LMFDB - Embedded newform 4016.2.a.m.1.19

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:	$23$
Twist minimal:	no (minimal twist has level 2008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
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)

$f(q)$	$=$	$q+2.21536 q^{3} +3.09704 q^{5} +4.67984 q^{7} +1.90784 q^{9} +O(q^{10})$	$q+2.21536 q^{3} +3.09704 q^{5} +4.67984 q^{7} +1.90784 q^{9} +4.76411 q^{11} -4.18229 q^{13} +6.86107 q^{15} -2.76575 q^{17} +4.43902 q^{19} +10.3676 q^{21} -8.88455 q^{23} +4.59165 q^{25} -2.41954 q^{27} +2.14992 q^{29} -7.46623 q^{31} +10.5542 q^{33} +14.4937 q^{35} +9.30620 q^{37} -9.26530 q^{39} -8.39778 q^{41} +3.77980 q^{43} +5.90865 q^{45} +3.41612 q^{47} +14.9009 q^{49} -6.12714 q^{51} +11.5655 q^{53} +14.7546 q^{55} +9.83404 q^{57} -0.620556 q^{59} -11.0665 q^{61} +8.92838 q^{63} -12.9527 q^{65} -3.88032 q^{67} -19.6825 q^{69} -5.36865 q^{71} +1.41033 q^{73} +10.1722 q^{75} +22.2953 q^{77} -6.15937 q^{79} -11.0837 q^{81} -4.20676 q^{83} -8.56563 q^{85} +4.76286 q^{87} -16.8333 q^{89} -19.5725 q^{91} -16.5404 q^{93} +13.7478 q^{95} +9.25212 q^{97} +9.08915 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	$ 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) $ 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.21536	1.27904	0.639520	−	0.768774i	$-0.279133\pi$
$3$	2.21536	1.27904	0.639520	+	0.768774i	$0.279133\pi$
$4$	0	0
$5$	3.09704	1.38504	0.692519	−	0.721400i	$-0.256501\pi$
$5$	3.09704	1.38504	0.692519	+	0.721400i	$0.256501\pi$
$6$	0	0
$7$	4.67984	1.76881	0.884407	−	0.466716i	$-0.154563\pi$
$7$	4.67984	1.76881	0.884407	+	0.466716i	$0.154563\pi$
$8$	0	0
$9$	1.90784	0.635946
$10$	0	0
$11$	4.76411	1.43643	0.718217	−	0.695819i	$-0.244959\pi$
$11$	4.76411	1.43643	0.718217	+	0.695819i	$0.244959\pi$
$12$	0	0
$13$	−4.18229	−1.15996	−0.579980	−	0.814631i	$-0.696940\pi$
$13$	−4.18229	−1.15996	−0.579980	+	0.814631i	$0.696940\pi$
$14$	0	0
$15$	6.86107	1.77152
$16$	0	0
$17$	−2.76575	−0.670792	−0.335396	−	0.942077i	$-0.608870\pi$
$17$	−2.76575	−0.670792	−0.335396	+	0.942077i	$0.608870\pi$
$18$	0	0
$19$	4.43902	1.01838	0.509190	−	0.860654i	$-0.329945\pi$
$19$	4.43902	1.01838	0.509190	+	0.860654i	$0.329945\pi$
$20$	0	0
$21$	10.3676	2.26239
$22$	0	0
$23$	−8.88455	−1.85256	−0.926279	−	0.376840i	$-0.877011\pi$
$23$	−8.88455	−1.85256	−0.926279	+	0.376840i	$0.877011\pi$
$24$	0	0
$25$	4.59165	0.918331
$26$	0	0
$27$	−2.41954	−0.465640
$28$	0	0
$29$	2.14992	0.399231	0.199615	−	0.979874i	$-0.436031\pi$
$29$	2.14992	0.399231	0.199615	+	0.979874i	$0.436031\pi$
$30$	0	0
$31$	−7.46623	−1.34098	−0.670488	−	0.741921i	$-0.733915\pi$
$31$	−7.46623	−1.34098	−0.670488	+	0.741921i	$0.733915\pi$
$32$	0	0
$33$	10.5542	1.83726
$34$	0	0
$35$	14.4937	2.44988
$36$	0	0
$37$	9.30620	1.52993	0.764965	−	0.644072i	$-0.222756\pi$
$37$	9.30620	1.52993	0.764965	+	0.644072i	$0.222756\pi$
$38$	0	0
$39$	−9.26530	−1.48364
$40$	0	0
$41$	−8.39778	−1.31151	−0.655756	−	0.754973i	$-0.727650\pi$
$41$	−8.39778	−1.31151	−0.655756	+	0.754973i	$0.727650\pi$
$42$	0	0
$43$	3.77980	0.576413	0.288207	−	0.957568i	$-0.406941\pi$
$43$	3.77980	0.576413	0.288207	+	0.957568i	$0.406941\pi$
$44$	0	0
$45$	5.90865	0.880809
$46$	0	0
$47$	3.41612	0.498293	0.249146	−	0.968466i	$-0.419850\pi$
$47$	3.41612	0.498293	0.249146	+	0.968466i	$0.419850\pi$
$48$	0	0
$49$	14.9009	2.12870
$50$	0	0
$51$	−6.12714	−0.857971
$52$	0	0
$53$	11.5655	1.58865	0.794323	−	0.607496i	$-0.207826\pi$
$53$	11.5655	1.58865	0.794323	+	0.607496i	$0.207826\pi$
$54$	0	0
$55$	14.7546	1.98952
$56$	0	0
$57$	9.83404	1.30255
$58$	0	0
$59$	−0.620556	−0.0807895	−0.0403947	−	0.999184i	$-0.512862\pi$
$59$	−0.620556	−0.0807895	−0.0403947	+	0.999184i	$0.512862\pi$
$60$	0	0
$61$	−11.0665	−1.41692	−0.708460	−	0.705751i	$-0.750610\pi$
$61$	−11.0665	−1.41692	−0.708460	+	0.705751i	$0.750610\pi$
$62$	0	0
$63$	8.92838	1.12487
$64$	0	0
$65$	−12.9527	−1.60659
$66$	0	0
$67$	−3.88032	−0.474056	−0.237028	−	0.971503i	$-0.576173\pi$
$67$	−3.88032	−0.474056	−0.237028	+	0.971503i	$0.576173\pi$
$68$	0	0
$69$	−19.6825	−2.36950
$70$	0	0
$71$	−5.36865	−0.637142	−0.318571	−	0.947899i	$-0.603203\pi$
$71$	−5.36865	−0.637142	−0.318571	+	0.947899i	$0.603203\pi$
$72$	0	0
$73$	1.41033	0.165066	0.0825331	−	0.996588i	$-0.473699\pi$
$73$	1.41033	0.165066	0.0825331	+	0.996588i	$0.473699\pi$
$74$	0	0
$75$	10.1722	1.17458
$76$	0	0
$77$	22.2953	2.54078
$78$	0	0
$79$	−6.15937	−0.692983	−0.346492	−	0.938053i	$-0.612627\pi$
$79$	−6.15937	−0.692983	−0.346492	+	0.938053i	$0.612627\pi$
$80$	0	0
$81$	−11.0837	−1.23152
$82$	0	0
$83$	−4.20676	−0.461752	−0.230876	−	0.972983i	$-0.574159\pi$
$83$	−4.20676	−0.461752	−0.230876	+	0.972983i	$0.574159\pi$
$84$	0	0
$85$	−8.56563	−0.929073
$86$	0	0
$87$	4.76286	0.510632
$88$	0	0
$89$	−16.8333	−1.78433	−0.892164	−	0.451712i	$-0.850814\pi$
$89$	−16.8333	−1.78433	−0.892164	+	0.451712i	$0.850814\pi$
$90$	0	0
$91$	−19.5725	−2.05175
$92$	0	0
$93$	−16.5404	−1.71516
$94$	0	0
$95$	13.7478	1.41050
$96$	0	0
$97$	9.25212	0.939411	0.469705	−	0.882823i	$-0.344360\pi$
$97$	9.25212	0.939411	0.469705	+	0.882823i	$0.344360\pi$
$98$	0	0
$99$	9.08915	0.913494
$100$	0	0
$101$	3.90761	0.388821	0.194411	−	0.980920i	$-0.437721\pi$
$101$	3.90761	0.388821	0.194411	+	0.980920i	$0.437721\pi$
$102$	0	0
$103$	0.244985	0.0241391	0.0120695	−	0.999927i	$-0.496158\pi$
$103$	0.244985	0.0241391	0.0120695	+	0.999927i	$0.496158\pi$
$104$	0	0
$105$	32.1087	3.13349
$106$	0	0
$107$	−2.64689	−0.255884	−0.127942	−	0.991782i	$-0.540837\pi$
$107$	−2.64689	−0.255884	−0.127942	+	0.991782i	$0.540837\pi$
$108$	0	0
$109$	19.7928	1.89581	0.947905	−	0.318554i	$-0.103197\pi$
$109$	19.7928	1.89581	0.947905	+	0.318554i	$0.103197\pi$
$110$	0	0
$111$	20.6166	1.95684
$112$	0	0
$113$	8.57936	0.807079	0.403539	−	0.914962i	$-0.367780\pi$
$113$	8.57936	0.807079	0.403539	+	0.914962i	$0.367780\pi$
$114$	0	0
$115$	−27.5158	−2.56586
$116$	0	0
$117$	−7.97914	−0.737671
$118$	0	0
$119$	−12.9433	−1.18651
$120$	0	0
$121$	11.6968	1.06334
$122$	0	0
$123$	−18.6041	−1.67748
$124$	0	0
$125$	−1.26466	−0.113115
$126$	0	0
$127$	2.05904	0.182710	0.0913552	−	0.995818i	$-0.470880\pi$
$127$	2.05904	0.182710	0.0913552	+	0.995818i	$0.470880\pi$
$128$	0	0
$129$	8.37362	0.737256
$130$	0	0
$131$	8.45887	0.739055	0.369527	−	0.929220i	$-0.379520\pi$
$131$	8.45887	0.739055	0.369527	+	0.929220i	$0.379520\pi$
$132$	0	0
$133$	20.7739	1.80133
$134$	0	0
$135$	−7.49341	−0.644930
$136$	0	0
$137$	−11.8914	−1.01595	−0.507977	−	0.861370i	$-0.669607\pi$
$137$	−11.8914	−1.01595	−0.507977	+	0.861370i	$0.669607\pi$
$138$	0	0
$139$	−17.0936	−1.44986	−0.724932	−	0.688821i	$-0.758129\pi$
$139$	−17.0936	−1.44986	−0.724932	+	0.688821i	$0.758129\pi$
$140$	0	0
$141$	7.56796	0.637337
$142$	0	0
$143$	−19.9249	−1.66621
$144$	0	0
$145$	6.65840	0.552950
$146$	0	0
$147$	33.0110	2.72270
$148$	0	0
$149$	−14.0639	−1.15216	−0.576079	−	0.817394i	$-0.695418\pi$
$149$	−14.0639	−1.15216	−0.576079	+	0.817394i	$0.695418\pi$
$150$	0	0
$151$	−17.7621	−1.44546	−0.722731	−	0.691129i	$-0.757113\pi$
$151$	−17.7621	−1.44546	−0.722731	+	0.691129i	$0.757113\pi$
$152$	0	0
$153$	−5.27659	−0.426587
$154$	0	0
$155$	−23.1232	−1.85730
$156$	0	0
$157$	−20.5184	−1.63755	−0.818774	−	0.574116i	$-0.805346\pi$
$157$	−20.5184	−1.63755	−0.818774	+	0.574116i	$0.805346\pi$
$158$	0	0
$159$	25.6218	2.03194
$160$	0	0
$161$	−41.5783	−3.27683
$162$	0	0
$163$	8.63658	0.676469	0.338235	−	0.941062i	$-0.390170\pi$
$163$	8.63658	0.676469	0.338235	+	0.941062i	$0.390170\pi$
$164$	0	0
$165$	32.6869	2.54467
$166$	0	0
$167$	−21.3671	−1.65344	−0.826719	−	0.562615i	$-0.809795\pi$
$167$	−21.3671	−1.65344	−0.826719	+	0.562615i	$0.809795\pi$
$168$	0	0
$169$	4.49159	0.345507
$170$	0	0
$171$	8.46892	0.647635
$172$	0	0
$173$	9.23508	0.702130	0.351065	−	0.936351i	$-0.385820\pi$
$173$	9.23508	0.702130	0.351065	+	0.936351i	$0.385820\pi$
$174$	0	0
$175$	21.4882	1.62436
$176$	0	0
$177$	−1.37476	−0.103333
$178$	0	0
$179$	20.4876	1.53131	0.765657	−	0.643249i	$-0.222414\pi$
$179$	20.4876	1.53131	0.765657	+	0.643249i	$0.222414\pi$
$180$	0	0
$181$	−8.31659	−0.618167	−0.309084	−	0.951035i	$-0.600022\pi$
$181$	−8.31659	−0.618167	−0.309084	+	0.951035i	$0.600022\pi$
$182$	0	0
$183$	−24.5163	−1.81230
$184$	0	0
$185$	28.8217	2.11901
$186$	0	0
$187$	−13.1763	−0.963548
$188$	0	0
$189$	−11.3231	−0.823631
$190$	0	0
$191$	14.9096	1.07882	0.539411	−	0.842043i	$-0.318647\pi$
$191$	14.9096	1.07882	0.539411	+	0.842043i	$0.318647\pi$
$192$	0	0
$193$	25.6297	1.84486	0.922432	−	0.386159i	$-0.126198\pi$
$193$	25.6297	1.84486	0.922432	+	0.386159i	$0.126198\pi$
$194$	0	0
$195$	−28.6950	−2.05489
$196$	0	0
$197$	5.72377	0.407802	0.203901	−	0.978992i	$-0.434638\pi$
$197$	5.72377	0.407802	0.203901	+	0.978992i	$0.434638\pi$
$198$	0	0
$199$	−12.6418	−0.896154	−0.448077	−	0.893995i	$-0.647891\pi$
$199$	−12.6418	−0.896154	−0.448077	+	0.893995i	$0.647891\pi$
$200$	0	0
$201$	−8.59631	−0.606337
$202$	0	0
$203$	10.0613	0.706165
$204$	0	0
$205$	−26.0082	−1.81649
$206$	0	0
$207$	−16.9503	−1.17813
$208$	0	0
$209$	21.1480	1.46284
$210$	0	0
$211$	16.1293	1.11039	0.555194	−	0.831721i	$-0.312644\pi$
$211$	16.1293	1.11039	0.555194	+	0.831721i	$0.312644\pi$
$212$	0	0
$213$	−11.8935	−0.814930
$214$	0	0
$215$	11.7062	0.798355
$216$	0	0
$217$	−34.9408	−2.37194
$218$	0	0
$219$	3.12439	0.211126
$220$	0	0
$221$	11.5672	0.778092
$222$	0	0
$223$	−11.9175	−0.798053	−0.399026	−	0.916939i	$-0.630652\pi$
$223$	−11.9175	−0.798053	−0.399026	+	0.916939i	$0.630652\pi$
$224$	0	0
$225$	8.76013	0.584009
$226$	0	0
$227$	5.85776	0.388793	0.194397	−	0.980923i	$-0.437725\pi$
$227$	5.85776	0.388793	0.194397	+	0.980923i	$0.437725\pi$
$228$	0	0
$229$	4.73938	0.313187	0.156594	−	0.987663i	$-0.449949\pi$
$229$	4.73938	0.313187	0.156594	+	0.987663i	$0.449949\pi$
$230$	0	0
$231$	49.3922	3.24977
$232$	0	0
$233$	3.31495	0.217169	0.108585	−	0.994087i	$-0.465368\pi$
$233$	3.31495	0.217169	0.108585	+	0.994087i	$0.465368\pi$
$234$	0	0
$235$	10.5799	0.690155
$236$	0	0
$237$	−13.6452	−0.886354
$238$	0	0
$239$	0.512422	0.0331458	0.0165729	−	0.999863i	$-0.494724\pi$
$239$	0.512422	0.0331458	0.0165729	+	0.999863i	$0.494724\pi$
$240$	0	0
$241$	−14.4997	−0.934011	−0.467005	−	0.884254i	$-0.654667\pi$
$241$	−14.4997	−0.934011	−0.467005	+	0.884254i	$0.654667\pi$
$242$	0	0
$243$	−17.2957	−1.10952
$244$	0	0
$245$	46.1488	2.94834
$246$	0	0
$247$	−18.5653	−1.18128
$248$	0	0
$249$	−9.31951	−0.590600
$250$	0	0
$251$	1.00000	0.0631194
$251$	1.00000	0.0631194
$252$	0	0
$253$	−42.3270	−2.66108
$254$	0	0
$255$	−18.9760	−1.18832
$256$	0	0
$257$	28.3085	1.76584	0.882918	−	0.469527i	$-0.155576\pi$
$257$	28.3085	1.76584	0.882918	+	0.469527i	$0.155576\pi$
$258$	0	0
$259$	43.5515	2.70616
$260$	0	0
$261$	4.10170	0.253889
$262$	0	0
$263$	−15.7409	−0.970625	−0.485312	−	0.874341i	$-0.661294\pi$
$263$	−15.7409	−0.970625	−0.485312	+	0.874341i	$0.661294\pi$
$264$	0	0
$265$	35.8189	2.20033
$266$	0	0
$267$	−37.2919	−2.28223
$268$	0	0
$269$	0.435839	0.0265736	0.0132868	−	0.999912i	$-0.495771\pi$
$269$	0.435839	0.0265736	0.0132868	+	0.999912i	$0.495771\pi$
$270$	0	0
$271$	−18.6737	−1.13434	−0.567172	−	0.823599i	$-0.691963\pi$
$271$	−18.6737	−1.13434	−0.567172	+	0.823599i	$0.691963\pi$
$272$	0	0
$273$	−43.3602	−2.62428
$274$	0	0
$275$	21.8752	1.31912
$276$	0	0
$277$	20.9307	1.25761	0.628803	−	0.777564i	$-0.283545\pi$
$277$	20.9307	1.25761	0.628803	+	0.777564i	$0.283545\pi$
$278$	0	0
$279$	−14.2444	−0.852787
$280$	0	0
$281$	28.1763	1.68086	0.840428	−	0.541924i	$-0.182304\pi$
$281$	28.1763	1.68086	0.840428	+	0.541924i	$0.182304\pi$
$282$	0	0
$283$	23.7841	1.41382	0.706910	−	0.707303i	$-0.250088\pi$
$283$	23.7841	1.41382	0.706910	+	0.707303i	$0.250088\pi$
$284$	0	0
$285$	30.4564	1.80408
$286$	0	0
$287$	−39.3003	−2.31982
$288$	0	0
$289$	−9.35064	−0.550038
$290$	0	0
$291$	20.4968	1.20154
$292$	0	0
$293$	−16.0129	−0.935482	−0.467741	−	0.883866i	$-0.654932\pi$
$293$	−16.0129	−0.935482	−0.467741	+	0.883866i	$0.654932\pi$
$294$	0	0
$295$	−1.92189	−0.111897
$296$	0	0
$297$	−11.5270	−0.668862
$298$	0	0
$299$	37.1578	2.14889
$300$	0	0
$301$	17.6888	1.01957
$302$	0	0
$303$	8.65677	0.497319
$304$	0	0
$305$	−34.2734	−1.96249
$306$	0	0
$307$	1.39255	0.0794772	0.0397386	−	0.999210i	$-0.487347\pi$
$307$	1.39255	0.0794772	0.0397386	+	0.999210i	$0.487347\pi$
$308$	0	0
$309$	0.542731	0.0308749
$310$	0	0
$311$	−29.9348	−1.69745	−0.848724	−	0.528836i	$-0.822629\pi$
$311$	−29.9348	−1.69745	−0.848724	+	0.528836i	$0.822629\pi$
$312$	0	0
$313$	−2.51730	−0.142286	−0.0711432	−	0.997466i	$-0.522665\pi$
$313$	−2.51730	−0.142286	−0.0711432	+	0.997466i	$0.522665\pi$
$314$	0	0
$315$	27.6515	1.55799
$316$	0	0
$317$	2.00168	0.112425	0.0562127	−	0.998419i	$-0.482098\pi$
$317$	2.00168	0.112425	0.0562127	+	0.998419i	$0.482098\pi$
$318$	0	0
$319$	10.2425	0.573468
$320$	0	0
$321$	−5.86382	−0.327286
$322$	0	0
$323$	−12.2772	−0.683121
$324$	0	0
$325$	−19.2036	−1.06523
$326$	0	0
$327$	43.8483	2.42482
$328$	0	0
$329$	15.9869	0.881388
$330$	0	0
$331$	4.32431	0.237686	0.118843	−	0.992913i	$-0.462082\pi$
$331$	4.32431	0.237686	0.118843	+	0.992913i	$0.462082\pi$
$332$	0	0
$333$	17.7547	0.972952
$334$	0	0
$335$	−12.0175	−0.656586
$336$	0	0
$337$	7.56938	0.412330	0.206165	−	0.978517i	$-0.433902\pi$
$337$	7.56938	0.412330	0.206165	+	0.978517i	$0.433902\pi$
$338$	0	0
$339$	19.0064	1.03229
$340$	0	0
$341$	−35.5700	−1.92622
$342$	0	0
$343$	36.9751	1.99647
$344$	0	0
$345$	−60.9575	−3.28184
$346$	0	0
$347$	−21.7995	−1.17026	−0.585128	−	0.810941i	$-0.698956\pi$
$347$	−21.7995	−1.17026	−0.585128	+	0.810941i	$0.698956\pi$
$348$	0	0
$349$	15.6011	0.835109	0.417554	−	0.908652i	$-0.362887\pi$
$349$	15.6011	0.835109	0.417554	+	0.908652i	$0.362887\pi$
$350$	0	0
$351$	10.1192	0.540124
$352$	0	0
$353$	7.89144	0.420019	0.210010	−	0.977699i	$-0.432650\pi$
$353$	7.89144	0.420019	0.210010	+	0.977699i	$0.432650\pi$
$354$	0	0
$355$	−16.6269	−0.882466
$356$	0	0
$357$	−28.6740	−1.51759
$358$	0	0
$359$	−3.07626	−0.162359	−0.0811794	−	0.996700i	$-0.525869\pi$
$359$	−3.07626	−0.162359	−0.0811794	+	0.996700i	$0.525869\pi$
$360$	0	0
$361$	0.704867	0.0370982
$362$	0	0
$363$	25.9126	1.36006
$364$	0	0
$365$	4.36784	0.228623
$366$	0	0
$367$	−9.13202	−0.476688	−0.238344	−	0.971181i	$-0.576605\pi$
$367$	−9.13202	−0.476688	−0.238344	+	0.971181i	$0.576605\pi$
$368$	0	0
$369$	−16.0216	−0.834050
$370$	0	0
$371$	54.1248	2.81002
$372$	0	0
$373$	−13.0955	−0.678060	−0.339030	−	0.940776i	$-0.610099\pi$
$373$	−13.0955	−0.678060	−0.339030	+	0.940776i	$0.610099\pi$
$374$	0	0
$375$	−2.80169	−0.144679
$376$	0	0
$377$	−8.99161	−0.463092
$378$	0	0
$379$	30.1576	1.54909	0.774547	−	0.632517i	$-0.217978\pi$
$379$	30.1576	1.54909	0.774547	+	0.632517i	$0.217978\pi$
$380$	0	0
$381$	4.56153	0.233694
$382$	0	0
$383$	11.8329	0.604635	0.302317	−	0.953207i	$-0.402240\pi$
$383$	11.8329	0.604635	0.302317	+	0.953207i	$0.402240\pi$
$384$	0	0
$385$	69.0494	3.51908
$386$	0	0
$387$	7.21123	0.366568
$388$	0	0
$389$	−15.7765	−0.799901	−0.399951	−	0.916537i	$-0.630973\pi$
$389$	−15.7765	−0.799901	−0.399951	+	0.916537i	$0.630973\pi$
$390$	0	0
$391$	24.5724	1.24268
$392$	0	0
$393$	18.7395	0.945281
$394$	0	0
$395$	−19.0758	−0.959808
$396$	0	0
$397$	20.2207	1.01485	0.507424	−	0.861697i	$-0.330598\pi$
$397$	20.2207	1.01485	0.507424	+	0.861697i	$0.330598\pi$
$398$	0	0
$399$	46.0217	2.30397
$400$	0	0
$401$	9.28020	0.463431	0.231716	−	0.972784i	$-0.425566\pi$
$401$	9.28020	0.463431	0.231716	+	0.972784i	$0.425566\pi$
$402$	0	0
$403$	31.2260	1.55548
$404$	0	0
$405$	−34.3266	−1.70570
$406$	0	0
$407$	44.3358	2.19764
$408$	0	0
$409$	13.6673	0.675804	0.337902	−	0.941181i	$-0.390283\pi$
$409$	13.6673	0.675804	0.337902	+	0.941181i	$0.390283\pi$
$410$	0	0
$411$	−26.3439	−1.29945
$412$	0	0
$413$	−2.90410	−0.142902
$414$	0	0
$415$	−13.0285	−0.639544
$416$	0	0
$417$	−37.8686	−1.85443
$418$	0	0
$419$	−10.3777	−0.506982	−0.253491	−	0.967338i	$-0.581579\pi$
$419$	−10.3777	−0.506982	−0.253491	+	0.967338i	$0.581579\pi$
$420$	0	0
$421$	14.2237	0.693222	0.346611	−	0.938009i	$-0.387332\pi$
$421$	14.2237	0.693222	0.346611	+	0.938009i	$0.387332\pi$
$422$	0	0
$423$	6.51741	0.316887
$424$	0	0
$425$	−12.6994	−0.616009
$426$	0	0
$427$	−51.7895	−2.50627
$428$	0	0
$429$	−44.1409	−2.13114
$430$	0	0
$431$	10.0895	0.485994	0.242997	−	0.970027i	$-0.421869\pi$
$431$	10.0895	0.485994	0.242997	+	0.970027i	$0.421869\pi$
$432$	0	0
$433$	−2.51477	−0.120852	−0.0604261	−	0.998173i	$-0.519246\pi$
$433$	−2.51477	−0.120852	−0.0604261	+	0.998173i	$0.519246\pi$
$434$	0	0
$435$	14.7508	0.707245
$436$	0	0
$437$	−39.4387	−1.88661
$438$	0	0
$439$	6.73881	0.321626	0.160813	−	0.986985i	$-0.448588\pi$
$439$	6.73881	0.321626	0.160813	+	0.986985i	$0.448588\pi$
$440$	0	0
$441$	28.4285	1.35374
$442$	0	0
$443$	−13.9799	−0.664204	−0.332102	−	0.943243i	$-0.607758\pi$
$443$	−13.9799	−0.664204	−0.332102	+	0.943243i	$0.607758\pi$
$444$	0	0
$445$	−52.1334	−2.47136
$446$	0	0
$447$	−31.1566	−1.47366
$448$	0	0
$449$	10.6552	0.502850	0.251425	−	0.967877i	$-0.419101\pi$
$449$	10.6552	0.502850	0.251425	+	0.967877i	$0.419101\pi$
$450$	0	0
$451$	−40.0079	−1.88390
$452$	0	0
$453$	−39.3496	−1.84881
$454$	0	0
$455$	−60.6168	−2.84176
$456$	0	0
$457$	−33.7899	−1.58062	−0.790312	−	0.612704i	$-0.790082\pi$
$457$	−33.7899	−1.58062	−0.790312	+	0.612704i	$0.790082\pi$
$458$	0	0
$459$	6.69183	0.312348
$460$	0	0
$461$	10.1644	0.473406	0.236703	−	0.971582i	$-0.423933\pi$
$461$	10.1644	0.473406	0.236703	+	0.971582i	$0.423933\pi$
$462$	0	0
$463$	16.8546	0.783299	0.391650	−	0.920114i	$-0.371905\pi$
$463$	16.8546	0.783299	0.391650	+	0.920114i	$0.371905\pi$
$464$	0	0
$465$	−51.2263	−2.37556
$466$	0	0
$467$	6.51332	0.301401	0.150700	−	0.988579i	$-0.451847\pi$
$467$	6.51332	0.301401	0.150700	+	0.988579i	$0.451847\pi$
$468$	0	0
$469$	−18.1593	−0.838517
$470$	0	0
$471$	−45.4558	−2.09449
$472$	0	0
$473$	18.0074	0.827980
$474$	0	0
$475$	20.3824	0.935210
$476$	0	0
$477$	22.0651	1.01029
$478$	0	0
$479$	10.9557	0.500579	0.250290	−	0.968171i	$-0.419474\pi$
$479$	10.9557	0.500579	0.250290	+	0.968171i	$0.419474\pi$
$480$	0	0
$481$	−38.9213	−1.77466
$482$	0	0
$483$	−92.1111	−4.19120
$484$	0	0
$485$	28.6542	1.30112
$486$	0	0
$487$	7.58900	0.343891	0.171945	−	0.985106i	$-0.444995\pi$
$487$	7.58900	0.343891	0.171945	+	0.985106i	$0.444995\pi$
$488$	0	0
$489$	19.1332	0.865232
$490$	0	0
$491$	1.97891	0.0893070	0.0446535	−	0.999003i	$-0.485782\pi$
$491$	1.97891	0.0893070	0.0446535	+	0.999003i	$0.485782\pi$
$492$	0	0
$493$	−5.94614	−0.267801
$494$	0	0
$495$	28.1495	1.26522
$496$	0	0
$497$	−25.1244	−1.12699
$498$	0	0
$499$	24.0373	1.07606	0.538029	−	0.842926i	$-0.319169\pi$
$499$	24.0373	1.07606	0.538029	+	0.842926i	$0.319169\pi$
$500$	0	0
$501$	−47.3360	−2.11482
$502$	0	0
$503$	36.9120	1.64582	0.822912	−	0.568169i	$-0.192348\pi$
$503$	36.9120	1.64582	0.822912	+	0.568169i	$0.192348\pi$
$504$	0	0
$505$	12.1020	0.538533
$506$	0	0
$507$	9.95050	0.441917
$508$	0	0
$509$	1.19623	0.0530218	0.0265109	−	0.999649i	$-0.491560\pi$
$509$	1.19623	0.0530218	0.0265109	+	0.999649i	$0.491560\pi$
$510$	0	0
$511$	6.60010	0.291971
$512$	0	0
$513$	−10.7404	−0.474199
$514$	0	0
$515$	0.758728	0.0334335
$516$	0	0
$517$	16.2748	0.715765
$518$	0	0
$519$	20.4591	0.898053
$520$	0	0
$521$	−42.0169	−1.84080	−0.920398	−	0.390984i	$-0.872135\pi$
$521$	−42.0169	−1.84080	−0.920398	+	0.390984i	$0.872135\pi$
$522$	0	0
$523$	−35.1386	−1.53650	−0.768252	−	0.640148i	$-0.778873\pi$
$523$	−35.1386	−1.53650	−0.768252	+	0.640148i	$0.778873\pi$
$524$	0	0
$525$	47.6042	2.07762
$526$	0	0
$527$	20.6497	0.899516
$528$	0	0
$529$	55.9353	2.43197
$530$	0	0
$531$	−1.18392	−0.0513777
$532$	0	0
$533$	35.1220	1.52130
$534$	0	0
$535$	−8.19751	−0.354409
$536$	0	0
$537$	45.3875	1.95861
$538$	0	0
$539$	70.9897	3.05774
$540$	0	0
$541$	14.9408	0.642357	0.321179	−	0.947019i	$-0.395921\pi$
$541$	14.9408	0.642357	0.321179	+	0.947019i	$0.395921\pi$
$542$	0	0
$543$	−18.4243	−0.790661
$544$	0	0
$545$	61.2992	2.62577
$546$	0	0
$547$	6.19872	0.265038	0.132519	−	0.991180i	$-0.457693\pi$
$547$	6.19872	0.265038	0.132519	+	0.991180i	$0.457693\pi$
$548$	0	0
$549$	−21.1131	−0.901085
$550$	0	0
$551$	9.54354	0.406569
$552$	0	0
$553$	−28.8249	−1.22576
$554$	0	0
$555$	63.8505	2.71030
$556$	0	0
$557$	14.9864	0.634993	0.317496	−	0.948259i	$-0.397158\pi$
$557$	14.9864	0.634993	0.317496	+	0.948259i	$0.397158\pi$
$558$	0	0
$559$	−15.8082	−0.668616
$560$	0	0
$561$	−29.1904	−1.23242
$562$	0	0
$563$	−26.1225	−1.10093	−0.550466	−	0.834858i	$-0.685550\pi$
$563$	−26.1225	−1.10093	−0.550466	+	0.834858i	$0.685550\pi$
$564$	0	0
$565$	26.5706	1.11783
$566$	0	0
$567$	−51.8698	−2.17833
$568$	0	0
$569$	28.5984	1.19891	0.599453	−	0.800410i	$-0.295385\pi$
$569$	28.5984	1.19891	0.599453	+	0.800410i	$0.295385\pi$
$570$	0	0
$571$	25.3749	1.06191	0.530953	−	0.847402i	$-0.321834\pi$
$571$	25.3749	1.06191	0.530953	+	0.847402i	$0.321834\pi$
$572$	0	0
$573$	33.0302	1.37986
$574$	0	0
$575$	−40.7948	−1.70126
$576$	0	0
$577$	27.4355	1.14216	0.571078	−	0.820896i	$-0.306525\pi$
$577$	27.4355	1.14216	0.571078	+	0.820896i	$0.306525\pi$
$578$	0	0
$579$	56.7791	2.35966
$580$	0	0
$581$	−19.6870	−0.816753
$582$	0	0
$583$	55.0994	2.28198
$584$	0	0
$585$	−24.7117	−1.02170
$586$	0	0
$587$	−26.4349	−1.09109	−0.545543	−	0.838083i	$-0.683676\pi$
$587$	−26.4349	−1.09109	−0.545543	+	0.838083i	$0.683676\pi$
$588$	0	0
$589$	−33.1427	−1.36562
$590$	0	0
$591$	12.6802	0.521595
$592$	0	0
$593$	−9.22666	−0.378893	−0.189447	−	0.981891i	$-0.560669\pi$
$593$	−9.22666	−0.378893	−0.189447	+	0.981891i	$0.560669\pi$
$594$	0	0
$595$	−40.0858	−1.64336
$596$	0	0
$597$	−28.0062	−1.14622
$598$	0	0
$599$	−47.6451	−1.94673	−0.973363	−	0.229268i	$-0.926367\pi$
$599$	−47.6451	−1.94673	−0.973363	+	0.229268i	$0.926367\pi$
$600$	0	0
$601$	−18.7911	−0.766506	−0.383253	−	0.923643i	$-0.625196\pi$
$601$	−18.7911	−0.766506	−0.383253	+	0.923643i	$0.625196\pi$
$602$	0	0
$603$	−7.40301	−0.301474
$604$	0	0
$605$	36.2253	1.47277
$606$	0	0
$607$	−9.16527	−0.372007	−0.186003	−	0.982549i	$-0.559554\pi$
$607$	−9.16527	−0.372007	−0.186003	+	0.982549i	$0.559554\pi$
$608$	0	0
$609$	22.2894	0.903214
$610$	0	0
$611$	−14.2872	−0.578000
$612$	0	0
$613$	10.2067	0.412243	0.206122	−	0.978526i	$-0.433916\pi$
$613$	10.2067	0.412243	0.206122	+	0.978526i	$0.433916\pi$
$614$	0	0
$615$	−57.6177	−2.32337
$616$	0	0
$617$	46.3435	1.86572	0.932860	−	0.360240i	$-0.117305\pi$
$617$	46.3435	1.86572	0.932860	+	0.360240i	$0.117305\pi$
$618$	0	0
$619$	20.6308	0.829221	0.414610	−	0.909999i	$-0.363918\pi$
$619$	20.6308	0.829221	0.414610	+	0.909999i	$0.363918\pi$
$620$	0	0
$621$	21.4965	0.862625
$622$	0	0
$623$	−78.7773	−3.15614
$624$	0	0
$625$	−26.8750	−1.07500
$626$	0	0
$627$	46.8504	1.87103
$628$	0	0
$629$	−25.7386	−1.02626
$630$	0	0
$631$	−0.503235	−0.0200335	−0.0100167	−	0.999950i	$-0.503188\pi$
$631$	−0.503235	−0.0200335	−0.0100167	+	0.999950i	$0.503188\pi$
$632$	0	0
$633$	35.7323	1.42023
$634$	0	0
$635$	6.37693	0.253061
$636$	0	0
$637$	−62.3201	−2.46921
$638$	0	0
$639$	−10.2425	−0.405187
$640$	0	0
$641$	37.3295	1.47443	0.737214	−	0.675659i	$-0.236141\pi$
$641$	37.3295	1.47443	0.737214	+	0.675659i	$0.236141\pi$
$642$	0	0
$643$	−0.106240	−0.00418969	−0.00209485	−	0.999998i	$-0.500667\pi$
$643$	−0.106240	−0.00418969	−0.00209485	+	0.999998i	$0.500667\pi$
$644$	0	0
$645$	25.9334	1.02113
$646$	0	0
$647$	−16.1931	−0.636616	−0.318308	−	0.947987i	$-0.603115\pi$
$647$	−16.1931	−0.636616	−0.318308	+	0.947987i	$0.603115\pi$
$648$	0	0
$649$	−2.95640	−0.116049
$650$	0	0
$651$	−77.4066	−3.03380
$652$	0	0
$653$	−17.0952	−0.668989	−0.334494	−	0.942398i	$-0.608565\pi$
$653$	−17.0952	−0.668989	−0.334494	+	0.942398i	$0.608565\pi$
$654$	0	0
$655$	26.1975	1.02362
$656$	0	0
$657$	2.69067	0.104973
$658$	0	0
$659$	14.4895	0.564431	0.282216	−	0.959351i	$-0.408931\pi$
$659$	14.4895	0.564431	0.282216	+	0.959351i	$0.408931\pi$
$660$	0	0
$661$	10.1919	0.396418	0.198209	−	0.980160i	$-0.436487\pi$
$661$	10.1919	0.396418	0.198209	+	0.980160i	$0.436487\pi$
$662$	0	0
$663$	25.6255	0.995211
$664$	0	0
$665$	64.3376	2.49490
$666$	0	0
$667$	−19.1011	−0.739598
$668$	0	0
$669$	−26.4015	−1.02074
$670$	0	0
$671$	−52.7221	−2.03531
$672$	0	0
$673$	38.0132	1.46530	0.732651	−	0.680604i	$-0.238283\pi$
$673$	38.0132	1.46530	0.732651	+	0.680604i	$0.238283\pi$
$674$	0	0
$675$	−11.1097	−0.427612
$676$	0	0
$677$	−42.3781	−1.62872	−0.814361	−	0.580358i	$-0.802913\pi$
$677$	−42.3781	−1.62872	−0.814361	+	0.580358i	$0.802913\pi$
$678$	0	0
$679$	43.2985	1.66164
$680$	0	0
$681$	12.9771	0.497282
$682$	0	0
$683$	−36.7923	−1.40782	−0.703908	−	0.710291i	$-0.748563\pi$
$683$	−36.7923	−1.40782	−0.703908	+	0.710291i	$0.748563\pi$
$684$	0	0
$685$	−36.8283	−1.40714
$686$	0	0
$687$	10.4995	0.400579
$688$	0	0
$689$	−48.3704	−1.84277
$690$	0	0
$691$	1.24309	0.0472893	0.0236446	−	0.999720i	$-0.492473\pi$
$691$	1.24309	0.0472893	0.0236446	+	0.999720i	$0.492473\pi$
$692$	0	0
$693$	42.5358	1.61580
$694$	0	0
$695$	−52.9397	−2.00812
$696$	0	0
$697$	23.2261	0.879752
$698$	0	0
$699$	7.34382	0.277769
$700$	0	0
$701$	13.7481	0.519257	0.259629	−	0.965709i	$-0.416400\pi$
$701$	13.7481	0.519257	0.259629	+	0.965709i	$0.416400\pi$
$702$	0	0
$703$	41.3104	1.55805
$704$	0	0
$705$	23.4383	0.882736
$706$	0	0
$707$	18.2870	0.687753
$708$	0	0
$709$	40.7096	1.52888	0.764441	−	0.644694i	$-0.223015\pi$
$709$	40.7096	1.52888	0.764441	+	0.644694i	$0.223015\pi$
$710$	0	0
$711$	−11.7511	−0.440700
$712$	0	0
$713$	66.3341	2.48423
$714$	0	0
$715$	−61.7083	−2.30776
$716$	0	0
$717$	1.13520	0.0423948
$718$	0	0
$719$	−21.7959	−0.812851	−0.406425	−	0.913684i	$-0.633225\pi$
$719$	−21.7959	−0.812851	−0.406425	+	0.913684i	$0.633225\pi$
$720$	0	0
$721$	1.14649	0.0426975
$722$	0	0
$723$	−32.1222	−1.19464
$724$	0	0
$725$	9.87170	0.366626
$726$	0	0
$727$	−44.6383	−1.65554	−0.827772	−	0.561064i	$-0.810392\pi$
$727$	−44.6383	−1.65554	−0.827772	+	0.561064i	$0.810392\pi$
$728$	0	0
$729$	−5.06536	−0.187606
$730$	0	0
$731$	−10.4540	−0.386654
$732$	0	0
$733$	−24.1455	−0.891836	−0.445918	−	0.895074i	$-0.647123\pi$
$733$	−24.1455	−0.891836	−0.445918	+	0.895074i	$0.647123\pi$
$734$	0	0
$735$	102.236	3.77104
$736$	0	0
$737$	−18.4863	−0.680950
$738$	0	0
$739$	−0.696924	−0.0256368	−0.0128184	−	0.999918i	$-0.504080\pi$
$739$	−0.696924	−0.0256368	−0.0128184	+	0.999918i	$0.504080\pi$
$740$	0	0
$741$	−41.1288	−1.51091
$742$	0	0
$743$	−11.1443	−0.408846	−0.204423	−	0.978883i	$-0.565532\pi$
$743$	−11.1443	−0.408846	−0.204423	+	0.978883i	$0.565532\pi$
$744$	0	0
$745$	−43.5564	−1.59578
$746$	0	0
$747$	−8.02581	−0.293649
$748$	0	0
$749$	−12.3870	−0.452612
$750$	0	0
$751$	24.8606	0.907174	0.453587	−	0.891212i	$-0.350144\pi$
$751$	24.8606	0.907174	0.453587	+	0.891212i	$0.350144\pi$
$752$	0	0
$753$	2.21536	0.0807323
$754$	0	0
$755$	−55.0100	−2.00202
$756$	0	0
$757$	−11.6288	−0.422657	−0.211329	−	0.977415i	$-0.567779\pi$
$757$	−11.6288	−0.422657	−0.211329	+	0.977415i	$0.567779\pi$
$758$	0	0
$759$	−93.7697	−3.40362
$760$	0	0
$761$	29.1632	1.05717	0.528583	−	0.848881i	$-0.322723\pi$
$761$	29.1632	1.05717	0.528583	+	0.848881i	$0.322723\pi$
$762$	0	0
$763$	92.6273	3.35333
$764$	0	0
$765$	−16.3418	−0.590840
$766$	0	0
$767$	2.59535	0.0937126
$768$	0	0
$769$	−24.3770	−0.879056	−0.439528	−	0.898229i	$-0.644854\pi$
$769$	−24.3770	−0.879056	−0.439528	+	0.898229i	$0.644854\pi$
$770$	0	0
$771$	62.7136	2.25858
$772$	0	0
$773$	−41.2500	−1.48366	−0.741830	−	0.670588i	$-0.766042\pi$
$773$	−41.2500	−1.48366	−0.741830	+	0.670588i	$0.766042\pi$
$774$	0	0
$775$	−34.2824	−1.23146
$776$	0	0
$777$	96.4825	3.46129
$778$	0	0
$779$	−37.2779	−1.33562
$780$	0	0
$781$	−25.5769	−0.915212
$782$	0	0
$783$	−5.20182	−0.185898
$784$	0	0
$785$	−63.5464	−2.26807
$786$	0	0
$787$	1.65386	0.0589537	0.0294769	−	0.999565i	$-0.490616\pi$
$787$	1.65386	0.0589537	0.0294769	+	0.999565i	$0.490616\pi$
$788$	0	0
$789$	−34.8718	−1.24147
$790$	0	0
$791$	40.1501	1.42757
$792$	0	0
$793$	46.2834	1.64357
$794$	0	0
$795$	79.3518	2.81432
$796$	0	0
$797$	2.53175	0.0896793	0.0448396	−	0.998994i	$-0.485722\pi$
$797$	2.53175	0.0896793	0.0448396	+	0.998994i	$0.485722\pi$
$798$	0	0
$799$	−9.44813	−0.334251
$800$	0	0
$801$	−32.1152	−1.13474
$802$	0	0
$803$	6.71895	0.237107
$804$	0	0
$805$	−128.770	−4.53853
$806$	0	0
$807$	0.965542	0.0339887
$808$	0	0
$809$	14.3444	0.504323	0.252161	−	0.967685i	$-0.418859\pi$
$809$	14.3444	0.504323	0.252161	+	0.967685i	$0.418859\pi$
$810$	0	0
$811$	25.5339	0.896618	0.448309	−	0.893879i	$-0.352026\pi$
$811$	25.5339	0.896618	0.448309	+	0.893879i	$0.352026\pi$
$812$	0	0
$813$	−41.3690	−1.45087
$814$	0	0
$815$	26.7478	0.936935
$816$	0	0
$817$	16.7786	0.587008
$818$	0	0
$819$	−37.3411	−1.30480
$820$	0	0
$821$	−3.58307	−0.125050	−0.0625249	−	0.998043i	$-0.519915\pi$
$821$	−3.58307	−0.125050	−0.0625249	+	0.998043i	$0.519915\pi$
$822$	0	0
$823$	24.1716	0.842568	0.421284	−	0.906929i	$-0.361580\pi$
$823$	24.1716	0.842568	0.421284	+	0.906929i	$0.361580\pi$
$824$	0	0
$825$	48.4614	1.68721
$826$	0	0
$827$	4.63507	0.161177	0.0805885	−	0.996747i	$-0.474320\pi$
$827$	4.63507	0.161177	0.0805885	+	0.996747i	$0.474320\pi$
$828$	0	0
$829$	−47.8954	−1.66348	−0.831738	−	0.555169i	$-0.812654\pi$
$829$	−47.8954	−1.66348	−0.831738	+	0.555169i	$0.812654\pi$
$830$	0	0
$831$	46.3692	1.60853
$832$	0	0
$833$	−41.2122	−1.42792
$834$	0	0
$835$	−66.1749	−2.29008
$836$	0	0
$837$	18.0648	0.624412
$838$	0	0
$839$	−32.6951	−1.12876	−0.564380	−	0.825515i	$-0.690885\pi$
$839$	−32.6951	−1.12876	−0.564380	+	0.825515i	$0.690885\pi$
$840$	0	0
$841$	−24.3778	−0.840615
$842$	0	0
$843$	62.4207	2.14988
$844$	0	0
$845$	13.9106	0.478540
$846$	0	0
$847$	54.7390	1.88085
$848$	0	0
$849$	52.6905	1.80833
$850$	0	0
$851$	−82.6814	−2.83428
$852$	0	0
$853$	4.10625	0.140595	0.0702977	−	0.997526i	$-0.477605\pi$
$853$	4.10625	0.140595	0.0702977	+	0.997526i	$0.477605\pi$
$854$	0	0
$855$	26.2286	0.896999
$856$	0	0
$857$	14.0056	0.478424	0.239212	−	0.970967i	$-0.423111\pi$
$857$	14.0056	0.478424	0.239212	+	0.970967i	$0.423111\pi$
$858$	0	0
$859$	17.1442	0.584954	0.292477	−	0.956273i	$-0.405520\pi$
$859$	17.1442	0.584954	0.292477	+	0.956273i	$0.405520\pi$
$860$	0	0
$861$	−87.0644	−2.96715
$862$	0	0
$863$	57.5292	1.95832	0.979158	−	0.203098i	$-0.0651010\pi$
$863$	57.5292	1.95832	0.979158	+	0.203098i	$0.0651010\pi$
$864$	0	0
$865$	28.6014	0.972477
$866$	0	0
$867$	−20.7151	−0.703521
$868$	0	0
$869$	−29.3439	−0.995425
$870$	0	0
$871$	16.2286	0.549886
$872$	0	0
$873$	17.6515	0.597414
$874$	0	0
$875$	−5.91843	−0.200079
$876$	0	0
$877$	46.5128	1.57063	0.785313	−	0.619099i	$-0.212502\pi$
$877$	46.5128	1.57063	0.785313	+	0.619099i	$0.212502\pi$
$878$	0	0
$879$	−35.4743	−1.19652
$880$	0	0
$881$	−11.0699	−0.372954	−0.186477	−	0.982459i	$-0.559707\pi$
$881$	−11.0699	−0.372954	−0.186477	+	0.982459i	$0.559707\pi$
$882$	0	0
$883$	−0.152523	−0.00513280	−0.00256640	−	0.999997i	$-0.500817\pi$
$883$	−0.152523	−0.00513280	−0.00256640	+	0.999997i	$0.500817\pi$
$884$	0	0
$885$	−4.25768	−0.143120
$886$	0	0
$887$	−9.69157	−0.325411	−0.162706	−	0.986675i	$-0.552022\pi$
$887$	−9.69157	−0.325411	−0.162706	+	0.986675i	$0.552022\pi$
$888$	0	0
$889$	9.63599	0.323181
$890$	0	0
$891$	−52.8038	−1.76900
$892$	0	0
$893$	15.1642	0.507452
$894$	0	0
$895$	63.4509	2.12093
$896$	0	0
$897$	82.3181	2.74852
$898$	0	0
$899$	−16.0518	−0.535358
$900$	0	0
$901$	−31.9873	−1.06565
$902$	0	0
$903$	39.1872	1.30407
$904$	0	0
$905$	−25.7568	−0.856185
$906$	0	0
$907$	22.2135	0.737587	0.368793	−	0.929511i	$-0.379771\pi$
$907$	22.2135	0.737587	0.368793	+	0.929511i	$0.379771\pi$
$908$	0	0
$909$	7.45508	0.247269
$910$	0	0
$911$	31.3519	1.03873	0.519367	−	0.854551i	$-0.326168\pi$
$911$	31.3519	1.03873	0.519367	+	0.854551i	$0.326168\pi$
$912$	0	0
$913$	−20.0415	−0.663276
$914$	0	0
$915$	−75.9281	−2.51010
$916$	0	0
$917$	39.5862	1.30725
$918$	0	0
$919$	−38.1544	−1.25860	−0.629299	−	0.777163i	$-0.716658\pi$
$919$	−38.1544	−1.25860	−0.629299	+	0.777163i	$0.716658\pi$
$920$	0	0
$921$	3.08501	0.101655
$922$	0	0
$923$	22.4533	0.739059
$924$	0	0
$925$	42.7308	1.40498
$926$	0	0
$927$	0.467391	0.0153511
$928$	0	0
$929$	−10.3278	−0.338843	−0.169421	−	0.985544i	$-0.554190\pi$
$929$	−10.3278	−0.338843	−0.169421	+	0.985544i	$0.554190\pi$
$930$	0	0
$931$	66.1455	2.16783
$932$	0	0
$933$	−66.3165	−2.17111
$934$	0	0
$935$	−40.8076	−1.33455
$936$	0	0
$937$	8.05077	0.263007	0.131504	−	0.991316i	$-0.458020\pi$
$937$	8.05077	0.263007	0.131504	+	0.991316i	$0.458020\pi$
$938$	0	0
$939$	−5.57674	−0.181990
$940$	0	0
$941$	28.8005	0.938870	0.469435	−	0.882967i	$-0.344458\pi$
$941$	28.8005	0.938870	0.469435	+	0.882967i	$0.344458\pi$
$942$	0	0
$943$	74.6105	2.42965
$944$	0	0
$945$	−35.0680	−1.14076
$946$	0	0
$947$	−19.4582	−0.632305	−0.316152	−	0.948708i	$-0.602391\pi$
$947$	−19.4582	−0.632305	−0.316152	+	0.948708i	$0.602391\pi$
$948$	0	0
$949$	−5.89840	−0.191470
$950$	0	0
$951$	4.43444	0.143797
$952$	0	0
$953$	−48.9075	−1.58427	−0.792135	−	0.610346i	$-0.791031\pi$
$953$	−48.9075	−1.58427	−0.792135	+	0.610346i	$0.791031\pi$
$954$	0	0
$955$	46.1757	1.49421
$956$	0	0
$957$	22.6908	0.733490
$958$	0	0
$959$	−55.6501	−1.79704
$960$	0	0
$961$	24.7446	0.798214
$962$	0	0
$963$	−5.04983	−0.162728
$964$	0	0
$965$	79.3761	2.55521
$966$	0	0
$967$	−44.7258	−1.43828	−0.719142	−	0.694863i	$-0.755465\pi$
$967$	−44.7258	−1.43828	−0.719142	+	0.694863i	$0.755465\pi$
$968$	0	0
$969$	−27.1985	−0.873740
$970$	0	0
$971$	2.73713	0.0878386	0.0439193	−	0.999035i	$-0.486016\pi$
$971$	2.73713	0.0878386	0.0439193	+	0.999035i	$0.486016\pi$
$972$	0	0
$973$	−79.9955	−2.56454
$974$	0	0
$975$	−42.5431	−1.36247
$976$	0	0
$977$	−14.8910	−0.476404	−0.238202	−	0.971216i	$-0.576558\pi$
$977$	−14.8910	−0.476404	−0.238202	+	0.971216i	$0.576558\pi$
$978$	0	0
$979$	−80.1958	−2.56307
$980$	0	0
$981$	37.7615	1.20563
$982$	0	0
$983$	28.3312	0.903626	0.451813	−	0.892113i	$-0.350777\pi$
$983$	28.3312	0.903626	0.451813	+	0.892113i	$0.350777\pi$
$984$	0	0
$985$	17.7267	0.564821
$986$	0	0
$987$	35.4169	1.12733
$988$	0	0
$989$	−33.5818	−1.06784
$990$	0	0
$991$	−20.3125	−0.645248	−0.322624	−	0.946527i	$-0.604565\pi$
$991$	−20.3125	−0.645248	−0.322624	+	0.946527i	$0.604565\pi$
$992$	0	0
$993$	9.57992	0.304010
$994$	0	0
$995$	−39.1522	−1.24121
$996$	0	0
$997$	24.2802	0.768961	0.384480	−	0.923133i	$-0.374381\pi$
$997$	24.2802	0.768961	0.384480	+	0.923133i	$0.374381\pi$
$998$	0	0
$999$	−22.5167	−0.712397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.m.1.19		23
4.3	odd	2		2008.2.a.d.1.5	✓	23

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2008.2.a.d.1.5	✓	23	4.3	odd	2
4016.2.a.m.1.19		23	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)

\(f(q)\)	\(=\)	\(q+2.21536 q^{3} +3.09704 q^{5} +4.67984 q^{7} +1.90784 q^{9} +O(q^{10})\)	\(q+2.21536 q^{3} +3.09704 q^{5} +4.67984 q^{7} +1.90784 q^{9} +4.76411 q^{11} -4.18229 q^{13} +6.86107 q^{15} -2.76575 q^{17} +4.43902 q^{19} +10.3676 q^{21} -8.88455 q^{23} +4.59165 q^{25} -2.41954 q^{27} +2.14992 q^{29} -7.46623 q^{31} +10.5542 q^{33} +14.4937 q^{35} +9.30620 q^{37} -9.26530 q^{39} -8.39778 q^{41} +3.77980 q^{43} +5.90865 q^{45} +3.41612 q^{47} +14.9009 q^{49} -6.12714 q^{51} +11.5655 q^{53} +14.7546 q^{55} +9.83404 q^{57} -0.620556 q^{59} -11.0665 q^{61} +8.92838 q^{63} -12.9527 q^{65} -3.88032 q^{67} -19.6825 q^{69} -5.36865 q^{71} +1.41033 q^{73} +10.1722 q^{75} +22.2953 q^{77} -6.15937 q^{79} -11.0837 q^{81} -4.20676 q^{83} -8.56563 q^{85} +4.76286 q^{87} -16.8333 q^{89} -19.5725 q^{91} -16.5404 q^{93} +13.7478 q^{95} +9.25212 q^{97} +9.08915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9}+O(q^{10}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9	\( 23 q - 2 q^{3} + 8 q^{5} - 2 q^{7} + 45 q^{9} - 8 q^{11} + 8 q^{13} - 7 q^{15} + 19 q^{17} + 9 q^{19} + 9 q^{21} - 21 q^{23} + 65 q^{25} - 5 q^{27} + 10 q^{29} + 9 q^{31} + 34 q^{33} - 12 q^{35} + 11 q^{37} + 9 q^{39} + 35 q^{41} + 9 q^{43} + 29 q^{45} - 37 q^{47} + 77 q^{49} + 17 q^{51} + 38 q^{53} + 20 q^{55} + 51 q^{57} - 17 q^{59} - 22 q^{63} + 41 q^{65} - 9 q^{67} + 8 q^{69} - 13 q^{71} + 41 q^{73} - 25 q^{75} + 36 q^{77} + 36 q^{79} + 127 q^{81} - 29 q^{83} + 34 q^{85} - 10 q^{87} + 36 q^{89} + 6 q^{91} + 36 q^{93} - 25 q^{95} + 40 q^{97} - 19 q^{99}+O(q^{100}) \) 23 * q - 2 * q^3 + 8 * q^5 - 2 * q^7 + 45 * q^9 - 8 * q^11 + 8 * q^13 - 7 * q^15 + 19 * q^17 + 9 * q^19 + 9 * q^21 - 21 * q^23 + 65 * q^25 - 5 * q^27 + 10 * q^29 + 9 * q^31 + 34 * q^33 - 12 * q^35 + 11 * q^37 + 9 * q^39 + 35 * q^41 + 9 * q^43 + 29 * q^45 - 37 * q^47 + 77 * q^49 + 17 * q^51 + 38 * q^53 + 20 * q^55 + 51 * q^57 - 17 * q^59 - 22 * q^63 + 41 * q^65 - 9 * q^67 + 8 * q^69 - 13 * q^71 + 41 * q^73 - 25 * q^75 + 36 * q^77 + 36 * q^79 + 127 * q^81 - 29 * q^83 + 34 * q^85 - 10 * q^87 + 36 * q^89 + 6 * q^91 + 36 * q^93 - 25 * q^95 + 40 * q^97 - 19 * q^99

Introduction

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Database

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