LMFDB - Embedded newform 405.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} -4.73205 q^{7} +2.53590 q^{8} +0.732051 q^{10} +5.73205 q^{11} +1.46410 q^{13} +3.46410 q^{14} +1.07180 q^{16} +2.73205 q^{17} +4.46410 q^{19} +1.46410 q^{20} -4.19615 q^{22} +3.46410 q^{23} +1.00000 q^{25} -1.07180 q^{26} +6.92820 q^{28} -3.19615 q^{29} -3.00000 q^{31} -5.85641 q^{32} -2.00000 q^{34} +4.73205 q^{35} -2.73205 q^{37} -3.26795 q^{38} -2.53590 q^{40} +7.19615 q^{41} +0.196152 q^{43} -8.39230 q^{44} -2.53590 q^{46} +8.73205 q^{47} +15.3923 q^{49} -0.732051 q^{50} -2.14359 q^{52} -6.73205 q^{53} -5.73205 q^{55} -12.0000 q^{56} +2.33975 q^{58} +8.26795 q^{59} +4.00000 q^{61} +2.19615 q^{62} +2.14359 q^{64} -1.46410 q^{65} +3.46410 q^{67} -4.00000 q^{68} -3.46410 q^{70} +3.73205 q^{71} -7.66025 q^{73} +2.00000 q^{74} -6.53590 q^{76} -27.1244 q^{77} +15.4641 q^{79} -1.07180 q^{80} -5.26795 q^{82} -2.19615 q^{83} -2.73205 q^{85} -0.143594 q^{86} +14.5359 q^{88} -5.19615 q^{89} -6.92820 q^{91} -5.07180 q^{92} -6.39230 q^{94} -4.46410 q^{95} -9.66025 q^{97} -11.2679 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} - 6 q^{7} + 12 q^{8} - 2 q^{10} + 8 q^{11} - 4 q^{13} + 16 q^{16} + 2 q^{17} + 2 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} - 16 q^{26} + 4 q^{29} - 6 q^{31} + 16 q^{32} - 4 q^{34}+ \cdots - 26 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 - 6 * q^7 + 12 * q^8 - 2 * q^10 + 8 * q^11 - 4 * q^13 + 16 * q^16 + 2 * q^17 + 2 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 - 16 * q^26 + 4 * q^29 - 6 * q^31 + 16 * q^32 - 4 * q^34 + 6 * q^35 - 2 * q^37 - 10 * q^38 - 12 * q^40 + 4 * q^41 - 10 * q^43 + 4 * q^44 - 12 * q^46 + 14 * q^47 + 10 * q^49 + 2 * q^50 - 32 * q^52 - 10 * q^53 - 8 * q^55 - 24 * q^56 + 22 * q^58 + 20 * q^59 + 8 * q^61 - 6 * q^62 + 32 * q^64 + 4 * q^65 - 8 * q^68 + 4 * q^71 + 2 * q^73 + 4 * q^74 - 20 * q^76 - 30 * q^77 + 24 * q^79 - 16 * q^80 - 14 * q^82 + 6 * q^83 - 2 * q^85 - 28 * q^86 + 36 * q^88 - 24 * q^92 + 8 * q^94 - 2 * q^95 - 2 * q^97 - 26 * 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 $

$2$	−0.732051	−0.517638	−0.258819	−	0.965926i	$-0.583333\pi$
$2$	−0.732051	−0.517638	−0.258819	+	0.965926i	$0.583333\pi$
$3$	0	0
$3$	0	0
$4$	−1.46410	−0.732051
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.73205	−1.78855	−0.894274	−	0.447521i	$-0.852307\pi$
$7$	−4.73205	−1.78855	−0.894274	+	0.447521i	$0.852307\pi$
$8$	2.53590	0.896575
$9$	0	0
$10$	0.732051	0.231495
$11$	5.73205	1.72828	0.864139	−	0.503253i	$-0.167864\pi$
$11$	5.73205	1.72828	0.864139	+	0.503253i	$0.167864\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	3.46410	0.925820
$15$	0	0
$16$	1.07180	0.267949
$17$	2.73205	0.662620	0.331310	−	0.943522i	$-0.392509\pi$
$17$	2.73205	0.662620	0.331310	+	0.943522i	$0.392509\pi$
$18$	0	0
$19$	4.46410	1.02414	0.512068	−	0.858945i	$-0.328880\pi$
$19$	4.46410	1.02414	0.512068	+	0.858945i	$0.328880\pi$
$20$	1.46410	0.327383
$21$	0	0
$22$	−4.19615	−0.894623
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.07180	−0.210197
$27$	0	0
$28$	6.92820	1.30931
$29$	−3.19615	−0.593511	−0.296755	−	0.954954i	$-0.595905\pi$
$29$	−3.19615	−0.593511	−0.296755	+	0.954954i	$0.595905\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−5.85641	−1.03528
$33$	0	0
$34$	−2.00000	−0.342997
$35$	4.73205	0.799863
$36$	0	0
$37$	−2.73205	−0.449146	−0.224573	−	0.974457i	$-0.572099\pi$
$37$	−2.73205	−0.449146	−0.224573	+	0.974457i	$0.572099\pi$
$38$	−3.26795	−0.530131
$39$	0	0
$40$	−2.53590	−0.400961
$41$	7.19615	1.12385	0.561925	−	0.827188i	$-0.310061\pi$
$41$	7.19615	1.12385	0.561925	+	0.827188i	$0.310061\pi$
$42$	0	0
$43$	0.196152	0.0299130	0.0149565	−	0.999888i	$-0.495239\pi$
$43$	0.196152	0.0299130	0.0149565	+	0.999888i	$0.495239\pi$
$44$	−8.39230	−1.26519
$45$	0	0
$46$	−2.53590	−0.373898
$47$	8.73205	1.27370	0.636850	−	0.770988i	$-0.280237\pi$
$47$	8.73205	1.27370	0.636850	+	0.770988i	$0.280237\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	−0.732051	−0.103528
$51$	0	0
$52$	−2.14359	−0.297263
$53$	−6.73205	−0.924718	−0.462359	−	0.886693i	$-0.652997\pi$
$53$	−6.73205	−0.924718	−0.462359	+	0.886693i	$0.652997\pi$
$54$	0	0
$55$	−5.73205	−0.772910
$56$	−12.0000	−1.60357
$57$	0	0
$58$	2.33975	0.307224
$59$	8.26795	1.07640	0.538198	−	0.842819i	$-0.319105\pi$
$59$	8.26795	1.07640	0.538198	+	0.842819i	$0.319105\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	2.19615	0.278912
$63$	0	0
$64$	2.14359	0.267949
$65$	−1.46410	−0.181599
$66$	0	0
$67$	3.46410	0.423207	0.211604	−	0.977356i	$-0.432131\pi$
$67$	3.46410	0.423207	0.211604	+	0.977356i	$0.432131\pi$
$68$	−4.00000	−0.485071
$69$	0	0
$70$	−3.46410	−0.414039
$71$	3.73205	0.442913	0.221456	−	0.975170i	$-0.428919\pi$
$71$	3.73205	0.442913	0.221456	+	0.975170i	$0.428919\pi$
$72$	0	0
$73$	−7.66025	−0.896565	−0.448282	−	0.893892i	$-0.647964\pi$
$73$	−7.66025	−0.896565	−0.448282	+	0.893892i	$0.647964\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	−6.53590	−0.749719
$77$	−27.1244	−3.09111
$78$	0	0
$79$	15.4641	1.73985	0.869924	−	0.493186i	$-0.164168\pi$
$79$	15.4641	1.73985	0.869924	+	0.493186i	$0.164168\pi$
$80$	−1.07180	−0.119831
$81$	0	0
$82$	−5.26795	−0.581748
$83$	−2.19615	−0.241059	−0.120530	−	0.992710i	$-0.538459\pi$
$83$	−2.19615	−0.241059	−0.120530	+	0.992710i	$0.538459\pi$
$84$	0	0
$85$	−2.73205	−0.296333
$86$	−0.143594	−0.0154841
$87$	0	0
$88$	14.5359	1.54953
$89$	−5.19615	−0.550791	−0.275396	−	0.961331i	$-0.588809\pi$
$89$	−5.19615	−0.550791	−0.275396	+	0.961331i	$0.588809\pi$
$90$	0	0
$91$	−6.92820	−0.726273
$92$	−5.07180	−0.528771
$93$	0	0
$94$	−6.39230	−0.659316
$95$	−4.46410	−0.458007
$96$	0	0
$97$	−9.66025	−0.980850	−0.490425	−	0.871483i	$-0.663158\pi$
$97$	−9.66025	−0.980850	−0.490425	+	0.871483i	$0.663158\pi$
$98$	−11.2679	−1.13823
$99$	0	0
$100$	−1.46410	−0.146410
$101$	2.66025	0.264705	0.132353	−	0.991203i	$-0.457747\pi$
$101$	2.66025	0.264705	0.132353	+	0.991203i	$0.457747\pi$
$102$	0	0
$103$	−0.535898	−0.0528036	−0.0264018	−	0.999651i	$-0.508405\pi$
$103$	−0.535898	−0.0528036	−0.0264018	+	0.999651i	$0.508405\pi$
$104$	3.71281	0.364071
$105$	0	0
$106$	4.92820	0.478669
$107$	8.53590	0.825196	0.412598	−	0.910913i	$-0.364621\pi$
$107$	8.53590	0.825196	0.412598	+	0.910913i	$0.364621\pi$
$108$	0	0
$109$	−6.07180	−0.581573	−0.290786	−	0.956788i	$-0.593917\pi$
$109$	−6.07180	−0.581573	−0.290786	+	0.956788i	$0.593917\pi$
$110$	4.19615	0.400087
$111$	0	0
$112$	−5.07180	−0.479240
$113$	19.1244	1.79907	0.899534	−	0.436851i	$-0.143906\pi$
$113$	19.1244	1.79907	0.899534	+	0.436851i	$0.143906\pi$
$114$	0	0
$115$	−3.46410	−0.323029
$116$	4.67949	0.434480
$117$	0	0
$118$	−6.05256	−0.557183
$119$	−12.9282	−1.18513
$120$	0	0
$121$	21.8564	1.98695
$122$	−2.92820	−0.265107
$123$	0	0
$124$	4.39230	0.394441
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.5885	−1.29452	−0.647258	−	0.762271i	$-0.724084\pi$
$127$	−14.5885	−1.29452	−0.647258	+	0.762271i	$0.724084\pi$
$128$	10.1436	0.896575
$129$	0	0
$130$	1.07180	0.0940028
$131$	−15.5885	−1.36197	−0.680985	−	0.732297i	$-0.738448\pi$
$131$	−15.5885	−1.36197	−0.680985	+	0.732297i	$0.738448\pi$
$132$	0	0
$133$	−21.1244	−1.83171
$134$	−2.53590	−0.219068
$135$	0	0
$136$	6.92820	0.594089
$137$	−2.53590	−0.216656	−0.108328	−	0.994115i	$-0.534550\pi$
$137$	−2.53590	−0.216656	−0.108328	+	0.994115i	$0.534550\pi$
$138$	0	0
$139$	0.607695	0.0515440	0.0257720	−	0.999668i	$-0.491796\pi$
$139$	0.607695	0.0515440	0.0257720	+	0.999668i	$0.491796\pi$
$140$	−6.92820	−0.585540
$141$	0	0
$142$	−2.73205	−0.229269
$143$	8.39230	0.701800
$144$	0	0
$145$	3.19615	0.265426
$146$	5.60770	0.464096
$147$	0	0
$148$	4.00000	0.328798
$149$	8.00000	0.655386	0.327693	−	0.944784i	$-0.393729\pi$
$149$	8.00000	0.655386	0.327693	+	0.944784i	$0.393729\pi$
$150$	0	0
$151$	5.39230	0.438820	0.219410	−	0.975633i	$-0.429587\pi$
$151$	5.39230	0.438820	0.219410	+	0.975633i	$0.429587\pi$
$152$	11.3205	0.918214
$153$	0	0
$154$	19.8564	1.60007
$155$	3.00000	0.240966
$156$	0	0
$157$	−19.1244	−1.52629	−0.763145	−	0.646227i	$-0.776346\pi$
$157$	−19.1244	−1.52629	−0.763145	+	0.646227i	$0.776346\pi$
$158$	−11.3205	−0.900611
$159$	0	0
$160$	5.85641	0.462990
$161$	−16.3923	−1.29189
$162$	0	0
$163$	12.7321	0.997251	0.498626	−	0.866817i	$-0.333838\pi$
$163$	12.7321	0.997251	0.498626	+	0.866817i	$0.333838\pi$
$164$	−10.5359	−0.822715
$165$	0	0
$166$	1.60770	0.124781
$167$	17.6603	1.36659	0.683296	−	0.730142i	$-0.260546\pi$
$167$	17.6603	1.36659	0.683296	+	0.730142i	$0.260546\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	2.00000	0.153393
$171$	0	0
$172$	−0.287187	−0.0218978
$173$	−8.53590	−0.648972	−0.324486	−	0.945890i	$-0.605191\pi$
$173$	−8.53590	−0.648972	−0.324486	+	0.945890i	$0.605191\pi$
$174$	0	0
$175$	−4.73205	−0.357709
$176$	6.14359	0.463091
$177$	0	0
$178$	3.80385	0.285110
$179$	−8.12436	−0.607243	−0.303621	−	0.952793i	$-0.598196\pi$
$179$	−8.12436	−0.607243	−0.303621	+	0.952793i	$0.598196\pi$
$180$	0	0
$181$	26.4641	1.96706	0.983531	−	0.180742i	$-0.0578498\pi$
$181$	26.4641	1.96706	0.983531	+	0.180742i	$0.0578498\pi$
$182$	5.07180	0.375947
$183$	0	0
$184$	8.78461	0.647610
$185$	2.73205	0.200864
$186$	0	0
$187$	15.6603	1.14519
$188$	−12.7846	−0.932413
$189$	0	0
$190$	3.26795	0.237082
$191$	8.12436	0.587858	0.293929	−	0.955827i	$-0.405037\pi$
$191$	8.12436	0.587858	0.293929	+	0.955827i	$0.405037\pi$
$192$	0	0
$193$	−5.26795	−0.379195	−0.189598	−	0.981862i	$-0.560718\pi$
$193$	−5.26795	−0.379195	−0.189598	+	0.981862i	$0.560718\pi$
$194$	7.07180	0.507725
$195$	0	0
$196$	−22.5359	−1.60971
$197$	13.8564	0.987228	0.493614	−	0.869681i	$-0.335676\pi$
$197$	13.8564	0.987228	0.493614	+	0.869681i	$0.335676\pi$
$198$	0	0
$199$	−2.00000	−0.141776	−0.0708881	−	0.997484i	$-0.522583\pi$
$199$	−2.00000	−0.141776	−0.0708881	+	0.997484i	$0.522583\pi$
$200$	2.53590	0.179315
$201$	0	0
$202$	−1.94744	−0.137021
$203$	15.1244	1.06152
$204$	0	0
$205$	−7.19615	−0.502601
$206$	0.392305	0.0273332
$207$	0	0
$208$	1.56922	0.108806
$209$	25.5885	1.76999
$210$	0	0
$211$	−8.85641	−0.609700	−0.304850	−	0.952400i	$-0.598606\pi$
$211$	−8.85641	−0.609700	−0.304850	+	0.952400i	$0.598606\pi$
$212$	9.85641	0.676941
$213$	0	0
$214$	−6.24871	−0.427153
$215$	−0.196152	−0.0133775
$216$	0	0
$217$	14.1962	0.963698
$218$	4.44486	0.301044
$219$	0	0
$220$	8.39230	0.565809
$221$	4.00000	0.269069
$222$	0	0
$223$	16.7846	1.12398	0.561990	−	0.827144i	$-0.310036\pi$
$223$	16.7846	1.12398	0.561990	+	0.827144i	$0.310036\pi$
$224$	27.7128	1.85164
$225$	0	0
$226$	−14.0000	−0.931266
$227$	18.0526	1.19819	0.599095	−	0.800678i	$-0.295527\pi$
$227$	18.0526	1.19819	0.599095	+	0.800678i	$0.295527\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	2.53590	0.167212
$231$	0	0
$232$	−8.10512	−0.532127
$233$	28.0526	1.83778	0.918892	−	0.394509i	$-0.129085\pi$
$233$	28.0526	1.83778	0.918892	+	0.394509i	$0.129085\pi$
$234$	0	0
$235$	−8.73205	−0.569616
$236$	−12.1051	−0.787976
$237$	0	0
$238$	9.46410	0.613467
$239$	0.535898	0.0346644	0.0173322	−	0.999850i	$-0.494483\pi$
$239$	0.535898	0.0346644	0.0173322	+	0.999850i	$0.494483\pi$
$240$	0	0
$241$	−16.3205	−1.05130	−0.525648	−	0.850702i	$-0.676177\pi$
$241$	−16.3205	−1.05130	−0.525648	+	0.850702i	$0.676177\pi$
$242$	−16.0000	−1.02852
$243$	0	0
$244$	−5.85641	−0.374918
$245$	−15.3923	−0.983378
$246$	0	0
$247$	6.53590	0.415869
$248$	−7.60770	−0.483089
$249$	0	0
$250$	0.732051	0.0462990
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	19.8564	1.24836
$254$	10.6795	0.670091
$255$	0	0
$256$	−11.7128	−0.732051
$257$	−10.3923	−0.648254	−0.324127	−	0.946014i	$-0.605071\pi$
$257$	−10.3923	−0.648254	−0.324127	+	0.946014i	$0.605071\pi$
$258$	0	0
$259$	12.9282	0.803319
$260$	2.14359	0.132940
$261$	0	0
$262$	11.4115	0.705007
$263$	−21.3205	−1.31468	−0.657339	−	0.753595i	$-0.728318\pi$
$263$	−21.3205	−1.31468	−0.657339	+	0.753595i	$0.728318\pi$
$264$	0	0
$265$	6.73205	0.413547
$266$	15.4641	0.948165
$267$	0	0
$268$	−5.07180	−0.309809
$269$	−10.6603	−0.649967	−0.324984	−	0.945720i	$-0.605359\pi$
$269$	−10.6603	−0.649967	−0.324984	+	0.945720i	$0.605359\pi$
$270$	0	0
$271$	−2.92820	−0.177876	−0.0889378	−	0.996037i	$-0.528347\pi$
$271$	−2.92820	−0.177876	−0.0889378	+	0.996037i	$0.528347\pi$
$272$	2.92820	0.177548
$273$	0	0
$274$	1.85641	0.112150
$275$	5.73205	0.345656
$276$	0	0
$277$	3.80385	0.228551	0.114276	−	0.993449i	$-0.463545\pi$
$277$	3.80385	0.228551	0.114276	+	0.993449i	$0.463545\pi$
$278$	−0.444864	−0.0266812
$279$	0	0
$280$	12.0000	0.717137
$281$	−15.4641	−0.922511	−0.461255	−	0.887267i	$-0.652601\pi$
$281$	−15.4641	−0.922511	−0.461255	+	0.887267i	$0.652601\pi$
$282$	0	0
$283$	−29.3205	−1.74292	−0.871462	−	0.490464i	$-0.836827\pi$
$283$	−29.3205	−1.74292	−0.871462	+	0.490464i	$0.836827\pi$
$284$	−5.46410	−0.324235
$285$	0	0
$286$	−6.14359	−0.363278
$287$	−34.0526	−2.01006
$288$	0	0
$289$	−9.53590	−0.560935
$290$	−2.33975	−0.137395
$291$	0	0
$292$	11.2154	0.656331
$293$	−28.7321	−1.67854	−0.839272	−	0.543712i	$-0.817019\pi$
$293$	−28.7321	−1.67854	−0.839272	+	0.543712i	$0.817019\pi$
$294$	0	0
$295$	−8.26795	−0.481379
$296$	−6.92820	−0.402694
$297$	0	0
$298$	−5.85641	−0.339253
$299$	5.07180	0.293310
$300$	0	0
$301$	−0.928203	−0.0535007
$302$	−3.94744	−0.227150
$303$	0	0
$304$	4.78461	0.274416
$305$	−4.00000	−0.229039
$306$	0	0
$307$	14.0526	0.802022	0.401011	−	0.916073i	$-0.368659\pi$
$307$	14.0526	0.802022	0.401011	+	0.916073i	$0.368659\pi$
$308$	39.7128	2.26285
$309$	0	0
$310$	−2.19615	−0.124733
$311$	−19.7321	−1.11890	−0.559451	−	0.828863i	$-0.688988\pi$
$311$	−19.7321	−1.11890	−0.559451	+	0.828863i	$0.688988\pi$
$312$	0	0
$313$	−9.07180	−0.512768	−0.256384	−	0.966575i	$-0.582531\pi$
$313$	−9.07180	−0.512768	−0.256384	+	0.966575i	$0.582531\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	−22.6410	−1.27366
$317$	−6.19615	−0.348011	−0.174005	−	0.984745i	$-0.555671\pi$
$317$	−6.19615	−0.348011	−0.174005	+	0.984745i	$0.555671\pi$
$318$	0	0
$319$	−18.3205	−1.02575
$320$	−2.14359	−0.119831
$321$	0	0
$322$	12.0000	0.668734
$323$	12.1962	0.678612
$324$	0	0
$325$	1.46410	0.0812137
$326$	−9.32051	−0.516215
$327$	0	0
$328$	18.2487	1.00762
$329$	−41.3205	−2.27807
$330$	0	0
$331$	−0.464102	−0.0255093	−0.0127547	−	0.999919i	$-0.504060\pi$
$331$	−0.464102	−0.0255093	−0.0127547	+	0.999919i	$0.504060\pi$
$332$	3.21539	0.176467
$333$	0	0
$334$	−12.9282	−0.707400
$335$	−3.46410	−0.189264
$336$	0	0
$337$	27.3205	1.48824	0.744121	−	0.668044i	$-0.232868\pi$
$337$	27.3205	1.48824	0.744121	+	0.668044i	$0.232868\pi$
$338$	7.94744	0.432284
$339$	0	0
$340$	4.00000	0.216930
$341$	−17.1962	−0.931224
$342$	0	0
$343$	−39.7128	−2.14429
$344$	0.497423	0.0268192
$345$	0	0
$346$	6.24871	0.335933
$347$	28.5885	1.53471	0.767354	−	0.641223i	$-0.221573\pi$
$347$	28.5885	1.53471	0.767354	+	0.641223i	$0.221573\pi$
$348$	0	0
$349$	−18.8564	−1.00936	−0.504680	−	0.863306i	$-0.668390\pi$
$349$	−18.8564	−1.00936	−0.504680	+	0.863306i	$0.668390\pi$
$350$	3.46410	0.185164
$351$	0	0
$352$	−33.5692	−1.78925
$353$	−25.5167	−1.35811	−0.679057	−	0.734085i	$-0.737611\pi$
$353$	−25.5167	−1.35811	−0.679057	+	0.734085i	$0.737611\pi$
$354$	0	0
$355$	−3.73205	−0.198077
$356$	7.60770	0.403207
$357$	0	0
$358$	5.94744	0.314332
$359$	6.12436	0.323231	0.161616	−	0.986854i	$-0.448330\pi$
$359$	6.12436	0.323231	0.161616	+	0.986854i	$0.448330\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	−19.3731	−1.01823
$363$	0	0
$364$	10.1436	0.531669
$365$	7.66025	0.400956
$366$	0	0
$367$	31.1769	1.62742	0.813711	−	0.581270i	$-0.197444\pi$
$367$	31.1769	1.62742	0.813711	+	0.581270i	$0.197444\pi$
$368$	3.71281	0.193544
$369$	0	0
$370$	−2.00000	−0.103975
$371$	31.8564	1.65390
$372$	0	0
$373$	20.0526	1.03828	0.519141	−	0.854689i	$-0.326252\pi$
$373$	20.0526	1.03828	0.519141	+	0.854689i	$0.326252\pi$
$374$	−11.4641	−0.592795
$375$	0	0
$376$	22.1436	1.14197
$377$	−4.67949	−0.241006
$378$	0	0
$379$	2.39230	0.122884	0.0614422	−	0.998111i	$-0.480430\pi$
$379$	2.39230	0.122884	0.0614422	+	0.998111i	$0.480430\pi$
$380$	6.53590	0.335285
$381$	0	0
$382$	−5.94744	−0.304298
$383$	−2.53590	−0.129578	−0.0647892	−	0.997899i	$-0.520637\pi$
$383$	−2.53590	−0.129578	−0.0647892	+	0.997899i	$0.520637\pi$
$384$	0	0
$385$	27.1244	1.38239
$386$	3.85641	0.196286
$387$	0	0
$388$	14.1436	0.718032
$389$	−27.4641	−1.39249	−0.696243	−	0.717807i	$-0.745146\pi$
$389$	−27.4641	−1.39249	−0.696243	+	0.717807i	$0.745146\pi$
$390$	0	0
$391$	9.46410	0.478620
$392$	39.0333	1.97148
$393$	0	0
$394$	−10.1436	−0.511027
$395$	−15.4641	−0.778083
$396$	0	0
$397$	14.3923	0.722329	0.361165	−	0.932502i	$-0.382379\pi$
$397$	14.3923	0.722329	0.361165	+	0.932502i	$0.382379\pi$
$398$	1.46410	0.0733888
$399$	0	0
$400$	1.07180	0.0535898
$401$	−24.9282	−1.24486	−0.622428	−	0.782677i	$-0.713853\pi$
$401$	−24.9282	−1.24486	−0.622428	+	0.782677i	$0.713853\pi$
$402$	0	0
$403$	−4.39230	−0.218796
$404$	−3.89488	−0.193778
$405$	0	0
$406$	−11.0718	−0.549484
$407$	−15.6603	−0.776250
$408$	0	0
$409$	−17.8564	−0.882942	−0.441471	−	0.897275i	$-0.645543\pi$
$409$	−17.8564	−0.882942	−0.441471	+	0.897275i	$0.645543\pi$
$410$	5.26795	0.260165
$411$	0	0
$412$	0.784610	0.0386549
$413$	−39.1244	−1.92518
$414$	0	0
$415$	2.19615	0.107805
$416$	−8.57437	−0.420393
$417$	0	0
$418$	−18.7321	−0.916215
$419$	−20.3923	−0.996229	−0.498115	−	0.867111i	$-0.665974\pi$
$419$	−20.3923	−0.996229	−0.498115	+	0.867111i	$0.665974\pi$
$420$	0	0
$421$	33.7846	1.64656	0.823281	−	0.567635i	$-0.192141\pi$
$421$	33.7846	1.64656	0.823281	+	0.567635i	$0.192141\pi$
$422$	6.48334	0.315604
$423$	0	0
$424$	−17.0718	−0.829080
$425$	2.73205	0.132524
$426$	0	0
$427$	−18.9282	−0.916000
$428$	−12.4974	−0.604086
$429$	0	0
$430$	0.143594	0.00692470
$431$	21.3397	1.02790	0.513950	−	0.857820i	$-0.328182\pi$
$431$	21.3397	1.02790	0.513950	+	0.857820i	$0.328182\pi$
$432$	0	0
$433$	−35.4641	−1.70430	−0.852148	−	0.523301i	$-0.824700\pi$
$433$	−35.4641	−1.70430	−0.852148	+	0.523301i	$0.824700\pi$
$434$	−10.3923	−0.498847
$435$	0	0
$436$	8.88973	0.425741
$437$	15.4641	0.739748
$438$	0	0
$439$	5.39230	0.257361	0.128680	−	0.991686i	$-0.458926\pi$
$439$	5.39230	0.257361	0.128680	+	0.991686i	$0.458926\pi$
$440$	−14.5359	−0.692972
$441$	0	0
$442$	−2.92820	−0.139280
$443$	0.339746	0.0161418	0.00807091	−	0.999967i	$-0.497431\pi$
$443$	0.339746	0.0161418	0.00807091	+	0.999967i	$0.497431\pi$
$444$	0	0
$445$	5.19615	0.246321
$446$	−12.2872	−0.581815
$447$	0	0
$448$	−10.1436	−0.479240
$449$	−8.12436	−0.383412	−0.191706	−	0.981452i	$-0.561402\pi$
$449$	−8.12436	−0.383412	−0.191706	+	0.981452i	$0.561402\pi$
$450$	0	0
$451$	41.2487	1.94233
$452$	−28.0000	−1.31701
$453$	0	0
$454$	−13.2154	−0.620229
$455$	6.92820	0.324799
$456$	0	0
$457$	2.73205	0.127800	0.0639000	−	0.997956i	$-0.479646\pi$
$457$	2.73205	0.127800	0.0639000	+	0.997956i	$0.479646\pi$
$458$	8.78461	0.410478
$459$	0	0
$460$	5.07180	0.236474
$461$	37.0526	1.72571	0.862855	−	0.505452i	$-0.168674\pi$
$461$	37.0526	1.72571	0.862855	+	0.505452i	$0.168674\pi$
$462$	0	0
$463$	−10.3923	−0.482971	−0.241486	−	0.970404i	$-0.577635\pi$
$463$	−10.3923	−0.482971	−0.241486	+	0.970404i	$0.577635\pi$
$464$	−3.42563	−0.159031
$465$	0	0
$466$	−20.5359	−0.951307
$467$	−37.3731	−1.72942	−0.864710	−	0.502272i	$-0.832498\pi$
$467$	−37.3731	−1.72942	−0.864710	+	0.502272i	$0.832498\pi$
$468$	0	0
$469$	−16.3923	−0.756926
$470$	6.39230	0.294855
$471$	0	0
$472$	20.9667	0.965070
$473$	1.12436	0.0516979
$474$	0	0
$475$	4.46410	0.204827
$476$	18.9282	0.867573
$477$	0	0
$478$	−0.392305	−0.0179436
$479$	11.8756	0.542612	0.271306	−	0.962493i	$-0.412544\pi$
$479$	11.8756	0.542612	0.271306	+	0.962493i	$0.412544\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	11.9474	0.544191
$483$	0	0
$484$	−32.0000	−1.45455
$485$	9.66025	0.438650
$486$	0	0
$487$	24.3923	1.10532	0.552660	−	0.833407i	$-0.313613\pi$
$487$	24.3923	1.10532	0.552660	+	0.833407i	$0.313613\pi$
$488$	10.1436	0.459179
$489$	0	0
$490$	11.2679	0.509034
$491$	13.8756	0.626199	0.313100	−	0.949720i	$-0.398633\pi$
$491$	13.8756	0.626199	0.313100	+	0.949720i	$0.398633\pi$
$492$	0	0
$493$	−8.73205	−0.393272
$494$	−4.78461	−0.215270
$495$	0	0
$496$	−3.21539	−0.144375
$497$	−17.6603	−0.792171
$498$	0	0
$499$	24.3205	1.08874	0.544368	−	0.838847i	$-0.316770\pi$
$499$	24.3205	1.08874	0.544368	+	0.838847i	$0.316770\pi$
$500$	1.46410	0.0654766
$501$	0	0
$502$	−7.60770	−0.339548
$503$	7.32051	0.326405	0.163203	−	0.986593i	$-0.447818\pi$
$503$	7.32051	0.326405	0.163203	+	0.986593i	$0.447818\pi$
$504$	0	0
$505$	−2.66025	−0.118380
$506$	−14.5359	−0.646200
$507$	0	0
$508$	21.3590	0.947652
$509$	6.78461	0.300723	0.150361	−	0.988631i	$-0.451956\pi$
$509$	6.78461	0.300723	0.150361	+	0.988631i	$0.451956\pi$
$510$	0	0
$511$	36.2487	1.60355
$512$	−11.7128	−0.517638
$513$	0	0
$514$	7.60770	0.335561
$515$	0.535898	0.0236145
$516$	0	0
$517$	50.0526	2.20131
$518$	−9.46410	−0.415829
$519$	0	0
$520$	−3.71281	−0.162818
$521$	−19.4641	−0.852738	−0.426369	−	0.904549i	$-0.640207\pi$
$521$	−19.4641	−0.852738	−0.426369	+	0.904549i	$0.640207\pi$
$522$	0	0
$523$	22.2487	0.972868	0.486434	−	0.873717i	$-0.338297\pi$
$523$	22.2487	0.972868	0.486434	+	0.873717i	$0.338297\pi$
$524$	22.8231	0.997031
$525$	0	0
$526$	15.6077	0.680528
$527$	−8.19615	−0.357030
$528$	0	0
$529$	−11.0000	−0.478261
$530$	−4.92820	−0.214067
$531$	0	0
$532$	30.9282	1.34091
$533$	10.5359	0.456360
$534$	0	0
$535$	−8.53590	−0.369039
$536$	8.78461	0.379437
$537$	0	0
$538$	7.80385	0.336448
$539$	88.2295	3.80031
$540$	0	0
$541$	−24.4641	−1.05179	−0.525897	−	0.850548i	$-0.676270\pi$
$541$	−24.4641	−1.05179	−0.525897	+	0.850548i	$0.676270\pi$
$542$	2.14359	0.0920752
$543$	0	0
$544$	−16.0000	−0.685994
$545$	6.07180	0.260087
$546$	0	0
$547$	−33.8564	−1.44760	−0.723798	−	0.690012i	$-0.757605\pi$
$547$	−33.8564	−1.44760	−0.723798	+	0.690012i	$0.757605\pi$
$548$	3.71281	0.158604
$549$	0	0
$550$	−4.19615	−0.178925
$551$	−14.2679	−0.607835
$552$	0	0
$553$	−73.1769	−3.11180
$554$	−2.78461	−0.118307
$555$	0	0
$556$	−0.889727	−0.0377328
$557$	2.53590	0.107449	0.0537247	−	0.998556i	$-0.482891\pi$
$557$	2.53590	0.107449	0.0537247	+	0.998556i	$0.482891\pi$
$558$	0	0
$559$	0.287187	0.0121467
$560$	5.07180	0.214323
$561$	0	0
$562$	11.3205	0.477527
$563$	16.7321	0.705172	0.352586	−	0.935779i	$-0.385302\pi$
$563$	16.7321	0.705172	0.352586	+	0.935779i	$0.385302\pi$
$564$	0	0
$565$	−19.1244	−0.804568
$566$	21.4641	0.902203
$567$	0	0
$568$	9.46410	0.397105
$569$	32.9090	1.37962	0.689808	−	0.723993i	$-0.257695\pi$
$569$	32.9090	1.37962	0.689808	+	0.723993i	$0.257695\pi$
$570$	0	0
$571$	−39.7846	−1.66493	−0.832467	−	0.554075i	$-0.813072\pi$
$571$	−39.7846	−1.66493	−0.832467	+	0.554075i	$0.813072\pi$
$572$	−12.2872	−0.513753
$573$	0	0
$574$	24.9282	1.04048
$575$	3.46410	0.144463
$576$	0	0
$577$	15.2679	0.635613	0.317807	−	0.948156i	$-0.397054\pi$
$577$	15.2679	0.635613	0.317807	+	0.948156i	$0.397054\pi$
$578$	6.98076	0.290361
$579$	0	0
$580$	−4.67949	−0.194305
$581$	10.3923	0.431145
$582$	0	0
$583$	−38.5885	−1.59817
$584$	−19.4256	−0.803838
$585$	0	0
$586$	21.0333	0.868878
$587$	11.6603	0.481270	0.240635	−	0.970616i	$-0.422644\pi$
$587$	11.6603	0.481270	0.240635	+	0.970616i	$0.422644\pi$
$588$	0	0
$589$	−13.3923	−0.551820
$590$	6.05256	0.249180
$591$	0	0
$592$	−2.92820	−0.120348
$593$	0.143594	0.00589668	0.00294834	−	0.999996i	$-0.499062\pi$
$593$	0.143594	0.00589668	0.00294834	+	0.999996i	$0.499062\pi$
$594$	0	0
$595$	12.9282	0.530005
$596$	−11.7128	−0.479776
$597$	0	0
$598$	−3.71281	−0.151828
$599$	−27.1962	−1.11120	−0.555602	−	0.831448i	$-0.687512\pi$
$599$	−27.1962	−1.11120	−0.555602	+	0.831448i	$0.687512\pi$
$600$	0	0
$601$	−31.2487	−1.27466	−0.637331	−	0.770590i	$-0.719961\pi$
$601$	−31.2487	−1.27466	−0.637331	+	0.770590i	$0.719961\pi$
$602$	0.679492	0.0276940
$603$	0	0
$604$	−7.89488	−0.321238
$605$	−21.8564	−0.888589
$606$	0	0
$607$	−16.1962	−0.657382	−0.328691	−	0.944438i	$-0.606607\pi$
$607$	−16.1962	−0.657382	−0.328691	+	0.944438i	$0.606607\pi$
$608$	−26.1436	−1.06026
$609$	0	0
$610$	2.92820	0.118559
$611$	12.7846	0.517210
$612$	0	0
$613$	1.46410	0.0591345	0.0295673	−	0.999563i	$-0.490587\pi$
$613$	1.46410	0.0591345	0.0295673	+	0.999563i	$0.490587\pi$
$614$	−10.2872	−0.415157
$615$	0	0
$616$	−68.7846	−2.77141
$617$	6.92820	0.278919	0.139459	−	0.990228i	$-0.455464\pi$
$617$	6.92820	0.278919	0.139459	+	0.990228i	$0.455464\pi$
$618$	0	0
$619$	−11.8564	−0.476549	−0.238275	−	0.971198i	$-0.576582\pi$
$619$	−11.8564	−0.476549	−0.238275	+	0.971198i	$0.576582\pi$
$620$	−4.39230	−0.176399
$621$	0	0
$622$	14.4449	0.579186
$623$	24.5885	0.985116
$624$	0	0
$625$	1.00000	0.0400000
$626$	6.64102	0.265428
$627$	0	0
$628$	28.0000	1.11732
$629$	−7.46410	−0.297613
$630$	0	0
$631$	−32.7128	−1.30228	−0.651138	−	0.758959i	$-0.725708\pi$
$631$	−32.7128	−1.30228	−0.651138	+	0.758959i	$0.725708\pi$
$632$	39.2154	1.55990
$633$	0	0
$634$	4.53590	0.180144
$635$	14.5885	0.578925
$636$	0	0
$637$	22.5359	0.892905
$638$	13.4115	0.530968
$639$	0	0
$640$	−10.1436	−0.400961
$641$	−9.33975	−0.368898	−0.184449	−	0.982842i	$-0.559050\pi$
$641$	−9.33975	−0.368898	−0.184449	+	0.982842i	$0.559050\pi$
$642$	0	0
$643$	41.6603	1.64292	0.821460	−	0.570266i	$-0.193160\pi$
$643$	41.6603	1.64292	0.821460	+	0.570266i	$0.193160\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	−8.92820	−0.351275
$647$	−11.4641	−0.450700	−0.225350	−	0.974278i	$-0.572353\pi$
$647$	−11.4641	−0.450700	−0.225350	+	0.974278i	$0.572353\pi$
$648$	0	0
$649$	47.3923	1.86031
$650$	−1.07180	−0.0420393
$651$	0	0
$652$	−18.6410	−0.730039
$653$	17.4641	0.683423	0.341712	−	0.939805i	$-0.388993\pi$
$653$	17.4641	0.683423	0.341712	+	0.939805i	$0.388993\pi$
$654$	0	0
$655$	15.5885	0.609091
$656$	7.71281	0.301135
$657$	0	0
$658$	30.2487	1.17922
$659$	1.46410	0.0570333	0.0285167	−	0.999593i	$-0.490922\pi$
$659$	1.46410	0.0570333	0.0285167	+	0.999593i	$0.490922\pi$
$660$	0	0
$661$	18.3205	0.712585	0.356293	−	0.934374i	$-0.384041\pi$
$661$	18.3205	0.712585	0.356293	+	0.934374i	$0.384041\pi$
$662$	0.339746	0.0132046
$663$	0	0
$664$	−5.56922	−0.216128
$665$	21.1244	0.819167
$666$	0	0
$667$	−11.0718	−0.428702
$668$	−25.8564	−1.00041
$669$	0	0
$670$	2.53590	0.0979703
$671$	22.9282	0.885133
$672$	0	0
$673$	−10.3923	−0.400594	−0.200297	−	0.979735i	$-0.564191\pi$
$673$	−10.3923	−0.400594	−0.200297	+	0.979735i	$0.564191\pi$
$674$	−20.0000	−0.770371
$675$	0	0
$676$	15.8949	0.611342
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	45.7128	1.75430
$680$	−6.92820	−0.265684
$681$	0	0
$682$	12.5885	0.482037
$683$	19.6077	0.750268	0.375134	−	0.926971i	$-0.377597\pi$
$683$	19.6077	0.750268	0.375134	+	0.926971i	$0.377597\pi$
$684$	0	0
$685$	2.53590	0.0968917
$686$	29.0718	1.10997
$687$	0	0
$688$	0.210236	0.00801515
$689$	−9.85641	−0.375499
$690$	0	0
$691$	−17.7128	−0.673827	−0.336914	−	0.941536i	$-0.609383\pi$
$691$	−17.7128	−0.673827	−0.336914	+	0.941536i	$0.609383\pi$
$692$	12.4974	0.475081
$693$	0	0
$694$	−20.9282	−0.794424
$695$	−0.607695	−0.0230512
$696$	0	0
$697$	19.6603	0.744685
$698$	13.8038	0.522483
$699$	0	0
$700$	6.92820	0.261861
$701$	20.8038	0.785750	0.392875	−	0.919592i	$-0.371480\pi$
$701$	20.8038	0.785750	0.392875	+	0.919592i	$0.371480\pi$
$702$	0	0
$703$	−12.1962	−0.459987
$704$	12.2872	0.463091
$705$	0	0
$706$	18.6795	0.703012
$707$	−12.5885	−0.473438
$708$	0	0
$709$	−22.5359	−0.846353	−0.423177	−	0.906047i	$-0.639085\pi$
$709$	−22.5359	−0.846353	−0.423177	+	0.906047i	$0.639085\pi$
$710$	2.73205	0.102532
$711$	0	0
$712$	−13.1769	−0.493826
$713$	−10.3923	−0.389195
$714$	0	0
$715$	−8.39230	−0.313854
$716$	11.8949	0.444533
$717$	0	0
$718$	−4.48334	−0.167317
$719$	−8.41154	−0.313698	−0.156849	−	0.987623i	$-0.550134\pi$
$719$	−8.41154	−0.313698	−0.156849	+	0.987623i	$0.550134\pi$
$720$	0	0
$721$	2.53590	0.0944418
$722$	−0.679492	−0.0252881
$723$	0	0
$724$	−38.7461	−1.43999
$725$	−3.19615	−0.118702
$726$	0	0
$727$	−8.39230	−0.311253	−0.155627	−	0.987816i	$-0.549740\pi$
$727$	−8.39230	−0.311253	−0.155627	+	0.987816i	$0.549740\pi$
$728$	−17.5692	−0.651159
$729$	0	0
$730$	−5.60770	−0.207550
$731$	0.535898	0.0198209
$732$	0	0
$733$	34.7846	1.28480	0.642399	−	0.766370i	$-0.277939\pi$
$733$	34.7846	1.28480	0.642399	+	0.766370i	$0.277939\pi$
$734$	−22.8231	−0.842415
$735$	0	0
$736$	−20.2872	−0.747796
$737$	19.8564	0.731420
$738$	0	0
$739$	22.4641	0.826355	0.413178	−	0.910650i	$-0.364419\pi$
$739$	22.4641	0.826355	0.413178	+	0.910650i	$0.364419\pi$
$740$	−4.00000	−0.147043
$741$	0	0
$742$	−23.3205	−0.856123
$743$	−19.9090	−0.730389	−0.365195	−	0.930931i	$-0.618998\pi$
$743$	−19.9090	−0.730389	−0.365195	+	0.930931i	$0.618998\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	−14.6795	−0.537454
$747$	0	0
$748$	−22.9282	−0.838338
$749$	−40.3923	−1.47590
$750$	0	0
$751$	48.7846	1.78018	0.890088	−	0.455789i	$-0.150643\pi$
$751$	48.7846	1.78018	0.890088	+	0.455789i	$0.150643\pi$
$752$	9.35898	0.341287
$753$	0	0
$754$	3.42563	0.124754
$755$	−5.39230	−0.196246
$756$	0	0
$757$	−9.17691	−0.333541	−0.166770	−	0.985996i	$-0.553334\pi$
$757$	−9.17691	−0.333541	−0.166770	+	0.985996i	$0.553334\pi$
$758$	−1.75129	−0.0636097
$759$	0	0
$760$	−11.3205	−0.410638
$761$	−23.4449	−0.849876	−0.424938	−	0.905223i	$-0.639704\pi$
$761$	−23.4449	−0.849876	−0.424938	+	0.905223i	$0.639704\pi$
$762$	0	0
$763$	28.7321	1.04017
$764$	−11.8949	−0.430342
$765$	0	0
$766$	1.85641	0.0670747
$767$	12.1051	0.437090
$768$	0	0
$769$	−9.53590	−0.343873	−0.171937	−	0.985108i	$-0.555002\pi$
$769$	−9.53590	−0.343873	−0.171937	+	0.985108i	$0.555002\pi$
$770$	−19.8564	−0.715575
$771$	0	0
$772$	7.71281	0.277590
$773$	−1.51666	−0.0545505	−0.0272752	−	0.999628i	$-0.508683\pi$
$773$	−1.51666	−0.0545505	−0.0272752	+	0.999628i	$0.508683\pi$
$774$	0	0
$775$	−3.00000	−0.107763
$776$	−24.4974	−0.879406
$777$	0	0
$778$	20.1051	0.720803
$779$	32.1244	1.15097
$780$	0	0
$781$	21.3923	0.765477
$782$	−6.92820	−0.247752
$783$	0	0
$784$	16.4974	0.589194
$785$	19.1244	0.682578
$786$	0	0
$787$	−48.0526	−1.71289	−0.856444	−	0.516239i	$-0.827332\pi$
$787$	−48.0526	−1.71289	−0.856444	+	0.516239i	$0.827332\pi$
$788$	−20.2872	−0.722701
$789$	0	0
$790$	11.3205	0.402766
$791$	−90.4974	−3.21772
$792$	0	0
$793$	5.85641	0.207967
$794$	−10.5359	−0.373905
$795$	0	0
$796$	2.92820	0.103787
$797$	15.6077	0.552853	0.276426	−	0.961035i	$-0.410850\pi$
$797$	15.6077	0.552853	0.276426	+	0.961035i	$0.410850\pi$
$798$	0	0
$799$	23.8564	0.843979
$800$	−5.85641	−0.207055
$801$	0	0
$802$	18.2487	0.644384
$803$	−43.9090	−1.54951
$804$	0	0
$805$	16.3923	0.577753
$806$	3.21539	0.113257
$807$	0	0
$808$	6.74613	0.237328
$809$	−45.4449	−1.59776	−0.798878	−	0.601493i	$-0.794573\pi$
$809$	−45.4449	−1.59776	−0.798878	+	0.601493i	$0.794573\pi$
$810$	0	0
$811$	−18.4641	−0.648362	−0.324181	−	0.945995i	$-0.605089\pi$
$811$	−18.4641	−0.648362	−0.324181	+	0.945995i	$0.605089\pi$
$812$	−22.1436	−0.777088
$813$	0	0
$814$	11.4641	0.401817
$815$	−12.7321	−0.445984
$816$	0	0
$817$	0.875644	0.0306349
$818$	13.0718	0.457045
$819$	0	0
$820$	10.5359	0.367930
$821$	37.7321	1.31686	0.658429	−	0.752643i	$-0.271221\pi$
$821$	37.7321	1.31686	0.658429	+	0.752643i	$0.271221\pi$
$822$	0	0
$823$	3.85641	0.134426	0.0672129	−	0.997739i	$-0.478589\pi$
$823$	3.85641	0.134426	0.0672129	+	0.997739i	$0.478589\pi$
$824$	−1.35898	−0.0473424
$825$	0	0
$826$	28.6410	0.996548
$827$	11.6077	0.403639	0.201820	−	0.979423i	$-0.435315\pi$
$827$	11.6077	0.403639	0.201820	+	0.979423i	$0.435315\pi$
$828$	0	0
$829$	−23.7846	−0.826074	−0.413037	−	0.910714i	$-0.635532\pi$
$829$	−23.7846	−0.826074	−0.413037	+	0.910714i	$0.635532\pi$
$830$	−1.60770	−0.0558039
$831$	0	0
$832$	3.13844	0.108806
$833$	42.0526	1.45703
$834$	0	0
$835$	−17.6603	−0.611158
$836$	−37.4641	−1.29572
$837$	0	0
$838$	14.9282	0.515686
$839$	−33.1962	−1.14606	−0.573029	−	0.819535i	$-0.694232\pi$
$839$	−33.1962	−1.14606	−0.573029	+	0.819535i	$0.694232\pi$
$840$	0	0
$841$	−18.7846	−0.647745
$842$	−24.7321	−0.852323
$843$	0	0
$844$	12.9667	0.446331
$845$	10.8564	0.373472
$846$	0	0
$847$	−103.426	−3.55375
$848$	−7.21539	−0.247778
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	−9.46410	−0.324425
$852$	0	0
$853$	10.4833	0.358943	0.179471	−	0.983763i	$-0.442561\pi$
$853$	10.4833	0.358943	0.179471	+	0.983763i	$0.442561\pi$
$854$	13.8564	0.474156
$855$	0	0
$856$	21.6462	0.739851
$857$	−44.4974	−1.52000	−0.760002	−	0.649921i	$-0.774802\pi$
$857$	−44.4974	−1.52000	−0.760002	+	0.649921i	$0.774802\pi$
$858$	0	0
$859$	−11.0000	−0.375315	−0.187658	−	0.982235i	$-0.560090\pi$
$859$	−11.0000	−0.375315	−0.187658	+	0.982235i	$0.560090\pi$
$860$	0.287187	0.00979300
$861$	0	0
$862$	−15.6218	−0.532080
$863$	23.1244	0.787162	0.393581	−	0.919290i	$-0.371236\pi$
$863$	23.1244	0.787162	0.393581	+	0.919290i	$0.371236\pi$
$864$	0	0
$865$	8.53590	0.290229
$866$	25.9615	0.882209
$867$	0	0
$868$	−20.7846	−0.705476
$869$	88.6410	3.00694
$870$	0	0
$871$	5.07180	0.171851
$872$	−15.3975	−0.521424
$873$	0	0
$874$	−11.3205	−0.382922
$875$	4.73205	0.159973
$876$	0	0
$877$	33.4641	1.13000	0.565001	−	0.825090i	$-0.308876\pi$
$877$	33.4641	1.13000	0.565001	+	0.825090i	$0.308876\pi$
$878$	−3.94744	−0.133220
$879$	0	0
$880$	−6.14359	−0.207100
$881$	47.0526	1.58524	0.792620	−	0.609715i	$-0.208716\pi$
$881$	47.0526	1.58524	0.792620	+	0.609715i	$0.208716\pi$
$882$	0	0
$883$	−30.1962	−1.01618	−0.508091	−	0.861304i	$-0.669649\pi$
$883$	−30.1962	−1.01618	−0.508091	+	0.861304i	$0.669649\pi$
$884$	−5.85641	−0.196972
$885$	0	0
$886$	−0.248711	−0.00835562
$887$	−33.8038	−1.13502	−0.567511	−	0.823366i	$-0.692094\pi$
$887$	−33.8038	−1.13502	−0.567511	+	0.823366i	$0.692094\pi$
$888$	0	0
$889$	69.0333	2.31530
$890$	−3.80385	−0.127505
$891$	0	0
$892$	−24.5744	−0.822811
$893$	38.9808	1.30444
$894$	0	0
$895$	8.12436	0.271567
$896$	−48.0000	−1.60357
$897$	0	0
$898$	5.94744	0.198469
$899$	9.58846	0.319793
$900$	0	0
$901$	−18.3923	−0.612737
$902$	−30.1962	−1.00542
$903$	0	0
$904$	48.4974	1.61300
$905$	−26.4641	−0.879697
$906$	0	0
$907$	−30.0000	−0.996134	−0.498067	−	0.867139i	$-0.665957\pi$
$907$	−30.0000	−0.996134	−0.498067	+	0.867139i	$0.665957\pi$
$908$	−26.4308	−0.877136
$909$	0	0
$910$	−5.07180	−0.168128
$911$	−23.5885	−0.781520	−0.390760	−	0.920493i	$-0.627788\pi$
$911$	−23.5885	−0.781520	−0.390760	+	0.920493i	$0.627788\pi$
$912$	0	0
$913$	−12.5885	−0.416617
$914$	−2.00000	−0.0661541
$915$	0	0
$916$	17.5692	0.580503
$917$	73.7654	2.43595
$918$	0	0
$919$	−56.9615	−1.87899	−0.939494	−	0.342566i	$-0.888704\pi$
$919$	−56.9615	−1.87899	−0.939494	+	0.342566i	$0.888704\pi$
$920$	−8.78461	−0.289620
$921$	0	0
$922$	−27.1244	−0.893293
$923$	5.46410	0.179853
$924$	0	0
$925$	−2.73205	−0.0898293
$926$	7.60770	0.250004
$927$	0	0
$928$	18.7180	0.614447
$929$	44.3731	1.45583	0.727917	−	0.685666i	$-0.240489\pi$
$929$	44.3731	1.45583	0.727917	+	0.685666i	$0.240489\pi$
$930$	0	0
$931$	68.7128	2.25197
$932$	−41.0718	−1.34535
$933$	0	0
$934$	27.3590	0.895213
$935$	−15.6603	−0.512145
$936$	0	0
$937$	−4.14359	−0.135365	−0.0676827	−	0.997707i	$-0.521561\pi$
$937$	−4.14359	−0.135365	−0.0676827	+	0.997707i	$0.521561\pi$
$938$	12.0000	0.391814
$939$	0	0
$940$	12.7846	0.416988
$941$	−35.1769	−1.14673	−0.573367	−	0.819298i	$-0.694363\pi$
$941$	−35.1769	−1.14673	−0.573367	+	0.819298i	$0.694363\pi$
$942$	0	0
$943$	24.9282	0.811774
$944$	8.86156	0.288419
$945$	0	0
$946$	−0.823085	−0.0267608
$947$	57.7128	1.87541	0.937707	−	0.347427i	$-0.112944\pi$
$947$	57.7128	1.87541	0.937707	+	0.347427i	$0.112944\pi$
$948$	0	0
$949$	−11.2154	−0.364067
$950$	−3.26795	−0.106026
$951$	0	0
$952$	−32.7846	−1.06256
$953$	−36.3923	−1.17886	−0.589431	−	0.807819i	$-0.700648\pi$
$953$	−36.3923	−1.17886	−0.589431	+	0.807819i	$0.700648\pi$
$954$	0	0
$955$	−8.12436	−0.262898
$956$	−0.784610	−0.0253761
$957$	0	0
$958$	−8.69358	−0.280877
$959$	12.0000	0.387500
$960$	0	0
$961$	−22.0000	−0.709677
$962$	2.92820	0.0944091
$963$	0	0
$964$	23.8949	0.769602
$965$	5.26795	0.169581
$966$	0	0
$967$	−25.8038	−0.829796	−0.414898	−	0.909868i	$-0.636183\pi$
$967$	−25.8038	−0.829796	−0.414898	+	0.909868i	$0.636183\pi$
$968$	55.4256	1.78145
$969$	0	0
$970$	−7.07180	−0.227062
$971$	−17.4449	−0.559832	−0.279916	−	0.960024i	$-0.590307\pi$
$971$	−17.4449	−0.559832	−0.279916	+	0.960024i	$0.590307\pi$
$972$	0	0
$973$	−2.87564	−0.0921889
$974$	−17.8564	−0.572156
$975$	0	0
$976$	4.28719	0.137230
$977$	−5.46410	−0.174812	−0.0874060	−	0.996173i	$-0.527858\pi$
$977$	−5.46410	−0.174812	−0.0874060	+	0.996173i	$0.527858\pi$
$978$	0	0
$979$	−29.7846	−0.951920
$980$	22.5359	0.719883
$981$	0	0
$982$	−10.1577	−0.324144
$983$	−48.5885	−1.54973	−0.774866	−	0.632126i	$-0.782182\pi$
$983$	−48.5885	−1.54973	−0.774866	+	0.632126i	$0.782182\pi$
$984$	0	0
$985$	−13.8564	−0.441502
$986$	6.39230	0.203572
$987$	0	0
$988$	−9.56922	−0.304437
$989$	0.679492	0.0216066
$990$	0	0
$991$	30.8564	0.980186	0.490093	−	0.871670i	$-0.336963\pi$
$991$	30.8564	0.980186	0.490093	+	0.871670i	$0.336963\pi$
$992$	17.5692	0.557823
$993$	0	0
$994$	12.9282	0.410058
$995$	2.00000	0.0634043
$996$	0	0
$997$	25.5167	0.808121	0.404060	−	0.914732i	$-0.367599\pi$
$997$	25.5167	0.808121	0.404060	+	0.914732i	$0.367599\pi$
$998$	−17.8038	−0.563571
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.a.h.1.1	yes	2
3.2	odd	2		405.2.a.g.1.2	✓	2
4.3	odd	2		6480.2.a.bi.1.2		2
5.2	odd	4		2025.2.b.h.649.2		4
5.3	odd	4		2025.2.b.h.649.3		4
5.4	even	2		2025.2.a.g.1.2		2
9.2	odd	6		405.2.e.l.271.1		4
9.4	even	3		405.2.e.i.136.2		4
9.5	odd	6		405.2.e.l.136.1		4
9.7	even	3		405.2.e.i.271.2		4
12.11	even	2		6480.2.a.br.1.2		2
15.2	even	4		2025.2.b.g.649.3		4
15.8	even	4		2025.2.b.g.649.2		4
15.14	odd	2		2025.2.a.m.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.2	✓	2	3.2	odd	2
405.2.a.h.1.1	yes	2	1.1	even	1	trivial
405.2.e.i.136.2		4	9.4	even	3
405.2.e.i.271.2		4	9.7	even	3
405.2.e.l.136.1		4	9.5	odd	6
405.2.e.l.271.1		4	9.2	odd	6
2025.2.a.g.1.2		2	5.4	even	2
2025.2.a.m.1.1		2	15.14	odd	2
2025.2.b.g.649.2		4	15.8	even	4
2025.2.b.g.649.3		4	15.2	even	4
2025.2.b.h.649.2		4	5.2	odd	4
2025.2.b.h.649.3		4	5.3	odd	4
6480.2.a.bi.1.2		2	4.3	odd	2
6480.2.a.br.1.2		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{2} -1.46410 q^{4} -1.00000 q^{5} -4.73205 q^{7} +2.53590 q^{8} +0.732051 q^{10} +5.73205 q^{11} +1.46410 q^{13} +3.46410 q^{14} +1.07180 q^{16} +2.73205 q^{17} +4.46410 q^{19} +1.46410 q^{20} -4.19615 q^{22} +3.46410 q^{23} +1.00000 q^{25} -1.07180 q^{26} +6.92820 q^{28} -3.19615 q^{29} -3.00000 q^{31} -5.85641 q^{32} -2.00000 q^{34} +4.73205 q^{35} -2.73205 q^{37} -3.26795 q^{38} -2.53590 q^{40} +7.19615 q^{41} +0.196152 q^{43} -8.39230 q^{44} -2.53590 q^{46} +8.73205 q^{47} +15.3923 q^{49} -0.732051 q^{50} -2.14359 q^{52} -6.73205 q^{53} -5.73205 q^{55} -12.0000 q^{56} +2.33975 q^{58} +8.26795 q^{59} +4.00000 q^{61} +2.19615 q^{62} +2.14359 q^{64} -1.46410 q^{65} +3.46410 q^{67} -4.00000 q^{68} -3.46410 q^{70} +3.73205 q^{71} -7.66025 q^{73} +2.00000 q^{74} -6.53590 q^{76} -27.1244 q^{77} +15.4641 q^{79} -1.07180 q^{80} -5.26795 q^{82} -2.19615 q^{83} -2.73205 q^{85} -0.143594 q^{86} +14.5359 q^{88} -5.19615 q^{89} -6.92820 q^{91} -5.07180 q^{92} -6.39230 q^{94} -4.46410 q^{95} -9.66025 q^{97} -11.2679 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 4 q^{4} - 2 q^{5} - 6 q^{7} + 12 q^{8} - 2 q^{10} + 8 q^{11} - 4 q^{13} + 16 q^{16} + 2 q^{17} + 2 q^{19} - 4 q^{20} + 2 q^{22} + 2 q^{25} - 16 q^{26} + 4 q^{29} - 6 q^{31} + 16 q^{32} - 4 q^{34}+ \cdots - 26 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 4 * q^4 - 2 * q^5 - 6 * q^7 + 12 * q^8 - 2 * q^10 + 8 * q^11 - 4 * q^13 + 16 * q^16 + 2 * q^17 + 2 * q^19 - 4 * q^20 + 2 * q^22 + 2 * q^25 - 16 * q^26 + 4 * q^29 - 6 * q^31 + 16 * q^32 - 4 * q^34 + 6 * q^35 - 2 * q^37 - 10 * q^38 - 12 * q^40 + 4 * q^41 - 10 * q^43 + 4 * q^44 - 12 * q^46 + 14 * q^47 + 10 * q^49 + 2 * q^50 - 32 * q^52 - 10 * q^53 - 8 * q^55 - 24 * q^56 + 22 * q^58 + 20 * q^59 + 8 * q^61 - 6 * q^62 + 32 * q^64 + 4 * q^65 - 8 * q^68 + 4 * q^71 + 2 * q^73 + 4 * q^74 - 20 * q^76 - 30 * q^77 + 24 * q^79 - 16 * q^80 - 14 * q^82 + 6 * q^83 - 2 * q^85 - 28 * q^86 + 36 * q^88 - 24 * q^92 + 8 * q^94 - 2 * q^95 - 2 * q^97 - 26 * q^98

Introduction

L-functions

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