LMFDB - Embedded newform 5610.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5610,2,Mod(1,5610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5610, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5610.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:	$44.7960755339$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -7.00000 q^{19} +1.00000 q^{20} -1.00000 q^{21} +1.00000 q^{22} +3.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} -1.00000 q^{28} -6.00000 q^{29} -1.00000 q^{30} +5.00000 q^{31} -1.00000 q^{32} -1.00000 q^{33} +1.00000 q^{34} -1.00000 q^{35} +1.00000 q^{36} +11.0000 q^{37} +7.00000 q^{38} -1.00000 q^{39} -1.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -1.00000 q^{44} +1.00000 q^{45} -3.00000 q^{46} -6.00000 q^{47} +1.00000 q^{48} -6.00000 q^{49} -1.00000 q^{50} -1.00000 q^{51} -1.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -1.00000 q^{55} +1.00000 q^{56} -7.00000 q^{57} +6.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} +5.00000 q^{61} -5.00000 q^{62} -1.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +1.00000 q^{66} -13.0000 q^{67} -1.00000 q^{68} +3.00000 q^{69} +1.00000 q^{70} -6.00000 q^{71} -1.00000 q^{72} -4.00000 q^{73} -11.0000 q^{74} +1.00000 q^{75} -7.00000 q^{76} +1.00000 q^{77} +1.00000 q^{78} -10.0000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -9.00000 q^{83} -1.00000 q^{84} -1.00000 q^{85} +4.00000 q^{86} -6.00000 q^{87} +1.00000 q^{88} +6.00000 q^{89} -1.00000 q^{90} +1.00000 q^{91} +3.00000 q^{92} +5.00000 q^{93} +6.00000 q^{94} -7.00000 q^{95} -1.00000 q^{96} -1.00000 q^{97} +6.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−1.00000	−0.408248
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	1.00000	0.288675
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	1.00000	0.267261
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	1.00000	0.223607
$21$	−1.00000	−0.218218
$22$	1.00000	0.213201
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	−1.00000	−0.188982
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	−1.00000	−0.182574
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	−1.00000	−0.176777
$33$	−1.00000	−0.174078
$34$	1.00000	0.171499
$35$	−1.00000	−0.169031
$36$	1.00000	0.166667
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	7.00000	1.13555
$39$	−1.00000	−0.160128
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	1.00000	0.154303
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	1.00000	0.149071
$46$	−3.00000	−0.442326
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	1.00000	0.144338
$49$	−6.00000	−0.857143
$50$	−1.00000	−0.141421
$51$	−1.00000	−0.140028
$52$	−1.00000	−0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	−1.00000	−0.134840
$56$	1.00000	0.133631
$57$	−7.00000	−0.927173
$58$	6.00000	0.787839
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	−5.00000	−0.635001
$63$	−1.00000	−0.125988
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	1.00000	0.123091
$67$	−13.0000	−1.58820	−0.794101	−	0.607785i	$-0.792058\pi$
$67$	−13.0000	−1.58820	−0.794101	+	0.607785i	$0.792058\pi$
$68$	−1.00000	−0.121268
$69$	3.00000	0.361158
$70$	1.00000	0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	−1.00000	−0.117851
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	−11.0000	−1.27872
$75$	1.00000	0.115470
$76$	−7.00000	−0.802955
$77$	1.00000	0.113961
$78$	1.00000	0.113228
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−9.00000	−0.987878	−0.493939	−	0.869496i	$-0.664443\pi$
$83$	−9.00000	−0.987878	−0.493939	+	0.869496i	$0.664443\pi$
$84$	−1.00000	−0.109109
$85$	−1.00000	−0.108465
$86$	4.00000	0.431331
$87$	−6.00000	−0.643268
$88$	1.00000	0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	1.00000	0.104828
$92$	3.00000	0.312772
$93$	5.00000	0.518476
$94$	6.00000	0.618853
$95$	−7.00000	−0.718185
$96$	−1.00000	−0.102062
$97$	−1.00000	−0.101535	−0.0507673	−	0.998711i	$-0.516167\pi$
$97$	−1.00000	−0.101535	−0.0507673	+	0.998711i	$0.516167\pi$
$98$	6.00000	0.606092
$99$	−1.00000	−0.100504
$100$	1.00000	0.100000
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	1.00000	0.0990148
$103$	−13.0000	−1.28093	−0.640464	−	0.767988i	$-0.721258\pi$
$103$	−13.0000	−1.28093	−0.640464	+	0.767988i	$0.721258\pi$
$104$	1.00000	0.0980581
$105$	−1.00000	−0.0975900
$106$	6.00000	0.582772
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	1.00000	0.0962250
$109$	−1.00000	−0.0957826	−0.0478913	−	0.998853i	$-0.515250\pi$
$109$	−1.00000	−0.0957826	−0.0478913	+	0.998853i	$0.515250\pi$
$110$	1.00000	0.0953463
$111$	11.0000	1.04407
$112$	−1.00000	−0.0944911
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	7.00000	0.655610
$115$	3.00000	0.279751
$116$	−6.00000	−0.557086
$117$	−1.00000	−0.0924500
$118$	−12.0000	−1.10469
$119$	1.00000	0.0916698
$120$	−1.00000	−0.0912871
$121$	1.00000	0.0909091
$122$	−5.00000	−0.452679
$123$	6.00000	0.541002
$124$	5.00000	0.449013
$125$	1.00000	0.0894427
$126$	1.00000	0.0890871
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	1.00000	0.0877058
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	−1.00000	−0.0870388
$133$	7.00000	0.606977
$134$	13.0000	1.12303
$135$	1.00000	0.0860663
$136$	1.00000	0.0857493
$137$	3.00000	0.256307	0.128154	−	0.991754i	$-0.459095\pi$
$137$	3.00000	0.256307	0.128154	+	0.991754i	$0.459095\pi$
$138$	−3.00000	−0.255377
$139$	−10.0000	−0.848189	−0.424094	−	0.905618i	$-0.639408\pi$
$139$	−10.0000	−0.848189	−0.424094	+	0.905618i	$0.639408\pi$
$140$	−1.00000	−0.0845154
$141$	−6.00000	−0.505291
$142$	6.00000	0.503509
$143$	1.00000	0.0836242
$144$	1.00000	0.0833333
$145$	−6.00000	−0.498273
$146$	4.00000	0.331042
$147$	−6.00000	−0.494872
$148$	11.0000	0.904194
$149$	−15.0000	−1.22885	−0.614424	−	0.788976i	$-0.710612\pi$
$149$	−15.0000	−1.22885	−0.614424	+	0.788976i	$0.710612\pi$
$150$	−1.00000	−0.0816497
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	7.00000	0.567775
$153$	−1.00000	−0.0808452
$154$	−1.00000	−0.0805823
$155$	5.00000	0.401610
$156$	−1.00000	−0.0800641
$157$	−16.0000	−1.27694	−0.638470	−	0.769647i	$-0.720432\pi$
$157$	−16.0000	−1.27694	−0.638470	+	0.769647i	$0.720432\pi$
$158$	10.0000	0.795557
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	−3.00000	−0.236433
$162$	−1.00000	−0.0785674
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	6.00000	0.468521
$165$	−1.00000	−0.0778499
$166$	9.00000	0.698535
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	1.00000	0.0771517
$169$	−12.0000	−0.923077
$170$	1.00000	0.0766965
$171$	−7.00000	−0.535303
$172$	−4.00000	−0.304997
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	6.00000	0.454859
$175$	−1.00000	−0.0755929
$176$	−1.00000	−0.0753778
$177$	12.0000	0.901975
$178$	−6.00000	−0.449719
$179$	3.00000	0.224231	0.112115	−	0.993695i	$-0.464237\pi$
$179$	3.00000	0.224231	0.112115	+	0.993695i	$0.464237\pi$
$180$	1.00000	0.0745356
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	−1.00000	−0.0741249
$183$	5.00000	0.369611
$184$	−3.00000	−0.221163
$185$	11.0000	0.808736
$186$	−5.00000	−0.366618
$187$	1.00000	0.0731272
$188$	−6.00000	−0.437595
$189$	−1.00000	−0.0727393
$190$	7.00000	0.507833
$191$	−24.0000	−1.73658	−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000	−1.73658	−0.868290	+	0.496058i	$0.834780\pi$
$192$	1.00000	0.0721688
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	1.00000	0.0717958
$195$	−1.00000	−0.0716115
$196$	−6.00000	−0.428571
$197$	21.0000	1.49619	0.748094	−	0.663593i	$-0.230969\pi$
$197$	21.0000	1.49619	0.748094	+	0.663593i	$0.230969\pi$
$198$	1.00000	0.0710669
$199$	5.00000	0.354441	0.177220	−	0.984171i	$-0.443289\pi$
$199$	5.00000	0.354441	0.177220	+	0.984171i	$0.443289\pi$
$200$	−1.00000	−0.0707107
$201$	−13.0000	−0.916949
$202$	18.0000	1.26648
$203$	6.00000	0.421117
$204$	−1.00000	−0.0700140
$205$	6.00000	0.419058
$206$	13.0000	0.905753
$207$	3.00000	0.208514
$208$	−1.00000	−0.0693375
$209$	7.00000	0.484200
$210$	1.00000	0.0690066
$211$	−22.0000	−1.51454	−0.757271	−	0.653101i	$-0.773468\pi$
$211$	−22.0000	−1.51454	−0.757271	+	0.653101i	$0.773468\pi$
$212$	−6.00000	−0.412082
$213$	−6.00000	−0.411113
$214$	−18.0000	−1.23045
$215$	−4.00000	−0.272798
$216$	−1.00000	−0.0680414
$217$	−5.00000	−0.339422
$218$	1.00000	0.0677285
$219$	−4.00000	−0.270295
$220$	−1.00000	−0.0674200
$221$	1.00000	0.0672673
$222$	−11.0000	−0.738272
$223$	−19.0000	−1.27233	−0.636167	−	0.771551i	$-0.719481\pi$
$223$	−19.0000	−1.27233	−0.636167	+	0.771551i	$0.719481\pi$
$224$	1.00000	0.0668153
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	−7.00000	−0.463586
$229$	−13.0000	−0.859064	−0.429532	−	0.903052i	$-0.641321\pi$
$229$	−13.0000	−0.859064	−0.429532	+	0.903052i	$0.641321\pi$
$230$	−3.00000	−0.197814
$231$	1.00000	0.0657952
$232$	6.00000	0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	1.00000	0.0653720
$235$	−6.00000	−0.391397
$236$	12.0000	0.781133
$237$	−10.0000	−0.649570
$238$	−1.00000	−0.0648204
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	1.00000	0.0645497
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	−1.00000	−0.0642824
$243$	1.00000	0.0641500
$244$	5.00000	0.320092
$245$	−6.00000	−0.383326
$246$	−6.00000	−0.382546
$247$	7.00000	0.445399
$248$	−5.00000	−0.317500
$249$	−9.00000	−0.570352
$250$	−1.00000	−0.0632456
$251$	15.0000	0.946792	0.473396	−	0.880850i	$-0.343028\pi$
$251$	15.0000	0.946792	0.473396	+	0.880850i	$0.343028\pi$
$252$	−1.00000	−0.0629941
$253$	−3.00000	−0.188608
$254$	−8.00000	−0.501965
$255$	−1.00000	−0.0626224
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	4.00000	0.249029
$259$	−11.0000	−0.683507
$260$	−1.00000	−0.0620174
$261$	−6.00000	−0.371391
$262$	15.0000	0.926703
$263$	−21.0000	−1.29492	−0.647458	−	0.762101i	$-0.724168\pi$
$263$	−21.0000	−1.29492	−0.647458	+	0.762101i	$0.724168\pi$
$264$	1.00000	0.0615457
$265$	−6.00000	−0.368577
$266$	−7.00000	−0.429198
$267$	6.00000	0.367194
$268$	−13.0000	−0.794101
$269$	9.00000	0.548740	0.274370	−	0.961624i	$-0.411531\pi$
$269$	9.00000	0.548740	0.274370	+	0.961624i	$0.411531\pi$
$270$	−1.00000	−0.0608581
$271$	−28.0000	−1.70088	−0.850439	−	0.526073i	$-0.823664\pi$
$271$	−28.0000	−1.70088	−0.850439	+	0.526073i	$0.823664\pi$
$272$	−1.00000	−0.0606339
$273$	1.00000	0.0605228
$274$	−3.00000	−0.181237
$275$	−1.00000	−0.0603023
$276$	3.00000	0.180579
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	10.0000	0.599760
$279$	5.00000	0.299342
$280$	1.00000	0.0597614
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	6.00000	0.357295
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−6.00000	−0.356034
$285$	−7.00000	−0.414644
$286$	−1.00000	−0.0591312
$287$	−6.00000	−0.354169
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	6.00000	0.352332
$291$	−1.00000	−0.0586210
$292$	−4.00000	−0.234082
$293$	0	0		−	1.00000i	$-0.5\pi$
$293$	0	0			1.00000i	$0.5\pi$
$294$	6.00000	0.349927
$295$	12.0000	0.698667
$296$	−11.0000	−0.639362
$297$	−1.00000	−0.0580259
$298$	15.0000	0.868927
$299$	−3.00000	−0.173494
$300$	1.00000	0.0577350
$301$	4.00000	0.230556
$302$	13.0000	0.748066
$303$	−18.0000	−1.03407
$304$	−7.00000	−0.401478
$305$	5.00000	0.286299
$306$	1.00000	0.0571662
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	1.00000	0.0569803
$309$	−13.0000	−0.739544
$310$	−5.00000	−0.283981
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	1.00000	0.0566139
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	16.0000	0.902932
$315$	−1.00000	−0.0563436
$316$	−10.0000	−0.562544
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	6.00000	0.336463
$319$	6.00000	0.335936
$320$	1.00000	0.0559017
$321$	18.0000	1.00466
$322$	3.00000	0.167183
$323$	7.00000	0.389490
$324$	1.00000	0.0555556
$325$	−1.00000	−0.0554700
$326$	4.00000	0.221540
$327$	−1.00000	−0.0553001
$328$	−6.00000	−0.331295
$329$	6.00000	0.330791
$330$	1.00000	0.0550482
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	−9.00000	−0.493939
$333$	11.0000	0.602796
$334$	−24.0000	−1.31322
$335$	−13.0000	−0.710266
$336$	−1.00000	−0.0545545
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	12.0000	0.652714
$339$	−6.00000	−0.325875
$340$	−1.00000	−0.0542326
$341$	−5.00000	−0.270765
$342$	7.00000	0.378517
$343$	13.0000	0.701934
$344$	4.00000	0.215666
$345$	3.00000	0.161515
$346$	15.0000	0.806405
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	−6.00000	−0.321634
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	1.00000	0.0534522
$351$	−1.00000	−0.0533761
$352$	1.00000	0.0533002
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	−12.0000	−0.637793
$355$	−6.00000	−0.318447
$356$	6.00000	0.317999
$357$	1.00000	0.0529256
$358$	−3.00000	−0.158555
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	−1.00000	−0.0527046
$361$	30.0000	1.57895
$362$	−8.00000	−0.420471
$363$	1.00000	0.0524864
$364$	1.00000	0.0524142
$365$	−4.00000	−0.209370
$366$	−5.00000	−0.261354
$367$	8.00000	0.417597	0.208798	−	0.977959i	$-0.433045\pi$
$367$	8.00000	0.417597	0.208798	+	0.977959i	$0.433045\pi$
$368$	3.00000	0.156386
$369$	6.00000	0.312348
$370$	−11.0000	−0.571863
$371$	6.00000	0.311504
$372$	5.00000	0.259238
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	−1.00000	−0.0517088
$375$	1.00000	0.0516398
$376$	6.00000	0.309426
$377$	6.00000	0.309016
$378$	1.00000	0.0514344
$379$	5.00000	0.256833	0.128416	−	0.991720i	$-0.459011\pi$
$379$	5.00000	0.256833	0.128416	+	0.991720i	$0.459011\pi$
$380$	−7.00000	−0.359092
$381$	8.00000	0.409852
$382$	24.0000	1.22795
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	−1.00000	−0.0510310
$385$	1.00000	0.0509647
$386$	4.00000	0.203595
$387$	−4.00000	−0.203331
$388$	−1.00000	−0.0507673
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	1.00000	0.0506370
$391$	−3.00000	−0.151717
$392$	6.00000	0.303046
$393$	−15.0000	−0.756650
$394$	−21.0000	−1.05796
$395$	−10.0000	−0.503155
$396$	−1.00000	−0.0502519
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−5.00000	−0.250627
$399$	7.00000	0.350438
$400$	1.00000	0.0500000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	13.0000	0.648381
$403$	−5.00000	−0.249068
$404$	−18.0000	−0.895533
$405$	1.00000	0.0496904
$406$	−6.00000	−0.297775
$407$	−11.0000	−0.545250
$408$	1.00000	0.0495074
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−6.00000	−0.296319
$411$	3.00000	0.147979
$412$	−13.0000	−0.640464
$413$	−12.0000	−0.590481
$414$	−3.00000	−0.147442
$415$	−9.00000	−0.441793
$416$	1.00000	0.0490290
$417$	−10.0000	−0.489702
$418$	−7.00000	−0.342381
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	−1.00000	−0.0487950
$421$	−13.0000	−0.633581	−0.316791	−	0.948495i	$-0.602605\pi$
$421$	−13.0000	−0.633581	−0.316791	+	0.948495i	$0.602605\pi$
$422$	22.0000	1.07094
$423$	−6.00000	−0.291730
$424$	6.00000	0.291386
$425$	−1.00000	−0.0485071
$426$	6.00000	0.290701
$427$	−5.00000	−0.241967
$428$	18.0000	0.870063
$429$	1.00000	0.0482805
$430$	4.00000	0.192897
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	1.00000	0.0481125
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	5.00000	0.240008
$435$	−6.00000	−0.287678
$436$	−1.00000	−0.0478913
$437$	−21.0000	−1.00457
$438$	4.00000	0.191127
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	1.00000	0.0476731
$441$	−6.00000	−0.285714
$442$	−1.00000	−0.0475651
$443$	6.00000	0.285069	0.142534	−	0.989790i	$-0.454475\pi$
$443$	6.00000	0.285069	0.142534	+	0.989790i	$0.454475\pi$
$444$	11.0000	0.522037
$445$	6.00000	0.284427
$446$	19.0000	0.899676
$447$	−15.0000	−0.709476
$448$	−1.00000	−0.0472456
$449$	−15.0000	−0.707894	−0.353947	−	0.935266i	$-0.615161\pi$
$449$	−15.0000	−0.707894	−0.353947	+	0.935266i	$0.615161\pi$
$450$	−1.00000	−0.0471405
$451$	−6.00000	−0.282529
$452$	−6.00000	−0.282216
$453$	−13.0000	−0.610793
$454$	−18.0000	−0.844782
$455$	1.00000	0.0468807
$456$	7.00000	0.327805
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	13.0000	0.607450
$459$	−1.00000	−0.0466760
$460$	3.00000	0.139876
$461$	−18.0000	−0.838344	−0.419172	−	0.907907i	$-0.637680\pi$
$461$	−18.0000	−0.838344	−0.419172	+	0.907907i	$0.637680\pi$
$462$	−1.00000	−0.0465242
$463$	−31.0000	−1.44069	−0.720346	−	0.693615i	$-0.756017\pi$
$463$	−31.0000	−1.44069	−0.720346	+	0.693615i	$0.756017\pi$
$464$	−6.00000	−0.278543
$465$	5.00000	0.231869
$466$	−18.0000	−0.833834
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	−1.00000	−0.0462250
$469$	13.0000	0.600284
$470$	6.00000	0.276759
$471$	−16.0000	−0.737241
$472$	−12.0000	−0.552345
$473$	4.00000	0.183920
$474$	10.0000	0.459315
$475$	−7.00000	−0.321182
$476$	1.00000	0.0458349
$477$	−6.00000	−0.274721
$478$	6.00000	0.274434
$479$	3.00000	0.137073	0.0685367	−	0.997649i	$-0.478167\pi$
$479$	3.00000	0.137073	0.0685367	+	0.997649i	$0.478167\pi$
$480$	−1.00000	−0.0456435
$481$	−11.0000	−0.501557
$482$	1.00000	0.0455488
$483$	−3.00000	−0.136505
$484$	1.00000	0.0454545
$485$	−1.00000	−0.0454077
$486$	−1.00000	−0.0453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−5.00000	−0.226339
$489$	−4.00000	−0.180886
$490$	6.00000	0.271052
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	6.00000	0.270501
$493$	6.00000	0.270226
$494$	−7.00000	−0.314945
$495$	−1.00000	−0.0449467
$496$	5.00000	0.224507
$497$	6.00000	0.269137
$498$	9.00000	0.403300
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	1.00000	0.0447214
$501$	24.0000	1.07224
$502$	−15.0000	−0.669483
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	1.00000	0.0445435
$505$	−18.0000	−0.800989
$506$	3.00000	0.133366
$507$	−12.0000	−0.532939
$508$	8.00000	0.354943
$509$	−42.0000	−1.86162	−0.930809	−	0.365507i	$-0.880896\pi$
$509$	−42.0000	−1.86162	−0.930809	+	0.365507i	$0.880896\pi$
$510$	1.00000	0.0442807
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	−7.00000	−0.309058
$514$	−6.00000	−0.264649
$515$	−13.0000	−0.572848
$516$	−4.00000	−0.176090
$517$	6.00000	0.263880
$518$	11.0000	0.483312
$519$	−15.0000	−0.658427
$520$	1.00000	0.0438529
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	6.00000	0.262613
$523$	32.0000	1.39926	0.699631	−	0.714504i	$-0.253348\pi$
$523$	32.0000	1.39926	0.699631	+	0.714504i	$0.253348\pi$
$524$	−15.0000	−0.655278
$525$	−1.00000	−0.0436436
$526$	21.0000	0.915644
$527$	−5.00000	−0.217803
$528$	−1.00000	−0.0435194
$529$	−14.0000	−0.608696
$530$	6.00000	0.260623
$531$	12.0000	0.520756
$532$	7.00000	0.303488
$533$	−6.00000	−0.259889
$534$	−6.00000	−0.259645
$535$	18.0000	0.778208
$536$	13.0000	0.561514
$537$	3.00000	0.129460
$538$	−9.00000	−0.388018
$539$	6.00000	0.258438
$540$	1.00000	0.0430331
$541$	38.0000	1.63375	0.816874	−	0.576816i	$-0.195705\pi$
$541$	38.0000	1.63375	0.816874	+	0.576816i	$0.195705\pi$
$542$	28.0000	1.20270
$543$	8.00000	0.343313
$544$	1.00000	0.0428746
$545$	−1.00000	−0.0428353
$546$	−1.00000	−0.0427960
$547$	29.0000	1.23995	0.619975	−	0.784621i	$-0.287143\pi$
$547$	29.0000	1.23995	0.619975	+	0.784621i	$0.287143\pi$
$548$	3.00000	0.128154
$549$	5.00000	0.213395
$550$	1.00000	0.0426401
$551$	42.0000	1.78926
$552$	−3.00000	−0.127688
$553$	10.0000	0.425243
$554$	−8.00000	−0.339887
$555$	11.0000	0.466924
$556$	−10.0000	−0.424094
$557$	−24.0000	−1.01691	−0.508456	−	0.861088i	$-0.669784\pi$
$557$	−24.0000	−1.01691	−0.508456	+	0.861088i	$0.669784\pi$
$558$	−5.00000	−0.211667
$559$	4.00000	0.169182
$560$	−1.00000	−0.0422577
$561$	1.00000	0.0422200
$562$	−6.00000	−0.253095
$563$	−27.0000	−1.13791	−0.568957	−	0.822367i	$-0.692653\pi$
$563$	−27.0000	−1.13791	−0.568957	+	0.822367i	$0.692653\pi$
$564$	−6.00000	−0.252646
$565$	−6.00000	−0.252422
$566$	4.00000	0.168133
$567$	−1.00000	−0.0419961
$568$	6.00000	0.251754
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	7.00000	0.293198
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	1.00000	0.0418121
$573$	−24.0000	−1.00261
$574$	6.00000	0.250435
$575$	3.00000	0.125109
$576$	1.00000	0.0416667
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	−1.00000	−0.0415945
$579$	−4.00000	−0.166234
$580$	−6.00000	−0.249136
$581$	9.00000	0.373383
$582$	1.00000	0.0414513
$583$	6.00000	0.248495
$584$	4.00000	0.165521
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	−6.00000	−0.247436
$589$	−35.0000	−1.44215
$590$	−12.0000	−0.494032
$591$	21.0000	0.863825
$592$	11.0000	0.452097
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	1.00000	0.0410305
$595$	1.00000	0.0409960
$596$	−15.0000	−0.614424
$597$	5.00000	0.204636
$598$	3.00000	0.122679
$599$	−15.0000	−0.612883	−0.306442	−	0.951889i	$-0.599138\pi$
$599$	−15.0000	−0.612883	−0.306442	+	0.951889i	$0.599138\pi$
$600$	−1.00000	−0.0408248
$601$	−1.00000	−0.0407909	−0.0203954	−	0.999792i	$-0.506493\pi$
$601$	−1.00000	−0.0407909	−0.0203954	+	0.999792i	$0.506493\pi$
$602$	−4.00000	−0.163028
$603$	−13.0000	−0.529401
$604$	−13.0000	−0.528962
$605$	1.00000	0.0406558
$606$	18.0000	0.731200
$607$	−1.00000	−0.0405887	−0.0202944	−	0.999794i	$-0.506460\pi$
$607$	−1.00000	−0.0405887	−0.0202944	+	0.999794i	$0.506460\pi$
$608$	7.00000	0.283887
$609$	6.00000	0.243132
$610$	−5.00000	−0.202444
$611$	6.00000	0.242734
$612$	−1.00000	−0.0404226
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	−8.00000	−0.322854
$615$	6.00000	0.241943
$616$	−1.00000	−0.0402911
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	13.0000	0.522937
$619$	−37.0000	−1.48716	−0.743578	−	0.668649i	$-0.766873\pi$
$619$	−37.0000	−1.48716	−0.743578	+	0.668649i	$0.766873\pi$
$620$	5.00000	0.200805
$621$	3.00000	0.120386
$622$	−18.0000	−0.721734
$623$	−6.00000	−0.240385
$624$	−1.00000	−0.0400320
$625$	1.00000	0.0400000
$626$	7.00000	0.279776
$627$	7.00000	0.279553
$628$	−16.0000	−0.638470
$629$	−11.0000	−0.438599
$630$	1.00000	0.0398410
$631$	26.0000	1.03504	0.517522	−	0.855670i	$-0.326855\pi$
$631$	26.0000	1.03504	0.517522	+	0.855670i	$0.326855\pi$
$632$	10.0000	0.397779
$633$	−22.0000	−0.874421
$634$	−6.00000	−0.238290
$635$	8.00000	0.317470
$636$	−6.00000	−0.237915
$637$	6.00000	0.237729
$638$	−6.00000	−0.237542
$639$	−6.00000	−0.237356
$640$	−1.00000	−0.0395285
$641$	−6.00000	−0.236986	−0.118493	−	0.992955i	$-0.537806\pi$
$641$	−6.00000	−0.236986	−0.118493	+	0.992955i	$0.537806\pi$
$642$	−18.0000	−0.710403
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	−3.00000	−0.118217
$645$	−4.00000	−0.157500
$646$	−7.00000	−0.275411
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	−1.00000	−0.0392837
$649$	−12.0000	−0.471041
$650$	1.00000	0.0392232
$651$	−5.00000	−0.195965
$652$	−4.00000	−0.156652
$653$	24.0000	0.939193	0.469596	−	0.882881i	$-0.344399\pi$
$653$	24.0000	0.939193	0.469596	+	0.882881i	$0.344399\pi$
$654$	1.00000	0.0391031
$655$	−15.0000	−0.586098
$656$	6.00000	0.234261
$657$	−4.00000	−0.156055
$658$	−6.00000	−0.233904
$659$	−30.0000	−1.16863	−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000	−1.16863	−0.584317	+	0.811525i	$0.698638\pi$
$660$	−1.00000	−0.0389249
$661$	−43.0000	−1.67251	−0.836253	−	0.548344i	$-0.815259\pi$
$661$	−43.0000	−1.67251	−0.836253	+	0.548344i	$0.815259\pi$
$662$	28.0000	1.08825
$663$	1.00000	0.0388368
$664$	9.00000	0.349268
$665$	7.00000	0.271448
$666$	−11.0000	−0.426241
$667$	−18.0000	−0.696963
$668$	24.0000	0.928588
$669$	−19.0000	−0.734582
$670$	13.0000	0.502234
$671$	−5.00000	−0.193023
$672$	1.00000	0.0385758
$673$	8.00000	0.308377	0.154189	−	0.988041i	$-0.450724\pi$
$673$	8.00000	0.308377	0.154189	+	0.988041i	$0.450724\pi$
$674$	−20.0000	−0.770371
$675$	1.00000	0.0384900
$676$	−12.0000	−0.461538
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	6.00000	0.230429
$679$	1.00000	0.0383765
$680$	1.00000	0.0383482
$681$	18.0000	0.689761
$682$	5.00000	0.191460
$683$	−9.00000	−0.344375	−0.172188	−	0.985064i	$-0.555084\pi$
$683$	−9.00000	−0.344375	−0.172188	+	0.985064i	$0.555084\pi$
$684$	−7.00000	−0.267652
$685$	3.00000	0.114624
$686$	−13.0000	−0.496342
$687$	−13.0000	−0.495981
$688$	−4.00000	−0.152499
$689$	6.00000	0.228582
$690$	−3.00000	−0.114208
$691$	−49.0000	−1.86405	−0.932024	−	0.362397i	$-0.881959\pi$
$691$	−49.0000	−1.86405	−0.932024	+	0.362397i	$0.881959\pi$
$692$	−15.0000	−0.570214
$693$	1.00000	0.0379869
$694$	18.0000	0.683271
$695$	−10.0000	−0.379322
$696$	6.00000	0.227429
$697$	−6.00000	−0.227266
$698$	−2.00000	−0.0757011
$699$	18.0000	0.680823
$700$	−1.00000	−0.0377964
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	1.00000	0.0377426
$703$	−77.0000	−2.90411
$704$	−1.00000	−0.0376889
$705$	−6.00000	−0.225973
$706$	−21.0000	−0.790345
$707$	18.0000	0.676960
$708$	12.0000	0.450988
$709$	−34.0000	−1.27690	−0.638448	−	0.769665i	$-0.720423\pi$
$709$	−34.0000	−1.27690	−0.638448	+	0.769665i	$0.720423\pi$
$710$	6.00000	0.225176
$711$	−10.0000	−0.375029
$712$	−6.00000	−0.224860
$713$	15.0000	0.561754
$714$	−1.00000	−0.0374241
$715$	1.00000	0.0373979
$716$	3.00000	0.112115
$717$	−6.00000	−0.224074
$718$	−30.0000	−1.11959
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	1.00000	0.0372678
$721$	13.0000	0.484145
$722$	−30.0000	−1.11648
$723$	−1.00000	−0.0371904
$724$	8.00000	0.297318
$725$	−6.00000	−0.222834
$726$	−1.00000	−0.0371135
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	−1.00000	−0.0370625
$729$	1.00000	0.0370370
$730$	4.00000	0.148047
$731$	4.00000	0.147945
$732$	5.00000	0.184805
$733$	29.0000	1.07114	0.535570	−	0.844491i	$-0.320097\pi$
$733$	29.0000	1.07114	0.535570	+	0.844491i	$0.320097\pi$
$734$	−8.00000	−0.295285
$735$	−6.00000	−0.221313
$736$	−3.00000	−0.110581
$737$	13.0000	0.478861
$738$	−6.00000	−0.220863
$739$	23.0000	0.846069	0.423034	−	0.906114i	$-0.360965\pi$
$739$	23.0000	0.846069	0.423034	+	0.906114i	$0.360965\pi$
$740$	11.0000	0.404368
$741$	7.00000	0.257151
$742$	−6.00000	−0.220267
$743$	36.0000	1.32071	0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000	1.32071	0.660356	+	0.750953i	$0.270405\pi$
$744$	−5.00000	−0.183309
$745$	−15.0000	−0.549557
$746$	−26.0000	−0.951928
$747$	−9.00000	−0.329293
$748$	1.00000	0.0365636
$749$	−18.0000	−0.657706
$750$	−1.00000	−0.0365148
$751$	8.00000	0.291924	0.145962	−	0.989290i	$-0.453372\pi$
$751$	8.00000	0.291924	0.145962	+	0.989290i	$0.453372\pi$
$752$	−6.00000	−0.218797
$753$	15.0000	0.546630
$754$	−6.00000	−0.218507
$755$	−13.0000	−0.473118
$756$	−1.00000	−0.0363696
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	−5.00000	−0.181608
$759$	−3.00000	−0.108893
$760$	7.00000	0.253917
$761$	−9.00000	−0.326250	−0.163125	−	0.986605i	$-0.552157\pi$
$761$	−9.00000	−0.326250	−0.163125	+	0.986605i	$0.552157\pi$
$762$	−8.00000	−0.289809
$763$	1.00000	0.0362024
$764$	−24.0000	−0.868290
$765$	−1.00000	−0.0361551
$766$	−18.0000	−0.650366
$767$	−12.0000	−0.433295
$768$	1.00000	0.0360844
$769$	−16.0000	−0.576975	−0.288487	−	0.957484i	$-0.593152\pi$
$769$	−16.0000	−0.576975	−0.288487	+	0.957484i	$0.593152\pi$
$770$	−1.00000	−0.0360375
$771$	6.00000	0.216085
$772$	−4.00000	−0.143963
$773$	33.0000	1.18693	0.593464	−	0.804861i	$-0.297760\pi$
$773$	33.0000	1.18693	0.593464	+	0.804861i	$0.297760\pi$
$774$	4.00000	0.143777
$775$	5.00000	0.179605
$776$	1.00000	0.0358979
$777$	−11.0000	−0.394623
$778$	−24.0000	−0.860442
$779$	−42.0000	−1.50481
$780$	−1.00000	−0.0358057
$781$	6.00000	0.214697
$782$	3.00000	0.107280
$783$	−6.00000	−0.214423
$784$	−6.00000	−0.214286
$785$	−16.0000	−0.571064
$786$	15.0000	0.535032
$787$	−25.0000	−0.891154	−0.445577	−	0.895244i	$-0.647001\pi$
$787$	−25.0000	−0.891154	−0.445577	+	0.895244i	$0.647001\pi$
$788$	21.0000	0.748094
$789$	−21.0000	−0.747620
$790$	10.0000	0.355784
$791$	6.00000	0.213335
$792$	1.00000	0.0355335
$793$	−5.00000	−0.177555
$794$	34.0000	1.20661
$795$	−6.00000	−0.212798
$796$	5.00000	0.177220
$797$	27.0000	0.956389	0.478195	−	0.878254i	$-0.341291\pi$
$797$	27.0000	0.956389	0.478195	+	0.878254i	$0.341291\pi$
$798$	−7.00000	−0.247797
$799$	6.00000	0.212265
$800$	−1.00000	−0.0353553
$801$	6.00000	0.212000
$802$	−3.00000	−0.105934
$803$	4.00000	0.141157
$804$	−13.0000	−0.458475
$805$	−3.00000	−0.105736
$806$	5.00000	0.176117
$807$	9.00000	0.316815
$808$	18.0000	0.633238
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−1.00000	−0.0351364
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	6.00000	0.210559
$813$	−28.0000	−0.982003
$814$	11.0000	0.385550
$815$	−4.00000	−0.140114
$816$	−1.00000	−0.0350070
$817$	28.0000	0.979596
$818$	22.0000	0.769212
$819$	1.00000	0.0349428
$820$	6.00000	0.209529
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	−3.00000	−0.104637
$823$	38.0000	1.32460	0.662298	−	0.749240i	$-0.269581\pi$
$823$	38.0000	1.32460	0.662298	+	0.749240i	$0.269581\pi$
$824$	13.0000	0.452876
$825$	−1.00000	−0.0348155
$826$	12.0000	0.417533
$827$	42.0000	1.46048	0.730242	−	0.683189i	$-0.239408\pi$
$827$	42.0000	1.46048	0.730242	+	0.683189i	$0.239408\pi$
$828$	3.00000	0.104257
$829$	11.0000	0.382046	0.191023	−	0.981586i	$-0.438820\pi$
$829$	11.0000	0.382046	0.191023	+	0.981586i	$0.438820\pi$
$830$	9.00000	0.312395
$831$	8.00000	0.277517
$832$	−1.00000	−0.0346688
$833$	6.00000	0.207888
$834$	10.0000	0.346272
$835$	24.0000	0.830554
$836$	7.00000	0.242100
$837$	5.00000	0.172825
$838$	24.0000	0.829066
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	1.00000	0.0345033
$841$	7.00000	0.241379
$842$	13.0000	0.448010
$843$	6.00000	0.206651
$844$	−22.0000	−0.757271
$845$	−12.0000	−0.412813
$846$	6.00000	0.206284
$847$	−1.00000	−0.0343604
$848$	−6.00000	−0.206041
$849$	−4.00000	−0.137280
$850$	1.00000	0.0342997
$851$	33.0000	1.13123
$852$	−6.00000	−0.205557
$853$	8.00000	0.273915	0.136957	−	0.990577i	$-0.456268\pi$
$853$	8.00000	0.273915	0.136957	+	0.990577i	$0.456268\pi$
$854$	5.00000	0.171096
$855$	−7.00000	−0.239395
$856$	−18.0000	−0.615227
$857$	33.0000	1.12726	0.563629	−	0.826028i	$-0.309405\pi$
$857$	33.0000	1.12726	0.563629	+	0.826028i	$0.309405\pi$
$858$	−1.00000	−0.0341394
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	−4.00000	−0.136399
$861$	−6.00000	−0.204479
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	−15.0000	−0.510015
$866$	10.0000	0.339814
$867$	1.00000	0.0339618
$868$	−5.00000	−0.169711
$869$	10.0000	0.339227
$870$	6.00000	0.203419
$871$	13.0000	0.440488
$872$	1.00000	0.0338643
$873$	−1.00000	−0.0338449
$874$	21.0000	0.710336
$875$	−1.00000	−0.0338062
$876$	−4.00000	−0.135147
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	−26.0000	−0.877457
$879$	0	0
$880$	−1.00000	−0.0337100
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	6.00000	0.202031
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	1.00000	0.0336336
$885$	12.0000	0.403376
$886$	−6.00000	−0.201574
$887$	36.0000	1.20876	0.604381	−	0.796696i	$-0.293421\pi$
$887$	36.0000	1.20876	0.604381	+	0.796696i	$0.293421\pi$
$888$	−11.0000	−0.369136
$889$	−8.00000	−0.268311
$890$	−6.00000	−0.201120
$891$	−1.00000	−0.0335013
$892$	−19.0000	−0.636167
$893$	42.0000	1.40548
$894$	15.0000	0.501675
$895$	3.00000	0.100279
$896$	1.00000	0.0334077
$897$	−3.00000	−0.100167
$898$	15.0000	0.500556
$899$	−30.0000	−1.00056
$900$	1.00000	0.0333333
$901$	6.00000	0.199889
$902$	6.00000	0.199778
$903$	4.00000	0.133112
$904$	6.00000	0.199557
$905$	8.00000	0.265929
$906$	13.0000	0.431896
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	18.0000	0.597351
$909$	−18.0000	−0.597022
$910$	−1.00000	−0.0331497
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	−7.00000	−0.231793
$913$	9.00000	0.297857
$914$	−17.0000	−0.562310
$915$	5.00000	0.165295
$916$	−13.0000	−0.429532
$917$	15.0000	0.495344
$918$	1.00000	0.0330049
$919$	−49.0000	−1.61636	−0.808180	−	0.588935i	$-0.799547\pi$
$919$	−49.0000	−1.61636	−0.808180	+	0.588935i	$0.799547\pi$
$920$	−3.00000	−0.0989071
$921$	8.00000	0.263609
$922$	18.0000	0.592798
$923$	6.00000	0.197492
$924$	1.00000	0.0328976
$925$	11.0000	0.361678
$926$	31.0000	1.01872
$927$	−13.0000	−0.426976
$928$	6.00000	0.196960
$929$	21.0000	0.688988	0.344494	−	0.938789i	$-0.388051\pi$
$929$	21.0000	0.688988	0.344494	+	0.938789i	$0.388051\pi$
$930$	−5.00000	−0.163956
$931$	42.0000	1.37649
$932$	18.0000	0.589610
$933$	18.0000	0.589294
$934$	12.0000	0.392652
$935$	1.00000	0.0327035
$936$	1.00000	0.0326860
$937$	−7.00000	−0.228680	−0.114340	−	0.993442i	$-0.536475\pi$
$937$	−7.00000	−0.228680	−0.114340	+	0.993442i	$0.536475\pi$
$938$	−13.0000	−0.424465
$939$	−7.00000	−0.228436
$940$	−6.00000	−0.195698
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	16.0000	0.521308
$943$	18.0000	0.586161
$944$	12.0000	0.390567
$945$	−1.00000	−0.0325300
$946$	−4.00000	−0.130051
$947$	−60.0000	−1.94974	−0.974869	−	0.222779i	$-0.928487\pi$
$947$	−60.0000	−1.94974	−0.974869	+	0.222779i	$0.928487\pi$
$948$	−10.0000	−0.324785
$949$	4.00000	0.129845
$950$	7.00000	0.227110
$951$	6.00000	0.194563
$952$	−1.00000	−0.0324102
$953$	−24.0000	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$953$	−24.0000	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$954$	6.00000	0.194257
$955$	−24.0000	−0.776622
$956$	−6.00000	−0.194054
$957$	6.00000	0.193952
$958$	−3.00000	−0.0969256
$959$	−3.00000	−0.0968751
$960$	1.00000	0.0322749
$961$	−6.00000	−0.193548
$962$	11.0000	0.354654
$963$	18.0000	0.580042
$964$	−1.00000	−0.0322078
$965$	−4.00000	−0.128765
$966$	3.00000	0.0965234
$967$	2.00000	0.0643157	0.0321578	−	0.999483i	$-0.489762\pi$
$967$	2.00000	0.0643157	0.0321578	+	0.999483i	$0.489762\pi$
$968$	−1.00000	−0.0321412
$969$	7.00000	0.224872
$970$	1.00000	0.0321081
$971$	57.0000	1.82922	0.914609	−	0.404341i	$-0.132499\pi$
$971$	57.0000	1.82922	0.914609	+	0.404341i	$0.132499\pi$
$972$	1.00000	0.0320750
$973$	10.0000	0.320585
$974$	4.00000	0.128168
$975$	−1.00000	−0.0320256
$976$	5.00000	0.160046
$977$	9.00000	0.287936	0.143968	−	0.989582i	$-0.454014\pi$
$977$	9.00000	0.287936	0.143968	+	0.989582i	$0.454014\pi$
$978$	4.00000	0.127906
$979$	−6.00000	−0.191761
$980$	−6.00000	−0.191663
$981$	−1.00000	−0.0319275
$982$	−30.0000	−0.957338
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	−6.00000	−0.191273
$985$	21.0000	0.669116
$986$	−6.00000	−0.191079
$987$	6.00000	0.190982
$988$	7.00000	0.222700
$989$	−12.0000	−0.381578
$990$	1.00000	0.0317821
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	−5.00000	−0.158750
$993$	−28.0000	−0.888553
$994$	−6.00000	−0.190308
$995$	5.00000	0.158511
$996$	−9.00000	−0.285176
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	4.00000	0.126618
$999$	11.0000	0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5610.2.a.r.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5610.2.a.r.1.1	✓	1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5610.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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