LMFDB - Embedded newform 4096.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4096,2,Mod(1,4096)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4096, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4096.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4096 = 2^{12} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4096.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:	$32.7067246679$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{16})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 2 $ x^4 - 4*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 32)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.84776$ of defining polynomial
Character	$\chi$	$=$	4096.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.84776 q^{3} +3.37849 q^{5} +1.41421 q^{7} +0.414214 q^{9} +O(q^{10})$	$q+1.84776 q^{3} +3.37849 q^{5} +1.41421 q^{7} +0.414214 q^{9} -0.317025 q^{11} -1.84776 q^{13} +6.24264 q^{15} +2.82843 q^{17} +5.99162 q^{19} +2.61313 q^{21} -0.242641 q^{23} +6.41421 q^{25} -4.77791 q^{27} +2.93015 q^{29} -4.00000 q^{31} -0.585786 q^{33} +4.77791 q^{35} +1.84776 q^{37} -3.41421 q^{39} -8.24264 q^{41} +8.60474 q^{43} +1.39942 q^{45} +11.6569 q^{47} -5.00000 q^{49} +5.22625 q^{51} +8.15640 q^{53} -1.07107 q^{55} +11.0711 q^{57} -6.62567 q^{59} -0.765367 q^{61} +0.585786 q^{63} -6.24264 q^{65} +4.01254 q^{67} -0.448342 q^{69} -0.242641 q^{71} -9.89949 q^{73} +11.8519 q^{75} -0.448342 q^{77} +6.00000 q^{79} -10.0711 q^{81} -6.62567 q^{83} +9.55582 q^{85} +5.41421 q^{87} -3.75736 q^{89} -2.61313 q^{91} -7.39104 q^{93} +20.2426 q^{95} +1.51472 q^{97} -0.131316 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^9	$ 4 q - 4 q^{9} + 8 q^{15} + 16 q^{23} + 20 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} - 16 q^{41} + 24 q^{47} - 20 q^{49} + 24 q^{55} + 16 q^{57} + 8 q^{63} - 8 q^{65} + 16 q^{71} + 24 q^{79} - 12 q^{81} + 16 q^{87} - 32 q^{89} + 64 q^{95} + 40 q^{97}+O(q^{100}) $ 4 * q - 4 * q^9 + 8 * q^15 + 16 * q^23 + 20 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 - 16 * q^41 + 24 * q^47 - 20 * q^49 + 24 * q^55 + 16 * q^57 + 8 * q^63 - 8 * q^65 + 16 * q^71 + 24 * q^79 - 12 * q^81 + 16 * q^87 - 32 * q^89 + 64 * q^95 + 40 * q^97

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
$3$	1.84776	1.06680	0.533402	−	0.845862i	$-0.320913\pi$
$3$	1.84776	1.06680	0.533402	+	0.845862i	$0.320913\pi$
$4$	0	0
$5$	3.37849	1.51091	0.755454	−	0.655202i	$-0.227416\pi$
$5$	3.37849	1.51091	0.755454	+	0.655202i	$0.227416\pi$
$6$	0	0
$7$	1.41421	0.534522	0.267261	−	0.963624i	$-0.413881\pi$
$7$	1.41421	0.534522	0.267261	+	0.963624i	$0.413881\pi$
$8$	0	0
$9$	0.414214	0.138071
$10$	0	0
$11$	−0.317025	−0.0955867	−0.0477934	−	0.998857i	$-0.515219\pi$
$11$	−0.317025	−0.0955867	−0.0477934	+	0.998857i	$0.515219\pi$
$12$	0	0
$13$	−1.84776	−0.512476	−0.256238	−	0.966614i	$-0.582483\pi$
$13$	−1.84776	−0.512476	−0.256238	+	0.966614i	$0.582483\pi$
$14$	0	0
$15$	6.24264	1.61184
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	5.99162	1.37457	0.687286	−	0.726387i	$-0.258802\pi$
$19$	5.99162	1.37457	0.687286	+	0.726387i	$0.258802\pi$
$20$	0	0
$21$	2.61313	0.570231
$22$	0	0
$23$	−0.242641	−0.0505941	−0.0252970	−	0.999680i	$-0.508053\pi$
$23$	−0.242641	−0.0505941	−0.0252970	+	0.999680i	$0.508053\pi$
$24$	0	0
$25$	6.41421	1.28284
$26$	0	0
$27$	−4.77791	−0.919509
$28$	0	0
$29$	2.93015	0.544115	0.272058	−	0.962281i	$-0.412296\pi$
$29$	2.93015	0.544115	0.272058	+	0.962281i	$0.412296\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	−0.585786	−0.101972
$34$	0	0
$35$	4.77791	0.807614
$36$	0	0
$37$	1.84776	0.303770	0.151885	−	0.988398i	$-0.451466\pi$
$37$	1.84776	0.303770	0.151885	+	0.988398i	$0.451466\pi$
$38$	0	0
$39$	−3.41421	−0.546712
$40$	0	0
$41$	−8.24264	−1.28728	−0.643642	−	0.765327i	$-0.722577\pi$
$41$	−8.24264	−1.28728	−0.643642	+	0.765327i	$0.722577\pi$
$42$	0	0
$43$	8.60474	1.31221	0.656106	−	0.754669i	$-0.272203\pi$
$43$	8.60474	1.31221	0.656106	+	0.754669i	$0.272203\pi$
$44$	0	0
$45$	1.39942	0.208613
$46$	0	0
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	5.22625	0.731822
$52$	0	0
$53$	8.15640	1.12037	0.560184	−	0.828368i	$-0.310730\pi$
$53$	8.15640	1.12037	0.560184	+	0.828368i	$0.310730\pi$
$54$	0	0
$55$	−1.07107	−0.144423
$56$	0	0
$57$	11.0711	1.46640
$58$	0	0
$59$	−6.62567	−0.862589	−0.431294	−	0.902211i	$-0.641943\pi$
$59$	−6.62567	−0.862589	−0.431294	+	0.902211i	$0.641943\pi$
$60$	0	0
$61$	−0.765367	−0.0979952	−0.0489976	−	0.998799i	$-0.515603\pi$
$61$	−0.765367	−0.0979952	−0.0489976	+	0.998799i	$0.515603\pi$
$62$	0	0
$63$	0.585786	0.0738022
$64$	0	0
$65$	−6.24264	−0.774304
$66$	0	0
$67$	4.01254	0.490210	0.245105	−	0.969497i	$-0.421178\pi$
$67$	4.01254	0.490210	0.245105	+	0.969497i	$0.421178\pi$
$68$	0	0
$69$	−0.448342	−0.0539740
$70$	0	0
$71$	−0.242641	−0.0287962	−0.0143981	−	0.999896i	$-0.504583\pi$
$71$	−0.242641	−0.0287962	−0.0143981	+	0.999896i	$0.504583\pi$
$72$	0	0
$73$	−9.89949	−1.15865	−0.579324	−	0.815097i	$-0.696683\pi$
$73$	−9.89949	−1.15865	−0.579324	+	0.815097i	$0.696683\pi$
$74$	0	0
$75$	11.8519	1.36854
$76$	0	0
$77$	−0.448342	−0.0510933
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	−10.0711	−1.11901
$82$	0	0
$83$	−6.62567	−0.727262	−0.363631	−	0.931543i	$-0.618463\pi$
$83$	−6.62567	−0.727262	−0.363631	+	0.931543i	$0.618463\pi$
$84$	0	0
$85$	9.55582	1.03647
$86$	0	0
$87$	5.41421	0.580465
$88$	0	0
$89$	−3.75736	−0.398279	−0.199140	−	0.979971i	$-0.563815\pi$
$89$	−3.75736	−0.398279	−0.199140	+	0.979971i	$0.563815\pi$
$90$	0	0
$91$	−2.61313	−0.273930
$92$	0	0
$93$	−7.39104	−0.766415
$94$	0	0
$95$	20.2426	2.07685
$96$	0	0
$97$	1.51472	0.153796	0.0768982	−	0.997039i	$-0.475498\pi$
$97$	1.51472	0.153796	0.0768982	+	0.997039i	$0.475498\pi$
$98$	0	0
$99$	−0.131316	−0.0131978
$100$	0	0
$101$	12.3003	1.22392	0.611961	−	0.790888i	$-0.290381\pi$
$101$	12.3003	1.22392	0.611961	+	0.790888i	$0.290381\pi$
$102$	0	0
$103$	10.5858	1.04305	0.521524	−	0.853236i	$-0.325364\pi$
$103$	10.5858	1.04305	0.521524	+	0.853236i	$0.325364\pi$
$104$	0	0
$105$	8.82843	0.861566
$106$	0	0
$107$	0.317025	0.0306480	0.0153240	−	0.999883i	$-0.495122\pi$
$107$	0.317025	0.0306480	0.0153240	+	0.999883i	$0.495122\pi$
$108$	0	0
$109$	−4.64659	−0.445063	−0.222532	−	0.974926i	$-0.571432\pi$
$109$	−4.64659	−0.445063	−0.222532	+	0.974926i	$0.571432\pi$
$110$	0	0
$111$	3.41421	0.324063
$112$	0	0
$113$	17.6569	1.66102	0.830509	−	0.557006i	$-0.188050\pi$
$113$	17.6569	1.66102	0.830509	+	0.557006i	$0.188050\pi$
$114$	0	0
$115$	−0.819760	−0.0764430
$116$	0	0
$117$	−0.765367	−0.0707582
$118$	0	0
$119$	4.00000	0.366679
$120$	0	0
$121$	−10.8995	−0.990863
$122$	0	0
$123$	−15.2304	−1.37328
$124$	0	0
$125$	4.77791	0.427349
$126$	0	0
$127$	20.9706	1.86084	0.930418	−	0.366499i	$-0.119444\pi$
$127$	20.9706	1.86084	0.930418	+	0.366499i	$0.119444\pi$
$128$	0	0
$129$	15.8995	1.39987
$130$	0	0
$131$	−9.50143	−0.830144	−0.415072	−	0.909789i	$-0.636244\pi$
$131$	−9.50143	−0.830144	−0.415072	+	0.909789i	$0.636244\pi$
$132$	0	0
$133$	8.47343	0.734739
$134$	0	0
$135$	−16.1421	−1.38929
$136$	0	0
$137$	−3.75736	−0.321013	−0.160506	−	0.987035i	$-0.551313\pi$
$137$	−3.75736	−0.321013	−0.160506	+	0.987035i	$0.551313\pi$
$138$	0	0
$139$	13.5684	1.15085	0.575427	−	0.817853i	$-0.304836\pi$
$139$	13.5684	1.15085	0.575427	+	0.817853i	$0.304836\pi$
$140$	0	0
$141$	21.5391	1.81392
$142$	0	0
$143$	0.585786	0.0489859
$144$	0	0
$145$	9.89949	0.822108
$146$	0	0
$147$	−9.23880	−0.762003
$148$	0	0
$149$	−14.6508	−1.20024	−0.600118	−	0.799911i	$-0.704880\pi$
$149$	−14.6508	−1.20024	−0.600118	+	0.799911i	$0.704880\pi$
$150$	0	0
$151$	−21.8995	−1.78216	−0.891078	−	0.453851i	$-0.850050\pi$
$151$	−21.8995	−1.78216	−0.891078	+	0.453851i	$0.850050\pi$
$152$	0	0
$153$	1.17157	0.0947161
$154$	0	0
$155$	−13.5140	−1.08547
$156$	0	0
$157$	0.765367	0.0610829	0.0305415	−	0.999534i	$-0.490277\pi$
$157$	0.765367	0.0610829	0.0305415	+	0.999534i	$0.490277\pi$
$158$	0	0
$159$	15.0711	1.19521
$160$	0	0
$161$	−0.343146	−0.0270437
$162$	0	0
$163$	−19.6913	−1.54234	−0.771171	−	0.636628i	$-0.780329\pi$
$163$	−19.6913	−1.54234	−0.771171	+	0.636628i	$0.780329\pi$
$164$	0	0
$165$	−1.97908	−0.154071
$166$	0	0
$167$	−4.72792	−0.365858	−0.182929	−	0.983126i	$-0.558558\pi$
$167$	−4.72792	−0.365858	−0.182929	+	0.983126i	$0.558558\pi$
$168$	0	0
$169$	−9.58579	−0.737368
$170$	0	0
$171$	2.48181	0.189789
$172$	0	0
$173$	−1.21371	−0.0922765	−0.0461383	−	0.998935i	$-0.514691\pi$
$173$	−1.21371	−0.0922765	−0.0461383	+	0.998935i	$0.514691\pi$
$174$	0	0
$175$	9.07107	0.685708
$176$	0	0
$177$	−12.2426	−0.920213
$178$	0	0
$179$	−15.5474	−1.16207	−0.581035	−	0.813879i	$-0.697352\pi$
$179$	−15.5474	−1.16207	−0.581035	+	0.813879i	$0.697352\pi$
$180$	0	0
$181$	−5.72899	−0.425832	−0.212916	−	0.977070i	$-0.568296\pi$
$181$	−5.72899	−0.425832	−0.212916	+	0.977070i	$0.568296\pi$
$182$	0	0
$183$	−1.41421	−0.104542
$184$	0	0
$185$	6.24264	0.458968
$186$	0	0
$187$	−0.896683	−0.0655720
$188$	0	0
$189$	−6.75699	−0.491498
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−18.4853	−1.33060	−0.665300	−	0.746576i	$-0.731696\pi$
$193$	−18.4853	−1.33060	−0.665300	+	0.746576i	$0.731696\pi$
$194$	0	0
$195$	−11.5349	−0.826031
$196$	0	0
$197$	−18.7946	−1.33906	−0.669530	−	0.742785i	$-0.733505\pi$
$197$	−18.7946	−1.33906	−0.669530	+	0.742785i	$0.733505\pi$
$198$	0	0
$199$	25.4142	1.80157	0.900783	−	0.434270i	$-0.142994\pi$
$199$	25.4142	1.80157	0.900783	+	0.434270i	$0.142994\pi$
$200$	0	0
$201$	7.41421	0.522958
$202$	0	0
$203$	4.14386	0.290842
$204$	0	0
$205$	−27.8477	−1.94497
$206$	0	0
$207$	−0.100505	−0.00698558
$208$	0	0
$209$	−1.89949	−0.131391
$210$	0	0
$211$	−0.502734	−0.0346097	−0.0173048	−	0.999850i	$-0.505509\pi$
$211$	−0.502734	−0.0346097	−0.0173048	+	0.999850i	$0.505509\pi$
$212$	0	0
$213$	−0.448342	−0.0307199
$214$	0	0
$215$	29.0711	1.98263
$216$	0	0
$217$	−5.65685	−0.384012
$218$	0	0
$219$	−18.2919	−1.23605
$220$	0	0
$221$	−5.22625	−0.351556
$222$	0	0
$223$	12.9706	0.868573	0.434287	−	0.900775i	$-0.357001\pi$
$223$	12.9706	0.868573	0.434287	+	0.900775i	$0.357001\pi$
$224$	0	0
$225$	2.65685	0.177124
$226$	0	0
$227$	6.81138	0.452087	0.226044	−	0.974117i	$-0.427421\pi$
$227$	6.81138	0.452087	0.226044	+	0.974117i	$0.427421\pi$
$228$	0	0
$229$	−26.8197	−1.77230	−0.886148	−	0.463402i	$-0.846628\pi$
$229$	−26.8197	−1.77230	−0.886148	+	0.463402i	$0.846628\pi$
$230$	0	0
$231$	−0.828427	−0.0545065
$232$	0	0
$233$	12.2426	0.802042	0.401021	−	0.916069i	$-0.368655\pi$
$233$	12.2426	0.802042	0.401021	+	0.916069i	$0.368655\pi$
$234$	0	0
$235$	39.3826	2.56904
$236$	0	0
$237$	11.0866	0.720149
$238$	0	0
$239$	−17.3137	−1.11993	−0.559965	−	0.828516i	$-0.689186\pi$
$239$	−17.3137	−1.11993	−0.559965	+	0.828516i	$0.689186\pi$
$240$	0	0
$241$	−8.48528	−0.546585	−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528	−0.546585	−0.273293	+	0.961931i	$0.588113\pi$
$242$	0	0
$243$	−4.27518	−0.274253
$244$	0	0
$245$	−16.8925	−1.07922
$246$	0	0
$247$	−11.0711	−0.704435
$248$	0	0
$249$	−12.2426	−0.775846
$250$	0	0
$251$	−15.8101	−0.997923	−0.498961	−	0.866624i	$-0.666285\pi$
$251$	−15.8101	−0.997923	−0.498961	+	0.866624i	$0.666285\pi$
$252$	0	0
$253$	0.0769232	0.00483612
$254$	0	0
$255$	17.6569	1.10572
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	2.61313	0.162372
$260$	0	0
$261$	1.21371	0.0751267
$262$	0	0
$263$	−0.242641	−0.0149619	−0.00748093	−	0.999972i	$-0.502381\pi$
$263$	−0.242641	−0.0149619	−0.00748093	+	0.999972i	$0.502381\pi$
$264$	0	0
$265$	27.5563	1.69277
$266$	0	0
$267$	−6.94269	−0.424886
$268$	0	0
$269$	−5.28064	−0.321967	−0.160983	−	0.986957i	$-0.551467\pi$
$269$	−5.28064	−0.321967	−0.160983	+	0.986957i	$0.551467\pi$
$270$	0	0
$271$	18.0000	1.09342	0.546711	−	0.837321i	$-0.315880\pi$
$271$	18.0000	1.09342	0.546711	+	0.837321i	$0.315880\pi$
$272$	0	0
$273$	−4.82843	−0.292230
$274$	0	0
$275$	−2.03347	−0.122623
$276$	0	0
$277$	−0.765367	−0.0459864	−0.0229932	−	0.999736i	$-0.507320\pi$
$277$	−0.765367	−0.0459864	−0.0229932	+	0.999736i	$0.507320\pi$
$278$	0	0
$279$	−1.65685	−0.0991933
$280$	0	0
$281$	−8.72792	−0.520664	−0.260332	−	0.965519i	$-0.583832\pi$
$281$	−8.72792	−0.520664	−0.260332	+	0.965519i	$0.583832\pi$
$282$	0	0
$283$	10.5838	0.629143	0.314571	−	0.949234i	$-0.398139\pi$
$283$	10.5838	0.629143	0.314571	+	0.949234i	$0.398139\pi$
$284$	0	0
$285$	37.4035	2.21559
$286$	0	0
$287$	−11.6569	−0.688082
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	2.79884	0.164071
$292$	0	0
$293$	−12.5629	−0.733932	−0.366966	−	0.930234i	$-0.619603\pi$
$293$	−12.5629	−0.733932	−0.366966	+	0.930234i	$0.619603\pi$
$294$	0	0
$295$	−22.3848	−1.30329
$296$	0	0
$297$	1.51472	0.0878929
$298$	0	0
$299$	0.448342	0.0259283
$300$	0	0
$301$	12.1689	0.701406
$302$	0	0
$303$	22.7279	1.30569
$304$	0	0
$305$	−2.58579	−0.148062
$306$	0	0
$307$	3.19278	0.182222	0.0911109	−	0.995841i	$-0.470958\pi$
$307$	3.19278	0.182222	0.0911109	+	0.995841i	$0.470958\pi$
$308$	0	0
$309$	19.5600	1.11273
$310$	0	0
$311$	12.2426	0.694216	0.347108	−	0.937825i	$-0.387164\pi$
$311$	12.2426	0.694216	0.347108	+	0.937825i	$0.387164\pi$
$312$	0	0
$313$	−13.4142	−0.758216	−0.379108	−	0.925352i	$-0.623769\pi$
$313$	−13.4142	−0.758216	−0.379108	+	0.925352i	$0.623769\pi$
$314$	0	0
$315$	1.97908	0.111508
$316$	0	0
$317$	−12.1146	−0.680421	−0.340211	−	0.940349i	$-0.610498\pi$
$317$	−12.1146	−0.680421	−0.340211	+	0.940349i	$0.610498\pi$
$318$	0	0
$319$	−0.928932	−0.0520102
$320$	0	0
$321$	0.585786	0.0326954
$322$	0	0
$323$	16.9469	0.942948
$324$	0	0
$325$	−11.8519	−0.657426
$326$	0	0
$327$	−8.58579	−0.474795
$328$	0	0
$329$	16.4853	0.908863
$330$	0	0
$331$	7.07401	0.388823	0.194411	−	0.980920i	$-0.437720\pi$
$331$	7.07401	0.388823	0.194411	+	0.980920i	$0.437720\pi$
$332$	0	0
$333$	0.765367	0.0419418
$334$	0	0
$335$	13.5563	0.740662
$336$	0	0
$337$	16.9706	0.924445	0.462223	−	0.886764i	$-0.347052\pi$
$337$	16.9706	0.924445	0.462223	+	0.886764i	$0.347052\pi$
$338$	0	0
$339$	32.6256	1.77198
$340$	0	0
$341$	1.26810	0.0686715
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	−1.51472	−0.0815497
$346$	0	0
$347$	−15.5474	−0.834630	−0.417315	−	0.908762i	$-0.637029\pi$
$347$	−15.5474	−0.834630	−0.417315	+	0.908762i	$0.637029\pi$
$348$	0	0
$349$	−27.9021	−1.49356	−0.746782	−	0.665068i	$-0.768402\pi$
$349$	−27.9021	−1.49356	−0.746782	+	0.665068i	$0.768402\pi$
$350$	0	0
$351$	8.82843	0.471227
$352$	0	0
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−0.819760	−0.0435083
$356$	0	0
$357$	7.39104	0.391175
$358$	0	0
$359$	17.2132	0.908478	0.454239	−	0.890880i	$-0.349911\pi$
$359$	17.2132	0.908478	0.454239	+	0.890880i	$0.349911\pi$
$360$	0	0
$361$	16.8995	0.889447
$362$	0	0
$363$	−20.1396	−1.05706
$364$	0	0
$365$	−33.4454	−1.75061
$366$	0	0
$367$	6.00000	0.313197	0.156599	−	0.987662i	$-0.449947\pi$
$367$	6.00000	0.313197	0.156599	+	0.987662i	$0.449947\pi$
$368$	0	0
$369$	−3.41421	−0.177737
$370$	0	0
$371$	11.5349	0.598862
$372$	0	0
$373$	−30.5921	−1.58400	−0.792001	−	0.610520i	$-0.790960\pi$
$373$	−30.5921	−1.58400	−0.792001	+	0.610520i	$0.790960\pi$
$374$	0	0
$375$	8.82843	0.455898
$376$	0	0
$377$	−5.41421	−0.278846
$378$	0	0
$379$	23.4637	1.20525	0.602626	−	0.798024i	$-0.294121\pi$
$379$	23.4637	1.20525	0.602626	+	0.798024i	$0.294121\pi$
$380$	0	0
$381$	38.7485	1.98515
$382$	0	0
$383$	−16.9706	−0.867155	−0.433578	−	0.901116i	$-0.642749\pi$
$383$	−16.9706	−0.867155	−0.433578	+	0.901116i	$0.642749\pi$
$384$	0	0
$385$	−1.51472	−0.0771972
$386$	0	0
$387$	3.56420	0.181179
$388$	0	0
$389$	32.0460	1.62479	0.812397	−	0.583104i	$-0.198162\pi$
$389$	32.0460	1.62479	0.812397	+	0.583104i	$0.198162\pi$
$390$	0	0
$391$	−0.686292	−0.0347073
$392$	0	0
$393$	−17.5563	−0.885601
$394$	0	0
$395$	20.2710	1.01994
$396$	0	0
$397$	26.8197	1.34604	0.673021	−	0.739623i	$-0.264996\pi$
$397$	26.8197	1.34604	0.673021	+	0.739623i	$0.264996\pi$
$398$	0	0
$399$	15.6569	0.783823
$400$	0	0
$401$	2.82843	0.141245	0.0706225	−	0.997503i	$-0.477501\pi$
$401$	2.82843	0.141245	0.0706225	+	0.997503i	$0.477501\pi$
$402$	0	0
$403$	7.39104	0.368174
$404$	0	0
$405$	−34.0250	−1.69072
$406$	0	0
$407$	−0.585786	−0.0290364
$408$	0	0
$409$	6.38478	0.315707	0.157853	−	0.987463i	$-0.449543\pi$
$409$	6.38478	0.315707	0.157853	+	0.987463i	$0.449543\pi$
$410$	0	0
$411$	−6.94269	−0.342458
$412$	0	0
$413$	−9.37011	−0.461073
$414$	0	0
$415$	−22.3848	−1.09883
$416$	0	0
$417$	25.0711	1.22774
$418$	0	0
$419$	22.4901	1.09871	0.549357	−	0.835587i	$-0.314873\pi$
$419$	22.4901	1.09871	0.549357	+	0.835587i	$0.314873\pi$
$420$	0	0
$421$	8.34211	0.406570	0.203285	−	0.979120i	$-0.434838\pi$
$421$	8.34211	0.406570	0.203285	+	0.979120i	$0.434838\pi$
$422$	0	0
$423$	4.82843	0.234766
$424$	0	0
$425$	18.1421	0.880023
$426$	0	0
$427$	−1.08239	−0.0523806
$428$	0	0
$429$	1.08239	0.0522584
$430$	0	0
$431$	23.6569	1.13951	0.569755	−	0.821814i	$-0.307038\pi$
$431$	23.6569	1.13951	0.569755	+	0.821814i	$0.307038\pi$
$432$	0	0
$433$	32.4853	1.56114	0.780571	−	0.625067i	$-0.214928\pi$
$433$	32.4853	1.56114	0.780571	+	0.625067i	$0.214928\pi$
$434$	0	0
$435$	18.2919	0.877029
$436$	0	0
$437$	−1.45381	−0.0695452
$438$	0	0
$439$	−24.0416	−1.14744	−0.573722	−	0.819050i	$-0.694501\pi$
$439$	−24.0416	−1.14744	−0.573722	+	0.819050i	$0.694501\pi$
$440$	0	0
$441$	−2.07107	−0.0986223
$442$	0	0
$443$	−22.3044	−1.05972	−0.529858	−	0.848087i	$-0.677755\pi$
$443$	−22.3044	−1.05972	−0.529858	+	0.848087i	$0.677755\pi$
$444$	0	0
$445$	−12.6942	−0.601763
$446$	0	0
$447$	−27.0711	−1.28042
$448$	0	0
$449$	31.4558	1.48449	0.742247	−	0.670127i	$-0.233760\pi$
$449$	31.4558	1.48449	0.742247	+	0.670127i	$0.233760\pi$
$450$	0	0
$451$	2.61313	0.123047
$452$	0	0
$453$	−40.4650	−1.90121
$454$	0	0
$455$	−8.82843	−0.413883
$456$	0	0
$457$	−13.4142	−0.627490	−0.313745	−	0.949507i	$-0.601584\pi$
$457$	−13.4142	−0.627490	−0.313745	+	0.949507i	$0.601584\pi$
$458$	0	0
$459$	−13.5140	−0.630778
$460$	0	0
$461$	−14.4650	−0.673704	−0.336852	−	0.941558i	$-0.609362\pi$
$461$	−14.4650	−0.673704	−0.336852	+	0.941558i	$0.609362\pi$
$462$	0	0
$463$	−10.9706	−0.509845	−0.254923	−	0.966961i	$-0.582050\pi$
$463$	−10.9706	−0.509845	−0.254923	+	0.966961i	$0.582050\pi$
$464$	0	0
$465$	−24.9706	−1.15798
$466$	0	0
$467$	−31.4888	−1.45713	−0.728565	−	0.684977i	$-0.759812\pi$
$467$	−31.4888	−1.45713	−0.728565	+	0.684977i	$0.759812\pi$
$468$	0	0
$469$	5.67459	0.262028
$470$	0	0
$471$	1.41421	0.0651635
$472$	0	0
$473$	−2.72792	−0.125430
$474$	0	0
$475$	38.4315	1.76336
$476$	0	0
$477$	3.37849	0.154691
$478$	0	0
$479$	−4.97056	−0.227111	−0.113555	−	0.993532i	$-0.536224\pi$
$479$	−4.97056	−0.227111	−0.113555	+	0.993532i	$0.536224\pi$
$480$	0	0
$481$	−3.41421	−0.155675
$482$	0	0
$483$	−0.634051	−0.0288503
$484$	0	0
$485$	5.11747	0.232372
$486$	0	0
$487$	15.5563	0.704925	0.352463	−	0.935826i	$-0.385344\pi$
$487$	15.5563	0.704925	0.352463	+	0.935826i	$0.385344\pi$
$488$	0	0
$489$	−36.3848	−1.64538
$490$	0	0
$491$	−19.1660	−0.864951	−0.432476	−	0.901646i	$-0.642360\pi$
$491$	−19.1660	−0.864951	−0.432476	+	0.901646i	$0.642360\pi$
$492$	0	0
$493$	8.28772	0.373260
$494$	0	0
$495$	−0.443651	−0.0199406
$496$	0	0
$497$	−0.343146	−0.0153922
$498$	0	0
$499$	9.68714	0.433656	0.216828	−	0.976210i	$-0.430429\pi$
$499$	9.68714	0.433656	0.216828	+	0.976210i	$0.430429\pi$
$500$	0	0
$501$	−8.73606	−0.390298
$502$	0	0
$503$	24.2426	1.08093	0.540463	−	0.841368i	$-0.318249\pi$
$503$	24.2426	1.08093	0.540463	+	0.841368i	$0.318249\pi$
$504$	0	0
$505$	41.5563	1.84923
$506$	0	0
$507$	−17.7122	−0.786627
$508$	0	0
$509$	−31.5976	−1.40054	−0.700270	−	0.713878i	$-0.746937\pi$
$509$	−31.5976	−1.40054	−0.700270	+	0.713878i	$0.746937\pi$
$510$	0	0
$511$	−14.0000	−0.619324
$512$	0	0
$513$	−28.6274	−1.26393
$514$	0	0
$515$	35.7640	1.57595
$516$	0	0
$517$	−3.69552	−0.162529
$518$	0	0
$519$	−2.24264	−0.0984410
$520$	0	0
$521$	−20.7279	−0.908107	−0.454053	−	0.890974i	$-0.650022\pi$
$521$	−20.7279	−0.908107	−0.454053	+	0.890974i	$0.650022\pi$
$522$	0	0
$523$	−2.11039	−0.0922810	−0.0461405	−	0.998935i	$-0.514692\pi$
$523$	−2.11039	−0.0922810	−0.0461405	+	0.998935i	$0.514692\pi$
$524$	0	0
$525$	16.7611	0.731516
$526$	0	0
$527$	−11.3137	−0.492833
$528$	0	0
$529$	−22.9411	−0.997440
$530$	0	0
$531$	−2.74444	−0.119099
$532$	0	0
$533$	15.2304	0.659702
$534$	0	0
$535$	1.07107	0.0463063
$536$	0	0
$537$	−28.7279	−1.23970
$538$	0	0
$539$	1.58513	0.0682762
$540$	0	0
$541$	13.7541	0.591334	0.295667	−	0.955291i	$-0.404458\pi$
$541$	13.7541	0.591334	0.295667	+	0.955291i	$0.404458\pi$
$542$	0	0
$543$	−10.5858	−0.454280
$544$	0	0
$545$	−15.6985	−0.672449
$546$	0	0
$547$	27.3450	1.16919	0.584593	−	0.811327i	$-0.301254\pi$
$547$	27.3450	1.16919	0.584593	+	0.811327i	$0.301254\pi$
$548$	0	0
$549$	−0.317025	−0.0135303
$550$	0	0
$551$	17.5563	0.747926
$552$	0	0
$553$	8.48528	0.360831
$554$	0	0
$555$	11.5349	0.489629
$556$	0	0
$557$	11.7750	0.498923	0.249461	−	0.968385i	$-0.419746\pi$
$557$	11.7750	0.498923	0.249461	+	0.968385i	$0.419746\pi$
$558$	0	0
$559$	−15.8995	−0.672477
$560$	0	0
$561$	−1.65685	−0.0699524
$562$	0	0
$563$	13.1200	0.552943	0.276472	−	0.961022i	$-0.410835\pi$
$563$	13.1200	0.552943	0.276472	+	0.961022i	$0.410835\pi$
$564$	0	0
$565$	59.6536	2.50964
$566$	0	0
$567$	−14.2426	−0.598135
$568$	0	0
$569$	−4.72792	−0.198205	−0.0991024	−	0.995077i	$-0.531597\pi$
$569$	−4.72792	−0.198205	−0.0991024	+	0.995077i	$0.531597\pi$
$570$	0	0
$571$	−1.39942	−0.0585638	−0.0292819	−	0.999571i	$-0.509322\pi$
$571$	−1.39942	−0.0585638	−0.0292819	+	0.999571i	$0.509322\pi$
$572$	0	0
$573$	22.1731	0.926295
$574$	0	0
$575$	−1.55635	−0.0649042
$576$	0	0
$577$	−14.9706	−0.623233	−0.311616	−	0.950208i	$-0.600870\pi$
$577$	−14.9706	−0.623233	−0.311616	+	0.950208i	$0.600870\pi$
$578$	0	0
$579$	−34.1563	−1.41949
$580$	0	0
$581$	−9.37011	−0.388738
$582$	0	0
$583$	−2.58579	−0.107092
$584$	0	0
$585$	−2.58579	−0.106909
$586$	0	0
$587$	−22.4901	−0.928267	−0.464134	−	0.885765i	$-0.653634\pi$
$587$	−22.4901	−0.928267	−0.464134	+	0.885765i	$0.653634\pi$
$588$	0	0
$589$	−23.9665	−0.987521
$590$	0	0
$591$	−34.7279	−1.42852
$592$	0	0
$593$	28.2843	1.16150	0.580748	−	0.814083i	$-0.302760\pi$
$593$	28.2843	1.16150	0.580748	+	0.814083i	$0.302760\pi$
$594$	0	0
$595$	13.5140	0.554019
$596$	0	0
$597$	46.9593	1.92192
$598$	0	0
$599$	21.6985	0.886576	0.443288	−	0.896379i	$-0.353812\pi$
$599$	21.6985	0.886576	0.443288	+	0.896379i	$0.353812\pi$
$600$	0	0
$601$	16.9289	0.690546	0.345273	−	0.938502i	$-0.387786\pi$
$601$	16.9289	0.690546	0.345273	+	0.938502i	$0.387786\pi$
$602$	0	0
$603$	1.66205	0.0676839
$604$	0	0
$605$	−36.8239	−1.49710
$606$	0	0
$607$	0.970563	0.0393939	0.0196970	−	0.999806i	$-0.493730\pi$
$607$	0.970563	0.0393939	0.0196970	+	0.999806i	$0.493730\pi$
$608$	0	0
$609$	7.65685	0.310271
$610$	0	0
$611$	−21.5391	−0.871377
$612$	0	0
$613$	39.6996	1.60345	0.801726	−	0.597691i	$-0.203915\pi$
$613$	39.6996	1.60345	0.801726	+	0.597691i	$0.203915\pi$
$614$	0	0
$615$	−51.4558	−2.07490
$616$	0	0
$617$	−23.7574	−0.956435	−0.478218	−	0.878241i	$-0.658717\pi$
$617$	−23.7574	−0.956435	−0.478218	+	0.878241i	$0.658717\pi$
$618$	0	0
$619$	−16.2584	−0.653481	−0.326740	−	0.945114i	$-0.605950\pi$
$619$	−16.2584	−0.653481	−0.326740	+	0.945114i	$0.605950\pi$
$620$	0	0
$621$	1.15932	0.0465217
$622$	0	0
$623$	−5.31371	−0.212889
$624$	0	0
$625$	−15.9289	−0.637157
$626$	0	0
$627$	−3.50981	−0.140168
$628$	0	0
$629$	5.22625	0.208384
$630$	0	0
$631$	−26.1005	−1.03904	−0.519522	−	0.854457i	$-0.673890\pi$
$631$	−26.1005	−1.03904	−0.519522	+	0.854457i	$0.673890\pi$
$632$	0	0
$633$	−0.928932	−0.0369217
$634$	0	0
$635$	70.8489	2.81155
$636$	0	0
$637$	9.23880	0.366054
$638$	0	0
$639$	−0.100505	−0.00397592
$640$	0	0
$641$	−43.4558	−1.71640	−0.858201	−	0.513313i	$-0.828418\pi$
$641$	−43.4558	−1.71640	−0.858201	+	0.513313i	$0.828418\pi$
$642$	0	0
$643$	−40.3337	−1.59060	−0.795302	−	0.606213i	$-0.792688\pi$
$643$	−40.3337	−1.59060	−0.795302	+	0.606213i	$0.792688\pi$
$644$	0	0
$645$	53.7163	2.11508
$646$	0	0
$647$	16.7279	0.657642	0.328821	−	0.944392i	$-0.393349\pi$
$647$	16.7279	0.657642	0.328821	+	0.944392i	$0.393349\pi$
$648$	0	0
$649$	2.10051	0.0824520
$650$	0	0
$651$	−10.4525	−0.409666
$652$	0	0
$653$	39.0656	1.52875	0.764377	−	0.644770i	$-0.223047\pi$
$653$	39.0656	1.52875	0.764377	+	0.644770i	$0.223047\pi$
$654$	0	0
$655$	−32.1005	−1.25427
$656$	0	0
$657$	−4.10051	−0.159976
$658$	0	0
$659$	6.36304	0.247869	0.123934	−	0.992290i	$-0.460449\pi$
$659$	6.36304	0.247869	0.123934	+	0.992290i	$0.460449\pi$
$660$	0	0
$661$	−20.2484	−0.787573	−0.393786	−	0.919202i	$-0.628835\pi$
$661$	−20.2484	−0.787573	−0.393786	+	0.919202i	$0.628835\pi$
$662$	0	0
$663$	−9.65685	−0.375041
$664$	0	0
$665$	28.6274	1.11012
$666$	0	0
$667$	−0.710974	−0.0275290
$668$	0	0
$669$	23.9665	0.926597
$670$	0	0
$671$	0.242641	0.00936704
$672$	0	0
$673$	−5.51472	−0.212577	−0.106288	−	0.994335i	$-0.533897\pi$
$673$	−5.51472	−0.212577	−0.106288	+	0.994335i	$0.533897\pi$
$674$	0	0
$675$	−30.6465	−1.17959
$676$	0	0
$677$	−6.06854	−0.233233	−0.116617	−	0.993177i	$-0.537205\pi$
$677$	−6.06854	−0.233233	−0.116617	+	0.993177i	$0.537205\pi$
$678$	0	0
$679$	2.14214	0.0822076
$680$	0	0
$681$	12.5858	0.482288
$682$	0	0
$683$	−15.3617	−0.587800	−0.293900	−	0.955836i	$-0.594953\pi$
$683$	−15.3617	−0.587800	−0.293900	+	0.955836i	$0.594953\pi$
$684$	0	0
$685$	−12.6942	−0.485021
$686$	0	0
$687$	−49.5563	−1.89069
$688$	0	0
$689$	−15.0711	−0.574162
$690$	0	0
$691$	−30.8548	−1.17377	−0.586886	−	0.809670i	$-0.699646\pi$
$691$	−30.8548	−1.17377	−0.586886	+	0.809670i	$0.699646\pi$
$692$	0	0
$693$	−0.185709	−0.00705451
$694$	0	0
$695$	45.8406	1.73883
$696$	0	0
$697$	−23.3137	−0.883070
$698$	0	0
$699$	22.6215	0.855622
$700$	0	0
$701$	−18.6089	−0.702849	−0.351424	−	0.936216i	$-0.614303\pi$
$701$	−18.6089	−0.702849	−0.351424	+	0.936216i	$0.614303\pi$
$702$	0	0
$703$	11.0711	0.417553
$704$	0	0
$705$	72.7696	2.74066
$706$	0	0
$707$	17.3952	0.654214
$708$	0	0
$709$	7.33664	0.275533	0.137767	−	0.990465i	$-0.456008\pi$
$709$	7.33664	0.275533	0.137767	+	0.990465i	$0.456008\pi$
$710$	0	0
$711$	2.48528	0.0932053
$712$	0	0
$713$	0.970563	0.0363479
$714$	0	0
$715$	1.97908	0.0740132
$716$	0	0
$717$	−31.9916	−1.19475
$718$	0	0
$719$	24.3431	0.907846	0.453923	−	0.891041i	$-0.350024\pi$
$719$	24.3431	0.907846	0.453923	+	0.891041i	$0.350024\pi$
$720$	0	0
$721$	14.9706	0.557533
$722$	0	0
$723$	−15.6788	−0.583099
$724$	0	0
$725$	18.7946	0.698015
$726$	0	0
$727$	−33.8995	−1.25726	−0.628631	−	0.777703i	$-0.716385\pi$
$727$	−33.8995	−1.25726	−0.628631	+	0.777703i	$0.716385\pi$
$728$	0	0
$729$	22.3137	0.826434
$730$	0	0
$731$	24.3379	0.900169
$732$	0	0
$733$	−1.92468	−0.0710898	−0.0355449	−	0.999368i	$-0.511317\pi$
$733$	−1.92468	−0.0710898	−0.0355449	+	0.999368i	$0.511317\pi$
$734$	0	0
$735$	−31.2132	−1.15132
$736$	0	0
$737$	−1.27208	−0.0468576
$738$	0	0
$739$	−19.6913	−0.724356	−0.362178	−	0.932109i	$-0.617967\pi$
$739$	−19.6913	−0.724356	−0.362178	+	0.932109i	$0.617967\pi$
$740$	0	0
$741$	−20.4567	−0.751494
$742$	0	0
$743$	19.2721	0.707024	0.353512	−	0.935430i	$-0.384987\pi$
$743$	19.2721	0.707024	0.353512	+	0.935430i	$0.384987\pi$
$744$	0	0
$745$	−49.4975	−1.81345
$746$	0	0
$747$	−2.74444	−0.100414
$748$	0	0
$749$	0.448342	0.0163820
$750$	0	0
$751$	−22.9706	−0.838208	−0.419104	−	0.907938i	$-0.637656\pi$
$751$	−22.9706	−0.838208	−0.419104	+	0.907938i	$0.637656\pi$
$752$	0	0
$753$	−29.2132	−1.06459
$754$	0	0
$755$	−73.9873	−2.69267
$756$	0	0
$757$	−1.92468	−0.0699538	−0.0349769	−	0.999388i	$-0.511136\pi$
$757$	−1.92468	−0.0699538	−0.0349769	+	0.999388i	$0.511136\pi$
$758$	0	0
$759$	0.142136	0.00515920
$760$	0	0
$761$	34.1838	1.23916	0.619580	−	0.784933i	$-0.287303\pi$
$761$	34.1838	1.23916	0.619580	+	0.784933i	$0.287303\pi$
$762$	0	0
$763$	−6.57128	−0.237896
$764$	0	0
$765$	3.95815	0.143107
$766$	0	0
$767$	12.2426	0.442056
$768$	0	0
$769$	22.4853	0.810840	0.405420	−	0.914131i	$-0.367125\pi$
$769$	22.4853	0.810840	0.405420	+	0.914131i	$0.367125\pi$
$770$	0	0
$771$	11.0866	0.399273
$772$	0	0
$773$	28.2417	1.01578	0.507891	−	0.861421i	$-0.330425\pi$
$773$	28.2417	1.01578	0.507891	+	0.861421i	$0.330425\pi$
$774$	0	0
$775$	−25.6569	−0.921621
$776$	0	0
$777$	4.82843	0.173219
$778$	0	0
$779$	−49.3868	−1.76946
$780$	0	0
$781$	0.0769232	0.00275253
$782$	0	0
$783$	−14.0000	−0.500319
$784$	0	0
$785$	2.58579	0.0922907
$786$	0	0
$787$	−9.68714	−0.345309	−0.172655	−	0.984982i	$-0.555234\pi$
$787$	−9.68714	−0.345309	−0.172655	+	0.984982i	$0.555234\pi$
$788$	0	0
$789$	−0.448342	−0.0159614
$790$	0	0
$791$	24.9706	0.887851
$792$	0	0
$793$	1.41421	0.0502202
$794$	0	0
$795$	50.9175	1.80586
$796$	0	0
$797$	31.5976	1.11924	0.559622	−	0.828748i	$-0.310946\pi$
$797$	31.5976	1.11924	0.559622	+	0.828748i	$0.310946\pi$
$798$	0	0
$799$	32.9706	1.16641
$800$	0	0
$801$	−1.55635	−0.0549909
$802$	0	0
$803$	3.13839	0.110751
$804$	0	0
$805$	−1.15932	−0.0408605
$806$	0	0
$807$	−9.75736	−0.343475
$808$	0	0
$809$	−1.21320	−0.0426540	−0.0213270	−	0.999773i	$-0.506789\pi$
$809$	−1.21320	−0.0426540	−0.0213270	+	0.999773i	$0.506789\pi$
$810$	0	0
$811$	−25.4428	−0.893418	−0.446709	−	0.894679i	$-0.647404\pi$
$811$	−25.4428	−0.893418	−0.446709	+	0.894679i	$0.647404\pi$
$812$	0	0
$813$	33.2597	1.16647
$814$	0	0
$815$	−66.5269	−2.33034
$816$	0	0
$817$	51.5563	1.80373
$818$	0	0
$819$	−1.08239	−0.0378218
$820$	0	0
$821$	−1.02800	−0.0358774	−0.0179387	−	0.999839i	$-0.505710\pi$
$821$	−1.02800	−0.0358774	−0.0179387	+	0.999839i	$0.505710\pi$
$822$	0	0
$823$	2.87006	0.100044	0.0500220	−	0.998748i	$-0.484071\pi$
$823$	2.87006	0.100044	0.0500220	+	0.998748i	$0.484071\pi$
$824$	0	0
$825$	−3.75736	−0.130814
$826$	0	0
$827$	12.8574	0.447095	0.223548	−	0.974693i	$-0.428236\pi$
$827$	12.8574	0.447095	0.223548	+	0.974693i	$0.428236\pi$
$828$	0	0
$829$	−41.3073	−1.43466	−0.717331	−	0.696733i	$-0.754636\pi$
$829$	−41.3073	−1.43466	−0.717331	+	0.696733i	$0.754636\pi$
$830$	0	0
$831$	−1.41421	−0.0490585
$832$	0	0
$833$	−14.1421	−0.489996
$834$	0	0
$835$	−15.9733	−0.552777
$836$	0	0
$837$	19.1116	0.660595
$838$	0	0
$839$	45.6985	1.57769	0.788843	−	0.614594i	$-0.210680\pi$
$839$	45.6985	1.57769	0.788843	+	0.614594i	$0.210680\pi$
$840$	0	0
$841$	−20.4142	−0.703938
$842$	0	0
$843$	−16.1271	−0.555447
$844$	0	0
$845$	−32.3855	−1.11410
$846$	0	0
$847$	−15.4142	−0.529639
$848$	0	0
$849$	19.5563	0.671172
$850$	0	0
$851$	−0.448342	−0.0153689
$852$	0	0
$853$	3.03894	0.104051	0.0520256	−	0.998646i	$-0.483432\pi$
$853$	3.03894	0.104051	0.0520256	+	0.998646i	$0.483432\pi$
$854$	0	0
$855$	8.38478	0.286753
$856$	0	0
$857$	45.6985	1.56103	0.780515	−	0.625137i	$-0.214957\pi$
$857$	45.6985	1.56103	0.780515	+	0.625137i	$0.214957\pi$
$858$	0	0
$859$	−36.4524	−1.24374	−0.621871	−	0.783120i	$-0.713627\pi$
$859$	−36.4524	−1.24374	−0.621871	+	0.783120i	$0.713627\pi$
$860$	0	0
$861$	−21.5391	−0.734049
$862$	0	0
$863$	−45.9411	−1.56385	−0.781927	−	0.623370i	$-0.785763\pi$
$863$	−45.9411	−1.56385	−0.781927	+	0.623370i	$0.785763\pi$
$864$	0	0
$865$	−4.10051	−0.139421
$866$	0	0
$867$	−16.6298	−0.564779
$868$	0	0
$869$	−1.90215	−0.0645261
$870$	0	0
$871$	−7.41421	−0.251221
$872$	0	0
$873$	0.627417	0.0212348
$874$	0	0
$875$	6.75699	0.228428
$876$	0	0
$877$	−36.0041	−1.21577	−0.607886	−	0.794024i	$-0.707982\pi$
$877$	−36.0041	−1.21577	−0.607886	+	0.794024i	$0.707982\pi$
$878$	0	0
$879$	−23.2132	−0.782962
$880$	0	0
$881$	22.6274	0.762337	0.381169	−	0.924506i	$-0.375522\pi$
$881$	22.6274	0.762337	0.381169	+	0.924506i	$0.375522\pi$
$882$	0	0
$883$	51.3433	1.72784	0.863920	−	0.503629i	$-0.168002\pi$
$883$	51.3433	1.72784	0.863920	+	0.503629i	$0.168002\pi$
$884$	0	0
$885$	−41.3617	−1.39036
$886$	0	0
$887$	−28.7279	−0.964589	−0.482295	−	0.876009i	$-0.660197\pi$
$887$	−28.7279	−0.964589	−0.482295	+	0.876009i	$0.660197\pi$
$888$	0	0
$889$	29.6569	0.994659
$890$	0	0
$891$	3.19278	0.106962
$892$	0	0
$893$	69.8434	2.33722
$894$	0	0
$895$	−52.5269	−1.75578
$896$	0	0
$897$	0.828427	0.0276604
$898$	0	0
$899$	−11.7206	−0.390904
$900$	0	0
$901$	23.0698	0.768566
$902$	0	0
$903$	22.4853	0.748263
$904$	0	0
$905$	−19.3553	−0.643393
$906$	0	0
$907$	0.579658	0.0192472	0.00962361	−	0.999954i	$-0.496937\pi$
$907$	0.579658	0.0192472	0.00962361	+	0.999954i	$0.496937\pi$
$908$	0	0
$909$	5.09494	0.168988
$910$	0	0
$911$	−45.5980	−1.51073	−0.755364	−	0.655305i	$-0.772540\pi$
$911$	−45.5980	−1.51073	−0.755364	+	0.655305i	$0.772540\pi$
$912$	0	0
$913$	2.10051	0.0695166
$914$	0	0
$915$	−4.77791	−0.157953
$916$	0	0
$917$	−13.4370	−0.443730
$918$	0	0
$919$	36.0416	1.18890	0.594452	−	0.804131i	$-0.297369\pi$
$919$	36.0416	1.18890	0.594452	+	0.804131i	$0.297369\pi$
$920$	0	0
$921$	5.89949	0.194395
$922$	0	0
$923$	0.448342	0.0147573
$924$	0	0
$925$	11.8519	0.389689
$926$	0	0
$927$	4.38478	0.144015
$928$	0	0
$929$	26.4853	0.868954	0.434477	−	0.900683i	$-0.356933\pi$
$929$	26.4853	0.868954	0.434477	+	0.900683i	$0.356933\pi$
$930$	0	0
$931$	−29.9581	−0.981837
$932$	0	0
$933$	22.6215	0.740593
$934$	0	0
$935$	−3.02944	−0.0990732
$936$	0	0
$937$	26.8701	0.877807	0.438903	−	0.898534i	$-0.355367\pi$
$937$	26.8701	0.877807	0.438903	+	0.898534i	$0.355367\pi$
$938$	0	0
$939$	−24.7862	−0.808868
$940$	0	0
$941$	14.4650	0.471547	0.235774	−	0.971808i	$-0.424238\pi$
$941$	14.4650	0.471547	0.235774	+	0.971808i	$0.424238\pi$
$942$	0	0
$943$	2.00000	0.0651290
$944$	0	0
$945$	−22.8284	−0.742609
$946$	0	0
$947$	−40.4106	−1.31317	−0.656584	−	0.754253i	$-0.727999\pi$
$947$	−40.4106	−1.31317	−0.656584	+	0.754253i	$0.727999\pi$
$948$	0	0
$949$	18.2919	0.593780
$950$	0	0
$951$	−22.3848	−0.725876
$952$	0	0
$953$	−4.72792	−0.153152	−0.0765762	−	0.997064i	$-0.524399\pi$
$953$	−4.72792	−0.153152	−0.0765762	+	0.997064i	$0.524399\pi$
$954$	0	0
$955$	40.5419	1.31191
$956$	0	0
$957$	−1.71644	−0.0554847
$958$	0	0
$959$	−5.31371	−0.171589
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0.131316	0.00423161
$964$	0	0
$965$	−62.4524	−2.01041
$966$	0	0
$967$	−56.5269	−1.81778	−0.908891	−	0.417033i	$-0.863070\pi$
$967$	−56.5269	−1.81778	−0.908891	+	0.417033i	$0.863070\pi$
$968$	0	0
$969$	31.3137	1.00594
$970$	0	0
$971$	−25.1802	−0.808071	−0.404035	−	0.914743i	$-0.632393\pi$
$971$	−25.1802	−0.808071	−0.404035	+	0.914743i	$0.632393\pi$
$972$	0	0
$973$	19.1886	0.615157
$974$	0	0
$975$	−21.8995	−0.701345
$976$	0	0
$977$	−14.1421	−0.452447	−0.226224	−	0.974075i	$-0.572638\pi$
$977$	−14.1421	−0.452447	−0.226224	+	0.974075i	$0.572638\pi$
$978$	0	0
$979$	1.19118	0.0380702
$980$	0	0
$981$	−1.92468	−0.0614504
$982$	0	0
$983$	−36.2426	−1.15596	−0.577980	−	0.816051i	$-0.696159\pi$
$983$	−36.2426	−1.15596	−0.577980	+	0.816051i	$0.696159\pi$
$984$	0	0
$985$	−63.4975	−2.02320
$986$	0	0
$987$	30.4608	0.969579
$988$	0	0
$989$	−2.08786	−0.0663901
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	13.0711	0.414798
$994$	0	0
$995$	85.8617	2.72200
$996$	0	0
$997$	−1.95654	−0.0619644	−0.0309822	−	0.999520i	$-0.509864\pi$
$997$	−1.95654	−0.0619644	−0.0309822	+	0.999520i	$0.509864\pi$
$998$	0	0
$999$	−8.82843	−0.279319

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4096.2.a.f.1.4		4
4.3	odd	2		4096.2.a.e.1.1		4
8.3	odd	2		4096.2.a.e.1.4		4
8.5	even	2	inner	4096.2.a.f.1.1		4
64.3	odd	16		512.2.g.d.321.1		4
64.5	even	16		256.2.g.a.225.1		4
64.11	odd	16		512.2.g.a.193.1		4
64.13	even	16		256.2.g.a.33.1		4
64.19	odd	16		32.2.g.a.13.1	yes	4
64.21	even	16		512.2.g.b.193.1		4
64.27	odd	16		32.2.g.a.5.1	✓	4
64.29	even	16		512.2.g.c.321.1		4
64.35	odd	16		512.2.g.a.321.1		4
64.37	even	16		128.2.g.a.113.1		4
64.43	odd	16		512.2.g.d.193.1		4
64.45	even	16		128.2.g.a.17.1		4
64.51	odd	16		256.2.g.b.33.1		4
64.53	even	16		512.2.g.c.193.1		4
64.59	odd	16		256.2.g.b.225.1		4
64.61	even	16		512.2.g.b.321.1		4
192.83	even	16		288.2.v.a.109.1		4
192.101	odd	16		1152.2.v.a.1009.1		4
192.155	even	16		288.2.v.a.37.1		4
192.173	odd	16		1152.2.v.a.145.1		4
320.19	odd	16		800.2.y.a.301.1		4
320.27	even	16		800.2.ba.b.549.1		4
320.83	even	16		800.2.ba.b.749.1		4
320.147	even	16		800.2.ba.a.749.1		4
320.219	odd	16		800.2.y.a.101.1		4
320.283	even	16		800.2.ba.a.549.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
32.2.g.a.5.1	✓	4	64.27	odd	16
32.2.g.a.13.1	yes	4	64.19	odd	16
128.2.g.a.17.1		4	64.45	even	16
128.2.g.a.113.1		4	64.37	even	16
256.2.g.a.33.1		4	64.13	even	16
256.2.g.a.225.1		4	64.5	even	16
256.2.g.b.33.1		4	64.51	odd	16
256.2.g.b.225.1		4	64.59	odd	16
288.2.v.a.37.1		4	192.155	even	16
288.2.v.a.109.1		4	192.83	even	16
512.2.g.a.193.1		4	64.11	odd	16
512.2.g.a.321.1		4	64.35	odd	16
512.2.g.b.193.1		4	64.21	even	16
512.2.g.b.321.1		4	64.61	even	16
512.2.g.c.193.1		4	64.53	even	16
512.2.g.c.321.1		4	64.29	even	16
512.2.g.d.193.1		4	64.43	odd	16
512.2.g.d.321.1		4	64.3	odd	16
800.2.y.a.101.1		4	320.219	odd	16
800.2.y.a.301.1		4	320.19	odd	16
800.2.ba.a.549.1		4	320.283	even	16
800.2.ba.a.749.1		4	320.147	even	16
800.2.ba.b.549.1		4	320.27	even	16
800.2.ba.b.749.1		4	320.83	even	16
1152.2.v.a.145.1		4	192.173	odd	16
1152.2.v.a.1009.1		4	192.101	odd	16
4096.2.a.e.1.1		4	4.3	odd	2
4096.2.a.e.1.4		4	8.3	odd	2
4096.2.a.f.1.1		4	8.5	even	2	inner
4096.2.a.f.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 4096 = 2^{12} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4096.a (trivial)

\(f(q)\)	\(=\)	\(q+1.84776 q^{3} +3.37849 q^{5} +1.41421 q^{7} +0.414214 q^{9} +O(q^{10})\)	\(q+1.84776 q^{3} +3.37849 q^{5} +1.41421 q^{7} +0.414214 q^{9} -0.317025 q^{11} -1.84776 q^{13} +6.24264 q^{15} +2.82843 q^{17} +5.99162 q^{19} +2.61313 q^{21} -0.242641 q^{23} +6.41421 q^{25} -4.77791 q^{27} +2.93015 q^{29} -4.00000 q^{31} -0.585786 q^{33} +4.77791 q^{35} +1.84776 q^{37} -3.41421 q^{39} -8.24264 q^{41} +8.60474 q^{43} +1.39942 q^{45} +11.6569 q^{47} -5.00000 q^{49} +5.22625 q^{51} +8.15640 q^{53} -1.07107 q^{55} +11.0711 q^{57} -6.62567 q^{59} -0.765367 q^{61} +0.585786 q^{63} -6.24264 q^{65} +4.01254 q^{67} -0.448342 q^{69} -0.242641 q^{71} -9.89949 q^{73} +11.8519 q^{75} -0.448342 q^{77} +6.00000 q^{79} -10.0711 q^{81} -6.62567 q^{83} +9.55582 q^{85} +5.41421 q^{87} -3.75736 q^{89} -2.61313 q^{91} -7.39104 q^{93} +20.2426 q^{95} +1.51472 q^{97} -0.131316 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^9	\( 4 q - 4 q^{9} + 8 q^{15} + 16 q^{23} + 20 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} - 16 q^{41} + 24 q^{47} - 20 q^{49} + 24 q^{55} + 16 q^{57} + 8 q^{63} - 8 q^{65} + 16 q^{71} + 24 q^{79} - 12 q^{81} + 16 q^{87} - 32 q^{89} + 64 q^{95} + 40 q^{97}+O(q^{100}) \) 4 * q - 4 * q^9 + 8 * q^15 + 16 * q^23 + 20 * q^25 - 16 * q^31 - 8 * q^33 - 8 * q^39 - 16 * q^41 + 24 * q^47 - 20 * q^49 + 24 * q^55 + 16 * q^57 + 8 * q^63 - 8 * q^65 + 16 * q^71 + 24 * q^79 - 12 * q^81 + 16 * q^87 - 32 * q^89 + 64 * q^95 + 40 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more