LMFDB - Embedded newform 8008.2.a.z.1.11

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:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - x^{14} - 35 x^{13} + 32 x^{12} + 477 x^{11} - 392 x^{10} - 3236 x^{9} + 2330 x^{8} + \cdots + 2560 $ x^15 - x^14 - 35x^13 + 32x^12 + 477x^11 - 392x^10 - 3236x^9 + 2330x^8 + 11690x^7 - 7119x^6 - 22246x^5 + 11137x^4 + 20034x^3 - 8392x^2 - 6016*x + 2560
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Root			$-1.58213$ 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)

$f(q)$	$=$	$q+1.58213 q^{3} -3.58630 q^{5} -1.00000 q^{7} -0.496866 q^{9} +O(q^{10})$	$q+1.58213 q^{3} -3.58630 q^{5} -1.00000 q^{7} -0.496866 q^{9} -1.00000 q^{11} +1.00000 q^{13} -5.67399 q^{15} +3.45661 q^{17} -2.87910 q^{19} -1.58213 q^{21} -3.76734 q^{23} +7.86156 q^{25} -5.53250 q^{27} +9.14363 q^{29} -10.0808 q^{31} -1.58213 q^{33} +3.58630 q^{35} +7.49165 q^{37} +1.58213 q^{39} -2.39472 q^{41} -9.24602 q^{43} +1.78191 q^{45} +3.77434 q^{47} +1.00000 q^{49} +5.46881 q^{51} -8.47308 q^{53} +3.58630 q^{55} -4.55511 q^{57} -13.9331 q^{59} +13.3801 q^{61} +0.496866 q^{63} -3.58630 q^{65} -4.43386 q^{67} -5.96041 q^{69} +1.43978 q^{71} -8.99167 q^{73} +12.4380 q^{75} +1.00000 q^{77} +12.6595 q^{79} -7.26252 q^{81} -0.700479 q^{83} -12.3965 q^{85} +14.4664 q^{87} +3.38773 q^{89} -1.00000 q^{91} -15.9491 q^{93} +10.3253 q^{95} +9.87800 q^{97} +0.496866 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	$ 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) $ 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * 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
$3$	1.58213	0.913443	0.456721	−	0.889610i	$-0.349024\pi$
$3$	1.58213	0.913443	0.456721	+	0.889610i	$0.349024\pi$
$4$	0	0
$5$	−3.58630	−1.60384	−0.801921	−	0.597430i	$-0.796189\pi$
$5$	−3.58630	−1.60384	−0.801921	+	0.597430i	$0.796189\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−0.496866	−0.165622
$10$	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
$15$	−5.67399	−1.46502
$16$	0	0
$17$	3.45661	0.838352	0.419176	−	0.907905i	$-0.362319\pi$
$17$	3.45661	0.838352	0.419176	+	0.907905i	$0.362319\pi$
$18$	0	0
$19$	−2.87910	−0.660511	−0.330256	−	0.943892i	$-0.607135\pi$
$19$	−2.87910	−0.660511	−0.330256	+	0.943892i	$0.607135\pi$
$20$	0	0
$21$	−1.58213	−0.345249
$22$	0	0
$23$	−3.76734	−0.785544	−0.392772	−	0.919636i	$-0.628484\pi$
$23$	−3.76734	−0.785544	−0.392772	+	0.919636i	$0.628484\pi$
$24$	0	0
$25$	7.86156	1.57231
$26$	0	0
$27$	−5.53250	−1.06473
$28$	0	0
$29$	9.14363	1.69793	0.848965	−	0.528450i	$-0.177227\pi$
$29$	9.14363	1.69793	0.848965	+	0.528450i	$0.177227\pi$
$30$	0	0
$31$	−10.0808	−1.81056	−0.905279	−	0.424818i	$-0.860338\pi$
$31$	−10.0808	−1.81056	−0.905279	+	0.424818i	$0.860338\pi$
$32$	0	0
$33$	−1.58213	−0.275413
$34$	0	0
$35$	3.58630	0.606196
$36$	0	0
$37$	7.49165	1.23162	0.615809	−	0.787895i	$-0.288829\pi$
$37$	7.49165	1.23162	0.615809	+	0.787895i	$0.288829\pi$
$38$	0	0
$39$	1.58213	0.253343
$40$	0	0
$41$	−2.39472	−0.373992	−0.186996	−	0.982361i	$-0.559875\pi$
$41$	−2.39472	−0.373992	−0.186996	+	0.982361i	$0.559875\pi$
$42$	0	0
$43$	−9.24602	−1.41000	−0.705002	−	0.709205i	$-0.749054\pi$
$43$	−9.24602	−1.41000	−0.705002	+	0.709205i	$0.749054\pi$
$44$	0	0
$45$	1.78191	0.265632
$46$	0	0
$47$	3.77434	0.550544	0.275272	−	0.961366i	$-0.411232\pi$
$47$	3.77434	0.550544	0.275272	+	0.961366i	$0.411232\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	5.46881	0.765786
$52$	0	0
$53$	−8.47308	−1.16387	−0.581933	−	0.813236i	$-0.697704\pi$
$53$	−8.47308	−1.16387	−0.581933	+	0.813236i	$0.697704\pi$
$54$	0	0
$55$	3.58630	0.483577
$56$	0	0
$57$	−4.55511	−0.603339
$58$	0	0
$59$	−13.9331	−1.81394	−0.906968	−	0.421200i	$-0.861609\pi$
$59$	−13.9331	−1.81394	−0.906968	+	0.421200i	$0.861609\pi$
$60$	0	0
$61$	13.3801	1.71314	0.856571	−	0.516030i	$-0.172591\pi$
$61$	13.3801	1.71314	0.856571	+	0.516030i	$0.172591\pi$
$62$	0	0
$63$	0.496866	0.0625993
$64$	0	0
$65$	−3.58630	−0.444826
$66$	0	0
$67$	−4.43386	−0.541682	−0.270841	−	0.962624i	$-0.587302\pi$
$67$	−4.43386	−0.541682	−0.270841	+	0.962624i	$0.587302\pi$
$68$	0	0
$69$	−5.96041	−0.717549
$70$	0	0
$71$	1.43978	0.170870	0.0854352	−	0.996344i	$-0.472772\pi$
$71$	1.43978	0.170870	0.0854352	+	0.996344i	$0.472772\pi$
$72$	0	0
$73$	−8.99167	−1.05240	−0.526198	−	0.850362i	$-0.676383\pi$
$73$	−8.99167	−1.05240	−0.526198	+	0.850362i	$0.676383\pi$
$74$	0	0
$75$	12.4380	1.43622
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	12.6595	1.42431	0.712155	−	0.702022i	$-0.247719\pi$
$79$	12.6595	1.42431	0.712155	+	0.702022i	$0.247719\pi$
$80$	0	0
$81$	−7.26252	−0.806947
$82$	0	0
$83$	−0.700479	−0.0768875	−0.0384438	−	0.999261i	$-0.512240\pi$
$83$	−0.700479	−0.0768875	−0.0384438	+	0.999261i	$0.512240\pi$
$84$	0	0
$85$	−12.3965	−1.34458
$86$	0	0
$87$	14.4664	1.55096
$88$	0	0
$89$	3.38773	0.359099	0.179550	−	0.983749i	$-0.442536\pi$
$89$	3.38773	0.359099	0.179550	+	0.983749i	$0.442536\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	−15.9491	−1.65384
$94$	0	0
$95$	10.3253	1.05936
$96$	0	0
$97$	9.87800	1.00296	0.501480	−	0.865170i	$-0.332789\pi$
$97$	9.87800	1.00296	0.501480	+	0.865170i	$0.332789\pi$
$98$	0	0
$99$	0.496866	0.0499370
$100$	0	0
$101$	8.07489	0.803481	0.401741	−	0.915753i	$-0.368405\pi$
$101$	8.07489	0.803481	0.401741	+	0.915753i	$0.368405\pi$
$102$	0	0
$103$	−17.9925	−1.77286	−0.886429	−	0.462864i	$-0.846822\pi$
$103$	−17.9925	−1.77286	−0.886429	+	0.462864i	$0.846822\pi$
$104$	0	0
$105$	5.67399	0.553725
$106$	0	0
$107$	14.9866	1.44881	0.724407	−	0.689373i	$-0.242114\pi$
$107$	14.9866	1.44881	0.724407	+	0.689373i	$0.242114\pi$
$108$	0	0
$109$	5.31169	0.508768	0.254384	−	0.967103i	$-0.418127\pi$
$109$	5.31169	0.508768	0.254384	+	0.967103i	$0.418127\pi$
$110$	0	0
$111$	11.8528	1.12501
$112$	0	0
$113$	9.99829	0.940560	0.470280	−	0.882517i	$-0.344153\pi$
$113$	9.99829	0.940560	0.470280	+	0.882517i	$0.344153\pi$
$114$	0	0
$115$	13.5108	1.25989
$116$	0	0
$117$	−0.496866	−0.0459353
$118$	0	0
$119$	−3.45661	−0.316867
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−3.78875	−0.341620
$124$	0	0
$125$	−10.2624	−0.917898
$126$	0	0
$127$	14.7384	1.30782	0.653912	−	0.756570i	$-0.273127\pi$
$127$	14.7384	1.30782	0.653912	+	0.756570i	$0.273127\pi$
$128$	0	0
$129$	−14.6284	−1.28796
$130$	0	0
$131$	6.08581	0.531720	0.265860	−	0.964012i	$-0.414344\pi$
$131$	6.08581	0.531720	0.265860	+	0.964012i	$0.414344\pi$
$132$	0	0
$133$	2.87910	0.249650
$134$	0	0
$135$	19.8412	1.70766
$136$	0	0
$137$	15.3167	1.30860	0.654298	−	0.756237i	$-0.272964\pi$
$137$	15.3167	1.30860	0.654298	+	0.756237i	$0.272964\pi$
$138$	0	0
$139$	8.30321	0.704269	0.352134	−	0.935949i	$-0.385456\pi$
$139$	8.30321	0.704269	0.352134	+	0.935949i	$0.385456\pi$
$140$	0	0
$141$	5.97149	0.502890
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−32.7918	−2.72321
$146$	0	0
$147$	1.58213	0.130492
$148$	0	0
$149$	−17.7243	−1.45204	−0.726018	−	0.687676i	$-0.758631\pi$
$149$	−17.7243	−1.45204	−0.726018	+	0.687676i	$0.758631\pi$
$150$	0	0
$151$	−11.4942	−0.935383	−0.467691	−	0.883892i	$-0.654914\pi$
$151$	−11.4942	−0.935383	−0.467691	+	0.883892i	$0.654914\pi$
$152$	0	0
$153$	−1.71748	−0.138850
$154$	0	0
$155$	36.1526	2.90385
$156$	0	0
$157$	−8.91403	−0.711417	−0.355708	−	0.934597i	$-0.615760\pi$
$157$	−8.91403	−0.711417	−0.355708	+	0.934597i	$0.615760\pi$
$158$	0	0
$159$	−13.4055	−1.06313
$160$	0	0
$161$	3.76734	0.296908
$162$	0	0
$163$	5.96166	0.466953	0.233477	−	0.972362i	$-0.424990\pi$
$163$	5.96166	0.466953	0.233477	+	0.972362i	$0.424990\pi$
$164$	0	0
$165$	5.67399	0.441720
$166$	0	0
$167$	18.9216	1.46420	0.732100	−	0.681198i	$-0.238541\pi$
$167$	18.9216	1.46420	0.732100	+	0.681198i	$0.238541\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	1.43053	0.109395
$172$	0	0
$173$	18.5196	1.40802	0.704010	−	0.710190i	$-0.251391\pi$
$173$	18.5196	1.40802	0.704010	+	0.710190i	$0.251391\pi$
$174$	0	0
$175$	−7.86156	−0.594278
$176$	0	0
$177$	−22.0440	−1.65693
$178$	0	0
$179$	−5.43032	−0.405881	−0.202941	−	0.979191i	$-0.565050\pi$
$179$	−5.43032	−0.405881	−0.202941	+	0.979191i	$0.565050\pi$
$180$	0	0
$181$	20.3401	1.51186	0.755932	−	0.654650i	$-0.227184\pi$
$181$	20.3401	1.51186	0.755932	+	0.654650i	$0.227184\pi$
$182$	0	0
$183$	21.1690	1.56486
$184$	0	0
$185$	−26.8673	−1.97532
$186$	0	0
$187$	−3.45661	−0.252773
$188$	0	0
$189$	5.53250	0.402430
$190$	0	0
$191$	−8.67525	−0.627719	−0.313860	−	0.949469i	$-0.601622\pi$
$191$	−8.67525	−0.627719	−0.313860	+	0.949469i	$0.601622\pi$
$192$	0	0
$193$	16.8054	1.20968	0.604841	−	0.796347i	$-0.293237\pi$
$193$	16.8054	1.20968	0.604841	+	0.796347i	$0.293237\pi$
$194$	0	0
$195$	−5.67399	−0.406323
$196$	0	0
$197$	16.7467	1.19315	0.596577	−	0.802556i	$-0.296527\pi$
$197$	16.7467	1.19315	0.596577	+	0.802556i	$0.296527\pi$
$198$	0	0
$199$	−9.98907	−0.708106	−0.354053	−	0.935225i	$-0.615197\pi$
$199$	−9.98907	−0.708106	−0.354053	+	0.935225i	$0.615197\pi$
$200$	0	0
$201$	−7.01494	−0.494796
$202$	0	0
$203$	−9.14363	−0.641757
$204$	0	0
$205$	8.58818	0.599824
$206$	0	0
$207$	1.87186	0.130103
$208$	0	0
$209$	2.87910	0.199152
$210$	0	0
$211$	15.8836	1.09347	0.546736	−	0.837305i	$-0.315870\pi$
$211$	15.8836	1.09347	0.546736	+	0.837305i	$0.315870\pi$
$212$	0	0
$213$	2.27792	0.156080
$214$	0	0
$215$	33.1590	2.26143
$216$	0	0
$217$	10.0808	0.684326
$218$	0	0
$219$	−14.2260	−0.961303
$220$	0	0
$221$	3.45661	0.232517
$222$	0	0
$223$	21.0179	1.40746	0.703730	−	0.710468i	$-0.251517\pi$
$223$	21.0179	1.40746	0.703730	+	0.710468i	$0.251517\pi$
$224$	0	0
$225$	−3.90614	−0.260410
$226$	0	0
$227$	−3.08318	−0.204638	−0.102319	−	0.994752i	$-0.532626\pi$
$227$	−3.08318	−0.204638	−0.102319	+	0.994752i	$0.532626\pi$
$228$	0	0
$229$	18.0874	1.19525	0.597625	−	0.801776i	$-0.296111\pi$
$229$	18.0874	1.19525	0.597625	+	0.801776i	$0.296111\pi$
$230$	0	0
$231$	1.58213	0.104096
$232$	0	0
$233$	−17.5020	−1.14659	−0.573296	−	0.819349i	$-0.694335\pi$
$233$	−17.5020	−1.14659	−0.573296	+	0.819349i	$0.694335\pi$
$234$	0	0
$235$	−13.5359	−0.882986
$236$	0	0
$237$	20.0290	1.30103
$238$	0	0
$239$	5.13919	0.332426	0.166213	−	0.986090i	$-0.446846\pi$
$239$	5.13919	0.332426	0.166213	+	0.986090i	$0.446846\pi$
$240$	0	0
$241$	−12.7678	−0.822448	−0.411224	−	0.911534i	$-0.634899\pi$
$241$	−12.7678	−0.822448	−0.411224	+	0.911534i	$0.634899\pi$
$242$	0	0
$243$	5.10723	0.327629
$244$	0	0
$245$	−3.58630	−0.229120
$246$	0	0
$247$	−2.87910	−0.183193
$248$	0	0
$249$	−1.10825	−0.0702324
$250$	0	0
$251$	−23.7038	−1.49617	−0.748087	−	0.663601i	$-0.769027\pi$
$251$	−23.7038	−1.49617	−0.748087	+	0.663601i	$0.769027\pi$
$252$	0	0
$253$	3.76734	0.236850
$254$	0	0
$255$	−19.6128	−1.22820
$256$	0	0
$257$	−8.88576	−0.554279	−0.277139	−	0.960830i	$-0.589386\pi$
$257$	−8.88576	−0.554279	−0.277139	+	0.960830i	$0.589386\pi$
$258$	0	0
$259$	−7.49165	−0.465508
$260$	0	0
$261$	−4.54316	−0.281215
$262$	0	0
$263$	28.8188	1.77704	0.888521	−	0.458836i	$-0.151734\pi$
$263$	28.8188	1.77704	0.888521	+	0.458836i	$0.151734\pi$
$264$	0	0
$265$	30.3870	1.86666
$266$	0	0
$267$	5.35983	0.328016
$268$	0	0
$269$	8.56587	0.522270	0.261135	−	0.965302i	$-0.415903\pi$
$269$	8.56587	0.522270	0.261135	+	0.965302i	$0.415903\pi$
$270$	0	0
$271$	11.2588	0.683922	0.341961	−	0.939714i	$-0.388909\pi$
$271$	11.2588	0.683922	0.341961	+	0.939714i	$0.388909\pi$
$272$	0	0
$273$	−1.58213	−0.0957548
$274$	0	0
$275$	−7.86156	−0.474070
$276$	0	0
$277$	15.1945	0.912947	0.456474	−	0.889737i	$-0.349112\pi$
$277$	15.1945	0.912947	0.456474	+	0.889737i	$0.349112\pi$
$278$	0	0
$279$	5.00879	0.299868
$280$	0	0
$281$	12.1169	0.722836	0.361418	−	0.932404i	$-0.382293\pi$
$281$	12.1169	0.722836	0.361418	+	0.932404i	$0.382293\pi$
$282$	0	0
$283$	−22.6802	−1.34820	−0.674100	−	0.738640i	$-0.735468\pi$
$283$	−22.6802	−1.34820	−0.674100	+	0.738640i	$0.735468\pi$
$284$	0	0
$285$	16.3360	0.967661
$286$	0	0
$287$	2.39472	0.141356
$288$	0	0
$289$	−5.05183	−0.297166
$290$	0	0
$291$	15.6283	0.916146
$292$	0	0
$293$	−3.32487	−0.194241	−0.0971206	−	0.995273i	$-0.530963\pi$
$293$	−3.32487	−0.194241	−0.0971206	+	0.995273i	$0.530963\pi$
$294$	0	0
$295$	49.9683	2.90927
$296$	0	0
$297$	5.53250	0.321028
$298$	0	0
$299$	−3.76734	−0.217871
$300$	0	0
$301$	9.24602	0.532932
$302$	0	0
$303$	12.7755	0.733934
$304$	0	0
$305$	−47.9849	−2.74761
$306$	0	0
$307$	−16.0054	−0.913474	−0.456737	−	0.889602i	$-0.650982\pi$
$307$	−16.0054	−0.913474	−0.456737	+	0.889602i	$0.650982\pi$
$308$	0	0
$309$	−28.4665	−1.61940
$310$	0	0
$311$	1.65715	0.0939681	0.0469841	−	0.998896i	$-0.485039\pi$
$311$	1.65715	0.0939681	0.0469841	+	0.998896i	$0.485039\pi$
$312$	0	0
$313$	32.7586	1.85163	0.925813	−	0.377982i	$-0.123382\pi$
$313$	32.7586	1.85163	0.925813	+	0.377982i	$0.123382\pi$
$314$	0	0
$315$	−1.78191	−0.100399
$316$	0	0
$317$	−7.13697	−0.400852	−0.200426	−	0.979709i	$-0.564233\pi$
$317$	−7.13697	−0.400852	−0.200426	+	0.979709i	$0.564233\pi$
$318$	0	0
$319$	−9.14363	−0.511945
$320$	0	0
$321$	23.7108	1.32341
$322$	0	0
$323$	−9.95194	−0.553741
$324$	0	0
$325$	7.86156	0.436081
$326$	0	0
$327$	8.40378	0.464730
$328$	0	0
$329$	−3.77434	−0.208086
$330$	0	0
$331$	−4.96954	−0.273150	−0.136575	−	0.990630i	$-0.543610\pi$
$331$	−4.96954	−0.273150	−0.136575	+	0.990630i	$0.543610\pi$
$332$	0	0
$333$	−3.72235	−0.203983
$334$	0	0
$335$	15.9012	0.868773
$336$	0	0
$337$	−16.7752	−0.913806	−0.456903	−	0.889517i	$-0.651041\pi$
$337$	−16.7752	−0.913806	−0.456903	+	0.889517i	$0.651041\pi$
$338$	0	0
$339$	15.8186	0.859148
$340$	0	0
$341$	10.0808	0.545904
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	21.3758	1.15084
$346$	0	0
$347$	3.72336	0.199881	0.0999403	−	0.994993i	$-0.468135\pi$
$347$	3.72336	0.199881	0.0999403	+	0.994993i	$0.468135\pi$
$348$	0	0
$349$	10.0426	0.537568	0.268784	−	0.963201i	$-0.413378\pi$
$349$	10.0426	0.537568	0.268784	+	0.963201i	$0.413378\pi$
$350$	0	0
$351$	−5.53250	−0.295303
$352$	0	0
$353$	26.6493	1.41840	0.709199	−	0.705009i	$-0.249057\pi$
$353$	26.6493	1.41840	0.709199	+	0.705009i	$0.249057\pi$
$354$	0	0
$355$	−5.16348	−0.274049
$356$	0	0
$357$	−5.46881	−0.289440
$358$	0	0
$359$	14.4866	0.764575	0.382288	−	0.924043i	$-0.375136\pi$
$359$	14.4866	0.764575	0.382288	+	0.924043i	$0.375136\pi$
$360$	0	0
$361$	−10.7108	−0.563725
$362$	0	0
$363$	1.58213	0.0830403
$364$	0	0
$365$	32.2468	1.68788
$366$	0	0
$367$	−1.94151	−0.101346	−0.0506731	−	0.998715i	$-0.516137\pi$
$367$	−1.94151	−0.101346	−0.0506731	+	0.998715i	$0.516137\pi$
$368$	0	0
$369$	1.18985	0.0619414
$370$	0	0
$371$	8.47308	0.439900
$372$	0	0
$373$	−36.5740	−1.89373	−0.946864	−	0.321633i	$-0.895768\pi$
$373$	−36.5740	−1.89373	−0.946864	+	0.321633i	$0.895768\pi$
$374$	0	0
$375$	−16.2365	−0.838447
$376$	0	0
$377$	9.14363	0.470921
$378$	0	0
$379$	−22.9670	−1.17973	−0.589867	−	0.807500i	$-0.700820\pi$
$379$	−22.9670	−1.17973	−0.589867	+	0.807500i	$0.700820\pi$
$380$	0	0
$381$	23.3181	1.19462
$382$	0	0
$383$	21.5521	1.10126	0.550631	−	0.834749i	$-0.314387\pi$
$383$	21.5521	1.10126	0.550631	+	0.834749i	$0.314387\pi$
$384$	0	0
$385$	−3.58630	−0.182775
$386$	0	0
$387$	4.59404	0.233528
$388$	0	0
$389$	19.7385	1.00078	0.500391	−	0.865799i	$-0.333189\pi$
$389$	19.7385	1.00078	0.500391	+	0.865799i	$0.333189\pi$
$390$	0	0
$391$	−13.0222	−0.658562
$392$	0	0
$393$	9.62854	0.485696
$394$	0	0
$395$	−45.4009	−2.28437
$396$	0	0
$397$	−15.4741	−0.776623	−0.388311	−	0.921528i	$-0.626941\pi$
$397$	−15.4741	−0.776623	−0.388311	+	0.921528i	$0.626941\pi$
$398$	0	0
$399$	4.55511	0.228041
$400$	0	0
$401$	−15.7999	−0.789009	−0.394505	−	0.918894i	$-0.629084\pi$
$401$	−15.7999	−0.789009	−0.394505	+	0.918894i	$0.629084\pi$
$402$	0	0
$403$	−10.0808	−0.502158
$404$	0	0
$405$	26.0456	1.29422
$406$	0	0
$407$	−7.49165	−0.371347
$408$	0	0
$409$	15.9933	0.790816	0.395408	−	0.918506i	$-0.370603\pi$
$409$	15.9933	0.790816	0.395408	+	0.918506i	$0.370603\pi$
$410$	0	0
$411$	24.2330	1.19533
$412$	0	0
$413$	13.9331	0.685603
$414$	0	0
$415$	2.51213	0.123316
$416$	0	0
$417$	13.1367	0.643309
$418$	0	0
$419$	23.8811	1.16667	0.583335	−	0.812232i	$-0.301748\pi$
$419$	23.8811	1.16667	0.583335	+	0.812232i	$0.301748\pi$
$420$	0	0
$421$	−1.13817	−0.0554711	−0.0277355	−	0.999615i	$-0.508830\pi$
$421$	−1.13817	−0.0554711	−0.0277355	+	0.999615i	$0.508830\pi$
$422$	0	0
$423$	−1.87534	−0.0911823
$424$	0	0
$425$	27.1744	1.31815
$426$	0	0
$427$	−13.3801	−0.647506
$428$	0	0
$429$	−1.58213	−0.0763859
$430$	0	0
$431$	36.2796	1.74753	0.873763	−	0.486353i	$-0.161673\pi$
$431$	36.2796	1.74753	0.873763	+	0.486353i	$0.161673\pi$
$432$	0	0
$433$	−2.58655	−0.124302	−0.0621509	−	0.998067i	$-0.519796\pi$
$433$	−2.58655	−0.124302	−0.0621509	+	0.998067i	$0.519796\pi$
$434$	0	0
$435$	−51.8809	−2.48750
$436$	0	0
$437$	10.8465	0.518860
$438$	0	0
$439$	−7.37690	−0.352080	−0.176040	−	0.984383i	$-0.556329\pi$
$439$	−7.37690	−0.352080	−0.176040	+	0.984383i	$0.556329\pi$
$440$	0	0
$441$	−0.496866	−0.0236603
$442$	0	0
$443$	−16.2978	−0.774330	−0.387165	−	0.922010i	$-0.626546\pi$
$443$	−16.2978	−0.774330	−0.387165	+	0.922010i	$0.626546\pi$
$444$	0	0
$445$	−12.1494	−0.575938
$446$	0	0
$447$	−28.0422	−1.32635
$448$	0	0
$449$	11.4024	0.538113	0.269056	−	0.963124i	$-0.413288\pi$
$449$	11.4024	0.538113	0.269056	+	0.963124i	$0.413288\pi$
$450$	0	0
$451$	2.39472	0.112763
$452$	0	0
$453$	−18.1853	−0.854419
$454$	0	0
$455$	3.58630	0.168128
$456$	0	0
$457$	6.87694	0.321690	0.160845	−	0.986980i	$-0.448578\pi$
$457$	6.87694	0.321690	0.160845	+	0.986980i	$0.448578\pi$
$458$	0	0
$459$	−19.1237	−0.892618
$460$	0	0
$461$	−37.9180	−1.76602	−0.883008	−	0.469358i	$-0.844486\pi$
$461$	−37.9180	−1.76602	−0.883008	+	0.469358i	$0.844486\pi$
$462$	0	0
$463$	−12.2458	−0.569110	−0.284555	−	0.958660i	$-0.591846\pi$
$463$	−12.2458	−0.569110	−0.284555	+	0.958660i	$0.591846\pi$
$464$	0	0
$465$	57.1982	2.65250
$466$	0	0
$467$	−26.6685	−1.23407	−0.617036	−	0.786935i	$-0.711667\pi$
$467$	−26.6685	−1.23407	−0.617036	+	0.786935i	$0.711667\pi$
$468$	0	0
$469$	4.43386	0.204737
$470$	0	0
$471$	−14.1031	−0.649839
$472$	0	0
$473$	9.24602	0.425132
$474$	0	0
$475$	−22.6342	−1.03853
$476$	0	0
$477$	4.20999	0.192762
$478$	0	0
$479$	18.2322	0.833049	0.416524	−	0.909125i	$-0.363248\pi$
$479$	18.2322	0.833049	0.416524	+	0.909125i	$0.363248\pi$
$480$	0	0
$481$	7.49165	0.341590
$482$	0	0
$483$	5.96041	0.271208
$484$	0	0
$485$	−35.4255	−1.60859
$486$	0	0
$487$	−29.5739	−1.34012	−0.670060	−	0.742307i	$-0.733732\pi$
$487$	−29.5739	−1.34012	−0.670060	+	0.742307i	$0.733732\pi$
$488$	0	0
$489$	9.43211	0.426535
$490$	0	0
$491$	41.4152	1.86904	0.934521	−	0.355907i	$-0.115828\pi$
$491$	41.4152	1.86904	0.934521	+	0.355907i	$0.115828\pi$
$492$	0	0
$493$	31.6060	1.42346
$494$	0	0
$495$	−1.78191	−0.0800910
$496$	0	0
$497$	−1.43978	−0.0645829
$498$	0	0
$499$	21.5565	0.965001	0.482501	−	0.875896i	$-0.339729\pi$
$499$	21.5565	0.965001	0.482501	+	0.875896i	$0.339729\pi$
$500$	0	0
$501$	29.9365	1.33746
$502$	0	0
$503$	29.6902	1.32382	0.661909	−	0.749584i	$-0.269746\pi$
$503$	29.6902	1.32382	0.661909	+	0.749584i	$0.269746\pi$
$504$	0	0
$505$	−28.9590	−1.28866
$506$	0	0
$507$	1.58213	0.0702648
$508$	0	0
$509$	15.9077	0.705096	0.352548	−	0.935794i	$-0.385315\pi$
$509$	15.9077	0.705096	0.352548	+	0.935794i	$0.385315\pi$
$510$	0	0
$511$	8.99167	0.397768
$512$	0	0
$513$	15.9286	0.703266
$514$	0	0
$515$	64.5267	2.84339
$516$	0	0
$517$	−3.77434	−0.165995
$518$	0	0
$519$	29.3004	1.28615
$520$	0	0
$521$	−12.0594	−0.528331	−0.264166	−	0.964477i	$-0.585097\pi$
$521$	−12.0594	−0.528331	−0.264166	+	0.964477i	$0.585097\pi$
$522$	0	0
$523$	32.2754	1.41131	0.705653	−	0.708557i	$-0.250654\pi$
$523$	32.2754	1.41131	0.705653	+	0.708557i	$0.250654\pi$
$524$	0	0
$525$	−12.4380	−0.542839
$526$	0	0
$527$	−34.8453	−1.51788
$528$	0	0
$529$	−8.80718	−0.382921
$530$	0	0
$531$	6.92289	0.300428
$532$	0	0
$533$	−2.39472	−0.103727
$534$	0	0
$535$	−53.7466	−2.32367
$536$	0	0
$537$	−8.59147	−0.370749
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	10.8302	0.465626	0.232813	−	0.972522i	$-0.425207\pi$
$541$	10.8302	0.465626	0.232813	+	0.972522i	$0.425207\pi$
$542$	0	0
$543$	32.1806	1.38100
$544$	0	0
$545$	−19.0493	−0.815983
$546$	0	0
$547$	23.2004	0.991978	0.495989	−	0.868329i	$-0.334806\pi$
$547$	23.2004	0.991978	0.495989	+	0.868329i	$0.334806\pi$
$548$	0	0
$549$	−6.64810	−0.283734
$550$	0	0
$551$	−26.3254	−1.12150
$552$	0	0
$553$	−12.6595	−0.538338
$554$	0	0
$555$	−42.5075	−1.80434
$556$	0	0
$557$	10.6448	0.451035	0.225518	−	0.974239i	$-0.427593\pi$
$557$	10.6448	0.451035	0.225518	+	0.974239i	$0.427593\pi$
$558$	0	0
$559$	−9.24602	−0.391065
$560$	0	0
$561$	−5.46881	−0.230893
$562$	0	0
$563$	6.95821	0.293254	0.146627	−	0.989192i	$-0.453158\pi$
$563$	6.95821	0.293254	0.146627	+	0.989192i	$0.453158\pi$
$564$	0	0
$565$	−35.8569	−1.50851
$566$	0	0
$567$	7.26252	0.304997
$568$	0	0
$569$	11.1448	0.467216	0.233608	−	0.972331i	$-0.424947\pi$
$569$	11.1448	0.467216	0.233608	+	0.972331i	$0.424947\pi$
$570$	0	0
$571$	−10.3135	−0.431607	−0.215804	−	0.976437i	$-0.569237\pi$
$571$	−10.3135	−0.431607	−0.215804	+	0.976437i	$0.569237\pi$
$572$	0	0
$573$	−13.7254	−0.573386
$574$	0	0
$575$	−29.6171	−1.23512
$576$	0	0
$577$	−43.8915	−1.82723	−0.913613	−	0.406584i	$-0.866720\pi$
$577$	−43.8915	−1.82723	−0.913613	+	0.406584i	$0.866720\pi$
$578$	0	0
$579$	26.5884	1.10497
$580$	0	0
$581$	0.700479	0.0290608
$582$	0	0
$583$	8.47308	0.350919
$584$	0	0
$585$	1.78191	0.0736730
$586$	0	0
$587$	−40.3718	−1.66632	−0.833162	−	0.553029i	$-0.813472\pi$
$587$	−40.3718	−1.66632	−0.833162	+	0.553029i	$0.813472\pi$
$588$	0	0
$589$	29.0235	1.19589
$590$	0	0
$591$	26.4955	1.08988
$592$	0	0
$593$	4.69087	0.192631	0.0963154	−	0.995351i	$-0.469294\pi$
$593$	4.69087	0.192631	0.0963154	+	0.995351i	$0.469294\pi$
$594$	0	0
$595$	12.3965	0.508205
$596$	0	0
$597$	−15.8040	−0.646815
$598$	0	0
$599$	−13.1135	−0.535804	−0.267902	−	0.963446i	$-0.586330\pi$
$599$	−13.1135	−0.535804	−0.267902	+	0.963446i	$0.586330\pi$
$600$	0	0
$601$	−41.9941	−1.71297	−0.856487	−	0.516169i	$-0.827358\pi$
$601$	−41.9941	−1.71297	−0.856487	+	0.516169i	$0.827358\pi$
$602$	0	0
$603$	2.20304	0.0897146
$604$	0	0
$605$	−3.58630	−0.145804
$606$	0	0
$607$	−15.6050	−0.633389	−0.316694	−	0.948528i	$-0.602573\pi$
$607$	−15.6050	−0.633389	−0.316694	+	0.948528i	$0.602573\pi$
$608$	0	0
$609$	−14.4664	−0.586208
$610$	0	0
$611$	3.77434	0.152693
$612$	0	0
$613$	−6.78984	−0.274239	−0.137120	−	0.990555i	$-0.543784\pi$
$613$	−6.78984	−0.274239	−0.137120	+	0.990555i	$0.543784\pi$
$614$	0	0
$615$	13.5876	0.547905
$616$	0	0
$617$	7.27227	0.292771	0.146385	−	0.989228i	$-0.453236\pi$
$617$	7.27227	0.292771	0.146385	+	0.989228i	$0.453236\pi$
$618$	0	0
$619$	−49.3247	−1.98253	−0.991263	−	0.131898i	$-0.957893\pi$
$619$	−49.3247	−1.98253	−0.991263	+	0.131898i	$0.957893\pi$
$620$	0	0
$621$	20.8428	0.836391
$622$	0	0
$623$	−3.38773	−0.135727
$624$	0	0
$625$	−2.50370	−0.100148
$626$	0	0
$627$	4.55511	0.181914
$628$	0	0
$629$	25.8957	1.03253
$630$	0	0
$631$	4.26453	0.169768	0.0848842	−	0.996391i	$-0.472948\pi$
$631$	4.26453	0.169768	0.0848842	+	0.996391i	$0.472948\pi$
$632$	0	0
$633$	25.1299	0.998824
$634$	0	0
$635$	−52.8565	−2.09754
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−0.715378	−0.0282999
$640$	0	0
$641$	45.3501	1.79122	0.895612	−	0.444837i	$-0.146738\pi$
$641$	45.3501	1.79122	0.895612	+	0.444837i	$0.146738\pi$
$642$	0	0
$643$	16.3559	0.645015	0.322507	−	0.946567i	$-0.395474\pi$
$643$	16.3559	0.645015	0.322507	+	0.946567i	$0.395474\pi$
$644$	0	0
$645$	52.4618	2.06568
$646$	0	0
$647$	24.3450	0.957100	0.478550	−	0.878060i	$-0.341162\pi$
$647$	24.3450	0.957100	0.478550	+	0.878060i	$0.341162\pi$
$648$	0	0
$649$	13.9331	0.546922
$650$	0	0
$651$	15.9491	0.625093
$652$	0	0
$653$	22.9254	0.897139	0.448570	−	0.893748i	$-0.351934\pi$
$653$	22.9254	0.897139	0.448570	+	0.893748i	$0.351934\pi$
$654$	0	0
$655$	−21.8255	−0.852795
$656$	0	0
$657$	4.46766	0.174300
$658$	0	0
$659$	12.6793	0.493917	0.246958	−	0.969026i	$-0.420569\pi$
$659$	12.6793	0.493917	0.246958	+	0.969026i	$0.420569\pi$
$660$	0	0
$661$	−21.4461	−0.834158	−0.417079	−	0.908870i	$-0.636946\pi$
$661$	−21.4461	−0.834158	−0.417079	+	0.908870i	$0.636946\pi$
$662$	0	0
$663$	5.46881	0.212391
$664$	0	0
$665$	−10.3253	−0.400399
$666$	0	0
$667$	−34.4471	−1.33380
$668$	0	0
$669$	33.2530	1.28563
$670$	0	0
$671$	−13.3801	−0.516531
$672$	0	0
$673$	−27.7282	−1.06884	−0.534422	−	0.845218i	$-0.679471\pi$
$673$	−27.7282	−1.06884	−0.534422	+	0.845218i	$0.679471\pi$
$674$	0	0
$675$	−43.4940	−1.67409
$676$	0	0
$677$	−17.5817	−0.675718	−0.337859	−	0.941197i	$-0.609703\pi$
$677$	−17.5817	−0.675718	−0.337859	+	0.941197i	$0.609703\pi$
$678$	0	0
$679$	−9.87800	−0.379083
$680$	0	0
$681$	−4.87799	−0.186925
$682$	0	0
$683$	−33.4717	−1.28076	−0.640379	−	0.768059i	$-0.721223\pi$
$683$	−33.4717	−1.28076	−0.640379	+	0.768059i	$0.721223\pi$
$684$	0	0
$685$	−54.9304	−2.09878
$686$	0	0
$687$	28.6166	1.09179
$688$	0	0
$689$	−8.47308	−0.322799
$690$	0	0
$691$	2.69505	0.102525	0.0512623	−	0.998685i	$-0.483676\pi$
$691$	2.69505	0.102525	0.0512623	+	0.998685i	$0.483676\pi$
$692$	0	0
$693$	−0.496866	−0.0188744
$694$	0	0
$695$	−29.7778	−1.12954
$696$	0	0
$697$	−8.27761	−0.313537
$698$	0	0
$699$	−27.6904	−1.04735
$700$	0	0
$701$	−24.8056	−0.936893	−0.468446	−	0.883492i	$-0.655186\pi$
$701$	−24.8056	−0.936893	−0.468446	+	0.883492i	$0.655186\pi$
$702$	0	0
$703$	−21.5692	−0.813498
$704$	0	0
$705$	−21.4156	−0.806557
$706$	0	0
$707$	−8.07489	−0.303687
$708$	0	0
$709$	10.3779	0.389749	0.194875	−	0.980828i	$-0.437570\pi$
$709$	10.3779	0.389749	0.194875	+	0.980828i	$0.437570\pi$
$710$	0	0
$711$	−6.29010	−0.235897
$712$	0	0
$713$	37.9776	1.42227
$714$	0	0
$715$	3.58630	0.134120
$716$	0	0
$717$	8.13086	0.303653
$718$	0	0
$719$	21.0878	0.786444	0.393222	−	0.919444i	$-0.371360\pi$
$719$	21.0878	0.786444	0.393222	+	0.919444i	$0.371360\pi$
$720$	0	0
$721$	17.9925	0.670078
$722$	0	0
$723$	−20.2004	−0.751260
$724$	0	0
$725$	71.8832	2.66967
$726$	0	0
$727$	−12.0276	−0.446078	−0.223039	−	0.974809i	$-0.571598\pi$
$727$	−12.0276	−0.446078	−0.223039	+	0.974809i	$0.571598\pi$
$728$	0	0
$729$	29.8679	1.10622
$730$	0	0
$731$	−31.9599	−1.18208
$732$	0	0
$733$	−32.9021	−1.21527	−0.607633	−	0.794218i	$-0.707881\pi$
$733$	−32.9021	−1.21527	−0.607633	+	0.794218i	$0.707881\pi$
$734$	0	0
$735$	−5.67399	−0.209288
$736$	0	0
$737$	4.43386	0.163323
$738$	0	0
$739$	−16.0613	−0.590823	−0.295411	−	0.955370i	$-0.595457\pi$
$739$	−16.0613	−0.590823	−0.295411	+	0.955370i	$0.595457\pi$
$740$	0	0
$741$	−4.55511	−0.167336
$742$	0	0
$743$	−14.4625	−0.530579	−0.265290	−	0.964169i	$-0.585468\pi$
$743$	−14.4625	−0.530579	−0.265290	+	0.964169i	$0.585468\pi$
$744$	0	0
$745$	63.5648	2.32884
$746$	0	0
$747$	0.348044	0.0127343
$748$	0	0
$749$	−14.9866	−0.547600
$750$	0	0
$751$	25.9241	0.945984	0.472992	−	0.881067i	$-0.343174\pi$
$751$	25.9241	0.945984	0.472992	+	0.881067i	$0.343174\pi$
$752$	0	0
$753$	−37.5026	−1.36667
$754$	0	0
$755$	41.2216	1.50021
$756$	0	0
$757$	−16.6084	−0.603642	−0.301821	−	0.953365i	$-0.597595\pi$
$757$	−16.6084	−0.603642	−0.301821	+	0.953365i	$0.597595\pi$
$758$	0	0
$759$	5.96041	0.216349
$760$	0	0
$761$	5.06636	0.183655	0.0918277	−	0.995775i	$-0.470729\pi$
$761$	5.06636	0.183655	0.0918277	+	0.995775i	$0.470729\pi$
$762$	0	0
$763$	−5.31169	−0.192296
$764$	0	0
$765$	6.15938	0.222693
$766$	0	0
$767$	−13.9331	−0.503095
$768$	0	0
$769$	19.8314	0.715137	0.357569	−	0.933887i	$-0.383606\pi$
$769$	19.8314	0.715137	0.357569	+	0.933887i	$0.383606\pi$
$770$	0	0
$771$	−14.0584	−0.506302
$772$	0	0
$773$	−40.9626	−1.47332	−0.736662	−	0.676261i	$-0.763599\pi$
$773$	−40.9626	−1.47332	−0.736662	+	0.676261i	$0.763599\pi$
$774$	0	0
$775$	−79.2505	−2.84676
$776$	0	0
$777$	−11.8528	−0.425215
$778$	0	0
$779$	6.89463	0.247026
$780$	0	0
$781$	−1.43978	−0.0515193
$782$	0	0
$783$	−50.5871	−1.80783
$784$	0	0
$785$	31.9684	1.14100
$786$	0	0
$787$	−51.8557	−1.84846	−0.924229	−	0.381839i	$-0.875291\pi$
$787$	−51.8557	−1.84846	−0.924229	+	0.381839i	$0.875291\pi$
$788$	0	0
$789$	45.5950	1.62323
$790$	0	0
$791$	−9.99829	−0.355498
$792$	0	0
$793$	13.3801	0.475140
$794$	0	0
$795$	48.0762	1.70509
$796$	0	0
$797$	42.4721	1.50444	0.752219	−	0.658913i	$-0.228984\pi$
$797$	42.4721	1.50444	0.752219	+	0.658913i	$0.228984\pi$
$798$	0	0
$799$	13.0464	0.461550
$800$	0	0
$801$	−1.68325	−0.0594748
$802$	0	0
$803$	8.99167	0.317309
$804$	0	0
$805$	−13.5108	−0.476193
$806$	0	0
$807$	13.5523	0.477064
$808$	0	0
$809$	−7.84433	−0.275792	−0.137896	−	0.990447i	$-0.544034\pi$
$809$	−7.84433	−0.275792	−0.137896	+	0.990447i	$0.544034\pi$
$810$	0	0
$811$	−37.8324	−1.32848	−0.664238	−	0.747521i	$-0.731244\pi$
$811$	−37.8324	−1.32848	−0.664238	+	0.747521i	$0.731244\pi$
$812$	0	0
$813$	17.8128	0.624724
$814$	0	0
$815$	−21.3803	−0.748919
$816$	0	0
$817$	26.6202	0.931324
$818$	0	0
$819$	0.496866	0.0173619
$820$	0	0
$821$	1.45200	0.0506751	0.0253376	−	0.999679i	$-0.491934\pi$
$821$	1.45200	0.0506751	0.0253376	+	0.999679i	$0.491934\pi$
$822$	0	0
$823$	−3.85478	−0.134369	−0.0671845	−	0.997741i	$-0.521402\pi$
$823$	−3.85478	−0.134369	−0.0671845	+	0.997741i	$0.521402\pi$
$824$	0	0
$825$	−12.4380	−0.433036
$826$	0	0
$827$	41.4597	1.44170	0.720848	−	0.693094i	$-0.243753\pi$
$827$	41.4597	1.44170	0.720848	+	0.693094i	$0.243753\pi$
$828$	0	0
$829$	−15.6302	−0.542858	−0.271429	−	0.962458i	$-0.587496\pi$
$829$	−15.6302	−0.542858	−0.271429	+	0.962458i	$0.587496\pi$
$830$	0	0
$831$	24.0396	0.833925
$832$	0	0
$833$	3.45661	0.119765
$834$	0	0
$835$	−67.8586	−2.34835
$836$	0	0
$837$	55.7718	1.92775
$838$	0	0
$839$	−16.7465	−0.578152	−0.289076	−	0.957306i	$-0.593348\pi$
$839$	−16.7465	−0.578152	−0.289076	+	0.957306i	$0.593348\pi$
$840$	0	0
$841$	54.6059	1.88296
$842$	0	0
$843$	19.1706	0.660269
$844$	0	0
$845$	−3.58630	−0.123373
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	0	0
$849$	−35.8831	−1.23150
$850$	0	0
$851$	−28.2235	−0.967491
$852$	0	0
$853$	41.0912	1.40694	0.703469	−	0.710726i	$-0.251633\pi$
$853$	41.0912	1.40694	0.703469	+	0.710726i	$0.251633\pi$
$854$	0	0
$855$	−5.13031	−0.175453
$856$	0	0
$857$	3.51486	0.120065	0.0600326	−	0.998196i	$-0.480880\pi$
$857$	3.51486	0.120065	0.0600326	+	0.998196i	$0.480880\pi$
$858$	0	0
$859$	5.02578	0.171477	0.0857387	−	0.996318i	$-0.472675\pi$
$859$	5.02578	0.171477	0.0857387	+	0.996318i	$0.472675\pi$
$860$	0	0
$861$	3.78875	0.129120
$862$	0	0
$863$	10.4144	0.354510	0.177255	−	0.984165i	$-0.443278\pi$
$863$	10.4144	0.354510	0.177255	+	0.984165i	$0.443278\pi$
$864$	0	0
$865$	−66.4169	−2.25824
$866$	0	0
$867$	−7.99264	−0.271444
$868$	0	0
$869$	−12.6595	−0.429446
$870$	0	0
$871$	−4.43386	−0.150236
$872$	0	0
$873$	−4.90805	−0.166112
$874$	0	0
$875$	10.2624	0.346933
$876$	0	0
$877$	−28.3820	−0.958391	−0.479196	−	0.877708i	$-0.659072\pi$
$877$	−28.3820	−0.958391	−0.479196	+	0.877708i	$0.659072\pi$
$878$	0	0
$879$	−5.26038	−0.177428
$880$	0	0
$881$	−25.7491	−0.867509	−0.433755	−	0.901031i	$-0.642811\pi$
$881$	−25.7491	−0.867509	−0.433755	+	0.901031i	$0.642811\pi$
$882$	0	0
$883$	6.93457	0.233367	0.116683	−	0.993169i	$-0.462774\pi$
$883$	6.93457	0.233367	0.116683	+	0.993169i	$0.462774\pi$
$884$	0	0
$885$	79.0563	2.65745
$886$	0	0
$887$	1.81750	0.0610257	0.0305128	−	0.999534i	$-0.490286\pi$
$887$	1.81750	0.0610257	0.0305128	+	0.999534i	$0.490286\pi$
$888$	0	0
$889$	−14.7384	−0.494311
$890$	0	0
$891$	7.26252	0.243304
$892$	0	0
$893$	−10.8667	−0.363640
$894$	0	0
$895$	19.4748	0.650970
$896$	0	0
$897$	−5.96041	−0.199012
$898$	0	0
$899$	−92.1747	−3.07420
$900$	0	0
$901$	−29.2882	−0.975730
$902$	0	0
$903$	14.6284	0.486803
$904$	0	0
$905$	−72.9456	−2.42479
$906$	0	0
$907$	−33.1955	−1.10224	−0.551119	−	0.834427i	$-0.685799\pi$
$907$	−33.1955	−1.10224	−0.551119	+	0.834427i	$0.685799\pi$
$908$	0	0
$909$	−4.01214	−0.133074
$910$	0	0
$911$	6.08052	0.201457	0.100728	−	0.994914i	$-0.467883\pi$
$911$	6.08052	0.201457	0.100728	+	0.994914i	$0.467883\pi$
$912$	0	0
$913$	0.700479	0.0231825
$914$	0	0
$915$	−75.9184	−2.50978
$916$	0	0
$917$	−6.08581	−0.200971
$918$	0	0
$919$	8.64342	0.285120	0.142560	−	0.989786i	$-0.454467\pi$
$919$	8.64342	0.285120	0.142560	+	0.989786i	$0.454467\pi$
$920$	0	0
$921$	−25.3225	−0.834406
$922$	0	0
$923$	1.43978	0.0473909
$924$	0	0
$925$	58.8960	1.93649
$926$	0	0
$927$	8.93989	0.293625
$928$	0	0
$929$	40.8783	1.34117	0.670587	−	0.741831i	$-0.266042\pi$
$929$	40.8783	1.34117	0.670587	+	0.741831i	$0.266042\pi$
$930$	0	0
$931$	−2.87910	−0.0943587
$932$	0	0
$933$	2.62182	0.0858345
$934$	0	0
$935$	12.3965	0.405407
$936$	0	0
$937$	−2.21726	−0.0724349	−0.0362174	−	0.999344i	$-0.511531\pi$
$937$	−2.21726	−0.0724349	−0.0362174	+	0.999344i	$0.511531\pi$
$938$	0	0
$939$	51.8283	1.69135
$940$	0	0
$941$	−28.5392	−0.930351	−0.465176	−	0.885218i	$-0.654009\pi$
$941$	−28.5392	−0.930351	−0.465176	+	0.885218i	$0.654009\pi$
$942$	0	0
$943$	9.02170	0.293787
$944$	0	0
$945$	−19.8412	−0.645434
$946$	0	0
$947$	23.9386	0.777900	0.388950	−	0.921259i	$-0.372838\pi$
$947$	23.9386	0.777900	0.388950	+	0.921259i	$0.372838\pi$
$948$	0	0
$949$	−8.99167	−0.291882
$950$	0	0
$951$	−11.2916	−0.366155
$952$	0	0
$953$	−9.62062	−0.311642	−0.155821	−	0.987785i	$-0.549802\pi$
$953$	−9.62062	−0.311642	−0.155821	+	0.987785i	$0.549802\pi$
$954$	0	0
$955$	31.1121	1.00676
$956$	0	0
$957$	−14.4664	−0.467632
$958$	0	0
$959$	−15.3167	−0.494603
$960$	0	0
$961$	70.6217	2.27812
$962$	0	0
$963$	−7.44636	−0.239956
$964$	0	0
$965$	−60.2693	−1.94014
$966$	0	0
$967$	−22.3083	−0.717386	−0.358693	−	0.933456i	$-0.616778\pi$
$967$	−22.3083	−0.717386	−0.358693	+	0.933456i	$0.616778\pi$
$968$	0	0
$969$	−15.7453	−0.505811
$970$	0	0
$971$	16.2238	0.520647	0.260324	−	0.965521i	$-0.416171\pi$
$971$	16.2238	0.520647	0.260324	+	0.965521i	$0.416171\pi$
$972$	0	0
$973$	−8.30321	−0.266189
$974$	0	0
$975$	12.4380	0.398335
$976$	0	0
$977$	26.5155	0.848305	0.424153	−	0.905591i	$-0.360572\pi$
$977$	26.5155	0.848305	0.424153	+	0.905591i	$0.360572\pi$
$978$	0	0
$979$	−3.38773	−0.108272
$980$	0	0
$981$	−2.63920	−0.0842632
$982$	0	0
$983$	5.30488	0.169199	0.0845997	−	0.996415i	$-0.473039\pi$
$983$	5.30488	0.169199	0.0845997	+	0.996415i	$0.473039\pi$
$984$	0	0
$985$	−60.0587	−1.91363
$986$	0	0
$987$	−5.97149	−0.190075
$988$	0	0
$989$	34.8328	1.10762
$990$	0	0
$991$	4.23261	0.134453	0.0672266	−	0.997738i	$-0.478585\pi$
$991$	4.23261	0.134453	0.0672266	+	0.997738i	$0.478585\pi$
$992$	0	0
$993$	−7.86245	−0.249507
$994$	0	0
$995$	35.8238	1.13569
$996$	0	0
$997$	6.22251	0.197069	0.0985345	−	0.995134i	$-0.468585\pi$
$997$	6.22251	0.197069	0.0985345	+	0.995134i	$0.468585\pi$
$998$	0	0
$999$	−41.4475	−1.31134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.z.1.11	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.z.1.11	✓	15	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.58213 q^{3} -3.58630 q^{5} -1.00000 q^{7} -0.496866 q^{9} +O(q^{10})\)	\(q+1.58213 q^{3} -3.58630 q^{5} -1.00000 q^{7} -0.496866 q^{9} -1.00000 q^{11} +1.00000 q^{13} -5.67399 q^{15} +3.45661 q^{17} -2.87910 q^{19} -1.58213 q^{21} -3.76734 q^{23} +7.86156 q^{25} -5.53250 q^{27} +9.14363 q^{29} -10.0808 q^{31} -1.58213 q^{33} +3.58630 q^{35} +7.49165 q^{37} +1.58213 q^{39} -2.39472 q^{41} -9.24602 q^{43} +1.78191 q^{45} +3.77434 q^{47} +1.00000 q^{49} +5.46881 q^{51} -8.47308 q^{53} +3.58630 q^{55} -4.55511 q^{57} -13.9331 q^{59} +13.3801 q^{61} +0.496866 q^{63} -3.58630 q^{65} -4.43386 q^{67} -5.96041 q^{69} +1.43978 q^{71} -8.99167 q^{73} +12.4380 q^{75} +1.00000 q^{77} +12.6595 q^{79} -7.26252 q^{81} -0.700479 q^{83} -12.3965 q^{85} +14.4664 q^{87} +3.38773 q^{89} -1.00000 q^{91} -15.9491 q^{93} +10.3253 q^{95} +9.87800 q^{97} +0.496866 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9}+O(q^{10}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9	\( 15 q - q^{3} + 4 q^{5} - 15 q^{7} + 26 q^{9} - 15 q^{11} + 15 q^{13} - 6 q^{15} + 8 q^{17} - 17 q^{19} + q^{21} + 7 q^{23} + 33 q^{25} - 4 q^{27} + 14 q^{29} - 4 q^{31} + q^{33} - 4 q^{35} + 3 q^{37} - q^{39} - 13 q^{43} + 20 q^{45} + 6 q^{47} + 15 q^{49} + 8 q^{51} + 38 q^{53} - 4 q^{55} + 24 q^{57} - 18 q^{59} + 23 q^{61} - 26 q^{63} + 4 q^{65} - 8 q^{67} + 43 q^{69} - 12 q^{71} + 11 q^{73} + 12 q^{75} + 15 q^{77} - q^{79} + 51 q^{81} - 16 q^{83} + 13 q^{85} - 25 q^{87} + 28 q^{89} - 15 q^{91} - 14 q^{93} + 49 q^{95} + 30 q^{97} - 26 q^{99}+O(q^{100}) \) 15 * q - q^3 + 4 * q^5 - 15 * q^7 + 26 * q^9 - 15 * q^11 + 15 * q^13 - 6 * q^15 + 8 * q^17 - 17 * q^19 + q^21 + 7 * q^23 + 33 * q^25 - 4 * q^27 + 14 * q^29 - 4 * q^31 + q^33 - 4 * q^35 + 3 * q^37 - q^39 - 13 * q^43 + 20 * q^45 + 6 * q^47 + 15 * q^49 + 8 * q^51 + 38 * q^53 - 4 * q^55 + 24 * q^57 - 18 * q^59 + 23 * q^61 - 26 * q^63 + 4 * q^65 - 8 * q^67 + 43 * q^69 - 12 * q^71 + 11 * q^73 + 12 * q^75 + 15 * q^77 - q^79 + 51 * q^81 - 16 * q^83 + 13 * q^85 - 25 * q^87 + 28 * q^89 - 15 * q^91 - 14 * q^93 + 49 * q^95 + 30 * q^97 - 26 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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