LMFDB - Embedded newform 2175.2.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2175,2,Mod(1,2175)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2175.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2175, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2175 = 3 \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2175.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,1,-2,7,0,-1,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$17.3674624396$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{21}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 5 $ x^2 - x - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.79129$ of defining polynomial
Character	$\chi$	$=$	2175.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} -2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +5.00000 q^{11} -5.79129 q^{12} -4.58258 q^{13} -2.79129 q^{14} +17.9564 q^{16} +3.00000 q^{17} +2.79129 q^{18} -5.58258 q^{19} +1.00000 q^{21} +13.9564 q^{22} +4.00000 q^{23} -10.5826 q^{24} -12.7913 q^{26} -1.00000 q^{27} -5.79129 q^{28} +1.00000 q^{29} +4.00000 q^{31} +28.9564 q^{32} -5.00000 q^{33} +8.37386 q^{34} +5.79129 q^{36} +4.00000 q^{37} -15.5826 q^{38} +4.58258 q^{39} +9.16515 q^{41} +2.79129 q^{42} +0.417424 q^{43} +28.9564 q^{44} +11.1652 q^{46} -1.41742 q^{47} -17.9564 q^{48} -6.00000 q^{49} -3.00000 q^{51} -26.5390 q^{52} -9.58258 q^{53} -2.79129 q^{54} -10.5826 q^{56} +5.58258 q^{57} +2.79129 q^{58} +1.58258 q^{59} -14.7477 q^{61} +11.1652 q^{62} -1.00000 q^{63} +44.9129 q^{64} -13.9564 q^{66} -14.1652 q^{67} +17.3739 q^{68} -4.00000 q^{69} -0.417424 q^{71} +10.5826 q^{72} -4.00000 q^{73} +11.1652 q^{74} -32.3303 q^{76} -5.00000 q^{77} +12.7913 q^{78} -1.58258 q^{79} +1.00000 q^{81} +25.5826 q^{82} +2.41742 q^{83} +5.79129 q^{84} +1.16515 q^{86} -1.00000 q^{87} +52.9129 q^{88} +10.5826 q^{89} +4.58258 q^{91} +23.1652 q^{92} -4.00000 q^{93} -3.95644 q^{94} -28.9564 q^{96} -2.41742 q^{97} -16.7477 q^{98} +5.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 10 q^{11} - 7 q^{12} - q^{14} + 13 q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 5 q^{22} + 8 q^{23} - 12 q^{24} - 21 q^{26}+ \cdots + 10 q^{99}+O(q^{100}) $ 2 * q + q^2 - 2 * q^3 + 7 * q^4 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 + 10 * q^11 - 7 * q^12 - q^14 + 13 * q^16 + 6 * q^17 + q^18 - 2 * q^19 + 2 * q^21 + 5 * q^22 + 8 * q^23 - 12 * q^24 - 21 * q^26 - 2 * q^27 - 7 * q^28 + 2 * q^29 + 8 * q^31 + 35 * q^32 - 10 * q^33 + 3 * q^34 + 7 * q^36 + 8 * q^37 - 22 * q^38 + q^42 + 10 * q^43 + 35 * q^44 + 4 * q^46 - 12 * q^47 - 13 * q^48 - 12 * q^49 - 6 * q^51 - 21 * q^52 - 10 * q^53 - q^54 - 12 * q^56 + 2 * q^57 + q^58 - 6 * q^59 - 2 * q^61 + 4 * q^62 - 2 * q^63 + 44 * q^64 - 5 * q^66 - 10 * q^67 + 21 * q^68 - 8 * q^69 - 10 * q^71 + 12 * q^72 - 8 * q^73 + 4 * q^74 - 28 * q^76 - 10 * q^77 + 21 * q^78 + 6 * q^79 + 2 * q^81 + 42 * q^82 + 14 * q^83 + 7 * q^84 - 16 * q^86 - 2 * q^87 + 60 * q^88 + 12 * q^89 + 28 * q^92 - 8 * q^93 + 15 * q^94 - 35 * q^96 - 14 * q^97 - 6 * q^98 + 10 * q^99

Coefficient data

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

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

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

$2$	2.79129	1.97374	0.986869	−	0.161521i	$-0.0516399\pi$
$2$	2.79129	1.97374	0.986869	+	0.161521i	$0.0516399\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	5.79129	2.89564
$5$	0	0
$5$	0	0
$6$	−2.79129	−1.13954
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	10.5826	3.74151
$9$	1.00000	0.333333
$10$	0	0
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	−5.79129	−1.67180
$13$	−4.58258	−1.27098	−0.635489	−	0.772110i	$-0.719201\pi$
$13$	−4.58258	−1.27098	−0.635489	+	0.772110i	$0.719201\pi$
$14$	−2.79129	−0.746003
$15$	0	0
$16$	17.9564	4.48911
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	2.79129	0.657913
$19$	−5.58258	−1.28073	−0.640365	−	0.768070i	$-0.721217\pi$
$19$	−5.58258	−1.28073	−0.640365	+	0.768070i	$0.721217\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	13.9564	2.97552
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	−10.5826	−2.16016
$25$	0	0
$26$	−12.7913	−2.50858
$27$	−1.00000	−0.192450
$28$	−5.79129	−1.09445
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	28.9564	5.11882
$33$	−5.00000	−0.870388
$34$	8.37386	1.43611
$35$	0	0
$36$	5.79129	0.965215
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	−15.5826	−2.52783
$39$	4.58258	0.733799
$40$	0	0
$41$	9.16515	1.43136	0.715678	−	0.698430i	$-0.246118\pi$
$41$	9.16515	1.43136	0.715678	+	0.698430i	$0.246118\pi$
$42$	2.79129	0.430705
$43$	0.417424	0.0636566	0.0318283	−	0.999493i	$-0.489867\pi$
$43$	0.417424	0.0636566	0.0318283	+	0.999493i	$0.489867\pi$
$44$	28.9564	4.36535
$45$	0	0
$46$	11.1652	1.64621
$47$	−1.41742	−0.206753	−0.103376	−	0.994642i	$-0.532965\pi$
$47$	−1.41742	−0.206753	−0.103376	+	0.994642i	$0.532965\pi$
$48$	−17.9564	−2.59179
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−3.00000	−0.420084
$52$	−26.5390	−3.68030
$53$	−9.58258	−1.31627	−0.658134	−	0.752901i	$-0.728654\pi$
$53$	−9.58258	−1.31627	−0.658134	+	0.752901i	$0.728654\pi$
$54$	−2.79129	−0.379846
$55$	0	0
$56$	−10.5826	−1.41416
$57$	5.58258	0.739430
$58$	2.79129	0.366514
$59$	1.58258	0.206034	0.103017	−	0.994680i	$-0.467150\pi$
$59$	1.58258	0.206034	0.103017	+	0.994680i	$0.467150\pi$
$60$	0	0
$61$	−14.7477	−1.88825	−0.944126	−	0.329583i	$-0.893092\pi$
$61$	−14.7477	−1.88825	−0.944126	+	0.329583i	$0.893092\pi$
$62$	11.1652	1.41798
$63$	−1.00000	−0.125988
$64$	44.9129	5.61411
$65$	0	0
$66$	−13.9564	−1.71792
$67$	−14.1652	−1.73055	−0.865274	−	0.501299i	$-0.832856\pi$
$67$	−14.1652	−1.73055	−0.865274	+	0.501299i	$0.832856\pi$
$68$	17.3739	2.10689
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−0.417424	−0.0495392	−0.0247696	−	0.999693i	$-0.507885\pi$
$71$	−0.417424	−0.0495392	−0.0247696	+	0.999693i	$0.507885\pi$
$72$	10.5826	1.24717
$73$	−4.00000	−0.468165	−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000	−0.468165	−0.234082	+	0.972217i	$0.575209\pi$
$74$	11.1652	1.29792
$75$	0	0
$76$	−32.3303	−3.70854
$77$	−5.00000	−0.569803
$78$	12.7913	1.44833
$79$	−1.58258	−0.178054	−0.0890268	−	0.996029i	$-0.528376\pi$
$79$	−1.58258	−0.178054	−0.0890268	+	0.996029i	$0.528376\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	25.5826	2.82512
$83$	2.41742	0.265347	0.132673	−	0.991160i	$-0.457644\pi$
$83$	2.41742	0.265347	0.132673	+	0.991160i	$0.457644\pi$
$84$	5.79129	0.631881
$85$	0	0
$86$	1.16515	0.125642
$87$	−1.00000	−0.107211
$88$	52.9129	5.64053
$89$	10.5826	1.12175	0.560875	−	0.827900i	$-0.310465\pi$
$89$	10.5826	1.12175	0.560875	+	0.827900i	$0.310465\pi$
$90$	0	0
$91$	4.58258	0.480384
$92$	23.1652	2.41513
$93$	−4.00000	−0.414781
$94$	−3.95644	−0.408076
$95$	0	0
$96$	−28.9564	−2.95535
$97$	−2.41742	−0.245452	−0.122726	−	0.992441i	$-0.539164\pi$
$97$	−2.41742	−0.245452	−0.122726	+	0.992441i	$0.539164\pi$
$98$	−16.7477	−1.69178
$99$	5.00000	0.502519
$100$	0	0
$101$	8.58258	0.853998	0.426999	−	0.904252i	$-0.359571\pi$
$101$	8.58258	0.853998	0.426999	+	0.904252i	$0.359571\pi$
$102$	−8.37386	−0.829136
$103$	3.16515	0.311872	0.155936	−	0.987767i	$-0.450161\pi$
$103$	3.16515	0.311872	0.155936	+	0.987767i	$0.450161\pi$
$104$	−48.4955	−4.75537
$105$	0	0
$106$	−26.7477	−2.59797
$107$	13.1652	1.27272	0.636362	−	0.771391i	$-0.280439\pi$
$107$	13.1652	1.27272	0.636362	+	0.771391i	$0.280439\pi$
$108$	−5.79129	−0.557267
$109$	−4.16515	−0.398949	−0.199475	−	0.979903i	$-0.563924\pi$
$109$	−4.16515	−0.398949	−0.199475	+	0.979903i	$0.563924\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	−17.9564	−1.69672
$113$	−4.16515	−0.391824	−0.195912	−	0.980621i	$-0.562767\pi$
$113$	−4.16515	−0.391824	−0.195912	+	0.980621i	$0.562767\pi$
$114$	15.5826	1.45944
$115$	0	0
$116$	5.79129	0.537708
$117$	−4.58258	−0.423659
$118$	4.41742	0.406657
$119$	−3.00000	−0.275010
$120$	0	0
$121$	14.0000	1.27273
$122$	−41.1652	−3.72692
$123$	−9.16515	−0.826394
$124$	23.1652	2.08029
$125$	0	0
$126$	−2.79129	−0.248668
$127$	−2.00000	−0.177471	−0.0887357	−	0.996055i	$-0.528283\pi$
$127$	−2.00000	−0.177471	−0.0887357	+	0.996055i	$0.528283\pi$
$128$	67.4519	5.96196
$129$	−0.417424	−0.0367522
$130$	0	0
$131$	−15.0000	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$131$	−15.0000	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$132$	−28.9564	−2.52033
$133$	5.58258	0.484071
$134$	−39.5390	−3.41565
$135$	0	0
$136$	31.7477	2.72235
$137$	20.3303	1.73693	0.868467	−	0.495746i	$-0.165105\pi$
$137$	20.3303	1.73693	0.868467	+	0.495746i	$0.165105\pi$
$138$	−11.1652	−0.950441
$139$	−18.5826	−1.57615	−0.788077	−	0.615577i	$-0.788923\pi$
$139$	−18.5826	−1.57615	−0.788077	+	0.615577i	$0.788923\pi$
$140$	0	0
$141$	1.41742	0.119369
$142$	−1.16515	−0.0977773
$143$	−22.9129	−1.91607
$144$	17.9564	1.49637
$145$	0	0
$146$	−11.1652	−0.924035
$147$	6.00000	0.494872
$148$	23.1652	1.90416
$149$	−10.7477	−0.880488	−0.440244	−	0.897878i	$-0.645108\pi$
$149$	−10.7477	−0.880488	−0.440244	+	0.897878i	$0.645108\pi$
$150$	0	0
$151$	−11.1652	−0.908607	−0.454304	−	0.890847i	$-0.650112\pi$
$151$	−11.1652	−0.908607	−0.454304	+	0.890847i	$0.650112\pi$
$152$	−59.0780	−4.79186
$153$	3.00000	0.242536
$154$	−13.9564	−1.12464
$155$	0	0
$156$	26.5390	2.12482
$157$	16.7477	1.33661	0.668307	−	0.743886i	$-0.267019\pi$
$157$	16.7477	1.33661	0.668307	+	0.743886i	$0.267019\pi$
$158$	−4.41742	−0.351431
$159$	9.58258	0.759948
$160$	0	0
$161$	−4.00000	−0.315244
$162$	2.79129	0.219304
$163$	1.58258	0.123957	0.0619784	−	0.998077i	$-0.480259\pi$
$163$	1.58258	0.123957	0.0619784	+	0.998077i	$0.480259\pi$
$164$	53.0780	4.14470
$165$	0	0
$166$	6.74773	0.523725
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	10.5826	0.816463
$169$	8.00000	0.615385
$170$	0	0
$171$	−5.58258	−0.426910
$172$	2.41742	0.184327
$173$	−15.1652	−1.15299	−0.576493	−	0.817102i	$-0.695579\pi$
$173$	−15.1652	−1.15299	−0.576493	+	0.817102i	$0.695579\pi$
$174$	−2.79129	−0.211607
$175$	0	0
$176$	89.7822	6.76759
$177$	−1.58258	−0.118954
$178$	29.5390	2.21404
$179$	−22.7477	−1.70024	−0.850122	−	0.526585i	$-0.823472\pi$
$179$	−22.7477	−1.70024	−0.850122	+	0.526585i	$0.823472\pi$
$180$	0	0
$181$	−2.16515	−0.160934	−0.0804672	−	0.996757i	$-0.525641\pi$
$181$	−2.16515	−0.160934	−0.0804672	+	0.996757i	$0.525641\pi$
$182$	12.7913	0.948153
$183$	14.7477	1.09018
$184$	42.3303	3.12063
$185$	0	0
$186$	−11.1652	−0.818669
$187$	15.0000	1.09691
$188$	−8.20871	−0.598682
$189$	1.00000	0.0727393
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	−44.9129	−3.24131
$193$	−16.3303	−1.17548	−0.587740	−	0.809050i	$-0.699982\pi$
$193$	−16.3303	−1.17548	−0.587740	+	0.809050i	$0.699982\pi$
$194$	−6.74773	−0.484459
$195$	0	0
$196$	−34.7477	−2.48198
$197$	−20.3303	−1.44847	−0.724237	−	0.689551i	$-0.757808\pi$
$197$	−20.3303	−1.44847	−0.724237	+	0.689551i	$0.757808\pi$
$198$	13.9564	0.991841
$199$	22.5826	1.60084	0.800418	−	0.599442i	$-0.204611\pi$
$199$	22.5826	1.60084	0.800418	+	0.599442i	$0.204611\pi$
$200$	0	0
$201$	14.1652	0.999133
$202$	23.9564	1.68557
$203$	−1.00000	−0.0701862
$204$	−17.3739	−1.21641
$205$	0	0
$206$	8.83485	0.615553
$207$	4.00000	0.278019
$208$	−82.2867	−5.70556
$209$	−27.9129	−1.93077
$210$	0	0
$211$	−16.3303	−1.12422	−0.562112	−	0.827061i	$-0.690011\pi$
$211$	−16.3303	−1.12422	−0.562112	+	0.827061i	$0.690011\pi$
$212$	−55.4955	−3.81144
$213$	0.417424	0.0286014
$214$	36.7477	2.51202
$215$	0	0
$216$	−10.5826	−0.720053
$217$	−4.00000	−0.271538
$218$	−11.6261	−0.787421
$219$	4.00000	0.270295
$220$	0	0
$221$	−13.7477	−0.924772
$222$	−11.1652	−0.749356
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	−28.9564	−1.93473
$225$	0	0
$226$	−11.6261	−0.773359
$227$	−1.58258	−0.105039	−0.0525196	−	0.998620i	$-0.516725\pi$
$227$	−1.58258	−0.105039	−0.0525196	+	0.998620i	$0.516725\pi$
$228$	32.3303	2.14113
$229$	1.16515	0.0769954	0.0384977	−	0.999259i	$-0.487743\pi$
$229$	1.16515	0.0769954	0.0384977	+	0.999259i	$0.487743\pi$
$230$	0	0
$231$	5.00000	0.328976
$232$	10.5826	0.694780
$233$	−5.16515	−0.338380	−0.169190	−	0.985583i	$-0.554115\pi$
$233$	−5.16515	−0.338380	−0.169190	+	0.985583i	$0.554115\pi$
$234$	−12.7913	−0.836193
$235$	0	0
$236$	9.16515	0.596601
$237$	1.58258	0.102799
$238$	−8.37386	−0.542797
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	0	0
$241$	−7.33030	−0.472186	−0.236093	−	0.971730i	$-0.575867\pi$
$241$	−7.33030	−0.472186	−0.236093	+	0.971730i	$0.575867\pi$
$242$	39.0780	2.51203
$243$	−1.00000	−0.0641500
$244$	−85.4083	−5.46771
$245$	0	0
$246$	−25.5826	−1.63109
$247$	25.5826	1.62778
$248$	42.3303	2.68798
$249$	−2.41742	−0.153198
$250$	0	0
$251$	−2.16515	−0.136663	−0.0683316	−	0.997663i	$-0.521768\pi$
$251$	−2.16515	−0.136663	−0.0683316	+	0.997663i	$0.521768\pi$
$252$	−5.79129	−0.364817
$253$	20.0000	1.25739
$254$	−5.58258	−0.350282
$255$	0	0
$256$	98.4519	6.15324
$257$	14.7477	0.919938	0.459969	−	0.887935i	$-0.347861\pi$
$257$	14.7477	0.919938	0.459969	+	0.887935i	$0.347861\pi$
$258$	−1.16515	−0.0725392
$259$	−4.00000	−0.248548
$260$	0	0
$261$	1.00000	0.0618984
$262$	−41.8693	−2.58670
$263$	6.33030	0.390343	0.195172	−	0.980769i	$-0.437474\pi$
$263$	6.33030	0.390343	0.195172	+	0.980769i	$0.437474\pi$
$264$	−52.9129	−3.25656
$265$	0	0
$266$	15.5826	0.955429
$267$	−10.5826	−0.647643
$268$	−82.0345	−5.01105
$269$	13.4174	0.818075	0.409037	−	0.912518i	$-0.365865\pi$
$269$	13.4174	0.818075	0.409037	+	0.912518i	$0.365865\pi$
$270$	0	0
$271$	−17.1652	−1.04271	−0.521354	−	0.853340i	$-0.674573\pi$
$271$	−17.1652	−1.04271	−0.521354	+	0.853340i	$0.674573\pi$
$272$	53.8693	3.26631
$273$	−4.58258	−0.277350
$274$	56.7477	3.42826
$275$	0	0
$276$	−23.1652	−1.39438
$277$	−20.9129	−1.25653	−0.628267	−	0.777998i	$-0.716235\pi$
$277$	−20.9129	−1.25653	−0.628267	+	0.777998i	$0.716235\pi$
$278$	−51.8693	−3.11091
$279$	4.00000	0.239474
$280$	0	0
$281$	−9.58258	−0.571649	−0.285824	−	0.958282i	$-0.592267\pi$
$281$	−9.58258	−0.571649	−0.285824	+	0.958282i	$0.592267\pi$
$282$	3.95644	0.235603
$283$	19.1652	1.13925	0.569625	−	0.821905i	$-0.307088\pi$
$283$	19.1652	1.13925	0.569625	+	0.821905i	$0.307088\pi$
$284$	−2.41742	−0.143448
$285$	0	0
$286$	−63.9564	−3.78182
$287$	−9.16515	−0.541002
$288$	28.9564	1.70627
$289$	−8.00000	−0.470588
$290$	0	0
$291$	2.41742	0.141712
$292$	−23.1652	−1.35564
$293$	−11.8348	−0.691399	−0.345700	−	0.938345i	$-0.612358\pi$
$293$	−11.8348	−0.691399	−0.345700	+	0.938345i	$0.612358\pi$
$294$	16.7477	0.976747
$295$	0	0
$296$	42.3303	2.46040
$297$	−5.00000	−0.290129
$298$	−30.0000	−1.73785
$299$	−18.3303	−1.06007
$300$	0	0
$301$	−0.417424	−0.0240599
$302$	−31.1652	−1.79335
$303$	−8.58258	−0.493056
$304$	−100.243	−5.74934
$305$	0	0
$306$	8.37386	0.478702
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	−28.9564	−1.64995
$309$	−3.16515	−0.180059
$310$	0	0
$311$	−3.00000	−0.170114	−0.0850572	−	0.996376i	$-0.527107\pi$
$311$	−3.00000	−0.170114	−0.0850572	+	0.996376i	$0.527107\pi$
$312$	48.4955	2.74551
$313$	−1.41742	−0.0801176	−0.0400588	−	0.999197i	$-0.512755\pi$
$313$	−1.41742	−0.0801176	−0.0400588	+	0.999197i	$0.512755\pi$
$314$	46.7477	2.63813
$315$	0	0
$316$	−9.16515	−0.515580
$317$	25.0000	1.40414	0.702070	−	0.712108i	$-0.252259\pi$
$317$	25.0000	1.40414	0.702070	+	0.712108i	$0.252259\pi$
$318$	26.7477	1.49994
$319$	5.00000	0.279946
$320$	0	0
$321$	−13.1652	−0.734807
$322$	−11.1652	−0.622210
$323$	−16.7477	−0.931868
$324$	5.79129	0.321738
$325$	0	0
$326$	4.41742	0.244659
$327$	4.16515	0.230333
$328$	96.9909	5.35543
$329$	1.41742	0.0781451
$330$	0	0
$331$	28.3303	1.55717	0.778587	−	0.627537i	$-0.215937\pi$
$331$	28.3303	1.55717	0.778587	+	0.627537i	$0.215937\pi$
$332$	14.0000	0.768350
$333$	4.00000	0.219199
$334$	0	0
$335$	0	0
$336$	17.9564	0.979604
$337$	2.83485	0.154424	0.0772120	−	0.997015i	$-0.475398\pi$
$337$	2.83485	0.154424	0.0772120	+	0.997015i	$0.475398\pi$
$338$	22.3303	1.21461
$339$	4.16515	0.226220
$340$	0	0
$341$	20.0000	1.08306
$342$	−15.5826	−0.842609
$343$	13.0000	0.701934
$344$	4.41742	0.238172
$345$	0	0
$346$	−42.3303	−2.27569
$347$	11.1652	0.599377	0.299688	−	0.954037i	$-0.403117\pi$
$347$	11.1652	0.599377	0.299688	+	0.954037i	$0.403117\pi$
$348$	−5.79129	−0.310446
$349$	32.3303	1.73060	0.865301	−	0.501253i	$-0.167127\pi$
$349$	32.3303	1.73060	0.865301	+	0.501253i	$0.167127\pi$
$350$	0	0
$351$	4.58258	0.244600
$352$	144.782	7.71692
$353$	33.1652	1.76520	0.882601	−	0.470122i	$-0.155790\pi$
$353$	33.1652	1.76520	0.882601	+	0.470122i	$0.155790\pi$
$354$	−4.41742	−0.234783
$355$	0	0
$356$	61.2867	3.24819
$357$	3.00000	0.158777
$358$	−63.4955	−3.35584
$359$	−8.83485	−0.466285	−0.233143	−	0.972443i	$-0.574901\pi$
$359$	−8.83485	−0.466285	−0.233143	+	0.972443i	$0.574901\pi$
$360$	0	0
$361$	12.1652	0.640271
$362$	−6.04356	−0.317643
$363$	−14.0000	−0.734809
$364$	26.5390	1.39102
$365$	0	0
$366$	41.1652	2.15174
$367$	−17.4955	−0.913255	−0.456628	−	0.889658i	$-0.650943\pi$
$367$	−17.4955	−0.913255	−0.456628	+	0.889658i	$0.650943\pi$
$368$	71.8258	3.74418
$369$	9.16515	0.477119
$370$	0	0
$371$	9.58258	0.497503
$372$	−23.1652	−1.20106
$373$	8.33030	0.431327	0.215663	−	0.976468i	$-0.430809\pi$
$373$	8.33030	0.431327	0.215663	+	0.976468i	$0.430809\pi$
$374$	41.8693	2.16501
$375$	0	0
$376$	−15.0000	−0.773566
$377$	−4.58258	−0.236015
$378$	2.79129	0.143568
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	2.00000	0.102463
$382$	−11.1652	−0.571259
$383$	5.58258	0.285256	0.142628	−	0.989776i	$-0.454445\pi$
$383$	5.58258	0.285256	0.142628	+	0.989776i	$0.454445\pi$
$384$	−67.4519	−3.44214
$385$	0	0
$386$	−45.5826	−2.32009
$387$	0.417424	0.0212189
$388$	−14.0000	−0.710742
$389$	2.58258	0.130942	0.0654709	−	0.997854i	$-0.479145\pi$
$389$	2.58258	0.130942	0.0654709	+	0.997854i	$0.479145\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	−63.4955	−3.20700
$393$	15.0000	0.756650
$394$	−56.7477	−2.85891
$395$	0	0
$396$	28.9564	1.45512
$397$	−29.1652	−1.46376	−0.731878	−	0.681435i	$-0.761356\pi$
$397$	−29.1652	−1.46376	−0.731878	+	0.681435i	$0.761356\pi$
$398$	63.0345	3.15963
$399$	−5.58258	−0.279478
$400$	0	0
$401$	21.5826	1.07778	0.538891	−	0.842375i	$-0.318843\pi$
$401$	21.5826	1.07778	0.538891	+	0.842375i	$0.318843\pi$
$402$	39.5390	1.97203
$403$	−18.3303	−0.913097
$404$	49.7042	2.47287
$405$	0	0
$406$	−2.79129	−0.138529
$407$	20.0000	0.991363
$408$	−31.7477	−1.57175
$409$	24.7477	1.22370	0.611848	−	0.790975i	$-0.290426\pi$
$409$	24.7477	1.22370	0.611848	+	0.790975i	$0.290426\pi$
$410$	0	0
$411$	−20.3303	−1.00282
$412$	18.3303	0.903069
$413$	−1.58258	−0.0778735
$414$	11.1652	0.548737
$415$	0	0
$416$	−132.695	−6.50591
$417$	18.5826	0.909993
$418$	−77.9129	−3.81084
$419$	−18.8348	−0.920143	−0.460071	−	0.887882i	$-0.652176\pi$
$419$	−18.8348	−0.920143	−0.460071	+	0.887882i	$0.652176\pi$
$420$	0	0
$421$	13.5826	0.661974	0.330987	−	0.943635i	$-0.392618\pi$
$421$	13.5826	0.661974	0.330987	+	0.943635i	$0.392618\pi$
$422$	−45.5826	−2.21893
$423$	−1.41742	−0.0689175
$424$	−101.408	−4.92482
$425$	0	0
$426$	1.16515	0.0564518
$427$	14.7477	0.713693
$428$	76.2432	3.68535
$429$	22.9129	1.10624
$430$	0	0
$431$	−20.8348	−1.00358	−0.501790	−	0.864990i	$-0.667325\pi$
$431$	−20.8348	−1.00358	−0.501790	+	0.864990i	$0.667325\pi$
$432$	−17.9564	−0.863930
$433$	39.0780	1.87797	0.938985	−	0.343958i	$-0.111768\pi$
$433$	39.0780	1.87797	0.938985	+	0.343958i	$0.111768\pi$
$434$	−11.1652	−0.535944
$435$	0	0
$436$	−24.1216	−1.15521
$437$	−22.3303	−1.06820
$438$	11.1652	0.533492
$439$	28.9129	1.37994	0.689968	−	0.723840i	$-0.257624\pi$
$439$	28.9129	1.37994	0.689968	+	0.723840i	$0.257624\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−38.3739	−1.82526
$443$	8.58258	0.407770	0.203885	−	0.978995i	$-0.434643\pi$
$443$	8.58258	0.407770	0.203885	+	0.978995i	$0.434643\pi$
$444$	−23.1652	−1.09937
$445$	0	0
$446$	−19.5390	−0.925199
$447$	10.7477	0.508350
$448$	−44.9129	−2.12193
$449$	34.0780	1.60824	0.804121	−	0.594466i	$-0.202636\pi$
$449$	34.0780	1.60824	0.804121	+	0.594466i	$0.202636\pi$
$450$	0	0
$451$	45.8258	2.15785
$452$	−24.1216	−1.13458
$453$	11.1652	0.524585
$454$	−4.41742	−0.207320
$455$	0	0
$456$	59.0780	2.76658
$457$	−23.7477	−1.11087	−0.555436	−	0.831559i	$-0.687449\pi$
$457$	−23.7477	−1.11087	−0.555436	+	0.831559i	$0.687449\pi$
$458$	3.25227	0.151969
$459$	−3.00000	−0.140028
$460$	0	0
$461$	9.16515	0.426864	0.213432	−	0.976958i	$-0.431536\pi$
$461$	9.16515	0.426864	0.213432	+	0.976958i	$0.431536\pi$
$462$	13.9564	0.649312
$463$	−18.1652	−0.844206	−0.422103	−	0.906548i	$-0.638708\pi$
$463$	−18.1652	−0.844206	−0.422103	+	0.906548i	$0.638708\pi$
$464$	17.9564	0.833607
$465$	0	0
$466$	−14.4174	−0.667874
$467$	8.00000	0.370196	0.185098	−	0.982720i	$-0.440740\pi$
$467$	8.00000	0.370196	0.185098	+	0.982720i	$0.440740\pi$
$468$	−26.5390	−1.22677
$469$	14.1652	0.654086
$470$	0	0
$471$	−16.7477	−0.771695
$472$	16.7477	0.770877
$473$	2.08712	0.0959659
$474$	4.41742	0.202899
$475$	0	0
$476$	−17.3739	−0.796330
$477$	−9.58258	−0.438756
$478$	−72.5735	−3.31943
$479$	−0.834849	−0.0381452	−0.0190726	−	0.999818i	$-0.506071\pi$
$479$	−0.834849	−0.0381452	−0.0190726	+	0.999818i	$0.506071\pi$
$480$	0	0
$481$	−18.3303	−0.835790
$482$	−20.4610	−0.931972
$483$	4.00000	0.182006
$484$	81.0780	3.68536
$485$	0	0
$486$	−2.79129	−0.126615
$487$	34.3303	1.55565	0.777827	−	0.628478i	$-0.216322\pi$
$487$	34.3303	1.55565	0.777827	+	0.628478i	$0.216322\pi$
$488$	−156.069	−7.06491
$489$	−1.58258	−0.0715665
$490$	0	0
$491$	16.0000	0.722070	0.361035	−	0.932552i	$-0.382424\pi$
$491$	16.0000	0.722070	0.361035	+	0.932552i	$0.382424\pi$
$492$	−53.0780	−2.39294
$493$	3.00000	0.135113
$494$	71.4083	3.21281
$495$	0	0
$496$	71.8258	3.22507
$497$	0.417424	0.0187240
$498$	−6.74773	−0.302373
$499$	−22.5826	−1.01093	−0.505467	−	0.862846i	$-0.668680\pi$
$499$	−22.5826	−1.01093	−0.505467	+	0.862846i	$0.668680\pi$
$500$	0	0
$501$	0	0
$502$	−6.04356	−0.269737
$503$	−22.9129	−1.02163	−0.510817	−	0.859689i	$-0.670657\pi$
$503$	−22.9129	−1.02163	−0.510817	+	0.859689i	$0.670657\pi$
$504$	−10.5826	−0.471385
$505$	0	0
$506$	55.8258	2.48176
$507$	−8.00000	−0.355292
$508$	−11.5826	−0.513894
$509$	−0.747727	−0.0331424	−0.0165712	−	0.999863i	$-0.505275\pi$
$509$	−0.747727	−0.0331424	−0.0165712	+	0.999863i	$0.505275\pi$
$510$	0	0
$511$	4.00000	0.176950
$512$	139.904	6.18293
$513$	5.58258	0.246477
$514$	41.1652	1.81572
$515$	0	0
$516$	−2.41742	−0.106421
$517$	−7.08712	−0.311691
$518$	−11.1652	−0.490569
$519$	15.1652	0.665676
$520$	0	0
$521$	−21.0780	−0.923445	−0.461723	−	0.887024i	$-0.652768\pi$
$521$	−21.0780	−0.923445	−0.461723	+	0.887024i	$0.652768\pi$
$522$	2.79129	0.122171
$523$	−3.33030	−0.145624	−0.0728120	−	0.997346i	$-0.523197\pi$
$523$	−3.33030	−0.145624	−0.0728120	+	0.997346i	$0.523197\pi$
$524$	−86.8693	−3.79490
$525$	0	0
$526$	17.6697	0.770435
$527$	12.0000	0.522728
$528$	−89.7822	−3.90727
$529$	−7.00000	−0.304348
$530$	0	0
$531$	1.58258	0.0686779
$532$	32.3303	1.40170
$533$	−42.0000	−1.81922
$534$	−29.5390	−1.27828
$535$	0	0
$536$	−149.904	−6.47486
$537$	22.7477	0.981637
$538$	37.4519	1.61467
$539$	−30.0000	−1.29219
$540$	0	0
$541$	23.4955	1.01015	0.505074	−	0.863076i	$-0.331465\pi$
$541$	23.4955	1.01015	0.505074	+	0.863076i	$0.331465\pi$
$542$	−47.9129	−2.05803
$543$	2.16515	0.0929155
$544$	86.8693	3.72449
$545$	0	0
$546$	−12.7913	−0.547417
$547$	14.1652	0.605658	0.302829	−	0.953045i	$-0.402069\pi$
$547$	14.1652	0.605658	0.302829	+	0.953045i	$0.402069\pi$
$548$	117.739	5.02955
$549$	−14.7477	−0.629418
$550$	0	0
$551$	−5.58258	−0.237826
$552$	−42.3303	−1.80170
$553$	1.58258	0.0672980
$554$	−58.3739	−2.48007
$555$	0	0
$556$	−107.617	−4.56398
$557$	−4.74773	−0.201168	−0.100584	−	0.994929i	$-0.532071\pi$
$557$	−4.74773	−0.201168	−0.100584	+	0.994929i	$0.532071\pi$
$558$	11.1652	0.472659
$559$	−1.91288	−0.0809061
$560$	0	0
$561$	−15.0000	−0.633300
$562$	−26.7477	−1.12828
$563$	5.41742	0.228317	0.114159	−	0.993463i	$-0.463583\pi$
$563$	5.41742	0.228317	0.114159	+	0.993463i	$0.463583\pi$
$564$	8.20871	0.345649
$565$	0	0
$566$	53.4955	2.24858
$567$	−1.00000	−0.0419961
$568$	−4.41742	−0.185351
$569$	19.4174	0.814021	0.407010	−	0.913424i	$-0.366571\pi$
$569$	19.4174	0.814021	0.407010	+	0.913424i	$0.366571\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−132.695	−5.54826
$573$	4.00000	0.167102
$574$	−25.5826	−1.06780
$575$	0	0
$576$	44.9129	1.87137
$577$	0.834849	0.0347552	0.0173776	−	0.999849i	$-0.494468\pi$
$577$	0.834849	0.0347552	0.0173776	+	0.999849i	$0.494468\pi$
$578$	−22.3303	−0.928818
$579$	16.3303	0.678664
$580$	0	0
$581$	−2.41742	−0.100292
$582$	6.74773	0.279702
$583$	−47.9129	−1.98435
$584$	−42.3303	−1.75164
$585$	0	0
$586$	−33.0345	−1.36464
$587$	2.41742	0.0997778	0.0498889	−	0.998755i	$-0.484113\pi$
$587$	2.41742	0.0997778	0.0498889	+	0.998755i	$0.484113\pi$
$588$	34.7477	1.43297
$589$	−22.3303	−0.920104
$590$	0	0
$591$	20.3303	0.836277
$592$	71.8258	2.95202
$593$	−17.5826	−0.722030	−0.361015	−	0.932560i	$-0.617570\pi$
$593$	−17.5826	−0.722030	−0.361015	+	0.932560i	$0.617570\pi$
$594$	−13.9564	−0.572640
$595$	0	0
$596$	−62.2432	−2.54958
$597$	−22.5826	−0.924243
$598$	−51.1652	−2.09230
$599$	−18.1652	−0.742208	−0.371104	−	0.928591i	$-0.621021\pi$
$599$	−18.1652	−0.742208	−0.371104	+	0.928591i	$0.621021\pi$
$600$	0	0
$601$	4.33030	0.176637	0.0883184	−	0.996092i	$-0.471851\pi$
$601$	4.33030	0.176637	0.0883184	+	0.996092i	$0.471851\pi$
$602$	−1.16515	−0.0474880
$603$	−14.1652	−0.576850
$604$	−64.6606	−2.63100
$605$	0	0
$606$	−23.9564	−0.973164
$607$	5.58258	0.226590	0.113295	−	0.993561i	$-0.463860\pi$
$607$	5.58258	0.226590	0.113295	+	0.993561i	$0.463860\pi$
$608$	−161.652	−6.55583
$609$	1.00000	0.0405220
$610$	0	0
$611$	6.49545	0.262778
$612$	17.3739	0.702297
$613$	16.2523	0.656423	0.328212	−	0.944604i	$-0.393554\pi$
$613$	16.2523	0.656423	0.328212	+	0.944604i	$0.393554\pi$
$614$	0	0
$615$	0	0
$616$	−52.9129	−2.13192
$617$	39.4955	1.59003	0.795014	−	0.606592i	$-0.207464\pi$
$617$	39.4955	1.59003	0.795014	+	0.606592i	$0.207464\pi$
$618$	−8.83485	−0.355390
$619$	−5.16515	−0.207605	−0.103802	−	0.994598i	$-0.533101\pi$
$619$	−5.16515	−0.207605	−0.103802	+	0.994598i	$0.533101\pi$
$620$	0	0
$621$	−4.00000	−0.160514
$622$	−8.37386	−0.335761
$623$	−10.5826	−0.423982
$624$	82.2867	3.29411
$625$	0	0
$626$	−3.95644	−0.158131
$627$	27.9129	1.11473
$628$	96.9909	3.87036
$629$	12.0000	0.478471
$630$	0	0
$631$	−34.9129	−1.38986	−0.694930	−	0.719078i	$-0.744565\pi$
$631$	−34.9129	−1.38986	−0.694930	+	0.719078i	$0.744565\pi$
$632$	−16.7477	−0.666189
$633$	16.3303	0.649071
$634$	69.7822	2.77141
$635$	0	0
$636$	55.4955	2.20054
$637$	27.4955	1.08941
$638$	13.9564	0.552541
$639$	−0.417424	−0.0165131
$640$	0	0
$641$	2.58258	0.102006	0.0510028	−	0.998699i	$-0.483758\pi$
$641$	2.58258	0.102006	0.0510028	+	0.998699i	$0.483758\pi$
$642$	−36.7477	−1.45032
$643$	29.6606	1.16970	0.584850	−	0.811141i	$-0.301153\pi$
$643$	29.6606	1.16970	0.584850	+	0.811141i	$0.301153\pi$
$644$	−23.1652	−0.912835
$645$	0	0
$646$	−46.7477	−1.83926
$647$	2.74773	0.108024	0.0540121	−	0.998540i	$-0.482799\pi$
$647$	2.74773	0.108024	0.0540121	+	0.998540i	$0.482799\pi$
$648$	10.5826	0.415723
$649$	7.91288	0.310608
$650$	0	0
$651$	4.00000	0.156772
$652$	9.16515	0.358935
$653$	−7.83485	−0.306601	−0.153301	−	0.988180i	$-0.548990\pi$
$653$	−7.83485	−0.306601	−0.153301	+	0.988180i	$0.548990\pi$
$654$	11.6261	0.454618
$655$	0	0
$656$	164.573	6.42552
$657$	−4.00000	−0.156055
$658$	3.95644	0.154238
$659$	−0.165151	−0.00643338	−0.00321669	−	0.999995i	$-0.501024\pi$
$659$	−0.165151	−0.00643338	−0.00321669	+	0.999995i	$0.501024\pi$
$660$	0	0
$661$	25.0000	0.972387	0.486194	−	0.873851i	$-0.338385\pi$
$661$	25.0000	0.972387	0.486194	+	0.873851i	$0.338385\pi$
$662$	79.0780	3.07345
$663$	13.7477	0.533917
$664$	25.5826	0.992796
$665$	0	0
$666$	11.1652	0.432641
$667$	4.00000	0.154881
$668$	0	0
$669$	7.00000	0.270636
$670$	0	0
$671$	−73.7386	−2.84665
$672$	28.9564	1.11702
$673$	25.7477	0.992502	0.496251	−	0.868179i	$-0.334710\pi$
$673$	25.7477	0.992502	0.496251	+	0.868179i	$0.334710\pi$
$674$	7.91288	0.304793
$675$	0	0
$676$	46.3303	1.78193
$677$	46.8258	1.79966	0.899830	−	0.436241i	$-0.143690\pi$
$677$	46.8258	1.79966	0.899830	+	0.436241i	$0.143690\pi$
$678$	11.6261	0.446499
$679$	2.41742	0.0927722
$680$	0	0
$681$	1.58258	0.0606444
$682$	55.8258	2.13768
$683$	11.1652	0.427223	0.213611	−	0.976919i	$-0.431477\pi$
$683$	11.1652	0.427223	0.213611	+	0.976919i	$0.431477\pi$
$684$	−32.3303	−1.23618
$685$	0	0
$686$	36.2867	1.38543
$687$	−1.16515	−0.0444533
$688$	7.49545	0.285762
$689$	43.9129	1.67295
$690$	0	0
$691$	−2.91288	−0.110811	−0.0554056	−	0.998464i	$-0.517645\pi$
$691$	−2.91288	−0.110811	−0.0554056	+	0.998464i	$0.517645\pi$
$692$	−87.8258	−3.33863
$693$	−5.00000	−0.189934
$694$	31.1652	1.18301
$695$	0	0
$696$	−10.5826	−0.401131
$697$	27.4955	1.04146
$698$	90.2432	3.41575
$699$	5.16515	0.195364
$700$	0	0
$701$	41.0780	1.55150	0.775748	−	0.631043i	$-0.217373\pi$
$701$	41.0780	1.55150	0.775748	+	0.631043i	$0.217373\pi$
$702$	12.7913	0.482776
$703$	−22.3303	−0.842203
$704$	224.564	8.46359
$705$	0	0
$706$	92.5735	3.48405
$707$	−8.58258	−0.322781
$708$	−9.16515	−0.344447
$709$	10.0000	0.375558	0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000	0.375558	0.187779	+	0.982211i	$0.439871\pi$
$710$	0	0
$711$	−1.58258	−0.0593512
$712$	111.991	4.19704
$713$	16.0000	0.599205
$714$	8.37386	0.313384
$715$	0	0
$716$	−131.739	−4.92330
$717$	26.0000	0.970988
$718$	−24.6606	−0.920326
$719$	−37.9129	−1.41391	−0.706956	−	0.707258i	$-0.749932\pi$
$719$	−37.9129	−1.41391	−0.706956	+	0.707258i	$0.749932\pi$
$720$	0	0
$721$	−3.16515	−0.117876
$722$	33.9564	1.26373
$723$	7.33030	0.272617
$724$	−12.5390	−0.466009
$725$	0	0
$726$	−39.0780	−1.45032
$727$	−46.3303	−1.71830	−0.859148	−	0.511727i	$-0.829006\pi$
$727$	−46.3303	−1.71830	−0.859148	+	0.511727i	$0.829006\pi$
$728$	48.4955	1.79736
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.25227	0.0463170
$732$	85.4083	3.15678
$733$	47.5826	1.75750	0.878751	−	0.477280i	$-0.158377\pi$
$733$	47.5826	1.75750	0.878751	+	0.477280i	$0.158377\pi$
$734$	−48.8348	−1.80253
$735$	0	0
$736$	115.826	4.26939
$737$	−70.8258	−2.60890
$738$	25.5826	0.941708
$739$	46.7477	1.71964	0.859821	−	0.510595i	$-0.170575\pi$
$739$	46.7477	1.71964	0.859821	+	0.510595i	$0.170575\pi$
$740$	0	0
$741$	−25.5826	−0.939799
$742$	26.7477	0.981940
$743$	−2.25227	−0.0826279	−0.0413139	−	0.999146i	$-0.513154\pi$
$743$	−2.25227	−0.0826279	−0.0413139	+	0.999146i	$0.513154\pi$
$744$	−42.3303	−1.55190
$745$	0	0
$746$	23.2523	0.851326
$747$	2.41742	0.0884489
$748$	86.8693	3.17626
$749$	−13.1652	−0.481044
$750$	0	0
$751$	37.4955	1.36823	0.684114	−	0.729375i	$-0.260189\pi$
$751$	37.4955	1.36823	0.684114	+	0.729375i	$0.260189\pi$
$752$	−25.4519	−0.928135
$753$	2.16515	0.0789025
$754$	−12.7913	−0.465831
$755$	0	0
$756$	5.79129	0.210627
$757$	−34.3303	−1.24776	−0.623878	−	0.781522i	$-0.714444\pi$
$757$	−34.3303	−1.24776	−0.623878	+	0.781522i	$0.714444\pi$
$758$	−72.5735	−2.63599
$759$	−20.0000	−0.725954
$760$	0	0
$761$	45.5826	1.65237	0.826184	−	0.563401i	$-0.190507\pi$
$761$	45.5826	1.65237	0.826184	+	0.563401i	$0.190507\pi$
$762$	5.58258	0.202235
$763$	4.16515	0.150789
$764$	−23.1652	−0.838086
$765$	0	0
$766$	15.5826	0.563021
$767$	−7.25227	−0.261864
$768$	−98.4519	−3.55258
$769$	16.4174	0.592027	0.296014	−	0.955184i	$-0.404343\pi$
$769$	16.4174	0.592027	0.296014	+	0.955184i	$0.404343\pi$
$770$	0	0
$771$	−14.7477	−0.531126
$772$	−94.5735	−3.40377
$773$	−13.1652	−0.473518	−0.236759	−	0.971568i	$-0.576085\pi$
$773$	−13.1652	−0.473518	−0.236759	+	0.971568i	$0.576085\pi$
$774$	1.16515	0.0418805
$775$	0	0
$776$	−25.5826	−0.918361
$777$	4.00000	0.143499
$778$	7.20871	0.258445
$779$	−51.1652	−1.83318
$780$	0	0
$781$	−2.08712	−0.0746831
$782$	33.4955	1.19779
$783$	−1.00000	−0.0357371
$784$	−107.739	−3.84781
$785$	0	0
$786$	41.8693	1.49343
$787$	24.0000	0.855508	0.427754	−	0.903895i	$-0.359305\pi$
$787$	24.0000	0.855508	0.427754	+	0.903895i	$0.359305\pi$
$788$	−117.739	−4.19427
$789$	−6.33030	−0.225365
$790$	0	0
$791$	4.16515	0.148096
$792$	52.9129	1.88018
$793$	67.5826	2.39993
$794$	−81.4083	−2.88907
$795$	0	0
$796$	130.782	4.63545
$797$	4.33030	0.153387	0.0766936	−	0.997055i	$-0.475564\pi$
$797$	4.33030	0.153387	0.0766936	+	0.997055i	$0.475564\pi$
$798$	−15.5826	−0.551617
$799$	−4.25227	−0.150435
$800$	0	0
$801$	10.5826	0.373917
$802$	60.2432	2.12726
$803$	−20.0000	−0.705785
$804$	82.0345	2.89313
$805$	0	0
$806$	−51.1652	−1.80222
$807$	−13.4174	−0.472316
$808$	90.8258	3.19524
$809$	−20.0780	−0.705906	−0.352953	−	0.935641i	$-0.614822\pi$
$809$	−20.0780	−0.705906	−0.352953	+	0.935641i	$0.614822\pi$
$810$	0	0
$811$	23.4174	0.822297	0.411148	−	0.911568i	$-0.365128\pi$
$811$	23.4174	0.822297	0.411148	+	0.911568i	$0.365128\pi$
$812$	−5.79129	−0.203234
$813$	17.1652	0.602008
$814$	55.8258	1.95669
$815$	0	0
$816$	−53.8693	−1.88580
$817$	−2.33030	−0.0815270
$818$	69.0780	2.41526
$819$	4.58258	0.160128
$820$	0	0
$821$	23.4955	0.819997	0.409999	−	0.912086i	$-0.365529\pi$
$821$	23.4955	0.819997	0.409999	+	0.912086i	$0.365529\pi$
$822$	−56.7477	−1.97930
$823$	13.0780	0.455871	0.227936	−	0.973676i	$-0.426802\pi$
$823$	13.0780	0.455871	0.227936	+	0.973676i	$0.426802\pi$
$824$	33.4955	1.16687
$825$	0	0
$826$	−4.41742	−0.153702
$827$	−47.8258	−1.66306	−0.831532	−	0.555476i	$-0.812536\pi$
$827$	−47.8258	−1.66306	−0.831532	+	0.555476i	$0.812536\pi$
$828$	23.1652	0.805045
$829$	4.00000	0.138926	0.0694629	−	0.997585i	$-0.477871\pi$
$829$	4.00000	0.138926	0.0694629	+	0.997585i	$0.477871\pi$
$830$	0	0
$831$	20.9129	0.725460
$832$	−205.817	−7.13541
$833$	−18.0000	−0.623663
$834$	51.8693	1.79609
$835$	0	0
$836$	−161.652	−5.59083
$837$	−4.00000	−0.138260
$838$	−52.5735	−1.81612
$839$	6.49545	0.224248	0.112124	−	0.993694i	$-0.464235\pi$
$839$	6.49545	0.224248	0.112124	+	0.993694i	$0.464235\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	37.9129	1.30656
$843$	9.58258	0.330041
$844$	−94.5735	−3.25535
$845$	0	0
$846$	−3.95644	−0.136025
$847$	−14.0000	−0.481046
$848$	−172.069	−5.90887
$849$	−19.1652	−0.657746
$850$	0	0
$851$	16.0000	0.548473
$852$	2.41742	0.0828196
$853$	6.74773	0.231038	0.115519	−	0.993305i	$-0.463147\pi$
$853$	6.74773	0.231038	0.115519	+	0.993305i	$0.463147\pi$
$854$	41.1652	1.40864
$855$	0	0
$856$	139.321	4.76190
$857$	−26.6606	−0.910709	−0.455354	−	0.890310i	$-0.650487\pi$
$857$	−26.6606	−0.910709	−0.455354	+	0.890310i	$0.650487\pi$
$858$	63.9564	2.18344
$859$	−38.4174	−1.31079	−0.655393	−	0.755288i	$-0.727497\pi$
$859$	−38.4174	−1.31079	−0.655393	+	0.755288i	$0.727497\pi$
$860$	0	0
$861$	9.16515	0.312348
$862$	−58.1561	−1.98080
$863$	15.5826	0.530437	0.265219	−	0.964188i	$-0.414556\pi$
$863$	15.5826	0.530437	0.265219	+	0.964188i	$0.414556\pi$
$864$	−28.9564	−0.985118
$865$	0	0
$866$	109.078	3.70662
$867$	8.00000	0.271694
$868$	−23.1652	−0.786276
$869$	−7.91288	−0.268426
$870$	0	0
$871$	64.9129	2.19949
$872$	−44.0780	−1.49267
$873$	−2.41742	−0.0818174
$874$	−62.3303	−2.10835
$875$	0	0
$876$	23.1652	0.782678
$877$	−56.3303	−1.90214	−0.951070	−	0.308977i	$-0.900013\pi$
$877$	−56.3303	−1.90214	−0.951070	+	0.308977i	$0.900013\pi$
$878$	80.7042	2.72363
$879$	11.8348	0.399180
$880$	0	0
$881$	32.0780	1.08074	0.540368	−	0.841429i	$-0.318285\pi$
$881$	32.0780	1.08074	0.540368	+	0.841429i	$0.318285\pi$
$882$	−16.7477	−0.563925
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	−79.6170	−2.67781
$885$	0	0
$886$	23.9564	0.804832
$887$	−0.912878	−0.0306515	−0.0153257	−	0.999883i	$-0.504879\pi$
$887$	−0.912878	−0.0306515	−0.0153257	+	0.999883i	$0.504879\pi$
$888$	−42.3303	−1.42051
$889$	2.00000	0.0670778
$890$	0	0
$891$	5.00000	0.167506
$892$	−40.5390	−1.35735
$893$	7.91288	0.264794
$894$	30.0000	1.00335
$895$	0	0
$896$	−67.4519	−2.25341
$897$	18.3303	0.612031
$898$	95.1216	3.17425
$899$	4.00000	0.133407
$900$	0	0
$901$	−28.7477	−0.957726
$902$	127.913	4.25903
$903$	0.417424	0.0138910
$904$	−44.0780	−1.46601
$905$	0	0
$906$	31.1652	1.03539
$907$	−14.4174	−0.478723	−0.239361	−	0.970931i	$-0.576938\pi$
$907$	−14.4174	−0.478723	−0.239361	+	0.970931i	$0.576938\pi$
$908$	−9.16515	−0.304156
$909$	8.58258	0.284666
$910$	0	0
$911$	40.8258	1.35262	0.676309	−	0.736618i	$-0.263578\pi$
$911$	40.8258	1.35262	0.676309	+	0.736618i	$0.263578\pi$
$912$	100.243	3.31938
$913$	12.0871	0.400025
$914$	−66.2867	−2.19257
$915$	0	0
$916$	6.74773	0.222951
$917$	15.0000	0.495344
$918$	−8.37386	−0.276379
$919$	−26.9129	−0.887774	−0.443887	−	0.896083i	$-0.646401\pi$
$919$	−26.9129	−0.887774	−0.443887	+	0.896083i	$0.646401\pi$
$920$	0	0
$921$	0	0
$922$	25.5826	0.842517
$923$	1.91288	0.0629632
$924$	28.9564	0.952597
$925$	0	0
$926$	−50.7042	−1.66624
$927$	3.16515	0.103957
$928$	28.9564	0.950542
$929$	52.3303	1.71690	0.858451	−	0.512896i	$-0.171427\pi$
$929$	52.3303	1.71690	0.858451	+	0.512896i	$0.171427\pi$
$930$	0	0
$931$	33.4955	1.09777
$932$	−29.9129	−0.979829
$933$	3.00000	0.0982156
$934$	22.3303	0.730670
$935$	0	0
$936$	−48.4955	−1.58512
$937$	9.41742	0.307654	0.153827	−	0.988098i	$-0.450840\pi$
$937$	9.41742	0.307654	0.153827	+	0.988098i	$0.450840\pi$
$938$	39.5390	1.29099
$939$	1.41742	0.0462559
$940$	0	0
$941$	−25.9129	−0.844736	−0.422368	−	0.906425i	$-0.638801\pi$
$941$	−25.9129	−0.844736	−0.422368	+	0.906425i	$0.638801\pi$
$942$	−46.7477	−1.52312
$943$	36.6606	1.19383
$944$	28.4174	0.924908
$945$	0	0
$946$	5.82576	0.189412
$947$	9.08712	0.295292	0.147646	−	0.989040i	$-0.452830\pi$
$947$	9.08712	0.295292	0.147646	+	0.989040i	$0.452830\pi$
$948$	9.16515	0.297670
$949$	18.3303	0.595027
$950$	0	0
$951$	−25.0000	−0.810681
$952$	−31.7477	−1.02895
$953$	28.4174	0.920531	0.460265	−	0.887781i	$-0.347754\pi$
$953$	28.4174	0.920531	0.460265	+	0.887781i	$0.347754\pi$
$954$	−26.7477	−0.865990
$955$	0	0
$956$	−150.573	−4.86989
$957$	−5.00000	−0.161627
$958$	−2.33030	−0.0752887
$959$	−20.3303	−0.656500
$960$	0	0
$961$	−15.0000	−0.483871
$962$	−51.1652	−1.64963
$963$	13.1652	0.424241
$964$	−42.4519	−1.36728
$965$	0	0
$966$	11.1652	0.359233
$967$	−36.7477	−1.18173	−0.590864	−	0.806771i	$-0.701213\pi$
$967$	−36.7477	−1.18173	−0.590864	+	0.806771i	$0.701213\pi$
$968$	148.156	4.76192
$969$	16.7477	0.538015
$970$	0	0
$971$	20.0000	0.641831	0.320915	−	0.947108i	$-0.396010\pi$
$971$	20.0000	0.641831	0.320915	+	0.947108i	$0.396010\pi$
$972$	−5.79129	−0.185756
$973$	18.5826	0.595730
$974$	95.8258	3.07046
$975$	0	0
$976$	−264.817	−8.47657
$977$	57.4955	1.83944	0.919721	−	0.392572i	$-0.128415\pi$
$977$	57.4955	1.83944	0.919721	+	0.392572i	$0.128415\pi$
$978$	−4.41742	−0.141254
$979$	52.9129	1.69110
$980$	0	0
$981$	−4.16515	−0.132983
$982$	44.6606	1.42518
$983$	−36.8348	−1.17485	−0.587425	−	0.809279i	$-0.699858\pi$
$983$	−36.8348	−1.17485	−0.587425	+	0.809279i	$0.699858\pi$
$984$	−96.9909	−3.09196
$985$	0	0
$986$	8.37386	0.266678
$987$	−1.41742	−0.0451171
$988$	148.156	4.71347
$989$	1.66970	0.0530933
$990$	0	0
$991$	−48.0780	−1.52725	−0.763624	−	0.645661i	$-0.776582\pi$
$991$	−48.0780	−1.52725	−0.763624	+	0.645661i	$0.776582\pi$
$992$	115.826	3.67747
$993$	−28.3303	−0.899035
$994$	1.16515	0.0369564
$995$	0	0
$996$	−14.0000	−0.443607
$997$	19.1652	0.606966	0.303483	−	0.952837i	$-0.401850\pi$
$997$	19.1652	0.606966	0.303483	+	0.952837i	$0.401850\pi$
$998$	−63.0345	−1.99532
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2175.2.a.r.1.2		2
3.2	odd	2		6525.2.a.t.1.1		2
5.2	odd	4		2175.2.c.f.349.4		4
5.3	odd	4		2175.2.c.f.349.1		4
5.4	even	2		435.2.a.f.1.1	✓	2
15.14	odd	2		1305.2.a.m.1.2		2
20.19	odd	2		6960.2.a.bw.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.a.f.1.1	✓	2	5.4	even	2
1305.2.a.m.1.2		2	15.14	odd	2
2175.2.a.r.1.2		2	1.1	even	1	trivial
2175.2.c.f.349.1		4	5.3	odd	4
2175.2.c.f.349.4		4	5.2	odd	4
6525.2.a.t.1.1		2	3.2	odd	2
6960.2.a.bw.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2175 = 3 \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2175.a (trivial)

\(f(q)\)	\(=\)	\(q+2.79129 q^{2} -1.00000 q^{3} +5.79129 q^{4} -2.79129 q^{6} -1.00000 q^{7} +10.5826 q^{8} +1.00000 q^{9} +5.00000 q^{11} -5.79129 q^{12} -4.58258 q^{13} -2.79129 q^{14} +17.9564 q^{16} +3.00000 q^{17} +2.79129 q^{18} -5.58258 q^{19} +1.00000 q^{21} +13.9564 q^{22} +4.00000 q^{23} -10.5826 q^{24} -12.7913 q^{26} -1.00000 q^{27} -5.79129 q^{28} +1.00000 q^{29} +4.00000 q^{31} +28.9564 q^{32} -5.00000 q^{33} +8.37386 q^{34} +5.79129 q^{36} +4.00000 q^{37} -15.5826 q^{38} +4.58258 q^{39} +9.16515 q^{41} +2.79129 q^{42} +0.417424 q^{43} +28.9564 q^{44} +11.1652 q^{46} -1.41742 q^{47} -17.9564 q^{48} -6.00000 q^{49} -3.00000 q^{51} -26.5390 q^{52} -9.58258 q^{53} -2.79129 q^{54} -10.5826 q^{56} +5.58258 q^{57} +2.79129 q^{58} +1.58258 q^{59} -14.7477 q^{61} +11.1652 q^{62} -1.00000 q^{63} +44.9129 q^{64} -13.9564 q^{66} -14.1652 q^{67} +17.3739 q^{68} -4.00000 q^{69} -0.417424 q^{71} +10.5826 q^{72} -4.00000 q^{73} +11.1652 q^{74} -32.3303 q^{76} -5.00000 q^{77} +12.7913 q^{78} -1.58258 q^{79} +1.00000 q^{81} +25.5826 q^{82} +2.41742 q^{83} +5.79129 q^{84} +1.16515 q^{86} -1.00000 q^{87} +52.9129 q^{88} +10.5826 q^{89} +4.58258 q^{91} +23.1652 q^{92} -4.00000 q^{93} -3.95644 q^{94} -28.9564 q^{96} -2.41742 q^{97} -16.7477 q^{98} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} - 2 q^{3} + 7 q^{4} - q^{6} - 2 q^{7} + 12 q^{8} + 2 q^{9} + 10 q^{11} - 7 q^{12} - q^{14} + 13 q^{16} + 6 q^{17} + q^{18} - 2 q^{19} + 2 q^{21} + 5 q^{22} + 8 q^{23} - 12 q^{24} - 21 q^{26}+ \cdots + 10 q^{99}+O(q^{100}) \) 2 * q + q^2 - 2 * q^3 + 7 * q^4 - q^6 - 2 * q^7 + 12 * q^8 + 2 * q^9 + 10 * q^11 - 7 * q^12 - q^14 + 13 * q^16 + 6 * q^17 + q^18 - 2 * q^19 + 2 * q^21 + 5 * q^22 + 8 * q^23 - 12 * q^24 - 21 * q^26 - 2 * q^27 - 7 * q^28 + 2 * q^29 + 8 * q^31 + 35 * q^32 - 10 * q^33 + 3 * q^34 + 7 * q^36 + 8 * q^37 - 22 * q^38 + q^42 + 10 * q^43 + 35 * q^44 + 4 * q^46 - 12 * q^47 - 13 * q^48 - 12 * q^49 - 6 * q^51 - 21 * q^52 - 10 * q^53 - q^54 - 12 * q^56 + 2 * q^57 + q^58 - 6 * q^59 - 2 * q^61 + 4 * q^62 - 2 * q^63 + 44 * q^64 - 5 * q^66 - 10 * q^67 + 21 * q^68 - 8 * q^69 - 10 * q^71 + 12 * q^72 - 8 * q^73 + 4 * q^74 - 28 * q^76 - 10 * q^77 + 21 * q^78 + 6 * q^79 + 2 * q^81 + 42 * q^82 + 14 * q^83 + 7 * q^84 - 16 * q^86 - 2 * q^87 + 60 * q^88 + 12 * q^89 + 28 * q^92 - 8 * q^93 + 15 * q^94 - 35 * q^96 - 14 * q^97 - 6 * q^98 + 10 * q^99

Introduction

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