LMFDB - Embedded newform 4016.2.a.h.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:	$1$
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			1.8
Root			$1.92428$ 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)

$f(q)$	$=$	$q+1.92428 q^{3} +0.603023 q^{5} -4.00229 q^{7} +0.702869 q^{9} +O(q^{10})$	$q+1.92428 q^{3} +0.603023 q^{5} -4.00229 q^{7} +0.702869 q^{9} +2.54052 q^{11} +0.384152 q^{13} +1.16039 q^{15} -4.88169 q^{17} -0.138707 q^{19} -7.70154 q^{21} +4.63393 q^{23} -4.63636 q^{25} -4.42033 q^{27} +0.641649 q^{29} -0.324579 q^{31} +4.88868 q^{33} -2.41347 q^{35} +2.71005 q^{37} +0.739217 q^{39} -11.1014 q^{41} -5.21463 q^{43} +0.423846 q^{45} -2.12420 q^{47} +9.01833 q^{49} -9.39376 q^{51} -3.74340 q^{53} +1.53199 q^{55} -0.266912 q^{57} -6.32385 q^{59} -7.81794 q^{61} -2.81309 q^{63} +0.231652 q^{65} -3.82671 q^{67} +8.91699 q^{69} +8.42635 q^{71} +1.50056 q^{73} -8.92168 q^{75} -10.1679 q^{77} +2.50191 q^{79} -10.6146 q^{81} +8.03974 q^{83} -2.94377 q^{85} +1.23471 q^{87} +0.594707 q^{89} -1.53749 q^{91} -0.624582 q^{93} -0.0836438 q^{95} -13.4617 q^{97} +1.78565 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	1.92428	1.11099	0.555493	−	0.831521i	$-0.312530\pi$
$3$	1.92428	1.11099	0.555493	+	0.831521i	$0.312530\pi$
$4$	0	0
$5$	0.603023	0.269680	0.134840	−	0.990867i	$-0.456948\pi$
$5$	0.603023	0.269680	0.134840	+	0.990867i	$0.456948\pi$
$6$	0	0
$7$	−4.00229	−1.51272	−0.756362	−	0.654153i	$-0.773025\pi$
$7$	−4.00229	−1.51272	−0.756362	+	0.654153i	$0.773025\pi$
$8$	0	0
$9$	0.702869	0.234290
$10$	0	0
$11$	2.54052	0.765996	0.382998	−	0.923749i	$-0.374892\pi$
$11$	2.54052	0.765996	0.382998	+	0.923749i	$0.374892\pi$
$12$	0	0
$13$	0.384152	0.106545	0.0532723	−	0.998580i	$-0.483035\pi$
$13$	0.384152	0.106545	0.0532723	+	0.998580i	$0.483035\pi$
$14$	0	0
$15$	1.16039	0.299611
$16$	0	0
$17$	−4.88169	−1.18398	−0.591992	−	0.805944i	$-0.701658\pi$
$17$	−4.88169	−1.18398	−0.591992	+	0.805944i	$0.701658\pi$
$18$	0	0
$19$	−0.138707	−0.0318217	−0.0159108	−	0.999873i	$-0.505065\pi$
$19$	−0.138707	−0.0318217	−0.0159108	+	0.999873i	$0.505065\pi$
$20$	0	0
$21$	−7.70154	−1.68061
$22$	0	0
$23$	4.63393	0.966241	0.483120	−	0.875554i	$-0.339503\pi$
$23$	4.63393	0.966241	0.483120	+	0.875554i	$0.339503\pi$
$24$	0	0
$25$	−4.63636	−0.927273
$26$	0	0
$27$	−4.42033	−0.850693
$28$	0	0
$29$	0.641649	0.119151	0.0595756	−	0.998224i	$-0.481025\pi$
$29$	0.641649	0.119151	0.0595756	+	0.998224i	$0.481025\pi$
$30$	0	0
$31$	−0.324579	−0.0582961	−0.0291481	−	0.999575i	$-0.509279\pi$
$31$	−0.324579	−0.0582961	−0.0291481	+	0.999575i	$0.509279\pi$
$32$	0	0
$33$	4.88868	0.851010
$34$	0	0
$35$	−2.41347	−0.407951
$36$	0	0
$37$	2.71005	0.445530	0.222765	−	0.974872i	$-0.428492\pi$
$37$	2.71005	0.445530	0.222765	+	0.974872i	$0.428492\pi$
$38$	0	0
$39$	0.739217	0.118369
$40$	0	0
$41$	−11.1014	−1.73374	−0.866871	−	0.498533i	$-0.833872\pi$
$41$	−11.1014	−1.73374	−0.866871	+	0.498533i	$0.833872\pi$
$42$	0	0
$43$	−5.21463	−0.795223	−0.397612	−	0.917554i	$-0.630161\pi$
$43$	−5.21463	−0.795223	−0.397612	+	0.917554i	$0.630161\pi$
$44$	0	0
$45$	0.423846	0.0631833
$46$	0	0
$47$	−2.12420	−0.309847	−0.154924	−	0.987926i	$-0.549513\pi$
$47$	−2.12420	−0.309847	−0.154924	+	0.987926i	$0.549513\pi$
$48$	0	0
$49$	9.01833	1.28833
$50$	0	0
$51$	−9.39376	−1.31539
$52$	0	0
$53$	−3.74340	−0.514196	−0.257098	−	0.966385i	$-0.582766\pi$
$53$	−3.74340	−0.514196	−0.257098	+	0.966385i	$0.582766\pi$
$54$	0	0
$55$	1.53199	0.206574
$56$	0	0
$57$	−0.266912	−0.0353534
$58$	0	0
$59$	−6.32385	−0.823295	−0.411647	−	0.911343i	$-0.635046\pi$
$59$	−6.32385	−0.823295	−0.411647	+	0.911343i	$0.635046\pi$
$60$	0	0
$61$	−7.81794	−1.00098	−0.500492	−	0.865741i	$-0.666848\pi$
$61$	−7.81794	−1.00098	−0.500492	+	0.865741i	$0.666848\pi$
$62$	0	0
$63$	−2.81309	−0.354416
$64$	0	0
$65$	0.231652	0.0287329
$66$	0	0
$67$	−3.82671	−0.467507	−0.233754	−	0.972296i	$-0.575101\pi$
$67$	−3.82671	−0.467507	−0.233754	+	0.972296i	$0.575101\pi$
$68$	0	0
$69$	8.91699	1.07348
$70$	0	0
$71$	8.42635	1.00002	0.500012	−	0.866018i	$-0.333329\pi$
$71$	8.42635	1.00002	0.500012	+	0.866018i	$0.333329\pi$
$72$	0	0
$73$	1.50056	0.175627	0.0878134	−	0.996137i	$-0.472012\pi$
$73$	1.50056	0.175627	0.0878134	+	0.996137i	$0.472012\pi$
$74$	0	0
$75$	−8.92168	−1.03019
$76$	0	0
$77$	−10.1679	−1.15874
$78$	0	0
$79$	2.50191	0.281487	0.140743	−	0.990046i	$-0.455051\pi$
$79$	2.50191	0.281487	0.140743	+	0.990046i	$0.455051\pi$
$80$	0	0
$81$	−10.6146	−1.17940
$82$	0	0
$83$	8.03974	0.882476	0.441238	−	0.897390i	$-0.354539\pi$
$83$	8.03974	0.882476	0.441238	+	0.897390i	$0.354539\pi$
$84$	0	0
$85$	−2.94377	−0.319297
$86$	0	0
$87$	1.23471	0.132375
$88$	0	0
$89$	0.594707	0.0630388	0.0315194	−	0.999503i	$-0.489965\pi$
$89$	0.594707	0.0630388	0.0315194	+	0.999503i	$0.489965\pi$
$90$	0	0
$91$	−1.53749	−0.161172
$92$	0	0
$93$	−0.624582	−0.0647662
$94$	0	0
$95$	−0.0836438	−0.00858167
$96$	0	0
$97$	−13.4617	−1.36683	−0.683413	−	0.730032i	$-0.739505\pi$
$97$	−13.4617	−1.36683	−0.683413	+	0.730032i	$0.739505\pi$
$98$	0	0
$99$	1.78565	0.179465
$100$	0	0
$101$	−9.55626	−0.950883	−0.475442	−	0.879747i	$-0.657712\pi$
$101$	−9.55626	−0.950883	−0.475442	+	0.879747i	$0.657712\pi$
$102$	0	0
$103$	11.5809	1.14110	0.570551	−	0.821262i	$-0.306730\pi$
$103$	11.5809	1.14110	0.570551	+	0.821262i	$0.306730\pi$
$104$	0	0
$105$	−4.64421	−0.453228
$106$	0	0
$107$	−2.97570	−0.287672	−0.143836	−	0.989602i	$-0.545944\pi$
$107$	−2.97570	−0.287672	−0.143836	+	0.989602i	$0.545944\pi$
$108$	0	0
$109$	7.18142	0.687856	0.343928	−	0.938996i	$-0.388242\pi$
$109$	7.18142	0.687856	0.343928	+	0.938996i	$0.388242\pi$
$110$	0	0
$111$	5.21491	0.494978
$112$	0	0
$113$	−8.66341	−0.814985	−0.407493	−	0.913208i	$-0.633597\pi$
$113$	−8.66341	−0.814985	−0.407493	+	0.913208i	$0.633597\pi$
$114$	0	0
$115$	2.79436	0.260576
$116$	0	0
$117$	0.270008	0.0249623
$118$	0	0
$119$	19.5379	1.79104
$120$	0	0
$121$	−4.54576	−0.413251
$122$	0	0
$123$	−21.3622	−1.92616
$124$	0	0
$125$	−5.81095	−0.519747
$126$	0	0
$127$	12.4637	1.10597	0.552987	−	0.833190i	$-0.313488\pi$
$127$	12.4637	1.10597	0.552987	+	0.833190i	$0.313488\pi$
$128$	0	0
$129$	−10.0344	−0.883482
$130$	0	0
$131$	−14.9630	−1.30733	−0.653664	−	0.756785i	$-0.726769\pi$
$131$	−14.9630	−1.30733	−0.653664	+	0.756785i	$0.726769\pi$
$132$	0	0
$133$	0.555147	0.0481374
$134$	0	0
$135$	−2.66556	−0.229415
$136$	0	0
$137$	5.02618	0.429415	0.214708	−	0.976678i	$-0.431120\pi$
$137$	5.02618	0.429415	0.214708	+	0.976678i	$0.431120\pi$
$138$	0	0
$139$	−0.397272	−0.0336962	−0.0168481	−	0.999858i	$-0.505363\pi$
$139$	−0.397272	−0.0336962	−0.0168481	+	0.999858i	$0.505363\pi$
$140$	0	0
$141$	−4.08757	−0.344236
$142$	0	0
$143$	0.975945	0.0816126
$144$	0	0
$145$	0.386929	0.0321327
$146$	0	0
$147$	17.3538	1.43132
$148$	0	0
$149$	−19.7072	−1.61448	−0.807240	−	0.590223i	$-0.799040\pi$
$149$	−19.7072	−1.61448	−0.807240	+	0.590223i	$0.799040\pi$
$150$	0	0
$151$	−5.35523	−0.435803	−0.217901	−	0.975971i	$-0.569921\pi$
$151$	−5.35523	−0.435803	−0.217901	+	0.975971i	$0.569921\pi$
$152$	0	0
$153$	−3.43119	−0.277395
$154$	0	0
$155$	−0.195729	−0.0157213
$156$	0	0
$157$	11.6285	0.928052	0.464026	−	0.885822i	$-0.346404\pi$
$157$	11.6285	0.928052	0.464026	+	0.885822i	$0.346404\pi$
$158$	0	0
$159$	−7.20337	−0.571264
$160$	0	0
$161$	−18.5463	−1.46166
$162$	0	0
$163$	−19.5645	−1.53241	−0.766204	−	0.642598i	$-0.777857\pi$
$163$	−19.5645	−1.53241	−0.766204	+	0.642598i	$0.777857\pi$
$164$	0	0
$165$	2.94799	0.229501
$166$	0	0
$167$	−2.92604	−0.226424	−0.113212	−	0.993571i	$-0.536114\pi$
$167$	−2.92604	−0.226424	−0.113212	+	0.993571i	$0.536114\pi$
$168$	0	0
$169$	−12.8524	−0.988648
$170$	0	0
$171$	−0.0974932	−0.00745549
$172$	0	0
$173$	−10.1084	−0.768530	−0.384265	−	0.923223i	$-0.625545\pi$
$173$	−10.1084	−0.768530	−0.384265	+	0.923223i	$0.625545\pi$
$174$	0	0
$175$	18.5561	1.40271
$176$	0	0
$177$	−12.1689	−0.914669
$178$	0	0
$179$	18.7844	1.40401	0.702005	−	0.712172i	$-0.252288\pi$
$179$	18.7844	1.40401	0.702005	+	0.712172i	$0.252288\pi$
$180$	0	0
$181$	−9.52007	−0.707622	−0.353811	−	0.935317i	$-0.615114\pi$
$181$	−9.52007	−0.707622	−0.353811	+	0.935317i	$0.615114\pi$
$182$	0	0
$183$	−15.0439	−1.11208
$184$	0	0
$185$	1.63422	0.120151
$186$	0	0
$187$	−12.4020	−0.906926
$188$	0	0
$189$	17.6915	1.28686
$190$	0	0
$191$	2.24590	0.162508	0.0812539	−	0.996693i	$-0.474108\pi$
$191$	2.24590	0.162508	0.0812539	+	0.996693i	$0.474108\pi$
$192$	0	0
$193$	−6.34804	−0.456942	−0.228471	−	0.973551i	$-0.573373\pi$
$193$	−6.34804	−0.456942	−0.228471	+	0.973551i	$0.573373\pi$
$194$	0	0
$195$	0.445765	0.0319219
$196$	0	0
$197$	−24.8509	−1.77055	−0.885276	−	0.465066i	$-0.846031\pi$
$197$	−24.8509	−1.77055	−0.885276	+	0.465066i	$0.846031\pi$
$198$	0	0
$199$	9.89221	0.701240	0.350620	−	0.936518i	$-0.385971\pi$
$199$	9.89221	0.701240	0.350620	+	0.936518i	$0.385971\pi$
$200$	0	0
$201$	−7.36368	−0.519394
$202$	0	0
$203$	−2.56806	−0.180243
$204$	0	0
$205$	−6.69438	−0.467555
$206$	0	0
$207$	3.25705	0.226380
$208$	0	0
$209$	−0.352389	−0.0243753
$210$	0	0
$211$	−0.100613	−0.00692647	−0.00346324	−	0.999994i	$-0.501102\pi$
$211$	−0.100613	−0.00692647	−0.00346324	+	0.999994i	$0.501102\pi$
$212$	0	0
$213$	16.2147	1.11101
$214$	0	0
$215$	−3.14454	−0.214456
$216$	0	0
$217$	1.29906	0.0881859
$218$	0	0
$219$	2.88750	0.195119
$220$	0	0
$221$	−1.87531	−0.126147
$222$	0	0
$223$	23.6921	1.58654	0.793271	−	0.608869i	$-0.208377\pi$
$223$	23.6921	1.58654	0.793271	+	0.608869i	$0.208377\pi$
$224$	0	0
$225$	−3.25876	−0.217251
$226$	0	0
$227$	1.76432	0.117102	0.0585510	−	0.998284i	$-0.481352\pi$
$227$	1.76432	0.117102	0.0585510	+	0.998284i	$0.481352\pi$
$228$	0	0
$229$	3.79914	0.251054	0.125527	−	0.992090i	$-0.459938\pi$
$229$	3.79914	0.251054	0.125527	+	0.992090i	$0.459938\pi$
$230$	0	0
$231$	−19.5659	−1.28734
$232$	0	0
$233$	11.9492	0.782818	0.391409	−	0.920217i	$-0.371988\pi$
$233$	11.9492	0.782818	0.391409	+	0.920217i	$0.371988\pi$
$234$	0	0
$235$	−1.28094	−0.0835596
$236$	0	0
$237$	4.81438	0.312728
$238$	0	0
$239$	−13.6752	−0.884576	−0.442288	−	0.896873i	$-0.645833\pi$
$239$	−13.6752	−0.884576	−0.442288	+	0.896873i	$0.645833\pi$
$240$	0	0
$241$	24.7093	1.59167	0.795834	−	0.605514i	$-0.207033\pi$
$241$	24.7093	1.59167	0.795834	+	0.605514i	$0.207033\pi$
$242$	0	0
$243$	−7.16448	−0.459601
$244$	0	0
$245$	5.43826	0.347438
$246$	0	0
$247$	−0.0532847	−0.00339042
$248$	0	0
$249$	15.4708	0.980419
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	11.7726	0.740136
$254$	0	0
$255$	−5.66465	−0.354734
$256$	0	0
$257$	12.3972	0.773315	0.386657	−	0.922223i	$-0.373630\pi$
$257$	12.3972	0.773315	0.386657	+	0.922223i	$0.373630\pi$
$258$	0	0
$259$	−10.8464	−0.673964
$260$	0	0
$261$	0.450995	0.0279159
$262$	0	0
$263$	−12.9721	−0.799894	−0.399947	−	0.916538i	$-0.630972\pi$
$263$	−12.9721	−0.799894	−0.399947	+	0.916538i	$0.630972\pi$
$264$	0	0
$265$	−2.25736	−0.138668
$266$	0	0
$267$	1.14438	0.0700352
$268$	0	0
$269$	26.6698	1.62609	0.813043	−	0.582203i	$-0.197809\pi$
$269$	26.6698	1.62609	0.813043	+	0.582203i	$0.197809\pi$
$270$	0	0
$271$	−1.57074	−0.0954154	−0.0477077	−	0.998861i	$-0.515192\pi$
$271$	−1.57074	−0.0954154	−0.0477077	+	0.998861i	$0.515192\pi$
$272$	0	0
$273$	−2.95856	−0.179060
$274$	0	0
$275$	−11.7788	−0.710287
$276$	0	0
$277$	14.7004	0.883260	0.441630	−	0.897197i	$-0.354400\pi$
$277$	14.7004	0.883260	0.441630	+	0.897197i	$0.354400\pi$
$278$	0	0
$279$	−0.228137	−0.0136582
$280$	0	0
$281$	15.4238	0.920106	0.460053	−	0.887892i	$-0.347830\pi$
$281$	15.4238	0.920106	0.460053	+	0.887892i	$0.347830\pi$
$282$	0	0
$283$	12.7896	0.760264	0.380132	−	0.924932i	$-0.375879\pi$
$283$	12.7896	0.760264	0.380132	+	0.924932i	$0.375879\pi$
$284$	0	0
$285$	−0.160954	−0.00953411
$286$	0	0
$287$	44.4309	2.62267
$288$	0	0
$289$	6.83090	0.401818
$290$	0	0
$291$	−25.9041	−1.51852
$292$	0	0
$293$	−21.5899	−1.26130	−0.630649	−	0.776068i	$-0.717211\pi$
$293$	−21.5899	−1.26130	−0.630649	+	0.776068i	$0.717211\pi$
$294$	0	0
$295$	−3.81343	−0.222026
$296$	0	0
$297$	−11.2299	−0.651627
$298$	0	0
$299$	1.78013	0.102948
$300$	0	0
$301$	20.8705	1.20295
$302$	0	0
$303$	−18.3890	−1.05642
$304$	0	0
$305$	−4.71440	−0.269946
$306$	0	0
$307$	13.4686	0.768691	0.384346	−	0.923189i	$-0.374427\pi$
$307$	13.4686	0.768691	0.384346	+	0.923189i	$0.374427\pi$
$308$	0	0
$309$	22.2850	1.26775
$310$	0	0
$311$	6.11970	0.347017	0.173508	−	0.984832i	$-0.444490\pi$
$311$	6.11970	0.347017	0.173508	+	0.984832i	$0.444490\pi$
$312$	0	0
$313$	12.3759	0.699530	0.349765	−	0.936838i	$-0.386261\pi$
$313$	12.3759	0.699530	0.349765	+	0.936838i	$0.386261\pi$
$314$	0	0
$315$	−1.69636	−0.0955789
$316$	0	0
$317$	20.0821	1.12792	0.563961	−	0.825801i	$-0.309277\pi$
$317$	20.0821	1.12792	0.563961	+	0.825801i	$0.309277\pi$
$318$	0	0
$319$	1.63012	0.0912693
$320$	0	0
$321$	−5.72610	−0.319600
$322$	0	0
$323$	0.677127	0.0376763
$324$	0	0
$325$	−1.78107	−0.0987958
$326$	0	0
$327$	13.8191	0.764198
$328$	0	0
$329$	8.50168	0.468713
$330$	0	0
$331$	10.5147	0.577939	0.288969	−	0.957338i	$-0.406687\pi$
$331$	10.5147	0.577939	0.288969	+	0.957338i	$0.406687\pi$
$332$	0	0
$333$	1.90481	0.104383
$334$	0	0
$335$	−2.30760	−0.126077
$336$	0	0
$337$	−7.08384	−0.385881	−0.192941	−	0.981210i	$-0.561802\pi$
$337$	−7.08384	−0.385881	−0.192941	+	0.981210i	$0.561802\pi$
$338$	0	0
$339$	−16.6709	−0.905437
$340$	0	0
$341$	−0.824600	−0.0446546
$342$	0	0
$343$	−8.07795	−0.436168
$344$	0	0
$345$	5.37715	0.289496
$346$	0	0
$347$	17.3433	0.931037	0.465519	−	0.885038i	$-0.345868\pi$
$347$	17.3433	0.931037	0.465519	+	0.885038i	$0.345868\pi$
$348$	0	0
$349$	7.16249	0.383399	0.191700	−	0.981454i	$-0.438600\pi$
$349$	7.16249	0.383399	0.191700	+	0.981454i	$0.438600\pi$
$350$	0	0
$351$	−1.69808	−0.0906367
$352$	0	0
$353$	23.5618	1.25407	0.627035	−	0.778991i	$-0.284268\pi$
$353$	23.5618	1.25407	0.627035	+	0.778991i	$0.284268\pi$
$354$	0	0
$355$	5.08128	0.269687
$356$	0	0
$357$	37.5966	1.98982
$358$	0	0
$359$	−31.5104	−1.66306	−0.831529	−	0.555482i	$-0.812534\pi$
$359$	−31.5104	−1.66306	−0.831529	+	0.555482i	$0.812534\pi$
$360$	0	0
$361$	−18.9808	−0.998987
$362$	0	0
$363$	−8.74733	−0.459116
$364$	0	0
$365$	0.904870	0.0473631
$366$	0	0
$367$	−24.1270	−1.25942	−0.629709	−	0.776832i	$-0.716826\pi$
$367$	−24.1270	−1.25942	−0.629709	+	0.776832i	$0.716826\pi$
$368$	0	0
$369$	−7.80281	−0.406198
$370$	0	0
$371$	14.9822	0.777836
$372$	0	0
$373$	14.1427	0.732279	0.366139	−	0.930560i	$-0.380679\pi$
$373$	14.1427	0.732279	0.366139	+	0.930560i	$0.380679\pi$
$374$	0	0
$375$	−11.1819	−0.577432
$376$	0	0
$377$	0.246490	0.0126949
$378$	0	0
$379$	6.30457	0.323844	0.161922	−	0.986804i	$-0.448231\pi$
$379$	6.30457	0.323844	0.161922	+	0.986804i	$0.448231\pi$
$380$	0	0
$381$	23.9837	1.22872
$382$	0	0
$383$	−0.400889	−0.0204845	−0.0102422	−	0.999948i	$-0.503260\pi$
$383$	−0.400889	−0.0204845	−0.0102422	+	0.999948i	$0.503260\pi$
$384$	0	0
$385$	−6.13148	−0.312489
$386$	0	0
$387$	−3.66520	−0.186313
$388$	0	0
$389$	−4.00381	−0.203001	−0.101501	−	0.994835i	$-0.532364\pi$
$389$	−4.00381	−0.203001	−0.101501	+	0.994835i	$0.532364\pi$
$390$	0	0
$391$	−22.6214	−1.14401
$392$	0	0
$393$	−28.7931	−1.45242
$394$	0	0
$395$	1.50871	0.0759114
$396$	0	0
$397$	−25.0004	−1.25473	−0.627367	−	0.778723i	$-0.715868\pi$
$397$	−25.0004	−1.25473	−0.627367	+	0.778723i	$0.715868\pi$
$398$	0	0
$399$	1.06826	0.0534800
$400$	0	0
$401$	−18.2154	−0.909633	−0.454817	−	0.890585i	$-0.650295\pi$
$401$	−18.2154	−0.909633	−0.454817	+	0.890585i	$0.650295\pi$
$402$	0	0
$403$	−0.124688	−0.00621113
$404$	0	0
$405$	−6.40084	−0.318060
$406$	0	0
$407$	6.88495	0.341274
$408$	0	0
$409$	−2.53089	−0.125145	−0.0625723	−	0.998040i	$-0.519930\pi$
$409$	−2.53089	−0.125145	−0.0625723	+	0.998040i	$0.519930\pi$
$410$	0	0
$411$	9.67179	0.477074
$412$	0	0
$413$	25.3099	1.24542
$414$	0	0
$415$	4.84815	0.237986
$416$	0	0
$417$	−0.764464	−0.0374360
$418$	0	0
$419$	5.36690	0.262190	0.131095	−	0.991370i	$-0.458151\pi$
$419$	5.36690	0.262190	0.131095	+	0.991370i	$0.458151\pi$
$420$	0	0
$421$	−25.0409	−1.22042	−0.610210	−	0.792239i	$-0.708915\pi$
$421$	−25.0409	−1.22042	−0.610210	+	0.792239i	$0.708915\pi$
$422$	0	0
$423$	−1.49304	−0.0725940
$424$	0	0
$425$	22.6333	1.09788
$426$	0	0
$427$	31.2897	1.51421
$428$	0	0
$429$	1.87800	0.0906705
$430$	0	0
$431$	−6.67102	−0.321332	−0.160666	−	0.987009i	$-0.551364\pi$
$431$	−6.67102	−0.321332	−0.160666	+	0.987009i	$0.551364\pi$
$432$	0	0
$433$	−7.97305	−0.383160	−0.191580	−	0.981477i	$-0.561361\pi$
$433$	−7.97305	−0.383160	−0.191580	+	0.981477i	$0.561361\pi$
$434$	0	0
$435$	0.744561	0.0356990
$436$	0	0
$437$	−0.642760	−0.0307474
$438$	0	0
$439$	−12.8390	−0.612773	−0.306386	−	0.951907i	$-0.599120\pi$
$439$	−12.8390	−0.612773	−0.306386	+	0.951907i	$0.599120\pi$
$440$	0	0
$441$	6.33871	0.301843
$442$	0	0
$443$	0.709218	0.0336960	0.0168480	−	0.999858i	$-0.494637\pi$
$443$	0.709218	0.0336960	0.0168480	+	0.999858i	$0.494637\pi$
$444$	0	0
$445$	0.358622	0.0170003
$446$	0	0
$447$	−37.9223	−1.79366
$448$	0	0
$449$	14.5427	0.686311	0.343155	−	0.939279i	$-0.388504\pi$
$449$	14.5427	0.686311	0.343155	+	0.939279i	$0.388504\pi$
$450$	0	0
$451$	−28.2032	−1.32804
$452$	0	0
$453$	−10.3050	−0.484171
$454$	0	0
$455$	−0.927140	−0.0434650
$456$	0	0
$457$	18.4798	0.864447	0.432224	−	0.901766i	$-0.357729\pi$
$457$	18.4798	0.864447	0.432224	+	0.901766i	$0.357729\pi$
$458$	0	0
$459$	21.5787	1.00721
$460$	0	0
$461$	22.7455	1.05936	0.529681	−	0.848197i	$-0.322312\pi$
$461$	22.7455	1.05936	0.529681	+	0.848197i	$0.322312\pi$
$462$	0	0
$463$	−18.9449	−0.880442	−0.440221	−	0.897889i	$-0.645100\pi$
$463$	−18.9449	−0.880442	−0.440221	+	0.897889i	$0.645100\pi$
$464$	0	0
$465$	−0.376637	−0.0174661
$466$	0	0
$467$	3.26958	0.151298	0.0756491	−	0.997135i	$-0.475897\pi$
$467$	3.26958	0.151298	0.0756491	+	0.997135i	$0.475897\pi$
$468$	0	0
$469$	15.3156	0.707209
$470$	0	0
$471$	22.3765	1.03105
$472$	0	0
$473$	−13.2479	−0.609138
$474$	0	0
$475$	0.643098	0.0295074
$476$	0	0
$477$	−2.63112	−0.120471
$478$	0	0
$479$	6.59467	0.301318	0.150659	−	0.988586i	$-0.451860\pi$
$479$	6.59467	0.301318	0.150659	+	0.988586i	$0.451860\pi$
$480$	0	0
$481$	1.04107	0.0474688
$482$	0	0
$483$	−35.6884	−1.62388
$484$	0	0
$485$	−8.11770	−0.368606
$486$	0	0
$487$	38.9971	1.76713	0.883563	−	0.468311i	$-0.155137\pi$
$487$	38.9971	1.76713	0.883563	+	0.468311i	$0.155137\pi$
$488$	0	0
$489$	−37.6476	−1.70248
$490$	0	0
$491$	−11.2203	−0.506364	−0.253182	−	0.967419i	$-0.581477\pi$
$491$	−11.2203	−0.506364	−0.253182	+	0.967419i	$0.581477\pi$
$492$	0	0
$493$	−3.13233	−0.141073
$494$	0	0
$495$	1.07679	0.0483981
$496$	0	0
$497$	−33.7247	−1.51276
$498$	0	0
$499$	−18.5343	−0.829711	−0.414855	−	0.909887i	$-0.636168\pi$
$499$	−18.5343	−0.829711	−0.414855	+	0.909887i	$0.636168\pi$
$500$	0	0
$501$	−5.63053	−0.251554
$502$	0	0
$503$	14.8410	0.661729	0.330864	−	0.943678i	$-0.392660\pi$
$503$	14.8410	0.661729	0.330864	+	0.943678i	$0.392660\pi$
$504$	0	0
$505$	−5.76264	−0.256434
$506$	0	0
$507$	−24.7317	−1.09837
$508$	0	0
$509$	11.7025	0.518703	0.259351	−	0.965783i	$-0.416491\pi$
$509$	11.7025	0.518703	0.259351	+	0.965783i	$0.416491\pi$
$510$	0	0
$511$	−6.00566	−0.265675
$512$	0	0
$513$	0.613133	0.0270705
$514$	0	0
$515$	6.98356	0.307732
$516$	0	0
$517$	−5.39658	−0.237341
$518$	0	0
$519$	−19.4515	−0.853827
$520$	0	0
$521$	24.0091	1.05186	0.525929	−	0.850528i	$-0.323718\pi$
$521$	24.0091	1.05186	0.525929	+	0.850528i	$0.323718\pi$
$522$	0	0
$523$	−26.0865	−1.14068	−0.570341	−	0.821408i	$-0.693189\pi$
$523$	−26.0865	−1.14068	−0.570341	+	0.821408i	$0.693189\pi$
$524$	0	0
$525$	35.7072	1.55839
$526$	0	0
$527$	1.58449	0.0690217
$528$	0	0
$529$	−1.52672	−0.0663791
$530$	0	0
$531$	−4.44484	−0.192890
$532$	0	0
$533$	−4.26461	−0.184721
$534$	0	0
$535$	−1.79442	−0.0775795
$536$	0	0
$537$	36.1465	1.55984
$538$	0	0
$539$	22.9112	0.986857
$540$	0	0
$541$	7.30995	0.314279	0.157140	−	0.987576i	$-0.449773\pi$
$541$	7.30995	0.314279	0.157140	+	0.987576i	$0.449773\pi$
$542$	0	0
$543$	−18.3193	−0.786158
$544$	0	0
$545$	4.33056	0.185501
$546$	0	0
$547$	29.8261	1.27527	0.637637	−	0.770337i	$-0.279912\pi$
$547$	29.8261	1.27527	0.637637	+	0.770337i	$0.279912\pi$
$548$	0	0
$549$	−5.49499	−0.234520
$550$	0	0
$551$	−0.0890014	−0.00379159
$552$	0	0
$553$	−10.0134	−0.425812
$554$	0	0
$555$	3.14471	0.133486
$556$	0	0
$557$	−12.8225	−0.543307	−0.271654	−	0.962395i	$-0.587571\pi$
$557$	−12.8225	−0.543307	−0.271654	+	0.962395i	$0.587571\pi$
$558$	0	0
$559$	−2.00321	−0.0847267
$560$	0	0
$561$	−23.8650	−1.00758
$562$	0	0
$563$	4.29127	0.180856	0.0904278	−	0.995903i	$-0.471177\pi$
$563$	4.29127	0.180856	0.0904278	+	0.995903i	$0.471177\pi$
$564$	0	0
$565$	−5.22424	−0.219785
$566$	0	0
$567$	42.4826	1.78410
$568$	0	0
$569$	−38.3876	−1.60929	−0.804646	−	0.593754i	$-0.797645\pi$
$569$	−38.3876	−1.60929	−0.804646	+	0.593754i	$0.797645\pi$
$570$	0	0
$571$	−37.9056	−1.58630	−0.793149	−	0.609027i	$-0.791560\pi$
$571$	−37.9056	−1.58630	−0.793149	+	0.609027i	$0.791560\pi$
$572$	0	0
$573$	4.32176	0.180544
$574$	0	0
$575$	−21.4846	−0.895968
$576$	0	0
$577$	−17.0575	−0.710114	−0.355057	−	0.934845i	$-0.615539\pi$
$577$	−17.0575	−0.710114	−0.355057	+	0.934845i	$0.615539\pi$
$578$	0	0
$579$	−12.2154	−0.507656
$580$	0	0
$581$	−32.1774	−1.33494
$582$	0	0
$583$	−9.51018	−0.393871
$584$	0	0
$585$	0.162821	0.00673183
$586$	0	0
$587$	22.0440	0.909853	0.454926	−	0.890529i	$-0.349666\pi$
$587$	22.0440	0.909853	0.454926	+	0.890529i	$0.349666\pi$
$588$	0	0
$589$	0.0450215	0.00185508
$590$	0	0
$591$	−47.8202	−1.96706
$592$	0	0
$593$	20.4614	0.840250	0.420125	−	0.907466i	$-0.361986\pi$
$593$	20.4614	0.840250	0.420125	+	0.907466i	$0.361986\pi$
$594$	0	0
$595$	11.7818	0.483008
$596$	0	0
$597$	19.0354	0.779068
$598$	0	0
$599$	−4.39979	−0.179771	−0.0898853	−	0.995952i	$-0.528650\pi$
$599$	−4.39979	−0.179771	−0.0898853	+	0.995952i	$0.528650\pi$
$600$	0	0
$601$	18.7704	0.765660	0.382830	−	0.923819i	$-0.374949\pi$
$601$	18.7704	0.765660	0.382830	+	0.923819i	$0.374949\pi$
$602$	0	0
$603$	−2.68968	−0.109532
$604$	0	0
$605$	−2.74120	−0.111445
$606$	0	0
$607$	24.7247	1.00355	0.501773	−	0.865000i	$-0.332681\pi$
$607$	24.7247	1.00355	0.501773	+	0.865000i	$0.332681\pi$
$608$	0	0
$609$	−4.94169	−0.200247
$610$	0	0
$611$	−0.816017	−0.0330125
$612$	0	0
$613$	−14.2935	−0.577311	−0.288655	−	0.957433i	$-0.593208\pi$
$613$	−14.2935	−0.577311	−0.288655	+	0.957433i	$0.593208\pi$
$614$	0	0
$615$	−12.8819	−0.519448
$616$	0	0
$617$	−22.8902	−0.921525	−0.460763	−	0.887523i	$-0.652424\pi$
$617$	−22.8902	−0.921525	−0.460763	+	0.887523i	$0.652424\pi$
$618$	0	0
$619$	−19.7725	−0.794724	−0.397362	−	0.917662i	$-0.630074\pi$
$619$	−19.7725	−0.794724	−0.397362	+	0.917662i	$0.630074\pi$
$620$	0	0
$621$	−20.4835	−0.821974
$622$	0	0
$623$	−2.38019	−0.0953602
$624$	0	0
$625$	19.6777	0.787107
$626$	0	0
$627$	−0.678097	−0.0270806
$628$	0	0
$629$	−13.2296	−0.527501
$630$	0	0
$631$	−34.9778	−1.39245	−0.696223	−	0.717826i	$-0.745138\pi$
$631$	−34.9778	−1.39245	−0.696223	+	0.717826i	$0.745138\pi$
$632$	0	0
$633$	−0.193608	−0.00769522
$634$	0	0
$635$	7.51589	0.298259
$636$	0	0
$637$	3.46441	0.137265
$638$	0	0
$639$	5.92263	0.234296
$640$	0	0
$641$	−29.1557	−1.15158	−0.575790	−	0.817597i	$-0.695306\pi$
$641$	−29.1557	−1.15158	−0.575790	+	0.817597i	$0.695306\pi$
$642$	0	0
$643$	−14.0858	−0.555491	−0.277745	−	0.960655i	$-0.589587\pi$
$643$	−14.0858	−0.555491	−0.277745	+	0.960655i	$0.589587\pi$
$644$	0	0
$645$	−6.05099	−0.238257
$646$	0	0
$647$	12.6674	0.498008	0.249004	−	0.968502i	$-0.419897\pi$
$647$	12.6674	0.498008	0.249004	+	0.968502i	$0.419897\pi$
$648$	0	0
$649$	−16.0659	−0.630640
$650$	0	0
$651$	2.49976	0.0979733
$652$	0	0
$653$	4.42639	0.173218	0.0866090	−	0.996242i	$-0.472397\pi$
$653$	4.42639	0.173218	0.0866090	+	0.996242i	$0.472397\pi$
$654$	0	0
$655$	−9.02306	−0.352560
$656$	0	0
$657$	1.05470	0.0411476
$658$	0	0
$659$	12.4817	0.486218	0.243109	−	0.969999i	$-0.421833\pi$
$659$	12.4817	0.486218	0.243109	+	0.969999i	$0.421833\pi$
$660$	0	0
$661$	20.7710	0.807899	0.403950	−	0.914781i	$-0.367637\pi$
$661$	20.7710	0.807899	0.403950	+	0.914781i	$0.367637\pi$
$662$	0	0
$663$	−3.60863	−0.140148
$664$	0	0
$665$	0.334767	0.0129817
$666$	0	0
$667$	2.97335	0.115129
$668$	0	0
$669$	45.5904	1.76263
$670$	0	0
$671$	−19.8616	−0.766750
$672$	0	0
$673$	49.9497	1.92542	0.962709	−	0.270538i	$-0.0872015\pi$
$673$	49.9497	1.92542	0.962709	+	0.270538i	$0.0872015\pi$
$674$	0	0
$675$	20.4943	0.788825
$676$	0	0
$677$	41.2022	1.58353	0.791765	−	0.610826i	$-0.209162\pi$
$677$	41.2022	1.58353	0.791765	+	0.610826i	$0.209162\pi$
$678$	0	0
$679$	53.8775	2.06763
$680$	0	0
$681$	3.39505	0.130099
$682$	0	0
$683$	−10.2590	−0.392551	−0.196275	−	0.980549i	$-0.562885\pi$
$683$	−10.2590	−0.392551	−0.196275	+	0.980549i	$0.562885\pi$
$684$	0	0
$685$	3.03090	0.115805
$686$	0	0
$687$	7.31063	0.278918
$688$	0	0
$689$	−1.43803	−0.0547847
$690$	0	0
$691$	−7.59351	−0.288871	−0.144435	−	0.989514i	$-0.546137\pi$
$691$	−7.59351	−0.288871	−0.144435	+	0.989514i	$0.546137\pi$
$692$	0	0
$693$	−7.14671	−0.271481
$694$	0	0
$695$	−0.239564	−0.00908718
$696$	0	0
$697$	54.1934	2.05272
$698$	0	0
$699$	22.9937	0.869700
$700$	0	0
$701$	−34.2324	−1.29294	−0.646470	−	0.762939i	$-0.723755\pi$
$701$	−34.2324	−1.29294	−0.646470	+	0.762939i	$0.723755\pi$
$702$	0	0
$703$	−0.375905	−0.0141775
$704$	0	0
$705$	−2.46490	−0.0928335
$706$	0	0
$707$	38.2469	1.43842
$708$	0	0
$709$	37.8905	1.42301	0.711504	−	0.702682i	$-0.248014\pi$
$709$	37.8905	1.42301	0.711504	+	0.702682i	$0.248014\pi$
$710$	0	0
$711$	1.75852	0.0659495
$712$	0	0
$713$	−1.50408	−0.0563281
$714$	0	0
$715$	0.588517	0.0220093
$716$	0	0
$717$	−26.3150	−0.982752
$718$	0	0
$719$	−1.35548	−0.0505510	−0.0252755	−	0.999681i	$-0.508046\pi$
$719$	−1.35548	−0.0505510	−0.0252755	+	0.999681i	$0.508046\pi$
$720$	0	0
$721$	−46.3502	−1.72617
$722$	0	0
$723$	47.5478	1.76832
$724$	0	0
$725$	−2.97492	−0.110486
$726$	0	0
$727$	46.7958	1.73556	0.867779	−	0.496950i	$-0.165547\pi$
$727$	46.7958	1.73556	0.867779	+	0.496950i	$0.165547\pi$
$728$	0	0
$729$	18.0573	0.668787
$730$	0	0
$731$	25.4562	0.941532
$732$	0	0
$733$	−15.8275	−0.584601	−0.292300	−	0.956327i	$-0.594421\pi$
$733$	−15.8275	−0.584601	−0.292300	+	0.956327i	$0.594421\pi$
$734$	0	0
$735$	10.4648	0.385998
$736$	0	0
$737$	−9.72184	−0.358109
$738$	0	0
$739$	5.13274	0.188811	0.0944055	−	0.995534i	$-0.469905\pi$
$739$	5.13274	0.188811	0.0944055	+	0.995534i	$0.469905\pi$
$740$	0	0
$741$	−0.102535	−0.00376671
$742$	0	0
$743$	33.0053	1.21085	0.605424	−	0.795903i	$-0.293003\pi$
$743$	33.0053	1.21085	0.605424	+	0.795903i	$0.293003\pi$
$744$	0	0
$745$	−11.8839	−0.435393
$746$	0	0
$747$	5.65089	0.206755
$748$	0	0
$749$	11.9096	0.435169
$750$	0	0
$751$	−39.5433	−1.44296	−0.721478	−	0.692437i	$-0.756537\pi$
$751$	−39.5433	−1.44296	−0.721478	+	0.692437i	$0.756537\pi$
$752$	0	0
$753$	−1.92428	−0.0701248
$754$	0	0
$755$	−3.22933	−0.117527
$756$	0	0
$757$	−40.0059	−1.45404	−0.727020	−	0.686617i	$-0.759095\pi$
$757$	−40.0059	−1.45404	−0.727020	+	0.686617i	$0.759095\pi$
$758$	0	0
$759$	22.6538	0.822281
$760$	0	0
$761$	19.2889	0.699223	0.349611	−	0.936895i	$-0.386314\pi$
$761$	19.2889	0.699223	0.349611	+	0.936895i	$0.386314\pi$
$762$	0	0
$763$	−28.7421	−1.04054
$764$	0	0
$765$	−2.06909	−0.0748080
$766$	0	0
$767$	−2.42932	−0.0877175
$768$	0	0
$769$	−15.4700	−0.557863	−0.278931	−	0.960311i	$-0.589980\pi$
$769$	−15.4700	−0.557863	−0.278931	+	0.960311i	$0.589980\pi$
$770$	0	0
$771$	23.8557	0.859142
$772$	0	0
$773$	27.6819	0.995648	0.497824	−	0.867278i	$-0.334132\pi$
$773$	27.6819	0.995648	0.497824	+	0.867278i	$0.334132\pi$
$774$	0	0
$775$	1.50487	0.0540564
$776$	0	0
$777$	−20.8716	−0.748765
$778$	0	0
$779$	1.53984	0.0551705
$780$	0	0
$781$	21.4073	0.766014
$782$	0	0
$783$	−2.83630	−0.101361
$784$	0	0
$785$	7.01223	0.250277
$786$	0	0
$787$	16.3519	0.582884	0.291442	−	0.956589i	$-0.405865\pi$
$787$	16.3519	0.582884	0.291442	+	0.956589i	$0.405865\pi$
$788$	0	0
$789$	−24.9620	−0.888671
$790$	0	0
$791$	34.6735	1.23285
$792$	0	0
$793$	−3.00327	−0.106649
$794$	0	0
$795$	−4.34379	−0.154059
$796$	0	0
$797$	51.4634	1.82293	0.911464	−	0.411380i	$-0.134953\pi$
$797$	51.4634	1.82293	0.911464	+	0.411380i	$0.134953\pi$
$798$	0	0
$799$	10.3697	0.366854
$800$	0	0
$801$	0.418001	0.0147693
$802$	0	0
$803$	3.81219	0.134529
$804$	0	0
$805$	−11.1839	−0.394179
$806$	0	0
$807$	51.3203	1.80656
$808$	0	0
$809$	−35.1922	−1.23729	−0.618646	−	0.785670i	$-0.712319\pi$
$809$	−35.1922	−1.23729	−0.618646	+	0.785670i	$0.712319\pi$
$810$	0	0
$811$	−39.0417	−1.37094	−0.685469	−	0.728102i	$-0.740403\pi$
$811$	−39.0417	−1.37094	−0.685469	+	0.728102i	$0.740403\pi$
$812$	0	0
$813$	−3.02254	−0.106005
$814$	0	0
$815$	−11.7978	−0.413260
$816$	0	0
$817$	0.723308	0.0253053
$818$	0	0
$819$	−1.08065	−0.0377611
$820$	0	0
$821$	−29.5854	−1.03254	−0.516268	−	0.856427i	$-0.672679\pi$
$821$	−29.5854	−1.03254	−0.516268	+	0.856427i	$0.672679\pi$
$822$	0	0
$823$	−39.1629	−1.36513	−0.682566	−	0.730824i	$-0.739136\pi$
$823$	−39.1629	−1.36513	−0.682566	+	0.730824i	$0.739136\pi$
$824$	0	0
$825$	−22.6657	−0.789119
$826$	0	0
$827$	−37.0268	−1.28755	−0.643774	−	0.765216i	$-0.722632\pi$
$827$	−37.0268	−1.28755	−0.643774	+	0.765216i	$0.722632\pi$
$828$	0	0
$829$	41.3337	1.43558	0.717789	−	0.696261i	$-0.245154\pi$
$829$	41.3337	1.43558	0.717789	+	0.696261i	$0.245154\pi$
$830$	0	0
$831$	28.2877	0.981290
$832$	0	0
$833$	−44.0247	−1.52537
$834$	0	0
$835$	−1.76447	−0.0610620
$836$	0	0
$837$	1.43475	0.0495921
$838$	0	0
$839$	19.8761	0.686200	0.343100	−	0.939299i	$-0.388523\pi$
$839$	19.8761	0.686200	0.343100	+	0.939299i	$0.388523\pi$
$840$	0	0
$841$	−28.5883	−0.985803
$842$	0	0
$843$	29.6797	1.02222
$844$	0	0
$845$	−7.75031	−0.266619
$846$	0	0
$847$	18.1934	0.625134
$848$	0	0
$849$	24.6109	0.844643
$850$	0	0
$851$	12.5582	0.430489
$852$	0	0
$853$	20.4716	0.700933	0.350466	−	0.936575i	$-0.386023\pi$
$853$	20.4716	0.700933	0.350466	+	0.936575i	$0.386023\pi$
$854$	0	0
$855$	−0.0587906	−0.00201060
$856$	0	0
$857$	−32.0871	−1.09607	−0.548037	−	0.836454i	$-0.684625\pi$
$857$	−32.0871	−1.09607	−0.548037	+	0.836454i	$0.684625\pi$
$858$	0	0
$859$	−35.9196	−1.22556	−0.612781	−	0.790253i	$-0.709949\pi$
$859$	−35.9196	−1.22556	−0.612781	+	0.790253i	$0.709949\pi$
$860$	0	0
$861$	85.4976	2.91375
$862$	0	0
$863$	−33.6033	−1.14387	−0.571934	−	0.820300i	$-0.693807\pi$
$863$	−33.6033	−1.14387	−0.571934	+	0.820300i	$0.693807\pi$
$864$	0	0
$865$	−6.09562	−0.207257
$866$	0	0
$867$	13.1446	0.446414
$868$	0	0
$869$	6.35615	0.215618
$870$	0	0
$871$	−1.47004	−0.0498103
$872$	0	0
$873$	−9.46180	−0.320233
$874$	0	0
$875$	23.2571	0.786234
$876$	0	0
$877$	0.927087	0.0313055	0.0156528	−	0.999877i	$-0.495017\pi$
$877$	0.927087	0.0313055	0.0156528	+	0.999877i	$0.495017\pi$
$878$	0	0
$879$	−41.5452	−1.40128
$880$	0	0
$881$	44.3087	1.49280	0.746400	−	0.665498i	$-0.231781\pi$
$881$	44.3087	1.49280	0.746400	+	0.665498i	$0.231781\pi$
$882$	0	0
$883$	24.0593	0.809661	0.404831	−	0.914392i	$-0.367330\pi$
$883$	24.0593	0.809661	0.404831	+	0.914392i	$0.367330\pi$
$884$	0	0
$885$	−7.33811	−0.246668
$886$	0	0
$887$	−20.3116	−0.681997	−0.340998	−	0.940064i	$-0.610765\pi$
$887$	−20.3116	−0.681997	−0.340998	+	0.940064i	$0.610765\pi$
$888$	0	0
$889$	−49.8833	−1.67303
$890$	0	0
$891$	−26.9666	−0.903414
$892$	0	0
$893$	0.294643	0.00985985
$894$	0	0
$895$	11.3274	0.378634
$896$	0	0
$897$	3.42548	0.114373
$898$	0	0
$899$	−0.208266	−0.00694605
$900$	0	0
$901$	18.2741	0.608799
$902$	0	0
$903$	40.1607	1.33646
$904$	0	0
$905$	−5.74082	−0.190831
$906$	0	0
$907$	38.0040	1.26190	0.630951	−	0.775822i	$-0.282665\pi$
$907$	38.0040	1.26190	0.630951	+	0.775822i	$0.282665\pi$
$908$	0	0
$909$	−6.71680	−0.222782
$910$	0	0
$911$	21.9105	0.725929	0.362964	−	0.931803i	$-0.381765\pi$
$911$	21.9105	0.725929	0.362964	+	0.931803i	$0.381765\pi$
$912$	0	0
$913$	20.4251	0.675973
$914$	0	0
$915$	−9.07184	−0.299906
$916$	0	0
$917$	59.8865	1.97762
$918$	0	0
$919$	−11.3579	−0.374661	−0.187330	−	0.982297i	$-0.559984\pi$
$919$	−11.3579	−0.374661	−0.187330	+	0.982297i	$0.559984\pi$
$920$	0	0
$921$	25.9173	0.854005
$922$	0	0
$923$	3.23700	0.106547
$924$	0	0
$925$	−12.5648	−0.413128
$926$	0	0
$927$	8.13987	0.267348
$928$	0	0
$929$	38.3826	1.25929	0.629646	−	0.776882i	$-0.283200\pi$
$929$	38.3826	1.25929	0.629646	+	0.776882i	$0.283200\pi$
$930$	0	0
$931$	−1.25091	−0.0409969
$932$	0	0
$933$	11.7760	0.385531
$934$	0	0
$935$	−7.47871	−0.244580
$936$	0	0
$937$	−39.5093	−1.29071	−0.645356	−	0.763882i	$-0.723291\pi$
$937$	−39.5093	−1.29071	−0.645356	+	0.763882i	$0.723291\pi$
$938$	0	0
$939$	23.8148	0.777168
$940$	0	0
$941$	28.6166	0.932874	0.466437	−	0.884554i	$-0.345537\pi$
$941$	28.6166	0.932874	0.466437	+	0.884554i	$0.345537\pi$
$942$	0	0
$943$	−51.4429	−1.67521
$944$	0	0
$945$	10.6684	0.347041
$946$	0	0
$947$	16.6198	0.540071	0.270035	−	0.962850i	$-0.412965\pi$
$947$	16.6198	0.540071	0.270035	+	0.962850i	$0.412965\pi$
$948$	0	0
$949$	0.576441	0.0187121
$950$	0	0
$951$	38.6436	1.25311
$952$	0	0
$953$	54.9975	1.78154	0.890772	−	0.454450i	$-0.150164\pi$
$953$	54.9975	1.78154	0.890772	+	0.454450i	$0.150164\pi$
$954$	0	0
$955$	1.35433	0.0438251
$956$	0	0
$957$	3.13682	0.101399
$958$	0	0
$959$	−20.1162	−0.649587
$960$	0	0
$961$	−30.8946	−0.996602
$962$	0	0
$963$	−2.09153	−0.0673987
$964$	0	0
$965$	−3.82801	−0.123228
$966$	0	0
$967$	−15.3308	−0.493005	−0.246503	−	0.969142i	$-0.579281\pi$
$967$	−15.3308	−0.493005	−0.246503	+	0.969142i	$0.579281\pi$
$968$	0	0
$969$	1.30298	0.0418579
$970$	0	0
$971$	−19.5706	−0.628049	−0.314024	−	0.949415i	$-0.601677\pi$
$971$	−19.5706	−0.628049	−0.314024	+	0.949415i	$0.601677\pi$
$972$	0	0
$973$	1.59000	0.0509730
$974$	0	0
$975$	−3.42728	−0.109761
$976$	0	0
$977$	−43.5920	−1.39463	−0.697316	−	0.716764i	$-0.745622\pi$
$977$	−43.5920	−1.39463	−0.697316	+	0.716764i	$0.745622\pi$
$978$	0	0
$979$	1.51086	0.0482874
$980$	0	0
$981$	5.04760	0.161158
$982$	0	0
$983$	21.4342	0.683644	0.341822	−	0.939765i	$-0.388956\pi$
$983$	21.4342	0.683644	0.341822	+	0.939765i	$0.388956\pi$
$984$	0	0
$985$	−14.9857	−0.477483
$986$	0	0
$987$	16.3597	0.520733
$988$	0	0
$989$	−24.1642	−0.768377
$990$	0	0
$991$	49.8476	1.58346	0.791730	−	0.610871i	$-0.209181\pi$
$991$	49.8476	1.58346	0.791730	+	0.610871i	$0.209181\pi$
$992$	0	0
$993$	20.2332	0.642082
$994$	0	0
$995$	5.96523	0.189111
$996$	0	0
$997$	4.58701	0.145272	0.0726361	−	0.997359i	$-0.476859\pi$
$997$	4.58701	0.145272	0.0726361	+	0.997359i	$0.476859\pi$
$998$	0	0
$999$	−11.9793	−0.379010

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.h.1.8		9
4.3	odd	2		2008.2.a.a.1.2	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.a.1.2	✓	9	4.3	odd	2
4016.2.a.h.1.8		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)

\(f(q)\)	\(=\)	\(q+1.92428 q^{3} +0.603023 q^{5} -4.00229 q^{7} +0.702869 q^{9} +O(q^{10})\)	\(q+1.92428 q^{3} +0.603023 q^{5} -4.00229 q^{7} +0.702869 q^{9} +2.54052 q^{11} +0.384152 q^{13} +1.16039 q^{15} -4.88169 q^{17} -0.138707 q^{19} -7.70154 q^{21} +4.63393 q^{23} -4.63636 q^{25} -4.42033 q^{27} +0.641649 q^{29} -0.324579 q^{31} +4.88868 q^{33} -2.41347 q^{35} +2.71005 q^{37} +0.739217 q^{39} -11.1014 q^{41} -5.21463 q^{43} +0.423846 q^{45} -2.12420 q^{47} +9.01833 q^{49} -9.39376 q^{51} -3.74340 q^{53} +1.53199 q^{55} -0.266912 q^{57} -6.32385 q^{59} -7.81794 q^{61} -2.81309 q^{63} +0.231652 q^{65} -3.82671 q^{67} +8.91699 q^{69} +8.42635 q^{71} +1.50056 q^{73} -8.92168 q^{75} -10.1679 q^{77} +2.50191 q^{79} -10.6146 q^{81} +8.03974 q^{83} -2.94377 q^{85} +1.23471 q^{87} +0.594707 q^{89} -1.53749 q^{91} -0.624582 q^{93} -0.0836438 q^{95} -13.4617 q^{97} +1.78565 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + q^{3} - 5 q^{5} - 4 q^{9}+O(q^{10}) \) 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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