LMFDB - Embedded newform 4016.2.a.k.1.8

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:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Root			$2.18124$ 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)

$f(q)$	$=$	$q-0.923498 q^{3} +1.20531 q^{5} +1.96022 q^{7} -2.14715 q^{9} +O(q^{10})$	$q-0.923498 q^{3} +1.20531 q^{5} +1.96022 q^{7} -2.14715 q^{9} -0.336128 q^{11} -7.20973 q^{13} -1.11310 q^{15} +5.33974 q^{17} -5.23206 q^{19} -1.81026 q^{21} -3.74762 q^{23} -3.54724 q^{25} +4.75338 q^{27} +6.55021 q^{29} -4.05430 q^{31} +0.310414 q^{33} +2.36266 q^{35} +10.0546 q^{37} +6.65817 q^{39} -1.65319 q^{41} +9.79171 q^{43} -2.58798 q^{45} +6.95907 q^{47} -3.15755 q^{49} -4.93124 q^{51} +9.58038 q^{53} -0.405138 q^{55} +4.83180 q^{57} +2.36093 q^{59} +4.90096 q^{61} -4.20888 q^{63} -8.68994 q^{65} +10.4025 q^{67} +3.46092 q^{69} +4.32072 q^{71} +6.12714 q^{73} +3.27586 q^{75} -0.658884 q^{77} -14.0037 q^{79} +2.05171 q^{81} +7.96857 q^{83} +6.43602 q^{85} -6.04911 q^{87} +9.79974 q^{89} -14.1326 q^{91} +3.74414 q^{93} -6.30624 q^{95} -9.23216 q^{97} +0.721718 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * 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
$3$	−0.923498	−0.533182	−0.266591	−	0.963810i	$-0.585897\pi$
$3$	−0.923498	−0.533182	−0.266591	+	0.963810i	$0.585897\pi$
$4$	0	0
$5$	1.20531	0.539030	0.269515	−	0.962996i	$-0.413137\pi$
$5$	1.20531	0.539030	0.269515	+	0.962996i	$0.413137\pi$
$6$	0	0
$7$	1.96022	0.740892	0.370446	−	0.928854i	$-0.379205\pi$
$7$	1.96022	0.740892	0.370446	+	0.928854i	$0.379205\pi$
$8$	0	0
$9$	−2.14715	−0.715717
$10$	0	0
$11$	−0.336128	−0.101347	−0.0506733	−	0.998715i	$-0.516137\pi$
$11$	−0.336128	−0.101347	−0.0506733	+	0.998715i	$0.516137\pi$
$12$	0	0
$13$	−7.20973	−1.99962	−0.999810	−	0.0194958i	$-0.993794\pi$
$13$	−7.20973	−1.99962	−0.999810	+	0.0194958i	$0.993794\pi$
$14$	0	0
$15$	−1.11310	−0.287401
$16$	0	0
$17$	5.33974	1.29508	0.647538	−	0.762033i	$-0.275799\pi$
$17$	5.33974	1.29508	0.647538	+	0.762033i	$0.275799\pi$
$18$	0	0
$19$	−5.23206	−1.20032	−0.600158	−	0.799881i	$-0.704896\pi$
$19$	−5.23206	−1.20032	−0.600158	+	0.799881i	$0.704896\pi$
$20$	0	0
$21$	−1.81026	−0.395030
$22$	0	0
$23$	−3.74762	−0.781434	−0.390717	−	0.920511i	$-0.627773\pi$
$23$	−3.74762	−0.781434	−0.390717	+	0.920511i	$0.627773\pi$
$24$	0	0
$25$	−3.54724	−0.709447
$26$	0	0
$27$	4.75338	0.914789
$28$	0	0
$29$	6.55021	1.21634	0.608172	−	0.793805i	$-0.291903\pi$
$29$	6.55021	1.21634	0.608172	+	0.793805i	$0.291903\pi$
$30$	0	0
$31$	−4.05430	−0.728174	−0.364087	−	0.931365i	$-0.618619\pi$
$31$	−4.05430	−0.728174	−0.364087	+	0.931365i	$0.618619\pi$
$32$	0	0
$33$	0.310414	0.0540361
$34$	0	0
$35$	2.36266	0.399363
$36$	0	0
$37$	10.0546	1.65297	0.826483	−	0.562962i	$-0.190338\pi$
$37$	10.0546	1.65297	0.826483	+	0.562962i	$0.190338\pi$
$38$	0	0
$39$	6.65817	1.06616
$40$	0	0
$41$	−1.65319	−0.258185	−0.129092	−	0.991633i	$-0.541206\pi$
$41$	−1.65319	−0.258185	−0.129092	+	0.991633i	$0.541206\pi$
$42$	0	0
$43$	9.79171	1.49322	0.746611	−	0.665261i	$-0.231680\pi$
$43$	9.79171	1.49322	0.746611	+	0.665261i	$0.231680\pi$
$44$	0	0
$45$	−2.58798	−0.385793
$46$	0	0
$47$	6.95907	1.01509	0.507543	−	0.861627i	$-0.330554\pi$
$47$	6.95907	1.01509	0.507543	+	0.861627i	$0.330554\pi$
$48$	0	0
$49$	−3.15755	−0.451079
$50$	0	0
$51$	−4.93124	−0.690511
$52$	0	0
$53$	9.58038	1.31597	0.657983	−	0.753033i	$-0.271410\pi$
$53$	9.58038	1.31597	0.657983	+	0.753033i	$0.271410\pi$
$54$	0	0
$55$	−0.405138	−0.0546288
$56$	0	0
$57$	4.83180	0.639987
$58$	0	0
$59$	2.36093	0.307367	0.153683	−	0.988120i	$-0.450886\pi$
$59$	2.36093	0.307367	0.153683	+	0.988120i	$0.450886\pi$
$60$	0	0
$61$	4.90096	0.627503	0.313752	−	0.949505i	$-0.398414\pi$
$61$	4.90096	0.627503	0.313752	+	0.949505i	$0.398414\pi$
$62$	0	0
$63$	−4.20888	−0.530269
$64$	0	0
$65$	−8.68994	−1.07785
$66$	0	0
$67$	10.4025	1.27087	0.635436	−	0.772153i	$-0.280820\pi$
$67$	10.4025	1.27087	0.635436	+	0.772153i	$0.280820\pi$
$68$	0	0
$69$	3.46092	0.416646
$70$	0	0
$71$	4.32072	0.512775	0.256388	−	0.966574i	$-0.417468\pi$
$71$	4.32072	0.512775	0.256388	+	0.966574i	$0.417468\pi$
$72$	0	0
$73$	6.12714	0.717127	0.358564	−	0.933505i	$-0.383267\pi$
$73$	6.12714	0.717127	0.358564	+	0.933505i	$0.383267\pi$
$74$	0	0
$75$	3.27586	0.378264
$76$	0	0
$77$	−0.658884	−0.0750868
$78$	0	0
$79$	−14.0037	−1.57554	−0.787769	−	0.615971i	$-0.788764\pi$
$79$	−14.0037	−1.57554	−0.787769	+	0.615971i	$0.788764\pi$
$80$	0	0
$81$	2.05171	0.227968
$82$	0	0
$83$	7.96857	0.874664	0.437332	−	0.899300i	$-0.355923\pi$
$83$	7.96857	0.874664	0.437332	+	0.899300i	$0.355923\pi$
$84$	0	0
$85$	6.43602	0.698084
$86$	0	0
$87$	−6.04911	−0.648533
$88$	0	0
$89$	9.79974	1.03877	0.519385	−	0.854540i	$-0.326161\pi$
$89$	9.79974	1.03877	0.519385	+	0.854540i	$0.326161\pi$
$90$	0	0
$91$	−14.1326	−1.48150
$92$	0	0
$93$	3.74414	0.388249
$94$	0	0
$95$	−6.30624	−0.647006
$96$	0	0
$97$	−9.23216	−0.937384	−0.468692	−	0.883362i	$-0.655275\pi$
$97$	−9.23216	−0.937384	−0.468692	+	0.883362i	$0.655275\pi$
$98$	0	0
$99$	0.721718	0.0725354
$100$	0	0
$101$	−3.40967	−0.339275	−0.169637	−	0.985507i	$-0.554260\pi$
$101$	−3.40967	−0.339275	−0.169637	+	0.985507i	$0.554260\pi$
$102$	0	0
$103$	9.15290	0.901862	0.450931	−	0.892559i	$-0.351092\pi$
$103$	9.15290	0.901862	0.450931	+	0.892559i	$0.351092\pi$
$104$	0	0
$105$	−2.18191	−0.212933
$106$	0	0
$107$	−2.12911	−0.205829	−0.102914	−	0.994690i	$-0.532817\pi$
$107$	−2.12911	−0.205829	−0.102914	+	0.994690i	$0.532817\pi$
$108$	0	0
$109$	2.57504	0.246644	0.123322	−	0.992367i	$-0.460645\pi$
$109$	2.57504	0.246644	0.123322	+	0.992367i	$0.460645\pi$
$110$	0	0
$111$	−9.28540	−0.881331
$112$	0	0
$113$	−0.864414	−0.0813172	−0.0406586	−	0.999173i	$-0.512946\pi$
$113$	−0.864414	−0.0813172	−0.0406586	+	0.999173i	$0.512946\pi$
$114$	0	0
$115$	−4.51704	−0.421216
$116$	0	0
$117$	15.4804	1.43116
$118$	0	0
$119$	10.4670	0.959512
$120$	0	0
$121$	−10.8870	−0.989729
$122$	0	0
$123$	1.52672	0.137659
$124$	0	0
$125$	−10.3020	−0.921443
$126$	0	0
$127$	−9.72900	−0.863309	−0.431655	−	0.902039i	$-0.642070\pi$
$127$	−9.72900	−0.863309	−0.431655	+	0.902039i	$0.642070\pi$
$128$	0	0
$129$	−9.04263	−0.796159
$130$	0	0
$131$	13.4033	1.17105	0.585525	−	0.810654i	$-0.300888\pi$
$131$	13.4033	1.17105	0.585525	+	0.810654i	$0.300888\pi$
$132$	0	0
$133$	−10.2560	−0.889305
$134$	0	0
$135$	5.72929	0.493098
$136$	0	0
$137$	13.9906	1.19530	0.597650	−	0.801757i	$-0.296101\pi$
$137$	13.9906	1.19530	0.597650	+	0.801757i	$0.296101\pi$
$138$	0	0
$139$	−3.37042	−0.285875	−0.142938	−	0.989732i	$-0.545655\pi$
$139$	−3.37042	−0.285875	−0.142938	+	0.989732i	$0.545655\pi$
$140$	0	0
$141$	−6.42669	−0.541225
$142$	0	0
$143$	2.42340	0.202654
$144$	0	0
$145$	7.89502	0.655645
$146$	0	0
$147$	2.91599	0.240507
$148$	0	0
$149$	19.1187	1.56627	0.783134	−	0.621853i	$-0.213620\pi$
$149$	19.1187	1.56627	0.783134	+	0.621853i	$0.213620\pi$
$150$	0	0
$151$	4.67694	0.380604	0.190302	−	0.981726i	$-0.439053\pi$
$151$	4.67694	0.380604	0.190302	+	0.981726i	$0.439053\pi$
$152$	0	0
$153$	−11.4652	−0.926908
$154$	0	0
$155$	−4.88668	−0.392507
$156$	0	0
$157$	3.56244	0.284314	0.142157	−	0.989844i	$-0.454596\pi$
$157$	3.56244	0.284314	0.142157	+	0.989844i	$0.454596\pi$
$158$	0	0
$159$	−8.84746	−0.701649
$160$	0	0
$161$	−7.34615	−0.578958
$162$	0	0
$163$	−9.71739	−0.761125	−0.380562	−	0.924755i	$-0.624270\pi$
$163$	−9.71739	−0.761125	−0.380562	+	0.924755i	$0.624270\pi$
$164$	0	0
$165$	0.374144	0.0291271
$166$	0	0
$167$	4.64669	0.359571	0.179786	−	0.983706i	$-0.442460\pi$
$167$	4.64669	0.359571	0.179786	+	0.983706i	$0.442460\pi$
$168$	0	0
$169$	38.9802	2.99848
$170$	0	0
$171$	11.2340	0.859087
$172$	0	0
$173$	−9.08071	−0.690394	−0.345197	−	0.938530i	$-0.612188\pi$
$173$	−9.08071	−0.690394	−0.345197	+	0.938530i	$0.612188\pi$
$174$	0	0
$175$	−6.95335	−0.525624
$176$	0	0
$177$	−2.18031	−0.163882
$178$	0	0
$179$	22.6394	1.69215	0.846073	−	0.533067i	$-0.178961\pi$
$179$	22.6394	1.69215	0.846073	+	0.533067i	$0.178961\pi$
$180$	0	0
$181$	−18.1038	−1.34565	−0.672823	−	0.739803i	$-0.734919\pi$
$181$	−18.1038	−1.34565	−0.672823	+	0.739803i	$0.734919\pi$
$182$	0	0
$183$	−4.52602	−0.334573
$184$	0	0
$185$	12.1189	0.890997
$186$	0	0
$187$	−1.79484	−0.131251
$188$	0	0
$189$	9.31766	0.677760
$190$	0	0
$191$	−20.5006	−1.48337	−0.741687	−	0.670746i	$-0.765974\pi$
$191$	−20.5006	−1.48337	−0.741687	+	0.670746i	$0.765974\pi$
$192$	0	0
$193$	14.5378	1.04646	0.523229	−	0.852192i	$-0.324727\pi$
$193$	14.5378	1.04646	0.523229	+	0.852192i	$0.324727\pi$
$194$	0	0
$195$	8.02514	0.574692
$196$	0	0
$197$	−8.32456	−0.593100	−0.296550	−	0.955017i	$-0.595836\pi$
$197$	−8.32456	−0.593100	−0.296550	+	0.955017i	$0.595836\pi$
$198$	0	0
$199$	−14.9423	−1.05923	−0.529614	−	0.848239i	$-0.677663\pi$
$199$	−14.9423	−1.05923	−0.529614	+	0.848239i	$0.677663\pi$
$200$	0	0
$201$	−9.60672	−0.677606
$202$	0	0
$203$	12.8398	0.901180
$204$	0	0
$205$	−1.99260	−0.139169
$206$	0	0
$207$	8.04672	0.559285
$208$	0	0
$209$	1.75864	0.121648
$210$	0	0
$211$	0.941415	0.0648097	0.0324048	−	0.999475i	$-0.489683\pi$
$211$	0.941415	0.0648097	0.0324048	+	0.999475i	$0.489683\pi$
$212$	0	0
$213$	−3.99018	−0.273402
$214$	0	0
$215$	11.8020	0.804891
$216$	0	0
$217$	−7.94731	−0.539499
$218$	0	0
$219$	−5.65840	−0.382359
$220$	0	0
$221$	−38.4981	−2.58966
$222$	0	0
$223$	12.8639	0.861431	0.430715	−	0.902488i	$-0.358261\pi$
$223$	12.8639	0.861431	0.430715	+	0.902488i	$0.358261\pi$
$224$	0	0
$225$	7.61645	0.507764
$226$	0	0
$227$	13.6829	0.908167	0.454083	−	0.890959i	$-0.349967\pi$
$227$	13.6829	0.908167	0.454083	+	0.890959i	$0.349967\pi$
$228$	0	0
$229$	2.79295	0.184563	0.0922816	−	0.995733i	$-0.470584\pi$
$229$	2.79295	0.184563	0.0922816	+	0.995733i	$0.470584\pi$
$230$	0	0
$231$	0.608478	0.0400349
$232$	0	0
$233$	−9.47362	−0.620637	−0.310319	−	0.950633i	$-0.600436\pi$
$233$	−9.47362	−0.620637	−0.310319	+	0.950633i	$0.600436\pi$
$234$	0	0
$235$	8.38782	0.547161
$236$	0	0
$237$	12.9324	0.840048
$238$	0	0
$239$	8.20097	0.530477	0.265238	−	0.964183i	$-0.414549\pi$
$239$	8.20097	0.530477	0.265238	+	0.964183i	$0.414549\pi$
$240$	0	0
$241$	−9.83224	−0.633350	−0.316675	−	0.948534i	$-0.602566\pi$
$241$	−9.83224	−0.633350	−0.316675	+	0.948534i	$0.602566\pi$
$242$	0	0
$243$	−16.1549	−1.03634
$244$	0	0
$245$	−3.80582	−0.243145
$246$	0	0
$247$	37.7217	2.40018
$248$	0	0
$249$	−7.35896	−0.466355
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	1.25968	0.0791956
$254$	0	0
$255$	−5.94365	−0.372206
$256$	0	0
$257$	−8.39462	−0.523642	−0.261821	−	0.965116i	$-0.584323\pi$
$257$	−8.39462	−0.523642	−0.261821	+	0.965116i	$0.584323\pi$
$258$	0	0
$259$	19.7092	1.22467
$260$	0	0
$261$	−14.0643	−0.870558
$262$	0	0
$263$	21.0433	1.29759	0.648793	−	0.760965i	$-0.275274\pi$
$263$	21.0433	1.29759	0.648793	+	0.760965i	$0.275274\pi$
$264$	0	0
$265$	11.5473	0.709344
$266$	0	0
$267$	−9.05004	−0.553854
$268$	0	0
$269$	−19.7667	−1.20520	−0.602598	−	0.798045i	$-0.705868\pi$
$269$	−19.7667	−1.20520	−0.602598	+	0.798045i	$0.705868\pi$
$270$	0	0
$271$	0.382653	0.0232445	0.0116223	−	0.999932i	$-0.496300\pi$
$271$	0.382653	0.0232445	0.0116223	+	0.999932i	$0.496300\pi$
$272$	0	0
$273$	13.0515	0.789910
$274$	0	0
$275$	1.19233	0.0719000
$276$	0	0
$277$	29.1930	1.75404	0.877020	−	0.480454i	$-0.159528\pi$
$277$	29.1930	1.75404	0.877020	+	0.480454i	$0.159528\pi$
$278$	0	0
$279$	8.70520	0.521167
$280$	0	0
$281$	−8.42816	−0.502782	−0.251391	−	0.967886i	$-0.580888\pi$
$281$	−8.42816	−0.502782	−0.251391	+	0.967886i	$0.580888\pi$
$282$	0	0
$283$	−18.2239	−1.08330	−0.541648	−	0.840605i	$-0.682200\pi$
$283$	−18.2239	−1.08330	−0.541648	+	0.840605i	$0.682200\pi$
$284$	0	0
$285$	5.82380	0.344972
$286$	0	0
$287$	−3.24061	−0.191287
$288$	0	0
$289$	11.5128	0.677223
$290$	0	0
$291$	8.52588	0.499796
$292$	0	0
$293$	5.52185	0.322590	0.161295	−	0.986906i	$-0.448433\pi$
$293$	5.52185	0.322590	0.161295	+	0.986906i	$0.448433\pi$
$294$	0	0
$295$	2.84564	0.165680
$296$	0	0
$297$	−1.59775	−0.0927107
$298$	0	0
$299$	27.0194	1.56257
$300$	0	0
$301$	19.1939	1.10632
$302$	0	0
$303$	3.14882	0.180895
$304$	0	0
$305$	5.90716	0.338243
$306$	0	0
$307$	3.64625	0.208102	0.104051	−	0.994572i	$-0.466819\pi$
$307$	3.64625	0.208102	0.104051	+	0.994572i	$0.466819\pi$
$308$	0	0
$309$	−8.45268	−0.480856
$310$	0	0
$311$	22.6489	1.28430	0.642152	−	0.766578i	$-0.278042\pi$
$311$	22.6489	1.28430	0.642152	+	0.766578i	$0.278042\pi$
$312$	0	0
$313$	−20.3561	−1.15059	−0.575297	−	0.817945i	$-0.695114\pi$
$313$	−20.3561	−1.15059	−0.575297	+	0.817945i	$0.695114\pi$
$314$	0	0
$315$	−5.07299	−0.285831
$316$	0	0
$317$	6.05127	0.339873	0.169937	−	0.985455i	$-0.445644\pi$
$317$	6.05127	0.339873	0.169937	+	0.985455i	$0.445644\pi$
$318$	0	0
$319$	−2.20171	−0.123272
$320$	0	0
$321$	1.96623	0.109744
$322$	0	0
$323$	−27.9378	−1.55450
$324$	0	0
$325$	25.5746	1.41862
$326$	0	0
$327$	−2.37804	−0.131506
$328$	0	0
$329$	13.6413	0.752069
$330$	0	0
$331$	28.9241	1.58981	0.794906	−	0.606732i	$-0.207520\pi$
$331$	28.9241	1.58981	0.794906	+	0.606732i	$0.207520\pi$
$332$	0	0
$333$	−21.5887	−1.18306
$334$	0	0
$335$	12.5383	0.685038
$336$	0	0
$337$	22.8919	1.24700	0.623499	−	0.781824i	$-0.285710\pi$
$337$	22.8919	1.24700	0.623499	+	0.781824i	$0.285710\pi$
$338$	0	0
$339$	0.798285	0.0433569
$340$	0	0
$341$	1.36277	0.0737979
$342$	0	0
$343$	−19.9110	−1.07509
$344$	0	0
$345$	4.17147	0.224585
$346$	0	0
$347$	−27.7632	−1.49041	−0.745203	−	0.666837i	$-0.767648\pi$
$347$	−27.7632	−1.49041	−0.745203	+	0.666837i	$0.767648\pi$
$348$	0	0
$349$	33.0581	1.76956	0.884781	−	0.466008i	$-0.154308\pi$
$349$	33.0581	1.76956	0.884781	+	0.466008i	$0.154308\pi$
$350$	0	0
$351$	−34.2706	−1.82923
$352$	0	0
$353$	−2.33647	−0.124358	−0.0621788	−	0.998065i	$-0.519805\pi$
$353$	−2.33647	−0.124358	−0.0621788	+	0.998065i	$0.519805\pi$
$354$	0	0
$355$	5.20779	0.276401
$356$	0	0
$357$	−9.66629	−0.511594
$358$	0	0
$359$	36.6220	1.93283	0.966417	−	0.256979i	$-0.0827272\pi$
$359$	36.6220	1.93283	0.966417	+	0.256979i	$0.0827272\pi$
$360$	0	0
$361$	8.37445	0.440760
$362$	0	0
$363$	10.0541	0.527705
$364$	0	0
$365$	7.38508	0.386553
$366$	0	0
$367$	15.5126	0.809750	0.404875	−	0.914372i	$-0.367315\pi$
$367$	15.5126	0.809750	0.404875	+	0.914372i	$0.367315\pi$
$368$	0	0
$369$	3.54965	0.184787
$370$	0	0
$371$	18.7796	0.974989
$372$	0	0
$373$	8.83566	0.457493	0.228747	−	0.973486i	$-0.426537\pi$
$373$	8.83566	0.457493	0.228747	+	0.973486i	$0.426537\pi$
$374$	0	0
$375$	9.51391	0.491296
$376$	0	0
$377$	−47.2253	−2.43223
$378$	0	0
$379$	−1.13787	−0.0584483	−0.0292242	−	0.999573i	$-0.509304\pi$
$379$	−1.13787	−0.0584483	−0.0292242	+	0.999573i	$0.509304\pi$
$380$	0	0
$381$	8.98471	0.460301
$382$	0	0
$383$	18.1329	0.926548	0.463274	−	0.886215i	$-0.346675\pi$
$383$	18.1329	0.926548	0.463274	+	0.886215i	$0.346675\pi$
$384$	0	0
$385$	−0.794158	−0.0404740
$386$	0	0
$387$	−21.0243	−1.06872
$388$	0	0
$389$	18.4550	0.935703	0.467852	−	0.883807i	$-0.345028\pi$
$389$	18.4550	0.935703	0.467852	+	0.883807i	$0.345028\pi$
$390$	0	0
$391$	−20.0113	−1.01202
$392$	0	0
$393$	−12.3779	−0.624383
$394$	0	0
$395$	−16.8787	−0.849261
$396$	0	0
$397$	16.1629	0.811195	0.405597	−	0.914052i	$-0.367064\pi$
$397$	16.1629	0.811195	0.405597	+	0.914052i	$0.367064\pi$
$398$	0	0
$399$	9.47137	0.474161
$400$	0	0
$401$	13.7114	0.684716	0.342358	−	0.939570i	$-0.388774\pi$
$401$	13.7114	0.684716	0.342358	+	0.939570i	$0.388774\pi$
$402$	0	0
$403$	29.2304	1.45607
$404$	0	0
$405$	2.47295	0.122882
$406$	0	0
$407$	−3.37963	−0.167522
$408$	0	0
$409$	22.8082	1.12779	0.563896	−	0.825846i	$-0.309302\pi$
$409$	22.8082	1.12779	0.563896	+	0.825846i	$0.309302\pi$
$410$	0	0
$411$	−12.9203	−0.637312
$412$	0	0
$413$	4.62793	0.227726
$414$	0	0
$415$	9.60457	0.471470
$416$	0	0
$417$	3.11258	0.152424
$418$	0	0
$419$	−35.4599	−1.73233	−0.866165	−	0.499759i	$-0.833422\pi$
$419$	−35.4599	−1.73233	−0.866165	+	0.499759i	$0.833422\pi$
$420$	0	0
$421$	6.42872	0.313317	0.156658	−	0.987653i	$-0.449928\pi$
$421$	6.42872	0.313317	0.156658	+	0.987653i	$0.449928\pi$
$422$	0	0
$423$	−14.9422	−0.726514
$424$	0	0
$425$	−18.9413	−0.918788
$426$	0	0
$427$	9.60694	0.464912
$428$	0	0
$429$	−2.23800	−0.108052
$430$	0	0
$431$	−38.0471	−1.83266	−0.916332	−	0.400419i	$-0.868865\pi$
$431$	−38.0471	−1.83266	−0.916332	+	0.400419i	$0.868865\pi$
$432$	0	0
$433$	−18.8500	−0.905874	−0.452937	−	0.891543i	$-0.649624\pi$
$433$	−18.8500	−0.905874	−0.452937	+	0.891543i	$0.649624\pi$
$434$	0	0
$435$	−7.29103	−0.349578
$436$	0	0
$437$	19.6078	0.937968
$438$	0	0
$439$	−1.61058	−0.0768689	−0.0384345	−	0.999261i	$-0.512237\pi$
$439$	−1.61058	−0.0768689	−0.0384345	+	0.999261i	$0.512237\pi$
$440$	0	0
$441$	6.77974	0.322845
$442$	0	0
$443$	−1.95924	−0.0930864	−0.0465432	−	0.998916i	$-0.514821\pi$
$443$	−1.95924	−0.0930864	−0.0465432	+	0.998916i	$0.514821\pi$
$444$	0	0
$445$	11.8117	0.559928
$446$	0	0
$447$	−17.6561	−0.835105
$448$	0	0
$449$	9.51798	0.449181	0.224591	−	0.974453i	$-0.427896\pi$
$449$	9.51798	0.449181	0.224591	+	0.974453i	$0.427896\pi$
$450$	0	0
$451$	0.555683	0.0261661
$452$	0	0
$453$	−4.31915	−0.202931
$454$	0	0
$455$	−17.0342	−0.798574
$456$	0	0
$457$	−26.7429	−1.25098	−0.625489	−	0.780233i	$-0.715101\pi$
$457$	−26.7429	−1.25098	−0.625489	+	0.780233i	$0.715101\pi$
$458$	0	0
$459$	25.3818	1.18472
$460$	0	0
$461$	21.7085	1.01107	0.505534	−	0.862807i	$-0.331296\pi$
$461$	21.7085	1.01107	0.505534	+	0.862807i	$0.331296\pi$
$462$	0	0
$463$	−6.05929	−0.281599	−0.140799	−	0.990038i	$-0.544967\pi$
$463$	−6.05929	−0.281599	−0.140799	+	0.990038i	$0.544967\pi$
$464$	0	0
$465$	4.51284	0.209278
$466$	0	0
$467$	−16.7757	−0.776289	−0.388145	−	0.921599i	$-0.626884\pi$
$467$	−16.7757	−0.776289	−0.388145	+	0.921599i	$0.626884\pi$
$468$	0	0
$469$	20.3912	0.941580
$470$	0	0
$471$	−3.28991	−0.151591
$472$	0	0
$473$	−3.29127	−0.151333
$474$	0	0
$475$	18.5593	0.851561
$476$	0	0
$477$	−20.5705	−0.941859
$478$	0	0
$479$	−16.0458	−0.733153	−0.366576	−	0.930388i	$-0.619470\pi$
$479$	−16.0458	−0.733153	−0.366576	+	0.930388i	$0.619470\pi$
$480$	0	0
$481$	−72.4909	−3.30530
$482$	0	0
$483$	6.78416	0.308690
$484$	0	0
$485$	−11.1276	−0.505278
$486$	0	0
$487$	0.953820	0.0432217	0.0216109	−	0.999766i	$-0.493121\pi$
$487$	0.953820	0.0432217	0.0216109	+	0.999766i	$0.493121\pi$
$488$	0	0
$489$	8.97399	0.405818
$490$	0	0
$491$	−3.98749	−0.179953	−0.0899764	−	0.995944i	$-0.528679\pi$
$491$	−3.98749	−0.179953	−0.0899764	+	0.995944i	$0.528679\pi$
$492$	0	0
$493$	34.9764	1.57526
$494$	0	0
$495$	0.869892	0.0390987
$496$	0	0
$497$	8.46955	0.379911
$498$	0	0
$499$	18.7447	0.839127	0.419564	−	0.907726i	$-0.362183\pi$
$499$	18.7447	0.839127	0.419564	+	0.907726i	$0.362183\pi$
$500$	0	0
$501$	−4.29121	−0.191717
$502$	0	0
$503$	−16.6402	−0.741951	−0.370975	−	0.928643i	$-0.620977\pi$
$503$	−16.6402	−0.741951	−0.370975	+	0.928643i	$0.620977\pi$
$504$	0	0
$505$	−4.10970	−0.182879
$506$	0	0
$507$	−35.9982	−1.59873
$508$	0	0
$509$	7.89623	0.349994	0.174997	−	0.984569i	$-0.444008\pi$
$509$	7.89623	0.349994	0.174997	+	0.984569i	$0.444008\pi$
$510$	0	0
$511$	12.0105	0.531314
$512$	0	0
$513$	−24.8700	−1.09804
$514$	0	0
$515$	11.0321	0.486130
$516$	0	0
$517$	−2.33914	−0.102875
$518$	0	0
$519$	8.38602	0.368105
$520$	0	0
$521$	−13.5986	−0.595766	−0.297883	−	0.954602i	$-0.596281\pi$
$521$	−13.5986	−0.595766	−0.297883	+	0.954602i	$0.596281\pi$
$522$	0	0
$523$	39.8445	1.74228	0.871139	−	0.491037i	$-0.163382\pi$
$523$	39.8445	1.74228	0.871139	+	0.491037i	$0.163382\pi$
$524$	0	0
$525$	6.42141	0.280253
$526$	0	0
$527$	−21.6489	−0.943041
$528$	0	0
$529$	−8.95532	−0.389362
$530$	0	0
$531$	−5.06927	−0.219988
$532$	0	0
$533$	11.9190	0.516271
$534$	0	0
$535$	−2.56623	−0.110948
$536$	0	0
$537$	−20.9074	−0.902221
$538$	0	0
$539$	1.06134	0.0457152
$540$	0	0
$541$	21.2045	0.911651	0.455826	−	0.890069i	$-0.349344\pi$
$541$	21.2045	0.911651	0.455826	+	0.890069i	$0.349344\pi$
$542$	0	0
$543$	16.7188	0.717474
$544$	0	0
$545$	3.10371	0.132948
$546$	0	0
$547$	−23.9161	−1.02258	−0.511289	−	0.859409i	$-0.670832\pi$
$547$	−23.9161	−1.02258	−0.511289	+	0.859409i	$0.670832\pi$
$548$	0	0
$549$	−10.5231	−0.449115
$550$	0	0
$551$	−34.2711	−1.46000
$552$	0	0
$553$	−27.4503	−1.16730
$554$	0	0
$555$	−11.1918	−0.475063
$556$	0	0
$557$	−19.3644	−0.820496	−0.410248	−	0.911974i	$-0.634558\pi$
$557$	−19.3644	−0.820496	−0.410248	+	0.911974i	$0.634558\pi$
$558$	0	0
$559$	−70.5956	−2.98588
$560$	0	0
$561$	1.65753	0.0699809
$562$	0	0
$563$	−25.2581	−1.06450	−0.532251	−	0.846587i	$-0.678654\pi$
$563$	−25.2581	−1.06450	−0.532251	+	0.846587i	$0.678654\pi$
$564$	0	0
$565$	−1.04188	−0.0438324
$566$	0	0
$567$	4.02181	0.168900
$568$	0	0
$569$	2.46847	0.103484	0.0517419	−	0.998660i	$-0.483523\pi$
$569$	2.46847	0.103484	0.0517419	+	0.998660i	$0.483523\pi$
$570$	0	0
$571$	−6.59362	−0.275934	−0.137967	−	0.990437i	$-0.544057\pi$
$571$	−6.59362	−0.275934	−0.137967	+	0.990437i	$0.544057\pi$
$572$	0	0
$573$	18.9323	0.790908
$574$	0	0
$575$	13.2937	0.554386
$576$	0	0
$577$	−33.9603	−1.41379	−0.706893	−	0.707321i	$-0.749904\pi$
$577$	−33.9603	−1.41379	−0.706893	+	0.707321i	$0.749904\pi$
$578$	0	0
$579$	−13.4257	−0.557952
$580$	0	0
$581$	15.6201	0.648032
$582$	0	0
$583$	−3.22024	−0.133369
$584$	0	0
$585$	18.6586	0.771439
$586$	0	0
$587$	7.73744	0.319358	0.159679	−	0.987169i	$-0.448954\pi$
$587$	7.73744	0.319358	0.159679	+	0.987169i	$0.448954\pi$
$588$	0	0
$589$	21.2124	0.874040
$590$	0	0
$591$	7.68771	0.316230
$592$	0	0
$593$	−29.5037	−1.21157	−0.605786	−	0.795628i	$-0.707141\pi$
$593$	−29.5037	−1.21157	−0.605786	+	0.795628i	$0.707141\pi$
$594$	0	0
$595$	12.6160	0.517205
$596$	0	0
$597$	13.7991	0.564761
$598$	0	0
$599$	−12.7885	−0.522522	−0.261261	−	0.965268i	$-0.584138\pi$
$599$	−12.7885	−0.522522	−0.261261	+	0.965268i	$0.584138\pi$
$600$	0	0
$601$	−12.5964	−0.513816	−0.256908	−	0.966436i	$-0.582704\pi$
$601$	−12.5964	−0.513816	−0.256908	+	0.966436i	$0.582704\pi$
$602$	0	0
$603$	−22.3358	−0.909585
$604$	0	0
$605$	−13.1222	−0.533493
$606$	0	0
$607$	31.1994	1.26634	0.633172	−	0.774011i	$-0.281753\pi$
$607$	31.1994	1.26634	0.633172	+	0.774011i	$0.281753\pi$
$608$	0	0
$609$	−11.8576	−0.480493
$610$	0	0
$611$	−50.1730	−2.02978
$612$	0	0
$613$	7.53670	0.304405	0.152202	−	0.988349i	$-0.451363\pi$
$613$	7.53670	0.304405	0.152202	+	0.988349i	$0.451363\pi$
$614$	0	0
$615$	1.84016	0.0742024
$616$	0	0
$617$	−34.6614	−1.39542	−0.697708	−	0.716382i	$-0.745797\pi$
$617$	−34.6614	−1.39542	−0.697708	+	0.716382i	$0.745797\pi$
$618$	0	0
$619$	−26.9664	−1.08387	−0.541936	−	0.840420i	$-0.682309\pi$
$619$	−26.9664	−1.08387	−0.541936	+	0.840420i	$0.682309\pi$
$620$	0	0
$621$	−17.8139	−0.714847
$622$	0	0
$623$	19.2096	0.769617
$624$	0	0
$625$	5.31906	0.212762
$626$	0	0
$627$	−1.62410	−0.0648604
$628$	0	0
$629$	53.6889	2.14072
$630$	0	0
$631$	−5.86488	−0.233477	−0.116739	−	0.993163i	$-0.537244\pi$
$631$	−5.86488	−0.233477	−0.116739	+	0.993163i	$0.537244\pi$
$632$	0	0
$633$	−0.869394	−0.0345553
$634$	0	0
$635$	−11.7264	−0.465349
$636$	0	0
$637$	22.7651	0.901986
$638$	0	0
$639$	−9.27724	−0.367002
$640$	0	0
$641$	25.2435	0.997060	0.498530	−	0.866872i	$-0.333873\pi$
$641$	25.2435	0.997060	0.498530	+	0.866872i	$0.333873\pi$
$642$	0	0
$643$	33.4918	1.32079	0.660394	−	0.750919i	$-0.270389\pi$
$643$	33.4918	1.32079	0.660394	+	0.750919i	$0.270389\pi$
$644$	0	0
$645$	−10.8991	−0.429153
$646$	0	0
$647$	8.47680	0.333257	0.166629	−	0.986020i	$-0.446712\pi$
$647$	8.47680	0.333257	0.166629	+	0.986020i	$0.446712\pi$
$648$	0	0
$649$	−0.793575	−0.0311505
$650$	0	0
$651$	7.33933	0.287651
$652$	0	0
$653$	−24.7050	−0.966780	−0.483390	−	0.875405i	$-0.660595\pi$
$653$	−24.7050	−0.966780	−0.483390	+	0.875405i	$0.660595\pi$
$654$	0	0
$655$	16.1551	0.631231
$656$	0	0
$657$	−13.1559	−0.513260
$658$	0	0
$659$	−44.3046	−1.72586	−0.862932	−	0.505320i	$-0.831374\pi$
$659$	−44.3046	−1.72586	−0.862932	+	0.505320i	$0.831374\pi$
$660$	0	0
$661$	8.11133	0.315494	0.157747	−	0.987480i	$-0.449577\pi$
$661$	8.11133	0.315494	0.157747	+	0.987480i	$0.449577\pi$
$662$	0	0
$663$	35.5529	1.38076
$664$	0	0
$665$	−12.3616	−0.479362
$666$	0	0
$667$	−24.5477	−0.950492
$668$	0	0
$669$	−11.8798	−0.459299
$670$	0	0
$671$	−1.64735	−0.0635952
$672$	0	0
$673$	34.2480	1.32016	0.660082	−	0.751193i	$-0.270521\pi$
$673$	34.2480	1.32016	0.660082	+	0.751193i	$0.270521\pi$
$674$	0	0
$675$	−16.8614	−0.648995
$676$	0	0
$677$	32.1371	1.23513	0.617564	−	0.786521i	$-0.288120\pi$
$677$	32.1371	1.23513	0.617564	+	0.786521i	$0.288120\pi$
$678$	0	0
$679$	−18.0970	−0.694500
$680$	0	0
$681$	−12.6361	−0.484218
$682$	0	0
$683$	48.6013	1.85968	0.929838	−	0.367969i	$-0.119947\pi$
$683$	48.6013	1.85968	0.929838	+	0.367969i	$0.119947\pi$
$684$	0	0
$685$	16.8630	0.644302
$686$	0	0
$687$	−2.57928	−0.0984057
$688$	0	0
$689$	−69.0719	−2.63143
$690$	0	0
$691$	−24.8010	−0.943473	−0.471736	−	0.881740i	$-0.656373\pi$
$691$	−24.8010	−0.943473	−0.471736	+	0.881740i	$0.656373\pi$
$692$	0	0
$693$	1.41472	0.0537409
$694$	0	0
$695$	−4.06239	−0.154095
$696$	0	0
$697$	−8.82759	−0.334369
$698$	0	0
$699$	8.74886	0.330912
$700$	0	0
$701$	−19.0603	−0.719897	−0.359948	−	0.932972i	$-0.617206\pi$
$701$	−19.0603	−0.719897	−0.359948	+	0.932972i	$0.617206\pi$
$702$	0	0
$703$	−52.6062	−1.98408
$704$	0	0
$705$	−7.74613	−0.291736
$706$	0	0
$707$	−6.68369	−0.251366
$708$	0	0
$709$	13.4836	0.506386	0.253193	−	0.967416i	$-0.418519\pi$
$709$	13.4836	0.506386	0.253193	+	0.967416i	$0.418519\pi$
$710$	0	0
$711$	30.0680	1.12764
$712$	0	0
$713$	15.1940	0.569020
$714$	0	0
$715$	2.92093	0.109237
$716$	0	0
$717$	−7.57358	−0.282840
$718$	0	0
$719$	−3.54113	−0.132062	−0.0660309	−	0.997818i	$-0.521034\pi$
$719$	−3.54113	−0.132062	−0.0660309	+	0.997818i	$0.521034\pi$
$720$	0	0
$721$	17.9417	0.668183
$722$	0	0
$723$	9.08005	0.337691
$724$	0	0
$725$	−23.2352	−0.862932
$726$	0	0
$727$	30.6007	1.13492	0.567458	−	0.823403i	$-0.307927\pi$
$727$	30.6007	1.13492	0.567458	+	0.823403i	$0.307927\pi$
$728$	0	0
$729$	8.76388	0.324588
$730$	0	0
$731$	52.2852	1.93384
$732$	0	0
$733$	−28.9617	−1.06973	−0.534863	−	0.844939i	$-0.679637\pi$
$733$	−28.9617	−1.06973	−0.534863	+	0.844939i	$0.679637\pi$
$734$	0	0
$735$	3.51466	0.129640
$736$	0	0
$737$	−3.49659	−0.128798
$738$	0	0
$739$	11.7200	0.431129	0.215564	−	0.976490i	$-0.430841\pi$
$739$	11.7200	0.431129	0.215564	+	0.976490i	$0.430841\pi$
$740$	0	0
$741$	−34.8360	−1.27973
$742$	0	0
$743$	31.1018	1.14101	0.570507	−	0.821293i	$-0.306747\pi$
$743$	31.1018	1.14101	0.570507	+	0.821293i	$0.306747\pi$
$744$	0	0
$745$	23.0439	0.844264
$746$	0	0
$747$	−17.1097	−0.626012
$748$	0	0
$749$	−4.17351	−0.152497
$750$	0	0
$751$	4.09394	0.149390	0.0746950	−	0.997206i	$-0.476202\pi$
$751$	4.09394	0.149390	0.0746950	+	0.997206i	$0.476202\pi$
$752$	0	0
$753$	0.923498	0.0336541
$754$	0	0
$755$	5.63715	0.205157
$756$	0	0
$757$	−4.43089	−0.161043	−0.0805217	−	0.996753i	$-0.525659\pi$
$757$	−4.43089	−0.161043	−0.0805217	+	0.996753i	$0.525659\pi$
$758$	0	0
$759$	−1.16331	−0.0422256
$760$	0	0
$761$	29.0057	1.05146	0.525729	−	0.850652i	$-0.323793\pi$
$761$	29.0057	1.05146	0.525729	+	0.850652i	$0.323793\pi$
$762$	0	0
$763$	5.04763	0.182736
$764$	0	0
$765$	−13.8191	−0.499631
$766$	0	0
$767$	−17.0217	−0.614616
$768$	0	0
$769$	27.0297	0.974717	0.487359	−	0.873202i	$-0.337960\pi$
$769$	27.0297	0.974717	0.487359	+	0.873202i	$0.337960\pi$
$770$	0	0
$771$	7.75242	0.279197
$772$	0	0
$773$	−6.78313	−0.243972	−0.121986	−	0.992532i	$-0.538926\pi$
$773$	−6.78313	−0.243972	−0.121986	+	0.992532i	$0.538926\pi$
$774$	0	0
$775$	14.3816	0.516601
$776$	0	0
$777$	−18.2014	−0.652971
$778$	0	0
$779$	8.64958	0.309903
$780$	0	0
$781$	−1.45232	−0.0519680
$782$	0	0
$783$	31.1357	1.11270
$784$	0	0
$785$	4.29383	0.153254
$786$	0	0
$787$	43.8347	1.56254	0.781269	−	0.624194i	$-0.214573\pi$
$787$	43.8347	1.56254	0.781269	+	0.624194i	$0.214573\pi$
$788$	0	0
$789$	−19.4334	−0.691849
$790$	0	0
$791$	−1.69444	−0.0602473
$792$	0	0
$793$	−35.3346	−1.25477
$794$	0	0
$795$	−10.6639	−0.378210
$796$	0	0
$797$	−9.61773	−0.340678	−0.170339	−	0.985386i	$-0.554486\pi$
$797$	−9.61773	−0.340678	−0.170339	+	0.985386i	$0.554486\pi$
$798$	0	0
$799$	37.1596	1.31461
$800$	0	0
$801$	−21.0415	−0.743466
$802$	0	0
$803$	−2.05951	−0.0726784
$804$	0	0
$805$	−8.85437	−0.312076
$806$	0	0
$807$	18.2545	0.642589
$808$	0	0
$809$	−22.2436	−0.782042	−0.391021	−	0.920382i	$-0.627878\pi$
$809$	−22.2436	−0.782042	−0.391021	+	0.920382i	$0.627878\pi$
$810$	0	0
$811$	−23.7718	−0.834740	−0.417370	−	0.908737i	$-0.637048\pi$
$811$	−23.7718	−0.834740	−0.417370	+	0.908737i	$0.637048\pi$
$812$	0	0
$813$	−0.353379	−0.0123936
$814$	0	0
$815$	−11.7124	−0.410269
$816$	0	0
$817$	−51.2308	−1.79234
$818$	0	0
$819$	30.3449	1.06034
$820$	0	0
$821$	2.89772	0.101131	0.0505655	−	0.998721i	$-0.483898\pi$
$821$	2.89772	0.101131	0.0505655	+	0.998721i	$0.483898\pi$
$822$	0	0
$823$	−17.2369	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$823$	−17.2369	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$824$	0	0
$825$	−1.10111	−0.0383358
$826$	0	0
$827$	−35.0687	−1.21946	−0.609729	−	0.792610i	$-0.708722\pi$
$827$	−35.0687	−1.21946	−0.609729	+	0.792610i	$0.708722\pi$
$828$	0	0
$829$	−22.3362	−0.775768	−0.387884	−	0.921708i	$-0.626794\pi$
$829$	−22.3362	−0.775768	−0.387884	+	0.921708i	$0.626794\pi$
$830$	0	0
$831$	−26.9597	−0.935222
$832$	0	0
$833$	−16.8605	−0.584181
$834$	0	0
$835$	5.60068	0.193820
$836$	0	0
$837$	−19.2717	−0.666126
$838$	0	0
$839$	57.6575	1.99056	0.995279	−	0.0970527i	$-0.0309415\pi$
$839$	57.6575	1.99056	0.995279	+	0.0970527i	$0.0309415\pi$
$840$	0	0
$841$	13.9053	0.479493
$842$	0	0
$843$	7.78339	0.268074
$844$	0	0
$845$	46.9831	1.61627
$846$	0	0
$847$	−21.3409	−0.733283
$848$	0	0
$849$	16.8297	0.577594
$850$	0	0
$851$	−37.6808	−1.29168
$852$	0	0
$853$	−25.2390	−0.864168	−0.432084	−	0.901833i	$-0.642222\pi$
$853$	−25.2390	−0.864168	−0.432084	+	0.901833i	$0.642222\pi$
$854$	0	0
$855$	13.5404	0.463073
$856$	0	0
$857$	−34.1323	−1.16594	−0.582969	−	0.812494i	$-0.698109\pi$
$857$	−34.1323	−1.16594	−0.582969	+	0.812494i	$0.698109\pi$
$858$	0	0
$859$	−38.3079	−1.30705	−0.653525	−	0.756905i	$-0.726710\pi$
$859$	−38.3079	−1.30705	−0.653525	+	0.756905i	$0.726710\pi$
$860$	0	0
$861$	2.99269	0.101991
$862$	0	0
$863$	−41.0831	−1.39849	−0.699243	−	0.714884i	$-0.746479\pi$
$863$	−41.0831	−1.39849	−0.699243	+	0.714884i	$0.746479\pi$
$864$	0	0
$865$	−10.9450	−0.372143
$866$	0	0
$867$	−10.6320	−0.361083
$868$	0	0
$869$	4.70703	0.159675
$870$	0	0
$871$	−74.9995	−2.54126
$872$	0	0
$873$	19.8228	0.670902
$874$	0	0
$875$	−20.1942	−0.682690
$876$	0	0
$877$	−29.8185	−1.00690	−0.503450	−	0.864024i	$-0.667936\pi$
$877$	−29.8185	−1.00690	−0.503450	+	0.864024i	$0.667936\pi$
$878$	0	0
$879$	−5.09941	−0.171999
$880$	0	0
$881$	28.1656	0.948923	0.474461	−	0.880276i	$-0.342643\pi$
$881$	28.1656	0.948923	0.474461	+	0.880276i	$0.342643\pi$
$882$	0	0
$883$	10.4599	0.352004	0.176002	−	0.984390i	$-0.443683\pi$
$883$	10.4599	0.352004	0.176002	+	0.984390i	$0.443683\pi$
$884$	0	0
$885$	−2.62794	−0.0883374
$886$	0	0
$887$	−3.88170	−0.130335	−0.0651674	−	0.997874i	$-0.520758\pi$
$887$	−3.88170	−0.130335	−0.0651674	+	0.997874i	$0.520758\pi$
$888$	0	0
$889$	−19.0710	−0.639619
$890$	0	0
$891$	−0.689639	−0.0231038
$892$	0	0
$893$	−36.4103	−1.21842
$894$	0	0
$895$	27.2874	0.912117
$896$	0	0
$897$	−24.9523	−0.833134
$898$	0	0
$899$	−26.5565	−0.885711
$900$	0	0
$901$	51.1567	1.70428
$902$	0	0
$903$	−17.7255	−0.589868
$904$	0	0
$905$	−21.8207	−0.725343
$906$	0	0
$907$	−25.5147	−0.847203	−0.423602	−	0.905849i	$-0.639234\pi$
$907$	−25.5147	−0.847203	−0.423602	+	0.905849i	$0.639234\pi$
$908$	0	0
$909$	7.32108	0.242825
$910$	0	0
$911$	36.7188	1.21655	0.608274	−	0.793728i	$-0.291862\pi$
$911$	36.7188	1.21655	0.608274	+	0.793728i	$0.291862\pi$
$912$	0	0
$913$	−2.67846	−0.0886442
$914$	0	0
$915$	−5.45525	−0.180345
$916$	0	0
$917$	26.2733	0.867622
$918$	0	0
$919$	−13.1987	−0.435384	−0.217692	−	0.976018i	$-0.569853\pi$
$919$	−13.1987	−0.435384	−0.217692	+	0.976018i	$0.569853\pi$
$920$	0	0
$921$	−3.36730	−0.110956
$922$	0	0
$923$	−31.1512	−1.02536
$924$	0	0
$925$	−35.6660	−1.17269
$926$	0	0
$927$	−19.6527	−0.645478
$928$	0	0
$929$	11.0489	0.362504	0.181252	−	0.983437i	$-0.441985\pi$
$929$	11.0489	0.362504	0.181252	+	0.983437i	$0.441985\pi$
$930$	0	0
$931$	16.5205	0.541437
$932$	0	0
$933$	−20.9162	−0.684767
$934$	0	0
$935$	−2.16333	−0.0707484
$936$	0	0
$937$	8.07355	0.263751	0.131876	−	0.991266i	$-0.457900\pi$
$937$	8.07355	0.263751	0.131876	+	0.991266i	$0.457900\pi$
$938$	0	0
$939$	18.7988	0.613476
$940$	0	0
$941$	27.0114	0.880547	0.440273	−	0.897864i	$-0.354882\pi$
$941$	27.0114	0.880547	0.440273	+	0.897864i	$0.354882\pi$
$942$	0	0
$943$	6.19553	0.201754
$944$	0	0
$945$	11.2306	0.365333
$946$	0	0
$947$	−3.62792	−0.117891	−0.0589457	−	0.998261i	$-0.518774\pi$
$947$	−3.62792	−0.117891	−0.0589457	+	0.998261i	$0.518774\pi$
$948$	0	0
$949$	−44.1750	−1.43398
$950$	0	0
$951$	−5.58834	−0.181214
$952$	0	0
$953$	42.0613	1.36250	0.681249	−	0.732052i	$-0.261437\pi$
$953$	42.0613	1.36250	0.681249	+	0.732052i	$0.261437\pi$
$954$	0	0
$955$	−24.7095	−0.799582
$956$	0	0
$957$	2.03328	0.0657265
$958$	0	0
$959$	27.4247	0.885589
$960$	0	0
$961$	−14.5626	−0.469762
$962$	0	0
$963$	4.57152	0.147315
$964$	0	0
$965$	17.5226	0.564071
$966$	0	0
$967$	−34.8255	−1.11991	−0.559957	−	0.828522i	$-0.689182\pi$
$967$	−34.8255	−1.11991	−0.559957	+	0.828522i	$0.689182\pi$
$968$	0	0
$969$	25.8005	0.828832
$970$	0	0
$971$	−57.8898	−1.85777	−0.928886	−	0.370366i	$-0.879232\pi$
$971$	−57.8898	−1.85777	−0.928886	+	0.370366i	$0.879232\pi$
$972$	0	0
$973$	−6.60676	−0.211803
$974$	0	0
$975$	−23.6181	−0.756385
$976$	0	0
$977$	−49.2739	−1.57641	−0.788206	−	0.615411i	$-0.788990\pi$
$977$	−49.2739	−1.57641	−0.788206	+	0.615411i	$0.788990\pi$
$978$	0	0
$979$	−3.29397	−0.105276
$980$	0	0
$981$	−5.52899	−0.176527
$982$	0	0
$983$	−24.5307	−0.782408	−0.391204	−	0.920304i	$-0.627941\pi$
$983$	−24.5307	−0.782408	−0.391204	+	0.920304i	$0.627941\pi$
$984$	0	0
$985$	−10.0336	−0.319699
$986$	0	0
$987$	−12.5977	−0.400989
$988$	0	0
$989$	−36.6957	−1.16685
$990$	0	0
$991$	15.2097	0.483151	0.241575	−	0.970382i	$-0.422336\pi$
$991$	15.2097	0.483151	0.241575	+	0.970382i	$0.422336\pi$
$992$	0	0
$993$	−26.7114	−0.847659
$994$	0	0
$995$	−18.0100	−0.570955
$996$	0	0
$997$	60.2651	1.90862	0.954308	−	0.298824i	$-0.0965944\pi$
$997$	60.2651	1.90862	0.954308	+	0.298824i	$0.0965944\pi$
$998$	0	0
$999$	47.7934	1.51211

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.8		17
4.3	odd	2		251.2.a.b.1.14	✓	17
12.11	even	2		2259.2.a.k.1.4		17
20.19	odd	2		6275.2.a.e.1.4		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.14	✓	17	4.3	odd	2
2259.2.a.k.1.4		17	12.11	even	2
4016.2.a.k.1.8		17	1.1	even	1	trivial
6275.2.a.e.1.4		17	20.19	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:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-0.923498 q^{3} +1.20531 q^{5} +1.96022 q^{7} -2.14715 q^{9} +O(q^{10})\)	\(q-0.923498 q^{3} +1.20531 q^{5} +1.96022 q^{7} -2.14715 q^{9} -0.336128 q^{11} -7.20973 q^{13} -1.11310 q^{15} +5.33974 q^{17} -5.23206 q^{19} -1.81026 q^{21} -3.74762 q^{23} -3.54724 q^{25} +4.75338 q^{27} +6.55021 q^{29} -4.05430 q^{31} +0.310414 q^{33} +2.36266 q^{35} +10.0546 q^{37} +6.65817 q^{39} -1.65319 q^{41} +9.79171 q^{43} -2.58798 q^{45} +6.95907 q^{47} -3.15755 q^{49} -4.93124 q^{51} +9.58038 q^{53} -0.405138 q^{55} +4.83180 q^{57} +2.36093 q^{59} +4.90096 q^{61} -4.20888 q^{63} -8.68994 q^{65} +10.4025 q^{67} +3.46092 q^{69} +4.32072 q^{71} +6.12714 q^{73} +3.27586 q^{75} -0.658884 q^{77} -14.0037 q^{79} +2.05171 q^{81} +7.96857 q^{83} +6.43602 q^{85} -6.04911 q^{87} +9.79974 q^{89} -14.1326 q^{91} +3.74414 q^{93} -6.30624 q^{95} -9.23216 q^{97} +0.721718 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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