LMFDB - Embedded newform 4016.2.a.i.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.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:	$32.0679214517$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 17 x^{10} + 49 x^{9} + 106 x^{8} - 277 x^{7} - 317 x^{6} + 644 x^{5} + 537 x^{4} + \cdots + 104 $ x^12 - 3x^11 - 17x^10 + 49x^9 + 106x^8 - 277x^7 - 317x^6 + 644x^5 + 537x^4 - 601x^3 - 462x^2 + 136*x + 104
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.69680$ of defining polynomial
Character	$\chi$	$=$	4016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.69680 q^{3} -1.00937 q^{5} +0.288703 q^{7} +4.27276 q^{9} +O(q^{10})$	$q-2.69680 q^{3} -1.00937 q^{5} +0.288703 q^{7} +4.27276 q^{9} +1.03273 q^{11} +4.06167 q^{13} +2.72209 q^{15} -1.28109 q^{17} -6.18702 q^{19} -0.778576 q^{21} -4.48944 q^{23} -3.98116 q^{25} -3.43238 q^{27} +4.25699 q^{29} -1.81811 q^{31} -2.78507 q^{33} -0.291410 q^{35} -1.80644 q^{37} -10.9535 q^{39} +9.25628 q^{41} +4.92555 q^{43} -4.31281 q^{45} -5.94149 q^{47} -6.91665 q^{49} +3.45485 q^{51} +7.90488 q^{53} -1.04241 q^{55} +16.6852 q^{57} +13.2791 q^{59} +10.7786 q^{61} +1.23356 q^{63} -4.09975 q^{65} +15.8430 q^{67} +12.1071 q^{69} -12.4700 q^{71} +7.47640 q^{73} +10.7364 q^{75} +0.298153 q^{77} +1.06251 q^{79} -3.56182 q^{81} -14.7001 q^{83} +1.29310 q^{85} -11.4803 q^{87} -5.36960 q^{89} +1.17262 q^{91} +4.90308 q^{93} +6.24501 q^{95} +0.0437136 q^{97} +4.41260 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	$ 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) $ 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * 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
$3$	−2.69680	−1.55700	−0.778501	−	0.627644i	$-0.784019\pi$
$3$	−2.69680	−1.55700	−0.778501	+	0.627644i	$0.784019\pi$
$4$	0	0
$5$	−1.00937	−0.451406	−0.225703	−	0.974196i	$-0.572468\pi$
$5$	−1.00937	−0.451406	−0.225703	+	0.974196i	$0.572468\pi$
$6$	0	0
$7$	0.288703	0.109120	0.0545598	−	0.998511i	$-0.482624\pi$
$7$	0.288703	0.109120	0.0545598	+	0.998511i	$0.482624\pi$
$8$	0	0
$9$	4.27276	1.42425
$10$	0	0
$11$	1.03273	0.311380	0.155690	−	0.987806i	$-0.450240\pi$
$11$	1.03273	0.311380	0.155690	+	0.987806i	$0.450240\pi$
$12$	0	0
$13$	4.06167	1.12651	0.563253	−	0.826285i	$-0.309550\pi$
$13$	4.06167	1.12651	0.563253	+	0.826285i	$0.309550\pi$
$14$	0	0
$15$	2.72209	0.702840
$16$	0	0
$17$	−1.28109	−0.310710	−0.155355	−	0.987859i	$-0.549652\pi$
$17$	−1.28109	−0.310710	−0.155355	+	0.987859i	$0.549652\pi$
$18$	0	0
$19$	−6.18702	−1.41940	−0.709699	−	0.704505i	$-0.751169\pi$
$19$	−6.18702	−1.41940	−0.709699	+	0.704505i	$0.751169\pi$
$20$	0	0
$21$	−0.778576	−0.169899
$22$	0	0
$23$	−4.48944	−0.936113	−0.468057	−	0.883698i	$-0.655046\pi$
$23$	−4.48944	−0.936113	−0.468057	+	0.883698i	$0.655046\pi$
$24$	0	0
$25$	−3.98116	−0.796233
$26$	0	0
$27$	−3.43238	−0.660561
$28$	0	0
$29$	4.25699	0.790504	0.395252	−	0.918573i	$-0.370657\pi$
$29$	4.25699	0.790504	0.395252	+	0.918573i	$0.370657\pi$
$30$	0	0
$31$	−1.81811	−0.326542	−0.163271	−	0.986581i	$-0.552204\pi$
$31$	−1.81811	−0.326542	−0.163271	+	0.986581i	$0.552204\pi$
$32$	0	0
$33$	−2.78507	−0.484819
$34$	0	0
$35$	−0.291410	−0.0492572
$36$	0	0
$37$	−1.80644	−0.296978	−0.148489	−	0.988914i	$-0.547441\pi$
$37$	−1.80644	−0.296978	−0.148489	+	0.988914i	$0.547441\pi$
$38$	0	0
$39$	−10.9535	−1.75397
$40$	0	0
$41$	9.25628	1.44559	0.722794	−	0.691063i	$-0.242857\pi$
$41$	9.25628	1.44559	0.722794	+	0.691063i	$0.242857\pi$
$42$	0	0
$43$	4.92555	0.751139	0.375569	−	0.926794i	$-0.377447\pi$
$43$	4.92555	0.751139	0.375569	+	0.926794i	$0.377447\pi$
$44$	0	0
$45$	−4.31281	−0.642916
$46$	0	0
$47$	−5.94149	−0.866656	−0.433328	−	0.901236i	$-0.642661\pi$
$47$	−5.94149	−0.866656	−0.433328	+	0.901236i	$0.642661\pi$
$48$	0	0
$49$	−6.91665	−0.988093
$50$	0	0
$51$	3.45485	0.483776
$52$	0	0
$53$	7.90488	1.08582	0.542909	−	0.839791i	$-0.317323\pi$
$53$	7.90488	1.08582	0.542909	+	0.839791i	$0.317323\pi$
$54$	0	0
$55$	−1.04241	−0.140559
$56$	0	0
$57$	16.6852	2.21001
$58$	0	0
$59$	13.2791	1.72879	0.864395	−	0.502814i	$-0.167702\pi$
$59$	13.2791	1.72879	0.864395	+	0.502814i	$0.167702\pi$
$60$	0	0
$61$	10.7786	1.38006	0.690030	−	0.723780i	$-0.257597\pi$
$61$	10.7786	1.38006	0.690030	+	0.723780i	$0.257597\pi$
$62$	0	0
$63$	1.23356	0.155414
$64$	0	0
$65$	−4.09975	−0.508511
$66$	0	0
$67$	15.8430	1.93554	0.967768	−	0.251845i	$-0.0810373\pi$
$67$	15.8430	1.93554	0.967768	+	0.251845i	$0.0810373\pi$
$68$	0	0
$69$	12.1071	1.45753
$70$	0	0
$71$	−12.4700	−1.47992	−0.739961	−	0.672650i	$-0.765156\pi$
$71$	−12.4700	−1.47992	−0.739961	+	0.672650i	$0.765156\pi$
$72$	0	0
$73$	7.47640	0.875047	0.437523	−	0.899207i	$-0.355856\pi$
$73$	7.47640	0.875047	0.437523	+	0.899207i	$0.355856\pi$
$74$	0	0
$75$	10.7364	1.23974
$76$	0	0
$77$	0.298153	0.0339776
$78$	0	0
$79$	1.06251	0.119542	0.0597710	−	0.998212i	$-0.480963\pi$
$79$	1.06251	0.119542	0.0597710	+	0.998212i	$0.480963\pi$
$80$	0	0
$81$	−3.56182	−0.395758
$82$	0	0
$83$	−14.7001	−1.61355	−0.806773	−	0.590862i	$-0.798788\pi$
$83$	−14.7001	−1.61355	−0.806773	+	0.590862i	$0.798788\pi$
$84$	0	0
$85$	1.29310	0.140256
$86$	0	0
$87$	−11.4803	−1.23082
$88$	0	0
$89$	−5.36960	−0.569176	−0.284588	−	0.958650i	$-0.591857\pi$
$89$	−5.36960	−0.569176	−0.284588	+	0.958650i	$0.591857\pi$
$90$	0	0
$91$	1.17262	0.122924
$92$	0	0
$93$	4.90308	0.508426
$94$	0	0
$95$	6.24501	0.640725
$96$	0	0
$97$	0.0437136	0.00443844	0.00221922	−	0.999998i	$-0.499294\pi$
$97$	0.0437136	0.00443844	0.00221922	+	0.999998i	$0.499294\pi$
$98$	0	0
$99$	4.41260	0.443483
$100$	0	0
$101$	3.61118	0.359326	0.179663	−	0.983728i	$-0.442499\pi$
$101$	3.61118	0.359326	0.179663	+	0.983728i	$0.442499\pi$
$102$	0	0
$103$	−3.97316	−0.391487	−0.195744	−	0.980655i	$-0.562712\pi$
$103$	−3.97316	−0.391487	−0.195744	+	0.980655i	$0.562712\pi$
$104$	0	0
$105$	0.785875	0.0766936
$106$	0	0
$107$	−4.88695	−0.472440	−0.236220	−	0.971700i	$-0.575909\pi$
$107$	−4.88695	−0.472440	−0.236220	+	0.971700i	$0.575909\pi$
$108$	0	0
$109$	−19.4454	−1.86253	−0.931267	−	0.364338i	$-0.881295\pi$
$109$	−19.4454	−1.86253	−0.931267	+	0.364338i	$0.881295\pi$
$110$	0	0
$111$	4.87163	0.462394
$112$	0	0
$113$	8.13361	0.765145	0.382573	−	0.923925i	$-0.375038\pi$
$113$	8.13361	0.765145	0.382573	+	0.923925i	$0.375038\pi$
$114$	0	0
$115$	4.53153	0.422567
$116$	0	0
$117$	17.3545	1.60443
$118$	0	0
$119$	−0.369855	−0.0339046
$120$	0	0
$121$	−9.93347	−0.903043
$122$	0	0
$123$	−24.9624	−2.25078
$124$	0	0
$125$	9.06536	0.810830
$126$	0	0
$127$	6.44714	0.572091	0.286046	−	0.958216i	$-0.407659\pi$
$127$	6.44714	0.572091	0.286046	+	0.958216i	$0.407659\pi$
$128$	0	0
$129$	−13.2832	−1.16952
$130$	0	0
$131$	−12.4394	−1.08684	−0.543418	−	0.839462i	$-0.682870\pi$
$131$	−12.4394	−1.08684	−0.543418	+	0.839462i	$0.682870\pi$
$132$	0	0
$133$	−1.78621	−0.154884
$134$	0	0
$135$	3.46455	0.298181
$136$	0	0
$137$	−11.0326	−0.942575	−0.471287	−	0.881980i	$-0.656211\pi$
$137$	−11.0326	−0.942575	−0.471287	+	0.881980i	$0.656211\pi$
$138$	0	0
$139$	10.9303	0.927092	0.463546	−	0.886073i	$-0.346577\pi$
$139$	10.9303	0.927092	0.463546	+	0.886073i	$0.346577\pi$
$140$	0	0
$141$	16.0230	1.34938
$142$	0	0
$143$	4.19461	0.350771
$144$	0	0
$145$	−4.29690	−0.356838
$146$	0	0
$147$	18.6529	1.53846
$148$	0	0
$149$	−7.74492	−0.634488	−0.317244	−	0.948344i	$-0.602757\pi$
$149$	−7.74492	−0.634488	−0.317244	+	0.948344i	$0.602757\pi$
$150$	0	0
$151$	−5.26685	−0.428610	−0.214305	−	0.976767i	$-0.568749\pi$
$151$	−5.26685	−0.428610	−0.214305	+	0.976767i	$0.568749\pi$
$152$	0	0
$153$	−5.47379	−0.442530
$154$	0	0
$155$	1.83515	0.147403
$156$	0	0
$157$	−14.3756	−1.14730	−0.573650	−	0.819101i	$-0.694473\pi$
$157$	−14.3756	−1.14730	−0.573650	+	0.819101i	$0.694473\pi$
$158$	0	0
$159$	−21.3179	−1.69062
$160$	0	0
$161$	−1.29612	−0.102148
$162$	0	0
$163$	−12.2193	−0.957090	−0.478545	−	0.878063i	$-0.658836\pi$
$163$	−12.2193	−0.957090	−0.478545	+	0.878063i	$0.658836\pi$
$164$	0	0
$165$	2.81118	0.218850
$166$	0	0
$167$	11.6457	0.901172	0.450586	−	0.892733i	$-0.351215\pi$
$167$	11.6457	0.901172	0.450586	+	0.892733i	$0.351215\pi$
$168$	0	0
$169$	3.49720	0.269016
$170$	0	0
$171$	−26.4356	−2.02158
$172$	0	0
$173$	−1.12380	−0.0854412	−0.0427206	−	0.999087i	$-0.513603\pi$
$173$	−1.12380	−0.0854412	−0.0427206	+	0.999087i	$0.513603\pi$
$174$	0	0
$175$	−1.14938	−0.0868846
$176$	0	0
$177$	−35.8111	−2.69173
$178$	0	0
$179$	−3.30904	−0.247330	−0.123665	−	0.992324i	$-0.539465\pi$
$179$	−3.30904	−0.247330	−0.123665	+	0.992324i	$0.539465\pi$
$180$	0	0
$181$	−22.0124	−1.63617	−0.818083	−	0.575100i	$-0.804963\pi$
$181$	−22.0124	−1.63617	−0.818083	+	0.575100i	$0.804963\pi$
$182$	0	0
$183$	−29.0678	−2.14876
$184$	0	0
$185$	1.82338	0.134057
$186$	0	0
$187$	−1.32302	−0.0967489
$188$	0	0
$189$	−0.990938	−0.0720802
$190$	0	0
$191$	20.7291	1.49991	0.749954	−	0.661490i	$-0.230076\pi$
$191$	20.7291	1.49991	0.749954	+	0.661490i	$0.230076\pi$
$192$	0	0
$193$	−16.4310	−1.18273	−0.591363	−	0.806405i	$-0.701410\pi$
$193$	−16.4310	−1.18273	−0.591363	+	0.806405i	$0.701410\pi$
$194$	0	0
$195$	11.0562	0.791753
$196$	0	0
$197$	−15.1156	−1.07695	−0.538473	−	0.842643i	$-0.680998\pi$
$197$	−15.1156	−1.07695	−0.538473	+	0.842643i	$0.680998\pi$
$198$	0	0
$199$	−2.62670	−0.186202	−0.0931010	−	0.995657i	$-0.529678\pi$
$199$	−2.62670	−0.186202	−0.0931010	+	0.995657i	$0.529678\pi$
$200$	0	0
$201$	−42.7256	−3.01363
$202$	0	0
$203$	1.22901	0.0862595
$204$	0	0
$205$	−9.34305	−0.652547
$206$	0	0
$207$	−19.1823	−1.33326
$208$	0	0
$209$	−6.38952	−0.441972
$210$	0	0
$211$	−9.81977	−0.676021	−0.338011	−	0.941142i	$-0.609754\pi$
$211$	−9.81977	−0.676021	−0.338011	+	0.941142i	$0.609754\pi$
$212$	0	0
$213$	33.6293	2.30424
$214$	0	0
$215$	−4.97172	−0.339069
$216$	0	0
$217$	−0.524894	−0.0356321
$218$	0	0
$219$	−20.1624	−1.36245
$220$	0	0
$221$	−5.20338	−0.350017
$222$	0	0
$223$	−0.163679	−0.0109607	−0.00548036	−	0.999985i	$-0.501744\pi$
$223$	−0.163679	−0.0109607	−0.00548036	+	0.999985i	$0.501744\pi$
$224$	0	0
$225$	−17.0105	−1.13404
$226$	0	0
$227$	23.9242	1.58790	0.793952	−	0.607981i	$-0.208020\pi$
$227$	23.9242	1.58790	0.793952	+	0.607981i	$0.208020\pi$
$228$	0	0
$229$	−8.56103	−0.565729	−0.282864	−	0.959160i	$-0.591285\pi$
$229$	−8.56103	−0.565729	−0.282864	+	0.959160i	$0.591285\pi$
$230$	0	0
$231$	−0.804059	−0.0529032
$232$	0	0
$233$	−15.8195	−1.03637	−0.518184	−	0.855269i	$-0.673392\pi$
$233$	−15.8195	−1.03637	−0.518184	+	0.855269i	$0.673392\pi$
$234$	0	0
$235$	5.99719	0.391214
$236$	0	0
$237$	−2.86539	−0.186127
$238$	0	0
$239$	4.67926	0.302676	0.151338	−	0.988482i	$-0.451642\pi$
$239$	4.67926	0.302676	0.151338	+	0.988482i	$0.451642\pi$
$240$	0	0
$241$	10.6813	0.688046	0.344023	−	0.938961i	$-0.388210\pi$
$241$	10.6813	0.688046	0.344023	+	0.938961i	$0.388210\pi$
$242$	0	0
$243$	19.9027	1.27676
$244$	0	0
$245$	6.98149	0.446031
$246$	0	0
$247$	−25.1296	−1.59896
$248$	0	0
$249$	39.6433	2.51229
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−4.63638	−0.291487
$254$	0	0
$255$	−3.48724	−0.218379
$256$	0	0
$257$	13.8086	0.861354	0.430677	−	0.902506i	$-0.358275\pi$
$257$	13.8086	0.861354	0.430677	+	0.902506i	$0.358275\pi$
$258$	0	0
$259$	−0.521526	−0.0324061
$260$	0	0
$261$	18.1891	1.12588
$262$	0	0
$263$	13.8052	0.851265	0.425632	−	0.904896i	$-0.360052\pi$
$263$	13.8052	0.851265	0.425632	+	0.904896i	$0.360052\pi$
$264$	0	0
$265$	−7.97898	−0.490145
$266$	0	0
$267$	14.4808	0.886208
$268$	0	0
$269$	0.984309	0.0600144	0.0300072	−	0.999550i	$-0.490447\pi$
$269$	0.984309	0.0600144	0.0300072	+	0.999550i	$0.490447\pi$
$270$	0	0
$271$	−25.6461	−1.55789	−0.778946	−	0.627091i	$-0.784245\pi$
$271$	−25.6461	−1.55789	−0.778946	+	0.627091i	$0.784245\pi$
$272$	0	0
$273$	−3.16232	−0.191393
$274$	0	0
$275$	−4.11147	−0.247931
$276$	0	0
$277$	−2.67136	−0.160506	−0.0802532	−	0.996775i	$-0.525573\pi$
$277$	−2.67136	−0.160506	−0.0802532	+	0.996775i	$0.525573\pi$
$278$	0	0
$279$	−7.76833	−0.465078
$280$	0	0
$281$	−7.20183	−0.429625	−0.214813	−	0.976655i	$-0.568914\pi$
$281$	−7.20183	−0.429625	−0.214813	+	0.976655i	$0.568914\pi$
$282$	0	0
$283$	−16.7017	−0.992813	−0.496407	−	0.868090i	$-0.665347\pi$
$283$	−16.7017	−0.992813	−0.496407	+	0.868090i	$0.665347\pi$
$284$	0	0
$285$	−16.8416	−0.997609
$286$	0	0
$287$	2.67232	0.157742
$288$	0	0
$289$	−15.3588	−0.903459
$290$	0	0
$291$	−0.117887	−0.00691066
$292$	0	0
$293$	6.01774	0.351560	0.175780	−	0.984429i	$-0.443755\pi$
$293$	6.01774	0.351560	0.175780	+	0.984429i	$0.443755\pi$
$294$	0	0
$295$	−13.4036	−0.780386
$296$	0	0
$297$	−3.54472	−0.205685
$298$	0	0
$299$	−18.2347	−1.05454
$300$	0	0
$301$	1.42202	0.0819640
$302$	0	0
$303$	−9.73865	−0.559471
$304$	0	0
$305$	−10.8797	−0.622968
$306$	0	0
$307$	12.6706	0.723148	0.361574	−	0.932343i	$-0.382239\pi$
$307$	12.6706	0.723148	0.361574	+	0.932343i	$0.382239\pi$
$308$	0	0
$309$	10.7148	0.609546
$310$	0	0
$311$	4.93352	0.279754	0.139877	−	0.990169i	$-0.455329\pi$
$311$	4.93352	0.279754	0.139877	+	0.990169i	$0.455329\pi$
$312$	0	0
$313$	13.9191	0.786755	0.393378	−	0.919377i	$-0.371306\pi$
$313$	13.9191	0.786755	0.393378	+	0.919377i	$0.371306\pi$
$314$	0	0
$315$	−1.24512	−0.0701547
$316$	0	0
$317$	27.7839	1.56050	0.780251	−	0.625467i	$-0.215092\pi$
$317$	27.7839	1.56050	0.780251	+	0.625467i	$0.215092\pi$
$318$	0	0
$319$	4.39632	0.246147
$320$	0	0
$321$	13.1792	0.735589
$322$	0	0
$323$	7.92613	0.441022
$324$	0	0
$325$	−16.1702	−0.896961
$326$	0	0
$327$	52.4405	2.89997
$328$	0	0
$329$	−1.71533	−0.0945691
$330$	0	0
$331$	−11.4171	−0.627539	−0.313769	−	0.949499i	$-0.601592\pi$
$331$	−11.4171	−0.627539	−0.313769	+	0.949499i	$0.601592\pi$
$332$	0	0
$333$	−7.71850	−0.422971
$334$	0	0
$335$	−15.9916	−0.873712
$336$	0	0
$337$	−23.4903	−1.27960	−0.639799	−	0.768542i	$-0.720982\pi$
$337$	−23.4903	−1.27960	−0.639799	+	0.768542i	$0.720982\pi$
$338$	0	0
$339$	−21.9347	−1.19133
$340$	0	0
$341$	−1.87761	−0.101679
$342$	0	0
$343$	−4.01778	−0.216940
$344$	0	0
$345$	−12.2206	−0.657937
$346$	0	0
$347$	−18.6516	−1.00127	−0.500636	−	0.865658i	$-0.666900\pi$
$347$	−18.6516	−1.00127	−0.500636	+	0.865658i	$0.666900\pi$
$348$	0	0
$349$	0.588035	0.0314768	0.0157384	−	0.999876i	$-0.494990\pi$
$349$	0.588035	0.0314768	0.0157384	+	0.999876i	$0.494990\pi$
$350$	0	0
$351$	−13.9412	−0.744126
$352$	0	0
$353$	19.2475	1.02444	0.512219	−	0.858855i	$-0.328824\pi$
$353$	19.2475	1.02444	0.512219	+	0.858855i	$0.328824\pi$
$354$	0	0
$355$	12.5869	0.668046
$356$	0	0
$357$	0.997428	0.0527895
$358$	0	0
$359$	−17.6652	−0.932334	−0.466167	−	0.884697i	$-0.654365\pi$
$359$	−17.6652	−0.932334	−0.466167	+	0.884697i	$0.654365\pi$
$360$	0	0
$361$	19.2792	1.01469
$362$	0	0
$363$	26.7886	1.40604
$364$	0	0
$365$	−7.54649	−0.395001
$366$	0	0
$367$	−27.4976	−1.43536	−0.717681	−	0.696372i	$-0.754796\pi$
$367$	−27.4976	−1.43536	−0.717681	+	0.696372i	$0.754796\pi$
$368$	0	0
$369$	39.5498	2.05888
$370$	0	0
$371$	2.28216	0.118484
$372$	0	0
$373$	−7.11956	−0.368637	−0.184318	−	0.982867i	$-0.559008\pi$
$373$	−7.11956	−0.368637	−0.184318	+	0.982867i	$0.559008\pi$
$374$	0	0
$375$	−24.4475	−1.26246
$376$	0	0
$377$	17.2905	0.890507
$378$	0	0
$379$	18.7739	0.964348	0.482174	−	0.876075i	$-0.339847\pi$
$379$	18.7739	0.964348	0.482174	+	0.876075i	$0.339847\pi$
$380$	0	0
$381$	−17.3867	−0.890747
$382$	0	0
$383$	−7.78814	−0.397955	−0.198978	−	0.980004i	$-0.563762\pi$
$383$	−7.78814	−0.397955	−0.198978	+	0.980004i	$0.563762\pi$
$384$	0	0
$385$	−0.300947	−0.0153377
$386$	0	0
$387$	21.0457	1.06981
$388$	0	0
$389$	−33.8918	−1.71838	−0.859192	−	0.511654i	$-0.829033\pi$
$389$	−33.8918	−1.71838	−0.859192	+	0.511654i	$0.829033\pi$
$390$	0	0
$391$	5.75138	0.290860
$392$	0	0
$393$	33.5466	1.69220
$394$	0	0
$395$	−1.07247	−0.0539620
$396$	0	0
$397$	−25.8246	−1.29610	−0.648050	−	0.761598i	$-0.724415\pi$
$397$	−25.8246	−1.29610	−0.648050	+	0.761598i	$0.724415\pi$
$398$	0	0
$399$	4.81706	0.241155
$400$	0	0
$401$	−15.7073	−0.784386	−0.392193	−	0.919883i	$-0.628283\pi$
$401$	−15.7073	−0.784386	−0.392193	+	0.919883i	$0.628283\pi$
$402$	0	0
$403$	−7.38457	−0.367851
$404$	0	0
$405$	3.59521	0.178647
$406$	0	0
$407$	−1.86557	−0.0924728
$408$	0	0
$409$	5.21682	0.257955	0.128978	−	0.991648i	$-0.458830\pi$
$409$	5.21682	0.257955	0.128978	+	0.991648i	$0.458830\pi$
$410$	0	0
$411$	29.7527	1.46759
$412$	0	0
$413$	3.83372	0.188645
$414$	0	0
$415$	14.8379	0.728364
$416$	0	0
$417$	−29.4768	−1.44348
$418$	0	0
$419$	−27.2603	−1.33175	−0.665877	−	0.746062i	$-0.731942\pi$
$419$	−27.2603	−1.33175	−0.665877	+	0.746062i	$0.731942\pi$
$420$	0	0
$421$	25.2693	1.23155	0.615774	−	0.787923i	$-0.288843\pi$
$421$	25.2693	1.23155	0.615774	+	0.787923i	$0.288843\pi$
$422$	0	0
$423$	−25.3866	−1.23434
$424$	0	0
$425$	5.10023	0.247398
$426$	0	0
$427$	3.11182	0.150592
$428$	0	0
$429$	−11.3121	−0.546151
$430$	0	0
$431$	24.4441	1.17743	0.588715	−	0.808341i	$-0.299634\pi$
$431$	24.4441	1.17743	0.588715	+	0.808341i	$0.299634\pi$
$432$	0	0
$433$	−27.7732	−1.33470	−0.667348	−	0.744746i	$-0.732571\pi$
$433$	−27.7732	−1.33470	−0.667348	+	0.744746i	$0.732571\pi$
$434$	0	0
$435$	11.5879	0.555597
$436$	0	0
$437$	27.7762	1.32872
$438$	0	0
$439$	7.91559	0.377791	0.188895	−	0.981997i	$-0.439509\pi$
$439$	7.91559	0.377791	0.188895	+	0.981997i	$0.439509\pi$
$440$	0	0
$441$	−29.5532	−1.40729
$442$	0	0
$443$	18.5041	0.879158	0.439579	−	0.898204i	$-0.355128\pi$
$443$	18.5041	0.879158	0.439579	+	0.898204i	$0.355128\pi$
$444$	0	0
$445$	5.41993	0.256929
$446$	0	0
$447$	20.8865	0.987899
$448$	0	0
$449$	−40.6933	−1.92044	−0.960218	−	0.279251i	$-0.909914\pi$
$449$	−40.6933	−1.92044	−0.960218	+	0.279251i	$0.909914\pi$
$450$	0	0
$451$	9.55924	0.450127
$452$	0	0
$453$	14.2037	0.667346
$454$	0	0
$455$	−1.18361	−0.0554886
$456$	0	0
$457$	−18.0572	−0.844678	−0.422339	−	0.906438i	$-0.638791\pi$
$457$	−18.0572	−0.844678	−0.422339	+	0.906438i	$0.638791\pi$
$458$	0	0
$459$	4.39719	0.205243
$460$	0	0
$461$	0.776649	0.0361722	0.0180861	−	0.999836i	$-0.494243\pi$
$461$	0.776649	0.0361722	0.0180861	+	0.999836i	$0.494243\pi$
$462$	0	0
$463$	−4.07384	−0.189328	−0.0946638	−	0.995509i	$-0.530178\pi$
$463$	−4.07384	−0.189328	−0.0946638	+	0.995509i	$0.530178\pi$
$464$	0	0
$465$	−4.94905	−0.229507
$466$	0	0
$467$	−2.95790	−0.136875	−0.0684376	−	0.997655i	$-0.521801\pi$
$467$	−2.95790	−0.136875	−0.0684376	+	0.997655i	$0.521801\pi$
$468$	0	0
$469$	4.57394	0.211205
$470$	0	0
$471$	38.7682	1.78635
$472$	0	0
$473$	5.08676	0.233889
$474$	0	0
$475$	24.6315	1.13017
$476$	0	0
$477$	33.7756	1.54648
$478$	0	0
$479$	−28.8615	−1.31872	−0.659358	−	0.751829i	$-0.729172\pi$
$479$	−28.8615	−1.31872	−0.659358	+	0.751829i	$0.729172\pi$
$480$	0	0
$481$	−7.33719	−0.334547
$482$	0	0
$483$	3.49537	0.159045
$484$	0	0
$485$	−0.0441234	−0.00200354
$486$	0	0
$487$	−15.6500	−0.709169	−0.354584	−	0.935024i	$-0.615378\pi$
$487$	−15.6500	−0.709169	−0.354584	+	0.935024i	$0.615378\pi$
$488$	0	0
$489$	32.9531	1.49019
$490$	0	0
$491$	−18.9880	−0.856916	−0.428458	−	0.903562i	$-0.640943\pi$
$491$	−18.9880	−0.856916	−0.428458	+	0.903562i	$0.640943\pi$
$492$	0	0
$493$	−5.45360	−0.245618
$494$	0	0
$495$	−4.45397	−0.200191
$496$	0	0
$497$	−3.60014	−0.161489
$498$	0	0
$499$	−23.3484	−1.04522	−0.522609	−	0.852572i	$-0.675041\pi$
$499$	−23.3484	−1.04522	−0.522609	+	0.852572i	$0.675041\pi$
$500$	0	0
$501$	−31.4062	−1.40313
$502$	0	0
$503$	−15.6239	−0.696635	−0.348317	−	0.937377i	$-0.613247\pi$
$503$	−15.6239	−0.696635	−0.348317	+	0.937377i	$0.613247\pi$
$504$	0	0
$505$	−3.64503	−0.162202
$506$	0	0
$507$	−9.43127	−0.418858
$508$	0	0
$509$	−12.7297	−0.564233	−0.282117	−	0.959380i	$-0.591036\pi$
$509$	−12.7297	−0.564233	−0.282117	+	0.959380i	$0.591036\pi$
$510$	0	0
$511$	2.15846	0.0954847
$512$	0	0
$513$	21.2362	0.937600
$514$	0	0
$515$	4.01041	0.176720
$516$	0	0
$517$	−6.13596	−0.269859
$518$	0	0
$519$	3.03068	0.133032
$520$	0	0
$521$	−12.8996	−0.565140	−0.282570	−	0.959247i	$-0.591187\pi$
$521$	−12.8996	−0.565140	−0.282570	+	0.959247i	$0.591187\pi$
$522$	0	0
$523$	40.5603	1.77358	0.886789	−	0.462175i	$-0.152931\pi$
$523$	40.5603	1.77358	0.886789	+	0.462175i	$0.152931\pi$
$524$	0	0
$525$	3.09964	0.135279
$526$	0	0
$527$	2.32916	0.101460
$528$	0	0
$529$	−2.84492	−0.123692
$530$	0	0
$531$	56.7383	2.46223
$532$	0	0
$533$	37.5960	1.62846
$534$	0	0
$535$	4.93276	0.213262
$536$	0	0
$537$	8.92384	0.385092
$538$	0	0
$539$	−7.14303	−0.307672
$540$	0	0
$541$	−5.39339	−0.231880	−0.115940	−	0.993256i	$-0.536988\pi$
$541$	−5.39339	−0.231880	−0.115940	+	0.993256i	$0.536988\pi$
$542$	0	0
$543$	59.3630	2.54751
$544$	0	0
$545$	19.6277	0.840759
$546$	0	0
$547$	−37.6382	−1.60929	−0.804647	−	0.593754i	$-0.797645\pi$
$547$	−37.6382	−1.60929	−0.804647	+	0.593754i	$0.797645\pi$
$548$	0	0
$549$	46.0544	1.96555
$550$	0	0
$551$	−26.3381	−1.12204
$552$	0	0
$553$	0.306751	0.0130444
$554$	0	0
$555$	−4.91730	−0.208728
$556$	0	0
$557$	−10.6932	−0.453084	−0.226542	−	0.974001i	$-0.572742\pi$
$557$	−10.6932	−0.453084	−0.226542	+	0.974001i	$0.572742\pi$
$558$	0	0
$559$	20.0060	0.846162
$560$	0	0
$561$	3.56793	0.150638
$562$	0	0
$563$	−1.46712	−0.0618319	−0.0309159	−	0.999522i	$-0.509842\pi$
$563$	−1.46712	−0.0618319	−0.0309159	+	0.999522i	$0.509842\pi$
$564$	0	0
$565$	−8.20985	−0.345391
$566$	0	0
$567$	−1.02831	−0.0431849
$568$	0	0
$569$	−37.6472	−1.57825	−0.789127	−	0.614230i	$-0.789467\pi$
$569$	−37.6472	−1.57825	−0.789127	+	0.614230i	$0.789467\pi$
$570$	0	0
$571$	27.5441	1.15269	0.576343	−	0.817208i	$-0.304479\pi$
$571$	27.5441	1.15269	0.576343	+	0.817208i	$0.304479\pi$
$572$	0	0
$573$	−55.9025	−2.33536
$574$	0	0
$575$	17.8732	0.745364
$576$	0	0
$577$	−17.9119	−0.745681	−0.372840	−	0.927896i	$-0.621616\pi$
$577$	−17.9119	−0.745681	−0.372840	+	0.927896i	$0.621616\pi$
$578$	0	0
$579$	44.3111	1.84151
$580$	0	0
$581$	−4.24397	−0.176069
$582$	0	0
$583$	8.16360	0.338102
$584$	0	0
$585$	−17.5172	−0.724249
$586$	0	0
$587$	−13.3934	−0.552807	−0.276403	−	0.961042i	$-0.589143\pi$
$587$	−13.3934	−0.552807	−0.276403	+	0.961042i	$0.589143\pi$
$588$	0	0
$589$	11.2487	0.463493
$590$	0	0
$591$	40.7640	1.67680
$592$	0	0
$593$	−6.27519	−0.257691	−0.128846	−	0.991665i	$-0.541127\pi$
$593$	−6.27519	−0.257691	−0.128846	+	0.991665i	$0.541127\pi$
$594$	0	0
$595$	0.373322	0.0153047
$596$	0	0
$597$	7.08370	0.289917
$598$	0	0
$599$	18.8224	0.769063	0.384531	−	0.923112i	$-0.374363\pi$
$599$	18.8224	0.769063	0.384531	+	0.923112i	$0.374363\pi$
$600$	0	0
$601$	−25.2664	−1.03064	−0.515319	−	0.856999i	$-0.672326\pi$
$601$	−25.2664	−1.03064	−0.515319	+	0.856999i	$0.672326\pi$
$602$	0	0
$603$	67.6934	2.75669
$604$	0	0
$605$	10.0266	0.407639
$606$	0	0
$607$	41.6865	1.69200	0.846002	−	0.533180i	$-0.179003\pi$
$607$	41.6865	1.69200	0.846002	+	0.533180i	$0.179003\pi$
$608$	0	0
$609$	−3.31439	−0.134306
$610$	0	0
$611$	−24.1324	−0.976293
$612$	0	0
$613$	−8.97877	−0.362649	−0.181324	−	0.983423i	$-0.558038\pi$
$613$	−8.97877	−0.362649	−0.181324	+	0.983423i	$0.558038\pi$
$614$	0	0
$615$	25.1964	1.01602
$616$	0	0
$617$	39.1638	1.57667	0.788337	−	0.615243i	$-0.210942\pi$
$617$	39.1638	1.57667	0.788337	+	0.615243i	$0.210942\pi$
$618$	0	0
$619$	−25.8011	−1.03703	−0.518517	−	0.855068i	$-0.673516\pi$
$619$	−25.8011	−1.03703	−0.518517	+	0.855068i	$0.673516\pi$
$620$	0	0
$621$	15.4095	0.618360
$622$	0	0
$623$	−1.55022	−0.0621083
$624$	0	0
$625$	10.7555	0.430219
$626$	0	0
$627$	17.2313	0.688151
$628$	0	0
$629$	2.31422	0.0922740
$630$	0	0
$631$	−13.1780	−0.524609	−0.262305	−	0.964985i	$-0.584483\pi$
$631$	−13.1780	−0.524609	−0.262305	+	0.964985i	$0.584483\pi$
$632$	0	0
$633$	26.4820	1.05257
$634$	0	0
$635$	−6.50758	−0.258245
$636$	0	0
$637$	−28.0932	−1.11309
$638$	0	0
$639$	−53.2815	−2.10778
$640$	0	0
$641$	0.405055	0.0159987	0.00799935	−	0.999968i	$-0.497454\pi$
$641$	0.405055	0.0159987	0.00799935	+	0.999968i	$0.497454\pi$
$642$	0	0
$643$	6.47002	0.255153	0.127576	−	0.991829i	$-0.459280\pi$
$643$	6.47002	0.255153	0.127576	+	0.991829i	$0.459280\pi$
$644$	0	0
$645$	13.4078	0.527930
$646$	0	0
$647$	9.19928	0.361661	0.180831	−	0.983514i	$-0.442121\pi$
$647$	9.19928	0.361661	0.180831	+	0.983514i	$0.442121\pi$
$648$	0	0
$649$	13.7137	0.538310
$650$	0	0
$651$	1.41554	0.0554792
$652$	0	0
$653$	−22.1285	−0.865956	−0.432978	−	0.901405i	$-0.642537\pi$
$653$	−22.1285	−0.865956	−0.432978	+	0.901405i	$0.642537\pi$
$654$	0	0
$655$	12.5560	0.490604
$656$	0	0
$657$	31.9448	1.24629
$658$	0	0
$659$	22.6686	0.883043	0.441522	−	0.897251i	$-0.354439\pi$
$659$	22.6686	0.883043	0.441522	+	0.897251i	$0.354439\pi$
$660$	0	0
$661$	26.4738	1.02971	0.514855	−	0.857277i	$-0.327846\pi$
$661$	26.4738	1.02971	0.514855	+	0.857277i	$0.327846\pi$
$662$	0	0
$663$	14.0325	0.544977
$664$	0	0
$665$	1.80296	0.0699156
$666$	0	0
$667$	−19.1115	−0.740001
$668$	0	0
$669$	0.441409	0.0170659
$670$	0	0
$671$	11.1314	0.429723
$672$	0	0
$673$	−24.2170	−0.933497	−0.466749	−	0.884390i	$-0.654575\pi$
$673$	−24.2170	−0.933497	−0.466749	+	0.884390i	$0.654575\pi$
$674$	0	0
$675$	13.6649	0.525960
$676$	0	0
$677$	27.2605	1.04771	0.523854	−	0.851808i	$-0.324494\pi$
$677$	27.2605	1.04771	0.523854	+	0.851808i	$0.324494\pi$
$678$	0	0
$679$	0.0126203	0.000484321 0
$680$	0	0
$681$	−64.5188	−2.47237
$682$	0	0
$683$	18.0331	0.690018	0.345009	−	0.938599i	$-0.387876\pi$
$683$	18.0331	0.690018	0.345009	+	0.938599i	$0.387876\pi$
$684$	0	0
$685$	11.1360	0.425484
$686$	0	0
$687$	23.0874	0.880841
$688$	0	0
$689$	32.1070	1.22318
$690$	0	0
$691$	−38.6322	−1.46964	−0.734820	−	0.678262i	$-0.762733\pi$
$691$	−38.6322	−1.46964	−0.734820	+	0.678262i	$0.762733\pi$
$692$	0	0
$693$	1.27393	0.0483927
$694$	0	0
$695$	−11.0327	−0.418495
$696$	0	0
$697$	−11.8581	−0.449159
$698$	0	0
$699$	42.6620	1.61363
$700$	0	0
$701$	34.9562	1.32028	0.660138	−	0.751144i	$-0.270498\pi$
$701$	34.9562	1.32028	0.660138	+	0.751144i	$0.270498\pi$
$702$	0	0
$703$	11.1765	0.421530
$704$	0	0
$705$	−16.1733	−0.609120
$706$	0	0
$707$	1.04256	0.0392095
$708$	0	0
$709$	24.8712	0.934056	0.467028	−	0.884242i	$-0.345325\pi$
$709$	24.8712	0.934056	0.467028	+	0.884242i	$0.345325\pi$
$710$	0	0
$711$	4.53986	0.170258
$712$	0	0
$713$	8.16229	0.305680
$714$	0	0
$715$	−4.23393	−0.158340
$716$	0	0
$717$	−12.6191	−0.471267
$718$	0	0
$719$	−28.5465	−1.06460	−0.532302	−	0.846554i	$-0.678673\pi$
$719$	−28.5465	−1.06460	−0.532302	+	0.846554i	$0.678673\pi$
$720$	0	0
$721$	−1.14706	−0.0427189
$722$	0	0
$723$	−28.8055	−1.07129
$724$	0	0
$725$	−16.9478	−0.629425
$726$	0	0
$727$	−4.20507	−0.155957	−0.0779787	−	0.996955i	$-0.524847\pi$
$727$	−4.20507	−0.155957	−0.0779787	+	0.996955i	$0.524847\pi$
$728$	0	0
$729$	−42.9881	−1.59215
$730$	0	0
$731$	−6.31008	−0.233387
$732$	0	0
$733$	20.4058	0.753704	0.376852	−	0.926274i	$-0.377007\pi$
$733$	20.4058	0.753704	0.376852	+	0.926274i	$0.377007\pi$
$734$	0	0
$735$	−18.8277	−0.694471
$736$	0	0
$737$	16.3616	0.602686
$738$	0	0
$739$	−30.0671	−1.10604	−0.553018	−	0.833169i	$-0.686524\pi$
$739$	−30.0671	−1.10604	−0.553018	+	0.833169i	$0.686524\pi$
$740$	0	0
$741$	67.7697	2.48958
$742$	0	0
$743$	34.1188	1.25170	0.625849	−	0.779944i	$-0.284752\pi$
$743$	34.1188	1.25170	0.625849	+	0.779944i	$0.284752\pi$
$744$	0	0
$745$	7.81752	0.286412
$746$	0	0
$747$	−62.8100	−2.29810
$748$	0	0
$749$	−1.41088	−0.0515524
$750$	0	0
$751$	−35.3085	−1.28843	−0.644214	−	0.764846i	$-0.722815\pi$
$751$	−35.3085	−1.28843	−0.644214	+	0.764846i	$0.722815\pi$
$752$	0	0
$753$	2.69680	0.0982770
$754$	0	0
$755$	5.31622	0.193477
$756$	0	0
$757$	−45.6466	−1.65906	−0.829528	−	0.558466i	$-0.811390\pi$
$757$	−45.6466	−1.65906	−0.829528	+	0.558466i	$0.811390\pi$
$758$	0	0
$759$	12.5034	0.453845
$760$	0	0
$761$	12.1422	0.440156	0.220078	−	0.975482i	$-0.429369\pi$
$761$	12.1422	0.440156	0.220078	+	0.975482i	$0.429369\pi$
$762$	0	0
$763$	−5.61396	−0.203239
$764$	0	0
$765$	5.52511	0.199761
$766$	0	0
$767$	53.9353	1.94749
$768$	0	0
$769$	−9.24418	−0.333354	−0.166677	−	0.986012i	$-0.553304\pi$
$769$	−9.24418	−0.333354	−0.166677	+	0.986012i	$0.553304\pi$
$770$	0	0
$771$	−37.2390	−1.34113
$772$	0	0
$773$	29.6524	1.06652	0.533261	−	0.845951i	$-0.320967\pi$
$773$	29.6524	1.06652	0.533261	+	0.845951i	$0.320967\pi$
$774$	0	0
$775$	7.23819	0.260003
$776$	0	0
$777$	1.40645	0.0504563
$778$	0	0
$779$	−57.2688	−2.05187
$780$	0	0
$781$	−12.8782	−0.460818
$782$	0	0
$783$	−14.6116	−0.522176
$784$	0	0
$785$	14.5104	0.517898
$786$	0	0
$787$	−13.3362	−0.475383	−0.237692	−	0.971341i	$-0.576391\pi$
$787$	−13.3362	−0.475383	−0.237692	+	0.971341i	$0.576391\pi$
$788$	0	0
$789$	−37.2299	−1.32542
$790$	0	0
$791$	2.34820	0.0834923
$792$	0	0
$793$	43.7792	1.55465
$794$	0	0
$795$	21.5177	0.763156
$796$	0	0
$797$	−7.88865	−0.279430	−0.139715	−	0.990192i	$-0.544619\pi$
$797$	−7.88865	−0.279430	−0.139715	+	0.990192i	$0.544619\pi$
$798$	0	0
$799$	7.61160	0.269279
$800$	0	0
$801$	−22.9430	−0.810650
$802$	0	0
$803$	7.72110	0.272472
$804$	0	0
$805$	1.30827	0.0461103
$806$	0	0
$807$	−2.65449	−0.0934424
$808$	0	0
$809$	50.9335	1.79073	0.895364	−	0.445336i	$-0.146916\pi$
$809$	50.9335	1.79073	0.895364	+	0.445336i	$0.146916\pi$
$810$	0	0
$811$	36.9831	1.29865	0.649327	−	0.760510i	$-0.275051\pi$
$811$	36.9831	1.29865	0.649327	+	0.760510i	$0.275051\pi$
$812$	0	0
$813$	69.1626	2.42564
$814$	0	0
$815$	12.3339	0.432036
$816$	0	0
$817$	−30.4744	−1.06617
$818$	0	0
$819$	5.01032	0.175075
$820$	0	0
$821$	18.5449	0.647222	0.323611	−	0.946190i	$-0.395103\pi$
$821$	18.5449	0.647222	0.323611	+	0.946190i	$0.395103\pi$
$822$	0	0
$823$	39.8020	1.38741	0.693705	−	0.720260i	$-0.255977\pi$
$823$	39.8020	1.38741	0.693705	+	0.720260i	$0.255977\pi$
$824$	0	0
$825$	11.0878	0.386028
$826$	0	0
$827$	19.0765	0.663356	0.331678	−	0.943393i	$-0.392385\pi$
$827$	19.0765	0.663356	0.331678	+	0.943393i	$0.392385\pi$
$828$	0	0
$829$	−19.8074	−0.687940	−0.343970	−	0.938981i	$-0.611772\pi$
$829$	−19.8074	−0.687940	−0.343970	+	0.938981i	$0.611772\pi$
$830$	0	0
$831$	7.20414	0.249909
$832$	0	0
$833$	8.86086	0.307011
$834$	0	0
$835$	−11.7549	−0.406794
$836$	0	0
$837$	6.24043	0.215701
$838$	0	0
$839$	38.9883	1.34603	0.673014	−	0.739630i	$-0.265000\pi$
$839$	38.9883	1.34603	0.673014	+	0.739630i	$0.265000\pi$
$840$	0	0
$841$	−10.8780	−0.375104
$842$	0	0
$843$	19.4219	0.668927
$844$	0	0
$845$	−3.52999	−0.121435
$846$	0	0
$847$	−2.86783	−0.0985396
$848$	0	0
$849$	45.0412	1.54581
$850$	0	0
$851$	8.10992	0.278005
$852$	0	0
$853$	2.35100	0.0804967	0.0402483	−	0.999190i	$-0.487185\pi$
$853$	2.35100	0.0804967	0.0402483	+	0.999190i	$0.487185\pi$
$854$	0	0
$855$	26.6834	0.912554
$856$	0	0
$857$	12.3305	0.421202	0.210601	−	0.977572i	$-0.432458\pi$
$857$	12.3305	0.421202	0.210601	+	0.977572i	$0.432458\pi$
$858$	0	0
$859$	−1.32101	−0.0450722	−0.0225361	−	0.999746i	$-0.507174\pi$
$859$	−1.32101	−0.0450722	−0.0225361	+	0.999746i	$0.507174\pi$
$860$	0	0
$861$	−7.20672	−0.245604
$862$	0	0
$863$	−54.0590	−1.84019	−0.920095	−	0.391695i	$-0.871889\pi$
$863$	−54.0590	−1.84019	−0.920095	+	0.391695i	$0.871889\pi$
$864$	0	0
$865$	1.13434	0.0385686
$866$	0	0
$867$	41.4197	1.40669
$868$	0	0
$869$	1.09729	0.0372230
$870$	0	0
$871$	64.3493	2.18039
$872$	0	0
$873$	0.186778	0.00632146
$874$	0	0
$875$	2.61720	0.0884774
$876$	0	0
$877$	12.6983	0.428792	0.214396	−	0.976747i	$-0.431222\pi$
$877$	12.6983	0.428792	0.214396	+	0.976747i	$0.431222\pi$
$878$	0	0
$879$	−16.2287	−0.547379
$880$	0	0
$881$	8.68845	0.292721	0.146361	−	0.989231i	$-0.453244\pi$
$881$	8.68845	0.292721	0.146361	+	0.989231i	$0.453244\pi$
$882$	0	0
$883$	7.20965	0.242624	0.121312	−	0.992614i	$-0.461290\pi$
$883$	7.20965	0.242624	0.121312	+	0.992614i	$0.461290\pi$
$884$	0	0
$885$	36.1468	1.21506
$886$	0	0
$887$	−3.47682	−0.116740	−0.0583700	−	0.998295i	$-0.518590\pi$
$887$	−3.47682	−0.116740	−0.0583700	+	0.998295i	$0.518590\pi$
$888$	0	0
$889$	1.86131	0.0624264
$890$	0	0
$891$	−3.67840	−0.123231
$892$	0	0
$893$	36.7601	1.23013
$894$	0	0
$895$	3.34006	0.111646
$896$	0	0
$897$	49.1753	1.64192
$898$	0	0
$899$	−7.73968	−0.258133
$900$	0	0
$901$	−10.1269	−0.337375
$902$	0	0
$903$	−3.83491	−0.127618
$904$	0	0
$905$	22.2187	0.738575
$906$	0	0
$907$	−52.4021	−1.73998	−0.869991	−	0.493067i	$-0.835876\pi$
$907$	−52.4021	−1.73998	−0.869991	+	0.493067i	$0.835876\pi$
$908$	0	0
$909$	15.4297	0.511771
$910$	0	0
$911$	−39.4971	−1.30860	−0.654299	−	0.756236i	$-0.727036\pi$
$911$	−39.4971	−1.30860	−0.654299	+	0.756236i	$0.727036\pi$
$912$	0	0
$913$	−15.1812	−0.502426
$914$	0	0
$915$	29.3403	0.969961
$916$	0	0
$917$	−3.59130	−0.118595
$918$	0	0
$919$	−9.18486	−0.302981	−0.151490	−	0.988459i	$-0.548407\pi$
$919$	−9.18486	−0.302981	−0.151490	+	0.988459i	$0.548407\pi$
$920$	0	0
$921$	−34.1701	−1.12594
$922$	0	0
$923$	−50.6493	−1.66714
$924$	0	0
$925$	7.19175	0.236463
$926$	0	0
$927$	−16.9763	−0.557576
$928$	0	0
$929$	−38.1618	−1.25205	−0.626024	−	0.779804i	$-0.715319\pi$
$929$	−38.1618	−1.25205	−0.626024	+	0.779804i	$0.715319\pi$
$930$	0	0
$931$	42.7934	1.40250
$932$	0	0
$933$	−13.3047	−0.435578
$934$	0	0
$935$	1.33542	0.0436730
$936$	0	0
$937$	−16.9986	−0.555319	−0.277660	−	0.960679i	$-0.589559\pi$
$937$	−16.9986	−0.555319	−0.277660	+	0.960679i	$0.589559\pi$
$938$	0	0
$939$	−37.5372	−1.22498
$940$	0	0
$941$	4.04432	0.131841	0.0659205	−	0.997825i	$-0.479002\pi$
$941$	4.04432	0.131841	0.0659205	+	0.997825i	$0.479002\pi$
$942$	0	0
$943$	−41.5555	−1.35323
$944$	0	0
$945$	1.00023	0.0325374
$946$	0	0
$947$	−25.0183	−0.812985	−0.406492	−	0.913654i	$-0.633248\pi$
$947$	−25.0183	−0.812985	−0.406492	+	0.913654i	$0.633248\pi$
$948$	0	0
$949$	30.3667	0.985745
$950$	0	0
$951$	−74.9278	−2.42970
$952$	0	0
$953$	−16.1448	−0.522980	−0.261490	−	0.965206i	$-0.584214\pi$
$953$	−16.1448	−0.522980	−0.261490	+	0.965206i	$0.584214\pi$
$954$	0	0
$955$	−20.9235	−0.677067
$956$	0	0
$957$	−11.8560	−0.383251
$958$	0	0
$959$	−3.18514	−0.102853
$960$	0	0
$961$	−27.6945	−0.893370
$962$	0	0
$963$	−20.8808	−0.672873
$964$	0	0
$965$	16.5850	0.533890
$966$	0	0
$967$	13.6494	0.438935	0.219468	−	0.975620i	$-0.429568\pi$
$967$	13.6494	0.438935	0.219468	+	0.975620i	$0.429568\pi$
$968$	0	0
$969$	−21.3752	−0.686671
$970$	0	0
$971$	2.65596	0.0852337	0.0426169	−	0.999091i	$-0.486431\pi$
$971$	2.65596	0.0852337	0.0426169	+	0.999091i	$0.486431\pi$
$972$	0	0
$973$	3.15560	0.101164
$974$	0	0
$975$	43.6079	1.39657
$976$	0	0
$977$	11.9401	0.381998	0.190999	−	0.981590i	$-0.438827\pi$
$977$	11.9401	0.381998	0.190999	+	0.981590i	$0.438827\pi$
$978$	0	0
$979$	−5.54534	−0.177230
$980$	0	0
$981$	−83.0856	−2.65272
$982$	0	0
$983$	6.33228	0.201968	0.100984	−	0.994888i	$-0.467801\pi$
$983$	6.33228	0.201968	0.100984	+	0.994888i	$0.467801\pi$
$984$	0	0
$985$	15.2573	0.486140
$986$	0	0
$987$	4.62591	0.147244
$988$	0	0
$989$	−22.1130	−0.703151
$990$	0	0
$991$	32.7733	1.04108	0.520539	−	0.853838i	$-0.325731\pi$
$991$	32.7733	1.04108	0.520539	+	0.853838i	$0.325731\pi$
$992$	0	0
$993$	30.7896	0.977078
$994$	0	0
$995$	2.65133	0.0840527
$996$	0	0
$997$	56.5223	1.79008	0.895039	−	0.445987i	$-0.147147\pi$
$997$	56.5223	1.79008	0.895039	+	0.445987i	$0.147147\pi$
$998$	0	0
$999$	6.20040	0.196172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.i.1.2		12
4.3	odd	2		2008.2.a.b.1.11	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.b.1.11	✓	12	4.3	odd	2
4016.2.a.i.1.2		12	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-2.69680 q^{3} -1.00937 q^{5} +0.288703 q^{7} +4.27276 q^{9} +O(q^{10})\)	\(q-2.69680 q^{3} -1.00937 q^{5} +0.288703 q^{7} +4.27276 q^{9} +1.03273 q^{11} +4.06167 q^{13} +2.72209 q^{15} -1.28109 q^{17} -6.18702 q^{19} -0.778576 q^{21} -4.48944 q^{23} -3.98116 q^{25} -3.43238 q^{27} +4.25699 q^{29} -1.81811 q^{31} -2.78507 q^{33} -0.291410 q^{35} -1.80644 q^{37} -10.9535 q^{39} +9.25628 q^{41} +4.92555 q^{43} -4.31281 q^{45} -5.94149 q^{47} -6.91665 q^{49} +3.45485 q^{51} +7.90488 q^{53} -1.04241 q^{55} +16.6852 q^{57} +13.2791 q^{59} +10.7786 q^{61} +1.23356 q^{63} -4.09975 q^{65} +15.8430 q^{67} +12.1071 q^{69} -12.4700 q^{71} +7.47640 q^{73} +10.7364 q^{75} +0.298153 q^{77} +1.06251 q^{79} -3.56182 q^{81} -14.7001 q^{83} +1.29310 q^{85} -11.4803 q^{87} -5.36960 q^{89} +1.17262 q^{91} +4.90308 q^{93} +6.24501 q^{95} +0.0437136 q^{97} +4.41260 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9}+O(q^{10}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9	\( 12 q - 3 q^{3} + 5 q^{5} - 5 q^{7} + 7 q^{9} - 10 q^{11} + 3 q^{13} - 11 q^{15} + 2 q^{17} - 15 q^{19} + 3 q^{21} - 20 q^{23} - 3 q^{25} - 15 q^{27} + 6 q^{29} - 14 q^{31} - 6 q^{33} - 16 q^{35} + 5 q^{37} - 21 q^{39} - 21 q^{43} + 10 q^{45} - 27 q^{47} - 13 q^{49} - 19 q^{51} + 22 q^{53} - 24 q^{55} + q^{57} - 23 q^{59} + 4 q^{61} - 21 q^{63} - q^{65} - 26 q^{67} + 10 q^{69} - 23 q^{71} - 8 q^{73} - 16 q^{75} + 22 q^{77} - 37 q^{79} - 20 q^{81} - 30 q^{83} + 2 q^{85} - 16 q^{87} + 3 q^{89} - 8 q^{91} + 20 q^{93} - 33 q^{95} - 4 q^{97} - 15 q^{99}+O(q^{100}) \) 12 * q - 3 * q^3 + 5 * q^5 - 5 * q^7 + 7 * q^9 - 10 * q^11 + 3 * q^13 - 11 * q^15 + 2 * q^17 - 15 * q^19 + 3 * q^21 - 20 * q^23 - 3 * q^25 - 15 * q^27 + 6 * q^29 - 14 * q^31 - 6 * q^33 - 16 * q^35 + 5 * q^37 - 21 * q^39 - 21 * q^43 + 10 * q^45 - 27 * q^47 - 13 * q^49 - 19 * q^51 + 22 * q^53 - 24 * q^55 + q^57 - 23 * q^59 + 4 * q^61 - 21 * q^63 - q^65 - 26 * q^67 + 10 * q^69 - 23 * q^71 - 8 * q^73 - 16 * q^75 + 22 * q^77 - 37 * q^79 - 20 * q^81 - 30 * q^83 + 2 * q^85 - 16 * q^87 + 3 * q^89 - 8 * q^91 + 20 * q^93 - 33 * q^95 - 4 * q^97 - 15 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more