LMFDB - Embedded newform 8016.2.a.t.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8016,2,Mod(1,8016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8016 = 2^{4} \cdot 3 \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8016.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:	$64.0080822603$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.149169.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 6x^{3} - 3x^{2} + 4x + 1 $ x^5 - 6x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2004)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.54970$ of defining polynomial
Character	$\chi$	$=$	8016.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +0.951283 q^{5} +1.28171 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.951283 q^{5} +1.28171 q^{7} +1.00000 q^{9} +0.951283 q^{11} -2.82707 q^{13} +0.951283 q^{15} +1.31500 q^{17} -5.94020 q^{19} +1.28171 q^{21} +2.56927 q^{23} -4.09506 q^{25} +1.00000 q^{27} -2.48910 q^{29} -0.984571 q^{31} +0.951283 q^{33} +1.21927 q^{35} -7.81553 q^{37} -2.82707 q^{39} -7.41944 q^{41} -7.65633 q^{43} +0.951283 q^{45} -10.1096 q^{47} -5.35721 q^{49} +1.31500 q^{51} -3.23734 q^{53} +0.904940 q^{55} -5.94020 q^{57} +10.3093 q^{59} -11.8766 q^{61} +1.28171 q^{63} -2.68935 q^{65} -8.14745 q^{67} +2.56927 q^{69} -1.71178 q^{71} -0.502340 q^{73} -4.09506 q^{75} +1.21927 q^{77} +14.6747 q^{79} +1.00000 q^{81} +11.3212 q^{83} +1.25094 q^{85} -2.48910 q^{87} +6.15777 q^{89} -3.62349 q^{91} -0.984571 q^{93} -5.65081 q^{95} +4.75139 q^{97} +0.951283 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 7 q^{17} + 2 q^{19} + 2 q^{21} - 13 q^{23} - 2 q^{25} + 5 q^{27} - 3 q^{29} + 12 q^{31} - 3 q^{33} - 10 q^{35} - 7 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{45} - q^{47} - 17 q^{49} - 7 q^{51} + 3 q^{53} + 23 q^{55} + 2 q^{57} - q^{59} - 22 q^{61} + 2 q^{63} - 20 q^{65} - 2 q^{67} - 13 q^{69} - 9 q^{71} - 28 q^{73} - 2 q^{75} - 10 q^{77} + 28 q^{79} + 5 q^{81} - 7 q^{83} - 11 q^{85} - 3 q^{87} - 30 q^{89} + 13 q^{91} + 12 q^{93} - 3 q^{95} - 33 q^{97} - 3 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 7 * q^17 + 2 * q^19 + 2 * q^21 - 13 * q^23 - 2 * q^25 + 5 * q^27 - 3 * q^29 + 12 * q^31 - 3 * q^33 - 10 * q^35 - 7 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^45 - q^47 - 17 * q^49 - 7 * q^51 + 3 * q^53 + 23 * q^55 + 2 * q^57 - q^59 - 22 * q^61 + 2 * q^63 - 20 * q^65 - 2 * q^67 - 13 * q^69 - 9 * q^71 - 28 * q^73 - 2 * q^75 - 10 * q^77 + 28 * q^79 + 5 * q^81 - 7 * q^83 - 11 * q^85 - 3 * q^87 - 30 * q^89 + 13 * q^91 + 12 * q^93 - 3 * q^95 - 33 * q^97 - 3 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0.951283	0.425427	0.212713	−	0.977115i	$-0.431770\pi$
$5$	0.951283	0.425427	0.212713	+	0.977115i	$0.431770\pi$
$6$	0	0
$7$	1.28171	0.484442	0.242221	−	0.970221i	$-0.422124\pi$
$7$	1.28171	0.484442	0.242221	+	0.970221i	$0.422124\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.951283	0.286823	0.143411	−	0.989663i	$-0.454193\pi$
$11$	0.951283	0.286823	0.143411	+	0.989663i	$0.454193\pi$
$12$	0	0
$13$	−2.82707	−0.784088	−0.392044	−	0.919946i	$-0.628232\pi$
$13$	−2.82707	−0.784088	−0.392044	+	0.919946i	$0.628232\pi$
$14$	0	0
$15$	0.951283	0.245620
$16$	0	0
$17$	1.31500	0.318935	0.159467	−	0.987203i	$-0.449022\pi$
$17$	1.31500	0.318935	0.159467	+	0.987203i	$0.449022\pi$
$18$	0	0
$19$	−5.94020	−1.36278	−0.681388	−	0.731923i	$-0.738623\pi$
$19$	−5.94020	−1.36278	−0.681388	+	0.731923i	$0.738623\pi$
$20$	0	0
$21$	1.28171	0.279693
$22$	0	0
$23$	2.56927	0.535729	0.267865	−	0.963457i	$-0.413682\pi$
$23$	2.56927	0.535729	0.267865	+	0.963457i	$0.413682\pi$
$24$	0	0
$25$	−4.09506	−0.819012
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.48910	−0.462214	−0.231107	−	0.972928i	$-0.574235\pi$
$29$	−2.48910	−0.462214	−0.231107	+	0.972928i	$0.574235\pi$
$30$	0	0
$31$	−0.984571	−0.176834	−0.0884171	−	0.996084i	$-0.528181\pi$
$31$	−0.984571	−0.176834	−0.0884171	+	0.996084i	$0.528181\pi$
$32$	0	0
$33$	0.951283	0.165597
$34$	0	0
$35$	1.21927	0.206095
$36$	0	0
$37$	−7.81553	−1.28487	−0.642433	−	0.766342i	$-0.722075\pi$
$37$	−7.81553	−1.28487	−0.642433	+	0.766342i	$0.722075\pi$
$38$	0	0
$39$	−2.82707	−0.452694
$40$	0	0
$41$	−7.41944	−1.15872	−0.579361	−	0.815071i	$-0.696698\pi$
$41$	−7.41944	−1.15872	−0.579361	+	0.815071i	$0.696698\pi$
$42$	0	0
$43$	−7.65633	−1.16758	−0.583789	−	0.811905i	$-0.698431\pi$
$43$	−7.65633	−1.16758	−0.583789	+	0.811905i	$0.698431\pi$
$44$	0	0
$45$	0.951283	0.141809
$46$	0	0
$47$	−10.1096	−1.47464	−0.737318	−	0.675546i	$-0.763908\pi$
$47$	−10.1096	−1.47464	−0.737318	+	0.675546i	$0.763908\pi$
$48$	0	0
$49$	−5.35721	−0.765316
$50$	0	0
$51$	1.31500	0.184137
$52$	0	0
$53$	−3.23734	−0.444683	−0.222342	−	0.974969i	$-0.571370\pi$
$53$	−3.23734	−0.444683	−0.222342	+	0.974969i	$0.571370\pi$
$54$	0	0
$55$	0.904940	0.122022
$56$	0	0
$57$	−5.94020	−0.786799
$58$	0	0
$59$	10.3093	1.34216	0.671078	−	0.741386i	$-0.265831\pi$
$59$	10.3093	1.34216	0.671078	+	0.741386i	$0.265831\pi$
$60$	0	0
$61$	−11.8766	−1.52064	−0.760321	−	0.649547i	$-0.774958\pi$
$61$	−11.8766	−1.52064	−0.760321	+	0.649547i	$0.774958\pi$
$62$	0	0
$63$	1.28171	0.161481
$64$	0	0
$65$	−2.68935	−0.333572
$66$	0	0
$67$	−8.14745	−0.995370	−0.497685	−	0.867358i	$-0.665816\pi$
$67$	−8.14745	−0.995370	−0.497685	+	0.867358i	$0.665816\pi$
$68$	0	0
$69$	2.56927	0.309303
$70$	0	0
$71$	−1.71178	−0.203151	−0.101575	−	0.994828i	$-0.532388\pi$
$71$	−1.71178	−0.203151	−0.101575	+	0.994828i	$0.532388\pi$
$72$	0	0
$73$	−0.502340	−0.0587944	−0.0293972	−	0.999568i	$-0.509359\pi$
$73$	−0.502340	−0.0587944	−0.0293972	+	0.999568i	$0.509359\pi$
$74$	0	0
$75$	−4.09506	−0.472857
$76$	0	0
$77$	1.21927	0.138949
$78$	0	0
$79$	14.6747	1.65104	0.825518	−	0.564376i	$-0.190883\pi$
$79$	14.6747	1.65104	0.825518	+	0.564376i	$0.190883\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	11.3212	1.24266	0.621331	−	0.783548i	$-0.286592\pi$
$83$	11.3212	1.24266	0.621331	+	0.783548i	$0.286592\pi$
$84$	0	0
$85$	1.25094	0.135683
$86$	0	0
$87$	−2.48910	−0.266859
$88$	0	0
$89$	6.15777	0.652722	0.326361	−	0.945245i	$-0.394177\pi$
$89$	6.15777	0.652722	0.326361	+	0.945245i	$0.394177\pi$
$90$	0	0
$91$	−3.62349	−0.379845
$92$	0	0
$93$	−0.984571	−0.102095
$94$	0	0
$95$	−5.65081	−0.579761
$96$	0	0
$97$	4.75139	0.482430	0.241215	−	0.970472i	$-0.422454\pi$
$97$	4.75139	0.482430	0.241215	+	0.970472i	$0.422454\pi$
$98$	0	0
$99$	0.951283	0.0956076
$100$	0	0
$101$	−2.52463	−0.251210	−0.125605	−	0.992080i	$-0.540087\pi$
$101$	−2.52463	−0.251210	−0.125605	+	0.992080i	$0.540087\pi$
$102$	0	0
$103$	10.1843	1.00349	0.501743	−	0.865016i	$-0.332692\pi$
$103$	10.1843	1.00349	0.501743	+	0.865016i	$0.332692\pi$
$104$	0	0
$105$	1.21927	0.118989
$106$	0	0
$107$	−13.1161	−1.26798	−0.633991	−	0.773341i	$-0.718584\pi$
$107$	−13.1161	−1.26798	−0.633991	+	0.773341i	$0.718584\pi$
$108$	0	0
$109$	−0.209627	−0.0200786	−0.0100393	−	0.999950i	$-0.503196\pi$
$109$	−0.209627	−0.0200786	−0.0100393	+	0.999950i	$0.503196\pi$
$110$	0	0
$111$	−7.81553	−0.741817
$112$	0	0
$113$	0.554507	0.0521636	0.0260818	−	0.999660i	$-0.491697\pi$
$113$	0.554507	0.0521636	0.0260818	+	0.999660i	$0.491697\pi$
$114$	0	0
$115$	2.44410	0.227914
$116$	0	0
$117$	−2.82707	−0.261363
$118$	0	0
$119$	1.68546	0.154505
$120$	0	0
$121$	−10.0951	−0.917733
$122$	0	0
$123$	−7.41944	−0.668988
$124$	0	0
$125$	−8.65198	−0.773857
$126$	0	0
$127$	17.5210	1.55474	0.777370	−	0.629044i	$-0.216553\pi$
$127$	17.5210	1.55474	0.777370	+	0.629044i	$0.216553\pi$
$128$	0	0
$129$	−7.65633	−0.674102
$130$	0	0
$131$	−6.30379	−0.550765	−0.275382	−	0.961335i	$-0.588804\pi$
$131$	−6.30379	−0.550765	−0.275382	+	0.961335i	$0.588804\pi$
$132$	0	0
$133$	−7.61364	−0.660186
$134$	0	0
$135$	0.951283	0.0818734
$136$	0	0
$137$	−6.12238	−0.523070	−0.261535	−	0.965194i	$-0.584229\pi$
$137$	−6.12238	−0.523070	−0.261535	+	0.965194i	$0.584229\pi$
$138$	0	0
$139$	7.59742	0.644405	0.322203	−	0.946671i	$-0.395577\pi$
$139$	7.59742	0.644405	0.322203	+	0.946671i	$0.395577\pi$
$140$	0	0
$141$	−10.1096	−0.851381
$142$	0	0
$143$	−2.68935	−0.224894
$144$	0	0
$145$	−2.36784	−0.196638
$146$	0	0
$147$	−5.35721	−0.441855
$148$	0	0
$149$	−17.8638	−1.46346	−0.731730	−	0.681594i	$-0.761287\pi$
$149$	−17.8638	−1.46346	−0.731730	+	0.681594i	$0.761287\pi$
$150$	0	0
$151$	4.74233	0.385925	0.192962	−	0.981206i	$-0.438190\pi$
$151$	4.74233	0.385925	0.192962	+	0.981206i	$0.438190\pi$
$152$	0	0
$153$	1.31500	0.106312
$154$	0	0
$155$	−0.936606	−0.0752300
$156$	0	0
$157$	−9.54024	−0.761394	−0.380697	−	0.924700i	$-0.624316\pi$
$157$	−9.54024	−0.761394	−0.380697	+	0.924700i	$0.624316\pi$
$158$	0	0
$159$	−3.23734	−0.256738
$160$	0	0
$161$	3.29306	0.259530
$162$	0	0
$163$	1.89939	0.148771	0.0743857	−	0.997230i	$-0.476300\pi$
$163$	1.89939	0.148771	0.0743857	+	0.997230i	$0.476300\pi$
$164$	0	0
$165$	0.904940	0.0704495
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−5.00767	−0.385206
$170$	0	0
$171$	−5.94020	−0.454259
$172$	0	0
$173$	13.8969	1.05656	0.528280	−	0.849070i	$-0.322837\pi$
$173$	13.8969	1.05656	0.528280	+	0.849070i	$0.322837\pi$
$174$	0	0
$175$	−5.24869	−0.396764
$176$	0	0
$177$	10.3093	0.774895
$178$	0	0
$179$	5.87857	0.439385	0.219692	−	0.975569i	$-0.429495\pi$
$179$	5.87857	0.439385	0.219692	+	0.975569i	$0.429495\pi$
$180$	0	0
$181$	−20.6955	−1.53828	−0.769140	−	0.639080i	$-0.779315\pi$
$181$	−20.6955	−1.53828	−0.769140	+	0.639080i	$0.779315\pi$
$182$	0	0
$183$	−11.8766	−0.877943
$184$	0	0
$185$	−7.43478	−0.546616
$186$	0	0
$187$	1.25094	0.0914777
$188$	0	0
$189$	1.28171	0.0932310
$190$	0	0
$191$	13.8727	1.00380	0.501898	−	0.864927i	$-0.332635\pi$
$191$	13.8727	1.00380	0.501898	+	0.864927i	$0.332635\pi$
$192$	0	0
$193$	−0.344963	−0.0248310	−0.0124155	−	0.999923i	$-0.503952\pi$
$193$	−0.344963	−0.0248310	−0.0124155	+	0.999923i	$0.503952\pi$
$194$	0	0
$195$	−2.68935	−0.192588
$196$	0	0
$197$	24.1397	1.71988	0.859942	−	0.510392i	$-0.170500\pi$
$197$	24.1397	1.71988	0.859942	+	0.510392i	$0.170500\pi$
$198$	0	0
$199$	3.97139	0.281524	0.140762	−	0.990043i	$-0.455045\pi$
$199$	3.97139	0.281524	0.140762	+	0.990043i	$0.455045\pi$
$200$	0	0
$201$	−8.14745	−0.574677
$202$	0	0
$203$	−3.19031	−0.223916
$204$	0	0
$205$	−7.05799	−0.492951
$206$	0	0
$207$	2.56927	0.178576
$208$	0	0
$209$	−5.65081	−0.390875
$210$	0	0
$211$	21.3487	1.46970	0.734852	−	0.678227i	$-0.237251\pi$
$211$	21.3487	1.46970	0.734852	+	0.678227i	$0.237251\pi$
$212$	0	0
$213$	−1.71178	−0.117289
$214$	0	0
$215$	−7.28334	−0.496719
$216$	0	0
$217$	−1.26194	−0.0856660
$218$	0	0
$219$	−0.502340	−0.0339450
$220$	0	0
$221$	−3.71760	−0.250073
$222$	0	0
$223$	−11.6936	−0.783063	−0.391531	−	0.920165i	$-0.628055\pi$
$223$	−11.6936	−0.783063	−0.391531	+	0.920165i	$0.628055\pi$
$224$	0	0
$225$	−4.09506	−0.273004
$226$	0	0
$227$	−14.3838	−0.954685	−0.477342	−	0.878717i	$-0.658400\pi$
$227$	−14.3838	−0.954685	−0.477342	+	0.878717i	$0.658400\pi$
$228$	0	0
$229$	9.30406	0.614830	0.307415	−	0.951576i	$-0.400536\pi$
$229$	9.30406	0.614830	0.307415	+	0.951576i	$0.400536\pi$
$230$	0	0
$231$	1.21927	0.0802223
$232$	0	0
$233$	−1.32737	−0.0869591	−0.0434795	−	0.999054i	$-0.513844\pi$
$233$	−1.32737	−0.0869591	−0.0434795	+	0.999054i	$0.513844\pi$
$234$	0	0
$235$	−9.61709	−0.627350
$236$	0	0
$237$	14.6747	0.953226
$238$	0	0
$239$	6.67440	0.431731	0.215865	−	0.976423i	$-0.430743\pi$
$239$	6.67440	0.431731	0.215865	+	0.976423i	$0.430743\pi$
$240$	0	0
$241$	21.3838	1.37745	0.688726	−	0.725021i	$-0.258170\pi$
$241$	21.3838	1.37745	0.688726	+	0.725021i	$0.258170\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−5.09622	−0.325586
$246$	0	0
$247$	16.7934	1.06854
$248$	0	0
$249$	11.3212	0.717451
$250$	0	0
$251$	−27.1243	−1.71207	−0.856034	−	0.516919i	$-0.827079\pi$
$251$	−27.1243	−1.71207	−0.856034	+	0.516919i	$0.827079\pi$
$252$	0	0
$253$	2.44410	0.153659
$254$	0	0
$255$	1.25094	0.0783368
$256$	0	0
$257$	11.2287	0.700429	0.350214	−	0.936670i	$-0.386109\pi$
$257$	11.2287	0.700429	0.350214	+	0.936670i	$0.386109\pi$
$258$	0	0
$259$	−10.0173	−0.622443
$260$	0	0
$261$	−2.48910	−0.154071
$262$	0	0
$263$	−25.5610	−1.57616	−0.788079	−	0.615574i	$-0.788924\pi$
$263$	−25.5610	−1.57616	−0.788079	+	0.615574i	$0.788924\pi$
$264$	0	0
$265$	−3.07963	−0.189180
$266$	0	0
$267$	6.15777	0.376849
$268$	0	0
$269$	20.1581	1.22906	0.614532	−	0.788892i	$-0.289345\pi$
$269$	20.1581	1.22906	0.614532	+	0.788892i	$0.289345\pi$
$270$	0	0
$271$	1.90126	0.115493	0.0577467	−	0.998331i	$-0.481608\pi$
$271$	1.90126	0.115493	0.0577467	+	0.998331i	$0.481608\pi$
$272$	0	0
$273$	−3.62349	−0.219304
$274$	0	0
$275$	−3.89556	−0.234911
$276$	0	0
$277$	31.1953	1.87435	0.937173	−	0.348865i	$-0.113433\pi$
$277$	31.1953	1.87435	0.937173	+	0.348865i	$0.113433\pi$
$278$	0	0
$279$	−0.984571	−0.0589447
$280$	0	0
$281$	−31.9013	−1.90307	−0.951535	−	0.307540i	$-0.900494\pi$
$281$	−31.9013	−1.90307	−0.951535	+	0.307540i	$0.900494\pi$
$282$	0	0
$283$	12.2935	0.730775	0.365387	−	0.930856i	$-0.380937\pi$
$283$	12.2935	0.730775	0.365387	+	0.930856i	$0.380937\pi$
$284$	0	0
$285$	−5.65081	−0.334725
$286$	0	0
$287$	−9.50960	−0.561334
$288$	0	0
$289$	−15.2708	−0.898281
$290$	0	0
$291$	4.75139	0.278531
$292$	0	0
$293$	−4.78588	−0.279594	−0.139797	−	0.990180i	$-0.544645\pi$
$293$	−4.78588	−0.279594	−0.139797	+	0.990180i	$0.544645\pi$
$294$	0	0
$295$	9.80707	0.570990
$296$	0	0
$297$	0.951283	0.0551991
$298$	0	0
$299$	−7.26350	−0.420059
$300$	0	0
$301$	−9.81322	−0.565624
$302$	0	0
$303$	−2.52463	−0.145036
$304$	0	0
$305$	−11.2980	−0.646922
$306$	0	0
$307$	19.7957	1.12980	0.564900	−	0.825159i	$-0.308915\pi$
$307$	19.7957	1.12980	0.564900	+	0.825159i	$0.308915\pi$
$308$	0	0
$309$	10.1843	0.579363
$310$	0	0
$311$	25.3327	1.43648	0.718242	−	0.695794i	$-0.244947\pi$
$311$	25.3327	1.43648	0.718242	+	0.695794i	$0.244947\pi$
$312$	0	0
$313$	2.86377	0.161870	0.0809349	−	0.996719i	$-0.474209\pi$
$313$	2.86377	0.161870	0.0809349	+	0.996719i	$0.474209\pi$
$314$	0	0
$315$	1.21927	0.0686982
$316$	0	0
$317$	4.73056	0.265695	0.132847	−	0.991137i	$-0.457588\pi$
$317$	4.73056	0.265695	0.132847	+	0.991137i	$0.457588\pi$
$318$	0	0
$319$	−2.36784	−0.132573
$320$	0	0
$321$	−13.1161	−0.732069
$322$	0	0
$323$	−7.81137	−0.434636
$324$	0	0
$325$	11.5770	0.642178
$326$	0	0
$327$	−0.209627	−0.0115924
$328$	0	0
$329$	−12.9576	−0.714376
$330$	0	0
$331$	15.1256	0.831379	0.415690	−	0.909506i	$-0.363540\pi$
$331$	15.1256	0.831379	0.415690	+	0.909506i	$0.363540\pi$
$332$	0	0
$333$	−7.81553	−0.428288
$334$	0	0
$335$	−7.75054	−0.423457
$336$	0	0
$337$	−4.31419	−0.235009	−0.117504	−	0.993072i	$-0.537489\pi$
$337$	−4.31419	−0.235009	−0.117504	+	0.993072i	$0.537489\pi$
$338$	0	0
$339$	0.554507	0.0301167
$340$	0	0
$341$	−0.936606	−0.0507201
$342$	0	0
$343$	−15.8384	−0.855193
$344$	0	0
$345$	2.44410	0.131586
$346$	0	0
$347$	−23.0258	−1.23609	−0.618045	−	0.786143i	$-0.712075\pi$
$347$	−23.0258	−1.23609	−0.618045	+	0.786143i	$0.712075\pi$
$348$	0	0
$349$	16.1595	0.864998	0.432499	−	0.901634i	$-0.357632\pi$
$349$	16.1595	0.864998	0.432499	+	0.901634i	$0.357632\pi$
$350$	0	0
$351$	−2.82707	−0.150898
$352$	0	0
$353$	29.4163	1.56567	0.782835	−	0.622229i	$-0.213773\pi$
$353$	29.4163	1.56567	0.782835	+	0.622229i	$0.213773\pi$
$354$	0	0
$355$	−1.62839	−0.0864258
$356$	0	0
$357$	1.68546	0.0892038
$358$	0	0
$359$	−27.2507	−1.43824	−0.719118	−	0.694888i	$-0.755454\pi$
$359$	−27.2507	−1.43824	−0.719118	+	0.694888i	$0.755454\pi$
$360$	0	0
$361$	16.2860	0.857157
$362$	0	0
$363$	−10.0951	−0.529853
$364$	0	0
$365$	−0.477867	−0.0250127
$366$	0	0
$367$	−2.46015	−0.128419	−0.0642095	−	0.997936i	$-0.520453\pi$
$367$	−2.46015	−0.128419	−0.0642095	+	0.997936i	$0.520453\pi$
$368$	0	0
$369$	−7.41944	−0.386241
$370$	0	0
$371$	−4.14935	−0.215423
$372$	0	0
$373$	−2.75950	−0.142881	−0.0714407	−	0.997445i	$-0.522760\pi$
$373$	−2.75950	−0.142881	−0.0714407	+	0.997445i	$0.522760\pi$
$374$	0	0
$375$	−8.65198	−0.446786
$376$	0	0
$377$	7.03685	0.362416
$378$	0	0
$379$	−29.6373	−1.52237	−0.761183	−	0.648537i	$-0.775381\pi$
$379$	−29.6373	−1.52237	−0.761183	+	0.648537i	$0.775381\pi$
$380$	0	0
$381$	17.5210	0.897630
$382$	0	0
$383$	19.3376	0.988104	0.494052	−	0.869432i	$-0.335515\pi$
$383$	19.3376	0.988104	0.494052	+	0.869432i	$0.335515\pi$
$384$	0	0
$385$	1.15987	0.0591127
$386$	0	0
$387$	−7.65633	−0.389193
$388$	0	0
$389$	−0.877537	−0.0444929	−0.0222465	−	0.999753i	$-0.507082\pi$
$389$	−0.877537	−0.0444929	−0.0222465	+	0.999753i	$0.507082\pi$
$390$	0	0
$391$	3.37859	0.170863
$392$	0	0
$393$	−6.30379	−0.317984
$394$	0	0
$395$	13.9598	0.702395
$396$	0	0
$397$	−24.0236	−1.20571	−0.602855	−	0.797851i	$-0.705970\pi$
$397$	−24.0236	−1.20571	−0.602855	+	0.797851i	$0.705970\pi$
$398$	0	0
$399$	−7.61364	−0.381159
$400$	0	0
$401$	22.0892	1.10308	0.551541	−	0.834148i	$-0.314040\pi$
$401$	22.0892	1.10308	0.551541	+	0.834148i	$0.314040\pi$
$402$	0	0
$403$	2.78345	0.138654
$404$	0	0
$405$	0.951283	0.0472697
$406$	0	0
$407$	−7.43478	−0.368529
$408$	0	0
$409$	9.50296	0.469891	0.234946	−	0.972009i	$-0.424509\pi$
$409$	9.50296	0.469891	0.234946	+	0.972009i	$0.424509\pi$
$410$	0	0
$411$	−6.12238	−0.301995
$412$	0	0
$413$	13.2136	0.650197
$414$	0	0
$415$	10.7697	0.528662
$416$	0	0
$417$	7.59742	0.372047
$418$	0	0
$419$	−24.0099	−1.17296	−0.586479	−	0.809964i	$-0.699486\pi$
$419$	−24.0099	−1.17296	−0.586479	+	0.809964i	$0.699486\pi$
$420$	0	0
$421$	39.8756	1.94342	0.971708	−	0.236187i	$-0.0758976\pi$
$421$	39.8756	1.94342	0.971708	+	0.236187i	$0.0758976\pi$
$422$	0	0
$423$	−10.1096	−0.491545
$424$	0	0
$425$	−5.38501	−0.261211
$426$	0	0
$427$	−15.2224	−0.736663
$428$	0	0
$429$	−2.68935	−0.129843
$430$	0	0
$431$	−18.6800	−0.899785	−0.449893	−	0.893083i	$-0.648538\pi$
$431$	−18.6800	−0.899785	−0.449893	+	0.893083i	$0.648538\pi$
$432$	0	0
$433$	−19.5622	−0.940099	−0.470049	−	0.882640i	$-0.655764\pi$
$433$	−19.5622	−0.940099	−0.470049	+	0.882640i	$0.655764\pi$
$434$	0	0
$435$	−2.36784	−0.113529
$436$	0	0
$437$	−15.2620	−0.730079
$438$	0	0
$439$	−11.5272	−0.550163	−0.275082	−	0.961421i	$-0.588705\pi$
$439$	−11.5272	−0.550163	−0.275082	+	0.961421i	$0.588705\pi$
$440$	0	0
$441$	−5.35721	−0.255105
$442$	0	0
$443$	−37.3814	−1.77604	−0.888022	−	0.459800i	$-0.847921\pi$
$443$	−37.3814	−1.77604	−0.888022	+	0.459800i	$0.847921\pi$
$444$	0	0
$445$	5.85778	0.277686
$446$	0	0
$447$	−17.8638	−0.844929
$448$	0	0
$449$	−0.0879786	−0.00415197	−0.00207598	−	0.999998i	$-0.500661\pi$
$449$	−0.0879786	−0.00415197	−0.00207598	+	0.999998i	$0.500661\pi$
$450$	0	0
$451$	−7.05799	−0.332348
$452$	0	0
$453$	4.74233	0.222814
$454$	0	0
$455$	−3.44697	−0.161596
$456$	0	0
$457$	−35.8125	−1.67524	−0.837619	−	0.546254i	$-0.816053\pi$
$457$	−35.8125	−1.67524	−0.837619	+	0.546254i	$0.816053\pi$
$458$	0	0
$459$	1.31500	0.0613790
$460$	0	0
$461$	19.4523	0.905985	0.452993	−	0.891514i	$-0.350356\pi$
$461$	19.4523	0.905985	0.452993	+	0.891514i	$0.350356\pi$
$462$	0	0
$463$	7.12623	0.331184	0.165592	−	0.986194i	$-0.447047\pi$
$463$	7.12623	0.331184	0.165592	+	0.986194i	$0.447047\pi$
$464$	0	0
$465$	−0.936606	−0.0434341
$466$	0	0
$467$	−1.30038	−0.0601746	−0.0300873	−	0.999547i	$-0.509579\pi$
$467$	−1.30038	−0.0601746	−0.0300873	+	0.999547i	$0.509579\pi$
$468$	0	0
$469$	−10.4427	−0.482199
$470$	0	0
$471$	−9.54024	−0.439591
$472$	0	0
$473$	−7.28334	−0.334888
$474$	0	0
$475$	24.3255	1.11613
$476$	0	0
$477$	−3.23734	−0.148228
$478$	0	0
$479$	−13.3269	−0.608921	−0.304461	−	0.952525i	$-0.598476\pi$
$479$	−13.3269	−0.608921	−0.304461	+	0.952525i	$0.598476\pi$
$480$	0	0
$481$	22.0951	1.00745
$482$	0	0
$483$	3.29306	0.149840
$484$	0	0
$485$	4.51991	0.205239
$486$	0	0
$487$	9.08890	0.411858	0.205929	−	0.978567i	$-0.433978\pi$
$487$	9.08890	0.411858	0.205929	+	0.978567i	$0.433978\pi$
$488$	0	0
$489$	1.89939	0.0858932
$490$	0	0
$491$	28.7483	1.29739	0.648697	−	0.761047i	$-0.275314\pi$
$491$	28.7483	1.29739	0.648697	+	0.761047i	$0.275314\pi$
$492$	0	0
$493$	−3.27316	−0.147416
$494$	0	0
$495$	0.904940	0.0406740
$496$	0	0
$497$	−2.19401	−0.0984148
$498$	0	0
$499$	−19.4053	−0.868702	−0.434351	−	0.900744i	$-0.643022\pi$
$499$	−19.4053	−0.868702	−0.434351	+	0.900744i	$0.643022\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	11.9825	0.534271	0.267136	−	0.963659i	$-0.413923\pi$
$503$	11.9825	0.534271	0.267136	+	0.963659i	$0.413923\pi$
$504$	0	0
$505$	−2.40164	−0.106871
$506$	0	0
$507$	−5.00767	−0.222399
$508$	0	0
$509$	−7.58018	−0.335986	−0.167993	−	0.985788i	$-0.553729\pi$
$509$	−7.58018	−0.335986	−0.167993	+	0.985788i	$0.553729\pi$
$510$	0	0
$511$	−0.643856	−0.0284825
$512$	0	0
$513$	−5.94020	−0.262266
$514$	0	0
$515$	9.68814	0.426910
$516$	0	0
$517$	−9.61709	−0.422959
$518$	0	0
$519$	13.8969	0.610005
$520$	0	0
$521$	−9.31483	−0.408090	−0.204045	−	0.978962i	$-0.565409\pi$
$521$	−9.31483	−0.408090	−0.204045	+	0.978962i	$0.565409\pi$
$522$	0	0
$523$	−8.16544	−0.357050	−0.178525	−	0.983935i	$-0.557133\pi$
$523$	−8.16544	−0.357050	−0.178525	+	0.983935i	$0.557133\pi$
$524$	0	0
$525$	−5.24869	−0.229072
$526$	0	0
$527$	−1.29471	−0.0563986
$528$	0	0
$529$	−16.3989	−0.712994
$530$	0	0
$531$	10.3093	0.447386
$532$	0	0
$533$	20.9753	0.908540
$534$	0	0
$535$	−12.4771	−0.539433
$536$	0	0
$537$	5.87857	0.253679
$538$	0	0
$539$	−5.09622	−0.219510
$540$	0	0
$541$	31.2426	1.34322	0.671611	−	0.740904i	$-0.265603\pi$
$541$	31.2426	1.34322	0.671611	+	0.740904i	$0.265603\pi$
$542$	0	0
$543$	−20.6955	−0.888127
$544$	0	0
$545$	−0.199414	−0.00854198
$546$	0	0
$547$	23.9718	1.02496	0.512481	−	0.858699i	$-0.328727\pi$
$547$	23.9718	1.02496	0.512481	+	0.858699i	$0.328727\pi$
$548$	0	0
$549$	−11.8766	−0.506881
$550$	0	0
$551$	14.7857	0.629893
$552$	0	0
$553$	18.8088	0.799831
$554$	0	0
$555$	−7.43478	−0.315589
$556$	0	0
$557$	−14.3707	−0.608907	−0.304454	−	0.952527i	$-0.598474\pi$
$557$	−14.3707	−0.608907	−0.304454	+	0.952527i	$0.598474\pi$
$558$	0	0
$559$	21.6450	0.915485
$560$	0	0
$561$	1.25094	0.0528147
$562$	0	0
$563$	−37.5575	−1.58286	−0.791431	−	0.611259i	$-0.790663\pi$
$563$	−37.5575	−1.58286	−0.791431	+	0.611259i	$0.790663\pi$
$564$	0	0
$565$	0.527493	0.0221918
$566$	0	0
$567$	1.28171	0.0538269
$568$	0	0
$569$	−20.9670	−0.878983	−0.439491	−	0.898247i	$-0.644841\pi$
$569$	−20.9670	−0.878983	−0.439491	+	0.898247i	$0.644841\pi$
$570$	0	0
$571$	−17.0570	−0.713815	−0.356907	−	0.934140i	$-0.616169\pi$
$571$	−17.0570	−0.713815	−0.356907	+	0.934140i	$0.616169\pi$
$572$	0	0
$573$	13.8727	0.579542
$574$	0	0
$575$	−10.5213	−0.438769
$576$	0	0
$577$	20.0594	0.835082	0.417541	−	0.908658i	$-0.362892\pi$
$577$	20.0594	0.835082	0.417541	+	0.908658i	$0.362892\pi$
$578$	0	0
$579$	−0.344963	−0.0143362
$580$	0	0
$581$	14.5105	0.601998
$582$	0	0
$583$	−3.07963	−0.127545
$584$	0	0
$585$	−2.68935	−0.111191
$586$	0	0
$587$	42.1301	1.73890	0.869449	−	0.494023i	$-0.164474\pi$
$587$	42.1301	1.73890	0.869449	+	0.494023i	$0.164474\pi$
$588$	0	0
$589$	5.84855	0.240985
$590$	0	0
$591$	24.1397	0.992975
$592$	0	0
$593$	−30.4857	−1.25190	−0.625950	−	0.779863i	$-0.715288\pi$
$593$	−30.4857	−1.25190	−0.625950	+	0.779863i	$0.715288\pi$
$594$	0	0
$595$	1.60335	0.0657308
$596$	0	0
$597$	3.97139	0.162538
$598$	0	0
$599$	9.37998	0.383256	0.191628	−	0.981468i	$-0.438623\pi$
$599$	9.37998	0.383256	0.191628	+	0.981468i	$0.438623\pi$
$600$	0	0
$601$	−11.7008	−0.477285	−0.238642	−	0.971108i	$-0.576702\pi$
$601$	−11.7008	−0.477285	−0.238642	+	0.971108i	$0.576702\pi$
$602$	0	0
$603$	−8.14745	−0.331790
$604$	0	0
$605$	−9.60326	−0.390428
$606$	0	0
$607$	12.2337	0.496549	0.248275	−	0.968690i	$-0.420136\pi$
$607$	12.2337	0.496549	0.248275	+	0.968690i	$0.420136\pi$
$608$	0	0
$609$	−3.19031	−0.129278
$610$	0	0
$611$	28.5805	1.15624
$612$	0	0
$613$	−44.6472	−1.80328	−0.901642	−	0.432483i	$-0.857637\pi$
$613$	−44.6472	−1.80328	−0.901642	+	0.432483i	$0.857637\pi$
$614$	0	0
$615$	−7.05799	−0.284606
$616$	0	0
$617$	36.7226	1.47840	0.739198	−	0.673488i	$-0.235205\pi$
$617$	36.7226	1.47840	0.739198	+	0.673488i	$0.235205\pi$
$618$	0	0
$619$	−5.35966	−0.215423	−0.107711	−	0.994182i	$-0.534352\pi$
$619$	−5.35966	−0.215423	−0.107711	+	0.994182i	$0.534352\pi$
$620$	0	0
$621$	2.56927	0.103101
$622$	0	0
$623$	7.89250	0.316206
$624$	0	0
$625$	12.2448	0.489793
$626$	0	0
$627$	−5.65081	−0.225672
$628$	0	0
$629$	−10.2774	−0.409788
$630$	0	0
$631$	7.52358	0.299509	0.149754	−	0.988723i	$-0.452152\pi$
$631$	7.52358	0.299509	0.149754	+	0.988723i	$0.452152\pi$
$632$	0	0
$633$	21.3487	0.848535
$634$	0	0
$635$	16.6675	0.661428
$636$	0	0
$637$	15.1452	0.600075
$638$	0	0
$639$	−1.71178	−0.0677169
$640$	0	0
$641$	−15.3947	−0.608056	−0.304028	−	0.952663i	$-0.598332\pi$
$641$	−15.3947	−0.608056	−0.304028	+	0.952663i	$0.598332\pi$
$642$	0	0
$643$	−40.2115	−1.58579	−0.792893	−	0.609361i	$-0.791426\pi$
$643$	−40.2115	−1.58579	−0.792893	+	0.609361i	$0.791426\pi$
$644$	0	0
$645$	−7.28334	−0.286781
$646$	0	0
$647$	38.1639	1.50038	0.750189	−	0.661224i	$-0.229963\pi$
$647$	38.1639	1.50038	0.750189	+	0.661224i	$0.229963\pi$
$648$	0	0
$649$	9.80707	0.384961
$650$	0	0
$651$	−1.26194	−0.0494593
$652$	0	0
$653$	−1.62256	−0.0634956	−0.0317478	−	0.999496i	$-0.510107\pi$
$653$	−1.62256	−0.0634956	−0.0317478	+	0.999496i	$0.510107\pi$
$654$	0	0
$655$	−5.99669	−0.234310
$656$	0	0
$657$	−0.502340	−0.0195981
$658$	0	0
$659$	12.5684	0.489595	0.244798	−	0.969574i	$-0.421278\pi$
$659$	12.5684	0.489595	0.244798	+	0.969574i	$0.421278\pi$
$660$	0	0
$661$	−35.0118	−1.36180	−0.680901	−	0.732375i	$-0.738412\pi$
$661$	−35.0118	−1.36180	−0.680901	+	0.732375i	$0.738412\pi$
$662$	0	0
$663$	−3.71760	−0.144380
$664$	0	0
$665$	−7.24273	−0.280861
$666$	0	0
$667$	−6.39515	−0.247621
$668$	0	0
$669$	−11.6936	−0.452101
$670$	0	0
$671$	−11.2980	−0.436155
$672$	0	0
$673$	15.1295	0.583199	0.291599	−	0.956541i	$-0.405813\pi$
$673$	15.1295	0.583199	0.291599	+	0.956541i	$0.405813\pi$
$674$	0	0
$675$	−4.09506	−0.157619
$676$	0	0
$677$	−33.9274	−1.30393	−0.651967	−	0.758247i	$-0.726056\pi$
$677$	−33.9274	−1.30393	−0.651967	+	0.758247i	$0.726056\pi$
$678$	0	0
$679$	6.08992	0.233710
$680$	0	0
$681$	−14.3838	−0.551188
$682$	0	0
$683$	−46.4231	−1.77633	−0.888164	−	0.459526i	$-0.848019\pi$
$683$	−46.4231	−1.77633	−0.888164	+	0.459526i	$0.848019\pi$
$684$	0	0
$685$	−5.82412	−0.222528
$686$	0	0
$687$	9.30406	0.354972
$688$	0	0
$689$	9.15220	0.348671
$690$	0	0
$691$	−17.4612	−0.664256	−0.332128	−	0.943234i	$-0.607767\pi$
$691$	−17.4612	−0.664256	−0.332128	+	0.943234i	$0.607767\pi$
$692$	0	0
$693$	1.21927	0.0463163
$694$	0	0
$695$	7.22730	0.274147
$696$	0	0
$697$	−9.75657	−0.369557
$698$	0	0
$699$	−1.32737	−0.0502058
$700$	0	0
$701$	33.9740	1.28318	0.641591	−	0.767047i	$-0.278275\pi$
$701$	33.9740	1.28318	0.641591	+	0.767047i	$0.278275\pi$
$702$	0	0
$703$	46.4258	1.75098
$704$	0	0
$705$	−9.61709	−0.362200
$706$	0	0
$707$	−3.23585	−0.121697
$708$	0	0
$709$	−23.0747	−0.866587	−0.433293	−	0.901253i	$-0.642649\pi$
$709$	−23.0747	−0.866587	−0.433293	+	0.901253i	$0.642649\pi$
$710$	0	0
$711$	14.6747	0.550345
$712$	0	0
$713$	−2.52963	−0.0947352
$714$	0	0
$715$	−2.55833	−0.0956761
$716$	0	0
$717$	6.67440	0.249260
$718$	0	0
$719$	−20.8956	−0.779273	−0.389637	−	0.920969i	$-0.627399\pi$
$719$	−20.8956	−0.779273	−0.389637	+	0.920969i	$0.627399\pi$
$720$	0	0
$721$	13.0533	0.486131
$722$	0	0
$723$	21.3838	0.795273
$724$	0	0
$725$	10.1930	0.378558
$726$	0	0
$727$	−8.15415	−0.302421	−0.151210	−	0.988502i	$-0.548317\pi$
$727$	−8.15415	−0.302421	−0.151210	+	0.988502i	$0.548317\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−10.0681	−0.372381
$732$	0	0
$733$	−19.2185	−0.709853	−0.354927	−	0.934894i	$-0.615494\pi$
$733$	−19.2185	−0.709853	−0.354927	+	0.934894i	$0.615494\pi$
$734$	0	0
$735$	−5.09622	−0.187977
$736$	0	0
$737$	−7.75054	−0.285495
$738$	0	0
$739$	−11.4850	−0.422482	−0.211241	−	0.977434i	$-0.567750\pi$
$739$	−11.4850	−0.422482	−0.211241	+	0.977434i	$0.567750\pi$
$740$	0	0
$741$	16.7934	0.616920
$742$	0	0
$743$	−44.3551	−1.62723	−0.813615	−	0.581404i	$-0.802504\pi$
$743$	−44.3551	−1.62723	−0.813615	+	0.581404i	$0.802504\pi$
$744$	0	0
$745$	−16.9935	−0.622595
$746$	0	0
$747$	11.3212	0.414221
$748$	0	0
$749$	−16.8111	−0.614264
$750$	0	0
$751$	−31.2517	−1.14039	−0.570195	−	0.821509i	$-0.693132\pi$
$751$	−31.2517	−1.14039	−0.570195	+	0.821509i	$0.693132\pi$
$752$	0	0
$753$	−27.1243	−0.988463
$754$	0	0
$755$	4.51130	0.164183
$756$	0	0
$757$	33.1738	1.20572	0.602862	−	0.797846i	$-0.294027\pi$
$757$	33.1738	1.20572	0.602862	+	0.797846i	$0.294027\pi$
$758$	0	0
$759$	2.44410	0.0887152
$760$	0	0
$761$	−50.1683	−1.81860	−0.909299	−	0.416143i	$-0.863382\pi$
$761$	−50.1683	−1.81860	−0.909299	+	0.416143i	$0.863382\pi$
$762$	0	0
$763$	−0.268682	−0.00972692
$764$	0	0
$765$	1.25094	0.0452278
$766$	0	0
$767$	−29.1451	−1.05237
$768$	0	0
$769$	39.9272	1.43981	0.719905	−	0.694073i	$-0.244185\pi$
$769$	39.9272	1.43981	0.719905	+	0.694073i	$0.244185\pi$
$770$	0	0
$771$	11.2287	0.404393
$772$	0	0
$773$	−22.0108	−0.791675	−0.395838	−	0.918321i	$-0.629546\pi$
$773$	−22.0108	−0.791675	−0.395838	+	0.918321i	$0.629546\pi$
$774$	0	0
$775$	4.03188	0.144829
$776$	0	0
$777$	−10.0173	−0.359368
$778$	0	0
$779$	44.0730	1.57908
$780$	0	0
$781$	−1.62839	−0.0582682
$782$	0	0
$783$	−2.48910	−0.0889530
$784$	0	0
$785$	−9.07547	−0.323918
$786$	0	0
$787$	27.7640	0.989681	0.494840	−	0.868984i	$-0.335227\pi$
$787$	27.7640	0.989681	0.494840	+	0.868984i	$0.335227\pi$
$788$	0	0
$789$	−25.5610	−0.909995
$790$	0	0
$791$	0.710719	0.0252702
$792$	0	0
$793$	33.5760	1.19232
$794$	0	0
$795$	−3.07963	−0.109223
$796$	0	0
$797$	−9.03196	−0.319928	−0.159964	−	0.987123i	$-0.551138\pi$
$797$	−9.03196	−0.319928	−0.159964	+	0.987123i	$0.551138\pi$
$798$	0	0
$799$	−13.2941	−0.470312
$800$	0	0
$801$	6.15777	0.217574
$802$	0	0
$803$	−0.477867	−0.0168636
$804$	0	0
$805$	3.13264	0.110411
$806$	0	0
$807$	20.1581	0.709600
$808$	0	0
$809$	27.3478	0.961496	0.480748	−	0.876859i	$-0.340365\pi$
$809$	27.3478	0.961496	0.480748	+	0.876859i	$0.340365\pi$
$810$	0	0
$811$	42.6726	1.49844	0.749219	−	0.662323i	$-0.230429\pi$
$811$	42.6726	1.49844	0.749219	+	0.662323i	$0.230429\pi$
$812$	0	0
$813$	1.90126	0.0666802
$814$	0	0
$815$	1.80685	0.0632913
$816$	0	0
$817$	45.4801	1.59115
$818$	0	0
$819$	−3.62349	−0.126615
$820$	0	0
$821$	28.5232	0.995468	0.497734	−	0.867330i	$-0.334166\pi$
$821$	28.5232	0.995468	0.497734	+	0.867330i	$0.334166\pi$
$822$	0	0
$823$	4.05092	0.141206	0.0706031	−	0.997504i	$-0.477508\pi$
$823$	4.05092	0.141206	0.0706031	+	0.997504i	$0.477508\pi$
$824$	0	0
$825$	−3.89556	−0.135626
$826$	0	0
$827$	6.63277	0.230644	0.115322	−	0.993328i	$-0.463210\pi$
$827$	6.63277	0.230644	0.115322	+	0.993328i	$0.463210\pi$
$828$	0	0
$829$	−7.97201	−0.276879	−0.138440	−	0.990371i	$-0.544209\pi$
$829$	−7.97201	−0.276879	−0.138440	+	0.990371i	$0.544209\pi$
$830$	0	0
$831$	31.1953	1.08215
$832$	0	0
$833$	−7.04474	−0.244086
$834$	0	0
$835$	−0.951283	−0.0329205
$836$	0	0
$837$	−0.984571	−0.0340318
$838$	0	0
$839$	−1.98285	−0.0684556	−0.0342278	−	0.999414i	$-0.510897\pi$
$839$	−1.98285	−0.0684556	−0.0342278	+	0.999414i	$0.510897\pi$
$840$	0	0
$841$	−22.8044	−0.786359
$842$	0	0
$843$	−31.9013	−1.09874
$844$	0	0
$845$	−4.76372	−0.163877
$846$	0	0
$847$	−12.9390	−0.444588
$848$	0	0
$849$	12.2935	0.421913
$850$	0	0
$851$	−20.0802	−0.688340
$852$	0	0
$853$	−40.0040	−1.36971	−0.684856	−	0.728678i	$-0.740135\pi$
$853$	−40.0040	−1.36971	−0.684856	+	0.728678i	$0.740135\pi$
$854$	0	0
$855$	−5.65081	−0.193254
$856$	0	0
$857$	−39.9386	−1.36428	−0.682138	−	0.731224i	$-0.738950\pi$
$857$	−39.9386	−1.36428	−0.682138	+	0.731224i	$0.738950\pi$
$858$	0	0
$859$	12.2669	0.418541	0.209271	−	0.977858i	$-0.432891\pi$
$859$	12.2669	0.418541	0.209271	+	0.977858i	$0.432891\pi$
$860$	0	0
$861$	−9.50960	−0.324086
$862$	0	0
$863$	40.6235	1.38284	0.691419	−	0.722454i	$-0.256986\pi$
$863$	40.6235	1.38284	0.691419	+	0.722454i	$0.256986\pi$
$864$	0	0
$865$	13.2199	0.449489
$866$	0	0
$867$	−15.2708	−0.518623
$868$	0	0
$869$	13.9598	0.473555
$870$	0	0
$871$	23.0334	0.780458
$872$	0	0
$873$	4.75139	0.160810
$874$	0	0
$875$	−11.0894	−0.374889
$876$	0	0
$877$	11.9961	0.405080	0.202540	−	0.979274i	$-0.435080\pi$
$877$	11.9961	0.405080	0.202540	+	0.979274i	$0.435080\pi$
$878$	0	0
$879$	−4.78588	−0.161424
$880$	0	0
$881$	13.4368	0.452697	0.226348	−	0.974046i	$-0.427321\pi$
$881$	13.4368	0.452697	0.226348	+	0.974046i	$0.427321\pi$
$882$	0	0
$883$	−32.1980	−1.08355	−0.541775	−	0.840524i	$-0.682248\pi$
$883$	−32.1980	−1.08355	−0.541775	+	0.840524i	$0.682248\pi$
$884$	0	0
$885$	9.80707	0.329661
$886$	0	0
$887$	−13.6383	−0.457928	−0.228964	−	0.973435i	$-0.573534\pi$
$887$	−13.6383	−0.457928	−0.228964	+	0.973435i	$0.573534\pi$
$888$	0	0
$889$	22.4569	0.753182
$890$	0	0
$891$	0.951283	0.0318692
$892$	0	0
$893$	60.0530	2.00960
$894$	0	0
$895$	5.59218	0.186926
$896$	0	0
$897$	−7.26350	−0.242521
$898$	0	0
$899$	2.45069	0.0817352
$900$	0	0
$901$	−4.25711	−0.141825
$902$	0	0
$903$	−9.81322	−0.326563
$904$	0	0
$905$	−19.6872	−0.654426
$906$	0	0
$907$	−15.0800	−0.500723	−0.250362	−	0.968152i	$-0.580550\pi$
$907$	−15.0800	−0.500723	−0.250362	+	0.968152i	$0.580550\pi$
$908$	0	0
$909$	−2.52463	−0.0837366
$910$	0	0
$911$	28.4366	0.942147	0.471074	−	0.882094i	$-0.343867\pi$
$911$	28.4366	0.942147	0.471074	+	0.882094i	$0.343867\pi$
$912$	0	0
$913$	10.7697	0.356424
$914$	0	0
$915$	−11.2980	−0.373501
$916$	0	0
$917$	−8.07965	−0.266814
$918$	0	0
$919$	26.8458	0.885562	0.442781	−	0.896630i	$-0.353992\pi$
$919$	26.8458	0.885562	0.442781	+	0.896630i	$0.353992\pi$
$920$	0	0
$921$	19.7957	0.652290
$922$	0	0
$923$	4.83932	0.159288
$924$	0	0
$925$	32.0051	1.05232
$926$	0	0
$927$	10.1843	0.334496
$928$	0	0
$929$	−16.0592	−0.526885	−0.263442	−	0.964675i	$-0.584858\pi$
$929$	−16.0592	−0.526885	−0.263442	+	0.964675i	$0.584858\pi$
$930$	0	0
$931$	31.8229	1.04295
$932$	0	0
$933$	25.3327	0.829354
$934$	0	0
$935$	1.19000	0.0389171
$936$	0	0
$937$	−21.3748	−0.698283	−0.349141	−	0.937070i	$-0.613527\pi$
$937$	−21.3748	−0.698283	−0.349141	+	0.937070i	$0.613527\pi$
$938$	0	0
$939$	2.86377	0.0934555
$940$	0	0
$941$	2.32835	0.0759020	0.0379510	−	0.999280i	$-0.487917\pi$
$941$	2.32835	0.0759020	0.0379510	+	0.999280i	$0.487917\pi$
$942$	0	0
$943$	−19.0625	−0.620761
$944$	0	0
$945$	1.21927	0.0396630
$946$	0	0
$947$	−42.7080	−1.38782	−0.693912	−	0.720059i	$-0.744115\pi$
$947$	−42.7080	−1.38782	−0.693912	+	0.720059i	$0.744115\pi$
$948$	0	0
$949$	1.42015	0.0461000
$950$	0	0
$951$	4.73056	0.153399
$952$	0	0
$953$	40.9000	1.32488	0.662440	−	0.749115i	$-0.269521\pi$
$953$	40.9000	1.32488	0.662440	+	0.749115i	$0.269521\pi$
$954$	0	0
$955$	13.1969	0.427042
$956$	0	0
$957$	−2.36784	−0.0765413
$958$	0	0
$959$	−7.84714	−0.253397
$960$	0	0
$961$	−30.0306	−0.968730
$962$	0	0
$963$	−13.1161	−0.422660
$964$	0	0
$965$	−0.328157	−0.0105638
$966$	0	0
$967$	54.1145	1.74021	0.870103	−	0.492870i	$-0.164052\pi$
$967$	54.1145	1.74021	0.870103	+	0.492870i	$0.164052\pi$
$968$	0	0
$969$	−7.81137	−0.250937
$970$	0	0
$971$	53.6916	1.72305	0.861523	−	0.507719i	$-0.169511\pi$
$971$	53.6916	1.72305	0.861523	+	0.507719i	$0.169511\pi$
$972$	0	0
$973$	9.73772	0.312177
$974$	0	0
$975$	11.5770	0.370761
$976$	0	0
$977$	−51.4253	−1.64524	−0.822621	−	0.568590i	$-0.807489\pi$
$977$	−51.4253	−1.64524	−0.822621	+	0.568590i	$0.807489\pi$
$978$	0	0
$979$	5.85778	0.187216
$980$	0	0
$981$	−0.209627	−0.00669287
$982$	0	0
$983$	−27.2398	−0.868816	−0.434408	−	0.900716i	$-0.643042\pi$
$983$	−27.2398	−0.868816	−0.434408	+	0.900716i	$0.643042\pi$
$984$	0	0
$985$	22.9637	0.731685
$986$	0	0
$987$	−12.9576	−0.412445
$988$	0	0
$989$	−19.6711	−0.625506
$990$	0	0
$991$	36.0301	1.14453	0.572267	−	0.820067i	$-0.306064\pi$
$991$	36.0301	1.14453	0.572267	+	0.820067i	$0.306064\pi$
$992$	0	0
$993$	15.1256	0.479997
$994$	0	0
$995$	3.77791	0.119768
$996$	0	0
$997$	−1.57982	−0.0500334	−0.0250167	−	0.999687i	$-0.507964\pi$
$997$	−1.57982	−0.0500334	−0.0250167	+	0.999687i	$0.507964\pi$
$998$	0	0
$999$	−7.81553	−0.247272

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.t.1.4		5
4.3	odd	2		2004.2.a.a.1.4	✓	5
12.11	even	2		6012.2.a.e.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2004.2.a.a.1.4	✓	5	4.3	odd	2
6012.2.a.e.1.2		5	12.11	even	2
8016.2.a.t.1.4		5	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:	\( N \)	\(=\)	\( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8016.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.951283 q^{5} +1.28171 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.951283 q^{5} +1.28171 q^{7} +1.00000 q^{9} +0.951283 q^{11} -2.82707 q^{13} +0.951283 q^{15} +1.31500 q^{17} -5.94020 q^{19} +1.28171 q^{21} +2.56927 q^{23} -4.09506 q^{25} +1.00000 q^{27} -2.48910 q^{29} -0.984571 q^{31} +0.951283 q^{33} +1.21927 q^{35} -7.81553 q^{37} -2.82707 q^{39} -7.41944 q^{41} -7.65633 q^{43} +0.951283 q^{45} -10.1096 q^{47} -5.35721 q^{49} +1.31500 q^{51} -3.23734 q^{53} +0.904940 q^{55} -5.94020 q^{57} +10.3093 q^{59} -11.8766 q^{61} +1.28171 q^{63} -2.68935 q^{65} -8.14745 q^{67} +2.56927 q^{69} -1.71178 q^{71} -0.502340 q^{73} -4.09506 q^{75} +1.21927 q^{77} +14.6747 q^{79} +1.00000 q^{81} +11.3212 q^{83} +1.25094 q^{85} -2.48910 q^{87} +6.15777 q^{89} -3.62349 q^{91} -0.984571 q^{93} -5.65081 q^{95} +4.75139 q^{97} +0.951283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} - 3 q^{5} + 2 q^{7} + 5 q^{9} - 3 q^{11} - 4 q^{13} - 3 q^{15} - 7 q^{17} + 2 q^{19} + 2 q^{21} - 13 q^{23} - 2 q^{25} + 5 q^{27} - 3 q^{29} + 12 q^{31} - 3 q^{33} - 10 q^{35} - 7 q^{37} - 4 q^{39} - 16 q^{41} - 3 q^{45} - q^{47} - 17 q^{49} - 7 q^{51} + 3 q^{53} + 23 q^{55} + 2 q^{57} - q^{59} - 22 q^{61} + 2 q^{63} - 20 q^{65} - 2 q^{67} - 13 q^{69} - 9 q^{71} - 28 q^{73} - 2 q^{75} - 10 q^{77} + 28 q^{79} + 5 q^{81} - 7 q^{83} - 11 q^{85} - 3 q^{87} - 30 q^{89} + 13 q^{91} + 12 q^{93} - 3 q^{95} - 33 q^{97} - 3 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 3 * q^5 + 2 * q^7 + 5 * q^9 - 3 * q^11 - 4 * q^13 - 3 * q^15 - 7 * q^17 + 2 * q^19 + 2 * q^21 - 13 * q^23 - 2 * q^25 + 5 * q^27 - 3 * q^29 + 12 * q^31 - 3 * q^33 - 10 * q^35 - 7 * q^37 - 4 * q^39 - 16 * q^41 - 3 * q^45 - q^47 - 17 * q^49 - 7 * q^51 + 3 * q^53 + 23 * q^55 + 2 * q^57 - q^59 - 22 * q^61 + 2 * q^63 - 20 * q^65 - 2 * q^67 - 13 * q^69 - 9 * q^71 - 28 * q^73 - 2 * q^75 - 10 * q^77 + 28 * q^79 + 5 * q^81 - 7 * q^83 - 11 * q^85 - 3 * q^87 - 30 * q^89 + 13 * q^91 + 12 * q^93 - 3 * q^95 - 33 * q^97 - 3 * q^99

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