559i	$0.417449\pi$
$620$	0	0
$621$	24.6044	0.987339
$622$	0	0
$623$	37.2888	1.49394
$624$	0	0
$625$	24.6700	0.986802
$626$	0	0
$627$	124.584	4.97541
$628$	0	0
$629$	4.30511	0.171656
$630$	0	0
$631$	−7.34600	−0.292440	−0.146220	−	0.989252i	$-0.546711\pi$
$631$	−7.34600	−0.292440	−0.146220	+	0.989252i	$0.546711\pi$
$632$	0	0
$633$	66.3266	2.63624
$634$	0	0
$635$	−0.129528	−0.00514014
$636$	0	0
$637$	33.1356	1.31288
$638$	0	0
$639$	59.3559	2.34808
$640$	0	0
$641$	38.0218	1.50177	0.750885	−	0.660433i	$-0.229627\pi$
$641$	38.0218	1.50177	0.750885	+	0.660433i	$0.229627\pi$
$642$	0	0
$643$	6.02505	0.237605	0.118802	−	0.992918i	$-0.462095\pi$
$643$	6.02505	0.237605	0.118802	+	0.992918i	$0.462095\pi$
$644$	0	0
$645$	0.340071	0.0133903
$646$	0	0
$647$	−31.4833	−1.23774	−0.618869	−	0.785495i	$-0.712409\pi$
$647$	−31.4833	−1.23774	−0.618869	+	0.785495i	$0.712409\pi$
$648$	0	0
$649$	−85.5255	−3.35717
$650$	0	0
$651$	67.9298	2.66238
$652$	0	0
$653$	1.16063	0.0454191	0.0227096	−	0.999742i	$-0.492771\pi$
$653$	1.16063	0.0454191	0.0227096	+	0.999742i	$0.492771\pi$
$654$	0	0
$655$	0.836471	0.0326836
$656$	0	0
$657$	12.2629	0.478420
$658$	0	0
$659$	−6.08026	−0.236853	−0.118427	−	0.992963i	$-0.537785\pi$
$659$	−6.08026	−0.236853	−0.118427	+	0.992963i	$0.537785\pi$
$660$	0	0
$661$	8.29926	0.322804	0.161402	−	0.986889i	$-0.448398\pi$
$661$	8.29926	0.322804	0.161402	+	0.986889i	$0.448398\pi$
$662$	0	0
$663$	11.6699	0.453222
$664$	0	0
$665$	−4.34932	−0.168659
$666$	0	0
$667$	10.8325	0.419435
$668$	0	0
$669$	33.2738	1.28644
$670$	0	0
$671$	−82.1415	−3.17104
$672$	0	0
$673$	−8.31993	−0.320710	−0.160355	−	0.987059i	$-0.551264\pi$
$673$	−8.31993	−0.320710	−0.160355	+	0.987059i	$0.551264\pi$
$674$	0	0
$675$	80.9766	3.11679
$676$	0	0
$677$	44.2891	1.70217	0.851083	−	0.525030i	$-0.175946\pi$
$677$	44.2891	1.70217	0.851083	+	0.525030i	$0.175946\pi$
$678$	0	0
$679$	66.3141	2.54490
$680$	0	0
$681$	−18.0620	−0.692139
$682$	0	0
$683$	4.78069	0.182928	0.0914641	−	0.995808i	$-0.470845\pi$
$683$	4.78069	0.182928	0.0914641	+	0.995808i	$0.470845\pi$
$684$	0	0
$685$	2.64706	0.101139
$686$	0	0
$687$	−21.0119	−0.801653
$688$	0	0
$689$	24.3988	0.929520
$690$	0	0
$691$	31.3183	1.19140	0.595702	−	0.803205i	$-0.296874\pi$
$691$	31.3183	1.19140	0.595702	+	0.803205i	$0.296874\pi$
$692$	0	0
$693$	204.888	7.78305
$694$	0	0
$695$	2.62503	0.0995729
$696$	0	0
$697$	−2.01845	−0.0764541
$698$	0	0
$699$	57.0631	2.15833
$700$	0	0
$701$	16.7524	0.632729	0.316365	−	0.948638i	$-0.397538\pi$
$701$	16.7524	0.632729	0.316365	+	0.948638i	$0.397538\pi$
$702$	0	0
$703$	−20.1482	−0.759902
$704$	0	0
$705$	−0.151867	−0.00571963
$706$	0	0
$707$	−55.4625	−2.08588
$708$	0	0
$709$	−13.4759	−0.506099	−0.253050	−	0.967453i	$-0.581434\pi$
$709$	−13.4759	−0.506099	−0.253050	+	0.967453i	$0.581434\pi$
$710$	0	0
$711$	−71.4604	−2.67998
$712$	0	0
$713$	−6.93377	−0.259672
$714$	0	0
$715$	−2.16482	−0.0809596
$716$	0	0
$717$	−38.7999	−1.44901
$718$	0	0
$719$	10.0649	0.375356	0.187678	−	0.982231i	$-0.439904\pi$
$719$	10.0649	0.375356	0.187678	+	0.982231i	$0.439904\pi$
$720$	0	0
$721$	−17.1809	−0.639849
$722$	0	0
$723$	31.2561	1.16243
$724$	0	0
$725$	35.6513	1.32406
$726$	0	0
$727$	3.58010	0.132779	0.0663893	−	0.997794i	$-0.478852\pi$
$727$	3.58010	0.132779	0.0663893	+	0.997794i	$0.478852\pi$
$728$	0	0
$729$	76.3344	2.82720
$730$	0	0
$731$	0.968145	0.0358081
$732$	0	0
$733$	−17.6416	−0.651608	−0.325804	−	0.945437i	$-0.605635\pi$
$733$	−17.6416	−0.651608	−0.325804	+	0.945437i	$0.605635\pi$
$734$	0	0
$735$	−6.42781	−0.237093
$736$	0	0
$737$	39.9678	1.47223
$738$	0	0
$739$	−1.28656	−0.0473269	−0.0236635	−	0.999720i	$-0.507533\pi$
$739$	−1.28656	−0.0473269	−0.0236635	+	0.999720i	$0.507533\pi$
$740$	0	0
$741$	−54.6158	−2.00636
$742$	0	0
$743$	7.44654	0.273187	0.136594	−	0.990627i	$-0.456385\pi$
$743$	7.44654	0.273187	0.136594	+	0.990627i	$0.456385\pi$
$744$	0	0
$745$	−0.826372	−0.0302759
$746$	0	0
$747$	−2.42197	−0.0886154
$748$	0	0
$749$	−40.4536	−1.47814
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	76.3178	2.78118
$754$	0	0
$755$	−1.44175	−0.0524707
$756$	0	0
$757$	−33.3750	−1.21303	−0.606517	−	0.795070i	$-0.707434\pi$
$757$	−33.3750	−1.21303	−0.606517	+	0.795070i	$0.707434\pi$
$758$	0	0
$759$	−28.8330	−1.04657
$760$	0	0
$761$	13.8705	0.502805	0.251403	−	0.967883i	$-0.419108\pi$
$761$	13.8705	0.502805	0.251403	+	0.967883i	$0.419108\pi$
$762$	0	0
$763$	61.5301	2.22754
$764$	0	0
$765$	−1.64199	−0.0593663
$766$	0	0
$767$	37.4931	1.35380
$768$	0	0
$769$	7.08639	0.255542	0.127771	−	0.991804i	$-0.459218\pi$
$769$	7.08639	0.255542	0.127771	+	0.991804i	$0.459218\pi$
$770$	0	0
$771$	−23.9392	−0.862151
$772$	0	0
$773$	23.5465	0.846908	0.423454	−	0.905917i	$-0.360817\pi$
$773$	23.5465	0.846908	0.423454	+	0.905917i	$0.360817\pi$
$774$	0	0
$775$	−22.8201	−0.819721
$776$	0	0
$777$	−45.6830	−1.63887
$778$	0	0
$779$	9.44643	0.338454
$780$	0	0
$781$	−43.2169	−1.54642
$782$	0	0
$783$	116.501	4.16341
$784$	0	0
$785$	−1.35203	−0.0482561
$786$	0	0
$787$	−38.8167	−1.38367	−0.691833	−	0.722057i	$-0.743197\pi$
$787$	−38.8167	−1.38367	−0.691833	+	0.722057i	$0.743197\pi$
$788$	0	0
$789$	3.10740	0.110626
$790$	0	0
$791$	−69.7614	−2.48043
$792$	0	0
$793$	36.0096	1.27874
$794$	0	0
$795$	−4.73301	−0.167862
$796$	0	0
$797$	−10.3686	−0.367274	−0.183637	−	0.982994i	$-0.558787\pi$
$797$	−10.3686	−0.367274	−0.183637	+	0.982994i	$0.558787\pi$
$798$	0	0
$799$	−0.432348	−0.0152954
$800$	0	0
$801$	−65.8837	−2.32789
$802$	0	0
$803$	−8.92856	−0.315082
$804$	0	0
$805$	1.00658	0.0354774
$806$	0	0
$807$	−86.9033	−3.05914
$808$	0	0
$809$	−43.6326	−1.53404	−0.767020	−	0.641624i	$-0.778261\pi$
$809$	−43.6326	−1.53404	−0.767020	+	0.641624i	$0.778261\pi$
$810$	0	0
$811$	39.1932	1.37626	0.688130	−	0.725587i	$-0.258432\pi$
$811$	39.1932	1.37626	0.688130	+	0.725587i	$0.258432\pi$
$812$	0	0
$813$	33.1337	1.16205
$814$	0	0
$815$	−0.244814	−0.00857548
$816$	0	0
$817$	−4.53097	−0.158519
$818$	0	0
$819$	−89.8197	−3.13856
$820$	0	0
$821$	−18.8792	−0.658890	−0.329445	−	0.944175i	$-0.606862\pi$
$821$	−18.8792	−0.658890	−0.329445	+	0.944175i	$0.606862\pi$
$822$	0	0
$823$	−50.7060	−1.76750	−0.883750	−	0.467959i	$-0.844989\pi$
$823$	−50.7060	−1.76750	−0.883750	+	0.467959i	$0.844989\pi$
$824$	0	0
$825$	−94.8938	−3.30378
$826$	0	0
$827$	−2.32122	−0.0807167	−0.0403583	−	0.999185i	$-0.512850\pi$
$827$	−2.32122	−0.0807167	−0.0403583	+	0.999185i	$0.512850\pi$
$828$	0	0
$829$	43.9550	1.52662	0.763311	−	0.646032i	$-0.223573\pi$
$829$	43.9550	1.52662	0.763311	+	0.646032i	$0.223573\pi$
$830$	0	0
$831$	−51.7747	−1.79604
$832$	0	0
$833$	−18.2993	−0.634032
$834$	0	0
$835$	−0.887178	−0.0307021
$836$	0	0
$837$	−74.5713	−2.57756
$838$	0	0
$839$	−18.6031	−0.642251	−0.321125	−	0.947037i	$-0.604061\pi$
$839$	−18.6031	−0.642251	−0.321125	+	0.947037i	$0.604061\pi$
$840$	0	0
$841$	22.2915	0.768673
$842$	0	0
$843$	91.2162	3.14165
$844$	0	0
$845$	−0.980480	−0.0337296
$846$	0	0
$847$	−99.8570	−3.43113
$848$	0	0
$849$	57.4396	1.97132
$850$	0	0
$851$	4.66298	0.159845
$852$	0	0
$853$	19.1768	0.656600	0.328300	−	0.944574i	$-0.393524\pi$
$853$	19.1768	0.656600	0.328300	+	0.944574i	$0.393524\pi$
$854$	0	0
$855$	7.68460	0.262808
$856$	0	0
$857$	57.3100	1.95767	0.978836	−	0.204645i	$-0.0656039\pi$
$857$	57.3100	1.95767	0.978836	+	0.204645i	$0.0656039\pi$
$858$	0	0
$859$	24.4731	0.835010	0.417505	−	0.908675i	$-0.362905\pi$
$859$	24.4731	0.835010	0.417505	+	0.908675i	$0.362905\pi$
$860$	0	0
$861$	21.4184	0.729937
$862$	0	0
$863$	−2.07967	−0.0707929	−0.0353965	−	0.999373i	$-0.511269\pi$
$863$	−2.07967	−0.0707929	−0.0353965	+	0.999373i	$0.511269\pi$
$864$	0	0
$865$	1.06662	0.0362663
$866$	0	0
$867$	49.7380	1.68919
$868$	0	0
$869$	52.0302	1.76500
$870$	0	0
$871$	−17.5213	−0.593686
$872$	0	0
$873$	−117.167	−3.96551
$874$	0	0
$875$	6.64028	0.224483
$876$	0	0
$877$	45.3804	1.53239	0.766194	−	0.642610i	$-0.222148\pi$
$877$	45.3804	1.53239	0.766194	+	0.642610i	$0.222148\pi$
$878$	0	0
$879$	36.1928	1.22075
$880$	0	0
$881$	10.4651	0.352578	0.176289	−	0.984338i	$-0.443591\pi$
$881$	10.4651	0.352578	0.176289	+	0.984338i	$0.443591\pi$
$882$	0	0
$883$	27.4005	0.922099	0.461049	−	0.887375i	$-0.347473\pi$
$883$	27.4005	0.922099	0.461049	+	0.887375i	$0.347473\pi$
$884$	0	0
$885$	−7.27311	−0.244483
$886$	0	0
$887$	−45.8910	−1.54087	−0.770435	−	0.637519i	$-0.779961\pi$
$887$	−45.8910	−1.54087	−0.770435	+	0.637519i	$0.779961\pi$
$888$	0	0
$889$	3.91293	0.131235
$890$	0	0
$891$	−173.007	−5.79595
$892$	0	0
$893$	2.02341	0.0677108
$894$	0	0
$895$	0.319127	0.0106672
$896$	0	0
$897$	12.6400	0.422036
$898$	0	0
$899$	−32.8312	−1.09498
$900$	0	0
$901$	−13.4743	−0.448895
$902$	0	0
$903$	−10.2733	−0.341874
$904$	0	0
$905$	1.30424	0.0433544
$906$	0	0
$907$	−3.41092	−0.113258	−0.0566290	−	0.998395i	$-0.518035\pi$
$907$	−3.41092	−0.113258	−0.0566290	+	0.998395i	$0.518035\pi$
$908$	0	0
$909$	97.9940	3.25026
$910$	0	0
$911$	22.2029	0.735616	0.367808	−	0.929902i	$-0.380108\pi$
$911$	22.2029	0.735616	0.367808	+	0.929902i	$0.380108\pi$
$912$	0	0
$913$	1.76343	0.0583611
$914$	0	0
$915$	−6.98533	−0.230928
$916$	0	0
$917$	−25.2692	−0.834461
$918$	0	0
$919$	−29.8312	−0.984042	−0.492021	−	0.870583i	$-0.663742\pi$
$919$	−29.8312	−0.984042	−0.492021	+	0.870583i	$0.663742\pi$
$920$	0	0
$921$	−68.3644	−2.25268
$922$	0	0
$923$	18.9456	0.623603
$924$	0	0
$925$	15.3465	0.504591
$926$	0	0
$927$	30.3560	0.997023
$928$	0	0
$929$	10.6696	0.350057	0.175029	−	0.984563i	$-0.443998\pi$
$929$	10.6696	0.350057	0.175029	+	0.984563i	$0.443998\pi$
$930$	0	0
$931$	85.6414	2.80678
$932$	0	0
$933$	−88.6491	−2.90224
$934$	0	0
$935$	1.19553	0.0390980
$936$	0	0
$937$	−2.29449	−0.0749578	−0.0374789	−	0.999297i	$-0.511933\pi$
$937$	−2.29449	−0.0749578	−0.0374789	+	0.999297i	$0.511933\pi$
$938$	0	0
$939$	−86.6935	−2.82913
$940$	0	0
$941$	15.9868	0.521155	0.260578	−	0.965453i	$-0.416087\pi$
$941$	15.9868	0.521155	0.260578	+	0.965453i	$0.416087\pi$
$942$	0	0
$943$	−2.18623	−0.0711934
$944$	0	0
$945$	10.8256	0.352157
$946$	0	0
$947$	−25.0989	−0.815606	−0.407803	−	0.913070i	$-0.633705\pi$
$947$	−25.0989	−0.815606	−0.407803	+	0.913070i	$0.633705\pi$
$948$	0	0
$949$	3.91414	0.127058
$950$	0	0
$951$	88.9713	2.88509
$952$	0	0
$953$	−6.77623	−0.219504	−0.109752	−	0.993959i	$-0.535006\pi$
$953$	−6.77623	−0.219504	−0.109752	+	0.993959i	$0.535006\pi$
$954$	0	0
$955$	0.369102	0.0119439
$956$	0	0
$957$	−136.524	−4.41319
$958$	0	0
$959$	−79.9656	−2.58222
$960$	0	0
$961$	−9.98502	−0.322097
$962$	0	0
$963$	71.4754	2.30326
$964$	0	0
$965$	−1.88114	−0.0605561
$966$	0	0
$967$	41.8228	1.34493	0.672465	−	0.740129i	$-0.265236\pi$
$967$	41.8228	1.34493	0.672465	+	0.740129i	$0.265236\pi$
$968$	0	0
$969$	30.1618	0.968936
$970$	0	0
$971$	17.8551	0.572999	0.286499	−	0.958080i	$-0.407508\pi$
$971$	17.8551	0.572999	0.286499	+	0.958080i	$0.407508\pi$
$972$	0	0
$973$	−79.3001	−2.54224
$974$	0	0
$975$	41.6000	1.33227
$976$	0	0
$977$	56.8503	1.81880	0.909401	−	0.415920i	$-0.136540\pi$
$977$	56.8503	1.81880	0.909401	+	0.415920i	$0.136540\pi$
$978$	0	0
$979$	47.9698	1.53312
$980$	0	0
$981$	−108.715	−3.47099
$982$	0	0
$983$	−39.0293	−1.24484	−0.622421	−	0.782682i	$-0.713851\pi$
$983$	−39.0293	−1.24484	−0.622421	+	0.782682i	$0.713851\pi$
$984$	0	0
$985$	−0.563768	−0.0179631
$986$	0	0
$987$	4.58778	0.146031
$988$	0	0
$989$	1.04862	0.0333442
$990$	0	0
$991$	32.4204	1.02987	0.514933	−	0.857230i	$-0.327817\pi$
$991$	32.4204	1.02987	0.514933	+	0.857230i	$0.327817\pi$
$992$	0	0
$993$	−112.873	−3.58191
$994$	0	0
$995$	0.280172	0.00888205
$996$	0	0
$997$	−35.9083	−1.13723	−0.568614	−	0.822605i	$-0.692520\pi$
$997$	−35.9083	−1.13723	−0.568614	+	0.822605i	$0.692520\pi$
$998$	0	0
$999$	50.1494	1.58666

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.1	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.1	✓	44	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6008 = 2^{3} \cdot 751 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6008.a (trivial)

\(f(q)\)	\(=\)	\(q-3.30487 q^{3} +0.148423 q^{5} -4.48376 q^{7} +7.92214 q^{9} +O(q^{10})\)	\(q-3.30487 q^{3} +0.148423 q^{5} -4.48376 q^{7} +7.92214 q^{9} -5.76809 q^{11} +2.52864 q^{13} -0.490519 q^{15} -1.39645 q^{17} +6.53547 q^{19} +14.8182 q^{21} -1.51253 q^{23} -4.97797 q^{25} -16.2670 q^{27} -7.16181 q^{29} +4.58421 q^{31} +19.0628 q^{33} -0.665494 q^{35} -3.08289 q^{37} -8.35682 q^{39} +1.44541 q^{41} -0.693289 q^{43} +1.17583 q^{45} +0.309604 q^{47} +13.1041 q^{49} +4.61509 q^{51} +9.64897 q^{53} -0.856119 q^{55} -21.5989 q^{57} +14.8274 q^{59} +14.2407 q^{61} -35.5209 q^{63} +0.375309 q^{65} -6.92913 q^{67} +4.99872 q^{69} +7.49241 q^{71} +1.54792 q^{73} +16.4515 q^{75} +25.8627 q^{77} -9.02035 q^{79} +29.9938 q^{81} -0.305722 q^{83} -0.207266 q^{85} +23.6688 q^{87} -8.31641 q^{89} -11.3378 q^{91} -15.1502 q^{93} +0.970016 q^{95} -14.7898 q^{97} -45.6956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9} - 19 q^{11} - 10 q^{13} - 17 q^{15} - 16 q^{17} - 25 q^{19} + 16 q^{21} - 29 q^{23} + 29 q^{25} - 50 q^{27} + 35 q^{29} - 49 q^{31} - 28 q^{33} - 37 q^{35} - 30 q^{37} - 28 q^{39} - 14 q^{41} - 35 q^{43} + 6 q^{45} - 45 q^{47} + 20 q^{49} - 17 q^{51} + 18 q^{53} - 53 q^{55} - 31 q^{57} - 57 q^{59} + 27 q^{61} - 77 q^{63} - 21 q^{65} - 56 q^{67} + 36 q^{69} - 52 q^{71} - 68 q^{73} - 77 q^{75} + 37 q^{77} - 55 q^{79} + 28 q^{81} - 51 q^{83} - 16 q^{85} - 67 q^{87} - 21 q^{89} - 51 q^{91} - 14 q^{93} - 56 q^{95} - 67 q^{97} - 58 q^{99}+O(q^{100}) \) 44 * q - 14 * q^3 + 7 * q^5 - 20 * q^7 + 38 * q^9 - 19 * q^11 - 10 * q^13 - 17 * q^15 - 16 * q^17 - 25 * q^19 + 16 * q^21 - 29 * q^23 + 29 * q^25 - 50 * q^27 + 35 * q^29 - 49 * q^31 - 28 * q^33 - 37 * q^35 - 30 * q^37 - 28 * q^39 - 14 * q^41 - 35 * q^43 + 6 * q^45 - 45 * q^47 + 20 * q^49 - 17 * q^51 + 18 * q^53 - 53 * q^55 - 31 * q^57 - 57 * q^59 + 27 * q^61 - 77 * q^63 - 21 * q^65 - 56 * q^67 + 36 * q^69 - 52 * q^71 - 68 * q^73 - 77 * q^75 + 37 * q^77 - 55 * q^79 + 28 * q^81 - 51 * q^83 - 16 * q^85 - 67 * q^87 - 21 * q^89 - 51 * q^91 - 14 * q^93 - 56 * q^95 - 67 * q^97 - 58 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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