LMFDB - Embedded newform 3450.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.76300$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.76300 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.76300 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.18291 q^{11} +1.00000 q^{12} -6.34310 q^{13} +3.76300 q^{14} +1.00000 q^{16} -2.58010 q^{17} -1.00000 q^{18} -6.34310 q^{19} -3.76300 q^{21} -3.18291 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.34310 q^{26} +1.00000 q^{27} -3.76300 q^{28} -1.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} +3.18291 q^{33} +2.58010 q^{34} +1.00000 q^{36} +2.58010 q^{37} +6.34310 q^{38} -6.34310 q^{39} +12.3431 q^{41} +3.76300 q^{42} +10.3431 q^{43} +3.18291 q^{44} +1.00000 q^{46} +3.34310 q^{47} +1.00000 q^{48} +7.16019 q^{49} -2.58010 q^{51} -6.34310 q^{52} +7.16019 q^{53} -1.00000 q^{54} +3.76300 q^{56} -6.34310 q^{57} +1.00000 q^{58} -4.36581 q^{59} -1.52601 q^{61} -6.00000 q^{62} -3.76300 q^{63} +1.00000 q^{64} -3.18291 q^{66} +10.6862 q^{67} -2.58010 q^{68} -1.00000 q^{69} +1.81709 q^{71} -1.00000 q^{72} +8.52601 q^{73} -2.58010 q^{74} -6.34310 q^{76} -11.9773 q^{77} +6.34310 q^{78} -3.18291 q^{79} +1.00000 q^{81} -12.3431 q^{82} -7.28901 q^{83} -3.76300 q^{84} -10.3431 q^{86} -1.00000 q^{87} -3.18291 q^{88} +4.58010 q^{89} +23.8691 q^{91} -1.00000 q^{92} +6.00000 q^{93} -3.34310 q^{94} -1.00000 q^{96} +11.5260 q^{97} -7.16019 q^{98} +3.18291 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} - q^{13} - q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} - q^{19} + q^{21} - 3 q^{22} - 3 q^{23} - 3 q^{24} + q^{26} + 3 q^{27} + q^{28} - 3 q^{29} + 18 q^{31} - 3 q^{32} + 3 q^{33} + 2 q^{34} + 3 q^{36} + 2 q^{37} + q^{38} - q^{39} + 19 q^{41} - q^{42} + 13 q^{43} + 3 q^{44} + 3 q^{46} - 8 q^{47} + 3 q^{48} + 10 q^{49} - 2 q^{51} - q^{52} + 10 q^{53} - 3 q^{54} - q^{56} - q^{57} + 3 q^{58} + 20 q^{61} - 18 q^{62} + q^{63} + 3 q^{64} - 3 q^{66} - 4 q^{67} - 2 q^{68} - 3 q^{69} + 12 q^{71} - 3 q^{72} + q^{73} - 2 q^{74} - q^{76} - 31 q^{77} + q^{78} - 3 q^{79} + 3 q^{81} - 19 q^{82} + 15 q^{83} + q^{84} - 13 q^{86} - 3 q^{87} - 3 q^{88} + 8 q^{89} + 29 q^{91} - 3 q^{92} + 18 q^{93} + 8 q^{94} - 3 q^{96} + 10 q^{97} - 10 q^{98} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 - q^13 - q^14 + 3 * q^16 - 2 * q^17 - 3 * q^18 - q^19 + q^21 - 3 * q^22 - 3 * q^23 - 3 * q^24 + q^26 + 3 * q^27 + q^28 - 3 * q^29 + 18 * q^31 - 3 * q^32 + 3 * q^33 + 2 * q^34 + 3 * q^36 + 2 * q^37 + q^38 - q^39 + 19 * q^41 - q^42 + 13 * q^43 + 3 * q^44 + 3 * q^46 - 8 * q^47 + 3 * q^48 + 10 * q^49 - 2 * q^51 - q^52 + 10 * q^53 - 3 * q^54 - q^56 - q^57 + 3 * q^58 + 20 * q^61 - 18 * q^62 + q^63 + 3 * q^64 - 3 * q^66 - 4 * q^67 - 2 * q^68 - 3 * q^69 + 12 * q^71 - 3 * q^72 + q^73 - 2 * q^74 - q^76 - 31 * q^77 + q^78 - 3 * q^79 + 3 * q^81 - 19 * q^82 + 15 * q^83 + q^84 - 13 * q^86 - 3 * q^87 - 3 * q^88 + 8 * q^89 + 29 * q^91 - 3 * q^92 + 18 * q^93 + 8 * q^94 - 3 * q^96 + 10 * q^97 - 10 * q^98 + 3 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−3.76300	−1.42228	−0.711141	−	0.703050i	$-0.751821\pi$
$7$	−3.76300	−1.42228	−0.711141	+	0.703050i	$0.751821\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	3.18291	0.959682	0.479841	−	0.877355i	$-0.340694\pi$
$11$	3.18291	0.959682	0.479841	+	0.877355i	$0.340694\pi$
$12$	1.00000	0.288675
$13$	−6.34310	−1.75926	−0.879630	−	0.475659i	$-0.842210\pi$
$13$	−6.34310	−1.75926	−0.879630	+	0.475659i	$0.842210\pi$
$14$	3.76300	1.00570
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.58010	−0.625765	−0.312883	−	0.949792i	$-0.601295\pi$
$17$	−2.58010	−0.625765	−0.312883	+	0.949792i	$0.601295\pi$
$18$	−1.00000	−0.235702
$19$	−6.34310	−1.45521	−0.727603	−	0.685998i	$-0.759366\pi$
$19$	−6.34310	−1.45521	−0.727603	+	0.685998i	$0.759366\pi$
$20$	0	0
$21$	−3.76300	−0.821155
$22$	−3.18291	−0.678598
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	6.34310	1.24398
$27$	1.00000	0.192450
$28$	−3.76300	−0.711141
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	3.18291	0.554073
$34$	2.58010	0.442483
$35$	0	0
$36$	1.00000	0.166667
$37$	2.58010	0.424165	0.212083	−	0.977252i	$-0.431975\pi$
$37$	2.58010	0.424165	0.212083	+	0.977252i	$0.431975\pi$
$38$	6.34310	1.02899
$39$	−6.34310	−1.01571
$40$	0	0
$41$	12.3431	1.92767	0.963834	−	0.266503i	$-0.0858681\pi$
$41$	12.3431	1.92767	0.963834	+	0.266503i	$0.0858681\pi$
$42$	3.76300	0.580644
$43$	10.3431	1.57731	0.788654	−	0.614837i	$-0.210778\pi$
$43$	10.3431	1.57731	0.788654	+	0.614837i	$0.210778\pi$
$44$	3.18291	0.479841
$45$	0	0
$46$	1.00000	0.147442
$47$	3.34310	0.487641	0.243821	−	0.969820i	$-0.421599\pi$
$47$	3.34310	0.487641	0.243821	+	0.969820i	$0.421599\pi$
$48$	1.00000	0.144338
$49$	7.16019	1.02288
$50$	0	0
$51$	−2.58010	−0.361286
$52$	−6.34310	−0.879630
$53$	7.16019	0.983528	0.491764	−	0.870728i	$-0.336352\pi$
$53$	7.16019	0.983528	0.491764	+	0.870728i	$0.336352\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	3.76300	0.502852
$57$	−6.34310	−0.840164
$58$	1.00000	0.131306
$59$	−4.36581	−0.568380	−0.284190	−	0.958768i	$-0.591725\pi$
$59$	−4.36581	−0.568380	−0.284190	+	0.958768i	$0.591725\pi$
$60$	0	0
$61$	−1.52601	−0.195385	−0.0976926	−	0.995217i	$-0.531146\pi$
$61$	−1.52601	−0.195385	−0.0976926	+	0.995217i	$0.531146\pi$
$62$	−6.00000	−0.762001
$63$	−3.76300	−0.474094
$64$	1.00000	0.125000
$65$	0	0
$66$	−3.18291	−0.391789
$67$	10.6862	1.30553	0.652764	−	0.757562i	$-0.273610\pi$
$67$	10.6862	1.30553	0.652764	+	0.757562i	$0.273610\pi$
$68$	−2.58010	−0.312883
$69$	−1.00000	−0.120386
$70$	0	0
$71$	1.81709	0.215649	0.107825	−	0.994170i	$-0.465611\pi$
$71$	1.81709	0.215649	0.107825	+	0.994170i	$0.465611\pi$
$72$	−1.00000	−0.117851
$73$	8.52601	0.997894	0.498947	−	0.866633i	$-0.333720\pi$
$73$	8.52601	0.997894	0.498947	+	0.866633i	$0.333720\pi$
$74$	−2.58010	−0.299930
$75$	0	0
$76$	−6.34310	−0.727603
$77$	−11.9773	−1.36494
$78$	6.34310	0.718215
$79$	−3.18291	−0.358105	−0.179052	−	0.983840i	$-0.557303\pi$
$79$	−3.18291	−0.358105	−0.179052	+	0.983840i	$0.557303\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−12.3431	−1.36307
$83$	−7.28901	−0.800073	−0.400036	−	0.916499i	$-0.631003\pi$
$83$	−7.28901	−0.800073	−0.400036	+	0.916499i	$0.631003\pi$
$84$	−3.76300	−0.410577
$85$	0	0
$86$	−10.3431	−1.11533
$87$	−1.00000	−0.107211
$88$	−3.18291	−0.339299
$89$	4.58010	0.485489	0.242745	−	0.970090i	$-0.421952\pi$
$89$	4.58010	0.485489	0.242745	+	0.970090i	$0.421952\pi$
$90$	0	0
$91$	23.8691	2.50216
$92$	−1.00000	−0.104257
$93$	6.00000	0.622171
$94$	−3.34310	−0.344814
$95$	0	0
$96$	−1.00000	−0.102062
$97$	11.5260	1.17029	0.585144	−	0.810929i	$-0.301038\pi$
$97$	11.5260	1.17029	0.585144	+	0.810929i	$0.301038\pi$
$98$	−7.16019	−0.723289
$99$	3.18291	0.319894
$100$	0	0
$101$	−7.70891	−0.767066	−0.383533	−	0.923527i	$-0.625293\pi$
$101$	−7.70891	−0.767066	−0.383533	+	0.923527i	$0.625293\pi$
$102$	2.58010	0.255468
$103$	3.39719	0.334735	0.167368	−	0.985895i	$-0.446473\pi$
$103$	3.39719	0.334735	0.167368	+	0.985895i	$0.446473\pi$
$104$	6.34310	0.621992
$105$	0	0
$106$	−7.16019	−0.695459
$107$	6.68620	0.646379	0.323190	−	0.946334i	$-0.395245\pi$
$107$	6.68620	0.646379	0.323190	+	0.946334i	$0.395245\pi$
$108$	1.00000	0.0962250
$109$	5.78572	0.554171	0.277086	−	0.960845i	$-0.410631\pi$
$109$	5.78572	0.554171	0.277086	+	0.960845i	$0.410631\pi$
$110$	0	0
$111$	2.58010	0.244892
$112$	−3.76300	−0.355570
$113$	−0.580097	−0.0545709	−0.0272855	−	0.999628i	$-0.508686\pi$
$113$	−0.580097	−0.0545709	−0.0272855	+	0.999628i	$0.508686\pi$
$114$	6.34310	0.594086
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	−6.34310	−0.586420
$118$	4.36581	0.401906
$119$	9.70891	0.890015
$120$	0	0
$121$	−0.869107	−0.0790097
$122$	1.52601	0.138158
$123$	12.3431	1.11294
$124$	6.00000	0.538816
$125$	0	0
$126$	3.76300	0.335235
$127$	−14.3204	−1.27073	−0.635364	−	0.772212i	$-0.719150\pi$
$127$	−14.3204	−1.27073	−0.635364	+	0.772212i	$0.719150\pi$
$128$	−1.00000	−0.0883883
$129$	10.3431	0.910659
$130$	0	0
$131$	15.8918	1.38847	0.694237	−	0.719746i	$-0.255742\pi$
$131$	15.8918	1.38847	0.694237	+	0.719746i	$0.255742\pi$
$132$	3.18291	0.277036
$133$	23.8691	2.06971
$134$	−10.6862	−0.923147
$135$	0	0
$136$	2.58010	0.221241
$137$	−15.4199	−1.31741	−0.658706	−	0.752401i	$-0.728896\pi$
$137$	−15.4199	−1.31741	−0.658706	+	0.752401i	$0.728896\pi$
$138$	1.00000	0.0851257
$139$	3.02271	0.256383	0.128192	−	0.991749i	$-0.459083\pi$
$139$	3.02271	0.256383	0.128192	+	0.991749i	$0.459083\pi$
$140$	0	0
$141$	3.34310	0.281540
$142$	−1.81709	−0.152487
$143$	−20.1895	−1.68833
$144$	1.00000	0.0833333
$145$	0	0
$146$	−8.52601	−0.705617
$147$	7.16019	0.590563
$148$	2.58010	0.212083
$149$	16.6862	1.36699	0.683493	−	0.729957i	$-0.260460\pi$
$149$	16.6862	1.36699	0.683493	+	0.729957i	$0.260460\pi$
$150$	0	0
$151$	−7.16019	−0.582688	−0.291344	−	0.956618i	$-0.594102\pi$
$151$	−7.16019	−0.582688	−0.291344	+	0.956618i	$0.594102\pi$
$152$	6.34310	0.514493
$153$	−2.58010	−0.208588
$154$	11.9773	0.965157
$155$	0	0
$156$	−6.34310	−0.507854
$157$	22.2122	1.77273	0.886364	−	0.462990i	$-0.153223\pi$
$157$	22.2122	1.77273	0.886364	+	0.462990i	$0.153223\pi$
$158$	3.18291	0.253218
$159$	7.16019	0.567840
$160$	0	0
$161$	3.76300	0.296566
$162$	−1.00000	−0.0785674
$163$	−8.68620	−0.680356	−0.340178	−	0.940361i	$-0.610487\pi$
$163$	−8.68620	−0.680356	−0.340178	+	0.940361i	$0.610487\pi$
$164$	12.3431	0.963834
$165$	0	0
$166$	7.28901	0.565737
$167$	−25.2349	−1.95274	−0.976368	−	0.216113i	$-0.930662\pi$
$167$	−25.2349	−1.95274	−0.976368	+	0.216113i	$0.930662\pi$
$168$	3.76300	0.290322
$169$	27.2349	2.09499
$170$	0	0
$171$	−6.34310	−0.485069
$172$	10.3431	0.788654
$173$	0.817094	0.0621225	0.0310612	−	0.999517i	$-0.490111\pi$
$173$	0.817094	0.0621225	0.0310612	+	0.999517i	$0.490111\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	3.18291	0.239921
$177$	−4.36581	−0.328155
$178$	−4.58010	−0.343293
$179$	−3.16019	−0.236204	−0.118102	−	0.993001i	$-0.537681\pi$
$179$	−3.16019	−0.236204	−0.118102	+	0.993001i	$0.537681\pi$
$180$	0	0
$181$	3.37448	0.250823	0.125411	−	0.992105i	$-0.459975\pi$
$181$	3.37448	0.250823	0.125411	+	0.992105i	$0.459975\pi$
$182$	−23.8691	−1.76930
$183$	−1.52601	−0.112806
$184$	1.00000	0.0737210
$185$	0	0
$186$	−6.00000	−0.439941
$187$	−8.21221	−0.600536
$188$	3.34310	0.243821
$189$	−3.76300	−0.273718
$190$	0	0
$191$	−11.8691	−0.858818	−0.429409	−	0.903110i	$-0.641278\pi$
$191$	−11.8691	−0.858818	−0.429409	+	0.903110i	$0.641278\pi$
$192$	1.00000	0.0721688
$193$	−2.97729	−0.214310	−0.107155	−	0.994242i	$-0.534174\pi$
$193$	−2.97729	−0.214310	−0.107155	+	0.994242i	$0.534174\pi$
$194$	−11.5260	−0.827519
$195$	0	0
$196$	7.16019	0.511442
$197$	−24.8918	−1.77347	−0.886734	−	0.462279i	$-0.847032\pi$
$197$	−24.8918	−1.77347	−0.886734	+	0.462279i	$0.847032\pi$
$198$	−3.18291	−0.226199
$199$	22.4492	1.59138	0.795691	−	0.605703i	$-0.207108\pi$
$199$	22.4492	1.59138	0.795691	+	0.605703i	$0.207108\pi$
$200$	0	0
$201$	10.6862	0.753746
$202$	7.70891	0.542397
$203$	3.76300	0.264111
$204$	−2.58010	−0.180643
$205$	0	0
$206$	−3.39719	−0.236693
$207$	−1.00000	−0.0695048
$208$	−6.34310	−0.439815
$209$	−20.1895	−1.39654
$210$	0	0
$211$	19.5553	1.34624	0.673121	−	0.739532i	$-0.264953\pi$
$211$	19.5553	1.34624	0.673121	+	0.739532i	$0.264953\pi$
$212$	7.16019	0.491764
$213$	1.81709	0.124505
$214$	−6.68620	−0.457059
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−22.5780	−1.53270
$218$	−5.78572	−0.391858
$219$	8.52601	0.576134
$220$	0	0
$221$	16.3658	1.10088
$222$	−2.58010	−0.173165
$223$	28.8984	1.93518	0.967590	−	0.252525i	$-0.0812612\pi$
$223$	28.8984	1.93518	0.967590	+	0.252525i	$0.0812612\pi$
$224$	3.76300	0.251426
$225$	0	0
$226$	0.580097	0.0385875
$227$	17.6321	1.17028	0.585142	−	0.810931i	$-0.301039\pi$
$227$	17.6321	1.17028	0.585142	+	0.810931i	$0.301039\pi$
$228$	−6.34310	−0.420082
$229$	−11.8918	−0.785834	−0.392917	−	0.919574i	$-0.628534\pi$
$229$	−11.8918	−0.785834	−0.392917	+	0.919574i	$0.628534\pi$
$230$	0	0
$231$	−11.9773	−0.788048
$232$	1.00000	0.0656532
$233$	9.02930	0.591529	0.295765	−	0.955261i	$-0.404426\pi$
$233$	9.02930	0.591529	0.295765	+	0.955261i	$0.404426\pi$
$234$	6.34310	0.414661
$235$	0	0
$236$	−4.36581	−0.284190
$237$	−3.18291	−0.206752
$238$	−9.70891	−0.629335
$239$	−20.0747	−1.29853	−0.649263	−	0.760564i	$-0.724923\pi$
$239$	−20.0747	−1.29853	−0.649263	+	0.760564i	$0.724923\pi$
$240$	0	0
$241$	−23.0066	−1.48198	−0.740992	−	0.671514i	$-0.765645\pi$
$241$	−23.0066	−1.48198	−0.740992	+	0.671514i	$0.765645\pi$
$242$	0.869107	0.0558683
$243$	1.00000	0.0641500
$244$	−1.52601	−0.0976926
$245$	0	0
$246$	−12.3431	−0.786967
$247$	40.2349	2.56009
$248$	−6.00000	−0.381000
$249$	−7.28901	−0.461922
$250$	0	0
$251$	−10.9459	−0.690900	−0.345450	−	0.938437i	$-0.612274\pi$
$251$	−10.9459	−0.690900	−0.345450	+	0.938437i	$0.612274\pi$
$252$	−3.76300	−0.237047
$253$	−3.18291	−0.200108
$254$	14.3204	0.898541
$255$	0	0
$256$	1.00000	0.0625000
$257$	−16.3204	−1.01804	−0.509019	−	0.860755i	$-0.669992\pi$
$257$	−16.3204	−1.01804	−0.509019	+	0.860755i	$0.669992\pi$
$258$	−10.3431	−0.643933
$259$	−9.70891	−0.603282
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	−15.8918	−0.981800
$263$	29.0066	1.78862	0.894311	−	0.447445i	$-0.147666\pi$
$263$	29.0066	1.78862	0.894311	+	0.447445i	$0.147666\pi$
$264$	−3.18291	−0.195894
$265$	0	0
$266$	−23.8691	−1.46351
$267$	4.58010	0.280297
$268$	10.6862	0.652764
$269$	−26.1895	−1.59680	−0.798401	−	0.602126i	$-0.794320\pi$
$269$	−26.1895	−1.59680	−0.798401	+	0.602126i	$0.794320\pi$
$270$	0	0
$271$	6.36581	0.386696	0.193348	−	0.981130i	$-0.438065\pi$
$271$	6.36581	0.386696	0.193348	+	0.981130i	$0.438065\pi$
$272$	−2.58010	−0.156441
$273$	23.8691	1.44462
$274$	15.4199	0.931550
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−9.50329	−0.570998	−0.285499	−	0.958379i	$-0.592159\pi$
$277$	−9.50329	−0.570998	−0.285499	+	0.958379i	$0.592159\pi$
$278$	−3.02271	−0.181290
$279$	6.00000	0.359211
$280$	0	0
$281$	27.2663	1.62657	0.813285	−	0.581865i	$-0.197677\pi$
$281$	27.2663	1.62657	0.813285	+	0.581865i	$0.197677\pi$
$282$	−3.34310	−0.199079
$283$	−10.3204	−0.613483	−0.306742	−	0.951793i	$-0.599239\pi$
$283$	−10.3204	−0.613483	−0.306742	+	0.951793i	$0.599239\pi$
$284$	1.81709	0.107825
$285$	0	0
$286$	20.1895	1.19383
$287$	−46.4471	−2.74169
$288$	−1.00000	−0.0589256
$289$	−10.3431	−0.608418
$290$	0	0
$291$	11.5260	0.675666
$292$	8.52601	0.498947
$293$	8.36581	0.488736	0.244368	−	0.969683i	$-0.421420\pi$
$293$	8.36581	0.488736	0.244368	+	0.969683i	$0.421420\pi$
$294$	−7.16019	−0.417591
$295$	0	0
$296$	−2.58010	−0.149965
$297$	3.18291	0.184691
$298$	−16.6862	−0.966606
$299$	6.34310	0.366831
$300$	0	0
$301$	−38.9211	−2.24338
$302$	7.16019	0.412023
$303$	−7.70891	−0.442865
$304$	−6.34310	−0.363802
$305$	0	0
$306$	2.58010	0.147494
$307$	33.7089	1.92387	0.961935	−	0.273280i	$-0.0881084\pi$
$307$	33.7089	1.92387	0.961935	+	0.273280i	$0.0881084\pi$
$308$	−11.9773	−0.682469
$309$	3.39719	0.193259
$310$	0	0
$311$	−1.86252	−0.105614	−0.0528069	−	0.998605i	$-0.516817\pi$
$311$	−1.86252	−0.105614	−0.0528069	+	0.998605i	$0.516817\pi$
$312$	6.34310	0.359107
$313$	−26.8984	−1.52039	−0.760194	−	0.649696i	$-0.774896\pi$
$313$	−26.8984	−1.52039	−0.760194	+	0.649696i	$0.774896\pi$
$314$	−22.2122	−1.25351
$315$	0	0
$316$	−3.18291	−0.179052
$317$	−11.0000	−0.617822	−0.308911	−	0.951091i	$-0.599964\pi$
$317$	−11.0000	−0.617822	−0.308911	+	0.951091i	$0.599964\pi$
$318$	−7.16019	−0.401524
$319$	−3.18291	−0.178209
$320$	0	0
$321$	6.68620	0.373187
$322$	−3.76300	−0.209704
$323$	16.3658	0.910618
$324$	1.00000	0.0555556
$325$	0	0
$326$	8.68620	0.481084
$327$	5.78572	0.319951
$328$	−12.3431	−0.681534
$329$	−12.5801	−0.693563
$330$	0	0
$331$	−6.18291	−0.339843	−0.169922	−	0.985458i	$-0.554351\pi$
$331$	−6.18291	−0.339843	−0.169922	+	0.985458i	$0.554351\pi$
$332$	−7.28901	−0.400036
$333$	2.58010	0.141388
$334$	25.2349	1.38079
$335$	0	0
$336$	−3.76300	−0.205289
$337$	−15.1602	−0.825828	−0.412914	−	0.910770i	$-0.635489\pi$
$337$	−15.1602	−0.825828	−0.412914	+	0.910770i	$0.635489\pi$
$338$	−27.2349	−1.48138
$339$	−0.580097	−0.0315065
$340$	0	0
$341$	19.0974	1.03418
$342$	6.34310	0.342996
$343$	−0.602810	−0.0325487
$344$	−10.3431	−0.557663
$345$	0	0
$346$	−0.817094	−0.0439272
$347$	29.8918	1.60468	0.802338	−	0.596869i	$-0.203589\pi$
$347$	29.8918	1.60468	0.802338	+	0.596869i	$0.203589\pi$
$348$	−1.00000	−0.0536056
$349$	7.02930	0.376270	0.188135	−	0.982143i	$-0.439756\pi$
$349$	7.02930	0.376270	0.188135	+	0.982143i	$0.439756\pi$
$350$	0	0
$351$	−6.34310	−0.338570
$352$	−3.18291	−0.169649
$353$	26.2349	1.39634	0.698172	−	0.715930i	$-0.253997\pi$
$353$	26.2349	1.39634	0.698172	+	0.715930i	$0.253997\pi$
$354$	4.36581	0.232040
$355$	0	0
$356$	4.58010	0.242745
$357$	9.70891	0.513850
$358$	3.16019	0.167021
$359$	19.5033	1.02934	0.514672	−	0.857387i	$-0.327914\pi$
$359$	19.5033	1.02934	0.514672	+	0.857387i	$0.327914\pi$
$360$	0	0
$361$	21.2349	1.11763
$362$	−3.37448	−0.177359
$363$	−0.869107	−0.0456163
$364$	23.8691	1.25108
$365$	0	0
$366$	1.52601	0.0797656
$367$	33.5033	1.74886	0.874429	−	0.485154i	$-0.161236\pi$
$367$	33.5033	1.74886	0.874429	+	0.485154i	$0.161236\pi$
$368$	−1.00000	−0.0521286
$369$	12.3431	0.642556
$370$	0	0
$371$	−26.9438	−1.39885
$372$	6.00000	0.311086
$373$	−10.1515	−0.525626	−0.262813	−	0.964847i	$-0.584650\pi$
$373$	−10.1515	−0.525626	−0.262813	+	0.964847i	$0.584650\pi$
$374$	8.21221	0.424643
$375$	0	0
$376$	−3.34310	−0.172407
$377$	6.34310	0.326686
$378$	3.76300	0.193548
$379$	20.5780	1.05702	0.528511	−	0.848926i	$-0.322751\pi$
$379$	20.5780	1.05702	0.528511	+	0.848926i	$0.322751\pi$
$380$	0	0
$381$	−14.3204	−0.733656
$382$	11.8691	0.607276
$383$	1.22833	0.0627648	0.0313824	−	0.999507i	$-0.490009\pi$
$383$	1.22833	0.0627648	0.0313824	+	0.999507i	$0.490009\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.97729	0.151540
$387$	10.3431	0.525769
$388$	11.5260	0.585144
$389$	35.3724	1.79345	0.896726	−	0.442586i	$-0.145939\pi$
$389$	35.3724	1.79345	0.896726	+	0.442586i	$0.145939\pi$
$390$	0	0
$391$	2.58010	0.130481
$392$	−7.16019	−0.361644
$393$	15.8918	0.801636
$394$	24.8918	1.25403
$395$	0	0
$396$	3.18291	0.159947
$397$	31.0066	1.55618	0.778088	−	0.628155i	$-0.216190\pi$
$397$	31.0066	1.55618	0.778088	+	0.628155i	$0.216190\pi$
$398$	−22.4492	−1.12528
$399$	23.8691	1.19495
$400$	0	0
$401$	15.2056	0.759332	0.379666	−	0.925124i	$-0.376039\pi$
$401$	15.2056	0.759332	0.379666	+	0.925124i	$0.376039\pi$
$402$	−10.6862	−0.532979
$403$	−38.0586	−1.89583
$404$	−7.70891	−0.383533
$405$	0	0
$406$	−3.76300	−0.186755
$407$	8.21221	0.407064
$408$	2.58010	0.127734
$409$	−2.57143	−0.127149	−0.0635746	−	0.997977i	$-0.520250\pi$
$409$	−2.57143	−0.127149	−0.0635746	+	0.997977i	$0.520250\pi$
$410$	0	0
$411$	−15.4199	−0.760608
$412$	3.39719	0.167368
$413$	16.4286	0.808397
$414$	1.00000	0.0491473
$415$	0	0
$416$	6.34310	0.310996
$417$	3.02271	0.148023
$418$	20.1895	0.987500
$419$	−19.0768	−0.931963	−0.465981	−	0.884795i	$-0.654299\pi$
$419$	−19.0768	−0.931963	−0.465981	+	0.884795i	$0.654299\pi$
$420$	0	0
$421$	−15.8918	−0.774520	−0.387260	−	0.921971i	$-0.626578\pi$
$421$	−15.8918	−0.774520	−0.387260	+	0.921971i	$0.626578\pi$
$422$	−19.5553	−0.951937
$423$	3.34310	0.162547
$424$	−7.16019	−0.347730
$425$	0	0
$426$	−1.81709	−0.0880385
$427$	5.74237	0.277893
$428$	6.68620	0.323190
$429$	−20.1895	−0.974758
$430$	0	0
$431$	−24.2122	−1.16626	−0.583130	−	0.812379i	$-0.698172\pi$
$431$	−24.2122	−1.16626	−0.583130	+	0.812379i	$0.698172\pi$
$432$	1.00000	0.0481125
$433$	−12.2122	−0.586881	−0.293441	−	0.955977i	$-0.594800\pi$
$433$	−12.2122	−0.586881	−0.293441	+	0.955977i	$0.594800\pi$
$434$	22.5780	1.08378
$435$	0	0
$436$	5.78572	0.277086
$437$	6.34310	0.303432
$438$	−8.52601	−0.407388
$439$	−0.794381	−0.0379137	−0.0189569	−	0.999820i	$-0.506035\pi$
$439$	−0.794381	−0.0379137	−0.0189569	+	0.999820i	$0.506035\pi$
$440$	0	0
$441$	7.16019	0.340962
$442$	−16.3658	−0.778442
$443$	14.2122	0.675242	0.337621	−	0.941282i	$-0.390378\pi$
$443$	14.2122	0.675242	0.337621	+	0.941282i	$0.390378\pi$
$444$	2.58010	0.122446
$445$	0	0
$446$	−28.8984	−1.36838
$447$	16.6862	0.789230
$448$	−3.76300	−0.177785
$449$	1.05201	0.0496476	0.0248238	−	0.999692i	$-0.492098\pi$
$449$	1.05201	0.0496476	0.0248238	+	0.999692i	$0.492098\pi$
$450$	0	0
$451$	39.2869	1.84995
$452$	−0.580097	−0.0272855
$453$	−7.16019	−0.336415
$454$	−17.6321	−0.827516
$455$	0	0
$456$	6.34310	0.297043
$457$	−0.686200	−0.0320991	−0.0160495	−	0.999871i	$-0.505109\pi$
$457$	−0.686200	−0.0320991	−0.0160495	+	0.999871i	$0.505109\pi$
$458$	11.8918	0.555668
$459$	−2.58010	−0.120429
$460$	0	0
$461$	−29.5780	−1.37759	−0.688793	−	0.724958i	$-0.741859\pi$
$461$	−29.5780	−1.37759	−0.688793	+	0.724958i	$0.741859\pi$
$462$	11.9773	0.557234
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	−9.02930	−0.418274
$467$	−29.2890	−1.35533	−0.677667	−	0.735369i	$-0.737009\pi$
$467$	−29.2890	−1.35533	−0.677667	+	0.735369i	$0.737009\pi$
$468$	−6.34310	−0.293210
$469$	−40.2122	−1.85683
$470$	0	0
$471$	22.2122	1.02348
$472$	4.36581	0.200953
$473$	32.9211	1.51371
$474$	3.18291	0.146196
$475$	0	0
$476$	9.70891	0.445007
$477$	7.16019	0.327843
$478$	20.0747	0.918197
$479$	15.9145	0.727154	0.363577	−	0.931564i	$-0.381555\pi$
$479$	15.9145	0.727154	0.363577	+	0.931564i	$0.381555\pi$
$480$	0	0
$481$	−16.3658	−0.746217
$482$	23.0066	1.04792
$483$	3.76300	0.171223
$484$	−0.869107	−0.0395049
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	38.9438	1.76471	0.882357	−	0.470581i	$-0.155956\pi$
$487$	38.9438	1.76471	0.882357	+	0.470581i	$0.155956\pi$
$488$	1.52601	0.0690791
$489$	−8.68620	−0.392804
$490$	0	0
$491$	27.4806	1.24018	0.620091	−	0.784530i	$-0.287096\pi$
$491$	27.4806	1.24018	0.620091	+	0.784530i	$0.287096\pi$
$492$	12.3431	0.556470
$493$	2.58010	0.116202
$494$	−40.2349	−1.81025
$495$	0	0
$496$	6.00000	0.269408
$497$	−6.83773	−0.306714
$498$	7.28901	0.326628
$499$	6.97729	0.312346	0.156173	−	0.987730i	$-0.450084\pi$
$499$	6.97729	0.312346	0.156173	+	0.987730i	$0.450084\pi$
$500$	0	0
$501$	−25.2349	−1.12741
$502$	10.9459	0.488540
$503$	7.65690	0.341404	0.170702	−	0.985323i	$-0.445396\pi$
$503$	7.65690	0.341404	0.170702	+	0.985323i	$0.445396\pi$
$504$	3.76300	0.167617
$505$	0	0
$506$	3.18291	0.141497
$507$	27.2349	1.20955
$508$	−14.3204	−0.635364
$509$	−34.6073	−1.53394	−0.766971	−	0.641681i	$-0.778237\pi$
$509$	−34.6073	−1.53394	−0.766971	+	0.641681i	$0.778237\pi$
$510$	0	0
$511$	−32.0834	−1.41929
$512$	−1.00000	−0.0441942
$513$	−6.34310	−0.280055
$514$	16.3204	0.719861
$515$	0	0
$516$	10.3431	0.455330
$517$	10.6408	0.467981
$518$	9.70891	0.426585
$519$	0.817094	0.0358664
$520$	0	0
$521$	25.4199	1.11367	0.556833	−	0.830624i	$-0.312016\pi$
$521$	25.4199	1.11367	0.556833	+	0.830624i	$0.312016\pi$
$522$	1.00000	0.0437688
$523$	25.8691	1.13118	0.565589	−	0.824688i	$-0.308649\pi$
$523$	25.8691	1.13118	0.565589	+	0.824688i	$0.308649\pi$
$524$	15.8918	0.694237
$525$	0	0
$526$	−29.0066	−1.26475
$527$	−15.4806	−0.674345
$528$	3.18291	0.138518
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.36581	−0.189460
$532$	23.8691	1.03486
$533$	−78.2935	−3.39127
$534$	−4.58010	−0.198200
$535$	0	0
$536$	−10.6862	−0.461574
$537$	−3.16019	−0.136372
$538$	26.1895	1.12911
$539$	22.7902	0.981645
$540$	0	0
$541$	34.6007	1.48760	0.743801	−	0.668401i	$-0.233021\pi$
$541$	34.6007	1.48760	0.743801	+	0.668401i	$0.233021\pi$
$542$	−6.36581	−0.273435
$543$	3.37448	0.144813
$544$	2.58010	0.110621
$545$	0	0
$546$	−23.8691	−1.02150
$547$	−3.55531	−0.152014	−0.0760070	−	0.997107i	$-0.524217\pi$
$547$	−3.55531	−0.152014	−0.0760070	+	0.997107i	$0.524217\pi$
$548$	−15.4199	−0.658706
$549$	−1.52601	−0.0651284
$550$	0	0
$551$	6.34310	0.270225
$552$	1.00000	0.0425628
$553$	11.9773	0.509326
$554$	9.50329	0.403756
$555$	0	0
$556$	3.02271	0.128192
$557$	15.2642	0.646766	0.323383	−	0.946268i	$-0.395180\pi$
$557$	15.2642	0.646766	0.323383	+	0.946268i	$0.395180\pi$
$558$	−6.00000	−0.254000
$559$	−65.6073	−2.77489
$560$	0	0
$561$	−8.21221	−0.346720
$562$	−27.2663	−1.15016
$563$	13.5487	0.571010	0.285505	−	0.958377i	$-0.407839\pi$
$563$	13.5487	0.571010	0.285505	+	0.958377i	$0.407839\pi$
$564$	3.34310	0.140770
$565$	0	0
$566$	10.3204	0.433798
$567$	−3.76300	−0.158031
$568$	−1.81709	−0.0762436
$569$	−18.4740	−0.774470	−0.387235	−	0.921981i	$-0.626570\pi$
$569$	−18.4740	−0.774470	−0.387235	+	0.921981i	$0.626570\pi$
$570$	0	0
$571$	−9.89182	−0.413960	−0.206980	−	0.978345i	$-0.566363\pi$
$571$	−9.89182	−0.413960	−0.206980	+	0.978345i	$0.566363\pi$
$572$	−20.1895	−0.844165
$573$	−11.8691	−0.495839
$574$	46.4471	1.93867
$575$	0	0
$576$	1.00000	0.0416667
$577$	29.5033	1.22824	0.614119	−	0.789213i	$-0.289511\pi$
$577$	29.5033	1.22824	0.614119	+	0.789213i	$0.289511\pi$
$578$	10.3431	0.430216
$579$	−2.97729	−0.123732
$580$	0	0
$581$	27.4286	1.13793
$582$	−11.5260	−0.477768
$583$	22.7902	0.943875
$584$	−8.52601	−0.352809
$585$	0	0
$586$	−8.36581	−0.345589
$587$	−14.4740	−0.597406	−0.298703	−	0.954346i	$-0.596554\pi$
$587$	−14.4740	−0.597406	−0.298703	+	0.954346i	$0.596554\pi$
$588$	7.16019	0.295281
$589$	−38.0586	−1.56818
$590$	0	0
$591$	−24.8918	−1.02391
$592$	2.58010	0.106041
$593$	−5.86911	−0.241015	−0.120508	−	0.992712i	$-0.538452\pi$
$593$	−5.86911	−0.241015	−0.120508	+	0.992712i	$0.538452\pi$
$594$	−3.18291	−0.130596
$595$	0	0
$596$	16.6862	0.683493
$597$	22.4492	0.918785
$598$	−6.34310	−0.259389
$599$	−3.95457	−0.161580	−0.0807898	−	0.996731i	$-0.525744\pi$
$599$	−3.95457	−0.161580	−0.0807898	+	0.996731i	$0.525744\pi$
$600$	0	0
$601$	9.23492	0.376700	0.188350	−	0.982102i	$-0.439686\pi$
$601$	9.23492	0.376700	0.188350	+	0.982102i	$0.439686\pi$
$602$	38.9211	1.58631
$603$	10.6862	0.435176
$604$	−7.16019	−0.291344
$605$	0	0
$606$	7.70891	0.313153
$607$	−48.7448	−1.97849	−0.989245	−	0.146266i	$-0.953274\pi$
$607$	−48.7448	−1.97849	−0.989245	+	0.146266i	$0.953274\pi$
$608$	6.34310	0.257247
$609$	3.76300	0.152485
$610$	0	0
$611$	−21.2056	−0.857888
$612$	−2.58010	−0.104294
$613$	−3.67754	−0.148534	−0.0742671	−	0.997238i	$-0.523662\pi$
$613$	−3.67754	−0.148534	−0.0742671	+	0.997238i	$0.523662\pi$
$614$	−33.7089	−1.36038
$615$	0	0
$616$	11.9773	0.482579
$617$	−0.794381	−0.0319806	−0.0159903	−	0.999872i	$-0.505090\pi$
$617$	−0.794381	−0.0319806	−0.0159903	+	0.999872i	$0.505090\pi$
$618$	−3.39719	−0.136655
$619$	−4.57802	−0.184006	−0.0920031	−	0.995759i	$-0.529327\pi$
$619$	−4.57802	−0.184006	−0.0920031	+	0.995759i	$0.529327\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	1.86252	0.0746802
$623$	−17.2349	−0.690502
$624$	−6.34310	−0.253927
$625$	0	0
$626$	26.8984	1.07508
$627$	−20.1895	−0.806291
$628$	22.2122	0.886364
$629$	−6.65690	−0.265428
$630$	0	0
$631$	−31.1808	−1.24129	−0.620645	−	0.784092i	$-0.713129\pi$
$631$	−31.1808	−1.24129	−0.620645	+	0.784092i	$0.713129\pi$
$632$	3.18291	0.126609
$633$	19.5553	0.777254
$634$	11.0000	0.436866
$635$	0	0
$636$	7.16019	0.283920
$637$	−45.4178	−1.79952
$638$	3.18291	0.126012
$639$	1.81709	0.0718831
$640$	0	0
$641$	46.7295	1.84571	0.922853	−	0.385152i	$-0.125851\pi$
$641$	46.7295	1.84571	0.922853	+	0.385152i	$0.125851\pi$
$642$	−6.68620	−0.263883
$643$	4.81709	0.189968	0.0949838	−	0.995479i	$-0.469720\pi$
$643$	4.81709	0.189968	0.0949838	+	0.995479i	$0.469720\pi$
$644$	3.76300	0.148283
$645$	0	0
$646$	−16.3658	−0.643904
$647$	19.4967	0.766495	0.383247	−	0.923646i	$-0.374806\pi$
$647$	19.4967	0.766495	0.383247	+	0.923646i	$0.374806\pi$
$648$	−1.00000	−0.0392837
$649$	−13.8960	−0.545465
$650$	0	0
$651$	−22.5780	−0.884902
$652$	−8.68620	−0.340178
$653$	−30.5260	−1.19457	−0.597287	−	0.802027i	$-0.703755\pi$
$653$	−30.5260	−1.19457	−0.597287	+	0.802027i	$0.703755\pi$
$654$	−5.78572	−0.226239
$655$	0	0
$656$	12.3431	0.481917
$657$	8.52601	0.332631
$658$	12.5801	0.490423
$659$	−2.49463	−0.0971769	−0.0485885	−	0.998819i	$-0.515472\pi$
$659$	−2.49463	−0.0971769	−0.0485885	+	0.998819i	$0.515472\pi$
$660$	0	0
$661$	−16.7337	−0.650866	−0.325433	−	0.945565i	$-0.605510\pi$
$661$	−16.7337	−0.650866	−0.325433	+	0.945565i	$0.605510\pi$
$662$	6.18291	0.240305
$663$	16.3658	0.635596
$664$	7.28901	0.282868
$665$	0	0
$666$	−2.58010	−0.0999767
$667$	1.00000	0.0387202
$668$	−25.2349	−0.976368
$669$	28.8984	1.11728
$670$	0	0
$671$	−4.85714	−0.187508
$672$	3.76300	0.145161
$673$	2.73821	0.105550	0.0527752	−	0.998606i	$-0.483193\pi$
$673$	2.73821	0.105550	0.0527752	+	0.998606i	$0.483193\pi$
$674$	15.1602	0.583949
$675$	0	0
$676$	27.2349	1.04750
$677$	38.1040	1.46446	0.732228	−	0.681059i	$-0.238480\pi$
$677$	38.1040	1.46446	0.732228	+	0.681059i	$0.238480\pi$
$678$	0.580097	0.0222785
$679$	−43.3724	−1.66448
$680$	0	0
$681$	17.6321	0.675664
$682$	−19.0974	−0.731279
$683$	−24.6234	−0.942190	−0.471095	−	0.882082i	$-0.656141\pi$
$683$	−24.6234	−0.942190	−0.471095	+	0.882082i	$0.656141\pi$
$684$	−6.34310	−0.242534
$685$	0	0
$686$	0.602810	0.0230154
$687$	−11.8918	−0.453701
$688$	10.3431	0.394327
$689$	−45.4178	−1.73028
$690$	0	0
$691$	43.2935	1.64696	0.823482	−	0.567343i	$-0.192029\pi$
$691$	43.2935	1.64696	0.823482	+	0.567343i	$0.192029\pi$
$692$	0.817094	0.0310612
$693$	−11.9773	−0.454980
$694$	−29.8918	−1.13468
$695$	0	0
$696$	1.00000	0.0379049
$697$	−31.8464	−1.20627
$698$	−7.02930	−0.266063
$699$	9.02930	0.341520
$700$	0	0
$701$	9.00659	0.340174	0.170087	−	0.985429i	$-0.445595\pi$
$701$	9.00659	0.340174	0.170087	+	0.985429i	$0.445595\pi$
$702$	6.34310	0.239405
$703$	−16.3658	−0.617248
$704$	3.18291	0.119960
$705$	0	0
$706$	−26.2349	−0.987364
$707$	29.0087	1.09098
$708$	−4.36581	−0.164077
$709$	22.3183	0.838182	0.419091	−	0.907944i	$-0.362349\pi$
$709$	22.3183	0.838182	0.419091	+	0.907944i	$0.362349\pi$
$710$	0	0
$711$	−3.18291	−0.119368
$712$	−4.58010	−0.171646
$713$	−6.00000	−0.224702
$714$	−9.70891	−0.363347
$715$	0	0
$716$	−3.16019	−0.118102
$717$	−20.0747	−0.749704
$718$	−19.5033	−0.727856
$719$	−32.1829	−1.20022	−0.600110	−	0.799918i	$-0.704877\pi$
$719$	−32.1829	−1.20022	−0.600110	+	0.799918i	$0.704877\pi$
$720$	0	0
$721$	−12.7836	−0.476088
$722$	−21.2349	−0.790282
$723$	−23.0066	−0.855624
$724$	3.37448	0.125411
$725$	0	0
$726$	0.869107	0.0322556
$727$	32.3658	1.20038	0.600191	−	0.799857i	$-0.295091\pi$
$727$	32.3658	1.20038	0.600191	+	0.799857i	$0.295091\pi$
$728$	−23.8691	−0.884648
$729$	1.00000	0.0370370
$730$	0	0
$731$	−26.6862	−0.987025
$732$	−1.52601	−0.0564028
$733$	−34.6841	−1.28109	−0.640544	−	0.767922i	$-0.721291\pi$
$733$	−34.6841	−1.28109	−0.640544	+	0.767922i	$0.721291\pi$
$734$	−33.5033	−1.23663
$735$	0	0
$736$	1.00000	0.0368605
$737$	34.0132	1.25289
$738$	−12.3431	−0.454356
$739$	45.1441	1.66065	0.830326	−	0.557278i	$-0.188154\pi$
$739$	45.1441	1.66065	0.830326	+	0.557278i	$0.188154\pi$
$740$	0	0
$741$	40.2349	1.47807
$742$	26.9438	0.989139
$743$	3.02930	0.111134	0.0555671	−	0.998455i	$-0.482303\pi$
$743$	3.02930	0.111134	0.0555671	+	0.998455i	$0.482303\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	10.1515	0.371674
$747$	−7.28901	−0.266691
$748$	−8.21221	−0.300268
$749$	−25.1602	−0.919333
$750$	0	0
$751$	−12.3410	−0.450330	−0.225165	−	0.974321i	$-0.572292\pi$
$751$	−12.3410	−0.450330	−0.225165	+	0.974321i	$0.572292\pi$
$752$	3.34310	0.121910
$753$	−10.9459	−0.398891
$754$	−6.34310	−0.231002
$755$	0	0
$756$	−3.76300	−0.136859
$757$	−6.00208	−0.218149	−0.109075	−	0.994034i	$-0.534789\pi$
$757$	−6.00208	−0.218149	−0.109075	+	0.994034i	$0.534789\pi$
$758$	−20.5780	−0.747427
$759$	−3.18291	−0.115532
$760$	0	0
$761$	−26.2977	−0.953290	−0.476645	−	0.879096i	$-0.658147\pi$
$761$	−26.2977	−0.953290	−0.476645	+	0.879096i	$0.658147\pi$
$762$	14.3204	0.518773
$763$	−21.7717	−0.788187
$764$	−11.8691	−0.429409
$765$	0	0
$766$	−1.22833	−0.0443814
$767$	27.6928	0.999929
$768$	1.00000	0.0360844
$769$	−45.8918	−1.65490	−0.827451	−	0.561538i	$-0.810210\pi$
$769$	−45.8918	−1.65490	−0.827451	+	0.561538i	$0.810210\pi$
$770$	0	0
$771$	−16.3204	−0.587764
$772$	−2.97729	−0.107155
$773$	−37.5846	−1.35182	−0.675912	−	0.736982i	$-0.736250\pi$
$773$	−37.5846	−1.35182	−0.675912	+	0.736982i	$0.736250\pi$
$774$	−10.3431	−0.371775
$775$	0	0
$776$	−11.5260	−0.413760
$777$	−9.70891	−0.348305
$778$	−35.3724	−1.26816
$779$	−78.2935	−2.80516
$780$	0	0
$781$	5.78364	0.206955
$782$	−2.58010	−0.0922641
$783$	−1.00000	−0.0357371
$784$	7.16019	0.255721
$785$	0	0
$786$	−15.8918	−0.566842
$787$	20.9211	0.745757	0.372879	−	0.927880i	$-0.378371\pi$
$787$	20.9211	0.745757	0.372879	+	0.927880i	$0.378371\pi$
$788$	−24.8918	−0.886734
$789$	29.0066	1.03266
$790$	0	0
$791$	2.18291	0.0776152
$792$	−3.18291	−0.113100
$793$	9.67961	0.343733
$794$	−31.0066	−1.10038
$795$	0	0
$796$	22.4492	0.795691
$797$	−46.5326	−1.64827	−0.824135	−	0.566394i	$-0.808338\pi$
$797$	−46.5326	−1.64827	−0.824135	+	0.566394i	$0.808338\pi$
$798$	−23.8691	−0.844957
$799$	−8.62552	−0.305149
$800$	0	0
$801$	4.58010	0.161830
$802$	−15.2056	−0.536929
$803$	27.1375	0.957661
$804$	10.6862	0.376873
$805$	0	0
$806$	38.0586	1.34056
$807$	−26.1895	−0.921914
$808$	7.70891	0.271199
$809$	26.1895	0.920774	0.460387	−	0.887718i	$-0.347711\pi$
$809$	26.1895	0.920774	0.460387	+	0.887718i	$0.347711\pi$
$810$	0	0
$811$	10.4112	0.365588	0.182794	−	0.983151i	$-0.441486\pi$
$811$	10.4112	0.365588	0.182794	+	0.983151i	$0.441486\pi$
$812$	3.76300	0.132056
$813$	6.36581	0.223259
$814$	−8.21221	−0.287838
$815$	0	0
$816$	−2.58010	−0.0903215
$817$	−65.6073	−2.29531
$818$	2.57143	0.0899080
$819$	23.8691	0.834054
$820$	0	0
$821$	32.8171	1.14532	0.572662	−	0.819791i	$-0.305911\pi$
$821$	32.8171	1.14532	0.572662	+	0.819791i	$0.305911\pi$
$822$	15.4199	0.537831
$823$	20.5780	0.717305	0.358652	−	0.933471i	$-0.383236\pi$
$823$	20.5780	0.717305	0.358652	+	0.933471i	$0.383236\pi$
$824$	−3.39719	−0.118347
$825$	0	0
$826$	−16.4286	−0.571623
$827$	−21.1982	−0.737132	−0.368566	−	0.929602i	$-0.620151\pi$
$827$	−21.1982	−0.737132	−0.368566	+	0.929602i	$0.620151\pi$
$828$	−1.00000	−0.0347524
$829$	−3.29109	−0.114304	−0.0571521	−	0.998365i	$-0.518202\pi$
$829$	−3.29109	−0.114304	−0.0571521	+	0.998365i	$0.518202\pi$
$830$	0	0
$831$	−9.50329	−0.329666
$832$	−6.34310	−0.219907
$833$	−18.4740	−0.640086
$834$	−3.02271	−0.104668
$835$	0	0
$836$	−20.1895	−0.698268
$837$	6.00000	0.207390
$838$	19.0768	0.658997
$839$	−20.0813	−0.693284	−0.346642	−	0.937998i	$-0.612678\pi$
$839$	−20.0813	−0.693284	−0.346642	+	0.937998i	$0.612678\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	15.8918	0.547668
$843$	27.2663	0.939101
$844$	19.5553	0.673121
$845$	0	0
$846$	−3.34310	−0.114938
$847$	3.27045	0.112374
$848$	7.16019	0.245882
$849$	−10.3204	−0.354195
$850$	0	0
$851$	−2.58010	−0.0884446
$852$	1.81709	0.0622526
$853$	32.9211	1.12720	0.563599	−	0.826049i	$-0.309416\pi$
$853$	32.9211	1.12720	0.563599	+	0.826049i	$0.309416\pi$
$854$	−5.74237	−0.196500
$855$	0	0
$856$	−6.68620	−0.228530
$857$	−46.5780	−1.59107	−0.795537	−	0.605904i	$-0.792811\pi$
$857$	−46.5780	−1.59107	−0.795537	+	0.605904i	$0.792811\pi$
$858$	20.1895	0.689258
$859$	−42.0132	−1.43347	−0.716736	−	0.697345i	$-0.754365\pi$
$859$	−42.0132	−1.43347	−0.716736	+	0.697345i	$0.754365\pi$
$860$	0	0
$861$	−46.4471	−1.58291
$862$	24.2122	0.824671
$863$	15.5553	0.529509	0.264754	−	0.964316i	$-0.414709\pi$
$863$	15.5553	0.529509	0.264754	+	0.964316i	$0.414709\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	12.2122	0.414988
$867$	−10.3431	−0.351270
$868$	−22.5780	−0.766348
$869$	−10.1309	−0.343667
$870$	0	0
$871$	−67.7836	−2.29676
$872$	−5.78572	−0.195929
$873$	11.5260	0.390096
$874$	−6.34310	−0.214559
$875$	0	0
$876$	8.52601	0.288067
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0.794381	0.0268091
$879$	8.36581	0.282172
$880$	0	0
$881$	10.4740	0.352878	0.176439	−	0.984312i	$-0.443542\pi$
$881$	10.4740	0.352878	0.176439	+	0.984312i	$0.443542\pi$
$882$	−7.16019	−0.241096
$883$	35.8757	1.20731	0.603657	−	0.797244i	$-0.293710\pi$
$883$	35.8757	1.20731	0.603657	+	0.797244i	$0.293710\pi$
$884$	16.3658	0.550442
$885$	0	0
$886$	−14.2122	−0.477468
$887$	7.29767	0.245032	0.122516	−	0.992467i	$-0.460904\pi$
$887$	7.29767	0.245032	0.122516	+	0.992467i	$0.460904\pi$
$888$	−2.58010	−0.0865824
$889$	53.8877	1.80733
$890$	0	0
$891$	3.18291	0.106631
$892$	28.8984	0.967590
$893$	−21.2056	−0.709619
$894$	−16.6862	−0.558070
$895$	0	0
$896$	3.76300	0.125713
$897$	6.34310	0.211790
$898$	−1.05201	−0.0351061
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−18.4740	−0.615458
$902$	−39.2869	−1.30811
$903$	−38.9211	−1.29521
$904$	0.580097	0.0192937
$905$	0	0
$906$	7.16019	0.237882
$907$	40.7089	1.35172	0.675859	−	0.737031i	$-0.263773\pi$
$907$	40.7089	1.35172	0.675859	+	0.737031i	$0.263773\pi$
$908$	17.6321	0.585142
$909$	−7.70891	−0.255689
$910$	0	0
$911$	22.7089	0.752380	0.376190	−	0.926543i	$-0.377234\pi$
$911$	22.7089	0.752380	0.376190	+	0.926543i	$0.377234\pi$
$912$	−6.34310	−0.210041
$913$	−23.2002	−0.767816
$914$	0.686200	0.0226975
$915$	0	0
$916$	−11.8918	−0.392917
$917$	−59.8010	−1.97480
$918$	2.58010	0.0851559
$919$	−26.5801	−0.876796	−0.438398	−	0.898781i	$-0.644454\pi$
$919$	−26.5801	−0.876796	−0.438398	+	0.898781i	$0.644454\pi$
$920$	0	0
$921$	33.7089	1.11075
$922$	29.5780	0.974100
$923$	−11.5260	−0.379383
$924$	−11.9773	−0.394024
$925$	0	0
$926$	12.0000	0.394344
$927$	3.39719	0.111578
$928$	1.00000	0.0328266
$929$	36.5553	1.19934	0.599670	−	0.800247i	$-0.295298\pi$
$929$	36.5553	1.19934	0.599670	+	0.800247i	$0.295298\pi$
$930$	0	0
$931$	−45.4178	−1.48851
$932$	9.02930	0.295765
$933$	−1.86252	−0.0609761
$934$	29.2890	0.958366
$935$	0	0
$936$	6.34310	0.207331
$937$	−19.3724	−0.632869	−0.316434	−	0.948614i	$-0.602486\pi$
$937$	−19.3724	−0.632869	−0.316434	+	0.948614i	$0.602486\pi$
$938$	40.2122	1.31298
$939$	−26.8984	−0.877796
$940$	0	0
$941$	−36.3204	−1.18401	−0.592005	−	0.805934i	$-0.701664\pi$
$941$	−36.3204	−1.18401	−0.592005	+	0.805934i	$0.701664\pi$
$942$	−22.2122	−0.723713
$943$	−12.3431	−0.401947
$944$	−4.36581	−0.142095
$945$	0	0
$946$	−32.9211	−1.07036
$947$	5.63419	0.183086	0.0915432	−	0.995801i	$-0.470820\pi$
$947$	5.63419	0.183086	0.0915432	+	0.995801i	$0.470820\pi$
$948$	−3.18291	−0.103376
$949$	−54.0813	−1.75555
$950$	0	0
$951$	−11.0000	−0.356699
$952$	−9.70891	−0.314668
$953$	−28.9005	−0.936179	−0.468089	−	0.883681i	$-0.655057\pi$
$953$	−28.9005	−0.936179	−0.468089	+	0.883681i	$0.655057\pi$
$954$	−7.16019	−0.231820
$955$	0	0
$956$	−20.0747	−0.649263
$957$	−3.18291	−0.102889
$958$	−15.9145	−0.514175
$959$	58.0251	1.87373
$960$	0	0
$961$	5.00000	0.161290
$962$	16.3658	0.527655
$963$	6.68620	0.215460
$964$	−23.0066	−0.740992
$965$	0	0
$966$	−3.76300	−0.121073
$967$	12.3658	0.397658	0.198829	−	0.980034i	$-0.436286\pi$
$967$	12.3658	0.397658	0.198829	+	0.980034i	$0.436286\pi$
$968$	0.869107	0.0279341
$969$	16.3658	0.525746
$970$	0	0
$971$	−33.9125	−1.08830	−0.544151	−	0.838987i	$-0.683148\pi$
$971$	−33.9125	−1.08830	−0.544151	+	0.838987i	$0.683148\pi$
$972$	1.00000	0.0320750
$973$	−11.3745	−0.364649
$974$	−38.9438	−1.24784
$975$	0	0
$976$	−1.52601	−0.0488463
$977$	41.3703	1.32355	0.661777	−	0.749701i	$-0.269803\pi$
$977$	41.3703	1.32355	0.661777	+	0.749701i	$0.269803\pi$
$978$	8.68620	0.277754
$979$	14.5780	0.465916
$980$	0	0
$981$	5.78572	0.184724
$982$	−27.4806	−0.876941
$983$	1.91453	0.0610641	0.0305320	−	0.999534i	$-0.490280\pi$
$983$	1.91453	0.0610641	0.0305320	+	0.999534i	$0.490280\pi$
$984$	−12.3431	−0.393484
$985$	0	0
$986$	−2.58010	−0.0821670
$987$	−12.5801	−0.400429
$988$	40.2349	1.28004
$989$	−10.3431	−0.328891
$990$	0	0
$991$	3.31380	0.105266	0.0526332	−	0.998614i	$-0.483239\pi$
$991$	3.31380	0.105266	0.0526332	+	0.998614i	$0.483239\pi$
$992$	−6.00000	−0.190500
$993$	−6.18291	−0.196209
$994$	6.83773	0.216880
$995$	0	0
$996$	−7.28901	−0.230961
$997$	−8.70891	−0.275814	−0.137907	−	0.990445i	$-0.544038\pi$
$997$	−8.70891	−0.275814	−0.137907	+	0.990445i	$0.544038\pi$
$998$	−6.97729	−0.220862
$999$	2.58010	0.0816307

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bp.1.1	✓	3
5.2	odd	4		3450.2.d.ba.2899.1		6
5.3	odd	4		3450.2.d.ba.2899.6		6
5.4	even	2		3450.2.a.bs.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.bp.1.1	✓	3	1.1	even	1	trivial
3450.2.a.bs.1.3	yes	3	5.4	even	2
3450.2.d.ba.2899.1		6	5.2	odd	4
3450.2.d.ba.2899.6		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.76300 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -3.76300 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.18291 q^{11} +1.00000 q^{12} -6.34310 q^{13} +3.76300 q^{14} +1.00000 q^{16} -2.58010 q^{17} -1.00000 q^{18} -6.34310 q^{19} -3.76300 q^{21} -3.18291 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.34310 q^{26} +1.00000 q^{27} -3.76300 q^{28} -1.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} +3.18291 q^{33} +2.58010 q^{34} +1.00000 q^{36} +2.58010 q^{37} +6.34310 q^{38} -6.34310 q^{39} +12.3431 q^{41} +3.76300 q^{42} +10.3431 q^{43} +3.18291 q^{44} +1.00000 q^{46} +3.34310 q^{47} +1.00000 q^{48} +7.16019 q^{49} -2.58010 q^{51} -6.34310 q^{52} +7.16019 q^{53} -1.00000 q^{54} +3.76300 q^{56} -6.34310 q^{57} +1.00000 q^{58} -4.36581 q^{59} -1.52601 q^{61} -6.00000 q^{62} -3.76300 q^{63} +1.00000 q^{64} -3.18291 q^{66} +10.6862 q^{67} -2.58010 q^{68} -1.00000 q^{69} +1.81709 q^{71} -1.00000 q^{72} +8.52601 q^{73} -2.58010 q^{74} -6.34310 q^{76} -11.9773 q^{77} +6.34310 q^{78} -3.18291 q^{79} +1.00000 q^{81} -12.3431 q^{82} -7.28901 q^{83} -3.76300 q^{84} -10.3431 q^{86} -1.00000 q^{87} -3.18291 q^{88} +4.58010 q^{89} +23.8691 q^{91} -1.00000 q^{92} +6.00000 q^{93} -3.34310 q^{94} -1.00000 q^{96} +11.5260 q^{97} -7.16019 q^{98} +3.18291 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} - q^{13} - q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} - q^{19} + q^{21} - 3 q^{22} - 3 q^{23} - 3 q^{24} + q^{26} + 3 q^{27} + q^{28} - 3 q^{29} + 18 q^{31} - 3 q^{32} + 3 q^{33} + 2 q^{34} + 3 q^{36} + 2 q^{37} + q^{38} - q^{39} + 19 q^{41} - q^{42} + 13 q^{43} + 3 q^{44} + 3 q^{46} - 8 q^{47} + 3 q^{48} + 10 q^{49} - 2 q^{51} - q^{52} + 10 q^{53} - 3 q^{54} - q^{56} - q^{57} + 3 q^{58} + 20 q^{61} - 18 q^{62} + q^{63} + 3 q^{64} - 3 q^{66} - 4 q^{67} - 2 q^{68} - 3 q^{69} + 12 q^{71} - 3 q^{72} + q^{73} - 2 q^{74} - q^{76} - 31 q^{77} + q^{78} - 3 q^{79} + 3 q^{81} - 19 q^{82} + 15 q^{83} + q^{84} - 13 q^{86} - 3 q^{87} - 3 q^{88} + 8 q^{89} + 29 q^{91} - 3 q^{92} + 18 q^{93} + 8 q^{94} - 3 q^{96} + 10 q^{97} - 10 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 - q^13 - q^14 + 3 * q^16 - 2 * q^17 - 3 * q^18 - q^19 + q^21 - 3 * q^22 - 3 * q^23 - 3 * q^24 + q^26 + 3 * q^27 + q^28 - 3 * q^29 + 18 * q^31 - 3 * q^32 + 3 * q^33 + 2 * q^34 + 3 * q^36 + 2 * q^37 + q^38 - q^39 + 19 * q^41 - q^42 + 13 * q^43 + 3 * q^44 + 3 * q^46 - 8 * q^47 + 3 * q^48 + 10 * q^49 - 2 * q^51 - q^52 + 10 * q^53 - 3 * q^54 - q^56 - q^57 + 3 * q^58 + 20 * q^61 - 18 * q^62 + q^63 + 3 * q^64 - 3 * q^66 - 4 * q^67 - 2 * q^68 - 3 * q^69 + 12 * q^71 - 3 * q^72 + q^73 - 2 * q^74 - q^76 - 31 * q^77 + q^78 - 3 * q^79 + 3 * q^81 - 19 * q^82 + 15 * q^83 + q^84 - 13 * q^86 - 3 * q^87 - 3 * q^88 + 8 * q^89 + 29 * q^91 - 3 * q^92 + 18 * q^93 + 8 * q^94 - 3 * q^96 + 10 * q^97 - 10 * q^98 + 3 * q^99

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