LMFDB - Embedded newform 2070.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2070,2,Mod(1,2070)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2070, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2070.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2070.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:	$16.5290332184$
Analytic rank:	$1$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2070.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.828427 q^{7} -1.00000 q^{8} +O(q^{10})$	$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.828427 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} +2.82843 q^{13} +0.828427 q^{14} +1.00000 q^{16} +0.828427 q^{17} -1.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} -2.82843 q^{26} -0.828427 q^{28} -4.82843 q^{29} +1.65685 q^{31} -1.00000 q^{32} -0.828427 q^{34} +0.828427 q^{35} -1.17157 q^{37} +1.00000 q^{40} +5.65685 q^{41} +8.48528 q^{43} -2.00000 q^{44} +1.00000 q^{46} -6.31371 q^{49} -1.00000 q^{50} +2.82843 q^{52} -13.3137 q^{53} +2.00000 q^{55} +0.828427 q^{56} +4.82843 q^{58} -10.0000 q^{59} -8.82843 q^{61} -1.65685 q^{62} +1.00000 q^{64} -2.82843 q^{65} +6.82843 q^{67} +0.828427 q^{68} -0.828427 q^{70} -2.82843 q^{71} -13.3137 q^{73} +1.17157 q^{74} +1.65685 q^{77} -2.82843 q^{79} -1.00000 q^{80} -5.65685 q^{82} +1.17157 q^{83} -0.828427 q^{85} -8.48528 q^{86} +2.00000 q^{88} -5.65685 q^{89} -2.34315 q^{91} -1.00000 q^{92} +3.17157 q^{97} +6.31371 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	$ 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{20} + 4 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{28} - 4 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{35} - 8 q^{37} + 2 q^{40} - 4 q^{44} + 2 q^{46} + 10 q^{49} - 2 q^{50} - 4 q^{53} + 4 q^{55} - 4 q^{56} + 4 q^{58} - 20 q^{59} - 12 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} - 4 q^{68} + 4 q^{70} - 4 q^{73} + 8 q^{74} - 8 q^{77} - 2 q^{80} + 8 q^{83} + 4 q^{85} + 4 q^{88} - 16 q^{91} - 2 q^{92} + 12 q^{97} - 10 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^20 + 4 * q^22 - 2 * q^23 + 2 * q^25 + 4 * q^28 - 4 * q^29 - 8 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^35 - 8 * q^37 + 2 * q^40 - 4 * q^44 + 2 * q^46 + 10 * q^49 - 2 * q^50 - 4 * q^53 + 4 * q^55 - 4 * q^56 + 4 * q^58 - 20 * q^59 - 12 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 - 4 * q^68 + 4 * q^70 - 4 * q^73 + 8 * q^74 - 8 * q^77 - 2 * q^80 + 8 * q^83 + 4 * q^85 + 4 * q^88 - 16 * q^91 - 2 * q^92 + 12 * q^97 - 10 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−0.828427	−0.313116	−0.156558	−	0.987669i	$-0.550040\pi$
$7$	−0.828427	−0.313116	−0.156558	+	0.987669i	$0.550040\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	0.828427	0.221406
$15$	0	0
$16$	1.00000	0.250000
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	2.00000	0.426401
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	−2.82843	−0.554700
$27$	0	0
$28$	−0.828427	−0.156558
$29$	−4.82843	−0.896616	−0.448308	−	0.893879i	$-0.647973\pi$
$29$	−4.82843	−0.896616	−0.448308	+	0.893879i	$0.647973\pi$
$30$	0	0
$31$	1.65685	0.297580	0.148790	−	0.988869i	$-0.452462\pi$
$31$	1.65685	0.297580	0.148790	+	0.988869i	$0.452462\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−0.828427	−0.142074
$35$	0.828427	0.140030
$36$	0	0
$37$	−1.17157	−0.192605	−0.0963027	−	0.995352i	$-0.530702\pi$
$37$	−1.17157	−0.192605	−0.0963027	+	0.995352i	$0.530702\pi$
$38$	0	0
$39$	0	0
$40$	1.00000	0.158114
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	8.48528	1.29399	0.646997	−	0.762493i	$-0.276025\pi$
$43$	8.48528	1.29399	0.646997	+	0.762493i	$0.276025\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	1.00000	0.147442
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	−1.00000	−0.141421
$51$	0	0
$52$	2.82843	0.392232
$53$	−13.3137	−1.82878	−0.914389	−	0.404836i	$-0.867329\pi$
$53$	−13.3137	−1.82878	−0.914389	+	0.404836i	$0.867329\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0.828427	0.110703
$57$	0	0
$58$	4.82843	0.634004
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−8.82843	−1.13036	−0.565182	−	0.824966i	$-0.691194\pi$
$61$	−8.82843	−1.13036	−0.565182	+	0.824966i	$0.691194\pi$
$62$	−1.65685	−0.210421
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.82843	−0.350823
$66$	0	0
$67$	6.82843	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$67$	6.82843	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$68$	0.828427	0.100462
$69$	0	0
$70$	−0.828427	−0.0990160
$71$	−2.82843	−0.335673	−0.167836	−	0.985815i	$-0.553678\pi$
$71$	−2.82843	−0.335673	−0.167836	+	0.985815i	$0.553678\pi$
$72$	0	0
$73$	−13.3137	−1.55825	−0.779126	−	0.626868i	$-0.784337\pi$
$73$	−13.3137	−1.55825	−0.779126	+	0.626868i	$0.784337\pi$
$74$	1.17157	0.136193
$75$	0	0
$76$	0	0
$77$	1.65685	0.188816
$78$	0	0
$79$	−2.82843	−0.318223	−0.159111	−	0.987261i	$-0.550863\pi$
$79$	−2.82843	−0.318223	−0.159111	+	0.987261i	$0.550863\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−5.65685	−0.624695
$83$	1.17157	0.128597	0.0642984	−	0.997931i	$-0.479519\pi$
$83$	1.17157	0.128597	0.0642984	+	0.997931i	$0.479519\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	−8.48528	−0.914991
$87$	0	0
$88$	2.00000	0.213201
$89$	−5.65685	−0.599625	−0.299813	−	0.953998i	$-0.596924\pi$
$89$	−5.65685	−0.599625	−0.299813	+	0.953998i	$0.596924\pi$
$90$	0	0
$91$	−2.34315	−0.245628
$92$	−1.00000	−0.104257
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.17157	0.322024	0.161012	−	0.986952i	$-0.448524\pi$
$97$	3.17157	0.322024	0.161012	+	0.986952i	$0.448524\pi$
$98$	6.31371	0.637781
$99$	0	0
$100$	1.00000	0.100000
$101$	−10.4853	−1.04332	−0.521662	−	0.853152i	$-0.674688\pi$
$101$	−10.4853	−1.04332	−0.521662	+	0.853152i	$0.674688\pi$
$102$	0	0
$103$	10.4853	1.03315	0.516573	−	0.856243i	$-0.327208\pi$
$103$	10.4853	1.03315	0.516573	+	0.856243i	$0.327208\pi$
$104$	−2.82843	−0.277350
$105$	0	0
$106$	13.3137	1.29314
$107$	−6.82843	−0.660129	−0.330064	−	0.943958i	$-0.607070\pi$
$107$	−6.82843	−0.660129	−0.330064	+	0.943958i	$0.607070\pi$
$108$	0	0
$109$	2.48528	0.238047	0.119023	−	0.992891i	$-0.462024\pi$
$109$	2.48528	0.238047	0.119023	+	0.992891i	$0.462024\pi$
$110$	−2.00000	−0.190693
$111$	0	0
$112$	−0.828427	−0.0782790
$113$	16.1421	1.51852	0.759262	−	0.650785i	$-0.225560\pi$
$113$	16.1421	1.51852	0.759262	+	0.650785i	$0.225560\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	−4.82843	−0.448308
$117$	0	0
$118$	10.0000	0.920575
$119$	−0.686292	−0.0629122
$120$	0	0
$121$	−7.00000	−0.636364
$122$	8.82843	0.799288
$123$	0	0
$124$	1.65685	0.148790
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.17157	−0.636374	−0.318187	−	0.948028i	$-0.603074\pi$
$127$	−7.17157	−0.636374	−0.318187	+	0.948028i	$0.603074\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	2.82843	0.248069
$131$	5.31371	0.464261	0.232130	−	0.972685i	$-0.425430\pi$
$131$	5.31371	0.464261	0.232130	+	0.972685i	$0.425430\pi$
$132$	0	0
$133$	0	0
$134$	−6.82843	−0.589886
$135$	0	0
$136$	−0.828427	−0.0710370
$137$	−11.1716	−0.954452	−0.477226	−	0.878781i	$-0.658358\pi$
$137$	−11.1716	−0.954452	−0.477226	+	0.878781i	$0.658358\pi$
$138$	0	0
$139$	−17.6569	−1.49763	−0.748817	−	0.662776i	$-0.769378\pi$
$139$	−17.6569	−1.49763	−0.748817	+	0.662776i	$0.769378\pi$
$140$	0.828427	0.0700149
$141$	0	0
$142$	2.82843	0.237356
$143$	−5.65685	−0.473050
$144$	0	0
$145$	4.82843	0.400979
$146$	13.3137	1.10185
$147$	0	0
$148$	−1.17157	−0.0963027
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−16.9706	−1.38104	−0.690522	−	0.723311i	$-0.742619\pi$
$151$	−16.9706	−1.38104	−0.690522	+	0.723311i	$0.742619\pi$
$152$	0	0
$153$	0	0
$154$	−1.65685	−0.133513
$155$	−1.65685	−0.133082
$156$	0	0
$157$	−8.48528	−0.677199	−0.338600	−	0.940931i	$-0.609953\pi$
$157$	−8.48528	−0.677199	−0.338600	+	0.940931i	$0.609953\pi$
$158$	2.82843	0.225018
$159$	0	0
$160$	1.00000	0.0790569
$161$	0.828427	0.0652892
$162$	0	0
$163$	9.65685	0.756383	0.378192	−	0.925727i	$-0.376546\pi$
$163$	9.65685	0.756383	0.378192	+	0.925727i	$0.376546\pi$
$164$	5.65685	0.441726
$165$	0	0
$166$	−1.17157	−0.0909317
$167$	−8.97056	−0.694163	−0.347081	−	0.937835i	$-0.612827\pi$
$167$	−8.97056	−0.694163	−0.347081	+	0.937835i	$0.612827\pi$
$168$	0	0
$169$	−5.00000	−0.384615
$170$	0.828427	0.0635375
$171$	0	0
$172$	8.48528	0.646997
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	0	0
$175$	−0.828427	−0.0626232
$176$	−2.00000	−0.150756
$177$	0	0
$178$	5.65685	0.423999
$179$	−12.3431	−0.922570	−0.461285	−	0.887252i	$-0.652611\pi$
$179$	−12.3431	−0.922570	−0.461285	+	0.887252i	$0.652611\pi$
$180$	0	0
$181$	−22.4853	−1.67132	−0.835659	−	0.549249i	$-0.814914\pi$
$181$	−22.4853	−1.67132	−0.835659	+	0.549249i	$0.814914\pi$
$182$	2.34315	0.173686
$183$	0	0
$184$	1.00000	0.0737210
$185$	1.17157	0.0861358
$186$	0	0
$187$	−1.65685	−0.121161
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.9706	1.51738	0.758688	−	0.651454i	$-0.225841\pi$
$191$	20.9706	1.51738	0.758688	+	0.651454i	$0.225841\pi$
$192$	0	0
$193$	0.343146	0.0247002	0.0123501	−	0.999924i	$-0.496069\pi$
$193$	0.343146	0.0247002	0.0123501	+	0.999924i	$0.496069\pi$
$194$	−3.17157	−0.227706
$195$	0	0
$196$	−6.31371	−0.450979
$197$	6.97056	0.496632	0.248316	−	0.968679i	$-0.420123\pi$
$197$	6.97056	0.496632	0.248316	+	0.968679i	$0.420123\pi$
$198$	0	0
$199$	4.48528	0.317953	0.158977	−	0.987282i	$-0.449181\pi$
$199$	4.48528	0.317953	0.158977	+	0.987282i	$0.449181\pi$
$200$	−1.00000	−0.0707107
$201$	0	0
$202$	10.4853	0.737742
$203$	4.00000	0.280745
$204$	0	0
$205$	−5.65685	−0.395092
$206$	−10.4853	−0.730544
$207$	0	0
$208$	2.82843	0.196116
$209$	0	0
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−13.3137	−0.914389
$213$	0	0
$214$	6.82843	0.466782
$215$	−8.48528	−0.578691
$216$	0	0
$217$	−1.37258	−0.0931770
$218$	−2.48528	−0.168324
$219$	0	0
$220$	2.00000	0.134840
$221$	2.34315	0.157617
$222$	0	0
$223$	−8.14214	−0.545238	−0.272619	−	0.962122i	$-0.587890\pi$
$223$	−8.14214	−0.545238	−0.272619	+	0.962122i	$0.587890\pi$
$224$	0.828427	0.0553516
$225$	0	0
$226$	−16.1421	−1.07376
$227$	2.82843	0.187729	0.0938647	−	0.995585i	$-0.470078\pi$
$227$	2.82843	0.187729	0.0938647	+	0.995585i	$0.470078\pi$
$228$	0	0
$229$	6.48528	0.428559	0.214280	−	0.976772i	$-0.431260\pi$
$229$	6.48528	0.428559	0.214280	+	0.976772i	$0.431260\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	4.82843	0.317002
$233$	2.00000	0.131024	0.0655122	−	0.997852i	$-0.479132\pi$
$233$	2.00000	0.131024	0.0655122	+	0.997852i	$0.479132\pi$
$234$	0	0
$235$	0	0
$236$	−10.0000	−0.650945
$237$	0	0
$238$	0.686292	0.0444857
$239$	−10.8284	−0.700433	−0.350216	−	0.936669i	$-0.613892\pi$
$239$	−10.8284	−0.700433	−0.350216	+	0.936669i	$0.613892\pi$
$240$	0	0
$241$	−18.0000	−1.15948	−0.579741	−	0.814801i	$-0.696846\pi$
$241$	−18.0000	−1.15948	−0.579741	+	0.814801i	$0.696846\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−8.82843	−0.565182
$245$	6.31371	0.403368
$246$	0	0
$247$	0	0
$248$	−1.65685	−0.105210
$249$	0	0
$250$	1.00000	0.0632456
$251$	−6.97056	−0.439978	−0.219989	−	0.975502i	$-0.570602\pi$
$251$	−6.97056	−0.439978	−0.219989	+	0.975502i	$0.570602\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	7.17157	0.449985
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.3137	−0.830486	−0.415243	−	0.909710i	$-0.636304\pi$
$257$	−13.3137	−0.830486	−0.415243	+	0.909710i	$0.636304\pi$
$258$	0	0
$259$	0.970563	0.0603078
$260$	−2.82843	−0.175412
$261$	0	0
$262$	−5.31371	−0.328282
$263$	−3.31371	−0.204332	−0.102166	−	0.994767i	$-0.532577\pi$
$263$	−3.31371	−0.204332	−0.102166	+	0.994767i	$0.532577\pi$
$264$	0	0
$265$	13.3137	0.817855
$266$	0	0
$267$	0	0
$268$	6.82843	0.417113
$269$	−8.14214	−0.496435	−0.248217	−	0.968704i	$-0.579845\pi$
$269$	−8.14214	−0.496435	−0.248217	+	0.968704i	$0.579845\pi$
$270$	0	0
$271$	5.65685	0.343629	0.171815	−	0.985129i	$-0.445037\pi$
$271$	5.65685	0.343629	0.171815	+	0.985129i	$0.445037\pi$
$272$	0.828427	0.0502308
$273$	0	0
$274$	11.1716	0.674899
$275$	−2.00000	−0.120605
$276$	0	0
$277$	−11.5147	−0.691852	−0.345926	−	0.938262i	$-0.612435\pi$
$277$	−11.5147	−0.691852	−0.345926	+	0.938262i	$0.612435\pi$
$278$	17.6569	1.05899
$279$	0	0
$280$	−0.828427	−0.0495080
$281$	13.6569	0.814700	0.407350	−	0.913272i	$-0.366453\pi$
$281$	13.6569	0.814700	0.407350	+	0.913272i	$0.366453\pi$
$282$	0	0
$283$	−2.14214	−0.127337	−0.0636684	−	0.997971i	$-0.520280\pi$
$283$	−2.14214	−0.127337	−0.0636684	+	0.997971i	$0.520280\pi$
$284$	−2.82843	−0.167836
$285$	0	0
$286$	5.65685	0.334497
$287$	−4.68629	−0.276623
$288$	0	0
$289$	−16.3137	−0.959630
$290$	−4.82843	−0.283535
$291$	0	0
$292$	−13.3137	−0.779126
$293$	−3.65685	−0.213636	−0.106818	−	0.994279i	$-0.534066\pi$
$293$	−3.65685	−0.213636	−0.106818	+	0.994279i	$0.534066\pi$
$294$	0	0
$295$	10.0000	0.582223
$296$	1.17157	0.0680963
$297$	0	0
$298$	6.00000	0.347571
$299$	−2.82843	−0.163572
$300$	0	0
$301$	−7.02944	−0.405170
$302$	16.9706	0.976546
$303$	0	0
$304$	0	0
$305$	8.82843	0.505514
$306$	0	0
$307$	−19.3137	−1.10229	−0.551146	−	0.834409i	$-0.685809\pi$
$307$	−19.3137	−1.10229	−0.551146	+	0.834409i	$0.685809\pi$
$308$	1.65685	0.0944080
$309$	0	0
$310$	1.65685	0.0941030
$311$	16.4853	0.934795	0.467397	−	0.884047i	$-0.345192\pi$
$311$	16.4853	0.934795	0.467397	+	0.884047i	$0.345192\pi$
$312$	0	0
$313$	16.1421	0.912407	0.456204	−	0.889875i	$-0.349209\pi$
$313$	16.1421	0.912407	0.456204	+	0.889875i	$0.349209\pi$
$314$	8.48528	0.478852
$315$	0	0
$316$	−2.82843	−0.159111
$317$	10.6863	0.600202	0.300101	−	0.953907i	$-0.402980\pi$
$317$	10.6863	0.600202	0.300101	+	0.953907i	$0.402980\pi$
$318$	0	0
$319$	9.65685	0.540680
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−0.828427	−0.0461664
$323$	0	0
$324$	0	0
$325$	2.82843	0.156893
$326$	−9.65685	−0.534844
$327$	0	0
$328$	−5.65685	−0.312348
$329$	0	0
$330$	0	0
$331$	18.6274	1.02386	0.511928	−	0.859029i	$-0.328932\pi$
$331$	18.6274	1.02386	0.511928	+	0.859029i	$0.328932\pi$
$332$	1.17157	0.0642984
$333$	0	0
$334$	8.97056	0.490847
$335$	−6.82843	−0.373077
$336$	0	0
$337$	−17.7990	−0.969573	−0.484786	−	0.874633i	$-0.661103\pi$
$337$	−17.7990	−0.969573	−0.484786	+	0.874633i	$0.661103\pi$
$338$	5.00000	0.271964
$339$	0	0
$340$	−0.828427	−0.0449278
$341$	−3.31371	−0.179447
$342$	0	0
$343$	11.0294	0.595534
$344$	−8.48528	−0.457496
$345$	0	0
$346$	−10.0000	−0.537603
$347$	5.65685	0.303676	0.151838	−	0.988405i	$-0.451481\pi$
$347$	5.65685	0.303676	0.151838	+	0.988405i	$0.451481\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0.828427	0.0442813
$351$	0	0
$352$	2.00000	0.106600
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	2.82843	0.150117
$356$	−5.65685	−0.299813
$357$	0	0
$358$	12.3431	0.652356
$359$	−4.68629	−0.247333	−0.123667	−	0.992324i	$-0.539465\pi$
$359$	−4.68629	−0.247333	−0.123667	+	0.992324i	$0.539465\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	22.4853	1.18180
$363$	0	0
$364$	−2.34315	−0.122814
$365$	13.3137	0.696871
$366$	0	0
$367$	4.82843	0.252042	0.126021	−	0.992028i	$-0.459779\pi$
$367$	4.82843	0.252042	0.126021	+	0.992028i	$0.459779\pi$
$368$	−1.00000	−0.0521286
$369$	0	0
$370$	−1.17157	−0.0609072
$371$	11.0294	0.572620
$372$	0	0
$373$	26.8284	1.38912	0.694562	−	0.719433i	$-0.255598\pi$
$373$	26.8284	1.38912	0.694562	+	0.719433i	$0.255598\pi$
$374$	1.65685	0.0856739
$375$	0	0
$376$	0	0
$377$	−13.6569	−0.703364
$378$	0	0
$379$	−2.34315	−0.120359	−0.0601797	−	0.998188i	$-0.519167\pi$
$379$	−2.34315	−0.120359	−0.0601797	+	0.998188i	$0.519167\pi$
$380$	0	0
$381$	0	0
$382$	−20.9706	−1.07295
$383$	18.3431	0.937291	0.468645	−	0.883386i	$-0.344742\pi$
$383$	18.3431	0.937291	0.468645	+	0.883386i	$0.344742\pi$
$384$	0	0
$385$	−1.65685	−0.0844411
$386$	−0.343146	−0.0174657
$387$	0	0
$388$	3.17157	0.161012
$389$	16.6274	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$389$	16.6274	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$390$	0	0
$391$	−0.828427	−0.0418954
$392$	6.31371	0.318890
$393$	0	0
$394$	−6.97056	−0.351172
$395$	2.82843	0.142314
$396$	0	0
$397$	−17.1716	−0.861817	−0.430908	−	0.902396i	$-0.641807\pi$
$397$	−17.1716	−0.861817	−0.430908	+	0.902396i	$0.641807\pi$
$398$	−4.48528	−0.224827
$399$	0	0
$400$	1.00000	0.0500000
$401$	16.0000	0.799002	0.399501	−	0.916733i	$-0.369183\pi$
$401$	16.0000	0.799002	0.399501	+	0.916733i	$0.369183\pi$
$402$	0	0
$403$	4.68629	0.233441
$404$	−10.4853	−0.521662
$405$	0	0
$406$	−4.00000	−0.198517
$407$	2.34315	0.116145
$408$	0	0
$409$	5.31371	0.262746	0.131373	−	0.991333i	$-0.458061\pi$
$409$	5.31371	0.262746	0.131373	+	0.991333i	$0.458061\pi$
$410$	5.65685	0.279372
$411$	0	0
$412$	10.4853	0.516573
$413$	8.28427	0.407642
$414$	0	0
$415$	−1.17157	−0.0575103
$416$	−2.82843	−0.138675
$417$	0	0
$418$	0	0
$419$	7.65685	0.374062	0.187031	−	0.982354i	$-0.440114\pi$
$419$	7.65685	0.374062	0.187031	+	0.982354i	$0.440114\pi$
$420$	0	0
$421$	−8.14214	−0.396823	−0.198412	−	0.980119i	$-0.563578\pi$
$421$	−8.14214	−0.396823	−0.198412	+	0.980119i	$0.563578\pi$
$422$	4.00000	0.194717
$423$	0	0
$424$	13.3137	0.646571
$425$	0.828427	0.0401846
$426$	0	0
$427$	7.31371	0.353935
$428$	−6.82843	−0.330064
$429$	0	0
$430$	8.48528	0.409197
$431$	30.6274	1.47527	0.737635	−	0.675199i	$-0.235942\pi$
$431$	30.6274	1.47527	0.737635	+	0.675199i	$0.235942\pi$
$432$	0	0
$433$	8.14214	0.391286	0.195643	−	0.980675i	$-0.437321\pi$
$433$	8.14214	0.391286	0.195643	+	0.980675i	$0.437321\pi$
$434$	1.37258	0.0658861
$435$	0	0
$436$	2.48528	0.119023
$437$	0	0
$438$	0	0
$439$	−14.6274	−0.698129	−0.349064	−	0.937099i	$-0.613501\pi$
$439$	−14.6274	−0.698129	−0.349064	+	0.937099i	$0.613501\pi$
$440$	−2.00000	−0.0953463
$441$	0	0
$442$	−2.34315	−0.111452
$443$	11.3137	0.537531	0.268765	−	0.963206i	$-0.413384\pi$
$443$	11.3137	0.537531	0.268765	+	0.963206i	$0.413384\pi$
$444$	0	0
$445$	5.65685	0.268161
$446$	8.14214	0.385541
$447$	0	0
$448$	−0.828427	−0.0391395
$449$	3.02944	0.142968	0.0714840	−	0.997442i	$-0.477227\pi$
$449$	3.02944	0.142968	0.0714840	+	0.997442i	$0.477227\pi$
$450$	0	0
$451$	−11.3137	−0.532742
$452$	16.1421	0.759262
$453$	0	0
$454$	−2.82843	−0.132745
$455$	2.34315	0.109848
$456$	0	0
$457$	0.828427	0.0387522	0.0193761	−	0.999812i	$-0.493832\pi$
$457$	0.828427	0.0387522	0.0193761	+	0.999812i	$0.493832\pi$
$458$	−6.48528	−0.303037
$459$	0	0
$460$	1.00000	0.0466252
$461$	24.1421	1.12441	0.562206	−	0.826997i	$-0.309953\pi$
$461$	24.1421	1.12441	0.562206	+	0.826997i	$0.309953\pi$
$462$	0	0
$463$	17.5147	0.813978	0.406989	−	0.913433i	$-0.366579\pi$
$463$	17.5147	0.813978	0.406989	+	0.913433i	$0.366579\pi$
$464$	−4.82843	−0.224154
$465$	0	0
$466$	−2.00000	−0.0926482
$467$	−20.4853	−0.947946	−0.473973	−	0.880539i	$-0.657181\pi$
$467$	−20.4853	−0.947946	−0.473973	+	0.880539i	$0.657181\pi$
$468$	0	0
$469$	−5.65685	−0.261209
$470$	0	0
$471$	0	0
$472$	10.0000	0.460287
$473$	−16.9706	−0.780307
$474$	0	0
$475$	0	0
$476$	−0.686292	−0.0314561
$477$	0	0
$478$	10.8284	0.495281
$479$	14.3431	0.655355	0.327678	−	0.944790i	$-0.393734\pi$
$479$	14.3431	0.655355	0.327678	+	0.944790i	$0.393734\pi$
$480$	0	0
$481$	−3.31371	−0.151092
$482$	18.0000	0.819878
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−3.17157	−0.144014
$486$	0	0
$487$	−29.1127	−1.31922	−0.659611	−	0.751607i	$-0.729279\pi$
$487$	−29.1127	−1.31922	−0.659611	+	0.751607i	$0.729279\pi$
$488$	8.82843	0.399644
$489$	0	0
$490$	−6.31371	−0.285224
$491$	14.6863	0.662783	0.331392	−	0.943493i	$-0.392482\pi$
$491$	14.6863	0.662783	0.331392	+	0.943493i	$0.392482\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	1.65685	0.0743950
$497$	2.34315	0.105104
$498$	0	0
$499$	25.6569	1.14856	0.574279	−	0.818659i	$-0.305282\pi$
$499$	25.6569	1.14856	0.574279	+	0.818659i	$0.305282\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	6.97056	0.311111
$503$	22.6274	1.00891	0.504453	−	0.863439i	$-0.331694\pi$
$503$	22.6274	1.00891	0.504453	+	0.863439i	$0.331694\pi$
$504$	0	0
$505$	10.4853	0.466589
$506$	−2.00000	−0.0889108
$507$	0	0
$508$	−7.17157	−0.318187
$509$	−8.14214	−0.360894	−0.180447	−	0.983585i	$-0.557754\pi$
$509$	−8.14214	−0.360894	−0.180447	+	0.983585i	$0.557754\pi$
$510$	0	0
$511$	11.0294	0.487914
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	13.3137	0.587243
$515$	−10.4853	−0.462037
$516$	0	0
$517$	0	0
$518$	−0.970563	−0.0426441
$519$	0	0
$520$	2.82843	0.124035
$521$	−3.31371	−0.145176	−0.0725881	−	0.997362i	$-0.523126\pi$
$521$	−3.31371	−0.145176	−0.0725881	+	0.997362i	$0.523126\pi$
$522$	0	0
$523$	36.4853	1.59539	0.797695	−	0.603061i	$-0.206053\pi$
$523$	36.4853	1.59539	0.797695	+	0.603061i	$0.206053\pi$
$524$	5.31371	0.232130
$525$	0	0
$526$	3.31371	0.144485
$527$	1.37258	0.0597907
$528$	0	0
$529$	1.00000	0.0434783
$530$	−13.3137	−0.578311
$531$	0	0
$532$	0	0
$533$	16.0000	0.693037
$534$	0	0
$535$	6.82843	0.295219
$536$	−6.82843	−0.294943
$537$	0	0
$538$	8.14214	0.351032
$539$	12.6274	0.543901
$540$	0	0
$541$	−5.02944	−0.216232	−0.108116	−	0.994138i	$-0.534482\pi$
$541$	−5.02944	−0.216232	−0.108116	+	0.994138i	$0.534482\pi$
$542$	−5.65685	−0.242983
$543$	0	0
$544$	−0.828427	−0.0355185
$545$	−2.48528	−0.106458
$546$	0	0
$547$	22.6274	0.967478	0.483739	−	0.875212i	$-0.339278\pi$
$547$	22.6274	0.967478	0.483739	+	0.875212i	$0.339278\pi$
$548$	−11.1716	−0.477226
$549$	0	0
$550$	2.00000	0.0852803
$551$	0	0
$552$	0	0
$553$	2.34315	0.0996407
$554$	11.5147	0.489214
$555$	0	0
$556$	−17.6569	−0.748817
$557$	6.97056	0.295352	0.147676	−	0.989036i	$-0.452821\pi$
$557$	6.97056	0.295352	0.147676	+	0.989036i	$0.452821\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0.828427	0.0350074
$561$	0	0
$562$	−13.6569	−0.576080
$563$	−13.1716	−0.555116	−0.277558	−	0.960709i	$-0.589525\pi$
$563$	−13.1716	−0.555116	−0.277558	+	0.960709i	$0.589525\pi$
$564$	0	0
$565$	−16.1421	−0.679105
$566$	2.14214	0.0900407
$567$	0	0
$568$	2.82843	0.118678
$569$	32.0000	1.34151	0.670755	−	0.741679i	$-0.265970\pi$
$569$	32.0000	1.34151	0.670755	+	0.741679i	$0.265970\pi$
$570$	0	0
$571$	0	0		−	1.00000i	$-0.5\pi$
$571$	0	0			1.00000i	$0.5\pi$
$572$	−5.65685	−0.236525
$573$	0	0
$574$	4.68629	0.195602
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−19.6569	−0.818326	−0.409163	−	0.912461i	$-0.634179\pi$
$577$	−19.6569	−0.818326	−0.409163	+	0.912461i	$0.634179\pi$
$578$	16.3137	0.678561
$579$	0	0
$580$	4.82843	0.200490
$581$	−0.970563	−0.0402657
$582$	0	0
$583$	26.6274	1.10279
$584$	13.3137	0.550925
$585$	0	0
$586$	3.65685	0.151063
$587$	43.5980	1.79948	0.899741	−	0.436425i	$-0.143756\pi$
$587$	43.5980	1.79948	0.899741	+	0.436425i	$0.143756\pi$
$588$	0	0
$589$	0	0
$590$	−10.0000	−0.411693
$591$	0	0
$592$	−1.17157	−0.0481513
$593$	1.31371	0.0539475	0.0269738	−	0.999636i	$-0.491413\pi$
$593$	1.31371	0.0539475	0.0269738	+	0.999636i	$0.491413\pi$
$594$	0	0
$595$	0.686292	0.0281352
$596$	−6.00000	−0.245770
$597$	0	0
$598$	2.82843	0.115663
$599$	2.14214	0.0875253	0.0437626	−	0.999042i	$-0.486065\pi$
$599$	2.14214	0.0875253	0.0437626	+	0.999042i	$0.486065\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	7.02944	0.286498
$603$	0	0
$604$	−16.9706	−0.690522
$605$	7.00000	0.284590
$606$	0	0
$607$	19.8579	0.806006	0.403003	−	0.915199i	$-0.367966\pi$
$607$	19.8579	0.806006	0.403003	+	0.915199i	$0.367966\pi$
$608$	0	0
$609$	0	0
$610$	−8.82843	−0.357453
$611$	0	0
$612$	0	0
$613$	−9.17157	−0.370436	−0.185218	−	0.982697i	$-0.559299\pi$
$613$	−9.17157	−0.370436	−0.185218	+	0.982697i	$0.559299\pi$
$614$	19.3137	0.779438
$615$	0	0
$616$	−1.65685	−0.0667566
$617$	1.79899	0.0724246	0.0362123	−	0.999344i	$-0.488471\pi$
$617$	1.79899	0.0724246	0.0362123	+	0.999344i	$0.488471\pi$
$618$	0	0
$619$	19.3137	0.776283	0.388142	−	0.921600i	$-0.373117\pi$
$619$	19.3137	0.776283	0.388142	+	0.921600i	$0.373117\pi$
$620$	−1.65685	−0.0665409
$621$	0	0
$622$	−16.4853	−0.661000
$623$	4.68629	0.187752
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.1421	−0.645169
$627$	0	0
$628$	−8.48528	−0.338600
$629$	−0.970563	−0.0386989
$630$	0	0
$631$	−27.7990	−1.10666	−0.553330	−	0.832962i	$-0.686643\pi$
$631$	−27.7990	−1.10666	−0.553330	+	0.832962i	$0.686643\pi$
$632$	2.82843	0.112509
$633$	0	0
$634$	−10.6863	−0.424407
$635$	7.17157	0.284595
$636$	0	0
$637$	−17.8579	−0.707554
$638$	−9.65685	−0.382319
$639$	0	0
$640$	1.00000	0.0395285
$641$	−1.65685	−0.0654418	−0.0327209	−	0.999465i	$-0.510417\pi$
$641$	−1.65685	−0.0654418	−0.0327209	+	0.999465i	$0.510417\pi$
$642$	0	0
$643$	44.4853	1.75433	0.877164	−	0.480191i	$-0.159433\pi$
$643$	44.4853	1.75433	0.877164	+	0.480191i	$0.159433\pi$
$644$	0.828427	0.0326446
$645$	0	0
$646$	0	0
$647$	−8.97056	−0.352669	−0.176335	−	0.984330i	$-0.556424\pi$
$647$	−8.97056	−0.352669	−0.176335	+	0.984330i	$0.556424\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	−2.82843	−0.110940
$651$	0	0
$652$	9.65685	0.378192
$653$	−48.6274	−1.90294	−0.951469	−	0.307745i	$-0.900426\pi$
$653$	−48.6274	−1.90294	−0.951469	+	0.307745i	$0.900426\pi$
$654$	0	0
$655$	−5.31371	−0.207624
$656$	5.65685	0.220863
$657$	0	0
$658$	0	0
$659$	−39.9411	−1.55589	−0.777943	−	0.628335i	$-0.783737\pi$
$659$	−39.9411	−1.55589	−0.777943	+	0.628335i	$0.783737\pi$
$660$	0	0
$661$	0.142136	0.00552844	0.00276422	−	0.999996i	$-0.499120\pi$
$661$	0.142136	0.00552844	0.00276422	+	0.999996i	$0.499120\pi$
$662$	−18.6274	−0.723975
$663$	0	0
$664$	−1.17157	−0.0454658
$665$	0	0
$666$	0	0
$667$	4.82843	0.186957
$668$	−8.97056	−0.347081
$669$	0	0
$670$	6.82843	0.263805
$671$	17.6569	0.681635
$672$	0	0
$673$	16.3431	0.629982	0.314991	−	0.949095i	$-0.397999\pi$
$673$	16.3431	0.629982	0.314991	+	0.949095i	$0.397999\pi$
$674$	17.7990	0.685591
$675$	0	0
$676$	−5.00000	−0.192308
$677$	−17.3137	−0.665420	−0.332710	−	0.943029i	$-0.607963\pi$
$677$	−17.3137	−0.665420	−0.332710	+	0.943029i	$0.607963\pi$
$678$	0	0
$679$	−2.62742	−0.100831
$680$	0.828427	0.0317687
$681$	0	0
$682$	3.31371	0.126888
$683$	−11.0294	−0.422030	−0.211015	−	0.977483i	$-0.567677\pi$
$683$	−11.0294	−0.422030	−0.211015	+	0.977483i	$0.567677\pi$
$684$	0	0
$685$	11.1716	0.426844
$686$	−11.0294	−0.421106
$687$	0	0
$688$	8.48528	0.323498
$689$	−37.6569	−1.43461
$690$	0	0
$691$	−41.6569	−1.58470	−0.792351	−	0.610066i	$-0.791143\pi$
$691$	−41.6569	−1.58470	−0.792351	+	0.610066i	$0.791143\pi$
$692$	10.0000	0.380143
$693$	0	0
$694$	−5.65685	−0.214731
$695$	17.6569	0.669763
$696$	0	0
$697$	4.68629	0.177506
$698$	14.0000	0.529908
$699$	0	0
$700$	−0.828427	−0.0313116
$701$	−31.6569	−1.19566	−0.597831	−	0.801622i	$-0.703971\pi$
$701$	−31.6569	−1.19566	−0.597831	+	0.801622i	$0.703971\pi$
$702$	0	0
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	−18.0000	−0.677439
$707$	8.68629	0.326682
$708$	0	0
$709$	−17.1127	−0.642681	−0.321340	−	0.946964i	$-0.604133\pi$
$709$	−17.1127	−0.642681	−0.321340	+	0.946964i	$0.604133\pi$
$710$	−2.82843	−0.106149
$711$	0	0
$712$	5.65685	0.212000
$713$	−1.65685	−0.0620497
$714$	0	0
$715$	5.65685	0.211554
$716$	−12.3431	−0.461285
$717$	0	0
$718$	4.68629	0.174891
$719$	−4.48528	−0.167273	−0.0836364	−	0.996496i	$-0.526653\pi$
$719$	−4.48528	−0.167273	−0.0836364	+	0.996496i	$0.526653\pi$
$720$	0	0
$721$	−8.68629	−0.323494
$722$	19.0000	0.707107
$723$	0	0
$724$	−22.4853	−0.835659
$725$	−4.82843	−0.179323
$726$	0	0
$727$	−15.1716	−0.562682	−0.281341	−	0.959608i	$-0.590779\pi$
$727$	−15.1716	−0.562682	−0.281341	+	0.959608i	$0.590779\pi$
$728$	2.34315	0.0868428
$729$	0	0
$730$	−13.3137	−0.492762
$731$	7.02944	0.259993
$732$	0	0
$733$	4.48528	0.165668	0.0828338	−	0.996563i	$-0.473603\pi$
$733$	4.48528	0.165668	0.0828338	+	0.996563i	$0.473603\pi$
$734$	−4.82843	−0.178220
$735$	0	0
$736$	1.00000	0.0368605
$737$	−13.6569	−0.503057
$738$	0	0
$739$	33.6569	1.23809	0.619044	−	0.785357i	$-0.287520\pi$
$739$	33.6569	1.23809	0.619044	+	0.785357i	$0.287520\pi$
$740$	1.17157	0.0430679
$741$	0	0
$742$	−11.0294	−0.404903
$743$	20.2843	0.744158	0.372079	−	0.928201i	$-0.378645\pi$
$743$	20.2843	0.744158	0.372079	+	0.928201i	$0.378645\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−26.8284	−0.982259
$747$	0	0
$748$	−1.65685	−0.0605806
$749$	5.65685	0.206697
$750$	0	0
$751$	31.1127	1.13532	0.567659	−	0.823264i	$-0.307849\pi$
$751$	31.1127	1.13532	0.567659	+	0.823264i	$0.307849\pi$
$752$	0	0
$753$	0	0
$754$	13.6569	0.497353
$755$	16.9706	0.617622
$756$	0	0
$757$	−14.8284	−0.538948	−0.269474	−	0.963008i	$-0.586850\pi$
$757$	−14.8284	−0.538948	−0.269474	+	0.963008i	$0.586850\pi$
$758$	2.34315	0.0851069
$759$	0	0
$760$	0	0
$761$	25.6569	0.930060	0.465030	−	0.885295i	$-0.346043\pi$
$761$	25.6569	0.930060	0.465030	+	0.885295i	$0.346043\pi$
$762$	0	0
$763$	−2.05887	−0.0745363
$764$	20.9706	0.758688
$765$	0	0
$766$	−18.3431	−0.662765
$767$	−28.2843	−1.02129
$768$	0	0
$769$	22.0000	0.793340	0.396670	−	0.917961i	$-0.370166\pi$
$769$	22.0000	0.793340	0.396670	+	0.917961i	$0.370166\pi$
$770$	1.65685	0.0597089
$771$	0	0
$772$	0.343146	0.0123501
$773$	−19.6569	−0.707008	−0.353504	−	0.935433i	$-0.615010\pi$
$773$	−19.6569	−0.707008	−0.353504	+	0.935433i	$0.615010\pi$
$774$	0	0
$775$	1.65685	0.0595160
$776$	−3.17157	−0.113853
$777$	0	0
$778$	−16.6274	−0.596122
$779$	0	0
$780$	0	0
$781$	5.65685	0.202418
$782$	0.828427	0.0296245
$783$	0	0
$784$	−6.31371	−0.225490
$785$	8.48528	0.302853
$786$	0	0
$787$	36.4853	1.30056	0.650280	−	0.759695i	$-0.274652\pi$
$787$	36.4853	1.30056	0.650280	+	0.759695i	$0.274652\pi$
$788$	6.97056	0.248316
$789$	0	0
$790$	−2.82843	−0.100631
$791$	−13.3726	−0.475474
$792$	0	0
$793$	−24.9706	−0.886731
$794$	17.1716	0.609396
$795$	0	0
$796$	4.48528	0.158977
$797$	−17.0294	−0.603214	−0.301607	−	0.953432i	$-0.597523\pi$
$797$	−17.0294	−0.603214	−0.301607	+	0.953432i	$0.597523\pi$
$798$	0	0
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	0	0
$802$	−16.0000	−0.564980
$803$	26.6274	0.939661
$804$	0	0
$805$	−0.828427	−0.0291982
$806$	−4.68629	−0.165068
$807$	0	0
$808$	10.4853	0.368871
$809$	4.28427	0.150627	0.0753135	−	0.997160i	$-0.476004\pi$
$809$	4.28427	0.150627	0.0753135	+	0.997160i	$0.476004\pi$
$810$	0	0
$811$	−16.2843	−0.571818	−0.285909	−	0.958257i	$-0.592296\pi$
$811$	−16.2843	−0.571818	−0.285909	+	0.958257i	$0.592296\pi$
$812$	4.00000	0.140372
$813$	0	0
$814$	−2.34315	−0.0821272
$815$	−9.65685	−0.338265
$816$	0	0
$817$	0	0
$818$	−5.31371	−0.185789
$819$	0	0
$820$	−5.65685	−0.197546
$821$	−32.1421	−1.12177	−0.560884	−	0.827894i	$-0.689539\pi$
$821$	−32.1421	−1.12177	−0.560884	+	0.827894i	$0.689539\pi$
$822$	0	0
$823$	26.4853	0.923219	0.461609	−	0.887083i	$-0.347272\pi$
$823$	26.4853	0.923219	0.461609	+	0.887083i	$0.347272\pi$
$824$	−10.4853	−0.365272
$825$	0	0
$826$	−8.28427	−0.288247
$827$	−49.4558	−1.71975	−0.859874	−	0.510506i	$-0.829458\pi$
$827$	−49.4558	−1.71975	−0.859874	+	0.510506i	$0.829458\pi$
$828$	0	0
$829$	−32.3431	−1.12332	−0.561662	−	0.827367i	$-0.689838\pi$
$829$	−32.3431	−1.12332	−0.561662	+	0.827367i	$0.689838\pi$
$830$	1.17157	0.0406659
$831$	0	0
$832$	2.82843	0.0980581
$833$	−5.23045	−0.181224
$834$	0	0
$835$	8.97056	0.310439
$836$	0	0
$837$	0	0
$838$	−7.65685	−0.264502
$839$	−20.2843	−0.700291	−0.350145	−	0.936695i	$-0.613868\pi$
$839$	−20.2843	−0.700291	−0.350145	+	0.936695i	$0.613868\pi$
$840$	0	0
$841$	−5.68629	−0.196079
$842$	8.14214	0.280597
$843$	0	0
$844$	−4.00000	−0.137686
$845$	5.00000	0.172005
$846$	0	0
$847$	5.79899	0.199256
$848$	−13.3137	−0.457195
$849$	0	0
$850$	−0.828427	−0.0284148
$851$	1.17157	0.0401610
$852$	0	0
$853$	−43.1127	−1.47615	−0.738075	−	0.674718i	$-0.764265\pi$
$853$	−43.1127	−1.47615	−0.738075	+	0.674718i	$0.764265\pi$
$854$	−7.31371	−0.250270
$855$	0	0
$856$	6.82843	0.233391
$857$	18.9706	0.648022	0.324011	−	0.946053i	$-0.394969\pi$
$857$	18.9706	0.648022	0.324011	+	0.946053i	$0.394969\pi$
$858$	0	0
$859$	−43.5980	−1.48754	−0.743772	−	0.668433i	$-0.766965\pi$
$859$	−43.5980	−1.48754	−0.743772	+	0.668433i	$0.766965\pi$
$860$	−8.48528	−0.289346
$861$	0	0
$862$	−30.6274	−1.04317
$863$	−53.2548	−1.81282	−0.906408	−	0.422404i	$-0.861186\pi$
$863$	−53.2548	−1.81282	−0.906408	+	0.422404i	$0.861186\pi$
$864$	0	0
$865$	−10.0000	−0.340010
$866$	−8.14214	−0.276681
$867$	0	0
$868$	−1.37258	−0.0465885
$869$	5.65685	0.191896
$870$	0	0
$871$	19.3137	0.654420
$872$	−2.48528	−0.0841622
$873$	0	0
$874$	0	0
$875$	0.828427	0.0280059
$876$	0	0
$877$	27.7990	0.938705	0.469353	−	0.883011i	$-0.344487\pi$
$877$	27.7990	0.938705	0.469353	+	0.883011i	$0.344487\pi$
$878$	14.6274	0.493651
$879$	0	0
$880$	2.00000	0.0674200
$881$	−8.97056	−0.302226	−0.151113	−	0.988516i	$-0.548286\pi$
$881$	−8.97056	−0.302226	−0.151113	+	0.988516i	$0.548286\pi$
$882$	0	0
$883$	44.9706	1.51338	0.756690	−	0.653774i	$-0.226815\pi$
$883$	44.9706	1.51338	0.756690	+	0.653774i	$0.226815\pi$
$884$	2.34315	0.0788085
$885$	0	0
$886$	−11.3137	−0.380091
$887$	−23.0294	−0.773253	−0.386626	−	0.922236i	$-0.626360\pi$
$887$	−23.0294	−0.773253	−0.386626	+	0.922236i	$0.626360\pi$
$888$	0	0
$889$	5.94113	0.199259
$890$	−5.65685	−0.189618
$891$	0	0
$892$	−8.14214	−0.272619
$893$	0	0
$894$	0	0
$895$	12.3431	0.412586
$896$	0.828427	0.0276758
$897$	0	0
$898$	−3.02944	−0.101094
$899$	−8.00000	−0.266815
$900$	0	0
$901$	−11.0294	−0.367444
$902$	11.3137	0.376705
$903$	0	0
$904$	−16.1421	−0.536879
$905$	22.4853	0.747436
$906$	0	0
$907$	16.4853	0.547385	0.273692	−	0.961817i	$-0.411755\pi$
$907$	16.4853	0.547385	0.273692	+	0.961817i	$0.411755\pi$
$908$	2.82843	0.0938647
$909$	0	0
$910$	−2.34315	−0.0776745
$911$	−21.6569	−0.717524	−0.358762	−	0.933429i	$-0.616801\pi$
$911$	−21.6569	−0.717524	−0.358762	+	0.933429i	$0.616801\pi$
$912$	0	0
$913$	−2.34315	−0.0775468
$914$	−0.828427	−0.0274019
$915$	0	0
$916$	6.48528	0.214280
$917$	−4.40202	−0.145368
$918$	0	0
$919$	−9.45584	−0.311920	−0.155960	−	0.987763i	$-0.549847\pi$
$919$	−9.45584	−0.311920	−0.155960	+	0.987763i	$0.549847\pi$
$920$	−1.00000	−0.0329690
$921$	0	0
$922$	−24.1421	−0.795079
$923$	−8.00000	−0.263323
$924$	0	0
$925$	−1.17157	−0.0385211
$926$	−17.5147	−0.575569
$927$	0	0
$928$	4.82843	0.158501
$929$	59.5980	1.95535	0.977673	−	0.210131i	$-0.0673892\pi$
$929$	59.5980	1.95535	0.977673	+	0.210131i	$0.0673892\pi$
$930$	0	0
$931$	0	0
$932$	2.00000	0.0655122
$933$	0	0
$934$	20.4853	0.670299
$935$	1.65685	0.0541849
$936$	0	0
$937$	−23.1716	−0.756982	−0.378491	−	0.925605i	$-0.623557\pi$
$937$	−23.1716	−0.756982	−0.378491	+	0.925605i	$0.623557\pi$
$938$	5.65685	0.184703
$939$	0	0
$940$	0	0
$941$	14.2843	0.465654	0.232827	−	0.972518i	$-0.425202\pi$
$941$	14.2843	0.465654	0.232827	+	0.972518i	$0.425202\pi$
$942$	0	0
$943$	−5.65685	−0.184213
$944$	−10.0000	−0.325472
$945$	0	0
$946$	16.9706	0.551761
$947$	15.0294	0.488391	0.244196	−	0.969726i	$-0.421476\pi$
$947$	15.0294	0.488391	0.244196	+	0.969726i	$0.421476\pi$
$948$	0	0
$949$	−37.6569	−1.22239
$950$	0	0
$951$	0	0
$952$	0.686292	0.0222428
$953$	−45.1127	−1.46134	−0.730672	−	0.682729i	$-0.760793\pi$
$953$	−45.1127	−1.46134	−0.730672	+	0.682729i	$0.760793\pi$
$954$	0	0
$955$	−20.9706	−0.678591
$956$	−10.8284	−0.350216
$957$	0	0
$958$	−14.3431	−0.463406
$959$	9.25483	0.298854
$960$	0	0
$961$	−28.2548	−0.911446
$962$	3.31371	0.106838
$963$	0	0
$964$	−18.0000	−0.579741
$965$	−0.343146	−0.0110463
$966$	0	0
$967$	39.4558	1.26881	0.634407	−	0.772999i	$-0.281244\pi$
$967$	39.4558	1.26881	0.634407	+	0.772999i	$0.281244\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	3.17157	0.101833
$971$	−31.9411	−1.02504	−0.512520	−	0.858675i	$-0.671288\pi$
$971$	−31.9411	−1.02504	−0.512520	+	0.858675i	$0.671288\pi$
$972$	0	0
$973$	14.6274	0.468933
$974$	29.1127	0.932831
$975$	0	0
$976$	−8.82843	−0.282591
$977$	−33.5147	−1.07223	−0.536115	−	0.844145i	$-0.680109\pi$
$977$	−33.5147	−1.07223	−0.536115	+	0.844145i	$0.680109\pi$
$978$	0	0
$979$	11.3137	0.361588
$980$	6.31371	0.201684
$981$	0	0
$982$	−14.6863	−0.468658
$983$	48.9706	1.56192	0.780959	−	0.624582i	$-0.214731\pi$
$983$	48.9706	1.56192	0.780959	+	0.624582i	$0.214731\pi$
$984$	0	0
$985$	−6.97056	−0.222101
$986$	4.00000	0.127386
$987$	0	0
$988$	0	0
$989$	−8.48528	−0.269816
$990$	0	0
$991$	−19.3137	−0.613520	−0.306760	−	0.951787i	$-0.599245\pi$
$991$	−19.3137	−0.613520	−0.306760	+	0.951787i	$0.599245\pi$
$992$	−1.65685	−0.0526052
$993$	0	0
$994$	−2.34315	−0.0743201
$995$	−4.48528	−0.142193
$996$	0	0
$997$	−28.7696	−0.911141	−0.455570	−	0.890200i	$-0.650565\pi$
$997$	−28.7696	−0.911141	−0.455570	+	0.890200i	$0.650565\pi$
$998$	−25.6569	−0.812154
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2070.2.a.v.1.1	✓	2
3.2	odd	2		2070.2.a.y.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2070.2.a.v.1.1	✓	2	1.1	even	1	trivial
2070.2.a.y.1.1	yes	2	3.2	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:	\( N \)	\(=\)	\( 2070 = 2 \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2070.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.828427 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -0.828427 q^{7} -1.00000 q^{8} +1.00000 q^{10} -2.00000 q^{11} +2.82843 q^{13} +0.828427 q^{14} +1.00000 q^{16} +0.828427 q^{17} -1.00000 q^{20} +2.00000 q^{22} -1.00000 q^{23} +1.00000 q^{25} -2.82843 q^{26} -0.828427 q^{28} -4.82843 q^{29} +1.65685 q^{31} -1.00000 q^{32} -0.828427 q^{34} +0.828427 q^{35} -1.17157 q^{37} +1.00000 q^{40} +5.65685 q^{41} +8.48528 q^{43} -2.00000 q^{44} +1.00000 q^{46} -6.31371 q^{49} -1.00000 q^{50} +2.82843 q^{52} -13.3137 q^{53} +2.00000 q^{55} +0.828427 q^{56} +4.82843 q^{58} -10.0000 q^{59} -8.82843 q^{61} -1.65685 q^{62} +1.00000 q^{64} -2.82843 q^{65} +6.82843 q^{67} +0.828427 q^{68} -0.828427 q^{70} -2.82843 q^{71} -13.3137 q^{73} +1.17157 q^{74} +1.65685 q^{77} -2.82843 q^{79} -1.00000 q^{80} -5.65685 q^{82} +1.17157 q^{83} -0.828427 q^{85} -8.48528 q^{86} +2.00000 q^{88} -5.65685 q^{89} -2.34315 q^{91} -1.00000 q^{92} +3.17157 q^{97} +6.31371 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8	\( 2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 4 q^{7} - 2 q^{8} + 2 q^{10} - 4 q^{11} - 4 q^{14} + 2 q^{16} - 4 q^{17} - 2 q^{20} + 4 q^{22} - 2 q^{23} + 2 q^{25} + 4 q^{28} - 4 q^{29} - 8 q^{31} - 2 q^{32} + 4 q^{34} - 4 q^{35} - 8 q^{37} + 2 q^{40} - 4 q^{44} + 2 q^{46} + 10 q^{49} - 2 q^{50} - 4 q^{53} + 4 q^{55} - 4 q^{56} + 4 q^{58} - 20 q^{59} - 12 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} - 4 q^{68} + 4 q^{70} - 4 q^{73} + 8 q^{74} - 8 q^{77} - 2 q^{80} + 8 q^{83} + 4 q^{85} + 4 q^{88} - 16 q^{91} - 2 q^{92} + 12 q^{97} - 10 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 4 * q^7 - 2 * q^8 + 2 * q^10 - 4 * q^11 - 4 * q^14 + 2 * q^16 - 4 * q^17 - 2 * q^20 + 4 * q^22 - 2 * q^23 + 2 * q^25 + 4 * q^28 - 4 * q^29 - 8 * q^31 - 2 * q^32 + 4 * q^34 - 4 * q^35 - 8 * q^37 + 2 * q^40 - 4 * q^44 + 2 * q^46 + 10 * q^49 - 2 * q^50 - 4 * q^53 + 4 * q^55 - 4 * q^56 + 4 * q^58 - 20 * q^59 - 12 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 - 4 * q^68 + 4 * q^70 - 4 * q^73 + 8 * q^74 - 8 * q^77 - 2 * q^80 + 8 * q^83 + 4 * q^85 + 4 * q^88 - 16 * q^91 - 2 * q^92 + 12 * q^97 - 10 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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