LMFDB - Embedded newform 1004.2.a.b.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1004,2,Mod(1,1004)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1004, 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("1004.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1004 = 2^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1004.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:	$8.01698036294$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - 3 x^{13} - 27 x^{12} + 79 x^{11} + 274 x^{10} - 747 x^{9} - 1422 x^{8} + 3287 x^{7} + \cdots - 196 $ x^14 - 3x^13 - 27x^12 + 79x^11 + 274x^10 - 747x^9 - 1422x^8 + 3287x^7 + 4161x^6 - 6861x^5 - 6676x^4 + 5599x^3 + 4627x^2 - 359*x - 196
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$0.224635$ of defining polynomial
Character	$\chi$	$=$	1004.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.224635 q^{3} -2.42875 q^{5} -2.24830 q^{7} -2.94954 q^{9} +O(q^{10})$	$q+0.224635 q^{3} -2.42875 q^{5} -2.24830 q^{7} -2.94954 q^{9} +1.64215 q^{11} +6.22381 q^{13} -0.545582 q^{15} -1.62535 q^{17} +7.83395 q^{19} -0.505047 q^{21} -6.98629 q^{23} +0.898832 q^{25} -1.33647 q^{27} +8.44758 q^{29} +7.71452 q^{31} +0.368883 q^{33} +5.46056 q^{35} +9.24659 q^{37} +1.39808 q^{39} +8.41207 q^{41} +3.97586 q^{43} +7.16370 q^{45} -0.735589 q^{47} -1.94514 q^{49} -0.365111 q^{51} -10.9903 q^{53} -3.98836 q^{55} +1.75978 q^{57} +8.63639 q^{59} -8.71829 q^{61} +6.63145 q^{63} -15.1161 q^{65} +4.17803 q^{67} -1.56936 q^{69} -4.49162 q^{71} -12.2570 q^{73} +0.201909 q^{75} -3.69204 q^{77} +3.66788 q^{79} +8.54840 q^{81} +1.17546 q^{83} +3.94758 q^{85} +1.89762 q^{87} +17.2264 q^{89} -13.9930 q^{91} +1.73295 q^{93} -19.0267 q^{95} +9.08104 q^{97} -4.84357 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 3 q^{3} - 2 q^{5} + 8 q^{7} + 21 q^{9}+O(q^{10}) $ 14 * q + 3 * q^3 - 2 * q^5 + 8 * q^7 + 21 * q^9	$ 14 q + 3 q^{3} - 2 q^{5} + 8 q^{7} + 21 q^{9} + 9 q^{11} - q^{13} + 14 q^{15} + 27 q^{19} - 3 q^{21} + 13 q^{23} + 26 q^{25} + 15 q^{27} + 25 q^{31} + 16 q^{33} + 21 q^{35} - q^{37} + 33 q^{39} + 10 q^{41} + 35 q^{43} - 4 q^{45} + 6 q^{47} + 36 q^{49} + 48 q^{51} - q^{53} + 41 q^{55} + 14 q^{57} + 30 q^{59} + 3 q^{61} + 31 q^{63} + 7 q^{65} + 22 q^{67} - 17 q^{69} + 6 q^{71} + 5 q^{73} + 4 q^{75} - 14 q^{77} + 56 q^{79} + 26 q^{81} - 28 q^{83} - 23 q^{85} + 11 q^{87} - 24 q^{89} + 38 q^{91} - 55 q^{93} - 4 q^{95} + 6 q^{97} + 12 q^{99}+O(q^{100}) $ 14 * q + 3 * q^3 - 2 * q^5 + 8 * q^7 + 21 * q^9 + 9 * q^11 - q^13 + 14 * q^15 + 27 * q^19 - 3 * q^21 + 13 * q^23 + 26 * q^25 + 15 * q^27 + 25 * q^31 + 16 * q^33 + 21 * q^35 - q^37 + 33 * q^39 + 10 * q^41 + 35 * q^43 - 4 * q^45 + 6 * q^47 + 36 * q^49 + 48 * q^51 - q^53 + 41 * q^55 + 14 * q^57 + 30 * q^59 + 3 * q^61 + 31 * q^63 + 7 * q^65 + 22 * q^67 - 17 * q^69 + 6 * q^71 + 5 * q^73 + 4 * q^75 - 14 * q^77 + 56 * q^79 + 26 * q^81 - 28 * q^83 - 23 * q^85 + 11 * q^87 - 24 * q^89 + 38 * q^91 - 55 * q^93 - 4 * q^95 + 6 * q^97 + 12 * 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
$3$	0.224635	0.129693	0.0648465	−	0.997895i	$-0.479344\pi$
$3$	0.224635	0.129693	0.0648465	+	0.997895i	$0.479344\pi$
$4$	0	0
$5$	−2.42875	−1.08617	−0.543085	−	0.839678i	$-0.682744\pi$
$5$	−2.42875	−1.08617	−0.543085	+	0.839678i	$0.682744\pi$
$6$	0	0
$7$	−2.24830	−0.849778	−0.424889	−	0.905245i	$-0.639687\pi$
$7$	−2.24830	−0.849778	−0.424889	+	0.905245i	$0.639687\pi$
$8$	0	0
$9$	−2.94954	−0.983180
$10$	0	0
$11$	1.64215	0.495125	0.247563	−	0.968872i	$-0.420370\pi$
$11$	1.64215	0.495125	0.247563	+	0.968872i	$0.420370\pi$
$12$	0	0
$13$	6.22381	1.72617	0.863087	−	0.505055i	$-0.168528\pi$
$13$	6.22381	1.72617	0.863087	+	0.505055i	$0.168528\pi$
$14$	0	0
$15$	−0.545582	−0.140869
$16$	0	0
$17$	−1.62535	−0.394206	−0.197103	−	0.980383i	$-0.563153\pi$
$17$	−1.62535	−0.394206	−0.197103	+	0.980383i	$0.563153\pi$
$18$	0	0
$19$	7.83395	1.79723	0.898615	−	0.438738i	$-0.144574\pi$
$19$	7.83395	1.79723	0.898615	+	0.438738i	$0.144574\pi$
$20$	0	0
$21$	−0.505047	−0.110210
$22$	0	0
$23$	−6.98629	−1.45674	−0.728371	−	0.685183i	$-0.759722\pi$
$23$	−6.98629	−1.45674	−0.728371	+	0.685183i	$0.759722\pi$
$24$	0	0
$25$	0.898832	0.179766
$26$	0	0
$27$	−1.33647	−0.257204
$28$	0	0
$29$	8.44758	1.56868	0.784338	−	0.620334i	$-0.213003\pi$
$29$	8.44758	1.56868	0.784338	+	0.620334i	$0.213003\pi$
$30$	0	0
$31$	7.71452	1.38557	0.692785	−	0.721145i	$-0.256384\pi$
$31$	7.71452	1.38557	0.692785	+	0.721145i	$0.256384\pi$
$32$	0	0
$33$	0.368883	0.0642143
$34$	0	0
$35$	5.46056	0.923004
$36$	0	0
$37$	9.24659	1.52013	0.760065	−	0.649847i	$-0.225167\pi$
$37$	9.24659	1.52013	0.760065	+	0.649847i	$0.225167\pi$
$38$	0	0
$39$	1.39808	0.223873
$40$	0	0
$41$	8.41207	1.31375	0.656873	−	0.754002i	$-0.271879\pi$
$41$	8.41207	1.31375	0.656873	+	0.754002i	$0.271879\pi$
$42$	0	0
$43$	3.97586	0.606313	0.303157	−	0.952941i	$-0.401959\pi$
$43$	3.97586	0.606313	0.303157	+	0.952941i	$0.401959\pi$
$44$	0	0
$45$	7.16370	1.06790
$46$	0	0
$47$	−0.735589	−0.107297	−0.0536484	−	0.998560i	$-0.517085\pi$
$47$	−0.735589	−0.107297	−0.0536484	+	0.998560i	$0.517085\pi$
$48$	0	0
$49$	−1.94514	−0.277877
$50$	0	0
$51$	−0.365111	−0.0511258
$52$	0	0
$53$	−10.9903	−1.50963	−0.754817	−	0.655936i	$-0.772274\pi$
$53$	−10.9903	−1.50963	−0.754817	+	0.655936i	$0.772274\pi$
$54$	0	0
$55$	−3.98836	−0.537791
$56$	0	0
$57$	1.75978	0.233088
$58$	0	0
$59$	8.63639	1.12436	0.562181	−	0.827014i	$-0.309962\pi$
$59$	8.63639	1.12436	0.562181	+	0.827014i	$0.309962\pi$
$60$	0	0
$61$	−8.71829	−1.11626	−0.558131	−	0.829753i	$-0.688481\pi$
$61$	−8.71829	−1.11626	−0.558131	+	0.829753i	$0.688481\pi$
$62$	0	0
$63$	6.63145	0.835485
$64$	0	0
$65$	−15.1161	−1.87492
$66$	0	0
$67$	4.17803	0.510428	0.255214	−	0.966885i	$-0.417854\pi$
$67$	4.17803	0.510428	0.255214	+	0.966885i	$0.417854\pi$
$68$	0	0
$69$	−1.56936	−0.188929
$70$	0	0
$71$	−4.49162	−0.533058	−0.266529	−	0.963827i	$-0.585877\pi$
$71$	−4.49162	−0.533058	−0.266529	+	0.963827i	$0.585877\pi$
$72$	0	0
$73$	−12.2570	−1.43458	−0.717288	−	0.696777i	$-0.754617\pi$
$73$	−12.2570	−1.43458	−0.717288	+	0.696777i	$0.754617\pi$
$74$	0	0
$75$	0.201909	0.0233144
$76$	0	0
$77$	−3.69204	−0.420747
$78$	0	0
$79$	3.66788	0.412669	0.206334	−	0.978482i	$-0.433847\pi$
$79$	3.66788	0.412669	0.206334	+	0.978482i	$0.433847\pi$
$80$	0	0
$81$	8.54840	0.949822
$82$	0	0
$83$	1.17546	0.129023	0.0645116	−	0.997917i	$-0.479451\pi$
$83$	1.17546	0.129023	0.0645116	+	0.997917i	$0.479451\pi$
$84$	0	0
$85$	3.94758	0.428175
$86$	0	0
$87$	1.89762	0.203446
$88$	0	0
$89$	17.2264	1.82599	0.912996	−	0.407968i	$-0.133762\pi$
$89$	17.2264	1.82599	0.912996	+	0.407968i	$0.133762\pi$
$90$	0	0
$91$	−13.9930	−1.46687
$92$	0	0
$93$	1.73295	0.179699
$94$	0	0
$95$	−19.0267	−1.95210
$96$	0	0
$97$	9.08104	0.922040	0.461020	−	0.887390i	$-0.347484\pi$
$97$	9.08104	0.922040	0.461020	+	0.887390i	$0.347484\pi$
$98$	0	0
$99$	−4.84357	−0.486797
$100$	0	0
$101$	1.31035	0.130385	0.0651924	−	0.997873i	$-0.479234\pi$
$101$	1.31035	0.130385	0.0651924	+	0.997873i	$0.479234\pi$
$102$	0	0
$103$	−5.17455	−0.509863	−0.254932	−	0.966959i	$-0.582053\pi$
$103$	−5.17455	−0.509863	−0.254932	+	0.966959i	$0.582053\pi$
$104$	0	0
$105$	1.22663	0.119707
$106$	0	0
$107$	15.5548	1.50374	0.751869	−	0.659313i	$-0.229153\pi$
$107$	15.5548	1.50374	0.751869	+	0.659313i	$0.229153\pi$
$108$	0	0
$109$	−7.94967	−0.761441	−0.380720	−	0.924690i	$-0.624324\pi$
$109$	−7.94967	−0.761441	−0.380720	+	0.924690i	$0.624324\pi$
$110$	0	0
$111$	2.07710	0.197150
$112$	0	0
$113$	−4.80047	−0.451590	−0.225795	−	0.974175i	$-0.572498\pi$
$113$	−4.80047	−0.451590	−0.225795	+	0.974175i	$0.572498\pi$
$114$	0	0
$115$	16.9680	1.58227
$116$	0	0
$117$	−18.3574	−1.69714
$118$	0	0
$119$	3.65429	0.334988
$120$	0	0
$121$	−8.30336	−0.754851
$122$	0	0
$123$	1.88964	0.170383
$124$	0	0
$125$	9.96072	0.890914
$126$	0	0
$127$	17.1925	1.52559	0.762795	−	0.646641i	$-0.223827\pi$
$127$	17.1925	1.52559	0.762795	+	0.646641i	$0.223827\pi$
$128$	0	0
$129$	0.893117	0.0786345
$130$	0	0
$131$	−4.30291	−0.375947	−0.187974	−	0.982174i	$-0.560192\pi$
$131$	−4.30291	−0.375947	−0.187974	+	0.982174i	$0.560192\pi$
$132$	0	0
$133$	−17.6131	−1.52725
$134$	0	0
$135$	3.24596	0.279368
$136$	0	0
$137$	0.00645063	0.000551114 0	0.000275557	−	1.00000i	$-0.499912\pi$
$137$	0.00645063	0.000551114 0	0.000275557		1.00000i	$0.499912\pi$
$138$	0	0
$139$	−0.181781	−0.0154185	−0.00770923	−	0.999970i	$-0.502454\pi$
$139$	−0.181781	−0.0154185	−0.00770923	+	0.999970i	$0.502454\pi$
$140$	0	0
$141$	−0.165239	−0.0139156
$142$	0	0
$143$	10.2204	0.854673
$144$	0	0
$145$	−20.5171	−1.70385
$146$	0	0
$147$	−0.436946	−0.0360387
$148$	0	0
$149$	−23.6239	−1.93535	−0.967673	−	0.252209i	$-0.918843\pi$
$149$	−23.6239	−1.93535	−0.967673	+	0.252209i	$0.918843\pi$
$150$	0	0
$151$	2.45736	0.199977	0.0999886	−	0.994989i	$-0.468119\pi$
$151$	2.45736	0.199977	0.0999886	+	0.994989i	$0.468119\pi$
$152$	0	0
$153$	4.79405	0.387576
$154$	0	0
$155$	−18.7367	−1.50496
$156$	0	0
$157$	12.7494	1.01751	0.508755	−	0.860912i	$-0.330106\pi$
$157$	12.7494	1.01751	0.508755	+	0.860912i	$0.330106\pi$
$158$	0	0
$159$	−2.46880	−0.195789
$160$	0	0
$161$	15.7073	1.23791
$162$	0	0
$163$	18.8119	1.47346	0.736730	−	0.676187i	$-0.236369\pi$
$163$	18.8119	1.47346	0.736730	+	0.676187i	$0.236369\pi$
$164$	0	0
$165$	−0.895925	−0.0697476
$166$	0	0
$167$	5.91688	0.457862	0.228931	−	0.973443i	$-0.426477\pi$
$167$	5.91688	0.457862	0.228931	+	0.973443i	$0.426477\pi$
$168$	0	0
$169$	25.7358	1.97968
$170$	0	0
$171$	−23.1065	−1.76700
$172$	0	0
$173$	−13.0221	−0.990053	−0.495027	−	0.868878i	$-0.664842\pi$
$173$	−13.0221	−0.990053	−0.495027	+	0.868878i	$0.664842\pi$
$174$	0	0
$175$	−2.02085	−0.152762
$176$	0	0
$177$	1.94003	0.145822
$178$	0	0
$179$	11.3014	0.844710	0.422355	−	0.906431i	$-0.361204\pi$
$179$	11.3014	0.844710	0.422355	+	0.906431i	$0.361204\pi$
$180$	0	0
$181$	−10.6066	−0.788379	−0.394190	−	0.919029i	$-0.628975\pi$
$181$	−10.6066	−0.788379	−0.394190	+	0.919029i	$0.628975\pi$
$182$	0	0
$183$	−1.95843	−0.144771
$184$	0	0
$185$	−22.4577	−1.65112
$186$	0	0
$187$	−2.66907	−0.195182
$188$	0	0
$189$	3.00479	0.218567
$190$	0	0
$191$	−10.1524	−0.734599	−0.367300	−	0.930103i	$-0.619718\pi$
$191$	−10.1524	−0.734599	−0.367300	+	0.930103i	$0.619718\pi$
$192$	0	0
$193$	−9.19678	−0.661999	−0.330999	−	0.943631i	$-0.607386\pi$
$193$	−9.19678	−0.661999	−0.330999	+	0.943631i	$0.607386\pi$
$194$	0	0
$195$	−3.39560	−0.243164
$196$	0	0
$197$	−20.1222	−1.43365	−0.716825	−	0.697253i	$-0.754405\pi$
$197$	−20.1222	−1.43365	−0.716825	+	0.697253i	$0.754405\pi$
$198$	0	0
$199$	−12.2239	−0.866531	−0.433266	−	0.901266i	$-0.642639\pi$
$199$	−12.2239	−0.866531	−0.433266	+	0.901266i	$0.642639\pi$
$200$	0	0
$201$	0.938532	0.0661989
$202$	0	0
$203$	−18.9927	−1.33303
$204$	0	0
$205$	−20.4308	−1.42695
$206$	0	0
$207$	20.6063	1.43224
$208$	0	0
$209$	12.8645	0.889854
$210$	0	0
$211$	24.4289	1.68175	0.840877	−	0.541227i	$-0.182040\pi$
$211$	24.4289	1.68175	0.840877	+	0.541227i	$0.182040\pi$
$212$	0	0
$213$	−1.00897	−0.0691338
$214$	0	0
$215$	−9.65638	−0.658560
$216$	0	0
$217$	−17.3446	−1.17743
$218$	0	0
$219$	−2.75335	−0.186054
$220$	0	0
$221$	−10.1159	−0.680469
$222$	0	0
$223$	19.9845	1.33826	0.669130	−	0.743145i	$-0.266667\pi$
$223$	19.9845	1.33826	0.669130	+	0.743145i	$0.266667\pi$
$224$	0	0
$225$	−2.65114	−0.176743
$226$	0	0
$227$	3.68676	0.244699	0.122349	−	0.992487i	$-0.460957\pi$
$227$	3.68676	0.244699	0.122349	+	0.992487i	$0.460957\pi$
$228$	0	0
$229$	1.96570	0.129897	0.0649485	−	0.997889i	$-0.479312\pi$
$229$	1.96570	0.129897	0.0649485	+	0.997889i	$0.479312\pi$
$230$	0	0
$231$	−0.829360	−0.0545679
$232$	0	0
$233$	−7.53560	−0.493674	−0.246837	−	0.969057i	$-0.579391\pi$
$233$	−7.53560	−0.493674	−0.246837	+	0.969057i	$0.579391\pi$
$234$	0	0
$235$	1.78656	0.116543
$236$	0	0
$237$	0.823933	0.0535202
$238$	0	0
$239$	−7.87974	−0.509698	−0.254849	−	0.966981i	$-0.582026\pi$
$239$	−7.87974	−0.509698	−0.254849	+	0.966981i	$0.582026\pi$
$240$	0	0
$241$	7.20915	0.464382	0.232191	−	0.972670i	$-0.425411\pi$
$241$	7.20915	0.464382	0.232191	+	0.972670i	$0.425411\pi$
$242$	0	0
$243$	5.92969	0.380390
$244$	0	0
$245$	4.72426	0.301822
$246$	0	0
$247$	48.7570	3.10233
$248$	0	0
$249$	0.264049	0.0167334
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−11.4725	−0.721270
$254$	0	0
$255$	0.886764	0.0555313
$256$	0	0
$257$	25.9988	1.62176	0.810879	−	0.585213i	$-0.198989\pi$
$257$	25.9988	1.62176	0.810879	+	0.585213i	$0.198989\pi$
$258$	0	0
$259$	−20.7891	−1.29177
$260$	0	0
$261$	−24.9165	−1.54229
$262$	0	0
$263$	−31.1491	−1.92074	−0.960369	−	0.278731i	$-0.910086\pi$
$263$	−31.1491	−1.92074	−0.960369	+	0.278731i	$0.910086\pi$
$264$	0	0
$265$	26.6927	1.63972
$266$	0	0
$267$	3.86964	0.236818
$268$	0	0
$269$	3.41438	0.208178	0.104089	−	0.994568i	$-0.466807\pi$
$269$	3.41438	0.208178	0.104089	+	0.994568i	$0.466807\pi$
$270$	0	0
$271$	22.8379	1.38730	0.693652	−	0.720310i	$-0.256001\pi$
$271$	22.8379	1.38730	0.693652	+	0.720310i	$0.256001\pi$
$272$	0	0
$273$	−3.14331	−0.190242
$274$	0	0
$275$	1.47601	0.0890069
$276$	0	0
$277$	−4.04269	−0.242902	−0.121451	−	0.992597i	$-0.538755\pi$
$277$	−4.04269	−0.242902	−0.121451	+	0.992597i	$0.538755\pi$
$278$	0	0
$279$	−22.7543	−1.36226
$280$	0	0
$281$	−2.35718	−0.140618	−0.0703088	−	0.997525i	$-0.522398\pi$
$281$	−2.35718	−0.140618	−0.0703088	+	0.997525i	$0.522398\pi$
$282$	0	0
$283$	20.1634	1.19859	0.599296	−	0.800528i	$-0.295447\pi$
$283$	20.1634	1.19859	0.599296	+	0.800528i	$0.295447\pi$
$284$	0	0
$285$	−4.27406	−0.253173
$286$	0	0
$287$	−18.9129	−1.11639
$288$	0	0
$289$	−14.3582	−0.844601
$290$	0	0
$291$	2.03992	0.119582
$292$	0	0
$293$	−12.9511	−0.756610	−0.378305	−	0.925681i	$-0.623493\pi$
$293$	−12.9511	−0.756610	−0.378305	+	0.925681i	$0.623493\pi$
$294$	0	0
$295$	−20.9757	−1.22125
$296$	0	0
$297$	−2.19468	−0.127348
$298$	0	0
$299$	−43.4814	−2.51459
$300$	0	0
$301$	−8.93894	−0.515232
$302$	0	0
$303$	0.294350	0.0169100
$304$	0	0
$305$	21.1746	1.21245
$306$	0	0
$307$	6.71716	0.383369	0.191684	−	0.981457i	$-0.438605\pi$
$307$	6.71716	0.383369	0.191684	+	0.981457i	$0.438605\pi$
$308$	0	0
$309$	−1.16238	−0.0661257
$310$	0	0
$311$	15.8087	0.896429	0.448214	−	0.893926i	$-0.352060\pi$
$311$	15.8087	0.896429	0.448214	+	0.893926i	$0.352060\pi$
$312$	0	0
$313$	−27.6720	−1.56411	−0.782057	−	0.623207i	$-0.785830\pi$
$313$	−27.6720	−1.56411	−0.782057	+	0.623207i	$0.785830\pi$
$314$	0	0
$315$	−16.1061	−0.907479
$316$	0	0
$317$	−2.14088	−0.120244	−0.0601218	−	0.998191i	$-0.519149\pi$
$317$	−2.14088	−0.120244	−0.0601218	+	0.998191i	$0.519149\pi$
$318$	0	0
$319$	13.8721	0.776691
$320$	0	0
$321$	3.49414	0.195024
$322$	0	0
$323$	−12.7329	−0.708479
$324$	0	0
$325$	5.59416	0.310308
$326$	0	0
$327$	−1.78577	−0.0987535
$328$	0	0
$329$	1.65383	0.0911784
$330$	0	0
$331$	−13.0310	−0.716246	−0.358123	−	0.933674i	$-0.616583\pi$
$331$	−13.0310	−0.716246	−0.358123	+	0.933674i	$0.616583\pi$
$332$	0	0
$333$	−27.2732	−1.49456
$334$	0	0
$335$	−10.1474	−0.554412
$336$	0	0
$337$	−2.70826	−0.147528	−0.0737642	−	0.997276i	$-0.523501\pi$
$337$	−2.70826	−0.147528	−0.0737642	+	0.997276i	$0.523501\pi$
$338$	0	0
$339$	−1.07835	−0.0585680
$340$	0	0
$341$	12.6684	0.686030
$342$	0	0
$343$	20.1114	1.08591
$344$	0	0
$345$	3.81159	0.205209
$346$	0	0
$347$	−26.4326	−1.41898	−0.709489	−	0.704716i	$-0.751074\pi$
$347$	−26.4326	−1.41898	−0.709489	+	0.704716i	$0.751074\pi$
$348$	0	0
$349$	−14.6941	−0.786559	−0.393280	−	0.919419i	$-0.628660\pi$
$349$	−14.6941	−0.786559	−0.393280	+	0.919419i	$0.628660\pi$
$350$	0	0
$351$	−8.31796	−0.443980
$352$	0	0
$353$	−12.0164	−0.639566	−0.319783	−	0.947491i	$-0.603610\pi$
$353$	−12.0164	−0.639566	−0.319783	+	0.947491i	$0.603610\pi$
$354$	0	0
$355$	10.9090	0.578992
$356$	0	0
$357$	0.820879	0.0434455
$358$	0	0
$359$	32.2776	1.70354	0.851772	−	0.523912i	$-0.175528\pi$
$359$	32.2776	1.70354	0.851772	+	0.523912i	$0.175528\pi$
$360$	0	0
$361$	42.3707	2.23004
$362$	0	0
$363$	−1.86522	−0.0978988
$364$	0	0
$365$	29.7693	1.55819
$366$	0	0
$367$	−27.7888	−1.45056	−0.725282	−	0.688452i	$-0.758291\pi$
$367$	−27.7888	−1.45056	−0.725282	+	0.688452i	$0.758291\pi$
$368$	0	0
$369$	−24.8117	−1.29165
$370$	0	0
$371$	24.7095	1.28285
$372$	0	0
$373$	23.2716	1.20496	0.602478	−	0.798135i	$-0.294180\pi$
$373$	23.2716	1.20496	0.602478	+	0.798135i	$0.294180\pi$
$374$	0	0
$375$	2.23752	0.115545
$376$	0	0
$377$	52.5761	2.70781
$378$	0	0
$379$	21.4700	1.10284	0.551420	−	0.834228i	$-0.314086\pi$
$379$	21.4700	1.10284	0.551420	+	0.834228i	$0.314086\pi$
$380$	0	0
$381$	3.86204	0.197858
$382$	0	0
$383$	−27.7055	−1.41569	−0.707843	−	0.706370i	$-0.750331\pi$
$383$	−27.7055	−1.41569	−0.707843	+	0.706370i	$0.750331\pi$
$384$	0	0
$385$	8.96704	0.457003
$386$	0	0
$387$	−11.7270	−0.596115
$388$	0	0
$389$	22.2832	1.12980	0.564901	−	0.825158i	$-0.308914\pi$
$389$	22.2832	1.12980	0.564901	+	0.825158i	$0.308914\pi$
$390$	0	0
$391$	11.3552	0.574257
$392$	0	0
$393$	−0.966584	−0.0487577
$394$	0	0
$395$	−8.90837	−0.448229
$396$	0	0
$397$	−2.54049	−0.127504	−0.0637518	−	0.997966i	$-0.520307\pi$
$397$	−2.54049	−0.127504	−0.0637518	+	0.997966i	$0.520307\pi$
$398$	0	0
$399$	−3.95651	−0.198073
$400$	0	0
$401$	1.42313	0.0710676	0.0355338	−	0.999368i	$-0.488687\pi$
$401$	1.42313	0.0710676	0.0355338	+	0.999368i	$0.488687\pi$
$402$	0	0
$403$	48.0137	2.39173
$404$	0	0
$405$	−20.7619	−1.03167
$406$	0	0
$407$	15.1842	0.752655
$408$	0	0
$409$	8.83345	0.436786	0.218393	−	0.975861i	$-0.429919\pi$
$409$	8.83345	0.436786	0.218393	+	0.975861i	$0.429919\pi$
$410$	0	0
$411$	0.00144903	7.14756e−5 0
$412$	0	0
$413$	−19.4172	−0.955459
$414$	0	0
$415$	−2.85490	−0.140141
$416$	0	0
$417$	−0.0408343	−0.00199967
$418$	0	0
$419$	34.0729	1.66457	0.832286	−	0.554346i	$-0.187032\pi$
$419$	34.0729	1.66457	0.832286	+	0.554346i	$0.187032\pi$
$420$	0	0
$421$	29.2308	1.42462	0.712311	−	0.701864i	$-0.247649\pi$
$421$	29.2308	1.42462	0.712311	+	0.701864i	$0.247649\pi$
$422$	0	0
$423$	2.16965	0.105492
$424$	0	0
$425$	−1.46092	−0.0708650
$426$	0	0
$427$	19.6013	0.948576
$428$	0	0
$429$	2.29586	0.110845
$430$	0	0
$431$	−4.82572	−0.232447	−0.116223	−	0.993223i	$-0.537079\pi$
$431$	−4.82572	−0.232447	−0.116223	+	0.993223i	$0.537079\pi$
$432$	0	0
$433$	−23.7354	−1.14065	−0.570326	−	0.821418i	$-0.693183\pi$
$433$	−23.7354	−1.14065	−0.570326	+	0.821418i	$0.693183\pi$
$434$	0	0
$435$	−4.60884	−0.220977
$436$	0	0
$437$	−54.7302	−2.61810
$438$	0	0
$439$	−30.8566	−1.47271	−0.736354	−	0.676597i	$-0.763454\pi$
$439$	−30.8566	−1.47271	−0.736354	+	0.676597i	$0.763454\pi$
$440$	0	0
$441$	5.73727	0.273203
$442$	0	0
$443$	37.8298	1.79735	0.898674	−	0.438616i	$-0.144531\pi$
$443$	37.8298	1.79735	0.898674	+	0.438616i	$0.144531\pi$
$444$	0	0
$445$	−41.8386	−1.98334
$446$	0	0
$447$	−5.30675	−0.251001
$448$	0	0
$449$	38.1436	1.80011	0.900054	−	0.435779i	$-0.143527\pi$
$449$	38.1436	1.80011	0.900054	+	0.435779i	$0.143527\pi$
$450$	0	0
$451$	13.8138	0.650469
$452$	0	0
$453$	0.552009	0.0259356
$454$	0	0
$455$	33.9855	1.59327
$456$	0	0
$457$	−20.1481	−0.942489	−0.471244	−	0.882003i	$-0.656195\pi$
$457$	−20.1481	−0.942489	−0.471244	+	0.882003i	$0.656195\pi$
$458$	0	0
$459$	2.17224	0.101392
$460$	0	0
$461$	−0.500312	−0.0233019	−0.0116509	−	0.999932i	$-0.503709\pi$
$461$	−0.500312	−0.0233019	−0.0116509	+	0.999932i	$0.503709\pi$
$462$	0	0
$463$	4.86536	0.226113	0.113056	−	0.993589i	$-0.463936\pi$
$463$	4.86536	0.226113	0.113056	+	0.993589i	$0.463936\pi$
$464$	0	0
$465$	−4.20890	−0.195183
$466$	0	0
$467$	−3.56899	−0.165153	−0.0825765	−	0.996585i	$-0.526315\pi$
$467$	−3.56899	−0.165153	−0.0825765	+	0.996585i	$0.526315\pi$
$468$	0	0
$469$	−9.39348	−0.433751
$470$	0	0
$471$	2.86395	0.131964
$472$	0	0
$473$	6.52894	0.300201
$474$	0	0
$475$	7.04140	0.323082
$476$	0	0
$477$	32.4163	1.48424
$478$	0	0
$479$	0.936786	0.0428028	0.0214014	−	0.999771i	$-0.493187\pi$
$479$	0.936786	0.0428028	0.0214014	+	0.999771i	$0.493187\pi$
$480$	0	0
$481$	57.5490	2.62401
$482$	0	0
$483$	3.52840	0.160548
$484$	0	0
$485$	−22.0556	−1.00149
$486$	0	0
$487$	−19.2487	−0.872242	−0.436121	−	0.899888i	$-0.643648\pi$
$487$	−19.2487	−0.872242	−0.436121	+	0.899888i	$0.643648\pi$
$488$	0	0
$489$	4.22580	0.191097
$490$	0	0
$491$	−15.5938	−0.703739	−0.351870	−	0.936049i	$-0.614454\pi$
$491$	−15.5938	−0.703739	−0.351870	+	0.936049i	$0.614454\pi$
$492$	0	0
$493$	−13.7303	−0.618382
$494$	0	0
$495$	11.7638	0.528745
$496$	0	0
$497$	10.0985	0.452981
$498$	0	0
$499$	−36.3922	−1.62914	−0.814570	−	0.580066i	$-0.803027\pi$
$499$	−36.3922	−1.62914	−0.814570	+	0.580066i	$0.803027\pi$
$500$	0	0
$501$	1.32914	0.0593814
$502$	0	0
$503$	9.65241	0.430380	0.215190	−	0.976572i	$-0.430963\pi$
$503$	9.65241	0.430380	0.215190	+	0.976572i	$0.430963\pi$
$504$	0	0
$505$	−3.18252	−0.141620
$506$	0	0
$507$	5.78116	0.256750
$508$	0	0
$509$	16.1978	0.717955	0.358977	−	0.933346i	$-0.383126\pi$
$509$	16.1978	0.717955	0.358977	+	0.933346i	$0.383126\pi$
$510$	0	0
$511$	27.5575	1.21907
$512$	0	0
$513$	−10.4699	−0.462256
$514$	0	0
$515$	12.5677	0.553799
$516$	0	0
$517$	−1.20794	−0.0531253
$518$	0	0
$519$	−2.92522	−0.128403
$520$	0	0
$521$	−14.2198	−0.622981	−0.311491	−	0.950249i	$-0.600828\pi$
$521$	−14.2198	−0.622981	−0.311491	+	0.950249i	$0.600828\pi$
$522$	0	0
$523$	3.66535	0.160275	0.0801373	−	0.996784i	$-0.474464\pi$
$523$	3.66535	0.160275	0.0801373	+	0.996784i	$0.474464\pi$
$524$	0	0
$525$	−0.453952	−0.0198121
$526$	0	0
$527$	−12.5388	−0.546200
$528$	0	0
$529$	25.8083	1.12210
$530$	0	0
$531$	−25.4734	−1.10545
$532$	0	0
$533$	52.3552	2.26775
$534$	0	0
$535$	−37.7787	−1.63332
$536$	0	0
$537$	2.53870	0.109553
$538$	0	0
$539$	−3.19420	−0.137584
$540$	0	0
$541$	−17.4901	−0.751959	−0.375980	−	0.926628i	$-0.622694\pi$
$541$	−17.4901	−0.751959	−0.375980	+	0.926628i	$0.622694\pi$
$542$	0	0
$543$	−2.38260	−0.102247
$544$	0	0
$545$	19.3078	0.827054
$546$	0	0
$547$	26.1597	1.11851	0.559254	−	0.828996i	$-0.311088\pi$
$547$	26.1597	1.11851	0.559254	+	0.828996i	$0.311088\pi$
$548$	0	0
$549$	25.7149	1.09749
$550$	0	0
$551$	66.1778	2.81927
$552$	0	0
$553$	−8.24650	−0.350677
$554$	0	0
$555$	−5.04477	−0.214139
$556$	0	0
$557$	−19.8690	−0.841879	−0.420939	−	0.907089i	$-0.638299\pi$
$557$	−19.8690	−0.841879	−0.420939	+	0.907089i	$0.638299\pi$
$558$	0	0
$559$	24.7450	1.04660
$560$	0	0
$561$	−0.599565	−0.0253137
$562$	0	0
$563$	−26.1613	−1.10257	−0.551284	−	0.834318i	$-0.685862\pi$
$563$	−26.1613	−1.10257	−0.551284	+	0.834318i	$0.685862\pi$
$564$	0	0
$565$	11.6591	0.490504
$566$	0	0
$567$	−19.2194	−0.807138
$568$	0	0
$569$	−20.8786	−0.875277	−0.437638	−	0.899151i	$-0.644185\pi$
$569$	−20.8786	−0.875277	−0.437638	+	0.899151i	$0.644185\pi$
$570$	0	0
$571$	43.1297	1.80492	0.902461	−	0.430772i	$-0.141759\pi$
$571$	43.1297	1.80492	0.902461	+	0.430772i	$0.141759\pi$
$572$	0	0
$573$	−2.28057	−0.0952723
$574$	0	0
$575$	−6.27950	−0.261873
$576$	0	0
$577$	−17.6062	−0.732958	−0.366479	−	0.930426i	$-0.619437\pi$
$577$	−17.6062	−0.732958	−0.366479	+	0.930426i	$0.619437\pi$
$578$	0	0
$579$	−2.06592	−0.0858566
$580$	0	0
$581$	−2.64278	−0.109641
$582$	0	0
$583$	−18.0477	−0.747458
$584$	0	0
$585$	44.5855	1.84338
$586$	0	0
$587$	−45.5328	−1.87934	−0.939670	−	0.342083i	$-0.888868\pi$
$587$	−45.5328	−1.87934	−0.939670	+	0.342083i	$0.888868\pi$
$588$	0	0
$589$	60.4351	2.49019
$590$	0	0
$591$	−4.52015	−0.185934
$592$	0	0
$593$	3.18374	0.130741	0.0653703	−	0.997861i	$-0.479177\pi$
$593$	3.18374	0.130741	0.0653703	+	0.997861i	$0.479177\pi$
$594$	0	0
$595$	−8.87535	−0.363854
$596$	0	0
$597$	−2.74592	−0.112383
$598$	0	0
$599$	9.93843	0.406073	0.203037	−	0.979171i	$-0.434919\pi$
$599$	9.93843	0.406073	0.203037	+	0.979171i	$0.434919\pi$
$600$	0	0
$601$	11.1826	0.456147	0.228074	−	0.973644i	$-0.426757\pi$
$601$	11.1826	0.456147	0.228074	+	0.973644i	$0.426757\pi$
$602$	0	0
$603$	−12.3233	−0.501843
$604$	0	0
$605$	20.1668	0.819897
$606$	0	0
$607$	−18.8713	−0.765961	−0.382981	−	0.923756i	$-0.625102\pi$
$607$	−18.8713	−0.765961	−0.382981	+	0.923756i	$0.625102\pi$
$608$	0	0
$609$	−4.26642	−0.172884
$610$	0	0
$611$	−4.57817	−0.185213
$612$	0	0
$613$	42.1894	1.70401	0.852007	−	0.523531i	$-0.175386\pi$
$613$	42.1894	1.70401	0.852007	+	0.523531i	$0.175386\pi$
$614$	0	0
$615$	−4.58948	−0.185065
$616$	0	0
$617$	16.7840	0.675696	0.337848	−	0.941201i	$-0.390301\pi$
$617$	16.7840	0.675696	0.337848	+	0.941201i	$0.390301\pi$
$618$	0	0
$619$	−31.7125	−1.27463	−0.637317	−	0.770602i	$-0.719956\pi$
$619$	−31.7125	−1.27463	−0.637317	+	0.770602i	$0.719956\pi$
$620$	0	0
$621$	9.33699	0.374681
$622$	0	0
$623$	−38.7301	−1.55169
$624$	0	0
$625$	−28.6863	−1.14745
$626$	0	0
$627$	2.88981	0.115408
$628$	0	0
$629$	−15.0290	−0.599245
$630$	0	0
$631$	−6.58208	−0.262028	−0.131014	−	0.991381i	$-0.541823\pi$
$631$	−6.58208	−0.262028	−0.131014	+	0.991381i	$0.541823\pi$
$632$	0	0
$633$	5.48757	0.218112
$634$	0	0
$635$	−41.7564	−1.65705
$636$	0	0
$637$	−12.1062	−0.479665
$638$	0	0
$639$	13.2482	0.524092
$640$	0	0
$641$	−40.0105	−1.58032	−0.790161	−	0.612900i	$-0.790003\pi$
$641$	−40.0105	−1.58032	−0.790161	+	0.612900i	$0.790003\pi$
$642$	0	0
$643$	−24.1750	−0.953370	−0.476685	−	0.879074i	$-0.658162\pi$
$643$	−24.1750	−0.953370	−0.476685	+	0.879074i	$0.658162\pi$
$644$	0	0
$645$	−2.16916	−0.0854105
$646$	0	0
$647$	−12.5910	−0.495004	−0.247502	−	0.968887i	$-0.579610\pi$
$647$	−12.5910	−0.495004	−0.247502	+	0.968887i	$0.579610\pi$
$648$	0	0
$649$	14.1822	0.556701
$650$	0	0
$651$	−3.89619	−0.152704
$652$	0	0
$653$	34.6001	1.35401	0.677003	−	0.735981i	$-0.263279\pi$
$653$	34.6001	1.35401	0.677003	+	0.735981i	$0.263279\pi$
$654$	0	0
$655$	10.4507	0.408343
$656$	0	0
$657$	36.1526	1.41045
$658$	0	0
$659$	31.6427	1.23263	0.616313	−	0.787502i	$-0.288626\pi$
$659$	31.6427	1.23263	0.616313	+	0.787502i	$0.288626\pi$
$660$	0	0
$661$	14.5221	0.564843	0.282422	−	0.959290i	$-0.408862\pi$
$661$	14.5221	0.564843	0.282422	+	0.959290i	$0.408862\pi$
$662$	0	0
$663$	−2.27238	−0.0882520
$664$	0	0
$665$	42.7778	1.65885
$666$	0	0
$667$	−59.0172	−2.28516
$668$	0	0
$669$	4.48921	0.173563
$670$	0	0
$671$	−14.3167	−0.552690
$672$	0	0
$673$	−11.7338	−0.452304	−0.226152	−	0.974092i	$-0.572615\pi$
$673$	−11.7338	−0.452304	−0.226152	+	0.974092i	$0.572615\pi$
$674$	0	0
$675$	−1.20126	−0.0462367
$676$	0	0
$677$	2.50545	0.0962923	0.0481461	−	0.998840i	$-0.484669\pi$
$677$	2.50545	0.0962923	0.0481461	+	0.998840i	$0.484669\pi$
$678$	0	0
$679$	−20.4169	−0.783529
$680$	0	0
$681$	0.828175	0.0317357
$682$	0	0
$683$	14.4053	0.551203	0.275602	−	0.961272i	$-0.411123\pi$
$683$	14.4053	0.551203	0.275602	+	0.961272i	$0.411123\pi$
$684$	0	0
$685$	−0.0156670	−0.000598604 0
$686$	0	0
$687$	0.441564	0.0168467
$688$	0	0
$689$	−68.4015	−2.60589
$690$	0	0
$691$	11.1265	0.423271	0.211635	−	0.977349i	$-0.432121\pi$
$691$	11.1265	0.423271	0.211635	+	0.977349i	$0.432121\pi$
$692$	0	0
$693$	10.8898	0.413670
$694$	0	0
$695$	0.441501	0.0167471
$696$	0	0
$697$	−13.6726	−0.517887
$698$	0	0
$699$	−1.69276	−0.0640260
$700$	0	0
$701$	14.5012	0.547704	0.273852	−	0.961772i	$-0.411702\pi$
$701$	14.5012	0.547704	0.273852	+	0.961772i	$0.411702\pi$
$702$	0	0
$703$	72.4373	2.73202
$704$	0	0
$705$	0.401324	0.0151147
$706$	0	0
$707$	−2.94606	−0.110798
$708$	0	0
$709$	19.8600	0.745859	0.372929	−	0.927860i	$-0.378353\pi$
$709$	19.8600	0.745859	0.372929	+	0.927860i	$0.378353\pi$
$710$	0	0
$711$	−10.8186	−0.405728
$712$	0	0
$713$	−53.8959	−2.01842
$714$	0	0
$715$	−24.8228	−0.928320
$716$	0	0
$717$	−1.77006	−0.0661042
$718$	0	0
$719$	6.68379	0.249263	0.124632	−	0.992203i	$-0.460225\pi$
$719$	6.68379	0.249263	0.124632	+	0.992203i	$0.460225\pi$
$720$	0	0
$721$	11.6339	0.433271
$722$	0	0
$723$	1.61943	0.0602271
$724$	0	0
$725$	7.59295	0.281995
$726$	0	0
$727$	−13.6007	−0.504421	−0.252210	−	0.967672i	$-0.581158\pi$
$727$	−13.6007	−0.504421	−0.252210	+	0.967672i	$0.581158\pi$
$728$	0	0
$729$	−24.3132	−0.900488
$730$	0	0
$731$	−6.46218	−0.239012
$732$	0	0
$733$	0.562255	0.0207674	0.0103837	−	0.999946i	$-0.496695\pi$
$733$	0.562255	0.0207674	0.0103837	+	0.999946i	$0.496695\pi$
$734$	0	0
$735$	1.06123	0.0391442
$736$	0	0
$737$	6.86094	0.252726
$738$	0	0
$739$	30.9406	1.13817	0.569085	−	0.822279i	$-0.307298\pi$
$739$	30.9406	1.13817	0.569085	+	0.822279i	$0.307298\pi$
$740$	0	0
$741$	10.9525	0.402351
$742$	0	0
$743$	10.2863	0.377369	0.188685	−	0.982038i	$-0.439578\pi$
$743$	10.2863	0.377369	0.188685	+	0.982038i	$0.439578\pi$
$744$	0	0
$745$	57.3766	2.10212
$746$	0	0
$747$	−3.46706	−0.126853
$748$	0	0
$749$	−34.9718	−1.27784
$750$	0	0
$751$	−23.1131	−0.843410	−0.421705	−	0.906733i	$-0.638568\pi$
$751$	−23.1131	−0.843410	−0.421705	+	0.906733i	$0.638568\pi$
$752$	0	0
$753$	−0.224635	−0.00818615
$754$	0	0
$755$	−5.96832	−0.217209
$756$	0	0
$757$	40.0512	1.45569	0.727843	−	0.685743i	$-0.240523\pi$
$757$	40.0512	1.45569	0.727843	+	0.685743i	$0.240523\pi$
$758$	0	0
$759$	−2.57712	−0.0935436
$760$	0	0
$761$	31.9526	1.15828	0.579141	−	0.815228i	$-0.303388\pi$
$761$	31.9526	1.15828	0.579141	+	0.815228i	$0.303388\pi$
$762$	0	0
$763$	17.8733	0.647055
$764$	0	0
$765$	−11.6435	−0.420973
$766$	0	0
$767$	53.7513	1.94085
$768$	0	0
$769$	−11.2993	−0.407463	−0.203732	−	0.979027i	$-0.565307\pi$
$769$	−11.2993	−0.407463	−0.203732	+	0.979027i	$0.565307\pi$
$770$	0	0
$771$	5.84023	0.210331
$772$	0	0
$773$	4.44963	0.160042	0.0800210	−	0.996793i	$-0.474501\pi$
$773$	4.44963	0.160042	0.0800210	+	0.996793i	$0.474501\pi$
$774$	0	0
$775$	6.93406	0.249079
$776$	0	0
$777$	−4.66996	−0.167534
$778$	0	0
$779$	65.8997	2.36110
$780$	0	0
$781$	−7.37590	−0.263930
$782$	0	0
$783$	−11.2900	−0.403470
$784$	0	0
$785$	−30.9650	−1.10519
$786$	0	0
$787$	−12.2070	−0.435133	−0.217566	−	0.976046i	$-0.569812\pi$
$787$	−12.2070	−0.435133	−0.217566	+	0.976046i	$0.569812\pi$
$788$	0	0
$789$	−6.99718	−0.249106
$790$	0	0
$791$	10.7929	0.383751
$792$	0	0
$793$	−54.2610	−1.92686
$794$	0	0
$795$	5.99611	0.212660
$796$	0	0
$797$	−27.5856	−0.977131	−0.488565	−	0.872527i	$-0.662480\pi$
$797$	−27.5856	−0.977131	−0.488565	+	0.872527i	$0.662480\pi$
$798$	0	0
$799$	1.19559	0.0422970
$800$	0	0
$801$	−50.8099	−1.79528
$802$	0	0
$803$	−20.1278	−0.710295
$804$	0	0
$805$	−38.1491	−1.34458
$806$	0	0
$807$	0.766988	0.0269993
$808$	0	0
$809$	−17.3522	−0.610071	−0.305036	−	0.952341i	$-0.598668\pi$
$809$	−17.3522	−0.610071	−0.305036	+	0.952341i	$0.598668\pi$
$810$	0	0
$811$	−22.4430	−0.788081	−0.394041	−	0.919093i	$-0.628923\pi$
$811$	−22.4430	−0.788081	−0.394041	+	0.919093i	$0.628923\pi$
$812$	0	0
$813$	5.13019	0.179924
$814$	0	0
$815$	−45.6894	−1.60043
$816$	0	0
$817$	31.1467	1.08968
$818$	0	0
$819$	41.2729	1.44219
$820$	0	0
$821$	2.18088	0.0761133	0.0380567	−	0.999276i	$-0.487883\pi$
$821$	2.18088	0.0761133	0.0380567	+	0.999276i	$0.487883\pi$
$822$	0	0
$823$	−6.22243	−0.216900	−0.108450	−	0.994102i	$-0.534589\pi$
$823$	−6.22243	−0.216900	−0.108450	+	0.994102i	$0.534589\pi$
$824$	0	0
$825$	0.331564	0.0115436
$826$	0	0
$827$	17.5574	0.610530	0.305265	−	0.952267i	$-0.401255\pi$
$827$	17.5574	0.610530	0.305265	+	0.952267i	$0.401255\pi$
$828$	0	0
$829$	−0.230530	−0.00800664	−0.00400332	−	0.999992i	$-0.501274\pi$
$829$	−0.230530	−0.00800664	−0.00400332	+	0.999992i	$0.501274\pi$
$830$	0	0
$831$	−0.908130	−0.0315027
$832$	0	0
$833$	3.16154	0.109541
$834$	0	0
$835$	−14.3706	−0.497316
$836$	0	0
$837$	−10.3103	−0.356374
$838$	0	0
$839$	−42.3974	−1.46372	−0.731860	−	0.681455i	$-0.761348\pi$
$839$	−42.3974	−1.46372	−0.731860	+	0.681455i	$0.761348\pi$
$840$	0	0
$841$	42.3615	1.46074
$842$	0	0
$843$	−0.529504	−0.0182371
$844$	0	0
$845$	−62.5059	−2.15027
$846$	0	0
$847$	18.6685	0.641456
$848$	0	0
$849$	4.52941	0.155449
$850$	0	0
$851$	−64.5994	−2.21444
$852$	0	0
$853$	−27.0749	−0.927026	−0.463513	−	0.886090i	$-0.653411\pi$
$853$	−27.0749	−0.927026	−0.463513	+	0.886090i	$0.653411\pi$
$854$	0	0
$855$	56.1200	1.91926
$856$	0	0
$857$	10.5979	0.362017	0.181008	−	0.983482i	$-0.442064\pi$
$857$	10.5979	0.362017	0.181008	+	0.983482i	$0.442064\pi$
$858$	0	0
$859$	−57.4152	−1.95898	−0.979491	−	0.201489i	$-0.935422\pi$
$859$	−57.4152	−1.95898	−0.979491	+	0.201489i	$0.935422\pi$
$860$	0	0
$861$	−4.24849	−0.144788
$862$	0	0
$863$	31.6763	1.07827	0.539136	−	0.842219i	$-0.318751\pi$
$863$	31.6763	1.07827	0.539136	+	0.842219i	$0.318751\pi$
$864$	0	0
$865$	31.6275	1.07537
$866$	0	0
$867$	−3.22536	−0.109539
$868$	0	0
$869$	6.02319	0.204323
$870$	0	0
$871$	26.0033	0.881088
$872$	0	0
$873$	−26.7849	−0.906531
$874$	0	0
$875$	−22.3947	−0.757079
$876$	0	0
$877$	23.1500	0.781719	0.390860	−	0.920450i	$-0.372178\pi$
$877$	23.1500	0.781719	0.390860	+	0.920450i	$0.372178\pi$
$878$	0	0
$879$	−2.90926	−0.0981270
$880$	0	0
$881$	9.26056	0.311996	0.155998	−	0.987757i	$-0.450141\pi$
$881$	9.26056	0.311996	0.155998	+	0.987757i	$0.450141\pi$
$882$	0	0
$883$	−34.1105	−1.14791	−0.573954	−	0.818887i	$-0.694591\pi$
$883$	−34.1105	−1.14791	−0.573954	+	0.818887i	$0.694591\pi$
$884$	0	0
$885$	−4.71186	−0.158387
$886$	0	0
$887$	36.0360	1.20997	0.604986	−	0.796236i	$-0.293179\pi$
$887$	36.0360	1.20997	0.604986	+	0.796236i	$0.293179\pi$
$888$	0	0
$889$	−38.6540	−1.29641
$890$	0	0
$891$	14.0377	0.470281
$892$	0	0
$893$	−5.76257	−0.192837
$894$	0	0
$895$	−27.4484	−0.917499
$896$	0	0
$897$	−9.76742	−0.326125
$898$	0	0
$899$	65.1690	2.17351
$900$	0	0
$901$	17.8631	0.595107
$902$	0	0
$903$	−2.00800	−0.0668219
$904$	0	0
$905$	25.7607	0.856314
$906$	0	0
$907$	41.7267	1.38551	0.692756	−	0.721172i	$-0.256396\pi$
$907$	41.7267	1.38551	0.692756	+	0.721172i	$0.256396\pi$
$908$	0	0
$909$	−3.86493	−0.128192
$910$	0	0
$911$	−32.2971	−1.07005	−0.535026	−	0.844836i	$-0.679698\pi$
$911$	−32.2971	−1.07005	−0.535026	+	0.844836i	$0.679698\pi$
$912$	0	0
$913$	1.93027	0.0638827
$914$	0	0
$915$	4.75654	0.157246
$916$	0	0
$917$	9.67425	0.319472
$918$	0	0
$919$	37.3668	1.23262	0.616308	−	0.787505i	$-0.288628\pi$
$919$	37.3668	1.23262	0.616308	+	0.787505i	$0.288628\pi$
$920$	0	0
$921$	1.50891	0.0497202
$922$	0	0
$923$	−27.9550	−0.920151
$924$	0	0
$925$	8.31113	0.273268
$926$	0	0
$927$	15.2625	0.501287
$928$	0	0
$929$	2.07810	0.0681804	0.0340902	−	0.999419i	$-0.489147\pi$
$929$	2.07810	0.0681804	0.0340902	+	0.999419i	$0.489147\pi$
$930$	0	0
$931$	−15.2381	−0.499410
$932$	0	0
$933$	3.55118	0.116260
$934$	0	0
$935$	6.48250	0.212000
$936$	0	0
$937$	−4.54512	−0.148483	−0.0742413	−	0.997240i	$-0.523654\pi$
$937$	−4.54512	−0.148483	−0.0742413	+	0.997240i	$0.523654\pi$
$938$	0	0
$939$	−6.21610	−0.202855
$940$	0	0
$941$	−27.5658	−0.898620	−0.449310	−	0.893376i	$-0.648330\pi$
$941$	−27.5658	−0.898620	−0.449310	+	0.893376i	$0.648330\pi$
$942$	0	0
$943$	−58.7692	−1.91379
$944$	0	0
$945$	−7.29790	−0.237401
$946$	0	0
$947$	−40.4811	−1.31546	−0.657729	−	0.753255i	$-0.728483\pi$
$947$	−40.4811	−1.31546	−0.657729	+	0.753255i	$0.728483\pi$
$948$	0	0
$949$	−76.2854	−2.47633
$950$	0	0
$951$	−0.480915	−0.0155948
$952$	0	0
$953$	−33.9206	−1.09880	−0.549398	−	0.835561i	$-0.685143\pi$
$953$	−33.9206	−1.09880	−0.549398	+	0.835561i	$0.685143\pi$
$954$	0	0
$955$	24.6576	0.797900
$956$	0	0
$957$	3.11617	0.100731
$958$	0	0
$959$	−0.0145030	−0.000468325 0
$960$	0	0
$961$	28.5139	0.919802
$962$	0	0
$963$	−45.8794	−1.47844
$964$	0	0
$965$	22.3367	0.719044
$966$	0	0
$967$	15.0645	0.484441	0.242221	−	0.970221i	$-0.422124\pi$
$967$	15.0645	0.484441	0.242221	+	0.970221i	$0.422124\pi$
$968$	0	0
$969$	−2.86026	−0.0918848
$970$	0	0
$971$	−3.61438	−0.115991	−0.0579956	−	0.998317i	$-0.518471\pi$
$971$	−3.61438	−0.115991	−0.0579956	+	0.998317i	$0.518471\pi$
$972$	0	0
$973$	0.408699	0.0131023
$974$	0	0
$975$	1.25664	0.0402448
$976$	0	0
$977$	40.6929	1.30188	0.650940	−	0.759129i	$-0.274375\pi$
$977$	40.6929	1.30188	0.650940	+	0.759129i	$0.274375\pi$
$978$	0	0
$979$	28.2882	0.904095
$980$	0	0
$981$	23.4479	0.748633
$982$	0	0
$983$	−17.5420	−0.559504	−0.279752	−	0.960072i	$-0.590252\pi$
$983$	−17.5420	−0.559504	−0.279752	+	0.960072i	$0.590252\pi$
$984$	0	0
$985$	48.8719	1.55719
$986$	0	0
$987$	0.371507	0.0118252
$988$	0	0
$989$	−27.7765	−0.883242
$990$	0	0
$991$	−47.1532	−1.49787	−0.748936	−	0.662643i	$-0.769435\pi$
$991$	−47.1532	−1.49787	−0.748936	+	0.662643i	$0.769435\pi$
$992$	0	0
$993$	−2.92721	−0.0928920
$994$	0	0
$995$	29.6889	0.941201
$996$	0	0
$997$	−53.5944	−1.69735	−0.848676	−	0.528912i	$-0.822600\pi$
$997$	−53.5944	−1.69735	−0.848676	+	0.528912i	$0.822600\pi$
$998$	0	0
$999$	−12.3578	−0.390984

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1004.2.a.b.1.8	✓	14
3.2	odd	2		9036.2.a.m.1.11		14
4.3	odd	2		4016.2.a.j.1.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1004.2.a.b.1.8	✓	14	1.1	even	1	trivial
4016.2.a.j.1.7		14	4.3	odd	2
9036.2.a.m.1.11		14	3.2	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:	\( N \)	\(=\)	\( 1004 = 2^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1004.a (trivial)

\(f(q)\)	\(=\)	\(q+0.224635 q^{3} -2.42875 q^{5} -2.24830 q^{7} -2.94954 q^{9} +O(q^{10})\)	\(q+0.224635 q^{3} -2.42875 q^{5} -2.24830 q^{7} -2.94954 q^{9} +1.64215 q^{11} +6.22381 q^{13} -0.545582 q^{15} -1.62535 q^{17} +7.83395 q^{19} -0.505047 q^{21} -6.98629 q^{23} +0.898832 q^{25} -1.33647 q^{27} +8.44758 q^{29} +7.71452 q^{31} +0.368883 q^{33} +5.46056 q^{35} +9.24659 q^{37} +1.39808 q^{39} +8.41207 q^{41} +3.97586 q^{43} +7.16370 q^{45} -0.735589 q^{47} -1.94514 q^{49} -0.365111 q^{51} -10.9903 q^{53} -3.98836 q^{55} +1.75978 q^{57} +8.63639 q^{59} -8.71829 q^{61} +6.63145 q^{63} -15.1161 q^{65} +4.17803 q^{67} -1.56936 q^{69} -4.49162 q^{71} -12.2570 q^{73} +0.201909 q^{75} -3.69204 q^{77} +3.66788 q^{79} +8.54840 q^{81} +1.17546 q^{83} +3.94758 q^{85} +1.89762 q^{87} +17.2264 q^{89} -13.9930 q^{91} +1.73295 q^{93} -19.0267 q^{95} +9.08104 q^{97} -4.84357 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 3 q^{3} - 2 q^{5} + 8 q^{7} + 21 q^{9}+O(q^{10}) \) 14 * q + 3 * q^3 - 2 * q^5 + 8 * q^7 + 21 * q^9	\( 14 q + 3 q^{3} - 2 q^{5} + 8 q^{7} + 21 q^{9} + 9 q^{11} - q^{13} + 14 q^{15} + 27 q^{19} - 3 q^{21} + 13 q^{23} + 26 q^{25} + 15 q^{27} + 25 q^{31} + 16 q^{33} + 21 q^{35} - q^{37} + 33 q^{39} + 10 q^{41} + 35 q^{43} - 4 q^{45} + 6 q^{47} + 36 q^{49} + 48 q^{51} - q^{53} + 41 q^{55} + 14 q^{57} + 30 q^{59} + 3 q^{61} + 31 q^{63} + 7 q^{65} + 22 q^{67} - 17 q^{69} + 6 q^{71} + 5 q^{73} + 4 q^{75} - 14 q^{77} + 56 q^{79} + 26 q^{81} - 28 q^{83} - 23 q^{85} + 11 q^{87} - 24 q^{89} + 38 q^{91} - 55 q^{93} - 4 q^{95} + 6 q^{97} + 12 q^{99}+O(q^{100}) \) 14 * q + 3 * q^3 - 2 * q^5 + 8 * q^7 + 21 * q^9 + 9 * q^11 - q^13 + 14 * q^15 + 27 * q^19 - 3 * q^21 + 13 * q^23 + 26 * q^25 + 15 * q^27 + 25 * q^31 + 16 * q^33 + 21 * q^35 - q^37 + 33 * q^39 + 10 * q^41 + 35 * q^43 - 4 * q^45 + 6 * q^47 + 36 * q^49 + 48 * q^51 - q^53 + 41 * q^55 + 14 * q^57 + 30 * q^59 + 3 * q^61 + 31 * q^63 + 7 * q^65 + 22 * q^67 - 17 * q^69 + 6 * q^71 + 5 * q^73 + 4 * q^75 - 14 * q^77 + 56 * q^79 + 26 * q^81 - 28 * q^83 - 23 * q^85 + 11 * q^87 - 24 * q^89 + 38 * q^91 - 55 * q^93 - 4 * q^95 + 6 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more