LMFDB - Embedded newform 6008.2.a.b.1.39

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			1.39
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)

$f(q)$	$=$	$q+1.94041 q^{3} +0.760740 q^{5} -0.889256 q^{7} +0.765188 q^{9} +O(q^{10})$	$q+1.94041 q^{3} +0.760740 q^{5} -0.889256 q^{7} +0.765188 q^{9} +5.58913 q^{11} -2.54312 q^{13} +1.47615 q^{15} -2.48725 q^{17} -7.62571 q^{19} -1.72552 q^{21} -5.94283 q^{23} -4.42127 q^{25} -4.33645 q^{27} -0.846897 q^{29} +4.09083 q^{31} +10.8452 q^{33} -0.676492 q^{35} -1.60686 q^{37} -4.93470 q^{39} -3.27310 q^{41} -5.80394 q^{43} +0.582109 q^{45} -5.13868 q^{47} -6.20922 q^{49} -4.82627 q^{51} -3.26645 q^{53} +4.25188 q^{55} -14.7970 q^{57} +2.72175 q^{59} +0.913771 q^{61} -0.680448 q^{63} -1.93466 q^{65} +8.89055 q^{67} -11.5315 q^{69} -0.635627 q^{71} +1.47952 q^{73} -8.57908 q^{75} -4.97017 q^{77} +9.39602 q^{79} -10.7101 q^{81} -5.77044 q^{83} -1.89215 q^{85} -1.64333 q^{87} -11.2838 q^{89} +2.26149 q^{91} +7.93789 q^{93} -5.80118 q^{95} -1.62968 q^{97} +4.27674 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.94041	1.12030	0.560148	−	0.828393i	$-0.310744\pi$
$3$	1.94041	1.12030	0.560148	+	0.828393i	$0.310744\pi$
$4$	0	0
$5$	0.760740	0.340213	0.170107	−	0.985426i	$-0.445589\pi$
$5$	0.760740	0.340213	0.170107	+	0.985426i	$0.445589\pi$
$6$	0	0
$7$	−0.889256	−0.336107	−0.168054	−	0.985778i	$-0.553748\pi$
$7$	−0.889256	−0.336107	−0.168054	+	0.985778i	$0.553748\pi$
$8$	0	0
$9$	0.765188	0.255063
$10$	0	0
$11$	5.58913	1.68519	0.842594	−	0.538550i	$-0.181028\pi$
$11$	5.58913	1.68519	0.842594	+	0.538550i	$0.181028\pi$
$12$	0	0
$13$	−2.54312	−0.705336	−0.352668	−	0.935749i	$-0.614725\pi$
$13$	−2.54312	−0.705336	−0.352668	+	0.935749i	$0.614725\pi$
$14$	0	0
$15$	1.47615	0.381140
$16$	0	0
$17$	−2.48725	−0.603246	−0.301623	−	0.953427i	$-0.597528\pi$
$17$	−2.48725	−0.603246	−0.301623	+	0.953427i	$0.597528\pi$
$18$	0	0
$19$	−7.62571	−1.74946	−0.874728	−	0.484613i	$-0.838960\pi$
$19$	−7.62571	−1.74946	−0.874728	+	0.484613i	$0.838960\pi$
$20$	0	0
$21$	−1.72552	−0.376539
$22$	0	0
$23$	−5.94283	−1.23917	−0.619583	−	0.784931i	$-0.712698\pi$
$23$	−5.94283	−1.23917	−0.619583	+	0.784931i	$0.712698\pi$
$24$	0	0
$25$	−4.42127	−0.884255
$26$	0	0
$27$	−4.33645	−0.834550
$28$	0	0
$29$	−0.846897	−0.157265	−0.0786325	−	0.996904i	$-0.525055\pi$
$29$	−0.846897	−0.157265	−0.0786325	+	0.996904i	$0.525055\pi$
$30$	0	0
$31$	4.09083	0.734735	0.367368	−	0.930076i	$-0.380259\pi$
$31$	4.09083	0.734735	0.367368	+	0.930076i	$0.380259\pi$
$32$	0	0
$33$	10.8452	1.88791
$34$	0	0
$35$	−0.676492	−0.114348
$36$	0	0
$37$	−1.60686	−0.264166	−0.132083	−	0.991239i	$-0.542167\pi$
$37$	−1.60686	−0.264166	−0.132083	+	0.991239i	$0.542167\pi$
$38$	0	0
$39$	−4.93470	−0.790185
$40$	0	0
$41$	−3.27310	−0.511173	−0.255586	−	0.966786i	$-0.582269\pi$
$41$	−3.27310	−0.511173	−0.255586	+	0.966786i	$0.582269\pi$
$42$	0	0
$43$	−5.80394	−0.885093	−0.442547	−	0.896746i	$-0.645925\pi$
$43$	−5.80394	−0.885093	−0.442547	+	0.896746i	$0.645925\pi$
$44$	0	0
$45$	0.582109	0.0867757
$46$	0	0
$47$	−5.13868	−0.749553	−0.374777	−	0.927115i	$-0.622281\pi$
$47$	−5.13868	−0.749553	−0.374777	+	0.927115i	$0.622281\pi$
$48$	0	0
$49$	−6.20922	−0.887032
$50$	0	0
$51$	−4.82627	−0.675814
$52$	0	0
$53$	−3.26645	−0.448681	−0.224341	−	0.974511i	$-0.572023\pi$
$53$	−3.26645	−0.448681	−0.224341	+	0.974511i	$0.572023\pi$
$54$	0	0
$55$	4.25188	0.573323
$56$	0	0
$57$	−14.7970	−1.95991
$58$	0	0
$59$	2.72175	0.354342	0.177171	−	0.984180i	$-0.443305\pi$
$59$	2.72175	0.354342	0.177171	+	0.984180i	$0.443305\pi$
$60$	0	0
$61$	0.913771	0.116996	0.0584982	−	0.998288i	$-0.481369\pi$
$61$	0.913771	0.116996	0.0584982	+	0.998288i	$0.481369\pi$
$62$	0	0
$63$	−0.680448	−0.0857284
$64$	0	0
$65$	−1.93466	−0.239965
$66$	0	0
$67$	8.89055	1.08615	0.543077	−	0.839683i	$-0.317259\pi$
$67$	8.89055	1.08615	0.543077	+	0.839683i	$0.317259\pi$
$68$	0	0
$69$	−11.5315	−1.38823
$70$	0	0
$71$	−0.635627	−0.0754350	−0.0377175	−	0.999288i	$-0.512009\pi$
$71$	−0.635627	−0.0754350	−0.0377175	+	0.999288i	$0.512009\pi$
$72$	0	0
$73$	1.47952	0.173165	0.0865825	−	0.996245i	$-0.472405\pi$
$73$	1.47952	0.173165	0.0865825	+	0.996245i	$0.472405\pi$
$74$	0	0
$75$	−8.57908	−0.990627
$76$	0	0
$77$	−4.97017	−0.566403
$78$	0	0
$79$	9.39602	1.05714	0.528568	−	0.848891i	$-0.322729\pi$
$79$	9.39602	1.05714	0.528568	+	0.848891i	$0.322729\pi$
$80$	0	0
$81$	−10.7101	−1.19001
$82$	0	0
$83$	−5.77044	−0.633388	−0.316694	−	0.948528i	$-0.602573\pi$
$83$	−5.77044	−0.633388	−0.316694	+	0.948528i	$0.602573\pi$
$84$	0	0
$85$	−1.89215	−0.205232
$86$	0	0
$87$	−1.64333	−0.176183
$88$	0	0
$89$	−11.2838	−1.19608	−0.598041	−	0.801466i	$-0.704054\pi$
$89$	−11.2838	−1.19608	−0.598041	+	0.801466i	$0.704054\pi$
$90$	0	0
$91$	2.26149	0.237068
$92$	0	0
$93$	7.93789	0.823121
$94$	0	0
$95$	−5.80118	−0.595189
$96$	0	0
$97$	−1.62968	−0.165469	−0.0827346	−	0.996572i	$-0.526365\pi$
$97$	−1.62968	−0.165469	−0.0827346	+	0.996572i	$0.526365\pi$
$98$	0	0
$99$	4.27674	0.429828
$100$	0	0
$101$	−1.98318	−0.197333	−0.0986667	−	0.995121i	$-0.531458\pi$
$101$	−1.98318	−0.197333	−0.0986667	+	0.995121i	$0.531458\pi$
$102$	0	0
$103$	−3.93741	−0.387964	−0.193982	−	0.981005i	$-0.562140\pi$
$103$	−3.93741	−0.387964	−0.193982	+	0.981005i	$0.562140\pi$
$104$	0	0
$105$	−1.31267	−0.128104
$106$	0	0
$107$	8.39359	0.811439	0.405720	−	0.913998i	$-0.367021\pi$
$107$	8.39359	0.811439	0.405720	+	0.913998i	$0.367021\pi$
$108$	0	0
$109$	−3.58941	−0.343803	−0.171901	−	0.985114i	$-0.554991\pi$
$109$	−3.58941	−0.343803	−0.171901	+	0.985114i	$0.554991\pi$
$110$	0	0
$111$	−3.11796	−0.295944
$112$	0	0
$113$	−6.02379	−0.566671	−0.283335	−	0.959021i	$-0.591441\pi$
$113$	−6.02379	−0.566671	−0.283335	+	0.959021i	$0.591441\pi$
$114$	0	0
$115$	−4.52095	−0.421581
$116$	0	0
$117$	−1.94597	−0.179905
$118$	0	0
$119$	2.21180	0.202755
$120$	0	0
$121$	20.2384	1.83986
$122$	0	0
$123$	−6.35116	−0.572665
$124$	0	0
$125$	−7.16714	−0.641049
$126$	0	0
$127$	0.546205	0.0484678	0.0242339	−	0.999706i	$-0.492285\pi$
$127$	0.546205	0.0484678	0.0242339	+	0.999706i	$0.492285\pi$
$128$	0	0
$129$	−11.2620	−0.991566
$130$	0	0
$131$	7.90479	0.690645	0.345322	−	0.938484i	$-0.387770\pi$
$131$	7.90479	0.690645	0.345322	+	0.938484i	$0.387770\pi$
$132$	0	0
$133$	6.78120	0.588005
$134$	0	0
$135$	−3.29891	−0.283925
$136$	0	0
$137$	17.4011	1.48667	0.743336	−	0.668918i	$-0.233242\pi$
$137$	17.4011	1.48667	0.743336	+	0.668918i	$0.233242\pi$
$138$	0	0
$139$	−3.16276	−0.268262	−0.134131	−	0.990964i	$-0.542824\pi$
$139$	−3.16276	−0.268262	−0.134131	+	0.990964i	$0.542824\pi$
$140$	0	0
$141$	−9.97114	−0.839721
$142$	0	0
$143$	−14.2139	−1.18862
$144$	0	0
$145$	−0.644269	−0.0535036
$146$	0	0
$147$	−12.0484	−0.993738
$148$	0	0
$149$	24.3175	1.99217	0.996085	−	0.0883961i	$-0.0281741\pi$
$149$	24.3175	1.99217	0.996085	+	0.0883961i	$0.0281741\pi$
$150$	0	0
$151$	21.7037	1.76622	0.883112	−	0.469163i	$-0.155444\pi$
$151$	21.7037	1.76622	0.883112	+	0.469163i	$0.155444\pi$
$152$	0	0
$153$	−1.90321	−0.153866
$154$	0	0
$155$	3.11206	0.249967
$156$	0	0
$157$	5.62629	0.449027	0.224514	−	0.974471i	$-0.427921\pi$
$157$	5.62629	0.449027	0.224514	+	0.974471i	$0.427921\pi$
$158$	0	0
$159$	−6.33825	−0.502656
$160$	0	0
$161$	5.28469	0.416492
$162$	0	0
$163$	−0.216070	−0.0169239	−0.00846197	−	0.999964i	$-0.502694\pi$
$163$	−0.216070	−0.0169239	−0.00846197	+	0.999964i	$0.502694\pi$
$164$	0	0
$165$	8.25038	0.642291
$166$	0	0
$167$	7.16553	0.554485	0.277243	−	0.960800i	$-0.410579\pi$
$167$	7.16553	0.554485	0.277243	+	0.960800i	$0.410579\pi$
$168$	0	0
$169$	−6.53252	−0.502502
$170$	0	0
$171$	−5.83510	−0.446221
$172$	0	0
$173$	−11.1074	−0.844478	−0.422239	−	0.906485i	$-0.638756\pi$
$173$	−11.1074	−0.844478	−0.422239	+	0.906485i	$0.638756\pi$
$174$	0	0
$175$	3.93164	0.297204
$176$	0	0
$177$	5.28131	0.396968
$178$	0	0
$179$	−15.7388	−1.17637	−0.588187	−	0.808725i	$-0.700158\pi$
$179$	−15.7388	−1.17637	−0.588187	+	0.808725i	$0.700158\pi$
$180$	0	0
$181$	16.3363	1.21427	0.607135	−	0.794598i	$-0.292319\pi$
$181$	16.3363	1.21427	0.607135	+	0.794598i	$0.292319\pi$
$182$	0	0
$183$	1.77309	0.131071
$184$	0	0
$185$	−1.22240	−0.0898728
$186$	0	0
$187$	−13.9015	−1.01658
$188$	0	0
$189$	3.85621	0.280498
$190$	0	0
$191$	−14.6581	−1.06062	−0.530312	−	0.847803i	$-0.677925\pi$
$191$	−14.6581	−1.06062	−0.530312	+	0.847803i	$0.677925\pi$
$192$	0	0
$193$	11.7349	0.844694	0.422347	−	0.906434i	$-0.361206\pi$
$193$	11.7349	0.844694	0.422347	+	0.906434i	$0.361206\pi$
$194$	0	0
$195$	−3.75402	−0.268831
$196$	0	0
$197$	17.2492	1.22895	0.614477	−	0.788934i	$-0.289367\pi$
$197$	17.2492	1.22895	0.614477	+	0.788934i	$0.289367\pi$
$198$	0	0
$199$	26.2438	1.86037	0.930186	−	0.367087i	$-0.119645\pi$
$199$	26.2438	1.86037	0.930186	+	0.367087i	$0.119645\pi$
$200$	0	0
$201$	17.2513	1.21681
$202$	0	0
$203$	0.753108	0.0528578
$204$	0	0
$205$	−2.48998	−0.173908
$206$	0	0
$207$	−4.54738	−0.316065
$208$	0	0
$209$	−42.6211	−2.94816
$210$	0	0
$211$	−11.9267	−0.821070	−0.410535	−	0.911845i	$-0.634658\pi$
$211$	−11.9267	−0.821070	−0.410535	+	0.911845i	$0.634658\pi$
$212$	0	0
$213$	−1.23338	−0.0845095
$214$	0	0
$215$	−4.41529	−0.301120
$216$	0	0
$217$	−3.63780	−0.246950
$218$	0	0
$219$	2.87088	0.193996
$220$	0	0
$221$	6.32537	0.425491
$222$	0	0
$223$	−2.42930	−0.162678	−0.0813390	−	0.996686i	$-0.525920\pi$
$223$	−2.42930	−0.162678	−0.0813390	+	0.996686i	$0.525920\pi$
$224$	0	0
$225$	−3.38311	−0.225540
$226$	0	0
$227$	−18.7788	−1.24639	−0.623197	−	0.782065i	$-0.714167\pi$
$227$	−18.7788	−1.24639	−0.623197	+	0.782065i	$0.714167\pi$
$228$	0	0
$229$	−0.853762	−0.0564182	−0.0282091	−	0.999602i	$-0.508980\pi$
$229$	−0.853762	−0.0564182	−0.0282091	+	0.999602i	$0.508980\pi$
$230$	0	0
$231$	−9.64416	−0.634539
$232$	0	0
$233$	−2.78016	−0.182134	−0.0910670	−	0.995845i	$-0.529028\pi$
$233$	−2.78016	−0.182134	−0.0910670	+	0.995845i	$0.529028\pi$
$234$	0	0
$235$	−3.90920	−0.255008
$236$	0	0
$237$	18.2321	1.18430
$238$	0	0
$239$	−18.3807	−1.18895	−0.594475	−	0.804114i	$-0.702640\pi$
$239$	−18.3807	−1.18895	−0.594475	+	0.804114i	$0.702640\pi$
$240$	0	0
$241$	1.39286	0.0897221	0.0448611	−	0.998993i	$-0.485715\pi$
$241$	1.39286	0.0897221	0.0448611	+	0.998993i	$0.485715\pi$
$242$	0	0
$243$	−7.77254	−0.498608
$244$	0	0
$245$	−4.72361	−0.301780
$246$	0	0
$247$	19.3931	1.23395
$248$	0	0
$249$	−11.1970	−0.709582
$250$	0	0
$251$	−0.00595589	−0.000375933 0	−0.000187966	−	1.00000i	$-0.500060\pi$
$251$	−0.00595589	−0.000375933 0	−0.000187966		1.00000i	$0.500060\pi$
$252$	0	0
$253$	−33.2153	−2.08823
$254$	0	0
$255$	−3.67154	−0.229921
$256$	0	0
$257$	0.646536	0.0403298	0.0201649	−	0.999797i	$-0.493581\pi$
$257$	0.646536	0.0403298	0.0201649	+	0.999797i	$0.493581\pi$
$258$	0	0
$259$	1.42891	0.0887881
$260$	0	0
$261$	−0.648036	−0.0401124
$262$	0	0
$263$	−8.75176	−0.539656	−0.269828	−	0.962908i	$-0.586967\pi$
$263$	−8.75176	−0.539656	−0.269828	+	0.962908i	$0.586967\pi$
$264$	0	0
$265$	−2.48492	−0.152647
$266$	0	0
$267$	−21.8952	−1.33997
$268$	0	0
$269$	−24.7747	−1.51054	−0.755270	−	0.655414i	$-0.772494\pi$
$269$	−24.7747	−1.51054	−0.755270	+	0.655414i	$0.772494\pi$
$270$	0	0
$271$	0.947238	0.0575406	0.0287703	−	0.999586i	$-0.490841\pi$
$271$	0.947238	0.0575406	0.0287703	+	0.999586i	$0.490841\pi$
$272$	0	0
$273$	4.38821	0.265587
$274$	0	0
$275$	−24.7111	−1.49014
$276$	0	0
$277$	9.33287	0.560758	0.280379	−	0.959889i	$-0.409540\pi$
$277$	9.33287	0.560758	0.280379	+	0.959889i	$0.409540\pi$
$278$	0	0
$279$	3.13026	0.187404
$280$	0	0
$281$	6.54062	0.390181	0.195090	−	0.980785i	$-0.437500\pi$
$281$	6.54062	0.390181	0.195090	+	0.980785i	$0.437500\pi$
$282$	0	0
$283$	−29.6081	−1.76002	−0.880010	−	0.474954i	$-0.842465\pi$
$283$	−29.6081	−1.76002	−0.880010	+	0.474954i	$0.842465\pi$
$284$	0	0
$285$	−11.2567	−0.666787
$286$	0	0
$287$	2.91063	0.171809
$288$	0	0
$289$	−10.8136	−0.636095
$290$	0	0
$291$	−3.16225	−0.185374
$292$	0	0
$293$	−14.4482	−0.844074	−0.422037	−	0.906579i	$-0.638685\pi$
$293$	−14.4482	−0.844074	−0.422037	+	0.906579i	$0.638685\pi$
$294$	0	0
$295$	2.07055	0.120552
$296$	0	0
$297$	−24.2370	−1.40637
$298$	0	0
$299$	15.1133	0.874028
$300$	0	0
$301$	5.16119	0.297486
$302$	0	0
$303$	−3.84817	−0.221072
$304$	0	0
$305$	0.695142	0.0398037
$306$	0	0
$307$	−0.0600235	−0.00342572	−0.00171286	−	0.999999i	$-0.500545\pi$
$307$	−0.0600235	−0.00342572	−0.00171286	+	0.999999i	$0.500545\pi$
$308$	0	0
$309$	−7.64018	−0.434635
$310$	0	0
$311$	6.48188	0.367554	0.183777	−	0.982968i	$-0.441168\pi$
$311$	6.48188	0.367554	0.183777	+	0.982968i	$0.441168\pi$
$312$	0	0
$313$	6.20051	0.350473	0.175237	−	0.984526i	$-0.443931\pi$
$313$	6.20051	0.350473	0.175237	+	0.984526i	$0.443931\pi$
$314$	0	0
$315$	−0.517644	−0.0291659
$316$	0	0
$317$	−8.47100	−0.475779	−0.237889	−	0.971292i	$-0.576456\pi$
$317$	−8.47100	−0.475779	−0.237889	+	0.971292i	$0.576456\pi$
$318$	0	0
$319$	−4.73342	−0.265021
$320$	0	0
$321$	16.2870	0.909052
$322$	0	0
$323$	18.9670	1.05535
$324$	0	0
$325$	11.2438	0.623696
$326$	0	0
$327$	−6.96492	−0.385161
$328$	0	0
$329$	4.56960	0.251930
$330$	0	0
$331$	−22.4479	−1.23385	−0.616924	−	0.787023i	$-0.711621\pi$
$331$	−22.4479	−1.23385	−0.616924	+	0.787023i	$0.711621\pi$
$332$	0	0
$333$	−1.22955	−0.0673789
$334$	0	0
$335$	6.76339	0.369524
$336$	0	0
$337$	−33.7773	−1.83997	−0.919984	−	0.391956i	$-0.871799\pi$
$337$	−33.7773	−1.83997	−0.919984	+	0.391956i	$0.871799\pi$
$338$	0	0
$339$	−11.6886	−0.634839
$340$	0	0
$341$	22.8642	1.23817
$342$	0	0
$343$	11.7464	0.634245
$344$	0	0
$345$	−8.77249	−0.472295
$346$	0	0
$347$	−17.8075	−0.955958	−0.477979	−	0.878371i	$-0.658630\pi$
$347$	−17.8075	−0.955958	−0.477979	+	0.878371i	$0.658630\pi$
$348$	0	0
$349$	−4.37841	−0.234371	−0.117185	−	0.993110i	$-0.537387\pi$
$349$	−4.37841	−0.234371	−0.117185	+	0.993110i	$0.537387\pi$
$350$	0	0
$351$	11.0281	0.588638
$352$	0	0
$353$	28.6094	1.52272	0.761361	−	0.648328i	$-0.224531\pi$
$353$	28.6094	1.52272	0.761361	+	0.648328i	$0.224531\pi$
$354$	0	0
$355$	−0.483547	−0.0256640
$356$	0	0
$357$	4.29179	0.227146
$358$	0	0
$359$	19.6096	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$359$	19.6096	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$360$	0	0
$361$	39.1514	2.06060
$362$	0	0
$363$	39.2708	2.06118
$364$	0	0
$365$	1.12553	0.0589130
$366$	0	0
$367$	−18.1778	−0.948871	−0.474436	−	0.880290i	$-0.657348\pi$
$367$	−18.1778	−0.948871	−0.474436	+	0.880290i	$0.657348\pi$
$368$	0	0
$369$	−2.50454	−0.130381
$370$	0	0
$371$	2.90471	0.150805
$372$	0	0
$373$	−18.6427	−0.965280	−0.482640	−	0.875819i	$-0.660322\pi$
$373$	−18.6427	−0.965280	−0.482640	+	0.875819i	$0.660322\pi$
$374$	0	0
$375$	−13.9072	−0.718164
$376$	0	0
$377$	2.15377	0.110925
$378$	0	0
$379$	13.3055	0.683458	0.341729	−	0.939799i	$-0.388987\pi$
$379$	13.3055	0.683458	0.341729	+	0.939799i	$0.388987\pi$
$380$	0	0
$381$	1.05986	0.0542983
$382$	0	0
$383$	32.7040	1.67110	0.835548	−	0.549417i	$-0.185150\pi$
$383$	32.7040	1.67110	0.835548	+	0.549417i	$0.185150\pi$
$384$	0	0
$385$	−3.78101	−0.192698
$386$	0	0
$387$	−4.44111	−0.225754
$388$	0	0
$389$	3.35251	0.169979	0.0849894	−	0.996382i	$-0.472914\pi$
$389$	3.35251	0.169979	0.0849894	+	0.996382i	$0.472914\pi$
$390$	0	0
$391$	14.7813	0.747521
$392$	0	0
$393$	15.3385	0.773726
$394$	0	0
$395$	7.14793	0.359651
$396$	0	0
$397$	37.1617	1.86509	0.932547	−	0.361049i	$-0.117581\pi$
$397$	37.1617	1.86509	0.932547	+	0.361049i	$0.117581\pi$
$398$	0	0
$399$	13.1583	0.658739
$400$	0	0
$401$	16.4617	0.822058	0.411029	−	0.911622i	$-0.365170\pi$
$401$	16.4617	0.822058	0.411029	+	0.911622i	$0.365170\pi$
$402$	0	0
$403$	−10.4035	−0.518235
$404$	0	0
$405$	−8.14757	−0.404856
$406$	0	0
$407$	−8.98095	−0.445169
$408$	0	0
$409$	22.7898	1.12688	0.563441	−	0.826157i	$-0.309477\pi$
$409$	22.7898	1.12688	0.563441	+	0.826157i	$0.309477\pi$
$410$	0	0
$411$	33.7652	1.66551
$412$	0	0
$413$	−2.42033	−0.119097
$414$	0	0
$415$	−4.38980	−0.215487
$416$	0	0
$417$	−6.13705	−0.300533
$418$	0	0
$419$	−26.4630	−1.29280	−0.646401	−	0.762997i	$-0.723727\pi$
$419$	−26.4630	−1.29280	−0.646401	+	0.762997i	$0.723727\pi$
$420$	0	0
$421$	−18.9057	−0.921405	−0.460703	−	0.887555i	$-0.652403\pi$
$421$	−18.9057	−0.921405	−0.460703	+	0.887555i	$0.652403\pi$
$422$	0	0
$423$	−3.93205	−0.191183
$424$	0	0
$425$	10.9968	0.533423
$426$	0	0
$427$	−0.812576	−0.0393233
$428$	0	0
$429$	−27.5807	−1.33161
$430$	0	0
$431$	−40.0800	−1.93059	−0.965294	−	0.261166i	$-0.915893\pi$
$431$	−40.0800	−1.93059	−0.965294	+	0.261166i	$0.915893\pi$
$432$	0	0
$433$	−16.2689	−0.781831	−0.390916	−	0.920427i	$-0.627842\pi$
$433$	−16.2689	−0.781831	−0.390916	+	0.920427i	$0.627842\pi$
$434$	0	0
$435$	−1.25015	−0.0599399
$436$	0	0
$437$	45.3183	2.16787
$438$	0	0
$439$	20.3794	0.972658	0.486329	−	0.873776i	$-0.338336\pi$
$439$	20.3794	0.972658	0.486329	+	0.873776i	$0.338336\pi$
$440$	0	0
$441$	−4.75123	−0.226249
$442$	0	0
$443$	−28.1415	−1.33704	−0.668522	−	0.743692i	$-0.733073\pi$
$443$	−28.1415	−1.33704	−0.668522	+	0.743692i	$0.733073\pi$
$444$	0	0
$445$	−8.58405	−0.406923
$446$	0	0
$447$	47.1860	2.23182
$448$	0	0
$449$	28.8287	1.36051	0.680256	−	0.732975i	$-0.261869\pi$
$449$	28.8287	1.36051	0.680256	+	0.732975i	$0.261869\pi$
$450$	0	0
$451$	−18.2938	−0.861422
$452$	0	0
$453$	42.1141	1.97869
$454$	0	0
$455$	1.72040	0.0806538
$456$	0	0
$457$	−28.3695	−1.32707	−0.663534	−	0.748146i	$-0.730944\pi$
$457$	−28.3695	−1.32707	−0.663534	+	0.748146i	$0.730944\pi$
$458$	0	0
$459$	10.7858	0.503439
$460$	0	0
$461$	8.27357	0.385339	0.192669	−	0.981264i	$-0.438286\pi$
$461$	8.27357	0.385339	0.192669	+	0.981264i	$0.438286\pi$
$462$	0	0
$463$	17.4972	0.813164	0.406582	−	0.913614i	$-0.366720\pi$
$463$	17.4972	0.813164	0.406582	+	0.913614i	$0.366720\pi$
$464$	0	0
$465$	6.03867	0.280037
$466$	0	0
$467$	−2.72307	−0.126009	−0.0630043	−	0.998013i	$-0.520068\pi$
$467$	−2.72307	−0.126009	−0.0630043	+	0.998013i	$0.520068\pi$
$468$	0	0
$469$	−7.90597	−0.365064
$470$	0	0
$471$	10.9173	0.503043
$472$	0	0
$473$	−32.4390	−1.49155
$474$	0	0
$475$	33.7153	1.54697
$476$	0	0
$477$	−2.49945	−0.114442
$478$	0	0
$479$	−18.2038	−0.831755	−0.415878	−	0.909421i	$-0.636526\pi$
$479$	−18.2038	−0.831755	−0.415878	+	0.909421i	$0.636526\pi$
$480$	0	0
$481$	4.08644	0.186326
$482$	0	0
$483$	10.2545	0.466594
$484$	0	0
$485$	−1.23976	−0.0562948
$486$	0	0
$487$	5.19719	0.235507	0.117754	−	0.993043i	$-0.462431\pi$
$487$	5.19719	0.235507	0.117754	+	0.993043i	$0.462431\pi$
$488$	0	0
$489$	−0.419265	−0.0189598
$490$	0	0
$491$	20.8190	0.939550	0.469775	−	0.882786i	$-0.344335\pi$
$491$	20.8190	0.939550	0.469775	+	0.882786i	$0.344335\pi$
$492$	0	0
$493$	2.10644	0.0948694
$494$	0	0
$495$	3.25349	0.146233
$496$	0	0
$497$	0.565234	0.0253542
$498$	0	0
$499$	−25.1995	−1.12808	−0.564042	−	0.825746i	$-0.690754\pi$
$499$	−25.1995	−1.12808	−0.564042	+	0.825746i	$0.690754\pi$
$500$	0	0
$501$	13.9041	0.621188
$502$	0	0
$503$	−11.4413	−0.510142	−0.255071	−	0.966922i	$-0.582099\pi$
$503$	−11.4413	−0.510142	−0.255071	+	0.966922i	$0.582099\pi$
$504$	0	0
$505$	−1.50868	−0.0671354
$506$	0	0
$507$	−12.6758	−0.562951
$508$	0	0
$509$	7.73511	0.342853	0.171426	−	0.985197i	$-0.445162\pi$
$509$	7.73511	0.342853	0.171426	+	0.985197i	$0.445162\pi$
$510$	0	0
$511$	−1.31567	−0.0582020
$512$	0	0
$513$	33.0685	1.46001
$514$	0	0
$515$	−2.99534	−0.131991
$516$	0	0
$517$	−28.7207	−1.26314
$518$	0	0
$519$	−21.5528	−0.946065
$520$	0	0
$521$	−16.0704	−0.704055	−0.352028	−	0.935990i	$-0.614508\pi$
$521$	−16.0704	−0.704055	−0.352028	+	0.935990i	$0.614508\pi$
$522$	0	0
$523$	−13.4267	−0.587109	−0.293554	−	0.955942i	$-0.594838\pi$
$523$	−13.4267	−0.587109	−0.293554	+	0.955942i	$0.594838\pi$
$524$	0	0
$525$	7.62900	0.332957
$526$	0	0
$527$	−10.1749	−0.443226
$528$	0	0
$529$	12.3172	0.535531
$530$	0	0
$531$	2.08265	0.0903794
$532$	0	0
$533$	8.32391	0.360548
$534$	0	0
$535$	6.38534	0.276062
$536$	0	0
$537$	−30.5398	−1.31789
$538$	0	0
$539$	−34.7042	−1.49482
$540$	0	0
$541$	9.55287	0.410710	0.205355	−	0.978688i	$-0.434165\pi$
$541$	9.55287	0.410710	0.205355	+	0.978688i	$0.434165\pi$
$542$	0	0
$543$	31.6992	1.36034
$544$	0	0
$545$	−2.73061	−0.116966
$546$	0	0
$547$	15.3514	0.656380	0.328190	−	0.944612i	$-0.393561\pi$
$547$	15.3514	0.656380	0.328190	+	0.944612i	$0.393561\pi$
$548$	0	0
$549$	0.699207	0.0298414
$550$	0	0
$551$	6.45819	0.275128
$552$	0	0
$553$	−8.35547	−0.355311
$554$	0	0
$555$	−2.37196	−0.100684
$556$	0	0
$557$	−9.75517	−0.413340	−0.206670	−	0.978411i	$-0.566263\pi$
$557$	−9.75517	−0.413340	−0.206670	+	0.978411i	$0.566263\pi$
$558$	0	0
$559$	14.7601	0.624288
$560$	0	0
$561$	−26.9747	−1.13887
$562$	0	0
$563$	−1.53778	−0.0648095	−0.0324048	−	0.999475i	$-0.510317\pi$
$563$	−1.53778	−0.0648095	−0.0324048	+	0.999475i	$0.510317\pi$
$564$	0	0
$565$	−4.58254	−0.192789
$566$	0	0
$567$	9.52397	0.399969
$568$	0	0
$569$	2.94761	0.123570	0.0617851	−	0.998089i	$-0.480321\pi$
$569$	2.94761	0.123570	0.0617851	+	0.998089i	$0.480321\pi$
$570$	0	0
$571$	7.30486	0.305699	0.152849	−	0.988249i	$-0.451155\pi$
$571$	7.30486	0.305699	0.152849	+	0.988249i	$0.451155\pi$
$572$	0	0
$573$	−28.4427	−1.18821
$574$	0	0
$575$	26.2749	1.09574
$576$	0	0
$577$	−25.3558	−1.05557	−0.527787	−	0.849377i	$-0.676978\pi$
$577$	−25.3558	−1.05557	−0.527787	+	0.849377i	$0.676978\pi$
$578$	0	0
$579$	22.7704	0.946307
$580$	0	0
$581$	5.13139	0.212886
$582$	0	0
$583$	−18.2566	−0.756112
$584$	0	0
$585$	−1.48038	−0.0612060
$586$	0	0
$587$	−12.7695	−0.527054	−0.263527	−	0.964652i	$-0.584886\pi$
$587$	−12.7695	−0.527054	−0.263527	+	0.964652i	$0.584886\pi$
$588$	0	0
$589$	−31.1955	−1.28539
$590$	0	0
$591$	33.4705	1.37679
$592$	0	0
$593$	−20.3188	−0.834394	−0.417197	−	0.908816i	$-0.636987\pi$
$593$	−20.3188	−0.834394	−0.417197	+	0.908816i	$0.636987\pi$
$594$	0	0
$595$	1.68260	0.0689800
$596$	0	0
$597$	50.9237	2.08417
$598$	0	0
$599$	−25.5588	−1.04430	−0.522152	−	0.852853i	$-0.674871\pi$
$599$	−25.5588	−1.04430	−0.522152	+	0.852853i	$0.674871\pi$
$600$	0	0
$601$	2.34401	0.0956143	0.0478072	−	0.998857i	$-0.484777\pi$
$601$	2.34401	0.0956143	0.0478072	+	0.998857i	$0.484777\pi$
$602$	0	0
$603$	6.80294	0.277037
$604$	0	0
$605$	15.3962	0.625943
$606$	0	0
$607$	−11.1210	−0.451386	−0.225693	−	0.974198i	$-0.572465\pi$
$607$	−11.1210	−0.451386	−0.225693	+	0.974198i	$0.572465\pi$
$608$	0	0
$609$	1.46134	0.0592164
$610$	0	0
$611$	13.0683	0.528686
$612$	0	0
$613$	26.2733	1.06117	0.530584	−	0.847632i	$-0.321973\pi$
$613$	26.2733	1.06117	0.530584	+	0.847632i	$0.321973\pi$
$614$	0	0
$615$	−4.83158	−0.194828
$616$	0	0
$617$	13.3346	0.536831	0.268415	−	0.963303i	$-0.413500\pi$
$617$	13.3346	0.536831	0.268415	+	0.963303i	$0.413500\pi$
$618$	0	0
$619$	13.1131	0.527061	0.263531	−	0.964651i	$-0.415113\pi$
$619$	13.1131	0.527061	0.263531	+	0.964651i	$0.415113\pi$
$620$	0	0
$621$	25.7708	1.03415
$622$	0	0
$623$	10.0342	0.402011
$624$	0	0
$625$	16.6540	0.666162
$626$	0	0
$627$	−82.7024	−3.30281
$628$	0	0
$629$	3.99665	0.159357
$630$	0	0
$631$	14.4912	0.576887	0.288444	−	0.957497i	$-0.406862\pi$
$631$	14.4912	0.576887	0.288444	+	0.957497i	$0.406862\pi$
$632$	0	0
$633$	−23.1427	−0.919841
$634$	0	0
$635$	0.415520	0.0164894
$636$	0	0
$637$	15.7908	0.625655
$638$	0	0
$639$	−0.486374	−0.0192407
$640$	0	0
$641$	40.0623	1.58237	0.791183	−	0.611579i	$-0.209466\pi$
$641$	40.0623	1.58237	0.791183	+	0.611579i	$0.209466\pi$
$642$	0	0
$643$	−41.1960	−1.62461	−0.812305	−	0.583232i	$-0.801788\pi$
$643$	−41.1960	−1.62461	−0.812305	+	0.583232i	$0.801788\pi$
$644$	0	0
$645$	−8.56747	−0.337344
$646$	0	0
$647$	−22.6970	−0.892311	−0.446155	−	0.894955i	$-0.647207\pi$
$647$	−22.6970	−0.892311	−0.446155	+	0.894955i	$0.647207\pi$
$648$	0	0
$649$	15.2122	0.597133
$650$	0	0
$651$	−7.05882	−0.276657
$652$	0	0
$653$	−1.20015	−0.0469656	−0.0234828	−	0.999724i	$-0.507475\pi$
$653$	−1.20015	−0.0469656	−0.0234828	+	0.999724i	$0.507475\pi$
$654$	0	0
$655$	6.01349	0.234966
$656$	0	0
$657$	1.13211	0.0441679
$658$	0	0
$659$	9.35596	0.364457	0.182228	−	0.983256i	$-0.441669\pi$
$659$	9.35596	0.364457	0.182228	+	0.983256i	$0.441669\pi$
$660$	0	0
$661$	−46.8278	−1.82139	−0.910696	−	0.413077i	$-0.864454\pi$
$661$	−46.8278	−1.82139	−0.910696	+	0.413077i	$0.864454\pi$
$662$	0	0
$663$	12.2738	0.476675
$664$	0	0
$665$	5.15873	0.200047
$666$	0	0
$667$	5.03297	0.194877
$668$	0	0
$669$	−4.71384	−0.182247
$670$	0	0
$671$	5.10719	0.197161
$672$	0	0
$673$	−26.1467	−1.00788	−0.503940	−	0.863738i	$-0.668117\pi$
$673$	−26.1467	−1.00788	−0.503940	+	0.863738i	$0.668117\pi$
$674$	0	0
$675$	19.1726	0.737955
$676$	0	0
$677$	−13.7646	−0.529016	−0.264508	−	0.964383i	$-0.585210\pi$
$677$	−13.7646	−0.529016	−0.264508	+	0.964383i	$0.585210\pi$
$678$	0	0
$679$	1.44920	0.0556154
$680$	0	0
$681$	−36.4386	−1.39633
$682$	0	0
$683$	−6.85441	−0.262277	−0.131138	−	0.991364i	$-0.541863\pi$
$683$	−6.85441	−0.262277	−0.131138	+	0.991364i	$0.541863\pi$
$684$	0	0
$685$	13.2377	0.505786
$686$	0	0
$687$	−1.65665	−0.0632051
$688$	0	0
$689$	8.30698	0.316471
$690$	0	0
$691$	−33.9403	−1.29115	−0.645574	−	0.763697i	$-0.723382\pi$
$691$	−33.9403	−1.29115	−0.645574	+	0.763697i	$0.723382\pi$
$692$	0	0
$693$	−3.80311	−0.144468
$694$	0	0
$695$	−2.40604	−0.0912663
$696$	0	0
$697$	8.14101	0.308363
$698$	0	0
$699$	−5.39464	−0.204044
$700$	0	0
$701$	23.6825	0.894477	0.447239	−	0.894415i	$-0.352407\pi$
$701$	23.6825	0.894477	0.447239	+	0.894415i	$0.352407\pi$
$702$	0	0
$703$	12.2534	0.462147
$704$	0	0
$705$	−7.58544	−0.285684
$706$	0	0
$707$	1.76355	0.0663251
$708$	0	0
$709$	47.5820	1.78698	0.893490	−	0.449083i	$-0.148249\pi$
$709$	47.5820	1.78698	0.893490	+	0.449083i	$0.148249\pi$
$710$	0	0
$711$	7.18973	0.269636
$712$	0	0
$713$	−24.3111	−0.910459
$714$	0	0
$715$	−10.8131	−0.404385
$716$	0	0
$717$	−35.6661	−1.33198
$718$	0	0
$719$	3.27158	0.122009	0.0610046	−	0.998137i	$-0.480570\pi$
$719$	3.27158	0.122009	0.0610046	+	0.998137i	$0.480570\pi$
$720$	0	0
$721$	3.50136	0.130398
$722$	0	0
$723$	2.70272	0.100515
$724$	0	0
$725$	3.74437	0.139062
$726$	0	0
$727$	−4.73998	−0.175796	−0.0878981	−	0.996129i	$-0.528015\pi$
$727$	−4.73998	−0.175796	−0.0878981	+	0.996129i	$0.528015\pi$
$728$	0	0
$729$	17.0483	0.631417
$730$	0	0
$731$	14.4358	0.533929
$732$	0	0
$733$	10.2577	0.378878	0.189439	−	0.981892i	$-0.439333\pi$
$733$	10.2577	0.378878	0.189439	+	0.981892i	$0.439333\pi$
$734$	0	0
$735$	−9.16573	−0.338083
$736$	0	0
$737$	49.6904	1.83037
$738$	0	0
$739$	−7.86962	−0.289489	−0.144744	−	0.989469i	$-0.546236\pi$
$739$	−7.86962	−0.289489	−0.144744	+	0.989469i	$0.546236\pi$
$740$	0	0
$741$	37.6306	1.38239
$742$	0	0
$743$	−5.31215	−0.194884	−0.0974419	−	0.995241i	$-0.531066\pi$
$743$	−5.31215	−0.194884	−0.0974419	+	0.995241i	$0.531066\pi$
$744$	0	0
$745$	18.4993	0.677763
$746$	0	0
$747$	−4.41547	−0.161554
$748$	0	0
$749$	−7.46405	−0.272730
$750$	0	0
$751$	−1.00000	−0.0364905
$751$	−1.00000	−0.0364905
$752$	0	0
$753$	−0.0115569	−0.000421156 0
$754$	0	0
$755$	16.5109	0.600893
$756$	0	0
$757$	32.3429	1.17552	0.587761	−	0.809035i	$-0.300010\pi$
$757$	32.3429	1.17552	0.587761	+	0.809035i	$0.300010\pi$
$758$	0	0
$759$	−64.4512	−2.33943
$760$	0	0
$761$	35.6228	1.29133	0.645663	−	0.763622i	$-0.276581\pi$
$761$	35.6228	1.29133	0.645663	+	0.763622i	$0.276581\pi$
$762$	0	0
$763$	3.19190	0.115555
$764$	0	0
$765$	−1.44785	−0.0523471
$766$	0	0
$767$	−6.92175	−0.249930
$768$	0	0
$769$	−6.76647	−0.244005	−0.122003	−	0.992530i	$-0.538932\pi$
$769$	−6.76647	−0.244005	−0.122003	+	0.992530i	$0.538932\pi$
$770$	0	0
$771$	1.25454	0.0451813
$772$	0	0
$773$	−23.0179	−0.827897	−0.413949	−	0.910300i	$-0.635851\pi$
$773$	−23.0179	−0.827897	−0.413949	+	0.910300i	$0.635851\pi$
$774$	0	0
$775$	−18.0867	−0.649693
$776$	0	0
$777$	2.77267	0.0994689
$778$	0	0
$779$	24.9597	0.894275
$780$	0	0
$781$	−3.55260	−0.127122
$782$	0	0
$783$	3.67253	0.131245
$784$	0	0
$785$	4.28015	0.152765
$786$	0	0
$787$	1.19416	0.0425674	0.0212837	−	0.999773i	$-0.493225\pi$
$787$	1.19416	0.0425674	0.0212837	+	0.999773i	$0.493225\pi$
$788$	0	0
$789$	−16.9820	−0.604575
$790$	0	0
$791$	5.35669	0.190462
$792$	0	0
$793$	−2.32383	−0.0825218
$794$	0	0
$795$	−4.82176	−0.171010
$796$	0	0
$797$	10.3521	0.366691	0.183346	−	0.983049i	$-0.441307\pi$
$797$	10.3521	0.366691	0.183346	+	0.983049i	$0.441307\pi$
$798$	0	0
$799$	12.7812	0.452165
$800$	0	0
$801$	−8.63424	−0.305076
$802$	0	0
$803$	8.26925	0.291815
$804$	0	0
$805$	4.02028	0.141696
$806$	0	0
$807$	−48.0731	−1.69225
$808$	0	0
$809$	42.7756	1.50391	0.751956	−	0.659213i	$-0.229111\pi$
$809$	42.7756	1.50391	0.751956	+	0.659213i	$0.229111\pi$
$810$	0	0
$811$	−40.0146	−1.40510	−0.702552	−	0.711632i	$-0.747956\pi$
$811$	−40.0146	−1.40510	−0.702552	+	0.711632i	$0.747956\pi$
$812$	0	0
$813$	1.83803	0.0644625
$814$	0	0
$815$	−0.164373	−0.00575775
$816$	0	0
$817$	44.2592	1.54843
$818$	0	0
$819$	1.73046	0.0604673
$820$	0	0
$821$	37.6264	1.31317	0.656585	−	0.754252i	$-0.272000\pi$
$821$	37.6264	1.31317	0.656585	+	0.754252i	$0.272000\pi$
$822$	0	0
$823$	−8.26833	−0.288216	−0.144108	−	0.989562i	$-0.546031\pi$
$823$	−8.26833	−0.288216	−0.144108	+	0.989562i	$0.546031\pi$
$824$	0	0
$825$	−47.9496	−1.66939
$826$	0	0
$827$	−26.4386	−0.919361	−0.459681	−	0.888084i	$-0.652036\pi$
$827$	−26.4386	−0.919361	−0.459681	+	0.888084i	$0.652036\pi$
$828$	0	0
$829$	−32.3975	−1.12521	−0.562606	−	0.826725i	$-0.690201\pi$
$829$	−32.3975	−1.12521	−0.562606	+	0.826725i	$0.690201\pi$
$830$	0	0
$831$	18.1096	0.628215
$832$	0	0
$833$	15.4439	0.535098
$834$	0	0
$835$	5.45111	0.188643
$836$	0	0
$837$	−17.7397	−0.613174
$838$	0	0
$839$	−11.3566	−0.392072	−0.196036	−	0.980597i	$-0.562807\pi$
$839$	−11.3566	−0.392072	−0.196036	+	0.980597i	$0.562807\pi$
$840$	0	0
$841$	−28.2828	−0.975268
$842$	0	0
$843$	12.6915	0.437118
$844$	0	0
$845$	−4.96955	−0.170958
$846$	0	0
$847$	−17.9971	−0.618388
$848$	0	0
$849$	−57.4519	−1.97174
$850$	0	0
$851$	9.54929	0.327345
$852$	0	0
$853$	27.9316	0.956360	0.478180	−	0.878262i	$-0.341297\pi$
$853$	27.9316	0.956360	0.478180	+	0.878262i	$0.341297\pi$
$854$	0	0
$855$	−4.43899	−0.151810
$856$	0	0
$857$	−19.1162	−0.652996	−0.326498	−	0.945198i	$-0.605869\pi$
$857$	−19.1162	−0.652996	−0.326498	+	0.945198i	$0.605869\pi$
$858$	0	0
$859$	−30.0727	−1.02607	−0.513033	−	0.858369i	$-0.671478\pi$
$859$	−30.0727	−1.02607	−0.513033	+	0.858369i	$0.671478\pi$
$860$	0	0
$861$	5.64781	0.192477
$862$	0	0
$863$	−12.9678	−0.441429	−0.220715	−	0.975338i	$-0.570839\pi$
$863$	−12.9678	−0.441429	−0.220715	+	0.975338i	$0.570839\pi$
$864$	0	0
$865$	−8.44982	−0.287303
$866$	0	0
$867$	−20.9828	−0.712614
$868$	0	0
$869$	52.5156	1.78147
$870$	0	0
$871$	−22.6098	−0.766102
$872$	0	0
$873$	−1.24701	−0.0422050
$874$	0	0
$875$	6.37342	0.215461
$876$	0	0
$877$	24.3252	0.821402	0.410701	−	0.911770i	$-0.365284\pi$
$877$	24.3252	0.821402	0.410701	+	0.911770i	$0.365284\pi$
$878$	0	0
$879$	−28.0355	−0.945613
$880$	0	0
$881$	11.4277	0.385009	0.192505	−	0.981296i	$-0.438339\pi$
$881$	11.4277	0.385009	0.192505	+	0.981296i	$0.438339\pi$
$882$	0	0
$883$	1.66153	0.0559150	0.0279575	−	0.999609i	$-0.491100\pi$
$883$	1.66153	0.0559150	0.0279575	+	0.999609i	$0.491100\pi$
$884$	0	0
$885$	4.01771	0.135054
$886$	0	0
$887$	−37.0453	−1.24386	−0.621929	−	0.783073i	$-0.713651\pi$
$887$	−37.0453	−1.24386	−0.621929	+	0.783073i	$0.713651\pi$
$888$	0	0
$889$	−0.485715	−0.0162904
$890$	0	0
$891$	−59.8599	−2.00538
$892$	0	0
$893$	39.1860	1.31131
$894$	0	0
$895$	−11.9732	−0.400218
$896$	0	0
$897$	29.3261	0.979169
$898$	0	0
$899$	−3.46452	−0.115548
$900$	0	0
$901$	8.12446	0.270665
$902$	0	0
$903$	10.0148	0.333272
$904$	0	0
$905$	12.4277	0.413111
$906$	0	0
$907$	−40.7204	−1.35210	−0.676050	−	0.736856i	$-0.736310\pi$
$907$	−40.7204	−1.35210	−0.676050	+	0.736856i	$0.736310\pi$
$908$	0	0
$909$	−1.51750	−0.0503324
$910$	0	0
$911$	−29.1509	−0.965811	−0.482906	−	0.875672i	$-0.660419\pi$
$911$	−29.1509	−0.965811	−0.482906	+	0.875672i	$0.660419\pi$
$912$	0	0
$913$	−32.2517	−1.06738
$914$	0	0
$915$	1.34886	0.0445920
$916$	0	0
$917$	−7.02938	−0.232131
$918$	0	0
$919$	−12.6637	−0.417736	−0.208868	−	0.977944i	$-0.566978\pi$
$919$	−12.6637	−0.417736	−0.208868	+	0.977944i	$0.566978\pi$
$920$	0	0
$921$	−0.116470	−0.00383782
$922$	0	0
$923$	1.61648	0.0532070
$924$	0	0
$925$	7.10437	0.233590
$926$	0	0
$927$	−3.01286	−0.0989552
$928$	0	0
$929$	6.35630	0.208543	0.104272	−	0.994549i	$-0.466749\pi$
$929$	6.35630	0.208543	0.104272	+	0.994549i	$0.466749\pi$
$930$	0	0
$931$	47.3497	1.55182
$932$	0	0
$933$	12.5775	0.411769
$934$	0	0
$935$	−10.5755	−0.345855
$936$	0	0
$937$	42.0539	1.37384	0.686921	−	0.726732i	$-0.258962\pi$
$937$	42.0539	1.37384	0.686921	+	0.726732i	$0.258962\pi$
$938$	0	0
$939$	12.0315	0.392634
$940$	0	0
$941$	25.0966	0.818124	0.409062	−	0.912507i	$-0.365856\pi$
$941$	25.0966	0.818124	0.409062	+	0.912507i	$0.365856\pi$
$942$	0	0
$943$	19.4515	0.633428
$944$	0	0
$945$	2.93357	0.0954292
$946$	0	0
$947$	36.0835	1.17256	0.586278	−	0.810110i	$-0.300593\pi$
$947$	36.0835	1.17256	0.586278	+	0.810110i	$0.300593\pi$
$948$	0	0
$949$	−3.76261	−0.122139
$950$	0	0
$951$	−16.4372	−0.533013
$952$	0	0
$953$	−39.7433	−1.28741	−0.643706	−	0.765273i	$-0.722604\pi$
$953$	−39.7433	−1.28741	−0.643706	+	0.765273i	$0.722604\pi$
$954$	0	0
$955$	−11.1510	−0.360838
$956$	0	0
$957$	−9.18478	−0.296902
$958$	0	0
$959$	−15.4740	−0.499681
$960$	0	0
$961$	−14.2651	−0.460164
$962$	0	0
$963$	6.42268	0.206968
$964$	0	0
$965$	8.92718	0.287376
$966$	0	0
$967$	−8.76146	−0.281750	−0.140875	−	0.990027i	$-0.544991\pi$
$967$	−8.76146	−0.281750	−0.140875	+	0.990027i	$0.544991\pi$
$968$	0	0
$969$	36.8038	1.18231
$970$	0	0
$971$	11.5792	0.371593	0.185796	−	0.982588i	$-0.440514\pi$
$971$	11.5792	0.371593	0.185796	+	0.982588i	$0.440514\pi$
$972$	0	0
$973$	2.81250	0.0901647
$974$	0	0
$975$	21.8177	0.698725
$976$	0	0
$977$	17.7690	0.568479	0.284240	−	0.958753i	$-0.408259\pi$
$977$	17.7690	0.568479	0.284240	+	0.958753i	$0.408259\pi$
$978$	0	0
$979$	−63.0667	−2.01562
$980$	0	0
$981$	−2.74657	−0.0876913
$982$	0	0
$983$	−5.88553	−0.187719	−0.0938597	−	0.995585i	$-0.529921\pi$
$983$	−5.88553	−0.187719	−0.0938597	+	0.995585i	$0.529921\pi$
$984$	0	0
$985$	13.1222	0.418107
$986$	0	0
$987$	8.86689	0.282236
$988$	0	0
$989$	34.4918	1.09678
$990$	0	0
$991$	−48.5494	−1.54222	−0.771111	−	0.636700i	$-0.780299\pi$
$991$	−48.5494	−1.54222	−0.771111	+	0.636700i	$0.780299\pi$
$992$	0	0
$993$	−43.5581	−1.38227
$994$	0	0
$995$	19.9647	0.632924
$996$	0	0
$997$	−26.0283	−0.824324	−0.412162	−	0.911111i	$-0.635226\pi$
$997$	−26.0283	−0.824324	−0.412162	+	0.911111i	$0.635226\pi$
$998$	0	0
$999$	6.96806	0.220460

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6008.2.a.b.1.39	✓	44

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6008.2.a.b.1.39	✓	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)

\(f(q)\)	\(=\)	\(q+1.94041 q^{3} +0.760740 q^{5} -0.889256 q^{7} +0.765188 q^{9} +O(q^{10})\)	\(q+1.94041 q^{3} +0.760740 q^{5} -0.889256 q^{7} +0.765188 q^{9} +5.58913 q^{11} -2.54312 q^{13} +1.47615 q^{15} -2.48725 q^{17} -7.62571 q^{19} -1.72552 q^{21} -5.94283 q^{23} -4.42127 q^{25} -4.33645 q^{27} -0.846897 q^{29} +4.09083 q^{31} +10.8452 q^{33} -0.676492 q^{35} -1.60686 q^{37} -4.93470 q^{39} -3.27310 q^{41} -5.80394 q^{43} +0.582109 q^{45} -5.13868 q^{47} -6.20922 q^{49} -4.82627 q^{51} -3.26645 q^{53} +4.25188 q^{55} -14.7970 q^{57} +2.72175 q^{59} +0.913771 q^{61} -0.680448 q^{63} -1.93466 q^{65} +8.89055 q^{67} -11.5315 q^{69} -0.635627 q^{71} +1.47952 q^{73} -8.57908 q^{75} -4.97017 q^{77} +9.39602 q^{79} -10.7101 q^{81} -5.77044 q^{83} -1.89215 q^{85} -1.64333 q^{87} -11.2838 q^{89} +2.26149 q^{91} +7.93789 q^{93} -5.80118 q^{95} -1.62968 q^{97} +4.27674 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 44 q - 14 q^{3} + 7 q^{5} - 20 q^{7} + 38 q^{9}+O(q^{10}) \) 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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