LMFDB - Embedded newform 4160.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.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:	$33.2177672409$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{3} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{9} +O(q^{10})$	$q+1.41421 q^{3} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{9} -3.41421 q^{11} +1.00000 q^{13} -1.41421 q^{15} +0.828427 q^{17} -0.585786 q^{19} +6.82843 q^{21} +1.41421 q^{23} +1.00000 q^{25} -5.65685 q^{27} +5.65685 q^{29} +1.75736 q^{31} -4.82843 q^{33} -4.82843 q^{35} +8.48528 q^{37} +1.41421 q^{39} -3.17157 q^{41} +11.0711 q^{43} +1.00000 q^{45} -4.82843 q^{47} +16.3137 q^{49} +1.17157 q^{51} -2.48528 q^{53} +3.41421 q^{55} -0.828427 q^{57} -1.75736 q^{59} +8.00000 q^{61} -4.82843 q^{63} -1.00000 q^{65} +2.00000 q^{67} +2.00000 q^{69} +11.8995 q^{71} +8.48528 q^{73} +1.41421 q^{75} -16.4853 q^{77} -8.48528 q^{79} -5.00000 q^{81} +3.17157 q^{83} -0.828427 q^{85} +8.00000 q^{87} +6.00000 q^{89} +4.82843 q^{91} +2.48528 q^{93} +0.585786 q^{95} -7.65685 q^{97} +3.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 + 4 * q^7 - 2 * q^9	$ 2 q - 2 q^{5} + 4 q^{7} - 2 q^{9} - 4 q^{11} + 2 q^{13} - 4 q^{17} - 4 q^{19} + 8 q^{21} + 2 q^{25} + 12 q^{31} - 4 q^{33} - 4 q^{35} - 12 q^{41} + 8 q^{43} + 2 q^{45} - 4 q^{47} + 10 q^{49} + 8 q^{51} + 12 q^{53} + 4 q^{55} + 4 q^{57} - 12 q^{59} + 16 q^{61} - 4 q^{63} - 2 q^{65} + 4 q^{67} + 4 q^{69} + 4 q^{71} - 16 q^{77} - 10 q^{81} + 12 q^{83} + 4 q^{85} + 16 q^{87} + 12 q^{89} + 4 q^{91} - 12 q^{93} + 4 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 + 4 * q^7 - 2 * q^9 - 4 * q^11 + 2 * q^13 - 4 * q^17 - 4 * q^19 + 8 * q^21 + 2 * q^25 + 12 * q^31 - 4 * q^33 - 4 * q^35 - 12 * q^41 + 8 * q^43 + 2 * q^45 - 4 * q^47 + 10 * q^49 + 8 * q^51 + 12 * q^53 + 4 * q^55 + 4 * q^57 - 12 * q^59 + 16 * q^61 - 4 * q^63 - 2 * q^65 + 4 * q^67 + 4 * q^69 + 4 * q^71 - 16 * q^77 - 10 * q^81 + 12 * q^83 + 4 * q^85 + 16 * q^87 + 12 * q^89 + 4 * q^91 - 12 * q^93 + 4 * 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
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.82843	1.82497	0.912487	−	0.409106i	$-0.134159\pi$
$7$	4.82843	1.82497	0.912487	+	0.409106i	$0.134159\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−3.41421	−1.02942	−0.514712	−	0.857363i	$-0.672101\pi$
$11$	−3.41421	−1.02942	−0.514712	+	0.857363i	$0.672101\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.41421	−0.365148
$16$	0	0
$17$	0.828427	0.200923	0.100462	−	0.994941i	$-0.467968\pi$
$17$	0.828427	0.200923	0.100462	+	0.994941i	$0.467968\pi$
$18$	0	0
$19$	−0.585786	−0.134389	−0.0671943	−	0.997740i	$-0.521405\pi$
$19$	−0.585786	−0.134389	−0.0671943	+	0.997740i	$0.521405\pi$
$20$	0	0
$21$	6.82843	1.49008
$22$	0	0
$23$	1.41421	0.294884	0.147442	−	0.989071i	$-0.452896\pi$
$23$	1.41421	0.294884	0.147442	+	0.989071i	$0.452896\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0	0
$31$	1.75736	0.315631	0.157816	−	0.987469i	$-0.449555\pi$
$31$	1.75736	0.315631	0.157816	+	0.987469i	$0.449555\pi$
$32$	0	0
$33$	−4.82843	−0.840521
$34$	0	0
$35$	−4.82843	−0.816153
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	0	0
$39$	1.41421	0.226455
$40$	0	0
$41$	−3.17157	−0.495316	−0.247658	−	0.968847i	$-0.579661\pi$
$41$	−3.17157	−0.495316	−0.247658	+	0.968847i	$0.579661\pi$
$42$	0	0
$43$	11.0711	1.68832	0.844161	−	0.536090i	$-0.180099\pi$
$43$	11.0711	1.68832	0.844161	+	0.536090i	$0.180099\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−4.82843	−0.704298	−0.352149	−	0.935944i	$-0.614549\pi$
$47$	−4.82843	−0.704298	−0.352149	+	0.935944i	$0.614549\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	0	0
$51$	1.17157	0.164053
$52$	0	0
$53$	−2.48528	−0.341380	−0.170690	−	0.985325i	$-0.554600\pi$
$53$	−2.48528	−0.341380	−0.170690	+	0.985325i	$0.554600\pi$
$54$	0	0
$55$	3.41421	0.460372
$56$	0	0
$57$	−0.828427	−0.109728
$58$	0	0
$59$	−1.75736	−0.228789	−0.114394	−	0.993435i	$-0.536493\pi$
$59$	−1.75736	−0.228789	−0.114394	+	0.993435i	$0.536493\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	−4.82843	−0.608325
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	2.00000	0.240772
$70$	0	0
$71$	11.8995	1.41221	0.706105	−	0.708107i	$-0.250451\pi$
$71$	11.8995	1.41221	0.706105	+	0.708107i	$0.250451\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	0	0
$75$	1.41421	0.163299
$76$	0	0
$77$	−16.4853	−1.87867
$78$	0	0
$79$	−8.48528	−0.954669	−0.477334	−	0.878722i	$-0.658397\pi$
$79$	−8.48528	−0.954669	−0.477334	+	0.878722i	$0.658397\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	3.17157	0.348125	0.174063	−	0.984735i	$-0.444310\pi$
$83$	3.17157	0.348125	0.174063	+	0.984735i	$0.444310\pi$
$84$	0	0
$85$	−0.828427	−0.0898555
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	4.82843	0.506157
$92$	0	0
$93$	2.48528	0.257712
$94$	0	0
$95$	0.585786	0.0601004
$96$	0	0
$97$	−7.65685	−0.777436	−0.388718	−	0.921357i	$-0.627082\pi$
$97$	−7.65685	−0.777436	−0.388718	+	0.921357i	$0.627082\pi$
$98$	0	0
$99$	3.41421	0.343141
$100$	0	0
$101$	3.65685	0.363871	0.181935	−	0.983311i	$-0.441764\pi$
$101$	3.65685	0.363871	0.181935	+	0.983311i	$0.441764\pi$
$102$	0	0
$103$	14.5858	1.43718	0.718590	−	0.695434i	$-0.244788\pi$
$103$	14.5858	1.43718	0.718590	+	0.695434i	$0.244788\pi$
$104$	0	0
$105$	−6.82843	−0.666386
$106$	0	0
$107$	9.41421	0.910106	0.455053	−	0.890464i	$-0.349620\pi$
$107$	9.41421	0.910106	0.455053	+	0.890464i	$0.349620\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	12.0000	1.13899
$112$	0	0
$113$	−8.82843	−0.830509	−0.415254	−	0.909705i	$-0.636307\pi$
$113$	−8.82843	−0.830509	−0.415254	+	0.909705i	$0.636307\pi$
$114$	0	0
$115$	−1.41421	−0.131876
$116$	0	0
$117$	−1.00000	−0.0924500
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	0.656854	0.0597140
$122$	0	0
$123$	−4.48528	−0.404424
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.58579	−0.584394	−0.292197	−	0.956358i	$-0.594386\pi$
$127$	−6.58579	−0.584394	−0.292197	+	0.956358i	$0.594386\pi$
$128$	0	0
$129$	15.6569	1.37851
$130$	0	0
$131$	16.9706	1.48272	0.741362	−	0.671105i	$-0.234180\pi$
$131$	16.9706	1.48272	0.741362	+	0.671105i	$0.234180\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	5.65685	0.486864
$136$	0	0
$137$	−17.3137	−1.47921	−0.739605	−	0.673041i	$-0.764988\pi$
$137$	−17.3137	−1.47921	−0.739605	+	0.673041i	$0.764988\pi$
$138$	0	0
$139$	−4.48528	−0.380437	−0.190218	−	0.981742i	$-0.560920\pi$
$139$	−4.48528	−0.380437	−0.190218	+	0.981742i	$0.560920\pi$
$140$	0	0
$141$	−6.82843	−0.575057
$142$	0	0
$143$	−3.41421	−0.285511
$144$	0	0
$145$	−5.65685	−0.469776
$146$	0	0
$147$	23.0711	1.90287
$148$	0	0
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	9.75736	0.794043	0.397021	−	0.917809i	$-0.370044\pi$
$151$	9.75736	0.794043	0.397021	+	0.917809i	$0.370044\pi$
$152$	0	0
$153$	−0.828427	−0.0669744
$154$	0	0
$155$	−1.75736	−0.141154
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	−3.51472	−0.278735
$160$	0	0
$161$	6.82843	0.538155
$162$	0	0
$163$	−18.9706	−1.48589	−0.742945	−	0.669353i	$-0.766571\pi$
$163$	−18.9706	−1.48589	−0.742945	+	0.669353i	$0.766571\pi$
$164$	0	0
$165$	4.82843	0.375893
$166$	0	0
$167$	−3.17157	−0.245424	−0.122712	−	0.992442i	$-0.539159\pi$
$167$	−3.17157	−0.245424	−0.122712	+	0.992442i	$0.539159\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.585786	0.0447962
$172$	0	0
$173$	−16.8284	−1.27944	−0.639721	−	0.768607i	$-0.720950\pi$
$173$	−16.8284	−1.27944	−0.639721	+	0.768607i	$0.720950\pi$
$174$	0	0
$175$	4.82843	0.364995
$176$	0	0
$177$	−2.48528	−0.186805
$178$	0	0
$179$	−5.65685	−0.422813	−0.211407	−	0.977398i	$-0.567804\pi$
$179$	−5.65685	−0.422813	−0.211407	+	0.977398i	$0.567804\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	0	0
$183$	11.3137	0.836333
$184$	0	0
$185$	−8.48528	−0.623850
$186$	0	0
$187$	−2.82843	−0.206835
$188$	0	0
$189$	−27.3137	−1.98678
$190$	0	0
$191$	2.34315	0.169544	0.0847720	−	0.996400i	$-0.472984\pi$
$191$	2.34315	0.169544	0.0847720	+	0.996400i	$0.472984\pi$
$192$	0	0
$193$	4.34315	0.312626	0.156313	−	0.987708i	$-0.450039\pi$
$193$	4.34315	0.312626	0.156313	+	0.987708i	$0.450039\pi$
$194$	0	0
$195$	−1.41421	−0.101274
$196$	0	0
$197$	−10.9706	−0.781620	−0.390810	−	0.920471i	$-0.627805\pi$
$197$	−10.9706	−0.781620	−0.390810	+	0.920471i	$0.627805\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	2.82843	0.199502
$202$	0	0
$203$	27.3137	1.91705
$204$	0	0
$205$	3.17157	0.221512
$206$	0	0
$207$	−1.41421	−0.0982946
$208$	0	0
$209$	2.00000	0.138343
$210$	0	0
$211$	−3.31371	−0.228125	−0.114063	−	0.993474i	$-0.536386\pi$
$211$	−3.31371	−0.228125	−0.114063	+	0.993474i	$0.536386\pi$
$212$	0	0
$213$	16.8284	1.15306
$214$	0	0
$215$	−11.0711	−0.755041
$216$	0	0
$217$	8.48528	0.576018
$218$	0	0
$219$	12.0000	0.810885
$220$	0	0
$221$	0.828427	0.0557260
$222$	0	0
$223$	9.51472	0.637153	0.318576	−	0.947897i	$-0.396795\pi$
$223$	9.51472	0.637153	0.318576	+	0.947897i	$0.396795\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	0	0
$227$	16.3431	1.08473	0.542366	−	0.840142i	$-0.317528\pi$
$227$	16.3431	1.08473	0.542366	+	0.840142i	$0.317528\pi$
$228$	0	0
$229$	4.82843	0.319071	0.159536	−	0.987192i	$-0.449000\pi$
$229$	4.82843	0.319071	0.159536	+	0.987192i	$0.449000\pi$
$230$	0	0
$231$	−23.3137	−1.53393
$232$	0	0
$233$	−20.6274	−1.35135	−0.675674	−	0.737201i	$-0.736147\pi$
$233$	−20.6274	−1.35135	−0.675674	+	0.737201i	$0.736147\pi$
$234$	0	0
$235$	4.82843	0.314972
$236$	0	0
$237$	−12.0000	−0.779484
$238$	0	0
$239$	−3.41421	−0.220847	−0.110424	−	0.993885i	$-0.535221\pi$
$239$	−3.41421	−0.220847	−0.110424	+	0.993885i	$0.535221\pi$
$240$	0	0
$241$	−14.4853	−0.933079	−0.466539	−	0.884500i	$-0.654499\pi$
$241$	−14.4853	−0.933079	−0.466539	+	0.884500i	$0.654499\pi$
$242$	0	0
$243$	9.89949	0.635053
$244$	0	0
$245$	−16.3137	−1.04224
$246$	0	0
$247$	−0.585786	−0.0372727
$248$	0	0
$249$	4.48528	0.284243
$250$	0	0
$251$	−19.7990	−1.24970	−0.624851	−	0.780744i	$-0.714840\pi$
$251$	−19.7990	−1.24970	−0.624851	+	0.780744i	$0.714840\pi$
$252$	0	0
$253$	−4.82843	−0.303561
$254$	0	0
$255$	−1.17157	−0.0733667
$256$	0	0
$257$	27.6569	1.72519	0.862594	−	0.505898i	$-0.168839\pi$
$257$	27.6569	1.72519	0.862594	+	0.505898i	$0.168839\pi$
$258$	0	0
$259$	40.9706	2.54579
$260$	0	0
$261$	−5.65685	−0.350150
$262$	0	0
$263$	−10.5858	−0.652748	−0.326374	−	0.945241i	$-0.605827\pi$
$263$	−10.5858	−0.652748	−0.326374	+	0.945241i	$0.605827\pi$
$264$	0	0
$265$	2.48528	0.152670
$266$	0	0
$267$	8.48528	0.519291
$268$	0	0
$269$	25.3137	1.54340	0.771702	−	0.635984i	$-0.219406\pi$
$269$	25.3137	1.54340	0.771702	+	0.635984i	$0.219406\pi$
$270$	0	0
$271$	26.7279	1.62361	0.811803	−	0.583932i	$-0.198486\pi$
$271$	26.7279	1.62361	0.811803	+	0.583932i	$0.198486\pi$
$272$	0	0
$273$	6.82843	0.413275
$274$	0	0
$275$	−3.41421	−0.205885
$276$	0	0
$277$	12.8284	0.770785	0.385393	−	0.922753i	$-0.374066\pi$
$277$	12.8284	0.770785	0.385393	+	0.922753i	$0.374066\pi$
$278$	0	0
$279$	−1.75736	−0.105210
$280$	0	0
$281$	21.7990	1.30042	0.650209	−	0.759755i	$-0.274681\pi$
$281$	21.7990	1.30042	0.650209	+	0.759755i	$0.274681\pi$
$282$	0	0
$283$	16.7279	0.994372	0.497186	−	0.867644i	$-0.334367\pi$
$283$	16.7279	0.994372	0.497186	+	0.867644i	$0.334367\pi$
$284$	0	0
$285$	0.828427	0.0490718
$286$	0	0
$287$	−15.3137	−0.903940
$288$	0	0
$289$	−16.3137	−0.959630
$290$	0	0
$291$	−10.8284	−0.634774
$292$	0	0
$293$	−26.1421	−1.52724	−0.763620	−	0.645666i	$-0.776580\pi$
$293$	−26.1421	−1.52724	−0.763620	+	0.645666i	$0.776580\pi$
$294$	0	0
$295$	1.75736	0.102317
$296$	0	0
$297$	19.3137	1.12070
$298$	0	0
$299$	1.41421	0.0817861
$300$	0	0
$301$	53.4558	3.08114
$302$	0	0
$303$	5.17157	0.297099
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	−24.8284	−1.41703	−0.708517	−	0.705694i	$-0.750635\pi$
$307$	−24.8284	−1.41703	−0.708517	+	0.705694i	$0.750635\pi$
$308$	0	0
$309$	20.6274	1.17345
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	0	0
$313$	−4.82843	−0.272919	−0.136459	−	0.990646i	$-0.543572\pi$
$313$	−4.82843	−0.272919	−0.136459	+	0.990646i	$0.543572\pi$
$314$	0	0
$315$	4.82843	0.272051
$316$	0	0
$317$	2.14214	0.120314	0.0601572	−	0.998189i	$-0.480840\pi$
$317$	2.14214	0.120314	0.0601572	+	0.998189i	$0.480840\pi$
$318$	0	0
$319$	−19.3137	−1.08136
$320$	0	0
$321$	13.3137	0.743099
$322$	0	0
$323$	−0.485281	−0.0270018
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	2.82843	0.156412
$328$	0	0
$329$	−23.3137	−1.28533
$330$	0	0
$331$	26.0416	1.43138	0.715689	−	0.698419i	$-0.246113\pi$
$331$	26.0416	1.43138	0.715689	+	0.698419i	$0.246113\pi$
$332$	0	0
$333$	−8.48528	−0.464991
$334$	0	0
$335$	−2.00000	−0.109272
$336$	0	0
$337$	12.8284	0.698809	0.349404	−	0.936972i	$-0.386384\pi$
$337$	12.8284	0.698809	0.349404	+	0.936972i	$0.386384\pi$
$338$	0	0
$339$	−12.4853	−0.678107
$340$	0	0
$341$	−6.00000	−0.324918
$342$	0	0
$343$	44.9706	2.42818
$344$	0	0
$345$	−2.00000	−0.107676
$346$	0	0
$347$	4.24264	0.227757	0.113878	−	0.993495i	$-0.463673\pi$
$347$	4.24264	0.227757	0.113878	+	0.993495i	$0.463673\pi$
$348$	0	0
$349$	−18.4853	−0.989494	−0.494747	−	0.869037i	$-0.664739\pi$
$349$	−18.4853	−0.989494	−0.494747	+	0.869037i	$0.664739\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	0	0
$353$	14.8284	0.789238	0.394619	−	0.918845i	$-0.370877\pi$
$353$	14.8284	0.789238	0.394619	+	0.918845i	$0.370877\pi$
$354$	0	0
$355$	−11.8995	−0.631560
$356$	0	0
$357$	5.65685	0.299392
$358$	0	0
$359$	−8.10051	−0.427528	−0.213764	−	0.976885i	$-0.568572\pi$
$359$	−8.10051	−0.427528	−0.213764	+	0.976885i	$0.568572\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	0	0
$363$	0.928932	0.0487563
$364$	0	0
$365$	−8.48528	−0.444140
$366$	0	0
$367$	35.5563	1.85603	0.928013	−	0.372547i	$-0.121516\pi$
$367$	35.5563	1.85603	0.928013	+	0.372547i	$0.121516\pi$
$368$	0	0
$369$	3.17157	0.165105
$370$	0	0
$371$	−12.0000	−0.623009
$372$	0	0
$373$	2.68629	0.139091	0.0695455	−	0.997579i	$-0.477845\pi$
$373$	2.68629	0.139091	0.0695455	+	0.997579i	$0.477845\pi$
$374$	0	0
$375$	−1.41421	−0.0730297
$376$	0	0
$377$	5.65685	0.291343
$378$	0	0
$379$	−29.0711	−1.49328	−0.746640	−	0.665228i	$-0.768334\pi$
$379$	−29.0711	−1.49328	−0.746640	+	0.665228i	$0.768334\pi$
$380$	0	0
$381$	−9.31371	−0.477156
$382$	0	0
$383$	−29.1127	−1.48759	−0.743795	−	0.668408i	$-0.766976\pi$
$383$	−29.1127	−1.48759	−0.743795	+	0.668408i	$0.766976\pi$
$384$	0	0
$385$	16.4853	0.840168
$386$	0	0
$387$	−11.0711	−0.562774
$388$	0	0
$389$	−28.6274	−1.45147	−0.725734	−	0.687976i	$-0.758500\pi$
$389$	−28.6274	−1.45147	−0.725734	+	0.687976i	$0.758500\pi$
$390$	0	0
$391$	1.17157	0.0592490
$392$	0	0
$393$	24.0000	1.21064
$394$	0	0
$395$	8.48528	0.426941
$396$	0	0
$397$	−11.7990	−0.592174	−0.296087	−	0.955161i	$-0.595682\pi$
$397$	−11.7990	−0.592174	−0.296087	+	0.955161i	$0.595682\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−5.31371	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$401$	−5.31371	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$402$	0	0
$403$	1.75736	0.0875403
$404$	0	0
$405$	5.00000	0.248452
$406$	0	0
$407$	−28.9706	−1.43602
$408$	0	0
$409$	7.17157	0.354611	0.177306	−	0.984156i	$-0.443262\pi$
$409$	7.17157	0.354611	0.177306	+	0.984156i	$0.443262\pi$
$410$	0	0
$411$	−24.4853	−1.20777
$412$	0	0
$413$	−8.48528	−0.417533
$414$	0	0
$415$	−3.17157	−0.155686
$416$	0	0
$417$	−6.34315	−0.310625
$418$	0	0
$419$	−10.8284	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$420$	0	0
$421$	34.9706	1.70436	0.852180	−	0.523248i	$-0.175280\pi$
$421$	34.9706	1.70436	0.852180	+	0.523248i	$0.175280\pi$
$422$	0	0
$423$	4.82843	0.234766
$424$	0	0
$425$	0.828427	0.0401846
$426$	0	0
$427$	38.6274	1.86931
$428$	0	0
$429$	−4.82843	−0.233119
$430$	0	0
$431$	40.3848	1.94527	0.972633	−	0.232346i	$-0.0746403\pi$
$431$	40.3848	1.94527	0.972633	+	0.232346i	$0.0746403\pi$
$432$	0	0
$433$	7.65685	0.367965	0.183982	−	0.982930i	$-0.441101\pi$
$433$	7.65685	0.367965	0.183982	+	0.982930i	$0.441101\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	0	0
$437$	−0.828427	−0.0396290
$438$	0	0
$439$	0.970563	0.0463224	0.0231612	−	0.999732i	$-0.492627\pi$
$439$	0.970563	0.0463224	0.0231612	+	0.999732i	$0.492627\pi$
$440$	0	0
$441$	−16.3137	−0.776843
$442$	0	0
$443$	9.41421	0.447283	0.223641	−	0.974671i	$-0.428206\pi$
$443$	9.41421	0.447283	0.223641	+	0.974671i	$0.428206\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	16.4853	0.779727
$448$	0	0
$449$	−33.1127	−1.56268	−0.781342	−	0.624103i	$-0.785465\pi$
$449$	−33.1127	−1.56268	−0.781342	+	0.624103i	$0.785465\pi$
$450$	0	0
$451$	10.8284	0.509891
$452$	0	0
$453$	13.7990	0.648333
$454$	0	0
$455$	−4.82843	−0.226360
$456$	0	0
$457$	−18.0000	−0.842004	−0.421002	−	0.907060i	$-0.638322\pi$
$457$	−18.0000	−0.842004	−0.421002	+	0.907060i	$0.638322\pi$
$458$	0	0
$459$	−4.68629	−0.218737
$460$	0	0
$461$	−9.51472	−0.443145	−0.221572	−	0.975144i	$-0.571119\pi$
$461$	−9.51472	−0.443145	−0.221572	+	0.975144i	$0.571119\pi$
$462$	0	0
$463$	−4.34315	−0.201843	−0.100922	−	0.994894i	$-0.532179\pi$
$463$	−4.34315	−0.201843	−0.100922	+	0.994894i	$0.532179\pi$
$464$	0	0
$465$	−2.48528	−0.115252
$466$	0	0
$467$	13.4142	0.620736	0.310368	−	0.950617i	$-0.399548\pi$
$467$	13.4142	0.620736	0.310368	+	0.950617i	$0.399548\pi$
$468$	0	0
$469$	9.65685	0.445912
$470$	0	0
$471$	−25.4558	−1.17294
$472$	0	0
$473$	−37.7990	−1.73800
$474$	0	0
$475$	−0.585786	−0.0268777
$476$	0	0
$477$	2.48528	0.113793
$478$	0	0
$479$	−30.7279	−1.40399	−0.701997	−	0.712180i	$-0.747708\pi$
$479$	−30.7279	−1.40399	−0.701997	+	0.712180i	$0.747708\pi$
$480$	0	0
$481$	8.48528	0.386896
$482$	0	0
$483$	9.65685	0.439402
$484$	0	0
$485$	7.65685	0.347680
$486$	0	0
$487$	−10.9706	−0.497124	−0.248562	−	0.968616i	$-0.579958\pi$
$487$	−10.9706	−0.497124	−0.248562	+	0.968616i	$0.579958\pi$
$488$	0	0
$489$	−26.8284	−1.21322
$490$	0	0
$491$	−5.17157	−0.233390	−0.116695	−	0.993168i	$-0.537230\pi$
$491$	−5.17157	−0.233390	−0.116695	+	0.993168i	$0.537230\pi$
$492$	0	0
$493$	4.68629	0.211060
$494$	0	0
$495$	−3.41421	−0.153457
$496$	0	0
$497$	57.4558	2.57725
$498$	0	0
$499$	−41.5563	−1.86032	−0.930159	−	0.367157i	$-0.880331\pi$
$499$	−41.5563	−1.86032	−0.930159	+	0.367157i	$0.880331\pi$
$500$	0	0
$501$	−4.48528	−0.200388
$502$	0	0
$503$	37.8995	1.68985	0.844927	−	0.534881i	$-0.179644\pi$
$503$	37.8995	1.68985	0.844927	+	0.534881i	$0.179644\pi$
$504$	0	0
$505$	−3.65685	−0.162728
$506$	0	0
$507$	1.41421	0.0628074
$508$	0	0
$509$	−41.1127	−1.82229	−0.911144	−	0.412088i	$-0.864800\pi$
$509$	−41.1127	−1.82229	−0.911144	+	0.412088i	$0.864800\pi$
$510$	0	0
$511$	40.9706	1.81243
$512$	0	0
$513$	3.31371	0.146304
$514$	0	0
$515$	−14.5858	−0.642727
$516$	0	0
$517$	16.4853	0.725022
$518$	0	0
$519$	−23.7990	−1.04466
$520$	0	0
$521$	−17.6569	−0.773561	−0.386780	−	0.922172i	$-0.626413\pi$
$521$	−17.6569	−0.773561	−0.386780	+	0.922172i	$0.626413\pi$
$522$	0	0
$523$	19.7574	0.863929	0.431965	−	0.901891i	$-0.357821\pi$
$523$	19.7574	0.863929	0.431965	+	0.901891i	$0.357821\pi$
$524$	0	0
$525$	6.82843	0.298017
$526$	0	0
$527$	1.45584	0.0634176
$528$	0	0
$529$	−21.0000	−0.913043
$530$	0	0
$531$	1.75736	0.0762629
$532$	0	0
$533$	−3.17157	−0.137376
$534$	0	0
$535$	−9.41421	−0.407012
$536$	0	0
$537$	−8.00000	−0.345225
$538$	0	0
$539$	−55.6985	−2.39910
$540$	0	0
$541$	7.17157	0.308330	0.154165	−	0.988045i	$-0.450731\pi$
$541$	7.17157	0.308330	0.154165	+	0.988045i	$0.450731\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−13.2132	−0.564956	−0.282478	−	0.959274i	$-0.591156\pi$
$547$	−13.2132	−0.564956	−0.282478	+	0.959274i	$0.591156\pi$
$548$	0	0
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−3.31371	−0.141169
$552$	0	0
$553$	−40.9706	−1.74225
$554$	0	0
$555$	−12.0000	−0.509372
$556$	0	0
$557$	35.7990	1.51685	0.758426	−	0.651759i	$-0.225969\pi$
$557$	35.7990	1.51685	0.758426	+	0.651759i	$0.225969\pi$
$558$	0	0
$559$	11.0711	0.468256
$560$	0	0
$561$	−4.00000	−0.168880
$562$	0	0
$563$	7.75736	0.326934	0.163467	−	0.986549i	$-0.447732\pi$
$563$	7.75736	0.326934	0.163467	+	0.986549i	$0.447732\pi$
$564$	0	0
$565$	8.82843	0.371415
$566$	0	0
$567$	−24.1421	−1.01387
$568$	0	0
$569$	−10.3431	−0.433607	−0.216804	−	0.976215i	$-0.569563\pi$
$569$	−10.3431	−0.433607	−0.216804	+	0.976215i	$0.569563\pi$
$570$	0	0
$571$	11.5147	0.481876	0.240938	−	0.970541i	$-0.422545\pi$
$571$	11.5147	0.481876	0.240938	+	0.970541i	$0.422545\pi$
$572$	0	0
$573$	3.31371	0.138432
$574$	0	0
$575$	1.41421	0.0589768
$576$	0	0
$577$	−34.8284	−1.44993	−0.724963	−	0.688788i	$-0.758143\pi$
$577$	−34.8284	−1.44993	−0.724963	+	0.688788i	$0.758143\pi$
$578$	0	0
$579$	6.14214	0.255258
$580$	0	0
$581$	15.3137	0.635320
$582$	0	0
$583$	8.48528	0.351424
$584$	0	0
$585$	1.00000	0.0413449
$586$	0	0
$587$	−20.3431	−0.839651	−0.419826	−	0.907605i	$-0.637909\pi$
$587$	−20.3431	−0.839651	−0.419826	+	0.907605i	$0.637909\pi$
$588$	0	0
$589$	−1.02944	−0.0424172
$590$	0	0
$591$	−15.5147	−0.638190
$592$	0	0
$593$	−24.6274	−1.01133	−0.505663	−	0.862731i	$-0.668752\pi$
$593$	−24.6274	−1.01133	−0.505663	+	0.862731i	$0.668752\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	5.65685	0.231520
$598$	0	0
$599$	25.4558	1.04010	0.520049	−	0.854137i	$-0.325914\pi$
$599$	25.4558	1.04010	0.520049	+	0.854137i	$0.325914\pi$
$600$	0	0
$601$	−44.6274	−1.82039	−0.910195	−	0.414180i	$-0.864069\pi$
$601$	−44.6274	−1.82039	−0.910195	+	0.414180i	$0.864069\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	0	0
$605$	−0.656854	−0.0267049
$606$	0	0
$607$	31.7574	1.28899	0.644496	−	0.764608i	$-0.277067\pi$
$607$	31.7574	1.28899	0.644496	+	0.764608i	$0.277067\pi$
$608$	0	0
$609$	38.6274	1.56526
$610$	0	0
$611$	−4.82843	−0.195337
$612$	0	0
$613$	14.6863	0.593174	0.296587	−	0.955006i	$-0.404152\pi$
$613$	14.6863	0.593174	0.296587	+	0.955006i	$0.404152\pi$
$614$	0	0
$615$	4.48528	0.180864
$616$	0	0
$617$	10.9706	0.441658	0.220829	−	0.975313i	$-0.429124\pi$
$617$	10.9706	0.441658	0.220829	+	0.975313i	$0.429124\pi$
$618$	0	0
$619$	−1.75736	−0.0706342	−0.0353171	−	0.999376i	$-0.511244\pi$
$619$	−1.75736	−0.0706342	−0.0353171	+	0.999376i	$0.511244\pi$
$620$	0	0
$621$	−8.00000	−0.321029
$622$	0	0
$623$	28.9706	1.16068
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.82843	0.112956
$628$	0	0
$629$	7.02944	0.280282
$630$	0	0
$631$	−9.75736	−0.388434	−0.194217	−	0.980959i	$-0.562217\pi$
$631$	−9.75736	−0.388434	−0.194217	+	0.980959i	$0.562217\pi$
$632$	0	0
$633$	−4.68629	−0.186263
$634$	0	0
$635$	6.58579	0.261349
$636$	0	0
$637$	16.3137	0.646373
$638$	0	0
$639$	−11.8995	−0.470737
$640$	0	0
$641$	47.6569	1.88233	0.941166	−	0.337944i	$-0.109731\pi$
$641$	47.6569	1.88233	0.941166	+	0.337944i	$0.109731\pi$
$642$	0	0
$643$	−9.51472	−0.375224	−0.187612	−	0.982243i	$-0.560075\pi$
$643$	−9.51472	−0.375224	−0.187612	+	0.982243i	$0.560075\pi$
$644$	0	0
$645$	−15.6569	−0.616488
$646$	0	0
$647$	9.41421	0.370111	0.185055	−	0.982728i	$-0.440754\pi$
$647$	9.41421	0.370111	0.185055	+	0.982728i	$0.440754\pi$
$648$	0	0
$649$	6.00000	0.235521
$650$	0	0
$651$	12.0000	0.470317
$652$	0	0
$653$	−46.9706	−1.83810	−0.919050	−	0.394141i	$-0.871042\pi$
$653$	−46.9706	−1.83810	−0.919050	+	0.394141i	$0.871042\pi$
$654$	0	0
$655$	−16.9706	−0.663095
$656$	0	0
$657$	−8.48528	−0.331042
$658$	0	0
$659$	−17.8579	−0.695644	−0.347822	−	0.937561i	$-0.613079\pi$
$659$	−17.8579	−0.695644	−0.347822	+	0.937561i	$0.613079\pi$
$660$	0	0
$661$	−29.5980	−1.15123	−0.575614	−	0.817722i	$-0.695237\pi$
$661$	−29.5980	−1.15123	−0.575614	+	0.817722i	$0.695237\pi$
$662$	0	0
$663$	1.17157	0.0455001
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	8.00000	0.309761
$668$	0	0
$669$	13.4558	0.520233
$670$	0	0
$671$	−27.3137	−1.05443
$672$	0	0
$673$	6.48528	0.249989	0.124995	−	0.992157i	$-0.460109\pi$
$673$	6.48528	0.249989	0.124995	+	0.992157i	$0.460109\pi$
$674$	0	0
$675$	−5.65685	−0.217732
$676$	0	0
$677$	20.1421	0.774125	0.387063	−	0.922053i	$-0.373490\pi$
$677$	20.1421	0.774125	0.387063	+	0.922053i	$0.373490\pi$
$678$	0	0
$679$	−36.9706	−1.41880
$680$	0	0
$681$	23.1127	0.885681
$682$	0	0
$683$	−10.6863	−0.408900	−0.204450	−	0.978877i	$-0.565541\pi$
$683$	−10.6863	−0.408900	−0.204450	+	0.978877i	$0.565541\pi$
$684$	0	0
$685$	17.3137	0.661523
$686$	0	0
$687$	6.82843	0.260521
$688$	0	0
$689$	−2.48528	−0.0946817
$690$	0	0
$691$	−6.92893	−0.263589	−0.131795	−	0.991277i	$-0.542074\pi$
$691$	−6.92893	−0.263589	−0.131795	+	0.991277i	$0.542074\pi$
$692$	0	0
$693$	16.4853	0.626224
$694$	0	0
$695$	4.48528	0.170136
$696$	0	0
$697$	−2.62742	−0.0995205
$698$	0	0
$699$	−29.1716	−1.10337
$700$	0	0
$701$	−14.6863	−0.554694	−0.277347	−	0.960770i	$-0.589455\pi$
$701$	−14.6863	−0.554694	−0.277347	+	0.960770i	$0.589455\pi$
$702$	0	0
$703$	−4.97056	−0.187468
$704$	0	0
$705$	6.82843	0.257173
$706$	0	0
$707$	17.6569	0.664054
$708$	0	0
$709$	−45.1127	−1.69424	−0.847121	−	0.531399i	$-0.821666\pi$
$709$	−45.1127	−1.69424	−0.847121	+	0.531399i	$0.821666\pi$
$710$	0	0
$711$	8.48528	0.318223
$712$	0	0
$713$	2.48528	0.0930745
$714$	0	0
$715$	3.41421	0.127684
$716$	0	0
$717$	−4.82843	−0.180321
$718$	0	0
$719$	28.9706	1.08042	0.540210	−	0.841530i	$-0.318345\pi$
$719$	28.9706	1.08042	0.540210	+	0.841530i	$0.318345\pi$
$720$	0	0
$721$	70.4264	2.62282
$722$	0	0
$723$	−20.4853	−0.761856
$724$	0	0
$725$	5.65685	0.210090
$726$	0	0
$727$	−51.3553	−1.90466	−0.952332	−	0.305063i	$-0.901322\pi$
$727$	−51.3553	−1.90466	−0.952332	+	0.305063i	$0.901322\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	9.17157	0.339223
$732$	0	0
$733$	−21.3137	−0.787240	−0.393620	−	0.919273i	$-0.628777\pi$
$733$	−21.3137	−0.787240	−0.393620	+	0.919273i	$0.628777\pi$
$734$	0	0
$735$	−23.0711	−0.850989
$736$	0	0
$737$	−6.82843	−0.251528
$738$	0	0
$739$	−5.27208	−0.193937	−0.0969683	−	0.995287i	$-0.530915\pi$
$739$	−5.27208	−0.193937	−0.0969683	+	0.995287i	$0.530915\pi$
$740$	0	0
$741$	−0.828427	−0.0304330
$742$	0	0
$743$	−21.5147	−0.789298	−0.394649	−	0.918832i	$-0.629134\pi$
$743$	−21.5147	−0.789298	−0.394649	+	0.918832i	$0.629134\pi$
$744$	0	0
$745$	−11.6569	−0.427074
$746$	0	0
$747$	−3.17157	−0.116042
$748$	0	0
$749$	45.4558	1.66092
$750$	0	0
$751$	−27.5147	−1.00403	−0.502013	−	0.864860i	$-0.667407\pi$
$751$	−27.5147	−1.00403	−0.502013	+	0.864860i	$0.667407\pi$
$752$	0	0
$753$	−28.0000	−1.02038
$754$	0	0
$755$	−9.75736	−0.355107
$756$	0	0
$757$	24.1421	0.877461	0.438730	−	0.898619i	$-0.355428\pi$
$757$	24.1421	0.877461	0.438730	+	0.898619i	$0.355428\pi$
$758$	0	0
$759$	−6.82843	−0.247856
$760$	0	0
$761$	8.62742	0.312744	0.156372	−	0.987698i	$-0.450020\pi$
$761$	8.62742	0.312744	0.156372	+	0.987698i	$0.450020\pi$
$762$	0	0
$763$	9.65685	0.349602
$764$	0	0
$765$	0.828427	0.0299518
$766$	0	0
$767$	−1.75736	−0.0634546
$768$	0	0
$769$	−22.9706	−0.828340	−0.414170	−	0.910200i	$-0.635928\pi$
$769$	−22.9706	−0.828340	−0.414170	+	0.910200i	$0.635928\pi$
$770$	0	0
$771$	39.1127	1.40861
$772$	0	0
$773$	22.1421	0.796397	0.398199	−	0.917299i	$-0.369635\pi$
$773$	22.1421	0.796397	0.398199	+	0.917299i	$0.369635\pi$
$774$	0	0
$775$	1.75736	0.0631262
$776$	0	0
$777$	57.9411	2.07863
$778$	0	0
$779$	1.85786	0.0665649
$780$	0	0
$781$	−40.6274	−1.45376
$782$	0	0
$783$	−32.0000	−1.14359
$784$	0	0
$785$	18.0000	0.642448
$786$	0	0
$787$	−22.4853	−0.801514	−0.400757	−	0.916184i	$-0.631253\pi$
$787$	−22.4853	−0.801514	−0.400757	+	0.916184i	$0.631253\pi$
$788$	0	0
$789$	−14.9706	−0.532966
$790$	0	0
$791$	−42.6274	−1.51566
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	3.51472	0.124654
$796$	0	0
$797$	22.9706	0.813659	0.406830	−	0.913504i	$-0.366634\pi$
$797$	22.9706	0.813659	0.406830	+	0.913504i	$0.366634\pi$
$798$	0	0
$799$	−4.00000	−0.141510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	0	0
$803$	−28.9706	−1.02235
$804$	0	0
$805$	−6.82843	−0.240670
$806$	0	0
$807$	35.7990	1.26018
$808$	0	0
$809$	45.2548	1.59108	0.795538	−	0.605904i	$-0.207189\pi$
$809$	45.2548	1.59108	0.795538	+	0.605904i	$0.207189\pi$
$810$	0	0
$811$	28.3848	0.996724	0.498362	−	0.866969i	$-0.333935\pi$
$811$	28.3848	0.996724	0.498362	+	0.866969i	$0.333935\pi$
$812$	0	0
$813$	37.7990	1.32567
$814$	0	0
$815$	18.9706	0.664510
$816$	0	0
$817$	−6.48528	−0.226891
$818$	0	0
$819$	−4.82843	−0.168719
$820$	0	0
$821$	51.2548	1.78881	0.894403	−	0.447262i	$-0.147601\pi$
$821$	51.2548	1.78881	0.894403	+	0.447262i	$0.147601\pi$
$822$	0	0
$823$	−2.38478	−0.0831281	−0.0415640	−	0.999136i	$-0.513234\pi$
$823$	−2.38478	−0.0831281	−0.0415640	+	0.999136i	$0.513234\pi$
$824$	0	0
$825$	−4.82843	−0.168104
$826$	0	0
$827$	−56.1421	−1.95225	−0.976127	−	0.217202i	$-0.930307\pi$
$827$	−56.1421	−1.95225	−0.976127	+	0.217202i	$0.930307\pi$
$828$	0	0
$829$	−40.9706	−1.42297	−0.711483	−	0.702703i	$-0.751976\pi$
$829$	−40.9706	−1.42297	−0.711483	+	0.702703i	$0.751976\pi$
$830$	0	0
$831$	18.1421	0.629344
$832$	0	0
$833$	13.5147	0.468257
$834$	0	0
$835$	3.17157	0.109757
$836$	0	0
$837$	−9.94113	−0.343616
$838$	0	0
$839$	6.72792	0.232274	0.116137	−	0.993233i	$-0.462949\pi$
$839$	6.72792	0.232274	0.116137	+	0.993233i	$0.462949\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	0	0
$843$	30.8284	1.06179
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	3.17157	0.108977
$848$	0	0
$849$	23.6569	0.811901
$850$	0	0
$851$	12.0000	0.411355
$852$	0	0
$853$	13.4558	0.460719	0.230360	−	0.973106i	$-0.426010\pi$
$853$	13.4558	0.460719	0.230360	+	0.973106i	$0.426010\pi$
$854$	0	0
$855$	−0.585786	−0.0200335
$856$	0	0
$857$	11.6569	0.398191	0.199095	−	0.979980i	$-0.436200\pi$
$857$	11.6569	0.398191	0.199095	+	0.979980i	$0.436200\pi$
$858$	0	0
$859$	27.7990	0.948489	0.474245	−	0.880393i	$-0.342721\pi$
$859$	27.7990	0.948489	0.474245	+	0.880393i	$0.342721\pi$
$860$	0	0
$861$	−21.6569	−0.738064
$862$	0	0
$863$	−31.4558	−1.07077	−0.535385	−	0.844608i	$-0.679833\pi$
$863$	−31.4558	−1.07077	−0.535385	+	0.844608i	$0.679833\pi$
$864$	0	0
$865$	16.8284	0.572184
$866$	0	0
$867$	−23.0711	−0.783535
$868$	0	0
$869$	28.9706	0.982759
$870$	0	0
$871$	2.00000	0.0677674
$872$	0	0
$873$	7.65685	0.259145
$874$	0	0
$875$	−4.82843	−0.163231
$876$	0	0
$877$	−25.3137	−0.854783	−0.427392	−	0.904067i	$-0.640567\pi$
$877$	−25.3137	−0.854783	−0.427392	+	0.904067i	$0.640567\pi$
$878$	0	0
$879$	−36.9706	−1.24699
$880$	0	0
$881$	19.0294	0.641118	0.320559	−	0.947229i	$-0.396129\pi$
$881$	19.0294	0.641118	0.320559	+	0.947229i	$0.396129\pi$
$882$	0	0
$883$	23.7574	0.799499	0.399749	−	0.916624i	$-0.369097\pi$
$883$	23.7574	0.799499	0.399749	+	0.916624i	$0.369097\pi$
$884$	0	0
$885$	2.48528	0.0835418
$886$	0	0
$887$	−22.3848	−0.751607	−0.375804	−	0.926699i	$-0.622633\pi$
$887$	−22.3848	−0.751607	−0.375804	+	0.926699i	$0.622633\pi$
$888$	0	0
$889$	−31.7990	−1.06650
$890$	0	0
$891$	17.0711	0.571902
$892$	0	0
$893$	2.82843	0.0946497
$894$	0	0
$895$	5.65685	0.189088
$896$	0	0
$897$	2.00000	0.0667781
$898$	0	0
$899$	9.94113	0.331555
$900$	0	0
$901$	−2.05887	−0.0685911
$902$	0	0
$903$	75.5980	2.51574
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−9.21320	−0.305919	−0.152960	−	0.988232i	$-0.548880\pi$
$907$	−9.21320	−0.305919	−0.152960	+	0.988232i	$0.548880\pi$
$908$	0	0
$909$	−3.65685	−0.121290
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−10.8284	−0.358369
$914$	0	0
$915$	−11.3137	−0.374020
$916$	0	0
$917$	81.9411	2.70593
$918$	0	0
$919$	0.485281	0.0160080	0.00800398	−	0.999968i	$-0.497452\pi$
$919$	0.485281	0.0160080	0.00800398	+	0.999968i	$0.497452\pi$
$920$	0	0
$921$	−35.1127	−1.15700
$922$	0	0
$923$	11.8995	0.391677
$924$	0	0
$925$	8.48528	0.278994
$926$	0	0
$927$	−14.5858	−0.479060
$928$	0	0
$929$	16.8284	0.552123	0.276061	−	0.961140i	$-0.410971\pi$
$929$	16.8284	0.552123	0.276061	+	0.961140i	$0.410971\pi$
$930$	0	0
$931$	−9.55635	−0.313197
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	2.82843	0.0924995
$936$	0	0
$937$	−22.9706	−0.750416	−0.375208	−	0.926941i	$-0.622429\pi$
$937$	−22.9706	−0.750416	−0.375208	+	0.926941i	$0.622429\pi$
$938$	0	0
$939$	−6.82843	−0.222837
$940$	0	0
$941$	−18.7696	−0.611870	−0.305935	−	0.952052i	$-0.598969\pi$
$941$	−18.7696	−0.611870	−0.305935	+	0.952052i	$0.598969\pi$
$942$	0	0
$943$	−4.48528	−0.146061
$944$	0	0
$945$	27.3137	0.888515
$946$	0	0
$947$	−17.1127	−0.556088	−0.278044	−	0.960568i	$-0.589686\pi$
$947$	−17.1127	−0.556088	−0.278044	+	0.960568i	$0.589686\pi$
$948$	0	0
$949$	8.48528	0.275444
$950$	0	0
$951$	3.02944	0.0982362
$952$	0	0
$953$	35.2548	1.14202	0.571008	−	0.820944i	$-0.306553\pi$
$953$	35.2548	1.14202	0.571008	+	0.820944i	$0.306553\pi$
$954$	0	0
$955$	−2.34315	−0.0758224
$956$	0	0
$957$	−27.3137	−0.882927
$958$	0	0
$959$	−83.5980	−2.69952
$960$	0	0
$961$	−27.9117	−0.900377
$962$	0	0
$963$	−9.41421	−0.303369
$964$	0	0
$965$	−4.34315	−0.139811
$966$	0	0
$967$	47.9411	1.54168	0.770841	−	0.637027i	$-0.219836\pi$
$967$	47.9411	1.54168	0.770841	+	0.637027i	$0.219836\pi$
$968$	0	0
$969$	−0.686292	−0.0220469
$970$	0	0
$971$	−44.2843	−1.42115	−0.710575	−	0.703622i	$-0.751565\pi$
$971$	−44.2843	−1.42115	−0.710575	+	0.703622i	$0.751565\pi$
$972$	0	0
$973$	−21.6569	−0.694287
$974$	0	0
$975$	1.41421	0.0452911
$976$	0	0
$977$	−39.5147	−1.26419	−0.632094	−	0.774892i	$-0.717804\pi$
$977$	−39.5147	−1.26419	−0.632094	+	0.774892i	$0.717804\pi$
$978$	0	0
$979$	−20.4853	−0.654712
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−1.02944	−0.0328339	−0.0164170	−	0.999865i	$-0.505226\pi$
$983$	−1.02944	−0.0328339	−0.0164170	+	0.999865i	$0.505226\pi$
$984$	0	0
$985$	10.9706	0.349551
$986$	0	0
$987$	−32.9706	−1.04946
$988$	0	0
$989$	15.6569	0.497859
$990$	0	0
$991$	48.9706	1.55560	0.777801	−	0.628511i	$-0.216335\pi$
$991$	48.9706	1.55560	0.777801	+	0.628511i	$0.216335\pi$
$992$	0	0
$993$	36.8284	1.16871
$994$	0	0
$995$	−4.00000	−0.126809
$996$	0	0
$997$	−28.8284	−0.913005	−0.456503	−	0.889722i	$-0.650898\pi$
$997$	−28.8284	−0.913005	−0.456503	+	0.889722i	$0.650898\pi$
$998$	0	0
$999$	−48.0000	−1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.bf.1.2		2
4.3	odd	2		4160.2.a.z.1.1		2
8.3	odd	2		1040.2.a.j.1.2		2
8.5	even	2		65.2.a.b.1.1	✓	2
24.5	odd	2		585.2.a.m.1.2		2
24.11	even	2		9360.2.a.cd.1.1		2
40.13	odd	4		325.2.b.f.274.4		4
40.19	odd	2		5200.2.a.bu.1.1		2
40.29	even	2		325.2.a.i.1.2		2
40.37	odd	4		325.2.b.f.274.1		4
56.13	odd	2		3185.2.a.j.1.1		2
88.21	odd	2		7865.2.a.j.1.2		2
104.5	odd	4		845.2.c.b.506.4		4
104.21	odd	4		845.2.c.b.506.1		4
104.29	even	6		845.2.e.h.191.2		4
104.37	odd	12		845.2.m.f.316.4		8
104.45	odd	12		845.2.m.f.361.4		8
104.61	even	6		845.2.e.h.146.2		4
104.69	even	6		845.2.e.c.146.1		4
104.77	even	2		845.2.a.g.1.2		2
104.85	odd	12		845.2.m.f.361.1		8
104.93	odd	12		845.2.m.f.316.1		8
104.101	even	6		845.2.e.c.191.1		4
120.29	odd	2		2925.2.a.u.1.1		2
120.53	even	4		2925.2.c.r.2224.1		4
120.77	even	4		2925.2.c.r.2224.4		4
312.77	odd	2		7605.2.a.x.1.1		2
520.389	even	2		4225.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.1	✓	2	8.5	even	2
325.2.a.i.1.2		2	40.29	even	2
325.2.b.f.274.1		4	40.37	odd	4
325.2.b.f.274.4		4	40.13	odd	4
585.2.a.m.1.2		2	24.5	odd	2
845.2.a.g.1.2		2	104.77	even	2
845.2.c.b.506.1		4	104.21	odd	4
845.2.c.b.506.4		4	104.5	odd	4
845.2.e.c.146.1		4	104.69	even	6
845.2.e.c.191.1		4	104.101	even	6
845.2.e.h.146.2		4	104.61	even	6
845.2.e.h.191.2		4	104.29	even	6
845.2.m.f.316.1		8	104.93	odd	12
845.2.m.f.316.4		8	104.37	odd	12
845.2.m.f.361.1		8	104.85	odd	12
845.2.m.f.361.4		8	104.45	odd	12
1040.2.a.j.1.2		2	8.3	odd	2
2925.2.a.u.1.1		2	120.29	odd	2
2925.2.c.r.2224.1		4	120.53	even	4
2925.2.c.r.2224.4		4	120.77	even	4
3185.2.a.j.1.1		2	56.13	odd	2
4160.2.a.z.1.1		2	4.3	odd	2
4160.2.a.bf.1.2		2	1.1	even	1	trivial
4225.2.a.r.1.1		2	520.389	even	2
5200.2.a.bu.1.1		2	40.19	odd	2
7605.2.a.x.1.1		2	312.77	odd	2
7865.2.a.j.1.2		2	88.21	odd	2
9360.2.a.cd.1.1		2	24.11	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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{3} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{3} -1.00000 q^{5} +4.82843 q^{7} -1.00000 q^{9} -3.41421 q^{11} +1.00000 q^{13} -1.41421 q^{15} +0.828427 q^{17} -0.585786 q^{19} +6.82843 q^{21} +1.41421 q^{23} +1.00000 q^{25} -5.65685 q^{27} +5.65685 q^{29} +1.75736 q^{31} -4.82843 q^{33} -4.82843 q^{35} +8.48528 q^{37} +1.41421 q^{39} -3.17157 q^{41} +11.0711 q^{43} +1.00000 q^{45} -4.82843 q^{47} +16.3137 q^{49} +1.17157 q^{51} -2.48528 q^{53} +3.41421 q^{55} -0.828427 q^{57} -1.75736 q^{59} +8.00000 q^{61} -4.82843 q^{63} -1.00000 q^{65} +2.00000 q^{67} +2.00000 q^{69} +11.8995 q^{71} +8.48528 q^{73} +1.41421 q^{75} -16.4853 q^{77} -8.48528 q^{79} -5.00000 q^{81} +3.17157 q^{83} -0.828427 q^{85} +8.00000 q^{87} +6.00000 q^{89} +4.82843 q^{91} +2.48528 q^{93} +0.585786 q^{95} -7.65685 q^{97} +3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 + 4 * q^7 - 2 * q^9	\( 2 q - 2 q^{5} + 4 q^{7} - 2 q^{9} - 4 q^{11} + 2 q^{13} - 4 q^{17} - 4 q^{19} + 8 q^{21} + 2 q^{25} + 12 q^{31} - 4 q^{33} - 4 q^{35} - 12 q^{41} + 8 q^{43} + 2 q^{45} - 4 q^{47} + 10 q^{49} + 8 q^{51} + 12 q^{53} + 4 q^{55} + 4 q^{57} - 12 q^{59} + 16 q^{61} - 4 q^{63} - 2 q^{65} + 4 q^{67} + 4 q^{69} + 4 q^{71} - 16 q^{77} - 10 q^{81} + 12 q^{83} + 4 q^{85} + 16 q^{87} + 12 q^{89} + 4 q^{91} - 12 q^{93} + 4 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 + 4 * q^7 - 2 * q^9 - 4 * q^11 + 2 * q^13 - 4 * q^17 - 4 * q^19 + 8 * q^21 + 2 * q^25 + 12 * q^31 - 4 * q^33 - 4 * q^35 - 12 * q^41 + 8 * q^43 + 2 * q^45 - 4 * q^47 + 10 * q^49 + 8 * q^51 + 12 * q^53 + 4 * q^55 + 4 * q^57 - 12 * q^59 + 16 * q^61 - 4 * q^63 - 2 * q^65 + 4 * q^67 + 4 * q^69 + 4 * q^71 - 16 * q^77 - 10 * q^81 + 12 * q^83 + 4 * q^85 + 16 * q^87 + 12 * q^89 + 4 * q^91 - 12 * q^93 + 4 * q^95 - 4 * q^97 + 4 * q^99

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