LMFDB - Embedded newform 8008.2.a.q.1.5

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 $ x^9 - 3x^8 - 15x^7 + 45x^6 + 64x^5 - 201x^4 - 53x^3 + 252x^2 - 69x - 26
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 5 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.827072$ 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)

$f(q)$	$=$	$q+0.827072 q^{3} -3.42555 q^{5} +1.00000 q^{7} -2.31595 q^{9} +O(q^{10})$	$q+0.827072 q^{3} -3.42555 q^{5} +1.00000 q^{7} -2.31595 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.83317 q^{15} -3.58099 q^{17} +5.27269 q^{19} +0.827072 q^{21} +4.25564 q^{23} +6.73436 q^{25} -4.39667 q^{27} -0.0839454 q^{29} -9.28245 q^{31} +0.827072 q^{33} -3.42555 q^{35} -7.88865 q^{37} -0.827072 q^{39} +6.40806 q^{41} -3.96441 q^{43} +7.93340 q^{45} -4.25564 q^{47} +1.00000 q^{49} -2.96173 q^{51} +5.49704 q^{53} -3.42555 q^{55} +4.36090 q^{57} +7.83579 q^{59} -5.37090 q^{61} -2.31595 q^{63} +3.42555 q^{65} -1.50319 q^{67} +3.51972 q^{69} -3.77911 q^{71} +9.42232 q^{73} +5.56980 q^{75} +1.00000 q^{77} -7.41114 q^{79} +3.31149 q^{81} -7.47488 q^{83} +12.2668 q^{85} -0.0694289 q^{87} +5.44418 q^{89} -1.00000 q^{91} -7.67725 q^{93} -18.0619 q^{95} -2.94013 q^{97} -2.31595 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	$ 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) $ 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * 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
$3$	0.827072	0.477510	0.238755	−	0.971080i	$-0.423261\pi$
$3$	0.827072	0.477510	0.238755	+	0.971080i	$0.423261\pi$
$4$	0	0
$5$	−3.42555	−1.53195	−0.765975	−	0.642870i	$-0.777744\pi$
$5$	−3.42555	−1.53195	−0.765975	+	0.642870i	$0.777744\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.31595	−0.771984
$10$	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
$15$	−2.83317	−0.731522
$16$	0	0
$17$	−3.58099	−0.868517	−0.434259	−	0.900788i	$-0.642990\pi$
$17$	−3.58099	−0.868517	−0.434259	+	0.900788i	$0.642990\pi$
$18$	0	0
$19$	5.27269	1.20964	0.604819	−	0.796363i	$-0.293245\pi$
$19$	5.27269	1.20964	0.604819	+	0.796363i	$0.293245\pi$
$20$	0	0
$21$	0.827072	0.180482
$22$	0	0
$23$	4.25564	0.887363	0.443681	−	0.896185i	$-0.353672\pi$
$23$	4.25564	0.887363	0.443681	+	0.896185i	$0.353672\pi$
$24$	0	0
$25$	6.73436	1.34687
$26$	0	0
$27$	−4.39667	−0.846140
$28$	0	0
$29$	−0.0839454	−0.0155883	−0.00779414	−	0.999970i	$-0.502481\pi$
$29$	−0.0839454	−0.0155883	−0.00779414	+	0.999970i	$0.502481\pi$
$30$	0	0
$31$	−9.28245	−1.66718	−0.833589	−	0.552386i	$-0.813718\pi$
$31$	−9.28245	−1.66718	−0.833589	+	0.552386i	$0.813718\pi$
$32$	0	0
$33$	0.827072	0.143975
$34$	0	0
$35$	−3.42555	−0.579023
$36$	0	0
$37$	−7.88865	−1.29689	−0.648443	−	0.761263i	$-0.724580\pi$
$37$	−7.88865	−1.29689	−0.648443	+	0.761263i	$0.724580\pi$
$38$	0	0
$39$	−0.827072	−0.132437
$40$	0	0
$41$	6.40806	1.00077	0.500385	−	0.865803i	$-0.333192\pi$
$41$	6.40806	1.00077	0.500385	+	0.865803i	$0.333192\pi$
$42$	0	0
$43$	−3.96441	−0.604567	−0.302283	−	0.953218i	$-0.597749\pi$
$43$	−3.96441	−0.604567	−0.302283	+	0.953218i	$0.597749\pi$
$44$	0	0
$45$	7.93340	1.18264
$46$	0	0
$47$	−4.25564	−0.620749	−0.310375	−	0.950614i	$-0.600454\pi$
$47$	−4.25564	−0.620749	−0.310375	+	0.950614i	$0.600454\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−2.96173	−0.414726
$52$	0	0
$53$	5.49704	0.755077	0.377538	−	0.925994i	$-0.376771\pi$
$53$	5.49704	0.755077	0.377538	+	0.925994i	$0.376771\pi$
$54$	0	0
$55$	−3.42555	−0.461900
$56$	0	0
$57$	4.36090	0.577615
$58$	0	0
$59$	7.83579	1.02013	0.510067	−	0.860135i	$-0.329621\pi$
$59$	7.83579	1.02013	0.510067	+	0.860135i	$0.329621\pi$
$60$	0	0
$61$	−5.37090	−0.687674	−0.343837	−	0.939029i	$-0.611727\pi$
$61$	−5.37090	−0.687674	−0.343837	+	0.939029i	$0.611727\pi$
$62$	0	0
$63$	−2.31595	−0.291783
$64$	0	0
$65$	3.42555	0.424887
$66$	0	0
$67$	−1.50319	−0.183644	−0.0918220	−	0.995775i	$-0.529269\pi$
$67$	−1.50319	−0.183644	−0.0918220	+	0.995775i	$0.529269\pi$
$68$	0	0
$69$	3.51972	0.423725
$70$	0	0
$71$	−3.77911	−0.448498	−0.224249	−	0.974532i	$-0.571993\pi$
$71$	−3.77911	−0.448498	−0.224249	+	0.974532i	$0.571993\pi$
$72$	0	0
$73$	9.42232	1.10280	0.551400	−	0.834241i	$-0.314094\pi$
$73$	9.42232	1.10280	0.551400	+	0.834241i	$0.314094\pi$
$74$	0	0
$75$	5.56980	0.643145
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	−7.41114	−0.833818	−0.416909	−	0.908948i	$-0.636887\pi$
$79$	−7.41114	−0.833818	−0.416909	+	0.908948i	$0.636887\pi$
$80$	0	0
$81$	3.31149	0.367943
$82$	0	0
$83$	−7.47488	−0.820474	−0.410237	−	0.911979i	$-0.634554\pi$
$83$	−7.47488	−0.820474	−0.410237	+	0.911979i	$0.634554\pi$
$84$	0	0
$85$	12.2668	1.33053
$86$	0	0
$87$	−0.0694289	−0.00744356
$88$	0	0
$89$	5.44418	0.577082	0.288541	−	0.957468i	$-0.406830\pi$
$89$	5.44418	0.577082	0.288541	+	0.957468i	$0.406830\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−7.67725	−0.796094
$94$	0	0
$95$	−18.0619	−1.85311
$96$	0	0
$97$	−2.94013	−0.298525	−0.149262	−	0.988798i	$-0.547690\pi$
$97$	−2.94013	−0.298525	−0.149262	+	0.988798i	$0.547690\pi$
$98$	0	0
$99$	−2.31595	−0.232762
$100$	0	0
$101$	10.1142	1.00640	0.503199	−	0.864170i	$-0.332156\pi$
$101$	10.1142	1.00640	0.503199	+	0.864170i	$0.332156\pi$
$102$	0	0
$103$	19.0309	1.87517	0.937584	−	0.347758i	$-0.113057\pi$
$103$	19.0309	1.87517	0.937584	+	0.347758i	$0.113057\pi$
$104$	0	0
$105$	−2.83317	−0.276489
$106$	0	0
$107$	−0.372686	−0.0360289	−0.0180144	−	0.999838i	$-0.505734\pi$
$107$	−0.372686	−0.0360289	−0.0180144	+	0.999838i	$0.505734\pi$
$108$	0	0
$109$	−4.47918	−0.429028	−0.214514	−	0.976721i	$-0.568817\pi$
$109$	−4.47918	−0.429028	−0.214514	+	0.976721i	$0.568817\pi$
$110$	0	0
$111$	−6.52448	−0.619277
$112$	0	0
$113$	−13.2505	−1.24651	−0.623253	−	0.782020i	$-0.714189\pi$
$113$	−13.2505	−1.24651	−0.623253	+	0.782020i	$0.714189\pi$
$114$	0	0
$115$	−14.5779	−1.35940
$116$	0	0
$117$	2.31595	0.214110
$118$	0	0
$119$	−3.58099	−0.328269
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	5.29993	0.477878
$124$	0	0
$125$	−5.94114	−0.531391
$126$	0	0
$127$	−12.3695	−1.09762	−0.548808	−	0.835949i	$-0.684918\pi$
$127$	−12.3695	−1.09762	−0.548808	+	0.835949i	$0.684918\pi$
$128$	0	0
$129$	−3.27885	−0.288687
$130$	0	0
$131$	1.18834	0.103826	0.0519130	−	0.998652i	$-0.483468\pi$
$131$	1.18834	0.103826	0.0519130	+	0.998652i	$0.483468\pi$
$132$	0	0
$133$	5.27269	0.457201
$134$	0	0
$135$	15.0610	1.29625
$136$	0	0
$137$	15.7079	1.34201	0.671007	−	0.741451i	$-0.265862\pi$
$137$	15.7079	1.34201	0.671007	+	0.741451i	$0.265862\pi$
$138$	0	0
$139$	10.9567	0.929339	0.464670	−	0.885484i	$-0.346173\pi$
$139$	10.9567	0.929339	0.464670	+	0.885484i	$0.346173\pi$
$140$	0	0
$141$	−3.51972	−0.296414
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	0.287559	0.0238805
$146$	0	0
$147$	0.827072	0.0682157
$148$	0	0
$149$	−15.4227	−1.26348	−0.631739	−	0.775181i	$-0.717658\pi$
$149$	−15.4227	−1.26348	−0.631739	+	0.775181i	$0.717658\pi$
$150$	0	0
$151$	13.7991	1.12296	0.561479	−	0.827491i	$-0.310233\pi$
$151$	13.7991	1.12296	0.561479	+	0.827491i	$0.310233\pi$
$152$	0	0
$153$	8.29340	0.670481
$154$	0	0
$155$	31.7975	2.55403
$156$	0	0
$157$	12.6096	1.00636	0.503179	−	0.864182i	$-0.332163\pi$
$157$	12.6096	1.00636	0.503179	+	0.864182i	$0.332163\pi$
$158$	0	0
$159$	4.54645	0.360557
$160$	0	0
$161$	4.25564	0.335392
$162$	0	0
$163$	13.0288	1.02049	0.510245	−	0.860029i	$-0.329555\pi$
$163$	13.0288	1.02049	0.510245	+	0.860029i	$0.329555\pi$
$164$	0	0
$165$	−2.83317	−0.220562
$166$	0	0
$167$	−7.41568	−0.573842	−0.286921	−	0.957954i	$-0.592632\pi$
$167$	−7.41568	−0.573842	−0.286921	+	0.957954i	$0.592632\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−12.2113	−0.933822
$172$	0	0
$173$	16.2370	1.23448	0.617240	−	0.786775i	$-0.288251\pi$
$173$	16.2370	1.23448	0.617240	+	0.786775i	$0.288251\pi$
$174$	0	0
$175$	6.73436	0.509070
$176$	0	0
$177$	6.48076	0.487124
$178$	0	0
$179$	−15.6019	−1.16614	−0.583071	−	0.812421i	$-0.698149\pi$
$179$	−15.6019	−1.16614	−0.583071	+	0.812421i	$0.698149\pi$
$180$	0	0
$181$	17.9191	1.33191	0.665957	−	0.745990i	$-0.268023\pi$
$181$	17.9191	1.33191	0.665957	+	0.745990i	$0.268023\pi$
$182$	0	0
$183$	−4.44212	−0.328371
$184$	0	0
$185$	27.0229	1.98677
$186$	0	0
$187$	−3.58099	−0.261868
$188$	0	0
$189$	−4.39667	−0.319811
$190$	0	0
$191$	−10.2812	−0.743920	−0.371960	−	0.928249i	$-0.621314\pi$
$191$	−10.2812	−0.743920	−0.371960	+	0.928249i	$0.621314\pi$
$192$	0	0
$193$	−1.65170	−0.118892	−0.0594459	−	0.998232i	$-0.518933\pi$
$193$	−1.65170	−0.118892	−0.0594459	+	0.998232i	$0.518933\pi$
$194$	0	0
$195$	2.83317	0.202888
$196$	0	0
$197$	19.1280	1.36281	0.681407	−	0.731904i	$-0.261368\pi$
$197$	19.1280	1.36281	0.681407	+	0.731904i	$0.261368\pi$
$198$	0	0
$199$	4.64289	0.329126	0.164563	−	0.986367i	$-0.447379\pi$
$199$	4.64289	0.329126	0.164563	+	0.986367i	$0.447379\pi$
$200$	0	0
$201$	−1.24325	−0.0876919
$202$	0	0
$203$	−0.0839454	−0.00589181
$204$	0	0
$205$	−21.9511	−1.53313
$206$	0	0
$207$	−9.85586	−0.685030
$208$	0	0
$209$	5.27269	0.364720
$210$	0	0
$211$	5.09057	0.350449	0.175225	−	0.984528i	$-0.443935\pi$
$211$	5.09057	0.350449	0.175225	+	0.984528i	$0.443935\pi$
$212$	0	0
$213$	−3.12559	−0.214162
$214$	0	0
$215$	13.5803	0.926166
$216$	0	0
$217$	−9.28245	−0.630134
$218$	0	0
$219$	7.79294	0.526598
$220$	0	0
$221$	3.58099	0.240883
$222$	0	0
$223$	8.93720	0.598479	0.299240	−	0.954178i	$-0.403267\pi$
$223$	8.93720	0.598479	0.299240	+	0.954178i	$0.403267\pi$
$224$	0	0
$225$	−15.5965	−1.03976
$226$	0	0
$227$	19.4523	1.29110	0.645548	−	0.763720i	$-0.276629\pi$
$227$	19.4523	1.29110	0.645548	+	0.763720i	$0.276629\pi$
$228$	0	0
$229$	5.03742	0.332882	0.166441	−	0.986051i	$-0.446772\pi$
$229$	5.03742	0.332882	0.166441	+	0.986051i	$0.446772\pi$
$230$	0	0
$231$	0.827072	0.0544173
$232$	0	0
$233$	11.6471	0.763027	0.381514	−	0.924363i	$-0.375403\pi$
$233$	11.6471	0.763027	0.381514	+	0.924363i	$0.375403\pi$
$234$	0	0
$235$	14.5779	0.950957
$236$	0	0
$237$	−6.12954	−0.398157
$238$	0	0
$239$	18.9889	1.22829	0.614146	−	0.789192i	$-0.289501\pi$
$239$	18.9889	1.22829	0.614146	+	0.789192i	$0.289501\pi$
$240$	0	0
$241$	−2.63850	−0.169961	−0.0849803	−	0.996383i	$-0.527083\pi$
$241$	−2.63850	−0.169961	−0.0849803	+	0.996383i	$0.527083\pi$
$242$	0	0
$243$	15.9289	1.02184
$244$	0	0
$245$	−3.42555	−0.218850
$246$	0	0
$247$	−5.27269	−0.335493
$248$	0	0
$249$	−6.18226	−0.391785
$250$	0	0
$251$	20.9637	1.32322	0.661608	−	0.749850i	$-0.269874\pi$
$251$	20.9637	1.32322	0.661608	+	0.749850i	$0.269874\pi$
$252$	0	0
$253$	4.25564	0.267550
$254$	0	0
$255$	10.1456	0.635339
$256$	0	0
$257$	15.8632	0.989521	0.494760	−	0.869030i	$-0.335256\pi$
$257$	15.8632	0.989521	0.494760	+	0.869030i	$0.335256\pi$
$258$	0	0
$259$	−7.88865	−0.490177
$260$	0	0
$261$	0.194414	0.0120339
$262$	0	0
$263$	27.9525	1.72363	0.861813	−	0.507227i	$-0.169329\pi$
$263$	27.9525	1.72363	0.861813	+	0.507227i	$0.169329\pi$
$264$	0	0
$265$	−18.8304	−1.15674
$266$	0	0
$267$	4.50273	0.275563
$268$	0	0
$269$	1.63153	0.0994761	0.0497380	−	0.998762i	$-0.484161\pi$
$269$	1.63153	0.0994761	0.0497380	+	0.998762i	$0.484161\pi$
$270$	0	0
$271$	−10.1039	−0.613766	−0.306883	−	0.951747i	$-0.599286\pi$
$271$	−10.1039	−0.613766	−0.306883	+	0.951747i	$0.599286\pi$
$272$	0	0
$273$	−0.827072	−0.0500567
$274$	0	0
$275$	6.73436	0.406097
$276$	0	0
$277$	−18.6766	−1.12217	−0.561085	−	0.827758i	$-0.689616\pi$
$277$	−18.6766	−1.12217	−0.561085	+	0.827758i	$0.689616\pi$
$278$	0	0
$279$	21.4977	1.28703
$280$	0	0
$281$	1.63327	0.0974329	0.0487164	−	0.998813i	$-0.484487\pi$
$281$	1.63327	0.0974329	0.0487164	+	0.998813i	$0.484487\pi$
$282$	0	0
$283$	3.88197	0.230759	0.115380	−	0.993321i	$-0.463192\pi$
$283$	3.88197	0.230759	0.115380	+	0.993321i	$0.463192\pi$
$284$	0	0
$285$	−14.9385	−0.884877
$286$	0	0
$287$	6.40806	0.378256
$288$	0	0
$289$	−4.17653	−0.245678
$290$	0	0
$291$	−2.43170	−0.142549
$292$	0	0
$293$	3.38292	0.197632	0.0988161	−	0.995106i	$-0.468494\pi$
$293$	3.38292	0.197632	0.0988161	+	0.995106i	$0.468494\pi$
$294$	0	0
$295$	−26.8419	−1.56279
$296$	0	0
$297$	−4.39667	−0.255121
$298$	0	0
$299$	−4.25564	−0.246110
$300$	0	0
$301$	−3.96441	−0.228505
$302$	0	0
$303$	8.36516	0.480566
$304$	0	0
$305$	18.3983	1.05348
$306$	0	0
$307$	−11.8955	−0.678914	−0.339457	−	0.940622i	$-0.610243\pi$
$307$	−11.8955	−0.678914	−0.339457	+	0.940622i	$0.610243\pi$
$308$	0	0
$309$	15.7399	0.895412
$310$	0	0
$311$	−17.6137	−0.998783	−0.499391	−	0.866377i	$-0.666443\pi$
$311$	−17.6137	−0.998783	−0.499391	+	0.866377i	$0.666443\pi$
$312$	0	0
$313$	−15.6699	−0.885717	−0.442859	−	0.896592i	$-0.646036\pi$
$313$	−15.6699	−0.885717	−0.442859	+	0.896592i	$0.646036\pi$
$314$	0	0
$315$	7.93340	0.446996
$316$	0	0
$317$	20.9965	1.17928	0.589639	−	0.807667i	$-0.299270\pi$
$317$	20.9965	1.17928	0.589639	+	0.807667i	$0.299270\pi$
$318$	0	0
$319$	−0.0839454	−0.00470004
$320$	0	0
$321$	−0.308238	−0.0172042
$322$	0	0
$323$	−18.8815	−1.05059
$324$	0	0
$325$	−6.73436	−0.373555
$326$	0	0
$327$	−3.70461	−0.204865
$328$	0	0
$329$	−4.25564	−0.234621
$330$	0	0
$331$	7.87225	0.432698	0.216349	−	0.976316i	$-0.430585\pi$
$331$	7.87225	0.432698	0.216349	+	0.976316i	$0.430585\pi$
$332$	0	0
$333$	18.2697	1.00118
$334$	0	0
$335$	5.14925	0.281334
$336$	0	0
$337$	−11.4512	−0.623787	−0.311893	−	0.950117i	$-0.600963\pi$
$337$	−11.4512	−0.623787	−0.311893	+	0.950117i	$0.600963\pi$
$338$	0	0
$339$	−10.9591	−0.595219
$340$	0	0
$341$	−9.28245	−0.502673
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−12.0570	−0.649125
$346$	0	0
$347$	−31.4824	−1.69006	−0.845031	−	0.534717i	$-0.820418\pi$
$347$	−31.4824	−1.69006	−0.845031	+	0.534717i	$0.820418\pi$
$348$	0	0
$349$	12.7918	0.684731	0.342366	−	0.939567i	$-0.388772\pi$
$349$	12.7918	0.684731	0.342366	+	0.939567i	$0.388772\pi$
$350$	0	0
$351$	4.39667	0.234677
$352$	0	0
$353$	26.1526	1.39196	0.695981	−	0.718060i	$-0.254970\pi$
$353$	26.1526	1.39196	0.695981	+	0.718060i	$0.254970\pi$
$354$	0	0
$355$	12.9455	0.687076
$356$	0	0
$357$	−2.96173	−0.156752
$358$	0	0
$359$	19.3638	1.02198	0.510990	−	0.859587i	$-0.329279\pi$
$359$	19.3638	1.02198	0.510990	+	0.859587i	$0.329279\pi$
$360$	0	0
$361$	8.80130	0.463226
$362$	0	0
$363$	0.827072	0.0434100
$364$	0	0
$365$	−32.2766	−1.68943
$366$	0	0
$367$	−2.16980	−0.113263	−0.0566314	−	0.998395i	$-0.518036\pi$
$367$	−2.16980	−0.113263	−0.0566314	+	0.998395i	$0.518036\pi$
$368$	0	0
$369$	−14.8408	−0.772579
$370$	0	0
$371$	5.49704	0.285392
$372$	0	0
$373$	7.99972	0.414210	0.207105	−	0.978319i	$-0.433596\pi$
$373$	7.99972	0.414210	0.207105	+	0.978319i	$0.433596\pi$
$374$	0	0
$375$	−4.91375	−0.253745
$376$	0	0
$377$	0.0839454	0.00432341
$378$	0	0
$379$	−20.0708	−1.03097	−0.515484	−	0.856899i	$-0.672388\pi$
$379$	−20.0708	−1.03097	−0.515484	+	0.856899i	$0.672388\pi$
$380$	0	0
$381$	−10.2305	−0.524123
$382$	0	0
$383$	14.9559	0.764212	0.382106	−	0.924119i	$-0.375199\pi$
$383$	14.9559	0.764212	0.382106	+	0.924119i	$0.375199\pi$
$384$	0	0
$385$	−3.42555	−0.174582
$386$	0	0
$387$	9.18138	0.466716
$388$	0	0
$389$	8.83799	0.448104	0.224052	−	0.974577i	$-0.428071\pi$
$389$	8.83799	0.448104	0.224052	+	0.974577i	$0.428071\pi$
$390$	0	0
$391$	−15.2394	−0.770690
$392$	0	0
$393$	0.982844	0.0495779
$394$	0	0
$395$	25.3872	1.27737
$396$	0	0
$397$	−12.5427	−0.629499	−0.314749	−	0.949175i	$-0.601920\pi$
$397$	−12.5427	−0.629499	−0.314749	+	0.949175i	$0.601920\pi$
$398$	0	0
$399$	4.36090	0.218318
$400$	0	0
$401$	26.7012	1.33339	0.666696	−	0.745330i	$-0.267708\pi$
$401$	26.7012	1.33339	0.666696	+	0.745330i	$0.267708\pi$
$402$	0	0
$403$	9.28245	0.462392
$404$	0	0
$405$	−11.3437	−0.563671
$406$	0	0
$407$	−7.88865	−0.391026
$408$	0	0
$409$	−11.8843	−0.587639	−0.293819	−	0.955861i	$-0.594926\pi$
$409$	−11.8843	−0.587639	−0.293819	+	0.955861i	$0.594926\pi$
$410$	0	0
$411$	12.9915	0.640826
$412$	0	0
$413$	7.83579	0.385574
$414$	0	0
$415$	25.6055	1.25693
$416$	0	0
$417$	9.06202	0.443769
$418$	0	0
$419$	−12.1657	−0.594332	−0.297166	−	0.954826i	$-0.596041\pi$
$419$	−12.1657	−0.594332	−0.297166	+	0.954826i	$0.596041\pi$
$420$	0	0
$421$	−20.3054	−0.989626	−0.494813	−	0.869000i	$-0.664763\pi$
$421$	−20.3054	−0.989626	−0.494813	+	0.869000i	$0.664763\pi$
$422$	0	0
$423$	9.85586	0.479208
$424$	0	0
$425$	−24.1157	−1.16978
$426$	0	0
$427$	−5.37090	−0.259916
$428$	0	0
$429$	−0.827072	−0.0399314
$430$	0	0
$431$	27.4436	1.32191	0.660957	−	0.750424i	$-0.270151\pi$
$431$	27.4436	1.32191	0.660957	+	0.750424i	$0.270151\pi$
$432$	0	0
$433$	17.6615	0.848757	0.424379	−	0.905485i	$-0.360493\pi$
$433$	17.6615	0.848757	0.424379	+	0.905485i	$0.360493\pi$
$434$	0	0
$435$	0.237832	0.0114032
$436$	0	0
$437$	22.4387	1.07339
$438$	0	0
$439$	−33.3791	−1.59310	−0.796548	−	0.604575i	$-0.793343\pi$
$439$	−33.3791	−1.59310	−0.796548	+	0.604575i	$0.793343\pi$
$440$	0	0
$441$	−2.31595	−0.110283
$442$	0	0
$443$	37.7037	1.79135	0.895677	−	0.444705i	$-0.146691\pi$
$443$	37.7037	1.79135	0.895677	+	0.444705i	$0.146691\pi$
$444$	0	0
$445$	−18.6493	−0.884061
$446$	0	0
$447$	−12.7557	−0.603323
$448$	0	0
$449$	2.88597	0.136197	0.0680987	−	0.997679i	$-0.478307\pi$
$449$	2.88597	0.136197	0.0680987	+	0.997679i	$0.478307\pi$
$450$	0	0
$451$	6.40806	0.301744
$452$	0	0
$453$	11.4129	0.536224
$454$	0	0
$455$	3.42555	0.160592
$456$	0	0
$457$	36.1061	1.68897	0.844487	−	0.535576i	$-0.179906\pi$
$457$	36.1061	1.68897	0.844487	+	0.535576i	$0.179906\pi$
$458$	0	0
$459$	15.7444	0.734887
$460$	0	0
$461$	−21.2607	−0.990208	−0.495104	−	0.868834i	$-0.664870\pi$
$461$	−21.2607	−0.990208	−0.495104	+	0.868834i	$0.664870\pi$
$462$	0	0
$463$	−39.1899	−1.82131	−0.910655	−	0.413167i	$-0.864423\pi$
$463$	−39.1899	−1.82131	−0.910655	+	0.413167i	$0.864423\pi$
$464$	0	0
$465$	26.2988	1.21958
$466$	0	0
$467$	13.4262	0.621289	0.310645	−	0.950526i	$-0.399455\pi$
$467$	13.4262	0.621289	0.310645	+	0.950526i	$0.399455\pi$
$468$	0	0
$469$	−1.50319	−0.0694109
$470$	0	0
$471$	10.4291	0.480547
$472$	0	0
$473$	−3.96441	−0.182284
$474$	0	0
$475$	35.5082	1.62923
$476$	0	0
$477$	−12.7309	−0.582907
$478$	0	0
$479$	−9.12464	−0.416916	−0.208458	−	0.978031i	$-0.566844\pi$
$479$	−9.12464	−0.416916	−0.208458	+	0.978031i	$0.566844\pi$
$480$	0	0
$481$	7.88865	0.359692
$482$	0	0
$483$	3.51972	0.160153
$484$	0	0
$485$	10.0715	0.457325
$486$	0	0
$487$	21.4165	0.970476	0.485238	−	0.874382i	$-0.338733\pi$
$487$	21.4165	0.970476	0.485238	+	0.874382i	$0.338733\pi$
$488$	0	0
$489$	10.7757	0.487295
$490$	0	0
$491$	6.49198	0.292979	0.146489	−	0.989212i	$-0.453203\pi$
$491$	6.49198	0.292979	0.146489	+	0.989212i	$0.453203\pi$
$492$	0	0
$493$	0.300608	0.0135387
$494$	0	0
$495$	7.93340	0.356580
$496$	0	0
$497$	−3.77911	−0.169516
$498$	0	0
$499$	40.7224	1.82298	0.911492	−	0.411319i	$-0.134932\pi$
$499$	40.7224	1.82298	0.911492	+	0.411319i	$0.134932\pi$
$500$	0	0
$501$	−6.13330	−0.274016
$502$	0	0
$503$	−31.7948	−1.41766	−0.708830	−	0.705379i	$-0.750777\pi$
$503$	−31.7948	−1.41766	−0.708830	+	0.705379i	$0.750777\pi$
$504$	0	0
$505$	−34.6466	−1.54175
$506$	0	0
$507$	0.827072	0.0367316
$508$	0	0
$509$	37.6159	1.66730	0.833648	−	0.552296i	$-0.186248\pi$
$509$	37.6159	1.66730	0.833648	+	0.552296i	$0.186248\pi$
$510$	0	0
$511$	9.42232	0.416819
$512$	0	0
$513$	−23.1823	−1.02352
$514$	0	0
$515$	−65.1912	−2.87267
$516$	0	0
$517$	−4.25564	−0.187163
$518$	0	0
$519$	13.4292	0.589476
$520$	0	0
$521$	33.5371	1.46929	0.734644	−	0.678453i	$-0.237349\pi$
$521$	33.5371	1.46929	0.734644	+	0.678453i	$0.237349\pi$
$522$	0	0
$523$	15.9717	0.698394	0.349197	−	0.937049i	$-0.386454\pi$
$523$	15.9717	0.698394	0.349197	+	0.937049i	$0.386454\pi$
$524$	0	0
$525$	5.56980	0.243086
$526$	0	0
$527$	33.2403	1.44797
$528$	0	0
$529$	−4.88952	−0.212588
$530$	0	0
$531$	−18.1473	−0.787527
$532$	0	0
$533$	−6.40806	−0.277564
$534$	0	0
$535$	1.27665	0.0551945
$536$	0	0
$537$	−12.9039	−0.556845
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	−5.22913	−0.224818	−0.112409	−	0.993662i	$-0.535857\pi$
$541$	−5.22913	−0.224818	−0.112409	+	0.993662i	$0.535857\pi$
$542$	0	0
$543$	14.8204	0.636003
$544$	0	0
$545$	15.3436	0.657249
$546$	0	0
$547$	−1.96620	−0.0840688	−0.0420344	−	0.999116i	$-0.513384\pi$
$547$	−1.96620	−0.0840688	−0.0420344	+	0.999116i	$0.513384\pi$
$548$	0	0
$549$	12.4388	0.530873
$550$	0	0
$551$	−0.442618	−0.0188562
$552$	0	0
$553$	−7.41114	−0.315154
$554$	0	0
$555$	22.3499	0.948701
$556$	0	0
$557$	24.1563	1.02354	0.511768	−	0.859124i	$-0.328991\pi$
$557$	24.1563	1.02354	0.511768	+	0.859124i	$0.328991\pi$
$558$	0	0
$559$	3.96441	0.167677
$560$	0	0
$561$	−2.96173	−0.125045
$562$	0	0
$563$	31.5939	1.33152	0.665762	−	0.746164i	$-0.268107\pi$
$563$	31.5939	1.33152	0.665762	+	0.746164i	$0.268107\pi$
$564$	0	0
$565$	45.3903	1.90959
$566$	0	0
$567$	3.31149	0.139070
$568$	0	0
$569$	−8.82135	−0.369810	−0.184905	−	0.982756i	$-0.559198\pi$
$569$	−8.82135	−0.369810	−0.184905	+	0.982756i	$0.559198\pi$
$570$	0	0
$571$	18.3773	0.769065	0.384532	−	0.923112i	$-0.374363\pi$
$571$	18.3773	0.769065	0.384532	+	0.923112i	$0.374363\pi$
$572$	0	0
$573$	−8.50327	−0.355229
$574$	0	0
$575$	28.6590	1.19516
$576$	0	0
$577$	38.6301	1.60819	0.804096	−	0.594500i	$-0.202650\pi$
$577$	38.6301	1.60819	0.804096	+	0.594500i	$0.202650\pi$
$578$	0	0
$579$	−1.36607	−0.0567720
$580$	0	0
$581$	−7.47488	−0.310110
$582$	0	0
$583$	5.49704	0.227664
$584$	0	0
$585$	−7.93340	−0.328006
$586$	0	0
$587$	−28.6191	−1.18124	−0.590618	−	0.806951i	$-0.701116\pi$
$587$	−28.6191	−1.18124	−0.590618	+	0.806951i	$0.701116\pi$
$588$	0	0
$589$	−48.9435	−2.01668
$590$	0	0
$591$	15.8202	0.650758
$592$	0	0
$593$	0.532845	0.0218813	0.0109407	−	0.999940i	$-0.496517\pi$
$593$	0.532845	0.0218813	0.0109407	+	0.999940i	$0.496517\pi$
$594$	0	0
$595$	12.2668	0.502891
$596$	0	0
$597$	3.84000	0.157161
$598$	0	0
$599$	−2.63491	−0.107660	−0.0538298	−	0.998550i	$-0.517143\pi$
$599$	−2.63491	−0.107660	−0.0538298	+	0.998550i	$0.517143\pi$
$600$	0	0
$601$	−27.3216	−1.11447	−0.557236	−	0.830354i	$-0.688138\pi$
$601$	−27.3216	−1.11447	−0.557236	+	0.830354i	$0.688138\pi$
$602$	0	0
$603$	3.48132	0.141770
$604$	0	0
$605$	−3.42555	−0.139268
$606$	0	0
$607$	−38.7450	−1.57261	−0.786305	−	0.617838i	$-0.788009\pi$
$607$	−38.7450	−1.57261	−0.786305	+	0.617838i	$0.788009\pi$
$608$	0	0
$609$	−0.0694289	−0.00281340
$610$	0	0
$611$	4.25564	0.172165
$612$	0	0
$613$	−38.9000	−1.57115	−0.785577	−	0.618764i	$-0.787634\pi$
$613$	−38.9000	−1.57115	−0.785577	+	0.618764i	$0.787634\pi$
$614$	0	0
$615$	−18.1551	−0.732086
$616$	0	0
$617$	−18.9138	−0.761439	−0.380720	−	0.924691i	$-0.624324\pi$
$617$	−18.9138	−0.761439	−0.380720	+	0.924691i	$0.624324\pi$
$618$	0	0
$619$	35.2316	1.41608	0.708038	−	0.706174i	$-0.249581\pi$
$619$	35.2316	1.41608	0.708038	+	0.706174i	$0.249581\pi$
$620$	0	0
$621$	−18.7107	−0.750833
$622$	0	0
$623$	5.44418	0.218117
$624$	0	0
$625$	−13.3202	−0.532807
$626$	0	0
$627$	4.36090	0.174157
$628$	0	0
$629$	28.2492	1.12637
$630$	0	0
$631$	−27.9496	−1.11266	−0.556329	−	0.830962i	$-0.687791\pi$
$631$	−27.9496	−1.11266	−0.556329	+	0.830962i	$0.687791\pi$
$632$	0	0
$633$	4.21027	0.167343
$634$	0	0
$635$	42.3723	1.68149
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	8.75223	0.346233
$640$	0	0
$641$	−9.63833	−0.380691	−0.190346	−	0.981717i	$-0.560961\pi$
$641$	−9.63833	−0.380691	−0.190346	+	0.981717i	$0.560961\pi$
$642$	0	0
$643$	43.6558	1.72162	0.860808	−	0.508929i	$-0.169959\pi$
$643$	43.6558	1.72162	0.860808	+	0.508929i	$0.169959\pi$
$644$	0	0
$645$	11.2319	0.442254
$646$	0	0
$647$	15.8425	0.622832	0.311416	−	0.950274i	$-0.399197\pi$
$647$	15.8425	0.622832	0.311416	+	0.950274i	$0.399197\pi$
$648$	0	0
$649$	7.83579	0.307582
$650$	0	0
$651$	−7.67725	−0.300895
$652$	0	0
$653$	−14.7459	−0.577051	−0.288526	−	0.957472i	$-0.593165\pi$
$653$	−14.7459	−0.577051	−0.288526	+	0.957472i	$0.593165\pi$
$654$	0	0
$655$	−4.07072	−0.159056
$656$	0	0
$657$	−21.8216	−0.851344
$658$	0	0
$659$	22.9422	0.893701	0.446851	−	0.894609i	$-0.352546\pi$
$659$	22.9422	0.893701	0.446851	+	0.894609i	$0.352546\pi$
$660$	0	0
$661$	−39.4834	−1.53573	−0.767864	−	0.640613i	$-0.778680\pi$
$661$	−39.4834	−1.53573	−0.767864	+	0.640613i	$0.778680\pi$
$662$	0	0
$663$	2.96173	0.115024
$664$	0	0
$665$	−18.0619	−0.700409
$666$	0	0
$667$	−0.357242	−0.0138324
$668$	0	0
$669$	7.39171	0.285780
$670$	0	0
$671$	−5.37090	−0.207341
$672$	0	0
$673$	−40.5416	−1.56276	−0.781382	−	0.624054i	$-0.785485\pi$
$673$	−40.5416	−1.56276	−0.781382	+	0.624054i	$0.785485\pi$
$674$	0	0
$675$	−29.6088	−1.13964
$676$	0	0
$677$	−1.86078	−0.0715156	−0.0357578	−	0.999360i	$-0.511384\pi$
$677$	−1.86078	−0.0715156	−0.0357578	+	0.999360i	$0.511384\pi$
$678$	0	0
$679$	−2.94013	−0.112832
$680$	0	0
$681$	16.0885	0.616511
$682$	0	0
$683$	−26.9144	−1.02985	−0.514925	−	0.857235i	$-0.672180\pi$
$683$	−26.9144	−1.02985	−0.514925	+	0.857235i	$0.672180\pi$
$684$	0	0
$685$	−53.8081	−2.05590
$686$	0	0
$687$	4.16631	0.158955
$688$	0	0
$689$	−5.49704	−0.209421
$690$	0	0
$691$	−12.5496	−0.477410	−0.238705	−	0.971092i	$-0.576723\pi$
$691$	−12.5496	−0.477410	−0.238705	+	0.971092i	$0.576723\pi$
$692$	0	0
$693$	−2.31595	−0.0879757
$694$	0	0
$695$	−37.5328	−1.42370
$696$	0	0
$697$	−22.9472	−0.869187
$698$	0	0
$699$	9.63300	0.364353
$700$	0	0
$701$	−23.9073	−0.902967	−0.451483	−	0.892280i	$-0.649105\pi$
$701$	−23.9073	−0.902967	−0.451483	+	0.892280i	$0.649105\pi$
$702$	0	0
$703$	−41.5945	−1.56876
$704$	0	0
$705$	12.0570	0.454092
$706$	0	0
$707$	10.1142	0.380383
$708$	0	0
$709$	−7.82124	−0.293733	−0.146866	−	0.989156i	$-0.546919\pi$
$709$	−7.82124	−0.293733	−0.146866	+	0.989156i	$0.546919\pi$
$710$	0	0
$711$	17.1638	0.643694
$712$	0	0
$713$	−39.5028	−1.47939
$714$	0	0
$715$	3.42555	0.128108
$716$	0	0
$717$	15.7052	0.586522
$718$	0	0
$719$	−41.8644	−1.56128	−0.780640	−	0.624981i	$-0.785107\pi$
$719$	−41.8644	−1.56128	−0.780640	+	0.624981i	$0.785107\pi$
$720$	0	0
$721$	19.0309	0.708747
$722$	0	0
$723$	−2.18223	−0.0811579
$724$	0	0
$725$	−0.565319	−0.0209954
$726$	0	0
$727$	9.56233	0.354647	0.177324	−	0.984153i	$-0.443256\pi$
$727$	9.56233	0.354647	0.177324	+	0.984153i	$0.443256\pi$
$728$	0	0
$729$	3.23985	0.119994
$730$	0	0
$731$	14.1965	0.525076
$732$	0	0
$733$	13.5392	0.500083	0.250041	−	0.968235i	$-0.419556\pi$
$733$	13.5392	0.500083	0.250041	+	0.968235i	$0.419556\pi$
$734$	0	0
$735$	−2.83317	−0.104503
$736$	0	0
$737$	−1.50319	−0.0553708
$738$	0	0
$739$	7.55711	0.277993	0.138996	−	0.990293i	$-0.455612\pi$
$739$	7.55711	0.277993	0.138996	+	0.990293i	$0.455612\pi$
$740$	0	0
$741$	−4.36090	−0.160202
$742$	0	0
$743$	−11.9418	−0.438102	−0.219051	−	0.975713i	$-0.570296\pi$
$743$	−11.9418	−0.438102	−0.219051	+	0.975713i	$0.570296\pi$
$744$	0	0
$745$	52.8312	1.93559
$746$	0	0
$747$	17.3115	0.633393
$748$	0	0
$749$	−0.372686	−0.0136176
$750$	0	0
$751$	20.9733	0.765326	0.382663	−	0.923888i	$-0.375007\pi$
$751$	20.9733	0.765326	0.382663	+	0.923888i	$0.375007\pi$
$752$	0	0
$753$	17.3385	0.631849
$754$	0	0
$755$	−47.2696	−1.72032
$756$	0	0
$757$	−24.3770	−0.885997	−0.442998	−	0.896522i	$-0.646085\pi$
$757$	−24.3770	−0.885997	−0.442998	+	0.896522i	$0.646085\pi$
$758$	0	0
$759$	3.51972	0.127758
$760$	0	0
$761$	−15.8875	−0.575922	−0.287961	−	0.957642i	$-0.592977\pi$
$761$	−15.8875	−0.575922	−0.287961	+	0.957642i	$0.592977\pi$
$762$	0	0
$763$	−4.47918	−0.162157
$764$	0	0
$765$	−28.4094	−1.02714
$766$	0	0
$767$	−7.83579	−0.282934
$768$	0	0
$769$	23.8252	0.859160	0.429580	−	0.903029i	$-0.358662\pi$
$769$	23.8252	0.859160	0.429580	+	0.903029i	$0.358662\pi$
$770$	0	0
$771$	13.1200	0.472506
$772$	0	0
$773$	−31.4593	−1.13151	−0.565757	−	0.824572i	$-0.691416\pi$
$773$	−31.4593	−1.13151	−0.565757	+	0.824572i	$0.691416\pi$
$774$	0	0
$775$	−62.5114	−2.24548
$776$	0	0
$777$	−6.52448	−0.234065
$778$	0	0
$779$	33.7877	1.21057
$780$	0	0
$781$	−3.77911	−0.135227
$782$	0	0
$783$	0.369081	0.0131899
$784$	0	0
$785$	−43.1949	−1.54169
$786$	0	0
$787$	−35.7963	−1.27600	−0.638000	−	0.770036i	$-0.720238\pi$
$787$	−35.7963	−1.27600	−0.638000	+	0.770036i	$0.720238\pi$
$788$	0	0
$789$	23.1187	0.823049
$790$	0	0
$791$	−13.2505	−0.471135
$792$	0	0
$793$	5.37090	0.190726
$794$	0	0
$795$	−15.5741	−0.552355
$796$	0	0
$797$	52.3172	1.85317	0.926586	−	0.376083i	$-0.122729\pi$
$797$	52.3172	1.85317	0.926586	+	0.376083i	$0.122729\pi$
$798$	0	0
$799$	15.2394	0.539131
$800$	0	0
$801$	−12.6085	−0.445498
$802$	0	0
$803$	9.42232	0.332507
$804$	0	0
$805$	−14.5779	−0.513803
$806$	0	0
$807$	1.34939	0.0475009
$808$	0	0
$809$	−4.28055	−0.150496	−0.0752481	−	0.997165i	$-0.523975\pi$
$809$	−4.28055	−0.150496	−0.0752481	+	0.997165i	$0.523975\pi$
$810$	0	0
$811$	−0.787980	−0.0276697	−0.0138349	−	0.999904i	$-0.504404\pi$
$811$	−0.787980	−0.0276697	−0.0138349	+	0.999904i	$0.504404\pi$
$812$	0	0
$813$	−8.35663	−0.293080
$814$	0	0
$815$	−44.6306	−1.56334
$816$	0	0
$817$	−20.9031	−0.731307
$818$	0	0
$819$	2.31595	0.0809259
$820$	0	0
$821$	11.0847	0.386858	0.193429	−	0.981114i	$-0.438039\pi$
$821$	11.0847	0.386858	0.193429	+	0.981114i	$0.438039\pi$
$822$	0	0
$823$	24.8490	0.866182	0.433091	−	0.901350i	$-0.357423\pi$
$823$	24.8490	0.866182	0.433091	+	0.901350i	$0.357423\pi$
$824$	0	0
$825$	5.56980	0.193916
$826$	0	0
$827$	17.2591	0.600159	0.300079	−	0.953914i	$-0.402987\pi$
$827$	17.2591	0.600159	0.300079	+	0.953914i	$0.402987\pi$
$828$	0	0
$829$	−25.2808	−0.878037	−0.439018	−	0.898478i	$-0.644674\pi$
$829$	−25.2808	−0.878037	−0.439018	+	0.898478i	$0.644674\pi$
$830$	0	0
$831$	−15.4469	−0.535848
$832$	0	0
$833$	−3.58099	−0.124074
$834$	0	0
$835$	25.4027	0.879098
$836$	0	0
$837$	40.8119	1.41067
$838$	0	0
$839$	−14.6037	−0.504175	−0.252088	−	0.967704i	$-0.581117\pi$
$839$	−14.6037	−0.504175	−0.252088	+	0.967704i	$0.581117\pi$
$840$	0	0
$841$	−28.9930	−0.999757
$842$	0	0
$843$	1.35083	0.0465252
$844$	0	0
$845$	−3.42555	−0.117842
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	3.21067	0.110190
$850$	0	0
$851$	−33.5713	−1.15081
$852$	0	0
$853$	34.2555	1.17289	0.586444	−	0.809990i	$-0.300527\pi$
$853$	34.2555	1.17289	0.586444	+	0.809990i	$0.300527\pi$
$854$	0	0
$855$	41.8304	1.43057
$856$	0	0
$857$	−40.1456	−1.37135	−0.685674	−	0.727909i	$-0.740492\pi$
$857$	−40.1456	−1.37135	−0.685674	+	0.727909i	$0.740492\pi$
$858$	0	0
$859$	33.4668	1.14187	0.570935	−	0.820995i	$-0.306581\pi$
$859$	33.4668	1.14187	0.570935	+	0.820995i	$0.306581\pi$
$860$	0	0
$861$	5.29993	0.180621
$862$	0	0
$863$	3.15604	0.107433	0.0537165	−	0.998556i	$-0.482893\pi$
$863$	3.15604	0.107433	0.0537165	+	0.998556i	$0.482893\pi$
$864$	0	0
$865$	−55.6207	−1.89116
$866$	0	0
$867$	−3.45429	−0.117314
$868$	0	0
$869$	−7.41114	−0.251406
$870$	0	0
$871$	1.50319	0.0509337
$872$	0	0
$873$	6.80920	0.230456
$874$	0	0
$875$	−5.94114	−0.200847
$876$	0	0
$877$	32.3336	1.09183	0.545914	−	0.837841i	$-0.316182\pi$
$877$	32.3336	1.09183	0.545914	+	0.837841i	$0.316182\pi$
$878$	0	0
$879$	2.79792	0.0943714
$880$	0	0
$881$	−38.6155	−1.30099	−0.650494	−	0.759511i	$-0.725438\pi$
$881$	−38.6155	−1.30099	−0.650494	+	0.759511i	$0.725438\pi$
$882$	0	0
$883$	24.7101	0.831562	0.415781	−	0.909465i	$-0.363508\pi$
$883$	24.7101	0.831562	0.415781	+	0.909465i	$0.363508\pi$
$884$	0	0
$885$	−22.2002	−0.746250
$886$	0	0
$887$	13.8722	0.465784	0.232892	−	0.972503i	$-0.425181\pi$
$887$	13.8722	0.465784	0.232892	+	0.972503i	$0.425181\pi$
$888$	0	0
$889$	−12.3695	−0.414860
$890$	0	0
$891$	3.31149	0.110939
$892$	0	0
$893$	−22.4387	−0.750882
$894$	0	0
$895$	53.4451	1.78647
$896$	0	0
$897$	−3.51972	−0.117520
$898$	0	0
$899$	0.779219	0.0259884
$900$	0	0
$901$	−19.6848	−0.655797
$902$	0	0
$903$	−3.27885	−0.109113
$904$	0	0
$905$	−61.3826	−2.04043
$906$	0	0
$907$	1.64171	0.0545122	0.0272561	−	0.999628i	$-0.491323\pi$
$907$	1.64171	0.0545122	0.0272561	+	0.999628i	$0.491323\pi$
$908$	0	0
$909$	−23.4240	−0.776924
$910$	0	0
$911$	−4.96053	−0.164350	−0.0821749	−	0.996618i	$-0.526187\pi$
$911$	−4.96053	−0.164350	−0.0821749	+	0.996618i	$0.526187\pi$
$912$	0	0
$913$	−7.47488	−0.247382
$914$	0	0
$915$	15.2167	0.503048
$916$	0	0
$917$	1.18834	0.0392425
$918$	0	0
$919$	8.62086	0.284376	0.142188	−	0.989840i	$-0.454586\pi$
$919$	8.62086	0.284376	0.142188	+	0.989840i	$0.454586\pi$
$920$	0	0
$921$	−9.83846	−0.324188
$922$	0	0
$923$	3.77911	0.124391
$924$	0	0
$925$	−53.1250	−1.74674
$926$	0	0
$927$	−44.0746	−1.44760
$928$	0	0
$929$	35.0126	1.14873	0.574363	−	0.818601i	$-0.305250\pi$
$929$	35.0126	1.14873	0.574363	+	0.818601i	$0.305250\pi$
$930$	0	0
$931$	5.27269	0.172806
$932$	0	0
$933$	−14.5678	−0.476929
$934$	0	0
$935$	12.2668	0.401168
$936$	0	0
$937$	9.69541	0.316735	0.158368	−	0.987380i	$-0.449377\pi$
$937$	9.69541	0.316735	0.158368	+	0.987380i	$0.449377\pi$
$938$	0	0
$939$	−12.9602	−0.422939
$940$	0	0
$941$	30.5683	0.996497	0.498249	−	0.867034i	$-0.333977\pi$
$941$	30.5683	0.996497	0.498249	+	0.867034i	$0.333977\pi$
$942$	0	0
$943$	27.2704	0.888047
$944$	0	0
$945$	15.0610	0.489935
$946$	0	0
$947$	46.8471	1.52233	0.761164	−	0.648560i	$-0.224628\pi$
$947$	46.8471	1.52233	0.761164	+	0.648560i	$0.224628\pi$
$948$	0	0
$949$	−9.42232	−0.305862
$950$	0	0
$951$	17.3656	0.563117
$952$	0	0
$953$	−24.6183	−0.797464	−0.398732	−	0.917067i	$-0.630550\pi$
$953$	−24.6183	−0.797464	−0.398732	+	0.917067i	$0.630550\pi$
$954$	0	0
$955$	35.2186	1.13965
$956$	0	0
$957$	−0.0694289	−0.00224432
$958$	0	0
$959$	15.7079	0.507234
$960$	0	0
$961$	55.1639	1.77948
$962$	0	0
$963$	0.863122	0.0278137
$964$	0	0
$965$	5.65796	0.182136
$966$	0	0
$967$	−53.0972	−1.70749	−0.853745	−	0.520692i	$-0.825674\pi$
$967$	−53.0972	−1.70749	−0.853745	+	0.520692i	$0.825674\pi$
$968$	0	0
$969$	−15.6163	−0.501668
$970$	0	0
$971$	33.9341	1.08900	0.544498	−	0.838762i	$-0.316720\pi$
$971$	33.9341	1.08900	0.544498	+	0.838762i	$0.316720\pi$
$972$	0	0
$973$	10.9567	0.351257
$974$	0	0
$975$	−5.56980	−0.178376
$976$	0	0
$977$	−44.6163	−1.42740	−0.713701	−	0.700451i	$-0.752982\pi$
$977$	−44.6163	−1.42740	−0.713701	+	0.700451i	$0.752982\pi$
$978$	0	0
$979$	5.44418	0.173997
$980$	0	0
$981$	10.3736	0.331203
$982$	0	0
$983$	16.9039	0.539151	0.269575	−	0.962979i	$-0.413117\pi$
$983$	16.9039	0.539151	0.269575	+	0.962979i	$0.413117\pi$
$984$	0	0
$985$	−65.5239	−2.08776
$986$	0	0
$987$	−3.51972	−0.112034
$988$	0	0
$989$	−16.8711	−0.536470
$990$	0	0
$991$	−23.0835	−0.733272	−0.366636	−	0.930364i	$-0.619491\pi$
$991$	−23.0835	−0.733272	−0.366636	+	0.930364i	$0.619491\pi$
$992$	0	0
$993$	6.51092	0.206618
$994$	0	0
$995$	−15.9044	−0.504204
$996$	0	0
$997$	39.1959	1.24135	0.620673	−	0.784070i	$-0.286859\pi$
$997$	39.1959	1.24135	0.620673	+	0.784070i	$0.286859\pi$
$998$	0	0
$999$	34.6838	1.09735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.q.1.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.q.1.5	✓	9	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)

\(f(q)\)	\(=\)	\(q+0.827072 q^{3} -3.42555 q^{5} +1.00000 q^{7} -2.31595 q^{9} +O(q^{10})\)	\(q+0.827072 q^{3} -3.42555 q^{5} +1.00000 q^{7} -2.31595 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.83317 q^{15} -3.58099 q^{17} +5.27269 q^{19} +0.827072 q^{21} +4.25564 q^{23} +6.73436 q^{25} -4.39667 q^{27} -0.0839454 q^{29} -9.28245 q^{31} +0.827072 q^{33} -3.42555 q^{35} -7.88865 q^{37} -0.827072 q^{39} +6.40806 q^{41} -3.96441 q^{43} +7.93340 q^{45} -4.25564 q^{47} +1.00000 q^{49} -2.96173 q^{51} +5.49704 q^{53} -3.42555 q^{55} +4.36090 q^{57} +7.83579 q^{59} -5.37090 q^{61} -2.31595 q^{63} +3.42555 q^{65} -1.50319 q^{67} +3.51972 q^{69} -3.77911 q^{71} +9.42232 q^{73} +5.56980 q^{75} +1.00000 q^{77} -7.41114 q^{79} +3.31149 q^{81} -7.47488 q^{83} +12.2668 q^{85} -0.0694289 q^{87} +5.44418 q^{89} -1.00000 q^{91} -7.67725 q^{93} -18.0619 q^{95} -2.94013 q^{97} -2.31595 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) 9 * q + 3 * q^3 + 8 * q^5 + 9 * q^7 + 12 * q^9 + 9 * q^11 - 9 * q^13 - 9 * q^15 + q^17 + 16 * q^19 + 3 * q^21 + 6 * q^23 + 11 * q^25 + 9 * q^27 - 2 * q^29 + 3 * q^31 + 3 * q^33 + 8 * q^35 - 4 * q^37 - 3 * q^39 + 20 * q^41 + 20 * q^43 + 8 * q^45 - 6 * q^47 + 9 * q^49 + 15 * q^51 + 15 * q^53 + 8 * q^55 + 16 * q^57 + 21 * q^59 + 12 * q^63 - 8 * q^65 + 18 * q^67 + 10 * q^69 + 12 * q^71 + 29 * q^73 + 12 * q^75 + 9 * q^77 - 6 * q^79 + 5 * q^81 + 25 * q^83 + 13 * q^85 + 17 * q^87 + 32 * q^89 - 9 * q^91 - 13 * q^93 + 38 * q^95 + 17 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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