LMFDB - Embedded newform 8670.2.a.bm.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8670,2,Mod(1,8670)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8670, 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("8670.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8670.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:	$69.2302985525$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
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			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	8670.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.87939 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.22668 q^{11} -1.00000 q^{12} +0.347296 q^{13} -1.87939 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +0.758770 q^{19} +1.00000 q^{20} -1.87939 q^{21} -2.22668 q^{22} +3.41147 q^{23} +1.00000 q^{24} +1.00000 q^{25} -0.347296 q^{26} -1.00000 q^{27} +1.87939 q^{28} -6.82295 q^{29} +1.00000 q^{30} +3.06418 q^{31} -1.00000 q^{32} -2.22668 q^{33} +1.87939 q^{35} +1.00000 q^{36} -3.12836 q^{37} -0.758770 q^{38} -0.347296 q^{39} -1.00000 q^{40} -0.630415 q^{41} +1.87939 q^{42} -0.694593 q^{43} +2.22668 q^{44} +1.00000 q^{45} -3.41147 q^{46} +10.0915 q^{47} -1.00000 q^{48} -3.46791 q^{49} -1.00000 q^{50} +0.347296 q^{52} +8.22668 q^{53} +1.00000 q^{54} +2.22668 q^{55} -1.87939 q^{56} -0.758770 q^{57} +6.82295 q^{58} +6.68004 q^{59} -1.00000 q^{60} +0.694593 q^{61} -3.06418 q^{62} +1.87939 q^{63} +1.00000 q^{64} +0.347296 q^{65} +2.22668 q^{66} -9.06418 q^{67} -3.41147 q^{69} -1.87939 q^{70} +4.45336 q^{71} -1.00000 q^{72} +4.00000 q^{73} +3.12836 q^{74} -1.00000 q^{75} +0.758770 q^{76} +4.18479 q^{77} +0.347296 q^{78} +6.08378 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.630415 q^{82} +8.36959 q^{83} -1.87939 q^{84} +0.694593 q^{86} +6.82295 q^{87} -2.22668 q^{88} -1.18479 q^{89} -1.00000 q^{90} +0.652704 q^{91} +3.41147 q^{92} -3.06418 q^{93} -10.0915 q^{94} +0.758770 q^{95} +1.00000 q^{96} -2.21213 q^{97} +3.46791 q^{98} +2.22668 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{12} - 3 q^{15} + 3 q^{16} - 3 q^{18} - 9 q^{19} + 3 q^{20} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{30} - 3 q^{32}+ \cdots + 15 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 - 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^12 - 3 * q^15 + 3 * q^16 - 3 * q^18 - 9 * q^19 + 3 * q^20 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^30 - 3 * q^32 + 3 * q^36 + 9 * q^37 + 9 * q^38 - 3 * q^40 - 9 * q^41 + 3 * q^45 - 3 * q^48 - 15 * q^49 - 3 * q^50 + 18 * q^53 + 3 * q^54 + 9 * q^57 - 3 * q^60 + 3 * q^64 - 18 * q^67 - 3 * q^72 + 12 * q^73 - 9 * q^74 - 3 * q^75 - 9 * q^76 + 9 * q^77 + 12 * q^79 + 3 * q^80 + 3 * q^81 + 9 * q^82 + 18 * q^83 - 3 * q^90 + 3 * q^91 - 9 * q^95 + 3 * q^96 + 18 * q^97 + 15 * q^98

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
$7$	1.87939	0.710341	0.355170	−	0.934802i	$-0.384423\pi$
$7$	1.87939	0.710341	0.355170	+	0.934802i	$0.384423\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.22668	0.671370	0.335685	−	0.941974i	$-0.391032\pi$
$11$	2.22668	0.671370	0.335685	+	0.941974i	$0.391032\pi$
$12$	−1.00000	−0.288675
$13$	0.347296	0.0963227	0.0481613	−	0.998840i	$-0.484664\pi$
$13$	0.347296	0.0963227	0.0481613	+	0.998840i	$0.484664\pi$
$14$	−1.87939	−0.502287
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0
$17$	0	0
$18$	−1.00000	−0.235702
$19$	0.758770	0.174074	0.0870369	−	0.996205i	$-0.472260\pi$
$19$	0.758770	0.174074	0.0870369	+	0.996205i	$0.472260\pi$
$20$	1.00000	0.223607
$21$	−1.87939	−0.410115
$22$	−2.22668	−0.474730
$23$	3.41147	0.711342	0.355671	−	0.934611i	$-0.384252\pi$
$23$	3.41147	0.711342	0.355671	+	0.934611i	$0.384252\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−0.347296	−0.0681104
$27$	−1.00000	−0.192450
$28$	1.87939	0.355170
$29$	−6.82295	−1.26699	−0.633495	−	0.773747i	$-0.718380\pi$
$29$	−6.82295	−1.26699	−0.633495	+	0.773747i	$0.718380\pi$
$30$	1.00000	0.182574
$31$	3.06418	0.550343	0.275171	−	0.961395i	$-0.411265\pi$
$31$	3.06418	0.550343	0.275171	+	0.961395i	$0.411265\pi$
$32$	−1.00000	−0.176777
$33$	−2.22668	−0.387616
$34$	0	0
$35$	1.87939	0.317674
$36$	1.00000	0.166667
$37$	−3.12836	−0.514298	−0.257149	−	0.966372i	$-0.582783\pi$
$37$	−3.12836	−0.514298	−0.257149	+	0.966372i	$0.582783\pi$
$38$	−0.758770	−0.123089
$39$	−0.347296	−0.0556119
$40$	−1.00000	−0.158114
$41$	−0.630415	−0.0984543	−0.0492271	−	0.998788i	$-0.515676\pi$
$41$	−0.630415	−0.0984543	−0.0492271	+	0.998788i	$0.515676\pi$
$42$	1.87939	0.289995
$43$	−0.694593	−0.105924	−0.0529622	−	0.998597i	$-0.516866\pi$
$43$	−0.694593	−0.105924	−0.0529622	+	0.998597i	$0.516866\pi$
$44$	2.22668	0.335685
$45$	1.00000	0.149071
$46$	−3.41147	−0.502994
$47$	10.0915	1.47200	0.736000	−	0.676982i	$-0.236712\pi$
$47$	10.0915	1.47200	0.736000	+	0.676982i	$0.236712\pi$
$48$	−1.00000	−0.144338
$49$	−3.46791	−0.495416
$50$	−1.00000	−0.141421
$51$	0	0
$52$	0.347296	0.0481613
$53$	8.22668	1.13002	0.565011	−	0.825084i	$-0.308872\pi$
$53$	8.22668	1.13002	0.565011	+	0.825084i	$0.308872\pi$
$54$	1.00000	0.136083
$55$	2.22668	0.300246
$56$	−1.87939	−0.251143
$57$	−0.758770	−0.100502
$58$	6.82295	0.895897
$59$	6.68004	0.869668	0.434834	−	0.900511i	$-0.356807\pi$
$59$	6.68004	0.869668	0.434834	+	0.900511i	$0.356807\pi$
$60$	−1.00000	−0.129099
$61$	0.694593	0.0889335	0.0444667	−	0.999011i	$-0.485841\pi$
$61$	0.694593	0.0889335	0.0444667	+	0.999011i	$0.485841\pi$
$62$	−3.06418	−0.389151
$63$	1.87939	0.236780
$64$	1.00000	0.125000
$65$	0.347296	0.0430768
$66$	2.22668	0.274086
$67$	−9.06418	−1.10737	−0.553683	−	0.832728i	$-0.686778\pi$
$67$	−9.06418	−1.10737	−0.553683	+	0.832728i	$0.686778\pi$
$68$	0	0
$69$	−3.41147	−0.410693
$70$	−1.87939	−0.224630
$71$	4.45336	0.528517	0.264258	−	0.964452i	$-0.414873\pi$
$71$	4.45336	0.528517	0.264258	+	0.964452i	$0.414873\pi$
$72$	−1.00000	−0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	3.12836	0.363664
$75$	−1.00000	−0.115470
$76$	0.758770	0.0870369
$77$	4.18479	0.476901
$78$	0.347296	0.0393236
$79$	6.08378	0.684479	0.342239	−	0.939613i	$-0.388815\pi$
$79$	6.08378	0.684479	0.342239	+	0.939613i	$0.388815\pi$
$80$	1.00000	0.111803
$81$	1.00000	0.111111
$82$	0.630415	0.0696177
$83$	8.36959	0.918681	0.459341	−	0.888260i	$-0.348086\pi$
$83$	8.36959	0.918681	0.459341	+	0.888260i	$0.348086\pi$
$84$	−1.87939	−0.205058
$85$	0	0
$86$	0.694593	0.0748999
$87$	6.82295	0.731497
$88$	−2.22668	−0.237365
$89$	−1.18479	−0.125588	−0.0627939	−	0.998027i	$-0.520001\pi$
$89$	−1.18479	−0.125588	−0.0627939	+	0.998027i	$0.520001\pi$
$90$	−1.00000	−0.105409
$91$	0.652704	0.0684219
$92$	3.41147	0.355671
$93$	−3.06418	−0.317740
$94$	−10.0915	−1.04086
$95$	0.758770	0.0778482
$96$	1.00000	0.102062
$97$	−2.21213	−0.224608	−0.112304	−	0.993674i	$-0.535823\pi$
$97$	−2.21213	−0.224608	−0.112304	+	0.993674i	$0.535823\pi$
$98$	3.46791	0.350312
$99$	2.22668	0.223790
$100$	1.00000	0.100000
$101$	−15.1925	−1.51171	−0.755857	−	0.654737i	$-0.772779\pi$
$101$	−15.1925	−1.51171	−0.755857	+	0.654737i	$0.772779\pi$
$102$	0	0
$103$	0.758770	0.0747639	0.0373819	−	0.999301i	$-0.488098\pi$
$103$	0.758770	0.0747639	0.0373819	+	0.999301i	$0.488098\pi$
$104$	−0.347296	−0.0340552
$105$	−1.87939	−0.183409
$106$	−8.22668	−0.799046
$107$	−8.90673	−0.861046	−0.430523	−	0.902580i	$-0.641671\pi$
$107$	−8.90673	−0.861046	−0.430523	+	0.902580i	$0.641671\pi$
$108$	−1.00000	−0.0962250
$109$	13.1925	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$109$	13.1925	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$110$	−2.22668	−0.212306
$111$	3.12836	0.296930
$112$	1.87939	0.177585
$113$	18.8229	1.77071	0.885357	−	0.464912i	$-0.153914\pi$
$113$	18.8229	1.77071	0.885357	+	0.464912i	$0.153914\pi$
$114$	0.758770	0.0710654
$115$	3.41147	0.318122
$116$	−6.82295	−0.633495
$117$	0.347296	0.0321076
$118$	−6.68004	−0.614948
$119$	0	0
$120$	1.00000	0.0912871
$121$	−6.04189	−0.549263
$122$	−0.694593	−0.0628855
$123$	0.630415	0.0568426
$124$	3.06418	0.275171
$125$	1.00000	0.0894427
$126$	−1.87939	−0.167429
$127$	11.4979	1.02028	0.510139	−	0.860092i	$-0.329594\pi$
$127$	11.4979	1.02028	0.510139	+	0.860092i	$0.329594\pi$
$128$	−1.00000	−0.0883883
$129$	0.694593	0.0611555
$130$	−0.347296	−0.0304599
$131$	−3.41147	−0.298062	−0.149031	−	0.988833i	$-0.547615\pi$
$131$	−3.41147	−0.298062	−0.149031	+	0.988833i	$0.547615\pi$
$132$	−2.22668	−0.193808
$133$	1.42602	0.123652
$134$	9.06418	0.783026
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	14.3696	1.22768	0.613838	−	0.789432i	$-0.289625\pi$
$137$	14.3696	1.22768	0.613838	+	0.789432i	$0.289625\pi$
$138$	3.41147	0.290404
$139$	−0.758770	−0.0643581	−0.0321790	−	0.999482i	$-0.510245\pi$
$139$	−0.758770	−0.0643581	−0.0321790	+	0.999482i	$0.510245\pi$
$140$	1.87939	0.158837
$141$	−10.0915	−0.849859
$142$	−4.45336	−0.373718
$143$	0.773318	0.0646681
$144$	1.00000	0.0833333
$145$	−6.82295	−0.566615
$146$	−4.00000	−0.331042
$147$	3.46791	0.286028
$148$	−3.12836	−0.257149
$149$	4.45336	0.364834	0.182417	−	0.983221i	$-0.441608\pi$
$149$	4.45336	0.364834	0.182417	+	0.983221i	$0.441608\pi$
$150$	1.00000	0.0816497
$151$	−0.157451	−0.0128132	−0.00640661	−	0.999979i	$-0.502039\pi$
$151$	−0.157451	−0.0128132	−0.00640661	+	0.999979i	$0.502039\pi$
$152$	−0.758770	−0.0615444
$153$	0	0
$154$	−4.18479	−0.337220
$155$	3.06418	0.246121
$156$	−0.347296	−0.0278060
$157$	−20.2645	−1.61728	−0.808640	−	0.588304i	$-0.799796\pi$
$157$	−20.2645	−1.61728	−0.808640	+	0.588304i	$0.799796\pi$
$158$	−6.08378	−0.483999
$159$	−8.22668	−0.652418
$160$	−1.00000	−0.0790569
$161$	6.41147	0.505295
$162$	−1.00000	−0.0785674
$163$	−9.10876	−0.713453	−0.356726	−	0.934209i	$-0.616107\pi$
$163$	−9.10876	−0.713453	−0.356726	+	0.934209i	$0.616107\pi$
$164$	−0.630415	−0.0492271
$165$	−2.22668	−0.173347
$166$	−8.36959	−0.649606
$167$	2.22668	0.172306	0.0861529	−	0.996282i	$-0.472543\pi$
$167$	2.22668	0.172306	0.0861529	+	0.996282i	$0.472543\pi$
$168$	1.87939	0.144998
$169$	−12.8794	−0.990722
$170$	0	0
$171$	0.758770	0.0580246
$172$	−0.694593	−0.0529622
$173$	19.0419	1.44773	0.723864	−	0.689943i	$-0.242364\pi$
$173$	19.0419	1.44773	0.723864	+	0.689943i	$0.242364\pi$
$174$	−6.82295	−0.517246
$175$	1.87939	0.142068
$176$	2.22668	0.167842
$177$	−6.68004	−0.502103
$178$	1.18479	0.0888040
$179$	−14.4020	−1.07645	−0.538227	−	0.842800i	$-0.680906\pi$
$179$	−14.4020	−1.07645	−0.538227	+	0.842800i	$0.680906\pi$
$180$	1.00000	0.0745356
$181$	13.5175	1.00475	0.502375	−	0.864650i	$-0.332460\pi$
$181$	13.5175	1.00475	0.502375	+	0.864650i	$0.332460\pi$
$182$	−0.652704	−0.0483816
$183$	−0.694593	−0.0513458
$184$	−3.41147	−0.251497
$185$	−3.12836	−0.230001
$186$	3.06418	0.224676
$187$	0	0
$188$	10.0915	0.736000
$189$	−1.87939	−0.136705
$190$	−0.758770	−0.0550470
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	−1.00000	−0.0721688
$193$	12.7939	0.920922	0.460461	−	0.887680i	$-0.347684\pi$
$193$	12.7939	0.920922	0.460461	+	0.887680i	$0.347684\pi$
$194$	2.21213	0.158822
$195$	−0.347296	−0.0248704
$196$	−3.46791	−0.247708
$197$	5.13341	0.365740	0.182870	−	0.983137i	$-0.441461\pi$
$197$	5.13341	0.365740	0.182870	+	0.983137i	$0.441461\pi$
$198$	−2.22668	−0.158243
$199$	7.91622	0.561166	0.280583	−	0.959830i	$-0.409472\pi$
$199$	7.91622	0.561166	0.280583	+	0.959830i	$0.409472\pi$
$200$	−1.00000	−0.0707107
$201$	9.06418	0.639338
$202$	15.1925	1.06894
$203$	−12.8229	−0.899995
$204$	0	0
$205$	−0.630415	−0.0440301
$206$	−0.758770	−0.0528660
$207$	3.41147	0.237114
$208$	0.347296	0.0240807
$209$	1.68954	0.116868
$210$	1.87939	0.129690
$211$	−2.28817	−0.157524	−0.0787621	−	0.996893i	$-0.525097\pi$
$211$	−2.28817	−0.157524	−0.0787621	+	0.996893i	$0.525097\pi$
$212$	8.22668	0.565011
$213$	−4.45336	−0.305139
$214$	8.90673	0.608851
$215$	−0.694593	−0.0473708
$216$	1.00000	0.0680414
$217$	5.75877	0.390931
$218$	−13.1925	−0.893511
$219$	−4.00000	−0.270295
$220$	2.22668	0.150123
$221$	0	0
$222$	−3.12836	−0.209961
$223$	14.3182	0.958818	0.479409	−	0.877592i	$-0.340851\pi$
$223$	14.3182	0.958818	0.479409	+	0.877592i	$0.340851\pi$
$224$	−1.87939	−0.125572
$225$	1.00000	0.0666667
$226$	−18.8229	−1.25208
$227$	13.6459	0.905710	0.452855	−	0.891584i	$-0.350406\pi$
$227$	13.6459	0.905710	0.452855	+	0.891584i	$0.350406\pi$
$228$	−0.758770	−0.0502508
$229$	3.10876	0.205432	0.102716	−	0.994711i	$-0.467247\pi$
$229$	3.10876	0.205432	0.102716	+	0.994711i	$0.467247\pi$
$230$	−3.41147	−0.224946
$231$	−4.18479	−0.275339
$232$	6.82295	0.447948
$233$	−9.47834	−0.620947	−0.310473	−	0.950582i	$-0.600488\pi$
$233$	−9.47834	−0.620947	−0.310473	+	0.950582i	$0.600488\pi$
$234$	−0.347296	−0.0227035
$235$	10.0915	0.658298
$236$	6.68004	0.434834
$237$	−6.08378	−0.395184
$238$	0	0
$239$	3.19253	0.206508	0.103254	−	0.994655i	$-0.467075\pi$
$239$	3.19253	0.206508	0.103254	+	0.994655i	$0.467075\pi$
$240$	−1.00000	−0.0645497
$241$	−13.6382	−0.878511	−0.439255	−	0.898362i	$-0.644758\pi$
$241$	−13.6382	−0.878511	−0.439255	+	0.898362i	$0.644758\pi$
$242$	6.04189	0.388387
$243$	−1.00000	−0.0641500
$244$	0.694593	0.0444667
$245$	−3.46791	−0.221557
$246$	−0.630415	−0.0401938
$247$	0.263518	0.0167673
$248$	−3.06418	−0.194575
$249$	−8.36959	−0.530401
$250$	−1.00000	−0.0632456
$251$	−29.5185	−1.86319	−0.931595	−	0.363498i	$-0.881582\pi$
$251$	−29.5185	−1.86319	−0.931595	+	0.363498i	$0.881582\pi$
$252$	1.87939	0.118390
$253$	7.59627	0.477573
$254$	−11.4979	−0.721445
$255$	0	0
$256$	1.00000	0.0625000
$257$	−2.08378	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$257$	−2.08378	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$258$	−0.694593	−0.0432435
$259$	−5.87939	−0.365327
$260$	0.347296	0.0215384
$261$	−6.82295	−0.422330
$262$	3.41147	0.210762
$263$	−17.0574	−1.05180	−0.525901	−	0.850546i	$-0.676272\pi$
$263$	−17.0574	−1.05180	−0.525901	+	0.850546i	$0.676272\pi$
$264$	2.22668	0.137043
$265$	8.22668	0.505361
$266$	−1.42602	−0.0874350
$267$	1.18479	0.0725081
$268$	−9.06418	−0.553683
$269$	14.9067	0.908879	0.454440	−	0.890778i	$-0.349840\pi$
$269$	14.9067	0.908879	0.454440	+	0.890778i	$0.349840\pi$
$270$	1.00000	0.0608581
$271$	−3.78787	−0.230096	−0.115048	−	0.993360i	$-0.536702\pi$
$271$	−3.78787	−0.230096	−0.115048	+	0.993360i	$0.536702\pi$
$272$	0	0
$273$	−0.652704	−0.0395034
$274$	−14.3696	−0.868098
$275$	2.22668	0.134274
$276$	−3.41147	−0.205347
$277$	−8.81614	−0.529711	−0.264855	−	0.964288i	$-0.585324\pi$
$277$	−8.81614	−0.529711	−0.264855	+	0.964288i	$0.585324\pi$
$278$	0.758770	0.0455080
$279$	3.06418	0.183448
$280$	−1.87939	−0.112315
$281$	−7.94087	−0.473713	−0.236856	−	0.971545i	$-0.576117\pi$
$281$	−7.94087	−0.473713	−0.236856	+	0.971545i	$0.576117\pi$
$282$	10.0915	0.600941
$283$	11.4338	0.679667	0.339833	−	0.940486i	$-0.389629\pi$
$283$	11.4338	0.679667	0.339833	+	0.940486i	$0.389629\pi$
$284$	4.45336	0.264258
$285$	−0.758770	−0.0449457
$286$	−0.773318	−0.0457273
$287$	−1.18479	−0.0699361
$288$	−1.00000	−0.0589256
$289$	0	0
$290$	6.82295	0.400657
$291$	2.21213	0.129678
$292$	4.00000	0.234082
$293$	2.12742	0.124285	0.0621427	−	0.998067i	$-0.480207\pi$
$293$	2.12742	0.124285	0.0621427	+	0.998067i	$0.480207\pi$
$294$	−3.46791	−0.202253
$295$	6.68004	0.388927
$296$	3.12836	0.181832
$297$	−2.22668	−0.129205
$298$	−4.45336	−0.257976
$299$	1.18479	0.0685183
$300$	−1.00000	−0.0577350
$301$	−1.30541	−0.0752424
$302$	0.157451	0.00906031
$303$	15.1925	0.872788
$304$	0.758770	0.0435185
$305$	0.694593	0.0397723
$306$	0	0
$307$	−28.3851	−1.62002	−0.810011	−	0.586415i	$-0.800539\pi$
$307$	−28.3851	−1.62002	−0.810011	+	0.586415i	$0.800539\pi$
$308$	4.18479	0.238451
$309$	−0.758770	−0.0431649
$310$	−3.06418	−0.174034
$311$	−16.5526	−0.938613	−0.469307	−	0.883035i	$-0.655496\pi$
$311$	−16.5526	−0.938613	−0.469307	+	0.883035i	$0.655496\pi$
$312$	0.347296	0.0196618
$313$	3.78787	0.214103	0.107051	−	0.994253i	$-0.465859\pi$
$313$	3.78787	0.214103	0.107051	+	0.994253i	$0.465859\pi$
$314$	20.2645	1.14359
$315$	1.87939	0.105891
$316$	6.08378	0.342239
$317$	21.5371	1.20965	0.604823	−	0.796360i	$-0.293244\pi$
$317$	21.5371	1.20965	0.604823	+	0.796360i	$0.293244\pi$
$318$	8.22668	0.461329
$319$	−15.1925	−0.850619
$320$	1.00000	0.0559017
$321$	8.90673	0.497125
$322$	−6.41147	−0.357297
$323$	0	0
$324$	1.00000	0.0555556
$325$	0.347296	0.0192645
$326$	9.10876	0.504487
$327$	−13.1925	−0.729549
$328$	0.630415	0.0348088
$329$	18.9659	1.04562
$330$	2.22668	0.122575
$331$	8.67324	0.476724	0.238362	−	0.971176i	$-0.423389\pi$
$331$	8.67324	0.476724	0.238362	+	0.971176i	$0.423389\pi$
$332$	8.36959	0.459341
$333$	−3.12836	−0.171433
$334$	−2.22668	−0.121839
$335$	−9.06418	−0.495229
$336$	−1.87939	−0.102529
$337$	26.3013	1.43272	0.716361	−	0.697730i	$-0.245806\pi$
$337$	26.3013	1.43272	0.716361	+	0.697730i	$0.245806\pi$
$338$	12.8794	0.700546
$339$	−18.8229	−1.02232
$340$	0	0
$341$	6.82295	0.369483
$342$	−0.758770	−0.0410296
$343$	−19.6732	−1.06225
$344$	0.694593	0.0374499
$345$	−3.41147	−0.183668
$346$	−19.0419	−1.02370
$347$	−3.34461	−0.179548	−0.0897740	−	0.995962i	$-0.528614\pi$
$347$	−3.34461	−0.179548	−0.0897740	+	0.995962i	$0.528614\pi$
$348$	6.82295	0.365748
$349$	−34.3851	−1.84059	−0.920295	−	0.391225i	$-0.872052\pi$
$349$	−34.3851	−1.84059	−0.920295	+	0.391225i	$0.872052\pi$
$350$	−1.87939	−0.100457
$351$	−0.347296	−0.0185373
$352$	−2.22668	−0.118683
$353$	28.0155	1.49111	0.745557	−	0.666442i	$-0.232184\pi$
$353$	28.0155	1.49111	0.745557	+	0.666442i	$0.232184\pi$
$354$	6.68004	0.355040
$355$	4.45336	0.236360
$356$	−1.18479	−0.0627939
$357$	0	0
$358$	14.4020	0.761168
$359$	7.64590	0.403535	0.201767	−	0.979433i	$-0.435331\pi$
$359$	7.64590	0.403535	0.201767	+	0.979433i	$0.435331\pi$
$360$	−1.00000	−0.0527046
$361$	−18.4243	−0.969698
$362$	−13.5175	−0.710466
$363$	6.04189	0.317117
$364$	0.652704	0.0342110
$365$	4.00000	0.209370
$366$	0.694593	0.0363069
$367$	−30.0847	−1.57041	−0.785205	−	0.619236i	$-0.787442\pi$
$367$	−30.0847	−1.57041	−0.785205	+	0.619236i	$0.787442\pi$
$368$	3.41147	0.177835
$369$	−0.630415	−0.0328181
$370$	3.12836	0.162635
$371$	15.4611	0.802701
$372$	−3.06418	−0.158870
$373$	5.80571	0.300608	0.150304	−	0.988640i	$-0.451975\pi$
$373$	5.80571	0.300608	0.150304	+	0.988640i	$0.451975\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−10.0915	−0.520430
$377$	−2.36959	−0.122040
$378$	1.87939	0.0966651
$379$	−1.96080	−0.100719	−0.0503597	−	0.998731i	$-0.516037\pi$
$379$	−1.96080	−0.100719	−0.0503597	+	0.998731i	$0.516037\pi$
$380$	0.758770	0.0389241
$381$	−11.4979	−0.589057
$382$	−12.0000	−0.613973
$383$	4.83244	0.246926	0.123463	−	0.992349i	$-0.460600\pi$
$383$	4.83244	0.246926	0.123463	+	0.992349i	$0.460600\pi$
$384$	1.00000	0.0510310
$385$	4.18479	0.213277
$386$	−12.7939	−0.651190
$387$	−0.694593	−0.0353081
$388$	−2.21213	−0.112304
$389$	−20.7547	−1.05230	−0.526151	−	0.850391i	$-0.676365\pi$
$389$	−20.7547	−1.05230	−0.526151	+	0.850391i	$0.676365\pi$
$390$	0.347296	0.0175860
$391$	0	0
$392$	3.46791	0.175156
$393$	3.41147	0.172086
$394$	−5.13341	−0.258617
$395$	6.08378	0.306108
$396$	2.22668	0.111895
$397$	25.5107	1.28035	0.640174	−	0.768230i	$-0.278862\pi$
$397$	25.5107	1.28035	0.640174	+	0.768230i	$0.278862\pi$
$398$	−7.91622	−0.396804
$399$	−1.42602	−0.0713904
$400$	1.00000	0.0500000
$401$	−9.87258	−0.493013	−0.246507	−	0.969141i	$-0.579283\pi$
$401$	−9.87258	−0.493013	−0.246507	+	0.969141i	$0.579283\pi$
$402$	−9.06418	−0.452080
$403$	1.06418	0.0530105
$404$	−15.1925	−0.755857
$405$	1.00000	0.0496904
$406$	12.8229	0.636392
$407$	−6.96585	−0.345284
$408$	0	0
$409$	16.6486	0.823220	0.411610	−	0.911360i	$-0.364967\pi$
$409$	16.6486	0.823220	0.411610	+	0.911360i	$0.364967\pi$
$410$	0.630415	0.0311340
$411$	−14.3696	−0.708799
$412$	0.758770	0.0373819
$413$	12.5544	0.617761
$414$	−3.41147	−0.167665
$415$	8.36959	0.410847
$416$	−0.347296	−0.0170276
$417$	0.758770	0.0371572
$418$	−1.68954	−0.0826381
$419$	−2.97359	−0.145270	−0.0726348	−	0.997359i	$-0.523141\pi$
$419$	−2.97359	−0.145270	−0.0726348	+	0.997359i	$0.523141\pi$
$420$	−1.87939	−0.0917046
$421$	−26.8776	−1.30994	−0.654968	−	0.755657i	$-0.727318\pi$
$421$	−26.8776	−1.30994	−0.654968	+	0.755657i	$0.727318\pi$
$422$	2.28817	0.111386
$423$	10.0915	0.490666
$424$	−8.22668	−0.399523
$425$	0	0
$426$	4.45336	0.215766
$427$	1.30541	0.0631731
$428$	−8.90673	−0.430523
$429$	−0.773318	−0.0373362
$430$	0.694593	0.0334962
$431$	−12.3851	−0.596568	−0.298284	−	0.954477i	$-0.596414\pi$
$431$	−12.3851	−0.596568	−0.298284	+	0.954477i	$0.596414\pi$
$432$	−1.00000	−0.0481125
$433$	−27.8871	−1.34017	−0.670085	−	0.742284i	$-0.733742\pi$
$433$	−27.8871	−1.34017	−0.670085	+	0.742284i	$0.733742\pi$
$434$	−5.75877	−0.276430
$435$	6.82295	0.327135
$436$	13.1925	0.631808
$437$	2.58853	0.123826
$438$	4.00000	0.191127
$439$	8.77837	0.418969	0.209484	−	0.977812i	$-0.432821\pi$
$439$	8.77837	0.418969	0.209484	+	0.977812i	$0.432821\pi$
$440$	−2.22668	−0.106153
$441$	−3.46791	−0.165139
$442$	0	0
$443$	38.3168	1.82048	0.910242	−	0.414076i	$-0.135895\pi$
$443$	38.3168	1.82048	0.910242	+	0.414076i	$0.135895\pi$
$444$	3.12836	0.148465
$445$	−1.18479	−0.0561646
$446$	−14.3182	−0.677986
$447$	−4.45336	−0.210637
$448$	1.87939	0.0887926
$449$	37.4593	1.76782	0.883908	−	0.467661i	$-0.154903\pi$
$449$	37.4593	1.76782	0.883908	+	0.467661i	$0.154903\pi$
$450$	−1.00000	−0.0471405
$451$	−1.40373	−0.0660992
$452$	18.8229	0.885357
$453$	0.157451	0.00739771
$454$	−13.6459	−0.640434
$455$	0.652704	0.0305992
$456$	0.758770	0.0355327
$457$	7.27631	0.340371	0.170186	−	0.985412i	$-0.445563\pi$
$457$	7.27631	0.340371	0.170186	+	0.985412i	$0.445563\pi$
$458$	−3.10876	−0.145263
$459$	0	0
$460$	3.41147	0.159061
$461$	15.0933	0.702964	0.351482	−	0.936195i	$-0.385678\pi$
$461$	15.0933	0.702964	0.351482	+	0.936195i	$0.385678\pi$
$462$	4.18479	0.194694
$463$	−26.0779	−1.21194	−0.605972	−	0.795486i	$-0.707215\pi$
$463$	−26.0779	−1.21194	−0.605972	+	0.795486i	$0.707215\pi$
$464$	−6.82295	−0.316747
$465$	−3.06418	−0.142098
$466$	9.47834	0.439076
$467$	32.7202	1.51411	0.757055	−	0.653352i	$-0.226638\pi$
$467$	32.7202	1.51411	0.757055	+	0.653352i	$0.226638\pi$
$468$	0.347296	0.0160538
$469$	−17.0351	−0.786607
$470$	−10.0915	−0.465487
$471$	20.2645	0.933737
$472$	−6.68004	−0.307474
$473$	−1.54664	−0.0711144
$474$	6.08378	0.279437
$475$	0.758770	0.0348148
$476$	0	0
$477$	8.22668	0.376674
$478$	−3.19253	−0.146023
$479$	28.4534	1.30007	0.650034	−	0.759905i	$-0.274755\pi$
$479$	28.4534	1.30007	0.650034	+	0.759905i	$0.274755\pi$
$480$	1.00000	0.0456435
$481$	−1.08647	−0.0495386
$482$	13.6382	0.621201
$483$	−6.41147	−0.291732
$484$	−6.04189	−0.274631
$485$	−2.21213	−0.100448
$486$	1.00000	0.0453609
$487$	6.31551	0.286183	0.143092	−	0.989709i	$-0.454296\pi$
$487$	6.31551	0.286183	0.143092	+	0.989709i	$0.454296\pi$
$488$	−0.694593	−0.0314427
$489$	9.10876	0.411912
$490$	3.46791	0.156664
$491$	−25.0327	−1.12971	−0.564855	−	0.825190i	$-0.691068\pi$
$491$	−25.0327	−1.12971	−0.564855	+	0.825190i	$0.691068\pi$
$492$	0.630415	0.0284213
$493$	0	0
$494$	−0.263518	−0.0118562
$495$	2.22668	0.100082
$496$	3.06418	0.137586
$497$	8.36959	0.375427
$498$	8.36959	0.375050
$499$	39.3732	1.76259	0.881293	−	0.472569i	$-0.156673\pi$
$499$	39.3732	1.76259	0.881293	+	0.472569i	$0.156673\pi$
$500$	1.00000	0.0447214
$501$	−2.22668	−0.0994808
$502$	29.5185	1.31747
$503$	−9.22493	−0.411319	−0.205660	−	0.978624i	$-0.565934\pi$
$503$	−9.22493	−0.411319	−0.205660	+	0.978624i	$0.565934\pi$
$504$	−1.87939	−0.0837145
$505$	−15.1925	−0.676059
$506$	−7.59627	−0.337695
$507$	12.8794	0.571994
$508$	11.4979	0.510139
$509$	25.8972	1.14787	0.573937	−	0.818899i	$-0.305415\pi$
$509$	25.8972	1.14787	0.573937	+	0.818899i	$0.305415\pi$
$510$	0	0
$511$	7.51754	0.332556
$512$	−1.00000	−0.0441942
$513$	−0.758770	−0.0335005
$514$	2.08378	0.0919115
$515$	0.758770	0.0334354
$516$	0.694593	0.0305777
$517$	22.4706	0.988256
$518$	5.87939	0.258325
$519$	−19.0419	−0.835846
$520$	−0.347296	−0.0152300
$521$	26.2763	1.15119	0.575593	−	0.817736i	$-0.304771\pi$
$521$	26.2763	1.15119	0.575593	+	0.817736i	$0.304771\pi$
$522$	6.82295	0.298632
$523$	−30.4688	−1.33231	−0.666155	−	0.745814i	$-0.732061\pi$
$523$	−30.4688	−1.33231	−0.666155	+	0.745814i	$0.732061\pi$
$524$	−3.41147	−0.149031
$525$	−1.87939	−0.0820231
$526$	17.0574	0.743737
$527$	0	0
$528$	−2.22668	−0.0969039
$529$	−11.3618	−0.493993
$530$	−8.22668	−0.357344
$531$	6.68004	0.289889
$532$	1.42602	0.0618259
$533$	−0.218941	−0.00948338
$534$	−1.18479	−0.0512710
$535$	−8.90673	−0.385071
$536$	9.06418	0.391513
$537$	14.4020	0.621491
$538$	−14.9067	−0.642675
$539$	−7.72193	−0.332607
$540$	−1.00000	−0.0430331
$541$	35.0215	1.50569	0.752845	−	0.658198i	$-0.228681\pi$
$541$	35.0215	1.50569	0.752845	+	0.658198i	$0.228681\pi$
$542$	3.78787	0.162703
$543$	−13.5175	−0.580093
$544$	0	0
$545$	13.1925	0.565106
$546$	0.652704	0.0279331
$547$	−17.0196	−0.727706	−0.363853	−	0.931456i	$-0.618539\pi$
$547$	−17.0196	−0.727706	−0.363853	+	0.931456i	$0.618539\pi$
$548$	14.3696	0.613838
$549$	0.694593	0.0296445
$550$	−2.22668	−0.0949460
$551$	−5.17705	−0.220550
$552$	3.41147	0.145202
$553$	11.4338	0.486213
$554$	8.81614	0.374562
$555$	3.12836	0.132791
$556$	−0.758770	−0.0321790
$557$	1.18479	0.0502013	0.0251006	−	0.999685i	$-0.492009\pi$
$557$	1.18479	0.0502013	0.0251006	+	0.999685i	$0.492009\pi$
$558$	−3.06418	−0.129717
$559$	−0.241230	−0.0102029
$560$	1.87939	0.0794185
$561$	0	0
$562$	7.94087	0.334966
$563$	42.6364	1.79691	0.898455	−	0.439066i	$-0.144690\pi$
$563$	42.6364	1.79691	0.898455	+	0.439066i	$0.144690\pi$
$564$	−10.0915	−0.424930
$565$	18.8229	0.791887
$566$	−11.4338	−0.480597
$567$	1.87939	0.0789268
$568$	−4.45336	−0.186859
$569$	25.0155	1.04870	0.524352	−	0.851502i	$-0.324308\pi$
$569$	25.0155	1.04870	0.524352	+	0.851502i	$0.324308\pi$
$570$	0.758770	0.0317814
$571$	−38.6641	−1.61804	−0.809020	−	0.587781i	$-0.800002\pi$
$571$	−38.6641	−1.61804	−0.809020	+	0.587781i	$0.800002\pi$
$572$	0.773318	0.0323341
$573$	−12.0000	−0.501307
$574$	1.18479	0.0494523
$575$	3.41147	0.142268
$576$	1.00000	0.0416667
$577$	16.1438	0.672077	0.336038	−	0.941848i	$-0.390913\pi$
$577$	16.1438	0.672077	0.336038	+	0.941848i	$0.390913\pi$
$578$	0	0
$579$	−12.7939	−0.531694
$580$	−6.82295	−0.283308
$581$	15.7297	0.652577
$582$	−2.21213	−0.0916959
$583$	18.3182	0.758662
$584$	−4.00000	−0.165521
$585$	0.347296	0.0143589
$586$	−2.12742	−0.0878830
$587$	−27.0060	−1.11466	−0.557328	−	0.830292i	$-0.688173\pi$
$587$	−27.0060	−1.11466	−0.557328	+	0.830292i	$0.688173\pi$
$588$	3.46791	0.143014
$589$	2.32501	0.0958003
$590$	−6.68004	−0.275013
$591$	−5.13341	−0.211160
$592$	−3.12836	−0.128575
$593$	−14.4688	−0.594164	−0.297082	−	0.954852i	$-0.596014\pi$
$593$	−14.4688	−0.594164	−0.297082	+	0.954852i	$0.596014\pi$
$594$	2.22668	0.0913619
$595$	0	0
$596$	4.45336	0.182417
$597$	−7.91622	−0.323989
$598$	−1.18479	−0.0484498
$599$	28.1147	1.14874	0.574369	−	0.818597i	$-0.305248\pi$
$599$	28.1147	1.14874	0.574369	+	0.818597i	$0.305248\pi$
$600$	1.00000	0.0408248
$601$	−21.2772	−0.867917	−0.433958	−	0.900933i	$-0.642884\pi$
$601$	−21.2772	−0.867917	−0.433958	+	0.900933i	$0.642884\pi$
$602$	1.30541	0.0532044
$603$	−9.06418	−0.369122
$604$	−0.157451	−0.00640661
$605$	−6.04189	−0.245638
$606$	−15.1925	−0.617154
$607$	20.0223	0.812680	0.406340	−	0.913722i	$-0.366805\pi$
$607$	20.0223	0.812680	0.406340	+	0.913722i	$0.366805\pi$
$608$	−0.758770	−0.0307722
$609$	12.8229	0.519612
$610$	−0.694593	−0.0281232
$611$	3.50475	0.141787
$612$	0	0
$613$	33.9685	1.37198	0.685988	−	0.727613i	$-0.259370\pi$
$613$	33.9685	1.37198	0.685988	+	0.727613i	$0.259370\pi$
$614$	28.3851	1.14553
$615$	0.630415	0.0254208
$616$	−4.18479	−0.168610
$617$	−1.64590	−0.0662613	−0.0331306	−	0.999451i	$-0.510548\pi$
$617$	−1.64590	−0.0662613	−0.0331306	+	0.999451i	$0.510548\pi$
$618$	0.758770	0.0305222
$619$	−8.74691	−0.351568	−0.175784	−	0.984429i	$-0.556246\pi$
$619$	−8.74691	−0.351568	−0.175784	+	0.984429i	$0.556246\pi$
$620$	3.06418	0.123060
$621$	−3.41147	−0.136898
$622$	16.5526	0.663700
$623$	−2.22668	−0.0892101
$624$	−0.347296	−0.0139030
$625$	1.00000	0.0400000
$626$	−3.78787	−0.151394
$627$	−1.68954	−0.0674737
$628$	−20.2645	−0.808640
$629$	0	0
$630$	−1.87939	−0.0748765
$631$	22.8675	0.910342	0.455171	−	0.890404i	$-0.349578\pi$
$631$	22.8675	0.910342	0.455171	+	0.890404i	$0.349578\pi$
$632$	−6.08378	−0.242000
$633$	2.28817	0.0909466
$634$	−21.5371	−0.855349
$635$	11.4979	0.456282
$636$	−8.22668	−0.326209
$637$	−1.20439	−0.0477198
$638$	15.1925	0.601478
$639$	4.45336	0.176172
$640$	−1.00000	−0.0395285
$641$	6.64765	0.262566	0.131283	−	0.991345i	$-0.458090\pi$
$641$	6.64765	0.262566	0.131283	+	0.991345i	$0.458090\pi$
$642$	−8.90673	−0.351520
$643$	−18.3797	−0.724824	−0.362412	−	0.932018i	$-0.618047\pi$
$643$	−18.3797	−0.724824	−0.362412	+	0.932018i	$0.618047\pi$
$644$	6.41147	0.252647
$645$	0.694593	0.0273496
$646$	0	0
$647$	23.4192	0.920704	0.460352	−	0.887736i	$-0.347723\pi$
$647$	23.4192	0.920704	0.460352	+	0.887736i	$0.347723\pi$
$648$	−1.00000	−0.0392837
$649$	14.8743	0.583869
$650$	−0.347296	−0.0136221
$651$	−5.75877	−0.225704
$652$	−9.10876	−0.356726
$653$	32.2935	1.26374	0.631872	−	0.775073i	$-0.282287\pi$
$653$	32.2935	1.26374	0.631872	+	0.775073i	$0.282287\pi$
$654$	13.1925	0.515869
$655$	−3.41147	−0.133297
$656$	−0.630415	−0.0246136
$657$	4.00000	0.156055
$658$	−18.9659	−0.739366
$659$	−14.3108	−0.557469	−0.278734	−	0.960368i	$-0.589915\pi$
$659$	−14.3108	−0.557469	−0.278734	+	0.960368i	$0.589915\pi$
$660$	−2.22668	−0.0866735
$661$	7.82707	0.304438	0.152219	−	0.988347i	$-0.451358\pi$
$661$	7.82707	0.304438	0.152219	+	0.988347i	$0.451358\pi$
$662$	−8.67324	−0.337095
$663$	0	0
$664$	−8.36959	−0.324803
$665$	1.42602	0.0552988
$666$	3.12836	0.121221
$667$	−23.2763	−0.901262
$668$	2.22668	0.0861529
$669$	−14.3182	−0.553574
$670$	9.06418	0.350180
$671$	1.54664	0.0597073
$672$	1.87939	0.0724989
$673$	16.3250	0.629283	0.314641	−	0.949211i	$-0.398116\pi$
$673$	16.3250	0.629283	0.314641	+	0.949211i	$0.398116\pi$
$674$	−26.3013	−1.01309
$675$	−1.00000	−0.0384900
$676$	−12.8794	−0.495361
$677$	−8.44562	−0.324592	−0.162296	−	0.986742i	$-0.551890\pi$
$677$	−8.44562	−0.324592	−0.162296	+	0.986742i	$0.551890\pi$
$678$	18.8229	0.722891
$679$	−4.15745	−0.159548
$680$	0	0
$681$	−13.6459	−0.522912
$682$	−6.82295	−0.261264
$683$	40.4005	1.54588	0.772942	−	0.634477i	$-0.218784\pi$
$683$	40.4005	1.54588	0.772942	+	0.634477i	$0.218784\pi$
$684$	0.758770	0.0290123
$685$	14.3696	0.549034
$686$	19.6732	0.751128
$687$	−3.10876	−0.118606
$688$	−0.694593	−0.0264811
$689$	2.85710	0.108847
$690$	3.41147	0.129873
$691$	−25.0182	−0.951736	−0.475868	−	0.879517i	$-0.657866\pi$
$691$	−25.0182	−0.951736	−0.475868	+	0.879517i	$0.657866\pi$
$692$	19.0419	0.723864
$693$	4.18479	0.158967
$694$	3.34461	0.126960
$695$	−0.758770	−0.0287818
$696$	−6.82295	−0.258623
$697$	0	0
$698$	34.3851	1.30149
$699$	9.47834	0.358504
$700$	1.87939	0.0710341
$701$	33.8634	1.27900	0.639502	−	0.768790i	$-0.279141\pi$
$701$	33.8634	1.27900	0.639502	+	0.768790i	$0.279141\pi$
$702$	0.347296	0.0131079
$703$	−2.37370	−0.0895259
$704$	2.22668	0.0839212
$705$	−10.0915	−0.380069
$706$	−28.0155	−1.05438
$707$	−28.5526	−1.07383
$708$	−6.68004	−0.251051
$709$	37.0797	1.39256	0.696278	−	0.717772i	$-0.254838\pi$
$709$	37.0797	1.39256	0.696278	+	0.717772i	$0.254838\pi$
$710$	−4.45336	−0.167132
$711$	6.08378	0.228160
$712$	1.18479	0.0444020
$713$	10.4534	0.391482
$714$	0	0
$715$	0.773318	0.0289205
$716$	−14.4020	−0.538227
$717$	−3.19253	−0.119227
$718$	−7.64590	−0.285342
$719$	−23.3756	−0.871762	−0.435881	−	0.900004i	$-0.643563\pi$
$719$	−23.3756	−0.871762	−0.435881	+	0.900004i	$0.643563\pi$
$720$	1.00000	0.0372678
$721$	1.42602	0.0531078
$722$	18.4243	0.685680
$723$	13.6382	0.507208
$724$	13.5175	0.502375
$725$	−6.82295	−0.253398
$726$	−6.04189	−0.224236
$727$	14.3318	0.531538	0.265769	−	0.964037i	$-0.414374\pi$
$727$	14.3318	0.531538	0.265769	+	0.964037i	$0.414374\pi$
$728$	−0.652704	−0.0241908
$729$	1.00000	0.0370370
$730$	−4.00000	−0.148047
$731$	0	0
$732$	−0.694593	−0.0256729
$733$	43.2959	1.59917	0.799585	−	0.600552i	$-0.205053\pi$
$733$	43.2959	1.59917	0.799585	+	0.600552i	$0.205053\pi$
$734$	30.0847	1.11045
$735$	3.46791	0.127916
$736$	−3.41147	−0.125749
$737$	−20.1830	−0.743452
$738$	0.630415	0.0232059
$739$	−43.7597	−1.60973	−0.804863	−	0.593460i	$-0.797761\pi$
$739$	−43.7597	−1.60973	−0.804863	+	0.593460i	$0.797761\pi$
$740$	−3.12836	−0.115001
$741$	−0.263518	−0.00968058
$742$	−15.4611	−0.567595
$743$	−8.61175	−0.315934	−0.157967	−	0.987444i	$-0.550494\pi$
$743$	−8.61175	−0.315934	−0.157967	+	0.987444i	$0.550494\pi$
$744$	3.06418	0.112338
$745$	4.45336	0.163159
$746$	−5.80571	−0.212562
$747$	8.36959	0.306227
$748$	0	0
$749$	−16.7392	−0.611636
$750$	1.00000	0.0365148
$751$	14.6263	0.533721	0.266861	−	0.963735i	$-0.414014\pi$
$751$	14.6263	0.533721	0.266861	+	0.963735i	$0.414014\pi$
$752$	10.0915	0.368000
$753$	29.5185	1.07571
$754$	2.36959	0.0862952
$755$	−0.157451	−0.00573024
$756$	−1.87939	−0.0683526
$757$	2.89218	0.105118	0.0525590	−	0.998618i	$-0.483262\pi$
$757$	2.89218	0.105118	0.0525590	+	0.998618i	$0.483262\pi$
$758$	1.96080	0.0712194
$759$	−7.59627	−0.275727
$760$	−0.758770	−0.0275235
$761$	6.07604	0.220256	0.110128	−	0.993917i	$-0.464874\pi$
$761$	6.07604	0.220256	0.110128	+	0.993917i	$0.464874\pi$
$762$	11.4979	0.416526
$763$	24.7939	0.897598
$764$	12.0000	0.434145
$765$	0	0
$766$	−4.83244	−0.174603
$767$	2.31996	0.0837687
$768$	−1.00000	−0.0360844
$769$	−20.9145	−0.754195	−0.377097	−	0.926174i	$-0.623078\pi$
$769$	−20.9145	−0.754195	−0.377097	+	0.926174i	$0.623078\pi$
$770$	−4.18479	−0.150809
$771$	2.08378	0.0750454
$772$	12.7939	0.460461
$773$	0.245348	0.00882457	0.00441228	−	0.999990i	$-0.498596\pi$
$773$	0.245348	0.00882457	0.00441228	+	0.999990i	$0.498596\pi$
$774$	0.694593	0.0249666
$775$	3.06418	0.110069
$776$	2.21213	0.0794110
$777$	5.87939	0.210922
$778$	20.7547	0.744090
$779$	−0.478340	−0.0171383
$780$	−0.347296	−0.0124352
$781$	9.91622	0.354830
$782$	0	0
$783$	6.82295	0.243832
$784$	−3.46791	−0.123854
$785$	−20.2645	−0.723269
$786$	−3.41147	−0.121683
$787$	7.66962	0.273392	0.136696	−	0.990613i	$-0.456352\pi$
$787$	7.66962	0.273392	0.136696	+	0.990613i	$0.456352\pi$
$788$	5.13341	0.182870
$789$	17.0574	0.607258
$790$	−6.08378	−0.216451
$791$	35.3756	1.25781
$792$	−2.22668	−0.0791217
$793$	0.241230	0.00856631
$794$	−25.5107	−0.905342
$795$	−8.22668	−0.291770
$796$	7.91622	0.280583
$797$	−46.2900	−1.63968	−0.819839	−	0.572595i	$-0.805937\pi$
$797$	−46.2900	−1.63968	−0.819839	+	0.572595i	$0.805937\pi$
$798$	1.42602	0.0504806
$799$	0	0
$800$	−1.00000	−0.0353553
$801$	−1.18479	−0.0418626
$802$	9.87258	0.348613
$803$	8.90673	0.314312
$804$	9.06418	0.319669
$805$	6.41147	0.225975
$806$	−1.06418	−0.0374841
$807$	−14.9067	−0.524742
$808$	15.1925	0.534471
$809$	20.8479	0.732974	0.366487	−	0.930423i	$-0.380560\pi$
$809$	20.8479	0.732974	0.366487	+	0.930423i	$0.380560\pi$
$810$	−1.00000	−0.0351364
$811$	46.4820	1.63220	0.816101	−	0.577909i	$-0.196131\pi$
$811$	46.4820	1.63220	0.816101	+	0.577909i	$0.196131\pi$
$812$	−12.8229	−0.449997
$813$	3.78787	0.132846
$814$	6.96585	0.244153
$815$	−9.10876	−0.319066
$816$	0	0
$817$	−0.527036	−0.0184387
$818$	−16.6486	−0.582104
$819$	0.652704	0.0228073
$820$	−0.630415	−0.0220150
$821$	−25.6459	−0.895048	−0.447524	−	0.894272i	$-0.647694\pi$
$821$	−25.6459	−0.895048	−0.447524	+	0.894272i	$0.647694\pi$
$822$	14.3696	0.501197
$823$	49.3310	1.71957	0.859785	−	0.510656i	$-0.170597\pi$
$823$	49.3310	1.71957	0.859785	+	0.510656i	$0.170597\pi$
$824$	−0.758770	−0.0264330
$825$	−2.22668	−0.0775231
$826$	−12.5544	−0.436823
$827$	25.0743	0.871918	0.435959	−	0.899966i	$-0.356409\pi$
$827$	25.0743	0.871918	0.435959	+	0.899966i	$0.356409\pi$
$828$	3.41147	0.118557
$829$	6.41416	0.222773	0.111387	−	0.993777i	$-0.464471\pi$
$829$	6.41416	0.222773	0.111387	+	0.993777i	$0.464471\pi$
$830$	−8.36959	−0.290513
$831$	8.81614	0.305829
$832$	0.347296	0.0120403
$833$	0	0
$834$	−0.758770	−0.0262741
$835$	2.22668	0.0770575
$836$	1.68954	0.0584340
$837$	−3.06418	−0.105913
$838$	2.97359	0.102721
$839$	19.5466	0.674825	0.337412	−	0.941357i	$-0.390448\pi$
$839$	19.5466	0.674825	0.337412	+	0.941357i	$0.390448\pi$
$840$	1.87939	0.0648450
$841$	17.5526	0.605263
$842$	26.8776	0.926264
$843$	7.94087	0.273498
$844$	−2.28817	−0.0787621
$845$	−12.8794	−0.443064
$846$	−10.0915	−0.346954
$847$	−11.3550	−0.390164
$848$	8.22668	0.282505
$849$	−11.4338	−0.392406
$850$	0	0
$851$	−10.6723	−0.365842
$852$	−4.45336	−0.152570
$853$	−18.5571	−0.635382	−0.317691	−	0.948194i	$-0.602907\pi$
$853$	−18.5571	−0.635382	−0.317691	+	0.948194i	$0.602907\pi$
$854$	−1.30541	−0.0446701
$855$	0.758770	0.0259494
$856$	8.90673	0.304426
$857$	14.6554	0.500619	0.250309	−	0.968166i	$-0.419468\pi$
$857$	14.6554	0.500619	0.250309	+	0.968166i	$0.419468\pi$
$858$	0.773318	0.0264007
$859$	−6.70058	−0.228621	−0.114310	−	0.993445i	$-0.536466\pi$
$859$	−6.70058	−0.228621	−0.114310	+	0.993445i	$0.536466\pi$
$860$	−0.694593	−0.0236854
$861$	1.18479	0.0403776
$862$	12.3851	0.421837
$863$	−3.58677	−0.122095	−0.0610476	−	0.998135i	$-0.519444\pi$
$863$	−3.58677	−0.122095	−0.0610476	+	0.998135i	$0.519444\pi$
$864$	1.00000	0.0340207
$865$	19.0419	0.647444
$866$	27.8871	0.947643
$867$	0	0
$868$	5.75877	0.195465
$869$	13.5466	0.459538
$870$	−6.82295	−0.231320
$871$	−3.14796	−0.106664
$872$	−13.1925	−0.446756
$873$	−2.21213	−0.0748694
$874$	−2.58853	−0.0875582
$875$	1.87939	0.0635348
$876$	−4.00000	−0.135147
$877$	−36.1908	−1.22208	−0.611038	−	0.791601i	$-0.709248\pi$
$877$	−36.1908	−1.22208	−0.611038	+	0.791601i	$0.709248\pi$
$878$	−8.77837	−0.296256
$879$	−2.12742	−0.0717562
$880$	2.22668	0.0750614
$881$	24.2837	0.818140	0.409070	−	0.912503i	$-0.365853\pi$
$881$	24.2837	0.818140	0.409070	+	0.912503i	$0.365853\pi$
$882$	3.46791	0.116771
$883$	43.5276	1.46482	0.732411	−	0.680863i	$-0.238395\pi$
$883$	43.5276	1.46482	0.732411	+	0.680863i	$0.238395\pi$
$884$	0	0
$885$	−6.68004	−0.224547
$886$	−38.3168	−1.28728
$887$	−16.6459	−0.558915	−0.279457	−	0.960158i	$-0.590155\pi$
$887$	−16.6459	−0.558915	−0.279457	+	0.960158i	$0.590155\pi$
$888$	−3.12836	−0.104981
$889$	21.6091	0.724745
$890$	1.18479	0.0397143
$891$	2.22668	0.0745966
$892$	14.3182	0.479409
$893$	7.65715	0.256237
$894$	4.45336	0.148943
$895$	−14.4020	−0.481405
$896$	−1.87939	−0.0627859
$897$	−1.18479	−0.0395591
$898$	−37.4593	−1.25003
$899$	−20.9067	−0.697278
$900$	1.00000	0.0333333
$901$	0	0
$902$	1.40373	0.0467392
$903$	1.30541	0.0434412
$904$	−18.8229	−0.626042
$905$	13.5175	0.449338
$906$	−0.157451	−0.00523097
$907$	25.0797	0.832756	0.416378	−	0.909192i	$-0.363299\pi$
$907$	25.0797	0.832756	0.416378	+	0.909192i	$0.363299\pi$
$908$	13.6459	0.452855
$909$	−15.1925	−0.503905
$910$	−0.652704	−0.0216369
$911$	−41.4748	−1.37412	−0.687061	−	0.726600i	$-0.741100\pi$
$911$	−41.4748	−1.37412	−0.687061	+	0.726600i	$0.741100\pi$
$912$	−0.758770	−0.0251254
$913$	18.6364	0.616775
$914$	−7.27631	−0.240679
$915$	−0.694593	−0.0229625
$916$	3.10876	0.102716
$917$	−6.41147	−0.211726
$918$	0	0
$919$	−50.8040	−1.67587	−0.837934	−	0.545772i	$-0.816237\pi$
$919$	−50.8040	−1.67587	−0.837934	+	0.545772i	$0.816237\pi$
$920$	−3.41147	−0.112473
$921$	28.3851	0.935320
$922$	−15.0933	−0.497070
$923$	1.54664	0.0509082
$924$	−4.18479	−0.137670
$925$	−3.12836	−0.102860
$926$	26.0779	0.856973
$927$	0.758770	0.0249213
$928$	6.82295	0.223974
$929$	−13.4593	−0.441587	−0.220793	−	0.975321i	$-0.570865\pi$
$929$	−13.4593	−0.441587	−0.220793	+	0.975321i	$0.570865\pi$
$930$	3.06418	0.100478
$931$	−2.63135	−0.0862390
$932$	−9.47834	−0.310473
$933$	16.5526	0.541909
$934$	−32.7202	−1.07064
$935$	0	0
$936$	−0.347296	−0.0113517
$937$	−20.5270	−0.670589	−0.335295	−	0.942113i	$-0.608836\pi$
$937$	−20.5270	−0.670589	−0.335295	+	0.942113i	$0.608836\pi$
$938$	17.0351	0.556215
$939$	−3.78787	−0.123612
$940$	10.0915	0.329149
$941$	−28.9377	−0.943342	−0.471671	−	0.881775i	$-0.656349\pi$
$941$	−28.9377	−0.943342	−0.471671	+	0.881775i	$0.656349\pi$
$942$	−20.2645	−0.660252
$943$	−2.15064	−0.0700346
$944$	6.68004	0.217417
$945$	−1.87939	−0.0611364
$946$	1.54664	0.0502855
$947$	42.8229	1.39156	0.695779	−	0.718256i	$-0.255059\pi$
$947$	42.8229	1.39156	0.695779	+	0.718256i	$0.255059\pi$
$948$	−6.08378	−0.197592
$949$	1.38919	0.0450949
$950$	−0.758770	−0.0246178
$951$	−21.5371	−0.698390
$952$	0	0
$953$	11.7142	0.379460	0.189730	−	0.981836i	$-0.439239\pi$
$953$	11.7142	0.379460	0.189730	+	0.981836i	$0.439239\pi$
$954$	−8.22668	−0.266349
$955$	12.0000	0.388311
$956$	3.19253	0.103254
$957$	15.1925	0.491105
$958$	−28.4534	−0.919286
$959$	27.0060	0.872069
$960$	−1.00000	−0.0322749
$961$	−21.6108	−0.697123
$962$	1.08647	0.0350291
$963$	−8.90673	−0.287015
$964$	−13.6382	−0.439255
$965$	12.7939	0.411849
$966$	6.41147	0.206286
$967$	24.4938	0.787668	0.393834	−	0.919182i	$-0.371148\pi$
$967$	24.4938	0.787668	0.393834	+	0.919182i	$0.371148\pi$
$968$	6.04189	0.194194
$969$	0	0
$970$	2.21213	0.0710273
$971$	−41.9067	−1.34485	−0.672425	−	0.740165i	$-0.734747\pi$
$971$	−41.9067	−1.34485	−0.672425	+	0.740165i	$0.734747\pi$
$972$	−1.00000	−0.0320750
$973$	−1.42602	−0.0457162
$974$	−6.31551	−0.202362
$975$	−0.347296	−0.0111224
$976$	0.694593	0.0222334
$977$	31.3601	1.00330	0.501649	−	0.865071i	$-0.332727\pi$
$977$	31.3601	1.00330	0.501649	+	0.865071i	$0.332727\pi$
$978$	−9.10876	−0.291266
$979$	−2.63816	−0.0843158
$980$	−3.46791	−0.110778
$981$	13.1925	0.421205
$982$	25.0327	0.798826
$983$	−51.4903	−1.64229	−0.821143	−	0.570723i	$-0.806663\pi$
$983$	−51.4903	−1.64229	−0.821143	+	0.570723i	$0.806663\pi$
$984$	−0.630415	−0.0200969
$985$	5.13341	0.163564
$986$	0	0
$987$	−18.9659	−0.603690
$988$	0.263518	0.00838363
$989$	−2.36959	−0.0753484
$990$	−2.22668	−0.0707686
$991$	5.36009	0.170269	0.0851344	−	0.996369i	$-0.472868\pi$
$991$	5.36009	0.170269	0.0851344	+	0.996369i	$0.472868\pi$
$992$	−3.06418	−0.0972877
$993$	−8.67324	−0.275237
$994$	−8.36959	−0.265467
$995$	7.91622	0.250961
$996$	−8.36959	−0.265200
$997$	4.28136	0.135592	0.0677961	−	0.997699i	$-0.478403\pi$
$997$	4.28136	0.135592	0.0677961	+	0.997699i	$0.478403\pi$
$998$	−39.3732	−1.24634
$999$	3.12836	0.0989768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8670.2.a.bm.1.3	✓	3
17.16	even	2		8670.2.a.bn.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8670.2.a.bm.1.3	✓	3	1.1	even	1	trivial
8670.2.a.bn.1.1	yes	3	17.16	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 8670 = 2 \cdot 3 \cdot 5 \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8670.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} +1.87939 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.22668 q^{11} -1.00000 q^{12} +0.347296 q^{13} -1.87939 q^{14} -1.00000 q^{15} +1.00000 q^{16} -1.00000 q^{18} +0.758770 q^{19} +1.00000 q^{20} -1.87939 q^{21} -2.22668 q^{22} +3.41147 q^{23} +1.00000 q^{24} +1.00000 q^{25} -0.347296 q^{26} -1.00000 q^{27} +1.87939 q^{28} -6.82295 q^{29} +1.00000 q^{30} +3.06418 q^{31} -1.00000 q^{32} -2.22668 q^{33} +1.87939 q^{35} +1.00000 q^{36} -3.12836 q^{37} -0.758770 q^{38} -0.347296 q^{39} -1.00000 q^{40} -0.630415 q^{41} +1.87939 q^{42} -0.694593 q^{43} +2.22668 q^{44} +1.00000 q^{45} -3.41147 q^{46} +10.0915 q^{47} -1.00000 q^{48} -3.46791 q^{49} -1.00000 q^{50} +0.347296 q^{52} +8.22668 q^{53} +1.00000 q^{54} +2.22668 q^{55} -1.87939 q^{56} -0.758770 q^{57} +6.82295 q^{58} +6.68004 q^{59} -1.00000 q^{60} +0.694593 q^{61} -3.06418 q^{62} +1.87939 q^{63} +1.00000 q^{64} +0.347296 q^{65} +2.22668 q^{66} -9.06418 q^{67} -3.41147 q^{69} -1.87939 q^{70} +4.45336 q^{71} -1.00000 q^{72} +4.00000 q^{73} +3.12836 q^{74} -1.00000 q^{75} +0.758770 q^{76} +4.18479 q^{77} +0.347296 q^{78} +6.08378 q^{79} +1.00000 q^{80} +1.00000 q^{81} +0.630415 q^{82} +8.36959 q^{83} -1.87939 q^{84} +0.694593 q^{86} +6.82295 q^{87} -2.22668 q^{88} -1.18479 q^{89} -1.00000 q^{90} +0.652704 q^{91} +3.41147 q^{92} -3.06418 q^{93} -10.0915 q^{94} +0.758770 q^{95} +1.00000 q^{96} -2.21213 q^{97} +3.46791 q^{98} +2.22668 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} + 3 q^{6} - 3 q^{8} + 3 q^{9} - 3 q^{10} - 3 q^{12} - 3 q^{15} + 3 q^{16} - 3 q^{18} - 9 q^{19} + 3 q^{20} + 3 q^{24} + 3 q^{25} - 3 q^{27} + 3 q^{30} - 3 q^{32}+ \cdots + 15 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 3 * q^4 + 3 * q^5 + 3 * q^6 - 3 * q^8 + 3 * q^9 - 3 * q^10 - 3 * q^12 - 3 * q^15 + 3 * q^16 - 3 * q^18 - 9 * q^19 + 3 * q^20 + 3 * q^24 + 3 * q^25 - 3 * q^27 + 3 * q^30 - 3 * q^32 + 3 * q^36 + 9 * q^37 + 9 * q^38 - 3 * q^40 - 9 * q^41 + 3 * q^45 - 3 * q^48 - 15 * q^49 - 3 * q^50 + 18 * q^53 + 3 * q^54 + 9 * q^57 - 3 * q^60 + 3 * q^64 - 18 * q^67 - 3 * q^72 + 12 * q^73 - 9 * q^74 - 3 * q^75 - 9 * q^76 + 9 * q^77 + 12 * q^79 + 3 * q^80 + 3 * q^81 + 9 * q^82 + 18 * q^83 - 3 * q^90 + 3 * q^91 - 9 * q^95 + 3 * q^96 + 18 * q^97 + 15 * q^98

Introduction

L-functions

Modular forms

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Database

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