LMFDB - Embedded newform 8016.2.a.y.1.1

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:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 23x^{6} - 3x^{5} + 163x^{4} + 13x^{3} - 418x^{2} + 4x + 269 $ x^8 - 23x^6 - 3x^5 + 163x^4 + 13x^3 - 418x^2 + 4x + 269
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 4008)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.24252$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.24252 q^{5} +3.21024 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.24252 q^{5} +3.21024 q^{7} +1.00000 q^{9} -2.82128 q^{11} -5.85040 q^{13} -3.24252 q^{15} +1.86063 q^{17} -5.37894 q^{19} +3.21024 q^{21} -4.21024 q^{23} +5.51393 q^{25} +1.00000 q^{27} +3.33063 q^{29} -1.27146 q^{31} -2.82128 q^{33} -10.4092 q^{35} -2.06358 q^{37} -5.85040 q^{39} +2.10689 q^{41} -6.96695 q^{43} -3.24252 q^{45} +6.68686 q^{47} +3.30561 q^{49} +1.86063 q^{51} +9.34827 q^{53} +9.14806 q^{55} -5.37894 q^{57} +11.2378 q^{59} +7.78735 q^{61} +3.21024 q^{63} +18.9700 q^{65} -3.51420 q^{67} -4.21024 q^{69} +11.6222 q^{71} -5.43704 q^{73} +5.51393 q^{75} -9.05698 q^{77} +4.16325 q^{79} +1.00000 q^{81} +1.27361 q^{83} -6.03313 q^{85} +3.33063 q^{87} -5.91690 q^{89} -18.7811 q^{91} -1.27146 q^{93} +17.4413 q^{95} +4.01613 q^{97} -2.82128 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + q^{7} + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + q^7 + 8 * q^9	$ 8 q + 8 q^{3} + q^{7} + 8 q^{9} + 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} - 9 q^{23} + 6 q^{25} + 8 q^{27} + 17 q^{29} + 23 q^{31} + 3 q^{33} + 15 q^{35} + 8 q^{37} - 8 q^{39} - 8 q^{41} + 2 q^{43} + 34 q^{47} + 5 q^{49} - 7 q^{51} + 12 q^{53} + 7 q^{55} + 16 q^{59} - 2 q^{61} + q^{63} - 14 q^{65} - 21 q^{67} - 9 q^{69} + 29 q^{71} - 38 q^{73} + 6 q^{75} + 20 q^{77} + 12 q^{79} + 8 q^{81} + 32 q^{83} + 23 q^{85} + 17 q^{87} + 11 q^{89} + 5 q^{91} + 23 q^{93} + 67 q^{95} + 8 q^{97} + 3 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + q^7 + 8 * q^9 + 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 - 9 * q^23 + 6 * q^25 + 8 * q^27 + 17 * q^29 + 23 * q^31 + 3 * q^33 + 15 * q^35 + 8 * q^37 - 8 * q^39 - 8 * q^41 + 2 * q^43 + 34 * q^47 + 5 * q^49 - 7 * q^51 + 12 * q^53 + 7 * q^55 + 16 * q^59 - 2 * q^61 + q^63 - 14 * q^65 - 21 * q^67 - 9 * q^69 + 29 * q^71 - 38 * q^73 + 6 * q^75 + 20 * q^77 + 12 * q^79 + 8 * q^81 + 32 * q^83 + 23 * q^85 + 17 * q^87 + 11 * q^89 + 5 * q^91 + 23 * q^93 + 67 * q^95 + 8 * 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
$5$	−3.24252	−1.45010	−0.725049	−	0.688697i	$-0.758183\pi$
$5$	−3.24252	−1.45010	−0.725049	+	0.688697i	$0.758183\pi$
$6$	0	0
$7$	3.21024	1.21335	0.606677	−	0.794948i	$-0.292502\pi$
$7$	3.21024	1.21335	0.606677	+	0.794948i	$0.292502\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.82128	−0.850649	−0.425324	−	0.905041i	$-0.639840\pi$
$11$	−2.82128	−0.850649	−0.425324	+	0.905041i	$0.639840\pi$
$12$	0	0
$13$	−5.85040	−1.62261	−0.811304	−	0.584625i	$-0.801242\pi$
$13$	−5.85040	−1.62261	−0.811304	+	0.584625i	$0.801242\pi$
$14$	0	0
$15$	−3.24252	−0.837215
$16$	0	0
$17$	1.86063	0.451269	0.225635	−	0.974212i	$-0.427554\pi$
$17$	1.86063	0.451269	0.225635	+	0.974212i	$0.427554\pi$
$18$	0	0
$19$	−5.37894	−1.23401	−0.617007	−	0.786958i	$-0.711655\pi$
$19$	−5.37894	−1.23401	−0.617007	+	0.786958i	$0.711655\pi$
$20$	0	0
$21$	3.21024	0.700531
$22$	0	0
$23$	−4.21024	−0.877895	−0.438947	−	0.898513i	$-0.644649\pi$
$23$	−4.21024	−0.877895	−0.438947	+	0.898513i	$0.644649\pi$
$24$	0	0
$25$	5.51393	1.10279
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	3.33063	0.618483	0.309241	−	0.950984i	$-0.399925\pi$
$29$	3.33063	0.618483	0.309241	+	0.950984i	$0.399925\pi$
$30$	0	0
$31$	−1.27146	−0.228361	−0.114180	−	0.993460i	$-0.536424\pi$
$31$	−1.27146	−0.228361	−0.114180	+	0.993460i	$0.536424\pi$
$32$	0	0
$33$	−2.82128	−0.491122
$34$	0	0
$35$	−10.4092	−1.75948
$36$	0	0
$37$	−2.06358	−0.339251	−0.169625	−	0.985509i	$-0.554256\pi$
$37$	−2.06358	−0.339251	−0.169625	+	0.985509i	$0.554256\pi$
$38$	0	0
$39$	−5.85040	−0.936813
$40$	0	0
$41$	2.10689	0.329041	0.164521	−	0.986374i	$-0.447392\pi$
$41$	2.10689	0.329041	0.164521	+	0.986374i	$0.447392\pi$
$42$	0	0
$43$	−6.96695	−1.06245	−0.531225	−	0.847231i	$-0.678268\pi$
$43$	−6.96695	−1.06245	−0.531225	+	0.847231i	$0.678268\pi$
$44$	0	0
$45$	−3.24252	−0.483366
$46$	0	0
$47$	6.68686	0.975380	0.487690	−	0.873017i	$-0.337840\pi$
$47$	6.68686	0.975380	0.487690	+	0.873017i	$0.337840\pi$
$48$	0	0
$49$	3.30561	0.472230
$50$	0	0
$51$	1.86063	0.260540
$52$	0	0
$53$	9.34827	1.28408	0.642042	−	0.766670i	$-0.278088\pi$
$53$	9.34827	1.28408	0.642042	+	0.766670i	$0.278088\pi$
$54$	0	0
$55$	9.14806	1.23352
$56$	0	0
$57$	−5.37894	−0.712458
$58$	0	0
$59$	11.2378	1.46304	0.731518	−	0.681822i	$-0.238812\pi$
$59$	11.2378	1.46304	0.731518	+	0.681822i	$0.238812\pi$
$60$	0	0
$61$	7.78735	0.997068	0.498534	−	0.866870i	$-0.333872\pi$
$61$	7.78735	0.997068	0.498534	+	0.866870i	$0.333872\pi$
$62$	0	0
$63$	3.21024	0.404452
$64$	0	0
$65$	18.9700	2.35294
$66$	0	0
$67$	−3.51420	−0.429327	−0.214664	−	0.976688i	$-0.568866\pi$
$67$	−3.51420	−0.429327	−0.214664	+	0.976688i	$0.568866\pi$
$68$	0	0
$69$	−4.21024	−0.506853
$70$	0	0
$71$	11.6222	1.37930	0.689651	−	0.724142i	$-0.257764\pi$
$71$	11.6222	1.37930	0.689651	+	0.724142i	$0.257764\pi$
$72$	0	0
$73$	−5.43704	−0.636357	−0.318179	−	0.948031i	$-0.603071\pi$
$73$	−5.43704	−0.636357	−0.318179	+	0.948031i	$0.603071\pi$
$74$	0	0
$75$	5.51393	0.636694
$76$	0	0
$77$	−9.05698	−1.03214
$78$	0	0
$79$	4.16325	0.468402	0.234201	−	0.972188i	$-0.424753\pi$
$79$	4.16325	0.468402	0.234201	+	0.972188i	$0.424753\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	1.27361	0.139797	0.0698987	−	0.997554i	$-0.477732\pi$
$83$	1.27361	0.139797	0.0698987	+	0.997554i	$0.477732\pi$
$84$	0	0
$85$	−6.03313	−0.654385
$86$	0	0
$87$	3.33063	0.357081
$88$	0	0
$89$	−5.91690	−0.627190	−0.313595	−	0.949557i	$-0.601533\pi$
$89$	−5.91690	−0.627190	−0.313595	+	0.949557i	$0.601533\pi$
$90$	0	0
$91$	−18.7811	−1.96880
$92$	0	0
$93$	−1.27146	−0.131844
$94$	0	0
$95$	17.4413	1.78944
$96$	0	0
$97$	4.01613	0.407776	0.203888	−	0.978994i	$-0.434642\pi$
$97$	4.01613	0.407776	0.203888	+	0.978994i	$0.434642\pi$
$98$	0	0
$99$	−2.82128	−0.283550
$100$	0	0
$101$	9.09153	0.904641	0.452321	−	0.891855i	$-0.350596\pi$
$101$	9.09153	0.904641	0.452321	+	0.891855i	$0.350596\pi$
$102$	0	0
$103$	15.5592	1.53309	0.766545	−	0.642190i	$-0.221974\pi$
$103$	15.5592	1.53309	0.766545	+	0.642190i	$0.221974\pi$
$104$	0	0
$105$	−10.4092	−1.01584
$106$	0	0
$107$	−4.95284	−0.478809	−0.239404	−	0.970920i	$-0.576952\pi$
$107$	−4.95284	−0.478809	−0.239404	+	0.970920i	$0.576952\pi$
$108$	0	0
$109$	9.59455	0.918991	0.459495	−	0.888180i	$-0.348030\pi$
$109$	9.59455	0.918991	0.459495	+	0.888180i	$0.348030\pi$
$110$	0	0
$111$	−2.06358	−0.195866
$112$	0	0
$113$	−14.1378	−1.32998	−0.664988	−	0.746854i	$-0.731563\pi$
$113$	−14.1378	−1.32998	−0.664988	+	0.746854i	$0.731563\pi$
$114$	0	0
$115$	13.6518	1.27303
$116$	0	0
$117$	−5.85040	−0.540869
$118$	0	0
$119$	5.97306	0.547550
$120$	0	0
$121$	−3.04036	−0.276397
$122$	0	0
$123$	2.10689	0.189972
$124$	0	0
$125$	−1.66643	−0.149050
$126$	0	0
$127$	−2.06152	−0.182930	−0.0914651	−	0.995808i	$-0.529155\pi$
$127$	−2.06152	−0.182930	−0.0914651	+	0.995808i	$0.529155\pi$
$128$	0	0
$129$	−6.96695	−0.613406
$130$	0	0
$131$	14.0106	1.22411	0.612055	−	0.790815i	$-0.290343\pi$
$131$	14.0106	1.22411	0.612055	+	0.790815i	$0.290343\pi$
$132$	0	0
$133$	−17.2677	−1.49730
$134$	0	0
$135$	−3.24252	−0.279072
$136$	0	0
$137$	9.83399	0.840174	0.420087	−	0.907484i	$-0.362000\pi$
$137$	9.83399	0.840174	0.420087	+	0.907484i	$0.362000\pi$
$138$	0	0
$139$	−4.61893	−0.391773	−0.195886	−	0.980627i	$-0.562758\pi$
$139$	−4.61893	−0.391773	−0.195886	+	0.980627i	$0.562758\pi$
$140$	0	0
$141$	6.68686	0.563136
$142$	0	0
$143$	16.5056	1.38027
$144$	0	0
$145$	−10.7996	−0.896861
$146$	0	0
$147$	3.30561	0.272642
$148$	0	0
$149$	11.6341	0.953105	0.476553	−	0.879146i	$-0.341886\pi$
$149$	11.6341	0.953105	0.476553	+	0.879146i	$0.341886\pi$
$150$	0	0
$151$	−13.1949	−1.07379	−0.536893	−	0.843651i	$-0.680402\pi$
$151$	−13.1949	−1.07379	−0.536893	+	0.843651i	$0.680402\pi$
$152$	0	0
$153$	1.86063	0.150423
$154$	0	0
$155$	4.12273	0.331146
$156$	0	0
$157$	−5.22817	−0.417254	−0.208627	−	0.977995i	$-0.566899\pi$
$157$	−5.22817	−0.417254	−0.208627	+	0.977995i	$0.566899\pi$
$158$	0	0
$159$	9.34827	0.741366
$160$	0	0
$161$	−13.5158	−1.06520
$162$	0	0
$163$	−0.261930	−0.0205160	−0.0102580	−	0.999947i	$-0.503265\pi$
$163$	−0.261930	−0.0205160	−0.0102580	+	0.999947i	$0.503265\pi$
$164$	0	0
$165$	9.14806	0.712176
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	21.2271	1.63286
$170$	0	0
$171$	−5.37894	−0.411338
$172$	0	0
$173$	−19.5287	−1.48474	−0.742370	−	0.669990i	$-0.766298\pi$
$173$	−19.5287	−1.48474	−0.742370	+	0.669990i	$0.766298\pi$
$174$	0	0
$175$	17.7010	1.33807
$176$	0	0
$177$	11.2378	0.844685
$178$	0	0
$179$	19.0968	1.42737	0.713683	−	0.700469i	$-0.247026\pi$
$179$	19.0968	1.42737	0.713683	+	0.700469i	$0.247026\pi$
$180$	0	0
$181$	−8.26626	−0.614427	−0.307213	−	0.951641i	$-0.599396\pi$
$181$	−8.26626	−0.614427	−0.307213	+	0.951641i	$0.599396\pi$
$182$	0	0
$183$	7.78735	0.575658
$184$	0	0
$185$	6.69120	0.491947
$186$	0	0
$187$	−5.24936	−0.383872
$188$	0	0
$189$	3.21024	0.233510
$190$	0	0
$191$	21.0760	1.52501	0.762504	−	0.646984i	$-0.223970\pi$
$191$	21.0760	1.52501	0.762504	+	0.646984i	$0.223970\pi$
$192$	0	0
$193$	22.3043	1.60550	0.802749	−	0.596318i	$-0.203370\pi$
$193$	22.3043	1.60550	0.802749	+	0.596318i	$0.203370\pi$
$194$	0	0
$195$	18.9700	1.35847
$196$	0	0
$197$	−10.7503	−0.765925	−0.382962	−	0.923764i	$-0.625096\pi$
$197$	−10.7503	−0.765925	−0.382962	+	0.923764i	$0.625096\pi$
$198$	0	0
$199$	12.7649	0.904877	0.452439	−	0.891796i	$-0.350554\pi$
$199$	12.7649	0.904877	0.452439	+	0.891796i	$0.350554\pi$
$200$	0	0
$201$	−3.51420	−0.247872
$202$	0	0
$203$	10.6921	0.750439
$204$	0	0
$205$	−6.83164	−0.477143
$206$	0	0
$207$	−4.21024	−0.292632
$208$	0	0
$209$	15.1755	1.04971
$210$	0	0
$211$	−16.9680	−1.16812	−0.584061	−	0.811710i	$-0.698537\pi$
$211$	−16.9680	−1.16812	−0.584061	+	0.811710i	$0.698537\pi$
$212$	0	0
$213$	11.6222	0.796340
$214$	0	0
$215$	22.5905	1.54066
$216$	0	0
$217$	−4.08168	−0.277083
$218$	0	0
$219$	−5.43704	−0.367401
$220$	0	0
$221$	−10.8854	−0.732233
$222$	0	0
$223$	14.7467	0.987514	0.493757	−	0.869600i	$-0.335623\pi$
$223$	14.7467	0.987514	0.493757	+	0.869600i	$0.335623\pi$
$224$	0	0
$225$	5.51393	0.367595
$226$	0	0
$227$	−0.120587	−0.00800367	−0.00400183	−	0.999992i	$-0.501274\pi$
$227$	−0.120587	−0.00800367	−0.00400183	+	0.999992i	$0.501274\pi$
$228$	0	0
$229$	1.33643	0.0883139	0.0441570	−	0.999025i	$-0.485940\pi$
$229$	1.33643	0.0883139	0.0441570	+	0.999025i	$0.485940\pi$
$230$	0	0
$231$	−9.05698	−0.595906
$232$	0	0
$233$	−8.77423	−0.574819	−0.287409	−	0.957808i	$-0.592794\pi$
$233$	−8.77423	−0.574819	−0.287409	+	0.957808i	$0.592794\pi$
$234$	0	0
$235$	−21.6823	−1.41440
$236$	0	0
$237$	4.16325	0.270432
$238$	0	0
$239$	5.99261	0.387630	0.193815	−	0.981038i	$-0.437914\pi$
$239$	5.99261	0.387630	0.193815	+	0.981038i	$0.437914\pi$
$240$	0	0
$241$	20.6690	1.33141	0.665704	−	0.746216i	$-0.268132\pi$
$241$	20.6690	1.33141	0.665704	+	0.746216i	$0.268132\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.7185	−0.684780
$246$	0	0
$247$	31.4689	2.00232
$248$	0	0
$249$	1.27361	0.0807121
$250$	0	0
$251$	−9.72679	−0.613950	−0.306975	−	0.951718i	$-0.599317\pi$
$251$	−9.72679	−0.613950	−0.306975	+	0.951718i	$0.599317\pi$
$252$	0	0
$253$	11.8783	0.746780
$254$	0	0
$255$	−6.03313	−0.377809
$256$	0	0
$257$	−6.72200	−0.419307	−0.209654	−	0.977776i	$-0.567234\pi$
$257$	−6.72200	−0.419307	−0.209654	+	0.977776i	$0.567234\pi$
$258$	0	0
$259$	−6.62458	−0.411631
$260$	0	0
$261$	3.33063	0.206161
$262$	0	0
$263$	20.9835	1.29390	0.646948	−	0.762534i	$-0.276045\pi$
$263$	20.9835	1.29390	0.646948	+	0.762534i	$0.276045\pi$
$264$	0	0
$265$	−30.3119	−1.86205
$266$	0	0
$267$	−5.91690	−0.362109
$268$	0	0
$269$	−2.08333	−0.127023	−0.0635115	−	0.997981i	$-0.520230\pi$
$269$	−2.08333	−0.127023	−0.0635115	+	0.997981i	$0.520230\pi$
$270$	0	0
$271$	−10.9907	−0.667639	−0.333819	−	0.942637i	$-0.608338\pi$
$271$	−10.9907	−0.667639	−0.333819	+	0.942637i	$0.608338\pi$
$272$	0	0
$273$	−18.7811	−1.13669
$274$	0	0
$275$	−15.5564	−0.938084
$276$	0	0
$277$	−25.5833	−1.53715	−0.768577	−	0.639758i	$-0.779035\pi$
$277$	−25.5833	−1.53715	−0.768577	+	0.639758i	$0.779035\pi$
$278$	0	0
$279$	−1.27146	−0.0761203
$280$	0	0
$281$	−11.2733	−0.672507	−0.336253	−	0.941772i	$-0.609160\pi$
$281$	−11.2733	−0.672507	−0.336253	+	0.941772i	$0.609160\pi$
$282$	0	0
$283$	−23.5911	−1.40235	−0.701174	−	0.712991i	$-0.747340\pi$
$283$	−23.5911	−1.40235	−0.701174	+	0.712991i	$0.747340\pi$
$284$	0	0
$285$	17.4413	1.03313
$286$	0	0
$287$	6.76362	0.399244
$288$	0	0
$289$	−13.5381	−0.796356
$290$	0	0
$291$	4.01613	0.235430
$292$	0	0
$293$	−9.21613	−0.538412	−0.269206	−	0.963083i	$-0.586761\pi$
$293$	−9.21613	−0.538412	−0.269206	+	0.963083i	$0.586761\pi$
$294$	0	0
$295$	−36.4388	−2.12155
$296$	0	0
$297$	−2.82128	−0.163707
$298$	0	0
$299$	24.6315	1.42448
$300$	0	0
$301$	−22.3656	−1.28913
$302$	0	0
$303$	9.09153	0.522295
$304$	0	0
$305$	−25.2506	−1.44585
$306$	0	0
$307$	13.3545	0.762183	0.381091	−	0.924537i	$-0.375548\pi$
$307$	13.3545	0.762183	0.381091	+	0.924537i	$0.375548\pi$
$308$	0	0
$309$	15.5592	0.885130
$310$	0	0
$311$	18.9770	1.07609	0.538043	−	0.842918i	$-0.319164\pi$
$311$	18.9770	1.07609	0.538043	+	0.842918i	$0.319164\pi$
$312$	0	0
$313$	0.865518	0.0489220	0.0244610	−	0.999701i	$-0.492213\pi$
$313$	0.865518	0.0489220	0.0244610	+	0.999701i	$0.492213\pi$
$314$	0	0
$315$	−10.4092	−0.586495
$316$	0	0
$317$	6.57367	0.369214	0.184607	−	0.982812i	$-0.440899\pi$
$317$	6.57367	0.369214	0.184607	+	0.982812i	$0.440899\pi$
$318$	0	0
$319$	−9.39666	−0.526112
$320$	0	0
$321$	−4.95284	−0.276440
$322$	0	0
$323$	−10.0082	−0.556872
$324$	0	0
$325$	−32.2587	−1.78939
$326$	0	0
$327$	9.59455	0.530580
$328$	0	0
$329$	21.4664	1.18348
$330$	0	0
$331$	−5.69256	−0.312892	−0.156446	−	0.987687i	$-0.550004\pi$
$331$	−5.69256	−0.312892	−0.156446	+	0.987687i	$0.550004\pi$
$332$	0	0
$333$	−2.06358	−0.113084
$334$	0	0
$335$	11.3948	0.622567
$336$	0	0
$337$	−10.1210	−0.551323	−0.275662	−	0.961255i	$-0.588897\pi$
$337$	−10.1210	−0.551323	−0.275662	+	0.961255i	$0.588897\pi$
$338$	0	0
$339$	−14.1378	−0.767862
$340$	0	0
$341$	3.58715	0.194255
$342$	0	0
$343$	−11.8599	−0.640372
$344$	0	0
$345$	13.6518	0.734987
$346$	0	0
$347$	35.4498	1.90304	0.951521	−	0.307583i	$-0.0995202\pi$
$347$	35.4498	1.90304	0.951521	+	0.307583i	$0.0995202\pi$
$348$	0	0
$349$	26.6545	1.42678	0.713392	−	0.700765i	$-0.247158\pi$
$349$	26.6545	1.42678	0.713392	+	0.700765i	$0.247158\pi$
$350$	0	0
$351$	−5.85040	−0.312271
$352$	0	0
$353$	7.94068	0.422640	0.211320	−	0.977417i	$-0.432224\pi$
$353$	7.94068	0.422640	0.211320	+	0.977417i	$0.432224\pi$
$354$	0	0
$355$	−37.6852	−2.00012
$356$	0	0
$357$	5.97306	0.316128
$358$	0	0
$359$	−15.7685	−0.832228	−0.416114	−	0.909312i	$-0.636608\pi$
$359$	−15.7685	−0.832228	−0.416114	+	0.909312i	$0.636608\pi$
$360$	0	0
$361$	9.93298	0.522789
$362$	0	0
$363$	−3.04036	−0.159578
$364$	0	0
$365$	17.6297	0.922781
$366$	0	0
$367$	−2.32595	−0.121414	−0.0607068	−	0.998156i	$-0.519335\pi$
$367$	−2.32595	−0.121414	−0.0607068	+	0.998156i	$0.519335\pi$
$368$	0	0
$369$	2.10689	0.109680
$370$	0	0
$371$	30.0101	1.55805
$372$	0	0
$373$	4.77773	0.247382	0.123691	−	0.992321i	$-0.460527\pi$
$373$	4.77773	0.247382	0.123691	+	0.992321i	$0.460527\pi$
$374$	0	0
$375$	−1.66643	−0.0860539
$376$	0	0
$377$	−19.4855	−1.00356
$378$	0	0
$379$	14.7452	0.757410	0.378705	−	0.925517i	$-0.376369\pi$
$379$	14.7452	0.757410	0.378705	+	0.925517i	$0.376369\pi$
$380$	0	0
$381$	−2.06152	−0.105615
$382$	0	0
$383$	−7.62816	−0.389781	−0.194890	−	0.980825i	$-0.562435\pi$
$383$	−7.62816	−0.389781	−0.194890	+	0.980825i	$0.562435\pi$
$384$	0	0
$385$	29.3674	1.49670
$386$	0	0
$387$	−6.96695	−0.354150
$388$	0	0
$389$	29.0716	1.47399	0.736993	−	0.675900i	$-0.236245\pi$
$389$	29.0716	1.47399	0.736993	+	0.675900i	$0.236245\pi$
$390$	0	0
$391$	−7.83369	−0.396167
$392$	0	0
$393$	14.0106	0.706740
$394$	0	0
$395$	−13.4994	−0.679229
$396$	0	0
$397$	−11.2476	−0.564499	−0.282249	−	0.959341i	$-0.591081\pi$
$397$	−11.2476	−0.564499	−0.282249	+	0.959341i	$0.591081\pi$
$398$	0	0
$399$	−17.2677	−0.864464
$400$	0	0
$401$	8.80206	0.439554	0.219777	−	0.975550i	$-0.429467\pi$
$401$	8.80206	0.439554	0.219777	+	0.975550i	$0.429467\pi$
$402$	0	0
$403$	7.43854	0.370540
$404$	0	0
$405$	−3.24252	−0.161122
$406$	0	0
$407$	5.82195	0.288583
$408$	0	0
$409$	35.7694	1.76868	0.884340	−	0.466843i	$-0.154609\pi$
$409$	35.7694	1.76868	0.884340	+	0.466843i	$0.154609\pi$
$410$	0	0
$411$	9.83399	0.485075
$412$	0	0
$413$	36.0760	1.77518
$414$	0	0
$415$	−4.12972	−0.202720
$416$	0	0
$417$	−4.61893	−0.226190
$418$	0	0
$419$	24.5130	1.19754	0.598770	−	0.800921i	$-0.295656\pi$
$419$	24.5130	1.19754	0.598770	+	0.800921i	$0.295656\pi$
$420$	0	0
$421$	4.85223	0.236483	0.118242	−	0.992985i	$-0.462274\pi$
$421$	4.85223	0.236483	0.118242	+	0.992985i	$0.462274\pi$
$422$	0	0
$423$	6.68686	0.325127
$424$	0	0
$425$	10.2594	0.497653
$426$	0	0
$427$	24.9992	1.20980
$428$	0	0
$429$	16.5056	0.796899
$430$	0	0
$431$	−16.5070	−0.795116	−0.397558	−	0.917577i	$-0.630142\pi$
$431$	−16.5070	−0.795116	−0.397558	+	0.917577i	$0.630142\pi$
$432$	0	0
$433$	−20.5559	−0.987851	−0.493926	−	0.869504i	$-0.664439\pi$
$433$	−20.5559	−0.987851	−0.493926	+	0.869504i	$0.664439\pi$
$434$	0	0
$435$	−10.7996	−0.517803
$436$	0	0
$437$	22.6466	1.08333
$438$	0	0
$439$	−3.71337	−0.177229	−0.0886147	−	0.996066i	$-0.528244\pi$
$439$	−3.71337	−0.177229	−0.0886147	+	0.996066i	$0.528244\pi$
$440$	0	0
$441$	3.30561	0.157410
$442$	0	0
$443$	4.80205	0.228153	0.114076	−	0.993472i	$-0.463609\pi$
$443$	4.80205	0.228153	0.114076	+	0.993472i	$0.463609\pi$
$444$	0	0
$445$	19.1857	0.909488
$446$	0	0
$447$	11.6341	0.550276
$448$	0	0
$449$	11.6862	0.551507	0.275753	−	0.961228i	$-0.411073\pi$
$449$	11.6862	0.551507	0.275753	+	0.961228i	$0.411073\pi$
$450$	0	0
$451$	−5.94414	−0.279899
$452$	0	0
$453$	−13.1949	−0.619950
$454$	0	0
$455$	60.8982	2.85495
$456$	0	0
$457$	6.66756	0.311895	0.155948	−	0.987765i	$-0.450157\pi$
$457$	6.66756	0.311895	0.155948	+	0.987765i	$0.450157\pi$
$458$	0	0
$459$	1.86063	0.0868468
$460$	0	0
$461$	−5.11682	−0.238314	−0.119157	−	0.992875i	$-0.538019\pi$
$461$	−5.11682	−0.238314	−0.119157	+	0.992875i	$0.538019\pi$
$462$	0	0
$463$	25.7320	1.19587	0.597934	−	0.801546i	$-0.295989\pi$
$463$	25.7320	1.19587	0.597934	+	0.801546i	$0.295989\pi$
$464$	0	0
$465$	4.12273	0.191187
$466$	0	0
$467$	10.1888	0.471482	0.235741	−	0.971816i	$-0.424248\pi$
$467$	10.1888	0.471482	0.235741	+	0.971816i	$0.424248\pi$
$468$	0	0
$469$	−11.2814	−0.520927
$470$	0	0
$471$	−5.22817	−0.240901
$472$	0	0
$473$	19.6557	0.903772
$474$	0	0
$475$	−29.6591	−1.36085
$476$	0	0
$477$	9.34827	0.428028
$478$	0	0
$479$	−22.1075	−1.01012	−0.505059	−	0.863085i	$-0.668529\pi$
$479$	−22.1075	−1.01012	−0.505059	+	0.863085i	$0.668529\pi$
$480$	0	0
$481$	12.0728	0.550471
$482$	0	0
$483$	−13.5158	−0.614992
$484$	0	0
$485$	−13.0224	−0.591316
$486$	0	0
$487$	31.9470	1.44766	0.723829	−	0.689980i	$-0.242381\pi$
$487$	31.9470	1.44766	0.723829	+	0.689980i	$0.242381\pi$
$488$	0	0
$489$	−0.261930	−0.0118449
$490$	0	0
$491$	−34.9820	−1.57871	−0.789357	−	0.613934i	$-0.789586\pi$
$491$	−34.9820	−1.57871	−0.789357	+	0.613934i	$0.789586\pi$
$492$	0	0
$493$	6.19708	0.279102
$494$	0	0
$495$	9.14806	0.411175
$496$	0	0
$497$	37.3100	1.67358
$498$	0	0
$499$	26.2314	1.17428	0.587140	−	0.809486i	$-0.300254\pi$
$499$	26.2314	1.17428	0.587140	+	0.809486i	$0.300254\pi$
$500$	0	0
$501$	−1.00000	−0.0446767
$502$	0	0
$503$	24.3106	1.08396	0.541979	−	0.840392i	$-0.317675\pi$
$503$	24.3106	1.08396	0.541979	+	0.840392i	$0.317675\pi$
$504$	0	0
$505$	−29.4795	−1.31182
$506$	0	0
$507$	21.2271	0.942730
$508$	0	0
$509$	−32.5371	−1.44218	−0.721090	−	0.692841i	$-0.756359\pi$
$509$	−32.5371	−1.44218	−0.721090	+	0.692841i	$0.756359\pi$
$510$	0	0
$511$	−17.4542	−0.772127
$512$	0	0
$513$	−5.37894	−0.237486
$514$	0	0
$515$	−50.4509	−2.22313
$516$	0	0
$517$	−18.8655	−0.829705
$518$	0	0
$519$	−19.5287	−0.857215
$520$	0	0
$521$	32.9927	1.44543	0.722717	−	0.691144i	$-0.242893\pi$
$521$	32.9927	1.44543	0.722717	+	0.691144i	$0.242893\pi$
$522$	0	0
$523$	19.4021	0.848393	0.424197	−	0.905570i	$-0.360557\pi$
$523$	19.4021	0.848393	0.424197	+	0.905570i	$0.360557\pi$
$524$	0	0
$525$	17.7010	0.772536
$526$	0	0
$527$	−2.36572	−0.103052
$528$	0	0
$529$	−5.27392	−0.229301
$530$	0	0
$531$	11.2378	0.487679
$532$	0	0
$533$	−12.3262	−0.533905
$534$	0	0
$535$	16.0597	0.694320
$536$	0	0
$537$	19.0968	0.824090
$538$	0	0
$539$	−9.32606	−0.401702
$540$	0	0
$541$	20.4863	0.880773	0.440386	−	0.897808i	$-0.354841\pi$
$541$	20.4863	0.880773	0.440386	+	0.897808i	$0.354841\pi$
$542$	0	0
$543$	−8.26626	−0.354739
$544$	0	0
$545$	−31.1105	−1.33263
$546$	0	0
$547$	2.16456	0.0925498	0.0462749	−	0.998929i	$-0.485265\pi$
$547$	2.16456	0.0925498	0.0462749	+	0.998929i	$0.485265\pi$
$548$	0	0
$549$	7.78735	0.332356
$550$	0	0
$551$	−17.9153	−0.763216
$552$	0	0
$553$	13.3650	0.568338
$554$	0	0
$555$	6.69120	0.284026
$556$	0	0
$557$	−1.20589	−0.0510952	−0.0255476	−	0.999674i	$-0.508133\pi$
$557$	−1.20589	−0.0510952	−0.0255476	+	0.999674i	$0.508133\pi$
$558$	0	0
$559$	40.7594	1.72394
$560$	0	0
$561$	−5.24936	−0.221628
$562$	0	0
$563$	1.05092	0.0442909	0.0221455	−	0.999755i	$-0.492950\pi$
$563$	1.05092	0.0442909	0.0221455	+	0.999755i	$0.492950\pi$
$564$	0	0
$565$	45.8422	1.92860
$566$	0	0
$567$	3.21024	0.134817
$568$	0	0
$569$	4.65108	0.194983	0.0974917	−	0.995236i	$-0.468918\pi$
$569$	4.65108	0.194983	0.0974917	+	0.995236i	$0.468918\pi$
$570$	0	0
$571$	23.8655	0.998739	0.499370	−	0.866389i	$-0.333565\pi$
$571$	23.8655	0.998739	0.499370	+	0.866389i	$0.333565\pi$
$572$	0	0
$573$	21.0760	0.880463
$574$	0	0
$575$	−23.2149	−0.968130
$576$	0	0
$577$	−44.4676	−1.85121	−0.925605	−	0.378490i	$-0.876443\pi$
$577$	−44.4676	−1.85121	−0.925605	+	0.378490i	$0.876443\pi$
$578$	0	0
$579$	22.3043	0.926934
$580$	0	0
$581$	4.08860	0.169624
$582$	0	0
$583$	−26.3741	−1.09230
$584$	0	0
$585$	18.9700	0.784314
$586$	0	0
$587$	48.2366	1.99094	0.995469	−	0.0950882i	$-0.0303133\pi$
$587$	48.2366	1.99094	0.995469	+	0.0950882i	$0.0303133\pi$
$588$	0	0
$589$	6.83910	0.281800
$590$	0	0
$591$	−10.7503	−0.442207
$592$	0	0
$593$	4.81080	0.197556	0.0987779	−	0.995110i	$-0.468507\pi$
$593$	4.81080	0.197556	0.0987779	+	0.995110i	$0.468507\pi$
$594$	0	0
$595$	−19.3678	−0.794001
$596$	0	0
$597$	12.7649	0.522431
$598$	0	0
$599$	23.7293	0.969552	0.484776	−	0.874638i	$-0.338901\pi$
$599$	23.7293	0.969552	0.484776	+	0.874638i	$0.338901\pi$
$600$	0	0
$601$	−15.1161	−0.616597	−0.308299	−	0.951290i	$-0.599760\pi$
$601$	−15.1161	−0.616597	−0.308299	+	0.951290i	$0.599760\pi$
$602$	0	0
$603$	−3.51420	−0.143109
$604$	0	0
$605$	9.85843	0.400802
$606$	0	0
$607$	37.6031	1.52626	0.763132	−	0.646242i	$-0.223660\pi$
$607$	37.6031	1.52626	0.763132	+	0.646242i	$0.223660\pi$
$608$	0	0
$609$	10.6921	0.433266
$610$	0	0
$611$	−39.1208	−1.58266
$612$	0	0
$613$	−13.0653	−0.527701	−0.263850	−	0.964564i	$-0.584992\pi$
$613$	−13.0653	−0.527701	−0.263850	+	0.964564i	$0.584992\pi$
$614$	0	0
$615$	−6.83164	−0.275478
$616$	0	0
$617$	−30.4263	−1.22492	−0.612459	−	0.790502i	$-0.709820\pi$
$617$	−30.4263	−1.22492	−0.612459	+	0.790502i	$0.709820\pi$
$618$	0	0
$619$	−22.9971	−0.924332	−0.462166	−	0.886793i	$-0.652928\pi$
$619$	−22.9971	−0.924332	−0.462166	+	0.886793i	$0.652928\pi$
$620$	0	0
$621$	−4.21024	−0.168951
$622$	0	0
$623$	−18.9947	−0.761005
$624$	0	0
$625$	−22.1662	−0.886649
$626$	0	0
$627$	15.1755	0.606051
$628$	0	0
$629$	−3.83956	−0.153093
$630$	0	0
$631$	45.3341	1.80472	0.902360	−	0.430982i	$-0.141833\pi$
$631$	45.3341	1.80472	0.902360	+	0.430982i	$0.141833\pi$
$632$	0	0
$633$	−16.9680	−0.674416
$634$	0	0
$635$	6.68452	0.265267
$636$	0	0
$637$	−19.3391	−0.766244
$638$	0	0
$639$	11.6222	0.459767
$640$	0	0
$641$	−4.60629	−0.181937	−0.0909687	−	0.995854i	$-0.528996\pi$
$641$	−4.60629	−0.181937	−0.0909687	+	0.995854i	$0.528996\pi$
$642$	0	0
$643$	−10.7066	−0.422227	−0.211114	−	0.977462i	$-0.567709\pi$
$643$	−10.7066	−0.422227	−0.211114	+	0.977462i	$0.567709\pi$
$644$	0	0
$645$	22.5905	0.889499
$646$	0	0
$647$	25.1350	0.988158	0.494079	−	0.869417i	$-0.335505\pi$
$647$	25.1350	0.988158	0.494079	+	0.869417i	$0.335505\pi$
$648$	0	0
$649$	−31.7050	−1.24453
$650$	0	0
$651$	−4.08168	−0.159974
$652$	0	0
$653$	−29.5769	−1.15743	−0.578717	−	0.815528i	$-0.696447\pi$
$653$	−29.5769	−1.15743	−0.578717	+	0.815528i	$0.696447\pi$
$654$	0	0
$655$	−45.4296	−1.77508
$656$	0	0
$657$	−5.43704	−0.212119
$658$	0	0
$659$	26.7918	1.04366	0.521831	−	0.853049i	$-0.325249\pi$
$659$	26.7918	1.04366	0.521831	+	0.853049i	$0.325249\pi$
$660$	0	0
$661$	5.89353	0.229232	0.114616	−	0.993410i	$-0.463436\pi$
$661$	5.89353	0.229232	0.114616	+	0.993410i	$0.463436\pi$
$662$	0	0
$663$	−10.8854	−0.422755
$664$	0	0
$665$	55.9907	2.17123
$666$	0	0
$667$	−14.0227	−0.542963
$668$	0	0
$669$	14.7467	0.570141
$670$	0	0
$671$	−21.9703	−0.848155
$672$	0	0
$673$	−3.26057	−0.125686	−0.0628428	−	0.998023i	$-0.520017\pi$
$673$	−3.26057	−0.125686	−0.0628428	+	0.998023i	$0.520017\pi$
$674$	0	0
$675$	5.51393	0.212231
$676$	0	0
$677$	4.15513	0.159695	0.0798474	−	0.996807i	$-0.474557\pi$
$677$	4.15513	0.159695	0.0798474	+	0.996807i	$0.474557\pi$
$678$	0	0
$679$	12.8927	0.494777
$680$	0	0
$681$	−0.120587	−0.00462092
$682$	0	0
$683$	−51.4838	−1.96997	−0.984987	−	0.172629i	$-0.944774\pi$
$683$	−51.4838	−1.96997	−0.984987	+	0.172629i	$0.944774\pi$
$684$	0	0
$685$	−31.8869	−1.21834
$686$	0	0
$687$	1.33643	0.0509881
$688$	0	0
$689$	−54.6911	−2.08356
$690$	0	0
$691$	−36.5334	−1.38980	−0.694899	−	0.719108i	$-0.744551\pi$
$691$	−36.5334	−1.38980	−0.694899	+	0.719108i	$0.744551\pi$
$692$	0	0
$693$	−9.05698	−0.344046
$694$	0	0
$695$	14.9770	0.568109
$696$	0	0
$697$	3.92015	0.148486
$698$	0	0
$699$	−8.77423	−0.331872
$700$	0	0
$701$	−20.6445	−0.779731	−0.389866	−	0.920872i	$-0.627479\pi$
$701$	−20.6445	−0.779731	−0.389866	+	0.920872i	$0.627479\pi$
$702$	0	0
$703$	11.0999	0.418640
$704$	0	0
$705$	−21.6823	−0.816602
$706$	0	0
$707$	29.1860	1.09765
$708$	0	0
$709$	33.5779	1.26105	0.630523	−	0.776171i	$-0.282840\pi$
$709$	33.5779	1.26105	0.630523	+	0.776171i	$0.282840\pi$
$710$	0	0
$711$	4.16325	0.156134
$712$	0	0
$713$	5.35314	0.200477
$714$	0	0
$715$	−53.5198	−2.00153
$716$	0	0
$717$	5.99261	0.223798
$718$	0	0
$719$	−22.2353	−0.829236	−0.414618	−	0.909996i	$-0.636085\pi$
$719$	−22.2353	−0.829236	−0.414618	+	0.909996i	$0.636085\pi$
$720$	0	0
$721$	49.9486	1.86018
$722$	0	0
$723$	20.6690	0.768688
$724$	0	0
$725$	18.3649	0.682054
$726$	0	0
$727$	−6.90158	−0.255966	−0.127983	−	0.991776i	$-0.540850\pi$
$727$	−6.90158	−0.255966	−0.127983	+	0.991776i	$0.540850\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−12.9629	−0.479451
$732$	0	0
$733$	47.0601	1.73821	0.869103	−	0.494632i	$-0.164697\pi$
$733$	47.0601	1.73821	0.869103	+	0.494632i	$0.164697\pi$
$734$	0	0
$735$	−10.7185	−0.395358
$736$	0	0
$737$	9.91454	0.365207
$738$	0	0
$739$	−3.09124	−0.113713	−0.0568565	−	0.998382i	$-0.518108\pi$
$739$	−3.09124	−0.113713	−0.0568565	+	0.998382i	$0.518108\pi$
$740$	0	0
$741$	31.4689	1.15604
$742$	0	0
$743$	26.7232	0.980380	0.490190	−	0.871616i	$-0.336927\pi$
$743$	26.7232	0.980380	0.490190	+	0.871616i	$0.336927\pi$
$744$	0	0
$745$	−37.7239	−1.38210
$746$	0	0
$747$	1.27361	0.0465991
$748$	0	0
$749$	−15.8998	−0.580965
$750$	0	0
$751$	−13.6355	−0.497566	−0.248783	−	0.968559i	$-0.580031\pi$
$751$	−13.6355	−0.497566	−0.248783	+	0.968559i	$0.580031\pi$
$752$	0	0
$753$	−9.72679	−0.354464
$754$	0	0
$755$	42.7847	1.55709
$756$	0	0
$757$	45.5231	1.65456	0.827282	−	0.561787i	$-0.189886\pi$
$757$	45.5231	1.65456	0.827282	+	0.561787i	$0.189886\pi$
$758$	0	0
$759$	11.8783	0.431154
$760$	0	0
$761$	−54.7160	−1.98345	−0.991727	−	0.128367i	$-0.959026\pi$
$761$	−54.7160	−1.98345	−0.991727	+	0.128367i	$0.959026\pi$
$762$	0	0
$763$	30.8007	1.11506
$764$	0	0
$765$	−6.03313	−0.218128
$766$	0	0
$767$	−65.7456	−2.37393
$768$	0	0
$769$	10.8454	0.391097	0.195548	−	0.980694i	$-0.437351\pi$
$769$	10.8454	0.391097	0.195548	+	0.980694i	$0.437351\pi$
$770$	0	0
$771$	−6.72200	−0.242087
$772$	0	0
$773$	−8.57334	−0.308362	−0.154181	−	0.988043i	$-0.549274\pi$
$773$	−8.57334	−0.308362	−0.154181	+	0.988043i	$0.549274\pi$
$774$	0	0
$775$	−7.01074	−0.251833
$776$	0	0
$777$	−6.62458	−0.237655
$778$	0	0
$779$	−11.3329	−0.406042
$780$	0	0
$781$	−32.7895	−1.17330
$782$	0	0
$783$	3.33063	0.119027
$784$	0	0
$785$	16.9524	0.605059
$786$	0	0
$787$	−54.0373	−1.92622	−0.963111	−	0.269104i	$-0.913272\pi$
$787$	−54.0373	−1.92622	−0.963111	+	0.269104i	$0.913272\pi$
$788$	0	0
$789$	20.9835	0.747031
$790$	0	0
$791$	−45.3858	−1.61373
$792$	0	0
$793$	−45.5591	−1.61785
$794$	0	0
$795$	−30.3119	−1.07505
$796$	0	0
$797$	22.5826	0.799917	0.399958	−	0.916533i	$-0.369025\pi$
$797$	22.5826	0.799917	0.399958	+	0.916533i	$0.369025\pi$
$798$	0	0
$799$	12.4418	0.440159
$800$	0	0
$801$	−5.91690	−0.209063
$802$	0	0
$803$	15.3394	0.541316
$804$	0	0
$805$	43.8254	1.54464
$806$	0	0
$807$	−2.08333	−0.0733368
$808$	0	0
$809$	−40.2340	−1.41455	−0.707277	−	0.706937i	$-0.750076\pi$
$809$	−40.2340	−1.41455	−0.707277	+	0.706937i	$0.750076\pi$
$810$	0	0
$811$	49.9998	1.75573	0.877865	−	0.478909i	$-0.158968\pi$
$811$	49.9998	1.75573	0.877865	+	0.478909i	$0.158968\pi$
$812$	0	0
$813$	−10.9907	−0.385462
$814$	0	0
$815$	0.849314	0.0297502
$816$	0	0
$817$	37.4748	1.31108
$818$	0	0
$819$	−18.7811	−0.656266
$820$	0	0
$821$	39.4031	1.37518	0.687589	−	0.726100i	$-0.258669\pi$
$821$	39.4031	1.37518	0.687589	+	0.726100i	$0.258669\pi$
$822$	0	0
$823$	43.3232	1.51015	0.755077	−	0.655636i	$-0.227600\pi$
$823$	43.3232	1.51015	0.755077	+	0.655636i	$0.227600\pi$
$824$	0	0
$825$	−15.5564	−0.541603
$826$	0	0
$827$	−42.1930	−1.46719	−0.733597	−	0.679585i	$-0.762160\pi$
$827$	−42.1930	−1.46719	−0.733597	+	0.679585i	$0.762160\pi$
$828$	0	0
$829$	27.1711	0.943693	0.471846	−	0.881681i	$-0.343588\pi$
$829$	27.1711	0.943693	0.471846	+	0.881681i	$0.343588\pi$
$830$	0	0
$831$	−25.5833	−0.887476
$832$	0	0
$833$	6.15052	0.213103
$834$	0	0
$835$	3.24252	0.112212
$836$	0	0
$837$	−1.27146	−0.0439481
$838$	0	0
$839$	35.5522	1.22740	0.613698	−	0.789541i	$-0.289681\pi$
$839$	35.5522	1.22740	0.613698	+	0.789541i	$0.289681\pi$
$840$	0	0
$841$	−17.9069	−0.617479
$842$	0	0
$843$	−11.2733	−0.388272
$844$	0	0
$845$	−68.8294	−2.36780
$846$	0	0
$847$	−9.76028	−0.335367
$848$	0	0
$849$	−23.5911	−0.809646
$850$	0	0
$851$	8.68816	0.297826
$852$	0	0
$853$	56.9119	1.94863	0.974313	−	0.225200i	$-0.0723034\pi$
$853$	56.9119	1.94863	0.974313	+	0.225200i	$0.0723034\pi$
$854$	0	0
$855$	17.4413	0.596480
$856$	0	0
$857$	−34.1803	−1.16758	−0.583788	−	0.811906i	$-0.698430\pi$
$857$	−34.1803	−1.16758	−0.583788	+	0.811906i	$0.698430\pi$
$858$	0	0
$859$	16.9625	0.578754	0.289377	−	0.957215i	$-0.406552\pi$
$859$	16.9625	0.578754	0.289377	+	0.957215i	$0.406552\pi$
$860$	0	0
$861$	6.76362	0.230504
$862$	0	0
$863$	−44.6654	−1.52043	−0.760214	−	0.649672i	$-0.774906\pi$
$863$	−44.6654	−1.52043	−0.760214	+	0.649672i	$0.774906\pi$
$864$	0	0
$865$	63.3222	2.15302
$866$	0	0
$867$	−13.5381	−0.459776
$868$	0	0
$869$	−11.7457	−0.398446
$870$	0	0
$871$	20.5594	0.696630
$872$	0	0
$873$	4.01613	0.135925
$874$	0	0
$875$	−5.34962	−0.180850
$876$	0	0
$877$	1.65783	0.0559808	0.0279904	−	0.999608i	$-0.491089\pi$
$877$	1.65783	0.0559808	0.0279904	+	0.999608i	$0.491089\pi$
$878$	0	0
$879$	−9.21613	−0.310852
$880$	0	0
$881$	−11.0757	−0.373150	−0.186575	−	0.982441i	$-0.559739\pi$
$881$	−11.0757	−0.373150	−0.186575	+	0.982441i	$0.559739\pi$
$882$	0	0
$883$	−6.29928	−0.211988	−0.105994	−	0.994367i	$-0.533802\pi$
$883$	−6.29928	−0.211988	−0.105994	+	0.994367i	$0.533802\pi$
$884$	0	0
$885$	−36.4388	−1.22488
$886$	0	0
$887$	13.7138	0.460463	0.230231	−	0.973136i	$-0.426052\pi$
$887$	13.7138	0.460463	0.230231	+	0.973136i	$0.426052\pi$
$888$	0	0
$889$	−6.61796	−0.221959
$890$	0	0
$891$	−2.82128	−0.0945165
$892$	0	0
$893$	−35.9682	−1.20363
$894$	0	0
$895$	−61.9219	−2.06982
$896$	0	0
$897$	24.6315	0.822423
$898$	0	0
$899$	−4.23476	−0.141237
$900$	0	0
$901$	17.3937	0.579467
$902$	0	0
$903$	−22.3656	−0.744279
$904$	0	0
$905$	26.8035	0.890979
$906$	0	0
$907$	35.5148	1.17925	0.589625	−	0.807677i	$-0.299275\pi$
$907$	35.5148	1.17925	0.589625	+	0.807677i	$0.299275\pi$
$908$	0	0
$909$	9.09153	0.301547
$910$	0	0
$911$	−38.4589	−1.27420	−0.637100	−	0.770781i	$-0.719866\pi$
$911$	−38.4589	−1.27420	−0.637100	+	0.770781i	$0.719866\pi$
$912$	0	0
$913$	−3.59323	−0.118918
$914$	0	0
$915$	−25.2506	−0.834760
$916$	0	0
$917$	44.9773	1.48528
$918$	0	0
$919$	−40.7347	−1.34371	−0.671856	−	0.740682i	$-0.734503\pi$
$919$	−40.7347	−1.34371	−0.671856	+	0.740682i	$0.734503\pi$
$920$	0	0
$921$	13.3545	0.440046
$922$	0	0
$923$	−67.9945	−2.23807
$924$	0	0
$925$	−11.3784	−0.374121
$926$	0	0
$927$	15.5592	0.511030
$928$	0	0
$929$	52.9417	1.73696	0.868481	−	0.495723i	$-0.165097\pi$
$929$	52.9417	1.73696	0.868481	+	0.495723i	$0.165097\pi$
$930$	0	0
$931$	−17.7807	−0.582738
$932$	0	0
$933$	18.9770	0.621278
$934$	0	0
$935$	17.0212	0.556652
$936$	0	0
$937$	22.6766	0.740811	0.370406	−	0.928870i	$-0.379219\pi$
$937$	22.6766	0.740811	0.370406	+	0.928870i	$0.379219\pi$
$938$	0	0
$939$	0.865518	0.0282451
$940$	0	0
$941$	−49.9912	−1.62967	−0.814834	−	0.579694i	$-0.803172\pi$
$941$	−49.9912	−1.62967	−0.814834	+	0.579694i	$0.803172\pi$
$942$	0	0
$943$	−8.87052	−0.288864
$944$	0	0
$945$	−10.4092	−0.338613
$946$	0	0
$947$	21.4522	0.697103	0.348552	−	0.937290i	$-0.386674\pi$
$947$	21.4522	0.697103	0.348552	+	0.937290i	$0.386674\pi$
$948$	0	0
$949$	31.8088	1.03256
$950$	0	0
$951$	6.57367	0.213166
$952$	0	0
$953$	−35.3902	−1.14640	−0.573201	−	0.819415i	$-0.694298\pi$
$953$	−35.3902	−1.14640	−0.573201	+	0.819415i	$0.694298\pi$
$954$	0	0
$955$	−68.3394	−2.21141
$956$	0	0
$957$	−9.39666	−0.303751
$958$	0	0
$959$	31.5694	1.01943
$960$	0	0
$961$	−29.3834	−0.947851
$962$	0	0
$963$	−4.95284	−0.159603
$964$	0	0
$965$	−72.3221	−2.32813
$966$	0	0
$967$	−41.2924	−1.32787	−0.663937	−	0.747789i	$-0.731116\pi$
$967$	−41.2924	−1.32787	−0.663937	+	0.747789i	$0.731116\pi$
$968$	0	0
$969$	−10.0082	−0.321510
$970$	0	0
$971$	−16.6855	−0.535464	−0.267732	−	0.963493i	$-0.586274\pi$
$971$	−16.6855	−0.535464	−0.267732	+	0.963493i	$0.586274\pi$
$972$	0	0
$973$	−14.8279	−0.475359
$974$	0	0
$975$	−32.2587	−1.03310
$976$	0	0
$977$	17.3143	0.553932	0.276966	−	0.960880i	$-0.410671\pi$
$977$	17.3143	0.553932	0.276966	+	0.960880i	$0.410671\pi$
$978$	0	0
$979$	16.6933	0.533519
$980$	0	0
$981$	9.59455	0.306330
$982$	0	0
$983$	−32.8464	−1.04764	−0.523819	−	0.851830i	$-0.675493\pi$
$983$	−32.8464	−1.04764	−0.523819	+	0.851830i	$0.675493\pi$
$984$	0	0
$985$	34.8579	1.11067
$986$	0	0
$987$	21.4664	0.683283
$988$	0	0
$989$	29.3325	0.932719
$990$	0	0
$991$	−25.3128	−0.804087	−0.402044	−	0.915621i	$-0.631700\pi$
$991$	−25.3128	−0.804087	−0.402044	+	0.915621i	$0.631700\pi$
$992$	0	0
$993$	−5.69256	−0.180648
$994$	0	0
$995$	−41.3903	−1.31216
$996$	0	0
$997$	−56.1380	−1.77791	−0.888954	−	0.457996i	$-0.848567\pi$
$997$	−56.1380	−1.77791	−0.888954	+	0.457996i	$0.848567\pi$
$998$	0	0
$999$	−2.06358	−0.0652888

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.y.1.1		8
4.3	odd	2		4008.2.a.h.1.1	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4008.2.a.h.1.1	✓	8	4.3	odd	2
8016.2.a.y.1.1		8	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.24252 q^{5} +3.21024 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.24252 q^{5} +3.21024 q^{7} +1.00000 q^{9} -2.82128 q^{11} -5.85040 q^{13} -3.24252 q^{15} +1.86063 q^{17} -5.37894 q^{19} +3.21024 q^{21} -4.21024 q^{23} +5.51393 q^{25} +1.00000 q^{27} +3.33063 q^{29} -1.27146 q^{31} -2.82128 q^{33} -10.4092 q^{35} -2.06358 q^{37} -5.85040 q^{39} +2.10689 q^{41} -6.96695 q^{43} -3.24252 q^{45} +6.68686 q^{47} +3.30561 q^{49} +1.86063 q^{51} +9.34827 q^{53} +9.14806 q^{55} -5.37894 q^{57} +11.2378 q^{59} +7.78735 q^{61} +3.21024 q^{63} +18.9700 q^{65} -3.51420 q^{67} -4.21024 q^{69} +11.6222 q^{71} -5.43704 q^{73} +5.51393 q^{75} -9.05698 q^{77} +4.16325 q^{79} +1.00000 q^{81} +1.27361 q^{83} -6.03313 q^{85} +3.33063 q^{87} -5.91690 q^{89} -18.7811 q^{91} -1.27146 q^{93} +17.4413 q^{95} +4.01613 q^{97} -2.82128 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + q^{7} + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + q^7 + 8 * q^9	\( 8 q + 8 q^{3} + q^{7} + 8 q^{9} + 3 q^{11} - 8 q^{13} - 7 q^{17} + q^{21} - 9 q^{23} + 6 q^{25} + 8 q^{27} + 17 q^{29} + 23 q^{31} + 3 q^{33} + 15 q^{35} + 8 q^{37} - 8 q^{39} - 8 q^{41} + 2 q^{43} + 34 q^{47} + 5 q^{49} - 7 q^{51} + 12 q^{53} + 7 q^{55} + 16 q^{59} - 2 q^{61} + q^{63} - 14 q^{65} - 21 q^{67} - 9 q^{69} + 29 q^{71} - 38 q^{73} + 6 q^{75} + 20 q^{77} + 12 q^{79} + 8 q^{81} + 32 q^{83} + 23 q^{85} + 17 q^{87} + 11 q^{89} + 5 q^{91} + 23 q^{93} + 67 q^{95} + 8 q^{97} + 3 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + q^7 + 8 * q^9 + 3 * q^11 - 8 * q^13 - 7 * q^17 + q^21 - 9 * q^23 + 6 * q^25 + 8 * q^27 + 17 * q^29 + 23 * q^31 + 3 * q^33 + 15 * q^35 + 8 * q^37 - 8 * q^39 - 8 * q^41 + 2 * q^43 + 34 * q^47 + 5 * q^49 - 7 * q^51 + 12 * q^53 + 7 * q^55 + 16 * q^59 - 2 * q^61 + q^63 - 14 * q^65 - 21 * q^67 - 9 * q^69 + 29 * q^71 - 38 * q^73 + 6 * q^75 + 20 * q^77 + 12 * q^79 + 8 * q^81 + 32 * q^83 + 23 * q^85 + 17 * q^87 + 11 * q^89 + 5 * q^91 + 23 * q^93 + 67 * q^95 + 8 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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