LMFDB - Embedded newform 8008.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8008,2,Mod(1,8008)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8008, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.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:	$63.9442019386$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.244558277.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 11x^{3} + 29x^{2} + 5x - 4 $ x^6 - x^5 - 13x^4 + 11x^3 + 29x^2 + 5x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.98317$ of defining polynomial
Character	$\chi$	$=$	8008.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.98317 q^{3} +3.19751 q^{5} +1.00000 q^{7} +5.89931 q^{9} +O(q^{10})$	$q-2.98317 q^{3} +3.19751 q^{5} +1.00000 q^{7} +5.89931 q^{9} -1.00000 q^{11} -1.00000 q^{13} -9.53872 q^{15} +1.18708 q^{17} +0.300378 q^{19} -2.98317 q^{21} -7.32438 q^{23} +5.22407 q^{25} -8.64913 q^{27} +3.24089 q^{29} -6.56527 q^{31} +2.98317 q^{33} +3.19751 q^{35} -4.73878 q^{37} +2.98317 q^{39} -4.72545 q^{41} +8.34121 q^{43} +18.8631 q^{45} -3.02365 q^{47} +1.00000 q^{49} -3.54127 q^{51} +9.50471 q^{53} -3.19751 q^{55} -0.896080 q^{57} +12.9166 q^{59} -1.00349 q^{61} +5.89931 q^{63} -3.19751 q^{65} -7.44408 q^{67} +21.8499 q^{69} -3.06703 q^{71} +10.3083 q^{73} -15.5843 q^{75} -1.00000 q^{77} +5.83010 q^{79} +8.10392 q^{81} +9.23729 q^{83} +3.79571 q^{85} -9.66814 q^{87} -1.07999 q^{89} -1.00000 q^{91} +19.5853 q^{93} +0.960462 q^{95} +12.0606 q^{97} -5.89931 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * q^97 - 9 * 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
$3$	−2.98317	−1.72233	−0.861167	−	0.508322i	$-0.830266\pi$
$3$	−2.98317	−1.72233	−0.861167	+	0.508322i	$0.830266\pi$
$4$	0	0
$5$	3.19751	1.42997	0.714985	−	0.699140i	$-0.246434\pi$
$5$	3.19751	1.42997	0.714985	+	0.699140i	$0.246434\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	5.89931	1.96644
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−9.53872	−2.46289
$16$	0	0
$17$	1.18708	0.287910	0.143955	−	0.989584i	$-0.454018\pi$
$17$	1.18708	0.287910	0.143955	+	0.989584i	$0.454018\pi$
$18$	0	0
$19$	0.300378	0.0689115	0.0344557	−	0.999406i	$-0.489030\pi$
$19$	0.300378	0.0689115	0.0344557	+	0.999406i	$0.489030\pi$
$20$	0	0
$21$	−2.98317	−0.650981
$22$	0	0
$23$	−7.32438	−1.52724	−0.763619	−	0.645667i	$-0.776579\pi$
$23$	−7.32438	−1.52724	−0.763619	+	0.645667i	$0.776579\pi$
$24$	0	0
$25$	5.22407	1.04481
$26$	0	0
$27$	−8.64913	−1.66453
$28$	0	0
$29$	3.24089	0.601819	0.300909	−	0.953653i	$-0.402710\pi$
$29$	3.24089	0.601819	0.300909	+	0.953653i	$0.402710\pi$
$30$	0	0
$31$	−6.56527	−1.17916	−0.589579	−	0.807711i	$-0.700706\pi$
$31$	−6.56527	−1.17916	−0.589579	+	0.807711i	$0.700706\pi$
$32$	0	0
$33$	2.98317	0.519303
$34$	0	0
$35$	3.19751	0.540478
$36$	0	0
$37$	−4.73878	−0.779051	−0.389525	−	0.921016i	$-0.627361\pi$
$37$	−4.73878	−0.779051	−0.389525	+	0.921016i	$0.627361\pi$
$38$	0	0
$39$	2.98317	0.477690
$40$	0	0
$41$	−4.72545	−0.737991	−0.368995	−	0.929431i	$-0.620298\pi$
$41$	−4.72545	−0.737991	−0.368995	+	0.929431i	$0.620298\pi$
$42$	0	0
$43$	8.34121	1.27202	0.636011	−	0.771680i	$-0.280583\pi$
$43$	8.34121	1.27202	0.636011	+	0.771680i	$0.280583\pi$
$44$	0	0
$45$	18.8631	2.81194
$46$	0	0
$47$	−3.02365	−0.441044	−0.220522	−	0.975382i	$-0.570776\pi$
$47$	−3.02365	−0.441044	−0.220522	+	0.975382i	$0.570776\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.54127	−0.495877
$52$	0	0
$53$	9.50471	1.30557	0.652786	−	0.757542i	$-0.273600\pi$
$53$	9.50471	1.30557	0.652786	+	0.757542i	$0.273600\pi$
$54$	0	0
$55$	−3.19751	−0.431152
$56$	0	0
$57$	−0.896080	−0.118689
$58$	0	0
$59$	12.9166	1.68159	0.840796	−	0.541352i	$-0.182087\pi$
$59$	12.9166	1.68159	0.840796	+	0.541352i	$0.182087\pi$
$60$	0	0
$61$	−1.00349	−0.128484	−0.0642422	−	0.997934i	$-0.520463\pi$
$61$	−1.00349	−0.128484	−0.0642422	+	0.997934i	$0.520463\pi$
$62$	0	0
$63$	5.89931	0.743243
$64$	0	0
$65$	−3.19751	−0.396602
$66$	0	0
$67$	−7.44408	−0.909439	−0.454719	−	0.890635i	$-0.650260\pi$
$67$	−7.44408	−0.909439	−0.454719	+	0.890635i	$0.650260\pi$
$68$	0	0
$69$	21.8499	2.63042
$70$	0	0
$71$	−3.06703	−0.363990	−0.181995	−	0.983299i	$-0.558255\pi$
$71$	−3.06703	−0.363990	−0.181995	+	0.983299i	$0.558255\pi$
$72$	0	0
$73$	10.3083	1.20649	0.603246	−	0.797555i	$-0.293874\pi$
$73$	10.3083	1.20649	0.603246	+	0.797555i	$0.293874\pi$
$74$	0	0
$75$	−15.5843	−1.79952
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.83010	0.655937	0.327969	−	0.944689i	$-0.393636\pi$
$79$	5.83010	0.655937	0.327969	+	0.944689i	$0.393636\pi$
$80$	0	0
$81$	8.10392	0.900436
$82$	0	0
$83$	9.23729	1.01392	0.506962	−	0.861968i	$-0.330768\pi$
$83$	9.23729	1.01392	0.506962	+	0.861968i	$0.330768\pi$
$84$	0	0
$85$	3.79571	0.411703
$86$	0	0
$87$	−9.66814	−1.03653
$88$	0	0
$89$	−1.07999	−0.114479	−0.0572393	−	0.998360i	$-0.518230\pi$
$89$	−1.07999	−0.114479	−0.0572393	+	0.998360i	$0.518230\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	19.5853	2.03090
$94$	0	0
$95$	0.960462	0.0985413
$96$	0	0
$97$	12.0606	1.22457	0.612286	−	0.790637i	$-0.290250\pi$
$97$	12.0606	1.22457	0.612286	+	0.790637i	$0.290250\pi$
$98$	0	0
$99$	−5.89931	−0.592903
$100$	0	0
$101$	4.75550	0.473190	0.236595	−	0.971608i	$-0.423969\pi$
$101$	4.75550	0.473190	0.236595	+	0.971608i	$0.423969\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−9.53872	−0.930883
$106$	0	0
$107$	−12.7893	−1.23639	−0.618196	−	0.786024i	$-0.712136\pi$
$107$	−12.7893	−1.23639	−0.618196	+	0.786024i	$0.712136\pi$
$108$	0	0
$109$	−2.45873	−0.235503	−0.117752	−	0.993043i	$-0.537569\pi$
$109$	−2.45873	−0.235503	−0.117752	+	0.993043i	$0.537569\pi$
$110$	0	0
$111$	14.1366	1.34179
$112$	0	0
$113$	10.4915	0.986956	0.493478	−	0.869758i	$-0.335725\pi$
$113$	10.4915	0.986956	0.493478	+	0.869758i	$0.335725\pi$
$114$	0	0
$115$	−23.4198	−2.18390
$116$	0	0
$117$	−5.89931	−0.545391
$118$	0	0
$119$	1.18708	0.108820
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	14.0968	1.27107
$124$	0	0
$125$	0.716451	0.0640813
$126$	0	0
$127$	20.1578	1.78872	0.894358	−	0.447352i	$-0.147633\pi$
$127$	20.1578	1.78872	0.894358	+	0.447352i	$0.147633\pi$
$128$	0	0
$129$	−24.8832	−2.19085
$130$	0	0
$131$	−0.644494	−0.0563097	−0.0281549	−	0.999604i	$-0.508963\pi$
$131$	−0.644494	−0.0563097	−0.0281549	+	0.999604i	$0.508963\pi$
$132$	0	0
$133$	0.300378	0.0260461
$134$	0	0
$135$	−27.6557	−2.38022
$136$	0	0
$137$	−0.565273	−0.0482945	−0.0241473	−	0.999708i	$-0.507687\pi$
$137$	−0.565273	−0.0482945	−0.0241473	+	0.999708i	$0.507687\pi$
$138$	0	0
$139$	−10.3614	−0.878839	−0.439420	−	0.898282i	$-0.644816\pi$
$139$	−10.3614	−0.878839	−0.439420	+	0.898282i	$0.644816\pi$
$140$	0	0
$141$	9.02006	0.759626
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	10.3628	0.860583
$146$	0	0
$147$	−2.98317	−0.246048
$148$	0	0
$149$	−5.47396	−0.448444	−0.224222	−	0.974538i	$-0.571984\pi$
$149$	−5.47396	−0.448444	−0.224222	+	0.974538i	$0.571984\pi$
$150$	0	0
$151$	19.8029	1.61154	0.805769	−	0.592230i	$-0.201752\pi$
$151$	19.8029	1.61154	0.805769	+	0.592230i	$0.201752\pi$
$152$	0	0
$153$	7.00297	0.566157
$154$	0	0
$155$	−20.9925	−1.68616
$156$	0	0
$157$	9.83045	0.784555	0.392278	−	0.919847i	$-0.371687\pi$
$157$	9.83045	0.784555	0.392278	+	0.919847i	$0.371687\pi$
$158$	0	0
$159$	−28.3542	−2.24863
$160$	0	0
$161$	−7.32438	−0.577242
$162$	0	0
$163$	−5.58857	−0.437730	−0.218865	−	0.975755i	$-0.570236\pi$
$163$	−5.58857	−0.437730	−0.218865	+	0.975755i	$0.570236\pi$
$164$	0	0
$165$	9.53872	0.742588
$166$	0	0
$167$	6.73196	0.520935	0.260467	−	0.965483i	$-0.416123\pi$
$167$	6.73196	0.520935	0.260467	+	0.965483i	$0.416123\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.77202	0.135510
$172$	0	0
$173$	10.2331	0.778005	0.389003	−	0.921237i	$-0.372820\pi$
$173$	10.2331	0.778005	0.389003	+	0.921237i	$0.372820\pi$
$174$	0	0
$175$	5.22407	0.394902
$176$	0	0
$177$	−38.5323	−2.89626
$178$	0	0
$179$	−4.01191	−0.299864	−0.149932	−	0.988696i	$-0.547905\pi$
$179$	−4.01191	−0.299864	−0.149932	+	0.988696i	$0.547905\pi$
$180$	0	0
$181$	−16.5881	−1.23299	−0.616493	−	0.787360i	$-0.711447\pi$
$181$	−16.5881	−1.23299	−0.616493	+	0.787360i	$0.711447\pi$
$182$	0	0
$183$	2.99360	0.221293
$184$	0	0
$185$	−15.1523	−1.11402
$186$	0	0
$187$	−1.18708	−0.0868081
$188$	0	0
$189$	−8.64913	−0.629132
$190$	0	0
$191$	0.00182569	0.000132103 0	6.60513e−5	−	1.00000i	$-0.499979\pi$
$191$	0.00182569	0.000132103 0	6.60513e−5		1.00000i	$0.499979\pi$
$192$	0	0
$193$	−9.55423	−0.687728	−0.343864	−	0.939019i	$-0.611736\pi$
$193$	−9.55423	−0.687728	−0.343864	+	0.939019i	$0.611736\pi$
$194$	0	0
$195$	9.53872	0.683082
$196$	0	0
$197$	0.975622	0.0695102	0.0347551	−	0.999396i	$-0.488935\pi$
$197$	0.975622	0.0695102	0.0347551	+	0.999396i	$0.488935\pi$
$198$	0	0
$199$	−3.43165	−0.243263	−0.121632	−	0.992575i	$-0.538813\pi$
$199$	−3.43165	−0.243263	−0.121632	+	0.992575i	$0.538813\pi$
$200$	0	0
$201$	22.2070	1.56636
$202$	0	0
$203$	3.24089	0.227466
$204$	0	0
$205$	−15.1097	−1.05530
$206$	0	0
$207$	−43.2088	−3.00322
$208$	0	0
$209$	−0.300378	−0.0207776
$210$	0	0
$211$	−0.582064	−0.0400709	−0.0200355	−	0.999799i	$-0.506378\pi$
$211$	−0.582064	−0.0400709	−0.0200355	+	0.999799i	$0.506378\pi$
$212$	0	0
$213$	9.14948	0.626912
$214$	0	0
$215$	26.6711	1.81895
$216$	0	0
$217$	−6.56527	−0.445680
$218$	0	0
$219$	−30.7514	−2.07798
$220$	0	0
$221$	−1.18708	−0.0798519
$222$	0	0
$223$	13.2528	0.887471	0.443735	−	0.896158i	$-0.353653\pi$
$223$	13.2528	0.887471	0.443735	+	0.896158i	$0.353653\pi$
$224$	0	0
$225$	30.8184	2.05456
$226$	0	0
$227$	−16.5019	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$227$	−16.5019	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$228$	0	0
$229$	−29.6994	−1.96259	−0.981296	−	0.192507i	$-0.938338\pi$
$229$	−29.6994	−1.96259	−0.981296	+	0.192507i	$0.938338\pi$
$230$	0	0
$231$	2.98317	0.196278
$232$	0	0
$233$	7.50544	0.491697	0.245849	−	0.969308i	$-0.420933\pi$
$233$	7.50544	0.491697	0.245849	+	0.969308i	$0.420933\pi$
$234$	0	0
$235$	−9.66814	−0.630680
$236$	0	0
$237$	−17.3922	−1.12974
$238$	0	0
$239$	23.2680	1.50508	0.752541	−	0.658545i	$-0.228828\pi$
$239$	23.2680	1.50508	0.752541	+	0.658545i	$0.228828\pi$
$240$	0	0
$241$	9.98221	0.643010	0.321505	−	0.946908i	$-0.395811\pi$
$241$	9.98221	0.643010	0.321505	+	0.946908i	$0.395811\pi$
$242$	0	0
$243$	1.77202	0.113675
$244$	0	0
$245$	3.19751	0.204281
$246$	0	0
$247$	−0.300378	−0.0191126
$248$	0	0
$249$	−27.5564	−1.74632
$250$	0	0
$251$	15.8463	1.00021	0.500104	−	0.865966i	$-0.333295\pi$
$251$	15.8463	1.00021	0.500104	+	0.865966i	$0.333295\pi$
$252$	0	0
$253$	7.32438	0.460480
$254$	0	0
$255$	−11.3233	−0.709090
$256$	0	0
$257$	19.9706	1.24573	0.622866	−	0.782328i	$-0.285968\pi$
$257$	19.9706	1.24573	0.622866	+	0.782328i	$0.285968\pi$
$258$	0	0
$259$	−4.73878	−0.294454
$260$	0	0
$261$	19.1190	1.18344
$262$	0	0
$263$	−19.5614	−1.20621	−0.603104	−	0.797663i	$-0.706070\pi$
$263$	−19.5614	−1.20621	−0.603104	+	0.797663i	$0.706070\pi$
$264$	0	0
$265$	30.3914	1.86693
$266$	0	0
$267$	3.22179	0.197171
$268$	0	0
$269$	−9.07317	−0.553201	−0.276600	−	0.960985i	$-0.589208\pi$
$269$	−9.07317	−0.553201	−0.276600	+	0.960985i	$0.589208\pi$
$270$	0	0
$271$	17.0135	1.03350	0.516750	−	0.856137i	$-0.327142\pi$
$271$	17.0135	1.03350	0.516750	+	0.856137i	$0.327142\pi$
$272$	0	0
$273$	2.98317	0.180550
$274$	0	0
$275$	−5.22407	−0.315023
$276$	0	0
$277$	3.99063	0.239773	0.119887	−	0.992788i	$-0.461747\pi$
$277$	3.99063	0.239773	0.119887	+	0.992788i	$0.461747\pi$
$278$	0	0
$279$	−38.7306	−2.31874
$280$	0	0
$281$	26.3835	1.57391	0.786954	−	0.617011i	$-0.211657\pi$
$281$	26.3835	1.57391	0.786954	+	0.617011i	$0.211657\pi$
$282$	0	0
$283$	−16.2739	−0.967384	−0.483692	−	0.875238i	$-0.660704\pi$
$283$	−16.2739	−0.967384	−0.483692	+	0.875238i	$0.660704\pi$
$284$	0	0
$285$	−2.86522	−0.169721
$286$	0	0
$287$	−4.72545	−0.278934
$288$	0	0
$289$	−15.5908	−0.917108
$290$	0	0
$291$	−35.9789	−2.10912
$292$	0	0
$293$	2.47530	0.144608	0.0723042	−	0.997383i	$-0.476965\pi$
$293$	2.47530	0.144608	0.0723042	+	0.997383i	$0.476965\pi$
$294$	0	0
$295$	41.3008	2.40463
$296$	0	0
$297$	8.64913	0.501874
$298$	0	0
$299$	7.32438	0.423580
$300$	0	0
$301$	8.34121	0.480779
$302$	0	0
$303$	−14.1865	−0.814991
$304$	0	0
$305$	−3.20868	−0.183729
$306$	0	0
$307$	7.12911	0.406880	0.203440	−	0.979087i	$-0.434788\pi$
$307$	7.12911	0.406880	0.203440	+	0.979087i	$0.434788\pi$
$308$	0	0
$309$	−23.8654	−1.35765
$310$	0	0
$311$	10.8766	0.616755	0.308378	−	0.951264i	$-0.400214\pi$
$311$	10.8766	0.616755	0.308378	+	0.951264i	$0.400214\pi$
$312$	0	0
$313$	−29.9258	−1.69151	−0.845754	−	0.533573i	$-0.820849\pi$
$313$	−29.9258	−1.69151	−0.845754	+	0.533573i	$0.820849\pi$
$314$	0	0
$315$	18.8631	1.06282
$316$	0	0
$317$	22.0112	1.23627	0.618136	−	0.786072i	$-0.287888\pi$
$317$	22.0112	1.23627	0.618136	+	0.786072i	$0.287888\pi$
$318$	0	0
$319$	−3.24089	−0.181455
$320$	0	0
$321$	38.1528	2.12948
$322$	0	0
$323$	0.356574	0.0198403
$324$	0	0
$325$	−5.22407	−0.289779
$326$	0	0
$327$	7.33480	0.405616
$328$	0	0
$329$	−3.02365	−0.166699
$330$	0	0
$331$	−29.1417	−1.60177	−0.800886	−	0.598817i	$-0.795637\pi$
$331$	−29.1417	−1.60177	−0.800886	+	0.598817i	$0.795637\pi$
$332$	0	0
$333$	−27.9555	−1.53195
$334$	0	0
$335$	−23.8025	−1.30047
$336$	0	0
$337$	15.3750	0.837527	0.418764	−	0.908095i	$-0.362464\pi$
$337$	15.3750	0.837527	0.418764	+	0.908095i	$0.362464\pi$
$338$	0	0
$339$	−31.2979	−1.69987
$340$	0	0
$341$	6.56527	0.355529
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	69.8652	3.76141
$346$	0	0
$347$	−22.5334	−1.20966	−0.604828	−	0.796356i	$-0.706758\pi$
$347$	−22.5334	−1.20966	−0.604828	+	0.796356i	$0.706758\pi$
$348$	0	0
$349$	21.4403	1.14767	0.573835	−	0.818971i	$-0.305455\pi$
$349$	21.4403	1.14767	0.573835	+	0.818971i	$0.305455\pi$
$350$	0	0
$351$	8.64913	0.461657
$352$	0	0
$353$	33.1167	1.76262	0.881312	−	0.472536i	$-0.156661\pi$
$353$	33.1167	1.76262	0.881312	+	0.472536i	$0.156661\pi$
$354$	0	0
$355$	−9.80687	−0.520494
$356$	0	0
$357$	−3.54127	−0.187424
$358$	0	0
$359$	−23.0681	−1.21749	−0.608743	−	0.793368i	$-0.708326\pi$
$359$	−23.0681	−1.21749	−0.608743	+	0.793368i	$0.708326\pi$
$360$	0	0
$361$	−18.9098	−0.995251
$362$	0	0
$363$	−2.98317	−0.156576
$364$	0	0
$365$	32.9608	1.72525
$366$	0	0
$367$	19.6427	1.02534	0.512671	−	0.858585i	$-0.328656\pi$
$367$	19.6427	1.02534	0.512671	+	0.858585i	$0.328656\pi$
$368$	0	0
$369$	−27.8769	−1.45121
$370$	0	0
$371$	9.50471	0.493460
$372$	0	0
$373$	16.9691	0.878629	0.439314	−	0.898333i	$-0.355221\pi$
$373$	16.9691	0.878629	0.439314	+	0.898333i	$0.355221\pi$
$374$	0	0
$375$	−2.13730	−0.110369
$376$	0	0
$377$	−3.24089	−0.166915
$378$	0	0
$379$	2.34873	0.120646	0.0603230	−	0.998179i	$-0.480787\pi$
$379$	2.34873	0.120646	0.0603230	+	0.998179i	$0.480787\pi$
$380$	0	0
$381$	−60.1342	−3.08077
$382$	0	0
$383$	28.8964	1.47654	0.738268	−	0.674508i	$-0.235644\pi$
$383$	28.8964	1.47654	0.738268	+	0.674508i	$0.235644\pi$
$384$	0	0
$385$	−3.19751	−0.162960
$386$	0	0
$387$	49.2074	2.50135
$388$	0	0
$389$	−17.3481	−0.879584	−0.439792	−	0.898100i	$-0.644948\pi$
$389$	−17.3481	−0.879584	−0.439792	+	0.898100i	$0.644948\pi$
$390$	0	0
$391$	−8.69465	−0.439707
$392$	0	0
$393$	1.92264	0.0969841
$394$	0	0
$395$	18.6418	0.937970
$396$	0	0
$397$	−9.97353	−0.500557	−0.250279	−	0.968174i	$-0.580522\pi$
$397$	−9.97353	−0.500557	−0.250279	+	0.968174i	$0.580522\pi$
$398$	0	0
$399$	−0.896080	−0.0448601
$400$	0	0
$401$	−30.2847	−1.51234	−0.756172	−	0.654373i	$-0.772933\pi$
$401$	−30.2847	−1.51234	−0.756172	+	0.654373i	$0.772933\pi$
$402$	0	0
$403$	6.56527	0.327040
$404$	0	0
$405$	25.9124	1.28760
$406$	0	0
$407$	4.73878	0.234893
$408$	0	0
$409$	11.9397	0.590379	0.295189	−	0.955439i	$-0.404617\pi$
$409$	11.9397	0.590379	0.295189	+	0.955439i	$0.404617\pi$
$410$	0	0
$411$	1.68631	0.0831793
$412$	0	0
$413$	12.9166	0.635582
$414$	0	0
$415$	29.5363	1.44988
$416$	0	0
$417$	30.9097	1.51365
$418$	0	0
$419$	2.53557	0.123871	0.0619353	−	0.998080i	$-0.480273\pi$
$419$	2.53557	0.123871	0.0619353	+	0.998080i	$0.480273\pi$
$420$	0	0
$421$	24.0444	1.17185	0.585926	−	0.810364i	$-0.300731\pi$
$421$	24.0444	1.17185	0.585926	+	0.810364i	$0.300731\pi$
$422$	0	0
$423$	−17.8374	−0.867286
$424$	0	0
$425$	6.20140	0.300812
$426$	0	0
$427$	−1.00349	−0.0485625
$428$	0	0
$429$	−2.98317	−0.144029
$430$	0	0
$431$	13.8931	0.669205	0.334602	−	0.942359i	$-0.391398\pi$
$431$	13.8931	0.669205	0.334602	+	0.942359i	$0.391398\pi$
$432$	0	0
$433$	4.65198	0.223560	0.111780	−	0.993733i	$-0.464345\pi$
$433$	4.65198	0.223560	0.111780	+	0.993733i	$0.464345\pi$
$434$	0	0
$435$	−30.9140	−1.48221
$436$	0	0
$437$	−2.20008	−0.105244
$438$	0	0
$439$	39.3177	1.87653	0.938266	−	0.345916i	$-0.112432\pi$
$439$	39.3177	1.87653	0.938266	+	0.345916i	$0.112432\pi$
$440$	0	0
$441$	5.89931	0.280919
$442$	0	0
$443$	12.7875	0.607553	0.303776	−	0.952743i	$-0.401752\pi$
$443$	12.7875	0.607553	0.303776	+	0.952743i	$0.401752\pi$
$444$	0	0
$445$	−3.45328	−0.163701
$446$	0	0
$447$	16.3297	0.772371
$448$	0	0
$449$	22.1827	1.04687	0.523433	−	0.852067i	$-0.324651\pi$
$449$	22.1827	1.04687	0.523433	+	0.852067i	$0.324651\pi$
$450$	0	0
$451$	4.72545	0.222513
$452$	0	0
$453$	−59.0755	−2.77561
$454$	0	0
$455$	−3.19751	−0.149902
$456$	0	0
$457$	2.48514	0.116250	0.0581250	−	0.998309i	$-0.481488\pi$
$457$	2.48514	0.116250	0.0581250	+	0.998309i	$0.481488\pi$
$458$	0	0
$459$	−10.2672	−0.479234
$460$	0	0
$461$	−23.0253	−1.07239	−0.536197	−	0.844093i	$-0.680140\pi$
$461$	−23.0253	−1.07239	−0.536197	+	0.844093i	$0.680140\pi$
$462$	0	0
$463$	20.0068	0.929796	0.464898	−	0.885364i	$-0.346091\pi$
$463$	20.0068	0.929796	0.464898	+	0.885364i	$0.346091\pi$
$464$	0	0
$465$	62.6243	2.90413
$466$	0	0
$467$	−4.94951	−0.229036	−0.114518	−	0.993421i	$-0.536532\pi$
$467$	−4.94951	−0.229036	−0.114518	+	0.993421i	$0.536532\pi$
$468$	0	0
$469$	−7.44408	−0.343736
$470$	0	0
$471$	−29.3259	−1.35127
$472$	0	0
$473$	−8.34121	−0.383529
$474$	0	0
$475$	1.56920	0.0719996
$476$	0	0
$477$	56.0712	2.56732
$478$	0	0
$479$	13.9476	0.637281	0.318640	−	0.947876i	$-0.396774\pi$
$479$	13.9476	0.637281	0.318640	+	0.947876i	$0.396774\pi$
$480$	0	0
$481$	4.73878	0.216070
$482$	0	0
$483$	21.8499	0.994204
$484$	0	0
$485$	38.5640	1.75110
$486$	0	0
$487$	15.5788	0.705945	0.352972	−	0.935634i	$-0.385171\pi$
$487$	15.5788	0.705945	0.352972	+	0.935634i	$0.385171\pi$
$488$	0	0
$489$	16.6717	0.753918
$490$	0	0
$491$	−38.7362	−1.74814	−0.874069	−	0.485801i	$-0.838528\pi$
$491$	−38.7362	−1.74814	−0.874069	+	0.485801i	$0.838528\pi$
$492$	0	0
$493$	3.84721	0.173270
$494$	0	0
$495$	−18.8631	−0.847833
$496$	0	0
$497$	−3.06703	−0.137575
$498$	0	0
$499$	30.4777	1.36437	0.682184	−	0.731181i	$-0.261031\pi$
$499$	30.4777	1.36437	0.682184	+	0.731181i	$0.261031\pi$
$500$	0	0
$501$	−20.0826	−0.897224
$502$	0	0
$503$	−21.5999	−0.963091	−0.481545	−	0.876421i	$-0.659924\pi$
$503$	−21.5999	−0.963091	−0.481545	+	0.876421i	$0.659924\pi$
$504$	0	0
$505$	15.2058	0.676647
$506$	0	0
$507$	−2.98317	−0.132487
$508$	0	0
$509$	37.8929	1.67957	0.839787	−	0.542916i	$-0.182680\pi$
$509$	37.8929	1.67957	0.839787	+	0.542916i	$0.182680\pi$
$510$	0	0
$511$	10.3083	0.456011
$512$	0	0
$513$	−2.59801	−0.114705
$514$	0	0
$515$	25.5801	1.12719
$516$	0	0
$517$	3.02365	0.132980
$518$	0	0
$519$	−30.5270	−1.33999
$520$	0	0
$521$	10.7631	0.471540	0.235770	−	0.971809i	$-0.424239\pi$
$521$	10.7631	0.471540	0.235770	+	0.971809i	$0.424239\pi$
$522$	0	0
$523$	13.5112	0.590804	0.295402	−	0.955373i	$-0.404546\pi$
$523$	13.5112	0.590804	0.295402	+	0.955373i	$0.404546\pi$
$524$	0	0
$525$	−15.5843	−0.680154
$526$	0	0
$527$	−7.79353	−0.339491
$528$	0	0
$529$	30.6465	1.33246
$530$	0	0
$531$	76.1987	3.30674
$532$	0	0
$533$	4.72545	0.204682
$534$	0	0
$535$	−40.8940	−1.76800
$536$	0	0
$537$	11.9682	0.516466
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−29.7148	−1.27754	−0.638770	−	0.769397i	$-0.720557\pi$
$541$	−29.7148	−1.27754	−0.638770	+	0.769397i	$0.720557\pi$
$542$	0	0
$543$	49.4852	2.12361
$544$	0	0
$545$	−7.86180	−0.336763
$546$	0	0
$547$	12.1263	0.518484	0.259242	−	0.965812i	$-0.416527\pi$
$547$	12.1263	0.518484	0.259242	+	0.965812i	$0.416527\pi$
$548$	0	0
$549$	−5.91993	−0.252656
$550$	0	0
$551$	0.973494	0.0414722
$552$	0	0
$553$	5.83010	0.247921
$554$	0	0
$555$	45.2019	1.91871
$556$	0	0
$557$	29.9648	1.26965	0.634825	−	0.772656i	$-0.281072\pi$
$557$	29.9648	1.26965	0.634825	+	0.772656i	$0.281072\pi$
$558$	0	0
$559$	−8.34121	−0.352795
$560$	0	0
$561$	3.54127	0.149513
$562$	0	0
$563$	25.9907	1.09538	0.547689	−	0.836682i	$-0.315508\pi$
$563$	25.9907	1.09538	0.547689	+	0.836682i	$0.315508\pi$
$564$	0	0
$565$	33.5466	1.41132
$566$	0	0
$567$	8.10392	0.340333
$568$	0	0
$569$	−1.30119	−0.0545487	−0.0272743	−	0.999628i	$-0.508683\pi$
$569$	−1.30119	−0.0545487	−0.0272743	+	0.999628i	$0.508683\pi$
$570$	0	0
$571$	−2.13173	−0.0892101	−0.0446051	−	0.999005i	$-0.514203\pi$
$571$	−2.13173	−0.0892101	−0.0446051	+	0.999005i	$0.514203\pi$
$572$	0	0
$573$	−0.00544636	−0.000227525 0
$574$	0	0
$575$	−38.2630	−1.59568
$576$	0	0
$577$	21.9561	0.914044	0.457022	−	0.889455i	$-0.348916\pi$
$577$	21.9561	0.914044	0.457022	+	0.889455i	$0.348916\pi$
$578$	0	0
$579$	28.5019	1.18450
$580$	0	0
$581$	9.23729	0.383227
$582$	0	0
$583$	−9.50471	−0.393645
$584$	0	0
$585$	−18.8631	−0.779893
$586$	0	0
$587$	16.8990	0.697496	0.348748	−	0.937217i	$-0.386607\pi$
$587$	16.8990	0.697496	0.348748	+	0.937217i	$0.386607\pi$
$588$	0	0
$589$	−1.97207	−0.0812575
$590$	0	0
$591$	−2.91045	−0.119720
$592$	0	0
$593$	−20.8869	−0.857720	−0.428860	−	0.903371i	$-0.641085\pi$
$593$	−20.8869	−0.857720	−0.428860	+	0.903371i	$0.641085\pi$
$594$	0	0
$595$	3.79571	0.155609
$596$	0	0
$597$	10.2372	0.418980
$598$	0	0
$599$	−28.9823	−1.18418	−0.592092	−	0.805870i	$-0.701698\pi$
$599$	−28.9823	−1.18418	−0.592092	+	0.805870i	$0.701698\pi$
$600$	0	0
$601$	9.34759	0.381296	0.190648	−	0.981658i	$-0.438941\pi$
$601$	9.34759	0.381296	0.190648	+	0.981658i	$0.438941\pi$
$602$	0	0
$603$	−43.9149	−1.78835
$604$	0	0
$605$	3.19751	0.129997
$606$	0	0
$607$	32.7917	1.33097	0.665487	−	0.746409i	$-0.268224\pi$
$607$	32.7917	1.33097	0.665487	+	0.746409i	$0.268224\pi$
$608$	0	0
$609$	−9.66814	−0.391773
$610$	0	0
$611$	3.02365	0.122324
$612$	0	0
$613$	23.1572	0.935311	0.467656	−	0.883911i	$-0.345099\pi$
$613$	23.1572	0.935311	0.467656	+	0.883911i	$0.345099\pi$
$614$	0	0
$615$	45.0747	1.81759
$616$	0	0
$617$	−29.3083	−1.17991	−0.589954	−	0.807437i	$-0.700854\pi$
$617$	−29.3083	−1.17991	−0.589954	+	0.807437i	$0.700854\pi$
$618$	0	0
$619$	−12.9366	−0.519966	−0.259983	−	0.965613i	$-0.583717\pi$
$619$	−12.9366	−0.519966	−0.259983	+	0.965613i	$0.583717\pi$
$620$	0	0
$621$	63.3495	2.54213
$622$	0	0
$623$	−1.07999	−0.0432689
$624$	0	0
$625$	−23.8295	−0.953179
$626$	0	0
$627$	0.896080	0.0357860
$628$	0	0
$629$	−5.62533	−0.224297
$630$	0	0
$631$	11.6680	0.464495	0.232247	−	0.972657i	$-0.425392\pi$
$631$	11.6680	0.464495	0.232247	+	0.972657i	$0.425392\pi$
$632$	0	0
$633$	1.73640	0.0690155
$634$	0	0
$635$	64.4548	2.55781
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−18.0934	−0.715763
$640$	0	0
$641$	30.1282	1.18999	0.594995	−	0.803729i	$-0.297154\pi$
$641$	30.1282	1.18999	0.594995	+	0.803729i	$0.297154\pi$
$642$	0	0
$643$	30.2577	1.19325	0.596623	−	0.802522i	$-0.296509\pi$
$643$	30.2577	1.19325	0.596623	+	0.802522i	$0.296509\pi$
$644$	0	0
$645$	−79.5644	−3.13285
$646$	0	0
$647$	−25.7746	−1.01331	−0.506653	−	0.862150i	$-0.669117\pi$
$647$	−25.7746	−1.01331	−0.506653	+	0.862150i	$0.669117\pi$
$648$	0	0
$649$	−12.9166	−0.507019
$650$	0	0
$651$	19.5853	0.767610
$652$	0	0
$653$	36.4729	1.42730	0.713648	−	0.700504i	$-0.247042\pi$
$653$	36.4729	1.42730	0.713648	+	0.700504i	$0.247042\pi$
$654$	0	0
$655$	−2.06078	−0.0805212
$656$	0	0
$657$	60.8117	2.37249
$658$	0	0
$659$	19.9993	0.779062	0.389531	−	0.921013i	$-0.372637\pi$
$659$	19.9993	0.779062	0.389531	+	0.921013i	$0.372637\pi$
$660$	0	0
$661$	−24.5502	−0.954891	−0.477445	−	0.878661i	$-0.658437\pi$
$661$	−24.5502	−0.954891	−0.477445	+	0.878661i	$0.658437\pi$
$662$	0	0
$663$	3.54127	0.137532
$664$	0	0
$665$	0.960462	0.0372451
$666$	0	0
$667$	−23.7375	−0.919121
$668$	0	0
$669$	−39.5353	−1.52852
$670$	0	0
$671$	1.00349	0.0387395
$672$	0	0
$673$	14.9857	0.577655	0.288828	−	0.957381i	$-0.406735\pi$
$673$	14.9857	0.577655	0.288828	+	0.957381i	$0.406735\pi$
$674$	0	0
$675$	−45.1836	−1.73912
$676$	0	0
$677$	−42.5415	−1.63500	−0.817501	−	0.575928i	$-0.804641\pi$
$677$	−42.5415	−1.63500	−0.817501	+	0.575928i	$0.804641\pi$
$678$	0	0
$679$	12.0606	0.462844
$680$	0	0
$681$	49.2280	1.88642
$682$	0	0
$683$	−16.6831	−0.638362	−0.319181	−	0.947694i	$-0.603408\pi$
$683$	−16.6831	−0.638362	−0.319181	+	0.947694i	$0.603408\pi$
$684$	0	0
$685$	−1.80746	−0.0690597
$686$	0	0
$687$	88.5984	3.38024
$688$	0	0
$689$	−9.50471	−0.362100
$690$	0	0
$691$	46.8600	1.78264	0.891319	−	0.453377i	$-0.149781\pi$
$691$	46.8600	1.78264	0.891319	+	0.453377i	$0.149781\pi$
$692$	0	0
$693$	−5.89931	−0.224096
$694$	0	0
$695$	−33.1305	−1.25671
$696$	0	0
$697$	−5.60950	−0.212475
$698$	0	0
$699$	−22.3900	−0.846868
$700$	0	0
$701$	−14.8465	−0.560746	−0.280373	−	0.959891i	$-0.590458\pi$
$701$	−14.8465	−0.560746	−0.280373	+	0.959891i	$0.590458\pi$
$702$	0	0
$703$	−1.42343	−0.0536856
$704$	0	0
$705$	28.8417	1.08624
$706$	0	0
$707$	4.75550	0.178849
$708$	0	0
$709$	−36.2226	−1.36037	−0.680184	−	0.733042i	$-0.738100\pi$
$709$	−36.2226	−1.36037	−0.680184	+	0.733042i	$0.738100\pi$
$710$	0	0
$711$	34.3936	1.28986
$712$	0	0
$713$	48.0865	1.80086
$714$	0	0
$715$	3.19751	0.119580
$716$	0	0
$717$	−69.4125	−2.59226
$718$	0	0
$719$	10.0712	0.375592	0.187796	−	0.982208i	$-0.439866\pi$
$719$	10.0712	0.375592	0.187796	+	0.982208i	$0.439866\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−29.7786	−1.10748
$724$	0	0
$725$	16.9306	0.628788
$726$	0	0
$727$	19.2979	0.715719	0.357859	−	0.933775i	$-0.383507\pi$
$727$	19.2979	0.715719	0.357859	+	0.933775i	$0.383507\pi$
$728$	0	0
$729$	−29.5980	−1.09622
$730$	0	0
$731$	9.90171	0.366228
$732$	0	0
$733$	8.72910	0.322417	0.161208	−	0.986920i	$-0.448461\pi$
$733$	8.72910	0.322417	0.161208	+	0.986920i	$0.448461\pi$
$734$	0	0
$735$	−9.53872	−0.351841
$736$	0	0
$737$	7.44408	0.274206
$738$	0	0
$739$	20.3871	0.749951	0.374975	−	0.927035i	$-0.377651\pi$
$739$	20.3871	0.749951	0.374975	+	0.927035i	$0.377651\pi$
$740$	0	0
$741$	0.896080	0.0329183
$742$	0	0
$743$	−40.1070	−1.47138	−0.735692	−	0.677316i	$-0.763143\pi$
$743$	−40.1070	−1.47138	−0.735692	+	0.677316i	$0.763143\pi$
$744$	0	0
$745$	−17.5030	−0.641261
$746$	0	0
$747$	54.4936	1.99382
$748$	0	0
$749$	−12.7893	−0.467312
$750$	0	0
$751$	−47.4794	−1.73255	−0.866273	−	0.499570i	$-0.833491\pi$
$751$	−47.4794	−1.73255	−0.866273	+	0.499570i	$0.833491\pi$
$752$	0	0
$753$	−47.2721	−1.72269
$754$	0	0
$755$	63.3200	2.30445
$756$	0	0
$757$	32.6137	1.18536	0.592682	−	0.805437i	$-0.298069\pi$
$757$	32.6137	1.18536	0.592682	+	0.805437i	$0.298069\pi$
$758$	0	0
$759$	−21.8499	−0.793100
$760$	0	0
$761$	23.8173	0.863376	0.431688	−	0.902023i	$-0.357918\pi$
$761$	23.8173	0.863376	0.431688	+	0.902023i	$0.357918\pi$
$762$	0	0
$763$	−2.45873	−0.0890119
$764$	0	0
$765$	22.3921	0.809587
$766$	0	0
$767$	−12.9166	−0.466390
$768$	0	0
$769$	7.65466	0.276034	0.138017	−	0.990430i	$-0.455927\pi$
$769$	7.65466	0.276034	0.138017	+	0.990430i	$0.455927\pi$
$770$	0	0
$771$	−59.5757	−2.14557
$772$	0	0
$773$	−28.4596	−1.02362	−0.511811	−	0.859098i	$-0.671025\pi$
$773$	−28.4596	−1.02362	−0.511811	+	0.859098i	$0.671025\pi$
$774$	0	0
$775$	−34.2974	−1.23200
$776$	0	0
$777$	14.1366	0.507148
$778$	0	0
$779$	−1.41942	−0.0508561
$780$	0	0
$781$	3.06703	0.109747
$782$	0	0
$783$	−28.0309	−1.00174
$784$	0	0
$785$	31.4330	1.12189
$786$	0	0
$787$	−10.3402	−0.368588	−0.184294	−	0.982871i	$-0.559000\pi$
$787$	−10.3402	−0.368588	−0.184294	+	0.982871i	$0.559000\pi$
$788$	0	0
$789$	58.3550	2.07749
$790$	0	0
$791$	10.4915	0.373034
$792$	0	0
$793$	1.00349	0.0356351
$794$	0	0
$795$	−90.6627	−3.21547
$796$	0	0
$797$	19.9378	0.706235	0.353117	−	0.935579i	$-0.385122\pi$
$797$	19.9378	0.706235	0.353117	+	0.935579i	$0.385122\pi$
$798$	0	0
$799$	−3.58932	−0.126981
$800$	0	0
$801$	−6.37119	−0.225115
$802$	0	0
$803$	−10.3083	−0.363771
$804$	0	0
$805$	−23.4198	−0.825438
$806$	0	0
$807$	27.0668	0.952797
$808$	0	0
$809$	−28.6798	−1.00833	−0.504164	−	0.863608i	$-0.668199\pi$
$809$	−28.6798	−1.00833	−0.504164	+	0.863608i	$0.668199\pi$
$810$	0	0
$811$	−25.1778	−0.884113	−0.442057	−	0.896987i	$-0.645751\pi$
$811$	−25.1778	−0.884113	−0.442057	+	0.896987i	$0.645751\pi$
$812$	0	0
$813$	−50.7543	−1.78003
$814$	0	0
$815$	−17.8695	−0.625941
$816$	0	0
$817$	2.50552	0.0876570
$818$	0	0
$819$	−5.89931	−0.206139
$820$	0	0
$821$	−22.9218	−0.799977	−0.399988	−	0.916520i	$-0.630986\pi$
$821$	−22.9218	−0.799977	−0.399988	+	0.916520i	$0.630986\pi$
$822$	0	0
$823$	−25.4907	−0.888549	−0.444275	−	0.895891i	$-0.646539\pi$
$823$	−25.4907	−0.888549	−0.444275	+	0.895891i	$0.646539\pi$
$824$	0	0
$825$	15.5843	0.542575
$826$	0	0
$827$	−5.50179	−0.191316	−0.0956579	−	0.995414i	$-0.530495\pi$
$827$	−5.50179	−0.191316	−0.0956579	+	0.995414i	$0.530495\pi$
$828$	0	0
$829$	40.0228	1.39005	0.695025	−	0.718986i	$-0.255393\pi$
$829$	40.0228	1.39005	0.695025	+	0.718986i	$0.255393\pi$
$830$	0	0
$831$	−11.9047	−0.412970
$832$	0	0
$833$	1.18708	0.0411300
$834$	0	0
$835$	21.5255	0.744921
$836$	0	0
$837$	56.7839	1.96274
$838$	0	0
$839$	16.9393	0.584811	0.292406	−	0.956294i	$-0.405544\pi$
$839$	16.9393	0.584811	0.292406	+	0.956294i	$0.405544\pi$
$840$	0	0
$841$	−18.4966	−0.637814
$842$	0	0
$843$	−78.7065	−2.71080
$844$	0	0
$845$	3.19751	0.109998
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	48.5479	1.66616
$850$	0	0
$851$	34.7086	1.18980
$852$	0	0
$853$	−12.8030	−0.438366	−0.219183	−	0.975684i	$-0.570339\pi$
$853$	−12.8030	−0.438366	−0.219183	+	0.975684i	$0.570339\pi$
$854$	0	0
$855$	5.66606	0.193775
$856$	0	0
$857$	17.9534	0.613276	0.306638	−	0.951826i	$-0.400796\pi$
$857$	17.9534	0.613276	0.306638	+	0.951826i	$0.400796\pi$
$858$	0	0
$859$	36.9107	1.25938	0.629689	−	0.776848i	$-0.283182\pi$
$859$	36.9107	1.25938	0.629689	+	0.776848i	$0.283182\pi$
$860$	0	0
$861$	14.0968	0.480418
$862$	0	0
$863$	−14.9728	−0.509680	−0.254840	−	0.966983i	$-0.582023\pi$
$863$	−14.9728	−0.509680	−0.254840	+	0.966983i	$0.582023\pi$
$864$	0	0
$865$	32.7203	1.11252
$866$	0	0
$867$	46.5101	1.57957
$868$	0	0
$869$	−5.83010	−0.197773
$870$	0	0
$871$	7.44408	0.252233
$872$	0	0
$873$	71.1494	2.40804
$874$	0	0
$875$	0.716451	0.0242205
$876$	0	0
$877$	−33.7707	−1.14036	−0.570178	−	0.821521i	$-0.693126\pi$
$877$	−33.7707	−1.14036	−0.570178	+	0.821521i	$0.693126\pi$
$878$	0	0
$879$	−7.38424	−0.249064
$880$	0	0
$881$	16.9144	0.569860	0.284930	−	0.958548i	$-0.408030\pi$
$881$	16.9144	0.569860	0.284930	+	0.958548i	$0.408030\pi$
$882$	0	0
$883$	10.4990	0.353319	0.176660	−	0.984272i	$-0.443471\pi$
$883$	10.4990	0.353319	0.176660	+	0.984272i	$0.443471\pi$
$884$	0	0
$885$	−123.207	−4.14157
$886$	0	0
$887$	4.11464	0.138156	0.0690781	−	0.997611i	$-0.477994\pi$
$887$	4.11464	0.138156	0.0690781	+	0.997611i	$0.477994\pi$
$888$	0	0
$889$	20.1578	0.676071
$890$	0	0
$891$	−8.10392	−0.271492
$892$	0	0
$893$	−0.908238	−0.0303930
$894$	0	0
$895$	−12.8281	−0.428796
$896$	0	0
$897$	−21.8499	−0.729546
$898$	0	0
$899$	−21.2774	−0.709640
$900$	0	0
$901$	11.2829	0.375887
$902$	0	0
$903$	−24.8832	−0.828063
$904$	0	0
$905$	−53.0407	−1.76313
$906$	0	0
$907$	1.26804	0.0421047	0.0210523	−	0.999778i	$-0.493298\pi$
$907$	1.26804	0.0421047	0.0210523	+	0.999778i	$0.493298\pi$
$908$	0	0
$909$	28.0542	0.930498
$910$	0	0
$911$	28.5742	0.946705	0.473353	−	0.880873i	$-0.343044\pi$
$911$	28.5742	0.946705	0.473353	+	0.880873i	$0.343044\pi$
$912$	0	0
$913$	−9.23729	−0.305710
$914$	0	0
$915$	9.57205	0.316442
$916$	0	0
$917$	−0.644494	−0.0212831
$918$	0	0
$919$	37.5286	1.23795	0.618977	−	0.785409i	$-0.287547\pi$
$919$	37.5286	1.23795	0.618977	+	0.785409i	$0.287547\pi$
$920$	0	0
$921$	−21.2674	−0.700783
$922$	0	0
$923$	3.06703	0.100953
$924$	0	0
$925$	−24.7557	−0.813963
$926$	0	0
$927$	47.1945	1.55007
$928$	0	0
$929$	−14.9227	−0.489599	−0.244800	−	0.969574i	$-0.578722\pi$
$929$	−14.9227	−0.489599	−0.244800	+	0.969574i	$0.578722\pi$
$930$	0	0
$931$	0.300378	0.00984450
$932$	0	0
$933$	−32.4468	−1.06226
$934$	0	0
$935$	−3.79571	−0.124133
$936$	0	0
$937$	−37.7125	−1.23201	−0.616007	−	0.787740i	$-0.711251\pi$
$937$	−37.7125	−1.23201	−0.616007	+	0.787740i	$0.711251\pi$
$938$	0	0
$939$	89.2739	2.91334
$940$	0	0
$941$	10.8319	0.353110	0.176555	−	0.984291i	$-0.443505\pi$
$941$	10.8319	0.353110	0.176555	+	0.984291i	$0.443505\pi$
$942$	0	0
$943$	34.6110	1.12709
$944$	0	0
$945$	−27.6557	−0.899640
$946$	0	0
$947$	−28.0606	−0.911848	−0.455924	−	0.890019i	$-0.650691\pi$
$947$	−28.0606	−0.911848	−0.455924	+	0.890019i	$0.650691\pi$
$948$	0	0
$949$	−10.3083	−0.334621
$950$	0	0
$951$	−65.6631	−2.12927
$952$	0	0
$953$	−41.9586	−1.35917	−0.679586	−	0.733596i	$-0.737840\pi$
$953$	−41.9586	−1.35917	−0.679586	+	0.733596i	$0.737840\pi$
$954$	0	0
$955$	0.00583767	0.000188903 0
$956$	0	0
$957$	9.66814	0.312527
$958$	0	0
$959$	−0.565273	−0.0182536
$960$	0	0
$961$	12.1028	0.390413
$962$	0	0
$963$	−75.4483	−2.43129
$964$	0	0
$965$	−30.5497	−0.983431
$966$	0	0
$967$	15.9842	0.514018	0.257009	−	0.966409i	$-0.417263\pi$
$967$	15.9842	0.514018	0.257009	+	0.966409i	$0.417263\pi$
$968$	0	0
$969$	−1.06372	−0.0341717
$970$	0	0
$971$	−37.9393	−1.21753	−0.608765	−	0.793351i	$-0.708335\pi$
$971$	−37.9393	−1.21753	−0.608765	+	0.793351i	$0.708335\pi$
$972$	0	0
$973$	−10.3614	−0.332170
$974$	0	0
$975$	15.5843	0.499096
$976$	0	0
$977$	26.8666	0.859537	0.429769	−	0.902939i	$-0.358595\pi$
$977$	26.8666	0.859537	0.429769	+	0.902939i	$0.358595\pi$
$978$	0	0
$979$	1.07999	0.0345166
$980$	0	0
$981$	−14.5048	−0.463102
$982$	0	0
$983$	42.1880	1.34559	0.672794	−	0.739830i	$-0.265094\pi$
$983$	42.1880	1.34559	0.672794	+	0.739830i	$0.265094\pi$
$984$	0	0
$985$	3.11956	0.0993975
$986$	0	0
$987$	9.02006	0.287112
$988$	0	0
$989$	−61.0942	−1.94268
$990$	0	0
$991$	0.198655	0.00631048	0.00315524	−	0.999995i	$-0.498996\pi$
$991$	0.198655	0.00631048	0.00315524	+	0.999995i	$0.498996\pi$
$992$	0	0
$993$	86.9346	2.75879
$994$	0	0
$995$	−10.9727	−0.347859
$996$	0	0
$997$	58.2283	1.84411	0.922055	−	0.387060i	$-0.126509\pi$
$997$	58.2283	1.84411	0.922055	+	0.387060i	$0.126509\pi$
$998$	0	0
$999$	40.9864	1.29675

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.k.1.1	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.k.1.1	✓	6	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:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q-2.98317 q^{3} +3.19751 q^{5} +1.00000 q^{7} +5.89931 q^{9} +O(q^{10})\)	\(q-2.98317 q^{3} +3.19751 q^{5} +1.00000 q^{7} +5.89931 q^{9} -1.00000 q^{11} -1.00000 q^{13} -9.53872 q^{15} +1.18708 q^{17} +0.300378 q^{19} -2.98317 q^{21} -7.32438 q^{23} +5.22407 q^{25} -8.64913 q^{27} +3.24089 q^{29} -6.56527 q^{31} +2.98317 q^{33} +3.19751 q^{35} -4.73878 q^{37} +2.98317 q^{39} -4.72545 q^{41} +8.34121 q^{43} +18.8631 q^{45} -3.02365 q^{47} +1.00000 q^{49} -3.54127 q^{51} +9.50471 q^{53} -3.19751 q^{55} -0.896080 q^{57} +12.9166 q^{59} -1.00349 q^{61} +5.89931 q^{63} -3.19751 q^{65} -7.44408 q^{67} +21.8499 q^{69} -3.06703 q^{71} +10.3083 q^{73} -15.5843 q^{75} -1.00000 q^{77} +5.83010 q^{79} +8.10392 q^{81} +9.23729 q^{83} +3.79571 q^{85} -9.66814 q^{87} -1.07999 q^{89} -1.00000 q^{91} +19.5853 q^{93} +0.960462 q^{95} +12.0606 q^{97} -5.89931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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