LMFDB - Embedded newform 688.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [688,2,Mod(1,688)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(688, 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("688.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 688 = 2^{4} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	688.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:	$5.49370765906$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 344)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	688.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} -2.73205 q^{5} +2.73205 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} -2.73205 q^{5} +2.73205 q^{7} -2.46410 q^{9} +3.00000 q^{11} -3.00000 q^{13} +2.00000 q^{15} +6.46410 q^{17} +0.535898 q^{19} -2.00000 q^{21} +1.00000 q^{23} +2.46410 q^{25} +4.00000 q^{27} +6.19615 q^{29} +5.00000 q^{31} -2.19615 q^{33} -7.46410 q^{35} -1.46410 q^{37} +2.19615 q^{39} +9.92820 q^{41} +1.00000 q^{43} +6.73205 q^{45} +10.3923 q^{47} +0.464102 q^{49} -4.73205 q^{51} -2.46410 q^{53} -8.19615 q^{55} -0.392305 q^{57} +8.92820 q^{59} -12.7321 q^{61} -6.73205 q^{63} +8.19615 q^{65} +7.92820 q^{67} -0.732051 q^{69} -3.46410 q^{71} +10.1962 q^{73} -1.80385 q^{75} +8.19615 q^{77} -3.07180 q^{79} +4.46410 q^{81} -16.8564 q^{83} -17.6603 q^{85} -4.53590 q^{87} -0.196152 q^{89} -8.19615 q^{91} -3.66025 q^{93} -1.46410 q^{95} -10.4641 q^{97} -7.39230 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 6 q^{13} + 4 q^{15} + 6 q^{17} + 8 q^{19} - 4 q^{21} + 2 q^{23} - 2 q^{25} + 8 q^{27} + 2 q^{29} + 10 q^{31} + 6 q^{33} - 8 q^{35} + 4 q^{37} - 6 q^{39} + 6 q^{41} + 2 q^{43} + 10 q^{45} - 6 q^{49} - 6 q^{51} + 2 q^{53} - 6 q^{55} + 20 q^{57} + 4 q^{59} - 22 q^{61} - 10 q^{63} + 6 q^{65} + 2 q^{67} + 2 q^{69} + 10 q^{73} - 14 q^{75} + 6 q^{77} - 20 q^{79} + 2 q^{81} - 6 q^{83} - 18 q^{85} - 16 q^{87} + 10 q^{89} - 6 q^{91} + 10 q^{93} + 4 q^{95} - 14 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 6 * q^13 + 4 * q^15 + 6 * q^17 + 8 * q^19 - 4 * q^21 + 2 * q^23 - 2 * q^25 + 8 * q^27 + 2 * q^29 + 10 * q^31 + 6 * q^33 - 8 * q^35 + 4 * q^37 - 6 * q^39 + 6 * q^41 + 2 * q^43 + 10 * q^45 - 6 * q^49 - 6 * q^51 + 2 * q^53 - 6 * q^55 + 20 * q^57 + 4 * q^59 - 22 * q^61 - 10 * q^63 + 6 * q^65 + 2 * q^67 + 2 * q^69 + 10 * q^73 - 14 * q^75 + 6 * q^77 - 20 * q^79 + 2 * q^81 - 6 * q^83 - 18 * q^85 - 16 * q^87 + 10 * q^89 - 6 * q^91 + 10 * q^93 + 4 * q^95 - 14 * q^97 + 6 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	−2.73205	−1.22181	−0.610905	−	0.791704i	$-0.709194\pi$
$5$	−2.73205	−1.22181	−0.610905	+	0.791704i	$0.709194\pi$
$6$	0	0
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	0	0
$17$	6.46410	1.56777	0.783887	−	0.620903i	$-0.213234\pi$
$17$	6.46410	1.56777	0.783887	+	0.620903i	$0.213234\pi$
$18$	0	0
$19$	0.535898	0.122944	0.0614718	−	0.998109i	$-0.480421\pi$
$19$	0.535898	0.122944	0.0614718	+	0.998109i	$0.480421\pi$
$20$	0	0
$21$	−2.00000	−0.436436
$22$	0	0
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	2.46410	0.492820
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	6.19615	1.15060	0.575298	−	0.817944i	$-0.304886\pi$
$29$	6.19615	1.15060	0.575298	+	0.817944i	$0.304886\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	−2.19615	−0.382301
$34$	0	0
$35$	−7.46410	−1.26166
$36$	0	0
$37$	−1.46410	−0.240697	−0.120348	−	0.992732i	$-0.538401\pi$
$37$	−1.46410	−0.240697	−0.120348	+	0.992732i	$0.538401\pi$
$38$	0	0
$39$	2.19615	0.351666
$40$	0	0
$41$	9.92820	1.55052	0.775262	−	0.631639i	$-0.217618\pi$
$41$	9.92820	1.55052	0.775262	+	0.631639i	$0.217618\pi$
$42$	0	0
$43$	1.00000	0.152499
$43$	1.00000	0.152499
$44$	0	0
$45$	6.73205	1.00355
$46$	0	0
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	−4.73205	−0.662620
$52$	0	0
$53$	−2.46410	−0.338470	−0.169235	−	0.985576i	$-0.554130\pi$
$53$	−2.46410	−0.338470	−0.169235	+	0.985576i	$0.554130\pi$
$54$	0	0
$55$	−8.19615	−1.10517
$56$	0	0
$57$	−0.392305	−0.0519620
$58$	0	0
$59$	8.92820	1.16235	0.581177	−	0.813778i	$-0.302593\pi$
$59$	8.92820	1.16235	0.581177	+	0.813778i	$0.302593\pi$
$60$	0	0
$61$	−12.7321	−1.63017	−0.815086	−	0.579340i	$-0.803310\pi$
$61$	−12.7321	−1.63017	−0.815086	+	0.579340i	$0.803310\pi$
$62$	0	0
$63$	−6.73205	−0.848159
$64$	0	0
$65$	8.19615	1.01661
$66$	0	0
$67$	7.92820	0.968584	0.484292	−	0.874906i	$-0.339077\pi$
$67$	7.92820	0.968584	0.484292	+	0.874906i	$0.339077\pi$
$68$	0	0
$69$	−0.732051	−0.0881286
$70$	0	0
$71$	−3.46410	−0.411113	−0.205557	−	0.978645i	$-0.565900\pi$
$71$	−3.46410	−0.411113	−0.205557	+	0.978645i	$0.565900\pi$
$72$	0	0
$73$	10.1962	1.19337	0.596685	−	0.802476i	$-0.296484\pi$
$73$	10.1962	1.19337	0.596685	+	0.802476i	$0.296484\pi$
$74$	0	0
$75$	−1.80385	−0.208290
$76$	0	0
$77$	8.19615	0.934038
$78$	0	0
$79$	−3.07180	−0.345604	−0.172802	−	0.984957i	$-0.555282\pi$
$79$	−3.07180	−0.345604	−0.172802	+	0.984957i	$0.555282\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−16.8564	−1.85023	−0.925115	−	0.379686i	$-0.876032\pi$
$83$	−16.8564	−1.85023	−0.925115	+	0.379686i	$0.876032\pi$
$84$	0	0
$85$	−17.6603	−1.91552
$86$	0	0
$87$	−4.53590	−0.486299
$88$	0	0
$89$	−0.196152	−0.0207921	−0.0103961	−	0.999946i	$-0.503309\pi$
$89$	−0.196152	−0.0207921	−0.0103961	+	0.999946i	$0.503309\pi$
$90$	0	0
$91$	−8.19615	−0.859190
$92$	0	0
$93$	−3.66025	−0.379551
$94$	0	0
$95$	−1.46410	−0.150214
$96$	0	0
$97$	−10.4641	−1.06247	−0.531234	−	0.847225i	$-0.678272\pi$
$97$	−10.4641	−1.06247	−0.531234	+	0.847225i	$0.678272\pi$
$98$	0	0
$99$	−7.39230	−0.742955
$100$	0	0
$101$	0.464102	0.0461798	0.0230899	−	0.999733i	$-0.492650\pi$
$101$	0.464102	0.0461798	0.0230899	+	0.999733i	$0.492650\pi$
$102$	0	0
$103$	−12.8564	−1.26678	−0.633390	−	0.773833i	$-0.718337\pi$
$103$	−12.8564	−1.26678	−0.633390	+	0.773833i	$0.718337\pi$
$104$	0	0
$105$	5.46410	0.533242
$106$	0	0
$107$	11.4641	1.10828	0.554138	−	0.832425i	$-0.313048\pi$
$107$	11.4641	1.10828	0.554138	+	0.832425i	$0.313048\pi$
$108$	0	0
$109$	−5.53590	−0.530243	−0.265121	−	0.964215i	$-0.585412\pi$
$109$	−5.53590	−0.530243	−0.265121	+	0.964215i	$0.585412\pi$
$110$	0	0
$111$	1.07180	0.101730
$112$	0	0
$113$	10.5359	0.991134	0.495567	−	0.868570i	$-0.334960\pi$
$113$	10.5359	0.991134	0.495567	+	0.868570i	$0.334960\pi$
$114$	0	0
$115$	−2.73205	−0.254765
$116$	0	0
$117$	7.39230	0.683419
$118$	0	0
$119$	17.6603	1.61891
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	−7.26795	−0.655329
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−12.8564	−1.14082	−0.570411	−	0.821360i	$-0.693216\pi$
$127$	−12.8564	−1.14082	−0.570411	+	0.821360i	$0.693216\pi$
$128$	0	0
$129$	−0.732051	−0.0644535
$130$	0	0
$131$	13.8564	1.21064	0.605320	−	0.795982i	$-0.293045\pi$
$131$	13.8564	1.21064	0.605320	+	0.795982i	$0.293045\pi$
$132$	0	0
$133$	1.46410	0.126954
$134$	0	0
$135$	−10.9282	−0.940550
$136$	0	0
$137$	−13.3205	−1.13805	−0.569024	−	0.822321i	$-0.692679\pi$
$137$	−13.3205	−1.13805	−0.569024	+	0.822321i	$0.692679\pi$
$138$	0	0
$139$	21.3923	1.81447	0.907236	−	0.420622i	$-0.138188\pi$
$139$	21.3923	1.81447	0.907236	+	0.420622i	$0.138188\pi$
$140$	0	0
$141$	−7.60770	−0.640684
$142$	0	0
$143$	−9.00000	−0.752618
$144$	0	0
$145$	−16.9282	−1.40581
$146$	0	0
$147$	−0.339746	−0.0280218
$148$	0	0
$149$	−16.3923	−1.34291	−0.671455	−	0.741045i	$-0.734330\pi$
$149$	−16.3923	−1.34291	−0.671455	+	0.741045i	$0.734330\pi$
$150$	0	0
$151$	8.58846	0.698919	0.349459	−	0.936952i	$-0.386365\pi$
$151$	8.58846	0.698919	0.349459	+	0.936952i	$0.386365\pi$
$152$	0	0
$153$	−15.9282	−1.28772
$154$	0	0
$155$	−13.6603	−1.09722
$156$	0	0
$157$	11.8564	0.946244	0.473122	−	0.880997i	$-0.343127\pi$
$157$	11.8564	0.946244	0.473122	+	0.880997i	$0.343127\pi$
$158$	0	0
$159$	1.80385	0.143054
$160$	0	0
$161$	2.73205	0.215316
$162$	0	0
$163$	10.5885	0.829352	0.414676	−	0.909969i	$-0.363895\pi$
$163$	10.5885	0.829352	0.414676	+	0.909969i	$0.363895\pi$
$164$	0	0
$165$	6.00000	0.467099
$166$	0	0
$167$	15.9282	1.23256	0.616281	−	0.787527i	$-0.288639\pi$
$167$	15.9282	1.23256	0.616281	+	0.787527i	$0.288639\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	−1.32051	−0.100982
$172$	0	0
$173$	−8.92820	−0.678799	−0.339399	−	0.940642i	$-0.610224\pi$
$173$	−8.92820	−0.678799	−0.339399	+	0.940642i	$0.610224\pi$
$174$	0	0
$175$	6.73205	0.508895
$176$	0	0
$177$	−6.53590	−0.491268
$178$	0	0
$179$	−22.7321	−1.69907	−0.849537	−	0.527530i	$-0.823118\pi$
$179$	−22.7321	−1.69907	−0.849537	+	0.527530i	$0.823118\pi$
$180$	0	0
$181$	−16.3923	−1.21843	−0.609215	−	0.793005i	$-0.708515\pi$
$181$	−16.3923	−1.21843	−0.609215	+	0.793005i	$0.708515\pi$
$182$	0	0
$183$	9.32051	0.688992
$184$	0	0
$185$	4.00000	0.294086
$186$	0	0
$187$	19.3923	1.41811
$188$	0	0
$189$	10.9282	0.794910
$190$	0	0
$191$	−23.3205	−1.68741	−0.843706	−	0.536805i	$-0.819631\pi$
$191$	−23.3205	−1.68741	−0.843706	+	0.536805i	$0.819631\pi$
$192$	0	0
$193$	12.8564	0.925424	0.462712	−	0.886509i	$-0.346876\pi$
$193$	12.8564	0.925424	0.462712	+	0.886509i	$0.346876\pi$
$194$	0	0
$195$	−6.00000	−0.429669
$196$	0	0
$197$	−5.07180	−0.361351	−0.180675	−	0.983543i	$-0.557828\pi$
$197$	−5.07180	−0.361351	−0.180675	+	0.983543i	$0.557828\pi$
$198$	0	0
$199$	3.07180	0.217754	0.108877	−	0.994055i	$-0.465275\pi$
$199$	3.07180	0.217754	0.108877	+	0.994055i	$0.465275\pi$
$200$	0	0
$201$	−5.80385	−0.409372
$202$	0	0
$203$	16.9282	1.18813
$204$	0	0
$205$	−27.1244	−1.89445
$206$	0	0
$207$	−2.46410	−0.171267
$208$	0	0
$209$	1.60770	0.111207
$210$	0	0
$211$	−3.46410	−0.238479	−0.119239	−	0.992866i	$-0.538046\pi$
$211$	−3.46410	−0.238479	−0.119239	+	0.992866i	$0.538046\pi$
$212$	0	0
$213$	2.53590	0.173757
$214$	0	0
$215$	−2.73205	−0.186324
$216$	0	0
$217$	13.6603	0.927318
$218$	0	0
$219$	−7.46410	−0.504377
$220$	0	0
$221$	−19.3923	−1.30447
$222$	0	0
$223$	5.26795	0.352768	0.176384	−	0.984321i	$-0.443560\pi$
$223$	5.26795	0.352768	0.176384	+	0.984321i	$0.443560\pi$
$224$	0	0
$225$	−6.07180	−0.404786
$226$	0	0
$227$	−6.92820	−0.459841	−0.229920	−	0.973209i	$-0.573847\pi$
$227$	−6.92820	−0.459841	−0.229920	+	0.973209i	$0.573847\pi$
$228$	0	0
$229$	−1.39230	−0.0920061	−0.0460030	−	0.998941i	$-0.514648\pi$
$229$	−1.39230	−0.0920061	−0.0460030	+	0.998941i	$0.514648\pi$
$230$	0	0
$231$	−6.00000	−0.394771
$232$	0	0
$233$	16.5359	1.08330	0.541651	−	0.840603i	$-0.317799\pi$
$233$	16.5359	1.08330	0.541651	+	0.840603i	$0.317799\pi$
$234$	0	0
$235$	−28.3923	−1.85211
$236$	0	0
$237$	2.24871	0.146069
$238$	0	0
$239$	11.0718	0.716175	0.358087	−	0.933688i	$-0.383429\pi$
$239$	11.0718	0.716175	0.358087	+	0.933688i	$0.383429\pi$
$240$	0	0
$241$	5.85641	0.377244	0.188622	−	0.982050i	$-0.439598\pi$
$241$	5.85641	0.377244	0.188622	+	0.982050i	$0.439598\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	−1.26795	−0.0810063
$246$	0	0
$247$	−1.60770	−0.102295
$248$	0	0
$249$	12.3397	0.782000
$250$	0	0
$251$	10.8564	0.685250	0.342625	−	0.939472i	$-0.388684\pi$
$251$	10.8564	0.685250	0.342625	+	0.939472i	$0.388684\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	0	0
$255$	12.9282	0.809595
$256$	0	0
$257$	−9.26795	−0.578119	−0.289059	−	0.957311i	$-0.593343\pi$
$257$	−9.26795	−0.578119	−0.289059	+	0.957311i	$0.593343\pi$
$258$	0	0
$259$	−4.00000	−0.248548
$260$	0	0
$261$	−15.2679	−0.945062
$262$	0	0
$263$	−14.3923	−0.887468	−0.443734	−	0.896159i	$-0.646346\pi$
$263$	−14.3923	−0.887468	−0.443734	+	0.896159i	$0.646346\pi$
$264$	0	0
$265$	6.73205	0.413547
$266$	0	0
$267$	0.143594	0.00878778
$268$	0	0
$269$	4.60770	0.280936	0.140468	−	0.990085i	$-0.455139\pi$
$269$	4.60770	0.280936	0.140468	+	0.990085i	$0.455139\pi$
$270$	0	0
$271$	24.4641	1.48609	0.743044	−	0.669242i	$-0.233381\pi$
$271$	24.4641	1.48609	0.743044	+	0.669242i	$0.233381\pi$
$272$	0	0
$273$	6.00000	0.363137
$274$	0	0
$275$	7.39230	0.445773
$276$	0	0
$277$	11.8038	0.709224	0.354612	−	0.935013i	$-0.384613\pi$
$277$	11.8038	0.709224	0.354612	+	0.935013i	$0.384613\pi$
$278$	0	0
$279$	−12.3205	−0.737610
$280$	0	0
$281$	−17.5359	−1.04610	−0.523052	−	0.852301i	$-0.675207\pi$
$281$	−17.5359	−1.04610	−0.523052	+	0.852301i	$0.675207\pi$
$282$	0	0
$283$	21.9282	1.30350	0.651748	−	0.758435i	$-0.274036\pi$
$283$	21.9282	1.30350	0.651748	+	0.758435i	$0.274036\pi$
$284$	0	0
$285$	1.07180	0.0634878
$286$	0	0
$287$	27.1244	1.60110
$288$	0	0
$289$	24.7846	1.45792
$290$	0	0
$291$	7.66025	0.449052
$292$	0	0
$293$	31.3205	1.82976	0.914882	−	0.403722i	$-0.132284\pi$
$293$	31.3205	1.82976	0.914882	+	0.403722i	$0.132284\pi$
$294$	0	0
$295$	−24.3923	−1.42017
$296$	0	0
$297$	12.0000	0.696311
$298$	0	0
$299$	−3.00000	−0.173494
$300$	0	0
$301$	2.73205	0.157473
$302$	0	0
$303$	−0.339746	−0.0195179
$304$	0	0
$305$	34.7846	1.99176
$306$	0	0
$307$	10.3205	0.589023	0.294511	−	0.955648i	$-0.404843\pi$
$307$	10.3205	0.589023	0.294511	+	0.955648i	$0.404843\pi$
$308$	0	0
$309$	9.41154	0.535404
$310$	0	0
$311$	9.53590	0.540731	0.270366	−	0.962758i	$-0.412855\pi$
$311$	9.53590	0.540731	0.270366	+	0.962758i	$0.412855\pi$
$312$	0	0
$313$	−3.66025	−0.206890	−0.103445	−	0.994635i	$-0.532987\pi$
$313$	−3.66025	−0.206890	−0.103445	+	0.994635i	$0.532987\pi$
$314$	0	0
$315$	18.3923	1.03629
$316$	0	0
$317$	0.0717968	0.00403251	0.00201625	−	0.999998i	$-0.499358\pi$
$317$	0.0717968	0.00403251	0.00201625	+	0.999998i	$0.499358\pi$
$318$	0	0
$319$	18.5885	1.04075
$320$	0	0
$321$	−8.39230	−0.468413
$322$	0	0
$323$	3.46410	0.192748
$324$	0	0
$325$	−7.39230	−0.410051
$326$	0	0
$327$	4.05256	0.224107
$328$	0	0
$329$	28.3923	1.56532
$330$	0	0
$331$	−7.66025	−0.421046	−0.210523	−	0.977589i	$-0.567517\pi$
$331$	−7.66025	−0.421046	−0.210523	+	0.977589i	$0.567517\pi$
$332$	0	0
$333$	3.60770	0.197700
$334$	0	0
$335$	−21.6603	−1.18343
$336$	0	0
$337$	28.4641	1.55054	0.775269	−	0.631631i	$-0.217614\pi$
$337$	28.4641	1.55054	0.775269	+	0.631631i	$0.217614\pi$
$338$	0	0
$339$	−7.71281	−0.418902
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	−17.8564	−0.964155
$344$	0	0
$345$	2.00000	0.107676
$346$	0	0
$347$	5.26795	0.282798	0.141399	−	0.989953i	$-0.454840\pi$
$347$	5.26795	0.282798	0.141399	+	0.989953i	$0.454840\pi$
$348$	0	0
$349$	−16.7321	−0.895646	−0.447823	−	0.894122i	$-0.647801\pi$
$349$	−16.7321	−0.895646	−0.447823	+	0.894122i	$0.647801\pi$
$350$	0	0
$351$	−12.0000	−0.640513
$352$	0	0
$353$	8.07180	0.429618	0.214809	−	0.976656i	$-0.431087\pi$
$353$	8.07180	0.429618	0.214809	+	0.976656i	$0.431087\pi$
$354$	0	0
$355$	9.46410	0.502302
$356$	0	0
$357$	−12.9282	−0.684233
$358$	0	0
$359$	−35.7846	−1.88864	−0.944320	−	0.329029i	$-0.893279\pi$
$359$	−35.7846	−1.88864	−0.944320	+	0.329029i	$0.893279\pi$
$360$	0	0
$361$	−18.7128	−0.984885
$362$	0	0
$363$	1.46410	0.0768454
$364$	0	0
$365$	−27.8564	−1.45807
$366$	0	0
$367$	−26.7846	−1.39815	−0.699073	−	0.715051i	$-0.746404\pi$
$367$	−26.7846	−1.39815	−0.699073	+	0.715051i	$0.746404\pi$
$368$	0	0
$369$	−24.4641	−1.27355
$370$	0	0
$371$	−6.73205	−0.349511
$372$	0	0
$373$	28.3923	1.47010	0.735049	−	0.678014i	$-0.237159\pi$
$373$	28.3923	1.47010	0.735049	+	0.678014i	$0.237159\pi$
$374$	0	0
$375$	−5.07180	−0.261906
$376$	0	0
$377$	−18.5885	−0.957354
$378$	0	0
$379$	−8.32051	−0.427396	−0.213698	−	0.976900i	$-0.568551\pi$
$379$	−8.32051	−0.427396	−0.213698	+	0.976900i	$0.568551\pi$
$380$	0	0
$381$	9.41154	0.482168
$382$	0	0
$383$	30.5359	1.56031	0.780156	−	0.625585i	$-0.215140\pi$
$383$	30.5359	1.56031	0.780156	+	0.625585i	$0.215140\pi$
$384$	0	0
$385$	−22.3923	−1.14122
$386$	0	0
$387$	−2.46410	−0.125257
$388$	0	0
$389$	23.8564	1.20957	0.604784	−	0.796390i	$-0.293259\pi$
$389$	23.8564	1.20957	0.604784	+	0.796390i	$0.293259\pi$
$390$	0	0
$391$	6.46410	0.326904
$392$	0	0
$393$	−10.1436	−0.511677
$394$	0	0
$395$	8.39230	0.422263
$396$	0	0
$397$	20.7846	1.04315	0.521575	−	0.853206i	$-0.325345\pi$
$397$	20.7846	1.04315	0.521575	+	0.853206i	$0.325345\pi$
$398$	0	0
$399$	−1.07180	−0.0536570
$400$	0	0
$401$	27.7846	1.38750	0.693749	−	0.720217i	$-0.255958\pi$
$401$	27.7846	1.38750	0.693749	+	0.720217i	$0.255958\pi$
$402$	0	0
$403$	−15.0000	−0.747203
$404$	0	0
$405$	−12.1962	−0.606032
$406$	0	0
$407$	−4.39230	−0.217718
$408$	0	0
$409$	−39.5167	−1.95397	−0.976987	−	0.213301i	$-0.931579\pi$
$409$	−39.5167	−1.95397	−0.976987	+	0.213301i	$0.931579\pi$
$410$	0	0
$411$	9.75129	0.480996
$412$	0	0
$413$	24.3923	1.20027
$414$	0	0
$415$	46.0526	2.26063
$416$	0	0
$417$	−15.6603	−0.766886
$418$	0	0
$419$	−20.5885	−1.00581	−0.502906	−	0.864341i	$-0.667736\pi$
$419$	−20.5885	−1.00581	−0.502906	+	0.864341i	$0.667736\pi$
$420$	0	0
$421$	−8.92820	−0.435134	−0.217567	−	0.976045i	$-0.569812\pi$
$421$	−8.92820	−0.435134	−0.217567	+	0.976045i	$0.569812\pi$
$422$	0	0
$423$	−25.6077	−1.24509
$424$	0	0
$425$	15.9282	0.772631
$426$	0	0
$427$	−34.7846	−1.68335
$428$	0	0
$429$	6.58846	0.318094
$430$	0	0
$431$	2.32051	0.111775	0.0558875	−	0.998437i	$-0.482201\pi$
$431$	2.32051	0.111775	0.0558875	+	0.998437i	$0.482201\pi$
$432$	0	0
$433$	−28.5885	−1.37387	−0.686937	−	0.726717i	$-0.741045\pi$
$433$	−28.5885	−1.37387	−0.686937	+	0.726717i	$0.741045\pi$
$434$	0	0
$435$	12.3923	0.594166
$436$	0	0
$437$	0.535898	0.0256355
$438$	0	0
$439$	23.3923	1.11645	0.558227	−	0.829688i	$-0.311482\pi$
$439$	23.3923	1.11645	0.558227	+	0.829688i	$0.311482\pi$
$440$	0	0
$441$	−1.14359	−0.0544568
$442$	0	0
$443$	30.0000	1.42534	0.712672	−	0.701498i	$-0.247485\pi$
$443$	30.0000	1.42534	0.712672	+	0.701498i	$0.247485\pi$
$444$	0	0
$445$	0.535898	0.0254040
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	−22.9808	−1.08453	−0.542265	−	0.840208i	$-0.682433\pi$
$449$	−22.9808	−1.08453	−0.542265	+	0.840208i	$0.682433\pi$
$450$	0	0
$451$	29.7846	1.40250
$452$	0	0
$453$	−6.28719	−0.295398
$454$	0	0
$455$	22.3923	1.04977
$456$	0	0
$457$	−9.80385	−0.458605	−0.229302	−	0.973355i	$-0.573644\pi$
$457$	−9.80385	−0.458605	−0.229302	+	0.973355i	$0.573644\pi$
$458$	0	0
$459$	25.8564	1.20687
$460$	0	0
$461$	11.3205	0.527249	0.263624	−	0.964625i	$-0.415082\pi$
$461$	11.3205	0.527249	0.263624	+	0.964625i	$0.415082\pi$
$462$	0	0
$463$	−22.0526	−1.02487	−0.512435	−	0.858726i	$-0.671256\pi$
$463$	−22.0526	−1.02487	−0.512435	+	0.858726i	$0.671256\pi$
$464$	0	0
$465$	10.0000	0.463739
$466$	0	0
$467$	27.0718	1.25273	0.626367	−	0.779529i	$-0.284541\pi$
$467$	27.0718	1.25273	0.626367	+	0.779529i	$0.284541\pi$
$468$	0	0
$469$	21.6603	1.00018
$470$	0	0
$471$	−8.67949	−0.399930
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	1.32051	0.0605891
$476$	0	0
$477$	6.07180	0.278008
$478$	0	0
$479$	33.9282	1.55022	0.775110	−	0.631827i	$-0.217695\pi$
$479$	33.9282	1.55022	0.775110	+	0.631827i	$0.217695\pi$
$480$	0	0
$481$	4.39230	0.200272
$482$	0	0
$483$	−2.00000	−0.0910032
$484$	0	0
$485$	28.5885	1.29813
$486$	0	0
$487$	−28.0000	−1.26880	−0.634401	−	0.773004i	$-0.718753\pi$
$487$	−28.0000	−1.26880	−0.634401	+	0.773004i	$0.718753\pi$
$488$	0	0
$489$	−7.75129	−0.350525
$490$	0	0
$491$	−24.4449	−1.10318	−0.551591	−	0.834115i	$-0.685979\pi$
$491$	−24.4449	−1.10318	−0.551591	+	0.834115i	$0.685979\pi$
$492$	0	0
$493$	40.0526	1.80388
$494$	0	0
$495$	20.1962	0.907750
$496$	0	0
$497$	−9.46410	−0.424523
$498$	0	0
$499$	−30.0526	−1.34534	−0.672669	−	0.739944i	$-0.734852\pi$
$499$	−30.0526	−1.34534	−0.672669	+	0.739944i	$0.734852\pi$
$500$	0	0
$501$	−11.6603	−0.520942
$502$	0	0
$503$	−22.3923	−0.998424	−0.499212	−	0.866480i	$-0.666377\pi$
$503$	−22.3923	−0.998424	−0.499212	+	0.866480i	$0.666377\pi$
$504$	0	0
$505$	−1.26795	−0.0564230
$506$	0	0
$507$	2.92820	0.130046
$508$	0	0
$509$	−12.4641	−0.552462	−0.276231	−	0.961091i	$-0.589085\pi$
$509$	−12.4641	−0.552462	−0.276231	+	0.961091i	$0.589085\pi$
$510$	0	0
$511$	27.8564	1.23229
$512$	0	0
$513$	2.14359	0.0946420
$514$	0	0
$515$	35.1244	1.54776
$516$	0	0
$517$	31.1769	1.37116
$518$	0	0
$519$	6.53590	0.286894
$520$	0	0
$521$	−31.2679	−1.36987	−0.684937	−	0.728602i	$-0.740170\pi$
$521$	−31.2679	−1.36987	−0.684937	+	0.728602i	$0.740170\pi$
$522$	0	0
$523$	−33.6603	−1.47186	−0.735930	−	0.677058i	$-0.763255\pi$
$523$	−33.6603	−1.47186	−0.735930	+	0.677058i	$0.763255\pi$
$524$	0	0
$525$	−4.92820	−0.215084
$526$	0	0
$527$	32.3205	1.40790
$528$	0	0
$529$	−22.0000	−0.956522
$530$	0	0
$531$	−22.0000	−0.954719
$532$	0	0
$533$	−29.7846	−1.29011
$534$	0	0
$535$	−31.3205	−1.35410
$536$	0	0
$537$	16.6410	0.718113
$538$	0	0
$539$	1.39230	0.0599708
$540$	0	0
$541$	−3.39230	−0.145847	−0.0729233	−	0.997338i	$-0.523233\pi$
$541$	−3.39230	−0.145847	−0.0729233	+	0.997338i	$0.523233\pi$
$542$	0	0
$543$	12.0000	0.514969
$544$	0	0
$545$	15.1244	0.647856
$546$	0	0
$547$	26.8564	1.14830	0.574149	−	0.818751i	$-0.305333\pi$
$547$	26.8564	1.14830	0.574149	+	0.818751i	$0.305333\pi$
$548$	0	0
$549$	31.3731	1.33897
$550$	0	0
$551$	3.32051	0.141458
$552$	0	0
$553$	−8.39230	−0.356877
$554$	0	0
$555$	−2.92820	−0.124295
$556$	0	0
$557$	−20.3205	−0.861008	−0.430504	−	0.902589i	$-0.641664\pi$
$557$	−20.3205	−0.861008	−0.430504	+	0.902589i	$0.641664\pi$
$558$	0	0
$559$	−3.00000	−0.126886
$560$	0	0
$561$	−14.1962	−0.599362
$562$	0	0
$563$	−12.4641	−0.525299	−0.262650	−	0.964891i	$-0.584596\pi$
$563$	−12.4641	−0.525299	−0.262650	+	0.964891i	$0.584596\pi$
$564$	0	0
$565$	−28.7846	−1.21098
$566$	0	0
$567$	12.1962	0.512190
$568$	0	0
$569$	−37.7846	−1.58401	−0.792007	−	0.610513i	$-0.790964\pi$
$569$	−37.7846	−1.58401	−0.792007	+	0.610513i	$0.790964\pi$
$570$	0	0
$571$	−21.1244	−0.884027	−0.442013	−	0.897008i	$-0.645736\pi$
$571$	−21.1244	−0.884027	−0.442013	+	0.897008i	$0.645736\pi$
$572$	0	0
$573$	17.0718	0.713185
$574$	0	0
$575$	2.46410	0.102760
$576$	0	0
$577$	−10.7321	−0.446781	−0.223391	−	0.974729i	$-0.571713\pi$
$577$	−10.7321	−0.446781	−0.223391	+	0.974729i	$0.571713\pi$
$578$	0	0
$579$	−9.41154	−0.391130
$580$	0	0
$581$	−46.0526	−1.91058
$582$	0	0
$583$	−7.39230	−0.306158
$584$	0	0
$585$	−20.1962	−0.835008
$586$	0	0
$587$	28.2487	1.16595	0.582975	−	0.812490i	$-0.301889\pi$
$587$	28.2487	1.16595	0.582975	+	0.812490i	$0.301889\pi$
$588$	0	0
$589$	2.67949	0.110407
$590$	0	0
$591$	3.71281	0.152725
$592$	0	0
$593$	29.6603	1.21800	0.609000	−	0.793170i	$-0.291571\pi$
$593$	29.6603	1.21800	0.609000	+	0.793170i	$0.291571\pi$
$594$	0	0
$595$	−48.2487	−1.97800
$596$	0	0
$597$	−2.24871	−0.0920336
$598$	0	0
$599$	43.3923	1.77296	0.886481	−	0.462765i	$-0.153143\pi$
$599$	43.3923	1.77296	0.886481	+	0.462765i	$0.153143\pi$
$600$	0	0
$601$	−33.8564	−1.38103	−0.690516	−	0.723317i	$-0.742616\pi$
$601$	−33.8564	−1.38103	−0.690516	+	0.723317i	$0.742616\pi$
$602$	0	0
$603$	−19.5359	−0.795563
$604$	0	0
$605$	5.46410	0.222147
$606$	0	0
$607$	9.85641	0.400059	0.200030	−	0.979790i	$-0.435896\pi$
$607$	9.85641	0.400059	0.200030	+	0.979790i	$0.435896\pi$
$608$	0	0
$609$	−12.3923	−0.502162
$610$	0	0
$611$	−31.1769	−1.26128
$612$	0	0
$613$	12.7846	0.516366	0.258183	−	0.966096i	$-0.416876\pi$
$613$	12.7846	0.516366	0.258183	+	0.966096i	$0.416876\pi$
$614$	0	0
$615$	19.8564	0.800688
$616$	0	0
$617$	−13.0000	−0.523360	−0.261680	−	0.965155i	$-0.584277\pi$
$617$	−13.0000	−0.523360	−0.261680	+	0.965155i	$0.584277\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	0	0
$623$	−0.535898	−0.0214703
$624$	0	0
$625$	−31.2487	−1.24995
$626$	0	0
$627$	−1.17691	−0.0470014
$628$	0	0
$629$	−9.46410	−0.377358
$630$	0	0
$631$	−13.3205	−0.530281	−0.265141	−	0.964210i	$-0.585418\pi$
$631$	−13.3205	−0.530281	−0.265141	+	0.964210i	$0.585418\pi$
$632$	0	0
$633$	2.53590	0.100793
$634$	0	0
$635$	35.1244	1.39387
$636$	0	0
$637$	−1.39230	−0.0551651
$638$	0	0
$639$	8.53590	0.337675
$640$	0	0
$641$	−30.4449	−1.20250	−0.601250	−	0.799061i	$-0.705330\pi$
$641$	−30.4449	−1.20250	−0.601250	+	0.799061i	$0.705330\pi$
$642$	0	0
$643$	−18.0000	−0.709851	−0.354925	−	0.934895i	$-0.615494\pi$
$643$	−18.0000	−0.709851	−0.354925	+	0.934895i	$0.615494\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	0	0
$647$	−28.1051	−1.10493	−0.552463	−	0.833537i	$-0.686312\pi$
$647$	−28.1051	−1.10493	−0.552463	+	0.833537i	$0.686312\pi$
$648$	0	0
$649$	26.7846	1.05139
$650$	0	0
$651$	−10.0000	−0.391931
$652$	0	0
$653$	12.2487	0.479329	0.239665	−	0.970856i	$-0.422963\pi$
$653$	12.2487	0.479329	0.239665	+	0.970856i	$0.422963\pi$
$654$	0	0
$655$	−37.8564	−1.47917
$656$	0	0
$657$	−25.1244	−0.980194
$658$	0	0
$659$	8.32051	0.324121	0.162060	−	0.986781i	$-0.448186\pi$
$659$	8.32051	0.324121	0.162060	+	0.986781i	$0.448186\pi$
$660$	0	0
$661$	19.7846	0.769532	0.384766	−	0.923014i	$-0.374282\pi$
$661$	19.7846	0.769532	0.384766	+	0.923014i	$0.374282\pi$
$662$	0	0
$663$	14.1962	0.551333
$664$	0	0
$665$	−4.00000	−0.155113
$666$	0	0
$667$	6.19615	0.239916
$668$	0	0
$669$	−3.85641	−0.149097
$670$	0	0
$671$	−38.1962	−1.47455
$672$	0	0
$673$	18.5885	0.716532	0.358266	−	0.933619i	$-0.383368\pi$
$673$	18.5885	0.716532	0.358266	+	0.933619i	$0.383368\pi$
$674$	0	0
$675$	9.85641	0.379373
$676$	0	0
$677$	15.1769	0.583296	0.291648	−	0.956526i	$-0.405796\pi$
$677$	15.1769	0.583296	0.291648	+	0.956526i	$0.405796\pi$
$678$	0	0
$679$	−28.5885	−1.09712
$680$	0	0
$681$	5.07180	0.194352
$682$	0	0
$683$	−14.8564	−0.568465	−0.284232	−	0.958755i	$-0.591739\pi$
$683$	−14.8564	−0.568465	−0.284232	+	0.958755i	$0.591739\pi$
$684$	0	0
$685$	36.3923	1.39048
$686$	0	0
$687$	1.01924	0.0388864
$688$	0	0
$689$	7.39230	0.281624
$690$	0	0
$691$	−23.1244	−0.879692	−0.439846	−	0.898073i	$-0.644967\pi$
$691$	−23.1244	−0.879692	−0.439846	+	0.898073i	$0.644967\pi$
$692$	0	0
$693$	−20.1962	−0.767188
$694$	0	0
$695$	−58.4449	−2.21694
$696$	0	0
$697$	64.1769	2.43087
$698$	0	0
$699$	−12.1051	−0.457858
$700$	0	0
$701$	−44.3923	−1.67667	−0.838337	−	0.545152i	$-0.816472\pi$
$701$	−44.3923	−1.67667	−0.838337	+	0.545152i	$0.816472\pi$
$702$	0	0
$703$	−0.784610	−0.0295921
$704$	0	0
$705$	20.7846	0.782794
$706$	0	0
$707$	1.26795	0.0476861
$708$	0	0
$709$	34.0718	1.27959	0.639797	−	0.768544i	$-0.279019\pi$
$709$	34.0718	1.27959	0.639797	+	0.768544i	$0.279019\pi$
$710$	0	0
$711$	7.56922	0.283868
$712$	0	0
$713$	5.00000	0.187251
$714$	0	0
$715$	24.5885	0.919556
$716$	0	0
$717$	−8.10512	−0.302691
$718$	0	0
$719$	−42.9282	−1.60095	−0.800476	−	0.599365i	$-0.795420\pi$
$719$	−42.9282	−1.60095	−0.800476	+	0.599365i	$0.795420\pi$
$720$	0	0
$721$	−35.1244	−1.30810
$722$	0	0
$723$	−4.28719	−0.159442
$724$	0	0
$725$	15.2679	0.567037
$726$	0	0
$727$	−22.9282	−0.850360	−0.425180	−	0.905109i	$-0.639789\pi$
$727$	−22.9282	−0.850360	−0.425180	+	0.905109i	$0.639789\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	6.46410	0.239083
$732$	0	0
$733$	−30.6410	−1.13175	−0.565876	−	0.824490i	$-0.691462\pi$
$733$	−30.6410	−1.13175	−0.565876	+	0.824490i	$0.691462\pi$
$734$	0	0
$735$	0.928203	0.0342373
$736$	0	0
$737$	23.7846	0.876117
$738$	0	0
$739$	−37.3205	−1.37286	−0.686429	−	0.727197i	$-0.740823\pi$
$739$	−37.3205	−1.37286	−0.686429	+	0.727197i	$0.740823\pi$
$740$	0	0
$741$	1.17691	0.0432350
$742$	0	0
$743$	4.39230	0.161138	0.0805690	−	0.996749i	$-0.474326\pi$
$743$	4.39230	0.161138	0.0805690	+	0.996749i	$0.474326\pi$
$744$	0	0
$745$	44.7846	1.64078
$746$	0	0
$747$	41.5359	1.51972
$748$	0	0
$749$	31.3205	1.14443
$750$	0	0
$751$	−39.2679	−1.43291	−0.716454	−	0.697634i	$-0.754236\pi$
$751$	−39.2679	−1.43291	−0.716454	+	0.697634i	$0.754236\pi$
$752$	0	0
$753$	−7.94744	−0.289621
$754$	0	0
$755$	−23.4641	−0.853946
$756$	0	0
$757$	−36.1051	−1.31226	−0.656131	−	0.754647i	$-0.727808\pi$
$757$	−36.1051	−1.31226	−0.656131	+	0.754647i	$0.727808\pi$
$758$	0	0
$759$	−2.19615	−0.0797153
$760$	0	0
$761$	−29.4641	−1.06807	−0.534036	−	0.845461i	$-0.679325\pi$
$761$	−29.4641	−1.06807	−0.534036	+	0.845461i	$0.679325\pi$
$762$	0	0
$763$	−15.1244	−0.547538
$764$	0	0
$765$	43.5167	1.57335
$766$	0	0
$767$	−26.7846	−0.967136
$768$	0	0
$769$	−24.7846	−0.893756	−0.446878	−	0.894595i	$-0.647464\pi$
$769$	−24.7846	−0.893756	−0.446878	+	0.894595i	$0.647464\pi$
$770$	0	0
$771$	6.78461	0.244342
$772$	0	0
$773$	−42.4449	−1.52664	−0.763318	−	0.646023i	$-0.776431\pi$
$773$	−42.4449	−1.52664	−0.763318	+	0.646023i	$0.776431\pi$
$774$	0	0
$775$	12.3205	0.442566
$776$	0	0
$777$	2.92820	0.105049
$778$	0	0
$779$	5.32051	0.190627
$780$	0	0
$781$	−10.3923	−0.371866
$782$	0	0
$783$	24.7846	0.885730
$784$	0	0
$785$	−32.3923	−1.15613
$786$	0	0
$787$	51.8564	1.84848	0.924241	−	0.381811i	$-0.124699\pi$
$787$	51.8564	1.84848	0.924241	+	0.381811i	$0.124699\pi$
$788$	0	0
$789$	10.5359	0.375088
$790$	0	0
$791$	28.7846	1.02346
$792$	0	0
$793$	38.1962	1.35639
$794$	0	0
$795$	−4.92820	−0.174785
$796$	0	0
$797$	−19.6077	−0.694540	−0.347270	−	0.937765i	$-0.612891\pi$
$797$	−19.6077	−0.694540	−0.347270	+	0.937765i	$0.612891\pi$
$798$	0	0
$799$	67.1769	2.37655
$800$	0	0
$801$	0.483340	0.0170780
$802$	0	0
$803$	30.5885	1.07944
$804$	0	0
$805$	−7.46410	−0.263075
$806$	0	0
$807$	−3.37307	−0.118738
$808$	0	0
$809$	−2.24871	−0.0790605	−0.0395302	−	0.999218i	$-0.512586\pi$
$809$	−2.24871	−0.0790605	−0.0395302	+	0.999218i	$0.512586\pi$
$810$	0	0
$811$	7.66025	0.268988	0.134494	−	0.990914i	$-0.457059\pi$
$811$	7.66025	0.268988	0.134494	+	0.990914i	$0.457059\pi$
$812$	0	0
$813$	−17.9090	−0.628095
$814$	0	0
$815$	−28.9282	−1.01331
$816$	0	0
$817$	0.535898	0.0187487
$818$	0	0
$819$	20.1962	0.705711
$820$	0	0
$821$	6.21539	0.216919	0.108459	−	0.994101i	$-0.465408\pi$
$821$	6.21539	0.216919	0.108459	+	0.994101i	$0.465408\pi$
$822$	0	0
$823$	−27.2487	−0.949830	−0.474915	−	0.880032i	$-0.657521\pi$
$823$	−27.2487	−0.949830	−0.474915	+	0.880032i	$0.657521\pi$
$824$	0	0
$825$	−5.41154	−0.188406
$826$	0	0
$827$	−17.0718	−0.593645	−0.296822	−	0.954933i	$-0.595927\pi$
$827$	−17.0718	−0.593645	−0.296822	+	0.954933i	$0.595927\pi$
$828$	0	0
$829$	14.5359	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$829$	14.5359	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$830$	0	0
$831$	−8.64102	−0.299754
$832$	0	0
$833$	3.00000	0.103944
$834$	0	0
$835$	−43.5167	−1.50596
$836$	0	0
$837$	20.0000	0.691301
$838$	0	0
$839$	−7.80385	−0.269419	−0.134709	−	0.990885i	$-0.543010\pi$
$839$	−7.80385	−0.269419	−0.134709	+	0.990885i	$0.543010\pi$
$840$	0	0
$841$	9.39230	0.323873
$842$	0	0
$843$	12.8372	0.442136
$844$	0	0
$845$	10.9282	0.375942
$846$	0	0
$847$	−5.46410	−0.187749
$848$	0	0
$849$	−16.0526	−0.550922
$850$	0	0
$851$	−1.46410	−0.0501888
$852$	0	0
$853$	27.7846	0.951327	0.475663	−	0.879627i	$-0.342208\pi$
$853$	27.7846	0.951327	0.475663	+	0.879627i	$0.342208\pi$
$854$	0	0
$855$	3.60770	0.123381
$856$	0	0
$857$	52.9282	1.80799	0.903996	−	0.427540i	$-0.140620\pi$
$857$	52.9282	1.80799	0.903996	+	0.427540i	$0.140620\pi$
$858$	0	0
$859$	34.6410	1.18194	0.590968	−	0.806695i	$-0.298746\pi$
$859$	34.6410	1.18194	0.590968	+	0.806695i	$0.298746\pi$
$860$	0	0
$861$	−19.8564	−0.676705
$862$	0	0
$863$	14.0000	0.476566	0.238283	−	0.971196i	$-0.423415\pi$
$863$	14.0000	0.476566	0.238283	+	0.971196i	$0.423415\pi$
$864$	0	0
$865$	24.3923	0.829363
$866$	0	0
$867$	−18.1436	−0.616189
$868$	0	0
$869$	−9.21539	−0.312611
$870$	0	0
$871$	−23.7846	−0.805911
$872$	0	0
$873$	25.7846	0.872677
$874$	0	0
$875$	18.9282	0.639890
$876$	0	0
$877$	37.1051	1.25295	0.626475	−	0.779441i	$-0.284497\pi$
$877$	37.1051	1.25295	0.626475	+	0.779441i	$0.284497\pi$
$878$	0	0
$879$	−22.9282	−0.773349
$880$	0	0
$881$	13.5359	0.456036	0.228018	−	0.973657i	$-0.426775\pi$
$881$	13.5359	0.456036	0.228018	+	0.973657i	$0.426775\pi$
$882$	0	0
$883$	35.1051	1.18138	0.590691	−	0.806898i	$-0.298855\pi$
$883$	35.1051	1.18138	0.590691	+	0.806898i	$0.298855\pi$
$884$	0	0
$885$	17.8564	0.600237
$886$	0	0
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	0	0
$889$	−35.1244	−1.17803
$890$	0	0
$891$	13.3923	0.448659
$892$	0	0
$893$	5.56922	0.186367
$894$	0	0
$895$	62.1051	2.07595
$896$	0	0
$897$	2.19615	0.0733274
$898$	0	0
$899$	30.9808	1.03327
$900$	0	0
$901$	−15.9282	−0.530645
$902$	0	0
$903$	−2.00000	−0.0665558
$904$	0	0
$905$	44.7846	1.48869
$906$	0	0
$907$	−31.7846	−1.05539	−0.527695	−	0.849434i	$-0.676944\pi$
$907$	−31.7846	−1.05539	−0.527695	+	0.849434i	$0.676944\pi$
$908$	0	0
$909$	−1.14359	−0.0379306
$910$	0	0
$911$	18.1962	0.602865	0.301433	−	0.953487i	$-0.402535\pi$
$911$	18.1962	0.602865	0.301433	+	0.953487i	$0.402535\pi$
$912$	0	0
$913$	−50.5692	−1.67360
$914$	0	0
$915$	−25.4641	−0.841817
$916$	0	0
$917$	37.8564	1.25013
$918$	0	0
$919$	41.6410	1.37361	0.686805	−	0.726842i	$-0.259013\pi$
$919$	41.6410	1.37361	0.686805	+	0.726842i	$0.259013\pi$
$920$	0	0
$921$	−7.55514	−0.248950
$922$	0	0
$923$	10.3923	0.342067
$924$	0	0
$925$	−3.60770	−0.118620
$926$	0	0
$927$	31.6795	1.04049
$928$	0	0
$929$	−13.6077	−0.446454	−0.223227	−	0.974766i	$-0.571659\pi$
$929$	−13.6077	−0.446454	−0.223227	+	0.974766i	$0.571659\pi$
$930$	0	0
$931$	0.248711	0.00815118
$932$	0	0
$933$	−6.98076	−0.228540
$934$	0	0
$935$	−52.9808	−1.73266
$936$	0	0
$937$	−39.9090	−1.30377	−0.651885	−	0.758318i	$-0.726021\pi$
$937$	−39.9090	−1.30377	−0.651885	+	0.758318i	$0.726021\pi$
$938$	0	0
$939$	2.67949	0.0874419
$940$	0	0
$941$	−37.1051	−1.20959	−0.604796	−	0.796380i	$-0.706745\pi$
$941$	−37.1051	−1.20959	−0.604796	+	0.796380i	$0.706745\pi$
$942$	0	0
$943$	9.92820	0.323307
$944$	0	0
$945$	−29.8564	−0.971229
$946$	0	0
$947$	−19.3923	−0.630165	−0.315083	−	0.949064i	$-0.602032\pi$
$947$	−19.3923	−0.630165	−0.315083	+	0.949064i	$0.602032\pi$
$948$	0	0
$949$	−30.5885	−0.992943
$950$	0	0
$951$	−0.0525589	−0.00170434
$952$	0	0
$953$	57.5167	1.86315	0.931574	−	0.363553i	$-0.118436\pi$
$953$	57.5167	1.86315	0.931574	+	0.363553i	$0.118436\pi$
$954$	0	0
$955$	63.7128	2.06170
$956$	0	0
$957$	−13.6077	−0.439874
$958$	0	0
$959$	−36.3923	−1.17517
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−28.2487	−0.910302
$964$	0	0
$965$	−35.1244	−1.13069
$966$	0	0
$967$	35.2487	1.13352	0.566761	−	0.823882i	$-0.308196\pi$
$967$	35.2487	1.13352	0.566761	+	0.823882i	$0.308196\pi$
$968$	0	0
$969$	−2.53590	−0.0814648
$970$	0	0
$971$	4.85641	0.155849	0.0779247	−	0.996959i	$-0.475171\pi$
$971$	4.85641	0.155849	0.0779247	+	0.996959i	$0.475171\pi$
$972$	0	0
$973$	58.4449	1.87366
$974$	0	0
$975$	5.41154	0.173308
$976$	0	0
$977$	13.4641	0.430755	0.215377	−	0.976531i	$-0.430902\pi$
$977$	13.4641	0.430755	0.215377	+	0.976531i	$0.430902\pi$
$978$	0	0
$979$	−0.588457	−0.0188072
$980$	0	0
$981$	13.6410	0.435524
$982$	0	0
$983$	29.2679	0.933503	0.466751	−	0.884389i	$-0.345424\pi$
$983$	29.2679	0.933503	0.466751	+	0.884389i	$0.345424\pi$
$984$	0	0
$985$	13.8564	0.441502
$986$	0	0
$987$	−20.7846	−0.661581
$988$	0	0
$989$	1.00000	0.0317982
$990$	0	0
$991$	9.80385	0.311429	0.155715	−	0.987802i	$-0.450232\pi$
$991$	9.80385	0.311429	0.155715	+	0.987802i	$0.450232\pi$
$992$	0	0
$993$	5.60770	0.177955
$994$	0	0
$995$	−8.39230	−0.266054
$996$	0	0
$997$	42.7321	1.35334	0.676669	−	0.736288i	$-0.263423\pi$
$997$	42.7321	1.35334	0.676669	+	0.736288i	$0.263423\pi$
$998$	0	0
$999$	−5.85641	−0.185289

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	688.2.a.h.1.1		2
3.2	odd	2		6192.2.a.bn.1.2		2
4.3	odd	2		344.2.a.b.1.2	✓	2
8.3	odd	2		2752.2.a.q.1.1		2
8.5	even	2		2752.2.a.h.1.2		2
12.11	even	2		3096.2.a.k.1.2		2
20.19	odd	2		8600.2.a.j.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
344.2.a.b.1.2	✓	2	4.3	odd	2
688.2.a.h.1.1		2	1.1	even	1	trivial
2752.2.a.h.1.2		2	8.5	even	2
2752.2.a.q.1.1		2	8.3	odd	2
3096.2.a.k.1.2		2	12.11	even	2
6192.2.a.bn.1.2		2	3.2	odd	2
8600.2.a.j.1.1		2	20.19	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:	\( N \)	\(=\)	\( 688 = 2^{4} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	688.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} -2.73205 q^{5} +2.73205 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} -2.73205 q^{5} +2.73205 q^{7} -2.46410 q^{9} +3.00000 q^{11} -3.00000 q^{13} +2.00000 q^{15} +6.46410 q^{17} +0.535898 q^{19} -2.00000 q^{21} +1.00000 q^{23} +2.46410 q^{25} +4.00000 q^{27} +6.19615 q^{29} +5.00000 q^{31} -2.19615 q^{33} -7.46410 q^{35} -1.46410 q^{37} +2.19615 q^{39} +9.92820 q^{41} +1.00000 q^{43} +6.73205 q^{45} +10.3923 q^{47} +0.464102 q^{49} -4.73205 q^{51} -2.46410 q^{53} -8.19615 q^{55} -0.392305 q^{57} +8.92820 q^{59} -12.7321 q^{61} -6.73205 q^{63} +8.19615 q^{65} +7.92820 q^{67} -0.732051 q^{69} -3.46410 q^{71} +10.1962 q^{73} -1.80385 q^{75} +8.19615 q^{77} -3.07180 q^{79} +4.46410 q^{81} -16.8564 q^{83} -17.6603 q^{85} -4.53590 q^{87} -0.196152 q^{89} -8.19615 q^{91} -3.66025 q^{93} -1.46410 q^{95} -10.4641 q^{97} -7.39230 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 6 q^{11} - 6 q^{13} + 4 q^{15} + 6 q^{17} + 8 q^{19} - 4 q^{21} + 2 q^{23} - 2 q^{25} + 8 q^{27} + 2 q^{29} + 10 q^{31} + 6 q^{33} - 8 q^{35} + 4 q^{37} - 6 q^{39} + 6 q^{41} + 2 q^{43} + 10 q^{45} - 6 q^{49} - 6 q^{51} + 2 q^{53} - 6 q^{55} + 20 q^{57} + 4 q^{59} - 22 q^{61} - 10 q^{63} + 6 q^{65} + 2 q^{67} + 2 q^{69} + 10 q^{73} - 14 q^{75} + 6 q^{77} - 20 q^{79} + 2 q^{81} - 6 q^{83} - 18 q^{85} - 16 q^{87} + 10 q^{89} - 6 q^{91} + 10 q^{93} + 4 q^{95} - 14 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 6 * q^11 - 6 * q^13 + 4 * q^15 + 6 * q^17 + 8 * q^19 - 4 * q^21 + 2 * q^23 - 2 * q^25 + 8 * q^27 + 2 * q^29 + 10 * q^31 + 6 * q^33 - 8 * q^35 + 4 * q^37 - 6 * q^39 + 6 * q^41 + 2 * q^43 + 10 * q^45 - 6 * q^49 - 6 * q^51 + 2 * q^53 - 6 * q^55 + 20 * q^57 + 4 * q^59 - 22 * q^61 - 10 * q^63 + 6 * q^65 + 2 * q^67 + 2 * q^69 + 10 * q^73 - 14 * q^75 + 6 * q^77 - 20 * q^79 + 2 * q^81 - 6 * q^83 - 18 * q^85 - 16 * q^87 + 10 * q^89 - 6 * q^91 + 10 * q^93 + 4 * q^95 - 14 * q^97 + 6 * q^99

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