LMFDB - Embedded newform 8016.2.a.v.1.2

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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 20x^{5} + 2x^{4} + 87x^{3} + 46x^{2} - 48x - 2 $ x^7 - 2x^6 - 20x^5 + 2x^4 + 87x^3 + 46x^2 - 48x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1002)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30288$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.30288 q^{5} -1.53952 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.30288 q^{5} -1.53952 q^{7} +1.00000 q^{9} -1.10693 q^{11} +4.17065 q^{13} +1.30288 q^{15} -4.29582 q^{17} -5.98298 q^{19} +1.53952 q^{21} +3.64268 q^{23} -3.30251 q^{25} -1.00000 q^{27} +4.89056 q^{29} +2.10316 q^{31} +1.10693 q^{33} +2.00580 q^{35} +5.69157 q^{37} -4.17065 q^{39} +2.38869 q^{41} -4.81041 q^{43} -1.30288 q^{45} -5.53701 q^{47} -4.62988 q^{49} +4.29582 q^{51} -8.47228 q^{53} +1.44220 q^{55} +5.98298 q^{57} -3.67819 q^{59} -0.549128 q^{61} -1.53952 q^{63} -5.43385 q^{65} -5.13636 q^{67} -3.64268 q^{69} +2.82810 q^{71} -2.95554 q^{73} +3.30251 q^{75} +1.70414 q^{77} +10.3862 q^{79} +1.00000 q^{81} +7.83252 q^{83} +5.59693 q^{85} -4.89056 q^{87} -16.1056 q^{89} -6.42080 q^{91} -2.10316 q^{93} +7.79509 q^{95} +4.52295 q^{97} -1.10693 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9}+O(q^{10}) $ 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9	$ 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9} + 6 q^{13} - 5 q^{15} + 6 q^{17} + 2 q^{19} + 7 q^{21} + 12 q^{25} - 7 q^{27} - 4 q^{29} - 7 q^{31} + 13 q^{35} - 3 q^{37} - 6 q^{39} - 12 q^{41} + 2 q^{43} + 5 q^{45} + 11 q^{47} + 10 q^{49} - 6 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 19 q^{59} + 12 q^{61} - 7 q^{63} - 10 q^{65} + 17 q^{67} + 20 q^{71} + 10 q^{73} - 12 q^{75} - 24 q^{77} - 2 q^{79} + 7 q^{81} + 7 q^{83} - 18 q^{85} + 4 q^{87} - 3 q^{89} - 4 q^{91} + 7 q^{93} + 24 q^{95} - 3 q^{97}+O(q^{100}) $ 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9 + 6 * q^13 - 5 * q^15 + 6 * q^17 + 2 * q^19 + 7 * q^21 + 12 * q^25 - 7 * q^27 - 4 * q^29 - 7 * q^31 + 13 * q^35 - 3 * q^37 - 6 * q^39 - 12 * q^41 + 2 * q^43 + 5 * q^45 + 11 * q^47 + 10 * q^49 - 6 * q^51 + q^53 + 2 * q^55 - 2 * q^57 + 19 * q^59 + 12 * q^61 - 7 * q^63 - 10 * q^65 + 17 * q^67 + 20 * q^71 + 10 * q^73 - 12 * q^75 - 24 * q^77 - 2 * q^79 + 7 * q^81 + 7 * q^83 - 18 * q^85 + 4 * q^87 - 3 * q^89 - 4 * q^91 + 7 * q^93 + 24 * q^95 - 3 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.30288	−0.582665	−0.291332	−	0.956622i	$-0.594099\pi$
$5$	−1.30288	−0.582665	−0.291332	+	0.956622i	$0.594099\pi$
$6$	0	0
$7$	−1.53952	−0.581883	−0.290942	−	0.956741i	$-0.593969\pi$
$7$	−1.53952	−0.581883	−0.290942	+	0.956741i	$0.593969\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.10693	−0.333752	−0.166876	−	0.985978i	$-0.553368\pi$
$11$	−1.10693	−0.333752	−0.166876	+	0.985978i	$0.553368\pi$
$12$	0	0
$13$	4.17065	1.15673	0.578366	−	0.815778i	$-0.303691\pi$
$13$	4.17065	1.15673	0.578366	+	0.815778i	$0.303691\pi$
$14$	0	0
$15$	1.30288	0.336402
$16$	0	0
$17$	−4.29582	−1.04189	−0.520945	−	0.853590i	$-0.674420\pi$
$17$	−4.29582	−1.04189	−0.520945	+	0.853590i	$0.674420\pi$
$18$	0	0
$19$	−5.98298	−1.37259	−0.686295	−	0.727324i	$-0.740764\pi$
$19$	−5.98298	−1.37259	−0.686295	+	0.727324i	$0.740764\pi$
$20$	0	0
$21$	1.53952	0.335950
$22$	0	0
$23$	3.64268	0.759552	0.379776	−	0.925078i	$-0.376001\pi$
$23$	3.64268	0.759552	0.379776	+	0.925078i	$0.376001\pi$
$24$	0	0
$25$	−3.30251	−0.660502
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.89056	0.908155	0.454078	−	0.890962i	$-0.349969\pi$
$29$	4.89056	0.908155	0.454078	+	0.890962i	$0.349969\pi$
$30$	0	0
$31$	2.10316	0.377740	0.188870	−	0.982002i	$-0.439518\pi$
$31$	2.10316	0.377740	0.188870	+	0.982002i	$0.439518\pi$
$32$	0	0
$33$	1.10693	0.192692
$34$	0	0
$35$	2.00580	0.339043
$36$	0	0
$37$	5.69157	0.935688	0.467844	−	0.883811i	$-0.345031\pi$
$37$	5.69157	0.935688	0.467844	+	0.883811i	$0.345031\pi$
$38$	0	0
$39$	−4.17065	−0.667839
$40$	0	0
$41$	2.38869	0.373051	0.186525	−	0.982450i	$-0.440277\pi$
$41$	2.38869	0.373051	0.186525	+	0.982450i	$0.440277\pi$
$42$	0	0
$43$	−4.81041	−0.733581	−0.366791	−	0.930303i	$-0.619543\pi$
$43$	−4.81041	−0.733581	−0.366791	+	0.930303i	$0.619543\pi$
$44$	0	0
$45$	−1.30288	−0.194222
$46$	0	0
$47$	−5.53701	−0.807657	−0.403828	−	0.914835i	$-0.632321\pi$
$47$	−5.53701	−0.807657	−0.403828	+	0.914835i	$0.632321\pi$
$48$	0	0
$49$	−4.62988	−0.661412
$50$	0	0
$51$	4.29582	0.601535
$52$	0	0
$53$	−8.47228	−1.16376	−0.581879	−	0.813276i	$-0.697682\pi$
$53$	−8.47228	−1.16376	−0.581879	+	0.813276i	$0.697682\pi$
$54$	0	0
$55$	1.44220	0.194466
$56$	0	0
$57$	5.98298	0.792465
$58$	0	0
$59$	−3.67819	−0.478860	−0.239430	−	0.970914i	$-0.576960\pi$
$59$	−3.67819	−0.478860	−0.239430	+	0.970914i	$0.576960\pi$
$60$	0	0
$61$	−0.549128	−0.0703087	−0.0351543	−	0.999382i	$-0.511192\pi$
$61$	−0.549128	−0.0703087	−0.0351543	+	0.999382i	$0.511192\pi$
$62$	0	0
$63$	−1.53952	−0.193961
$64$	0	0
$65$	−5.43385	−0.673986
$66$	0	0
$67$	−5.13636	−0.627506	−0.313753	−	0.949505i	$-0.601586\pi$
$67$	−5.13636	−0.627506	−0.313753	+	0.949505i	$0.601586\pi$
$68$	0	0
$69$	−3.64268	−0.438528
$70$	0	0
$71$	2.82810	0.335633	0.167817	−	0.985818i	$-0.446328\pi$
$71$	2.82810	0.335633	0.167817	+	0.985818i	$0.446328\pi$
$72$	0	0
$73$	−2.95554	−0.345920	−0.172960	−	0.984929i	$-0.555333\pi$
$73$	−2.95554	−0.345920	−0.172960	+	0.984929i	$0.555333\pi$
$74$	0	0
$75$	3.30251	0.381341
$76$	0	0
$77$	1.70414	0.194205
$78$	0	0
$79$	10.3862	1.16854	0.584269	−	0.811560i	$-0.301381\pi$
$79$	10.3862	1.16854	0.584269	+	0.811560i	$0.301381\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.83252	0.859730	0.429865	−	0.902893i	$-0.358561\pi$
$83$	7.83252	0.859730	0.429865	+	0.902893i	$0.358561\pi$
$84$	0	0
$85$	5.59693	0.607072
$86$	0	0
$87$	−4.89056	−0.524324
$88$	0	0
$89$	−16.1056	−1.70719	−0.853593	−	0.520940i	$-0.825581\pi$
$89$	−16.1056	−1.70719	−0.853593	+	0.520940i	$0.825581\pi$
$90$	0	0
$91$	−6.42080	−0.673082
$92$	0	0
$93$	−2.10316	−0.218088
$94$	0	0
$95$	7.79509	0.799759
$96$	0	0
$97$	4.52295	0.459236	0.229618	−	0.973281i	$-0.426252\pi$
$97$	4.52295	0.459236	0.229618	+	0.973281i	$0.426252\pi$
$98$	0	0
$99$	−1.10693	−0.111251
$100$	0	0
$101$	−9.78513	−0.973657	−0.486829	−	0.873498i	$-0.661846\pi$
$101$	−9.78513	−0.973657	−0.486829	+	0.873498i	$0.661846\pi$
$102$	0	0
$103$	12.0844	1.19071	0.595353	−	0.803464i	$-0.297012\pi$
$103$	12.0844	1.19071	0.595353	+	0.803464i	$0.297012\pi$
$104$	0	0
$105$	−2.00580	−0.195746
$106$	0	0
$107$	1.73454	0.167684	0.0838422	−	0.996479i	$-0.473281\pi$
$107$	1.73454	0.167684	0.0838422	+	0.996479i	$0.473281\pi$
$108$	0	0
$109$	−3.98492	−0.381686	−0.190843	−	0.981621i	$-0.561122\pi$
$109$	−3.98492	−0.381686	−0.190843	+	0.981621i	$0.561122\pi$
$110$	0	0
$111$	−5.69157	−0.540220
$112$	0	0
$113$	11.2489	1.05820	0.529102	−	0.848558i	$-0.322529\pi$
$113$	11.2489	1.05820	0.529102	+	0.848558i	$0.322529\pi$
$114$	0	0
$115$	−4.74597	−0.442564
$116$	0	0
$117$	4.17065	0.385577
$118$	0	0
$119$	6.61350	0.606258
$120$	0	0
$121$	−9.77470	−0.888609
$122$	0	0
$123$	−2.38869	−0.215381
$124$	0	0
$125$	10.8172	0.967516
$126$	0	0
$127$	4.28622	0.380341	0.190170	−	0.981751i	$-0.439096\pi$
$127$	4.28622	0.380341	0.190170	+	0.981751i	$0.439096\pi$
$128$	0	0
$129$	4.81041	0.423533
$130$	0	0
$131$	2.08626	0.182278	0.0911388	−	0.995838i	$-0.470949\pi$
$131$	2.08626	0.182278	0.0911388	+	0.995838i	$0.470949\pi$
$132$	0	0
$133$	9.21091	0.798687
$134$	0	0
$135$	1.30288	0.112134
$136$	0	0
$137$	−10.8365	−0.925824	−0.462912	−	0.886404i	$-0.653195\pi$
$137$	−10.8365	−0.925824	−0.462912	+	0.886404i	$0.653195\pi$
$138$	0	0
$139$	22.2333	1.88581	0.942903	−	0.333068i	$-0.108084\pi$
$139$	22.2333	1.88581	0.942903	+	0.333068i	$0.108084\pi$
$140$	0	0
$141$	5.53701	0.466301
$142$	0	0
$143$	−4.61663	−0.386062
$144$	0	0
$145$	−6.37181	−0.529150
$146$	0	0
$147$	4.62988	0.381866
$148$	0	0
$149$	−7.91637	−0.648535	−0.324267	−	0.945965i	$-0.605118\pi$
$149$	−7.91637	−0.648535	−0.324267	+	0.945965i	$0.605118\pi$
$150$	0	0
$151$	−1.14387	−0.0930869	−0.0465435	−	0.998916i	$-0.514821\pi$
$151$	−1.14387	−0.0930869	−0.0465435	+	0.998916i	$0.514821\pi$
$152$	0	0
$153$	−4.29582	−0.347297
$154$	0	0
$155$	−2.74017	−0.220095
$156$	0	0
$157$	0.864535	0.0689974	0.0344987	−	0.999405i	$-0.489017\pi$
$157$	0.864535	0.0689974	0.0344987	+	0.999405i	$0.489017\pi$
$158$	0	0
$159$	8.47228	0.671895
$160$	0	0
$161$	−5.60798	−0.441971
$162$	0	0
$163$	−0.610097	−0.0477865	−0.0238933	−	0.999715i	$-0.507606\pi$
$163$	−0.610097	−0.0477865	−0.0238933	+	0.999715i	$0.507606\pi$
$164$	0	0
$165$	−1.44220	−0.112275
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	4.39435	0.338027
$170$	0	0
$171$	−5.98298	−0.457530
$172$	0	0
$173$	6.26195	0.476087	0.238043	−	0.971255i	$-0.423494\pi$
$173$	6.26195	0.476087	0.238043	+	0.971255i	$0.423494\pi$
$174$	0	0
$175$	5.08428	0.384335
$176$	0	0
$177$	3.67819	0.276470
$178$	0	0
$179$	6.07291	0.453911	0.226955	−	0.973905i	$-0.427123\pi$
$179$	6.07291	0.453911	0.226955	+	0.973905i	$0.427123\pi$
$180$	0	0
$181$	20.5447	1.52708	0.763539	−	0.645762i	$-0.223460\pi$
$181$	20.5447	1.52708	0.763539	+	0.645762i	$0.223460\pi$
$182$	0	0
$183$	0.549128	0.0405927
$184$	0	0
$185$	−7.41542	−0.545192
$186$	0	0
$187$	4.75518	0.347733
$188$	0	0
$189$	1.53952	0.111983
$190$	0	0
$191$	−16.9039	−1.22312	−0.611560	−	0.791198i	$-0.709458\pi$
$191$	−16.9039	−1.22312	−0.611560	+	0.791198i	$0.709458\pi$
$192$	0	0
$193$	12.0710	0.868893	0.434447	−	0.900698i	$-0.356944\pi$
$193$	12.0710	0.868893	0.434447	+	0.900698i	$0.356944\pi$
$194$	0	0
$195$	5.43385	0.389126
$196$	0	0
$197$	−17.4635	−1.24422	−0.622112	−	0.782928i	$-0.713725\pi$
$197$	−17.4635	−1.24422	−0.622112	+	0.782928i	$0.713725\pi$
$198$	0	0
$199$	−17.5886	−1.24682	−0.623410	−	0.781895i	$-0.714253\pi$
$199$	−17.5886	−1.24682	−0.623410	+	0.781895i	$0.714253\pi$
$200$	0	0
$201$	5.13636	0.362291
$202$	0	0
$203$	−7.52911	−0.528440
$204$	0	0
$205$	−3.11217	−0.217363
$206$	0	0
$207$	3.64268	0.253184
$208$	0	0
$209$	6.62275	0.458105
$210$	0	0
$211$	11.0535	0.760957	0.380478	−	0.924790i	$-0.375759\pi$
$211$	11.0535	0.760957	0.380478	+	0.924790i	$0.375759\pi$
$212$	0	0
$213$	−2.82810	−0.193778
$214$	0	0
$215$	6.26738	0.427432
$216$	0	0
$217$	−3.23786	−0.219800
$218$	0	0
$219$	2.95554	0.199717
$220$	0	0
$221$	−17.9164	−1.20519
$222$	0	0
$223$	−15.3933	−1.03081	−0.515404	−	0.856947i	$-0.672358\pi$
$223$	−15.3933	−1.03081	−0.515404	+	0.856947i	$0.672358\pi$
$224$	0	0
$225$	−3.30251	−0.220167
$226$	0	0
$227$	9.45775	0.627733	0.313866	−	0.949467i	$-0.398376\pi$
$227$	9.45775	0.627733	0.313866	+	0.949467i	$0.398376\pi$
$228$	0	0
$229$	21.1324	1.39647	0.698235	−	0.715869i	$-0.253969\pi$
$229$	21.1324	1.39647	0.698235	+	0.715869i	$0.253969\pi$
$230$	0	0
$231$	−1.70414	−0.112124
$232$	0	0
$233$	−7.93738	−0.519995	−0.259998	−	0.965609i	$-0.583722\pi$
$233$	−7.93738	−0.519995	−0.259998	+	0.965609i	$0.583722\pi$
$234$	0	0
$235$	7.21405	0.470593
$236$	0	0
$237$	−10.3862	−0.674655
$238$	0	0
$239$	−5.51828	−0.356948	−0.178474	−	0.983945i	$-0.557116\pi$
$239$	−5.51828	−0.356948	−0.178474	+	0.983945i	$0.557116\pi$
$240$	0	0
$241$	24.7655	1.59529	0.797643	−	0.603130i	$-0.206080\pi$
$241$	24.7655	1.59529	0.797643	+	0.603130i	$0.206080\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.03217	0.385381
$246$	0	0
$247$	−24.9529	−1.58772
$248$	0	0
$249$	−7.83252	−0.496366
$250$	0	0
$251$	7.39915	0.467030	0.233515	−	0.972353i	$-0.424977\pi$
$251$	7.39915	0.467030	0.233515	+	0.972353i	$0.424977\pi$
$252$	0	0
$253$	−4.03220	−0.253502
$254$	0	0
$255$	−5.59693	−0.350493
$256$	0	0
$257$	−14.4033	−0.898454	−0.449227	−	0.893418i	$-0.648301\pi$
$257$	−14.4033	−0.898454	−0.449227	+	0.893418i	$0.648301\pi$
$258$	0	0
$259$	−8.76227	−0.544461
$260$	0	0
$261$	4.89056	0.302718
$262$	0	0
$263$	11.5284	0.710872	0.355436	−	0.934701i	$-0.384332\pi$
$263$	11.5284	0.710872	0.355436	+	0.934701i	$0.384332\pi$
$264$	0	0
$265$	11.0383	0.678080
$266$	0	0
$267$	16.1056	0.985644
$268$	0	0
$269$	−5.00858	−0.305379	−0.152689	−	0.988274i	$-0.548793\pi$
$269$	−5.00858	−0.305379	−0.152689	+	0.988274i	$0.548793\pi$
$270$	0	0
$271$	−2.89720	−0.175992	−0.0879962	−	0.996121i	$-0.528046\pi$
$271$	−2.89720	−0.175992	−0.0879962	+	0.996121i	$0.528046\pi$
$272$	0	0
$273$	6.42080	0.388604
$274$	0	0
$275$	3.65565	0.220444
$276$	0	0
$277$	−12.3999	−0.745038	−0.372519	−	0.928024i	$-0.621506\pi$
$277$	−12.3999	−0.745038	−0.372519	+	0.928024i	$0.621506\pi$
$278$	0	0
$279$	2.10316	0.125913
$280$	0	0
$281$	7.57409	0.451833	0.225916	−	0.974147i	$-0.427462\pi$
$281$	7.57409	0.451833	0.225916	+	0.974147i	$0.427462\pi$
$282$	0	0
$283$	26.4522	1.57242	0.786211	−	0.617958i	$-0.212040\pi$
$283$	26.4522	1.57242	0.786211	+	0.617958i	$0.212040\pi$
$284$	0	0
$285$	−7.79509	−0.461741
$286$	0	0
$287$	−3.67743	−0.217072
$288$	0	0
$289$	1.45408	0.0855344
$290$	0	0
$291$	−4.52295	−0.265140
$292$	0	0
$293$	26.4005	1.54234	0.771168	−	0.636632i	$-0.219673\pi$
$293$	26.4005	1.54234	0.771168	+	0.636632i	$0.219673\pi$
$294$	0	0
$295$	4.79223	0.279015
$296$	0	0
$297$	1.10693	0.0642307
$298$	0	0
$299$	15.1924	0.878597
$300$	0	0
$301$	7.40572	0.426859
$302$	0	0
$303$	9.78513	0.562141
$304$	0	0
$305$	0.715447	0.0409664
$306$	0	0
$307$	25.8586	1.47583	0.737914	−	0.674895i	$-0.235811\pi$
$307$	25.8586	1.47583	0.737914	+	0.674895i	$0.235811\pi$
$308$	0	0
$309$	−12.0844	−0.687455
$310$	0	0
$311$	18.0842	1.02546	0.512732	−	0.858549i	$-0.328634\pi$
$311$	18.0842	1.02546	0.512732	+	0.858549i	$0.328634\pi$
$312$	0	0
$313$	5.63952	0.318764	0.159382	−	0.987217i	$-0.449050\pi$
$313$	5.63952	0.318764	0.159382	+	0.987217i	$0.449050\pi$
$314$	0	0
$315$	2.00580	0.113014
$316$	0	0
$317$	−11.3814	−0.639243	−0.319622	−	0.947545i	$-0.603556\pi$
$317$	−11.3814	−0.639243	−0.319622	+	0.947545i	$0.603556\pi$
$318$	0	0
$319$	−5.41352	−0.303099
$320$	0	0
$321$	−1.73454	−0.0968126
$322$	0	0
$323$	25.7018	1.43009
$324$	0	0
$325$	−13.7736	−0.764023
$326$	0	0
$327$	3.98492	0.220366
$328$	0	0
$329$	8.52434	0.469962
$330$	0	0
$331$	8.43848	0.463821	0.231910	−	0.972737i	$-0.425502\pi$
$331$	8.43848	0.463821	0.231910	+	0.972737i	$0.425502\pi$
$332$	0	0
$333$	5.69157	0.311896
$334$	0	0
$335$	6.69205	0.365626
$336$	0	0
$337$	6.38116	0.347604	0.173802	−	0.984781i	$-0.444395\pi$
$337$	6.38116	0.347604	0.173802	+	0.984781i	$0.444395\pi$
$338$	0	0
$339$	−11.2489	−0.610954
$340$	0	0
$341$	−2.32806	−0.126072
$342$	0	0
$343$	17.9044	0.966748
$344$	0	0
$345$	4.74597	0.255514
$346$	0	0
$347$	−1.75364	−0.0941404	−0.0470702	−	0.998892i	$-0.514988\pi$
$347$	−1.75364	−0.0941404	−0.0470702	+	0.998892i	$0.514988\pi$
$348$	0	0
$349$	31.8272	1.70367	0.851836	−	0.523809i	$-0.175490\pi$
$349$	31.8272	1.70367	0.851836	+	0.523809i	$0.175490\pi$
$350$	0	0
$351$	−4.17065	−0.222613
$352$	0	0
$353$	28.7216	1.52870	0.764349	−	0.644802i	$-0.223060\pi$
$353$	28.7216	1.52870	0.764349	+	0.644802i	$0.223060\pi$
$354$	0	0
$355$	−3.68466	−0.195562
$356$	0	0
$357$	−6.61350	−0.350023
$358$	0	0
$359$	−5.63246	−0.297270	−0.148635	−	0.988892i	$-0.547488\pi$
$359$	−5.63246	−0.297270	−0.148635	+	0.988892i	$0.547488\pi$
$360$	0	0
$361$	16.7960	0.884002
$362$	0	0
$363$	9.77470	0.513039
$364$	0	0
$365$	3.85070	0.201555
$366$	0	0
$367$	−2.22538	−0.116164	−0.0580818	−	0.998312i	$-0.518498\pi$
$367$	−2.22538	−0.116164	−0.0580818	+	0.998312i	$0.518498\pi$
$368$	0	0
$369$	2.38869	0.124350
$370$	0	0
$371$	13.0432	0.677171
$372$	0	0
$373$	16.2135	0.839505	0.419753	−	0.907639i	$-0.362117\pi$
$373$	16.2135	0.839505	0.419753	+	0.907639i	$0.362117\pi$
$374$	0	0
$375$	−10.8172	−0.558595
$376$	0	0
$377$	20.3968	1.05049
$378$	0	0
$379$	6.67122	0.342677	0.171339	−	0.985212i	$-0.445191\pi$
$379$	6.67122	0.342677	0.171339	+	0.985212i	$0.445191\pi$
$380$	0	0
$381$	−4.28622	−0.219590
$382$	0	0
$383$	−31.1170	−1.59001	−0.795003	−	0.606606i	$-0.792531\pi$
$383$	−31.1170	−1.59001	−0.795003	+	0.606606i	$0.792531\pi$
$384$	0	0
$385$	−2.22029	−0.113156
$386$	0	0
$387$	−4.81041	−0.244527
$388$	0	0
$389$	7.71800	0.391318	0.195659	−	0.980672i	$-0.437315\pi$
$389$	7.71800	0.391318	0.195659	+	0.980672i	$0.437315\pi$
$390$	0	0
$391$	−15.6483	−0.791369
$392$	0	0
$393$	−2.08626	−0.105238
$394$	0	0
$395$	−13.5319	−0.680865
$396$	0	0
$397$	13.3106	0.668039	0.334020	−	0.942566i	$-0.391595\pi$
$397$	13.3106	0.668039	0.334020	+	0.942566i	$0.391595\pi$
$398$	0	0
$399$	−9.21091	−0.461122
$400$	0	0
$401$	36.5536	1.82540	0.912699	−	0.408632i	$-0.133994\pi$
$401$	36.5536	1.82540	0.912699	+	0.408632i	$0.133994\pi$
$402$	0	0
$403$	8.77157	0.436943
$404$	0	0
$405$	−1.30288	−0.0647405
$406$	0	0
$407$	−6.30018	−0.312288
$408$	0	0
$409$	11.0315	0.545473	0.272737	−	0.962089i	$-0.412071\pi$
$409$	11.0315	0.545473	0.272737	+	0.962089i	$0.412071\pi$
$410$	0	0
$411$	10.8365	0.534525
$412$	0	0
$413$	5.66264	0.278640
$414$	0	0
$415$	−10.2048	−0.500934
$416$	0	0
$417$	−22.2333	−1.08877
$418$	0	0
$419$	33.4711	1.63517	0.817586	−	0.575807i	$-0.195312\pi$
$419$	33.4711	1.63517	0.817586	+	0.575807i	$0.195312\pi$
$420$	0	0
$421$	−16.4341	−0.800949	−0.400475	−	0.916308i	$-0.631155\pi$
$421$	−16.4341	−0.800949	−0.400475	+	0.916308i	$0.631155\pi$
$422$	0	0
$423$	−5.53701	−0.269219
$424$	0	0
$425$	14.1870	0.688170
$426$	0	0
$427$	0.845393	0.0409114
$428$	0	0
$429$	4.61663	0.222893
$430$	0	0
$431$	15.9352	0.767569	0.383785	−	0.923423i	$-0.374620\pi$
$431$	15.9352	0.767569	0.383785	+	0.923423i	$0.374620\pi$
$432$	0	0
$433$	40.3283	1.93805	0.969027	−	0.246954i	$-0.0794296\pi$
$433$	40.3283	1.93805	0.969027	+	0.246954i	$0.0794296\pi$
$434$	0	0
$435$	6.37181	0.305505
$436$	0	0
$437$	−21.7941	−1.04255
$438$	0	0
$439$	3.10899	0.148384	0.0741921	−	0.997244i	$-0.476362\pi$
$439$	3.10899	0.148384	0.0741921	+	0.997244i	$0.476362\pi$
$440$	0	0
$441$	−4.62988	−0.220471
$442$	0	0
$443$	−1.77872	−0.0845097	−0.0422548	−	0.999107i	$-0.513454\pi$
$443$	−1.77872	−0.0845097	−0.0422548	+	0.999107i	$0.513454\pi$
$444$	0	0
$445$	20.9836	0.994717
$446$	0	0
$447$	7.91637	0.374432
$448$	0	0
$449$	−28.2427	−1.33286	−0.666428	−	0.745569i	$-0.732178\pi$
$449$	−28.2427	−1.33286	−0.666428	+	0.745569i	$0.732178\pi$
$450$	0	0
$451$	−2.64412	−0.124507
$452$	0	0
$453$	1.14387	0.0537438
$454$	0	0
$455$	8.36551	0.392181
$456$	0	0
$457$	−19.3448	−0.904911	−0.452456	−	0.891787i	$-0.649452\pi$
$457$	−19.3448	−0.904911	−0.452456	+	0.891787i	$0.649452\pi$
$458$	0	0
$459$	4.29582	0.200512
$460$	0	0
$461$	34.4963	1.60665	0.803326	−	0.595540i	$-0.203062\pi$
$461$	34.4963	1.60665	0.803326	+	0.595540i	$0.203062\pi$
$462$	0	0
$463$	−38.8017	−1.80327	−0.901633	−	0.432501i	$-0.857631\pi$
$463$	−38.8017	−1.80327	−0.901633	+	0.432501i	$0.857631\pi$
$464$	0	0
$465$	2.74017	0.127072
$466$	0	0
$467$	−8.88226	−0.411022	−0.205511	−	0.978655i	$-0.565886\pi$
$467$	−8.88226	−0.411022	−0.205511	+	0.978655i	$0.565886\pi$
$468$	0	0
$469$	7.90752	0.365135
$470$	0	0
$471$	−0.864535	−0.0398357
$472$	0	0
$473$	5.32480	0.244835
$474$	0	0
$475$	19.7588	0.906598
$476$	0	0
$477$	−8.47228	−0.387919
$478$	0	0
$479$	12.8938	0.589133	0.294566	−	0.955631i	$-0.404825\pi$
$479$	12.8938	0.589133	0.294566	+	0.955631i	$0.404825\pi$
$480$	0	0
$481$	23.7376	1.08234
$482$	0	0
$483$	5.60798	0.255172
$484$	0	0
$485$	−5.89285	−0.267581
$486$	0	0
$487$	21.5928	0.978463	0.489231	−	0.872154i	$-0.337277\pi$
$487$	21.5928	0.978463	0.489231	+	0.872154i	$0.337277\pi$
$488$	0	0
$489$	0.610097	0.0275896
$490$	0	0
$491$	8.93194	0.403093	0.201546	−	0.979479i	$-0.435403\pi$
$491$	8.93194	0.403093	0.201546	+	0.979479i	$0.435403\pi$
$492$	0	0
$493$	−21.0090	−0.946198
$494$	0	0
$495$	1.44220	0.0648219
$496$	0	0
$497$	−4.35390	−0.195299
$498$	0	0
$499$	12.5333	0.561069	0.280534	−	0.959844i	$-0.409488\pi$
$499$	12.5333	0.561069	0.280534	+	0.959844i	$0.409488\pi$
$500$	0	0
$501$	1.00000	0.0446767
$502$	0	0
$503$	−10.2827	−0.458484	−0.229242	−	0.973370i	$-0.573625\pi$
$503$	−10.2827	−0.458484	−0.229242	+	0.973370i	$0.573625\pi$
$504$	0	0
$505$	12.7488	0.567316
$506$	0	0
$507$	−4.39435	−0.195160
$508$	0	0
$509$	39.8485	1.76625	0.883126	−	0.469135i	$-0.155434\pi$
$509$	39.8485	1.76625	0.883126	+	0.469135i	$0.155434\pi$
$510$	0	0
$511$	4.55011	0.201285
$512$	0	0
$513$	5.98298	0.264155
$514$	0	0
$515$	−15.7444	−0.693783
$516$	0	0
$517$	6.12910	0.269557
$518$	0	0
$519$	−6.26195	−0.274869
$520$	0	0
$521$	−34.7106	−1.52070	−0.760349	−	0.649515i	$-0.774972\pi$
$521$	−34.7106	−1.52070	−0.760349	+	0.649515i	$0.774972\pi$
$522$	0	0
$523$	−9.44247	−0.412891	−0.206445	−	0.978458i	$-0.566190\pi$
$523$	−9.44247	−0.412891	−0.206445	+	0.978458i	$0.566190\pi$
$524$	0	0
$525$	−5.08428	−0.221896
$526$	0	0
$527$	−9.03482	−0.393563
$528$	0	0
$529$	−9.73086	−0.423081
$530$	0	0
$531$	−3.67819	−0.159620
$532$	0	0
$533$	9.96240	0.431519
$534$	0	0
$535$	−2.25989	−0.0977037
$536$	0	0
$537$	−6.07291	−0.262066
$538$	0	0
$539$	5.12496	0.220748
$540$	0	0
$541$	−9.17006	−0.394252	−0.197126	−	0.980378i	$-0.563161\pi$
$541$	−9.17006	−0.394252	−0.197126	+	0.980378i	$0.563161\pi$
$542$	0	0
$543$	−20.5447	−0.881659
$544$	0	0
$545$	5.19186	0.222395
$546$	0	0
$547$	−4.80140	−0.205293	−0.102646	−	0.994718i	$-0.532731\pi$
$547$	−4.80140	−0.205293	−0.102646	+	0.994718i	$0.532731\pi$
$548$	0	0
$549$	−0.549128	−0.0234362
$550$	0	0
$551$	−29.2601	−1.24652
$552$	0	0
$553$	−15.9897	−0.679952
$554$	0	0
$555$	7.41542	0.314767
$556$	0	0
$557$	40.6540	1.72257	0.861283	−	0.508126i	$-0.169662\pi$
$557$	40.6540	1.72257	0.861283	+	0.508126i	$0.169662\pi$
$558$	0	0
$559$	−20.0626	−0.848556
$560$	0	0
$561$	−4.75518	−0.200764
$562$	0	0
$563$	−17.4032	−0.733458	−0.366729	−	0.930328i	$-0.619522\pi$
$563$	−17.4032	−0.733458	−0.366729	+	0.930328i	$0.619522\pi$
$564$	0	0
$565$	−14.6559	−0.616578
$566$	0	0
$567$	−1.53952	−0.0646537
$568$	0	0
$569$	−5.26087	−0.220547	−0.110274	−	0.993901i	$-0.535173\pi$
$569$	−5.26087	−0.220547	−0.110274	+	0.993901i	$0.535173\pi$
$570$	0	0
$571$	−7.19886	−0.301263	−0.150631	−	0.988590i	$-0.548131\pi$
$571$	−7.19886	−0.301263	−0.150631	+	0.988590i	$0.548131\pi$
$572$	0	0
$573$	16.9039	0.706169
$574$	0	0
$575$	−12.0300	−0.501686
$576$	0	0
$577$	−15.0302	−0.625716	−0.312858	−	0.949800i	$-0.601286\pi$
$577$	−15.0302	−0.625716	−0.312858	+	0.949800i	$0.601286\pi$
$578$	0	0
$579$	−12.0710	−0.501656
$580$	0	0
$581$	−12.0583	−0.500263
$582$	0	0
$583$	9.37823	0.388407
$584$	0	0
$585$	−5.43385	−0.224662
$586$	0	0
$587$	−6.21714	−0.256609	−0.128304	−	0.991735i	$-0.540953\pi$
$587$	−6.21714	−0.256609	−0.128304	+	0.991735i	$0.540953\pi$
$588$	0	0
$589$	−12.5832	−0.518481
$590$	0	0
$591$	17.4635	0.718353
$592$	0	0
$593$	−1.43834	−0.0590655	−0.0295328	−	0.999564i	$-0.509402\pi$
$593$	−1.43834	−0.0590655	−0.0295328	+	0.999564i	$0.509402\pi$
$594$	0	0
$595$	−8.61658	−0.353245
$596$	0	0
$597$	17.5886	0.719852
$598$	0	0
$599$	−9.23991	−0.377533	−0.188766	−	0.982022i	$-0.560449\pi$
$599$	−9.23991	−0.377533	−0.188766	+	0.982022i	$0.560449\pi$
$600$	0	0
$601$	35.7542	1.45844	0.729221	−	0.684278i	$-0.239883\pi$
$601$	35.7542	1.45844	0.729221	+	0.684278i	$0.239883\pi$
$602$	0	0
$603$	−5.13636	−0.209169
$604$	0	0
$605$	12.7352	0.517761
$606$	0	0
$607$	−25.8671	−1.04991	−0.524956	−	0.851129i	$-0.675918\pi$
$607$	−25.8671	−1.04991	−0.524956	+	0.851129i	$0.675918\pi$
$608$	0	0
$609$	7.52911	0.305095
$610$	0	0
$611$	−23.0930	−0.934242
$612$	0	0
$613$	−3.40268	−0.137433	−0.0687165	−	0.997636i	$-0.521890\pi$
$613$	−3.40268	−0.137433	−0.0687165	+	0.997636i	$0.521890\pi$
$614$	0	0
$615$	3.11217	0.125495
$616$	0	0
$617$	29.0332	1.16883	0.584417	−	0.811453i	$-0.301323\pi$
$617$	29.0332	1.16883	0.584417	+	0.811453i	$0.301323\pi$
$618$	0	0
$619$	6.79266	0.273020	0.136510	−	0.990639i	$-0.456411\pi$
$619$	6.79266	0.273020	0.136510	+	0.990639i	$0.456411\pi$
$620$	0	0
$621$	−3.64268	−0.146176
$622$	0	0
$623$	24.7948	0.993383
$624$	0	0
$625$	2.41912	0.0967650
$626$	0	0
$627$	−6.62275	−0.264487
$628$	0	0
$629$	−24.4500	−0.974884
$630$	0	0
$631$	48.4976	1.93066	0.965329	−	0.261038i	$-0.0840648\pi$
$631$	48.4976	1.93066	0.965329	+	0.261038i	$0.0840648\pi$
$632$	0	0
$633$	−11.0535	−0.439339
$634$	0	0
$635$	−5.58442	−0.221611
$636$	0	0
$637$	−19.3096	−0.765076
$638$	0	0
$639$	2.82810	0.111878
$640$	0	0
$641$	30.5429	1.20637	0.603185	−	0.797601i	$-0.293898\pi$
$641$	30.5429	1.20637	0.603185	+	0.797601i	$0.293898\pi$
$642$	0	0
$643$	−19.9534	−0.786883	−0.393442	−	0.919350i	$-0.628716\pi$
$643$	−19.9534	−0.786883	−0.393442	+	0.919350i	$0.628716\pi$
$644$	0	0
$645$	−6.26738	−0.246778
$646$	0	0
$647$	−8.38906	−0.329808	−0.164904	−	0.986310i	$-0.552731\pi$
$647$	−8.38906	−0.329808	−0.164904	+	0.986310i	$0.552731\pi$
$648$	0	0
$649$	4.07151	0.159821
$650$	0	0
$651$	3.23786	0.126902
$652$	0	0
$653$	5.86996	0.229709	0.114855	−	0.993382i	$-0.463360\pi$
$653$	5.86996	0.229709	0.114855	+	0.993382i	$0.463360\pi$
$654$	0	0
$655$	−2.71814	−0.106207
$656$	0	0
$657$	−2.95554	−0.115307
$658$	0	0
$659$	−36.5609	−1.42421	−0.712105	−	0.702072i	$-0.752258\pi$
$659$	−36.5609	−1.42421	−0.712105	+	0.702072i	$0.752258\pi$
$660$	0	0
$661$	29.5500	1.14936	0.574680	−	0.818378i	$-0.305126\pi$
$661$	29.5500	1.14936	0.574680	+	0.818378i	$0.305126\pi$
$662$	0	0
$663$	17.9164	0.695815
$664$	0	0
$665$	−12.0007	−0.465366
$666$	0	0
$667$	17.8148	0.689791
$668$	0	0
$669$	15.3933	0.595138
$670$	0	0
$671$	0.607847	0.0234657
$672$	0	0
$673$	−35.1107	−1.35342	−0.676709	−	0.736251i	$-0.736594\pi$
$673$	−35.1107	−1.35342	−0.676709	+	0.736251i	$0.736594\pi$
$674$	0	0
$675$	3.30251	0.127114
$676$	0	0
$677$	27.4779	1.05606	0.528030	−	0.849226i	$-0.322931\pi$
$677$	27.4779	1.05606	0.528030	+	0.849226i	$0.322931\pi$
$678$	0	0
$679$	−6.96317	−0.267222
$680$	0	0
$681$	−9.45775	−0.362422
$682$	0	0
$683$	−29.3191	−1.12186	−0.560931	−	0.827862i	$-0.689557\pi$
$683$	−29.3191	−1.12186	−0.560931	+	0.827862i	$0.689557\pi$
$684$	0	0
$685$	14.1186	0.539445
$686$	0	0
$687$	−21.1324	−0.806252
$688$	0	0
$689$	−35.3349	−1.34615
$690$	0	0
$691$	29.4518	1.12040	0.560199	−	0.828358i	$-0.310725\pi$
$691$	29.4518	1.12040	0.560199	+	0.828358i	$0.310725\pi$
$692$	0	0
$693$	1.70414	0.0647350
$694$	0	0
$695$	−28.9673	−1.09879
$696$	0	0
$697$	−10.2614	−0.388678
$698$	0	0
$699$	7.93738	0.300219
$700$	0	0
$701$	−16.4762	−0.622297	−0.311149	−	0.950361i	$-0.600714\pi$
$701$	−16.4762	−0.622297	−0.311149	+	0.950361i	$0.600714\pi$
$702$	0	0
$703$	−34.0525	−1.28432
$704$	0	0
$705$	−7.21405	−0.271697
$706$	0	0
$707$	15.0644	0.566555
$708$	0	0
$709$	−3.16626	−0.118911	−0.0594557	−	0.998231i	$-0.518937\pi$
$709$	−3.16626	−0.118911	−0.0594557	+	0.998231i	$0.518937\pi$
$710$	0	0
$711$	10.3862	0.389512
$712$	0	0
$713$	7.66116	0.286913
$714$	0	0
$715$	6.01490	0.224945
$716$	0	0
$717$	5.51828	0.206084
$718$	0	0
$719$	−15.1758	−0.565964	−0.282982	−	0.959125i	$-0.591324\pi$
$719$	−15.1758	−0.565964	−0.282982	+	0.959125i	$0.591324\pi$
$720$	0	0
$721$	−18.6041	−0.692852
$722$	0	0
$723$	−24.7655	−0.921039
$724$	0	0
$725$	−16.1511	−0.599838
$726$	0	0
$727$	34.1595	1.26690	0.633452	−	0.773782i	$-0.281637\pi$
$727$	34.1595	1.26690	0.633452	+	0.773782i	$0.281637\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	20.6647	0.764311
$732$	0	0
$733$	53.0393	1.95905	0.979526	−	0.201316i	$-0.0645217\pi$
$733$	53.0393	1.95905	0.979526	+	0.201316i	$0.0645217\pi$
$734$	0	0
$735$	−6.03217	−0.222500
$736$	0	0
$737$	5.68560	0.209432
$738$	0	0
$739$	46.3794	1.70609	0.853047	−	0.521835i	$-0.174752\pi$
$739$	46.3794	1.70609	0.853047	+	0.521835i	$0.174752\pi$
$740$	0	0
$741$	24.9529	0.916669
$742$	0	0
$743$	26.8843	0.986291	0.493145	−	0.869947i	$-0.335847\pi$
$743$	26.8843	0.986291	0.493145	+	0.869947i	$0.335847\pi$
$744$	0	0
$745$	10.3141	0.377878
$746$	0	0
$747$	7.83252	0.286577
$748$	0	0
$749$	−2.67036	−0.0975727
$750$	0	0
$751$	8.24273	0.300782	0.150391	−	0.988627i	$-0.451947\pi$
$751$	8.24273	0.300782	0.150391	+	0.988627i	$0.451947\pi$
$752$	0	0
$753$	−7.39915	−0.269640
$754$	0	0
$755$	1.49032	0.0542384
$756$	0	0
$757$	−16.5864	−0.602843	−0.301422	−	0.953491i	$-0.597461\pi$
$757$	−16.5864	−0.602843	−0.301422	+	0.953491i	$0.597461\pi$
$758$	0	0
$759$	4.03220	0.146360
$760$	0	0
$761$	1.70698	0.0618779	0.0309389	−	0.999521i	$-0.490150\pi$
$761$	1.70698	0.0618779	0.0309389	+	0.999521i	$0.490150\pi$
$762$	0	0
$763$	6.13485	0.222097
$764$	0	0
$765$	5.59693	0.202357
$766$	0	0
$767$	−15.3405	−0.553912
$768$	0	0
$769$	−17.6207	−0.635420	−0.317710	−	0.948188i	$-0.602914\pi$
$769$	−17.6207	−0.635420	−0.317710	+	0.948188i	$0.602914\pi$
$770$	0	0
$771$	14.4033	0.518723
$772$	0	0
$773$	−33.0421	−1.18844	−0.594221	−	0.804302i	$-0.702540\pi$
$773$	−33.0421	−1.18844	−0.594221	+	0.804302i	$0.702540\pi$
$774$	0	0
$775$	−6.94572	−0.249498
$776$	0	0
$777$	8.76227	0.314345
$778$	0	0
$779$	−14.2915	−0.512045
$780$	0	0
$781$	−3.13051	−0.112018
$782$	0	0
$783$	−4.89056	−0.174775
$784$	0	0
$785$	−1.12638	−0.0402023
$786$	0	0
$787$	19.3133	0.688445	0.344223	−	0.938888i	$-0.388142\pi$
$787$	19.3133	0.688445	0.344223	+	0.938888i	$0.388142\pi$
$788$	0	0
$789$	−11.5284	−0.410422
$790$	0	0
$791$	−17.3178	−0.615751
$792$	0	0
$793$	−2.29022	−0.0813282
$794$	0	0
$795$	−11.0383	−0.391490
$796$	0	0
$797$	−5.30149	−0.187788	−0.0938941	−	0.995582i	$-0.529932\pi$
$797$	−5.30149	−0.187788	−0.0938941	+	0.995582i	$0.529932\pi$
$798$	0	0
$799$	23.7860	0.841489
$800$	0	0
$801$	−16.1056	−0.569062
$802$	0	0
$803$	3.27158	0.115452
$804$	0	0
$805$	7.30651	0.257521
$806$	0	0
$807$	5.00858	0.176310
$808$	0	0
$809$	−2.40674	−0.0846166	−0.0423083	−	0.999105i	$-0.513471\pi$
$809$	−2.40674	−0.0846166	−0.0423083	+	0.999105i	$0.513471\pi$
$810$	0	0
$811$	8.23859	0.289296	0.144648	−	0.989483i	$-0.453795\pi$
$811$	8.23859	0.289296	0.144648	+	0.989483i	$0.453795\pi$
$812$	0	0
$813$	2.89720	0.101609
$814$	0	0
$815$	0.794882	0.0278435
$816$	0	0
$817$	28.7806	1.00691
$818$	0	0
$819$	−6.42080	−0.224361
$820$	0	0
$821$	16.5069	0.576095	0.288047	−	0.957616i	$-0.406994\pi$
$821$	16.5069	0.576095	0.288047	+	0.957616i	$0.406994\pi$
$822$	0	0
$823$	40.8950	1.42551	0.712755	−	0.701413i	$-0.247447\pi$
$823$	40.8950	1.42551	0.712755	+	0.701413i	$0.247447\pi$
$824$	0	0
$825$	−3.65565	−0.127274
$826$	0	0
$827$	15.9428	0.554385	0.277193	−	0.960814i	$-0.410596\pi$
$827$	15.9428	0.554385	0.277193	+	0.960814i	$0.410596\pi$
$828$	0	0
$829$	−3.06849	−0.106573	−0.0532866	−	0.998579i	$-0.516970\pi$
$829$	−3.06849	−0.106573	−0.0532866	+	0.998579i	$0.516970\pi$
$830$	0	0
$831$	12.3999	0.430148
$832$	0	0
$833$	19.8892	0.689118
$834$	0	0
$835$	1.30288	0.0450879
$836$	0	0
$837$	−2.10316	−0.0726960
$838$	0	0
$839$	−46.5149	−1.60587	−0.802936	−	0.596065i	$-0.796730\pi$
$839$	−46.5149	−1.60587	−0.802936	+	0.596065i	$0.796730\pi$
$840$	0	0
$841$	−5.08238	−0.175254
$842$	0	0
$843$	−7.57409	−0.260866
$844$	0	0
$845$	−5.72530	−0.196956
$846$	0	0
$847$	15.0483	0.517067
$848$	0	0
$849$	−26.4522	−0.907838
$850$	0	0
$851$	20.7326	0.710704
$852$	0	0
$853$	12.1374	0.415575	0.207788	−	0.978174i	$-0.433374\pi$
$853$	12.1374	0.415575	0.207788	+	0.978174i	$0.433374\pi$
$854$	0	0
$855$	7.79509	0.266586
$856$	0	0
$857$	−30.7919	−1.05183	−0.525916	−	0.850537i	$-0.676277\pi$
$857$	−30.7919	−1.05183	−0.525916	+	0.850537i	$0.676277\pi$
$858$	0	0
$859$	4.93062	0.168230	0.0841152	−	0.996456i	$-0.473194\pi$
$859$	4.93062	0.168230	0.0841152	+	0.996456i	$0.473194\pi$
$860$	0	0
$861$	3.67743	0.125327
$862$	0	0
$863$	14.6281	0.497945	0.248972	−	0.968511i	$-0.419907\pi$
$863$	14.6281	0.497945	0.248972	+	0.968511i	$0.419907\pi$
$864$	0	0
$865$	−8.15855	−0.277399
$866$	0	0
$867$	−1.45408	−0.0493833
$868$	0	0
$869$	−11.4968	−0.390002
$870$	0	0
$871$	−21.4220	−0.725856
$872$	0	0
$873$	4.52295	0.153079
$874$	0	0
$875$	−16.6532	−0.562981
$876$	0	0
$877$	−5.93656	−0.200463	−0.100232	−	0.994964i	$-0.531958\pi$
$877$	−5.93656	−0.200463	−0.100232	+	0.994964i	$0.531958\pi$
$878$	0	0
$879$	−26.4005	−0.890468
$880$	0	0
$881$	−11.3114	−0.381091	−0.190546	−	0.981678i	$-0.561026\pi$
$881$	−11.3114	−0.381091	−0.190546	+	0.981678i	$0.561026\pi$
$882$	0	0
$883$	6.55955	0.220746	0.110373	−	0.993890i	$-0.464795\pi$
$883$	6.55955	0.220746	0.110373	+	0.993890i	$0.464795\pi$
$884$	0	0
$885$	−4.79223	−0.161089
$886$	0	0
$887$	−54.4122	−1.82698	−0.913491	−	0.406860i	$-0.866624\pi$
$887$	−54.4122	−1.82698	−0.913491	+	0.406860i	$0.866624\pi$
$888$	0	0
$889$	−6.59872	−0.221314
$890$	0	0
$891$	−1.10693	−0.0370836
$892$	0	0
$893$	33.1278	1.10858
$894$	0	0
$895$	−7.91226	−0.264478
$896$	0	0
$897$	−15.1924	−0.507258
$898$	0	0
$899$	10.2857	0.343046
$900$	0	0
$901$	36.3954	1.21251
$902$	0	0
$903$	−7.40572	−0.246447
$904$	0	0
$905$	−26.7673	−0.889774
$906$	0	0
$907$	42.4227	1.40862	0.704311	−	0.709891i	$-0.251256\pi$
$907$	42.4227	1.40862	0.704311	+	0.709891i	$0.251256\pi$
$908$	0	0
$909$	−9.78513	−0.324552
$910$	0	0
$911$	−33.1126	−1.09707	−0.548535	−	0.836128i	$-0.684814\pi$
$911$	−33.1126	−1.09707	−0.548535	+	0.836128i	$0.684814\pi$
$912$	0	0
$913$	−8.67006	−0.286937
$914$	0	0
$915$	−0.715447	−0.0236519
$916$	0	0
$917$	−3.21184	−0.106064
$918$	0	0
$919$	26.2190	0.864884	0.432442	−	0.901662i	$-0.357652\pi$
$919$	26.2190	0.864884	0.432442	+	0.901662i	$0.357652\pi$
$920$	0	0
$921$	−25.8586	−0.852069
$922$	0	0
$923$	11.7950	0.388237
$924$	0	0
$925$	−18.7965	−0.618024
$926$	0	0
$927$	12.0844	0.396902
$928$	0	0
$929$	−37.1832	−1.21994	−0.609971	−	0.792424i	$-0.708819\pi$
$929$	−37.1832	−1.21994	−0.609971	+	0.792424i	$0.708819\pi$
$930$	0	0
$931$	27.7005	0.907847
$932$	0	0
$933$	−18.0842	−0.592052
$934$	0	0
$935$	−6.19542	−0.202612
$936$	0	0
$937$	−40.8669	−1.33506	−0.667532	−	0.744581i	$-0.732649\pi$
$937$	−40.8669	−1.33506	−0.667532	+	0.744581i	$0.732649\pi$
$938$	0	0
$939$	−5.63952	−0.184039
$940$	0	0
$941$	16.7066	0.544619	0.272309	−	0.962210i	$-0.412213\pi$
$941$	16.7066	0.544619	0.272309	+	0.962210i	$0.412213\pi$
$942$	0	0
$943$	8.70124	0.283351
$944$	0	0
$945$	−2.00580	−0.0652488
$946$	0	0
$947$	16.3046	0.529827	0.264914	−	0.964272i	$-0.414657\pi$
$947$	16.3046	0.529827	0.264914	+	0.964272i	$0.414657\pi$
$948$	0	0
$949$	−12.3265	−0.400136
$950$	0	0
$951$	11.3814	0.369067
$952$	0	0
$953$	6.82539	0.221096	0.110548	−	0.993871i	$-0.464739\pi$
$953$	6.82539	0.221096	0.110548	+	0.993871i	$0.464739\pi$
$954$	0	0
$955$	22.0237	0.712669
$956$	0	0
$957$	5.41352	0.174994
$958$	0	0
$959$	16.6830	0.538722
$960$	0	0
$961$	−26.5767	−0.857313
$962$	0	0
$963$	1.73454	0.0558948
$964$	0	0
$965$	−15.7271	−0.506273
$966$	0	0
$967$	−27.1142	−0.871933	−0.435966	−	0.899963i	$-0.643593\pi$
$967$	−27.1142	−0.871933	−0.435966	+	0.899963i	$0.643593\pi$
$968$	0	0
$969$	−25.7018	−0.825661
$970$	0	0
$971$	−53.2285	−1.70818	−0.854092	−	0.520122i	$-0.825887\pi$
$971$	−53.2285	−1.70818	−0.854092	+	0.520122i	$0.825887\pi$
$972$	0	0
$973$	−34.2286	−1.09732
$974$	0	0
$975$	13.7736	0.441109
$976$	0	0
$977$	28.2581	0.904056	0.452028	−	0.892004i	$-0.350701\pi$
$977$	28.2581	0.904056	0.452028	+	0.892004i	$0.350701\pi$
$978$	0	0
$979$	17.8278	0.569778
$980$	0	0
$981$	−3.98492	−0.127229
$982$	0	0
$983$	55.0688	1.75642	0.878211	−	0.478273i	$-0.158737\pi$
$983$	55.0688	1.75642	0.878211	+	0.478273i	$0.158737\pi$
$984$	0	0
$985$	22.7528	0.724965
$986$	0	0
$987$	−8.52434	−0.271333
$988$	0	0
$989$	−17.5228	−0.557193
$990$	0	0
$991$	−2.61680	−0.0831252	−0.0415626	−	0.999136i	$-0.513234\pi$
$991$	−2.61680	−0.0831252	−0.0415626	+	0.999136i	$0.513234\pi$
$992$	0	0
$993$	−8.43848	−0.267787
$994$	0	0
$995$	22.9157	0.726478
$996$	0	0
$997$	11.2570	0.356511	0.178256	−	0.983984i	$-0.442955\pi$
$997$	11.2570	0.356511	0.178256	+	0.983984i	$0.442955\pi$
$998$	0	0
$999$	−5.69157	−0.180073

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8016.2.a.v.1.2		7
4.3	odd	2		1002.2.a.k.1.2	✓	7
12.11	even	2		3006.2.a.u.1.6		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1002.2.a.k.1.2	✓	7	4.3	odd	2
3006.2.a.u.1.6		7	12.11	even	2
8016.2.a.v.1.2		7	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.30288 q^{5} -1.53952 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.30288 q^{5} -1.53952 q^{7} +1.00000 q^{9} -1.10693 q^{11} +4.17065 q^{13} +1.30288 q^{15} -4.29582 q^{17} -5.98298 q^{19} +1.53952 q^{21} +3.64268 q^{23} -3.30251 q^{25} -1.00000 q^{27} +4.89056 q^{29} +2.10316 q^{31} +1.10693 q^{33} +2.00580 q^{35} +5.69157 q^{37} -4.17065 q^{39} +2.38869 q^{41} -4.81041 q^{43} -1.30288 q^{45} -5.53701 q^{47} -4.62988 q^{49} +4.29582 q^{51} -8.47228 q^{53} +1.44220 q^{55} +5.98298 q^{57} -3.67819 q^{59} -0.549128 q^{61} -1.53952 q^{63} -5.43385 q^{65} -5.13636 q^{67} -3.64268 q^{69} +2.82810 q^{71} -2.95554 q^{73} +3.30251 q^{75} +1.70414 q^{77} +10.3862 q^{79} +1.00000 q^{81} +7.83252 q^{83} +5.59693 q^{85} -4.89056 q^{87} -16.1056 q^{89} -6.42080 q^{91} -2.10316 q^{93} +7.79509 q^{95} +4.52295 q^{97} -1.10693 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9}+O(q^{10}) \) 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9	\( 7 q - 7 q^{3} + 5 q^{5} - 7 q^{7} + 7 q^{9} + 6 q^{13} - 5 q^{15} + 6 q^{17} + 2 q^{19} + 7 q^{21} + 12 q^{25} - 7 q^{27} - 4 q^{29} - 7 q^{31} + 13 q^{35} - 3 q^{37} - 6 q^{39} - 12 q^{41} + 2 q^{43} + 5 q^{45} + 11 q^{47} + 10 q^{49} - 6 q^{51} + q^{53} + 2 q^{55} - 2 q^{57} + 19 q^{59} + 12 q^{61} - 7 q^{63} - 10 q^{65} + 17 q^{67} + 20 q^{71} + 10 q^{73} - 12 q^{75} - 24 q^{77} - 2 q^{79} + 7 q^{81} + 7 q^{83} - 18 q^{85} + 4 q^{87} - 3 q^{89} - 4 q^{91} + 7 q^{93} + 24 q^{95} - 3 q^{97}+O(q^{100}) \) 7 * q - 7 * q^3 + 5 * q^5 - 7 * q^7 + 7 * q^9 + 6 * q^13 - 5 * q^15 + 6 * q^17 + 2 * q^19 + 7 * q^21 + 12 * q^25 - 7 * q^27 - 4 * q^29 - 7 * q^31 + 13 * q^35 - 3 * q^37 - 6 * q^39 - 12 * q^41 + 2 * q^43 + 5 * q^45 + 11 * q^47 + 10 * q^49 - 6 * q^51 + q^53 + 2 * q^55 - 2 * q^57 + 19 * q^59 + 12 * q^61 - 7 * q^63 - 10 * q^65 + 17 * q^67 + 20 * q^71 + 10 * q^73 - 12 * q^75 - 24 * q^77 - 2 * q^79 + 7 * q^81 + 7 * q^83 - 18 * q^85 + 4 * q^87 - 3 * q^89 - 4 * q^91 + 7 * q^93 + 24 * q^95 - 3 * q^97

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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