LMFDB - Embedded newform 77.2.a.d.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [77,2,Mod(1,77)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(77, 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("77.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 77 = 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	77.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:	$0.614848095564$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	77.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} -2.00000 q^{5} -2.76393 q^{6} +1.00000 q^{7} +2.23607 q^{8} -1.47214 q^{9} +O(q^{10})$	\(q+2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} -2.00000 q^{5} -2.76393 q^{6} +1.00000 q^{7} +2.23607 q^{8} -1.47214 q^{9} -4.47214 q^{10} -1.00000 q^{11} -3.70820 q^{12} +3.23607 q^{13} +2.23607 q^{14} +2.47214 q^{15} -1.00000 q^{16} -3.23607 q^{17} -3.29180 q^{18} +6.47214 q^{19} -6.00000 q^{20} -1.23607 q^{21} -2.23607 q^{22} +2.47214 q^{23} -2.76393 q^{24} -1.00000 q^{25} +7.23607 q^{26} +5.52786 q^{27} +3.00000 q^{28} +8.47214 q^{29} +5.52786 q^{30} -2.76393 q^{31} -6.70820 q^{32} +1.23607 q^{33} -7.23607 q^{34} -2.00000 q^{35} -4.41641 q^{36} -8.47214 q^{37} +14.4721 q^{38} -4.00000 q^{39} -4.47214 q^{40} -11.2361 q^{41} -2.76393 q^{42} +8.00000 q^{43} -3.00000 q^{44} +2.94427 q^{45} +5.52786 q^{46} +2.76393 q^{47} +1.23607 q^{48} +1.00000 q^{49} -2.23607 q^{50} +4.00000 q^{51} +9.70820 q^{52} -0.472136 q^{53} +12.3607 q^{54} +2.00000 q^{55} +2.23607 q^{56} -8.00000 q^{57} +18.9443 q^{58} -1.23607 q^{59} +7.41641 q^{60} -7.23607 q^{61} -6.18034 q^{62} -1.47214 q^{63} -13.0000 q^{64} -6.47214 q^{65} +2.76393 q^{66} +14.4721 q^{67} -9.70820 q^{68} -3.05573 q^{69} -4.47214 q^{70} -10.4721 q^{71} -3.29180 q^{72} -0.763932 q^{73} -18.9443 q^{74} +1.23607 q^{75} +19.4164 q^{76} -1.00000 q^{77} -8.94427 q^{78} -8.94427 q^{79} +2.00000 q^{80} -2.41641 q^{81} -25.1246 q^{82} -11.4164 q^{83} -3.70820 q^{84} +6.47214 q^{85} +17.8885 q^{86} -10.4721 q^{87} -2.23607 q^{88} +2.00000 q^{89} +6.58359 q^{90} +3.23607 q^{91} +7.41641 q^{92} +3.41641 q^{93} +6.18034 q^{94} -12.9443 q^{95} +8.29180 q^{96} +17.4164 q^{97} +2.23607 q^{98} +1.47214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{4} - 4 q^{5} - 10 q^{6} + 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 6 * q^4 - 4 * q^5 - 10 * q^6 + 2 * q^7 + 6 * q^9	$ 2 q + 2 q^{3} + 6 q^{4} - 4 q^{5} - 10 q^{6} + 2 q^{7} + 6 q^{9} - 2 q^{11} + 6 q^{12} + 2 q^{13} - 4 q^{15} - 2 q^{16} - 2 q^{17} - 20 q^{18} + 4 q^{19} - 12 q^{20} + 2 q^{21} - 4 q^{23} - 10 q^{24} - 2 q^{25} + 10 q^{26} + 20 q^{27} + 6 q^{28} + 8 q^{29} + 20 q^{30} - 10 q^{31} - 2 q^{33} - 10 q^{34} - 4 q^{35} + 18 q^{36} - 8 q^{37} + 20 q^{38} - 8 q^{39} - 18 q^{41} - 10 q^{42} + 16 q^{43} - 6 q^{44} - 12 q^{45} + 20 q^{46} + 10 q^{47} - 2 q^{48} + 2 q^{49} + 8 q^{51} + 6 q^{52} + 8 q^{53} - 20 q^{54} + 4 q^{55} - 16 q^{57} + 20 q^{58} + 2 q^{59} - 12 q^{60} - 10 q^{61} + 10 q^{62} + 6 q^{63} - 26 q^{64} - 4 q^{65} + 10 q^{66} + 20 q^{67} - 6 q^{68} - 24 q^{69} - 12 q^{71} - 20 q^{72} - 6 q^{73} - 20 q^{74} - 2 q^{75} + 12 q^{76} - 2 q^{77} + 4 q^{80} + 22 q^{81} - 10 q^{82} + 4 q^{83} + 6 q^{84} + 4 q^{85} - 12 q^{87} + 4 q^{89} + 40 q^{90} + 2 q^{91} - 12 q^{92} - 20 q^{93} - 10 q^{94} - 8 q^{95} + 30 q^{96} + 8 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^4 - 4 * q^5 - 10 * q^6 + 2 * q^7 + 6 * q^9 - 2 * q^11 + 6 * q^12 + 2 * q^13 - 4 * q^15 - 2 * q^16 - 2 * q^17 - 20 * q^18 + 4 * q^19 - 12 * q^20 + 2 * q^21 - 4 * q^23 - 10 * q^24 - 2 * q^25 + 10 * q^26 + 20 * q^27 + 6 * q^28 + 8 * q^29 + 20 * q^30 - 10 * q^31 - 2 * q^33 - 10 * q^34 - 4 * q^35 + 18 * q^36 - 8 * q^37 + 20 * q^38 - 8 * q^39 - 18 * q^41 - 10 * q^42 + 16 * q^43 - 6 * q^44 - 12 * q^45 + 20 * q^46 + 10 * q^47 - 2 * q^48 + 2 * q^49 + 8 * q^51 + 6 * q^52 + 8 * q^53 - 20 * q^54 + 4 * q^55 - 16 * q^57 + 20 * q^58 + 2 * q^59 - 12 * q^60 - 10 * q^61 + 10 * q^62 + 6 * q^63 - 26 * q^64 - 4 * q^65 + 10 * q^66 + 20 * q^67 - 6 * q^68 - 24 * q^69 - 12 * q^71 - 20 * q^72 - 6 * q^73 - 20 * q^74 - 2 * q^75 + 12 * q^76 - 2 * q^77 + 4 * q^80 + 22 * q^81 - 10 * q^82 + 4 * q^83 + 6 * q^84 + 4 * q^85 - 12 * q^87 + 4 * q^89 + 40 * q^90 + 2 * q^91 - 12 * q^92 - 20 * q^93 - 10 * q^94 - 8 * q^95 + 30 * q^96 + 8 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	2.23607	1.58114	0.790569	−	0.612372i	$-0.209785\pi$
$2$	2.23607	1.58114	0.790569	+	0.612372i	$0.209785\pi$
$3$	−1.23607	−0.713644	−0.356822	−	0.934172i	$-0.616140\pi$
$3$	−1.23607	−0.713644	−0.356822	+	0.934172i	$0.616140\pi$
$4$	3.00000	1.50000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−2.76393	−1.12837
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.23607	0.790569
$9$	−1.47214	−0.490712
$10$	−4.47214	−1.41421
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−3.70820	−1.07047
$13$	3.23607	0.897524	0.448762	−	0.893651i	$-0.351865\pi$
$13$	3.23607	0.897524	0.448762	+	0.893651i	$0.351865\pi$
$14$	2.23607	0.597614
$15$	2.47214	0.638303
$16$	−1.00000	−0.250000
$17$	−3.23607	−0.784862	−0.392431	−	0.919781i	$-0.628366\pi$
$17$	−3.23607	−0.784862	−0.392431	+	0.919781i	$0.628366\pi$
$18$	−3.29180	−0.775884
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	−6.00000	−1.34164
$21$	−1.23607	−0.269732
$22$	−2.23607	−0.476731
$23$	2.47214	0.515476	0.257738	−	0.966215i	$-0.417023\pi$
$23$	2.47214	0.515476	0.257738	+	0.966215i	$0.417023\pi$
$24$	−2.76393	−0.564185
$25$	−1.00000	−0.200000
$26$	7.23607	1.41911
$27$	5.52786	1.06384
$28$	3.00000	0.566947
$29$	8.47214	1.57324	0.786618	−	0.617440i	$-0.211830\pi$
$29$	8.47214	1.57324	0.786618	+	0.617440i	$0.211830\pi$
$30$	5.52786	1.00925
$31$	−2.76393	−0.496417	−0.248208	−	0.968707i	$-0.579842\pi$
$31$	−2.76393	−0.496417	−0.248208	+	0.968707i	$0.579842\pi$
$32$	−6.70820	−1.18585
$33$	1.23607	0.215172
$34$	−7.23607	−1.24098
$35$	−2.00000	−0.338062
$36$	−4.41641	−0.736068
$37$	−8.47214	−1.39281	−0.696405	−	0.717649i	$-0.745218\pi$
$37$	−8.47214	−1.39281	−0.696405	+	0.717649i	$0.745218\pi$
$38$	14.4721	2.34769
$39$	−4.00000	−0.640513
$40$	−4.47214	−0.707107
$41$	−11.2361	−1.75478	−0.877390	−	0.479779i	$-0.840717\pi$
$41$	−11.2361	−1.75478	−0.877390	+	0.479779i	$0.840717\pi$
$42$	−2.76393	−0.426484
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−3.00000	−0.452267
$45$	2.94427	0.438906
$46$	5.52786	0.815039
$47$	2.76393	0.403161	0.201580	−	0.979472i	$-0.435392\pi$
$47$	2.76393	0.403161	0.201580	+	0.979472i	$0.435392\pi$
$48$	1.23607	0.178411
$49$	1.00000	0.142857
$50$	−2.23607	−0.316228
$51$	4.00000	0.560112
$52$	9.70820	1.34629
$53$	−0.472136	−0.0648529	−0.0324264	−	0.999474i	$-0.510323\pi$
$53$	−0.472136	−0.0648529	−0.0324264	+	0.999474i	$0.510323\pi$
$54$	12.3607	1.68208
$55$	2.00000	0.269680
$56$	2.23607	0.298807
$57$	−8.00000	−1.05963
$58$	18.9443	2.48750
$59$	−1.23607	−0.160922	−0.0804612	−	0.996758i	$-0.525639\pi$
$59$	−1.23607	−0.160922	−0.0804612	+	0.996758i	$0.525639\pi$
$60$	7.41641	0.957454
$61$	−7.23607	−0.926484	−0.463242	−	0.886232i	$-0.653314\pi$
$61$	−7.23607	−0.926484	−0.463242	+	0.886232i	$0.653314\pi$
$62$	−6.18034	−0.784904
$63$	−1.47214	−0.185472
$64$	−13.0000	−1.62500
$65$	−6.47214	−0.802770
$66$	2.76393	0.340217
$67$	14.4721	1.76805	0.884026	−	0.467437i	$-0.154823\pi$
$67$	14.4721	1.76805	0.884026	+	0.467437i	$0.154823\pi$
$68$	−9.70820	−1.17729
$69$	−3.05573	−0.367866
$70$	−4.47214	−0.534522
$71$	−10.4721	−1.24281	−0.621407	−	0.783488i	$-0.713439\pi$
$71$	−10.4721	−1.24281	−0.621407	+	0.783488i	$0.713439\pi$
$72$	−3.29180	−0.387942
$73$	−0.763932	−0.0894115	−0.0447057	−	0.999000i	$-0.514235\pi$
$73$	−0.763932	−0.0894115	−0.0447057	+	0.999000i	$0.514235\pi$
$74$	−18.9443	−2.20223
$75$	1.23607	0.142729
$76$	19.4164	2.22721
$77$	−1.00000	−0.113961
$78$	−8.94427	−1.01274
$79$	−8.94427	−1.00631	−0.503155	−	0.864196i	$-0.667827\pi$
$79$	−8.94427	−1.00631	−0.503155	+	0.864196i	$0.667827\pi$
$80$	2.00000	0.223607
$81$	−2.41641	−0.268490
$82$	−25.1246	−2.77455
$83$	−11.4164	−1.25311	−0.626557	−	0.779376i	$-0.715536\pi$
$83$	−11.4164	−1.25311	−0.626557	+	0.779376i	$0.715536\pi$
$84$	−3.70820	−0.404598
$85$	6.47214	0.702002
$86$	17.8885	1.92897
$87$	−10.4721	−1.12273
$88$	−2.23607	−0.238366
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	6.58359	0.693972
$91$	3.23607	0.339232
$92$	7.41641	0.773214
$93$	3.41641	0.354265
$94$	6.18034	0.637453
$95$	−12.9443	−1.32805
$96$	8.29180	0.846278
$97$	17.4164	1.76837	0.884184	−	0.467139i	$-0.154715\pi$
$97$	17.4164	1.76837	0.884184	+	0.467139i	$0.154715\pi$
$98$	2.23607	0.225877
$99$	1.47214	0.147955
$100$	−3.00000	−0.300000
$101$	−4.76393	−0.474029	−0.237014	−	0.971506i	$-0.576169\pi$
$101$	−4.76393	−0.474029	−0.237014	+	0.971506i	$0.576169\pi$
$102$	8.94427	0.885615
$103$	7.70820	0.759512	0.379756	−	0.925087i	$-0.376008\pi$
$103$	7.70820	0.759512	0.379756	+	0.925087i	$0.376008\pi$
$104$	7.23607	0.709555
$105$	2.47214	0.241256
$106$	−1.05573	−0.102541
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	16.5836	1.59576
$109$	−4.47214	−0.428353	−0.214176	−	0.976795i	$-0.568707\pi$
$109$	−4.47214	−0.428353	−0.214176	+	0.976795i	$0.568707\pi$
$110$	4.47214	0.426401
$111$	10.4721	0.993971
$112$	−1.00000	−0.0944911
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	−17.8885	−1.67542
$115$	−4.94427	−0.461056
$116$	25.4164	2.35985
$117$	−4.76393	−0.440426
$118$	−2.76393	−0.254441
$119$	−3.23607	−0.296650
$120$	5.52786	0.504623
$121$	1.00000	0.0909091
$122$	−16.1803	−1.46490
$123$	13.8885	1.25229
$124$	−8.29180	−0.744625
$125$	12.0000	1.07331
$126$	−3.29180	−0.293257
$127$	3.05573	0.271152	0.135576	−	0.990767i	$-0.456712\pi$
$127$	3.05573	0.271152	0.135576	+	0.990767i	$0.456712\pi$
$128$	−15.6525	−1.38350
$129$	−9.88854	−0.870638
$130$	−14.4721	−1.26929
$131$	21.8885	1.91241	0.956205	−	0.292696i	$-0.0945525\pi$
$131$	21.8885	1.91241	0.956205	+	0.292696i	$0.0945525\pi$
$132$	3.70820	0.322758
$133$	6.47214	0.561205
$134$	32.3607	2.79554
$135$	−11.0557	−0.951526
$136$	−7.23607	−0.620488
$137$	16.4721	1.40731	0.703655	−	0.710542i	$-0.251550\pi$
$137$	16.4721	1.40731	0.703655	+	0.710542i	$0.251550\pi$
$138$	−6.83282	−0.581648
$139$	−1.52786	−0.129592	−0.0647959	−	0.997899i	$-0.520640\pi$
$139$	−1.52786	−0.129592	−0.0647959	+	0.997899i	$0.520640\pi$
$140$	−6.00000	−0.507093
$141$	−3.41641	−0.287713
$142$	−23.4164	−1.96506
$143$	−3.23607	−0.270614
$144$	1.47214	0.122678
$145$	−16.9443	−1.40715
$146$	−1.70820	−0.141372
$147$	−1.23607	−0.101949
$148$	−25.4164	−2.08922
$149$	14.0000	1.14692	0.573462	−	0.819232i	$-0.305600\pi$
$149$	14.0000	1.14692	0.573462	+	0.819232i	$0.305600\pi$
$150$	2.76393	0.225674
$151$	8.94427	0.727875	0.363937	−	0.931423i	$-0.381432\pi$
$151$	8.94427	0.727875	0.363937	+	0.931423i	$0.381432\pi$
$152$	14.4721	1.17385
$153$	4.76393	0.385141
$154$	−2.23607	−0.180187
$155$	5.52786	0.444009
$156$	−12.0000	−0.960769
$157$	10.9443	0.873448	0.436724	−	0.899596i	$-0.356139\pi$
$157$	10.9443	0.873448	0.436724	+	0.899596i	$0.356139\pi$
$158$	−20.0000	−1.59111
$159$	0.583592	0.0462819
$160$	13.4164	1.06066
$161$	2.47214	0.194832
$162$	−5.40325	−0.424520
$163$	−3.41641	−0.267594	−0.133797	−	0.991009i	$-0.542717\pi$
$163$	−3.41641	−0.267594	−0.133797	+	0.991009i	$0.542717\pi$
$164$	−33.7082	−2.63217
$165$	−2.47214	−0.192456
$166$	−25.5279	−1.98135
$167$	4.94427	0.382599	0.191300	−	0.981532i	$-0.438730\pi$
$167$	4.94427	0.382599	0.191300	+	0.981532i	$0.438730\pi$
$168$	−2.76393	−0.213242
$169$	−2.52786	−0.194451
$170$	14.4721	1.10996
$171$	−9.52786	−0.728614
$172$	24.0000	1.82998
$173$	−12.7639	−0.970424	−0.485212	−	0.874397i	$-0.661258\pi$
$173$	−12.7639	−0.970424	−0.485212	+	0.874397i	$0.661258\pi$
$174$	−23.4164	−1.77519
$175$	−1.00000	−0.0755929
$176$	1.00000	0.0753778
$177$	1.52786	0.114841
$178$	4.47214	0.335201
$179$	−8.94427	−0.668526	−0.334263	−	0.942480i	$-0.608487\pi$
$179$	−8.94427	−0.668526	−0.334263	+	0.942480i	$0.608487\pi$
$180$	8.83282	0.658359
$181$	−25.4164	−1.88919	−0.944593	−	0.328243i	$-0.893544\pi$
$181$	−25.4164	−1.88919	−0.944593	+	0.328243i	$0.893544\pi$
$182$	7.23607	0.536373
$183$	8.94427	0.661180
$184$	5.52786	0.407520
$185$	16.9443	1.24577
$186$	7.63932	0.560142
$187$	3.23607	0.236645
$188$	8.29180	0.604741
$189$	5.52786	0.402093
$190$	−28.9443	−2.09984
$191$	−3.05573	−0.221105	−0.110552	−	0.993870i	$-0.535262\pi$
$191$	−3.05573	−0.221105	−0.110552	+	0.993870i	$0.535262\pi$
$192$	16.0689	1.15967
$193$	11.8885	0.855756	0.427878	−	0.903836i	$-0.359261\pi$
$193$	11.8885	0.855756	0.427878	+	0.903836i	$0.359261\pi$
$194$	38.9443	2.79604
$195$	8.00000	0.572892
$196$	3.00000	0.214286
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	3.29180	0.233938
$199$	−2.18034	−0.154560	−0.0772801	−	0.997009i	$-0.524624\pi$
$199$	−2.18034	−0.154560	−0.0772801	+	0.997009i	$0.524624\pi$
$200$	−2.23607	−0.158114
$201$	−17.8885	−1.26176
$202$	−10.6525	−0.749506
$203$	8.47214	0.594627
$204$	12.0000	0.840168
$205$	22.4721	1.56952
$206$	17.2361	1.20089
$207$	−3.63932	−0.252950
$208$	−3.23607	−0.224381
$209$	−6.47214	−0.447687
$210$	5.52786	0.381459
$211$	−13.8885	−0.956127	−0.478063	−	0.878325i	$-0.658661\pi$
$211$	−13.8885	−0.956127	−0.478063	+	0.878325i	$0.658661\pi$
$212$	−1.41641	−0.0972793
$213$	12.9443	0.886927
$214$	−8.94427	−0.611418
$215$	−16.0000	−1.09119
$216$	12.3607	0.841038
$217$	−2.76393	−0.187628
$218$	−10.0000	−0.677285
$219$	0.944272	0.0638080
$220$	6.00000	0.404520
$221$	−10.4721	−0.704432
$222$	23.4164	1.57161
$223$	−10.1803	−0.681726	−0.340863	−	0.940113i	$-0.610719\pi$
$223$	−10.1803	−0.681726	−0.340863	+	0.940113i	$0.610719\pi$
$224$	−6.70820	−0.448211
$225$	1.47214	0.0981424
$226$	4.47214	0.297482
$227$	5.88854	0.390836	0.195418	−	0.980720i	$-0.437394\pi$
$227$	5.88854	0.390836	0.195418	+	0.980720i	$0.437394\pi$
$228$	−24.0000	−1.58944
$229$	−4.47214	−0.295527	−0.147764	−	0.989023i	$-0.547207\pi$
$229$	−4.47214	−0.295527	−0.147764	+	0.989023i	$0.547207\pi$
$230$	−11.0557	−0.728993
$231$	1.23607	0.0813273
$232$	18.9443	1.24375
$233$	9.41641	0.616889	0.308445	−	0.951242i	$-0.400192\pi$
$233$	9.41641	0.616889	0.308445	+	0.951242i	$0.400192\pi$
$234$	−10.6525	−0.696374
$235$	−5.52786	−0.360598
$236$	−3.70820	−0.241384
$237$	11.0557	0.718147
$238$	−7.23607	−0.469045
$239$	−9.88854	−0.639637	−0.319818	−	0.947479i	$-0.603622\pi$
$239$	−9.88854	−0.639637	−0.319818	+	0.947479i	$0.603622\pi$
$240$	−2.47214	−0.159576
$241$	−13.1246	−0.845431	−0.422715	−	0.906263i	$-0.638923\pi$
$241$	−13.1246	−0.845431	−0.422715	+	0.906263i	$0.638923\pi$
$242$	2.23607	0.143740
$243$	−13.5967	−0.872232
$244$	−21.7082	−1.38973
$245$	−2.00000	−0.127775
$246$	31.0557	1.98004
$247$	20.9443	1.33265
$248$	−6.18034	−0.392452
$249$	14.1115	0.894277
$250$	26.8328	1.69706
$251$	4.29180	0.270896	0.135448	−	0.990784i	$-0.456753\pi$
$251$	4.29180	0.270896	0.135448	+	0.990784i	$0.456753\pi$
$252$	−4.41641	−0.278208
$253$	−2.47214	−0.155422
$254$	6.83282	0.428729
$255$	−8.00000	−0.500979
$256$	−9.00000	−0.562500
$257$	−6.00000	−0.374270	−0.187135	−	0.982334i	$-0.559920\pi$
$257$	−6.00000	−0.374270	−0.187135	+	0.982334i	$0.559920\pi$
$258$	−22.1115	−1.37660
$259$	−8.47214	−0.526433
$260$	−19.4164	−1.20415
$261$	−12.4721	−0.772006
$262$	48.9443	3.02379
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	2.76393	0.170108
$265$	0.944272	0.0580062
$266$	14.4721	0.887344
$267$	−2.47214	−0.151292
$268$	43.4164	2.65208
$269$	13.4164	0.818013	0.409006	−	0.912532i	$-0.365875\pi$
$269$	13.4164	0.818013	0.409006	+	0.912532i	$0.365875\pi$
$270$	−24.7214	−1.50449
$271$	−10.4721	−0.636137	−0.318068	−	0.948068i	$-0.603034\pi$
$271$	−10.4721	−0.636137	−0.318068	+	0.948068i	$0.603034\pi$
$272$	3.23607	0.196215
$273$	−4.00000	−0.242091
$274$	36.8328	2.22515
$275$	1.00000	0.0603023
$276$	−9.16718	−0.551800
$277$	−19.8885	−1.19499	−0.597493	−	0.801874i	$-0.703837\pi$
$277$	−19.8885	−1.19499	−0.597493	+	0.801874i	$0.703837\pi$
$278$	−3.41641	−0.204903
$279$	4.06888	0.243598
$280$	−4.47214	−0.267261
$281$	−3.52786	−0.210455	−0.105227	−	0.994448i	$-0.533557\pi$
$281$	−3.52786	−0.210455	−0.105227	+	0.994448i	$0.533557\pi$
$282$	−7.63932	−0.454915
$283$	29.8885	1.77669	0.888345	−	0.459177i	$-0.151856\pi$
$283$	29.8885	1.77669	0.888345	+	0.459177i	$0.151856\pi$
$284$	−31.4164	−1.86422
$285$	16.0000	0.947758
$286$	−7.23607	−0.427878
$287$	−11.2361	−0.663244
$288$	9.87539	0.581913
$289$	−6.52786	−0.383992
$290$	−37.8885	−2.22489
$291$	−21.5279	−1.26199
$292$	−2.29180	−0.134117
$293$	−25.1246	−1.46780	−0.733898	−	0.679260i	$-0.762301\pi$
$293$	−25.1246	−1.46780	−0.733898	+	0.679260i	$0.762301\pi$
$294$	−2.76393	−0.161196
$295$	2.47214	0.143933
$296$	−18.9443	−1.10111
$297$	−5.52786	−0.320759
$298$	31.3050	1.81345
$299$	8.00000	0.462652
$300$	3.70820	0.214093
$301$	8.00000	0.461112
$302$	20.0000	1.15087
$303$	5.88854	0.338288
$304$	−6.47214	−0.371202
$305$	14.4721	0.828672
$306$	10.6525	0.608962
$307$	−8.94427	−0.510477	−0.255238	−	0.966878i	$-0.582154\pi$
$307$	−8.94427	−0.510477	−0.255238	+	0.966878i	$0.582154\pi$
$308$	−3.00000	−0.170941
$309$	−9.52786	−0.542021
$310$	12.3607	0.702039
$311$	−8.29180	−0.470185	−0.235092	−	0.971973i	$-0.575539\pi$
$311$	−8.29180	−0.470185	−0.235092	+	0.971973i	$0.575539\pi$
$312$	−8.94427	−0.506370
$313$	14.9443	0.844700	0.422350	−	0.906433i	$-0.361205\pi$
$313$	14.9443	0.844700	0.422350	+	0.906433i	$0.361205\pi$
$314$	24.4721	1.38104
$315$	2.94427	0.165891
$316$	−26.8328	−1.50946
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	1.30495	0.0731781
$319$	−8.47214	−0.474349
$320$	26.0000	1.45344
$321$	4.94427	0.275962
$322$	5.52786	0.308056
$323$	−20.9443	−1.16537
$324$	−7.24922	−0.402735
$325$	−3.23607	−0.179505
$326$	−7.63932	−0.423103
$327$	5.52786	0.305692
$328$	−25.1246	−1.38727
$329$	2.76393	0.152381
$330$	−5.52786	−0.304299
$331$	−13.8885	−0.763383	−0.381692	−	0.924290i	$-0.624658\pi$
$331$	−13.8885	−0.763383	−0.381692	+	0.924290i	$0.624658\pi$
$332$	−34.2492	−1.87967
$333$	12.4721	0.683469
$334$	11.0557	0.604943
$335$	−28.9443	−1.58139
$336$	1.23607	0.0674330
$337$	−11.5279	−0.627963	−0.313981	−	0.949429i	$-0.601663\pi$
$337$	−11.5279	−0.627963	−0.313981	+	0.949429i	$0.601663\pi$
$338$	−5.65248	−0.307454
$339$	−2.47214	−0.134268
$340$	19.4164	1.05300
$341$	2.76393	0.149675
$342$	−21.3050	−1.15204
$343$	1.00000	0.0539949
$344$	17.8885	0.964486
$345$	6.11146	0.329030
$346$	−28.5410	−1.53437
$347$	20.9443	1.12435	0.562174	−	0.827019i	$-0.309965\pi$
$347$	20.9443	1.12435	0.562174	+	0.827019i	$0.309965\pi$
$348$	−31.4164	−1.68410
$349$	−7.23607	−0.387338	−0.193669	−	0.981067i	$-0.562039\pi$
$349$	−7.23607	−0.387338	−0.193669	+	0.981067i	$0.562039\pi$
$350$	−2.23607	−0.119523
$351$	17.8885	0.954820
$352$	6.70820	0.357548
$353$	19.8885	1.05856	0.529280	−	0.848447i	$-0.322462\pi$
$353$	19.8885	1.05856	0.529280	+	0.848447i	$0.322462\pi$
$354$	3.41641	0.181580
$355$	20.9443	1.11161
$356$	6.00000	0.317999
$357$	4.00000	0.211702
$358$	−20.0000	−1.05703
$359$	−24.9443	−1.31651	−0.658254	−	0.752796i	$-0.728705\pi$
$359$	−24.9443	−1.31651	−0.658254	+	0.752796i	$0.728705\pi$
$360$	6.58359	0.346986
$361$	22.8885	1.20466
$362$	−56.8328	−2.98707
$363$	−1.23607	−0.0648767
$364$	9.70820	0.508848
$365$	1.52786	0.0799721
$366$	20.0000	1.04542
$367$	−23.1246	−1.20709	−0.603547	−	0.797327i	$-0.706247\pi$
$367$	−23.1246	−1.20709	−0.603547	+	0.797327i	$0.706247\pi$
$368$	−2.47214	−0.128869
$369$	16.5410	0.861091
$370$	37.8885	1.96973
$371$	−0.472136	−0.0245121
$372$	10.2492	0.531397
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	7.23607	0.374168
$375$	−14.8328	−0.765963
$376$	6.18034	0.318727
$377$	27.4164	1.41202
$378$	12.3607	0.635765
$379$	37.3050	1.91623	0.958113	−	0.286389i	$-0.0924551\pi$
$379$	37.3050	1.91623	0.958113	+	0.286389i	$0.0924551\pi$
$380$	−38.8328	−1.99208
$381$	−3.77709	−0.193506
$382$	−6.83282	−0.349597
$383$	−4.65248	−0.237730	−0.118865	−	0.992910i	$-0.537926\pi$
$383$	−4.65248	−0.237730	−0.118865	+	0.992910i	$0.537926\pi$
$384$	19.3475	0.987324
$385$	2.00000	0.101929
$386$	26.5836	1.35307
$387$	−11.7771	−0.598663
$388$	52.2492	2.65255
$389$	15.8885	0.805581	0.402791	−	0.915292i	$-0.368040\pi$
$389$	15.8885	0.805581	0.402791	+	0.915292i	$0.368040\pi$
$390$	17.8885	0.905822
$391$	−8.00000	−0.404577
$392$	2.23607	0.112938
$393$	−27.0557	−1.36478
$394$	−4.47214	−0.225303
$395$	17.8885	0.900070
$396$	4.41641	0.221933
$397$	−35.8885	−1.80119	−0.900597	−	0.434655i	$-0.856870\pi$
$397$	−35.8885	−1.80119	−0.900597	+	0.434655i	$0.856870\pi$
$398$	−4.87539	−0.244381
$399$	−8.00000	−0.400501
$400$	1.00000	0.0500000
$401$	22.9443	1.14578	0.572891	−	0.819631i	$-0.305822\pi$
$401$	22.9443	1.14578	0.572891	+	0.819631i	$0.305822\pi$
$402$	−40.0000	−1.99502
$403$	−8.94427	−0.445546
$404$	−14.2918	−0.711043
$405$	4.83282	0.240145
$406$	18.9443	0.940188
$407$	8.47214	0.419948
$408$	8.94427	0.442807
$409$	9.12461	0.451183	0.225592	−	0.974222i	$-0.427569\pi$
$409$	9.12461	0.451183	0.225592	+	0.974222i	$0.427569\pi$
$410$	50.2492	2.48163
$411$	−20.3607	−1.00432
$412$	23.1246	1.13927
$413$	−1.23607	−0.0608229
$414$	−8.13777	−0.399949
$415$	22.8328	1.12082
$416$	−21.7082	−1.06433
$417$	1.88854	0.0924824
$418$	−14.4721	−0.707855
$419$	24.6525	1.20435	0.602176	−	0.798363i	$-0.294300\pi$
$419$	24.6525	1.20435	0.602176	+	0.798363i	$0.294300\pi$
$420$	7.41641	0.361884
$421$	22.3607	1.08979	0.544896	−	0.838503i	$-0.316569\pi$
$421$	22.3607	1.08979	0.544896	+	0.838503i	$0.316569\pi$
$422$	−31.0557	−1.51177
$423$	−4.06888	−0.197836
$424$	−1.05573	−0.0512707
$425$	3.23607	0.156972
$426$	28.9443	1.40235
$427$	−7.23607	−0.350178
$428$	−12.0000	−0.580042
$429$	4.00000	0.193122
$430$	−35.7771	−1.72532
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−5.52786	−0.265959
$433$	−8.47214	−0.407145	−0.203572	−	0.979060i	$-0.565255\pi$
$433$	−8.47214	−0.407145	−0.203572	+	0.979060i	$0.565255\pi$
$434$	−6.18034	−0.296666
$435$	20.9443	1.00420
$436$	−13.4164	−0.642529
$437$	16.0000	0.765384
$438$	2.11146	0.100889
$439$	10.4721	0.499808	0.249904	−	0.968271i	$-0.419601\pi$
$439$	10.4721	0.499808	0.249904	+	0.968271i	$0.419601\pi$
$440$	4.47214	0.213201
$441$	−1.47214	−0.0701017
$442$	−23.4164	−1.11380
$443$	−24.9443	−1.18514	−0.592569	−	0.805520i	$-0.701886\pi$
$443$	−24.9443	−1.18514	−0.592569	+	0.805520i	$0.701886\pi$
$444$	31.4164	1.49096
$445$	−4.00000	−0.189618
$446$	−22.7639	−1.07790
$447$	−17.3050	−0.818496
$448$	−13.0000	−0.614192
$449$	−28.4721	−1.34368	−0.671842	−	0.740695i	$-0.734496\pi$
$449$	−28.4721	−1.34368	−0.671842	+	0.740695i	$0.734496\pi$
$450$	3.29180	0.155177
$451$	11.2361	0.529086
$452$	6.00000	0.282216
$453$	−11.0557	−0.519443
$454$	13.1672	0.617967
$455$	−6.47214	−0.303418
$456$	−17.8885	−0.837708
$457$	−28.8328	−1.34874	−0.674371	−	0.738393i	$-0.735585\pi$
$457$	−28.8328	−1.34874	−0.674371	+	0.738393i	$0.735585\pi$
$458$	−10.0000	−0.467269
$459$	−17.8885	−0.834966
$460$	−14.8328	−0.691584
$461$	−12.1803	−0.567295	−0.283647	−	0.958929i	$-0.591545\pi$
$461$	−12.1803	−0.567295	−0.283647	+	0.958929i	$0.591545\pi$
$462$	2.76393	0.128590
$463$	−5.52786	−0.256902	−0.128451	−	0.991716i	$-0.541000\pi$
$463$	−5.52786	−0.256902	−0.128451	+	0.991716i	$0.541000\pi$
$464$	−8.47214	−0.393309
$465$	−6.83282	−0.316864
$466$	21.0557	0.975388
$467$	24.0689	1.11378	0.556888	−	0.830588i	$-0.311995\pi$
$467$	24.0689	1.11378	0.556888	+	0.830588i	$0.311995\pi$
$468$	−14.2918	−0.660639
$469$	14.4721	0.668261
$470$	−12.3607	−0.570156
$471$	−13.5279	−0.623331
$472$	−2.76393	−0.127220
$473$	−8.00000	−0.367840
$474$	24.7214	1.13549
$475$	−6.47214	−0.296962
$476$	−9.70820	−0.444975
$477$	0.695048	0.0318241
$478$	−22.1115	−1.01135
$479$	13.5279	0.618104	0.309052	−	0.951045i	$-0.399988\pi$
$479$	13.5279	0.618104	0.309052	+	0.951045i	$0.399988\pi$
$480$	−16.5836	−0.756934
$481$	−27.4164	−1.25008
$482$	−29.3475	−1.33674
$483$	−3.05573	−0.139040
$484$	3.00000	0.136364
$485$	−34.8328	−1.58168
$486$	−30.4033	−1.37912
$487$	−36.3607	−1.64766	−0.823830	−	0.566837i	$-0.808167\pi$
$487$	−36.3607	−1.64766	−0.823830	+	0.566837i	$0.808167\pi$
$488$	−16.1803	−0.732450
$489$	4.22291	0.190967
$490$	−4.47214	−0.202031
$491$	0	0		−	1.00000i	$-0.5\pi$
$491$	0	0			1.00000i	$0.5\pi$
$492$	41.6656	1.87843
$493$	−27.4164	−1.23477
$494$	46.8328	2.10711
$495$	−2.94427	−0.132335
$496$	2.76393	0.124104
$497$	−10.4721	−0.469739
$498$	31.5542	1.41398
$499$	1.52786	0.0683966	0.0341983	−	0.999415i	$-0.489112\pi$
$499$	1.52786	0.0683966	0.0341983	+	0.999415i	$0.489112\pi$
$500$	36.0000	1.60997
$501$	−6.11146	−0.273040
$502$	9.59675	0.428324
$503$	23.4164	1.04409	0.522043	−	0.852919i	$-0.325170\pi$
$503$	23.4164	1.04409	0.522043	+	0.852919i	$0.325170\pi$
$504$	−3.29180	−0.146628
$505$	9.52786	0.423984
$506$	−5.52786	−0.245744
$507$	3.12461	0.138769
$508$	9.16718	0.406728
$509$	40.4721	1.79390	0.896948	−	0.442136i	$-0.145779\pi$
$509$	40.4721	1.79390	0.896948	+	0.442136i	$0.145779\pi$
$510$	−17.8885	−0.792118
$511$	−0.763932	−0.0337944
$512$	11.1803	0.494106
$513$	35.7771	1.57960
$514$	−13.4164	−0.591772
$515$	−15.4164	−0.679328
$516$	−29.6656	−1.30596
$517$	−2.76393	−0.121558
$518$	−18.9443	−0.832364
$519$	15.7771	0.692537
$520$	−14.4721	−0.634645
$521$	30.3607	1.33013	0.665063	−	0.746787i	$-0.268405\pi$
$521$	30.3607	1.33013	0.665063	+	0.746787i	$0.268405\pi$
$522$	−27.8885	−1.22065
$523$	44.0000	1.92399	0.961993	−	0.273075i	$-0.0880406\pi$
$523$	44.0000	1.92399	0.961993	+	0.273075i	$0.0880406\pi$
$524$	65.6656	2.86862
$525$	1.23607	0.0539464
$526$	0	0
$527$	8.94427	0.389619
$528$	−1.23607	−0.0537930
$529$	−16.8885	−0.734285
$530$	2.11146	0.0917158
$531$	1.81966	0.0789665
$532$	19.4164	0.841808
$533$	−36.3607	−1.57496
$534$	−5.52786	−0.239214
$535$	8.00000	0.345870
$536$	32.3607	1.39777
$537$	11.0557	0.477090
$538$	30.0000	1.29339
$539$	−1.00000	−0.0430730
$540$	−33.1672	−1.42729
$541$	20.8328	0.895673	0.447836	−	0.894116i	$-0.352195\pi$
$541$	20.8328	0.895673	0.447836	+	0.894116i	$0.352195\pi$
$542$	−23.4164	−1.00582
$543$	31.4164	1.34821
$544$	21.7082	0.930732
$545$	8.94427	0.383131
$546$	−8.94427	−0.382780
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	49.4164	2.11096
$549$	10.6525	0.454637
$550$	2.23607	0.0953463
$551$	54.8328	2.33596
$552$	−6.83282	−0.290824
$553$	−8.94427	−0.380349
$554$	−44.4721	−1.88944
$555$	−20.9443	−0.889035
$556$	−4.58359	−0.194388
$557$	−38.9443	−1.65012	−0.825061	−	0.565044i	$-0.808859\pi$
$557$	−38.9443	−1.65012	−0.825061	+	0.565044i	$0.808859\pi$
$558$	9.09830	0.385162
$559$	25.8885	1.09497
$560$	2.00000	0.0845154
$561$	−4.00000	−0.168880
$562$	−7.88854	−0.332758
$563$	12.5836	0.530335	0.265168	−	0.964202i	$-0.414573\pi$
$563$	12.5836	0.530335	0.265168	+	0.964202i	$0.414573\pi$
$564$	−10.2492	−0.431570
$565$	−4.00000	−0.168281
$566$	66.8328	2.80919
$567$	−2.41641	−0.101480
$568$	−23.4164	−0.982531
$569$	7.52786	0.315584	0.157792	−	0.987472i	$-0.449562\pi$
$569$	7.52786	0.315584	0.157792	+	0.987472i	$0.449562\pi$
$570$	35.7771	1.49854
$571$	−15.0557	−0.630063	−0.315031	−	0.949081i	$-0.602015\pi$
$571$	−15.0557	−0.630063	−0.315031	+	0.949081i	$0.602015\pi$
$572$	−9.70820	−0.405920
$573$	3.77709	0.157790
$574$	−25.1246	−1.04868
$575$	−2.47214	−0.103095
$576$	19.1378	0.797407
$577$	−19.5279	−0.812956	−0.406478	−	0.913661i	$-0.633243\pi$
$577$	−19.5279	−0.812956	−0.406478	+	0.913661i	$0.633243\pi$
$578$	−14.5967	−0.607145
$579$	−14.6950	−0.610705
$580$	−50.8328	−2.11072
$581$	−11.4164	−0.473632
$582$	−48.1378	−1.99537
$583$	0.472136	0.0195539
$584$	−1.70820	−0.0706860
$585$	9.52786	0.393929
$586$	−56.1803	−2.32079
$587$	27.1246	1.11955	0.559776	−	0.828644i	$-0.310887\pi$
$587$	27.1246	1.11955	0.559776	+	0.828644i	$0.310887\pi$
$588$	−3.70820	−0.152924
$589$	−17.8885	−0.737085
$590$	5.52786	0.227579
$591$	2.47214	0.101690
$592$	8.47214	0.348203
$593$	−45.7082	−1.87701	−0.938505	−	0.345264i	$-0.887789\pi$
$593$	−45.7082	−1.87701	−0.938505	+	0.345264i	$0.887789\pi$
$594$	−12.3607	−0.507165
$595$	6.47214	0.265332
$596$	42.0000	1.72039
$597$	2.69505	0.110301
$598$	17.8885	0.731517
$599$	−23.4164	−0.956768	−0.478384	−	0.878151i	$-0.658777\pi$
$599$	−23.4164	−0.956768	−0.478384	+	0.878151i	$0.658777\pi$
$600$	2.76393	0.112837
$601$	−37.1246	−1.51434	−0.757172	−	0.653215i	$-0.773420\pi$
$601$	−37.1246	−1.51434	−0.757172	+	0.653215i	$0.773420\pi$
$602$	17.8885	0.729083
$603$	−21.3050	−0.867605
$604$	26.8328	1.09181
$605$	−2.00000	−0.0813116
$606$	13.1672	0.534880
$607$	12.9443	0.525392	0.262696	−	0.964879i	$-0.415388\pi$
$607$	12.9443	0.525392	0.262696	+	0.964879i	$0.415388\pi$
$608$	−43.4164	−1.76077
$609$	−10.4721	−0.424352
$610$	32.3607	1.31025
$611$	8.94427	0.361847
$612$	14.2918	0.577712
$613$	15.3050	0.618161	0.309081	−	0.951036i	$-0.399979\pi$
$613$	15.3050	0.618161	0.309081	+	0.951036i	$0.399979\pi$
$614$	−20.0000	−0.807134
$615$	−27.7771	−1.12008
$616$	−2.23607	−0.0900937
$617$	6.58359	0.265045	0.132523	−	0.991180i	$-0.457692\pi$
$617$	6.58359	0.265045	0.132523	+	0.991180i	$0.457692\pi$
$618$	−21.3050	−0.857011
$619$	11.1246	0.447136	0.223568	−	0.974688i	$-0.428230\pi$
$619$	11.1246	0.447136	0.223568	+	0.974688i	$0.428230\pi$
$620$	16.5836	0.666013
$621$	13.6656	0.548383
$622$	−18.5410	−0.743427
$623$	2.00000	0.0801283
$624$	4.00000	0.160128
$625$	−19.0000	−0.760000
$626$	33.4164	1.33559
$627$	8.00000	0.319489
$628$	32.8328	1.31017
$629$	27.4164	1.09316
$630$	6.58359	0.262297
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	−20.0000	−0.795557
$633$	17.1672	0.682334
$634$	31.3050	1.24328
$635$	−6.11146	−0.242526
$636$	1.75078	0.0694228
$637$	3.23607	0.128218
$638$	−18.9443	−0.750011
$639$	15.4164	0.609864
$640$	31.3050	1.23744
$641$	−15.5279	−0.613314	−0.306657	−	0.951820i	$-0.599210\pi$
$641$	−15.5279	−0.613314	−0.306657	+	0.951820i	$0.599210\pi$
$642$	11.0557	0.436335
$643$	11.1246	0.438712	0.219356	−	0.975645i	$-0.429604\pi$
$643$	11.1246	0.438712	0.219356	+	0.975645i	$0.429604\pi$
$644$	7.41641	0.292247
$645$	19.7771	0.778722
$646$	−46.8328	−1.84261
$647$	36.0689	1.41801	0.709007	−	0.705201i	$-0.249143\pi$
$647$	36.0689	1.41801	0.709007	+	0.705201i	$0.249143\pi$
$648$	−5.40325	−0.212260
$649$	1.23607	0.0485199
$650$	−7.23607	−0.283822
$651$	3.41641	0.133900
$652$	−10.2492	−0.401391
$653$	25.0557	0.980506	0.490253	−	0.871580i	$-0.336904\pi$
$653$	25.0557	0.980506	0.490253	+	0.871580i	$0.336904\pi$
$654$	12.3607	0.483341
$655$	−43.7771	−1.71051
$656$	11.2361	0.438695
$657$	1.12461	0.0438753
$658$	6.18034	0.240935
$659$	−17.8885	−0.696839	−0.348419	−	0.937339i	$-0.613281\pi$
$659$	−17.8885	−0.696839	−0.348419	+	0.937339i	$0.613281\pi$
$660$	−7.41641	−0.288683
$661$	−40.8328	−1.58821	−0.794106	−	0.607779i	$-0.792061\pi$
$661$	−40.8328	−1.58821	−0.794106	+	0.607779i	$0.792061\pi$
$662$	−31.0557	−1.20702
$663$	12.9443	0.502714
$664$	−25.5279	−0.990673
$665$	−12.9443	−0.501957
$666$	27.8885	1.08066
$667$	20.9443	0.810965
$668$	14.8328	0.573899
$669$	12.5836	0.486510
$670$	−64.7214	−2.50040
$671$	7.23607	0.279345
$672$	8.29180	0.319863
$673$	−21.4164	−0.825542	−0.412771	−	0.910835i	$-0.635439\pi$
$673$	−21.4164	−0.825542	−0.412771	+	0.910835i	$0.635439\pi$
$674$	−25.7771	−0.992896
$675$	−5.52786	−0.212768
$676$	−7.58359	−0.291677
$677$	−9.70820	−0.373117	−0.186558	−	0.982444i	$-0.559733\pi$
$677$	−9.70820	−0.373117	−0.186558	+	0.982444i	$0.559733\pi$
$678$	−5.52786	−0.212296
$679$	17.4164	0.668380
$680$	14.4721	0.554981
$681$	−7.27864	−0.278918
$682$	6.18034	0.236657
$683$	−5.88854	−0.225319	−0.112659	−	0.993634i	$-0.535937\pi$
$683$	−5.88854	−0.225319	−0.112659	+	0.993634i	$0.535937\pi$
$684$	−28.5836	−1.09292
$685$	−32.9443	−1.25874
$686$	2.23607	0.0853735
$687$	5.52786	0.210901
$688$	−8.00000	−0.304997
$689$	−1.52786	−0.0582070
$690$	13.6656	0.520242
$691$	18.5410	0.705334	0.352667	−	0.935749i	$-0.385275\pi$
$691$	18.5410	0.705334	0.352667	+	0.935749i	$0.385275\pi$
$692$	−38.2918	−1.45564
$693$	1.47214	0.0559218
$694$	46.8328	1.77775
$695$	3.05573	0.115910
$696$	−23.4164	−0.887597
$697$	36.3607	1.37726
$698$	−16.1803	−0.612435
$699$	−11.6393	−0.440240
$700$	−3.00000	−0.113389
$701$	−15.5279	−0.586479	−0.293240	−	0.956039i	$-0.594733\pi$
$701$	−15.5279	−0.586479	−0.293240	+	0.956039i	$0.594733\pi$
$702$	40.0000	1.50970
$703$	−54.8328	−2.06806
$704$	13.0000	0.489956
$705$	6.83282	0.257339
$706$	44.4721	1.67373
$707$	−4.76393	−0.179166
$708$	4.58359	0.172262
$709$	−14.9443	−0.561244	−0.280622	−	0.959818i	$-0.590541\pi$
$709$	−14.9443	−0.561244	−0.280622	+	0.959818i	$0.590541\pi$
$710$	46.8328	1.75760
$711$	13.1672	0.493808
$712$	4.47214	0.167600
$713$	−6.83282	−0.255891
$714$	8.94427	0.334731
$715$	6.47214	0.242044
$716$	−26.8328	−1.00279
$717$	12.2229	0.456473
$718$	−55.7771	−2.08158
$719$	−51.4853	−1.92008	−0.960039	−	0.279867i	$-0.909710\pi$
$719$	−51.4853	−1.92008	−0.960039	+	0.279867i	$0.909710\pi$
$720$	−2.94427	−0.109727
$721$	7.70820	0.287069
$722$	51.1803	1.90474
$723$	16.2229	0.603337
$724$	−76.2492	−2.83378
$725$	−8.47214	−0.314647
$726$	−2.76393	−0.102579
$727$	25.0132	0.927687	0.463843	−	0.885917i	$-0.346470\pi$
$727$	25.0132	0.927687	0.463843	+	0.885917i	$0.346470\pi$
$728$	7.23607	0.268187
$729$	24.0557	0.890953
$730$	3.41641	0.126447
$731$	−25.8885	−0.957522
$732$	26.8328	0.991769
$733$	8.76393	0.323703	0.161852	−	0.986815i	$-0.448253\pi$
$733$	8.76393	0.323703	0.161852	+	0.986815i	$0.448253\pi$
$734$	−51.7082	−1.90858
$735$	2.47214	0.0911861
$736$	−16.5836	−0.611279
$737$	−14.4721	−0.533088
$738$	36.9868	1.36150
$739$	24.9443	0.917590	0.458795	−	0.888542i	$-0.348281\pi$
$739$	24.9443	0.917590	0.458795	+	0.888542i	$0.348281\pi$
$740$	50.8328	1.86865
$741$	−25.8885	−0.951039
$742$	−1.05573	−0.0387570
$743$	1.88854	0.0692840	0.0346420	−	0.999400i	$-0.488971\pi$
$743$	1.88854	0.0692840	0.0346420	+	0.999400i	$0.488971\pi$
$744$	7.63932	0.280071
$745$	−28.0000	−1.02584
$746$	13.4164	0.491210
$747$	16.8065	0.614918
$748$	9.70820	0.354967
$749$	−4.00000	−0.146157
$750$	−33.1672	−1.21109
$751$	29.5279	1.07749	0.538744	−	0.842470i	$-0.318899\pi$
$751$	29.5279	1.07749	0.538744	+	0.842470i	$0.318899\pi$
$752$	−2.76393	−0.100790
$753$	−5.30495	−0.193323
$754$	61.3050	2.23259
$755$	−17.8885	−0.651031
$756$	16.5836	0.603139
$757$	15.8885	0.577479	0.288739	−	0.957408i	$-0.406764\pi$
$757$	15.8885	0.577479	0.288739	+	0.957408i	$0.406764\pi$
$758$	83.4164	3.02982
$759$	3.05573	0.110916
$760$	−28.9443	−1.04992
$761$	−31.5967	−1.14538	−0.572691	−	0.819772i	$-0.694100\pi$
$761$	−31.5967	−1.14538	−0.572691	+	0.819772i	$0.694100\pi$
$762$	−8.44582	−0.305960
$763$	−4.47214	−0.161902
$764$	−9.16718	−0.331657
$765$	−9.52786	−0.344481
$766$	−10.4033	−0.375885
$767$	−4.00000	−0.144432
$768$	11.1246	0.401425
$769$	18.2918	0.659619	0.329810	−	0.944047i	$-0.393015\pi$
$769$	18.2918	0.659619	0.329810	+	0.944047i	$0.393015\pi$
$770$	4.47214	0.161165
$771$	7.41641	0.267095
$772$	35.6656	1.28363
$773$	−38.3607	−1.37974	−0.689869	−	0.723934i	$-0.742332\pi$
$773$	−38.3607	−1.37974	−0.689869	+	0.723934i	$0.742332\pi$
$774$	−26.3344	−0.946569
$775$	2.76393	0.0992834
$776$	38.9443	1.39802
$777$	10.4721	0.375686
$778$	35.5279	1.27374
$779$	−72.7214	−2.60551
$780$	24.0000	0.859338
$781$	10.4721	0.374722
$782$	−17.8885	−0.639693
$783$	46.8328	1.67367
$784$	−1.00000	−0.0357143
$785$	−21.8885	−0.781236
$786$	−60.4984	−2.15791
$787$	−43.4164	−1.54763	−0.773814	−	0.633413i	$-0.781653\pi$
$787$	−43.4164	−1.54763	−0.773814	+	0.633413i	$0.781653\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	40.0000	1.42314
$791$	2.00000	0.0711118
$792$	3.29180	0.116969
$793$	−23.4164	−0.831541
$794$	−80.2492	−2.84794
$795$	−1.16718	−0.0413958
$796$	−6.54102	−0.231840
$797$	−14.9443	−0.529353	−0.264677	−	0.964337i	$-0.585265\pi$
$797$	−14.9443	−0.529353	−0.264677	+	0.964337i	$0.585265\pi$
$798$	−17.8885	−0.633248
$799$	−8.94427	−0.316426
$800$	6.70820	0.237171
$801$	−2.94427	−0.104031
$802$	51.3050	1.81164
$803$	0.763932	0.0269586
$804$	−53.6656	−1.89264
$805$	−4.94427	−0.174263
$806$	−20.0000	−0.704470
$807$	−16.5836	−0.583770
$808$	−10.6525	−0.374753
$809$	21.0557	0.740280	0.370140	−	0.928976i	$-0.379310\pi$
$809$	21.0557	0.740280	0.370140	+	0.928976i	$0.379310\pi$
$810$	10.8065	0.379702
$811$	−34.8328	−1.22315	−0.611573	−	0.791188i	$-0.709463\pi$
$811$	−34.8328	−1.22315	−0.611573	+	0.791188i	$0.709463\pi$
$812$	25.4164	0.891941
$813$	12.9443	0.453975
$814$	18.9443	0.663996
$815$	6.83282	0.239343
$816$	−4.00000	−0.140028
$817$	51.7771	1.81145
$818$	20.4033	0.713383
$819$	−4.76393	−0.166465
$820$	67.4164	2.35428
$821$	44.8328	1.56468	0.782338	−	0.622854i	$-0.214027\pi$
$821$	44.8328	1.56468	0.782338	+	0.622854i	$0.214027\pi$
$822$	−45.5279	−1.58797
$823$	−14.1115	−0.491894	−0.245947	−	0.969283i	$-0.579099\pi$
$823$	−14.1115	−0.491894	−0.245947	+	0.969283i	$0.579099\pi$
$824$	17.2361	0.600447
$825$	−1.23607	−0.0430344
$826$	−2.76393	−0.0961695
$827$	−12.9443	−0.450116	−0.225058	−	0.974345i	$-0.572257\pi$
$827$	−12.9443	−0.450116	−0.225058	+	0.974345i	$0.572257\pi$
$828$	−10.9180	−0.379425
$829$	36.8328	1.27926	0.639628	−	0.768684i	$-0.279088\pi$
$829$	36.8328	1.27926	0.639628	+	0.768684i	$0.279088\pi$
$830$	51.0557	1.77217
$831$	24.5836	0.852795
$832$	−42.0689	−1.45848
$833$	−3.23607	−0.112123
$834$	4.22291	0.146227
$835$	−9.88854	−0.342207
$836$	−19.4164	−0.671531
$837$	−15.2786	−0.528107
$838$	55.1246	1.90425
$839$	44.0689	1.52143	0.760713	−	0.649088i	$-0.224849\pi$
$839$	44.0689	1.52143	0.760713	+	0.649088i	$0.224849\pi$
$840$	5.52786	0.190729
$841$	42.7771	1.47507
$842$	50.0000	1.72311
$843$	4.36068	0.150190
$844$	−41.6656	−1.43419
$845$	5.05573	0.173922
$846$	−9.09830	−0.312806
$847$	1.00000	0.0343604
$848$	0.472136	0.0162132
$849$	−36.9443	−1.26792
$850$	7.23607	0.248195
$851$	−20.9443	−0.717960
$852$	38.8328	1.33039
$853$	−30.6525	−1.04952	−0.524760	−	0.851250i	$-0.675845\pi$
$853$	−30.6525	−1.04952	−0.524760	+	0.851250i	$0.675845\pi$
$854$	−16.1803	−0.553680
$855$	19.0557	0.651692
$856$	−8.94427	−0.305709
$857$	15.2361	0.520454	0.260227	−	0.965547i	$-0.416203\pi$
$857$	15.2361	0.520454	0.260227	+	0.965547i	$0.416203\pi$
$858$	8.94427	0.305352
$859$	−26.5410	−0.905568	−0.452784	−	0.891620i	$-0.649569\pi$
$859$	−26.5410	−0.905568	−0.452784	+	0.891620i	$0.649569\pi$
$860$	−48.0000	−1.63679
$861$	13.8885	0.473320
$862$	26.8328	0.913929
$863$	−3.05573	−0.104018	−0.0520091	−	0.998647i	$-0.516562\pi$
$863$	−3.05573	−0.104018	−0.0520091	+	0.998647i	$0.516562\pi$
$864$	−37.0820	−1.26156
$865$	25.5279	0.867973
$866$	−18.9443	−0.643753
$867$	8.06888	0.274034
$868$	−8.29180	−0.281442
$869$	8.94427	0.303414
$870$	46.8328	1.58778
$871$	46.8328	1.58687
$872$	−10.0000	−0.338643
$873$	−25.6393	−0.867760
$874$	35.7771	1.21018
$875$	12.0000	0.405674
$876$	2.83282	0.0957120
$877$	14.5836	0.492453	0.246226	−	0.969212i	$-0.420809\pi$
$877$	14.5836	0.492453	0.246226	+	0.969212i	$0.420809\pi$
$878$	23.4164	0.790265
$879$	31.0557	1.04748
$880$	−2.00000	−0.0674200
$881$	2.58359	0.0870434	0.0435217	−	0.999052i	$-0.486142\pi$
$881$	2.58359	0.0870434	0.0435217	+	0.999052i	$0.486142\pi$
$882$	−3.29180	−0.110841
$883$	−8.94427	−0.300999	−0.150499	−	0.988610i	$-0.548088\pi$
$883$	−8.94427	−0.300999	−0.150499	+	0.988610i	$0.548088\pi$
$884$	−31.4164	−1.05665
$885$	−3.05573	−0.102717
$886$	−55.7771	−1.87387
$887$	−4.36068	−0.146417	−0.0732086	−	0.997317i	$-0.523324\pi$
$887$	−4.36068	−0.146417	−0.0732086	+	0.997317i	$0.523324\pi$
$888$	23.4164	0.785803
$889$	3.05573	0.102486
$890$	−8.94427	−0.299813
$891$	2.41641	0.0809527
$892$	−30.5410	−1.02259
$893$	17.8885	0.598617
$894$	−38.6950	−1.29416
$895$	17.8885	0.597948
$896$	−15.6525	−0.522913
$897$	−9.88854	−0.330169
$898$	−63.6656	−2.12455
$899$	−23.4164	−0.780981
$900$	4.41641	0.147214
$901$	1.52786	0.0509005
$902$	25.1246	0.836558
$903$	−9.88854	−0.329070
$904$	4.47214	0.148741
$905$	50.8328	1.68974
$906$	−24.7214	−0.821312
$907$	22.4721	0.746175	0.373088	−	0.927796i	$-0.378299\pi$
$907$	22.4721	0.746175	0.373088	+	0.927796i	$0.378299\pi$
$908$	17.6656	0.586255
$909$	7.01316	0.232612
$910$	−14.4721	−0.479747
$911$	42.4721	1.40716	0.703582	−	0.710614i	$-0.251583\pi$
$911$	42.4721	1.40716	0.703582	+	0.710614i	$0.251583\pi$
$912$	8.00000	0.264906
$913$	11.4164	0.377828
$914$	−64.4721	−2.13255
$915$	−17.8885	−0.591377
$916$	−13.4164	−0.443291
$917$	21.8885	0.722823
$918$	−40.0000	−1.32020
$919$	41.8885	1.38178	0.690888	−	0.722962i	$-0.257220\pi$
$919$	41.8885	1.38178	0.690888	+	0.722962i	$0.257220\pi$
$920$	−11.0557	−0.364497
$921$	11.0557	0.364299
$922$	−27.2361	−0.896972
$923$	−33.8885	−1.11546
$924$	3.70820	0.121991
$925$	8.47214	0.278562
$926$	−12.3607	−0.406197
$927$	−11.3475	−0.372702
$928$	−56.8328	−1.86563
$929$	−52.2492	−1.71424	−0.857121	−	0.515116i	$-0.827749\pi$
$929$	−52.2492	−1.71424	−0.857121	+	0.515116i	$0.827749\pi$
$930$	−15.2786	−0.501006
$931$	6.47214	0.212116
$932$	28.2492	0.925334
$933$	10.2492	0.335545
$934$	53.8197	1.76103
$935$	−6.47214	−0.211661
$936$	−10.6525	−0.348187
$937$	−10.6525	−0.348001	−0.174001	−	0.984746i	$-0.555669\pi$
$937$	−10.6525	−0.348001	−0.174001	+	0.984746i	$0.555669\pi$
$938$	32.3607	1.05661
$939$	−18.4721	−0.602815
$940$	−16.5836	−0.540897
$941$	7.59675	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$941$	7.59675	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$942$	−30.2492	−0.985573
$943$	−27.7771	−0.904546
$944$	1.23607	0.0402306
$945$	−11.0557	−0.359643
$946$	−17.8885	−0.581607
$947$	−5.16718	−0.167911	−0.0839555	−	0.996470i	$-0.526755\pi$
$947$	−5.16718	−0.167911	−0.0839555	+	0.996470i	$0.526755\pi$
$948$	33.1672	1.07722
$949$	−2.47214	−0.0802489
$950$	−14.4721	−0.469538
$951$	−17.3050	−0.561152
$952$	−7.23607	−0.234522
$953$	22.9443	0.743238	0.371619	−	0.928385i	$-0.378803\pi$
$953$	22.9443	0.743238	0.371619	+	0.928385i	$0.378803\pi$
$954$	1.55418	0.0503183
$955$	6.11146	0.197762
$956$	−29.6656	−0.959455
$957$	10.4721	0.338516
$958$	30.2492	0.977308
$959$	16.4721	0.531913
$960$	−32.1378	−1.03724
$961$	−23.3607	−0.753570
$962$	−61.3050	−1.97655
$963$	5.88854	0.189756
$964$	−39.3738	−1.26815
$965$	−23.7771	−0.765412
$966$	−6.83282	−0.219842
$967$	−13.8885	−0.446625	−0.223313	−	0.974747i	$-0.571687\pi$
$967$	−13.8885	−0.446625	−0.223313	+	0.974747i	$0.571687\pi$
$968$	2.23607	0.0718699
$969$	25.8885	0.831660
$970$	−77.8885	−2.50085
$971$	11.1246	0.357006	0.178503	−	0.983939i	$-0.442875\pi$
$971$	11.1246	0.357006	0.178503	+	0.983939i	$0.442875\pi$
$972$	−40.7902	−1.30835
$973$	−1.52786	−0.0489811
$974$	−81.3050	−2.60518
$975$	4.00000	0.128103
$976$	7.23607	0.231621
$977$	22.9443	0.734052	0.367026	−	0.930211i	$-0.380376\pi$
$977$	22.9443	0.734052	0.367026	+	0.930211i	$0.380376\pi$
$978$	9.44272	0.301945
$979$	−2.00000	−0.0639203
$980$	−6.00000	−0.191663
$981$	6.58359	0.210198
$982$	0	0
$983$	21.8197	0.695939	0.347970	−	0.937506i	$-0.386871\pi$
$983$	21.8197	0.695939	0.347970	+	0.937506i	$0.386871\pi$
$984$	31.0557	0.990020
$985$	4.00000	0.127451
$986$	−61.3050	−1.95235
$987$	−3.41641	−0.108745
$988$	62.8328	1.99898
$989$	19.7771	0.628875
$990$	−6.58359	−0.209240
$991$	54.2492	1.72328	0.861642	−	0.507517i	$-0.169437\pi$
$991$	54.2492	1.72328	0.861642	+	0.507517i	$0.169437\pi$
$992$	18.5410	0.588678
$993$	17.1672	0.544784
$994$	−23.4164	−0.742723
$995$	4.36068	0.138243
$996$	42.3344	1.34142
$997$	1.34752	0.0426765	0.0213383	−	0.999772i	$-0.493207\pi$
$997$	1.34752	0.0426765	0.0213383	+	0.999772i	$0.493207\pi$
$998$	3.41641	0.108145
$999$	−46.8328	−1.48172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	77.2.a.d.1.2	✓	2
3.2	odd	2		693.2.a.h.1.1		2
4.3	odd	2		1232.2.a.m.1.2		2
5.2	odd	4		1925.2.b.h.1849.3		4
5.3	odd	4		1925.2.b.h.1849.2		4
5.4	even	2		1925.2.a.r.1.1		2
7.2	even	3		539.2.e.i.67.1		4
7.3	odd	6		539.2.e.j.177.1		4
7.4	even	3		539.2.e.i.177.1		4
7.5	odd	6		539.2.e.j.67.1		4
7.6	odd	2		539.2.a.f.1.2		2
8.3	odd	2		4928.2.a.bv.1.1		2
8.5	even	2		4928.2.a.bm.1.2		2
11.2	odd	10		847.2.f.m.323.1		4
11.3	even	5		847.2.f.n.372.1		4
11.4	even	5		847.2.f.n.148.1		4
11.5	even	5		847.2.f.a.729.1		4
11.6	odd	10		847.2.f.m.729.1		4
11.7	odd	10		847.2.f.b.148.1		4
11.8	odd	10		847.2.f.b.372.1		4
11.9	even	5		847.2.f.a.323.1		4
11.10	odd	2		847.2.a.f.1.1		2
21.20	even	2		4851.2.a.y.1.1		2
28.27	even	2		8624.2.a.ce.1.1		2
33.32	even	2		7623.2.a.bl.1.2		2
77.76	even	2		5929.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.a.d.1.2	✓	2	1.1	even	1	trivial
539.2.a.f.1.2		2	7.6	odd	2
539.2.e.i.67.1		4	7.2	even	3
539.2.e.i.177.1		4	7.4	even	3
539.2.e.j.67.1		4	7.5	odd	6
539.2.e.j.177.1		4	7.3	odd	6
693.2.a.h.1.1		2	3.2	odd	2
847.2.a.f.1.1		2	11.10	odd	2
847.2.f.a.323.1		4	11.9	even	5
847.2.f.a.729.1		4	11.5	even	5
847.2.f.b.148.1		4	11.7	odd	10
847.2.f.b.372.1		4	11.8	odd	10
847.2.f.m.323.1		4	11.2	odd	10
847.2.f.m.729.1		4	11.6	odd	10
847.2.f.n.148.1		4	11.4	even	5
847.2.f.n.372.1		4	11.3	even	5
1232.2.a.m.1.2		2	4.3	odd	2
1925.2.a.r.1.1		2	5.4	even	2
1925.2.b.h.1849.2		4	5.3	odd	4
1925.2.b.h.1849.3		4	5.2	odd	4
4851.2.a.y.1.1		2	21.20	even	2
4928.2.a.bm.1.2		2	8.5	even	2
4928.2.a.bv.1.1		2	8.3	odd	2
5929.2.a.m.1.1		2	77.76	even	2
7623.2.a.bl.1.2		2	33.32	even	2
8624.2.a.ce.1.1		2	28.27	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:	\( N \)	\(=\)	\( 77 = 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	77.a (trivial)

\(f(q)\)	\(=\)	\(q+2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} -2.00000 q^{5} -2.76393 q^{6} +1.00000 q^{7} +2.23607 q^{8} -1.47214 q^{9} +O(q^{10})\)	\(q+2.23607 q^{2} -1.23607 q^{3} +3.00000 q^{4} -2.00000 q^{5} -2.76393 q^{6} +1.00000 q^{7} +2.23607 q^{8} -1.47214 q^{9} -4.47214 q^{10} -1.00000 q^{11} -3.70820 q^{12} +3.23607 q^{13} +2.23607 q^{14} +2.47214 q^{15} -1.00000 q^{16} -3.23607 q^{17} -3.29180 q^{18} +6.47214 q^{19} -6.00000 q^{20} -1.23607 q^{21} -2.23607 q^{22} +2.47214 q^{23} -2.76393 q^{24} -1.00000 q^{25} +7.23607 q^{26} +5.52786 q^{27} +3.00000 q^{28} +8.47214 q^{29} +5.52786 q^{30} -2.76393 q^{31} -6.70820 q^{32} +1.23607 q^{33} -7.23607 q^{34} -2.00000 q^{35} -4.41641 q^{36} -8.47214 q^{37} +14.4721 q^{38} -4.00000 q^{39} -4.47214 q^{40} -11.2361 q^{41} -2.76393 q^{42} +8.00000 q^{43} -3.00000 q^{44} +2.94427 q^{45} +5.52786 q^{46} +2.76393 q^{47} +1.23607 q^{48} +1.00000 q^{49} -2.23607 q^{50} +4.00000 q^{51} +9.70820 q^{52} -0.472136 q^{53} +12.3607 q^{54} +2.00000 q^{55} +2.23607 q^{56} -8.00000 q^{57} +18.9443 q^{58} -1.23607 q^{59} +7.41641 q^{60} -7.23607 q^{61} -6.18034 q^{62} -1.47214 q^{63} -13.0000 q^{64} -6.47214 q^{65} +2.76393 q^{66} +14.4721 q^{67} -9.70820 q^{68} -3.05573 q^{69} -4.47214 q^{70} -10.4721 q^{71} -3.29180 q^{72} -0.763932 q^{73} -18.9443 q^{74} +1.23607 q^{75} +19.4164 q^{76} -1.00000 q^{77} -8.94427 q^{78} -8.94427 q^{79} +2.00000 q^{80} -2.41641 q^{81} -25.1246 q^{82} -11.4164 q^{83} -3.70820 q^{84} +6.47214 q^{85} +17.8885 q^{86} -10.4721 q^{87} -2.23607 q^{88} +2.00000 q^{89} +6.58359 q^{90} +3.23607 q^{91} +7.41641 q^{92} +3.41641 q^{93} +6.18034 q^{94} -12.9443 q^{95} +8.29180 q^{96} +17.4164 q^{97} +2.23607 q^{98} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{4} - 4 q^{5} - 10 q^{6} + 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 6 * q^4 - 4 * q^5 - 10 * q^6 + 2 * q^7 + 6 * q^9	\( 2 q + 2 q^{3} + 6 q^{4} - 4 q^{5} - 10 q^{6} + 2 q^{7} + 6 q^{9} - 2 q^{11} + 6 q^{12} + 2 q^{13} - 4 q^{15} - 2 q^{16} - 2 q^{17} - 20 q^{18} + 4 q^{19} - 12 q^{20} + 2 q^{21} - 4 q^{23} - 10 q^{24} - 2 q^{25} + 10 q^{26} + 20 q^{27} + 6 q^{28} + 8 q^{29} + 20 q^{30} - 10 q^{31} - 2 q^{33} - 10 q^{34} - 4 q^{35} + 18 q^{36} - 8 q^{37} + 20 q^{38} - 8 q^{39} - 18 q^{41} - 10 q^{42} + 16 q^{43} - 6 q^{44} - 12 q^{45} + 20 q^{46} + 10 q^{47} - 2 q^{48} + 2 q^{49} + 8 q^{51} + 6 q^{52} + 8 q^{53} - 20 q^{54} + 4 q^{55} - 16 q^{57} + 20 q^{58} + 2 q^{59} - 12 q^{60} - 10 q^{61} + 10 q^{62} + 6 q^{63} - 26 q^{64} - 4 q^{65} + 10 q^{66} + 20 q^{67} - 6 q^{68} - 24 q^{69} - 12 q^{71} - 20 q^{72} - 6 q^{73} - 20 q^{74} - 2 q^{75} + 12 q^{76} - 2 q^{77} + 4 q^{80} + 22 q^{81} - 10 q^{82} + 4 q^{83} + 6 q^{84} + 4 q^{85} - 12 q^{87} + 4 q^{89} + 40 q^{90} + 2 q^{91} - 12 q^{92} - 20 q^{93} - 10 q^{94} - 8 q^{95} + 30 q^{96} + 8 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^4 - 4 * q^5 - 10 * q^6 + 2 * q^7 + 6 * q^9 - 2 * q^11 + 6 * q^12 + 2 * q^13 - 4 * q^15 - 2 * q^16 - 2 * q^17 - 20 * q^18 + 4 * q^19 - 12 * q^20 + 2 * q^21 - 4 * q^23 - 10 * q^24 - 2 * q^25 + 10 * q^26 + 20 * q^27 + 6 * q^28 + 8 * q^29 + 20 * q^30 - 10 * q^31 - 2 * q^33 - 10 * q^34 - 4 * q^35 + 18 * q^36 - 8 * q^37 + 20 * q^38 - 8 * q^39 - 18 * q^41 - 10 * q^42 + 16 * q^43 - 6 * q^44 - 12 * q^45 + 20 * q^46 + 10 * q^47 - 2 * q^48 + 2 * q^49 + 8 * q^51 + 6 * q^52 + 8 * q^53 - 20 * q^54 + 4 * q^55 - 16 * q^57 + 20 * q^58 + 2 * q^59 - 12 * q^60 - 10 * q^61 + 10 * q^62 + 6 * q^63 - 26 * q^64 - 4 * q^65 + 10 * q^66 + 20 * q^67 - 6 * q^68 - 24 * q^69 - 12 * q^71 - 20 * q^72 - 6 * q^73 - 20 * q^74 - 2 * q^75 + 12 * q^76 - 2 * q^77 + 4 * q^80 + 22 * q^81 - 10 * q^82 + 4 * q^83 + 6 * q^84 + 4 * q^85 - 12 * q^87 + 4 * q^89 + 40 * q^90 + 2 * q^91 - 12 * q^92 - 20 * q^93 - 10 * q^94 - 8 * q^95 + 30 * q^96 + 8 * q^97 - 6 * q^99

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