LMFDB - Embedded newform 4016.2.a.h.1.1

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 11x^{7} + 7x^{6} + 40x^{5} - 11x^{4} - 53x^{3} - 2x^{2} + 13x + 1 $ x^9 - x^8 - 11x^7 + 7x^6 + 40x^5 - 11x^4 - 53x^3 - 2x^2 + 13*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.1
Root			$-2.17295$ 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)

$f(q)$	$=$	$q-2.17295 q^{3} -1.48785 q^{5} +1.36809 q^{7} +1.72170 q^{9} +O(q^{10})$	$q-2.17295 q^{3} -1.48785 q^{5} +1.36809 q^{7} +1.72170 q^{9} +3.99440 q^{11} +0.869492 q^{13} +3.23303 q^{15} -3.28029 q^{17} -2.89232 q^{19} -2.97279 q^{21} -4.05912 q^{23} -2.78629 q^{25} +2.77768 q^{27} -3.25588 q^{29} +2.60980 q^{31} -8.67963 q^{33} -2.03552 q^{35} +3.86704 q^{37} -1.88936 q^{39} -0.539329 q^{41} +11.6470 q^{43} -2.56164 q^{45} +5.24207 q^{47} -5.12833 q^{49} +7.12791 q^{51} -1.98148 q^{53} -5.94309 q^{55} +6.28487 q^{57} +6.43719 q^{59} -7.60329 q^{61} +2.35544 q^{63} -1.29368 q^{65} -12.0530 q^{67} +8.82026 q^{69} +11.8919 q^{71} +6.22464 q^{73} +6.05446 q^{75} +5.46470 q^{77} -4.06674 q^{79} -11.2008 q^{81} +13.3267 q^{83} +4.88060 q^{85} +7.07485 q^{87} +4.72315 q^{89} +1.18954 q^{91} -5.67096 q^{93} +4.30335 q^{95} -9.96261 q^{97} +6.87717 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9	$ 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) $ 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * 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
$3$	−2.17295	−1.25455	−0.627276	−	0.778797i	$-0.715830\pi$
$3$	−2.17295	−1.25455	−0.627276	+	0.778797i	$0.715830\pi$
$4$	0	0
$5$	−1.48785	−0.665388	−0.332694	−	0.943035i	$-0.607958\pi$
$5$	−1.48785	−0.665388	−0.332694	+	0.943035i	$0.607958\pi$
$6$	0	0
$7$	1.36809	0.517089	0.258545	−	0.965999i	$-0.416757\pi$
$7$	1.36809	0.517089	0.258545	+	0.965999i	$0.416757\pi$
$8$	0	0
$9$	1.72170	0.573901
$10$	0	0
$11$	3.99440	1.20436	0.602179	−	0.798361i	$-0.294299\pi$
$11$	3.99440	1.20436	0.602179	+	0.798361i	$0.294299\pi$
$12$	0	0
$13$	0.869492	0.241154	0.120577	−	0.992704i	$-0.461526\pi$
$13$	0.869492	0.241154	0.120577	+	0.992704i	$0.461526\pi$
$14$	0	0
$15$	3.23303	0.834764
$16$	0	0
$17$	−3.28029	−0.795588	−0.397794	−	0.917475i	$-0.630224\pi$
$17$	−3.28029	−0.795588	−0.397794	+	0.917475i	$0.630224\pi$
$18$	0	0
$19$	−2.89232	−0.663544	−0.331772	−	0.943360i	$-0.607647\pi$
$19$	−2.89232	−0.663544	−0.331772	+	0.943360i	$0.607647\pi$
$20$	0	0
$21$	−2.97279	−0.648715
$22$	0	0
$23$	−4.05912	−0.846385	−0.423193	−	0.906040i	$-0.639091\pi$
$23$	−4.05912	−0.846385	−0.423193	+	0.906040i	$0.639091\pi$
$24$	0	0
$25$	−2.78629	−0.557258
$26$	0	0
$27$	2.77768	0.534564
$28$	0	0
$29$	−3.25588	−0.604601	−0.302301	−	0.953213i	$-0.597755\pi$
$29$	−3.25588	−0.604601	−0.302301	+	0.953213i	$0.597755\pi$
$30$	0	0
$31$	2.60980	0.468734	0.234367	−	0.972148i	$-0.424698\pi$
$31$	2.60980	0.468734	0.234367	+	0.972148i	$0.424698\pi$
$32$	0	0
$33$	−8.67963	−1.51093
$34$	0	0
$35$	−2.03552	−0.344065
$36$	0	0
$37$	3.86704	0.635737	0.317869	−	0.948135i	$-0.397033\pi$
$37$	3.86704	0.635737	0.317869	+	0.948135i	$0.397033\pi$
$38$	0	0
$39$	−1.88936	−0.302540
$40$	0	0
$41$	−0.539329	−0.0842290	−0.0421145	−	0.999113i	$-0.513409\pi$
$41$	−0.539329	−0.0842290	−0.0421145	+	0.999113i	$0.513409\pi$
$42$	0	0
$43$	11.6470	1.77615	0.888076	−	0.459697i	$-0.152042\pi$
$43$	11.6470	1.77615	0.888076	+	0.459697i	$0.152042\pi$
$44$	0	0
$45$	−2.56164	−0.381867
$46$	0	0
$47$	5.24207	0.764635	0.382318	−	0.924031i	$-0.375126\pi$
$47$	5.24207	0.764635	0.382318	+	0.924031i	$0.375126\pi$
$48$	0	0
$49$	−5.12833	−0.732619
$50$	0	0
$51$	7.12791	0.998107
$52$	0	0
$53$	−1.98148	−0.272177	−0.136088	−	0.990697i	$-0.543453\pi$
$53$	−1.98148	−0.272177	−0.136088	+	0.990697i	$0.543453\pi$
$54$	0	0
$55$	−5.94309	−0.801365
$56$	0	0
$57$	6.28487	0.832451
$58$	0	0
$59$	6.43719	0.838051	0.419026	−	0.907974i	$-0.362372\pi$
$59$	6.43719	0.838051	0.419026	+	0.907974i	$0.362372\pi$
$60$	0	0
$61$	−7.60329	−0.973502	−0.486751	−	0.873541i	$-0.661818\pi$
$61$	−7.60329	−0.973502	−0.486751	+	0.873541i	$0.661818\pi$
$62$	0	0
$63$	2.35544	0.296758
$64$	0	0
$65$	−1.29368	−0.160461
$66$	0	0
$67$	−12.0530	−1.47250	−0.736252	−	0.676707i	$-0.763406\pi$
$67$	−12.0530	−1.47250	−0.736252	+	0.676707i	$0.763406\pi$
$68$	0	0
$69$	8.82026	1.06183
$70$	0	0
$71$	11.8919	1.41131	0.705654	−	0.708556i	$-0.250653\pi$
$71$	11.8919	1.41131	0.705654	+	0.708556i	$0.250653\pi$
$72$	0	0
$73$	6.22464	0.728539	0.364269	−	0.931294i	$-0.381319\pi$
$73$	6.22464	0.728539	0.364269	+	0.931294i	$0.381319\pi$
$74$	0	0
$75$	6.05446	0.699109
$76$	0	0
$77$	5.46470	0.622760
$78$	0	0
$79$	−4.06674	−0.457544	−0.228772	−	0.973480i	$-0.573471\pi$
$79$	−4.06674	−0.457544	−0.228772	+	0.973480i	$0.573471\pi$
$80$	0	0
$81$	−11.2008	−1.24454
$82$	0	0
$83$	13.3267	1.46280	0.731399	−	0.681950i	$-0.238868\pi$
$83$	13.3267	1.46280	0.731399	+	0.681950i	$0.238868\pi$
$84$	0	0
$85$	4.88060	0.529375
$86$	0	0
$87$	7.07485	0.758504
$88$	0	0
$89$	4.72315	0.500653	0.250327	−	0.968161i	$-0.419462\pi$
$89$	4.72315	0.500653	0.250327	+	0.968161i	$0.419462\pi$
$90$	0	0
$91$	1.18954	0.124698
$92$	0	0
$93$	−5.67096	−0.588051
$94$	0	0
$95$	4.30335	0.441515
$96$	0	0
$97$	−9.96261	−1.01155	−0.505775	−	0.862666i	$-0.668793\pi$
$97$	−9.96261	−1.01155	−0.505775	+	0.862666i	$0.668793\pi$
$98$	0	0
$99$	6.87717	0.691181
$100$	0	0
$101$	13.8741	1.38052	0.690262	−	0.723559i	$-0.257495\pi$
$101$	13.8741	1.38052	0.690262	+	0.723559i	$0.257495\pi$
$102$	0	0
$103$	−10.7371	−1.05796	−0.528978	−	0.848635i	$-0.677425\pi$
$103$	−10.7371	−1.05796	−0.528978	+	0.848635i	$0.677425\pi$
$104$	0	0
$105$	4.42307	0.431648
$106$	0	0
$107$	−1.34430	−0.129958	−0.0649791	−	0.997887i	$-0.520698\pi$
$107$	−1.34430	−0.129958	−0.0649791	+	0.997887i	$0.520698\pi$
$108$	0	0
$109$	2.15100	0.206029	0.103014	−	0.994680i	$-0.467151\pi$
$109$	2.15100	0.206029	0.103014	+	0.994680i	$0.467151\pi$
$110$	0	0
$111$	−8.40287	−0.797565
$112$	0	0
$113$	−5.55916	−0.522962	−0.261481	−	0.965209i	$-0.584211\pi$
$113$	−5.55916	−0.522962	−0.261481	+	0.965209i	$0.584211\pi$
$114$	0	0
$115$	6.03938	0.563175
$116$	0	0
$117$	1.49701	0.138398
$118$	0	0
$119$	−4.48774	−0.411390
$120$	0	0
$121$	4.95524	0.450477
$122$	0	0
$123$	1.17193	0.105670
$124$	0	0
$125$	11.5849	1.03618
$126$	0	0
$127$	−10.4773	−0.929707	−0.464853	−	0.885388i	$-0.653893\pi$
$127$	−10.4773	−0.929707	−0.464853	+	0.885388i	$0.653893\pi$
$128$	0	0
$129$	−25.3083	−2.22827
$130$	0	0
$131$	3.44018	0.300570	0.150285	−	0.988643i	$-0.451981\pi$
$131$	3.44018	0.300570	0.150285	+	0.988643i	$0.451981\pi$
$132$	0	0
$133$	−3.95696	−0.343112
$134$	0	0
$135$	−4.13278	−0.355693
$136$	0	0
$137$	−11.8181	−1.00969	−0.504843	−	0.863211i	$-0.668450\pi$
$137$	−11.8181	−1.00969	−0.504843	+	0.863211i	$0.668450\pi$
$138$	0	0
$139$	−14.7062	−1.24736	−0.623680	−	0.781679i	$-0.714363\pi$
$139$	−14.7062	−1.24736	−0.623680	+	0.781679i	$0.714363\pi$
$140$	0	0
$141$	−11.3908	−0.959274
$142$	0	0
$143$	3.47310	0.290435
$144$	0	0
$145$	4.84427	0.402295
$146$	0	0
$147$	11.1436	0.919108
$148$	0	0
$149$	−16.5423	−1.35520	−0.677598	−	0.735433i	$-0.736979\pi$
$149$	−16.5423	−1.35520	−0.677598	+	0.735433i	$0.736979\pi$
$150$	0	0
$151$	−15.1717	−1.23466	−0.617328	−	0.786706i	$-0.711785\pi$
$151$	−15.1717	−1.23466	−0.617328	+	0.786706i	$0.711785\pi$
$152$	0	0
$153$	−5.64769	−0.456589
$154$	0	0
$155$	−3.88300	−0.311890
$156$	0	0
$157$	11.4849	0.916598	0.458299	−	0.888798i	$-0.348459\pi$
$157$	11.4849	0.916598	0.458299	+	0.888798i	$0.348459\pi$
$158$	0	0
$159$	4.30564	0.341460
$160$	0	0
$161$	−5.55324	−0.437657
$162$	0	0
$163$	−8.91778	−0.698495	−0.349247	−	0.937031i	$-0.613563\pi$
$163$	−8.91778	−0.698495	−0.349247	+	0.937031i	$0.613563\pi$
$164$	0	0
$165$	12.9140	1.00535
$166$	0	0
$167$	−15.5042	−1.19975	−0.599877	−	0.800092i	$-0.704784\pi$
$167$	−15.5042	−1.19975	−0.599877	+	0.800092i	$0.704784\pi$
$168$	0	0
$169$	−12.2440	−0.941845
$170$	0	0
$171$	−4.97972	−0.380808
$172$	0	0
$173$	−14.9805	−1.13895	−0.569473	−	0.822010i	$-0.692853\pi$
$173$	−14.9805	−1.13895	−0.569473	+	0.822010i	$0.692853\pi$
$174$	0	0
$175$	−3.81190	−0.288152
$176$	0	0
$177$	−13.9877	−1.05138
$178$	0	0
$179$	−19.1858	−1.43401	−0.717007	−	0.697066i	$-0.754488\pi$
$179$	−19.1858	−1.43401	−0.717007	+	0.697066i	$0.754488\pi$
$180$	0	0
$181$	5.17553	0.384694	0.192347	−	0.981327i	$-0.438390\pi$
$181$	5.17553	0.384694	0.192347	+	0.981327i	$0.438390\pi$
$182$	0	0
$183$	16.5216	1.22131
$184$	0	0
$185$	−5.75359	−0.423012
$186$	0	0
$187$	−13.1028	−0.958173
$188$	0	0
$189$	3.80011	0.276417
$190$	0	0
$191$	−17.0995	−1.23728	−0.618639	−	0.785676i	$-0.712315\pi$
$191$	−17.0995	−1.23728	−0.618639	+	0.785676i	$0.712315\pi$
$192$	0	0
$193$	−2.47095	−0.177863	−0.0889313	−	0.996038i	$-0.528345\pi$
$193$	−2.47095	−0.177863	−0.0889313	+	0.996038i	$0.528345\pi$
$194$	0	0
$195$	2.81109	0.201307
$196$	0	0
$197$	18.3286	1.30586	0.652928	−	0.757420i	$-0.273540\pi$
$197$	18.3286	1.30586	0.652928	+	0.757420i	$0.273540\pi$
$198$	0	0
$199$	−8.60184	−0.609769	−0.304884	−	0.952389i	$-0.598618\pi$
$199$	−8.60184	−0.609769	−0.304884	+	0.952389i	$0.598618\pi$
$200$	0	0
$201$	26.1905	1.84733
$202$	0	0
$203$	−4.45433	−0.312633
$204$	0	0
$205$	0.802442	0.0560450
$206$	0	0
$207$	−6.98859	−0.485741
$208$	0	0
$209$	−11.5531	−0.799145
$210$	0	0
$211$	5.90564	0.406561	0.203280	−	0.979121i	$-0.434840\pi$
$211$	5.90564	0.406561	0.203280	+	0.979121i	$0.434840\pi$
$212$	0	0
$213$	−25.8405	−1.77056
$214$	0	0
$215$	−17.3290	−1.18183
$216$	0	0
$217$	3.57044	0.242377
$218$	0	0
$219$	−13.5258	−0.913990
$220$	0	0
$221$	−2.85219	−0.191859
$222$	0	0
$223$	4.67384	0.312983	0.156492	−	0.987679i	$-0.449982\pi$
$223$	4.67384	0.312983	0.156492	+	0.987679i	$0.449982\pi$
$224$	0	0
$225$	−4.79716	−0.319811
$226$	0	0
$227$	−13.2731	−0.880966	−0.440483	−	0.897761i	$-0.645193\pi$
$227$	−13.2731	−0.880966	−0.440483	+	0.897761i	$0.645193\pi$
$228$	0	0
$229$	0.352051	0.0232642	0.0116321	−	0.999932i	$-0.496297\pi$
$229$	0.352051	0.0232642	0.0116321	+	0.999932i	$0.496297\pi$
$230$	0	0
$231$	−11.8745	−0.781285
$232$	0	0
$233$	4.85831	0.318278	0.159139	−	0.987256i	$-0.449128\pi$
$233$	4.85831	0.318278	0.159139	+	0.987256i	$0.449128\pi$
$234$	0	0
$235$	−7.79944	−0.508779
$236$	0	0
$237$	8.83681	0.574012
$238$	0	0
$239$	10.3873	0.671897	0.335948	−	0.941880i	$-0.390943\pi$
$239$	10.3873	0.671897	0.335948	+	0.941880i	$0.390943\pi$
$240$	0	0
$241$	−5.92995	−0.381982	−0.190991	−	0.981592i	$-0.561170\pi$
$241$	−5.92995	−0.381982	−0.190991	+	0.981592i	$0.561170\pi$
$242$	0	0
$243$	16.0058	1.02677
$244$	0	0
$245$	7.63021	0.487476
$246$	0	0
$247$	−2.51485	−0.160016
$248$	0	0
$249$	−28.9583	−1.83516
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−16.2138	−1.01935
$254$	0	0
$255$	−10.6053	−0.664129
$256$	0	0
$257$	10.3490	0.645552	0.322776	−	0.946475i	$-0.395384\pi$
$257$	10.3490	0.645552	0.322776	+	0.946475i	$0.395384\pi$
$258$	0	0
$259$	5.29046	0.328733
$260$	0	0
$261$	−5.60565	−0.346981
$262$	0	0
$263$	−9.48709	−0.584999	−0.292500	−	0.956266i	$-0.594487\pi$
$263$	−9.48709	−0.584999	−0.292500	+	0.956266i	$0.594487\pi$
$264$	0	0
$265$	2.94815	0.181103
$266$	0	0
$267$	−10.2632	−0.628095
$268$	0	0
$269$	−15.9439	−0.972119	−0.486060	−	0.873926i	$-0.661566\pi$
$269$	−15.9439	−0.972119	−0.486060	+	0.873926i	$0.661566\pi$
$270$	0	0
$271$	10.2563	0.623026	0.311513	−	0.950242i	$-0.399164\pi$
$271$	10.2563	0.623026	0.311513	+	0.950242i	$0.399164\pi$
$272$	0	0
$273$	−2.58482	−0.156440
$274$	0	0
$275$	−11.1296	−0.671138
$276$	0	0
$277$	−16.3349	−0.981469	−0.490734	−	0.871309i	$-0.663271\pi$
$277$	−16.3349	−0.981469	−0.490734	+	0.871309i	$0.663271\pi$
$278$	0	0
$279$	4.49330	0.269007
$280$	0	0
$281$	−23.8487	−1.42269	−0.711347	−	0.702841i	$-0.751914\pi$
$281$	−23.8487	−1.42269	−0.711347	+	0.702841i	$0.751914\pi$
$282$	0	0
$283$	25.1280	1.49371	0.746853	−	0.664989i	$-0.231564\pi$
$283$	25.1280	1.49371	0.746853	+	0.664989i	$0.231564\pi$
$284$	0	0
$285$	−9.35096	−0.553903
$286$	0	0
$287$	−0.737850	−0.0435539
$288$	0	0
$289$	−6.23967	−0.367039
$290$	0	0
$291$	21.6482	1.26904
$292$	0	0
$293$	4.11048	0.240137	0.120068	−	0.992766i	$-0.461689\pi$
$293$	4.11048	0.240137	0.120068	+	0.992766i	$0.461689\pi$
$294$	0	0
$295$	−9.57760	−0.557630
$296$	0	0
$297$	11.0952	0.643806
$298$	0	0
$299$	−3.52937	−0.204109
$300$	0	0
$301$	15.9341	0.918429
$302$	0	0
$303$	−30.1477	−1.73194
$304$	0	0
$305$	11.3126	0.647757
$306$	0	0
$307$	20.4410	1.16663	0.583315	−	0.812246i	$-0.301755\pi$
$307$	20.4410	1.16663	0.583315	+	0.812246i	$0.301755\pi$
$308$	0	0
$309$	23.3311	1.32726
$310$	0	0
$311$	5.27959	0.299378	0.149689	−	0.988733i	$-0.452173\pi$
$311$	5.27959	0.299378	0.149689	+	0.988733i	$0.452173\pi$
$312$	0	0
$313$	−8.81977	−0.498523	−0.249261	−	0.968436i	$-0.580188\pi$
$313$	−8.81977	−0.498523	−0.249261	+	0.968436i	$0.580188\pi$
$314$	0	0
$315$	−3.50455	−0.197459
$316$	0	0
$317$	1.56563	0.0879343	0.0439672	−	0.999033i	$-0.486000\pi$
$317$	1.56563	0.0879343	0.0439672	+	0.999033i	$0.486000\pi$
$318$	0	0
$319$	−13.0053	−0.728156
$320$	0	0
$321$	2.92109	0.163039
$322$	0	0
$323$	9.48767	0.527908
$324$	0	0
$325$	−2.42266	−0.134385
$326$	0	0
$327$	−4.67402	−0.258474
$328$	0	0
$329$	7.17163	0.395385
$330$	0	0
$331$	5.44477	0.299272	0.149636	−	0.988741i	$-0.452190\pi$
$331$	5.44477	0.299272	0.149636	+	0.988741i	$0.452190\pi$
$332$	0	0
$333$	6.65789	0.364850
$334$	0	0
$335$	17.9331	0.979788
$336$	0	0
$337$	24.7789	1.34979	0.674896	−	0.737913i	$-0.264188\pi$
$337$	24.7789	1.34979	0.674896	+	0.737913i	$0.264188\pi$
$338$	0	0
$339$	12.0798	0.656083
$340$	0	0
$341$	10.4246	0.564523
$342$	0	0
$343$	−16.5926	−0.895919
$344$	0	0
$345$	−13.1233	−0.706532
$346$	0	0
$347$	16.5531	0.888616	0.444308	−	0.895874i	$-0.353450\pi$
$347$	16.5531	0.888616	0.444308	+	0.895874i	$0.353450\pi$
$348$	0	0
$349$	−6.22486	−0.333209	−0.166604	−	0.986024i	$-0.553280\pi$
$349$	−6.22486	−0.333209	−0.166604	+	0.986024i	$0.553280\pi$
$350$	0	0
$351$	2.41517	0.128912
$352$	0	0
$353$	−23.0700	−1.22789	−0.613947	−	0.789348i	$-0.710419\pi$
$353$	−23.0700	−1.22789	−0.613947	+	0.789348i	$0.710419\pi$
$354$	0	0
$355$	−17.6934	−0.939068
$356$	0	0
$357$	9.75162	0.516110
$358$	0	0
$359$	12.7592	0.673405	0.336702	−	0.941611i	$-0.390688\pi$
$359$	12.7592	0.673405	0.336702	+	0.941611i	$0.390688\pi$
$360$	0	0
$361$	−10.6345	−0.559709
$362$	0	0
$363$	−10.7675	−0.565146
$364$	0	0
$365$	−9.26135	−0.484761
$366$	0	0
$367$	−17.9200	−0.935415	−0.467707	−	0.883883i	$-0.654920\pi$
$367$	−17.9200	−0.935415	−0.467707	+	0.883883i	$0.654920\pi$
$368$	0	0
$369$	−0.928563	−0.0483390
$370$	0	0
$371$	−2.71084	−0.140740
$372$	0	0
$373$	6.74891	0.349446	0.174723	−	0.984618i	$-0.444097\pi$
$373$	6.74891	0.349446	0.174723	+	0.984618i	$0.444097\pi$
$374$	0	0
$375$	−25.1733	−1.29994
$376$	0	0
$377$	−2.83096	−0.145802
$378$	0	0
$379$	−13.3945	−0.688028	−0.344014	−	0.938964i	$-0.611787\pi$
$379$	−13.3945	−0.688028	−0.344014	+	0.938964i	$0.611787\pi$
$380$	0	0
$381$	22.7665	1.16637
$382$	0	0
$383$	33.3707	1.70516	0.852582	−	0.522593i	$-0.175035\pi$
$383$	33.3707	1.70516	0.852582	+	0.522593i	$0.175035\pi$
$384$	0	0
$385$	−8.13067	−0.414378
$386$	0	0
$387$	20.0527	1.01933
$388$	0	0
$389$	1.32748	0.0673060	0.0336530	−	0.999434i	$-0.489286\pi$
$389$	1.32748	0.0673060	0.0336530	+	0.999434i	$0.489286\pi$
$390$	0	0
$391$	13.3151	0.673374
$392$	0	0
$393$	−7.47532	−0.377080
$394$	0	0
$395$	6.05071	0.304444
$396$	0	0
$397$	−30.1536	−1.51337	−0.756683	−	0.653781i	$-0.773182\pi$
$397$	−30.1536	−1.51337	−0.756683	+	0.653781i	$0.773182\pi$
$398$	0	0
$399$	8.59826	0.430451
$400$	0	0
$401$	−30.8260	−1.53938	−0.769689	−	0.638419i	$-0.779589\pi$
$401$	−30.8260	−1.53938	−0.769689	+	0.638419i	$0.779589\pi$
$402$	0	0
$403$	2.26920	0.113037
$404$	0	0
$405$	16.6652	0.828102
$406$	0	0
$407$	15.4465	0.765655
$408$	0	0
$409$	−3.26597	−0.161492	−0.0807458	−	0.996735i	$-0.525730\pi$
$409$	−3.26597	−0.161492	−0.0807458	+	0.996735i	$0.525730\pi$
$410$	0	0
$411$	25.6801	1.26670
$412$	0	0
$413$	8.80666	0.433347
$414$	0	0
$415$	−19.8282	−0.973329
$416$	0	0
$417$	31.9557	1.56488
$418$	0	0
$419$	−13.4942	−0.659237	−0.329618	−	0.944114i	$-0.606920\pi$
$419$	−13.4942	−0.659237	−0.329618	+	0.944114i	$0.606920\pi$
$420$	0	0
$421$	4.90129	0.238874	0.119437	−	0.992842i	$-0.461891\pi$
$421$	4.90129	0.238874	0.119437	+	0.992842i	$0.461891\pi$
$422$	0	0
$423$	9.02529	0.438824
$424$	0	0
$425$	9.13986	0.443348
$426$	0	0
$427$	−10.4020	−0.503387
$428$	0	0
$429$	−7.54687	−0.364366
$430$	0	0
$431$	−21.1623	−1.01935	−0.509677	−	0.860366i	$-0.670235\pi$
$431$	−21.1623	−1.01935	−0.509677	+	0.860366i	$0.670235\pi$
$432$	0	0
$433$	14.5982	0.701545	0.350772	−	0.936461i	$-0.385919\pi$
$433$	14.5982	0.701545	0.350772	+	0.936461i	$0.385919\pi$
$434$	0	0
$435$	−10.5263	−0.504700
$436$	0	0
$437$	11.7403	0.561614
$438$	0	0
$439$	−39.4531	−1.88299	−0.941496	−	0.337025i	$-0.890579\pi$
$439$	−39.4531	−1.88299	−0.941496	+	0.337025i	$0.890579\pi$
$440$	0	0
$441$	−8.82945	−0.420450
$442$	0	0
$443$	3.25608	0.154701	0.0773505	−	0.997004i	$-0.475354\pi$
$443$	3.25608	0.154701	0.0773505	+	0.997004i	$0.475354\pi$
$444$	0	0
$445$	−7.02736	−0.333129
$446$	0	0
$447$	35.9455	1.70016
$448$	0	0
$449$	−35.9251	−1.69541	−0.847706	−	0.530466i	$-0.822017\pi$
$449$	−35.9251	−1.69541	−0.847706	+	0.530466i	$0.822017\pi$
$450$	0	0
$451$	−2.15429	−0.101442
$452$	0	0
$453$	32.9673	1.54894
$454$	0	0
$455$	−1.76987	−0.0829726
$456$	0	0
$457$	−17.4753	−0.817459	−0.408730	−	0.912655i	$-0.634028\pi$
$457$	−17.4753	−0.817459	−0.408730	+	0.912655i	$0.634028\pi$
$458$	0	0
$459$	−9.11160	−0.425293
$460$	0	0
$461$	−12.0370	−0.560620	−0.280310	−	0.959910i	$-0.590437\pi$
$461$	−12.0370	−0.560620	−0.280310	+	0.959910i	$0.590437\pi$
$462$	0	0
$463$	−21.5140	−0.999841	−0.499921	−	0.866071i	$-0.666638\pi$
$463$	−21.5140	−0.999841	−0.499921	+	0.866071i	$0.666638\pi$
$464$	0	0
$465$	8.43756	0.391283
$466$	0	0
$467$	−42.0421	−1.94547	−0.972737	−	0.231909i	$-0.925503\pi$
$467$	−42.0421	−1.94547	−0.972737	+	0.231909i	$0.925503\pi$
$468$	0	0
$469$	−16.4895	−0.761417
$470$	0	0
$471$	−24.9562	−1.14992
$472$	0	0
$473$	46.5228	2.13912
$474$	0	0
$475$	8.05885	0.369766
$476$	0	0
$477$	−3.41151	−0.156202
$478$	0	0
$479$	3.64862	0.166710	0.0833550	−	0.996520i	$-0.473436\pi$
$479$	3.64862	0.166710	0.0833550	+	0.996520i	$0.473436\pi$
$480$	0	0
$481$	3.36236	0.153310
$482$	0	0
$483$	12.0669	0.549063
$484$	0	0
$485$	14.8229	0.673073
$486$	0	0
$487$	−4.58648	−0.207833	−0.103917	−	0.994586i	$-0.533138\pi$
$487$	−4.58648	−0.207833	−0.103917	+	0.994586i	$0.533138\pi$
$488$	0	0
$489$	19.3779	0.876298
$490$	0	0
$491$	−17.6943	−0.798534	−0.399267	−	0.916835i	$-0.630735\pi$
$491$	−17.6943	−0.798534	−0.399267	+	0.916835i	$0.630735\pi$
$492$	0	0
$493$	10.6802	0.481014
$494$	0	0
$495$	−10.2322	−0.459904
$496$	0	0
$497$	16.2692	0.729773
$498$	0	0
$499$	0.122607	0.00548865	0.00274433	−	0.999996i	$-0.499126\pi$
$499$	0.122607	0.00548865	0.00274433	+	0.999996i	$0.499126\pi$
$500$	0	0
$501$	33.6899	1.50515
$502$	0	0
$503$	36.9075	1.64562	0.822812	−	0.568314i	$-0.192404\pi$
$503$	36.9075	1.64562	0.822812	+	0.568314i	$0.192404\pi$
$504$	0	0
$505$	−20.6426	−0.918585
$506$	0	0
$507$	26.6055	1.18159
$508$	0	0
$509$	3.58446	0.158879	0.0794393	−	0.996840i	$-0.474687\pi$
$509$	3.58446	0.158879	0.0794393	+	0.996840i	$0.474687\pi$
$510$	0	0
$511$	8.51586	0.376720
$512$	0	0
$513$	−8.03394	−0.354707
$514$	0	0
$515$	15.9752	0.703952
$516$	0	0
$517$	20.9389	0.920894
$518$	0	0
$519$	32.5518	1.42887
$520$	0	0
$521$	−14.7008	−0.644055	−0.322027	−	0.946730i	$-0.604364\pi$
$521$	−14.7008	−0.644055	−0.322027	+	0.946730i	$0.604364\pi$
$522$	0	0
$523$	−27.0481	−1.18273	−0.591364	−	0.806404i	$-0.701411\pi$
$523$	−27.0481	−1.18273	−0.591364	+	0.806404i	$0.701411\pi$
$524$	0	0
$525$	8.28305	0.361502
$526$	0	0
$527$	−8.56092	−0.372919
$528$	0	0
$529$	−6.52354	−0.283632
$530$	0	0
$531$	11.0829	0.480958
$532$	0	0
$533$	−0.468942	−0.0203121
$534$	0	0
$535$	2.00012	0.0864727
$536$	0	0
$537$	41.6897	1.79904
$538$	0	0
$539$	−20.4846	−0.882335
$540$	0	0
$541$	0.670287	0.0288179	0.0144090	−	0.999896i	$-0.495413\pi$
$541$	0.670287	0.0288179	0.0144090	+	0.999896i	$0.495413\pi$
$542$	0	0
$543$	−11.2461	−0.482618
$544$	0	0
$545$	−3.20038	−0.137089
$546$	0	0
$547$	−29.0819	−1.24345	−0.621726	−	0.783235i	$-0.713568\pi$
$547$	−29.0819	−1.24345	−0.621726	+	0.783235i	$0.713568\pi$
$548$	0	0
$549$	−13.0906	−0.558693
$550$	0	0
$551$	9.41705	0.401180
$552$	0	0
$553$	−5.56366	−0.236591
$554$	0	0
$555$	12.5022	0.530691
$556$	0	0
$557$	−10.5154	−0.445553	−0.222777	−	0.974870i	$-0.571512\pi$
$557$	−10.5154	−0.445553	−0.222777	+	0.974870i	$0.571512\pi$
$558$	0	0
$559$	10.1270	0.428326
$560$	0	0
$561$	28.4717	1.20208
$562$	0	0
$563$	15.6299	0.658723	0.329361	−	0.944204i	$-0.393167\pi$
$563$	15.6299	0.658723	0.329361	+	0.944204i	$0.393167\pi$
$564$	0	0
$565$	8.27122	0.347973
$566$	0	0
$567$	−15.3238	−0.643538
$568$	0	0
$569$	25.5346	1.07047	0.535233	−	0.844704i	$-0.320224\pi$
$569$	25.5346	1.07047	0.535233	+	0.844704i	$0.320224\pi$
$570$	0	0
$571$	−40.0171	−1.67466	−0.837331	−	0.546696i	$-0.815885\pi$
$571$	−40.0171	−1.67466	−0.837331	+	0.546696i	$0.815885\pi$
$572$	0	0
$573$	37.1563	1.55223
$574$	0	0
$575$	11.3099	0.471655
$576$	0	0
$577$	−9.32937	−0.388387	−0.194193	−	0.980963i	$-0.562209\pi$
$577$	−9.32937	−0.388387	−0.194193	+	0.980963i	$0.562209\pi$
$578$	0	0
$579$	5.36924	0.223138
$580$	0	0
$581$	18.2321	0.756397
$582$	0	0
$583$	−7.91481	−0.327798
$584$	0	0
$585$	−2.22733	−0.0920886
$586$	0	0
$587$	19.1007	0.788369	0.394185	−	0.919031i	$-0.371027\pi$
$587$	19.1007	0.788369	0.394185	+	0.919031i	$0.371027\pi$
$588$	0	0
$589$	−7.54839	−0.311026
$590$	0	0
$591$	−39.8270	−1.63827
$592$	0	0
$593$	33.8715	1.39093	0.695467	−	0.718558i	$-0.255197\pi$
$593$	33.8715	1.39093	0.695467	+	0.718558i	$0.255197\pi$
$594$	0	0
$595$	6.67710	0.273734
$596$	0	0
$597$	18.6914	0.764986
$598$	0	0
$599$	20.6886	0.845315	0.422657	−	0.906290i	$-0.361097\pi$
$599$	20.6886	0.845315	0.422657	+	0.906290i	$0.361097\pi$
$600$	0	0
$601$	15.3156	0.624738	0.312369	−	0.949961i	$-0.398877\pi$
$601$	15.3156	0.624738	0.312369	+	0.949961i	$0.398877\pi$
$602$	0	0
$603$	−20.7516	−0.845071
$604$	0	0
$605$	−7.37268	−0.299742
$606$	0	0
$607$	15.0609	0.611305	0.305653	−	0.952143i	$-0.401125\pi$
$607$	15.0609	0.611305	0.305653	+	0.952143i	$0.401125\pi$
$608$	0	0
$609$	9.67903	0.392214
$610$	0	0
$611$	4.55794	0.184395
$612$	0	0
$613$	27.6949	1.11859	0.559294	−	0.828969i	$-0.311072\pi$
$613$	27.6949	1.11859	0.559294	+	0.828969i	$0.311072\pi$
$614$	0	0
$615$	−1.74366	−0.0703113
$616$	0	0
$617$	4.99909	0.201256	0.100628	−	0.994924i	$-0.467915\pi$
$617$	4.99909	0.201256	0.100628	+	0.994924i	$0.467915\pi$
$618$	0	0
$619$	−0.599857	−0.0241103	−0.0120551	−	0.999927i	$-0.503837\pi$
$619$	−0.599857	−0.0241103	−0.0120551	+	0.999927i	$0.503837\pi$
$620$	0	0
$621$	−11.2749	−0.452447
$622$	0	0
$623$	6.46170	0.258882
$624$	0	0
$625$	−3.30513	−0.132205
$626$	0	0
$627$	25.1043	1.00257
$628$	0	0
$629$	−12.6850	−0.505785
$630$	0	0
$631$	−38.6770	−1.53971	−0.769854	−	0.638220i	$-0.779671\pi$
$631$	−38.6770	−1.53971	−0.769854	+	0.638220i	$0.779671\pi$
$632$	0	0
$633$	−12.8326	−0.510052
$634$	0	0
$635$	15.5886	0.618616
$636$	0	0
$637$	−4.45904	−0.176674
$638$	0	0
$639$	20.4743	0.809951
$640$	0	0
$641$	10.5638	0.417244	0.208622	−	0.977996i	$-0.433102\pi$
$641$	10.5638	0.417244	0.208622	+	0.977996i	$0.433102\pi$
$642$	0	0
$643$	4.58024	0.180627	0.0903134	−	0.995913i	$-0.471213\pi$
$643$	4.58024	0.180627	0.0903134	+	0.995913i	$0.471213\pi$
$644$	0	0
$645$	37.6551	1.48267
$646$	0	0
$647$	44.3021	1.74170	0.870848	−	0.491553i	$-0.163571\pi$
$647$	44.3021	1.74170	0.870848	+	0.491553i	$0.163571\pi$
$648$	0	0
$649$	25.7127	1.00931
$650$	0	0
$651$	−7.75838	−0.304075
$652$	0	0
$653$	−12.3081	−0.481653	−0.240827	−	0.970568i	$-0.577419\pi$
$653$	−12.3081	−0.481653	−0.240827	+	0.970568i	$0.577419\pi$
$654$	0	0
$655$	−5.11848	−0.199996
$656$	0	0
$657$	10.7170	0.418109
$658$	0	0
$659$	7.44920	0.290180	0.145090	−	0.989418i	$-0.453653\pi$
$659$	7.44920	0.290180	0.145090	+	0.989418i	$0.453653\pi$
$660$	0	0
$661$	−11.3520	−0.441542	−0.220771	−	0.975326i	$-0.570857\pi$
$661$	−11.3520	−0.441542	−0.220771	+	0.975326i	$0.570857\pi$
$662$	0	0
$663$	6.19766	0.240697
$664$	0	0
$665$	5.88737	0.228303
$666$	0	0
$667$	13.2160	0.511726
$668$	0	0
$669$	−10.1560	−0.392654
$670$	0	0
$671$	−30.3706	−1.17244
$672$	0	0
$673$	−8.62418	−0.332438	−0.166219	−	0.986089i	$-0.553156\pi$
$673$	−8.62418	−0.332438	−0.166219	+	0.986089i	$0.553156\pi$
$674$	0	0
$675$	−7.73941	−0.297890
$676$	0	0
$677$	−22.1250	−0.850333	−0.425166	−	0.905115i	$-0.639784\pi$
$677$	−22.1250	−0.850333	−0.425166	+	0.905115i	$0.639784\pi$
$678$	0	0
$679$	−13.6297	−0.523061
$680$	0	0
$681$	28.8417	1.10522
$682$	0	0
$683$	−6.46164	−0.247248	−0.123624	−	0.992329i	$-0.539452\pi$
$683$	−6.46164	−0.247248	−0.123624	+	0.992329i	$0.539452\pi$
$684$	0	0
$685$	17.5836	0.671834
$686$	0	0
$687$	−0.764987	−0.0291861
$688$	0	0
$689$	−1.72288	−0.0656364
$690$	0	0
$691$	18.5679	0.706357	0.353179	−	0.935556i	$-0.385101\pi$
$691$	18.5679	0.706357	0.353179	+	0.935556i	$0.385101\pi$
$692$	0	0
$693$	9.40858	0.357402
$694$	0	0
$695$	21.8806	0.829980
$696$	0	0
$697$	1.76916	0.0670116
$698$	0	0
$699$	−10.5568	−0.399297
$700$	0	0
$701$	14.7555	0.557307	0.278654	−	0.960392i	$-0.410112\pi$
$701$	14.7555	0.557307	0.278654	+	0.960392i	$0.410112\pi$
$702$	0	0
$703$	−11.1847	−0.421840
$704$	0	0
$705$	16.9478	0.638290
$706$	0	0
$707$	18.9810	0.713854
$708$	0	0
$709$	−28.8389	−1.08307	−0.541533	−	0.840679i	$-0.682156\pi$
$709$	−28.8389	−1.08307	−0.541533	+	0.840679i	$0.682156\pi$
$710$	0	0
$711$	−7.00171	−0.262585
$712$	0	0
$713$	−10.5935	−0.396730
$714$	0	0
$715$	−5.16747	−0.193252
$716$	0	0
$717$	−22.5710	−0.842929
$718$	0	0
$719$	18.8534	0.703113	0.351556	−	0.936167i	$-0.385653\pi$
$719$	18.8534	0.703113	0.351556	+	0.936167i	$0.385653\pi$
$720$	0	0
$721$	−14.6893	−0.547058
$722$	0	0
$723$	12.8855	0.479216
$724$	0	0
$725$	9.07182	0.336919
$726$	0	0
$727$	6.88365	0.255300	0.127650	−	0.991819i	$-0.459257\pi$
$727$	6.88365	0.255300	0.127650	+	0.991819i	$0.459257\pi$
$728$	0	0
$729$	−1.17729	−0.0436032
$730$	0	0
$731$	−38.2056	−1.41309
$732$	0	0
$733$	5.91466	0.218463	0.109231	−	0.994016i	$-0.465161\pi$
$733$	5.91466	0.218463	0.109231	+	0.994016i	$0.465161\pi$
$734$	0	0
$735$	−16.5800	−0.611564
$736$	0	0
$737$	−48.1444	−1.77342
$738$	0	0
$739$	−2.04425	−0.0751988	−0.0375994	−	0.999293i	$-0.511971\pi$
$739$	−2.04425	−0.0751988	−0.0375994	+	0.999293i	$0.511971\pi$
$740$	0	0
$741$	5.46464	0.200749
$742$	0	0
$743$	26.2568	0.963267	0.481634	−	0.876373i	$-0.340044\pi$
$743$	26.2568	0.963267	0.481634	+	0.876373i	$0.340044\pi$
$744$	0	0
$745$	24.6125	0.901731
$746$	0	0
$747$	22.9446	0.839500
$748$	0	0
$749$	−1.83912	−0.0672000
$750$	0	0
$751$	33.4968	1.22231	0.611157	−	0.791509i	$-0.290704\pi$
$751$	33.4968	1.22231	0.611157	+	0.791509i	$0.290704\pi$
$752$	0	0
$753$	2.17295	0.0791866
$754$	0	0
$755$	22.5733	0.821526
$756$	0	0
$757$	−7.08029	−0.257338	−0.128669	−	0.991688i	$-0.541070\pi$
$757$	−7.08029	−0.257338	−0.128669	+	0.991688i	$0.541070\pi$
$758$	0	0
$759$	35.2316	1.27883
$760$	0	0
$761$	−28.6625	−1.03901	−0.519507	−	0.854466i	$-0.673884\pi$
$761$	−28.6625	−1.03901	−0.519507	+	0.854466i	$0.673884\pi$
$762$	0	0
$763$	2.94277	0.106535
$764$	0	0
$765$	8.40294	0.303809
$766$	0	0
$767$	5.59709	0.202099
$768$	0	0
$769$	−24.3092	−0.876611	−0.438306	−	0.898826i	$-0.644421\pi$
$769$	−24.3092	−0.876611	−0.438306	+	0.898826i	$0.644421\pi$
$770$	0	0
$771$	−22.4878	−0.809879
$772$	0	0
$773$	−32.6130	−1.17301	−0.586504	−	0.809946i	$-0.699496\pi$
$773$	−32.6130	−1.17301	−0.586504	+	0.809946i	$0.699496\pi$
$774$	0	0
$775$	−7.27166	−0.261206
$776$	0	0
$777$	−11.4959	−0.412413
$778$	0	0
$779$	1.55991	0.0558897
$780$	0	0
$781$	47.5010	1.69972
$782$	0	0
$783$	−9.04377	−0.323198
$784$	0	0
$785$	−17.0879	−0.609894
$786$	0	0
$787$	−25.1579	−0.896782	−0.448391	−	0.893838i	$-0.648003\pi$
$787$	−25.1579	−0.896782	−0.448391	+	0.893838i	$0.648003\pi$
$788$	0	0
$789$	20.6150	0.733912
$790$	0	0
$791$	−7.60543	−0.270418
$792$	0	0
$793$	−6.61100	−0.234764
$794$	0	0
$795$	−6.40617	−0.227203
$796$	0	0
$797$	14.1616	0.501630	0.250815	−	0.968035i	$-0.419301\pi$
$797$	14.1616	0.501630	0.250815	+	0.968035i	$0.419301\pi$
$798$	0	0
$799$	−17.1955	−0.608335
$800$	0	0
$801$	8.13186	0.287325
$802$	0	0
$803$	24.8637	0.877421
$804$	0	0
$805$	8.26241	0.291212
$806$	0	0
$807$	34.6454	1.21957
$808$	0	0
$809$	−19.6386	−0.690456	−0.345228	−	0.938519i	$-0.612198\pi$
$809$	−19.6386	−0.690456	−0.345228	+	0.938519i	$0.612198\pi$
$810$	0	0
$811$	6.28241	0.220605	0.110303	−	0.993898i	$-0.464818\pi$
$811$	6.28241	0.220605	0.110303	+	0.993898i	$0.464818\pi$
$812$	0	0
$813$	−22.2864	−0.781618
$814$	0	0
$815$	13.2684	0.464770
$816$	0	0
$817$	−33.6869	−1.17856
$818$	0	0
$819$	2.04804	0.0715643
$820$	0	0
$821$	32.8538	1.14660	0.573302	−	0.819344i	$-0.305662\pi$
$821$	32.8538	1.14660	0.573302	+	0.819344i	$0.305662\pi$
$822$	0	0
$823$	26.9543	0.939569	0.469785	−	0.882781i	$-0.344332\pi$
$823$	26.9543	0.939569	0.469785	+	0.882781i	$0.344332\pi$
$824$	0	0
$825$	24.1840	0.841977
$826$	0	0
$827$	−51.4384	−1.78869	−0.894343	−	0.447381i	$-0.852357\pi$
$827$	−51.4384	−1.78869	−0.894343	+	0.447381i	$0.852357\pi$
$828$	0	0
$829$	36.8291	1.27913	0.639563	−	0.768738i	$-0.279115\pi$
$829$	36.8291	1.27913	0.639563	+	0.768738i	$0.279115\pi$
$830$	0	0
$831$	35.4949	1.23130
$832$	0	0
$833$	16.8224	0.582863
$834$	0	0
$835$	23.0680	0.798302
$836$	0	0
$837$	7.24918	0.250568
$838$	0	0
$839$	−24.1933	−0.835245	−0.417623	−	0.908621i	$-0.637137\pi$
$839$	−24.1933	−0.835245	−0.417623	+	0.908621i	$0.637137\pi$
$840$	0	0
$841$	−18.3993	−0.634457
$842$	0	0
$843$	51.8219	1.78484
$844$	0	0
$845$	18.2173	0.626693
$846$	0	0
$847$	6.77922	0.232937
$848$	0	0
$849$	−54.6019	−1.87393
$850$	0	0
$851$	−15.6968	−0.538079
$852$	0	0
$853$	11.9133	0.407905	0.203953	−	0.978981i	$-0.434621\pi$
$853$	11.9133	0.407905	0.203953	+	0.978981i	$0.434621\pi$
$854$	0	0
$855$	7.40909	0.253386
$856$	0	0
$857$	17.3175	0.591555	0.295777	−	0.955257i	$-0.404421\pi$
$857$	17.3175	0.591555	0.295777	+	0.955257i	$0.404421\pi$
$858$	0	0
$859$	21.3995	0.730141	0.365070	−	0.930980i	$-0.381045\pi$
$859$	21.3995	0.730141	0.365070	+	0.930980i	$0.381045\pi$
$860$	0	0
$861$	1.60331	0.0546406
$862$	0	0
$863$	55.7127	1.89648	0.948241	−	0.317551i	$-0.102860\pi$
$863$	55.7127	1.89648	0.948241	+	0.317551i	$0.102860\pi$
$864$	0	0
$865$	22.2888	0.757841
$866$	0	0
$867$	13.5585	0.460470
$868$	0	0
$869$	−16.2442	−0.551046
$870$	0	0
$871$	−10.4800	−0.355100
$872$	0	0
$873$	−17.1526	−0.580529
$874$	0	0
$875$	15.8491	0.535798
$876$	0	0
$877$	0.486259	0.0164198	0.00820990	−	0.999966i	$-0.497387\pi$
$877$	0.486259	0.0164198	0.00820990	+	0.999966i	$0.497387\pi$
$878$	0	0
$879$	−8.93185	−0.301264
$880$	0	0
$881$	25.9437	0.874066	0.437033	−	0.899445i	$-0.356029\pi$
$881$	25.9437	0.874066	0.437033	+	0.899445i	$0.356029\pi$
$882$	0	0
$883$	−41.7294	−1.40430	−0.702152	−	0.712027i	$-0.747778\pi$
$883$	−41.7294	−1.40430	−0.702152	+	0.712027i	$0.747778\pi$
$884$	0	0
$885$	20.8116	0.699575
$886$	0	0
$887$	49.6412	1.66679	0.833394	−	0.552679i	$-0.186394\pi$
$887$	49.6412	1.66679	0.833394	+	0.552679i	$0.186394\pi$
$888$	0	0
$889$	−14.3338	−0.480741
$890$	0	0
$891$	−44.7407	−1.49887
$892$	0	0
$893$	−15.1618	−0.507369
$894$	0	0
$895$	28.5456	0.954176
$896$	0	0
$897$	7.66914	0.256065
$898$	0	0
$899$	−8.49719	−0.283397
$900$	0	0
$901$	6.49982	0.216541
$902$	0	0
$903$	−34.6241	−1.15222
$904$	0	0
$905$	−7.70043	−0.255971
$906$	0	0
$907$	−41.5363	−1.37919	−0.689595	−	0.724195i	$-0.742211\pi$
$907$	−41.5363	−1.37919	−0.689595	+	0.724195i	$0.742211\pi$
$908$	0	0
$909$	23.8871	0.792284
$910$	0	0
$911$	38.5592	1.27752	0.638761	−	0.769405i	$-0.279447\pi$
$911$	38.5592	1.27752	0.638761	+	0.769405i	$0.279447\pi$
$912$	0	0
$913$	53.2323	1.76173
$914$	0	0
$915$	−24.5817	−0.812645
$916$	0	0
$917$	4.70647	0.155421
$918$	0	0
$919$	8.85566	0.292121	0.146061	−	0.989276i	$-0.453341\pi$
$919$	8.85566	0.292121	0.146061	+	0.989276i	$0.453341\pi$
$920$	0	0
$921$	−44.4172	−1.46360
$922$	0	0
$923$	10.3399	0.340342
$924$	0	0
$925$	−10.7747	−0.354270
$926$	0	0
$927$	−18.4861	−0.607162
$928$	0	0
$929$	−19.8268	−0.650497	−0.325248	−	0.945629i	$-0.605448\pi$
$929$	−19.8268	−0.650497	−0.325248	+	0.945629i	$0.605448\pi$
$930$	0	0
$931$	14.8328	0.486125
$932$	0	0
$933$	−11.4723	−0.375585
$934$	0	0
$935$	19.4951	0.637557
$936$	0	0
$937$	14.1858	0.463428	0.231714	−	0.972784i	$-0.425567\pi$
$937$	14.1858	0.463428	0.231714	+	0.972784i	$0.425567\pi$
$938$	0	0
$939$	19.1649	0.625423
$940$	0	0
$941$	−25.6372	−0.835750	−0.417875	−	0.908505i	$-0.637225\pi$
$941$	−25.6372	−0.835750	−0.417875	+	0.908505i	$0.637225\pi$
$942$	0	0
$943$	2.18920	0.0712901
$944$	0	0
$945$	−5.65401	−0.183925
$946$	0	0
$947$	12.1825	0.395877	0.197939	−	0.980214i	$-0.436575\pi$
$947$	12.1825	0.395877	0.197939	+	0.980214i	$0.436575\pi$
$948$	0	0
$949$	5.41228	0.175690
$950$	0	0
$951$	−3.40202	−0.110318
$952$	0	0
$953$	12.0522	0.390410	0.195205	−	0.980762i	$-0.437463\pi$
$953$	12.0522	0.390410	0.195205	+	0.980762i	$0.437463\pi$
$954$	0	0
$955$	25.4416	0.823270
$956$	0	0
$957$	28.2598	0.913510
$958$	0	0
$959$	−16.1682	−0.522098
$960$	0	0
$961$	−24.1889	−0.780288
$962$	0	0
$963$	−2.31448	−0.0745831
$964$	0	0
$965$	3.67641	0.118348
$966$	0	0
$967$	54.4000	1.74939	0.874693	−	0.484677i	$-0.161063\pi$
$967$	54.4000	1.74939	0.874693	+	0.484677i	$0.161063\pi$
$968$	0	0
$969$	−20.6162	−0.662288
$970$	0	0
$971$	12.4456	0.399399	0.199699	−	0.979857i	$-0.436003\pi$
$971$	12.4456	0.399399	0.199699	+	0.979857i	$0.436003\pi$
$972$	0	0
$973$	−20.1194	−0.644997
$974$	0	0
$975$	5.26431	0.168593
$976$	0	0
$977$	32.7023	1.04624	0.523119	−	0.852260i	$-0.324768\pi$
$977$	32.7023	1.04624	0.523119	+	0.852260i	$0.324768\pi$
$978$	0	0
$979$	18.8662	0.602965
$980$	0	0
$981$	3.70339	0.118240
$982$	0	0
$983$	20.6533	0.658737	0.329369	−	0.944201i	$-0.393164\pi$
$983$	20.6533	0.658737	0.329369	+	0.944201i	$0.393164\pi$
$984$	0	0
$985$	−27.2702	−0.868902
$986$	0	0
$987$	−15.5836	−0.496031
$988$	0	0
$989$	−47.2766	−1.50331
$990$	0	0
$991$	53.0310	1.68459	0.842293	−	0.539020i	$-0.181205\pi$
$991$	53.0310	1.68459	0.842293	+	0.539020i	$0.181205\pi$
$992$	0	0
$993$	−11.8312	−0.375452
$994$	0	0
$995$	12.7983	0.405733
$996$	0	0
$997$	−42.9199	−1.35929	−0.679643	−	0.733543i	$-0.737865\pi$
$997$	−42.9199	−1.35929	−0.679643	+	0.733543i	$0.737865\pi$
$998$	0	0
$999$	10.7414	0.339842

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.1		9
4.3	odd	2		2008.2.a.a.1.9	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.9	✓	9	4.3	odd	2
4016.2.a.h.1.1		9	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)

\(f(q)\)	\(=\)	\(q-2.17295 q^{3} -1.48785 q^{5} +1.36809 q^{7} +1.72170 q^{9} +O(q^{10})\)	\(q-2.17295 q^{3} -1.48785 q^{5} +1.36809 q^{7} +1.72170 q^{9} +3.99440 q^{11} +0.869492 q^{13} +3.23303 q^{15} -3.28029 q^{17} -2.89232 q^{19} -2.97279 q^{21} -4.05912 q^{23} -2.78629 q^{25} +2.77768 q^{27} -3.25588 q^{29} +2.60980 q^{31} -8.67963 q^{33} -2.03552 q^{35} +3.86704 q^{37} -1.88936 q^{39} -0.539329 q^{41} +11.6470 q^{43} -2.56164 q^{45} +5.24207 q^{47} -5.12833 q^{49} +7.12791 q^{51} -1.98148 q^{53} -5.94309 q^{55} +6.28487 q^{57} +6.43719 q^{59} -7.60329 q^{61} +2.35544 q^{63} -1.29368 q^{65} -12.0530 q^{67} +8.82026 q^{69} +11.8919 q^{71} +6.22464 q^{73} +6.05446 q^{75} +5.46470 q^{77} -4.06674 q^{79} -11.2008 q^{81} +13.3267 q^{83} +4.88060 q^{85} +7.07485 q^{87} +4.72315 q^{89} +1.18954 q^{91} -5.67096 q^{93} +4.30335 q^{95} -9.96261 q^{97} +6.87717 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9} + 3 q^{11} - 3 q^{13} + q^{15} - 11 q^{17} - 4 q^{19} - 3 q^{21} + 9 q^{23} - 12 q^{25} + 7 q^{27} - 9 q^{29} - 3 q^{31} - 14 q^{33} + 8 q^{35} - 10 q^{37} + q^{39} - 23 q^{41} - 10 q^{45} + 11 q^{47} - 21 q^{49} + 3 q^{51} - 21 q^{53} + 4 q^{55} - 21 q^{57} + 4 q^{59} - 11 q^{61} + 2 q^{63} - 29 q^{65} + 4 q^{67} - 14 q^{69} + 19 q^{71} - 31 q^{73} - 16 q^{75} - 26 q^{77} - 4 q^{79} - 27 q^{81} + 22 q^{83} + 4 q^{85} + 6 q^{87} - 36 q^{89} - 14 q^{91} - 32 q^{93} + 3 q^{95} - 38 q^{97}+O(q^{100}) \) 9 * q + q^3 - 5 * q^5 - 4 * q^9 + 3 * q^11 - 3 * q^13 + q^15 - 11 * q^17 - 4 * q^19 - 3 * q^21 + 9 * q^23 - 12 * q^25 + 7 * q^27 - 9 * q^29 - 3 * q^31 - 14 * q^33 + 8 * q^35 - 10 * q^37 + q^39 - 23 * q^41 - 10 * q^45 + 11 * q^47 - 21 * q^49 + 3 * q^51 - 21 * q^53 + 4 * q^55 - 21 * q^57 + 4 * q^59 - 11 * q^61 + 2 * q^63 - 29 * q^65 + 4 * q^67 - 14 * q^69 + 19 * q^71 - 31 * q^73 - 16 * q^75 - 26 * q^77 - 4 * q^79 - 27 * q^81 + 22 * q^83 + 4 * q^85 + 6 * q^87 - 36 * q^89 - 14 * q^91 - 32 * q^93 + 3 * q^95 - 38 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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