LMFDB - Embedded newform 4016.2.a.k.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4016,2,Mod(1,4016)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4016, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4016.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4016 = 2^{4} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4016.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$32.0679214517$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.64128$ 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)

$f(q)$	$=$	$q-1.66988 q^{3} -3.99830 q^{5} +2.26241 q^{7} -0.211508 q^{9} +O(q^{10})$	$q-1.66988 q^{3} -3.99830 q^{5} +2.26241 q^{7} -0.211508 q^{9} +4.12088 q^{11} +5.91171 q^{13} +6.67667 q^{15} -1.43412 q^{17} -0.204854 q^{19} -3.77795 q^{21} -6.71463 q^{23} +10.9864 q^{25} +5.36283 q^{27} -1.38888 q^{29} +3.06652 q^{31} -6.88137 q^{33} -9.04579 q^{35} +4.37440 q^{37} -9.87183 q^{39} -7.17966 q^{41} -9.92677 q^{43} +0.845671 q^{45} -6.19940 q^{47} -1.88150 q^{49} +2.39480 q^{51} +2.85440 q^{53} -16.4765 q^{55} +0.342082 q^{57} +10.5814 q^{59} +1.35511 q^{61} -0.478517 q^{63} -23.6368 q^{65} +4.56380 q^{67} +11.2126 q^{69} -0.472215 q^{71} -3.38350 q^{73} -18.3460 q^{75} +9.32312 q^{77} +4.84515 q^{79} -8.32074 q^{81} +4.30359 q^{83} +5.73403 q^{85} +2.31926 q^{87} +2.35465 q^{89} +13.3747 q^{91} -5.12071 q^{93} +0.819069 q^{95} -6.51100 q^{97} -0.871598 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	$ 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) $ 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.66988	−0.964104	−0.482052	−	0.876142i	$-0.660108\pi$
$3$	−1.66988	−0.964104	−0.482052	+	0.876142i	$0.660108\pi$
$4$	0	0
$5$	−3.99830	−1.78809	−0.894047	−	0.447973i	$-0.852146\pi$
$5$	−3.99830	−1.78809	−0.894047	+	0.447973i	$0.852146\pi$
$6$	0	0
$7$	2.26241	0.855111	0.427555	−	0.903989i	$-0.359375\pi$
$7$	2.26241	0.855111	0.427555	+	0.903989i	$0.359375\pi$
$8$	0	0
$9$	−0.211508	−0.0705026
$10$	0	0
$11$	4.12088	1.24249	0.621246	−	0.783615i	$-0.286627\pi$
$11$	4.12088	1.24249	0.621246	+	0.783615i	$0.286627\pi$
$12$	0	0
$13$	5.91171	1.63961	0.819807	−	0.572641i	$-0.194081\pi$
$13$	5.91171	1.63961	0.819807	+	0.572641i	$0.194081\pi$
$14$	0	0
$15$	6.67667	1.72391
$16$	0	0
$17$	−1.43412	−0.347824	−0.173912	−	0.984761i	$-0.555641\pi$
$17$	−1.43412	−0.347824	−0.173912	+	0.984761i	$0.555641\pi$
$18$	0	0
$19$	−0.204854	−0.0469968	−0.0234984	−	0.999724i	$-0.507480\pi$
$19$	−0.204854	−0.0469968	−0.0234984	+	0.999724i	$0.507480\pi$
$20$	0	0
$21$	−3.77795	−0.824416
$22$	0	0
$23$	−6.71463	−1.40010	−0.700049	−	0.714095i	$-0.746839\pi$
$23$	−6.71463	−1.40010	−0.700049	+	0.714095i	$0.746839\pi$
$24$	0	0
$25$	10.9864	2.19728
$26$	0	0
$27$	5.36283	1.03208
$28$	0	0
$29$	−1.38888	−0.257909	−0.128954	−	0.991651i	$-0.541162\pi$
$29$	−1.38888	−0.257909	−0.128954	+	0.991651i	$0.541162\pi$
$30$	0	0
$31$	3.06652	0.550763	0.275382	−	0.961335i	$-0.411196\pi$
$31$	3.06652	0.550763	0.275382	+	0.961335i	$0.411196\pi$
$32$	0	0
$33$	−6.88137	−1.19789
$34$	0	0
$35$	−9.04579	−1.52902
$36$	0	0
$37$	4.37440	0.719147	0.359574	−	0.933117i	$-0.382922\pi$
$37$	4.37440	0.719147	0.359574	+	0.933117i	$0.382922\pi$
$38$	0	0
$39$	−9.87183	−1.58076
$40$	0	0
$41$	−7.17966	−1.12128	−0.560638	−	0.828061i	$-0.689444\pi$
$41$	−7.17966	−1.12128	−0.560638	+	0.828061i	$0.689444\pi$
$42$	0	0
$43$	−9.92677	−1.51382	−0.756909	−	0.653520i	$-0.773292\pi$
$43$	−9.92677	−1.51382	−0.756909	+	0.653520i	$0.773292\pi$
$44$	0	0
$45$	0.845671	0.126065
$46$	0	0
$47$	−6.19940	−0.904275	−0.452137	−	0.891948i	$-0.649338\pi$
$47$	−6.19940	−0.904275	−0.452137	+	0.891948i	$0.649338\pi$
$48$	0	0
$49$	−1.88150	−0.268786
$50$	0	0
$51$	2.39480	0.335339
$52$	0	0
$53$	2.85440	0.392082	0.196041	−	0.980596i	$-0.437191\pi$
$53$	2.85440	0.392082	0.196041	+	0.980596i	$0.437191\pi$
$54$	0	0
$55$	−16.4765	−2.22169
$56$	0	0
$57$	0.342082	0.0453098
$58$	0	0
$59$	10.5814	1.37758	0.688788	−	0.724962i	$-0.258143\pi$
$59$	10.5814	1.37758	0.688788	+	0.724962i	$0.258143\pi$
$60$	0	0
$61$	1.35511	0.173504	0.0867519	−	0.996230i	$-0.472351\pi$
$61$	1.35511	0.173504	0.0867519	+	0.996230i	$0.472351\pi$
$62$	0	0
$63$	−0.478517	−0.0602875
$64$	0	0
$65$	−23.6368	−2.93178
$66$	0	0
$67$	4.56380	0.557557	0.278779	−	0.960355i	$-0.410070\pi$
$67$	4.56380	0.557557	0.278779	+	0.960355i	$0.410070\pi$
$68$	0	0
$69$	11.2126	1.34984
$70$	0	0
$71$	−0.472215	−0.0560416	−0.0280208	−	0.999607i	$-0.508920\pi$
$71$	−0.472215	−0.0560416	−0.0280208	+	0.999607i	$0.508920\pi$
$72$	0	0
$73$	−3.38350	−0.396008	−0.198004	−	0.980201i	$-0.563446\pi$
$73$	−3.38350	−0.396008	−0.198004	+	0.980201i	$0.563446\pi$
$74$	0	0
$75$	−18.3460	−2.11841
$76$	0	0
$77$	9.32312	1.06247
$78$	0	0
$79$	4.84515	0.545122	0.272561	−	0.962138i	$-0.412129\pi$
$79$	4.84515	0.545122	0.272561	+	0.962138i	$0.412129\pi$
$80$	0	0
$81$	−8.32074	−0.924527
$82$	0	0
$83$	4.30359	0.472380	0.236190	−	0.971707i	$-0.424101\pi$
$83$	4.30359	0.472380	0.236190	+	0.971707i	$0.424101\pi$
$84$	0	0
$85$	5.73403	0.621943
$86$	0	0
$87$	2.31926	0.248651
$88$	0	0
$89$	2.35465	0.249593	0.124796	−	0.992182i	$-0.460172\pi$
$89$	2.35465	0.249593	0.124796	+	0.992182i	$0.460172\pi$
$90$	0	0
$91$	13.3747	1.40205
$92$	0	0
$93$	−5.12071	−0.530993
$94$	0	0
$95$	0.819069	0.0840347
$96$	0	0
$97$	−6.51100	−0.661092	−0.330546	−	0.943790i	$-0.607233\pi$
$97$	−6.51100	−0.661092	−0.330546	+	0.943790i	$0.607233\pi$
$98$	0	0
$99$	−0.871598	−0.0875989
$100$	0	0
$101$	−16.3106	−1.62297	−0.811484	−	0.584374i	$-0.801340\pi$
$101$	−16.3106	−1.62297	−0.811484	+	0.584374i	$0.801340\pi$
$102$	0	0
$103$	8.09062	0.797193	0.398596	−	0.917126i	$-0.369497\pi$
$103$	8.09062	0.797193	0.398596	+	0.917126i	$0.369497\pi$
$104$	0	0
$105$	15.1054	1.47413
$106$	0	0
$107$	−2.76868	−0.267659	−0.133829	−	0.991004i	$-0.542727\pi$
$107$	−2.76868	−0.267659	−0.133829	+	0.991004i	$0.542727\pi$
$108$	0	0
$109$	14.9773	1.43456	0.717282	−	0.696783i	$-0.245386\pi$
$109$	14.9773	1.43456	0.717282	+	0.696783i	$0.245386\pi$
$110$	0	0
$111$	−7.30472	−0.693333
$112$	0	0
$113$	6.74339	0.634365	0.317182	−	0.948365i	$-0.397263\pi$
$113$	6.74339	0.634365	0.317182	+	0.948365i	$0.397263\pi$
$114$	0	0
$115$	26.8471	2.50351
$116$	0	0
$117$	−1.25037	−0.115597
$118$	0	0
$119$	−3.24456	−0.297428
$120$	0	0
$121$	5.98166	0.543787
$122$	0	0
$123$	11.9892	1.08103
$124$	0	0
$125$	−23.9354	−2.14085
$126$	0	0
$127$	−5.75724	−0.510873	−0.255436	−	0.966826i	$-0.582219\pi$
$127$	−5.75724	−0.510873	−0.255436	+	0.966826i	$0.582219\pi$
$128$	0	0
$129$	16.5765	1.45948
$130$	0	0
$131$	−18.7599	−1.63906	−0.819529	−	0.573037i	$-0.805765\pi$
$131$	−18.7599	−1.63906	−0.819529	+	0.573037i	$0.805765\pi$
$132$	0	0
$133$	−0.463464	−0.0401875
$134$	0	0
$135$	−21.4422	−1.84545
$136$	0	0
$137$	20.7895	1.77617	0.888085	−	0.459678i	$-0.152035\pi$
$137$	20.7895	1.77617	0.888085	+	0.459678i	$0.152035\pi$
$138$	0	0
$139$	4.33518	0.367705	0.183852	−	0.982954i	$-0.441143\pi$
$139$	4.33518	0.367705	0.183852	+	0.982954i	$0.441143\pi$
$140$	0	0
$141$	10.3522	0.871815
$142$	0	0
$143$	24.3615	2.03721
$144$	0	0
$145$	5.55317	0.461165
$146$	0	0
$147$	3.14188	0.259138
$148$	0	0
$149$	2.15232	0.176325	0.0881623	−	0.996106i	$-0.471901\pi$
$149$	2.15232	0.176325	0.0881623	+	0.996106i	$0.471901\pi$
$150$	0	0
$151$	6.28712	0.511639	0.255819	−	0.966725i	$-0.417655\pi$
$151$	6.28712	0.511639	0.255819	+	0.966725i	$0.417655\pi$
$152$	0	0
$153$	0.303327	0.0245225
$154$	0	0
$155$	−12.2609	−0.984817
$156$	0	0
$157$	1.43414	0.114457	0.0572286	−	0.998361i	$-0.481774\pi$
$157$	1.43414	0.114457	0.0572286	+	0.998361i	$0.481774\pi$
$158$	0	0
$159$	−4.76650	−0.378008
$160$	0	0
$161$	−15.1913	−1.19724
$162$	0	0
$163$	14.0073	1.09714	0.548568	−	0.836106i	$-0.315173\pi$
$163$	14.0073	1.09714	0.548568	+	0.836106i	$0.315173\pi$
$164$	0	0
$165$	27.5138	2.14194
$166$	0	0
$167$	−11.1128	−0.859934	−0.429967	−	0.902845i	$-0.641475\pi$
$167$	−11.1128	−0.859934	−0.429967	+	0.902845i	$0.641475\pi$
$168$	0	0
$169$	21.9483	1.68833
$170$	0	0
$171$	0.0433283	0.00331339
$172$	0	0
$173$	14.6669	1.11511	0.557553	−	0.830141i	$-0.311740\pi$
$173$	14.6669	1.11511	0.557553	+	0.830141i	$0.311740\pi$
$174$	0	0
$175$	24.8558	1.87892
$176$	0	0
$177$	−17.6696	−1.32813
$178$	0	0
$179$	5.37248	0.401558	0.200779	−	0.979637i	$-0.435653\pi$
$179$	5.37248	0.401558	0.200779	+	0.979637i	$0.435653\pi$
$180$	0	0
$181$	17.9739	1.33599	0.667996	−	0.744165i	$-0.267152\pi$
$181$	17.9739	1.33599	0.667996	+	0.744165i	$0.267152\pi$
$182$	0	0
$183$	−2.26287	−0.167276
$184$	0	0
$185$	−17.4902	−1.28590
$186$	0	0
$187$	−5.90983	−0.432169
$188$	0	0
$189$	12.1329	0.882539
$190$	0	0
$191$	22.6723	1.64051	0.820254	−	0.571999i	$-0.193832\pi$
$191$	22.6723	1.64051	0.820254	+	0.571999i	$0.193832\pi$
$192$	0	0
$193$	−4.89351	−0.352243	−0.176121	−	0.984368i	$-0.556355\pi$
$193$	−4.89351	−0.352243	−0.176121	+	0.984368i	$0.556355\pi$
$194$	0	0
$195$	39.4706	2.82654
$196$	0	0
$197$	19.8914	1.41720	0.708601	−	0.705609i	$-0.249327\pi$
$197$	19.8914	1.41720	0.708601	+	0.705609i	$0.249327\pi$
$198$	0	0
$199$	1.63401	0.115832	0.0579161	−	0.998321i	$-0.481554\pi$
$199$	1.63401	0.115832	0.0579161	+	0.998321i	$0.481554\pi$
$200$	0	0
$201$	−7.62099	−0.537544
$202$	0	0
$203$	−3.14222	−0.220541
$204$	0	0
$205$	28.7065	2.00495
$206$	0	0
$207$	1.42020	0.0987105
$208$	0	0
$209$	−0.844180	−0.0583932
$210$	0	0
$211$	−8.95153	−0.616249	−0.308125	−	0.951346i	$-0.599701\pi$
$211$	−8.95153	−0.616249	−0.308125	+	0.951346i	$0.599701\pi$
$212$	0	0
$213$	0.788541	0.0540299
$214$	0	0
$215$	39.6902	2.70685
$216$	0	0
$217$	6.93773	0.470964
$218$	0	0
$219$	5.65003	0.381793
$220$	0	0
$221$	−8.47808	−0.570298
$222$	0	0
$223$	−23.7770	−1.59223	−0.796114	−	0.605147i	$-0.793114\pi$
$223$	−23.7770	−1.59223	−0.796114	+	0.605147i	$0.793114\pi$
$224$	0	0
$225$	−2.32371	−0.154914
$226$	0	0
$227$	24.1597	1.60353	0.801767	−	0.597637i	$-0.203894\pi$
$227$	24.1597	1.60353	0.801767	+	0.597637i	$0.203894\pi$
$228$	0	0
$229$	18.0972	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$229$	18.0972	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$230$	0	0
$231$	−15.5685	−1.02433
$232$	0	0
$233$	6.96996	0.456617	0.228309	−	0.973589i	$-0.426680\pi$
$233$	6.96996	0.456617	0.228309	+	0.973589i	$0.426680\pi$
$234$	0	0
$235$	24.7870	1.61693
$236$	0	0
$237$	−8.09082	−0.525555
$238$	0	0
$239$	19.6467	1.27084	0.635419	−	0.772168i	$-0.280828\pi$
$239$	19.6467	1.27084	0.635419	+	0.772168i	$0.280828\pi$
$240$	0	0
$241$	−10.7963	−0.695453	−0.347726	−	0.937596i	$-0.613046\pi$
$241$	−10.7963	−0.695453	−0.347726	+	0.937596i	$0.613046\pi$
$242$	0	0
$243$	−2.19385	−0.140736
$244$	0	0
$245$	7.52280	0.480614
$246$	0	0
$247$	−1.21104	−0.0770566
$248$	0	0
$249$	−7.18646	−0.455424
$250$	0	0
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−27.6702	−1.73961
$254$	0	0
$255$	−9.57513	−0.599618
$256$	0	0
$257$	−10.6298	−0.663067	−0.331533	−	0.943443i	$-0.607566\pi$
$257$	−10.6298	−0.663067	−0.331533	+	0.943443i	$0.607566\pi$
$258$	0	0
$259$	9.89669	0.614951
$260$	0	0
$261$	0.293759	0.0181832
$262$	0	0
$263$	17.2119	1.06133	0.530667	−	0.847581i	$-0.321942\pi$
$263$	17.2119	1.06133	0.530667	+	0.847581i	$0.321942\pi$
$264$	0	0
$265$	−11.4127	−0.701079
$266$	0	0
$267$	−3.93198	−0.240633
$268$	0	0
$269$	27.2641	1.66232	0.831161	−	0.556032i	$-0.187677\pi$
$269$	27.2641	1.66232	0.831161	+	0.556032i	$0.187677\pi$
$270$	0	0
$271$	17.2661	1.04884	0.524421	−	0.851459i	$-0.324282\pi$
$271$	17.2661	1.04884	0.524421	+	0.851459i	$0.324282\pi$
$272$	0	0
$273$	−22.3341	−1.35172
$274$	0	0
$275$	45.2737	2.73010
$276$	0	0
$277$	−15.2465	−0.916071	−0.458035	−	0.888934i	$-0.651447\pi$
$277$	−15.2465	−0.916071	−0.458035	+	0.888934i	$0.651447\pi$
$278$	0	0
$279$	−0.648593	−0.0388302
$280$	0	0
$281$	−28.2512	−1.68532	−0.842661	−	0.538444i	$-0.819012\pi$
$281$	−28.2512	−1.68532	−0.842661	+	0.538444i	$0.819012\pi$
$282$	0	0
$283$	1.97805	0.117583	0.0587916	−	0.998270i	$-0.481275\pi$
$283$	1.97805	0.117583	0.0587916	+	0.998270i	$0.481275\pi$
$284$	0	0
$285$	−1.36775	−0.0810182
$286$	0	0
$287$	−16.2433	−0.958814
$288$	0	0
$289$	−14.9433	−0.879018
$290$	0	0
$291$	10.8726	0.637362
$292$	0	0
$293$	−17.4457	−1.01919	−0.509593	−	0.860416i	$-0.670204\pi$
$293$	−17.4457	−1.01919	−0.509593	+	0.860416i	$0.670204\pi$
$294$	0	0
$295$	−42.3075	−2.46324
$296$	0	0
$297$	22.0996	1.28235
$298$	0	0
$299$	−39.6950	−2.29562
$300$	0	0
$301$	−22.4584	−1.29448
$302$	0	0
$303$	27.2368	1.56471
$304$	0	0
$305$	−5.41813	−0.310241
$306$	0	0
$307$	1.43476	0.0818858	0.0409429	−	0.999161i	$-0.486964\pi$
$307$	1.43476	0.0818858	0.0409429	+	0.999161i	$0.486964\pi$
$308$	0	0
$309$	−13.5104	−0.768577
$310$	0	0
$311$	−23.0447	−1.30675	−0.653374	−	0.757036i	$-0.726647\pi$
$311$	−23.0447	−1.30675	−0.653374	+	0.757036i	$0.726647\pi$
$312$	0	0
$313$	6.04912	0.341917	0.170958	−	0.985278i	$-0.445314\pi$
$313$	6.04912	0.341917	0.170958	+	0.985278i	$0.445314\pi$
$314$	0	0
$315$	1.91326	0.107800
$316$	0	0
$317$	−18.1963	−1.02201	−0.511003	−	0.859579i	$-0.670726\pi$
$317$	−18.1963	−1.02201	−0.511003	+	0.859579i	$0.670726\pi$
$318$	0	0
$319$	−5.72342	−0.320450
$320$	0	0
$321$	4.62337	0.258051
$322$	0	0
$323$	0.293785	0.0163466
$324$	0	0
$325$	64.9484	3.60269
$326$	0	0
$327$	−25.0102	−1.38307
$328$	0	0
$329$	−14.0256	−0.773255
$330$	0	0
$331$	9.53948	0.524337	0.262169	−	0.965022i	$-0.415562\pi$
$331$	9.53948	0.524337	0.262169	+	0.965022i	$0.415562\pi$
$332$	0	0
$333$	−0.925220	−0.0507017
$334$	0	0
$335$	−18.2475	−0.996965
$336$	0	0
$337$	−1.32177	−0.0720015	−0.0360008	−	0.999352i	$-0.511462\pi$
$337$	−1.32177	−0.0720015	−0.0360008	+	0.999352i	$0.511462\pi$
$338$	0	0
$339$	−11.2606	−0.611594
$340$	0	0
$341$	12.6368	0.684319
$342$	0	0
$343$	−20.0936	−1.08495
$344$	0	0
$345$	−44.8314	−2.41364
$346$	0	0
$347$	−13.3801	−0.718279	−0.359140	−	0.933284i	$-0.616930\pi$
$347$	−13.3801	−0.718279	−0.359140	+	0.933284i	$0.616930\pi$
$348$	0	0
$349$	7.87732	0.421663	0.210832	−	0.977522i	$-0.432383\pi$
$349$	7.87732	0.421663	0.210832	+	0.977522i	$0.432383\pi$
$350$	0	0
$351$	31.7035	1.69221
$352$	0	0
$353$	28.2912	1.50579	0.752893	−	0.658143i	$-0.228658\pi$
$353$	28.2912	1.50579	0.752893	+	0.658143i	$0.228658\pi$
$354$	0	0
$355$	1.88806	0.100208
$356$	0	0
$357$	5.41802	0.286752
$358$	0	0
$359$	13.8511	0.731035	0.365517	−	0.930805i	$-0.380892\pi$
$359$	13.8511	0.731035	0.365517	+	0.930805i	$0.380892\pi$
$360$	0	0
$361$	−18.9580	−0.997791
$362$	0	0
$363$	−9.98864	−0.524268
$364$	0	0
$365$	13.5282	0.708100
$366$	0	0
$367$	−2.14261	−0.111843	−0.0559217	−	0.998435i	$-0.517810\pi$
$367$	−2.14261	−0.111843	−0.0559217	+	0.998435i	$0.517810\pi$
$368$	0	0
$369$	1.51855	0.0790528
$370$	0	0
$371$	6.45782	0.335273
$372$	0	0
$373$	35.1954	1.82235	0.911175	−	0.412019i	$-0.135176\pi$
$373$	35.1954	1.82235	0.911175	+	0.412019i	$0.135176\pi$
$374$	0	0
$375$	39.9693	2.06400
$376$	0	0
$377$	−8.21066	−0.422871
$378$	0	0
$379$	−19.2843	−0.990567	−0.495283	−	0.868732i	$-0.664936\pi$
$379$	−19.2843	−0.990567	−0.495283	+	0.868732i	$0.664936\pi$
$380$	0	0
$381$	9.61389	0.492535
$382$	0	0
$383$	18.2315	0.931589	0.465794	−	0.884893i	$-0.345769\pi$
$383$	18.2315	0.931589	0.465794	+	0.884893i	$0.345769\pi$
$384$	0	0
$385$	−37.2766	−1.89979
$386$	0	0
$387$	2.09959	0.106728
$388$	0	0
$389$	−28.2291	−1.43127	−0.715636	−	0.698474i	$-0.753863\pi$
$389$	−28.2291	−1.43127	−0.715636	+	0.698474i	$0.753863\pi$
$390$	0	0
$391$	9.62957	0.486988
$392$	0	0
$393$	31.3267	1.58022
$394$	0	0
$395$	−19.3724	−0.974730
$396$	0	0
$397$	−3.30716	−0.165982	−0.0829908	−	0.996550i	$-0.526447\pi$
$397$	−3.30716	−0.165982	−0.0829908	+	0.996550i	$0.526447\pi$
$398$	0	0
$399$	0.773929	0.0387449
$400$	0	0
$401$	19.3611	0.966847	0.483424	−	0.875386i	$-0.339393\pi$
$401$	19.3611	0.966847	0.483424	+	0.875386i	$0.339393\pi$
$402$	0	0
$403$	18.1284	0.903039
$404$	0	0
$405$	33.2688	1.65314
$406$	0	0
$407$	18.0264	0.893535
$408$	0	0
$409$	−6.66759	−0.329691	−0.164846	−	0.986319i	$-0.552713\pi$
$409$	−6.66759	−0.329691	−0.164846	+	0.986319i	$0.552713\pi$
$410$	0	0
$411$	−34.7160	−1.71241
$412$	0	0
$413$	23.9394	1.17798
$414$	0	0
$415$	−17.2070	−0.844660
$416$	0	0
$417$	−7.23921	−0.354506
$418$	0	0
$419$	−0.698838	−0.0341405	−0.0170702	−	0.999854i	$-0.505434\pi$
$419$	−0.698838	−0.0341405	−0.0170702	+	0.999854i	$0.505434\pi$
$420$	0	0
$421$	26.3118	1.28236	0.641180	−	0.767390i	$-0.278445\pi$
$421$	26.3118	1.28236	0.641180	+	0.767390i	$0.278445\pi$
$422$	0	0
$423$	1.31122	0.0637537
$424$	0	0
$425$	−15.7558	−0.764268
$426$	0	0
$427$	3.06581	0.148365
$428$	0	0
$429$	−40.6806	−1.96408
$430$	0	0
$431$	−3.35623	−0.161664	−0.0808319	−	0.996728i	$-0.525758\pi$
$431$	−3.35623	−0.161664	−0.0808319	+	0.996728i	$0.525758\pi$
$432$	0	0
$433$	14.5887	0.701087	0.350544	−	0.936546i	$-0.385997\pi$
$433$	14.5887	0.701087	0.350544	+	0.936546i	$0.385997\pi$
$434$	0	0
$435$	−9.27311	−0.444612
$436$	0	0
$437$	1.37552	0.0658001
$438$	0	0
$439$	8.45608	0.403587	0.201793	−	0.979428i	$-0.435323\pi$
$439$	8.45608	0.403587	0.201793	+	0.979428i	$0.435323\pi$
$440$	0	0
$441$	0.397952	0.0189501
$442$	0	0
$443$	16.3156	0.775176	0.387588	−	0.921833i	$-0.373308\pi$
$443$	16.3156	0.775176	0.387588	+	0.921833i	$0.373308\pi$
$444$	0	0
$445$	−9.41461	−0.446295
$446$	0	0
$447$	−3.59411	−0.169995
$448$	0	0
$449$	28.8432	1.36119	0.680597	−	0.732658i	$-0.261721\pi$
$449$	28.8432	1.36119	0.680597	+	0.732658i	$0.261721\pi$
$450$	0	0
$451$	−29.5865	−1.39318
$452$	0	0
$453$	−10.4987	−0.493273
$454$	0	0
$455$	−53.4761	−2.50700
$456$	0	0
$457$	24.3382	1.13849	0.569246	−	0.822167i	$-0.307235\pi$
$457$	24.3382	1.13849	0.569246	+	0.822167i	$0.307235\pi$
$458$	0	0
$459$	−7.69092	−0.358981
$460$	0	0
$461$	12.4524	0.579965	0.289982	−	0.957032i	$-0.406351\pi$
$461$	12.4524	0.579965	0.289982	+	0.957032i	$0.406351\pi$
$462$	0	0
$463$	−4.87682	−0.226645	−0.113323	−	0.993558i	$-0.536149\pi$
$463$	−4.87682	−0.226645	−0.113323	+	0.993558i	$0.536149\pi$
$464$	0	0
$465$	20.4742	0.949466
$466$	0	0
$467$	−4.52604	−0.209440	−0.104720	−	0.994502i	$-0.533395\pi$
$467$	−4.52604	−0.209440	−0.104720	+	0.994502i	$0.533395\pi$
$468$	0	0
$469$	10.3252	0.476773
$470$	0	0
$471$	−2.39485	−0.110349
$472$	0	0
$473$	−40.9070	−1.88091
$474$	0	0
$475$	−2.25061	−0.103265
$476$	0	0
$477$	−0.603727	−0.0276428
$478$	0	0
$479$	37.4665	1.71189	0.855943	−	0.517070i	$-0.172977\pi$
$479$	37.4665	1.71189	0.855943	+	0.517070i	$0.172977\pi$
$480$	0	0
$481$	25.8602	1.17912
$482$	0	0
$483$	25.3675	1.15426
$484$	0	0
$485$	26.0329	1.18209
$486$	0	0
$487$	9.22972	0.418238	0.209119	−	0.977890i	$-0.432940\pi$
$487$	9.22972	0.418238	0.209119	+	0.977890i	$0.432940\pi$
$488$	0	0
$489$	−23.3905	−1.05775
$490$	0	0
$491$	4.75442	0.214564	0.107282	−	0.994229i	$-0.465785\pi$
$491$	4.75442	0.214564	0.107282	+	0.994229i	$0.465785\pi$
$492$	0	0
$493$	1.99182	0.0897070
$494$	0	0
$495$	3.48491	0.156635
$496$	0	0
$497$	−1.06834	−0.0479218
$498$	0	0
$499$	4.18286	0.187250	0.0936252	−	0.995608i	$-0.470154\pi$
$499$	4.18286	0.187250	0.0936252	+	0.995608i	$0.470154\pi$
$500$	0	0
$501$	18.5570	0.829066
$502$	0	0
$503$	−1.26808	−0.0565407	−0.0282704	−	0.999600i	$-0.509000\pi$
$503$	−1.26808	−0.0565407	−0.0282704	+	0.999600i	$0.509000\pi$
$504$	0	0
$505$	65.2148	2.90202
$506$	0	0
$507$	−36.6510	−1.62773
$508$	0	0
$509$	23.7633	1.05329	0.526646	−	0.850085i	$-0.323449\pi$
$509$	23.7633	1.05329	0.526646	+	0.850085i	$0.323449\pi$
$510$	0	0
$511$	−7.65486	−0.338631
$512$	0	0
$513$	−1.09860	−0.0485043
$514$	0	0
$515$	−32.3487	−1.42546
$516$	0	0
$517$	−25.5470	−1.12355
$518$	0	0
$519$	−24.4920	−1.07508
$520$	0	0
$521$	−41.9607	−1.83833	−0.919167	−	0.393869i	$-0.871136\pi$
$521$	−41.9607	−1.83833	−0.919167	+	0.393869i	$0.871136\pi$
$522$	0	0
$523$	−26.0315	−1.13828	−0.569138	−	0.822242i	$-0.692723\pi$
$523$	−26.0315	−1.13828	−0.569138	+	0.822242i	$0.692723\pi$
$524$	0	0
$525$	−41.5061	−1.81147
$526$	0	0
$527$	−4.39775	−0.191569
$528$	0	0
$529$	22.0863	0.960275
$530$	0	0
$531$	−2.23804	−0.0971227
$532$	0	0
$533$	−42.4441	−1.83846
$534$	0	0
$535$	11.0700	0.478599
$536$	0	0
$537$	−8.97138	−0.387144
$538$	0	0
$539$	−7.75344	−0.333964
$540$	0	0
$541$	14.1415	0.607990	0.303995	−	0.952674i	$-0.401679\pi$
$541$	14.1415	0.607990	0.303995	+	0.952674i	$0.401679\pi$
$542$	0	0
$543$	−30.0143	−1.28804
$544$	0	0
$545$	−59.8837	−2.56513
$546$	0	0
$547$	32.1825	1.37602	0.688011	−	0.725700i	$-0.258484\pi$
$547$	32.1825	1.37602	0.688011	+	0.725700i	$0.258484\pi$
$548$	0	0
$549$	−0.286616	−0.0122325
$550$	0	0
$551$	0.284518	0.0121209
$552$	0	0
$553$	10.9617	0.466140
$554$	0	0
$555$	29.2065	1.23975
$556$	0	0
$557$	29.5890	1.25373	0.626863	−	0.779130i	$-0.284339\pi$
$557$	29.5890	1.25373	0.626863	+	0.779130i	$0.284339\pi$
$558$	0	0
$559$	−58.6842	−2.48208
$560$	0	0
$561$	9.86869	0.416656
$562$	0	0
$563$	8.83396	0.372307	0.186154	−	0.982521i	$-0.440398\pi$
$563$	8.83396	0.372307	0.186154	+	0.982521i	$0.440398\pi$
$564$	0	0
$565$	−26.9621	−1.13430
$566$	0	0
$567$	−18.8249	−0.790573
$568$	0	0
$569$	−18.9193	−0.793139	−0.396570	−	0.918005i	$-0.629799\pi$
$569$	−18.9193	−0.793139	−0.396570	+	0.918005i	$0.629799\pi$
$570$	0	0
$571$	7.35664	0.307866	0.153933	−	0.988081i	$-0.450806\pi$
$571$	7.35664	0.307866	0.153933	+	0.988081i	$0.450806\pi$
$572$	0	0
$573$	−37.8599	−1.58162
$574$	0	0
$575$	−73.7697	−3.07641
$576$	0	0
$577$	3.61792	0.150616	0.0753079	−	0.997160i	$-0.476006\pi$
$577$	3.61792	0.150616	0.0753079	+	0.997160i	$0.476006\pi$
$578$	0	0
$579$	8.17156	0.339599
$580$	0	0
$581$	9.73648	0.403937
$582$	0	0
$583$	11.7626	0.487159
$584$	0	0
$585$	4.99936	0.206698
$586$	0	0
$587$	−43.2403	−1.78472	−0.892360	−	0.451324i	$-0.850952\pi$
$587$	−43.2403	−1.78472	−0.892360	+	0.451324i	$0.850952\pi$
$588$	0	0
$589$	−0.628190	−0.0258841
$590$	0	0
$591$	−33.2162	−1.36633
$592$	0	0
$593$	−13.1959	−0.541891	−0.270945	−	0.962595i	$-0.587336\pi$
$593$	−13.1959	−0.541891	−0.270945	+	0.962595i	$0.587336\pi$
$594$	0	0
$595$	12.9727	0.531830
$596$	0	0
$597$	−2.72860	−0.111674
$598$	0	0
$599$	−6.39530	−0.261305	−0.130652	−	0.991428i	$-0.541707\pi$
$599$	−6.39530	−0.261305	−0.130652	+	0.991428i	$0.541707\pi$
$600$	0	0
$601$	31.4204	1.28166	0.640831	−	0.767682i	$-0.278590\pi$
$601$	31.4204	1.28166	0.640831	+	0.767682i	$0.278590\pi$
$602$	0	0
$603$	−0.965279	−0.0393092
$604$	0	0
$605$	−23.9165	−0.972343
$606$	0	0
$607$	4.02795	0.163490	0.0817448	−	0.996653i	$-0.473951\pi$
$607$	4.02795	0.163490	0.0817448	+	0.996653i	$0.473951\pi$
$608$	0	0
$609$	5.24712	0.212624
$610$	0	0
$611$	−36.6490	−1.48266
$612$	0	0
$613$	−29.7607	−1.20202	−0.601011	−	0.799241i	$-0.705235\pi$
$613$	−29.7607	−1.20202	−0.601011	+	0.799241i	$0.705235\pi$
$614$	0	0
$615$	−47.9363	−1.93298
$616$	0	0
$617$	22.2513	0.895803	0.447901	−	0.894083i	$-0.352172\pi$
$617$	22.2513	0.895803	0.447901	+	0.894083i	$0.352172\pi$
$618$	0	0
$619$	32.4844	1.30566	0.652829	−	0.757505i	$-0.273582\pi$
$619$	32.4844	1.30566	0.652829	+	0.757505i	$0.273582\pi$
$620$	0	0
$621$	−36.0094	−1.44501
$622$	0	0
$623$	5.32719	0.213429
$624$	0	0
$625$	40.7691	1.63076
$626$	0	0
$627$	1.40968	0.0562971
$628$	0	0
$629$	−6.27341	−0.250137
$630$	0	0
$631$	28.2517	1.12468	0.562341	−	0.826906i	$-0.309901\pi$
$631$	28.2517	1.12468	0.562341	+	0.826906i	$0.309901\pi$
$632$	0	0
$633$	14.9480	0.594128
$634$	0	0
$635$	23.0192	0.913488
$636$	0	0
$637$	−11.1229	−0.440705
$638$	0	0
$639$	0.0998771	0.00395108
$640$	0	0
$641$	0.0850478	0.00335918	0.00167959	−	0.999999i	$-0.499465\pi$
$641$	0.0850478	0.00335918	0.00167959	+	0.999999i	$0.499465\pi$
$642$	0	0
$643$	−18.3367	−0.723130	−0.361565	−	0.932347i	$-0.617757\pi$
$643$	−18.3367	−0.723130	−0.361565	+	0.932347i	$0.617757\pi$
$644$	0	0
$645$	−66.2778	−2.60969
$646$	0	0
$647$	27.9945	1.10058	0.550289	−	0.834974i	$-0.314518\pi$
$647$	27.9945	1.10058	0.550289	+	0.834974i	$0.314518\pi$
$648$	0	0
$649$	43.6046	1.71163
$650$	0	0
$651$	−11.5852	−0.454058
$652$	0	0
$653$	−12.8763	−0.503889	−0.251945	−	0.967742i	$-0.581070\pi$
$653$	−12.8763	−0.503889	−0.251945	+	0.967742i	$0.581070\pi$
$654$	0	0
$655$	75.0077	2.93079
$656$	0	0
$657$	0.715636	0.0279196
$658$	0	0
$659$	7.67979	0.299162	0.149581	−	0.988749i	$-0.452208\pi$
$659$	7.67979	0.299162	0.149581	+	0.988749i	$0.452208\pi$
$660$	0	0
$661$	−37.9087	−1.47448	−0.737238	−	0.675633i	$-0.763871\pi$
$661$	−37.9087	−1.47448	−0.737238	+	0.675633i	$0.763871\pi$
$662$	0	0
$663$	14.1574	0.549826
$664$	0	0
$665$	1.85307	0.0718590
$666$	0	0
$667$	9.32583	0.361098
$668$	0	0
$669$	39.7047	1.53507
$670$	0	0
$671$	5.58424	0.215577
$672$	0	0
$673$	−39.1700	−1.50989	−0.754946	−	0.655786i	$-0.772337\pi$
$673$	−39.1700	−1.50989	−0.754946	+	0.655786i	$0.772337\pi$
$674$	0	0
$675$	58.9182	2.26776
$676$	0	0
$677$	−1.79517	−0.0689938	−0.0344969	−	0.999405i	$-0.510983\pi$
$677$	−1.79517	−0.0689938	−0.0344969	+	0.999405i	$0.510983\pi$
$678$	0	0
$679$	−14.7306	−0.565307
$680$	0	0
$681$	−40.3437	−1.54597
$682$	0	0
$683$	−5.50696	−0.210718	−0.105359	−	0.994434i	$-0.533599\pi$
$683$	−5.50696	−0.210718	−0.105359	+	0.994434i	$0.533599\pi$
$684$	0	0
$685$	−83.1228	−3.17596
$686$	0	0
$687$	−30.2201	−1.15297
$688$	0	0
$689$	16.8744	0.642862
$690$	0	0
$691$	46.2011	1.75757	0.878786	−	0.477216i	$-0.158354\pi$
$691$	46.2011	1.75757	0.878786	+	0.477216i	$0.158354\pi$
$692$	0	0
$693$	−1.97191	−0.0749067
$694$	0	0
$695$	−17.3333	−0.657491
$696$	0	0
$697$	10.2965	0.390007
$698$	0	0
$699$	−11.6390	−0.440227
$700$	0	0
$701$	40.4921	1.52937	0.764684	−	0.644406i	$-0.222895\pi$
$701$	40.4921	1.52937	0.764684	+	0.644406i	$0.222895\pi$
$702$	0	0
$703$	−0.896115	−0.0337976
$704$	0	0
$705$	−41.3913	−1.55889
$706$	0	0
$707$	−36.9013	−1.38782
$708$	0	0
$709$	17.7851	0.667935	0.333967	−	0.942585i	$-0.391612\pi$
$709$	17.7851	0.667935	0.333967	+	0.942585i	$0.391612\pi$
$710$	0	0
$711$	−1.02479	−0.0384325
$712$	0	0
$713$	−20.5906	−0.771123
$714$	0	0
$715$	−97.4044	−3.64272
$716$	0	0
$717$	−32.8075	−1.22522
$718$	0	0
$719$	−45.9669	−1.71428	−0.857139	−	0.515085i	$-0.827760\pi$
$719$	−45.9669	−1.71428	−0.857139	+	0.515085i	$0.827760\pi$
$720$	0	0
$721$	18.3043	0.681688
$722$	0	0
$723$	18.0286	0.670489
$724$	0	0
$725$	−15.2588	−0.566698
$726$	0	0
$727$	18.5294	0.687218	0.343609	−	0.939113i	$-0.388351\pi$
$727$	18.5294	0.687218	0.343609	+	0.939113i	$0.388351\pi$
$728$	0	0
$729$	28.6257	1.06021
$730$	0	0
$731$	14.2362	0.526543
$732$	0	0
$733$	−7.21557	−0.266513	−0.133257	−	0.991082i	$-0.542543\pi$
$733$	−7.21557	−0.266513	−0.133257	+	0.991082i	$0.542543\pi$
$734$	0	0
$735$	−12.5622	−0.463362
$736$	0	0
$737$	18.8069	0.692761
$738$	0	0
$739$	2.18433	0.0803519	0.0401759	−	0.999193i	$-0.487208\pi$
$739$	2.18433	0.0803519	0.0401759	+	0.999193i	$0.487208\pi$
$740$	0	0
$741$	2.02229	0.0742906
$742$	0	0
$743$	32.9240	1.20786	0.603932	−	0.797036i	$-0.293600\pi$
$743$	32.9240	1.20786	0.603932	+	0.797036i	$0.293600\pi$
$744$	0	0
$745$	−8.60561	−0.315285
$746$	0	0
$747$	−0.910241	−0.0333040
$748$	0	0
$749$	−6.26390	−0.228878
$750$	0	0
$751$	7.25344	0.264682	0.132341	−	0.991204i	$-0.457751\pi$
$751$	7.25344	0.264682	0.132341	+	0.991204i	$0.457751\pi$
$752$	0	0
$753$	1.66988	0.0608537
$754$	0	0
$755$	−25.1378	−0.914858
$756$	0	0
$757$	33.2428	1.20823	0.604115	−	0.796897i	$-0.293527\pi$
$757$	33.2428	1.20823	0.604115	+	0.796897i	$0.293527\pi$
$758$	0	0
$759$	46.2059	1.67717
$760$	0	0
$761$	31.8325	1.15393	0.576963	−	0.816770i	$-0.304238\pi$
$761$	31.8325	1.15393	0.576963	+	0.816770i	$0.304238\pi$
$762$	0	0
$763$	33.8848	1.22671
$764$	0	0
$765$	−1.21279	−0.0438486
$766$	0	0
$767$	62.5540	2.25869
$768$	0	0
$769$	33.5647	1.21037	0.605187	−	0.796083i	$-0.293098\pi$
$769$	33.5647	1.21037	0.605187	+	0.796083i	$0.293098\pi$
$770$	0	0
$771$	17.7504	0.639266
$772$	0	0
$773$	40.9906	1.47433	0.737165	−	0.675713i	$-0.236164\pi$
$773$	40.9906	1.47433	0.737165	+	0.675713i	$0.236164\pi$
$774$	0	0
$775$	33.6900	1.21018
$776$	0	0
$777$	−16.5263	−0.592877
$778$	0	0
$779$	1.47079	0.0526963
$780$	0	0
$781$	−1.94594	−0.0696312
$782$	0	0
$783$	−7.44833	−0.266182
$784$	0	0
$785$	−5.73414	−0.204660
$786$	0	0
$787$	−9.09783	−0.324303	−0.162151	−	0.986766i	$-0.551843\pi$
$787$	−9.09783	−0.324303	−0.162151	+	0.986766i	$0.551843\pi$
$788$	0	0
$789$	−28.7418	−1.02324
$790$	0	0
$791$	15.2563	0.542452
$792$	0	0
$793$	8.01101	0.284479
$794$	0	0
$795$	19.0579	0.675914
$796$	0	0
$797$	2.24596	0.0795561	0.0397781	−	0.999209i	$-0.487335\pi$
$797$	2.24596	0.0795561	0.0397781	+	0.999209i	$0.487335\pi$
$798$	0	0
$799$	8.89066	0.314529
$800$	0	0
$801$	−0.498027	−0.0175969
$802$	0	0
$803$	−13.9430	−0.492037
$804$	0	0
$805$	60.7392	2.14078
$806$	0	0
$807$	−45.5277	−1.60265
$808$	0	0
$809$	26.5247	0.932558	0.466279	−	0.884638i	$-0.345594\pi$
$809$	26.5247	0.932558	0.466279	+	0.884638i	$0.345594\pi$
$810$	0	0
$811$	−38.1022	−1.33795	−0.668974	−	0.743286i	$-0.733266\pi$
$811$	−38.1022	−1.33795	−0.668974	+	0.743286i	$0.733266\pi$
$812$	0	0
$813$	−28.8323	−1.01119
$814$	0	0
$815$	−56.0054	−1.96178
$816$	0	0
$817$	2.03354	0.0711446
$818$	0	0
$819$	−2.82885	−0.0988482
$820$	0	0
$821$	51.8976	1.81124	0.905620	−	0.424090i	$-0.139406\pi$
$821$	51.8976	1.81124	0.905620	+	0.424090i	$0.139406\pi$
$822$	0	0
$823$	25.2537	0.880288	0.440144	−	0.897927i	$-0.354927\pi$
$823$	25.2537	0.880288	0.440144	+	0.897927i	$0.354927\pi$
$824$	0	0
$825$	−75.6015	−2.63211
$826$	0	0
$827$	−22.3782	−0.778167	−0.389083	−	0.921203i	$-0.627208\pi$
$827$	−22.3782	−0.778167	−0.389083	+	0.921203i	$0.627208\pi$
$828$	0	0
$829$	−24.0763	−0.836204	−0.418102	−	0.908400i	$-0.637304\pi$
$829$	−24.0763	−0.836204	−0.418102	+	0.908400i	$0.637304\pi$
$830$	0	0
$831$	25.4597	0.883188
$832$	0	0
$833$	2.69829	0.0934903
$834$	0	0
$835$	44.4323	1.53764
$836$	0	0
$837$	16.4452	0.568430
$838$	0	0
$839$	54.3943	1.87790	0.938950	−	0.344053i	$-0.111800\pi$
$839$	54.3943	1.87790	0.938950	+	0.344053i	$0.111800\pi$
$840$	0	0
$841$	−27.0710	−0.933483
$842$	0	0
$843$	47.1760	1.62483
$844$	0	0
$845$	−87.7559	−3.01890
$846$	0	0
$847$	13.5330	0.464998
$848$	0	0
$849$	−3.30311	−0.113362
$850$	0	0
$851$	−29.3725	−1.00688
$852$	0	0
$853$	−7.60714	−0.260464	−0.130232	−	0.991484i	$-0.541572\pi$
$853$	−7.60714	−0.260464	−0.130232	+	0.991484i	$0.541572\pi$
$854$	0	0
$855$	−0.173239	−0.00592466
$856$	0	0
$857$	−15.0700	−0.514783	−0.257391	−	0.966307i	$-0.582863\pi$
$857$	−15.0700	−0.514783	−0.257391	+	0.966307i	$0.582863\pi$
$858$	0	0
$859$	−16.3535	−0.557975	−0.278988	−	0.960295i	$-0.589999\pi$
$859$	−16.3535	−0.557975	−0.278988	+	0.960295i	$0.589999\pi$
$860$	0	0
$861$	27.1244	0.924397
$862$	0	0
$863$	23.0445	0.784445	0.392222	−	0.919870i	$-0.371706\pi$
$863$	23.0445	0.784445	0.392222	+	0.919870i	$0.371706\pi$
$864$	0	0
$865$	−58.6428	−1.99391
$866$	0	0
$867$	24.9535	0.847465
$868$	0	0
$869$	19.9663	0.677310
$870$	0	0
$871$	26.9799	0.914178
$872$	0	0
$873$	1.37713	0.0466087
$874$	0	0
$875$	−54.1518	−1.83066
$876$	0	0
$877$	−45.9518	−1.55168	−0.775841	−	0.630928i	$-0.782674\pi$
$877$	−45.9518	−1.55168	−0.775841	+	0.630928i	$0.782674\pi$
$878$	0	0
$879$	29.1321	0.982602
$880$	0	0
$881$	−21.0771	−0.710107	−0.355053	−	0.934846i	$-0.615537\pi$
$881$	−21.0771	−0.710107	−0.355053	+	0.934846i	$0.615537\pi$
$882$	0	0
$883$	−30.4789	−1.02570	−0.512848	−	0.858480i	$-0.671409\pi$
$883$	−30.4789	−1.02570	−0.512848	+	0.858480i	$0.671409\pi$
$884$	0	0
$885$	70.6483	2.37482
$886$	0	0
$887$	−37.5105	−1.25948	−0.629739	−	0.776807i	$-0.716838\pi$
$887$	−37.5105	−1.25948	−0.629739	+	0.776807i	$0.716838\pi$
$888$	0	0
$889$	−13.0252	−0.436853
$890$	0	0
$891$	−34.2888	−1.14872
$892$	0	0
$893$	1.26997	0.0424980
$894$	0	0
$895$	−21.4808	−0.718023
$896$	0	0
$897$	66.2858	2.21322
$898$	0	0
$899$	−4.25903	−0.142047
$900$	0	0
$901$	−4.09354	−0.136376
$902$	0	0
$903$	37.5028	1.24802
$904$	0	0
$905$	−71.8652	−2.38888
$906$	0	0
$907$	21.8787	0.726469	0.363235	−	0.931698i	$-0.381672\pi$
$907$	21.8787	0.726469	0.363235	+	0.931698i	$0.381672\pi$
$908$	0	0
$909$	3.44982	0.114423
$910$	0	0
$911$	5.75740	0.190751	0.0953755	−	0.995441i	$-0.469595\pi$
$911$	5.75740	0.190751	0.0953755	+	0.995441i	$0.469595\pi$
$912$	0	0
$913$	17.7346	0.586928
$914$	0	0
$915$	9.04761	0.299105
$916$	0	0
$917$	−42.4426	−1.40158
$918$	0	0
$919$	2.63033	0.0867666	0.0433833	−	0.999059i	$-0.486186\pi$
$919$	2.63033	0.0867666	0.0433833	+	0.999059i	$0.486186\pi$
$920$	0	0
$921$	−2.39587	−0.0789465
$922$	0	0
$923$	−2.79160	−0.0918865
$924$	0	0
$925$	48.0590	1.58017
$926$	0	0
$927$	−1.71123	−0.0562041
$928$	0	0
$929$	−11.8023	−0.387219	−0.193610	−	0.981079i	$-0.562020\pi$
$929$	−11.8023	−0.387219	−0.193610	+	0.981079i	$0.562020\pi$
$930$	0	0
$931$	0.385433	0.0126321
$932$	0	0
$933$	38.4819	1.25984
$934$	0	0
$935$	23.6293	0.772759
$936$	0	0
$937$	−32.6464	−1.06651	−0.533256	−	0.845954i	$-0.679032\pi$
$937$	−32.6464	−1.06651	−0.533256	+	0.845954i	$0.679032\pi$
$938$	0	0
$939$	−10.1013	−0.329643
$940$	0	0
$941$	54.2934	1.76991	0.884957	−	0.465673i	$-0.154188\pi$
$941$	54.2934	1.76991	0.884957	+	0.465673i	$0.154188\pi$
$942$	0	0
$943$	48.2088	1.56990
$944$	0	0
$945$	−48.5110	−1.57806
$946$	0	0
$947$	−37.1795	−1.20817	−0.604086	−	0.796919i	$-0.706462\pi$
$947$	−37.1795	−1.20817	−0.604086	+	0.796919i	$0.706462\pi$
$948$	0	0
$949$	−20.0023	−0.649301
$950$	0	0
$951$	30.3856	0.985320
$952$	0	0
$953$	2.80816	0.0909651	0.0454826	−	0.998965i	$-0.485517\pi$
$953$	2.80816	0.0909651	0.0454826	+	0.998965i	$0.485517\pi$
$954$	0	0
$955$	−90.6506	−2.93338
$956$	0	0
$957$	9.55741	0.308947
$958$	0	0
$959$	47.0345	1.51882
$960$	0	0
$961$	−21.5965	−0.696660
$962$	0	0
$963$	0.585598	0.0188706
$964$	0	0
$965$	19.5657	0.629843
$966$	0	0
$967$	−11.1523	−0.358634	−0.179317	−	0.983791i	$-0.557389\pi$
$967$	−11.1523	−0.358634	−0.179317	+	0.983791i	$0.557389\pi$
$968$	0	0
$969$	−0.490585	−0.0157599
$970$	0	0
$971$	−53.7152	−1.72380	−0.861901	−	0.507077i	$-0.830726\pi$
$971$	−53.7152	−1.72380	−0.861901	+	0.507077i	$0.830726\pi$
$972$	0	0
$973$	9.80795	0.314428
$974$	0	0
$975$	−108.456	−3.47337
$976$	0	0
$977$	−37.9210	−1.21320	−0.606599	−	0.795008i	$-0.707467\pi$
$977$	−37.9210	−1.21320	−0.606599	+	0.795008i	$0.707467\pi$
$978$	0	0
$979$	9.70324	0.310117
$980$	0	0
$981$	−3.16781	−0.101140
$982$	0	0
$983$	8.88319	0.283330	0.141665	−	0.989915i	$-0.454754\pi$
$983$	8.88319	0.283330	0.141665	+	0.989915i	$0.454754\pi$
$984$	0	0
$985$	−79.5317	−2.53409
$986$	0	0
$987$	23.4210	0.745499
$988$	0	0
$989$	66.6547	2.11949
$990$	0	0
$991$	−14.6530	−0.465469	−0.232735	−	0.972540i	$-0.574767\pi$
$991$	−14.6530	−0.465469	−0.232735	+	0.972540i	$0.574767\pi$
$992$	0	0
$993$	−15.9298	−0.505516
$994$	0	0
$995$	−6.53328	−0.207119
$996$	0	0
$997$	5.76977	0.182731	0.0913653	−	0.995817i	$-0.470877\pi$
$997$	5.76977	0.182731	0.0913653	+	0.995817i	$0.470877\pi$
$998$	0	0
$999$	23.4592	0.742215

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4016.2.a.k.1.6		17
4.3	odd	2		251.2.a.b.1.16	✓	17
12.11	even	2		2259.2.a.k.1.2		17
20.19	odd	2		6275.2.a.e.1.2		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.16	✓	17	4.3	odd	2
2259.2.a.k.1.2		17	12.11	even	2
4016.2.a.k.1.6		17	1.1	even	1	trivial
6275.2.a.e.1.2		17	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4016 = 2^{4} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4016.a (trivial)

\(f(q)\)	\(=\)	\(q-1.66988 q^{3} -3.99830 q^{5} +2.26241 q^{7} -0.211508 q^{9} +O(q^{10})\)	\(q-1.66988 q^{3} -3.99830 q^{5} +2.26241 q^{7} -0.211508 q^{9} +4.12088 q^{11} +5.91171 q^{13} +6.67667 q^{15} -1.43412 q^{17} -0.204854 q^{19} -3.77795 q^{21} -6.71463 q^{23} +10.9864 q^{25} +5.36283 q^{27} -1.38888 q^{29} +3.06652 q^{31} -6.88137 q^{33} -9.04579 q^{35} +4.37440 q^{37} -9.87183 q^{39} -7.17966 q^{41} -9.92677 q^{43} +0.845671 q^{45} -6.19940 q^{47} -1.88150 q^{49} +2.39480 q^{51} +2.85440 q^{53} -16.4765 q^{55} +0.342082 q^{57} +10.5814 q^{59} +1.35511 q^{61} -0.478517 q^{63} -23.6368 q^{65} +4.56380 q^{67} +11.2126 q^{69} -0.472215 q^{71} -3.38350 q^{73} -18.3460 q^{75} +9.32312 q^{77} +4.84515 q^{79} -8.32074 q^{81} +4.30359 q^{83} +5.73403 q^{85} +2.31926 q^{87} +2.35465 q^{89} +13.3747 q^{91} -5.12071 q^{93} +0.819069 q^{95} -6.51100 q^{97} -0.871598 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9}+O(q^{10}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9	\( 17 q + 3 q^{5} - 3 q^{7} + 25 q^{9} + q^{11} + 22 q^{13} + 8 q^{15} - q^{17} - 13 q^{19} + 25 q^{21} + 2 q^{23} + 32 q^{25} + 15 q^{27} + 28 q^{29} - 12 q^{31} - 16 q^{33} + 15 q^{35} + 27 q^{37} - 13 q^{39} - q^{41} - 9 q^{43} - 7 q^{45} + 20 q^{47} + 32 q^{49} + 2 q^{51} + q^{53} + 11 q^{55} - 24 q^{57} + 20 q^{59} + 59 q^{61} + 41 q^{63} - 14 q^{65} - 15 q^{67} + 38 q^{69} + 26 q^{71} + 8 q^{73} + 20 q^{75} - 33 q^{79} + 29 q^{81} + 67 q^{85} + 11 q^{87} + 11 q^{89} + 2 q^{91} + 28 q^{93} + 8 q^{95} - 10 q^{97} + 36 q^{99}+O(q^{100}) \) 17 * q + 3 * q^5 - 3 * q^7 + 25 * q^9 + q^11 + 22 * q^13 + 8 * q^15 - q^17 - 13 * q^19 + 25 * q^21 + 2 * q^23 + 32 * q^25 + 15 * q^27 + 28 * q^29 - 12 * q^31 - 16 * q^33 + 15 * q^35 + 27 * q^37 - 13 * q^39 - q^41 - 9 * q^43 - 7 * q^45 + 20 * q^47 + 32 * q^49 + 2 * q^51 + q^53 + 11 * q^55 - 24 * q^57 + 20 * q^59 + 59 * q^61 + 41 * q^63 - 14 * q^65 - 15 * q^67 + 38 * q^69 + 26 * q^71 + 8 * q^73 + 20 * q^75 - 33 * q^79 + 29 * q^81 + 67 * q^85 + 11 * q^87 + 11 * q^89 + 2 * q^91 + 28 * q^93 + 8 * q^95 - 10 * q^97 + 36 * q^99

Introduction

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