LMFDB - Embedded newform 8012.2.a.a.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8012,2,Mod(1,8012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8012, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8012 = 2^{2} \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8012.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:	$63.9761420994$
Analytic rank:	$1$
Dimension:	$79$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	8012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.92727 q^{3} -2.52260 q^{5} -3.66775 q^{7} +0.714366 q^{9} +O(q^{10})$	$q-1.92727 q^{3} -2.52260 q^{5} -3.66775 q^{7} +0.714366 q^{9} -5.84871 q^{11} +0.920033 q^{13} +4.86172 q^{15} +5.25652 q^{17} -3.85929 q^{19} +7.06873 q^{21} -0.515982 q^{23} +1.36350 q^{25} +4.40503 q^{27} -3.27119 q^{29} -6.69926 q^{31} +11.2720 q^{33} +9.25224 q^{35} +9.00428 q^{37} -1.77315 q^{39} -4.30823 q^{41} +3.54313 q^{43} -1.80206 q^{45} +5.69628 q^{47} +6.45236 q^{49} -10.1307 q^{51} +11.2910 q^{53} +14.7539 q^{55} +7.43789 q^{57} -11.1681 q^{59} +4.38715 q^{61} -2.62011 q^{63} -2.32087 q^{65} +3.59123 q^{67} +0.994435 q^{69} +1.27562 q^{71} -4.99523 q^{73} -2.62782 q^{75} +21.4516 q^{77} -2.85853 q^{79} -10.6328 q^{81} +13.8710 q^{83} -13.2601 q^{85} +6.30447 q^{87} +7.34407 q^{89} -3.37445 q^{91} +12.9113 q^{93} +9.73544 q^{95} +16.1869 q^{97} -4.17811 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9}+O(q^{10}) $ 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9	$ 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9} - 14 q^{11} - 3 q^{13} - 19 q^{15} - 30 q^{17} - 49 q^{19} + 7 q^{21} - 40 q^{23} + 51 q^{25} - 70 q^{27} + 2 q^{29} - 48 q^{31} - 25 q^{33} - 34 q^{35} - 35 q^{39} - 20 q^{41} - 104 q^{43} + 12 q^{45} - 38 q^{47} + 51 q^{49} - 41 q^{51} - q^{53} - 112 q^{55} - 34 q^{57} - 24 q^{59} - 120 q^{63} - 21 q^{65} - 67 q^{67} + 15 q^{69} - 28 q^{71} - 88 q^{73} - 103 q^{75} + 4 q^{77} - 99 q^{79} + 47 q^{81} - 70 q^{83} + 7 q^{85} - 109 q^{87} - 50 q^{89} - 83 q^{91} - 7 q^{93} - 61 q^{95} - 93 q^{97} - 78 q^{99}+O(q^{100}) $ 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9 - 14 * q^11 - 3 * q^13 - 19 * q^15 - 30 * q^17 - 49 * q^19 + 7 * q^21 - 40 * q^23 + 51 * q^25 - 70 * q^27 + 2 * q^29 - 48 * q^31 - 25 * q^33 - 34 * q^35 - 35 * q^39 - 20 * q^41 - 104 * q^43 + 12 * q^45 - 38 * q^47 + 51 * q^49 - 41 * q^51 - q^53 - 112 * q^55 - 34 * q^57 - 24 * q^59 - 120 * q^63 - 21 * q^65 - 67 * q^67 + 15 * q^69 - 28 * q^71 - 88 * q^73 - 103 * q^75 + 4 * q^77 - 99 * q^79 + 47 * q^81 - 70 * q^83 + 7 * q^85 - 109 * q^87 - 50 * q^89 - 83 * q^91 - 7 * q^93 - 61 * q^95 - 93 * q^97 - 78 * 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
$3$	−1.92727	−1.11271	−0.556355	−	0.830945i	$-0.687800\pi$
$3$	−1.92727	−1.11271	−0.556355	+	0.830945i	$0.687800\pi$
$4$	0	0
$5$	−2.52260	−1.12814	−0.564070	−	0.825727i	$-0.690765\pi$
$5$	−2.52260	−1.12814	−0.564070	+	0.825727i	$0.690765\pi$
$6$	0	0
$7$	−3.66775	−1.38628	−0.693139	−	0.720804i	$-0.743773\pi$
$7$	−3.66775	−1.38628	−0.693139	+	0.720804i	$0.743773\pi$
$8$	0	0
$9$	0.714366	0.238122
$10$	0	0
$11$	−5.84871	−1.76345	−0.881726	−	0.471763i	$-0.843618\pi$
$11$	−5.84871	−1.76345	−0.881726	+	0.471763i	$0.843618\pi$
$12$	0	0
$13$	0.920033	0.255171	0.127586	−	0.991828i	$-0.459277\pi$
$13$	0.920033	0.255171	0.127586	+	0.991828i	$0.459277\pi$
$14$	0	0
$15$	4.86172	1.25529
$16$	0	0
$17$	5.25652	1.27489	0.637447	−	0.770495i	$-0.279991\pi$
$17$	5.25652	1.27489	0.637447	+	0.770495i	$0.279991\pi$
$18$	0	0
$19$	−3.85929	−0.885382	−0.442691	−	0.896674i	$-0.645976\pi$
$19$	−3.85929	−0.885382	−0.442691	+	0.896674i	$0.645976\pi$
$20$	0	0
$21$	7.06873	1.54252
$22$	0	0
$23$	−0.515982	−0.107590	−0.0537948	−	0.998552i	$-0.517132\pi$
$23$	−0.515982	−0.107590	−0.0537948	+	0.998552i	$0.517132\pi$
$24$	0	0
$25$	1.36350	0.272699
$26$	0	0
$27$	4.40503	0.847749
$28$	0	0
$29$	−3.27119	−0.607446	−0.303723	−	0.952760i	$-0.598230\pi$
$29$	−3.27119	−0.607446	−0.303723	+	0.952760i	$0.598230\pi$
$30$	0	0
$31$	−6.69926	−1.20322	−0.601612	−	0.798789i	$-0.705474\pi$
$31$	−6.69926	−1.20322	−0.601612	+	0.798789i	$0.705474\pi$
$32$	0	0
$33$	11.2720	1.96221
$34$	0	0
$35$	9.25224	1.56391
$36$	0	0
$37$	9.00428	1.48030	0.740148	−	0.672444i	$-0.234756\pi$
$37$	9.00428	1.48030	0.740148	+	0.672444i	$0.234756\pi$
$38$	0	0
$39$	−1.77315	−0.283932
$40$	0	0
$41$	−4.30823	−0.672833	−0.336416	−	0.941713i	$-0.609215\pi$
$41$	−4.30823	−0.672833	−0.336416	+	0.941713i	$0.609215\pi$
$42$	0	0
$43$	3.54313	0.540322	0.270161	−	0.962815i	$-0.412923\pi$
$43$	3.54313	0.540322	0.270161	+	0.962815i	$0.412923\pi$
$44$	0	0
$45$	−1.80206	−0.268635
$46$	0	0
$47$	5.69628	0.830888	0.415444	−	0.909619i	$-0.363626\pi$
$47$	5.69628	0.830888	0.415444	+	0.909619i	$0.363626\pi$
$48$	0	0
$49$	6.45236	0.921765
$50$	0	0
$51$	−10.1307	−1.41859
$52$	0	0
$53$	11.2910	1.55094	0.775472	−	0.631381i	$-0.217512\pi$
$53$	11.2910	1.55094	0.775472	+	0.631381i	$0.217512\pi$
$54$	0	0
$55$	14.7539	1.98942
$56$	0	0
$57$	7.43789	0.985173
$58$	0	0
$59$	−11.1681	−1.45396	−0.726979	−	0.686659i	$-0.759076\pi$
$59$	−11.1681	−1.45396	−0.726979	+	0.686659i	$0.759076\pi$
$60$	0	0
$61$	4.38715	0.561717	0.280859	−	0.959749i	$-0.409381\pi$
$61$	4.38715	0.561717	0.280859	+	0.959749i	$0.409381\pi$
$62$	0	0
$63$	−2.62011	−0.330103
$64$	0	0
$65$	−2.32087	−0.287869
$66$	0	0
$67$	3.59123	0.438739	0.219370	−	0.975642i	$-0.429600\pi$
$67$	3.59123	0.438739	0.219370	+	0.975642i	$0.429600\pi$
$68$	0	0
$69$	0.994435	0.119716
$70$	0	0
$71$	1.27562	0.151388	0.0756940	−	0.997131i	$-0.475883\pi$
$71$	1.27562	0.151388	0.0756940	+	0.997131i	$0.475883\pi$
$72$	0	0
$73$	−4.99523	−0.584648	−0.292324	−	0.956319i	$-0.594429\pi$
$73$	−4.99523	−0.584648	−0.292324	+	0.956319i	$0.594429\pi$
$74$	0	0
$75$	−2.62782	−0.303435
$76$	0	0
$77$	21.4516	2.44463
$78$	0	0
$79$	−2.85853	−0.321609	−0.160805	−	0.986986i	$-0.551409\pi$
$79$	−2.85853	−0.321609	−0.160805	+	0.986986i	$0.551409\pi$
$80$	0	0
$81$	−10.6328	−1.18142
$82$	0	0
$83$	13.8710	1.52253	0.761267	−	0.648438i	$-0.224577\pi$
$83$	13.8710	1.52253	0.761267	+	0.648438i	$0.224577\pi$
$84$	0	0
$85$	−13.2601	−1.43826
$86$	0	0
$87$	6.30447	0.675910
$88$	0	0
$89$	7.34407	0.778470	0.389235	−	0.921138i	$-0.372739\pi$
$89$	7.34407	0.778470	0.389235	+	0.921138i	$0.372739\pi$
$90$	0	0
$91$	−3.37445	−0.353738
$92$	0	0
$93$	12.9113	1.33884
$94$	0	0
$95$	9.73544	0.998835
$96$	0	0
$97$	16.1869	1.64353	0.821765	−	0.569827i	$-0.192990\pi$
$97$	16.1869	1.64353	0.821765	+	0.569827i	$0.192990\pi$
$98$	0	0
$99$	−4.17811	−0.419916
$100$	0	0
$101$	−4.88936	−0.486509	−0.243255	−	0.969962i	$-0.578215\pi$
$101$	−4.88936	−0.486509	−0.243255	+	0.969962i	$0.578215\pi$
$102$	0	0
$103$	−2.75937	−0.271889	−0.135944	−	0.990716i	$-0.543407\pi$
$103$	−2.75937	−0.271889	−0.135944	+	0.990716i	$0.543407\pi$
$104$	0	0
$105$	−17.8316	−1.74018
$106$	0	0
$107$	−3.74426	−0.361971	−0.180986	−	0.983486i	$-0.557929\pi$
$107$	−3.74426	−0.361971	−0.180986	+	0.983486i	$0.557929\pi$
$108$	0	0
$109$	5.92461	0.567475	0.283738	−	0.958902i	$-0.408426\pi$
$109$	5.92461	0.567475	0.283738	+	0.958902i	$0.408426\pi$
$110$	0	0
$111$	−17.3537	−1.64714
$112$	0	0
$113$	−6.44091	−0.605910	−0.302955	−	0.953005i	$-0.597973\pi$
$113$	−6.44091	−0.605910	−0.302955	+	0.953005i	$0.597973\pi$
$114$	0	0
$115$	1.30161	0.121376
$116$	0	0
$117$	0.657240	0.0607619
$118$	0	0
$119$	−19.2796	−1.76736
$120$	0	0
$121$	23.2074	2.10976
$122$	0	0
$123$	8.30312	0.748667
$124$	0	0
$125$	9.17344	0.820497
$126$	0	0
$127$	−17.0405	−1.51210	−0.756052	−	0.654512i	$-0.772874\pi$
$127$	−17.0405	−1.51210	−0.756052	+	0.654512i	$0.772874\pi$
$128$	0	0
$129$	−6.82856	−0.601222
$130$	0	0
$131$	6.53532	0.570994	0.285497	−	0.958380i	$-0.407841\pi$
$131$	6.53532	0.570994	0.285497	+	0.958380i	$0.407841\pi$
$132$	0	0
$133$	14.1549	1.22739
$134$	0	0
$135$	−11.1121	−0.956379
$136$	0	0
$137$	17.6419	1.50725	0.753623	−	0.657307i	$-0.228305\pi$
$137$	17.6419	1.50725	0.753623	+	0.657307i	$0.228305\pi$
$138$	0	0
$139$	−4.83477	−0.410080	−0.205040	−	0.978754i	$-0.565732\pi$
$139$	−4.83477	−0.410080	−0.205040	+	0.978754i	$0.565732\pi$
$140$	0	0
$141$	−10.9783	−0.924536
$142$	0	0
$143$	−5.38100	−0.449982
$144$	0	0
$145$	8.25190	0.685283
$146$	0	0
$147$	−12.4354	−1.02566
$148$	0	0
$149$	−4.25177	−0.348319	−0.174159	−	0.984717i	$-0.555721\pi$
$149$	−4.25177	−0.348319	−0.174159	+	0.984717i	$0.555721\pi$
$150$	0	0
$151$	14.9770	1.21881	0.609404	−	0.792860i	$-0.291409\pi$
$151$	14.9770	1.21881	0.609404	+	0.792860i	$0.291409\pi$
$152$	0	0
$153$	3.75508	0.303580
$154$	0	0
$155$	16.8995	1.35740
$156$	0	0
$157$	−8.43565	−0.673238	−0.336619	−	0.941641i	$-0.609284\pi$
$157$	−8.43565	−0.673238	−0.336619	+	0.941641i	$0.609284\pi$
$158$	0	0
$159$	−21.7609	−1.72575
$160$	0	0
$161$	1.89249	0.149149
$162$	0	0
$163$	14.9337	1.16970	0.584850	−	0.811142i	$-0.301153\pi$
$163$	14.9337	1.16970	0.584850	+	0.811142i	$0.301153\pi$
$164$	0	0
$165$	−28.4348	−2.21364
$166$	0	0
$167$	11.1771	0.864913	0.432456	−	0.901655i	$-0.357647\pi$
$167$	11.1771	0.864913	0.432456	+	0.901655i	$0.357647\pi$
$168$	0	0
$169$	−12.1535	−0.934888
$170$	0	0
$171$	−2.75695	−0.210829
$172$	0	0
$173$	−4.48262	−0.340808	−0.170404	−	0.985374i	$-0.554507\pi$
$173$	−4.48262	−0.340808	−0.170404	+	0.985374i	$0.554507\pi$
$174$	0	0
$175$	−5.00095	−0.378037
$176$	0	0
$177$	21.5239	1.61783
$178$	0	0
$179$	−5.93868	−0.443878	−0.221939	−	0.975061i	$-0.571239\pi$
$179$	−5.93868	−0.443878	−0.221939	+	0.975061i	$0.571239\pi$
$180$	0	0
$181$	−5.78300	−0.429847	−0.214924	−	0.976631i	$-0.568950\pi$
$181$	−5.78300	−0.429847	−0.214924	+	0.976631i	$0.568950\pi$
$182$	0	0
$183$	−8.45522	−0.625028
$184$	0	0
$185$	−22.7142	−1.66998
$186$	0	0
$187$	−30.7438	−2.24821
$188$	0	0
$189$	−16.1565	−1.17522
$190$	0	0
$191$	−7.31512	−0.529304	−0.264652	−	0.964344i	$-0.585257\pi$
$191$	−7.31512	−0.529304	−0.264652	+	0.964344i	$0.585257\pi$
$192$	0	0
$193$	−19.3938	−1.39599	−0.697997	−	0.716101i	$-0.745925\pi$
$193$	−19.3938	−1.39599	−0.697997	+	0.716101i	$0.745925\pi$
$194$	0	0
$195$	4.47295	0.320314
$196$	0	0
$197$	0.862664	0.0614623	0.0307311	−	0.999528i	$-0.490216\pi$
$197$	0.862664	0.0614623	0.0307311	+	0.999528i	$0.490216\pi$
$198$	0	0
$199$	−5.78520	−0.410102	−0.205051	−	0.978751i	$-0.565736\pi$
$199$	−5.78520	−0.410102	−0.205051	+	0.978751i	$0.565736\pi$
$200$	0	0
$201$	−6.92127	−0.488189
$202$	0	0
$203$	11.9979	0.842088
$204$	0	0
$205$	10.8679	0.759049
$206$	0	0
$207$	−0.368600	−0.0256194
$208$	0	0
$209$	22.5719	1.56133
$210$	0	0
$211$	24.3249	1.67459	0.837297	−	0.546748i	$-0.184135\pi$
$211$	24.3249	1.67459	0.837297	+	0.546748i	$0.184135\pi$
$212$	0	0
$213$	−2.45846	−0.168451
$214$	0	0
$215$	−8.93789	−0.609559
$216$	0	0
$217$	24.5712	1.66800
$218$	0	0
$219$	9.62716	0.650543
$220$	0	0
$221$	4.83617	0.325316
$222$	0	0
$223$	−20.9089	−1.40016	−0.700082	−	0.714063i	$-0.746853\pi$
$223$	−20.9089	−1.40016	−0.700082	+	0.714063i	$0.746853\pi$
$224$	0	0
$225$	0.974034	0.0649356
$226$	0	0
$227$	13.3329	0.884938	0.442469	−	0.896784i	$-0.354103\pi$
$227$	13.3329	0.884938	0.442469	+	0.896784i	$0.354103\pi$
$228$	0	0
$229$	21.4242	1.41575	0.707875	−	0.706338i	$-0.249654\pi$
$229$	21.4242	1.41575	0.707875	+	0.706338i	$0.249654\pi$
$230$	0	0
$231$	−41.3429	−2.72017
$232$	0	0
$233$	−4.36767	−0.286136	−0.143068	−	0.989713i	$-0.545697\pi$
$233$	−4.36767	−0.286136	−0.143068	+	0.989713i	$0.545697\pi$
$234$	0	0
$235$	−14.3694	−0.937357
$236$	0	0
$237$	5.50915	0.357858
$238$	0	0
$239$	−4.71653	−0.305087	−0.152543	−	0.988297i	$-0.548746\pi$
$239$	−4.71653	−0.305087	−0.152543	+	0.988297i	$0.548746\pi$
$240$	0	0
$241$	−6.91515	−0.445444	−0.222722	−	0.974882i	$-0.571494\pi$
$241$	−6.91515	−0.445444	−0.222722	+	0.974882i	$0.571494\pi$
$242$	0	0
$243$	7.27713	0.466828
$244$	0	0
$245$	−16.2767	−1.03988
$246$	0	0
$247$	−3.55068	−0.225924
$248$	0	0
$249$	−26.7331	−1.69414
$250$	0	0
$251$	−9.45231	−0.596625	−0.298312	−	0.954468i	$-0.596424\pi$
$251$	−9.45231	−0.596625	−0.298312	+	0.954468i	$0.596424\pi$
$252$	0	0
$253$	3.01782	0.189729
$254$	0	0
$255$	25.5557	1.60036
$256$	0	0
$257$	−18.6127	−1.16103	−0.580513	−	0.814251i	$-0.697148\pi$
$257$	−18.6127	−1.16103	−0.580513	+	0.814251i	$0.697148\pi$
$258$	0	0
$259$	−33.0254	−2.05210
$260$	0	0
$261$	−2.33683	−0.144646
$262$	0	0
$263$	−9.63001	−0.593812	−0.296906	−	0.954907i	$-0.595955\pi$
$263$	−9.63001	−0.593812	−0.296906	+	0.954907i	$0.595955\pi$
$264$	0	0
$265$	−28.4828	−1.74968
$266$	0	0
$267$	−14.1540	−0.866211
$268$	0	0
$269$	−31.2816	−1.90727	−0.953637	−	0.300960i	$-0.902693\pi$
$269$	−31.2816	−1.90727	−0.953637	+	0.300960i	$0.902693\pi$
$270$	0	0
$271$	−30.6389	−1.86118	−0.930591	−	0.366062i	$-0.880706\pi$
$271$	−30.6389	−1.86118	−0.930591	+	0.366062i	$0.880706\pi$
$272$	0	0
$273$	6.50347	0.393608
$274$	0	0
$275$	−7.97468	−0.480891
$276$	0	0
$277$	−2.37191	−0.142514	−0.0712570	−	0.997458i	$-0.522701\pi$
$277$	−2.37191	−0.142514	−0.0712570	+	0.997458i	$0.522701\pi$
$278$	0	0
$279$	−4.78572	−0.286514
$280$	0	0
$281$	6.03998	0.360315	0.180158	−	0.983638i	$-0.442339\pi$
$281$	6.03998	0.360315	0.180158	+	0.983638i	$0.442339\pi$
$282$	0	0
$283$	−4.68266	−0.278355	−0.139178	−	0.990267i	$-0.544446\pi$
$283$	−4.68266	−0.278355	−0.139178	+	0.990267i	$0.544446\pi$
$284$	0	0
$285$	−18.7628	−1.11141
$286$	0	0
$287$	15.8015	0.932733
$288$	0	0
$289$	10.6310	0.625353
$290$	0	0
$291$	−31.1965	−1.82877
$292$	0	0
$293$	25.3383	1.48028	0.740140	−	0.672453i	$-0.234759\pi$
$293$	25.3383	1.48028	0.740140	+	0.672453i	$0.234759\pi$
$294$	0	0
$295$	28.1725	1.64027
$296$	0	0
$297$	−25.7637	−1.49496
$298$	0	0
$299$	−0.474720	−0.0274538
$300$	0	0
$301$	−12.9953	−0.749037
$302$	0	0
$303$	9.42311	0.541343
$304$	0	0
$305$	−11.0670	−0.633695
$306$	0	0
$307$	15.7259	0.897526	0.448763	−	0.893651i	$-0.351865\pi$
$307$	15.7259	0.897526	0.448763	+	0.893651i	$0.351865\pi$
$308$	0	0
$309$	5.31805	0.302533
$310$	0	0
$311$	14.9193	0.845998	0.422999	−	0.906130i	$-0.360977\pi$
$311$	14.9193	0.845998	0.422999	+	0.906130i	$0.360977\pi$
$312$	0	0
$313$	−18.5551	−1.04880	−0.524398	−	0.851473i	$-0.675710\pi$
$313$	−18.5551	−1.04880	−0.524398	+	0.851473i	$0.675710\pi$
$314$	0	0
$315$	6.60948	0.372402
$316$	0	0
$317$	−15.3050	−0.859612	−0.429806	−	0.902921i	$-0.641418\pi$
$317$	−15.3050	−0.859612	−0.429806	+	0.902921i	$0.641418\pi$
$318$	0	0
$319$	19.1323	1.07120
$320$	0	0
$321$	7.21619	0.402769
$322$	0	0
$323$	−20.2864	−1.12877
$324$	0	0
$325$	1.25446	0.0695850
$326$	0	0
$327$	−11.4183	−0.631435
$328$	0	0
$329$	−20.8925	−1.15184
$330$	0	0
$331$	32.6914	1.79688	0.898442	−	0.439092i	$-0.144700\pi$
$331$	32.6914	1.79688	0.898442	+	0.439092i	$0.144700\pi$
$332$	0	0
$333$	6.43235	0.352491
$334$	0	0
$335$	−9.05924	−0.494959
$336$	0	0
$337$	−30.9224	−1.68445	−0.842225	−	0.539126i	$-0.818755\pi$
$337$	−30.9224	−1.68445	−0.842225	+	0.539126i	$0.818755\pi$
$338$	0	0
$339$	12.4134	0.674202
$340$	0	0
$341$	39.1820	2.12183
$342$	0	0
$343$	2.00862	0.108455
$344$	0	0
$345$	−2.50856	−0.135056
$346$	0	0
$347$	4.64586	0.249403	0.124701	−	0.992194i	$-0.460203\pi$
$347$	4.64586	0.249403	0.124701	+	0.992194i	$0.460203\pi$
$348$	0	0
$349$	5.73629	0.307057	0.153528	−	0.988144i	$-0.450936\pi$
$349$	5.73629	0.307057	0.153528	+	0.988144i	$0.450936\pi$
$350$	0	0
$351$	4.05278	0.216321
$352$	0	0
$353$	14.7856	0.786960	0.393480	−	0.919333i	$-0.371271\pi$
$353$	14.7856	0.786960	0.393480	+	0.919333i	$0.371271\pi$
$354$	0	0
$355$	−3.21787	−0.170787
$356$	0	0
$357$	37.1569	1.96655
$358$	0	0
$359$	0.669182	0.0353181	0.0176590	−	0.999844i	$-0.494379\pi$
$359$	0.669182	0.0353181	0.0176590	+	0.999844i	$0.494379\pi$
$360$	0	0
$361$	−4.10587	−0.216099
$362$	0	0
$363$	−44.7268	−2.34755
$364$	0	0
$365$	12.6010	0.659564
$366$	0	0
$367$	−0.282568	−0.0147500	−0.00737498	−	0.999973i	$-0.502348\pi$
$367$	−0.282568	−0.0147500	−0.00737498	+	0.999973i	$0.502348\pi$
$368$	0	0
$369$	−3.07765	−0.160216
$370$	0	0
$371$	−41.4127	−2.15004
$372$	0	0
$373$	0.197701	0.0102366	0.00511829	−	0.999987i	$-0.498371\pi$
$373$	0.197701	0.0102366	0.00511829	+	0.999987i	$0.498371\pi$
$374$	0	0
$375$	−17.6797	−0.912975
$376$	0	0
$377$	−3.00961	−0.155003
$378$	0	0
$379$	−11.7114	−0.601576	−0.300788	−	0.953691i	$-0.597250\pi$
$379$	−11.7114	−0.601576	−0.300788	+	0.953691i	$0.597250\pi$
$380$	0	0
$381$	32.8417	1.68253
$382$	0	0
$383$	20.8311	1.06442	0.532210	−	0.846612i	$-0.321362\pi$
$383$	20.8311	1.06442	0.532210	+	0.846612i	$0.321362\pi$
$384$	0	0
$385$	−54.1136	−2.75789
$386$	0	0
$387$	2.53109	0.128663
$388$	0	0
$389$	−17.3147	−0.877893	−0.438946	−	0.898513i	$-0.644648\pi$
$389$	−17.3147	−0.877893	−0.438946	+	0.898513i	$0.644648\pi$
$390$	0	0
$391$	−2.71227	−0.137165
$392$	0	0
$393$	−12.5953	−0.635350
$394$	0	0
$395$	7.21091	0.362820
$396$	0	0
$397$	28.2007	1.41535	0.707677	−	0.706536i	$-0.249743\pi$
$397$	28.2007	1.41535	0.707677	+	0.706536i	$0.249743\pi$
$398$	0	0
$399$	−27.2803	−1.36572
$400$	0	0
$401$	0.0901861	0.00450368	0.00225184	−	0.999997i	$-0.499283\pi$
$401$	0.0901861	0.00450368	0.00225184	+	0.999997i	$0.499283\pi$
$402$	0	0
$403$	−6.16355	−0.307028
$404$	0	0
$405$	26.8222	1.33281
$406$	0	0
$407$	−52.6634	−2.61043
$408$	0	0
$409$	−9.03914	−0.446957	−0.223478	−	0.974709i	$-0.571741\pi$
$409$	−9.03914	−0.446957	−0.223478	+	0.974709i	$0.571741\pi$
$410$	0	0
$411$	−34.0006	−1.67713
$412$	0	0
$413$	40.9616	2.01559
$414$	0	0
$415$	−34.9908	−1.71763
$416$	0	0
$417$	9.31791	0.456300
$418$	0	0
$419$	25.1459	1.22846	0.614230	−	0.789127i	$-0.289467\pi$
$419$	25.1459	1.22846	0.614230	+	0.789127i	$0.289467\pi$
$420$	0	0
$421$	23.9315	1.16635	0.583174	−	0.812347i	$-0.301810\pi$
$421$	23.9315	1.16635	0.583174	+	0.812347i	$0.301810\pi$
$422$	0	0
$423$	4.06923	0.197853
$424$	0	0
$425$	7.16724	0.347662
$426$	0	0
$427$	−16.0910	−0.778696
$428$	0	0
$429$	10.3706	0.500699
$430$	0	0
$431$	−11.7954	−0.568163	−0.284082	−	0.958800i	$-0.591689\pi$
$431$	−11.7954	−0.568163	−0.284082	+	0.958800i	$0.591689\pi$
$432$	0	0
$433$	29.4212	1.41389	0.706947	−	0.707267i	$-0.250072\pi$
$433$	29.4212	1.41389	0.706947	+	0.707267i	$0.250072\pi$
$434$	0	0
$435$	−15.9036	−0.762521
$436$	0	0
$437$	1.99132	0.0952579
$438$	0	0
$439$	5.92619	0.282841	0.141421	−	0.989950i	$-0.454833\pi$
$439$	5.92619	0.282841	0.141421	+	0.989950i	$0.454833\pi$
$440$	0	0
$441$	4.60934	0.219492
$442$	0	0
$443$	−0.949059	−0.0450911	−0.0225456	−	0.999746i	$-0.507177\pi$
$443$	−0.949059	−0.0450911	−0.0225456	+	0.999746i	$0.507177\pi$
$444$	0	0
$445$	−18.5261	−0.878223
$446$	0	0
$447$	8.19431	0.387578
$448$	0	0
$449$	−0.901857	−0.0425613	−0.0212806	−	0.999774i	$-0.506774\pi$
$449$	−0.901857	−0.0425613	−0.0212806	+	0.999774i	$0.506774\pi$
$450$	0	0
$451$	25.1976	1.18651
$452$	0	0
$453$	−28.8646	−1.35618
$454$	0	0
$455$	8.51237	0.399066
$456$	0	0
$457$	16.6422	0.778489	0.389245	−	0.921134i	$-0.372736\pi$
$457$	16.6422	0.778489	0.389245	+	0.921134i	$0.372736\pi$
$458$	0	0
$459$	23.1551	1.08079
$460$	0	0
$461$	−4.93518	−0.229854	−0.114927	−	0.993374i	$-0.536663\pi$
$461$	−4.93518	−0.229854	−0.114927	+	0.993374i	$0.536663\pi$
$462$	0	0
$463$	−1.70936	−0.0794407	−0.0397204	−	0.999211i	$-0.512647\pi$
$463$	−1.70936	−0.0794407	−0.0397204	+	0.999211i	$0.512647\pi$
$464$	0	0
$465$	−32.5700	−1.51040
$466$	0	0
$467$	15.2081	0.703746	0.351873	−	0.936048i	$-0.385545\pi$
$467$	15.2081	0.703746	0.351873	+	0.936048i	$0.385545\pi$
$468$	0	0
$469$	−13.1717	−0.608214
$470$	0	0
$471$	16.2578	0.749119
$472$	0	0
$473$	−20.7227	−0.952832
$474$	0	0
$475$	−5.26212	−0.241443
$476$	0	0
$477$	8.06594	0.369314
$478$	0	0
$479$	36.7636	1.67977	0.839886	−	0.542763i	$-0.182622\pi$
$479$	36.7636	1.67977	0.839886	+	0.542763i	$0.182622\pi$
$480$	0	0
$481$	8.28424	0.377729
$482$	0	0
$483$	−3.64734	−0.165960
$484$	0	0
$485$	−40.8330	−1.85413
$486$	0	0
$487$	3.90618	0.177006	0.0885029	−	0.996076i	$-0.471792\pi$
$487$	3.90618	0.177006	0.0885029	+	0.996076i	$0.471792\pi$
$488$	0	0
$489$	−28.7813	−1.30154
$490$	0	0
$491$	−27.7979	−1.25450	−0.627251	−	0.778817i	$-0.715820\pi$
$491$	−27.7979	−1.25450	−0.627251	+	0.778817i	$0.715820\pi$
$492$	0	0
$493$	−17.1951	−0.774428
$494$	0	0
$495$	10.5397	0.473724
$496$	0	0
$497$	−4.67864	−0.209866
$498$	0	0
$499$	11.9798	0.536292	0.268146	−	0.963378i	$-0.413589\pi$
$499$	11.9798	0.536292	0.268146	+	0.963378i	$0.413589\pi$
$500$	0	0
$501$	−21.5413	−0.962396
$502$	0	0
$503$	29.7049	1.32448	0.662238	−	0.749294i	$-0.269607\pi$
$503$	29.7049	1.32448	0.662238	+	0.749294i	$0.269607\pi$
$504$	0	0
$505$	12.3339	0.548850
$506$	0	0
$507$	23.4231	1.04026
$508$	0	0
$509$	−20.6898	−0.917057	−0.458529	−	0.888680i	$-0.651623\pi$
$509$	−20.6898	−0.917057	−0.458529	+	0.888680i	$0.651623\pi$
$510$	0	0
$511$	18.3212	0.810484
$512$	0	0
$513$	−17.0003	−0.750582
$514$	0	0
$515$	6.96077	0.306728
$516$	0	0
$517$	−33.3159	−1.46523
$518$	0	0
$519$	8.63922	0.379220
$520$	0	0
$521$	34.2437	1.50024	0.750122	−	0.661300i	$-0.229995\pi$
$521$	34.2437	1.50024	0.750122	+	0.661300i	$0.229995\pi$
$522$	0	0
$523$	22.5398	0.985596	0.492798	−	0.870144i	$-0.335974\pi$
$523$	22.5398	0.985596	0.492798	+	0.870144i	$0.335974\pi$
$524$	0	0
$525$	9.63818	0.420645
$526$	0	0
$527$	−35.2148	−1.53398
$528$	0	0
$529$	−22.7338	−0.988424
$530$	0	0
$531$	−7.97809	−0.346219
$532$	0	0
$533$	−3.96372	−0.171688
$534$	0	0
$535$	9.44525	0.408354
$536$	0	0
$537$	11.4454	0.493907
$538$	0	0
$539$	−37.7379	−1.62549
$540$	0	0
$541$	26.1690	1.12509	0.562547	−	0.826765i	$-0.309821\pi$
$541$	26.1690	1.12509	0.562547	+	0.826765i	$0.309821\pi$
$542$	0	0
$543$	11.1454	0.478295
$544$	0	0
$545$	−14.9454	−0.640191
$546$	0	0
$547$	−37.0047	−1.58221	−0.791104	−	0.611681i	$-0.790493\pi$
$547$	−37.0047	−1.58221	−0.791104	+	0.611681i	$0.790493\pi$
$548$	0	0
$549$	3.13403	0.133757
$550$	0	0
$551$	12.6245	0.537821
$552$	0	0
$553$	10.4843	0.445840
$554$	0	0
$555$	43.7763	1.85820
$556$	0	0
$557$	13.7039	0.580655	0.290327	−	0.956927i	$-0.406236\pi$
$557$	13.7039	0.580655	0.290327	+	0.956927i	$0.406236\pi$
$558$	0	0
$559$	3.25980	0.137875
$560$	0	0
$561$	59.2516	2.50161
$562$	0	0
$563$	3.57633	0.150724	0.0753622	−	0.997156i	$-0.475989\pi$
$563$	3.57633	0.150724	0.0753622	+	0.997156i	$0.475989\pi$
$564$	0	0
$565$	16.2478	0.683551
$566$	0	0
$567$	38.9983	1.63778
$568$	0	0
$569$	19.3542	0.811371	0.405686	−	0.914013i	$-0.367033\pi$
$569$	19.3542	0.811371	0.405686	+	0.914013i	$0.367033\pi$
$570$	0	0
$571$	−30.0696	−1.25837	−0.629186	−	0.777255i	$-0.716612\pi$
$571$	−30.0696	−1.25837	−0.629186	+	0.777255i	$0.716612\pi$
$572$	0	0
$573$	14.0982	0.588961
$574$	0	0
$575$	−0.703538	−0.0293396
$576$	0	0
$577$	−27.2067	−1.13263	−0.566314	−	0.824189i	$-0.691631\pi$
$577$	−27.2067	−1.13263	−0.566314	+	0.824189i	$0.691631\pi$
$578$	0	0
$579$	37.3770	1.55334
$580$	0	0
$581$	−50.8751	−2.11066
$582$	0	0
$583$	−66.0380	−2.73502
$584$	0	0
$585$	−1.65795	−0.0685479
$586$	0	0
$587$	16.2389	0.670250	0.335125	−	0.942174i	$-0.391221\pi$
$587$	16.2389	0.670250	0.335125	+	0.942174i	$0.391221\pi$
$588$	0	0
$589$	25.8544	1.06531
$590$	0	0
$591$	−1.66259	−0.0683897
$592$	0	0
$593$	−8.94424	−0.367296	−0.183648	−	0.982992i	$-0.558791\pi$
$593$	−8.94424	−0.367296	−0.183648	+	0.982992i	$0.558791\pi$
$594$	0	0
$595$	48.6346	1.99382
$596$	0	0
$597$	11.1496	0.456324
$598$	0	0
$599$	34.0296	1.39041	0.695206	−	0.718811i	$-0.255313\pi$
$599$	34.0296	1.39041	0.695206	+	0.718811i	$0.255313\pi$
$600$	0	0
$601$	−17.1409	−0.699194	−0.349597	−	0.936900i	$-0.613681\pi$
$601$	−17.1409	−0.699194	−0.349597	+	0.936900i	$0.613681\pi$
$602$	0	0
$603$	2.56545	0.104473
$604$	0	0
$605$	−58.5428	−2.38010
$606$	0	0
$607$	−19.8652	−0.806304	−0.403152	−	0.915133i	$-0.632085\pi$
$607$	−19.8652	−0.806304	−0.403152	+	0.915133i	$0.632085\pi$
$608$	0	0
$609$	−23.1232	−0.936999
$610$	0	0
$611$	5.24077	0.212019
$612$	0	0
$613$	1.51088	0.0610240	0.0305120	−	0.999534i	$-0.490286\pi$
$613$	1.51088	0.0610240	0.0305120	+	0.999534i	$0.490286\pi$
$614$	0	0
$615$	−20.9454	−0.844601
$616$	0	0
$617$	18.0238	0.725613	0.362806	−	0.931865i	$-0.381819\pi$
$617$	18.0238	0.725613	0.362806	+	0.931865i	$0.381819\pi$
$618$	0	0
$619$	−25.7405	−1.03460	−0.517300	−	0.855804i	$-0.673063\pi$
$619$	−25.7405	−1.03460	−0.517300	+	0.855804i	$0.673063\pi$
$620$	0	0
$621$	−2.27292	−0.0912090
$622$	0	0
$623$	−26.9362	−1.07918
$624$	0	0
$625$	−29.9584	−1.19833
$626$	0	0
$627$	−43.5020	−1.73730
$628$	0	0
$629$	47.3312	1.88722
$630$	0	0
$631$	1.81778	0.0723646	0.0361823	−	0.999345i	$-0.488480\pi$
$631$	1.81778	0.0723646	0.0361823	+	0.999345i	$0.488480\pi$
$632$	0	0
$633$	−46.8806	−1.86334
$634$	0	0
$635$	42.9864	1.70586
$636$	0	0
$637$	5.93638	0.235208
$638$	0	0
$639$	0.911258	0.0360488
$640$	0	0
$641$	−2.33432	−0.0921999	−0.0461000	−	0.998937i	$-0.514679\pi$
$641$	−2.33432	−0.0921999	−0.0461000	+	0.998937i	$0.514679\pi$
$642$	0	0
$643$	−35.9256	−1.41677	−0.708385	−	0.705827i	$-0.750576\pi$
$643$	−35.9256	−1.41677	−0.708385	+	0.705827i	$0.750576\pi$
$644$	0	0
$645$	17.2257	0.678262
$646$	0	0
$647$	−12.0823	−0.475002	−0.237501	−	0.971387i	$-0.576328\pi$
$647$	−12.0823	−0.475002	−0.237501	+	0.971387i	$0.576328\pi$
$648$	0	0
$649$	65.3187	2.56399
$650$	0	0
$651$	−47.3553	−1.85600
$652$	0	0
$653$	−23.9137	−0.935816	−0.467908	−	0.883777i	$-0.654992\pi$
$653$	−23.9137	−0.935816	−0.467908	+	0.883777i	$0.654992\pi$
$654$	0	0
$655$	−16.4860	−0.644161
$656$	0	0
$657$	−3.56842	−0.139217
$658$	0	0
$659$	2.08753	0.0813186	0.0406593	−	0.999173i	$-0.487054\pi$
$659$	2.08753	0.0813186	0.0406593	+	0.999173i	$0.487054\pi$
$660$	0	0
$661$	−27.2864	−1.06132	−0.530659	−	0.847586i	$-0.678055\pi$
$661$	−27.2864	−1.06132	−0.530659	+	0.847586i	$0.678055\pi$
$662$	0	0
$663$	−9.32061	−0.361982
$664$	0	0
$665$	−35.7071	−1.38466
$666$	0	0
$667$	1.68788	0.0653548
$668$	0	0
$669$	40.2971	1.55798
$670$	0	0
$671$	−25.6592	−0.990561
$672$	0	0
$673$	0.784060	0.0302233	0.0151117	−	0.999886i	$-0.495190\pi$
$673$	0.784060	0.0302233	0.0151117	+	0.999886i	$0.495190\pi$
$674$	0	0
$675$	6.00624	0.231180
$676$	0	0
$677$	−11.2708	−0.433171	−0.216586	−	0.976264i	$-0.569492\pi$
$677$	−11.2708	−0.433171	−0.216586	+	0.976264i	$0.569492\pi$
$678$	0	0
$679$	−59.3694	−2.27839
$680$	0	0
$681$	−25.6962	−0.984679
$682$	0	0
$683$	26.8267	1.02649	0.513247	−	0.858241i	$-0.328442\pi$
$683$	26.8267	1.02649	0.513247	+	0.858241i	$0.328442\pi$
$684$	0	0
$685$	−44.5033	−1.70038
$686$	0	0
$687$	−41.2902	−1.57532
$688$	0	0
$689$	10.3881	0.395757
$690$	0	0
$691$	3.86556	0.147053	0.0735264	−	0.997293i	$-0.476575\pi$
$691$	3.86556	0.147053	0.0735264	+	0.997293i	$0.476575\pi$
$692$	0	0
$693$	15.3243	0.582120
$694$	0	0
$695$	12.1962	0.462628
$696$	0	0
$697$	−22.6463	−0.857790
$698$	0	0
$699$	8.41768	0.318386
$700$	0	0
$701$	−0.258770	−0.00977362	−0.00488681	−	0.999988i	$-0.501556\pi$
$701$	−0.258770	−0.00977362	−0.00488681	+	0.999988i	$0.501556\pi$
$702$	0	0
$703$	−34.7502	−1.31063
$704$	0	0
$705$	27.6937	1.04301
$706$	0	0
$707$	17.9329	0.674437
$708$	0	0
$709$	11.8811	0.446203	0.223101	−	0.974795i	$-0.428382\pi$
$709$	11.8811	0.446203	0.223101	+	0.974795i	$0.428382\pi$
$710$	0	0
$711$	−2.04203	−0.0765822
$712$	0	0
$713$	3.45670	0.129454
$714$	0	0
$715$	13.5741	0.507643
$716$	0	0
$717$	9.09002	0.339473
$718$	0	0
$719$	−17.2546	−0.643488	−0.321744	−	0.946827i	$-0.604269\pi$
$719$	−17.2546	−0.643488	−0.321744	+	0.946827i	$0.604269\pi$
$720$	0	0
$721$	10.1207	0.376913
$722$	0	0
$723$	13.3274	0.495650
$724$	0	0
$725$	−4.46026	−0.165650
$726$	0	0
$727$	−10.9648	−0.406662	−0.203331	−	0.979110i	$-0.565177\pi$
$727$	−10.9648	−0.406662	−0.203331	+	0.979110i	$0.565177\pi$
$728$	0	0
$729$	17.8734	0.661976
$730$	0	0
$731$	18.6245	0.688853
$732$	0	0
$733$	−16.4716	−0.608393	−0.304197	−	0.952609i	$-0.598388\pi$
$733$	−16.4716	−0.608393	−0.304197	+	0.952609i	$0.598388\pi$
$734$	0	0
$735$	31.3696	1.15708
$736$	0	0
$737$	−21.0041	−0.773695
$738$	0	0
$739$	−27.6131	−1.01577	−0.507883	−	0.861426i	$-0.669572\pi$
$739$	−27.6131	−1.01577	−0.507883	+	0.861426i	$0.669572\pi$
$740$	0	0
$741$	6.84311	0.251388
$742$	0	0
$743$	−13.3653	−0.490325	−0.245162	−	0.969482i	$-0.578841\pi$
$743$	−13.3653	−0.490325	−0.245162	+	0.969482i	$0.578841\pi$
$744$	0	0
$745$	10.7255	0.392952
$746$	0	0
$747$	9.90893	0.362549
$748$	0	0
$749$	13.7330	0.501792
$750$	0	0
$751$	−31.0344	−1.13246	−0.566230	−	0.824247i	$-0.691599\pi$
$751$	−31.0344	−1.13246	−0.566230	+	0.824247i	$0.691599\pi$
$752$	0	0
$753$	18.2171	0.663870
$754$	0	0
$755$	−37.7808	−1.37498
$756$	0	0
$757$	−5.85179	−0.212687	−0.106343	−	0.994329i	$-0.533914\pi$
$757$	−5.85179	−0.212687	−0.106343	+	0.994329i	$0.533914\pi$
$758$	0	0
$759$	−5.81616	−0.211113
$760$	0	0
$761$	−26.1525	−0.948028	−0.474014	−	0.880517i	$-0.657195\pi$
$761$	−26.1525	−0.948028	−0.474014	+	0.880517i	$0.657195\pi$
$762$	0	0
$763$	−21.7300	−0.786678
$764$	0	0
$765$	−9.47255	−0.342481
$766$	0	0
$767$	−10.2750	−0.371009
$768$	0	0
$769$	−35.1975	−1.26925	−0.634627	−	0.772819i	$-0.718846\pi$
$769$	−35.1975	−1.26925	−0.634627	+	0.772819i	$0.718846\pi$
$770$	0	0
$771$	35.8716	1.29189
$772$	0	0
$773$	44.9204	1.61568	0.807838	−	0.589405i	$-0.200638\pi$
$773$	44.9204	1.61568	0.807838	+	0.589405i	$0.200638\pi$
$774$	0	0
$775$	−9.13441	−0.328118
$776$	0	0
$777$	63.6489	2.28339
$778$	0	0
$779$	16.6267	0.595714
$780$	0	0
$781$	−7.46071	−0.266965
$782$	0	0
$783$	−14.4097	−0.514961
$784$	0	0
$785$	21.2798	0.759507
$786$	0	0
$787$	−28.5906	−1.01914	−0.509572	−	0.860428i	$-0.670196\pi$
$787$	−28.5906	−1.01914	−0.509572	+	0.860428i	$0.670196\pi$
$788$	0	0
$789$	18.5596	0.660740
$790$	0	0
$791$	23.6236	0.839960
$792$	0	0
$793$	4.03632	0.143334
$794$	0	0
$795$	54.8939	1.94689
$796$	0	0
$797$	5.44783	0.192972	0.0964859	−	0.995334i	$-0.469240\pi$
$797$	5.44783	0.192972	0.0964859	+	0.995334i	$0.469240\pi$
$798$	0	0
$799$	29.9426	1.05929
$800$	0	0
$801$	5.24635	0.185371
$802$	0	0
$803$	29.2156	1.03100
$804$	0	0
$805$	−4.77399	−0.168261
$806$	0	0
$807$	60.2881	2.12224
$808$	0	0
$809$	5.19371	0.182601	0.0913005	−	0.995823i	$-0.470898\pi$
$809$	5.19371	0.182601	0.0913005	+	0.995823i	$0.470898\pi$
$810$	0	0
$811$	−47.4238	−1.66528	−0.832638	−	0.553818i	$-0.813170\pi$
$811$	−47.4238	−1.66528	−0.832638	+	0.553818i	$0.813170\pi$
$812$	0	0
$813$	59.0494	2.07095
$814$	0	0
$815$	−37.6718	−1.31958
$816$	0	0
$817$	−13.6740	−0.478392
$818$	0	0
$819$	−2.41059	−0.0842328
$820$	0	0
$821$	16.7091	0.583153	0.291576	−	0.956548i	$-0.405820\pi$
$821$	16.7091	0.583153	0.291576	+	0.956548i	$0.405820\pi$
$822$	0	0
$823$	−56.8302	−1.98098	−0.990488	−	0.137599i	$-0.956061\pi$
$823$	−56.8302	−1.98098	−0.990488	+	0.137599i	$0.956061\pi$
$824$	0	0
$825$	15.3694	0.535092
$826$	0	0
$827$	−36.4941	−1.26903	−0.634513	−	0.772913i	$-0.718799\pi$
$827$	−36.4941	−1.26903	−0.634513	+	0.772913i	$0.718799\pi$
$828$	0	0
$829$	48.3873	1.68056	0.840281	−	0.542152i	$-0.182390\pi$
$829$	48.3873	1.68056	0.840281	+	0.542152i	$0.182390\pi$
$830$	0	0
$831$	4.57130	0.158577
$832$	0	0
$833$	33.9169	1.17515
$834$	0	0
$835$	−28.1954	−0.975742
$836$	0	0
$837$	−29.5105	−1.02003
$838$	0	0
$839$	−8.04224	−0.277649	−0.138825	−	0.990317i	$-0.544332\pi$
$839$	−8.04224	−0.277649	−0.138825	+	0.990317i	$0.544332\pi$
$840$	0	0
$841$	−18.2993	−0.631010
$842$	0	0
$843$	−11.6407	−0.400926
$844$	0	0
$845$	30.6585	1.05468
$846$	0	0
$847$	−85.1187	−2.92471
$848$	0	0
$849$	9.02475	0.309729
$850$	0	0
$851$	−4.64605	−0.159264
$852$	0	0
$853$	−32.5678	−1.11510	−0.557550	−	0.830144i	$-0.688258\pi$
$853$	−32.5678	−1.11510	−0.557550	+	0.830144i	$0.688258\pi$
$854$	0	0
$855$	6.95466	0.237844
$856$	0	0
$857$	−32.8830	−1.12326	−0.561631	−	0.827388i	$-0.689826\pi$
$857$	−32.8830	−1.12326	−0.561631	+	0.827388i	$0.689826\pi$
$858$	0	0
$859$	−5.36366	−0.183006	−0.0915028	−	0.995805i	$-0.529167\pi$
$859$	−5.36366	−0.183006	−0.0915028	+	0.995805i	$0.529167\pi$
$860$	0	0
$861$	−30.4537	−1.03786
$862$	0	0
$863$	20.7642	0.706821	0.353410	−	0.935468i	$-0.385022\pi$
$863$	20.7642	0.706821	0.353410	+	0.935468i	$0.385022\pi$
$864$	0	0
$865$	11.3079	0.384479
$866$	0	0
$867$	−20.4888	−0.695836
$868$	0	0
$869$	16.7187	0.567142
$870$	0	0
$871$	3.30406	0.111954
$872$	0	0
$873$	11.5634	0.391360
$874$	0	0
$875$	−33.6458	−1.13744
$876$	0	0
$877$	44.3963	1.49916	0.749578	−	0.661916i	$-0.230256\pi$
$877$	44.3963	1.49916	0.749578	+	0.661916i	$0.230256\pi$
$878$	0	0
$879$	−48.8337	−1.64712
$880$	0	0
$881$	14.9230	0.502768	0.251384	−	0.967887i	$-0.419114\pi$
$881$	14.9230	0.502768	0.251384	+	0.967887i	$0.419114\pi$
$882$	0	0
$883$	20.3727	0.685596	0.342798	−	0.939409i	$-0.388625\pi$
$883$	20.3727	0.685596	0.342798	+	0.939409i	$0.388625\pi$
$884$	0	0
$885$	−54.2961	−1.82514
$886$	0	0
$887$	15.6764	0.526364	0.263182	−	0.964746i	$-0.415228\pi$
$887$	15.6764	0.526364	0.263182	+	0.964746i	$0.415228\pi$
$888$	0	0
$889$	62.5004	2.09619
$890$	0	0
$891$	62.1880	2.08338
$892$	0	0
$893$	−21.9836	−0.735653
$894$	0	0
$895$	14.9809	0.500756
$896$	0	0
$897$	0.914914	0.0305481
$898$	0	0
$899$	21.9146	0.730893
$900$	0	0
$901$	59.3516	1.97729
$902$	0	0
$903$	25.0454	0.833460
$904$	0	0
$905$	14.5882	0.484928
$906$	0	0
$907$	−0.378049	−0.0125529	−0.00627645	−	0.999980i	$-0.501998\pi$
$907$	−0.378049	−0.0125529	−0.00627645	+	0.999980i	$0.501998\pi$
$908$	0	0
$909$	−3.49279	−0.115849
$910$	0	0
$911$	27.5267	0.912000	0.456000	−	0.889980i	$-0.349282\pi$
$911$	27.5267	0.912000	0.456000	+	0.889980i	$0.349282\pi$
$912$	0	0
$913$	−81.1271	−2.68492
$914$	0	0
$915$	21.3291	0.705119
$916$	0	0
$917$	−23.9699	−0.791556
$918$	0	0
$919$	25.8038	0.851189	0.425595	−	0.904914i	$-0.360065\pi$
$919$	25.8038	0.851189	0.425595	+	0.904914i	$0.360065\pi$
$920$	0	0
$921$	−30.3081	−0.998685
$922$	0	0
$923$	1.17361	0.0386299
$924$	0	0
$925$	12.2773	0.403675
$926$	0	0
$927$	−1.97120	−0.0647426
$928$	0	0
$929$	−30.9808	−1.01645	−0.508223	−	0.861225i	$-0.669697\pi$
$929$	−30.9808	−1.01645	−0.508223	+	0.861225i	$0.669697\pi$
$930$	0	0
$931$	−24.9015	−0.816114
$932$	0	0
$933$	−28.7536	−0.941350
$934$	0	0
$935$	77.5543	2.53630
$936$	0	0
$937$	3.54371	0.115768	0.0578839	−	0.998323i	$-0.481565\pi$
$937$	3.54371	0.115768	0.0578839	+	0.998323i	$0.481565\pi$
$938$	0	0
$939$	35.7607	1.16701
$940$	0	0
$941$	−24.2534	−0.790639	−0.395319	−	0.918544i	$-0.629366\pi$
$941$	−24.2534	−0.790639	−0.395319	+	0.918544i	$0.629366\pi$
$942$	0	0
$943$	2.22297	0.0723898
$944$	0	0
$945$	40.7564	1.32581
$946$	0	0
$947$	21.6740	0.704309	0.352155	−	0.935942i	$-0.385449\pi$
$947$	21.6740	0.704309	0.352155	+	0.935942i	$0.385449\pi$
$948$	0	0
$949$	−4.59578	−0.149185
$950$	0	0
$951$	29.4968	0.956498
$952$	0	0
$953$	11.7882	0.381856	0.190928	−	0.981604i	$-0.438850\pi$
$953$	11.7882	0.381856	0.190928	+	0.981604i	$0.438850\pi$
$954$	0	0
$955$	18.4531	0.597128
$956$	0	0
$957$	−36.8730	−1.19193
$958$	0	0
$959$	−64.7058	−2.08946
$960$	0	0
$961$	13.8801	0.447746
$962$	0	0
$963$	−2.67477	−0.0861932
$964$	0	0
$965$	48.9227	1.57488
$966$	0	0
$967$	44.4462	1.42929	0.714647	−	0.699486i	$-0.246588\pi$
$967$	44.4462	1.42929	0.714647	+	0.699486i	$0.246588\pi$
$968$	0	0
$969$	39.0974	1.25599
$970$	0	0
$971$	37.4028	1.20031	0.600156	−	0.799883i	$-0.295105\pi$
$971$	37.4028	1.20031	0.600156	+	0.799883i	$0.295105\pi$
$972$	0	0
$973$	17.7327	0.568485
$974$	0	0
$975$	−2.41768	−0.0774278
$976$	0	0
$977$	57.5581	1.84145	0.920723	−	0.390216i	$-0.127600\pi$
$977$	57.5581	1.84145	0.920723	+	0.390216i	$0.127600\pi$
$978$	0	0
$979$	−42.9533	−1.37279
$980$	0	0
$981$	4.23234	0.135128
$982$	0	0
$983$	32.9674	1.05150	0.525748	−	0.850641i	$-0.323786\pi$
$983$	32.9674	1.05150	0.525748	+	0.850641i	$0.323786\pi$
$984$	0	0
$985$	−2.17615	−0.0693381
$986$	0	0
$987$	40.2655	1.28166
$988$	0	0
$989$	−1.82819	−0.0581331
$990$	0	0
$991$	22.4571	0.713374	0.356687	−	0.934224i	$-0.383906\pi$
$991$	22.4571	0.713374	0.356687	+	0.934224i	$0.383906\pi$
$992$	0	0
$993$	−63.0052	−1.99941
$994$	0	0
$995$	14.5937	0.462652
$996$	0	0
$997$	−7.25854	−0.229880	−0.114940	−	0.993372i	$-0.536668\pi$
$997$	−7.25854	−0.229880	−0.114940	+	0.993372i	$0.536668\pi$
$998$	0	0
$999$	39.6642	1.25492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8012.2.a.a.1.19	✓	79

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8012.2.a.a.1.19	✓	79	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 8012 = 2^{2} \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8012.a (trivial)

\(f(q)\)	\(=\)	\(q-1.92727 q^{3} -2.52260 q^{5} -3.66775 q^{7} +0.714366 q^{9} +O(q^{10})\)	\(q-1.92727 q^{3} -2.52260 q^{5} -3.66775 q^{7} +0.714366 q^{9} -5.84871 q^{11} +0.920033 q^{13} +4.86172 q^{15} +5.25652 q^{17} -3.85929 q^{19} +7.06873 q^{21} -0.515982 q^{23} +1.36350 q^{25} +4.40503 q^{27} -3.27119 q^{29} -6.69926 q^{31} +11.2720 q^{33} +9.25224 q^{35} +9.00428 q^{37} -1.77315 q^{39} -4.30823 q^{41} +3.54313 q^{43} -1.80206 q^{45} +5.69628 q^{47} +6.45236 q^{49} -10.1307 q^{51} +11.2910 q^{53} +14.7539 q^{55} +7.43789 q^{57} -11.1681 q^{59} +4.38715 q^{61} -2.62011 q^{63} -2.32087 q^{65} +3.59123 q^{67} +0.994435 q^{69} +1.27562 q^{71} -4.99523 q^{73} -2.62782 q^{75} +21.4516 q^{77} -2.85853 q^{79} -10.6328 q^{81} +13.8710 q^{83} -13.2601 q^{85} +6.30447 q^{87} +7.34407 q^{89} -3.37445 q^{91} +12.9113 q^{93} +9.73544 q^{95} +16.1869 q^{97} -4.17811 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9}+O(q^{10}) \) 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9	\( 79 q - 19 q^{3} - 40 q^{7} + 72 q^{9} - 14 q^{11} - 3 q^{13} - 19 q^{15} - 30 q^{17} - 49 q^{19} + 7 q^{21} - 40 q^{23} + 51 q^{25} - 70 q^{27} + 2 q^{29} - 48 q^{31} - 25 q^{33} - 34 q^{35} - 35 q^{39} - 20 q^{41} - 104 q^{43} + 12 q^{45} - 38 q^{47} + 51 q^{49} - 41 q^{51} - q^{53} - 112 q^{55} - 34 q^{57} - 24 q^{59} - 120 q^{63} - 21 q^{65} - 67 q^{67} + 15 q^{69} - 28 q^{71} - 88 q^{73} - 103 q^{75} + 4 q^{77} - 99 q^{79} + 47 q^{81} - 70 q^{83} + 7 q^{85} - 109 q^{87} - 50 q^{89} - 83 q^{91} - 7 q^{93} - 61 q^{95} - 93 q^{97} - 78 q^{99}+O(q^{100}) \) 79 * q - 19 * q^3 - 40 * q^7 + 72 * q^9 - 14 * q^11 - 3 * q^13 - 19 * q^15 - 30 * q^17 - 49 * q^19 + 7 * q^21 - 40 * q^23 + 51 * q^25 - 70 * q^27 + 2 * q^29 - 48 * q^31 - 25 * q^33 - 34 * q^35 - 35 * q^39 - 20 * q^41 - 104 * q^43 + 12 * q^45 - 38 * q^47 + 51 * q^49 - 41 * q^51 - q^53 - 112 * q^55 - 34 * q^57 - 24 * q^59 - 120 * q^63 - 21 * q^65 - 67 * q^67 + 15 * q^69 - 28 * q^71 - 88 * q^73 - 103 * q^75 + 4 * q^77 - 99 * q^79 + 47 * q^81 - 70 * q^83 + 7 * q^85 - 109 * q^87 - 50 * q^89 - 83 * q^91 - 7 * q^93 - 61 * q^95 - 93 * q^97 - 78 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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