LMFDB - Embedded newform 686.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [686,2,Mod(1,686)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(686, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("686.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 686 = 2 \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	686.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:	$5.47773757866$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	686.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.24698 q^{3} +1.00000 q^{4} +1.35690 q^{5} +1.24698 q^{6} -1.00000 q^{8} -1.44504 q^{9} -1.35690 q^{10} -0.335126 q^{11} -1.24698 q^{12} +6.38404 q^{13} -1.69202 q^{15} +1.00000 q^{16} -2.96077 q^{17} +1.44504 q^{18} +4.10992 q^{19} +1.35690 q^{20} +0.335126 q^{22} -3.65279 q^{23} +1.24698 q^{24} -3.15883 q^{25} -6.38404 q^{26} +5.54288 q^{27} +4.04892 q^{29} +1.69202 q^{30} +0.704103 q^{31} -1.00000 q^{32} +0.417895 q^{33} +2.96077 q^{34} -1.44504 q^{36} +6.74094 q^{37} -4.10992 q^{38} -7.96077 q^{39} -1.35690 q^{40} +6.60388 q^{41} -4.91185 q^{43} -0.335126 q^{44} -1.96077 q^{45} +3.65279 q^{46} -0.862937 q^{47} -1.24698 q^{48} +3.15883 q^{50} +3.69202 q^{51} +6.38404 q^{52} +12.8877 q^{53} -5.54288 q^{54} -0.454731 q^{55} -5.12498 q^{57} -4.04892 q^{58} +9.05861 q^{59} -1.69202 q^{60} +13.1588 q^{61} -0.704103 q^{62} +1.00000 q^{64} +8.66248 q^{65} -0.417895 q^{66} +3.14914 q^{67} -2.96077 q^{68} +4.55496 q^{69} +10.5700 q^{71} +1.44504 q^{72} +10.0218 q^{73} -6.74094 q^{74} +3.93900 q^{75} +4.10992 q^{76} +7.96077 q^{78} -8.86294 q^{79} +1.35690 q^{80} -2.57673 q^{81} -6.60388 q^{82} -17.4426 q^{83} -4.01746 q^{85} +4.91185 q^{86} -5.04892 q^{87} +0.335126 q^{88} -0.625646 q^{89} +1.96077 q^{90} -3.65279 q^{92} -0.878002 q^{93} +0.862937 q^{94} +5.57673 q^{95} +1.24698 q^{96} +3.24698 q^{97} +0.484271 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 3 q^{8} - 4 q^{9} + q^{12} + 9 q^{13} + 3 q^{16} + 4 q^{17} + 4 q^{18} + 13 q^{19} + 7 q^{23} - q^{24} - q^{25} - 9 q^{26} - 2 q^{27} + 3 q^{29} + 16 q^{31}+ \cdots + 14 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 3 * q^8 - 4 * q^9 + q^12 + 9 * q^13 + 3 * q^16 + 4 * q^17 + 4 * q^18 + 13 * q^19 + 7 * q^23 - q^24 - q^25 - 9 * q^26 - 2 * q^27 + 3 * q^29 + 16 * q^31 - 3 * q^32 + 7 * q^33 - 4 * q^34 - 4 * q^36 + 6 * q^37 - 13 * q^38 - 11 * q^39 + 11 * q^41 - 11 * q^43 + 7 * q^45 - 7 * q^46 - 8 * q^47 + q^48 + q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 + 2 * q^54 + 21 * q^55 + 9 * q^57 - 3 * q^58 - 4 * q^59 + 31 * q^61 - 16 * q^62 + 3 * q^64 - 14 * q^65 - 7 * q^66 + 23 * q^67 + 4 * q^68 + 14 * q^69 + 7 * q^71 + 4 * q^72 + 27 * q^73 - 6 * q^74 + 2 * q^75 + 13 * q^76 + 11 * q^78 - 32 * q^79 - 5 * q^81 - 11 * q^82 - 11 * q^83 - 28 * q^85 + 11 * q^86 - 6 * q^87 + 10 * q^89 - 7 * q^90 + 7 * q^92 + 17 * q^93 + 8 * q^94 + 14 * q^95 - q^96 + 5 * q^97 + 14 * q^99

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
$3$	−1.24698	−0.719944	−0.359972	−	0.932963i	$-0.617214\pi$
$3$	−1.24698	−0.719944	−0.359972	+	0.932963i	$0.617214\pi$
$4$	1.00000	0.500000
$5$	1.35690	0.606822	0.303411	−	0.952860i	$-0.401874\pi$
$5$	1.35690	0.606822	0.303411	+	0.952860i	$0.401874\pi$
$6$	1.24698	0.509077
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.44504	−0.481681
$10$	−1.35690	−0.429088
$11$	−0.335126	−0.101044	−0.0505221	−	0.998723i	$-0.516089\pi$
$11$	−0.335126	−0.101044	−0.0505221	+	0.998723i	$0.516089\pi$
$12$	−1.24698	−0.359972
$13$	6.38404	1.77061	0.885307	−	0.465006i	$-0.153948\pi$
$13$	6.38404	1.77061	0.885307	+	0.465006i	$0.153948\pi$
$14$	0	0
$15$	−1.69202	−0.436878
$16$	1.00000	0.250000
$17$	−2.96077	−0.718093	−0.359046	−	0.933320i	$-0.616898\pi$
$17$	−2.96077	−0.718093	−0.359046	+	0.933320i	$0.616898\pi$
$18$	1.44504	0.340600
$19$	4.10992	0.942879	0.471440	−	0.881898i	$-0.343734\pi$
$19$	4.10992	0.942879	0.471440	+	0.881898i	$0.343734\pi$
$20$	1.35690	0.303411
$21$	0	0
$22$	0.335126	0.0714490
$23$	−3.65279	−0.761660	−0.380830	−	0.924645i	$-0.624362\pi$
$23$	−3.65279	−0.761660	−0.380830	+	0.924645i	$0.624362\pi$
$24$	1.24698	0.254539
$25$	−3.15883	−0.631767
$26$	−6.38404	−1.25201
$27$	5.54288	1.06673
$28$	0	0
$29$	4.04892	0.751865	0.375933	−	0.926647i	$-0.377322\pi$
$29$	4.04892	0.751865	0.375933	+	0.926647i	$0.377322\pi$
$30$	1.69202	0.308919
$31$	0.704103	0.126461	0.0632303	−	0.997999i	$-0.479860\pi$
$31$	0.704103	0.126461	0.0632303	+	0.997999i	$0.479860\pi$
$32$	−1.00000	−0.176777
$33$	0.417895	0.0727461
$34$	2.96077	0.507768
$35$	0	0
$36$	−1.44504	−0.240840
$37$	6.74094	1.10820	0.554102	−	0.832449i	$-0.313062\pi$
$37$	6.74094	1.10820	0.554102	+	0.832449i	$0.313062\pi$
$38$	−4.10992	−0.666716
$39$	−7.96077	−1.27474
$40$	−1.35690	−0.214544
$41$	6.60388	1.03135	0.515676	−	0.856784i	$-0.327541\pi$
$41$	6.60388	1.03135	0.515676	+	0.856784i	$0.327541\pi$
$42$	0	0
$43$	−4.91185	−0.749051	−0.374525	−	0.927217i	$-0.622194\pi$
$43$	−4.91185	−0.749051	−0.374525	+	0.927217i	$0.622194\pi$
$44$	−0.335126	−0.0505221
$45$	−1.96077	−0.292295
$46$	3.65279	0.538575
$47$	−0.862937	−0.125872	−0.0629361	−	0.998018i	$-0.520046\pi$
$47$	−0.862937	−0.125872	−0.0629361	+	0.998018i	$0.520046\pi$
$48$	−1.24698	−0.179986
$49$	0	0
$50$	3.15883	0.446727
$51$	3.69202	0.516986
$52$	6.38404	0.885307
$53$	12.8877	1.77026	0.885130	−	0.465343i	$-0.154069\pi$
$53$	12.8877	1.77026	0.885130	+	0.465343i	$0.154069\pi$
$54$	−5.54288	−0.754290
$55$	−0.454731	−0.0613159
$56$	0	0
$57$	−5.12498	−0.678820
$58$	−4.04892	−0.531649
$59$	9.05861	1.17933	0.589665	−	0.807648i	$-0.299260\pi$
$59$	9.05861	1.17933	0.589665	+	0.807648i	$0.299260\pi$
$60$	−1.69202	−0.218439
$61$	13.1588	1.68482	0.842408	−	0.538840i	$-0.181137\pi$
$61$	13.1588	1.68482	0.842408	+	0.538840i	$0.181137\pi$
$62$	−0.704103	−0.0894212
$63$	0	0
$64$	1.00000	0.125000
$65$	8.66248	1.07445
$66$	−0.417895	−0.0514393
$67$	3.14914	0.384729	0.192365	−	0.981324i	$-0.438384\pi$
$67$	3.14914	0.384729	0.192365	+	0.981324i	$0.438384\pi$
$68$	−2.96077	−0.359046
$69$	4.55496	0.548353
$70$	0	0
$71$	10.5700	1.25443	0.627216	−	0.778846i	$-0.284195\pi$
$71$	10.5700	1.25443	0.627216	+	0.778846i	$0.284195\pi$
$72$	1.44504	0.170300
$73$	10.0218	1.17296	0.586480	−	0.809964i	$-0.300513\pi$
$73$	10.0218	1.17296	0.586480	+	0.809964i	$0.300513\pi$
$74$	−6.74094	−0.783618
$75$	3.93900	0.454837
$76$	4.10992	0.471440
$77$	0	0
$78$	7.96077	0.901380
$79$	−8.86294	−0.997158	−0.498579	−	0.866844i	$-0.666145\pi$
$79$	−8.86294	−0.997158	−0.498579	+	0.866844i	$0.666145\pi$
$80$	1.35690	0.151706
$81$	−2.57673	−0.286303
$82$	−6.60388	−0.729276
$83$	−17.4426	−1.91458	−0.957290	−	0.289130i	$-0.906634\pi$
$83$	−17.4426	−1.91458	−0.957290	+	0.289130i	$0.906634\pi$
$84$	0	0
$85$	−4.01746	−0.435755
$86$	4.91185	0.529659
$87$	−5.04892	−0.541301
$88$	0.335126	0.0357245
$89$	−0.625646	−0.0663183	−0.0331592	−	0.999450i	$-0.510557\pi$
$89$	−0.625646	−0.0663183	−0.0331592	+	0.999450i	$0.510557\pi$
$90$	1.96077	0.206683
$91$	0	0
$92$	−3.65279	−0.380830
$93$	−0.878002	−0.0910446
$94$	0.862937	0.0890051
$95$	5.57673	0.572160
$96$	1.24698	0.127269
$97$	3.24698	0.329681	0.164840	−	0.986320i	$-0.447289\pi$
$97$	3.24698	0.329681	0.164840	+	0.986320i	$0.447289\pi$
$98$	0	0
$99$	0.484271	0.0486710
$100$	−3.15883	−0.315883
$101$	1.77479	0.176598	0.0882991	−	0.996094i	$-0.471857\pi$
$101$	1.77479	0.176598	0.0882991	+	0.996094i	$0.471857\pi$
$102$	−3.69202	−0.365565
$103$	15.7168	1.54862	0.774310	−	0.632807i	$-0.218097\pi$
$103$	15.7168	1.54862	0.774310	+	0.632807i	$0.218097\pi$
$104$	−6.38404	−0.626007
$105$	0	0
$106$	−12.8877	−1.25176
$107$	−17.4306	−1.68508	−0.842538	−	0.538636i	$-0.818940\pi$
$107$	−17.4306	−1.68508	−0.842538	+	0.538636i	$0.818940\pi$
$108$	5.54288	0.533364
$109$	7.09246	0.679334	0.339667	−	0.940546i	$-0.389686\pi$
$109$	7.09246	0.679334	0.339667	+	0.940546i	$0.389686\pi$
$110$	0.454731	0.0433569
$111$	−8.40581	−0.797844
$112$	0	0
$113$	−12.9433	−1.21760	−0.608802	−	0.793322i	$-0.708350\pi$
$113$	−12.9433	−1.21760	−0.608802	+	0.793322i	$0.708350\pi$
$114$	5.12498	0.479999
$115$	−4.95646	−0.462192
$116$	4.04892	0.375933
$117$	−9.22521	−0.852871
$118$	−9.05861	−0.833912
$119$	0	0
$120$	1.69202	0.154460
$121$	−10.8877	−0.989790
$122$	−13.1588	−1.19134
$123$	−8.23490	−0.742516
$124$	0.704103	0.0632303
$125$	−11.0707	−0.990192
$126$	0	0
$127$	10.9487	0.971539	0.485770	−	0.874087i	$-0.338539\pi$
$127$	10.9487	0.971539	0.485770	+	0.874087i	$0.338539\pi$
$128$	−1.00000	−0.0883883
$129$	6.12498	0.539275
$130$	−8.66248	−0.759750
$131$	−4.43967	−0.387895	−0.193948	−	0.981012i	$-0.562129\pi$
$131$	−4.43967	−0.387895	−0.193948	+	0.981012i	$0.562129\pi$
$132$	0.417895	0.0363731
$133$	0	0
$134$	−3.14914	−0.272045
$135$	7.52111	0.647314
$136$	2.96077	0.253884
$137$	−15.7627	−1.34670	−0.673350	−	0.739324i	$-0.735145\pi$
$137$	−15.7627	−1.34670	−0.673350	+	0.739324i	$0.735145\pi$
$138$	−4.55496	−0.387744
$139$	−17.9584	−1.52321	−0.761605	−	0.648042i	$-0.775588\pi$
$139$	−17.9584	−1.52321	−0.761605	+	0.648042i	$0.775588\pi$
$140$	0	0
$141$	1.07606	0.0906210
$142$	−10.5700	−0.887017
$143$	−2.13946	−0.178910
$144$	−1.44504	−0.120420
$145$	5.49396	0.456248
$146$	−10.0218	−0.829408
$147$	0	0
$148$	6.74094	0.554102
$149$	14.0911	1.15439	0.577195	−	0.816606i	$-0.304147\pi$
$149$	14.0911	1.15439	0.577195	+	0.816606i	$0.304147\pi$
$150$	−3.93900	−0.321618
$151$	12.2567	0.997434	0.498717	−	0.866765i	$-0.333805\pi$
$151$	12.2567	0.997434	0.498717	+	0.866765i	$0.333805\pi$
$152$	−4.10992	−0.333358
$153$	4.27844	0.345891
$154$	0	0
$155$	0.955395	0.0767391
$156$	−7.96077	−0.637372
$157$	−21.2054	−1.69237	−0.846186	−	0.532888i	$-0.821107\pi$
$157$	−21.2054	−1.69237	−0.846186	+	0.532888i	$0.821107\pi$
$158$	8.86294	0.705097
$159$	−16.0707	−1.27449
$160$	−1.35690	−0.107272
$161$	0	0
$162$	2.57673	0.202447
$163$	18.4862	1.44795	0.723975	−	0.689826i	$-0.242313\pi$
$163$	18.4862	1.44795	0.723975	+	0.689826i	$0.242313\pi$
$164$	6.60388	0.515676
$165$	0.567040	0.0441440
$166$	17.4426	1.35381
$167$	−20.2741	−1.56886	−0.784430	−	0.620218i	$-0.787044\pi$
$167$	−20.2741	−1.56886	−0.784430	+	0.620218i	$0.787044\pi$
$168$	0	0
$169$	27.7560	2.13508
$170$	4.01746	0.308125
$171$	−5.93900	−0.454167
$172$	−4.91185	−0.374525
$173$	−6.35690	−0.483306	−0.241653	−	0.970363i	$-0.577690\pi$
$173$	−6.35690	−0.483306	−0.241653	+	0.970363i	$0.577690\pi$
$174$	5.04892	0.382757
$175$	0	0
$176$	−0.335126	−0.0252610
$177$	−11.2959	−0.849052
$178$	0.625646	0.0468941
$179$	3.46011	0.258621	0.129310	−	0.991604i	$-0.458724\pi$
$179$	3.46011	0.258621	0.129310	+	0.991604i	$0.458724\pi$
$180$	−1.96077	−0.146147
$181$	13.7071	1.01884	0.509420	−	0.860518i	$-0.329860\pi$
$181$	13.7071	1.01884	0.509420	+	0.860518i	$0.329860\pi$
$182$	0	0
$183$	−16.4088	−1.21297
$184$	3.65279	0.269287
$185$	9.14675	0.672483
$186$	0.878002	0.0643782
$187$	0.992230	0.0725591
$188$	−0.862937	−0.0629361
$189$	0	0
$190$	−5.57673	−0.404578
$191$	−6.92931	−0.501387	−0.250694	−	0.968066i	$-0.580659\pi$
$191$	−6.92931	−0.501387	−0.250694	+	0.968066i	$0.580659\pi$
$192$	−1.24698	−0.0899930
$193$	−0.716185	−0.0515521	−0.0257760	−	0.999668i	$-0.508206\pi$
$193$	−0.716185	−0.0515521	−0.0257760	+	0.999668i	$0.508206\pi$
$194$	−3.24698	−0.233120
$195$	−10.8019	−0.773543
$196$	0	0
$197$	2.06100	0.146840	0.0734200	−	0.997301i	$-0.476609\pi$
$197$	2.06100	0.146840	0.0734200	+	0.997301i	$0.476609\pi$
$198$	−0.484271	−0.0344156
$199$	0.0217703	0.00154325	0.000771627	−	1.00000i	$-0.499754\pi$
$199$	0.0217703	0.00154325	0.000771627		1.00000i	$0.499754\pi$
$200$	3.15883	0.223363
$201$	−3.92692	−0.276984
$202$	−1.77479	−0.124874
$203$	0	0
$204$	3.69202	0.258493
$205$	8.96077	0.625847
$206$	−15.7168	−1.09504
$207$	5.27844	0.366877
$208$	6.38404	0.442654
$209$	−1.37734	−0.0952725
$210$	0	0
$211$	−27.7875	−1.91297	−0.956484	−	0.291785i	$-0.905751\pi$
$211$	−27.7875	−1.91297	−0.956484	+	0.291785i	$0.905751\pi$
$212$	12.8877	0.885130
$213$	−13.1806	−0.903120
$214$	17.4306	1.19153
$215$	−6.66487	−0.454541
$216$	−5.54288	−0.377145
$217$	0	0
$218$	−7.09246	−0.480362
$219$	−12.4969	−0.844465
$220$	−0.454731	−0.0306579
$221$	−18.9017	−1.27147
$222$	8.40581	0.564161
$223$	−12.7017	−0.850569	−0.425285	−	0.905060i	$-0.639826\pi$
$223$	−12.7017	−0.850569	−0.425285	+	0.905060i	$0.639826\pi$
$224$	0	0
$225$	4.56465	0.304310
$226$	12.9433	0.860976
$227$	−6.49635	−0.431178	−0.215589	−	0.976484i	$-0.569167\pi$
$227$	−6.49635	−0.431178	−0.215589	+	0.976484i	$0.569167\pi$
$228$	−5.12498	−0.339410
$229$	9.70171	0.641107	0.320554	−	0.947230i	$-0.396131\pi$
$229$	9.70171	0.641107	0.320554	+	0.947230i	$0.396131\pi$
$230$	4.95646	0.326819
$231$	0	0
$232$	−4.04892	−0.265824
$233$	1.81940	0.119193	0.0595963	−	0.998223i	$-0.481019\pi$
$233$	1.81940	0.119193	0.0595963	+	0.998223i	$0.481019\pi$
$234$	9.22521	0.603071
$235$	−1.17092	−0.0763821
$236$	9.05861	0.589665
$237$	11.0519	0.717898
$238$	0	0
$239$	8.19567	0.530134	0.265067	−	0.964230i	$-0.414606\pi$
$239$	8.19567	0.530134	0.265067	+	0.964230i	$0.414606\pi$
$240$	−1.69202	−0.109220
$241$	5.23729	0.337364	0.168682	−	0.985671i	$-0.446049\pi$
$241$	5.23729	0.337364	0.168682	+	0.985671i	$0.446049\pi$
$242$	10.8877	0.699887
$243$	−13.4155	−0.860605
$244$	13.1588	0.842408
$245$	0	0
$246$	8.23490	0.525038
$247$	26.2379	1.66948
$248$	−0.704103	−0.0447106
$249$	21.7506	1.37839
$250$	11.0707	0.700172
$251$	0.972853	0.0614059	0.0307030	−	0.999529i	$-0.490225\pi$
$251$	0.972853	0.0614059	0.0307030	+	0.999529i	$0.490225\pi$
$252$	0	0
$253$	1.22414	0.0769613
$254$	−10.9487	−0.686982
$255$	5.00969	0.313719
$256$	1.00000	0.0625000
$257$	−15.7168	−0.980386	−0.490193	−	0.871614i	$-0.663074\pi$
$257$	−15.7168	−0.980386	−0.490193	+	0.871614i	$0.663074\pi$
$258$	−6.12498	−0.381325
$259$	0	0
$260$	8.66248	0.537224
$261$	−5.85086	−0.362159
$262$	4.43967	0.274283
$263$	−18.5623	−1.14460	−0.572299	−	0.820045i	$-0.693948\pi$
$263$	−18.5623	−1.14460	−0.572299	+	0.820045i	$0.693948\pi$
$264$	−0.417895	−0.0257196
$265$	17.4873	1.07423
$266$	0	0
$267$	0.780167	0.0477455
$268$	3.14914	0.192365
$269$	11.1304	0.678630	0.339315	−	0.940673i	$-0.389805\pi$
$269$	11.1304	0.678630	0.339315	+	0.940673i	$0.389805\pi$
$270$	−7.52111	−0.457720
$271$	20.1250	1.22251	0.611253	−	0.791435i	$-0.290666\pi$
$271$	20.1250	1.22251	0.611253	+	0.791435i	$0.290666\pi$
$272$	−2.96077	−0.179523
$273$	0	0
$274$	15.7627	0.952260
$275$	1.05861	0.0638363
$276$	4.55496	0.274176
$277$	−25.8485	−1.55308	−0.776542	−	0.630066i	$-0.783028\pi$
$277$	−25.8485	−1.55308	−0.776542	+	0.630066i	$0.783028\pi$
$278$	17.9584	1.07707
$279$	−1.01746	−0.0609136
$280$	0	0
$281$	−13.0737	−0.779910	−0.389955	−	0.920834i	$-0.627509\pi$
$281$	−13.0737	−0.779910	−0.389955	+	0.920834i	$0.627509\pi$
$282$	−1.07606	−0.0640787
$283$	5.77718	0.343418	0.171709	−	0.985148i	$-0.445071\pi$
$283$	5.77718	0.343418	0.171709	+	0.985148i	$0.445071\pi$
$284$	10.5700	0.627216
$285$	−6.95407	−0.411923
$286$	2.13946	0.126509
$287$	0	0
$288$	1.44504	0.0851499
$289$	−8.23383	−0.484343
$290$	−5.49396	−0.322616
$291$	−4.04892	−0.237352
$292$	10.0218	0.586480
$293$	20.5200	1.19879	0.599397	−	0.800452i	$-0.295407\pi$
$293$	20.5200	1.19879	0.599397	+	0.800452i	$0.295407\pi$
$294$	0	0
$295$	12.2916	0.715644
$296$	−6.74094	−0.391809
$297$	−1.85756	−0.107787
$298$	−14.0911	−0.816277
$299$	−23.3196	−1.34861
$300$	3.93900	0.227418
$301$	0	0
$302$	−12.2567	−0.705292
$303$	−2.21313	−0.127141
$304$	4.10992	0.235720
$305$	17.8552	1.02238
$306$	−4.27844	−0.244582
$307$	10.2446	0.584689	0.292345	−	0.956313i	$-0.405565\pi$
$307$	10.2446	0.584689	0.292345	+	0.956313i	$0.405565\pi$
$308$	0	0
$309$	−19.5985	−1.11492
$310$	−0.955395	−0.0542628
$311$	34.6679	1.96583	0.982917	−	0.184050i	$-0.0589207\pi$
$311$	34.6679	1.96583	0.982917	+	0.184050i	$0.0589207\pi$
$312$	7.96077	0.450690
$313$	−11.6993	−0.661285	−0.330642	−	0.943756i	$-0.607265\pi$
$313$	−11.6993	−0.661285	−0.330642	+	0.943756i	$0.607265\pi$
$314$	21.2054	1.19669
$315$	0	0
$316$	−8.86294	−0.498579
$317$	−6.82908	−0.383560	−0.191780	−	0.981438i	$-0.561426\pi$
$317$	−6.82908	−0.383560	−0.191780	+	0.981438i	$0.561426\pi$
$318$	16.0707	0.901199
$319$	−1.35690	−0.0759716
$320$	1.35690	0.0758528
$321$	21.7356	1.21316
$322$	0	0
$323$	−12.1685	−0.677075
$324$	−2.57673	−0.143152
$325$	−20.1661	−1.11862
$326$	−18.4862	−1.02386
$327$	−8.84415	−0.489083
$328$	−6.60388	−0.364638
$329$	0	0
$330$	−0.567040	−0.0312145
$331$	−15.2664	−0.839115	−0.419557	−	0.907729i	$-0.637815\pi$
$331$	−15.2664	−0.839115	−0.419557	+	0.907729i	$0.637815\pi$
$332$	−17.4426	−0.957290
$333$	−9.74094	−0.533800
$334$	20.2741	1.10935
$335$	4.27306	0.233462
$336$	0	0
$337$	−0.0163935	−0.000893008 0	−0.000446504	−	1.00000i	$-0.500142\pi$
$337$	−0.0163935	−0.000893008 0	−0.000446504		1.00000i	$0.500142\pi$
$338$	−27.7560	−1.50973
$339$	16.1400	0.876607
$340$	−4.01746	−0.217877
$341$	−0.235963	−0.0127781
$342$	5.93900	0.321144
$343$	0	0
$344$	4.91185	0.264829
$345$	6.18060	0.332753
$346$	6.35690	0.341749
$347$	19.5133	1.04753	0.523765	−	0.851863i	$-0.324527\pi$
$347$	19.5133	1.04753	0.523765	+	0.851863i	$0.324527\pi$
$348$	−5.04892	−0.270650
$349$	11.7778	0.630450	0.315225	−	0.949017i	$-0.397920\pi$
$349$	11.7778	0.630450	0.315225	+	0.949017i	$0.397920\pi$
$350$	0	0
$351$	35.3860	1.88876
$352$	0.335126	0.0178623
$353$	34.1672	1.81854	0.909268	−	0.416211i	$-0.136642\pi$
$353$	34.1672	1.81854	0.909268	+	0.416211i	$0.136642\pi$
$354$	11.2959	0.600370
$355$	14.3424	0.761217
$356$	−0.625646	−0.0331592
$357$	0	0
$358$	−3.46011	−0.182872
$359$	10.6558	0.562390	0.281195	−	0.959651i	$-0.409269\pi$
$359$	10.6558	0.562390	0.281195	+	0.959651i	$0.409269\pi$
$360$	1.96077	0.103342
$361$	−2.10859	−0.110978
$362$	−13.7071	−0.720428
$363$	13.5767	0.712593
$364$	0	0
$365$	13.5985	0.711778
$366$	16.4088	0.857702
$367$	−33.0978	−1.72769	−0.863846	−	0.503755i	$-0.831951\pi$
$367$	−33.0978	−1.72769	−0.863846	+	0.503755i	$0.831951\pi$
$368$	−3.65279	−0.190415
$369$	−9.54288	−0.496782
$370$	−9.14675	−0.475517
$371$	0	0
$372$	−0.878002	−0.0455223
$373$	−18.5754	−0.961798	−0.480899	−	0.876776i	$-0.659690\pi$
$373$	−18.5754	−0.961798	−0.480899	+	0.876776i	$0.659690\pi$
$374$	−0.992230	−0.0513070
$375$	13.8049	0.712883
$376$	0.862937	0.0445026
$377$	25.8485	1.33126
$378$	0	0
$379$	8.85517	0.454859	0.227430	−	0.973795i	$-0.426968\pi$
$379$	8.85517	0.454859	0.227430	+	0.973795i	$0.426968\pi$
$380$	5.57673	0.286080
$381$	−13.6528	−0.699454
$382$	6.92931	0.354534
$383$	−30.6969	−1.56854	−0.784270	−	0.620420i	$-0.786962\pi$
$383$	−30.6969	−1.56854	−0.784270	+	0.620420i	$0.786962\pi$
$384$	1.24698	0.0636347
$385$	0	0
$386$	0.716185	0.0364528
$387$	7.09783	0.360803
$388$	3.24698	0.164840
$389$	−1.26205	−0.0639882	−0.0319941	−	0.999488i	$-0.510186\pi$
$389$	−1.26205	−0.0639882	−0.0319941	+	0.999488i	$0.510186\pi$
$390$	10.8019	0.546977
$391$	10.8151	0.546942
$392$	0	0
$393$	5.53617	0.279263
$394$	−2.06100	−0.103832
$395$	−12.0261	−0.605098
$396$	0.484271	0.0243355
$397$	−12.4558	−0.625138	−0.312569	−	0.949895i	$-0.601190\pi$
$397$	−12.4558	−0.625138	−0.312569	+	0.949895i	$0.601190\pi$
$398$	−0.0217703	−0.00109124
$399$	0	0
$400$	−3.15883	−0.157942
$401$	−11.8267	−0.590597	−0.295298	−	0.955405i	$-0.595419\pi$
$401$	−11.8267	−0.590597	−0.295298	+	0.955405i	$0.595419\pi$
$402$	3.92692	0.195857
$403$	4.49502	0.223913
$404$	1.77479	0.0882991
$405$	−3.49635	−0.173735
$406$	0	0
$407$	−2.25906	−0.111978
$408$	−3.69202	−0.182782
$409$	18.4058	0.910109	0.455054	−	0.890464i	$-0.349620\pi$
$409$	18.4058	0.910109	0.455054	+	0.890464i	$0.349620\pi$
$410$	−8.96077	−0.442541
$411$	19.6558	0.969548
$412$	15.7168	0.774310
$413$	0	0
$414$	−5.27844	−0.259421
$415$	−23.6679	−1.16181
$416$	−6.38404	−0.313003
$417$	22.3937	1.09663
$418$	1.37734	0.0673678
$419$	−5.12067	−0.250161	−0.125081	−	0.992147i	$-0.539919\pi$
$419$	−5.12067	−0.250161	−0.125081	+	0.992147i	$0.539919\pi$
$420$	0	0
$421$	−16.4034	−0.799454	−0.399727	−	0.916634i	$-0.630895\pi$
$421$	−16.4034	−0.799454	−0.399727	+	0.916634i	$0.630895\pi$
$422$	27.7875	1.35267
$423$	1.24698	0.0606302
$424$	−12.8877	−0.625882
$425$	9.35258	0.453667
$426$	13.1806	0.638602
$427$	0	0
$428$	−17.4306	−0.842538
$429$	2.66786	0.128805
$430$	6.66487	0.321409
$431$	−0.330814	−0.0159347	−0.00796737	−	0.999968i	$-0.502536\pi$
$431$	−0.330814	−0.0159347	−0.00796737	+	0.999968i	$0.502536\pi$
$432$	5.54288	0.266682
$433$	−2.45042	−0.117760	−0.0588798	−	0.998265i	$-0.518753\pi$
$433$	−2.45042	−0.117760	−0.0588798	+	0.998265i	$0.518753\pi$
$434$	0	0
$435$	−6.85086	−0.328473
$436$	7.09246	0.339667
$437$	−15.0127	−0.718154
$438$	12.4969	0.597127
$439$	2.69202	0.128483	0.0642416	−	0.997934i	$-0.479537\pi$
$439$	2.69202	0.128483	0.0642416	+	0.997934i	$0.479537\pi$
$440$	0.454731	0.0216784
$441$	0	0
$442$	18.9017	0.899062
$443$	36.5526	1.73666	0.868332	−	0.495983i	$-0.165192\pi$
$443$	36.5526	1.73666	0.868332	+	0.495983i	$0.165192\pi$
$444$	−8.40581	−0.398922
$445$	−0.848936	−0.0402434
$446$	12.7017	0.601443
$447$	−17.5714	−0.831096
$448$	0	0
$449$	−3.48427	−0.164433	−0.0822164	−	0.996614i	$-0.526200\pi$
$449$	−3.48427	−0.164433	−0.0822164	+	0.996614i	$0.526200\pi$
$450$	−4.56465	−0.215180
$451$	−2.21313	−0.104212
$452$	−12.9433	−0.608802
$453$	−15.2838	−0.718096
$454$	6.49635	0.304889
$455$	0	0
$456$	5.12498	0.239999
$457$	−23.1183	−1.08143	−0.540714	−	0.841207i	$-0.681846\pi$
$457$	−23.1183	−1.08143	−0.540714	+	0.841207i	$0.681846\pi$
$458$	−9.70171	−0.453331
$459$	−16.4112	−0.766009
$460$	−4.95646	−0.231096
$461$	9.69501	0.451541	0.225771	−	0.974180i	$-0.427510\pi$
$461$	9.69501	0.451541	0.225771	+	0.974180i	$0.427510\pi$
$462$	0	0
$463$	−5.84654	−0.271712	−0.135856	−	0.990729i	$-0.543378\pi$
$463$	−5.84654	−0.271712	−0.135856	+	0.990729i	$0.543378\pi$
$464$	4.04892	0.187966
$465$	−1.19136	−0.0552479
$466$	−1.81940	−0.0842819
$467$	−25.7832	−1.19310	−0.596551	−	0.802575i	$-0.703463\pi$
$467$	−25.7832	−1.19310	−0.596551	+	0.802575i	$0.703463\pi$
$468$	−9.22521	−0.426435
$469$	0	0
$470$	1.17092	0.0540103
$471$	26.4426	1.21841
$472$	−9.05861	−0.416956
$473$	1.64609	0.0756872
$474$	−11.0519	−0.507631
$475$	−12.9825	−0.595680
$476$	0	0
$477$	−18.6233	−0.852700
$478$	−8.19567	−0.374861
$479$	20.8799	0.954028	0.477014	−	0.878896i	$-0.341719\pi$
$479$	20.8799	0.954028	0.477014	+	0.878896i	$0.341719\pi$
$480$	1.69202	0.0772299
$481$	43.0344	1.96220
$482$	−5.23729	−0.238552
$483$	0	0
$484$	−10.8877	−0.494895
$485$	4.40581	0.200058
$486$	13.4155	0.608540
$487$	−0.629958	−0.0285461	−0.0142731	−	0.999898i	$-0.504543\pi$
$487$	−0.629958	−0.0285461	−0.0142731	+	0.999898i	$0.504543\pi$
$488$	−13.1588	−0.595672
$489$	−23.0519	−1.04244
$490$	0	0
$491$	−7.19269	−0.324601	−0.162301	−	0.986741i	$-0.551891\pi$
$491$	−7.19269	−0.324601	−0.162301	+	0.986741i	$0.551891\pi$
$492$	−8.23490	−0.371258
$493$	−11.9879	−0.539909
$494$	−26.2379	−1.18050
$495$	0.657105	0.0295347
$496$	0.704103	0.0316152
$497$	0	0
$498$	−21.7506	−0.974669
$499$	21.2567	0.951579	0.475790	−	0.879559i	$-0.342162\pi$
$499$	21.2567	0.951579	0.475790	+	0.879559i	$0.342162\pi$
$500$	−11.0707	−0.495096
$501$	25.2814	1.12949
$502$	−0.972853	−0.0434206
$503$	25.7614	1.14864	0.574322	−	0.818630i	$-0.305266\pi$
$503$	25.7614	1.14864	0.574322	+	0.818630i	$0.305266\pi$
$504$	0	0
$505$	2.40821	0.107164
$506$	−1.22414	−0.0544199
$507$	−34.6112	−1.53714
$508$	10.9487	0.485770
$509$	17.1967	0.762232	0.381116	−	0.924527i	$-0.375540\pi$
$509$	17.1967	0.762232	0.381116	+	0.924527i	$0.375540\pi$
$510$	−5.00969	−0.221833
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	22.7808	1.00580
$514$	15.7168	0.693237
$515$	21.3260	0.939737
$516$	6.12498	0.269637
$517$	0.289192	0.0127187
$518$	0	0
$519$	7.92692	0.347953
$520$	−8.66248	−0.379875
$521$	−0.432960	−0.0189683	−0.00948417	−	0.999955i	$-0.503019\pi$
$521$	−0.432960	−0.0189683	−0.00948417	+	0.999955i	$0.503019\pi$
$522$	5.85086	0.256085
$523$	32.9071	1.43893	0.719463	−	0.694531i	$-0.244388\pi$
$523$	32.9071	1.43893	0.719463	+	0.694531i	$0.244388\pi$
$524$	−4.43967	−0.193948
$525$	0	0
$526$	18.5623	0.809353
$527$	−2.08469	−0.0908104
$528$	0.417895	0.0181865
$529$	−9.65710	−0.419874
$530$	−17.4873	−0.759598
$531$	−13.0901	−0.568060
$532$	0	0
$533$	42.1594	1.82613
$534$	−0.780167	−0.0337611
$535$	−23.6515	−1.02254
$536$	−3.14914	−0.136022
$537$	−4.31468	−0.186192
$538$	−11.1304	−0.479864
$539$	0	0
$540$	7.52111	0.323657
$541$	10.3056	0.443072	0.221536	−	0.975152i	$-0.428893\pi$
$541$	10.3056	0.443072	0.221536	+	0.975152i	$0.428893\pi$
$542$	−20.1250	−0.864442
$543$	−17.0925	−0.733508
$544$	2.96077	0.126942
$545$	9.62373	0.412235
$546$	0	0
$547$	4.55363	0.194699	0.0973496	−	0.995250i	$-0.468964\pi$
$547$	4.55363	0.194699	0.0973496	+	0.995250i	$0.468964\pi$
$548$	−15.7627	−0.673350
$549$	−19.0151	−0.811543
$550$	−1.05861	−0.0451391
$551$	16.6407	0.708918
$552$	−4.55496	−0.193872
$553$	0	0
$554$	25.8485	1.09820
$555$	−11.4058	−0.484150
$556$	−17.9584	−0.761605
$557$	−28.2911	−1.19873	−0.599366	−	0.800475i	$-0.704581\pi$
$557$	−28.2911	−1.19873	−0.599366	+	0.800475i	$0.704581\pi$
$558$	1.01746	0.0430724
$559$	−31.3575	−1.32628
$560$	0	0
$561$	−1.23729	−0.0522385
$562$	13.0737	0.551480
$563$	26.1062	1.10024	0.550122	−	0.835084i	$-0.314581\pi$
$563$	26.1062	1.10024	0.550122	+	0.835084i	$0.314581\pi$
$564$	1.07606	0.0453105
$565$	−17.5627	−0.738870
$566$	−5.77718	−0.242833
$567$	0	0
$568$	−10.5700	−0.443508
$569$	−9.66248	−0.405072	−0.202536	−	0.979275i	$-0.564918\pi$
$569$	−9.66248	−0.405072	−0.202536	+	0.979275i	$0.564918\pi$
$570$	6.95407	0.291274
$571$	−17.5415	−0.734091	−0.367045	−	0.930203i	$-0.619631\pi$
$571$	−17.5415	−0.734091	−0.367045	+	0.930203i	$0.619631\pi$
$572$	−2.13946	−0.0894552
$573$	8.64071	0.360971
$574$	0	0
$575$	11.5386	0.481191
$576$	−1.44504	−0.0602101
$577$	−5.04593	−0.210065	−0.105032	−	0.994469i	$-0.533495\pi$
$577$	−5.04593	−0.210065	−0.105032	+	0.994469i	$0.533495\pi$
$578$	8.23383	0.342482
$579$	0.893068	0.0371146
$580$	5.49396	0.228124
$581$	0	0
$582$	4.04892	0.167833
$583$	−4.31900	−0.178875
$584$	−10.0218	−0.414704
$585$	−12.5176	−0.517541
$586$	−20.5200	−0.847675
$587$	−23.3787	−0.964941	−0.482470	−	0.875912i	$-0.660260\pi$
$587$	−23.3787	−0.964941	−0.482470	+	0.875912i	$0.660260\pi$
$588$	0	0
$589$	2.89380	0.119237
$590$	−12.2916	−0.506037
$591$	−2.57002	−0.105717
$592$	6.74094	0.277051
$593$	21.4330	0.880146	0.440073	−	0.897962i	$-0.354953\pi$
$593$	21.4330	0.880146	0.440073	+	0.897962i	$0.354953\pi$
$594$	1.85756	0.0762166
$595$	0	0
$596$	14.0911	0.577195
$597$	−0.0271471	−0.00111106
$598$	23.3196	0.953609
$599$	−33.8170	−1.38173	−0.690863	−	0.722986i	$-0.742769\pi$
$599$	−33.8170	−1.38173	−0.690863	+	0.722986i	$0.742769\pi$
$600$	−3.93900	−0.160809
$601$	−44.4161	−1.81177	−0.905885	−	0.423524i	$-0.860793\pi$
$601$	−44.4161	−1.81177	−0.905885	+	0.423524i	$0.860793\pi$
$602$	0	0
$603$	−4.55065	−0.185317
$604$	12.2567	0.498717
$605$	−14.7735	−0.600627
$606$	2.21313	0.0899022
$607$	5.74525	0.233193	0.116596	−	0.993179i	$-0.462802\pi$
$607$	5.74525	0.233193	0.116596	+	0.993179i	$0.462802\pi$
$608$	−4.10992	−0.166679
$609$	0	0
$610$	−17.8552	−0.722935
$611$	−5.50902	−0.222871
$612$	4.27844	0.172946
$613$	40.4741	1.63473	0.817367	−	0.576117i	$-0.195433\pi$
$613$	40.4741	1.63473	0.817367	+	0.576117i	$0.195433\pi$
$614$	−10.2446	−0.413438
$615$	−11.1739	−0.450575
$616$	0	0
$617$	−35.3706	−1.42397	−0.711984	−	0.702196i	$-0.752203\pi$
$617$	−35.3706	−1.42397	−0.711984	+	0.702196i	$0.752203\pi$
$618$	19.5985	0.788367
$619$	27.1282	1.09038	0.545188	−	0.838314i	$-0.316458\pi$
$619$	27.1282	1.09038	0.545188	+	0.838314i	$0.316458\pi$
$620$	0.955395	0.0383696
$621$	−20.2470	−0.812483
$622$	−34.6679	−1.39005
$623$	0	0
$624$	−7.96077	−0.318686
$625$	0.772398	0.0308959
$626$	11.6993	0.467599
$627$	1.71751	0.0685908
$628$	−21.2054	−0.846186
$629$	−19.9584	−0.795793
$630$	0	0
$631$	−43.8418	−1.74531	−0.872656	−	0.488335i	$-0.837605\pi$
$631$	−43.8418	−1.74531	−0.872656	+	0.488335i	$0.837605\pi$
$632$	8.86294	0.352549
$633$	34.6504	1.37723
$634$	6.82908	0.271218
$635$	14.8562	0.589552
$636$	−16.0707	−0.637244
$637$	0	0
$638$	1.35690	0.0537200
$639$	−15.2741	−0.604235
$640$	−1.35690	−0.0536360
$641$	25.4765	1.00626	0.503131	−	0.864210i	$-0.332181\pi$
$641$	25.4765	1.00626	0.503131	+	0.864210i	$0.332181\pi$
$642$	−21.7356	−0.857834
$643$	−4.20344	−0.165767	−0.0828837	−	0.996559i	$-0.526413\pi$
$643$	−4.20344	−0.165767	−0.0828837	+	0.996559i	$0.526413\pi$
$644$	0	0
$645$	8.31096	0.327244
$646$	12.1685	0.478764
$647$	−36.4282	−1.43214	−0.716070	−	0.698029i	$-0.754061\pi$
$647$	−36.4282	−1.43214	−0.716070	+	0.698029i	$0.754061\pi$
$648$	2.57673	0.101223
$649$	−3.03577	−0.119164
$650$	20.1661	0.790981
$651$	0	0
$652$	18.4862	0.723975
$653$	−36.0538	−1.41090	−0.705448	−	0.708762i	$-0.749254\pi$
$653$	−36.0538	−1.41090	−0.705448	+	0.708762i	$0.749254\pi$
$654$	8.84415	0.345834
$655$	−6.02416	−0.235384
$656$	6.60388	0.257838
$657$	−14.4819	−0.564992
$658$	0	0
$659$	−29.5881	−1.15259	−0.576294	−	0.817243i	$-0.695502\pi$
$659$	−29.5881	−1.15259	−0.576294	+	0.817243i	$0.695502\pi$
$660$	0.567040	0.0220720
$661$	−13.3405	−0.518885	−0.259443	−	0.965759i	$-0.583539\pi$
$661$	−13.3405	−0.518885	−0.259443	+	0.965759i	$0.583539\pi$
$662$	15.2664	0.593344
$663$	23.5700	0.915384
$664$	17.4426	0.676906
$665$	0	0
$666$	9.74094	0.377454
$667$	−14.7899	−0.572666
$668$	−20.2741	−0.784430
$669$	15.8388	0.612362
$670$	−4.27306	−0.165083
$671$	−4.40986	−0.170241
$672$	0	0
$673$	22.5332	0.868591	0.434295	−	0.900771i	$-0.356997\pi$
$673$	22.5332	0.868591	0.434295	+	0.900771i	$0.356997\pi$
$674$	0.0163935	0.000631452 0
$675$	−17.5090	−0.673923
$676$	27.7560	1.06754
$677$	25.5924	0.983595	0.491798	−	0.870710i	$-0.336340\pi$
$677$	25.5924	0.983595	0.491798	+	0.870710i	$0.336340\pi$
$678$	−16.1400	−0.619855
$679$	0	0
$680$	4.01746	0.154062
$681$	8.10082	0.310424
$682$	0.235963	0.00903549
$683$	−2.17390	−0.0831819	−0.0415910	−	0.999135i	$-0.513243\pi$
$683$	−2.17390	−0.0831819	−0.0415910	+	0.999135i	$0.513243\pi$
$684$	−5.93900	−0.227083
$685$	−21.3884	−0.817207
$686$	0	0
$687$	−12.0978	−0.461561
$688$	−4.91185	−0.187263
$689$	82.2756	3.13445
$690$	−6.18060	−0.235292
$691$	−20.5515	−0.781816	−0.390908	−	0.920430i	$-0.627839\pi$
$691$	−20.5515	−0.781816	−0.390908	+	0.920430i	$0.627839\pi$
$692$	−6.35690	−0.241653
$693$	0	0
$694$	−19.5133	−0.740716
$695$	−24.3676	−0.924318
$696$	5.04892	0.191379
$697$	−19.5526	−0.740606
$698$	−11.7778	−0.445795
$699$	−2.26875	−0.0858120
$700$	0	0
$701$	20.8170	0.786247	0.393124	−	0.919486i	$-0.371394\pi$
$701$	20.8170	0.786247	0.393124	+	0.919486i	$0.371394\pi$
$702$	−35.3860	−1.33556
$703$	27.7047	1.04490
$704$	−0.335126	−0.0126305
$705$	1.46011	0.0549908
$706$	−34.1672	−1.28590
$707$	0	0
$708$	−11.2959	−0.424526
$709$	8.66679	0.325488	0.162744	−	0.986668i	$-0.447965\pi$
$709$	8.66679	0.325488	0.162744	+	0.986668i	$0.447965\pi$
$710$	−14.3424	−0.538261
$711$	12.8073	0.480312
$712$	0.625646	0.0234471
$713$	−2.57194	−0.0963200
$714$	0	0
$715$	−2.90302	−0.108567
$716$	3.46011	0.129310
$717$	−10.2198	−0.381667
$718$	−10.6558	−0.397670
$719$	−26.3545	−0.982857	−0.491429	−	0.870918i	$-0.663525\pi$
$719$	−26.3545	−0.982857	−0.491429	+	0.870918i	$0.663525\pi$
$720$	−1.96077	−0.0730736
$721$	0	0
$722$	2.10859	0.0784735
$723$	−6.53079	−0.242883
$724$	13.7071	0.509420
$725$	−12.7899	−0.475003
$726$	−13.5767	−0.503880
$727$	16.8931	0.626529	0.313265	−	0.949666i	$-0.398577\pi$
$727$	16.8931	0.626529	0.313265	+	0.949666i	$0.398577\pi$
$728$	0	0
$729$	24.4590	0.905890
$730$	−13.5985	−0.503303
$731$	14.5429	0.537888
$732$	−16.4088	−0.606487
$733$	17.0586	0.630074	0.315037	−	0.949079i	$-0.397983\pi$
$733$	17.0586	0.630074	0.315037	+	0.949079i	$0.397983\pi$
$734$	33.0978	1.22166
$735$	0	0
$736$	3.65279	0.134644
$737$	−1.05536	−0.0388747
$738$	9.54288	0.351278
$739$	−36.3443	−1.33695	−0.668474	−	0.743735i	$-0.733052\pi$
$739$	−36.3443	−1.33695	−0.668474	+	0.743735i	$0.733052\pi$
$740$	9.14675	0.336241
$741$	−32.7181	−1.20193
$742$	0	0
$743$	34.1890	1.25427	0.627136	−	0.778910i	$-0.284227\pi$
$743$	34.1890	1.25427	0.627136	+	0.778910i	$0.284227\pi$
$744$	0.878002	0.0321891
$745$	19.1202	0.700510
$746$	18.5754	0.680094
$747$	25.2054	0.922216
$748$	0.992230	0.0362795
$749$	0	0
$750$	−13.8049	−0.504084
$751$	−9.66547	−0.352698	−0.176349	−	0.984328i	$-0.556429\pi$
$751$	−9.66547	−0.352698	−0.176349	+	0.984328i	$0.556429\pi$
$752$	−0.862937	−0.0314681
$753$	−1.21313	−0.0442088
$754$	−25.8485	−0.941345
$755$	16.6310	0.605265
$756$	0	0
$757$	2.97584	0.108159	0.0540793	−	0.998537i	$-0.482778\pi$
$757$	2.97584	0.108159	0.0540793	+	0.998537i	$0.482778\pi$
$758$	−8.85517	−0.321634
$759$	−1.52648	−0.0554078
$760$	−5.57673	−0.202289
$761$	22.1884	0.804328	0.402164	−	0.915568i	$-0.368258\pi$
$761$	22.1884	0.804328	0.402164	+	0.915568i	$0.368258\pi$
$762$	13.6528	0.494588
$763$	0	0
$764$	−6.92931	−0.250694
$765$	5.80540	0.209895
$766$	30.6969	1.10912
$767$	57.8305	2.08814
$768$	−1.24698	−0.0449965
$769$	−27.5803	−0.994571	−0.497286	−	0.867587i	$-0.665670\pi$
$769$	−27.5803	−0.994571	−0.497286	+	0.867587i	$0.665670\pi$
$770$	0	0
$771$	19.5985	0.705823
$772$	−0.716185	−0.0257760
$773$	−36.5327	−1.31399	−0.656995	−	0.753895i	$-0.728173\pi$
$773$	−36.5327	−1.31399	−0.656995	+	0.753895i	$0.728173\pi$
$774$	−7.09783	−0.255126
$775$	−2.22414	−0.0798936
$776$	−3.24698	−0.116560
$777$	0	0
$778$	1.26205	0.0452465
$779$	27.1414	0.972441
$780$	−10.8019	−0.386771
$781$	−3.54229	−0.126753
$782$	−10.8151	−0.386747
$783$	22.4426	0.802035
$784$	0	0
$785$	−28.7735	−1.02697
$786$	−5.53617	−0.197469
$787$	−14.3870	−0.512842	−0.256421	−	0.966565i	$-0.582543\pi$
$787$	−14.3870	−0.512842	−0.256421	+	0.966565i	$0.582543\pi$
$788$	2.06100	0.0734200
$789$	23.1468	0.824046
$790$	12.0261	0.427869
$791$	0	0
$792$	−0.484271	−0.0172078
$793$	84.0066	2.98316
$794$	12.4558	0.442040
$795$	−21.8062	−0.773388
$796$	0.0217703	0.000771627 0
$797$	−19.1564	−0.678556	−0.339278	−	0.940686i	$-0.610183\pi$
$797$	−19.1564	−0.678556	−0.339278	+	0.940686i	$0.610183\pi$
$798$	0	0
$799$	2.55496	0.0903879
$800$	3.15883	0.111682
$801$	0.904084	0.0319442
$802$	11.8267	0.417615
$803$	−3.35855	−0.118521
$804$	−3.92692	−0.138492
$805$	0	0
$806$	−4.49502	−0.158330
$807$	−13.8793	−0.488576
$808$	−1.77479	−0.0624369
$809$	−47.2771	−1.66217	−0.831087	−	0.556142i	$-0.812281\pi$
$809$	−47.2771	−1.66217	−0.831087	+	0.556142i	$0.812281\pi$
$810$	3.49635	0.122849
$811$	35.9691	1.26305	0.631524	−	0.775357i	$-0.282430\pi$
$811$	35.9691	1.26305	0.631524	+	0.775357i	$0.282430\pi$
$812$	0	0
$813$	−25.0954	−0.880136
$814$	2.25906	0.0791801
$815$	25.0838	0.878648
$816$	3.69202	0.129247
$817$	−20.1873	−0.706265
$818$	−18.4058	−0.643544
$819$	0	0
$820$	8.96077	0.312924
$821$	−15.9554	−0.556847	−0.278424	−	0.960458i	$-0.589812\pi$
$821$	−15.9554	−0.556847	−0.278424	+	0.960458i	$0.589812\pi$
$822$	−19.6558	−0.685574
$823$	32.5810	1.13570	0.567852	−	0.823131i	$-0.307775\pi$
$823$	32.5810	1.13570	0.567852	+	0.823131i	$0.307775\pi$
$824$	−15.7168	−0.547520
$825$	−1.32006	−0.0459586
$826$	0	0
$827$	18.6300	0.647827	0.323914	−	0.946087i	$-0.395001\pi$
$827$	18.6300	0.647827	0.323914	+	0.946087i	$0.395001\pi$
$828$	5.27844	0.183438
$829$	−16.9661	−0.589259	−0.294629	−	0.955612i	$-0.595196\pi$
$829$	−16.9661	−0.589259	−0.294629	+	0.955612i	$0.595196\pi$
$830$	23.6679	0.821523
$831$	32.2325	1.11813
$832$	6.38404	0.221327
$833$	0	0
$834$	−22.3937	−0.775432
$835$	−27.5099	−0.952019
$836$	−1.37734	−0.0476362
$837$	3.90276	0.134899
$838$	5.12067	0.176891
$839$	−9.12365	−0.314984	−0.157492	−	0.987520i	$-0.550341\pi$
$839$	−9.12365	−0.314984	−0.157492	+	0.987520i	$0.550341\pi$
$840$	0	0
$841$	−12.6063	−0.434699
$842$	16.4034	0.565299
$843$	16.3026	0.561491
$844$	−27.7875	−0.956484
$845$	37.6620	1.29561
$846$	−1.24698	−0.0428720
$847$	0	0
$848$	12.8877	0.442565
$849$	−7.20403	−0.247242
$850$	−9.35258	−0.320791
$851$	−24.6233	−0.844074
$852$	−13.1806	−0.451560
$853$	−10.2620	−0.351366	−0.175683	−	0.984447i	$-0.556213\pi$
$853$	−10.2620	−0.351366	−0.175683	+	0.984447i	$0.556213\pi$
$854$	0	0
$855$	−8.05861	−0.275599
$856$	17.4306	0.595765
$857$	11.2832	0.385428	0.192714	−	0.981255i	$-0.438271\pi$
$857$	11.2832	0.385428	0.192714	+	0.981255i	$0.438271\pi$
$858$	−2.66786	−0.0910792
$859$	20.0242	0.683216	0.341608	−	0.939843i	$-0.389029\pi$
$859$	20.0242	0.683216	0.341608	+	0.939843i	$0.389029\pi$
$860$	−6.66487	−0.227270
$861$	0	0
$862$	0.330814	0.0112676
$863$	41.4596	1.41130	0.705651	−	0.708559i	$-0.250655\pi$
$863$	41.4596	1.41130	0.705651	+	0.708559i	$0.250655\pi$
$864$	−5.54288	−0.188572
$865$	−8.62565	−0.293281
$866$	2.45042	0.0832686
$867$	10.2674	0.348700
$868$	0	0
$869$	2.97020	0.100757
$870$	6.85086	0.232266
$871$	20.1043	0.681207
$872$	−7.09246	−0.240181
$873$	−4.69202	−0.158801
$874$	15.0127	0.507811
$875$	0	0
$876$	−12.4969	−0.422233
$877$	24.2034	0.817292	0.408646	−	0.912693i	$-0.366001\pi$
$877$	24.2034	0.817292	0.408646	+	0.912693i	$0.366001\pi$
$878$	−2.69202	−0.0908513
$879$	−25.5881	−0.863064
$880$	−0.454731	−0.0153290
$881$	37.3793	1.25934	0.629670	−	0.776863i	$-0.283190\pi$
$881$	37.3793	1.25934	0.629670	+	0.776863i	$0.283190\pi$
$882$	0	0
$883$	0.550172	0.0185148	0.00925739	−	0.999957i	$-0.497053\pi$
$883$	0.550172	0.0185148	0.00925739	+	0.999957i	$0.497053\pi$
$884$	−18.9017	−0.635733
$885$	−15.3274	−0.515223
$886$	−36.5526	−1.22801
$887$	25.3521	0.851241	0.425620	−	0.904902i	$-0.360056\pi$
$887$	25.3521	0.851241	0.425620	+	0.904902i	$0.360056\pi$
$888$	8.40581	0.282081
$889$	0	0
$890$	0.848936	0.0284564
$891$	0.863528	0.0289293
$892$	−12.7017	−0.425285
$893$	−3.54660	−0.118682
$894$	17.5714	0.587674
$895$	4.69501	0.156937
$896$	0	0
$897$	29.0790	0.970921
$898$	3.48427	0.116272
$899$	2.85086	0.0950813
$900$	4.56465	0.152155
$901$	−38.1575	−1.27121
$902$	2.21313	0.0736891
$903$	0	0
$904$	12.9433	0.430488
$905$	18.5991	0.618255
$906$	15.2838	0.507771
$907$	53.9512	1.79142	0.895710	−	0.444639i	$-0.146668\pi$
$907$	53.9512	1.79142	0.895710	+	0.444639i	$0.146668\pi$
$908$	−6.49635	−0.215589
$909$	−2.56465	−0.0850640
$910$	0	0
$911$	13.2433	0.438769	0.219384	−	0.975639i	$-0.429595\pi$
$911$	13.2433	0.438769	0.219384	+	0.975639i	$0.429595\pi$
$912$	−5.12498	−0.169705
$913$	5.84548	0.193457
$914$	23.1183	0.764685
$915$	−22.2650	−0.736059
$916$	9.70171	0.320554
$917$	0	0
$918$	16.4112	0.541650
$919$	−24.2389	−0.799569	−0.399785	−	0.916609i	$-0.630915\pi$
$919$	−24.2389	−0.799569	−0.399785	+	0.916609i	$0.630915\pi$
$920$	4.95646	0.163410
$921$	−12.7748	−0.420944
$922$	−9.69501	−0.319288
$923$	67.4795	2.22111
$924$	0	0
$925$	−21.2935	−0.700126
$926$	5.84654	0.192129
$927$	−22.7114	−0.745940
$928$	−4.04892	−0.132912
$929$	−12.4295	−0.407799	−0.203899	−	0.978992i	$-0.565362\pi$
$929$	−12.4295	−0.407799	−0.203899	+	0.978992i	$0.565362\pi$
$930$	1.19136	0.0390662
$931$	0	0
$932$	1.81940	0.0595963
$933$	−43.2301	−1.41529
$934$	25.7832	0.843650
$935$	1.34635	0.0440305
$936$	9.22521	0.301535
$937$	25.0672	0.818911	0.409455	−	0.912330i	$-0.365719\pi$
$937$	25.0672	0.818911	0.409455	+	0.912330i	$0.365719\pi$
$938$	0	0
$939$	14.5888	0.476088
$940$	−1.17092	−0.0381910
$941$	−41.8961	−1.36577	−0.682886	−	0.730525i	$-0.739275\pi$
$941$	−41.8961	−1.36577	−0.682886	+	0.730525i	$0.739275\pi$
$942$	−26.4426	−0.861548
$943$	−24.1226	−0.785540
$944$	9.05861	0.294833
$945$	0	0
$946$	−1.64609	−0.0535189
$947$	13.9788	0.454251	0.227125	−	0.973866i	$-0.427067\pi$
$947$	13.9788	0.454251	0.227125	+	0.973866i	$0.427067\pi$
$948$	11.0519	0.358949
$949$	63.9794	2.07686
$950$	12.9825	0.421209
$951$	8.51573	0.276141
$952$	0	0
$953$	−14.9038	−0.482782	−0.241391	−	0.970428i	$-0.577604\pi$
$953$	−14.9038	−0.482782	−0.241391	+	0.970428i	$0.577604\pi$
$954$	18.6233	0.602950
$955$	−9.40236	−0.304253
$956$	8.19567	0.265067
$957$	1.69202	0.0546953
$958$	−20.8799	−0.674600
$959$	0	0
$960$	−1.69202	−0.0546098
$961$	−30.5042	−0.984008
$962$	−43.0344	−1.38749
$963$	25.1879	0.811669
$964$	5.23729	0.168682
$965$	−0.971788	−0.0312830
$966$	0	0
$967$	42.6469	1.37143	0.685717	−	0.727869i	$-0.259489\pi$
$967$	42.6469	1.37143	0.685717	+	0.727869i	$0.259489\pi$
$968$	10.8877	0.349944
$969$	15.1739	0.487456
$970$	−4.40581	−0.141462
$971$	−3.03790	−0.0974909	−0.0487454	−	0.998811i	$-0.515522\pi$
$971$	−3.03790	−0.0974909	−0.0487454	+	0.998811i	$0.515522\pi$
$972$	−13.4155	−0.430302
$973$	0	0
$974$	0.629958	0.0201851
$975$	25.1468	0.805341
$976$	13.1588	0.421204
$977$	5.60089	0.179188	0.0895942	−	0.995978i	$-0.471443\pi$
$977$	5.60089	0.179188	0.0895942	+	0.995978i	$0.471443\pi$
$978$	23.0519	0.737119
$979$	0.209670	0.00670108
$980$	0	0
$981$	−10.2489	−0.327222
$982$	7.19269	0.229528
$983$	−8.13600	−0.259498	−0.129749	−	0.991547i	$-0.541417\pi$
$983$	−8.13600	−0.259498	−0.129749	+	0.991547i	$0.541417\pi$
$984$	8.23490	0.262519
$985$	2.79656	0.0891058
$986$	11.9879	0.381773
$987$	0	0
$988$	26.2379	0.834738
$989$	17.9420	0.570522
$990$	−0.657105	−0.0208842
$991$	−16.1637	−0.513458	−0.256729	−	0.966483i	$-0.582645\pi$
$991$	−16.1637	−0.513458	−0.256729	+	0.966483i	$0.582645\pi$
$992$	−0.704103	−0.0223553
$993$	19.0368	0.604116
$994$	0	0
$995$	0.0295400	0.000936480 0
$996$	21.7506	0.689195
$997$	−31.6614	−1.00273	−0.501364	−	0.865237i	$-0.667168\pi$
$997$	−31.6614	−1.00273	−0.501364	+	0.865237i	$0.667168\pi$
$998$	−21.2567	−0.672868
$999$	37.3642	1.18215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	686.2.a.b.1.1	yes	3
3.2	odd	2		6174.2.a.l.1.2		3
4.3	odd	2		5488.2.a.b.1.3		3
7.2	even	3		686.2.c.c.361.3		6
7.3	odd	6		686.2.c.d.667.1		6
7.4	even	3		686.2.c.c.667.3		6
7.5	odd	6		686.2.c.d.361.1		6
7.6	odd	2		686.2.a.a.1.3	✓	3
21.20	even	2		6174.2.a.k.1.2		3
28.27	even	2		5488.2.a.e.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
686.2.a.a.1.3	✓	3	7.6	odd	2
686.2.a.b.1.1	yes	3	1.1	even	1	trivial
686.2.c.c.361.3		6	7.2	even	3
686.2.c.c.667.3		6	7.4	even	3
686.2.c.d.361.1		6	7.5	odd	6
686.2.c.d.667.1		6	7.3	odd	6
5488.2.a.b.1.3		3	4.3	odd	2
5488.2.a.e.1.1		3	28.27	even	2
6174.2.a.k.1.2		3	21.20	even	2
6174.2.a.l.1.2		3	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:	\( N \)	\(=\)	\( 686 = 2 \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	686.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.24698 q^{3} +1.00000 q^{4} +1.35690 q^{5} +1.24698 q^{6} -1.00000 q^{8} -1.44504 q^{9} -1.35690 q^{10} -0.335126 q^{11} -1.24698 q^{12} +6.38404 q^{13} -1.69202 q^{15} +1.00000 q^{16} -2.96077 q^{17} +1.44504 q^{18} +4.10992 q^{19} +1.35690 q^{20} +0.335126 q^{22} -3.65279 q^{23} +1.24698 q^{24} -3.15883 q^{25} -6.38404 q^{26} +5.54288 q^{27} +4.04892 q^{29} +1.69202 q^{30} +0.704103 q^{31} -1.00000 q^{32} +0.417895 q^{33} +2.96077 q^{34} -1.44504 q^{36} +6.74094 q^{37} -4.10992 q^{38} -7.96077 q^{39} -1.35690 q^{40} +6.60388 q^{41} -4.91185 q^{43} -0.335126 q^{44} -1.96077 q^{45} +3.65279 q^{46} -0.862937 q^{47} -1.24698 q^{48} +3.15883 q^{50} +3.69202 q^{51} +6.38404 q^{52} +12.8877 q^{53} -5.54288 q^{54} -0.454731 q^{55} -5.12498 q^{57} -4.04892 q^{58} +9.05861 q^{59} -1.69202 q^{60} +13.1588 q^{61} -0.704103 q^{62} +1.00000 q^{64} +8.66248 q^{65} -0.417895 q^{66} +3.14914 q^{67} -2.96077 q^{68} +4.55496 q^{69} +10.5700 q^{71} +1.44504 q^{72} +10.0218 q^{73} -6.74094 q^{74} +3.93900 q^{75} +4.10992 q^{76} +7.96077 q^{78} -8.86294 q^{79} +1.35690 q^{80} -2.57673 q^{81} -6.60388 q^{82} -17.4426 q^{83} -4.01746 q^{85} +4.91185 q^{86} -5.04892 q^{87} +0.335126 q^{88} -0.625646 q^{89} +1.96077 q^{90} -3.65279 q^{92} -0.878002 q^{93} +0.862937 q^{94} +5.57673 q^{95} +1.24698 q^{96} +3.24698 q^{97} +0.484271 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + q^{3} + 3 q^{4} - q^{6} - 3 q^{8} - 4 q^{9} + q^{12} + 9 q^{13} + 3 q^{16} + 4 q^{17} + 4 q^{18} + 13 q^{19} + 7 q^{23} - q^{24} - q^{25} - 9 q^{26} - 2 q^{27} + 3 q^{29} + 16 q^{31}+ \cdots + 14 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + q^3 + 3 * q^4 - q^6 - 3 * q^8 - 4 * q^9 + q^12 + 9 * q^13 + 3 * q^16 + 4 * q^17 + 4 * q^18 + 13 * q^19 + 7 * q^23 - q^24 - q^25 - 9 * q^26 - 2 * q^27 + 3 * q^29 + 16 * q^31 - 3 * q^32 + 7 * q^33 - 4 * q^34 - 4 * q^36 + 6 * q^37 - 13 * q^38 - 11 * q^39 + 11 * q^41 - 11 * q^43 + 7 * q^45 - 7 * q^46 - 8 * q^47 + q^48 + q^50 + 6 * q^51 + 9 * q^52 - 3 * q^53 + 2 * q^54 + 21 * q^55 + 9 * q^57 - 3 * q^58 - 4 * q^59 + 31 * q^61 - 16 * q^62 + 3 * q^64 - 14 * q^65 - 7 * q^66 + 23 * q^67 + 4 * q^68 + 14 * q^69 + 7 * q^71 + 4 * q^72 + 27 * q^73 - 6 * q^74 + 2 * q^75 + 13 * q^76 + 11 * q^78 - 32 * q^79 - 5 * q^81 - 11 * q^82 - 11 * q^83 - 28 * q^85 + 11 * q^86 - 6 * q^87 + 10 * q^89 - 7 * q^90 + 7 * q^92 + 17 * q^93 + 8 * q^94 + 14 * q^95 - q^96 + 5 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

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Database

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