LMFDB - Embedded newform 165.2.a.c.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [165,2,Mod(1,165)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(165, 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("165.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 165 = 3 \cdot 5 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	165.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:	$1.31753163335$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.48119$ of defining polynomial
Character	$\chi$	$=$	165.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} +1.00000 q^{5} -0.193937 q^{6} +3.35026 q^{7} +0.768452 q^{8} +1.00000 q^{9} -0.193937 q^{10} +1.00000 q^{11} -1.96239 q^{12} +2.96239 q^{13} -0.649738 q^{14} +1.00000 q^{15} +3.77575 q^{16} -4.57452 q^{17} -0.193937 q^{18} -4.31265 q^{19} -1.96239 q^{20} +3.35026 q^{21} -0.193937 q^{22} -6.70052 q^{23} +0.768452 q^{24} +1.00000 q^{25} -0.574515 q^{26} +1.00000 q^{27} -6.57452 q^{28} -3.61213 q^{29} -0.193937 q^{30} +9.92478 q^{31} -2.26916 q^{32} +1.00000 q^{33} +0.887166 q^{34} +3.35026 q^{35} -1.96239 q^{36} -2.00000 q^{37} +0.836381 q^{38} +2.96239 q^{39} +0.768452 q^{40} -4.38787 q^{41} -0.649738 q^{42} -9.27504 q^{43} -1.96239 q^{44} +1.00000 q^{45} +1.29948 q^{46} -9.92478 q^{47} +3.77575 q^{48} +4.22425 q^{49} -0.193937 q^{50} -4.57452 q^{51} -5.81336 q^{52} +4.70052 q^{53} -0.193937 q^{54} +1.00000 q^{55} +2.57452 q^{56} -4.31265 q^{57} +0.700523 q^{58} +10.7005 q^{59} -1.96239 q^{60} -8.70052 q^{61} -1.92478 q^{62} +3.35026 q^{63} -7.11142 q^{64} +2.96239 q^{65} -0.193937 q^{66} +5.92478 q^{67} +8.97698 q^{68} -6.70052 q^{69} -0.649738 q^{70} +9.92478 q^{71} +0.768452 q^{72} -7.73813 q^{73} +0.387873 q^{74} +1.00000 q^{75} +8.46310 q^{76} +3.35026 q^{77} -0.574515 q^{78} +11.5369 q^{79} +3.77575 q^{80} +1.00000 q^{81} +0.850969 q^{82} +10.8872 q^{83} -6.57452 q^{84} -4.57452 q^{85} +1.79877 q^{86} -3.61213 q^{87} +0.768452 q^{88} -2.77575 q^{89} -0.193937 q^{90} +9.92478 q^{91} +13.1490 q^{92} +9.92478 q^{93} +1.92478 q^{94} -4.31265 q^{95} -2.26916 q^{96} +0.0752228 q^{97} -0.819237 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 9 q^{8} + 3 q^{9} - q^{10} + 3 q^{11} + 5 q^{12} - 2 q^{13} - 12 q^{14} + 3 q^{15} + 13 q^{16} - 2 q^{17} - q^{18} + 8 q^{19} + 5 q^{20} - q^{22}+ \cdots + 3 q^{99}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 - 9 * q^8 + 3 * q^9 - q^10 + 3 * q^11 + 5 * q^12 - 2 * q^13 - 12 * q^14 + 3 * q^15 + 13 * q^16 - 2 * q^17 - q^18 + 8 * q^19 + 5 * q^20 - q^22 - 9 * q^24 + 3 * q^25 + 10 * q^26 + 3 * q^27 - 8 * q^28 - 10 * q^29 - q^30 + 8 * q^31 - 29 * q^32 + 3 * q^33 - 30 * q^34 + 5 * q^36 - 6 * q^37 - 2 * q^39 - 9 * q^40 - 14 * q^41 - 12 * q^42 + 4 * q^43 + 5 * q^44 + 3 * q^45 + 24 * q^46 - 8 * q^47 + 13 * q^48 + 11 * q^49 - q^50 - 2 * q^51 - 30 * q^52 - 6 * q^53 - q^54 + 3 * q^55 - 4 * q^56 + 8 * q^57 - 18 * q^58 + 12 * q^59 + 5 * q^60 - 6 * q^61 + 16 * q^62 + 13 * q^64 - 2 * q^65 - q^66 - 4 * q^67 + 42 * q^68 - 12 * q^70 + 8 * q^71 - 9 * q^72 - 14 * q^73 + 2 * q^74 + 3 * q^75 + 48 * q^76 + 10 * q^78 + 12 * q^79 + 13 * q^80 + 3 * q^81 + 26 * q^82 - 8 * q^84 - 2 * q^85 - 8 * q^86 - 10 * q^87 - 9 * q^88 - 10 * q^89 - q^90 + 8 * q^91 + 16 * q^92 + 8 * q^93 - 16 * q^94 + 8 * q^95 - 29 * q^96 + 22 * q^97 + 39 * q^98 + 3 * 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 $

$2$	−0.193937	−0.137134	−0.0685669	−	0.997647i	$-0.521843\pi$
$2$	−0.193937	−0.137134	−0.0685669	+	0.997647i	$0.521843\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.96239	−0.981194
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.193937	−0.0791743
$7$	3.35026	1.26628	0.633140	−	0.774037i	$-0.281766\pi$
$7$	3.35026	1.26628	0.633140	+	0.774037i	$0.281766\pi$
$8$	0.768452	0.271689
$9$	1.00000	0.333333
$10$	−0.193937	−0.0613281
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	−1.96239	−0.566493
$13$	2.96239	0.821619	0.410809	−	0.911721i	$-0.365246\pi$
$13$	2.96239	0.821619	0.410809	+	0.911721i	$0.365246\pi$
$14$	−0.649738	−0.173650
$15$	1.00000	0.258199
$16$	3.77575	0.943937
$17$	−4.57452	−1.10948	−0.554741	−	0.832023i	$-0.687183\pi$
$17$	−4.57452	−1.10948	−0.554741	+	0.832023i	$0.687183\pi$
$18$	−0.193937	−0.0457113
$19$	−4.31265	−0.989390	−0.494695	−	0.869067i	$-0.664720\pi$
$19$	−4.31265	−0.989390	−0.494695	+	0.869067i	$0.664720\pi$
$20$	−1.96239	−0.438803
$21$	3.35026	0.731087
$22$	−0.193937	−0.0413474
$23$	−6.70052	−1.39716	−0.698578	−	0.715534i	$-0.746183\pi$
$23$	−6.70052	−1.39716	−0.698578	+	0.715534i	$0.746183\pi$
$24$	0.768452	0.156860
$25$	1.00000	0.200000
$26$	−0.574515	−0.112672
$27$	1.00000	0.192450
$28$	−6.57452	−1.24247
$29$	−3.61213	−0.670755	−0.335378	−	0.942084i	$-0.608864\pi$
$29$	−3.61213	−0.670755	−0.335378	+	0.942084i	$0.608864\pi$
$30$	−0.193937	−0.0354078
$31$	9.92478	1.78254	0.891271	−	0.453470i	$-0.149814\pi$
$31$	9.92478	1.78254	0.891271	+	0.453470i	$0.149814\pi$
$32$	−2.26916	−0.401134
$33$	1.00000	0.174078
$34$	0.887166	0.152148
$35$	3.35026	0.566298
$36$	−1.96239	−0.327065
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0.836381	0.135679
$39$	2.96239	0.474362
$40$	0.768452	0.121503
$41$	−4.38787	−0.685271	−0.342635	−	0.939468i	$-0.611320\pi$
$41$	−4.38787	−0.685271	−0.342635	+	0.939468i	$0.611320\pi$
$42$	−0.649738	−0.100257
$43$	−9.27504	−1.41443	−0.707215	−	0.706998i	$-0.750049\pi$
$43$	−9.27504	−1.41443	−0.707215	+	0.706998i	$0.750049\pi$
$44$	−1.96239	−0.295841
$45$	1.00000	0.149071
$46$	1.29948	0.191597
$47$	−9.92478	−1.44768	−0.723839	−	0.689969i	$-0.757624\pi$
$47$	−9.92478	−1.44768	−0.723839	+	0.689969i	$0.757624\pi$
$48$	3.77575	0.544982
$49$	4.22425	0.603465
$50$	−0.193937	−0.0274268
$51$	−4.57452	−0.640560
$52$	−5.81336	−0.806168
$53$	4.70052	0.645667	0.322833	−	0.946456i	$-0.395365\pi$
$53$	4.70052	0.645667	0.322833	+	0.946456i	$0.395365\pi$
$54$	−0.193937	−0.0263914
$55$	1.00000	0.134840
$56$	2.57452	0.344034
$57$	−4.31265	−0.571224
$58$	0.700523	0.0919832
$59$	10.7005	1.39309	0.696545	−	0.717513i	$-0.254720\pi$
$59$	10.7005	1.39309	0.696545	+	0.717513i	$0.254720\pi$
$60$	−1.96239	−0.253343
$61$	−8.70052	−1.11399	−0.556994	−	0.830517i	$-0.688045\pi$
$61$	−8.70052	−1.11399	−0.556994	+	0.830517i	$0.688045\pi$
$62$	−1.92478	−0.244447
$63$	3.35026	0.422093
$64$	−7.11142	−0.888927
$65$	2.96239	0.367439
$66$	−0.193937	−0.0238719
$67$	5.92478	0.723827	0.361913	−	0.932212i	$-0.382124\pi$
$67$	5.92478	0.723827	0.361913	+	0.932212i	$0.382124\pi$
$68$	8.97698	1.08862
$69$	−6.70052	−0.806648
$70$	−0.649738	−0.0776586
$71$	9.92478	1.17785	0.588927	−	0.808186i	$-0.299550\pi$
$71$	9.92478	1.17785	0.588927	+	0.808186i	$0.299550\pi$
$72$	0.768452	0.0905629
$73$	−7.73813	−0.905680	−0.452840	−	0.891592i	$-0.649589\pi$
$73$	−7.73813	−0.905680	−0.452840	+	0.891592i	$0.649589\pi$
$74$	0.387873	0.0450893
$75$	1.00000	0.115470
$76$	8.46310	0.970784
$77$	3.35026	0.381798
$78$	−0.574515	−0.0650511
$79$	11.5369	1.29800	0.649002	−	0.760787i	$-0.275187\pi$
$79$	11.5369	1.29800	0.649002	+	0.760787i	$0.275187\pi$
$80$	3.77575	0.422141
$81$	1.00000	0.111111
$82$	0.850969	0.0939738
$83$	10.8872	1.19502	0.597511	−	0.801861i	$-0.296156\pi$
$83$	10.8872	1.19502	0.597511	+	0.801861i	$0.296156\pi$
$84$	−6.57452	−0.717338
$85$	−4.57452	−0.496176
$86$	1.79877	0.193966
$87$	−3.61213	−0.387261
$88$	0.768452	0.0819173
$89$	−2.77575	−0.294229	−0.147114	−	0.989120i	$-0.546999\pi$
$89$	−2.77575	−0.294229	−0.147114	+	0.989120i	$0.546999\pi$
$90$	−0.193937	−0.0204427
$91$	9.92478	1.04040
$92$	13.1490	1.37088
$93$	9.92478	1.02915
$94$	1.92478	0.198526
$95$	−4.31265	−0.442469
$96$	−2.26916	−0.231595
$97$	0.0752228	0.00763772	0.00381886	−	0.999993i	$-0.498784\pi$
$97$	0.0752228	0.00763772	0.00381886	+	0.999993i	$0.498784\pi$
$98$	−0.819237	−0.0827555
$99$	1.00000	0.100504
$100$	−1.96239	−0.196239
$101$	−15.0884	−1.50135	−0.750676	−	0.660671i	$-0.770272\pi$
$101$	−15.0884	−1.50135	−0.750676	+	0.660671i	$0.770272\pi$
$102$	0.887166	0.0878425
$103$	−3.22425	−0.317695	−0.158848	−	0.987303i	$-0.550778\pi$
$103$	−3.22425	−0.317695	−0.158848	+	0.987303i	$0.550778\pi$
$104$	2.27645	0.223225
$105$	3.35026	0.326952
$106$	−0.911603	−0.0885427
$107$	−0.962389	−0.0930376	−0.0465188	−	0.998917i	$-0.514813\pi$
$107$	−0.962389	−0.0930376	−0.0465188	+	0.998917i	$0.514813\pi$
$108$	−1.96239	−0.188831
$109$	11.4010	1.09202	0.546011	−	0.837778i	$-0.316146\pi$
$109$	11.4010	1.09202	0.546011	+	0.837778i	$0.316146\pi$
$110$	−0.193937	−0.0184911
$111$	−2.00000	−0.189832
$112$	12.6497	1.19529
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0.836381	0.0783342
$115$	−6.70052	−0.624827
$116$	7.08840	0.658141
$117$	2.96239	0.273873
$118$	−2.07522	−0.191040
$119$	−15.3258	−1.40492
$120$	0.768452	0.0701498
$121$	1.00000	0.0909091
$122$	1.68735	0.152765
$123$	−4.38787	−0.395641
$124$	−19.4763	−1.74902
$125$	1.00000	0.0894427
$126$	−0.649738	−0.0578833
$127$	−14.5745	−1.29328	−0.646640	−	0.762796i	$-0.723826\pi$
$127$	−14.5745	−1.29328	−0.646640	+	0.762796i	$0.723826\pi$
$128$	5.91748	0.523037
$129$	−9.27504	−0.816622
$130$	−0.574515	−0.0503883
$131$	−5.92478	−0.517650	−0.258825	−	0.965924i	$-0.583335\pi$
$131$	−5.92478	−0.517650	−0.258825	+	0.965924i	$0.583335\pi$
$132$	−1.96239	−0.170804
$133$	−14.4485	−1.25284
$134$	−1.14903	−0.0992612
$135$	1.00000	0.0860663
$136$	−3.51530	−0.301434
$137$	13.8496	1.18325	0.591624	−	0.806214i	$-0.298487\pi$
$137$	13.8496	1.18325	0.591624	+	0.806214i	$0.298487\pi$
$138$	1.29948	0.110619
$139$	13.6121	1.15457	0.577283	−	0.816544i	$-0.304113\pi$
$139$	13.6121	1.15457	0.577283	+	0.816544i	$0.304113\pi$
$140$	−6.57452	−0.555648
$141$	−9.92478	−0.835817
$142$	−1.92478	−0.161524
$143$	2.96239	0.247727
$144$	3.77575	0.314646
$145$	−3.61213	−0.299971
$146$	1.50071	0.124199
$147$	4.22425	0.348411
$148$	3.92478	0.322615
$149$	1.53690	0.125908	0.0629540	−	0.998016i	$-0.479948\pi$
$149$	1.53690	0.125908	0.0629540	+	0.998016i	$0.479948\pi$
$150$	−0.193937	−0.0158349
$151$	−6.76116	−0.550215	−0.275108	−	0.961413i	$-0.588713\pi$
$151$	−6.76116	−0.550215	−0.275108	+	0.961413i	$0.588713\pi$
$152$	−3.31406	−0.268806
$153$	−4.57452	−0.369828
$154$	−0.649738	−0.0523574
$155$	9.92478	0.797177
$156$	−5.81336	−0.465441
$157$	−5.47627	−0.437054	−0.218527	−	0.975831i	$-0.570125\pi$
$157$	−5.47627	−0.437054	−0.218527	+	0.975831i	$0.570125\pi$
$158$	−2.23743	−0.178000
$159$	4.70052	0.372776
$160$	−2.26916	−0.179393
$161$	−22.4485	−1.76919
$162$	−0.193937	−0.0152371
$163$	12.6253	0.988890	0.494445	−	0.869209i	$-0.335371\pi$
$163$	12.6253	0.988890	0.494445	+	0.869209i	$0.335371\pi$
$164$	8.61071	0.672384
$165$	1.00000	0.0778499
$166$	−2.11142	−0.163878
$167$	18.3634	1.42101	0.710503	−	0.703695i	$-0.248468\pi$
$167$	18.3634	1.42101	0.710503	+	0.703695i	$0.248468\pi$
$168$	2.57452	0.198628
$169$	−4.22425	−0.324943
$170$	0.887166	0.0680425
$171$	−4.31265	−0.329797
$172$	18.2012	1.38783
$173$	−8.57452	−0.651908	−0.325954	−	0.945386i	$-0.605686\pi$
$173$	−8.57452	−0.651908	−0.325954	+	0.945386i	$0.605686\pi$
$174$	0.700523	0.0531065
$175$	3.35026	0.253256
$176$	3.77575	0.284608
$177$	10.7005	0.804301
$178$	0.538319	0.0403487
$179$	14.1768	1.05962	0.529812	−	0.848115i	$-0.322263\pi$
$179$	14.1768	1.05962	0.529812	+	0.848115i	$0.322263\pi$
$180$	−1.96239	−0.146268
$181$	−5.22425	−0.388316	−0.194158	−	0.980970i	$-0.562197\pi$
$181$	−5.22425	−0.388316	−0.194158	+	0.980970i	$0.562197\pi$
$182$	−1.92478	−0.142674
$183$	−8.70052	−0.643161
$184$	−5.14903	−0.379592
$185$	−2.00000	−0.147043
$186$	−1.92478	−0.141132
$187$	−4.57452	−0.334522
$188$	19.4763	1.42045
$189$	3.35026	0.243696
$190$	0.836381	0.0606774
$191$	−16.6253	−1.20296	−0.601482	−	0.798886i	$-0.705423\pi$
$191$	−16.6253	−1.20296	−0.601482	+	0.798886i	$0.705423\pi$
$192$	−7.11142	−0.513222
$193$	−16.3634	−1.17787	−0.588933	−	0.808182i	$-0.700452\pi$
$193$	−16.3634	−1.17787	−0.588933	+	0.808182i	$0.700452\pi$
$194$	−0.0145884	−0.00104739
$195$	2.96239	0.212141
$196$	−8.28963	−0.592116
$197$	−20.4241	−1.45515	−0.727577	−	0.686026i	$-0.759354\pi$
$197$	−20.4241	−1.45515	−0.727577	+	0.686026i	$0.759354\pi$
$198$	−0.193937	−0.0137825
$199$	−8.62530	−0.611431	−0.305716	−	0.952123i	$-0.598896\pi$
$199$	−8.62530	−0.611431	−0.305716	+	0.952123i	$0.598896\pi$
$200$	0.768452	0.0543378
$201$	5.92478	0.417902
$202$	2.92619	0.205886
$203$	−12.1016	−0.849364
$204$	8.97698	0.628514
$205$	−4.38787	−0.306462
$206$	0.625301	0.0435668
$207$	−6.70052	−0.465719
$208$	11.1852	0.775556
$209$	−4.31265	−0.298312
$210$	−0.649738	−0.0448362
$211$	9.08840	0.625671	0.312836	−	0.949807i	$-0.398721\pi$
$211$	9.08840	0.625671	0.312836	+	0.949807i	$0.398721\pi$
$212$	−9.22425	−0.633524
$213$	9.92478	0.680035
$214$	0.186642	0.0127586
$215$	−9.27504	−0.632552
$216$	0.768452	0.0522865
$217$	33.2506	2.25720
$218$	−2.21108	−0.149753
$219$	−7.73813	−0.522895
$220$	−1.96239	−0.132304
$221$	−13.5515	−0.911572
$222$	0.387873	0.0260323
$223$	−6.70052	−0.448700	−0.224350	−	0.974509i	$-0.572026\pi$
$223$	−6.70052	−0.448700	−0.224350	+	0.974509i	$0.572026\pi$
$224$	−7.60228	−0.507949
$225$	1.00000	0.0666667
$226$	1.16362	0.0774028
$227$	16.9624	1.12583	0.562917	−	0.826514i	$-0.309679\pi$
$227$	16.9624	1.12583	0.562917	+	0.826514i	$0.309679\pi$
$228$	8.46310	0.560482
$229$	25.8496	1.70819	0.854093	−	0.520120i	$-0.174113\pi$
$229$	25.8496	1.70819	0.854093	+	0.520120i	$0.174113\pi$
$230$	1.29948	0.0856849
$231$	3.35026	0.220431
$232$	−2.77575	−0.182237
$233$	−19.2750	−1.26275	−0.631375	−	0.775478i	$-0.717509\pi$
$233$	−19.2750	−1.26275	−0.631375	+	0.775478i	$0.717509\pi$
$234$	−0.574515	−0.0375573
$235$	−9.92478	−0.647421
$236$	−20.9986	−1.36689
$237$	11.5369	0.749402
$238$	2.97224	0.192662
$239$	26.5501	1.71738	0.858691	−	0.512494i	$-0.171278\pi$
$239$	26.5501	1.71738	0.858691	+	0.512494i	$0.171278\pi$
$240$	3.77575	0.243723
$241$	28.5501	1.83907	0.919536	−	0.393006i	$-0.128565\pi$
$241$	28.5501	1.83907	0.919536	+	0.393006i	$0.128565\pi$
$242$	−0.193937	−0.0124667
$243$	1.00000	0.0641500
$244$	17.0738	1.09304
$245$	4.22425	0.269878
$246$	0.850969	0.0542558
$247$	−12.7757	−0.812901
$248$	7.62672	0.484297
$249$	10.8872	0.689946
$250$	−0.193937	−0.0122656
$251$	29.9248	1.88884	0.944418	−	0.328748i	$-0.106627\pi$
$251$	29.9248	1.88884	0.944418	+	0.328748i	$0.106627\pi$
$252$	−6.57452	−0.414156
$253$	−6.70052	−0.421258
$254$	2.82653	0.177352
$255$	−4.57452	−0.286467
$256$	13.0752	0.817201
$257$	8.70052	0.542724	0.271362	−	0.962477i	$-0.412526\pi$
$257$	8.70052	0.542724	0.271362	+	0.962477i	$0.412526\pi$
$258$	1.79877	0.111986
$259$	−6.70052	−0.416350
$260$	−5.81336	−0.360529
$261$	−3.61213	−0.223585
$262$	1.14903	0.0709874
$263$	12.2882	0.757724	0.378862	−	0.925453i	$-0.376316\pi$
$263$	12.2882	0.757724	0.378862	+	0.925453i	$0.376316\pi$
$264$	0.768452	0.0472950
$265$	4.70052	0.288751
$266$	2.80209	0.171807
$267$	−2.77575	−0.169873
$268$	−11.6267	−0.710215
$269$	−5.84955	−0.356654	−0.178327	−	0.983971i	$-0.557068\pi$
$269$	−5.84955	−0.356654	−0.178327	+	0.983971i	$0.557068\pi$
$270$	−0.193937	−0.0118026
$271$	−5.08840	−0.309098	−0.154549	−	0.987985i	$-0.549392\pi$
$271$	−5.08840	−0.309098	−0.154549	+	0.987985i	$0.549392\pi$
$272$	−17.2722	−1.04728
$273$	9.92478	0.600675
$274$	−2.68594	−0.162263
$275$	1.00000	0.0603023
$276$	13.1490	0.791479
$277$	1.41090	0.0847725	0.0423863	−	0.999101i	$-0.486504\pi$
$277$	1.41090	0.0847725	0.0423863	+	0.999101i	$0.486504\pi$
$278$	−2.63989	−0.158330
$279$	9.92478	0.594181
$280$	2.57452	0.153857
$281$	−4.38787	−0.261759	−0.130879	−	0.991398i	$-0.541780\pi$
$281$	−4.38787	−0.261759	−0.130879	+	0.991398i	$0.541780\pi$
$282$	1.92478	0.114619
$283$	26.5745	1.57969	0.789845	−	0.613306i	$-0.210161\pi$
$283$	26.5745	1.57969	0.789845	+	0.613306i	$0.210161\pi$
$284$	−19.4763	−1.15570
$285$	−4.31265	−0.255459
$286$	−0.574515	−0.0339718
$287$	−14.7005	−0.867744
$288$	−2.26916	−0.133711
$289$	3.92619	0.230952
$290$	0.700523	0.0411362
$291$	0.0752228	0.00440964
$292$	15.1852	0.888648
$293$	−3.42548	−0.200119	−0.100059	−	0.994981i	$-0.531903\pi$
$293$	−3.42548	−0.200119	−0.100059	+	0.994981i	$0.531903\pi$
$294$	−0.819237	−0.0477789
$295$	10.7005	0.623009
$296$	−1.53690	−0.0893307
$297$	1.00000	0.0580259
$298$	−0.298062	−0.0172663
$299$	−19.8496	−1.14793
$300$	−1.96239	−0.113299
$301$	−31.0738	−1.79106
$302$	1.31124	0.0754531
$303$	−15.0884	−0.866806
$304$	−16.2835	−0.933921
$305$	−8.70052	−0.498191
$306$	0.887166	0.0507159
$307$	−16.6497	−0.950251	−0.475125	−	0.879918i	$-0.657597\pi$
$307$	−16.6497	−0.950251	−0.475125	+	0.879918i	$0.657597\pi$
$308$	−6.57452	−0.374618
$309$	−3.22425	−0.183421
$310$	−1.92478	−0.109320
$311$	32.9986	1.87118	0.935589	−	0.353091i	$-0.114869\pi$
$311$	32.9986	1.87118	0.935589	+	0.353091i	$0.114869\pi$
$312$	2.27645	0.128879
$313$	15.4010	0.870519	0.435259	−	0.900305i	$-0.356657\pi$
$313$	15.4010	0.870519	0.435259	+	0.900305i	$0.356657\pi$
$314$	1.06205	0.0599349
$315$	3.35026	0.188766
$316$	−22.6399	−1.27359
$317$	2.15045	0.120781	0.0603905	−	0.998175i	$-0.480765\pi$
$317$	2.15045	0.120781	0.0603905	+	0.998175i	$0.480765\pi$
$318$	−0.911603	−0.0511202
$319$	−3.61213	−0.202240
$320$	−7.11142	−0.397540
$321$	−0.962389	−0.0537153
$322$	4.35359	0.242616
$323$	19.7283	1.09771
$324$	−1.96239	−0.109022
$325$	2.96239	0.164324
$326$	−2.44851	−0.135610
$327$	11.4010	0.630479
$328$	−3.37187	−0.186180
$329$	−33.2506	−1.83316
$330$	−0.193937	−0.0106759
$331$	−14.5501	−0.799745	−0.399872	−	0.916571i	$-0.630946\pi$
$331$	−14.5501	−0.799745	−0.399872	+	0.916571i	$0.630946\pi$
$332$	−21.3649	−1.17255
$333$	−2.00000	−0.109599
$334$	−3.56134	−0.194868
$335$	5.92478	0.323705
$336$	12.6497	0.690100
$337$	16.2619	0.885840	0.442920	−	0.896561i	$-0.353943\pi$
$337$	16.2619	0.885840	0.442920	+	0.896561i	$0.353943\pi$
$338$	0.819237	0.0445606
$339$	−6.00000	−0.325875
$340$	8.97698	0.486845
$341$	9.92478	0.537457
$342$	0.836381	0.0452263
$343$	−9.29948	−0.502125
$344$	−7.12742	−0.384285
$345$	−6.70052	−0.360744
$346$	1.66291	0.0893987
$347$	−0.962389	−0.0516637	−0.0258319	−	0.999666i	$-0.508223\pi$
$347$	−0.962389	−0.0516637	−0.0258319	+	0.999666i	$0.508223\pi$
$348$	7.08840	0.379978
$349$	20.7005	1.10807	0.554037	−	0.832492i	$-0.313087\pi$
$349$	20.7005	1.10807	0.554037	+	0.832492i	$0.313087\pi$
$350$	−0.649738	−0.0347300
$351$	2.96239	0.158121
$352$	−2.26916	−0.120947
$353$	20.5501	1.09377	0.546885	−	0.837208i	$-0.315813\pi$
$353$	20.5501	1.09377	0.546885	+	0.837208i	$0.315813\pi$
$354$	−2.07522	−0.110297
$355$	9.92478	0.526752
$356$	5.44709	0.288695
$357$	−15.3258	−0.811129
$358$	−2.74940	−0.145310
$359$	17.9248	0.946034	0.473017	−	0.881053i	$-0.343165\pi$
$359$	17.9248	0.946034	0.473017	+	0.881053i	$0.343165\pi$
$360$	0.768452	0.0405010
$361$	−0.401047	−0.0211077
$362$	1.01317	0.0532512
$363$	1.00000	0.0524864
$364$	−19.4763	−1.02083
$365$	−7.73813	−0.405032
$366$	1.68735	0.0881992
$367$	−29.6531	−1.54788	−0.773939	−	0.633261i	$-0.781716\pi$
$367$	−29.6531	−1.54788	−0.773939	+	0.633261i	$0.781716\pi$
$368$	−25.2995	−1.31883
$369$	−4.38787	−0.228424
$370$	0.387873	0.0201646
$371$	15.7480	0.817595
$372$	−19.4763	−1.00980
$373$	−9.13918	−0.473209	−0.236604	−	0.971606i	$-0.576035\pi$
$373$	−9.13918	−0.473209	−0.236604	+	0.971606i	$0.576035\pi$
$374$	0.887166	0.0458743
$375$	1.00000	0.0516398
$376$	−7.62672	−0.393318
$377$	−10.7005	−0.551105
$378$	−0.649738	−0.0334189
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	8.46310	0.434148
$381$	−14.5745	−0.746675
$382$	3.22425	0.164967
$383$	−34.9234	−1.78450	−0.892250	−	0.451541i	$-0.850874\pi$
$383$	−34.9234	−1.78450	−0.892250	+	0.451541i	$0.850874\pi$
$384$	5.91748	0.301975
$385$	3.35026	0.170745
$386$	3.17347	0.161525
$387$	−9.27504	−0.471477
$388$	−0.147616	−0.00749408
$389$	2.77575	0.140736	0.0703680	−	0.997521i	$-0.477583\pi$
$389$	2.77575	0.140736	0.0703680	+	0.997521i	$0.477583\pi$
$390$	−0.574515	−0.0290917
$391$	30.6516	1.55012
$392$	3.24614	0.163955
$393$	−5.92478	−0.298865
$394$	3.96097	0.199551
$395$	11.5369	0.580485
$396$	−1.96239	−0.0986137
$397$	−19.9248	−0.999996	−0.499998	−	0.866027i	$-0.666666\pi$
$397$	−19.9248	−0.999996	−0.499998	+	0.866027i	$0.666666\pi$
$398$	1.67276	0.0838479
$399$	−14.4485	−0.723330
$400$	3.77575	0.188787
$401$	2.00000	0.0998752	0.0499376	−	0.998752i	$-0.484098\pi$
$401$	2.00000	0.0998752	0.0499376	+	0.998752i	$0.484098\pi$
$402$	−1.14903	−0.0573085
$403$	29.4010	1.46457
$404$	29.6093	1.47312
$405$	1.00000	0.0496904
$406$	2.34694	0.116477
$407$	−2.00000	−0.0991363
$408$	−3.51530	−0.174033
$409$	−13.0738	−0.646458	−0.323229	−	0.946321i	$-0.604768\pi$
$409$	−13.0738	−0.646458	−0.323229	+	0.946321i	$0.604768\pi$
$410$	0.850969	0.0420264
$411$	13.8496	0.683148
$412$	6.32724	0.311721
$413$	35.8496	1.76404
$414$	1.29948	0.0638658
$415$	10.8872	0.534430
$416$	−6.72213	−0.329580
$417$	13.6121	0.666589
$418$	0.836381	0.0409087
$419$	7.22425	0.352928	0.176464	−	0.984307i	$-0.443534\pi$
$419$	7.22425	0.352928	0.176464	+	0.984307i	$0.443534\pi$
$420$	−6.57452	−0.320804
$421$	30.6253	1.49259	0.746293	−	0.665618i	$-0.231832\pi$
$421$	30.6253	1.49259	0.746293	+	0.665618i	$0.231832\pi$
$422$	−1.76257	−0.0858007
$423$	−9.92478	−0.482559
$424$	3.61213	0.175420
$425$	−4.57452	−0.221897
$426$	−1.92478	−0.0932558
$427$	−29.1490	−1.41062
$428$	1.88858	0.0912880
$429$	2.96239	0.143025
$430$	1.79877	0.0867444
$431$	−33.8759	−1.63174	−0.815872	−	0.578232i	$-0.803743\pi$
$431$	−33.8759	−1.63174	−0.815872	+	0.578232i	$0.803743\pi$
$432$	3.77575	0.181661
$433$	−9.47627	−0.455400	−0.227700	−	0.973731i	$-0.573121\pi$
$433$	−9.47627	−0.455400	−0.227700	+	0.973731i	$0.573121\pi$
$434$	−6.44851	−0.309538
$435$	−3.61213	−0.173188
$436$	−22.3733	−1.07149
$437$	28.8970	1.38233
$438$	1.50071	0.0717066
$439$	−29.4617	−1.40613	−0.703065	−	0.711126i	$-0.748186\pi$
$439$	−29.4617	−1.40613	−0.703065	+	0.711126i	$0.748186\pi$
$440$	0.768452	0.0366345
$441$	4.22425	0.201155
$442$	2.62813	0.125007
$443$	−19.0738	−0.906224	−0.453112	−	0.891454i	$-0.649686\pi$
$443$	−19.0738	−0.906224	−0.453112	+	0.891454i	$0.649686\pi$
$444$	3.92478	0.186262
$445$	−2.77575	−0.131583
$446$	1.29948	0.0615320
$447$	1.53690	0.0726931
$448$	−23.8251	−1.12563
$449$	35.8759	1.69309	0.846544	−	0.532318i	$-0.178679\pi$
$449$	35.8759	1.69309	0.846544	+	0.532318i	$0.178679\pi$
$450$	−0.193937	−0.00914226
$451$	−4.38787	−0.206617
$452$	11.7743	0.553818
$453$	−6.76116	−0.317667
$454$	−3.28963	−0.154390
$455$	9.92478	0.465281
$456$	−3.31406	−0.155195
$457$	5.28963	0.247438	0.123719	−	0.992317i	$-0.460518\pi$
$457$	5.28963	0.247438	0.123719	+	0.992317i	$0.460518\pi$
$458$	−5.01317	−0.234250
$459$	−4.57452	−0.213520
$460$	13.1490	0.613077
$461$	36.3390	1.69248	0.846238	−	0.532805i	$-0.178862\pi$
$461$	36.3390	1.69248	0.846238	+	0.532805i	$0.178862\pi$
$462$	−0.649738	−0.0302286
$463$	10.5501	0.490304	0.245152	−	0.969485i	$-0.421162\pi$
$463$	10.5501	0.490304	0.245152	+	0.969485i	$0.421162\pi$
$464$	−13.6385	−0.633150
$465$	9.92478	0.460251
$466$	3.73813	0.173166
$467$	18.7005	0.865357	0.432679	−	0.901548i	$-0.357569\pi$
$467$	18.7005	0.865357	0.432679	+	0.901548i	$0.357569\pi$
$468$	−5.81336	−0.268723
$469$	19.8496	0.916567
$470$	1.92478	0.0887834
$471$	−5.47627	−0.252333
$472$	8.22284	0.378487
$473$	−9.27504	−0.426467
$474$	−2.23743	−0.102768
$475$	−4.31265	−0.197878
$476$	30.0752	1.37850
$477$	4.70052	0.215222
$478$	−5.14903	−0.235511
$479$	−9.29948	−0.424904	−0.212452	−	0.977172i	$-0.568145\pi$
$479$	−9.29948	−0.424904	−0.212452	+	0.977172i	$0.568145\pi$
$480$	−2.26916	−0.103572
$481$	−5.92478	−0.270147
$482$	−5.53690	−0.252199
$483$	−22.4485	−1.02144
$484$	−1.96239	−0.0891995
$485$	0.0752228	0.00341569
$486$	−0.193937	−0.00879714
$487$	−35.4763	−1.60758	−0.803792	−	0.594911i	$-0.797187\pi$
$487$	−35.4763	−1.60758	−0.803792	+	0.594911i	$0.797187\pi$
$488$	−6.68594	−0.302658
$489$	12.6253	0.570936
$490$	−0.819237	−0.0370094
$491$	24.7757	1.11811	0.559057	−	0.829129i	$-0.311163\pi$
$491$	24.7757	1.11811	0.559057	+	0.829129i	$0.311163\pi$
$492$	8.61071	0.388201
$493$	16.5237	0.744191
$494$	2.47768	0.111476
$495$	1.00000	0.0449467
$496$	37.4734	1.68261
$497$	33.2506	1.49149
$498$	−2.11142	−0.0946150
$499$	14.1768	0.634640	0.317320	−	0.948318i	$-0.397217\pi$
$499$	14.1768	0.634640	0.317320	+	0.948318i	$0.397217\pi$
$500$	−1.96239	−0.0877607
$501$	18.3634	0.820418
$502$	−5.80351	−0.259023
$503$	−8.43866	−0.376261	−0.188131	−	0.982144i	$-0.560243\pi$
$503$	−8.43866	−0.376261	−0.188131	+	0.982144i	$0.560243\pi$
$504$	2.57452	0.114678
$505$	−15.0884	−0.671425
$506$	1.29948	0.0577688
$507$	−4.22425	−0.187606
$508$	28.6009	1.26896
$509$	1.10299	0.0488890	0.0244445	−	0.999701i	$-0.492218\pi$
$509$	1.10299	0.0488890	0.0244445	+	0.999701i	$0.492218\pi$
$510$	0.887166	0.0392844
$511$	−25.9248	−1.14684
$512$	−14.3707	−0.635103
$513$	−4.31265	−0.190408
$514$	−1.68735	−0.0744258
$515$	−3.22425	−0.142078
$516$	18.2012	0.801265
$517$	−9.92478	−0.436491
$518$	1.29948	0.0570957
$519$	−8.57452	−0.376379
$520$	2.27645	0.0998291
$521$	−12.4485	−0.545379	−0.272690	−	0.962102i	$-0.587913\pi$
$521$	−12.4485	−0.545379	−0.272690	+	0.962102i	$0.587913\pi$
$522$	0.700523	0.0306611
$523$	30.0508	1.31403	0.657015	−	0.753878i	$-0.271819\pi$
$523$	30.0508	1.31403	0.657015	+	0.753878i	$0.271819\pi$
$524$	11.6267	0.507915
$525$	3.35026	0.146217
$526$	−2.38313	−0.103910
$527$	−45.4010	−1.97770
$528$	3.77575	0.164318
$529$	21.8970	0.952044
$530$	−0.911603	−0.0395975
$531$	10.7005	0.464363
$532$	28.3536	1.22928
$533$	−12.9986	−0.563031
$534$	0.538319	0.0232953
$535$	−0.962389	−0.0416077
$536$	4.55291	0.196656
$537$	14.1768	0.611774
$538$	1.13444	0.0489093
$539$	4.22425	0.181951
$540$	−1.96239	−0.0844478
$541$	−18.0000	−0.773880	−0.386940	−	0.922105i	$-0.626468\pi$
$541$	−18.0000	−0.773880	−0.386940	+	0.922105i	$0.626468\pi$
$542$	0.986826	0.0423878
$543$	−5.22425	−0.224194
$544$	10.3803	0.445052
$545$	11.4010	0.488367
$546$	−1.92478	−0.0823729
$547$	−14.3028	−0.611544	−0.305772	−	0.952105i	$-0.598914\pi$
$547$	−14.3028	−0.611544	−0.305772	+	0.952105i	$0.598914\pi$
$548$	−27.1782	−1.16100
$549$	−8.70052	−0.371329
$550$	−0.193937	−0.00826948
$551$	15.5778	0.663638
$552$	−5.14903	−0.219157
$553$	38.6516	1.64364
$554$	−0.273624	−0.0116252
$555$	−2.00000	−0.0848953
$556$	−26.7123	−1.13285
$557$	−11.7988	−0.499930	−0.249965	−	0.968255i	$-0.580419\pi$
$557$	−11.7988	−0.499930	−0.249965	+	0.968255i	$0.580419\pi$
$558$	−1.92478	−0.0814823
$559$	−27.4763	−1.16212
$560$	12.6497	0.534549
$561$	−4.57452	−0.193136
$562$	0.850969	0.0358960
$563$	30.4847	1.28478	0.642389	−	0.766379i	$-0.277944\pi$
$563$	30.4847	1.28478	0.642389	+	0.766379i	$0.277944\pi$
$564$	19.4763	0.820099
$565$	−6.00000	−0.252422
$566$	−5.15377	−0.216629
$567$	3.35026	0.140698
$568$	7.62672	0.320010
$569$	−27.0884	−1.13560	−0.567802	−	0.823165i	$-0.692206\pi$
$569$	−27.0884	−1.13560	−0.567802	+	0.823165i	$0.692206\pi$
$570$	0.836381	0.0350321
$571$	7.28489	0.304863	0.152432	−	0.988314i	$-0.451290\pi$
$571$	7.28489	0.304863	0.152432	+	0.988314i	$0.451290\pi$
$572$	−5.81336	−0.243069
$573$	−16.6253	−0.694532
$574$	2.85097	0.118997
$575$	−6.70052	−0.279431
$576$	−7.11142	−0.296309
$577$	−31.6239	−1.31652	−0.658260	−	0.752791i	$-0.728707\pi$
$577$	−31.6239	−1.31652	−0.658260	+	0.752791i	$0.728707\pi$
$578$	−0.761432	−0.0316714
$579$	−16.3634	−0.680041
$580$	7.08840	0.294330
$581$	36.4749	1.51323
$582$	−0.0145884	−0.000604711 0
$583$	4.70052	0.194676
$584$	−5.94639	−0.246063
$585$	2.96239	0.122480
$586$	0.664327	0.0274431
$587$	33.1490	1.36821	0.684103	−	0.729385i	$-0.260194\pi$
$587$	33.1490	1.36821	0.684103	+	0.729385i	$0.260194\pi$
$588$	−8.28963	−0.341858
$589$	−42.8021	−1.76363
$590$	−2.07522	−0.0854356
$591$	−20.4241	−0.840134
$592$	−7.55149	−0.310364
$593$	34.4993	1.41672	0.708358	−	0.705853i	$-0.249436\pi$
$593$	34.4993	1.41672	0.708358	+	0.705853i	$0.249436\pi$
$594$	−0.193937	−0.00795731
$595$	−15.3258	−0.628298
$596$	−3.01600	−0.123540
$597$	−8.62530	−0.353010
$598$	3.84955	0.157420
$599$	−14.4485	−0.590350	−0.295175	−	0.955443i	$-0.595378\pi$
$599$	−14.4485	−0.590350	−0.295175	+	0.955443i	$0.595378\pi$
$600$	0.768452	0.0313719
$601$	−15.9248	−0.649585	−0.324793	−	0.945785i	$-0.605295\pi$
$601$	−15.9248	−0.649585	−0.324793	+	0.945785i	$0.605295\pi$
$602$	6.02635	0.245616
$603$	5.92478	0.241276
$604$	13.2680	0.539868
$605$	1.00000	0.0406558
$606$	2.92619	0.118868
$607$	−14.5745	−0.591561	−0.295781	−	0.955256i	$-0.595580\pi$
$607$	−14.5745	−0.591561	−0.295781	+	0.955256i	$0.595580\pi$
$608$	9.78609	0.396878
$609$	−12.1016	−0.490380
$610$	1.68735	0.0683188
$611$	−29.4010	−1.18944
$612$	8.97698	0.362873
$613$	16.4123	0.662887	0.331443	−	0.943475i	$-0.392464\pi$
$613$	16.4123	0.662887	0.331443	+	0.943475i	$0.392464\pi$
$614$	3.22899	0.130312
$615$	−4.38787	−0.176936
$616$	2.57452	0.103730
$617$	−17.8496	−0.718596	−0.359298	−	0.933223i	$-0.616984\pi$
$617$	−17.8496	−0.718596	−0.359298	+	0.933223i	$0.616984\pi$
$618$	0.625301	0.0251533
$619$	−0.402462	−0.0161763	−0.00808815	−	0.999967i	$-0.502575\pi$
$619$	−0.402462	−0.0161763	−0.00808815	+	0.999967i	$0.502575\pi$
$620$	−19.4763	−0.782186
$621$	−6.70052	−0.268883
$622$	−6.39963	−0.256602
$623$	−9.29948	−0.372576
$624$	11.1852	0.447767
$625$	1.00000	0.0400000
$626$	−2.98683	−0.119378
$627$	−4.31265	−0.172231
$628$	10.7466	0.428835
$629$	9.14903	0.364796
$630$	−0.649738	−0.0258862
$631$	−38.0263	−1.51380	−0.756902	−	0.653528i	$-0.773288\pi$
$631$	−38.0263	−1.51380	−0.756902	+	0.653528i	$0.773288\pi$
$632$	8.86556	0.352653
$633$	9.08840	0.361231
$634$	−0.417050	−0.0165632
$635$	−14.5745	−0.578372
$636$	−9.22425	−0.365765
$637$	12.5139	0.495818
$638$	0.700523	0.0277340
$639$	9.92478	0.392618
$640$	5.91748	0.233909
$641$	−28.0263	−1.10697	−0.553487	−	0.832858i	$-0.686703\pi$
$641$	−28.0263	−1.10697	−0.553487	+	0.832858i	$0.686703\pi$
$642$	0.186642	0.00736619
$643$	−4.62530	−0.182404	−0.0912020	−	0.995832i	$-0.529071\pi$
$643$	−4.62530	−0.182404	−0.0912020	+	0.995832i	$0.529071\pi$
$644$	44.0527	1.73592
$645$	−9.27504	−0.365204
$646$	−3.82604	−0.150533
$647$	23.5778	0.926941	0.463470	−	0.886112i	$-0.346604\pi$
$647$	23.5778	0.926941	0.463470	+	0.886112i	$0.346604\pi$
$648$	0.768452	0.0301876
$649$	10.7005	0.420032
$650$	−0.574515	−0.0225344
$651$	33.2506	1.30319
$652$	−24.7757	−0.970293
$653$	−2.25202	−0.0881282	−0.0440641	−	0.999029i	$-0.514031\pi$
$653$	−2.25202	−0.0881282	−0.0440641	+	0.999029i	$0.514031\pi$
$654$	−2.21108	−0.0864601
$655$	−5.92478	−0.231500
$656$	−16.5675	−0.646852
$657$	−7.73813	−0.301893
$658$	6.44851	0.251389
$659$	−41.4010	−1.61276	−0.806378	−	0.591401i	$-0.798575\pi$
$659$	−41.4010	−1.61276	−0.806378	+	0.591401i	$0.798575\pi$
$660$	−1.96239	−0.0763859
$661$	3.40105	0.132285	0.0661427	−	0.997810i	$-0.478931\pi$
$661$	3.40105	0.132285	0.0661427	+	0.997810i	$0.478931\pi$
$662$	2.82179	0.109672
$663$	−13.5515	−0.526296
$664$	8.36626	0.324674
$665$	−14.4485	−0.560289
$666$	0.387873	0.0150298
$667$	24.2031	0.937149
$668$	−36.0362	−1.39428
$669$	−6.70052	−0.259057
$670$	−1.14903	−0.0443909
$671$	−8.70052	−0.335880
$672$	−7.60228	−0.293264
$673$	0.887166	0.0341977	0.0170989	−	0.999854i	$-0.494557\pi$
$673$	0.887166	0.0341977	0.0170989	+	0.999854i	$0.494557\pi$
$674$	−3.15377	−0.121479
$675$	1.00000	0.0384900
$676$	8.28963	0.318832
$677$	18.9018	0.726453	0.363227	−	0.931701i	$-0.381675\pi$
$677$	18.9018	0.726453	0.363227	+	0.931701i	$0.381675\pi$
$678$	1.16362	0.0446885
$679$	0.252016	0.00967149
$680$	−3.51530	−0.134805
$681$	16.9624	0.650000
$682$	−1.92478	−0.0737035
$683$	−20.8773	−0.798848	−0.399424	−	0.916766i	$-0.630790\pi$
$683$	−20.8773	−0.798848	−0.399424	+	0.916766i	$0.630790\pi$
$684$	8.46310	0.323595
$685$	13.8496	0.529164
$686$	1.80351	0.0688583
$687$	25.8496	0.986222
$688$	−35.0202	−1.33513
$689$	13.9248	0.530492
$690$	1.29948	0.0494702
$691$	−2.44851	−0.0931456	−0.0465728	−	0.998915i	$-0.514830\pi$
$691$	−2.44851	−0.0931456	−0.0465728	+	0.998915i	$0.514830\pi$
$692$	16.8265	0.639649
$693$	3.35026	0.127266
$694$	0.186642	0.00708485
$695$	13.6121	0.516337
$696$	−2.77575	−0.105214
$697$	20.0724	0.760296
$698$	−4.01459	−0.151954
$699$	−19.2750	−0.729049
$700$	−6.57452	−0.248493
$701$	−2.98683	−0.112811	−0.0564054	−	0.998408i	$-0.517964\pi$
$701$	−2.98683	−0.112811	−0.0564054	+	0.998408i	$0.517964\pi$
$702$	−0.574515	−0.0216837
$703$	8.62530	0.325309
$704$	−7.11142	−0.268022
$705$	−9.92478	−0.373789
$706$	−3.98541	−0.149993
$707$	−50.5501	−1.90113
$708$	−20.9986	−0.789175
$709$	24.1768	0.907979	0.453989	−	0.891007i	$-0.350000\pi$
$709$	24.1768	0.907979	0.453989	+	0.891007i	$0.350000\pi$
$710$	−1.92478	−0.0722356
$711$	11.5369	0.432668
$712$	−2.13303	−0.0799386
$713$	−66.5012	−2.49049
$714$	2.97224	0.111233
$715$	2.96239	0.110787
$716$	−27.8204	−1.03970
$717$	26.5501	0.991531
$718$	−3.47627	−0.129733
$719$	−30.0263	−1.11979	−0.559897	−	0.828562i	$-0.689159\pi$
$719$	−30.0263	−1.11979	−0.559897	+	0.828562i	$0.689159\pi$
$720$	3.77575	0.140714
$721$	−10.8021	−0.402291
$722$	0.0777777	0.00289459
$723$	28.5501	1.06179
$724$	10.2520	0.381013
$725$	−3.61213	−0.134151
$726$	−0.193937	−0.00719766
$727$	14.9525	0.554559	0.277279	−	0.960789i	$-0.410567\pi$
$727$	14.9525	0.554559	0.277279	+	0.960789i	$0.410567\pi$
$728$	7.62672	0.282665
$729$	1.00000	0.0370370
$730$	1.50071	0.0555437
$731$	42.4288	1.56929
$732$	17.0738	0.631066
$733$	−19.1128	−0.705949	−0.352974	−	0.935633i	$-0.614830\pi$
$733$	−19.1128	−0.705949	−0.352974	+	0.935633i	$0.614830\pi$
$734$	5.75081	0.212266
$735$	4.22425	0.155814
$736$	15.2046	0.560447
$737$	5.92478	0.218242
$738$	0.850969	0.0313246
$739$	−3.31406	−0.121910	−0.0609549	−	0.998141i	$-0.519415\pi$
$739$	−3.31406	−0.121910	−0.0609549	+	0.998141i	$0.519415\pi$
$740$	3.92478	0.144278
$741$	−12.7757	−0.469329
$742$	−3.05411	−0.112120
$743$	−34.9887	−1.28361	−0.641806	−	0.766867i	$-0.721815\pi$
$743$	−34.9887	−1.28361	−0.641806	+	0.766867i	$0.721815\pi$
$744$	7.62672	0.279609
$745$	1.53690	0.0563078
$746$	1.77242	0.0648930
$747$	10.8872	0.398341
$748$	8.97698	0.328231
$749$	−3.22425	−0.117812
$750$	−0.193937	−0.00708156
$751$	−26.9234	−0.982447	−0.491224	−	0.871033i	$-0.663450\pi$
$751$	−26.9234	−0.982447	−0.491224	+	0.871033i	$0.663450\pi$
$752$	−37.4734	−1.36652
$753$	29.9248	1.09052
$754$	2.07522	0.0755752
$755$	−6.76116	−0.246064
$756$	−6.57452	−0.239113
$757$	15.9248	0.578796	0.289398	−	0.957209i	$-0.406545\pi$
$757$	15.9248	0.578796	0.289398	+	0.957209i	$0.406545\pi$
$758$	3.87873	0.140882
$759$	−6.70052	−0.243214
$760$	−3.31406	−0.120214
$761$	−30.9380	−1.12150	−0.560750	−	0.827985i	$-0.689487\pi$
$761$	−30.9380	−1.12150	−0.560750	+	0.827985i	$0.689487\pi$
$762$	2.82653	0.102394
$763$	38.1965	1.38281
$764$	32.6253	1.18034
$765$	−4.57452	−0.165392
$766$	6.77292	0.244715
$767$	31.6991	1.14459
$768$	13.0752	0.471811
$769$	9.32582	0.336298	0.168149	−	0.985762i	$-0.446221\pi$
$769$	9.32582	0.336298	0.168149	+	0.985762i	$0.446221\pi$
$770$	−0.649738	−0.0234149
$771$	8.70052	0.313342
$772$	32.1114	1.15572
$773$	44.7005	1.60777	0.803883	−	0.594787i	$-0.202764\pi$
$773$	44.7005	1.60777	0.803883	+	0.594787i	$0.202764\pi$
$774$	1.79877	0.0646554
$775$	9.92478	0.356509
$776$	0.0578051	0.00207508
$777$	−6.70052	−0.240380
$778$	−0.538319	−0.0192997
$779$	18.9234	0.678000
$780$	−5.81336	−0.208152
$781$	9.92478	0.355136
$782$	−5.94448	−0.212574
$783$	−3.61213	−0.129087
$784$	15.9497	0.569633
$785$	−5.47627	−0.195456
$786$	1.14903	0.0409846
$787$	21.6775	0.772719	0.386360	−	0.922348i	$-0.373732\pi$
$787$	21.6775	0.772719	0.386360	+	0.922348i	$0.373732\pi$
$788$	40.0800	1.42779
$789$	12.2882	0.437472
$790$	−2.23743	−0.0796041
$791$	−20.1016	−0.714730
$792$	0.768452	0.0273058
$793$	−25.7743	−0.915273
$794$	3.86414	0.137133
$795$	4.70052	0.166710
$796$	16.9262	0.599933
$797$	22.7466	0.805725	0.402862	−	0.915261i	$-0.368015\pi$
$797$	22.7466	0.805725	0.402862	+	0.915261i	$0.368015\pi$
$798$	2.80209	0.0991930
$799$	45.4010	1.60617
$800$	−2.26916	−0.0802269
$801$	−2.77575	−0.0980762
$802$	−0.387873	−0.0136963
$803$	−7.73813	−0.273073
$804$	−11.6267	−0.410043
$805$	−22.4485	−0.791206
$806$	−5.70194	−0.200842
$807$	−5.84955	−0.205914
$808$	−11.5947	−0.407900
$809$	−23.6121	−0.830158	−0.415079	−	0.909785i	$-0.636246\pi$
$809$	−23.6121	−0.830158	−0.415079	+	0.909785i	$0.636246\pi$
$810$	−0.193937	−0.00681424
$811$	−26.0870	−0.916038	−0.458019	−	0.888942i	$-0.651441\pi$
$811$	−26.0870	−0.916038	−0.458019	+	0.888942i	$0.651441\pi$
$812$	23.7480	0.833391
$813$	−5.08840	−0.178458
$814$	0.387873	0.0135949
$815$	12.6253	0.442245
$816$	−17.2722	−0.604648
$817$	40.0000	1.39942
$818$	2.53549	0.0886513
$819$	9.92478	0.346800
$820$	8.61071	0.300699
$821$	−54.4142	−1.89907	−0.949535	−	0.313662i	$-0.898444\pi$
$821$	−54.4142	−1.89907	−0.949535	+	0.313662i	$0.898444\pi$
$822$	−2.68594	−0.0936827
$823$	0.121269	0.00422716	0.00211358	−	0.999998i	$-0.499327\pi$
$823$	0.121269	0.00422716	0.00211358	+	0.999998i	$0.499327\pi$
$824$	−2.47768	−0.0863142
$825$	1.00000	0.0348155
$826$	−6.95254	−0.241910
$827$	−18.2130	−0.633328	−0.316664	−	0.948538i	$-0.602563\pi$
$827$	−18.2130	−0.633328	−0.316664	+	0.948538i	$0.602563\pi$
$828$	13.1490	0.456960
$829$	13.0738	0.454072	0.227036	−	0.973886i	$-0.427096\pi$
$829$	13.0738	0.454072	0.227036	+	0.973886i	$0.427096\pi$
$830$	−2.11142	−0.0732884
$831$	1.41090	0.0489434
$832$	−21.0668	−0.730359
$833$	−19.3239	−0.669534
$834$	−2.63989	−0.0914119
$835$	18.3634	0.635493
$836$	8.46310	0.292702
$837$	9.92478	0.343050
$838$	−1.40105	−0.0483984
$839$	−26.5501	−0.916610	−0.458305	−	0.888795i	$-0.651543\pi$
$839$	−26.5501	−0.916610	−0.458305	+	0.888795i	$0.651543\pi$
$840$	2.57452	0.0888292
$841$	−15.9525	−0.550088
$842$	−5.93937	−0.204684
$843$	−4.38787	−0.151126
$844$	−17.8350	−0.613905
$845$	−4.22425	−0.145319
$846$	1.92478	0.0661752
$847$	3.35026	0.115116
$848$	17.7480	0.609468
$849$	26.5745	0.912035
$850$	0.887166	0.0304295
$851$	13.4010	0.459382
$852$	−19.4763	−0.667246
$853$	40.6155	1.39065	0.695323	−	0.718697i	$-0.255261\pi$
$853$	40.6155	1.39065	0.695323	+	0.718697i	$0.255261\pi$
$854$	5.65306	0.193444
$855$	−4.31265	−0.147490
$856$	−0.739549	−0.0252773
$857$	20.1721	0.689064	0.344532	−	0.938775i	$-0.388038\pi$
$857$	20.1721	0.689064	0.344532	+	0.938775i	$0.388038\pi$
$858$	−0.574515	−0.0196136
$859$	21.8035	0.743926	0.371963	−	0.928248i	$-0.378685\pi$
$859$	21.8035	0.743926	0.371963	+	0.928248i	$0.378685\pi$
$860$	18.2012	0.620657
$861$	−14.7005	−0.500993
$862$	6.56978	0.223767
$863$	35.4274	1.20596	0.602981	−	0.797755i	$-0.293979\pi$
$863$	35.4274	1.20596	0.602981	+	0.797755i	$0.293979\pi$
$864$	−2.26916	−0.0771984
$865$	−8.57452	−0.291542
$866$	1.83780	0.0624508
$867$	3.92619	0.133340
$868$	−65.2506	−2.21475
$869$	11.5369	0.391363
$870$	0.700523	0.0237500
$871$	17.5515	0.594710
$872$	8.76116	0.296690
$873$	0.0752228	0.00254591
$874$	−5.60419	−0.189564
$875$	3.35026	0.113260
$876$	15.1852	0.513061
$877$	−14.0362	−0.473969	−0.236984	−	0.971513i	$-0.576159\pi$
$877$	−14.0362	−0.473969	−0.236984	+	0.971513i	$0.576159\pi$
$878$	5.71370	0.192828
$879$	−3.42548	−0.115539
$880$	3.77575	0.127280
$881$	−21.0738	−0.709995	−0.354997	−	0.934867i	$-0.615518\pi$
$881$	−21.0738	−0.709995	−0.354997	+	0.934867i	$0.615518\pi$
$882$	−0.819237	−0.0275852
$883$	−42.1476	−1.41838	−0.709190	−	0.705017i	$-0.750939\pi$
$883$	−42.1476	−1.41838	−0.709190	+	0.705017i	$0.750939\pi$
$884$	26.5933	0.894429
$885$	10.7005	0.359694
$886$	3.69911	0.124274
$887$	6.93604	0.232889	0.116445	−	0.993197i	$-0.462850\pi$
$887$	6.93604	0.232889	0.116445	+	0.993197i	$0.462850\pi$
$888$	−1.53690	−0.0515751
$889$	−48.8284	−1.63765
$890$	0.538319	0.0180445
$891$	1.00000	0.0335013
$892$	13.1490	0.440262
$893$	42.8021	1.43232
$894$	−0.298062	−0.00996868
$895$	14.1768	0.473878
$896$	19.8251	0.662311
$897$	−19.8496	−0.662757
$898$	−6.95765	−0.232180
$899$	−35.8496	−1.19565
$900$	−1.96239	−0.0654130
$901$	−21.5026	−0.716356
$902$	0.850969	0.0283342
$903$	−31.0738	−1.03407
$904$	−4.61071	−0.153350
$905$	−5.22425	−0.173660
$906$	1.31124	0.0435629
$907$	53.2017	1.76653	0.883267	−	0.468870i	$-0.155339\pi$
$907$	53.2017	1.76653	0.883267	+	0.468870i	$0.155339\pi$
$908$	−33.2868	−1.10466
$909$	−15.0884	−0.500451
$910$	−1.92478	−0.0638057
$911$	36.4749	1.20847	0.604233	−	0.796808i	$-0.293480\pi$
$911$	36.4749	1.20847	0.604233	+	0.796808i	$0.293480\pi$
$912$	−16.2835	−0.539200
$913$	10.8872	0.360313
$914$	−1.02585	−0.0339322
$915$	−8.70052	−0.287630
$916$	−50.7269	−1.67606
$917$	−19.8496	−0.655490
$918$	0.887166	0.0292808
$919$	9.73340	0.321075	0.160538	−	0.987030i	$-0.448677\pi$
$919$	9.73340	0.321075	0.160538	+	0.987030i	$0.448677\pi$
$920$	−5.14903	−0.169759
$921$	−16.6497	−0.548628
$922$	−7.04746	−0.232096
$923$	29.4010	0.967747
$924$	−6.57452	−0.216286
$925$	−2.00000	−0.0657596
$926$	−2.04605	−0.0672372
$927$	−3.22425	−0.105898
$928$	8.19649	0.269063
$929$	−24.1768	−0.793215	−0.396607	−	0.917988i	$-0.629813\pi$
$929$	−24.1768	−0.793215	−0.396607	+	0.917988i	$0.629813\pi$
$930$	−1.92478	−0.0631159
$931$	−18.2177	−0.597062
$932$	37.8251	1.23900
$933$	32.9986	1.08033
$934$	−3.62672	−0.118670
$935$	−4.57452	−0.149603
$936$	2.27645	0.0744082
$937$	−7.48612	−0.244561	−0.122280	−	0.992496i	$-0.539021\pi$
$937$	−7.48612	−0.244561	−0.122280	+	0.992496i	$0.539021\pi$
$938$	−3.84955	−0.125692
$939$	15.4010	0.502594
$940$	19.4763	0.635246
$941$	−21.2360	−0.692274	−0.346137	−	0.938184i	$-0.612507\pi$
$941$	−21.2360	−0.692274	−0.346137	+	0.938184i	$0.612507\pi$
$942$	1.06205	0.0346034
$943$	29.4010	0.957430
$944$	40.4025	1.31499
$945$	3.35026	0.108984
$946$	1.79877	0.0584830
$947$	−15.4763	−0.502911	−0.251456	−	0.967869i	$-0.580909\pi$
$947$	−15.4763	−0.502911	−0.251456	+	0.967869i	$0.580909\pi$
$948$	−22.6399	−0.735309
$949$	−22.9234	−0.744124
$950$	0.836381	0.0271358
$951$	2.15045	0.0697330
$952$	−11.7772	−0.381700
$953$	−32.0508	−1.03823	−0.519113	−	0.854705i	$-0.673738\pi$
$953$	−32.0508	−1.03823	−0.519113	+	0.854705i	$0.673738\pi$
$954$	−0.911603	−0.0295142
$955$	−16.6253	−0.537982
$956$	−52.1016	−1.68509
$957$	−3.61213	−0.116763
$958$	1.80351	0.0582687
$959$	46.3996	1.49832
$960$	−7.11142	−0.229520
$961$	67.5012	2.17746
$962$	1.14903	0.0370462
$963$	−0.962389	−0.0310125
$964$	−56.0263	−1.80449
$965$	−16.3634	−0.526758
$966$	4.35359	0.140074
$967$	−17.3766	−0.558794	−0.279397	−	0.960176i	$-0.590135\pi$
$967$	−17.3766	−0.558794	−0.279397	+	0.960176i	$0.590135\pi$
$968$	0.768452	0.0246990
$969$	19.7283	0.633764
$970$	−0.0145884	−0.000468407 0
$971$	36.2031	1.16181	0.580907	−	0.813970i	$-0.302698\pi$
$971$	36.2031	1.16181	0.580907	+	0.813970i	$0.302698\pi$
$972$	−1.96239	−0.0629436
$973$	45.6042	1.46200
$974$	6.88015	0.220454
$975$	2.96239	0.0948724
$976$	−32.8510	−1.05153
$977$	−28.1476	−0.900522	−0.450261	−	0.892897i	$-0.648669\pi$
$977$	−28.1476	−0.900522	−0.450261	+	0.892897i	$0.648669\pi$
$978$	−2.44851	−0.0782946
$979$	−2.77575	−0.0887132
$980$	−8.28963	−0.264802
$981$	11.4010	0.364007
$982$	−4.80492	−0.153331
$983$	7.07381	0.225619	0.112810	−	0.993617i	$-0.464015\pi$
$983$	7.07381	0.225619	0.112810	+	0.993617i	$0.464015\pi$
$984$	−3.37187	−0.107491
$985$	−20.4241	−0.650765
$986$	−3.20456	−0.102054
$987$	−33.2506	−1.05838
$988$	25.0710	0.797614
$989$	62.1476	1.97618
$990$	−0.193937	−0.00616371
$991$	44.4260	1.41124	0.705619	−	0.708592i	$-0.250669\pi$
$991$	44.4260	1.41124	0.705619	+	0.708592i	$0.250669\pi$
$992$	−22.5209	−0.715039
$993$	−14.5501	−0.461733
$994$	−6.44851	−0.204534
$995$	−8.62530	−0.273440
$996$	−21.3649	−0.676971
$997$	−28.4847	−0.902120	−0.451060	−	0.892494i	$-0.648954\pi$
$997$	−28.4847	−0.902120	−0.451060	+	0.892494i	$0.648954\pi$
$998$	−2.74940	−0.0870307
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	165.2.a.c.1.2	✓	3
3.2	odd	2		495.2.a.e.1.2		3
4.3	odd	2		2640.2.a.be.1.1		3
5.2	odd	4		825.2.c.g.199.3		6
5.3	odd	4		825.2.c.g.199.4		6
5.4	even	2		825.2.a.k.1.2		3
7.6	odd	2		8085.2.a.bk.1.2		3
11.10	odd	2		1815.2.a.m.1.2		3
12.11	even	2		7920.2.a.cj.1.1		3
15.2	even	4		2475.2.c.r.199.4		6
15.8	even	4		2475.2.c.r.199.3		6
15.14	odd	2		2475.2.a.bb.1.2		3
33.32	even	2		5445.2.a.z.1.2		3
55.54	odd	2		9075.2.a.cf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
165.2.a.c.1.2	✓	3	1.1	even	1	trivial
495.2.a.e.1.2		3	3.2	odd	2
825.2.a.k.1.2		3	5.4	even	2
825.2.c.g.199.3		6	5.2	odd	4
825.2.c.g.199.4		6	5.3	odd	4
1815.2.a.m.1.2		3	11.10	odd	2
2475.2.a.bb.1.2		3	15.14	odd	2
2475.2.c.r.199.3		6	15.8	even	4
2475.2.c.r.199.4		6	15.2	even	4
2640.2.a.be.1.1		3	4.3	odd	2
5445.2.a.z.1.2		3	33.32	even	2
7920.2.a.cj.1.1		3	12.11	even	2
8085.2.a.bk.1.2		3	7.6	odd	2
9075.2.a.cf.1.2		3	55.54	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 165 = 3 \cdot 5 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	165.a (trivial)

\(f(q)\)	\(=\)	\(q-0.193937 q^{2} +1.00000 q^{3} -1.96239 q^{4} +1.00000 q^{5} -0.193937 q^{6} +3.35026 q^{7} +0.768452 q^{8} +1.00000 q^{9} -0.193937 q^{10} +1.00000 q^{11} -1.96239 q^{12} +2.96239 q^{13} -0.649738 q^{14} +1.00000 q^{15} +3.77575 q^{16} -4.57452 q^{17} -0.193937 q^{18} -4.31265 q^{19} -1.96239 q^{20} +3.35026 q^{21} -0.193937 q^{22} -6.70052 q^{23} +0.768452 q^{24} +1.00000 q^{25} -0.574515 q^{26} +1.00000 q^{27} -6.57452 q^{28} -3.61213 q^{29} -0.193937 q^{30} +9.92478 q^{31} -2.26916 q^{32} +1.00000 q^{33} +0.887166 q^{34} +3.35026 q^{35} -1.96239 q^{36} -2.00000 q^{37} +0.836381 q^{38} +2.96239 q^{39} +0.768452 q^{40} -4.38787 q^{41} -0.649738 q^{42} -9.27504 q^{43} -1.96239 q^{44} +1.00000 q^{45} +1.29948 q^{46} -9.92478 q^{47} +3.77575 q^{48} +4.22425 q^{49} -0.193937 q^{50} -4.57452 q^{51} -5.81336 q^{52} +4.70052 q^{53} -0.193937 q^{54} +1.00000 q^{55} +2.57452 q^{56} -4.31265 q^{57} +0.700523 q^{58} +10.7005 q^{59} -1.96239 q^{60} -8.70052 q^{61} -1.92478 q^{62} +3.35026 q^{63} -7.11142 q^{64} +2.96239 q^{65} -0.193937 q^{66} +5.92478 q^{67} +8.97698 q^{68} -6.70052 q^{69} -0.649738 q^{70} +9.92478 q^{71} +0.768452 q^{72} -7.73813 q^{73} +0.387873 q^{74} +1.00000 q^{75} +8.46310 q^{76} +3.35026 q^{77} -0.574515 q^{78} +11.5369 q^{79} +3.77575 q^{80} +1.00000 q^{81} +0.850969 q^{82} +10.8872 q^{83} -6.57452 q^{84} -4.57452 q^{85} +1.79877 q^{86} -3.61213 q^{87} +0.768452 q^{88} -2.77575 q^{89} -0.193937 q^{90} +9.92478 q^{91} +13.1490 q^{92} +9.92478 q^{93} +1.92478 q^{94} -4.31265 q^{95} -2.26916 q^{96} +0.0752228 q^{97} -0.819237 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 3 q^{5} - q^{6} - 9 q^{8} + 3 q^{9} - q^{10} + 3 q^{11} + 5 q^{12} - 2 q^{13} - 12 q^{14} + 3 q^{15} + 13 q^{16} - 2 q^{17} - q^{18} + 8 q^{19} + 5 q^{20} - q^{22}+ \cdots + 3 q^{99}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 3 * q^5 - q^6 - 9 * q^8 + 3 * q^9 - q^10 + 3 * q^11 + 5 * q^12 - 2 * q^13 - 12 * q^14 + 3 * q^15 + 13 * q^16 - 2 * q^17 - q^18 + 8 * q^19 + 5 * q^20 - q^22 - 9 * q^24 + 3 * q^25 + 10 * q^26 + 3 * q^27 - 8 * q^28 - 10 * q^29 - q^30 + 8 * q^31 - 29 * q^32 + 3 * q^33 - 30 * q^34 + 5 * q^36 - 6 * q^37 - 2 * q^39 - 9 * q^40 - 14 * q^41 - 12 * q^42 + 4 * q^43 + 5 * q^44 + 3 * q^45 + 24 * q^46 - 8 * q^47 + 13 * q^48 + 11 * q^49 - q^50 - 2 * q^51 - 30 * q^52 - 6 * q^53 - q^54 + 3 * q^55 - 4 * q^56 + 8 * q^57 - 18 * q^58 + 12 * q^59 + 5 * q^60 - 6 * q^61 + 16 * q^62 + 13 * q^64 - 2 * q^65 - q^66 - 4 * q^67 + 42 * q^68 - 12 * q^70 + 8 * q^71 - 9 * q^72 - 14 * q^73 + 2 * q^74 + 3 * q^75 + 48 * q^76 + 10 * q^78 + 12 * q^79 + 13 * q^80 + 3 * q^81 + 26 * q^82 - 8 * q^84 - 2 * q^85 - 8 * q^86 - 10 * q^87 - 9 * q^88 - 10 * q^89 - q^90 + 8 * q^91 + 16 * q^92 + 8 * q^93 - 16 * q^94 + 8 * q^95 - 29 * q^96 + 22 * q^97 + 39 * q^98 + 3 * q^99

Introduction

L-functions

Modular forms

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