LMFDB - Embedded newform 4923.2.a.l.1.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4923,2,Mod(1,4923)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4923, 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("4923.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4923 = 3^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4923.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:	$39.3103529151$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.13
Root			$1.35726$ of defining polynomial
Character	$\chi$	$=$	4923.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35726 q^{2} -0.157846 q^{4} +1.46929 q^{5} +0.194696 q^{7} -2.92876 q^{8} +O(q^{10})$	\(q+1.35726 q^{2} -0.157846 q^{4} +1.46929 q^{5} +0.194696 q^{7} -2.92876 q^{8} +1.99421 q^{10} -1.87259 q^{11} +6.15611 q^{13} +0.264253 q^{14} -3.65939 q^{16} +5.37804 q^{17} -3.78340 q^{19} -0.231922 q^{20} -2.54160 q^{22} +3.43075 q^{23} -2.84119 q^{25} +8.35544 q^{26} -0.0307320 q^{28} -0.0833365 q^{29} -3.95043 q^{31} +0.890769 q^{32} +7.29940 q^{34} +0.286065 q^{35} +3.24239 q^{37} -5.13506 q^{38} -4.30319 q^{40} +12.5362 q^{41} +2.32367 q^{43} +0.295581 q^{44} +4.65642 q^{46} +8.88803 q^{47} -6.96209 q^{49} -3.85623 q^{50} -0.971717 q^{52} -7.04808 q^{53} -2.75138 q^{55} -0.570218 q^{56} -0.113109 q^{58} -7.89744 q^{59} -2.95043 q^{61} -5.36176 q^{62} +8.52779 q^{64} +9.04511 q^{65} -1.41048 q^{67} -0.848903 q^{68} +0.388265 q^{70} +11.1843 q^{71} -0.418335 q^{73} +4.40076 q^{74} +0.597195 q^{76} -0.364587 q^{77} +0.962128 q^{79} -5.37671 q^{80} +17.0149 q^{82} +15.7469 q^{83} +7.90190 q^{85} +3.15383 q^{86} +5.48437 q^{88} +9.44358 q^{89} +1.19857 q^{91} -0.541530 q^{92} +12.0634 q^{94} -5.55892 q^{95} +11.6561 q^{97} -9.44937 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * 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 $

$2$	1.35726	0.959728	0.479864	−	0.877343i	$-0.340686\pi$
$2$	1.35726	0.959728	0.479864	+	0.877343i	$0.340686\pi$
$3$	0	0
$3$	0	0
$4$	−0.157846	−0.0789230
$5$	1.46929	0.657086	0.328543	−	0.944489i	$-0.393442\pi$
$5$	1.46929	0.657086	0.328543	+	0.944489i	$0.393442\pi$
$6$	0	0
$7$	0.194696	0.0735883	0.0367941	−	0.999323i	$-0.488285\pi$
$7$	0.194696	0.0735883	0.0367941	+	0.999323i	$0.488285\pi$
$8$	−2.92876	−1.03547
$9$	0	0
$10$	1.99421	0.630624
$11$	−1.87259	−0.564608	−0.282304	−	0.959325i	$-0.591099\pi$
$11$	−1.87259	−0.564608	−0.282304	+	0.959325i	$0.591099\pi$
$12$	0	0
$13$	6.15611	1.70740	0.853698	−	0.520768i	$-0.174354\pi$
$13$	6.15611	1.70740	0.853698	+	0.520768i	$0.174354\pi$
$14$	0.264253	0.0706247
$15$	0	0
$16$	−3.65939	−0.914848
$17$	5.37804	1.30437	0.652183	−	0.758061i	$-0.273853\pi$
$17$	5.37804	1.30437	0.652183	+	0.758061i	$0.273853\pi$
$18$	0	0
$19$	−3.78340	−0.867972	−0.433986	−	0.900920i	$-0.642893\pi$
$19$	−3.78340	−0.867972	−0.433986	+	0.900920i	$0.642893\pi$
$20$	−0.231922	−0.0518593
$21$	0	0
$22$	−2.54160	−0.541870
$23$	3.43075	0.715361	0.357680	−	0.933844i	$-0.383568\pi$
$23$	3.43075	0.715361	0.357680	+	0.933844i	$0.383568\pi$
$24$	0	0
$25$	−2.84119	−0.568237
$26$	8.35544	1.63864
$27$	0	0
$28$	−0.0307320	−0.00580781
$29$	−0.0833365	−0.0154752	−0.00773760	−	0.999970i	$-0.502463\pi$
$29$	−0.0833365	−0.0154752	−0.00773760	+	0.999970i	$0.502463\pi$
$30$	0	0
$31$	−3.95043	−0.709518	−0.354759	−	0.934958i	$-0.615437\pi$
$31$	−3.95043	−0.709518	−0.354759	+	0.934958i	$0.615437\pi$
$32$	0.890769	0.157467
$33$	0	0
$34$	7.29940	1.25184
$35$	0.286065	0.0483539
$36$	0	0
$37$	3.24239	0.533045	0.266522	−	0.963829i	$-0.414125\pi$
$37$	3.24239	0.533045	0.266522	+	0.963829i	$0.414125\pi$
$38$	−5.13506	−0.833017
$39$	0	0
$40$	−4.30319	−0.680395
$41$	12.5362	1.95783	0.978915	−	0.204269i	$-0.0654816\pi$
$41$	12.5362	1.95783	0.978915	+	0.204269i	$0.0654816\pi$
$42$	0	0
$43$	2.32367	0.354357	0.177178	−	0.984179i	$-0.443303\pi$
$43$	2.32367	0.354357	0.177178	+	0.984179i	$0.443303\pi$
$44$	0.295581	0.0445606
$45$	0	0
$46$	4.65642	0.686552
$47$	8.88803	1.29645	0.648226	−	0.761448i	$-0.275511\pi$
$47$	8.88803	1.29645	0.648226	+	0.761448i	$0.275511\pi$
$48$	0	0
$49$	−6.96209	−0.994585
$50$	−3.85623	−0.545353
$51$	0	0
$52$	−0.971717	−0.134753
$53$	−7.04808	−0.968128	−0.484064	−	0.875033i	$-0.660840\pi$
$53$	−7.04808	−0.968128	−0.484064	+	0.875033i	$0.660840\pi$
$54$	0	0
$55$	−2.75138	−0.370996
$56$	−0.570218	−0.0761986
$57$	0	0
$58$	−0.113109	−0.0148520
$59$	−7.89744	−1.02816	−0.514080	−	0.857742i	$-0.671866\pi$
$59$	−7.89744	−1.02816	−0.514080	+	0.857742i	$0.671866\pi$
$60$	0	0
$61$	−2.95043	−0.377764	−0.188882	−	0.982000i	$-0.560486\pi$
$61$	−2.95043	−0.377764	−0.188882	+	0.982000i	$0.560486\pi$
$62$	−5.36176	−0.680944
$63$	0	0
$64$	8.52779	1.06597
$65$	9.04511	1.12191
$66$	0	0
$67$	−1.41048	−0.172318	−0.0861589	−	0.996281i	$-0.527459\pi$
$67$	−1.41048	−0.172318	−0.0861589	+	0.996281i	$0.527459\pi$
$68$	−0.848903	−0.102945
$69$	0	0
$70$	0.388265	0.0464065
$71$	11.1843	1.32733	0.663667	−	0.748029i	$-0.268999\pi$
$71$	11.1843	1.32733	0.663667	+	0.748029i	$0.268999\pi$
$72$	0	0
$73$	−0.418335	−0.0489624	−0.0244812	−	0.999700i	$-0.507793\pi$
$73$	−0.418335	−0.0489624	−0.0244812	+	0.999700i	$0.507793\pi$
$74$	4.40076	0.511578
$75$	0	0
$76$	0.597195	0.0685030
$77$	−0.364587	−0.0415485
$78$	0	0
$79$	0.962128	0.108248	0.0541239	−	0.998534i	$-0.482763\pi$
$79$	0.962128	0.108248	0.0541239	+	0.998534i	$0.482763\pi$
$80$	−5.37671	−0.601134
$81$	0	0
$82$	17.0149	1.87898
$83$	15.7469	1.72844	0.864222	−	0.503110i	$-0.167811\pi$
$83$	15.7469	1.72844	0.864222	+	0.503110i	$0.167811\pi$
$84$	0	0
$85$	7.90190	0.857082
$86$	3.15383	0.340086
$87$	0	0
$88$	5.48437	0.584636
$89$	9.44358	1.00102	0.500509	−	0.865732i	$-0.333146\pi$
$89$	9.44358	1.00102	0.500509	+	0.865732i	$0.333146\pi$
$90$	0	0
$91$	1.19857	0.125644
$92$	−0.541530	−0.0564585
$93$	0	0
$94$	12.0634	1.24424
$95$	−5.55892	−0.570333
$96$	0	0
$97$	11.6561	1.18350	0.591749	−	0.806122i	$-0.298438\pi$
$97$	11.6561	1.18350	0.591749	+	0.806122i	$0.298438\pi$
$98$	−9.44937	−0.954530
$99$	0	0
$100$	0.448470	0.0448470
$101$	6.90248	0.686822	0.343411	−	0.939185i	$-0.388418\pi$
$101$	6.90248	0.686822	0.343411	+	0.939185i	$0.388418\pi$
$102$	0	0
$103$	2.77350	0.273281	0.136640	−	0.990621i	$-0.456370\pi$
$103$	2.77350	0.273281	0.136640	+	0.990621i	$0.456370\pi$
$104$	−18.0297	−1.76796
$105$	0	0
$106$	−9.56607	−0.929139
$107$	3.43676	0.332244	0.166122	−	0.986105i	$-0.446875\pi$
$107$	3.43676	0.332244	0.166122	+	0.986105i	$0.446875\pi$
$108$	0	0
$109$	14.3014	1.36982	0.684912	−	0.728626i	$-0.259841\pi$
$109$	14.3014	1.36982	0.684912	+	0.728626i	$0.259841\pi$
$110$	−3.73434	−0.356055
$111$	0	0
$112$	−0.712470	−0.0673221
$113$	−5.25192	−0.494059	−0.247029	−	0.969008i	$-0.579454\pi$
$113$	−5.25192	−0.494059	−0.247029	+	0.969008i	$0.579454\pi$
$114$	0	0
$115$	5.04077	0.470054
$116$	0.0131543	0.00122135
$117$	0	0
$118$	−10.7189	−0.986753
$119$	1.04708	0.0959861
$120$	0	0
$121$	−7.49340	−0.681218
$122$	−4.00450	−0.362550
$123$	0	0
$124$	0.623560	0.0559973
$125$	−11.5210	−1.03047
$126$	0	0
$127$	−6.29760	−0.558822	−0.279411	−	0.960172i	$-0.590139\pi$
$127$	−6.29760	−0.558822	−0.279411	+	0.960172i	$0.590139\pi$
$128$	9.79289	0.865577
$129$	0	0
$130$	12.2766	1.07673
$131$	−13.4030	−1.17102	−0.585512	−	0.810663i	$-0.699107\pi$
$131$	−13.4030	−1.17102	−0.585512	+	0.810663i	$0.699107\pi$
$132$	0	0
$133$	−0.736615	−0.0638726
$134$	−1.91439	−0.165378
$135$	0	0
$136$	−15.7510	−1.35064
$137$	0.171047	0.0146136	0.00730678	−	0.999973i	$-0.497674\pi$
$137$	0.171047	0.0146136	0.00730678	+	0.999973i	$0.497674\pi$
$138$	0	0
$139$	13.8883	1.17799	0.588995	−	0.808137i	$-0.299524\pi$
$139$	13.8883	1.17799	0.588995	+	0.808137i	$0.299524\pi$
$140$	−0.0451543	−0.00381623
$141$	0	0
$142$	15.1800	1.27388
$143$	−11.5279	−0.964010
$144$	0	0
$145$	−0.122445	−0.0101685
$146$	−0.567790	−0.0469906
$147$	0	0
$148$	−0.511798	−0.0420695
$149$	12.9947	1.06457	0.532286	−	0.846565i	$-0.321333\pi$
$149$	12.9947	1.06457	0.532286	+	0.846565i	$0.321333\pi$
$150$	0	0
$151$	17.0458	1.38716	0.693582	−	0.720378i	$-0.256031\pi$
$151$	17.0458	1.38716	0.693582	+	0.720378i	$0.256031\pi$
$152$	11.0807	0.898761
$153$	0	0
$154$	−0.494839	−0.0398753
$155$	−5.80433	−0.466215
$156$	0	0
$157$	3.46375	0.276437	0.138219	−	0.990402i	$-0.455862\pi$
$157$	3.46375	0.276437	0.138219	+	0.990402i	$0.455862\pi$
$158$	1.30586	0.103888
$159$	0	0
$160$	1.30880	0.103470
$161$	0.667954	0.0526422
$162$	0	0
$163$	8.68759	0.680464	0.340232	−	0.940341i	$-0.389494\pi$
$163$	8.68759	0.680464	0.340232	+	0.940341i	$0.389494\pi$
$164$	−1.97879	−0.154518
$165$	0	0
$166$	21.3726	1.65884
$167$	−6.93900	−0.536956	−0.268478	−	0.963286i	$-0.586521\pi$
$167$	−6.93900	−0.536956	−0.268478	+	0.963286i	$0.586521\pi$
$168$	0	0
$169$	24.8977	1.91520
$170$	10.7249	0.822565
$171$	0	0
$172$	−0.366783	−0.0279669
$173$	8.06411	0.613103	0.306552	−	0.951854i	$-0.400825\pi$
$173$	8.06411	0.613103	0.306552	+	0.951854i	$0.400825\pi$
$174$	0	0
$175$	−0.553169	−0.0418156
$176$	6.85255	0.516531
$177$	0	0
$178$	12.8174	0.960704
$179$	4.43362	0.331384	0.165692	−	0.986178i	$-0.447014\pi$
$179$	4.43362	0.331384	0.165692	+	0.986178i	$0.447014\pi$
$180$	0	0
$181$	15.5528	1.15603	0.578015	−	0.816026i	$-0.303828\pi$
$181$	15.5528	1.15603	0.578015	+	0.816026i	$0.303828\pi$
$182$	1.62677	0.120584
$183$	0	0
$184$	−10.0478	−0.740736
$185$	4.76400	0.350257
$186$	0	0
$187$	−10.0709	−0.736456
$188$	−1.40294	−0.102320
$189$	0	0
$190$	−7.54489	−0.547364
$191$	−22.5547	−1.63200	−0.816000	−	0.578052i	$-0.803813\pi$
$191$	−22.5547	−1.63200	−0.816000	+	0.578052i	$0.803813\pi$
$192$	0	0
$193$	−21.8042	−1.56950	−0.784751	−	0.619811i	$-0.787209\pi$
$193$	−21.8042	−1.56950	−0.784751	+	0.619811i	$0.787209\pi$
$194$	15.8204	1.13584
$195$	0	0
$196$	1.09894	0.0784956
$197$	17.3331	1.23493	0.617465	−	0.786598i	$-0.288160\pi$
$197$	17.3331	1.23493	0.617465	+	0.786598i	$0.288160\pi$
$198$	0	0
$199$	−8.74283	−0.619763	−0.309881	−	0.950775i	$-0.600289\pi$
$199$	−8.74283	−0.619763	−0.309881	+	0.950775i	$0.600289\pi$
$200$	8.32115	0.588394
$201$	0	0
$202$	9.36845	0.659162
$203$	−0.0162253	−0.00113879
$204$	0	0
$205$	18.4194	1.28646
$206$	3.76435	0.262275
$207$	0	0
$208$	−22.5276	−1.56201
$209$	7.08477	0.490064
$210$	0	0
$211$	−15.9290	−1.09659	−0.548297	−	0.836284i	$-0.684724\pi$
$211$	−15.9290	−1.09659	−0.548297	+	0.836284i	$0.684724\pi$
$212$	1.11251	0.0764076
$213$	0	0
$214$	4.66458	0.318864
$215$	3.41415	0.232843
$216$	0	0
$217$	−0.769134	−0.0522122
$218$	19.4107	1.31466
$219$	0	0
$220$	0.434295	0.0292802
$221$	33.1078	2.22707
$222$	0	0
$223$	−14.0059	−0.937903	−0.468952	−	0.883224i	$-0.655368\pi$
$223$	−14.0059	−0.937903	−0.468952	+	0.883224i	$0.655368\pi$
$224$	0.173429	0.0115877
$225$	0	0
$226$	−7.12821	−0.474162
$227$	0.284670	0.0188942	0.00944712	−	0.999955i	$-0.496993\pi$
$227$	0.284670	0.0188942	0.00944712	+	0.999955i	$0.496993\pi$
$228$	0	0
$229$	−20.0021	−1.32177	−0.660887	−	0.750486i	$-0.729820\pi$
$229$	−20.0021	−1.32177	−0.660887	+	0.750486i	$0.729820\pi$
$230$	6.84163	0.451124
$231$	0	0
$232$	0.244072	0.0160241
$233$	11.1194	0.728459	0.364229	−	0.931309i	$-0.381332\pi$
$233$	11.1194	0.728459	0.364229	+	0.931309i	$0.381332\pi$
$234$	0	0
$235$	13.0591	0.851881
$236$	1.24658	0.0811454
$237$	0	0
$238$	1.42117	0.0921205
$239$	2.37885	0.153875	0.0769375	−	0.997036i	$-0.475486\pi$
$239$	2.37885	0.153875	0.0769375	+	0.997036i	$0.475486\pi$
$240$	0	0
$241$	9.34293	0.601831	0.300915	−	0.953651i	$-0.402708\pi$
$241$	9.34293	0.601831	0.300915	+	0.953651i	$0.402708\pi$
$242$	−10.1705	−0.653783
$243$	0	0
$244$	0.465714	0.0298143
$245$	−10.2293	−0.653528
$246$	0	0
$247$	−23.2910	−1.48197
$248$	11.5698	0.734686
$249$	0	0
$250$	−15.6370	−0.988968
$251$	−18.0060	−1.13653	−0.568265	−	0.822846i	$-0.692385\pi$
$251$	−18.0060	−1.13653	−0.568265	+	0.822846i	$0.692385\pi$
$252$	0	0
$253$	−6.42440	−0.403899
$254$	−8.54748	−0.536316
$255$	0	0
$256$	−3.76409	−0.235255
$257$	27.8759	1.73885	0.869425	−	0.494065i	$-0.164490\pi$
$257$	27.8759	1.73885	0.869425	+	0.494065i	$0.164490\pi$
$258$	0	0
$259$	0.631280	0.0392259
$260$	−1.42773	−0.0885443
$261$	0	0
$262$	−18.1913	−1.12386
$263$	7.92612	0.488745	0.244373	−	0.969681i	$-0.421418\pi$
$263$	7.92612	0.488745	0.244373	+	0.969681i	$0.421418\pi$
$264$	0	0
$265$	−10.3557	−0.636144
$266$	−0.999777	−0.0613003
$267$	0	0
$268$	0.222639	0.0135998
$269$	11.9964	0.731436	0.365718	−	0.930726i	$-0.380823\pi$
$269$	11.9964	0.731436	0.365718	+	0.930726i	$0.380823\pi$
$270$	0	0
$271$	15.2592	0.926929	0.463465	−	0.886115i	$-0.346606\pi$
$271$	15.2592	0.926929	0.463465	+	0.886115i	$0.346606\pi$
$272$	−19.6804	−1.19330
$273$	0	0
$274$	0.232156	0.0140250
$275$	5.32039	0.320831
$276$	0	0
$277$	−12.3368	−0.741247	−0.370624	−	0.928783i	$-0.620856\pi$
$277$	−12.3368	−0.741247	−0.370624	+	0.928783i	$0.620856\pi$
$278$	18.8500	1.13055
$279$	0	0
$280$	−0.837816	−0.0500691
$281$	−3.53084	−0.210632	−0.105316	−	0.994439i	$-0.533585\pi$
$281$	−3.53084	−0.210632	−0.105316	+	0.994439i	$0.533585\pi$
$282$	0	0
$283$	−19.8205	−1.17820	−0.589102	−	0.808059i	$-0.700518\pi$
$283$	−19.8205	−1.17820	−0.589102	+	0.808059i	$0.700518\pi$
$284$	−1.76540	−0.104757
$285$	0	0
$286$	−15.6463	−0.925187
$287$	2.44076	0.144073
$288$	0	0
$289$	11.9233	0.701372
$290$	−0.166190	−0.00975903
$291$	0	0
$292$	0.0660326	0.00386426
$293$	−2.84565	−0.166244	−0.0831222	−	0.996539i	$-0.526489\pi$
$293$	−2.84565	−0.166244	−0.0831222	+	0.996539i	$0.526489\pi$
$294$	0	0
$295$	−11.6036	−0.675590
$296$	−9.49616	−0.551953
$297$	0	0
$298$	17.6372	1.02170
$299$	21.1201	1.22141
$300$	0	0
$301$	0.452411	0.0260765
$302$	23.1355	1.33130
$303$	0	0
$304$	13.8450	0.794063
$305$	−4.33504	−0.248224
$306$	0	0
$307$	14.8470	0.847362	0.423681	−	0.905811i	$-0.360738\pi$
$307$	14.8470	0.847362	0.423681	+	0.905811i	$0.360738\pi$
$308$	0.0575486	0.00327914
$309$	0	0
$310$	−7.87798	−0.447439
$311$	−27.9972	−1.58758	−0.793789	−	0.608194i	$-0.791894\pi$
$311$	−27.9972	−1.58758	−0.793789	+	0.608194i	$0.791894\pi$
$312$	0	0
$313$	5.53601	0.312914	0.156457	−	0.987685i	$-0.449993\pi$
$313$	5.53601	0.312914	0.156457	+	0.987685i	$0.449993\pi$
$314$	4.70121	0.265305
$315$	0	0
$316$	−0.151868	−0.00854325
$317$	4.87056	0.273558	0.136779	−	0.990602i	$-0.456325\pi$
$317$	4.87056	0.273558	0.136779	+	0.990602i	$0.456325\pi$
$318$	0	0
$319$	0.156055	0.00873742
$320$	12.5298	0.700437
$321$	0	0
$322$	0.906588	0.0505222
$323$	−20.3473	−1.13215
$324$	0	0
$325$	−17.4907	−0.970207
$326$	11.7913	0.653060
$327$	0	0
$328$	−36.7156	−2.02728
$329$	1.73047	0.0954037
$330$	0	0
$331$	33.0775	1.81810	0.909051	−	0.416686i	$-0.136808\pi$
$331$	33.0775	1.81810	0.909051	+	0.416686i	$0.136808\pi$
$332$	−2.48558	−0.136414
$333$	0	0
$334$	−9.41802	−0.515331
$335$	−2.07241	−0.113228
$336$	0	0
$337$	−4.05736	−0.221018	−0.110509	−	0.993875i	$-0.535248\pi$
$337$	−4.05736	−0.221018	−0.110509	+	0.993875i	$0.535248\pi$
$338$	33.7926	1.83807
$339$	0	0
$340$	−1.24728	−0.0676435
$341$	7.39755	0.400600
$342$	0	0
$343$	−2.71837	−0.146778
$344$	−6.80548	−0.366927
$345$	0	0
$346$	10.9451	0.588412
$347$	−16.2823	−0.874080	−0.437040	−	0.899442i	$-0.643973\pi$
$347$	−16.2823	−0.874080	−0.437040	+	0.899442i	$0.643973\pi$
$348$	0	0
$349$	1.78004	0.0952835	0.0476418	−	0.998864i	$-0.484829\pi$
$349$	1.78004	0.0952835	0.0476418	+	0.998864i	$0.484829\pi$
$350$	−0.750793	−0.0401316
$351$	0	0
$352$	−1.66805	−0.0889073
$353$	−30.6765	−1.63274	−0.816372	−	0.577527i	$-0.804018\pi$
$353$	−30.6765	−1.63274	−0.816372	+	0.577527i	$0.804018\pi$
$354$	0	0
$355$	16.4330	0.872173
$356$	−1.49063	−0.0790033
$357$	0	0
$358$	6.01757	0.318039
$359$	−33.7830	−1.78300	−0.891499	−	0.453024i	$-0.850345\pi$
$359$	−33.7830	−1.78300	−0.891499	+	0.453024i	$0.850345\pi$
$360$	0	0
$361$	−4.68586	−0.246624
$362$	21.1092	1.10947
$363$	0	0
$364$	−0.189190	−0.00991624
$365$	−0.614656	−0.0321726
$366$	0	0
$367$	−35.3670	−1.84614	−0.923070	−	0.384632i	$-0.874328\pi$
$367$	−35.3670	−1.84614	−0.923070	+	0.384632i	$0.874328\pi$
$368$	−12.5545	−0.654447
$369$	0	0
$370$	6.46599	0.336151
$371$	−1.37223	−0.0712429
$372$	0	0
$373$	31.9814	1.65594	0.827968	−	0.560775i	$-0.189497\pi$
$373$	31.9814	1.65594	0.827968	+	0.560775i	$0.189497\pi$
$374$	−13.6688	−0.706797
$375$	0	0
$376$	−26.0309	−1.34244
$377$	−0.513028	−0.0264223
$378$	0	0
$379$	33.1441	1.70250	0.851249	−	0.524761i	$-0.175845\pi$
$379$	33.1441	1.70250	0.851249	+	0.524761i	$0.175845\pi$
$380$	0.877453	0.0450124
$381$	0	0
$382$	−30.6126	−1.56627
$383$	15.2158	0.777490	0.388745	−	0.921345i	$-0.372909\pi$
$383$	15.2158	0.777490	0.388745	+	0.921345i	$0.372909\pi$
$384$	0	0
$385$	−0.535684	−0.0273010
$386$	−29.5940	−1.50629
$387$	0	0
$388$	−1.83987	−0.0934053
$389$	−6.55899	−0.332554	−0.166277	−	0.986079i	$-0.553175\pi$
$389$	−6.55899	−0.332554	−0.166277	+	0.986079i	$0.553175\pi$
$390$	0	0
$391$	18.4507	0.933093
$392$	20.3903	1.02986
$393$	0	0
$394$	23.5255	1.18520
$395$	1.41365	0.0711282
$396$	0	0
$397$	−19.5788	−0.982632	−0.491316	−	0.870982i	$-0.663484\pi$
$397$	−19.5788	−0.982632	−0.491316	+	0.870982i	$0.663484\pi$
$398$	−11.8663	−0.594803
$399$	0	0
$400$	10.3970	0.519851
$401$	36.0834	1.80192	0.900959	−	0.433905i	$-0.142865\pi$
$401$	36.0834	1.80192	0.900959	+	0.433905i	$0.142865\pi$
$402$	0	0
$403$	−24.3193	−1.21143
$404$	−1.08953	−0.0542061
$405$	0	0
$406$	−0.0220219	−0.00109293
$407$	−6.07167	−0.300961
$408$	0	0
$409$	−19.0011	−0.939545	−0.469773	−	0.882787i	$-0.655664\pi$
$409$	−19.0011	−0.939545	−0.469773	+	0.882787i	$0.655664\pi$
$410$	24.9998	1.23465
$411$	0	0
$412$	−0.437785	−0.0215681
$413$	−1.53760	−0.0756605
$414$	0	0
$415$	23.1367	1.13574
$416$	5.48367	0.268859
$417$	0	0
$418$	9.61588	0.470328
$419$	−31.5429	−1.54097	−0.770485	−	0.637458i	$-0.779986\pi$
$419$	−31.5429	−1.54097	−0.770485	+	0.637458i	$0.779986\pi$
$420$	0	0
$421$	−22.9117	−1.11665	−0.558325	−	0.829623i	$-0.688556\pi$
$421$	−22.9117	−1.11665	−0.558325	+	0.829623i	$0.688556\pi$
$422$	−21.6197	−1.05243
$423$	0	0
$424$	20.6421	1.00247
$425$	−15.2800	−0.741190
$426$	0	0
$427$	−0.574438	−0.0277990
$428$	−0.542479	−0.0262217
$429$	0	0
$430$	4.63389	0.223466
$431$	26.6379	1.28310	0.641552	−	0.767079i	$-0.278291\pi$
$431$	26.6379	1.28310	0.641552	+	0.767079i	$0.278291\pi$
$432$	0	0
$433$	−25.7160	−1.23583	−0.617916	−	0.786244i	$-0.712023\pi$
$433$	−25.7160	−1.23583	−0.617916	+	0.786244i	$0.712023\pi$
$434$	−1.04391	−0.0501095
$435$	0	0
$436$	−2.25742	−0.108111
$437$	−12.9799	−0.620913
$438$	0	0
$439$	−25.6733	−1.22532	−0.612661	−	0.790346i	$-0.709901\pi$
$439$	−25.6733	−1.22532	−0.612661	+	0.790346i	$0.709901\pi$
$440$	8.05813	0.384156
$441$	0	0
$442$	44.9359	2.13738
$443$	−34.7224	−1.64971	−0.824857	−	0.565342i	$-0.808744\pi$
$443$	−34.7224	−1.64971	−0.824857	+	0.565342i	$0.808744\pi$
$444$	0	0
$445$	13.8754	0.657755
$446$	−19.0096	−0.900132
$447$	0	0
$448$	1.66033	0.0784432
$449$	−18.0522	−0.851936	−0.425968	−	0.904738i	$-0.640066\pi$
$449$	−18.0522	−0.851936	−0.425968	+	0.904738i	$0.640066\pi$
$450$	0	0
$451$	−23.4753	−1.10541
$452$	0.828994	0.0389926
$453$	0	0
$454$	0.386372	0.0181333
$455$	1.76105	0.0825592
$456$	0	0
$457$	13.7905	0.645092	0.322546	−	0.946554i	$-0.395461\pi$
$457$	13.7905	0.645092	0.322546	+	0.946554i	$0.395461\pi$
$458$	−27.1480	−1.26854
$459$	0	0
$460$	−0.795665	−0.0370981
$461$	4.41465	0.205611	0.102805	−	0.994701i	$-0.467218\pi$
$461$	4.41465	0.205611	0.102805	+	0.994701i	$0.467218\pi$
$462$	0	0
$463$	−31.5089	−1.46434	−0.732171	−	0.681121i	$-0.761493\pi$
$463$	−31.5089	−1.46434	−0.732171	+	0.681121i	$0.761493\pi$
$464$	0.304961	0.0141575
$465$	0	0
$466$	15.0920	0.699122
$467$	−13.6967	−0.633810	−0.316905	−	0.948457i	$-0.602644\pi$
$467$	−13.6967	−0.633810	−0.316905	+	0.948457i	$0.602644\pi$
$468$	0	0
$469$	−0.274616	−0.0126806
$470$	17.7246	0.817574
$471$	0	0
$472$	23.1297	1.06463
$473$	−4.35130	−0.200073
$474$	0	0
$475$	10.7494	0.493214
$476$	−0.165278	−0.00757551
$477$	0	0
$478$	3.22871	0.147678
$479$	−24.1143	−1.10181	−0.550905	−	0.834568i	$-0.685717\pi$
$479$	−24.1143	−1.10181	−0.550905	+	0.834568i	$0.685717\pi$
$480$	0	0
$481$	19.9605	0.910119
$482$	12.6808	0.577594
$483$	0	0
$484$	1.18280	0.0537638
$485$	17.1262	0.777661
$486$	0	0
$487$	9.17271	0.415655	0.207828	−	0.978165i	$-0.433361\pi$
$487$	9.17271	0.415655	0.207828	+	0.978165i	$0.433361\pi$
$488$	8.64110	0.391164
$489$	0	0
$490$	−13.8839	−0.627209
$491$	32.9154	1.48545	0.742727	−	0.669595i	$-0.233532\pi$
$491$	32.9154	1.48545	0.742727	+	0.669595i	$0.233532\pi$
$492$	0	0
$493$	−0.448187	−0.0201853
$494$	−31.6120	−1.42229
$495$	0	0
$496$	14.4562	0.649101
$497$	2.17754	0.0976762
$498$	0	0
$499$	−1.24204	−0.0556016	−0.0278008	−	0.999613i	$-0.508850\pi$
$499$	−1.24204	−0.0556016	−0.0278008	+	0.999613i	$0.508850\pi$
$500$	1.81854	0.0813276
$501$	0	0
$502$	−24.4388	−1.09076
$503$	−27.6666	−1.23359	−0.616796	−	0.787123i	$-0.711570\pi$
$503$	−27.6666	−1.23359	−0.616796	+	0.787123i	$0.711570\pi$
$504$	0	0
$505$	10.1417	0.451301
$506$	−8.71958	−0.387633
$507$	0	0
$508$	0.994051	0.0441039
$509$	17.4776	0.774682	0.387341	−	0.921937i	$-0.373394\pi$
$509$	17.4776	0.774682	0.387341	+	0.921937i	$0.373394\pi$
$510$	0	0
$511$	−0.0814483	−0.00360306
$512$	−24.6946	−1.09136
$513$	0	0
$514$	37.8348	1.66882
$515$	4.07507	0.179569
$516$	0	0
$517$	−16.6437	−0.731988
$518$	0.856811	0.0376461
$519$	0	0
$520$	−26.4909	−1.16170
$521$	25.9169	1.13544	0.567719	−	0.823222i	$-0.307826\pi$
$521$	25.9169	1.13544	0.567719	+	0.823222i	$0.307826\pi$
$522$	0	0
$523$	2.02186	0.0884096	0.0442048	−	0.999022i	$-0.485925\pi$
$523$	2.02186	0.0884096	0.0442048	+	0.999022i	$0.485925\pi$
$524$	2.11561	0.0924208
$525$	0	0
$526$	10.7578	0.469062
$527$	−21.2456	−0.925472
$528$	0	0
$529$	−11.2299	−0.488259
$530$	−14.0553	−0.610525
$531$	0	0
$532$	0.116272	0.00504102
$533$	77.1744	3.34279
$534$	0	0
$535$	5.04960	0.218313
$536$	4.13096	0.178430
$537$	0	0
$538$	16.2823	0.701979
$539$	13.0372	0.561551
$540$	0	0
$541$	−5.39281	−0.231855	−0.115927	−	0.993258i	$-0.536984\pi$
$541$	−5.39281	−0.231855	−0.115927	+	0.993258i	$0.536984\pi$
$542$	20.7107	0.889599
$543$	0	0
$544$	4.79059	0.205395
$545$	21.0129	0.900093
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−0.0269991	−0.00115335
$549$	0	0
$550$	7.22115	0.307911
$551$	0.315295	0.0134320
$552$	0	0
$553$	0.187323	0.00796578
$554$	−16.7443	−0.711396
$555$	0	0
$556$	−2.19221	−0.0929705
$557$	24.8723	1.05387	0.526937	−	0.849904i	$-0.323340\pi$
$557$	24.8723	1.05387	0.526937	+	0.849904i	$0.323340\pi$
$558$	0	0
$559$	14.3048	0.605028
$560$	−1.04683	−0.0442364
$561$	0	0
$562$	−4.79227	−0.202150
$563$	32.9750	1.38973	0.694865	−	0.719140i	$-0.255464\pi$
$563$	32.9750	1.38973	0.694865	+	0.719140i	$0.255464\pi$
$564$	0	0
$565$	−7.71659	−0.324639
$566$	−26.9015	−1.13075
$567$	0	0
$568$	−32.7561	−1.37442
$569$	−31.7599	−1.33145	−0.665723	−	0.746199i	$-0.731877\pi$
$569$	−31.7599	−1.33145	−0.665723	+	0.746199i	$0.731877\pi$
$570$	0	0
$571$	23.0226	0.963466	0.481733	−	0.876318i	$-0.340008\pi$
$571$	23.0226	0.963466	0.481733	+	0.876318i	$0.340008\pi$
$572$	1.81963	0.0760826
$573$	0	0
$574$	3.31274	0.138271
$575$	−9.74741	−0.406495
$576$	0	0
$577$	9.46259	0.393933	0.196966	−	0.980410i	$-0.436891\pi$
$577$	9.46259	0.393933	0.196966	+	0.980410i	$0.436891\pi$
$578$	16.1831	0.673126
$579$	0	0
$580$	0.0193275	0.000802532 0
$581$	3.06586	0.127193
$582$	0	0
$583$	13.1982	0.546613
$584$	1.22520	0.0506992
$585$	0	0
$586$	−3.86228	−0.159549
$587$	−32.6862	−1.34911	−0.674553	−	0.738227i	$-0.735664\pi$
$587$	−32.6862	−1.34911	−0.674553	+	0.738227i	$0.735664\pi$
$588$	0	0
$589$	14.9461	0.615842
$590$	−15.7491	−0.648382
$591$	0	0
$592$	−11.8652	−0.487655
$593$	−26.7627	−1.09901	−0.549506	−	0.835490i	$-0.685184\pi$
$593$	−26.7627	−1.09901	−0.549506	+	0.835490i	$0.685184\pi$
$594$	0	0
$595$	1.53847	0.0630712
$596$	−2.05117	−0.0840192
$597$	0	0
$598$	28.6654	1.17222
$599$	28.9681	1.18361	0.591803	−	0.806083i	$-0.298416\pi$
$599$	28.9681	1.18361	0.591803	+	0.806083i	$0.298416\pi$
$600$	0	0
$601$	37.2161	1.51808	0.759038	−	0.651046i	$-0.225670\pi$
$601$	37.2161	1.51808	0.759038	+	0.651046i	$0.225670\pi$
$602$	0.614039	0.0250264
$603$	0	0
$604$	−2.69061	−0.109479
$605$	−11.0100	−0.447619
$606$	0	0
$607$	−21.4512	−0.870678	−0.435339	−	0.900267i	$-0.643371\pi$
$607$	−21.4512	−0.870678	−0.435339	+	0.900267i	$0.643371\pi$
$608$	−3.37014	−0.136677
$609$	0	0
$610$	−5.88377	−0.238227
$611$	54.7157	2.21356
$612$	0	0
$613$	38.1315	1.54012	0.770058	−	0.637973i	$-0.220227\pi$
$613$	38.1315	1.54012	0.770058	+	0.637973i	$0.220227\pi$
$614$	20.1512	0.813237
$615$	0	0
$616$	1.06779	0.0430224
$617$	−3.35306	−0.134989	−0.0674946	−	0.997720i	$-0.521501\pi$
$617$	−3.35306	−0.134989	−0.0674946	+	0.997720i	$0.521501\pi$
$618$	0	0
$619$	−20.8491	−0.837995	−0.418998	−	0.907987i	$-0.637618\pi$
$619$	−20.8491	−0.837995	−0.418998	+	0.907987i	$0.637618\pi$
$620$	0.916190	0.0367951
$621$	0	0
$622$	−37.9995	−1.52364
$623$	1.83863	0.0736631
$624$	0	0
$625$	−2.72172	−0.108869
$626$	7.51380	0.300312
$627$	0	0
$628$	−0.546739	−0.0218173
$629$	17.4377	0.695286
$630$	0	0
$631$	44.4501	1.76953	0.884765	−	0.466037i	$-0.154319\pi$
$631$	44.4501	1.76953	0.884765	+	0.466037i	$0.154319\pi$
$632$	−2.81784	−0.112088
$633$	0	0
$634$	6.61061	0.262541
$635$	−9.25300	−0.367194
$636$	0	0
$637$	−42.8594	−1.69815
$638$	0.211808	0.00838554
$639$	0	0
$640$	14.3886	0.568759
$641$	−15.0918	−0.596092	−0.298046	−	0.954551i	$-0.596335\pi$
$641$	−15.0918	−0.596092	−0.298046	+	0.954551i	$0.596335\pi$
$642$	0	0
$643$	−23.5111	−0.927186	−0.463593	−	0.886048i	$-0.653440\pi$
$643$	−23.5111	−0.927186	−0.463593	+	0.886048i	$0.653440\pi$
$644$	−0.105434	−0.00415468
$645$	0	0
$646$	−27.6166	−1.08656
$647$	−26.2736	−1.03292	−0.516461	−	0.856311i	$-0.672751\pi$
$647$	−26.2736	−1.03292	−0.516461	+	0.856311i	$0.672751\pi$
$648$	0	0
$649$	14.7887	0.580507
$650$	−23.7394	−0.931134
$651$	0	0
$652$	−1.37130	−0.0537043
$653$	5.29541	0.207225	0.103613	−	0.994618i	$-0.466960\pi$
$653$	5.29541	0.207225	0.103613	+	0.994618i	$0.466960\pi$
$654$	0	0
$655$	−19.6929	−0.769465
$656$	−45.8750	−1.79112
$657$	0	0
$658$	2.34869	0.0915616
$659$	24.9443	0.971693	0.485846	−	0.874044i	$-0.338511\pi$
$659$	24.9443	0.971693	0.485846	+	0.874044i	$0.338511\pi$
$660$	0	0
$661$	−15.4362	−0.600400	−0.300200	−	0.953876i	$-0.597053\pi$
$661$	−15.4362	−0.600400	−0.300200	+	0.953876i	$0.597053\pi$
$662$	44.8947	1.74488
$663$	0	0
$664$	−46.1188	−1.78976
$665$	−1.08230	−0.0419698
$666$	0	0
$667$	−0.285907	−0.0110703
$668$	1.09529	0.0423782
$669$	0	0
$670$	−2.81279	−0.108668
$671$	5.52496	0.213289
$672$	0	0
$673$	−8.36186	−0.322326	−0.161163	−	0.986928i	$-0.551524\pi$
$673$	−8.36186	−0.322326	−0.161163	+	0.986928i	$0.551524\pi$
$674$	−5.50689	−0.212117
$675$	0	0
$676$	−3.93000	−0.151154
$677$	−23.5347	−0.904513	−0.452257	−	0.891888i	$-0.649381\pi$
$677$	−23.5347	−0.904513	−0.452257	+	0.891888i	$0.649381\pi$
$678$	0	0
$679$	2.26940	0.0870916
$680$	−23.1428	−0.887484
$681$	0	0
$682$	10.0404	0.384466
$683$	−42.2912	−1.61823	−0.809114	−	0.587651i	$-0.800053\pi$
$683$	−42.2912	−1.61823	−0.809114	+	0.587651i	$0.800053\pi$
$684$	0	0
$685$	0.251318	0.00960237
$686$	−3.68953	−0.140867
$687$	0	0
$688$	−8.50323	−0.324183
$689$	−43.3887	−1.65298
$690$	0	0
$691$	7.89039	0.300165	0.150082	−	0.988674i	$-0.452046\pi$
$691$	7.89039	0.300165	0.150082	+	0.988674i	$0.452046\pi$
$692$	−1.27289	−0.0483880
$693$	0	0
$694$	−22.0993	−0.838879
$695$	20.4059	0.774041
$696$	0	0
$697$	67.4204	2.55373
$698$	2.41598	0.0914462
$699$	0	0
$700$	0.0873155	0.00330022
$701$	7.76122	0.293137	0.146569	−	0.989200i	$-0.453177\pi$
$701$	7.76122	0.293137	0.146569	+	0.989200i	$0.453177\pi$
$702$	0	0
$703$	−12.2673	−0.462668
$704$	−15.9691	−0.601857
$705$	0	0
$706$	−41.6359	−1.56699
$707$	1.34389	0.0505421
$708$	0	0
$709$	−29.9358	−1.12426	−0.562131	−	0.827048i	$-0.690019\pi$
$709$	−29.9358	−1.12426	−0.562131	+	0.827048i	$0.690019\pi$
$710$	22.3038	0.837048
$711$	0	0
$712$	−27.6580	−1.03653
$713$	−13.5529	−0.507561
$714$	0	0
$715$	−16.9378	−0.633438
$716$	−0.699830	−0.0261539
$717$	0	0
$718$	−45.8523	−1.71119
$719$	38.2117	1.42506	0.712528	−	0.701644i	$-0.247550\pi$
$719$	38.2117	1.42506	0.712528	+	0.701644i	$0.247550\pi$
$720$	0	0
$721$	0.539989	0.0201103
$722$	−6.35993	−0.236692
$723$	0	0
$724$	−2.45495	−0.0912373
$725$	0.236774	0.00879358
$726$	0	0
$727$	−34.4040	−1.27597	−0.637986	−	0.770048i	$-0.720232\pi$
$727$	−34.4040	−1.27597	−0.637986	+	0.770048i	$0.720232\pi$
$728$	−3.51032	−0.130101
$729$	0	0
$730$	−0.834248	−0.0308769
$731$	12.4968	0.462211
$732$	0	0
$733$	−32.2174	−1.18998	−0.594988	−	0.803735i	$-0.702843\pi$
$733$	−32.2174	−1.18998	−0.594988	+	0.803735i	$0.702843\pi$
$734$	−48.0021	−1.77179
$735$	0	0
$736$	3.05601	0.112646
$737$	2.64126	0.0972920
$738$	0	0
$739$	39.5093	1.45337	0.726686	−	0.686969i	$-0.241059\pi$
$739$	39.5093	1.45337	0.726686	+	0.686969i	$0.241059\pi$
$740$	−0.751979	−0.0276433
$741$	0	0
$742$	−1.86248	−0.0683737
$743$	5.63707	0.206804	0.103402	−	0.994640i	$-0.467027\pi$
$743$	5.63707	0.206804	0.103402	+	0.994640i	$0.467027\pi$
$744$	0	0
$745$	19.0930	0.699515
$746$	43.4071	1.58925
$747$	0	0
$748$	1.58965	0.0581233
$749$	0.669125	0.0244493
$750$	0	0
$751$	−30.4166	−1.10992	−0.554958	−	0.831878i	$-0.687266\pi$
$751$	−30.4166	−1.10992	−0.554958	+	0.831878i	$0.687266\pi$
$752$	−32.5248	−1.18606
$753$	0	0
$754$	−0.696312	−0.0253582
$755$	25.0452	0.911487
$756$	0	0
$757$	−5.69640	−0.207039	−0.103520	−	0.994627i	$-0.533010\pi$
$757$	−5.69640	−0.207039	−0.103520	+	0.994627i	$0.533010\pi$
$758$	44.9852	1.63394
$759$	0	0
$760$	16.2807	0.590564
$761$	46.0415	1.66900	0.834501	−	0.551007i	$-0.185756\pi$
$761$	46.0415	1.66900	0.834501	+	0.551007i	$0.185756\pi$
$762$	0	0
$763$	2.78443	0.100803
$764$	3.56017	0.128802
$765$	0	0
$766$	20.6518	0.746178
$767$	−48.6175	−1.75548
$768$	0	0
$769$	24.4948	0.883304	0.441652	−	0.897186i	$-0.354393\pi$
$769$	24.4948	0.883304	0.441652	+	0.897186i	$0.354393\pi$
$770$	−0.727062	−0.0262015
$771$	0	0
$772$	3.44171	0.123870
$773$	−9.93793	−0.357442	−0.178721	−	0.983900i	$-0.557196\pi$
$773$	−9.93793	−0.357442	−0.178721	+	0.983900i	$0.557196\pi$
$774$	0	0
$775$	11.2239	0.403175
$776$	−34.1379	−1.22548
$777$	0	0
$778$	−8.90225	−0.319161
$779$	−47.4296	−1.69934
$780$	0	0
$781$	−20.9437	−0.749423
$782$	25.0424	0.895515
$783$	0	0
$784$	25.4770	0.909894
$785$	5.08925	0.181643
$786$	0	0
$787$	−29.5730	−1.05416	−0.527082	−	0.849814i	$-0.676714\pi$
$787$	−29.5730	−1.05416	−0.527082	+	0.849814i	$0.676714\pi$
$788$	−2.73596	−0.0974644
$789$	0	0
$790$	1.91868	0.0682637
$791$	−1.02253	−0.0363569
$792$	0	0
$793$	−18.1632	−0.644993
$794$	−26.5735	−0.943059
$795$	0	0
$796$	1.38002	0.0489136
$797$	18.9234	0.670302	0.335151	−	0.942165i	$-0.391213\pi$
$797$	18.9234	0.670302	0.335151	+	0.942165i	$0.391213\pi$
$798$	0	0
$799$	47.8002	1.69105
$800$	−2.53084	−0.0894788
$801$	0	0
$802$	48.9745	1.72935
$803$	0.783372	0.0276446
$804$	0	0
$805$	0.981419	0.0345905
$806$	−33.0076	−1.16264
$807$	0	0
$808$	−20.2157	−0.711185
$809$	−16.3223	−0.573860	−0.286930	−	0.957952i	$-0.592635\pi$
$809$	−16.3223	−0.573860	−0.286930	+	0.957952i	$0.592635\pi$
$810$	0	0
$811$	37.4100	1.31364	0.656821	−	0.754047i	$-0.271901\pi$
$811$	37.4100	1.31364	0.656821	+	0.754047i	$0.271901\pi$
$812$	0.00256110	8.98770e−5 0
$813$	0	0
$814$	−8.24083	−0.288841
$815$	12.7646	0.447124
$816$	0	0
$817$	−8.79140	−0.307572
$818$	−25.7895	−0.901708
$819$	0	0
$820$	−2.90742	−0.101532
$821$	−24.6390	−0.859906	−0.429953	−	0.902851i	$-0.641470\pi$
$821$	−24.6390	−0.859906	−0.429953	+	0.902851i	$0.641470\pi$
$822$	0	0
$823$	−13.5452	−0.472155	−0.236078	−	0.971734i	$-0.575862\pi$
$823$	−13.5452	−0.472155	−0.236078	+	0.971734i	$0.575862\pi$
$824$	−8.12290	−0.282975
$825$	0	0
$826$	−2.08693	−0.0726134
$827$	−17.7461	−0.617093	−0.308547	−	0.951209i	$-0.599843\pi$
$827$	−17.7461	−0.617093	−0.308547	+	0.951209i	$0.599843\pi$
$828$	0	0
$829$	−26.4419	−0.918365	−0.459182	−	0.888342i	$-0.651858\pi$
$829$	−26.4419	−0.918365	−0.459182	+	0.888342i	$0.651858\pi$
$830$	31.4026	1.09000
$831$	0	0
$832$	52.4980	1.82004
$833$	−37.4424	−1.29730
$834$	0	0
$835$	−10.1954	−0.352826
$836$	−1.11830	−0.0386773
$837$	0	0
$838$	−42.8119	−1.47891
$839$	−5.72384	−0.197609	−0.0988044	−	0.995107i	$-0.531502\pi$
$839$	−5.72384	−0.197609	−0.0988044	+	0.995107i	$0.531502\pi$
$840$	0	0
$841$	−28.9931	−0.999761
$842$	−31.0972	−1.07168
$843$	0	0
$844$	2.51432	0.0865465
$845$	36.5819	1.25845
$846$	0	0
$847$	−1.45894	−0.0501296
$848$	25.7917	0.885690
$849$	0	0
$850$	−20.7390	−0.711340
$851$	11.1238	0.381319
$852$	0	0
$853$	5.06635	0.173469	0.0867343	−	0.996231i	$-0.472357\pi$
$853$	5.06635	0.173469	0.0867343	+	0.996231i	$0.472357\pi$
$854$	−0.779661	−0.0266795
$855$	0	0
$856$	−10.0654	−0.344030
$857$	52.1200	1.78038	0.890192	−	0.455586i	$-0.150570\pi$
$857$	52.1200	1.78038	0.890192	+	0.455586i	$0.150570\pi$
$858$	0	0
$859$	9.02910	0.308069	0.154034	−	0.988065i	$-0.450773\pi$
$859$	9.02910	0.308069	0.154034	+	0.988065i	$0.450773\pi$
$860$	−0.538910	−0.0183767
$861$	0	0
$862$	36.1546	1.23143
$863$	−11.6103	−0.395220	−0.197610	−	0.980281i	$-0.563318\pi$
$863$	−11.6103	−0.395220	−0.197610	+	0.980281i	$0.563318\pi$
$864$	0	0
$865$	11.8485	0.402862
$866$	−34.9033	−1.18606
$867$	0	0
$868$	0.121405	0.00412075
$869$	−1.80167	−0.0611176
$870$	0	0
$871$	−8.68308	−0.294215
$872$	−41.8853	−1.41842
$873$	0	0
$874$	−17.6171	−0.595908
$875$	−2.24309	−0.0758303
$876$	0	0
$877$	−7.52717	−0.254174	−0.127087	−	0.991892i	$-0.540563\pi$
$877$	−7.52717	−0.254174	−0.127087	+	0.991892i	$0.540563\pi$
$878$	−34.8454	−1.17597
$879$	0	0
$880$	10.0684	0.339405
$881$	−33.5613	−1.13071	−0.565354	−	0.824848i	$-0.691261\pi$
$881$	−33.5613	−1.13071	−0.565354	+	0.824848i	$0.691261\pi$
$882$	0	0
$883$	12.5358	0.421863	0.210932	−	0.977501i	$-0.432350\pi$
$883$	12.5358	0.421863	0.210932	+	0.977501i	$0.432350\pi$
$884$	−5.22594	−0.175767
$885$	0	0
$886$	−47.1274	−1.58328
$887$	−3.55658	−0.119418	−0.0597091	−	0.998216i	$-0.519017\pi$
$887$	−3.55658	−0.119418	−0.0597091	+	0.998216i	$0.519017\pi$
$888$	0	0
$889$	−1.22612	−0.0411227
$890$	18.8325	0.631266
$891$	0	0
$892$	2.21077	0.0740222
$893$	−33.6270	−1.12528
$894$	0	0
$895$	6.51427	0.217748
$896$	1.90664	0.0636963
$897$	0	0
$898$	−24.5015	−0.817626
$899$	0.329215	0.0109799
$900$	0	0
$901$	−37.9049	−1.26279
$902$	−31.8620	−1.06089
$903$	0	0
$904$	15.3816	0.511584
$905$	22.8515	0.759611
$906$	0	0
$907$	−5.91904	−0.196539	−0.0982693	−	0.995160i	$-0.531331\pi$
$907$	−5.91904	−0.196539	−0.0982693	+	0.995160i	$0.531331\pi$
$908$	−0.0449341	−0.00149119
$909$	0	0
$910$	2.39020	0.0792344
$911$	−14.3654	−0.475948	−0.237974	−	0.971271i	$-0.576483\pi$
$911$	−14.3654	−0.475948	−0.237974	+	0.971271i	$0.576483\pi$
$912$	0	0
$913$	−29.4875	−0.975894
$914$	18.7173	0.619112
$915$	0	0
$916$	3.15725	0.104318
$917$	−2.60951	−0.0861737
$918$	0	0
$919$	7.41560	0.244618	0.122309	−	0.992492i	$-0.460970\pi$
$919$	7.41560	0.244618	0.122309	+	0.992492i	$0.460970\pi$
$920$	−14.7632	−0.486728
$921$	0	0
$922$	5.99183	0.197330
$923$	68.8518	2.26628
$924$	0	0
$925$	−9.21222	−0.302896
$926$	−42.7657	−1.40537
$927$	0	0
$928$	−0.0742335	−0.00243684
$929$	−9.71228	−0.318650	−0.159325	−	0.987226i	$-0.550932\pi$
$929$	−9.71228	−0.318650	−0.159325	+	0.987226i	$0.550932\pi$
$930$	0	0
$931$	26.3404	0.863272
$932$	−1.75516	−0.0574922
$933$	0	0
$934$	−18.5900	−0.608285
$935$	−14.7970	−0.483915
$936$	0	0
$937$	−28.3847	−0.927287	−0.463643	−	0.886022i	$-0.653458\pi$
$937$	−28.3847	−0.927287	−0.463643	+	0.886022i	$0.653458\pi$
$938$	−0.372725	−0.0121699
$939$	0	0
$940$	−2.06133	−0.0672331
$941$	7.54245	0.245877	0.122938	−	0.992414i	$-0.460768\pi$
$941$	7.54245	0.245877	0.122938	+	0.992414i	$0.460768\pi$
$942$	0	0
$943$	43.0087	1.40056
$944$	28.8998	0.940610
$945$	0	0
$946$	−5.90584	−0.192015
$947$	32.4877	1.05571	0.527854	−	0.849335i	$-0.322997\pi$
$947$	32.4877	1.05571	0.527854	+	0.849335i	$0.322997\pi$
$948$	0	0
$949$	−2.57532	−0.0835983
$950$	14.5897	0.473351
$951$	0	0
$952$	−3.06666	−0.0993909
$953$	6.79285	0.220042	0.110021	−	0.993929i	$-0.464908\pi$
$953$	6.79285	0.220042	0.110021	+	0.993929i	$0.464908\pi$
$954$	0	0
$955$	−33.1394	−1.07236
$956$	−0.375492	−0.0121443
$957$	0	0
$958$	−32.7293	−1.05744
$959$	0.0333023	0.00107539
$960$	0	0
$961$	−15.3941	−0.496584
$962$	27.0915	0.873466
$963$	0	0
$964$	−1.47474	−0.0474983
$965$	−32.0367	−1.03130
$966$	0	0
$967$	−3.50671	−0.112768	−0.0563840	−	0.998409i	$-0.517957\pi$
$967$	−3.50671	−0.112768	−0.0563840	+	0.998409i	$0.517957\pi$
$968$	21.9463	0.705382
$969$	0	0
$970$	23.2447	0.746342
$971$	31.1697	1.00028	0.500142	−	0.865943i	$-0.333281\pi$
$971$	31.1697	1.00028	0.500142	+	0.865943i	$0.333281\pi$
$972$	0	0
$973$	2.70400	0.0866862
$974$	12.4498	0.398916
$975$	0	0
$976$	10.7968	0.345597
$977$	27.2061	0.870400	0.435200	−	0.900334i	$-0.356678\pi$
$977$	27.2061	0.870400	0.435200	+	0.900334i	$0.356678\pi$
$978$	0	0
$979$	−17.6840	−0.565182
$980$	1.61466	0.0515784
$981$	0	0
$982$	44.6748	1.42563
$983$	−9.22401	−0.294200	−0.147100	−	0.989122i	$-0.546994\pi$
$983$	−9.22401	−0.294200	−0.147100	+	0.989122i	$0.546994\pi$
$984$	0	0
$985$	25.4673	0.811456
$986$	−0.608306	−0.0193724
$987$	0	0
$988$	3.67640	0.116962
$989$	7.97195	0.253493
$990$	0	0
$991$	−37.7461	−1.19904	−0.599522	−	0.800358i	$-0.704643\pi$
$991$	−37.7461	−1.19904	−0.599522	+	0.800358i	$0.704643\pi$
$992$	−3.51892	−0.111726
$993$	0	0
$994$	2.95549	0.0937425
$995$	−12.8458	−0.407238
$996$	0	0
$997$	−45.6962	−1.44721	−0.723606	−	0.690213i	$-0.757517\pi$
$997$	−45.6962	−1.44721	−0.723606	+	0.690213i	$0.757517\pi$
$998$	−1.68578	−0.0533623
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.13		18
3.2	odd	2		547.2.a.b.1.6	✓	18
12.11	even	2		8752.2.a.s.1.7		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.6	✓	18	3.2	odd	2
4923.2.a.l.1.13		18	1.1	even	1	trivial
8752.2.a.s.1.7		18	12.11	even	2

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:	\( N \)	\(=\)	\( 4923 = 3^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4923.a (trivial)

\(f(q)\)	\(=\)	\(q+1.35726 q^{2} -0.157846 q^{4} +1.46929 q^{5} +0.194696 q^{7} -2.92876 q^{8} +O(q^{10})\)	\(q+1.35726 q^{2} -0.157846 q^{4} +1.46929 q^{5} +0.194696 q^{7} -2.92876 q^{8} +1.99421 q^{10} -1.87259 q^{11} +6.15611 q^{13} +0.264253 q^{14} -3.65939 q^{16} +5.37804 q^{17} -3.78340 q^{19} -0.231922 q^{20} -2.54160 q^{22} +3.43075 q^{23} -2.84119 q^{25} +8.35544 q^{26} -0.0307320 q^{28} -0.0833365 q^{29} -3.95043 q^{31} +0.890769 q^{32} +7.29940 q^{34} +0.286065 q^{35} +3.24239 q^{37} -5.13506 q^{38} -4.30319 q^{40} +12.5362 q^{41} +2.32367 q^{43} +0.295581 q^{44} +4.65642 q^{46} +8.88803 q^{47} -6.96209 q^{49} -3.85623 q^{50} -0.971717 q^{52} -7.04808 q^{53} -2.75138 q^{55} -0.570218 q^{56} -0.113109 q^{58} -7.89744 q^{59} -2.95043 q^{61} -5.36176 q^{62} +8.52779 q^{64} +9.04511 q^{65} -1.41048 q^{67} -0.848903 q^{68} +0.388265 q^{70} +11.1843 q^{71} -0.418335 q^{73} +4.40076 q^{74} +0.597195 q^{76} -0.364587 q^{77} +0.962128 q^{79} -5.37671 q^{80} +17.0149 q^{82} +15.7469 q^{83} +7.90190 q^{85} +3.15383 q^{86} +5.48437 q^{88} +9.44358 q^{89} +1.19857 q^{91} -0.541530 q^{92} +12.0634 q^{94} -5.55892 q^{95} +11.6561 q^{97} -9.44937 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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