LMFDB - Embedded newform 639.2.a.j.1.3

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:	$1$
Dimension:	$4$
Coefficient field:	4.4.8112.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 5x^{2} + 3 $ x^4 - 5*x^2 + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.835000$ 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)

$f(q)$	$=$	$q+0.835000 q^{2} -1.30278 q^{4} -2.83500 q^{5} +4.22559 q^{7} -2.75782 q^{8} +O(q^{10})$	\(q+0.835000 q^{2} -1.30278 q^{4} -2.83500 q^{5} +4.22559 q^{7} -2.75782 q^{8} -2.36722 q^{10} +0.137775 q^{11} -3.68063 q^{13} +3.52837 q^{14} +0.302776 q^{16} -6.13778 q^{17} -5.27555 q^{19} +3.69337 q^{20} +0.115042 q^{22} -5.39059 q^{23} +3.03722 q^{25} -3.07333 q^{26} -5.50500 q^{28} +4.83290 q^{29} -7.51563 q^{31} +5.76845 q^{32} -5.12504 q^{34} -11.9796 q^{35} +7.42396 q^{37} -4.40508 q^{38} +7.81841 q^{40} -11.5029 q^{41} +5.88286 q^{43} -0.179490 q^{44} -4.50114 q^{46} -11.8039 q^{47} +10.8556 q^{49} +2.53608 q^{50} +4.79504 q^{52} +6.10845 q^{53} -0.390593 q^{55} -11.6534 q^{56} +4.03547 q^{58} +9.67888 q^{59} -5.46567 q^{61} -6.27555 q^{62} +4.21110 q^{64} +10.4346 q^{65} -4.01063 q^{67} +7.99614 q^{68} -10.0029 q^{70} -1.00000 q^{71} +7.84563 q^{73} +6.19900 q^{74} +6.87286 q^{76} +0.582182 q^{77} +8.22285 q^{79} -0.858369 q^{80} -9.60492 q^{82} -1.82226 q^{83} +17.4006 q^{85} +4.91218 q^{86} -0.379959 q^{88} +9.81455 q^{89} -15.5529 q^{91} +7.02273 q^{92} -9.85627 q^{94} +14.9562 q^{95} +11.1173 q^{97} +9.06445 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{4} - 8 q^{5} + 2 q^{7}+O(q^{10}) $ 4 * q + 2 * q^4 - 8 * q^5 + 2 * q^7	$ 4 q + 2 q^{4} - 8 q^{5} + 2 q^{7} - 10 q^{10} - 10 q^{11} + 4 q^{13} - 8 q^{14} - 6 q^{16} - 14 q^{17} - 4 q^{20} + 10 q^{22} - 10 q^{23} + 6 q^{25} + 6 q^{26} - 12 q^{28} + 4 q^{29} - 8 q^{31} - 10 q^{34} + 4 q^{35} - 14 q^{37} - 20 q^{38} + 2 q^{40} - 24 q^{41} + 2 q^{43} - 18 q^{44} + 18 q^{46} - 4 q^{47} + 8 q^{49} + 40 q^{50} + 2 q^{52} - 12 q^{53} + 10 q^{55} - 14 q^{56} - 4 q^{58} + 6 q^{59} - 6 q^{61} - 4 q^{62} - 12 q^{64} - 14 q^{65} - 4 q^{67} + 6 q^{68} + 24 q^{70} - 4 q^{71} + 16 q^{73} + 4 q^{74} + 26 q^{76} - 2 q^{79} + 12 q^{80} - 16 q^{82} - 4 q^{83} + 38 q^{85} + 24 q^{86} - 2 q^{88} - 16 q^{89} - 34 q^{91} + 8 q^{92} - 12 q^{94} + 20 q^{95} - 18 q^{97} + 44 q^{98}+O(q^{100}) $ 4 * q + 2 * q^4 - 8 * q^5 + 2 * q^7 - 10 * q^10 - 10 * q^11 + 4 * q^13 - 8 * q^14 - 6 * q^16 - 14 * q^17 - 4 * q^20 + 10 * q^22 - 10 * q^23 + 6 * q^25 + 6 * q^26 - 12 * q^28 + 4 * q^29 - 8 * q^31 - 10 * q^34 + 4 * q^35 - 14 * q^37 - 20 * q^38 + 2 * q^40 - 24 * q^41 + 2 * q^43 - 18 * q^44 + 18 * q^46 - 4 * q^47 + 8 * q^49 + 40 * q^50 + 2 * q^52 - 12 * q^53 + 10 * q^55 - 14 * q^56 - 4 * q^58 + 6 * q^59 - 6 * q^61 - 4 * q^62 - 12 * q^64 - 14 * q^65 - 4 * q^67 + 6 * q^68 + 24 * q^70 - 4 * q^71 + 16 * q^73 + 4 * q^74 + 26 * q^76 - 2 * q^79 + 12 * q^80 - 16 * q^82 - 4 * q^83 + 38 * q^85 + 24 * q^86 - 2 * q^88 - 16 * q^89 - 34 * q^91 + 8 * q^92 - 12 * q^94 + 20 * q^95 - 18 * q^97 + 44 * 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 $

$2$	0.835000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$2$	0.835000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$3$	0	0
$3$	0	0
$4$	−1.30278	−0.651388
$5$	−2.83500	−1.26785	−0.633925	−	0.773394i	$-0.718557\pi$
$5$	−2.83500	−1.26785	−0.633925	+	0.773394i	$0.718557\pi$
$6$	0	0
$7$	4.22559	1.59712	0.798562	−	0.601913i	$-0.205594\pi$
$7$	4.22559	1.59712	0.798562	+	0.601913i	$0.205594\pi$
$8$	−2.75782	−0.975035
$9$	0	0
$10$	−2.36722	−0.748582
$11$	0.137775	0.0415408	0.0207704	−	0.999784i	$-0.493388\pi$
$11$	0.137775	0.0415408	0.0207704	+	0.999784i	$0.493388\pi$
$12$	0	0
$13$	−3.68063	−1.02082	−0.510412	−	0.859930i	$-0.670507\pi$
$13$	−3.68063	−1.02082	−0.510412	+	0.859930i	$0.670507\pi$
$14$	3.52837	0.942996
$15$	0	0
$16$	0.302776	0.0756939
$17$	−6.13778	−1.48863	−0.744315	−	0.667829i	$-0.767224\pi$
$17$	−6.13778	−1.48863	−0.744315	+	0.667829i	$0.767224\pi$
$18$	0	0
$19$	−5.27555	−1.21029	−0.605147	−	0.796114i	$-0.706886\pi$
$19$	−5.27555	−1.21029	−0.605147	+	0.796114i	$0.706886\pi$
$20$	3.69337	0.825862
$21$	0	0
$22$	0.115042	0.0245271
$23$	−5.39059	−1.12402	−0.562008	−	0.827132i	$-0.689971\pi$
$23$	−5.39059	−1.12402	−0.562008	+	0.827132i	$0.689971\pi$
$24$	0	0
$25$	3.03722	0.607445
$26$	−3.07333	−0.602729
$27$	0	0
$28$	−5.50500	−1.04035
$29$	4.83290	0.897447	0.448723	−	0.893671i	$-0.351879\pi$
$29$	4.83290	0.897447	0.448723	+	0.893671i	$0.351879\pi$
$30$	0	0
$31$	−7.51563	−1.34985	−0.674924	−	0.737887i	$-0.735823\pi$
$31$	−7.51563	−1.34985	−0.674924	+	0.737887i	$0.735823\pi$
$32$	5.76845	1.01973
$33$	0	0
$34$	−5.12504	−0.878937
$35$	−11.9796	−2.02491
$36$	0	0
$37$	7.42396	1.22049	0.610246	−	0.792212i	$-0.291071\pi$
$37$	7.42396	1.22049	0.610246	+	0.792212i	$0.291071\pi$
$38$	−4.40508	−0.714599
$39$	0	0
$40$	7.81841	1.23620
$41$	−11.5029	−1.79645	−0.898225	−	0.439535i	$-0.855143\pi$
$41$	−11.5029	−1.79645	−0.898225	+	0.439535i	$0.855143\pi$
$42$	0	0
$43$	5.88286	0.897127	0.448564	−	0.893751i	$-0.351936\pi$
$43$	5.88286	0.897127	0.448564	+	0.893751i	$0.351936\pi$
$44$	−0.179490	−0.0270592
$45$	0	0
$46$	−4.50114	−0.663657
$47$	−11.8039	−1.72178	−0.860889	−	0.508792i	$-0.830092\pi$
$47$	−11.8039	−1.72178	−0.860889	+	0.508792i	$0.830092\pi$
$48$	0	0
$49$	10.8556	1.55080
$50$	2.53608	0.358656
$51$	0	0
$52$	4.79504	0.664952
$53$	6.10845	0.839060	0.419530	−	0.907741i	$-0.362195\pi$
$53$	6.10845	0.839060	0.419530	+	0.907741i	$0.362195\pi$
$54$	0	0
$55$	−0.390593	−0.0526675
$56$	−11.6534	−1.55725
$57$	0	0
$58$	4.03547	0.529883
$59$	9.67888	1.26008	0.630041	−	0.776562i	$-0.283038\pi$
$59$	9.67888	1.26008	0.630041	+	0.776562i	$0.283038\pi$
$60$	0	0
$61$	−5.46567	−0.699808	−0.349904	−	0.936786i	$-0.613786\pi$
$61$	−5.46567	−0.699808	−0.349904	+	0.936786i	$0.613786\pi$
$62$	−6.27555	−0.796996
$63$	0	0
$64$	4.21110	0.526388
$65$	10.4346	1.29425
$66$	0	0
$67$	−4.01063	−0.489977	−0.244988	−	0.969526i	$-0.578784\pi$
$67$	−4.01063	−0.489977	−0.244988	+	0.969526i	$0.578784\pi$
$68$	7.99614	0.969675
$69$	0	0
$70$	−10.0029	−1.19558
$71$	−1.00000	−0.118678
$71$	−1.00000	−0.118678
$72$	0	0
$73$	7.84563	0.918262	0.459131	−	0.888369i	$-0.348161\pi$
$73$	7.84563	0.918262	0.459131	+	0.888369i	$0.348161\pi$
$74$	6.19900	0.720620
$75$	0	0
$76$	6.87286	0.788371
$77$	0.582182	0.0663458
$78$	0	0
$79$	8.22285	0.925143	0.462572	−	0.886582i	$-0.346927\pi$
$79$	8.22285	0.925143	0.462572	+	0.886582i	$0.346927\pi$
$80$	−0.858369	−0.0959686
$81$	0	0
$82$	−9.60492	−1.06069
$83$	−1.82226	−0.200019	−0.100010	−	0.994986i	$-0.531887\pi$
$83$	−1.82226	−0.200019	−0.100010	+	0.994986i	$0.531887\pi$
$84$	0	0
$85$	17.4006	1.88736
$86$	4.91218	0.529694
$87$	0	0
$88$	−0.379959	−0.0405038
$89$	9.81455	1.04034	0.520170	−	0.854063i	$-0.325868\pi$
$89$	9.81455	1.04034	0.520170	+	0.854063i	$0.325868\pi$
$90$	0	0
$91$	−15.5529	−1.63038
$92$	7.02273	0.732171
$93$	0	0
$94$	−9.85627	−1.01660
$95$	14.9562	1.53447
$96$	0	0
$97$	11.1173	1.12879	0.564397	−	0.825504i	$-0.309109\pi$
$97$	11.1173	1.12879	0.564397	+	0.825504i	$0.309109\pi$
$98$	9.06445	0.915648
$99$	0	0
$100$	−3.95682	−0.395682
$101$	2.25218	0.224100	0.112050	−	0.993703i	$-0.464258\pi$
$101$	2.25218	0.224100	0.112050	+	0.993703i	$0.464258\pi$
$102$	0	0
$103$	−4.82437	−0.475359	−0.237679	−	0.971344i	$-0.576387\pi$
$103$	−4.82437	−0.475359	−0.237679	+	0.971344i	$0.576387\pi$
$104$	10.1505	0.995339
$105$	0	0
$106$	5.10055	0.495409
$107$	2.09845	0.202865	0.101432	−	0.994842i	$-0.467657\pi$
$107$	2.09845	0.202865	0.101432	+	0.994842i	$0.467657\pi$
$108$	0	0
$109$	−14.3867	−1.37800	−0.689000	−	0.724762i	$-0.741950\pi$
$109$	−14.3867	−1.37800	−0.689000	+	0.724762i	$0.741950\pi$
$110$	−0.326145	−0.0310967
$111$	0	0
$112$	1.27941	0.120893
$113$	4.15437	0.390810	0.195405	−	0.980723i	$-0.437398\pi$
$113$	4.15437	0.390810	0.195405	+	0.980723i	$0.437398\pi$
$114$	0	0
$115$	15.2823	1.42508
$116$	−6.29618	−0.584586
$117$	0	0
$118$	8.08186	0.743996
$119$	−25.9357	−2.37753
$120$	0	0
$121$	−10.9810	−0.998274
$122$	−4.56384	−0.413190
$123$	0	0
$124$	9.79118	0.879274
$125$	5.56447	0.497702
$126$	0	0
$127$	−13.1102	−1.16334	−0.581671	−	0.813424i	$-0.697601\pi$
$127$	−13.1102	−1.16334	−0.581671	+	0.813424i	$0.697601\pi$
$128$	−8.02063	−0.708930
$129$	0	0
$130$	8.71288	0.764170
$131$	4.45118	0.388902	0.194451	−	0.980912i	$-0.437708\pi$
$131$	4.45118	0.388902	0.194451	+	0.980912i	$0.437708\pi$
$132$	0	0
$133$	−22.2923	−1.93299
$134$	−3.34888	−0.289299
$135$	0	0
$136$	16.9269	1.45147
$137$	−18.7267	−1.59993	−0.799966	−	0.600045i	$-0.795149\pi$
$137$	−18.7267	−1.59993	−0.799966	+	0.600045i	$0.795149\pi$
$138$	0	0
$139$	−5.42606	−0.460232	−0.230116	−	0.973163i	$-0.573911\pi$
$139$	−5.42606	−0.460232	−0.230116	+	0.973163i	$0.573911\pi$
$140$	15.6067	1.31900
$141$	0	0
$142$	−0.835000	−0.0700716
$143$	−0.507100	−0.0424058
$144$	0	0
$145$	−13.7013	−1.13783
$146$	6.55110	0.542173
$147$	0	0
$148$	−9.67175	−0.795013
$149$	−9.06059	−0.742273	−0.371136	−	0.928578i	$-0.621032\pi$
$149$	−9.06059	−0.742273	−0.371136	+	0.928578i	$0.621032\pi$
$150$	0	0
$151$	−6.75396	−0.549630	−0.274815	−	0.961497i	$-0.588617\pi$
$151$	−6.75396	−0.549630	−0.274815	+	0.961497i	$0.588617\pi$
$152$	14.5490	1.18008
$153$	0	0
$154$	0.486122	0.0391728
$155$	21.3068	1.71140
$156$	0	0
$157$	−2.90657	−0.231970	−0.115985	−	0.993251i	$-0.537002\pi$
$157$	−2.90657	−0.231970	−0.115985	+	0.993251i	$0.537002\pi$
$158$	6.86608	0.546236
$159$	0	0
$160$	−16.3536	−1.29286
$161$	−22.7784	−1.79519
$162$	0	0
$163$	−3.08122	−0.241340	−0.120670	−	0.992693i	$-0.538504\pi$
$163$	−3.08122	−0.241340	−0.120670	+	0.992693i	$0.538504\pi$
$164$	14.9857	1.17019
$165$	0	0
$166$	−1.52159	−0.118098
$167$	7.12100	0.551039	0.275520	−	0.961295i	$-0.411150\pi$
$167$	7.12100	0.551039	0.275520	+	0.961295i	$0.411150\pi$
$168$	0	0
$169$	0.547061	0.0420816
$170$	14.5295	1.11436
$171$	0	0
$172$	−7.66404	−0.584378
$173$	−20.7846	−1.58022	−0.790112	−	0.612963i	$-0.789977\pi$
$173$	−20.7846	−1.58022	−0.790112	+	0.612963i	$0.789977\pi$
$174$	0	0
$175$	12.8341	0.970164
$176$	0.0417150	0.00314439
$177$	0	0
$178$	8.19515	0.614252
$179$	15.7182	1.17483	0.587417	−	0.809285i	$-0.300145\pi$
$179$	15.7182	1.17483	0.587417	+	0.809285i	$0.300145\pi$
$180$	0	0
$181$	22.3829	1.66371	0.831853	−	0.554996i	$-0.187280\pi$
$181$	22.3829	1.66371	0.831853	+	0.554996i	$0.187280\pi$
$182$	−12.9866	−0.962633
$183$	0	0
$184$	14.8663	1.09596
$185$	−21.0469	−1.54740
$186$	0	0
$187$	−0.845634	−0.0618389
$188$	15.3779	1.12155
$189$	0	0
$190$	12.4884	0.906004
$191$	14.5235	1.05089	0.525443	−	0.850829i	$-0.323900\pi$
$191$	14.5235	1.05089	0.525443	+	0.850829i	$0.323900\pi$
$192$	0	0
$193$	−7.26457	−0.522915	−0.261458	−	0.965215i	$-0.584203\pi$
$193$	−7.26457	−0.522915	−0.261458	+	0.965215i	$0.584203\pi$
$194$	9.28296	0.666478
$195$	0	0
$196$	−14.1425	−1.01018
$197$	14.6761	1.04563	0.522816	−	0.852446i	$-0.324882\pi$
$197$	14.6761	1.04563	0.522816	+	0.852446i	$0.324882\pi$
$198$	0	0
$199$	18.1170	1.28428	0.642139	−	0.766588i	$-0.278047\pi$
$199$	18.1170	1.28428	0.642139	+	0.766588i	$0.278047\pi$
$200$	−8.37610	−0.592280
$201$	0	0
$202$	1.88057	0.132316
$203$	20.4219	1.43333
$204$	0	0
$205$	32.6107	2.27763
$206$	−4.02834	−0.280668
$207$	0	0
$208$	−1.11441	−0.0772702
$209$	−0.726840	−0.0502766
$210$	0	0
$211$	−19.9668	−1.37457	−0.687286	−	0.726387i	$-0.741198\pi$
$211$	−19.9668	−1.37457	−0.687286	+	0.726387i	$0.741198\pi$
$212$	−7.95794	−0.546553
$213$	0	0
$214$	1.75221	0.119778
$215$	−16.6779	−1.13742
$216$	0	0
$217$	−31.7580	−2.15587
$218$	−12.0129	−0.813618
$219$	0	0
$220$	0.508855	0.0343070
$221$	22.5909	1.51963
$222$	0	0
$223$	−16.9378	−1.13424	−0.567121	−	0.823635i	$-0.691943\pi$
$223$	−16.9378	−1.13424	−0.567121	+	0.823635i	$0.691943\pi$
$224$	24.3751	1.62863
$225$	0	0
$226$	3.46889	0.230747
$227$	−7.85451	−0.521322	−0.260661	−	0.965430i	$-0.583941\pi$
$227$	−7.85451	−0.521322	−0.260661	+	0.965430i	$0.583941\pi$
$228$	0	0
$229$	−4.97453	−0.328726	−0.164363	−	0.986400i	$-0.552557\pi$
$229$	−4.97453	−0.328726	−0.164363	+	0.986400i	$0.552557\pi$
$230$	12.7607	0.841418
$231$	0	0
$232$	−13.3282	−0.875042
$233$	−15.4384	−1.01141	−0.505703	−	0.862708i	$-0.668767\pi$
$233$	−15.4384	−1.01141	−0.505703	+	0.862708i	$0.668767\pi$
$234$	0	0
$235$	33.4641	2.18296
$236$	−12.6094	−0.820802
$237$	0	0
$238$	−21.6563	−1.40377
$239$	5.17324	0.334629	0.167315	−	0.985904i	$-0.446490\pi$
$239$	5.17324	0.334629	0.167315	+	0.985904i	$0.446490\pi$
$240$	0	0
$241$	−5.10826	−0.329052	−0.164526	−	0.986373i	$-0.552609\pi$
$241$	−5.10826	−0.329052	−0.164526	+	0.986373i	$0.552609\pi$
$242$	−9.16915	−0.589415
$243$	0	0
$244$	7.12055	0.455846
$245$	−30.7757	−1.96619
$246$	0	0
$247$	19.4174	1.23550
$248$	20.7267	1.31615
$249$	0	0
$250$	4.64633	0.293860
$251$	25.8076	1.62896	0.814481	−	0.580190i	$-0.197022\pi$
$251$	25.8076	1.62896	0.814481	+	0.580190i	$0.197022\pi$
$252$	0	0
$253$	−0.742690	−0.0466925
$254$	−10.9470	−0.686877
$255$	0	0
$256$	−15.1194	−0.944964
$257$	5.21461	0.325279	0.162639	−	0.986686i	$-0.447999\pi$
$257$	5.21461	0.325279	0.162639	+	0.986686i	$0.447999\pi$
$258$	0	0
$259$	31.3706	1.94928
$260$	−13.5939	−0.843060
$261$	0	0
$262$	3.71674	0.229621
$263$	6.96039	0.429196	0.214598	−	0.976702i	$-0.431156\pi$
$263$	6.96039	0.429196	0.214598	+	0.976702i	$0.431156\pi$
$264$	0	0
$265$	−17.3175	−1.06380
$266$	−18.6141	−1.14130
$267$	0	0
$268$	5.22496	0.319165
$269$	−19.1856	−1.16977	−0.584884	−	0.811117i	$-0.698860\pi$
$269$	−19.1856	−1.16977	−0.584884	+	0.811117i	$0.698860\pi$
$270$	0	0
$271$	−14.1702	−0.860776	−0.430388	−	0.902644i	$-0.641623\pi$
$271$	−14.1702	−0.860776	−0.430388	+	0.902644i	$0.641623\pi$
$272$	−1.85837	−0.112680
$273$	0	0
$274$	−15.6368	−0.944655
$275$	0.418454	0.0252337
$276$	0	0
$277$	28.7952	1.73014	0.865069	−	0.501653i	$-0.167275\pi$
$277$	28.7952	1.73014	0.865069	+	0.501653i	$0.167275\pi$
$278$	−4.53076	−0.271737
$279$	0	0
$280$	33.0374	1.97436
$281$	−24.7229	−1.47484	−0.737422	−	0.675433i	$-0.763957\pi$
$281$	−24.7229	−1.47484	−0.737422	+	0.675433i	$0.763957\pi$
$282$	0	0
$283$	−27.0023	−1.60512	−0.802560	−	0.596572i	$-0.796529\pi$
$283$	−27.0023	−1.60512	−0.802560	+	0.596572i	$0.796529\pi$
$284$	1.30278	0.0773055
$285$	0	0
$286$	−0.423428	−0.0250378
$287$	−48.6066	−2.86915
$288$	0	0
$289$	20.6723	1.21602
$290$	−11.4406	−0.671812
$291$	0	0
$292$	−10.2211	−0.598145
$293$	0.985163	0.0575539	0.0287769	−	0.999586i	$-0.490839\pi$
$293$	0.985163	0.0575539	0.0287769	+	0.999586i	$0.490839\pi$
$294$	0	0
$295$	−27.4396	−1.59760
$296$	−20.4739	−1.19002
$297$	0	0
$298$	−7.56559	−0.438263
$299$	19.8408	1.14742
$300$	0	0
$301$	24.8586	1.43282
$302$	−5.63955	−0.324520
$303$	0	0
$304$	−1.59731	−0.0916119
$305$	15.4952	0.887252
$306$	0	0
$307$	−4.03161	−0.230096	−0.115048	−	0.993360i	$-0.536702\pi$
$307$	−4.03161	−0.230096	−0.115048	+	0.993360i	$0.536702\pi$
$308$	−0.758453	−0.0432168
$309$	0	0
$310$	17.7912	1.01047
$311$	1.94829	0.110477	0.0552386	−	0.998473i	$-0.482408\pi$
$311$	1.94829	0.110477	0.0552386	+	0.998473i	$0.482408\pi$
$312$	0	0
$313$	−20.0123	−1.13116	−0.565581	−	0.824693i	$-0.691348\pi$
$313$	−20.0123	−1.13116	−0.565581	+	0.824693i	$0.691348\pi$
$314$	−2.42699	−0.136963
$315$	0	0
$316$	−10.7125	−0.602627
$317$	0.467429	0.0262534	0.0131267	−	0.999914i	$-0.495822\pi$
$317$	0.467429	0.0262534	0.0131267	+	0.999914i	$0.495822\pi$
$318$	0	0
$319$	0.665854	0.0372807
$320$	−11.9385	−0.667381
$321$	0	0
$322$	−19.0200	−1.05994
$323$	32.3801	1.80168
$324$	0	0
$325$	−11.1789	−0.620094
$326$	−2.57282	−0.142495
$327$	0	0
$328$	31.7229	1.75160
$329$	−49.8785	−2.74989
$330$	0	0
$331$	−15.8305	−0.870123	−0.435062	−	0.900401i	$-0.643273\pi$
$331$	−15.8305	−0.870123	−0.435062	+	0.900401i	$0.643273\pi$
$332$	2.37400	0.130290
$333$	0	0
$334$	5.94603	0.325352
$335$	11.3701	0.621217
$336$	0	0
$337$	−5.92106	−0.322541	−0.161270	−	0.986910i	$-0.551559\pi$
$337$	−5.92106	−0.322541	−0.161270	+	0.986910i	$0.551559\pi$
$338$	0.456795	0.0248464
$339$	0	0
$340$	−22.6691	−1.22940
$341$	−1.03547	−0.0560737
$342$	0	0
$343$	16.2923	0.879703
$344$	−16.2238	−0.874731
$345$	0	0
$346$	−17.3551	−0.933017
$347$	12.4022	0.665785	0.332893	−	0.942965i	$-0.391975\pi$
$347$	12.4022	0.665785	0.332893	+	0.942965i	$0.391975\pi$
$348$	0	0
$349$	2.09670	0.112234	0.0561168	−	0.998424i	$-0.482128\pi$
$349$	2.09670	0.112234	0.0561168	+	0.998424i	$0.482128\pi$
$350$	10.7164	0.572818
$351$	0	0
$352$	0.794750	0.0423603
$353$	28.2875	1.50559	0.752796	−	0.658254i	$-0.228705\pi$
$353$	28.2875	1.50559	0.752796	+	0.658254i	$0.228705\pi$
$354$	0	0
$355$	2.83500	0.150466
$356$	−12.7862	−0.677665
$357$	0	0
$358$	13.1247	0.693662
$359$	−9.58447	−0.505849	−0.252924	−	0.967486i	$-0.581392\pi$
$359$	−9.58447	−0.505849	−0.252924	+	0.967486i	$0.581392\pi$
$360$	0	0
$361$	8.83143	0.464812
$362$	18.6897	0.982309
$363$	0	0
$364$	20.2619	1.06201
$365$	−22.2424	−1.16422
$366$	0	0
$367$	21.6682	1.13107	0.565537	−	0.824723i	$-0.308669\pi$
$367$	21.6682	1.13107	0.565537	+	0.824723i	$0.308669\pi$
$368$	−1.63214	−0.0850812
$369$	0	0
$370$	−17.5742	−0.913638
$371$	25.8118	1.34008
$372$	0	0
$373$	23.3595	1.20951	0.604755	−	0.796412i	$-0.293271\pi$
$373$	23.3595	1.20951	0.604755	+	0.796412i	$0.293271\pi$
$374$	−0.706104	−0.0365118
$375$	0	0
$376$	32.5530	1.67879
$377$	−17.7881	−0.916135
$378$	0	0
$379$	−16.5824	−0.851779	−0.425889	−	0.904775i	$-0.640039\pi$
$379$	−16.5824	−0.851779	−0.425889	+	0.904775i	$0.640039\pi$
$380$	−19.4846	−0.999536
$381$	0	0
$382$	12.1271	0.620479
$383$	21.0189	1.07401	0.537007	−	0.843578i	$-0.319555\pi$
$383$	21.0189	1.07401	0.537007	+	0.843578i	$0.319555\pi$
$384$	0	0
$385$	−1.65049	−0.0841166
$386$	−6.06591	−0.308747
$387$	0	0
$388$	−14.4834	−0.735282
$389$	−6.60135	−0.334702	−0.167351	−	0.985897i	$-0.553521\pi$
$389$	−6.60135	−0.334702	−0.167351	+	0.985897i	$0.553521\pi$
$390$	0	0
$391$	33.0862	1.67324
$392$	−29.9378	−1.51209
$393$	0	0
$394$	12.2546	0.617376
$395$	−23.3118	−1.17294
$396$	0	0
$397$	−8.86627	−0.444985	−0.222493	−	0.974934i	$-0.571419\pi$
$397$	−8.86627	−0.444985	−0.222493	+	0.974934i	$0.571419\pi$
$398$	15.1277	0.758282
$399$	0	0
$400$	0.919597	0.0459799
$401$	−16.0251	−0.800256	−0.400128	−	0.916459i	$-0.631034\pi$
$401$	−16.0251	−0.800256	−0.400128	+	0.916459i	$0.631034\pi$
$402$	0	0
$403$	27.6623	1.37796
$404$	−2.93409	−0.145976
$405$	0	0
$406$	17.0522	0.846289
$407$	1.02284	0.0507002
$408$	0	0
$409$	−7.71270	−0.381368	−0.190684	−	0.981651i	$-0.561071\pi$
$409$	−7.71270	−0.381368	−0.190684	+	0.981651i	$0.561071\pi$
$410$	27.2299	1.34479
$411$	0	0
$412$	6.28507	0.309643
$413$	40.8990	2.01251
$414$	0	0
$415$	5.16612	0.253595
$416$	−21.2315	−1.04096
$417$	0	0
$418$	−0.606911	−0.0296850
$419$	−14.2864	−0.697935	−0.348967	−	0.937135i	$-0.613468\pi$
$419$	−14.2864	−0.697935	−0.348967	+	0.937135i	$0.613468\pi$
$420$	0	0
$421$	11.2569	0.548626	0.274313	−	0.961641i	$-0.411550\pi$
$421$	11.2569	0.548626	0.274313	+	0.961641i	$0.411550\pi$
$422$	−16.6723	−0.811594
$423$	0	0
$424$	−16.8460	−0.818113
$425$	−18.6418	−0.904260
$426$	0	0
$427$	−23.0957	−1.11768
$428$	−2.73381	−0.132144
$429$	0	0
$430$	−13.9260	−0.671573
$431$	−10.4823	−0.504913	−0.252456	−	0.967608i	$-0.581238\pi$
$431$	−10.4823	−0.504913	−0.252456	+	0.967608i	$0.581238\pi$
$432$	0	0
$433$	−28.4363	−1.36656	−0.683282	−	0.730155i	$-0.739448\pi$
$433$	−28.4363	−1.36656	−0.683282	+	0.730155i	$0.739448\pi$
$434$	−26.5179	−1.27290
$435$	0	0
$436$	18.7427	0.897612
$437$	28.4383	1.36039
$438$	0	0
$439$	−40.5985	−1.93766	−0.968830	−	0.247725i	$-0.920317\pi$
$439$	−40.5985	−1.93766	−0.968830	+	0.247725i	$0.920317\pi$
$440$	1.07718	0.0513527
$441$	0	0
$442$	18.8634	0.897240
$443$	−36.4997	−1.73415	−0.867077	−	0.498175i	$-0.834004\pi$
$443$	−36.4997	−1.73415	−0.867077	+	0.498175i	$0.834004\pi$
$444$	0	0
$445$	−27.8243	−1.31900
$446$	−14.1431	−0.669695
$447$	0	0
$448$	17.7944	0.840707
$449$	16.3168	0.770038	0.385019	−	0.922909i	$-0.374195\pi$
$449$	16.3168	0.770038	0.385019	+	0.922909i	$0.374195\pi$
$450$	0	0
$451$	−1.58481	−0.0746260
$452$	−5.41221	−0.254569
$453$	0	0
$454$	−6.55852	−0.307806
$455$	44.0923	2.06708
$456$	0	0
$457$	−16.4502	−0.769508	−0.384754	−	0.923019i	$-0.625714\pi$
$457$	−16.4502	−0.769508	−0.384754	+	0.923019i	$0.625714\pi$
$458$	−4.15373	−0.194091
$459$	0	0
$460$	−19.9094	−0.928283
$461$	17.7588	0.827110	0.413555	−	0.910479i	$-0.364287\pi$
$461$	17.7588	0.827110	0.413555	+	0.910479i	$0.364287\pi$
$462$	0	0
$463$	−19.0395	−0.884841	−0.442421	−	0.896808i	$-0.645880\pi$
$463$	−19.0395	−0.884841	−0.442421	+	0.896808i	$0.645880\pi$
$464$	1.46328	0.0679312
$465$	0	0
$466$	−12.8911	−0.597169
$467$	−19.6492	−0.909257	−0.454628	−	0.890681i	$-0.650228\pi$
$467$	−19.6492	−0.909257	−0.454628	+	0.890681i	$0.650228\pi$
$468$	0	0
$469$	−16.9473	−0.782554
$470$	27.9425	1.28889
$471$	0	0
$472$	−26.6926	−1.22863
$473$	0.810512	0.0372674
$474$	0	0
$475$	−16.0230	−0.735187
$476$	33.7884	1.54869
$477$	0	0
$478$	4.31966	0.197576
$479$	26.3597	1.20441	0.602203	−	0.798343i	$-0.294290\pi$
$479$	26.3597	1.20441	0.602203	+	0.798343i	$0.294290\pi$
$480$	0	0
$481$	−27.3249	−1.24591
$482$	−4.26540	−0.194284
$483$	0	0
$484$	14.3058	0.650264
$485$	−31.5176	−1.43114
$486$	0	0
$487$	12.8576	0.582632	0.291316	−	0.956627i	$-0.405907\pi$
$487$	12.8576	0.582632	0.291316	+	0.956627i	$0.405907\pi$
$488$	15.0733	0.682337
$489$	0	0
$490$	−25.6977	−1.16090
$491$	2.00771	0.0906068	0.0453034	−	0.998973i	$-0.485575\pi$
$491$	2.00771	0.0906068	0.0453034	+	0.998973i	$0.485575\pi$
$492$	0	0
$493$	−29.6632	−1.33597
$494$	16.2135	0.729480
$495$	0	0
$496$	−2.27555	−0.102175
$497$	−4.22559	−0.189544
$498$	0	0
$499$	18.6255	0.833794	0.416897	−	0.908954i	$-0.363118\pi$
$499$	18.6255	0.833794	0.416897	+	0.908954i	$0.363118\pi$
$500$	−7.24926	−0.324197
$501$	0	0
$502$	21.5493	0.961794
$503$	42.4628	1.89332	0.946662	−	0.322228i	$-0.104432\pi$
$503$	42.4628	1.89332	0.946662	+	0.322228i	$0.104432\pi$
$504$	0	0
$505$	−6.38493	−0.284126
$506$	−0.620146	−0.0275689
$507$	0	0
$508$	17.0797	0.757787
$509$	−31.4577	−1.39434	−0.697168	−	0.716907i	$-0.745557\pi$
$509$	−31.4577	−1.39434	−0.697168	+	0.716907i	$0.745557\pi$
$510$	0	0
$511$	33.1524	1.46658
$512$	3.41655	0.150991
$513$	0	0
$514$	4.35420	0.192055
$515$	13.6771	0.602684
$516$	0	0
$517$	−1.62629	−0.0715241
$518$	26.1945	1.15092
$519$	0	0
$520$	−28.7767	−1.26194
$521$	−16.7055	−0.731880	−0.365940	−	0.930638i	$-0.619252\pi$
$521$	−16.7055	−0.731880	−0.365940	+	0.930638i	$0.619252\pi$
$522$	0	0
$523$	17.6984	0.773897	0.386948	−	0.922101i	$-0.373529\pi$
$523$	17.6984	0.773897	0.386948	+	0.922101i	$0.373529\pi$
$524$	−5.79890	−0.253326
$525$	0	0
$526$	5.81192	0.253412
$527$	46.1293	2.00942
$528$	0	0
$529$	6.05849	0.263413
$530$	−14.4601	−0.628105
$531$	0	0
$532$	29.0419	1.25913
$533$	42.3379	1.83386
$534$	0	0
$535$	−5.94911	−0.257202
$536$	11.0606	0.477745
$537$	0	0
$538$	−16.0200	−0.690671
$539$	1.49564	0.0644217
$540$	0	0
$541$	9.56979	0.411438	0.205719	−	0.978611i	$-0.434047\pi$
$541$	9.56979	0.411438	0.205719	+	0.978611i	$0.434047\pi$
$542$	−11.8321	−0.508231
$543$	0	0
$544$	−35.4055	−1.51800
$545$	40.7864	1.74710
$546$	0	0
$547$	−6.98277	−0.298562	−0.149281	−	0.988795i	$-0.547696\pi$
$547$	−6.98277	−0.298562	−0.149281	+	0.988795i	$0.547696\pi$
$548$	24.3967	1.04218
$549$	0	0
$550$	0.349409	0.0148989
$551$	−25.4962	−1.08617
$552$	0	0
$553$	34.7464	1.47757
$554$	24.0440	1.02153
$555$	0	0
$556$	7.06894	0.299790
$557$	2.76329	0.117084	0.0585422	−	0.998285i	$-0.481355\pi$
$557$	2.76329	0.117084	0.0585422	+	0.998285i	$0.481355\pi$
$558$	0	0
$559$	−21.6526	−0.915809
$560$	−3.62712	−0.153274
$561$	0	0
$562$	−20.6436	−0.870797
$563$	−19.5407	−0.823540	−0.411770	−	0.911288i	$-0.635089\pi$
$563$	−19.5407	−0.823540	−0.411770	+	0.911288i	$0.635089\pi$
$564$	0	0
$565$	−11.7776	−0.495489
$566$	−22.5469	−0.947717
$567$	0	0
$568$	2.75782	0.115715
$569$	−25.0257	−1.04913	−0.524565	−	0.851370i	$-0.675772\pi$
$569$	−25.0257	−1.04913	−0.524565	+	0.851370i	$0.675772\pi$
$570$	0	0
$571$	−8.95927	−0.374934	−0.187467	−	0.982271i	$-0.560028\pi$
$571$	−8.95927	−0.374934	−0.187467	+	0.982271i	$0.560028\pi$
$572$	0.660638	0.0276227
$573$	0	0
$574$	−40.5865	−1.69405
$575$	−16.3724	−0.682778
$576$	0	0
$577$	−20.5294	−0.854648	−0.427324	−	0.904099i	$-0.640544\pi$
$577$	−20.5294	−0.854648	−0.427324	+	0.904099i	$0.640544\pi$
$578$	17.2614	0.717978
$579$	0	0
$580$	17.8497	0.741167
$581$	−7.70015	−0.319456
$582$	0	0
$583$	0.841593	0.0348552
$584$	−21.6368	−0.895338
$585$	0	0
$586$	0.822611	0.0339818
$587$	25.6913	1.06039	0.530196	−	0.847875i	$-0.322118\pi$
$587$	25.6913	1.06039	0.530196	+	0.847875i	$0.322118\pi$
$588$	0	0
$589$	39.6491	1.63371
$590$	−22.9121	−0.943275
$591$	0	0
$592$	2.24779	0.0923838
$593$	16.8500	0.691947	0.345974	−	0.938244i	$-0.387549\pi$
$593$	16.8500	0.691947	0.345974	+	0.938244i	$0.387549\pi$
$594$	0	0
$595$	73.5278	3.01435
$596$	11.8039	0.483507
$597$	0	0
$598$	16.5671	0.677477
$599$	17.7171	0.723900	0.361950	−	0.932197i	$-0.382111\pi$
$599$	17.7171	0.723900	0.361950	+	0.932197i	$0.382111\pi$
$600$	0	0
$601$	−2.09377	−0.0854068	−0.0427034	−	0.999088i	$-0.513597\pi$
$601$	−2.09377	−0.0854068	−0.0427034	+	0.999088i	$0.513597\pi$
$602$	20.7569	0.845987
$603$	0	0
$604$	8.79890	0.358022
$605$	31.1312	1.26566
$606$	0	0
$607$	40.3680	1.63849	0.819244	−	0.573445i	$-0.194393\pi$
$607$	40.3680	1.63849	0.819244	+	0.573445i	$0.194393\pi$
$608$	−30.4318	−1.23417
$609$	0	0
$610$	12.9385	0.523864
$611$	43.4459	1.75763
$612$	0	0
$613$	−1.28262	−0.0518045	−0.0259022	−	0.999664i	$-0.508246\pi$
$613$	−1.28262	−0.0518045	−0.0259022	+	0.999664i	$0.508246\pi$
$614$	−3.36639	−0.135857
$615$	0	0
$616$	−1.60555	−0.0646895
$617$	38.8602	1.56445	0.782226	−	0.622994i	$-0.214084\pi$
$617$	38.8602	1.56445	0.782226	+	0.622994i	$0.214084\pi$
$618$	0	0
$619$	40.8785	1.64305	0.821524	−	0.570174i	$-0.193124\pi$
$619$	40.8785	1.64305	0.821524	+	0.570174i	$0.193124\pi$
$620$	−27.7580	−1.11479
$621$	0	0
$622$	1.62682	0.0652295
$623$	41.4723	1.66155
$624$	0	0
$625$	−30.9614	−1.23846
$626$	−16.7102	−0.667876
$627$	0	0
$628$	3.78661	0.151102
$629$	−45.5666	−1.81686
$630$	0	0
$631$	−36.9894	−1.47252	−0.736262	−	0.676696i	$-0.763411\pi$
$631$	−36.9894	−1.47252	−0.736262	+	0.676696i	$0.763411\pi$
$632$	−22.6771	−0.902048
$633$	0	0
$634$	0.390303	0.0155009
$635$	37.1674	1.47494
$636$	0	0
$637$	−39.9556	−1.58310
$638$	0.555988	0.0220118
$639$	0	0
$640$	22.7385	0.898818
$641$	−24.4967	−0.967560	−0.483780	−	0.875190i	$-0.660736\pi$
$641$	−24.4967	−0.967560	−0.483780	+	0.875190i	$0.660736\pi$
$642$	0	0
$643$	38.0653	1.50115	0.750574	−	0.660786i	$-0.229777\pi$
$643$	38.0653	1.50115	0.750574	+	0.660786i	$0.229777\pi$
$644$	29.6752	1.16937
$645$	0	0
$646$	27.0374	1.06377
$647$	−33.3521	−1.31121	−0.655603	−	0.755106i	$-0.727585\pi$
$647$	−33.3521	−1.31121	−0.655603	+	0.755106i	$0.727585\pi$
$648$	0	0
$649$	1.33351	0.0523448
$650$	−9.33438	−0.366124
$651$	0	0
$652$	4.01414	0.157206
$653$	41.9552	1.64184	0.820918	−	0.571046i	$-0.193462\pi$
$653$	41.9552	1.64184	0.820918	+	0.571046i	$0.193462\pi$
$654$	0	0
$655$	−12.6191	−0.493069
$656$	−3.48280	−0.135980
$657$	0	0
$658$	−41.6486	−1.62363
$659$	−20.2699	−0.789605	−0.394802	−	0.918766i	$-0.629187\pi$
$659$	−20.2699	−0.789605	−0.394802	+	0.918766i	$0.629187\pi$
$660$	0	0
$661$	0.184040	0.00715831	0.00357916	−	0.999994i	$-0.498861\pi$
$661$	0.184040	0.00715831	0.00357916	+	0.999994i	$0.498861\pi$
$662$	−13.2185	−0.513750
$663$	0	0
$664$	5.02547	0.195026
$665$	63.1987	2.45074
$666$	0	0
$667$	−26.0522	−1.00874
$668$	−9.27706	−0.358940
$669$	0	0
$670$	9.49407	0.366788
$671$	−0.753035	−0.0290706
$672$	0	0
$673$	−16.5917	−0.639563	−0.319781	−	0.947491i	$-0.603609\pi$
$673$	−16.5917	−0.639563	−0.319781	+	0.947491i	$0.603609\pi$
$674$	−4.94408	−0.190439
$675$	0	0
$676$	−0.712697	−0.0274114
$677$	−25.7311	−0.988925	−0.494463	−	0.869199i	$-0.664635\pi$
$677$	−25.7311	−0.988925	−0.494463	+	0.869199i	$0.664635\pi$
$678$	0	0
$679$	46.9773	1.80282
$680$	−47.9876	−1.84024
$681$	0	0
$682$	−0.864616	−0.0331078
$683$	−43.8039	−1.67611	−0.838056	−	0.545585i	$-0.816308\pi$
$683$	−43.8039	−1.67611	−0.838056	+	0.545585i	$0.816308\pi$
$684$	0	0
$685$	53.0903	2.02848
$686$	13.6041	0.519406
$687$	0	0
$688$	1.78119	0.0679071
$689$	−22.4830	−0.856533
$690$	0	0
$691$	22.4902	0.855568	0.427784	−	0.903881i	$-0.359294\pi$
$691$	22.4902	0.855568	0.427784	+	0.903881i	$0.359294\pi$
$692$	27.0777	1.02934
$693$	0	0
$694$	10.3558	0.393102
$695$	15.3829	0.583506
$696$	0	0
$697$	70.6022	2.67425
$698$	1.75074	0.0662665
$699$	0	0
$700$	−16.7199	−0.631953
$701$	−13.2346	−0.499864	−0.249932	−	0.968263i	$-0.580408\pi$
$701$	−13.2346	−0.499864	−0.249932	+	0.968263i	$0.580408\pi$
$702$	0	0
$703$	−39.1655	−1.47715
$704$	0.580186	0.0218666
$705$	0	0
$706$	23.6200	0.888952
$707$	9.51680	0.357916
$708$	0	0
$709$	−13.1653	−0.494435	−0.247217	−	0.968960i	$-0.579516\pi$
$709$	−13.1653	−0.494435	−0.247217	+	0.968960i	$0.579516\pi$
$710$	2.36722	0.0888403
$711$	0	0
$712$	−27.0667	−1.01437
$713$	40.5137	1.51725
$714$	0	0
$715$	1.43763	0.0537643
$716$	−20.4773	−0.765272
$717$	0	0
$718$	−8.00303	−0.298670
$719$	2.42508	0.0904402	0.0452201	−	0.998977i	$-0.485601\pi$
$719$	2.42508	0.0904402	0.0452201	+	0.998977i	$0.485601\pi$
$720$	0	0
$721$	−20.3858	−0.759207
$722$	7.37424	0.274441
$723$	0	0
$724$	−29.1599	−1.08372
$725$	14.6786	0.545149
$726$	0	0
$727$	−22.4563	−0.832860	−0.416430	−	0.909168i	$-0.636719\pi$
$727$	−22.4563	−0.832860	−0.416430	+	0.909168i	$0.636719\pi$
$728$	42.8919	1.58968
$729$	0	0
$730$	−18.5724	−0.687394
$731$	−36.1077	−1.33549
$732$	0	0
$733$	8.39702	0.310151	0.155076	−	0.987903i	$-0.450438\pi$
$733$	8.39702	0.310151	0.155076	+	0.987903i	$0.450438\pi$
$734$	18.0930	0.667824
$735$	0	0
$736$	−31.0954	−1.14619
$737$	−0.552566	−0.0203540
$738$	0	0
$739$	−4.99404	−0.183709	−0.0918544	−	0.995772i	$-0.529279\pi$
$739$	−4.99404	−0.183709	−0.0918544	+	0.995772i	$0.529279\pi$
$740$	27.4194	1.00796
$741$	0	0
$742$	21.5529	0.791230
$743$	−8.55041	−0.313684	−0.156842	−	0.987624i	$-0.550131\pi$
$743$	−8.55041	−0.313684	−0.156842	+	0.987624i	$0.550131\pi$
$744$	0	0
$745$	25.6868	0.941091
$746$	19.5052	0.714136
$747$	0	0
$748$	1.10167	0.0402811
$749$	8.86720	0.324000
$750$	0	0
$751$	−33.9952	−1.24050	−0.620252	−	0.784403i	$-0.712969\pi$
$751$	−33.9952	−1.24050	−0.620252	+	0.784403i	$0.712969\pi$
$752$	−3.57394	−0.130328
$753$	0	0
$754$	−14.8531	−0.540917
$755$	19.1475	0.696848
$756$	0	0
$757$	50.2933	1.82794	0.913970	−	0.405781i	$-0.133001\pi$
$757$	50.2933	1.82794	0.913970	+	0.405781i	$0.133001\pi$
$758$	−13.8463	−0.502919
$759$	0	0
$760$	−41.2464	−1.49616
$761$	−48.0498	−1.74180	−0.870902	−	0.491457i	$-0.836465\pi$
$761$	−48.0498	−1.74180	−0.870902	+	0.491457i	$0.836465\pi$
$762$	0	0
$763$	−60.7925	−2.20084
$764$	−18.9209	−0.684534
$765$	0	0
$766$	17.5508	0.634135
$767$	−35.6244	−1.28632
$768$	0	0
$769$	−16.0458	−0.578625	−0.289312	−	0.957235i	$-0.593427\pi$
$769$	−16.0458	−0.578625	−0.289312	+	0.957235i	$0.593427\pi$
$770$	−1.37816	−0.0496653
$771$	0	0
$772$	9.46410	0.340621
$773$	−19.8668	−0.714559	−0.357280	−	0.933998i	$-0.616296\pi$
$773$	−19.8668	−0.714559	−0.357280	+	0.933998i	$0.616296\pi$
$774$	0	0
$775$	−22.8267	−0.819958
$776$	−30.6595	−1.10061
$777$	0	0
$778$	−5.51212	−0.197619
$779$	60.6841	2.17423
$780$	0	0
$781$	−0.137775	−0.00492999
$782$	27.6270	0.987940
$783$	0	0
$784$	3.28682	0.117386
$785$	8.24013	0.294103
$786$	0	0
$787$	38.4462	1.37046	0.685230	−	0.728327i	$-0.259702\pi$
$787$	38.4462	1.37046	0.685230	+	0.728327i	$0.259702\pi$
$788$	−19.1197	−0.681112
$789$	0	0
$790$	−19.4653	−0.692546
$791$	17.5547	0.624172
$792$	0	0
$793$	20.1171	0.714381
$794$	−7.40333	−0.262734
$795$	0	0
$796$	−23.6024	−0.836564
$797$	34.9472	1.23789	0.618947	−	0.785433i	$-0.287560\pi$
$797$	34.9472	1.23789	0.618947	+	0.785433i	$0.287560\pi$
$798$	0	0
$799$	72.4498	2.56309
$800$	17.5201	0.619428
$801$	0	0
$802$	−13.3810	−0.472499
$803$	1.08093	0.0381453
$804$	0	0
$805$	64.5769	2.27604
$806$	23.0980	0.813592
$807$	0	0
$808$	−6.21110	−0.218506
$809$	−15.8787	−0.558266	−0.279133	−	0.960252i	$-0.590047\pi$
$809$	−15.8787	−0.558266	−0.279133	+	0.960252i	$0.590047\pi$
$810$	0	0
$811$	22.6911	0.796792	0.398396	−	0.917213i	$-0.369567\pi$
$811$	22.6911	0.796792	0.398396	+	0.917213i	$0.369567\pi$
$812$	−26.6051	−0.933656
$813$	0	0
$814$	0.854069	0.0299351
$815$	8.73527	0.305983
$816$	0	0
$817$	−31.0353	−1.08579
$818$	−6.44010	−0.225173
$819$	0	0
$820$	−42.4844	−1.48362
$821$	−14.9377	−0.521328	−0.260664	−	0.965430i	$-0.583941\pi$
$821$	−14.9377	−0.521328	−0.260664	+	0.965430i	$0.583941\pi$
$822$	0	0
$823$	−15.8894	−0.553871	−0.276936	−	0.960888i	$-0.589319\pi$
$823$	−15.8894	−0.553871	−0.276936	+	0.960888i	$0.589319\pi$
$824$	13.3047	0.463492
$825$	0	0
$826$	34.1506	1.18825
$827$	0.608714	0.0211671	0.0105835	−	0.999944i	$-0.496631\pi$
$827$	0.608714	0.0211671	0.0105835	+	0.999944i	$0.496631\pi$
$828$	0	0
$829$	5.78119	0.200789	0.100394	−	0.994948i	$-0.467990\pi$
$829$	5.78119	0.200789	0.100394	+	0.994948i	$0.467990\pi$
$830$	4.31371	0.149731
$831$	0	0
$832$	−15.4995	−0.537349
$833$	−66.6294	−2.30857
$834$	0	0
$835$	−20.1880	−0.698636
$836$	0.946910	0.0327496
$837$	0	0
$838$	−11.9291	−0.412084
$839$	25.0780	0.865789	0.432894	−	0.901445i	$-0.357492\pi$
$839$	25.0780	0.865789	0.432894	+	0.901445i	$0.357492\pi$
$840$	0	0
$841$	−5.64309	−0.194589
$842$	9.39947	0.323927
$843$	0	0
$844$	26.0123	0.895380
$845$	−1.55092	−0.0533531
$846$	0	0
$847$	−46.4013	−1.59437
$848$	1.84949	0.0635117
$849$	0	0
$850$	−15.5659	−0.533906
$851$	−40.0195	−1.37185
$852$	0	0
$853$	−0.109432	−0.00374687	−0.00187344	−	0.999998i	$-0.500596\pi$
$853$	−0.109432	−0.00374687	−0.00187344	+	0.999998i	$0.500596\pi$
$854$	−19.2849	−0.659916
$855$	0	0
$856$	−5.78714	−0.197800
$857$	27.9997	0.956453	0.478226	−	0.878237i	$-0.341280\pi$
$857$	27.9997	0.956453	0.478226	+	0.878237i	$0.341280\pi$
$858$	0	0
$859$	−2.83338	−0.0966737	−0.0483368	−	0.998831i	$-0.515392\pi$
$859$	−2.83338	−0.0966737	−0.0483368	+	0.998831i	$0.515392\pi$
$860$	21.7276	0.740904
$861$	0	0
$862$	−8.75269	−0.298118
$863$	−10.9915	−0.374156	−0.187078	−	0.982345i	$-0.559902\pi$
$863$	−10.9915	−0.374156	−0.187078	+	0.982345i	$0.559902\pi$
$864$	0	0
$865$	58.9243	2.00349
$866$	−23.7443	−0.806866
$867$	0	0
$868$	41.3735	1.40431
$869$	1.13291	0.0384312
$870$	0	0
$871$	14.7617	0.500180
$872$	39.6760	1.34360
$873$	0	0
$874$	23.7460	0.803221
$875$	23.5132	0.794891
$876$	0	0
$877$	0.893016	0.0301550	0.0150775	−	0.999886i	$-0.495200\pi$
$877$	0.893016	0.0301550	0.0150775	+	0.999886i	$0.495200\pi$
$878$	−33.8997	−1.14406
$879$	0	0
$880$	−0.118262	−0.00398661
$881$	20.0584	0.675784	0.337892	−	0.941185i	$-0.390286\pi$
$881$	20.0584	0.675784	0.337892	+	0.941185i	$0.390286\pi$
$882$	0	0
$883$	−16.6645	−0.560805	−0.280402	−	0.959883i	$-0.590468\pi$
$883$	−16.6645	−0.560805	−0.280402	+	0.959883i	$0.590468\pi$
$884$	−29.4309	−0.989867
$885$	0	0
$886$	−30.4772	−1.02390
$887$	13.5527	0.455055	0.227527	−	0.973772i	$-0.426936\pi$
$887$	13.5527	0.455055	0.227527	+	0.973772i	$0.426936\pi$
$888$	0	0
$889$	−55.3984	−1.85800
$890$	−23.2332	−0.778780
$891$	0	0
$892$	22.0662	0.738831
$893$	62.2722	2.08386
$894$	0	0
$895$	−44.5611	−1.48951
$896$	−33.8919	−1.13225
$897$	0	0
$898$	13.6245	0.454657
$899$	−36.3223	−1.21142
$900$	0	0
$901$	−37.4923	−1.24905
$902$	−1.32332	−0.0440617
$903$	0	0
$904$	−11.4570	−0.381053
$905$	−63.4555	−2.10933
$906$	0	0
$907$	−13.5817	−0.450973	−0.225486	−	0.974246i	$-0.572397\pi$
$907$	−13.5817	−0.450973	−0.225486	+	0.974246i	$0.572397\pi$
$908$	10.2327	0.339583
$909$	0	0
$910$	36.8171	1.22047
$911$	25.4165	0.842087	0.421044	−	0.907040i	$-0.361664\pi$
$911$	25.4165	0.842087	0.421044	+	0.907040i	$0.361664\pi$
$912$	0	0
$913$	−0.251063	−0.00830897
$914$	−13.7359	−0.454344
$915$	0	0
$916$	6.48070	0.214128
$917$	18.8089	0.621124
$918$	0	0
$919$	36.1267	1.19171	0.595856	−	0.803092i	$-0.296813\pi$
$919$	36.1267	1.19171	0.595856	+	0.803092i	$0.296813\pi$
$920$	−42.1459	−1.38951
$921$	0	0
$922$	14.8286	0.488354
$923$	3.68063	0.121150
$924$	0	0
$925$	22.5482	0.741381
$926$	−15.8980	−0.522440
$927$	0	0
$928$	27.8783	0.915151
$929$	1.51016	0.0495467	0.0247733	−	0.999693i	$-0.492114\pi$
$929$	1.51016	0.0495467	0.0247733	+	0.999693i	$0.492114\pi$
$930$	0	0
$931$	−57.2694	−1.87693
$932$	20.1128	0.658818
$933$	0	0
$934$	−16.4071	−0.536856
$935$	2.39737	0.0784024
$936$	0	0
$937$	32.0706	1.04770	0.523851	−	0.851810i	$-0.324495\pi$
$937$	32.0706	1.04770	0.523851	+	0.851810i	$0.324495\pi$
$938$	−14.1510	−0.462046
$939$	0	0
$940$	−43.5962	−1.42195
$941$	−58.2891	−1.90017	−0.950085	−	0.311990i	$-0.899004\pi$
$941$	−58.2891	−1.90017	−0.950085	+	0.311990i	$0.899004\pi$
$942$	0	0
$943$	62.0074	2.01924
$944$	2.93053	0.0953806
$945$	0	0
$946$	0.676777	0.0220039
$947$	−14.4138	−0.468386	−0.234193	−	0.972190i	$-0.575245\pi$
$947$	−14.4138	−0.468386	−0.234193	+	0.972190i	$0.575245\pi$
$948$	0	0
$949$	−28.8769	−0.937384
$950$	−13.3792	−0.434079
$951$	0	0
$952$	71.5260	2.31817
$953$	−9.92766	−0.321588	−0.160794	−	0.986988i	$-0.551406\pi$
$953$	−9.92766	−0.321588	−0.160794	+	0.986988i	$0.551406\pi$
$954$	0	0
$955$	−41.1742	−1.33237
$956$	−6.73958	−0.217973
$957$	0	0
$958$	22.0103	0.711122
$959$	−79.1316	−2.55529
$960$	0	0
$961$	25.4847	0.822088
$962$	−22.8163	−0.735626
$963$	0	0
$964$	6.65492	0.214341
$965$	20.5951	0.662978
$966$	0	0
$967$	48.4361	1.55760	0.778799	−	0.627273i	$-0.215829\pi$
$967$	48.4361	1.55760	0.778799	+	0.627273i	$0.215829\pi$
$968$	30.2836	0.973353
$969$	0	0
$970$	−26.3172	−0.844994
$971$	−31.3434	−1.00586	−0.502928	−	0.864328i	$-0.667744\pi$
$971$	−31.3434	−1.00586	−0.502928	+	0.864328i	$0.667744\pi$
$972$	0	0
$973$	−22.9283	−0.735048
$974$	10.7361	0.344006
$975$	0	0
$976$	−1.65487	−0.0529712
$977$	−49.3537	−1.57896	−0.789482	−	0.613773i	$-0.789651\pi$
$977$	−49.3537	−1.57896	−0.789482	+	0.613773i	$0.789651\pi$
$978$	0	0
$979$	1.35220	0.0432166
$980$	40.0938	1.28075
$981$	0	0
$982$	1.67644	0.0534973
$983$	−8.87192	−0.282970	−0.141485	−	0.989940i	$-0.545188\pi$
$983$	−8.87192	−0.282970	−0.141485	+	0.989940i	$0.545188\pi$
$984$	0	0
$985$	−41.6069	−1.32570
$986$	−24.7688	−0.788799
$987$	0	0
$988$	−25.2965	−0.804788
$989$	−31.7121	−1.00839
$990$	0	0
$991$	−16.1640	−0.513467	−0.256733	−	0.966482i	$-0.582646\pi$
$991$	−16.1640	−0.513467	−0.256733	+	0.966482i	$0.582646\pi$
$992$	−43.3536	−1.37648
$993$	0	0
$994$	−3.52837	−0.111913
$995$	−51.3616	−1.62827
$996$	0	0
$997$	−36.8597	−1.16736	−0.583679	−	0.811985i	$-0.698387\pi$
$997$	−36.8597	−1.16736	−0.583679	+	0.811985i	$0.698387\pi$
$998$	15.5523	0.492300
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	639.2.a.j.1.3	✓	4
3.2	odd	2		639.2.a.k.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
639.2.a.j.1.3	✓	4	1.1	even	1	trivial
639.2.a.k.1.2	yes	4	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:	\( N \)	\(=\)	\( 639 = 3^{2} \cdot 71 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	639.a (trivial)

\(f(q)\)	\(=\)	\(q+0.835000 q^{2} -1.30278 q^{4} -2.83500 q^{5} +4.22559 q^{7} -2.75782 q^{8} +O(q^{10})\)	\(q+0.835000 q^{2} -1.30278 q^{4} -2.83500 q^{5} +4.22559 q^{7} -2.75782 q^{8} -2.36722 q^{10} +0.137775 q^{11} -3.68063 q^{13} +3.52837 q^{14} +0.302776 q^{16} -6.13778 q^{17} -5.27555 q^{19} +3.69337 q^{20} +0.115042 q^{22} -5.39059 q^{23} +3.03722 q^{25} -3.07333 q^{26} -5.50500 q^{28} +4.83290 q^{29} -7.51563 q^{31} +5.76845 q^{32} -5.12504 q^{34} -11.9796 q^{35} +7.42396 q^{37} -4.40508 q^{38} +7.81841 q^{40} -11.5029 q^{41} +5.88286 q^{43} -0.179490 q^{44} -4.50114 q^{46} -11.8039 q^{47} +10.8556 q^{49} +2.53608 q^{50} +4.79504 q^{52} +6.10845 q^{53} -0.390593 q^{55} -11.6534 q^{56} +4.03547 q^{58} +9.67888 q^{59} -5.46567 q^{61} -6.27555 q^{62} +4.21110 q^{64} +10.4346 q^{65} -4.01063 q^{67} +7.99614 q^{68} -10.0029 q^{70} -1.00000 q^{71} +7.84563 q^{73} +6.19900 q^{74} +6.87286 q^{76} +0.582182 q^{77} +8.22285 q^{79} -0.858369 q^{80} -9.60492 q^{82} -1.82226 q^{83} +17.4006 q^{85} +4.91218 q^{86} -0.379959 q^{88} +9.81455 q^{89} -15.5529 q^{91} +7.02273 q^{92} -9.85627 q^{94} +14.9562 q^{95} +11.1173 q^{97} +9.06445 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{4} - 8 q^{5} + 2 q^{7}+O(q^{10}) \) 4 * q + 2 * q^4 - 8 * q^5 + 2 * q^7	\( 4 q + 2 q^{4} - 8 q^{5} + 2 q^{7} - 10 q^{10} - 10 q^{11} + 4 q^{13} - 8 q^{14} - 6 q^{16} - 14 q^{17} - 4 q^{20} + 10 q^{22} - 10 q^{23} + 6 q^{25} + 6 q^{26} - 12 q^{28} + 4 q^{29} - 8 q^{31} - 10 q^{34} + 4 q^{35} - 14 q^{37} - 20 q^{38} + 2 q^{40} - 24 q^{41} + 2 q^{43} - 18 q^{44} + 18 q^{46} - 4 q^{47} + 8 q^{49} + 40 q^{50} + 2 q^{52} - 12 q^{53} + 10 q^{55} - 14 q^{56} - 4 q^{58} + 6 q^{59} - 6 q^{61} - 4 q^{62} - 12 q^{64} - 14 q^{65} - 4 q^{67} + 6 q^{68} + 24 q^{70} - 4 q^{71} + 16 q^{73} + 4 q^{74} + 26 q^{76} - 2 q^{79} + 12 q^{80} - 16 q^{82} - 4 q^{83} + 38 q^{85} + 24 q^{86} - 2 q^{88} - 16 q^{89} - 34 q^{91} + 8 q^{92} - 12 q^{94} + 20 q^{95} - 18 q^{97} + 44 q^{98}+O(q^{100}) \) 4 * q + 2 * q^4 - 8 * q^5 + 2 * q^7 - 10 * q^10 - 10 * q^11 + 4 * q^13 - 8 * q^14 - 6 * q^16 - 14 * q^17 - 4 * q^20 + 10 * q^22 - 10 * q^23 + 6 * q^25 + 6 * q^26 - 12 * q^28 + 4 * q^29 - 8 * q^31 - 10 * q^34 + 4 * q^35 - 14 * q^37 - 20 * q^38 + 2 * q^40 - 24 * q^41 + 2 * q^43 - 18 * q^44 + 18 * q^46 - 4 * q^47 + 8 * q^49 + 40 * q^50 + 2 * q^52 - 12 * q^53 + 10 * q^55 - 14 * q^56 - 4 * q^58 + 6 * q^59 - 6 * q^61 - 4 * q^62 - 12 * q^64 - 14 * q^65 - 4 * q^67 + 6 * q^68 + 24 * q^70 - 4 * q^71 + 16 * q^73 + 4 * q^74 + 26 * q^76 - 2 * q^79 + 12 * q^80 - 16 * q^82 - 4 * q^83 + 38 * q^85 + 24 * q^86 - 2 * q^88 - 16 * q^89 - 34 * q^91 + 8 * q^92 - 12 * q^94 + 20 * q^95 - 18 * q^97 + 44 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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