LMFDB - Embedded newform 639.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [639,2,Mod(1,639)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(639, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("639.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 639 = 3^{2} \cdot 71 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	639.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:	$5.10244068916$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.257.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 3 $ x^3 - x^2 - 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 71)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.91223$ of defining polynomial
Character	$\chi$	$=$	639.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.91223 q^{2} +1.65662 q^{4} -3.25561 q^{5} -3.82446 q^{7} +0.656620 q^{8} +O(q^{10})$	\(q-1.91223 q^{2} +1.65662 q^{4} -3.25561 q^{5} -3.82446 q^{7} +0.656620 q^{8} +6.22547 q^{10} -1.31324 q^{11} -3.31324 q^{13} +7.31324 q^{14} -4.56885 q^{16} -5.13770 q^{17} +5.48108 q^{19} -5.39331 q^{20} +2.51122 q^{22} +4.00000 q^{23} +5.59899 q^{25} +6.33568 q^{26} -6.33568 q^{28} -4.59899 q^{29} +4.00000 q^{31} +7.42345 q^{32} +9.82446 q^{34} +12.4509 q^{35} -5.65662 q^{37} -10.4811 q^{38} -2.13770 q^{40} +9.64892 q^{41} +1.43115 q^{43} -2.17554 q^{44} -7.64892 q^{46} -7.13770 q^{47} +7.62648 q^{49} -10.7065 q^{50} -5.48878 q^{52} +8.62648 q^{53} +4.27540 q^{55} -2.51122 q^{56} +8.79432 q^{58} +4.51122 q^{59} -3.64892 q^{61} -7.64892 q^{62} -5.05763 q^{64} +10.7866 q^{65} -5.31324 q^{67} -8.51122 q^{68} -23.8091 q^{70} -1.00000 q^{71} +16.3933 q^{73} +10.8168 q^{74} +9.08007 q^{76} +5.02243 q^{77} +5.08007 q^{79} +14.8744 q^{80} -18.4509 q^{82} +10.1678 q^{83} +16.7263 q^{85} -2.73669 q^{86} -0.862301 q^{88} +3.22547 q^{89} +12.6714 q^{91} +6.62648 q^{92} +13.6489 q^{94} -17.8442 q^{95} +11.8245 q^{97} -14.5836 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 2 * q^7	$ 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 2 q^{7} + 8 q^{10} - 6 q^{13} + 18 q^{14} - 5 q^{16} + 2 q^{17} + q^{19} + 6 q^{20} - 2 q^{22} + 12 q^{23} + 14 q^{25} - 4 q^{26} + 4 q^{28} - 11 q^{29} + 12 q^{31} + 6 q^{32} + 16 q^{34} + 16 q^{35} - 15 q^{37} - 16 q^{38} + 11 q^{40} + 2 q^{41} + 13 q^{43} - 20 q^{44} + 4 q^{46} - 4 q^{47} + 15 q^{49} - 6 q^{50} - 26 q^{52} + 18 q^{53} - 22 q^{55} + 2 q^{56} + 7 q^{58} + 4 q^{59} + 16 q^{61} + 4 q^{62} - 16 q^{64} - 12 q^{65} - 12 q^{67} - 16 q^{68} - 8 q^{70} - 3 q^{71} + 27 q^{73} - 6 q^{74} + 9 q^{76} - 4 q^{77} - 3 q^{79} + 7 q^{80} - 34 q^{82} + 19 q^{83} - 6 q^{85} + 12 q^{86} - 20 q^{88} - q^{89} - 8 q^{91} + 12 q^{92} + 14 q^{94} - 10 q^{95} + 22 q^{97} + 9 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 2 * q^7 + 8 * q^10 - 6 * q^13 + 18 * q^14 - 5 * q^16 + 2 * q^17 + q^19 + 6 * q^20 - 2 * q^22 + 12 * q^23 + 14 * q^25 - 4 * q^26 + 4 * q^28 - 11 * q^29 + 12 * q^31 + 6 * q^32 + 16 * q^34 + 16 * q^35 - 15 * q^37 - 16 * q^38 + 11 * q^40 + 2 * q^41 + 13 * q^43 - 20 * q^44 + 4 * q^46 - 4 * q^47 + 15 * q^49 - 6 * q^50 - 26 * q^52 + 18 * q^53 - 22 * q^55 + 2 * q^56 + 7 * q^58 + 4 * q^59 + 16 * q^61 + 4 * q^62 - 16 * q^64 - 12 * q^65 - 12 * q^67 - 16 * q^68 - 8 * q^70 - 3 * q^71 + 27 * q^73 - 6 * q^74 + 9 * q^76 - 4 * q^77 - 3 * q^79 + 7 * q^80 - 34 * q^82 + 19 * q^83 - 6 * q^85 + 12 * q^86 - 20 * q^88 - q^89 - 8 * q^91 + 12 * q^92 + 14 * q^94 - 10 * q^95 + 22 * q^97 + 9 * 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 $

$2$	−1.91223	−1.35215	−0.676075	−	0.736833i	$-0.736321\pi$
$2$	−1.91223	−1.35215	−0.676075	+	0.736833i	$0.736321\pi$
$3$	0	0
$3$	0	0
$4$	1.65662	0.828310
$5$	−3.25561	−1.45595	−0.727976	−	0.685602i	$-0.759539\pi$
$5$	−3.25561	−1.45595	−0.727976	+	0.685602i	$0.759539\pi$
$6$	0	0
$7$	−3.82446	−1.44551	−0.722755	−	0.691105i	$-0.757124\pi$
$7$	−3.82446	−1.44551	−0.722755	+	0.691105i	$0.757124\pi$
$8$	0.656620	0.232150
$9$	0	0
$10$	6.22547	1.96867
$11$	−1.31324	−0.395957	−0.197979	−	0.980206i	$-0.563438\pi$
$11$	−1.31324	−0.395957	−0.197979	+	0.980206i	$0.563438\pi$
$12$	0	0
$13$	−3.31324	−0.918928	−0.459464	−	0.888196i	$-0.651958\pi$
$13$	−3.31324	−0.918928	−0.459464	+	0.888196i	$0.651958\pi$
$14$	7.31324	1.95455
$15$	0	0
$16$	−4.56885	−1.14221
$17$	−5.13770	−1.24608	−0.623038	−	0.782192i	$-0.714102\pi$
$17$	−5.13770	−1.24608	−0.623038	+	0.782192i	$0.714102\pi$
$18$	0	0
$19$	5.48108	1.25745	0.628723	−	0.777629i	$-0.283578\pi$
$19$	5.48108	1.25745	0.628723	+	0.777629i	$0.283578\pi$
$20$	−5.39331	−1.20598
$21$	0	0
$22$	2.51122	0.535393
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	5.59899	1.11980
$26$	6.33568	1.24253
$27$	0	0
$28$	−6.33568	−1.19733
$29$	−4.59899	−0.854011	−0.427005	−	0.904249i	$-0.640431\pi$
$29$	−4.59899	−0.854011	−0.427005	+	0.904249i	$0.640431\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	7.42345	1.31229
$33$	0	0
$34$	9.82446	1.68488
$35$	12.4509	2.10459
$36$	0	0
$37$	−5.65662	−0.929943	−0.464971	−	0.885326i	$-0.653935\pi$
$37$	−5.65662	−0.929943	−0.464971	+	0.885326i	$0.653935\pi$
$38$	−10.4811	−1.70026
$39$	0	0
$40$	−2.13770	−0.338000
$41$	9.64892	1.50691	0.753454	−	0.657501i	$-0.228386\pi$
$41$	9.64892	1.50691	0.753454	+	0.657501i	$0.228386\pi$
$42$	0	0
$43$	1.43115	0.218248	0.109124	−	0.994028i	$-0.465195\pi$
$43$	1.43115	0.218248	0.109124	+	0.994028i	$0.465195\pi$
$44$	−2.17554	−0.327975
$45$	0	0
$46$	−7.64892	−1.12777
$47$	−7.13770	−1.04114	−0.520570	−	0.853819i	$-0.674281\pi$
$47$	−7.13770	−1.04114	−0.520570	+	0.853819i	$0.674281\pi$
$48$	0	0
$49$	7.62648	1.08950
$50$	−10.7065	−1.51413
$51$	0	0
$52$	−5.48878	−0.761157
$53$	8.62648	1.18494	0.592469	−	0.805593i	$-0.298153\pi$
$53$	8.62648	1.18494	0.592469	+	0.805593i	$0.298153\pi$
$54$	0	0
$55$	4.27540	0.576495
$56$	−2.51122	−0.335576
$57$	0	0
$58$	8.79432	1.15475
$59$	4.51122	0.587310	0.293655	−	0.955911i	$-0.405128\pi$
$59$	4.51122	0.587310	0.293655	+	0.955911i	$0.405128\pi$
$60$	0	0
$61$	−3.64892	−0.467196	−0.233598	−	0.972333i	$-0.575050\pi$
$61$	−3.64892	−0.467196	−0.233598	+	0.972333i	$0.575050\pi$
$62$	−7.64892	−0.971413
$63$	0	0
$64$	−5.05763	−0.632204
$65$	10.7866	1.33792
$66$	0	0
$67$	−5.31324	−0.649116	−0.324558	−	0.945866i	$-0.605215\pi$
$67$	−5.31324	−0.649116	−0.324558	+	0.945866i	$0.605215\pi$
$68$	−8.51122	−1.03214
$69$	0	0
$70$	−23.8091	−2.84573
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	0	0
$73$	16.3933	1.91869	0.959346	−	0.282233i	$-0.0910752\pi$
$73$	16.3933	1.91869	0.959346	+	0.282233i	$0.0910752\pi$
$74$	10.8168	1.25742
$75$	0	0
$76$	9.08007	1.04156
$77$	5.02243	0.572360
$78$	0	0
$79$	5.08007	0.571552	0.285776	−	0.958296i	$-0.407749\pi$
$79$	5.08007	0.571552	0.285776	+	0.958296i	$0.407749\pi$
$80$	14.8744	1.66301
$81$	0	0
$82$	−18.4509	−2.03757
$83$	10.1678	1.11607	0.558033	−	0.829819i	$-0.311556\pi$
$83$	10.1678	1.11607	0.558033	+	0.829819i	$0.311556\pi$
$84$	0	0
$85$	16.7263	1.81423
$86$	−2.73669	−0.295105
$87$	0	0
$88$	−0.862301	−0.0919216
$89$	3.22547	0.341899	0.170950	−	0.985280i	$-0.445316\pi$
$89$	3.22547	0.341899	0.170950	+	0.985280i	$0.445316\pi$
$90$	0	0
$91$	12.6714	1.32832
$92$	6.62648	0.690858
$93$	0	0
$94$	13.6489	1.40778
$95$	−17.8442	−1.83078
$96$	0	0
$97$	11.8245	1.20059	0.600296	−	0.799778i	$-0.295049\pi$
$97$	11.8245	1.20059	0.600296	+	0.799778i	$0.295049\pi$
$98$	−14.5836	−1.47316
$99$	0	0
$100$	9.27540	0.927540
$101$	−2.34338	−0.233175	−0.116587	−	0.993180i	$-0.537196\pi$
$101$	−2.34338	−0.233175	−0.116587	+	0.993180i	$0.537196\pi$
$102$	0	0
$103$	−16.1876	−1.59501	−0.797507	−	0.603309i	$-0.793848\pi$
$103$	−16.1876	−1.59501	−0.797507	+	0.603309i	$0.793848\pi$
$104$	−2.17554	−0.213329
$105$	0	0
$106$	−16.4958	−1.60221
$107$	−0.351083	−0.0339405	−0.0169703	−	0.999856i	$-0.505402\pi$
$107$	−0.351083	−0.0339405	−0.0169703	+	0.999856i	$0.505402\pi$
$108$	0	0
$109$	0.598988	0.0573727	0.0286863	−	0.999588i	$-0.490868\pi$
$109$	0.598988	0.0573727	0.0286863	+	0.999588i	$0.490868\pi$
$110$	−8.17554	−0.779507
$111$	0	0
$112$	17.4734	1.65108
$113$	−3.13770	−0.295170	−0.147585	−	0.989049i	$-0.547150\pi$
$113$	−3.13770	−0.295170	−0.147585	+	0.989049i	$0.547150\pi$
$114$	0	0
$115$	−13.0224	−1.21435
$116$	−7.61878	−0.707386
$117$	0	0
$118$	−8.62648	−0.794132
$119$	19.6489	1.80121
$120$	0	0
$121$	−9.27540	−0.843218
$122$	6.97757	0.631719
$123$	0	0
$124$	6.62648	0.595076
$125$	−1.95007	−0.174420
$126$	0	0
$127$	−4.33568	−0.384729	−0.192365	−	0.981324i	$-0.561616\pi$
$127$	−4.33568	−0.384729	−0.192365	+	0.981324i	$0.561616\pi$
$128$	−5.17554	−0.457458
$129$	0	0
$130$	−20.6265	−1.80906
$131$	−14.3330	−1.25228	−0.626141	−	0.779710i	$-0.715367\pi$
$131$	−14.3330	−1.25228	−0.626141	+	0.779710i	$0.715367\pi$
$132$	0	0
$133$	−20.9622	−1.81765
$134$	10.1601	0.877702
$135$	0	0
$136$	−3.37352	−0.289277
$137$	−14.2754	−1.21963	−0.609815	−	0.792544i	$-0.708756\pi$
$137$	−14.2754	−1.21963	−0.609815	+	0.792544i	$0.708756\pi$
$138$	0	0
$139$	−4.45094	−0.377524	−0.188762	−	0.982023i	$-0.560447\pi$
$139$	−4.45094	−0.377524	−0.188762	+	0.982023i	$0.560447\pi$
$140$	20.6265	1.74326
$141$	0	0
$142$	1.91223	0.160471
$143$	4.35108	0.363856
$144$	0	0
$145$	14.9725	1.24340
$146$	−31.3478	−2.59436
$147$	0	0
$148$	−9.37087	−0.770281
$149$	8.45094	0.692328	0.346164	−	0.938174i	$-0.387484\pi$
$149$	8.45094	0.692328	0.346164	+	0.938174i	$0.387484\pi$
$150$	0	0
$151$	17.9122	1.45768	0.728838	−	0.684686i	$-0.240061\pi$
$151$	17.9122	1.45768	0.728838	+	0.684686i	$0.240061\pi$
$152$	3.59899	0.291916
$153$	0	0
$154$	−9.60405	−0.773916
$155$	−13.0224	−1.04599
$156$	0	0
$157$	−5.97251	−0.476658	−0.238329	−	0.971184i	$-0.576600\pi$
$157$	−5.97251	−0.476658	−0.238329	+	0.971184i	$0.576600\pi$
$158$	−9.71425	−0.772824
$159$	0	0
$160$	−24.1678	−1.91064
$161$	−15.2978	−1.20564
$162$	0	0
$163$	−11.3735	−0.890843	−0.445421	−	0.895321i	$-0.646946\pi$
$163$	−11.3735	−0.890843	−0.445421	+	0.895321i	$0.646946\pi$
$164$	15.9846	1.24819
$165$	0	0
$166$	−19.4432	−1.50909
$167$	−9.08007	−0.702637	−0.351318	−	0.936256i	$-0.614267\pi$
$167$	−9.08007	−0.702637	−0.351318	+	0.936256i	$0.614267\pi$
$168$	0	0
$169$	−2.02243	−0.155572
$170$	−31.9846	−2.45311
$171$	0	0
$172$	2.37087	0.180777
$173$	−1.64892	−0.125365	−0.0626824	−	0.998034i	$-0.519966\pi$
$173$	−1.64892	−0.125365	−0.0626824	+	0.998034i	$0.519966\pi$
$174$	0	0
$175$	−21.4131	−1.61868
$176$	6.00000	0.452267
$177$	0	0
$178$	−6.16784	−0.462299
$179$	−5.19533	−0.388317	−0.194159	−	0.980970i	$-0.562198\pi$
$179$	−5.19533	−0.388317	−0.194159	+	0.980970i	$0.562198\pi$
$180$	0	0
$181$	14.0999	1.04803	0.524017	−	0.851708i	$-0.324433\pi$
$181$	14.0999	1.04803	0.524017	+	0.851708i	$0.324433\pi$
$182$	−24.2305	−1.79609
$183$	0	0
$184$	2.62648	0.193627
$185$	18.4157	1.35395
$186$	0	0
$187$	6.74704	0.493392
$188$	−11.8245	−0.862387
$189$	0	0
$190$	34.1223	2.47549
$191$	6.98965	0.505753	0.252877	−	0.967499i	$-0.418623\pi$
$191$	6.98965	0.505753	0.252877	+	0.967499i	$0.418623\pi$
$192$	0	0
$193$	7.93972	0.571514	0.285757	−	0.958302i	$-0.407755\pi$
$193$	7.93972	0.571514	0.285757	+	0.958302i	$0.407755\pi$
$194$	−22.6111	−1.62338
$195$	0	0
$196$	12.6342	0.902442
$197$	3.48878	0.248565	0.124283	−	0.992247i	$-0.460337\pi$
$197$	3.48878	0.248565	0.124283	+	0.992247i	$0.460337\pi$
$198$	0	0
$199$	17.4811	1.23920	0.619600	−	0.784917i	$-0.287295\pi$
$199$	17.4811	1.23920	0.619600	+	0.784917i	$0.287295\pi$
$200$	3.67641	0.259961
$201$	0	0
$202$	4.48108	0.315288
$203$	17.5886	1.23448
$204$	0	0
$205$	−31.4131	−2.19399
$206$	30.9545	2.15670
$207$	0	0
$208$	15.1377	1.04961
$209$	−7.19798	−0.497894
$210$	0	0
$211$	−4.23582	−0.291606	−0.145803	−	0.989314i	$-0.546577\pi$
$211$	−4.23582	−0.291606	−0.145803	+	0.989314i	$0.546577\pi$
$212$	14.2908	0.981497
$213$	0	0
$214$	0.671352	0.0458927
$215$	−4.65927	−0.317759
$216$	0	0
$217$	−15.2978	−1.03848
$218$	−1.14540	−0.0775765
$219$	0	0
$220$	7.08271	0.477516
$221$	17.0224	1.14505
$222$	0	0
$223$	−19.5259	−1.30755	−0.653777	−	0.756687i	$-0.726817\pi$
$223$	−19.5259	−1.30755	−0.653777	+	0.756687i	$0.726817\pi$
$224$	−28.3907	−1.89693
$225$	0	0
$226$	6.00000	0.399114
$227$	20.0396	1.33007	0.665037	−	0.746811i	$-0.268416\pi$
$227$	20.0396	1.33007	0.665037	+	0.746811i	$0.268416\pi$
$228$	0	0
$229$	−1.65156	−0.109138	−0.0545691	−	0.998510i	$-0.517379\pi$
$229$	−1.65156	−0.109138	−0.0545691	+	0.998510i	$0.517379\pi$
$230$	24.9019	1.64198
$231$	0	0
$232$	−3.01979	−0.198259
$233$	−23.5809	−1.54484	−0.772419	−	0.635113i	$-0.780954\pi$
$233$	−23.5809	−1.54484	−0.772419	+	0.635113i	$0.780954\pi$
$234$	0	0
$235$	23.2376	1.51585
$236$	7.47338	0.486475
$237$	0	0
$238$	−37.5732	−2.43551
$239$	−0.571495	−0.0369669	−0.0184835	−	0.999829i	$-0.505884\pi$
$239$	−0.571495	−0.0369669	−0.0184835	+	0.999829i	$0.505884\pi$
$240$	0	0
$241$	25.4734	1.64088	0.820442	−	0.571730i	$-0.193727\pi$
$241$	25.4734	1.64088	0.820442	+	0.571730i	$0.193727\pi$
$242$	17.7367	1.14016
$243$	0	0
$244$	−6.04487	−0.386983
$245$	−24.8288	−1.58626
$246$	0	0
$247$	−18.1601	−1.15550
$248$	2.62648	0.166782
$249$	0	0
$250$	3.72898	0.235842
$251$	−20.1076	−1.26918	−0.634589	−	0.772850i	$-0.718831\pi$
$251$	−20.1076	−1.26918	−0.634589	+	0.772850i	$0.718831\pi$
$252$	0	0
$253$	−5.25296	−0.330251
$254$	8.29081	0.520211
$255$	0	0
$256$	20.0121	1.25076
$257$	24.2754	1.51426	0.757129	−	0.653266i	$-0.226601\pi$
$257$	24.2754	1.51426	0.757129	+	0.653266i	$0.226601\pi$
$258$	0	0
$259$	21.6335	1.34424
$260$	17.8693	1.10821
$261$	0	0
$262$	27.4080	1.69327
$263$	−9.54641	−0.588657	−0.294329	−	0.955704i	$-0.595096\pi$
$263$	−9.54641	−0.588657	−0.294329	+	0.955704i	$0.595096\pi$
$264$	0	0
$265$	−28.0844	−1.72521
$266$	40.0844	2.45774
$267$	0	0
$268$	−8.80202	−0.537669
$269$	11.8847	0.724625	0.362313	−	0.932057i	$-0.381987\pi$
$269$	11.8847	0.724625	0.362313	+	0.932057i	$0.381987\pi$
$270$	0	0
$271$	−3.51628	−0.213599	−0.106799	−	0.994281i	$-0.534060\pi$
$271$	−3.51628	−0.213599	−0.106799	+	0.994281i	$0.534060\pi$
$272$	23.4734	1.42328
$273$	0	0
$274$	27.2978	1.64912
$275$	−7.35282	−0.443392
$276$	0	0
$277$	5.99230	0.360042	0.180021	−	0.983663i	$-0.442383\pi$
$277$	5.99230	0.360042	0.180021	+	0.983663i	$0.442383\pi$
$278$	8.51122	0.510469
$279$	0	0
$280$	8.17554	0.488582
$281$	6.51122	0.388427	0.194213	−	0.980959i	$-0.437785\pi$
$281$	6.51122	0.388427	0.194213	+	0.980959i	$0.437785\pi$
$282$	0	0
$283$	11.4734	0.682021	0.341011	−	0.940059i	$-0.389231\pi$
$283$	11.4734	0.682021	0.341011	+	0.940059i	$0.389231\pi$
$284$	−1.65662	−0.0983023
$285$	0	0
$286$	−8.32027	−0.491988
$287$	−36.9019	−2.17825
$288$	0	0
$289$	9.39595	0.552703
$290$	−28.6309	−1.68126
$291$	0	0
$292$	27.1575	1.58927
$293$	20.6265	1.20501	0.602506	−	0.798114i	$-0.294169\pi$
$293$	20.6265	1.20501	0.602506	+	0.798114i	$0.294169\pi$
$294$	0	0
$295$	−14.6868	−0.855096
$296$	−3.71425	−0.215887
$297$	0	0
$298$	−16.1601	−0.936131
$299$	−13.2530	−0.766439
$300$	0	0
$301$	−5.47338	−0.315480
$302$	−34.2523	−1.97100
$303$	0	0
$304$	−25.0422	−1.43627
$305$	11.8794	0.680215
$306$	0	0
$307$	11.8245	0.674857	0.337429	−	0.941351i	$-0.390443\pi$
$307$	11.8245	0.674857	0.337429	+	0.941351i	$0.390443\pi$
$308$	8.32027	0.474091
$309$	0	0
$310$	24.9019	1.41433
$311$	18.3830	1.04240	0.521201	−	0.853434i	$-0.325484\pi$
$311$	18.3830	1.04240	0.521201	+	0.853434i	$0.325484\pi$
$312$	0	0
$313$	−5.30554	−0.299887	−0.149943	−	0.988695i	$-0.547909\pi$
$313$	−5.30554	−0.299887	−0.149943	+	0.988695i	$0.547909\pi$
$314$	11.4208	0.644513
$315$	0	0
$316$	8.41574	0.473423
$317$	5.88474	0.330520	0.165260	−	0.986250i	$-0.447154\pi$
$317$	5.88474	0.330520	0.165260	+	0.986250i	$0.447154\pi$
$318$	0	0
$319$	6.03958	0.338152
$320$	16.4657	0.920459
$321$	0	0
$322$	29.2530	1.63020
$323$	−28.1601	−1.56687
$324$	0	0
$325$	−18.5508	−1.02901
$326$	21.7488	1.20455
$327$	0	0
$328$	6.33568	0.349829
$329$	27.2978	1.50498
$330$	0	0
$331$	12.7316	0.699794	0.349897	−	0.936788i	$-0.386217\pi$
$331$	12.7316	0.699794	0.349897	+	0.936788i	$0.386217\pi$
$332$	16.8442	0.924448
$333$	0	0
$334$	17.3632	0.950070
$335$	17.2978	0.945082
$336$	0	0
$337$	28.7712	1.56727	0.783634	−	0.621223i	$-0.213364\pi$
$337$	28.7712	1.56727	0.783634	+	0.621223i	$0.213364\pi$
$338$	3.86736	0.210357
$339$	0	0
$340$	27.7092	1.50274
$341$	−5.25296	−0.284464
$342$	0	0
$343$	−2.39595	−0.129369
$344$	0.939723	0.0506664
$345$	0	0
$346$	3.15311	0.169512
$347$	−1.02243	−0.0548872	−0.0274436	−	0.999623i	$-0.508737\pi$
$347$	−1.02243	−0.0548872	−0.0274436	+	0.999623i	$0.508737\pi$
$348$	0	0
$349$	2.46635	0.132021	0.0660103	−	0.997819i	$-0.478973\pi$
$349$	2.46635	0.132021	0.0660103	+	0.997819i	$0.478973\pi$
$350$	40.9468	2.18870
$351$	0	0
$352$	−9.74877	−0.519611
$353$	−8.74175	−0.465276	−0.232638	−	0.972563i	$-0.574736\pi$
$353$	−8.74175	−0.465276	−0.232638	+	0.972563i	$0.574736\pi$
$354$	0	0
$355$	3.25561	0.172790
$356$	5.34338	0.283199
$357$	0	0
$358$	9.93466	0.525063
$359$	7.59128	0.400653	0.200326	−	0.979729i	$-0.435800\pi$
$359$	7.59128	0.400653	0.200326	+	0.979729i	$0.435800\pi$
$360$	0	0
$361$	11.0422	0.581170
$362$	−26.9622	−1.41710
$363$	0	0
$364$	20.9916	1.10026
$365$	−53.3702	−2.79352
$366$	0	0
$367$	30.1025	1.57134	0.785669	−	0.618647i	$-0.212319\pi$
$367$	30.1025	1.57134	0.785669	+	0.618647i	$0.212319\pi$
$368$	−18.2754	−0.952671
$369$	0	0
$370$	−35.2151	−1.83075
$371$	−32.9916	−1.71284
$372$	0	0
$373$	8.39331	0.434589	0.217295	−	0.976106i	$-0.430277\pi$
$373$	8.39331	0.434589	0.217295	+	0.976106i	$0.430277\pi$
$374$	−12.9019	−0.667140
$375$	0	0
$376$	−4.68676	−0.241701
$377$	15.2376	0.784774
$378$	0	0
$379$	−10.4389	−0.536208	−0.268104	−	0.963390i	$-0.586397\pi$
$379$	−10.4389	−0.536208	−0.268104	+	0.963390i	$0.586397\pi$
$380$	−29.5611	−1.51645
$381$	0	0
$382$	−13.3658	−0.683855
$383$	4.97757	0.254342	0.127171	−	0.991881i	$-0.459410\pi$
$383$	4.97757	0.254342	0.127171	+	0.991881i	$0.459410\pi$
$384$	0	0
$385$	−16.3511	−0.833328
$386$	−15.1826	−0.772772
$387$	0	0
$388$	19.5886	0.994462
$389$	15.3528	0.778419	0.389209	−	0.921149i	$-0.372748\pi$
$389$	15.3528	0.778419	0.389209	+	0.921149i	$0.372748\pi$
$390$	0	0
$391$	−20.5508	−1.03930
$392$	5.00770	0.252927
$393$	0	0
$394$	−6.67135	−0.336098
$395$	−16.5387	−0.832153
$396$	0	0
$397$	−3.93972	−0.197729	−0.0988645	−	0.995101i	$-0.531521\pi$
$397$	−3.93972	−0.197729	−0.0988645	+	0.995101i	$0.531521\pi$
$398$	−33.4278	−1.67559
$399$	0	0
$400$	−25.5809	−1.27905
$401$	11.4734	0.572953	0.286477	−	0.958087i	$-0.407516\pi$
$401$	11.4734	0.572953	0.286477	+	0.958087i	$0.407516\pi$
$402$	0	0
$403$	−13.2530	−0.660177
$404$	−3.88209	−0.193141
$405$	0	0
$406$	−33.6335	−1.66920
$407$	7.42851	0.368217
$408$	0	0
$409$	−24.6232	−1.21754	−0.608768	−	0.793348i	$-0.708336\pi$
$409$	−24.6232	−1.21754	−0.608768	+	0.793348i	$0.708336\pi$
$410$	60.0690	2.96660
$411$	0	0
$412$	−26.8168	−1.32117
$413$	−17.2530	−0.848963
$414$	0	0
$415$	−33.1025	−1.62494
$416$	−24.5957	−1.20590
$417$	0	0
$418$	13.7642	0.673228
$419$	5.06534	0.247458	0.123729	−	0.992316i	$-0.460515\pi$
$419$	5.06534	0.247458	0.123729	+	0.992316i	$0.460515\pi$
$420$	0	0
$421$	−22.0396	−1.07414	−0.537072	−	0.843537i	$-0.680470\pi$
$421$	−22.0396	−1.07414	−0.537072	+	0.843537i	$0.680470\pi$
$422$	8.09986	0.394295
$423$	0	0
$424$	5.66432	0.275084
$425$	−28.7659	−1.39535
$426$	0	0
$427$	13.9551	0.675336
$428$	−0.581612	−0.0281133
$429$	0	0
$430$	8.90958	0.429658
$431$	16.4278	0.791301	0.395650	−	0.918401i	$-0.370519\pi$
$431$	16.4278	0.791301	0.395650	+	0.918401i	$0.370519\pi$
$432$	0	0
$433$	33.6885	1.61897	0.809483	−	0.587143i	$-0.199748\pi$
$433$	33.6885	1.61897	0.809483	+	0.587143i	$0.199748\pi$
$434$	29.2530	1.40419
$435$	0	0
$436$	0.992296	0.0475224
$437$	21.9243	1.04878
$438$	0	0
$439$	9.70390	0.463142	0.231571	−	0.972818i	$-0.425613\pi$
$439$	9.70390	0.463142	0.231571	+	0.972818i	$0.425613\pi$
$440$	2.80731	0.133833
$441$	0	0
$442$	−32.5508	−1.54828
$443$	22.1601	1.05286	0.526430	−	0.850219i	$-0.323530\pi$
$443$	22.1601	1.05286	0.526430	+	0.850219i	$0.323530\pi$
$444$	0	0
$445$	−10.5009	−0.497789
$446$	37.3381	1.76801
$447$	0	0
$448$	19.3427	0.913857
$449$	29.0224	1.36965	0.684827	−	0.728706i	$-0.259878\pi$
$449$	29.0224	1.36965	0.684827	+	0.728706i	$0.259878\pi$
$450$	0	0
$451$	−12.6714	−0.596671
$452$	−5.19798	−0.244492
$453$	0	0
$454$	−38.3203	−1.79846
$455$	−41.2530	−1.93397
$456$	0	0
$457$	19.5491	0.914466	0.457233	−	0.889347i	$-0.348840\pi$
$457$	19.5491	0.914466	0.457233	+	0.889347i	$0.348840\pi$
$458$	3.15817	0.147571
$459$	0	0
$460$	−21.5732	−1.00586
$461$	−6.04487	−0.281538	−0.140769	−	0.990042i	$-0.544957\pi$
$461$	−6.04487	−0.281538	−0.140769	+	0.990042i	$0.544957\pi$
$462$	0	0
$463$	−21.1300	−0.981994	−0.490997	−	0.871161i	$-0.663367\pi$
$463$	−21.1300	−0.981994	−0.490997	+	0.871161i	$0.663367\pi$
$464$	21.0121	0.975462
$465$	0	0
$466$	45.0922	2.08885
$467$	−40.8262	−1.88921	−0.944606	−	0.328208i	$-0.893555\pi$
$467$	−40.8262	−1.88921	−0.944606	+	0.328208i	$0.893555\pi$
$468$	0	0
$469$	20.3203	0.938303
$470$	−44.4355	−2.04966
$471$	0	0
$472$	2.96216	0.136344
$473$	−1.87945	−0.0864170
$474$	0	0
$475$	30.6885	1.40808
$476$	32.5508	1.49196
$477$	0	0
$478$	1.09283	0.0499848
$479$	24.0000	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	24.0000	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	18.7417	0.854550
$482$	−48.7109	−2.21872
$483$	0	0
$484$	−15.3658	−0.698446
$485$	−38.4958	−1.74800
$486$	0	0
$487$	10.9622	0.496743	0.248371	−	0.968665i	$-0.420105\pi$
$487$	10.9622	0.496743	0.248371	+	0.968665i	$0.420105\pi$
$488$	−2.39595	−0.108460
$489$	0	0
$490$	47.4784	2.14486
$491$	−23.4734	−1.05934	−0.529669	−	0.848204i	$-0.677684\pi$
$491$	−23.4734	−1.05934	−0.529669	+	0.848204i	$0.677684\pi$
$492$	0	0
$493$	23.6282	1.06416
$494$	34.7263	1.56241
$495$	0	0
$496$	−18.2754	−0.820590
$497$	3.82446	0.171550
$498$	0	0
$499$	7.11021	0.318297	0.159148	−	0.987255i	$-0.449125\pi$
$499$	7.11021	0.318297	0.159148	+	0.987255i	$0.449125\pi$
$500$	−3.23053	−0.144474
$501$	0	0
$502$	38.4503	1.71612
$503$	31.8365	1.41952	0.709761	−	0.704443i	$-0.248803\pi$
$503$	31.8365	1.41952	0.709761	+	0.704443i	$0.248803\pi$
$504$	0	0
$505$	7.62913	0.339492
$506$	10.0449	0.446549
$507$	0	0
$508$	−7.18257	−0.318675
$509$	11.0224	0.488561	0.244280	−	0.969705i	$-0.421448\pi$
$509$	11.0224	0.488561	0.244280	+	0.969705i	$0.421448\pi$
$510$	0	0
$511$	−62.6955	−2.77349
$512$	−27.9166	−1.23375
$513$	0	0
$514$	−46.4201	−2.04750
$515$	52.7006	2.32227
$516$	0	0
$517$	9.37352	0.412247
$518$	−41.3682	−1.81762
$519$	0	0
$520$	7.08271	0.310597
$521$	29.4105	1.28850	0.644248	−	0.764817i	$-0.277171\pi$
$521$	29.4105	1.28850	0.644248	+	0.764817i	$0.277171\pi$
$522$	0	0
$523$	−20.8623	−0.912245	−0.456122	−	0.889917i	$-0.650762\pi$
$523$	−20.8623	−0.912245	−0.456122	+	0.889917i	$0.650762\pi$
$524$	−23.7444	−1.03728
$525$	0	0
$526$	18.2549	0.795953
$527$	−20.5508	−0.895207
$528$	0	0
$529$	−7.00000	−0.304348
$530$	53.7039	2.33275
$531$	0	0
$532$	−34.7263	−1.50558
$533$	−31.9692	−1.38474
$534$	0	0
$535$	1.14299	0.0494158
$536$	−3.48878	−0.150692
$537$	0	0
$538$	−22.7263	−0.979802
$539$	−10.0154	−0.431394
$540$	0	0
$541$	14.7417	0.633797	0.316899	−	0.948459i	$-0.397359\pi$
$541$	14.7417	0.633797	0.316899	+	0.948459i	$0.397359\pi$
$542$	6.72393	0.288817
$543$	0	0
$544$	−38.1394	−1.63521
$545$	−1.95007	−0.0835319
$546$	0	0
$547$	−26.5189	−1.13387	−0.566934	−	0.823763i	$-0.691870\pi$
$547$	−26.5189	−1.13387	−0.566934	+	0.823763i	$0.691870\pi$
$548$	−23.6489	−1.01023
$549$	0	0
$550$	14.0603	0.599532
$551$	−25.2074	−1.07387
$552$	0	0
$553$	−19.4285	−0.826184
$554$	−11.4586	−0.486831
$555$	0	0
$556$	−7.37352	−0.312707
$557$	−44.1274	−1.86974	−0.934868	−	0.354996i	$-0.884482\pi$
$557$	−44.1274	−1.86974	−0.934868	+	0.354996i	$0.884482\pi$
$558$	0	0
$559$	−4.74175	−0.200554
$560$	−56.8865	−2.40389
$561$	0	0
$562$	−12.4509	−0.525211
$563$	−4.84689	−0.204272	−0.102136	−	0.994770i	$-0.532568\pi$
$563$	−4.84689	−0.204272	−0.102136	+	0.994770i	$0.532568\pi$
$564$	0	0
$565$	10.2151	0.429753
$566$	−21.9397	−0.922195
$567$	0	0
$568$	−0.656620	−0.0275512
$569$	37.9665	1.59164	0.795820	−	0.605533i	$-0.207040\pi$
$569$	37.9665	1.59164	0.795820	+	0.605533i	$0.207040\pi$
$570$	0	0
$571$	−20.7789	−0.869570	−0.434785	−	0.900534i	$-0.643176\pi$
$571$	−20.7789	−0.869570	−0.434785	+	0.900534i	$0.643176\pi$
$572$	7.20809	0.301386
$573$	0	0
$574$	70.5649	2.94532
$575$	22.3960	0.933976
$576$	0	0
$577$	37.9665	1.58057	0.790284	−	0.612741i	$-0.209933\pi$
$577$	37.9665	1.58057	0.790284	+	0.612741i	$0.209933\pi$
$578$	−17.9672	−0.747338
$579$	0	0
$580$	24.8038	1.02992
$581$	−38.8865	−1.61328
$582$	0	0
$583$	−11.3286	−0.469185
$584$	10.7642	0.445425
$585$	0	0
$586$	−39.4426	−1.62936
$587$	9.03452	0.372895	0.186447	−	0.982465i	$-0.440303\pi$
$587$	9.03452	0.372895	0.186447	+	0.982465i	$0.440303\pi$
$588$	0	0
$589$	21.9243	0.903376
$590$	28.0844	1.15622
$591$	0	0
$592$	25.8442	1.06219
$593$	−9.44588	−0.387896	−0.193948	−	0.981012i	$-0.562129\pi$
$593$	−9.44588	−0.387896	−0.193948	+	0.981012i	$0.562129\pi$
$594$	0	0
$595$	−63.9692	−2.62248
$596$	14.0000	0.573462
$597$	0	0
$598$	25.3427	1.03634
$599$	−39.9089	−1.63063	−0.815317	−	0.579015i	$-0.803437\pi$
$599$	−39.9089	−1.63063	−0.815317	+	0.579015i	$0.803437\pi$
$600$	0	0
$601$	45.5130	1.85651	0.928256	−	0.371942i	$-0.121308\pi$
$601$	45.5130	1.85651	0.928256	+	0.371942i	$0.121308\pi$
$602$	10.4663	0.426576
$603$	0	0
$604$	29.6738	1.20741
$605$	30.1971	1.22769
$606$	0	0
$607$	−40.4509	−1.64185	−0.820927	−	0.571034i	$-0.806542\pi$
$607$	−40.4509	−1.64185	−0.820927	+	0.571034i	$0.806542\pi$
$608$	40.6885	1.65014
$609$	0	0
$610$	−22.7162	−0.919753
$611$	23.6489	0.956733
$612$	0	0
$613$	−22.4683	−0.907487	−0.453743	−	0.891132i	$-0.649912\pi$
$613$	−22.4683	−0.907487	−0.453743	+	0.891132i	$0.649912\pi$
$614$	−22.6111	−0.912509
$615$	0	0
$616$	3.29783	0.132873
$617$	3.95778	0.159334	0.0796670	−	0.996822i	$-0.474614\pi$
$617$	3.95778	0.159334	0.0796670	+	0.996822i	$0.474614\pi$
$618$	0	0
$619$	−4.80202	−0.193010	−0.0965048	−	0.995333i	$-0.530766\pi$
$619$	−4.80202	−0.193010	−0.0965048	+	0.995333i	$0.530766\pi$
$620$	−21.5732	−0.866402
$621$	0	0
$622$	−35.1524	−1.40948
$623$	−12.3357	−0.494218
$624$	0	0
$625$	−21.6463	−0.865851
$626$	10.1454	0.405492
$627$	0	0
$628$	−9.89418	−0.394821
$629$	29.0620	1.15878
$630$	0	0
$631$	−32.0449	−1.27569	−0.637843	−	0.770166i	$-0.720173\pi$
$631$	−32.0449	−1.27569	−0.637843	+	0.770166i	$0.720173\pi$
$632$	3.33568	0.132686
$633$	0	0
$634$	−11.2530	−0.446912
$635$	14.1153	0.560147
$636$	0	0
$637$	−25.2684	−1.00117
$638$	−11.5491	−0.457232
$639$	0	0
$640$	16.8495	0.666036
$641$	40.2677	1.59048	0.795239	−	0.606296i	$-0.207345\pi$
$641$	40.2677	1.59048	0.795239	+	0.606296i	$0.207345\pi$
$642$	0	0
$643$	12.6714	0.499709	0.249855	−	0.968283i	$-0.419617\pi$
$643$	12.6714	0.499709	0.249855	+	0.968283i	$0.419617\pi$
$644$	−25.3427	−0.998642
$645$	0	0
$646$	53.8486	2.11865
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−5.92432	−0.232550
$650$	35.4734	1.39138
$651$	0	0
$652$	−18.8416	−0.737894
$653$	−31.6643	−1.23912	−0.619560	−	0.784949i	$-0.712689\pi$
$653$	−31.6643	−1.23912	−0.619560	+	0.784949i	$0.712689\pi$
$654$	0	0
$655$	46.6627	1.82326
$656$	−44.0844	−1.72121
$657$	0	0
$658$	−52.1997	−2.03496
$659$	38.0570	1.48249	0.741244	−	0.671235i	$-0.234236\pi$
$659$	38.0570	1.48249	0.741244	+	0.671235i	$0.234236\pi$
$660$	0	0
$661$	41.1672	1.60122	0.800609	−	0.599188i	$-0.204510\pi$
$661$	41.1672	1.60122	0.800609	+	0.599188i	$0.204510\pi$
$662$	−24.3458	−0.946226
$663$	0	0
$664$	6.67641	0.259095
$665$	68.2446	2.64641
$666$	0	0
$667$	−18.3960	−0.712294
$668$	−15.0422	−0.582001
$669$	0	0
$670$	−33.0774	−1.27789
$671$	4.79191	0.184989
$672$	0	0
$673$	14.9622	0.576749	0.288374	−	0.957518i	$-0.406885\pi$
$673$	14.9622	0.576749	0.288374	+	0.957518i	$0.406885\pi$
$674$	−55.0171	−2.11918
$675$	0	0
$676$	−3.35041	−0.128862
$677$	39.5457	1.51987	0.759933	−	0.650001i	$-0.225232\pi$
$677$	39.5457	1.51987	0.759933	+	0.650001i	$0.225232\pi$
$678$	0	0
$679$	−45.2221	−1.73547
$680$	10.9829	0.421173
$681$	0	0
$682$	10.0449	0.384638
$683$	25.2925	0.967792	0.483896	−	0.875125i	$-0.339221\pi$
$683$	25.2925	0.967792	0.483896	+	0.875125i	$0.339221\pi$
$684$	0	0
$685$	46.4751	1.77572
$686$	4.58161	0.174927
$687$	0	0
$688$	−6.53871	−0.249286
$689$	−28.5816	−1.08887
$690$	0	0
$691$	−12.2600	−0.466392	−0.233196	−	0.972430i	$-0.574918\pi$
$691$	−12.2600	−0.466392	−0.233196	+	0.972430i	$0.574918\pi$
$692$	−2.73163	−0.103841
$693$	0	0
$694$	1.95513	0.0742157
$695$	14.4905	0.549657
$696$	0	0
$697$	−49.5732	−1.87772
$698$	−4.71622	−0.178512
$699$	0	0
$700$	−35.4734	−1.34077
$701$	7.83987	0.296108	0.148054	−	0.988979i	$-0.452699\pi$
$701$	7.83987	0.296108	0.148054	+	0.988979i	$0.452699\pi$
$702$	0	0
$703$	−31.0044	−1.16935
$704$	6.64189	0.250326
$705$	0	0
$706$	16.7162	0.629123
$707$	8.96216	0.337057
$708$	0	0
$709$	−52.6352	−1.97676	−0.988379	−	0.152009i	$-0.951426\pi$
$709$	−52.6352	−1.97676	−0.988379	+	0.152009i	$0.951426\pi$
$710$	−6.22547	−0.233638
$711$	0	0
$712$	2.11791	0.0793720
$713$	16.0000	0.599205
$714$	0	0
$715$	−14.1654	−0.529757
$716$	−8.60669	−0.321647
$717$	0	0
$718$	−14.5163	−0.541743
$719$	−1.71161	−0.0638322	−0.0319161	−	0.999491i	$-0.510161\pi$
$719$	−1.71161	−0.0638322	−0.0319161	+	0.999491i	$0.510161\pi$
$720$	0	0
$721$	61.9089	2.30561
$722$	−21.1153	−0.785829
$723$	0	0
$724$	23.3581	0.868097
$725$	−25.7497	−0.956319
$726$	0	0
$727$	15.0378	0.557723	0.278861	−	0.960331i	$-0.410043\pi$
$727$	15.0378	0.557723	0.278861	+	0.960331i	$0.410043\pi$
$728$	8.32027	0.308370
$729$	0	0
$730$	102.056	3.77726
$731$	−7.35282	−0.271954
$732$	0	0
$733$	45.7334	1.68920	0.844600	−	0.535398i	$-0.179838\pi$
$733$	45.7334	1.68920	0.844600	+	0.535398i	$0.179838\pi$
$734$	−57.5629	−2.12468
$735$	0	0
$736$	29.6938	1.09453
$737$	6.97757	0.257022
$738$	0	0
$739$	−31.5284	−1.15979	−0.579895	−	0.814691i	$-0.696906\pi$
$739$	−31.5284	−1.15979	−0.579895	+	0.814691i	$0.696906\pi$
$740$	30.5079	1.12149
$741$	0	0
$742$	63.0875	2.31602
$743$	−0.471638	−0.0173027	−0.00865136	−	0.999963i	$-0.502754\pi$
$743$	−0.471638	−0.0173027	−0.00865136	+	0.999963i	$0.502754\pi$
$744$	0	0
$745$	−27.5130	−1.00800
$746$	−16.0499	−0.587630
$747$	0	0
$748$	11.1773	0.408682
$749$	1.34270	0.0490613
$750$	0	0
$751$	−16.5059	−0.602310	−0.301155	−	0.953575i	$-0.597372\pi$
$751$	−16.5059	−0.602310	−0.301155	+	0.953575i	$0.597372\pi$
$752$	32.6111	1.18920
$753$	0	0
$754$	−29.1377	−1.06113
$755$	−58.3152	−2.12231
$756$	0	0
$757$	−32.8918	−1.19547	−0.597736	−	0.801693i	$-0.703933\pi$
$757$	−32.8918	−1.19547	−0.597736	+	0.801693i	$0.703933\pi$
$758$	19.9615	0.725034
$759$	0	0
$760$	−11.7169	−0.425017
$761$	−2.39595	−0.0868532	−0.0434266	−	0.999057i	$-0.513827\pi$
$761$	−2.39595	−0.0868532	−0.0434266	+	0.999057i	$0.513827\pi$
$762$	0	0
$763$	−2.29081	−0.0829327
$764$	11.5792	0.418921
$765$	0	0
$766$	−9.51825	−0.343908
$767$	−14.9468	−0.539696
$768$	0	0
$769$	53.6335	1.93407	0.967037	−	0.254636i	$-0.0819557\pi$
$769$	53.6335	1.93407	0.967037	+	0.254636i	$0.0819557\pi$
$770$	31.2670	1.12679
$771$	0	0
$772$	13.1531	0.473391
$773$	−30.5059	−1.09722	−0.548611	−	0.836078i	$-0.684843\pi$
$773$	−30.5059	−1.09722	−0.548611	+	0.836078i	$0.684843\pi$
$774$	0	0
$775$	22.3960	0.804486
$776$	7.76418	0.278718
$777$	0	0
$778$	−29.3581	−1.05254
$779$	52.8865	1.89485
$780$	0	0
$781$	1.31324	0.0469915
$782$	39.2978	1.40529
$783$	0	0
$784$	−34.8442	−1.24444
$785$	19.4441	0.693991
$786$	0	0
$787$	38.6936	1.37928	0.689638	−	0.724154i	$-0.257770\pi$
$787$	38.6936	1.37928	0.689638	+	0.724154i	$0.257770\pi$
$788$	5.77959	0.205889
$789$	0	0
$790$	31.6258	1.12520
$791$	12.0000	0.426671
$792$	0	0
$793$	12.0897	0.429319
$794$	7.53365	0.267359
$795$	0	0
$796$	28.9595	1.02644
$797$	35.4450	1.25553	0.627763	−	0.778405i	$-0.283971\pi$
$797$	35.4450	1.25553	0.627763	+	0.778405i	$0.283971\pi$
$798$	0	0
$799$	36.6714	1.29734
$800$	41.5638	1.46950
$801$	0	0
$802$	−21.9397	−0.774719
$803$	−21.5284	−0.759719
$804$	0	0
$805$	49.8038	1.75535
$806$	25.3427	0.892659
$807$	0	0
$808$	−1.53871	−0.0541317
$809$	−38.9622	−1.36984	−0.684918	−	0.728620i	$-0.740162\pi$
$809$	−38.9622	−1.36984	−0.684918	+	0.728620i	$0.740162\pi$
$810$	0	0
$811$	36.1421	1.26912	0.634560	−	0.772874i	$-0.281181\pi$
$811$	36.1421	1.26912	0.634560	+	0.772874i	$0.281181\pi$
$812$	29.1377	1.02253
$813$	0	0
$814$	−14.2050	−0.497885
$815$	37.0277	1.29702
$816$	0	0
$817$	7.84425	0.274436
$818$	47.0851	1.64629
$819$	0	0
$820$	−52.0396	−1.81730
$821$	−14.8788	−0.519273	−0.259636	−	0.965706i	$-0.583603\pi$
$821$	−14.8788	−0.519273	−0.259636	+	0.965706i	$0.583603\pi$
$822$	0	0
$823$	24.2151	0.844086	0.422043	−	0.906576i	$-0.361313\pi$
$823$	24.2151	0.844086	0.422043	+	0.906576i	$0.361313\pi$
$824$	−10.6291	−0.370283
$825$	0	0
$826$	32.9916	1.14793
$827$	−24.4114	−0.848866	−0.424433	−	0.905459i	$-0.639527\pi$
$827$	−24.4114	−0.848866	−0.424433	+	0.905459i	$0.639527\pi$
$828$	0	0
$829$	−14.2228	−0.493979	−0.246990	−	0.969018i	$-0.579441\pi$
$829$	−14.2228	−0.493979	−0.246990	+	0.969018i	$0.579441\pi$
$830$	63.2996	2.19716
$831$	0	0
$832$	16.7572	0.580950
$833$	−39.1826	−1.35760
$834$	0	0
$835$	29.5611	1.02301
$836$	−11.9243	−0.412411
$837$	0	0
$838$	−9.68608	−0.334600
$839$	−10.6241	−0.366784	−0.183392	−	0.983040i	$-0.558708\pi$
$839$	−10.6241	−0.366784	−0.183392	+	0.983040i	$0.558708\pi$
$840$	0	0
$841$	−7.84931	−0.270666
$842$	42.1447	1.45240
$843$	0	0
$844$	−7.01714	−0.241540
$845$	6.58426	0.226505
$846$	0	0
$847$	35.4734	1.21888
$848$	−39.4131	−1.35345
$849$	0	0
$850$	55.0070	1.88673
$851$	−22.6265	−0.775626
$852$	0	0
$853$	−56.0415	−1.91883	−0.959413	−	0.282005i	$-0.909001\pi$
$853$	−56.0415	−1.91883	−0.959413	+	0.282005i	$0.909001\pi$
$854$	−26.6854	−0.913156
$855$	0	0
$856$	−0.230528	−0.00787930
$857$	−11.8968	−0.406388	−0.203194	−	0.979139i	$-0.565132\pi$
$857$	−11.8968	−0.406388	−0.203194	+	0.979139i	$0.565132\pi$
$858$	0	0
$859$	−53.2221	−1.81592	−0.907958	−	0.419061i	$-0.862359\pi$
$859$	−53.2221	−1.81592	−0.907958	+	0.419061i	$0.862359\pi$
$860$	−7.71863	−0.263203
$861$	0	0
$862$	−31.4138	−1.06996
$863$	45.0620	1.53393	0.766964	−	0.641690i	$-0.221766\pi$
$863$	45.0620	1.53393	0.766964	+	0.641690i	$0.221766\pi$
$864$	0	0
$865$	5.36823	0.182525
$866$	−64.4201	−2.18908
$867$	0	0
$868$	−25.3427	−0.860187
$869$	−6.67135	−0.226310
$870$	0	0
$871$	17.6040	0.596490
$872$	0.393308	0.0133191
$873$	0	0
$874$	−41.9243	−1.41811
$875$	7.45797	0.252125
$876$	0	0
$877$	−57.0647	−1.92694	−0.963468	−	0.267822i	$-0.913696\pi$
$877$	−57.0647	−1.92694	−0.963468	+	0.267822i	$0.913696\pi$
$878$	−18.5561	−0.626238
$879$	0	0
$880$	−19.5337	−0.658479
$881$	−22.4329	−0.755783	−0.377892	−	0.925850i	$-0.623351\pi$
$881$	−22.4329	−0.755783	−0.377892	+	0.925850i	$0.623351\pi$
$882$	0	0
$883$	−10.3805	−0.349333	−0.174667	−	0.984628i	$-0.555885\pi$
$883$	−10.3805	−0.349333	−0.174667	+	0.984628i	$0.555885\pi$
$884$	28.1997	0.948459
$885$	0	0
$886$	−42.3753	−1.42362
$887$	29.4734	0.989619	0.494810	−	0.869001i	$-0.335238\pi$
$887$	29.4734	0.989619	0.494810	+	0.869001i	$0.335238\pi$
$888$	0	0
$889$	16.5816	0.556129
$890$	20.0801	0.673085
$891$	0	0
$892$	−32.3471	−1.08306
$893$	−39.1223	−1.30918
$894$	0	0
$895$	16.9140	0.565372
$896$	19.7936	0.661259
$897$	0	0
$898$	−55.4975	−1.85198
$899$	−18.3960	−0.613539
$900$	0	0
$901$	−44.3203	−1.47652
$902$	24.2305	0.806788
$903$	0	0
$904$	−2.06028	−0.0685238
$905$	−45.9036	−1.52589
$906$	0	0
$907$	13.5939	0.451379	0.225690	−	0.974199i	$-0.427537\pi$
$907$	13.5939	0.451379	0.225690	+	0.974199i	$0.427537\pi$
$908$	33.1980	1.10171
$909$	0	0
$910$	78.8851	2.61502
$911$	20.3203	0.673241	0.336620	−	0.941640i	$-0.390716\pi$
$911$	20.3203	0.673241	0.336620	+	0.941640i	$0.390716\pi$
$912$	0	0
$913$	−13.3528	−0.441914
$914$	−37.3823	−1.23650
$915$	0	0
$916$	−2.73601	−0.0904004
$917$	54.8161	1.81019
$918$	0	0
$919$	17.4734	0.576393	0.288197	−	0.957571i	$-0.406944\pi$
$919$	17.4734	0.576393	0.288197	+	0.957571i	$0.406944\pi$
$920$	−8.55080	−0.281911
$921$	0	0
$922$	11.5592	0.380681
$923$	3.31324	0.109057
$924$	0	0
$925$	−31.6714	−1.04135
$926$	40.4054	1.32780
$927$	0	0
$928$	−34.1403	−1.12071
$929$	−7.36581	−0.241665	−0.120832	−	0.992673i	$-0.538556\pi$
$929$	−7.36581	−0.241665	−0.120832	+	0.992673i	$0.538556\pi$
$930$	0	0
$931$	41.8013	1.36998
$932$	−39.0647	−1.27961
$933$	0	0
$934$	78.0690	2.55450
$935$	−21.9657	−0.718356
$936$	0	0
$937$	8.96745	0.292954	0.146477	−	0.989214i	$-0.453207\pi$
$937$	8.96745	0.292954	0.146477	+	0.989214i	$0.453207\pi$
$938$	−38.8570	−1.26873
$939$	0	0
$940$	38.4958	1.25559
$941$	−11.0455	−0.360075	−0.180037	−	0.983660i	$-0.557622\pi$
$941$	−11.0455	−0.360075	−0.180037	+	0.983660i	$0.557622\pi$
$942$	0	0
$943$	38.5957	1.25685
$944$	−20.6111	−0.670833
$945$	0	0
$946$	3.59393	0.116849
$947$	36.1729	1.17546	0.587731	−	0.809057i	$-0.300022\pi$
$947$	36.1729	1.17546	0.587731	+	0.809057i	$0.300022\pi$
$948$	0	0
$949$	−54.3150	−1.76314
$950$	−58.6834	−1.90394
$951$	0	0
$952$	12.9019	0.418152
$953$	25.7307	0.833500	0.416750	−	0.909021i	$-0.363169\pi$
$953$	25.7307	0.833500	0.416750	+	0.909021i	$0.363169\pi$
$954$	0	0
$955$	−22.7556	−0.736353
$956$	−0.946750	−0.0306201
$957$	0	0
$958$	−45.8935	−1.48275
$959$	54.5957	1.76299
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−35.8385	−1.15548
$963$	0	0
$964$	42.1997	1.35916
$965$	−25.8486	−0.832097
$966$	0	0
$967$	0.862301	0.0277297	0.0138649	−	0.999904i	$-0.495587\pi$
$967$	0.862301	0.0277297	0.0138649	+	0.999904i	$0.495587\pi$
$968$	−6.09042	−0.195753
$969$	0	0
$970$	73.6128	2.36356
$971$	−52.7505	−1.69284	−0.846422	−	0.532512i	$-0.821248\pi$
$971$	−52.7505	−1.69284	−0.846422	+	0.532512i	$0.821248\pi$
$972$	0	0
$973$	17.0224	0.545714
$974$	−20.9622	−0.671671
$975$	0	0
$976$	16.6714	0.533637
$977$	−25.8761	−0.827851	−0.413925	−	0.910311i	$-0.635843\pi$
$977$	−25.8761	−0.827851	−0.413925	+	0.910311i	$0.635843\pi$
$978$	0	0
$979$	−4.23582	−0.135377
$980$	−41.1320	−1.31391
$981$	0	0
$982$	44.8865	1.43238
$983$	−50.4881	−1.61032	−0.805160	−	0.593057i	$-0.797921\pi$
$983$	−50.4881	−1.61032	−0.805160	+	0.593057i	$0.797921\pi$
$984$	0	0
$985$	−11.3581	−0.361900
$986$	−45.1826	−1.43891
$987$	0	0
$988$	−30.0844	−0.957114
$989$	5.72460	0.182032
$990$	0	0
$991$	−6.74175	−0.214159	−0.107079	−	0.994250i	$-0.534150\pi$
$991$	−6.74175	−0.214159	−0.107079	+	0.994250i	$0.534150\pi$
$992$	29.6938	0.942779
$993$	0	0
$994$	−7.31324	−0.231962
$995$	−56.9116	−1.80422
$996$	0	0
$997$	55.5706	1.75994	0.879969	−	0.475031i	$-0.157563\pi$
$997$	55.5706	1.75994	0.879969	+	0.475031i	$0.157563\pi$
$998$	−13.5963	−0.430385
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.h.1.1		3
3.2	odd	2		71.2.a.a.1.3	✓	3
12.11	even	2		1136.2.a.h.1.3		3
15.14	odd	2		1775.2.a.f.1.1		3
21.20	even	2		3479.2.a.k.1.3		3
24.5	odd	2		4544.2.a.r.1.3		3
24.11	even	2		4544.2.a.u.1.1		3
33.32	even	2		8591.2.a.g.1.1		3
213.212	even	2		5041.2.a.a.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
71.2.a.a.1.3	✓	3	3.2	odd	2
639.2.a.h.1.1		3	1.1	even	1	trivial
1136.2.a.h.1.3		3	12.11	even	2
1775.2.a.f.1.1		3	15.14	odd	2
3479.2.a.k.1.3		3	21.20	even	2
4544.2.a.r.1.3		3	24.5	odd	2
4544.2.a.u.1.1		3	24.11	even	2
5041.2.a.a.1.3		3	213.212	even	2
8591.2.a.g.1.1		3	33.32	even	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:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	639.a (trivial)

\(f(q)\)	\(=\)	\(q-1.91223 q^{2} +1.65662 q^{4} -3.25561 q^{5} -3.82446 q^{7} +0.656620 q^{8} +O(q^{10})\)	\(q-1.91223 q^{2} +1.65662 q^{4} -3.25561 q^{5} -3.82446 q^{7} +0.656620 q^{8} +6.22547 q^{10} -1.31324 q^{11} -3.31324 q^{13} +7.31324 q^{14} -4.56885 q^{16} -5.13770 q^{17} +5.48108 q^{19} -5.39331 q^{20} +2.51122 q^{22} +4.00000 q^{23} +5.59899 q^{25} +6.33568 q^{26} -6.33568 q^{28} -4.59899 q^{29} +4.00000 q^{31} +7.42345 q^{32} +9.82446 q^{34} +12.4509 q^{35} -5.65662 q^{37} -10.4811 q^{38} -2.13770 q^{40} +9.64892 q^{41} +1.43115 q^{43} -2.17554 q^{44} -7.64892 q^{46} -7.13770 q^{47} +7.62648 q^{49} -10.7065 q^{50} -5.48878 q^{52} +8.62648 q^{53} +4.27540 q^{55} -2.51122 q^{56} +8.79432 q^{58} +4.51122 q^{59} -3.64892 q^{61} -7.64892 q^{62} -5.05763 q^{64} +10.7866 q^{65} -5.31324 q^{67} -8.51122 q^{68} -23.8091 q^{70} -1.00000 q^{71} +16.3933 q^{73} +10.8168 q^{74} +9.08007 q^{76} +5.02243 q^{77} +5.08007 q^{79} +14.8744 q^{80} -18.4509 q^{82} +10.1678 q^{83} +16.7263 q^{85} -2.73669 q^{86} -0.862301 q^{88} +3.22547 q^{89} +12.6714 q^{91} +6.62648 q^{92} +13.6489 q^{94} -17.8442 q^{95} +11.8245 q^{97} -14.5836 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 2 * q^7	\( 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 2 q^{7} + 8 q^{10} - 6 q^{13} + 18 q^{14} - 5 q^{16} + 2 q^{17} + q^{19} + 6 q^{20} - 2 q^{22} + 12 q^{23} + 14 q^{25} - 4 q^{26} + 4 q^{28} - 11 q^{29} + 12 q^{31} + 6 q^{32} + 16 q^{34} + 16 q^{35} - 15 q^{37} - 16 q^{38} + 11 q^{40} + 2 q^{41} + 13 q^{43} - 20 q^{44} + 4 q^{46} - 4 q^{47} + 15 q^{49} - 6 q^{50} - 26 q^{52} + 18 q^{53} - 22 q^{55} + 2 q^{56} + 7 q^{58} + 4 q^{59} + 16 q^{61} + 4 q^{62} - 16 q^{64} - 12 q^{65} - 12 q^{67} - 16 q^{68} - 8 q^{70} - 3 q^{71} + 27 q^{73} - 6 q^{74} + 9 q^{76} - 4 q^{77} - 3 q^{79} + 7 q^{80} - 34 q^{82} + 19 q^{83} - 6 q^{85} + 12 q^{86} - 20 q^{88} - q^{89} - 8 q^{91} + 12 q^{92} + 14 q^{94} - 10 q^{95} + 22 q^{97} + 9 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 2 * q^7 + 8 * q^10 - 6 * q^13 + 18 * q^14 - 5 * q^16 + 2 * q^17 + q^19 + 6 * q^20 - 2 * q^22 + 12 * q^23 + 14 * q^25 - 4 * q^26 + 4 * q^28 - 11 * q^29 + 12 * q^31 + 6 * q^32 + 16 * q^34 + 16 * q^35 - 15 * q^37 - 16 * q^38 + 11 * q^40 + 2 * q^41 + 13 * q^43 - 20 * q^44 + 4 * q^46 - 4 * q^47 + 15 * q^49 - 6 * q^50 - 26 * q^52 + 18 * q^53 - 22 * q^55 + 2 * q^56 + 7 * q^58 + 4 * q^59 + 16 * q^61 + 4 * q^62 - 16 * q^64 - 12 * q^65 - 12 * q^67 - 16 * q^68 - 8 * q^70 - 3 * q^71 + 27 * q^73 - 6 * q^74 + 9 * q^76 - 4 * q^77 - 3 * q^79 + 7 * q^80 - 34 * q^82 + 19 * q^83 - 6 * q^85 + 12 * q^86 - 20 * q^88 - q^89 - 8 * q^91 + 12 * q^92 + 14 * q^94 - 10 * q^95 + 22 * q^97 + 9 * q^98

Introduction

L-functions

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