LMFDB - Embedded newform 777.2.a.l.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [777,2,Mod(1,777)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(777, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("777.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$N$	$=$	$777 = 3 \cdot 7 \cdot 37$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	777.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,-5,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$6.20437623705$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.643312.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$x^{5} - 2x^{4} - 7x^{3} + 10x^{2} + 12x - 4$ x^5 - 2x^4 - 7x^3 + 10x^2 + 12x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.09979$ of defining polynomial
Character	$\chi$	$=$	777.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.09979 q^{2} -1.00000 q^{3} +2.40912 q^{4} -3.89695 q^{5} -2.09979 q^{6} +1.00000 q^{7} +0.859065 q^{8} +1.00000 q^{9} -8.18278 q^{10} -2.92767 q^{11} -2.40912 q^{12} +3.41550 q^{13} +2.09979 q^{14} +3.89695 q^{15} -3.01438 q^{16} -6.66423 q^{17} +2.09979 q^{18} -6.95886 q^{19} -9.38822 q^{20} -1.00000 q^{21} -6.14749 q^{22} -6.75927 q^{23} -0.859065 q^{24} +10.1862 q^{25} +7.17183 q^{26} -1.00000 q^{27} +2.40912 q^{28} +8.13069 q^{29} +8.18278 q^{30} +1.12087 q^{31} -8.04770 q^{32} +2.92767 q^{33} -13.9935 q^{34} -3.89695 q^{35} +2.40912 q^{36} +1.00000 q^{37} -14.6121 q^{38} -3.41550 q^{39} -3.34773 q^{40} -10.1307 q^{41} -2.09979 q^{42} +2.77608 q^{43} -7.05310 q^{44} -3.89695 q^{45} -14.1931 q^{46} +1.17183 q^{47} +3.01438 q^{48} +1.00000 q^{49} +21.3889 q^{50} +6.66423 q^{51} +8.22835 q^{52} +4.72157 q^{53} -2.09979 q^{54} +11.4090 q^{55} +0.859065 q^{56} +6.95886 q^{57} +17.0727 q^{58} +6.57798 q^{59} +9.38822 q^{60} +4.29383 q^{61} +2.35359 q^{62} +1.00000 q^{63} -10.8697 q^{64} -13.3100 q^{65} +6.14749 q^{66} +2.59088 q^{67} -16.0549 q^{68} +6.75927 q^{69} -8.18278 q^{70} +3.11790 q^{71} +0.859065 q^{72} -3.37793 q^{73} +2.09979 q^{74} -10.1862 q^{75} -16.7647 q^{76} -2.92767 q^{77} -7.17183 q^{78} +0.388878 q^{79} +11.7469 q^{80} +1.00000 q^{81} -21.2723 q^{82} +8.59533 q^{83} -2.40912 q^{84} +25.9702 q^{85} +5.82918 q^{86} -8.13069 q^{87} -2.51506 q^{88} +1.50975 q^{89} -8.18278 q^{90} +3.41550 q^{91} -16.2839 q^{92} -1.12087 q^{93} +2.46060 q^{94} +27.1183 q^{95} +8.04770 q^{96} -14.9569 q^{97} +2.09979 q^{98} -2.92767 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$5 q - 2 q^{2} - 5 q^{3} + 8 q^{4} - 5 q^{5} + 2 q^{6} + 5 q^{7} - 12 q^{8} + 5 q^{9} - 8 q^{10} - 3 q^{11} - 8 q^{12} - 5 q^{13} - 2 q^{14} + 5 q^{15} + 10 q^{16} - 8 q^{17} - 2 q^{18} - 6 q^{19} - 14 q^{20}+ \cdots - 3 q^{99}+O(q^{100})$ 5 * q - 2 * q^2 - 5 * q^3 + 8 * q^4 - 5 * q^5 + 2 * q^6 + 5 * q^7 - 12 * q^8 + 5 * q^9 - 8 * q^10 - 3 * q^11 - 8 * q^12 - 5 * q^13 - 2 * q^14 + 5 * q^15 + 10 * q^16 - 8 * q^17 - 2 * q^18 - 6 * q^19 - 14 * q^20 - 5 * q^21 + 6 * q^22 - 30 * q^23 + 12 * q^24 + 10 * q^25 + 20 * q^26 - 5 * q^27 + 8 * q^28 - 4 * q^29 + 8 * q^30 - 13 * q^31 - 16 * q^32 + 3 * q^33 - 16 * q^34 - 5 * q^35 + 8 * q^36 + 5 * q^37 - 36 * q^38 + 5 * q^39 + 16 * q^40 - 6 * q^41 + 2 * q^42 + 18 * q^43 - 6 * q^44 - 5 * q^45 + 8 * q^46 - 10 * q^47 - 10 * q^48 + 5 * q^49 - 4 * q^50 + 8 * q^51 - 34 * q^52 - 17 * q^53 + 2 * q^54 - 19 * q^55 - 12 * q^56 + 6 * q^57 + 4 * q^58 + q^59 + 14 * q^60 - 4 * q^61 + 4 * q^62 + 5 * q^63 - 6 * q^64 + 3 * q^65 - 6 * q^66 + 17 * q^67 - 32 * q^68 + 30 * q^69 - 8 * q^70 - 15 * q^71 - 12 * q^72 - 30 * q^73 - 2 * q^74 - 10 * q^75 + 32 * q^76 - 3 * q^77 - 20 * q^78 - 2 * q^79 - 24 * q^80 + 5 * q^81 - 2 * q^83 - 8 * q^84 + 14 * q^85 + 4 * q^86 + 4 * q^87 - 36 * q^88 - 15 * q^89 - 8 * q^90 - 5 * q^91 - 32 * q^92 + 13 * q^93 - 32 * q^94 + 12 * q^95 + 16 * q^96 - 15 * q^97 - 2 * q^98 - 3 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\alpha_p$ , and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$ .

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$\alpha_n$			$\theta_n$
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$\alpha_p$			$\theta_p$

$2$	2.09979	1.48478	0.742388	−	0.669970i	$-0.233693\pi$
$2$	2.09979	1.48478	0.742388	+	0.669970i	$0.233693\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	2.40912	1.20456
$5$	−3.89695	−1.74277	−0.871384	−	0.490601i	$-0.836777\pi$
$5$	−3.89695	−1.74277	−0.871384	+	0.490601i	$0.836777\pi$
$6$	−2.09979	−0.857236
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.859065	0.303725
$9$	1.00000	0.333333
$10$	−8.18278	−2.58762
$11$	−2.92767	−0.882725	−0.441363	−	0.897329i	$-0.645505\pi$
$11$	−2.92767	−0.882725	−0.441363	+	0.897329i	$0.645505\pi$
$12$	−2.40912	−0.695453
$13$	3.41550	0.947289	0.473644	−	0.880716i	$-0.342938\pi$
$13$	3.41550	0.947289	0.473644	+	0.880716i	$0.342938\pi$
$14$	2.09979	0.561193
$15$	3.89695	1.00619
$16$	−3.01438	−0.753596
$17$	−6.66423	−1.61631	−0.808156	−	0.588968i	$-0.799534\pi$
$17$	−6.66423	−1.61631	−0.808156	+	0.588968i	$0.799534\pi$
$18$	2.09979	0.494925
$19$	−6.95886	−1.59647	−0.798235	−	0.602346i	$-0.794233\pi$
$19$	−6.95886	−1.59647	−0.798235	+	0.602346i	$0.794233\pi$
$20$	−9.38822	−2.09927
$21$	−1.00000	−0.218218
$22$	−6.14749	−1.31065
$23$	−6.75927	−1.40941	−0.704703	−	0.709502i	$-0.748920\pi$
$23$	−6.75927	−1.40941	−0.704703	+	0.709502i	$0.748920\pi$
$24$	−0.859065	−0.175356
$25$	10.1862	2.03724
$26$	7.17183	1.40651
$27$	−1.00000	−0.192450
$28$	2.40912	0.455281
$29$	8.13069	1.50983	0.754915	−	0.655822i	$-0.227678\pi$
$29$	8.13069	1.50983	0.754915	+	0.655822i	$0.227678\pi$
$30$	8.18278	1.49396
$31$	1.12087	0.201314	0.100657	−	0.994921i	$-0.467905\pi$
$31$	1.12087	0.201314	0.100657	+	0.994921i	$0.467905\pi$
$32$	−8.04770	−1.42265
$33$	2.92767	0.509642
$34$	−13.9935	−2.39986
$35$	−3.89695	−0.658705
$36$	2.40912	0.401520
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	−14.6121	−2.37040
$39$	−3.41550	−0.546918
$40$	−3.34773	−0.529323
$41$	−10.1307	−1.58215	−0.791074	−	0.611721i	$-0.790478\pi$
$41$	−10.1307	−1.58215	−0.791074	+	0.611721i	$0.790478\pi$
$42$	−2.09979	−0.324005
$43$	2.77608	0.423348	0.211674	−	0.977340i	$-0.432108\pi$
$43$	2.77608	0.423348	0.211674	+	0.977340i	$0.432108\pi$
$44$	−7.05310	−1.06330
$45$	−3.89695	−0.580923
$46$	−14.1931	−2.09265
$47$	1.17183	0.170929	0.0854646	−	0.996341i	$-0.472763\pi$
$47$	1.17183	0.170929	0.0854646	+	0.996341i	$0.472763\pi$
$48$	3.01438	0.435089
$49$	1.00000	0.142857
$50$	21.3889	3.02485
$51$	6.66423	0.933178
$52$	8.22835	1.14107
$53$	4.72157	0.648557	0.324279	−	0.945962i	$-0.394878\pi$
$53$	4.72157	0.648557	0.324279	+	0.945962i	$0.394878\pi$
$54$	−2.09979	−0.285745
$55$	11.4090	1.53839
$56$	0.859065	0.114797
$57$	6.95886	0.921723
$58$	17.0727	2.24176
$59$	6.57798	0.856380	0.428190	−	0.903689i	$-0.359151\pi$
$59$	6.57798	0.856380	0.428190	+	0.903689i	$0.359151\pi$
$60$	9.38822	1.21201
$61$	4.29383	0.549769	0.274884	−	0.961477i	$-0.411360\pi$
$61$	4.29383	0.549769	0.274884	+	0.961477i	$0.411360\pi$
$62$	2.35359	0.298907
$63$	1.00000	0.125988
$64$	−10.8697	−1.35872
$65$	−13.3100	−1.65091
$66$	6.14749	0.756704
$67$	2.59088	0.316526	0.158263	−	0.987397i	$-0.449411\pi$
$67$	2.59088	0.316526	0.158263	+	0.987397i	$0.449411\pi$
$68$	−16.0549	−1.94694
$69$	6.75927	0.813721
$70$	−8.18278	−0.978029
$71$	3.11790	0.370027	0.185013	−	0.982736i	$-0.440767\pi$
$71$	3.11790	0.370027	0.185013	+	0.982736i	$0.440767\pi$
$72$	0.859065	0.101242
$73$	−3.37793	−0.395357	−0.197679	−	0.980267i	$-0.563340\pi$
$73$	−3.37793	−0.395357	−0.197679	+	0.980267i	$0.563340\pi$
$74$	2.09979	0.244096
$75$	−10.1862	−1.17620
$76$	−16.7647	−1.92304
$77$	−2.92767	−0.333639
$78$	−7.17183	−0.812050
$79$	0.388878	0.0437522	0.0218761	−	0.999761i	$-0.493036\pi$
$79$	0.388878	0.0437522	0.0218761	+	0.999761i	$0.493036\pi$
$80$	11.7469	1.31334
$81$	1.00000	0.111111
$82$	−21.2723	−2.34913
$83$	8.59533	0.943460	0.471730	−	0.881743i	$-0.343630\pi$
$83$	8.59533	0.943460	0.471730	+	0.881743i	$0.343630\pi$
$84$	−2.40912	−0.262856
$85$	25.9702	2.81686
$86$	5.82918	0.628577
$87$	−8.13069	−0.871701
$88$	−2.51506	−0.268106
$89$	1.50975	0.160033	0.0800165	−	0.996794i	$-0.474503\pi$
$89$	1.50975	0.160033	0.0800165	+	0.996794i	$0.474503\pi$
$90$	−8.18278	−0.862540
$91$	3.41550	0.358042
$92$	−16.2839	−1.69771
$93$	−1.12087	−0.116229
$94$	2.46060	0.253792
$95$	27.1183	2.78228
$96$	8.04770	0.821365
$97$	−14.9569	−1.51865	−0.759323	−	0.650714i	$-0.774470\pi$
$97$	−14.9569	−1.51865	−0.759323	+	0.650714i	$0.774470\pi$
$98$	2.09979	0.212111
$99$	−2.92767	−0.294242
$100$	24.5398	2.45398
$101$	3.40670	0.338979	0.169490	−	0.985532i	$-0.445788\pi$
$101$	3.40670	0.338979	0.169490	+	0.985532i	$0.445788\pi$
$102$	13.9935	1.38556
$103$	−5.24873	−0.517173	−0.258586	−	0.965988i	$-0.583257\pi$
$103$	−5.24873	−0.517173	−0.258586	+	0.965988i	$0.583257\pi$
$104$	2.93414	0.287716
$105$	3.89695	0.380303
$106$	9.91430	0.962962
$107$	−6.47903	−0.626351	−0.313176	−	0.949695i	$-0.601393\pi$
$107$	−6.47903	−0.626351	−0.313176	+	0.949695i	$0.601393\pi$
$108$	−2.40912	−0.231818
$109$	−14.4934	−1.38822	−0.694108	−	0.719870i	$-0.744201\pi$
$109$	−14.4934	−1.38822	−0.694108	+	0.719870i	$0.744201\pi$
$110$	23.9565	2.28416
$111$	−1.00000	−0.0949158
$112$	−3.01438	−0.284832
$113$	−4.24254	−0.399105	−0.199552	−	0.979887i	$-0.563949\pi$
$113$	−4.24254	−0.399105	−0.199552	+	0.979887i	$0.563949\pi$
$114$	14.6121	1.36855
$115$	26.3406	2.45627
$116$	19.5878	1.81868
$117$	3.41550	0.315763
$118$	13.8124	1.27153
$119$	−6.66423	−0.610909
$120$	3.34773	0.305605
$121$	−2.42875	−0.220796
$122$	9.01614	0.816283
$123$	10.1307	0.913453
$124$	2.70031	0.242495
$125$	−20.2104	−1.80767
$126$	2.09979	0.187064
$127$	−19.5230	−1.73239	−0.866193	−	0.499709i	$-0.833440\pi$
$127$	−19.5230	−1.73239	−0.866193	+	0.499709i	$0.833440\pi$
$128$	−6.72873	−0.594742
$129$	−2.77608	−0.244420
$130$	−27.9483	−2.45122
$131$	14.7869	1.29194	0.645969	−	0.763363i	$-0.276453\pi$
$131$	14.7869	1.29194	0.645969	+	0.763363i	$0.276453\pi$
$132$	7.05310	0.613894
$133$	−6.95886	−0.603409
$134$	5.44031	0.469971
$135$	3.89695	0.335396
$136$	−5.72501	−0.490915
$137$	12.3412	1.05438	0.527192	−	0.849746i	$-0.323245\pi$
$137$	12.3412	1.05438	0.527192	+	0.849746i	$0.323245\pi$
$138$	14.1931	1.20819
$139$	−12.0364	−1.02092	−0.510459	−	0.859902i	$-0.670524\pi$
$139$	−12.0364	−1.02092	−0.510459	+	0.859902i	$0.670524\pi$
$140$	−9.38822	−0.793449
$141$	−1.17183	−0.0986860
$142$	6.54694	0.549407
$143$	−9.99945	−0.836196
$144$	−3.01438	−0.251199
$145$	−31.6849	−2.63129
$146$	−7.09295	−0.587017
$147$	−1.00000	−0.0824786
$148$	2.40912	0.198028
$149$	−18.5298	−1.51802	−0.759012	−	0.651076i	$-0.774318\pi$
$149$	−18.5298	−1.51802	−0.759012	+	0.651076i	$0.774318\pi$
$150$	−21.3889	−1.74640
$151$	−8.97324	−0.730232	−0.365116	−	0.930962i	$-0.618971\pi$
$151$	−8.97324	−0.730232	−0.365116	+	0.930962i	$0.618971\pi$
$152$	−5.97811	−0.484889
$153$	−6.66423	−0.538771
$154$	−6.14749	−0.495379
$155$	−4.36798	−0.350844
$156$	−8.22835	−0.658795
$157$	23.5710	1.88117	0.940585	−	0.339558i	$-0.110278\pi$
$157$	23.5710	1.88117	0.940585	+	0.339558i	$0.110278\pi$
$158$	0.816562	0.0649622
$159$	−4.72157	−0.374445
$160$	31.3615	2.47934
$161$	−6.75927	−0.532705
$162$	2.09979	0.164975
$163$	−13.2428	−1.03725	−0.518626	−	0.855001i	$-0.673556\pi$
$163$	−13.2428	−1.03725	−0.518626	+	0.855001i	$0.673556\pi$
$164$	−24.4060	−1.90579
$165$	−11.4090	−0.888188
$166$	18.0484	1.40083
$167$	6.59074	0.510007	0.255003	−	0.966940i	$-0.417923\pi$
$167$	6.59074	0.510007	0.255003	+	0.966940i	$0.417923\pi$
$168$	−0.859065	−0.0662783
$169$	−1.33437	−0.102644
$170$	54.5319	4.18240
$171$	−6.95886	−0.532157
$172$	6.68791	0.509948
$173$	−13.4859	−1.02531	−0.512656	−	0.858594i	$-0.671338\pi$
$173$	−13.4859	−1.02531	−0.512656	+	0.858594i	$0.671338\pi$
$174$	−17.0727	−1.29428
$175$	10.1862	0.770005
$176$	8.82511	0.665218
$177$	−6.57798	−0.494431
$178$	3.17015	0.237613
$179$	−20.3463	−1.52075	−0.760376	−	0.649483i	$-0.774985\pi$
$179$	−20.3463	−1.52075	−0.760376	+	0.649483i	$0.774985\pi$
$180$	−9.38822	−0.699756
$181$	23.1847	1.72331	0.861654	−	0.507497i	$-0.169429\pi$
$181$	23.1847	1.72331	0.861654	+	0.507497i	$0.169429\pi$
$182$	7.17183	0.531612
$183$	−4.29383	−0.317409
$184$	−5.80666	−0.428072
$185$	−3.89695	−0.286509
$186$	−2.35359	−0.172574
$187$	19.5107	1.42676
$188$	2.82308	0.205894
$189$	−1.00000	−0.0727393
$190$	56.9428	4.13106
$191$	−23.6755	−1.71310	−0.856550	−	0.516063i	$-0.827397\pi$
$191$	−23.6755	−1.71310	−0.856550	+	0.516063i	$0.827397\pi$
$192$	10.8697	0.784455
$193$	0.609308	0.0438589	0.0219295	−	0.999760i	$-0.493019\pi$
$193$	0.609308	0.0438589	0.0219295	+	0.999760i	$0.493019\pi$
$194$	−31.4064	−2.25485
$195$	13.3100	0.953151
$196$	2.40912	0.172080
$197$	18.9396	1.34939	0.674696	−	0.738095i	$-0.264275\pi$
$197$	18.9396	1.34939	0.674696	+	0.738095i	$0.264275\pi$
$198$	−6.14749	−0.436883
$199$	−22.6136	−1.60304	−0.801519	−	0.597970i	$-0.795974\pi$
$199$	−22.6136	−1.60304	−0.801519	+	0.597970i	$0.795974\pi$
$200$	8.75062	0.618762
$201$	−2.59088	−0.182747
$202$	7.15335	0.503308
$203$	8.13069	0.570662
$204$	16.0549	1.12407
$205$	39.4788	2.75732
$206$	−11.0212	−0.767885
$207$	−6.75927	−0.469802
$208$	−10.2956	−0.713873
$209$	20.3732	1.40925
$210$	8.18278	0.564665
$211$	−13.1063	−0.902278	−0.451139	−	0.892454i	$-0.648982\pi$
$211$	−13.1063	−0.902278	−0.451139	+	0.892454i	$0.648982\pi$
$212$	11.3748	0.781226
$213$	−3.11790	−0.213635
$214$	−13.6046	−0.929991
$215$	−10.8182	−0.737798
$216$	−0.859065	−0.0584520
$217$	1.12087	0.0760896
$218$	−30.4331	−2.06119
$219$	3.37793	0.228260
$220$	27.4856	1.85308
$221$	−22.7617	−1.53111
$222$	−2.09979	−0.140929
$223$	−2.22291	−0.148857	−0.0744284	−	0.997226i	$-0.523713\pi$
$223$	−2.22291	−0.148857	−0.0744284	+	0.997226i	$0.523713\pi$
$224$	−8.04770	−0.537710
$225$	10.1862	0.679081
$226$	−8.90844	−0.592581
$227$	−6.96573	−0.462332	−0.231166	−	0.972914i	$-0.574254\pi$
$227$	−6.96573	−0.462332	−0.231166	+	0.972914i	$0.574254\pi$
$228$	16.7647	1.11027
$229$	12.4160	0.820470	0.410235	−	0.911980i	$-0.365447\pi$
$229$	12.4160	0.820470	0.410235	+	0.911980i	$0.365447\pi$
$230$	55.3096	3.64701
$231$	2.92767	0.192626
$232$	6.98479	0.458574
$233$	1.10797	0.0725855	0.0362928	−	0.999341i	$-0.488445\pi$
$233$	1.10797	0.0725855	0.0362928	+	0.999341i	$0.488445\pi$
$234$	7.17183	0.468837
$235$	−4.56657	−0.297890
$236$	15.8471	1.03156
$237$	−0.388878	−0.0252603
$238$	−13.9935	−0.907063
$239$	20.0906	1.29955	0.649775	−	0.760127i	$-0.274863\pi$
$239$	20.0906	1.29955	0.649775	+	0.760127i	$0.274863\pi$
$240$	−11.7469	−0.758259
$241$	22.4505	1.44616	0.723081	−	0.690763i	$-0.242725\pi$
$241$	22.4505	1.44616	0.723081	+	0.690763i	$0.242725\pi$
$242$	−5.09987	−0.327832
$243$	−1.00000	−0.0641500
$244$	10.3443	0.662229
$245$	−3.89695	−0.248967
$246$	21.2723	1.35627
$247$	−23.7680	−1.51232
$248$	0.962901	0.0611443
$249$	−8.59533	−0.544707
$250$	−42.4376	−2.68399
$251$	−4.22291	−0.266547	−0.133274	−	0.991079i	$-0.542549\pi$
$251$	−4.22291	−0.266547	−0.133274	+	0.991079i	$0.542549\pi$
$252$	2.40912	0.151760
$253$	19.7889	1.24412
$254$	−40.9942	−2.57221
$255$	−25.9702	−1.62631
$256$	7.61051	0.475657
$257$	1.98355	0.123730	0.0618652	−	0.998085i	$-0.480295\pi$
$257$	1.98355	0.123730	0.0618652	+	0.998085i	$0.480295\pi$
$258$	−5.82918	−0.362909
$259$	1.00000	0.0621370
$260$	−32.0654	−1.98861
$261$	8.13069	0.503277
$262$	31.0494	1.91824
$263$	−19.3193	−1.19128	−0.595640	−	0.803251i	$-0.703102\pi$
$263$	−19.3193	−1.19128	−0.595640	+	0.803251i	$0.703102\pi$
$264$	2.51506	0.154791
$265$	−18.3997	−1.13029
$266$	−14.6121	−0.895928
$267$	−1.50975	−0.0923951
$268$	6.24174	0.381275
$269$	−21.7185	−1.32420	−0.662100	−	0.749416i	$-0.730334\pi$
$269$	−21.7185	−1.32420	−0.662100	+	0.749416i	$0.730334\pi$
$270$	8.18278	0.497988
$271$	22.6932	1.37851	0.689257	−	0.724517i	$-0.257937\pi$
$271$	22.6932	1.37851	0.689257	+	0.724517i	$0.257937\pi$
$272$	20.0885	1.21805
$273$	−3.41550	−0.206715
$274$	25.9140	1.56552
$275$	−29.8219	−1.79833
$276$	16.2839	0.980176
$277$	6.51426	0.391404	0.195702	−	0.980663i	$-0.437301\pi$
$277$	6.51426	0.391404	0.195702	+	0.980663i	$0.437301\pi$
$278$	−25.2740	−1.51583
$279$	1.12087	0.0671048
$280$	−3.34773	−0.200065
$281$	−10.0324	−0.598482	−0.299241	−	0.954178i	$-0.596734\pi$
$281$	−10.0324	−0.598482	−0.299241	+	0.954178i	$0.596734\pi$
$282$	−2.46060	−0.146527
$283$	−15.5891	−0.926678	−0.463339	−	0.886181i	$-0.653349\pi$
$283$	−15.5891	−0.926678	−0.463339	+	0.886181i	$0.653349\pi$
$284$	7.51139	0.445719
$285$	−27.1183	−1.60635
$286$	−20.9968	−1.24156
$287$	−10.1307	−0.597996
$288$	−8.04770	−0.474215
$289$	27.4119	1.61247
$290$	−66.5316	−3.90687
$291$	14.9569	0.876791
$292$	−8.13784	−0.476231
$293$	25.6660	1.49943	0.749713	−	0.661763i	$-0.230192\pi$
$293$	25.6660	1.49943	0.749713	+	0.661763i	$0.230192\pi$
$294$	−2.09979	−0.122462
$295$	−25.6341	−1.49247
$296$	0.859065	0.0499321
$297$	2.92767	0.169881
$298$	−38.9088	−2.25393
$299$	−23.0863	−1.33511
$300$	−24.5398	−1.41681
$301$	2.77608	0.160011
$302$	−18.8419	−1.08423
$303$	−3.40670	−0.195710
$304$	20.9767	1.20309
$305$	−16.7328	−0.958120
$306$	−13.9935	−0.799954
$307$	24.5635	1.40191	0.700955	−	0.713205i	$-0.252757\pi$
$307$	24.5635	1.40191	0.700955	+	0.713205i	$0.252757\pi$
$308$	−7.05310	−0.401888
$309$	5.24873	0.298590
$310$	−9.17183	−0.520925
$311$	10.6276	0.602636	0.301318	−	0.953524i	$-0.402574\pi$
$311$	10.6276	0.602636	0.301318	+	0.953524i	$0.402574\pi$
$312$	−2.93414	−0.166113
$313$	−20.5665	−1.16249	−0.581243	−	0.813730i	$-0.697433\pi$
$313$	−20.5665	−1.16249	−0.581243	+	0.813730i	$0.697433\pi$
$314$	49.4941	2.79312
$315$	−3.89695	−0.219568
$316$	0.936853	0.0527021
$317$	3.89645	0.218847	0.109423	−	0.993995i	$-0.465100\pi$
$317$	3.89645	0.218847	0.109423	+	0.993995i	$0.465100\pi$
$318$	−9.91430	−0.555966
$319$	−23.8040	−1.33277
$320$	42.3588	2.36793
$321$	6.47903	0.361624
$322$	−14.1931	−0.790948
$323$	46.3754	2.58040
$324$	2.40912	0.133840
$325$	34.7910	1.92986
$326$	−27.8070	−1.54009
$327$	14.4934	0.801487
$328$	−8.70292	−0.480538
$329$	1.17183	0.0646052
$330$	−23.9565	−1.31876
$331$	17.7665	0.976537	0.488268	−	0.872694i	$-0.337629\pi$
$331$	17.7665	0.976537	0.488268	+	0.872694i	$0.337629\pi$
$332$	20.7072	1.13645
$333$	1.00000	0.0547997
$334$	13.8392	0.757246
$335$	−10.0965	−0.551632
$336$	3.01438	0.164448
$337$	−14.8998	−0.811646	−0.405823	−	0.913952i	$-0.633015\pi$
$337$	−14.8998	−0.811646	−0.405823	+	0.913952i	$0.633015\pi$
$338$	−2.80189	−0.152403
$339$	4.24254	0.230423
$340$	62.5652	3.39307
$341$	−3.28154	−0.177705
$342$	−14.6121	−0.790134
$343$	1.00000	0.0539949
$344$	2.38483	0.128582
$345$	−26.3406	−1.41813
$346$	−28.3175	−1.52236
$347$	−28.9179	−1.55240	−0.776198	−	0.630489i	$-0.782854\pi$
$347$	−28.9179	−1.55240	−0.776198	+	0.630489i	$0.782854\pi$
$348$	−19.5878	−1.05002
$349$	−4.65850	−0.249364	−0.124682	−	0.992197i	$-0.539791\pi$
$349$	−4.65850	−0.249364	−0.124682	+	0.992197i	$0.539791\pi$
$350$	21.3889	1.14329
$351$	−3.41550	−0.182306
$352$	23.5610	1.25581
$353$	−12.1478	−0.646563	−0.323282	−	0.946303i	$-0.604786\pi$
$353$	−12.1478	−0.646563	−0.323282	+	0.946303i	$0.604786\pi$
$354$	−13.8124	−0.734120
$355$	−12.1503	−0.644871
$356$	3.63716	0.192769
$357$	6.66423	0.352708
$358$	−42.7229	−2.25798
$359$	26.8993	1.41969	0.709846	−	0.704357i	$-0.248765\pi$
$359$	26.8993	1.41969	0.709846	+	0.704357i	$0.248765\pi$
$360$	−3.34773	−0.176441
$361$	29.4257	1.54872
$362$	48.6831	2.55872
$363$	2.42875	0.127476
$364$	8.22835	0.431282
$365$	13.1636	0.689016
$366$	−9.01614	−0.471281
$367$	−1.18943	−0.0620880	−0.0310440	−	0.999518i	$-0.509883\pi$
$367$	−1.18943	−0.0620880	−0.0310440	+	0.999518i	$0.509883\pi$
$368$	20.3750	1.06212
$369$	−10.1307	−0.527383
$370$	−8.18278	−0.425402
$371$	4.72157	0.245132
$372$	−2.70031	−0.140005
$373$	−19.2682	−0.997671	−0.498835	−	0.866697i	$-0.666239\pi$
$373$	−19.2682	−0.997671	−0.498835	+	0.866697i	$0.666239\pi$
$374$	40.9683	2.11842
$375$	20.2104	1.04366
$376$	1.00668	0.0519155
$377$	27.7704	1.43025
$378$	−2.09979	−0.108002
$379$	7.37631	0.378896	0.189448	−	0.981891i	$-0.439330\pi$
$379$	7.37631	0.378896	0.189448	+	0.981891i	$0.439330\pi$
$380$	65.3312	3.35142
$381$	19.5230	1.00019
$382$	−49.7136	−2.54357
$383$	−9.32925	−0.476703	−0.238351	−	0.971179i	$-0.576607\pi$
$383$	−9.32925	−0.476703	−0.238351	+	0.971179i	$0.576607\pi$
$384$	6.72873	0.343374
$385$	11.4090	0.581455
$386$	1.27942	0.0651207
$387$	2.77608	0.141116
$388$	−36.0330	−1.82930
$389$	−7.61684	−0.386189	−0.193095	−	0.981180i	$-0.561852\pi$
$389$	−7.61684	−0.386189	−0.193095	+	0.981180i	$0.561852\pi$
$390$	27.9483	1.41522
$391$	45.0453	2.27804
$392$	0.859065	0.0433893
$393$	−14.7869	−0.745901
$394$	39.7693	2.00355
$395$	−1.51544	−0.0762499
$396$	−7.05310	−0.354432
$397$	6.41503	0.321961	0.160981	−	0.986958i	$-0.448534\pi$
$397$	6.41503	0.321961	0.160981	+	0.986958i	$0.448534\pi$
$398$	−47.4839	−2.38015
$399$	6.95886	0.348379
$400$	−30.7051	−1.53526
$401$	26.4734	1.32202	0.661008	−	0.750379i	$-0.270129\pi$
$401$	26.4734	1.32202	0.661008	+	0.750379i	$0.270129\pi$
$402$	−5.44031	−0.271338
$403$	3.82833	0.190703
$404$	8.20714	0.408321
$405$	−3.89695	−0.193641
$406$	17.0727	0.847306
$407$	−2.92767	−0.145119
$408$	5.72501	0.283430
$409$	3.84962	0.190351	0.0951756	−	0.995460i	$-0.469659\pi$
$409$	3.84962	0.190351	0.0951756	+	0.995460i	$0.469659\pi$
$410$	82.8972	4.09400
$411$	−12.3412	−0.608749
$412$	−12.6448	−0.622965
$413$	6.57798	0.323681
$414$	−14.1931	−0.697551
$415$	−33.4956	−1.64423
$416$	−27.4869	−1.34766
$417$	12.0364	0.589427
$418$	42.7795	2.09241
$419$	−28.8855	−1.41115	−0.705574	−	0.708636i	$-0.749311\pi$
$419$	−28.8855	−1.41115	−0.705574	+	0.708636i	$0.749311\pi$
$420$	9.38822	0.458098
$421$	−20.0485	−0.977106	−0.488553	−	0.872534i	$-0.662475\pi$
$421$	−20.0485	−0.977106	−0.488553	+	0.872534i	$0.662475\pi$
$422$	−27.5206	−1.33968
$423$	1.17183	0.0569764
$424$	4.05613	0.196983
$425$	−67.8832	−3.29282
$426$	−6.54694	−0.317200
$427$	4.29383	0.207793
$428$	−15.6088	−0.754478
$429$	9.99945	0.482778
$430$	−22.7160	−1.09546
$431$	8.06179	0.388323	0.194161	−	0.980970i	$-0.437801\pi$
$431$	8.06179	0.388323	0.194161	+	0.980970i	$0.437801\pi$
$432$	3.01438	0.145030
$433$	−23.9801	−1.15241	−0.576205	−	0.817305i	$-0.695467\pi$
$433$	−23.9801	−1.15241	−0.576205	+	0.817305i	$0.695467\pi$
$434$	2.35359	0.112976
$435$	31.6849	1.51917
$436$	−34.9164	−1.67219
$437$	47.0368	2.25008
$438$	7.09295	0.338914
$439$	−34.7382	−1.65797	−0.828983	−	0.559274i	$-0.811080\pi$
$439$	−34.7382	−1.65797	−0.828983	+	0.559274i	$0.811080\pi$
$440$	9.80106	0.467247
$441$	1.00000	0.0476190
$442$	−47.7947	−2.27336
$443$	14.4355	0.685851	0.342925	−	0.939363i	$-0.388582\pi$
$443$	14.4355	0.685851	0.342925	+	0.939363i	$0.388582\pi$
$444$	−2.40912	−0.114332
$445$	−5.88341	−0.278901
$446$	−4.66764	−0.221019
$447$	18.5298	0.876432
$448$	−10.8697	−0.513546
$449$	35.1708	1.65981	0.829905	−	0.557904i	$-0.188394\pi$
$449$	35.1708	1.65981	0.829905	+	0.557904i	$0.188394\pi$
$450$	21.3889	1.00828
$451$	29.6593	1.39660
$452$	−10.2208	−0.480745
$453$	8.97324	0.421599
$454$	−14.6266	−0.686459
$455$	−13.3100	−0.623984
$456$	5.97811	0.279951
$457$	26.5525	1.24208	0.621038	−	0.783781i	$-0.286711\pi$
$457$	26.5525	1.24208	0.621038	+	0.783781i	$0.286711\pi$
$458$	26.0709	1.21821
$459$	6.66423	0.311059
$460$	63.4575	2.95872
$461$	−17.2253	−0.802261	−0.401130	−	0.916021i	$-0.631383\pi$
$461$	−17.2253	−0.802261	−0.401130	+	0.916021i	$0.631383\pi$
$462$	6.14749	0.286007
$463$	−13.4605	−0.625563	−0.312782	−	0.949825i	$-0.601261\pi$
$463$	−13.4605	−0.625563	−0.312782	+	0.949825i	$0.601261\pi$
$464$	−24.5090	−1.13780
$465$	4.36798	0.202560
$466$	2.32651	0.107773
$467$	−40.6969	−1.88323	−0.941615	−	0.336691i	$-0.890692\pi$
$467$	−40.6969	−1.88323	−0.941615	+	0.336691i	$0.890692\pi$
$468$	8.22835	0.380355
$469$	2.59088	0.119636
$470$	−9.58884	−0.442300
$471$	−23.5710	−1.08609
$472$	5.65091	0.260104
$473$	−8.12744	−0.373700
$474$	−0.816562	−0.0375059
$475$	−70.8844	−3.25240
$476$	−16.0549	−0.735876
$477$	4.72157	0.216186
$478$	42.1860	1.92954
$479$	−1.74577	−0.0797661	−0.0398831	−	0.999204i	$-0.512699\pi$
$479$	−1.74577	−0.0797661	−0.0398831	+	0.999204i	$0.512699\pi$
$480$	−31.3615	−1.43145
$481$	3.41550	0.155733
$482$	47.1413	2.14723
$483$	6.75927	0.307558
$484$	−5.85116	−0.265962
$485$	58.2864	2.64665
$486$	−2.09979	−0.0952484
$487$	−38.2520	−1.73336	−0.866682	−	0.498860i	$-0.833752\pi$
$487$	−38.2520	−1.73336	−0.866682	+	0.498860i	$0.833752\pi$
$488$	3.68868	0.166979
$489$	13.2428	0.598858
$490$	−8.18278	−0.369660
$491$	27.3112	1.23254	0.616269	−	0.787535i	$-0.288643\pi$
$491$	27.3112	1.23254	0.616269	+	0.787535i	$0.288643\pi$
$492$	24.4060	1.10031
$493$	−54.1847	−2.44036
$494$	−49.9077	−2.24546
$495$	11.4090	0.512795
$496$	−3.37873	−0.151710
$497$	3.11790	0.139857
$498$	−18.0484	−0.808768
$499$	29.1149	1.30336	0.651681	−	0.758493i	$-0.274064\pi$
$499$	29.1149	1.30336	0.651681	+	0.758493i	$0.274064\pi$
$500$	−48.6893	−2.17745
$501$	−6.59074	−0.294453
$502$	−8.86722	−0.395763
$503$	−0.0182862	−0.000815343 0	−0.000407671	−	1.00000i	$-0.500130\pi$
$503$	−0.0182862	−0.000815343 0	−0.000407671		1.00000i	$0.500130\pi$
$504$	0.859065	0.0382658
$505$	−13.2757	−0.590762
$506$	41.5526	1.84724
$507$	1.33437	0.0592613
$508$	−47.0332	−2.08676
$509$	−2.30519	−0.102176	−0.0510880	−	0.998694i	$-0.516269\pi$
$509$	−2.30519	−0.102176	−0.0510880	+	0.998694i	$0.516269\pi$
$510$	−54.5319	−2.41471
$511$	−3.37793	−0.149431
$512$	29.4379	1.30099
$513$	6.95886	0.307241
$514$	4.16504	0.183712
$515$	20.4540	0.901312
$516$	−6.68791	−0.294419
$517$	−3.43074	−0.150884
$518$	2.09979	0.0922595
$519$	13.4859	0.591964
$520$	−11.4342	−0.501422
$521$	−9.96144	−0.436419	−0.218209	−	0.975902i	$-0.570022\pi$
$521$	−9.96144	−0.436419	−0.218209	+	0.975902i	$0.570022\pi$
$522$	17.0727	0.747253
$523$	−0.331454	−0.0144935	−0.00724673	−	0.999974i	$-0.502307\pi$
$523$	−0.331454	−0.0144935	−0.00724673	+	0.999974i	$0.502307\pi$
$524$	35.6234	1.55622
$525$	−10.1862	−0.444563
$526$	−40.5665	−1.76878
$527$	−7.46973	−0.325387
$528$	−8.82511	−0.384064
$529$	22.6878	0.986426
$530$	−38.6355	−1.67822
$531$	6.57798	0.285460
$532$	−16.7647	−0.726843
$533$	−34.6014	−1.49875
$534$	−3.17015	−0.137186
$535$	25.2484	1.09159
$536$	2.22574	0.0961371
$537$	20.3463	0.878007
$538$	−45.6043	−1.96614
$539$	−2.92767	−0.126104
$540$	9.38822	0.404005
$541$	18.1572	0.780639	0.390319	−	0.920680i	$-0.372365\pi$
$541$	18.1572	0.780639	0.390319	+	0.920680i	$0.372365\pi$
$542$	47.6510	2.04679
$543$	−23.1847	−0.994952
$544$	53.6317	2.29944
$545$	56.4801	2.41934
$546$	−7.17183	−0.306926
$547$	20.6839	0.884378	0.442189	−	0.896922i	$-0.354202\pi$
$547$	20.6839	0.884378	0.442189	+	0.896922i	$0.354202\pi$
$548$	29.7315	1.27007
$549$	4.29383	0.183256
$550$	−62.6197	−2.67011
$551$	−56.5803	−2.41040
$552$	5.80666	0.247148
$553$	0.388878	0.0165368
$554$	13.6786	0.581147
$555$	3.89695	0.165416
$556$	−28.9972	−1.22976
$557$	−45.1618	−1.91356	−0.956782	−	0.290805i	$-0.906077\pi$
$557$	−45.1618	−1.91356	−0.956782	+	0.290805i	$0.906077\pi$
$558$	2.35359	0.0996355
$559$	9.48169	0.401033
$560$	11.7469	0.496397
$561$	−19.5107	−0.823740
$562$	−21.0659	−0.888612
$563$	15.1369	0.637943	0.318972	−	0.947764i	$-0.396663\pi$
$563$	15.1369	0.637943	0.318972	+	0.947764i	$0.396663\pi$
$564$	−2.82308	−0.118873
$565$	16.5330	0.695547
$566$	−32.7339	−1.37591
$567$	1.00000	0.0419961
$568$	2.67848	0.112386
$569$	−31.0528	−1.30180	−0.650901	−	0.759163i	$-0.725609\pi$
$569$	−31.0528	−1.30180	−0.650901	+	0.759163i	$0.725609\pi$
$570$	−56.9428	−2.38507
$571$	−8.52619	−0.356810	−0.178405	−	0.983957i	$-0.557094\pi$
$571$	−8.52619	−0.356810	−0.178405	+	0.983957i	$0.557094\pi$
$572$	−24.0899	−1.00725
$573$	23.6755	0.989059
$574$	−21.2723	−0.887889
$575$	−68.8514	−2.87130
$576$	−10.8697	−0.452905
$577$	34.7713	1.44755	0.723773	−	0.690038i	$-0.242406\pi$
$577$	34.7713	1.44755	0.723773	+	0.690038i	$0.242406\pi$
$578$	57.5593	2.39415
$579$	−0.609308	−0.0253220
$580$	−76.3327	−3.16954
$581$	8.59533	0.356595
$582$	31.4064	1.30184
$583$	−13.8232	−0.572498
$584$	−2.90186	−0.120080
$585$	−13.3100	−0.550302
$586$	53.8933	2.22631
$587$	−12.6167	−0.520748	−0.260374	−	0.965508i	$-0.583846\pi$
$587$	−12.6167	−0.520748	−0.260374	+	0.965508i	$0.583846\pi$
$588$	−2.40912	−0.0993504
$589$	−7.79997	−0.321392
$590$	−53.8261	−2.21599
$591$	−18.9396	−0.779072
$592$	−3.01438	−0.123890
$593$	15.2788	0.627427	0.313713	−	0.949518i	$-0.398427\pi$
$593$	15.2788	0.627427	0.313713	+	0.949518i	$0.398427\pi$
$594$	6.14749	0.252235
$595$	25.9702	1.06467
$596$	−44.6406	−1.82855
$597$	22.6136	0.925514
$598$	−48.4764	−1.98235
$599$	0.764997	0.0312569	0.0156285	−	0.999878i	$-0.495025\pi$
$599$	0.764997	0.0312569	0.0156285	+	0.999878i	$0.495025\pi$
$600$	−8.75062	−0.357243
$601$	−7.52848	−0.307093	−0.153547	−	0.988141i	$-0.549069\pi$
$601$	−7.52848	−0.307093	−0.153547	+	0.988141i	$0.549069\pi$
$602$	5.82918	0.237580
$603$	2.59088	0.105509
$604$	−21.6176	−0.879608
$605$	9.46473	0.384796
$606$	−7.15335	−0.290585
$607$	−10.7946	−0.438138	−0.219069	−	0.975709i	$-0.570302\pi$
$607$	−10.7946	−0.438138	−0.219069	+	0.975709i	$0.570302\pi$
$608$	56.0028	2.27121
$609$	−8.13069	−0.329472
$610$	−35.1355	−1.42259
$611$	4.00239	0.161919
$612$	−16.0549	−0.648982
$613$	22.1620	0.895115	0.447558	−	0.894255i	$-0.352294\pi$
$613$	22.1620	0.895115	0.447558	+	0.894255i	$0.352294\pi$
$614$	51.5781	2.08152
$615$	−39.4788	−1.59194
$616$	−2.51506	−0.101335
$617$	−26.8273	−1.08002	−0.540012	−	0.841657i	$-0.681580\pi$
$617$	−26.8273	−1.08002	−0.540012	+	0.841657i	$0.681580\pi$
$618$	11.0212	0.443339
$619$	−29.5998	−1.18971	−0.594857	−	0.803831i	$-0.702791\pi$
$619$	−29.5998	−1.18971	−0.594857	+	0.803831i	$0.702791\pi$
$620$	−10.5230	−0.422613
$621$	6.75927	0.271240
$622$	22.3157	0.894779
$623$	1.50975	0.0604868
$624$	10.2956	0.412155
$625$	27.8279	1.11312
$626$	−43.1853	−1.72603
$627$	−20.3732	−0.813628
$628$	56.7853	2.26598
$629$	−6.66423	−0.265720
$630$	−8.18278	−0.326010
$631$	−12.1013	−0.481745	−0.240872	−	0.970557i	$-0.577433\pi$
$631$	−12.1013	−0.481745	−0.240872	+	0.970557i	$0.577433\pi$
$632$	0.334071	0.0132886
$633$	13.1063	0.520931
$634$	8.18174	0.324938
$635$	76.0802	3.01915
$636$	−11.3748	−0.451041
$637$	3.41550	0.135327
$638$	−49.9833	−1.97886
$639$	3.11790	0.123342
$640$	26.2215	1.03650
$641$	49.4792	1.95431	0.977155	−	0.212527i	$-0.0681695\pi$
$641$	49.4792	1.95431	0.977155	+	0.212527i	$0.0681695\pi$
$642$	13.6046	0.536931
$643$	8.16122	0.321847	0.160923	−	0.986967i	$-0.448553\pi$
$643$	8.16122	0.321847	0.160923	+	0.986967i	$0.448553\pi$
$644$	−16.2839	−0.641676
$645$	10.8182	0.425968
$646$	97.3786	3.83131
$647$	−23.6826	−0.931059	−0.465529	−	0.885032i	$-0.654136\pi$
$647$	−23.6826	−0.931059	−0.465529	+	0.885032i	$0.654136\pi$
$648$	0.859065	0.0337473
$649$	−19.2582	−0.755949
$650$	73.0538	2.86541
$651$	−1.12087	−0.0439304
$652$	−31.9034	−1.24943
$653$	−37.9906	−1.48669	−0.743343	−	0.668910i	$-0.766761\pi$
$653$	−37.9906	−1.48669	−0.743343	+	0.668910i	$0.766761\pi$
$654$	30.4331	1.19003
$655$	−57.6238	−2.25155
$656$	30.5378	1.19230
$657$	−3.37793	−0.131786
$658$	2.46060	0.0959242
$659$	29.0926	1.13329	0.566643	−	0.823963i	$-0.308242\pi$
$659$	29.0926	1.13329	0.566643	+	0.823963i	$0.308242\pi$
$660$	−27.4856	−1.06988
$661$	−4.58100	−0.178180	−0.0890902	−	0.996024i	$-0.528396\pi$
$661$	−4.58100	−0.178180	−0.0890902	+	0.996024i	$0.528396\pi$
$662$	37.3060	1.44994
$663$	22.7617	0.883990
$664$	7.38395	0.286553
$665$	27.1183	1.05160
$666$	2.09979	0.0813652
$667$	−54.9575	−2.12796
$668$	15.8779	0.614334
$669$	2.22291	0.0859425
$670$	−21.2006	−0.819050
$671$	−12.5709	−0.485295
$672$	8.04770	0.310447
$673$	20.0873	0.774307	0.387153	−	0.922015i	$-0.373458\pi$
$673$	20.0873	0.774307	0.387153	+	0.922015i	$0.373458\pi$
$674$	−31.2865	−1.20511
$675$	−10.1862	−0.392068
$676$	−3.21465	−0.123640
$677$	35.7293	1.37319	0.686595	−	0.727040i	$-0.259105\pi$
$677$	35.7293	1.37319	0.686595	+	0.727040i	$0.259105\pi$
$678$	8.90844	0.342127
$679$	−14.9569	−0.573994
$680$	22.3101	0.855552
$681$	6.96573	0.266927
$682$	−6.89054	−0.263852
$683$	0.902205	0.0345219	0.0172610	−	0.999851i	$-0.494505\pi$
$683$	0.902205	0.0345219	0.0172610	+	0.999851i	$0.494505\pi$
$684$	−16.7647	−0.641015
$685$	−48.0932	−1.83755
$686$	2.09979	0.0801704
$687$	−12.4160	−0.473699
$688$	−8.36816	−0.319033
$689$	16.1265	0.614371
$690$	−55.3096	−2.10560
$691$	19.7472	0.751221	0.375610	−	0.926778i	$-0.377433\pi$
$691$	19.7472	0.751221	0.375610	+	0.926778i	$0.377433\pi$
$692$	−32.4891	−1.23505
$693$	−2.92767	−0.111213
$694$	−60.7216	−2.30496
$695$	46.9054	1.77922
$696$	−6.98479	−0.264758
$697$	67.5132	2.55724
$698$	−9.78188	−0.370250
$699$	−1.10797	−0.0419073
$700$	24.5398	0.927518
$701$	16.7548	0.632820	0.316410	−	0.948623i	$-0.397523\pi$
$701$	16.7548	0.632820	0.316410	+	0.948623i	$0.397523\pi$
$702$	−7.17183	−0.270683
$703$	−6.95886	−0.262458
$704$	31.8229	1.19937
$705$	4.56657	0.171987
$706$	−25.5079	−0.960002
$707$	3.40670	0.128122
$708$	−15.8471	−0.595572
$709$	−31.5211	−1.18380	−0.591900	−	0.806012i	$-0.701622\pi$
$709$	−31.5211	−1.18380	−0.591900	+	0.806012i	$0.701622\pi$
$710$	−25.5131	−0.957489
$711$	0.388878	0.0145841
$712$	1.29697	0.0486061
$713$	−7.57627	−0.283734
$714$	13.9935	0.523693
$715$	38.9674	1.45730
$716$	−49.0166	−1.83184
$717$	−20.0906	−0.750296
$718$	56.4829	2.10792
$719$	−25.7959	−0.962026	−0.481013	−	0.876714i	$-0.659731\pi$
$719$	−25.7959	−0.962026	−0.481013	+	0.876714i	$0.659731\pi$
$720$	11.7469	0.437781
$721$	−5.24873	−0.195473
$722$	61.7877	2.29950
$723$	−22.4505	−0.834942
$724$	55.8548	2.07583
$725$	82.8209	3.07589
$726$	5.09987	0.189274
$727$	−48.8955	−1.81344	−0.906718	−	0.421738i	$-0.861420\pi$
$727$	−48.8955	−1.81344	−0.906718	+	0.421738i	$0.861420\pi$
$728$	2.93414	0.108746
$729$	1.00000	0.0370370
$730$	27.6409	1.02303
$731$	−18.5004	−0.684263
$732$	−10.3443	−0.382338
$733$	−30.5168	−1.12716	−0.563582	−	0.826060i	$-0.690577\pi$
$733$	−30.5168	−1.12716	−0.563582	+	0.826060i	$0.690577\pi$
$734$	−2.49756	−0.0921867
$735$	3.89695	0.143741
$736$	54.3966	2.00509
$737$	−7.58524	−0.279406
$738$	−21.2723	−0.783045
$739$	4.68593	0.172375	0.0861873	−	0.996279i	$-0.472532\pi$
$739$	4.68593	0.172375	0.0861873	+	0.996279i	$0.472532\pi$
$740$	−9.38822	−0.345118
$741$	23.7680	0.873138
$742$	9.91430	0.363965
$743$	10.7108	0.392942	0.196471	−	0.980510i	$-0.437052\pi$
$743$	10.7108	0.392942	0.196471	+	0.980510i	$0.437052\pi$
$744$	−0.962901	−0.0353017
$745$	72.2099	2.64557
$746$	−40.4592	−1.48132
$747$	8.59533	0.314487
$748$	47.0035	1.71862
$749$	−6.47903	−0.236739
$750$	42.4376	1.54960
$751$	37.7954	1.37917	0.689586	−	0.724203i	$-0.257792\pi$
$751$	37.7954	1.37917	0.689586	+	0.724203i	$0.257792\pi$
$752$	−3.53235	−0.128812
$753$	4.22291	0.153891
$754$	58.3119	2.12359
$755$	34.9683	1.27263
$756$	−2.40912	−0.0876188
$757$	46.6004	1.69372	0.846861	−	0.531815i	$-0.178490\pi$
$757$	46.6004	1.69372	0.846861	+	0.531815i	$0.178490\pi$
$758$	15.4887	0.562575
$759$	−19.7889	−0.718292
$760$	23.2964	0.845049
$761$	22.9240	0.830996	0.415498	−	0.909594i	$-0.363607\pi$
$761$	22.9240	0.830996	0.415498	+	0.909594i	$0.363607\pi$
$762$	40.9942	1.48506
$763$	−14.4934	−0.524697
$764$	−57.0372	−2.06353
$765$	25.9702	0.938953
$766$	−19.5895	−0.707796
$767$	22.4671	0.811239
$768$	−7.61051	−0.274621
$769$	16.6161	0.599192	0.299596	−	0.954066i	$-0.403148\pi$
$769$	16.6161	0.599192	0.299596	+	0.954066i	$0.403148\pi$
$770$	23.9565	0.863331
$771$	−1.98355	−0.0714358
$772$	1.46790	0.0528307
$773$	34.4703	1.23981	0.619905	−	0.784677i	$-0.287171\pi$
$773$	34.4703	1.23981	0.619905	+	0.784677i	$0.287171\pi$
$774$	5.82918	0.209526
$775$	11.4174	0.410126
$776$	−12.8490	−0.461251
$777$	−1.00000	−0.0358748
$778$	−15.9938	−0.573405
$779$	70.4980	2.52585
$780$	32.0654	1.14813
$781$	−9.12818	−0.326632
$782$	94.5858	3.38238
$783$	−8.13069	−0.290567
$784$	−3.01438	−0.107657
$785$	−91.8550	−3.27844
$786$	−31.0494	−1.10750
$787$	−36.7648	−1.31053	−0.655263	−	0.755401i	$-0.727442\pi$
$787$	−36.7648	−1.31053	−0.655263	+	0.755401i	$0.727442\pi$
$788$	45.6278	1.62542
$789$	19.3193	0.687786
$790$	−3.18210	−0.113214
$791$	−4.24254	−0.150847
$792$	−2.51506	−0.0893687
$793$	14.6656	0.520790
$794$	13.4702	0.478040
$795$	18.3997	0.652570
$796$	−54.4789	−1.93095
$797$	−44.1041	−1.56225	−0.781124	−	0.624376i	$-0.785353\pi$
$797$	−44.1041	−1.56225	−0.781124	+	0.624376i	$0.785353\pi$
$798$	14.6121	0.517264
$799$	−7.80935	−0.276275
$800$	−81.9756	−2.89828
$801$	1.50975	0.0533443
$802$	55.5885	1.96290
$803$	9.88947	0.348992
$804$	−6.24174	−0.220129
$805$	26.3406	0.928382
$806$	8.03869	0.283151
$807$	21.7185	0.764527
$808$	2.92658	0.102957
$809$	14.8282	0.521331	0.260665	−	0.965429i	$-0.416058\pi$
$809$	14.8282	0.521331	0.260665	+	0.965429i	$0.416058\pi$
$810$	−8.18278	−0.287513
$811$	0.853878	0.0299837	0.0149919	−	0.999888i	$-0.495228\pi$
$811$	0.853878	0.0299837	0.0149919	+	0.999888i	$0.495228\pi$
$812$	19.5878	0.687397
$813$	−22.6932	−0.795886
$814$	−6.14749	−0.215469
$815$	51.6064	1.80769
$816$	−20.0885	−0.703239
$817$	−19.3183	−0.675863
$818$	8.08339	0.282629
$819$	3.41550	0.119347
$820$	95.1091	3.32135
$821$	19.1353	0.667826	0.333913	−	0.942604i	$-0.391631\pi$
$821$	19.1353	0.667826	0.333913	+	0.942604i	$0.391631\pi$
$822$	−25.9140	−0.903855
$823$	1.79712	0.0626436	0.0313218	−	0.999509i	$-0.490028\pi$
$823$	1.79712	0.0626436	0.0313218	+	0.999509i	$0.490028\pi$
$824$	−4.50900	−0.157078
$825$	29.8219	1.03826
$826$	13.8124	0.480594
$827$	13.0782	0.454772	0.227386	−	0.973805i	$-0.426982\pi$
$827$	13.0782	0.454772	0.227386	+	0.973805i	$0.426982\pi$
$828$	−16.2839	−0.565905
$829$	30.2257	1.04978	0.524891	−	0.851170i	$-0.324106\pi$
$829$	30.2257	1.04978	0.524891	+	0.851170i	$0.324106\pi$
$830$	−70.3337	−2.44132
$831$	−6.51426	−0.225977
$832$	−37.1255	−1.28710
$833$	−6.66423	−0.230902
$834$	25.2740	0.875167
$835$	−25.6838	−0.888824
$836$	49.0815	1.69752
$837$	−1.12087	−0.0387429
$838$	−60.6534	−2.09524
$839$	21.7117	0.749573	0.374786	−	0.927111i	$-0.377716\pi$
$839$	21.7117	0.749573	0.374786	+	0.927111i	$0.377716\pi$
$840$	3.34773	0.115508
$841$	37.1081	1.27959
$842$	−42.0977	−1.45078
$843$	10.0324	0.345534
$844$	−31.5748	−1.08685
$845$	5.19996	0.178884
$846$	2.46060	0.0845972
$847$	−2.42875	−0.0834530
$848$	−14.2326	−0.488750
$849$	15.5891	0.535018
$850$	−142.541	−4.88910
$851$	−6.75927	−0.231705
$852$	−7.51139	−0.257336
$853$	−8.33190	−0.285279	−0.142639	−	0.989775i	$-0.545559\pi$
$853$	−8.33190	−0.285279	−0.142639	+	0.989775i	$0.545559\pi$
$854$	9.01614	0.308526
$855$	27.1183	0.927427
$856$	−5.56591	−0.190239
$857$	48.9854	1.67331	0.836655	−	0.547730i	$-0.184508\pi$
$857$	48.9854	1.67331	0.836655	+	0.547730i	$0.184508\pi$
$858$	20.9968	0.716817
$859$	49.0612	1.67395	0.836973	−	0.547244i	$-0.184323\pi$
$859$	49.0612	1.67395	0.836973	+	0.547244i	$0.184323\pi$
$860$	−26.0624	−0.888722
$861$	10.1307	0.345253
$862$	16.9281	0.576573
$863$	−9.00242	−0.306446	−0.153223	−	0.988192i	$-0.548965\pi$
$863$	−9.00242	−0.306446	−0.153223	+	0.988192i	$0.548965\pi$
$864$	8.04770	0.273788
$865$	52.5538	1.78688
$866$	−50.3531	−1.71107
$867$	−27.4119	−0.930958
$868$	2.70031	0.0916545
$869$	−1.13851	−0.0386212
$870$	66.5316	2.25563
$871$	8.84915	0.299842
$872$	−12.4508	−0.421637
$873$	−14.9569	−0.506215
$874$	98.7675	3.34086
$875$	−20.2104	−0.683237
$876$	8.13784	0.274952
$877$	−3.31798	−0.112040	−0.0560202	−	0.998430i	$-0.517841\pi$
$877$	−3.31798	−0.112040	−0.0560202	+	0.998430i	$0.517841\pi$
$878$	−72.9430	−2.46171
$879$	−25.6660	−0.865694
$880$	−34.3910	−1.15932
$881$	−10.7838	−0.363317	−0.181658	−	0.983362i	$-0.558146\pi$
$881$	−10.7838	−0.363317	−0.181658	+	0.983362i	$0.558146\pi$
$882$	2.09979	0.0707036
$883$	−22.2785	−0.749732	−0.374866	−	0.927079i	$-0.622311\pi$
$883$	−22.2785	−0.749732	−0.374866	+	0.927079i	$0.622311\pi$
$884$	−54.8356	−1.84432
$885$	25.6341	0.861679
$886$	30.3115	1.01833
$887$	33.3345	1.11926	0.559632	−	0.828741i	$-0.310942\pi$
$887$	33.3345	1.11926	0.559632	+	0.828741i	$0.310942\pi$
$888$	−0.859065	−0.0288283
$889$	−19.5230	−0.654780
$890$	−12.3539	−0.414105
$891$	−2.92767	−0.0980806
$892$	−5.35524	−0.179307
$893$	−8.15461	−0.272884
$894$	38.9088	1.30130
$895$	79.2884	2.65032
$896$	−6.72873	−0.224791
$897$	23.0863	0.770829
$898$	73.8512	2.46445
$899$	9.11345	0.303950
$900$	24.5398	0.817994
$901$	−31.4656	−1.04827
$902$	62.2783	2.07364
$903$	−2.77608	−0.0923821
$904$	−3.64462	−0.121218
$905$	−90.3497	−3.00333
$906$	18.8419	0.625981
$907$	3.14606	0.104463	0.0522316	−	0.998635i	$-0.483367\pi$
$907$	3.14606	0.104463	0.0522316	+	0.998635i	$0.483367\pi$
$908$	−16.7813	−0.556906
$909$	3.40670	0.112993
$910$	−27.9483	−0.926476
$911$	7.56838	0.250752	0.125376	−	0.992109i	$-0.459986\pi$
$911$	7.56838	0.250752	0.125376	+	0.992109i	$0.459986\pi$
$912$	−20.9767	−0.694606
$913$	−25.1643	−0.832817
$914$	55.7548	1.84420
$915$	16.7328	0.553171
$916$	29.9115	0.988305
$917$	14.7869	0.488307
$918$	13.9935	0.461854
$919$	−18.3766	−0.606189	−0.303095	−	0.952960i	$-0.598020\pi$
$919$	−18.3766	−0.606189	−0.303095	+	0.952960i	$0.598020\pi$
$920$	22.6283	0.746031
$921$	−24.5635	−0.809393
$922$	−36.1695	−1.19118
$923$	10.6492	0.350522
$924$	7.05310	0.232030
$925$	10.1862	0.334921
$926$	−28.2643	−0.928821
$927$	−5.24873	−0.172391
$928$	−65.4333	−2.14795
$929$	−20.7974	−0.682340	−0.341170	−	0.940002i	$-0.610823\pi$
$929$	−20.7974	−0.682340	−0.341170	+	0.940002i	$0.610823\pi$
$930$	9.17183	0.300756
$931$	−6.95886	−0.228067
$932$	2.66923	0.0874336
$933$	−10.6276	−0.347932
$934$	−85.4551	−2.79617
$935$	−76.0320	−2.48651
$936$	2.93414	0.0959052
$937$	−41.2717	−1.34829	−0.674144	−	0.738600i	$-0.735487\pi$
$937$	−41.2717	−1.34829	−0.674144	+	0.738600i	$0.735487\pi$
$938$	5.44031	0.177632
$939$	20.5665	0.671162
$940$	−11.0014	−0.358826
$941$	−4.29914	−0.140148	−0.0700739	−	0.997542i	$-0.522324\pi$
$941$	−4.29914	−0.140148	−0.0700739	+	0.997542i	$0.522324\pi$
$942$	−49.4941	−1.61261
$943$	68.4761	2.22989
$944$	−19.8285	−0.645364
$945$	3.89695	0.126768
$946$	−17.0659	−0.554861
$947$	−40.5297	−1.31704	−0.658519	−	0.752564i	$-0.728817\pi$
$947$	−40.5297	−1.31704	−0.658519	+	0.752564i	$0.728817\pi$
$948$	−0.936853	−0.0304276
$949$	−11.5373	−0.374517
$950$	−148.842	−4.82908
$951$	−3.89645	−0.126351
$952$	−5.72501	−0.185548
$953$	−3.19342	−0.103445	−0.0517225	−	0.998661i	$-0.516471\pi$
$953$	−3.19342	−0.103445	−0.0517225	+	0.998661i	$0.516471\pi$
$954$	9.91430	0.320987
$955$	92.2623	2.98554
$956$	48.4006	1.56539
$957$	23.8040	0.769473
$958$	−3.66574	−0.118435
$959$	12.3412	0.398519
$960$	−42.3588	−1.36712
$961$	−29.7436	−0.959473
$962$	7.17183	0.231229
$963$	−6.47903	−0.208784
$964$	54.0859	1.74199
$965$	−2.37444	−0.0764360
$966$	14.1931	0.456654
$967$	−3.76450	−0.121058	−0.0605290	−	0.998166i	$-0.519279\pi$
$967$	−3.76450	−0.121058	−0.0605290	+	0.998166i	$0.519279\pi$
$968$	−2.08646	−0.0670613
$969$	−46.3754	−1.48979
$970$	122.389	3.92968
$971$	40.0522	1.28534	0.642668	−	0.766145i	$-0.277828\pi$
$971$	40.0522	1.28534	0.642668	+	0.766145i	$0.277828\pi$
$972$	−2.40912	−0.0772725
$973$	−12.0364	−0.385870
$974$	−80.3212	−2.57366
$975$	−34.7910	−1.11420
$976$	−12.9432	−0.414303
$977$	18.0313	0.576872	0.288436	−	0.957499i	$-0.406865\pi$
$977$	18.0313	0.576872	0.288436	+	0.957499i	$0.406865\pi$
$978$	27.8070	0.889170
$979$	−4.42004	−0.141265
$980$	−9.38822	−0.299896
$981$	−14.4934	−0.462739
$982$	57.3479	1.83004
$983$	−47.5628	−1.51702	−0.758509	−	0.651662i	$-0.774072\pi$
$983$	−47.5628	−1.51702	−0.758509	+	0.651662i	$0.774072\pi$
$984$	8.70292	0.277439
$985$	−73.8068	−2.35168
$986$	−113.777	−3.62339
$987$	−1.17183	−0.0372998
$988$	−57.2599	−1.82168
$989$	−18.7643	−0.596669
$990$	23.9565	0.761386
$991$	−33.4805	−1.06354	−0.531771	−	0.846888i	$-0.678473\pi$
$991$	−33.4805	−1.06354	−0.531771	+	0.846888i	$0.678473\pi$
$992$	−9.02043	−0.286399
$993$	−17.7665	−0.563804
$994$	6.54694	0.207656
$995$	88.1241	2.79372
$996$	−20.7072	−0.656132
$997$	9.40975	0.298010	0.149005	−	0.988836i	$-0.452393\pi$
$997$	9.40975	0.298010	0.149005	+	0.988836i	$0.452393\pi$
$998$	61.1352	1.93520
$999$	−1.00000	−0.0316386

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Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	777.2.a.l.1.5	✓	5
3.2	odd	2		2331.2.a.w.1.1		5
7.6	odd	2		5439.2.a.x.1.5		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
777.2.a.l.1.5	✓	5	1.1	even	1	trivial
2331.2.a.w.1.1		5	3.2	odd	2
5439.2.a.x.1.5		5	7.6	odd	2

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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Varieties

Fields

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Properties

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Newspace parameters

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Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	777.2.a.l.1.5

Level	$777$
Weight	$2$
Character	777.1
Self dual	yes
Analytic conductor	$6.204$
Analytic rank	$1$
Dimension	$5$
CM	no
Inner twists	$1$