LMFDB - Embedded newform 4664.2.a.k.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4664,2,Mod(1,4664)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4664, 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("4664.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4664 = 2^{3} \cdot 11 \cdot 53 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4664.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:	$37.2422275027$
Analytic rank:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 5x^{10} - 7x^{9} + 53x^{8} + 13x^{7} - 189x^{6} - 16x^{5} + 260x^{4} + 32x^{3} - 118x^{2} - 8x + 16 $ x^11 - 5x^10 - 7x^9 + 53x^8 + 13x^7 - 189x^6 - 16x^5 + 260x^4 + 32x^3 - 118x^2 - 8x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$2.92319$ of defining polynomial
Character	$\chi$	$=$	4664.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.92319 q^{3} -2.25184 q^{5} +2.05673 q^{7} +0.698658 q^{9} +O(q^{10})$	$q+1.92319 q^{3} -2.25184 q^{5} +2.05673 q^{7} +0.698658 q^{9} +1.00000 q^{11} -0.869052 q^{13} -4.33071 q^{15} +2.41006 q^{17} -5.91409 q^{19} +3.95548 q^{21} -2.27460 q^{23} +0.0707739 q^{25} -4.42592 q^{27} -3.46157 q^{29} -2.48145 q^{31} +1.92319 q^{33} -4.63142 q^{35} -0.292187 q^{37} -1.67135 q^{39} -9.11298 q^{41} +2.13082 q^{43} -1.57326 q^{45} +3.45473 q^{47} -2.76987 q^{49} +4.63500 q^{51} +1.00000 q^{53} -2.25184 q^{55} -11.3739 q^{57} -3.53532 q^{59} -6.97041 q^{61} +1.43695 q^{63} +1.95696 q^{65} +8.28766 q^{67} -4.37448 q^{69} +3.45919 q^{71} -4.39722 q^{73} +0.136112 q^{75} +2.05673 q^{77} +7.14377 q^{79} -10.6079 q^{81} +14.2194 q^{83} -5.42707 q^{85} -6.65726 q^{87} -7.00321 q^{89} -1.78740 q^{91} -4.77230 q^{93} +13.3176 q^{95} -15.5203 q^{97} +0.698658 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9	$ 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9} + 11 q^{11} - 13 q^{13} - 8 q^{15} - 7 q^{17} - 21 q^{19} + 6 q^{21} + 11 q^{23} + 4 q^{25} - 6 q^{27} - 5 q^{31} - 6 q^{33} - 25 q^{35} - 4 q^{37} - 19 q^{39} - 11 q^{41} + 16 q^{45} - 17 q^{47} - 2 q^{49} - 18 q^{51} + 11 q^{53} + 3 q^{55} - 5 q^{57} - 19 q^{59} - 2 q^{61} - 36 q^{63} - 13 q^{65} + 25 q^{67} + 3 q^{69} - 30 q^{71} + 5 q^{73} - 5 q^{75} - 5 q^{77} - 23 q^{79} - 9 q^{81} - 19 q^{83} + 2 q^{85} - 7 q^{87} + 6 q^{89} - 20 q^{91} + 43 q^{93} - 50 q^{95} - 35 q^{97} + 7 q^{99}+O(q^{100}) $ 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9 + 11 * q^11 - 13 * q^13 - 8 * q^15 - 7 * q^17 - 21 * q^19 + 6 * q^21 + 11 * q^23 + 4 * q^25 - 6 * q^27 - 5 * q^31 - 6 * q^33 - 25 * q^35 - 4 * q^37 - 19 * q^39 - 11 * q^41 + 16 * q^45 - 17 * q^47 - 2 * q^49 - 18 * q^51 + 11 * q^53 + 3 * q^55 - 5 * q^57 - 19 * q^59 - 2 * q^61 - 36 * q^63 - 13 * q^65 + 25 * q^67 + 3 * q^69 - 30 * q^71 + 5 * q^73 - 5 * q^75 - 5 * q^77 - 23 * q^79 - 9 * q^81 - 19 * q^83 + 2 * q^85 - 7 * q^87 + 6 * q^89 - 20 * q^91 + 43 * q^93 - 50 * q^95 - 35 * 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
$3$	1.92319	1.11035	0.555177	−	0.831732i	$-0.312651\pi$
$3$	1.92319	1.11035	0.555177	+	0.831732i	$0.312651\pi$
$4$	0	0
$5$	−2.25184	−1.00705	−0.503526	−	0.863980i	$-0.667964\pi$
$5$	−2.25184	−1.00705	−0.503526	+	0.863980i	$0.667964\pi$
$6$	0	0
$7$	2.05673	0.777370	0.388685	−	0.921371i	$-0.372929\pi$
$7$	2.05673	0.777370	0.388685	+	0.921371i	$0.372929\pi$
$8$	0	0
$9$	0.698658	0.232886
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.869052	−0.241032	−0.120516	−	0.992711i	$-0.538455\pi$
$13$	−0.869052	−0.241032	−0.120516	+	0.992711i	$0.538455\pi$
$14$	0	0
$15$	−4.33071	−1.11818
$16$	0	0
$17$	2.41006	0.584526	0.292263	−	0.956338i	$-0.405592\pi$
$17$	2.41006	0.584526	0.292263	+	0.956338i	$0.405592\pi$
$18$	0	0
$19$	−5.91409	−1.35678	−0.678392	−	0.734700i	$-0.737323\pi$
$19$	−5.91409	−1.35678	−0.678392	+	0.734700i	$0.737323\pi$
$20$	0	0
$21$	3.95548	0.863156
$22$	0	0
$23$	−2.27460	−0.474286	−0.237143	−	0.971475i	$-0.576211\pi$
$23$	−2.27460	−0.474286	−0.237143	+	0.971475i	$0.576211\pi$
$24$	0	0
$25$	0.0707739	0.0141548
$26$	0	0
$27$	−4.42592	−0.851768
$28$	0	0
$29$	−3.46157	−0.642798	−0.321399	−	0.946944i	$-0.604153\pi$
$29$	−3.46157	−0.642798	−0.321399	+	0.946944i	$0.604153\pi$
$30$	0	0
$31$	−2.48145	−0.445682	−0.222841	−	0.974855i	$-0.571533\pi$
$31$	−2.48145	−0.445682	−0.222841	+	0.974855i	$0.571533\pi$
$32$	0	0
$33$	1.92319	0.334784
$34$	0	0
$35$	−4.63142	−0.782852
$36$	0	0
$37$	−0.292187	−0.0480353	−0.0240176	−	0.999712i	$-0.507646\pi$
$37$	−0.292187	−0.0480353	−0.0240176	+	0.999712i	$0.507646\pi$
$38$	0	0
$39$	−1.67135	−0.267630
$40$	0	0
$41$	−9.11298	−1.42321	−0.711604	−	0.702581i	$-0.752031\pi$
$41$	−9.11298	−1.42321	−0.711604	+	0.702581i	$0.752031\pi$
$42$	0	0
$43$	2.13082	0.324946	0.162473	−	0.986713i	$-0.448053\pi$
$43$	2.13082	0.324946	0.162473	+	0.986713i	$0.448053\pi$
$44$	0	0
$45$	−1.57326	−0.234528
$46$	0	0
$47$	3.45473	0.503925	0.251962	−	0.967737i	$-0.418924\pi$
$47$	3.45473	0.503925	0.251962	+	0.967737i	$0.418924\pi$
$48$	0	0
$49$	−2.76987	−0.395696
$50$	0	0
$51$	4.63500	0.649030
$52$	0	0
$53$	1.00000	0.137361
$53$	1.00000	0.137361
$54$	0	0
$55$	−2.25184	−0.303638
$56$	0	0
$57$	−11.3739	−1.50651
$58$	0	0
$59$	−3.53532	−0.460259	−0.230130	−	0.973160i	$-0.573915\pi$
$59$	−3.53532	−0.460259	−0.230130	+	0.973160i	$0.573915\pi$
$60$	0	0
$61$	−6.97041	−0.892470	−0.446235	−	0.894916i	$-0.647235\pi$
$61$	−6.97041	−0.892470	−0.446235	+	0.894916i	$0.647235\pi$
$62$	0	0
$63$	1.43695	0.181039
$64$	0	0
$65$	1.95696	0.242732
$66$	0	0
$67$	8.28766	1.01250	0.506250	−	0.862387i	$-0.331031\pi$
$67$	8.28766	1.01250	0.506250	+	0.862387i	$0.331031\pi$
$68$	0	0
$69$	−4.37448	−0.526626
$70$	0	0
$71$	3.45919	0.410530	0.205265	−	0.978706i	$-0.434194\pi$
$71$	3.45919	0.410530	0.205265	+	0.978706i	$0.434194\pi$
$72$	0	0
$73$	−4.39722	−0.514655	−0.257328	−	0.966324i	$-0.582842\pi$
$73$	−4.39722	−0.514655	−0.257328	+	0.966324i	$0.582842\pi$
$74$	0	0
$75$	0.136112	0.0157168
$76$	0	0
$77$	2.05673	0.234386
$78$	0	0
$79$	7.14377	0.803737	0.401869	−	0.915697i	$-0.368361\pi$
$79$	7.14377	0.803737	0.401869	+	0.915697i	$0.368361\pi$
$80$	0	0
$81$	−10.6079	−1.17865
$82$	0	0
$83$	14.2194	1.56078	0.780389	−	0.625295i	$-0.215021\pi$
$83$	14.2194	1.56078	0.780389	+	0.625295i	$0.215021\pi$
$84$	0	0
$85$	−5.42707	−0.588648
$86$	0	0
$87$	−6.65726	−0.713733
$88$	0	0
$89$	−7.00321	−0.742338	−0.371169	−	0.928565i	$-0.621043\pi$
$89$	−7.00321	−0.742338	−0.371169	+	0.928565i	$0.621043\pi$
$90$	0	0
$91$	−1.78740	−0.187371
$92$	0	0
$93$	−4.77230	−0.494864
$94$	0	0
$95$	13.3176	1.36635
$96$	0	0
$97$	−15.5203	−1.57585	−0.787924	−	0.615773i	$-0.788844\pi$
$97$	−15.5203	−1.57585	−0.787924	+	0.615773i	$0.788844\pi$
$98$	0	0
$99$	0.698658	0.0702178
$100$	0	0
$101$	5.52739	0.549996	0.274998	−	0.961445i	$-0.411323\pi$
$101$	5.52739	0.549996	0.274998	+	0.961445i	$0.411323\pi$
$102$	0	0
$103$	−4.54354	−0.447689	−0.223844	−	0.974625i	$-0.571861\pi$
$103$	−4.54354	−0.447689	−0.223844	+	0.974625i	$0.571861\pi$
$104$	0	0
$105$	−8.90709	−0.869243
$106$	0	0
$107$	−16.5073	−1.59582	−0.797910	−	0.602777i	$-0.794061\pi$
$107$	−16.5073	−1.59582	−0.797910	+	0.602777i	$0.794061\pi$
$108$	0	0
$109$	4.13442	0.396006	0.198003	−	0.980201i	$-0.436554\pi$
$109$	4.13442	0.396006	0.198003	+	0.980201i	$0.436554\pi$
$110$	0	0
$111$	−0.561931	−0.0533362
$112$	0	0
$113$	2.49342	0.234561	0.117280	−	0.993099i	$-0.462582\pi$
$113$	2.49342	0.234561	0.117280	+	0.993099i	$0.462582\pi$
$114$	0	0
$115$	5.12203	0.477631
$116$	0	0
$117$	−0.607170	−0.0561329
$118$	0	0
$119$	4.95684	0.454393
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−17.5260	−1.58027
$124$	0	0
$125$	11.0998	0.992798
$126$	0	0
$127$	−3.00612	−0.266750	−0.133375	−	0.991066i	$-0.542582\pi$
$127$	−3.00612	−0.266750	−0.133375	+	0.991066i	$0.542582\pi$
$128$	0	0
$129$	4.09796	0.360806
$130$	0	0
$131$	3.32518	0.290522	0.145261	−	0.989393i	$-0.453598\pi$
$131$	3.32518	0.290522	0.145261	+	0.989393i	$0.453598\pi$
$132$	0	0
$133$	−12.1637	−1.05472
$134$	0	0
$135$	9.96645	0.857775
$136$	0	0
$137$	−16.7155	−1.42810	−0.714052	−	0.700092i	$-0.753142\pi$
$137$	−16.7155	−1.42810	−0.714052	+	0.700092i	$0.753142\pi$
$138$	0	0
$139$	7.10402	0.602555	0.301277	−	0.953537i	$-0.402587\pi$
$139$	7.10402	0.602555	0.301277	+	0.953537i	$0.402587\pi$
$140$	0	0
$141$	6.64411	0.559535
$142$	0	0
$143$	−0.869052	−0.0726738
$144$	0	0
$145$	7.79490	0.647331
$146$	0	0
$147$	−5.32699	−0.439363
$148$	0	0
$149$	10.8327	0.887447	0.443723	−	0.896164i	$-0.353657\pi$
$149$	10.8327	0.887447	0.443723	+	0.896164i	$0.353657\pi$
$150$	0	0
$151$	−6.50902	−0.529697	−0.264848	−	0.964290i	$-0.585322\pi$
$151$	−6.50902	−0.529697	−0.264848	+	0.964290i	$0.585322\pi$
$152$	0	0
$153$	1.68381	0.136128
$154$	0	0
$155$	5.58782	0.448825
$156$	0	0
$157$	−9.24944	−0.738186	−0.369093	−	0.929393i	$-0.620332\pi$
$157$	−9.24944	−0.738186	−0.369093	+	0.929393i	$0.620332\pi$
$158$	0	0
$159$	1.92319	0.152519
$160$	0	0
$161$	−4.67823	−0.368696
$162$	0	0
$163$	−3.64024	−0.285125	−0.142563	−	0.989786i	$-0.545534\pi$
$163$	−3.64024	−0.285125	−0.142563	+	0.989786i	$0.545534\pi$
$164$	0	0
$165$	−4.33071	−0.337145
$166$	0	0
$167$	−14.6058	−1.13023	−0.565114	−	0.825013i	$-0.691168\pi$
$167$	−14.6058	−1.13023	−0.565114	+	0.825013i	$0.691168\pi$
$168$	0	0
$169$	−12.2447	−0.941904
$170$	0	0
$171$	−4.13192	−0.315976
$172$	0	0
$173$	−21.5679	−1.63978	−0.819890	−	0.572521i	$-0.805965\pi$
$173$	−21.5679	−1.63978	−0.819890	+	0.572521i	$0.805965\pi$
$174$	0	0
$175$	0.145563	0.0110035
$176$	0	0
$177$	−6.79909	−0.511051
$178$	0	0
$179$	7.04669	0.526694	0.263347	−	0.964701i	$-0.415174\pi$
$179$	7.04669	0.526694	0.263347	+	0.964701i	$0.415174\pi$
$180$	0	0
$181$	16.5893	1.23307	0.616536	−	0.787327i	$-0.288535\pi$
$181$	16.5893	1.23307	0.616536	+	0.787327i	$0.288535\pi$
$182$	0	0
$183$	−13.4054	−0.990958
$184$	0	0
$185$	0.657958	0.0483741
$186$	0	0
$187$	2.41006	0.176241
$188$	0	0
$189$	−9.10291	−0.662139
$190$	0	0
$191$	−5.78580	−0.418646	−0.209323	−	0.977847i	$-0.567126\pi$
$191$	−5.78580	−0.418646	−0.209323	+	0.977847i	$0.567126\pi$
$192$	0	0
$193$	0.391053	0.0281486	0.0140743	−	0.999901i	$-0.495520\pi$
$193$	0.391053	0.0281486	0.0140743	+	0.999901i	$0.495520\pi$
$194$	0	0
$195$	3.76361	0.269518
$196$	0	0
$197$	15.5721	1.10947	0.554733	−	0.832029i	$-0.312820\pi$
$197$	15.5721	1.10947	0.554733	+	0.832029i	$0.312820\pi$
$198$	0	0
$199$	−3.20863	−0.227454	−0.113727	−	0.993512i	$-0.536279\pi$
$199$	−3.20863	−0.227454	−0.113727	+	0.993512i	$0.536279\pi$
$200$	0	0
$201$	15.9387	1.12423
$202$	0	0
$203$	−7.11951	−0.499692
$204$	0	0
$205$	20.5210	1.43325
$206$	0	0
$207$	−1.58917	−0.110455
$208$	0	0
$209$	−5.91409	−0.409086
$210$	0	0
$211$	−7.68012	−0.528722	−0.264361	−	0.964424i	$-0.585161\pi$
$211$	−7.68012	−0.528722	−0.264361	+	0.964424i	$0.585161\pi$
$212$	0	0
$213$	6.65267	0.455834
$214$	0	0
$215$	−4.79825	−0.327238
$216$	0	0
$217$	−5.10367	−0.346460
$218$	0	0
$219$	−8.45668	−0.571450
$220$	0	0
$221$	−2.09447	−0.140889
$222$	0	0
$223$	−15.9526	−1.06827	−0.534134	−	0.845400i	$-0.679362\pi$
$223$	−15.9526	−1.06827	−0.534134	+	0.845400i	$0.679362\pi$
$224$	0	0
$225$	0.0494468	0.00329645
$226$	0	0
$227$	−14.1256	−0.937549	−0.468775	−	0.883318i	$-0.655304\pi$
$227$	−14.1256	−0.937549	−0.468775	+	0.883318i	$0.655304\pi$
$228$	0	0
$229$	3.69353	0.244075	0.122038	−	0.992525i	$-0.461057\pi$
$229$	3.69353	0.244075	0.122038	+	0.992525i	$0.461057\pi$
$230$	0	0
$231$	3.95548	0.260251
$232$	0	0
$233$	25.4727	1.66877	0.834387	−	0.551179i	$-0.185822\pi$
$233$	25.4727	1.66877	0.834387	+	0.551179i	$0.185822\pi$
$234$	0	0
$235$	−7.77950	−0.507479
$236$	0	0
$237$	13.7388	0.892433
$238$	0	0
$239$	3.50293	0.226586	0.113293	−	0.993562i	$-0.463860\pi$
$239$	3.50293	0.226586	0.113293	+	0.993562i	$0.463860\pi$
$240$	0	0
$241$	−21.1022	−1.35931	−0.679657	−	0.733531i	$-0.737871\pi$
$241$	−21.1022	−1.35931	−0.679657	+	0.733531i	$0.737871\pi$
$242$	0	0
$243$	−7.12316	−0.456951
$244$	0	0
$245$	6.23730	0.398487
$246$	0	0
$247$	5.13965	0.327028
$248$	0	0
$249$	27.3465	1.73302
$250$	0	0
$251$	−23.0408	−1.45432	−0.727160	−	0.686468i	$-0.759160\pi$
$251$	−23.0408	−1.45432	−0.727160	+	0.686468i	$0.759160\pi$
$252$	0	0
$253$	−2.27460	−0.143003
$254$	0	0
$255$	−10.4373	−0.653608
$256$	0	0
$257$	−16.4012	−1.02308	−0.511539	−	0.859260i	$-0.670924\pi$
$257$	−16.4012	−1.02308	−0.511539	+	0.859260i	$0.670924\pi$
$258$	0	0
$259$	−0.600950	−0.0373412
$260$	0	0
$261$	−2.41846	−0.149699
$262$	0	0
$263$	2.44644	0.150854	0.0754271	−	0.997151i	$-0.475968\pi$
$263$	2.44644	0.150854	0.0754271	+	0.997151i	$0.475968\pi$
$264$	0	0
$265$	−2.25184	−0.138329
$266$	0	0
$267$	−13.4685	−0.824258
$268$	0	0
$269$	1.46060	0.0890545	0.0445272	−	0.999008i	$-0.485822\pi$
$269$	1.46060	0.0890545	0.0445272	+	0.999008i	$0.485822\pi$
$270$	0	0
$271$	17.8437	1.08393	0.541964	−	0.840401i	$-0.317681\pi$
$271$	17.8437	1.08393	0.541964	+	0.840401i	$0.317681\pi$
$272$	0	0
$273$	−3.43751	−0.208048
$274$	0	0
$275$	0.0707739	0.00426783
$276$	0	0
$277$	−26.4781	−1.59091	−0.795457	−	0.606011i	$-0.792769\pi$
$277$	−26.4781	−1.59091	−0.795457	+	0.606011i	$0.792769\pi$
$278$	0	0
$279$	−1.73369	−0.103793
$280$	0	0
$281$	−4.28748	−0.255770	−0.127885	−	0.991789i	$-0.540819\pi$
$281$	−4.28748	−0.255770	−0.127885	+	0.991789i	$0.540819\pi$
$282$	0	0
$283$	18.0919	1.07545	0.537726	−	0.843120i	$-0.319283\pi$
$283$	18.0919	1.07545	0.537726	+	0.843120i	$0.319283\pi$
$284$	0	0
$285$	25.6122	1.51714
$286$	0	0
$287$	−18.7429	−1.10636
$288$	0	0
$289$	−11.1916	−0.658330
$290$	0	0
$291$	−29.8485	−1.74975
$292$	0	0
$293$	2.44773	0.142998	0.0714990	−	0.997441i	$-0.477222\pi$
$293$	2.44773	0.142998	0.0714990	+	0.997441i	$0.477222\pi$
$294$	0	0
$295$	7.96096	0.463505
$296$	0	0
$297$	−4.42592	−0.256818
$298$	0	0
$299$	1.97674	0.114318
$300$	0	0
$301$	4.38251	0.252604
$302$	0	0
$303$	10.6302	0.610690
$304$	0	0
$305$	15.6962	0.898764
$306$	0	0
$307$	−27.3815	−1.56274	−0.781371	−	0.624066i	$-0.785480\pi$
$307$	−27.3815	−1.56274	−0.781371	+	0.624066i	$0.785480\pi$
$308$	0	0
$309$	−8.73810	−0.497093
$310$	0	0
$311$	−2.78058	−0.157672	−0.0788360	−	0.996888i	$-0.525120\pi$
$311$	−2.78058	−0.157672	−0.0788360	+	0.996888i	$0.525120\pi$
$312$	0	0
$313$	11.5254	0.651455	0.325727	−	0.945464i	$-0.394391\pi$
$313$	11.5254	0.651455	0.325727	+	0.945464i	$0.394391\pi$
$314$	0	0
$315$	−3.23578	−0.182315
$316$	0	0
$317$	28.7676	1.61575	0.807875	−	0.589353i	$-0.200617\pi$
$317$	28.7676	1.61575	0.807875	+	0.589353i	$0.200617\pi$
$318$	0	0
$319$	−3.46157	−0.193811
$320$	0	0
$321$	−31.7466	−1.77192
$322$	0	0
$323$	−14.2533	−0.793075
$324$	0	0
$325$	−0.0615062	−0.00341175
$326$	0	0
$327$	7.95128	0.439707
$328$	0	0
$329$	7.10545	0.391736
$330$	0	0
$331$	−10.0838	−0.554255	−0.277128	−	0.960833i	$-0.589383\pi$
$331$	−10.0838	−0.554255	−0.277128	+	0.960833i	$0.589383\pi$
$332$	0	0
$333$	−0.204139	−0.0111867
$334$	0	0
$335$	−18.6625	−1.01964
$336$	0	0
$337$	−19.0054	−1.03529	−0.517645	−	0.855595i	$-0.673191\pi$
$337$	−19.0054	−1.03529	−0.517645	+	0.855595i	$0.673191\pi$
$338$	0	0
$339$	4.79531	0.260446
$340$	0	0
$341$	−2.48145	−0.134378
$342$	0	0
$343$	−20.0940	−1.08497
$344$	0	0
$345$	9.85063	0.530340
$346$	0	0
$347$	11.3302	0.608235	0.304117	−	0.952635i	$-0.401638\pi$
$347$	11.3302	0.608235	0.304117	+	0.952635i	$0.401638\pi$
$348$	0	0
$349$	−11.1485	−0.596764	−0.298382	−	0.954447i	$-0.596447\pi$
$349$	−11.1485	−0.596764	−0.298382	+	0.954447i	$0.596447\pi$
$350$	0	0
$351$	3.84635	0.205303
$352$	0	0
$353$	−7.62384	−0.405776	−0.202888	−	0.979202i	$-0.565033\pi$
$353$	−7.62384	−0.405776	−0.202888	+	0.979202i	$0.565033\pi$
$354$	0	0
$355$	−7.78953	−0.413425
$356$	0	0
$357$	9.53294	0.504537
$358$	0	0
$359$	30.0253	1.58467	0.792336	−	0.610084i	$-0.208865\pi$
$359$	30.0253	1.58467	0.792336	+	0.610084i	$0.208865\pi$
$360$	0	0
$361$	15.9764	0.840864
$362$	0	0
$363$	1.92319	0.100941
$364$	0	0
$365$	9.90182	0.518285
$366$	0	0
$367$	11.9791	0.625306	0.312653	−	0.949867i	$-0.398782\pi$
$367$	11.9791	0.625306	0.312653	+	0.949867i	$0.398782\pi$
$368$	0	0
$369$	−6.36686	−0.331445
$370$	0	0
$371$	2.05673	0.106780
$372$	0	0
$373$	23.4907	1.21630	0.608150	−	0.793822i	$-0.291912\pi$
$373$	23.4907	1.21630	0.608150	+	0.793822i	$0.291912\pi$
$374$	0	0
$375$	21.3471	1.10236
$376$	0	0
$377$	3.00829	0.154935
$378$	0	0
$379$	4.17050	0.214224	0.107112	−	0.994247i	$-0.465840\pi$
$379$	4.17050	0.214224	0.107112	+	0.994247i	$0.465840\pi$
$380$	0	0
$381$	−5.78135	−0.296187
$382$	0	0
$383$	17.1098	0.874270	0.437135	−	0.899396i	$-0.355993\pi$
$383$	17.1098	0.874270	0.437135	+	0.899396i	$0.355993\pi$
$384$	0	0
$385$	−4.63142	−0.236039
$386$	0	0
$387$	1.48871	0.0756755
$388$	0	0
$389$	30.3319	1.53789	0.768943	−	0.639317i	$-0.220783\pi$
$389$	30.3319	1.53789	0.768943	+	0.639317i	$0.220783\pi$
$390$	0	0
$391$	−5.48192	−0.277233
$392$	0	0
$393$	6.39495	0.322583
$394$	0	0
$395$	−16.0866	−0.809406
$396$	0	0
$397$	8.31492	0.417314	0.208657	−	0.977989i	$-0.433091\pi$
$397$	8.31492	0.417314	0.208657	+	0.977989i	$0.433091\pi$
$398$	0	0
$399$	−23.3930	−1.17112
$400$	0	0
$401$	24.2001	1.20850	0.604248	−	0.796796i	$-0.293474\pi$
$401$	24.2001	1.20850	0.604248	+	0.796796i	$0.293474\pi$
$402$	0	0
$403$	2.15651	0.107423
$404$	0	0
$405$	23.8872	1.18696
$406$	0	0
$407$	−0.292187	−0.0144832
$408$	0	0
$409$	1.68731	0.0834320	0.0417160	−	0.999130i	$-0.486718\pi$
$409$	1.68731	0.0834320	0.0417160	+	0.999130i	$0.486718\pi$
$410$	0	0
$411$	−32.1471	−1.58570
$412$	0	0
$413$	−7.27119	−0.357792
$414$	0	0
$415$	−32.0197	−1.57178
$416$	0	0
$417$	13.6624	0.669049
$418$	0	0
$419$	31.5315	1.54041	0.770207	−	0.637794i	$-0.220153\pi$
$419$	31.5315	1.54041	0.770207	+	0.637794i	$0.220153\pi$
$420$	0	0
$421$	−7.38006	−0.359682	−0.179841	−	0.983696i	$-0.557558\pi$
$421$	−7.38006	−0.359682	−0.179841	+	0.983696i	$0.557558\pi$
$422$	0	0
$423$	2.41368	0.117357
$424$	0	0
$425$	0.170570	0.00827384
$426$	0	0
$427$	−14.3362	−0.693779
$428$	0	0
$429$	−1.67135	−0.0806936
$430$	0	0
$431$	−6.80113	−0.327599	−0.163799	−	0.986494i	$-0.552375\pi$
$431$	−6.80113	−0.327599	−0.163799	+	0.986494i	$0.552375\pi$
$432$	0	0
$433$	13.9979	0.672698	0.336349	−	0.941737i	$-0.390808\pi$
$433$	13.9979	0.672698	0.336349	+	0.941737i	$0.390808\pi$
$434$	0	0
$435$	14.9911	0.718767
$436$	0	0
$437$	13.4522	0.643504
$438$	0	0
$439$	−29.7545	−1.42011	−0.710053	−	0.704148i	$-0.751329\pi$
$439$	−29.7545	−1.42011	−0.710053	+	0.704148i	$0.751329\pi$
$440$	0	0
$441$	−1.93519	−0.0921520
$442$	0	0
$443$	12.6464	0.600851	0.300425	−	0.953805i	$-0.402871\pi$
$443$	12.6464	0.600851	0.300425	+	0.953805i	$0.402871\pi$
$444$	0	0
$445$	15.7701	0.747574
$446$	0	0
$447$	20.8333	0.985380
$448$	0	0
$449$	−24.9886	−1.17928	−0.589642	−	0.807665i	$-0.700731\pi$
$449$	−24.9886	−1.17928	−0.589642	+	0.807665i	$0.700731\pi$
$450$	0	0
$451$	−9.11298	−0.429113
$452$	0	0
$453$	−12.5181	−0.588151
$454$	0	0
$455$	4.02494	0.188692
$456$	0	0
$457$	0.333274	0.0155899	0.00779495	−	0.999970i	$-0.497519\pi$
$457$	0.333274	0.0155899	0.00779495	+	0.999970i	$0.497519\pi$
$458$	0	0
$459$	−10.6667	−0.497880
$460$	0	0
$461$	−3.55756	−0.165692	−0.0828460	−	0.996562i	$-0.526401\pi$
$461$	−3.55756	−0.165692	−0.0828460	+	0.996562i	$0.526401\pi$
$462$	0	0
$463$	9.21397	0.428210	0.214105	−	0.976811i	$-0.431317\pi$
$463$	9.21397	0.428210	0.214105	+	0.976811i	$0.431317\pi$
$464$	0	0
$465$	10.7464	0.498354
$466$	0	0
$467$	34.6866	1.60511	0.802553	−	0.596581i	$-0.203474\pi$
$467$	34.6866	1.60511	0.802553	+	0.596581i	$0.203474\pi$
$468$	0	0
$469$	17.0455	0.787086
$470$	0	0
$471$	−17.7884	−0.819647
$472$	0	0
$473$	2.13082	0.0979750
$474$	0	0
$475$	−0.418563	−0.0192050
$476$	0	0
$477$	0.698658	0.0319893
$478$	0	0
$479$	28.2914	1.29267	0.646334	−	0.763054i	$-0.276301\pi$
$479$	28.2914	1.29267	0.646334	+	0.763054i	$0.276301\pi$
$480$	0	0
$481$	0.253926	0.0115780
$482$	0	0
$483$	−8.99712	−0.409383
$484$	0	0
$485$	34.9492	1.58696
$486$	0	0
$487$	9.33342	0.422938	0.211469	−	0.977385i	$-0.432175\pi$
$487$	9.33342	0.422938	0.211469	+	0.977385i	$0.432175\pi$
$488$	0	0
$489$	−7.00086	−0.316590
$490$	0	0
$491$	22.6616	1.02270	0.511352	−	0.859371i	$-0.329145\pi$
$491$	22.6616	1.02270	0.511352	+	0.859371i	$0.329145\pi$
$492$	0	0
$493$	−8.34260	−0.375732
$494$	0	0
$495$	−1.57326	−0.0707130
$496$	0	0
$497$	7.11461	0.319134
$498$	0	0
$499$	14.5583	0.651718	0.325859	−	0.945418i	$-0.394346\pi$
$499$	14.5583	0.651718	0.325859	+	0.945418i	$0.394346\pi$
$500$	0	0
$501$	−28.0897	−1.25495
$502$	0	0
$503$	−19.9008	−0.887333	−0.443667	−	0.896192i	$-0.646323\pi$
$503$	−19.9008	−0.887333	−0.443667	+	0.896192i	$0.646323\pi$
$504$	0	0
$505$	−12.4468	−0.553875
$506$	0	0
$507$	−23.5490	−1.04585
$508$	0	0
$509$	31.1691	1.38154	0.690772	−	0.723072i	$-0.257271\pi$
$509$	31.1691	1.38154	0.690772	+	0.723072i	$0.257271\pi$
$510$	0	0
$511$	−9.04388	−0.400078
$512$	0	0
$513$	26.1753	1.15567
$514$	0	0
$515$	10.2313	0.450846
$516$	0	0
$517$	3.45473	0.151939
$518$	0	0
$519$	−41.4792	−1.82074
$520$	0	0
$521$	13.1236	0.574956	0.287478	−	0.957787i	$-0.407183\pi$
$521$	13.1236	0.574956	0.287478	+	0.957787i	$0.407183\pi$
$522$	0	0
$523$	4.68678	0.204938	0.102469	−	0.994736i	$-0.467326\pi$
$523$	4.68678	0.204938	0.102469	+	0.994736i	$0.467326\pi$
$524$	0	0
$525$	0.279945	0.0122178
$526$	0	0
$527$	−5.98045	−0.260512
$528$	0	0
$529$	−17.8262	−0.775052
$530$	0	0
$531$	−2.46998	−0.107188
$532$	0	0
$533$	7.91965	0.343038
$534$	0	0
$535$	37.1717	1.60707
$536$	0	0
$537$	13.5521	0.584817
$538$	0	0
$539$	−2.76987	−0.119307
$540$	0	0
$541$	−16.8416	−0.724075	−0.362038	−	0.932163i	$-0.617919\pi$
$541$	−16.8416	−0.724075	−0.362038	+	0.932163i	$0.617919\pi$
$542$	0	0
$543$	31.9044	1.36915
$544$	0	0
$545$	−9.31005	−0.398799
$546$	0	0
$547$	4.39498	0.187916	0.0939579	−	0.995576i	$-0.470048\pi$
$547$	4.39498	0.187916	0.0939579	+	0.995576i	$0.470048\pi$
$548$	0	0
$549$	−4.86993	−0.207844
$550$	0	0
$551$	20.4720	0.872138
$552$	0	0
$553$	14.6928	0.624801
$554$	0	0
$555$	1.26538	0.0537123
$556$	0	0
$557$	−9.16413	−0.388296	−0.194148	−	0.980972i	$-0.562194\pi$
$557$	−9.16413	−0.388296	−0.194148	+	0.980972i	$0.562194\pi$
$558$	0	0
$559$	−1.85179	−0.0783224
$560$	0	0
$561$	4.63500	0.195690
$562$	0	0
$563$	0.233036	0.00982129	0.00491065	−	0.999988i	$-0.498437\pi$
$563$	0.233036	0.00982129	0.00491065	+	0.999988i	$0.498437\pi$
$564$	0	0
$565$	−5.61477	−0.236215
$566$	0	0
$567$	−21.8175	−0.916247
$568$	0	0
$569$	−15.0522	−0.631021	−0.315510	−	0.948922i	$-0.602176\pi$
$569$	−15.0522	−0.631021	−0.315510	+	0.948922i	$0.602176\pi$
$570$	0	0
$571$	−8.03065	−0.336072	−0.168036	−	0.985781i	$-0.553743\pi$
$571$	−8.03065	−0.336072	−0.168036	+	0.985781i	$0.553743\pi$
$572$	0	0
$573$	−11.1272	−0.464845
$574$	0	0
$575$	−0.160982	−0.00671342
$576$	0	0
$577$	−8.79892	−0.366304	−0.183152	−	0.983085i	$-0.558630\pi$
$577$	−8.79892	−0.366304	−0.183152	+	0.983085i	$0.558630\pi$
$578$	0	0
$579$	0.752069	0.0312549
$580$	0	0
$581$	29.2454	1.21330
$582$	0	0
$583$	1.00000	0.0414158
$584$	0	0
$585$	1.36725	0.0565288
$586$	0	0
$587$	9.74961	0.402409	0.201205	−	0.979549i	$-0.435514\pi$
$587$	9.74961	0.402409	0.201205	+	0.979549i	$0.435514\pi$
$588$	0	0
$589$	14.6755	0.604694
$590$	0	0
$591$	29.9481	1.23190
$592$	0	0
$593$	43.2939	1.77787	0.888934	−	0.458036i	$-0.151447\pi$
$593$	43.2939	1.77787	0.888934	+	0.458036i	$0.151447\pi$
$594$	0	0
$595$	−11.1620	−0.457597
$596$	0	0
$597$	−6.17080	−0.252554
$598$	0	0
$599$	9.56975	0.391009	0.195505	−	0.980703i	$-0.437366\pi$
$599$	9.56975	0.391009	0.195505	+	0.980703i	$0.437366\pi$
$600$	0	0
$601$	15.3448	0.625928	0.312964	−	0.949765i	$-0.398678\pi$
$601$	15.3448	0.625928	0.312964	+	0.949765i	$0.398678\pi$
$602$	0	0
$603$	5.79024	0.235797
$604$	0	0
$605$	−2.25184	−0.0915502
$606$	0	0
$607$	−34.4264	−1.39732	−0.698662	−	0.715452i	$-0.746221\pi$
$607$	−34.4264	−1.39732	−0.698662	+	0.715452i	$0.746221\pi$
$608$	0	0
$609$	−13.6922	−0.554835
$610$	0	0
$611$	−3.00234	−0.121462
$612$	0	0
$613$	31.4154	1.26885	0.634427	−	0.772982i	$-0.281236\pi$
$613$	31.4154	1.26885	0.634427	+	0.772982i	$0.281236\pi$
$614$	0	0
$615$	39.4657	1.59141
$616$	0	0
$617$	24.0747	0.969209	0.484605	−	0.874733i	$-0.338963\pi$
$617$	24.0747	0.969209	0.484605	+	0.874733i	$0.338963\pi$
$618$	0	0
$619$	−33.8256	−1.35957	−0.679783	−	0.733413i	$-0.737926\pi$
$619$	−33.8256	−1.35957	−0.679783	+	0.733413i	$0.737926\pi$
$620$	0	0
$621$	10.0672	0.403982
$622$	0	0
$623$	−14.4037	−0.577072
$624$	0	0
$625$	−25.3489	−1.01395
$626$	0	0
$627$	−11.3739	−0.454230
$628$	0	0
$629$	−0.704189	−0.0280779
$630$	0	0
$631$	32.2735	1.28479	0.642395	−	0.766374i	$-0.277941\pi$
$631$	32.2735	1.28479	0.642395	+	0.766374i	$0.277941\pi$
$632$	0	0
$633$	−14.7703	−0.587068
$634$	0	0
$635$	6.76931	0.268632
$636$	0	0
$637$	2.40716	0.0953752
$638$	0	0
$639$	2.41679	0.0956067
$640$	0	0
$641$	13.8632	0.547563	0.273782	−	0.961792i	$-0.411725\pi$
$641$	13.8632	0.547563	0.273782	+	0.961792i	$0.411725\pi$
$642$	0	0
$643$	−8.10417	−0.319597	−0.159799	−	0.987150i	$-0.551085\pi$
$643$	−8.10417	−0.319597	−0.159799	+	0.987150i	$0.551085\pi$
$644$	0	0
$645$	−9.22795	−0.363350
$646$	0	0
$647$	−38.6856	−1.52089	−0.760445	−	0.649403i	$-0.775019\pi$
$647$	−38.6856	−1.52089	−0.760445	+	0.649403i	$0.775019\pi$
$648$	0	0
$649$	−3.53532	−0.138773
$650$	0	0
$651$	−9.81532	−0.384693
$652$	0	0
$653$	24.8453	0.972270	0.486135	−	0.873884i	$-0.338406\pi$
$653$	24.8453	0.972270	0.486135	+	0.873884i	$0.338406\pi$
$654$	0	0
$655$	−7.48776	−0.292571
$656$	0	0
$657$	−3.07215	−0.119856
$658$	0	0
$659$	−38.4593	−1.49816	−0.749081	−	0.662478i	$-0.769505\pi$
$659$	−38.4593	−1.49816	−0.749081	+	0.662478i	$0.769505\pi$
$660$	0	0
$661$	1.41946	0.0552104	0.0276052	−	0.999619i	$-0.491212\pi$
$661$	1.41946	0.0552104	0.0276052	+	0.999619i	$0.491212\pi$
$662$	0	0
$663$	−4.02806	−0.156437
$664$	0	0
$665$	27.3906	1.06216
$666$	0	0
$667$	7.87369	0.304870
$668$	0	0
$669$	−30.6799	−1.18616
$670$	0	0
$671$	−6.97041	−0.269090
$672$	0	0
$673$	−48.9873	−1.88832	−0.944160	−	0.329486i	$-0.893124\pi$
$673$	−48.9873	−1.88832	−0.944160	+	0.329486i	$0.893124\pi$
$674$	0	0
$675$	−0.313240	−0.0120566
$676$	0	0
$677$	7.82433	0.300713	0.150357	−	0.988632i	$-0.451958\pi$
$677$	7.82433	0.300713	0.150357	+	0.988632i	$0.451958\pi$
$678$	0	0
$679$	−31.9210	−1.22502
$680$	0	0
$681$	−27.1662	−1.04101
$682$	0	0
$683$	−11.9703	−0.458033	−0.229016	−	0.973423i	$-0.573551\pi$
$683$	−11.9703	−0.458033	−0.229016	+	0.973423i	$0.573551\pi$
$684$	0	0
$685$	37.6407	1.43818
$686$	0	0
$687$	7.10335	0.271010
$688$	0	0
$689$	−0.869052	−0.0331082
$690$	0	0
$691$	−24.6473	−0.937626	−0.468813	−	0.883297i	$-0.655318\pi$
$691$	−24.6473	−0.937626	−0.468813	+	0.883297i	$0.655318\pi$
$692$	0	0
$693$	1.43695	0.0545852
$694$	0	0
$695$	−15.9971	−0.606804
$696$	0	0
$697$	−21.9628	−0.831902
$698$	0	0
$699$	48.9889	1.85293
$700$	0	0
$701$	−7.30213	−0.275798	−0.137899	−	0.990446i	$-0.544035\pi$
$701$	−7.30213	−0.275798	−0.137899	+	0.990446i	$0.544035\pi$
$702$	0	0
$703$	1.72802	0.0651735
$704$	0	0
$705$	−14.9615	−0.563481
$706$	0	0
$707$	11.3683	0.427550
$708$	0	0
$709$	30.2620	1.13651	0.568256	−	0.822852i	$-0.307618\pi$
$709$	30.2620	1.13651	0.568256	+	0.822852i	$0.307618\pi$
$710$	0	0
$711$	4.99105	0.187179
$712$	0	0
$713$	5.64430	0.211381
$714$	0	0
$715$	1.95696	0.0731863
$716$	0	0
$717$	6.73679	0.251590
$718$	0	0
$719$	11.9369	0.445171	0.222585	−	0.974913i	$-0.428550\pi$
$719$	11.9369	0.445171	0.222585	+	0.974913i	$0.428550\pi$
$720$	0	0
$721$	−9.34483	−0.348020
$722$	0	0
$723$	−40.5836	−1.50932
$724$	0	0
$725$	−0.244989	−0.00909867
$726$	0	0
$727$	−19.5502	−0.725078	−0.362539	−	0.931969i	$-0.618090\pi$
$727$	−19.5502	−0.725078	−0.362539	+	0.931969i	$0.618090\pi$
$728$	0	0
$729$	18.1244	0.671273
$730$	0	0
$731$	5.13540	0.189940
$732$	0	0
$733$	−30.8520	−1.13955	−0.569773	−	0.821802i	$-0.692969\pi$
$733$	−30.8520	−1.13955	−0.569773	+	0.821802i	$0.692969\pi$
$734$	0	0
$735$	11.9955	0.442461
$736$	0	0
$737$	8.28766	0.305280
$738$	0	0
$739$	−44.2432	−1.62751	−0.813756	−	0.581207i	$-0.802581\pi$
$739$	−44.2432	−1.62751	−0.813756	+	0.581207i	$0.802581\pi$
$740$	0	0
$741$	9.88452	0.363117
$742$	0	0
$743$	−37.5531	−1.37769	−0.688846	−	0.724908i	$-0.741882\pi$
$743$	−37.5531	−1.37769	−0.688846	+	0.724908i	$0.741882\pi$
$744$	0	0
$745$	−24.3934	−0.893705
$746$	0	0
$747$	9.93447	0.363483
$748$	0	0
$749$	−33.9510	−1.24054
$750$	0	0
$751$	33.4846	1.22187	0.610935	−	0.791681i	$-0.290794\pi$
$751$	33.4846	1.22187	0.610935	+	0.791681i	$0.290794\pi$
$752$	0	0
$753$	−44.3117	−1.61481
$754$	0	0
$755$	14.6573	0.533432
$756$	0	0
$757$	27.8876	1.01359	0.506797	−	0.862066i	$-0.330829\pi$
$757$	27.8876	1.01359	0.506797	+	0.862066i	$0.330829\pi$
$758$	0	0
$759$	−4.37448	−0.158784
$760$	0	0
$761$	15.6228	0.566326	0.283163	−	0.959072i	$-0.408616\pi$
$761$	15.6228	0.566326	0.283163	+	0.959072i	$0.408616\pi$
$762$	0	0
$763$	8.50338	0.307843
$764$	0	0
$765$	−3.79166	−0.137088
$766$	0	0
$767$	3.07238	0.110937
$768$	0	0
$769$	19.6981	0.710332	0.355166	−	0.934803i	$-0.384424\pi$
$769$	19.6981	0.710332	0.355166	+	0.934803i	$0.384424\pi$
$770$	0	0
$771$	−31.5426	−1.13598
$772$	0	0
$773$	−3.92883	−0.141310	−0.0706551	−	0.997501i	$-0.522509\pi$
$773$	−3.92883	−0.141310	−0.0706551	+	0.997501i	$0.522509\pi$
$774$	0	0
$775$	−0.175622	−0.00630853
$776$	0	0
$777$	−1.15574	−0.0414619
$778$	0	0
$779$	53.8949	1.93099
$780$	0	0
$781$	3.45919	0.123779
$782$	0	0
$783$	15.3206	0.547515
$784$	0	0
$785$	20.8282	0.743392
$786$	0	0
$787$	−45.4173	−1.61895	−0.809476	−	0.587153i	$-0.800249\pi$
$787$	−45.4173	−1.61895	−0.809476	+	0.587153i	$0.800249\pi$
$788$	0	0
$789$	4.70498	0.167502
$790$	0	0
$791$	5.12828	0.182341
$792$	0	0
$793$	6.05765	0.215114
$794$	0	0
$795$	−4.33071	−0.153594
$796$	0	0
$797$	11.4315	0.404924	0.202462	−	0.979290i	$-0.435106\pi$
$797$	11.4315	0.404924	0.202462	+	0.979290i	$0.435106\pi$
$798$	0	0
$799$	8.32612	0.294557
$800$	0	0
$801$	−4.89284	−0.172880
$802$	0	0
$803$	−4.39722	−0.155174
$804$	0	0
$805$	10.5346	0.371296
$806$	0	0
$807$	2.80901	0.0988820
$808$	0	0
$809$	5.55610	0.195342	0.0976710	−	0.995219i	$-0.468861\pi$
$809$	5.55610	0.195342	0.0976710	+	0.995219i	$0.468861\pi$
$810$	0	0
$811$	38.0120	1.33478	0.667391	−	0.744707i	$-0.267411\pi$
$811$	38.0120	1.33478	0.667391	+	0.744707i	$0.267411\pi$
$812$	0	0
$813$	34.3169	1.20354
$814$	0	0
$815$	8.19722	0.287136
$816$	0	0
$817$	−12.6018	−0.440882
$818$	0	0
$819$	−1.24878	−0.0436360
$820$	0	0
$821$	13.8662	0.483935	0.241968	−	0.970284i	$-0.422207\pi$
$821$	13.8662	0.483935	0.241968	+	0.970284i	$0.422207\pi$
$822$	0	0
$823$	−20.0732	−0.699707	−0.349853	−	0.936804i	$-0.613769\pi$
$823$	−20.0732	−0.699707	−0.349853	+	0.936804i	$0.613769\pi$
$824$	0	0
$825$	0.136112	0.00473880
$826$	0	0
$827$	−39.6242	−1.37787	−0.688933	−	0.724825i	$-0.741921\pi$
$827$	−39.6242	−1.37787	−0.688933	+	0.724825i	$0.741921\pi$
$828$	0	0
$829$	14.0626	0.488416	0.244208	−	0.969723i	$-0.421472\pi$
$829$	14.0626	0.488416	0.244208	+	0.969723i	$0.421472\pi$
$830$	0	0
$831$	−50.9224	−1.76648
$832$	0	0
$833$	−6.67556	−0.231294
$834$	0	0
$835$	32.8898	1.13820
$836$	0	0
$837$	10.9827	0.379617
$838$	0	0
$839$	−21.8664	−0.754914	−0.377457	−	0.926027i	$-0.623201\pi$
$839$	−21.8664	−0.754914	−0.377457	+	0.926027i	$0.623201\pi$
$840$	0	0
$841$	−17.0175	−0.586811
$842$	0	0
$843$	−8.24564	−0.283995
$844$	0	0
$845$	27.5732	0.948547
$846$	0	0
$847$	2.05673	0.0706700
$848$	0	0
$849$	34.7942	1.19413
$850$	0	0
$851$	0.664609	0.0227825
$852$	0	0
$853$	24.9425	0.854016	0.427008	−	0.904248i	$-0.359568\pi$
$853$	24.9425	0.854016	0.427008	+	0.904248i	$0.359568\pi$
$854$	0	0
$855$	9.30442	0.318204
$856$	0	0
$857$	1.79570	0.0613400	0.0306700	−	0.999530i	$-0.490236\pi$
$857$	1.79570	0.0613400	0.0306700	+	0.999530i	$0.490236\pi$
$858$	0	0
$859$	10.4820	0.357642	0.178821	−	0.983882i	$-0.442772\pi$
$859$	10.4820	0.357642	0.178821	+	0.983882i	$0.442772\pi$
$860$	0	0
$861$	−36.0462	−1.22845
$862$	0	0
$863$	34.8850	1.18750	0.593749	−	0.804650i	$-0.297647\pi$
$863$	34.8850	1.18750	0.593749	+	0.804650i	$0.297647\pi$
$864$	0	0
$865$	48.5675	1.65134
$866$	0	0
$867$	−21.5236	−0.730979
$868$	0	0
$869$	7.14377	0.242336
$870$	0	0
$871$	−7.20241	−0.244044
$872$	0	0
$873$	−10.8434	−0.366993
$874$	0	0
$875$	22.8293	0.771771
$876$	0	0
$877$	−39.3051	−1.32724	−0.663620	−	0.748070i	$-0.730981\pi$
$877$	−39.3051	−1.32724	−0.663620	+	0.748070i	$0.730981\pi$
$878$	0	0
$879$	4.70745	0.158778
$880$	0	0
$881$	−6.55739	−0.220924	−0.110462	−	0.993880i	$-0.535233\pi$
$881$	−6.55739	−0.220924	−0.110462	+	0.993880i	$0.535233\pi$
$882$	0	0
$883$	−20.5735	−0.692355	−0.346177	−	0.938169i	$-0.612520\pi$
$883$	−20.5735	−0.692355	−0.346177	+	0.938169i	$0.612520\pi$
$884$	0	0
$885$	15.3104	0.514655
$886$	0	0
$887$	−33.3081	−1.11838	−0.559189	−	0.829040i	$-0.688887\pi$
$887$	−33.3081	−1.11838	−0.559189	+	0.829040i	$0.688887\pi$
$888$	0	0
$889$	−6.18278	−0.207364
$890$	0	0
$891$	−10.6079	−0.355376
$892$	0	0
$893$	−20.4316	−0.683717
$894$	0	0
$895$	−15.8680	−0.530409
$896$	0	0
$897$	3.80165	0.126933
$898$	0	0
$899$	8.58972	0.286483
$900$	0	0
$901$	2.41006	0.0802908
$902$	0	0
$903$	8.42839	0.280479
$904$	0	0
$905$	−37.3564	−1.24177
$906$	0	0
$907$	−42.9742	−1.42693	−0.713467	−	0.700689i	$-0.752876\pi$
$907$	−42.9742	−1.42693	−0.713467	+	0.700689i	$0.752876\pi$
$908$	0	0
$909$	3.86175	0.128086
$910$	0	0
$911$	6.81135	0.225670	0.112835	−	0.993614i	$-0.464007\pi$
$911$	6.81135	0.225670	0.112835	+	0.993614i	$0.464007\pi$
$912$	0	0
$913$	14.2194	0.470592
$914$	0	0
$915$	30.1868	0.997946
$916$	0	0
$917$	6.83899	0.225843
$918$	0	0
$919$	34.5579	1.13996	0.569980	−	0.821658i	$-0.306951\pi$
$919$	34.5579	1.13996	0.569980	+	0.821658i	$0.306951\pi$
$920$	0	0
$921$	−52.6597	−1.73520
$922$	0	0
$923$	−3.00621	−0.0989507
$924$	0	0
$925$	−0.0206792	−0.000679929 0
$926$	0	0
$927$	−3.17438	−0.104260
$928$	0	0
$929$	13.1365	0.430993	0.215497	−	0.976505i	$-0.430863\pi$
$929$	13.1365	0.430993	0.215497	+	0.976505i	$0.430863\pi$
$930$	0	0
$931$	16.3813	0.536874
$932$	0	0
$933$	−5.34757	−0.175072
$934$	0	0
$935$	−5.42707	−0.177484
$936$	0	0
$937$	27.5463	0.899898	0.449949	−	0.893054i	$-0.351442\pi$
$937$	27.5463	0.899898	0.449949	+	0.893054i	$0.351442\pi$
$938$	0	0
$939$	22.1655	0.723345
$940$	0	0
$941$	3.79057	0.123569	0.0617846	−	0.998090i	$-0.480321\pi$
$941$	3.79057	0.123569	0.0617846	+	0.998090i	$0.480321\pi$
$942$	0	0
$943$	20.7284	0.675008
$944$	0	0
$945$	20.4983	0.666809
$946$	0	0
$947$	−16.2169	−0.526979	−0.263490	−	0.964662i	$-0.584873\pi$
$947$	−16.2169	−0.526979	−0.263490	+	0.964662i	$0.584873\pi$
$948$	0	0
$949$	3.82141	0.124048
$950$	0	0
$951$	55.3256	1.79406
$952$	0	0
$953$	−45.7992	−1.48358	−0.741790	−	0.670632i	$-0.766023\pi$
$953$	−45.7992	−1.48358	−0.741790	+	0.670632i	$0.766023\pi$
$954$	0	0
$955$	13.0287	0.421598
$956$	0	0
$957$	−6.65726	−0.215199
$958$	0	0
$959$	−34.3793	−1.11017
$960$	0	0
$961$	−24.8424	−0.801368
$962$	0	0
$963$	−11.5329	−0.371644
$964$	0	0
$965$	−0.880588	−0.0283471
$966$	0	0
$967$	19.7722	0.635832	0.317916	−	0.948119i	$-0.397017\pi$
$967$	19.7722	0.635832	0.317916	+	0.948119i	$0.397017\pi$
$968$	0	0
$969$	−27.4118	−0.880594
$970$	0	0
$971$	12.7828	0.410221	0.205111	−	0.978739i	$-0.434245\pi$
$971$	12.7828	0.410221	0.205111	+	0.978739i	$0.434245\pi$
$972$	0	0
$973$	14.6110	0.468408
$974$	0	0
$975$	−0.118288	−0.00378825
$976$	0	0
$977$	48.6353	1.55598	0.777991	−	0.628276i	$-0.216239\pi$
$977$	48.6353	1.55598	0.777991	+	0.628276i	$0.216239\pi$
$978$	0	0
$979$	−7.00321	−0.223823
$980$	0	0
$981$	2.88855	0.0922242
$982$	0	0
$983$	−12.4344	−0.396596	−0.198298	−	0.980142i	$-0.563541\pi$
$983$	−12.4344	−0.396596	−0.198298	+	0.980142i	$0.563541\pi$
$984$	0	0
$985$	−35.0658	−1.11729
$986$	0	0
$987$	13.6651	0.434965
$988$	0	0
$989$	−4.84675	−0.154118
$990$	0	0
$991$	18.9603	0.602292	0.301146	−	0.953578i	$-0.402631\pi$
$991$	18.9603	0.602292	0.301146	+	0.953578i	$0.402631\pi$
$992$	0	0
$993$	−19.3930	−0.615420
$994$	0	0
$995$	7.22531	0.229058
$996$	0	0
$997$	−34.5786	−1.09511	−0.547557	−	0.836768i	$-0.684442\pi$
$997$	−34.5786	−1.09511	−0.547557	+	0.836768i	$0.684442\pi$
$998$	0	0
$999$	1.29320	0.0409149

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4664.2.a.k.1.10	✓	11
4.3	odd	2		9328.2.a.bm.1.2		11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4664.2.a.k.1.10	✓	11	1.1	even	1	trivial
9328.2.a.bm.1.2		11	4.3	odd	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:	\( N \)	\(=\)	\( 4664 = 2^{3} \cdot 11 \cdot 53 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4664.a (trivial)

\(f(q)\)	\(=\)	\(q+1.92319 q^{3} -2.25184 q^{5} +2.05673 q^{7} +0.698658 q^{9} +O(q^{10})\)	\(q+1.92319 q^{3} -2.25184 q^{5} +2.05673 q^{7} +0.698658 q^{9} +1.00000 q^{11} -0.869052 q^{13} -4.33071 q^{15} +2.41006 q^{17} -5.91409 q^{19} +3.95548 q^{21} -2.27460 q^{23} +0.0707739 q^{25} -4.42592 q^{27} -3.46157 q^{29} -2.48145 q^{31} +1.92319 q^{33} -4.63142 q^{35} -0.292187 q^{37} -1.67135 q^{39} -9.11298 q^{41} +2.13082 q^{43} -1.57326 q^{45} +3.45473 q^{47} -2.76987 q^{49} +4.63500 q^{51} +1.00000 q^{53} -2.25184 q^{55} -11.3739 q^{57} -3.53532 q^{59} -6.97041 q^{61} +1.43695 q^{63} +1.95696 q^{65} +8.28766 q^{67} -4.37448 q^{69} +3.45919 q^{71} -4.39722 q^{73} +0.136112 q^{75} +2.05673 q^{77} +7.14377 q^{79} -10.6079 q^{81} +14.2194 q^{83} -5.42707 q^{85} -6.65726 q^{87} -7.00321 q^{89} -1.78740 q^{91} -4.77230 q^{93} +13.3176 q^{95} -15.5203 q^{97} +0.698658 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9	\( 11 q - 6 q^{3} + 3 q^{5} - 5 q^{7} + 7 q^{9} + 11 q^{11} - 13 q^{13} - 8 q^{15} - 7 q^{17} - 21 q^{19} + 6 q^{21} + 11 q^{23} + 4 q^{25} - 6 q^{27} - 5 q^{31} - 6 q^{33} - 25 q^{35} - 4 q^{37} - 19 q^{39} - 11 q^{41} + 16 q^{45} - 17 q^{47} - 2 q^{49} - 18 q^{51} + 11 q^{53} + 3 q^{55} - 5 q^{57} - 19 q^{59} - 2 q^{61} - 36 q^{63} - 13 q^{65} + 25 q^{67} + 3 q^{69} - 30 q^{71} + 5 q^{73} - 5 q^{75} - 5 q^{77} - 23 q^{79} - 9 q^{81} - 19 q^{83} + 2 q^{85} - 7 q^{87} + 6 q^{89} - 20 q^{91} + 43 q^{93} - 50 q^{95} - 35 q^{97} + 7 q^{99}+O(q^{100}) \) 11 * q - 6 * q^3 + 3 * q^5 - 5 * q^7 + 7 * q^9 + 11 * q^11 - 13 * q^13 - 8 * q^15 - 7 * q^17 - 21 * q^19 + 6 * q^21 + 11 * q^23 + 4 * q^25 - 6 * q^27 - 5 * q^31 - 6 * q^33 - 25 * q^35 - 4 * q^37 - 19 * q^39 - 11 * q^41 + 16 * q^45 - 17 * q^47 - 2 * q^49 - 18 * q^51 + 11 * q^53 + 3 * q^55 - 5 * q^57 - 19 * q^59 - 2 * q^61 - 36 * q^63 - 13 * q^65 + 25 * q^67 + 3 * q^69 - 30 * q^71 + 5 * q^73 - 5 * q^75 - 5 * q^77 - 23 * q^79 - 9 * q^81 - 19 * q^83 + 2 * q^85 - 7 * q^87 + 6 * q^89 - 20 * q^91 + 43 * q^93 - 50 * q^95 - 35 * q^97 + 7 * q^99

Introduction

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