LMFDB - Embedded newform 164.2.b.a.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [164,2,Mod(81,164)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(164, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("164.81");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 164 = 2^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	164.b (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$1.30954659315$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.25088.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 6x^{2} + 2 $ x^4 + 6*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			81.1
Root			$-2.37608i$ of defining polynomial
Character	$\chi$	$=$	164.81
Dual form			164.2.b.a.81.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.37608i q^{3} -3.64575 q^{5} -1.53436i q^{7} -2.64575 q^{9} +O(q^{10})$	$q-2.37608i q^{3} -3.64575 q^{5} -1.53436i q^{7} -2.64575 q^{9} -3.21780i q^{11} +3.06871i q^{13} +8.66259i q^{15} -7.82087i q^{17} +4.05952i q^{19} -3.64575 q^{21} +7.29150 q^{23} +8.29150 q^{25} -0.841723i q^{27} -4.75216i q^{29} +3.29150 q^{31} -7.64575 q^{33} +5.59388i q^{35} -0.354249 q^{37} +7.29150 q^{39} +(-4.29150 + 4.75216i) q^{41} +3.29150 q^{43} +9.64575 q^{45} +5.44479i q^{47} +4.64575 q^{49} -18.5830 q^{51} +6.43560i q^{53} +11.7313i q^{55} +9.64575 q^{57} -7.29150 q^{59} -11.2915 q^{61} +4.05952i q^{63} -11.1878i q^{65} -12.7221i q^{67} -17.3252i q^{69} +2.37608i q^{71} +6.93725 q^{73} -19.7013i q^{75} -4.93725 q^{77} -10.1969i q^{79} -9.93725 q^{81} +12.0000 q^{83} +28.5129i q^{85} -11.2915 q^{87} +4.45398i q^{89} +4.70850 q^{91} -7.82087i q^{93} -14.8000i q^{95} -6.13742i q^{97} +8.51350i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5}+O(q^{10}) $ 4 * q - 4 * q^5	$ 4 q - 4 q^{5} - 4 q^{21} + 8 q^{23} + 12 q^{25} - 8 q^{31} - 20 q^{33} - 12 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 28 q^{45} + 8 q^{49} - 32 q^{51} + 28 q^{57} - 8 q^{59} - 24 q^{61} - 4 q^{73} + 12 q^{77} - 8 q^{81} + 48 q^{83} - 24 q^{87} + 40 q^{91}+O(q^{100}) $ 4 * q - 4 * q^5 - 4 * q^21 + 8 * q^23 + 12 * q^25 - 8 * q^31 - 20 * q^33 - 12 * q^37 + 8 * q^39 + 4 * q^41 - 8 * q^43 + 28 * q^45 + 8 * q^49 - 32 * q^51 + 28 * q^57 - 8 * q^59 - 24 * q^61 - 4 * q^73 + 12 * q^77 - 8 * q^81 + 48 * q^83 - 24 * q^87 + 40 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/164\mathbb{Z}\right)^\times$.

$n$	$83$	$129$
$\chi(n)$	$1$	$-1$

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
$3$		−	2.37608i		−	1.37183i	−0.727682	−	0.685915i	$-0.759402\pi$
$3$		−	2.37608i		−	1.37183i	0.727682	−	0.685915i	$-0.240598\pi$
$4$		0			0
$5$	−3.64575			−1.63043			−0.815215	−	0.579159i	$-0.803381\pi$
$5$	−3.64575			−1.63043			−0.815215	+	0.579159i	$0.803381\pi$
$6$		0			0
$7$		−	1.53436i		−	0.579932i	−0.957037	−	0.289966i	$-0.906356\pi$
$7$		−	1.53436i		−	0.579932i	0.957037	−	0.289966i	$-0.0936440\pi$
$8$		0			0
$9$	−2.64575			−0.881917
$10$		0			0
$11$		−	3.21780i		−	0.970204i	−0.874458	−	0.485102i	$-0.838783\pi$
$11$		−	3.21780i		−	0.970204i	0.874458	−	0.485102i	$-0.161217\pi$
$12$		0			0
$13$			3.06871i			0.851108i	0.904933	+	0.425554i	$0.139921\pi$
$13$			3.06871i			0.851108i	−0.904933	+	0.425554i	$0.860079\pi$
$14$		0			0
$15$			8.66259i			2.23667i
$16$		0			0
$17$		−	7.82087i		−	1.89684i	−0.317017	−	0.948420i	$-0.602681\pi$
$17$		−	7.82087i		−	1.89684i	0.317017	−	0.948420i	$-0.397319\pi$
$18$		0			0
$19$			4.05952i			0.931319i	0.884964	+	0.465659i	$0.154183\pi$
$19$			4.05952i			0.931319i	−0.884964	+	0.465659i	$0.845817\pi$
$20$		0			0
$21$	−3.64575			−0.795568
$22$		0			0
$23$	7.29150			1.52038			0.760192	−	0.649699i	$-0.225105\pi$
$23$	7.29150			1.52038			0.760192	+	0.649699i	$0.225105\pi$
$24$		0			0
$25$	8.29150			1.65830
$26$		0			0
$27$		−	0.841723i		−	0.161990i
$28$		0			0
$29$		−	4.75216i		−	0.882454i	−0.897396	−	0.441227i	$-0.854543\pi$
$29$		−	4.75216i		−	0.882454i	0.897396	−	0.441227i	$-0.145457\pi$
$30$		0			0
$31$	3.29150			0.591171			0.295586	−	0.955316i	$-0.404485\pi$
$31$	3.29150			0.591171			0.295586	+	0.955316i	$0.404485\pi$
$32$		0			0
$33$	−7.64575			−1.33095
$34$		0			0
$35$			5.59388i			0.945538i
$36$		0			0
$37$	−0.354249			−0.0582381			−0.0291191	−	0.999576i	$-0.509270\pi$
$37$	−0.354249			−0.0582381			−0.0291191	+	0.999576i	$0.509270\pi$
$38$		0			0
$39$	7.29150			1.16757
$40$		0			0
$41$	−4.29150	+	4.75216i	−0.670220	+	0.742162i
$41$	−4.29150	+	4.75216i	−0.670220	+	0.742162i
$42$		0			0
$43$	3.29150			0.501949			0.250975	−	0.967994i	$-0.419249\pi$
$43$	3.29150			0.501949			0.250975	+	0.967994i	$0.419249\pi$
$44$		0			0
$45$	9.64575			1.43790
$46$		0			0
$47$			5.44479i			0.794204i	0.917774	+	0.397102i	$0.129984\pi$
$47$			5.44479i			0.794204i	−0.917774	+	0.397102i	$0.870016\pi$
$48$		0			0
$49$	4.64575			0.663679
$50$		0			0
$51$	−18.5830			−2.60214
$52$		0			0
$53$			6.43560i			0.883998i	0.897016	+	0.441999i	$0.145731\pi$
$53$			6.43560i			0.883998i	−0.897016	+	0.441999i	$0.854269\pi$
$54$		0			0
$55$			11.7313i			1.58185i
$56$		0			0
$57$	9.64575			1.27761
$58$		0			0
$59$	−7.29150			−0.949273			−0.474636	−	0.880182i	$-0.657420\pi$
$59$	−7.29150			−0.949273			−0.474636	+	0.880182i	$0.657420\pi$
$60$		0			0
$61$	−11.2915			−1.44573			−0.722864	−	0.690990i	$-0.757175\pi$
$61$	−11.2915			−1.44573			−0.722864	+	0.690990i	$0.757175\pi$
$62$		0			0
$63$			4.05952i			0.511452i
$64$		0			0
$65$		−	11.1878i		−	1.38767i
$66$		0			0
$67$		−	12.7221i		−	1.55425i	−0.629344	−	0.777127i	$-0.716676\pi$
$67$		−	12.7221i		−	1.55425i	0.629344	−	0.777127i	$-0.283324\pi$
$68$		0			0
$69$		−	17.3252i		−	2.08571i
$70$		0			0
$71$			2.37608i			0.281989i	0.990010	+	0.140994i	$0.0450299\pi$
$71$			2.37608i			0.281989i	−0.990010	+	0.140994i	$0.954970\pi$
$72$		0			0
$73$	6.93725			0.811944			0.405972	−	0.913885i	$-0.366933\pi$
$73$	6.93725			0.811944			0.405972	+	0.913885i	$0.366933\pi$
$74$		0			0
$75$		−	19.7013i		−	2.27491i
$76$		0			0
$77$	−4.93725			−0.562652
$78$		0			0
$79$		−	10.1969i		−	1.14725i	−0.819119	−	0.573623i	$-0.805537\pi$
$79$		−	10.1969i		−	1.14725i	0.819119	−	0.573623i	$-0.194463\pi$
$80$		0			0
$81$	−9.93725			−1.10414
$82$		0			0
$83$	12.0000			1.31717			0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000			1.31717			0.658586	+	0.752506i	$0.271155\pi$
$84$		0			0
$85$			28.5129i			3.09266i
$86$		0			0
$87$	−11.2915			−1.21058
$88$		0			0
$89$			4.45398i			0.472121i	0.971738	+	0.236060i	$0.0758563\pi$
$89$			4.45398i			0.472121i	−0.971738	+	0.236060i	$0.924144\pi$
$90$		0			0
$91$	4.70850			0.493585
$92$		0			0
$93$		−	7.82087i		−	0.810986i
$94$		0			0
$95$		−	14.8000i		−	1.51845i
$96$		0			0
$97$		−	6.13742i		−	0.623161i	−0.950220	−	0.311581i	$-0.899142\pi$
$97$		−	6.13742i		−	0.623161i	0.950220	−	0.311581i	$-0.100858\pi$
$98$		0			0
$99$			8.51350i			0.855639i
$100$		0			0
$101$			1.38527i			0.137839i	0.997622	+	0.0689196i	$0.0219552\pi$
$101$			1.38527i			0.137839i	−0.997622	+	0.0689196i	$0.978045\pi$
$102$		0			0
$103$	−4.00000			−0.394132			−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000			−0.394132			−0.197066	+	0.980390i	$0.563141\pi$
$104$		0			0
$105$	13.2915			1.29712
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$			14.2565i			1.36552i	0.730642	+	0.682761i	$0.239221\pi$
$109$			14.2565i			1.36552i	−0.730642	+	0.682761i	$0.760779\pi$
$110$		0			0
$111$			0.841723i			0.0798928i
$112$		0			0
$113$	−15.6458			−1.47183			−0.735914	−	0.677075i	$-0.763247\pi$
$113$	−15.6458			−1.47183			−0.735914	+	0.677075i	$0.763247\pi$
$114$		0			0
$115$	−26.5830			−2.47888
$116$		0			0
$117$		−	8.11905i		−	0.750606i
$118$		0			0
$119$	−12.0000			−1.10004
$120$		0			0
$121$	0.645751			0.0587047
$122$		0			0
$123$	11.2915	+	10.1969i	1.01812	+	0.919428i
$124$		0			0
$125$	−12.0000			−1.07331
$126$		0			0
$127$	15.2915			1.35690			0.678451	−	0.734646i	$-0.262652\pi$
$127$	15.2915			1.35690			0.678451	+	0.734646i	$0.262652\pi$
$128$		0			0
$129$		−	7.82087i		−	0.688589i
$130$		0			0
$131$	−2.58301			−0.225678			−0.112839	−	0.993613i	$-0.535994\pi$
$131$	−2.58301			−0.225678			−0.112839	+	0.993613i	$0.535994\pi$
$132$		0			0
$133$	6.22876			0.540102
$134$		0			0
$135$			3.06871i			0.264113i
$136$		0			0
$137$			14.5547i			1.24349i	0.783221	+	0.621744i	$0.213576\pi$
$137$			14.5547i			1.24349i	−0.783221	+	0.621744i	$0.786424\pi$
$138$		0			0
$139$	0.708497			0.0600940			0.0300470	−	0.999548i	$-0.490434\pi$
$139$	0.708497			0.0600940			0.0300470	+	0.999548i	$0.490434\pi$
$140$		0			0
$141$	12.9373			1.08951
$142$		0			0
$143$	9.87451			0.825748
$144$		0			0
$145$			17.3252i			1.43878i
$146$		0			0
$147$		−	11.0387i		−	0.910454i
$148$		0			0
$149$		−	10.8896i		−	0.892109i	−0.895006	−	0.446055i	$-0.852829\pi$
$149$		−	10.8896i		−	0.892109i	0.895006	−	0.446055i	$-0.147171\pi$
$150$		0			0
$151$		−	0.990812i		−	0.0806312i	−0.999187	−	0.0403156i	$-0.987164\pi$
$151$		−	0.990812i		−	0.0806312i	0.999187	−	0.0403156i	$-0.0128363\pi$
$152$		0			0
$153$			20.6921i			1.67286i
$154$		0			0
$155$	−12.0000			−0.963863
$156$		0			0
$157$			14.2565i			1.13779i	0.822410	+	0.568895i	$0.192629\pi$
$157$			14.2565i			1.13779i	−0.822410	+	0.568895i	$0.807371\pi$
$158$		0			0
$159$	15.2915			1.21270
$160$		0			0
$161$		−	11.1878i		−	0.881719i
$162$		0			0
$163$	22.5830			1.76884			0.884419	−	0.466694i	$-0.154555\pi$
$163$	22.5830			1.76884			0.884419	+	0.466694i	$0.154555\pi$
$164$		0			0
$165$	27.8745			2.17003
$166$		0			0
$167$		−	6.28651i		−	0.486465i	−0.969968	−	0.243233i	$-0.921792\pi$
$167$		−	6.28651i		−	0.486465i	0.969968	−	0.243233i	$-0.0782078\pi$
$168$		0			0
$169$	3.58301			0.275616
$170$		0			0
$171$		−	10.7405i		−	0.821346i
$172$		0			0
$173$	6.00000			0.456172			0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000			0.456172			0.228086	+	0.973641i	$0.426753\pi$
$174$		0			0
$175$		−	12.7221i		−	0.961702i
$176$		0			0
$177$			17.3252i			1.30224i
$178$		0			0
$179$			17.1761i			1.28380i	0.766788	+	0.641901i	$0.221854\pi$
$179$			17.1761i			1.28380i	−0.766788	+	0.641901i	$0.778146\pi$
$180$		0			0
$181$		−	25.4442i		−	1.89126i	−0.325251	−	0.945628i	$-0.605449\pi$
$181$		−	25.4442i		−	1.89126i	0.325251	−	0.945628i	$-0.394551\pi$
$182$		0			0
$183$			26.8295i			1.98329i
$184$		0			0
$185$	1.29150			0.0949532
$186$		0			0
$187$	−25.1660			−1.84032
$188$		0			0
$189$	−1.29150			−0.0939430
$190$		0			0
$191$		−	12.4239i		−	0.898965i	−0.893289	−	0.449482i	$-0.851608\pi$
$191$		−	12.4239i		−	0.898965i	0.893289	−	0.449482i	$-0.148392\pi$
$192$		0			0
$193$			6.13742i			0.441781i	0.975299	+	0.220891i	$0.0708964\pi$
$193$			6.13742i			0.441781i	−0.975299	+	0.220891i	$0.929104\pi$
$194$		0			0
$195$	−26.5830			−1.90365
$196$		0			0
$197$	9.87451			0.703530			0.351765	−	0.936088i	$-0.385582\pi$
$197$	9.87451			0.703530			0.351765	+	0.936088i	$0.385582\pi$
$198$		0			0
$199$			12.7221i			0.901847i	0.892563	+	0.450924i	$0.148905\pi$
$199$			12.7221i			0.901847i	−0.892563	+	0.450924i	$0.851095\pi$
$200$		0			0
$201$	−30.2288			−2.13217
$202$		0			0
$203$	−7.29150			−0.511763
$204$		0			0
$205$	15.6458	−	17.3252i	1.09275	−	1.21004i
$206$		0			0
$207$	−19.2915			−1.34085
$208$		0			0
$209$	13.0627			0.903569
$210$		0			0
$211$		−	10.7405i		−	0.739406i	−0.929150	−	0.369703i	$-0.879459\pi$
$211$		−	10.7405i		−	0.739406i	0.929150	−	0.369703i	$-0.120541\pi$
$212$		0			0
$213$	5.64575			0.386841
$214$		0			0
$215$	−12.0000			−0.818393
$216$		0			0
$217$		−	5.05034i		−	0.342839i
$218$		0			0
$219$		−	16.4835i		−	1.11385i
$220$		0			0
$221$	24.0000			1.61441
$222$		0			0
$223$	−18.5830			−1.24441			−0.622205	−	0.782854i	$-0.713763\pi$
$223$	−18.5830			−1.24441			−0.622205	+	0.782854i	$0.713763\pi$
$224$		0			0
$225$	−21.9373			−1.46248
$226$		0			0
$227$		−	5.74297i		−	0.381174i	−0.981670	−	0.190587i	$-0.938961\pi$
$227$		−	5.74297i		−	0.381174i	0.981670	−	0.190587i	$-0.0610392\pi$
$228$		0			0
$229$			1.98162i			0.130949i	0.997854	+	0.0654747i	$0.0208562\pi$
$229$			1.98162i			0.130949i	−0.997854	+	0.0654747i	$0.979144\pi$
$230$		0			0
$231$			11.7313i			0.771863i
$232$		0			0
$233$			26.8295i			1.75766i	0.477136	+	0.878830i	$0.341675\pi$
$233$			26.8295i			1.75766i	−0.477136	+	0.878830i	$0.658325\pi$
$234$		0			0
$235$		−	19.8504i		−	1.29489i
$236$		0			0
$237$	−24.2288			−1.57383
$238$		0			0
$239$			17.7196i			1.14619i	0.819490	+	0.573094i	$0.194257\pi$
$239$			17.7196i			1.14619i	−0.819490	+	0.573094i	$0.805743\pi$
$240$		0			0
$241$	12.7085			0.818626			0.409313	−	0.912394i	$-0.365768\pi$
$241$	12.7085			0.818626			0.409313	+	0.912394i	$0.365768\pi$
$242$		0			0
$243$			21.0865i			1.35270i
$244$		0			0
$245$	−16.9373			−1.08208
$246$		0			0
$247$	−12.4575			−0.792653
$248$		0			0
$249$		−	28.5129i		−	1.80693i
$250$		0			0
$251$	14.5830			0.920471			0.460236	−	0.887797i	$-0.347765\pi$
$251$	14.5830			0.920471			0.460236	+	0.887797i	$0.347765\pi$
$252$		0			0
$253$		−	23.4626i		−	1.47508i
$254$		0			0
$255$	67.7490			4.24261
$256$		0			0
$257$		−	7.82087i		−	0.487852i	−0.969794	−	0.243926i	$-0.921565\pi$
$257$		−	7.82087i		−	0.487852i	0.969794	−	0.243926i	$-0.0784355\pi$
$258$		0			0
$259$			0.543544i			0.0337742i
$260$		0			0
$261$			12.5730i			0.778251i
$262$		0			0
$263$		−	9.89877i		−	0.610384i	−0.952291	−	0.305192i	$-0.901279\pi$
$263$		−	9.89877i		−	0.610384i	0.952291	−	0.305192i	$-0.0987207\pi$
$264$		0			0
$265$		−	23.4626i		−	1.44130i
$266$		0			0
$267$	10.5830			0.647669
$268$		0			0
$269$	4.70850			0.287082			0.143541	−	0.989644i	$-0.454151\pi$
$269$	4.70850			0.287082			0.143541	+	0.989644i	$0.454151\pi$
$270$		0			0
$271$	−1.87451			−0.113868			−0.0569341	−	0.998378i	$-0.518132\pi$
$271$	−1.87451			−0.113868			−0.0569341	+	0.998378i	$0.518132\pi$
$272$		0			0
$273$		−	11.1878i		−	0.677114i
$274$		0			0
$275$		−	26.6804i		−	1.60889i
$276$		0			0
$277$	−19.6458			−1.18040			−0.590199	−	0.807257i	$-0.700951\pi$
$277$	−19.6458			−1.18040			−0.590199	+	0.807257i	$0.700951\pi$
$278$		0			0
$279$	−8.70850			−0.521364
$280$		0			0
$281$		−	2.77053i		−	0.165276i	−0.996580	−	0.0826381i	$-0.973665\pi$
$281$		−	2.77053i		−	0.165276i	0.996580	−	0.0826381i	$-0.0263345\pi$
$282$		0			0
$283$	5.87451			0.349203			0.174602	−	0.984639i	$-0.444136\pi$
$283$	5.87451			0.349203			0.174602	+	0.984639i	$0.444136\pi$
$284$		0			0
$285$	−35.1660			−2.08305
$286$		0			0
$287$	7.29150	+	6.58469i	0.430404	+	0.388682i
$288$		0			0
$289$	−44.1660			−2.59800
$290$		0			0
$291$	−14.5830			−0.854871
$292$		0			0
$293$			7.52269i			0.439480i	0.975558	+	0.219740i	$0.0705210\pi$
$293$			7.52269i			0.439480i	−0.975558	+	0.219740i	$0.929479\pi$
$294$		0			0
$295$	26.5830			1.54772
$296$		0			0
$297$	−2.70850			−0.157163
$298$		0			0
$299$			22.3755i			1.29401i
$300$		0			0
$301$		−	5.05034i		−	0.291097i
$302$		0			0
$303$	3.29150			0.189092
$304$		0			0
$305$	41.1660			2.35716
$306$		0			0
$307$	−18.5830			−1.06059			−0.530294	−	0.847814i	$-0.677918\pi$
$307$	−18.5830			−1.06059			−0.530294	+	0.847814i	$0.677918\pi$
$308$		0			0
$309$			9.50432i			0.540682i
$310$		0			0
$311$			14.1074i			0.799956i	0.916525	+	0.399978i	$0.130982\pi$
$311$			14.1074i			0.799956i	−0.916525	+	0.399978i	$0.869018\pi$
$312$		0			0
$313$			6.13742i			0.346908i	0.984842	+	0.173454i	$0.0554928\pi$
$313$			6.13742i			0.346908i	−0.984842	+	0.173454i	$0.944507\pi$
$314$		0			0
$315$		−	14.8000i		−	0.833887i
$316$		0			0
$317$		−	3.66507i		−	0.205851i	−0.994689	−	0.102925i	$-0.967180\pi$
$317$		−	3.66507i		−	0.205851i	0.994689	−	0.102925i	$-0.0328203\pi$
$318$		0			0
$319$	−15.2915			−0.856160
$320$		0			0
$321$		−	28.5129i		−	1.59144i
$322$		0			0
$323$	31.7490			1.76656
$324$		0			0
$325$			25.4442i			1.41139i
$326$		0			0
$327$	33.8745			1.87326
$328$		0			0
$329$	8.35425			0.460585
$330$		0			0
$331$			7.67178i			0.421679i	0.977521	+	0.210840i	$0.0676198\pi$
$331$			7.67178i			0.421679i	−0.977521	+	0.210840i	$0.932380\pi$
$332$		0			0
$333$	0.937254			0.0513612
$334$		0			0
$335$			46.3817i			2.53410i
$336$		0			0
$337$	−2.93725			−0.160002			−0.0800012	−	0.996795i	$-0.525492\pi$
$337$	−2.93725			−0.160002			−0.0800012	+	0.996795i	$0.525492\pi$
$338$		0			0
$339$			37.1755i			2.01910i
$340$		0			0
$341$		−	10.5914i		−	0.573557i
$342$		0			0
$343$		−	17.8687i		−	0.964821i
$344$		0			0
$345$			63.1633i			3.40060i
$346$		0			0
$347$		−	8.81168i		−	0.473036i	−0.971627	−	0.236518i	$-0.923994\pi$
$347$		−	8.81168i		−	0.473036i	0.971627	−	0.236518i	$-0.0760062\pi$
$348$		0			0
$349$	−10.2288			−0.547533			−0.273766	−	0.961796i	$-0.588270\pi$
$349$	−10.2288			−0.547533			−0.273766	+	0.961796i	$0.588270\pi$
$350$		0			0
$351$	2.58301			0.137871
$352$		0			0
$353$	−10.9373			−0.582131			−0.291066	−	0.956703i	$-0.594010\pi$
$353$	−10.9373			−0.582131			−0.291066	+	0.956703i	$0.594010\pi$
$354$		0			0
$355$		−	8.66259i		−	0.459763i
$356$		0			0
$357$			28.5129i			1.50907i
$358$		0			0
$359$	−9.41699			−0.497010			−0.248505	−	0.968631i	$-0.579939\pi$
$359$	−9.41699			−0.497010			−0.248505	+	0.968631i	$0.579939\pi$
$360$		0			0
$361$	2.52026			0.132645
$362$		0			0
$363$		−	1.53436i		−	0.0805328i
$364$		0			0
$365$	−25.2915			−1.32382
$366$		0			0
$367$	0.708497			0.0369833			0.0184916	−	0.999829i	$-0.494114\pi$
$367$	0.708497			0.0369833			0.0184916	+	0.999829i	$0.494114\pi$
$368$		0			0
$369$	11.3542	−	12.5730i	0.591079	−	0.654526i
$370$		0			0
$371$	9.87451			0.512659
$372$		0			0
$373$	26.0000			1.34623			0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000			1.34623			0.673114	+	0.739538i	$0.264956\pi$
$374$		0			0
$375$			28.5129i			1.47240i
$376$		0			0
$377$	14.5830			0.751063
$378$		0			0
$379$	5.87451			0.301753			0.150877	−	0.988553i	$-0.451790\pi$
$379$	5.87451			0.301753			0.150877	+	0.988553i	$0.451790\pi$
$380$		0			0
$381$		−	36.3338i		−	1.86144i
$382$		0			0
$383$		−	36.7811i		−	1.87943i	−0.341965	−	0.939713i	$-0.611092\pi$
$383$		−	36.7811i		−	1.87943i	0.341965	−	0.939713i	$-0.388908\pi$
$384$		0			0
$385$	18.0000			0.917365
$386$		0			0
$387$	−8.70850			−0.442678
$388$		0			0
$389$	−18.0000			−0.912636			−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000			−0.912636			−0.456318	+	0.889817i	$0.650832\pi$
$390$		0			0
$391$		−	57.0259i		−	2.88392i
$392$		0			0
$393$			6.13742i			0.309592i
$394$		0			0
$395$			37.1755i			1.87050i
$396$		0			0
$397$		−	3.06871i		−	0.154014i	−0.997031	−	0.0770071i	$-0.975464\pi$
$397$		−	3.06871i		−	0.154014i	0.997031	−	0.0770071i	$-0.0245364\pi$
$398$		0			0
$399$		−	14.8000i		−	0.740928i
$400$		0			0
$401$	−25.5203			−1.27442			−0.637210	−	0.770690i	$-0.719912\pi$
$401$	−25.5203			−1.27442			−0.637210	+	0.770690i	$0.719912\pi$
$402$		0			0
$403$			10.1007i			0.503150i
$404$		0			0
$405$	36.2288			1.80022
$406$		0			0
$407$			1.13990i			0.0565028i
$408$		0			0
$409$	21.5203			1.06411			0.532054	−	0.846710i	$-0.321420\pi$
$409$	21.5203			1.06411			0.532054	+	0.846710i	$0.321420\pi$
$410$		0			0
$411$	34.5830			1.70585
$412$		0			0
$413$			11.1878i			0.550514i
$414$		0			0
$415$	−43.7490			−2.14755
$416$		0			0
$417$		−	1.68345i		−	0.0824387i
$418$		0			0
$419$	4.70850			0.230025			0.115013	−	0.993364i	$-0.463309\pi$
$419$	4.70850			0.230025			0.115013	+	0.993364i	$0.463309\pi$
$420$		0			0
$421$			31.5817i			1.53920i	0.638529	+	0.769598i	$0.279543\pi$
$421$			31.5817i			1.53920i	−0.638529	+	0.769598i	$0.720457\pi$
$422$		0			0
$423$		−	14.4056i		−	0.700422i
$424$		0			0
$425$		−	64.8468i		−	3.14553i
$426$		0			0
$427$			17.3252i			0.838425i
$428$		0			0
$429$		−	23.4626i		−	1.13279i
$430$		0			0
$431$	−31.2915			−1.50726			−0.753629	−	0.657300i	$-0.771699\pi$
$431$	−31.2915			−1.50726			−0.753629	+	0.657300i	$0.771699\pi$
$432$		0			0
$433$	−0.583005			−0.0280174			−0.0140087	−	0.999902i	$-0.504459\pi$
$433$	−0.583005			−0.0280174			−0.0140087	+	0.999902i	$0.504459\pi$
$434$		0			0
$435$	41.1660			1.97376
$436$		0			0
$437$			29.6000i			1.41596i
$438$		0			0
$439$		−	9.65341i		−	0.460732i	−0.973104	−	0.230366i	$-0.926008\pi$
$439$		−	9.65341i		−	0.460732i	0.973104	−	0.230366i	$-0.0739923\pi$
$440$		0			0
$441$	−12.2915			−0.585310
$442$		0			0
$443$		0			0			−	1.00000i	$-0.5\pi$
$443$		0			0				1.00000i	$0.5\pi$
$444$		0			0
$445$		−	16.2381i		−	0.769760i
$446$		0			0
$447$	−25.8745			−1.22382
$448$		0			0
$449$	−4.70850			−0.222208			−0.111104	−	0.993809i	$-0.535439\pi$
$449$	−4.70850			−0.222208			−0.111104	+	0.993809i	$0.535439\pi$
$450$		0			0
$451$	15.2915	+	13.8092i	0.720049	+	0.650250i
$452$		0			0
$453$	−2.35425			−0.110612
$454$		0			0
$455$	−17.1660			−0.804755
$456$		0			0
$457$		−	11.1878i		−	0.523341i	−0.965157	−	0.261671i	$-0.915727\pi$
$457$		−	11.1878i		−	0.523341i	0.965157	−	0.261671i	$-0.0842734\pi$
$458$		0			0
$459$	−6.58301			−0.307268
$460$		0			0
$461$	1.06275			0.0494970			0.0247485	−	0.999694i	$-0.492122\pi$
$461$	1.06275			0.0494970			0.0247485	+	0.999694i	$0.492122\pi$
$462$		0			0
$463$			15.2473i			0.708601i	0.935132	+	0.354301i	$0.115281\pi$
$463$			15.2473i			0.708601i	−0.935132	+	0.354301i	$0.884719\pi$
$464$		0			0
$465$			28.5129i			1.32226i
$466$		0			0
$467$	−9.87451			−0.456938			−0.228469	−	0.973551i	$-0.573372\pi$
$467$	−9.87451			−0.456938			−0.228469	+	0.973551i	$0.573372\pi$
$468$		0			0
$469$	−19.5203			−0.901362
$470$		0			0
$471$	33.8745			1.56085
$472$		0			0
$473$		−	10.5914i		−	0.486993i
$474$		0			0
$475$			33.6596i			1.54441i
$476$		0			0
$477$		−	17.0270i		−	0.779613i
$478$		0			0
$479$			7.96996i			0.364157i	0.983284	+	0.182078i	$0.0582825\pi$
$479$			7.96996i			0.364157i	−0.983284	+	0.182078i	$0.941718\pi$
$480$		0			0
$481$		−	1.08709i		−	0.0495669i
$482$		0			0
$483$	−26.5830			−1.20957
$484$		0			0
$485$			22.3755i			1.01602i
$486$		0			0
$487$	−35.7490			−1.61994			−0.809971	−	0.586470i	$-0.800517\pi$
$487$	−35.7490			−1.61994			−0.809971	+	0.586470i	$0.800517\pi$
$488$		0			0
$489$		−	53.6590i		−	2.42654i
$490$		0			0
$491$	−21.8745			−0.987183			−0.493591	−	0.869694i	$-0.664316\pi$
$491$	−21.8745			−0.987183			−0.493591	+	0.869694i	$0.664316\pi$
$492$		0			0
$493$	−37.1660			−1.67387
$494$		0			0
$495$		−	31.0381i		−	1.39506i
$496$		0			0
$497$	3.64575			0.163534
$498$		0			0
$499$			25.5405i			1.14335i	0.820480	+	0.571675i	$0.193706\pi$
$499$			25.5405i			1.14335i	−0.820480	+	0.571675i	$0.806294\pi$
$500$		0			0
$501$	−14.9373			−0.667347
$502$		0			0
$503$		−	0.692633i		−	0.0308830i	−0.999881	−	0.0154415i	$-0.995085\pi$
$503$		−	0.692633i		−	0.0308830i	0.999881	−	0.0154415i	$-0.00491538\pi$
$504$		0			0
$505$		−	5.05034i		−	0.224737i
$506$		0			0
$507$		−	8.51350i		−	0.378098i
$508$		0			0
$509$		−	15.9399i		−	0.706525i	−0.935524	−	0.353262i	$-0.885072\pi$
$509$		−	15.9399i		−	0.706525i	0.935524	−	0.353262i	$-0.114928\pi$
$510$		0			0
$511$		−	10.6442i		−	0.470872i
$512$		0			0
$513$	3.41699			0.150864
$514$		0			0
$515$	14.5830			0.642604
$516$		0			0
$517$	17.5203			0.770540
$518$		0			0
$519$		−	14.2565i		−	0.625790i
$520$		0			0
$521$			38.0173i			1.66557i	0.553599	+	0.832783i	$0.313254\pi$
$521$			38.0173i			1.66557i	−0.553599	+	0.832783i	$0.686746\pi$
$522$		0			0
$523$	10.5830			0.462763			0.231381	−	0.972863i	$-0.425676\pi$
$523$	10.5830			0.462763			0.231381	+	0.972863i	$0.425676\pi$
$524$		0			0
$525$	−30.2288			−1.31929
$526$		0			0
$527$		−	25.7424i		−	1.12136i
$528$		0			0
$529$	30.1660			1.31157
$530$		0			0
$531$	19.2915			0.837180
$532$		0			0
$533$	−14.5830	−	13.1694i	−0.631660	−	0.570429i
$534$		0			0
$535$	−43.7490			−1.89143
$536$		0			0
$537$	40.8118			1.76116
$538$		0			0
$539$		−	14.9491i		−	0.643904i
$540$		0			0
$541$	−5.06275			−0.217664			−0.108832	−	0.994060i	$-0.534711\pi$
$541$	−5.06275			−0.217664			−0.108832	+	0.994060i	$0.534711\pi$
$542$		0			0
$543$	−60.4575			−2.59448
$544$		0			0
$545$		−	51.9756i		−	2.22639i
$546$		0			0
$547$		−	42.8657i		−	1.83280i	−0.400258	−	0.916402i	$-0.631080\pi$
$547$		−	42.8657i		−	1.83280i	0.400258	−	0.916402i	$-0.368920\pi$
$548$		0			0
$549$	29.8745			1.27501
$550$		0			0
$551$	19.2915			0.821846
$552$		0			0
$553$	−15.6458			−0.665325
$554$		0			0
$555$		−	3.06871i		−	0.130260i
$556$		0			0
$557$			34.9486i			1.48082i	0.672157	+	0.740409i	$0.265368\pi$
$557$			34.9486i			1.48082i	−0.672157	+	0.740409i	$0.734632\pi$
$558$		0			0
$559$			10.1007i			0.427213i
$560$		0			0
$561$			59.7964i			2.52461i
$562$		0			0
$563$		−	9.35523i		−	0.394276i	−0.980376	−	0.197138i	$-0.936835\pi$
$563$		−	9.35523i		−	0.394276i	0.980376	−	0.197138i	$-0.0631647\pi$
$564$		0			0
$565$	57.0405			2.39971
$566$		0			0
$567$			15.2473i			0.640326i
$568$		0			0
$569$	−10.9373			−0.458514			−0.229257	−	0.973366i	$-0.573630\pi$
$569$	−10.9373			−0.458514			−0.229257	+	0.973366i	$0.573630\pi$
$570$		0			0
$571$			15.7908i			0.660826i	0.943837	+	0.330413i	$0.107188\pi$
$571$			15.7908i			0.660826i	−0.943837	+	0.330413i	$0.892812\pi$
$572$		0			0
$573$	−29.5203			−1.23323
$574$		0			0
$575$	60.4575			2.52125
$576$		0			0
$577$		−	29.6000i		−	1.23227i	−0.787642	−	0.616133i	$-0.788698\pi$
$577$		−	29.6000i		−	1.23227i	0.787642	−	0.616133i	$-0.211302\pi$
$578$		0			0
$579$	14.5830			0.606049
$580$		0			0
$581$		−	18.4123i		−	0.763870i
$582$		0			0
$583$	20.7085			0.857658
$584$		0			0
$585$			29.6000i			1.22381i
$586$		0			0
$587$			45.6890i			1.88579i	0.333092	+	0.942894i	$0.391908\pi$
$587$			45.6890i			1.88579i	−0.333092	+	0.942894i	$0.608092\pi$
$588$		0			0
$589$			13.3619i			0.550569i
$590$		0			0
$591$		−	23.4626i		−	0.965123i
$592$		0			0
$593$		−	1.68345i		−	0.0691308i	−0.999402	−	0.0345654i	$-0.988995\pi$
$593$		−	1.68345i		−	0.0691308i	0.999402	−	0.0345654i	$-0.0110047\pi$
$594$		0			0
$595$	43.7490			1.79353
$596$		0			0
$597$	30.2288			1.23718
$598$		0			0
$599$	2.12549			0.0868453			0.0434226	−	0.999057i	$-0.486174\pi$
$599$	2.12549			0.0868453			0.0434226	+	0.999057i	$0.486174\pi$
$600$		0			0
$601$			27.4259i			1.11872i	0.828923	+	0.559362i	$0.188954\pi$
$601$			27.4259i			1.11872i	−0.828923	+	0.559362i	$0.811046\pi$
$602$		0			0
$603$			33.6596i			1.37072i
$604$		0			0
$605$	−2.35425			−0.0957138
$606$		0			0
$607$	−25.8745			−1.05021			−0.525107	−	0.851036i	$-0.675975\pi$
$607$	−25.8745			−1.05021			−0.525107	+	0.851036i	$0.675975\pi$
$608$		0			0
$609$			17.3252i			0.702052i
$610$		0			0
$611$	−16.7085			−0.675953
$612$		0			0
$613$	38.6863			1.56252			0.781262	−	0.624203i	$-0.214576\pi$
$613$	38.6863			1.56252			0.781262	+	0.624203i	$0.214576\pi$
$614$		0			0
$615$	−41.1660	−	37.1755i	−1.65997	−	1.49906i
$616$		0			0
$617$	40.3320			1.62371			0.811853	−	0.583862i	$-0.198459\pi$
$617$	40.3320			1.62371			0.811853	+	0.583862i	$0.198459\pi$
$618$		0			0
$619$	8.45751			0.339936			0.169968	−	0.985450i	$-0.445634\pi$
$619$	8.45751			0.339936			0.169968	+	0.985450i	$0.445634\pi$
$620$		0			0
$621$		−	6.13742i		−	0.246286i
$622$		0			0
$623$	6.83399			0.273798
$624$		0			0
$625$	2.29150			0.0916601
$626$		0			0
$627$		−	31.0381i		−	1.23954i
$628$		0			0
$629$			2.77053i			0.110468i
$630$		0			0
$631$	−13.4170			−0.534122			−0.267061	−	0.963680i	$-0.586053\pi$
$631$	−13.4170			−0.534122			−0.267061	+	0.963680i	$0.586053\pi$
$632$		0			0
$633$	−25.5203			−1.01434
$634$		0			0
$635$	−55.7490			−2.21233
$636$		0			0
$637$			14.2565i			0.564862i
$638$		0			0
$639$		−	6.28651i		−	0.248691i
$640$		0			0
$641$			34.0540i			1.34505i	0.740073	+	0.672526i	$0.234791\pi$
$641$			34.0540i			1.34505i	−0.740073	+	0.672526i	$0.765209\pi$
$642$		0			0
$643$			23.9099i			0.942914i	0.881889	+	0.471457i	$0.156272\pi$
$643$			23.9099i			0.942914i	−0.881889	+	0.471457i	$0.843728\pi$
$644$		0			0
$645$			28.5129i			1.12270i
$646$		0			0
$647$	46.3320			1.82150			0.910750	−	0.412958i	$-0.135505\pi$
$647$	46.3320			1.82150			0.910750	+	0.412958i	$0.135505\pi$
$648$		0			0
$649$			23.4626i			0.920988i
$650$		0			0
$651$	−12.0000			−0.470317
$652$		0			0
$653$			23.7608i			0.929832i	0.885355	+	0.464916i	$0.153915\pi$
$653$			23.7608i			0.929832i	−0.885355	+	0.464916i	$0.846085\pi$
$654$		0			0
$655$	9.41699			0.367952
$656$		0			0
$657$	−18.3542			−0.716067
$658$		0			0
$659$		−	27.2240i		−	1.06049i	−0.847843	−	0.530247i	$-0.822099\pi$
$659$		−	27.2240i		−	1.06049i	0.847843	−	0.530247i	$-0.177901\pi$
$660$		0			0
$661$	−24.3542			−0.947270			−0.473635	−	0.880721i	$-0.657058\pi$
$661$	−24.3542			−0.947270			−0.473635	+	0.880721i	$0.657058\pi$
$662$		0			0
$663$		−	57.0259i		−	2.21470i
$664$		0			0
$665$	−22.7085			−0.880598
$666$		0			0
$667$		−	34.6504i		−	1.34167i
$668$		0			0
$669$			44.1547i			1.70712i
$670$		0			0
$671$			36.3338i			1.40265i
$672$		0			0
$673$		−	1.08709i		−	0.0419041i	−0.999780	−	0.0209521i	$-0.993330\pi$
$673$		−	1.08709i		−	0.0419041i	0.999780	−	0.0209521i	$-0.00666974\pi$
$674$		0			0
$675$		−	6.97915i		−	0.268628i
$676$		0			0
$677$	−8.35425			−0.321080			−0.160540	−	0.987029i	$-0.551324\pi$
$677$	−8.35425			−0.321080			−0.160540	+	0.987029i	$0.551324\pi$
$678$		0			0
$679$	−9.41699			−0.361391
$680$		0			0
$681$	−13.6458			−0.522906
$682$		0			0
$683$			3.46317i			0.132514i	0.997803	+	0.0662572i	$0.0211058\pi$
$683$			3.46317i			0.132514i	−0.997803	+	0.0662572i	$0.978894\pi$
$684$		0			0
$685$		−	53.0626i		−	2.02742i
$686$		0			0
$687$	4.70850			0.179640
$688$		0			0
$689$	−19.7490			−0.752378
$690$		0			0
$691$			26.4350i			1.00564i	0.864392	+	0.502818i	$0.167704\pi$
$691$			26.4350i			1.00564i	−0.864392	+	0.502818i	$0.832296\pi$
$692$		0			0
$693$	13.0627			0.496213
$694$		0			0
$695$	−2.58301			−0.0979790
$696$		0			0
$697$	37.1660	+	33.5633i	1.40776	+	1.27130i
$698$		0			0
$699$	63.7490			2.41121
$700$		0			0
$701$	30.2288			1.14172			0.570862	−	0.821046i	$-0.306609\pi$
$701$	30.2288			1.14172			0.570862	+	0.821046i	$0.306609\pi$
$702$		0			0
$703$		−	1.43808i		−	0.0542383i
$704$		0			0
$705$	−47.1660			−1.77637
$706$		0			0
$707$	2.12549			0.0799374
$708$		0			0
$709$			26.5313i			0.996405i	0.867061	+	0.498202i	$0.166006\pi$
$709$			26.5313i			0.996405i	−0.867061	+	0.498202i	$0.833994\pi$
$710$		0			0
$711$			26.9786i			1.01178i
$712$		0			0
$713$	24.0000			0.898807
$714$		0			0
$715$	−36.0000			−1.34632
$716$		0			0
$717$	42.1033			1.57238
$718$		0			0
$719$			27.8203i			1.03752i	0.854919	+	0.518761i	$0.173607\pi$
$719$			27.8203i			1.03752i	−0.854919	+	0.518761i	$0.826393\pi$
$720$		0			0
$721$			6.13742i			0.228570i
$722$		0			0
$723$		−	30.1964i		−	1.12302i
$724$		0			0
$725$		−	39.4025i		−	1.46337i
$726$		0			0
$727$		−	26.4350i		−	0.980422i	−0.871604	−	0.490211i	$-0.836920\pi$
$727$		−	26.4350i		−	0.980422i	0.871604	−	0.490211i	$-0.163080\pi$
$728$		0			0
$729$	20.2915			0.751537
$730$		0			0
$731$		−	25.7424i		−	0.952118i
$732$		0			0
$733$	−16.4575			−0.607872			−0.303936	−	0.952692i	$-0.598301\pi$
$733$	−16.4575			−0.607872			−0.303936	+	0.952692i	$0.598301\pi$
$734$		0			0
$735$			40.2443i			1.48443i
$736$		0			0
$737$	−40.9373			−1.50794
$738$		0			0
$739$	25.1660			0.925747			0.462873	−	0.886424i	$-0.346818\pi$
$739$	25.1660			0.925747			0.462873	+	0.886424i	$0.346818\pi$
$740$		0			0
$741$			29.6000i			1.08738i
$742$		0			0
$743$	36.0000			1.32071			0.660356	−	0.750953i	$-0.270405\pi$
$743$	36.0000			1.32071			0.660356	+	0.750953i	$0.270405\pi$
$744$		0			0
$745$			39.7007i			1.45452i
$746$		0			0
$747$	−31.7490			−1.16164
$748$		0			0
$749$		−	18.4123i		−	0.672770i
$750$		0			0
$751$		−	5.69016i		−	0.207637i	−0.994596	−	0.103818i	$-0.966894\pi$
$751$		−	5.69016i		−	0.207637i	0.994596	−	0.103818i	$-0.0331061\pi$
$752$		0			0
$753$		−	34.6504i		−	1.26273i
$754$		0			0
$755$			3.61226i			0.131463i
$756$		0			0
$757$		−	8.11905i		−	0.295092i	−0.989055	−	0.147546i	$-0.952863\pi$
$757$		−	8.11905i		−	0.295092i	0.989055	−	0.147546i	$-0.0471374\pi$
$758$		0			0
$759$	−55.7490			−2.02356
$760$		0			0
$761$	22.9373			0.831475			0.415737	−	0.909485i	$-0.363524\pi$
$761$	22.9373			0.831475			0.415737	+	0.909485i	$0.363524\pi$
$762$		0			0
$763$	21.8745			0.791910
$764$		0			0
$765$		−	75.4382i		−	2.72747i
$766$		0			0
$767$		−	22.3755i		−	0.807933i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	−18.5830			−0.669251
$772$		0			0
$773$		−	34.3522i		−	1.23556i	−0.786350	−	0.617781i	$-0.788032\pi$
$773$		−	34.3522i		−	1.23556i	0.786350	−	0.617781i	$-0.211968\pi$
$774$		0			0
$775$	27.2915			0.980340
$776$		0			0
$777$	1.29150			0.0463324
$778$		0			0
$779$	−19.2915	−	17.4215i	−0.691190	−	0.624189i
$780$		0			0
$781$	7.64575			0.273586
$782$		0			0
$783$	−4.00000			−0.142948
$784$		0			0
$785$		−	51.9756i		−	1.85509i
$786$		0			0
$787$	39.7490			1.41690			0.708450	−	0.705761i	$-0.249395\pi$
$787$	39.7490			1.41690			0.708450	+	0.705761i	$0.249395\pi$
$788$		0			0
$789$	−23.5203			−0.837343
$790$		0			0
$791$			24.0062i			0.853561i
$792$		0			0
$793$		−	34.6504i		−	1.23047i
$794$		0			0
$795$	−55.7490			−1.97721
$796$		0			0
$797$	28.7085			1.01691			0.508454	−	0.861089i	$-0.330217\pi$
$797$	28.7085			1.01691			0.508454	+	0.861089i	$0.330217\pi$
$798$		0			0
$799$	42.5830			1.50648
$800$		0			0
$801$		−	11.7841i		−	0.416371i
$802$		0			0
$803$		−	22.3227i		−	0.787751i
$804$		0			0
$805$			40.7878i			1.43758i
$806$		0			0
$807$		−	11.1878i		−	0.393828i
$808$		0			0
$809$		−	17.9215i		−	0.630088i	−0.949077	−	0.315044i	$-0.897981\pi$
$809$		−	17.9215i		−	0.630088i	0.949077	−	0.315044i	$-0.102019\pi$
$810$		0			0
$811$	−16.0000			−0.561836			−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000			−0.561836			−0.280918	+	0.959732i	$0.590639\pi$
$812$		0			0
$813$			4.45398i			0.156208i
$814$		0			0
$815$	−82.3320			−2.88396
$816$		0			0
$817$			13.3619i			0.467475i
$818$		0			0
$819$	−12.4575			−0.435301
$820$		0			0
$821$	−8.35425			−0.291565			−0.145783	−	0.989317i	$-0.546570\pi$
$821$	−8.35425			−0.291565			−0.145783	+	0.989317i	$0.546570\pi$
$822$		0			0
$823$		−	4.05952i		−	0.141506i	−0.997494	−	0.0707531i	$-0.977460\pi$
$823$		−	4.05952i		−	0.141506i	0.997494	−	0.0707531i	$-0.0225402\pi$
$824$		0			0
$825$	−63.3948			−2.20712
$826$		0			0
$827$		−	1.23618i		−	0.0429861i	−0.999769	−	0.0214930i	$-0.993158\pi$
$827$		−	1.23618i		−	0.0429861i	0.999769	−	0.0214930i	$-0.00684197\pi$
$828$		0			0
$829$	−34.2288			−1.18881			−0.594407	−	0.804164i	$-0.702613\pi$
$829$	−34.2288			−1.18881			−0.594407	+	0.804164i	$0.702613\pi$
$830$		0			0
$831$			46.6799i			1.61931i
$832$		0			0
$833$		−	36.3338i		−	1.25889i
$834$		0			0
$835$			22.9191i			0.793147i
$836$		0			0
$837$		−	2.77053i		−	0.0957636i
$838$		0			0
$839$		−	10.4423i		−	0.360509i	−0.983620	−	0.180254i	$-0.942308\pi$
$839$		−	10.4423i		−	0.360509i	0.983620	−	0.180254i	$-0.0576921\pi$
$840$		0			0
$841$	6.41699			0.221276
$842$		0			0
$843$	−6.58301			−0.226731
$844$		0			0
$845$	−13.0627			−0.449372
$846$		0			0
$847$		−	0.990812i		−	0.0340447i
$848$		0			0
$849$		−	13.9583i		−	0.479047i
$850$		0			0
$851$	−2.58301			−0.0885443
$852$		0			0
$853$	−27.1660			−0.930146			−0.465073	−	0.885272i	$-0.653972\pi$
$853$	−27.1660			−0.930146			−0.465073	+	0.885272i	$0.653972\pi$
$854$		0			0
$855$			39.1572i			1.33915i
$856$		0			0
$857$	−25.5203			−0.871755			−0.435878	−	0.900006i	$-0.643562\pi$
$857$	−25.5203			−0.871755			−0.435878	+	0.900006i	$0.643562\pi$
$858$		0			0
$859$	51.7490			1.76565			0.882827	−	0.469699i	$-0.155637\pi$
$859$	51.7490			1.76565			0.882827	+	0.469699i	$0.155637\pi$
$860$		0			0
$861$	15.6458	−	17.3252i	0.533206	−	0.590441i
$862$		0			0
$863$	−24.4575			−0.832543			−0.416272	−	0.909240i	$-0.636663\pi$
$863$	−24.4575			−0.832543			−0.416272	+	0.909240i	$0.636663\pi$
$864$		0			0
$865$	−21.8745			−0.743756
$866$		0			0
$867$			104.942i			3.56401i
$868$		0			0
$869$	−32.8118			−1.11306
$870$		0			0
$871$	39.0405			1.32284
$872$		0			0
$873$			16.2381i			0.549576i
$874$		0			0
$875$			18.4123i			0.622448i
$876$		0			0
$877$	−11.2915			−0.381287			−0.190643	−	0.981659i	$-0.561057\pi$
$877$	−11.2915			−0.381287			−0.190643	+	0.981659i	$0.561057\pi$
$878$		0			0
$879$	17.8745			0.602892
$880$		0			0
$881$	37.7490			1.27180			0.635898	−	0.771773i	$-0.280630\pi$
$881$	37.7490			1.27180			0.635898	+	0.771773i	$0.280630\pi$
$882$		0			0
$883$			13.2657i			0.446425i	0.974770	+	0.223213i	$0.0716544\pi$
$883$			13.2657i			0.446425i	−0.974770	+	0.223213i	$0.928346\pi$
$884$		0			0
$885$		−	63.1633i		−	2.12321i
$886$		0			0
$887$		−	28.6620i		−	0.962377i	−0.876617	−	0.481189i	$-0.840205\pi$
$887$		−	28.6620i		−	0.962377i	0.876617	−	0.481189i	$-0.159795\pi$
$888$		0			0
$889$		−	23.4626i		−	0.786911i
$890$		0			0
$891$			31.9761i			1.07124i
$892$		0			0
$893$	−22.1033			−0.739658
$894$		0			0
$895$		−	62.6198i		−	2.09315i
$896$		0			0
$897$	53.1660			1.77516
$898$		0			0
$899$		−	15.6417i		−	0.521681i
$900$		0			0
$901$	50.3320			1.67680
$902$		0			0
$903$	−12.0000			−0.399335
$904$		0			0
$905$			92.7634i			3.08356i
$906$		0			0
$907$	8.00000			0.265636			0.132818	−	0.991140i	$-0.457597\pi$
$907$	8.00000			0.265636			0.132818	+	0.991140i	$0.457597\pi$
$908$		0			0
$909$		−	3.66507i		−	0.121563i
$910$		0			0
$911$	−28.7085			−0.951155			−0.475577	−	0.879674i	$-0.657761\pi$
$911$	−28.7085			−0.951155			−0.475577	+	0.879674i	$0.657761\pi$
$912$		0			0
$913$		−	38.6136i		−	1.27792i
$914$		0			0
$915$		−	97.8137i		−	3.23362i
$916$		0			0
$917$			3.96325i			0.130878i
$918$		0			0
$919$			57.6657i			1.90222i	0.308858	+	0.951108i	$0.400053\pi$
$919$			57.6657i			1.90222i	−0.308858	+	0.951108i	$0.599947\pi$
$920$		0			0
$921$			44.1547i			1.45495i
$922$		0			0
$923$	−7.29150			−0.240003
$924$		0			0
$925$	−2.93725			−0.0965763
$926$		0			0
$927$	10.5830			0.347591
$928$		0			0
$929$			36.9302i			1.21164i	0.795602	+	0.605820i	$0.207155\pi$
$929$			36.9302i			1.21164i	−0.795602	+	0.605820i	$0.792845\pi$
$930$		0			0
$931$			18.8595i			0.618097i
$932$		0			0
$933$	33.5203			1.09740
$934$		0			0
$935$	91.7490			3.00051
$936$		0			0
$937$		−	6.13742i		−	0.200501i	−0.994962	−	0.100250i	$-0.968036\pi$
$937$		−	6.13742i		−	0.200501i	0.994962	−	0.100250i	$-0.0319644\pi$
$938$		0			0
$939$	14.5830			0.475898
$940$		0			0
$941$	−6.00000			−0.195594			−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000			−0.195594			−0.0977972	+	0.995206i	$0.531180\pi$
$942$		0			0
$943$	−31.2915	+	34.6504i	−1.01899	+	1.12837i
$944$		0			0
$945$	4.70850			0.153167
$946$		0			0
$947$	−40.7085			−1.32285			−0.661424	−	0.750012i	$-0.730048\pi$
$947$	−40.7085			−1.32285			−0.661424	+	0.750012i	$0.730048\pi$
$948$		0			0
$949$			21.2884i			0.691052i
$950$		0			0
$951$	−8.70850			−0.282392
$952$		0			0
$953$	3.18824			0.103277			0.0516386	−	0.998666i	$-0.483556\pi$
$953$	3.18824			0.103277			0.0516386	+	0.998666i	$0.483556\pi$
$954$		0			0
$955$			45.2946i			1.46570i
$956$		0			0
$957$			36.3338i			1.17451i
$958$		0			0
$959$	22.3320			0.721139
$960$		0			0
$961$	−20.1660			−0.650516
$962$		0			0
$963$	−31.7490			−1.02310
$964$		0			0
$965$		−	22.3755i		−	0.720294i
$966$		0			0
$967$		−	34.2031i		−	1.09990i	−0.835198	−	0.549949i	$-0.814647\pi$
$967$		−	34.2031i		−	1.09990i	0.835198	−	0.549949i	$-0.185353\pi$
$968$		0			0
$969$		−	75.4382i		−	2.42342i
$970$		0			0
$971$		−	34.2559i		−	1.09932i	−0.835387	−	0.549662i	$-0.814756\pi$
$971$		−	34.2559i		−	1.09932i	0.835387	−	0.549662i	$-0.185244\pi$
$972$		0			0
$973$		−	1.08709i		−	0.0348504i
$974$		0			0
$975$	60.4575			1.93619
$976$		0			0
$977$			20.6921i			0.661998i	0.943631	+	0.330999i	$0.107386\pi$
$977$			20.6921i			0.661998i	−0.943631	+	0.330999i	$0.892614\pi$
$978$		0			0
$979$	14.3320			0.458053
$980$		0			0
$981$		−	37.7191i		−	1.20428i
$982$		0			0
$983$	−2.12549			−0.0677927			−0.0338963	−	0.999425i	$-0.510792\pi$
$983$	−2.12549			−0.0677927			−0.0338963	+	0.999425i	$0.510792\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$		0			0
$987$		−	19.8504i		−	0.631844i
$988$		0			0
$989$	24.0000			0.763156
$990$		0			0
$991$			37.6228i			1.19513i	0.801821	+	0.597564i	$0.203865\pi$
$991$			37.6228i			1.19513i	−0.801821	+	0.597564i	$0.796135\pi$
$992$		0			0
$993$	18.2288			0.578472
$994$		0			0
$995$		−	46.3817i		−	1.47040i
$996$		0			0
$997$		−	9.20614i		−	0.291561i	−0.989317	−	0.145781i	$-0.953431\pi$
$997$		−	9.20614i		−	0.291561i	0.989317	−	0.145781i	$-0.0465694\pi$
$998$		0			0
$999$			0.298179i			0.00943397i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	164.2.b.a.81.1	✓	4
3.2	odd	2		1476.2.f.d.901.3		4
4.3	odd	2		656.2.d.d.81.4		4
5.2	odd	4		4100.2.g.c.2049.1		8
5.3	odd	4		4100.2.g.c.2049.7		8
5.4	even	2		4100.2.b.e.901.4		4
8.3	odd	2		2624.2.d.l.2049.1		4
8.5	even	2		2624.2.d.m.2049.4		4
41.9	even	4		6724.2.a.d.1.4		4
41.32	even	4		6724.2.a.d.1.1		4
41.40	even	2	inner	164.2.b.a.81.4	yes	4
123.122	odd	2		1476.2.f.d.901.4		4
164.163	odd	2		656.2.d.d.81.1		4
205.122	odd	4		4100.2.g.c.2049.8		8
205.163	odd	4		4100.2.g.c.2049.2		8
205.204	even	2		4100.2.b.e.901.1		4
328.163	odd	2		2624.2.d.l.2049.4		4
328.245	even	2		2624.2.d.m.2049.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
164.2.b.a.81.1	✓	4	1.1	even	1	trivial
164.2.b.a.81.4	yes	4	41.40	even	2	inner
656.2.d.d.81.1		4	164.163	odd	2
656.2.d.d.81.4		4	4.3	odd	2
1476.2.f.d.901.3		4	3.2	odd	2
1476.2.f.d.901.4		4	123.122	odd	2
2624.2.d.l.2049.1		4	8.3	odd	2
2624.2.d.l.2049.4		4	328.163	odd	2
2624.2.d.m.2049.1		4	328.245	even	2
2624.2.d.m.2049.4		4	8.5	even	2
4100.2.b.e.901.1		4	205.204	even	2
4100.2.b.e.901.4		4	5.4	even	2
4100.2.g.c.2049.1		8	5.2	odd	4
4100.2.g.c.2049.2		8	205.163	odd	4
4100.2.g.c.2049.7		8	5.3	odd	4
4100.2.g.c.2049.8		8	205.122	odd	4
6724.2.a.d.1.1		4	41.32	even	4
6724.2.a.d.1.4		4	41.9	even	4

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:	\( N \)	\(=\)	\( 164 = 2^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	164.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.37608i q^{3} -3.64575 q^{5} -1.53436i q^{7} -2.64575 q^{9} +O(q^{10})\)	\(q-2.37608i q^{3} -3.64575 q^{5} -1.53436i q^{7} -2.64575 q^{9} -3.21780i q^{11} +3.06871i q^{13} +8.66259i q^{15} -7.82087i q^{17} +4.05952i q^{19} -3.64575 q^{21} +7.29150 q^{23} +8.29150 q^{25} -0.841723i q^{27} -4.75216i q^{29} +3.29150 q^{31} -7.64575 q^{33} +5.59388i q^{35} -0.354249 q^{37} +7.29150 q^{39} +(-4.29150 + 4.75216i) q^{41} +3.29150 q^{43} +9.64575 q^{45} +5.44479i q^{47} +4.64575 q^{49} -18.5830 q^{51} +6.43560i q^{53} +11.7313i q^{55} +9.64575 q^{57} -7.29150 q^{59} -11.2915 q^{61} +4.05952i q^{63} -11.1878i q^{65} -12.7221i q^{67} -17.3252i q^{69} +2.37608i q^{71} +6.93725 q^{73} -19.7013i q^{75} -4.93725 q^{77} -10.1969i q^{79} -9.93725 q^{81} +12.0000 q^{83} +28.5129i q^{85} -11.2915 q^{87} +4.45398i q^{89} +4.70850 q^{91} -7.82087i q^{93} -14.8000i q^{95} -6.13742i q^{97} +8.51350i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5}+O(q^{10}) \) 4 * q - 4 * q^5	\( 4 q - 4 q^{5} - 4 q^{21} + 8 q^{23} + 12 q^{25} - 8 q^{31} - 20 q^{33} - 12 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 28 q^{45} + 8 q^{49} - 32 q^{51} + 28 q^{57} - 8 q^{59} - 24 q^{61} - 4 q^{73} + 12 q^{77} - 8 q^{81} + 48 q^{83} - 24 q^{87} + 40 q^{91}+O(q^{100}) \) 4 * q - 4 * q^5 - 4 * q^21 + 8 * q^23 + 12 * q^25 - 8 * q^31 - 20 * q^33 - 12 * q^37 + 8 * q^39 + 4 * q^41 - 8 * q^43 + 28 * q^45 + 8 * q^49 - 32 * q^51 + 28 * q^57 - 8 * q^59 - 24 * q^61 - 4 * q^73 + 12 * q^77 - 8 * q^81 + 48 * q^83 - 24 * q^87 + 40 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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