LMFDB - Embedded newform 3450.2.a.bl.1.2

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:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.44949$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -2.89898 q^{13} +1.44949 q^{14} +1.00000 q^{16} -5.44949 q^{17} +1.00000 q^{18} +6.89898 q^{19} +1.44949 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.89898 q^{26} +1.00000 q^{27} +1.44949 q^{28} +5.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -5.44949 q^{34} +1.00000 q^{36} +8.34847 q^{37} +6.89898 q^{38} -2.89898 q^{39} -4.89898 q^{41} +1.44949 q^{42} +10.8990 q^{43} +2.00000 q^{44} +1.00000 q^{46} -3.89898 q^{47} +1.00000 q^{48} -4.89898 q^{49} -5.44949 q^{51} -2.89898 q^{52} +0.898979 q^{53} +1.00000 q^{54} +1.44949 q^{56} +6.89898 q^{57} +5.00000 q^{58} +10.0000 q^{59} -4.89898 q^{61} +2.00000 q^{62} +1.44949 q^{63} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} -5.44949 q^{68} +1.00000 q^{69} +10.7980 q^{71} +1.00000 q^{72} +2.10102 q^{73} +8.34847 q^{74} +6.89898 q^{76} +2.89898 q^{77} -2.89898 q^{78} +10.0000 q^{79} +1.00000 q^{81} -4.89898 q^{82} -2.55051 q^{83} +1.44949 q^{84} +10.8990 q^{86} +5.00000 q^{87} +2.00000 q^{88} -10.3485 q^{89} -4.20204 q^{91} +1.00000 q^{92} +2.00000 q^{93} -3.89898 q^{94} +1.00000 q^{96} -12.6969 q^{97} -4.89898 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 6 q^{17} + 2 q^{18} + 4 q^{19} - 2 q^{21} + 4 q^{22} + 2 q^{23} + 2 q^{24} + 4 q^{26} + 2 q^{27} - 2 q^{28} + 10 q^{29} + 4 q^{31} + 2 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{36} + 2 q^{37} + 4 q^{38} + 4 q^{39} - 2 q^{42} + 12 q^{43} + 4 q^{44} + 2 q^{46} + 2 q^{47} + 2 q^{48} - 6 q^{51} + 4 q^{52} - 8 q^{53} + 2 q^{54} - 2 q^{56} + 4 q^{57} + 10 q^{58} + 20 q^{59} + 4 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{66} - 4 q^{67} - 6 q^{68} + 2 q^{69} + 2 q^{71} + 2 q^{72} + 14 q^{73} + 2 q^{74} + 4 q^{76} - 4 q^{77} + 4 q^{78} + 20 q^{79} + 2 q^{81} - 10 q^{83} - 2 q^{84} + 12 q^{86} + 10 q^{87} + 4 q^{88} - 6 q^{89} - 28 q^{91} + 2 q^{92} + 4 q^{93} + 2 q^{94} + 2 q^{96} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 6 * q^17 + 2 * q^18 + 4 * q^19 - 2 * q^21 + 4 * q^22 + 2 * q^23 + 2 * q^24 + 4 * q^26 + 2 * q^27 - 2 * q^28 + 10 * q^29 + 4 * q^31 + 2 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^36 + 2 * q^37 + 4 * q^38 + 4 * q^39 - 2 * q^42 + 12 * q^43 + 4 * q^44 + 2 * q^46 + 2 * q^47 + 2 * q^48 - 6 * q^51 + 4 * q^52 - 8 * q^53 + 2 * q^54 - 2 * q^56 + 4 * q^57 + 10 * q^58 + 20 * q^59 + 4 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^66 - 4 * q^67 - 6 * q^68 + 2 * q^69 + 2 * q^71 + 2 * q^72 + 14 * q^73 + 2 * q^74 + 4 * q^76 - 4 * q^77 + 4 * q^78 + 20 * q^79 + 2 * q^81 - 10 * q^83 - 2 * q^84 + 12 * q^86 + 10 * q^87 + 4 * q^88 - 6 * q^89 - 28 * q^91 + 2 * q^92 + 4 * q^93 + 2 * q^94 + 2 * q^96 + 4 * q^97 + 4 * 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
$7$	1.44949	0.547856	0.273928	−	0.961750i	$-0.411677\pi$
$7$	1.44949	0.547856	0.273928	+	0.961750i	$0.411677\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	−2.89898	−0.804032	−0.402016	−	0.915633i	$-0.631690\pi$
$13$	−2.89898	−0.804032	−0.402016	+	0.915633i	$0.631690\pi$
$14$	1.44949	0.387392
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.44949	−1.32170	−0.660848	−	0.750520i	$-0.729803\pi$
$17$	−5.44949	−1.32170	−0.660848	+	0.750520i	$0.729803\pi$
$18$	1.00000	0.235702
$19$	6.89898	1.58273	0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898	1.58273	0.791367	+	0.611341i	$0.209370\pi$
$20$	0	0
$21$	1.44949	0.316305
$22$	2.00000	0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	0	0
$26$	−2.89898	−0.568537
$27$	1.00000	0.192450
$28$	1.44949	0.273928
$29$	5.00000	0.928477	0.464238	−	0.885710i	$-0.346328\pi$
$29$	5.00000	0.928477	0.464238	+	0.885710i	$0.346328\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	−5.44949	−0.934580
$35$	0	0
$36$	1.00000	0.166667
$37$	8.34847	1.37248	0.686240	−	0.727375i	$-0.259260\pi$
$37$	8.34847	1.37248	0.686240	+	0.727375i	$0.259260\pi$
$38$	6.89898	1.11916
$39$	−2.89898	−0.464208
$40$	0	0
$41$	−4.89898	−0.765092	−0.382546	−	0.923936i	$-0.624953\pi$
$41$	−4.89898	−0.765092	−0.382546	+	0.923936i	$0.624953\pi$
$42$	1.44949	0.223661
$43$	10.8990	1.66208	0.831039	−	0.556214i	$-0.187746\pi$
$43$	10.8990	1.66208	0.831039	+	0.556214i	$0.187746\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.89898	−0.568725	−0.284362	−	0.958717i	$-0.591782\pi$
$47$	−3.89898	−0.568725	−0.284362	+	0.958717i	$0.591782\pi$
$48$	1.00000	0.144338
$49$	−4.89898	−0.699854
$50$	0	0
$51$	−5.44949	−0.763081
$52$	−2.89898	−0.402016
$53$	0.898979	0.123484	0.0617422	−	0.998092i	$-0.480334\pi$
$53$	0.898979	0.123484	0.0617422	+	0.998092i	$0.480334\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	1.44949	0.193696
$57$	6.89898	0.913792
$58$	5.00000	0.656532
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−4.89898	−0.627250	−0.313625	−	0.949547i	$-0.601543\pi$
$61$	−4.89898	−0.627250	−0.313625	+	0.949547i	$0.601543\pi$
$62$	2.00000	0.254000
$63$	1.44949	0.182619
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−5.44949	−0.660848
$69$	1.00000	0.120386
$70$	0	0
$71$	10.7980	1.28148	0.640741	−	0.767757i	$-0.278627\pi$
$71$	10.7980	1.28148	0.640741	+	0.767757i	$0.278627\pi$
$72$	1.00000	0.117851
$73$	2.10102	0.245906	0.122953	−	0.992413i	$-0.460764\pi$
$73$	2.10102	0.245906	0.122953	+	0.992413i	$0.460764\pi$
$74$	8.34847	0.970490
$75$	0	0
$76$	6.89898	0.791367
$77$	2.89898	0.330369
$78$	−2.89898	−0.328245
$79$	10.0000	1.12509	0.562544	−	0.826767i	$-0.309823\pi$
$79$	10.0000	1.12509	0.562544	+	0.826767i	$0.309823\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−4.89898	−0.541002
$83$	−2.55051	−0.279955	−0.139977	−	0.990155i	$-0.544703\pi$
$83$	−2.55051	−0.279955	−0.139977	+	0.990155i	$0.544703\pi$
$84$	1.44949	0.158152
$85$	0	0
$86$	10.8990	1.17527
$87$	5.00000	0.536056
$88$	2.00000	0.213201
$89$	−10.3485	−1.09694	−0.548468	−	0.836172i	$-0.684789\pi$
$89$	−10.3485	−1.09694	−0.548468	+	0.836172i	$0.684789\pi$
$90$	0	0
$91$	−4.20204	−0.440494
$92$	1.00000	0.104257
$93$	2.00000	0.207390
$94$	−3.89898	−0.402149
$95$	0	0
$96$	1.00000	0.102062
$97$	−12.6969	−1.28918	−0.644589	−	0.764529i	$-0.722972\pi$
$97$	−12.6969	−1.28918	−0.644589	+	0.764529i	$0.722972\pi$
$98$	−4.89898	−0.494872
$99$	2.00000	0.201008
$100$	0	0
$101$	−13.6969	−1.36290	−0.681448	−	0.731866i	$-0.738650\pi$
$101$	−13.6969	−1.36290	−0.681448	+	0.731866i	$0.738650\pi$
$102$	−5.44949	−0.539580
$103$	−13.2474	−1.30531	−0.652655	−	0.757655i	$-0.726345\pi$
$103$	−13.2474	−1.30531	−0.652655	+	0.757655i	$0.726345\pi$
$104$	−2.89898	−0.284268
$105$	0	0
$106$	0.898979	0.0873166
$107$	−2.00000	−0.193347	−0.0966736	−	0.995316i	$-0.530820\pi$
$107$	−2.00000	−0.193347	−0.0966736	+	0.995316i	$0.530820\pi$
$108$	1.00000	0.0962250
$109$	3.44949	0.330401	0.165201	−	0.986260i	$-0.447173\pi$
$109$	3.44949	0.330401	0.165201	+	0.986260i	$0.447173\pi$
$110$	0	0
$111$	8.34847	0.792402
$112$	1.44949	0.136964
$113$	14.3485	1.34979	0.674895	−	0.737914i	$-0.264189\pi$
$113$	14.3485	1.34979	0.674895	+	0.737914i	$0.264189\pi$
$114$	6.89898	0.646149
$115$	0	0
$116$	5.00000	0.464238
$117$	−2.89898	−0.268011
$118$	10.0000	0.920575
$119$	−7.89898	−0.724098
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−4.89898	−0.443533
$123$	−4.89898	−0.441726
$124$	2.00000	0.179605
$125$	0	0
$126$	1.44949	0.129131
$127$	1.79796	0.159543	0.0797715	−	0.996813i	$-0.474581\pi$
$127$	1.79796	0.159543	0.0797715	+	0.996813i	$0.474581\pi$
$128$	1.00000	0.0883883
$129$	10.8990	0.959602
$130$	0	0
$131$	8.89898	0.777507	0.388754	−	0.921342i	$-0.372906\pi$
$131$	8.89898	0.777507	0.388754	+	0.921342i	$0.372906\pi$
$132$	2.00000	0.174078
$133$	10.0000	0.867110
$134$	−2.00000	−0.172774
$135$	0	0
$136$	−5.44949	−0.467290
$137$	−22.3485	−1.90936	−0.954679	−	0.297636i	$-0.903802\pi$
$137$	−22.3485	−1.90936	−0.954679	+	0.297636i	$0.903802\pi$
$138$	1.00000	0.0851257
$139$	15.6969	1.33140	0.665698	−	0.746221i	$-0.268134\pi$
$139$	15.6969	1.33140	0.665698	+	0.746221i	$0.268134\pi$
$140$	0	0
$141$	−3.89898	−0.328353
$142$	10.7980	0.906145
$143$	−5.79796	−0.484850
$144$	1.00000	0.0833333
$145$	0	0
$146$	2.10102	0.173882
$147$	−4.89898	−0.404061
$148$	8.34847	0.686240
$149$	20.0000	1.63846	0.819232	−	0.573462i	$-0.194400\pi$
$149$	20.0000	1.63846	0.819232	+	0.573462i	$0.194400\pi$
$150$	0	0
$151$	−18.6969	−1.52154	−0.760768	−	0.649024i	$-0.775177\pi$
$151$	−18.6969	−1.52154	−0.760768	+	0.649024i	$0.775177\pi$
$152$	6.89898	0.559581
$153$	−5.44949	−0.440565
$154$	2.89898	0.233606
$155$	0	0
$156$	−2.89898	−0.232104
$157$	−8.89898	−0.710216	−0.355108	−	0.934825i	$-0.615556\pi$
$157$	−8.89898	−0.710216	−0.355108	+	0.934825i	$0.615556\pi$
$158$	10.0000	0.795557
$159$	0.898979	0.0712937
$160$	0	0
$161$	1.44949	0.114236
$162$	1.00000	0.0785674
$163$	17.7980	1.39404	0.697022	−	0.717050i	$-0.254508\pi$
$163$	17.7980	1.39404	0.697022	+	0.717050i	$0.254508\pi$
$164$	−4.89898	−0.382546
$165$	0	0
$166$	−2.55051	−0.197958
$167$	−7.00000	−0.541676	−0.270838	−	0.962625i	$-0.587301\pi$
$167$	−7.00000	−0.541676	−0.270838	+	0.962625i	$0.587301\pi$
$168$	1.44949	0.111831
$169$	−4.59592	−0.353532
$170$	0	0
$171$	6.89898	0.527578
$172$	10.8990	0.831039
$173$	−19.7980	−1.50521	−0.752605	−	0.658472i	$-0.771203\pi$
$173$	−19.7980	−1.50521	−0.752605	+	0.658472i	$0.771203\pi$
$174$	5.00000	0.379049
$175$	0	0
$176$	2.00000	0.150756
$177$	10.0000	0.751646
$178$	−10.3485	−0.775651
$179$	3.10102	0.231781	0.115891	−	0.993262i	$-0.463028\pi$
$179$	3.10102	0.231781	0.115891	+	0.993262i	$0.463028\pi$
$180$	0	0
$181$	−18.3485	−1.36383	−0.681915	−	0.731431i	$-0.738853\pi$
$181$	−18.3485	−1.36383	−0.681915	+	0.731431i	$0.738853\pi$
$182$	−4.20204	−0.311476
$183$	−4.89898	−0.362143
$184$	1.00000	0.0737210
$185$	0	0
$186$	2.00000	0.146647
$187$	−10.8990	−0.797012
$188$	−3.89898	−0.284362
$189$	1.44949	0.105435
$190$	0	0
$191$	2.00000	0.144715	0.0723575	−	0.997379i	$-0.476948\pi$
$191$	2.00000	0.144715	0.0723575	+	0.997379i	$0.476948\pi$
$192$	1.00000	0.0721688
$193$	9.69694	0.698001	0.349000	−	0.937123i	$-0.386521\pi$
$193$	9.69694	0.698001	0.349000	+	0.937123i	$0.386521\pi$
$194$	−12.6969	−0.911587
$195$	0	0
$196$	−4.89898	−0.349927
$197$	−3.89898	−0.277791	−0.138895	−	0.990307i	$-0.544355\pi$
$197$	−3.89898	−0.277791	−0.138895	+	0.990307i	$0.544355\pi$
$198$	2.00000	0.142134
$199$	24.1464	1.71169	0.855847	−	0.517228i	$-0.173036\pi$
$199$	24.1464	1.71169	0.855847	+	0.517228i	$0.173036\pi$
$200$	0	0
$201$	−2.00000	−0.141069
$202$	−13.6969	−0.963713
$203$	7.24745	0.508671
$204$	−5.44949	−0.381541
$205$	0	0
$206$	−13.2474	−0.922993
$207$	1.00000	0.0695048
$208$	−2.89898	−0.201008
$209$	13.7980	0.954425
$210$	0	0
$211$	10.7980	0.743362	0.371681	−	0.928360i	$-0.378781\pi$
$211$	10.7980	0.743362	0.371681	+	0.928360i	$0.378781\pi$
$212$	0.898979	0.0617422
$213$	10.7980	0.739864
$214$	−2.00000	−0.136717
$215$	0	0
$216$	1.00000	0.0680414
$217$	2.89898	0.196796
$218$	3.44949	0.233629
$219$	2.10102	0.141974
$220$	0	0
$221$	15.7980	1.06269
$222$	8.34847	0.560313
$223$	24.6969	1.65383	0.826915	−	0.562327i	$-0.190094\pi$
$223$	24.6969	1.65383	0.826915	+	0.562327i	$0.190094\pi$
$224$	1.44949	0.0968481
$225$	0	0
$226$	14.3485	0.954446
$227$	−4.75255	−0.315438	−0.157719	−	0.987484i	$-0.550414\pi$
$227$	−4.75255	−0.315438	−0.157719	+	0.987484i	$0.550414\pi$
$228$	6.89898	0.456896
$229$	3.10102	0.204921	0.102461	−	0.994737i	$-0.467328\pi$
$229$	3.10102	0.204921	0.102461	+	0.994737i	$0.467328\pi$
$230$	0	0
$231$	2.89898	0.190739
$232$	5.00000	0.328266
$233$	−26.6969	−1.74897	−0.874487	−	0.485048i	$-0.838802\pi$
$233$	−26.6969	−1.74897	−0.874487	+	0.485048i	$0.838802\pi$
$234$	−2.89898	−0.189512
$235$	0	0
$236$	10.0000	0.650945
$237$	10.0000	0.649570
$238$	−7.89898	−0.512015
$239$	−8.10102	−0.524011	−0.262006	−	0.965066i	$-0.584384\pi$
$239$	−8.10102	−0.524011	−0.262006	+	0.965066i	$0.584384\pi$
$240$	0	0
$241$	−8.00000	−0.515325	−0.257663	−	0.966235i	$-0.582952\pi$
$241$	−8.00000	−0.515325	−0.257663	+	0.966235i	$0.582952\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−4.89898	−0.313625
$245$	0	0
$246$	−4.89898	−0.312348
$247$	−20.0000	−1.27257
$248$	2.00000	0.127000
$249$	−2.55051	−0.161632
$250$	0	0
$251$	−5.24745	−0.331216	−0.165608	−	0.986192i	$-0.552959\pi$
$251$	−5.24745	−0.331216	−0.165608	+	0.986192i	$0.552959\pi$
$252$	1.44949	0.0913093
$253$	2.00000	0.125739
$254$	1.79796	0.112814
$255$	0	0
$256$	1.00000	0.0625000
$257$	−8.20204	−0.511629	−0.255815	−	0.966726i	$-0.582344\pi$
$257$	−8.20204	−0.511629	−0.255815	+	0.966726i	$0.582344\pi$
$258$	10.8990	0.678541
$259$	12.1010	0.751921
$260$	0	0
$261$	5.00000	0.309492
$262$	8.89898	0.549781
$263$	−19.7980	−1.22079	−0.610397	−	0.792095i	$-0.708990\pi$
$263$	−19.7980	−1.22079	−0.610397	+	0.792095i	$0.708990\pi$
$264$	2.00000	0.123091
$265$	0	0
$266$	10.0000	0.613139
$267$	−10.3485	−0.633316
$268$	−2.00000	−0.122169
$269$	3.79796	0.231566	0.115783	−	0.993275i	$-0.463062\pi$
$269$	3.79796	0.231566	0.115783	+	0.993275i	$0.463062\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−5.44949	−0.330424
$273$	−4.20204	−0.254319
$274$	−22.3485	−1.35012
$275$	0	0
$276$	1.00000	0.0601929
$277$	25.5959	1.53791	0.768955	−	0.639303i	$-0.220777\pi$
$277$	25.5959	1.53791	0.768955	+	0.639303i	$0.220777\pi$
$278$	15.6969	0.941440
$279$	2.00000	0.119737
$280$	0	0
$281$	−0.752551	−0.0448934	−0.0224467	−	0.999748i	$-0.507146\pi$
$281$	−0.752551	−0.0448934	−0.0224467	+	0.999748i	$0.507146\pi$
$282$	−3.89898	−0.232181
$283$	11.5959	0.689306	0.344653	−	0.938730i	$-0.387997\pi$
$283$	11.5959	0.689306	0.344653	+	0.938730i	$0.387997\pi$
$284$	10.7980	0.640741
$285$	0	0
$286$	−5.79796	−0.342841
$287$	−7.10102	−0.419160
$288$	1.00000	0.0589256
$289$	12.6969	0.746879
$290$	0	0
$291$	−12.6969	−0.744308
$292$	2.10102	0.122953
$293$	7.79796	0.455562	0.227781	−	0.973712i	$-0.426853\pi$
$293$	7.79796	0.455562	0.227781	+	0.973712i	$0.426853\pi$
$294$	−4.89898	−0.285714
$295$	0	0
$296$	8.34847	0.485245
$297$	2.00000	0.116052
$298$	20.0000	1.15857
$299$	−2.89898	−0.167652
$300$	0	0
$301$	15.7980	0.910579
$302$	−18.6969	−1.07589
$303$	−13.6969	−0.786869
$304$	6.89898	0.395684
$305$	0	0
$306$	−5.44949	−0.311527
$307$	−20.1010	−1.14723	−0.573613	−	0.819126i	$-0.694459\pi$
$307$	−20.1010	−1.14723	−0.573613	+	0.819126i	$0.694459\pi$
$308$	2.89898	0.165185
$309$	−13.2474	−0.753621
$310$	0	0
$311$	14.5959	0.827659	0.413829	−	0.910355i	$-0.364191\pi$
$311$	14.5959	0.827659	0.413829	+	0.910355i	$0.364191\pi$
$312$	−2.89898	−0.164122
$313$	−12.8990	−0.729093	−0.364547	−	0.931185i	$-0.618776\pi$
$313$	−12.8990	−0.729093	−0.364547	+	0.931185i	$0.618776\pi$
$314$	−8.89898	−0.502198
$315$	0	0
$316$	10.0000	0.562544
$317$	10.5959	0.595126	0.297563	−	0.954702i	$-0.403826\pi$
$317$	10.5959	0.595126	0.297563	+	0.954702i	$0.403826\pi$
$318$	0.898979	0.0504123
$319$	10.0000	0.559893
$320$	0	0
$321$	−2.00000	−0.111629
$322$	1.44949	0.0807769
$323$	−37.5959	−2.09189
$324$	1.00000	0.0555556
$325$	0	0
$326$	17.7980	0.985738
$327$	3.44949	0.190757
$328$	−4.89898	−0.270501
$329$	−5.65153	−0.311579
$330$	0	0
$331$	−6.79796	−0.373650	−0.186825	−	0.982393i	$-0.559820\pi$
$331$	−6.79796	−0.373650	−0.186825	+	0.982393i	$0.559820\pi$
$332$	−2.55051	−0.139977
$333$	8.34847	0.457493
$334$	−7.00000	−0.383023
$335$	0	0
$336$	1.44949	0.0790761
$337$	−16.4949	−0.898534	−0.449267	−	0.893397i	$-0.648315\pi$
$337$	−16.4949	−0.898534	−0.449267	+	0.893397i	$0.648315\pi$
$338$	−4.59592	−0.249985
$339$	14.3485	0.779302
$340$	0	0
$341$	4.00000	0.216612
$342$	6.89898	0.373054
$343$	−17.2474	−0.931275
$344$	10.8990	0.587634
$345$	0	0
$346$	−19.7980	−1.06434
$347$	−32.6969	−1.75526	−0.877632	−	0.479335i	$-0.840878\pi$
$347$	−32.6969	−1.75526	−0.877632	+	0.479335i	$0.840878\pi$
$348$	5.00000	0.268028
$349$	13.1010	0.701282	0.350641	−	0.936510i	$-0.385964\pi$
$349$	13.1010	0.701282	0.350641	+	0.936510i	$0.385964\pi$
$350$	0	0
$351$	−2.89898	−0.154736
$352$	2.00000	0.106600
$353$	−26.0000	−1.38384	−0.691920	−	0.721974i	$-0.743235\pi$
$353$	−26.0000	−1.38384	−0.691920	+	0.721974i	$0.743235\pi$
$354$	10.0000	0.531494
$355$	0	0
$356$	−10.3485	−0.548468
$357$	−7.89898	−0.418058
$358$	3.10102	0.163894
$359$	−6.20204	−0.327331	−0.163666	−	0.986516i	$-0.552332\pi$
$359$	−6.20204	−0.327331	−0.163666	+	0.986516i	$0.552332\pi$
$360$	0	0
$361$	28.5959	1.50505
$362$	−18.3485	−0.964374
$363$	−7.00000	−0.367405
$364$	−4.20204	−0.220247
$365$	0	0
$366$	−4.89898	−0.256074
$367$	−8.20204	−0.428143	−0.214072	−	0.976818i	$-0.568673\pi$
$367$	−8.20204	−0.428143	−0.214072	+	0.976818i	$0.568673\pi$
$368$	1.00000	0.0521286
$369$	−4.89898	−0.255031
$370$	0	0
$371$	1.30306	0.0676516
$372$	2.00000	0.103695
$373$	−3.24745	−0.168147	−0.0840733	−	0.996460i	$-0.526793\pi$
$373$	−3.24745	−0.168147	−0.0840733	+	0.996460i	$0.526793\pi$
$374$	−10.8990	−0.563573
$375$	0	0
$376$	−3.89898	−0.201075
$377$	−14.4949	−0.746525
$378$	1.44949	0.0745537
$379$	−38.2929	−1.96697	−0.983486	−	0.180984i	$-0.942072\pi$
$379$	−38.2929	−1.96697	−0.983486	+	0.180984i	$0.942072\pi$
$380$	0	0
$381$	1.79796	0.0921122
$382$	2.00000	0.102329
$383$	−9.79796	−0.500652	−0.250326	−	0.968162i	$-0.580538\pi$
$383$	−9.79796	−0.500652	−0.250326	+	0.968162i	$0.580538\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	9.69694	0.493561
$387$	10.8990	0.554026
$388$	−12.6969	−0.644589
$389$	−23.7980	−1.20660	−0.603302	−	0.797513i	$-0.706149\pi$
$389$	−23.7980	−1.20660	−0.603302	+	0.797513i	$0.706149\pi$
$390$	0	0
$391$	−5.44949	−0.275593
$392$	−4.89898	−0.247436
$393$	8.89898	0.448894
$394$	−3.89898	−0.196428
$395$	0	0
$396$	2.00000	0.100504
$397$	28.0000	1.40528	0.702640	−	0.711546i	$-0.252005\pi$
$397$	28.0000	1.40528	0.702640	+	0.711546i	$0.252005\pi$
$398$	24.1464	1.21035
$399$	10.0000	0.500626
$400$	0	0
$401$	−18.6969	−0.933681	−0.466840	−	0.884342i	$-0.654608\pi$
$401$	−18.6969	−0.933681	−0.466840	+	0.884342i	$0.654608\pi$
$402$	−2.00000	−0.0997509
$403$	−5.79796	−0.288817
$404$	−13.6969	−0.681448
$405$	0	0
$406$	7.24745	0.359685
$407$	16.6969	0.827637
$408$	−5.44949	−0.269790
$409$	25.6969	1.27063	0.635316	−	0.772252i	$-0.280870\pi$
$409$	25.6969	1.27063	0.635316	+	0.772252i	$0.280870\pi$
$410$	0	0
$411$	−22.3485	−1.10237
$412$	−13.2474	−0.652655
$413$	14.4949	0.713247
$414$	1.00000	0.0491473
$415$	0	0
$416$	−2.89898	−0.142134
$417$	15.6969	0.768682
$418$	13.7980	0.674880
$419$	14.1464	0.691098	0.345549	−	0.938401i	$-0.387693\pi$
$419$	14.1464	0.691098	0.345549	+	0.938401i	$0.387693\pi$
$420$	0	0
$421$	−12.4949	−0.608964	−0.304482	−	0.952518i	$-0.598483\pi$
$421$	−12.4949	−0.608964	−0.304482	+	0.952518i	$0.598483\pi$
$422$	10.7980	0.525636
$423$	−3.89898	−0.189575
$424$	0.898979	0.0436583
$425$	0	0
$426$	10.7980	0.523163
$427$	−7.10102	−0.343642
$428$	−2.00000	−0.0966736
$429$	−5.79796	−0.279928
$430$	0	0
$431$	11.3031	0.544449	0.272225	−	0.962234i	$-0.412241\pi$
$431$	11.3031	0.544449	0.272225	+	0.962234i	$0.412241\pi$
$432$	1.00000	0.0481125
$433$	−16.6969	−0.802404	−0.401202	−	0.915990i	$-0.631407\pi$
$433$	−16.6969	−0.802404	−0.401202	+	0.915990i	$0.631407\pi$
$434$	2.89898	0.139155
$435$	0	0
$436$	3.44949	0.165201
$437$	6.89898	0.330023
$438$	2.10102	0.100391
$439$	−10.6969	−0.510537	−0.255269	−	0.966870i	$-0.582164\pi$
$439$	−10.6969	−0.510537	−0.255269	+	0.966870i	$0.582164\pi$
$440$	0	0
$441$	−4.89898	−0.233285
$442$	15.7980	0.751432
$443$	−5.30306	−0.251956	−0.125978	−	0.992033i	$-0.540207\pi$
$443$	−5.30306	−0.251956	−0.125978	+	0.992033i	$0.540207\pi$
$444$	8.34847	0.396201
$445$	0	0
$446$	24.6969	1.16943
$447$	20.0000	0.945968
$448$	1.44949	0.0684820
$449$	3.79796	0.179237	0.0896184	−	0.995976i	$-0.471435\pi$
$449$	3.79796	0.179237	0.0896184	+	0.995976i	$0.471435\pi$
$450$	0	0
$451$	−9.79796	−0.461368
$452$	14.3485	0.674895
$453$	−18.6969	−0.878459
$454$	−4.75255	−0.223048
$455$	0	0
$456$	6.89898	0.323074
$457$	0.404082	0.0189022	0.00945108	−	0.999955i	$-0.496992\pi$
$457$	0.404082	0.0189022	0.00945108	+	0.999955i	$0.496992\pi$
$458$	3.10102	0.144901
$459$	−5.44949	−0.254360
$460$	0	0
$461$	21.4949	1.00112	0.500559	−	0.865703i	$-0.333128\pi$
$461$	21.4949	1.00112	0.500559	+	0.865703i	$0.333128\pi$
$462$	2.89898	0.134873
$463$	4.00000	0.185896	0.0929479	−	0.995671i	$-0.470371\pi$
$463$	4.00000	0.185896	0.0929479	+	0.995671i	$0.470371\pi$
$464$	5.00000	0.232119
$465$	0	0
$466$	−26.6969	−1.23671
$467$	−29.9444	−1.38566	−0.692830	−	0.721100i	$-0.743637\pi$
$467$	−29.9444	−1.38566	−0.692830	+	0.721100i	$0.743637\pi$
$468$	−2.89898	−0.134005
$469$	−2.89898	−0.133862
$470$	0	0
$471$	−8.89898	−0.410043
$472$	10.0000	0.460287
$473$	21.7980	1.00227
$474$	10.0000	0.459315
$475$	0	0
$476$	−7.89898	−0.362049
$477$	0.898979	0.0411614
$478$	−8.10102	−0.370532
$479$	−30.0000	−1.37073	−0.685367	−	0.728197i	$-0.740358\pi$
$479$	−30.0000	−1.37073	−0.685367	+	0.728197i	$0.740358\pi$
$480$	0	0
$481$	−24.2020	−1.10352
$482$	−8.00000	−0.364390
$483$	1.44949	0.0659541
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	−15.1010	−0.684293	−0.342146	−	0.939647i	$-0.611154\pi$
$487$	−15.1010	−0.684293	−0.342146	+	0.939647i	$0.611154\pi$
$488$	−4.89898	−0.221766
$489$	17.7980	0.804852
$490$	0	0
$491$	25.1010	1.13279	0.566397	−	0.824133i	$-0.308337\pi$
$491$	25.1010	1.13279	0.566397	+	0.824133i	$0.308337\pi$
$492$	−4.89898	−0.220863
$493$	−27.2474	−1.22716
$494$	−20.0000	−0.899843
$495$	0	0
$496$	2.00000	0.0898027
$497$	15.6515	0.702067
$498$	−2.55051	−0.114291
$499$	21.8990	0.980333	0.490166	−	0.871629i	$-0.336936\pi$
$499$	21.8990	0.980333	0.490166	+	0.871629i	$0.336936\pi$
$500$	0	0
$501$	−7.00000	−0.312737
$502$	−5.24745	−0.234205
$503$	14.6969	0.655304	0.327652	−	0.944798i	$-0.393743\pi$
$503$	14.6969	0.655304	0.327652	+	0.944798i	$0.393743\pi$
$504$	1.44949	0.0645654
$505$	0	0
$506$	2.00000	0.0889108
$507$	−4.59592	−0.204112
$508$	1.79796	0.0797715
$509$	−15.0000	−0.664863	−0.332432	−	0.943127i	$-0.607869\pi$
$509$	−15.0000	−0.664863	−0.332432	+	0.943127i	$0.607869\pi$
$510$	0	0
$511$	3.04541	0.134721
$512$	1.00000	0.0441942
$513$	6.89898	0.304597
$514$	−8.20204	−0.361777
$515$	0	0
$516$	10.8990	0.479801
$517$	−7.79796	−0.342954
$518$	12.1010	0.531688
$519$	−19.7980	−0.869034
$520$	0	0
$521$	26.1464	1.14550	0.572748	−	0.819732i	$-0.305877\pi$
$521$	26.1464	1.14550	0.572748	+	0.819732i	$0.305877\pi$
$522$	5.00000	0.218844
$523$	−16.0000	−0.699631	−0.349816	−	0.936819i	$-0.613756\pi$
$523$	−16.0000	−0.699631	−0.349816	+	0.936819i	$0.613756\pi$
$524$	8.89898	0.388754
$525$	0	0
$526$	−19.7980	−0.863232
$527$	−10.8990	−0.474767
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	0	0
$531$	10.0000	0.433963
$532$	10.0000	0.433555
$533$	14.2020	0.615159
$534$	−10.3485	−0.447822
$535$	0	0
$536$	−2.00000	−0.0863868
$537$	3.10102	0.133819
$538$	3.79796	0.163742
$539$	−9.79796	−0.422028
$540$	0	0
$541$	−15.5959	−0.670521	−0.335260	−	0.942125i	$-0.608824\pi$
$541$	−15.5959	−0.670521	−0.335260	+	0.942125i	$0.608824\pi$
$542$	−8.00000	−0.343629
$543$	−18.3485	−0.787408
$544$	−5.44949	−0.233645
$545$	0	0
$546$	−4.20204	−0.179831
$547$	−10.7980	−0.461687	−0.230844	−	0.972991i	$-0.574149\pi$
$547$	−10.7980	−0.461687	−0.230844	+	0.972991i	$0.574149\pi$
$548$	−22.3485	−0.954679
$549$	−4.89898	−0.209083
$550$	0	0
$551$	34.4949	1.46953
$552$	1.00000	0.0425628
$553$	14.4949	0.616386
$554$	25.5959	1.08747
$555$	0	0
$556$	15.6969	0.665698
$557$	28.6969	1.21593	0.607964	−	0.793964i	$-0.291986\pi$
$557$	28.6969	1.21593	0.607964	+	0.793964i	$0.291986\pi$
$558$	2.00000	0.0846668
$559$	−31.5959	−1.33636
$560$	0	0
$561$	−10.8990	−0.460155
$562$	−0.752551	−0.0317445
$563$	−6.00000	−0.252870	−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000	−0.252870	−0.126435	+	0.991975i	$0.540353\pi$
$564$	−3.89898	−0.164177
$565$	0	0
$566$	11.5959	0.487413
$567$	1.44949	0.0608728
$568$	10.7980	0.453072
$569$	−44.4949	−1.86532	−0.932662	−	0.360753i	$-0.882520\pi$
$569$	−44.4949	−1.86532	−0.932662	+	0.360753i	$0.882520\pi$
$570$	0	0
$571$	32.6969	1.36832	0.684162	−	0.729330i	$-0.260168\pi$
$571$	32.6969	1.36832	0.684162	+	0.729330i	$0.260168\pi$
$572$	−5.79796	−0.242425
$573$	2.00000	0.0835512
$574$	−7.10102	−0.296391
$575$	0	0
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	12.6969	0.528123
$579$	9.69694	0.402991
$580$	0	0
$581$	−3.69694	−0.153375
$582$	−12.6969	−0.526305
$583$	1.79796	0.0744639
$584$	2.10102	0.0869408
$585$	0	0
$586$	7.79796	0.322131
$587$	4.89898	0.202203	0.101101	−	0.994876i	$-0.467763\pi$
$587$	4.89898	0.202203	0.101101	+	0.994876i	$0.467763\pi$
$588$	−4.89898	−0.202031
$589$	13.7980	0.568535
$590$	0	0
$591$	−3.89898	−0.160383
$592$	8.34847	0.343120
$593$	−43.5959	−1.79027	−0.895135	−	0.445795i	$-0.852921\pi$
$593$	−43.5959	−1.79027	−0.895135	+	0.445795i	$0.852921\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	20.0000	0.819232
$597$	24.1464	0.988248
$598$	−2.89898	−0.118548
$599$	−21.3939	−0.874130	−0.437065	−	0.899430i	$-0.643982\pi$
$599$	−21.3939	−0.874130	−0.437065	+	0.899430i	$0.643982\pi$
$600$	0	0
$601$	37.0000	1.50926	0.754631	−	0.656150i	$-0.227816\pi$
$601$	37.0000	1.50926	0.754631	+	0.656150i	$0.227816\pi$
$602$	15.7980	0.643877
$603$	−2.00000	−0.0814463
$604$	−18.6969	−0.760768
$605$	0	0
$606$	−13.6969	−0.556400
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	6.89898	0.279791
$609$	7.24745	0.293681
$610$	0	0
$611$	11.3031	0.457273
$612$	−5.44949	−0.220283
$613$	−0.146428	−0.00591418	−0.00295709	−	0.999996i	$-0.500941\pi$
$613$	−0.146428	−0.00591418	−0.00295709	+	0.999996i	$0.500941\pi$
$614$	−20.1010	−0.811211
$615$	0	0
$616$	2.89898	0.116803
$617$	12.4949	0.503026	0.251513	−	0.967854i	$-0.419072\pi$
$617$	12.4949	0.503026	0.251513	+	0.967854i	$0.419072\pi$
$618$	−13.2474	−0.532891
$619$	−44.4949	−1.78840	−0.894200	−	0.447667i	$-0.852255\pi$
$619$	−44.4949	−1.78840	−0.894200	+	0.447667i	$0.852255\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	14.5959	0.585243
$623$	−15.0000	−0.600962
$624$	−2.89898	−0.116052
$625$	0	0
$626$	−12.8990	−0.515547
$627$	13.7980	0.551037
$628$	−8.89898	−0.355108
$629$	−45.4949	−1.81400
$630$	0	0
$631$	15.4495	0.615034	0.307517	−	0.951543i	$-0.400502\pi$
$631$	15.4495	0.615034	0.307517	+	0.951543i	$0.400502\pi$
$632$	10.0000	0.397779
$633$	10.7980	0.429180
$634$	10.5959	0.420818
$635$	0	0
$636$	0.898979	0.0356469
$637$	14.2020	0.562705
$638$	10.0000	0.395904
$639$	10.7980	0.427161
$640$	0	0
$641$	−24.5505	−0.969687	−0.484843	−	0.874601i	$-0.661123\pi$
$641$	−24.5505	−0.969687	−0.484843	+	0.874601i	$0.661123\pi$
$642$	−2.00000	−0.0789337
$643$	−19.7980	−0.780755	−0.390378	−	0.920655i	$-0.627656\pi$
$643$	−19.7980	−0.780755	−0.390378	+	0.920655i	$0.627656\pi$
$644$	1.44949	0.0571179
$645$	0	0
$646$	−37.5959	−1.47919
$647$	20.5959	0.809709	0.404855	−	0.914381i	$-0.367322\pi$
$647$	20.5959	0.809709	0.404855	+	0.914381i	$0.367322\pi$
$648$	1.00000	0.0392837
$649$	20.0000	0.785069
$650$	0	0
$651$	2.89898	0.113620
$652$	17.7980	0.697022
$653$	−0.303062	−0.0118597	−0.00592986	−	0.999982i	$-0.501888\pi$
$653$	−0.303062	−0.0118597	−0.00592986	+	0.999982i	$0.501888\pi$
$654$	3.44949	0.134886
$655$	0	0
$656$	−4.89898	−0.191273
$657$	2.10102	0.0819686
$658$	−5.65153	−0.220320
$659$	−23.4495	−0.913462	−0.456731	−	0.889605i	$-0.650980\pi$
$659$	−23.4495	−0.913462	−0.456731	+	0.889605i	$0.650980\pi$
$660$	0	0
$661$	−21.4495	−0.834288	−0.417144	−	0.908840i	$-0.636969\pi$
$661$	−21.4495	−0.834288	−0.417144	+	0.908840i	$0.636969\pi$
$662$	−6.79796	−0.264210
$663$	15.7980	0.613542
$664$	−2.55051	−0.0989790
$665$	0	0
$666$	8.34847	0.323497
$667$	5.00000	0.193601
$668$	−7.00000	−0.270838
$669$	24.6969	0.954839
$670$	0	0
$671$	−9.79796	−0.378246
$672$	1.44949	0.0559153
$673$	26.5959	1.02520	0.512599	−	0.858628i	$-0.328683\pi$
$673$	26.5959	1.02520	0.512599	+	0.858628i	$0.328683\pi$
$674$	−16.4949	−0.635360
$675$	0	0
$676$	−4.59592	−0.176766
$677$	−33.3939	−1.28343	−0.641715	−	0.766943i	$-0.721777\pi$
$677$	−33.3939	−1.28343	−0.641715	+	0.766943i	$0.721777\pi$
$678$	14.3485	0.551050
$679$	−18.4041	−0.706284
$680$	0	0
$681$	−4.75255	−0.182118
$682$	4.00000	0.153168
$683$	7.10102	0.271713	0.135856	−	0.990729i	$-0.456621\pi$
$683$	7.10102	0.271713	0.135856	+	0.990729i	$0.456621\pi$
$684$	6.89898	0.263789
$685$	0	0
$686$	−17.2474	−0.658511
$687$	3.10102	0.118311
$688$	10.8990	0.415520
$689$	−2.60612	−0.0992854
$690$	0	0
$691$	−9.20204	−0.350062	−0.175031	−	0.984563i	$-0.556003\pi$
$691$	−9.20204	−0.350062	−0.175031	+	0.984563i	$0.556003\pi$
$692$	−19.7980	−0.752605
$693$	2.89898	0.110123
$694$	−32.6969	−1.24116
$695$	0	0
$696$	5.00000	0.189525
$697$	26.6969	1.01122
$698$	13.1010	0.495881
$699$	−26.6969	−1.00977
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	−2.89898	−0.109415
$703$	57.5959	2.17227
$704$	2.00000	0.0753778
$705$	0	0
$706$	−26.0000	−0.978523
$707$	−19.8536	−0.746670
$708$	10.0000	0.375823
$709$	27.9444	1.04947	0.524737	−	0.851265i	$-0.324164\pi$
$709$	27.9444	1.04947	0.524737	+	0.851265i	$0.324164\pi$
$710$	0	0
$711$	10.0000	0.375029
$712$	−10.3485	−0.387825
$713$	2.00000	0.0749006
$714$	−7.89898	−0.295612
$715$	0	0
$716$	3.10102	0.115891
$717$	−8.10102	−0.302538
$718$	−6.20204	−0.231458
$719$	28.7980	1.07398	0.536991	−	0.843588i	$-0.319561\pi$
$719$	28.7980	1.07398	0.536991	+	0.843588i	$0.319561\pi$
$720$	0	0
$721$	−19.2020	−0.715121
$722$	28.5959	1.06423
$723$	−8.00000	−0.297523
$724$	−18.3485	−0.681915
$725$	0	0
$726$	−7.00000	−0.259794
$727$	39.3939	1.46104	0.730519	−	0.682892i	$-0.239278\pi$
$727$	39.3939	1.46104	0.730519	+	0.682892i	$0.239278\pi$
$728$	−4.20204	−0.155738
$729$	1.00000	0.0370370
$730$	0	0
$731$	−59.3939	−2.19676
$732$	−4.89898	−0.181071
$733$	−40.1464	−1.48284	−0.741421	−	0.671040i	$-0.765848\pi$
$733$	−40.1464	−1.48284	−0.741421	+	0.671040i	$0.765848\pi$
$734$	−8.20204	−0.302743
$735$	0	0
$736$	1.00000	0.0368605
$737$	−4.00000	−0.147342
$738$	−4.89898	−0.180334
$739$	12.5959	0.463348	0.231674	−	0.972793i	$-0.425580\pi$
$739$	12.5959	0.463348	0.231674	+	0.972793i	$0.425580\pi$
$740$	0	0
$741$	−20.0000	−0.734718
$742$	1.30306	0.0478369
$743$	−24.2929	−0.891218	−0.445609	−	0.895228i	$-0.647013\pi$
$743$	−24.2929	−0.891218	−0.445609	+	0.895228i	$0.647013\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	−3.24745	−0.118898
$747$	−2.55051	−0.0933183
$748$	−10.8990	−0.398506
$749$	−2.89898	−0.105926
$750$	0	0
$751$	−25.2474	−0.921292	−0.460646	−	0.887584i	$-0.652382\pi$
$751$	−25.2474	−0.921292	−0.460646	+	0.887584i	$0.652382\pi$
$752$	−3.89898	−0.142181
$753$	−5.24745	−0.191228
$754$	−14.4949	−0.527873
$755$	0	0
$756$	1.44949	0.0527174
$757$	10.7526	0.390808	0.195404	−	0.980723i	$-0.437398\pi$
$757$	10.7526	0.390808	0.195404	+	0.980723i	$0.437398\pi$
$758$	−38.2929	−1.39086
$759$	2.00000	0.0725954
$760$	0	0
$761$	−50.0908	−1.81579	−0.907895	−	0.419197i	$-0.862312\pi$
$761$	−50.0908	−1.81579	−0.907895	+	0.419197i	$0.862312\pi$
$762$	1.79796	0.0651332
$763$	5.00000	0.181012
$764$	2.00000	0.0723575
$765$	0	0
$766$	−9.79796	−0.354015
$767$	−28.9898	−1.04676
$768$	1.00000	0.0360844
$769$	48.2929	1.74148	0.870742	−	0.491739i	$-0.163639\pi$
$769$	48.2929	1.74148	0.870742	+	0.491739i	$0.163639\pi$
$770$	0	0
$771$	−8.20204	−0.295389
$772$	9.69694	0.349000
$773$	10.8990	0.392009	0.196005	−	0.980603i	$-0.437203\pi$
$773$	10.8990	0.392009	0.196005	+	0.980603i	$0.437203\pi$
$774$	10.8990	0.391756
$775$	0	0
$776$	−12.6969	−0.455794
$777$	12.1010	0.434122
$778$	−23.7980	−0.853198
$779$	−33.7980	−1.21094
$780$	0	0
$781$	21.5959	0.772763
$782$	−5.44949	−0.194873
$783$	5.00000	0.178685
$784$	−4.89898	−0.174964
$785$	0	0
$786$	8.89898	0.317416
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	−3.89898	−0.138895
$789$	−19.7980	−0.704826
$790$	0	0
$791$	20.7980	0.739490
$792$	2.00000	0.0710669
$793$	14.2020	0.504329
$794$	28.0000	0.993683
$795$	0	0
$796$	24.1464	0.855847
$797$	34.8990	1.23619	0.618093	−	0.786105i	$-0.287906\pi$
$797$	34.8990	1.23619	0.618093	+	0.786105i	$0.287906\pi$
$798$	10.0000	0.353996
$799$	21.2474	0.751681
$800$	0	0
$801$	−10.3485	−0.365645
$802$	−18.6969	−0.660212
$803$	4.20204	0.148287
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	−5.79796	−0.204224
$807$	3.79796	0.133694
$808$	−13.6969	−0.481857
$809$	11.3939	0.400587	0.200294	−	0.979736i	$-0.435810\pi$
$809$	11.3939	0.400587	0.200294	+	0.979736i	$0.435810\pi$
$810$	0	0
$811$	45.7980	1.60818	0.804092	−	0.594505i	$-0.202652\pi$
$811$	45.7980	1.60818	0.804092	+	0.594505i	$0.202652\pi$
$812$	7.24745	0.254336
$813$	−8.00000	−0.280572
$814$	16.6969	0.585227
$815$	0	0
$816$	−5.44949	−0.190770
$817$	75.1918	2.63063
$818$	25.6969	0.898472
$819$	−4.20204	−0.146831
$820$	0	0
$821$	−51.7980	−1.80776	−0.903881	−	0.427785i	$-0.859294\pi$
$821$	−51.7980	−1.80776	−0.903881	+	0.427785i	$0.859294\pi$
$822$	−22.3485	−0.779492
$823$	56.0908	1.95520	0.977601	−	0.210465i	$-0.0674977\pi$
$823$	56.0908	1.95520	0.977601	+	0.210465i	$0.0674977\pi$
$824$	−13.2474	−0.461497
$825$	0	0
$826$	14.4949	0.504342
$827$	−14.7526	−0.512996	−0.256498	−	0.966545i	$-0.582569\pi$
$827$	−14.7526	−0.512996	−0.256498	+	0.966545i	$0.582569\pi$
$828$	1.00000	0.0347524
$829$	−33.1010	−1.14965	−0.574823	−	0.818278i	$-0.694929\pi$
$829$	−33.1010	−1.14965	−0.574823	+	0.818278i	$0.694929\pi$
$830$	0	0
$831$	25.5959	0.887913
$832$	−2.89898	−0.100504
$833$	26.6969	0.924994
$834$	15.6969	0.543541
$835$	0	0
$836$	13.7980	0.477212
$837$	2.00000	0.0691301
$838$	14.1464	0.488680
$839$	−9.30306	−0.321177	−0.160589	−	0.987021i	$-0.551339\pi$
$839$	−9.30306	−0.321177	−0.160589	+	0.987021i	$0.551339\pi$
$840$	0	0
$841$	−4.00000	−0.137931
$842$	−12.4949	−0.430603
$843$	−0.752551	−0.0259192
$844$	10.7980	0.371681
$845$	0	0
$846$	−3.89898	−0.134050
$847$	−10.1464	−0.348635
$848$	0.898979	0.0308711
$849$	11.5959	0.397971
$850$	0	0
$851$	8.34847	0.286182
$852$	10.7980	0.369932
$853$	−44.9898	−1.54042	−0.770211	−	0.637790i	$-0.779849\pi$
$853$	−44.9898	−1.54042	−0.770211	+	0.637790i	$0.779849\pi$
$854$	−7.10102	−0.242992
$855$	0	0
$856$	−2.00000	−0.0683586
$857$	42.4949	1.45160	0.725799	−	0.687907i	$-0.241470\pi$
$857$	42.4949	1.45160	0.725799	+	0.687907i	$0.241470\pi$
$858$	−5.79796	−0.197939
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	0	0
$861$	−7.10102	−0.242002
$862$	11.3031	0.384984
$863$	20.3939	0.694216	0.347108	−	0.937825i	$-0.387164\pi$
$863$	20.3939	0.694216	0.347108	+	0.937825i	$0.387164\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	−16.6969	−0.567385
$867$	12.6969	0.431211
$868$	2.89898	0.0983978
$869$	20.0000	0.678454
$870$	0	0
$871$	5.79796	0.196456
$872$	3.44949	0.116814
$873$	−12.6969	−0.429726
$874$	6.89898	0.233361
$875$	0	0
$876$	2.10102	0.0709869
$877$	−9.59592	−0.324031	−0.162016	−	0.986788i	$-0.551799\pi$
$877$	−9.59592	−0.324031	−0.162016	+	0.986788i	$0.551799\pi$
$878$	−10.6969	−0.361004
$879$	7.79796	0.263019
$880$	0	0
$881$	−31.1010	−1.04782	−0.523910	−	0.851774i	$-0.675527\pi$
$881$	−31.1010	−1.04782	−0.523910	+	0.851774i	$0.675527\pi$
$882$	−4.89898	−0.164957
$883$	−48.5959	−1.63538	−0.817691	−	0.575657i	$-0.804746\pi$
$883$	−48.5959	−1.63538	−0.817691	+	0.575657i	$0.804746\pi$
$884$	15.7980	0.531343
$885$	0	0
$886$	−5.30306	−0.178160
$887$	−10.1010	−0.339159	−0.169580	−	0.985517i	$-0.554241\pi$
$887$	−10.1010	−0.339159	−0.169580	+	0.985517i	$0.554241\pi$
$888$	8.34847	0.280156
$889$	2.60612	0.0874066
$890$	0	0
$891$	2.00000	0.0670025
$892$	24.6969	0.826915
$893$	−26.8990	−0.900140
$894$	20.0000	0.668900
$895$	0	0
$896$	1.44949	0.0484241
$897$	−2.89898	−0.0967941
$898$	3.79796	0.126740
$899$	10.0000	0.333519
$900$	0	0
$901$	−4.89898	−0.163209
$902$	−9.79796	−0.326236
$903$	15.7980	0.525723
$904$	14.3485	0.477223
$905$	0	0
$906$	−18.6969	−0.621164
$907$	2.49490	0.0828417	0.0414209	−	0.999142i	$-0.486812\pi$
$907$	2.49490	0.0828417	0.0414209	+	0.999142i	$0.486812\pi$
$908$	−4.75255	−0.157719
$909$	−13.6969	−0.454299
$910$	0	0
$911$	21.3031	0.705802	0.352901	−	0.935661i	$-0.385195\pi$
$911$	21.3031	0.705802	0.352901	+	0.935661i	$0.385195\pi$
$912$	6.89898	0.228448
$913$	−5.10102	−0.168819
$914$	0.404082	0.0133658
$915$	0	0
$916$	3.10102	0.102461
$917$	12.8990	0.425962
$918$	−5.44949	−0.179860
$919$	−27.9444	−0.921800	−0.460900	−	0.887452i	$-0.652473\pi$
$919$	−27.9444	−0.921800	−0.460900	+	0.887452i	$0.652473\pi$
$920$	0	0
$921$	−20.1010	−0.662351
$922$	21.4949	0.707897
$923$	−31.3031	−1.03035
$924$	2.89898	0.0953694
$925$	0	0
$926$	4.00000	0.131448
$927$	−13.2474	−0.435103
$928$	5.00000	0.164133
$929$	−3.79796	−0.124607	−0.0623035	−	0.998057i	$-0.519845\pi$
$929$	−3.79796	−0.124607	−0.0623035	+	0.998057i	$0.519845\pi$
$930$	0	0
$931$	−33.7980	−1.10768
$932$	−26.6969	−0.874487
$933$	14.5959	0.477849
$934$	−29.9444	−0.979810
$935$	0	0
$936$	−2.89898	−0.0947561
$937$	−15.7980	−0.516097	−0.258048	−	0.966132i	$-0.583079\pi$
$937$	−15.7980	−0.516097	−0.258048	+	0.966132i	$0.583079\pi$
$938$	−2.89898	−0.0946550
$939$	−12.8990	−0.420942
$940$	0	0
$941$	2.00000	0.0651981	0.0325991	−	0.999469i	$-0.489622\pi$
$941$	2.00000	0.0651981	0.0325991	+	0.999469i	$0.489622\pi$
$942$	−8.89898	−0.289944
$943$	−4.89898	−0.159533
$944$	10.0000	0.325472
$945$	0	0
$946$	21.7980	0.708713
$947$	0.404082	0.0131309	0.00656545	−	0.999978i	$-0.497910\pi$
$947$	0.404082	0.0131309	0.00656545	+	0.999978i	$0.497910\pi$
$948$	10.0000	0.324785
$949$	−6.09082	−0.197716
$950$	0	0
$951$	10.5959	0.343596
$952$	−7.89898	−0.256007
$953$	−30.8434	−0.999115	−0.499557	−	0.866281i	$-0.666504\pi$
$953$	−30.8434	−0.999115	−0.499557	+	0.866281i	$0.666504\pi$
$954$	0.898979	0.0291055
$955$	0	0
$956$	−8.10102	−0.262006
$957$	10.0000	0.323254
$958$	−30.0000	−0.969256
$959$	−32.3939	−1.04605
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−24.2020	−0.780305
$963$	−2.00000	−0.0644491
$964$	−8.00000	−0.257663
$965$	0	0
$966$	1.44949	0.0466366
$967$	−29.5959	−0.951741	−0.475870	−	0.879515i	$-0.657867\pi$
$967$	−29.5959	−0.951741	−0.475870	+	0.879515i	$0.657867\pi$
$968$	−7.00000	−0.224989
$969$	−37.5959	−1.20775
$970$	0	0
$971$	26.1464	0.839079	0.419539	−	0.907737i	$-0.362192\pi$
$971$	26.1464	0.839079	0.419539	+	0.907737i	$0.362192\pi$
$972$	1.00000	0.0320750
$973$	22.7526	0.729413
$974$	−15.1010	−0.483868
$975$	0	0
$976$	−4.89898	−0.156813
$977$	−24.7526	−0.791904	−0.395952	−	0.918271i	$-0.629585\pi$
$977$	−24.7526	−0.791904	−0.395952	+	0.918271i	$0.629585\pi$
$978$	17.7980	0.569116
$979$	−20.6969	−0.661477
$980$	0	0
$981$	3.44949	0.110134
$982$	25.1010	0.801006
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−4.89898	−0.156174
$985$	0	0
$986$	−27.2474	−0.867736
$987$	−5.65153	−0.179890
$988$	−20.0000	−0.636285
$989$	10.8990	0.346567
$990$	0	0
$991$	18.2020	0.578207	0.289103	−	0.957298i	$-0.406643\pi$
$991$	18.2020	0.578207	0.289103	+	0.957298i	$0.406643\pi$
$992$	2.00000	0.0635001
$993$	−6.79796	−0.215727
$994$	15.6515	0.496436
$995$	0	0
$996$	−2.55051	−0.0808160
$997$	21.1010	0.668276	0.334138	−	0.942524i	$-0.391555\pi$
$997$	21.1010	0.668276	0.334138	+	0.942524i	$0.391555\pi$
$998$	21.8990	0.693200
$999$	8.34847	0.264134

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bl.1.2	yes	2
5.2	odd	4		3450.2.d.y.2899.4		4
5.3	odd	4		3450.2.d.y.2899.1		4
5.4	even	2		3450.2.a.bf.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.bf.1.1	✓	2	5.4	even	2
3450.2.a.bl.1.2	yes	2	1.1	even	1	trivial
3450.2.d.y.2899.1		4	5.3	odd	4
3450.2.d.y.2899.4		4	5.2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +1.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.00000 q^{12} -2.89898 q^{13} +1.44949 q^{14} +1.00000 q^{16} -5.44949 q^{17} +1.00000 q^{18} +6.89898 q^{19} +1.44949 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} -2.89898 q^{26} +1.00000 q^{27} +1.44949 q^{28} +5.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} -5.44949 q^{34} +1.00000 q^{36} +8.34847 q^{37} +6.89898 q^{38} -2.89898 q^{39} -4.89898 q^{41} +1.44949 q^{42} +10.8990 q^{43} +2.00000 q^{44} +1.00000 q^{46} -3.89898 q^{47} +1.00000 q^{48} -4.89898 q^{49} -5.44949 q^{51} -2.89898 q^{52} +0.898979 q^{53} +1.00000 q^{54} +1.44949 q^{56} +6.89898 q^{57} +5.00000 q^{58} +10.0000 q^{59} -4.89898 q^{61} +2.00000 q^{62} +1.44949 q^{63} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} -5.44949 q^{68} +1.00000 q^{69} +10.7980 q^{71} +1.00000 q^{72} +2.10102 q^{73} +8.34847 q^{74} +6.89898 q^{76} +2.89898 q^{77} -2.89898 q^{78} +10.0000 q^{79} +1.00000 q^{81} -4.89898 q^{82} -2.55051 q^{83} +1.44949 q^{84} +10.8990 q^{86} +5.00000 q^{87} +2.00000 q^{88} -10.3485 q^{89} -4.20204 q^{91} +1.00000 q^{92} +2.00000 q^{93} -3.89898 q^{94} +1.00000 q^{96} -12.6969 q^{97} -4.89898 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} - 2 q^{7} + 2 q^{8} + 2 q^{9} + 4 q^{11} + 2 q^{12} + 4 q^{13} - 2 q^{14} + 2 q^{16} - 6 q^{17} + 2 q^{18} + 4 q^{19} - 2 q^{21} + 4 q^{22} + 2 q^{23} + 2 q^{24} + 4 q^{26} + 2 q^{27} - 2 q^{28} + 10 q^{29} + 4 q^{31} + 2 q^{32} + 4 q^{33} - 6 q^{34} + 2 q^{36} + 2 q^{37} + 4 q^{38} + 4 q^{39} - 2 q^{42} + 12 q^{43} + 4 q^{44} + 2 q^{46} + 2 q^{47} + 2 q^{48} - 6 q^{51} + 4 q^{52} - 8 q^{53} + 2 q^{54} - 2 q^{56} + 4 q^{57} + 10 q^{58} + 20 q^{59} + 4 q^{62} - 2 q^{63} + 2 q^{64} + 4 q^{66} - 4 q^{67} - 6 q^{68} + 2 q^{69} + 2 q^{71} + 2 q^{72} + 14 q^{73} + 2 q^{74} + 4 q^{76} - 4 q^{77} + 4 q^{78} + 20 q^{79} + 2 q^{81} - 10 q^{83} - 2 q^{84} + 12 q^{86} + 10 q^{87} + 4 q^{88} - 6 q^{89} - 28 q^{91} + 2 q^{92} + 4 q^{93} + 2 q^{94} + 2 q^{96} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 - 2 * q^7 + 2 * q^8 + 2 * q^9 + 4 * q^11 + 2 * q^12 + 4 * q^13 - 2 * q^14 + 2 * q^16 - 6 * q^17 + 2 * q^18 + 4 * q^19 - 2 * q^21 + 4 * q^22 + 2 * q^23 + 2 * q^24 + 4 * q^26 + 2 * q^27 - 2 * q^28 + 10 * q^29 + 4 * q^31 + 2 * q^32 + 4 * q^33 - 6 * q^34 + 2 * q^36 + 2 * q^37 + 4 * q^38 + 4 * q^39 - 2 * q^42 + 12 * q^43 + 4 * q^44 + 2 * q^46 + 2 * q^47 + 2 * q^48 - 6 * q^51 + 4 * q^52 - 8 * q^53 + 2 * q^54 - 2 * q^56 + 4 * q^57 + 10 * q^58 + 20 * q^59 + 4 * q^62 - 2 * q^63 + 2 * q^64 + 4 * q^66 - 4 * q^67 - 6 * q^68 + 2 * q^69 + 2 * q^71 + 2 * q^72 + 14 * q^73 + 2 * q^74 + 4 * q^76 - 4 * q^77 + 4 * q^78 + 20 * q^79 + 2 * q^81 - 10 * q^83 - 2 * q^84 + 12 * q^86 + 10 * q^87 + 4 * q^88 - 6 * q^89 - 28 * q^91 + 2 * q^92 + 4 * q^93 + 2 * q^94 + 2 * q^96 + 4 * q^97 + 4 * q^99

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