LMFDB - Embedded newform 4560.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +0.585786 q^{11} +0.585786 q^{13} -1.00000 q^{15} -2.82843 q^{17} -1.00000 q^{19} -1.41421 q^{21} +4.82843 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.07107 q^{29} +4.82843 q^{31} +0.585786 q^{33} +1.41421 q^{35} +6.24264 q^{37} +0.585786 q^{39} +9.89949 q^{41} -11.0711 q^{43} -1.00000 q^{45} +3.17157 q^{47} -5.00000 q^{49} -2.82843 q^{51} +8.48528 q^{53} -0.585786 q^{55} -1.00000 q^{57} +1.17157 q^{59} -1.65685 q^{61} -1.41421 q^{63} -0.585786 q^{65} +11.3137 q^{67} +4.82843 q^{69} +14.8284 q^{71} -11.6569 q^{73} +1.00000 q^{75} -0.828427 q^{77} +13.6569 q^{79} +1.00000 q^{81} +7.65685 q^{83} +2.82843 q^{85} -7.07107 q^{87} -12.2426 q^{89} -0.828427 q^{91} +4.82843 q^{93} +1.00000 q^{95} +11.8995 q^{97} +0.585786 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{31} + 4 q^{33} + 4 q^{37} + 4 q^{39} - 8 q^{43} - 2 q^{45} + 12 q^{47} - 10 q^{49} - 4 q^{55} - 2 q^{57} + 8 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{69} + 24 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 16 q^{79} + 2 q^{81} + 4 q^{83} - 16 q^{89} + 4 q^{91} + 4 q^{93} + 2 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^31 + 4 * q^33 + 4 * q^37 + 4 * q^39 - 8 * q^43 - 2 * q^45 + 12 * q^47 - 10 * q^49 - 4 * q^55 - 2 * q^57 + 8 * q^59 + 8 * q^61 - 4 * q^65 + 4 * q^69 + 24 * q^71 - 12 * q^73 + 2 * q^75 + 4 * q^77 + 16 * q^79 + 2 * q^81 + 4 * q^83 - 16 * q^89 + 4 * q^91 + 4 * q^93 + 2 * q^95 + 4 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

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
$7$	−1.41421	−0.534522	−0.267261	−	0.963624i	$-0.586119\pi$
$7$	−1.41421	−0.534522	−0.267261	+	0.963624i	$0.586119\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.585786	0.176621	0.0883106	−	0.996093i	$-0.471853\pi$
$11$	0.585786	0.176621	0.0883106	+	0.996093i	$0.471853\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−1.41421	−0.308607
$22$	0	0
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.07107	−1.31306	−0.656532	−	0.754298i	$-0.727977\pi$
$29$	−7.07107	−1.31306	−0.656532	+	0.754298i	$0.727977\pi$
$30$	0	0
$31$	4.82843	0.867211	0.433606	−	0.901103i	$-0.357241\pi$
$31$	4.82843	0.867211	0.433606	+	0.901103i	$0.357241\pi$
$32$	0	0
$33$	0.585786	0.101972
$34$	0	0
$35$	1.41421	0.239046
$36$	0	0
$37$	6.24264	1.02628	0.513142	−	0.858304i	$-0.328481\pi$
$37$	6.24264	1.02628	0.513142	+	0.858304i	$0.328481\pi$
$38$	0	0
$39$	0.585786	0.0938009
$40$	0	0
$41$	9.89949	1.54604	0.773021	−	0.634381i	$-0.218745\pi$
$41$	9.89949	1.54604	0.773021	+	0.634381i	$0.218745\pi$
$42$	0	0
$43$	−11.0711	−1.68832	−0.844161	−	0.536090i	$-0.819901\pi$
$43$	−11.0711	−1.68832	−0.844161	+	0.536090i	$0.819901\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	3.17157	0.462621	0.231311	−	0.972880i	$-0.425699\pi$
$47$	3.17157	0.462621	0.231311	+	0.972880i	$0.425699\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	−2.82843	−0.396059
$52$	0	0
$53$	8.48528	1.16554	0.582772	−	0.812636i	$-0.301968\pi$
$53$	8.48528	1.16554	0.582772	+	0.812636i	$0.301968\pi$
$54$	0	0
$55$	−0.585786	−0.0789874
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	0	0
$61$	−1.65685	−0.212138	−0.106069	−	0.994359i	$-0.533827\pi$
$61$	−1.65685	−0.212138	−0.106069	+	0.994359i	$0.533827\pi$
$62$	0	0
$63$	−1.41421	−0.178174
$64$	0	0
$65$	−0.585786	−0.0726579
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	4.82843	0.581274
$70$	0	0
$71$	14.8284	1.75981	0.879905	−	0.475149i	$-0.157606\pi$
$71$	14.8284	1.75981	0.879905	+	0.475149i	$0.157606\pi$
$72$	0	0
$73$	−11.6569	−1.36433	−0.682166	−	0.731198i	$-0.738962\pi$
$73$	−11.6569	−1.36433	−0.682166	+	0.731198i	$0.738962\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−0.828427	−0.0944080
$78$	0	0
$79$	13.6569	1.53652	0.768258	−	0.640140i	$-0.221124\pi$
$79$	13.6569	1.53652	0.768258	+	0.640140i	$0.221124\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.65685	0.840449	0.420224	−	0.907420i	$-0.361951\pi$
$83$	7.65685	0.840449	0.420224	+	0.907420i	$0.361951\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	0	0
$87$	−7.07107	−0.758098
$88$	0	0
$89$	−12.2426	−1.29772	−0.648859	−	0.760909i	$-0.724753\pi$
$89$	−12.2426	−1.29772	−0.648859	+	0.760909i	$0.724753\pi$
$90$	0	0
$91$	−0.828427	−0.0868428
$92$	0	0
$93$	4.82843	0.500685
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	11.8995	1.20821	0.604105	−	0.796904i	$-0.293531\pi$
$97$	11.8995	1.20821	0.604105	+	0.796904i	$0.293531\pi$
$98$	0	0
$99$	0.585786	0.0588738
$100$	0	0
$101$	−0.828427	−0.0824316	−0.0412158	−	0.999150i	$-0.513123\pi$
$101$	−0.828427	−0.0824316	−0.0412158	+	0.999150i	$0.513123\pi$
$102$	0	0
$103$	−3.31371	−0.326509	−0.163255	−	0.986584i	$-0.552199\pi$
$103$	−3.31371	−0.326509	−0.163255	+	0.986584i	$0.552199\pi$
$104$	0	0
$105$	1.41421	0.138013
$106$	0	0
$107$	13.6569	1.32026	0.660129	−	0.751152i	$-0.270502\pi$
$107$	13.6569	1.32026	0.660129	+	0.751152i	$0.270502\pi$
$108$	0	0
$109$	8.14214	0.779875	0.389938	−	0.920841i	$-0.372497\pi$
$109$	8.14214	0.779875	0.389938	+	0.920841i	$0.372497\pi$
$110$	0	0
$111$	6.24264	0.592525
$112$	0	0
$113$	−15.3137	−1.44059	−0.720296	−	0.693667i	$-0.755994\pi$
$113$	−15.3137	−1.44059	−0.720296	+	0.693667i	$0.755994\pi$
$114$	0	0
$115$	−4.82843	−0.450253
$116$	0	0
$117$	0.585786	0.0541560
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.6569	−0.968805
$122$	0	0
$123$	9.89949	0.892607
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	5.65685	0.501965	0.250982	−	0.967992i	$-0.419246\pi$
$127$	5.65685	0.501965	0.250982	+	0.967992i	$0.419246\pi$
$128$	0	0
$129$	−11.0711	−0.974753
$130$	0	0
$131$	−6.72792	−0.587821	−0.293911	−	0.955833i	$-0.594957\pi$
$131$	−6.72792	−0.587821	−0.293911	+	0.955833i	$0.594957\pi$
$132$	0	0
$133$	1.41421	0.122628
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	15.6569	1.33766	0.668828	−	0.743417i	$-0.266796\pi$
$137$	15.6569	1.33766	0.668828	+	0.743417i	$0.266796\pi$
$138$	0	0
$139$	2.82843	0.239904	0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843	0.239904	0.119952	+	0.992780i	$0.461726\pi$
$140$	0	0
$141$	3.17157	0.267095
$142$	0	0
$143$	0.343146	0.0286953
$144$	0	0
$145$	7.07107	0.587220
$146$	0	0
$147$	−5.00000	−0.412393
$148$	0	0
$149$	−4.34315	−0.355804	−0.177902	−	0.984048i	$-0.556931\pi$
$149$	−4.34315	−0.355804	−0.177902	+	0.984048i	$0.556931\pi$
$150$	0	0
$151$	−0.828427	−0.0674164	−0.0337082	−	0.999432i	$-0.510732\pi$
$151$	−0.828427	−0.0674164	−0.0337082	+	0.999432i	$0.510732\pi$
$152$	0	0
$153$	−2.82843	−0.228665
$154$	0	0
$155$	−4.82843	−0.387829
$156$	0	0
$157$	−4.14214	−0.330578	−0.165289	−	0.986245i	$-0.552856\pi$
$157$	−4.14214	−0.330578	−0.165289	+	0.986245i	$0.552856\pi$
$158$	0	0
$159$	8.48528	0.672927
$160$	0	0
$161$	−6.82843	−0.538155
$162$	0	0
$163$	23.5563	1.84508	0.922538	−	0.385907i	$-0.126111\pi$
$163$	23.5563	1.84508	0.922538	+	0.385907i	$0.126111\pi$
$164$	0	0
$165$	−0.585786	−0.0456034
$166$	0	0
$167$	17.3137	1.33977	0.669887	−	0.742463i	$-0.266342\pi$
$167$	17.3137	1.33977	0.669887	+	0.742463i	$0.266342\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−11.3137	−0.860165	−0.430083	−	0.902790i	$-0.641516\pi$
$173$	−11.3137	−0.860165	−0.430083	+	0.902790i	$0.641516\pi$
$174$	0	0
$175$	−1.41421	−0.106904
$176$	0	0
$177$	1.17157	0.0880608
$178$	0	0
$179$	13.1716	0.984490	0.492245	−	0.870457i	$-0.336176\pi$
$179$	13.1716	0.984490	0.492245	+	0.870457i	$0.336176\pi$
$180$	0	0
$181$	20.1421	1.49715	0.748577	−	0.663048i	$-0.230738\pi$
$181$	20.1421	1.49715	0.748577	+	0.663048i	$0.230738\pi$
$182$	0	0
$183$	−1.65685	−0.122478
$184$	0	0
$185$	−6.24264	−0.458968
$186$	0	0
$187$	−1.65685	−0.121161
$188$	0	0
$189$	−1.41421	−0.102869
$190$	0	0
$191$	13.7574	0.995448	0.497724	−	0.867336i	$-0.334169\pi$
$191$	13.7574	0.995448	0.497724	+	0.867336i	$0.334169\pi$
$192$	0	0
$193$	13.0711	0.940876	0.470438	−	0.882433i	$-0.344096\pi$
$193$	13.0711	0.940876	0.470438	+	0.882433i	$0.344096\pi$
$194$	0	0
$195$	−0.585786	−0.0419490
$196$	0	0
$197$	22.1421	1.57756	0.788781	−	0.614674i	$-0.210713\pi$
$197$	22.1421	1.57756	0.788781	+	0.614674i	$0.210713\pi$
$198$	0	0
$199$	9.17157	0.650156	0.325078	−	0.945687i	$-0.394610\pi$
$199$	9.17157	0.650156	0.325078	+	0.945687i	$0.394610\pi$
$200$	0	0
$201$	11.3137	0.798007
$202$	0	0
$203$	10.0000	0.701862
$204$	0	0
$205$	−9.89949	−0.691411
$206$	0	0
$207$	4.82843	0.335599
$208$	0	0
$209$	−0.585786	−0.0405197
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	14.8284	1.01603
$214$	0	0
$215$	11.0711	0.755041
$216$	0	0
$217$	−6.82843	−0.463544
$218$	0	0
$219$	−11.6569	−0.787697
$220$	0	0
$221$	−1.65685	−0.111452
$222$	0	0
$223$	−11.3137	−0.757622	−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137	−0.757622	−0.378811	+	0.925474i	$0.623667\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−11.6569	−0.773693	−0.386846	−	0.922144i	$-0.626436\pi$
$227$	−11.6569	−0.773693	−0.386846	+	0.922144i	$0.626436\pi$
$228$	0	0
$229$	−21.6569	−1.43113	−0.715563	−	0.698549i	$-0.753830\pi$
$229$	−21.6569	−1.43113	−0.715563	+	0.698549i	$0.753830\pi$
$230$	0	0
$231$	−0.828427	−0.0545065
$232$	0	0
$233$	15.6569	1.02571	0.512857	−	0.858474i	$-0.328587\pi$
$233$	15.6569	1.02571	0.512857	+	0.858474i	$0.328587\pi$
$234$	0	0
$235$	−3.17157	−0.206891
$236$	0	0
$237$	13.6569	0.887108
$238$	0	0
$239$	17.7574	1.14863	0.574314	−	0.818635i	$-0.305269\pi$
$239$	17.7574	1.14863	0.574314	+	0.818635i	$0.305269\pi$
$240$	0	0
$241$	12.3431	0.795092	0.397546	−	0.917582i	$-0.369862\pi$
$241$	12.3431	0.795092	0.397546	+	0.917582i	$0.369862\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.00000	0.319438
$246$	0	0
$247$	−0.585786	−0.0372727
$248$	0	0
$249$	7.65685	0.485233
$250$	0	0
$251$	5.75736	0.363401	0.181701	−	0.983354i	$-0.441840\pi$
$251$	5.75736	0.363401	0.181701	+	0.983354i	$0.441840\pi$
$252$	0	0
$253$	2.82843	0.177822
$254$	0	0
$255$	2.82843	0.177123
$256$	0	0
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	0	0
$259$	−8.82843	−0.548572
$260$	0	0
$261$	−7.07107	−0.437688
$262$	0	0
$263$	2.00000	0.123325	0.0616626	−	0.998097i	$-0.480360\pi$
$263$	2.00000	0.123325	0.0616626	+	0.998097i	$0.480360\pi$
$264$	0	0
$265$	−8.48528	−0.521247
$266$	0	0
$267$	−12.2426	−0.749237
$268$	0	0
$269$	14.3848	0.877055	0.438528	−	0.898718i	$-0.355500\pi$
$269$	14.3848	0.877055	0.438528	+	0.898718i	$0.355500\pi$
$270$	0	0
$271$	2.82843	0.171815	0.0859074	−	0.996303i	$-0.472621\pi$
$271$	2.82843	0.171815	0.0859074	+	0.996303i	$0.472621\pi$
$272$	0	0
$273$	−0.828427	−0.0501387
$274$	0	0
$275$	0.585786	0.0353243
$276$	0	0
$277$	−24.6274	−1.47972	−0.739859	−	0.672762i	$-0.765108\pi$
$277$	−24.6274	−1.47972	−0.739859	+	0.672762i	$0.765108\pi$
$278$	0	0
$279$	4.82843	0.289070
$280$	0	0
$281$	−30.3848	−1.81260	−0.906302	−	0.422631i	$-0.861107\pi$
$281$	−30.3848	−1.81260	−0.906302	+	0.422631i	$0.861107\pi$
$282$	0	0
$283$	−7.75736	−0.461127	−0.230564	−	0.973057i	$-0.574057\pi$
$283$	−7.75736	−0.461127	−0.230564	+	0.973057i	$0.574057\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	−14.0000	−0.826394
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	11.8995	0.697561
$292$	0	0
$293$	1.65685	0.0967945	0.0483972	−	0.998828i	$-0.484589\pi$
$293$	1.65685	0.0967945	0.0483972	+	0.998828i	$0.484589\pi$
$294$	0	0
$295$	−1.17157	−0.0682116
$296$	0	0
$297$	0.585786	0.0339908
$298$	0	0
$299$	2.82843	0.163572
$300$	0	0
$301$	15.6569	0.902446
$302$	0	0
$303$	−0.828427	−0.0475919
$304$	0	0
$305$	1.65685	0.0948712
$306$	0	0
$307$	−30.1421	−1.72030	−0.860151	−	0.510039i	$-0.829631\pi$
$307$	−30.1421	−1.72030	−0.860151	+	0.510039i	$0.829631\pi$
$308$	0	0
$309$	−3.31371	−0.188510
$310$	0	0
$311$	23.2132	1.31630	0.658150	−	0.752887i	$-0.271339\pi$
$311$	23.2132	1.31630	0.658150	+	0.752887i	$0.271339\pi$
$312$	0	0
$313$	25.7990	1.45825	0.729123	−	0.684383i	$-0.239928\pi$
$313$	25.7990	1.45825	0.729123	+	0.684383i	$0.239928\pi$
$314$	0	0
$315$	1.41421	0.0796819
$316$	0	0
$317$	26.1421	1.46829	0.734144	−	0.678993i	$-0.237584\pi$
$317$	26.1421	1.46829	0.734144	+	0.678993i	$0.237584\pi$
$318$	0	0
$319$	−4.14214	−0.231915
$320$	0	0
$321$	13.6569	0.762251
$322$	0	0
$323$	2.82843	0.157378
$324$	0	0
$325$	0.585786	0.0324936
$326$	0	0
$327$	8.14214	0.450261
$328$	0	0
$329$	−4.48528	−0.247282
$330$	0	0
$331$	12.1421	0.667392	0.333696	−	0.942681i	$-0.391704\pi$
$331$	12.1421	0.667392	0.333696	+	0.942681i	$0.391704\pi$
$332$	0	0
$333$	6.24264	0.342095
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	17.7574	0.967305	0.483652	−	0.875260i	$-0.339310\pi$
$337$	17.7574	0.967305	0.483652	+	0.875260i	$0.339310\pi$
$338$	0	0
$339$	−15.3137	−0.831726
$340$	0	0
$341$	2.82843	0.153168
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	−4.82843	−0.259954
$346$	0	0
$347$	−21.3137	−1.14418	−0.572090	−	0.820191i	$-0.693867\pi$
$347$	−21.3137	−1.14418	−0.572090	+	0.820191i	$0.693867\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	0.585786	0.0312670
$352$	0	0
$353$	−6.68629	−0.355875	−0.177938	−	0.984042i	$-0.556943\pi$
$353$	−6.68629	−0.355875	−0.177938	+	0.984042i	$0.556943\pi$
$354$	0	0
$355$	−14.8284	−0.787011
$356$	0	0
$357$	4.00000	0.211702
$358$	0	0
$359$	−13.5563	−0.715477	−0.357738	−	0.933822i	$-0.616452\pi$
$359$	−13.5563	−0.715477	−0.357738	+	0.933822i	$0.616452\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−10.6569	−0.559340
$364$	0	0
$365$	11.6569	0.610148
$366$	0	0
$367$	−24.7279	−1.29079	−0.645394	−	0.763850i	$-0.723307\pi$
$367$	−24.7279	−1.29079	−0.645394	+	0.763850i	$0.723307\pi$
$368$	0	0
$369$	9.89949	0.515347
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	−17.0711	−0.883906	−0.441953	−	0.897038i	$-0.645714\pi$
$373$	−17.0711	−0.883906	−0.441953	+	0.897038i	$0.645714\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−4.14214	−0.213331
$378$	0	0
$379$	−9.51472	−0.488738	−0.244369	−	0.969682i	$-0.578581\pi$
$379$	−9.51472	−0.488738	−0.244369	+	0.969682i	$0.578581\pi$
$380$	0	0
$381$	5.65685	0.289809
$382$	0	0
$383$	34.6274	1.76938	0.884689	−	0.466181i	$-0.154371\pi$
$383$	34.6274	1.76938	0.884689	+	0.466181i	$0.154371\pi$
$384$	0	0
$385$	0.828427	0.0422206
$386$	0	0
$387$	−11.0711	−0.562774
$388$	0	0
$389$	−34.9706	−1.77308	−0.886539	−	0.462654i	$-0.846897\pi$
$389$	−34.9706	−1.77308	−0.886539	+	0.462654i	$0.846897\pi$
$390$	0	0
$391$	−13.6569	−0.690657
$392$	0	0
$393$	−6.72792	−0.339379
$394$	0	0
$395$	−13.6569	−0.687151
$396$	0	0
$397$	16.6274	0.834506	0.417253	−	0.908790i	$-0.362993\pi$
$397$	16.6274	0.834506	0.417253	+	0.908790i	$0.362993\pi$
$398$	0	0
$399$	1.41421	0.0707992
$400$	0	0
$401$	18.3848	0.918092	0.459046	−	0.888413i	$-0.348191\pi$
$401$	18.3848	0.918092	0.459046	+	0.888413i	$0.348191\pi$
$402$	0	0
$403$	2.82843	0.140894
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	3.65685	0.181264
$408$	0	0
$409$	−4.82843	−0.238750	−0.119375	−	0.992849i	$-0.538089\pi$
$409$	−4.82843	−0.238750	−0.119375	+	0.992849i	$0.538089\pi$
$410$	0	0
$411$	15.6569	0.772296
$412$	0	0
$413$	−1.65685	−0.0815285
$414$	0	0
$415$	−7.65685	−0.375860
$416$	0	0
$417$	2.82843	0.138509
$418$	0	0
$419$	1.55635	0.0760326	0.0380163	−	0.999277i	$-0.487896\pi$
$419$	1.55635	0.0760326	0.0380163	+	0.999277i	$0.487896\pi$
$420$	0	0
$421$	−16.6274	−0.810371	−0.405185	−	0.914235i	$-0.632793\pi$
$421$	−16.6274	−0.810371	−0.405185	+	0.914235i	$0.632793\pi$
$422$	0	0
$423$	3.17157	0.154207
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	2.34315	0.113393
$428$	0	0
$429$	0.343146	0.0165672
$430$	0	0
$431$	−6.14214	−0.295856	−0.147928	−	0.988998i	$-0.547260\pi$
$431$	−6.14214	−0.295856	−0.147928	+	0.988998i	$0.547260\pi$
$432$	0	0
$433$	−14.2426	−0.684458	−0.342229	−	0.939617i	$-0.611182\pi$
$433$	−14.2426	−0.684458	−0.342229	+	0.939617i	$0.611182\pi$
$434$	0	0
$435$	7.07107	0.339032
$436$	0	0
$437$	−4.82843	−0.230975
$438$	0	0
$439$	−4.68629	−0.223664	−0.111832	−	0.993727i	$-0.535672\pi$
$439$	−4.68629	−0.223664	−0.111832	+	0.993727i	$0.535672\pi$
$440$	0	0
$441$	−5.00000	−0.238095
$442$	0	0
$443$	−23.4558	−1.11442	−0.557210	−	0.830371i	$-0.688128\pi$
$443$	−23.4558	−1.11442	−0.557210	+	0.830371i	$0.688128\pi$
$444$	0	0
$445$	12.2426	0.580357
$446$	0	0
$447$	−4.34315	−0.205424
$448$	0	0
$449$	−14.3848	−0.678860	−0.339430	−	0.940631i	$-0.610234\pi$
$449$	−14.3848	−0.678860	−0.339430	+	0.940631i	$0.610234\pi$
$450$	0	0
$451$	5.79899	0.273064
$452$	0	0
$453$	−0.828427	−0.0389229
$454$	0	0
$455$	0.828427	0.0388373
$456$	0	0
$457$	16.1421	0.755097	0.377549	−	0.925990i	$-0.376767\pi$
$457$	16.1421	0.755097	0.377549	+	0.925990i	$0.376767\pi$
$458$	0	0
$459$	−2.82843	−0.132020
$460$	0	0
$461$	6.68629	0.311412	0.155706	−	0.987803i	$-0.450235\pi$
$461$	6.68629	0.311412	0.155706	+	0.987803i	$0.450235\pi$
$462$	0	0
$463$	−24.7279	−1.14920	−0.574602	−	0.818433i	$-0.694843\pi$
$463$	−24.7279	−1.14920	−0.574602	+	0.818433i	$0.694843\pi$
$464$	0	0
$465$	−4.82843	−0.223913
$466$	0	0
$467$	16.8284	0.778727	0.389363	−	0.921084i	$-0.372695\pi$
$467$	16.8284	0.778727	0.389363	+	0.921084i	$0.372695\pi$
$468$	0	0
$469$	−16.0000	−0.738811
$470$	0	0
$471$	−4.14214	−0.190860
$472$	0	0
$473$	−6.48528	−0.298194
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	8.48528	0.388514
$478$	0	0
$479$	−5.75736	−0.263060	−0.131530	−	0.991312i	$-0.541989\pi$
$479$	−5.75736	−0.263060	−0.131530	+	0.991312i	$0.541989\pi$
$480$	0	0
$481$	3.65685	0.166738
$482$	0	0
$483$	−6.82843	−0.310704
$484$	0	0
$485$	−11.8995	−0.540328
$486$	0	0
$487$	−23.7990	−1.07844	−0.539218	−	0.842166i	$-0.681280\pi$
$487$	−23.7990	−1.07844	−0.539218	+	0.842166i	$0.681280\pi$
$488$	0	0
$489$	23.5563	1.06525
$490$	0	0
$491$	12.3848	0.558917	0.279459	−	0.960158i	$-0.409845\pi$
$491$	12.3848	0.558917	0.279459	+	0.960158i	$0.409845\pi$
$492$	0	0
$493$	20.0000	0.900755
$494$	0	0
$495$	−0.585786	−0.0263291
$496$	0	0
$497$	−20.9706	−0.940658
$498$	0	0
$499$	−1.85786	−0.0831694	−0.0415847	−	0.999135i	$-0.513241\pi$
$499$	−1.85786	−0.0831694	−0.0415847	+	0.999135i	$0.513241\pi$
$500$	0	0
$501$	17.3137	0.773519
$502$	0	0
$503$	6.68629	0.298127	0.149064	−	0.988828i	$-0.452374\pi$
$503$	6.68629	0.298127	0.149064	+	0.988828i	$0.452374\pi$
$504$	0	0
$505$	0.828427	0.0368645
$506$	0	0
$507$	−12.6569	−0.562111
$508$	0	0
$509$	−4.92893	−0.218471	−0.109236	−	0.994016i	$-0.534840\pi$
$509$	−4.92893	−0.218471	−0.109236	+	0.994016i	$0.534840\pi$
$510$	0	0
$511$	16.4853	0.729266
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	3.31371	0.146019
$516$	0	0
$517$	1.85786	0.0817088
$518$	0	0
$519$	−11.3137	−0.496617
$520$	0	0
$521$	7.55635	0.331050	0.165525	−	0.986206i	$-0.447068\pi$
$521$	7.55635	0.331050	0.165525	+	0.986206i	$0.447068\pi$
$522$	0	0
$523$	−23.7990	−1.04066	−0.520329	−	0.853966i	$-0.674191\pi$
$523$	−23.7990	−1.04066	−0.520329	+	0.853966i	$0.674191\pi$
$524$	0	0
$525$	−1.41421	−0.0617213
$526$	0	0
$527$	−13.6569	−0.594902
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	1.17157	0.0508419
$532$	0	0
$533$	5.79899	0.251182
$534$	0	0
$535$	−13.6569	−0.590437
$536$	0	0
$537$	13.1716	0.568395
$538$	0	0
$539$	−2.92893	−0.126158
$540$	0	0
$541$	−13.3137	−0.572401	−0.286201	−	0.958170i	$-0.592392\pi$
$541$	−13.3137	−0.572401	−0.286201	+	0.958170i	$0.592392\pi$
$542$	0	0
$543$	20.1421	0.864382
$544$	0	0
$545$	−8.14214	−0.348771
$546$	0	0
$547$	8.48528	0.362804	0.181402	−	0.983409i	$-0.441936\pi$
$547$	8.48528	0.362804	0.181402	+	0.983409i	$0.441936\pi$
$548$	0	0
$549$	−1.65685	−0.0707128
$550$	0	0
$551$	7.07107	0.301238
$552$	0	0
$553$	−19.3137	−0.821302
$554$	0	0
$555$	−6.24264	−0.264985
$556$	0	0
$557$	12.3431	0.522996	0.261498	−	0.965204i	$-0.415784\pi$
$557$	12.3431	0.522996	0.261498	+	0.965204i	$0.415784\pi$
$558$	0	0
$559$	−6.48528	−0.274298
$560$	0	0
$561$	−1.65685	−0.0699524
$562$	0	0
$563$	−11.6569	−0.491278	−0.245639	−	0.969361i	$-0.578998\pi$
$563$	−11.6569	−0.491278	−0.245639	+	0.969361i	$0.578998\pi$
$564$	0	0
$565$	15.3137	0.644253
$566$	0	0
$567$	−1.41421	−0.0593914
$568$	0	0
$569$	34.1838	1.43306	0.716529	−	0.697557i	$-0.245730\pi$
$569$	34.1838	1.43306	0.716529	+	0.697557i	$0.245730\pi$
$570$	0	0
$571$	−26.1421	−1.09401	−0.547007	−	0.837128i	$-0.684233\pi$
$571$	−26.1421	−1.09401	−0.547007	+	0.837128i	$0.684233\pi$
$572$	0	0
$573$	13.7574	0.574722
$574$	0	0
$575$	4.82843	0.201359
$576$	0	0
$577$	−26.4853	−1.10260	−0.551298	−	0.834308i	$-0.685867\pi$
$577$	−26.4853	−1.10260	−0.551298	+	0.834308i	$0.685867\pi$
$578$	0	0
$579$	13.0711	0.543215
$580$	0	0
$581$	−10.8284	−0.449239
$582$	0	0
$583$	4.97056	0.205860
$584$	0	0
$585$	−0.585786	−0.0242193
$586$	0	0
$587$	23.4558	0.968126	0.484063	−	0.875033i	$-0.339160\pi$
$587$	23.4558	0.968126	0.484063	+	0.875033i	$0.339160\pi$
$588$	0	0
$589$	−4.82843	−0.198952
$590$	0	0
$591$	22.1421	0.910806
$592$	0	0
$593$	−0.627417	−0.0257649	−0.0128825	−	0.999917i	$-0.504101\pi$
$593$	−0.627417	−0.0257649	−0.0128825	+	0.999917i	$0.504101\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	9.17157	0.375367
$598$	0	0
$599$	16.9706	0.693398	0.346699	−	0.937976i	$-0.387302\pi$
$599$	16.9706	0.693398	0.346699	+	0.937976i	$0.387302\pi$
$600$	0	0
$601$	45.7990	1.86818	0.934090	−	0.357038i	$-0.116213\pi$
$601$	45.7990	1.86818	0.934090	+	0.357038i	$0.116213\pi$
$602$	0	0
$603$	11.3137	0.460730
$604$	0	0
$605$	10.6569	0.433263
$606$	0	0
$607$	17.4558	0.708511	0.354255	−	0.935149i	$-0.384734\pi$
$607$	17.4558	0.708511	0.354255	+	0.935149i	$0.384734\pi$
$608$	0	0
$609$	10.0000	0.405220
$610$	0	0
$611$	1.85786	0.0751611
$612$	0	0
$613$	−7.17157	−0.289657	−0.144829	−	0.989457i	$-0.546263\pi$
$613$	−7.17157	−0.289657	−0.144829	+	0.989457i	$0.546263\pi$
$614$	0	0
$615$	−9.89949	−0.399186
$616$	0	0
$617$	47.1127	1.89669	0.948343	−	0.317247i	$-0.102758\pi$
$617$	47.1127	1.89669	0.948343	+	0.317247i	$0.102758\pi$
$618$	0	0
$619$	−20.4853	−0.823373	−0.411686	−	0.911326i	$-0.635060\pi$
$619$	−20.4853	−0.823373	−0.411686	+	0.911326i	$0.635060\pi$
$620$	0	0
$621$	4.82843	0.193758
$622$	0	0
$623$	17.3137	0.693659
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−0.585786	−0.0233941
$628$	0	0
$629$	−17.6569	−0.704025
$630$	0	0
$631$	−20.2843	−0.807504	−0.403752	−	0.914868i	$-0.632294\pi$
$631$	−20.2843	−0.807504	−0.403752	+	0.914868i	$0.632294\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	0	0
$635$	−5.65685	−0.224485
$636$	0	0
$637$	−2.92893	−0.116049
$638$	0	0
$639$	14.8284	0.586604
$640$	0	0
$641$	31.0711	1.22723	0.613617	−	0.789604i	$-0.289714\pi$
$641$	31.0711	1.22723	0.613617	+	0.789604i	$0.289714\pi$
$642$	0	0
$643$	43.0711	1.69856	0.849279	−	0.527945i	$-0.177037\pi$
$643$	43.0711	1.69856	0.849279	+	0.527945i	$0.177037\pi$
$644$	0	0
$645$	11.0711	0.435923
$646$	0	0
$647$	−2.68629	−0.105609	−0.0528045	−	0.998605i	$-0.516816\pi$
$647$	−2.68629	−0.105609	−0.0528045	+	0.998605i	$0.516816\pi$
$648$	0	0
$649$	0.686292	0.0269393
$650$	0	0
$651$	−6.82843	−0.267627
$652$	0	0
$653$	−18.1421	−0.709957	−0.354978	−	0.934875i	$-0.615512\pi$
$653$	−18.1421	−0.709957	−0.354978	+	0.934875i	$0.615512\pi$
$654$	0	0
$655$	6.72792	0.262882
$656$	0	0
$657$	−11.6569	−0.454777
$658$	0	0
$659$	4.97056	0.193626	0.0968128	−	0.995303i	$-0.469135\pi$
$659$	4.97056	0.193626	0.0968128	+	0.995303i	$0.469135\pi$
$660$	0	0
$661$	13.1127	0.510025	0.255012	−	0.966938i	$-0.417920\pi$
$661$	13.1127	0.510025	0.255012	+	0.966938i	$0.417920\pi$
$662$	0	0
$663$	−1.65685	−0.0643469
$664$	0	0
$665$	−1.41421	−0.0548408
$666$	0	0
$667$	−34.1421	−1.32199
$668$	0	0
$669$	−11.3137	−0.437413
$670$	0	0
$671$	−0.970563	−0.0374682
$672$	0	0
$673$	5.75736	0.221930	0.110965	−	0.993824i	$-0.464606\pi$
$673$	5.75736	0.221930	0.110965	+	0.993824i	$0.464606\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	34.8284	1.33857	0.669283	−	0.743008i	$-0.266602\pi$
$677$	34.8284	1.33857	0.669283	+	0.743008i	$0.266602\pi$
$678$	0	0
$679$	−16.8284	−0.645816
$680$	0	0
$681$	−11.6569	−0.446692
$682$	0	0
$683$	−12.6863	−0.485427	−0.242714	−	0.970098i	$-0.578038\pi$
$683$	−12.6863	−0.485427	−0.242714	+	0.970098i	$0.578038\pi$
$684$	0	0
$685$	−15.6569	−0.598218
$686$	0	0
$687$	−21.6569	−0.826261
$688$	0	0
$689$	4.97056	0.189363
$690$	0	0
$691$	−47.7990	−1.81836	−0.909180	−	0.416404i	$-0.863290\pi$
$691$	−47.7990	−1.81836	−0.909180	+	0.416404i	$0.863290\pi$
$692$	0	0
$693$	−0.828427	−0.0314693
$694$	0	0
$695$	−2.82843	−0.107288
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	15.6569	0.592197
$700$	0	0
$701$	−10.9706	−0.414352	−0.207176	−	0.978304i	$-0.566427\pi$
$701$	−10.9706	−0.414352	−0.207176	+	0.978304i	$0.566427\pi$
$702$	0	0
$703$	−6.24264	−0.235446
$704$	0	0
$705$	−3.17157	−0.119448
$706$	0	0
$707$	1.17157	0.0440615
$708$	0	0
$709$	−20.0000	−0.751116	−0.375558	−	0.926799i	$-0.622549\pi$
$709$	−20.0000	−0.751116	−0.375558	+	0.926799i	$0.622549\pi$
$710$	0	0
$711$	13.6569	0.512172
$712$	0	0
$713$	23.3137	0.873105
$714$	0	0
$715$	−0.343146	−0.0128329
$716$	0	0
$717$	17.7574	0.663161
$718$	0	0
$719$	−10.2426	−0.381986	−0.190993	−	0.981591i	$-0.561171\pi$
$719$	−10.2426	−0.381986	−0.190993	+	0.981591i	$0.561171\pi$
$720$	0	0
$721$	4.68629	0.174527
$722$	0	0
$723$	12.3431	0.459047
$724$	0	0
$725$	−7.07107	−0.262613
$726$	0	0
$727$	−24.2426	−0.899110	−0.449555	−	0.893253i	$-0.648417\pi$
$727$	−24.2426	−0.899110	−0.449555	+	0.893253i	$0.648417\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.3137	1.15818
$732$	0	0
$733$	−10.6863	−0.394707	−0.197354	−	0.980332i	$-0.563235\pi$
$733$	−10.6863	−0.394707	−0.197354	+	0.980332i	$0.563235\pi$
$734$	0	0
$735$	5.00000	0.184428
$736$	0	0
$737$	6.62742	0.244124
$738$	0	0
$739$	−7.31371	−0.269039	−0.134520	−	0.990911i	$-0.542949\pi$
$739$	−7.31371	−0.269039	−0.134520	+	0.990911i	$0.542949\pi$
$740$	0	0
$741$	−0.585786	−0.0215194
$742$	0	0
$743$	−41.2548	−1.51349	−0.756747	−	0.653708i	$-0.773212\pi$
$743$	−41.2548	−1.51349	−0.756747	+	0.653708i	$0.773212\pi$
$744$	0	0
$745$	4.34315	0.159121
$746$	0	0
$747$	7.65685	0.280150
$748$	0	0
$749$	−19.3137	−0.705708
$750$	0	0
$751$	44.1421	1.61077	0.805385	−	0.592752i	$-0.201959\pi$
$751$	44.1421	1.61077	0.805385	+	0.592752i	$0.201959\pi$
$752$	0	0
$753$	5.75736	0.209810
$754$	0	0
$755$	0.828427	0.0301496
$756$	0	0
$757$	−26.4853	−0.962624	−0.481312	−	0.876549i	$-0.659840\pi$
$757$	−26.4853	−0.962624	−0.481312	+	0.876549i	$0.659840\pi$
$758$	0	0
$759$	2.82843	0.102665
$760$	0	0
$761$	−39.6569	−1.43756	−0.718780	−	0.695238i	$-0.755299\pi$
$761$	−39.6569	−1.43756	−0.718780	+	0.695238i	$0.755299\pi$
$762$	0	0
$763$	−11.5147	−0.416861
$764$	0	0
$765$	2.82843	0.102262
$766$	0	0
$767$	0.686292	0.0247805
$768$	0	0
$769$	−34.2843	−1.23632	−0.618161	−	0.786051i	$-0.712122\pi$
$769$	−34.2843	−1.23632	−0.618161	+	0.786051i	$0.712122\pi$
$770$	0	0
$771$	12.0000	0.432169
$772$	0	0
$773$	5.45584	0.196233	0.0981165	−	0.995175i	$-0.468718\pi$
$773$	5.45584	0.196233	0.0981165	+	0.995175i	$0.468718\pi$
$774$	0	0
$775$	4.82843	0.173442
$776$	0	0
$777$	−8.82843	−0.316718
$778$	0	0
$779$	−9.89949	−0.354686
$780$	0	0
$781$	8.68629	0.310820
$782$	0	0
$783$	−7.07107	−0.252699
$784$	0	0
$785$	4.14214	0.147839
$786$	0	0
$787$	6.82843	0.243407	0.121704	−	0.992566i	$-0.461164\pi$
$787$	6.82843	0.243407	0.121704	+	0.992566i	$0.461164\pi$
$788$	0	0
$789$	2.00000	0.0712019
$790$	0	0
$791$	21.6569	0.770029
$792$	0	0
$793$	−0.970563	−0.0344657
$794$	0	0
$795$	−8.48528	−0.300942
$796$	0	0
$797$	4.68629	0.165997	0.0829985	−	0.996550i	$-0.473550\pi$
$797$	4.68629	0.165997	0.0829985	+	0.996550i	$0.473550\pi$
$798$	0	0
$799$	−8.97056	−0.317356
$800$	0	0
$801$	−12.2426	−0.432572
$802$	0	0
$803$	−6.82843	−0.240970
$804$	0	0
$805$	6.82843	0.240670
$806$	0	0
$807$	14.3848	0.506368
$808$	0	0
$809$	−53.3137	−1.87441	−0.937205	−	0.348779i	$-0.886596\pi$
$809$	−53.3137	−1.87441	−0.937205	+	0.348779i	$0.886596\pi$
$810$	0	0
$811$	−52.9706	−1.86005	−0.930024	−	0.367499i	$-0.880214\pi$
$811$	−52.9706	−1.86005	−0.930024	+	0.367499i	$0.880214\pi$
$812$	0	0
$813$	2.82843	0.0991973
$814$	0	0
$815$	−23.5563	−0.825143
$816$	0	0
$817$	11.0711	0.387328
$818$	0	0
$819$	−0.828427	−0.0289476
$820$	0	0
$821$	−21.7990	−0.760790	−0.380395	−	0.924824i	$-0.624212\pi$
$821$	−21.7990	−0.760790	−0.380395	+	0.924824i	$0.624212\pi$
$822$	0	0
$823$	50.8701	1.77322	0.886609	−	0.462519i	$-0.153054\pi$
$823$	50.8701	1.77322	0.886609	+	0.462519i	$0.153054\pi$
$824$	0	0
$825$	0.585786	0.0203945
$826$	0	0
$827$	−7.65685	−0.266255	−0.133127	−	0.991099i	$-0.542502\pi$
$827$	−7.65685	−0.266255	−0.133127	+	0.991099i	$0.542502\pi$
$828$	0	0
$829$	−0.544156	−0.0188993	−0.00944966	−	0.999955i	$-0.503008\pi$
$829$	−0.544156	−0.0188993	−0.00944966	+	0.999955i	$0.503008\pi$
$830$	0	0
$831$	−24.6274	−0.854316
$832$	0	0
$833$	14.1421	0.489996
$834$	0	0
$835$	−17.3137	−0.599166
$836$	0	0
$837$	4.82843	0.166895
$838$	0	0
$839$	−34.1421	−1.17872	−0.589359	−	0.807871i	$-0.700620\pi$
$839$	−34.1421	−1.17872	−0.589359	+	0.807871i	$0.700620\pi$
$840$	0	0
$841$	21.0000	0.724138
$842$	0	0
$843$	−30.3848	−1.04651
$844$	0	0
$845$	12.6569	0.435409
$846$	0	0
$847$	15.0711	0.517848
$848$	0	0
$849$	−7.75736	−0.266232
$850$	0	0
$851$	30.1421	1.03326
$852$	0	0
$853$	3.17157	0.108593	0.0542963	−	0.998525i	$-0.482708\pi$
$853$	3.17157	0.108593	0.0542963	+	0.998525i	$0.482708\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−25.4558	−0.869555	−0.434778	−	0.900538i	$-0.643173\pi$
$857$	−25.4558	−0.869555	−0.434778	+	0.900538i	$0.643173\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	0	0
$861$	−14.0000	−0.477119
$862$	0	0
$863$	57.2548	1.94898	0.974489	−	0.224437i	$-0.0720543\pi$
$863$	57.2548	1.94898	0.974489	+	0.224437i	$0.0720543\pi$
$864$	0	0
$865$	11.3137	0.384678
$866$	0	0
$867$	−9.00000	−0.305656
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	6.62742	0.224561
$872$	0	0
$873$	11.8995	0.402737
$874$	0	0
$875$	1.41421	0.0478091
$876$	0	0
$877$	2.04163	0.0689410	0.0344705	−	0.999406i	$-0.489026\pi$
$877$	2.04163	0.0689410	0.0344705	+	0.999406i	$0.489026\pi$
$878$	0	0
$879$	1.65685	0.0558843
$880$	0	0
$881$	−17.5147	−0.590086	−0.295043	−	0.955484i	$-0.595334\pi$
$881$	−17.5147	−0.590086	−0.295043	+	0.955484i	$0.595334\pi$
$882$	0	0
$883$	44.2426	1.48888	0.744442	−	0.667687i	$-0.232716\pi$
$883$	44.2426	1.48888	0.744442	+	0.667687i	$0.232716\pi$
$884$	0	0
$885$	−1.17157	−0.0393820
$886$	0	0
$887$	10.3431	0.347289	0.173644	−	0.984808i	$-0.444446\pi$
$887$	10.3431	0.347289	0.173644	+	0.984808i	$0.444446\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	0.585786	0.0196246
$892$	0	0
$893$	−3.17157	−0.106133
$894$	0	0
$895$	−13.1716	−0.440277
$896$	0	0
$897$	2.82843	0.0944384
$898$	0	0
$899$	−34.1421	−1.13870
$900$	0	0
$901$	−24.0000	−0.799556
$902$	0	0
$903$	15.6569	0.521027
$904$	0	0
$905$	−20.1421	−0.669547
$906$	0	0
$907$	20.7696	0.689642	0.344821	−	0.938669i	$-0.387940\pi$
$907$	20.7696	0.689642	0.344821	+	0.938669i	$0.387940\pi$
$908$	0	0
$909$	−0.828427	−0.0274772
$910$	0	0
$911$	−34.6274	−1.14726	−0.573629	−	0.819115i	$-0.694465\pi$
$911$	−34.6274	−1.14726	−0.573629	+	0.819115i	$0.694465\pi$
$912$	0	0
$913$	4.48528	0.148441
$914$	0	0
$915$	1.65685	0.0547739
$916$	0	0
$917$	9.51472	0.314204
$918$	0	0
$919$	51.5980	1.70206	0.851030	−	0.525117i	$-0.175978\pi$
$919$	51.5980	1.70206	0.851030	+	0.525117i	$0.175978\pi$
$920$	0	0
$921$	−30.1421	−0.993217
$922$	0	0
$923$	8.68629	0.285913
$924$	0	0
$925$	6.24264	0.205257
$926$	0	0
$927$	−3.31371	−0.108836
$928$	0	0
$929$	−14.2010	−0.465920	−0.232960	−	0.972486i	$-0.574841\pi$
$929$	−14.2010	−0.465920	−0.232960	+	0.972486i	$0.574841\pi$
$930$	0	0
$931$	5.00000	0.163868
$932$	0	0
$933$	23.2132	0.759966
$934$	0	0
$935$	1.65685	0.0541849
$936$	0	0
$937$	−29.7990	−0.973491	−0.486745	−	0.873544i	$-0.661816\pi$
$937$	−29.7990	−0.973491	−0.486745	+	0.873544i	$0.661816\pi$
$938$	0	0
$939$	25.7990	0.841918
$940$	0	0
$941$	7.75736	0.252883	0.126441	−	0.991974i	$-0.459644\pi$
$941$	7.75736	0.252883	0.126441	+	0.991974i	$0.459644\pi$
$942$	0	0
$943$	47.7990	1.55655
$944$	0	0
$945$	1.41421	0.0460044
$946$	0	0
$947$	−38.2843	−1.24407	−0.622036	−	0.782989i	$-0.713694\pi$
$947$	−38.2843	−1.24407	−0.622036	+	0.782989i	$0.713694\pi$
$948$	0	0
$949$	−6.82843	−0.221660
$950$	0	0
$951$	26.1421	0.847717
$952$	0	0
$953$	36.9706	1.19759	0.598797	−	0.800901i	$-0.295646\pi$
$953$	36.9706	1.19759	0.598797	+	0.800901i	$0.295646\pi$
$954$	0	0
$955$	−13.7574	−0.445178
$956$	0	0
$957$	−4.14214	−0.133896
$958$	0	0
$959$	−22.1421	−0.715007
$960$	0	0
$961$	−7.68629	−0.247945
$962$	0	0
$963$	13.6569	0.440086
$964$	0	0
$965$	−13.0711	−0.420773
$966$	0	0
$967$	17.4142	0.560003	0.280002	−	0.960000i	$-0.409665\pi$
$967$	17.4142	0.560003	0.280002	+	0.960000i	$0.409665\pi$
$968$	0	0
$969$	2.82843	0.0908622
$970$	0	0
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	0	0
$975$	0.585786	0.0187602
$976$	0	0
$977$	3.51472	0.112446	0.0562229	−	0.998418i	$-0.482094\pi$
$977$	3.51472	0.112446	0.0562229	+	0.998418i	$0.482094\pi$
$978$	0	0
$979$	−7.17157	−0.229204
$980$	0	0
$981$	8.14214	0.259958
$982$	0	0
$983$	51.9411	1.65666	0.828332	−	0.560237i	$-0.189290\pi$
$983$	51.9411	1.65666	0.828332	+	0.560237i	$0.189290\pi$
$984$	0	0
$985$	−22.1421	−0.705507
$986$	0	0
$987$	−4.48528	−0.142768
$988$	0	0
$989$	−53.4558	−1.69980
$990$	0	0
$991$	21.6569	0.687953	0.343976	−	0.938978i	$-0.388226\pi$
$991$	21.6569	0.687953	0.343976	+	0.938978i	$0.388226\pi$
$992$	0	0
$993$	12.1421	0.385319
$994$	0	0
$995$	−9.17157	−0.290758
$996$	0	0
$997$	−23.4558	−0.742854	−0.371427	−	0.928462i	$-0.621131\pi$
$997$	−23.4558	−0.742854	−0.371427	+	0.928462i	$0.621131\pi$
$998$	0	0
$999$	6.24264	0.197508

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bn.1.1		2
4.3	odd	2		2280.2.a.k.1.2	✓	2
12.11	even	2		6840.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.k.1.2	✓	2	4.3	odd	2
4560.2.a.bn.1.1		2	1.1	even	1	trivial
6840.2.a.bb.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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} -1.41421 q^{7} +1.00000 q^{9} +0.585786 q^{11} +0.585786 q^{13} -1.00000 q^{15} -2.82843 q^{17} -1.00000 q^{19} -1.41421 q^{21} +4.82843 q^{23} +1.00000 q^{25} +1.00000 q^{27} -7.07107 q^{29} +4.82843 q^{31} +0.585786 q^{33} +1.41421 q^{35} +6.24264 q^{37} +0.585786 q^{39} +9.89949 q^{41} -11.0711 q^{43} -1.00000 q^{45} +3.17157 q^{47} -5.00000 q^{49} -2.82843 q^{51} +8.48528 q^{53} -0.585786 q^{55} -1.00000 q^{57} +1.17157 q^{59} -1.65685 q^{61} -1.41421 q^{63} -0.585786 q^{65} +11.3137 q^{67} +4.82843 q^{69} +14.8284 q^{71} -11.6569 q^{73} +1.00000 q^{75} -0.828427 q^{77} +13.6569 q^{79} +1.00000 q^{81} +7.65685 q^{83} +2.82843 q^{85} -7.07107 q^{87} -12.2426 q^{89} -0.828427 q^{91} +4.82843 q^{93} +1.00000 q^{95} +11.8995 q^{97} +0.585786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{15} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 2 q^{27} + 4 q^{31} + 4 q^{33} + 4 q^{37} + 4 q^{39} - 8 q^{43} - 2 q^{45} + 12 q^{47} - 10 q^{49} - 4 q^{55} - 2 q^{57} + 8 q^{59} + 8 q^{61} - 4 q^{65} + 4 q^{69} + 24 q^{71} - 12 q^{73} + 2 q^{75} + 4 q^{77} + 16 q^{79} + 2 q^{81} + 4 q^{83} - 16 q^{89} + 4 q^{91} + 4 q^{93} + 2 q^{95} + 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^5 + 2 * q^9 + 4 * q^11 + 4 * q^13 - 2 * q^15 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 2 * q^27 + 4 * q^31 + 4 * q^33 + 4 * q^37 + 4 * q^39 - 8 * q^43 - 2 * q^45 + 12 * q^47 - 10 * q^49 - 4 * q^55 - 2 * q^57 + 8 * q^59 + 8 * q^61 - 4 * q^65 + 4 * q^69 + 24 * q^71 - 12 * q^73 + 2 * q^75 + 4 * q^77 + 16 * q^79 + 2 * q^81 + 4 * q^83 - 16 * q^89 + 4 * q^91 + 4 * q^93 + 2 * q^95 + 4 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

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