LMFDB - Embedded newform 4014.2.a.m.1.1

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:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1338)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -0.532089 q^{5} -2.87939 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -0.532089 q^{5} -2.87939 q^{7} -1.00000 q^{8} +0.532089 q^{10} +1.18479 q^{11} -0.773318 q^{13} +2.87939 q^{14} +1.00000 q^{16} +6.59627 q^{17} -1.34730 q^{19} -0.532089 q^{20} -1.18479 q^{22} -2.94356 q^{23} -4.71688 q^{25} +0.773318 q^{26} -2.87939 q^{28} +4.26857 q^{29} +4.17024 q^{31} -1.00000 q^{32} -6.59627 q^{34} +1.53209 q^{35} -9.02734 q^{37} +1.34730 q^{38} +0.532089 q^{40} -5.17024 q^{41} +1.04189 q^{43} +1.18479 q^{44} +2.94356 q^{46} +8.61587 q^{47} +1.29086 q^{49} +4.71688 q^{50} -0.773318 q^{52} +11.8452 q^{53} -0.630415 q^{55} +2.87939 q^{56} -4.26857 q^{58} -7.29086 q^{59} -0.0564370 q^{61} -4.17024 q^{62} +1.00000 q^{64} +0.411474 q^{65} +9.43882 q^{67} +6.59627 q^{68} -1.53209 q^{70} -3.00000 q^{71} -9.29086 q^{73} +9.02734 q^{74} -1.34730 q^{76} -3.41147 q^{77} -2.57398 q^{79} -0.532089 q^{80} +5.17024 q^{82} +7.84524 q^{83} -3.50980 q^{85} -1.04189 q^{86} -1.18479 q^{88} +10.0719 q^{89} +2.22668 q^{91} -2.94356 q^{92} -8.61587 q^{94} +0.716881 q^{95} -10.0419 q^{97} -1.29086 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8} - 3 q^{10} - 9 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{19} + 3 q^{20} + 6 q^{23} - 6 q^{25} + 9 q^{26} - 3 q^{28} + 3 q^{29} - 9 q^{31} - 3 q^{32} - 6 q^{34} - 6 q^{37} + 3 q^{38} - 3 q^{40} + 6 q^{41} - 6 q^{46} + 15 q^{47} - 12 q^{49} + 6 q^{50} - 9 q^{52} + 9 q^{53} - 9 q^{55} + 3 q^{56} - 3 q^{58} - 6 q^{59} - 15 q^{61} + 9 q^{62} + 3 q^{64} - 9 q^{65} - 3 q^{67} + 6 q^{68} - 9 q^{71} - 12 q^{73} + 6 q^{74} - 3 q^{76} + 3 q^{80} - 6 q^{82} - 3 q^{83} - 12 q^{85} - 3 q^{89} + 6 q^{92} - 15 q^{94} - 6 q^{95} - 27 q^{97} + 12 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8 - 3 * q^10 - 9 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 - 3 * q^19 + 3 * q^20 + 6 * q^23 - 6 * q^25 + 9 * q^26 - 3 * q^28 + 3 * q^29 - 9 * q^31 - 3 * q^32 - 6 * q^34 - 6 * q^37 + 3 * q^38 - 3 * q^40 + 6 * q^41 - 6 * q^46 + 15 * q^47 - 12 * q^49 + 6 * q^50 - 9 * q^52 + 9 * q^53 - 9 * q^55 + 3 * q^56 - 3 * q^58 - 6 * q^59 - 15 * q^61 + 9 * q^62 + 3 * q^64 - 9 * q^65 - 3 * q^67 + 6 * q^68 - 9 * q^71 - 12 * q^73 + 6 * q^74 - 3 * q^76 + 3 * q^80 - 6 * q^82 - 3 * q^83 - 12 * q^85 - 3 * q^89 + 6 * q^92 - 15 * q^94 - 6 * q^95 - 27 * q^97 + 12 * q^98

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
$5$	−0.532089	−0.237957	−0.118979	−	0.992897i	$-0.537962\pi$
$5$	−0.532089	−0.237957	−0.118979	+	0.992897i	$0.537962\pi$
$6$	0	0
$7$	−2.87939	−1.08831	−0.544153	−	0.838986i	$-0.683149\pi$
$7$	−2.87939	−1.08831	−0.544153	+	0.838986i	$0.683149\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	0.532089	0.168261
$11$	1.18479	0.357228	0.178614	−	0.983919i	$-0.442839\pi$
$11$	1.18479	0.357228	0.178614	+	0.983919i	$0.442839\pi$
$12$	0	0
$13$	−0.773318	−0.214480	−0.107240	−	0.994233i	$-0.534201\pi$
$13$	−0.773318	−0.214480	−0.107240	+	0.994233i	$0.534201\pi$
$14$	2.87939	0.769548
$15$	0	0
$16$	1.00000	0.250000
$17$	6.59627	1.59983	0.799915	−	0.600114i	$-0.204878\pi$
$17$	6.59627	1.59983	0.799915	+	0.600114i	$0.204878\pi$
$18$	0	0
$19$	−1.34730	−0.309091	−0.154545	−	0.987986i	$-0.549391\pi$
$19$	−1.34730	−0.309091	−0.154545	+	0.987986i	$0.549391\pi$
$20$	−0.532089	−0.118979
$21$	0	0
$22$	−1.18479	−0.252599
$23$	−2.94356	−0.613775	−0.306888	−	0.951746i	$-0.599288\pi$
$23$	−2.94356	−0.613775	−0.306888	+	0.951746i	$0.599288\pi$
$24$	0	0
$25$	−4.71688	−0.943376
$26$	0.773318	0.151660
$27$	0	0
$28$	−2.87939	−0.544153
$29$	4.26857	0.792654	0.396327	−	0.918109i	$-0.370285\pi$
$29$	4.26857	0.792654	0.396327	+	0.918109i	$0.370285\pi$
$30$	0	0
$31$	4.17024	0.748998	0.374499	−	0.927227i	$-0.377815\pi$
$31$	4.17024	0.748998	0.374499	+	0.927227i	$0.377815\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−6.59627	−1.13125
$35$	1.53209	0.258970
$36$	0	0
$37$	−9.02734	−1.48409	−0.742043	−	0.670352i	$-0.766143\pi$
$37$	−9.02734	−1.48409	−0.742043	+	0.670352i	$0.766143\pi$
$38$	1.34730	0.218560
$39$	0	0
$40$	0.532089	0.0841306
$41$	−5.17024	−0.807457	−0.403728	−	0.914879i	$-0.632286\pi$
$41$	−5.17024	−0.807457	−0.403728	+	0.914879i	$0.632286\pi$
$42$	0	0
$43$	1.04189	0.158887	0.0794433	−	0.996839i	$-0.474686\pi$
$43$	1.04189	0.158887	0.0794433	+	0.996839i	$0.474686\pi$
$44$	1.18479	0.178614
$45$	0	0
$46$	2.94356	0.434005
$47$	8.61587	1.25675	0.628377	−	0.777909i	$-0.283720\pi$
$47$	8.61587	1.25675	0.628377	+	0.777909i	$0.283720\pi$
$48$	0	0
$49$	1.29086	0.184408
$50$	4.71688	0.667068
$51$	0	0
$52$	−0.773318	−0.107240
$53$	11.8452	1.62707	0.813534	−	0.581517i	$-0.197541\pi$
$53$	11.8452	1.62707	0.813534	+	0.581517i	$0.197541\pi$
$54$	0	0
$55$	−0.630415	−0.0850051
$56$	2.87939	0.384774
$57$	0	0
$58$	−4.26857	−0.560491
$59$	−7.29086	−0.949189	−0.474595	−	0.880205i	$-0.657405\pi$
$59$	−7.29086	−0.949189	−0.474595	+	0.880205i	$0.657405\pi$
$60$	0	0
$61$	−0.0564370	−0.00722602	−0.00361301	−	0.999993i	$-0.501150\pi$
$61$	−0.0564370	−0.00722602	−0.00361301	+	0.999993i	$0.501150\pi$
$62$	−4.17024	−0.529622
$63$	0	0
$64$	1.00000	0.125000
$65$	0.411474	0.0510371
$66$	0	0
$67$	9.43882	1.15313	0.576567	−	0.817050i	$-0.304392\pi$
$67$	9.43882	1.15313	0.576567	+	0.817050i	$0.304392\pi$
$68$	6.59627	0.799915
$69$	0	0
$70$	−1.53209	−0.183120
$71$	−3.00000	−0.356034	−0.178017	−	0.984027i	$-0.556968\pi$
$71$	−3.00000	−0.356034	−0.178017	+	0.984027i	$0.556968\pi$
$72$	0	0
$73$	−9.29086	−1.08741	−0.543706	−	0.839275i	$-0.682979\pi$
$73$	−9.29086	−1.08741	−0.543706	+	0.839275i	$0.682979\pi$
$74$	9.02734	1.04941
$75$	0	0
$76$	−1.34730	−0.154545
$77$	−3.41147	−0.388774
$78$	0	0
$79$	−2.57398	−0.289595	−0.144798	−	0.989461i	$-0.546253\pi$
$79$	−2.57398	−0.289595	−0.144798	+	0.989461i	$0.546253\pi$
$80$	−0.532089	−0.0594893
$81$	0	0
$82$	5.17024	0.570958
$83$	7.84524	0.861127	0.430563	−	0.902560i	$-0.358315\pi$
$83$	7.84524	0.861127	0.430563	+	0.902560i	$0.358315\pi$
$84$	0	0
$85$	−3.50980	−0.380691
$86$	−1.04189	−0.112350
$87$	0	0
$88$	−1.18479	−0.126299
$89$	10.0719	1.06762	0.533811	−	0.845604i	$-0.320760\pi$
$89$	10.0719	1.06762	0.533811	+	0.845604i	$0.320760\pi$
$90$	0	0
$91$	2.22668	0.233420
$92$	−2.94356	−0.306888
$93$	0	0
$94$	−8.61587	−0.888659
$95$	0.716881	0.0735505
$96$	0	0
$97$	−10.0419	−1.01960	−0.509800	−	0.860293i	$-0.670280\pi$
$97$	−10.0419	−1.01960	−0.509800	+	0.860293i	$0.670280\pi$
$98$	−1.29086	−0.130396
$99$	0	0
$100$	−4.71688	−0.471688
$101$	−11.3327	−1.12765	−0.563825	−	0.825894i	$-0.690671\pi$
$101$	−11.3327	−1.12765	−0.563825	+	0.825894i	$0.690671\pi$
$102$	0	0
$103$	0.128356	0.0126472	0.00632362	−	0.999980i	$-0.497987\pi$
$103$	0.128356	0.0126472	0.00632362	+	0.999980i	$0.497987\pi$
$104$	0.773318	0.0758301
$105$	0	0
$106$	−11.8452	−1.15051
$107$	−8.32770	−0.805069	−0.402534	−	0.915405i	$-0.631871\pi$
$107$	−8.32770	−0.805069	−0.402534	+	0.915405i	$0.631871\pi$
$108$	0	0
$109$	−1.71688	−0.164447	−0.0822237	−	0.996614i	$-0.526202\pi$
$109$	−1.71688	−0.164447	−0.0822237	+	0.996614i	$0.526202\pi$
$110$	0.630415	0.0601077
$111$	0	0
$112$	−2.87939	−0.272076
$113$	−1.20439	−0.113300	−0.0566499	−	0.998394i	$-0.518042\pi$
$113$	−1.20439	−0.113300	−0.0566499	+	0.998394i	$0.518042\pi$
$114$	0	0
$115$	1.56624	0.146052
$116$	4.26857	0.396327
$117$	0	0
$118$	7.29086	0.671178
$119$	−18.9932	−1.74110
$120$	0	0
$121$	−9.59627	−0.872388
$122$	0.0564370	0.00510956
$123$	0	0
$124$	4.17024	0.374499
$125$	5.17024	0.462441
$126$	0	0
$127$	16.6878	1.48080	0.740401	−	0.672166i	$-0.234636\pi$
$127$	16.6878	1.48080	0.740401	+	0.672166i	$0.234636\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−0.411474	−0.0360887
$131$	−13.4979	−1.17932	−0.589660	−	0.807651i	$-0.700738\pi$
$131$	−13.4979	−1.17932	−0.589660	+	0.807651i	$0.700738\pi$
$132$	0	0
$133$	3.87939	0.336385
$134$	−9.43882	−0.815389
$135$	0	0
$136$	−6.59627	−0.565625
$137$	−17.5398	−1.49853	−0.749264	−	0.662271i	$-0.769593\pi$
$137$	−17.5398	−1.49853	−0.749264	+	0.662271i	$0.769593\pi$
$138$	0	0
$139$	−15.6313	−1.32583	−0.662917	−	0.748693i	$-0.730682\pi$
$139$	−15.6313	−1.32583	−0.662917	+	0.748693i	$0.730682\pi$
$140$	1.53209	0.129485
$141$	0	0
$142$	3.00000	0.251754
$143$	−0.916222	−0.0766183
$144$	0	0
$145$	−2.27126	−0.188618
$146$	9.29086	0.768917
$147$	0	0
$148$	−9.02734	−0.742043
$149$	−15.4757	−1.26781	−0.633907	−	0.773409i	$-0.718550\pi$
$149$	−15.4757	−1.26781	−0.633907	+	0.773409i	$0.718550\pi$
$150$	0	0
$151$	3.58853	0.292030	0.146015	−	0.989282i	$-0.453355\pi$
$151$	3.58853	0.292030	0.146015	+	0.989282i	$0.453355\pi$
$152$	1.34730	0.109280
$153$	0	0
$154$	3.41147	0.274904
$155$	−2.21894	−0.178230
$156$	0	0
$157$	−16.6578	−1.32943	−0.664717	−	0.747095i	$-0.731448\pi$
$157$	−16.6578	−1.32943	−0.664717	+	0.747095i	$0.731448\pi$
$158$	2.57398	0.204775
$159$	0	0
$160$	0.532089	0.0420653
$161$	8.47565	0.667975
$162$	0	0
$163$	6.61081	0.517799	0.258899	−	0.965904i	$-0.416640\pi$
$163$	6.61081	0.517799	0.258899	+	0.965904i	$0.416640\pi$
$164$	−5.17024	−0.403728
$165$	0	0
$166$	−7.84524	−0.608908
$167$	14.5594	1.12664	0.563321	−	0.826238i	$-0.309523\pi$
$167$	14.5594	1.12664	0.563321	+	0.826238i	$0.309523\pi$
$168$	0	0
$169$	−12.4020	−0.953998
$170$	3.50980	0.269189
$171$	0	0
$172$	1.04189	0.0794433
$173$	−20.7793	−1.57982	−0.789911	−	0.613222i	$-0.789873\pi$
$173$	−20.7793	−1.57982	−0.789911	+	0.613222i	$0.789873\pi$
$174$	0	0
$175$	13.5817	1.02668
$176$	1.18479	0.0893071
$177$	0	0
$178$	−10.0719	−0.754922
$179$	9.58946	0.716750	0.358375	−	0.933578i	$-0.383331\pi$
$179$	9.58946	0.716750	0.358375	+	0.933578i	$0.383331\pi$
$180$	0	0
$181$	1.79292	0.133267	0.0666333	−	0.997778i	$-0.478774\pi$
$181$	1.79292	0.133267	0.0666333	+	0.997778i	$0.478774\pi$
$182$	−2.22668	−0.165053
$183$	0	0
$184$	2.94356	0.217002
$185$	4.80335	0.353149
$186$	0	0
$187$	7.81521	0.571505
$188$	8.61587	0.628377
$189$	0	0
$190$	−0.716881	−0.0520080
$191$	−14.6973	−1.06346	−0.531729	−	0.846915i	$-0.678457\pi$
$191$	−14.6973	−1.06346	−0.531729	+	0.846915i	$0.678457\pi$
$192$	0	0
$193$	20.0428	1.44271	0.721357	−	0.692563i	$-0.243519\pi$
$193$	20.0428	1.44271	0.721357	+	0.692563i	$0.243519\pi$
$194$	10.0419	0.720966
$195$	0	0
$196$	1.29086	0.0922042
$197$	−1.74422	−0.124271	−0.0621354	−	0.998068i	$-0.519791\pi$
$197$	−1.74422	−0.124271	−0.0621354	+	0.998068i	$0.519791\pi$
$198$	0	0
$199$	−5.00774	−0.354989	−0.177495	−	0.984122i	$-0.556799\pi$
$199$	−5.00774	−0.354989	−0.177495	+	0.984122i	$0.556799\pi$
$200$	4.71688	0.333534
$201$	0	0
$202$	11.3327	0.797369
$203$	−12.2909	−0.862649
$204$	0	0
$205$	2.75103	0.192140
$206$	−0.128356	−0.00894295
$207$	0	0
$208$	−0.773318	−0.0536200
$209$	−1.59627	−0.110416
$210$	0	0
$211$	−23.8776	−1.64380	−0.821902	−	0.569629i	$-0.807087\pi$
$211$	−23.8776	−1.64380	−0.821902	+	0.569629i	$0.807087\pi$
$212$	11.8452	0.813534
$213$	0	0
$214$	8.32770	0.569270
$215$	−0.554378	−0.0378082
$216$	0	0
$217$	−12.0077	−0.815139
$218$	1.71688	0.116282
$219$	0	0
$220$	−0.630415	−0.0425026
$221$	−5.10101	−0.343131
$222$	0	0
$223$	1.00000	0.0669650
$223$	1.00000	0.0669650
$224$	2.87939	0.192387
$225$	0	0
$226$	1.20439	0.0801150
$227$	25.2841	1.67816	0.839081	−	0.544007i	$-0.183093\pi$
$227$	25.2841	1.67816	0.839081	+	0.544007i	$0.183093\pi$
$228$	0	0
$229$	−9.02229	−0.596210	−0.298105	−	0.954533i	$-0.596354\pi$
$229$	−9.02229	−0.596210	−0.298105	+	0.954533i	$0.596354\pi$
$230$	−1.56624	−0.103275
$231$	0	0
$232$	−4.26857	−0.280245
$233$	−21.6117	−1.41583	−0.707916	−	0.706296i	$-0.750365\pi$
$233$	−21.6117	−1.41583	−0.707916	+	0.706296i	$0.750365\pi$
$234$	0	0
$235$	−4.58441	−0.299054
$236$	−7.29086	−0.474595
$237$	0	0
$238$	18.9932	1.23115
$239$	−11.8203	−0.764589	−0.382295	−	0.924041i	$-0.624866\pi$
$239$	−11.8203	−0.764589	−0.382295	+	0.924041i	$0.624866\pi$
$240$	0	0
$241$	9.22762	0.594403	0.297201	−	0.954815i	$-0.403947\pi$
$241$	9.22762	0.594403	0.297201	+	0.954815i	$0.403947\pi$
$242$	9.59627	0.616871
$243$	0	0
$244$	−0.0564370	−0.00361301
$245$	−0.686852	−0.0438814
$246$	0	0
$247$	1.04189	0.0662938
$248$	−4.17024	−0.264811
$249$	0	0
$250$	−5.17024	−0.326995
$251$	−19.3354	−1.22044	−0.610221	−	0.792231i	$-0.708919\pi$
$251$	−19.3354	−1.22044	−0.610221	+	0.792231i	$0.708919\pi$
$252$	0	0
$253$	−3.48751	−0.219258
$254$	−16.6878	−1.04708
$255$	0	0
$256$	1.00000	0.0625000
$257$	−26.6759	−1.66400	−0.831999	−	0.554777i	$-0.812804\pi$
$257$	−26.6759	−1.66400	−0.831999	+	0.554777i	$0.812804\pi$
$258$	0	0
$259$	25.9932	1.61514
$260$	0.411474	0.0255185
$261$	0	0
$262$	13.4979	0.833906
$263$	5.77601	0.356164	0.178082	−	0.984016i	$-0.443011\pi$
$263$	5.77601	0.356164	0.178082	+	0.984016i	$0.443011\pi$
$264$	0	0
$265$	−6.30272	−0.387173
$266$	−3.87939	−0.237860
$267$	0	0
$268$	9.43882	0.576567
$269$	5.29860	0.323061	0.161531	−	0.986868i	$-0.448357\pi$
$269$	5.29860	0.323061	0.161531	+	0.986868i	$0.448357\pi$
$270$	0	0
$271$	−3.30541	−0.200789	−0.100395	−	0.994948i	$-0.532011\pi$
$271$	−3.30541	−0.200789	−0.100395	+	0.994948i	$0.532011\pi$
$272$	6.59627	0.399957
$273$	0	0
$274$	17.5398	1.05962
$275$	−5.58853	−0.337001
$276$	0	0
$277$	15.8384	0.951639	0.475820	−	0.879543i	$-0.342152\pi$
$277$	15.8384	0.951639	0.475820	+	0.879543i	$0.342152\pi$
$278$	15.6313	0.937506
$279$	0	0
$280$	−1.53209	−0.0915598
$281$	−8.40642	−0.501485	−0.250743	−	0.968054i	$-0.580675\pi$
$281$	−8.40642	−0.501485	−0.250743	+	0.968054i	$0.580675\pi$
$282$	0	0
$283$	−13.6459	−0.811164	−0.405582	−	0.914059i	$-0.632931\pi$
$283$	−13.6459	−0.811164	−0.405582	+	0.914059i	$0.632931\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	0.916222	0.0541773
$287$	14.8871	0.878759
$288$	0	0
$289$	26.5107	1.55945
$290$	2.27126	0.133373
$291$	0	0
$292$	−9.29086	−0.543706
$293$	12.8280	0.749420	0.374710	−	0.927142i	$-0.377742\pi$
$293$	12.8280	0.749420	0.374710	+	0.927142i	$0.377742\pi$
$294$	0	0
$295$	3.87939	0.225867
$296$	9.02734	0.524704
$297$	0	0
$298$	15.4757	0.896480
$299$	2.27631	0.131642
$300$	0	0
$301$	−3.00000	−0.172917
$302$	−3.58853	−0.206496
$303$	0	0
$304$	−1.34730	−0.0772727
$305$	0.0300295	0.00171948
$306$	0	0
$307$	−7.25671	−0.414162	−0.207081	−	0.978324i	$-0.566396\pi$
$307$	−7.25671	−0.414162	−0.207081	+	0.978324i	$0.566396\pi$
$308$	−3.41147	−0.194387
$309$	0	0
$310$	2.21894	0.126027
$311$	−2.65507	−0.150555	−0.0752775	−	0.997163i	$-0.523984\pi$
$311$	−2.65507	−0.150555	−0.0752775	+	0.997163i	$0.523984\pi$
$312$	0	0
$313$	7.20945	0.407502	0.203751	−	0.979023i	$-0.434687\pi$
$313$	7.20945	0.407502	0.203751	+	0.979023i	$0.434687\pi$
$314$	16.6578	0.940052
$315$	0	0
$316$	−2.57398	−0.144798
$317$	4.70233	0.264109	0.132055	−	0.991242i	$-0.457843\pi$
$317$	4.70233	0.264109	0.132055	+	0.991242i	$0.457843\pi$
$318$	0	0
$319$	5.05737	0.283158
$320$	−0.532089	−0.0297447
$321$	0	0
$322$	−8.47565	−0.472330
$323$	−8.88713	−0.494493
$324$	0	0
$325$	3.64765	0.202335
$326$	−6.61081	−0.366139
$327$	0	0
$328$	5.17024	0.285479
$329$	−24.8084	−1.36773
$330$	0	0
$331$	3.98276	0.218912	0.109456	−	0.993992i	$-0.465089\pi$
$331$	3.98276	0.218912	0.109456	+	0.993992i	$0.465089\pi$
$332$	7.84524	0.430563
$333$	0	0
$334$	−14.5594	−0.796657
$335$	−5.02229	−0.274397
$336$	0	0
$337$	−25.0746	−1.36590	−0.682950	−	0.730465i	$-0.739304\pi$
$337$	−25.0746	−1.36590	−0.682950	+	0.730465i	$0.739304\pi$
$338$	12.4020	0.674579
$339$	0	0
$340$	−3.50980	−0.190346
$341$	4.94087	0.267563
$342$	0	0
$343$	16.4388	0.887613
$344$	−1.04189	−0.0561749
$345$	0	0
$346$	20.7793	1.11710
$347$	6.48070	0.347902	0.173951	−	0.984754i	$-0.444346\pi$
$347$	6.48070	0.347902	0.173951	+	0.984754i	$0.444346\pi$
$348$	0	0
$349$	5.68685	0.304410	0.152205	−	0.988349i	$-0.451363\pi$
$349$	5.68685	0.304410	0.152205	+	0.988349i	$0.451363\pi$
$350$	−13.5817	−0.725973
$351$	0	0
$352$	−1.18479	−0.0631497
$353$	−10.2739	−0.546827	−0.273414	−	0.961897i	$-0.588153\pi$
$353$	−10.2739	−0.546827	−0.273414	+	0.961897i	$0.588153\pi$
$354$	0	0
$355$	1.59627	0.0847210
$356$	10.0719	0.533811
$357$	0	0
$358$	−9.58946	−0.506819
$359$	−34.5080	−1.82126	−0.910632	−	0.413217i	$-0.864405\pi$
$359$	−34.5080	−1.82126	−0.910632	+	0.413217i	$0.864405\pi$
$360$	0	0
$361$	−17.1848	−0.904463
$362$	−1.79292	−0.0942337
$363$	0	0
$364$	2.22668	0.116710
$365$	4.94356	0.258758
$366$	0	0
$367$	−20.5449	−1.07243	−0.536217	−	0.844080i	$-0.680147\pi$
$367$	−20.5449	−1.07243	−0.536217	+	0.844080i	$0.680147\pi$
$368$	−2.94356	−0.153444
$369$	0	0
$370$	−4.80335	−0.249714
$371$	−34.1070	−1.77075
$372$	0	0
$373$	4.91891	0.254692	0.127346	−	0.991858i	$-0.459354\pi$
$373$	4.91891	0.254692	0.127346	+	0.991858i	$0.459354\pi$
$374$	−7.81521	−0.404115
$375$	0	0
$376$	−8.61587	−0.444329
$377$	−3.30096	−0.170008
$378$	0	0
$379$	13.0692	0.671321	0.335661	−	0.941983i	$-0.391040\pi$
$379$	13.0692	0.671321	0.335661	+	0.941983i	$0.391040\pi$
$380$	0.716881	0.0367752
$381$	0	0
$382$	14.6973	0.751978
$383$	11.9581	0.611031	0.305515	−	0.952187i	$-0.401171\pi$
$383$	11.9581	0.611031	0.305515	+	0.952187i	$0.401171\pi$
$384$	0	0
$385$	1.81521	0.0925115
$386$	−20.0428	−1.02015
$387$	0	0
$388$	−10.0419	−0.509800
$389$	−11.7219	−0.594326	−0.297163	−	0.954827i	$-0.596040\pi$
$389$	−11.7219	−0.594326	−0.297163	+	0.954827i	$0.596040\pi$
$390$	0	0
$391$	−19.4165	−0.981936
$392$	−1.29086	−0.0651982
$393$	0	0
$394$	1.74422	0.0878727
$395$	1.36959	0.0689113
$396$	0	0
$397$	−15.5398	−0.779922	−0.389961	−	0.920831i	$-0.627511\pi$
$397$	−15.5398	−0.779922	−0.389961	+	0.920831i	$0.627511\pi$
$398$	5.00774	0.251015
$399$	0	0
$400$	−4.71688	−0.235844
$401$	3.69553	0.184546	0.0922729	−	0.995734i	$-0.470587\pi$
$401$	3.69553	0.184546	0.0922729	+	0.995734i	$0.470587\pi$
$402$	0	0
$403$	−3.22493	−0.160645
$404$	−11.3327	−0.563825
$405$	0	0
$406$	12.2909	0.609985
$407$	−10.6955	−0.530158
$408$	0	0
$409$	0.763823	0.0377686	0.0188843	−	0.999822i	$-0.493989\pi$
$409$	0.763823	0.0377686	0.0188843	+	0.999822i	$0.493989\pi$
$410$	−2.75103	−0.135864
$411$	0	0
$412$	0.128356	0.00632362
$413$	20.9932	1.03301
$414$	0	0
$415$	−4.17436	−0.204911
$416$	0.773318	0.0379151
$417$	0	0
$418$	1.59627	0.0780760
$419$	−10.6604	−0.520797	−0.260398	−	0.965501i	$-0.583854\pi$
$419$	−10.6604	−0.520797	−0.260398	+	0.965501i	$0.583854\pi$
$420$	0	0
$421$	−22.8675	−1.11450	−0.557248	−	0.830346i	$-0.688143\pi$
$421$	−22.8675	−1.11450	−0.557248	+	0.830346i	$0.688143\pi$
$422$	23.8776	1.16234
$423$	0	0
$424$	−11.8452	−0.575256
$425$	−31.1138	−1.50924
$426$	0	0
$427$	0.162504	0.00786411
$428$	−8.32770	−0.402534
$429$	0	0
$430$	0.554378	0.0267345
$431$	39.1780	1.88714	0.943569	−	0.331177i	$-0.107446\pi$
$431$	39.1780	1.88714	0.943569	+	0.331177i	$0.107446\pi$
$432$	0	0
$433$	−20.6483	−0.992292	−0.496146	−	0.868239i	$-0.665252\pi$
$433$	−20.6483	−0.992292	−0.496146	+	0.868239i	$0.665252\pi$
$434$	12.0077	0.576390
$435$	0	0
$436$	−1.71688	−0.0822237
$437$	3.96585	0.189712
$438$	0	0
$439$	−37.5158	−1.79053	−0.895265	−	0.445533i	$-0.853014\pi$
$439$	−37.5158	−1.79053	−0.895265	+	0.445533i	$0.853014\pi$
$440$	0.630415	0.0300539
$441$	0	0
$442$	5.10101	0.242631
$443$	12.1138	0.575544	0.287772	−	0.957699i	$-0.407085\pi$
$443$	12.1138	0.575544	0.287772	+	0.957699i	$0.407085\pi$
$444$	0	0
$445$	−5.35916	−0.254048
$446$	−1.00000	−0.0473514
$447$	0	0
$448$	−2.87939	−0.136038
$449$	4.43376	0.209242	0.104621	−	0.994512i	$-0.466637\pi$
$449$	4.43376	0.209242	0.104621	+	0.994512i	$0.466637\pi$
$450$	0	0
$451$	−6.12567	−0.288446
$452$	−1.20439	−0.0566499
$453$	0	0
$454$	−25.2841	−1.18664
$455$	−1.18479	−0.0555439
$456$	0	0
$457$	−20.5621	−0.961855	−0.480928	−	0.876760i	$-0.659700\pi$
$457$	−20.5621	−0.961855	−0.480928	+	0.876760i	$0.659700\pi$
$458$	9.02229	0.421584
$459$	0	0
$460$	1.56624	0.0730262
$461$	9.31996	0.434074	0.217037	−	0.976163i	$-0.430361\pi$
$461$	9.31996	0.434074	0.217037	+	0.976163i	$0.430361\pi$
$462$	0	0
$463$	−32.8881	−1.52844	−0.764219	−	0.644957i	$-0.776875\pi$
$463$	−32.8881	−1.52844	−0.764219	+	0.644957i	$0.776875\pi$
$464$	4.26857	0.198163
$465$	0	0
$466$	21.6117	1.00114
$467$	−18.2618	−0.845054	−0.422527	−	0.906350i	$-0.638857\pi$
$467$	−18.2618	−0.845054	−0.422527	+	0.906350i	$0.638857\pi$
$468$	0	0
$469$	−27.1780	−1.25496
$470$	4.58441	0.211463
$471$	0	0
$472$	7.29086	0.335589
$473$	1.23442	0.0567588
$474$	0	0
$475$	6.35504	0.291589
$476$	−18.9932	−0.870552
$477$	0	0
$478$	11.8203	0.540646
$479$	41.6742	1.90414	0.952071	−	0.305878i	$-0.0989499\pi$
$479$	41.6742	1.90414	0.952071	+	0.305878i	$0.0989499\pi$
$480$	0	0
$481$	6.98101	0.318307
$482$	−9.22762	−0.420306
$483$	0	0
$484$	−9.59627	−0.436194
$485$	5.34318	0.242621
$486$	0	0
$487$	−20.1584	−0.913464	−0.456732	−	0.889604i	$-0.650980\pi$
$487$	−20.1584	−0.913464	−0.456732	+	0.889604i	$0.650980\pi$
$488$	0.0564370	0.00255478
$489$	0	0
$490$	0.686852	0.0310288
$491$	−19.7547	−0.891515	−0.445757	−	0.895154i	$-0.647066\pi$
$491$	−19.7547	−0.891515	−0.445757	+	0.895154i	$0.647066\pi$
$492$	0	0
$493$	28.1566	1.26811
$494$	−1.04189	−0.0468768
$495$	0	0
$496$	4.17024	0.187250
$497$	8.63816	0.387474
$498$	0	0
$499$	−5.21894	−0.233632	−0.116816	−	0.993154i	$-0.537269\pi$
$499$	−5.21894	−0.233632	−0.116816	+	0.993154i	$0.537269\pi$
$500$	5.17024	0.231220
$501$	0	0
$502$	19.3354	0.862983
$503$	17.9463	0.800184	0.400092	−	0.916475i	$-0.368978\pi$
$503$	17.9463	0.800184	0.400092	+	0.916475i	$0.368978\pi$
$504$	0	0
$505$	6.03003	0.268333
$506$	3.48751	0.155039
$507$	0	0
$508$	16.6878	0.740401
$509$	17.7743	0.787830	0.393915	−	0.919147i	$-0.371120\pi$
$509$	17.7743	0.787830	0.393915	+	0.919147i	$0.371120\pi$
$510$	0	0
$511$	26.7520	1.18344
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	26.6759	1.17662
$515$	−0.0682966	−0.00300951
$516$	0	0
$517$	10.2080	0.448948
$518$	−25.9932	−1.14208
$519$	0	0
$520$	−0.411474	−0.0180443
$521$	20.5749	0.901403	0.450702	−	0.892675i	$-0.351174\pi$
$521$	20.5749	0.901403	0.450702	+	0.892675i	$0.351174\pi$
$522$	0	0
$523$	−4.17799	−0.182691	−0.0913453	−	0.995819i	$-0.529117\pi$
$523$	−4.17799	−0.182691	−0.0913453	+	0.995819i	$0.529117\pi$
$524$	−13.4979	−0.589660
$525$	0	0
$526$	−5.77601	−0.251846
$527$	27.5080	1.19827
$528$	0	0
$529$	−14.3354	−0.623280
$530$	6.30272	0.273773
$531$	0	0
$532$	3.87939	0.168193
$533$	3.99825	0.173183
$534$	0	0
$535$	4.43107	0.191572
$536$	−9.43882	−0.407695
$537$	0	0
$538$	−5.29860	−0.228439
$539$	1.52940	0.0658759
$540$	0	0
$541$	40.0856	1.72342	0.861708	−	0.507404i	$-0.169395\pi$
$541$	40.0856	1.72342	0.861708	+	0.507404i	$0.169395\pi$
$542$	3.30541	0.141979
$543$	0	0
$544$	−6.59627	−0.282813
$545$	0.913534	0.0391315
$546$	0	0
$547$	36.1935	1.54752	0.773760	−	0.633478i	$-0.218373\pi$
$547$	36.1935	1.54752	0.773760	+	0.633478i	$0.218373\pi$
$548$	−17.5398	−0.749264
$549$	0	0
$550$	5.58853	0.238296
$551$	−5.75103	−0.245002
$552$	0	0
$553$	7.41147	0.315168
$554$	−15.8384	−0.672910
$555$	0	0
$556$	−15.6313	−0.662917
$557$	23.4679	0.994367	0.497184	−	0.867645i	$-0.334368\pi$
$557$	23.4679	0.994367	0.497184	+	0.867645i	$0.334368\pi$
$558$	0	0
$559$	−0.805712	−0.0340780
$560$	1.53209	0.0647426
$561$	0	0
$562$	8.40642	0.354603
$563$	−39.5681	−1.66760	−0.833798	−	0.552069i	$-0.813838\pi$
$563$	−39.5681	−1.66760	−0.833798	+	0.552069i	$0.813838\pi$
$564$	0	0
$565$	0.640844	0.0269605
$566$	13.6459	0.573580
$567$	0	0
$568$	3.00000	0.125877
$569$	5.72699	0.240088	0.120044	−	0.992769i	$-0.461696\pi$
$569$	5.72699	0.240088	0.120044	+	0.992769i	$0.461696\pi$
$570$	0	0
$571$	−9.52528	−0.398621	−0.199310	−	0.979936i	$-0.563870\pi$
$571$	−9.52528	−0.398621	−0.199310	+	0.979936i	$0.563870\pi$
$572$	−0.916222	−0.0383092
$573$	0	0
$574$	−14.8871	−0.621377
$575$	13.8844	0.579021
$576$	0	0
$577$	−16.7716	−0.698209	−0.349105	−	0.937084i	$-0.613514\pi$
$577$	−16.7716	−0.698209	−0.349105	+	0.937084i	$0.613514\pi$
$578$	−26.5107	−1.10270
$579$	0	0
$580$	−2.27126	−0.0943089
$581$	−22.5895	−0.937169
$582$	0	0
$583$	14.0341	0.581235
$584$	9.29086	0.384458
$585$	0	0
$586$	−12.8280	−0.529920
$587$	9.49020	0.391702	0.195851	−	0.980634i	$-0.437253\pi$
$587$	9.49020	0.391702	0.195851	+	0.980634i	$0.437253\pi$
$588$	0	0
$589$	−5.61856	−0.231509
$590$	−3.87939	−0.159712
$591$	0	0
$592$	−9.02734	−0.371021
$593$	13.9504	0.572873	0.286437	−	0.958099i	$-0.407529\pi$
$593$	13.9504	0.572873	0.286437	+	0.958099i	$0.407529\pi$
$594$	0	0
$595$	10.1061	0.414308
$596$	−15.4757	−0.633907
$597$	0	0
$598$	−2.27631	−0.0930853
$599$	−30.4911	−1.24583	−0.622917	−	0.782288i	$-0.714053\pi$
$599$	−30.4911	−1.24583	−0.622917	+	0.782288i	$0.714053\pi$
$600$	0	0
$601$	−28.1557	−1.14849	−0.574247	−	0.818682i	$-0.694705\pi$
$601$	−28.1557	−1.14849	−0.574247	+	0.818682i	$0.694705\pi$
$602$	3.00000	0.122271
$603$	0	0
$604$	3.58853	0.146015
$605$	5.10607	0.207591
$606$	0	0
$607$	−27.9240	−1.13340	−0.566699	−	0.823925i	$-0.691780\pi$
$607$	−27.9240	−1.13340	−0.566699	+	0.823925i	$0.691780\pi$
$608$	1.34730	0.0546401
$609$	0	0
$610$	−0.0300295	−0.00121586
$611$	−6.66281	−0.269548
$612$	0	0
$613$	24.7101	0.998031	0.499015	−	0.866593i	$-0.333695\pi$
$613$	24.7101	0.998031	0.499015	+	0.866593i	$0.333695\pi$
$614$	7.25671	0.292857
$615$	0	0
$616$	3.41147	0.137452
$617$	19.5936	0.788808	0.394404	−	0.918937i	$-0.370951\pi$
$617$	19.5936	0.788808	0.394404	+	0.918937i	$0.370951\pi$
$618$	0	0
$619$	−9.35679	−0.376081	−0.188041	−	0.982161i	$-0.560214\pi$
$619$	−9.35679	−0.376081	−0.188041	+	0.982161i	$0.560214\pi$
$620$	−2.21894	−0.0891148
$621$	0	0
$622$	2.65507	0.106459
$623$	−29.0009	−1.16190
$624$	0	0
$625$	20.8334	0.833335
$626$	−7.20945	−0.288147
$627$	0	0
$628$	−16.6578	−0.664717
$629$	−59.5467	−2.37428
$630$	0	0
$631$	3.63041	0.144525	0.0722623	−	0.997386i	$-0.476978\pi$
$631$	3.63041	0.144525	0.0722623	+	0.997386i	$0.476978\pi$
$632$	2.57398	0.102387
$633$	0	0
$634$	−4.70233	−0.186754
$635$	−8.87939	−0.352368
$636$	0	0
$637$	−0.998245	−0.0395519
$638$	−5.05737	−0.200223
$639$	0	0
$640$	0.532089	0.0210327
$641$	28.1388	1.11142	0.555708	−	0.831378i	$-0.312447\pi$
$641$	28.1388	1.11142	0.555708	+	0.831378i	$0.312447\pi$
$642$	0	0
$643$	8.51073	0.335631	0.167815	−	0.985818i	$-0.446329\pi$
$643$	8.51073	0.335631	0.167815	+	0.985818i	$0.446329\pi$
$644$	8.47565	0.333987
$645$	0	0
$646$	8.88713	0.349659
$647$	23.2445	0.913837	0.456918	−	0.889509i	$-0.348953\pi$
$647$	23.2445	0.913837	0.456918	+	0.889509i	$0.348953\pi$
$648$	0	0
$649$	−8.63816	−0.339077
$650$	−3.64765	−0.143073
$651$	0	0
$652$	6.61081	0.258899
$653$	3.13104	0.122527	0.0612636	−	0.998122i	$-0.480487\pi$
$653$	3.13104	0.122527	0.0612636	+	0.998122i	$0.480487\pi$
$654$	0	0
$655$	7.18210	0.280628
$656$	−5.17024	−0.201864
$657$	0	0
$658$	24.8084	0.967132
$659$	−36.9546	−1.43955	−0.719773	−	0.694209i	$-0.755754\pi$
$659$	−36.9546	−1.43955	−0.719773	+	0.694209i	$0.755754\pi$
$660$	0	0
$661$	−14.9017	−0.579608	−0.289804	−	0.957086i	$-0.593590\pi$
$661$	−14.9017	−0.579608	−0.289804	+	0.957086i	$0.593590\pi$
$662$	−3.98276	−0.154795
$663$	0	0
$664$	−7.84524	−0.304454
$665$	−2.06418	−0.0800454
$666$	0	0
$667$	−12.5648	−0.486511
$668$	14.5594	0.563321
$669$	0	0
$670$	5.02229	0.194028
$671$	−0.0668661	−0.00258134
$672$	0	0
$673$	−35.9344	−1.38517	−0.692585	−	0.721337i	$-0.743528\pi$
$673$	−35.9344	−1.38517	−0.692585	+	0.721337i	$0.743528\pi$
$674$	25.0746	0.965838
$675$	0	0
$676$	−12.4020	−0.476999
$677$	−0.463792	−0.0178250	−0.00891249	−	0.999960i	$-0.502837\pi$
$677$	−0.463792	−0.0178250	−0.00891249	+	0.999960i	$0.502837\pi$
$678$	0	0
$679$	28.9145	1.10964
$680$	3.50980	0.134595
$681$	0	0
$682$	−4.94087	−0.189196
$683$	10.5030	0.401886	0.200943	−	0.979603i	$-0.435599\pi$
$683$	10.5030	0.401886	0.200943	+	0.979603i	$0.435599\pi$
$684$	0	0
$685$	9.33275	0.356586
$686$	−16.4388	−0.627637
$687$	0	0
$688$	1.04189	0.0397216
$689$	−9.16014	−0.348974
$690$	0	0
$691$	−0.409719	−0.0155865	−0.00779323	−	0.999970i	$-0.502481\pi$
$691$	−0.409719	−0.0155865	−0.00779323	+	0.999970i	$0.502481\pi$
$692$	−20.7793	−0.789911
$693$	0	0
$694$	−6.48070	−0.246004
$695$	8.31727	0.315492
$696$	0	0
$697$	−34.1043	−1.29179
$698$	−5.68685	−0.215251
$699$	0	0
$700$	13.5817	0.513341
$701$	29.9050	1.12950	0.564748	−	0.825264i	$-0.308974\pi$
$701$	29.9050	1.12950	0.564748	+	0.825264i	$0.308974\pi$
$702$	0	0
$703$	12.1625	0.458718
$704$	1.18479	0.0446535
$705$	0	0
$706$	10.2739	0.386665
$707$	32.6313	1.22723
$708$	0	0
$709$	−8.35504	−0.313780	−0.156890	−	0.987616i	$-0.550147\pi$
$709$	−8.35504	−0.313780	−0.156890	+	0.987616i	$0.550147\pi$
$710$	−1.59627	−0.0599068
$711$	0	0
$712$	−10.0719	−0.377461
$713$	−12.2754	−0.459717
$714$	0	0
$715$	0.487511	0.0182319
$716$	9.58946	0.358375
$717$	0	0
$718$	34.5080	1.28783
$719$	21.8990	0.816694	0.408347	−	0.912827i	$-0.366105\pi$
$719$	21.8990	0.816694	0.408347	+	0.912827i	$0.366105\pi$
$720$	0	0
$721$	−0.369585	−0.0137641
$722$	17.1848	0.639552
$723$	0	0
$724$	1.79292	0.0666333
$725$	−20.1343	−0.747771
$726$	0	0
$727$	9.41653	0.349240	0.174620	−	0.984636i	$-0.444130\pi$
$727$	9.41653	0.349240	0.174620	+	0.984636i	$0.444130\pi$
$728$	−2.22668	−0.0825263
$729$	0	0
$730$	−4.94356	−0.182969
$731$	6.87258	0.254191
$732$	0	0
$733$	−15.3637	−0.567472	−0.283736	−	0.958902i	$-0.591574\pi$
$733$	−15.3637	−0.567472	−0.283736	+	0.958902i	$0.591574\pi$
$734$	20.5449	0.758325
$735$	0	0
$736$	2.94356	0.108501
$737$	11.1830	0.411932
$738$	0	0
$739$	21.0564	0.774574	0.387287	−	0.921959i	$-0.373412\pi$
$739$	21.0564	0.774574	0.387287	+	0.921959i	$0.373412\pi$
$740$	4.80335	0.176575
$741$	0	0
$742$	34.1070	1.25211
$743$	33.6076	1.23294	0.616472	−	0.787377i	$-0.288561\pi$
$743$	33.6076	1.23294	0.616472	+	0.787377i	$0.288561\pi$
$744$	0	0
$745$	8.23442	0.301686
$746$	−4.91891	−0.180094
$747$	0	0
$748$	7.81521	0.285752
$749$	23.9786	0.876161
$750$	0	0
$751$	31.2823	1.14151	0.570754	−	0.821121i	$-0.306651\pi$
$751$	31.2823	1.14151	0.570754	+	0.821121i	$0.306651\pi$
$752$	8.61587	0.314188
$753$	0	0
$754$	3.30096	0.120214
$755$	−1.90941	−0.0694907
$756$	0	0
$757$	−18.8135	−0.683787	−0.341893	−	0.939739i	$-0.611068\pi$
$757$	−18.8135	−0.683787	−0.341893	+	0.939739i	$0.611068\pi$
$758$	−13.0692	−0.474696
$759$	0	0
$760$	−0.716881	−0.0260040
$761$	19.4507	0.705086	0.352543	−	0.935796i	$-0.385317\pi$
$761$	19.4507	0.705086	0.352543	+	0.935796i	$0.385317\pi$
$762$	0	0
$763$	4.94356	0.178969
$764$	−14.6973	−0.531729
$765$	0	0
$766$	−11.9581	−0.432064
$767$	5.63816	0.203582
$768$	0	0
$769$	−44.9299	−1.62022	−0.810108	−	0.586281i	$-0.800591\pi$
$769$	−44.9299	−1.62022	−0.810108	+	0.586281i	$0.800591\pi$
$770$	−1.81521	−0.0654155
$771$	0	0
$772$	20.0428	0.721357
$773$	−15.1016	−0.543168	−0.271584	−	0.962415i	$-0.587547\pi$
$773$	−15.1016	−0.543168	−0.271584	+	0.962415i	$0.587547\pi$
$774$	0	0
$775$	−19.6705	−0.706587
$776$	10.0419	0.360483
$777$	0	0
$778$	11.7219	0.420252
$779$	6.96585	0.249578
$780$	0	0
$781$	−3.55438	−0.127186
$782$	19.4165	0.694334
$783$	0	0
$784$	1.29086	0.0461021
$785$	8.86341	0.316349
$786$	0	0
$787$	34.2550	1.22106	0.610529	−	0.791994i	$-0.290957\pi$
$787$	34.2550	1.22106	0.610529	+	0.791994i	$0.290957\pi$
$788$	−1.74422	−0.0621354
$789$	0	0
$790$	−1.36959	−0.0487276
$791$	3.46791	0.123305
$792$	0	0
$793$	0.0436438	0.00154984
$794$	15.5398	0.551488
$795$	0	0
$796$	−5.00774	−0.177495
$797$	4.34493	0.153905	0.0769527	−	0.997035i	$-0.475481\pi$
$797$	4.34493	0.153905	0.0769527	+	0.997035i	$0.475481\pi$
$798$	0	0
$799$	56.8326	2.01059
$800$	4.71688	0.166767
$801$	0	0
$802$	−3.69553	−0.130494
$803$	−11.0077	−0.388455
$804$	0	0
$805$	−4.50980	−0.158950
$806$	3.22493	0.113593
$807$	0	0
$808$	11.3327	0.398685
$809$	−25.5458	−0.898143	−0.449072	−	0.893496i	$-0.648245\pi$
$809$	−25.5458	−0.898143	−0.449072	+	0.893496i	$0.648245\pi$
$810$	0	0
$811$	18.1679	0.637961	0.318980	−	0.947761i	$-0.396660\pi$
$811$	18.1679	0.637961	0.318980	+	0.947761i	$0.396660\pi$
$812$	−12.2909	−0.431325
$813$	0	0
$814$	10.6955	0.374878
$815$	−3.51754	−0.123214
$816$	0	0
$817$	−1.40373	−0.0491104
$818$	−0.763823	−0.0267064
$819$	0	0
$820$	2.75103	0.0960701
$821$	−8.34318	−0.291179	−0.145589	−	0.989345i	$-0.546508\pi$
$821$	−8.34318	−0.291179	−0.145589	+	0.989345i	$0.546508\pi$
$822$	0	0
$823$	27.7888	0.968657	0.484328	−	0.874886i	$-0.339064\pi$
$823$	27.7888	0.968657	0.484328	+	0.874886i	$0.339064\pi$
$824$	−0.128356	−0.00447148
$825$	0	0
$826$	−20.9932	−0.730447
$827$	−5.43470	−0.188983	−0.0944915	−	0.995526i	$-0.530123\pi$
$827$	−5.43470	−0.188983	−0.0944915	+	0.995526i	$0.530123\pi$
$828$	0	0
$829$	−20.3946	−0.708332	−0.354166	−	0.935182i	$-0.615235\pi$
$829$	−20.3946	−0.708332	−0.354166	+	0.935182i	$0.615235\pi$
$830$	4.17436	0.144894
$831$	0	0
$832$	−0.773318	−0.0268100
$833$	8.51485	0.295022
$834$	0	0
$835$	−7.74691	−0.268093
$836$	−1.59627	−0.0552080
$837$	0	0
$838$	10.6604	0.368259
$839$	14.5909	0.503733	0.251867	−	0.967762i	$-0.418956\pi$
$839$	14.5909	0.503733	0.251867	+	0.967762i	$0.418956\pi$
$840$	0	0
$841$	−10.7793	−0.371700
$842$	22.8675	0.788067
$843$	0	0
$844$	−23.8776	−0.821902
$845$	6.59896	0.227011
$846$	0	0
$847$	27.6313	0.949424
$848$	11.8452	0.406767
$849$	0	0
$850$	31.1138	1.06719
$851$	26.5725	0.910895
$852$	0	0
$853$	42.8539	1.46729	0.733645	−	0.679533i	$-0.237818\pi$
$853$	42.8539	1.46729	0.733645	+	0.679533i	$0.237818\pi$
$854$	−0.162504	−0.00556077
$855$	0	0
$856$	8.32770	0.284635
$857$	−27.0292	−0.923300	−0.461650	−	0.887062i	$-0.652742\pi$
$857$	−27.0292	−0.923300	−0.461650	+	0.887062i	$0.652742\pi$
$858$	0	0
$859$	24.2449	0.827224	0.413612	−	0.910453i	$-0.364267\pi$
$859$	24.2449	0.827224	0.413612	+	0.910453i	$0.364267\pi$
$860$	−0.554378	−0.0189041
$861$	0	0
$862$	−39.1780	−1.33441
$863$	37.4115	1.27350	0.636751	−	0.771070i	$-0.280278\pi$
$863$	37.4115	1.27350	0.636751	+	0.771070i	$0.280278\pi$
$864$	0	0
$865$	11.0564	0.375930
$866$	20.6483	0.701656
$867$	0	0
$868$	−12.0077	−0.407569
$869$	−3.04963	−0.103452
$870$	0	0
$871$	−7.29921	−0.247324
$872$	1.71688	0.0581409
$873$	0	0
$874$	−3.96585	−0.134147
$875$	−14.8871	−0.503277
$876$	0	0
$877$	−41.5699	−1.40371	−0.701857	−	0.712318i	$-0.747646\pi$
$877$	−41.5699	−1.40371	−0.701857	+	0.712318i	$0.747646\pi$
$878$	37.5158	1.26610
$879$	0	0
$880$	−0.630415	−0.0212513
$881$	6.85616	0.230990	0.115495	−	0.993308i	$-0.463155\pi$
$881$	6.85616	0.230990	0.115495	+	0.993308i	$0.463155\pi$
$882$	0	0
$883$	24.4442	0.822613	0.411306	−	0.911497i	$-0.365073\pi$
$883$	24.4442	0.822613	0.411306	+	0.911497i	$0.365073\pi$
$884$	−5.10101	−0.171566
$885$	0	0
$886$	−12.1138	−0.406971
$887$	37.7992	1.26917	0.634587	−	0.772851i	$-0.281170\pi$
$887$	37.7992	1.26917	0.634587	+	0.772851i	$0.281170\pi$
$888$	0	0
$889$	−48.0506	−1.61156
$890$	5.35916	0.179639
$891$	0	0
$892$	1.00000	0.0334825
$893$	−11.6081	−0.388451
$894$	0	0
$895$	−5.10244	−0.170556
$896$	2.87939	0.0961935
$897$	0	0
$898$	−4.43376	−0.147957
$899$	17.8010	0.593696
$900$	0	0
$901$	78.1343	2.60303
$902$	6.12567	0.203962
$903$	0	0
$904$	1.20439	0.0400575
$905$	−0.953992	−0.0317118
$906$	0	0
$907$	−56.9323	−1.89041	−0.945203	−	0.326483i	$-0.894137\pi$
$907$	−56.9323	−1.89041	−0.945203	+	0.326483i	$0.894137\pi$
$908$	25.2841	0.839081
$909$	0	0
$910$	1.18479	0.0392755
$911$	28.6272	0.948462	0.474231	−	0.880400i	$-0.342726\pi$
$911$	28.6272	0.948462	0.474231	+	0.880400i	$0.342726\pi$
$912$	0	0
$913$	9.29498	0.307619
$914$	20.5621	0.680134
$915$	0	0
$916$	−9.02229	−0.298105
$917$	38.8658	1.28346
$918$	0	0
$919$	48.5948	1.60300	0.801498	−	0.597998i	$-0.204037\pi$
$919$	48.5948	1.60300	0.801498	+	0.597998i	$0.204037\pi$
$920$	−1.56624	−0.0516373
$921$	0	0
$922$	−9.31996	−0.306936
$923$	2.31996	0.0763623
$924$	0	0
$925$	42.5809	1.40005
$926$	32.8881	1.08077
$927$	0	0
$928$	−4.26857	−0.140123
$929$	−37.3806	−1.22642	−0.613209	−	0.789921i	$-0.710122\pi$
$929$	−37.3806	−1.22642	−0.613209	+	0.789921i	$0.710122\pi$
$930$	0	0
$931$	−1.73917	−0.0569990
$932$	−21.6117	−0.707916
$933$	0	0
$934$	18.2618	0.597543
$935$	−4.15839	−0.135994
$936$	0	0
$937$	56.5886	1.84867	0.924335	−	0.381582i	$-0.124621\pi$
$937$	56.5886	1.84867	0.924335	+	0.381582i	$0.124621\pi$
$938$	27.1780	0.887393
$939$	0	0
$940$	−4.58441	−0.149527
$941$	−29.5841	−0.964414	−0.482207	−	0.876057i	$-0.660165\pi$
$941$	−29.5841	−0.964414	−0.482207	+	0.876057i	$0.660165\pi$
$942$	0	0
$943$	15.2189	0.495597
$944$	−7.29086	−0.237297
$945$	0	0
$946$	−1.23442	−0.0401345
$947$	20.6922	0.672407	0.336204	−	0.941789i	$-0.390857\pi$
$947$	20.6922	0.672407	0.336204	+	0.941789i	$0.390857\pi$
$948$	0	0
$949$	7.18479	0.233228
$950$	−6.35504	−0.206185
$951$	0	0
$952$	18.9932	0.615573
$953$	23.7306	0.768710	0.384355	−	0.923185i	$-0.374424\pi$
$953$	23.7306	0.768710	0.384355	+	0.923185i	$0.374424\pi$
$954$	0	0
$955$	7.82026	0.253058
$956$	−11.8203	−0.382295
$957$	0	0
$958$	−41.6742	−1.34643
$959$	50.5039	1.63086
$960$	0	0
$961$	−13.6091	−0.439002
$962$	−6.98101	−0.225077
$963$	0	0
$964$	9.22762	0.297201
$965$	−10.6646	−0.343304
$966$	0	0
$967$	−8.04963	−0.258859	−0.129429	−	0.991589i	$-0.541315\pi$
$967$	−8.04963	−0.258859	−0.129429	+	0.991589i	$0.541315\pi$
$968$	9.59627	0.308436
$969$	0	0
$970$	−5.34318	−0.171559
$971$	−34.3560	−1.10254	−0.551268	−	0.834328i	$-0.685856\pi$
$971$	−34.3560	−1.10254	−0.551268	+	0.834328i	$0.685856\pi$
$972$	0	0
$973$	45.0087	1.44291
$974$	20.1584	0.645916
$975$	0	0
$976$	−0.0564370	−0.00180650
$977$	44.9195	1.43710	0.718551	−	0.695474i	$-0.244806\pi$
$977$	44.9195	1.43710	0.718551	+	0.695474i	$0.244806\pi$
$978$	0	0
$979$	11.9331	0.381385
$980$	−0.686852	−0.0219407
$981$	0	0
$982$	19.7547	0.630396
$983$	6.01361	0.191805	0.0959023	−	0.995391i	$-0.469426\pi$
$983$	6.01361	0.191805	0.0959023	+	0.995391i	$0.469426\pi$
$984$	0	0
$985$	0.928081	0.0295711
$986$	−28.1566	−0.896690
$987$	0	0
$988$	1.04189	0.0331469
$989$	−3.06687	−0.0975207
$990$	0	0
$991$	24.0583	0.764237	0.382119	−	0.924113i	$-0.375195\pi$
$991$	24.0583	0.764237	0.382119	+	0.924113i	$0.375195\pi$
$992$	−4.17024	−0.132405
$993$	0	0
$994$	−8.63816	−0.273986
$995$	2.66456	0.0844723
$996$	0	0
$997$	0.127422	0.00403549	0.00201775	−	0.999998i	$-0.499358\pi$
$997$	0.127422	0.00403549	0.00201775	+	0.999998i	$0.499358\pi$
$998$	5.21894	0.165203
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4014.2.a.m.1.1		3
3.2	odd	2		1338.2.a.e.1.3	✓	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1338.2.a.e.1.3	✓	3	3.2	odd	2
4014.2.a.m.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.532089 q^{5} -2.87939 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -0.532089 q^{5} -2.87939 q^{7} -1.00000 q^{8} +0.532089 q^{10} +1.18479 q^{11} -0.773318 q^{13} +2.87939 q^{14} +1.00000 q^{16} +6.59627 q^{17} -1.34730 q^{19} -0.532089 q^{20} -1.18479 q^{22} -2.94356 q^{23} -4.71688 q^{25} +0.773318 q^{26} -2.87939 q^{28} +4.26857 q^{29} +4.17024 q^{31} -1.00000 q^{32} -6.59627 q^{34} +1.53209 q^{35} -9.02734 q^{37} +1.34730 q^{38} +0.532089 q^{40} -5.17024 q^{41} +1.04189 q^{43} +1.18479 q^{44} +2.94356 q^{46} +8.61587 q^{47} +1.29086 q^{49} +4.71688 q^{50} -0.773318 q^{52} +11.8452 q^{53} -0.630415 q^{55} +2.87939 q^{56} -4.26857 q^{58} -7.29086 q^{59} -0.0564370 q^{61} -4.17024 q^{62} +1.00000 q^{64} +0.411474 q^{65} +9.43882 q^{67} +6.59627 q^{68} -1.53209 q^{70} -3.00000 q^{71} -9.29086 q^{73} +9.02734 q^{74} -1.34730 q^{76} -3.41147 q^{77} -2.57398 q^{79} -0.532089 q^{80} +5.17024 q^{82} +7.84524 q^{83} -3.50980 q^{85} -1.04189 q^{86} -1.18479 q^{88} +10.0719 q^{89} +2.22668 q^{91} -2.94356 q^{92} -8.61587 q^{94} +0.716881 q^{95} -10.0419 q^{97} -1.29086 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} - 3 q^{8} - 3 q^{10} - 9 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} - 3 q^{19} + 3 q^{20} + 6 q^{23} - 6 q^{25} + 9 q^{26} - 3 q^{28} + 3 q^{29} - 9 q^{31} - 3 q^{32} - 6 q^{34} - 6 q^{37} + 3 q^{38} - 3 q^{40} + 6 q^{41} - 6 q^{46} + 15 q^{47} - 12 q^{49} + 6 q^{50} - 9 q^{52} + 9 q^{53} - 9 q^{55} + 3 q^{56} - 3 q^{58} - 6 q^{59} - 15 q^{61} + 9 q^{62} + 3 q^{64} - 9 q^{65} - 3 q^{67} + 6 q^{68} - 9 q^{71} - 12 q^{73} + 6 q^{74} - 3 q^{76} + 3 q^{80} - 6 q^{82} - 3 q^{83} - 12 q^{85} - 3 q^{89} + 6 q^{92} - 15 q^{94} - 6 q^{95} - 27 q^{97} + 12 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 - 3 * q^8 - 3 * q^10 - 9 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 - 3 * q^19 + 3 * q^20 + 6 * q^23 - 6 * q^25 + 9 * q^26 - 3 * q^28 + 3 * q^29 - 9 * q^31 - 3 * q^32 - 6 * q^34 - 6 * q^37 + 3 * q^38 - 3 * q^40 + 6 * q^41 - 6 * q^46 + 15 * q^47 - 12 * q^49 + 6 * q^50 - 9 * q^52 + 9 * q^53 - 9 * q^55 + 3 * q^56 - 3 * q^58 - 6 * q^59 - 15 * q^61 + 9 * q^62 + 3 * q^64 - 9 * q^65 - 3 * q^67 + 6 * q^68 - 9 * q^71 - 12 * q^73 + 6 * q^74 - 3 * q^76 + 3 * q^80 - 6 * q^82 - 3 * q^83 - 12 * q^85 - 3 * q^89 + 6 * q^92 - 15 * q^94 - 6 * q^95 - 27 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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