LMFDB - Embedded newform 1881.2.a.d.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1881,2,Mod(1,1881)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1881.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1881, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1881 = 3^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1881.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$15.0198606202$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 209)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1881.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} +1.00000 q^{5} -3.41421 q^{7} +2.82843 q^{8} -1.41421 q^{10} +1.00000 q^{11} +2.24264 q^{13} +4.82843 q^{14} -4.00000 q^{16} -3.41421 q^{17} -1.00000 q^{19} -1.41421 q^{22} +3.00000 q^{23} -4.00000 q^{25} -3.17157 q^{26} +6.24264 q^{29} -6.41421 q^{31} +4.82843 q^{34} -3.41421 q^{35} +10.0711 q^{37} +1.41421 q^{38} +2.82843 q^{40} +1.65685 q^{41} +0.343146 q^{43} -4.24264 q^{46} -8.82843 q^{47} +4.65685 q^{49} +5.65685 q^{50} +4.48528 q^{53} +1.00000 q^{55} -9.65685 q^{56} -8.82843 q^{58} +1.58579 q^{59} -11.0711 q^{61} +9.07107 q^{62} +8.00000 q^{64} +2.24264 q^{65} -10.4142 q^{67} +4.82843 q^{70} +12.4142 q^{71} -4.48528 q^{73} -14.2426 q^{74} -3.41421 q^{77} -14.5858 q^{79} -4.00000 q^{80} -2.34315 q^{82} -3.41421 q^{83} -3.41421 q^{85} -0.485281 q^{86} +2.82843 q^{88} -4.89949 q^{89} -7.65685 q^{91} +12.4853 q^{94} -1.00000 q^{95} +2.41421 q^{97} -6.58579 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{14} - 8 q^{16} - 4 q^{17} - 2 q^{19} + 6 q^{23} - 8 q^{25} - 12 q^{26} + 4 q^{29} - 10 q^{31} + 4 q^{34} - 4 q^{35} + 6 q^{37} - 8 q^{41} + 12 q^{43}+ \cdots - 16 q^{98}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 + 2 * q^11 - 4 * q^13 + 4 * q^14 - 8 * q^16 - 4 * q^17 - 2 * q^19 + 6 * q^23 - 8 * q^25 - 12 * q^26 + 4 * q^29 - 10 * q^31 + 4 * q^34 - 4 * q^35 + 6 * q^37 - 8 * q^41 + 12 * q^43 - 12 * q^47 - 2 * q^49 - 8 * q^53 + 2 * q^55 - 8 * q^56 - 12 * q^58 + 6 * q^59 - 8 * q^61 + 4 * q^62 + 16 * q^64 - 4 * q^65 - 18 * q^67 + 4 * q^70 + 22 * q^71 + 8 * q^73 - 20 * q^74 - 4 * q^77 - 32 * q^79 - 8 * q^80 - 16 * q^82 - 4 * q^83 - 4 * q^85 + 16 * q^86 + 10 * q^89 - 4 * q^91 + 8 * q^94 - 2 * q^95 + 2 * q^97 - 16 * 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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	2.82843	1.00000
$9$	0	0
$10$	−1.41421	−0.447214
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.24264	0.621997	0.310998	−	0.950410i	$-0.399337\pi$
$13$	2.24264	0.621997	0.310998	+	0.950410i	$0.399337\pi$
$14$	4.82843	1.29045
$15$	0	0
$16$	−4.00000	−1.00000
$17$	−3.41421	−0.828068	−0.414034	−	0.910261i	$-0.635881\pi$
$17$	−3.41421	−0.828068	−0.414034	+	0.910261i	$0.635881\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	−1.41421	−0.301511
$23$	3.00000	0.625543	0.312772	−	0.949828i	$-0.398743\pi$
$23$	3.00000	0.625543	0.312772	+	0.949828i	$0.398743\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	−3.17157	−0.621997
$27$	0	0
$28$	0	0
$29$	6.24264	1.15923	0.579615	−	0.814891i	$-0.303203\pi$
$29$	6.24264	1.15923	0.579615	+	0.814891i	$0.303203\pi$
$30$	0	0
$31$	−6.41421	−1.15203	−0.576013	−	0.817440i	$-0.695392\pi$
$31$	−6.41421	−1.15203	−0.576013	+	0.817440i	$0.695392\pi$
$32$	0	0
$33$	0	0
$34$	4.82843	0.828068
$35$	−3.41421	−0.577107
$36$	0	0
$37$	10.0711	1.65567	0.827837	−	0.560969i	$-0.189571\pi$
$37$	10.0711	1.65567	0.827837	+	0.560969i	$0.189571\pi$
$38$	1.41421	0.229416
$39$	0	0
$40$	2.82843	0.447214
$41$	1.65685	0.258757	0.129379	−	0.991595i	$-0.458702\pi$
$41$	1.65685	0.258757	0.129379	+	0.991595i	$0.458702\pi$
$42$	0	0
$43$	0.343146	0.0523292	0.0261646	−	0.999658i	$-0.491671\pi$
$43$	0.343146	0.0523292	0.0261646	+	0.999658i	$0.491671\pi$
$44$	0	0
$45$	0	0
$46$	−4.24264	−0.625543
$47$	−8.82843	−1.28776	−0.643879	−	0.765127i	$-0.722676\pi$
$47$	−8.82843	−1.28776	−0.643879	+	0.765127i	$0.722676\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	5.65685	0.800000
$51$	0	0
$52$	0	0
$53$	4.48528	0.616101	0.308050	−	0.951370i	$-0.400323\pi$
$53$	4.48528	0.616101	0.308050	+	0.951370i	$0.400323\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−9.65685	−1.29045
$57$	0	0
$58$	−8.82843	−1.15923
$59$	1.58579	0.206452	0.103226	−	0.994658i	$-0.467084\pi$
$59$	1.58579	0.206452	0.103226	+	0.994658i	$0.467084\pi$
$60$	0	0
$61$	−11.0711	−1.41750	−0.708752	−	0.705457i	$-0.750742\pi$
$61$	−11.0711	−1.41750	−0.708752	+	0.705457i	$0.750742\pi$
$62$	9.07107	1.15203
$63$	0	0
$64$	8.00000	1.00000
$65$	2.24264	0.278165
$66$	0	0
$67$	−10.4142	−1.27230	−0.636149	−	0.771566i	$-0.719474\pi$
$67$	−10.4142	−1.27230	−0.636149	+	0.771566i	$0.719474\pi$
$68$	0	0
$69$	0	0
$70$	4.82843	0.577107
$71$	12.4142	1.47330	0.736648	−	0.676276i	$-0.236407\pi$
$71$	12.4142	1.47330	0.736648	+	0.676276i	$0.236407\pi$
$72$	0	0
$73$	−4.48528	−0.524962	−0.262481	−	0.964937i	$-0.584541\pi$
$73$	−4.48528	−0.524962	−0.262481	+	0.964937i	$0.584541\pi$
$74$	−14.2426	−1.65567
$75$	0	0
$76$	0	0
$77$	−3.41421	−0.389086
$78$	0	0
$79$	−14.5858	−1.64103	−0.820515	−	0.571626i	$-0.806313\pi$
$79$	−14.5858	−1.64103	−0.820515	+	0.571626i	$0.806313\pi$
$80$	−4.00000	−0.447214
$81$	0	0
$82$	−2.34315	−0.258757
$83$	−3.41421	−0.374759	−0.187379	−	0.982288i	$-0.559999\pi$
$83$	−3.41421	−0.374759	−0.187379	+	0.982288i	$0.559999\pi$
$84$	0	0
$85$	−3.41421	−0.370323
$86$	−0.485281	−0.0523292
$87$	0	0
$88$	2.82843	0.301511
$89$	−4.89949	−0.519345	−0.259673	−	0.965697i	$-0.583615\pi$
$89$	−4.89949	−0.519345	−0.259673	+	0.965697i	$0.583615\pi$
$90$	0	0
$91$	−7.65685	−0.802656
$92$	0	0
$93$	0	0
$94$	12.4853	1.28776
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.41421	0.245126	0.122563	−	0.992461i	$-0.460889\pi$
$97$	2.41421	0.245126	0.122563	+	0.992461i	$0.460889\pi$
$98$	−6.58579	−0.665265
$99$	0	0
$100$	0	0
$101$	−6.24264	−0.621166	−0.310583	−	0.950546i	$-0.600524\pi$
$101$	−6.24264	−0.621166	−0.310583	+	0.950546i	$0.600524\pi$
$102$	0	0
$103$	13.6569	1.34565	0.672825	−	0.739802i	$-0.265081\pi$
$103$	13.6569	1.34565	0.672825	+	0.739802i	$0.265081\pi$
$104$	6.34315	0.621997
$105$	0	0
$106$	−6.34315	−0.616101
$107$	−15.6569	−1.51361	−0.756803	−	0.653643i	$-0.773240\pi$
$107$	−15.6569	−1.51361	−0.756803	+	0.653643i	$0.773240\pi$
$108$	0	0
$109$	−0.343146	−0.0328674	−0.0164337	−	0.999865i	$-0.505231\pi$
$109$	−0.343146	−0.0328674	−0.0164337	+	0.999865i	$0.505231\pi$
$110$	−1.41421	−0.134840
$111$	0	0
$112$	13.6569	1.29045
$113$	3.58579	0.337322	0.168661	−	0.985674i	$-0.446056\pi$
$113$	3.58579	0.337322	0.168661	+	0.985674i	$0.446056\pi$
$114$	0	0
$115$	3.00000	0.279751
$116$	0	0
$117$	0	0
$118$	−2.24264	−0.206452
$119$	11.6569	1.06858
$120$	0	0
$121$	1.00000	0.0909091
$122$	15.6569	1.41750
$123$	0	0
$124$	0	0
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−19.4142	−1.72273	−0.861366	−	0.507984i	$-0.830391\pi$
$127$	−19.4142	−1.72273	−0.861366	+	0.507984i	$0.830391\pi$
$128$	−11.3137	−1.00000
$129$	0	0
$130$	−3.17157	−0.278165
$131$	−14.4853	−1.26558	−0.632792	−	0.774321i	$-0.718091\pi$
$131$	−14.4853	−1.26558	−0.632792	+	0.774321i	$0.718091\pi$
$132$	0	0
$133$	3.41421	0.296050
$134$	14.7279	1.27230
$135$	0	0
$136$	−9.65685	−0.828068
$137$	2.65685	0.226990	0.113495	−	0.993539i	$-0.463795\pi$
$137$	2.65685	0.226990	0.113495	+	0.993539i	$0.463795\pi$
$138$	0	0
$139$	20.3848	1.72901	0.864507	−	0.502621i	$-0.167631\pi$
$139$	20.3848	1.72901	0.864507	+	0.502621i	$0.167631\pi$
$140$	0	0
$141$	0	0
$142$	−17.5563	−1.47330
$143$	2.24264	0.187539
$144$	0	0
$145$	6.24264	0.518423
$146$	6.34315	0.524962
$147$	0	0
$148$	0	0
$149$	−21.3137	−1.74609	−0.873044	−	0.487642i	$-0.837857\pi$
$149$	−21.3137	−1.74609	−0.873044	+	0.487642i	$0.837857\pi$
$150$	0	0
$151$	−10.4853	−0.853280	−0.426640	−	0.904422i	$-0.640303\pi$
$151$	−10.4853	−0.853280	−0.426640	+	0.904422i	$0.640303\pi$
$152$	−2.82843	−0.229416
$153$	0	0
$154$	4.82843	0.389086
$155$	−6.41421	−0.515202
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	20.6274	1.64103
$159$	0	0
$160$	0	0
$161$	−10.2426	−0.807233
$162$	0	0
$163$	−20.1421	−1.57765	−0.788827	−	0.614615i	$-0.789311\pi$
$163$	−20.1421	−1.57765	−0.788827	+	0.614615i	$0.789311\pi$
$164$	0	0
$165$	0	0
$166$	4.82843	0.374759
$167$	18.7279	1.44921	0.724605	−	0.689164i	$-0.242022\pi$
$167$	18.7279	1.44921	0.724605	+	0.689164i	$0.242022\pi$
$168$	0	0
$169$	−7.97056	−0.613120
$170$	4.82843	0.370323
$171$	0	0
$172$	0	0
$173$	−1.89949	−0.144416	−0.0722080	−	0.997390i	$-0.523005\pi$
$173$	−1.89949	−0.144416	−0.0722080	+	0.997390i	$0.523005\pi$
$174$	0	0
$175$	13.6569	1.03236
$176$	−4.00000	−0.301511
$177$	0	0
$178$	6.92893	0.519345
$179$	−16.0711	−1.20121	−0.600604	−	0.799547i	$-0.705073\pi$
$179$	−16.0711	−1.20121	−0.600604	+	0.799547i	$0.705073\pi$
$180$	0	0
$181$	−19.3848	−1.44086	−0.720430	−	0.693528i	$-0.756055\pi$
$181$	−19.3848	−1.44086	−0.720430	+	0.693528i	$0.756055\pi$
$182$	10.8284	0.802656
$183$	0	0
$184$	8.48528	0.625543
$185$	10.0711	0.740440
$186$	0	0
$187$	−3.41421	−0.249672
$188$	0	0
$189$	0	0
$190$	1.41421	0.102598
$191$	−8.31371	−0.601559	−0.300779	−	0.953694i	$-0.597247\pi$
$191$	−8.31371	−0.601559	−0.300779	+	0.953694i	$0.597247\pi$
$192$	0	0
$193$	1.17157	0.0843317	0.0421658	−	0.999111i	$-0.486574\pi$
$193$	1.17157	0.0843317	0.0421658	+	0.999111i	$0.486574\pi$
$194$	−3.41421	−0.245126
$195$	0	0
$196$	0	0
$197$	−15.8995	−1.13279	−0.566396	−	0.824133i	$-0.691663\pi$
$197$	−15.8995	−1.13279	−0.566396	+	0.824133i	$0.691663\pi$
$198$	0	0
$199$	12.1421	0.860733	0.430367	−	0.902654i	$-0.358384\pi$
$199$	12.1421	0.860733	0.430367	+	0.902654i	$0.358384\pi$
$200$	−11.3137	−0.800000
$201$	0	0
$202$	8.82843	0.621166
$203$	−21.3137	−1.49593
$204$	0	0
$205$	1.65685	0.115720
$206$	−19.3137	−1.34565
$207$	0	0
$208$	−8.97056	−0.621997
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−12.5858	−0.866441	−0.433221	−	0.901288i	$-0.642623\pi$
$211$	−12.5858	−0.866441	−0.433221	+	0.901288i	$0.642623\pi$
$212$	0	0
$213$	0	0
$214$	22.1421	1.51361
$215$	0.343146	0.0234023
$216$	0	0
$217$	21.8995	1.48663
$218$	0.485281	0.0328674
$219$	0	0
$220$	0	0
$221$	−7.65685	−0.515056
$222$	0	0
$223$	−16.4142	−1.09918	−0.549589	−	0.835435i	$-0.685215\pi$
$223$	−16.4142	−1.09918	−0.549589	+	0.835435i	$0.685215\pi$
$224$	0	0
$225$	0	0
$226$	−5.07107	−0.337322
$227$	−4.92893	−0.327145	−0.163572	−	0.986531i	$-0.552302\pi$
$227$	−4.92893	−0.327145	−0.163572	+	0.986531i	$0.552302\pi$
$228$	0	0
$229$	−14.3137	−0.945876	−0.472938	−	0.881096i	$-0.656807\pi$
$229$	−14.3137	−0.945876	−0.472938	+	0.881096i	$0.656807\pi$
$230$	−4.24264	−0.279751
$231$	0	0
$232$	17.6569	1.15923
$233$	0.242641	0.0158959	0.00794796	−	0.999968i	$-0.497470\pi$
$233$	0.242641	0.0158959	0.00794796	+	0.999968i	$0.497470\pi$
$234$	0	0
$235$	−8.82843	−0.575903
$236$	0	0
$237$	0	0
$238$	−16.4853	−1.06858
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−16.9706	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$242$	−1.41421	−0.0909091
$243$	0	0
$244$	0	0
$245$	4.65685	0.297516
$246$	0	0
$247$	−2.24264	−0.142696
$248$	−18.1421	−1.15203
$249$	0	0
$250$	12.7279	0.804984
$251$	11.3431	0.715973	0.357987	−	0.933727i	$-0.383463\pi$
$251$	11.3431	0.715973	0.357987	+	0.933727i	$0.383463\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	27.4558	1.72273
$255$	0	0
$256$	0	0
$257$	19.1716	1.19589	0.597945	−	0.801537i	$-0.295984\pi$
$257$	19.1716	1.19589	0.597945	+	0.801537i	$0.295984\pi$
$258$	0	0
$259$	−34.3848	−2.13657
$260$	0	0
$261$	0	0
$262$	20.4853	1.26558
$263$	9.51472	0.586703	0.293351	−	0.956005i	$-0.405229\pi$
$263$	9.51472	0.586703	0.293351	+	0.956005i	$0.405229\pi$
$264$	0	0
$265$	4.48528	0.275529
$266$	−4.82843	−0.296050
$267$	0	0
$268$	0	0
$269$	2.00000	0.121942	0.0609711	−	0.998140i	$-0.480580\pi$
$269$	2.00000	0.121942	0.0609711	+	0.998140i	$0.480580\pi$
$270$	0	0
$271$	6.14214	0.373108	0.186554	−	0.982445i	$-0.440268\pi$
$271$	6.14214	0.373108	0.186554	+	0.982445i	$0.440268\pi$
$272$	13.6569	0.828068
$273$	0	0
$274$	−3.75736	−0.226990
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−2.48528	−0.149326	−0.0746630	−	0.997209i	$-0.523788\pi$
$277$	−2.48528	−0.149326	−0.0746630	+	0.997209i	$0.523788\pi$
$278$	−28.8284	−1.72901
$279$	0	0
$280$	−9.65685	−0.577107
$281$	−23.6569	−1.41125	−0.705625	−	0.708586i	$-0.749334\pi$
$281$	−23.6569	−1.41125	−0.705625	+	0.708586i	$0.749334\pi$
$282$	0	0
$283$	−16.7279	−0.994372	−0.497186	−	0.867644i	$-0.665633\pi$
$283$	−16.7279	−0.994372	−0.497186	+	0.867644i	$0.665633\pi$
$284$	0	0
$285$	0	0
$286$	−3.17157	−0.187539
$287$	−5.65685	−0.333914
$288$	0	0
$289$	−5.34315	−0.314303
$290$	−8.82843	−0.518423
$291$	0	0
$292$	0	0
$293$	31.6985	1.85185	0.925923	−	0.377713i	$-0.123289\pi$
$293$	31.6985	1.85185	0.925923	+	0.377713i	$0.123289\pi$
$294$	0	0
$295$	1.58579	0.0923281
$296$	28.4853	1.65567
$297$	0	0
$298$	30.1421	1.74609
$299$	6.72792	0.389086
$300$	0	0
$301$	−1.17157	−0.0675283
$302$	14.8284	0.853280
$303$	0	0
$304$	4.00000	0.229416
$305$	−11.0711	−0.633927
$306$	0	0
$307$	7.41421	0.423152	0.211576	−	0.977362i	$-0.432140\pi$
$307$	7.41421	0.423152	0.211576	+	0.977362i	$0.432140\pi$
$308$	0	0
$309$	0	0
$310$	9.07107	0.515202
$311$	11.6569	0.661000	0.330500	−	0.943806i	$-0.392783\pi$
$311$	11.6569	0.661000	0.330500	+	0.943806i	$0.392783\pi$
$312$	0	0
$313$	−13.9706	−0.789663	−0.394831	−	0.918754i	$-0.629197\pi$
$313$	−13.9706	−0.789663	−0.394831	+	0.918754i	$0.629197\pi$
$314$	−7.07107	−0.399043
$315$	0	0
$316$	0	0
$317$	28.4142	1.59590	0.797951	−	0.602723i	$-0.205918\pi$
$317$	28.4142	1.59590	0.797951	+	0.602723i	$0.205918\pi$
$318$	0	0
$319$	6.24264	0.349521
$320$	8.00000	0.447214
$321$	0	0
$322$	14.4853	0.807233
$323$	3.41421	0.189972
$324$	0	0
$325$	−8.97056	−0.497597
$326$	28.4853	1.57765
$327$	0	0
$328$	4.68629	0.258757
$329$	30.1421	1.66179
$330$	0	0
$331$	34.2132	1.88053	0.940264	−	0.340447i	$-0.110578\pi$
$331$	34.2132	1.88053	0.940264	+	0.340447i	$0.110578\pi$
$332$	0	0
$333$	0	0
$334$	−26.4853	−1.44921
$335$	−10.4142	−0.568989
$336$	0	0
$337$	16.7279	0.911228	0.455614	−	0.890177i	$-0.349420\pi$
$337$	16.7279	0.911228	0.455614	+	0.890177i	$0.349420\pi$
$338$	11.2721	0.613120
$339$	0	0
$340$	0	0
$341$	−6.41421	−0.347349
$342$	0	0
$343$	8.00000	0.431959
$344$	0.970563	0.0523292
$345$	0	0
$346$	2.68629	0.144416
$347$	−33.6985	−1.80903	−0.904515	−	0.426442i	$-0.859767\pi$
$347$	−33.6985	−1.80903	−0.904515	+	0.426442i	$0.859767\pi$
$348$	0	0
$349$	23.2132	1.24257	0.621287	−	0.783583i	$-0.286610\pi$
$349$	23.2132	1.24257	0.621287	+	0.783583i	$0.286610\pi$
$350$	−19.3137	−1.03236
$351$	0	0
$352$	0	0
$353$	24.3137	1.29409	0.647044	−	0.762453i	$-0.276005\pi$
$353$	24.3137	1.29409	0.647044	+	0.762453i	$0.276005\pi$
$354$	0	0
$355$	12.4142	0.658878
$356$	0	0
$357$	0	0
$358$	22.7279	1.20121
$359$	−16.4853	−0.870060	−0.435030	−	0.900416i	$-0.643262\pi$
$359$	−16.4853	−0.870060	−0.435030	+	0.900416i	$0.643262\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	27.4142	1.44086
$363$	0	0
$364$	0	0
$365$	−4.48528	−0.234770
$366$	0	0
$367$	−9.82843	−0.513040	−0.256520	−	0.966539i	$-0.582576\pi$
$367$	−9.82843	−0.513040	−0.256520	+	0.966539i	$0.582576\pi$
$368$	−12.0000	−0.625543
$369$	0	0
$370$	−14.2426	−0.740440
$371$	−15.3137	−0.795048
$372$	0	0
$373$	−2.68629	−0.139091	−0.0695455	−	0.997579i	$-0.522155\pi$
$373$	−2.68629	−0.139091	−0.0695455	+	0.997579i	$0.522155\pi$
$374$	4.82843	0.249672
$375$	0	0
$376$	−24.9706	−1.28776
$377$	14.0000	0.721037
$378$	0	0
$379$	−30.6985	−1.57688	−0.788438	−	0.615115i	$-0.789110\pi$
$379$	−30.6985	−1.57688	−0.788438	+	0.615115i	$0.789110\pi$
$380$	0	0
$381$	0	0
$382$	11.7574	0.601559
$383$	−33.5269	−1.71315	−0.856573	−	0.516027i	$-0.827411\pi$
$383$	−33.5269	−1.71315	−0.856573	+	0.516027i	$0.827411\pi$
$384$	0	0
$385$	−3.41421	−0.174004
$386$	−1.65685	−0.0843317
$387$	0	0
$388$	0	0
$389$	28.3137	1.43556	0.717781	−	0.696269i	$-0.245158\pi$
$389$	28.3137	1.43556	0.717781	+	0.696269i	$0.245158\pi$
$390$	0	0
$391$	−10.2426	−0.517993
$392$	13.1716	0.665265
$393$	0	0
$394$	22.4853	1.13279
$395$	−14.5858	−0.733891
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	−17.1716	−0.860733
$399$	0	0
$400$	16.0000	0.800000
$401$	34.9706	1.74635	0.873173	−	0.487410i	$-0.162058\pi$
$401$	34.9706	1.74635	0.873173	+	0.487410i	$0.162058\pi$
$402$	0	0
$403$	−14.3848	−0.716557
$404$	0	0
$405$	0	0
$406$	30.1421	1.49593
$407$	10.0711	0.499204
$408$	0	0
$409$	0.727922	0.0359934	0.0179967	−	0.999838i	$-0.494271\pi$
$409$	0.727922	0.0359934	0.0179967	+	0.999838i	$0.494271\pi$
$410$	−2.34315	−0.115720
$411$	0	0
$412$	0	0
$413$	−5.41421	−0.266416
$414$	0	0
$415$	−3.41421	−0.167597
$416$	0	0
$417$	0	0
$418$	1.41421	0.0691714
$419$	27.4558	1.34131	0.670653	−	0.741771i	$-0.266014\pi$
$419$	27.4558	1.34131	0.670653	+	0.741771i	$0.266014\pi$
$420$	0	0
$421$	−28.1421	−1.37156	−0.685782	−	0.727807i	$-0.740540\pi$
$421$	−28.1421	−1.37156	−0.685782	+	0.727807i	$0.740540\pi$
$422$	17.7990	0.866441
$423$	0	0
$424$	12.6863	0.616101
$425$	13.6569	0.662455
$426$	0	0
$427$	37.7990	1.82922
$428$	0	0
$429$	0	0
$430$	−0.485281	−0.0234023
$431$	−30.4853	−1.46842	−0.734212	−	0.678920i	$-0.762448\pi$
$431$	−30.4853	−1.46842	−0.734212	+	0.678920i	$0.762448\pi$
$432$	0	0
$433$	11.3848	0.547117	0.273559	−	0.961855i	$-0.411799\pi$
$433$	11.3848	0.547117	0.273559	+	0.961855i	$0.411799\pi$
$434$	−30.9706	−1.48663
$435$	0	0
$436$	0	0
$437$	−3.00000	−0.143509
$438$	0	0
$439$	−14.2426	−0.679764	−0.339882	−	0.940468i	$-0.610387\pi$
$439$	−14.2426	−0.679764	−0.339882	+	0.940468i	$0.610387\pi$
$440$	2.82843	0.134840
$441$	0	0
$442$	10.8284	0.515056
$443$	11.9706	0.568739	0.284369	−	0.958715i	$-0.408216\pi$
$443$	11.9706	0.568739	0.284369	+	0.958715i	$0.408216\pi$
$444$	0	0
$445$	−4.89949	−0.232258
$446$	23.2132	1.09918
$447$	0	0
$448$	−27.3137	−1.29045
$449$	13.3848	0.631667	0.315833	−	0.948815i	$-0.397716\pi$
$449$	13.3848	0.631667	0.315833	+	0.948815i	$0.397716\pi$
$450$	0	0
$451$	1.65685	0.0780182
$452$	0	0
$453$	0	0
$454$	6.97056	0.327145
$455$	−7.65685	−0.358959
$456$	0	0
$457$	16.9289	0.791902	0.395951	−	0.918272i	$-0.370415\pi$
$457$	16.9289	0.791902	0.395951	+	0.918272i	$0.370415\pi$
$458$	20.2426	0.945876
$459$	0	0
$460$	0	0
$461$	−12.9706	−0.604099	−0.302050	−	0.953292i	$-0.597671\pi$
$461$	−12.9706	−0.604099	−0.302050	+	0.953292i	$0.597671\pi$
$462$	0	0
$463$	12.4558	0.578872	0.289436	−	0.957197i	$-0.406532\pi$
$463$	12.4558	0.578872	0.289436	+	0.957197i	$0.406532\pi$
$464$	−24.9706	−1.15923
$465$	0	0
$466$	−0.343146	−0.0158959
$467$	−9.68629	−0.448228	−0.224114	−	0.974563i	$-0.571949\pi$
$467$	−9.68629	−0.448228	−0.224114	+	0.974563i	$0.571949\pi$
$468$	0	0
$469$	35.5563	1.64184
$470$	12.4853	0.575903
$471$	0	0
$472$	4.48528	0.206452
$473$	0.343146	0.0157779
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	8.48528	0.388108
$479$	40.5269	1.85172	0.925861	−	0.377864i	$-0.123341\pi$
$479$	40.5269	1.85172	0.925861	+	0.377864i	$0.123341\pi$
$480$	0	0
$481$	22.5858	1.02982
$482$	24.0000	1.09317
$483$	0	0
$484$	0	0
$485$	2.41421	0.109624
$486$	0	0
$487$	12.5563	0.568982	0.284491	−	0.958679i	$-0.408175\pi$
$487$	12.5563	0.568982	0.284491	+	0.958679i	$0.408175\pi$
$488$	−31.3137	−1.41750
$489$	0	0
$490$	−6.58579	−0.297516
$491$	−27.7574	−1.25267	−0.626336	−	0.779553i	$-0.715446\pi$
$491$	−27.7574	−1.25267	−0.626336	+	0.779553i	$0.715446\pi$
$492$	0	0
$493$	−21.3137	−0.959921
$494$	3.17157	0.142696
$495$	0	0
$496$	25.6569	1.15203
$497$	−42.3848	−1.90122
$498$	0	0
$499$	−32.6274	−1.46060	−0.730302	−	0.683125i	$-0.760621\pi$
$499$	−32.6274	−1.46060	−0.730302	+	0.683125i	$0.760621\pi$
$500$	0	0
$501$	0	0
$502$	−16.0416	−0.715973
$503$	28.1421	1.25480	0.627398	−	0.778699i	$-0.284120\pi$
$503$	28.1421	1.25480	0.627398	+	0.778699i	$0.284120\pi$
$504$	0	0
$505$	−6.24264	−0.277794
$506$	−4.24264	−0.188608
$507$	0	0
$508$	0	0
$509$	8.21320	0.364044	0.182022	−	0.983294i	$-0.441736\pi$
$509$	8.21320	0.364044	0.182022	+	0.983294i	$0.441736\pi$
$510$	0	0
$511$	15.3137	0.677439
$512$	22.6274	1.00000
$513$	0	0
$514$	−27.1127	−1.19589
$515$	13.6569	0.601793
$516$	0	0
$517$	−8.82843	−0.388274
$518$	48.6274	2.13657
$519$	0	0
$520$	6.34315	0.278165
$521$	−0.556349	−0.0243741	−0.0121871	−	0.999926i	$-0.503879\pi$
$521$	−0.556349	−0.0243741	−0.0121871	+	0.999926i	$0.503879\pi$
$522$	0	0
$523$	2.34315	0.102459	0.0512293	−	0.998687i	$-0.483686\pi$
$523$	2.34315	0.102459	0.0512293	+	0.998687i	$0.483686\pi$
$524$	0	0
$525$	0	0
$526$	−13.4558	−0.586703
$527$	21.8995	0.953957
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−6.34315	−0.275529
$531$	0	0
$532$	0	0
$533$	3.71573	0.160946
$534$	0	0
$535$	−15.6569	−0.676905
$536$	−29.4558	−1.27230
$537$	0	0
$538$	−2.82843	−0.121942
$539$	4.65685	0.200585
$540$	0	0
$541$	9.21320	0.396107	0.198053	−	0.980191i	$-0.436538\pi$
$541$	9.21320	0.396107	0.198053	+	0.980191i	$0.436538\pi$
$542$	−8.68629	−0.373108
$543$	0	0
$544$	0	0
$545$	−0.343146	−0.0146987
$546$	0	0
$547$	−14.7279	−0.629720	−0.314860	−	0.949138i	$-0.601958\pi$
$547$	−14.7279	−0.629720	−0.314860	+	0.949138i	$0.601958\pi$
$548$	0	0
$549$	0	0
$550$	5.65685	0.241209
$551$	−6.24264	−0.265945
$552$	0	0
$553$	49.7990	2.11767
$554$	3.51472	0.149326
$555$	0	0
$556$	0	0
$557$	27.9411	1.18390	0.591952	−	0.805973i	$-0.298358\pi$
$557$	27.9411	1.18390	0.591952	+	0.805973i	$0.298358\pi$
$558$	0	0
$559$	0.769553	0.0325486
$560$	13.6569	0.577107
$561$	0	0
$562$	33.4558	1.41125
$563$	23.2132	0.978320	0.489160	−	0.872194i	$-0.337303\pi$
$563$	23.2132	0.978320	0.489160	+	0.872194i	$0.337303\pi$
$564$	0	0
$565$	3.58579	0.150855
$566$	23.6569	0.994372
$567$	0	0
$568$	35.1127	1.47330
$569$	26.2426	1.10015	0.550074	−	0.835116i	$-0.314599\pi$
$569$	26.2426	1.10015	0.550074	+	0.835116i	$0.314599\pi$
$570$	0	0
$571$	33.6985	1.41024	0.705119	−	0.709089i	$-0.250894\pi$
$571$	33.6985	1.41024	0.705119	+	0.709089i	$0.250894\pi$
$572$	0	0
$573$	0	0
$574$	8.00000	0.333914
$575$	−12.0000	−0.500435
$576$	0	0
$577$	29.9706	1.24769	0.623845	−	0.781548i	$-0.285569\pi$
$577$	29.9706	1.24769	0.623845	+	0.781548i	$0.285569\pi$
$578$	7.55635	0.314303
$579$	0	0
$580$	0	0
$581$	11.6569	0.483608
$582$	0	0
$583$	4.48528	0.185761
$584$	−12.6863	−0.524962
$585$	0	0
$586$	−44.8284	−1.85185
$587$	−7.65685	−0.316032	−0.158016	−	0.987437i	$-0.550510\pi$
$587$	−7.65685	−0.316032	−0.158016	+	0.987437i	$0.550510\pi$
$588$	0	0
$589$	6.41421	0.264293
$590$	−2.24264	−0.0923281
$591$	0	0
$592$	−40.2843	−1.65567
$593$	−36.9706	−1.51820	−0.759100	−	0.650975i	$-0.774360\pi$
$593$	−36.9706	−1.51820	−0.759100	+	0.650975i	$0.774360\pi$
$594$	0	0
$595$	11.6569	0.477884
$596$	0	0
$597$	0	0
$598$	−9.51472	−0.389086
$599$	−13.3137	−0.543983	−0.271992	−	0.962300i	$-0.587682\pi$
$599$	−13.3137	−0.543983	−0.271992	+	0.962300i	$0.587682\pi$
$600$	0	0
$601$	11.5563	0.471393	0.235697	−	0.971827i	$-0.424263\pi$
$601$	11.5563	0.471393	0.235697	+	0.971827i	$0.424263\pi$
$602$	1.65685	0.0675283
$603$	0	0
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	−2.10051	−0.0852569	−0.0426284	−	0.999091i	$-0.513573\pi$
$607$	−2.10051	−0.0852569	−0.0426284	+	0.999091i	$0.513573\pi$
$608$	0	0
$609$	0	0
$610$	15.6569	0.633927
$611$	−19.7990	−0.800981
$612$	0	0
$613$	−13.4142	−0.541795	−0.270897	−	0.962608i	$-0.587320\pi$
$613$	−13.4142	−0.541795	−0.270897	+	0.962608i	$0.587320\pi$
$614$	−10.4853	−0.423152
$615$	0	0
$616$	−9.65685	−0.389086
$617$	−22.8284	−0.919038	−0.459519	−	0.888168i	$-0.651978\pi$
$617$	−22.8284	−0.919038	−0.459519	+	0.888168i	$0.651978\pi$
$618$	0	0
$619$	23.4853	0.943953	0.471977	−	0.881611i	$-0.343541\pi$
$619$	23.4853	0.943953	0.471977	+	0.881611i	$0.343541\pi$
$620$	0	0
$621$	0	0
$622$	−16.4853	−0.661000
$623$	16.7279	0.670190
$624$	0	0
$625$	11.0000	0.440000
$626$	19.7574	0.789663
$627$	0	0
$628$	0	0
$629$	−34.3848	−1.37101
$630$	0	0
$631$	−8.02944	−0.319647	−0.159823	−	0.987146i	$-0.551092\pi$
$631$	−8.02944	−0.319647	−0.159823	+	0.987146i	$0.551092\pi$
$632$	−41.2548	−1.64103
$633$	0	0
$634$	−40.1838	−1.59590
$635$	−19.4142	−0.770430
$636$	0	0
$637$	10.4437	0.413793
$638$	−8.82843	−0.349521
$639$	0	0
$640$	−11.3137	−0.447214
$641$	−0.899495	−0.0355279	−0.0177640	−	0.999842i	$-0.505655\pi$
$641$	−0.899495	−0.0355279	−0.0177640	+	0.999842i	$0.505655\pi$
$642$	0	0
$643$	24.6569	0.972371	0.486186	−	0.873856i	$-0.338388\pi$
$643$	24.6569	0.972371	0.486186	+	0.873856i	$0.338388\pi$
$644$	0	0
$645$	0	0
$646$	−4.82843	−0.189972
$647$	30.9411	1.21642	0.608211	−	0.793776i	$-0.291888\pi$
$647$	30.9411	1.21642	0.608211	+	0.793776i	$0.291888\pi$
$648$	0	0
$649$	1.58579	0.0622476
$650$	12.6863	0.497597
$651$	0	0
$652$	0	0
$653$	8.51472	0.333207	0.166603	−	0.986024i	$-0.446720\pi$
$653$	8.51472	0.333207	0.166603	+	0.986024i	$0.446720\pi$
$654$	0	0
$655$	−14.4853	−0.565987
$656$	−6.62742	−0.258757
$657$	0	0
$658$	−42.6274	−1.66179
$659$	−21.7990	−0.849168	−0.424584	−	0.905389i	$-0.639580\pi$
$659$	−21.7990	−0.849168	−0.424584	+	0.905389i	$0.639580\pi$
$660$	0	0
$661$	−49.8701	−1.93972	−0.969860	−	0.243662i	$-0.921651\pi$
$661$	−49.8701	−1.93972	−0.969860	+	0.243662i	$0.921651\pi$
$662$	−48.3848	−1.88053
$663$	0	0
$664$	−9.65685	−0.374759
$665$	3.41421	0.132398
$666$	0	0
$667$	18.7279	0.725148
$668$	0	0
$669$	0	0
$670$	14.7279	0.568989
$671$	−11.0711	−0.427394
$672$	0	0
$673$	−3.85786	−0.148710	−0.0743549	−	0.997232i	$-0.523690\pi$
$673$	−3.85786	−0.148710	−0.0743549	+	0.997232i	$0.523690\pi$
$674$	−23.6569	−0.911228
$675$	0	0
$676$	0	0
$677$	−21.3137	−0.819152	−0.409576	−	0.912276i	$-0.634323\pi$
$677$	−21.3137	−0.819152	−0.409576	+	0.912276i	$0.634323\pi$
$678$	0	0
$679$	−8.24264	−0.316324
$680$	−9.65685	−0.370323
$681$	0	0
$682$	9.07107	0.347349
$683$	−23.1127	−0.884383	−0.442191	−	0.896921i	$-0.645799\pi$
$683$	−23.1127	−0.884383	−0.442191	+	0.896921i	$0.645799\pi$
$684$	0	0
$685$	2.65685	0.101513
$686$	−11.3137	−0.431959
$687$	0	0
$688$	−1.37258	−0.0523292
$689$	10.0589	0.383213
$690$	0	0
$691$	25.9706	0.987967	0.493983	−	0.869471i	$-0.335540\pi$
$691$	25.9706	0.987967	0.493983	+	0.869471i	$0.335540\pi$
$692$	0	0
$693$	0	0
$694$	47.6569	1.80903
$695$	20.3848	0.773239
$696$	0	0
$697$	−5.65685	−0.214269
$698$	−32.8284	−1.24257
$699$	0	0
$700$	0	0
$701$	1.31371	0.0496181	0.0248090	−	0.999692i	$-0.492102\pi$
$701$	1.31371	0.0496181	0.0248090	+	0.999692i	$0.492102\pi$
$702$	0	0
$703$	−10.0711	−0.379838
$704$	8.00000	0.301511
$705$	0	0
$706$	−34.3848	−1.29409
$707$	21.3137	0.801585
$708$	0	0
$709$	−25.2843	−0.949571	−0.474785	−	0.880102i	$-0.657474\pi$
$709$	−25.2843	−0.949571	−0.474785	+	0.880102i	$0.657474\pi$
$710$	−17.5563	−0.658878
$711$	0	0
$712$	−13.8579	−0.519345
$713$	−19.2426	−0.720643
$714$	0	0
$715$	2.24264	0.0838700
$716$	0	0
$717$	0	0
$718$	23.3137	0.870060
$719$	−10.0294	−0.374035	−0.187017	−	0.982357i	$-0.559882\pi$
$719$	−10.0294	−0.374035	−0.187017	+	0.982357i	$0.559882\pi$
$720$	0	0
$721$	−46.6274	−1.73650
$722$	−1.41421	−0.0526316
$723$	0	0
$724$	0	0
$725$	−24.9706	−0.927383
$726$	0	0
$727$	−18.1127	−0.671763	−0.335881	−	0.941904i	$-0.609034\pi$
$727$	−18.1127	−0.671763	−0.335881	+	0.941904i	$0.609034\pi$
$728$	−21.6569	−0.802656
$729$	0	0
$730$	6.34315	0.234770
$731$	−1.17157	−0.0433322
$732$	0	0
$733$	13.4142	0.495465	0.247733	−	0.968828i	$-0.420315\pi$
$733$	13.4142	0.495465	0.247733	+	0.968828i	$0.420315\pi$
$734$	13.8995	0.513040
$735$	0	0
$736$	0	0
$737$	−10.4142	−0.383612
$738$	0	0
$739$	−27.4142	−1.00845	−0.504224	−	0.863573i	$-0.668221\pi$
$739$	−27.4142	−1.00845	−0.504224	+	0.863573i	$0.668221\pi$
$740$	0	0
$741$	0	0
$742$	21.6569	0.795048
$743$	6.92893	0.254198	0.127099	−	0.991890i	$-0.459433\pi$
$743$	6.92893	0.254198	0.127099	+	0.991890i	$0.459433\pi$
$744$	0	0
$745$	−21.3137	−0.780874
$746$	3.79899	0.139091
$747$	0	0
$748$	0	0
$749$	53.4558	1.95323
$750$	0	0
$751$	3.44365	0.125661	0.0628303	−	0.998024i	$-0.479987\pi$
$751$	3.44365	0.125661	0.0628303	+	0.998024i	$0.479987\pi$
$752$	35.3137	1.28776
$753$	0	0
$754$	−19.7990	−0.721037
$755$	−10.4853	−0.381598
$756$	0	0
$757$	41.9411	1.52438	0.762188	−	0.647356i	$-0.224125\pi$
$757$	41.9411	1.52438	0.762188	+	0.647356i	$0.224125\pi$
$758$	43.4142	1.57688
$759$	0	0
$760$	−2.82843	−0.102598
$761$	−5.85786	−0.212347	−0.106174	−	0.994348i	$-0.533860\pi$
$761$	−5.85786	−0.212347	−0.106174	+	0.994348i	$0.533860\pi$
$762$	0	0
$763$	1.17157	0.0424138
$764$	0	0
$765$	0	0
$766$	47.4142	1.71315
$767$	3.55635	0.128412
$768$	0	0
$769$	1.85786	0.0669963	0.0334982	−	0.999439i	$-0.489335\pi$
$769$	1.85786	0.0669963	0.0334982	+	0.999439i	$0.489335\pi$
$770$	4.82843	0.174004
$771$	0	0
$772$	0	0
$773$	−19.6569	−0.707008	−0.353504	−	0.935433i	$-0.615010\pi$
$773$	−19.6569	−0.707008	−0.353504	+	0.935433i	$0.615010\pi$
$774$	0	0
$775$	25.6569	0.921621
$776$	6.82843	0.245126
$777$	0	0
$778$	−40.0416	−1.43556
$779$	−1.65685	−0.0593630
$780$	0	0
$781$	12.4142	0.444215
$782$	14.4853	0.517993
$783$	0	0
$784$	−18.6274	−0.665265
$785$	5.00000	0.178458
$786$	0	0
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	0	0
$789$	0	0
$790$	20.6274	0.733891
$791$	−12.2426	−0.435298
$792$	0	0
$793$	−24.8284	−0.881683
$794$	−36.7696	−1.30490
$795$	0	0
$796$	0	0
$797$	13.2426	0.469078	0.234539	−	0.972107i	$-0.424642\pi$
$797$	13.2426	0.469078	0.234539	+	0.972107i	$0.424642\pi$
$798$	0	0
$799$	30.1421	1.06635
$800$	0	0
$801$	0	0
$802$	−49.4558	−1.74635
$803$	−4.48528	−0.158282
$804$	0	0
$805$	−10.2426	−0.361006
$806$	20.3431	0.716557
$807$	0	0
$808$	−17.6569	−0.621166
$809$	−35.5147	−1.24863	−0.624316	−	0.781172i	$-0.714622\pi$
$809$	−35.5147	−1.24863	−0.624316	+	0.781172i	$0.714622\pi$
$810$	0	0
$811$	43.7574	1.53653	0.768264	−	0.640133i	$-0.221121\pi$
$811$	43.7574	1.53653	0.768264	+	0.640133i	$0.221121\pi$
$812$	0	0
$813$	0	0
$814$	−14.2426	−0.499204
$815$	−20.1421	−0.705548
$816$	0	0
$817$	−0.343146	−0.0120052
$818$	−1.02944	−0.0359934
$819$	0	0
$820$	0	0
$821$	38.5269	1.34460	0.672299	−	0.740279i	$-0.265307\pi$
$821$	38.5269	1.34460	0.672299	+	0.740279i	$0.265307\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	38.6274	1.34565
$825$	0	0
$826$	7.65685	0.266416
$827$	−23.8995	−0.831067	−0.415533	−	0.909578i	$-0.636405\pi$
$827$	−23.8995	−0.831067	−0.415533	+	0.909578i	$0.636405\pi$
$828$	0	0
$829$	−37.0416	−1.28651	−0.643255	−	0.765652i	$-0.722416\pi$
$829$	−37.0416	−1.28651	−0.643255	+	0.765652i	$0.722416\pi$
$830$	4.82843	0.167597
$831$	0	0
$832$	17.9411	0.621997
$833$	−15.8995	−0.550885
$834$	0	0
$835$	18.7279	0.648106
$836$	0	0
$837$	0	0
$838$	−38.8284	−1.34131
$839$	−9.44365	−0.326031	−0.163016	−	0.986624i	$-0.552122\pi$
$839$	−9.44365	−0.326031	−0.163016	+	0.986624i	$0.552122\pi$
$840$	0	0
$841$	9.97056	0.343813
$842$	39.7990	1.37156
$843$	0	0
$844$	0	0
$845$	−7.97056	−0.274196
$846$	0	0
$847$	−3.41421	−0.117314
$848$	−17.9411	−0.616101
$849$	0	0
$850$	−19.3137	−0.662455
$851$	30.2132	1.03570
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	−53.4558	−1.82922
$855$	0	0
$856$	−44.2843	−1.51361
$857$	11.6985	0.399613	0.199806	−	0.979835i	$-0.435969\pi$
$857$	11.6985	0.399613	0.199806	+	0.979835i	$0.435969\pi$
$858$	0	0
$859$	27.2843	0.930927	0.465464	−	0.885067i	$-0.345888\pi$
$859$	27.2843	0.930927	0.465464	+	0.885067i	$0.345888\pi$
$860$	0	0
$861$	0	0
$862$	43.1127	1.46842
$863$	31.5980	1.07561	0.537804	−	0.843070i	$-0.319254\pi$
$863$	31.5980	1.07561	0.537804	+	0.843070i	$0.319254\pi$
$864$	0	0
$865$	−1.89949	−0.0645848
$866$	−16.1005	−0.547117
$867$	0	0
$868$	0	0
$869$	−14.5858	−0.494789
$870$	0	0
$871$	−23.3553	−0.791365
$872$	−0.970563	−0.0328674
$873$	0	0
$874$	4.24264	0.143509
$875$	30.7279	1.03879
$876$	0	0
$877$	−38.1421	−1.28797	−0.643984	−	0.765039i	$-0.722720\pi$
$877$	−38.1421	−1.28797	−0.643984	+	0.765039i	$0.722720\pi$
$878$	20.1421	0.679764
$879$	0	0
$880$	−4.00000	−0.134840
$881$	51.7696	1.74416	0.872080	−	0.489363i	$-0.162771\pi$
$881$	51.7696	1.74416	0.872080	+	0.489363i	$0.162771\pi$
$882$	0	0
$883$	50.4853	1.69896	0.849482	−	0.527617i	$-0.176914\pi$
$883$	50.4853	1.69896	0.849482	+	0.527617i	$0.176914\pi$
$884$	0	0
$885$	0	0
$886$	−16.9289	−0.568739
$887$	−39.2548	−1.31805	−0.659024	−	0.752122i	$-0.729031\pi$
$887$	−39.2548	−1.31805	−0.659024	+	0.752122i	$0.729031\pi$
$888$	0	0
$889$	66.2843	2.22310
$890$	6.92893	0.232258
$891$	0	0
$892$	0	0
$893$	8.82843	0.295432
$894$	0	0
$895$	−16.0711	−0.537197
$896$	38.6274	1.29045
$897$	0	0
$898$	−18.9289	−0.631667
$899$	−40.0416	−1.33546
$900$	0	0
$901$	−15.3137	−0.510174
$902$	−2.34315	−0.0780182
$903$	0	0
$904$	10.1421	0.337322
$905$	−19.3848	−0.644372
$906$	0	0
$907$	−21.8579	−0.725778	−0.362889	−	0.931832i	$-0.618210\pi$
$907$	−21.8579	−0.725778	−0.362889	+	0.931832i	$0.618210\pi$
$908$	0	0
$909$	0	0
$910$	10.8284	0.358959
$911$	−38.4264	−1.27312	−0.636562	−	0.771226i	$-0.719644\pi$
$911$	−38.4264	−1.27312	−0.636562	+	0.771226i	$0.719644\pi$
$912$	0	0
$913$	−3.41421	−0.112994
$914$	−23.9411	−0.791902
$915$	0	0
$916$	0	0
$917$	49.4558	1.63318
$918$	0	0
$919$	−56.3848	−1.85996	−0.929981	−	0.367607i	$-0.880177\pi$
$919$	−56.3848	−1.85996	−0.929981	+	0.367607i	$0.880177\pi$
$920$	8.48528	0.279751
$921$	0	0
$922$	18.3431	0.604099
$923$	27.8406	0.916385
$924$	0	0
$925$	−40.2843	−1.32454
$926$	−17.6152	−0.578872
$927$	0	0
$928$	0	0
$929$	−0.284271	−0.00932664	−0.00466332	−	0.999989i	$-0.501484\pi$
$929$	−0.284271	−0.00932664	−0.00466332	+	0.999989i	$0.501484\pi$
$930$	0	0
$931$	−4.65685	−0.152622
$932$	0	0
$933$	0	0
$934$	13.6985	0.448228
$935$	−3.41421	−0.111657
$936$	0	0
$937$	−7.55635	−0.246855	−0.123428	−	0.992354i	$-0.539389\pi$
$937$	−7.55635	−0.246855	−0.123428	+	0.992354i	$0.539389\pi$
$938$	−50.2843	−1.64184
$939$	0	0
$940$	0	0
$941$	−28.2426	−0.920684	−0.460342	−	0.887742i	$-0.652273\pi$
$941$	−28.2426	−0.920684	−0.460342	+	0.887742i	$0.652273\pi$
$942$	0	0
$943$	4.97056	0.161864
$944$	−6.34315	−0.206452
$945$	0	0
$946$	−0.485281	−0.0157779
$947$	27.9706	0.908921	0.454461	−	0.890767i	$-0.349832\pi$
$947$	27.9706	0.908921	0.454461	+	0.890767i	$0.349832\pi$
$948$	0	0
$949$	−10.0589	−0.326525
$950$	−5.65685	−0.183533
$951$	0	0
$952$	32.9706	1.06858
$953$	−6.10051	−0.197615	−0.0988074	−	0.995107i	$-0.531503\pi$
$953$	−6.10051	−0.197615	−0.0988074	+	0.995107i	$0.531503\pi$
$954$	0	0
$955$	−8.31371	−0.269025
$956$	0	0
$957$	0	0
$958$	−57.3137	−1.85172
$959$	−9.07107	−0.292920
$960$	0	0
$961$	10.1421	0.327166
$962$	−31.9411	−1.02982
$963$	0	0
$964$	0	0
$965$	1.17157	0.0377143
$966$	0	0
$967$	3.55635	0.114364	0.0571822	−	0.998364i	$-0.481788\pi$
$967$	3.55635	0.114364	0.0571822	+	0.998364i	$0.481788\pi$
$968$	2.82843	0.0909091
$969$	0	0
$970$	−3.41421	−0.109624
$971$	−0.272078	−0.00873140	−0.00436570	−	0.999990i	$-0.501390\pi$
$971$	−0.272078	−0.00873140	−0.00436570	+	0.999990i	$0.501390\pi$
$972$	0	0
$973$	−69.5980	−2.23121
$974$	−17.7574	−0.568982
$975$	0	0
$976$	44.2843	1.41750
$977$	−42.8406	−1.37059	−0.685296	−	0.728264i	$-0.740327\pi$
$977$	−42.8406	−1.37059	−0.685296	+	0.728264i	$0.740327\pi$
$978$	0	0
$979$	−4.89949	−0.156589
$980$	0	0
$981$	0	0
$982$	39.2548	1.25267
$983$	−32.8995	−1.04933	−0.524665	−	0.851308i	$-0.675810\pi$
$983$	−32.8995	−1.04933	−0.524665	+	0.851308i	$0.675810\pi$
$984$	0	0
$985$	−15.8995	−0.506600
$986$	30.1421	0.959921
$987$	0	0
$988$	0	0
$989$	1.02944	0.0327342
$990$	0	0
$991$	−31.3137	−0.994713	−0.497356	−	0.867546i	$-0.665696\pi$
$991$	−31.3137	−0.994713	−0.497356	+	0.867546i	$0.665696\pi$
$992$	0	0
$993$	0	0
$994$	59.9411	1.90122
$995$	12.1421	0.384932
$996$	0	0
$997$	−6.44365	−0.204072	−0.102036	−	0.994781i	$-0.532536\pi$
$997$	−6.44365	−0.204072	−0.102036	+	0.994781i	$0.532536\pi$
$998$	46.1421	1.46060
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1881.2.a.d.1.1		2
3.2	odd	2		209.2.a.b.1.2	✓	2
12.11	even	2		3344.2.a.n.1.2		2
15.14	odd	2		5225.2.a.f.1.1		2
33.32	even	2		2299.2.a.f.1.1		2
57.56	even	2		3971.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.b.1.2	✓	2	3.2	odd	2
1881.2.a.d.1.1		2	1.1	even	1	trivial
2299.2.a.f.1.1		2	33.32	even	2
3344.2.a.n.1.2		2	12.11	even	2
3971.2.a.d.1.1		2	57.56	even	2
5225.2.a.f.1.1		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1881 = 3^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1881.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +1.00000 q^{5} -3.41421 q^{7} +2.82843 q^{8} -1.41421 q^{10} +1.00000 q^{11} +2.24264 q^{13} +4.82843 q^{14} -4.00000 q^{16} -3.41421 q^{17} -1.00000 q^{19} -1.41421 q^{22} +3.00000 q^{23} -4.00000 q^{25} -3.17157 q^{26} +6.24264 q^{29} -6.41421 q^{31} +4.82843 q^{34} -3.41421 q^{35} +10.0711 q^{37} +1.41421 q^{38} +2.82843 q^{40} +1.65685 q^{41} +0.343146 q^{43} -4.24264 q^{46} -8.82843 q^{47} +4.65685 q^{49} +5.65685 q^{50} +4.48528 q^{53} +1.00000 q^{55} -9.65685 q^{56} -8.82843 q^{58} +1.58579 q^{59} -11.0711 q^{61} +9.07107 q^{62} +8.00000 q^{64} +2.24264 q^{65} -10.4142 q^{67} +4.82843 q^{70} +12.4142 q^{71} -4.48528 q^{73} -14.2426 q^{74} -3.41421 q^{77} -14.5858 q^{79} -4.00000 q^{80} -2.34315 q^{82} -3.41421 q^{83} -3.41421 q^{85} -0.485281 q^{86} +2.82843 q^{88} -4.89949 q^{89} -7.65685 q^{91} +12.4853 q^{94} -1.00000 q^{95} +2.41421 q^{97} -6.58579 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7} + 2 q^{11} - 4 q^{13} + 4 q^{14} - 8 q^{16} - 4 q^{17} - 2 q^{19} + 6 q^{23} - 8 q^{25} - 12 q^{26} + 4 q^{29} - 10 q^{31} + 4 q^{34} - 4 q^{35} + 6 q^{37} - 8 q^{41} + 12 q^{43}+ \cdots - 16 q^{98}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 + 2 * q^11 - 4 * q^13 + 4 * q^14 - 8 * q^16 - 4 * q^17 - 2 * q^19 + 6 * q^23 - 8 * q^25 - 12 * q^26 + 4 * q^29 - 10 * q^31 + 4 * q^34 - 4 * q^35 + 6 * q^37 - 8 * q^41 + 12 * q^43 - 12 * q^47 - 2 * q^49 - 8 * q^53 + 2 * q^55 - 8 * q^56 - 12 * q^58 + 6 * q^59 - 8 * q^61 + 4 * q^62 + 16 * q^64 - 4 * q^65 - 18 * q^67 + 4 * q^70 + 22 * q^71 + 8 * q^73 - 20 * q^74 - 4 * q^77 - 32 * q^79 - 8 * q^80 - 16 * q^82 - 4 * q^83 - 4 * q^85 + 16 * q^86 + 10 * q^89 - 4 * q^91 + 8 * q^94 - 2 * q^95 + 2 * q^97 - 16 * q^98

Introduction

L-functions

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