LMFDB - Embedded newform 4080.2.a.bt.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4080,2,Mod(1,4080)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4080, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4080.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4080.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,-4,0,-4,0,-4,0,4,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$32.5789640247$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13768.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 2 $ x^4 - x^3 - 5x^2 + 2x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 255)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.849256$ of defining polynomial
Character	$\chi$	$=$	4080.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +4.48302 q^{7} +1.00000 q^{9} -1.47153 q^{11} -4.55753 q^{13} +1.00000 q^{15} -1.00000 q^{17} -8.48302 q^{19} -4.48302 q^{21} +6.40852 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.48302 q^{29} +3.01149 q^{31} +1.47153 q^{33} -4.48302 q^{35} +9.04055 q^{37} +4.55753 q^{39} -3.53996 q^{41} -3.39702 q^{43} -1.00000 q^{45} +1.47153 q^{47} +13.0975 q^{49} +1.00000 q^{51} +3.53996 q^{53} +1.47153 q^{55} +8.48302 q^{57} +6.55753 q^{59} -4.55753 q^{61} +4.48302 q^{63} +4.55753 q^{65} -7.16050 q^{67} -6.40852 q^{69} +7.39702 q^{71} -3.92549 q^{73} -1.00000 q^{75} -6.59690 q^{77} -0.453964 q^{79} +1.00000 q^{81} +7.39702 q^{83} +1.00000 q^{85} -6.48302 q^{87} -3.50059 q^{89} -20.4315 q^{91} -3.01149 q^{93} +8.48302 q^{95} +12.0230 q^{97} -1.47153 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} - 2 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 4 q^{29} - 6 q^{31} + 2 q^{33} + 4 q^{35} - 2 q^{37} + 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 - 2 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 4 * q^29 - 6 * q^31 + 2 * q^33 + 4 * q^35 - 2 * q^37 + 2 * q^39 - 4 * q^43 - 4 * q^45 + 2 * q^47 + 22 * q^49 + 4 * q^51 + 2 * q^55 + 12 * q^57 + 10 * q^59 - 2 * q^61 - 4 * q^63 + 2 * q^65 - 22 * q^67 + 2 * q^69 + 20 * q^71 - 10 * q^73 - 4 * q^75 - 20 * q^77 + 4 * q^81 + 20 * q^83 + 4 * q^85 - 4 * q^87 + 10 * q^89 - 18 * q^91 + 6 * q^93 + 12 * q^95 + 12 * q^97 - 2 * 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	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.48302	1.69442	0.847212	−	0.531256i	$-0.178280\pi$
$7$	4.48302	1.69442	0.847212	+	0.531256i	$0.178280\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.47153	−0.443683	−0.221842	−	0.975083i	$-0.571207\pi$
$11$	−1.47153	−0.443683	−0.221842	+	0.975083i	$0.571207\pi$
$12$	0	0
$13$	−4.55753	−1.26403	−0.632015	−	0.774956i	$-0.717772\pi$
$13$	−4.55753	−1.26403	−0.632015	+	0.774956i	$0.717772\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−8.48302	−1.94614	−0.973069	−	0.230512i	$-0.925960\pi$
$19$	−8.48302	−1.94614	−0.973069	+	0.230512i	$0.925960\pi$
$20$	0	0
$21$	−4.48302	−0.978276
$22$	0	0
$23$	6.40852	1.33627	0.668134	−	0.744041i	$-0.267093\pi$
$23$	6.40852	1.33627	0.668134	+	0.744041i	$0.267093\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	6.48302	1.20387	0.601934	−	0.798546i	$-0.294397\pi$
$29$	6.48302	1.20387	0.601934	+	0.798546i	$0.294397\pi$
$30$	0	0
$31$	3.01149	0.540880	0.270440	−	0.962737i	$-0.412831\pi$
$31$	3.01149	0.540880	0.270440	+	0.962737i	$0.412831\pi$
$32$	0	0
$33$	1.47153	0.256161
$34$	0	0
$35$	−4.48302	−0.757769
$36$	0	0
$37$	9.04055	1.48626	0.743129	−	0.669149i	$-0.233341\pi$
$37$	9.04055	1.48626	0.743129	+	0.669149i	$0.233341\pi$
$38$	0	0
$39$	4.55753	0.729789
$40$	0	0
$41$	−3.53996	−0.552849	−0.276425	−	0.961036i	$-0.589150\pi$
$41$	−3.53996	−0.552849	−0.276425	+	0.961036i	$0.589150\pi$
$42$	0	0
$43$	−3.39702	−0.518041	−0.259021	−	0.965872i	$-0.583400\pi$
$43$	−3.39702	−0.518041	−0.259021	+	0.965872i	$0.583400\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	1.47153	0.214645	0.107322	−	0.994224i	$-0.465772\pi$
$47$	1.47153	0.214645	0.107322	+	0.994224i	$0.465772\pi$
$48$	0	0
$49$	13.0975	1.87107
$50$	0	0
$51$	1.00000	0.140028
$52$	0	0
$53$	3.53996	0.486251	0.243126	−	0.969995i	$-0.421827\pi$
$53$	3.53996	0.486251	0.243126	+	0.969995i	$0.421827\pi$
$54$	0	0
$55$	1.47153	0.198421
$56$	0	0
$57$	8.48302	1.12360
$58$	0	0
$59$	6.55753	0.853717	0.426859	−	0.904318i	$-0.359620\pi$
$59$	6.55753	0.853717	0.426859	+	0.904318i	$0.359620\pi$
$60$	0	0
$61$	−4.55753	−0.583532	−0.291766	−	0.956490i	$-0.594243\pi$
$61$	−4.55753	−0.583532	−0.291766	+	0.956490i	$0.594243\pi$
$62$	0	0
$63$	4.48302	0.564808
$64$	0	0
$65$	4.55753	0.565292
$66$	0	0
$67$	−7.16050	−0.874795	−0.437397	−	0.899268i	$-0.644100\pi$
$67$	−7.16050	−0.874795	−0.437397	+	0.899268i	$0.644100\pi$
$68$	0	0
$69$	−6.40852	−0.771495
$70$	0	0
$71$	7.39702	0.877865	0.438933	−	0.898520i	$-0.355357\pi$
$71$	7.39702	0.877865	0.438933	+	0.898520i	$0.355357\pi$
$72$	0	0
$73$	−3.92549	−0.459444	−0.229722	−	0.973256i	$-0.573782\pi$
$73$	−3.92549	−0.459444	−0.229722	+	0.973256i	$0.573782\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−6.59690	−0.751787
$78$	0	0
$79$	−0.453964	−0.0510750	−0.0255375	−	0.999674i	$-0.508130\pi$
$79$	−0.453964	−0.0510750	−0.0255375	+	0.999674i	$0.508130\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.39702	0.811929	0.405964	−	0.913889i	$-0.366936\pi$
$83$	7.39702	0.811929	0.405964	+	0.913889i	$0.366936\pi$
$84$	0	0
$85$	1.00000	0.108465
$86$	0	0
$87$	−6.48302	−0.695053
$88$	0	0
$89$	−3.50059	−0.371062	−0.185531	−	0.982638i	$-0.559400\pi$
$89$	−3.50059	−0.371062	−0.185531	+	0.982638i	$0.559400\pi$
$90$	0	0
$91$	−20.4315	−2.14180
$92$	0	0
$93$	−3.01149	−0.312277
$94$	0	0
$95$	8.48302	0.870340
$96$	0	0
$97$	12.0230	1.22075	0.610375	−	0.792113i	$-0.291019\pi$
$97$	12.0230	1.22075	0.610375	+	0.792113i	$0.291019\pi$
$98$	0	0
$99$	−1.47153	−0.147894
$100$	0	0
$101$	−7.11506	−0.707975	−0.353987	−	0.935250i	$-0.615174\pi$
$101$	−7.11506	−0.707975	−0.353987	+	0.935250i	$0.615174\pi$
$102$	0	0
$103$	−4.51208	−0.444589	−0.222294	−	0.974980i	$-0.571355\pi$
$103$	−4.51208	−0.444589	−0.222294	+	0.974980i	$0.571355\pi$
$104$	0	0
$105$	4.48302	0.437498
$106$	0	0
$107$	10.0230	0.968959	0.484479	−	0.874803i	$-0.339009\pi$
$107$	10.0230	0.968959	0.484479	+	0.874803i	$0.339009\pi$
$108$	0	0
$109$	12.0230	1.15159	0.575797	−	0.817593i	$-0.304692\pi$
$109$	12.0230	1.15159	0.575797	+	0.817593i	$0.304692\pi$
$110$	0	0
$111$	−9.04055	−0.858091
$112$	0	0
$113$	15.3516	1.44415	0.722077	−	0.691812i	$-0.243187\pi$
$113$	15.3516	1.44415	0.722077	+	0.691812i	$0.243187\pi$
$114$	0	0
$115$	−6.40852	−0.597597
$116$	0	0
$117$	−4.55753	−0.421344
$118$	0	0
$119$	−4.48302	−0.410958
$120$	0	0
$121$	−8.83460	−0.803145
$122$	0	0
$123$	3.53996	0.319188
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.10356	0.541604	0.270802	−	0.962635i	$-0.412711\pi$
$127$	6.10356	0.541604	0.270802	+	0.962635i	$0.412711\pi$
$128$	0	0
$129$	3.39702	0.299091
$130$	0	0
$131$	4.60298	0.402164	0.201082	−	0.979574i	$-0.435554\pi$
$131$	4.60298	0.402164	0.201082	+	0.979574i	$0.435554\pi$
$132$	0	0
$133$	−38.0296	−3.29758
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.44907	−0.636417	−0.318208	−	0.948021i	$-0.603081\pi$
$137$	−7.44907	−0.636417	−0.318208	+	0.948021i	$0.603081\pi$
$138$	0	0
$139$	−1.56902	−0.133083	−0.0665413	−	0.997784i	$-0.521196\pi$
$139$	−1.56902	−0.133083	−0.0665413	+	0.997784i	$0.521196\pi$
$140$	0	0
$141$	−1.47153	−0.123925
$142$	0	0
$143$	6.70654	0.560829
$144$	0	0
$145$	−6.48302	−0.538386
$146$	0	0
$147$	−13.0975	−1.08026
$148$	0	0
$149$	22.5805	1.84987	0.924934	−	0.380128i	$-0.124120\pi$
$149$	22.5805	1.84987	0.924934	+	0.380128i	$0.124120\pi$
$150$	0	0
$151$	−18.5060	−1.50600	−0.752999	−	0.658022i	$-0.771393\pi$
$151$	−18.5060	−1.50600	−0.752999	+	0.658022i	$0.771393\pi$
$152$	0	0
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−3.01149	−0.241889
$156$	0	0
$157$	−7.05694	−0.563205	−0.281603	−	0.959531i	$-0.590866\pi$
$157$	−7.05694	−0.563205	−0.281603	+	0.959531i	$0.590866\pi$
$158$	0	0
$159$	−3.53996	−0.280737
$160$	0	0
$161$	28.7295	2.26420
$162$	0	0
$163$	0.424906	0.0332812	0.0166406	−	0.999862i	$-0.494703\pi$
$163$	0.424906	0.0332812	0.0166406	+	0.999862i	$0.494703\pi$
$164$	0	0
$165$	−1.47153	−0.114558
$166$	0	0
$167$	−2.94306	−0.227741	−0.113870	−	0.993496i	$-0.536325\pi$
$167$	−2.94306	−0.227741	−0.113870	+	0.993496i	$0.536325\pi$
$168$	0	0
$169$	7.77106	0.597774
$170$	0	0
$171$	−8.48302	−0.648713
$172$	0	0
$173$	11.5466	0.877869	0.438934	−	0.898519i	$-0.355356\pi$
$173$	11.5466	0.877869	0.438934	+	0.898519i	$0.355356\pi$
$174$	0	0
$175$	4.48302	0.338885
$176$	0	0
$177$	−6.55753	−0.492894
$178$	0	0
$179$	14.9431	1.11690	0.558448	−	0.829539i	$-0.311397\pi$
$179$	14.9431	1.11690	0.558448	+	0.829539i	$0.311397\pi$
$180$	0	0
$181$	8.55753	0.636076	0.318038	−	0.948078i	$-0.396976\pi$
$181$	8.55753	0.636076	0.318038	+	0.948078i	$0.396976\pi$
$182$	0	0
$183$	4.55753	0.336902
$184$	0	0
$185$	−9.04055	−0.664675
$186$	0	0
$187$	1.47153	0.107609
$188$	0	0
$189$	−4.48302	−0.326092
$190$	0	0
$191$	5.44247	0.393803	0.196902	−	0.980423i	$-0.436912\pi$
$191$	5.44247	0.393803	0.196902	+	0.980423i	$0.436912\pi$
$192$	0	0
$193$	−1.09207	−0.0786090	−0.0393045	−	0.999227i	$-0.512514\pi$
$193$	−1.09207	−0.0786090	−0.0393045	+	0.999227i	$0.512514\pi$
$194$	0	0
$195$	−4.55753	−0.326371
$196$	0	0
$197$	13.0799	0.931906	0.465953	−	0.884809i	$-0.345712\pi$
$197$	13.0799	0.931906	0.465953	+	0.884809i	$0.345712\pi$
$198$	0	0
$199$	21.2710	1.50786	0.753931	−	0.656954i	$-0.228156\pi$
$199$	21.2710	1.50786	0.753931	+	0.656954i	$0.228156\pi$
$200$	0	0
$201$	7.16050	0.505063
$202$	0	0
$203$	29.0635	2.03986
$204$	0	0
$205$	3.53996	0.247242
$206$	0	0
$207$	6.40852	0.445423
$208$	0	0
$209$	12.4830	0.863469
$210$	0	0
$211$	19.4430	1.33851	0.669255	−	0.743032i	$-0.266613\pi$
$211$	19.4430	1.33851	0.669255	+	0.743032i	$0.266613\pi$
$212$	0	0
$213$	−7.39702	−0.506836
$214$	0	0
$215$	3.39702	0.231675
$216$	0	0
$217$	13.5006	0.916480
$218$	0	0
$219$	3.92549	0.265260
$220$	0	0
$221$	4.55753	0.306573
$222$	0	0
$223$	−27.5011	−1.84161	−0.920805	−	0.390023i	$-0.872467\pi$
$223$	−27.5011	−1.84161	−0.920805	+	0.390023i	$0.872467\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	16.7295	1.11038	0.555189	−	0.831724i	$-0.312646\pi$
$227$	16.7295	1.11038	0.555189	+	0.831724i	$0.312646\pi$
$228$	0	0
$229$	10.8686	0.718214	0.359107	−	0.933296i	$-0.383081\pi$
$229$	10.8686	0.718214	0.359107	+	0.933296i	$0.383081\pi$
$230$	0	0
$231$	6.59690	0.434044
$232$	0	0
$233$	−23.3975	−1.53282	−0.766412	−	0.642349i	$-0.777960\pi$
$233$	−23.3975	−1.53282	−0.766412	+	0.642349i	$0.777960\pi$
$234$	0	0
$235$	−1.47153	−0.0959920
$236$	0	0
$237$	0.453964	0.0294881
$238$	0	0
$239$	28.4315	1.83908	0.919540	−	0.392995i	$-0.128561\pi$
$239$	28.4315	1.83908	0.919540	+	0.392995i	$0.128561\pi$
$240$	0	0
$241$	−1.76348	−0.113596	−0.0567979	−	0.998386i	$-0.518089\pi$
$241$	−1.76348	−0.113596	−0.0567979	+	0.998386i	$0.518089\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−13.0975	−0.836768
$246$	0	0
$247$	38.6616	2.45998
$248$	0	0
$249$	−7.39702	−0.468767
$250$	0	0
$251$	−13.4884	−0.851383	−0.425691	−	0.904868i	$-0.639969\pi$
$251$	−13.4884	−0.851383	−0.425691	+	0.904868i	$0.639969\pi$
$252$	0	0
$253$	−9.43032	−0.592879
$254$	0	0
$255$	−1.00000	−0.0626224
$256$	0	0
$257$	−20.2301	−1.26192	−0.630960	−	0.775815i	$-0.717339\pi$
$257$	−20.2301	−1.26192	−0.630960	+	0.775815i	$0.717339\pi$
$258$	0	0
$259$	40.5290	2.51835
$260$	0	0
$261$	6.48302	0.401289
$262$	0	0
$263$	0.505485	0.0311695	0.0155848	−	0.999879i	$-0.495039\pi$
$263$	0.505485	0.0311695	0.0155848	+	0.999879i	$0.495039\pi$
$264$	0	0
$265$	−3.53996	−0.217458
$266$	0	0
$267$	3.50059	0.214233
$268$	0	0
$269$	20.5641	1.25382	0.626908	−	0.779093i	$-0.284320\pi$
$269$	20.5641	1.25382	0.626908	+	0.779093i	$0.284320\pi$
$270$	0	0
$271$	25.1610	1.52842	0.764212	−	0.644965i	$-0.223128\pi$
$271$	25.1610	1.52842	0.764212	+	0.644965i	$0.223128\pi$
$272$	0	0
$273$	20.4315	1.23657
$274$	0	0
$275$	−1.47153	−0.0887366
$276$	0	0
$277$	26.2531	1.57740	0.788698	−	0.614781i	$-0.210756\pi$
$277$	26.2531	1.57740	0.788698	+	0.614781i	$0.210756\pi$
$278$	0	0
$279$	3.01149	0.180293
$280$	0	0
$281$	−14.1950	−0.846802	−0.423401	−	0.905942i	$-0.639164\pi$
$281$	−14.1950	−0.846802	−0.423401	+	0.905942i	$0.639164\pi$
$282$	0	0
$283$	20.3921	1.21219	0.606093	−	0.795394i	$-0.292736\pi$
$283$	20.3921	1.21219	0.606093	+	0.795394i	$0.292736\pi$
$284$	0	0
$285$	−8.48302	−0.502491
$286$	0	0
$287$	−15.8697	−0.936761
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−12.0230	−0.704800
$292$	0	0
$293$	−16.5060	−0.964291	−0.482146	−	0.876091i	$-0.660142\pi$
$293$	−16.5060	−0.964291	−0.482146	+	0.876091i	$0.660142\pi$
$294$	0	0
$295$	−6.55753	−0.381794
$296$	0	0
$297$	1.47153	0.0853868
$298$	0	0
$299$	−29.2070	−1.68908
$300$	0	0
$301$	−15.2289	−0.877781
$302$	0	0
$303$	7.11506	0.408749
$304$	0	0
$305$	4.55753	0.260963
$306$	0	0
$307$	−10.6260	−0.606456	−0.303228	−	0.952918i	$-0.598064\pi$
$307$	−10.6260	−0.606456	−0.303228	+	0.952918i	$0.598064\pi$
$308$	0	0
$309$	4.51208	0.256683
$310$	0	0
$311$	10.5285	0.597015	0.298507	−	0.954407i	$-0.403511\pi$
$311$	10.5285	0.597015	0.298507	+	0.954407i	$0.403511\pi$
$312$	0	0
$313$	3.30344	0.186722	0.0933608	−	0.995632i	$-0.470239\pi$
$313$	3.30344	0.186722	0.0933608	+	0.995632i	$0.470239\pi$
$314$	0	0
$315$	−4.48302	−0.252590
$316$	0	0
$317$	23.6956	1.33088	0.665438	−	0.746453i	$-0.268245\pi$
$317$	23.6956	1.33088	0.665438	+	0.746453i	$0.268245\pi$
$318$	0	0
$319$	−9.53996	−0.534135
$320$	0	0
$321$	−10.0230	−0.559428
$322$	0	0
$323$	8.48302	0.472008
$324$	0	0
$325$	−4.55753	−0.252806
$326$	0	0
$327$	−12.0230	−0.664873
$328$	0	0
$329$	6.59690	0.363699
$330$	0	0
$331$	15.4721	0.850421	0.425210	−	0.905095i	$-0.360200\pi$
$331$	15.4721	0.850421	0.425210	+	0.905095i	$0.360200\pi$
$332$	0	0
$333$	9.04055	0.495419
$334$	0	0
$335$	7.16050	0.391220
$336$	0	0
$337$	−17.0406	−0.928258	−0.464129	−	0.885768i	$-0.653633\pi$
$337$	−17.0406	−0.928258	−0.464129	+	0.885768i	$0.653633\pi$
$338$	0	0
$339$	−15.3516	−0.833783
$340$	0	0
$341$	−4.43150	−0.239979
$342$	0	0
$343$	27.3352	1.47596
$344$	0	0
$345$	6.40852	0.345023
$346$	0	0
$347$	−8.53454	−0.458158	−0.229079	−	0.973408i	$-0.573571\pi$
$347$	−8.53454	−0.458158	−0.229079	+	0.973408i	$0.573571\pi$
$348$	0	0
$349$	−15.0635	−0.806333	−0.403166	−	0.915127i	$-0.632090\pi$
$349$	−15.0635	−0.806333	−0.403166	+	0.915127i	$0.632090\pi$
$350$	0	0
$351$	4.55753	0.243263
$352$	0	0
$353$	31.5981	1.68180	0.840898	−	0.541194i	$-0.182027\pi$
$353$	31.5981	1.68180	0.840898	+	0.541194i	$0.182027\pi$
$354$	0	0
$355$	−7.39702	−0.392593
$356$	0	0
$357$	4.48302	0.237267
$358$	0	0
$359$	−7.37456	−0.389214	−0.194607	−	0.980881i	$-0.562343\pi$
$359$	−7.37456	−0.389214	−0.194607	+	0.980881i	$0.562343\pi$
$360$	0	0
$361$	52.9617	2.78746
$362$	0	0
$363$	8.83460	0.463696
$364$	0	0
$365$	3.92549	0.205470
$366$	0	0
$367$	−8.00000	−0.417597	−0.208798	−	0.977959i	$-0.566955\pi$
$367$	−8.00000	−0.417597	−0.208798	+	0.977959i	$0.566955\pi$
$368$	0	0
$369$	−3.53996	−0.184283
$370$	0	0
$371$	15.8697	0.823915
$372$	0	0
$373$	−12.9431	−0.670166	−0.335083	−	0.942189i	$-0.608764\pi$
$373$	−12.9431	−0.670166	−0.335083	+	0.942189i	$0.608764\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−29.5466	−1.52173
$378$	0	0
$379$	12.5121	0.642702	0.321351	−	0.946960i	$-0.395863\pi$
$379$	12.5121	0.642702	0.321351	+	0.946960i	$0.395863\pi$
$380$	0	0
$381$	−6.10356	−0.312695
$382$	0	0
$383$	−25.7246	−1.31447	−0.657234	−	0.753687i	$-0.728273\pi$
$383$	−25.7246	−1.31447	−0.657234	+	0.753687i	$0.728273\pi$
$384$	0	0
$385$	6.59690	0.336209
$386$	0	0
$387$	−3.39702	−0.172680
$388$	0	0
$389$	−8.08110	−0.409728	−0.204864	−	0.978790i	$-0.565675\pi$
$389$	−8.08110	−0.409728	−0.204864	+	0.978790i	$0.565675\pi$
$390$	0	0
$391$	−6.40852	−0.324093
$392$	0	0
$393$	−4.60298	−0.232189
$394$	0	0
$395$	0.453964	0.0228414
$396$	0	0
$397$	−9.96063	−0.499909	−0.249955	−	0.968258i	$-0.580416\pi$
$397$	−9.96063	−0.499909	−0.249955	+	0.968258i	$0.580416\pi$
$398$	0	0
$399$	38.0296	1.90386
$400$	0	0
$401$	28.6550	1.43096	0.715482	−	0.698631i	$-0.246207\pi$
$401$	28.6550	1.43096	0.715482	+	0.698631i	$0.246207\pi$
$402$	0	0
$403$	−13.7250	−0.683689
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−13.3034	−0.659427
$408$	0	0
$409$	23.8346	1.17855	0.589273	−	0.807934i	$-0.299414\pi$
$409$	23.8346	1.17855	0.589273	+	0.807934i	$0.299414\pi$
$410$	0	0
$411$	7.44907	0.367435
$412$	0	0
$413$	29.3975	1.44656
$414$	0	0
$415$	−7.39702	−0.363106
$416$	0	0
$417$	1.56902	0.0768353
$418$	0	0
$419$	8.70047	0.425046	0.212523	−	0.977156i	$-0.431832\pi$
$419$	8.70047	0.425046	0.212523	+	0.977156i	$0.431832\pi$
$420$	0	0
$421$	−9.09867	−0.443442	−0.221721	−	0.975110i	$-0.571167\pi$
$421$	−9.09867	−0.443442	−0.221721	+	0.975110i	$0.571167\pi$
$422$	0	0
$423$	1.47153	0.0715482
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−20.4315	−0.988750
$428$	0	0
$429$	−6.70654	−0.323795
$430$	0	0
$431$	−36.4606	−1.75624	−0.878122	−	0.478437i	$-0.841203\pi$
$431$	−36.4606	−1.75624	−0.878122	+	0.478437i	$0.841203\pi$
$432$	0	0
$433$	20.7032	0.994930	0.497465	−	0.867484i	$-0.334264\pi$
$433$	20.7032	0.994930	0.497465	+	0.867484i	$0.334264\pi$
$434$	0	0
$435$	6.48302	0.310837
$436$	0	0
$437$	−54.3636	−2.60056
$438$	0	0
$439$	−27.1277	−1.29474	−0.647368	−	0.762178i	$-0.724130\pi$
$439$	−27.1277	−1.29474	−0.647368	+	0.762178i	$0.724130\pi$
$440$	0	0
$441$	13.0975	0.623690
$442$	0	0
$443$	−0.168088	−0.00798610	−0.00399305	−	0.999992i	$-0.501271\pi$
$443$	−0.168088	−0.00798610	−0.00399305	+	0.999992i	$0.501271\pi$
$444$	0	0
$445$	3.50059	0.165944
$446$	0	0
$447$	−22.5805	−1.06802
$448$	0	0
$449$	2.09089	0.0986754	0.0493377	−	0.998782i	$-0.484289\pi$
$449$	2.09089	0.0986754	0.0493377	+	0.998782i	$0.484289\pi$
$450$	0	0
$451$	5.20916	0.245290
$452$	0	0
$453$	18.5060	0.869488
$454$	0	0
$455$	20.4315	0.957844
$456$	0	0
$457$	−22.7755	−1.06539	−0.532696	−	0.846306i	$-0.678821\pi$
$457$	−22.7755	−1.06539	−0.532696	+	0.846306i	$0.678821\pi$
$458$	0	0
$459$	1.00000	0.0466760
$460$	0	0
$461$	−2.38553	−0.111105	−0.0555526	−	0.998456i	$-0.517692\pi$
$461$	−2.38553	−0.111105	−0.0555526	+	0.998456i	$0.517692\pi$
$462$	0	0
$463$	20.4540	0.950576	0.475288	−	0.879830i	$-0.342344\pi$
$463$	20.4540	0.950576	0.475288	+	0.879830i	$0.342344\pi$
$464$	0	0
$465$	3.01149	0.139655
$466$	0	0
$467$	1.32252	0.0611989	0.0305994	−	0.999532i	$-0.490258\pi$
$467$	1.32252	0.0611989	0.0305994	+	0.999532i	$0.490258\pi$
$468$	0	0
$469$	−32.1007	−1.48227
$470$	0	0
$471$	7.05694	0.325167
$472$	0	0
$473$	4.99882	0.229846
$474$	0	0
$475$	−8.48302	−0.389228
$476$	0	0
$477$	3.53996	0.162084
$478$	0	0
$479$	11.3061	0.516590	0.258295	−	0.966066i	$-0.416839\pi$
$479$	11.3061	0.516590	0.258295	+	0.966066i	$0.416839\pi$
$480$	0	0
$481$	−41.2026	−1.87868
$482$	0	0
$483$	−28.7295	−1.30724
$484$	0	0
$485$	−12.0230	−0.545936
$486$	0	0
$487$	−33.9321	−1.53761	−0.768805	−	0.639483i	$-0.779148\pi$
$487$	−33.9321	−1.53761	−0.768805	+	0.639483i	$0.779148\pi$
$488$	0	0
$489$	−0.424906	−0.0192149
$490$	0	0
$491$	−0.149012	−0.00672480	−0.00336240	−	0.999994i	$-0.501070\pi$
$491$	−0.149012	−0.00672480	−0.00336240	+	0.999994i	$0.501070\pi$
$492$	0	0
$493$	−6.48302	−0.291981
$494$	0	0
$495$	1.47153	0.0661404
$496$	0	0
$497$	33.1610	1.48748
$498$	0	0
$499$	−34.2077	−1.53134	−0.765672	−	0.643231i	$-0.777594\pi$
$499$	−34.2077	−1.53134	−0.765672	+	0.643231i	$0.777594\pi$
$500$	0	0
$501$	2.94306	0.131486
$502$	0	0
$503$	−9.87397	−0.440259	−0.220129	−	0.975471i	$-0.570648\pi$
$503$	−9.87397	−0.440259	−0.220129	+	0.975471i	$0.570648\pi$
$504$	0	0
$505$	7.11506	0.316616
$506$	0	0
$507$	−7.77106	−0.345125
$508$	0	0
$509$	−4.69439	−0.208075	−0.104038	−	0.994573i	$-0.533176\pi$
$509$	−4.69439	−0.208075	−0.104038	+	0.994573i	$0.533176\pi$
$510$	0	0
$511$	−17.5981	−0.778493
$512$	0	0
$513$	8.48302	0.374535
$514$	0	0
$515$	4.51208	0.198826
$516$	0	0
$517$	−2.16540	−0.0952342
$518$	0	0
$519$	−11.5466	−0.506838
$520$	0	0
$521$	34.2431	1.50022	0.750109	−	0.661314i	$-0.230001\pi$
$521$	34.2431	1.50022	0.750109	+	0.661314i	$0.230001\pi$
$522$	0	0
$523$	32.7071	1.43018	0.715090	−	0.699032i	$-0.246386\pi$
$523$	32.7071	1.43018	0.715090	+	0.699032i	$0.246386\pi$
$524$	0	0
$525$	−4.48302	−0.195655
$526$	0	0
$527$	−3.01149	−0.131183
$528$	0	0
$529$	18.0691	0.785612
$530$	0	0
$531$	6.55753	0.284572
$532$	0	0
$533$	16.1335	0.698819
$534$	0	0
$535$	−10.0230	−0.433331
$536$	0	0
$537$	−14.9431	−0.644841
$538$	0	0
$539$	−19.2734	−0.830162
$540$	0	0
$541$	36.4217	1.56589	0.782946	−	0.622090i	$-0.213716\pi$
$541$	36.4217	1.56589	0.782946	+	0.622090i	$0.213716\pi$
$542$	0	0
$543$	−8.55753	−0.367239
$544$	0	0
$545$	−12.0230	−0.515008
$546$	0	0
$547$	−37.3582	−1.59732	−0.798660	−	0.601782i	$-0.794457\pi$
$547$	−37.3582	−1.59732	−0.798660	+	0.601782i	$0.794457\pi$
$548$	0	0
$549$	−4.55753	−0.194511
$550$	0	0
$551$	−54.9956	−2.34289
$552$	0	0
$553$	−2.03513	−0.0865426
$554$	0	0
$555$	9.04055	0.383750
$556$	0	0
$557$	−36.1392	−1.53127	−0.765634	−	0.643277i	$-0.777575\pi$
$557$	−36.1392	−1.53127	−0.765634	+	0.643277i	$0.777575\pi$
$558$	0	0
$559$	15.4820	0.654820
$560$	0	0
$561$	−1.47153	−0.0621280
$562$	0	0
$563$	−21.9645	−0.925695	−0.462847	−	0.886438i	$-0.653172\pi$
$563$	−21.9645	−0.925695	−0.462847	+	0.886438i	$0.653172\pi$
$564$	0	0
$565$	−15.3516	−0.645846
$566$	0	0
$567$	4.48302	0.188269
$568$	0	0
$569$	−6.96604	−0.292032	−0.146016	−	0.989282i	$-0.546645\pi$
$569$	−6.96604	−0.292032	−0.146016	+	0.989282i	$0.546645\pi$
$570$	0	0
$571$	−5.03448	−0.210686	−0.105343	−	0.994436i	$-0.533594\pi$
$571$	−5.03448	−0.210686	−0.105343	+	0.994436i	$0.533594\pi$
$572$	0	0
$573$	−5.44247	−0.227363
$574$	0	0
$575$	6.40852	0.267254
$576$	0	0
$577$	30.2410	1.25895	0.629474	−	0.777022i	$-0.283271\pi$
$577$	30.2410	1.25895	0.629474	+	0.777022i	$0.283271\pi$
$578$	0	0
$579$	1.09207	0.0453849
$580$	0	0
$581$	33.1610	1.37575
$582$	0	0
$583$	−5.20916	−0.215741
$584$	0	0
$585$	4.55753	0.188431
$586$	0	0
$587$	34.4835	1.42329	0.711644	−	0.702540i	$-0.247951\pi$
$587$	34.4835	1.42329	0.711644	+	0.702540i	$0.247951\pi$
$588$	0	0
$589$	−25.5466	−1.05263
$590$	0	0
$591$	−13.0799	−0.538036
$592$	0	0
$593$	5.66599	0.232674	0.116337	−	0.993210i	$-0.462885\pi$
$593$	5.66599	0.232674	0.116337	+	0.993210i	$0.462885\pi$
$594$	0	0
$595$	4.48302	0.183786
$596$	0	0
$597$	−21.2710	−0.870564
$598$	0	0
$599$	1.26407	0.0516484	0.0258242	−	0.999666i	$-0.491779\pi$
$599$	1.26407	0.0516484	0.0258242	+	0.999666i	$0.491779\pi$
$600$	0	0
$601$	−44.1686	−1.80168	−0.900838	−	0.434156i	$-0.857047\pi$
$601$	−44.1686	−1.80168	−0.900838	+	0.434156i	$0.857047\pi$
$602$	0	0
$603$	−7.16050	−0.291598
$604$	0	0
$605$	8.83460	0.359178
$606$	0	0
$607$	−18.4273	−0.747939	−0.373970	−	0.927441i	$-0.622004\pi$
$607$	−18.4273	−0.747939	−0.373970	+	0.927441i	$0.622004\pi$
$608$	0	0
$609$	−29.0635	−1.17771
$610$	0	0
$611$	−6.70654	−0.271318
$612$	0	0
$613$	−1.76009	−0.0710895	−0.0355448	−	0.999368i	$-0.511317\pi$
$613$	−1.76009	−0.0710895	−0.0355448	+	0.999368i	$0.511317\pi$
$614$	0	0
$615$	−3.53996	−0.142745
$616$	0	0
$617$	2.23652	0.0900389	0.0450195	−	0.998986i	$-0.485665\pi$
$617$	2.23652	0.0900389	0.0450195	+	0.998986i	$0.485665\pi$
$618$	0	0
$619$	−21.3619	−0.858607	−0.429303	−	0.903160i	$-0.641241\pi$
$619$	−21.3619	−0.858607	−0.429303	+	0.903160i	$0.641241\pi$
$620$	0	0
$621$	−6.40852	−0.257165
$622$	0	0
$623$	−15.6932	−0.628735
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−12.4830	−0.498524
$628$	0	0
$629$	−9.04055	−0.360470
$630$	0	0
$631$	8.76890	0.349084	0.174542	−	0.984650i	$-0.444155\pi$
$631$	8.76890	0.349084	0.174542	+	0.984650i	$0.444155\pi$
$632$	0	0
$633$	−19.4430	−0.772790
$634$	0	0
$635$	−6.10356	−0.242213
$636$	0	0
$637$	−59.6922	−2.36509
$638$	0	0
$639$	7.39702	0.292622
$640$	0	0
$641$	−23.5072	−0.928478	−0.464239	−	0.885710i	$-0.653672\pi$
$641$	−23.5072	−0.928478	−0.464239	+	0.885710i	$0.653672\pi$
$642$	0	0
$643$	−0.586064	−0.0231121	−0.0115561	−	0.999933i	$-0.503678\pi$
$643$	−0.586064	−0.0231121	−0.0115561	+	0.999933i	$0.503678\pi$
$644$	0	0
$645$	−3.39702	−0.133758
$646$	0	0
$647$	−19.3970	−0.762576	−0.381288	−	0.924456i	$-0.624519\pi$
$647$	−19.3970	−0.762576	−0.381288	+	0.924456i	$0.624519\pi$
$648$	0	0
$649$	−9.64960	−0.378780
$650$	0	0
$651$	−13.5006	−0.529130
$652$	0	0
$653$	7.11506	0.278434	0.139217	−	0.990262i	$-0.455541\pi$
$653$	7.11506	0.278434	0.139217	+	0.990262i	$0.455541\pi$
$654$	0	0
$655$	−4.60298	−0.179853
$656$	0	0
$657$	−3.92549	−0.153148
$658$	0	0
$659$	23.8510	0.929103	0.464551	−	0.885546i	$-0.346216\pi$
$659$	23.8510	0.929103	0.464551	+	0.885546i	$0.346216\pi$
$660$	0	0
$661$	40.0756	1.55876	0.779379	−	0.626553i	$-0.215535\pi$
$661$	40.0756	1.55876	0.779379	+	0.626553i	$0.215535\pi$
$662$	0	0
$663$	−4.55753	−0.177000
$664$	0	0
$665$	38.0296	1.47472
$666$	0	0
$667$	41.5466	1.60869
$668$	0	0
$669$	27.5011	1.06325
$670$	0	0
$671$	6.70654	0.258903
$672$	0	0
$673$	−3.02416	−0.116573	−0.0582864	−	0.998300i	$-0.518564\pi$
$673$	−3.02416	−0.116573	−0.0582864	+	0.998300i	$0.518564\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−0.113880	−0.00437676	−0.00218838	−	0.999998i	$-0.500697\pi$
$677$	−0.113880	−0.00437676	−0.00218838	+	0.999998i	$0.500697\pi$
$678$	0	0
$679$	53.8993	2.06847
$680$	0	0
$681$	−16.7295	−0.641077
$682$	0	0
$683$	−17.5881	−0.672990	−0.336495	−	0.941685i	$-0.609241\pi$
$683$	−17.5881	−0.672990	−0.336495	+	0.941685i	$0.609241\pi$
$684$	0	0
$685$	7.44907	0.284614
$686$	0	0
$687$	−10.8686	−0.414661
$688$	0	0
$689$	−16.1335	−0.614637
$690$	0	0
$691$	−0.602976	−0.0229383	−0.0114691	−	0.999934i	$-0.503651\pi$
$691$	−0.602976	−0.0229383	−0.0114691	+	0.999934i	$0.503651\pi$
$692$	0	0
$693$	−6.59690	−0.250596
$694$	0	0
$695$	1.56902	0.0595163
$696$	0	0
$697$	3.53996	0.134086
$698$	0	0
$699$	23.3975	0.884976
$700$	0	0
$701$	8.20139	0.309762	0.154881	−	0.987933i	$-0.450501\pi$
$701$	8.20139	0.309762	0.154881	+	0.987933i	$0.450501\pi$
$702$	0	0
$703$	−76.6912	−2.89246
$704$	0	0
$705$	1.47153	0.0554210
$706$	0	0
$707$	−31.8970	−1.19961
$708$	0	0
$709$	−33.9551	−1.27521	−0.637605	−	0.770364i	$-0.720075\pi$
$709$	−33.9551	−1.27521	−0.637605	+	0.770364i	$0.720075\pi$
$710$	0	0
$711$	−0.453964	−0.0170250
$712$	0	0
$713$	19.2992	0.722761
$714$	0	0
$715$	−6.70654	−0.250810
$716$	0	0
$717$	−28.4315	−1.06179
$718$	0	0
$719$	28.5187	1.06357	0.531784	−	0.846880i	$-0.321522\pi$
$719$	28.5187	1.06357	0.531784	+	0.846880i	$0.321522\pi$
$720$	0	0
$721$	−20.2278	−0.753321
$722$	0	0
$723$	1.76348	0.0655845
$724$	0	0
$725$	6.48302	0.240773
$726$	0	0
$727$	−14.6841	−0.544602	−0.272301	−	0.962212i	$-0.587785\pi$
$727$	−14.6841	−0.544602	−0.272301	+	0.962212i	$0.587785\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	3.39702	0.125643
$732$	0	0
$733$	−8.51156	−0.314382	−0.157191	−	0.987568i	$-0.550244\pi$
$733$	−8.51156	−0.314382	−0.157191	+	0.987568i	$0.550244\pi$
$734$	0	0
$735$	13.0975	0.483108
$736$	0	0
$737$	10.5369	0.388132
$738$	0	0
$739$	6.23011	0.229178	0.114589	−	0.993413i	$-0.463445\pi$
$739$	6.23011	0.229178	0.114589	+	0.993413i	$0.463445\pi$
$740$	0	0
$741$	−38.6616	−1.42027
$742$	0	0
$743$	14.3791	0.527519	0.263759	−	0.964588i	$-0.415037\pi$
$743$	14.3791	0.527519	0.263759	+	0.964588i	$0.415037\pi$
$744$	0	0
$745$	−22.5805	−0.827286
$746$	0	0
$747$	7.39702	0.270643
$748$	0	0
$749$	44.9333	1.64183
$750$	0	0
$751$	11.1277	0.406056	0.203028	−	0.979173i	$-0.434922\pi$
$751$	11.1277	0.406056	0.203028	+	0.979173i	$0.434922\pi$
$752$	0	0
$753$	13.4884	0.491546
$754$	0	0
$755$	18.5060	0.673503
$756$	0	0
$757$	18.5771	0.675197	0.337599	−	0.941290i	$-0.390385\pi$
$757$	18.5771	0.675197	0.337599	+	0.941290i	$0.390385\pi$
$758$	0	0
$759$	9.43032	0.342299
$760$	0	0
$761$	23.6956	0.858964	0.429482	−	0.903075i	$-0.358696\pi$
$761$	23.6956	0.858964	0.429482	+	0.903075i	$0.358696\pi$
$762$	0	0
$763$	53.8993	1.95129
$764$	0	0
$765$	1.00000	0.0361551
$766$	0	0
$767$	−29.8861	−1.07913
$768$	0	0
$769$	17.8697	0.644399	0.322199	−	0.946672i	$-0.395578\pi$
$769$	17.8697	0.644399	0.322199	+	0.946672i	$0.395578\pi$
$770$	0	0
$771$	20.2301	0.728570
$772$	0	0
$773$	2.27589	0.0818582	0.0409291	−	0.999162i	$-0.486968\pi$
$773$	2.27589	0.0818582	0.0409291	+	0.999162i	$0.486968\pi$
$774$	0	0
$775$	3.01149	0.108176
$776$	0	0
$777$	−40.5290	−1.45397
$778$	0	0
$779$	30.0296	1.07592
$780$	0	0
$781$	−10.8849	−0.389494
$782$	0	0
$783$	−6.48302	−0.231684
$784$	0	0
$785$	7.05694	0.251873
$786$	0	0
$787$	1.68897	0.0602054	0.0301027	−	0.999547i	$-0.490417\pi$
$787$	1.68897	0.0602054	0.0301027	+	0.999547i	$0.490417\pi$
$788$	0	0
$789$	−0.505485	−0.0179957
$790$	0	0
$791$	68.8215	2.44701
$792$	0	0
$793$	20.7711	0.737602
$794$	0	0
$795$	3.53996	0.125550
$796$	0	0
$797$	−47.0601	−1.66696	−0.833478	−	0.552553i	$-0.813654\pi$
$797$	−47.0601	−1.66696	−0.833478	+	0.552553i	$0.813654\pi$
$798$	0	0
$799$	−1.47153	−0.0520590
$800$	0	0
$801$	−3.50059	−0.123687
$802$	0	0
$803$	5.77648	0.203848
$804$	0	0
$805$	−28.7295	−1.01258
$806$	0	0
$807$	−20.5641	−0.723891
$808$	0	0
$809$	−5.13804	−0.180644	−0.0903220	−	0.995913i	$-0.528790\pi$
$809$	−5.13804	−0.180644	−0.0903220	+	0.995913i	$0.528790\pi$
$810$	0	0
$811$	4.54147	0.159473	0.0797363	−	0.996816i	$-0.474592\pi$
$811$	4.54147	0.159473	0.0797363	+	0.996816i	$0.474592\pi$
$812$	0	0
$813$	−25.1610	−0.882436
$814$	0	0
$815$	−0.424906	−0.0148838
$816$	0	0
$817$	28.8170	1.00818
$818$	0	0
$819$	−20.4315	−0.713934
$820$	0	0
$821$	−41.0931	−1.43416	−0.717080	−	0.696991i	$-0.754522\pi$
$821$	−41.0931	−1.43416	−0.717080	+	0.696991i	$0.754522\pi$
$822$	0	0
$823$	−12.9179	−0.450290	−0.225145	−	0.974325i	$-0.572286\pi$
$823$	−12.9179	−0.450290	−0.225145	+	0.974325i	$0.572286\pi$
$824$	0	0
$825$	1.47153	0.0512321
$826$	0	0
$827$	27.2377	0.947148	0.473574	−	0.880754i	$-0.342964\pi$
$827$	27.2377	0.947148	0.473574	+	0.880754i	$0.342964\pi$
$828$	0	0
$829$	37.1774	1.29123	0.645613	−	0.763665i	$-0.276602\pi$
$829$	37.1774	1.29123	0.645613	+	0.763665i	$0.276602\pi$
$830$	0	0
$831$	−26.2531	−0.910710
$832$	0	0
$833$	−13.0975	−0.453801
$834$	0	0
$835$	2.94306	0.101849
$836$	0	0
$837$	−3.01149	−0.104092
$838$	0	0
$839$	44.6555	1.54168	0.770840	−	0.637029i	$-0.219837\pi$
$839$	44.6555	1.54168	0.770840	+	0.637029i	$0.219837\pi$
$840$	0	0
$841$	13.0296	0.449296
$842$	0	0
$843$	14.1950	0.488901
$844$	0	0
$845$	−7.77106	−0.267333
$846$	0	0
$847$	−39.6057	−1.36087
$848$	0	0
$849$	−20.3921	−0.699856
$850$	0	0
$851$	57.9365	1.98604
$852$	0	0
$853$	12.8400	0.439634	0.219817	−	0.975541i	$-0.429454\pi$
$853$	12.8400	0.439634	0.219817	+	0.975541i	$0.429454\pi$
$854$	0	0
$855$	8.48302	0.290113
$856$	0	0
$857$	−17.0340	−0.581869	−0.290934	−	0.956743i	$-0.593966\pi$
$857$	−17.0340	−0.581869	−0.290934	+	0.956743i	$0.593966\pi$
$858$	0	0
$859$	−35.6792	−1.21736	−0.608679	−	0.793417i	$-0.708300\pi$
$859$	−35.6792	−1.21736	−0.608679	+	0.793417i	$0.708300\pi$
$860$	0	0
$861$	15.8697	0.540839
$862$	0	0
$863$	20.5065	0.698050	0.349025	−	0.937113i	$-0.386513\pi$
$863$	20.5065	0.698050	0.349025	+	0.937113i	$0.386513\pi$
$864$	0	0
$865$	−11.5466	−0.392595
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	0.668022	0.0226611
$870$	0	0
$871$	32.6342	1.10577
$872$	0	0
$873$	12.0230	0.406916
$874$	0	0
$875$	−4.48302	−0.151554
$876$	0	0
$877$	1.81161	0.0611739	0.0305869	−	0.999532i	$-0.490262\pi$
$877$	1.81161	0.0611739	0.0305869	+	0.999532i	$0.490262\pi$
$878$	0	0
$879$	16.5060	0.556734
$880$	0	0
$881$	8.97905	0.302512	0.151256	−	0.988495i	$-0.451668\pi$
$881$	8.97905	0.302512	0.151256	+	0.988495i	$0.451668\pi$
$882$	0	0
$883$	−3.54265	−0.119220	−0.0596098	−	0.998222i	$-0.518986\pi$
$883$	−3.54265	−0.119220	−0.0596098	+	0.998222i	$0.518986\pi$
$884$	0	0
$885$	6.55753	0.220429
$886$	0	0
$887$	−12.8620	−0.431862	−0.215931	−	0.976409i	$-0.569279\pi$
$887$	−12.8620	−0.431862	−0.215931	+	0.976409i	$0.569279\pi$
$888$	0	0
$889$	27.3624	0.917706
$890$	0	0
$891$	−1.47153	−0.0492981
$892$	0	0
$893$	−12.4830	−0.417728
$894$	0	0
$895$	−14.9431	−0.499491
$896$	0	0
$897$	29.2070	0.975193
$898$	0	0
$899$	19.5236	0.651148
$900$	0	0
$901$	−3.53996	−0.117933
$902$	0	0
$903$	15.2289	0.506787
$904$	0	0
$905$	−8.55753	−0.284462
$906$	0	0
$907$	−3.56295	−0.118306	−0.0591529	−	0.998249i	$-0.518840\pi$
$907$	−3.56295	−0.118306	−0.0591529	+	0.998249i	$0.518840\pi$
$908$	0	0
$909$	−7.11506	−0.235992
$910$	0	0
$911$	14.4376	0.478338	0.239169	−	0.970978i	$-0.423125\pi$
$911$	14.4376	0.478338	0.239169	+	0.970978i	$0.423125\pi$
$912$	0	0
$913$	−10.8849	−0.360239
$914$	0	0
$915$	−4.55753	−0.150667
$916$	0	0
$917$	20.6352	0.681436
$918$	0	0
$919$	20.2431	0.667759	0.333879	−	0.942616i	$-0.391642\pi$
$919$	20.2431	0.667759	0.333879	+	0.942616i	$0.391642\pi$
$920$	0	0
$921$	10.6260	0.350137
$922$	0	0
$923$	−33.7121	−1.10965
$924$	0	0
$925$	9.04055	0.297251
$926$	0	0
$927$	−4.51208	−0.148196
$928$	0	0
$929$	47.4369	1.55635	0.778177	−	0.628044i	$-0.216144\pi$
$929$	47.4369	1.55635	0.778177	+	0.628044i	$0.216144\pi$
$930$	0	0
$931$	−111.106	−3.64136
$932$	0	0
$933$	−10.5285	−0.344687
$934$	0	0
$935$	−1.47153	−0.0481242
$936$	0	0
$937$	−19.8706	−0.649144	−0.324572	−	0.945861i	$-0.605220\pi$
$937$	−19.8706	−0.649144	−0.324572	+	0.945861i	$0.605220\pi$
$938$	0	0
$939$	−3.30344	−0.107804
$940$	0	0
$941$	−36.6329	−1.19420	−0.597099	−	0.802168i	$-0.703680\pi$
$941$	−36.6329	−1.19420	−0.597099	+	0.802168i	$0.703680\pi$
$942$	0	0
$943$	−22.6859	−0.738755
$944$	0	0
$945$	4.48302	0.145833
$946$	0	0
$947$	−6.36698	−0.206899	−0.103449	−	0.994635i	$-0.532988\pi$
$947$	−6.36698	−0.206899	−0.103449	+	0.994635i	$0.532988\pi$
$948$	0	0
$949$	17.8906	0.580752
$950$	0	0
$951$	−23.6956	−0.768381
$952$	0	0
$953$	−35.6113	−1.15356	−0.576781	−	0.816899i	$-0.695691\pi$
$953$	−35.6113	−1.15356	−0.576781	+	0.816899i	$0.695691\pi$
$954$	0	0
$955$	−5.44247	−0.176114
$956$	0	0
$957$	9.53996	0.308383
$958$	0	0
$959$	−33.3943	−1.07836
$960$	0	0
$961$	−21.9309	−0.707449
$962$	0	0
$963$	10.0230	0.322986
$964$	0	0
$965$	1.09207	0.0351550
$966$	0	0
$967$	−12.9764	−0.417292	−0.208646	−	0.977991i	$-0.566906\pi$
$967$	−12.9764	−0.417292	−0.208646	+	0.977991i	$0.566906\pi$
$968$	0	0
$969$	−8.48302	−0.272514
$970$	0	0
$971$	21.6496	0.694769	0.347384	−	0.937723i	$-0.387070\pi$
$971$	21.6496	0.694769	0.347384	+	0.937723i	$0.387070\pi$
$972$	0	0
$973$	−7.03396	−0.225498
$974$	0	0
$975$	4.55753	0.145958
$976$	0	0
$977$	13.2289	0.423231	0.211616	−	0.977353i	$-0.432128\pi$
$977$	13.2289	0.423231	0.211616	+	0.977353i	$0.432128\pi$
$978$	0	0
$979$	5.15122	0.164634
$980$	0	0
$981$	12.0230	0.383864
$982$	0	0
$983$	−34.3176	−1.09456	−0.547281	−	0.836949i	$-0.684337\pi$
$983$	−34.3176	−1.09456	−0.547281	+	0.836949i	$0.684337\pi$
$984$	0	0
$985$	−13.0799	−0.416761
$986$	0	0
$987$	−6.59690	−0.209982
$988$	0	0
$989$	−21.7699	−0.692242
$990$	0	0
$991$	6.66750	0.211800	0.105900	−	0.994377i	$-0.466228\pi$
$991$	6.66750	0.211800	0.105900	+	0.994377i	$0.466228\pi$
$992$	0	0
$993$	−15.4721	−0.490991
$994$	0	0
$995$	−21.2710	−0.674336
$996$	0	0
$997$	−44.3857	−1.40571	−0.702855	−	0.711333i	$-0.748092\pi$
$997$	−44.3857	−1.40571	−0.702855	+	0.711333i	$0.748092\pi$
$998$	0	0
$999$	−9.04055	−0.286030

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4080.2.a.bt.1.4		4
4.3	odd	2		255.2.a.d.1.4	✓	4
12.11	even	2		765.2.a.m.1.1		4
20.3	even	4		1275.2.b.k.1174.2		8
20.7	even	4		1275.2.b.k.1174.7		8
20.19	odd	2		1275.2.a.t.1.1		4
60.59	even	2		3825.2.a.bi.1.4		4
68.67	odd	2		4335.2.a.z.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
255.2.a.d.1.4	✓	4	4.3	odd	2
765.2.a.m.1.1		4	12.11	even	2
1275.2.a.t.1.1		4	20.19	odd	2
1275.2.b.k.1174.2		8	20.3	even	4
1275.2.b.k.1174.7		8	20.7	even	4
3825.2.a.bi.1.4		4	60.59	even	2
4080.2.a.bt.1.4		4	1.1	even	1	trivial
4335.2.a.z.1.4		4	68.67	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:	\( N \)	\(=\)	\( 4080 = 2^{4} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4080.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +4.48302 q^{7} +1.00000 q^{9} -1.47153 q^{11} -4.55753 q^{13} +1.00000 q^{15} -1.00000 q^{17} -8.48302 q^{19} -4.48302 q^{21} +6.40852 q^{23} +1.00000 q^{25} -1.00000 q^{27} +6.48302 q^{29} +3.01149 q^{31} +1.47153 q^{33} -4.48302 q^{35} +9.04055 q^{37} +4.55753 q^{39} -3.53996 q^{41} -3.39702 q^{43} -1.00000 q^{45} +1.47153 q^{47} +13.0975 q^{49} +1.00000 q^{51} +3.53996 q^{53} +1.47153 q^{55} +8.48302 q^{57} +6.55753 q^{59} -4.55753 q^{61} +4.48302 q^{63} +4.55753 q^{65} -7.16050 q^{67} -6.40852 q^{69} +7.39702 q^{71} -3.92549 q^{73} -1.00000 q^{75} -6.59690 q^{77} -0.453964 q^{79} +1.00000 q^{81} +7.39702 q^{83} +1.00000 q^{85} -6.48302 q^{87} -3.50059 q^{89} -20.4315 q^{91} -3.01149 q^{93} +8.48302 q^{95} +12.0230 q^{97} -1.47153 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} - 4 q^{7} + 4 q^{9} - 2 q^{11} - 2 q^{13} + 4 q^{15} - 4 q^{17} - 12 q^{19} + 4 q^{21} - 2 q^{23} + 4 q^{25} - 4 q^{27} + 4 q^{29} - 6 q^{31} + 2 q^{33} + 4 q^{35} - 2 q^{37} + 2 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 - 4 * q^7 + 4 * q^9 - 2 * q^11 - 2 * q^13 + 4 * q^15 - 4 * q^17 - 12 * q^19 + 4 * q^21 - 2 * q^23 + 4 * q^25 - 4 * q^27 + 4 * q^29 - 6 * q^31 + 2 * q^33 + 4 * q^35 - 2 * q^37 + 2 * q^39 - 4 * q^43 - 4 * q^45 + 2 * q^47 + 22 * q^49 + 4 * q^51 + 2 * q^55 + 12 * q^57 + 10 * q^59 - 2 * q^61 - 4 * q^63 + 2 * q^65 - 22 * q^67 + 2 * q^69 + 20 * q^71 - 10 * q^73 - 4 * q^75 - 20 * q^77 + 4 * q^81 + 20 * q^83 + 4 * q^85 - 4 * q^87 + 10 * q^89 - 18 * q^91 + 6 * q^93 + 12 * q^95 + 12 * q^97 - 2 * q^99

Introduction

L-functions

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