LMFDB - Embedded newform 8008.2.a.k.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8008.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8008.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

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

Embedding invariants

Embedding label			1.4
Root			$-0.676795$ 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)

$f(q)$	$=$	$q+0.676795 q^{3} +3.59312 q^{5} +1.00000 q^{7} -2.54195 q^{9} +O(q^{10})$	$q+0.676795 q^{3} +3.59312 q^{5} +1.00000 q^{7} -2.54195 q^{9} -1.00000 q^{11} -1.00000 q^{13} +2.43181 q^{15} -5.62384 q^{17} -5.78115 q^{19} +0.676795 q^{21} +8.70173 q^{23} +7.91054 q^{25} -3.75076 q^{27} +9.58734 q^{29} +3.11439 q^{31} -0.676795 q^{33} +3.59312 q^{35} -5.39931 q^{37} -0.676795 q^{39} +8.94093 q^{41} -4.02493 q^{43} -9.13354 q^{45} +1.80586 q^{47} +1.00000 q^{49} -3.80619 q^{51} -4.67067 q^{53} -3.59312 q^{55} -3.91265 q^{57} +3.19237 q^{59} +9.66345 q^{61} -2.54195 q^{63} -3.59312 q^{65} +11.3992 q^{67} +5.88929 q^{69} -4.18836 q^{71} +12.6610 q^{73} +5.35382 q^{75} -1.00000 q^{77} +11.1859 q^{79} +5.08735 q^{81} -0.112281 q^{83} -20.2072 q^{85} +6.48866 q^{87} +10.6256 q^{89} -1.00000 q^{91} +2.10780 q^{93} -20.7724 q^{95} +16.7285 q^{97} +2.54195 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	$ 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) $ 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * q^97 - 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0.676795	0.390748	0.195374	−	0.980729i	$-0.437408\pi$
$3$	0.676795	0.390748	0.195374	+	0.980729i	$0.437408\pi$
$4$	0	0
$5$	3.59312	1.60689	0.803447	−	0.595376i	$-0.202997\pi$
$5$	3.59312	1.60689	0.803447	+	0.595376i	$0.202997\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.54195	−0.847316
$10$	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
$15$	2.43181	0.627890
$16$	0	0
$17$	−5.62384	−1.36398	−0.681991	−	0.731360i	$-0.738886\pi$
$17$	−5.62384	−1.36398	−0.681991	+	0.731360i	$0.738886\pi$
$18$	0	0
$19$	−5.78115	−1.32629	−0.663143	−	0.748493i	$-0.730778\pi$
$19$	−5.78115	−1.32629	−0.663143	+	0.748493i	$0.730778\pi$
$20$	0	0
$21$	0.676795	0.147689
$22$	0	0
$23$	8.70173	1.81444	0.907218	−	0.420661i	$-0.138202\pi$
$23$	8.70173	1.81444	0.907218	+	0.420661i	$0.138202\pi$
$24$	0	0
$25$	7.91054	1.58211
$26$	0	0
$27$	−3.75076	−0.721835
$28$	0	0
$29$	9.58734	1.78032	0.890162	−	0.455644i	$-0.150591\pi$
$29$	9.58734	1.78032	0.890162	+	0.455644i	$0.150591\pi$
$30$	0	0
$31$	3.11439	0.559361	0.279680	−	0.960093i	$-0.409771\pi$
$31$	3.11439	0.559361	0.279680	+	0.960093i	$0.409771\pi$
$32$	0	0
$33$	−0.676795	−0.117815
$34$	0	0
$35$	3.59312	0.607349
$36$	0	0
$37$	−5.39931	−0.887642	−0.443821	−	0.896116i	$-0.646377\pi$
$37$	−5.39931	−0.887642	−0.443821	+	0.896116i	$0.646377\pi$
$38$	0	0
$39$	−0.676795	−0.108374
$40$	0	0
$41$	8.94093	1.39634	0.698169	−	0.715933i	$-0.253998\pi$
$41$	8.94093	1.39634	0.698169	+	0.715933i	$0.253998\pi$
$42$	0	0
$43$	−4.02493	−0.613797	−0.306898	−	0.951742i	$-0.599291\pi$
$43$	−4.02493	−0.613797	−0.306898	+	0.951742i	$0.599291\pi$
$44$	0	0
$45$	−9.13354	−1.36155
$46$	0	0
$47$	1.80586	0.263411	0.131706	−	0.991289i	$-0.457955\pi$
$47$	1.80586	0.263411	0.131706	+	0.991289i	$0.457955\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.80619	−0.532973
$52$	0	0
$53$	−4.67067	−0.641566	−0.320783	−	0.947153i	$-0.603946\pi$
$53$	−4.67067	−0.641566	−0.320783	+	0.947153i	$0.603946\pi$
$54$	0	0
$55$	−3.59312	−0.484497
$56$	0	0
$57$	−3.91265	−0.518244
$58$	0	0
$59$	3.19237	0.415611	0.207806	−	0.978170i	$-0.433368\pi$
$59$	3.19237	0.415611	0.207806	+	0.978170i	$0.433368\pi$
$60$	0	0
$61$	9.66345	1.23728	0.618639	−	0.785676i	$-0.287684\pi$
$61$	9.66345	1.23728	0.618639	+	0.785676i	$0.287684\pi$
$62$	0	0
$63$	−2.54195	−0.320255
$64$	0	0
$65$	−3.59312	−0.445672
$66$	0	0
$67$	11.3992	1.39263	0.696317	−	0.717734i	$-0.254821\pi$
$67$	11.3992	1.39263	0.696317	+	0.717734i	$0.254821\pi$
$68$	0	0
$69$	5.88929	0.708987
$70$	0	0
$71$	−4.18836	−0.497067	−0.248533	−	0.968623i	$-0.579949\pi$
$71$	−4.18836	−0.497067	−0.248533	+	0.968623i	$0.579949\pi$
$72$	0	0
$73$	12.6610	1.48185	0.740927	−	0.671585i	$-0.234386\pi$
$73$	12.6610	1.48185	0.740927	+	0.671585i	$0.234386\pi$
$74$	0	0
$75$	5.35382	0.618205
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	11.1859	1.25851	0.629257	−	0.777197i	$-0.283359\pi$
$79$	11.1859	1.25851	0.629257	+	0.777197i	$0.283359\pi$
$80$	0	0
$81$	5.08735	0.565261
$82$	0	0
$83$	−0.112281	−0.0123244	−0.00616220	−	0.999981i	$-0.501962\pi$
$83$	−0.112281	−0.0123244	−0.00616220	+	0.999981i	$0.501962\pi$
$84$	0	0
$85$	−20.2072	−2.19178
$86$	0	0
$87$	6.48866	0.695658
$88$	0	0
$89$	10.6256	1.12631	0.563157	−	0.826350i	$-0.309587\pi$
$89$	10.6256	1.12631	0.563157	+	0.826350i	$0.309587\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	2.10780	0.218569
$94$	0	0
$95$	−20.7724	−2.13120
$96$	0	0
$97$	16.7285	1.69853	0.849263	−	0.527971i	$-0.177047\pi$
$97$	16.7285	1.69853	0.849263	+	0.527971i	$0.177047\pi$
$98$	0	0
$99$	2.54195	0.255475
$100$	0	0
$101$	−17.2870	−1.72012	−0.860058	−	0.510196i	$-0.829573\pi$
$101$	−17.2870	−1.72012	−0.860058	+	0.510196i	$0.829573\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	2.43181	0.237320
$106$	0	0
$107$	−5.79615	−0.560335	−0.280168	−	0.959951i	$-0.590390\pi$
$107$	−5.79615	−0.560335	−0.280168	+	0.959951i	$0.590390\pi$
$108$	0	0
$109$	−2.19381	−0.210129	−0.105064	−	0.994465i	$-0.533505\pi$
$109$	−2.19381	−0.210129	−0.105064	+	0.994465i	$0.533505\pi$
$110$	0	0
$111$	−3.65423	−0.346844
$112$	0	0
$113$	8.35215	0.785704	0.392852	−	0.919602i	$-0.371488\pi$
$113$	8.35215	0.785704	0.392852	+	0.919602i	$0.371488\pi$
$114$	0	0
$115$	31.2664	2.91561
$116$	0	0
$117$	2.54195	0.235003
$118$	0	0
$119$	−5.62384	−0.515537
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	6.05118	0.545616
$124$	0	0
$125$	10.4579	0.935387
$126$	0	0
$127$	11.6650	1.03510	0.517550	−	0.855653i	$-0.326844\pi$
$127$	11.6650	1.03510	0.517550	+	0.855653i	$0.326844\pi$
$128$	0	0
$129$	−2.72405	−0.239840
$130$	0	0
$131$	10.6828	0.933361	0.466681	−	0.884426i	$-0.345450\pi$
$131$	10.6828	0.933361	0.466681	+	0.884426i	$0.345450\pi$
$132$	0	0
$133$	−5.78115	−0.501289
$134$	0	0
$135$	−13.4770	−1.15991
$136$	0	0
$137$	9.11439	0.778695	0.389347	−	0.921091i	$-0.372701\pi$
$137$	9.11439	0.778695	0.389347	+	0.921091i	$0.372701\pi$
$138$	0	0
$139$	−3.83266	−0.325082	−0.162541	−	0.986702i	$-0.551969\pi$
$139$	−3.83266	−0.325082	−0.162541	+	0.986702i	$0.551969\pi$
$140$	0	0
$141$	1.22219	0.102927
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	34.4485	2.86079
$146$	0	0
$147$	0.676795	0.0558211
$148$	0	0
$149$	8.90099	0.729198	0.364599	−	0.931165i	$-0.381206\pi$
$149$	8.90099	0.729198	0.364599	+	0.931165i	$0.381206\pi$
$150$	0	0
$151$	−4.34140	−0.353298	−0.176649	−	0.984274i	$-0.556526\pi$
$151$	−4.34140	−0.353298	−0.176649	+	0.984274i	$0.556526\pi$
$152$	0	0
$153$	14.2955	1.15572
$154$	0	0
$155$	11.1904	0.898834
$156$	0	0
$157$	10.0712	0.803768	0.401884	−	0.915690i	$-0.368355\pi$
$157$	10.0712	0.803768	0.401884	+	0.915690i	$0.368355\pi$
$158$	0	0
$159$	−3.16109	−0.250691
$160$	0	0
$161$	8.70173	0.685792
$162$	0	0
$163$	3.80552	0.298071	0.149036	−	0.988832i	$-0.452383\pi$
$163$	3.80552	0.298071	0.149036	+	0.988832i	$0.452383\pi$
$164$	0	0
$165$	−2.43181	−0.189316
$166$	0	0
$167$	15.8820	1.22898	0.614492	−	0.788923i	$-0.289361\pi$
$167$	15.8820	1.22898	0.614492	+	0.788923i	$0.289361\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	14.6954	1.12378
$172$	0	0
$173$	−10.4883	−0.797413	−0.398706	−	0.917079i	$-0.630541\pi$
$173$	−10.4883	−0.797413	−0.398706	+	0.917079i	$0.630541\pi$
$174$	0	0
$175$	7.91054	0.597981
$176$	0	0
$177$	2.16058	0.162399
$178$	0	0
$179$	12.2756	0.917521	0.458761	−	0.888560i	$-0.348294\pi$
$179$	12.2756	0.917521	0.458761	+	0.888560i	$0.348294\pi$
$180$	0	0
$181$	20.1320	1.49640	0.748199	−	0.663474i	$-0.230919\pi$
$181$	20.1320	1.49640	0.748199	+	0.663474i	$0.230919\pi$
$182$	0	0
$183$	6.54017	0.483464
$184$	0	0
$185$	−19.4004	−1.42635
$186$	0	0
$187$	5.62384	0.411256
$188$	0	0
$189$	−3.75076	−0.272828
$190$	0	0
$191$	−9.80150	−0.709211	−0.354606	−	0.935016i	$-0.615385\pi$
$191$	−9.80150	−0.709211	−0.354606	+	0.935016i	$0.615385\pi$
$192$	0	0
$193$	3.00779	0.216505	0.108253	−	0.994123i	$-0.465474\pi$
$193$	3.00779	0.216505	0.108253	+	0.994123i	$0.465474\pi$
$194$	0	0
$195$	−2.43181	−0.174145
$196$	0	0
$197$	−16.2336	−1.15660	−0.578299	−	0.815825i	$-0.696283\pi$
$197$	−16.2336	−1.15660	−0.578299	+	0.815825i	$0.696283\pi$
$198$	0	0
$199$	−18.8354	−1.33520	−0.667602	−	0.744519i	$-0.732679\pi$
$199$	−18.8354	−1.33520	−0.667602	+	0.744519i	$0.732679\pi$
$200$	0	0
$201$	7.71493	0.544169
$202$	0	0
$203$	9.58734	0.672899
$204$	0	0
$205$	32.1259	2.24377
$206$	0	0
$207$	−22.1193	−1.53740
$208$	0	0
$209$	5.78115	0.399890
$210$	0	0
$211$	−26.6591	−1.83529	−0.917644	−	0.397403i	$-0.869911\pi$
$211$	−26.6591	−1.83529	−0.917644	+	0.397403i	$0.869911\pi$
$212$	0	0
$213$	−2.83466	−0.194228
$214$	0	0
$215$	−14.4621	−0.986306
$216$	0	0
$217$	3.11439	0.211419
$218$	0	0
$219$	8.56888	0.579031
$220$	0	0
$221$	5.62384	0.378301
$222$	0	0
$223$	−10.1514	−0.679789	−0.339894	−	0.940464i	$-0.610391\pi$
$223$	−10.1514	−0.679789	−0.339894	+	0.940464i	$0.610391\pi$
$224$	0	0
$225$	−20.1082	−1.34055
$226$	0	0
$227$	−21.5691	−1.43159	−0.715796	−	0.698309i	$-0.753936\pi$
$227$	−21.5691	−1.43159	−0.715796	+	0.698309i	$0.753936\pi$
$228$	0	0
$229$	−2.89330	−0.191194	−0.0955972	−	0.995420i	$-0.530476\pi$
$229$	−2.89330	−0.191194	−0.0955972	+	0.995420i	$0.530476\pi$
$230$	0	0
$231$	−0.676795	−0.0445298
$232$	0	0
$233$	15.3688	1.00684	0.503422	−	0.864040i	$-0.332074\pi$
$233$	15.3688	1.00684	0.503422	+	0.864040i	$0.332074\pi$
$234$	0	0
$235$	6.48866	0.423274
$236$	0	0
$237$	7.57057	0.491761
$238$	0	0
$239$	−18.1509	−1.17408	−0.587042	−	0.809556i	$-0.699708\pi$
$239$	−18.1509	−1.17408	−0.587042	+	0.809556i	$0.699708\pi$
$240$	0	0
$241$	−3.04429	−0.196100	−0.0980501	−	0.995181i	$-0.531261\pi$
$241$	−3.04429	−0.196100	−0.0980501	+	0.995181i	$0.531261\pi$
$242$	0	0
$243$	14.6954	0.942709
$244$	0	0
$245$	3.59312	0.229556
$246$	0	0
$247$	5.78115	0.367846
$248$	0	0
$249$	−0.0759910	−0.00481573
$250$	0	0
$251$	−15.8103	−0.997939	−0.498970	−	0.866619i	$-0.666288\pi$
$251$	−15.8103	−0.997939	−0.498970	+	0.866619i	$0.666288\pi$
$252$	0	0
$253$	−8.70173	−0.547073
$254$	0	0
$255$	−13.6761	−0.856431
$256$	0	0
$257$	−8.07424	−0.503657	−0.251829	−	0.967772i	$-0.581032\pi$
$257$	−8.07424	−0.503657	−0.251829	+	0.967772i	$0.581032\pi$
$258$	0	0
$259$	−5.39931	−0.335497
$260$	0	0
$261$	−24.3705	−1.50850
$262$	0	0
$263$	6.60516	0.407292	0.203646	−	0.979045i	$-0.434721\pi$
$263$	6.60516	0.407292	0.203646	+	0.979045i	$0.434721\pi$
$264$	0	0
$265$	−16.7823	−1.03093
$266$	0	0
$267$	7.19137	0.440104
$268$	0	0
$269$	−5.85703	−0.357110	−0.178555	−	0.983930i	$-0.557142\pi$
$269$	−5.85703	−0.357110	−0.178555	+	0.983930i	$0.557142\pi$
$270$	0	0
$271$	−13.6007	−0.826184	−0.413092	−	0.910689i	$-0.635551\pi$
$271$	−13.6007	−0.826184	−0.413092	+	0.910689i	$0.635551\pi$
$272$	0	0
$273$	−0.676795	−0.0409615
$274$	0	0
$275$	−7.91054	−0.477024
$276$	0	0
$277$	0.244652	0.0146997	0.00734985	−	0.999973i	$-0.497660\pi$
$277$	0.244652	0.0146997	0.00734985	+	0.999973i	$0.497660\pi$
$278$	0	0
$279$	−7.91662	−0.473956
$280$	0	0
$281$	−15.7613	−0.940239	−0.470120	−	0.882603i	$-0.655789\pi$
$281$	−15.7613	−0.940239	−0.470120	+	0.882603i	$0.655789\pi$
$282$	0	0
$283$	−21.5368	−1.28023	−0.640116	−	0.768278i	$-0.721114\pi$
$283$	−21.5368	−1.28023	−0.640116	+	0.768278i	$0.721114\pi$
$284$	0	0
$285$	−14.0586	−0.832762
$286$	0	0
$287$	8.94093	0.527766
$288$	0	0
$289$	14.6276	0.860448
$290$	0	0
$291$	11.3218	0.663695
$292$	0	0
$293$	24.6207	1.43836	0.719179	−	0.694825i	$-0.244518\pi$
$293$	24.6207	1.43836	0.719179	+	0.694825i	$0.244518\pi$
$294$	0	0
$295$	11.4706	0.667843
$296$	0	0
$297$	3.75076	0.217641
$298$	0	0
$299$	−8.70173	−0.503234
$300$	0	0
$301$	−4.02493	−0.231993
$302$	0	0
$303$	−11.6997	−0.672132
$304$	0	0
$305$	34.7220	1.98817
$306$	0	0
$307$	−24.0388	−1.37197	−0.685984	−	0.727616i	$-0.740628\pi$
$307$	−24.0388	−1.37197	−0.685984	+	0.727616i	$0.740628\pi$
$308$	0	0
$309$	5.41436	0.308012
$310$	0	0
$311$	−15.6375	−0.886723	−0.443362	−	0.896343i	$-0.646214\pi$
$311$	−15.6375	−0.886723	−0.443362	+	0.896343i	$0.646214\pi$
$312$	0	0
$313$	8.49347	0.480080	0.240040	−	0.970763i	$-0.422840\pi$
$313$	8.49347	0.480080	0.240040	+	0.970763i	$0.422840\pi$
$314$	0	0
$315$	−9.13354	−0.514617
$316$	0	0
$317$	−35.1209	−1.97259	−0.986293	−	0.165005i	$-0.947236\pi$
$317$	−35.1209	−1.97259	−0.986293	+	0.165005i	$0.947236\pi$
$318$	0	0
$319$	−9.58734	−0.536788
$320$	0	0
$321$	−3.92281	−0.218950
$322$	0	0
$323$	32.5123	1.80903
$324$	0	0
$325$	−7.91054	−0.438798
$326$	0	0
$327$	−1.48476	−0.0821074
$328$	0	0
$329$	1.80586	0.0995600
$330$	0	0
$331$	−25.5645	−1.40515	−0.702577	−	0.711608i	$-0.747967\pi$
$331$	−25.5645	−1.40515	−0.702577	+	0.711608i	$0.747967\pi$
$332$	0	0
$333$	13.7248	0.752113
$334$	0	0
$335$	40.9588	2.23782
$336$	0	0
$337$	23.9641	1.30541	0.652703	−	0.757614i	$-0.273635\pi$
$337$	23.9641	1.30541	0.652703	+	0.757614i	$0.273635\pi$
$338$	0	0
$339$	5.65269	0.307012
$340$	0	0
$341$	−3.11439	−0.168654
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	21.1609	1.13927
$346$	0	0
$347$	−31.0331	−1.66595	−0.832973	−	0.553314i	$-0.813363\pi$
$347$	−31.0331	−1.66595	−0.832973	+	0.553314i	$0.813363\pi$
$348$	0	0
$349$	35.0480	1.87608	0.938039	−	0.346530i	$-0.112640\pi$
$349$	35.0480	1.87608	0.938039	+	0.346530i	$0.112640\pi$
$350$	0	0
$351$	3.75076	0.200201
$352$	0	0
$353$	−2.15133	−0.114504	−0.0572519	−	0.998360i	$-0.518234\pi$
$353$	−2.15133	−0.114504	−0.0572519	+	0.998360i	$0.518234\pi$
$354$	0	0
$355$	−15.0493	−0.798734
$356$	0	0
$357$	−3.80619	−0.201445
$358$	0	0
$359$	4.61897	0.243780	0.121890	−	0.992544i	$-0.461105\pi$
$359$	4.61897	0.243780	0.121890	+	0.992544i	$0.461105\pi$
$360$	0	0
$361$	14.4217	0.759036
$362$	0	0
$363$	0.676795	0.0355225
$364$	0	0
$365$	45.4924	2.38118
$366$	0	0
$367$	18.2909	0.954779	0.477390	−	0.878692i	$-0.341583\pi$
$367$	18.2909	0.954779	0.477390	+	0.878692i	$0.341583\pi$
$368$	0	0
$369$	−22.7274	−1.18314
$370$	0	0
$371$	−4.67067	−0.242489
$372$	0	0
$373$	−9.59338	−0.496726	−0.248363	−	0.968667i	$-0.579893\pi$
$373$	−9.59338	−0.496726	−0.248363	+	0.968667i	$0.579893\pi$
$374$	0	0
$375$	7.07788	0.365500
$376$	0	0
$377$	−9.58734	−0.493773
$378$	0	0
$379$	−9.93124	−0.510133	−0.255067	−	0.966923i	$-0.582097\pi$
$379$	−9.93124	−0.510133	−0.255067	+	0.966923i	$0.582097\pi$
$380$	0	0
$381$	7.89480	0.404463
$382$	0	0
$383$	−18.9342	−0.967491	−0.483745	−	0.875209i	$-0.660724\pi$
$383$	−18.9342	−0.967491	−0.483745	+	0.875209i	$0.660724\pi$
$384$	0	0
$385$	−3.59312	−0.183123
$386$	0	0
$387$	10.2312	0.520080
$388$	0	0
$389$	9.41346	0.477281	0.238641	−	0.971108i	$-0.423298\pi$
$389$	9.41346	0.477281	0.238641	+	0.971108i	$0.423298\pi$
$390$	0	0
$391$	−48.9372	−2.47486
$392$	0	0
$393$	7.23007	0.364709
$394$	0	0
$395$	40.1924	2.02230
$396$	0	0
$397$	−1.77670	−0.0891702	−0.0445851	−	0.999006i	$-0.514197\pi$
$397$	−1.77670	−0.0891702	−0.0445851	+	0.999006i	$0.514197\pi$
$398$	0	0
$399$	−3.91265	−0.195878
$400$	0	0
$401$	−5.37013	−0.268172	−0.134086	−	0.990970i	$-0.542810\pi$
$401$	−5.37013	−0.268172	−0.134086	+	0.990970i	$0.542810\pi$
$402$	0	0
$403$	−3.11439	−0.155139
$404$	0	0
$405$	18.2795	0.908314
$406$	0	0
$407$	5.39931	0.267634
$408$	0	0
$409$	−24.0341	−1.18841	−0.594204	−	0.804314i	$-0.702533\pi$
$409$	−24.0341	−1.18841	−0.594204	+	0.804314i	$0.702533\pi$
$410$	0	0
$411$	6.16857	0.304273
$412$	0	0
$413$	3.19237	0.157086
$414$	0	0
$415$	−0.403438	−0.0198040
$416$	0	0
$417$	−2.59392	−0.127025
$418$	0	0
$419$	14.9227	0.729022	0.364511	−	0.931199i	$-0.381236\pi$
$419$	14.9227	0.729022	0.364511	+	0.931199i	$0.381236\pi$
$420$	0	0
$421$	1.18689	0.0578455	0.0289227	−	0.999582i	$-0.490792\pi$
$421$	1.18689	0.0578455	0.0289227	+	0.999582i	$0.490792\pi$
$422$	0	0
$423$	−4.59039	−0.223192
$424$	0	0
$425$	−44.4877	−2.15797
$426$	0	0
$427$	9.66345	0.467647
$428$	0	0
$429$	0.676795	0.0326760
$430$	0	0
$431$	−36.1150	−1.73960	−0.869798	−	0.493407i	$-0.835751\pi$
$431$	−36.1150	−1.73960	−0.869798	+	0.493407i	$0.835751\pi$
$432$	0	0
$433$	−20.0523	−0.963650	−0.481825	−	0.876267i	$-0.660026\pi$
$433$	−20.0523	−0.963650	−0.481825	+	0.876267i	$0.660026\pi$
$434$	0	0
$435$	23.3146	1.11785
$436$	0	0
$437$	−50.3060	−2.40646
$438$	0	0
$439$	13.3196	0.635711	0.317855	−	0.948139i	$-0.397037\pi$
$439$	13.3196	0.635711	0.317855	+	0.948139i	$0.397037\pi$
$440$	0	0
$441$	−2.54195	−0.121045
$442$	0	0
$443$	34.4034	1.63456	0.817278	−	0.576243i	$-0.195482\pi$
$443$	34.4034	1.63456	0.817278	+	0.576243i	$0.195482\pi$
$444$	0	0
$445$	38.1792	1.80987
$446$	0	0
$447$	6.02415	0.284932
$448$	0	0
$449$	16.8444	0.794934	0.397467	−	0.917616i	$-0.369889\pi$
$449$	16.8444	0.794934	0.397467	+	0.917616i	$0.369889\pi$
$450$	0	0
$451$	−8.94093	−0.421012
$452$	0	0
$453$	−2.93823	−0.138050
$454$	0	0
$455$	−3.59312	−0.168448
$456$	0	0
$457$	−35.3150	−1.65197	−0.825983	−	0.563695i	$-0.809379\pi$
$457$	−35.3150	−1.65197	−0.825983	+	0.563695i	$0.809379\pi$
$458$	0	0
$459$	21.0937	0.984570
$460$	0	0
$461$	11.0321	0.513814	0.256907	−	0.966436i	$-0.417297\pi$
$461$	11.0321	0.513814	0.256907	+	0.966436i	$0.417297\pi$
$462$	0	0
$463$	−7.28844	−0.338723	−0.169361	−	0.985554i	$-0.554170\pi$
$463$	−7.28844	−0.338723	−0.169361	+	0.985554i	$0.554170\pi$
$464$	0	0
$465$	7.57360	0.351217
$466$	0	0
$467$	6.03039	0.279053	0.139526	−	0.990218i	$-0.455442\pi$
$467$	6.03039	0.279053	0.139526	+	0.990218i	$0.455442\pi$
$468$	0	0
$469$	11.3992	0.526366
$470$	0	0
$471$	6.81613	0.314071
$472$	0	0
$473$	4.02493	0.185067
$474$	0	0
$475$	−45.7320	−2.09833
$476$	0	0
$477$	11.8726	0.543610
$478$	0	0
$479$	−13.5920	−0.621032	−0.310516	−	0.950568i	$-0.600502\pi$
$479$	−13.5920	−0.621032	−0.310516	+	0.950568i	$0.600502\pi$
$480$	0	0
$481$	5.39931	0.246188
$482$	0	0
$483$	5.88929	0.267972
$484$	0	0
$485$	60.1077	2.72935
$486$	0	0
$487$	7.55386	0.342298	0.171149	−	0.985245i	$-0.445252\pi$
$487$	7.55386	0.342298	0.171149	+	0.985245i	$0.445252\pi$
$488$	0	0
$489$	2.57556	0.116471
$490$	0	0
$491$	18.5708	0.838088	0.419044	−	0.907966i	$-0.362365\pi$
$491$	18.5708	0.838088	0.419044	+	0.907966i	$0.362365\pi$
$492$	0	0
$493$	−53.9177	−2.42833
$494$	0	0
$495$	9.13354	0.410522
$496$	0	0
$497$	−4.18836	−0.187874
$498$	0	0
$499$	25.8928	1.15912	0.579559	−	0.814930i	$-0.303225\pi$
$499$	25.8928	1.15912	0.579559	+	0.814930i	$0.303225\pi$
$500$	0	0
$501$	10.7488	0.480223
$502$	0	0
$503$	3.66225	0.163291	0.0816457	−	0.996661i	$-0.473982\pi$
$503$	3.66225	0.163291	0.0816457	+	0.996661i	$0.473982\pi$
$504$	0	0
$505$	−62.1142	−2.76405
$506$	0	0
$507$	0.676795	0.0300575
$508$	0	0
$509$	32.9982	1.46262	0.731310	−	0.682045i	$-0.238909\pi$
$509$	32.9982	1.46262	0.731310	+	0.682045i	$0.238909\pi$
$510$	0	0
$511$	12.6610	0.560088
$512$	0	0
$513$	21.6837	0.957360
$514$	0	0
$515$	28.7450	1.26666
$516$	0	0
$517$	−1.80586	−0.0794214
$518$	0	0
$519$	−7.09845	−0.311587
$520$	0	0
$521$	28.4607	1.24689	0.623443	−	0.781869i	$-0.285733\pi$
$521$	28.4607	1.24689	0.623443	+	0.781869i	$0.285733\pi$
$522$	0	0
$523$	22.1522	0.968649	0.484324	−	0.874888i	$-0.339065\pi$
$523$	22.1522	0.968649	0.484324	+	0.874888i	$0.339065\pi$
$524$	0	0
$525$	5.35382	0.233660
$526$	0	0
$527$	−17.5148	−0.762959
$528$	0	0
$529$	52.7201	2.29218
$530$	0	0
$531$	−8.11484	−0.352154
$532$	0	0
$533$	−8.94093	−0.387275
$534$	0	0
$535$	−20.8263	−0.900399
$536$	0	0
$537$	8.30806	0.358519
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−24.8923	−1.07020	−0.535101	−	0.844788i	$-0.679726\pi$
$541$	−24.8923	−1.07020	−0.535101	+	0.844788i	$0.679726\pi$
$542$	0	0
$543$	13.6252	0.584715
$544$	0	0
$545$	−7.88263	−0.337655
$546$	0	0
$547$	−18.6048	−0.795485	−0.397743	−	0.917497i	$-0.630206\pi$
$547$	−18.6048	−0.795485	−0.397743	+	0.917497i	$0.630206\pi$
$548$	0	0
$549$	−24.5640	−1.04837
$550$	0	0
$551$	−55.4258	−2.36122
$552$	0	0
$553$	11.1859	0.475673
$554$	0	0
$555$	−13.1301	−0.557342
$556$	0	0
$557$	13.8700	0.587692	0.293846	−	0.955853i	$-0.405065\pi$
$557$	13.8700	0.587692	0.293846	+	0.955853i	$0.405065\pi$
$558$	0	0
$559$	4.02493	0.170237
$560$	0	0
$561$	3.80619	0.160697
$562$	0	0
$563$	−28.4856	−1.20052	−0.600262	−	0.799803i	$-0.704937\pi$
$563$	−28.4856	−1.20052	−0.600262	+	0.799803i	$0.704937\pi$
$564$	0	0
$565$	30.0103	1.26254
$566$	0	0
$567$	5.08735	0.213649
$568$	0	0
$569$	−30.1583	−1.26430	−0.632150	−	0.774846i	$-0.717827\pi$
$569$	−30.1583	−1.26430	−0.632150	+	0.774846i	$0.717827\pi$
$570$	0	0
$571$	−5.48529	−0.229552	−0.114776	−	0.993391i	$-0.536615\pi$
$571$	−5.48529	−0.229552	−0.114776	+	0.993391i	$0.536615\pi$
$572$	0	0
$573$	−6.63360	−0.277123
$574$	0	0
$575$	68.8354	2.87063
$576$	0	0
$577$	33.9648	1.41397	0.706986	−	0.707228i	$-0.250055\pi$
$577$	33.9648	1.41397	0.706986	+	0.707228i	$0.250055\pi$
$578$	0	0
$579$	2.03566	0.0845990
$580$	0	0
$581$	−0.112281	−0.00465818
$582$	0	0
$583$	4.67067	0.193440
$584$	0	0
$585$	9.13354	0.377625
$586$	0	0
$587$	27.0359	1.11589	0.557946	−	0.829877i	$-0.311590\pi$
$587$	27.0359	1.11589	0.557946	+	0.829877i	$0.311590\pi$
$588$	0	0
$589$	−18.0048	−0.741873
$590$	0	0
$591$	−10.9868	−0.451939
$592$	0	0
$593$	−14.4237	−0.592309	−0.296155	−	0.955140i	$-0.595704\pi$
$593$	−14.4237	−0.592309	−0.296155	+	0.955140i	$0.595704\pi$
$594$	0	0
$595$	−20.2072	−0.828413
$596$	0	0
$597$	−12.7477	−0.521728
$598$	0	0
$599$	2.67784	0.109414	0.0547068	−	0.998502i	$-0.482578\pi$
$599$	2.67784	0.109414	0.0547068	+	0.998502i	$0.482578\pi$
$600$	0	0
$601$	−19.2928	−0.786969	−0.393484	−	0.919331i	$-0.628730\pi$
$601$	−19.2928	−0.786969	−0.393484	+	0.919331i	$0.628730\pi$
$602$	0	0
$603$	−28.9762	−1.18000
$604$	0	0
$605$	3.59312	0.146081
$606$	0	0
$607$	18.6095	0.755334	0.377667	−	0.925941i	$-0.376726\pi$
$607$	18.6095	0.755334	0.377667	+	0.925941i	$0.376726\pi$
$608$	0	0
$609$	6.48866	0.262934
$610$	0	0
$611$	−1.80586	−0.0730571
$612$	0	0
$613$	32.4517	1.31071	0.655356	−	0.755320i	$-0.272519\pi$
$613$	32.4517	1.31071	0.655356	+	0.755320i	$0.272519\pi$
$614$	0	0
$615$	21.7426	0.876747
$616$	0	0
$617$	0.435725	0.0175416	0.00877081	−	0.999962i	$-0.497208\pi$
$617$	0.435725	0.0175416	0.00877081	+	0.999962i	$0.497208\pi$
$618$	0	0
$619$	−40.9744	−1.64690	−0.823450	−	0.567388i	$-0.807954\pi$
$619$	−40.9744	−1.64690	−0.823450	+	0.567388i	$0.807954\pi$
$620$	0	0
$621$	−32.6381	−1.30972
$622$	0	0
$623$	10.6256	0.425706
$624$	0	0
$625$	−1.97602	−0.0790409
$626$	0	0
$627$	3.91265	0.156256
$628$	0	0
$629$	30.3649	1.21073
$630$	0	0
$631$	8.35558	0.332630	0.166315	−	0.986073i	$-0.446813\pi$
$631$	8.35558	0.332630	0.166315	+	0.986073i	$0.446813\pi$
$632$	0	0
$633$	−18.0427	−0.717135
$634$	0	0
$635$	41.9137	1.66330
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	10.6466	0.421173
$640$	0	0
$641$	−29.2121	−1.15381	−0.576905	−	0.816811i	$-0.695740\pi$
$641$	−29.2121	−1.15381	−0.576905	+	0.816811i	$0.695740\pi$
$642$	0	0
$643$	−5.46796	−0.215635	−0.107818	−	0.994171i	$-0.534386\pi$
$643$	−5.46796	−0.215635	−0.107818	+	0.994171i	$0.534386\pi$
$644$	0	0
$645$	−9.78787	−0.385397
$646$	0	0
$647$	−35.5057	−1.39588	−0.697938	−	0.716159i	$-0.745899\pi$
$647$	−35.5057	−1.39588	−0.697938	+	0.716159i	$0.745899\pi$
$648$	0	0
$649$	−3.19237	−0.125311
$650$	0	0
$651$	2.10780	0.0826113
$652$	0	0
$653$	−13.4593	−0.526703	−0.263352	−	0.964700i	$-0.584828\pi$
$653$	−13.4593	−0.526703	−0.263352	+	0.964700i	$0.584828\pi$
$654$	0	0
$655$	38.3847	1.49981
$656$	0	0
$657$	−32.1835	−1.25560
$658$	0	0
$659$	11.4237	0.445002	0.222501	−	0.974932i	$-0.428578\pi$
$659$	11.4237	0.445002	0.222501	+	0.974932i	$0.428578\pi$
$660$	0	0
$661$	−36.5377	−1.42115	−0.710576	−	0.703621i	$-0.751565\pi$
$661$	−36.5377	−1.42115	−0.710576	+	0.703621i	$0.751565\pi$
$662$	0	0
$663$	3.80619	0.147820
$664$	0	0
$665$	−20.7724	−0.805519
$666$	0	0
$667$	83.4264	3.23028
$668$	0	0
$669$	−6.87042	−0.265626
$670$	0	0
$671$	−9.66345	−0.373053
$672$	0	0
$673$	−40.9767	−1.57954	−0.789768	−	0.613406i	$-0.789799\pi$
$673$	−40.9767	−1.57954	−0.789768	+	0.613406i	$0.789799\pi$
$674$	0	0
$675$	−29.6706	−1.14202
$676$	0	0
$677$	−4.97586	−0.191238	−0.0956190	−	0.995418i	$-0.530483\pi$
$677$	−4.97586	−0.191238	−0.0956190	+	0.995418i	$0.530483\pi$
$678$	0	0
$679$	16.7285	0.641982
$680$	0	0
$681$	−14.5979	−0.559392
$682$	0	0
$683$	−0.526478	−0.0201451	−0.0100726	−	0.999949i	$-0.503206\pi$
$683$	−0.526478	−0.0201451	−0.0100726	+	0.999949i	$0.503206\pi$
$684$	0	0
$685$	32.7491	1.25128
$686$	0	0
$687$	−1.95817	−0.0747088
$688$	0	0
$689$	4.67067	0.177939
$690$	0	0
$691$	−7.68151	−0.292218	−0.146109	−	0.989268i	$-0.546675\pi$
$691$	−7.68151	−0.292218	−0.146109	+	0.989268i	$0.546675\pi$
$692$	0	0
$693$	2.54195	0.0965606
$694$	0	0
$695$	−13.7712	−0.522372
$696$	0	0
$697$	−50.2824	−1.90458
$698$	0	0
$699$	10.4015	0.393422
$700$	0	0
$701$	3.29151	0.124319	0.0621593	−	0.998066i	$-0.480201\pi$
$701$	3.29151	0.124319	0.0621593	+	0.998066i	$0.480201\pi$
$702$	0	0
$703$	31.2142	1.17727
$704$	0	0
$705$	4.39149	0.165393
$706$	0	0
$707$	−17.2870	−0.650143
$708$	0	0
$709$	37.4374	1.40599	0.702995	−	0.711194i	$-0.251845\pi$
$709$	37.4374	1.40599	0.702995	+	0.711194i	$0.251845\pi$
$710$	0	0
$711$	−28.4340	−1.06636
$712$	0	0
$713$	27.1006	1.01492
$714$	0	0
$715$	3.59312	0.134375
$716$	0	0
$717$	−12.2844	−0.458771
$718$	0	0
$719$	−35.8954	−1.33867	−0.669335	−	0.742960i	$-0.733421\pi$
$719$	−35.8954	−1.33867	−0.669335	+	0.742960i	$0.733421\pi$
$720$	0	0
$721$	8.00000	0.297936
$722$	0	0
$723$	−2.06036	−0.0766257
$724$	0	0
$725$	75.8411	2.81667
$726$	0	0
$727$	1.15310	0.0427661	0.0213831	−	0.999771i	$-0.493193\pi$
$727$	1.15310	0.0427661	0.0213831	+	0.999771i	$0.493193\pi$
$728$	0	0
$729$	−5.31628	−0.196899
$730$	0	0
$731$	22.6356	0.837208
$732$	0	0
$733$	−24.5439	−0.906550	−0.453275	−	0.891371i	$-0.649745\pi$
$733$	−24.5439	−0.906550	−0.453275	+	0.891371i	$0.649745\pi$
$734$	0	0
$735$	2.43181	0.0896986
$736$	0	0
$737$	−11.3992	−0.419895
$738$	0	0
$739$	48.0467	1.76743	0.883714	−	0.468027i	$-0.155035\pi$
$739$	48.0467	1.76743	0.883714	+	0.468027i	$0.155035\pi$
$740$	0	0
$741$	3.91265	0.143735
$742$	0	0
$743$	−25.9216	−0.950970	−0.475485	−	0.879724i	$-0.657727\pi$
$743$	−25.9216	−0.950970	−0.475485	+	0.879724i	$0.657727\pi$
$744$	0	0
$745$	31.9824	1.17174
$746$	0	0
$747$	0.285412	0.0104427
$748$	0	0
$749$	−5.79615	−0.211787
$750$	0	0
$751$	12.0776	0.440718	0.220359	−	0.975419i	$-0.429277\pi$
$751$	12.0776	0.440718	0.220359	+	0.975419i	$0.429277\pi$
$752$	0	0
$753$	−10.7004	−0.389943
$754$	0	0
$755$	−15.5992	−0.567712
$756$	0	0
$757$	11.3929	0.414081	0.207041	−	0.978332i	$-0.433617\pi$
$757$	11.3929	0.414081	0.207041	+	0.978332i	$0.433617\pi$
$758$	0	0
$759$	−5.88929	−0.213768
$760$	0	0
$761$	23.5386	0.853273	0.426637	−	0.904423i	$-0.359698\pi$
$761$	23.5386	0.853273	0.426637	+	0.904423i	$0.359698\pi$
$762$	0	0
$763$	−2.19381	−0.0794213
$764$	0	0
$765$	51.3656	1.85713
$766$	0	0
$767$	−3.19237	−0.115270
$768$	0	0
$769$	22.2843	0.803592	0.401796	−	0.915729i	$-0.368386\pi$
$769$	22.2843	0.803592	0.401796	+	0.915729i	$0.368386\pi$
$770$	0	0
$771$	−5.46460	−0.196803
$772$	0	0
$773$	22.3364	0.803384	0.401692	−	0.915775i	$-0.368422\pi$
$773$	22.3364	0.803384	0.401692	+	0.915775i	$0.368422\pi$
$774$	0	0
$775$	24.6365	0.884970
$776$	0	0
$777$	−3.65423	−0.131095
$778$	0	0
$779$	−51.6888	−1.85194
$780$	0	0
$781$	4.18836	0.149871
$782$	0	0
$783$	−35.9598	−1.28510
$784$	0	0
$785$	36.1870	1.29157
$786$	0	0
$787$	−47.3562	−1.68806	−0.844032	−	0.536292i	$-0.819824\pi$
$787$	−47.3562	−1.68806	−0.844032	+	0.536292i	$0.819824\pi$
$788$	0	0
$789$	4.47034	0.159148
$790$	0	0
$791$	8.35215	0.296968
$792$	0	0
$793$	−9.66345	−0.343159
$794$	0	0
$795$	−11.3582	−0.402833
$796$	0	0
$797$	−6.23258	−0.220769	−0.110385	−	0.993889i	$-0.535208\pi$
$797$	−6.23258	−0.220769	−0.110385	+	0.993889i	$0.535208\pi$
$798$	0	0
$799$	−10.1558	−0.359288
$800$	0	0
$801$	−27.0098	−0.954344
$802$	0	0
$803$	−12.6610	−0.446796
$804$	0	0
$805$	31.2664	1.10200
$806$	0	0
$807$	−3.96401	−0.139540
$808$	0	0
$809$	4.55541	0.160160	0.0800799	−	0.996788i	$-0.474482\pi$
$809$	4.55541	0.160160	0.0800799	+	0.996788i	$0.474482\pi$
$810$	0	0
$811$	4.57597	0.160684	0.0803421	−	0.996767i	$-0.474399\pi$
$811$	4.57597	0.160684	0.0803421	+	0.996767i	$0.474399\pi$
$812$	0	0
$813$	−9.20489	−0.322829
$814$	0	0
$815$	13.6737	0.478969
$816$	0	0
$817$	23.2687	0.814070
$818$	0	0
$819$	2.54195	0.0888229
$820$	0	0
$821$	26.2506	0.916153	0.458076	−	0.888913i	$-0.348539\pi$
$821$	26.2506	0.916153	0.458076	+	0.888913i	$0.348539\pi$
$822$	0	0
$823$	11.5873	0.403907	0.201953	−	0.979395i	$-0.435271\pi$
$823$	11.5873	0.403907	0.201953	+	0.979395i	$0.435271\pi$
$824$	0	0
$825$	−5.35382	−0.186396
$826$	0	0
$827$	−32.9718	−1.14654	−0.573271	−	0.819366i	$-0.694326\pi$
$827$	−32.9718	−1.14654	−0.573271	+	0.819366i	$0.694326\pi$
$828$	0	0
$829$	7.36535	0.255809	0.127905	−	0.991786i	$-0.459175\pi$
$829$	7.36535	0.255809	0.127905	+	0.991786i	$0.459175\pi$
$830$	0	0
$831$	0.165579	0.00574388
$832$	0	0
$833$	−5.62384	−0.194855
$834$	0	0
$835$	57.0659	1.97485
$836$	0	0
$837$	−11.6813	−0.403766
$838$	0	0
$839$	2.74972	0.0949307	0.0474654	−	0.998873i	$-0.484886\pi$
$839$	2.74972	0.0949307	0.0474654	+	0.998873i	$0.484886\pi$
$840$	0	0
$841$	62.9171	2.16955
$842$	0	0
$843$	−10.6672	−0.367396
$844$	0	0
$845$	3.59312	0.123607
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	−14.5760	−0.500248
$850$	0	0
$851$	−46.9834	−1.61057
$852$	0	0
$853$	−21.0998	−0.722443	−0.361222	−	0.932480i	$-0.617640\pi$
$853$	−21.0998	−0.722443	−0.361222	+	0.932480i	$0.617640\pi$
$854$	0	0
$855$	52.8023	1.80580
$856$	0	0
$857$	4.65459	0.158998	0.0794989	−	0.996835i	$-0.474668\pi$
$857$	4.65459	0.158998	0.0794989	+	0.996835i	$0.474668\pi$
$858$	0	0
$859$	−38.1289	−1.30094	−0.650471	−	0.759531i	$-0.725428\pi$
$859$	−38.1289	−1.30094	−0.650471	+	0.759531i	$0.725428\pi$
$860$	0	0
$861$	6.05118	0.206224
$862$	0	0
$863$	−26.2902	−0.894929	−0.447464	−	0.894302i	$-0.647673\pi$
$863$	−26.2902	−0.894929	−0.447464	+	0.894302i	$0.647673\pi$
$864$	0	0
$865$	−37.6859	−1.28136
$866$	0	0
$867$	9.89990	0.336218
$868$	0	0
$869$	−11.1859	−0.379456
$870$	0	0
$871$	−11.3992	−0.386247
$872$	0	0
$873$	−42.5231	−1.43919
$874$	0	0
$875$	10.4579	0.353543
$876$	0	0
$877$	21.9230	0.740288	0.370144	−	0.928974i	$-0.379308\pi$
$877$	21.9230	0.740288	0.370144	+	0.928974i	$0.379308\pi$
$878$	0	0
$879$	16.6632	0.562035
$880$	0	0
$881$	34.8363	1.17366	0.586832	−	0.809709i	$-0.300375\pi$
$881$	34.8363	1.17366	0.586832	+	0.809709i	$0.300375\pi$
$882$	0	0
$883$	57.2116	1.92532	0.962662	−	0.270707i	$-0.0872573\pi$
$883$	57.2116	1.92532	0.962662	+	0.270707i	$0.0872573\pi$
$884$	0	0
$885$	7.76323	0.260958
$886$	0	0
$887$	22.5586	0.757444	0.378722	−	0.925510i	$-0.376364\pi$
$887$	22.5586	0.757444	0.378722	+	0.925510i	$0.376364\pi$
$888$	0	0
$889$	11.6650	0.391231
$890$	0	0
$891$	−5.08735	−0.170433
$892$	0	0
$893$	−10.4399	−0.349359
$894$	0	0
$895$	44.1077	1.47436
$896$	0	0
$897$	−5.88929	−0.196638
$898$	0	0
$899$	29.8587	0.995844
$900$	0	0
$901$	26.2671	0.875085
$902$	0	0
$903$	−2.72405	−0.0906509
$904$	0	0
$905$	72.3367	2.40455
$906$	0	0
$907$	−26.6878	−0.886152	−0.443076	−	0.896484i	$-0.646113\pi$
$907$	−26.6878	−0.886152	−0.443076	+	0.896484i	$0.646113\pi$
$908$	0	0
$909$	43.9426	1.45748
$910$	0	0
$911$	−1.47959	−0.0490211	−0.0245106	−	0.999700i	$-0.507803\pi$
$911$	−1.47959	−0.0490211	−0.0245106	+	0.999700i	$0.507803\pi$
$912$	0	0
$913$	0.112281	0.00371595
$914$	0	0
$915$	23.4997	0.776875
$916$	0	0
$917$	10.6828	0.352777
$918$	0	0
$919$	39.1529	1.29154	0.645768	−	0.763534i	$-0.276537\pi$
$919$	39.1529	1.29154	0.645768	+	0.763534i	$0.276537\pi$
$920$	0	0
$921$	−16.2694	−0.536094
$922$	0	0
$923$	4.18836	0.137861
$924$	0	0
$925$	−42.7115	−1.40435
$926$	0	0
$927$	−20.3356	−0.667908
$928$	0	0
$929$	53.2474	1.74699	0.873495	−	0.486834i	$-0.161848\pi$
$929$	53.2474	1.74699	0.873495	+	0.486834i	$0.161848\pi$
$930$	0	0
$931$	−5.78115	−0.189469
$932$	0	0
$933$	−10.5834	−0.346485
$934$	0	0
$935$	20.2072	0.660845
$936$	0	0
$937$	14.0294	0.458322	0.229161	−	0.973389i	$-0.426402\pi$
$937$	14.0294	0.458322	0.229161	+	0.973389i	$0.426402\pi$
$938$	0	0
$939$	5.74834	0.187590
$940$	0	0
$941$	59.0549	1.92513	0.962567	−	0.271044i	$-0.0873688\pi$
$941$	59.0549	1.92513	0.962567	+	0.271044i	$0.0873688\pi$
$942$	0	0
$943$	77.8015	2.53357
$944$	0	0
$945$	−13.4770	−0.438406
$946$	0	0
$947$	−13.9227	−0.452428	−0.226214	−	0.974078i	$-0.572635\pi$
$947$	−13.9227	−0.452428	−0.226214	+	0.974078i	$0.572635\pi$
$948$	0	0
$949$	−12.6610	−0.410993
$950$	0	0
$951$	−23.7696	−0.770783
$952$	0	0
$953$	−37.7030	−1.22132	−0.610659	−	0.791893i	$-0.709096\pi$
$953$	−37.7030	−1.22132	−0.610659	+	0.791893i	$0.709096\pi$
$954$	0	0
$955$	−35.2180	−1.13963
$956$	0	0
$957$	−6.48866	−0.209749
$958$	0	0
$959$	9.11439	0.294319
$960$	0	0
$961$	−21.3006	−0.687115
$962$	0	0
$963$	14.7335	0.474781
$964$	0	0
$965$	10.8074	0.347901
$966$	0	0
$967$	−42.5027	−1.36679	−0.683397	−	0.730047i	$-0.739498\pi$
$967$	−42.5027	−1.36679	−0.683397	+	0.730047i	$0.739498\pi$
$968$	0	0
$969$	22.0041	0.706875
$970$	0	0
$971$	57.4572	1.84389	0.921945	−	0.387320i	$-0.126599\pi$
$971$	57.4572	1.84389	0.921945	+	0.387320i	$0.126599\pi$
$972$	0	0
$973$	−3.83266	−0.122869
$974$	0	0
$975$	−5.35382	−0.171459
$976$	0	0
$977$	−20.6200	−0.659691	−0.329845	−	0.944035i	$-0.606997\pi$
$977$	−20.6200	−0.659691	−0.329845	+	0.944035i	$0.606997\pi$
$978$	0	0
$979$	−10.6256	−0.339596
$980$	0	0
$981$	5.57655	0.178046
$982$	0	0
$983$	−33.2574	−1.06075	−0.530373	−	0.847764i	$-0.677948\pi$
$983$	−33.2574	−1.06075	−0.530373	+	0.847764i	$0.677948\pi$
$984$	0	0
$985$	−58.3295	−1.85853
$986$	0	0
$987$	1.22219	0.0389029
$988$	0	0
$989$	−35.0239	−1.11369
$990$	0	0
$991$	17.6901	0.561946	0.280973	−	0.959716i	$-0.409343\pi$
$991$	17.6901	0.561946	0.280973	+	0.959716i	$0.409343\pi$
$992$	0	0
$993$	−17.3019	−0.549061
$994$	0	0
$995$	−67.6778	−2.14553
$996$	0	0
$997$	28.4947	0.902437	0.451218	−	0.892414i	$-0.350990\pi$
$997$	28.4947	0.902437	0.451218	+	0.892414i	$0.350990\pi$
$998$	0	0
$999$	20.2515	0.640731

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8008.2.a.k.1.4	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8008.2.a.k.1.4	✓	6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8008.a (trivial)

\(f(q)\)	\(=\)	\(q+0.676795 q^{3} +3.59312 q^{5} +1.00000 q^{7} -2.54195 q^{9} +O(q^{10})\)	\(q+0.676795 q^{3} +3.59312 q^{5} +1.00000 q^{7} -2.54195 q^{9} -1.00000 q^{11} -1.00000 q^{13} +2.43181 q^{15} -5.62384 q^{17} -5.78115 q^{19} +0.676795 q^{21} +8.70173 q^{23} +7.91054 q^{25} -3.75076 q^{27} +9.58734 q^{29} +3.11439 q^{31} -0.676795 q^{33} +3.59312 q^{35} -5.39931 q^{37} -0.676795 q^{39} +8.94093 q^{41} -4.02493 q^{43} -9.13354 q^{45} +1.80586 q^{47} +1.00000 q^{49} -3.80619 q^{51} -4.67067 q^{53} -3.59312 q^{55} -3.91265 q^{57} +3.19237 q^{59} +9.66345 q^{61} -2.54195 q^{63} -3.59312 q^{65} +11.3992 q^{67} +5.88929 q^{69} -4.18836 q^{71} +12.6610 q^{73} +5.35382 q^{75} -1.00000 q^{77} +11.1859 q^{79} +5.08735 q^{81} -0.112281 q^{83} -20.2072 q^{85} +6.48866 q^{87} +10.6256 q^{89} -1.00000 q^{91} +2.10780 q^{93} -20.7724 q^{95} +16.7285 q^{97} +2.54195 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9	\( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) \) 6 * q - q^3 + q^5 + 6 * q^7 + 9 * q^9 - 6 * q^11 - 6 * q^13 - q^15 + 2 * q^19 - q^21 + 11 * q^23 + 9 * q^25 - q^27 + 14 * q^29 + 21 * q^31 + q^33 + q^35 - 5 * q^37 + q^39 + 12 * q^43 + 2 * q^45 + 2 * q^47 + 6 * q^49 - 16 * q^51 + 2 * q^53 - q^55 - 12 * q^57 + 3 * q^59 - 18 * q^61 + 9 * q^63 - q^65 - 25 * q^67 + 29 * q^69 - 11 * q^71 + 14 * q^73 - 42 * q^75 - 6 * q^77 + 24 * q^79 + 42 * q^81 + 24 * q^83 - 46 * q^85 - 16 * q^87 + 55 * q^89 - 6 * q^91 + 41 * q^93 - 30 * q^95 + 37 * q^97 - 9 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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