LMFDB - Embedded newform 207.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [207,2,Mod(1,207)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(207, 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("207.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 207 = 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	207.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:	$1.65290332184$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	207.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} -3.41421 q^{7} +4.41421 q^{8} +O(q^{10})$	$q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} -3.41421 q^{7} +4.41421 q^{8} +1.41421 q^{10} -2.82843 q^{11} -8.24264 q^{14} +3.00000 q^{16} +7.41421 q^{17} -6.24264 q^{19} +2.24264 q^{20} -6.82843 q^{22} +1.00000 q^{23} -4.65685 q^{25} -13.0711 q^{28} +8.48528 q^{29} +8.48528 q^{31} -1.58579 q^{32} +17.8995 q^{34} -2.00000 q^{35} -4.82843 q^{37} -15.0711 q^{38} +2.58579 q^{40} -1.65685 q^{41} -1.75736 q^{43} -10.8284 q^{44} +2.41421 q^{46} -0.343146 q^{47} +4.65685 q^{49} -11.2426 q^{50} -5.07107 q^{53} -1.65685 q^{55} -15.0711 q^{56} +20.4853 q^{58} +7.65685 q^{59} -0.828427 q^{61} +20.4853 q^{62} -9.82843 q^{64} +8.58579 q^{67} +28.3848 q^{68} -4.82843 q^{70} -13.6569 q^{71} +13.3137 q^{73} -11.6569 q^{74} -23.8995 q^{76} +9.65685 q^{77} +7.89949 q^{79} +1.75736 q^{80} -4.00000 q^{82} +6.82843 q^{83} +4.34315 q^{85} -4.24264 q^{86} -12.4853 q^{88} +13.0711 q^{89} +3.82843 q^{92} -0.828427 q^{94} -3.65685 q^{95} -10.0000 q^{97} +11.2426 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} - 8 q^{14} + 6 q^{16} + 12 q^{17} - 4 q^{19} - 4 q^{20} - 8 q^{22} + 2 q^{23} + 2 q^{25} - 12 q^{28} - 6 q^{32} + 16 q^{34} - 4 q^{35} - 4 q^{37} - 16 q^{38} + 8 q^{40} + 8 q^{41} - 12 q^{43} - 16 q^{44} + 2 q^{46} - 12 q^{47} - 2 q^{49} - 14 q^{50} + 4 q^{53} + 8 q^{55} - 16 q^{56} + 24 q^{58} + 4 q^{59} + 4 q^{61} + 24 q^{62} - 14 q^{64} + 20 q^{67} + 20 q^{68} - 4 q^{70} - 16 q^{71} + 4 q^{73} - 12 q^{74} - 28 q^{76} + 8 q^{77} - 4 q^{79} + 12 q^{80} - 8 q^{82} + 8 q^{83} + 20 q^{85} - 8 q^{88} + 12 q^{89} + 2 q^{92} + 4 q^{94} + 4 q^{95} - 20 q^{97} + 14 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8 - 8 * q^14 + 6 * q^16 + 12 * q^17 - 4 * q^19 - 4 * q^20 - 8 * q^22 + 2 * q^23 + 2 * q^25 - 12 * q^28 - 6 * q^32 + 16 * q^34 - 4 * q^35 - 4 * q^37 - 16 * q^38 + 8 * q^40 + 8 * q^41 - 12 * q^43 - 16 * q^44 + 2 * q^46 - 12 * q^47 - 2 * q^49 - 14 * q^50 + 4 * q^53 + 8 * q^55 - 16 * q^56 + 24 * q^58 + 4 * q^59 + 4 * q^61 + 24 * q^62 - 14 * q^64 + 20 * q^67 + 20 * q^68 - 4 * q^70 - 16 * q^71 + 4 * q^73 - 12 * q^74 - 28 * q^76 + 8 * q^77 - 4 * q^79 + 12 * q^80 - 8 * q^82 + 8 * q^83 + 20 * q^85 - 8 * q^88 + 12 * q^89 + 2 * q^92 + 4 * q^94 + 4 * q^95 - 20 * q^97 + 14 * 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	0.585786	0.261972	0.130986	−	0.991384i	$-0.458186\pi$
$5$	0.585786	0.261972	0.130986	+	0.991384i	$0.458186\pi$
$6$	0	0
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	1.41421	0.447214
$11$	−2.82843	−0.852803	−0.426401	−	0.904534i	$-0.640219\pi$
$11$	−2.82843	−0.852803	−0.426401	+	0.904534i	$0.640219\pi$
$12$	0	0
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−8.24264	−2.20294
$15$	0	0
$16$	3.00000	0.750000
$17$	7.41421	1.79821	0.899105	−	0.437732i	$-0.144218\pi$
$17$	7.41421	1.79821	0.899105	+	0.437732i	$0.144218\pi$
$18$	0	0
$19$	−6.24264	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$20$	2.24264	0.501470
$21$	0	0
$22$	−6.82843	−1.45583
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.65685	−0.931371
$26$	0	0
$27$	0	0
$28$	−13.0711	−2.47020
$29$	8.48528	1.57568	0.787839	−	0.615882i	$-0.211200\pi$
$29$	8.48528	1.57568	0.787839	+	0.615882i	$0.211200\pi$
$30$	0	0
$31$	8.48528	1.52400	0.762001	−	0.647576i	$-0.224217\pi$
$31$	8.48528	1.52400	0.762001	+	0.647576i	$0.224217\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	17.8995	3.06974
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−4.82843	−0.793789	−0.396894	−	0.917864i	$-0.629912\pi$
$37$	−4.82843	−0.793789	−0.396894	+	0.917864i	$0.629912\pi$
$38$	−15.0711	−2.44485
$39$	0	0
$40$	2.58579	0.408849
$41$	−1.65685	−0.258757	−0.129379	−	0.991595i	$-0.541298\pi$
$41$	−1.65685	−0.258757	−0.129379	+	0.991595i	$0.541298\pi$
$42$	0	0
$43$	−1.75736	−0.267995	−0.133997	−	0.990982i	$-0.542781\pi$
$43$	−1.75736	−0.267995	−0.133997	+	0.990982i	$0.542781\pi$
$44$	−10.8284	−1.63245
$45$	0	0
$46$	2.41421	0.355956
$47$	−0.343146	−0.0500530	−0.0250265	−	0.999687i	$-0.507967\pi$
$47$	−0.343146	−0.0500530	−0.0250265	+	0.999687i	$0.507967\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	−11.2426	−1.58995
$51$	0	0
$52$	0	0
$53$	−5.07107	−0.696565	−0.348282	−	0.937390i	$-0.613235\pi$
$53$	−5.07107	−0.696565	−0.348282	+	0.937390i	$0.613235\pi$
$54$	0	0
$55$	−1.65685	−0.223410
$56$	−15.0711	−2.01396
$57$	0	0
$58$	20.4853	2.68985
$59$	7.65685	0.996838	0.498419	−	0.866936i	$-0.333914\pi$
$59$	7.65685	0.996838	0.498419	+	0.866936i	$0.333914\pi$
$60$	0	0
$61$	−0.828427	−0.106069	−0.0530346	−	0.998593i	$-0.516889\pi$
$61$	−0.828427	−0.106069	−0.0530346	+	0.998593i	$0.516889\pi$
$62$	20.4853	2.60163
$63$	0	0
$64$	−9.82843	−1.22855
$65$	0	0
$66$	0	0
$67$	8.58579	1.04892	0.524460	−	0.851435i	$-0.324267\pi$
$67$	8.58579	1.04892	0.524460	+	0.851435i	$0.324267\pi$
$68$	28.3848	3.44216
$69$	0	0
$70$	−4.82843	−0.577107
$71$	−13.6569	−1.62077	−0.810385	−	0.585897i	$-0.800742\pi$
$71$	−13.6569	−1.62077	−0.810385	+	0.585897i	$0.800742\pi$
$72$	0	0
$73$	13.3137	1.55825	0.779126	−	0.626868i	$-0.215663\pi$
$73$	13.3137	1.55825	0.779126	+	0.626868i	$0.215663\pi$
$74$	−11.6569	−1.35508
$75$	0	0
$76$	−23.8995	−2.74146
$77$	9.65685	1.10050
$78$	0	0
$79$	7.89949	0.888763	0.444381	−	0.895838i	$-0.353424\pi$
$79$	7.89949	0.888763	0.444381	+	0.895838i	$0.353424\pi$
$80$	1.75736	0.196479
$81$	0	0
$82$	−4.00000	−0.441726
$83$	6.82843	0.749517	0.374759	−	0.927122i	$-0.377726\pi$
$83$	6.82843	0.749517	0.374759	+	0.927122i	$0.377726\pi$
$84$	0	0
$85$	4.34315	0.471080
$86$	−4.24264	−0.457496
$87$	0	0
$88$	−12.4853	−1.33094
$89$	13.0711	1.38553	0.692765	−	0.721163i	$-0.256392\pi$
$89$	13.0711	1.38553	0.692765	+	0.721163i	$0.256392\pi$
$90$	0	0
$91$	0	0
$92$	3.82843	0.399141
$93$	0	0
$94$	−0.828427	−0.0854457
$95$	−3.65685	−0.375185
$96$	0	0
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	11.2426	1.13568
$99$	0	0
$100$	−17.8284	−1.78284
$101$	−11.3137	−1.12576	−0.562878	−	0.826540i	$-0.690306\pi$
$101$	−11.3137	−1.12576	−0.562878	+	0.826540i	$0.690306\pi$
$102$	0	0
$103$	−3.41421	−0.336412	−0.168206	−	0.985752i	$-0.553797\pi$
$103$	−3.41421	−0.336412	−0.168206	+	0.985752i	$0.553797\pi$
$104$	0	0
$105$	0	0
$106$	−12.2426	−1.18911
$107$	−8.48528	−0.820303	−0.410152	−	0.912017i	$-0.634524\pi$
$107$	−8.48528	−0.820303	−0.410152	+	0.912017i	$0.634524\pi$
$108$	0	0
$109$	−2.48528	−0.238047	−0.119023	−	0.992891i	$-0.537976\pi$
$109$	−2.48528	−0.238047	−0.119023	+	0.992891i	$0.537976\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−10.2426	−0.967839
$113$	15.4142	1.45005	0.725024	−	0.688724i	$-0.241829\pi$
$113$	15.4142	1.45005	0.725024	+	0.688724i	$0.241829\pi$
$114$	0	0
$115$	0.585786	0.0546249
$116$	32.4853	3.01618
$117$	0	0
$118$	18.4853	1.70171
$119$	−25.3137	−2.32050
$120$	0	0
$121$	−3.00000	−0.272727
$122$	−2.00000	−0.181071
$123$	0	0
$124$	32.4853	2.91726
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−4.48528	−0.398004	−0.199002	−	0.979999i	$-0.563770\pi$
$127$	−4.48528	−0.398004	−0.199002	+	0.979999i	$0.563770\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	21.3137	1.84813
$134$	20.7279	1.79062
$135$	0	0
$136$	32.7279	2.80640
$137$	0.585786	0.0500471	0.0250236	−	0.999687i	$-0.492034\pi$
$137$	0.585786	0.0500471	0.0250236	+	0.999687i	$0.492034\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−7.65685	−0.647122
$141$	0	0
$142$	−32.9706	−2.76683
$143$	0	0
$144$	0	0
$145$	4.97056	0.412783
$146$	32.1421	2.66010
$147$	0	0
$148$	−18.4853	−1.51948
$149$	−16.5858	−1.35876	−0.679380	−	0.733786i	$-0.737751\pi$
$149$	−16.5858	−1.35876	−0.679380	+	0.733786i	$0.737751\pi$
$150$	0	0
$151$	−9.65685	−0.785864	−0.392932	−	0.919568i	$-0.628539\pi$
$151$	−9.65685	−0.785864	−0.392932	+	0.919568i	$0.628539\pi$
$152$	−27.5563	−2.23512
$153$	0	0
$154$	23.3137	1.87867
$155$	4.97056	0.399245
$156$	0	0
$157$	−1.51472	−0.120888	−0.0604439	−	0.998172i	$-0.519252\pi$
$157$	−1.51472	−0.120888	−0.0604439	+	0.998172i	$0.519252\pi$
$158$	19.0711	1.51721
$159$	0	0
$160$	−0.928932	−0.0734385
$161$	−3.41421	−0.269078
$162$	0	0
$163$	18.8284	1.47476	0.737378	−	0.675480i	$-0.236064\pi$
$163$	18.8284	1.47476	0.737378	+	0.675480i	$0.236064\pi$
$164$	−6.34315	−0.495316
$165$	0	0
$166$	16.4853	1.27951
$167$	−5.65685	−0.437741	−0.218870	−	0.975754i	$-0.570237\pi$
$167$	−5.65685	−0.437741	−0.218870	+	0.975754i	$0.570237\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	10.4853	0.804184
$171$	0	0
$172$	−6.72792	−0.512999
$173$	10.8284	0.823270	0.411635	−	0.911349i	$-0.364958\pi$
$173$	10.8284	0.823270	0.411635	+	0.911349i	$0.364958\pi$
$174$	0	0
$175$	15.8995	1.20189
$176$	−8.48528	−0.639602
$177$	0	0
$178$	31.5563	2.36525
$179$	−18.0000	−1.34538	−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000	−1.34538	−0.672692	+	0.739923i	$0.734862\pi$
$180$	0	0
$181$	−11.1716	−0.830376	−0.415188	−	0.909736i	$-0.636284\pi$
$181$	−11.1716	−0.830376	−0.415188	+	0.909736i	$0.636284\pi$
$182$	0	0
$183$	0	0
$184$	4.41421	0.325420
$185$	−2.82843	−0.207950
$186$	0	0
$187$	−20.9706	−1.53352
$188$	−1.31371	−0.0958120
$189$	0	0
$190$	−8.82843	−0.640481
$191$	−0.686292	−0.0496583	−0.0248292	−	0.999692i	$-0.507904\pi$
$191$	−0.686292	−0.0496583	−0.0248292	+	0.999692i	$0.507904\pi$
$192$	0	0
$193$	−1.65685	−0.119263	−0.0596315	−	0.998220i	$-0.518993\pi$
$193$	−1.65685	−0.119263	−0.0596315	+	0.998220i	$0.518993\pi$
$194$	−24.1421	−1.73330
$195$	0	0
$196$	17.8284	1.27346
$197$	−9.17157	−0.653448	−0.326724	−	0.945120i	$-0.605945\pi$
$197$	−9.17157	−0.653448	−0.326724	+	0.945120i	$0.605945\pi$
$198$	0	0
$199$	−2.24264	−0.158977	−0.0794883	−	0.996836i	$-0.525329\pi$
$199$	−2.24264	−0.158977	−0.0794883	+	0.996836i	$0.525329\pi$
$200$	−20.5563	−1.45355
$201$	0	0
$202$	−27.3137	−1.92179
$203$	−28.9706	−2.03333
$204$	0	0
$205$	−0.970563	−0.0677870
$206$	−8.24264	−0.574292
$207$	0	0
$208$	0	0
$209$	17.6569	1.22135
$210$	0	0
$211$	−12.4853	−0.859522	−0.429761	−	0.902943i	$-0.641402\pi$
$211$	−12.4853	−0.859522	−0.429761	+	0.902943i	$0.641402\pi$
$212$	−19.4142	−1.33337
$213$	0	0
$214$	−20.4853	−1.40035
$215$	−1.02944	−0.0702070
$216$	0	0
$217$	−28.9706	−1.96665
$218$	−6.00000	−0.406371
$219$	0	0
$220$	−6.34315	−0.427655
$221$	0	0
$222$	0	0
$223$	−12.9706	−0.868573	−0.434287	−	0.900775i	$-0.642999\pi$
$223$	−12.9706	−0.868573	−0.434287	+	0.900775i	$0.642999\pi$
$224$	5.41421	0.361752
$225$	0	0
$226$	37.2132	2.47539
$227$	−6.14214	−0.407668	−0.203834	−	0.979005i	$-0.565340\pi$
$227$	−6.14214	−0.407668	−0.203834	+	0.979005i	$0.565340\pi$
$228$	0	0
$229$	14.4853	0.957214	0.478607	−	0.878029i	$-0.341142\pi$
$229$	14.4853	0.957214	0.478607	+	0.878029i	$0.341142\pi$
$230$	1.41421	0.0932505
$231$	0	0
$232$	37.4558	2.45910
$233$	21.6569	1.41879	0.709394	−	0.704812i	$-0.248969\pi$
$233$	21.6569	1.41879	0.709394	+	0.704812i	$0.248969\pi$
$234$	0	0
$235$	−0.201010	−0.0131125
$236$	29.3137	1.90816
$237$	0	0
$238$	−61.1127	−3.96135
$239$	−19.3137	−1.24930	−0.624650	−	0.780905i	$-0.714758\pi$
$239$	−19.3137	−1.24930	−0.624650	+	0.780905i	$0.714758\pi$
$240$	0	0
$241$	−22.9706	−1.47966	−0.739832	−	0.672792i	$-0.765095\pi$
$241$	−22.9706	−1.47966	−0.739832	+	0.672792i	$0.765095\pi$
$242$	−7.24264	−0.465575
$243$	0	0
$244$	−3.17157	−0.203039
$245$	2.72792	0.174281
$246$	0	0
$247$	0	0
$248$	37.4558	2.37845
$249$	0	0
$250$	−13.6569	−0.863735
$251$	−20.4853	−1.29302	−0.646510	−	0.762906i	$-0.723772\pi$
$251$	−20.4853	−1.29302	−0.646510	+	0.762906i	$0.723772\pi$
$252$	0	0
$253$	−2.82843	−0.177822
$254$	−10.8284	−0.679436
$255$	0	0
$256$	−29.9706	−1.87316
$257$	6.34315	0.395675	0.197837	−	0.980235i	$-0.436608\pi$
$257$	6.34315	0.395675	0.197837	+	0.980235i	$0.436608\pi$
$258$	0	0
$259$	16.4853	1.02435
$260$	0	0
$261$	0	0
$262$	40.9706	2.53117
$263$	23.3137	1.43758	0.718792	−	0.695225i	$-0.244695\pi$
$263$	23.3137	1.43758	0.718792	+	0.695225i	$0.244695\pi$
$264$	0	0
$265$	−2.97056	−0.182480
$266$	51.4558	3.15496
$267$	0	0
$268$	32.8701	2.00786
$269$	18.1421	1.10615	0.553073	−	0.833133i	$-0.313455\pi$
$269$	18.1421	1.10615	0.553073	+	0.833133i	$0.313455\pi$
$270$	0	0
$271$	−12.4853	−0.758427	−0.379213	−	0.925309i	$-0.623805\pi$
$271$	−12.4853	−0.758427	−0.379213	+	0.925309i	$0.623805\pi$
$272$	22.2426	1.34866
$273$	0	0
$274$	1.41421	0.0854358
$275$	13.1716	0.794276
$276$	0	0
$277$	7.65685	0.460056	0.230028	−	0.973184i	$-0.426118\pi$
$277$	7.65685	0.460056	0.230028	+	0.973184i	$0.426118\pi$
$278$	−9.65685	−0.579180
$279$	0	0
$280$	−8.82843	−0.527599
$281$	8.58579	0.512185	0.256093	−	0.966652i	$-0.417565\pi$
$281$	8.58579	0.512185	0.256093	+	0.966652i	$0.417565\pi$
$282$	0	0
$283$	15.2132	0.904331	0.452166	−	0.891934i	$-0.350652\pi$
$283$	15.2132	0.904331	0.452166	+	0.891934i	$0.350652\pi$
$284$	−52.2843	−3.10250
$285$	0	0
$286$	0	0
$287$	5.65685	0.333914
$288$	0	0
$289$	37.9706	2.23356
$290$	12.0000	0.704664
$291$	0	0
$292$	50.9706	2.98283
$293$	7.41421	0.433143	0.216571	−	0.976267i	$-0.430513\pi$
$293$	7.41421	0.433143	0.216571	+	0.976267i	$0.430513\pi$
$294$	0	0
$295$	4.48528	0.261143
$296$	−21.3137	−1.23883
$297$	0	0
$298$	−40.0416	−2.31955
$299$	0	0
$300$	0	0
$301$	6.00000	0.345834
$302$	−23.3137	−1.34155
$303$	0	0
$304$	−18.7279	−1.07412
$305$	−0.485281	−0.0277871
$306$	0	0
$307$	10.8284	0.618011	0.309005	−	0.951060i	$-0.400004\pi$
$307$	10.8284	0.618011	0.309005	+	0.951060i	$0.400004\pi$
$308$	36.9706	2.10659
$309$	0	0
$310$	12.0000	0.681554
$311$	−9.31371	−0.528132	−0.264066	−	0.964505i	$-0.585064\pi$
$311$	−9.31371	−0.528132	−0.264066	+	0.964505i	$0.585064\pi$
$312$	0	0
$313$	1.31371	0.0742552	0.0371276	−	0.999311i	$-0.488179\pi$
$313$	1.31371	0.0742552	0.0371276	+	0.999311i	$0.488179\pi$
$314$	−3.65685	−0.206368
$315$	0	0
$316$	30.2426	1.70128
$317$	−25.4558	−1.42974	−0.714871	−	0.699256i	$-0.753515\pi$
$317$	−25.4558	−1.42974	−0.714871	+	0.699256i	$0.753515\pi$
$318$	0	0
$319$	−24.0000	−1.34374
$320$	−5.75736	−0.321846
$321$	0	0
$322$	−8.24264	−0.459344
$323$	−46.2843	−2.57533
$324$	0	0
$325$	0	0
$326$	45.4558	2.51757
$327$	0	0
$328$	−7.31371	−0.403832
$329$	1.17157	0.0645909
$330$	0	0
$331$	−6.82843	−0.375324	−0.187662	−	0.982234i	$-0.560091\pi$
$331$	−6.82843	−0.375324	−0.187662	+	0.982234i	$0.560091\pi$
$332$	26.1421	1.43474
$333$	0	0
$334$	−13.6569	−0.747270
$335$	5.02944	0.274788
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	−31.3848	−1.70711
$339$	0	0
$340$	16.6274	0.901748
$341$	−24.0000	−1.29967
$342$	0	0
$343$	8.00000	0.431959
$344$	−7.75736	−0.418249
$345$	0	0
$346$	26.1421	1.40541
$347$	25.3137	1.35891	0.679456	−	0.733717i	$-0.262216\pi$
$347$	25.3137	1.35891	0.679456	+	0.733717i	$0.262216\pi$
$348$	0	0
$349$	−20.9706	−1.12253	−0.561264	−	0.827637i	$-0.689685\pi$
$349$	−20.9706	−1.12253	−0.561264	+	0.827637i	$0.689685\pi$
$350$	38.3848	2.05175
$351$	0	0
$352$	4.48528	0.239066
$353$	5.85786	0.311783	0.155891	−	0.987774i	$-0.450175\pi$
$353$	5.85786	0.311783	0.155891	+	0.987774i	$0.450175\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	50.0416	2.65220
$357$	0	0
$358$	−43.4558	−2.29671
$359$	32.2843	1.70390	0.851949	−	0.523624i	$-0.175420\pi$
$359$	32.2843	1.70390	0.851949	+	0.523624i	$0.175420\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	−26.9706	−1.41754
$363$	0	0
$364$	0	0
$365$	7.79899	0.408218
$366$	0	0
$367$	13.5563	0.707636	0.353818	−	0.935314i	$-0.384883\pi$
$367$	13.5563	0.707636	0.353818	+	0.935314i	$0.384883\pi$
$368$	3.00000	0.156386
$369$	0	0
$370$	−6.82843	−0.354993
$371$	17.3137	0.898883
$372$	0	0
$373$	6.48528	0.335795	0.167898	−	0.985804i	$-0.446302\pi$
$373$	6.48528	0.335795	0.167898	+	0.985804i	$0.446302\pi$
$374$	−50.6274	−2.61788
$375$	0	0
$376$	−1.51472	−0.0781156
$377$	0	0
$378$	0	0
$379$	0.585786	0.0300898	0.0150449	−	0.999887i	$-0.495211\pi$
$379$	0.585786	0.0300898	0.0150449	+	0.999887i	$0.495211\pi$
$380$	−14.0000	−0.718185
$381$	0	0
$382$	−1.65685	−0.0847720
$383$	21.6569	1.10661	0.553307	−	0.832978i	$-0.313366\pi$
$383$	21.6569	1.10661	0.553307	+	0.832978i	$0.313366\pi$
$384$	0	0
$385$	5.65685	0.288300
$386$	−4.00000	−0.203595
$387$	0	0
$388$	−38.2843	−1.94359
$389$	−3.89949	−0.197712	−0.0988561	−	0.995102i	$-0.531518\pi$
$389$	−3.89949	−0.197712	−0.0988561	+	0.995102i	$0.531518\pi$
$390$	0	0
$391$	7.41421	0.374953
$392$	20.5563	1.03825
$393$	0	0
$394$	−22.1421	−1.11550
$395$	4.62742	0.232831
$396$	0	0
$397$	−26.0000	−1.30490	−0.652451	−	0.757831i	$-0.726259\pi$
$397$	−26.0000	−1.30490	−0.652451	+	0.757831i	$0.726259\pi$
$398$	−5.41421	−0.271390
$399$	0	0
$400$	−13.9706	−0.698528
$401$	30.0416	1.50021	0.750104	−	0.661320i	$-0.230004\pi$
$401$	30.0416	1.50021	0.750104	+	0.661320i	$0.230004\pi$
$402$	0	0
$403$	0	0
$404$	−43.3137	−2.15494
$405$	0	0
$406$	−69.9411	−3.47112
$407$	13.6569	0.676945
$408$	0	0
$409$	14.3431	0.709223	0.354611	−	0.935014i	$-0.384613\pi$
$409$	14.3431	0.709223	0.354611	+	0.935014i	$0.384613\pi$
$410$	−2.34315	−0.115720
$411$	0	0
$412$	−13.0711	−0.643965
$413$	−26.1421	−1.28637
$414$	0	0
$415$	4.00000	0.196352
$416$	0	0
$417$	0	0
$418$	42.6274	2.08498
$419$	20.4853	1.00077	0.500386	−	0.865803i	$-0.333192\pi$
$419$	20.4853	1.00077	0.500386	+	0.865803i	$0.333192\pi$
$420$	0	0
$421$	−39.4558	−1.92296	−0.961480	−	0.274875i	$-0.911364\pi$
$421$	−39.4558	−1.92296	−0.961480	+	0.274875i	$0.911364\pi$
$422$	−30.1421	−1.46730
$423$	0	0
$424$	−22.3848	−1.08710
$425$	−34.5269	−1.67480
$426$	0	0
$427$	2.82843	0.136877
$428$	−32.4853	−1.57024
$429$	0	0
$430$	−2.48528	−0.119851
$431$	8.68629	0.418404	0.209202	−	0.977872i	$-0.432913\pi$
$431$	8.68629	0.418404	0.209202	+	0.977872i	$0.432913\pi$
$432$	0	0
$433$	−3.65685	−0.175737	−0.0878686	−	0.996132i	$-0.528006\pi$
$433$	−3.65685	−0.175737	−0.0878686	+	0.996132i	$0.528006\pi$
$434$	−69.9411	−3.35728
$435$	0	0
$436$	−9.51472	−0.455672
$437$	−6.24264	−0.298626
$438$	0	0
$439$	24.0000	1.14546	0.572729	−	0.819745i	$-0.305885\pi$
$439$	24.0000	1.14546	0.572729	+	0.819745i	$0.305885\pi$
$440$	−7.31371	−0.348667
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	7.65685	0.362970
$446$	−31.3137	−1.48275
$447$	0	0
$448$	33.5563	1.58539
$449$	10.6274	0.501539	0.250769	−	0.968047i	$-0.419316\pi$
$449$	10.6274	0.501539	0.250769	+	0.968047i	$0.419316\pi$
$450$	0	0
$451$	4.68629	0.220669
$452$	59.0122	2.77570
$453$	0	0
$454$	−14.8284	−0.695933
$455$	0	0
$456$	0	0
$457$	34.9706	1.63585	0.817927	−	0.575322i	$-0.195123\pi$
$457$	34.9706	1.63585	0.817927	+	0.575322i	$0.195123\pi$
$458$	34.9706	1.63407
$459$	0	0
$460$	2.24264	0.104564
$461$	−9.17157	−0.427163	−0.213581	−	0.976925i	$-0.568513\pi$
$461$	−9.17157	−0.427163	−0.213581	+	0.976925i	$0.568513\pi$
$462$	0	0
$463$	8.97056	0.416897	0.208449	−	0.978033i	$-0.433159\pi$
$463$	8.97056	0.416897	0.208449	+	0.978033i	$0.433159\pi$
$464$	25.4558	1.18176
$465$	0	0
$466$	52.2843	2.42202
$467$	5.85786	0.271070	0.135535	−	0.990773i	$-0.456725\pi$
$467$	5.85786	0.271070	0.135535	+	0.990773i	$0.456725\pi$
$468$	0	0
$469$	−29.3137	−1.35358
$470$	−0.485281	−0.0223844
$471$	0	0
$472$	33.7990	1.55572
$473$	4.97056	0.228547
$474$	0	0
$475$	29.0711	1.33387
$476$	−96.9117	−4.44194
$477$	0	0
$478$	−46.6274	−2.13269
$479$	16.6863	0.762416	0.381208	−	0.924489i	$-0.375508\pi$
$479$	16.6863	0.762416	0.381208	+	0.924489i	$0.375508\pi$
$480$	0	0
$481$	0	0
$482$	−55.4558	−2.52594
$483$	0	0
$484$	−11.4853	−0.522058
$485$	−5.85786	−0.265992
$486$	0	0
$487$	8.97056	0.406495	0.203247	−	0.979127i	$-0.434850\pi$
$487$	8.97056	0.406495	0.203247	+	0.979127i	$0.434850\pi$
$488$	−3.65685	−0.165538
$489$	0	0
$490$	6.58579	0.297516
$491$	6.68629	0.301748	0.150874	−	0.988553i	$-0.451791\pi$
$491$	6.68629	0.301748	0.150874	+	0.988553i	$0.451791\pi$
$492$	0	0
$493$	62.9117	2.83340
$494$	0	0
$495$	0	0
$496$	25.4558	1.14300
$497$	46.6274	2.09153
$498$	0	0
$499$	−24.4853	−1.09611	−0.548056	−	0.836442i	$-0.684632\pi$
$499$	−24.4853	−1.09611	−0.548056	+	0.836442i	$0.684632\pi$
$500$	−21.6569	−0.968524
$501$	0	0
$502$	−49.4558	−2.20732
$503$	17.6569	0.787280	0.393640	−	0.919265i	$-0.371216\pi$
$503$	17.6569	0.787280	0.393640	+	0.919265i	$0.371216\pi$
$504$	0	0
$505$	−6.62742	−0.294916
$506$	−6.82843	−0.303561
$507$	0	0
$508$	−17.1716	−0.761865
$509$	−38.1421	−1.69062	−0.845310	−	0.534276i	$-0.820584\pi$
$509$	−38.1421	−1.69062	−0.845310	+	0.534276i	$0.820584\pi$
$510$	0	0
$511$	−45.4558	−2.01085
$512$	−31.2426	−1.38074
$513$	0	0
$514$	15.3137	0.675459
$515$	−2.00000	−0.0881305
$516$	0	0
$517$	0.970563	0.0426853
$518$	39.7990	1.74867
$519$	0	0
$520$	0	0
$521$	18.7279	0.820485	0.410243	−	0.911976i	$-0.365444\pi$
$521$	18.7279	0.820485	0.410243	+	0.911976i	$0.365444\pi$
$522$	0	0
$523$	−35.6985	−1.56099	−0.780493	−	0.625165i	$-0.785032\pi$
$523$	−35.6985	−1.56099	−0.780493	+	0.625165i	$0.785032\pi$
$524$	64.9706	2.83825
$525$	0	0
$526$	56.2843	2.45411
$527$	62.9117	2.74048
$528$	0	0
$529$	1.00000	0.0434783
$530$	−7.17157	−0.311513
$531$	0	0
$532$	81.5980	3.53772
$533$	0	0
$534$	0	0
$535$	−4.97056	−0.214896
$536$	37.8995	1.63701
$537$	0	0
$538$	43.7990	1.88831
$539$	−13.1716	−0.567340
$540$	0	0
$541$	4.97056	0.213701	0.106851	−	0.994275i	$-0.465923\pi$
$541$	4.97056	0.213701	0.106851	+	0.994275i	$0.465923\pi$
$542$	−30.1421	−1.29472
$543$	0	0
$544$	−11.7574	−0.504093
$545$	−1.45584	−0.0623615
$546$	0	0
$547$	12.4853	0.533832	0.266916	−	0.963720i	$-0.413995\pi$
$547$	12.4853	0.533832	0.266916	+	0.963720i	$0.413995\pi$
$548$	2.24264	0.0958009
$549$	0	0
$550$	31.7990	1.35591
$551$	−52.9706	−2.25662
$552$	0	0
$553$	−26.9706	−1.14690
$554$	18.4853	0.785364
$555$	0	0
$556$	−15.3137	−0.649446
$557$	−22.2426	−0.942451	−0.471225	−	0.882013i	$-0.656188\pi$
$557$	−22.2426	−0.942451	−0.471225	+	0.882013i	$0.656188\pi$
$558$	0	0
$559$	0	0
$560$	−6.00000	−0.253546
$561$	0	0
$562$	20.7279	0.874355
$563$	21.1716	0.892275	0.446138	−	0.894964i	$-0.352799\pi$
$563$	21.1716	0.892275	0.446138	+	0.894964i	$0.352799\pi$
$564$	0	0
$565$	9.02944	0.379871
$566$	36.7279	1.54379
$567$	0	0
$568$	−60.2843	−2.52947
$569$	8.58579	0.359935	0.179967	−	0.983673i	$-0.442401\pi$
$569$	8.58579	0.359935	0.179967	+	0.983673i	$0.442401\pi$
$570$	0	0
$571$	1.75736	0.0735432	0.0367716	−	0.999324i	$-0.488293\pi$
$571$	1.75736	0.0735432	0.0367716	+	0.999324i	$0.488293\pi$
$572$	0	0
$573$	0	0
$574$	13.6569	0.570026
$575$	−4.65685	−0.194204
$576$	0	0
$577$	−32.2843	−1.34401	−0.672006	−	0.740546i	$-0.734567\pi$
$577$	−32.2843	−1.34401	−0.672006	+	0.740546i	$0.734567\pi$
$578$	91.6690	3.81293
$579$	0	0
$580$	19.0294	0.790154
$581$	−23.3137	−0.967216
$582$	0	0
$583$	14.3431	0.594032
$584$	58.7696	2.43190
$585$	0	0
$586$	17.8995	0.739421
$587$	7.02944	0.290136	0.145068	−	0.989422i	$-0.453660\pi$
$587$	7.02944	0.290136	0.145068	+	0.989422i	$0.453660\pi$
$588$	0	0
$589$	−52.9706	−2.18261
$590$	10.8284	0.445799
$591$	0	0
$592$	−14.4853	−0.595341
$593$	0.686292	0.0281826	0.0140913	−	0.999901i	$-0.495514\pi$
$593$	0.686292	0.0281826	0.0140913	+	0.999901i	$0.495514\pi$
$594$	0	0
$595$	−14.8284	−0.607906
$596$	−63.4975	−2.60096
$597$	0	0
$598$	0	0
$599$	4.68629	0.191477	0.0957383	−	0.995407i	$-0.469479\pi$
$599$	4.68629	0.191477	0.0957383	+	0.995407i	$0.469479\pi$
$600$	0	0
$601$	1.65685	0.0675845	0.0337922	−	0.999429i	$-0.489242\pi$
$601$	1.65685	0.0675845	0.0337922	+	0.999429i	$0.489242\pi$
$602$	14.4853	0.590376
$603$	0	0
$604$	−36.9706	−1.50431
$605$	−1.75736	−0.0714468
$606$	0	0
$607$	−14.3431	−0.582170	−0.291085	−	0.956697i	$-0.594016\pi$
$607$	−14.3431	−0.582170	−0.291085	+	0.956697i	$0.594016\pi$
$608$	9.89949	0.401478
$609$	0	0
$610$	−1.17157	−0.0474356
$611$	0	0
$612$	0	0
$613$	39.4558	1.59361	0.796803	−	0.604239i	$-0.206523\pi$
$613$	39.4558	1.59361	0.796803	+	0.604239i	$0.206523\pi$
$614$	26.1421	1.05501
$615$	0	0
$616$	42.6274	1.71751
$617$	−33.5563	−1.35093	−0.675464	−	0.737393i	$-0.736057\pi$
$617$	−33.5563	−1.35093	−0.675464	+	0.737393i	$0.736057\pi$
$618$	0	0
$619$	−22.2426	−0.894007	−0.447004	−	0.894532i	$-0.647509\pi$
$619$	−22.2426	−0.894007	−0.447004	+	0.894532i	$0.647509\pi$
$620$	19.0294	0.764241
$621$	0	0
$622$	−22.4853	−0.901578
$623$	−44.6274	−1.78796
$624$	0	0
$625$	19.9706	0.798823
$626$	3.17157	0.126762
$627$	0	0
$628$	−5.79899	−0.231405
$629$	−35.7990	−1.42740
$630$	0	0
$631$	−24.8701	−0.990061	−0.495031	−	0.868875i	$-0.664843\pi$
$631$	−24.8701	−0.990061	−0.495031	+	0.868875i	$0.664843\pi$
$632$	34.8701	1.38706
$633$	0	0
$634$	−61.4558	−2.44072
$635$	−2.62742	−0.104266
$636$	0	0
$637$	0	0
$638$	−57.9411	−2.29391
$639$	0	0
$640$	−12.0416	−0.475987
$641$	−2.92893	−0.115686	−0.0578429	−	0.998326i	$-0.518422\pi$
$641$	−2.92893	−0.115686	−0.0578429	+	0.998326i	$0.518422\pi$
$642$	0	0
$643$	−8.38478	−0.330663	−0.165332	−	0.986238i	$-0.552869\pi$
$643$	−8.38478	−0.330663	−0.165332	+	0.986238i	$0.552869\pi$
$644$	−13.0711	−0.515072
$645$	0	0
$646$	−111.740	−4.39636
$647$	6.00000	0.235884	0.117942	−	0.993020i	$-0.462370\pi$
$647$	6.00000	0.235884	0.117942	+	0.993020i	$0.462370\pi$
$648$	0	0
$649$	−21.6569	−0.850106
$650$	0	0
$651$	0	0
$652$	72.0833	2.82300
$653$	−21.6569	−0.847498	−0.423749	−	0.905780i	$-0.639286\pi$
$653$	−21.6569	−0.847498	−0.423749	+	0.905780i	$0.639286\pi$
$654$	0	0
$655$	9.94113	0.388432
$656$	−4.97056	−0.194068
$657$	0	0
$658$	2.82843	0.110264
$659$	34.1421	1.32999	0.664994	−	0.746848i	$-0.268434\pi$
$659$	34.1421	1.32999	0.664994	+	0.746848i	$0.268434\pi$
$660$	0	0
$661$	−11.8579	−0.461217	−0.230609	−	0.973047i	$-0.574072\pi$
$661$	−11.8579	−0.461217	−0.230609	+	0.973047i	$0.574072\pi$
$662$	−16.4853	−0.640719
$663$	0	0
$664$	30.1421	1.16974
$665$	12.4853	0.484158
$666$	0	0
$667$	8.48528	0.328551
$668$	−21.6569	−0.837929
$669$	0	0
$670$	12.1421	0.469092
$671$	2.34315	0.0904561
$672$	0	0
$673$	−40.9706	−1.57930	−0.789650	−	0.613558i	$-0.789738\pi$
$673$	−40.9706	−1.57930	−0.789650	+	0.613558i	$0.789738\pi$
$674$	53.1127	2.04582
$675$	0	0
$676$	−49.7696	−1.91421
$677$	35.6985	1.37200	0.686002	−	0.727600i	$-0.259364\pi$
$677$	35.6985	1.37200	0.686002	+	0.727600i	$0.259364\pi$
$678$	0	0
$679$	34.1421	1.31025
$680$	19.1716	0.735196
$681$	0	0
$682$	−57.9411	−2.21868
$683$	−10.3431	−0.395769	−0.197885	−	0.980225i	$-0.563407\pi$
$683$	−10.3431	−0.395769	−0.197885	+	0.980225i	$0.563407\pi$
$684$	0	0
$685$	0.343146	0.0131109
$686$	19.3137	0.737401
$687$	0	0
$688$	−5.27208	−0.200996
$689$	0	0
$690$	0	0
$691$	−22.8284	−0.868434	−0.434217	−	0.900808i	$-0.642975\pi$
$691$	−22.8284	−0.868434	−0.434217	+	0.900808i	$0.642975\pi$
$692$	41.4558	1.57591
$693$	0	0
$694$	61.1127	2.31981
$695$	−2.34315	−0.0888806
$696$	0	0
$697$	−12.2843	−0.465300
$698$	−50.6274	−1.91628
$699$	0	0
$700$	60.8701	2.30067
$701$	3.89949	0.147282	0.0736409	−	0.997285i	$-0.476538\pi$
$701$	3.89949	0.147282	0.0736409	+	0.997285i	$0.476538\pi$
$702$	0	0
$703$	30.1421	1.13683
$704$	27.7990	1.04771
$705$	0	0
$706$	14.1421	0.532246
$707$	38.6274	1.45273
$708$	0	0
$709$	−29.7990	−1.11912	−0.559562	−	0.828788i	$-0.689031\pi$
$709$	−29.7990	−1.11912	−0.559562	+	0.828788i	$0.689031\pi$
$710$	−19.3137	−0.724831
$711$	0	0
$712$	57.6985	2.16234
$713$	8.48528	0.317776
$714$	0	0
$715$	0	0
$716$	−68.9117	−2.57535
$717$	0	0
$718$	77.9411	2.90874
$719$	41.3137	1.54074	0.770371	−	0.637596i	$-0.220071\pi$
$719$	41.3137	1.54074	0.770371	+	0.637596i	$0.220071\pi$
$720$	0	0
$721$	11.6569	0.434124
$722$	48.2132	1.79431
$723$	0	0
$724$	−42.7696	−1.58952
$725$	−39.5147	−1.46754
$726$	0	0
$727$	37.7574	1.40034	0.700171	−	0.713975i	$-0.253107\pi$
$727$	37.7574	1.40034	0.700171	+	0.713975i	$0.253107\pi$
$728$	0	0
$729$	0	0
$730$	18.8284	0.696871
$731$	−13.0294	−0.481911
$732$	0	0
$733$	−9.79899	−0.361934	−0.180967	−	0.983489i	$-0.557923\pi$
$733$	−9.79899	−0.361934	−0.180967	+	0.983489i	$0.557923\pi$
$734$	32.7279	1.20801
$735$	0	0
$736$	−1.58579	−0.0584529
$737$	−24.2843	−0.894523
$738$	0	0
$739$	43.3137	1.59332	0.796660	−	0.604427i	$-0.206598\pi$
$739$	43.3137	1.59332	0.796660	+	0.604427i	$0.206598\pi$
$740$	−10.8284	−0.398061
$741$	0	0
$742$	41.7990	1.53449
$743$	−7.02944	−0.257885	−0.128943	−	0.991652i	$-0.541158\pi$
$743$	−7.02944	−0.257885	−0.128943	+	0.991652i	$0.541158\pi$
$744$	0	0
$745$	−9.71573	−0.355957
$746$	15.6569	0.573238
$747$	0	0
$748$	−80.2843	−2.93548
$749$	28.9706	1.05856
$750$	0	0
$751$	−21.3553	−0.779267	−0.389634	−	0.920970i	$-0.627398\pi$
$751$	−21.3553	−0.779267	−0.389634	+	0.920970i	$0.627398\pi$
$752$	−1.02944	−0.0375397
$753$	0	0
$754$	0	0
$755$	−5.65685	−0.205874
$756$	0	0
$757$	24.1421	0.877461	0.438730	−	0.898619i	$-0.355428\pi$
$757$	24.1421	0.877461	0.438730	+	0.898619i	$0.355428\pi$
$758$	1.41421	0.0513665
$759$	0	0
$760$	−16.1421	−0.585537
$761$	−31.7990	−1.15271	−0.576356	−	0.817199i	$-0.695526\pi$
$761$	−31.7990	−1.15271	−0.576356	+	0.817199i	$0.695526\pi$
$762$	0	0
$763$	8.48528	0.307188
$764$	−2.62742	−0.0950566
$765$	0	0
$766$	52.2843	1.88911
$767$	0	0
$768$	0	0
$769$	−34.9706	−1.26107	−0.630535	−	0.776161i	$-0.717165\pi$
$769$	−34.9706	−1.26107	−0.630535	+	0.776161i	$0.717165\pi$
$770$	13.6569	0.492159
$771$	0	0
$772$	−6.34315	−0.228295
$773$	8.38478	0.301579	0.150790	−	0.988566i	$-0.451818\pi$
$773$	8.38478	0.301579	0.150790	+	0.988566i	$0.451818\pi$
$774$	0	0
$775$	−39.5147	−1.41941
$776$	−44.1421	−1.58461
$777$	0	0
$778$	−9.41421	−0.337516
$779$	10.3431	0.370582
$780$	0	0
$781$	38.6274	1.38220
$782$	17.8995	0.640085
$783$	0	0
$784$	13.9706	0.498949
$785$	−0.887302	−0.0316692
$786$	0	0
$787$	−27.6985	−0.987344	−0.493672	−	0.869648i	$-0.664346\pi$
$787$	−27.6985	−0.987344	−0.493672	+	0.869648i	$0.664346\pi$
$788$	−35.1127	−1.25084
$789$	0	0
$790$	11.1716	0.397467
$791$	−52.6274	−1.87122
$792$	0	0
$793$	0	0
$794$	−62.7696	−2.22761
$795$	0	0
$796$	−8.58579	−0.304315
$797$	−37.0711	−1.31312	−0.656562	−	0.754272i	$-0.727990\pi$
$797$	−37.0711	−1.31312	−0.656562	+	0.754272i	$0.727990\pi$
$798$	0	0
$799$	−2.54416	−0.0900058
$800$	7.38478	0.261091
$801$	0	0
$802$	72.5269	2.56101
$803$	−37.6569	−1.32888
$804$	0	0
$805$	−2.00000	−0.0704907
$806$	0	0
$807$	0	0
$808$	−49.9411	−1.75692
$809$	−19.1127	−0.671967	−0.335983	−	0.941868i	$-0.609069\pi$
$809$	−19.1127	−0.671967	−0.335983	+	0.941868i	$0.609069\pi$
$810$	0	0
$811$	56.7696	1.99345	0.996724	−	0.0808743i	$-0.0257712\pi$
$811$	56.7696	1.99345	0.996724	+	0.0808743i	$0.0257712\pi$
$812$	−110.912	−3.89224
$813$	0	0
$814$	32.9706	1.15562
$815$	11.0294	0.386344
$816$	0	0
$817$	10.9706	0.383811
$818$	34.6274	1.21072
$819$	0	0
$820$	−3.71573	−0.129759
$821$	−11.3137	−0.394851	−0.197426	−	0.980318i	$-0.563258\pi$
$821$	−11.3137	−0.394851	−0.197426	+	0.980318i	$0.563258\pi$
$822$	0	0
$823$	15.5147	0.540809	0.270405	−	0.962747i	$-0.412843\pi$
$823$	15.5147	0.540809	0.270405	+	0.962747i	$0.412843\pi$
$824$	−15.0711	−0.525026
$825$	0	0
$826$	−63.1127	−2.19597
$827$	7.79899	0.271197	0.135599	−	0.990764i	$-0.456704\pi$
$827$	7.79899	0.271197	0.135599	+	0.990764i	$0.456704\pi$
$828$	0	0
$829$	52.9706	1.83974	0.919872	−	0.392219i	$-0.128292\pi$
$829$	52.9706	1.83974	0.919872	+	0.392219i	$0.128292\pi$
$830$	9.65685	0.335194
$831$	0	0
$832$	0	0
$833$	34.5269	1.19629
$834$	0	0
$835$	−3.31371	−0.114676
$836$	67.5980	2.33793
$837$	0	0
$838$	49.4558	1.70842
$839$	−52.9706	−1.82875	−0.914373	−	0.404872i	$-0.867316\pi$
$839$	−52.9706	−1.82875	−0.914373	+	0.404872i	$0.867316\pi$
$840$	0	0
$841$	43.0000	1.48276
$842$	−95.2548	−3.28270
$843$	0	0
$844$	−47.7990	−1.64531
$845$	−7.61522	−0.261972
$846$	0	0
$847$	10.2426	0.351941
$848$	−15.2132	−0.522424
$849$	0	0
$850$	−83.3553	−2.85906
$851$	−4.82843	−0.165516
$852$	0	0
$853$	51.9411	1.77843	0.889215	−	0.457489i	$-0.151251\pi$
$853$	51.9411	1.77843	0.889215	+	0.457489i	$0.151251\pi$
$854$	6.82843	0.233664
$855$	0	0
$856$	−37.4558	−1.28021
$857$	29.6569	1.01306	0.506529	−	0.862223i	$-0.330928\pi$
$857$	29.6569	1.01306	0.506529	+	0.862223i	$0.330928\pi$
$858$	0	0
$859$	15.0294	0.512798	0.256399	−	0.966571i	$-0.417464\pi$
$859$	15.0294	0.512798	0.256399	+	0.966571i	$0.417464\pi$
$860$	−3.94113	−0.134391
$861$	0	0
$862$	20.9706	0.714260
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	6.34315	0.215673
$866$	−8.82843	−0.300002
$867$	0	0
$868$	−110.912	−3.76459
$869$	−22.3431	−0.757939
$870$	0	0
$871$	0	0
$872$	−10.9706	−0.371510
$873$	0	0
$874$	−15.0711	−0.509786
$875$	19.3137	0.652923
$876$	0	0
$877$	−43.6569	−1.47419	−0.737094	−	0.675791i	$-0.763802\pi$
$877$	−43.6569	−1.47419	−0.737094	+	0.675791i	$0.763802\pi$
$878$	57.9411	1.95542
$879$	0	0
$880$	−4.97056	−0.167558
$881$	−13.0711	−0.440375	−0.220188	−	0.975458i	$-0.570667\pi$
$881$	−13.0711	−0.440375	−0.220188	+	0.975458i	$0.570667\pi$
$882$	0	0
$883$	−11.5147	−0.387501	−0.193751	−	0.981051i	$-0.562065\pi$
$883$	−11.5147	−0.387501	−0.193751	+	0.981051i	$0.562065\pi$
$884$	0	0
$885$	0	0
$886$	−57.9411	−1.94657
$887$	0.343146	0.0115217	0.00576085	−	0.999983i	$-0.498166\pi$
$887$	0.343146	0.0115217	0.00576085	+	0.999983i	$0.498166\pi$
$888$	0	0
$889$	15.3137	0.513605
$890$	18.4853	0.619628
$891$	0	0
$892$	−49.6569	−1.66263
$893$	2.14214	0.0716838
$894$	0	0
$895$	−10.5442	−0.352452
$896$	70.1838	2.34468
$897$	0	0
$898$	25.6569	0.856180
$899$	72.0000	2.40133
$900$	0	0
$901$	−37.5980	−1.25257
$902$	11.3137	0.376705
$903$	0	0
$904$	68.0416	2.26303
$905$	−6.54416	−0.217535
$906$	0	0
$907$	−35.6985	−1.18535	−0.592674	−	0.805442i	$-0.701928\pi$
$907$	−35.6985	−1.18535	−0.592674	+	0.805442i	$0.701928\pi$
$908$	−23.5147	−0.780363
$909$	0	0
$910$	0	0
$911$	−13.3726	−0.443053	−0.221527	−	0.975154i	$-0.571104\pi$
$911$	−13.3726	−0.443053	−0.221527	+	0.975154i	$0.571104\pi$
$912$	0	0
$913$	−19.3137	−0.639190
$914$	84.4264	2.79258
$915$	0	0
$916$	55.4558	1.83231
$917$	−57.9411	−1.91338
$918$	0	0
$919$	−3.21320	−0.105994	−0.0529969	−	0.998595i	$-0.516877\pi$
$919$	−3.21320	−0.105994	−0.0529969	+	0.998595i	$0.516877\pi$
$920$	2.58579	0.0852509
$921$	0	0
$922$	−22.1421	−0.729212
$923$	0	0
$924$	0	0
$925$	22.4853	0.739311
$926$	21.6569	0.711688
$927$	0	0
$928$	−13.4558	−0.441710
$929$	23.3137	0.764898	0.382449	−	0.923977i	$-0.375081\pi$
$929$	23.3137	0.764898	0.382449	+	0.923977i	$0.375081\pi$
$930$	0	0
$931$	−29.0711	−0.952766
$932$	82.9117	2.71586
$933$	0	0
$934$	14.1421	0.462745
$935$	−12.2843	−0.401739
$936$	0	0
$937$	31.6569	1.03418	0.517092	−	0.855930i	$-0.327014\pi$
$937$	31.6569	1.03418	0.517092	+	0.855930i	$0.327014\pi$
$938$	−70.7696	−2.31071
$939$	0	0
$940$	−0.769553	−0.0251000
$941$	−24.3848	−0.794921	−0.397460	−	0.917619i	$-0.630108\pi$
$941$	−24.3848	−0.794921	−0.397460	+	0.917619i	$0.630108\pi$
$942$	0	0
$943$	−1.65685	−0.0539546
$944$	22.9706	0.747628
$945$	0	0
$946$	12.0000	0.390154
$947$	2.68629	0.0872927	0.0436464	−	0.999047i	$-0.486103\pi$
$947$	2.68629	0.0872927	0.0436464	+	0.999047i	$0.486103\pi$
$948$	0	0
$949$	0	0
$950$	70.1838	2.27706
$951$	0	0
$952$	−111.740	−3.62152
$953$	−31.0122	−1.00458	−0.502292	−	0.864698i	$-0.667510\pi$
$953$	−31.0122	−1.00458	−0.502292	+	0.864698i	$0.667510\pi$
$954$	0	0
$955$	−0.402020	−0.0130091
$956$	−73.9411	−2.39143
$957$	0	0
$958$	40.2843	1.30153
$959$	−2.00000	−0.0645834
$960$	0	0
$961$	41.0000	1.32258
$962$	0	0
$963$	0	0
$964$	−87.9411	−2.83239
$965$	−0.970563	−0.0312435
$966$	0	0
$967$	−15.5147	−0.498920	−0.249460	−	0.968385i	$-0.580253\pi$
$967$	−15.5147	−0.498920	−0.249460	+	0.968385i	$0.580253\pi$
$968$	−13.2426	−0.425635
$969$	0	0
$970$	−14.1421	−0.454077
$971$	7.11270	0.228257	0.114129	−	0.993466i	$-0.463592\pi$
$971$	7.11270	0.228257	0.114129	+	0.993466i	$0.463592\pi$
$972$	0	0
$973$	13.6569	0.437819
$974$	21.6569	0.693930
$975$	0	0
$976$	−2.48528	−0.0795519
$977$	30.0416	0.961117	0.480558	−	0.876963i	$-0.340434\pi$
$977$	30.0416	0.961117	0.480558	+	0.876963i	$0.340434\pi$
$978$	0	0
$979$	−36.9706	−1.18158
$980$	10.4437	0.333610
$981$	0	0
$982$	16.1421	0.515116
$983$	23.3137	0.743592	0.371796	−	0.928314i	$-0.378742\pi$
$983$	23.3137	0.743592	0.371796	+	0.928314i	$0.378742\pi$
$984$	0	0
$985$	−5.37258	−0.171185
$986$	151.882	4.83692
$987$	0	0
$988$	0	0
$989$	−1.75736	−0.0558808
$990$	0	0
$991$	25.4558	0.808632	0.404316	−	0.914619i	$-0.367510\pi$
$991$	25.4558	0.808632	0.404316	+	0.914619i	$0.367510\pi$
$992$	−13.4558	−0.427223
$993$	0	0
$994$	112.569	3.57046
$995$	−1.31371	−0.0416474
$996$	0	0
$997$	3.02944	0.0959432	0.0479716	−	0.998849i	$-0.484724\pi$
$997$	3.02944	0.0959432	0.0479716	+	0.998849i	$0.484724\pi$
$998$	−59.1127	−1.87118
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	207.2.a.e.1.2	yes	2
3.2	odd	2		207.2.a.b.1.1	✓	2
4.3	odd	2		3312.2.a.be.1.1		2
5.4	even	2		5175.2.a.bc.1.1		2
12.11	even	2		3312.2.a.u.1.2		2
15.14	odd	2		5175.2.a.bo.1.2		2
23.22	odd	2		4761.2.a.z.1.2		2
69.68	even	2		4761.2.a.k.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
207.2.a.b.1.1	✓	2	3.2	odd	2
207.2.a.e.1.2	yes	2	1.1	even	1	trivial
3312.2.a.u.1.2		2	12.11	even	2
3312.2.a.be.1.1		2	4.3	odd	2
4761.2.a.k.1.1		2	69.68	even	2
4761.2.a.z.1.2		2	23.22	odd	2
5175.2.a.bc.1.1		2	5.4	even	2
5175.2.a.bo.1.2		2	15.14	odd	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:	\( N \)	\(=\)	\( 207 = 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	207.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} -3.41421 q^{7} +4.41421 q^{8} +O(q^{10})\)	\(q+2.41421 q^{2} +3.82843 q^{4} +0.585786 q^{5} -3.41421 q^{7} +4.41421 q^{8} +1.41421 q^{10} -2.82843 q^{11} -8.24264 q^{14} +3.00000 q^{16} +7.41421 q^{17} -6.24264 q^{19} +2.24264 q^{20} -6.82843 q^{22} +1.00000 q^{23} -4.65685 q^{25} -13.0711 q^{28} +8.48528 q^{29} +8.48528 q^{31} -1.58579 q^{32} +17.8995 q^{34} -2.00000 q^{35} -4.82843 q^{37} -15.0711 q^{38} +2.58579 q^{40} -1.65685 q^{41} -1.75736 q^{43} -10.8284 q^{44} +2.41421 q^{46} -0.343146 q^{47} +4.65685 q^{49} -11.2426 q^{50} -5.07107 q^{53} -1.65685 q^{55} -15.0711 q^{56} +20.4853 q^{58} +7.65685 q^{59} -0.828427 q^{61} +20.4853 q^{62} -9.82843 q^{64} +8.58579 q^{67} +28.3848 q^{68} -4.82843 q^{70} -13.6569 q^{71} +13.3137 q^{73} -11.6569 q^{74} -23.8995 q^{76} +9.65685 q^{77} +7.89949 q^{79} +1.75736 q^{80} -4.00000 q^{82} +6.82843 q^{83} +4.34315 q^{85} -4.24264 q^{86} -12.4853 q^{88} +13.0711 q^{89} +3.82843 q^{92} -0.828427 q^{94} -3.65685 q^{95} -10.0000 q^{97} +11.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} - 4 q^{7} + 6 q^{8} - 8 q^{14} + 6 q^{16} + 12 q^{17} - 4 q^{19} - 4 q^{20} - 8 q^{22} + 2 q^{23} + 2 q^{25} - 12 q^{28} - 6 q^{32} + 16 q^{34} - 4 q^{35} - 4 q^{37} - 16 q^{38} + 8 q^{40} + 8 q^{41} - 12 q^{43} - 16 q^{44} + 2 q^{46} - 12 q^{47} - 2 q^{49} - 14 q^{50} + 4 q^{53} + 8 q^{55} - 16 q^{56} + 24 q^{58} + 4 q^{59} + 4 q^{61} + 24 q^{62} - 14 q^{64} + 20 q^{67} + 20 q^{68} - 4 q^{70} - 16 q^{71} + 4 q^{73} - 12 q^{74} - 28 q^{76} + 8 q^{77} - 4 q^{79} + 12 q^{80} - 8 q^{82} + 8 q^{83} + 20 q^{85} - 8 q^{88} + 12 q^{89} + 2 q^{92} + 4 q^{94} + 4 q^{95} - 20 q^{97} + 14 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 - 4 * q^7 + 6 * q^8 - 8 * q^14 + 6 * q^16 + 12 * q^17 - 4 * q^19 - 4 * q^20 - 8 * q^22 + 2 * q^23 + 2 * q^25 - 12 * q^28 - 6 * q^32 + 16 * q^34 - 4 * q^35 - 4 * q^37 - 16 * q^38 + 8 * q^40 + 8 * q^41 - 12 * q^43 - 16 * q^44 + 2 * q^46 - 12 * q^47 - 2 * q^49 - 14 * q^50 + 4 * q^53 + 8 * q^55 - 16 * q^56 + 24 * q^58 + 4 * q^59 + 4 * q^61 + 24 * q^62 - 14 * q^64 + 20 * q^67 + 20 * q^68 - 4 * q^70 - 16 * q^71 + 4 * q^73 - 12 * q^74 - 28 * q^76 + 8 * q^77 - 4 * q^79 + 12 * q^80 - 8 * q^82 + 8 * q^83 + 20 * q^85 - 8 * q^88 + 12 * q^89 + 2 * q^92 + 4 * q^94 + 4 * q^95 - 20 * q^97 + 14 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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