LMFDB - Embedded newform 2673.2.a.p.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2673,2,Mod(1,2673)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2673, 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("2673.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2673 = 3^{5} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2673.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:	$21.3440124603$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.864654912.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 8x^{4} + 14x^{3} + 14x^{2} - 16x - 9 $ x^6 - 2x^5 - 8x^4 + 14x^3 + 14x^2 - 16*x - 9
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.93306$ of defining polynomial
Character	$\chi$	$=$	2673.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.93306 q^{2} +1.73670 q^{4} +0.736703 q^{5} +2.75195 q^{7} -0.508967 q^{8} +O(q^{10})$	\(q+1.93306 q^{2} +1.73670 q^{4} +0.736703 q^{5} +2.75195 q^{7} -0.508967 q^{8} +1.42409 q^{10} +1.00000 q^{11} +6.25273 q^{13} +5.31967 q^{14} -4.45727 q^{16} -1.05875 q^{17} -1.94459 q^{19} +1.27943 q^{20} +1.93306 q^{22} +8.24060 q^{23} -4.45727 q^{25} +12.0869 q^{26} +4.77932 q^{28} +8.56265 q^{29} -9.88470 q^{31} -7.59821 q^{32} -2.04663 q^{34} +2.02737 q^{35} +6.73222 q^{37} -3.75900 q^{38} -0.374958 q^{40} -9.62141 q^{41} -2.08278 q^{43} +1.73670 q^{44} +15.9295 q^{46} +4.69007 q^{47} +0.573223 q^{49} -8.61615 q^{50} +10.8591 q^{52} +0.823963 q^{53} +0.736703 q^{55} -1.40065 q^{56} +16.5521 q^{58} +10.6393 q^{59} -4.12063 q^{61} -19.1077 q^{62} -5.77323 q^{64} +4.60640 q^{65} +8.19397 q^{67} -1.83874 q^{68} +3.91902 q^{70} +11.1821 q^{71} +15.8613 q^{73} +13.0138 q^{74} -3.37718 q^{76} +2.75195 q^{77} -9.67681 q^{79} -3.28368 q^{80} -18.5987 q^{82} -3.68711 q^{83} -0.779987 q^{85} -4.02613 q^{86} -0.508967 q^{88} -4.60281 q^{89} +17.2072 q^{91} +14.3115 q^{92} +9.06617 q^{94} -1.43259 q^{95} -8.56874 q^{97} +1.10807 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	$ 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * q^97 - 6 * 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 $

$2$	1.93306	1.36688	0.683438	−	0.730008i	$-0.260484\pi$
$2$	1.93306	1.36688	0.683438	+	0.730008i	$0.260484\pi$
$3$	0	0
$3$	0	0
$4$	1.73670	0.868352
$5$	0.736703	0.329464	0.164732	−	0.986338i	$-0.447324\pi$
$5$	0.736703	0.329464	0.164732	+	0.986338i	$0.447324\pi$
$6$	0	0
$7$	2.75195	1.04014	0.520069	−	0.854124i	$-0.325906\pi$
$7$	2.75195	1.04014	0.520069	+	0.854124i	$0.325906\pi$
$8$	−0.508967	−0.179947
$9$	0	0
$10$	1.42409	0.450336
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	6.25273	1.73419	0.867097	−	0.498139i	$-0.165983\pi$
$13$	6.25273	1.73419	0.867097	+	0.498139i	$0.165983\pi$
$14$	5.31967	1.42174
$15$	0	0
$16$	−4.45727	−1.11432
$17$	−1.05875	−0.256785	−0.128393	−	0.991723i	$-0.540982\pi$
$17$	−1.05875	−0.256785	−0.128393	+	0.991723i	$0.540982\pi$
$18$	0	0
$19$	−1.94459	−0.446120	−0.223060	−	0.974805i	$-0.571605\pi$
$19$	−1.94459	−0.446120	−0.223060	+	0.974805i	$0.571605\pi$
$20$	1.27943	0.286090
$21$	0	0
$22$	1.93306	0.412129
$23$	8.24060	1.71828	0.859142	−	0.511737i	$-0.170998\pi$
$23$	8.24060	1.71828	0.859142	+	0.511737i	$0.170998\pi$
$24$	0	0
$25$	−4.45727	−0.891454
$26$	12.0869	2.37043
$27$	0	0
$28$	4.77932	0.903206
$29$	8.56265	1.59004	0.795022	−	0.606580i	$-0.207459\pi$
$29$	8.56265	1.59004	0.795022	+	0.606580i	$0.207459\pi$
$30$	0	0
$31$	−9.88470	−1.77534	−0.887672	−	0.460475i	$-0.847679\pi$
$31$	−9.88470	−1.77534	−0.887672	+	0.460475i	$0.847679\pi$
$32$	−7.59821	−1.34319
$33$	0	0
$34$	−2.04663	−0.350994
$35$	2.02737	0.342688
$36$	0	0
$37$	6.73222	1.10677	0.553385	−	0.832925i	$-0.313336\pi$
$37$	6.73222	1.10677	0.553385	+	0.832925i	$0.313336\pi$
$38$	−3.75900	−0.609791
$39$	0	0
$40$	−0.374958	−0.0592860
$41$	−9.62141	−1.50261	−0.751306	−	0.659955i	$-0.770576\pi$
$41$	−9.62141	−1.50261	−0.751306	+	0.659955i	$0.770576\pi$
$42$	0	0
$43$	−2.08278	−0.317621	−0.158810	−	0.987309i	$-0.550766\pi$
$43$	−2.08278	−0.317621	−0.158810	+	0.987309i	$0.550766\pi$
$44$	1.73670	0.261818
$45$	0	0
$46$	15.9295	2.34868
$47$	4.69007	0.684118	0.342059	−	0.939679i	$-0.388876\pi$
$47$	4.69007	0.684118	0.342059	+	0.939679i	$0.388876\pi$
$48$	0	0
$49$	0.573223	0.0818890
$50$	−8.61615	−1.21851
$51$	0	0
$52$	10.8591	1.50589
$53$	0.823963	0.113180	0.0565900	−	0.998398i	$-0.481977\pi$
$53$	0.823963	0.113180	0.0565900	+	0.998398i	$0.481977\pi$
$54$	0	0
$55$	0.736703	0.0993370
$56$	−1.40065	−0.187170
$57$	0	0
$58$	16.5521	2.17339
$59$	10.6393	1.38512	0.692562	−	0.721358i	$-0.256482\pi$
$59$	10.6393	1.38512	0.692562	+	0.721358i	$0.256482\pi$
$60$	0	0
$61$	−4.12063	−0.527592	−0.263796	−	0.964578i	$-0.584975\pi$
$61$	−4.12063	−0.527592	−0.263796	+	0.964578i	$0.584975\pi$
$62$	−19.1077	−2.42668
$63$	0	0
$64$	−5.77323	−0.721654
$65$	4.60640	0.571354
$66$	0	0
$67$	8.19397	1.00105	0.500526	−	0.865721i	$-0.333140\pi$
$67$	8.19397	1.00105	0.500526	+	0.865721i	$0.333140\pi$
$68$	−1.83874	−0.222980
$69$	0	0
$70$	3.91902	0.468412
$71$	11.1821	1.32707	0.663534	−	0.748146i	$-0.269056\pi$
$71$	11.1821	1.32707	0.663534	+	0.748146i	$0.269056\pi$
$72$	0	0
$73$	15.8613	1.85643	0.928215	−	0.372044i	$-0.121343\pi$
$73$	15.8613	1.85643	0.928215	+	0.372044i	$0.121343\pi$
$74$	13.0138	1.51282
$75$	0	0
$76$	−3.37718	−0.387389
$77$	2.75195	0.313614
$78$	0	0
$79$	−9.67681	−1.08873	−0.544363	−	0.838850i	$-0.683229\pi$
$79$	−9.67681	−1.08873	−0.544363	+	0.838850i	$0.683229\pi$
$80$	−3.28368	−0.367127
$81$	0	0
$82$	−18.5987	−2.05388
$83$	−3.68711	−0.404712	−0.202356	−	0.979312i	$-0.564860\pi$
$83$	−3.68711	−0.404712	−0.202356	+	0.979312i	$0.564860\pi$
$84$	0	0
$85$	−0.779987	−0.0846015
$86$	−4.02613	−0.434148
$87$	0	0
$88$	−0.508967	−0.0542561
$89$	−4.60281	−0.487897	−0.243949	−	0.969788i	$-0.578443\pi$
$89$	−4.60281	−0.487897	−0.243949	+	0.969788i	$0.578443\pi$
$90$	0	0
$91$	17.2072	1.80380
$92$	14.3115	1.49207
$93$	0	0
$94$	9.06617	0.935104
$95$	−1.43259	−0.146980
$96$	0	0
$97$	−8.56874	−0.870024	−0.435012	−	0.900425i	$-0.643256\pi$
$97$	−8.56874	−0.870024	−0.435012	+	0.900425i	$0.643256\pi$
$98$	1.10807	0.111932
$99$	0	0
$100$	−7.74095	−0.774095
$101$	15.1460	1.50708	0.753541	−	0.657401i	$-0.228344\pi$
$101$	15.1460	1.50708	0.753541	+	0.657401i	$0.228344\pi$
$102$	0	0
$103$	−7.97282	−0.785586	−0.392793	−	0.919627i	$-0.628491\pi$
$103$	−7.97282	−0.785586	−0.392793	+	0.919627i	$0.628491\pi$
$104$	−3.18243	−0.312063
$105$	0	0
$106$	1.59277	0.154703
$107$	−0.478081	−0.0462178	−0.0231089	−	0.999733i	$-0.507356\pi$
$107$	−0.478081	−0.0462178	−0.0231089	+	0.999733i	$0.507356\pi$
$108$	0	0
$109$	−3.64632	−0.349254	−0.174627	−	0.984635i	$-0.555872\pi$
$109$	−3.64632	−0.349254	−0.174627	+	0.984635i	$0.555872\pi$
$110$	1.42409	0.135781
$111$	0	0
$112$	−12.2662	−1.15904
$113$	−19.3240	−1.81785	−0.908926	−	0.416957i	$-0.863097\pi$
$113$	−19.3240	−1.81785	−0.908926	+	0.416957i	$0.863097\pi$
$114$	0	0
$115$	6.07088	0.566112
$116$	14.8708	1.38072
$117$	0	0
$118$	20.5664	1.89329
$119$	−2.91364	−0.267093
$120$	0	0
$121$	1.00000	0.0909091
$122$	−7.96540	−0.721154
$123$	0	0
$124$	−17.1668	−1.54162
$125$	−6.96720	−0.623165
$126$	0	0
$127$	−7.30049	−0.647814	−0.323907	−	0.946089i	$-0.604996\pi$
$127$	−7.30049	−0.647814	−0.323907	+	0.946089i	$0.604996\pi$
$128$	4.03645	0.356775
$129$	0	0
$130$	8.90443	0.780970
$131$	13.0814	1.14292	0.571462	−	0.820629i	$-0.306376\pi$
$131$	13.0814	1.14292	0.571462	+	0.820629i	$0.306376\pi$
$132$	0	0
$133$	−5.35142	−0.464027
$134$	15.8394	1.36832
$135$	0	0
$136$	0.538871	0.0462078
$137$	−8.75016	−0.747576	−0.373788	−	0.927514i	$-0.621941\pi$
$137$	−8.75016	−0.747576	−0.373788	+	0.927514i	$0.621941\pi$
$138$	0	0
$139$	3.98853	0.338303	0.169151	−	0.985590i	$-0.445897\pi$
$139$	3.98853	0.338303	0.169151	+	0.985590i	$0.445897\pi$
$140$	3.52094	0.297574
$141$	0	0
$142$	21.6156	1.81394
$143$	6.25273	0.522879
$144$	0	0
$145$	6.30813	0.523862
$146$	30.6609	2.53751
$147$	0	0
$148$	11.6919	0.961066
$149$	−1.71023	−0.140107	−0.0700537	−	0.997543i	$-0.522317\pi$
$149$	−1.71023	−0.140107	−0.0700537	+	0.997543i	$0.522317\pi$
$150$	0	0
$151$	1.47653	0.120158	0.0600791	−	0.998194i	$-0.480865\pi$
$151$	1.47653	0.120158	0.0600791	+	0.998194i	$0.480865\pi$
$152$	0.989733	0.0802780
$153$	0	0
$154$	5.31967	0.428671
$155$	−7.28209	−0.584912
$156$	0	0
$157$	−18.3536	−1.46478	−0.732389	−	0.680886i	$-0.761595\pi$
$157$	−18.3536	−1.46478	−0.732389	+	0.680886i	$0.761595\pi$
$158$	−18.7058	−1.48815
$159$	0	0
$160$	−5.59763	−0.442531
$161$	22.6777	1.78725
$162$	0	0
$163$	−4.76206	−0.372993	−0.186497	−	0.982456i	$-0.559713\pi$
$163$	−4.76206	−0.372993	−0.186497	+	0.982456i	$0.559713\pi$
$164$	−16.7095	−1.30479
$165$	0	0
$166$	−7.12738	−0.553192
$167$	−9.39426	−0.726949	−0.363475	−	0.931604i	$-0.618410\pi$
$167$	−9.39426	−0.726949	−0.363475	+	0.931604i	$0.618410\pi$
$168$	0	0
$169$	26.0966	2.00743
$170$	−1.50776	−0.115640
$171$	0	0
$172$	−3.61717	−0.275806
$173$	−2.75931	−0.209786	−0.104893	−	0.994483i	$-0.533450\pi$
$173$	−2.75931	−0.209786	−0.104893	+	0.994483i	$0.533450\pi$
$174$	0	0
$175$	−12.2662	−0.927236
$176$	−4.45727	−0.335979
$177$	0	0
$178$	−8.89749	−0.666895
$179$	20.7696	1.55239	0.776197	−	0.630491i	$-0.217146\pi$
$179$	20.7696	1.55239	0.776197	+	0.630491i	$0.217146\pi$
$180$	0	0
$181$	−6.55590	−0.487297	−0.243648	−	0.969864i	$-0.578344\pi$
$181$	−6.55590	−0.487297	−0.243648	+	0.969864i	$0.578344\pi$
$182$	33.2624	2.46558
$183$	0	0
$184$	−4.19420	−0.309200
$185$	4.95965	0.364641
$186$	0	0
$187$	−1.05875	−0.0774237
$188$	8.14527	0.594055
$189$	0	0
$190$	−2.76927	−0.200904
$191$	−21.6355	−1.56549	−0.782745	−	0.622342i	$-0.786181\pi$
$191$	−21.6355	−1.56549	−0.782745	+	0.622342i	$0.786181\pi$
$192$	0	0
$193$	−7.25495	−0.522222	−0.261111	−	0.965309i	$-0.584089\pi$
$193$	−7.25495	−0.522222	−0.261111	+	0.965309i	$0.584089\pi$
$194$	−16.5639	−1.18922
$195$	0	0
$196$	0.995518	0.0711085
$197$	2.02507	0.144280	0.0721401	−	0.997395i	$-0.477017\pi$
$197$	2.02507	0.144280	0.0721401	+	0.997395i	$0.477017\pi$
$198$	0	0
$199$	5.99279	0.424817	0.212409	−	0.977181i	$-0.431869\pi$
$199$	5.99279	0.424817	0.212409	+	0.977181i	$0.431869\pi$
$200$	2.26860	0.160414
$201$	0	0
$202$	29.2780	2.05999
$203$	23.5640	1.65387
$204$	0	0
$205$	−7.08812	−0.495056
$206$	−15.4119	−1.07380
$207$	0	0
$208$	−27.8701	−1.93244
$209$	−1.94459	−0.134510
$210$	0	0
$211$	−9.36741	−0.644879	−0.322440	−	0.946590i	$-0.604503\pi$
$211$	−9.36741	−0.644879	−0.322440	+	0.946590i	$0.604503\pi$
$212$	1.43098	0.0982800
$213$	0	0
$214$	−0.924157	−0.0631740
$215$	−1.53439	−0.104644
$216$	0	0
$217$	−27.2022	−1.84661
$218$	−7.04854	−0.477388
$219$	0	0
$220$	1.27943	0.0862595
$221$	−6.62010	−0.445316
$222$	0	0
$223$	−17.2933	−1.15804	−0.579021	−	0.815312i	$-0.696565\pi$
$223$	−17.2933	−1.15804	−0.579021	+	0.815312i	$0.696565\pi$
$224$	−20.9099	−1.39710
$225$	0	0
$226$	−37.3544	−2.48478
$227$	−23.0188	−1.52781	−0.763905	−	0.645329i	$-0.776720\pi$
$227$	−23.0188	−1.52781	−0.763905	+	0.645329i	$0.776720\pi$
$228$	0	0
$229$	−25.8407	−1.70760	−0.853802	−	0.520598i	$-0.825709\pi$
$229$	−25.8407	−1.70760	−0.853802	+	0.520598i	$0.825709\pi$
$230$	11.7353	0.773806
$231$	0	0
$232$	−4.35811	−0.286124
$233$	17.2902	1.13272	0.566359	−	0.824158i	$-0.308351\pi$
$233$	17.2902	1.13272	0.566359	+	0.824158i	$0.308351\pi$
$234$	0	0
$235$	3.45519	0.225392
$236$	18.4774	1.20277
$237$	0	0
$238$	−5.63222	−0.365083
$239$	−14.6554	−0.947982	−0.473991	−	0.880530i	$-0.657187\pi$
$239$	−14.6554	−0.947982	−0.473991	+	0.880530i	$0.657187\pi$
$240$	0	0
$241$	13.2520	0.853635	0.426817	−	0.904338i	$-0.359635\pi$
$241$	13.2520	0.853635	0.426817	+	0.904338i	$0.359635\pi$
$242$	1.93306	0.124262
$243$	0	0
$244$	−7.15631	−0.458136
$245$	0.422295	0.0269795
$246$	0	0
$247$	−12.1590	−0.773659
$248$	5.03099	0.319468
$249$	0	0
$250$	−13.4680	−0.851790
$251$	4.15268	0.262115	0.131058	−	0.991375i	$-0.458163\pi$
$251$	4.15268	0.262115	0.131058	+	0.991375i	$0.458163\pi$
$252$	0	0
$253$	8.24060	0.518082
$254$	−14.1123	−0.885482
$255$	0	0
$256$	19.3491	1.20932
$257$	−1.80192	−0.112401	−0.0562005	−	0.998420i	$-0.517899\pi$
$257$	−1.80192	−0.112401	−0.0562005	+	0.998420i	$0.517899\pi$
$258$	0	0
$259$	18.5267	1.15120
$260$	7.99996	0.496136
$261$	0	0
$262$	25.2870	1.56224
$263$	−11.0177	−0.679378	−0.339689	−	0.940538i	$-0.610322\pi$
$263$	−11.0177	−0.679378	−0.339689	+	0.940538i	$0.610322\pi$
$264$	0	0
$265$	0.607016	0.0372887
$266$	−10.3446	−0.634267
$267$	0	0
$268$	14.2305	0.869266
$269$	−4.89542	−0.298479	−0.149240	−	0.988801i	$-0.547683\pi$
$269$	−4.89542	−0.298479	−0.149240	+	0.988801i	$0.547683\pi$
$270$	0	0
$271$	0.101758	0.00618137	0.00309068	−	0.999995i	$-0.499016\pi$
$271$	0.101758	0.00618137	0.00309068	+	0.999995i	$0.499016\pi$
$272$	4.71915	0.286140
$273$	0	0
$274$	−16.9145	−1.02184
$275$	−4.45727	−0.268783
$276$	0	0
$277$	7.91364	0.475484	0.237742	−	0.971328i	$-0.423593\pi$
$277$	7.91364	0.475484	0.237742	+	0.971328i	$0.423593\pi$
$278$	7.71005	0.462418
$279$	0	0
$280$	−1.03186	−0.0616657
$281$	18.2334	1.08771	0.543856	−	0.839179i	$-0.316964\pi$
$281$	18.2334	1.08771	0.543856	+	0.839179i	$0.316964\pi$
$282$	0	0
$283$	−1.20185	−0.0714428	−0.0357214	−	0.999362i	$-0.511373\pi$
$283$	−1.20185	−0.0714428	−0.0357214	+	0.999362i	$0.511373\pi$
$284$	19.4199	1.15236
$285$	0	0
$286$	12.0869	0.714711
$287$	−26.4776	−1.56292
$288$	0	0
$289$	−15.8790	−0.934061
$290$	12.1940	0.716055
$291$	0	0
$292$	27.5465	1.61203
$293$	11.2682	0.658296	0.329148	−	0.944278i	$-0.393239\pi$
$293$	11.2682	0.658296	0.329148	+	0.944278i	$0.393239\pi$
$294$	0	0
$295$	7.83804	0.456348
$296$	−3.42648	−0.199160
$297$	0	0
$298$	−3.30597	−0.191510
$299$	51.5262	2.97984
$300$	0	0
$301$	−5.73170	−0.330370
$302$	2.85421	0.164241
$303$	0	0
$304$	8.66757	0.497119
$305$	−3.03568	−0.173823
$306$	0	0
$307$	10.9993	0.627763	0.313882	−	0.949462i	$-0.398371\pi$
$307$	10.9993	0.627763	0.313882	+	0.949462i	$0.398371\pi$
$308$	4.77932	0.272327
$309$	0	0
$310$	−14.0767	−0.799502
$311$	12.2397	0.694050	0.347025	−	0.937856i	$-0.387192\pi$
$311$	12.2397	0.694050	0.347025	+	0.937856i	$0.387192\pi$
$312$	0	0
$313$	−14.9627	−0.845743	−0.422871	−	0.906190i	$-0.638978\pi$
$313$	−14.9627	−0.845743	−0.422871	+	0.906190i	$0.638978\pi$
$314$	−35.4786	−2.00217
$315$	0	0
$316$	−16.8058	−0.945398
$317$	−14.7197	−0.826739	−0.413369	−	0.910563i	$-0.635648\pi$
$317$	−14.7197	−0.826739	−0.413369	+	0.910563i	$0.635648\pi$
$318$	0	0
$319$	8.56265	0.479416
$320$	−4.25316	−0.237759
$321$	0	0
$322$	43.8373	2.44296
$323$	2.05884	0.114557
$324$	0	0
$325$	−27.8701	−1.54595
$326$	−9.20532	−0.509835
$327$	0	0
$328$	4.89698	0.270390
$329$	12.9068	0.711577
$330$	0	0
$331$	−14.0706	−0.773388	−0.386694	−	0.922208i	$-0.626383\pi$
$331$	−14.0706	−0.773388	−0.386694	+	0.922208i	$0.626383\pi$
$332$	−6.40341	−0.351433
$333$	0	0
$334$	−18.1596	−0.993650
$335$	6.03653	0.329811
$336$	0	0
$337$	4.23385	0.230633	0.115316	−	0.993329i	$-0.463212\pi$
$337$	4.23385	0.230633	0.115316	+	0.993329i	$0.463212\pi$
$338$	50.4461	2.74391
$339$	0	0
$340$	−1.35461	−0.0734639
$341$	−9.88470	−0.535287
$342$	0	0
$343$	−17.6862	−0.954963
$344$	1.06007	0.0571549
$345$	0	0
$346$	−5.33390	−0.286752
$347$	11.7729	0.632000	0.316000	−	0.948759i	$-0.397660\pi$
$347$	11.7729	0.632000	0.316000	+	0.948759i	$0.397660\pi$
$348$	0	0
$349$	12.6815	0.678824	0.339412	−	0.940638i	$-0.389772\pi$
$349$	12.6815	0.678824	0.339412	+	0.940638i	$0.389772\pi$
$350$	−23.7112	−1.26742
$351$	0	0
$352$	−7.59821	−0.404986
$353$	−27.3855	−1.45759	−0.728793	−	0.684735i	$-0.759918\pi$
$353$	−27.3855	−1.45759	−0.728793	+	0.684735i	$0.759918\pi$
$354$	0	0
$355$	8.23787	0.437221
$356$	−7.99372	−0.423666
$357$	0	0
$358$	40.1488	2.12193
$359$	6.29963	0.332482	0.166241	−	0.986085i	$-0.446837\pi$
$359$	6.29963	0.332482	0.166241	+	0.986085i	$0.446837\pi$
$360$	0	0
$361$	−15.2186	−0.800977
$362$	−12.6729	−0.666074
$363$	0	0
$364$	29.8838	1.56634
$365$	11.6851	0.611627
$366$	0	0
$367$	−25.8065	−1.34709	−0.673544	−	0.739147i	$-0.735229\pi$
$367$	−25.8065	−1.34709	−0.673544	+	0.739147i	$0.735229\pi$
$368$	−36.7306	−1.91471
$369$	0	0
$370$	9.58728	0.498419
$371$	2.26750	0.117723
$372$	0	0
$373$	−31.5247	−1.63229	−0.816144	−	0.577849i	$-0.803892\pi$
$373$	−31.5247	−1.63229	−0.816144	+	0.577849i	$0.803892\pi$
$374$	−2.04663	−0.105829
$375$	0	0
$376$	−2.38709	−0.123105
$377$	53.5399	2.75745
$378$	0	0
$379$	−0.812547	−0.0417377	−0.0208689	−	0.999782i	$-0.506643\pi$
$379$	−0.812547	−0.0417377	−0.0208689	+	0.999782i	$0.506643\pi$
$380$	−2.48798	−0.127631
$381$	0	0
$382$	−41.8226	−2.13983
$383$	−13.8221	−0.706275	−0.353137	−	0.935571i	$-0.614885\pi$
$383$	−13.8221	−0.706275	−0.353137	+	0.935571i	$0.614885\pi$
$384$	0	0
$385$	2.02737	0.103324
$386$	−14.0242	−0.713814
$387$	0	0
$388$	−14.8814	−0.755487
$389$	14.9663	0.758820	0.379410	−	0.925229i	$-0.376127\pi$
$389$	14.9663	0.758820	0.379410	+	0.925229i	$0.376127\pi$
$390$	0	0
$391$	−8.72477	−0.441230
$392$	−0.291752	−0.0147357
$393$	0	0
$394$	3.91457	0.197213
$395$	−7.12894	−0.358696
$396$	0	0
$397$	34.3962	1.72630	0.863148	−	0.504952i	$-0.168490\pi$
$397$	34.3962	1.72630	0.863148	+	0.504952i	$0.168490\pi$
$398$	11.5844	0.580673
$399$	0	0
$400$	19.8672	0.993362
$401$	−17.0784	−0.852856	−0.426428	−	0.904521i	$-0.640228\pi$
$401$	−17.0784	−0.852856	−0.426428	+	0.904521i	$0.640228\pi$
$402$	0	0
$403$	−61.8063	−3.07879
$404$	26.3041	1.30868
$405$	0	0
$406$	45.5505	2.26063
$407$	6.73222	0.333704
$408$	0	0
$409$	−14.7739	−0.730522	−0.365261	−	0.930905i	$-0.619020\pi$
$409$	−14.7739	−0.730522	−0.365261	+	0.930905i	$0.619020\pi$
$410$	−13.7017	−0.676680
$411$	0	0
$412$	−13.8464	−0.682165
$413$	29.2789	1.44072
$414$	0	0
$415$	−2.71630	−0.133338
$416$	−47.5095	−2.32935
$417$	0	0
$418$	−3.75900	−0.183859
$419$	22.2765	1.08828	0.544138	−	0.838995i	$-0.316857\pi$
$419$	22.2765	1.08828	0.544138	+	0.838995i	$0.316857\pi$
$420$	0	0
$421$	−2.09367	−0.102039	−0.0510196	−	0.998698i	$-0.516247\pi$
$421$	−2.09367	−0.102039	−0.0510196	+	0.998698i	$0.516247\pi$
$422$	−18.1077	−0.881470
$423$	0	0
$424$	−0.419370	−0.0203664
$425$	4.71915	0.228912
$426$	0	0
$427$	−11.3398	−0.548769
$428$	−0.830284	−0.0401333
$429$	0	0
$430$	−2.96606	−0.143036
$431$	6.98387	0.336401	0.168201	−	0.985753i	$-0.446204\pi$
$431$	6.98387	0.336401	0.168201	+	0.985753i	$0.446204\pi$
$432$	0	0
$433$	−13.7024	−0.658493	−0.329246	−	0.944244i	$-0.606795\pi$
$433$	−13.7024	−0.658493	−0.329246	+	0.944244i	$0.606795\pi$
$434$	−52.5834	−2.52408
$435$	0	0
$436$	−6.33258	−0.303276
$437$	−16.0246	−0.766561
$438$	0	0
$439$	−15.4711	−0.738395	−0.369197	−	0.929351i	$-0.620367\pi$
$439$	−15.4711	−0.738395	−0.369197	+	0.929351i	$0.620367\pi$
$440$	−0.374958	−0.0178754
$441$	0	0
$442$	−12.7970	−0.608692
$443$	14.8215	0.704192	0.352096	−	0.935964i	$-0.385469\pi$
$443$	14.8215	0.704192	0.352096	+	0.935964i	$0.385469\pi$
$444$	0	0
$445$	−3.39091	−0.160744
$446$	−33.4288	−1.58290
$447$	0	0
$448$	−15.8876	−0.750620
$449$	−1.42009	−0.0670180	−0.0335090	−	0.999438i	$-0.510668\pi$
$449$	−1.42009	−0.0670180	−0.0335090	+	0.999438i	$0.510668\pi$
$450$	0	0
$451$	−9.62141	−0.453054
$452$	−33.5601	−1.57854
$453$	0	0
$454$	−44.4966	−2.08833
$455$	12.6766	0.594288
$456$	0	0
$457$	−37.1299	−1.73686	−0.868432	−	0.495808i	$-0.834872\pi$
$457$	−37.1299	−1.73686	−0.868432	+	0.495808i	$0.834872\pi$
$458$	−49.9516	−2.33408
$459$	0	0
$460$	10.5433	0.491585
$461$	−16.8852	−0.786423	−0.393212	−	0.919448i	$-0.628636\pi$
$461$	−16.8852	−0.786423	−0.393212	+	0.919448i	$0.628636\pi$
$462$	0	0
$463$	41.4741	1.92747	0.963733	−	0.266870i	$-0.0859894\pi$
$463$	41.4741	1.92747	0.963733	+	0.266870i	$0.0859894\pi$
$464$	−38.1660	−1.77181
$465$	0	0
$466$	33.4229	1.54829
$467$	−35.3509	−1.63585	−0.817923	−	0.575327i	$-0.804875\pi$
$467$	−35.3509	−1.63585	−0.817923	+	0.575327i	$0.804875\pi$
$468$	0	0
$469$	22.5494	1.04123
$470$	6.67908	0.308083
$471$	0	0
$472$	−5.41507	−0.249249
$473$	−2.08278	−0.0957662
$474$	0	0
$475$	8.66757	0.397695
$476$	−5.06012	−0.231930
$477$	0	0
$478$	−28.3298	−1.29577
$479$	−27.5748	−1.25993	−0.629963	−	0.776625i	$-0.716930\pi$
$479$	−27.5748	−1.25993	−0.629963	+	0.776625i	$0.716930\pi$
$480$	0	0
$481$	42.0947	1.91935
$482$	25.6168	1.16681
$483$	0	0
$484$	1.73670	0.0789411
$485$	−6.31262	−0.286641
$486$	0	0
$487$	24.7794	1.12286	0.561432	−	0.827523i	$-0.310251\pi$
$487$	24.7794	1.12286	0.561432	+	0.827523i	$0.310251\pi$
$488$	2.09726	0.0949387
$489$	0	0
$490$	0.816320	0.0368776
$491$	8.23701	0.371731	0.185866	−	0.982575i	$-0.440491\pi$
$491$	8.23701	0.371731	0.185866	+	0.982575i	$0.440491\pi$
$492$	0	0
$493$	−9.06574	−0.408300
$494$	−23.5040	−1.05750
$495$	0	0
$496$	44.0588	1.97830
$497$	30.7725	1.38033
$498$	0	0
$499$	−2.97438	−0.133151	−0.0665757	−	0.997781i	$-0.521207\pi$
$499$	−2.97438	−0.133151	−0.0665757	+	0.997781i	$0.521207\pi$
$500$	−12.1000	−0.541127
$501$	0	0
$502$	8.02737	0.358279
$503$	37.4439	1.66954	0.834770	−	0.550599i	$-0.185601\pi$
$503$	37.4439	1.66954	0.834770	+	0.550599i	$0.185601\pi$
$504$	0	0
$505$	11.1581	0.496529
$506$	15.9295	0.708154
$507$	0	0
$508$	−12.6788	−0.562530
$509$	−2.30389	−0.102118	−0.0510590	−	0.998696i	$-0.516260\pi$
$509$	−2.30389	−0.102118	−0.0510590	+	0.998696i	$0.516260\pi$
$510$	0	0
$511$	43.6496	1.93095
$512$	29.3301	1.29622
$513$	0	0
$514$	−3.48322	−0.153638
$515$	−5.87360	−0.258822
$516$	0	0
$517$	4.69007	0.206269
$518$	35.8132	1.57354
$519$	0	0
$520$	−2.34451	−0.102813
$521$	35.0799	1.53688	0.768439	−	0.639923i	$-0.221034\pi$
$521$	35.0799	1.53688	0.768439	+	0.639923i	$0.221034\pi$
$522$	0	0
$523$	−9.47653	−0.414380	−0.207190	−	0.978301i	$-0.566432\pi$
$523$	−9.47653	−0.414380	−0.207190	+	0.978301i	$0.566432\pi$
$524$	22.7184	0.992460
$525$	0	0
$526$	−21.2977	−0.928626
$527$	10.4655	0.455883
$528$	0	0
$529$	44.9075	1.95250
$530$	1.17340	0.0509691
$531$	0	0
$532$	−9.29382	−0.402938
$533$	−60.1600	−2.60582
$534$	0	0
$535$	−0.352204	−0.0152271
$536$	−4.17046	−0.180137
$537$	0	0
$538$	−9.46312	−0.407984
$539$	0.573223	0.0246905
$540$	0	0
$541$	−1.82789	−0.0785871	−0.0392936	−	0.999228i	$-0.512511\pi$
$541$	−1.82789	−0.0785871	−0.0392936	+	0.999228i	$0.512511\pi$
$542$	0.196704	0.00844917
$543$	0	0
$544$	8.04464	0.344911
$545$	−2.68626	−0.115067
$546$	0	0
$547$	31.6160	1.35180	0.675902	−	0.736992i	$-0.263754\pi$
$547$	31.6160	1.35180	0.675902	+	0.736992i	$0.263754\pi$
$548$	−15.1964	−0.649159
$549$	0	0
$550$	−8.61615	−0.367394
$551$	−16.6509	−0.709351
$552$	0	0
$553$	−26.6301	−1.13243
$554$	15.2975	0.649928
$555$	0	0
$556$	6.92690	0.293766
$557$	−26.1551	−1.10823	−0.554113	−	0.832441i	$-0.686943\pi$
$557$	−26.1551	−1.10823	−0.554113	+	0.832441i	$0.686943\pi$
$558$	0	0
$559$	−13.0230	−0.550816
$560$	−9.03653	−0.381863
$561$	0	0
$562$	35.2461	1.48677
$563$	13.4745	0.567884	0.283942	−	0.958841i	$-0.408358\pi$
$563$	13.4745	0.567884	0.283942	+	0.958841i	$0.408358\pi$
$564$	0	0
$565$	−14.2361	−0.598916
$566$	−2.32325	−0.0976535
$567$	0	0
$568$	−5.69131	−0.238802
$569$	−43.7411	−1.83372	−0.916860	−	0.399208i	$-0.869285\pi$
$569$	−43.7411	−1.83372	−0.916860	+	0.399208i	$0.869285\pi$
$570$	0	0
$571$	31.6565	1.32478	0.662391	−	0.749158i	$-0.269542\pi$
$571$	31.6565	1.32478	0.662391	+	0.749158i	$0.269542\pi$
$572$	10.8591	0.454043
$573$	0	0
$574$	−51.1827	−2.13632
$575$	−36.7306	−1.53177
$576$	0	0
$577$	10.1485	0.422487	0.211243	−	0.977433i	$-0.432249\pi$
$577$	10.1485	0.422487	0.211243	+	0.977433i	$0.432249\pi$
$578$	−30.6951	−1.27675
$579$	0	0
$580$	10.9554	0.454896
$581$	−10.1467	−0.420957
$582$	0	0
$583$	0.823963	0.0341251
$584$	−8.07291	−0.334059
$585$	0	0
$586$	21.7821	0.899809
$587$	−10.1222	−0.417788	−0.208894	−	0.977938i	$-0.566986\pi$
$587$	−10.1222	−0.417788	−0.208894	+	0.977938i	$0.566986\pi$
$588$	0	0
$589$	19.2217	0.792017
$590$	15.1514	0.623772
$591$	0	0
$592$	−30.0073	−1.23329
$593$	12.4265	0.510294	0.255147	−	0.966902i	$-0.417876\pi$
$593$	12.4265	0.510294	0.255147	+	0.966902i	$0.417876\pi$
$594$	0	0
$595$	−2.14649	−0.0879973
$596$	−2.97016	−0.121663
$597$	0	0
$598$	99.6030	4.07307
$599$	−13.6848	−0.559145	−0.279573	−	0.960125i	$-0.590193\pi$
$599$	−13.6848	−0.559145	−0.279573	+	0.960125i	$0.590193\pi$
$600$	0	0
$601$	3.39554	0.138507	0.0692535	−	0.997599i	$-0.477938\pi$
$601$	3.39554	0.138507	0.0692535	+	0.997599i	$0.477938\pi$
$602$	−11.0797	−0.451574
$603$	0	0
$604$	2.56429	0.104340
$605$	0.736703	0.0299512
$606$	0	0
$607$	−30.3380	−1.23138	−0.615690	−	0.787989i	$-0.711123\pi$
$607$	−30.3380	−1.23138	−0.615690	+	0.787989i	$0.711123\pi$
$608$	14.7754	0.599222
$609$	0	0
$610$	−5.86814	−0.237594
$611$	29.3257	1.18639
$612$	0	0
$613$	−8.49026	−0.342918	−0.171459	−	0.985191i	$-0.554848\pi$
$613$	−8.49026	−0.342918	−0.171459	+	0.985191i	$0.554848\pi$
$614$	21.2623	0.858075
$615$	0	0
$616$	−1.40065	−0.0564339
$617$	40.1385	1.61591	0.807957	−	0.589241i	$-0.200573\pi$
$617$	40.1385	1.61591	0.807957	+	0.589241i	$0.200573\pi$
$618$	0	0
$619$	1.56667	0.0629696	0.0314848	−	0.999504i	$-0.489976\pi$
$619$	1.56667	0.0629696	0.0314848	+	0.999504i	$0.489976\pi$
$620$	−12.6468	−0.507909
$621$	0	0
$622$	23.6600	0.948680
$623$	−12.6667	−0.507481
$624$	0	0
$625$	17.1536	0.686143
$626$	−28.9238	−1.15603
$627$	0	0
$628$	−31.8748	−1.27194
$629$	−7.12776	−0.284203
$630$	0	0
$631$	−0.543836	−0.0216498	−0.0108249	−	0.999941i	$-0.503446\pi$
$631$	−0.543836	−0.0216498	−0.0108249	+	0.999941i	$0.503446\pi$
$632$	4.92518	0.195913
$633$	0	0
$634$	−28.4539	−1.13005
$635$	−5.37830	−0.213431
$636$	0	0
$637$	3.58421	0.142011
$638$	16.5521	0.655303
$639$	0	0
$640$	2.97367	0.117545
$641$	36.4416	1.43936	0.719679	−	0.694307i	$-0.244289\pi$
$641$	36.4416	1.43936	0.719679	+	0.694307i	$0.244289\pi$
$642$	0	0
$643$	0.0803249	0.00316771	0.00158385	−	0.999999i	$-0.499496\pi$
$643$	0.0803249	0.00316771	0.00158385	+	0.999999i	$0.499496\pi$
$644$	39.3845	1.55197
$645$	0	0
$646$	3.97986	0.156585
$647$	14.7544	0.580055	0.290028	−	0.957018i	$-0.406335\pi$
$647$	14.7544	0.580055	0.290028	+	0.957018i	$0.406335\pi$
$648$	0	0
$649$	10.6393	0.417631
$650$	−53.8744	−2.11313
$651$	0	0
$652$	−8.27028	−0.323889
$653$	−23.4290	−0.916846	−0.458423	−	0.888734i	$-0.651585\pi$
$653$	−23.4290	−0.916846	−0.458423	+	0.888734i	$0.651585\pi$
$654$	0	0
$655$	9.63708	0.376552
$656$	42.8852	1.67439
$657$	0	0
$658$	24.9496	0.972638
$659$	−33.5313	−1.30619	−0.653096	−	0.757275i	$-0.726530\pi$
$659$	−33.5313	−1.30619	−0.653096	+	0.757275i	$0.726530\pi$
$660$	0	0
$661$	19.8038	0.770277	0.385138	−	0.922859i	$-0.374154\pi$
$661$	19.8038	0.770277	0.385138	+	0.922859i	$0.374154\pi$
$662$	−27.1992	−1.05713
$663$	0	0
$664$	1.87662	0.0728268
$665$	−3.94241	−0.152880
$666$	0	0
$667$	70.5614	2.73215
$668$	−16.3150	−0.631248
$669$	0	0
$670$	11.6689	0.450810
$671$	−4.12063	−0.159075
$672$	0	0
$673$	−18.5743	−0.715987	−0.357993	−	0.933724i	$-0.616539\pi$
$673$	−18.5743	−0.715987	−0.357993	+	0.933724i	$0.616539\pi$
$674$	8.18428	0.315247
$675$	0	0
$676$	45.3220	1.74315
$677$	45.9678	1.76669	0.883343	−	0.468727i	$-0.155287\pi$
$677$	45.9678	1.76669	0.883343	+	0.468727i	$0.155287\pi$
$678$	0	0
$679$	−23.5807	−0.904946
$680$	0.396988	0.0152238
$681$	0	0
$682$	−19.1077	−0.731671
$683$	−15.1447	−0.579496	−0.289748	−	0.957103i	$-0.593571\pi$
$683$	−15.1447	−0.579496	−0.289748	+	0.957103i	$0.593571\pi$
$684$	0	0
$685$	−6.44627	−0.246299
$686$	−34.1883	−1.30532
$687$	0	0
$688$	9.28350	0.353930
$689$	5.15201	0.196276
$690$	0	0
$691$	8.39960	0.319536	0.159768	−	0.987155i	$-0.448925\pi$
$691$	8.39960	0.319536	0.159768	+	0.987155i	$0.448925\pi$
$692$	−4.79210	−0.182168
$693$	0	0
$694$	22.7576	0.863866
$695$	2.93837	0.111459
$696$	0	0
$697$	10.1867	0.385849
$698$	24.5140	0.927869
$699$	0	0
$700$	−21.3027	−0.805167
$701$	−16.3583	−0.617845	−0.308922	−	0.951087i	$-0.599968\pi$
$701$	−16.3583	−0.617845	−0.308922	+	0.951087i	$0.599968\pi$
$702$	0	0
$703$	−13.0914	−0.493752
$704$	−5.77323	−0.217587
$705$	0	0
$706$	−52.9378	−1.99234
$707$	41.6810	1.56757
$708$	0	0
$709$	−23.6512	−0.888239	−0.444119	−	0.895968i	$-0.646483\pi$
$709$	−23.6512	−0.888239	−0.444119	+	0.895968i	$0.646483\pi$
$710$	15.9243	0.597627
$711$	0	0
$712$	2.34268	0.0877957
$713$	−81.4559	−3.05055
$714$	0	0
$715$	4.60640	0.172270
$716$	36.0706	1.34802
$717$	0	0
$718$	12.1775	0.454462
$719$	19.2429	0.717638	0.358819	−	0.933407i	$-0.383179\pi$
$719$	19.2429	0.717638	0.358819	+	0.933407i	$0.383179\pi$
$720$	0	0
$721$	−21.9408	−0.817118
$722$	−29.4183	−1.09484
$723$	0	0
$724$	−11.3857	−0.423145
$725$	−38.1660	−1.41745
$726$	0	0
$727$	11.0721	0.410641	0.205320	−	0.978695i	$-0.434176\pi$
$727$	11.0721	0.410641	0.205320	+	0.978695i	$0.434176\pi$
$728$	−8.75789	−0.324589
$729$	0	0
$730$	22.5880	0.836018
$731$	2.20515	0.0815604
$732$	0	0
$733$	−37.1941	−1.37380	−0.686898	−	0.726754i	$-0.741028\pi$
$733$	−37.1941	−1.37380	−0.686898	+	0.726754i	$0.741028\pi$
$734$	−49.8854	−1.84130
$735$	0	0
$736$	−62.6138	−2.30798
$737$	8.19397	0.301829
$738$	0	0
$739$	23.9087	0.879497	0.439748	−	0.898121i	$-0.355068\pi$
$739$	23.9087	0.879497	0.439748	+	0.898121i	$0.355068\pi$
$740$	8.61344	0.316636
$741$	0	0
$742$	4.38321	0.160913
$743$	−33.4343	−1.22659	−0.613293	−	0.789855i	$-0.710156\pi$
$743$	−33.4343	−1.22659	−0.613293	+	0.789855i	$0.710156\pi$
$744$	0	0
$745$	−1.25993	−0.0461603
$746$	−60.9390	−2.23114
$747$	0	0
$748$	−1.83874	−0.0672310
$749$	−1.31565	−0.0480729
$750$	0	0
$751$	13.9300	0.508313	0.254156	−	0.967163i	$-0.418202\pi$
$751$	13.9300	0.508313	0.254156	+	0.967163i	$0.418202\pi$
$752$	−20.9049	−0.762324
$753$	0	0
$754$	103.496	3.76909
$755$	1.08776	0.0395878
$756$	0	0
$757$	45.1704	1.64175	0.820873	−	0.571110i	$-0.193487\pi$
$757$	45.1704	1.64175	0.820873	+	0.571110i	$0.193487\pi$
$758$	−1.57070	−0.0570503
$759$	0	0
$760$	0.729140	0.0264487
$761$	−20.8657	−0.756382	−0.378191	−	0.925728i	$-0.623454\pi$
$761$	−20.8657	−0.756382	−0.378191	+	0.925728i	$0.623454\pi$
$762$	0	0
$763$	−10.0345	−0.363273
$764$	−37.5745	−1.35940
$765$	0	0
$766$	−26.7188	−0.965390
$767$	66.5249	2.40207
$768$	0	0
$769$	−3.08716	−0.111326	−0.0556630	−	0.998450i	$-0.517727\pi$
$769$	−3.08716	−0.111326	−0.0556630	+	0.998450i	$0.517727\pi$
$770$	3.91902	0.141232
$771$	0	0
$772$	−12.5997	−0.453473
$773$	32.2571	1.16021	0.580103	−	0.814543i	$-0.303012\pi$
$773$	32.2571	1.16021	0.580103	+	0.814543i	$0.303012\pi$
$774$	0	0
$775$	44.0588	1.58264
$776$	4.36121	0.156558
$777$	0	0
$778$	28.9306	1.03721
$779$	18.7097	0.670345
$780$	0	0
$781$	11.1821	0.400126
$782$	−16.8655	−0.603108
$783$	0	0
$784$	−2.55501	−0.0912503
$785$	−13.5212	−0.482592
$786$	0	0
$787$	34.7069	1.23717	0.618583	−	0.785720i	$-0.287707\pi$
$787$	34.7069	1.23717	0.618583	+	0.785720i	$0.287707\pi$
$788$	3.51695	0.125286
$789$	0	0
$790$	−13.7806	−0.490293
$791$	−53.1788	−1.89082
$792$	0	0
$793$	−25.7652	−0.914948
$794$	66.4897	2.35963
$795$	0	0
$796$	10.4077	0.368891
$797$	17.6044	0.623580	0.311790	−	0.950151i	$-0.399071\pi$
$797$	17.6044	0.623580	0.311790	+	0.950151i	$0.399071\pi$
$798$	0	0
$799$	−4.96563	−0.175671
$800$	33.8673	1.19739
$801$	0	0
$802$	−33.0136	−1.16575
$803$	15.8613	0.559735
$804$	0	0
$805$	16.7067	0.588835
$806$	−119.475	−4.20833
$807$	0	0
$808$	−7.70881	−0.271195
$809$	−5.46716	−0.192215	−0.0961076	−	0.995371i	$-0.530639\pi$
$809$	−5.46716	−0.192215	−0.0961076	+	0.995371i	$0.530639\pi$
$810$	0	0
$811$	8.50715	0.298726	0.149363	−	0.988782i	$-0.452278\pi$
$811$	8.50715	0.298726	0.149363	+	0.988782i	$0.452278\pi$
$812$	40.9236	1.43614
$813$	0	0
$814$	13.0138	0.456132
$815$	−3.50822	−0.122888
$816$	0	0
$817$	4.05015	0.141697
$818$	−28.5588	−0.998534
$819$	0	0
$820$	−12.3100	−0.429883
$821$	−22.8349	−0.796945	−0.398472	−	0.917180i	$-0.630460\pi$
$821$	−22.8349	−0.796945	−0.398472	+	0.917180i	$0.630460\pi$
$822$	0	0
$823$	22.8026	0.794847	0.397424	−	0.917635i	$-0.369904\pi$
$823$	22.8026	0.794847	0.397424	+	0.917635i	$0.369904\pi$
$824$	4.05791	0.141364
$825$	0	0
$826$	56.5978	1.96929
$827$	−14.5887	−0.507297	−0.253649	−	0.967296i	$-0.581631\pi$
$827$	−14.5887	−0.507297	−0.253649	+	0.967296i	$0.581631\pi$
$828$	0	0
$829$	−2.39874	−0.0833118	−0.0416559	−	0.999132i	$-0.513263\pi$
$829$	−2.39874	−0.0833118	−0.0416559	+	0.999132i	$0.513263\pi$
$830$	−5.25076	−0.182257
$831$	0	0
$832$	−36.0984	−1.25149
$833$	−0.606902	−0.0210279
$834$	0	0
$835$	−6.92078	−0.239503
$836$	−3.37718	−0.116802
$837$	0	0
$838$	43.0617	1.48754
$839$	37.8328	1.30613	0.653066	−	0.757301i	$-0.273482\pi$
$839$	37.8328	1.30613	0.653066	+	0.757301i	$0.273482\pi$
$840$	0	0
$841$	44.3190	1.52824
$842$	−4.04718	−0.139475
$843$	0	0
$844$	−16.2684	−0.559982
$845$	19.2254	0.661375
$846$	0	0
$847$	2.75195	0.0945581
$848$	−3.67262	−0.126118
$849$	0	0
$850$	9.12238	0.312895
$851$	55.4776	1.90175
$852$	0	0
$853$	−15.7973	−0.540889	−0.270444	−	0.962736i	$-0.587171\pi$
$853$	−15.7973	−0.540889	−0.270444	+	0.962736i	$0.587171\pi$
$854$	−21.9204	−0.750100
$855$	0	0
$856$	0.243327	0.00831676
$857$	−34.0118	−1.16182	−0.580910	−	0.813968i	$-0.697303\pi$
$857$	−34.0118	−1.16182	−0.580910	+	0.813968i	$0.697303\pi$
$858$	0	0
$859$	−31.7770	−1.08422	−0.542108	−	0.840309i	$-0.682374\pi$
$859$	−31.7770	−1.08422	−0.542108	+	0.840309i	$0.682374\pi$
$860$	−2.66478	−0.0908682
$861$	0	0
$862$	13.5002	0.459819
$863$	34.7324	1.18230	0.591152	−	0.806560i	$-0.298674\pi$
$863$	34.7324	1.18230	0.591152	+	0.806560i	$0.298674\pi$
$864$	0	0
$865$	−2.03279	−0.0691170
$866$	−26.4874	−0.900079
$867$	0	0
$868$	−47.2421	−1.60350
$869$	−9.67681	−0.328263
$870$	0	0
$871$	51.2347	1.73602
$872$	1.85586	0.0628473
$873$	0	0
$874$	−30.9765	−1.04779
$875$	−19.1734	−0.648179
$876$	0	0
$877$	−1.78993	−0.0604416	−0.0302208	−	0.999543i	$-0.509621\pi$
$877$	−1.78993	−0.0604416	−0.0302208	+	0.999543i	$0.509621\pi$
$878$	−29.9065	−1.00929
$879$	0	0
$880$	−3.28368	−0.110693
$881$	−18.9540	−0.638575	−0.319288	−	0.947658i	$-0.603444\pi$
$881$	−18.9540	−0.638575	−0.319288	+	0.947658i	$0.603444\pi$
$882$	0	0
$883$	−29.8688	−1.00516	−0.502582	−	0.864530i	$-0.667616\pi$
$883$	−29.8688	−1.00516	−0.502582	+	0.864530i	$0.667616\pi$
$884$	−11.4971	−0.386691
$885$	0	0
$886$	28.6508	0.962543
$887$	30.9847	1.04036	0.520182	−	0.854056i	$-0.325864\pi$
$887$	30.9847	1.04036	0.520182	+	0.854056i	$0.325864\pi$
$888$	0	0
$889$	−20.0906	−0.673816
$890$	−6.55481	−0.219718
$891$	0	0
$892$	−30.0333	−1.00559
$893$	−9.12028	−0.305198
$894$	0	0
$895$	15.3010	0.511457
$896$	11.1081	0.371096
$897$	0	0
$898$	−2.74510	−0.0916053
$899$	−84.6393	−2.82288
$900$	0	0
$901$	−0.872374	−0.0290630
$902$	−18.5987	−0.619269
$903$	0	0
$904$	9.83530	0.327117
$905$	−4.82976	−0.160547
$906$	0	0
$907$	49.4860	1.64316	0.821578	−	0.570096i	$-0.193094\pi$
$907$	49.4860	1.64316	0.821578	+	0.570096i	$0.193094\pi$
$908$	−39.9768	−1.32668
$909$	0	0
$910$	24.5045	0.812318
$911$	−16.6726	−0.552387	−0.276193	−	0.961102i	$-0.589073\pi$
$911$	−16.6726	−0.552387	−0.276193	+	0.961102i	$0.589073\pi$
$912$	0	0
$913$	−3.68711	−0.122025
$914$	−71.7742	−2.37408
$915$	0	0
$916$	−44.8777	−1.48280
$917$	35.9992	1.18880
$918$	0	0
$919$	43.1328	1.42282	0.711410	−	0.702777i	$-0.248057\pi$
$919$	43.1328	1.42282	0.711410	+	0.702777i	$0.248057\pi$
$920$	−3.08988	−0.101870
$921$	0	0
$922$	−32.6401	−1.07494
$923$	69.9184	2.30139
$924$	0	0
$925$	−30.0073	−0.986635
$926$	80.1717	2.63461
$927$	0	0
$928$	−65.0608	−2.13573
$929$	2.24678	0.0737145	0.0368572	−	0.999321i	$-0.488265\pi$
$929$	2.24678	0.0737145	0.0368572	+	0.999321i	$0.488265\pi$
$930$	0	0
$931$	−1.11468	−0.0365323
$932$	30.0279	0.983598
$933$	0	0
$934$	−68.3353	−2.23600
$935$	−0.779987	−0.0255083
$936$	0	0
$937$	−3.68848	−0.120497	−0.0602487	−	0.998183i	$-0.519189\pi$
$937$	−3.68848	−0.120497	−0.0602487	+	0.998183i	$0.519189\pi$
$938$	43.5892	1.42324
$939$	0	0
$940$	6.00064	0.195719
$941$	44.8597	1.46239	0.731193	−	0.682171i	$-0.238964\pi$
$941$	44.8597	1.46239	0.731193	+	0.682171i	$0.238964\pi$
$942$	0	0
$943$	−79.2862	−2.58191
$944$	−47.4224	−1.54347
$945$	0	0
$946$	−4.02613	−0.130901
$947$	−13.1381	−0.426930	−0.213465	−	0.976951i	$-0.568475\pi$
$947$	−13.1381	−0.426930	−0.213465	+	0.976951i	$0.568475\pi$
$948$	0	0
$949$	99.1767	3.21941
$950$	16.7549	0.543600
$951$	0	0
$952$	1.48295	0.0480625
$953$	19.7125	0.638550	0.319275	−	0.947662i	$-0.396561\pi$
$953$	19.7125	0.638550	0.319275	+	0.947662i	$0.396561\pi$
$954$	0	0
$955$	−15.9390	−0.515772
$956$	−25.4522	−0.823182
$957$	0	0
$958$	−53.3037	−1.72216
$959$	−24.0800	−0.777583
$960$	0	0
$961$	66.7073	2.15185
$962$	81.3715	2.62352
$963$	0	0
$964$	23.0148	0.741255
$965$	−5.34474	−0.172053
$966$	0	0
$967$	15.3762	0.494464	0.247232	−	0.968956i	$-0.420479\pi$
$967$	15.3762	0.494464	0.247232	+	0.968956i	$0.420479\pi$
$968$	−0.508967	−0.0163588
$969$	0	0
$970$	−12.2026	−0.391803
$971$	17.3286	0.556102	0.278051	−	0.960566i	$-0.410312\pi$
$971$	17.3286	0.556102	0.278051	+	0.960566i	$0.410312\pi$
$972$	0	0
$973$	10.9762	0.351882
$974$	47.9000	1.53482
$975$	0	0
$976$	18.3667	0.587905
$977$	61.1978	1.95789	0.978946	−	0.204121i	$-0.0654337\pi$
$977$	61.1978	1.95789	0.978946	+	0.204121i	$0.0654337\pi$
$978$	0	0
$979$	−4.60281	−0.147107
$980$	0.733402	0.0234277
$981$	0	0
$982$	15.9226	0.508111
$983$	54.8293	1.74878	0.874391	−	0.485222i	$-0.161261\pi$
$983$	54.8293	1.74878	0.874391	+	0.485222i	$0.161261\pi$
$984$	0	0
$985$	1.49188	0.0475351
$986$	−17.5246	−0.558096
$987$	0	0
$988$	−21.1166	−0.671808
$989$	−17.1633	−0.545763
$990$	0	0
$991$	−40.2449	−1.27842	−0.639211	−	0.769031i	$-0.720739\pi$
$991$	−40.2449	−1.27842	−0.639211	+	0.769031i	$0.720739\pi$
$992$	75.1061	2.38462
$993$	0	0
$994$	59.4849	1.88675
$995$	4.41490	0.139962
$996$	0	0
$997$	11.2747	0.357072	0.178536	−	0.983933i	$-0.442864\pi$
$997$	11.2747	0.357072	0.178536	+	0.983933i	$0.442864\pi$
$998$	−5.74963	−0.182001
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2673.2.a.p.1.5	yes	6
3.2	odd	2		2673.2.a.j.1.2	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2673.2.a.j.1.2	✓	6	3.2	odd	2
2673.2.a.p.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:	\( N \)	\(=\)	\( 2673 = 3^{5} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2673.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93306 q^{2} +1.73670 q^{4} +0.736703 q^{5} +2.75195 q^{7} -0.508967 q^{8} +O(q^{10})\)	\(q+1.93306 q^{2} +1.73670 q^{4} +0.736703 q^{5} +2.75195 q^{7} -0.508967 q^{8} +1.42409 q^{10} +1.00000 q^{11} +6.25273 q^{13} +5.31967 q^{14} -4.45727 q^{16} -1.05875 q^{17} -1.94459 q^{19} +1.27943 q^{20} +1.93306 q^{22} +8.24060 q^{23} -4.45727 q^{25} +12.0869 q^{26} +4.77932 q^{28} +8.56265 q^{29} -9.88470 q^{31} -7.59821 q^{32} -2.04663 q^{34} +2.02737 q^{35} +6.73222 q^{37} -3.75900 q^{38} -0.374958 q^{40} -9.62141 q^{41} -2.08278 q^{43} +1.73670 q^{44} +15.9295 q^{46} +4.69007 q^{47} +0.573223 q^{49} -8.61615 q^{50} +10.8591 q^{52} +0.823963 q^{53} +0.736703 q^{55} -1.40065 q^{56} +16.5521 q^{58} +10.6393 q^{59} -4.12063 q^{61} -19.1077 q^{62} -5.77323 q^{64} +4.60640 q^{65} +8.19397 q^{67} -1.83874 q^{68} +3.91902 q^{70} +11.1821 q^{71} +15.8613 q^{73} +13.0138 q^{74} -3.37718 q^{76} +2.75195 q^{77} -9.67681 q^{79} -3.28368 q^{80} -18.5987 q^{82} -3.68711 q^{83} -0.779987 q^{85} -4.02613 q^{86} -0.508967 q^{88} -4.60281 q^{89} +17.2072 q^{91} +14.3115 q^{92} +9.06617 q^{94} -1.43259 q^{95} -8.56874 q^{97} +1.10807 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8	\( 6 q + 2 q^{2} + 8 q^{4} + 2 q^{5} - 2 q^{7} + 6 q^{8} + 8 q^{10} + 6 q^{11} - 8 q^{13} - 4 q^{14} + 8 q^{16} + 2 q^{17} - 4 q^{19} + 40 q^{20} + 2 q^{22} + 10 q^{23} + 8 q^{25} + 2 q^{26} - 12 q^{28} + 6 q^{29} - 8 q^{31} + 4 q^{32} - 10 q^{34} - 10 q^{35} + 2 q^{37} + 30 q^{38} + 18 q^{40} - 4 q^{41} + 2 q^{43} + 8 q^{44} + 4 q^{46} + 28 q^{47} + 6 q^{49} + 16 q^{50} - 12 q^{52} + 24 q^{53} + 2 q^{55} - 6 q^{56} + 6 q^{58} - 8 q^{59} + 2 q^{61} - 10 q^{62} + 18 q^{64} - 4 q^{65} + 12 q^{67} - 14 q^{68} - 10 q^{70} + 30 q^{71} + 16 q^{73} + 78 q^{74} + 2 q^{76} - 2 q^{77} - 12 q^{79} + 58 q^{80} - 16 q^{82} - 16 q^{85} - 24 q^{86} + 6 q^{88} - 6 q^{89} + 6 q^{91} + 32 q^{92} + 10 q^{94} + 6 q^{95} - 18 q^{97} - 6 q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 8 * q^4 + 2 * q^5 - 2 * q^7 + 6 * q^8 + 8 * q^10 + 6 * q^11 - 8 * q^13 - 4 * q^14 + 8 * q^16 + 2 * q^17 - 4 * q^19 + 40 * q^20 + 2 * q^22 + 10 * q^23 + 8 * q^25 + 2 * q^26 - 12 * q^28 + 6 * q^29 - 8 * q^31 + 4 * q^32 - 10 * q^34 - 10 * q^35 + 2 * q^37 + 30 * q^38 + 18 * q^40 - 4 * q^41 + 2 * q^43 + 8 * q^44 + 4 * q^46 + 28 * q^47 + 6 * q^49 + 16 * q^50 - 12 * q^52 + 24 * q^53 + 2 * q^55 - 6 * q^56 + 6 * q^58 - 8 * q^59 + 2 * q^61 - 10 * q^62 + 18 * q^64 - 4 * q^65 + 12 * q^67 - 14 * q^68 - 10 * q^70 + 30 * q^71 + 16 * q^73 + 78 * q^74 + 2 * q^76 - 2 * q^77 - 12 * q^79 + 58 * q^80 - 16 * q^82 - 16 * q^85 - 24 * q^86 + 6 * q^88 - 6 * q^89 + 6 * q^91 + 32 * q^92 + 10 * q^94 + 6 * q^95 - 18 * q^97 - 6 * q^98

Introduction

L-functions

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Varieties

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Database

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