LMFDB - Embedded newform 6160.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6160,2,Mod(1,6160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6160, 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("6160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6160.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:	$49.1878476451$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 770)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	6160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.14637 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.68585 q^{9} +O(q^{10})$	$q-1.14637 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.68585 q^{9} -1.00000 q^{11} +4.68585 q^{13} -1.14637 q^{15} -0.292731 q^{17} -6.51806 q^{19} -1.14637 q^{21} -2.85363 q^{23} +1.00000 q^{25} +5.37169 q^{27} -1.43910 q^{29} -0.978577 q^{31} +1.14637 q^{33} +1.00000 q^{35} +0.853635 q^{37} -5.37169 q^{39} -6.22533 q^{41} +10.3503 q^{43} -1.68585 q^{45} +9.95715 q^{47} +1.00000 q^{49} +0.335577 q^{51} +5.43910 q^{53} -1.00000 q^{55} +7.47208 q^{57} +9.37169 q^{59} -11.9572 q^{61} -1.68585 q^{63} +4.68585 q^{65} +0.585462 q^{67} +3.27131 q^{69} +0.335577 q^{71} -3.70727 q^{73} -1.14637 q^{75} -1.00000 q^{77} +2.51806 q^{79} -1.10038 q^{81} +1.70727 q^{83} -0.292731 q^{85} +1.64973 q^{87} +13.0790 q^{89} +4.68585 q^{91} +1.12181 q^{93} -6.51806 q^{95} +9.10352 q^{97} +1.68585 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9	$ 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99}+O(q^{100}) $ 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9 - 3 * q^11 + 2 * q^13 - 2 * q^15 + 2 * q^17 + 6 * q^19 - 2 * q^21 - 10 * q^23 + 3 * q^25 - 8 * q^27 + 12 * q^31 + 2 * q^33 + 3 * q^35 + 4 * q^37 + 8 * q^39 + 4 * q^41 - 8 * q^43 + 7 * q^45 + 3 * q^49 + 28 * q^51 + 12 * q^53 - 3 * q^55 - 8 * q^57 + 4 * q^59 - 6 * q^61 + 7 * q^63 + 2 * q^65 - 4 * q^67 - 8 * q^69 + 28 * q^71 - 14 * q^73 - 2 * q^75 - 3 * q^77 - 18 * q^79 + 3 * q^81 + 8 * q^83 + 2 * q^85 + 44 * q^87 + 18 * q^89 + 2 * q^91 + 12 * q^93 + 6 * q^95 - 4 * q^97 - 7 * 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
$3$	−1.14637	−0.661854	−0.330927	−	0.943656i	$-0.607361\pi$
$3$	−1.14637	−0.661854	−0.330927	+	0.943656i	$0.607361\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.68585	−0.561949
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.68585	1.29962	0.649810	−	0.760097i	$-0.274848\pi$
$13$	4.68585	1.29962	0.649810	+	0.760097i	$0.274848\pi$
$14$	0	0
$15$	−1.14637	−0.295990
$16$	0	0
$17$	−0.292731	−0.0709977	−0.0354988	−	0.999370i	$-0.511302\pi$
$17$	−0.292731	−0.0709977	−0.0354988	+	0.999370i	$0.511302\pi$
$18$	0	0
$19$	−6.51806	−1.49535	−0.747673	−	0.664068i	$-0.768829\pi$
$19$	−6.51806	−1.49535	−0.747673	+	0.664068i	$0.768829\pi$
$20$	0	0
$21$	−1.14637	−0.250157
$22$	0	0
$23$	−2.85363	−0.595024	−0.297512	−	0.954718i	$-0.596157\pi$
$23$	−2.85363	−0.595024	−0.297512	+	0.954718i	$0.596157\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.37169	1.03378
$28$	0	0
$29$	−1.43910	−0.267234	−0.133617	−	0.991033i	$-0.542659\pi$
$29$	−1.43910	−0.267234	−0.133617	+	0.991033i	$0.542659\pi$
$30$	0	0
$31$	−0.978577	−0.175758	−0.0878788	−	0.996131i	$-0.528009\pi$
$31$	−0.978577	−0.175758	−0.0878788	+	0.996131i	$0.528009\pi$
$32$	0	0
$33$	1.14637	0.199557
$34$	0	0
$35$	1.00000	0.169031
$36$	0	0
$37$	0.853635	0.140337	0.0701683	−	0.997535i	$-0.477646\pi$
$37$	0.853635	0.140337	0.0701683	+	0.997535i	$0.477646\pi$
$38$	0	0
$39$	−5.37169	−0.860159
$40$	0	0
$41$	−6.22533	−0.972233	−0.486116	−	0.873894i	$-0.661587\pi$
$41$	−6.22533	−0.972233	−0.486116	+	0.873894i	$0.661587\pi$
$42$	0	0
$43$	10.3503	1.57840	0.789201	−	0.614135i	$-0.210495\pi$
$43$	10.3503	1.57840	0.789201	+	0.614135i	$0.210495\pi$
$44$	0	0
$45$	−1.68585	−0.251311
$46$	0	0
$47$	9.95715	1.45240	0.726200	−	0.687483i	$-0.241285\pi$
$47$	9.95715	1.45240	0.726200	+	0.687483i	$0.241285\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.335577	0.0469901
$52$	0	0
$53$	5.43910	0.747117	0.373559	−	0.927607i	$-0.378137\pi$
$53$	5.43910	0.747117	0.373559	+	0.927607i	$0.378137\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	0	0
$57$	7.47208	0.989701
$58$	0	0
$59$	9.37169	1.22009	0.610045	−	0.792367i	$-0.291151\pi$
$59$	9.37169	1.22009	0.610045	+	0.792367i	$0.291151\pi$
$60$	0	0
$61$	−11.9572	−1.53096	−0.765478	−	0.643462i	$-0.777498\pi$
$61$	−11.9572	−1.53096	−0.765478	+	0.643462i	$0.777498\pi$
$62$	0	0
$63$	−1.68585	−0.212397
$64$	0	0
$65$	4.68585	0.581208
$66$	0	0
$67$	0.585462	0.0715256	0.0357628	−	0.999360i	$-0.488614\pi$
$67$	0.585462	0.0715256	0.0357628	+	0.999360i	$0.488614\pi$
$68$	0	0
$69$	3.27131	0.393819
$70$	0	0
$71$	0.335577	0.0398256	0.0199128	−	0.999802i	$-0.493661\pi$
$71$	0.335577	0.0398256	0.0199128	+	0.999802i	$0.493661\pi$
$72$	0	0
$73$	−3.70727	−0.433903	−0.216952	−	0.976182i	$-0.569611\pi$
$73$	−3.70727	−0.433903	−0.216952	+	0.976182i	$0.569611\pi$
$74$	0	0
$75$	−1.14637	−0.132371
$76$	0	0
$77$	−1.00000	−0.113961
$78$	0	0
$79$	2.51806	0.283304	0.141652	−	0.989917i	$-0.454759\pi$
$79$	2.51806	0.283304	0.141652	+	0.989917i	$0.454759\pi$
$80$	0	0
$81$	−1.10038	−0.122265
$82$	0	0
$83$	1.70727	0.187397	0.0936986	−	0.995601i	$-0.470131\pi$
$83$	1.70727	0.187397	0.0936986	+	0.995601i	$0.470131\pi$
$84$	0	0
$85$	−0.292731	−0.0317511
$86$	0	0
$87$	1.64973	0.176870
$88$	0	0
$89$	13.0790	1.38637	0.693184	−	0.720761i	$-0.256208\pi$
$89$	13.0790	1.38637	0.693184	+	0.720761i	$0.256208\pi$
$90$	0	0
$91$	4.68585	0.491210
$92$	0	0
$93$	1.12181	0.116326
$94$	0	0
$95$	−6.51806	−0.668739
$96$	0	0
$97$	9.10352	0.924322	0.462161	−	0.886796i	$-0.347074\pi$
$97$	9.10352	0.924322	0.462161	+	0.886796i	$0.347074\pi$
$98$	0	0
$99$	1.68585	0.169434
$100$	0	0
$101$	−10.7862	−1.07327	−0.536635	−	0.843814i	$-0.680305\pi$
$101$	−10.7862	−1.07327	−0.536635	+	0.843814i	$0.680305\pi$
$102$	0	0
$103$	−7.66442	−0.755198	−0.377599	−	0.925969i	$-0.623250\pi$
$103$	−7.66442	−0.755198	−0.377599	+	0.925969i	$0.623250\pi$
$104$	0	0
$105$	−1.14637	−0.111874
$106$	0	0
$107$	−9.56404	−0.924591	−0.462295	−	0.886726i	$-0.652974\pi$
$107$	−9.56404	−0.924591	−0.462295	+	0.886726i	$0.652974\pi$
$108$	0	0
$109$	3.14637	0.301367	0.150684	−	0.988582i	$-0.451853\pi$
$109$	3.14637	0.301367	0.150684	+	0.988582i	$0.451853\pi$
$110$	0	0
$111$	−0.978577	−0.0928824
$112$	0	0
$113$	−2.58546	−0.243220	−0.121610	−	0.992578i	$-0.538806\pi$
$113$	−2.58546	−0.243220	−0.121610	+	0.992578i	$0.538806\pi$
$114$	0	0
$115$	−2.85363	−0.266103
$116$	0	0
$117$	−7.89962	−0.730320
$118$	0	0
$119$	−0.292731	−0.0268346
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	7.13650	0.643477
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	13.3717	1.18655	0.593273	−	0.805001i	$-0.297836\pi$
$127$	13.3717	1.18655	0.593273	+	0.805001i	$0.297836\pi$
$128$	0	0
$129$	−11.8652	−1.04467
$130$	0	0
$131$	11.4391	0.999438	0.499719	−	0.866187i	$-0.333437\pi$
$131$	11.4391	0.999438	0.499719	+	0.866187i	$0.333437\pi$
$132$	0	0
$133$	−6.51806	−0.565187
$134$	0	0
$135$	5.37169	0.462322
$136$	0	0
$137$	10.0000	0.854358	0.427179	−	0.904167i	$-0.359507\pi$
$137$	10.0000	0.854358	0.427179	+	0.904167i	$0.359507\pi$
$138$	0	0
$139$	14.1825	1.20294	0.601471	−	0.798895i	$-0.294581\pi$
$139$	14.1825	1.20294	0.601471	+	0.798895i	$0.294581\pi$
$140$	0	0
$141$	−11.4145	−0.961278
$142$	0	0
$143$	−4.68585	−0.391850
$144$	0	0
$145$	−1.43910	−0.119510
$146$	0	0
$147$	−1.14637	−0.0945506
$148$	0	0
$149$	11.5970	0.950065	0.475032	−	0.879968i	$-0.342436\pi$
$149$	11.5970	0.950065	0.475032	+	0.879968i	$0.342436\pi$
$150$	0	0
$151$	−1.73183	−0.140934	−0.0704671	−	0.997514i	$-0.522449\pi$
$151$	−1.73183	−0.140934	−0.0704671	+	0.997514i	$0.522449\pi$
$152$	0	0
$153$	0.493499	0.0398971
$154$	0	0
$155$	−0.978577	−0.0786012
$156$	0	0
$157$	−12.2927	−0.981067	−0.490533	−	0.871422i	$-0.663198\pi$
$157$	−12.2927	−0.981067	−0.490533	+	0.871422i	$0.663198\pi$
$158$	0	0
$159$	−6.23519	−0.494483
$160$	0	0
$161$	−2.85363	−0.224898
$162$	0	0
$163$	−23.9143	−1.87311	−0.936557	−	0.350516i	$-0.886006\pi$
$163$	−23.9143	−1.87311	−0.936557	+	0.350516i	$0.886006\pi$
$164$	0	0
$165$	1.14637	0.0892444
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	8.95715	0.689012
$170$	0	0
$171$	10.9884	0.840307
$172$	0	0
$173$	−7.37169	−0.560459	−0.280230	−	0.959933i	$-0.590411\pi$
$173$	−7.37169	−0.560459	−0.280230	+	0.959933i	$0.590411\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−10.7434	−0.807522
$178$	0	0
$179$	−1.56404	−0.116902	−0.0584509	−	0.998290i	$-0.518616\pi$
$179$	−1.56404	−0.116902	−0.0584509	+	0.998290i	$0.518616\pi$
$180$	0	0
$181$	14.7862	1.09905	0.549526	−	0.835477i	$-0.314808\pi$
$181$	14.7862	1.09905	0.549526	+	0.835477i	$0.314808\pi$
$182$	0	0
$183$	13.7073	1.01327
$184$	0	0
$185$	0.853635	0.0627605
$186$	0	0
$187$	0.292731	0.0214066
$188$	0	0
$189$	5.37169	0.390733
$190$	0	0
$191$	17.9572	1.29933	0.649667	−	0.760219i	$-0.274908\pi$
$191$	17.9572	1.29933	0.649667	+	0.760219i	$0.274908\pi$
$192$	0	0
$193$	18.6430	1.34195	0.670976	−	0.741479i	$-0.265875\pi$
$193$	18.6430	1.34195	0.670976	+	0.741479i	$0.265875\pi$
$194$	0	0
$195$	−5.37169	−0.384675
$196$	0	0
$197$	−15.0361	−1.07128	−0.535639	−	0.844447i	$-0.679929\pi$
$197$	−15.0361	−1.07128	−0.535639	+	0.844447i	$0.679929\pi$
$198$	0	0
$199$	−15.3288	−1.08663	−0.543317	−	0.839528i	$-0.682832\pi$
$199$	−15.3288	−1.08663	−0.543317	+	0.839528i	$0.682832\pi$
$200$	0	0
$201$	−0.671153	−0.0473395
$202$	0	0
$203$	−1.43910	−0.101005
$204$	0	0
$205$	−6.22533	−0.434796
$206$	0	0
$207$	4.81079	0.334373
$208$	0	0
$209$	6.51806	0.450863
$210$	0	0
$211$	−2.04285	−0.140635	−0.0703176	−	0.997525i	$-0.522401\pi$
$211$	−2.04285	−0.140635	−0.0703176	+	0.997525i	$0.522401\pi$
$212$	0	0
$213$	−0.384694	−0.0263588
$214$	0	0
$215$	10.3503	0.705883
$216$	0	0
$217$	−0.978577	−0.0664301
$218$	0	0
$219$	4.24989	0.287181
$220$	0	0
$221$	−1.37169	−0.0922700
$222$	0	0
$223$	−11.0790	−0.741902	−0.370951	−	0.928652i	$-0.620968\pi$
$223$	−11.0790	−0.741902	−0.370951	+	0.928652i	$0.620968\pi$
$224$	0	0
$225$	−1.68585	−0.112390
$226$	0	0
$227$	18.5426	1.23072	0.615358	−	0.788248i	$-0.289011\pi$
$227$	18.5426	1.23072	0.615358	+	0.788248i	$0.289011\pi$
$228$	0	0
$229$	−8.01469	−0.529626	−0.264813	−	0.964300i	$-0.585310\pi$
$229$	−8.01469	−0.529626	−0.264813	+	0.964300i	$0.585310\pi$
$230$	0	0
$231$	1.14637	0.0754253
$232$	0	0
$233$	27.9572	1.83153	0.915767	−	0.401710i	$-0.131584\pi$
$233$	27.9572	1.83153	0.915767	+	0.401710i	$0.131584\pi$
$234$	0	0
$235$	9.95715	0.649533
$236$	0	0
$237$	−2.88661	−0.187506
$238$	0	0
$239$	28.8108	1.86362	0.931808	−	0.362953i	$-0.118231\pi$
$239$	28.8108	1.86362	0.931808	+	0.362953i	$0.118231\pi$
$240$	0	0
$241$	−8.51806	−0.548696	−0.274348	−	0.961630i	$-0.588462\pi$
$241$	−8.51806	−0.548696	−0.274348	+	0.961630i	$0.588462\pi$
$242$	0	0
$243$	−14.8536	−0.952861
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−30.5426	−1.94338
$248$	0	0
$249$	−1.95715	−0.124030
$250$	0	0
$251$	23.1281	1.45983	0.729916	−	0.683537i	$-0.239559\pi$
$251$	23.1281	1.45983	0.729916	+	0.683537i	$0.239559\pi$
$252$	0	0
$253$	2.85363	0.179406
$254$	0	0
$255$	0.335577	0.0210146
$256$	0	0
$257$	14.8108	0.923872	0.461936	−	0.886913i	$-0.347155\pi$
$257$	14.8108	0.923872	0.461936	+	0.886913i	$0.347155\pi$
$258$	0	0
$259$	0.853635	0.0530423
$260$	0	0
$261$	2.42610	0.150172
$262$	0	0
$263$	−18.7434	−1.15577	−0.577883	−	0.816119i	$-0.696121\pi$
$263$	−18.7434	−1.15577	−0.577883	+	0.816119i	$0.696121\pi$
$264$	0	0
$265$	5.43910	0.334121
$266$	0	0
$267$	−14.9933	−0.917573
$268$	0	0
$269$	12.6858	0.773470	0.386735	−	0.922191i	$-0.373603\pi$
$269$	12.6858	0.773470	0.386735	+	0.922191i	$0.373603\pi$
$270$	0	0
$271$	−1.12181	−0.0681449	−0.0340725	−	0.999419i	$-0.510848\pi$
$271$	−1.12181	−0.0681449	−0.0340725	+	0.999419i	$0.510848\pi$
$272$	0	0
$273$	−5.37169	−0.325110
$274$	0	0
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	30.1151	1.80944	0.904720	−	0.426007i	$-0.140080\pi$
$277$	30.1151	1.80944	0.904720	+	0.426007i	$0.140080\pi$
$278$	0	0
$279$	1.64973	0.0987668
$280$	0	0
$281$	−14.3356	−0.855189	−0.427594	−	0.903971i	$-0.640639\pi$
$281$	−14.3356	−0.855189	−0.427594	+	0.903971i	$0.640639\pi$
$282$	0	0
$283$	−3.21377	−0.191039	−0.0955194	−	0.995428i	$-0.530451\pi$
$283$	−3.21377	−0.191039	−0.0955194	+	0.995428i	$0.530451\pi$
$284$	0	0
$285$	7.47208	0.442608
$286$	0	0
$287$	−6.22533	−0.367469
$288$	0	0
$289$	−16.9143	−0.994959
$290$	0	0
$291$	−10.4360	−0.611767
$292$	0	0
$293$	−16.6858	−0.974798	−0.487399	−	0.873179i	$-0.662054\pi$
$293$	−16.6858	−0.974798	−0.487399	+	0.873179i	$0.662054\pi$
$294$	0	0
$295$	9.37169	0.545641
$296$	0	0
$297$	−5.37169	−0.311697
$298$	0	0
$299$	−13.3717	−0.773305
$300$	0	0
$301$	10.3503	0.596580
$302$	0	0
$303$	12.3650	0.710349
$304$	0	0
$305$	−11.9572	−0.684665
$306$	0	0
$307$	6.29273	0.359145	0.179573	−	0.983745i	$-0.442529\pi$
$307$	6.29273	0.359145	0.179573	+	0.983745i	$0.442529\pi$
$308$	0	0
$309$	8.78623	0.499831
$310$	0	0
$311$	20.3931	1.15639	0.578194	−	0.815900i	$-0.303758\pi$
$311$	20.3931	1.15639	0.578194	+	0.815900i	$0.303758\pi$
$312$	0	0
$313$	12.0674	0.682090	0.341045	−	0.940047i	$-0.389219\pi$
$313$	12.0674	0.682090	0.341045	+	0.940047i	$0.389219\pi$
$314$	0	0
$315$	−1.68585	−0.0949867
$316$	0	0
$317$	28.7679	1.61577	0.807884	−	0.589341i	$-0.200613\pi$
$317$	28.7679	1.61577	0.807884	+	0.589341i	$0.200613\pi$
$318$	0	0
$319$	1.43910	0.0805739
$320$	0	0
$321$	10.9639	0.611944
$322$	0	0
$323$	1.90804	0.106166
$324$	0	0
$325$	4.68585	0.259924
$326$	0	0
$327$	−3.60688	−0.199461
$328$	0	0
$329$	9.95715	0.548956
$330$	0	0
$331$	3.80765	0.209288	0.104644	−	0.994510i	$-0.466630\pi$
$331$	3.80765	0.209288	0.104644	+	0.994510i	$0.466630\pi$
$332$	0	0
$333$	−1.43910	−0.0788620
$334$	0	0
$335$	0.585462	0.0319872
$336$	0	0
$337$	15.3717	0.837349	0.418675	−	0.908136i	$-0.362495\pi$
$337$	15.3717	0.837349	0.418675	+	0.908136i	$0.362495\pi$
$338$	0	0
$339$	2.96388	0.160976
$340$	0	0
$341$	0.978577	0.0529929
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.27131	0.176121
$346$	0	0
$347$	−3.02142	−0.162198	−0.0810992	−	0.996706i	$-0.525843\pi$
$347$	−3.02142	−0.162198	−0.0810992	+	0.996706i	$0.525843\pi$
$348$	0	0
$349$	11.0361	0.590750	0.295375	−	0.955381i	$-0.404555\pi$
$349$	11.0361	0.590750	0.295375	+	0.955381i	$0.404555\pi$
$350$	0	0
$351$	25.1709	1.34352
$352$	0	0
$353$	23.9817	1.27642	0.638209	−	0.769863i	$-0.279676\pi$
$353$	23.9817	1.27642	0.638209	+	0.769863i	$0.279676\pi$
$354$	0	0
$355$	0.335577	0.0178106
$356$	0	0
$357$	0.335577	0.0177606
$358$	0	0
$359$	10.5181	0.555122	0.277561	−	0.960708i	$-0.410474\pi$
$359$	10.5181	0.555122	0.277561	+	0.960708i	$0.410474\pi$
$360$	0	0
$361$	23.4851	1.23606
$362$	0	0
$363$	−1.14637	−0.0601686
$364$	0	0
$365$	−3.70727	−0.194047
$366$	0	0
$367$	35.5787	1.85719	0.928597	−	0.371089i	$-0.121015\pi$
$367$	35.5787	1.85719	0.928597	+	0.371089i	$0.121015\pi$
$368$	0	0
$369$	10.4949	0.546345
$370$	0	0
$371$	5.43910	0.282384
$372$	0	0
$373$	7.50650	0.388672	0.194336	−	0.980935i	$-0.437745\pi$
$373$	7.50650	0.388672	0.194336	+	0.980935i	$0.437745\pi$
$374$	0	0
$375$	−1.14637	−0.0591981
$376$	0	0
$377$	−6.74338	−0.347302
$378$	0	0
$379$	−33.4868	−1.72010	−0.860050	−	0.510210i	$-0.829568\pi$
$379$	−33.4868	−1.72010	−0.860050	+	0.510210i	$0.829568\pi$
$380$	0	0
$381$	−15.3288	−0.785321
$382$	0	0
$383$	−15.6644	−0.800415	−0.400207	−	0.916425i	$-0.631062\pi$
$383$	−15.6644	−0.800415	−0.400207	+	0.916425i	$0.631062\pi$
$384$	0	0
$385$	−1.00000	−0.0509647
$386$	0	0
$387$	−17.4490	−0.886981
$388$	0	0
$389$	−19.6216	−0.994853	−0.497427	−	0.867506i	$-0.665722\pi$
$389$	−19.6216	−0.994853	−0.497427	+	0.867506i	$0.665722\pi$
$390$	0	0
$391$	0.835347	0.0422453
$392$	0	0
$393$	−13.1134	−0.661483
$394$	0	0
$395$	2.51806	0.126697
$396$	0	0
$397$	10.2499	0.514427	0.257213	−	0.966355i	$-0.417196\pi$
$397$	10.2499	0.514427	0.257213	+	0.966355i	$0.417196\pi$
$398$	0	0
$399$	7.47208	0.374072
$400$	0	0
$401$	−16.8866	−0.843277	−0.421639	−	0.906764i	$-0.638545\pi$
$401$	−16.8866	−0.843277	−0.421639	+	0.906764i	$0.638545\pi$
$402$	0	0
$403$	−4.58546	−0.228418
$404$	0	0
$405$	−1.10038	−0.0546785
$406$	0	0
$407$	−0.853635	−0.0423131
$408$	0	0
$409$	18.2744	0.903613	0.451807	−	0.892116i	$-0.350780\pi$
$409$	18.2744	0.903613	0.451807	+	0.892116i	$0.350780\pi$
$410$	0	0
$411$	−11.4637	−0.565460
$412$	0	0
$413$	9.37169	0.461151
$414$	0	0
$415$	1.70727	0.0838065
$416$	0	0
$417$	−16.2583	−0.796173
$418$	0	0
$419$	37.2860	1.82154	0.910770	−	0.412914i	$-0.135489\pi$
$419$	37.2860	1.82154	0.910770	+	0.412914i	$0.135489\pi$
$420$	0	0
$421$	−2.78623	−0.135793	−0.0678963	−	0.997692i	$-0.521629\pi$
$421$	−2.78623	−0.135793	−0.0678963	+	0.997692i	$0.521629\pi$
$422$	0	0
$423$	−16.7862	−0.816174
$424$	0	0
$425$	−0.292731	−0.0141995
$426$	0	0
$427$	−11.9572	−0.578647
$428$	0	0
$429$	5.37169	0.259348
$430$	0	0
$431$	−38.0477	−1.83269	−0.916346	−	0.400387i	$-0.868876\pi$
$431$	−38.0477	−1.83269	−0.916346	+	0.400387i	$0.868876\pi$
$432$	0	0
$433$	−10.8108	−0.519533	−0.259767	−	0.965671i	$-0.583646\pi$
$433$	−10.8108	−0.519533	−0.259767	+	0.965671i	$0.583646\pi$
$434$	0	0
$435$	1.64973	0.0790985
$436$	0	0
$437$	18.6002	0.889766
$438$	0	0
$439$	−30.9933	−1.47923	−0.739614	−	0.673031i	$-0.764992\pi$
$439$	−30.9933	−1.47923	−0.739614	+	0.673031i	$0.764992\pi$
$440$	0	0
$441$	−1.68585	−0.0802784
$442$	0	0
$443$	−25.3717	−1.20545	−0.602723	−	0.797951i	$-0.705917\pi$
$443$	−25.3717	−1.20545	−0.602723	+	0.797951i	$0.705917\pi$
$444$	0	0
$445$	13.0790	0.620002
$446$	0	0
$447$	−13.2944	−0.628805
$448$	0	0
$449$	−0.886615	−0.0418419	−0.0209210	−	0.999781i	$-0.506660\pi$
$449$	−0.886615	−0.0418419	−0.0209210	+	0.999781i	$0.506660\pi$
$450$	0	0
$451$	6.22533	0.293139
$452$	0	0
$453$	1.98531	0.0932779
$454$	0	0
$455$	4.68585	0.219676
$456$	0	0
$457$	29.9143	1.39933	0.699666	−	0.714470i	$-0.253332\pi$
$457$	29.9143	1.39933	0.699666	+	0.714470i	$0.253332\pi$
$458$	0	0
$459$	−1.57246	−0.0733962
$460$	0	0
$461$	−2.33558	−0.108779	−0.0543893	−	0.998520i	$-0.517321\pi$
$461$	−2.33558	−0.108779	−0.0543893	+	0.998520i	$0.517321\pi$
$462$	0	0
$463$	−0.110250	−0.00512374	−0.00256187	−	0.999997i	$-0.500815\pi$
$463$	−0.110250	−0.00512374	−0.00256187	+	0.999997i	$0.500815\pi$
$464$	0	0
$465$	1.12181	0.0520226
$466$	0	0
$467$	−36.6760	−1.69716	−0.848581	−	0.529066i	$-0.822543\pi$
$467$	−36.6760	−1.69716	−0.848581	+	0.529066i	$0.822543\pi$
$468$	0	0
$469$	0.585462	0.0270341
$470$	0	0
$471$	14.0920	0.649323
$472$	0	0
$473$	−10.3503	−0.475906
$474$	0	0
$475$	−6.51806	−0.299069
$476$	0	0
$477$	−9.16948	−0.419842
$478$	0	0
$479$	42.7434	1.95300	0.976498	−	0.215529i	$-0.0691474\pi$
$479$	42.7434	1.95300	0.976498	+	0.215529i	$0.0691474\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	3.27131	0.148850
$484$	0	0
$485$	9.10352	0.413370
$486$	0	0
$487$	36.1396	1.63764	0.818822	−	0.574048i	$-0.194628\pi$
$487$	36.1396	1.63764	0.818822	+	0.574048i	$0.194628\pi$
$488$	0	0
$489$	27.4145	1.23973
$490$	0	0
$491$	−2.54262	−0.114747	−0.0573733	−	0.998353i	$-0.518273\pi$
$491$	−2.54262	−0.114747	−0.0573733	+	0.998353i	$0.518273\pi$
$492$	0	0
$493$	0.421268	0.0189730
$494$	0	0
$495$	1.68585	0.0757732
$496$	0	0
$497$	0.335577	0.0150527
$498$	0	0
$499$	−3.13650	−0.140409	−0.0702045	−	0.997533i	$-0.522365\pi$
$499$	−3.13650	−0.140409	−0.0702045	+	0.997533i	$0.522365\pi$
$500$	0	0
$501$	−9.17092	−0.409727
$502$	0	0
$503$	28.5855	1.27456	0.637281	−	0.770631i	$-0.280059\pi$
$503$	28.5855	1.27456	0.637281	+	0.770631i	$0.280059\pi$
$504$	0	0
$505$	−10.7862	−0.479981
$506$	0	0
$507$	−10.2682	−0.456026
$508$	0	0
$509$	2.20077	0.0975473	0.0487737	−	0.998810i	$-0.484469\pi$
$509$	2.20077	0.0975473	0.0487737	+	0.998810i	$0.484469\pi$
$510$	0	0
$511$	−3.70727	−0.164000
$512$	0	0
$513$	−35.0130	−1.54586
$514$	0	0
$515$	−7.66442	−0.337735
$516$	0	0
$517$	−9.95715	−0.437915
$518$	0	0
$519$	8.45065	0.370943
$520$	0	0
$521$	37.1940	1.62950	0.814750	−	0.579812i	$-0.196874\pi$
$521$	37.1940	1.62950	0.814750	+	0.579812i	$0.196874\pi$
$522$	0	0
$523$	−30.4078	−1.32964	−0.664820	−	0.747003i	$-0.731492\pi$
$523$	−30.4078	−1.32964	−0.664820	+	0.747003i	$0.731492\pi$
$524$	0	0
$525$	−1.14637	−0.0500315
$526$	0	0
$527$	0.286460	0.0124784
$528$	0	0
$529$	−14.8568	−0.645947
$530$	0	0
$531$	−15.7992	−0.685628
$532$	0	0
$533$	−29.1709	−1.26353
$534$	0	0
$535$	−9.56404	−0.413489
$536$	0	0
$537$	1.79296	0.0773720
$538$	0	0
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	39.0607	1.67935	0.839675	−	0.543090i	$-0.182746\pi$
$541$	39.0607	1.67935	0.839675	+	0.543090i	$0.182746\pi$
$542$	0	0
$543$	−16.9504	−0.727412
$544$	0	0
$545$	3.14637	0.134775
$546$	0	0
$547$	0.585462	0.0250325	0.0125163	−	0.999922i	$-0.496016\pi$
$547$	0.585462	0.0250325	0.0125163	+	0.999922i	$0.496016\pi$
$548$	0	0
$549$	20.1579	0.860319
$550$	0	0
$551$	9.38011	0.399606
$552$	0	0
$553$	2.51806	0.107079
$554$	0	0
$555$	−0.978577	−0.0415383
$556$	0	0
$557$	17.4637	0.739959	0.369979	−	0.929040i	$-0.379365\pi$
$557$	17.4637	0.739959	0.369979	+	0.929040i	$0.379365\pi$
$558$	0	0
$559$	48.4998	2.05132
$560$	0	0
$561$	−0.335577	−0.0141681
$562$	0	0
$563$	18.1579	0.765265	0.382633	−	0.923901i	$-0.375018\pi$
$563$	18.1579	0.765265	0.382633	+	0.923901i	$0.375018\pi$
$564$	0	0
$565$	−2.58546	−0.108771
$566$	0	0
$567$	−1.10038	−0.0462118
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	9.03612	0.378150	0.189075	−	0.981963i	$-0.439451\pi$
$571$	9.03612	0.378150	0.189075	+	0.981963i	$0.439451\pi$
$572$	0	0
$573$	−20.5855	−0.859970
$574$	0	0
$575$	−2.85363	−0.119005
$576$	0	0
$577$	19.3963	0.807476	0.403738	−	0.914875i	$-0.367711\pi$
$577$	19.3963	0.807476	0.403738	+	0.914875i	$0.367711\pi$
$578$	0	0
$579$	−21.3717	−0.888177
$580$	0	0
$581$	1.70727	0.0708295
$582$	0	0
$583$	−5.43910	−0.225264
$584$	0	0
$585$	−7.89962	−0.326609
$586$	0	0
$587$	−17.2614	−0.712456	−0.356228	−	0.934399i	$-0.615937\pi$
$587$	−17.2614	−0.712456	−0.356228	+	0.934399i	$0.615937\pi$
$588$	0	0
$589$	6.37842	0.262818
$590$	0	0
$591$	17.2369	0.709031
$592$	0	0
$593$	−11.0361	−0.453199	−0.226599	−	0.973988i	$-0.572761\pi$
$593$	−11.0361	−0.453199	−0.226599	+	0.973988i	$0.572761\pi$
$594$	0	0
$595$	−0.292731	−0.0120008
$596$	0	0
$597$	17.5725	0.719193
$598$	0	0
$599$	41.7367	1.70531	0.852657	−	0.522472i	$-0.174990\pi$
$599$	41.7367	1.70531	0.852657	+	0.522472i	$0.174990\pi$
$600$	0	0
$601$	39.3106	1.60351	0.801756	−	0.597652i	$-0.203900\pi$
$601$	39.3106	1.60351	0.801756	+	0.597652i	$0.203900\pi$
$602$	0	0
$603$	−0.986999	−0.0401937
$604$	0	0
$605$	1.00000	0.0406558
$606$	0	0
$607$	6.35027	0.257749	0.128875	−	0.991661i	$-0.458863\pi$
$607$	6.35027	0.257749	0.128875	+	0.991661i	$0.458863\pi$
$608$	0	0
$609$	1.64973	0.0668505
$610$	0	0
$611$	46.6577	1.88757
$612$	0	0
$613$	−42.4507	−1.71457	−0.857283	−	0.514846i	$-0.827849\pi$
$613$	−42.4507	−1.71457	−0.857283	+	0.514846i	$0.827849\pi$
$614$	0	0
$615$	7.13650	0.287771
$616$	0	0
$617$	−0.677425	−0.0272721	−0.0136360	−	0.999907i	$-0.504341\pi$
$617$	−0.677425	−0.0272721	−0.0136360	+	0.999907i	$0.504341\pi$
$618$	0	0
$619$	18.5426	0.745291	0.372645	−	0.927974i	$-0.378451\pi$
$619$	18.5426	0.745291	0.372645	+	0.927974i	$0.378451\pi$
$620$	0	0
$621$	−15.3288	−0.615125
$622$	0	0
$623$	13.0790	0.523998
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−7.47208	−0.298406
$628$	0	0
$629$	−0.249885	−0.00996358
$630$	0	0
$631$	−21.3717	−0.850794	−0.425397	−	0.905007i	$-0.639865\pi$
$631$	−21.3717	−0.850794	−0.425397	+	0.905007i	$0.639865\pi$
$632$	0	0
$633$	2.34185	0.0930801
$634$	0	0
$635$	13.3717	0.530639
$636$	0	0
$637$	4.68585	0.185660
$638$	0	0
$639$	−0.565731	−0.0223800
$640$	0	0
$641$	23.8715	0.942866	0.471433	−	0.881902i	$-0.343737\pi$
$641$	23.8715	0.942866	0.471433	+	0.881902i	$0.343737\pi$
$642$	0	0
$643$	−0.311018	−0.0122654	−0.00613268	−	0.999981i	$-0.501952\pi$
$643$	−0.311018	−0.0122654	−0.00613268	+	0.999981i	$0.501952\pi$
$644$	0	0
$645$	−11.8652	−0.467191
$646$	0	0
$647$	9.62158	0.378263	0.189132	−	0.981952i	$-0.439433\pi$
$647$	9.62158	0.378263	0.189132	+	0.981952i	$0.439433\pi$
$648$	0	0
$649$	−9.37169	−0.367871
$650$	0	0
$651$	1.12181	0.0439671
$652$	0	0
$653$	−27.0607	−1.05897	−0.529483	−	0.848321i	$-0.677614\pi$
$653$	−27.0607	−1.05897	−0.529483	+	0.848321i	$0.677614\pi$
$654$	0	0
$655$	11.4391	0.446962
$656$	0	0
$657$	6.24989	0.243831
$658$	0	0
$659$	26.4935	1.03204	0.516020	−	0.856576i	$-0.327413\pi$
$659$	26.4935	1.03204	0.516020	+	0.856576i	$0.327413\pi$
$660$	0	0
$661$	35.0852	1.36466	0.682329	−	0.731046i	$-0.260967\pi$
$661$	35.0852	1.36466	0.682329	+	0.731046i	$0.260967\pi$
$662$	0	0
$663$	1.57246	0.0610693
$664$	0	0
$665$	−6.51806	−0.252759
$666$	0	0
$667$	4.10666	0.159010
$668$	0	0
$669$	12.7005	0.491031
$670$	0	0
$671$	11.9572	0.461601
$672$	0	0
$673$	32.6577	1.25886	0.629431	−	0.777057i	$-0.283288\pi$
$673$	32.6577	1.25886	0.629431	+	0.777057i	$0.283288\pi$
$674$	0	0
$675$	5.37169	0.206757
$676$	0	0
$677$	9.12808	0.350821	0.175410	−	0.984495i	$-0.443875\pi$
$677$	9.12808	0.350821	0.175410	+	0.984495i	$0.443875\pi$
$678$	0	0
$679$	9.10352	0.349361
$680$	0	0
$681$	−21.2566	−0.814555
$682$	0	0
$683$	−10.4935	−0.401523	−0.200761	−	0.979640i	$-0.564342\pi$
$683$	−10.4935	−0.401523	−0.200761	+	0.979640i	$0.564342\pi$
$684$	0	0
$685$	10.0000	0.382080
$686$	0	0
$687$	9.18777	0.350535
$688$	0	0
$689$	25.4868	0.970969
$690$	0	0
$691$	−1.70727	−0.0649476	−0.0324738	−	0.999473i	$-0.510339\pi$
$691$	−1.70727	−0.0649476	−0.0324738	+	0.999473i	$0.510339\pi$
$692$	0	0
$693$	1.68585	0.0640400
$694$	0	0
$695$	14.1825	0.537972
$696$	0	0
$697$	1.82235	0.0690263
$698$	0	0
$699$	−32.0491	−1.21221
$700$	0	0
$701$	−37.3534	−1.41082	−0.705409	−	0.708800i	$-0.749237\pi$
$701$	−37.3534	−1.41082	−0.705409	+	0.708800i	$0.749237\pi$
$702$	0	0
$703$	−5.56404	−0.209852
$704$	0	0
$705$	−11.4145	−0.429896
$706$	0	0
$707$	−10.7862	−0.405658
$708$	0	0
$709$	−8.20704	−0.308222	−0.154111	−	0.988054i	$-0.549251\pi$
$709$	−8.20704	−0.308222	−0.154111	+	0.988054i	$0.549251\pi$
$710$	0	0
$711$	−4.24506	−0.159202
$712$	0	0
$713$	2.79250	0.104580
$714$	0	0
$715$	−4.68585	−0.175241
$716$	0	0
$717$	−33.0277	−1.23344
$718$	0	0
$719$	8.00000	0.298350	0.149175	−	0.988811i	$-0.452338\pi$
$719$	8.00000	0.298350	0.149175	+	0.988811i	$0.452338\pi$
$720$	0	0
$721$	−7.66442	−0.285438
$722$	0	0
$723$	9.76481	0.363157
$724$	0	0
$725$	−1.43910	−0.0534467
$726$	0	0
$727$	46.9442	1.74106	0.870531	−	0.492113i	$-0.163775\pi$
$727$	46.9442	1.74106	0.870531	+	0.492113i	$0.163775\pi$
$728$	0	0
$729$	20.3288	0.752920
$730$	0	0
$731$	−3.02984	−0.112063
$732$	0	0
$733$	−42.0000	−1.55131	−0.775653	−	0.631160i	$-0.782579\pi$
$733$	−42.0000	−1.55131	−0.775653	+	0.631160i	$0.782579\pi$
$734$	0	0
$735$	−1.14637	−0.0422843
$736$	0	0
$737$	−0.585462	−0.0215658
$738$	0	0
$739$	0.871922	0.0320742	0.0160371	−	0.999871i	$-0.494895\pi$
$739$	0.871922	0.0320742	0.0160371	+	0.999871i	$0.494895\pi$
$740$	0	0
$741$	35.0130	1.28623
$742$	0	0
$743$	24.6712	0.905097	0.452548	−	0.891740i	$-0.350515\pi$
$743$	24.6712	0.905097	0.452548	+	0.891740i	$0.350515\pi$
$744$	0	0
$745$	11.5970	0.424882
$746$	0	0
$747$	−2.87819	−0.105308
$748$	0	0
$749$	−9.56404	−0.349462
$750$	0	0
$751$	3.75011	0.136844	0.0684218	−	0.997656i	$-0.478204\pi$
$751$	3.75011	0.136844	0.0684218	+	0.997656i	$0.478204\pi$
$752$	0	0
$753$	−26.5132	−0.966196
$754$	0	0
$755$	−1.73183	−0.0630277
$756$	0	0
$757$	15.7318	0.571783	0.285891	−	0.958262i	$-0.407710\pi$
$757$	15.7318	0.571783	0.285891	+	0.958262i	$0.407710\pi$
$758$	0	0
$759$	−3.27131	−0.118741
$760$	0	0
$761$	15.2614	0.553227	0.276613	−	0.960981i	$-0.410788\pi$
$761$	15.2614	0.553227	0.276613	+	0.960981i	$0.410788\pi$
$762$	0	0
$763$	3.14637	0.113906
$764$	0	0
$765$	0.493499	0.0178425
$766$	0	0
$767$	43.9143	1.58565
$768$	0	0
$769$	−9.63986	−0.347622	−0.173811	−	0.984779i	$-0.555608\pi$
$769$	−9.63986	−0.347622	−0.173811	+	0.984779i	$0.555608\pi$
$770$	0	0
$771$	−16.9786	−0.611469
$772$	0	0
$773$	12.8291	0.461430	0.230715	−	0.973021i	$-0.425894\pi$
$773$	12.8291	0.461430	0.230715	+	0.973021i	$0.425894\pi$
$774$	0	0
$775$	−0.978577	−0.0351515
$776$	0	0
$777$	−0.978577	−0.0351063
$778$	0	0
$779$	40.5770	1.45382
$780$	0	0
$781$	−0.335577	−0.0120079
$782$	0	0
$783$	−7.73038	−0.276261
$784$	0	0
$785$	−12.2927	−0.438746
$786$	0	0
$787$	−10.8291	−0.386015	−0.193007	−	0.981197i	$-0.561824\pi$
$787$	−10.8291	−0.386015	−0.193007	+	0.981197i	$0.561824\pi$
$788$	0	0
$789$	21.4868	0.764949
$790$	0	0
$791$	−2.58546	−0.0919284
$792$	0	0
$793$	−56.0294	−1.98966
$794$	0	0
$795$	−6.23519	−0.221139
$796$	0	0
$797$	−21.0790	−0.746655	−0.373328	−	0.927700i	$-0.621783\pi$
$797$	−21.0790	−0.746655	−0.373328	+	0.927700i	$0.621783\pi$
$798$	0	0
$799$	−2.91477	−0.103117
$800$	0	0
$801$	−22.0491	−0.779067
$802$	0	0
$803$	3.70727	0.130827
$804$	0	0
$805$	−2.85363	−0.100577
$806$	0	0
$807$	−14.5426	−0.511924
$808$	0	0
$809$	3.62158	0.127328	0.0636639	−	0.997971i	$-0.479721\pi$
$809$	3.62158	0.127328	0.0636639	+	0.997971i	$0.479721\pi$
$810$	0	0
$811$	−34.7188	−1.21914	−0.609571	−	0.792731i	$-0.708658\pi$
$811$	−34.7188	−1.21914	−0.609571	+	0.792731i	$0.708658\pi$
$812$	0	0
$813$	1.28600	0.0451020
$814$	0	0
$815$	−23.9143	−0.837682
$816$	0	0
$817$	−67.4637	−2.36025
$818$	0	0
$819$	−7.89962	−0.276035
$820$	0	0
$821$	−37.7549	−1.31766	−0.658828	−	0.752293i	$-0.728948\pi$
$821$	−37.7549	−1.31766	−0.658828	+	0.752293i	$0.728948\pi$
$822$	0	0
$823$	−11.1892	−0.390031	−0.195016	−	0.980800i	$-0.562476\pi$
$823$	−11.1892	−0.390031	−0.195016	+	0.980800i	$0.562476\pi$
$824$	0	0
$825$	1.14637	0.0399113
$826$	0	0
$827$	7.91431	0.275207	0.137604	−	0.990487i	$-0.456060\pi$
$827$	7.91431	0.275207	0.137604	+	0.990487i	$0.456060\pi$
$828$	0	0
$829$	−44.7152	−1.55302	−0.776512	−	0.630102i	$-0.783013\pi$
$829$	−44.7152	−1.55302	−0.776512	+	0.630102i	$0.783013\pi$
$830$	0	0
$831$	−34.5229	−1.19759
$832$	0	0
$833$	−0.292731	−0.0101425
$834$	0	0
$835$	8.00000	0.276851
$836$	0	0
$837$	−5.25662	−0.181695
$838$	0	0
$839$	−21.6791	−0.748446	−0.374223	−	0.927339i	$-0.622091\pi$
$839$	−21.6791	−0.748446	−0.374223	+	0.927339i	$0.622091\pi$
$840$	0	0
$841$	−26.9290	−0.928586
$842$	0	0
$843$	16.4338	0.566010
$844$	0	0
$845$	8.95715	0.308135
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	3.68415	0.126440
$850$	0	0
$851$	−2.43596	−0.0835037
$852$	0	0
$853$	9.41454	0.322348	0.161174	−	0.986926i	$-0.448472\pi$
$853$	9.41454	0.322348	0.161174	+	0.986926i	$0.448472\pi$
$854$	0	0
$855$	10.9884	0.375797
$856$	0	0
$857$	−25.8652	−0.883538	−0.441769	−	0.897129i	$-0.645649\pi$
$857$	−25.8652	−0.883538	−0.441769	+	0.897129i	$0.645649\pi$
$858$	0	0
$859$	29.2860	0.999225	0.499613	−	0.866249i	$-0.333476\pi$
$859$	29.2860	0.999225	0.499613	+	0.866249i	$0.333476\pi$
$860$	0	0
$861$	7.13650	0.243211
$862$	0	0
$863$	20.3110	0.691395	0.345698	−	0.938346i	$-0.387642\pi$
$863$	20.3110	0.691395	0.345698	+	0.938346i	$0.387642\pi$
$864$	0	0
$865$	−7.37169	−0.250645
$866$	0	0
$867$	19.3900	0.658518
$868$	0	0
$869$	−2.51806	−0.0854193
$870$	0	0
$871$	2.74338	0.0929560
$872$	0	0
$873$	−15.3471	−0.519422
$874$	0	0
$875$	1.00000	0.0338062
$876$	0	0
$877$	−38.3650	−1.29549	−0.647746	−	0.761856i	$-0.724288\pi$
$877$	−38.3650	−1.29549	−0.647746	+	0.761856i	$0.724288\pi$
$878$	0	0
$879$	19.1281	0.645174
$880$	0	0
$881$	40.5426	1.36592	0.682958	−	0.730458i	$-0.260693\pi$
$881$	40.5426	1.36592	0.682958	+	0.730458i	$0.260693\pi$
$882$	0	0
$883$	−28.0491	−0.943928	−0.471964	−	0.881618i	$-0.656455\pi$
$883$	−28.0491	−0.943928	−0.471964	+	0.881618i	$0.656455\pi$
$884$	0	0
$885$	−10.7434	−0.361135
$886$	0	0
$887$	6.65769	0.223543	0.111772	−	0.993734i	$-0.464347\pi$
$887$	6.65769	0.223543	0.111772	+	0.993734i	$0.464347\pi$
$888$	0	0
$889$	13.3717	0.448472
$890$	0	0
$891$	1.10038	0.0368643
$892$	0	0
$893$	−64.9013	−2.17184
$894$	0	0
$895$	−1.56404	−0.0522801
$896$	0	0
$897$	15.3288	0.511815
$898$	0	0
$899$	1.40827	0.0469683
$900$	0	0
$901$	−1.59219	−0.0530436
$902$	0	0
$903$	−11.8652	−0.394849
$904$	0	0
$905$	14.7862	0.491511
$906$	0	0
$907$	−39.5787	−1.31419	−0.657095	−	0.753808i	$-0.728215\pi$
$907$	−39.5787	−1.31419	−0.657095	+	0.753808i	$0.728215\pi$
$908$	0	0
$909$	18.1839	0.603123
$910$	0	0
$911$	24.3356	0.806274	0.403137	−	0.915140i	$-0.367920\pi$
$911$	24.3356	0.806274	0.403137	+	0.915140i	$0.367920\pi$
$912$	0	0
$913$	−1.70727	−0.0565024
$914$	0	0
$915$	13.7073	0.453148
$916$	0	0
$917$	11.4391	0.377752
$918$	0	0
$919$	−50.6331	−1.67023	−0.835117	−	0.550073i	$-0.814600\pi$
$919$	−50.6331	−1.67023	−0.835117	+	0.550073i	$0.814600\pi$
$920$	0	0
$921$	−7.21377	−0.237702
$922$	0	0
$923$	1.57246	0.0517582
$924$	0	0
$925$	0.853635	0.0280673
$926$	0	0
$927$	12.9210	0.424383
$928$	0	0
$929$	47.1512	1.54698	0.773490	−	0.633808i	$-0.218509\pi$
$929$	47.1512	1.54698	0.773490	+	0.633808i	$0.218509\pi$
$930$	0	0
$931$	−6.51806	−0.213621
$932$	0	0
$933$	−23.3780	−0.765360
$934$	0	0
$935$	0.292731	0.00957333
$936$	0	0
$937$	−42.0294	−1.37304	−0.686520	−	0.727111i	$-0.740863\pi$
$937$	−42.0294	−1.37304	−0.686520	+	0.727111i	$0.740863\pi$
$938$	0	0
$939$	−13.8337	−0.451444
$940$	0	0
$941$	−36.7434	−1.19780	−0.598900	−	0.800824i	$-0.704395\pi$
$941$	−36.7434	−1.19780	−0.598900	+	0.800824i	$0.704395\pi$
$942$	0	0
$943$	17.7648	0.578502
$944$	0	0
$945$	5.37169	0.174741
$946$	0	0
$947$	−18.8782	−0.613459	−0.306729	−	0.951797i	$-0.599235\pi$
$947$	−18.8782	−0.613459	−0.306729	+	0.951797i	$0.599235\pi$
$948$	0	0
$949$	−17.3717	−0.563909
$950$	0	0
$951$	−32.9786	−1.06940
$952$	0	0
$953$	−43.2285	−1.40031	−0.700154	−	0.713992i	$-0.746885\pi$
$953$	−43.2285	−1.40031	−0.700154	+	0.713992i	$0.746885\pi$
$954$	0	0
$955$	17.9572	0.581080
$956$	0	0
$957$	−1.64973	−0.0533282
$958$	0	0
$959$	10.0000	0.322917
$960$	0	0
$961$	−30.0424	−0.969109
$962$	0	0
$963$	16.1235	0.519572
$964$	0	0
$965$	18.6430	0.600139
$966$	0	0
$967$	−48.7299	−1.56705	−0.783524	−	0.621361i	$-0.786580\pi$
$967$	−48.7299	−1.56705	−0.783524	+	0.621361i	$0.786580\pi$
$968$	0	0
$969$	−2.18731	−0.0702665
$970$	0	0
$971$	25.9865	0.833948	0.416974	−	0.908918i	$-0.363091\pi$
$971$	25.9865	0.833948	0.416974	+	0.908918i	$0.363091\pi$
$972$	0	0
$973$	14.1825	0.454669
$974$	0	0
$975$	−5.37169	−0.172032
$976$	0	0
$977$	2.67115	0.0854578	0.0427289	−	0.999087i	$-0.486395\pi$
$977$	2.67115	0.0854578	0.0427289	+	0.999087i	$0.486395\pi$
$978$	0	0
$979$	−13.0790	−0.418005
$980$	0	0
$981$	−5.30429	−0.169353
$982$	0	0
$983$	8.33558	0.265864	0.132932	−	0.991125i	$-0.457561\pi$
$983$	8.33558	0.265864	0.132932	+	0.991125i	$0.457561\pi$
$984$	0	0
$985$	−15.0361	−0.479090
$986$	0	0
$987$	−11.4145	−0.363329
$988$	0	0
$989$	−29.5359	−0.939187
$990$	0	0
$991$	−46.2730	−1.46991	−0.734955	−	0.678116i	$-0.762797\pi$
$991$	−46.2730	−1.46991	−0.734955	+	0.678116i	$0.762797\pi$
$992$	0	0
$993$	−4.36496	−0.138518
$994$	0	0
$995$	−15.3288	−0.485957
$996$	0	0
$997$	35.8715	1.13606	0.568030	−	0.823008i	$-0.307706\pi$
$997$	35.8715	1.13606	0.568030	+	0.823008i	$0.307706\pi$
$998$	0	0
$999$	4.58546	0.145078

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6160.2.a.bf.1.2		3
4.3	odd	2		770.2.a.m.1.2	✓	3
12.11	even	2		6930.2.a.ce.1.3		3
20.3	even	4		3850.2.c.ba.1849.2		6
20.7	even	4		3850.2.c.ba.1849.5		6
20.19	odd	2		3850.2.a.bt.1.2		3
28.27	even	2		5390.2.a.ca.1.2		3
44.43	even	2		8470.2.a.ci.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
770.2.a.m.1.2	✓	3	4.3	odd	2
3850.2.a.bt.1.2		3	20.19	odd	2
3850.2.c.ba.1849.2		6	20.3	even	4
3850.2.c.ba.1849.5		6	20.7	even	4
5390.2.a.ca.1.2		3	28.27	even	2
6160.2.a.bf.1.2		3	1.1	even	1	trivial
6930.2.a.ce.1.3		3	12.11	even	2
8470.2.a.ci.1.2		3	44.43	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:	\( N \)	\(=\)	\( 6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6160.a (trivial)

\(f(q)\)	\(=\)	\(q-1.14637 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.68585 q^{9} +O(q^{10})\)	\(q-1.14637 q^{3} +1.00000 q^{5} +1.00000 q^{7} -1.68585 q^{9} -1.00000 q^{11} +4.68585 q^{13} -1.14637 q^{15} -0.292731 q^{17} -6.51806 q^{19} -1.14637 q^{21} -2.85363 q^{23} +1.00000 q^{25} +5.37169 q^{27} -1.43910 q^{29} -0.978577 q^{31} +1.14637 q^{33} +1.00000 q^{35} +0.853635 q^{37} -5.37169 q^{39} -6.22533 q^{41} +10.3503 q^{43} -1.68585 q^{45} +9.95715 q^{47} +1.00000 q^{49} +0.335577 q^{51} +5.43910 q^{53} -1.00000 q^{55} +7.47208 q^{57} +9.37169 q^{59} -11.9572 q^{61} -1.68585 q^{63} +4.68585 q^{65} +0.585462 q^{67} +3.27131 q^{69} +0.335577 q^{71} -3.70727 q^{73} -1.14637 q^{75} -1.00000 q^{77} +2.51806 q^{79} -1.10038 q^{81} +1.70727 q^{83} -0.292731 q^{85} +1.64973 q^{87} +13.0790 q^{89} +4.68585 q^{91} +1.12181 q^{93} -6.51806 q^{95} +9.10352 q^{97} +1.68585 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9}+O(q^{10}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9	\( 3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99}+O(q^{100}) \) 3 * q - 2 * q^3 + 3 * q^5 + 3 * q^7 + 7 * q^9 - 3 * q^11 + 2 * q^13 - 2 * q^15 + 2 * q^17 + 6 * q^19 - 2 * q^21 - 10 * q^23 + 3 * q^25 - 8 * q^27 + 12 * q^31 + 2 * q^33 + 3 * q^35 + 4 * q^37 + 8 * q^39 + 4 * q^41 - 8 * q^43 + 7 * q^45 + 3 * q^49 + 28 * q^51 + 12 * q^53 - 3 * q^55 - 8 * q^57 + 4 * q^59 - 6 * q^61 + 7 * q^63 + 2 * q^65 - 4 * q^67 - 8 * q^69 + 28 * q^71 - 14 * q^73 - 2 * q^75 - 3 * q^77 - 18 * q^79 + 3 * q^81 + 8 * q^83 + 2 * q^85 + 44 * q^87 + 18 * q^89 + 2 * q^91 + 12 * q^93 + 6 * q^95 - 4 * q^97 - 7 * q^99

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