LMFDB - Embedded newform 4014.2.a.u.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4014,2,Mod(1,4014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4014, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4014.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4014 = 2 \cdot 3^{2} \cdot 223 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4014.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:	$32.0519513713$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.103354048.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 13x^{2} - 16x + 3 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 13x^2 - 16*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.35901$ of defining polynomial
Character	$\chi$	$=$	4014.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +2.56494 q^{5} +2.27923 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +2.56494 q^{5} +2.27923 q^{7} +1.00000 q^{8} +2.56494 q^{10} -0.680471 q^{11} +3.64472 q^{13} +2.27923 q^{14} +1.00000 q^{16} -1.93795 q^{17} -1.65872 q^{19} +2.56494 q^{20} -0.680471 q^{22} +6.35901 q^{23} +1.57894 q^{25} +3.64472 q^{26} +2.27923 q^{28} +8.02443 q^{29} -0.995451 q^{31} +1.00000 q^{32} -1.93795 q^{34} +5.84611 q^{35} -8.32997 q^{37} -1.65872 q^{38} +2.56494 q^{40} -2.25667 q^{41} +0.273398 q^{43} -0.680471 q^{44} +6.35901 q^{46} +2.71219 q^{47} -1.80509 q^{49} +1.57894 q^{50} +3.64472 q^{52} -7.97469 q^{53} -1.74537 q^{55} +2.27923 q^{56} +8.02443 q^{58} -3.85520 q^{59} +10.6871 q^{61} -0.995451 q^{62} +1.00000 q^{64} +9.34851 q^{65} +3.40771 q^{67} -1.93795 q^{68} +5.84611 q^{70} -6.56955 q^{71} -2.15139 q^{73} -8.32997 q^{74} -1.65872 q^{76} -1.55095 q^{77} +16.5396 q^{79} +2.56494 q^{80} -2.25667 q^{82} -1.47011 q^{83} -4.97073 q^{85} +0.273398 q^{86} -0.680471 q^{88} +18.7660 q^{89} +8.30718 q^{91} +6.35901 q^{92} +2.71219 q^{94} -4.25451 q^{95} -8.56500 q^{97} -1.80509 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8	$ 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{20} + 2 q^{22} + 22 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 8 q^{29} - 12 q^{31} + 6 q^{32} + 2 q^{34} + 20 q^{35} - 2 q^{38} + 2 q^{40} + 28 q^{41} + 14 q^{43} + 2 q^{44} + 22 q^{46} + 2 q^{47} + 12 q^{50} - 2 q^{52} + 26 q^{53} + 6 q^{55} + 8 q^{56} + 8 q^{58} + 4 q^{59} - 6 q^{61} - 12 q^{62} + 6 q^{64} + 16 q^{65} + 18 q^{67} + 2 q^{68} + 20 q^{70} + 12 q^{71} - 20 q^{73} - 2 q^{76} + 20 q^{77} + 12 q^{79} + 2 q^{80} + 28 q^{82} + 12 q^{83} - 10 q^{85} + 14 q^{86} + 2 q^{88} - 2 q^{89} - 22 q^{91} + 22 q^{92} + 2 q^{94} + 6 q^{95} - 6 q^{97}+O(q^{100}) $ 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8 + 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^20 + 2 * q^22 + 22 * q^23 + 12 * q^25 - 2 * q^26 + 8 * q^28 + 8 * q^29 - 12 * q^31 + 6 * q^32 + 2 * q^34 + 20 * q^35 - 2 * q^38 + 2 * q^40 + 28 * q^41 + 14 * q^43 + 2 * q^44 + 22 * q^46 + 2 * q^47 + 12 * q^50 - 2 * q^52 + 26 * q^53 + 6 * q^55 + 8 * q^56 + 8 * q^58 + 4 * q^59 - 6 * q^61 - 12 * q^62 + 6 * q^64 + 16 * q^65 + 18 * q^67 + 2 * q^68 + 20 * q^70 + 12 * q^71 - 20 * q^73 - 2 * q^76 + 20 * q^77 + 12 * q^79 + 2 * q^80 + 28 * q^82 + 12 * q^83 - 10 * q^85 + 14 * q^86 + 2 * q^88 - 2 * q^89 - 22 * q^91 + 22 * q^92 + 2 * q^94 + 6 * q^95 - 6 * q^97

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	2.56494	1.14708	0.573539	−	0.819178i	$-0.305570\pi$
$5$	2.56494	1.14708	0.573539	+	0.819178i	$0.305570\pi$
$6$	0	0
$7$	2.27923	0.861469	0.430735	−	0.902479i	$-0.358254\pi$
$7$	2.27923	0.861469	0.430735	+	0.902479i	$0.358254\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	2.56494	0.811106
$11$	−0.680471	−0.205170	−0.102585	−	0.994724i	$-0.532711\pi$
$11$	−0.680471	−0.205170	−0.102585	+	0.994724i	$0.532711\pi$
$12$	0	0
$13$	3.64472	1.01086	0.505432	−	0.862866i	$-0.331333\pi$
$13$	3.64472	1.01086	0.505432	+	0.862866i	$0.331333\pi$
$14$	2.27923	0.609151
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.93795	−0.470022	−0.235011	−	0.971993i	$-0.575513\pi$
$17$	−1.93795	−0.470022	−0.235011	+	0.971993i	$0.575513\pi$
$18$	0	0
$19$	−1.65872	−0.380535	−0.190268	−	0.981732i	$-0.560936\pi$
$19$	−1.65872	−0.380535	−0.190268	+	0.981732i	$0.560936\pi$
$20$	2.56494	0.573539
$21$	0	0
$22$	−0.680471	−0.145077
$23$	6.35901	1.32595	0.662973	−	0.748643i	$-0.269294\pi$
$23$	6.35901	1.32595	0.662973	+	0.748643i	$0.269294\pi$
$24$	0	0
$25$	1.57894	0.315787
$26$	3.64472	0.714789
$27$	0	0
$28$	2.27923	0.430735
$29$	8.02443	1.49010	0.745050	−	0.667009i	$-0.232426\pi$
$29$	8.02443	1.49010	0.745050	+	0.667009i	$0.232426\pi$
$30$	0	0
$31$	−0.995451	−0.178788	−0.0893942	−	0.995996i	$-0.528493\pi$
$31$	−0.995451	−0.178788	−0.0893942	+	0.995996i	$0.528493\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−1.93795	−0.332355
$35$	5.84611	0.988172
$36$	0	0
$37$	−8.32997	−1.36944	−0.684719	−	0.728807i	$-0.740075\pi$
$37$	−8.32997	−1.36944	−0.684719	+	0.728807i	$0.740075\pi$
$38$	−1.65872	−0.269079
$39$	0	0
$40$	2.56494	0.405553
$41$	−2.25667	−0.352432	−0.176216	−	0.984352i	$-0.556386\pi$
$41$	−2.25667	−0.352432	−0.176216	+	0.984352i	$0.556386\pi$
$42$	0	0
$43$	0.273398	0.0416928	0.0208464	−	0.999783i	$-0.493364\pi$
$43$	0.273398	0.0416928	0.0208464	+	0.999783i	$0.493364\pi$
$44$	−0.680471	−0.102585
$45$	0	0
$46$	6.35901	0.937585
$47$	2.71219	0.395614	0.197807	−	0.980241i	$-0.436618\pi$
$47$	2.71219	0.395614	0.197807	+	0.980241i	$0.436618\pi$
$48$	0	0
$49$	−1.80509	−0.257871
$50$	1.57894	0.223295
$51$	0	0
$52$	3.64472	0.505432
$53$	−7.97469	−1.09541	−0.547704	−	0.836672i	$-0.684498\pi$
$53$	−7.97469	−1.09541	−0.547704	+	0.836672i	$0.684498\pi$
$54$	0	0
$55$	−1.74537	−0.235346
$56$	2.27923	0.304575
$57$	0	0
$58$	8.02443	1.05366
$59$	−3.85520	−0.501905	−0.250952	−	0.967999i	$-0.580744\pi$
$59$	−3.85520	−0.501905	−0.250952	+	0.967999i	$0.580744\pi$
$60$	0	0
$61$	10.6871	1.36834	0.684169	−	0.729324i	$-0.260165\pi$
$61$	10.6871	1.36834	0.684169	+	0.729324i	$0.260165\pi$
$62$	−0.995451	−0.126422
$63$	0	0
$64$	1.00000	0.125000
$65$	9.34851	1.15954
$66$	0	0
$67$	3.40771	0.416319	0.208159	−	0.978095i	$-0.433253\pi$
$67$	3.40771	0.416319	0.208159	+	0.978095i	$0.433253\pi$
$68$	−1.93795	−0.235011
$69$	0	0
$70$	5.84611	0.698743
$71$	−6.56955	−0.779662	−0.389831	−	0.920886i	$-0.627467\pi$
$71$	−6.56955	−0.779662	−0.389831	+	0.920886i	$0.627467\pi$
$72$	0	0
$73$	−2.15139	−0.251802	−0.125901	−	0.992043i	$-0.540182\pi$
$73$	−2.15139	−0.251802	−0.125901	+	0.992043i	$0.540182\pi$
$74$	−8.32997	−0.968339
$75$	0	0
$76$	−1.65872	−0.190268
$77$	−1.55095	−0.176747
$78$	0	0
$79$	16.5396	1.86085	0.930427	−	0.366477i	$-0.119436\pi$
$79$	16.5396	1.86085	0.930427	+	0.366477i	$0.119436\pi$
$80$	2.56494	0.286769
$81$	0	0
$82$	−2.25667	−0.249207
$83$	−1.47011	−0.161366	−0.0806829	−	0.996740i	$-0.525710\pi$
$83$	−1.47011	−0.161366	−0.0806829	+	0.996740i	$0.525710\pi$
$84$	0	0
$85$	−4.97073	−0.539151
$86$	0.273398	0.0294813
$87$	0	0
$88$	−0.680471	−0.0725384
$89$	18.7660	1.98919	0.994597	−	0.103809i	$-0.0331032\pi$
$89$	18.7660	1.98919	0.994597	+	0.103809i	$0.0331032\pi$
$90$	0	0
$91$	8.30718	0.870829
$92$	6.35901	0.662973
$93$	0	0
$94$	2.71219	0.279741
$95$	−4.25451	−0.436504
$96$	0	0
$97$	−8.56500	−0.869644	−0.434822	−	0.900516i	$-0.643189\pi$
$97$	−8.56500	−0.869644	−0.434822	+	0.900516i	$0.643189\pi$
$98$	−1.80509	−0.182342
$99$	0	0
$100$	1.57894	0.157894
$101$	5.63882	0.561083	0.280542	−	0.959842i	$-0.409486\pi$
$101$	5.63882	0.561083	0.280542	+	0.959842i	$0.409486\pi$
$102$	0	0
$103$	−13.6169	−1.34171	−0.670855	−	0.741589i	$-0.734073\pi$
$103$	−13.6169	−1.34171	−0.670855	+	0.741589i	$0.734073\pi$
$104$	3.64472	0.357395
$105$	0	0
$106$	−7.97469	−0.774571
$107$	7.11150	0.687495	0.343747	−	0.939062i	$-0.388304\pi$
$107$	7.11150	0.687495	0.343747	+	0.939062i	$0.388304\pi$
$108$	0	0
$109$	3.27276	0.313473	0.156737	−	0.987640i	$-0.449903\pi$
$109$	3.27276	0.313473	0.156737	+	0.987640i	$0.449903\pi$
$110$	−1.74537	−0.166414
$111$	0	0
$112$	2.27923	0.215367
$113$	−14.5262	−1.36651	−0.683256	−	0.730179i	$-0.739437\pi$
$113$	−14.5262	−1.36651	−0.683256	+	0.730179i	$0.739437\pi$
$114$	0	0
$115$	16.3105	1.52096
$116$	8.02443	0.745050
$117$	0	0
$118$	−3.85520	−0.354900
$119$	−4.41704	−0.404909
$120$	0	0
$121$	−10.5370	−0.957905
$122$	10.6871	0.967560
$123$	0	0
$124$	−0.995451	−0.0893942
$125$	−8.77484	−0.784845
$126$	0	0
$127$	2.23882	0.198663	0.0993315	−	0.995054i	$-0.468330\pi$
$127$	2.23882	0.198663	0.0993315	+	0.995054i	$0.468330\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	9.34851	0.819919
$131$	3.28962	0.287415	0.143708	−	0.989620i	$-0.454098\pi$
$131$	3.28962	0.287415	0.143708	+	0.989620i	$0.454098\pi$
$132$	0	0
$133$	−3.78060	−0.327820
$134$	3.40771	0.294382
$135$	0	0
$136$	−1.93795	−0.166178
$137$	−6.53719	−0.558510	−0.279255	−	0.960217i	$-0.590087\pi$
$137$	−6.53719	−0.558510	−0.279255	+	0.960217i	$0.590087\pi$
$138$	0	0
$139$	−20.9846	−1.77989	−0.889944	−	0.456071i	$-0.849256\pi$
$139$	−20.9846	−1.77989	−0.889944	+	0.456071i	$0.849256\pi$
$140$	5.84611	0.494086
$141$	0	0
$142$	−6.56955	−0.551304
$143$	−2.48013	−0.207399
$144$	0	0
$145$	20.5822	1.70926
$146$	−2.15139	−0.178051
$147$	0	0
$148$	−8.32997	−0.684719
$149$	−3.77226	−0.309036	−0.154518	−	0.987990i	$-0.549382\pi$
$149$	−3.77226	−0.309036	−0.154518	+	0.987990i	$0.549382\pi$
$150$	0	0
$151$	−17.0023	−1.38363	−0.691813	−	0.722077i	$-0.743188\pi$
$151$	−17.0023	−1.38363	−0.691813	+	0.722077i	$0.743188\pi$
$152$	−1.65872	−0.134540
$153$	0	0
$154$	−1.55095	−0.124979
$155$	−2.55328	−0.205084
$156$	0	0
$157$	−16.8411	−1.34407	−0.672035	−	0.740519i	$-0.734580\pi$
$157$	−16.8411	−1.34407	−0.672035	+	0.740519i	$0.734580\pi$
$158$	16.5396	1.31582
$159$	0	0
$160$	2.56494	0.202777
$161$	14.4937	1.14226
$162$	0	0
$163$	−4.61493	−0.361469	−0.180735	−	0.983532i	$-0.557848\pi$
$163$	−4.61493	−0.361469	−0.180735	+	0.983532i	$0.557848\pi$
$164$	−2.25667	−0.176216
$165$	0	0
$166$	−1.47011	−0.114103
$167$	24.5842	1.90238	0.951190	−	0.308607i	$-0.0998628\pi$
$167$	24.5842	1.90238	0.951190	+	0.308607i	$0.0998628\pi$
$168$	0	0
$169$	0.284009	0.0218468
$170$	−4.97073	−0.381238
$171$	0	0
$172$	0.273398	0.0208464
$173$	−18.0558	−1.37275	−0.686377	−	0.727245i	$-0.740800\pi$
$173$	−18.0558	−1.37275	−0.686377	+	0.727245i	$0.740800\pi$
$174$	0	0
$175$	3.59876	0.272041
$176$	−0.680471	−0.0512924
$177$	0	0
$178$	18.7660	1.40657
$179$	16.6636	1.24549	0.622747	−	0.782423i	$-0.286017\pi$
$179$	16.6636	1.24549	0.622747	+	0.782423i	$0.286017\pi$
$180$	0	0
$181$	−23.0383	−1.71243	−0.856213	−	0.516624i	$-0.827189\pi$
$181$	−23.0383	−1.71243	−0.856213	+	0.516624i	$0.827189\pi$
$182$	8.30718	0.615769
$183$	0	0
$184$	6.35901	0.468793
$185$	−21.3659	−1.57085
$186$	0	0
$187$	1.31872	0.0964342
$188$	2.71219	0.197807
$189$	0	0
$190$	−4.25451	−0.308655
$191$	9.93738	0.719043	0.359522	−	0.933137i	$-0.382940\pi$
$191$	9.93738	0.719043	0.359522	+	0.933137i	$0.382940\pi$
$192$	0	0
$193$	10.3568	0.745502	0.372751	−	0.927931i	$-0.378415\pi$
$193$	10.3568	0.745502	0.372751	+	0.927931i	$0.378415\pi$
$194$	−8.56500	−0.614931
$195$	0	0
$196$	−1.80509	−0.128935
$197$	17.2488	1.22893	0.614463	−	0.788945i	$-0.289373\pi$
$197$	17.2488	1.22893	0.614463	+	0.788945i	$0.289373\pi$
$198$	0	0
$199$	8.36903	0.593265	0.296633	−	0.954992i	$-0.404136\pi$
$199$	8.36903	0.593265	0.296633	+	0.954992i	$0.404136\pi$
$200$	1.57894	0.111648
$201$	0	0
$202$	5.63882	0.396746
$203$	18.2896	1.28368
$204$	0	0
$205$	−5.78822	−0.404267
$206$	−13.6169	−0.948732
$207$	0	0
$208$	3.64472	0.252716
$209$	1.12871	0.0780743
$210$	0	0
$211$	−6.83480	−0.470527	−0.235264	−	0.971932i	$-0.575595\pi$
$211$	−6.83480	−0.470527	−0.235264	+	0.971932i	$0.575595\pi$
$212$	−7.97469	−0.547704
$213$	0	0
$214$	7.11150	0.486132
$215$	0.701251	0.0478249
$216$	0	0
$217$	−2.26887	−0.154021
$218$	3.27276	0.221659
$219$	0	0
$220$	−1.74537	−0.117673
$221$	−7.06329	−0.475128
$222$	0	0
$223$	−1.00000	−0.0669650
$223$	−1.00000	−0.0669650
$224$	2.27923	0.152288
$225$	0	0
$226$	−14.5262	−0.966270
$227$	23.3729	1.55131	0.775657	−	0.631154i	$-0.217419\pi$
$227$	23.3729	1.55131	0.775657	+	0.631154i	$0.217419\pi$
$228$	0	0
$229$	21.2287	1.40283	0.701417	−	0.712752i	$-0.252551\pi$
$229$	21.2287	1.40283	0.701417	+	0.712752i	$0.252551\pi$
$230$	16.3105	1.07548
$231$	0	0
$232$	8.02443	0.526830
$233$	25.3454	1.66043	0.830217	−	0.557440i	$-0.188216\pi$
$233$	25.3454	1.66043	0.830217	+	0.557440i	$0.188216\pi$
$234$	0	0
$235$	6.95662	0.453800
$236$	−3.85520	−0.250952
$237$	0	0
$238$	−4.41704	−0.286314
$239$	11.6628	0.754404	0.377202	−	0.926131i	$-0.376886\pi$
$239$	11.6628	0.754404	0.377202	+	0.926131i	$0.376886\pi$
$240$	0	0
$241$	9.32707	0.600810	0.300405	−	0.953812i	$-0.402878\pi$
$241$	9.32707	0.600810	0.300405	+	0.953812i	$0.402878\pi$
$242$	−10.5370	−0.677341
$243$	0	0
$244$	10.6871	0.684169
$245$	−4.62997	−0.295798
$246$	0	0
$247$	−6.04556	−0.384670
$248$	−0.995451	−0.0632112
$249$	0	0
$250$	−8.77484	−0.554969
$251$	−11.7576	−0.742134	−0.371067	−	0.928606i	$-0.621008\pi$
$251$	−11.7576	−0.742134	−0.371067	+	0.928606i	$0.621008\pi$
$252$	0	0
$253$	−4.32712	−0.272044
$254$	2.23882	0.140476
$255$	0	0
$256$	1.00000	0.0625000
$257$	17.8744	1.11497	0.557487	−	0.830186i	$-0.311766\pi$
$257$	17.8744	1.11497	0.557487	+	0.830186i	$0.311766\pi$
$258$	0	0
$259$	−18.9859	−1.17973
$260$	9.34851	0.579770
$261$	0	0
$262$	3.28962	0.203233
$263$	10.8388	0.668347	0.334173	−	0.942512i	$-0.391543\pi$
$263$	10.8388	0.668347	0.334173	+	0.942512i	$0.391543\pi$
$264$	0	0
$265$	−20.4546	−1.25652
$266$	−3.78060	−0.231803
$267$	0	0
$268$	3.40771	0.208159
$269$	−13.7447	−0.838028	−0.419014	−	0.907980i	$-0.637624\pi$
$269$	−13.7447	−0.838028	−0.419014	+	0.907980i	$0.637624\pi$
$270$	0	0
$271$	−12.2577	−0.744599	−0.372300	−	0.928113i	$-0.621431\pi$
$271$	−12.2577	−0.744599	−0.372300	+	0.928113i	$0.621431\pi$
$272$	−1.93795	−0.117505
$273$	0	0
$274$	−6.53719	−0.394926
$275$	−1.07442	−0.0647899
$276$	0	0
$277$	8.98069	0.539598	0.269799	−	0.962917i	$-0.413043\pi$
$277$	8.98069	0.539598	0.269799	+	0.962917i	$0.413043\pi$
$278$	−20.9846	−1.25857
$279$	0	0
$280$	5.84611	0.349372
$281$	−12.9385	−0.771849	−0.385925	−	0.922530i	$-0.626117\pi$
$281$	−12.9385	−0.771849	−0.385925	+	0.922530i	$0.626117\pi$
$282$	0	0
$283$	28.4962	1.69392	0.846961	−	0.531655i	$-0.178430\pi$
$283$	28.4962	1.69392	0.846961	+	0.531655i	$0.178430\pi$
$284$	−6.56955	−0.389831
$285$	0	0
$286$	−2.48013	−0.146653
$287$	−5.14347	−0.303609
$288$	0	0
$289$	−13.2444	−0.779080
$290$	20.5822	1.20863
$291$	0	0
$292$	−2.15139	−0.125901
$293$	4.11215	0.240234	0.120117	−	0.992760i	$-0.461673\pi$
$293$	4.11215	0.240234	0.120117	+	0.992760i	$0.461673\pi$
$294$	0	0
$295$	−9.88838	−0.575724
$296$	−8.32997	−0.484170
$297$	0	0
$298$	−3.77226	−0.218521
$299$	23.1768	1.34035
$300$	0	0
$301$	0.623138	0.0359171
$302$	−17.0023	−0.978371
$303$	0	0
$304$	−1.65872	−0.0951339
$305$	27.4117	1.56959
$306$	0	0
$307$	−2.85733	−0.163077	−0.0815384	−	0.996670i	$-0.525983\pi$
$307$	−2.85733	−0.163077	−0.0815384	+	0.996670i	$0.525983\pi$
$308$	−1.55095	−0.0883737
$309$	0	0
$310$	−2.55328	−0.145016
$311$	−14.5423	−0.824617	−0.412308	−	0.911044i	$-0.635277\pi$
$311$	−14.5423	−0.824617	−0.412308	+	0.911044i	$0.635277\pi$
$312$	0	0
$313$	−24.1576	−1.36547	−0.682733	−	0.730668i	$-0.739209\pi$
$313$	−24.1576	−1.36547	−0.682733	+	0.730668i	$0.739209\pi$
$314$	−16.8411	−0.950401
$315$	0	0
$316$	16.5396	0.930427
$317$	−18.6517	−1.04759	−0.523793	−	0.851845i	$-0.675484\pi$
$317$	−18.6517	−1.04759	−0.523793	+	0.851845i	$0.675484\pi$
$318$	0	0
$319$	−5.46039	−0.305723
$320$	2.56494	0.143385
$321$	0	0
$322$	14.4937	0.807701
$323$	3.21451	0.178860
$324$	0	0
$325$	5.75478	0.319218
$326$	−4.61493	−0.255597
$327$	0	0
$328$	−2.25667	−0.124604
$329$	6.18172	0.340809
$330$	0	0
$331$	34.4242	1.89213	0.946064	−	0.323981i	$-0.105021\pi$
$331$	34.4242	1.89213	0.946064	+	0.323981i	$0.105021\pi$
$332$	−1.47011	−0.0806829
$333$	0	0
$334$	24.5842	1.34519
$335$	8.74059	0.477550
$336$	0	0
$337$	−6.60308	−0.359693	−0.179846	−	0.983695i	$-0.557560\pi$
$337$	−6.60308	−0.359693	−0.179846	+	0.983695i	$0.557560\pi$
$338$	0.284009	0.0154480
$339$	0	0
$340$	−4.97073	−0.269576
$341$	0.677376	0.0366819
$342$	0	0
$343$	−20.0689	−1.08362
$344$	0.273398	0.0147406
$345$	0	0
$346$	−18.0558	−0.970684
$347$	14.9907	0.804745	0.402373	−	0.915476i	$-0.368186\pi$
$347$	14.9907	0.804745	0.402373	+	0.915476i	$0.368186\pi$
$348$	0	0
$349$	−0.908100	−0.0486095	−0.0243047	−	0.999705i	$-0.507737\pi$
$349$	−0.908100	−0.0486095	−0.0243047	+	0.999705i	$0.507737\pi$
$350$	3.59876	0.192362
$351$	0	0
$352$	−0.680471	−0.0362692
$353$	−5.51004	−0.293270	−0.146635	−	0.989191i	$-0.546844\pi$
$353$	−5.51004	−0.293270	−0.146635	+	0.989191i	$0.546844\pi$
$354$	0	0
$355$	−16.8505	−0.894333
$356$	18.7660	0.994597
$357$	0	0
$358$	16.6636	0.880697
$359$	−23.2727	−1.22829	−0.614144	−	0.789194i	$-0.710499\pi$
$359$	−23.2727	−1.22829	−0.614144	+	0.789194i	$0.710499\pi$
$360$	0	0
$361$	−16.2487	−0.855193
$362$	−23.0383	−1.21087
$363$	0	0
$364$	8.30718	0.435414
$365$	−5.51821	−0.288836
$366$	0	0
$367$	16.0029	0.835347	0.417674	−	0.908597i	$-0.362846\pi$
$367$	16.0029	0.835347	0.417674	+	0.908597i	$0.362846\pi$
$368$	6.35901	0.331486
$369$	0	0
$370$	−21.3659	−1.11076
$371$	−18.1762	−0.943661
$372$	0	0
$373$	6.43338	0.333108	0.166554	−	0.986032i	$-0.446736\pi$
$373$	6.43338	0.333108	0.166554	+	0.986032i	$0.446736\pi$
$374$	1.31872	0.0681893
$375$	0	0
$376$	2.71219	0.139871
$377$	29.2468	1.50629
$378$	0	0
$379$	0.359831	0.0184833	0.00924164	−	0.999957i	$-0.497058\pi$
$379$	0.359831	0.0184833	0.00924164	+	0.999957i	$0.497058\pi$
$380$	−4.25451	−0.218252
$381$	0	0
$382$	9.93738	0.508440
$383$	−12.7676	−0.652394	−0.326197	−	0.945302i	$-0.605767\pi$
$383$	−12.7676	−0.652394	−0.326197	+	0.945302i	$0.605767\pi$
$384$	0	0
$385$	−3.97810	−0.202743
$386$	10.3568	0.527150
$387$	0	0
$388$	−8.56500	−0.434822
$389$	−22.9846	−1.16537	−0.582683	−	0.812700i	$-0.697997\pi$
$389$	−22.9846	−1.16537	−0.582683	+	0.812700i	$0.697997\pi$
$390$	0	0
$391$	−12.3234	−0.623223
$392$	−1.80509	−0.0911711
$393$	0	0
$394$	17.2488	0.868982
$395$	42.4232	2.13454
$396$	0	0
$397$	−19.1949	−0.963365	−0.481683	−	0.876346i	$-0.659974\pi$
$397$	−19.1949	−0.963365	−0.481683	+	0.876346i	$0.659974\pi$
$398$	8.36903	0.419502
$399$	0	0
$400$	1.57894	0.0789468
$401$	14.2817	0.713194	0.356597	−	0.934258i	$-0.383937\pi$
$401$	14.2817	0.713194	0.356597	+	0.934258i	$0.383937\pi$
$402$	0	0
$403$	−3.62815	−0.180731
$404$	5.63882	0.280542
$405$	0	0
$406$	18.2896	0.907695
$407$	5.66830	0.280967
$408$	0	0
$409$	−16.1060	−0.796389	−0.398194	−	0.917301i	$-0.630363\pi$
$409$	−16.1060	−0.796389	−0.398194	+	0.917301i	$0.630363\pi$
$410$	−5.78822	−0.285860
$411$	0	0
$412$	−13.6169	−0.670855
$413$	−8.78691	−0.432375
$414$	0	0
$415$	−3.77075	−0.185099
$416$	3.64472	0.178697
$417$	0	0
$418$	1.12871	0.0552069
$419$	−27.1797	−1.32782	−0.663909	−	0.747814i	$-0.731104\pi$
$419$	−27.1797	−1.32782	−0.663909	+	0.747814i	$0.731104\pi$
$420$	0	0
$421$	0.913384	0.0445156	0.0222578	−	0.999752i	$-0.492915\pi$
$421$	0.913384	0.0445156	0.0222578	+	0.999752i	$0.492915\pi$
$422$	−6.83480	−0.332713
$423$	0	0
$424$	−7.97469	−0.387285
$425$	−3.05990	−0.148427
$426$	0	0
$427$	24.3583	1.17878
$428$	7.11150	0.343747
$429$	0	0
$430$	0.701251	0.0338173
$431$	−1.94158	−0.0935227	−0.0467614	−	0.998906i	$-0.514890\pi$
$431$	−1.94158	−0.0935227	−0.0467614	+	0.998906i	$0.514890\pi$
$432$	0	0
$433$	8.23032	0.395524	0.197762	−	0.980250i	$-0.436633\pi$
$433$	8.23032	0.395524	0.197762	+	0.980250i	$0.436633\pi$
$434$	−2.26887	−0.108909
$435$	0	0
$436$	3.27276	0.156737
$437$	−10.5478	−0.504569
$438$	0	0
$439$	−24.8607	−1.18653	−0.593267	−	0.805006i	$-0.702162\pi$
$439$	−24.8607	−1.18653	−0.593267	+	0.805006i	$0.702162\pi$
$440$	−1.74537	−0.0832072
$441$	0	0
$442$	−7.06329	−0.335966
$443$	13.9332	0.661986	0.330993	−	0.943633i	$-0.392616\pi$
$443$	13.9332	0.661986	0.330993	+	0.943633i	$0.392616\pi$
$444$	0	0
$445$	48.1338	2.28176
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	2.27923	0.107684
$449$	12.7694	0.602625	0.301313	−	0.953525i	$-0.402575\pi$
$449$	12.7694	0.602625	0.301313	+	0.953525i	$0.402575\pi$
$450$	0	0
$451$	1.53560	0.0723084
$452$	−14.5262	−0.683256
$453$	0	0
$454$	23.3729	1.09694
$455$	21.3074	0.998908
$456$	0	0
$457$	5.48399	0.256530	0.128265	−	0.991740i	$-0.459059\pi$
$457$	5.48399	0.256530	0.128265	+	0.991740i	$0.459059\pi$
$458$	21.2287	0.991953
$459$	0	0
$460$	16.3105	0.760481
$461$	29.7359	1.38494	0.692469	−	0.721447i	$-0.256523\pi$
$461$	29.7359	1.38494	0.692469	+	0.721447i	$0.256523\pi$
$462$	0	0
$463$	−8.69723	−0.404195	−0.202097	−	0.979365i	$-0.564776\pi$
$463$	−8.69723	−0.404195	−0.202097	+	0.979365i	$0.564776\pi$
$464$	8.02443	0.372525
$465$	0	0
$466$	25.3454	1.17410
$467$	14.5706	0.674245	0.337123	−	0.941461i	$-0.390546\pi$
$467$	14.5706	0.674245	0.337123	+	0.941461i	$0.390546\pi$
$468$	0	0
$469$	7.76698	0.358646
$470$	6.95662	0.320885
$471$	0	0
$472$	−3.85520	−0.177450
$473$	−0.186040	−0.00855411
$474$	0	0
$475$	−2.61900	−0.120168
$476$	−4.41704	−0.202455
$477$	0	0
$478$	11.6628	0.533444
$479$	1.64304	0.0750726	0.0375363	−	0.999295i	$-0.488049\pi$
$479$	1.64304	0.0750726	0.0375363	+	0.999295i	$0.488049\pi$
$480$	0	0
$481$	−30.3604	−1.38432
$482$	9.32707	0.424836
$483$	0	0
$484$	−10.5370	−0.478953
$485$	−21.9687	−0.997549
$486$	0	0
$487$	−5.76211	−0.261106	−0.130553	−	0.991441i	$-0.541675\pi$
$487$	−5.76211	−0.261106	−0.130553	+	0.991441i	$0.541675\pi$
$488$	10.6871	0.483780
$489$	0	0
$490$	−4.62997	−0.209161
$491$	−4.11710	−0.185802	−0.0929010	−	0.995675i	$-0.529614\pi$
$491$	−4.11710	−0.185802	−0.0929010	+	0.995675i	$0.529614\pi$
$492$	0	0
$493$	−15.5509	−0.700379
$494$	−6.04556	−0.272003
$495$	0	0
$496$	−0.995451	−0.0446971
$497$	−14.9735	−0.671655
$498$	0	0
$499$	−31.3933	−1.40536	−0.702679	−	0.711507i	$-0.748013\pi$
$499$	−31.3933	−1.40536	−0.702679	+	0.711507i	$0.748013\pi$
$500$	−8.77484	−0.392423
$501$	0	0
$502$	−11.7576	−0.524768
$503$	5.39533	0.240566	0.120283	−	0.992740i	$-0.461620\pi$
$503$	5.39533	0.240566	0.120283	+	0.992740i	$0.461620\pi$
$504$	0	0
$505$	14.4633	0.643606
$506$	−4.32712	−0.192364
$507$	0	0
$508$	2.23882	0.0993315
$509$	−0.262996	−0.0116571	−0.00582855	−	0.999983i	$-0.501855\pi$
$509$	−0.262996	−0.0116571	−0.00582855	+	0.999983i	$0.501855\pi$
$510$	0	0
$511$	−4.90353	−0.216919
$512$	1.00000	0.0441942
$513$	0	0
$514$	17.8744	0.788406
$515$	−34.9265	−1.53904
$516$	0	0
$517$	−1.84557	−0.0811679
$518$	−18.9859	−0.834195
$519$	0	0
$520$	9.34851	0.409959
$521$	31.3174	1.37204	0.686019	−	0.727583i	$-0.259356\pi$
$521$	31.3174	1.37204	0.686019	+	0.727583i	$0.259356\pi$
$522$	0	0
$523$	−24.8532	−1.08676	−0.543378	−	0.839488i	$-0.682855\pi$
$523$	−24.8532	−1.08676	−0.543378	+	0.839488i	$0.682855\pi$
$524$	3.28962	0.143708
$525$	0	0
$526$	10.8388	0.472592
$527$	1.92913	0.0840344
$528$	0	0
$529$	17.4370	0.758133
$530$	−20.4546	−0.888493
$531$	0	0
$532$	−3.78060	−0.163910
$533$	−8.22492	−0.356261
$534$	0	0
$535$	18.2406	0.788610
$536$	3.40771	0.147191
$537$	0	0
$538$	−13.7447	−0.592575
$539$	1.22831	0.0529072
$540$	0	0
$541$	−12.6580	−0.544209	−0.272105	−	0.962268i	$-0.587720\pi$
$541$	−12.6580	−0.544209	−0.272105	+	0.962268i	$0.587720\pi$
$542$	−12.2577	−0.526511
$543$	0	0
$544$	−1.93795	−0.0830889
$545$	8.39444	0.359578
$546$	0	0
$547$	−34.6915	−1.48330	−0.741650	−	0.670787i	$-0.765956\pi$
$547$	−34.6915	−1.48330	−0.741650	+	0.670787i	$0.765956\pi$
$548$	−6.53719	−0.279255
$549$	0	0
$550$	−1.07442	−0.0458134
$551$	−13.3102	−0.567036
$552$	0	0
$553$	37.6977	1.60307
$554$	8.98069	0.381553
$555$	0	0
$556$	−20.9846	−0.889944
$557$	−42.7547	−1.81158	−0.905788	−	0.423732i	$-0.860720\pi$
$557$	−42.7547	−1.81158	−0.905788	+	0.423732i	$0.860720\pi$
$558$	0	0
$559$	0.996461	0.0421458
$560$	5.84611	0.247043
$561$	0	0
$562$	−12.9385	−0.545780
$563$	−9.55525	−0.402706	−0.201353	−	0.979519i	$-0.564534\pi$
$563$	−9.55525	−0.402706	−0.201353	+	0.979519i	$0.564534\pi$
$564$	0	0
$565$	−37.2589	−1.56749
$566$	28.4962	1.19778
$567$	0	0
$568$	−6.56955	−0.275652
$569$	−18.6808	−0.783141	−0.391570	−	0.920148i	$-0.628068\pi$
$569$	−18.6808	−0.783141	−0.391570	+	0.920148i	$0.628068\pi$
$570$	0	0
$571$	11.4837	0.480578	0.240289	−	0.970701i	$-0.422758\pi$
$571$	11.4837	0.480578	0.240289	+	0.970701i	$0.422758\pi$
$572$	−2.48013	−0.103699
$573$	0	0
$574$	−5.14347	−0.214684
$575$	10.0405	0.418717
$576$	0	0
$577$	−6.14834	−0.255959	−0.127979	−	0.991777i	$-0.540849\pi$
$577$	−6.14834	−0.255959	−0.127979	+	0.991777i	$0.540849\pi$
$578$	−13.2444	−0.550893
$579$	0	0
$580$	20.5822	0.854630
$581$	−3.35073	−0.139012
$582$	0	0
$583$	5.42654	0.224745
$584$	−2.15139	−0.0890253
$585$	0	0
$586$	4.11215	0.169871
$587$	−20.5971	−0.850133	−0.425067	−	0.905162i	$-0.639749\pi$
$587$	−20.5971	−0.850133	−0.425067	+	0.905162i	$0.639749\pi$
$588$	0	0
$589$	1.65117	0.0680353
$590$	−9.88838	−0.407098
$591$	0	0
$592$	−8.32997	−0.342360
$593$	−30.3567	−1.24660	−0.623299	−	0.781983i	$-0.714208\pi$
$593$	−30.3567	−1.24660	−0.623299	+	0.781983i	$0.714208\pi$
$594$	0	0
$595$	−11.3295	−0.464462
$596$	−3.77226	−0.154518
$597$	0	0
$598$	23.1768	0.947772
$599$	−18.3763	−0.750834	−0.375417	−	0.926856i	$-0.622501\pi$
$599$	−18.3763	−0.750834	−0.375417	+	0.926856i	$0.622501\pi$
$600$	0	0
$601$	−20.4200	−0.832948	−0.416474	−	0.909148i	$-0.636734\pi$
$601$	−20.4200	−0.832948	−0.416474	+	0.909148i	$0.636734\pi$
$602$	0.623138	0.0253972
$603$	0	0
$604$	−17.0023	−0.691813
$605$	−27.0267	−1.09879
$606$	0	0
$607$	20.0264	0.812847	0.406424	−	0.913685i	$-0.366776\pi$
$607$	20.0264	0.812847	0.406424	+	0.913685i	$0.366776\pi$
$608$	−1.65872	−0.0672698
$609$	0	0
$610$	27.4117	1.10987
$611$	9.88519	0.399912
$612$	0	0
$613$	−0.715794	−0.0289106	−0.0144553	−	0.999896i	$-0.504601\pi$
$613$	−0.715794	−0.0289106	−0.0144553	+	0.999896i	$0.504601\pi$
$614$	−2.85733	−0.115313
$615$	0	0
$616$	−1.55095	−0.0624896
$617$	−21.8981	−0.881585	−0.440792	−	0.897609i	$-0.645303\pi$
$617$	−21.8981	−0.881585	−0.440792	+	0.897609i	$0.645303\pi$
$618$	0	0
$619$	13.8508	0.556712	0.278356	−	0.960478i	$-0.410210\pi$
$619$	13.8508	0.556712	0.278356	+	0.960478i	$0.410210\pi$
$620$	−2.55328	−0.102542
$621$	0	0
$622$	−14.5423	−0.583092
$623$	42.7721	1.71363
$624$	0	0
$625$	−30.4016	−1.21607
$626$	−24.1576	−0.965531
$627$	0	0
$628$	−16.8411	−0.672035
$629$	16.1431	0.643666
$630$	0	0
$631$	−13.6451	−0.543201	−0.271601	−	0.962410i	$-0.587553\pi$
$631$	−13.6451	−0.543201	−0.271601	+	0.962410i	$0.587553\pi$
$632$	16.5396	0.657911
$633$	0	0
$634$	−18.6517	−0.740755
$635$	5.74244	0.227882
$636$	0	0
$637$	−6.57907	−0.260672
$638$	−5.46039	−0.216179
$639$	0	0
$640$	2.56494	0.101388
$641$	17.7279	0.700209	0.350104	−	0.936711i	$-0.386146\pi$
$641$	17.7279	0.700209	0.350104	+	0.936711i	$0.386146\pi$
$642$	0	0
$643$	21.2618	0.838482	0.419241	−	0.907875i	$-0.362296\pi$
$643$	21.2618	0.838482	0.419241	+	0.907875i	$0.362296\pi$
$644$	14.4937	0.571131
$645$	0	0
$646$	3.21451	0.126473
$647$	7.84923	0.308585	0.154293	−	0.988025i	$-0.450690\pi$
$647$	7.84923	0.308585	0.154293	+	0.988025i	$0.450690\pi$
$648$	0	0
$649$	2.62335	0.102976
$650$	5.75478	0.225721
$651$	0	0
$652$	−4.61493	−0.180735
$653$	−14.3690	−0.562301	−0.281150	−	0.959664i	$-0.590716\pi$
$653$	−14.3690	−0.562301	−0.281150	+	0.959664i	$0.590716\pi$
$654$	0	0
$655$	8.43768	0.329688
$656$	−2.25667	−0.0881080
$657$	0	0
$658$	6.18172	0.240988
$659$	−3.68197	−0.143429	−0.0717146	−	0.997425i	$-0.522847\pi$
$659$	−3.68197	−0.143429	−0.0717146	+	0.997425i	$0.522847\pi$
$660$	0	0
$661$	−32.7297	−1.27304	−0.636519	−	0.771261i	$-0.719626\pi$
$661$	−32.7297	−1.27304	−0.636519	+	0.771261i	$0.719626\pi$
$662$	34.4242	1.33794
$663$	0	0
$664$	−1.47011	−0.0570514
$665$	−9.69702	−0.376034
$666$	0	0
$667$	51.0275	1.97579
$668$	24.5842	0.951190
$669$	0	0
$670$	8.74059	0.337679
$671$	−7.27223	−0.280741
$672$	0	0
$673$	23.5559	0.908012	0.454006	−	0.890999i	$-0.349994\pi$
$673$	23.5559	0.908012	0.454006	+	0.890999i	$0.349994\pi$
$674$	−6.60308	−0.254341
$675$	0	0
$676$	0.284009	0.0109234
$677$	−24.6102	−0.945847	−0.472924	−	0.881103i	$-0.656801\pi$
$677$	−24.6102	−0.945847	−0.472924	+	0.881103i	$0.656801\pi$
$678$	0	0
$679$	−19.5216	−0.749172
$680$	−4.97073	−0.190619
$681$	0	0
$682$	0.677376	0.0259381
$683$	10.4170	0.398596	0.199298	−	0.979939i	$-0.436134\pi$
$683$	10.4170	0.398596	0.199298	+	0.979939i	$0.436134\pi$
$684$	0	0
$685$	−16.7675	−0.640654
$686$	−20.0689	−0.766233
$687$	0	0
$688$	0.273398	0.0104232
$689$	−29.0655	−1.10731
$690$	0	0
$691$	19.6114	0.746053	0.373027	−	0.927821i	$-0.378320\pi$
$691$	19.6114	0.746053	0.373027	+	0.927821i	$0.378320\pi$
$692$	−18.0558	−0.686377
$693$	0	0
$694$	14.9907	0.569041
$695$	−53.8242	−2.04167
$696$	0	0
$697$	4.37330	0.165651
$698$	−0.908100	−0.0343721
$699$	0	0
$700$	3.59876	0.136020
$701$	27.9358	1.05512	0.527560	−	0.849518i	$-0.323107\pi$
$701$	27.9358	1.05512	0.527560	+	0.849518i	$0.323107\pi$
$702$	0	0
$703$	13.8170	0.521120
$704$	−0.680471	−0.0256462
$705$	0	0
$706$	−5.51004	−0.207373
$707$	12.8522	0.483356
$708$	0	0
$709$	−47.0568	−1.76725	−0.883627	−	0.468191i	$-0.844906\pi$
$709$	−47.0568	−1.76725	−0.883627	+	0.468191i	$0.844906\pi$
$710$	−16.8505	−0.632389
$711$	0	0
$712$	18.7660	0.703286
$713$	−6.33009	−0.237064
$714$	0	0
$715$	−6.36139	−0.237902
$716$	16.6636	0.622747
$717$	0	0
$718$	−23.2727	−0.868531
$719$	6.63601	0.247481	0.123741	−	0.992315i	$-0.460511\pi$
$719$	6.63601	0.247481	0.123741	+	0.992315i	$0.460511\pi$
$720$	0	0
$721$	−31.0360	−1.15584
$722$	−16.2487	−0.604713
$723$	0	0
$724$	−23.0383	−0.856213
$725$	12.6701	0.470554
$726$	0	0
$727$	15.6811	0.581579	0.290790	−	0.956787i	$-0.406082\pi$
$727$	15.6811	0.581579	0.290790	+	0.956787i	$0.406082\pi$
$728$	8.30718	0.307884
$729$	0	0
$730$	−5.51821	−0.204238
$731$	−0.529832	−0.0195965
$732$	0	0
$733$	−33.8629	−1.25075	−0.625377	−	0.780323i	$-0.715055\pi$
$733$	−33.8629	−1.25075	−0.625377	+	0.780323i	$0.715055\pi$
$734$	16.0029	0.590680
$735$	0	0
$736$	6.35901	0.234396
$737$	−2.31885	−0.0854159
$738$	0	0
$739$	−50.7641	−1.86739	−0.933694	−	0.358073i	$-0.883434\pi$
$739$	−50.7641	−1.86739	−0.933694	+	0.358073i	$0.883434\pi$
$740$	−21.3659	−0.785426
$741$	0	0
$742$	−18.1762	−0.667269
$743$	13.3841	0.491016	0.245508	−	0.969395i	$-0.421045\pi$
$743$	13.3841	0.491016	0.245508	+	0.969395i	$0.421045\pi$
$744$	0	0
$745$	−9.67564	−0.354488
$746$	6.43338	0.235543
$747$	0	0
$748$	1.31872	0.0482171
$749$	16.2088	0.592256
$750$	0	0
$751$	6.82115	0.248907	0.124454	−	0.992225i	$-0.460282\pi$
$751$	6.82115	0.248907	0.124454	+	0.992225i	$0.460282\pi$
$752$	2.71219	0.0989034
$753$	0	0
$754$	29.2468	1.06511
$755$	−43.6099	−1.58713
$756$	0	0
$757$	−46.1353	−1.67682	−0.838408	−	0.545043i	$-0.816513\pi$
$757$	−46.1353	−1.67682	−0.838408	+	0.545043i	$0.816513\pi$
$758$	0.359831	0.0130697
$759$	0	0
$760$	−4.25451	−0.154327
$761$	31.5747	1.14458	0.572291	−	0.820050i	$-0.306055\pi$
$761$	31.5747	1.14458	0.572291	+	0.820050i	$0.306055\pi$
$762$	0	0
$763$	7.45938	0.270048
$764$	9.93738	0.359522
$765$	0	0
$766$	−12.7676	−0.461312
$767$	−14.0511	−0.507357
$768$	0	0
$769$	−15.7344	−0.567399	−0.283699	−	0.958913i	$-0.591562\pi$
$769$	−15.7344	−0.567399	−0.283699	+	0.958913i	$0.591562\pi$
$770$	−3.97810	−0.143361
$771$	0	0
$772$	10.3568	0.372751
$773$	8.89500	0.319931	0.159966	−	0.987123i	$-0.448862\pi$
$773$	8.89500	0.319931	0.159966	+	0.987123i	$0.448862\pi$
$774$	0	0
$775$	−1.57175	−0.0564591
$776$	−8.56500	−0.307466
$777$	0	0
$778$	−22.9846	−0.824038
$779$	3.74317	0.134113
$780$	0	0
$781$	4.47039	0.159963
$782$	−12.3234	−0.440685
$783$	0	0
$784$	−1.80509	−0.0644677
$785$	−43.1966	−1.54175
$786$	0	0
$787$	33.6219	1.19849	0.599246	−	0.800565i	$-0.295467\pi$
$787$	33.6219	1.19849	0.599246	+	0.800565i	$0.295467\pi$
$788$	17.2488	0.614463
$789$	0	0
$790$	42.4232	1.50935
$791$	−33.1086	−1.17721
$792$	0	0
$793$	38.9514	1.38320
$794$	−19.1949	−0.681202
$795$	0	0
$796$	8.36903	0.296633
$797$	−21.4773	−0.760764	−0.380382	−	0.924829i	$-0.624208\pi$
$797$	−21.4773	−0.760764	−0.380382	+	0.924829i	$0.624208\pi$
$798$	0	0
$799$	−5.25609	−0.185947
$800$	1.57894	0.0558238
$801$	0	0
$802$	14.2817	0.504305
$803$	1.46396	0.0516621
$804$	0	0
$805$	37.1755	1.31026
$806$	−3.62815	−0.127796
$807$	0	0
$808$	5.63882	0.198373
$809$	22.0571	0.775488	0.387744	−	0.921767i	$-0.373254\pi$
$809$	22.0571	0.775488	0.387744	+	0.921767i	$0.373254\pi$
$810$	0	0
$811$	8.33568	0.292705	0.146353	−	0.989232i	$-0.453247\pi$
$811$	8.33568	0.292705	0.146353	+	0.989232i	$0.453247\pi$
$812$	18.2896	0.641838
$813$	0	0
$814$	5.66830	0.198674
$815$	−11.8370	−0.414633
$816$	0	0
$817$	−0.453490	−0.0158656
$818$	−16.1060	−0.563132
$819$	0	0
$820$	−5.78822	−0.202133
$821$	8.77712	0.306324	0.153162	−	0.988201i	$-0.451054\pi$
$821$	8.77712	0.306324	0.153162	+	0.988201i	$0.451054\pi$
$822$	0	0
$823$	19.5000	0.679727	0.339863	−	0.940475i	$-0.389619\pi$
$823$	19.5000	0.679727	0.339863	+	0.940475i	$0.389619\pi$
$824$	−13.6169	−0.474366
$825$	0	0
$826$	−8.78691	−0.305736
$827$	−18.6182	−0.647420	−0.323710	−	0.946156i	$-0.604930\pi$
$827$	−18.6182	−0.647420	−0.323710	+	0.946156i	$0.604930\pi$
$828$	0	0
$829$	−40.4836	−1.40605	−0.703027	−	0.711163i	$-0.748169\pi$
$829$	−40.4836	−1.40605	−0.703027	+	0.711163i	$0.748169\pi$
$830$	−3.77075	−0.130885
$831$	0	0
$832$	3.64472	0.126358
$833$	3.49818	0.121205
$834$	0	0
$835$	63.0570	2.18218
$836$	1.12871	0.0390372
$837$	0	0
$838$	−27.1797	−0.938909
$839$	−14.7119	−0.507910	−0.253955	−	0.967216i	$-0.581732\pi$
$839$	−14.7119	−0.507910	−0.253955	+	0.967216i	$0.581732\pi$
$840$	0	0
$841$	35.3915	1.22040
$842$	0.913384	0.0314773
$843$	0	0
$844$	−6.83480	−0.235264
$845$	0.728466	0.0250600
$846$	0	0
$847$	−24.0162	−0.825206
$848$	−7.97469	−0.273852
$849$	0	0
$850$	−3.05990	−0.104954
$851$	−52.9704	−1.81580
$852$	0	0
$853$	10.7687	0.368712	0.184356	−	0.982860i	$-0.440980\pi$
$853$	10.7687	0.368712	0.184356	+	0.982860i	$0.440980\pi$
$854$	24.3583	0.833524
$855$	0	0
$856$	7.11150	0.243066
$857$	12.2786	0.419428	0.209714	−	0.977763i	$-0.432747\pi$
$857$	12.2786	0.419428	0.209714	+	0.977763i	$0.432747\pi$
$858$	0	0
$859$	6.30189	0.215018	0.107509	−	0.994204i	$-0.465713\pi$
$859$	6.30189	0.215018	0.107509	+	0.994204i	$0.465713\pi$
$860$	0.701251	0.0239125
$861$	0	0
$862$	−1.94158	−0.0661305
$863$	−42.6926	−1.45327	−0.726637	−	0.687021i	$-0.758918\pi$
$863$	−42.6926	−1.45327	−0.726637	+	0.687021i	$0.758918\pi$
$864$	0	0
$865$	−46.3120	−1.57466
$866$	8.23032	0.279678
$867$	0	0
$868$	−2.26887	−0.0770103
$869$	−11.2547	−0.381791
$870$	0	0
$871$	12.4202	0.420842
$872$	3.27276	0.110830
$873$	0	0
$874$	−10.5478	−0.356784
$875$	−19.9999	−0.676120
$876$	0	0
$877$	19.2636	0.650486	0.325243	−	0.945631i	$-0.394554\pi$
$877$	19.2636	0.650486	0.325243	+	0.945631i	$0.394554\pi$
$878$	−24.8607	−0.839007
$879$	0	0
$880$	−1.74537	−0.0588364
$881$	12.3211	0.415108	0.207554	−	0.978224i	$-0.433450\pi$
$881$	12.3211	0.415108	0.207554	+	0.978224i	$0.433450\pi$
$882$	0	0
$883$	15.4544	0.520081	0.260041	−	0.965598i	$-0.416264\pi$
$883$	15.4544	0.520081	0.260041	+	0.965598i	$0.416264\pi$
$884$	−7.06329	−0.237564
$885$	0	0
$886$	13.9332	0.468094
$887$	3.80551	0.127777	0.0638883	−	0.997957i	$-0.479650\pi$
$887$	3.80551	0.127777	0.0638883	+	0.997957i	$0.479650\pi$
$888$	0	0
$889$	5.10279	0.171142
$890$	48.1338	1.61345
$891$	0	0
$892$	−1.00000	−0.0334825
$893$	−4.49875	−0.150545
$894$	0	0
$895$	42.7411	1.42868
$896$	2.27923	0.0761438
$897$	0	0
$898$	12.7694	0.426121
$899$	−7.98793	−0.266412
$900$	0	0
$901$	15.4545	0.514866
$902$	1.53560	0.0511297
$903$	0	0
$904$	−14.5262	−0.483135
$905$	−59.0920	−1.96428
$906$	0	0
$907$	4.59267	0.152497	0.0762485	−	0.997089i	$-0.475706\pi$
$907$	4.59267	0.152497	0.0762485	+	0.997089i	$0.475706\pi$
$908$	23.3729	0.775657
$909$	0	0
$910$	21.3074	0.706335
$911$	24.2595	0.803752	0.401876	−	0.915694i	$-0.368358\pi$
$911$	24.2595	0.803752	0.401876	+	0.915694i	$0.368358\pi$
$912$	0	0
$913$	1.00037	0.0331074
$914$	5.48399	0.181394
$915$	0	0
$916$	21.2287	0.701417
$917$	7.49781	0.247599
$918$	0	0
$919$	4.62978	0.152722	0.0763612	−	0.997080i	$-0.475670\pi$
$919$	4.62978	0.152722	0.0763612	+	0.997080i	$0.475670\pi$
$920$	16.3105	0.537742
$921$	0	0
$922$	29.7359	0.979299
$923$	−23.9442	−0.788133
$924$	0	0
$925$	−13.1525	−0.432451
$926$	−8.69723	−0.285809
$927$	0	0
$928$	8.02443	0.263415
$929$	12.5881	0.413003	0.206501	−	0.978446i	$-0.433792\pi$
$929$	12.5881	0.413003	0.206501	+	0.978446i	$0.433792\pi$
$930$	0	0
$931$	2.99414	0.0981289
$932$	25.3454	0.830217
$933$	0	0
$934$	14.5706	0.476763
$935$	3.38244	0.110617
$936$	0	0
$937$	−39.2518	−1.28230	−0.641151	−	0.767415i	$-0.721543\pi$
$937$	−39.2518	−1.28230	−0.641151	+	0.767415i	$0.721543\pi$
$938$	7.76698	0.253601
$939$	0	0
$940$	6.95662	0.226900
$941$	21.3756	0.696823	0.348412	−	0.937342i	$-0.386721\pi$
$941$	21.3756	0.696823	0.348412	+	0.937342i	$0.386721\pi$
$942$	0	0
$943$	−14.3502	−0.467306
$944$	−3.85520	−0.125476
$945$	0	0
$946$	−0.186040	−0.00604867
$947$	−15.4699	−0.502705	−0.251352	−	0.967896i	$-0.580875\pi$
$947$	−15.4699	−0.502705	−0.251352	+	0.967896i	$0.580875\pi$
$948$	0	0
$949$	−7.84124	−0.254537
$950$	−2.61900	−0.0849717
$951$	0	0
$952$	−4.41704	−0.143157
$953$	−12.3847	−0.401179	−0.200590	−	0.979675i	$-0.564286\pi$
$953$	−12.3847	−0.401179	−0.200590	+	0.979675i	$0.564286\pi$
$954$	0	0
$955$	25.4888	0.824799
$956$	11.6628	0.377202
$957$	0	0
$958$	1.64304	0.0530843
$959$	−14.8998	−0.481139
$960$	0	0
$961$	−30.0091	−0.968035
$962$	−30.3604	−0.978860
$963$	0	0
$964$	9.32707	0.300405
$965$	26.5647	0.855149
$966$	0	0
$967$	−7.69167	−0.247348	−0.123674	−	0.992323i	$-0.539468\pi$
$967$	−7.69167	−0.247348	−0.123674	+	0.992323i	$0.539468\pi$
$968$	−10.5370	−0.338671
$969$	0	0
$970$	−21.9687	−0.705374
$971$	0.864067	0.0277292	0.0138646	−	0.999904i	$-0.495587\pi$
$971$	0.864067	0.0277292	0.0138646	+	0.999904i	$0.495587\pi$
$972$	0	0
$973$	−47.8287	−1.53332
$974$	−5.76211	−0.184630
$975$	0	0
$976$	10.6871	0.342084
$977$	42.9435	1.37388	0.686942	−	0.726712i	$-0.258953\pi$
$977$	42.9435	1.37388	0.686942	+	0.726712i	$0.258953\pi$
$978$	0	0
$979$	−12.7697	−0.408122
$980$	−4.62997	−0.147899
$981$	0	0
$982$	−4.11710	−0.131382
$983$	−35.2495	−1.12428	−0.562141	−	0.827041i	$-0.690022\pi$
$983$	−35.2495	−1.12428	−0.562141	+	0.827041i	$0.690022\pi$
$984$	0	0
$985$	44.2422	1.40967
$986$	−15.5509	−0.495243
$987$	0	0
$988$	−6.04556	−0.192335
$989$	1.73854	0.0552825
$990$	0	0
$991$	−3.97460	−0.126257	−0.0631287	−	0.998005i	$-0.520108\pi$
$991$	−3.97460	−0.126257	−0.0631287	+	0.998005i	$0.520108\pi$
$992$	−0.995451	−0.0316056
$993$	0	0
$994$	−14.9735	−0.474932
$995$	21.4661	0.680521
$996$	0	0
$997$	13.3980	0.424319	0.212160	−	0.977235i	$-0.431950\pi$
$997$	13.3980	0.424319	0.212160	+	0.977235i	$0.431950\pi$
$998$	−31.3933	−0.993739
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.u.1.5	yes	6
3.2	odd	2		4014.2.a.t.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4014.2.a.t.1.2	✓	6	3.2	odd	2
4014.2.a.u.1.5	yes	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:	\( N \)	\(=\)	\( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4014.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.56494 q^{5} +2.27923 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +2.56494 q^{5} +2.27923 q^{7} +1.00000 q^{8} +2.56494 q^{10} -0.680471 q^{11} +3.64472 q^{13} +2.27923 q^{14} +1.00000 q^{16} -1.93795 q^{17} -1.65872 q^{19} +2.56494 q^{20} -0.680471 q^{22} +6.35901 q^{23} +1.57894 q^{25} +3.64472 q^{26} +2.27923 q^{28} +8.02443 q^{29} -0.995451 q^{31} +1.00000 q^{32} -1.93795 q^{34} +5.84611 q^{35} -8.32997 q^{37} -1.65872 q^{38} +2.56494 q^{40} -2.25667 q^{41} +0.273398 q^{43} -0.680471 q^{44} +6.35901 q^{46} +2.71219 q^{47} -1.80509 q^{49} +1.57894 q^{50} +3.64472 q^{52} -7.97469 q^{53} -1.74537 q^{55} +2.27923 q^{56} +8.02443 q^{58} -3.85520 q^{59} +10.6871 q^{61} -0.995451 q^{62} +1.00000 q^{64} +9.34851 q^{65} +3.40771 q^{67} -1.93795 q^{68} +5.84611 q^{70} -6.56955 q^{71} -2.15139 q^{73} -8.32997 q^{74} -1.65872 q^{76} -1.55095 q^{77} +16.5396 q^{79} +2.56494 q^{80} -2.25667 q^{82} -1.47011 q^{83} -4.97073 q^{85} +0.273398 q^{86} -0.680471 q^{88} +18.7660 q^{89} +8.30718 q^{91} +6.35901 q^{92} +2.71219 q^{94} -4.25451 q^{95} -8.56500 q^{97} -1.80509 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8	\( 6 q + 6 q^{2} + 6 q^{4} + 2 q^{5} + 8 q^{7} + 6 q^{8} + 2 q^{10} + 2 q^{11} - 2 q^{13} + 8 q^{14} + 6 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{20} + 2 q^{22} + 22 q^{23} + 12 q^{25} - 2 q^{26} + 8 q^{28} + 8 q^{29} - 12 q^{31} + 6 q^{32} + 2 q^{34} + 20 q^{35} - 2 q^{38} + 2 q^{40} + 28 q^{41} + 14 q^{43} + 2 q^{44} + 22 q^{46} + 2 q^{47} + 12 q^{50} - 2 q^{52} + 26 q^{53} + 6 q^{55} + 8 q^{56} + 8 q^{58} + 4 q^{59} - 6 q^{61} - 12 q^{62} + 6 q^{64} + 16 q^{65} + 18 q^{67} + 2 q^{68} + 20 q^{70} + 12 q^{71} - 20 q^{73} - 2 q^{76} + 20 q^{77} + 12 q^{79} + 2 q^{80} + 28 q^{82} + 12 q^{83} - 10 q^{85} + 14 q^{86} + 2 q^{88} - 2 q^{89} - 22 q^{91} + 22 q^{92} + 2 q^{94} + 6 q^{95} - 6 q^{97}+O(q^{100}) \) 6 * q + 6 * q^2 + 6 * q^4 + 2 * q^5 + 8 * q^7 + 6 * q^8 + 2 * q^10 + 2 * q^11 - 2 * q^13 + 8 * q^14 + 6 * q^16 + 2 * q^17 - 2 * q^19 + 2 * q^20 + 2 * q^22 + 22 * q^23 + 12 * q^25 - 2 * q^26 + 8 * q^28 + 8 * q^29 - 12 * q^31 + 6 * q^32 + 2 * q^34 + 20 * q^35 - 2 * q^38 + 2 * q^40 + 28 * q^41 + 14 * q^43 + 2 * q^44 + 22 * q^46 + 2 * q^47 + 12 * q^50 - 2 * q^52 + 26 * q^53 + 6 * q^55 + 8 * q^56 + 8 * q^58 + 4 * q^59 - 6 * q^61 - 12 * q^62 + 6 * q^64 + 16 * q^65 + 18 * q^67 + 2 * q^68 + 20 * q^70 + 12 * q^71 - 20 * q^73 - 2 * q^76 + 20 * q^77 + 12 * q^79 + 2 * q^80 + 28 * q^82 + 12 * q^83 - 10 * q^85 + 14 * q^86 + 2 * q^88 - 2 * q^89 - 22 * q^91 + 22 * q^92 + 2 * q^94 + 6 * q^95 - 6 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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