LMFDB - Embedded newform 8008.2.a.q.1.3

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			1.3
Root			$-1.45026$ 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)

$f(q)$	$=$	$q-1.45026 q^{3} +1.74448 q^{5} +1.00000 q^{7} -0.896739 q^{9} +O(q^{10})$	$q-1.45026 q^{3} +1.74448 q^{5} +1.00000 q^{7} -0.896739 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.52995 q^{15} +2.73773 q^{17} -7.15582 q^{19} -1.45026 q^{21} +5.24277 q^{23} -1.95680 q^{25} +5.65129 q^{27} -4.92838 q^{29} +1.16162 q^{31} -1.45026 q^{33} +1.74448 q^{35} +4.00652 q^{37} +1.45026 q^{39} -2.18799 q^{41} +0.322898 q^{43} -1.56434 q^{45} -5.24277 q^{47} +1.00000 q^{49} -3.97043 q^{51} -5.66611 q^{53} +1.74448 q^{55} +10.3778 q^{57} +7.41510 q^{59} +12.2603 q^{61} -0.896739 q^{63} -1.74448 q^{65} +5.83521 q^{67} -7.60340 q^{69} +2.51462 q^{71} -9.22089 q^{73} +2.83787 q^{75} +1.00000 q^{77} +7.64529 q^{79} -5.50564 q^{81} +13.1119 q^{83} +4.77591 q^{85} +7.14744 q^{87} +5.75551 q^{89} -1.00000 q^{91} -1.68465 q^{93} -12.4832 q^{95} -7.10275 q^{97} -0.896739 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.45026	−0.837309	−0.418655	−	0.908145i	$-0.637498\pi$
$3$	−1.45026	−0.837309	−0.418655	+	0.908145i	$0.637498\pi$
$4$	0	0
$5$	1.74448	0.780154	0.390077	−	0.920782i	$-0.372448\pi$
$5$	1.74448	0.780154	0.390077	+	0.920782i	$0.372448\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−0.896739	−0.298913
$10$	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
$15$	−2.52995	−0.653231
$16$	0	0
$17$	2.73773	0.663997	0.331999	−	0.943280i	$-0.392277\pi$
$17$	2.73773	0.663997	0.331999	+	0.943280i	$0.392277\pi$
$18$	0	0
$19$	−7.15582	−1.64166	−0.820829	−	0.571174i	$-0.806488\pi$
$19$	−7.15582	−1.64166	−0.820829	+	0.571174i	$0.806488\pi$
$20$	0	0
$21$	−1.45026	−0.316473
$22$	0	0
$23$	5.24277	1.09319	0.546597	−	0.837396i	$-0.315923\pi$
$23$	5.24277	1.09319	0.546597	+	0.837396i	$0.315923\pi$
$24$	0	0
$25$	−1.95680	−0.391359
$26$	0	0
$27$	5.65129	1.08759
$28$	0	0
$29$	−4.92838	−0.915177	−0.457589	−	0.889164i	$-0.651287\pi$
$29$	−4.92838	−0.915177	−0.457589	+	0.889164i	$0.651287\pi$
$30$	0	0
$31$	1.16162	0.208632	0.104316	−	0.994544i	$-0.466735\pi$
$31$	1.16162	0.208632	0.104316	+	0.994544i	$0.466735\pi$
$32$	0	0
$33$	−1.45026	−0.252458
$34$	0	0
$35$	1.74448	0.294871
$36$	0	0
$37$	4.00652	0.658668	0.329334	−	0.944214i	$-0.393176\pi$
$37$	4.00652	0.658668	0.329334	+	0.944214i	$0.393176\pi$
$38$	0	0
$39$	1.45026	0.232228
$40$	0	0
$41$	−2.18799	−0.341707	−0.170854	−	0.985296i	$-0.554653\pi$
$41$	−2.18799	−0.341707	−0.170854	+	0.985296i	$0.554653\pi$
$42$	0	0
$43$	0.322898	0.0492415	0.0246207	−	0.999697i	$-0.492162\pi$
$43$	0.322898	0.0492415	0.0246207	+	0.999697i	$0.492162\pi$
$44$	0	0
$45$	−1.56434	−0.233198
$46$	0	0
$47$	−5.24277	−0.764737	−0.382369	−	0.924010i	$-0.624892\pi$
$47$	−5.24277	−0.764737	−0.382369	+	0.924010i	$0.624892\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.97043	−0.555971
$52$	0	0
$53$	−5.66611	−0.778300	−0.389150	−	0.921174i	$-0.627231\pi$
$53$	−5.66611	−0.778300	−0.389150	+	0.921174i	$0.627231\pi$
$54$	0	0
$55$	1.74448	0.235225
$56$	0	0
$57$	10.3778	1.37458
$58$	0	0
$59$	7.41510	0.965364	0.482682	−	0.875796i	$-0.339663\pi$
$59$	7.41510	0.965364	0.482682	+	0.875796i	$0.339663\pi$
$60$	0	0
$61$	12.2603	1.56978	0.784888	−	0.619638i	$-0.212721\pi$
$61$	12.2603	1.56978	0.784888	+	0.619638i	$0.212721\pi$
$62$	0	0
$63$	−0.896739	−0.112978
$64$	0	0
$65$	−1.74448	−0.216376
$66$	0	0
$67$	5.83521	0.712884	0.356442	−	0.934317i	$-0.383990\pi$
$67$	5.83521	0.712884	0.356442	+	0.934317i	$0.383990\pi$
$68$	0	0
$69$	−7.60340	−0.915342
$70$	0	0
$71$	2.51462	0.298430	0.149215	−	0.988805i	$-0.452325\pi$
$71$	2.51462	0.298430	0.149215	+	0.988805i	$0.452325\pi$
$72$	0	0
$73$	−9.22089	−1.07922	−0.539612	−	0.841914i	$-0.681429\pi$
$73$	−9.22089	−1.07922	−0.539612	+	0.841914i	$0.681429\pi$
$74$	0	0
$75$	2.83787	0.327689
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	7.64529	0.860163	0.430081	−	0.902790i	$-0.358485\pi$
$79$	7.64529	0.860163	0.430081	+	0.902790i	$0.358485\pi$
$80$	0	0
$81$	−5.50564	−0.611738
$82$	0	0
$83$	13.1119	1.43921	0.719606	−	0.694382i	$-0.244322\pi$
$83$	13.1119	1.43921	0.719606	+	0.694382i	$0.244322\pi$
$84$	0	0
$85$	4.77591	0.518020
$86$	0	0
$87$	7.14744	0.766286
$88$	0	0
$89$	5.75551	0.610083	0.305041	−	0.952339i	$-0.401330\pi$
$89$	5.75551	0.610083	0.305041	+	0.952339i	$0.401330\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−1.68465	−0.174690
$94$	0	0
$95$	−12.4832	−1.28075
$96$	0	0
$97$	−7.10275	−0.721175	−0.360588	−	0.932725i	$-0.617424\pi$
$97$	−7.10275	−0.721175	−0.360588	+	0.932725i	$0.617424\pi$
$98$	0	0
$99$	−0.896739	−0.0901256
$100$	0	0
$101$	9.82940	0.978062	0.489031	−	0.872266i	$-0.337350\pi$
$101$	9.82940	0.978062	0.489031	+	0.872266i	$0.337350\pi$
$102$	0	0
$103$	−0.939197	−0.0925419	−0.0462709	−	0.998929i	$-0.514734\pi$
$103$	−0.939197	−0.0925419	−0.0462709	+	0.998929i	$0.514734\pi$
$104$	0	0
$105$	−2.52995	−0.246898
$106$	0	0
$107$	−6.67714	−0.645504	−0.322752	−	0.946484i	$-0.604608\pi$
$107$	−6.67714	−0.645504	−0.322752	+	0.946484i	$0.604608\pi$
$108$	0	0
$109$	−0.312219	−0.0299052	−0.0149526	−	0.999888i	$-0.504760\pi$
$109$	−0.312219	−0.0299052	−0.0149526	+	0.999888i	$0.504760\pi$
$110$	0	0
$111$	−5.81051	−0.551509
$112$	0	0
$113$	−5.64945	−0.531456	−0.265728	−	0.964048i	$-0.585612\pi$
$113$	−5.64945	−0.531456	−0.265728	+	0.964048i	$0.585612\pi$
$114$	0	0
$115$	9.14591	0.852860
$116$	0	0
$117$	0.896739	0.0829035
$118$	0	0
$119$	2.73773	0.250967
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.17316	0.286115
$124$	0	0
$125$	−12.1360	−1.08547
$126$	0	0
$127$	8.14536	0.722784	0.361392	−	0.932414i	$-0.382302\pi$
$127$	8.14536	0.722784	0.361392	+	0.932414i	$0.382302\pi$
$128$	0	0
$129$	−0.468287	−0.0412304
$130$	0	0
$131$	−4.52616	−0.395453	−0.197726	−	0.980257i	$-0.563356\pi$
$131$	−4.52616	−0.395453	−0.197726	+	0.980257i	$0.563356\pi$
$132$	0	0
$133$	−7.15582	−0.620488
$134$	0	0
$135$	9.85856	0.848490
$136$	0	0
$137$	−0.865596	−0.0739529	−0.0369764	−	0.999316i	$-0.511773\pi$
$137$	−0.865596	−0.0739529	−0.0369764	+	0.999316i	$0.511773\pi$
$138$	0	0
$139$	14.1985	1.20430	0.602148	−	0.798384i	$-0.294312\pi$
$139$	14.1985	1.20430	0.602148	+	0.798384i	$0.294312\pi$
$140$	0	0
$141$	7.60340	0.640322
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−8.59745	−0.713980
$146$	0	0
$147$	−1.45026	−0.119616
$148$	0	0
$149$	5.71163	0.467915	0.233958	−	0.972247i	$-0.424832\pi$
$149$	5.71163	0.467915	0.233958	+	0.972247i	$0.424832\pi$
$150$	0	0
$151$	20.9857	1.70779	0.853894	−	0.520446i	$-0.174234\pi$
$151$	20.9857	1.70779	0.853894	+	0.520446i	$0.174234\pi$
$152$	0	0
$153$	−2.45503	−0.198477
$154$	0	0
$155$	2.02641	0.162765
$156$	0	0
$157$	1.40698	0.112290	0.0561448	−	0.998423i	$-0.482119\pi$
$157$	1.40698	0.112290	0.0561448	+	0.998423i	$0.482119\pi$
$158$	0	0
$159$	8.21735	0.651678
$160$	0	0
$161$	5.24277	0.413188
$162$	0	0
$163$	7.08547	0.554977	0.277488	−	0.960729i	$-0.410498\pi$
$163$	7.08547	0.554977	0.277488	+	0.960729i	$0.410498\pi$
$164$	0	0
$165$	−2.52995	−0.196956
$166$	0	0
$167$	−0.471583	−0.0364922	−0.0182461	−	0.999834i	$-0.505808\pi$
$167$	−0.471583	−0.0364922	−0.0182461	+	0.999834i	$0.505808\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.41690	0.490713
$172$	0	0
$173$	4.94713	0.376124	0.188062	−	0.982157i	$-0.439779\pi$
$173$	4.94713	0.376124	0.188062	+	0.982157i	$0.439779\pi$
$174$	0	0
$175$	−1.95680	−0.147920
$176$	0	0
$177$	−10.7538	−0.808308
$178$	0	0
$179$	−4.69400	−0.350846	−0.175423	−	0.984493i	$-0.556129\pi$
$179$	−4.69400	−0.350846	−0.175423	+	0.984493i	$0.556129\pi$
$180$	0	0
$181$	2.25858	0.167879	0.0839393	−	0.996471i	$-0.473250\pi$
$181$	2.25858	0.167879	0.0839393	+	0.996471i	$0.473250\pi$
$182$	0	0
$183$	−17.7807	−1.31439
$184$	0	0
$185$	6.98929	0.513863
$186$	0	0
$187$	2.73773	0.200203
$188$	0	0
$189$	5.65129	0.411071
$190$	0	0
$191$	−22.9938	−1.66378	−0.831888	−	0.554943i	$-0.812740\pi$
$191$	−22.9938	−1.66378	−0.831888	+	0.554943i	$0.812740\pi$
$192$	0	0
$193$	3.89862	0.280629	0.140314	−	0.990107i	$-0.455189\pi$
$193$	3.89862	0.280629	0.140314	+	0.990107i	$0.455189\pi$
$194$	0	0
$195$	2.52995	0.181174
$196$	0	0
$197$	2.14852	0.153076	0.0765380	−	0.997067i	$-0.475613\pi$
$197$	2.14852	0.153076	0.0765380	+	0.997067i	$0.475613\pi$
$198$	0	0
$199$	25.0089	1.77284	0.886418	−	0.462885i	$-0.153186\pi$
$199$	25.0089	1.77284	0.886418	+	0.462885i	$0.153186\pi$
$200$	0	0
$201$	−8.46259	−0.596905
$202$	0	0
$203$	−4.92838	−0.345904
$204$	0	0
$205$	−3.81691	−0.266584
$206$	0	0
$207$	−4.70140	−0.326770
$208$	0	0
$209$	−7.15582	−0.494978
$210$	0	0
$211$	2.27819	0.156837	0.0784185	−	0.996921i	$-0.475013\pi$
$211$	2.27819	0.156837	0.0784185	+	0.996921i	$0.475013\pi$
$212$	0	0
$213$	−3.64685	−0.249878
$214$	0	0
$215$	0.563289	0.0384160
$216$	0	0
$217$	1.16162	0.0788556
$218$	0	0
$219$	13.3727	0.903644
$220$	0	0
$221$	−2.73773	−0.184160
$222$	0	0
$223$	−7.55627	−0.506005	−0.253003	−	0.967466i	$-0.581418\pi$
$223$	−7.55627	−0.506005	−0.253003	+	0.967466i	$0.581418\pi$
$224$	0	0
$225$	1.75473	0.116982
$226$	0	0
$227$	12.0321	0.798596	0.399298	−	0.916821i	$-0.369254\pi$
$227$	12.0321	0.798596	0.399298	+	0.916821i	$0.369254\pi$
$228$	0	0
$229$	−4.59704	−0.303781	−0.151890	−	0.988397i	$-0.548536\pi$
$229$	−4.59704	−0.303781	−0.151890	+	0.988397i	$0.548536\pi$
$230$	0	0
$231$	−1.45026	−0.0954203
$232$	0	0
$233$	−7.33937	−0.480818	−0.240409	−	0.970672i	$-0.577282\pi$
$233$	−7.33937	−0.480818	−0.240409	+	0.970672i	$0.577282\pi$
$234$	0	0
$235$	−9.14591	−0.596613
$236$	0	0
$237$	−11.0877	−0.720222
$238$	0	0
$239$	2.14665	0.138855	0.0694275	−	0.997587i	$-0.477883\pi$
$239$	2.14665	0.138855	0.0694275	+	0.997587i	$0.477883\pi$
$240$	0	0
$241$	−19.7404	−1.27159	−0.635795	−	0.771858i	$-0.719328\pi$
$241$	−19.7404	−1.27159	−0.635795	+	0.771858i	$0.719328\pi$
$242$	0	0
$243$	−8.96926	−0.575378
$244$	0	0
$245$	1.74448	0.111451
$246$	0	0
$247$	7.15582	0.455314
$248$	0	0
$249$	−19.0156	−1.20507
$250$	0	0
$251$	16.3773	1.03373	0.516864	−	0.856067i	$-0.327099\pi$
$251$	16.3773	1.03373	0.516864	+	0.856067i	$0.327099\pi$
$252$	0	0
$253$	5.24277	0.329610
$254$	0	0
$255$	−6.92633	−0.433743
$256$	0	0
$257$	10.2172	0.637328	0.318664	−	0.947868i	$-0.396766\pi$
$257$	10.2172	0.637328	0.318664	+	0.947868i	$0.396766\pi$
$258$	0	0
$259$	4.00652	0.248953
$260$	0	0
$261$	4.41947	0.273558
$262$	0	0
$263$	2.84009	0.175127	0.0875636	−	0.996159i	$-0.472092\pi$
$263$	2.84009	0.175127	0.0875636	+	0.996159i	$0.472092\pi$
$264$	0	0
$265$	−9.88441	−0.607194
$266$	0	0
$267$	−8.34700	−0.510828
$268$	0	0
$269$	−12.8875	−0.785763	−0.392882	−	0.919589i	$-0.628522\pi$
$269$	−12.8875	−0.785763	−0.392882	+	0.919589i	$0.628522\pi$
$270$	0	0
$271$	25.1856	1.52991	0.764957	−	0.644082i	$-0.222760\pi$
$271$	25.1856	1.52991	0.764957	+	0.644082i	$0.222760\pi$
$272$	0	0
$273$	1.45026	0.0877739
$274$	0	0
$275$	−1.95680	−0.117999
$276$	0	0
$277$	17.4038	1.04569	0.522847	−	0.852427i	$-0.324870\pi$
$277$	17.4038	1.04569	0.522847	+	0.852427i	$0.324870\pi$
$278$	0	0
$279$	−1.04167	−0.0623629
$280$	0	0
$281$	−1.35047	−0.0805624	−0.0402812	−	0.999188i	$-0.512825\pi$
$281$	−1.35047	−0.0805624	−0.0402812	+	0.999188i	$0.512825\pi$
$282$	0	0
$283$	−26.6522	−1.58431	−0.792153	−	0.610323i	$-0.791040\pi$
$283$	−26.6522	−1.58431	−0.792153	+	0.610323i	$0.791040\pi$
$284$	0	0
$285$	18.1039	1.07238
$286$	0	0
$287$	−2.18799	−0.129153
$288$	0	0
$289$	−9.50483	−0.559108
$290$	0	0
$291$	10.3009	0.603847
$292$	0	0
$293$	10.0521	0.587250	0.293625	−	0.955921i	$-0.405138\pi$
$293$	10.0521	0.587250	0.293625	+	0.955921i	$0.405138\pi$
$294$	0	0
$295$	12.9355	0.753133
$296$	0	0
$297$	5.65129	0.327921
$298$	0	0
$299$	−5.24277	−0.303197
$300$	0	0
$301$	0.322898	0.0186115
$302$	0	0
$303$	−14.2552	−0.818941
$304$	0	0
$305$	21.3879	1.22467
$306$	0	0
$307$	13.4557	0.767960	0.383980	−	0.923341i	$-0.374553\pi$
$307$	13.4557	0.767960	0.383980	+	0.923341i	$0.374553\pi$
$308$	0	0
$309$	1.36208	0.0774862
$310$	0	0
$311$	−21.8344	−1.23812	−0.619059	−	0.785345i	$-0.712486\pi$
$311$	−21.8344	−1.23812	−0.619059	+	0.785345i	$0.712486\pi$
$312$	0	0
$313$	−17.9575	−1.01502	−0.507508	−	0.861647i	$-0.669433\pi$
$313$	−17.9575	−1.01502	−0.507508	+	0.861647i	$0.669433\pi$
$314$	0	0
$315$	−1.56434	−0.0881407
$316$	0	0
$317$	3.76496	0.211461	0.105731	−	0.994395i	$-0.466282\pi$
$317$	3.76496	0.211461	0.105731	+	0.994395i	$0.466282\pi$
$318$	0	0
$319$	−4.92838	−0.275936
$320$	0	0
$321$	9.68361	0.540486
$322$	0	0
$323$	−19.5907	−1.09006
$324$	0	0
$325$	1.95680	0.108543
$326$	0	0
$327$	0.452800	0.0250399
$328$	0	0
$329$	−5.24277	−0.289043
$330$	0	0
$331$	−2.10561	−0.115735	−0.0578673	−	0.998324i	$-0.518430\pi$
$331$	−2.10561	−0.115735	−0.0578673	+	0.998324i	$0.518430\pi$
$332$	0	0
$333$	−3.59280	−0.196884
$334$	0	0
$335$	10.1794	0.556160
$336$	0	0
$337$	14.5240	0.791171	0.395586	−	0.918429i	$-0.370542\pi$
$337$	14.5240	0.791171	0.395586	+	0.918429i	$0.370542\pi$
$338$	0	0
$339$	8.19319	0.444993
$340$	0	0
$341$	1.16162	0.0629050
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−13.2640	−0.714108
$346$	0	0
$347$	−17.2133	−0.924061	−0.462030	−	0.886864i	$-0.652879\pi$
$347$	−17.2133	−0.924061	−0.462030	+	0.886864i	$0.652879\pi$
$348$	0	0
$349$	30.1518	1.61399	0.806995	−	0.590559i	$-0.201093\pi$
$349$	30.1518	1.61399	0.806995	+	0.590559i	$0.201093\pi$
$350$	0	0
$351$	−5.65129	−0.301644
$352$	0	0
$353$	−13.7411	−0.731367	−0.365684	−	0.930739i	$-0.619165\pi$
$353$	−13.7411	−0.731367	−0.365684	+	0.930739i	$0.619165\pi$
$354$	0	0
$355$	4.38669	0.232822
$356$	0	0
$357$	−3.97043	−0.210137
$358$	0	0
$359$	26.3817	1.39237	0.696187	−	0.717861i	$-0.254879\pi$
$359$	26.3817	1.39237	0.696187	+	0.717861i	$0.254879\pi$
$360$	0	0
$361$	32.2057	1.69504
$362$	0	0
$363$	−1.45026	−0.0761190
$364$	0	0
$365$	−16.0856	−0.841961
$366$	0	0
$367$	−9.05688	−0.472765	−0.236383	−	0.971660i	$-0.575962\pi$
$367$	−9.05688	−0.472765	−0.236383	+	0.971660i	$0.575962\pi$
$368$	0	0
$369$	1.96206	0.102141
$370$	0	0
$371$	−5.66611	−0.294170
$372$	0	0
$373$	−11.1195	−0.575745	−0.287872	−	0.957669i	$-0.592948\pi$
$373$	−11.1195	−0.575745	−0.287872	+	0.957669i	$0.592948\pi$
$374$	0	0
$375$	17.6004	0.908878
$376$	0	0
$377$	4.92838	0.253824
$378$	0	0
$379$	22.6327	1.16256	0.581282	−	0.813702i	$-0.302551\pi$
$379$	22.6327	1.16256	0.581282	+	0.813702i	$0.302551\pi$
$380$	0	0
$381$	−11.8129	−0.605194
$382$	0	0
$383$	11.7212	0.598927	0.299464	−	0.954108i	$-0.403192\pi$
$383$	11.7212	0.598927	0.299464	+	0.954108i	$0.403192\pi$
$384$	0	0
$385$	1.74448	0.0889069
$386$	0	0
$387$	−0.289555	−0.0147189
$388$	0	0
$389$	1.55548	0.0788660	0.0394330	−	0.999222i	$-0.487445\pi$
$389$	1.55548	0.0788660	0.0394330	+	0.999222i	$0.487445\pi$
$390$	0	0
$391$	14.3533	0.725878
$392$	0	0
$393$	6.56412	0.331116
$394$	0	0
$395$	13.3371	0.671060
$396$	0	0
$397$	29.1100	1.46099	0.730496	−	0.682917i	$-0.239289\pi$
$397$	29.1100	1.46099	0.730496	+	0.682917i	$0.239289\pi$
$398$	0	0
$399$	10.3778	0.519541
$400$	0	0
$401$	7.88564	0.393790	0.196895	−	0.980425i	$-0.436914\pi$
$401$	7.88564	0.393790	0.196895	+	0.980425i	$0.436914\pi$
$402$	0	0
$403$	−1.16162	−0.0578642
$404$	0	0
$405$	−9.60447	−0.477250
$406$	0	0
$407$	4.00652	0.198596
$408$	0	0
$409$	−30.7161	−1.51881	−0.759406	−	0.650617i	$-0.774510\pi$
$409$	−30.7161	−1.51881	−0.759406	+	0.650617i	$0.774510\pi$
$410$	0	0
$411$	1.25534	0.0619214
$412$	0	0
$413$	7.41510	0.364873
$414$	0	0
$415$	22.8733	1.12281
$416$	0	0
$417$	−20.5915	−1.00837
$418$	0	0
$419$	33.2821	1.62594	0.812969	−	0.582307i	$-0.197850\pi$
$419$	33.2821	1.62594	0.812969	+	0.582307i	$0.197850\pi$
$420$	0	0
$421$	−15.0199	−0.732026	−0.366013	−	0.930610i	$-0.619277\pi$
$421$	−15.0199	−0.732026	−0.366013	+	0.930610i	$0.619277\pi$
$422$	0	0
$423$	4.70140	0.228590
$424$	0	0
$425$	−5.35718	−0.259861
$426$	0	0
$427$	12.2603	0.593319
$428$	0	0
$429$	1.45026	0.0700193
$430$	0	0
$431$	26.9234	1.29685	0.648427	−	0.761277i	$-0.275427\pi$
$431$	26.9234	1.29685	0.648427	+	0.761277i	$0.275427\pi$
$432$	0	0
$433$	14.3085	0.687624	0.343812	−	0.939039i	$-0.388282\pi$
$433$	14.3085	0.687624	0.343812	+	0.939039i	$0.388282\pi$
$434$	0	0
$435$	12.4686	0.597822
$436$	0	0
$437$	−37.5163	−1.79465
$438$	0	0
$439$	37.6181	1.79541	0.897707	−	0.440592i	$-0.145232\pi$
$439$	37.6181	1.79541	0.897707	+	0.440592i	$0.145232\pi$
$440$	0	0
$441$	−0.896739	−0.0427019
$442$	0	0
$443$	−4.43702	−0.210809	−0.105404	−	0.994429i	$-0.533614\pi$
$443$	−4.43702	−0.210809	−0.105404	+	0.994429i	$0.533614\pi$
$444$	0	0
$445$	10.0404	0.475959
$446$	0	0
$447$	−8.28337	−0.391790
$448$	0	0
$449$	11.7161	0.552917	0.276459	−	0.961026i	$-0.410839\pi$
$449$	11.7161	0.552917	0.276459	+	0.961026i	$0.410839\pi$
$450$	0	0
$451$	−2.18799	−0.103029
$452$	0	0
$453$	−30.4347	−1.42995
$454$	0	0
$455$	−1.74448	−0.0817824
$456$	0	0
$457$	0.0813106	0.00380355	0.00190177	−	0.999998i	$-0.499395\pi$
$457$	0.0813106	0.00380355	0.00190177	+	0.999998i	$0.499395\pi$
$458$	0	0
$459$	15.4717	0.722158
$460$	0	0
$461$	−24.2359	−1.12878	−0.564388	−	0.825510i	$-0.690888\pi$
$461$	−24.2359	−1.12878	−0.564388	+	0.825510i	$0.690888\pi$
$462$	0	0
$463$	−33.4541	−1.55475	−0.777373	−	0.629040i	$-0.783448\pi$
$463$	−33.4541	−1.55475	−0.777373	+	0.629040i	$0.783448\pi$
$464$	0	0
$465$	−2.93883	−0.136285
$466$	0	0
$467$	30.9964	1.43434	0.717171	−	0.696898i	$-0.245437\pi$
$467$	30.9964	1.43434	0.717171	+	0.696898i	$0.245437\pi$
$468$	0	0
$469$	5.83521	0.269445
$470$	0	0
$471$	−2.04050	−0.0940211
$472$	0	0
$473$	0.322898	0.0148469
$474$	0	0
$475$	14.0025	0.642477
$476$	0	0
$477$	5.08102	0.232644
$478$	0	0
$479$	5.22832	0.238888	0.119444	−	0.992841i	$-0.461889\pi$
$479$	5.22832	0.238888	0.119444	+	0.992841i	$0.461889\pi$
$480$	0	0
$481$	−4.00652	−0.182682
$482$	0	0
$483$	−7.60340	−0.345967
$484$	0	0
$485$	−12.3906	−0.562628
$486$	0	0
$487$	36.7685	1.66614	0.833070	−	0.553167i	$-0.186581\pi$
$487$	36.7685	1.66614	0.833070	+	0.553167i	$0.186581\pi$
$488$	0	0
$489$	−10.2758	−0.464687
$490$	0	0
$491$	4.79028	0.216182	0.108091	−	0.994141i	$-0.465526\pi$
$491$	4.79028	0.216182	0.108091	+	0.994141i	$0.465526\pi$
$492$	0	0
$493$	−13.4926	−0.607675
$494$	0	0
$495$	−1.56434	−0.0703119
$496$	0	0
$497$	2.51462	0.112796
$498$	0	0
$499$	19.9545	0.893287	0.446643	−	0.894712i	$-0.352619\pi$
$499$	19.9545	0.893287	0.446643	+	0.894712i	$0.352619\pi$
$500$	0	0
$501$	0.683919	0.0305552
$502$	0	0
$503$	22.6923	1.01180	0.505901	−	0.862592i	$-0.331160\pi$
$503$	22.6923	1.01180	0.505901	+	0.862592i	$0.331160\pi$
$504$	0	0
$505$	17.1472	0.763039
$506$	0	0
$507$	−1.45026	−0.0644084
$508$	0	0
$509$	33.4370	1.48207	0.741034	−	0.671468i	$-0.234336\pi$
$509$	33.4370	1.48207	0.741034	+	0.671468i	$0.234336\pi$
$510$	0	0
$511$	−9.22089	−0.407908
$512$	0	0
$513$	−40.4396	−1.78545
$514$	0	0
$515$	−1.63841	−0.0721969
$516$	0	0
$517$	−5.24277	−0.230577
$518$	0	0
$519$	−7.17464	−0.314932
$520$	0	0
$521$	−11.4086	−0.499818	−0.249909	−	0.968269i	$-0.580401\pi$
$521$	−11.4086	−0.499818	−0.249909	+	0.968269i	$0.580401\pi$
$522$	0	0
$523$	22.2298	0.972041	0.486020	−	0.873947i	$-0.338448\pi$
$523$	22.2298	0.972041	0.486020	+	0.873947i	$0.338448\pi$
$524$	0	0
$525$	2.83787	0.123855
$526$	0	0
$527$	3.18019	0.138531
$528$	0	0
$529$	4.48668	0.195073
$530$	0	0
$531$	−6.64941	−0.288560
$532$	0	0
$533$	2.18799	0.0947725
$534$	0	0
$535$	−11.6481	−0.503593
$536$	0	0
$537$	6.80753	0.293767
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	13.1752	0.566445	0.283222	−	0.959054i	$-0.408597\pi$
$541$	13.1752	0.566445	0.283222	+	0.959054i	$0.408597\pi$
$542$	0	0
$543$	−3.27553	−0.140566
$544$	0	0
$545$	−0.544660	−0.0233307
$546$	0	0
$547$	−3.08477	−0.131895	−0.0659477	−	0.997823i	$-0.521007\pi$
$547$	−3.08477	−0.131895	−0.0659477	+	0.997823i	$0.521007\pi$
$548$	0	0
$549$	−10.9943	−0.469226
$550$	0	0
$551$	35.2666	1.50241
$552$	0	0
$553$	7.64529	0.325111
$554$	0	0
$555$	−10.1363	−0.430262
$556$	0	0
$557$	−4.14312	−0.175549	−0.0877747	−	0.996140i	$-0.527976\pi$
$557$	−4.14312	−0.175549	−0.0877747	+	0.996140i	$0.527976\pi$
$558$	0	0
$559$	−0.322898	−0.0136571
$560$	0	0
$561$	−3.97043	−0.167632
$562$	0	0
$563$	21.5360	0.907635	0.453818	−	0.891095i	$-0.350062\pi$
$563$	21.5360	0.907635	0.453818	+	0.891095i	$0.350062\pi$
$564$	0	0
$565$	−9.85534	−0.414617
$566$	0	0
$567$	−5.50564	−0.231215
$568$	0	0
$569$	−12.5203	−0.524878	−0.262439	−	0.964949i	$-0.584527\pi$
$569$	−12.5203	−0.524878	−0.262439	+	0.964949i	$0.584527\pi$
$570$	0	0
$571$	−25.6831	−1.07481	−0.537403	−	0.843326i	$-0.680594\pi$
$571$	−25.6831	−1.07481	−0.537403	+	0.843326i	$0.680594\pi$
$572$	0	0
$573$	33.3471	1.39310
$574$	0	0
$575$	−10.2590	−0.427831
$576$	0	0
$577$	−18.2696	−0.760574	−0.380287	−	0.924869i	$-0.624175\pi$
$577$	−18.2696	−0.760574	−0.380287	+	0.924869i	$0.624175\pi$
$578$	0	0
$579$	−5.65402	−0.234973
$580$	0	0
$581$	13.1119	0.543971
$582$	0	0
$583$	−5.66611	−0.234666
$584$	0	0
$585$	1.56434	0.0646776
$586$	0	0
$587$	2.42032	0.0998975	0.0499487	−	0.998752i	$-0.484094\pi$
$587$	2.42032	0.0998975	0.0499487	+	0.998752i	$0.484094\pi$
$588$	0	0
$589$	−8.31231	−0.342503
$590$	0	0
$591$	−3.11592	−0.128172
$592$	0	0
$593$	21.8404	0.896878	0.448439	−	0.893813i	$-0.351980\pi$
$593$	21.8404	0.896878	0.448439	+	0.893813i	$0.351980\pi$
$594$	0	0
$595$	4.77591	0.195793
$596$	0	0
$597$	−36.2695	−1.48441
$598$	0	0
$599$	−7.64089	−0.312198	−0.156099	−	0.987741i	$-0.549892\pi$
$599$	−7.64089	−0.312198	−0.156099	+	0.987741i	$0.549892\pi$
$600$	0	0
$601$	19.1337	0.780480	0.390240	−	0.920713i	$-0.372392\pi$
$601$	19.1337	0.780480	0.390240	+	0.920713i	$0.372392\pi$
$602$	0	0
$603$	−5.23266	−0.213090
$604$	0	0
$605$	1.74448	0.0709231
$606$	0	0
$607$	31.4950	1.27834	0.639172	−	0.769064i	$-0.279277\pi$
$607$	31.4950	1.27834	0.639172	+	0.769064i	$0.279277\pi$
$608$	0	0
$609$	7.14744	0.289629
$610$	0	0
$611$	5.24277	0.212100
$612$	0	0
$613$	−24.8013	−1.00172	−0.500858	−	0.865530i	$-0.666982\pi$
$613$	−24.8013	−1.00172	−0.500858	+	0.865530i	$0.666982\pi$
$614$	0	0
$615$	5.53552	0.223214
$616$	0	0
$617$	6.38526	0.257061	0.128530	−	0.991706i	$-0.458974\pi$
$617$	6.38526	0.257061	0.128530	+	0.991706i	$0.458974\pi$
$618$	0	0
$619$	−30.5111	−1.22634	−0.613172	−	0.789950i	$-0.710107\pi$
$619$	−30.5111	−1.22634	−0.613172	+	0.789950i	$0.710107\pi$
$620$	0	0
$621$	29.6285	1.18895
$622$	0	0
$623$	5.75551	0.230590
$624$	0	0
$625$	−11.3870	−0.455479
$626$	0	0
$627$	10.3778	0.414450
$628$	0	0
$629$	10.9688	0.437354
$630$	0	0
$631$	−16.2755	−0.647918	−0.323959	−	0.946071i	$-0.605014\pi$
$631$	−16.2755	−0.647918	−0.323959	+	0.946071i	$0.605014\pi$
$632$	0	0
$633$	−3.30397	−0.131321
$634$	0	0
$635$	14.2094	0.563883
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−2.25495	−0.0892046
$640$	0	0
$641$	20.2773	0.800904	0.400452	−	0.916318i	$-0.368853\pi$
$641$	20.2773	0.800904	0.400452	+	0.916318i	$0.368853\pi$
$642$	0	0
$643$	27.9945	1.10399	0.551997	−	0.833846i	$-0.313866\pi$
$643$	27.9945	1.10399	0.551997	+	0.833846i	$0.313866\pi$
$644$	0	0
$645$	−0.816916	−0.0321661
$646$	0	0
$647$	14.2849	0.561599	0.280800	−	0.959766i	$-0.409400\pi$
$647$	14.2849	0.561599	0.280800	+	0.959766i	$0.409400\pi$
$648$	0	0
$649$	7.41510	0.291068
$650$	0	0
$651$	−1.68465	−0.0660265
$652$	0	0
$653$	21.8387	0.854615	0.427308	−	0.904106i	$-0.359462\pi$
$653$	21.8387	0.854615	0.427308	+	0.904106i	$0.359462\pi$
$654$	0	0
$655$	−7.89579	−0.308514
$656$	0	0
$657$	8.26873	0.322594
$658$	0	0
$659$	18.4667	0.719361	0.359681	−	0.933075i	$-0.382886\pi$
$659$	18.4667	0.719361	0.359681	+	0.933075i	$0.382886\pi$
$660$	0	0
$661$	24.7164	0.961355	0.480677	−	0.876898i	$-0.340391\pi$
$661$	24.7164	0.961355	0.480677	+	0.876898i	$0.340391\pi$
$662$	0	0
$663$	3.97043	0.154199
$664$	0	0
$665$	−12.4832	−0.484077
$666$	0	0
$667$	−25.8384	−1.00047
$668$	0	0
$669$	10.9586	0.423683
$670$	0	0
$671$	12.2603	0.473305
$672$	0	0
$673$	18.1696	0.700387	0.350194	−	0.936677i	$-0.386116\pi$
$673$	18.1696	0.700387	0.350194	+	0.936677i	$0.386116\pi$
$674$	0	0
$675$	−11.0584	−0.425639
$676$	0	0
$677$	27.0144	1.03825	0.519124	−	0.854699i	$-0.326258\pi$
$677$	27.0144	1.03825	0.519124	+	0.854699i	$0.326258\pi$
$678$	0	0
$679$	−7.10275	−0.272579
$680$	0	0
$681$	−17.4496	−0.668672
$682$	0	0
$683$	35.1340	1.34437	0.672183	−	0.740385i	$-0.265357\pi$
$683$	35.1340	1.34437	0.672183	+	0.740385i	$0.265357\pi$
$684$	0	0
$685$	−1.51001	−0.0576947
$686$	0	0
$687$	6.66691	0.254359
$688$	0	0
$689$	5.66611	0.215862
$690$	0	0
$691$	−19.6531	−0.747639	−0.373820	−	0.927501i	$-0.621952\pi$
$691$	−19.6531	−0.747639	−0.373820	+	0.927501i	$0.621952\pi$
$692$	0	0
$693$	−0.896739	−0.0340643
$694$	0	0
$695$	24.7689	0.939538
$696$	0	0
$697$	−5.99014	−0.226893
$698$	0	0
$699$	10.6440	0.402593
$700$	0	0
$701$	−5.92112	−0.223638	−0.111819	−	0.993729i	$-0.535668\pi$
$701$	−5.92112	−0.223638	−0.111819	+	0.993729i	$0.535668\pi$
$702$	0	0
$703$	−28.6699	−1.08131
$704$	0	0
$705$	13.2640	0.499550
$706$	0	0
$707$	9.82940	0.369673
$708$	0	0
$709$	40.1545	1.50803	0.754017	−	0.656855i	$-0.228114\pi$
$709$	40.1545	1.50803	0.754017	+	0.656855i	$0.228114\pi$
$710$	0	0
$711$	−6.85583	−0.257114
$712$	0	0
$713$	6.09009	0.228076
$714$	0	0
$715$	−1.74448	−0.0652398
$716$	0	0
$717$	−3.11320	−0.116265
$718$	0	0
$719$	−40.2455	−1.50090	−0.750452	−	0.660925i	$-0.770164\pi$
$719$	−40.2455	−1.50090	−0.750452	+	0.660925i	$0.770164\pi$
$720$	0	0
$721$	−0.939197	−0.0349775
$722$	0	0
$723$	28.6287	1.06471
$724$	0	0
$725$	9.64383	0.358163
$726$	0	0
$727$	41.8789	1.55320	0.776602	−	0.629992i	$-0.216942\pi$
$727$	41.8789	1.55320	0.776602	+	0.629992i	$0.216942\pi$
$728$	0	0
$729$	29.5247	1.09351
$730$	0	0
$731$	0.884008	0.0326962
$732$	0	0
$733$	8.17576	0.301979	0.150989	−	0.988535i	$-0.451754\pi$
$733$	8.17576	0.301979	0.150989	+	0.988535i	$0.451754\pi$
$734$	0	0
$735$	−2.52995	−0.0933187
$736$	0	0
$737$	5.83521	0.214943
$738$	0	0
$739$	−35.2634	−1.29719	−0.648593	−	0.761136i	$-0.724642\pi$
$739$	−35.2634	−1.29719	−0.648593	+	0.761136i	$0.724642\pi$
$740$	0	0
$741$	−10.3778	−0.381239
$742$	0	0
$743$	15.3424	0.562857	0.281429	−	0.959582i	$-0.409192\pi$
$743$	15.3424	0.562857	0.281429	+	0.959582i	$0.409192\pi$
$744$	0	0
$745$	9.96382	0.365046
$746$	0	0
$747$	−11.7579	−0.430199
$748$	0	0
$749$	−6.67714	−0.243977
$750$	0	0
$751$	8.64943	0.315622	0.157811	−	0.987469i	$-0.449556\pi$
$751$	8.64943	0.315622	0.157811	+	0.987469i	$0.449556\pi$
$752$	0	0
$753$	−23.7514	−0.865551
$754$	0	0
$755$	36.6090	1.33234
$756$	0	0
$757$	−19.1818	−0.697176	−0.348588	−	0.937276i	$-0.613339\pi$
$757$	−19.1818	−0.697176	−0.348588	+	0.937276i	$0.613339\pi$
$758$	0	0
$759$	−7.60340	−0.275986
$760$	0	0
$761$	−1.12307	−0.0407113	−0.0203557	−	0.999793i	$-0.506480\pi$
$761$	−1.12307	−0.0407113	−0.0203557	+	0.999793i	$0.506480\pi$
$762$	0	0
$763$	−0.312219	−0.0113031
$764$	0	0
$765$	−4.28275	−0.154843
$766$	0	0
$767$	−7.41510	−0.267744
$768$	0	0
$769$	−40.5900	−1.46371	−0.731856	−	0.681460i	$-0.761345\pi$
$769$	−40.5900	−1.46371	−0.731856	+	0.681460i	$0.761345\pi$
$770$	0	0
$771$	−14.8176	−0.533641
$772$	0	0
$773$	12.4564	0.448026	0.224013	−	0.974586i	$-0.428084\pi$
$773$	12.4564	0.448026	0.224013	+	0.974586i	$0.428084\pi$
$774$	0	0
$775$	−2.27304	−0.0816501
$776$	0	0
$777$	−5.81051	−0.208451
$778$	0	0
$779$	15.6569	0.560966
$780$	0	0
$781$	2.51462	0.0899801
$782$	0	0
$783$	−27.8517	−0.995339
$784$	0	0
$785$	2.45445	0.0876032
$786$	0	0
$787$	−37.0359	−1.32019	−0.660093	−	0.751184i	$-0.729483\pi$
$787$	−37.0359	−1.32019	−0.660093	+	0.751184i	$0.729483\pi$
$788$	0	0
$789$	−4.11887	−0.146636
$790$	0	0
$791$	−5.64945	−0.200871
$792$	0	0
$793$	−12.2603	−0.435377
$794$	0	0
$795$	14.3350	0.508410
$796$	0	0
$797$	27.2988	0.966972	0.483486	−	0.875352i	$-0.339370\pi$
$797$	27.2988	0.966972	0.483486	+	0.875352i	$0.339370\pi$
$798$	0	0
$799$	−14.3533	−0.507783
$800$	0	0
$801$	−5.16119	−0.182362
$802$	0	0
$803$	−9.22089	−0.325398
$804$	0	0
$805$	9.14591	0.322351
$806$	0	0
$807$	18.6902	0.657927
$808$	0	0
$809$	−17.7300	−0.623354	−0.311677	−	0.950188i	$-0.600891\pi$
$809$	−17.7300	−0.623354	−0.311677	+	0.950188i	$0.600891\pi$
$810$	0	0
$811$	−21.6418	−0.759945	−0.379972	−	0.924998i	$-0.624067\pi$
$811$	−21.6418	−0.759945	−0.379972	+	0.924998i	$0.624067\pi$
$812$	0	0
$813$	−36.5257	−1.28101
$814$	0	0
$815$	12.3604	0.432968
$816$	0	0
$817$	−2.31060	−0.0808377
$818$	0	0
$819$	0.896739	0.0313346
$820$	0	0
$821$	−6.81733	−0.237926	−0.118963	−	0.992899i	$-0.537957\pi$
$821$	−6.81733	−0.237926	−0.118963	+	0.992899i	$0.537957\pi$
$822$	0	0
$823$	−20.0062	−0.697372	−0.348686	−	0.937240i	$-0.613372\pi$
$823$	−20.0062	−0.697372	−0.348686	+	0.937240i	$0.613372\pi$
$824$	0	0
$825$	2.83787	0.0988018
$826$	0	0
$827$	−24.5743	−0.854532	−0.427266	−	0.904126i	$-0.640523\pi$
$827$	−24.5743	−0.854532	−0.427266	+	0.904126i	$0.640523\pi$
$828$	0	0
$829$	11.0231	0.382847	0.191423	−	0.981508i	$-0.438690\pi$
$829$	11.0231	0.382847	0.191423	+	0.981508i	$0.438690\pi$
$830$	0	0
$831$	−25.2401	−0.875569
$832$	0	0
$833$	2.73773	0.0948567
$834$	0	0
$835$	−0.822666	−0.0284695
$836$	0	0
$837$	6.56463	0.226907
$838$	0	0
$839$	−17.1953	−0.593648	−0.296824	−	0.954932i	$-0.595928\pi$
$839$	−17.1953	−0.593648	−0.296824	+	0.954932i	$0.595928\pi$
$840$	0	0
$841$	−4.71107	−0.162451
$842$	0	0
$843$	1.95854	0.0674556
$844$	0	0
$845$	1.74448	0.0600119
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	38.6526	1.32655
$850$	0	0
$851$	21.0053	0.720052
$852$	0	0
$853$	−10.2163	−0.349801	−0.174900	−	0.984586i	$-0.555960\pi$
$853$	−10.2163	−0.349801	−0.174900	+	0.984586i	$0.555960\pi$
$854$	0	0
$855$	11.1941	0.382832
$856$	0	0
$857$	−5.05861	−0.172799	−0.0863994	−	0.996261i	$-0.527536\pi$
$857$	−5.05861	−0.172799	−0.0863994	+	0.996261i	$0.527536\pi$
$858$	0	0
$859$	−0.834091	−0.0284588	−0.0142294	−	0.999899i	$-0.504530\pi$
$859$	−0.834091	−0.0284588	−0.0142294	+	0.999899i	$0.504530\pi$
$860$	0	0
$861$	3.17316	0.108141
$862$	0	0
$863$	−58.3907	−1.98764	−0.993822	−	0.110990i	$-0.964598\pi$
$863$	−58.3907	−1.98764	−0.993822	+	0.110990i	$0.964598\pi$
$864$	0	0
$865$	8.63017	0.293435
$866$	0	0
$867$	13.7845	0.468146
$868$	0	0
$869$	7.64529	0.259349
$870$	0	0
$871$	−5.83521	−0.197719
$872$	0	0
$873$	6.36931	0.215569
$874$	0	0
$875$	−12.1360	−0.410271
$876$	0	0
$877$	−2.76782	−0.0934626	−0.0467313	−	0.998907i	$-0.514880\pi$
$877$	−2.76782	−0.0934626	−0.0467313	+	0.998907i	$0.514880\pi$
$878$	0	0
$879$	−14.5782	−0.491710
$880$	0	0
$881$	−40.6457	−1.36939	−0.684694	−	0.728831i	$-0.740064\pi$
$881$	−40.6457	−1.36939	−0.684694	+	0.728831i	$0.740064\pi$
$882$	0	0
$883$	12.2123	0.410976	0.205488	−	0.978660i	$-0.434122\pi$
$883$	12.2123	0.410976	0.205488	+	0.978660i	$0.434122\pi$
$884$	0	0
$885$	−18.7598	−0.630605
$886$	0	0
$887$	54.2285	1.82082	0.910408	−	0.413711i	$-0.135768\pi$
$887$	54.2285	1.82082	0.910408	+	0.413711i	$0.135768\pi$
$888$	0	0
$889$	8.14536	0.273187
$890$	0	0
$891$	−5.50564	−0.184446
$892$	0	0
$893$	37.5163	1.25544
$894$	0	0
$895$	−8.18858	−0.273714
$896$	0	0
$897$	7.60340	0.253870
$898$	0	0
$899$	−5.72488	−0.190935
$900$	0	0
$901$	−15.5123	−0.516789
$902$	0	0
$903$	−0.468287	−0.0155836
$904$	0	0
$905$	3.94004	0.130971
$906$	0	0
$907$	−13.5629	−0.450350	−0.225175	−	0.974318i	$-0.572295\pi$
$907$	−13.5629	−0.450350	−0.225175	+	0.974318i	$0.572295\pi$
$908$	0	0
$909$	−8.81441	−0.292355
$910$	0	0
$911$	14.1919	0.470198	0.235099	−	0.971971i	$-0.424459\pi$
$911$	14.1919	0.470198	0.235099	+	0.971971i	$0.424459\pi$
$912$	0	0
$913$	13.1119	0.433939
$914$	0	0
$915$	−31.0181	−1.02543
$916$	0	0
$917$	−4.52616	−0.149467
$918$	0	0
$919$	38.7092	1.27690	0.638449	−	0.769664i	$-0.279576\pi$
$919$	38.7092	1.27690	0.638449	+	0.769664i	$0.279576\pi$
$920$	0	0
$921$	−19.5144	−0.643020
$922$	0	0
$923$	−2.51462	−0.0827696
$924$	0	0
$925$	−7.83994	−0.257776
$926$	0	0
$927$	0.842215	0.0276620
$928$	0	0
$929$	28.5291	0.936010	0.468005	−	0.883726i	$-0.344973\pi$
$929$	28.5291	0.936010	0.468005	+	0.883726i	$0.344973\pi$
$930$	0	0
$931$	−7.15582	−0.234522
$932$	0	0
$933$	31.6657	1.03669
$934$	0	0
$935$	4.77591	0.156189
$936$	0	0
$937$	12.6319	0.412665	0.206332	−	0.978482i	$-0.433847\pi$
$937$	12.6319	0.412665	0.206332	+	0.978482i	$0.433847\pi$
$938$	0	0
$939$	26.0430	0.849883
$940$	0	0
$941$	20.4687	0.667260	0.333630	−	0.942704i	$-0.391726\pi$
$941$	20.4687	0.667260	0.333630	+	0.942704i	$0.391726\pi$
$942$	0	0
$943$	−11.4712	−0.373552
$944$	0	0
$945$	9.85856	0.320699
$946$	0	0
$947$	−20.9077	−0.679408	−0.339704	−	0.940532i	$-0.610327\pi$
$947$	−20.9077	−0.679408	−0.339704	+	0.940532i	$0.610327\pi$
$948$	0	0
$949$	9.22089	0.299323
$950$	0	0
$951$	−5.46018	−0.177058
$952$	0	0
$953$	52.2460	1.69241	0.846207	−	0.532855i	$-0.178881\pi$
$953$	52.2460	1.69241	0.846207	+	0.532855i	$0.178881\pi$
$954$	0	0
$955$	−40.1123	−1.29800
$956$	0	0
$957$	7.14744	0.231044
$958$	0	0
$959$	−0.865596	−0.0279516
$960$	0	0
$961$	−29.6507	−0.956473
$962$	0	0
$963$	5.98765	0.192949
$964$	0	0
$965$	6.80106	0.218934
$966$	0	0
$967$	2.81447	0.0905072	0.0452536	−	0.998976i	$-0.485590\pi$
$967$	2.81447	0.0905072	0.0452536	+	0.998976i	$0.485590\pi$
$968$	0	0
$969$	28.4117	0.912714
$970$	0	0
$971$	46.5725	1.49458	0.747292	−	0.664496i	$-0.231354\pi$
$971$	46.5725	1.49458	0.747292	+	0.664496i	$0.231354\pi$
$972$	0	0
$973$	14.1985	0.455181
$974$	0	0
$975$	−2.83787	−0.0908845
$976$	0	0
$977$	37.0137	1.18417	0.592087	−	0.805874i	$-0.298304\pi$
$977$	37.0137	1.18417	0.592087	+	0.805874i	$0.298304\pi$
$978$	0	0
$979$	5.75551	0.183947
$980$	0	0
$981$	0.279979	0.00893905
$982$	0	0
$983$	−30.9675	−0.987711	−0.493856	−	0.869544i	$-0.664413\pi$
$983$	−30.9675	−0.987711	−0.493856	+	0.869544i	$0.664413\pi$
$984$	0	0
$985$	3.74805	0.119423
$986$	0	0
$987$	7.60340	0.242019
$988$	0	0
$989$	1.69288	0.0538305
$990$	0	0
$991$	−34.2580	−1.08824	−0.544121	−	0.839007i	$-0.683137\pi$
$991$	−34.2580	−1.08824	−0.544121	+	0.839007i	$0.683137\pi$
$992$	0	0
$993$	3.05368	0.0969057
$994$	0	0
$995$	43.6276	1.38309
$996$	0	0
$997$	−6.95210	−0.220175	−0.110088	−	0.993922i	$-0.535113\pi$
$997$	−6.95210	−0.220175	−0.110088	+	0.993922i	$0.535113\pi$
$998$	0	0
$999$	22.6420	0.716362

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.q.1.3	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.q.1.3	✓	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)

\(f(q)\)	\(=\)	\(q-1.45026 q^{3} +1.74448 q^{5} +1.00000 q^{7} -0.896739 q^{9} +O(q^{10})\)	\(q-1.45026 q^{3} +1.74448 q^{5} +1.00000 q^{7} -0.896739 q^{9} +1.00000 q^{11} -1.00000 q^{13} -2.52995 q^{15} +2.73773 q^{17} -7.15582 q^{19} -1.45026 q^{21} +5.24277 q^{23} -1.95680 q^{25} +5.65129 q^{27} -4.92838 q^{29} +1.16162 q^{31} -1.45026 q^{33} +1.74448 q^{35} +4.00652 q^{37} +1.45026 q^{39} -2.18799 q^{41} +0.322898 q^{43} -1.56434 q^{45} -5.24277 q^{47} +1.00000 q^{49} -3.97043 q^{51} -5.66611 q^{53} +1.74448 q^{55} +10.3778 q^{57} +7.41510 q^{59} +12.2603 q^{61} -0.896739 q^{63} -1.74448 q^{65} +5.83521 q^{67} -7.60340 q^{69} +2.51462 q^{71} -9.22089 q^{73} +2.83787 q^{75} +1.00000 q^{77} +7.64529 q^{79} -5.50564 q^{81} +13.1119 q^{83} +4.77591 q^{85} +7.14744 q^{87} +5.75551 q^{89} -1.00000 q^{91} -1.68465 q^{93} -12.4832 q^{95} -7.10275 q^{97} -0.896739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) 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

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