LMFDB - Embedded newform 693.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [693,2,Mod(1,693)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(693, 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("693.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.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:	$5.53363286007$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +O(q^{10})$	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +1.61803 q^{10} -1.00000 q^{11} -5.47214 q^{13} -1.61803 q^{14} -4.85410 q^{16} -0.763932 q^{17} +6.70820 q^{19} -0.618034 q^{20} +1.61803 q^{22} +7.70820 q^{23} -4.00000 q^{25} +8.85410 q^{26} +0.618034 q^{28} -5.00000 q^{29} -0.763932 q^{31} +3.38197 q^{32} +1.23607 q^{34} -1.00000 q^{35} -7.00000 q^{37} -10.8541 q^{38} -2.23607 q^{40} -6.47214 q^{41} -7.70820 q^{43} -0.618034 q^{44} -12.4721 q^{46} +4.23607 q^{47} +1.00000 q^{49} +6.47214 q^{50} -3.38197 q^{52} -10.1803 q^{53} +1.00000 q^{55} +2.23607 q^{56} +8.09017 q^{58} -11.1803 q^{59} +2.00000 q^{61} +1.23607 q^{62} +4.23607 q^{64} +5.47214 q^{65} -14.2361 q^{67} -0.472136 q^{68} +1.61803 q^{70} -6.47214 q^{71} +13.4721 q^{73} +11.3262 q^{74} +4.14590 q^{76} -1.00000 q^{77} -5.52786 q^{79} +4.85410 q^{80} +10.4721 q^{82} -11.2361 q^{83} +0.763932 q^{85} +12.4721 q^{86} -2.23607 q^{88} -4.47214 q^{89} -5.47214 q^{91} +4.76393 q^{92} -6.85410 q^{94} -6.70820 q^{95} -3.70820 q^{97} -1.61803 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7	$ 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} + q^{10} - 2 q^{11} - 2 q^{13} - q^{14} - 3 q^{16} - 6 q^{17} + q^{20} + q^{22} + 2 q^{23} - 8 q^{25} + 11 q^{26} - q^{28} - 10 q^{29} - 6 q^{31} + 9 q^{32} - 2 q^{34} - 2 q^{35} - 14 q^{37} - 15 q^{38} - 4 q^{41} - 2 q^{43} + q^{44} - 16 q^{46} + 4 q^{47} + 2 q^{49} + 4 q^{50} - 9 q^{52} + 2 q^{53} + 2 q^{55} + 5 q^{58} + 4 q^{61} - 2 q^{62} + 4 q^{64} + 2 q^{65} - 24 q^{67} + 8 q^{68} + q^{70} - 4 q^{71} + 18 q^{73} + 7 q^{74} + 15 q^{76} - 2 q^{77} - 20 q^{79} + 3 q^{80} + 12 q^{82} - 18 q^{83} + 6 q^{85} + 16 q^{86} - 2 q^{91} + 14 q^{92} - 7 q^{94} + 6 q^{97} - q^{98}+O(q^{100}) $ 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7 + q^10 - 2 * q^11 - 2 * q^13 - q^14 - 3 * q^16 - 6 * q^17 + q^20 + q^22 + 2 * q^23 - 8 * q^25 + 11 * q^26 - q^28 - 10 * q^29 - 6 * q^31 + 9 * q^32 - 2 * q^34 - 2 * q^35 - 14 * q^37 - 15 * q^38 - 4 * q^41 - 2 * q^43 + q^44 - 16 * q^46 + 4 * q^47 + 2 * q^49 + 4 * q^50 - 9 * q^52 + 2 * q^53 + 2 * q^55 + 5 * q^58 + 4 * q^61 - 2 * q^62 + 4 * q^64 + 2 * q^65 - 24 * q^67 + 8 * q^68 + q^70 - 4 * q^71 + 18 * q^73 + 7 * q^74 + 15 * q^76 - 2 * q^77 - 20 * q^79 + 3 * q^80 + 12 * q^82 - 18 * q^83 + 6 * q^85 + 16 * q^86 - 2 * q^91 + 14 * q^92 - 7 * q^94 + 6 * q^97 - 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 $

$2$	−1.61803	−1.14412	−0.572061	−	0.820211i	$-0.693856\pi$
$2$	−1.61803	−1.14412	−0.572061	+	0.820211i	$0.693856\pi$
$3$	0	0
$3$	0	0
$4$	0.618034	0.309017
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.23607	0.790569
$9$	0	0
$10$	1.61803	0.511667
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−5.47214	−1.51770	−0.758849	−	0.651267i	$-0.774238\pi$
$13$	−5.47214	−1.51770	−0.758849	+	0.651267i	$0.774238\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	−4.85410	−1.21353
$17$	−0.763932	−0.185281	−0.0926404	−	0.995700i	$-0.529531\pi$
$17$	−0.763932	−0.185281	−0.0926404	+	0.995700i	$0.529531\pi$
$18$	0	0
$19$	6.70820	1.53897	0.769484	−	0.638666i	$-0.220514\pi$
$19$	6.70820	1.53897	0.769484	+	0.638666i	$0.220514\pi$
$20$	−0.618034	−0.138197
$21$	0	0
$22$	1.61803	0.344966
$23$	7.70820	1.60727	0.803636	−	0.595121i	$-0.202896\pi$
$23$	7.70820	1.60727	0.803636	+	0.595121i	$0.202896\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	8.85410	1.73643
$27$	0	0
$28$	0.618034	0.116797
$29$	−5.00000	−0.928477	−0.464238	−	0.885710i	$-0.653672\pi$
$29$	−5.00000	−0.928477	−0.464238	+	0.885710i	$0.653672\pi$
$30$	0	0
$31$	−0.763932	−0.137206	−0.0686031	−	0.997644i	$-0.521854\pi$
$31$	−0.763932	−0.137206	−0.0686031	+	0.997644i	$0.521854\pi$
$32$	3.38197	0.597853
$33$	0	0
$34$	1.23607	0.211984
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−7.00000	−1.15079	−0.575396	−	0.817875i	$-0.695152\pi$
$37$	−7.00000	−1.15079	−0.575396	+	0.817875i	$0.695152\pi$
$38$	−10.8541	−1.76077
$39$	0	0
$40$	−2.23607	−0.353553
$41$	−6.47214	−1.01078	−0.505389	−	0.862892i	$-0.668651\pi$
$41$	−6.47214	−1.01078	−0.505389	+	0.862892i	$0.668651\pi$
$42$	0	0
$43$	−7.70820	−1.17549	−0.587745	−	0.809046i	$-0.699984\pi$
$43$	−7.70820	−1.17549	−0.587745	+	0.809046i	$0.699984\pi$
$44$	−0.618034	−0.0931721
$45$	0	0
$46$	−12.4721	−1.83892
$47$	4.23607	0.617894	0.308947	−	0.951079i	$-0.400023\pi$
$47$	4.23607	0.617894	0.308947	+	0.951079i	$0.400023\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	6.47214	0.915298
$51$	0	0
$52$	−3.38197	−0.468994
$53$	−10.1803	−1.39838	−0.699189	−	0.714937i	$-0.746455\pi$
$53$	−10.1803	−1.39838	−0.699189	+	0.714937i	$0.746455\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	2.23607	0.298807
$57$	0	0
$58$	8.09017	1.06229
$59$	−11.1803	−1.45556	−0.727778	−	0.685813i	$-0.759447\pi$
$59$	−11.1803	−1.45556	−0.727778	+	0.685813i	$0.759447\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	1.23607	0.156981
$63$	0	0
$64$	4.23607	0.529508
$65$	5.47214	0.678735
$66$	0	0
$67$	−14.2361	−1.73921	−0.869606	−	0.493746i	$-0.835627\pi$
$67$	−14.2361	−1.73921	−0.869606	+	0.493746i	$0.835627\pi$
$68$	−0.472136	−0.0572549
$69$	0	0
$70$	1.61803	0.193392
$71$	−6.47214	−0.768101	−0.384051	−	0.923312i	$-0.625471\pi$
$71$	−6.47214	−0.768101	−0.384051	+	0.923312i	$0.625471\pi$
$72$	0	0
$73$	13.4721	1.57679	0.788397	−	0.615167i	$-0.210911\pi$
$73$	13.4721	1.57679	0.788397	+	0.615167i	$0.210911\pi$
$74$	11.3262	1.31665
$75$	0	0
$76$	4.14590	0.475567
$77$	−1.00000	−0.113961
$78$	0	0
$79$	−5.52786	−0.621933	−0.310967	−	0.950421i	$-0.600653\pi$
$79$	−5.52786	−0.621933	−0.310967	+	0.950421i	$0.600653\pi$
$80$	4.85410	0.542705
$81$	0	0
$82$	10.4721	1.15645
$83$	−11.2361	−1.23332	−0.616659	−	0.787230i	$-0.711514\pi$
$83$	−11.2361	−1.23332	−0.616659	+	0.787230i	$0.711514\pi$
$84$	0	0
$85$	0.763932	0.0828601
$86$	12.4721	1.34491
$87$	0	0
$88$	−2.23607	−0.238366
$89$	−4.47214	−0.474045	−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214	−0.474045	−0.237023	+	0.971504i	$0.576172\pi$
$90$	0	0
$91$	−5.47214	−0.573636
$92$	4.76393	0.496674
$93$	0	0
$94$	−6.85410	−0.706947
$95$	−6.70820	−0.688247
$96$	0	0
$97$	−3.70820	−0.376511	−0.188256	−	0.982120i	$-0.560283\pi$
$97$	−3.70820	−0.376511	−0.188256	+	0.982120i	$0.560283\pi$
$98$	−1.61803	−0.163446
$99$	0	0
$100$	−2.47214	−0.247214
$101$	4.18034	0.415959	0.207980	−	0.978133i	$-0.433311\pi$
$101$	4.18034	0.415959	0.207980	+	0.978133i	$0.433311\pi$
$102$	0	0
$103$	−9.41641	−0.927826	−0.463913	−	0.885881i	$-0.653555\pi$
$103$	−9.41641	−0.927826	−0.463913	+	0.885881i	$0.653555\pi$
$104$	−12.2361	−1.19985
$105$	0	0
$106$	16.4721	1.59992
$107$	−0.236068	−0.0228216	−0.0114108	−	0.999935i	$-0.503632\pi$
$107$	−0.236068	−0.0228216	−0.0114108	+	0.999935i	$0.503632\pi$
$108$	0	0
$109$	7.23607	0.693090	0.346545	−	0.938033i	$-0.387355\pi$
$109$	7.23607	0.693090	0.346545	+	0.938033i	$0.387355\pi$
$110$	−1.61803	−0.154273
$111$	0	0
$112$	−4.85410	−0.458670
$113$	−8.47214	−0.796992	−0.398496	−	0.917170i	$-0.630468\pi$
$113$	−8.47214	−0.796992	−0.398496	+	0.917170i	$0.630468\pi$
$114$	0	0
$115$	−7.70820	−0.718794
$116$	−3.09017	−0.286915
$117$	0	0
$118$	18.0902	1.66534
$119$	−0.763932	−0.0700295
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.23607	−0.292980
$123$	0	0
$124$	−0.472136	−0.0423991
$125$	9.00000	0.804984
$126$	0	0
$127$	18.6525	1.65514	0.827570	−	0.561363i	$-0.189723\pi$
$127$	18.6525	1.65514	0.827570	+	0.561363i	$0.189723\pi$
$128$	−13.6180	−1.20368
$129$	0	0
$130$	−8.85410	−0.776556
$131$	16.9443	1.48043	0.740214	−	0.672371i	$-0.234724\pi$
$131$	16.9443	1.48043	0.740214	+	0.672371i	$0.234724\pi$
$132$	0	0
$133$	6.70820	0.581675
$134$	23.0344	1.98987
$135$	0	0
$136$	−1.70820	−0.146477
$137$	−6.29180	−0.537544	−0.268772	−	0.963204i	$-0.586618\pi$
$137$	−6.29180	−0.537544	−0.268772	+	0.963204i	$0.586618\pi$
$138$	0	0
$139$	5.52786	0.468867	0.234434	−	0.972132i	$-0.424676\pi$
$139$	5.52786	0.468867	0.234434	+	0.972132i	$0.424676\pi$
$140$	−0.618034	−0.0522334
$141$	0	0
$142$	10.4721	0.878802
$143$	5.47214	0.457603
$144$	0	0
$145$	5.00000	0.415227
$146$	−21.7984	−1.80405
$147$	0	0
$148$	−4.32624	−0.355615
$149$	−5.00000	−0.409616	−0.204808	−	0.978802i	$-0.565657\pi$
$149$	−5.00000	−0.409616	−0.204808	+	0.978802i	$0.565657\pi$
$150$	0	0
$151$	8.18034	0.665707	0.332853	−	0.942979i	$-0.391989\pi$
$151$	8.18034	0.665707	0.332853	+	0.942979i	$0.391989\pi$
$152$	15.0000	1.21666
$153$	0	0
$154$	1.61803	0.130385
$155$	0.763932	0.0613605
$156$	0	0
$157$	11.4164	0.911129	0.455564	−	0.890203i	$-0.349438\pi$
$157$	11.4164	0.911129	0.455564	+	0.890203i	$0.349438\pi$
$158$	8.94427	0.711568
$159$	0	0
$160$	−3.38197	−0.267368
$161$	7.70820	0.607492
$162$	0	0
$163$	−9.29180	−0.727790	−0.363895	−	0.931440i	$-0.618553\pi$
$163$	−9.29180	−0.727790	−0.363895	+	0.931440i	$0.618553\pi$
$164$	−4.00000	−0.312348
$165$	0	0
$166$	18.1803	1.41107
$167$	−8.65248	−0.669549	−0.334774	−	0.942298i	$-0.608660\pi$
$167$	−8.65248	−0.669549	−0.334774	+	0.942298i	$0.608660\pi$
$168$	0	0
$169$	16.9443	1.30341
$170$	−1.23607	−0.0948021
$171$	0	0
$172$	−4.76393	−0.363246
$173$	10.4721	0.796182	0.398091	−	0.917346i	$-0.369673\pi$
$173$	10.4721	0.796182	0.398091	+	0.917346i	$0.369673\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	4.85410	0.365892
$177$	0	0
$178$	7.23607	0.542366
$179$	8.94427	0.668526	0.334263	−	0.942480i	$-0.391513\pi$
$179$	8.94427	0.668526	0.334263	+	0.942480i	$0.391513\pi$
$180$	0	0
$181$	−5.23607	−0.389194	−0.194597	−	0.980883i	$-0.562340\pi$
$181$	−5.23607	−0.389194	−0.194597	+	0.980883i	$0.562340\pi$
$182$	8.85410	0.656310
$183$	0	0
$184$	17.2361	1.27066
$185$	7.00000	0.514650
$186$	0	0
$187$	0.763932	0.0558642
$188$	2.61803	0.190940
$189$	0	0
$190$	10.8541	0.787439
$191$	15.2361	1.10244	0.551222	−	0.834359i	$-0.314162\pi$
$191$	15.2361	1.10244	0.551222	+	0.834359i	$0.314162\pi$
$192$	0	0
$193$	−16.6525	−1.19867	−0.599336	−	0.800498i	$-0.704569\pi$
$193$	−16.6525	−1.19867	−0.599336	+	0.800498i	$0.704569\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	0.618034	0.0441453
$197$	7.52786	0.536338	0.268169	−	0.963372i	$-0.413581\pi$
$197$	7.52786	0.536338	0.268169	+	0.963372i	$0.413581\pi$
$198$	0	0
$199$	−26.1803	−1.85588	−0.927938	−	0.372736i	$-0.878420\pi$
$199$	−26.1803	−1.85588	−0.927938	+	0.372736i	$0.878420\pi$
$200$	−8.94427	−0.632456
$201$	0	0
$202$	−6.76393	−0.475909
$203$	−5.00000	−0.350931
$204$	0	0
$205$	6.47214	0.452034
$206$	15.2361	1.06155
$207$	0	0
$208$	26.5623	1.84176
$209$	−6.70820	−0.464016
$210$	0	0
$211$	−21.4164	−1.47437	−0.737183	−	0.675693i	$-0.763845\pi$
$211$	−21.4164	−1.47437	−0.737183	+	0.675693i	$0.763845\pi$
$212$	−6.29180	−0.432122
$213$	0	0
$214$	0.381966	0.0261107
$215$	7.70820	0.525695
$216$	0	0
$217$	−0.763932	−0.0518591
$218$	−11.7082	−0.792980
$219$	0	0
$220$	0.618034	0.0416678
$221$	4.18034	0.281200
$222$	0	0
$223$	−6.00000	−0.401790	−0.200895	−	0.979613i	$-0.564385\pi$
$223$	−6.00000	−0.401790	−0.200895	+	0.979613i	$0.564385\pi$
$224$	3.38197	0.225967
$225$	0	0
$226$	13.7082	0.911856
$227$	2.00000	0.132745	0.0663723	−	0.997795i	$-0.478857\pi$
$227$	2.00000	0.132745	0.0663723	+	0.997795i	$0.478857\pi$
$228$	0	0
$229$	2.76393	0.182646	0.0913229	−	0.995821i	$-0.470890\pi$
$229$	2.76393	0.182646	0.0913229	+	0.995821i	$0.470890\pi$
$230$	12.4721	0.822388
$231$	0	0
$232$	−11.1803	−0.734025
$233$	−2.94427	−0.192886	−0.0964428	−	0.995339i	$-0.530746\pi$
$233$	−2.94427	−0.192886	−0.0964428	+	0.995339i	$0.530746\pi$
$234$	0	0
$235$	−4.23607	−0.276331
$236$	−6.90983	−0.449792
$237$	0	0
$238$	1.23607	0.0801224
$239$	30.1246	1.94860	0.974300	−	0.225256i	$-0.0723219\pi$
$239$	30.1246	1.94860	0.974300	+	0.225256i	$0.0723219\pi$
$240$	0	0
$241$	8.05573	0.518915	0.259458	−	0.965755i	$-0.416456\pi$
$241$	8.05573	0.518915	0.259458	+	0.965755i	$0.416456\pi$
$242$	−1.61803	−0.104011
$243$	0	0
$244$	1.23607	0.0791311
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−36.7082	−2.33569
$248$	−1.70820	−0.108471
$249$	0	0
$250$	−14.5623	−0.921001
$251$	28.1246	1.77521	0.887605	−	0.460606i	$-0.152368\pi$
$251$	28.1246	1.77521	0.887605	+	0.460606i	$0.152368\pi$
$252$	0	0
$253$	−7.70820	−0.484611
$254$	−30.1803	−1.89368
$255$	0	0
$256$	13.5623	0.847644
$257$	7.00000	0.436648	0.218324	−	0.975876i	$-0.429941\pi$
$257$	7.00000	0.436648	0.218324	+	0.975876i	$0.429941\pi$
$258$	0	0
$259$	−7.00000	−0.434959
$260$	3.38197	0.209741
$261$	0	0
$262$	−27.4164	−1.69379
$263$	−14.1246	−0.870961	−0.435480	−	0.900198i	$-0.643421\pi$
$263$	−14.1246	−0.870961	−0.435480	+	0.900198i	$0.643421\pi$
$264$	0	0
$265$	10.1803	0.625373
$266$	−10.8541	−0.665508
$267$	0	0
$268$	−8.79837	−0.537446
$269$	−18.9443	−1.15505	−0.577526	−	0.816372i	$-0.695982\pi$
$269$	−18.9443	−1.15505	−0.577526	+	0.816372i	$0.695982\pi$
$270$	0	0
$271$	18.7082	1.13644	0.568221	−	0.822876i	$-0.307632\pi$
$271$	18.7082	1.13644	0.568221	+	0.822876i	$0.307632\pi$
$272$	3.70820	0.224843
$273$	0	0
$274$	10.1803	0.615017
$275$	4.00000	0.241209
$276$	0	0
$277$	2.47214	0.148536	0.0742681	−	0.997238i	$-0.476338\pi$
$277$	2.47214	0.148536	0.0742681	+	0.997238i	$0.476338\pi$
$278$	−8.94427	−0.536442
$279$	0	0
$280$	−2.23607	−0.133631
$281$	−2.52786	−0.150800	−0.0753999	−	0.997153i	$-0.524023\pi$
$281$	−2.52786	−0.150800	−0.0753999	+	0.997153i	$0.524023\pi$
$282$	0	0
$283$	−18.2361	−1.08402	−0.542011	−	0.840371i	$-0.682337\pi$
$283$	−18.2361	−1.08402	−0.542011	+	0.840371i	$0.682337\pi$
$284$	−4.00000	−0.237356
$285$	0	0
$286$	−8.85410	−0.523554
$287$	−6.47214	−0.382038
$288$	0	0
$289$	−16.4164	−0.965671
$290$	−8.09017	−0.475071
$291$	0	0
$292$	8.32624	0.487256
$293$	16.0000	0.934730	0.467365	−	0.884064i	$-0.345203\pi$
$293$	16.0000	0.934730	0.467365	+	0.884064i	$0.345203\pi$
$294$	0	0
$295$	11.1803	0.650945
$296$	−15.6525	−0.909782
$297$	0	0
$298$	8.09017	0.468651
$299$	−42.1803	−2.43935
$300$	0	0
$301$	−7.70820	−0.444293
$302$	−13.2361	−0.761650
$303$	0	0
$304$	−32.5623	−1.86758
$305$	−2.00000	−0.114520
$306$	0	0
$307$	−3.05573	−0.174400	−0.0871998	−	0.996191i	$-0.527792\pi$
$307$	−3.05573	−0.174400	−0.0871998	+	0.996191i	$0.527792\pi$
$308$	−0.618034	−0.0352158
$309$	0	0
$310$	−1.23607	−0.0702039
$311$	25.8885	1.46800	0.734002	−	0.679147i	$-0.237650\pi$
$311$	25.8885	1.46800	0.734002	+	0.679147i	$0.237650\pi$
$312$	0	0
$313$	−6.65248	−0.376020	−0.188010	−	0.982167i	$-0.560204\pi$
$313$	−6.65248	−0.376020	−0.188010	+	0.982167i	$0.560204\pi$
$314$	−18.4721	−1.04244
$315$	0	0
$316$	−3.41641	−0.192188
$317$	−1.81966	−0.102202	−0.0511011	−	0.998693i	$-0.516273\pi$
$317$	−1.81966	−0.102202	−0.0511011	+	0.998693i	$0.516273\pi$
$318$	0	0
$319$	5.00000	0.279946
$320$	−4.23607	−0.236803
$321$	0	0
$322$	−12.4721	−0.695045
$323$	−5.12461	−0.285141
$324$	0	0
$325$	21.8885	1.21416
$326$	15.0344	0.832681
$327$	0	0
$328$	−14.4721	−0.799090
$329$	4.23607	0.233542
$330$	0	0
$331$	15.4164	0.847362	0.423681	−	0.905811i	$-0.360738\pi$
$331$	15.4164	0.847362	0.423681	+	0.905811i	$0.360738\pi$
$332$	−6.94427	−0.381116
$333$	0	0
$334$	14.0000	0.766046
$335$	14.2361	0.777799
$336$	0	0
$337$	14.1803	0.772452	0.386226	−	0.922404i	$-0.373778\pi$
$337$	14.1803	0.772452	0.386226	+	0.922404i	$0.373778\pi$
$338$	−27.4164	−1.49126
$339$	0	0
$340$	0.472136	0.0256052
$341$	0.763932	0.0413692
$342$	0	0
$343$	1.00000	0.0539949
$344$	−17.2361	−0.929307
$345$	0	0
$346$	−16.9443	−0.910930
$347$	−28.0000	−1.50312	−0.751559	−	0.659665i	$-0.770698\pi$
$347$	−28.0000	−1.50312	−0.751559	+	0.659665i	$0.770698\pi$
$348$	0	0
$349$	28.4164	1.52110	0.760548	−	0.649282i	$-0.224931\pi$
$349$	28.4164	1.52110	0.760548	+	0.649282i	$0.224931\pi$
$350$	6.47214	0.345950
$351$	0	0
$352$	−3.38197	−0.180259
$353$	−33.4721	−1.78154	−0.890771	−	0.454452i	$-0.849835\pi$
$353$	−33.4721	−1.78154	−0.890771	+	0.454452i	$0.849835\pi$
$354$	0	0
$355$	6.47214	0.343505
$356$	−2.76393	−0.146488
$357$	0	0
$358$	−14.4721	−0.764876
$359$	3.41641	0.180311	0.0901556	−	0.995928i	$-0.471264\pi$
$359$	3.41641	0.180311	0.0901556	+	0.995928i	$0.471264\pi$
$360$	0	0
$361$	26.0000	1.36842
$362$	8.47214	0.445286
$363$	0	0
$364$	−3.38197	−0.177263
$365$	−13.4721	−0.705164
$366$	0	0
$367$	15.8885	0.829375	0.414688	−	0.909964i	$-0.363891\pi$
$367$	15.8885	0.829375	0.414688	+	0.909964i	$0.363891\pi$
$368$	−37.4164	−1.95047
$369$	0	0
$370$	−11.3262	−0.588823
$371$	−10.1803	−0.528537
$372$	0	0
$373$	−26.6525	−1.38001	−0.690006	−	0.723803i	$-0.742392\pi$
$373$	−26.6525	−1.38001	−0.690006	+	0.723803i	$0.742392\pi$
$374$	−1.23607	−0.0639156
$375$	0	0
$376$	9.47214	0.488488
$377$	27.3607	1.40915
$378$	0	0
$379$	−8.81966	−0.453036	−0.226518	−	0.974007i	$-0.572734\pi$
$379$	−8.81966	−0.453036	−0.226518	+	0.974007i	$0.572734\pi$
$380$	−4.14590	−0.212680
$381$	0	0
$382$	−24.6525	−1.26133
$383$	−15.0557	−0.769312	−0.384656	−	0.923060i	$-0.625680\pi$
$383$	−15.0557	−0.769312	−0.384656	+	0.923060i	$0.625680\pi$
$384$	0	0
$385$	1.00000	0.0509647
$386$	26.9443	1.37143
$387$	0	0
$388$	−2.29180	−0.116348
$389$	−28.9443	−1.46753	−0.733766	−	0.679402i	$-0.762239\pi$
$389$	−28.9443	−1.46753	−0.733766	+	0.679402i	$0.762239\pi$
$390$	0	0
$391$	−5.88854	−0.297796
$392$	2.23607	0.112938
$393$	0	0
$394$	−12.1803	−0.613637
$395$	5.52786	0.278137
$396$	0	0
$397$	−17.1246	−0.859460	−0.429730	−	0.902958i	$-0.641391\pi$
$397$	−17.1246	−0.859460	−0.429730	+	0.902958i	$0.641391\pi$
$398$	42.3607	2.12335
$399$	0	0
$400$	19.4164	0.970820
$401$	16.2918	0.813573	0.406787	−	0.913523i	$-0.366649\pi$
$401$	16.2918	0.813573	0.406787	+	0.913523i	$0.366649\pi$
$402$	0	0
$403$	4.18034	0.208238
$404$	2.58359	0.128539
$405$	0	0
$406$	8.09017	0.401508
$407$	7.00000	0.346977
$408$	0	0
$409$	38.9443	1.92567	0.962835	−	0.270090i	$-0.0870534\pi$
$409$	38.9443	1.92567	0.962835	+	0.270090i	$0.0870534\pi$
$410$	−10.4721	−0.517182
$411$	0	0
$412$	−5.81966	−0.286714
$413$	−11.1803	−0.550149
$414$	0	0
$415$	11.2361	0.551557
$416$	−18.5066	−0.907360
$417$	0	0
$418$	10.8541	0.530891
$419$	−21.1803	−1.03473	−0.517364	−	0.855766i	$-0.673087\pi$
$419$	−21.1803	−1.03473	−0.517364	+	0.855766i	$0.673087\pi$
$420$	0	0
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	34.6525	1.68686
$423$	0	0
$424$	−22.7639	−1.10551
$425$	3.05573	0.148225
$426$	0	0
$427$	2.00000	0.0967868
$428$	−0.145898	−0.00705225
$429$	0	0
$430$	−12.4721	−0.601460
$431$	4.70820	0.226786	0.113393	−	0.993550i	$-0.463828\pi$
$431$	4.70820	0.226786	0.113393	+	0.993550i	$0.463828\pi$
$432$	0	0
$433$	−1.52786	−0.0734245	−0.0367122	−	0.999326i	$-0.511688\pi$
$433$	−1.52786	−0.0734245	−0.0367122	+	0.999326i	$0.511688\pi$
$434$	1.23607	0.0593332
$435$	0	0
$436$	4.47214	0.214176
$437$	51.7082	2.47354
$438$	0	0
$439$	−11.1803	−0.533609	−0.266804	−	0.963751i	$-0.585968\pi$
$439$	−11.1803	−0.533609	−0.266804	+	0.963751i	$0.585968\pi$
$440$	2.23607	0.106600
$441$	0	0
$442$	−6.76393	−0.321727
$443$	−9.52786	−0.452682	−0.226341	−	0.974048i	$-0.572676\pi$
$443$	−9.52786	−0.452682	−0.226341	+	0.974048i	$0.572676\pi$
$444$	0	0
$445$	4.47214	0.212000
$446$	9.70820	0.459697
$447$	0	0
$448$	4.23607	0.200135
$449$	−20.0000	−0.943858	−0.471929	−	0.881636i	$-0.656442\pi$
$449$	−20.0000	−0.943858	−0.471929	+	0.881636i	$0.656442\pi$
$450$	0	0
$451$	6.47214	0.304761
$452$	−5.23607	−0.246284
$453$	0	0
$454$	−3.23607	−0.151876
$455$	5.47214	0.256538
$456$	0	0
$457$	10.7639	0.503516	0.251758	−	0.967790i	$-0.418991\pi$
$457$	10.7639	0.503516	0.251758	+	0.967790i	$0.418991\pi$
$458$	−4.47214	−0.208969
$459$	0	0
$460$	−4.76393	−0.222119
$461$	28.0000	1.30409	0.652045	−	0.758180i	$-0.273911\pi$
$461$	28.0000	1.30409	0.652045	+	0.758180i	$0.273911\pi$
$462$	0	0
$463$	−27.1803	−1.26318	−0.631589	−	0.775304i	$-0.717597\pi$
$463$	−27.1803	−1.26318	−0.631589	+	0.775304i	$0.717597\pi$
$464$	24.2705	1.12673
$465$	0	0
$466$	4.76393	0.220685
$467$	−6.81966	−0.315576	−0.157788	−	0.987473i	$-0.550436\pi$
$467$	−6.81966	−0.315576	−0.157788	+	0.987473i	$0.550436\pi$
$468$	0	0
$469$	−14.2361	−0.657361
$470$	6.85410	0.316156
$471$	0	0
$472$	−25.0000	−1.15072
$473$	7.70820	0.354424
$474$	0	0
$475$	−26.8328	−1.23117
$476$	−0.472136	−0.0216403
$477$	0	0
$478$	−48.7426	−2.22944
$479$	1.70820	0.0780498	0.0390249	−	0.999238i	$-0.487575\pi$
$479$	1.70820	0.0780498	0.0390249	+	0.999238i	$0.487575\pi$
$480$	0	0
$481$	38.3050	1.74656
$482$	−13.0344	−0.593703
$483$	0	0
$484$	0.618034	0.0280925
$485$	3.70820	0.168381
$486$	0	0
$487$	−0.944272	−0.0427890	−0.0213945	−	0.999771i	$-0.506811\pi$
$487$	−0.944272	−0.0427890	−0.0213945	+	0.999771i	$0.506811\pi$
$488$	4.47214	0.202444
$489$	0	0
$490$	1.61803	0.0730953
$491$	−22.1246	−0.998470	−0.499235	−	0.866467i	$-0.666386\pi$
$491$	−22.1246	−0.998470	−0.499235	+	0.866467i	$0.666386\pi$
$492$	0	0
$493$	3.81966	0.172029
$494$	59.3951	2.67231
$495$	0	0
$496$	3.70820	0.166503
$497$	−6.47214	−0.290315
$498$	0	0
$499$	2.23607	0.100100	0.0500501	−	0.998747i	$-0.484062\pi$
$499$	2.23607	0.100100	0.0500501	+	0.998747i	$0.484062\pi$
$500$	5.56231	0.248754
$501$	0	0
$502$	−45.5066	−2.03106
$503$	−15.7082	−0.700394	−0.350197	−	0.936676i	$-0.613885\pi$
$503$	−15.7082	−0.700394	−0.350197	+	0.936676i	$0.613885\pi$
$504$	0	0
$505$	−4.18034	−0.186023
$506$	12.4721	0.554454
$507$	0	0
$508$	11.5279	0.511466
$509$	30.0000	1.32973	0.664863	−	0.746965i	$-0.268490\pi$
$509$	30.0000	1.32973	0.664863	+	0.746965i	$0.268490\pi$
$510$	0	0
$511$	13.4721	0.595972
$512$	5.29180	0.233867
$513$	0	0
$514$	−11.3262	−0.499579
$515$	9.41641	0.414937
$516$	0	0
$517$	−4.23607	−0.186302
$518$	11.3262	0.497646
$519$	0	0
$520$	12.2361	0.536587
$521$	24.3050	1.06482	0.532410	−	0.846487i	$-0.321287\pi$
$521$	24.3050	1.06482	0.532410	+	0.846487i	$0.321287\pi$
$522$	0	0
$523$	9.65248	0.422073	0.211037	−	0.977478i	$-0.432316\pi$
$523$	9.65248	0.422073	0.211037	+	0.977478i	$0.432316\pi$
$524$	10.4721	0.457477
$525$	0	0
$526$	22.8541	0.996486
$527$	0.583592	0.0254217
$528$	0	0
$529$	36.4164	1.58332
$530$	−16.4721	−0.715504
$531$	0	0
$532$	4.14590	0.179747
$533$	35.4164	1.53405
$534$	0	0
$535$	0.236068	0.0102061
$536$	−31.8328	−1.37497
$537$	0	0
$538$	30.6525	1.32152
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−19.0557	−0.819270	−0.409635	−	0.912250i	$-0.634344\pi$
$541$	−19.0557	−0.819270	−0.409635	+	0.912250i	$0.634344\pi$
$542$	−30.2705	−1.30023
$543$	0	0
$544$	−2.58359	−0.110771
$545$	−7.23607	−0.309959
$546$	0	0
$547$	−38.8328	−1.66037	−0.830186	−	0.557487i	$-0.811766\pi$
$547$	−38.8328	−1.66037	−0.830186	+	0.557487i	$0.811766\pi$
$548$	−3.88854	−0.166110
$549$	0	0
$550$	−6.47214	−0.275973
$551$	−33.5410	−1.42890
$552$	0	0
$553$	−5.52786	−0.235069
$554$	−4.00000	−0.169944
$555$	0	0
$556$	3.41641	0.144888
$557$	21.4721	0.909804	0.454902	−	0.890542i	$-0.349674\pi$
$557$	21.4721	0.909804	0.454902	+	0.890542i	$0.349674\pi$
$558$	0	0
$559$	42.1803	1.78404
$560$	4.85410	0.205123
$561$	0	0
$562$	4.09017	0.172533
$563$	−3.34752	−0.141081	−0.0705407	−	0.997509i	$-0.522472\pi$
$563$	−3.34752	−0.141081	−0.0705407	+	0.997509i	$0.522472\pi$
$564$	0	0
$565$	8.47214	0.356425
$566$	29.5066	1.24025
$567$	0	0
$568$	−14.4721	−0.607237
$569$	−30.0000	−1.25767	−0.628833	−	0.777541i	$-0.716467\pi$
$569$	−30.0000	−1.25767	−0.628833	+	0.777541i	$0.716467\pi$
$570$	0	0
$571$	26.4721	1.10782	0.553912	−	0.832575i	$-0.313134\pi$
$571$	26.4721	1.10782	0.553912	+	0.832575i	$0.313134\pi$
$572$	3.38197	0.141407
$573$	0	0
$574$	10.4721	0.437099
$575$	−30.8328	−1.28582
$576$	0	0
$577$	38.6525	1.60912	0.804562	−	0.593869i	$-0.202400\pi$
$577$	38.6525	1.60912	0.804562	+	0.593869i	$0.202400\pi$
$578$	26.5623	1.10485
$579$	0	0
$580$	3.09017	0.128312
$581$	−11.2361	−0.466151
$582$	0	0
$583$	10.1803	0.421627
$584$	30.1246	1.24657
$585$	0	0
$586$	−25.8885	−1.06945
$587$	30.0132	1.23878	0.619388	−	0.785085i	$-0.287381\pi$
$587$	30.0132	1.23878	0.619388	+	0.785085i	$0.287381\pi$
$588$	0	0
$589$	−5.12461	−0.211156
$590$	−18.0902	−0.744761
$591$	0	0
$592$	33.9787	1.39652
$593$	−30.8328	−1.26615	−0.633076	−	0.774090i	$-0.718208\pi$
$593$	−30.8328	−1.26615	−0.633076	+	0.774090i	$0.718208\pi$
$594$	0	0
$595$	0.763932	0.0313182
$596$	−3.09017	−0.126578
$597$	0	0
$598$	68.2492	2.79092
$599$	34.4721	1.40849	0.704247	−	0.709955i	$-0.251285\pi$
$599$	34.4721	1.40849	0.704247	+	0.709955i	$0.251285\pi$
$600$	0	0
$601$	27.0000	1.10135	0.550676	−	0.834719i	$-0.314370\pi$
$601$	27.0000	1.10135	0.550676	+	0.834719i	$0.314370\pi$
$602$	12.4721	0.508326
$603$	0	0
$604$	5.05573	0.205715
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	29.1803	1.18439	0.592197	−	0.805793i	$-0.298261\pi$
$607$	29.1803	1.18439	0.592197	+	0.805793i	$0.298261\pi$
$608$	22.6869	0.920076
$609$	0	0
$610$	3.23607	0.131025
$611$	−23.1803	−0.937776
$612$	0	0
$613$	−35.5967	−1.43774	−0.718870	−	0.695145i	$-0.755340\pi$
$613$	−35.5967	−1.43774	−0.718870	+	0.695145i	$0.755340\pi$
$614$	4.94427	0.199535
$615$	0	0
$616$	−2.23607	−0.0900937
$617$	−23.5279	−0.947196	−0.473598	−	0.880741i	$-0.657045\pi$
$617$	−23.5279	−0.947196	−0.473598	+	0.880741i	$0.657045\pi$
$618$	0	0
$619$	14.0689	0.565476	0.282738	−	0.959197i	$-0.408757\pi$
$619$	14.0689	0.565476	0.282738	+	0.959197i	$0.408757\pi$
$620$	0.472136	0.0189614
$621$	0	0
$622$	−41.8885	−1.67958
$623$	−4.47214	−0.179172
$624$	0	0
$625$	11.0000	0.440000
$626$	10.7639	0.430213
$627$	0	0
$628$	7.05573	0.281554
$629$	5.34752	0.213220
$630$	0	0
$631$	−0.360680	−0.0143584	−0.00717922	−	0.999974i	$-0.502285\pi$
$631$	−0.360680	−0.0143584	−0.00717922	+	0.999974i	$0.502285\pi$
$632$	−12.3607	−0.491681
$633$	0	0
$634$	2.94427	0.116932
$635$	−18.6525	−0.740201
$636$	0	0
$637$	−5.47214	−0.216814
$638$	−8.09017	−0.320293
$639$	0	0
$640$	13.6180	0.538300
$641$	−20.5410	−0.811321	−0.405661	−	0.914024i	$-0.632959\pi$
$641$	−20.5410	−0.811321	−0.405661	+	0.914024i	$0.632959\pi$
$642$	0	0
$643$	45.9574	1.81238	0.906192	−	0.422866i	$-0.138976\pi$
$643$	45.9574	1.81238	0.906192	+	0.422866i	$0.138976\pi$
$644$	4.76393	0.187725
$645$	0	0
$646$	8.29180	0.326236
$647$	−43.6525	−1.71616	−0.858078	−	0.513519i	$-0.828341\pi$
$647$	−43.6525	−1.71616	−0.858078	+	0.513519i	$0.828341\pi$
$648$	0	0
$649$	11.1803	0.438867
$650$	−35.4164	−1.38915
$651$	0	0
$652$	−5.74265	−0.224899
$653$	27.0557	1.05877	0.529386	−	0.848381i	$-0.322422\pi$
$653$	27.0557	1.05877	0.529386	+	0.848381i	$0.322422\pi$
$654$	0	0
$655$	−16.9443	−0.662067
$656$	31.4164	1.22660
$657$	0	0
$658$	−6.85410	−0.267201
$659$	43.5410	1.69612	0.848059	−	0.529902i	$-0.177771\pi$
$659$	43.5410	1.69612	0.848059	+	0.529902i	$0.177771\pi$
$660$	0	0
$661$	−26.5410	−1.03233	−0.516163	−	0.856490i	$-0.672640\pi$
$661$	−26.5410	−1.03233	−0.516163	+	0.856490i	$0.672640\pi$
$662$	−24.9443	−0.969487
$663$	0	0
$664$	−25.1246	−0.975024
$665$	−6.70820	−0.260133
$666$	0	0
$667$	−38.5410	−1.49231
$668$	−5.34752	−0.206902
$669$	0	0
$670$	−23.0344	−0.889898
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	−45.5967	−1.75763	−0.878813	−	0.477167i	$-0.841664\pi$
$673$	−45.5967	−1.75763	−0.878813	+	0.477167i	$0.841664\pi$
$674$	−22.9443	−0.883780
$675$	0	0
$676$	10.4721	0.402774
$677$	43.3050	1.66434	0.832172	−	0.554517i	$-0.187097\pi$
$677$	43.3050	1.66434	0.832172	+	0.554517i	$0.187097\pi$
$678$	0	0
$679$	−3.70820	−0.142308
$680$	1.70820	0.0655066
$681$	0	0
$682$	−1.23607	−0.0473315
$683$	39.0132	1.49280	0.746398	−	0.665499i	$-0.231781\pi$
$683$	39.0132	1.49280	0.746398	+	0.665499i	$0.231781\pi$
$684$	0	0
$685$	6.29180	0.240397
$686$	−1.61803	−0.0617768
$687$	0	0
$688$	37.4164	1.42649
$689$	55.7082	2.12231
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	6.47214	0.246034
$693$	0	0
$694$	45.3050	1.71975
$695$	−5.52786	−0.209684
$696$	0	0
$697$	4.94427	0.187278
$698$	−45.9787	−1.74032
$699$	0	0
$700$	−2.47214	−0.0934380
$701$	−39.8885	−1.50657	−0.753285	−	0.657695i	$-0.771532\pi$
$701$	−39.8885	−1.50657	−0.753285	+	0.657695i	$0.771532\pi$
$702$	0	0
$703$	−46.9574	−1.77103
$704$	−4.23607	−0.159653
$705$	0	0
$706$	54.1591	2.03830
$707$	4.18034	0.157218
$708$	0	0
$709$	39.7214	1.49177	0.745883	−	0.666076i	$-0.232028\pi$
$709$	39.7214	1.49177	0.745883	+	0.666076i	$0.232028\pi$
$710$	−10.4721	−0.393012
$711$	0	0
$712$	−10.0000	−0.374766
$713$	−5.88854	−0.220528
$714$	0	0
$715$	−5.47214	−0.204646
$716$	5.52786	0.206586
$717$	0	0
$718$	−5.52786	−0.206298
$719$	−12.2361	−0.456328	−0.228164	−	0.973623i	$-0.573272\pi$
$719$	−12.2361	−0.456328	−0.228164	+	0.973623i	$0.573272\pi$
$720$	0	0
$721$	−9.41641	−0.350685
$722$	−42.0689	−1.56564
$723$	0	0
$724$	−3.23607	−0.120268
$725$	20.0000	0.742781
$726$	0	0
$727$	1.81966	0.0674875	0.0337437	−	0.999431i	$-0.489257\pi$
$727$	1.81966	0.0674875	0.0337437	+	0.999431i	$0.489257\pi$
$728$	−12.2361	−0.453499
$729$	0	0
$730$	21.7984	0.806794
$731$	5.88854	0.217796
$732$	0	0
$733$	−43.8885	−1.62106	−0.810530	−	0.585697i	$-0.800821\pi$
$733$	−43.8885	−1.62106	−0.810530	+	0.585697i	$0.800821\pi$
$734$	−25.7082	−0.948907
$735$	0	0
$736$	26.0689	0.960912
$737$	14.2361	0.524392
$738$	0	0
$739$	−24.0689	−0.885388	−0.442694	−	0.896673i	$-0.645977\pi$
$739$	−24.0689	−0.885388	−0.442694	+	0.896673i	$0.645977\pi$
$740$	4.32624	0.159036
$741$	0	0
$742$	16.4721	0.604711
$743$	−25.1803	−0.923777	−0.461889	−	0.886938i	$-0.652828\pi$
$743$	−25.1803	−0.923777	−0.461889	+	0.886938i	$0.652828\pi$
$744$	0	0
$745$	5.00000	0.183186
$746$	43.1246	1.57890
$747$	0	0
$748$	0.472136	0.0172630
$749$	−0.236068	−0.00862574
$750$	0	0
$751$	44.2361	1.61420	0.807099	−	0.590417i	$-0.201037\pi$
$751$	44.2361	1.61420	0.807099	+	0.590417i	$0.201037\pi$
$752$	−20.5623	−0.749830
$753$	0	0
$754$	−44.2705	−1.61224
$755$	−8.18034	−0.297713
$756$	0	0
$757$	37.7214	1.37101	0.685503	−	0.728070i	$-0.259582\pi$
$757$	37.7214	1.37101	0.685503	+	0.728070i	$0.259582\pi$
$758$	14.2705	0.518328
$759$	0	0
$760$	−15.0000	−0.544107
$761$	43.7771	1.58692	0.793459	−	0.608624i	$-0.208278\pi$
$761$	43.7771	1.58692	0.793459	+	0.608624i	$0.208278\pi$
$762$	0	0
$763$	7.23607	0.261963
$764$	9.41641	0.340674
$765$	0	0
$766$	24.3607	0.880187
$767$	61.1803	2.20909
$768$	0	0
$769$	−3.94427	−0.142234	−0.0711170	−	0.997468i	$-0.522656\pi$
$769$	−3.94427	−0.142234	−0.0711170	+	0.997468i	$0.522656\pi$
$770$	−1.61803	−0.0583099
$771$	0	0
$772$	−10.2918	−0.370410
$773$	−3.47214	−0.124884	−0.0624420	−	0.998049i	$-0.519889\pi$
$773$	−3.47214	−0.124884	−0.0624420	+	0.998049i	$0.519889\pi$
$774$	0	0
$775$	3.05573	0.109765
$776$	−8.29180	−0.297658
$777$	0	0
$778$	46.8328	1.67904
$779$	−43.4164	−1.55555
$780$	0	0
$781$	6.47214	0.231591
$782$	9.52786	0.340716
$783$	0	0
$784$	−4.85410	−0.173361
$785$	−11.4164	−0.407469
$786$	0	0
$787$	−27.6525	−0.985704	−0.492852	−	0.870113i	$-0.664046\pi$
$787$	−27.6525	−0.985704	−0.492852	+	0.870113i	$0.664046\pi$
$788$	4.65248	0.165738
$789$	0	0
$790$	−8.94427	−0.318223
$791$	−8.47214	−0.301234
$792$	0	0
$793$	−10.9443	−0.388642
$794$	27.7082	0.983327
$795$	0	0
$796$	−16.1803	−0.573497
$797$	11.4721	0.406364	0.203182	−	0.979141i	$-0.434872\pi$
$797$	11.4721	0.406364	0.203182	+	0.979141i	$0.434872\pi$
$798$	0	0
$799$	−3.23607	−0.114484
$800$	−13.5279	−0.478282
$801$	0	0
$802$	−26.3607	−0.930828
$803$	−13.4721	−0.475421
$804$	0	0
$805$	−7.70820	−0.271678
$806$	−6.76393	−0.238249
$807$	0	0
$808$	9.34752	0.328845
$809$	−6.30495	−0.221670	−0.110835	−	0.993839i	$-0.535353\pi$
$809$	−6.30495	−0.221670	−0.110835	+	0.993839i	$0.535353\pi$
$810$	0	0
$811$	28.7082	1.00808	0.504041	−	0.863680i	$-0.331846\pi$
$811$	28.7082	1.00808	0.504041	+	0.863680i	$0.331846\pi$
$812$	−3.09017	−0.108444
$813$	0	0
$814$	−11.3262	−0.396984
$815$	9.29180	0.325477
$816$	0	0
$817$	−51.7082	−1.80904
$818$	−63.0132	−2.20320
$819$	0	0
$820$	4.00000	0.139686
$821$	−1.47214	−0.0513779	−0.0256889	−	0.999670i	$-0.508178\pi$
$821$	−1.47214	−0.0513779	−0.0256889	+	0.999670i	$0.508178\pi$
$822$	0	0
$823$	5.18034	0.180575	0.0902876	−	0.995916i	$-0.471221\pi$
$823$	5.18034	0.180575	0.0902876	+	0.995916i	$0.471221\pi$
$824$	−21.0557	−0.733511
$825$	0	0
$826$	18.0902	0.629438
$827$	−43.6525	−1.51795	−0.758973	−	0.651122i	$-0.774298\pi$
$827$	−43.6525	−1.51795	−0.758973	+	0.651122i	$0.774298\pi$
$828$	0	0
$829$	−35.7771	−1.24259	−0.621295	−	0.783577i	$-0.713393\pi$
$829$	−35.7771	−1.24259	−0.621295	+	0.783577i	$0.713393\pi$
$830$	−18.1803	−0.631049
$831$	0	0
$832$	−23.1803	−0.803634
$833$	−0.763932	−0.0264687
$834$	0	0
$835$	8.65248	0.299431
$836$	−4.14590	−0.143389
$837$	0	0
$838$	34.2705	1.18386
$839$	−43.5410	−1.50320	−0.751601	−	0.659617i	$-0.770718\pi$
$839$	−43.5410	−1.50320	−0.751601	+	0.659617i	$0.770718\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	21.0344	0.724895
$843$	0	0
$844$	−13.2361	−0.455604
$845$	−16.9443	−0.582901
$846$	0	0
$847$	1.00000	0.0343604
$848$	49.4164	1.69697
$849$	0	0
$850$	−4.94427	−0.169587
$851$	−53.9574	−1.84964
$852$	0	0
$853$	−2.58359	−0.0884605	−0.0442303	−	0.999021i	$-0.514084\pi$
$853$	−2.58359	−0.0884605	−0.0442303	+	0.999021i	$0.514084\pi$
$854$	−3.23607	−0.110736
$855$	0	0
$856$	−0.527864	−0.0180420
$857$	−35.8885	−1.22593	−0.612965	−	0.790110i	$-0.710023\pi$
$857$	−35.8885	−1.22593	−0.612965	+	0.790110i	$0.710023\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	4.76393	0.162449
$861$	0	0
$862$	−7.61803	−0.259471
$863$	38.7639	1.31954	0.659770	−	0.751468i	$-0.270654\pi$
$863$	38.7639	1.31954	0.659770	+	0.751468i	$0.270654\pi$
$864$	0	0
$865$	−10.4721	−0.356063
$866$	2.47214	0.0840066
$867$	0	0
$868$	−0.472136	−0.0160253
$869$	5.52786	0.187520
$870$	0	0
$871$	77.9017	2.63960
$872$	16.1803	0.547935
$873$	0	0
$874$	−83.6656	−2.83003
$875$	9.00000	0.304256
$876$	0	0
$877$	31.4164	1.06086	0.530428	−	0.847730i	$-0.322031\pi$
$877$	31.4164	1.06086	0.530428	+	0.847730i	$0.322031\pi$
$878$	18.0902	0.610514
$879$	0	0
$880$	−4.85410	−0.163632
$881$	−19.1115	−0.643881	−0.321941	−	0.946760i	$-0.604335\pi$
$881$	−19.1115	−0.643881	−0.321941	+	0.946760i	$0.604335\pi$
$882$	0	0
$883$	−37.1803	−1.25122	−0.625609	−	0.780137i	$-0.715149\pi$
$883$	−37.1803	−1.25122	−0.625609	+	0.780137i	$0.715149\pi$
$884$	2.58359	0.0868956
$885$	0	0
$886$	15.4164	0.517924
$887$	−15.2361	−0.511577	−0.255789	−	0.966733i	$-0.582335\pi$
$887$	−15.2361	−0.511577	−0.255789	+	0.966733i	$0.582335\pi$
$888$	0	0
$889$	18.6525	0.625584
$890$	−7.23607	−0.242554
$891$	0	0
$892$	−3.70820	−0.124160
$893$	28.4164	0.950919
$894$	0	0
$895$	−8.94427	−0.298974
$896$	−13.6180	−0.454947
$897$	0	0
$898$	32.3607	1.07989
$899$	3.81966	0.127393
$900$	0	0
$901$	7.77709	0.259092
$902$	−10.4721	−0.348684
$903$	0	0
$904$	−18.9443	−0.630077
$905$	5.23607	0.174053
$906$	0	0
$907$	28.0000	0.929725	0.464862	−	0.885383i	$-0.346104\pi$
$907$	28.0000	0.929725	0.464862	+	0.885383i	$0.346104\pi$
$908$	1.23607	0.0410204
$909$	0	0
$910$	−8.85410	−0.293511
$911$	41.4164	1.37219	0.686093	−	0.727513i	$-0.259324\pi$
$911$	41.4164	1.37219	0.686093	+	0.727513i	$0.259324\pi$
$912$	0	0
$913$	11.2361	0.371860
$914$	−17.4164	−0.576084
$915$	0	0
$916$	1.70820	0.0564406
$917$	16.9443	0.559549
$918$	0	0
$919$	−9.59675	−0.316567	−0.158284	−	0.987394i	$-0.550596\pi$
$919$	−9.59675	−0.316567	−0.158284	+	0.987394i	$0.550596\pi$
$920$	−17.2361	−0.568256
$921$	0	0
$922$	−45.3050	−1.49204
$923$	35.4164	1.16575
$924$	0	0
$925$	28.0000	0.920634
$926$	43.9787	1.44523
$927$	0	0
$928$	−16.9098	−0.555092
$929$	22.8885	0.750949	0.375474	−	0.926833i	$-0.377480\pi$
$929$	22.8885	0.750949	0.375474	+	0.926833i	$0.377480\pi$
$930$	0	0
$931$	6.70820	0.219853
$932$	−1.81966	−0.0596049
$933$	0	0
$934$	11.0344	0.361058
$935$	−0.763932	−0.0249832
$936$	0	0
$937$	−4.11146	−0.134315	−0.0671577	−	0.997742i	$-0.521393\pi$
$937$	−4.11146	−0.134315	−0.0671577	+	0.997742i	$0.521393\pi$
$938$	23.0344	0.752101
$939$	0	0
$940$	−2.61803	−0.0853909
$941$	15.6393	0.509827	0.254914	−	0.966964i	$-0.417953\pi$
$941$	15.6393	0.509827	0.254914	+	0.966964i	$0.417953\pi$
$942$	0	0
$943$	−49.8885	−1.62459
$944$	54.2705	1.76635
$945$	0	0
$946$	−12.4721	−0.405504
$947$	−21.4164	−0.695940	−0.347970	−	0.937506i	$-0.613129\pi$
$947$	−21.4164	−0.695940	−0.347970	+	0.937506i	$0.613129\pi$
$948$	0	0
$949$	−73.7214	−2.39310
$950$	43.4164	1.40861
$951$	0	0
$952$	−1.70820	−0.0553632
$953$	26.7771	0.867395	0.433697	−	0.901059i	$-0.357209\pi$
$953$	26.7771	0.867395	0.433697	+	0.901059i	$0.357209\pi$
$954$	0	0
$955$	−15.2361	−0.493028
$956$	18.6180	0.602150
$957$	0	0
$958$	−2.76393	−0.0892986
$959$	−6.29180	−0.203173
$960$	0	0
$961$	−30.4164	−0.981174
$962$	−61.9787	−1.99827
$963$	0	0
$964$	4.97871	0.160354
$965$	16.6525	0.536062
$966$	0	0
$967$	37.1935	1.19606	0.598031	−	0.801473i	$-0.295950\pi$
$967$	37.1935	1.19606	0.598031	+	0.801473i	$0.295950\pi$
$968$	2.23607	0.0718699
$969$	0	0
$970$	−6.00000	−0.192648
$971$	8.12461	0.260731	0.130366	−	0.991466i	$-0.458385\pi$
$971$	8.12461	0.260731	0.130366	+	0.991466i	$0.458385\pi$
$972$	0	0
$973$	5.52786	0.177215
$974$	1.52786	0.0489559
$975$	0	0
$976$	−9.70820	−0.310752
$977$	18.1803	0.581641	0.290820	−	0.956778i	$-0.406072\pi$
$977$	18.1803	0.581641	0.290820	+	0.956778i	$0.406072\pi$
$978$	0	0
$979$	4.47214	0.142930
$980$	−0.618034	−0.0197424
$981$	0	0
$982$	35.7984	1.14237
$983$	−0.583592	−0.0186137	−0.00930685	−	0.999957i	$-0.502963\pi$
$983$	−0.583592	−0.0186137	−0.00930685	+	0.999957i	$0.502963\pi$
$984$	0	0
$985$	−7.52786	−0.239858
$986$	−6.18034	−0.196822
$987$	0	0
$988$	−22.6869	−0.721767
$989$	−59.4164	−1.88933
$990$	0	0
$991$	−6.81966	−0.216634	−0.108317	−	0.994116i	$-0.534546\pi$
$991$	−6.81966	−0.216634	−0.108317	+	0.994116i	$0.534546\pi$
$992$	−2.58359	−0.0820291
$993$	0	0
$994$	10.4721	0.332156
$995$	26.1803	0.829973
$996$	0	0
$997$	9.05573	0.286798	0.143399	−	0.989665i	$-0.454197\pi$
$997$	9.05573	0.286798	0.143399	+	0.989665i	$0.454197\pi$
$998$	−3.61803	−0.114527
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.f.1.1		2
3.2	odd	2		231.2.a.c.1.2	✓	2
7.6	odd	2		4851.2.a.w.1.1		2
11.10	odd	2		7623.2.a.bm.1.2		2
12.11	even	2		3696.2.a.be.1.1		2
15.14	odd	2		5775.2.a.be.1.1		2
21.20	even	2		1617.2.a.p.1.2		2
33.32	even	2		2541.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.c.1.2	✓	2	3.2	odd	2
693.2.a.f.1.1		2	1.1	even	1	trivial
1617.2.a.p.1.2		2	21.20	even	2
2541.2.a.t.1.1		2	33.32	even	2
3696.2.a.be.1.1		2	12.11	even	2
4851.2.a.w.1.1		2	7.6	odd	2
5775.2.a.be.1.1		2	15.14	odd	2
7623.2.a.bm.1.2		2	11.10	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:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +O(q^{10})\)	\(q-1.61803 q^{2} +0.618034 q^{4} -1.00000 q^{5} +1.00000 q^{7} +2.23607 q^{8} +1.61803 q^{10} -1.00000 q^{11} -5.47214 q^{13} -1.61803 q^{14} -4.85410 q^{16} -0.763932 q^{17} +6.70820 q^{19} -0.618034 q^{20} +1.61803 q^{22} +7.70820 q^{23} -4.00000 q^{25} +8.85410 q^{26} +0.618034 q^{28} -5.00000 q^{29} -0.763932 q^{31} +3.38197 q^{32} +1.23607 q^{34} -1.00000 q^{35} -7.00000 q^{37} -10.8541 q^{38} -2.23607 q^{40} -6.47214 q^{41} -7.70820 q^{43} -0.618034 q^{44} -12.4721 q^{46} +4.23607 q^{47} +1.00000 q^{49} +6.47214 q^{50} -3.38197 q^{52} -10.1803 q^{53} +1.00000 q^{55} +2.23607 q^{56} +8.09017 q^{58} -11.1803 q^{59} +2.00000 q^{61} +1.23607 q^{62} +4.23607 q^{64} +5.47214 q^{65} -14.2361 q^{67} -0.472136 q^{68} +1.61803 q^{70} -6.47214 q^{71} +13.4721 q^{73} +11.3262 q^{74} +4.14590 q^{76} -1.00000 q^{77} -5.52786 q^{79} +4.85410 q^{80} +10.4721 q^{82} -11.2361 q^{83} +0.763932 q^{85} +12.4721 q^{86} -2.23607 q^{88} -4.47214 q^{89} -5.47214 q^{91} +4.76393 q^{92} -6.85410 q^{94} -6.70820 q^{95} -3.70820 q^{97} -1.61803 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7	\( 2 q - q^{2} - q^{4} - 2 q^{5} + 2 q^{7} + q^{10} - 2 q^{11} - 2 q^{13} - q^{14} - 3 q^{16} - 6 q^{17} + q^{20} + q^{22} + 2 q^{23} - 8 q^{25} + 11 q^{26} - q^{28} - 10 q^{29} - 6 q^{31} + 9 q^{32} - 2 q^{34} - 2 q^{35} - 14 q^{37} - 15 q^{38} - 4 q^{41} - 2 q^{43} + q^{44} - 16 q^{46} + 4 q^{47} + 2 q^{49} + 4 q^{50} - 9 q^{52} + 2 q^{53} + 2 q^{55} + 5 q^{58} + 4 q^{61} - 2 q^{62} + 4 q^{64} + 2 q^{65} - 24 q^{67} + 8 q^{68} + q^{70} - 4 q^{71} + 18 q^{73} + 7 q^{74} + 15 q^{76} - 2 q^{77} - 20 q^{79} + 3 q^{80} + 12 q^{82} - 18 q^{83} + 6 q^{85} + 16 q^{86} - 2 q^{91} + 14 q^{92} - 7 q^{94} + 6 q^{97} - q^{98}+O(q^{100}) \) 2 * q - q^2 - q^4 - 2 * q^5 + 2 * q^7 + q^10 - 2 * q^11 - 2 * q^13 - q^14 - 3 * q^16 - 6 * q^17 + q^20 + q^22 + 2 * q^23 - 8 * q^25 + 11 * q^26 - q^28 - 10 * q^29 - 6 * q^31 + 9 * q^32 - 2 * q^34 - 2 * q^35 - 14 * q^37 - 15 * q^38 - 4 * q^41 - 2 * q^43 + q^44 - 16 * q^46 + 4 * q^47 + 2 * q^49 + 4 * q^50 - 9 * q^52 + 2 * q^53 + 2 * q^55 + 5 * q^58 + 4 * q^61 - 2 * q^62 + 4 * q^64 + 2 * q^65 - 24 * q^67 + 8 * q^68 + q^70 - 4 * q^71 + 18 * q^73 + 7 * q^74 + 15 * q^76 - 2 * q^77 - 20 * q^79 + 3 * q^80 + 12 * q^82 - 18 * q^83 + 6 * q^85 + 16 * q^86 - 2 * q^91 + 14 * q^92 - 7 * q^94 + 6 * q^97 - q^98

Introduction

L-functions

Modular forms

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